# Math 2.0

> Math 2.0 is a versioned knowledge graph of mathematics. Each node is a concept, a definition, a statement, a conjecture, a problem, an example, a counterexample, an object, a proof, a computation, a topic or an exercise. Each node has a permanent tag, sources, typed links to other nodes and a status that the build computes from the evidence.

Each node page has a Markdown copy at node/<TAG>.md and a JSON copy at api/node/<TAG>.json. The URI of a node is m2://<TAG>. The front matter follows the JSON Schema at https://tamnd.github.io/math2/schema/node.v1.json. The site has 14183 nodes now.

## Data

- [All nodes](https://tamnd.github.io/math2/api/nodes.json): the tag, kind, name, aliases and MSC classes of each node
- [Reviewers](https://tamnd.github.io/math2/api/reviewers.json): the reviewers and their fidelity reviews
- [Node schema](https://tamnd.github.io/math2/schema/node.v1.json): JSON Schema 2020-12 for the front matter
- [Edges](https://tamnd.github.io/math2/schema/edges.md): the 25 link types between nodes
- [Macros](https://tamnd.github.io/math2/macros/m2.katex.json): the TeX macros that node bodies use
- [Contribute](https://tamnd.github.io/math2/contribute.md): how to add a node
- [Curation](https://tamnd.github.io/math2/curate.md): how to decide on the alignment proposals of m2 align

## Specification

- [Math 2.0: specification](https://tamnd.github.io/math2/spec/00-README.md)
- [01. The state of the field, October 2026](https://tamnd.github.io/math2/spec/01-research-2026.md)
- [02. The goal](https://tamnd.github.io/math2/spec/02-the-goal.md)
- [03. Architecture](https://tamnd.github.io/math2/spec/03-architecture.md)
- [04. The object model](https://tamnd.github.io/math2/spec/04-object-model.md)
- [05. The knowledge graph](https://tamnd.github.io/math2/spec/05-knowledge-graph.md)
- [06. Versioned storage: metadata, artefacts, provenance](https://tamnd.github.io/math2/spec/06-versioned-storage.md)
- [07. The formal engine: Lean 4, Mathlib, proof checking](https://tamnd.github.io/math2/spec/07-formal-engine.md)
- [08. The computational engine: CAS, SMT, numerical experiments](https://tamnd.github.io/math2/spec/08-computational-engine.md)
- [09. Literature ingestion: papers, books, existing libraries](https://tamnd.github.io/math2/spec/09-literature-ingestion.md)
- [10. Autoformalization and statement fidelity](https://tamnd.github.io/math2/spec/10-autoformalization-and-fidelity.md)
- [11. The Mathematical API and the agent orchestrator](https://tamnd.github.io/math2/spec/11-api-and-orchestrator.md)
- [12. Research agents](https://tamnd.github.io/math2/spec/12-research-agents.md)
- [13. The research workspace: search, explore, explain, collaborate](https://tamnd.github.io/math2/spec/13-research-workspace.md)
- [14. Search](https://tamnd.github.io/math2/spec/14-search.md)
- [15. Trust and the audit](https://tamnd.github.io/math2/spec/15-trust-and-audit.md)
- [16. Collaboration and governance](https://tamnd.github.io/math2/spec/16-collaboration-and-governance.md)
- [17. Evaluation](https://tamnd.github.io/math2/spec/17-evaluation.md)
- [18. Coverage: what "all mathematics" means](https://tamnd.github.io/math2/spec/18-coverage-all-math.md)
- [19. Milestones](https://tamnd.github.io/math2/spec/19-milestones.md)
- [20. Repository layout](https://tamnd.github.io/math2/spec/20-repository-layout.md)
- [21. Open questions](https://tamnd.github.io/math2/spec/21-open-questions.md)
- [Math 2.0: the corpus specification](https://tamnd.github.io/math2/spec/corpus/00-README.md)
- [K01. Research](https://tamnd.github.io/math2/spec/corpus/01-research.md)
- [K01, part A. Encyclopedic, reference and database sources of mathematics written by people](https://tamnd.github.io/math2/spec/corpus/01a-reference-works.md)
- [K01, part B. Literature indexes, open textbooks and syllabi: sizes, licences, channels (research for math2)](https://tamnd.github.io/math2/spec/corpus/01b-literature-and-books.md)
- [K01, part C. Formal libraries and formal / informal-formal datasets](https://tamnd.github.io/math2/spec/corpus/01c-formal-libraries.md)
- [K01, part D. LLM generation at scale: quality, prior projects, memorization, law, terms and cost](https://tamnd.github.io/math2/spec/corpus/01d-model-generation.md)
- [K01, part E. Representation, pedagogy, exercises, packaging and notation](https://tamnd.github.io/math2/spec/corpus/01e-representation-and-packaging.md)
- [K02. The goal of the corpus](https://tamnd.github.io/math2/spec/corpus/02-the-goal.md)
- [K03. Identifiers: tags of ten characters](https://tamnd.github.io/math2/spec/corpus/03-identifiers.md)
- [K04. The inventory](https://tamnd.github.io/math2/spec/corpus/04-inventory.md)
- [K05. Human sources](https://tamnd.github.io/math2/spec/corpus/05-human-sources.md)
- [K06. Formal sources](https://tamnd.github.io/math2/spec/corpus/06-formal-sources.md)
- [K07. Machine generation: what models write, and how](https://tamnd.github.io/math2/spec/corpus/07-machine-generation.md)
- [K08. The node card: what a good node contains](https://tamnd.github.io/math2/spec/corpus/08-the-node-card.md)
- [K09. Quality: how the corpus stays correct](https://tamnd.github.io/math2/spec/corpus/09-quality.md)
- [K10. Structure, notation and conventions](https://tamnd.github.io/math2/spec/corpus/10-structure-and-notation.md)
- [K11. Packaging: from nodes to courses, books and articles](https://tamnd.github.io/math2/spec/corpus/11-packaging.md)
- [K12. Pipeline and operations](https://tamnd.github.io/math2/spec/corpus/12-pipeline-and-operations.md)
- [K13. Law and ethics](https://tamnd.github.io/math2/spec/corpus/13-legal-and-ethics.md)
- [K14. Roadmap](https://tamnd.github.io/math2/spec/corpus/14-roadmap.md)
- [K15. Open questions](https://tamnd.github.io/math2/spec/corpus/15-open-questions.md)

## Nodes

- [Vector space](https://tamnd.github.io/math2/node/000AYPA9AJ.md): concept 000AYPA9AJ
- [The integers are a Euclidean domain](https://tamnd.github.io/math2/node/000JFQGDXG.md): statement 000JFQGDXG
- [Sequential characterization of limits](https://tamnd.github.io/math2/node/000MXHEY22.md): statement 000MXHEY22
- [Singular value decomposition](https://tamnd.github.io/math2/node/000NPTMX6J.md): statement 000NPTMX6J
- [Quotient set](https://tamnd.github.io/math2/node/0011MTQMR6.md): concept 0011MTQMR6
- [Mutual independence of events](https://tamnd.github.io/math2/node/001B78R0DJ.md): concept 001B78R0DJ
- [Partition function](https://tamnd.github.io/math2/node/001CX45SWQ.md): definition 001CX45SWQ
- [Block diagonal matrix](https://tamnd.github.io/math2/node/001T5FJDC1.md): concept 001T5FJDC1
- [Power series are differentiable inside the radius of convergence](https://tamnd.github.io/math2/node/002X7EK91H.md): statement 002X7EK91H
- [A function with zero derivative is constant](https://tamnd.github.io/math2/node/003QDZ4T16.md): statement 003QDZ4T16
- [Closure is the set of limits of sequences](https://tamnd.github.io/math2/node/003QP7A8HG.md): statement 003QP7A8HG
- [Differential equation](https://tamnd.github.io/math2/node/003R0V5M7V.md): concept 003R0V5M7V
- [Natural logarithm](https://tamnd.github.io/math2/node/0044CPHJBV.md): definition 0044CPHJBV
- [Jump discontinuity](https://tamnd.github.io/math2/node/004FH91BN5.md): concept 004FH91BN5
- [Exterior derivative](https://tamnd.github.io/math2/node/004FRCXH6M.md): definition 004FRCXH6M
- [Power set Boolean algebra](https://tamnd.github.io/math2/node/004J7ZEFAT.md): example 004J7ZEFAT
- [Square-free integer](https://tamnd.github.io/math2/node/004JW7A2N5.md): definition 004JW7A2N5
- [Sylow p-subgroup](https://tamnd.github.io/math2/node/0053CQE9Y2.md): concept 0053CQE9Y2
- [A differentiable function is convex iff its derivative is increasing](https://tamnd.github.io/math2/node/005JT76HCK.md): statement 005JT76HCK
- [Taylor's theorem with the Cauchy remainder](https://tamnd.github.io/math2/node/005Q84BMAD.md): statement 005Q84BMAD
- [Row equivalence](https://tamnd.github.io/math2/node/006BHFWMAP.md): concept 006BHFWMAP
- [The characteristic of an integral domain is zero or prime](https://tamnd.github.io/math2/node/00758QM6JJ.md): statement 00758QM6JJ
- [Quotient ring](https://tamnd.github.io/math2/node/0076HWHGHQ.md): concept 0076HWHGHQ
- [Least common multiple](https://tamnd.github.io/math2/node/007NPQX34K.md): concept 007NPQX34K
- [Variance](https://tamnd.github.io/math2/node/007QED3E0Y.md): definition 007QED3E0Y
- [Dimension of the dual space](https://tamnd.github.io/math2/node/00889YD15K.md): statement 00889YD15K
- [Composite number](https://tamnd.github.io/math2/node/008BH74D7R.md): concept 008BH74D7R
- [Zero ring](https://tamnd.github.io/math2/node/008TBCYDPX.md): example 008TBCYDPX
- [Linear map of a matrix](https://tamnd.github.io/math2/node/009892PZ30.md): definition 009892PZ30
- [Local maximum](https://tamnd.github.io/math2/node/00999TYPXB.md): definition 00999TYPXB
- [Scaling a row scales the determinant](https://tamnd.github.io/math2/node/009CY8JXFX.md): statement 009CY8JXFX
- [Sum of divisors](https://tamnd.github.io/math2/node/009QCP5T3Z.md): definition 009QCP5T3Z
- [Stars and bars](https://tamnd.github.io/math2/node/009TEMTQKH.md): statement 009TEMTQKH
- [Normed vector space](https://tamnd.github.io/math2/node/00A0RPAQ8A.md): definition 00A0RPAQ8A
- [Principal ideal](https://tamnd.github.io/math2/node/00A2EDFT9Q.md): definition 00A2EDFT9Q
- [One-sided limit](https://tamnd.github.io/math2/node/00BE2J8H05.md): concept 00BE2J8H05
- [Platonic solid](https://tamnd.github.io/math2/node/00BETAJNZ4.md): definition 00BETAJNZ4
- [Composition of linear maps](https://tamnd.github.io/math2/node/00BKRZKMSS.md): concept 00BKRZKMSS
- [Picard–Lindelöf theorem in integral form](https://tamnd.github.io/math2/node/00BT70ND50.md): statement 00BT70ND50
- [Finite intersection property](https://tamnd.github.io/math2/node/00BWV5CKCF.md): statement 00BWV5CKCF
- [Graph](https://tamnd.github.io/math2/node/00C0BR7D7H.md): concept 00C0BR7D7H
- [Polyhedron](https://tamnd.github.io/math2/node/00C2WN1H21.md): definition 00C2WN1H21
- [Euler–Mascheroni constant](https://tamnd.github.io/math2/node/00C5P9JZBK.md): object 00C5P9JZBK
- [Addition formula for Fibonacci numbers](https://tamnd.github.io/math2/node/00CN2F997Q.md): statement 00CN2F997Q
- [Countably infinite set](https://tamnd.github.io/math2/node/00CR00C9DN.md): concept 00CR00C9DN
- [Graph isomorphism](https://tamnd.github.io/math2/node/00CT0PTH74.md): definition 00CT0PTH74
- [The logarithm grows slower than any power](https://tamnd.github.io/math2/node/00DGYY10FV.md): statement 00DGYY10FV
- [Laws of exponents in a group](https://tamnd.github.io/math2/node/00DRK90Z6K.md): statement 00DRK90Z6K
- [−1 is a square modulo p if and only if p is not 3 modulo 4](https://tamnd.github.io/math2/node/00DW862T78.md): statement 00DW862T78
- [Power set](https://tamnd.github.io/math2/node/00E0YZCERH.md): concept 00E0YZCERH
- [Euclidean vector](https://tamnd.github.io/math2/node/00EBGSR5NY.md): definition 00EBGSR5NY
- [Root of a polynomial](https://tamnd.github.io/math2/node/00EHX6FWC0.md): definition 00EHX6FWC0
- [Hasse diagram](https://tamnd.github.io/math2/node/00EYM7MDKS.md): concept 00EYM7MDKS
- [Orbit](https://tamnd.github.io/math2/node/00F2JFA41J.md): definition 00F2JFA41J
- [Closed map](https://tamnd.github.io/math2/node/00FSJTXPJM.md): concept 00FSJTXPJM
- [Congruence is compatible with addition](https://tamnd.github.io/math2/node/00FWBN7F0Y.md): statement 00FWBN7F0Y
- [Contour integral](https://tamnd.github.io/math2/node/00FWNP9TXY.md): definition 00FWNP9TXY
- [Zero of a holomorphic function](https://tamnd.github.io/math2/node/00GC8ZQP4Q.md): concept 00GC8ZQP4Q
- [Modular multiplicative inverse](https://tamnd.github.io/math2/node/00GXTQMY8Z.md): definition 00GXTQMY8Z
- [Direct product of groups](https://tamnd.github.io/math2/node/00HF4RJ5QN.md): concept 00HF4RJ5QN
- [Conic section](https://tamnd.github.io/math2/node/00HJCMC2CD.md): concept 00HJCMC2CD
- [Norm of a Gaussian integer](https://tamnd.github.io/math2/node/00HPT88ZX0.md): definition 00HPT88ZX0
- [Trisecting the angle](https://tamnd.github.io/math2/node/00J3DCCEK5.md): problem 00J3DCCEK5
- [Lower triangular matrix](https://tamnd.github.io/math2/node/00J3KZP8Y0.md): concept 00J3KZP8Y0
- [Divisibility rule](https://tamnd.github.io/math2/node/00J8E3HX0M.md): concept 00J8E3HX0M
- [Multinomial coefficient](https://tamnd.github.io/math2/node/00K0RF9BFY.md): definition 00K0RF9BFY
- [Nullity](https://tamnd.github.io/math2/node/00K4FK4NBH.md): definition 00K4FK4NBH
- [Non-uniqueness for a non-Lipschitz equation](https://tamnd.github.io/math2/node/00K7ASZ1PR.md): counterexample 00K7ASZ1PR
- [Intermediate value theorem](https://tamnd.github.io/math2/node/00KAFBR32E.md): statement 00KAFBR32E
- [Determinant](https://tamnd.github.io/math2/node/00KDZRW4C0.md): concept 00KDZRW4C0
- [Modulus of the conjugate](https://tamnd.github.io/math2/node/00KRCVE7CK.md): statement 00KRCVE7CK
- [Prime ideal](https://tamnd.github.io/math2/node/00MHHZ9SNE.md): concept 00MHHZ9SNE
- [Independent and identically distributed random variables](https://tamnd.github.io/math2/node/00MRN0FARV.md): concept 00MRN0FARV
- [Every vector space has a basis](https://tamnd.github.io/math2/node/00MS9VMQ9Z.md): statement 00MS9VMQ9Z
- [Saddle point](https://tamnd.github.io/math2/node/00N3T18QNA.md): definition 00N3T18QNA
- [Change of variables in multiple integrals](https://tamnd.github.io/math2/node/00N5FHP1CV.md): statement 00N5FHP1CV
- [Prior probability](https://tamnd.github.io/math2/node/00NBBSXRQS.md): concept 00NBBSXRQS
- [Complex sine and cosine](https://tamnd.github.io/math2/node/00NZKRSWNX.md): concept 00NZKRSWNX
- [Power of a point](https://tamnd.github.io/math2/node/00PPY3RPNB.md): definition 00PPY3RPNB
- [Positive definite quadratic form](https://tamnd.github.io/math2/node/00QN1N6MNT.md): definition 00QN1N6MNT
- [Primitive element of a finite field](https://tamnd.github.io/math2/node/00QSK85Y2X.md): concept 00QSK85Y2X
- [Compact set](https://tamnd.github.io/math2/node/00R0X1WGPX.md): definition 00R0X1WGPX
- [Arc length parametrisation](https://tamnd.github.io/math2/node/00R1JDDV5X.md): definition 00R1JDDV5X
- [Derivative of sine](https://tamnd.github.io/math2/node/00R4MWW532.md): statement 00R4MWW532
- [Divergence of the sum of the reciprocals of the primes](https://tamnd.github.io/math2/node/00RA98X2CP.md): statement 00RA98X2CP
- [Second-countable space](https://tamnd.github.io/math2/node/00RE1R4W7N.md): definition 00RE1R4W7N
- [Absolute value](https://tamnd.github.io/math2/node/00S7C68D51.md): definition 00S7C68D51
- [Asymptotic equivalence](https://tamnd.github.io/math2/node/00SB5EA3VW.md): definition 00SB5EA3VW
- [Gram matrix](https://tamnd.github.io/math2/node/00SPJSDT3Z.md): concept 00SPJSDT3Z
- [Event](https://tamnd.github.io/math2/node/00SQGJNRTB.md): definition 00SQGJNRTB
- [Exponential growth and decay](https://tamnd.github.io/math2/node/00SVP6W3CB.md): example 00SVP6W3CB
- [Transcendental element](https://tamnd.github.io/math2/node/00SW8B91E7.md): definition 00SW8B91E7
- [Little o notation](https://tamnd.github.io/math2/node/00T5EPFQB9.md): concept 00T5EPFQB9
- [Ramsey's theorem](https://tamnd.github.io/math2/node/00TQV3F5S4.md): statement 00TQV3F5S4
- [Picard operator](https://tamnd.github.io/math2/node/00WAVB68F9.md): definition 00WAVB68F9
- [Irreducible polynomial](https://tamnd.github.io/math2/node/00X2A9G9W3.md): definition 00X2A9G9W3
- [Lotka–Volterra equations](https://tamnd.github.io/math2/node/00Y5QZ82AA.md): example 00Y5QZ82AA
- [A finite group is nilpotent iff it is the product of its Sylow subgroups](https://tamnd.github.io/math2/node/00Y9N80VSS.md): statement 00Y9N80VSS
- [Linear subspace](https://tamnd.github.io/math2/node/00YC1D652K.md): concept 00YC1D652K
- [Volume of a ball in R^n](https://tamnd.github.io/math2/node/00YH28R457.md): statement 00YH28R457
- [Arcsine](https://tamnd.github.io/math2/node/00Z1YBJJW8.md): concept 00Z1YBJJW8
- [Additivity of line integrals under concatenation](https://tamnd.github.io/math2/node/00ZEBSMNNV.md): statement 00ZEBSMNNV
- [Topological embedding](https://tamnd.github.io/math2/node/00ZG9Y9TR4.md): concept 00ZG9Y9TR4
- [Ordered field](https://tamnd.github.io/math2/node/010JCV9J4E.md): definition 010JCV9J4E
- [Uniqueness of finite fields of a given order](https://tamnd.github.io/math2/node/010PKCQPXP.md): statement 010PKCQPXP
- [Hensel's lemma](https://tamnd.github.io/math2/node/010QVB71T2.md): statement 010QVB71T2
- [Recurrence for Bell numbers](https://tamnd.github.io/math2/node/0110GBHPPB.md): statement 0110GBHPPB
- [Image of a homomorphism](https://tamnd.github.io/math2/node/0112DTS464.md): definition 0112DTS464
- [Partial order](https://tamnd.github.io/math2/node/011C29Q1S1.md): definition 011C29Q1S1
- [Strictly monotone function](https://tamnd.github.io/math2/node/011EJQWWJS.md): concept 011EJQWWJS
- [A convex function on an open interval is continuous](https://tamnd.github.io/math2/node/011HV2GAWF.md): statement 011HV2GAWF
- [Jordan–Chevalley decomposition](https://tamnd.github.io/math2/node/011K11VPJN.md): statement 011K11VPJN
- [Evaluation homomorphism](https://tamnd.github.io/math2/node/0125QVNCFQ.md): definition 0125QVNCFQ
- [1 + x is at most exp x](https://tamnd.github.io/math2/node/012C7GFF20.md): statement 012C7GFF20
- [Hölder continuity](https://tamnd.github.io/math2/node/012PBA0FMD.md): concept 012PBA0FMD
- [Reduction formula](https://tamnd.github.io/math2/node/012RYET5E2.md): concept 012RYET5E2
- [Differentiable function of several variables](https://tamnd.github.io/math2/node/0135XZZQ8M.md): definition 0135XZZQ8M
- [exp z = 1 iff z is an integer multiple of 2πi](https://tamnd.github.io/math2/node/013AHC4DRT.md): statement 013AHC4DRT
- [Existence of splitting fields](https://tamnd.github.io/math2/node/01441KDGBY.md): statement 01441KDGBY
- [Strict differentiability](https://tamnd.github.io/math2/node/0146MBQNNE.md): definition 0146MBQNNE
- [Semidirect product](https://tamnd.github.io/math2/node/014E1DZQE8.md): definition 014E1DZQE8
- [Chernoff bound](https://tamnd.github.io/math2/node/014PEBGDYF.md): statement 014PEBGDYF
- [Cayley's theorem](https://tamnd.github.io/math2/node/0151VYVX27.md): statement 0151VYVX27
- [Integral equation form of an initial value problem](https://tamnd.github.io/math2/node/015KK993C5.md): statement 015KK993C5
- [Glide reflection](https://tamnd.github.io/math2/node/0167F9B344.md): concept 0167F9B344
- [Leibniz formula for determinants](https://tamnd.github.io/math2/node/016PE4VZ8Q.md): statement 016PE4VZ8Q
- [Principal branch](https://tamnd.github.io/math2/node/017BPGVEE8.md): concept 017BPGVEE8
- [Alternating series](https://tamnd.github.io/math2/node/017BR1M98J.md): definition 017BR1M98J
- [Riemann's criterion for integrability](https://tamnd.github.io/math2/node/017DH8WXTE.md): statement 017DH8WXTE
- [Platonic solid](https://tamnd.github.io/math2/node/017E5PRS1Q.md): concept 017E5PRS1Q
- [Dyck words are counted by Catalan numbers](https://tamnd.github.io/math2/node/017SRZ7NE9.md): statement 017SRZ7NE9
- [Prime number](https://tamnd.github.io/math2/node/01826KMG2H.md): concept 01826KMG2H
- [Vector-valued function](https://tamnd.github.io/math2/node/018GTRS7A4.md): concept 018GTRS7A4
- [Finite-dimensional vector space](https://tamnd.github.io/math2/node/018W3V3TCT.md): definition 018W3V3TCT
- [Rolle's theorem](https://tamnd.github.io/math2/node/0194DY35BG.md): statement 0194DY35BG
- [Residue at a simple pole](https://tamnd.github.io/math2/node/01991NCP7H.md): statement 01991NCP7H
- [Derivative of the complex exponential](https://tamnd.github.io/math2/node/019H1Z6419.md): statement 019H1Z6419
- [Root of a polynomial](https://tamnd.github.io/math2/node/019MA1VTR2.md): concept 019MA1VTR2
- [Unit of a ring](https://tamnd.github.io/math2/node/019NFGFPR5.md): definition 019NFGFPR5
- [Integral test](https://tamnd.github.io/math2/node/019QZ6RRVH.md): statement 019QZ6RRVH
- [Maximal solution](https://tamnd.github.io/math2/node/019VSXSFMQ.md): definition 019VSXSFMQ
- [Weierstrass function](https://tamnd.github.io/math2/node/01A0MK1730.md): counterexample 01A0MK1730
- [First derivative test](https://tamnd.github.io/math2/node/01A2EGXJEM.md): statement 01A2EGXJEM
- [cos pi = -1](https://tamnd.github.io/math2/node/01A8JGR4JM.md): statement 01A8JGR4JM
- [Distance from a point to a line](https://tamnd.github.io/math2/node/01AAYNSBN6.md): statement 01AAYNSBN6
- [Preimage of a set](https://tamnd.github.io/math2/node/01AHG4G9RT.md): concept 01AHG4G9RT
- [Chapman–Kolmogorov equation](https://tamnd.github.io/math2/node/01APZ8GDQH.md): statement 01APZ8GDQH
- [Block matrix](https://tamnd.github.io/math2/node/01AXG9KZQF.md): definition 01AXG9KZQF
- [Bias of an estimator](https://tamnd.github.io/math2/node/01BD2TFFP4.md): definition 01BD2TFFP4
- [Well-ordering principle](https://tamnd.github.io/math2/node/01BX15HSRR.md): statement 01BX15HSRR
- [Well-order](https://tamnd.github.io/math2/node/01CGSJ0GNQ.md): concept 01CGSJ0GNQ
- [The complex numbers cannot be ordered](https://tamnd.github.io/math2/node/01CJRJKDAA.md): statement 01CJRJKDAA
- [Characteristic subgroup](https://tamnd.github.io/math2/node/01CS8Q2APC.md): definition 01CS8Q2APC
- [Totally disconnected space](https://tamnd.github.io/math2/node/01CYW8S04D.md): definition 01CYW8S04D
- [Periodic orbit](https://tamnd.github.io/math2/node/01DSNP0B4R.md): definition 01DSNP0B4R
- [Ramsey number](https://tamnd.github.io/math2/node/01DSVCRF6X.md): definition 01DSVCRF6X
- [Normality via left and right cosets](https://tamnd.github.io/math2/node/01E7SWYJX3.md): statement 01E7SWYJX3
- [Public-key cryptography](https://tamnd.github.io/math2/node/01E9MNJYEX.md): concept 01E9MNJYEX
- [Inner product space](https://tamnd.github.io/math2/node/01EK923SKP.md): definition 01EK923SKP
- [Law of a random variable](https://tamnd.github.io/math2/node/01EQPZ6RYZ.md): concept 01EQPZ6RYZ
- [Determinant of a scalar multiple](https://tamnd.github.io/math2/node/01EXSMK864.md): statement 01EXSMK864
- [Extensions of solvable groups are solvable](https://tamnd.github.io/math2/node/01EY0EM7RP.md): statement 01EY0EM7RP
- [Urysohn's lemma](https://tamnd.github.io/math2/node/01EY3VR7ZD.md): statement 01EY3VR7ZD
- [Composition of injections is injective](https://tamnd.github.io/math2/node/01FPXMTBX1.md): statement 01FPXMTBX1
- [Pythagorean triple](https://tamnd.github.io/math2/node/01G1WY1GVX.md): definition 01G1WY1GVX
- [Periodic continued fraction](https://tamnd.github.io/math2/node/01GBHQ5ABF.md): concept 01GBHQ5ABF
- [Order of cardinals](https://tamnd.github.io/math2/node/01GTP8M5RG.md): definition 01GTP8M5RG
- [Generalised eigenvector](https://tamnd.github.io/math2/node/01H6QKWGDH.md): concept 01H6QKWGDH
- [Divergent sequence](https://tamnd.github.io/math2/node/01HR2AH0BX.md): definition 01HR2AH0BX
- [Equivalence classes form a partition](https://tamnd.github.io/math2/node/01HT2CH8BH.md): statement 01HT2CH8BH
- [Maximal solution](https://tamnd.github.io/math2/node/01J7FZ65J0.md): concept 01J7FZ65J0
- [Legendre symbol](https://tamnd.github.io/math2/node/01J8AMV9RS.md): definition 01J8AMV9RS
- [Vandermonde's identity](https://tamnd.github.io/math2/node/01JADB9VW5.md): statement 01JADB9VW5
- [Regular polygon](https://tamnd.github.io/math2/node/01KJJ9A400.md): concept 01KJJ9A400
- [Multigraph](https://tamnd.github.io/math2/node/01M5T0KDG7.md): definition 01M5T0KDG7
- [Triangle inequality for a metric](https://tamnd.github.io/math2/node/01M6CW9VM6.md): statement 01M6CW9VM6
- [Antisymmetric relation](https://tamnd.github.io/math2/node/01MGYYBB9R.md): definition 01MGYYBB9R
- [Infinite set](https://tamnd.github.io/math2/node/01MKW4R283.md): definition 01MKW4R283
- [Positive semidefinite matrix](https://tamnd.github.io/math2/node/01MMVW8CKC.md): concept 01MMVW8CKC
- [Cauchy's integral theorem](https://tamnd.github.io/math2/node/01MPZXYVYF.md): statement 01MPZXYVYF
- [Integers modulo n](https://tamnd.github.io/math2/node/01N3AYAA6N.md): definition 01N3AYAA6N
- [Nondegenerate bilinear form](https://tamnd.github.io/math2/node/01NHFZF2SK.md): concept 01NHFZF2SK
- [Function with unequal mixed partial derivatives](https://tamnd.github.io/math2/node/01NN8NY497.md): counterexample 01NN8NY497
- [Isolated point](https://tamnd.github.io/math2/node/01PKQYTTMT.md): definition 01PKQYTTMT
- [Similar triangles](https://tamnd.github.io/math2/node/01PNAVFMYZ.md): definition 01PNAVFMYZ
- [Hyperbolic functions](https://tamnd.github.io/math2/node/01PPDSDP2N.md): concept 01PPDSDP2N
- [Spherical coordinates](https://tamnd.github.io/math2/node/01QFQAQ2PA.md): concept 01QFQAQ2PA
- [sin x / x tends to 1](https://tamnd.github.io/math2/node/01QSSKS5C1.md): statement 01QSSKS5C1
- [Sum of the first n natural numbers](https://tamnd.github.io/math2/node/01QZNHQWYD.md): statement 01QZNHQWYD
- [A graph is bipartite iff it has no odd cycle](https://tamnd.github.io/math2/node/01R1HPEK6S.md): statement 01R1HPEK6S
- [Tangent line](https://tamnd.github.io/math2/node/01R1HW3Y42.md): concept 01R1HW3Y42
- [Definite and indefinite quadratic forms](https://tamnd.github.io/math2/node/01RBSSGMY5.md): concept 01RBSSGMY5
- [Elementary matrix](https://tamnd.github.io/math2/node/01REXW3WWE.md): concept 01REXW3WWE
- [Lyapunov function](https://tamnd.github.io/math2/node/01REYCJ371.md): concept 01REYCJ371
- [Residue class](https://tamnd.github.io/math2/node/01RSTGNTXG.md): concept 01RSTGNTXG
- [Surjective function](https://tamnd.github.io/math2/node/01RVCMSN5N.md): concept 01RVCMSN5N
- [Composition of relations](https://tamnd.github.io/math2/node/01SWBKDAWA.md): definition 01SWBKDAWA
- [Neighbourhood](https://tamnd.github.io/math2/node/01T3TMGTTE.md): definition 01T3TMGTTE
- [Integrating factor](https://tamnd.github.io/math2/node/01THH3Y3YP.md): definition 01THH3Y3YP
- [Diffie–Hellman key exchange](https://tamnd.github.io/math2/node/01TKKESHSB.md): example 01TKKESHSB
- [Hyperbolic equilibrium point](https://tamnd.github.io/math2/node/01TPMXM6NM.md): concept 01TPMXM6NM
- [Countable set](https://tamnd.github.io/math2/node/01TQB88H4C.md): concept 01TQB88H4C
- [Orthogonal family for a bilinear form](https://tamnd.github.io/math2/node/01TXN7ETH2.md): concept 01TXN7ETH2
- [Symmetry of second derivatives](https://tamnd.github.io/math2/node/01V9FPBBVK.md): statement 01V9FPBBVK
- [Bijective iff it has a two-sided inverse](https://tamnd.github.io/math2/node/01VG2A0BHW.md): statement 01VG2A0BHW
- [Prime element](https://tamnd.github.io/math2/node/01VH21EV6F.md): concept 01VH21EV6F
- [Quadratic integer](https://tamnd.github.io/math2/node/01VN79T2PR.md): definition 01VN79T2PR
- [Uniqueness of the algebraic closure](https://tamnd.github.io/math2/node/01VQ7C4FTB.md): statement 01VQ7C4FTB
- [Cantor set is closed](https://tamnd.github.io/math2/node/01VT24J1SD.md): statement 01VT24J1SD
- [Connected component](https://tamnd.github.io/math2/node/01VVP10MWJ.md): definition 01VVP10MWJ
- [Arguments add under multiplication](https://tamnd.github.io/math2/node/01W6RKJP01.md): statement 01W6RKJP01
- [Central binomial coefficient](https://tamnd.github.io/math2/node/01WHD9T0V6.md): definition 01WHD9T0V6
- [Legendre's formula](https://tamnd.github.io/math2/node/01WPV9PHN7.md): statement 01WPV9PHN7
- [Cassini's identity](https://tamnd.github.io/math2/node/01WQZK97Z5.md): statement 01WQZK97Z5
- [Geometric series](https://tamnd.github.io/math2/node/01X5CVV1Q5.md): example 01X5CVV1Q5
- [Identically distributed random variables](https://tamnd.github.io/math2/node/01X8YSDTBW.md): definition 01X8YSDTBW
- [Moment](https://tamnd.github.io/math2/node/01XAHPADBA.md): concept 01XAHPADBA
- [Uniformly Cauchy sequence of functions](https://tamnd.github.io/math2/node/01XAY17QA3.md): definition 01XAY17QA3
- [Constructive proof](https://tamnd.github.io/math2/node/01XM7JEKHY.md): concept 01XM7JEKHY
- [Limit at infinity](https://tamnd.github.io/math2/node/01YYRV7S9Z.md): concept 01YYRV7S9Z
- [Open set](https://tamnd.github.io/math2/node/01ZQQDCERP.md): definition 01ZQQDCERP
- [Adding a multiple of one row to another preserves the determinant](https://tamnd.github.io/math2/node/01ZTQBK64Q.md): statement 01ZTQBK64Q
- [Law of the unconscious statistician](https://tamnd.github.io/math2/node/01ZWX875AT.md): statement 01ZWX875AT
- [Carathéodory's theorem](https://tamnd.github.io/math2/node/0201Q4P2B5.md): statement 0201Q4P2B5
- [Euler's theorem on Eulerian circuits](https://tamnd.github.io/math2/node/020422MDQF.md): statement 020422MDQF
- [Existence of prime factorization](https://tamnd.github.io/math2/node/020K0X7HVR.md): statement 020K0X7HVR
- [Matrix exponential](https://tamnd.github.io/math2/node/020M04FN1H.md): concept 020M04FN1H
- [Totally bounded set](https://tamnd.github.io/math2/node/020Q06TYYH.md): concept 020Q06TYYH
- [Free group](https://tamnd.github.io/math2/node/020Z5SEN27.md): definition 020Z5SEN27
- [Definite integral](https://tamnd.github.io/math2/node/021WEM680W.md): definition 021WEM680W
- [Darboux and Riemann integrals agree](https://tamnd.github.io/math2/node/021XJAQVDC.md): statement 021XJAQVDC
- [Taylor series](https://tamnd.github.io/math2/node/0225QPWRRE.md): definition 0225QPWRRE
- [Order of cardinals](https://tamnd.github.io/math2/node/022RSX714V.md): concept 022RSX714V
- [Well-founded relation](https://tamnd.github.io/math2/node/0230ECX1Q5.md): definition 0230ECX1Q5
- [Euclidean algorithm in a Euclidean domain](https://tamnd.github.io/math2/node/023VY493K0.md): concept 023VY493K0
- [Left inverse](https://tamnd.github.io/math2/node/02486SD328.md): definition 02486SD328
- [Reflection](https://tamnd.github.io/math2/node/025158VEYZ.md): definition 025158VEYZ
- [Uniqueness of solutions for Lipschitz vector fields](https://tamnd.github.io/math2/node/025PKW8AMR.md): statement 025PKW8AMR
- [Conjunction](https://tamnd.github.io/math2/node/025YZN3VJW.md): concept 025YZN3VJW
- [First integral](https://tamnd.github.io/math2/node/0262JP0NV9.md): concept 0262JP0NV9
- [A function with nonnegative derivative is increasing](https://tamnd.github.io/math2/node/026BJBQNGJ.md): statement 026BJBQNGJ
- [Uniformly continuous functions map Cauchy sequences to Cauchy sequences](https://tamnd.github.io/math2/node/026FDZFRXS.md): statement 026FDZFRXS
- [Parallelogram law](https://tamnd.github.io/math2/node/026HPGYGS4.md): statement 026HPGYGS4
- [Flux through a surface](https://tamnd.github.io/math2/node/026J1MG2NW.md): concept 026J1MG2NW
- [Density of the normal distribution](https://tamnd.github.io/math2/node/027JF1FBB0.md): definition 027JF1FBB0
- [Derivative function](https://tamnd.github.io/math2/node/027K9RMZ43.md): definition 027K9RMZ43
- [Similar matrices](https://tamnd.github.io/math2/node/027KDW2K72.md): definition 027KDW2K72
- [Reflection in a line or plane](https://tamnd.github.io/math2/node/027MBSVPY3.md): definition 027MBSVPY3
- [Quotients of solvable groups are solvable](https://tamnd.github.io/math2/node/027ZQS04SJ.md): statement 027ZQS04SJ
- [Associate elements](https://tamnd.github.io/math2/node/02879QH7BY.md): concept 02879QH7BY
- [Angle addition formulas](https://tamnd.github.io/math2/node/028XEQCE88.md): statement 028XEQCE88
- [Exponential distribution](https://tamnd.github.io/math2/node/028Z8WC5Y9.md): definition 028Z8WC5Y9
- [Flow of a vector field](https://tamnd.github.io/math2/node/02903SSA48.md): definition 02903SSA48
- [Euclid's theorem](https://tamnd.github.io/math2/node/0292EPPPT0.md): statement 0292EPPPT0
- [Vector field](https://tamnd.github.io/math2/node/0299VCW1TH.md): concept 0299VCW1TH
- [Negation](https://tamnd.github.io/math2/node/029JBBF693.md): concept 029JBBF693
- [Planar graph](https://tamnd.github.io/math2/node/029JQEK15B.md): definition 029JQEK15B
- [Binary tree](https://tamnd.github.io/math2/node/02A8EMC83C.md): definition 02A8EMC83C
- [Holomorphic functions on a disc have primitives](https://tamnd.github.io/math2/node/02AC7JR6G6.md): statement 02AC7JR6G6
- [Convex set](https://tamnd.github.io/math2/node/02AS45T3H9.md): concept 02AS45T3H9
- [Infinite cyclic group](https://tamnd.github.io/math2/node/02B1T50JD1.md): example 02B1T50JD1
- [Open set](https://tamnd.github.io/math2/node/02BH3MCBQB.md): concept 02BH3MCBQB
- [Eisenstein's criterion](https://tamnd.github.io/math2/node/02BPKD7TMD.md): statement 02BPKD7TMD
- [Covariance matrix](https://tamnd.github.io/math2/node/02CBHV4ZZQ.md): definition 02CBHV4ZZQ
- [Edge colouring](https://tamnd.github.io/math2/node/02CDYWSWSY.md): definition 02CDYWSWSY
- [Cardinality of a finite set](https://tamnd.github.io/math2/node/02CK6QMXYD.md): concept 02CK6QMXYD
- [Mean value inequality](https://tamnd.github.io/math2/node/02CYB184RC.md): statement 02CYB184RC
- [Sign of a permutation](https://tamnd.github.io/math2/node/02DKV69ZH0.md): concept 02DKV69ZH0
- [Transposition](https://tamnd.github.io/math2/node/02DTRYKGE8.md): definition 02DTRYKGE8
- [Free module](https://tamnd.github.io/math2/node/02E1T12PP4.md): definition 02E1T12PP4
- [Power of a point](https://tamnd.github.io/math2/node/02E9VT1RVQ.md): concept 02E9VT1RVQ
- [Open balls form a basis of the metric topology](https://tamnd.github.io/math2/node/02EEX96DFS.md): statement 02EEX96DFS
- [Prime factorization](https://tamnd.github.io/math2/node/02EKEPPJ3R.md): concept 02EKEPPJ3R
- [Rank of a product of matrices](https://tamnd.github.io/math2/node/02FDWY55AS.md): statement 02FDWY55AS
- [Transitive action](https://tamnd.github.io/math2/node/02GK6HZYBT.md): concept 02GK6HZYBT
- [Newton's law of cooling](https://tamnd.github.io/math2/node/02HB5F40VR.md): example 02HB5F40VR
- [Casorati–Weierstrass theorem](https://tamnd.github.io/math2/node/02HHNPC8ED.md): statement 02HHNPC8ED
- [Existence of maximal ideals](https://tamnd.github.io/math2/node/02HJYT53F7.md): statement 02HJYT53F7
- [Isometry](https://tamnd.github.io/math2/node/02HPP5VCZH.md): concept 02HPP5VCZH
- [Nullcline](https://tamnd.github.io/math2/node/02HR4C0B04.md): concept 02HR4C0B04
- [Inverse trigonometric functions](https://tamnd.github.io/math2/node/02HTDK1FBD.md): definition 02HTDK1FBD
- [Chebyshev's inequality](https://tamnd.github.io/math2/node/02HZ1BK3MW.md): statement 02HZ1BK3MW
- [Isometries are injective](https://tamnd.github.io/math2/node/02J4G4J4TJ.md): statement 02J4G4J4TJ
- [Monotone sequence](https://tamnd.github.io/math2/node/02JBEMG8R4.md): concept 02JBEMG8R4
- [Characteristic function of a sum of independent random variables](https://tamnd.github.io/math2/node/02JGYG0T7M.md): statement 02JGYG0T7M
- [Power series of sine](https://tamnd.github.io/math2/node/02KTRHQMHN.md): statement 02KTRHQMHN
- [Stationary distribution](https://tamnd.github.io/math2/node/02KYGSCTCN.md): definition 02KYGSCTCN
- [Fermat pseudoprime](https://tamnd.github.io/math2/node/02M2K4QH9P.md): definition 02M2K4QH9P
- [Polynomial evaluation](https://tamnd.github.io/math2/node/02MAKWDKGT.md): concept 02MAKWDKGT
- [Discrete uniform distribution](https://tamnd.github.io/math2/node/02MY6WZXEQ.md): concept 02MY6WZXEQ
- [Intermediate field](https://tamnd.github.io/math2/node/02NBKVQFRQ.md): definition 02NBKVQFRQ
- [Divergence](https://tamnd.github.io/math2/node/02NSMQVQWK.md): definition 02NSMQVQWK
- [Splitting field](https://tamnd.github.io/math2/node/02NZZ1J0FH.md): definition 02NZZ1J0FH
- [The complex exponential is 2πi-periodic](https://tamnd.github.io/math2/node/02P2PYH3X3.md): statement 02P2PYH3X3
- [Sum of the roots of unity](https://tamnd.github.io/math2/node/02PETK9PQ4.md): statement 02PETK9PQ4
- [Complex logarithm](https://tamnd.github.io/math2/node/02QGGD5G87.md): concept 02QGGD5G87
- [Line integral of a scalar field](https://tamnd.github.io/math2/node/02QJVFCNYD.md): concept 02QJVFCNYD
- [Multiplicative order](https://tamnd.github.io/math2/node/02RRHJ7R98.md): concept 02RRHJ7R98
- [Higher partial derivatives](https://tamnd.github.io/math2/node/02RSXVM1ZN.md): definition 02RSXVM1ZN
- [Algebraically closed field](https://tamnd.github.io/math2/node/02SJG1HC32.md): definition 02SJG1HC32
- [Topological property](https://tamnd.github.io/math2/node/02SMVGC690.md): concept 02SMVGC690
- [Reduction of order](https://tamnd.github.io/math2/node/02SRXDAM3X.md): concept 02SRXDAM3X
- [Polar form of a quadratic form](https://tamnd.github.io/math2/node/02STTYFBVY.md): definition 02STTYFBVY
- [Cauchy mean value theorem](https://tamnd.github.io/math2/node/02SYZAWM2K.md): statement 02SYZAWM2K
- [Closure is closed](https://tamnd.github.io/math2/node/02TZDKG50Q.md): statement 02TZDKG50Q
- [Decreasing sequence bounded below converges](https://tamnd.github.io/math2/node/02V2GYAC8P.md): statement 02V2GYAC8P
- [Greatest common divisor](https://tamnd.github.io/math2/node/02WF878BH1.md): definition 02WF878BH1
- [Riemann mapping theorem](https://tamnd.github.io/math2/node/02WHMEG5ZR.md): statement 02WHMEG5ZR
- [External direct sum](https://tamnd.github.io/math2/node/02WZB422CE.md): definition 02WZB422CE
- [Bernoulli process](https://tamnd.github.io/math2/node/02XPXR1FBK.md): definition 02XPXR1FBK
- [Kernel of a linear map](https://tamnd.github.io/math2/node/02YFME3RGW.md): definition 02YFME3RGW
- [Conditional convergence](https://tamnd.github.io/math2/node/02YRQXZDCS.md): definition 02YRQXZDCS
- [Branch of the logarithm](https://tamnd.github.io/math2/node/02YSJPQYQH.md): concept 02YSJPQYQH
- [Centraliser](https://tamnd.github.io/math2/node/02YYV36BFM.md): concept 02YYV36BFM
- [One-form](https://tamnd.github.io/math2/node/02Z5B2HA3F.md): concept 02Z5B2HA3F
- [Real and imaginary parts](https://tamnd.github.io/math2/node/02ZE6JHJ7G.md): definition 02ZE6JHJ7G
- [K5 and K3,3 are not planar](https://tamnd.github.io/math2/node/030A3FHN14.md): statement 030A3FHN14
- [Ideal generated by a subset](https://tamnd.github.io/math2/node/030TZ0DQWT.md): definition 030TZ0DQWT
- [Partial fraction decomposition](https://tamnd.github.io/math2/node/030VFEREKG.md): concept 030VFEREKG
- [Commutative ring](https://tamnd.github.io/math2/node/0310REJJY8.md): concept 0310REJJY8
- [Total order](https://tamnd.github.io/math2/node/0316AYH1SC.md): definition 0316AYH1SC
- [The real part of a holomorphic function is harmonic](https://tamnd.github.io/math2/node/031BG625Y3.md): statement 031BG625Y3
- [A tangent is perpendicular to the radius](https://tamnd.github.io/math2/node/031JV5S54D.md): statement 031JV5S54D
- [Order of a cyclic subgroup equals the order of its generator](https://tamnd.github.io/math2/node/031M8HX7DA.md): statement 031M8HX7DA
- [Event](https://tamnd.github.io/math2/node/031TPJVS74.md): concept 031TPJVS74
- [Polyhedron](https://tamnd.github.io/math2/node/0325SGSJAR.md): concept 0325SGSJAR
- [Residue theorem](https://tamnd.github.io/math2/node/032HKYS329.md): statement 032HKYS329
- [Matrix multiplication](https://tamnd.github.io/math2/node/032Q9FB8MT.md): concept 032Q9FB8MT
- [Order on the reals](https://tamnd.github.io/math2/node/0337CYB0VD.md): definition 0337CYB0VD
- [Projection](https://tamnd.github.io/math2/node/033DA1WE8Y.md): concept 033DA1WE8Y
- [Nine-point circle](https://tamnd.github.io/math2/node/033H8MQRC3.md): concept 033H8MQRC3
- [Factorial formula for binomial coefficients](https://tamnd.github.io/math2/node/033KTKKRV6.md): statement 033KTKKRV6
- [Limit inferior](https://tamnd.github.io/math2/node/033RZ32T4Q.md): concept 033RZ32T4Q
- [Markov chain](https://tamnd.github.io/math2/node/033TJBZDG0.md): definition 033TJBZDG0
- [Formal power series](https://tamnd.github.io/math2/node/033WGN00BF.md): concept 033WGN00BF
- [Formula for the modulus](https://tamnd.github.io/math2/node/034FGCEYX4.md): statement 034FGCEYX4
- [Tangent plane to a graph](https://tamnd.github.io/math2/node/035V802J58.md): concept 035V802J58
- [Triangularisation](https://tamnd.github.io/math2/node/0363JMQ42Y.md): statement 0363JMQ42Y
- [Hyperbolic cosine](https://tamnd.github.io/math2/node/036696EMG2.md): concept 036696EMG2
- [Statistic](https://tamnd.github.io/math2/node/036756NFD6.md): definition 036756NFD6
- [Gauss's lemma on quadratic residues](https://tamnd.github.io/math2/node/037ACYXB5Z.md): statement 037ACYXB5Z
- [Dirichlet convolution](https://tamnd.github.io/math2/node/037F589D1D.md): concept 037F589D1D
- [Leibniz formula for pi](https://tamnd.github.io/math2/node/037FD5NBRT.md): statement 037FD5NBRT
- [System of linear differential equations](https://tamnd.github.io/math2/node/037ZNY7NAG.md): concept 037ZNY7NAG
- [Arzelà–Ascoli theorem](https://tamnd.github.io/math2/node/0381S8Z5P0.md): statement 0381S8Z5P0
- [Bernoulli distribution](https://tamnd.github.io/math2/node/0386Y0Z7Y7.md): concept 0386Y0Z7Y7
- [Chromatic polynomial](https://tamnd.github.io/math2/node/038YGY0HSC.md): definition 038YGY0HSC
- [Freshman's dream](https://tamnd.github.io/math2/node/0396VGFA78.md): statement 0396VGFA78
- [Parametric equation of a line](https://tamnd.github.io/math2/node/0399QFDFS8.md): concept 0399QFDFS8
- [Characteristic polynomial](https://tamnd.github.io/math2/node/039HF58KQ5.md): definition 039HF58KQ5
- [Integral of exp(-x) over the half line](https://tamnd.github.io/math2/node/039JAF7ZAN.md): example 039JAF7ZAN
- [Independence of two events](https://tamnd.github.io/math2/node/03AHF0XRVG.md): concept 03AHF0XRVG
- [Taylor's theorem with the Peano remainder](https://tamnd.github.io/math2/node/03AJHX431A.md): statement 03AJHX431A
- [Column space](https://tamnd.github.io/math2/node/03AJKXH1AV.md): concept 03AJKXH1AV
- [Norm on the Gaussian integers](https://tamnd.github.io/math2/node/03AN1V4ZFJ.md): definition 03AN1V4ZFJ
- [Absorbing Markov chain](https://tamnd.github.io/math2/node/03AN9X29R7.md): concept 03AN9X29R7
- [Generalized Euclid's lemma](https://tamnd.github.io/math2/node/03BAREW1Z5.md): statement 03BAREW1Z5
- [Modulo operation](https://tamnd.github.io/math2/node/03BR7SAX0Q.md): concept 03BR7SAX0Q
- [Law of total variance](https://tamnd.github.io/math2/node/03BY5XTNA4.md): statement 03BY5XTNA4
- [Multivalued function](https://tamnd.github.io/math2/node/03CPD1HGF3.md): concept 03CPD1HGF3
- [Left coset](https://tamnd.github.io/math2/node/03DEH43F4H.md): concept 03DEH43F4H
- [Variance of the normal distribution](https://tamnd.github.io/math2/node/03EESZDB9A.md): statement 03EESZDB9A
- [Determinant of a linear map](https://tamnd.github.io/math2/node/03F5KP94K0.md): definition 03F5KP94K0
- [Cauchy–Euler equation](https://tamnd.github.io/math2/node/03F7N3915D.md): concept 03F7N3915D
- [Range of a function](https://tamnd.github.io/math2/node/03FHD14RKC.md): concept 03FHD14RKC
- [Irreducible polynomial](https://tamnd.github.io/math2/node/03FKX7Z5DC.md): concept 03FKX7Z5DC
- [Vector triple product](https://tamnd.github.io/math2/node/03FTB26QYD.md): statement 03FTB26QYD
- [Chain rule for Jacobian matrices](https://tamnd.github.io/math2/node/03GGQEAG9S.md): statement 03GGQEAG9S
- [Absolutely convergent improper integral](https://tamnd.github.io/math2/node/03GP7KMBYM.md): definition 03GP7KMBYM
- [Dimension](https://tamnd.github.io/math2/node/03GSZ8RTZJ.md): concept 03GSZ8RTZJ
- [Pythagorean trigonometric identity](https://tamnd.github.io/math2/node/03H7WZ4T3K.md): statement 03H7WZ4T3K
- [P-adic valuation](https://tamnd.github.io/math2/node/03HDF50VN4.md): concept 03HDF50VN4
- [Antichain](https://tamnd.github.io/math2/node/03HF498P1J.md): concept 03HF498P1J
- [Triangle inequality](https://tamnd.github.io/math2/node/03HF5PMY5T.md): statement 03HF5PMY5T
- [Mean value theorem in several variables](https://tamnd.github.io/math2/node/03HGK7EW6R.md): statement 03HGK7EW6R
- [Regular space](https://tamnd.github.io/math2/node/03HMJASQBQ.md): concept 03HMJASQBQ
- [Perpendicular bisector](https://tamnd.github.io/math2/node/03J4NZFYRX.md): definition 03J4NZFYRX
- [Normed vector space](https://tamnd.github.io/math2/node/03J5RAKB30.md): concept 03J5RAKB30
- [Separable polynomial](https://tamnd.github.io/math2/node/03J8BS7FMV.md): definition 03J8BS7FMV
- [Wilson's theorem](https://tamnd.github.io/math2/node/03KGGKTB2E.md): statement 03KGGKTB2E
- [Equivalence of norms in finite dimension](https://tamnd.github.io/math2/node/03MC70ZRNR.md): statement 03MC70ZRNR
- [de Moivre's formula](https://tamnd.github.io/math2/node/03MTBVG3VE.md): statement 03MTBVG3VE
- [Type I and type II errors](https://tamnd.github.io/math2/node/03N8C8ZR4B.md): concept 03N8C8ZR4B
- [Path](https://tamnd.github.io/math2/node/03NAMV1DHJ.md): concept 03NAMV1DHJ
- [Minimal polynomial](https://tamnd.github.io/math2/node/03NDVCSHSB.md): concept 03NDVCSHSB
- [Vectors are perpendicular if and only if their dot product is zero](https://tamnd.github.io/math2/node/03NK6RSRQ1.md): statement 03NK6RSRQ1
- [Torus](https://tamnd.github.io/math2/node/03NM7DZXXC.md): example 03NM7DZXXC
- [Cauchy–Schwarz inequality for expectations](https://tamnd.github.io/math2/node/03PGHKAJG3.md): statement 03PGHKAJG3
- [Second-countable space](https://tamnd.github.io/math2/node/03PPNCM9GG.md): concept 03PPNCM9GG
- [A matrix is invertible iff its determinant is a unit](https://tamnd.github.io/math2/node/03Q7RE2HXR.md): statement 03Q7RE2HXR
- [Complement principle](https://tamnd.github.io/math2/node/03QC59M99B.md): statement 03QC59M99B
- [Taylor's theorem](https://tamnd.github.io/math2/node/03QHBVW4TT.md): statement 03QHBVW4TT
- [Skew-symmetric matrix](https://tamnd.github.io/math2/node/03RNVFDEAN.md): definition 03RNVFDEAN
- [Formula for the number of derangements](https://tamnd.github.io/math2/node/03RSJXF728.md): statement 03RSJXF728
- [Limit of a quotient of sequences](https://tamnd.github.io/math2/node/03S6GWWKEZ.md): statement 03S6GWWKEZ
- [Continuous linear map](https://tamnd.github.io/math2/node/03TM135G0N.md): concept 03TM135G0N
- [Integers modulo n](https://tamnd.github.io/math2/node/03VFZ2WETT.md): concept 03VFZ2WETT
- [Indefinite integral](https://tamnd.github.io/math2/node/03VHANA6ZK.md): definition 03VHANA6ZK
- [Circumcircle](https://tamnd.github.io/math2/node/03VZJE6C8C.md): concept 03VZJE6C8C
- [Jacobian determinant](https://tamnd.github.io/math2/node/03W6Q1MFR5.md): definition 03W6Q1MFR5
- [Complete graph](https://tamnd.github.io/math2/node/03WKB2H474.md): example 03WKB2H474
- [Number of derangements](https://tamnd.github.io/math2/node/03WY4HHMT1.md): definition 03WY4HHMT1
- [Sum of a finite geometric progression](https://tamnd.github.io/math2/node/03X8V7D64M.md): statement 03X8V7D64M
- [Multiplicity of a root](https://tamnd.github.io/math2/node/03XVP8Q4GY.md): definition 03XVP8Q4GY
- [Sample mean](https://tamnd.github.io/math2/node/03Y6AQ2S0J.md): concept 03Y6AQ2S0J
- [Open mapping theorem](https://tamnd.github.io/math2/node/03YEM7JEDN.md): statement 03YEM7JEDN
- [Oriented angle](https://tamnd.github.io/math2/node/03YN70NK0Z.md): concept 03YN70NK0Z
- [The Dirichlet function is not Riemann integrable](https://tamnd.github.io/math2/node/03Z4JC4YRJ.md): counterexample 03Z4JC4YRJ
- [C^1 vector fields satisfy the Picard–Lindelöf hypotheses](https://tamnd.github.io/math2/node/03Z9V0Y78A.md): statement 03Z9V0Y78A
- [Binet's formula](https://tamnd.github.io/math2/node/0404D8TN7Y.md): statement 0404D8TN7Y
- [Integer partition](https://tamnd.github.io/math2/node/0404R9P9J3.md): definition 0404R9P9J3
- [Lebesgue number lemma](https://tamnd.github.io/math2/node/040CFNJEAP.md): statement 040CFNJEAP
- [Lattice path](https://tamnd.github.io/math2/node/040WY0TT2G.md): concept 040WY0TT2G
- [Entire function](https://tamnd.github.io/math2/node/0419794R0T.md): definition 0419794R0T
- [Nilpotent matrix](https://tamnd.github.io/math2/node/0419GJC84P.md): definition 0419GJC84P
- [Direct sum of subspaces](https://tamnd.github.io/math2/node/041JTZCK9Q.md): definition 041JTZCK9Q
- [Subgroups of a finite cyclic group correspond to divisors of its order](https://tamnd.github.io/math2/node/0423TXR9AN.md): statement 0423TXR9AN
- [Rational canonical form](https://tamnd.github.io/math2/node/042JWAR3Q0.md): statement 042JWAR3Q0
- [Continuous bijection from a compact space to a Hausdorff space is a homeomorphism](https://tamnd.github.io/math2/node/0430ETCMKD.md): statement 0430ETCMKD
- [Uniform limits of holomorphic functions are holomorphic](https://tamnd.github.io/math2/node/0431RN0R1N.md): statement 0431RN0R1N
- [Fermat's theorem on sums of two squares](https://tamnd.github.io/math2/node/043B3ENSTY.md): statement 043B3ENSTY
- [Maclaurin series](https://tamnd.github.io/math2/node/043SJ5RF59.md): definition 043SJ5RF59
- [Multilinear map](https://tamnd.github.io/math2/node/0443XNARDV.md): concept 0443XNARDV
- [Product metric](https://tamnd.github.io/math2/node/044D4AQV8V.md): definition 044D4AQV8V
- [Diagonal matrix](https://tamnd.github.io/math2/node/044JNC7ZAA.md): concept 044JNC7ZAA
- [Random variable with a density](https://tamnd.github.io/math2/node/045D8P03E0.md): concept 045D8P03E0
- [Circle](https://tamnd.github.io/math2/node/045DN06F3S.md): definition 045DN06F3S
- [The derangement numbers count derangements](https://tamnd.github.io/math2/node/045ENVCCZV.md): statement 045ENVCCZV
- [Disc of convergence](https://tamnd.github.io/math2/node/045EWAXY9P.md): definition 045EWAXY9P
- [Closed curve](https://tamnd.github.io/math2/node/045YFKRW0P.md): concept 045YFKRW0P
- [Monotonicity of probability](https://tamnd.github.io/math2/node/0464Z0JP2E.md): statement 0464Z0JP2E
- [Homogeneous linear system with constant coefficients](https://tamnd.github.io/math2/node/0469V0EN5N.md): concept 0469V0EN5N
- [Evaluation of real integrals by residues](https://tamnd.github.io/math2/node/0476QH9846.md): concept 0476QH9846
- [Variance](https://tamnd.github.io/math2/node/0477R9PYAE.md): concept 0477R9PYAE
- [Every real number has a decimal expansion](https://tamnd.github.io/math2/node/047KEVEZ9Q.md): statement 047KEVEZ9Q
- [Jacobian matrix](https://tamnd.github.io/math2/node/047NDTEPCV.md): concept 047NDTEPCV
- [Parseval's identity](https://tamnd.github.io/math2/node/047RFCNVSK.md): statement 047RFCNVSK
- [Multiplicity of a root](https://tamnd.github.io/math2/node/047SX3TD69.md): concept 047SX3TD69
- [Maximum modulus principle](https://tamnd.github.io/math2/node/048F2P90WR.md): statement 048F2P90WR
- [Concyclic points](https://tamnd.github.io/math2/node/049DCJZ448.md): concept 049DCJZ448
- [Open cover](https://tamnd.github.io/math2/node/049PXYHW1T.md): concept 049PXYHW1T
- [Lagrange multiplier with one constraint](https://tamnd.github.io/math2/node/04A2XEX4D4.md): statement 04A2XEX4D4
- [Greedy colouring](https://tamnd.github.io/math2/node/04AAKZNXHD.md): concept 04AAKZNXHD
- [Möbius strip](https://tamnd.github.io/math2/node/04AJE0M271.md): example 04AJE0M271
- [Exponential of a sum of commuting matrices](https://tamnd.github.io/math2/node/04BD23PSK5.md): statement 04BD23PSK5
- [Complementary subspaces](https://tamnd.github.io/math2/node/04C3BAHV03.md): definition 04C3BAHV03
- [Field](https://tamnd.github.io/math2/node/04C5ZBG1WT.md): concept 04C5ZBG1WT
- [log x is at most x - 1](https://tamnd.github.io/math2/node/04CBRA4JRK.md): statement 04CBRA4JRK
- [Real sequence](https://tamnd.github.io/math2/node/04CC6K5VDT.md): concept 04CC6K5VDT
- [Local maximum and minimum](https://tamnd.github.io/math2/node/04CPG9K1C4.md): definition 04CPG9K1C4
- [Determinant is multiplicative](https://tamnd.github.io/math2/node/04CPJ6AF00.md): statement 04CPJ6AF00
- [Correlation coefficient](https://tamnd.github.io/math2/node/04CTY6XYWN.md): definition 04CTY6XYWN
- [Spectral theorem for normal matrices](https://tamnd.github.io/math2/node/04CW1P44VD.md): statement 04CW1P44VD
- [Variance under translation](https://tamnd.github.io/math2/node/04CYN5G6RY.md): statement 04CYN5G6RY
- [Coordinate vector](https://tamnd.github.io/math2/node/04CYQHN6MC.md): concept 04CYQHN6MC
- [Inverse hyperbolic functions](https://tamnd.github.io/math2/node/04D7CHT5V1.md): definition 04D7CHT5V1
- [Order of a group is the index times the order of the subgroup](https://tamnd.github.io/math2/node/04DC0SRN2K.md): statement 04DC0SRN2K
- [Modular exponentiation](https://tamnd.github.io/math2/node/04DR5PM25N.md): concept 04DR5PM25N
- [Natural logarithm](https://tamnd.github.io/math2/node/04DXAFN2ES.md): concept 04DXAFN2ES
- [Commutativity](https://tamnd.github.io/math2/node/04E4V9G5FR.md): definition 04E4V9G5FR
- [Circle group](https://tamnd.github.io/math2/node/04EAFVFTF7.md): object 04EAFVFTF7
- [Spectrum of a matrix](https://tamnd.github.io/math2/node/04EHB7YDQS.md): definition 04EHB7YDQS
- [Uniqueness of the supremum](https://tamnd.github.io/math2/node/04ETX30HT2.md): statement 04ETX30HT2
- [Modulo operation](https://tamnd.github.io/math2/node/04FHQSJCQ5.md): definition 04FHQSJCQ5
- [Mesh of a partition](https://tamnd.github.io/math2/node/04GM4NYXJB.md): definition 04GM4NYXJB
- [Divisibility in a commutative ring](https://tamnd.github.io/math2/node/04GNK7K4VP.md): definition 04GNK7K4VP
- [Riesz representation theorem in finite dimensions](https://tamnd.github.io/math2/node/04GYPXDXNH.md): statement 04GYPXDXNH
- [Separation axioms](https://tamnd.github.io/math2/node/04J4BYB0XE.md): concept 04J4BYB0XE
- [Divisor function](https://tamnd.github.io/math2/node/04J4Z73J5B.md): concept 04J4Z73J5B
- [Density of the irrationals](https://tamnd.github.io/math2/node/04J6G4H78J.md): statement 04J6G4H78J
- [Consistent and inconsistent systems](https://tamnd.github.io/math2/node/04JAAVHDDM.md): definition 04JAAVHDDM
- [Dirichlet's approximation theorem](https://tamnd.github.io/math2/node/04JAY9DGB7.md): statement 04JAY9DGB7
- [Hockey-stick identity](https://tamnd.github.io/math2/node/04JHQNNBFE.md): statement 04JHQNNBFE
- [Infimum](https://tamnd.github.io/math2/node/04KFTA34C5.md): definition 04KFTA34C5
- [Homogeneous system of linear equations](https://tamnd.github.io/math2/node/04KGXJG1A2.md): concept 04KGXJG1A2
- [Clique number](https://tamnd.github.io/math2/node/04MQSQ1F5Q.md): concept 04MQSQ1F5Q
- [Sum of two squares theorem](https://tamnd.github.io/math2/node/04NPWJMA6H.md): statement 04NPWJMA6H
- [Continued fraction](https://tamnd.github.io/math2/node/04P2RJ4WPS.md): concept 04P2RJ4WPS
- [Geometric distribution](https://tamnd.github.io/math2/node/04P2SR252N.md): concept 04P2SR252N
- [Lyapunov function](https://tamnd.github.io/math2/node/04P7B5XC22.md): definition 04P7B5XC22
- [Almost sure convergence](https://tamnd.github.io/math2/node/04P7D63W8P.md): definition 04P7D63W8P
- [Index of a subgroup](https://tamnd.github.io/math2/node/04QEBF8S49.md): definition 04QEBF8S49
- [Dedekind cut](https://tamnd.github.io/math2/node/04QNZZ4HHY.md): definition 04QNZZ4HHY
- [Euler's formula](https://tamnd.github.io/math2/node/04R3CY41VJ.md): statement 04R3CY41VJ
- [Classification of discontinuities](https://tamnd.github.io/math2/node/04R9K5H562.md): concept 04R9K5H562
- [Cyclic groups are abelian](https://tamnd.github.io/math2/node/04RG02S6BA.md): statement 04RG02S6BA
- [Law of total expectation](https://tamnd.github.io/math2/node/04RRB6HM00.md): statement 04RRB6HM00
- [Zorn's lemma](https://tamnd.github.io/math2/node/04RXTQA5ZA.md): statement 04RXTQA5ZA
- [Parabola](https://tamnd.github.io/math2/node/04S2GMHX82.md): concept 04S2GMHX82
- [Product topology on an arbitrary product](https://tamnd.github.io/math2/node/04S9HX3GAP.md): definition 04S9HX3GAP
- [Doubling the cube](https://tamnd.github.io/math2/node/04SD3F4FCR.md): problem 04SD3F4FCR
- [Galois field](https://tamnd.github.io/math2/node/04SGPNT8R9.md): object 04SGPNT8R9
- [Chinese remainder theorem for non-coprime moduli](https://tamnd.github.io/math2/node/04SJTX9P4N.md): statement 04SJTX9P4N
- [Solution of an ordinary differential equation](https://tamnd.github.io/math2/node/04SW7WECS8.md): definition 04SW7WECS8
- [Contrapositive](https://tamnd.github.io/math2/node/04T1CT7212.md): statement 04T1CT7212
- [Higher derivative](https://tamnd.github.io/math2/node/04TKTF8KDK.md): concept 04TKTF8KDK
- [Definite integral](https://tamnd.github.io/math2/node/04TMP0XGMJ.md): concept 04TMP0XGMJ
- [Little o notation](https://tamnd.github.io/math2/node/04TSHB9H8D.md): definition 04TSHB9H8D
- [Continued fraction expansion of a real number](https://tamnd.github.io/math2/node/04VA28F3HB.md): concept 04VA28F3HB
- [P-adic valuation](https://tamnd.github.io/math2/node/04VCPS30ZE.md): definition 04VCPS30ZE
- [Random walk](https://tamnd.github.io/math2/node/04VZE3FZXF.md): concept 04VZE3FZXF
- [Multiplicativity of the divisor function](https://tamnd.github.io/math2/node/04WC36XKV6.md): statement 04WC36XKV6
- [Sigma-algebra](https://tamnd.github.io/math2/node/04WSW8T3SY.md): definition 04WSW8T3SY
- [Integral of cosine](https://tamnd.github.io/math2/node/04WTBQ7APQ.md): statement 04WTBQ7APQ
- [Determinant as signed volume](https://tamnd.github.io/math2/node/04X3PR72VJ.md): concept 04X3PR72VJ
- [Existence of primitive roots modulo a prime](https://tamnd.github.io/math2/node/04XGKK8WQW.md): statement 04XGKK8WQW
- [Every connected graph has a spanning tree](https://tamnd.github.io/math2/node/04XPQT2XTC.md): statement 04XPQT2XTC
- [Elementary row operations](https://tamnd.github.io/math2/node/04XWH28452.md): concept 04XWH28452
- [Probability density function](https://tamnd.github.io/math2/node/04YG690N5V.md): definition 04YG690N5V
- [Monotone functions are Riemann integrable](https://tamnd.github.io/math2/node/04YKAYDQPW.md): statement 04YKAYDQPW
- [Comparison test](https://tamnd.github.io/math2/node/04Z5G0THN6.md): statement 04Z5G0THN6
- [Tautology](https://tamnd.github.io/math2/node/05006AJ7DZ.md): concept 05006AJ7DZ
- [Cardinality of a finite set](https://tamnd.github.io/math2/node/05099E1NMJ.md): definition 05099E1NMJ
- [Harmonic conjugate](https://tamnd.github.io/math2/node/050F2FMTKF.md): concept 050F2FMTKF
- [Symmetric group](https://tamnd.github.io/math2/node/0515A6D2Y1.md): concept 0515A6D2Y1
- [The real numbers are uncountable](https://tamnd.github.io/math2/node/0518MM6BE1.md): statement 0518MM6BE1
- [Classical definition of probability](https://tamnd.github.io/math2/node/051CYS71H5.md): concept 051CYS71H5
- [Euclid's lemma](https://tamnd.github.io/math2/node/051VYKA1EX.md): statement 051VYKA1EX
- [Product of a matrix and its adjugate](https://tamnd.github.io/math2/node/0528HKWZ1Z.md): statement 0528HKWZ1Z
- [Graph homomorphism](https://tamnd.github.io/math2/node/052JJ9YVNM.md): concept 052JJ9YVNM
- [Annihilating polynomial](https://tamnd.github.io/math2/node/052M4SX57M.md): definition 052M4SX57M
- [Irreducible element](https://tamnd.github.io/math2/node/052SDEPZH7.md): definition 052SDEPZH7
- [Associate elements](https://tamnd.github.io/math2/node/053E86NWVK.md): definition 053E86NWVK
- [Eigenvalues are the roots of the characteristic polynomial](https://tamnd.github.io/math2/node/053QVHF6N7.md): statement 053QVHF6N7
- [Converse relation](https://tamnd.github.io/math2/node/054K3HNN5Z.md): concept 054K3HNN5Z
- [Compact iff complete and totally bounded](https://tamnd.github.io/math2/node/054Q5PW5GX.md): statement 054Q5PW5GX
- [The dihedral group of order 8](https://tamnd.github.io/math2/node/054SBCQ93T.md): example 054SBCQ93T
- [Complex exponential](https://tamnd.github.io/math2/node/054VXHMCA3.md): definition 054VXHMCA3
- [Sesquilinear form](https://tamnd.github.io/math2/node/054YEQ0Y8M.md): concept 054YEQ0Y8M
- [Fermat numbers are pairwise coprime](https://tamnd.github.io/math2/node/05598J0Y8R.md): statement 05598J0Y8R
- [PID implies UFD](https://tamnd.github.io/math2/node/055HT85T3S.md): statement 055HT85T3S
- [Perpendicular bisector](https://tamnd.github.io/math2/node/055VNYT0PB.md): concept 055VNYT0PB
- [Lipschitz condition](https://tamnd.github.io/math2/node/055VVSN00V.md): definition 055VVSN00V
- [Sequential continuity](https://tamnd.github.io/math2/node/055Z15H69G.md): concept 055Z15H69G
- [Jordan measure](https://tamnd.github.io/math2/node/0565DC8RMK.md): definition 0565DC8RMK
- [Lebesgue's criterion for Riemann integrability](https://tamnd.github.io/math2/node/0565EGK6XP.md): statement 0565EGK6XP
- [Pointwise convergence](https://tamnd.github.io/math2/node/056FSKG32J.md): definition 056FSKG32J
- [A matrix is diagonalisable iff it has a basis of eigenvectors](https://tamnd.github.io/math2/node/056SW1E0AC.md): statement 056SW1E0AC
- [Isosceles triangle theorem](https://tamnd.github.io/math2/node/056VS31A0J.md): statement 056VS31A0J
- [Parametrised surface](https://tamnd.github.io/math2/node/05788J51WZ.md): definition 05788J51WZ
- [Generating function for partitions](https://tamnd.github.io/math2/node/0578PA53VA.md): statement 0578PA53VA
- [The nth roots of unity](https://tamnd.github.io/math2/node/057BH3JC9S.md): statement 057BH3JC9S
- [Number of compositions](https://tamnd.github.io/math2/node/057H72B2MB.md): statement 057H72B2MB
- [Binary operation](https://tamnd.github.io/math2/node/0580ZJSA3Q.md): definition 0580ZJSA3Q
- [Ptolemy's theorem](https://tamnd.github.io/math2/node/058YS05YP5.md): statement 058YS05YP5
- [Scalar triple product](https://tamnd.github.io/math2/node/059Q2XWJDC.md): statement 059Q2XWJDC
- [Isomorphism with the coordinate space](https://tamnd.github.io/math2/node/059THXBRBB.md): statement 059THXBRBB
- [Probability measure](https://tamnd.github.io/math2/node/059TJN0NZ1.md): concept 059TJN0NZ1
- [Partition of an interval](https://tamnd.github.io/math2/node/05A6PKMAJ9.md): concept 05A6PKMAJ9
- [Sum of a finite geometric series](https://tamnd.github.io/math2/node/05APWSX53S.md): statement 05APWSX53S
- [Normal distribution](https://tamnd.github.io/math2/node/05AQE0QR0D.md): concept 05AQE0QR0D
- [Binary relation](https://tamnd.github.io/math2/node/05AVARPXA2.md): concept 05AVARPXA2
- [Expectation of a product of independent random variables](https://tamnd.github.io/math2/node/05AW1G0WE1.md): statement 05AW1G0WE1
- [Partial derivative](https://tamnd.github.io/math2/node/05AXG3QRKR.md): definition 05AXG3QRKR
- [Jump discontinuity](https://tamnd.github.io/math2/node/05B0FKFXQA.md): definition 05B0FKFXQA
- [Space of solutions of a linear recurrence](https://tamnd.github.io/math2/node/05BDYVQQ0T.md): definition 05BDYVQQ0T
- [Stabilisers of points in the same orbit are conjugate](https://tamnd.github.io/math2/node/05BM304NRX.md): statement 05BM304NRX
- [Tangent function](https://tamnd.github.io/math2/node/05CG0S20S3.md): concept 05CG0S20S3
- [Determinant of a family of vectors in a basis](https://tamnd.github.io/math2/node/05CJSSXPR3.md): concept 05CJSSXPR3
- [Boundary](https://tamnd.github.io/math2/node/05D5EGMFYV.md): concept 05D5EGMFYV
- [Zeckendorf's theorem](https://tamnd.github.io/math2/node/05DEJR6TGF.md): statement 05DEJR6TGF
- [Pascal's triangle](https://tamnd.github.io/math2/node/05DME4TZ3K.md): object 05DME4TZ3K
- [Closed ball](https://tamnd.github.io/math2/node/05DSV40EB9.md): definition 05DSV40EB9
- [Having a directional derivative](https://tamnd.github.io/math2/node/05EGFBZTGC.md): definition 05EGFBZTGC
- [Vector space](https://tamnd.github.io/math2/node/05EGHEQZAW.md): definition 05EGHEQZAW
- [Left coset](https://tamnd.github.io/math2/node/05FD317V2W.md): definition 05FD317V2W
- [Polarisation identity](https://tamnd.github.io/math2/node/05FKHRT0B4.md): statement 05FKHRT0B4
- [Bounded sequence](https://tamnd.github.io/math2/node/05FQ3VGAR9.md): definition 05FQ3VGAR9
- [Likelihood function](https://tamnd.github.io/math2/node/05FSTXM92W.md): definition 05FSTXM92W
- [Dimension](https://tamnd.github.io/math2/node/05G2G0P224.md): definition 05G2G0P224
- [Union of a family of sets](https://tamnd.github.io/math2/node/05GK32H72H.md): definition 05GK32H72H
- [T1 space](https://tamnd.github.io/math2/node/05GQJXBP3F.md): definition 05GQJXBP3F
- [Homeomorphism](https://tamnd.github.io/math2/node/05GZ11PWRY.md): definition 05GZ11PWRY
- [Root test](https://tamnd.github.io/math2/node/05H4DDDYBZ.md): statement 05H4DDDYBZ
- [Dihedral angle](https://tamnd.github.io/math2/node/05HN08ZSRJ.md): concept 05HN08ZSRJ
- [Dot product](https://tamnd.github.io/math2/node/05HQYM0R4G.md): definition 05HQYM0R4G
- [Inner automorphism group is isomorphic to G modulo its centre](https://tamnd.github.io/math2/node/05HYGWFG63.md): statement 05HYGWFG63
- [Transitive closure](https://tamnd.github.io/math2/node/05J9NWX07X.md): concept 05J9NWX07X
- [The rationals as the field of fractions of the integers](https://tamnd.github.io/math2/node/05K3J8J8T2.md): example 05K3J8J8T2
- [Improper integral](https://tamnd.github.io/math2/node/05KQ3F99BQ.md): concept 05KQ3F99BQ
- [Limit of a sequence](https://tamnd.github.io/math2/node/05M5SWSZZD.md): concept 05M5SWSZZD
- [Poincaré lemma for one-forms on convex sets](https://tamnd.github.io/math2/node/05MC3HND79.md): statement 05MC3HND79
- [Angle sum of a triangle](https://tamnd.github.io/math2/node/05MDKQBEAF.md): statement 05MDKQBEAF
- [Groups of prime order are cyclic](https://tamnd.github.io/math2/node/05MM4EK4SA.md): statement 05MM4EK4SA
- [Cross-ratio](https://tamnd.github.io/math2/node/05MW7MPRBS.md): definition 05MW7MPRBS
- [Subgroup criterion](https://tamnd.github.io/math2/node/05N4TQKPH0.md): statement 05N4TQKPH0
- [Field generated by adjoining elements](https://tamnd.github.io/math2/node/05NAS7AGMZ.md): concept 05NAS7AGMZ
- [Composition of continuous maps is continuous](https://tamnd.github.io/math2/node/05NK26G0DR.md): statement 05NK26G0DR
- [Eulerian trail](https://tamnd.github.io/math2/node/05NWGBHKSZ.md): concept 05NWGBHKSZ
- [Mean squared error](https://tamnd.github.io/math2/node/05NZ1HHQMV.md): concept 05NZ1HHQMV
- [Derivative of arctangent](https://tamnd.github.io/math2/node/05P4B4Q22V.md): statement 05P4B4Q22V
- [Markov's inequality](https://tamnd.github.io/math2/node/05PWWHYBX5.md): statement 05PWWHYBX5
- [Matrix of a composition is the product of matrices](https://tamnd.github.io/math2/node/05Q1B7MH0M.md): statement 05Q1B7MH0M
- [Orthogonal family for a bilinear form](https://tamnd.github.io/math2/node/05Q4JD6XEH.md): definition 05Q4JD6XEH
- [Family of sets](https://tamnd.github.io/math2/node/05Q6G4EH2G.md): definition 05Q6G4EH2G
- [Finitely generated module](https://tamnd.github.io/math2/node/05QY1VEBYD.md): concept 05QY1VEBYD
- [Conservative vector field](https://tamnd.github.io/math2/node/05R9NF2J49.md): definition 05R9NF2J49
- [Additivity of the integral over intervals](https://tamnd.github.io/math2/node/05RCDKHZM6.md): statement 05RCDKHZM6
- [Laplacian as a sum of second partial derivatives](https://tamnd.github.io/math2/node/05RCT1522R.md): statement 05RCT1522R
- [Number of edges of a complete graph](https://tamnd.github.io/math2/node/05RN90E14Y.md): statement 05RN90E14Y
- [Truth table](https://tamnd.github.io/math2/node/05RSK4Q657.md): concept 05RSK4Q657
- [Permutation](https://tamnd.github.io/math2/node/05RWFKVTJ7.md): concept 05RWFKVTJ7
- [Standard score](https://tamnd.github.io/math2/node/05RZ4YZG4E.md): definition 05RZ4YZG4E
- [Harmonic numbers grow like log n](https://tamnd.github.io/math2/node/05S2GSPMGA.md): statement 05S2GSPMGA
- [Left inverse](https://tamnd.github.io/math2/node/05SG20AMXH.md): concept 05SG20AMXH
- [Negation of an existential statement](https://tamnd.github.io/math2/node/05SZQKH75K.md): statement 05SZQKH75K
- [Poisson point process](https://tamnd.github.io/math2/node/05TG0SE4HT.md): definition 05TG0SE4HT
- [Group of units of a ring](https://tamnd.github.io/math2/node/05THZKR9AN.md): definition 05THZKR9AN
- [Equality of complex numbers](https://tamnd.github.io/math2/node/05V7J4WD1C.md): statement 05V7J4WD1C
- [Phase plane](https://tamnd.github.io/math2/node/05VDGH129F.md): concept 05VDGH129F
- [Maximal ideals are prime](https://tamnd.github.io/math2/node/05WAF5T0GZ.md): statement 05WAF5T0GZ
- [Submodule](https://tamnd.github.io/math2/node/05WDSN597C.md): concept 05WDSN597C
- [Inverse trigonometric functions](https://tamnd.github.io/math2/node/05WEYX99X4.md): concept 05WEYX99X4
- [Jacobian of spherical coordinates](https://tamnd.github.io/math2/node/05WKRC7HWT.md): statement 05WKRC7HWT
- [Number of derangements](https://tamnd.github.io/math2/node/05WR29H2KH.md): concept 05WR29H2KH
- [Set difference](https://tamnd.github.io/math2/node/05WTQ6FFPS.md): concept 05WTQ6FFPS
- [Laplace transform](https://tamnd.github.io/math2/node/05XT5AVJBK.md): definition 05XT5AVJBK
- [Subgroups of solvable groups are solvable](https://tamnd.github.io/math2/node/05Y0F9M0TS.md): statement 05Y0F9M0TS
- [Asymptotic stability](https://tamnd.github.io/math2/node/05Y1JAM3Y5.md): definition 05Y1JAM3Y5
- [Composition of linear maps](https://tamnd.github.io/math2/node/05YBKMRVAR.md): definition 05YBKMRVAR
- [Isolated singularity](https://tamnd.github.io/math2/node/05Z3FC0154.md): concept 05Z3FC0154
- [Perfect matching](https://tamnd.github.io/math2/node/05ZPH8KF9K.md): definition 05ZPH8KF9K
- [Circle](https://tamnd.github.io/math2/node/060W5CBZP6.md): concept 060W5CBZP6
- [Right coset](https://tamnd.github.io/math2/node/060Z37VVB6.md): definition 060Z37VVB6
- [Law of cosines](https://tamnd.github.io/math2/node/060ZVZBV3E.md): statement 060ZVZBV3E
- [Carmichael number](https://tamnd.github.io/math2/node/060ZX7C32T.md): concept 060ZX7C32T
- [Lucas–Lehmer test](https://tamnd.github.io/math2/node/061BTYA0RN.md): statement 061BTYA0RN
- [Symmetric relation](https://tamnd.github.io/math2/node/061J5VGSPT.md): concept 061J5VGSPT
- [Tree](https://tamnd.github.io/math2/node/062D9KXJB1.md): concept 062D9KXJB1
- [Trail](https://tamnd.github.io/math2/node/062MQGP8M5.md): definition 062MQGP8M5
- [Gamma distribution](https://tamnd.github.io/math2/node/06395GBNWD.md): definition 06395GBNWD
- [Nilpotent endomorphism](https://tamnd.github.io/math2/node/063D5Z2C70.md): concept 063D5Z2C70
- [First fundamental theorem of calculus](https://tamnd.github.io/math2/node/063H5KJC69.md): statement 063H5KJC69
- [Goldbach's conjecture](https://tamnd.github.io/math2/node/063JNBSMJ0.md): conjecture 063JNBSMJ0
- [Metric space](https://tamnd.github.io/math2/node/063W5JMY85.md): definition 063W5JMY85
- [Order of a zero](https://tamnd.github.io/math2/node/064N002TX4.md): definition 064N002TX4
- [LaSalle's invariance principle](https://tamnd.github.io/math2/node/0659J5X7B5.md): statement 0659J5X7B5
- [Coprime ideals](https://tamnd.github.io/math2/node/065CK367F9.md): concept 065CK367F9
- [Surface integral](https://tamnd.github.io/math2/node/065HE5V98E.md): concept 065HE5V98E
- [Inclusion of a subspace is continuous](https://tamnd.github.io/math2/node/065J822CKN.md): statement 065J822CKN
- [Area of a triangle](https://tamnd.github.io/math2/node/065JWMZW86.md): concept 065JWMZW86
- [Repeating decimals are rational](https://tamnd.github.io/math2/node/065S5B9HF1.md): statement 065S5B9HF1
- [De Moivre–Laplace theorem](https://tamnd.github.io/math2/node/065X9S8CM3.md): statement 065X9S8CM3
- [Arctangent](https://tamnd.github.io/math2/node/065XW2V6C3.md): definition 065XW2V6C3
- [Radius of convergence](https://tamnd.github.io/math2/node/0660QX1VPR.md): definition 0660QX1VPR
- [Pole](https://tamnd.github.io/math2/node/066E7PJCYE.md): concept 066E7PJCYE
- [Closed subsets of compact sets are compact](https://tamnd.github.io/math2/node/066HW6D9A7.md): statement 066HW6D9A7
- [The integers are countable](https://tamnd.github.io/math2/node/066RJJVWH9.md): statement 066RJJVWH9
- [Noetherian ring](https://tamnd.github.io/math2/node/0670T59HGE.md): concept 0670T59HGE
- [Flow of a vector field](https://tamnd.github.io/math2/node/06738GGWWK.md): concept 06738GGWWK
- [Autonomous system](https://tamnd.github.io/math2/node/0673D85QKZ.md): concept 0673D85QKZ
- [Repeated roots and the derivative](https://tamnd.github.io/math2/node/067F8YVYMH.md): statement 067F8YVYMH
- [Scalar matrix](https://tamnd.github.io/math2/node/067NKSBEWT.md): concept 067NKSBEWT
- [Tonelli's theorem](https://tamnd.github.io/math2/node/067SYD2J4B.md): statement 067SYD2J4B
- [Existence of square roots of nonnegative reals](https://tamnd.github.io/math2/node/068BZBXE0C.md): statement 068BZBXE0C
- [Closure in a metric space](https://tamnd.github.io/math2/node/068P0P32DW.md): statement 068P0P32DW
- [Diameter of a set](https://tamnd.github.io/math2/node/0692CGQF00.md): definition 0692CGQF00
- [Hessian matrix](https://tamnd.github.io/math2/node/0693Y2METT.md): concept 0693Y2METT
- [Triple integral](https://tamnd.github.io/math2/node/0698M7C7AX.md): concept 0698M7C7AX
- [Uniqueness of splitting fields](https://tamnd.github.io/math2/node/069A3BC3EP.md): statement 069A3BC3EP
- [Comparison of a series and an integral](https://tamnd.github.io/math2/node/069HKV4X1E.md): statement 069HKV4X1E
- [Row space](https://tamnd.github.io/math2/node/069MCGX635.md): concept 069MCGX635
- [Partition of an interval](https://tamnd.github.io/math2/node/069P20KYQT.md): definition 069P20KYQT
- [Laplace expansion along a column](https://tamnd.github.io/math2/node/06AMCJMMVM.md): statement 06AMCJMMVM
- [Heine–Cantor theorem](https://tamnd.github.io/math2/node/06B8BNJCD6.md): statement 06B8BNJCD6
- [Solution formula for linear first-order equations](https://tamnd.github.io/math2/node/06B9W45XV0.md): statement 06B9W45XV0
- [Strictly convex function](https://tamnd.github.io/math2/node/06BA5JZKTQ.md): definition 06BA5JZKTQ
- [Reflection](https://tamnd.github.io/math2/node/06BRDF3F0E.md): concept 06BRDF3F0E
- [Five colour theorem](https://tamnd.github.io/math2/node/06C0JBE52D.md): statement 06C0JBE52D
- [Symmetric polynomial](https://tamnd.github.io/math2/node/06CTFY2DEZ.md): definition 06CTFY2DEZ
- [Integral of the exponential function](https://tamnd.github.io/math2/node/06D0EWR2DV.md): statement 06D0EWR2DV
- [P-group](https://tamnd.github.io/math2/node/06DAH3VS6M.md): definition 06DAH3VS6M
- [Identity matrix](https://tamnd.github.io/math2/node/06DW9BAWB1.md): object 06DW9BAWB1
- [Z[x] is not a principal ideal domain](https://tamnd.github.io/math2/node/06E3AG9TM2.md): counterexample 06E3AG9TM2
- [Fréchet derivative](https://tamnd.github.io/math2/node/06EH97V2A7.md): concept 06EH97V2A7
- [Operator norm](https://tamnd.github.io/math2/node/06EKW1AKXF.md): concept 06EKW1AKXF
- [Basis of a topology](https://tamnd.github.io/math2/node/06EX0FHBTF.md): definition 06EX0FHBTF
- [Midpoint](https://tamnd.github.io/math2/node/06F3Y6CMG9.md): definition 06F3Y6CMG9
- [Boolean algebra](https://tamnd.github.io/math2/node/06GKBTAA5E.md): definition 06GKBTAA5E
- [Mersenne prime](https://tamnd.github.io/math2/node/06GPJF9V7M.md): definition 06GPJF9V7M
- [Logical equivalence](https://tamnd.github.io/math2/node/06GYGS41JK.md): definition 06GYGS41JK
- [Banach space](https://tamnd.github.io/math2/node/06GZ956657.md): concept 06GZ956657
- [Prime ideal](https://tamnd.github.io/math2/node/06HC6HK0Z8.md): definition 06HC6HK0Z8
- [Uniform metric on bounded functions](https://tamnd.github.io/math2/node/06HE9J7DXF.md): example 06HE9J7DXF
- [Linear map of a matrix](https://tamnd.github.io/math2/node/06HMCTAQH3.md): concept 06HMCTAQH3
- [Dimension of a direct product](https://tamnd.github.io/math2/node/06HN47VPTH.md): statement 06HN47VPTH
- [Grassmann's formula](https://tamnd.github.io/math2/node/06HTJNCMPA.md): statement 06HTJNCMPA
- [Affine combination](https://tamnd.github.io/math2/node/06HYY6DK6X.md): definition 06HYY6DK6X
- [Spectral theorem](https://tamnd.github.io/math2/node/06J4PXRG22.md): statement 06J4PXRG22
- [Multiplication of natural numbers](https://tamnd.github.io/math2/node/06J5DE801P.md): concept 06J5DE801P
- [Classification of finite simple groups](https://tamnd.github.io/math2/node/06J6P03MG7.md): statement 06J6P03MG7
- [Real and imaginary parts](https://tamnd.github.io/math2/node/06J8TEPJPY.md): concept 06J8TEPJPY
- [Regular curve](https://tamnd.github.io/math2/node/06JMR2XWXW.md): definition 06JMR2XWXW
- [Homogeneous system of linear equations](https://tamnd.github.io/math2/node/06JT7KDHXM.md): definition 06JT7KDHXM
- [Branch cut](https://tamnd.github.io/math2/node/06JYC06JHY.md): definition 06JYC06JHY
- [Convergence of a sequence in a metric space](https://tamnd.github.io/math2/node/06JYPJ4WHH.md): definition 06JYPJ4WHH
- [Power series of the exponential function](https://tamnd.github.io/math2/node/06JZGMYTYD.md): statement 06JZGMYTYD
- [The minimal polynomial divides every polynomial vanishing at the element](https://tamnd.github.io/math2/node/06K7SARZKR.md): statement 06K7SARZKR
- [Directional derivative](https://tamnd.github.io/math2/node/06KWGHDN2J.md): concept 06KWGHDN2J
- [Inflection point](https://tamnd.github.io/math2/node/06MG208MGR.md): concept 06MG208MGR
- [Solution of a linear recurrence](https://tamnd.github.io/math2/node/06MJC8YDYR.md): concept 06MJC8YDYR
- [Every spanning set contains a basis](https://tamnd.github.io/math2/node/06MRNP4BNT.md): statement 06MRNP4BNT
- [Fermat prime](https://tamnd.github.io/math2/node/06N6JAZF44.md): concept 06N6JAZF44
- [Cauchy criterion for series](https://tamnd.github.io/math2/node/06PVXMY0TH.md): statement 06PVXMY0TH
- [Dimension of the annihilator](https://tamnd.github.io/math2/node/06Q1EFVH0Q.md): statement 06Q1EFVH0Q
- [cosh^2 - sinh^2 = 1](https://tamnd.github.io/math2/node/06QCTRFBM0.md): statement 06QCTRFBM0
- [P-group](https://tamnd.github.io/math2/node/06QYRTX9KQ.md): concept 06QYRTX9KQ
- [Commutator of two elements](https://tamnd.github.io/math2/node/06RGT4V6Y6.md): definition 06RGT4V6Y6
- [Units of the Gaussian integers](https://tamnd.github.io/math2/node/06RH4XAGRY.md): statement 06RH4XAGRY
- [Number of invertible residues modulo n](https://tamnd.github.io/math2/node/06RJWZCG95.md): statement 06RJWZCG95
- [Completion of a metric space](https://tamnd.github.io/math2/node/06RW4RKWH7.md): concept 06RW4RKWH7
- [Disjunctive normal form](https://tamnd.github.io/math2/node/06S4WGSEQA.md): definition 06S4WGSEQA
- [Graph homomorphism](https://tamnd.github.io/math2/node/06SCB3R6BN.md): definition 06SCB3R6BN
- [Continuous functions are Riemann integrable](https://tamnd.github.io/math2/node/06SESV7GQY.md): statement 06SESV7GQY
- [Continuously differentiable function](https://tamnd.github.io/math2/node/06SKCHDQT5.md): definition 06SKCHDQT5
- [Tangent function](https://tamnd.github.io/math2/node/06T0HS14PP.md): definition 06T0HS14PP
- [Sum of divisors](https://tamnd.github.io/math2/node/06TK51263Q.md): concept 06TK51263Q
- [Rayleigh quotient](https://tamnd.github.io/math2/node/06WHP8SE98.md): concept 06WHP8SE98
- [Existence of Laurent expansions on an annulus](https://tamnd.github.io/math2/node/06WHRARZKY.md): statement 06WHRARZKY
- [Converse of Wilson's theorem](https://tamnd.github.io/math2/node/06WRV8HFDY.md): statement 06WRV8HFDY
- [Buffon's needle problem](https://tamnd.github.io/math2/node/06X5PF6YFP.md): problem 06X5PF6YFP
- [Simple continued fraction](https://tamnd.github.io/math2/node/06XF7BCFFJ.md): definition 06XF7BCFFJ
- [Decimal expansion](https://tamnd.github.io/math2/node/06XY4X9430.md): definition 06XY4X9430
- [Unique factorisation domain](https://tamnd.github.io/math2/node/06YMXJMXYS.md): definition 06YMXJMXYS
- [Geometric multiplicity](https://tamnd.github.io/math2/node/06YSJVR356.md): definition 06YSJVR356
- [Continuous image of a closed interval is a closed interval](https://tamnd.github.io/math2/node/06ZARTQTQE.md): statement 06ZARTQTQE
- [Contour integral](https://tamnd.github.io/math2/node/06ZGPYTWK8.md): concept 06ZGPYTWK8
- [Argument of a complex number](https://tamnd.github.io/math2/node/070PFSYJJR.md): concept 070PFSYJJR
- [Finite set](https://tamnd.github.io/math2/node/070R982G4B.md): concept 070R982G4B
- [Geometric formula for the dot product](https://tamnd.github.io/math2/node/070TXS9AXM.md): statement 070TXS9AXM
- [Linear isometry](https://tamnd.github.io/math2/node/0717XXPWM6.md): definition 0717XXPWM6
- [Abelian group](https://tamnd.github.io/math2/node/071DDFVG4J.md): definition 071DDFVG4J
- [Congruence is compatible with multiplication](https://tamnd.github.io/math2/node/072D5G4SK6.md): statement 072D5G4SK6
- [Maximal element](https://tamnd.github.io/math2/node/072G59HW59.md): concept 072G59HW59
- [Comparison of topologies](https://tamnd.github.io/math2/node/072J68M6HX.md): concept 072J68M6HX
- [Formal power series](https://tamnd.github.io/math2/node/072W67KFYK.md): definition 072W67KFYK
- [Limit of a product of sequences](https://tamnd.github.io/math2/node/0734RSVF2V.md): statement 0734RSVF2V
- [Cyclic groups of the same order are isomorphic](https://tamnd.github.io/math2/node/0736AXH2ME.md): statement 0736AXH2ME
- [Third isomorphism theorem for rings](https://tamnd.github.io/math2/node/073C80EC63.md): statement 073C80EC63
- [Associativity of matrix multiplication](https://tamnd.github.io/math2/node/073E4XC016.md): statement 073E4XC016
- [Tangent plane and normal vector of a surface](https://tamnd.github.io/math2/node/073JKP99K2.md): concept 073JKP99K2
- [Order of a pole](https://tamnd.github.io/math2/node/073MWACF1K.md): concept 073MWACF1K
- [Kolmogorov's zero–one law](https://tamnd.github.io/math2/node/073Q8Z94KG.md): statement 073Q8Z94KG
- [Correspondence between equivalence relations and partitions](https://tamnd.github.io/math2/node/074482Q7CD.md): statement 074482Q7CD
- [Symmetric bilinear form](https://tamnd.github.io/math2/node/0744CN4MDA.md): definition 0744CN4MDA
- [Separable equation](https://tamnd.github.io/math2/node/074E9A8ZJK.md): concept 074E9A8ZJK
- [Gaussian integer](https://tamnd.github.io/math2/node/074P7C85XH.md): concept 074P7C85XH
- [One-sided inverse of a square matrix is two-sided](https://tamnd.github.io/math2/node/0753Y2X5K6.md): statement 0753Y2X5K6
- [Simply connected domain](https://tamnd.github.io/math2/node/075C629XTH.md): definition 075C629XTH
- [The reals have the cardinality of the continuum](https://tamnd.github.io/math2/node/075PM9FH7A.md): statement 075PM9FH7A
- [Initial value problem](https://tamnd.github.io/math2/node/075VBY4K4X.md): definition 075VBY4K4X
- [Path graph](https://tamnd.github.io/math2/node/075WR6W3Y3.md): example 075WR6W3Y3
- [Uniqueness of the derivative](https://tamnd.github.io/math2/node/0762WAE49P.md): statement 0762WAE49P
- [Matrix multiplication](https://tamnd.github.io/math2/node/076JGP6R3T.md): definition 076JGP6R3T
- [Even and odd permutations](https://tamnd.github.io/math2/node/0771AJP4BX.md): definition 0771AJP4BX
- [Independent random variables are uncorrelated](https://tamnd.github.io/math2/node/077GN9HDR6.md): statement 077GN9HDR6
- [Determinant of the inverse](https://tamnd.github.io/math2/node/0783S9D0ZH.md): statement 0783S9D0ZH
- [Orthogonal complement](https://tamnd.github.io/math2/node/079D68S3AW.md): definition 079D68S3AW
- [Transcendental number](https://tamnd.github.io/math2/node/079EGSD32Y.md): definition 079EGSD32Y
- [Autonomous system](https://tamnd.github.io/math2/node/07BNRFQ0PJ.md): definition 07BNRFQ0PJ
- [Nilpotent matrix](https://tamnd.github.io/math2/node/07BSPZPSMF.md): concept 07BSPZPSMF
- [Continuity of the inverse of a strictly monotone continuous function](https://tamnd.github.io/math2/node/07BY47HG5S.md): statement 07BY47HG5S
- [Conjunctive normal form](https://tamnd.github.io/math2/node/07C4A8Y18K.md): definition 07C4A8Y18K
- [Group of units of a ring](https://tamnd.github.io/math2/node/07C6ECNJET.md): concept 07C6ECNJET
- [Convolution of probability distributions](https://tamnd.github.io/math2/node/07CHJ5YX28.md): definition 07CHJ5YX28
- [Taxicab metric](https://tamnd.github.io/math2/node/07CJNE6ZAQ.md): example 07CJNE6ZAQ
- [Classification of finite abelian groups](https://tamnd.github.io/math2/node/07CQR9BW3R.md): statement 07CQR9BW3R
- [Cycle](https://tamnd.github.io/math2/node/07CT9RDY7Q.md): definition 07CT9RDY7Q
- [Lcm of coprime integers](https://tamnd.github.io/math2/node/07CZYE06ZJ.md): statement 07CZYE06ZJ
- [Lipschitz maps are uniformly continuous](https://tamnd.github.io/math2/node/07DCKDXRBP.md): statement 07DCKDXRBP
- [Incircle](https://tamnd.github.io/math2/node/07DFG5JYPT.md): concept 07DFG5JYPT
- [Integrating factor](https://tamnd.github.io/math2/node/07DJAZW32T.md): concept 07DJAZW32T
- [Bipartite graph](https://tamnd.github.io/math2/node/07DVHJ8N0A.md): definition 07DVHJ8N0A
- [Cauchy–Hadamard theorem](https://tamnd.github.io/math2/node/07E5MB1CA2.md): statement 07E5MB1CA2
- [Disjoint cycles commute](https://tamnd.github.io/math2/node/07E6D8R4D6.md): statement 07E6D8R4D6
- [Limit point of a set](https://tamnd.github.io/math2/node/07EGCNHAVW.md): definition 07EGCNHAVW
- [Composition of surjections is surjective](https://tamnd.github.io/math2/node/07EH4R3RVQ.md): statement 07EH4R3RVQ
- [Grönwall's inequality](https://tamnd.github.io/math2/node/07EJHQYR9A.md): statement 07EJHQYR9A
- [Newton's method](https://tamnd.github.io/math2/node/07EXVNF9VP.md): concept 07EXVNF9VP
- [Signature of a quadratic form](https://tamnd.github.io/math2/node/07EZA80NEH.md): concept 07EZA80NEH
- [Complex exponential](https://tamnd.github.io/math2/node/07GJV36896.md): concept 07GJV36896
- [Polynomials in several variables](https://tamnd.github.io/math2/node/07GNX8GCYP.md): concept 07GNX8GCYP
- [Convex function on R^n](https://tamnd.github.io/math2/node/07GR3TNW4H.md): concept 07GR3TNW4H
- [Function composition](https://tamnd.github.io/math2/node/07HF9YCWPP.md): definition 07HF9YCWPP
- [Circulant matrix](https://tamnd.github.io/math2/node/07HFRH2AWY.md): example 07HFRH2AWY
- [Pépin's test](https://tamnd.github.io/math2/node/07HXA8KV6F.md): statement 07HXA8KV6F
- [A function with positive derivative is strictly increasing](https://tamnd.github.io/math2/node/07J5X2T1ZX.md): statement 07J5X2T1ZX
- [Order on the integers](https://tamnd.github.io/math2/node/07J9ZJFECP.md): concept 07J9ZJFECP
- [Dual space](https://tamnd.github.io/math2/node/07JBSZE0V4.md): concept 07JBSZE0V4
- [Linear first-order equation](https://tamnd.github.io/math2/node/07JH131FVP.md): concept 07JH131FVP
- [i squared is minus one](https://tamnd.github.io/math2/node/07KBQCG9KV.md): statement 07KBQCG9KV
- [Union](https://tamnd.github.io/math2/node/07KVB4JSC7.md): definition 07KVB4JSC7
- [If 2^p − 1 is prime then p is prime](https://tamnd.github.io/math2/node/07MMWAXMYD.md): statement 07MMWAXMYD
- [Continuity at a point of a topological space](https://tamnd.github.io/math2/node/07N7W5C29X.md): definition 07N7W5C29X
- [Fermat's Last Theorem for n = 3](https://tamnd.github.io/math2/node/07NKG38VEG.md): statement 07NKG38VEG
- [Liouville function](https://tamnd.github.io/math2/node/07NN4VNYWM.md): concept 07NN4VNYWM
- [Menelaus's theorem](https://tamnd.github.io/math2/node/07NSKDGS7S.md): statement 07NSKDGS7S
- [Picard operator](https://tamnd.github.io/math2/node/07PB7HH1T9.md): concept 07PB7HH1T9
- [Heine–Borel theorem](https://tamnd.github.io/math2/node/07PFYV65MB.md): statement 07PFYV65MB
- [Jordan block](https://tamnd.github.io/math2/node/07PH4RX0X5.md): concept 07PH4RX0X5
- [Generalised eigenspace](https://tamnd.github.io/math2/node/07PNDWA7CE.md): definition 07PNDWA7CE
- [Trace of a matrix](https://tamnd.github.io/math2/node/07PTPSZPME.md): definition 07PTPSZPME
- [Möbius inversion formula](https://tamnd.github.io/math2/node/07Q4TBTFVB.md): statement 07Q4TBTFVB
- [Convex function](https://tamnd.github.io/math2/node/07Q7CNM5WF.md): concept 07Q7CNM5WF
- [Subfield](https://tamnd.github.io/math2/node/07QY5RPFBR.md): definition 07QY5RPFBR
- [Third isomorphism theorem for vector spaces](https://tamnd.github.io/math2/node/07RB3EMBWW.md): statement 07RB3EMBWW
- [Uniform continuity on a set](https://tamnd.github.io/math2/node/07RB7B04HX.md): concept 07RB7B04HX
- [Divisibility in a commutative ring](https://tamnd.github.io/math2/node/07RGNK57ED.md): concept 07RGNK57ED
- [Normal distribution](https://tamnd.github.io/math2/node/07RXBVTVZB.md): definition 07RXBVTVZB
- [Equation of a plane](https://tamnd.github.io/math2/node/07SDJ5DNE4.md): concept 07SDJ5DNE4
- [T0 space](https://tamnd.github.io/math2/node/07SG12TQH3.md): concept 07SG12TQH3
- [Linear isometry](https://tamnd.github.io/math2/node/07SJXTWVQW.md): concept 07SJXTWVQW
- [Diagonalisable matrix](https://tamnd.github.io/math2/node/07SQCNV233.md): concept 07SQCNV233
- [Derivatives of the hyperbolic functions](https://tamnd.github.io/math2/node/07T2TKGZ4J.md): statement 07T2TKGZ4J
- [Distance from a point to a set](https://tamnd.github.io/math2/node/07TXYYVYZR.md): definition 07TXYYVYZR
- [Trajectory](https://tamnd.github.io/math2/node/07TZ2Q4C1M.md): concept 07TZ2Q4C1M
- [Recurrence relation](https://tamnd.github.io/math2/node/07TZV87Y0T.md): definition 07TZV87Y0T
- [Young's inequality for products](https://tamnd.github.io/math2/node/07VHQ31EYJ.md): statement 07VHQ31EYJ
- [Solution of an ordinary differential equation](https://tamnd.github.io/math2/node/07VX34QXAV.md): concept 07VX34QXAV
- [Adjugate matrix](https://tamnd.github.io/math2/node/07W9H8QF38.md): concept 07W9H8QF38
- [Inverse Laplace transform](https://tamnd.github.io/math2/node/07WBTX1MHA.md): concept 07WBTX1MHA
- [Identity element](https://tamnd.github.io/math2/node/07WYQPB8E3.md): concept 07WYQPB8E3
- [Derivative of the logarithm](https://tamnd.github.io/math2/node/07X1QA6BRW.md): statement 07X1QA6BRW
- [Algebra of limits](https://tamnd.github.io/math2/node/07XBM3A9J5.md): concept 07XBM3A9J5
- [Image of a set](https://tamnd.github.io/math2/node/07XF282NWH.md): concept 07XF282NWH
- [Median](https://tamnd.github.io/math2/node/07Y2E7P9NP.md): concept 07Y2E7P9NP
- [Linear form](https://tamnd.github.io/math2/node/07YQ1C78G0.md): concept 07YQ1C78G0
- [Differentiable functions are continuous](https://tamnd.github.io/math2/node/07ZMX4ANJS.md): statement 07ZMX4ANJS
- [Anticommutativity of the cross product](https://tamnd.github.io/math2/node/07ZS685JJC.md): statement 07ZS685JJC
- [Every integer is a square times a square-free integer](https://tamnd.github.io/math2/node/080Q3RPKY6.md): statement 080Q3RPKY6
- [Linearity of the integral](https://tamnd.github.io/math2/node/080RWVVSB6.md): statement 080RWVVSB6
- [Irrational number](https://tamnd.github.io/math2/node/080T2TXVQV.md): definition 080T2TXVQV
- [Rational root theorem](https://tamnd.github.io/math2/node/0819CQ1V8H.md): statement 0819CQ1V8H
- [Phase line](https://tamnd.github.io/math2/node/081QSEAGEY.md): concept 081QSEAGEY
- [Linear recurrence with constant coefficients](https://tamnd.github.io/math2/node/081THG13JZ.md): definition 081THG13JZ
- [Analytic function](https://tamnd.github.io/math2/node/081WKEDK6C.md): concept 081WKEDK6C
- [Finite p-groups are nilpotent](https://tamnd.github.io/math2/node/081WTKYX6C.md): statement 081WTKYX6C
- [Directrix](https://tamnd.github.io/math2/node/081YA6RCX6.md): definition 081YA6RCX6
- [Holomorphic functions are analytic](https://tamnd.github.io/math2/node/0826SGV3R0.md): statement 0826SGV3R0
- [Finitely generated group](https://tamnd.github.io/math2/node/082DWWXMRJ.md): concept 082DWWXMRJ
- [Abelian group](https://tamnd.github.io/math2/node/082GHNPJ53.md): concept 082GHNPJ53
- [Absolutely convergent series in a Banach space converge](https://tamnd.github.io/math2/node/082MGNWNA8.md): statement 082MGNWNA8
- [Equivalence class](https://tamnd.github.io/math2/node/082RMRSY26.md): definition 082RMRSY26
- [Successor function](https://tamnd.github.io/math2/node/0832XVR512.md): definition 0832XVR512
- [Intersection of a family of sets](https://tamnd.github.io/math2/node/0839XNQXDK.md): definition 0839XNQXDK
- [Number of prime factors counted with multiplicity](https://tamnd.github.io/math2/node/08420828EY.md): definition 08420828EY
- [Fundamental theorem of arithmetic](https://tamnd.github.io/math2/node/084HH1NDCB.md): statement 084HH1NDCB
- [Higher derivative](https://tamnd.github.io/math2/node/084NGSTWM7.md): definition 084NGSTWM7
- [P-adic valuation of a product](https://tamnd.github.io/math2/node/0850JJMJYZ.md): statement 0850JJMJYZ
- [Subring](https://tamnd.github.io/math2/node/0859D58WPD.md): definition 0859D58WPD
- [Angle bisector theorem](https://tamnd.github.io/math2/node/0859R2V6ES.md): statement 0859R2V6ES
- [Walk](https://tamnd.github.io/math2/node/085CH3CYMX.md): concept 085CH3CYMX
- [Jordan normal form](https://tamnd.github.io/math2/node/085HQ5G5TE.md): statement 085HQ5G5TE
- [Law of quadratic reciprocity](https://tamnd.github.io/math2/node/085MJS6907.md): statement 085MJS6907
- [Group isomorphism](https://tamnd.github.io/math2/node/085PND4KNT.md): definition 085PND4KNT
- [Fermat's right triangle theorem](https://tamnd.github.io/math2/node/086BE0J3FZ.md): statement 086BE0J3FZ
- [Mean of the Poisson distribution](https://tamnd.github.io/math2/node/086HEVXR9Z.md): statement 086HEVXR9Z
- [Predicate logic](https://tamnd.github.io/math2/node/086QKPF0A6.md): concept 086QKPF0A6
- [Graph colouring](https://tamnd.github.io/math2/node/0874138YP6.md): definition 0874138YP6
- [Isometry](https://tamnd.github.io/math2/node/0878DN3WCS.md): definition 0878DN3WCS
- [von Mangoldt function](https://tamnd.github.io/math2/node/087EMZTHWN.md): definition 087EMZTHWN
- [Statistical hypothesis test](https://tamnd.github.io/math2/node/087P03NEND.md): concept 087P03NEND
- [Translation](https://tamnd.github.io/math2/node/087VJM5Q77.md): concept 087VJM5Q77
- [Positive semidefinite matrix](https://tamnd.github.io/math2/node/0880F9EQCP.md): definition 0880F9EQCP
- [Chevalley–Warning theorem](https://tamnd.github.io/math2/node/08895Q5W96.md): statement 08895Q5W96
- [Spanning set](https://tamnd.github.io/math2/node/088RYKAWK9.md): concept 088RYKAWK9
- [Ring](https://tamnd.github.io/math2/node/0894J980RK.md): definition 0894J980RK
- [Periodic function](https://tamnd.github.io/math2/node/089KWVYD69.md): definition 089KWVYD69
- [Cyclic subgroup generated by an element](https://tamnd.github.io/math2/node/08ABT85339.md): concept 08ABT85339
- [Divergence part of the ratio test](https://tamnd.github.io/math2/node/08AM0KRCZ1.md): statement 08AM0KRCZ1
- [Moment of inertia](https://tamnd.github.io/math2/node/08ANASGBXV.md): concept 08ANASGBXV
- [Independent set](https://tamnd.github.io/math2/node/08ANQHHG3P.md): concept 08ANQHHG3P
- [Diophantine equation](https://tamnd.github.io/math2/node/08B1RTW7VZ.md): concept 08B1RTW7VZ
- [Adjacency matrix](https://tamnd.github.io/math2/node/08B9CX109G.md): concept 08B9CX109G
- [Class equation](https://tamnd.github.io/math2/node/08BPSC6YTE.md): statement 08BPSC6YTE
- [Product of primitive polynomials is primitive](https://tamnd.github.io/math2/node/08C1NWBNHA.md): statement 08C1NWBNHA
- [Existence of an eigenvalue over an algebraically closed field](https://tamnd.github.io/math2/node/08CJB2NQ92.md): statement 08CJB2NQ92
- [Binary relation](https://tamnd.github.io/math2/node/08EJC6Q25H.md): definition 08EJC6Q25H
- [Equinumerous sets](https://tamnd.github.io/math2/node/08EJHHMB5R.md): definition 08EJHHMB5R
- [Fréchet derivative](https://tamnd.github.io/math2/node/08EVNYZF42.md): definition 08EVNYZF42
- [Complete metric space](https://tamnd.github.io/math2/node/08EWT7J0SC.md): definition 08EWT7J0SC
- [Product of cyclic groups is cyclic iff the orders are coprime](https://tamnd.github.io/math2/node/08FDFEF8D1.md): statement 08FDFEF8D1
- [Group of units of a monoid](https://tamnd.github.io/math2/node/08FHNR8Z5D.md): definition 08FHNR8Z5D
- [Homomorphisms preserve identity and inverses](https://tamnd.github.io/math2/node/08FKQD7NJ3.md): statement 08FKQD7NJ3
- [Uniform convergence](https://tamnd.github.io/math2/node/08FPABDQ4C.md): definition 08FPABDQ4C
- [Product of countable sets is countable](https://tamnd.github.io/math2/node/08G4XH86JC.md): statement 08G4XH86JC
- [Floor function](https://tamnd.github.io/math2/node/08GR54CKEA.md): concept 08GR54CKEA
- [Uniform convergence on a set](https://tamnd.github.io/math2/node/08HJRTG5N7.md): definition 08HJRTG5N7
- [Exterior algebra](https://tamnd.github.io/math2/node/08HK38GCSM.md): definition 08HK38GCSM
- [Real numbers](https://tamnd.github.io/math2/node/08J2M1JYAG.md): object 08J2M1JYAG
- [Tutte's theorem](https://tamnd.github.io/math2/node/08J7R35B6Q.md): statement 08J7R35B6Q
- [Fixed points of an action](https://tamnd.github.io/math2/node/08JF1K4DW1.md): definition 08JF1K4DW1
- [Cyclotomic polynomial](https://tamnd.github.io/math2/node/08KEYFAH66.md): concept 08KEYFAH66
- [Group isomorphism](https://tamnd.github.io/math2/node/08MG7H8JW5.md): concept 08MG7H8JW5
- [Distributive laws for sets](https://tamnd.github.io/math2/node/08N6EVA7DV.md): statement 08N6EVA7DV
- [Infinite limit](https://tamnd.github.io/math2/node/08NM8KQPDQ.md): definition 08NM8KQPDQ
- [Norm](https://tamnd.github.io/math2/node/08NYZ2FS6B.md): definition 08NYZ2FS6B
- [Sequence spaces lp](https://tamnd.github.io/math2/node/08P6T2WH6R.md): example 08P6T2WH6R
- [Lower central series](https://tamnd.github.io/math2/node/08PA437800.md): definition 08PA437800
- [Product formula for the order of HK](https://tamnd.github.io/math2/node/08PY4GNAPK.md): statement 08PY4GNAPK
- [Picard's great theorem](https://tamnd.github.io/math2/node/08QA8QTM71.md): statement 08QA8QTM71
- [The sequence 1/n tends to zero](https://tamnd.github.io/math2/node/08QAX4H1PF.md): example 08QAX4H1PF
- [Independence means the joint law is the product of the marginals](https://tamnd.github.io/math2/node/08QCA2F1NX.md): statement 08QCA2F1NX
- [Partition of a set](https://tamnd.github.io/math2/node/08QMKTWF9T.md): concept 08QMKTWF9T
- [Number of functions between finite sets](https://tamnd.github.io/math2/node/08QQH9F7G0.md): statement 08QQH9F7G0
- [Cesàro mean](https://tamnd.github.io/math2/node/08R2V9GZF7.md): concept 08R2V9GZF7
- [Number of distinct prime factors](https://tamnd.github.io/math2/node/08RQ62VAVA.md): concept 08RQ62VAVA
- [Liouville function](https://tamnd.github.io/math2/node/08RQAPXF1T.md): definition 08RQAPXF1T
- [Power series solution](https://tamnd.github.io/math2/node/08RT9GDDPQ.md): concept 08RT9GDDPQ
- [Finite field](https://tamnd.github.io/math2/node/08RTDHX1SK.md): definition 08RTDHX1SK
- [Normal extension](https://tamnd.github.io/math2/node/08S0NW8SES.md): concept 08S0NW8SES
- [Differentiable on an open set](https://tamnd.github.io/math2/node/08S7FDRS5Z.md): definition 08S7FDRS5Z
- [Commutator subgroup](https://tamnd.github.io/math2/node/08SJ5K2CBG.md): definition 08SJ5K2CBG
- [Permutation matrix](https://tamnd.github.io/math2/node/08SP3DF8NB.md): concept 08SP3DF8NB
- [Topological embedding](https://tamnd.github.io/math2/node/08SVRVHJNY.md): definition 08SVRVHJNY
- [Order of an element divides the order of the group](https://tamnd.github.io/math2/node/08V7JHS3QZ.md): statement 08V7JHS3QZ
- [Semisimple endomorphism](https://tamnd.github.io/math2/node/08VRX3Q4PB.md): concept 08VRX3Q4PB
- [Content of a polynomial](https://tamnd.github.io/math2/node/08VV7YJRB7.md): definition 08VV7YJRB7
- [Uniform convergence](https://tamnd.github.io/math2/node/08VZ57PT26.md): concept 08VZ57PT26
- [Median](https://tamnd.github.io/math2/node/08W2CW8PKW.md): definition 08W2CW8PKW
- [Power series of cosine](https://tamnd.github.io/math2/node/08W5RN375X.md): statement 08W5RN375X
- [Hall's marriage theorem](https://tamnd.github.io/math2/node/08WCB90WND.md): statement 08WCB90WND
- [Cauchy–Binet formula](https://tamnd.github.io/math2/node/08WE2W7DYS.md): statement 08WE2W7DYS
- [Sine and cosine](https://tamnd.github.io/math2/node/08WFZP6P89.md): concept 08WFZP6P89
- [Euler's four-square identity](https://tamnd.github.io/math2/node/08WTN6SCQ6.md): statement 08WTN6SCQ6
- [Explicit and implicit differential equations](https://tamnd.github.io/math2/node/08WWTN738R.md): concept 08WWTN738R
- [Square-free integer](https://tamnd.github.io/math2/node/08X0VCR2NW.md): concept 08X0VCR2NW
- [An iterate of the Picard operator is a contraction](https://tamnd.github.io/math2/node/08XDXP86XE.md): statement 08XDXP86XE
- [Comparison of topologies](https://tamnd.github.io/math2/node/08XFJJ8EYW.md): definition 08XFJJ8EYW
- [Integration by parts](https://tamnd.github.io/math2/node/08Y1BBQTK9.md): statement 08Y1BBQTK9
- [Subfields of a finite field](https://tamnd.github.io/math2/node/08YERZDM9X.md): statement 08YERZDM9X
- [Steinitz exchange lemma](https://tamnd.github.io/math2/node/08ZDRN6H41.md): statement 08ZDRN6H41
- [Identically distributed random variables](https://tamnd.github.io/math2/node/08ZDW5SA3B.md): concept 08ZDW5SA3B
- [Alternating bilinear form](https://tamnd.github.io/math2/node/08ZQAKBKHF.md): definition 08ZQAKBKHF
- [Eigenvalue method for linear systems](https://tamnd.github.io/math2/node/08ZWFDJP3H.md): concept 08ZWFDJP3H
- [Tower of Hanoi](https://tamnd.github.io/math2/node/0906Z03EMN.md): example 0906Z03EMN
- [One-sided limit](https://tamnd.github.io/math2/node/0909PWZR62.md): definition 0909PWZR62
- [Convergence of a sequence in a metric space](https://tamnd.github.io/math2/node/090AA928Y7.md): concept 090AA928Y7
- [Hamiltonian graph](https://tamnd.github.io/math2/node/090WPKFABX.md): definition 090WPKFABX
- [Arithmetic function](https://tamnd.github.io/math2/node/09115NQZ07.md): concept 09115NQZ07
- [An injective endomorphism of a finite-dimensional space is surjective](https://tamnd.github.io/math2/node/0912EEEVCT.md): statement 0912EEEVCT
- [Real exponentiation](https://tamnd.github.io/math2/node/091AV9YHZY.md): concept 091AV9YHZY
- [Metrizable space](https://tamnd.github.io/math2/node/091GJ3FYD7.md): concept 091GJ3FYD7
- [Geometric probability](https://tamnd.github.io/math2/node/0924BCD257.md): concept 0924BCD257
- [Euler's identity](https://tamnd.github.io/math2/node/09285XX7AE.md): statement 09285XX7AE
- [Catalan numbers](https://tamnd.github.io/math2/node/092DRBFKMG.md): definition 092DRBFKMG
- [Binomial coefficient](https://tamnd.github.io/math2/node/092GEF38YD.md): definition 092GEF38YD
- [Lyapunov stability](https://tamnd.github.io/math2/node/092S6VBECQ.md): definition 092S6VBECQ
- [Domain](https://tamnd.github.io/math2/node/093CM69AGS.md): definition 093CM69AGS
- [Congruent triangles](https://tamnd.github.io/math2/node/093EDENVA1.md): concept 093EDENVA1
- [Mesh of a partition](https://tamnd.github.io/math2/node/093SAXPJ9P.md): concept 093SAXPJ9P
- [Minimal polynomial divides the characteristic polynomial](https://tamnd.github.io/math2/node/093X6MN6Y6.md): statement 093X6MN6Y6
- [Multiplication by a matrix as a linear map](https://tamnd.github.io/math2/node/094V2VBJXJ.md): concept 094V2VBJXJ
- [Row equivalence](https://tamnd.github.io/math2/node/095DDGZF16.md): definition 095DDGZF16
- [In a PID a nonzero element is irreducible iff it generates a maximal ideal](https://tamnd.github.io/math2/node/095ZHQ7WCJ.md): statement 095ZHQ7WCJ
- [Extended Euclidean algorithm](https://tamnd.github.io/math2/node/0964YQFSA8.md): concept 0964YQFSA8
- [Young diagram](https://tamnd.github.io/math2/node/096AHMB3P4.md): concept 096AHMB3P4
- [Unique paths in trees](https://tamnd.github.io/math2/node/096B7Q5742.md): statement 096B7Q5742
- [Subnormal series](https://tamnd.github.io/math2/node/096HN851YZ.md): definition 096HN851YZ
- [Divisibility](https://tamnd.github.io/math2/node/096PW7Q014.md): definition 096PW7Q014
- [Geometric solutions come from roots of the characteristic polynomial](https://tamnd.github.io/math2/node/096SD4ERQH.md): statement 096SD4ERQH
- [Linearly independent set of the right size is a basis](https://tamnd.github.io/math2/node/096V7H13H9.md): statement 096V7H13H9
- [Ideal](https://tamnd.github.io/math2/node/096Z9MP2ME.md): concept 096Z9MP2ME
- [Central moment](https://tamnd.github.io/math2/node/09753QQDCK.md): concept 09753QQDCK
- [Cramer's rule](https://tamnd.github.io/math2/node/0977KZ6PQB.md): statement 0977KZ6PQB
- [Riesz's lemma](https://tamnd.github.io/math2/node/097C7KCAT5.md): statement 097C7KCAT5
- [Domain in the complex plane](https://tamnd.github.io/math2/node/097KNJFJ7V.md): definition 097KNJFJ7V
- [Intermediate value theorem for connected spaces](https://tamnd.github.io/math2/node/098C0KZZAP.md): statement 098C0KZZAP
- [Fixed points of a p-group action](https://tamnd.github.io/math2/node/098D2E7DEH.md): statement 098D2E7DEH
- [Gauss–Jordan elimination](https://tamnd.github.io/math2/node/098D6FQN3N.md): concept 098D6FQN3N
- [Combination](https://tamnd.github.io/math2/node/098PHYBTV9.md): concept 098PHYBTV9
- [Peano existence theorem](https://tamnd.github.io/math2/node/098YP831QY.md): statement 098YP831QY
- [Conditional probability](https://tamnd.github.io/math2/node/0990W5PN13.md): concept 0990W5PN13
- [Commutativity](https://tamnd.github.io/math2/node/099E72KTF8.md): concept 099E72KTF8
- [Space of solutions of a linear recurrence](https://tamnd.github.io/math2/node/099GXMHV40.md): concept 099GXMHV40
- [Dirac measure](https://tamnd.github.io/math2/node/099SXAG014.md): definition 099SXAG014
- [Geometric multiplicity](https://tamnd.github.io/math2/node/099TWVJ909.md): concept 099TWVJ909
- [Converse](https://tamnd.github.io/math2/node/09A3N7ZN5G.md): concept 09A3N7ZN5G
- [Cross-ratio](https://tamnd.github.io/math2/node/09ASMRMD67.md): concept 09ASMRMD67
- [Sequence tending to infinity](https://tamnd.github.io/math2/node/09AZCV6GMC.md): definition 09AZCV6GMC
- [Discrete logarithm](https://tamnd.github.io/math2/node/09B4X51J09.md): definition 09B4X51J09
- [Power series of the complex logarithm](https://tamnd.github.io/math2/node/09BG0Y7SZD.md): statement 09BG0Y7SZD
- [Necessary and sufficient conditions](https://tamnd.github.io/math2/node/09BJQQB3VS.md): concept 09BJQQB3VS
- [Heaviside step function](https://tamnd.github.io/math2/node/09C4JP7Y0M.md): object 09C4JP7Y0M
- [Area of a region as a line integral](https://tamnd.github.io/math2/node/09CBQB6V67.md): statement 09CBQB6V67
- [Partition of a set](https://tamnd.github.io/math2/node/09CCE86A6D.md): definition 09CCE86A6D
- [Cartesian product](https://tamnd.github.io/math2/node/09CEDJYBKP.md): concept 09CEDJYBKP
- [Open sets characterised by neighbourhoods](https://tamnd.github.io/math2/node/09CJ00H3NW.md): statement 09CJ00H3NW
- [Transpose](https://tamnd.github.io/math2/node/09DXWYRD43.md): concept 09DXWYRD43
- [Coordinates in an orthonormal basis](https://tamnd.github.io/math2/node/09DZ8CD1AK.md): statement 09DZ8CD1AK
- [Trace of a transpose](https://tamnd.github.io/math2/node/09E1RCZ7YT.md): statement 09E1RCZ7YT
- [Complex differentiability](https://tamnd.github.io/math2/node/09E6T7VYAG.md): concept 09E6T7VYAG
- [Partial derivative](https://tamnd.github.io/math2/node/09EGC6F8PT.md): concept 09EGC6F8PT
- [Algebraic extension](https://tamnd.github.io/math2/node/09EYMKMSB9.md): definition 09EYMKMSB9
- [Circuit](https://tamnd.github.io/math2/node/09F2QPVQY7.md): concept 09F2QPVQY7
- [Upper and lower Darboux integrals](https://tamnd.github.io/math2/node/09FQT057T7.md): definition 09FQT057T7
- [Complex numbers](https://tamnd.github.io/math2/node/09FTAET5Z4.md): object 09FTAET5Z4
- [Epsilon-N definition of the limit of a sequence](https://tamnd.github.io/math2/node/09G6KHS6GY.md): statement 09G6KHS6GY
- [Second Sylow theorem](https://tamnd.github.io/math2/node/09GCQ5STVJ.md): statement 09GCQ5STVJ
- [Dihedral group](https://tamnd.github.io/math2/node/09GK3DE0WV.md): definition 09GK3DE0WV
- [Transpose of a product](https://tamnd.github.io/math2/node/09GK5PS824.md): statement 09GK5PS824
- [Closed sets contain their limits](https://tamnd.github.io/math2/node/09GQ3ZTQJ9.md): statement 09GQ3ZTQJ9
- [Completeness of the space of bounded continuous functions](https://tamnd.github.io/math2/node/09GRCZVR5E.md): statement 09GRCZVR5E
- [The symmetric group S3](https://tamnd.github.io/math2/node/09H9Y0Q11W.md): example 09H9Y0Q11W
- [Outer automorphism group](https://tamnd.github.io/math2/node/09HEXDQ0CQ.md): definition 09HEXDQ0CQ
- [Euler's polyhedron formula](https://tamnd.github.io/math2/node/09HT3P35YP.md): statement 09HT3P35YP
- [Method of moments](https://tamnd.github.io/math2/node/09J1H07VAH.md): concept 09J1H07VAH
- [Maximal ideal](https://tamnd.github.io/math2/node/09J6KGT65G.md): definition 09J6KGT65G
- [Forest](https://tamnd.github.io/math2/node/09JP6G9FTA.md): definition 09JP6G9FTA
- [Sum of the totient over the divisors](https://tamnd.github.io/math2/node/09JXE6XMCC.md): statement 09JXE6XMCC
- [Groups of order p squared are abelian](https://tamnd.github.io/math2/node/09JZ9P4899.md): statement 09JZ9P4899
- [Propositional logic](https://tamnd.github.io/math2/node/09K9MWD2QQ.md): concept 09K9MWD2QQ
- [Metrizable space](https://tamnd.github.io/math2/node/09KDHB82SG.md): definition 09KDHB82SG
- [Order of a differential equation](https://tamnd.github.io/math2/node/09M2RDB61G.md): definition 09M2RDB61G
- [Ring homomorphisms from a field are injective](https://tamnd.github.io/math2/node/09N7DZ7SDB.md): statement 09N7DZ7SDB
- [Finite intersection of open sets is open](https://tamnd.github.io/math2/node/09NDE9V7X1.md): statement 09NDE9V7X1
- [Injective function](https://tamnd.github.io/math2/node/09NWETVTNY.md): concept 09NWETVTNY
- [De Morgan's laws](https://tamnd.github.io/math2/node/09PKJKA6J0.md): statement 09PKJKA6J0
- [Mean of the normal distribution](https://tamnd.github.io/math2/node/09PR1V6RWR.md): statement 09PR1V6RWR
- [Liouville's formula](https://tamnd.github.io/math2/node/09PS3S0EF6.md): statement 09PS3S0EF6
- [Prüfer sequence](https://tamnd.github.io/math2/node/09PV9JSY6E.md): definition 09PV9JSY6E
- [Bounded set](https://tamnd.github.io/math2/node/09QSZVVM26.md): concept 09QSZVVM26
- [Removable singularity](https://tamnd.github.io/math2/node/09R8GC85RB.md): concept 09R8GC85RB
- [Cancellation in congruences](https://tamnd.github.io/math2/node/09RDZZGJZ9.md): statement 09RDZZGJZ9
- [The rationals form an ordered field](https://tamnd.github.io/math2/node/09RFKMBQ63.md): statement 09RFKMBQ63
- [Partial sum](https://tamnd.github.io/math2/node/09RJ859N7A.md): concept 09RJ859N7A
- [Student's t-distribution](https://tamnd.github.io/math2/node/09RTQRK5PC.md): concept 09RTQRK5PC
- [Smooth non-analytic function](https://tamnd.github.io/math2/node/09RTYYHXKA.md): counterexample 09RTYYHXKA
- [Integral domain](https://tamnd.github.io/math2/node/09S3PFBEPM.md): definition 09S3PFBEPM
- [Least common multiple](https://tamnd.github.io/math2/node/09S5SN35DJ.md): definition 09S5SN35DJ
- [Subbasis](https://tamnd.github.io/math2/node/09S8DV34Y0.md): concept 09S8DV34Y0
- [Independent sigma-algebras](https://tamnd.github.io/math2/node/09T8ZHST1V.md): definition 09T8ZHST1V
- [Equivalent metrics](https://tamnd.github.io/math2/node/09V3NASXND.md): concept 09V3NASXND
- [Scalar multiplication of matrices](https://tamnd.github.io/math2/node/09V6DDACXR.md): concept 09V6DDACXR
- [Module over a ring](https://tamnd.github.io/math2/node/09VDS2W1E8.md): definition 09VDS2W1E8
- [Singular value](https://tamnd.github.io/math2/node/09VMDSRXWJ.md): concept 09VMDSRXWJ
- [Orthonormal family](https://tamnd.github.io/math2/node/09VYHEF2VT.md): definition 09VYHEF2VT
- [Every walk contains a path](https://tamnd.github.io/math2/node/09VZBHB33V.md): statement 09VZBHB33V
- [Local minimum](https://tamnd.github.io/math2/node/09WRNGPTMV.md): concept 09WRNGPTMV
- [Affine space](https://tamnd.github.io/math2/node/09XNAPTJQ7.md): concept 09XNAPTJQ7
- [Permuting rows multiplies the determinant by the sign](https://tamnd.github.io/math2/node/09YB3WQG38.md): statement 09YB3WQG38
- [Divisibility by 11](https://tamnd.github.io/math2/node/09YBMJTNBM.md): statement 09YBMJTNBM
- [Intersection of closed sets is closed](https://tamnd.github.io/math2/node/09YMV8H4X5.md): statement 09YMV8H4X5
- [The number of Sylow subgroups is the index of the normaliser](https://tamnd.github.io/math2/node/09ZJ6CAK7S.md): statement 09ZJ6CAK7S
- [Product of compact sets is compact](https://tamnd.github.io/math2/node/09ZKE5XQ74.md): statement 09ZKE5XQ74
- [Linearisation at an equilibrium](https://tamnd.github.io/math2/node/09ZWYRQ41J.md): concept 09ZWYRQ41J
- [Higher partial derivatives](https://tamnd.github.io/math2/node/09ZXGX55DM.md): concept 09ZXGX55DM
- [Determinant of a matrix exponential](https://tamnd.github.io/math2/node/0A1QQN333A.md): statement 0A1QQN333A
- [Isolated point](https://tamnd.github.io/math2/node/0A1YXSKB48.md): concept 0A1YXSKB48
- [Open sets of a metric space](https://tamnd.github.io/math2/node/0A26X79RTB.md): statement 0A26X79RTB
- [Convergence in mean](https://tamnd.github.io/math2/node/0A2HNZC44K.md): concept 0A2HNZC44K
- [Uniform continuity on a set](https://tamnd.github.io/math2/node/0A3CK6DVVB.md): definition 0A3CK6DVVB
- [The reals are Archimedean](https://tamnd.github.io/math2/node/0A3GWZ86AY.md): statement 0A3GWZ86AY
- [Pseudometric space](https://tamnd.github.io/math2/node/0A3HJX3KTF.md): concept 0A3HJX3KTF
- [Boolean function](https://tamnd.github.io/math2/node/0A58THH5XV.md): definition 0A58THH5XV
- [Differentiable function](https://tamnd.github.io/math2/node/0A61B60B5R.md): definition 0A61B60B5R
- [Linear equation with constant coefficients](https://tamnd.github.io/math2/node/0A6PGTTP2A.md): definition 0A6PGTTP2A
- [Derivative gives the directional derivatives](https://tamnd.github.io/math2/node/0A6PMR20N4.md): statement 0A6PMR20N4
- [System of linear differential equations](https://tamnd.github.io/math2/node/0A6QZ367GX.md): definition 0A6QZ367GX
- [Metric topology](https://tamnd.github.io/math2/node/0A6YRNZB2R.md): definition 0A6YRNZB2R
- [Archimedean property](https://tamnd.github.io/math2/node/0A74SX55QP.md): statement 0A74SX55QP
- [Baire category theorem](https://tamnd.github.io/math2/node/0A7KGAYHGV.md): statement 0A7KGAYHGV
- [Tower law](https://tamnd.github.io/math2/node/0A7KJ68XHT.md): statement 0A7KJ68XHT
- [Monotone function](https://tamnd.github.io/math2/node/0A7KV2WB7A.md): concept 0A7KV2WB7A
- [Complex sine and cosine](https://tamnd.github.io/math2/node/0A7YDKRNR6.md): definition 0A7YDKRNR6
- [Bounded sequence](https://tamnd.github.io/math2/node/0A8JNBGT2Z.md): concept 0A8JNBGT2Z
- [x^2 is not uniformly continuous on the real line](https://tamnd.github.io/math2/node/0A8JXHYTJC.md): counterexample 0A8JXHYTJC
- [Chain rule for a composition with a curve](https://tamnd.github.io/math2/node/0A8TJ0VHC7.md): statement 0A8TJ0VHC7
- [Complete ordered field](https://tamnd.github.io/math2/node/0A9E6X0S3E.md): concept 0A9E6X0S3E
- [Chi-squared distribution](https://tamnd.github.io/math2/node/0A9EB0KQBC.md): concept 0A9EB0KQBC
- [Schröder–Bernstein theorem](https://tamnd.github.io/math2/node/0A9V6RKQG6.md): statement 0A9V6RKQG6
- [Periodic orbit](https://tamnd.github.io/math2/node/0AA97XA70K.md): concept 0AA97XA70K
- [Kernel of a ring homomorphism](https://tamnd.github.io/math2/node/0AAHE7GEW6.md): definition 0AAHE7GEW6
- [Least element](https://tamnd.github.io/math2/node/0AANKKNHS4.md): concept 0AANKKNHS4
- [Limit cycle](https://tamnd.github.io/math2/node/0AAQ73WRFB.md): concept 0AAQ73WRFB
- [Almost sure convergence](https://tamnd.github.io/math2/node/0AAXWGT5TD.md): concept 0AAXWGT5TD
- [Gradient](https://tamnd.github.io/math2/node/0ABK9V9Y64.md): concept 0ABK9V9Y64
- [Finite-dimensional normed spaces are complete](https://tamnd.github.io/math2/node/0ABP2F2ZP7.md): statement 0ABP2F2ZP7
- [Dirac measure](https://tamnd.github.io/math2/node/0ACRGYNHSR.md): concept 0ACRGYNHSR
- [Invariant subspace](https://tamnd.github.io/math2/node/0ACX5QT8TE.md): concept 0ACX5QT8TE
- [Metrizable spaces are normal](https://tamnd.github.io/math2/node/0AD0BZ49DW.md): statement 0AD0BZ49DW
- [Formal derivative](https://tamnd.github.io/math2/node/0AD4VS02H3.md): definition 0AD4VS02H3
- [Jacobi symbol](https://tamnd.github.io/math2/node/0ADTQBXRMC.md): concept 0ADTQBXRMC
- [Strictly convex function](https://tamnd.github.io/math2/node/0AEAHMFSCA.md): concept 0AEAHMFSCA
- [Prime factorization](https://tamnd.github.io/math2/node/0AEKDFCYFA.md): definition 0AEKDFCYFA
- [Pell's equation](https://tamnd.github.io/math2/node/0AF3DANJ76.md): definition 0AF3DANJ76
- [Convergent series](https://tamnd.github.io/math2/node/0AF7X08PCR.md): concept 0AF7X08PCR
- [Collinear points](https://tamnd.github.io/math2/node/0AFNNTRTJE.md): concept 0AFNNTRTJE
- [sin(1/x) has no limit at 0](https://tamnd.github.io/math2/node/0AFWQZ1PKF.md): counterexample 0AFWQZ1PKF
- [If G modulo its centre is cyclic then G is abelian](https://tamnd.github.io/math2/node/0AH9Q6YNMT.md): statement 0AH9Q6YNMT
- [Vertex cover](https://tamnd.github.io/math2/node/0AHKWBQSWV.md): concept 0AHKWBQSWV
- [Generalised eigenspace](https://tamnd.github.io/math2/node/0AJ035ZTZE.md): concept 0AJ035ZTZE
- [Multiplication by a matrix as a linear map](https://tamnd.github.io/math2/node/0AJ0KP1B31.md): definition 0AJ0KP1B31
- [Annulus](https://tamnd.github.io/math2/node/0AJ0YVK743.md): definition 0AJ0YVK743
- [Normal space](https://tamnd.github.io/math2/node/0AJ646N339.md): definition 0AJ646N339
- [Independent and identically distributed random variables](https://tamnd.github.io/math2/node/0AJN7WHBJZ.md): definition 0AJN7WHBJZ
- [Step function](https://tamnd.github.io/math2/node/0AJXYH4DDQ.md): concept 0AJXYH4DDQ
- [Successor function](https://tamnd.github.io/math2/node/0AK9TGSNXY.md): concept 0AK9TGSNXY
- [Classical definition of probability](https://tamnd.github.io/math2/node/0AKNPYQ04R.md): definition 0AKNPYQ04R
- [Stokes' theorem](https://tamnd.github.io/math2/node/0AKRKVHW94.md): statement 0AKRKVHW94
- [Gram matrices are positive semidefinite](https://tamnd.github.io/math2/node/0AM050KCAW.md): statement 0AM050KCAW
- [Cauchy product of absolutely convergent series](https://tamnd.github.io/math2/node/0AN28XQ45Z.md): statement 0AN28XQ45Z
- [Circle integral](https://tamnd.github.io/math2/node/0AP750DHVP.md): concept 0AP750DHVP
- [Multiplicative function](https://tamnd.github.io/math2/node/0AP80TT660.md): definition 0AP80TT660
- [Nullity](https://tamnd.github.io/math2/node/0APCFW7639.md): concept 0APCFW7639
- [Second isomorphism theorem for vector spaces](https://tamnd.github.io/math2/node/0APN65D6Y6.md): statement 0APN65D6Y6
- [Edge set](https://tamnd.github.io/math2/node/0AQBSVMX9P.md): concept 0AQBSVMX9P
- [The symmetric group is generated by transpositions](https://tamnd.github.io/math2/node/0AQNX7G443.md): statement 0AQNX7G443
- [Abelianisation](https://tamnd.github.io/math2/node/0AQZV0905N.md): concept 0AQZV0905N
- [Differentiation under the integral sign](https://tamnd.github.io/math2/node/0AR06W1E1B.md): statement 0AR06W1E1B
- [Pairwise independent events that are not mutually independent](https://tamnd.github.io/math2/node/0AR4B3K5QD.md): counterexample 0AR4B3K5QD
- [Unitary matrix](https://tamnd.github.io/math2/node/0AR68JTMDT.md): concept 0AR68JTMDT
- [Continuous uniform distribution](https://tamnd.github.io/math2/node/0AREFQ9Z65.md): concept 0AREFQ9Z65
- [Simplicity of the alternating group](https://tamnd.github.io/math2/node/0ARPRJQWZ2.md): statement 0ARPRJQWZ2
- [Power sum symmetric polynomials](https://tamnd.github.io/math2/node/0AS1FZT6BM.md): definition 0AS1FZT6BM
- [Consistent estimator](https://tamnd.github.io/math2/node/0AS800VAYX.md): concept 0AS800VAYX
- [The Möbius function is the inverse of the constant function 1](https://tamnd.github.io/math2/node/0ASC73QEQT.md): statement 0ASC73QEQT
- [Block matrix](https://tamnd.github.io/math2/node/0ASDFEQS6P.md): concept 0ASDFEQS6P
- [Commutator subgroup](https://tamnd.github.io/math2/node/0ASGSTXSBW.md): concept 0ASGSTXSBW
- [Annihilator of a subspace](https://tamnd.github.io/math2/node/0ASHNYF7AG.md): concept 0ASHNYF7AG
- [Mathematical induction](https://tamnd.github.io/math2/node/0ASJVEFA9J.md): concept 0ASJVEFA9J
- [Ceiling function](https://tamnd.github.io/math2/node/0ASM4EPMB2.md): definition 0ASM4EPMB2
- [Normaliser](https://tamnd.github.io/math2/node/0AT1ZSNM25.md): definition 0AT1ZSNM25
- [Periodic function](https://tamnd.github.io/math2/node/0ATF6Q0633.md): concept 0ATF6Q0633
- [Back substitution](https://tamnd.github.io/math2/node/0ATFVE08MW.md): concept 0ATFVE08MW
- [Boundedness theorem](https://tamnd.github.io/math2/node/0ATVXC3NCC.md): statement 0ATVXC3NCC
- [The nth root of n tends to 1](https://tamnd.github.io/math2/node/0AVEYZ2T0Q.md): example 0AVEYZ2T0Q
- [R/P is an integral domain iff P is prime](https://tamnd.github.io/math2/node/0AVXBEC05W.md): statement 0AVXBEC05W
- [Totient of a prime power](https://tamnd.github.io/math2/node/0AVY8HKBGD.md): statement 0AVY8HKBGD
- [Lipschitz condition](https://tamnd.github.io/math2/node/0AW1E6YTJY.md): concept 0AW1E6YTJY
- [K[x]/(f) is a field iff f is irreducible](https://tamnd.github.io/math2/node/0AWMPX35CW.md): statement 0AWMPX35CW
- [Marginal distribution](https://tamnd.github.io/math2/node/0AXP2Z5N9H.md): definition 0AXP2Z5N9H
- [Completeness axiom](https://tamnd.github.io/math2/node/0AXWET80YW.md): statement 0AXWET80YW
- [Symmetric bilinear form](https://tamnd.github.io/math2/node/0AXZNE5XNH.md): concept 0AXZNE5XNH
- [Eigenspace](https://tamnd.github.io/math2/node/0AYEPBQMAW.md): definition 0AYEPBQMAW
- [Stabiliser](https://tamnd.github.io/math2/node/0AYMP41THN.md): concept 0AYMP41THN
- [Product metric](https://tamnd.github.io/math2/node/0AYV6VGKF5.md): concept 0AYV6VGKF5
- [Convergent sequence](https://tamnd.github.io/math2/node/0AZ3R52593.md): definition 0AZ3R52593
- [Dual map](https://tamnd.github.io/math2/node/0AZ5ARP287.md): concept 0AZ5ARP287
- [Action on left cosets](https://tamnd.github.io/math2/node/0AZG100DKH.md): example 0AZG100DKH
- [Equinumerous sets](https://tamnd.github.io/math2/node/0B0KR7FH2Q.md): concept 0B0KR7FH2Q
- [Faithful action](https://tamnd.github.io/math2/node/0B0WE1GKTK.md): definition 0B0WE1GKTK
- [Ring](https://tamnd.github.io/math2/node/0B1261B5ED.md): concept 0B1261B5ED
- [Complement](https://tamnd.github.io/math2/node/0B1FSS1VZW.md): definition 0B1FSS1VZW
- [Moment-generating function of the normal distribution](https://tamnd.github.io/math2/node/0B1G4E21G7.md): statement 0B1G4E21G7
- [Cauchy product](https://tamnd.github.io/math2/node/0B1K9PVWE9.md): definition 0B1K9PVWE9
- [Curl-free vector field on a convex set is conservative](https://tamnd.github.io/math2/node/0B22P010D6.md): statement 0B22P010D6
- [Open balls are open](https://tamnd.github.io/math2/node/0B2305GNAZ.md): statement 0B2305GNAZ
- [Gaussian integer](https://tamnd.github.io/math2/node/0B268GVX95.md): definition 0B268GVX95
- [Level set](https://tamnd.github.io/math2/node/0B2RZ0778N.md): concept 0B2RZ0778N
- [Möbius transformation](https://tamnd.github.io/math2/node/0B346DDAM5.md): definition 0B346DDAM5
- [Conjunctive normal form](https://tamnd.github.io/math2/node/0B3D4C8YF2.md): concept 0B3D4C8YF2
- [Quantile function](https://tamnd.github.io/math2/node/0B3Q3XDM5X.md): definition 0B3Q3XDM5X
- [Independent sigma-algebras](https://tamnd.github.io/math2/node/0B5BA4FRWS.md): concept 0B5BA4FRWS
- [Set bounded above](https://tamnd.github.io/math2/node/0B5H2V4JW8.md): definition 0B5H2V4JW8
- [Eigenvector](https://tamnd.github.io/math2/node/0B5H4208Z9.md): definition 0B5H4208Z9
- [Edge colouring](https://tamnd.github.io/math2/node/0B5WCC02AN.md): concept 0B5WCC02AN
- [R[x] is an integral domain iff R is](https://tamnd.github.io/math2/node/0B5Z1EC66J.md): statement 0B5Z1EC66J
- [Chromatic polynomial](https://tamnd.github.io/math2/node/0B621RBVS9.md): concept 0B621RBVS9
- [Legendre's three-square theorem](https://tamnd.github.io/math2/node/0B64Q7XNE3.md): statement 0B64Q7XNE3
- [Lexicographic order](https://tamnd.github.io/math2/node/0B68KNPGCG.md): concept 0B68KNPGCG
- [Ring isomorphism](https://tamnd.github.io/math2/node/0B6QC0C0G1.md): definition 0B6QC0C0G1
- [Strict differentiability](https://tamnd.github.io/math2/node/0B6QWX3086.md): concept 0B6QWX3086
- [Markov property](https://tamnd.github.io/math2/node/0B77KZCQ51.md): concept 0B77KZCQ51
- [Gamma distribution](https://tamnd.github.io/math2/node/0B80PAQD3C.md): concept 0B80PAQD3C
- [Cyclic group](https://tamnd.github.io/math2/node/0B85Q41RP6.md): definition 0B85Q41RP6
- [Order of a group](https://tamnd.github.io/math2/node/0B85S2V2DR.md): concept 0B85S2V2DR
- [Essential singularity](https://tamnd.github.io/math2/node/0B8837S5T3.md): concept 0B8837S5T3
- [Universal quantifier](https://tamnd.github.io/math2/node/0B88PQTJ66.md): definition 0B88PQTJ66
- [Arcsine](https://tamnd.github.io/math2/node/0B89D93YA1.md): definition 0B89D93YA1
- [Lattice of subgroups](https://tamnd.github.io/math2/node/0B8BW4YWTF.md): concept 0B8BW4YWTF
- [Continuity via closed sets](https://tamnd.github.io/math2/node/0B8GPXFSP7.md): statement 0B8GPXFSP7
- [Orthogonal projection](https://tamnd.github.io/math2/node/0B8KZRT28M.md): concept 0B8KZRT28M
- [Linear isomorphism](https://tamnd.github.io/math2/node/0B8T10CE7N.md): concept 0B8T10CE7N
- [Cofinite topology](https://tamnd.github.io/math2/node/0B906ZBTMZ.md): example 0B906ZBTMZ
- [Mathematical proof](https://tamnd.github.io/math2/node/0B94TKFQJX.md): concept 0B94TKFQJX
- [Cauchy condensation test](https://tamnd.github.io/math2/node/0B96KMDDD1.md): statement 0B96KMDDD1
- [Rotation matrix](https://tamnd.github.io/math2/node/0B9QNFGTHM.md): example 0B9QNFGTHM
- [Connected subsets of the real line are intervals](https://tamnd.github.io/math2/node/0B9ZQFMQNV.md): statement 0B9ZQFMQNV
- [Bridge](https://tamnd.github.io/math2/node/0BANA3XYPF.md): concept 0BANA3XYPF
- [Mean value theorem](https://tamnd.github.io/math2/node/0BARQY52N5.md): statement 0BARQY52N5
- [Proof by contrapositive](https://tamnd.github.io/math2/node/0BB5TPGQ1N.md): concept 0BB5TPGQ1N
- [Finite set](https://tamnd.github.io/math2/node/0BBTH3YSZA.md): definition 0BBTH3YSZA
- [Matrix determinant lemma](https://tamnd.github.io/math2/node/0BC3Q6K92H.md): statement 0BC3Q6K92H
- [Power set](https://tamnd.github.io/math2/node/0BC7KX51Z8.md): definition 0BC7KX51Z8
- [Cycle type](https://tamnd.github.io/math2/node/0BCHJ30REG.md): definition 0BCHJ30REG
- [Pivot](https://tamnd.github.io/math2/node/0BCMGDHZGR.md): definition 0BCMGDHZGR
- [Order of the Galois group of a Galois extension equals its degree](https://tamnd.github.io/math2/node/0BDJBH8B4Z.md): statement 0BDJBH8B4Z
- [Banach space](https://tamnd.github.io/math2/node/0BDTTM24DP.md): definition 0BDTTM24DP
- [Monic polynomial](https://tamnd.github.io/math2/node/0BDX4MBKT4.md): definition 0BDX4MBKT4
- [Smooth function](https://tamnd.github.io/math2/node/0BE3QDH73A.md): definition 0BE3QDH73A
- [Solutions for complex conjugate roots](https://tamnd.github.io/math2/node/0BEZX8F2XW.md): statement 0BEZX8F2XW
- [Tensor product of vector spaces](https://tamnd.github.io/math2/node/0BF2DE2SYK.md): concept 0BF2DE2SYK
- [Global existence for globally Lipschitz vector fields](https://tamnd.github.io/math2/node/0BFD3A6CRH.md): statement 0BFD3A6CRH
- [Weak law of large numbers](https://tamnd.github.io/math2/node/0BFDHF2H7N.md): statement 0BFDHF2H7N
- [Number of k-element subsets](https://tamnd.github.io/math2/node/0BFMR2FQ8B.md): statement 0BFMR2FQ8B
- [Path independence of line integrals](https://tamnd.github.io/math2/node/0BFN21TF5F.md): statement 0BFN21TF5F
- [Infinite limit](https://tamnd.github.io/math2/node/0BH9JH7HBY.md): concept 0BH9JH7HBY
- [Limit along different paths](https://tamnd.github.io/math2/node/0BJ6H85DRQ.md): concept 0BJ6H85DRQ
- [A linear map is injective iff its kernel is trivial](https://tamnd.github.io/math2/node/0BJBYCGMWE.md): statement 0BJBYCGMWE
- [Tangent plane and normal vector of a surface](https://tamnd.github.io/math2/node/0BJDQ2HBH1.md): definition 0BJDQ2HBH1
- [Spanning set](https://tamnd.github.io/math2/node/0BJKVF90Z6.md): definition 0BJKVF90Z6
- [Subset](https://tamnd.github.io/math2/node/0BJR9QR08E.md): concept 0BJR9QR08E
- [Mertens' theorem on Cauchy products](https://tamnd.github.io/math2/node/0BJSWZJG18.md): statement 0BJSWZJG18
- [Complex conjugate](https://tamnd.github.io/math2/node/0BJV24C4SY.md): definition 0BJV24C4SY
- [Square root of a positive semidefinite matrix](https://tamnd.github.io/math2/node/0BJVDCF38Y.md): concept 0BJVDCF38Y
- [Continued fraction](https://tamnd.github.io/math2/node/0BK6JBK58W.md): definition 0BK6JBK58W
- [Hölder's inequality for sums](https://tamnd.github.io/math2/node/0BKDAGMGX9.md): statement 0BKDAGMGX9
- [Number of divisors](https://tamnd.github.io/math2/node/0BKJT6WG96.md): definition 0BKJT6WG96
- [Coordinate vector](https://tamnd.github.io/math2/node/0BMVBHTWV2.md): definition 0BMVBHTWV2
- [Fermat's little theorem in the form a^p = a](https://tamnd.github.io/math2/node/0BMXWXDVJZ.md): statement 0BMXWXDVJZ
- [Uniformly Cauchy sequence of functions](https://tamnd.github.io/math2/node/0BMY28TBSF.md): concept 0BMY28TBSF
- [Line](https://tamnd.github.io/math2/node/0BMYPJBJH4.md): definition 0BMYPJBJH4
- [Trees are bipartite](https://tamnd.github.io/math2/node/0BN2A6EBYA.md): statement 0BN2A6EBYA
- [Polynomial of an endomorphism](https://tamnd.github.io/math2/node/0BNEXSKNRV.md): definition 0BNEXSKNRV
- [Trivial group](https://tamnd.github.io/math2/node/0BNPC9850Y.md): example 0BNPC9850Y
- [Universal set](https://tamnd.github.io/math2/node/0BPYRMHJ48.md): definition 0BPYRMHJ48
- [Eigenvalue](https://tamnd.github.io/math2/node/0BQAA2BK3Q.md): concept 0BQAA2BK3Q
- [Permutation](https://tamnd.github.io/math2/node/0BQMWCNS8S.md): definition 0BQMWCNS8S
- [Quotient space](https://tamnd.github.io/math2/node/0BR3FY5Q8M.md): concept 0BR3FY5Q8M
- [Side-side-side congruence](https://tamnd.github.io/math2/node/0BRBYRK6PA.md): statement 0BRBYRK6PA
- [Goursat's theorem](https://tamnd.github.io/math2/node/0BRG85WXYC.md): statement 0BRG85WXYC
- [Inverse element](https://tamnd.github.io/math2/node/0BRHKKP333.md): definition 0BRHKKP333
- [Polar form](https://tamnd.github.io/math2/node/0BRXFNQ5JV.md): statement 0BRXFNQ5JV
- [Quotient rule](https://tamnd.github.io/math2/node/0BS0NJPKVE.md): statement 0BS0NJPKVE
- [Multiplication rule](https://tamnd.github.io/math2/node/0BS3RMX8HM.md): statement 0BS3RMX8HM
- [Additive group of integers](https://tamnd.github.io/math2/node/0BSCJA5G53.md): example 0BSCJA5G53
- [Set](https://tamnd.github.io/math2/node/0BSJ7RQ9BW.md): concept 0BSJ7RQ9BW
- [Dense set](https://tamnd.github.io/math2/node/0BSJSX7ME2.md): concept 0BSJSX7ME2
- [Z[√-5] is not a UFD](https://tamnd.github.io/math2/node/0BSJYCG7NM.md): counterexample 0BSJYCG7NM
- [Riemann's theorem on removable singularities](https://tamnd.github.io/math2/node/0BSQDZWF34.md): statement 0BSQDZWF34
- [Polar coordinates](https://tamnd.github.io/math2/node/0BSW4T7THS.md): concept 0BSW4T7THS
- [Squaring the circle](https://tamnd.github.io/math2/node/0BSXRER81B.md): problem 0BSXRER81B
- [Minor](https://tamnd.github.io/math2/node/0BT9BYWVRY.md): concept 0BT9BYWVRY
- [Supremum](https://tamnd.github.io/math2/node/0BTMQ04556.md): definition 0BTMQ04556
- [Borel sigma-algebra](https://tamnd.github.io/math2/node/0BVHZXWKCD.md): definition 0BVHZXWKCD
- [Trajectory](https://tamnd.github.io/math2/node/0BVT6VSB3S.md): definition 0BVT6VSB3S
- [Pythagorean theorem in an inner product space](https://tamnd.github.io/math2/node/0BVTDM872V.md): statement 0BVTDM872V
- [Matrix addition](https://tamnd.github.io/math2/node/0BVXADM10P.md): definition 0BVXADM10P
- [Antiderivative](https://tamnd.github.io/math2/node/0BW9N5CDSB.md): definition 0BW9N5CDSB
- [Characteristic polynomial of a linear recurrence](https://tamnd.github.io/math2/node/0BWDJGQYST.md): definition 0BWDJGQYST
- [Bias of an estimator](https://tamnd.github.io/math2/node/0BWGTN92FG.md): concept 0BWGTN92FG
- [Uniform limits commute with integration](https://tamnd.github.io/math2/node/0BWGW68B66.md): statement 0BWGW68B66
- [Box topology](https://tamnd.github.io/math2/node/0BWJTRE3X3.md): concept 0BWJTRE3X3
- [Power series converge uniformly on smaller discs inside the radius of convergence](https://tamnd.github.io/math2/node/0BWXV7GADW.md): statement 0BWXV7GADW
- [Ring of integers](https://tamnd.github.io/math2/node/0BXA4Q0JHW.md): statement 0BXA4Q0JHW
- [Implicit function theorem](https://tamnd.github.io/math2/node/0BXBADR94C.md): statement 0BXBADR94C
- [Having a gradient](https://tamnd.github.io/math2/node/0BXE0S1XER.md): definition 0BXE0S1XER
- [Minimal element](https://tamnd.github.io/math2/node/0BXEZV83GY.md): concept 0BXEZV83GY
- [Cantor set](https://tamnd.github.io/math2/node/0BXNDZFQME.md): example 0BXNDZFQME
- [Complete metric space](https://tamnd.github.io/math2/node/0BY52JW6NK.md): concept 0BY52JW6NK
- [Slopes of a convex function are increasing](https://tamnd.github.io/math2/node/0BY5FTHEEE.md): statement 0BY5FTHEEE
- [Generalized pigeonhole principle](https://tamnd.github.io/math2/node/0BY7GN3449.md): statement 0BY7GN3449
- [Least squares](https://tamnd.github.io/math2/node/0BYTQ4X1HC.md): concept 0BYTQ4X1HC
- [Antiderivative](https://tamnd.github.io/math2/node/0BYWRWFT4F.md): concept 0BYWRWFT4F
- [Clique](https://tamnd.github.io/math2/node/0BYX1D7C2E.md): concept 0BYX1D7C2E
- [Zero derivative on a connected open set implies constant](https://tamnd.github.io/math2/node/0BYZTFM0DJ.md): statement 0BYZTFM0DJ
- [Characteristic polynomial of a linear map](https://tamnd.github.io/math2/node/0BZ81AJNWF.md): definition 0BZ81AJNWF
- [Addition rule for two events](https://tamnd.github.io/math2/node/0BZQ6ZX584.md): statement 0BZQ6ZX584
- [Order of the dihedral group](https://tamnd.github.io/math2/node/0BZVNWJ352.md): statement 0BZVNWJ352
- [Recurrence for Stirling numbers of the second kind](https://tamnd.github.io/math2/node/0C019SDTER.md): statement 0C019SDTER
- [Laplace expansion along a row](https://tamnd.github.io/math2/node/0C08BQ3M05.md): statement 0C08BQ3M05
- [Random walk](https://tamnd.github.io/math2/node/0C08YMV95H.md): definition 0C08YMV95H
- [Simple group](https://tamnd.github.io/math2/node/0C093D1T4M.md): definition 0C093D1T4M
- [Separately continuous but not continuous function](https://tamnd.github.io/math2/node/0C0KPAKK9Y.md): counterexample 0C0KPAKK9Y
- [Density of the normal distribution](https://tamnd.github.io/math2/node/0C12P4W4RH.md): concept 0C12P4W4RH
- [Linear maps on finite-dimensional spaces are continuous](https://tamnd.github.io/math2/node/0C13E623WD.md): statement 0C13E623WD
- [pi](https://tamnd.github.io/math2/node/0C1VYVNV2N.md): object 0C1VYVNV2N
- [Finite extension](https://tamnd.github.io/math2/node/0C1Y04SZ1W.md): concept 0C1Y04SZ1W
- [Family of sets](https://tamnd.github.io/math2/node/0C21Q694AG.md): concept 0C21Q694AG
- [Random variable with a density](https://tamnd.github.io/math2/node/0C21RZG5V2.md): definition 0C21RZG5V2
- [Rectifiable curve](https://tamnd.github.io/math2/node/0C28RT5BG7.md): definition 0C28RT5BG7
- [Hamiltonian system](https://tamnd.github.io/math2/node/0C2AKPDCD9.md): definition 0C2AKPDCD9
- [Determinant is the product of the eigenvalues](https://tamnd.github.io/math2/node/0C2F40MEBC.md): statement 0C2F40MEBC
- [Dual map](https://tamnd.github.io/math2/node/0C2TQMW5A1.md): definition 0C2TQMW5A1
- [Associativity of addition](https://tamnd.github.io/math2/node/0C32J27WQC.md): statement 0C32J27WQC
- [Interior](https://tamnd.github.io/math2/node/0C38S0KMJK.md): definition 0C38S0KMJK
- [Determinant](https://tamnd.github.io/math2/node/0C3CFHDN6R.md): definition 0C3CFHDN6R
- [Euler product formula](https://tamnd.github.io/math2/node/0C3PN4075R.md): statement 0C3PN4075R
- [Monic polynomial](https://tamnd.github.io/math2/node/0C3QBCR46M.md): concept 0C3QBCR46M
- [Integers](https://tamnd.github.io/math2/node/0C4JS9RF5M.md): object 0C4JS9RF5M
- [Eigenvalues of a self-adjoint operator are real](https://tamnd.github.io/math2/node/0C4NT4GT3X.md): statement 0C4NT4GT3X
- [Bernstein polynomial](https://tamnd.github.io/math2/node/0C4VWBFY73.md): definition 0C4VWBFY73
- [Inclusion–exclusion for two sets](https://tamnd.github.io/math2/node/0C4YEQ9H5E.md): statement 0C4YEQ9H5E
- [Recurrence for Stirling numbers of the first kind](https://tamnd.github.io/math2/node/0C5BFX94J9.md): statement 0C5BFX94J9
- [Chromatic number](https://tamnd.github.io/math2/node/0C5M7AM6FH.md): definition 0C5M7AM6FH
- [Hyperbolic equilibrium point](https://tamnd.github.io/math2/node/0C5PREPZWG.md): definition 0C5PREPZWG
- [Number of subsets of a finite set](https://tamnd.github.io/math2/node/0C5RDJE73X.md): statement 0C5RDJE73X
- [Codomain](https://tamnd.github.io/math2/node/0C5SR4FC74.md): concept 0C5SR4FC74
- [Polynomial functions are continuous](https://tamnd.github.io/math2/node/0C5TP6RYSV.md): statement 0C5TP6RYSV
- [Multigraph](https://tamnd.github.io/math2/node/0C62PZXX02.md): concept 0C62PZXX02
- [Uncorrelated random variables that are not independent](https://tamnd.github.io/math2/node/0C6J8BQXVR.md): counterexample 0C6J8BQXVR
- [Tangent line to a circle](https://tamnd.github.io/math2/node/0C6THFRMZR.md): definition 0C6THFRMZR
- [Cartesian product](https://tamnd.github.io/math2/node/0C81YBVTWH.md): definition 0C81YBVTWH
- [Normal operator](https://tamnd.github.io/math2/node/0C8BR8XRTR.md): concept 0C8BR8XRTR
- [Möbius function](https://tamnd.github.io/math2/node/0C8QENHYZW.md): definition 0C8QENHYZW
- [Greatest common divisor](https://tamnd.github.io/math2/node/0C8WYW4KYQ.md): concept 0C8WYW4KYQ
- [Euclidean distance](https://tamnd.github.io/math2/node/0C95TG69SG.md): definition 0C95TG69SG
- [Contractive sequence](https://tamnd.github.io/math2/node/0C9CMCN9BQ.md): concept 0C9CMCN9BQ
- [Rooted tree](https://tamnd.github.io/math2/node/0C9J6W2A8A.md): definition 0C9J6W2A8A
- [Altitude of a triangle](https://tamnd.github.io/math2/node/0C9MZY2RNA.md): concept 0C9MZY2RNA
- [Reverse triangle inequality](https://tamnd.github.io/math2/node/0C9S13WKMF.md): statement 0C9S13WKMF
- [The kernel is a normal subgroup](https://tamnd.github.io/math2/node/0C9ZKYSPYR.md): statement 0C9ZKYSPYR
- [Sequential compactness](https://tamnd.github.io/math2/node/0CARW59WD2.md): definition 0CARW59WD2
- [Normal subgroup](https://tamnd.github.io/math2/node/0CB0DX5BVV.md): concept 0CB0DX5BVV
- [Complex numbers as pairs of reals](https://tamnd.github.io/math2/node/0CBAB9QQ9X.md): concept 0CBAB9QQ9X
- [Kurtosis](https://tamnd.github.io/math2/node/0CBQVVNBMP.md): definition 0CBQVVNBMP
- [Normal extension](https://tamnd.github.io/math2/node/0CBRFQ7DVZ.md): definition 0CBRFQ7DVZ
- [Logistic equation](https://tamnd.github.io/math2/node/0CC4VC7KR3.md): example 0CC4VC7KR3
- [Harmonic conjugate](https://tamnd.github.io/math2/node/0CC56MJ4FG.md): definition 0CC56MJ4FG
- [Discriminant of a polynomial](https://tamnd.github.io/math2/node/0CCPN09EMJ.md): definition 0CCPN09EMJ
- [Matrix–vector product](https://tamnd.github.io/math2/node/0CCQV3884B.md): definition 0CCQV3884B
- [Two-sided ideal](https://tamnd.github.io/math2/node/0CD87SFM2N.md): concept 0CD87SFM2N
- [A finite field of order q is the splitting field of x^q - x](https://tamnd.github.io/math2/node/0CDAHW576M.md): statement 0CDAHW576M
- [Riemann sphere](https://tamnd.github.io/math2/node/0CDMXGPSES.md): definition 0CDMXGPSES
- [Degrees along an Eulerian trail](https://tamnd.github.io/math2/node/0CDVDGWGEG.md): statement 0CDVDGWGEG
- [Martingale](https://tamnd.github.io/math2/node/0CDVK9RQBH.md): definition 0CDVK9RQBH
- [Universal property of the free group](https://tamnd.github.io/math2/node/0CDWKDYWN6.md): statement 0CDWKDYWN6
- [Work done by a force field](https://tamnd.github.io/math2/node/0CDZ684TCD.md): concept 0CDZ684TCD
- [Arctangent](https://tamnd.github.io/math2/node/0CE768XNVP.md): concept 0CE768XNVP
- [Conic section](https://tamnd.github.io/math2/node/0CEAT3418X.md): definition 0CEAT3418X
- [Number of permutations](https://tamnd.github.io/math2/node/0CEBCS0E0E.md): statement 0CEBCS0E0E
- [Modular multiplicative inverse](https://tamnd.github.io/math2/node/0CEC45PRJN.md): concept 0CEC45PRJN
- [Zero of a holomorphic function](https://tamnd.github.io/math2/node/0CEGEASY49.md): definition 0CEGEASY49
- [Polynomials in several variables](https://tamnd.github.io/math2/node/0CES91M79A.md): definition 0CES91M79A
- [Classification of complex quadratic forms](https://tamnd.github.io/math2/node/0CEX9BRXRZ.md): statement 0CEX9BRXRZ
- [exp(2 pi i/n) is a primitive root of unity](https://tamnd.github.io/math2/node/0CF0XGYHNJ.md): statement 0CF0XGYHNJ
- [Frobenius automorphism of a finite field](https://tamnd.github.io/math2/node/0CGCPX8Z70.md): concept 0CGCPX8Z70
- [Special orthogonal group](https://tamnd.github.io/math2/node/0CGYNDCWRR.md): object 0CGYNDCWRR
- [Disc of convergence](https://tamnd.github.io/math2/node/0CH4VG15RE.md): concept 0CH4VG15RE
- [Irreducible elements need not be prime](https://tamnd.github.io/math2/node/0CHE3J7QMX.md): counterexample 0CHE3J7QMX
- [Division algorithm](https://tamnd.github.io/math2/node/0CHQKZFK5S.md): statement 0CHQKZFK5S
- [Subbasis](https://tamnd.github.io/math2/node/0CHR35YNJQ.md): definition 0CHR35YNJQ
- [Euclidean domain](https://tamnd.github.io/math2/node/0CHS7RBD39.md): concept 0CHS7RBD39
- [Finite subgroups of the multiplicative group of a field are cyclic](https://tamnd.github.io/math2/node/0CKGBEJX6B.md): statement 0CKGBEJX6B
- [Cyclic quadrilateral](https://tamnd.github.io/math2/node/0CMC2KAXFS.md): definition 0CMC2KAXFS
- [Principal ideal domain](https://tamnd.github.io/math2/node/0CMD0BCH8T.md): concept 0CMD0BCH8T
- [Equivalent matrices](https://tamnd.github.io/math2/node/0CMHSGTWXB.md): concept 0CMHSGTWXB
- [Abel's test](https://tamnd.github.io/math2/node/0CN97GADE5.md): statement 0CN97GADE5
- [Topological space](https://tamnd.github.io/math2/node/0CNQXVBHCX.md): definition 0CNQXVBHCX
- [First-countable space](https://tamnd.github.io/math2/node/0CP5NDD21C.md): concept 0CP5NDD21C
- [Square matrix](https://tamnd.github.io/math2/node/0CP7YY6NAG.md): definition 0CP7YY6NAG
- [Impossibility of doubling the cube](https://tamnd.github.io/math2/node/0CQ357M0QS.md): statement 0CQ357M0QS
- [Absolute value is continuous but not differentiable at 0](https://tamnd.github.io/math2/node/0CQ9ZM383N.md): counterexample 0CQ9ZM383N
- [Topological space](https://tamnd.github.io/math2/node/0CQHFPGJ45.md): concept 0CQHFPGJ45
- [Bernoulli process](https://tamnd.github.io/math2/node/0CQQG3DEW9.md): concept 0CQQG3DEW9
- [Parabola](https://tamnd.github.io/math2/node/0CQW20XGFW.md): definition 0CQW20XGFW
- [Linear independence](https://tamnd.github.io/math2/node/0CRMGPAH83.md): concept 0CRMGPAH83
- [Uniform convergence of derivatives allows term-by-term differentiation](https://tamnd.github.io/math2/node/0CS42KEM5C.md): statement 0CS42KEM5C
- [Exponential growth beats polynomial growth](https://tamnd.github.io/math2/node/0CT6CBGTYY.md): statement 0CT6CBGTYY
- [Complex plane](https://tamnd.github.io/math2/node/0CTE7A31XQ.md): concept 0CTE7A31XQ
- [Box topology](https://tamnd.github.io/math2/node/0CTM1EHG9P.md): definition 0CTM1EHG9P
- [Triangle](https://tamnd.github.io/math2/node/0CTNFGYMM2.md): concept 0CTNFGYMM2
- [Real sequence](https://tamnd.github.io/math2/node/0CTV783HYT.md): definition 0CTV783HYT
- [Partition function](https://tamnd.github.io/math2/node/0CV69WDRXW.md): concept 0CV69WDRXW
- [Convergent series](https://tamnd.github.io/math2/node/0CVVKY7GY2.md): definition 0CVVKY7GY2
- [Annulus](https://tamnd.github.io/math2/node/0CW52NV5TC.md): concept 0CW52NV5TC
- [Arccosine](https://tamnd.github.io/math2/node/0CWDVN132S.md): concept 0CWDVN132S
- [Improper multiple integral](https://tamnd.github.io/math2/node/0CWK2NFSEW.md): concept 0CWK2NFSEW
- [Automorphism group](https://tamnd.github.io/math2/node/0CWXCWP0Y0.md): definition 0CWXCWP0Y0
- [Integrating factor for a non-exact equation](https://tamnd.github.io/math2/node/0CX86CR8JV.md): concept 0CX86CR8JV
- [Harmonic series](https://tamnd.github.io/math2/node/0CXTMAY5PX.md): example 0CXTMAY5PX
- [First isomorphism theorem](https://tamnd.github.io/math2/node/0CXXJNG5B6.md): statement 0CXXJNG5B6
- [Linearity of the derivative](https://tamnd.github.io/math2/node/0CXZ217QDP.md): statement 0CXZ217QDP
- [Transitive relation](https://tamnd.github.io/math2/node/0CYRECG2YQ.md): definition 0CYRECG2YQ
- [Derivative of a continuous linear map](https://tamnd.github.io/math2/node/0CYYAF8AD2.md): statement 0CYYAF8AD2
- [Singular solution](https://tamnd.github.io/math2/node/0CZA0NCFWP.md): definition 0CZA0NCFWP
- [Contractive sequence](https://tamnd.github.io/math2/node/0CZRT49QNG.md): definition 0CZRT49QNG
- [Closed curve](https://tamnd.github.io/math2/node/0D032NSZKY.md): definition 0D032NSZKY
- [Martingale](https://tamnd.github.io/math2/node/0D0GBNDKDH.md): concept 0D0GBNDKDH
- [Fibonacci recurrence](https://tamnd.github.io/math2/node/0D0SMQYEJA.md): statement 0D0SMQYEJA
- [Dense set](https://tamnd.github.io/math2/node/0D198GPMTJ.md): definition 0D198GPMTJ
- [Tautology](https://tamnd.github.io/math2/node/0D1AHYPM8N.md): definition 0D1AHYPM8N
- [Complex differentiability](https://tamnd.github.io/math2/node/0D1C9Y1EXA.md): definition 0D1C9Y1EXA
- [Geometric sequence](https://tamnd.github.io/math2/node/0D1NY1ARDR.md): concept 0D1NY1ARDR
- [Correspondence theorem for rings](https://tamnd.github.io/math2/node/0D1R4SVMBQ.md): statement 0D1R4SVMBQ
- [Conjugate elements](https://tamnd.github.io/math2/node/0D1VWQGZZ7.md): definition 0D1VWQGZZ7
- [Restriction of a function](https://tamnd.github.io/math2/node/0D2MXBV21T.md): definition 0D2MXBV21T
- [A set is closed iff it equals its closure](https://tamnd.github.io/math2/node/0D2W7FS52T.md): statement 0D2W7FS52T
- [Harmonic number](https://tamnd.github.io/math2/node/0D2Y6G4C4S.md): definition 0D2Y6G4C4S
- [Integration by substitution](https://tamnd.github.io/math2/node/0D37QFPEZE.md): statement 0D37QFPEZE
- [Consecutive Fibonacci numbers are coprime](https://tamnd.github.io/math2/node/0D3PFXEAKF.md): statement 0D3PFXEAKF
- [Order isomorphism](https://tamnd.github.io/math2/node/0D3WA36K2Q.md): concept 0D3WA36K2Q
- [Rouché's theorem](https://tamnd.github.io/math2/node/0D430QR02W.md): statement 0D430QR02W
- [Convergents of a continued fraction](https://tamnd.github.io/math2/node/0D474C7ES1.md): concept 0D474C7ES1
- [Baire space](https://tamnd.github.io/math2/node/0D5KR33H7X.md): definition 0D5KR33H7X
- [Subspace topology](https://tamnd.github.io/math2/node/0D5YHQGDJY.md): definition 0D5YHQGDJY
- [A continuous injective function on an interval is strictly monotone](https://tamnd.github.io/math2/node/0D6FCW80A0.md): statement 0D6FCW80A0
- [Probability mass function](https://tamnd.github.io/math2/node/0D6J4B671D.md): concept 0D6J4B671D
- [Polynomial ring over a UFD is a UFD](https://tamnd.github.io/math2/node/0D6KH8Y59N.md): statement 0D6KH8Y59N
- [Power series in the complex plane](https://tamnd.github.io/math2/node/0D6KXBHKC0.md): concept 0D6KXBHKC0
- [Unitary matrix](https://tamnd.github.io/math2/node/0D6RMJ10M9.md): definition 0D6RMJ10M9
- [Conjugate subgroup](https://tamnd.github.io/math2/node/0D6WHTTH7T.md): definition 0D6WHTTH7T
- [Zero is not a successor](https://tamnd.github.io/math2/node/0D752BG155.md): statement 0D752BG155
- [Binary tree](https://tamnd.github.io/math2/node/0D7EZ10V2R.md): concept 0D7EZ10V2R
- [The exponential function is convex](https://tamnd.github.io/math2/node/0D7GRZ2ADX.md): statement 0D7GRZ2ADX
- [Bridge](https://tamnd.github.io/math2/node/0D81KWH9ZM.md): definition 0D81KWH9ZM
- [Variance of a sum](https://tamnd.github.io/math2/node/0D86GJEZ39.md): statement 0D86GJEZ39
- [Construction of the reals from Cauchy sequences](https://tamnd.github.io/math2/node/0D8HE681SC.md): concept 0D8HE681SC
- [Limits are unique in Hausdorff spaces](https://tamnd.github.io/math2/node/0D8T984XEE.md): statement 0D8T984XEE
- [Connected space](https://tamnd.github.io/math2/node/0D939CMME7.md): definition 0D939CMME7
- [Tangent vector of a curve](https://tamnd.github.io/math2/node/0D98771SM0.md): definition 0D98771SM0
- [Uniqueness of inverses](https://tamnd.github.io/math2/node/0D9NDGE9KP.md): statement 0D9NDGE9KP
- [Straightedge and compass construction](https://tamnd.github.io/math2/node/0DAECMZVRT.md): concept 0DAECMZVRT
- [Euler's totient function](https://tamnd.github.io/math2/node/0DAMHCT17R.md): concept 0DAMHCT17R
- [Probability axioms](https://tamnd.github.io/math2/node/0DAX8PMAJ4.md): concept 0DAX8PMAJ4
- [Mean of the geometric distribution](https://tamnd.github.io/math2/node/0DB0R4N5XX.md): statement 0DB0R4N5XX
- [Matrix of the adjoint is the conjugate transpose](https://tamnd.github.io/math2/node/0DBTE6HVBD.md): statement 0DBTE6HVBD
- [Torsion subgroup](https://tamnd.github.io/math2/node/0DBWXQRXQJ.md): concept 0DBWXQRXQJ
- [Open ball](https://tamnd.github.io/math2/node/0DC3MPR7C9.md): definition 0DC3MPR7C9
- [Expected value](https://tamnd.github.io/math2/node/0DCT0QZVQ1.md): definition 0DCT0QZVQ1
- [Two-sided ideal](https://tamnd.github.io/math2/node/0DDA6CY7GJ.md): definition 0DDA6CY7GJ
- [Right inverse](https://tamnd.github.io/math2/node/0DDPZAG5ER.md): definition 0DDPZAG5ER
- [Elementary row operations](https://tamnd.github.io/math2/node/0DE5R5EB4D.md): definition 0DE5R5EB4D
- [Harmonic function](https://tamnd.github.io/math2/node/0DEB0JMN0W.md): concept 0DEB0JMN0W
- [Conjugacy class](https://tamnd.github.io/math2/node/0DEBXAJ5SA.md): concept 0DEBXAJ5SA
- [Gaussian integral](https://tamnd.github.io/math2/node/0DFT1GP1JP.md): example 0DFT1GP1JP
- [Strong law of large numbers](https://tamnd.github.io/math2/node/0DFWHQ839T.md): statement 0DFWHQ839T
- [Finite extensions are algebraic](https://tamnd.github.io/math2/node/0DFYTNZ6AM.md): statement 0DFYTNZ6AM
- [Commutativity of addition](https://tamnd.github.io/math2/node/0DFZNHJ0S2.md): statement 0DFZNHJ0S2
- [Holomorphic function](https://tamnd.github.io/math2/node/0DG0Z72CQD.md): definition 0DG0Z72CQD
- [Cauchy's integral formula](https://tamnd.github.io/math2/node/0DG10TVYPN.md): statement 0DG10TVYPN
- [Alternating bilinear form](https://tamnd.github.io/math2/node/0DGDS5XY84.md): concept 0DGDS5XY84
- [Spectral theorem for Hermitian matrices](https://tamnd.github.io/math2/node/0DGZBQWZWN.md): statement 0DGZBQWZWN
- [Order of a permutation is the lcm of its cycle lengths](https://tamnd.github.io/math2/node/0DH7HMXP9F.md): statement 0DH7HMXP9F
- [Covariance](https://tamnd.github.io/math2/node/0DJ8W3HP2D.md): definition 0DJ8W3HP2D
- [Upper central series](https://tamnd.github.io/math2/node/0DJ9Y9V610.md): definition 0DJ9Y9V610
- [Estimation lemma](https://tamnd.github.io/math2/node/0DJMGPN8K9.md): statement 0DJMGPN8K9
- [Quadratic residue](https://tamnd.github.io/math2/node/0DKCECSYCF.md): concept 0DKCECSYCF
- [Cosets have the same cardinality as the subgroup](https://tamnd.github.io/math2/node/0DKMBWTD7P.md): statement 0DKMBWTD7P
- [Random variable](https://tamnd.github.io/math2/node/0DKRR22AEQ.md): definition 0DKRR22AEQ
- [Basel problem](https://tamnd.github.io/math2/node/0DMY75D6FV.md): statement 0DMY75D6FV
- [Correlation coefficient](https://tamnd.github.io/math2/node/0DNN9B6Q2F.md): concept 0DNN9B6Q2F
- [Linear equation](https://tamnd.github.io/math2/node/0DNNT7DBFM.md): definition 0DNNT7DBFM
- [Degree of a polynomial](https://tamnd.github.io/math2/node/0DPF6WRZW6.md): definition 0DPF6WRZW6
- [Surface integral](https://tamnd.github.io/math2/node/0DQ8ERJZNS.md): definition 0DQ8ERJZNS
- [Slope field](https://tamnd.github.io/math2/node/0DQEHHVRBX.md): concept 0DQEHHVRBX
- [K[x] is a Euclidean domain](https://tamnd.github.io/math2/node/0DQG39K2NH.md): statement 0DQG39K2NH
- [Universal property of the polynomial ring](https://tamnd.github.io/math2/node/0DQMDSVSXJ.md): statement 0DQMDSVSXJ
- [Well-founded relation](https://tamnd.github.io/math2/node/0DQPNJNPYK.md): concept 0DQPNJNPYK
- [Bézout's identity](https://tamnd.github.io/math2/node/0DQVHMYAGA.md): statement 0DQVHMYAGA
- [Joint probability distribution](https://tamnd.github.io/math2/node/0DR81R1F2B.md): concept 0DR81R1F2B
- [Inequality of arithmetic and geometric means](https://tamnd.github.io/math2/node/0DRF0F3XC9.md): statement 0DRF0F3XC9
- [Closure](https://tamnd.github.io/math2/node/0DRKT1QJR7.md): definition 0DRKT1QJR7
- [Galois extension](https://tamnd.github.io/math2/node/0DS2F7GK05.md): definition 0DS2F7GK05
- [Fundamental system of solutions](https://tamnd.github.io/math2/node/0DS2QFYJHE.md): concept 0DS2QFYJHE
- [The rationals are countable](https://tamnd.github.io/math2/node/0DS9KWFDK2.md): statement 0DS9KWFDK2
- [Definition by recursion](https://tamnd.github.io/math2/node/0DSHAVH45F.md): concept 0DSHAVH45F
- [Set difference](https://tamnd.github.io/math2/node/0DT0TCV75N.md): definition 0DT0TCV75N
- [Transvection](https://tamnd.github.io/math2/node/0DTCBCKEJP.md): concept 0DTCBCKEJP
- [Index of a subgroup](https://tamnd.github.io/math2/node/0DTXBQPA2S.md): concept 0DTXBQPA2S
- [Linear span](https://tamnd.github.io/math2/node/0DTY8B9HMS.md): concept 0DTY8B9HMS
- [Probability distribution](https://tamnd.github.io/math2/node/0DVDCMA2DK.md): concept 0DVDCMA2DK
- [Conjugate elements](https://tamnd.github.io/math2/node/0DXD4QSZ8A.md): concept 0DXD4QSZ8A
- [Conjugate transpose](https://tamnd.github.io/math2/node/0DXKE10PHF.md): concept 0DXKE10PHF
- [Stone's representation theorem for finite Boolean algebras](https://tamnd.github.io/math2/node/0DYS9C3HH5.md): statement 0DYS9C3HH5
- [Centre of a group](https://tamnd.github.io/math2/node/0DZ93493VN.md): concept 0DZ93493VN
- [Free module](https://tamnd.github.io/math2/node/0DZ9DKAKZR.md): concept 0DZ9DKAKZR
- [Irrationality of the square root of 2](https://tamnd.github.io/math2/node/0DZGSDHD92.md): statement 0DZGSDHD92
- [Nilpotent group](https://tamnd.github.io/math2/node/0DZPD2MG33.md): definition 0DZPD2MG33
- [Length of a vector](https://tamnd.github.io/math2/node/0DZS7ZG0QY.md): definition 0DZS7ZG0QY
- [Irrational number](https://tamnd.github.io/math2/node/0DZXAXEJQ9.md): concept 0DZXAXEJQ9
- [Number of nth roots of unity](https://tamnd.github.io/math2/node/0E021GF2B8.md): statement 0E021GF2B8
- [Jordan–Hölder theorem](https://tamnd.github.io/math2/node/0E088ZP533.md): statement 0E088ZP533
- [Laplacian](https://tamnd.github.io/math2/node/0E0B6RA8K0.md): concept 0E0B6RA8K0
- [Primitive root theorem](https://tamnd.github.io/math2/node/0E0DNQK22P.md): statement 0E0DNQK22P
- [Square root function](https://tamnd.github.io/math2/node/0E10E0NW9V.md): concept 0E10E0NW9V
- [Euclidean metric](https://tamnd.github.io/math2/node/0E1MMCAY5B.md): example 0E1MMCAY5B
- [Riccati equation](https://tamnd.github.io/math2/node/0E1ZZBHZ3Z.md): definition 0E1ZZBHZ3Z
- [Complete residue system](https://tamnd.github.io/math2/node/0E22JKS4W4.md): definition 0E22JKS4W4
- [Orbit-stabiliser theorem](https://tamnd.github.io/math2/node/0E2HSCKZ6C.md): statement 0E2HSCKZ6C
- [Dimension of the orthogonal complement](https://tamnd.github.io/math2/node/0E2NYPVSP9.md): statement 0E2NYPVSP9
- [Differential form](https://tamnd.github.io/math2/node/0E3G0GRPZ6.md): concept 0E3G0GRPZ6
- [Polynomial of an endomorphism](https://tamnd.github.io/math2/node/0E3MV6E6MG.md): concept 0E3MV6E6MG
- [Algebra of endomorphisms](https://tamnd.github.io/math2/node/0E3X2G1ZQB.md): object 0E3X2G1ZQB
- [Limit point](https://tamnd.github.io/math2/node/0E3X7TXS67.md): concept 0E3X7TXS67
- [Base rate fallacy](https://tamnd.github.io/math2/node/0E44JMG6NV.md): example 0E44JMG6NV
- [Blow-up in finite time](https://tamnd.github.io/math2/node/0E4GXX7SJP.md): example 0E4GXX7SJP
- [Cayley table](https://tamnd.github.io/math2/node/0E4WH0R7HG.md): definition 0E4WH0R7HG
- [Directrix](https://tamnd.github.io/math2/node/0E5CXYNEPX.md): concept 0E5CXYNEPX
- [Law of total probability](https://tamnd.github.io/math2/node/0E5G5FMT2R.md): statement 0E5G5FMT2R
- [Convergents converge to the number](https://tamnd.github.io/math2/node/0E70M92FE3.md): statement 0E70M92FE3
- [Fermat's Last Theorem for n = 4](https://tamnd.github.io/math2/node/0E741HSMJ1.md): statement 0E741HSMJ1
- [Riemann rearrangement theorem](https://tamnd.github.io/math2/node/0E7VBYGFF7.md): statement 0E7VBYGFF7
- [Casting out nines](https://tamnd.github.io/math2/node/0E7W0EM6J6.md): concept 0E7W0EM6J6
- [Similar matrices](https://tamnd.github.io/math2/node/0E89HQATM8.md): concept 0E89HQATM8
- [Exponential of a sum](https://tamnd.github.io/math2/node/0E8HNVBCSP.md): statement 0E8HNVBCSP
- [Independence of two random variables](https://tamnd.github.io/math2/node/0E9502NT92.md): definition 0E9502NT92
- [Exponential distribution](https://tamnd.github.io/math2/node/0E99F51GVB.md): concept 0E99F51GVB
- [Range of a function](https://tamnd.github.io/math2/node/0E9GTAV478.md): definition 0E9GTAV478
- [Harmonic series diverges](https://tamnd.github.io/math2/node/0E9M24KS9J.md): statement 0E9M24KS9J
- [Ceiling function](https://tamnd.github.io/math2/node/0EA4XFB0EY.md): concept 0EA4XFB0EY
- [Division ring](https://tamnd.github.io/math2/node/0EA5BTA0CX.md): concept 0EA5BTA0CX
- [Chain rule in several variables](https://tamnd.github.io/math2/node/0EAAS2EHDR.md): statement 0EAAS2EHDR
- [Degree of a field extension](https://tamnd.github.io/math2/node/0EAJ2XXADC.md): concept 0EAJ2XXADC
- [Trigonometric substitution](https://tamnd.github.io/math2/node/0EAR3KNRTN.md): concept 0EAR3KNRTN
- [A composite number has a prime factor at most its square root](https://tamnd.github.io/math2/node/0EB6BBMQMC.md): statement 0EB6BBMQMC
- [Invertible matrix](https://tamnd.github.io/math2/node/0EB9N0S33C.md): concept 0EB9N0S33C
- [Cardinal number](https://tamnd.github.io/math2/node/0EBACQFAZW.md): concept 0EBACQFAZW
- [Probability-generating function](https://tamnd.github.io/math2/node/0EBCWQ8BVE.md): concept 0EBCWQ8BVE
- [A finite integral domain is a field](https://tamnd.github.io/math2/node/0EBE031B8E.md): statement 0EBE031B8E
- [The multiplicative group of a finite field is cyclic](https://tamnd.github.io/math2/node/0EBZ3G455G.md): statement 0EBZ3G455G
- [Exact equation](https://tamnd.github.io/math2/node/0EC7AGJ0PX.md): definition 0EC7AGJ0PX
- [Empty set](https://tamnd.github.io/math2/node/0ECVK59Y03.md): object 0ECVK59Y03
- [Interval of convergence](https://tamnd.github.io/math2/node/0ED4HWNDGW.md): concept 0ED4HWNDGW
- [Counterexample](https://tamnd.github.io/math2/node/0ED72CTBNP.md): concept 0ED72CTBNP
- [Existence of the infimum](https://tamnd.github.io/math2/node/0EDH403M21.md): statement 0EDH403M21
- [Hamiltonian system](https://tamnd.github.io/math2/node/0EDKGA2FAK.md): concept 0EDKGA2FAK
- [Neyman–Pearson lemma](https://tamnd.github.io/math2/node/0EE19CFEEF.md): statement 0EE19CFEEF
- [Rule of sum](https://tamnd.github.io/math2/node/0EE2V3CHFK.md): statement 0EE2V3CHFK
- [Continuity at a point](https://tamnd.github.io/math2/node/0EEE2CZB2A.md): definition 0EEE2CZB2A
- [Refinement of a partition](https://tamnd.github.io/math2/node/0EEKF7YFBW.md): definition 0EEKF7YFBW
- [Limit cycle](https://tamnd.github.io/math2/node/0EERDDPSVT.md): definition 0EERDDPSVT
- [Concave function](https://tamnd.github.io/math2/node/0EF2VG8XKQ.md): definition 0EF2VG8XKQ
- [Convergence in probability](https://tamnd.github.io/math2/node/0EFDDXTVD6.md): definition 0EFDDXTVD6
- [Proof by infinite descent](https://tamnd.github.io/math2/node/0EFYCFHYFG.md): concept 0EFYCFHYFG
- [Primality test](https://tamnd.github.io/math2/node/0EGJZHZ02C.md): concept 0EGJZHZ02C
- [Formal derivative](https://tamnd.github.io/math2/node/0EGN1E1JPB.md): concept 0EGN1E1JPB
- [Boole's inequality](https://tamnd.github.io/math2/node/0EGSEE9J9R.md): statement 0EGSEE9J9R
- [Prime number theorem](https://tamnd.github.io/math2/node/0EHBY5Y84N.md): statement 0EHBY5Y84N
- [Liouville's theorem](https://tamnd.github.io/math2/node/0EHKVRF4J6.md): statement 0EHKVRF4J6
- [Primitive root of unity](https://tamnd.github.io/math2/node/0EHM5ZP5QH.md): concept 0EHM5ZP5QH
- [Binomial theorem](https://tamnd.github.io/math2/node/0EHWPWGHB2.md): statement 0EHWPWGHB2
- [P-value](https://tamnd.github.io/math2/node/0EHYZFZ4PK.md): definition 0EHYZFZ4PK
- [Universal property of the abelianisation](https://tamnd.github.io/math2/node/0EJ4PC8NJ6.md): statement 0EJ4PC8NJ6
- [Normal convergence](https://tamnd.github.io/math2/node/0EJVD9K0K1.md): concept 0EJVD9K0K1
- [Epsilon-delta continuity in metric spaces](https://tamnd.github.io/math2/node/0EKGKY5TGH.md): statement 0EKGKY5TGH
- [Hermitian matrix](https://tamnd.github.io/math2/node/0EKZCX4K3P.md): definition 0EKZCX4K3P
- [Orientable surface](https://tamnd.github.io/math2/node/0EKZW08FQ5.md): definition 0EKZW08FQ5
- [Existence of nontrivial solutions of Pell's equation](https://tamnd.github.io/math2/node/0EM6C9SZN5.md): statement 0EM6C9SZN5
- [Chi-squared test](https://tamnd.github.io/math2/node/0EN0V3NY20.md): concept 0EN0V3NY20
- [Inverse in terms of the adjugate](https://tamnd.github.io/math2/node/0ENVM493VK.md): statement 0ENVM493VK
- [Normal operator](https://tamnd.github.io/math2/node/0ENZN5G4CV.md): definition 0ENZN5G4CV
- [Invariant subspace](https://tamnd.github.io/math2/node/0EP59ESQ94.md): definition 0EP59ESQ94
- [Row space](https://tamnd.github.io/math2/node/0EP73HK4C4.md): definition 0EP73HK4C4
- [Function](https://tamnd.github.io/math2/node/0EPAT51DXQ.md): concept 0EPAT51DXQ
- [Continuous image of a connected set is connected](https://tamnd.github.io/math2/node/0EPD87YVG8.md): statement 0EPD87YVG8
- [Convergence in probability implies convergence in distribution](https://tamnd.github.io/math2/node/0EPTDZPJ83.md): statement 0EPTDZPJ83
- [Linear subspace](https://tamnd.github.io/math2/node/0EPTJA5QV4.md): definition 0EPTJA5QV4
- [Constrained optimisation](https://tamnd.github.io/math2/node/0EPW94MJ1P.md): concept 0EPW94MJ1P
- [Central limit theorem](https://tamnd.github.io/math2/node/0EPZC4Z1FC.md): statement 0EPZC4Z1FC
- [Hyperbolic cosine](https://tamnd.github.io/math2/node/0EQFWKJ37J.md): definition 0EQFWKJ37J
- [Locally compact space](https://tamnd.github.io/math2/node/0EQMWQ4WMJ.md): definition 0EQMWQ4WMJ
- [Field extension](https://tamnd.github.io/math2/node/0EQWVHF3W5.md): concept 0EQWVHF3W5
- [Homogeneous linear equation](https://tamnd.github.io/math2/node/0EQZSAB9EC.md): definition 0EQZSAB9EC
- [Induced subgraph](https://tamnd.github.io/math2/node/0EREPBVG2G.md): concept 0EREPBVG2G
- [QR decomposition](https://tamnd.github.io/math2/node/0ERWG4NXQ9.md): statement 0ERWG4NXQ9
- [Imaginary unit](https://tamnd.github.io/math2/node/0ESCX36MMC.md): object 0ESCX36MMC
- [Divisibility by 9](https://tamnd.github.io/math2/node/0ESHD7ZQDX.md): statement 0ESHD7ZQDX
- [Collinear points](https://tamnd.github.io/math2/node/0ETCTBTJM1.md): definition 0ETCTBTJM1
- [Existence of an algebraic closure](https://tamnd.github.io/math2/node/0ETGBAK2CC.md): statement 0ETGBAK2CC
- [Gershgorin circle theorem](https://tamnd.github.io/math2/node/0ETGP0XVCJ.md): statement 0ETGP0XVCJ
- [Linear independence](https://tamnd.github.io/math2/node/0EVHJFTG9Z.md): definition 0EVHJFTG9Z
- [Poincaré–Bendixson theorem](https://tamnd.github.io/math2/node/0EW1X61ZT1.md): statement 0EW1X61ZT1
- [Beta distribution](https://tamnd.github.io/math2/node/0EW824Z7G2.md): concept 0EW824Z7G2
- [Domain in the complex plane](https://tamnd.github.io/math2/node/0EWMDD6C14.md): concept 0EWMDD6C14
- [Rank of a matrix](https://tamnd.github.io/math2/node/0EWQEYRVA8.md): definition 0EWQEYRVA8
- [Impossibility of trisecting the angle](https://tamnd.github.io/math2/node/0EWXPZCSZ0.md): statement 0EWXPZCSZ0
- [Conjugate transpose](https://tamnd.github.io/math2/node/0EX045E2BP.md): definition 0EX045E2BP
- [Product of vector spaces](https://tamnd.github.io/math2/node/0EX2HPXPJ6.md): concept 0EX2HPXPJ6
- [The exponential function is its own derivative](https://tamnd.github.io/math2/node/0EX9GFJ0B3.md): statement 0EX9GFJ0B3
- [Iterated derivative](https://tamnd.github.io/math2/node/0EY05Z9RPN.md): definition 0EY05Z9RPN
- [68–95–99.7 rule](https://tamnd.github.io/math2/node/0EYQ03XCR3.md): example 0EYQ03XCR3
- [Element](https://tamnd.github.io/math2/node/0EZ758XE73.md): definition 0EZ758XE73
- [Cantor's intersection theorem](https://tamnd.github.io/math2/node/0EZBG8SKJM.md): statement 0EZBG8SKJM
- [x sin(1/x) is continuous at 0](https://tamnd.github.io/math2/node/0EZMAY1FQD.md): example 0EZMAY1FQD
- [Lower triangular matrix](https://tamnd.github.io/math2/node/0EZSD6T9E0.md): definition 0EZSD6T9E0
- [Cauchy's integral formula for derivatives](https://tamnd.github.io/math2/node/0F0H5P5DED.md): statement 0F0H5P5DED
- [Set bounded above](https://tamnd.github.io/math2/node/0F0J193T76.md): concept 0F0J193T76
- [Prime elements are irreducible in an integral domain](https://tamnd.github.io/math2/node/0F0K8DB9TD.md): statement 0F0K8DB9TD
- [Column space](https://tamnd.github.io/math2/node/0F0RPPJ2TT.md): definition 0F0RPPJ2TT
- [Closed and exact forms](https://tamnd.github.io/math2/node/0F1AVGX26N.md): definition 0F1AVGX26N
- [Well-order](https://tamnd.github.io/math2/node/0F1D6PK9RZ.md): definition 0F1D6PK9RZ
- [Transition matrix](https://tamnd.github.io/math2/node/0F1KBVY161.md): definition 0F1KBVY161
- [Subsequence](https://tamnd.github.io/math2/node/0F1PR74BRX.md): concept 0F1PR74BRX
- [Conditional variance](https://tamnd.github.io/math2/node/0F1SN93KXE.md): definition 0F1SN93KXE
- [Derivative](https://tamnd.github.io/math2/node/0F1WR9C83K.md): definition 0F1WR9C83K
- [Row echelon form](https://tamnd.github.io/math2/node/0F27PBRWD7.md): definition 0F27PBRWD7
- [Field of fractions](https://tamnd.github.io/math2/node/0F2PTGX1A2.md): concept 0F2PTGX1A2
- [Riemann integrable function](https://tamnd.github.io/math2/node/0F2QCWVFT7.md): concept 0F2QCWVFT7
- [Lyapunov stability](https://tamnd.github.io/math2/node/0F2SYCKDT2.md): concept 0F2SYCKDT2
- [Superposition principle](https://tamnd.github.io/math2/node/0F2VBPXNP9.md): statement 0F2VBPXNP9
- [Iterated derivative](https://tamnd.github.io/math2/node/0F32S73FJN.md): concept 0F32S73FJN
- [Primitive root modulo n](https://tamnd.github.io/math2/node/0F33KX9Q6F.md): concept 0F33KX9Q6F
- [Bounded linear maps are continuous](https://tamnd.github.io/math2/node/0F3K4C7XYW.md): statement 0F3K4C7XYW
- [Mersenne prime](https://tamnd.github.io/math2/node/0F3Z22M2GT.md): concept 0F3Z22M2GT
- [Derivative vanishes at a local extremum](https://tamnd.github.io/math2/node/0F405477F8.md): statement 0F405477F8
- [Transition matrix](https://tamnd.github.io/math2/node/0F462493EA.md): concept 0F462493EA
- [Divisors of a natural number](https://tamnd.github.io/math2/node/0F4JSQE2EK.md): concept 0F4JSQE2EK
- [Quadratic integer](https://tamnd.github.io/math2/node/0F4WRWBZZC.md): concept 0F4WRWBZZC
- [Outcome](https://tamnd.github.io/math2/node/0F50N2P3FC.md): concept 0F50N2P3FC
- [Divergence of a curl is zero](https://tamnd.github.io/math2/node/0F56CMMC5P.md): statement 0F56CMMC5P
- [g to the n equals 1 iff the order divides n](https://tamnd.github.io/math2/node/0F5QC101Q7.md): statement 0F5QC101Q7
- [Characteristic of a ring](https://tamnd.github.io/math2/node/0F6ES3BRYP.md): concept 0F6ES3BRYP
- [Order on the integers](https://tamnd.github.io/math2/node/0F6FM9Y4Y6.md): definition 0F6FM9Y4Y6
- [Disjunction](https://tamnd.github.io/math2/node/0F72FK8EKB.md): concept 0F72FK8EKB
- [Cayley–Hamilton theorem](https://tamnd.github.io/math2/node/0F765KFC2J.md): statement 0F765KFC2J
- [Square root](https://tamnd.github.io/math2/node/0F7FQS6XY6.md): concept 0F7FQS6XY6
- [Greatest common divisor in an integral domain](https://tamnd.github.io/math2/node/0F7KNJ1ZYN.md): definition 0F7KNJ1ZYN
- [Graph isomorphism](https://tamnd.github.io/math2/node/0F7QKGJS6B.md): concept 0F7QKGJS6B
- [Convolution of probability distributions](https://tamnd.github.io/math2/node/0F8GDKM0A7.md): concept 0F8GDKM0A7
- [Group of units of Z/nZ](https://tamnd.github.io/math2/node/0F8X8ZRQ53.md): object 0F8X8ZRQ53
- [Double counting](https://tamnd.github.io/math2/node/0F8YBHF2W3.md): concept 0F8YBHF2W3
- [Circumcentre](https://tamnd.github.io/math2/node/0F98FZK14A.md): concept 0F98FZK14A
- [Codomain](https://tamnd.github.io/math2/node/0F9JHP8AJN.md): definition 0F9JHP8AJN
- [Absolutely convergent improper integral](https://tamnd.github.io/math2/node/0F9VKJPWCA.md): concept 0F9VKJPWCA
- [Möbius function](https://tamnd.github.io/math2/node/0FANE7P0DY.md): concept 0FANE7P0DY
- [Local maximum](https://tamnd.github.io/math2/node/0FBETJR0JG.md): concept 0FBETJR0JG
- [Affine subspace](https://tamnd.github.io/math2/node/0FBXFBZBH8.md): definition 0FBXFBZBH8
- [Closed subsets of complete spaces are complete](https://tamnd.github.io/math2/node/0FBXJZX3MK.md): statement 0FBXJZX3MK
- [Evaluation homomorphism](https://tamnd.github.io/math2/node/0FCCDXH97W.md): concept 0FCCDXH97W
- [Exponent of a group](https://tamnd.github.io/math2/node/0FCCTFA275.md): definition 0FCCTFA275
- [General linear group](https://tamnd.github.io/math2/node/0FCHE902D7.md): object 0FCHE902D7
- [Divisors of a natural number](https://tamnd.github.io/math2/node/0FCWA16FZE.md): definition 0FCWA16FZE
- [The gamma function extends the factorial](https://tamnd.github.io/math2/node/0FDCW77H44.md): statement 0FDCW77H44
- [Reachability](https://tamnd.github.io/math2/node/0FDKRX9M9Y.md): concept 0FDKRX9M9Y
- [Locally compact space](https://tamnd.github.io/math2/node/0FE2SB8WVJ.md): concept 0FE2SB8WVJ
- [Subset](https://tamnd.github.io/math2/node/0FE8Q5S42Y.md): definition 0FE8Q5S42Y
- [A subgroup whose index is the smallest prime divisor of the order is normal](https://tamnd.github.io/math2/node/0FEBE9NC0E.md): statement 0FEBE9NC0E
- [e as the limit of (1 + 1/n)^n](https://tamnd.github.io/math2/node/0FEMF8JKAF.md): statement 0FEMF8JKAF
- [Twin prime conjecture](https://tamnd.github.io/math2/node/0FER36N8SB.md): conjecture 0FER36N8SB
- [Universal property of the quotient topology](https://tamnd.github.io/math2/node/0FF2FJ8CH2.md): statement 0FF2FJ8CH2
- [Ellipse](https://tamnd.github.io/math2/node/0FF6CAQT2Y.md): definition 0FF6CAQT2Y
- [Inclusion–exclusion principle](https://tamnd.github.io/math2/node/0FF7726C7P.md): statement 0FF7726C7P
- [Affine span](https://tamnd.github.io/math2/node/0FFAE5P6R4.md): definition 0FFAE5P6R4
- [Connected graph](https://tamnd.github.io/math2/node/0FFAYNMAJZ.md): definition 0FFAYNMAJZ
- [Construction of the rationals as fractions](https://tamnd.github.io/math2/node/0FFH3HSRBA.md): concept 0FFH3HSRBA
- [Absolute value](https://tamnd.github.io/math2/node/0FFXKTKKQC.md): concept 0FFXKTKKQC
- [Kurtosis](https://tamnd.github.io/math2/node/0FGAJSPX8E.md): concept 0FGAJSPX8E
- [Preimage of a set](https://tamnd.github.io/math2/node/0FGCFY1AZX.md): definition 0FGCFY1AZX
- [The reals form a field](https://tamnd.github.io/math2/node/0FGENPMPJX.md): statement 0FGENPMPJX
- [Characterization of bridges](https://tamnd.github.io/math2/node/0FHM8DB6PM.md): statement 0FHM8DB6PM
- [Epsilon-delta uniform continuity](https://tamnd.github.io/math2/node/0FHSDCQACY.md): statement 0FHSDCQACY
- [Matrix ring](https://tamnd.github.io/math2/node/0FHVPRAE60.md): object 0FHVPRAE60
- [Group of units of a monoid](https://tamnd.github.io/math2/node/0FJ90Y9926.md): concept 0FJ90Y9926
- [Conditional probability](https://tamnd.github.io/math2/node/0FJB06AD2P.md): definition 0FJB06AD2P
- [Chromatic number of a complete bipartite graph](https://tamnd.github.io/math2/node/0FJBBCXYPS.md): statement 0FJBBCXYPS
- [Bertrand's postulate](https://tamnd.github.io/math2/node/0FJTGZA4GS.md): statement 0FJTGZA4GS
- [Maximum degree](https://tamnd.github.io/math2/node/0FJY6G7FNA.md): concept 0FJY6G7FNA
- [Triangle inequality for norms](https://tamnd.github.io/math2/node/0FJZMBX0ZZ.md): statement 0FJZMBX0ZZ
- [Simultaneous diagonalisation](https://tamnd.github.io/math2/node/0FK17N4DKA.md): statement 0FK17N4DKA
- [Monotone convergence theorem for sequences](https://tamnd.github.io/math2/node/0FM6N4R0VB.md): statement 0FM6N4R0VB
- [Burnside's p^a q^b theorem](https://tamnd.github.io/math2/node/0FMAQSB1SC.md): statement 0FMAQSB1SC
- [Constructible number](https://tamnd.github.io/math2/node/0FMN2MN7TM.md): definition 0FMN2MN7TM
- [Total derivative](https://tamnd.github.io/math2/node/0FNKFT8H6E.md): concept 0FNKFT8H6E
- [Chinese remainder theorem](https://tamnd.github.io/math2/node/0FNN9B4W64.md): statement 0FNN9B4W64
- [Even and odd functions](https://tamnd.github.io/math2/node/0FP31SX5EA.md): concept 0FP31SX5EA
- [Scalar potential](https://tamnd.github.io/math2/node/0FP4BVDAAQ.md): definition 0FP4BVDAAQ
- [Lévy's continuity theorem](https://tamnd.github.io/math2/node/0FPWF7TTMB.md): statement 0FPWF7TTMB
- [Reduced word](https://tamnd.github.io/math2/node/0FPXK5CHQ3.md): definition 0FPXK5CHQ3
- [Spanning tree](https://tamnd.github.io/math2/node/0FQ5B8S1ZC.md): definition 0FQ5B8S1ZC
- [Convergence of random variables](https://tamnd.github.io/math2/node/0FQ968YKA1.md): concept 0FQ968YKA1
- [Meagre set](https://tamnd.github.io/math2/node/0FQPYAY9PJ.md): definition 0FQPYAY9PJ
- [Inverse of a complex number](https://tamnd.github.io/math2/node/0FQRAV5SKH.md): statement 0FQRAV5SKH
- [Convex hull](https://tamnd.github.io/math2/node/0FQSDC2EQ2.md): concept 0FQSDC2EQ2
- [Comparison test for improper integrals](https://tamnd.github.io/math2/node/0FR1BDKK10.md): statement 0FR1BDKK10
- [Heron's formula](https://tamnd.github.io/math2/node/0FRBZ2X4HJ.md): statement 0FRBZ2X4HJ
- [Maximum likelihood estimation](https://tamnd.github.io/math2/node/0FREPFKNS7.md): concept 0FREPFKNS7
- [Scalar matrix](https://tamnd.github.io/math2/node/0FRM62SG7D.md): definition 0FRM62SG7D
- [Sequential compactness](https://tamnd.github.io/math2/node/0FRMJH3CY7.md): concept 0FRMJH3CY7
- [The complex numbers form a field](https://tamnd.github.io/math2/node/0FRSR10DWF.md): statement 0FRSR10DWF
- [Grönwall bound](https://tamnd.github.io/math2/node/0FRX8JD7QH.md): definition 0FRX8JD7QH
- [Total order](https://tamnd.github.io/math2/node/0FRXBVVR03.md): concept 0FRXBVVR03
- [Trace is a coefficient of the characteristic polynomial](https://tamnd.github.io/math2/node/0FRYPZ6D0X.md): statement 0FRYPZ6D0X
- [Infinite pigeonhole principle](https://tamnd.github.io/math2/node/0FS3RHF6KE.md): statement 0FS3RHF6KE
- [Determinant of a matrix with a zero row](https://tamnd.github.io/math2/node/0FS6B46E5T.md): statement 0FS6B46E5T
- [Bertrand's ballot theorem](https://tamnd.github.io/math2/node/0FS972T0ZH.md): statement 0FS972T0ZH
- [Reversing a path changes the sign of a line integral](https://tamnd.github.io/math2/node/0FSCXYG856.md): statement 0FSCXYG856
- [Multiplicativity of the totient function](https://tamnd.github.io/math2/node/0FSJA9M43W.md): statement 0FSJA9M43W
- [Identity function](https://tamnd.github.io/math2/node/0FSNQ0GNCC.md): definition 0FSNQ0GNCC
- [Complex conjugate](https://tamnd.github.io/math2/node/0FTH2E9TAS.md): concept 0FTH2E9TAS
- [Choice function](https://tamnd.github.io/math2/node/0FWBQ53MPJ.md): concept 0FWBQ53MPJ
- [Orthonormal basis](https://tamnd.github.io/math2/node/0FWJ0VW561.md): definition 0FWJ0VW561
- [Transposition](https://tamnd.github.io/math2/node/0FWVRHBRWM.md): concept 0FWVRHBRWM
- [Initial value problem](https://tamnd.github.io/math2/node/0FXDW992DS.md): concept 0FXDW992DS
- [Solvable group](https://tamnd.github.io/math2/node/0FXJ7YFMMX.md): concept 0FXJ7YFMMX
- [Algebraic multiplicity](https://tamnd.github.io/math2/node/0FY42N02DV.md): concept 0FY42N02DV
- [Order of quantifiers](https://tamnd.github.io/math2/node/0FYQMKFE77.md): concept 0FYQMKFE77
- [Recurrence relation](https://tamnd.github.io/math2/node/0FYX44GP92.md): concept 0FYX44GP92
- [Second derivative test](https://tamnd.github.io/math2/node/0FZFHK97P7.md): statement 0FZFHK97P7
- [Deficient number](https://tamnd.github.io/math2/node/0G01TS1RRP.md): definition 0G01TS1RRP
- [Cardinality](https://tamnd.github.io/math2/node/0G0ASKNA6Z.md): concept 0G0ASKNA6Z
- [Cyclic group of order n](https://tamnd.github.io/math2/node/0G0B3DRG65.md): example 0G0B3DRG65
- [Transcendence of pi](https://tamnd.github.io/math2/node/0G0F7FGCGP.md): statement 0G0F7FGCGP
- [Eigenspace](https://tamnd.github.io/math2/node/0G0RFPG8HR.md): concept 0G0RFPG8HR
- [Galton–Watson process](https://tamnd.github.io/math2/node/0G1022YXVQ.md): concept 0G1022YXVQ
- [Sylvester's determinant theorem](https://tamnd.github.io/math2/node/0G24VQ1088.md): statement 0G24VQ1088
- [The complex exponential is entire](https://tamnd.github.io/math2/node/0G2YJN0H5S.md): statement 0G2YJN0H5S
- [Tensor product of vector spaces](https://tamnd.github.io/math2/node/0G31VJ5YS5.md): definition 0G31VJ5YS5
- [Continuous dependence on initial data](https://tamnd.github.io/math2/node/0G3DBXRRQJ.md): statement 0G3DBXRRQJ
- [Dedekind cut](https://tamnd.github.io/math2/node/0G3RRM41X9.md): concept 0G3RRM41X9
- [Markov chain](https://tamnd.github.io/math2/node/0G48PYM96Q.md): concept 0G48PYM96Q
- [Automorphism group of a cyclic group](https://tamnd.github.io/math2/node/0G4C8SJ6PV.md): statement 0G4C8SJ6PV
- [Sum of the reciprocals of the factorials](https://tamnd.github.io/math2/node/0G4D3ZKTP8.md): example 0G4D3ZKTP8
- [Entire function](https://tamnd.github.io/math2/node/0G4F0XB9X2.md): concept 0G4F0XB9X2
- [Completion of a metric space](https://tamnd.github.io/math2/node/0G4J7Z7WPJ.md): definition 0G4J7Z7WPJ
- [Cylindrical coordinates](https://tamnd.github.io/math2/node/0G4MFRQSGN.md): definition 0G4MFRQSGN
- [Oriented angle](https://tamnd.github.io/math2/node/0G4W89YJRK.md): definition 0G4W89YJRK
- [Equilibrium point](https://tamnd.github.io/math2/node/0G4X36AE0X.md): definition 0G4X36AE0X
- [Clique](https://tamnd.github.io/math2/node/0G52BPNPKG.md): definition 0G52BPNPKG
- [Multivariate normal distribution](https://tamnd.github.io/math2/node/0G558GJQMW.md): definition 0G558GJQMW
- [Closed set](https://tamnd.github.io/math2/node/0G5AV9Q4RR.md): definition 0G5AV9Q4RR
- [Derivative of the inverse function](https://tamnd.github.io/math2/node/0G65751VSW.md): statement 0G65751VSW
- [Cauchy sequences are bounded](https://tamnd.github.io/math2/node/0G6D8E7CTA.md): statement 0G6D8E7CTA
- [Lexicographic order](https://tamnd.github.io/math2/node/0G77XFBV00.md): definition 0G77XFBV00
- [Vector projection](https://tamnd.github.io/math2/node/0G78GP5TF1.md): definition 0G78GP5TF1
- [Parametrised curve](https://tamnd.github.io/math2/node/0G7WM8DX4P.md): concept 0G7WM8DX4P
- [Uniqueness of the complete ordered field](https://tamnd.github.io/math2/node/0G8G1TH5WK.md): statement 0G8G1TH5WK
- [Equivalence relation](https://tamnd.github.io/math2/node/0G8VEZ3WTK.md): concept 0G8VEZ3WTK
- [Sequence](https://tamnd.github.io/math2/node/0G9EKM0E15.md): concept 0G9EKM0E15
- [Inscribed angle theorem](https://tamnd.github.io/math2/node/0GA1V8NN8C.md): statement 0GA1V8NN8C
- [Derivative of arcsine](https://tamnd.github.io/math2/node/0GA6G43EA9.md): statement 0GA6G43EA9
- [Fibonacci sequence](https://tamnd.github.io/math2/node/0GA9K57A39.md): example 0GA9K57A39
- [Product topology](https://tamnd.github.io/math2/node/0GBG8B49AA.md): concept 0GBG8B49AA
- [Independence of two random variables](https://tamnd.github.io/math2/node/0GBJACNAP6.md): concept 0GBJACNAP6
- [Epsilon-delta definition of a limit](https://tamnd.github.io/math2/node/0GBZDAFPX8.md): statement 0GBZDAFPX8
- [Binomial coefficient](https://tamnd.github.io/math2/node/0GBZM0MT0F.md): concept 0GBZM0MT0F
- [Concave function](https://tamnd.github.io/math2/node/0GCM7B6PBK.md): concept 0GCM7B6PBK
- [Mutual independence of events](https://tamnd.github.io/math2/node/0GCVEV6BJA.md): definition 0GCVEV6BJA
- [Eccentricity](https://tamnd.github.io/math2/node/0GCWJS2EGB.md): definition 0GCWJS2EGB
- [Curl](https://tamnd.github.io/math2/node/0GDNHS9J2X.md): definition 0GDNHS9J2X
- [Prüfer sequence](https://tamnd.github.io/math2/node/0GDQRETR59.md): concept 0GDQRETR59
- [The quaternions](https://tamnd.github.io/math2/node/0GE4DVQKR8.md): example 0GE4DVQKR8
- [Critical point](https://tamnd.github.io/math2/node/0GEQ42PTGB.md): definition 0GEQ42PTGB
- [Normal core](https://tamnd.github.io/math2/node/0GETX79CJN.md): concept 0GETX79CJN
- [Induction from an arbitrary base case](https://tamnd.github.io/math2/node/0GF6AJEBZ0.md): statement 0GF6AJEBZ0
- [Subfield](https://tamnd.github.io/math2/node/0GFGHRT2FJ.md): concept 0GFGHRT2FJ
- [Pointwise convergence](https://tamnd.github.io/math2/node/0GFGK40582.md): concept 0GFGK40582
- [Coprime integers](https://tamnd.github.io/math2/node/0GFKP8F8AG.md): definition 0GFKP8F8AG
- [Bijective function](https://tamnd.github.io/math2/node/0GG9TVW90M.md): concept 0GG9TVW90M
- [Ring homomorphism](https://tamnd.github.io/math2/node/0GGE0Z0JQ1.md): definition 0GGE0Z0JQ1
- [P-value](https://tamnd.github.io/math2/node/0GGKJGZF6D.md): concept 0GGKJGZF6D
- [Forest](https://tamnd.github.io/math2/node/0GGXWDNKBR.md): concept 0GGXWDNKBR
- [Frobenius endomorphism](https://tamnd.github.io/math2/node/0GHDDVAH8P.md): definition 0GHDDVAH8P
- [Universal quantifier](https://tamnd.github.io/math2/node/0GHS47PDAQ.md): concept 0GHS47PDAQ
- [General solution of an inhomogeneous linear equation](https://tamnd.github.io/math2/node/0GHWEH6H15.md): statement 0GHWEH6H15
- [Euclidean domain](https://tamnd.github.io/math2/node/0GJ2HRG95T.md): definition 0GJ2HRG95T
- [Polynomial evaluation](https://tamnd.github.io/math2/node/0GJ7T0NK49.md): definition 0GJ7T0NK49
- [Universal property of the quotient group](https://tamnd.github.io/math2/node/0GJ83E78A5.md): statement 0GJ83E78A5
- [Angle at a point](https://tamnd.github.io/math2/node/0GJAJC6TNZ.md): concept 0GJAJC6TNZ
- [Mersenne number](https://tamnd.github.io/math2/node/0GJBJ19NXR.md): definition 0GJBJ19NXR
- [Cancellation in an integral domain](https://tamnd.github.io/math2/node/0GJM24EXP9.md): statement 0GJM24EXP9
- [Trace is linear](https://tamnd.github.io/math2/node/0GJWQZCPZM.md): statement 0GJWQZCPZM
- [Addition formulas for hyperbolic functions](https://tamnd.github.io/math2/node/0GJZCE87WP.md): statement 0GJZCE87WP
- [Function for which the iterated integrals differ](https://tamnd.github.io/math2/node/0GJZE2E4RX.md): counterexample 0GJZE2E4RX
- [Transitive relation](https://tamnd.github.io/math2/node/0GJZW9JFZQ.md): concept 0GJZW9JFZQ
- [Minimal polynomial of an algebraic element](https://tamnd.github.io/math2/node/0GK1WB9R98.md): definition 0GK1WB9R98
- [Solution space of a homogeneous linear equation has dimension n](https://tamnd.github.io/math2/node/0GK68PZ6KH.md): statement 0GK68PZ6KH
- [Separable space](https://tamnd.github.io/math2/node/0GKJ78948V.md): definition 0GKJ78948V
- [The alternating group A5 is simple](https://tamnd.github.io/math2/node/0GKQKF0FXJ.md): example 0GKQKF0FXJ
- [Adjugate matrix](https://tamnd.github.io/math2/node/0GMPYM0NSR.md): definition 0GMPYM0NSR
- [Absolute value is multiplicative](https://tamnd.github.io/math2/node/0GMV7GP8W0.md): statement 0GMV7GP8W0
- [Subring](https://tamnd.github.io/math2/node/0GNCRPYXYV.md): concept 0GNCRPYXYV
- [Nilpotent groups are solvable](https://tamnd.github.io/math2/node/0GNG2KVPZJ.md): statement 0GNG2KVPZJ
- [Antichain](https://tamnd.github.io/math2/node/0GNGPF7KS5.md): definition 0GNGPF7KS5
- [Dirichlet's theorem on arithmetic progressions](https://tamnd.github.io/math2/node/0GNVN3CA2P.md): statement 0GNVN3CA2P
- [Chain](https://tamnd.github.io/math2/node/0GP9ASVQ1X.md): definition 0GP9ASVQ1X
- [Complete residue system](https://tamnd.github.io/math2/node/0GPBVKE31D.md): concept 0GPBVKE31D
- [Cardinality of the power set](https://tamnd.github.io/math2/node/0GPFKS96T7.md): statement 0GPFKS96T7
- [Cauchy distribution](https://tamnd.github.io/math2/node/0GPG24Q3GR.md): concept 0GPG24Q3GR
- [The medians meet at the centroid](https://tamnd.github.io/math2/node/0GPQ67XV7N.md): statement 0GPQ67XV7N
- [Skewness](https://tamnd.github.io/math2/node/0GQ3H7VW3V.md): definition 0GQ3H7VW3V
- [Independence of random variables](https://tamnd.github.io/math2/node/0GQ938GDE0.md): concept 0GQ938GDE0
- [Quotient of continuous functions is continuous](https://tamnd.github.io/math2/node/0GR2APWTAC.md): statement 0GR2APWTAC
- [Wedge product of forms](https://tamnd.github.io/math2/node/0GRAATMGP1.md): concept 0GRAATMGP1
- [Curl-free field that is not conservative](https://tamnd.github.io/math2/node/0GRM3BS0DD.md): counterexample 0GRM3BS0DD
- [Angles in the same segment are equal](https://tamnd.github.io/math2/node/0GS0AW6JRY.md): statement 0GS0AW6JRY
- [Completely multiplicative function](https://tamnd.github.io/math2/node/0GS27TKF1V.md): definition 0GS27TKF1V
- [Degree of a field extension](https://tamnd.github.io/math2/node/0GS74FGFEK.md): definition 0GS74FGFEK
- [Thales's theorem](https://tamnd.github.io/math2/node/0GSDHYMA1T.md): statement 0GSDHYMA1T
- [Ideal generated by a subset](https://tamnd.github.io/math2/node/0GSW76Q14N.md): concept 0GSW76Q14N
- [Extended real number line](https://tamnd.github.io/math2/node/0GSZ7B0CGR.md): object 0GSZ7B0CGR
- [System of linear equations](https://tamnd.github.io/math2/node/0GT4WWGWH2.md): concept 0GT4WWGWH2
- [Decimal expansion as a series](https://tamnd.github.io/math2/node/0GT85FBPTS.md): example 0GT85FBPTS
- [Triple integral](https://tamnd.github.io/math2/node/0GTBGMZXV8.md): definition 0GTBGMZXV8
- [Gaussian prime](https://tamnd.github.io/math2/node/0GTEHJQRFK.md): definition 0GTEHJQRFK
- [Cavalieri's principle](https://tamnd.github.io/math2/node/0GTZ2F3PQY.md): statement 0GTZ2F3PQY
- [Bijection principle](https://tamnd.github.io/math2/node/0GV2573FP4.md): statement 0GV2573FP4
- [Disjunctive normal form](https://tamnd.github.io/math2/node/0GVDFAY4KW.md): concept 0GVDFAY4KW
- [Moment of inertia](https://tamnd.github.io/math2/node/0GVMR0C3S6.md): definition 0GVMR0C3S6
- [Fundamental theorem of symmetric polynomials](https://tamnd.github.io/math2/node/0GW1ZZH0R4.md): statement 0GW1ZZH0R4
- [Hyperbolic sine](https://tamnd.github.io/math2/node/0GWD21EWRW.md): definition 0GWD21EWRW
- [One-point compactification](https://tamnd.github.io/math2/node/0GWN3CMRK8.md): concept 0GWN3CMRK8
- [Area under a curve](https://tamnd.github.io/math2/node/0GXA8Z205E.md): concept 0GXA8Z205E
- [Integral of sine](https://tamnd.github.io/math2/node/0GXCXCX3QN.md): statement 0GXCXCX3QN
- [Telescoping series](https://tamnd.github.io/math2/node/0GXKM1R7E4.md): concept 0GXKM1R7E4
- [Laurent series](https://tamnd.github.io/math2/node/0GXMPDNYX3.md): concept 0GXMPDNYX3
- [Units of Z/nZ are the residues coprime to n](https://tamnd.github.io/math2/node/0GXNHEHQNA.md): statement 0GXNHEHQNA
- [Second derivative test in several variables](https://tamnd.github.io/math2/node/0GYDV9FE98.md): statement 0GYDV9FE98
- [Commutative ring](https://tamnd.github.io/math2/node/0GYGHKX834.md): definition 0GYGHKX834
- [Cayley graph](https://tamnd.github.io/math2/node/0GYQK9NDJF.md): concept 0GYQK9NDJF
- [Divergence theorem on a rectangle in the plane](https://tamnd.github.io/math2/node/0GZ3D8F921.md): statement 0GZ3D8F921
- [Ascending chain condition](https://tamnd.github.io/math2/node/0GZ6Y1H4RN.md): concept 0GZ6Y1H4RN
- [Function space](https://tamnd.github.io/math2/node/0GZB2SN0PH.md): example 0GZB2SN0PH
- [Dirichlet integral](https://tamnd.github.io/math2/node/0GZNSHMG4Q.md): example 0GZNSHMG4Q
- [Characteristic equation](https://tamnd.github.io/math2/node/0GZWXM2DKE.md): concept 0GZWXM2DKE
- [Conjugate subgroup](https://tamnd.github.io/math2/node/0H11ZPX4ND.md): concept 0H11ZPX4ND
- [Fundamental matrix](https://tamnd.github.io/math2/node/0H1GN6P9R5.md): concept 0H1GN6P9R5
- [Hyperbolic sine](https://tamnd.github.io/math2/node/0H1JFKE7H4.md): concept 0H1JFKE7H4
- [Arithmetic function](https://tamnd.github.io/math2/node/0H2EPRZC0D.md): definition 0H2EPRZC0D
- [Pseudometric space](https://tamnd.github.io/math2/node/0H3W614GEC.md): definition 0H3W614GEC
- [Semidirect product](https://tamnd.github.io/math2/node/0H3WZXCZNF.md): concept 0H3WZXCZNF
- [Euclidean group](https://tamnd.github.io/math2/node/0H3YY8CPKH.md): object 0H3YY8CPKH
- [Inner automorphism](https://tamnd.github.io/math2/node/0H47DMFN3S.md): definition 0H47DMFN3S
- [Fundamental system of solutions](https://tamnd.github.io/math2/node/0H4KNNCHPY.md): definition 0H4KNNCHPY
- [Limit inferior](https://tamnd.github.io/math2/node/0H4R3RESGV.md): definition 0H4R3RESGV
- [Disjunction](https://tamnd.github.io/math2/node/0H5GKDTB01.md): definition 0H5GKDTB01
- [Upper bound](https://tamnd.github.io/math2/node/0H5J02P8S9.md): definition 0H5J02P8S9
- [Line integral of a vector field](https://tamnd.github.io/math2/node/0H5MR9B6GT.md): concept 0H5MR9B6GT
- [Gamma function](https://tamnd.github.io/math2/node/0H5SYBSED9.md): concept 0H5SYBSED9
- [The order of a finite field is a prime power](https://tamnd.github.io/math2/node/0H5WWC8RZX.md): statement 0H5WWC8RZX
- [Taylor's theorem with the Lagrange remainder](https://tamnd.github.io/math2/node/0H5ZAZCX2T.md): statement 0H5ZAZCX2T
- [Changing the order of integration](https://tamnd.github.io/math2/node/0H60H1Q0SX.md): statement 0H60H1Q0SX
- [Algebraic element](https://tamnd.github.io/math2/node/0H628RSGWX.md): concept 0H628RSGWX
- [Galton–Watson process](https://tamnd.github.io/math2/node/0H631CZP5K.md): definition 0H631CZP5K
- [Del operator](https://tamnd.github.io/math2/node/0H6DP8CHND.md): concept 0H6DP8CHND
- [Product of power series](https://tamnd.github.io/math2/node/0H6KWAYS87.md): statement 0H6KWAYS87
- [Scalar potential](https://tamnd.github.io/math2/node/0H7C3T1GJY.md): concept 0H7C3T1GJY
- [Cumulative distribution function](https://tamnd.github.io/math2/node/0H8H7ZTHW2.md): concept 0H8H7ZTHW2
- [Every finite tree with at least two vertices has a leaf](https://tamnd.github.io/math2/node/0H8Q4MANBD.md): statement 0H8Q4MANBD
- [Rule of Sarrus](https://tamnd.github.io/math2/node/0H8SZFFDXJ.md): statement 0H8SZFFDXJ
- [Plane](https://tamnd.github.io/math2/node/0H97Q4Z2SF.md): definition 0H97Q4Z2SF
- [Abelianisation](https://tamnd.github.io/math2/node/0H9K5D3RDV.md): definition 0H9K5D3RDV
- [Rectifiable curve](https://tamnd.github.io/math2/node/0HATW2VWAZ.md): concept 0HATW2VWAZ
- [Interior is the largest open subset](https://tamnd.github.io/math2/node/0HAZEFH137.md): statement 0HAZEFH137
- [Binomial series](https://tamnd.github.io/math2/node/0HB0XKDXHH.md): example 0HB0XKDXHH
- [Markov property](https://tamnd.github.io/math2/node/0HB8SXHPFA.md): definition 0HB8SXHPFA
- [Limit point](https://tamnd.github.io/math2/node/0HBQC2GJXT.md): definition 0HBQC2GJXT
- [Intersection](https://tamnd.github.io/math2/node/0HBRW4E6G1.md): definition 0HBRW4E6G1
- [Induction on the integers](https://tamnd.github.io/math2/node/0HBS1F00DN.md): statement 0HBS1F00DN
- [Riemann sum](https://tamnd.github.io/math2/node/0HBY48G8G1.md): definition 0HBY48G8G1
- [Zermelo–Fraenkel set theory](https://tamnd.github.io/math2/node/0HE5WDQ9FJ.md): concept 0HE5WDQ9FJ
- [Union of open sets is open](https://tamnd.github.io/math2/node/0HEF46RPEK.md): statement 0HEF46RPEK
- [Russell's paradox](https://tamnd.github.io/math2/node/0HESMMSYEG.md): statement 0HESMMSYEG
- [Hamiltonian cycle](https://tamnd.github.io/math2/node/0HF30P75G4.md): concept 0HF30P75G4
- [Rotation](https://tamnd.github.io/math2/node/0HFREYRSZP.md): concept 0HFREYRSZP
- [Taylor polynomial](https://tamnd.github.io/math2/node/0HFSFVMWQB.md): definition 0HFSFVMWQB
- [First-countable space](https://tamnd.github.io/math2/node/0HG11FW8MJ.md): definition 0HG11FW8MJ
- [Pigeonhole principle](https://tamnd.github.io/math2/node/0HGVCWQGY3.md): statement 0HGVCWQGY3
- [Sum of a series](https://tamnd.github.io/math2/node/0HH4Y7HSRQ.md): concept 0HH4Y7HSRQ
- [Generating function](https://tamnd.github.io/math2/node/0HH6JX5WZZ.md): definition 0HH6JX5WZZ
- [Differentiable function of several variables](https://tamnd.github.io/math2/node/0HHHDCXZV4.md): concept 0HHHDCXZV4
- [Solutions for distinct real roots](https://tamnd.github.io/math2/node/0HJ0HDA6W7.md): statement 0HJ0HDA6W7
- [Thomae's function](https://tamnd.github.io/math2/node/0HJ28H9Y95.md): example 0HJ28H9Y95
- [Second isomorphism theorem for rings](https://tamnd.github.io/math2/node/0HJ4DTW6FN.md): statement 0HJ4DTW6FN
- [Discrete logarithm](https://tamnd.github.io/math2/node/0HJHYMGN3C.md): concept 0HJHYMGN3C
- [Hamiltonian cycle](https://tamnd.github.io/math2/node/0HKXNG2RS1.md): definition 0HKXNG2RS1
- [Random experiment](https://tamnd.github.io/math2/node/0HM05N3FJD.md): concept 0HM05N3FJD
- [Distributive lattice](https://tamnd.github.io/math2/node/0HM0HQCK6N.md): concept 0HM0HQCK6N
- [RSA cryptosystem](https://tamnd.github.io/math2/node/0HM584C465.md): example 0HM584C465
- [A sequence converges iff its liminf and limsup agree](https://tamnd.github.io/math2/node/0HM732ZNVC.md): statement 0HM732ZNVC
- [Order on the natural numbers](https://tamnd.github.io/math2/node/0HNATXFEV5.md): concept 0HNATXFEV5
- [Centre of mass](https://tamnd.github.io/math2/node/0HNK6KT05K.md): concept 0HNK6KT05K
- [Eigenvectors with distinct eigenvalues are linearly independent](https://tamnd.github.io/math2/node/0HNMFG8F8K.md): statement 0HNMFG8F8K
- [Generalised Stokes theorem](https://tamnd.github.io/math2/node/0HPHKCGEQ0.md): statement 0HPHKCGEQ0
- [Archimedean ordered field](https://tamnd.github.io/math2/node/0HQARMKEV5.md): definition 0HQARMKEV5
- [Coefficient matrix](https://tamnd.github.io/math2/node/0HQFZAP3ND.md): concept 0HQFZAP3ND
- [Green's theorem](https://tamnd.github.io/math2/node/0HR3N02317.md): statement 0HR3N02317
- [Linear span](https://tamnd.github.io/math2/node/0HR573CHNM.md): definition 0HR573CHNM
- [Rearrangement of a series](https://tamnd.github.io/math2/node/0HRJ2Y74T3.md): definition 0HRJ2Y74T3
- [Solutions for repeated roots](https://tamnd.github.io/math2/node/0HRVY67CNV.md): statement 0HRVY67CNV
- [Axiom of choice](https://tamnd.github.io/math2/node/0HTGB13FE2.md): statement 0HTGB13FE2
- [Reduced residue system](https://tamnd.github.io/math2/node/0HV01NH20W.md): concept 0HV01NH20W
- [Rouché–Capelli theorem](https://tamnd.github.io/math2/node/0HV1SKWPMS.md): statement 0HV1SKWPMS
- [Group homomorphism](https://tamnd.github.io/math2/node/0HV9Y7WG5R.md): definition 0HV9Y7WG5R
- [Eccentricity](https://tamnd.github.io/math2/node/0HVX863SAJ.md): concept 0HVX863SAJ
- [Limit comparison test](https://tamnd.github.io/math2/node/0HW7BNHHGF.md): statement 0HW7BNHHGF
- [Gram matrix](https://tamnd.github.io/math2/node/0HX8HZ1WTQ.md): definition 0HX8HZ1WTQ
- [One-sided derivative](https://tamnd.github.io/math2/node/0HXEX0DPRA.md): definition 0HXEX0DPRA
- [Finitely generated group](https://tamnd.github.io/math2/node/0HY15D6WTX.md): definition 0HY15D6WTX
- [Totient of a prime](https://tamnd.github.io/math2/node/0HY1MXMEN9.md): statement 0HY1MXMEN9
- [Eigenvalues of a positive definite matrix are positive](https://tamnd.github.io/math2/node/0HYC2RHD3M.md): statement 0HYC2RHD3M
- [Injective iff it has a left inverse](https://tamnd.github.io/math2/node/0HYFEEZAEM.md): statement 0HYFEEZAEM
- [Polar form of a quadratic form](https://tamnd.github.io/math2/node/0HYFWKKY3C.md): concept 0HYFWKKY3C
- [Modulus is multiplicative](https://tamnd.github.io/math2/node/0HYNZE940T.md): statement 0HYNZE940T
- [Student's t-test](https://tamnd.github.io/math2/node/0HYP606RYF.md): concept 0HYP606RYF
- [Mean value property](https://tamnd.github.io/math2/node/0HYPR64JM2.md): statement 0HYPR64JM2
- [Norm of a Gaussian integer](https://tamnd.github.io/math2/node/0HYRJF4WNQ.md): concept 0HYRJF4WNQ
- [Hamiltonian graph](https://tamnd.github.io/math2/node/0HYV2NME07.md): concept 0HYV2NME07
- [Quadratic form](https://tamnd.github.io/math2/node/0HZNE5QZDJ.md): definition 0HZNE5QZDJ
- [Pivot](https://tamnd.github.io/math2/node/0J0BBCMJ5W.md): concept 0J0BBCMJ5W
- [Exponential function](https://tamnd.github.io/math2/node/0J0FCWV32D.md): concept 0J0FCWV32D
- [Reduced residue system](https://tamnd.github.io/math2/node/0J0QEN1DZR.md): definition 0J0QEN1DZR
- [Quotient map](https://tamnd.github.io/math2/node/0J0SK5FF4M.md): definition 0J0SK5FF4M
- [Proof by contradiction](https://tamnd.github.io/math2/node/0J0XBKG64S.md): concept 0J0XBKG64S
- [Fixed field](https://tamnd.github.io/math2/node/0J1E5ZRZRM.md): concept 0J1E5ZRZRM
- [Euclidean isometry](https://tamnd.github.io/math2/node/0J1N3WTKM4.md): concept 0J1N3WTKM4
- [Liouville number](https://tamnd.github.io/math2/node/0J1R5NP36S.md): definition 0J1R5NP36S
- [Solutions of Pell's equation are powers of the fundamental solution](https://tamnd.github.io/math2/node/0J1TT091TN.md): statement 0J1TT091TN
- [Existence of nth roots by the intermediate value theorem](https://tamnd.github.io/math2/node/0J29GM30Z7.md): example 0J29GM30Z7
- [Polar decomposition](https://tamnd.github.io/math2/node/0J2C7BDJQA.md): statement 0J2C7BDJQA
- [Inverse function](https://tamnd.github.io/math2/node/0J2WAR83WP.md): concept 0J2WAR83WP
- [Equivalent norms](https://tamnd.github.io/math2/node/0J3578562S.md): definition 0J3578562S
- [Cycle type](https://tamnd.github.io/math2/node/0J3K4Y1KPD.md): concept 0J3K4Y1KPD
- [Conjugacy classes of the symmetric group correspond to cycle types](https://tamnd.github.io/math2/node/0J438GAC8K.md): statement 0J438GAC8K
- [The symmetric group on five or more letters is not solvable](https://tamnd.github.io/math2/node/0J53NE674D.md): statement 0J53NE674D
- [Characteristic polynomial of a linear map](https://tamnd.github.io/math2/node/0J5EMJRRFM.md): concept 0J5EMJRRFM
- [Congruence is compatible with powers](https://tamnd.github.io/math2/node/0J5VGJYYC3.md): statement 0J5VGJYYC3
- [Intersecting secants theorem](https://tamnd.github.io/math2/node/0J5XR7BW2S.md): statement 0J5XR7BW2S
- [Lyapunov's stability theorem](https://tamnd.github.io/math2/node/0J6A752DTZ.md): statement 0J6A752DTZ
- [Associativity of composition](https://tamnd.github.io/math2/node/0J6AKYZQT8.md): statement 0J6AKYZQT8
- [Kuratowski's theorem](https://tamnd.github.io/math2/node/0J6G5NJC5R.md): statement 0J6G5NJC5R
- [Hyperbolic functions](https://tamnd.github.io/math2/node/0J6M03WSS6.md): definition 0J6M03WSS6
- [Discrete random variable](https://tamnd.github.io/math2/node/0J6Z6H5VQJ.md): definition 0J6Z6H5VQJ
- [Nine-point circle](https://tamnd.github.io/math2/node/0J768YF1ZE.md): definition 0J768YF1ZE
- [Rearrangements of an absolutely convergent series have the same sum](https://tamnd.github.io/math2/node/0J76TP9FGD.md): statement 0J76TP9FGD
- [Riemann sphere](https://tamnd.github.io/math2/node/0J78KBPFZB.md): concept 0J78KBPFZB
- [Endomorphism](https://tamnd.github.io/math2/node/0J7H0YBFDJ.md): concept 0J7H0YBFDJ
- [Picard–Lindelöf hypotheses](https://tamnd.github.io/math2/node/0J7HF2PF1H.md): definition 0J7HF2PF1H
- [Logarithm of a quotient](https://tamnd.github.io/math2/node/0J7PHRBNNS.md): statement 0J7PHRBNNS
- [Primitive root of unity](https://tamnd.github.io/math2/node/0J817RTVG6.md): definition 0J817RTVG6
- [Sperner's theorem](https://tamnd.github.io/math2/node/0J87V9K1GE.md): statement 0J87V9K1GE
- [Composition of an integer](https://tamnd.github.io/math2/node/0J8ANSYKAC.md): definition 0J8ANSYKAC
- [Taylor's theorem in several variables](https://tamnd.github.io/math2/node/0J8T4K4YTE.md): statement 0J8T4K4YTE
- [Reachability](https://tamnd.github.io/math2/node/0J8XE7W68F.md): definition 0J8XE7W68F
- [Limit of a function of several variables](https://tamnd.github.io/math2/node/0J8Y1NCY0K.md): concept 0J8Y1NCY0K
- [Bernoulli distribution](https://tamnd.github.io/math2/node/0J97P5KZY0.md): definition 0J97P5KZY0
- [Determinant of a block triangular matrix](https://tamnd.github.io/math2/node/0J9TVTNB4K.md): statement 0J9TVTNB4K
- [Riemann sum](https://tamnd.github.io/math2/node/0JAEHXTVV5.md): concept 0JAEHXTVV5
- [Euler method](https://tamnd.github.io/math2/node/0JARTXX4XP.md): concept 0JARTXX4XP
- [Image of a linear map](https://tamnd.github.io/math2/node/0JB228DCWB.md): definition 0JB228DCWB
- [Linear combination](https://tamnd.github.io/math2/node/0JB3DP1BMF.md): definition 0JB3DP1BMF
- [Graph of a function](https://tamnd.github.io/math2/node/0JBJYT69ZV.md): concept 0JBJYT69ZV
- [Ratio formula for the radius of convergence](https://tamnd.github.io/math2/node/0JBMYAA9QH.md): statement 0JBMYAA9QH
- [Vector calculus identities](https://tamnd.github.io/math2/node/0JC5X4HWR2.md): statement 0JC5X4HWR2
- [Stirling numbers of the second kind](https://tamnd.github.io/math2/node/0JCQGPY8Z4.md): concept 0JCQGPY8Z4
- [Converse relation](https://tamnd.github.io/math2/node/0JCSQNNP2M.md): definition 0JCSQNNP2M
- [Graph](https://tamnd.github.io/math2/node/0JD5C82QMB.md): definition 0JD5C82QMB
- [Analytic function](https://tamnd.github.io/math2/node/0JD5ZNW0F4.md): definition 0JD5ZNW0F4
- [Convergence in distribution](https://tamnd.github.io/math2/node/0JDAQHWA9N.md): definition 0JDAQHWA9N
- [Triangle](https://tamnd.github.io/math2/node/0JDB8YJBSR.md): definition 0JDB8YJBSR
- [Path](https://tamnd.github.io/math2/node/0JDE56HWQ8.md): definition 0JDE56HWQ8
- [Continuity in R^n](https://tamnd.github.io/math2/node/0JEJATBVD7.md): concept 0JEJATBVD7
- [Sign is a homomorphism](https://tamnd.github.io/math2/node/0JEJJCB0EX.md): statement 0JEJJCB0EX
- [The cumulative distribution function determines the distribution](https://tamnd.github.io/math2/node/0JEVZAFDY6.md): statement 0JEVZAFDY6
- [Affine map](https://tamnd.github.io/math2/node/0JF0TVYGWT.md): concept 0JF0TVYGWT
- [Parametrised curve](https://tamnd.github.io/math2/node/0JF1XXGWB4.md): definition 0JF1XXGWB4
- [Regular space](https://tamnd.github.io/math2/node/0JF9KW2TXB.md): definition 0JF9KW2TXB
- [Residue](https://tamnd.github.io/math2/node/0JFGKKWWFK.md): definition 0JFGKKWWFK
- [Turán's theorem](https://tamnd.github.io/math2/node/0JG1KZSPFZ.md): statement 0JG1KZSPFZ
- [Connected component](https://tamnd.github.io/math2/node/0JG9W3KA6B.md): concept 0JG9W3KA6B
- [Rank of a linear map](https://tamnd.github.io/math2/node/0JGB7A6GTS.md): definition 0JGB7A6GTS
- [Presentation of a group](https://tamnd.github.io/math2/node/0JGEQXQ4AC.md): concept 0JGEQXQ4AC
- [Space of polynomials of bounded degree](https://tamnd.github.io/math2/node/0JGZCDT5DN.md): example 0JGZCDT5DN
- [Series](https://tamnd.github.io/math2/node/0JH4MN4K1C.md): definition 0JH4MN4K1C
- [Eigenspaces of a self-adjoint operator are orthogonal](https://tamnd.github.io/math2/node/0JHGAV9H25.md): statement 0JHGAV9H25
- [The ring of integers](https://tamnd.github.io/math2/node/0JHM6JK7FD.md): example 0JHM6JK7FD
- [Exponential generating function](https://tamnd.github.io/math2/node/0JHXPVFKT0.md): concept 0JHXPVFKT0
- [Underdamped](https://tamnd.github.io/math2/node/0JJ6GDDE7Z.md): concept 0JJ6GDDE7Z
- [Boundary](https://tamnd.github.io/math2/node/0JJ9J1XZ8G.md): definition 0JJ9J1XZ8G
- [Cauchy's estimates](https://tamnd.github.io/math2/node/0JJRENWMP6.md): statement 0JJRENWMP6
- [Abelian simple groups have prime order](https://tamnd.github.io/math2/node/0JJV80RXHD.md): statement 0JJV80RXHD
- [Kronecker product](https://tamnd.github.io/math2/node/0JK6M18QZH.md): definition 0JK6M18QZH
- [Combinatorics](https://tamnd.github.io/math2/node/0JKW7W7XFR.md): concept 0JKW7W7XFR
- [Cauchy's theorem](https://tamnd.github.io/math2/node/0JM7PA288B.md): statement 0JM7PA288B
- [Logical equivalence](https://tamnd.github.io/math2/node/0JMR2HM2N2.md): concept 0JMR2HM2N2
- [Sum of subspaces](https://tamnd.github.io/math2/node/0JMTY8MF54.md): concept 0JMTY8MF54
- [Kernel of a ring homomorphism](https://tamnd.github.io/math2/node/0JN0SQG2JH.md): concept 0JN0SQG2JH
- [Sample variance](https://tamnd.github.io/math2/node/0JN9P4A944.md): definition 0JN9P4A944
- [Lindemann–Weierstrass theorem](https://tamnd.github.io/math2/node/0JNCBFMPXR.md): statement 0JNCBFMPXR
- [Convolution theorem for the Laplace transform](https://tamnd.github.io/math2/node/0JNMJEDD93.md): statement 0JNMJEDD93
- [Universal set](https://tamnd.github.io/math2/node/0JNXSS9TJ9.md): concept 0JNXSS9TJ9
- [Fundamental theorem of algebra](https://tamnd.github.io/math2/node/0JP0W07Y9A.md): statement 0JP0W07Y9A
- [Kőnig's theorem](https://tamnd.github.io/math2/node/0JP0ZF57PN.md): statement 0JP0ZF57PN
- [Function of several variables](https://tamnd.github.io/math2/node/0JPCHTXWG0.md): concept 0JPCHTXWG0
- [Subgroups of abelian groups are normal](https://tamnd.github.io/math2/node/0JPDTEDCFD.md): statement 0JPDTEDCFD
- [Central binomial coefficient](https://tamnd.github.io/math2/node/0JPH2Y4DFV.md): concept 0JPH2Y4DFV
- [Order of an element](https://tamnd.github.io/math2/node/0JPYSD5E0B.md): definition 0JPYSD5E0B
- [Order of the general linear group over a finite field](https://tamnd.github.io/math2/node/0JQ1KNQW2K.md): statement 0JQ1KNQW2K
- [Alternating harmonic series](https://tamnd.github.io/math2/node/0JQ5Y3WWVE.md): example 0JQ5Y3WWVE
- [Weierstrass approximation theorem](https://tamnd.github.io/math2/node/0JQE130DHW.md): statement 0JQE130DHW
- [Ordered pair](https://tamnd.github.io/math2/node/0JQWWFCS9R.md): definition 0JQWWFCS9R
- [Multiplication of natural numbers](https://tamnd.github.io/math2/node/0JRC32D6K2.md): definition 0JRC32D6K2
- [Inverse Laplace transform](https://tamnd.github.io/math2/node/0JRW4X1MDC.md): definition 0JRW4X1MDC
- [Bernstein polynomial](https://tamnd.github.io/math2/node/0JT92VK307.md): concept 0JT92VK307
- [Legendre symbol](https://tamnd.github.io/math2/node/0JTC045NNZ.md): concept 0JTC045NNZ
- [Cardinality](https://tamnd.github.io/math2/node/0JTGRV4WT4.md): definition 0JTGRV4WT4
- [Estimate for approximate solutions](https://tamnd.github.io/math2/node/0JTPMCCZT2.md): statement 0JTPMCCZT2
- [System of linear equations](https://tamnd.github.io/math2/node/0JTRMBQ1SA.md): definition 0JTRMBQ1SA
- [Tychonoff's theorem](https://tamnd.github.io/math2/node/0JTRSJKCFB.md): statement 0JTRSJKCFB
- [Integer partition](https://tamnd.github.io/math2/node/0JV30XN73Z.md): concept 0JV30XN73Z
- [Gaussian elimination](https://tamnd.github.io/math2/node/0JVJ2WCXZH.md): concept 0JVJ2WCXZH
- [Split polynomial](https://tamnd.github.io/math2/node/0JVSNJCN68.md): concept 0JVSNJCN68
- [Vector projection](https://tamnd.github.io/math2/node/0JW13B48YP.md): concept 0JW13B48YP
- [Existential quantifier](https://tamnd.github.io/math2/node/0JW2FHW7G3.md): definition 0JW2FHW7G3
- [Saddle point](https://tamnd.github.io/math2/node/0JW6MDHAZD.md): concept 0JW6MDHAZD
- [Minimum spanning tree](https://tamnd.github.io/math2/node/0JW7KT6PQW.md): concept 0JW7KT6PQW
- [Concyclic points](https://tamnd.github.io/math2/node/0JWCX0TW2S.md): definition 0JWCX0TW2S
- [An orthonormal family is linearly independent](https://tamnd.github.io/math2/node/0JWM6GJNQW.md): statement 0JWM6GJNQW
- [Divergent series](https://tamnd.github.io/math2/node/0JWMJSTBSG.md): concept 0JWMJSTBSG
- [Bayesian inference](https://tamnd.github.io/math2/node/0JWRJX2RMP.md): concept 0JWRJX2RMP
- [L'Hôpital's rule](https://tamnd.github.io/math2/node/0JWT4PPQ7B.md): statement 0JWT4PPQ7B
- [Sylow p-subgroup](https://tamnd.github.io/math2/node/0JWZX6FKSH.md): definition 0JWZX6FKSH
- [Divergence theorem](https://tamnd.github.io/math2/node/0JX86PSSA9.md): statement 0JX86PSSA9
- [Legendre's theorem on continued fractions](https://tamnd.github.io/math2/node/0JXTQ392ZM.md): statement 0JXTQ392ZM
- [Square matrix](https://tamnd.github.io/math2/node/0JXZKX526R.md): concept 0JXZKX526R
- [Lifting-the-exponent lemma](https://tamnd.github.io/math2/node/0JYF6PTKEW.md): statement 0JYF6PTKEW
- [Group homomorphism](https://tamnd.github.io/math2/node/0JYTG5K9Y4.md): concept 0JYTG5K9Y4
- [Dihedral group](https://tamnd.github.io/math2/node/0JZ27DMJ5D.md): concept 0JZ27DMJ5D
- [Power rule](https://tamnd.github.io/math2/node/0JZGW75C5Z.md): statement 0JZGW75C5Z
- [Combination](https://tamnd.github.io/math2/node/0JZSRWRBZE.md): definition 0JZSRWRBZE
- [Galois group of a polynomial](https://tamnd.github.io/math2/node/0JZY3R908H.md): definition 0JZY3R908H
- [Convex set](https://tamnd.github.io/math2/node/0K06X9QXDX.md): definition 0K06X9QXDX
- [Laplace transform of a derivative](https://tamnd.github.io/math2/node/0K090GVHVN.md): statement 0K090GVHVN
- [Upper and lower Darboux sums](https://tamnd.github.io/math2/node/0K0A06X2MQ.md): concept 0K0A06X2MQ
- [Path-connected space](https://tamnd.github.io/math2/node/0K0GE6BCYW.md): definition 0K0GE6BCYW
- [Sum of a series](https://tamnd.github.io/math2/node/0K0H5E3Z5N.md): definition 0K0H5E3Z5N
- [Distance from a point to a plane](https://tamnd.github.io/math2/node/0K1E2S6Z69.md): statement 0K1E2S6Z69
- [Lipschitz functions are uniformly continuous](https://tamnd.github.io/math2/node/0K1EWAR6AF.md): statement 0K1EWAR6AF
- [Determinant of a triangular matrix](https://tamnd.github.io/math2/node/0K1KKMFHYH.md): statement 0K1KKMFHYH
- [Integral of 1/x](https://tamnd.github.io/math2/node/0K1N1Y4EGZ.md): statement 0K1N1Y4EGZ
- [Automorphism group](https://tamnd.github.io/math2/node/0K1SCPBNKJ.md): concept 0K1SCPBNKJ
- [Zero divisor](https://tamnd.github.io/math2/node/0K26YKNBSA.md): definition 0K26YKNBSA
- [Scalar multiplication of matrices](https://tamnd.github.io/math2/node/0K28YXMS7Q.md): definition 0K28YXMS7Q
- [Brahmagupta–Fibonacci identity](https://tamnd.github.io/math2/node/0K2DMACEZ4.md): statement 0K2DMACEZ4
- [Logarithm of a product](https://tamnd.github.io/math2/node/0K2FNY6DYV.md): statement 0K2FNY6DYV
- [Singular value](https://tamnd.github.io/math2/node/0K2TX12X58.md): definition 0K2TX12X58
- [Minkowski's theorem](https://tamnd.github.io/math2/node/0K387S2PKQ.md): statement 0K387S2PKQ
- [Solving initial value problems with the Laplace transform](https://tamnd.github.io/math2/node/0K3HFZJK5Q.md): concept 0K3HFZJK5Q
- [Factor theorem](https://tamnd.github.io/math2/node/0K3MWA7DMC.md): statement 0K3MWA7DMC
- [Topologist's sine curve](https://tamnd.github.io/math2/node/0K3T4A659R.md): counterexample 0K3T4A659R
- [Confidence interval](https://tamnd.github.io/math2/node/0K4NMMJBG2.md): definition 0K4NMMJBG2
- [Determinant of a matrix with two equal rows](https://tamnd.github.io/math2/node/0K4VQB7HWR.md): statement 0K4VQB7HWR
- [Rayleigh quotient](https://tamnd.github.io/math2/node/0K4WCNTMHG.md): definition 0K4WCNTMHG
- [Homogeneous first-order equation](https://tamnd.github.io/math2/node/0K4ZWTFSV5.md): definition 0K4ZWTFSV5
- [Differentiable function](https://tamnd.github.io/math2/node/0K57YG3BB8.md): concept 0K57YG3BB8
- [Action by left multiplication](https://tamnd.github.io/math2/node/0K5TWJAGPD.md): example 0K5TWJAGPD
- [Gaussian integers](https://tamnd.github.io/math2/node/0K5W69AC0J.md): object 0K5W69AC0J
- [Matrix exponential](https://tamnd.github.io/math2/node/0K665H5V1S.md): definition 0K665H5V1S
- [Complement of an open set is closed](https://tamnd.github.io/math2/node/0K67Q60Z8P.md): statement 0K67Q60Z8P
- [Alternating multilinear map](https://tamnd.github.io/math2/node/0K6CH0MT0F.md): concept 0K6CH0MT0F
- [Young diagram](https://tamnd.github.io/math2/node/0K78D81EF7.md): definition 0K78D81EF7
- [Maclaurin series](https://tamnd.github.io/math2/node/0K7P6PR838.md): concept 0K7P6PR838
- [Idempotent matrix](https://tamnd.github.io/math2/node/0K7TD9BV2D.md): definition 0K7TD9BV2D
- [Isolated zeros theorem](https://tamnd.github.io/math2/node/0K83KEJ43P.md): statement 0K83KEJ43P
- [Associativity](https://tamnd.github.io/math2/node/0K8F7PVDMX.md): concept 0K8F7PVDMX
- [Compactness and sequential compactness agree in metric spaces](https://tamnd.github.io/math2/node/0K8SGQSV7G.md): statement 0K8SGQSV7G
- [Removable discontinuity](https://tamnd.github.io/math2/node/0K96Z2S699.md): concept 0K96Z2S699
- [Lower central series](https://tamnd.github.io/math2/node/0K98R7A4VC.md): concept 0K98R7A4VC
- [Continuous image of a compact set is compact](https://tamnd.github.io/math2/node/0K9AAJ8G51.md): statement 0K9AAJ8G51
- [Parallelepiped](https://tamnd.github.io/math2/node/0K9GBTZ1FX.md): definition 0K9GBTZ1FX
- [Inverse function theorem](https://tamnd.github.io/math2/node/0K9HSJFCMJ.md): statement 0K9HSJFCMJ
- [General Leibniz rule](https://tamnd.github.io/math2/node/0K9QCK6WJC.md): statement 0K9QCK6WJC
- [Ramsey number](https://tamnd.github.io/math2/node/0K9QVPJYJF.md): concept 0K9QVPJYJF
- [Regular graph](https://tamnd.github.io/math2/node/0KA4S8R39M.md): definition 0KA4S8R39M
- [Determinant via the Schur complement](https://tamnd.github.io/math2/node/0KA7A4ENVJ.md): statement 0KA7A4ENVJ
- [Upper and lower Darboux integrals](https://tamnd.github.io/math2/node/0KABP2HWMQ.md): concept 0KABP2HWMQ
- [Characteristic subgroup](https://tamnd.github.io/math2/node/0KABPMRHM2.md): concept 0KABPMRHM2
- [Sum of binomial coefficients](https://tamnd.github.io/math2/node/0KAMF19NV8.md): statement 0KAMF19NV8
- [Nondegenerate bilinear form](https://tamnd.github.io/math2/node/0KAN330PAH.md): definition 0KAN330PAH
- [Subgroups of cyclic groups are cyclic](https://tamnd.github.io/math2/node/0KB1ND7XDS.md): statement 0KB1ND7XDS
- [Distributive law](https://tamnd.github.io/math2/node/0KBEMAJAYA.md): statement 0KBEMAJAYA
- [Covariance](https://tamnd.github.io/math2/node/0KBFAWYRTE.md): concept 0KBFAWYRTE
- [Reduced word](https://tamnd.github.io/math2/node/0KBTP8Q2FS.md): concept 0KBTP8Q2FS
- [Bernoulli number](https://tamnd.github.io/math2/node/0KC00K7KY4.md): concept 0KC00K7KY4
- [Stirling's formula](https://tamnd.github.io/math2/node/0KC0F51T6A.md): statement 0KC0F51T6A
- [Wronskian](https://tamnd.github.io/math2/node/0KC1NMG90B.md): concept 0KC1NMG90B
- [Idempotent matrix](https://tamnd.github.io/math2/node/0KCE8CG86P.md): concept 0KCE8CG86P
- [Basis](https://tamnd.github.io/math2/node/0KCQ9Y3QKA.md): concept 0KCQ9Y3QKA
- [Venn diagram](https://tamnd.github.io/math2/node/0KCT6DPXJK.md): concept 0KCT6DPXJK
- [Fundamental solution of Pell's equation](https://tamnd.github.io/math2/node/0KCYJSB9FW.md): definition 0KCYJSB9FW
- [Vertex cover](https://tamnd.github.io/math2/node/0KDADE9RQ8.md): definition 0KDADE9RQ8
- [Bolzano–Weierstrass theorem](https://tamnd.github.io/math2/node/0KDJD249DT.md): statement 0KDJD249DT
- [Exponential generating function](https://tamnd.github.io/math2/node/0KE6NBTVY8.md): definition 0KE6NBTVY8
- [Classification of Pythagorean triples](https://tamnd.github.io/math2/node/0KF3J2KAER.md): statement 0KF3J2KAER
- [Directed graph](https://tamnd.github.io/math2/node/0KFYGSHHF5.md): concept 0KFYGSHHF5
- [The alternating group is generated by three-cycles](https://tamnd.github.io/math2/node/0KFZ0QX83G.md): statement 0KFZ0QX83G
- [Norm induced by an inner product](https://tamnd.github.io/math2/node/0KG4D8AP6G.md): definition 0KG4D8AP6G
- [Dirichlet convolution](https://tamnd.github.io/math2/node/0KH1N66QWW.md): definition 0KH1N66QWW
- [Posterior probability](https://tamnd.github.io/math2/node/0KJ7ZS8AJA.md): concept 0KJ7ZS8AJA
- [Affine combination](https://tamnd.github.io/math2/node/0KJH6SRFZK.md): concept 0KJH6SRFZK
- [A polynomial over a field has at most as many roots as its degree](https://tamnd.github.io/math2/node/0KJVQV3E0F.md): statement 0KJVQV3E0F
- [One-sided limits of a monotone function exist](https://tamnd.github.io/math2/node/0KJXHN0TW0.md): statement 0KJXHN0TW0
- [Primitive root modulo n](https://tamnd.github.io/math2/node/0KM2CJ80G7.md): definition 0KM2CJ80G7
- [Hausdorff maximal principle](https://tamnd.github.io/math2/node/0KM4KVR9MV.md): statement 0KM4KVR9MV
- [Fixed field](https://tamnd.github.io/math2/node/0KMJ6919WS.md): definition 0KMJ6919WS
- [Determinant of a linear map is the determinant of its matrix](https://tamnd.github.io/math2/node/0KMN32XGPF.md): statement 0KMN32XGPF
- [Strong induction](https://tamnd.github.io/math2/node/0KMVFSHFGZ.md): statement 0KMVFSHFGZ
- [Monotone sequence](https://tamnd.github.io/math2/node/0KN22AAW5R.md): definition 0KN22AAW5R
- [Ring isomorphism](https://tamnd.github.io/math2/node/0KN2S81K72.md): concept 0KN2S81K72
- [The exponential function is strictly increasing](https://tamnd.github.io/math2/node/0KN2V7840E.md): statement 0KN2V7840E
- [Restriction of an endomorphism to an invariant subspace](https://tamnd.github.io/math2/node/0KN7EPH7Z0.md): concept 0KN7EPH7Z0
- [Ceva's theorem](https://tamnd.github.io/math2/node/0KN93VDSXA.md): statement 0KN93VDSXA
- [Maximum degree](https://tamnd.github.io/math2/node/0KNAN9F27P.md): definition 0KNAN9F27P
- [p-norm on R^n](https://tamnd.github.io/math2/node/0KNJWEVSMG.md): example 0KNJWEVSMG
- [Homogeneous linear equation](https://tamnd.github.io/math2/node/0KP0QNXPRT.md): concept 0KP0QNXPRT
- [Dirac delta function](https://tamnd.github.io/math2/node/0KPHENZ5S7.md): object 0KPHENZ5S7
- [Signature of a quadratic form](https://tamnd.github.io/math2/node/0KQZYGEJTF.md): definition 0KQZYGEJTF
- [Stationary distribution](https://tamnd.github.io/math2/node/0KRH5157WP.md): concept 0KRH5157WP
- [Continuity at a point](https://tamnd.github.io/math2/node/0KRMJRDRZT.md): concept 0KRMJRDRZT
- [Symmetric bilinear form associated with a quadratic form](https://tamnd.github.io/math2/node/0KRVRV5V80.md): concept 0KRVRV5V80
- [Degree of a vertex](https://tamnd.github.io/math2/node/0KS0KRDV4Y.md): concept 0KS0KRDV4Y
- [Nowhere dense set](https://tamnd.github.io/math2/node/0KSM23SN3H.md): definition 0KSM23SN3H
- [Equivalence relation induced by a function](https://tamnd.github.io/math2/node/0KT8TGXD8K.md): concept 0KT8TGXD8K
- [Jordan measure](https://tamnd.github.io/math2/node/0KTGJW1ZTH.md): concept 0KTGJW1ZTH
- [Free group](https://tamnd.github.io/math2/node/0KTW26C80J.md): concept 0KTW26C80J
- [Cyclic group](https://tamnd.github.io/math2/node/0KVGH0J5B3.md): concept 0KVGH0J5B3
- [Variance under scaling](https://tamnd.github.io/math2/node/0KVQ9D86BM.md): statement 0KVQ9D86BM
- [Modulus of a complex number](https://tamnd.github.io/math2/node/0KW2X0TFBX.md): definition 0KW2X0TFBX
- [Diameter of a set](https://tamnd.github.io/math2/node/0KW3PN547K.md): concept 0KW3PN547K
- [Connected graph](https://tamnd.github.io/math2/node/0KWFFMDF8T.md): concept 0KWFFMDF8T
- [Reduction modulo p test for irreducibility](https://tamnd.github.io/math2/node/0KWWEXW0RN.md): statement 0KWWEXW0RN
- [Change of basis matrix](https://tamnd.github.io/math2/node/0KWXYZPJKN.md): definition 0KWXYZPJKN
- [Radius of convergence of the binomial series](https://tamnd.github.io/math2/node/0KX1MXJ5MR.md): statement 0KX1MXJ5MR
- [Positive definite matrix](https://tamnd.github.io/math2/node/0KX66CZMBB.md): concept 0KX66CZMBB
- [Usual metric on the real line](https://tamnd.github.io/math2/node/0KXJTZSW9B.md): example 0KXJTZSW9B
- [Number of prime factors counted with multiplicity](https://tamnd.github.io/math2/node/0KXQKES075.md): concept 0KXQKES075
- [Splitting field](https://tamnd.github.io/math2/node/0KXQKX9P63.md): concept 0KXQKX9P63
- [Continuous partial derivatives imply differentiability](https://tamnd.github.io/math2/node/0KXSK8TEEB.md): statement 0KXSK8TEEB
- [Derivative of x^a](https://tamnd.github.io/math2/node/0KY08TWDMQ.md): statement 0KY08TWDMQ
- [Stirling numbers of the first kind](https://tamnd.github.io/math2/node/0KY7GS695G.md): concept 0KY7GS695G
- [Discrete uniform distribution](https://tamnd.github.io/math2/node/0KYP951QWH.md): definition 0KYP951QWH
- [Quotient group](https://tamnd.github.io/math2/node/0KYX2N69J0.md): concept 0KYX2N69J0
- [Lagrange's four-square theorem](https://tamnd.github.io/math2/node/0KYYSMEVHH.md): statement 0KYYSMEVHH
- [Quadratic formula](https://tamnd.github.io/math2/node/0KYZXYDGQZ.md): statement 0KYZXYDGQZ
- [Trace of a linear map](https://tamnd.github.io/math2/node/0KZDNGA561.md): concept 0KZDNGA561
- [Independent set](https://tamnd.github.io/math2/node/0KZJ82WEPP.md): definition 0KZJ82WEPP
- [Diffeomorphism](https://tamnd.github.io/math2/node/0KZZ70ZHXP.md): concept 0KZZ70ZHXP
- [Antisymmetric relation](https://tamnd.github.io/math2/node/0M0H1ZH8TT.md): concept 0M0H1ZH8TT
- [Derivative of an inverse function](https://tamnd.github.io/math2/node/0M0RZ3HSB0.md): statement 0M0RZ3HSB0
- [Block diagonal matrix](https://tamnd.github.io/math2/node/0M0SNKZ482.md): definition 0M0SNKZ482
- [Linearity of differentiation](https://tamnd.github.io/math2/node/0M0TKFRX2A.md): statement 0M0TKFRX2A
- [Intercept theorem](https://tamnd.github.io/math2/node/0M1GFS6PVA.md): statement 0M1GFS6PVA
- [Convex function on R^n](https://tamnd.github.io/math2/node/0M1GJSWZDA.md): definition 0M1GJSWZDA
- [Bernoulli's inequality](https://tamnd.github.io/math2/node/0M1MX28VXK.md): statement 0M1MX28VXK
- [Fermat's little theorem](https://tamnd.github.io/math2/node/0M25PRN3WQ.md): statement 0M25PRN3WQ
- [Conjunction](https://tamnd.github.io/math2/node/0M28ASWG8T.md): definition 0M28ASWG8T
- [Curl of a gradient is zero](https://tamnd.github.io/math2/node/0M2JRW1J0N.md): statement 0M2JRW1J0N
- [Cayley's formula](https://tamnd.github.io/math2/node/0M2SSGSYPE.md): statement 0M2SSGSYPE
- [The number of Sylow subgroups divides the index](https://tamnd.github.io/math2/node/0M3PYJ3TSE.md): statement 0M3PYJ3TSE
- [Analytic continuation](https://tamnd.github.io/math2/node/0M3YQJ81NB.md): concept 0M3YQJ81NB
- [Four colour theorem](https://tamnd.github.io/math2/node/0M406JC8J1.md): statement 0M406JC8J1
- [Binary operation](https://tamnd.github.io/math2/node/0M418FE7BB.md): concept 0M418FE7BB
- [Uniform continuity](https://tamnd.github.io/math2/node/0M4JJT780X.md): concept 0M4JJT780X
- [Every linearly independent set extends to a basis](https://tamnd.github.io/math2/node/0M4SXE5WW3.md): statement 0M4SXE5WW3
- [Classification of the Platonic solids](https://tamnd.github.io/math2/node/0M4TCS0RRG.md): statement 0M4TCS0RRG
- [Line integral of a scalar field](https://tamnd.github.io/math2/node/0M50E4WYWJ.md): definition 0M50E4WYWJ
- [Existence of inverses modulo n](https://tamnd.github.io/math2/node/0M67FWZWPQ.md): statement 0M67FWZWPQ
- [The only ideals of a field are zero and the whole field](https://tamnd.github.io/math2/node/0M6G75E95E.md): statement 0M6G75E95E
- [Universal property of the product topology](https://tamnd.github.io/math2/node/0M6W4XNST6.md): statement 0M6W4XNST6
- [Elementary matrix](https://tamnd.github.io/math2/node/0M79HRJCRR.md): definition 0M79HRJCRR
- [Pythagorean triple](https://tamnd.github.io/math2/node/0M7ARED5Q7.md): concept 0M7ARED5Q7
- [Euler's criterion](https://tamnd.github.io/math2/node/0M7DEDKSYB.md): statement 0M7DEDKSYB
- [Complementary subspaces](https://tamnd.github.io/math2/node/0M83V17KHX.md): concept 0M83V17KHX
- [Dual space](https://tamnd.github.io/math2/node/0M856Q1AMZ.md): definition 0M856Q1AMZ
- [Equivalent norms](https://tamnd.github.io/math2/node/0M85DXB5RM.md): concept 0M85DXB5RM
- [Adjoining a root of a polynomial](https://tamnd.github.io/math2/node/0M893X7YY7.md): definition 0M893X7YY7
- [Vieta's formulas](https://tamnd.github.io/math2/node/0M8EWQJ1YV.md): statement 0M8EWQJ1YV
- [Nilpotent endomorphism](https://tamnd.github.io/math2/node/0M8MHYERTY.md): definition 0M8MHYERTY
- [Approximation property of the supremum](https://tamnd.github.io/math2/node/0M8TGGP9JP.md): statement 0M8TGGP9JP
- [Coefficient of a power series](https://tamnd.github.io/math2/node/0M98XYVX6S.md): definition 0M98XYVX6S
- [Incentre](https://tamnd.github.io/math2/node/0M9MX8YW0A.md): concept 0M9MX8YW0A
- [Step function](https://tamnd.github.io/math2/node/0M9XEDF1HJ.md): definition 0M9XEDF1HJ
- [Field generated by adjoining elements](https://tamnd.github.io/math2/node/0MA5B744K3.md): definition 0MA5B744K3
- [Schwarz lemma](https://tamnd.github.io/math2/node/0MB48XDBY3.md): statement 0MB48XDBY3
- [Algebraic number](https://tamnd.github.io/math2/node/0MB4TGMXG6.md): definition 0MB4TGMXG6
- [Self-adjoint operator](https://tamnd.github.io/math2/node/0MB8HQW495.md): definition 0MB8HQW495
- [Sum of ideals](https://tamnd.github.io/math2/node/0MBJVDSK1R.md): concept 0MBJVDSK1R
- [Generators of a group](https://tamnd.github.io/math2/node/0MBV2VHMXA.md): concept 0MBV2VHMXA
- [Monotonicity of the integral](https://tamnd.github.io/math2/node/0MBWQKM8QR.md): statement 0MBWQKM8QR
- [A finite abelian group is cyclic iff its exponent equals its order](https://tamnd.github.io/math2/node/0MC0AQZSHE.md): statement 0MC0AQZSHE
- [Orthogonal projection is the closest point](https://tamnd.github.io/math2/node/0MCDDSCB9Z.md): statement 0MCDDSCB9Z
- [Epsilon characterisation of Cauchy sequences](https://tamnd.github.io/math2/node/0MCK8K2D7Z.md): statement 0MCK8K2D7Z
- [Minimal element](https://tamnd.github.io/math2/node/0MCR8ZAXQD.md): definition 0MCR8ZAXQD
- [Algebraic closure](https://tamnd.github.io/math2/node/0MCXX29HWY.md): definition 0MCXX29HWY
- [Complement graph](https://tamnd.github.io/math2/node/0MCYJ72VGA.md): concept 0MCYJ72VGA
- [Coprime ideals](https://tamnd.github.io/math2/node/0MDJP82TEH.md): definition 0MDJP82TEH
- [Exterior algebra](https://tamnd.github.io/math2/node/0MDMZ965VE.md): concept 0MDMZ965VE
- [Symmetric difference](https://tamnd.github.io/math2/node/0ME21QN125.md): concept 0ME21QN125
- [Division algorithm for polynomials](https://tamnd.github.io/math2/node/0MES3KS0ZC.md): statement 0MES3KS0ZC
- [Norm induced by an inner product](https://tamnd.github.io/math2/node/0MFJJV97RJ.md): concept 0MFJJV97RJ
- [Derivative of a polynomial](https://tamnd.github.io/math2/node/0MH1WK89PP.md): statement 0MH1WK89PP
- [Wronskian test for linear independence of solutions](https://tamnd.github.io/math2/node/0MHE4B1QCQ.md): statement 0MHE4B1QCQ
- [Every sequence has a monotone subsequence](https://tamnd.github.io/math2/node/0MHGJRZMTM.md): statement 0MHGJRZMTM
- [Matching](https://tamnd.github.io/math2/node/0MJ04A50QW.md): concept 0MJ04A50QW
- [Tree](https://tamnd.github.io/math2/node/0MJ0JS72DZ.md): definition 0MJ0JS72DZ
- [Upper triangular matrix](https://tamnd.github.io/math2/node/0MKABWDRNC.md): definition 0MKABWDRNC
- [Change of basis matrix](https://tamnd.github.io/math2/node/0MKRHZB3WS.md): concept 0MKRHZB3WS
- [Product of vector spaces](https://tamnd.github.io/math2/node/0MM085RX00.md): definition 0MM085RX00
- [One-point compactification](https://tamnd.github.io/math2/node/0MM45BT6HC.md): definition 0MM45BT6HC
- [A ring homomorphism is injective iff its kernel is zero](https://tamnd.github.io/math2/node/0MMCB8RXGA.md): statement 0MMCB8RXGA
- [Continuum hypothesis](https://tamnd.github.io/math2/node/0MN2MER9WN.md): statement 0MN2MER9WN
- [Derivative as a little-o approximation](https://tamnd.github.io/math2/node/0MN882VJ4P.md): statement 0MN882VJ4P
- [Algebraic number](https://tamnd.github.io/math2/node/0MNF0BYFQC.md): concept 0MNF0BYFQC
- [Prime element](https://tamnd.github.io/math2/node/0MNHXBSX3M.md): definition 0MNHXBSX3M
- [Geometric sequence](https://tamnd.github.io/math2/node/0MNM54WB99.md): definition 0MNM54WB99
- [Product of rings](https://tamnd.github.io/math2/node/0MNZCN4BQK.md): definition 0MNZCN4BQK
- [Line](https://tamnd.github.io/math2/node/0MP9BCF6DP.md): concept 0MP9BCF6DP
- [Diagonalisable iff the minimal polynomial has distinct roots](https://tamnd.github.io/math2/node/0MPAJRB23T.md): statement 0MPAJRB23T
- [Centre of mass](https://tamnd.github.io/math2/node/0MPZGAZ2GK.md): definition 0MPZGAZ2GK
- [Fundamental matrix](https://tamnd.github.io/math2/node/0MQ24RBG93.md): definition 0MQ24RBG93
- [Compact Hausdorff spaces are normal](https://tamnd.github.io/math2/node/0MQQQ5NA2S.md): statement 0MQQQ5NA2S
- [Change of basis formula](https://tamnd.github.io/math2/node/0MQXVZEBMT.md): statement 0MQXVZEBMT
- [Normal matrix](https://tamnd.github.io/math2/node/0MR0BYF0K2.md): concept 0MR0BYF0K2
- [Series of functions](https://tamnd.github.io/math2/node/0MRE6XRE5V.md): definition 0MRE6XRE5V
- [A quotient is abelian iff the subgroup contains the commutator subgroup](https://tamnd.github.io/math2/node/0MS8QGA6S5.md): statement 0MS8QGA6S5
- [Test for exactness](https://tamnd.github.io/math2/node/0MSCQNFTM8.md): statement 0MSCQNFTM8
- [Existential quantifier](https://tamnd.github.io/math2/node/0MSRQHCNN9.md): concept 0MSRQHCNN9
- [Order on the reals](https://tamnd.github.io/math2/node/0MSSD1VCW9.md): concept 0MSSD1VCW9
- [Alternating multilinear map](https://tamnd.github.io/math2/node/0MSWYWHYF5.md): definition 0MSWYWHYF5
- [Contraction mapping](https://tamnd.github.io/math2/node/0MT2S0ZX4Y.md): definition 0MT2S0ZX4Y
- [First isomorphism theorem for linear maps](https://tamnd.github.io/math2/node/0MTBGVZ4XJ.md): statement 0MTBGVZ4XJ
- [One-form](https://tamnd.github.io/math2/node/0MTENFBW3Q.md): definition 0MTENFBW3Q
- [Poisson point process](https://tamnd.github.io/math2/node/0MTNR28K60.md): concept 0MTNR28K60
- [e](https://tamnd.github.io/math2/node/0MVCFMW7S4.md): object 0MVCFMW7S4
- [Sequential continuity](https://tamnd.github.io/math2/node/0MVDQEAB60.md): definition 0MVDQEAB60
- [Noetherian ring](https://tamnd.github.io/math2/node/0MVH4Y1CRZ.md): definition 0MVH4Y1CRZ
- [Exponentials dominate powers in sequences](https://tamnd.github.io/math2/node/0MVTCFW42K.md): statement 0MVTCFW42K
- [Equicontinuity](https://tamnd.github.io/math2/node/0MW2WXTHMC.md): definition 0MW2WXTHMC
- [Natural numbers](https://tamnd.github.io/math2/node/0MWKJ44V5A.md): object 0MWKJ44V5A
- [Divergence](https://tamnd.github.io/math2/node/0MWV6ZTK6S.md): concept 0MWV6ZTK6S
- [Hyperbolic tangent](https://tamnd.github.io/math2/node/0MWYQTZ07Z.md): definition 0MWYQTZ07Z
- [A series of nonnegative terms converges iff its partial sums are bounded](https://tamnd.github.io/math2/node/0MX1HZBBXJ.md): statement 0MX1HZBBXJ
- [Hausdorff space](https://tamnd.github.io/math2/node/0MXB4CKHFW.md): definition 0MXB4CKHFW
- [Coupon collector's problem](https://tamnd.github.io/math2/node/0MXCP7MXWJ.md): problem 0MXCP7MXWJ
- [Mantel's theorem](https://tamnd.github.io/math2/node/0MXEHSCYCS.md): statement 0MXEHSCYCS
- [Deficient number](https://tamnd.github.io/math2/node/0MXFJ0J00R.md): concept 0MXFJ0J00R
- [Perfect number](https://tamnd.github.io/math2/node/0MXGYCH4SY.md): concept 0MXGYCH4SY
- [Cumulative distribution function](https://tamnd.github.io/math2/node/0MXMS3AE13.md): definition 0MXMS3AE13
- [Open connected subsets of R^n are path-connected](https://tamnd.github.io/math2/node/0MXN7AFBPE.md): statement 0MXN7AFBPE
- [Marginal distribution](https://tamnd.github.io/math2/node/0MXT6PRZZ6.md): concept 0MXT6PRZZ6
- [Characteristic function](https://tamnd.github.io/math2/node/0MYDNRT7PP.md): definition 0MYDNRT7PP
- [Compact subsets of a Hausdorff space are closed](https://tamnd.github.io/math2/node/0MYFJVMN7M.md): statement 0MYFJVMN7M
- [Ideal](https://tamnd.github.io/math2/node/0MYSA6MYNA.md): definition 0MYSA6MYNA
- [Power series](https://tamnd.github.io/math2/node/0MZB06ST2B.md): concept 0MZB06ST2B
- [Separable extension](https://tamnd.github.io/math2/node/0N019F6RE9.md): definition 0N019F6RE9
- [Subgroup](https://tamnd.github.io/math2/node/0N0PXCTYA7.md): definition 0N0PXCTYA7
- [Well-ordering theorem](https://tamnd.github.io/math2/node/0N109NG5ZX.md): statement 0N109NG5ZX
- [Irrationality of square roots of non-squares](https://tamnd.github.io/math2/node/0N10YRNNWV.md): statement 0N10YRNNWV
- [Riemann integrable function](https://tamnd.github.io/math2/node/0N1JQXZXHC.md): definition 0N1JQXZXHC
- [Matrix](https://tamnd.github.io/math2/node/0N1MY1T0W9.md): definition 0N1MY1T0W9
- [Completeness via Cauchy sequences](https://tamnd.github.io/math2/node/0N1ZWFJDN5.md): statement 0N1ZWFJDN5
- [Radius of convergence](https://tamnd.github.io/math2/node/0N288P2TB5.md): concept 0N288P2TB5
- [Symmetric matrix](https://tamnd.github.io/math2/node/0N28QN4CYA.md): definition 0N28QN4CYA
- [A homomorphism is injective iff its kernel is trivial](https://tamnd.github.io/math2/node/0N2P0XRTDF.md): statement 0N2P0XRTDF
- [Cumulant generating function](https://tamnd.github.io/math2/node/0N2SP6K9ZC.md): concept 0N2SP6K9ZC
- [The rationals form a field](https://tamnd.github.io/math2/node/0N2WPCC00T.md): statement 0N2WPCC00T
- [Limits preserve non-strict inequalities](https://tamnd.github.io/math2/node/0N32V0E3XJ.md): statement 0N32V0E3XJ
- [Uniqueness quantifier](https://tamnd.github.io/math2/node/0N3ADS4CHH.md): definition 0N3ADS4CHH
- [Composite number](https://tamnd.github.io/math2/node/0N3ADXHNTQ.md): definition 0N3ADXHNTQ
- [Liouville's theorem on Diophantine approximation](https://tamnd.github.io/math2/node/0N3WFZEFPM.md): statement 0N3WFZEFPM
- [Hilbert's hotel](https://tamnd.github.io/math2/node/0N3Y8G340X.md): example 0N3Y8G340X
- [Quotient map](https://tamnd.github.io/math2/node/0N4F5FMQZW.md): concept 0N4F5FMQZW
- [Upper triangular matrix](https://tamnd.github.io/math2/node/0N4XY7Z3NN.md): concept 0N4XY7Z3NN
- [Pairwise independence](https://tamnd.github.io/math2/node/0N57D1XY6T.md): definition 0N57D1XY6T
- [Boolean ring](https://tamnd.github.io/math2/node/0N5FCV2YV6.md): definition 0N5FCV2YV6
- [Square root of a positive semidefinite matrix](https://tamnd.github.io/math2/node/0N5GP3YT8T.md): definition 0N5GP3YT8T
- [Falling factorial](https://tamnd.github.io/math2/node/0N5M924N3P.md): definition 0N5M924N3P
- [Cross product](https://tamnd.github.io/math2/node/0N5N15HWZ7.md): concept 0N5N15HWZ7
- [Wronskian](https://tamnd.github.io/math2/node/0N5NF1RGBP.md): definition 0N5NF1RGBP
- [Idempotent element](https://tamnd.github.io/math2/node/0N5NJZKKN3.md): concept 0N5NJZKKN3
- [Classification of conics by the discriminant](https://tamnd.github.io/math2/node/0N5NR32NQG.md): statement 0N5NR32NQG
- [Degree of a simple extension is the degree of the minimal polynomial](https://tamnd.github.io/math2/node/0N5QW9K7MH.md): statement 0N5QW9K7MH
- [Fermat primality test](https://tamnd.github.io/math2/node/0N5ZSZMYH6.md): concept 0N5ZSZMYH6
- [Vector field](https://tamnd.github.io/math2/node/0N67NBKQ5T.md): definition 0N67NBKQ5T
- [Simple extension](https://tamnd.github.io/math2/node/0N681SSTHC.md): concept 0N681SSTHC
- [Eigenvector](https://tamnd.github.io/math2/node/0N6960W0BV.md): concept 0N6960W0BV
- [Monoid](https://tamnd.github.io/math2/node/0N6R902FG1.md): concept 0N6R902FG1
- [Classification of equilibria of planar linear systems](https://tamnd.github.io/math2/node/0N7ATVF2X8.md): concept 0N7ATVF2X8
- [Kronecker product](https://tamnd.github.io/math2/node/0N7ENQRCXN.md): concept 0N7ENQRCXN
- [Equilibrium point](https://tamnd.github.io/math2/node/0N8NGKDZDZ.md): concept 0N8NGKDZDZ
- [Mean value theorem for integrals](https://tamnd.github.io/math2/node/0N91N98EG5.md): statement 0N91N98EG5
- [Fermat prime](https://tamnd.github.io/math2/node/0N9E4G2P97.md): definition 0N9E4G2P97
- [Nilpotent element](https://tamnd.github.io/math2/node/0N9R5NN6TW.md): definition 0N9R5NN6TW
- [Symmetric relation](https://tamnd.github.io/math2/node/0N9SA25Z3V.md): definition 0N9SA25Z3V
- [Basis of a topology](https://tamnd.github.io/math2/node/0NA0732YAG.md): concept 0NA0732YAG
- [Discrete metric](https://tamnd.github.io/math2/node/0NA4FK7B8S.md): example 0NA4FK7B8S
- [Cauchy distribution](https://tamnd.github.io/math2/node/0NA6Q82YR7.md): definition 0NA6Q82YR7
- [Ore's theorem](https://tamnd.github.io/math2/node/0NACHJYPKW.md): statement 0NACHJYPKW
- [Holomorphic function](https://tamnd.github.io/math2/node/0NAHXR1FRH.md): concept 0NAHXR1FRH
- [Apollonius's theorem](https://tamnd.github.io/math2/node/0NAKHQ9XYY.md): statement 0NAKHQ9XYY
- [Harmonic number](https://tamnd.github.io/math2/node/0NANDX7V02.md): concept 0NANDX7V02
- [Inner product](https://tamnd.github.io/math2/node/0NB0AWGJBW.md): concept 0NB0AWGJBW
- [exp(1/z) has an essential singularity at 0](https://tamnd.github.io/math2/node/0NB31CC4SN.md): example 0NB31CC4SN
- [Partial sum](https://tamnd.github.io/math2/node/0NB643K60Q.md): definition 0NB643K60Q
- [Asymptotic stability](https://tamnd.github.io/math2/node/0NB689782W.md): concept 0NB689782W
- [Composition series](https://tamnd.github.io/math2/node/0NB8VPB93V.md): definition 0NB8VPB93V
- [Divergent series](https://tamnd.github.io/math2/node/0NBRP6093V.md): definition 0NBRP6093V
- [The integral of a continuous function is an antiderivative](https://tamnd.github.io/math2/node/0NC2A3HWGT.md): statement 0NC2A3HWGT
- [Planar graph](https://tamnd.github.io/math2/node/0NC9R64TXP.md): concept 0NC9R64TXP
- [Rational numbers](https://tamnd.github.io/math2/node/0NCJ5WXN6Z.md): object 0NCJ5WXN6Z
- [Graph colouring](https://tamnd.github.io/math2/node/0NCPXH194Y.md): concept 0NCPXH194Y
- [Principal part](https://tamnd.github.io/math2/node/0NDV5R6Z6X.md): definition 0NDV5R6Z6X
- [Linear recurrence with constant coefficients](https://tamnd.github.io/math2/node/0NDVHBVBXB.md): concept 0NDVHBVBXB
- [Right coset](https://tamnd.github.io/math2/node/0NE1DK7H4T.md): concept 0NE1DK7H4T
- [Conditional convergence](https://tamnd.github.io/math2/node/0NE233WGD1.md): concept 0NE233WGD1
- [Implicit function theorem for a surjective derivative](https://tamnd.github.io/math2/node/0NE2W2XPQJ.md): statement 0NE2W2XPQJ
- [Self-adjoint operator](https://tamnd.github.io/math2/node/0NEGN5CRSC.md): concept 0NEGN5CRSC
- [Restriction of a function](https://tamnd.github.io/math2/node/0NF1KGDXX3.md): concept 0NF1KGDXX3
- [Curl](https://tamnd.github.io/math2/node/0NF6GQNXMY.md): concept 0NF6GQNXMY
- [Double negation law](https://tamnd.github.io/math2/node/0NFAGWH84K.md): statement 0NFAGWH84K
- [The minimal polynomial is irreducible](https://tamnd.github.io/math2/node/0NFEG48RDR.md): statement 0NFEG48RDR
- [Probability space](https://tamnd.github.io/math2/node/0NFPJVJJHX.md): concept 0NFPJVJJHX
- [Fundamental theorem of calculus](https://tamnd.github.io/math2/node/0NHGY0N0JB.md): statement 0NHGY0N0JB
- [Perfect matching](https://tamnd.github.io/math2/node/0NHK9MFFW9.md): concept 0NHK9MFFW9
- [Sample variance](https://tamnd.github.io/math2/node/0NHXGY2C5J.md): concept 0NHXGY2C5J
- [von Mangoldt function](https://tamnd.github.io/math2/node/0NK8JTE1NN.md): concept 0NK8JTE1NN
- [Double integral](https://tamnd.github.io/math2/node/0NKBMFC18R.md): definition 0NKBMFC18R
- [Series](https://tamnd.github.io/math2/node/0NM626FH6Q.md): concept 0NM626FH6Q
- [Bayes' theorem](https://tamnd.github.io/math2/node/0NM6E9VW04.md): statement 0NM6E9VW04
- [Variance as the mean of the square minus the square of the mean](https://tamnd.github.io/math2/node/0NMF95YK8W.md): statement 0NMF95YK8W
- [Implication](https://tamnd.github.io/math2/node/0NMG7W7Q11.md): concept 0NMG7W7Q11
- [Epsilon-net characterisation of total boundedness](https://tamnd.github.io/math2/node/0NMGJJ19HD.md): statement 0NMGJJ19HD
- [Transvection](https://tamnd.github.io/math2/node/0NMN00FNW1.md): definition 0NMN00FNW1
- [Birthday problem](https://tamnd.github.io/math2/node/0NMWG2PHJ0.md): example 0NMWG2PHJ0
- [Trace is the sum of the eigenvalues](https://tamnd.github.io/math2/node/0NMXDCNNKR.md): statement 0NMXDCNNKR
- [0.999... equals 1](https://tamnd.github.io/math2/node/0NN5JEMTVE.md): example 0NN5JEMTVE
- [Closed formula for Catalan numbers](https://tamnd.github.io/math2/node/0NNDQVJ54Q.md): statement 0NNDQVJ54Q
- [Associativity](https://tamnd.github.io/math2/node/0NNQ4KFEX7.md): definition 0NNQ4KFEX7
- [Schwarz's theorem for C^2 functions](https://tamnd.github.io/math2/node/0NNQY8WCW6.md): statement 0NNQY8WCW6
- [Reduced row echelon form](https://tamnd.github.io/math2/node/0NNR68BFRY.md): definition 0NNR68BFRY
- [Zero divisor](https://tamnd.github.io/math2/node/0NNRDW10RZ.md): concept 0NNRDW10RZ
- [Moment-generating function](https://tamnd.github.io/math2/node/0NNRNGKMFM.md): definition 0NNRNGKMFM
- [Eulerian trail](https://tamnd.github.io/math2/node/0NNYDD1AVR.md): definition 0NNYDD1AVR
- [Cantor pairing function](https://tamnd.github.io/math2/node/0NP5N5PYW5.md): example 0NP5N5PYW5
- [Euclidean distance](https://tamnd.github.io/math2/node/0NPPGHQNXQ.md): concept 0NPPGHQNXQ
- [Algebraic element](https://tamnd.github.io/math2/node/0NPW1DWECM.md): definition 0NPW1DWECM
- [Quotient topology](https://tamnd.github.io/math2/node/0NQ1H2X495.md): concept 0NQ1H2X495
- [Student's t-distribution](https://tamnd.github.io/math2/node/0NQ5798SB1.md): definition 0NQ5798SB1
- [Transcendental element](https://tamnd.github.io/math2/node/0NQMZ89X4A.md): concept 0NQMZ89X4A
- [Winding number](https://tamnd.github.io/math2/node/0NQSV2EWDR.md): concept 0NQSV2EWDR
- [Triangle inequality for integrals](https://tamnd.github.io/math2/node/0NR82KMYKD.md): statement 0NR82KMYKD
- [Petersen graph](https://tamnd.github.io/math2/node/0NR95CTFCZ.md): example 0NR95CTFCZ
- [Quaternion group](https://tamnd.github.io/math2/node/0NRJGW864D.md): object 0NRJGW864D
- [Composition of continuous functions is continuous](https://tamnd.github.io/math2/node/0NRQPQX3VA.md): statement 0NRQPQX3VA
- [Connected components are closed](https://tamnd.github.io/math2/node/0NRT0E2QT2.md): statement 0NRT0E2QT2
- [Linear map](https://tamnd.github.io/math2/node/0NRYHJPQG2.md): concept 0NRYHJPQG2
- [Finite extension](https://tamnd.github.io/math2/node/0NRYPM9Y27.md): definition 0NRYPM9Y27
- [Characteristic function of the normal distribution](https://tamnd.github.io/math2/node/0NS0TDA4E2.md): statement 0NS0TDA4E2
- [Derivative of a map into R^n by components](https://tamnd.github.io/math2/node/0NS0X75RS1.md): statement 0NS0X75RS1
- [Product of two subgroups](https://tamnd.github.io/math2/node/0NS1YTGM19.md): definition 0NS1YTGM19
- [Gradient](https://tamnd.github.io/math2/node/0NS2QYVC4Y.md): definition 0NS2QYVC4Y
- [Construction of the integers from the natural numbers](https://tamnd.github.io/math2/node/0NS58JP68E.md): concept 0NS58JP68E
- [Local minimum](https://tamnd.github.io/math2/node/0NS8BP96QX.md): definition 0NS8BP96QX
- [Meromorphic function](https://tamnd.github.io/math2/node/0NSD1TPHR0.md): definition 0NSD1TPHR0
- [Normal space](https://tamnd.github.io/math2/node/0NSR9V8TED.md): concept 0NSR9V8TED
- [Ordinary differential equation](https://tamnd.github.io/math2/node/0NT64JV96A.md): concept 0NT64JV96A
- [Direct proof](https://tamnd.github.io/math2/node/0NTF6DMVAP.md): concept 0NTF6DMVAP
- [Discrete Grönwall inequality](https://tamnd.github.io/math2/node/0NTM3J5MFE.md): statement 0NTM3J5MFE
- [Exterior angle theorem](https://tamnd.github.io/math2/node/0NTSE7EREA.md): statement 0NTSE7EREA
- [Multiplicative function](https://tamnd.github.io/math2/node/0NTWN57GM3.md): concept 0NTWN57GM3
- [Z/pZ is a field for p prime](https://tamnd.github.io/math2/node/0NV6E43P4A.md): statement 0NV6E43P4A
- [Open cover](https://tamnd.github.io/math2/node/0NVPNC12JY.md): definition 0NVPNC12JY
- [Multiplicativity of the Legendre symbol](https://tamnd.github.io/math2/node/0NW1JP7TBR.md): statement 0NW1JP7TBR
- [T1 space](https://tamnd.github.io/math2/node/0NW6CH38YR.md): concept 0NW6CH38YR
- [Second Borel–Cantelli lemma](https://tamnd.github.io/math2/node/0NWXRAMEE7.md): statement 0NWXRAMEE7
- [Weierstrass M-test](https://tamnd.github.io/math2/node/0NX5NWAFC9.md): statement 0NX5NWAFC9
- [Sum of independent normal random variables](https://tamnd.github.io/math2/node/0NXGPJWNFN.md): statement 0NXGPJWNFN
- [Euclidean isometry](https://tamnd.github.io/math2/node/0NXJM01QX1.md): definition 0NXJM01QX1
- [Adjoining a root of a polynomial](https://tamnd.github.io/math2/node/0NXWQK2Y56.md): concept 0NXWQK2Y56
- [Theorem on friends and strangers](https://tamnd.github.io/math2/node/0NYM0ZGAK9.md): statement 0NYM0ZGAK9
- [Closed map](https://tamnd.github.io/math2/node/0NYZ2M45G9.md): definition 0NYZ2M45G9
- [Module over a ring](https://tamnd.github.io/math2/node/0NZ8JA25WK.md): concept 0NZ8JA25WK
- [Adjacency matrix](https://tamnd.github.io/math2/node/0NZ8MDBN6Q.md): definition 0NZ8MDBN6Q
- [Angle-angle similarity](https://tamnd.github.io/math2/node/0NZVJYZVTZ.md): statement 0NZVJYZVTZ
- [Leading coefficient](https://tamnd.github.io/math2/node/0P05RJC5XA.md): definition 0P05RJC5XA
- [Cesàro mean](https://tamnd.github.io/math2/node/0P0DJHBGBB.md): definition 0P0DJHBGBB
- [Independence of two events](https://tamnd.github.io/math2/node/0P0MZ4X682.md): definition 0P0MZ4X682
- [Classification of isometries of the plane](https://tamnd.github.io/math2/node/0P0XDY9FZN.md): statement 0P0XDY9FZN
- [Subgraph](https://tamnd.github.io/math2/node/0P11X5HKAX.md): definition 0P11X5HKAX
- [Hermitian matrix](https://tamnd.github.io/math2/node/0P17NGQQMQ.md): concept 0P17NGQQMQ
- [Removable singularity](https://tamnd.github.io/math2/node/0P18JC72RE.md): definition 0P18JC72RE
- [Equivalent matrices](https://tamnd.github.io/math2/node/0P21AVFFSP.md): definition 0P21AVFFSP
- [Hyperbola](https://tamnd.github.io/math2/node/0P234N3M28.md): definition 0P234N3M28
- [Matrix of a bilinear form](https://tamnd.github.io/math2/node/0P24YW565F.md): definition 0P24YW565F
- [Continuous function](https://tamnd.github.io/math2/node/0P27W9RW0G.md): definition 0P27W9RW0G
- [Second-order Taylor expansion](https://tamnd.github.io/math2/node/0P30YYZN85.md): statement 0P30YYZN85
- [Negation of a universal statement](https://tamnd.github.io/math2/node/0P3A06QNDP.md): statement 0P3A06QNDP
- [Solvable group](https://tamnd.github.io/math2/node/0P3KZK0FTV.md): definition 0P3KZK0FTV
- [Divergent sequence](https://tamnd.github.io/math2/node/0P3M4ZY9X8.md): concept 0P3M4ZY9X8
- [Ordered field](https://tamnd.github.io/math2/node/0P3VCBDEE2.md): concept 0P3VCBDEE2
- [Glide reflection](https://tamnd.github.io/math2/node/0P42JB4E3M.md): definition 0P42JB4E3M
- [Standard deviation](https://tamnd.github.io/math2/node/0P4FTHHMYS.md): concept 0P4FTHHMYS
- [Degree of a polynomial](https://tamnd.github.io/math2/node/0P4JRMSWNH.md): concept 0P4JRMSWNH
- [Sine is 2π-periodic](https://tamnd.github.io/math2/node/0P4NQD746K.md): statement 0P4NQD746K
- [Iterated integral](https://tamnd.github.io/math2/node/0P4WX7Y502.md): definition 0P4WX7Y502
- [Non-commutativity of matrix multiplication](https://tamnd.github.io/math2/node/0P506SPZP2.md): counterexample 0P506SPZP2
- [A function with bounded derivative is Lipschitz](https://tamnd.github.io/math2/node/0P64VE79X3.md): statement 0P64VE79X3
- [Picard–Lindelöf theorem](https://tamnd.github.io/math2/node/0P6KRD5B7A.md): statement 0P6KRD5B7A
- [Compactness via open covers](https://tamnd.github.io/math2/node/0P6PMJB4MA.md): statement 0P6PMJB4MA
- [Simple extension](https://tamnd.github.io/math2/node/0P6SBR7YM2.md): definition 0P6SBR7YM2
- [Bilinear form](https://tamnd.github.io/math2/node/0P799D6AMN.md): definition 0P799D6AMN
- [Coefficient matrix](https://tamnd.github.io/math2/node/0P7A2MJD24.md): definition 0P7A2MJD24
- [Characteristic of a ring](https://tamnd.github.io/math2/node/0P7CH1CND5.md): definition 0P7CH1CND5
- [Composition of an integer](https://tamnd.github.io/math2/node/0P7FZHXZ5S.md): concept 0P7FZHXZ5S
- [Faithful action](https://tamnd.github.io/math2/node/0P7G9NGG2G.md): concept 0P7G9NGG2G
- [Linearity of expectation](https://tamnd.github.io/math2/node/0P7PTC8DDE.md): statement 0P7PTC8DDE
- [Vandermonde determinant](https://tamnd.github.io/math2/node/0P84B554AS.md): statement 0P84B554AS
- [Limit superior](https://tamnd.github.io/math2/node/0P8BA7T7S5.md): definition 0P8BA7T7S5
- [Action by conjugation](https://tamnd.github.io/math2/node/0P8CY9V437.md): example 0P8CY9V437
- [Bounded set](https://tamnd.github.io/math2/node/0P953MG05E.md): definition 0P953MG05E
- [Ordinary differential equation](https://tamnd.github.io/math2/node/0P97V20RW7.md): definition 0P97V20RW7
- [The exponential function is positive](https://tamnd.github.io/math2/node/0P9BNBW2W3.md): statement 0P9BNBW2W3
- [Metric spaces are Hausdorff](https://tamnd.github.io/math2/node/0P9DCDES73.md): statement 0P9DCDES73
- [Existence of a complementary subspace](https://tamnd.github.io/math2/node/0P9T8WHNMH.md): statement 0P9T8WHNMH
- [Continuity is equivalent to sequential continuity](https://tamnd.github.io/math2/node/0PAQM3H57X.md): statement 0PAQM3H57X
- [Sup norm on bounded continuous functions](https://tamnd.github.io/math2/node/0PARCKE8WP.md): example 0PARCKE8WP
- [Law of sines](https://tamnd.github.io/math2/node/0PASQ34GHG.md): statement 0PASQ34GHG
- [Linear differential equation of order n](https://tamnd.github.io/math2/node/0PB7K8M7FG.md): definition 0PB7K8M7FG
- [Jordan block](https://tamnd.github.io/math2/node/0PBFS92AJP.md): definition 0PBFS92AJP
- [Annihilating polynomial](https://tamnd.github.io/math2/node/0PBN8VEKY8.md): concept 0PBN8VEKY8
- [Monty Hall problem](https://tamnd.github.io/math2/node/0PBWWQD9KY.md): problem 0PBWWQD9KY
- [Skew-symmetric matrix](https://tamnd.github.io/math2/node/0PBY4E9GQF.md): concept 0PBY4E9GQF
- [Order of the symmetric group](https://tamnd.github.io/math2/node/0PCKBBMEXC.md): statement 0PCKBBMEXC
- [Multinomial coefficient](https://tamnd.github.io/math2/node/0PCV9EDJM2.md): concept 0PCV9EDJM2
- [Fermat's theorem on stationary points](https://tamnd.github.io/math2/node/0PCVBRV9VJ.md): statement 0PCVBRV9VJ
- [Circuit](https://tamnd.github.io/math2/node/0PD57PSHNR.md): definition 0PD57PSHNR
- [Converse of Lagrange's theorem fails for the alternating group A4](https://tamnd.github.io/math2/node/0PDBVPP233.md): counterexample 0PDBVPP233
- [Matrix–vector product](https://tamnd.github.io/math2/node/0PDVQEP0ZC.md): concept 0PDVQEP0ZC
- [Transitive closure](https://tamnd.github.io/math2/node/0PEJH7AC74.md): definition 0PEJH7AC74
- [Algebraically closed field](https://tamnd.github.io/math2/node/0PF6TM8TWT.md): concept 0PF6TM8TWT
- [Restriction of an endomorphism to an invariant subspace](https://tamnd.github.io/math2/node/0PFPWKBDTJ.md): definition 0PFPWKBDTJ
- [Divisibility](https://tamnd.github.io/math2/node/0PG2H7WKGG.md): concept 0PG2H7WKGG
- [Moment](https://tamnd.github.io/math2/node/0PH0VHTQS0.md): definition 0PH0VHTQS0
- [Generating function](https://tamnd.github.io/math2/node/0PHAC833JF.md): concept 0PHAC833JF
- [First supplement to quadratic reciprocity](https://tamnd.github.io/math2/node/0PHBS78HKT.md): statement 0PHBS78HKT
- [Bernoulli trial](https://tamnd.github.io/math2/node/0PHC9FE2V5.md): concept 0PHC9FE2V5
- [Null space](https://tamnd.github.io/math2/node/0PHDYNR24D.md): concept 0PHDYNR24D
- [Residue](https://tamnd.github.io/math2/node/0PHN07KVQT.md): concept 0PHN07KVQT
- [Heine–Borel theorem in finite-dimensional normed spaces](https://tamnd.github.io/math2/node/0PHT6NBYPB.md): statement 0PHT6NBYPB
- [Composition factors](https://tamnd.github.io/math2/node/0PHTC93M58.md): concept 0PHTC93M58
- [Multi-index notation](https://tamnd.github.io/math2/node/0PJXBR6QA3.md): concept 0PJXBR6QA3
- [Alternating series test](https://tamnd.github.io/math2/node/0PK8QH1W1V.md): statement 0PK8QH1W1V
- [Conservative vector field](https://tamnd.github.io/math2/node/0PKG97PX9Y.md): concept 0PKG97PX9Y
- [Limit of a function](https://tamnd.github.io/math2/node/0PKRYB8ATM.md): concept 0PKRYB8ATM
- [Projections are continuous](https://tamnd.github.io/math2/node/0PM88Z9W4M.md): statement 0PM88Z9W4M
- [Klein four-group](https://tamnd.github.io/math2/node/0PMRQJVTME.md): object 0PMRQJVTME
- [Order of an inverse](https://tamnd.github.io/math2/node/0PMT9YJZVW.md): statement 0PMT9YJZVW
- [The continued fraction of a rational number terminates](https://tamnd.github.io/math2/node/0PMY0YQYXX.md): statement 0PMY0YQYXX
- [The Gaussian integers are a Euclidean domain](https://tamnd.github.io/math2/node/0PN13BB7BT.md): statement 0PN13BB7BT
- [Argument principle](https://tamnd.github.io/math2/node/0PN2BXRS8G.md): statement 0PN2BXRS8G
- [Farey sequence](https://tamnd.github.io/math2/node/0PN4QJE969.md): definition 0PN4QJE969
- [The constructible numbers form a field closed under square roots](https://tamnd.github.io/math2/node/0PN9WMRWT7.md): statement 0PN9WMRWT7
- [Jordan's lemma](https://tamnd.github.io/math2/node/0PNPG2PP9H.md): statement 0PNPG2PP9H
- [Random variable](https://tamnd.github.io/math2/node/0PNRKMRC55.md): concept 0PNRKMRC55
- [Interior is open](https://tamnd.github.io/math2/node/0PNWCM559P.md): statement 0PNWCM559P
- [Beta distribution](https://tamnd.github.io/math2/node/0PP5ZAJPP7.md): definition 0PP5ZAJPP7
- [Determinant is multilinear in the rows](https://tamnd.github.io/math2/node/0PPG793642.md): statement 0PPG793642
- [Second isomorphism theorem](https://tamnd.github.io/math2/node/0PPP2J2DJV.md): statement 0PPP2J2DJV
- [Null space](https://tamnd.github.io/math2/node/0PQ2GGY1QV.md): definition 0PQ2GGY1QV
- [Probability of the complement](https://tamnd.github.io/math2/node/0PQEBTYDZM.md): statement 0PQEBTYDZM
- [Alternating sum of binomial coefficients](https://tamnd.github.io/math2/node/0PQQGFAWSH.md): statement 0PQQGFAWSH
- [Principal axis theorem](https://tamnd.github.io/math2/node/0PR5P9Z4KM.md): statement 0PR5P9Z4KM
- [Dirichlet convolution of multiplicative functions is multiplicative](https://tamnd.github.io/math2/node/0PR88XHDMB.md): statement 0PR88XHDMB
- [Angle between two vectors](https://tamnd.github.io/math2/node/0PRAS55318.md): definition 0PRAS55318
- [Identity function](https://tamnd.github.io/math2/node/0PSKT5W817.md): concept 0PSKT5W817
- [Moore–Penrose inverse](https://tamnd.github.io/math2/node/0PSRCMCM5X.md): definition 0PSRCMCM5X
- [Product of two subgroups](https://tamnd.github.io/math2/node/0PT52XQZD7.md): concept 0PT52XQZD7
- [Algebraic extension](https://tamnd.github.io/math2/node/0PTBM7K0R2.md): concept 0PTBM7K0R2
- [Endomorphism](https://tamnd.github.io/math2/node/0PTF1PTJK0.md): definition 0PTF1PTJK0
- [Linear congruence](https://tamnd.github.io/math2/node/0PTVNH4GPS.md): concept 0PTVNH4GPS
- [Maximal ideal](https://tamnd.github.io/math2/node/0PV74TR5FP.md): concept 0PV74TR5FP
- [Expectation with respect to the binomial distribution](https://tamnd.github.io/math2/node/0PVCTCKD3C.md): statement 0PVCTCKD3C
- [Injective function](https://tamnd.github.io/math2/node/0PVG1CRAJN.md): definition 0PVG1CRAJN
- [The exponential series has infinite radius of convergence](https://tamnd.github.io/math2/node/0PVJRACB9H.md): statement 0PVJRACB9H
- [Infinite set](https://tamnd.github.io/math2/node/0PWE3BCHVJ.md): concept 0PWE3BCHVJ
- [Cauchy sequence](https://tamnd.github.io/math2/node/0PWX18FM4G.md): concept 0PWX18FM4G
- [Orthogonal complement of the orthogonal complement](https://tamnd.github.io/math2/node/0PXN9G4MX1.md): statement 0PXN9G4MX1
- [Commutator of two elements](https://tamnd.github.io/math2/node/0PXW0BN7DW.md): concept 0PXW0BN7DW
- [Liouville number](https://tamnd.github.io/math2/node/0PYEZM6CN9.md): concept 0PYEZM6CN9
- [Composition of relations](https://tamnd.github.io/math2/node/0PYSR20HGF.md): concept 0PYSR20HGF
- [Affine subspace](https://tamnd.github.io/math2/node/0PYW26VNT6.md): concept 0PYW26VNT6
- [Closed ball](https://tamnd.github.io/math2/node/0Q0H13G8ZB.md): concept 0Q0H13G8ZB
- [Centraliser](https://tamnd.github.io/math2/node/0Q0Z1BEXTF.md): definition 0Q0Z1BEXTF
- [Interval](https://tamnd.github.io/math2/node/0Q1D85B8G6.md): concept 0Q1D85B8G6
- [Exact equation](https://tamnd.github.io/math2/node/0Q1FKQKTKJ.md): concept 0Q1FKQKTKJ
- [Singular solution](https://tamnd.github.io/math2/node/0Q2MDD13G4.md): concept 0Q2MDD13G4
- [Internal direct product](https://tamnd.github.io/math2/node/0Q2MVQ7JMT.md): definition 0Q2MVQ7JMT
- [Cumulant generating function](https://tamnd.github.io/math2/node/0Q2QC81VTW.md): definition 0Q2QC81VTW
- [Stolz–Cesàro theorem](https://tamnd.github.io/math2/node/0Q3AKZV74D.md): statement 0Q3AKZV74D
- [Image of a set](https://tamnd.github.io/math2/node/0Q3F20XB0A.md): definition 0Q3F20XB0A
- [Integral of a power](https://tamnd.github.io/math2/node/0Q3HYFT9RQ.md): statement 0Q3HYFT9RQ
- [Linear equation](https://tamnd.github.io/math2/node/0Q3ZT8J05R.md): concept 0Q3ZT8J05R
- [Domain](https://tamnd.github.io/math2/node/0Q4CBD3HX0.md): concept 0Q4CBD3HX0
- [Arc length parametrisation](https://tamnd.github.io/math2/node/0Q4CJ5NAT1.md): concept 0Q4CJ5NAT1
- [Uniform continuity](https://tamnd.github.io/math2/node/0Q4DR37DVE.md): definition 0Q4DR37DVE
- [Second derivative](https://tamnd.github.io/math2/node/0Q4QJWEHW3.md): concept 0Q4QJWEHW3
- [Darboux's theorem](https://tamnd.github.io/math2/node/0Q5A47J48D.md): statement 0Q5A47J48D
- [Statistic](https://tamnd.github.io/math2/node/0Q5CG2NEQ8.md): concept 0Q5CG2NEQ8
- [Derived series](https://tamnd.github.io/math2/node/0Q5G5DJ4P2.md): concept 0Q5G5DJ4P2
- [Galois group of a polynomial](https://tamnd.github.io/math2/node/0Q5NA8Y362.md): concept 0Q5NA8Y362
- [Permutation matrix](https://tamnd.github.io/math2/node/0Q5SM60HTH.md): definition 0Q5SM60HTH
- [Regular graph](https://tamnd.github.io/math2/node/0Q70YFB6ZM.md): concept 0Q70YFB6ZM
- [The sequence r^n tends to zero for |r| < 1](https://tamnd.github.io/math2/node/0Q7K4AXE53.md): example 0Q7K4AXE53
- [Holomorphic functions with nonzero derivative are conformal](https://tamnd.github.io/math2/node/0Q7M6K6VD7.md): statement 0Q7M6K6VD7
- [Nilpotent group](https://tamnd.github.io/math2/node/0Q7MMQHC1P.md): concept 0Q7MMQHC1P
- [Conjugacy class](https://tamnd.github.io/math2/node/0Q88KQD17A.md): definition 0Q88KQD17A
- [Orientation of a real vector space](https://tamnd.github.io/math2/node/0Q8BZXVT1Z.md): definition 0Q8BZXVT1Z
- [Third Sylow theorem](https://tamnd.github.io/math2/node/0Q8DGAZSJ4.md): statement 0Q8DGAZSJ4
- [Orthogonal projection](https://tamnd.github.io/math2/node/0Q8HZTJC7G.md): definition 0Q8HZTJC7G
- [Outer automorphism group](https://tamnd.github.io/math2/node/0Q8Y862DTZ.md): concept 0Q8Y862DTZ
- [Polynomial ring](https://tamnd.github.io/math2/node/0Q9EZS90GY.md): definition 0Q9EZS90GY
- [Diagonal matrix](https://tamnd.github.io/math2/node/0Q9PGXD9E4.md): definition 0Q9PGXD9E4
- [Convergence in probability](https://tamnd.github.io/math2/node/0QA1HBD9Z4.md): concept 0QA1HBD9Z4
- [Field](https://tamnd.github.io/math2/node/0QA8CHH8MG.md): definition 0QA8CHH8MG
- [Uniqueness quantifier](https://tamnd.github.io/math2/node/0QAJHFGZZJ.md): concept 0QAJHFGZZJ
- [Joint probability distribution](https://tamnd.github.io/math2/node/0QAR0XDZH4.md): definition 0QAR0XDZH4
- [Probability distribution](https://tamnd.github.io/math2/node/0QB913WM5E.md): definition 0QB913WM5E
- [Subspace metric](https://tamnd.github.io/math2/node/0QBYFZV91P.md): concept 0QBYFZV91P
- [Reciprocal rule](https://tamnd.github.io/math2/node/0QD4E5JYPT.md): statement 0QD4E5JYPT
- [Contour line](https://tamnd.github.io/math2/node/0QD83RAN8G.md): concept 0QD83RAN8G
- [Characteristic equation](https://tamnd.github.io/math2/node/0QDXMG46QW.md): definition 0QDXMG46QW
- [Equivalence relation induced by a function](https://tamnd.github.io/math2/node/0QE5PFJY7V.md): definition 0QE5PFJY7V
- [Sequence](https://tamnd.github.io/math2/node/0QE8STGJH2.md): definition 0QE8STGJH2
- [Closure](https://tamnd.github.io/math2/node/0QEAJB91GZ.md): concept 0QEAJB91GZ
- [Holomorphic functions are infinitely differentiable](https://tamnd.github.io/math2/node/0QEBN8DGVE.md): statement 0QEBN8DGVE
- [Alternating series](https://tamnd.github.io/math2/node/0QED0GPYKY.md): concept 0QED0GPYKY
- [Cayley–Hamilton theorem for linear maps](https://tamnd.github.io/math2/node/0QEM68Y0DE.md): statement 0QEM68Y0DE
- [Infimum](https://tamnd.github.io/math2/node/0QF1MZ9FRG.md): concept 0QF1MZ9FRG
- [Eigenvalues are the roots of the minimal polynomial](https://tamnd.github.io/math2/node/0QF4ZEFG52.md): statement 0QF4ZEFG52
- [Peano axioms](https://tamnd.github.io/math2/node/0QFT7Z76N5.md): concept 0QFT7Z76N5
- [The algebraic numbers form a field](https://tamnd.github.io/math2/node/0QFTETCHKB.md): statement 0QFTETCHKB
- [Gauss's lemma](https://tamnd.github.io/math2/node/0QFV5Y1Z5X.md): statement 0QFV5Y1Z5X
- [Fundamental solution of Pell's equation](https://tamnd.github.io/math2/node/0QG0Q3NDT9.md): concept 0QG0Q3NDT9
- [Removable discontinuity](https://tamnd.github.io/math2/node/0QG9C579SR.md): definition 0QG9C579SR
- [Stochastic process](https://tamnd.github.io/math2/node/0QH5ENKZM1.md): concept 0QH5ENKZM1
- [Predicate](https://tamnd.github.io/math2/node/0QHEWQD30B.md): concept 0QHEWQD30B
- [Differential form](https://tamnd.github.io/math2/node/0QHTX7WQEY.md): definition 0QHTX7WQEY
- [Affine map](https://tamnd.github.io/math2/node/0QHX5GGABE.md): definition 0QHX5GGABE
- [Irrationality of pi](https://tamnd.github.io/math2/node/0QHZX1PK8V.md): statement 0QHZX1PK8V
- [Orientation of a real vector space](https://tamnd.github.io/math2/node/0QJJAQ6ZYZ.md): concept 0QJJAQ6ZYZ
- [Matrix of a bilinear form](https://tamnd.github.io/math2/node/0QK6K857HS.md): concept 0QK6K857HS
- [Dimension of the coordinate space](https://tamnd.github.io/math2/node/0QK7FPE6VE.md): statement 0QK7FPE6VE
- [Implicit differentiation](https://tamnd.github.io/math2/node/0QKA345AW1.md): concept 0QKA345AW1
- [Derivative function](https://tamnd.github.io/math2/node/0QKJ6NT0R9.md): concept 0QKJ6NT0R9
- [Cardinal number](https://tamnd.github.io/math2/node/0QKPNZKNEA.md): definition 0QKPNZKNEA
- [Euclidean domains are principal ideal domains](https://tamnd.github.io/math2/node/0QKVE1GD9A.md): statement 0QKVE1GD9A
- [Element](https://tamnd.github.io/math2/node/0QM0T7T3PZ.md): concept 0QM0T7T3PZ
- [Product of rings](https://tamnd.github.io/math2/node/0QM2G72DXA.md): concept 0QM2G72DXA
- [Riemann integral](https://tamnd.github.io/math2/node/0QM4TXT3DR.md): definition 0QM4TXT3DR
- [The Galois group has order at most the degree](https://tamnd.github.io/math2/node/0QM98DNWVW.md): statement 0QM98DNWVW
- [Orthocentre](https://tamnd.github.io/math2/node/0QMYQ1X03H.md): definition 0QMYQ1X03H
- [Galois extension](https://tamnd.github.io/math2/node/0QMZAQ6QQ4.md): concept 0QMZAQ6QQ4
- [Euclidean vector](https://tamnd.github.io/math2/node/0QN3D80AWP.md): concept 0QN3D80AWP
- [Contraction mapping](https://tamnd.github.io/math2/node/0QN69YA357.md): concept 0QN69YA357
- [Constructible number](https://tamnd.github.io/math2/node/0QNFRBJF4Q.md): concept 0QNFRBJF4Q
- [Sample mean](https://tamnd.github.io/math2/node/0QNRBHWV35.md): definition 0QNRBHWV35
- [Lipschitz continuity](https://tamnd.github.io/math2/node/0QNVQ4VD9T.md): definition 0QNVQ4VD9T
- [Ratio test](https://tamnd.github.io/math2/node/0QNY6ZBXPN.md): statement 0QNY6ZBXPN
- [Transitive action](https://tamnd.github.io/math2/node/0QP2YYR79N.md): definition 0QP2YYR79N
- [Compact space](https://tamnd.github.io/math2/node/0QPHW703PV.md): definition 0QPHW703PV
- [Von Dyck's theorem](https://tamnd.github.io/math2/node/0QPYHHNRXC.md): statement 0QPYHHNRXC
- [Cycle graph](https://tamnd.github.io/math2/node/0QQ0F12962.md): example 0QQ0F12962
- [Positive definite matrix](https://tamnd.github.io/math2/node/0QQCG3JWJ4.md): definition 0QQCG3JWJ4
- [Derivative of a map into a product](https://tamnd.github.io/math2/node/0QQMXFTX0A.md): statement 0QQMXFTX0A
- [Number of odd-degree vertices is even](https://tamnd.github.io/math2/node/0QQQQCPZA8.md): statement 0QQQQCPZA8
- [Linear differential equation of order n](https://tamnd.github.io/math2/node/0QQRP17Z7S.md): concept 0QQRP17Z7S
- [Asymptotic equivalence](https://tamnd.github.io/math2/node/0QR92XBV6H.md): concept 0QR92XBV6H
- [Orthogonal matrix](https://tamnd.github.io/math2/node/0QRBVFJ50X.md): definition 0QRBVFJ50X
- [In a UFD irreducible elements are prime](https://tamnd.github.io/math2/node/0QRCFZKXH8.md): statement 0QRCFZKXH8
- [Upper and lower Darboux sums](https://tamnd.github.io/math2/node/0QRQVH0CRW.md): definition 0QRQVH0CRW
- [Recurrence for derangements](https://tamnd.github.io/math2/node/0QRRYXG7V7.md): statement 0QRRYXG7V7
- [Residue class](https://tamnd.github.io/math2/node/0QRV8FPK5M.md): definition 0QRV8FPK5M
- [Semisimple endomorphism](https://tamnd.github.io/math2/node/0QS8P74F2Z.md): definition 0QS8P74F2Z
- [The reals form an ordered field](https://tamnd.github.io/math2/node/0QSPNASE50.md): statement 0QSPNASE50
- [Homeomorphism](https://tamnd.github.io/math2/node/0QT00PB5G2.md): concept 0QT00PB5G2
- [The group of units of Z/pZ is cyclic](https://tamnd.github.io/math2/node/0QTHB6PE52.md): statement 0QTHB6PE52
- [Frattini argument](https://tamnd.github.io/math2/node/0QTZDKJEDV.md): statement 0QTZDKJEDV
- [Partial order](https://tamnd.github.io/math2/node/0QV2JS9RJ3.md): concept 0QV2JS9RJ3
- [Distance in a graph](https://tamnd.github.io/math2/node/0QV9VNNB4Z.md): concept 0QV9VNNB4Z
- [Integral domain](https://tamnd.github.io/math2/node/0QVD7WBFGN.md): concept 0QVD7WBFGN
- [Homothety](https://tamnd.github.io/math2/node/0QVDWMKQ0T.md): definition 0QVDWMKQ0T
- [Inner product space](https://tamnd.github.io/math2/node/0QVNCJXRY3.md): concept 0QVNCJXRY3
- [Dimension of a subspace](https://tamnd.github.io/math2/node/0QW084460E.md): statement 0QW084460E
- [Simply connected domain](https://tamnd.github.io/math2/node/0QW64ARSHB.md): concept 0QW64ARSHB
- [Every element of a finite field of order q satisfies x^q = x](https://tamnd.github.io/math2/node/0QWR8WTZEW.md): statement 0QWR8WTZEW
- [Order of a zero](https://tamnd.github.io/math2/node/0QX19C7K1C.md): concept 0QX19C7K1C
- [Tangent–secant theorem](https://tamnd.github.io/math2/node/0QX7E9KAAX.md): statement 0QX7E9KAAX
- [Picard–Lindelöf hypotheses](https://tamnd.github.io/math2/node/0QXM0A9878.md): concept 0QXM0A9878
- [Convergent sequences are Cauchy](https://tamnd.github.io/math2/node/0QY87R9CY3.md): statement 0QY87R9CY3
- [Presentation of a group](https://tamnd.github.io/math2/node/0QZCY9PE3Q.md): definition 0QZCY9PE3Q
- [Cauchy sequence](https://tamnd.github.io/math2/node/0QZMTN5AXX.md): definition 0QZMTN5AXX
- [Fermat number](https://tamnd.github.io/math2/node/0R03B54V4Z.md): definition 0R03B54V4Z
- [Coefficient of a power series](https://tamnd.github.io/math2/node/0R0D0XY4HT.md): concept 0R0D0XY4HT
- [Continuity of the sum of a uniformly convergent series of functions](https://tamnd.github.io/math2/node/0R0MYT0BJD.md): statement 0R0MYT0BJD
- [Fubini's theorem](https://tamnd.github.io/math2/node/0R0T1VWH5X.md): statement 0R0T1VWH5X
- [Abelian groups are solvable](https://tamnd.github.io/math2/node/0R0X73XHAT.md): statement 0R0X73XHAT
- [Content of a polynomial](https://tamnd.github.io/math2/node/0R2X8K9QA7.md): concept 0R2X8K9QA7
- [Connected set](https://tamnd.github.io/math2/node/0R37DFJKFW.md): definition 0R37DFJKFW
- [T0 space](https://tamnd.github.io/math2/node/0R3B7DX55J.md): definition 0R3B7DX55J
- [Derived series](https://tamnd.github.io/math2/node/0R3H35G96G.md): definition 0R3H35G96G
- [Supremum](https://tamnd.github.io/math2/node/0R3H5PANRQ.md): concept 0R3H5PANRQ
- [Type I and type II errors](https://tamnd.github.io/math2/node/0R41PVFA39.md): definition 0R41PVFA39
- [Almost surely](https://tamnd.github.io/math2/node/0R46A1XY60.md): definition 0R46A1XY60
- [Monoid](https://tamnd.github.io/math2/node/0R47MAD57S.md): definition 0R47MAD57S
- [Directed graph](https://tamnd.github.io/math2/node/0R5HTV7REV.md): definition 0R5HTV7REV
- [Translation](https://tamnd.github.io/math2/node/0R5QPQ3AGJ.md): definition 0R5QPQ3AGJ
- [Fraction in lowest terms](https://tamnd.github.io/math2/node/0R64A2CVN4.md): statement 0R64A2CVN4
- [Compact set](https://tamnd.github.io/math2/node/0R6EWFKRCF.md): concept 0R6EWFKRCF
- [Annihilator of a subspace](https://tamnd.github.io/math2/node/0R6M158SQ9.md): definition 0R6M158SQ9
- [Universal property of the quotient ring](https://tamnd.github.io/math2/node/0R74MZYNRB.md): statement 0R74MZYNRB
- [Compact subsets of a metric space are bounded](https://tamnd.github.io/math2/node/0R7BTEWCJ2.md): statement 0R7BTEWCJ2
- [Symmetry of binomial coefficients](https://tamnd.github.io/math2/node/0R7JS3DP3W.md): statement 0R7JS3DP3W
- [pi is greater than 3](https://tamnd.github.io/math2/node/0R7SVNYTJS.md): statement 0R7SVNYTJS
- [A linear isometry preserves the inner product](https://tamnd.github.io/math2/node/0R7VXTYHYZ.md): statement 0R7VXTYHYZ
- [Dimension of a tensor product](https://tamnd.github.io/math2/node/0R84PF4JV7.md): statement 0R84PF4JV7
- [Subsequence](https://tamnd.github.io/math2/node/0R8J3ZQMJS.md): definition 0R8J3ZQMJS
- [Geometric series as a power series](https://tamnd.github.io/math2/node/0R8JR0ARJ4.md): example 0R8JR0ARJ4
- [A linear map is determined by its values on a basis](https://tamnd.github.io/math2/node/0R8X9XA83A.md): statement 0R8X9XA83A
- [Integers modulo a prime form a field](https://tamnd.github.io/math2/node/0R8YXRC9AJ.md): statement 0R8YXRC9AJ
- [Characterization of the floor](https://tamnd.github.io/math2/node/0R9B0B5458.md): statement 0R9B0B5458
- [Divisibility by 3](https://tamnd.github.io/math2/node/0R9D4CEJEB.md): statement 0R9D4CEJEB
- [Gcd of Fibonacci numbers](https://tamnd.github.io/math2/node/0R9F850CTH.md): statement 0R9F850CTH
- [Linear Diophantine equation](https://tamnd.github.io/math2/node/0R9GF3JD0K.md): concept 0R9GF3JD0K
- [Dyck word](https://tamnd.github.io/math2/node/0R9JR0465V.md): definition 0R9JR0465V
- [Adjoint of a linear map](https://tamnd.github.io/math2/node/0R9NMSM1ZP.md): concept 0R9NMSM1ZP
- [One-sided derivative](https://tamnd.github.io/math2/node/0R9PHTGVNW.md): concept 0R9PHTGVNW
- [Unitary group](https://tamnd.github.io/math2/node/0R9R0Y1HDK.md): object 0R9R0Y1HDK
- [Limit of a sum of sequences](https://tamnd.github.io/math2/node/0RAM4E07NR.md): statement 0RAM4E07NR
- [Bernoulli number](https://tamnd.github.io/math2/node/0RAS24E1R5.md): definition 0RAS24E1R5
- [Focus of a conic](https://tamnd.github.io/math2/node/0RBKKNH4GV.md): concept 0RBKKNH4GV
- [Quadratic form](https://tamnd.github.io/math2/node/0RBQS1JJCR.md): concept 0RBQS1JJCR
- [Bessel's inequality](https://tamnd.github.io/math2/node/0RBT5PB74K.md): statement 0RBT5PB74K
- [Cluster point of a sequence](https://tamnd.github.io/math2/node/0RCKQN1E09.md): definition 0RCKQN1E09
- [Cyclic quadrilateral](https://tamnd.github.io/math2/node/0RCP929Y1M.md): concept 0RCP929Y1M
- [Indicator function](https://tamnd.github.io/math2/node/0RCQZ36MME.md): concept 0RCQZ36MME
- [Boolean ring](https://tamnd.github.io/math2/node/0RCV1ZFP8G.md): concept 0RCV1ZFP8G
- [Ordinal number](https://tamnd.github.io/math2/node/0RDJK7EQYY.md): definition 0RDJK7EQYY
- [Arccosine](https://tamnd.github.io/math2/node/0RDP2HEFEW.md): definition 0RDP2HEFEW
- [Cesàro means of a convergent sequence converge](https://tamnd.github.io/math2/node/0RDRPBMNAC.md): statement 0RDRPBMNAC
- [Space of matrices](https://tamnd.github.io/math2/node/0RE6X3BGPN.md): example 0RE6X3BGPN
- [Generating function of a sum of independent random variables](https://tamnd.github.io/math2/node/0REMT01N36.md): statement 0REMT01N36
- [Chi-squared distribution](https://tamnd.github.io/math2/node/0RFMMBHFTV.md): definition 0RFMMBHFTV
- [Composition factors](https://tamnd.github.io/math2/node/0RFP7F3SEW.md): definition 0RFP7F3SEW
- [Cauchy–Schwarz inequality](https://tamnd.github.io/math2/node/0RFWW56Y22.md): statement 0RFWW56Y22
- [Subgroup generated by a subset](https://tamnd.github.io/math2/node/0RG9SPQSWM.md): definition 0RG9SPQSWM
- [Stabiliser](https://tamnd.github.io/math2/node/0RGGF8ZS6D.md): definition 0RGGF8ZS6D
- [Incircle](https://tamnd.github.io/math2/node/0RGWYWWBM5.md): definition 0RGWYWWBM5
- [Submodule](https://tamnd.github.io/math2/node/0RGZ10DFPG.md): definition 0RGZ10DFPG
- [Torsion subgroup](https://tamnd.github.io/math2/node/0RH1TPDGCQ.md): definition 0RH1TPDGCQ
- [Projection](https://tamnd.github.io/math2/node/0RH4E3CAV9.md): definition 0RH4E3CAV9
- [Picard iteration](https://tamnd.github.io/math2/node/0RJ5NHWYTP.md): concept 0RJ5NHWYTP
- [Limit of the cumulative distribution function at infinity](https://tamnd.github.io/math2/node/0RJ5TZH6G0.md): statement 0RJ5TZH6G0
- [Linear map](https://tamnd.github.io/math2/node/0RJF668P1T.md): definition 0RJF668P1T
- [Symmetric group](https://tamnd.github.io/math2/node/0RJMNEN3BJ.md): definition 0RJMNEN3BJ
- [Totally disconnected space](https://tamnd.github.io/math2/node/0RJSBR9GW1.md): concept 0RJSBR9GW1
- [Polar coordinates](https://tamnd.github.io/math2/node/0RJZV05MJA.md): definition 0RJZV05MJA
- [Circumcircle](https://tamnd.github.io/math2/node/0RKB5MKC05.md): definition 0RKB5MKC05
- [Complement](https://tamnd.github.io/math2/node/0RKF1ZT5JJ.md): concept 0RKF1ZT5JJ
- [LU decomposition](https://tamnd.github.io/math2/node/0RKG5YW66Y.md): concept 0RKG5YW66Y
- [Slope field](https://tamnd.github.io/math2/node/0RKQQQACY7.md): definition 0RKQQQACY7
- [Logarithm is the inverse of the exponential](https://tamnd.github.io/math2/node/0RKW2X63YR.md): statement 0RKW2X63YR
- [Sum of subspaces](https://tamnd.github.io/math2/node/0RMKABS2S5.md): definition 0RMKABS2S5
- [Exponential function](https://tamnd.github.io/math2/node/0RMVS7RXZX.md): definition 0RMVS7RXZX
- [Inner automorphism](https://tamnd.github.io/math2/node/0RN00NTM3S.md): concept 0RN00NTM3S
- [Proper metric space](https://tamnd.github.io/math2/node/0RN765S7DH.md): concept 0RN765S7DH
- [Homogeneous first-order equation](https://tamnd.github.io/math2/node/0RN88KZBJ3.md): concept 0RN88KZBJ3
- [Complex power](https://tamnd.github.io/math2/node/0RNGDMHT7M.md): concept 0RNGDMHT7M
- [Complement graph](https://tamnd.github.io/math2/node/0RNMP6VCR2.md): definition 0RNMP6VCR2
- [Damped harmonic oscillator](https://tamnd.github.io/math2/node/0RNXVVTJTJ.md): example 0RNXVVTJTJ
- [Positive definite quadratic form](https://tamnd.github.io/math2/node/0RP20HFYM3.md): concept 0RP20HFYM3
- [Subgroup generated by a subset](https://tamnd.github.io/math2/node/0RP7VWQ0YN.md): concept 0RP7VWQ0YN
- [Trace of a linear map](https://tamnd.github.io/math2/node/0RPDTWY2JS.md): definition 0RPDTWY2JS
- [Cantor set is homeomorphic to the space of binary sequences](https://tamnd.github.io/math2/node/0RQ8DHHR8V.md): statement 0RQ8DHHR8V
- [Identity element](https://tamnd.github.io/math2/node/0RQPZZ44T4.md): definition 0RQPZZ44T4
- [Prime-counting function](https://tamnd.github.io/math2/node/0RQSNVTFVZ.md): concept 0RQSNVTFVZ
- [Wedderburn's little theorem](https://tamnd.github.io/math2/node/0RRNVX1VKA.md): statement 0RRNVX1VKA
- [Fixed points of an action](https://tamnd.github.io/math2/node/0RRWNMYRXC.md): concept 0RRWNMYRXC
- [Linear form](https://tamnd.github.io/math2/node/0RT96J4D3H.md): definition 0RT96J4D3H
- [Focus of a conic](https://tamnd.github.io/math2/node/0RV3AEXPT6.md): definition 0RV3AEXPT6
- [Cauchy product](https://tamnd.github.io/math2/node/0RVD9R2V99.md): concept 0RVD9R2V99
- [Gaussian prime](https://tamnd.github.io/math2/node/0RVRNR7KA1.md): concept 0RVRNR7KA1
- [Converse](https://tamnd.github.io/math2/node/0RVYE75E2M.md): definition 0RVYE75E2M
- [Spherical coordinates](https://tamnd.github.io/math2/node/0RW8KPMXF6.md): definition 0RW8KPMXF6
- [Orthonormal family](https://tamnd.github.io/math2/node/0RW9TNNT87.md): concept 0RW9TNNT87
- [Probability theory](https://tamnd.github.io/math2/node/0RWE7CPJ32.md): concept 0RWE7CPJ32
- [Cartesian coordinate system](https://tamnd.github.io/math2/node/0RWKYX8339.md): concept 0RWKYX8339
- [Jensen's inequality](https://tamnd.github.io/math2/node/0RWMRTPC87.md): statement 0RWMRTPC87
- [Every infinite set has a countably infinite subset](https://tamnd.github.io/math2/node/0RWQ5VZ6VQ.md): statement 0RWQ5VZ6VQ
- [Image of an intersection](https://tamnd.github.io/math2/node/0RX7DDBYCK.md): statement 0RX7DDBYCK
- [Distributivity of matrix multiplication](https://tamnd.github.io/math2/node/0RXD6EZYBK.md): statement 0RXD6EZYBK
- [A continuous nowhere differentiable function](https://tamnd.github.io/math2/node/0RXK2Z808Q.md): counterexample 0RXK2Z808Q
- [Completely multiplicative function](https://tamnd.github.io/math2/node/0RXKW044R4.md): concept 0RXKW044R4
- [Preorder](https://tamnd.github.io/math2/node/0RXS8R6K37.md): definition 0RXS8R6K37
- [Chain rule](https://tamnd.github.io/math2/node/0RXX2PT1MQ.md): statement 0RXX2PT1MQ
- [Structure theorem for finitely generated abelian groups](https://tamnd.github.io/math2/node/0RY0B72XAS.md): statement 0RY0B72XAS
- [Lucas's theorem](https://tamnd.github.io/math2/node/0RY4GGMEQG.md): statement 0RY4GGMEQG
- [Big O notation](https://tamnd.github.io/math2/node/0RYTGBJAGN.md): definition 0RYTGBJAGN
- [Law of excluded middle](https://tamnd.github.io/math2/node/0RYYK8PG4S.md): statement 0RYYK8PG4S
- [Separable polynomial](https://tamnd.github.io/math2/node/0RZ89QWTSJ.md): concept 0RZ89QWTSJ
- [A nontrivial finite p-group has nontrivial centre](https://tamnd.github.io/math2/node/0RZCTT30H5.md): statement 0RZCTT30H5
- [Interior](https://tamnd.github.io/math2/node/0RZH6W98HK.md): concept 0RZH6W98HK
- [Principal ideal domain](https://tamnd.github.io/math2/node/0RZYDVZ475.md): definition 0RZYDVZ475
- [Power-reduction formula for cosine](https://tamnd.github.io/math2/node/0S049HG1W5.md): statement 0S049HG1W5
- [Addition of natural numbers](https://tamnd.github.io/math2/node/0S13V32CPB.md): concept 0S13V32CPB
- [Wedge product of forms](https://tamnd.github.io/math2/node/0S166D363M.md): definition 0S166D363M
- [Size of an orbit times the size of the stabiliser](https://tamnd.github.io/math2/node/0S19CWFZM2.md): statement 0S19CWFZM2
- [Hausdorff space](https://tamnd.github.io/math2/node/0S1FJ8VEX9.md): concept 0S1FJ8VEX9
- [Variation of constants formula for linear systems](https://tamnd.github.io/math2/node/0S1KG0XE40.md): statement 0S1KG0XE40
- [Image of a homomorphism](https://tamnd.github.io/math2/node/0S21GNW9ZB.md): concept 0S21GNW9ZB
- [Kernel of a linear map](https://tamnd.github.io/math2/node/0S23E6YE83.md): concept 0S23E6YE83
- [Surface area](https://tamnd.github.io/math2/node/0S29Y4ZW0Y.md): concept 0S29Y4ZW0Y
- [Smallest prime factor](https://tamnd.github.io/math2/node/0S2SYHTWPY.md): definition 0S2SYHTWPY
- [Discrete topology](https://tamnd.github.io/math2/node/0S2TEHX4E1.md): example 0S2TEHX4E1
- [Continuous linear map](https://tamnd.github.io/math2/node/0S3SMRH0Z5.md): definition 0S3SMRH0Z5
- [Abel's theorem](https://tamnd.github.io/math2/node/0S4EMDQS2F.md): statement 0S4EMDQS2F
- [Convergent sequences are bounded](https://tamnd.github.io/math2/node/0S4JF619V7.md): statement 0S4JF619V7
- [Inverse of a product](https://tamnd.github.io/math2/node/0S4WTRE9GG.md): statement 0S4WTRE9GG
- [Spanning tree](https://tamnd.github.io/math2/node/0S4ZEZM7KE.md): concept 0S4ZEZM7KE
- [Dot product](https://tamnd.github.io/math2/node/0S52TNNTH9.md): concept 0S52TNNTH9
- [Upper bound](https://tamnd.github.io/math2/node/0S60DVMVPN.md): concept 0S60DVMVPN
- [Root of unity](https://tamnd.github.io/math2/node/0S61N75H45.md): definition 0S61N75H45
- [Biconditional](https://tamnd.github.io/math2/node/0S6GMNJ62Y.md): concept 0S6GMNJ62Y
- [Confidence interval](https://tamnd.github.io/math2/node/0S6T9KQNC0.md): concept 0S6T9KQNC0
- [Midpoint](https://tamnd.github.io/math2/node/0S706JBZBG.md): concept 0S706JBZBG
- [Order of the group of units of Z/nZ is Euler's totient](https://tamnd.github.io/math2/node/0S7495H4HM.md): statement 0S7495H4HM
- [Algebraic multiplicity](https://tamnd.github.io/math2/node/0S7A0EMPF4.md): definition 0S7A0EMPF4
- [Indeterminate form](https://tamnd.github.io/math2/node/0S7G92KKA7.md): concept 0S7G92KKA7
- [External direct sum](https://tamnd.github.io/math2/node/0S7W6EG32V.md): concept 0S7W6EG32V
- [Centre of a group](https://tamnd.github.io/math2/node/0S7Y9Q990Q.md): definition 0S7Y9Q990Q
- [Chain](https://tamnd.github.io/math2/node/0S8FE7ZMFM.md): concept 0S8FE7ZMFM
- [Hartman–Grobman theorem](https://tamnd.github.io/math2/node/0S8WX8A4V9.md): statement 0S8WX8A4V9
- [Continuity in R^n](https://tamnd.github.io/math2/node/0S9EK5MQPE.md): definition 0S9EK5MQPE
- [Orthonormal basis](https://tamnd.github.io/math2/node/0S9GGTR0EX.md): concept 0S9GGTR0EX
- [Correspondence theorem](https://tamnd.github.io/math2/node/0S9ZVZ34MD.md): statement 0S9ZVZ34MD
- [Generalised eigenvector](https://tamnd.github.io/math2/node/0SAAHC36J9.md): definition 0SAAHC36J9
- [Derivative of tangent](https://tamnd.github.io/math2/node/0SAMR34WXC.md): statement 0SAMR34WXC
- [Dirichlet convolution formula](https://tamnd.github.io/math2/node/0SAR26CKPS.md): statement 0SAR26CKPS
- [Perfect number](https://tamnd.github.io/math2/node/0SB7Q8SYSF.md): definition 0SB7Q8SYSF
- [Path component](https://tamnd.github.io/math2/node/0SB9FNZECG.md): definition 0SB9FNZECG
- [Multiplicative order](https://tamnd.github.io/math2/node/0SBRX5CZ5R.md): definition 0SBRX5CZ5R
- [Golden ratio](https://tamnd.github.io/math2/node/0SC8T7548D.md): object 0SC8T7548D
- [Hessian matrix](https://tamnd.github.io/math2/node/0SCSQWXRA4.md): definition 0SCSQWXRA4
- [Gradient theorem](https://tamnd.github.io/math2/node/0SD237FEJD.md): statement 0SD237FEJD
- [Minor](https://tamnd.github.io/math2/node/0SD528P0EK.md): definition 0SD528P0EK
- [Polynomial ring](https://tamnd.github.io/math2/node/0SDC934R52.md): concept 0SDC934R52
- [Dihedral angle](https://tamnd.github.io/math2/node/0SDFXN1VDQ.md): definition 0SDFXN1VDQ
- [Multilinear map](https://tamnd.github.io/math2/node/0SDMPMCN43.md): definition 0SDMPMCN43
- [Cross product](https://tamnd.github.io/math2/node/0SEDGXG9S3.md): definition 0SEDGXG9S3
- [Order of a group](https://tamnd.github.io/math2/node/0SEX9F9J5Z.md): definition 0SEX9F9J5Z
- [Countably infinite set](https://tamnd.github.io/math2/node/0SEZ3JGZP7.md): definition 0SEZ3JGZP7
- [Product rule in several variables](https://tamnd.github.io/math2/node/0SF85PY0QN.md): statement 0SF85PY0QN
- [Möbius transformation](https://tamnd.github.io/math2/node/0SG74WBFA2.md): concept 0SG74WBFA2
- [Equivalence relation](https://tamnd.github.io/math2/node/0SGTMBEWRF.md): definition 0SGTMBEWRF
- [Basis of a tensor product](https://tamnd.github.io/math2/node/0SH5VT8QYA.md): statement 0SH5VT8QYA
- [Closure is the smallest closed superset](https://tamnd.github.io/math2/node/0SHC6Z304Z.md): statement 0SHC6Z304Z
- [Split polynomial](https://tamnd.github.io/math2/node/0SHMC00RJV.md): definition 0SHMC00RJV
- [Pendulum equation](https://tamnd.github.io/math2/node/0SHPXM2HWH.md): example 0SHPXM2HWH
- [Set](https://tamnd.github.io/math2/node/0SJ3RX9TXA.md): definition 0SJ3RX9TXA
- [Continued fraction expansion of a real number](https://tamnd.github.io/math2/node/0SJ7Z6VKWK.md): definition 0SJ7Z6VKWK
- [Falling factorial](https://tamnd.github.io/math2/node/0SJNCJCCQG.md): concept 0SJNCJCCQG
- [Irreducibility of the cyclotomic polynomials over the rationals](https://tamnd.github.io/math2/node/0SK1VHZ5R2.md): statement 0SK1VHZ5R2
- [Sign of a permutation](https://tamnd.github.io/math2/node/0SK90238BH.md): definition 0SK90238BH
- [Stability of linear systems via eigenvalues](https://tamnd.github.io/math2/node/0SKVEMXJWF.md): statement 0SKVEMXJWF
- [General solution and particular solution](https://tamnd.github.io/math2/node/0SKYFQAFYD.md): concept 0SKYFQAFYD
- [Edge bound for planar graphs](https://tamnd.github.io/math2/node/0SM1REAVYJ.md): statement 0SM1REAVYJ
- [Prime number](https://tamnd.github.io/math2/node/0SN0MAEKYX.md): definition 0SN0MAEKYX
- [Erdős–Szekeres theorem](https://tamnd.github.io/math2/node/0SNJM8HSD3.md): statement 0SNJM8HSD3
- [Bijective function](https://tamnd.github.io/math2/node/0SNQEF7Z53.md): definition 0SNQEF7Z53
- [Primitive polynomial](https://tamnd.github.io/math2/node/0SP14DZN0G.md): definition 0SP14DZN0G
- [A space is the direct sum of a subspace and its orthogonal complement](https://tamnd.github.io/math2/node/0SP7P0G7KQ.md): statement 0SP7P0G7KQ
- [Existence of a prime divisor](https://tamnd.github.io/math2/node/0SPBEDVNW1.md): statement 0SPBEDVNW1
- [Hypergeometric distribution](https://tamnd.github.io/math2/node/0SPBPW3K61.md): definition 0SPBPW3K61
- [Connected set](https://tamnd.github.io/math2/node/0SPE8WNWFV.md): concept 0SPE8WNWFV
- [Change of basis for a bilinear form](https://tamnd.github.io/math2/node/0SPYAGE7T0.md): statement 0SPYAGE7T0
- [The Gaussian integers form a Euclidean domain](https://tamnd.github.io/math2/node/0SPYEFH7H4.md): statement 0SPYEFH7H4
- [Refinement of a partition](https://tamnd.github.io/math2/node/0SPZ1H3N9V.md): concept 0SPZ1H3N9V
- [Branch of the logarithm](https://tamnd.github.io/math2/node/0SQ1F2BEDS.md): definition 0SQ1F2BEDS
- [Open map](https://tamnd.github.io/math2/node/0SQ2PC596Y.md): definition 0SQ2PC596Y
- [Quadratic residue](https://tamnd.github.io/math2/node/0SQDB87Z3T.md): definition 0SQDB87Z3T
- [Principal value of the argument](https://tamnd.github.io/math2/node/0SQSQP748R.md): statement 0SQSQP748R
- [Even and odd permutations](https://tamnd.github.io/math2/node/0SQTMQTSRZ.md): concept 0SQTMQTSRZ
- [Parallelepiped](https://tamnd.github.io/math2/node/0SQV2VJTGY.md): concept 0SQV2VJTGY
- [Bilinear form](https://tamnd.github.io/math2/node/0SQZMXB6GP.md): concept 0SQZMXB6GP
- [Subgroups of the integers](https://tamnd.github.io/math2/node/0SRATCFF1W.md): statement 0SRATCFF1W
- [Characteristic subgroups are normal](https://tamnd.github.io/math2/node/0SRE7K4E49.md): statement 0SRE7K4E49
- [Convergents of a continued fraction](https://tamnd.github.io/math2/node/0SRWNR42BQ.md): definition 0SRWNR42BQ
- [Convergence in distribution](https://tamnd.github.io/math2/node/0SRYJGXKR4.md): concept 0SRYJGXKR4
- [First integral](https://tamnd.github.io/math2/node/0SS1P1YC9A.md): definition 0SS1P1YC9A
- [Differentiation as a linear map](https://tamnd.github.io/math2/node/0SS770NS8B.md): example 0SS770NS8B
- [Hurwitz's theorem](https://tamnd.github.io/math2/node/0SSKBQN58M.md): statement 0SSKBQN58M
- [Estimator](https://tamnd.github.io/math2/node/0SSS6ERQ0S.md): concept 0SSS6ERQ0S
- [The degree of a constructible number is a power of two](https://tamnd.github.io/math2/node/0SSSM1D475.md): statement 0SSSM1D475
- [Almost surely](https://tamnd.github.io/math2/node/0ST21PGDAP.md): concept 0ST21PGDAP
- [Adjoint of a linear map](https://tamnd.github.io/math2/node/0ST8QF9T3T.md): definition 0ST8QF9T3T
- [Moments are derivatives of the moment-generating function at zero](https://tamnd.github.io/math2/node/0STK8KEY9G.md): statement 0STK8KEY9G
- [C^1 functions are strictly differentiable](https://tamnd.github.io/math2/node/0STPPDXGME.md): statement 0STPPDXGME
- [Density of the rationals](https://tamnd.github.io/math2/node/0STX0WRDNG.md): statement 0STX0WRDNG
- [Sigma-algebra](https://tamnd.github.io/math2/node/0SV8K0TDGY.md): concept 0SV8K0TDGY
- [Almost sure convergence implies convergence in probability](https://tamnd.github.io/math2/node/0SVN6X695K.md): statement 0SVN6X695K
- [Quotient space](https://tamnd.github.io/math2/node/0SVR29SV1V.md): definition 0SVR29SV1V
- [Primitive element theorem](https://tamnd.github.io/math2/node/0SVX98HYV1.md): statement 0SVX98HYV1
- [Principal ideal](https://tamnd.github.io/math2/node/0SVYSP8HCG.md): concept 0SVYSP8HCG
- [Linear equation with constant coefficients](https://tamnd.github.io/math2/node/0SWTJ5VC07.md): concept 0SWTJ5VC07
- [Sample space](https://tamnd.github.io/math2/node/0SWXJHKJ02.md): concept 0SWXJHKJ02
- [Möbius transformations map circles and lines to circles and lines](https://tamnd.github.io/math2/node/0SXDHQM2MH.md): statement 0SXDHQM2MH
- [Dimension of a quotient space](https://tamnd.github.io/math2/node/0SXDQ6TN00.md): statement 0SXDQ6TN00
- [Acyclic graph](https://tamnd.github.io/math2/node/0SXHBTNM9T.md): concept 0SXHBTNM9T
- [Integral of 1/z around the unit circle](https://tamnd.github.io/math2/node/0SXZ68HV6F.md): example 0SXZ68HV6F
- [Continuous map of topological spaces](https://tamnd.github.io/math2/node/0SY9ECQ19Q.md): definition 0SY9ECQ19Q
- [Cauchy principal value](https://tamnd.github.io/math2/node/0SYR10HD4R.md): concept 0SYR10HD4R
- [Order of the alternating group](https://tamnd.github.io/math2/node/0SZ1KMNWVE.md): statement 0SZ1KMNWVE
- [Miller–Rabin primality test](https://tamnd.github.io/math2/node/0SZCSYHW60.md): concept 0SZCSYHW60
- [Normal convergence](https://tamnd.github.io/math2/node/0T0EJDGHMB.md): definition 0T0EJDGHMB
- [Degree of a vertex](https://tamnd.github.io/math2/node/0T0F6CCS9W.md): definition 0T0F6CCS9W
- [Bertrand paradox](https://tamnd.github.io/math2/node/0T0JWJ3M47.md): example 0T0JWJ3M47
- [Normal core](https://tamnd.github.io/math2/node/0T0KMZPYV9.md): definition 0T0KMZPYV9
- [Determinant is a coefficient of the characteristic polynomial](https://tamnd.github.io/math2/node/0T0X8GTYYS.md): statement 0T0X8GTYYS
- [Alternating group](https://tamnd.github.io/math2/node/0T13HM377F.md): definition 0T13HM377F
- [Infinitely many primes congruent to 1 modulo n](https://tamnd.github.io/math2/node/0T1BSD7H3M.md): statement 0T1BSD7H3M
- [Fundamental theorem of Galois theory](https://tamnd.github.io/math2/node/0T1Q8D0FW8.md): statement 0T1Q8D0FW8
- [Degree of a product](https://tamnd.github.io/math2/node/0T233AZ1GE.md): statement 0T233AZ1GE
- [Least element](https://tamnd.github.io/math2/node/0T2NTJCV83.md): definition 0T2NTJCV83
- [Subspace metric](https://tamnd.github.io/math2/node/0T2WYDTRP2.md): definition 0T2WYDTRP2
- [Power series of arctangent](https://tamnd.github.io/math2/node/0T32ME0A6H.md): statement 0T32ME0A6H
- [Distributive lattice](https://tamnd.github.io/math2/node/0T36P6TXQ6.md): definition 0T36P6TXQ6
- [Predicate](https://tamnd.github.io/math2/node/0T3DPBG981.md): definition 0T3DPBG981
- [Phase portrait](https://tamnd.github.io/math2/node/0T3GK0XCBC.md): concept 0T3GK0XCBC
- [Limit superior](https://tamnd.github.io/math2/node/0T3RZQXDYD.md): concept 0T3RZQXDYD
- [Frobenius endomorphism](https://tamnd.github.io/math2/node/0T3WVTD675.md): concept 0T3WVTD675
- [Dirichlet function](https://tamnd.github.io/math2/node/0T4EJ8ZJM9.md): counterexample 0T4EJ8ZJM9
- [The cross product is perpendicular to its factors](https://tamnd.github.io/math2/node/0T4QHJXSD4.md): statement 0T4QHJXSD4
- [Clairaut's equation](https://tamnd.github.io/math2/node/0T6D4PE51C.md): example 0T6D4PE51C
- [Augmented matrix](https://tamnd.github.io/math2/node/0T6MWGZSXA.md): concept 0T6MWGZSXA
- [First Borel–Cantelli lemma](https://tamnd.github.io/math2/node/0T6WS7TT77.md): statement 0T6WS7TT77
- [Cauchy–Riemann equations](https://tamnd.github.io/math2/node/0T71SSYZ39.md): statement 0T71SSYZ39
- [Circle integral](https://tamnd.github.io/math2/node/0T7ABX4CWE.md): definition 0T7ABX4CWE
- [Tangent line to a circle](https://tamnd.github.io/math2/node/0T7DX8NY1X.md): concept 0T7DX8NY1X
- [Integration in polar coordinates](https://tamnd.github.io/math2/node/0T89W1QZ9K.md): statement 0T89W1QZ9K
- [The characteristic function determines the distribution](https://tamnd.github.io/math2/node/0T8BYAFEXX.md): statement 0T8BYAFEXX
- [Existence of finite fields of every prime power order](https://tamnd.github.io/math2/node/0T8M3YC5RG.md): statement 0T8M3YC5RG
- [Bernoulli equation](https://tamnd.github.io/math2/node/0T92VGGVWP.md): concept 0T92VGGVWP
- [Carmichael number](https://tamnd.github.io/math2/node/0TA7WT68RK.md): definition 0TA7WT68RK
- [Axiom of extensionality](https://tamnd.github.io/math2/node/0TB16T2C97.md): statement 0TB16T2C97
- [The successor function is injective](https://tamnd.github.io/math2/node/0TBYMWPWWB.md): statement 0TBYMWPWWB
- [Dirac's theorem on Hamiltonian cycles](https://tamnd.github.io/math2/node/0TC99BRSSB.md): statement 0TC99BRSSB
- [C^k function characterised by continuous derivatives](https://tamnd.github.io/math2/node/0TCBZMEYGN.md): statement 0TCBZMEYGN
- [Convex hull](https://tamnd.github.io/math2/node/0TCHAS04WK.md): definition 0TCHAS04WK
- [Orientable surface](https://tamnd.github.io/math2/node/0TD904MG06.md): concept 0TD904MG06
- [Product of the gcd and the lcm](https://tamnd.github.io/math2/node/0TDY6JZHKS.md): statement 0TDY6JZHKS
- [Remainder theorem](https://tamnd.github.io/math2/node/0TEDQH56RW.md): statement 0TEDQH56RW
- [Affine space](https://tamnd.github.io/math2/node/0TEJ6QF7K9.md): definition 0TEJ6QF7K9
- [p-metric on R^n](https://tamnd.github.io/math2/node/0TFDZN6AXP.md): example 0TFDZN6AXP
- [Big O notation](https://tamnd.github.io/math2/node/0TGV80E6A4.md): concept 0TGV80E6A4
- [Derangement](https://tamnd.github.io/math2/node/0TH61JR04G.md): definition 0TH61JR04G
- [Arc length](https://tamnd.github.io/math2/node/0TH78S7GBZ.md): concept 0TH78S7GBZ
- [Reduction of a higher-order equation to a first-order system](https://tamnd.github.io/math2/node/0TH9CPW3VS.md): statement 0TH9CPW3VS
- [Function composition](https://tamnd.github.io/math2/node/0THBK9V825.md): concept 0THBK9V825
- [Conditional independence](https://tamnd.github.io/math2/node/0THJ7416DX.md): concept 0THJ7416DX
- [Third isomorphism theorem](https://tamnd.github.io/math2/node/0TJ1V9DDPC.md): statement 0TJ1V9DDPC
- [Term-by-term differentiation](https://tamnd.github.io/math2/node/0TJ8QBJX5K.md): statement 0TJ8QBJX5K
- [Interval](https://tamnd.github.io/math2/node/0TK8Y3J5RW.md): definition 0TK8Y3J5RW
- [Cylindrical coordinates](https://tamnd.github.io/math2/node/0TK9MCFHBW.md): concept 0TK9MCFHBW
- [Bounded derivative on a convex set implies Lipschitz](https://tamnd.github.io/math2/node/0TKG7T3K0E.md): statement 0TKG7T3K0E
- [Bell numbers](https://tamnd.github.io/math2/node/0TKP3TDD02.md): concept 0TKP3TDD02
- [Orthogonal group](https://tamnd.github.io/math2/node/0TKYAEJ8N7.md): object 0TKYAEJ8N7
- [Minimum degree](https://tamnd.github.io/math2/node/0TKZCSGCCA.md): definition 0TKZCSGCCA
- [Lipschitz map](https://tamnd.github.io/math2/node/0TMZ21WKCN.md): definition 0TMZ21WKCN
- [Set-builder notation](https://tamnd.github.io/math2/node/0TMZ89Q5KA.md): concept 0TMZ89Q5KA
- [Powers of a diagonalisable matrix](https://tamnd.github.io/math2/node/0TNR9BQ871.md): concept 0TNR9BQ871
- [Prime subfield](https://tamnd.github.io/math2/node/0TNS65QB87.md): concept 0TNS65QB87
- [Partial derivatives do not imply differentiability](https://tamnd.github.io/math2/node/0TNZC55X3J.md): counterexample 0TNZC55X3J
- [Circumcentre](https://tamnd.github.io/math2/node/0TP2QKJ8DE.md): definition 0TP2QKJ8DE
- [Congruence modulo n](https://tamnd.github.io/math2/node/0TPW25CBN6.md): definition 0TPW25CBN6
- [An algebraic extension of an algebraic extension is algebraic](https://tamnd.github.io/math2/node/0TPWS0K1BC.md): statement 0TPWS0K1BC
- [Strictly monotone function](https://tamnd.github.io/math2/node/0TR9SR9FTB.md): definition 0TR9SR9FTB
- [Euclidean algorithm](https://tamnd.github.io/math2/node/0TRNT4X7J0.md): concept 0TRNT4X7J0
- [Orthogonal vectors](https://tamnd.github.io/math2/node/0TS7VZJBWD.md): definition 0TS7VZJBWD
- [Law of a random variable](https://tamnd.github.io/math2/node/0TST4MHA3A.md): definition 0TST4MHA3A
- [Convergence in mean](https://tamnd.github.io/math2/node/0TTA57SEJG.md): definition 0TTA57SEJG
- [Conditional expectation](https://tamnd.github.io/math2/node/0TTDGTEFMN.md): concept 0TTDGTEFMN
- [Estimator](https://tamnd.github.io/math2/node/0TTNRP5PFQ.md): definition 0TTNRP5PFQ
- [Real exponentiation](https://tamnd.github.io/math2/node/0TV07AQT62.md): definition 0TV07AQT62
- [Compact space](https://tamnd.github.io/math2/node/0TVJZYB6SH.md): concept 0TVJZYB6SH
- [Incentre](https://tamnd.github.io/math2/node/0TWPTQ0AD7.md): definition 0TWPTQ0AD7
- [Equivalent metrics](https://tamnd.github.io/math2/node/0TWQ5PRKWS.md): definition 0TWQ5PRKWS
- [Invertible matrix](https://tamnd.github.io/math2/node/0TWTS88TWN.md): definition 0TWTS88TWN
- [Symmetric bilinear form associated with a quadratic form](https://tamnd.github.io/math2/node/0TX2YH2FQS.md): definition 0TX2YH2FQS
- [Integration over a non-rectangular region](https://tamnd.github.io/math2/node/0TXCQ9CST1.md): concept 0TXCQ9CST1
- [Primitive polynomial](https://tamnd.github.io/math2/node/0TXNRQZY0H.md): concept 0TXNRQZY0H
- [Limit of a function](https://tamnd.github.io/math2/node/0TXTSEMHR5.md): definition 0TXTSEMHR5
- [Product of a subgroup and a normal subgroup is a subgroup](https://tamnd.github.io/math2/node/0TXWV1JGG1.md): statement 0TXWV1JGG1
- [Isolated singularity](https://tamnd.github.io/math2/node/0TYB3HM9KE.md): definition 0TYB3HM9KE
- [Characteristic polynomial does not depend on the basis](https://tamnd.github.io/math2/node/0TYMASG33Y.md): statement 0TYMASG33Y
- [Surjective iff it has a right inverse](https://tamnd.github.io/math2/node/0TYQNHVB4C.md): statement 0TYQNHVB4C
- [Principal part](https://tamnd.github.io/math2/node/0TZ2DPT0JV.md): concept 0TZ2DPT0JV
- [Row rank equals column rank](https://tamnd.github.io/math2/node/0TZAG60ZYT.md): statement 0TZAG60ZYT
- [Maximal element](https://tamnd.github.io/math2/node/0TZEEFTNCC.md): definition 0TZEEFTNCC
- [The sequence r^n tends to infinity for r > 1](https://tamnd.github.io/math2/node/0TZK91NQE3.md): example 0TZK91NQE3
- [Connected space](https://tamnd.github.io/math2/node/0V032VWTHD.md): concept 0V032VWTHD
- [Induced subgraph](https://tamnd.github.io/math2/node/0V05FC4GKQ.md): definition 0V05FC4GKQ
- [Variance of a sum of independent random variables](https://tamnd.github.io/math2/node/0V0778DCW9.md): statement 0V0778DCW9
- [Sign of a cycle](https://tamnd.github.io/math2/node/0V0E04DWX2.md): statement 0V0E04DWX2
- [Complete bipartite graph](https://tamnd.github.io/math2/node/0V0GC7BQAN.md): example 0V0GC7BQAN
- [Frobenius coin problem](https://tamnd.github.io/math2/node/0V11FEXK08.md): statement 0V11FEXK08
- [Derivative](https://tamnd.github.io/math2/node/0V1EJCT8G6.md): concept 0V1EJCT8G6
- [Derivative of cosine](https://tamnd.github.io/math2/node/0V2DV29EGT.md): statement 0V2DV29EGT
- [Aleph-null](https://tamnd.github.io/math2/node/0V2H5PGJ9B.md): object 0V2H5PGJ9B
- [Norm](https://tamnd.github.io/math2/node/0V2WP9B1MF.md): concept 0V2WP9B1MF
- [Smooth function](https://tamnd.github.io/math2/node/0V33BNKZ9S.md): concept 0V33BNKZ9S
- [Intersection of subspaces](https://tamnd.github.io/math2/node/0V33SPH7RR.md): statement 0V33SPH7RR
- [Semigroup](https://tamnd.github.io/math2/node/0V3CFG1KK9.md): definition 0V3CFG1KK9
- [Gamma function](https://tamnd.github.io/math2/node/0V3FZMVB4K.md): definition 0V3FZMVB4K
- [Cauchy principal value](https://tamnd.github.io/math2/node/0V3RDQV3ZX.md): definition 0V3RDQV3ZX
- [Plane](https://tamnd.github.io/math2/node/0V3XAMF3C1.md): concept 0V3XAMF3C1
- [Laplace transform](https://tamnd.github.io/math2/node/0V49JEZW06.md): concept 0V49JEZW06
- [Continuous real function on a compact set attains its bounds](https://tamnd.github.io/math2/node/0V4DET59NW.md): statement 0V4DET59NW
- [Separable equation](https://tamnd.github.io/math2/node/0V4E623THY.md): definition 0V4E623THY
- [Vacuous truth](https://tamnd.github.io/math2/node/0V4QXPGJQK.md): concept 0V4QXPGJQK
- [Fibonacci numbers as sums of binomial coefficients](https://tamnd.github.io/math2/node/0V4VB6MQDC.md): statement 0V4VB6MQDC
- [Mean squared error](https://tamnd.github.io/math2/node/0V4VSCBXP3.md): definition 0V4VSCBXP3
- [Subgroups of index two are normal](https://tamnd.github.io/math2/node/0V547YQRMV.md): statement 0V547YQRMV
- [Minimal polynomial](https://tamnd.github.io/math2/node/0V58FPM3EQ.md): definition 0V58FPM3EQ
- [Characteristic function](https://tamnd.github.io/math2/node/0V5B6D6BQH.md): concept 0V5B6D6BQH
- [Leading coefficient](https://tamnd.github.io/math2/node/0V5GYWR8R8.md): concept 0V5GYWR8R8
- [Regular polygon](https://tamnd.github.io/math2/node/0V63D9V0FY.md): definition 0V63D9V0FY
- [Biconditional](https://tamnd.github.io/math2/node/0V6GAWTBYW.md): definition 0V6GAWTBYW
- [Regular curve](https://tamnd.github.io/math2/node/0V7090SSPC.md): concept 0V7090SSPC
- [Square root function](https://tamnd.github.io/math2/node/0V7E9KP6Q2.md): definition 0V7E9KP6Q2
- [Covariance matrix](https://tamnd.github.io/math2/node/0V7G6WMFK0.md): concept 0V7G6WMFK0
- [Standard basis](https://tamnd.github.io/math2/node/0V7ZD6P7RK.md): object 0V7ZD6P7RK
- [Quotient topology](https://tamnd.github.io/math2/node/0V8046R0P9.md): definition 0V8046R0P9
- [Linear combination](https://tamnd.github.io/math2/node/0V8F3E48E6.md): concept 0V8F3E48E6
- [Logarithm of a power](https://tamnd.github.io/math2/node/0V948DE5KA.md): statement 0V948DE5KA
- [Determinant of a family of vectors in a basis](https://tamnd.github.io/math2/node/0V94H674MB.md): definition 0V94H674MB
- [First Sylow theorem](https://tamnd.github.io/math2/node/0V98WY94B8.md): statement 0V98WY94B8
- [Product rule](https://tamnd.github.io/math2/node/0V9A43H7WV.md): statement 0V9A43H7WV
- [Dirichlet's test](https://tamnd.github.io/math2/node/0V9ME8GQE7.md): statement 0V9ME8GQE7
- [Special linear group](https://tamnd.github.io/math2/node/0V9PMN0RBH.md): object 0V9PMN0RBH
- [Preorder](https://tamnd.github.io/math2/node/0VABYWJWH9.md): concept 0VABYWJWH9
- [Power series in the complex plane](https://tamnd.github.io/math2/node/0VARVQ9ZW1.md): definition 0VARVQ9ZW1
- [Square root](https://tamnd.github.io/math2/node/0VB8BFRABH.md): definition 0VB8BFRABH
- [Multiplicativity of the Möbius function](https://tamnd.github.io/math2/node/0VB9G5B034.md): statement 0VB9G5B034
- [Kummer's theorem](https://tamnd.github.io/math2/node/0VBWK9XYDG.md): statement 0VBWK9XYDG
- [Internal direct product](https://tamnd.github.io/math2/node/0VBZH90R1A.md): concept 0VBZH90R1A
- [Euler's theorem](https://tamnd.github.io/math2/node/0VC4DVE7DQ.md): statement 0VC4DVE7DQ
- [Improper integral](https://tamnd.github.io/math2/node/0VCTBJMEJF.md): definition 0VCTBJMEJF
- [Matrix](https://tamnd.github.io/math2/node/0VD9ZSWWJG.md): concept 0VD9ZSWWJG
- [Morera's theorem](https://tamnd.github.io/math2/node/0VDD7W0EKZ.md): statement 0VDD7W0EKZ
- [Poisson distribution](https://tamnd.github.io/math2/node/0VEFSSKX18.md): definition 0VEFSSKX18
- [Volume of a box](https://tamnd.github.io/math2/node/0VEKNG02W0.md): statement 0VEKNG02W0
- [Field of fractions](https://tamnd.github.io/math2/node/0VF11ZTTMD.md): definition 0VF11ZTTMD
- [Quotient ring](https://tamnd.github.io/math2/node/0VFW4Y69BT.md): definition 0VFW4Y69BT
- [Every polynomial has a root in some extension field](https://tamnd.github.io/math2/node/0VG02M1FEZ.md): statement 0VG02M1FEZ
- [Abundant number](https://tamnd.github.io/math2/node/0VG7VNEWMG.md): concept 0VG7VNEWMG
- [Continuous map](https://tamnd.github.io/math2/node/0VGFHTM75Y.md): concept 0VGFHTM75Y
- [Improper integral of x^p on an unbounded interval](https://tamnd.github.io/math2/node/0VGKB2NC4T.md): example 0VGKB2NC4T
- [Probability measure](https://tamnd.github.io/math2/node/0VGMEBVW1M.md): definition 0VGMEBVW1M
- [Product of continuous functions is continuous](https://tamnd.github.io/math2/node/0VGMHCF4SC.md): statement 0VGMHCF4SC
- [Continuity is equivalent to sequential continuity in metric spaces](https://tamnd.github.io/math2/node/0VGPKZ8MRK.md): statement 0VGPKZ8MRK
- [Conditional variance](https://tamnd.github.io/math2/node/0VGPX25NXQ.md): concept 0VGPX25NXQ
- [Inner product](https://tamnd.github.io/math2/node/0VGSYBWHV2.md): definition 0VGSYBWHV2
- [Abel's identity](https://tamnd.github.io/math2/node/0VGWKX2JHB.md): statement 0VGWKX2JHB
- [Companion matrix](https://tamnd.github.io/math2/node/0VH1TC1PWF.md): definition 0VH1TC1PWF
- [Subsequences of a convergent sequence converge to the same limit](https://tamnd.github.io/math2/node/0VHAVXW08V.md): statement 0VHAVXW08V
- [Chromatic number](https://tamnd.github.io/math2/node/0VHJRXNAAS.md): concept 0VHJRXNAAS
- [Conformal map](https://tamnd.github.io/math2/node/0VHNKP1N3K.md): definition 0VHNKP1N3K
- [Symmetric polynomial](https://tamnd.github.io/math2/node/0VHSHB9B6J.md): concept 0VHSHB9B6J
- [Finite-dimensional spaces are isomorphic iff they have equal dimension](https://tamnd.github.io/math2/node/0VJBSK15SN.md): statement 0VJBSK15SN
- [Homogeneous linear system with constant coefficients](https://tamnd.github.io/math2/node/0VJF99CZ0Z.md): definition 0VJF99CZ0Z
- [Walk](https://tamnd.github.io/math2/node/0VJZM0F9YM.md): definition 0VJZM0F9YM
- [De Morgan's laws for sets](https://tamnd.github.io/math2/node/0VK0W4D5Y5.md): statement 0VK0W4D5Y5
- [Size of a conjugacy class is the index of the centraliser](https://tamnd.github.io/math2/node/0VK5V9P8G3.md): statement 0VK5V9P8G3
- [Banach fixed point theorem](https://tamnd.github.io/math2/node/0VK6VEH345.md): statement 0VK6VEH345
- [Ellipse](https://tamnd.github.io/math2/node/0VKCCEXBJ1.md): concept 0VKCCEXBJ1
- [Metric space](https://tamnd.github.io/math2/node/0VKD1TP7F3.md): concept 0VKD1TP7F3
- [Lagrange multipliers](https://tamnd.github.io/math2/node/0VKKP8X6Q6.md): statement 0VKKP8X6Q6
- [Decimal expansion](https://tamnd.github.io/math2/node/0VM8B2SB4P.md): concept 0VM8B2SB4P
- [Limit of a function of several variables](https://tamnd.github.io/math2/node/0VMG8TA8VF.md): definition 0VMG8TA8VF
- [Cartan–Dieudonné theorem](https://tamnd.github.io/math2/node/0VMWGYY35Q.md): statement 0VMWGYY35Q
- [Smallest prime factor](https://tamnd.github.io/math2/node/0VMWPGGG3C.md): concept 0VMWPGGG3C
- [If 2^n + 1 is prime then n is a power of 2](https://tamnd.github.io/math2/node/0VN4H8M12X.md): statement 0VN4H8M12X
- [Polynomials of degree two or three are irreducible iff they have no roots](https://tamnd.github.io/math2/node/0VNE161QCK.md): statement 0VNE161QCK
- [Pythagorean theorem](https://tamnd.github.io/math2/node/0VP3A1290K.md): statement 0VP3A1290K
- [Frobenius automorphism of a finite field](https://tamnd.github.io/math2/node/0VP4TBVJ19.md): definition 0VP4TBVJ19
- [Power series](https://tamnd.github.io/math2/node/0VP56B4EMX.md): definition 0VP56B4EMX
- [Space of linear maps](https://tamnd.github.io/math2/node/0VPCJWGMP9.md): definition 0VPCJWGMP9
- [Prime subfield](https://tamnd.github.io/math2/node/0VPFBAY7MW.md): definition 0VPFBAY7MW
- [Rule of product](https://tamnd.github.io/math2/node/0VPK4X73Q0.md): statement 0VPK4X73Q0
- [Sequence tending to infinity](https://tamnd.github.io/math2/node/0VQA7F720Z.md): concept 0VQA7F720Z
- [Combinations with repetition](https://tamnd.github.io/math2/node/0VQQCVG8WA.md): concept 0VQQCVG8WA
- [Nowhere dense set](https://tamnd.github.io/math2/node/0VQYCCQVJ5.md): concept 0VQYCCQVJ5
- [Symmetric matrix](https://tamnd.github.io/math2/node/0VR671PHE5.md): concept 0VR671PHE5
- [Isometric isomorphism](https://tamnd.github.io/math2/node/0VRD8GC5BM.md): definition 0VRD8GC5BM
- [Binomial distribution](https://tamnd.github.io/math2/node/0VRWZDWW6B.md): definition 0VRWZDWW6B
- [Abundant number](https://tamnd.github.io/math2/node/0VS0MP1P5H.md): definition 0VS0MP1P5H
- [Indicator function](https://tamnd.github.io/math2/node/0VSB0XF4G7.md): definition 0VSB0XF4G7
- [A tree on n vertices has n − 1 edges](https://tamnd.github.io/math2/node/0VSNHTHK8T.md): statement 0VSNHTHK8T
- [Cancellation laws](https://tamnd.github.io/math2/node/0VSNX4P2Q0.md): statement 0VSNX4P2Q0
- [Characteristic polynomial](https://tamnd.github.io/math2/node/0VSSPAFM2C.md): concept 0VSSPAFM2C
- [Riccati equation](https://tamnd.github.io/math2/node/0VT2QK6X6Q.md): concept 0VT2QK6X6Q
- [Gram–Schmidt process](https://tamnd.github.io/math2/node/0VT6P0GNQR.md): concept 0VT6P0GNQR
- [Minimal polynomial divides every annihilating polynomial](https://tamnd.github.io/math2/node/0VTAD25A3G.md): statement 0VTAD25A3G
- [Sup metric on R^n](https://tamnd.github.io/math2/node/0VTAXQEWQD.md): example 0VTAXQEWQD
- [Turán graph](https://tamnd.github.io/math2/node/0VTCABADBE.md): example 0VTCABADBE
- [Ring homomorphism](https://tamnd.github.io/math2/node/0VTH8KPCG2.md): concept 0VTH8KPCG2
- [Standard deviation](https://tamnd.github.io/math2/node/0VTN5RKNC5.md): definition 0VTN5RKNC5
- [Basis](https://tamnd.github.io/math2/node/0VTQTPQGAV.md): definition 0VTQTPQGAV
- [Transcendental number](https://tamnd.github.io/math2/node/0VTQXTCR2G.md): concept 0VTQXTCR2G
- [Euler's totient function](https://tamnd.github.io/math2/node/0VV04M41AN.md): definition 0VV04M41AN
- [Convex function](https://tamnd.github.io/math2/node/0VV0HSW1ZS.md): definition 0VV0HSW1ZS
- [Number of divisors](https://tamnd.github.io/math2/node/0VV3ZMRF88.md): concept 0VV3ZMRF88
- [Discriminant of a polynomial](https://tamnd.github.io/math2/node/0VVK64Q4JK.md): concept 0VVK64Q4JK
- [Order of a power](https://tamnd.github.io/math2/node/0VVPQNVM8S.md): statement 0VVPQNVM8S
- [Term test](https://tamnd.github.io/math2/node/0VVY4H92G2.md): statement 0VVY4H92G2
- [Continuously differentiable function](https://tamnd.github.io/math2/node/0VW90BBSJ4.md): concept 0VW90BBSJ4
- [Likelihood function](https://tamnd.github.io/math2/node/0VW9E4QXR6.md): concept 0VW9E4QXR6
- [Euler line](https://tamnd.github.io/math2/node/0VX0CPRQTZ.md): statement 0VX0CPRQTZ
- [Minimal polynomial (field theory)](https://tamnd.github.io/math2/node/0VX9EP2GS8.md): concept 0VX9EP2GS8
- [Standard score](https://tamnd.github.io/math2/node/0VXJHZ1G9J.md): concept 0VXJHZ1G9J
- [Gradient is orthogonal to level sets](https://tamnd.github.io/math2/node/0VXYV3NQQ1.md): statement 0VXYV3NQQ1
- [Bézout's identity in a Euclidean domain](https://tamnd.github.io/math2/node/0VY3WPP9Y3.md): statement 0VY3WPP9Y3
- [Row echelon form](https://tamnd.github.io/math2/node/0VYCDGMJJJ.md): concept 0VYCDGMJJJ
- [Coprime integers](https://tamnd.github.io/math2/node/0VZ48N083F.md): concept 0VZ48N083F
- [Compact closed unit ball implies finite dimension](https://tamnd.github.io/math2/node/0VZVV8VME5.md): statement 0VZVV8VME5
- [Orthogonal matrix](https://tamnd.github.io/math2/node/0W02QESPJQ.md): concept 0W02QESPJQ
- [Cyclotomic polynomial](https://tamnd.github.io/math2/node/0W0KVBM54S.md): definition 0W0KVBM54S
- [Continuous functions on a closed bounded interval are uniformly continuous](https://tamnd.github.io/math2/node/0W0TZDRF9C.md): statement 0W0TZDRF9C
- [Euler's formula for planar graphs](https://tamnd.github.io/math2/node/0W1C598HMS.md): statement 0W1C598HMS
- [Rearrangement of a series](https://tamnd.github.io/math2/node/0W1K48V5SG.md): concept 0W1K48V5SG
- [Uniform convergence on a set](https://tamnd.github.io/math2/node/0W1QMKJ9W8.md): concept 0W1QMKJ9W8
- [Linear approximation](https://tamnd.github.io/math2/node/0W276SSSCG.md): concept 0W276SSSCG
- [Riemann integral](https://tamnd.github.io/math2/node/0W29TBTY12.md): concept 0W29TBTY12
- [Cofactor](https://tamnd.github.io/math2/node/0W2QAVJQ1W.md): definition 0W2QAVJQ1W
- [Absolute convergence](https://tamnd.github.io/math2/node/0W3K3QPEWT.md): definition 0W3K3QPEWT
- [Line integral as an integral over the parameter interval](https://tamnd.github.io/math2/node/0W3W1VYWBP.md): statement 0W3W1VYWBP
- [Complete subspaces are closed](https://tamnd.github.io/math2/node/0W41XG4W60.md): statement 0W41XG4W60
- [Sesquilinear form](https://tamnd.github.io/math2/node/0W44K2VNBD.md): definition 0W44K2VNBD
- [Convergent sequence](https://tamnd.github.io/math2/node/0W4CN9E2W5.md): concept 0W4CN9E2W5
- [Rank–nullity theorem](https://tamnd.github.io/math2/node/0W4FSW1TVB.md): statement 0W4FSW1TVB
- [Irrationality of e](https://tamnd.github.io/math2/node/0W4GQSG7B6.md): statement 0W4GQSG7B6
- [Disjoint sets](https://tamnd.github.io/math2/node/0W4RPA8QH3.md): definition 0W4RPA8QH3
- [Unique factorisation domain](https://tamnd.github.io/math2/node/0W4X9G6263.md): concept 0W4X9G6263
- [Trail](https://tamnd.github.io/math2/node/0W50MT1C6X.md): concept 0W50MT1C6X
- [Addition of natural numbers](https://tamnd.github.io/math2/node/0W5DSDFQ4G.md): definition 0W5DSDFQ4G
- [Harmonic function](https://tamnd.github.io/math2/node/0W5NJQ8MKY.md): definition 0W5NJQ8MKY
- [Law of total probability for an event and its complement](https://tamnd.github.io/math2/node/0W62TKMW7V.md): statement 0W62TKMW7V
- [Dual basis](https://tamnd.github.io/math2/node/0W6M3PSD08.md): definition 0W6M3PSD08
- [Inverse function](https://tamnd.github.io/math2/node/0W766JAPSN.md): definition 0W766JAPSN
- [Normal subgroup](https://tamnd.github.io/math2/node/0W77CVBCHQ.md): definition 0W77CVBCHQ
- [Uncountable set](https://tamnd.github.io/math2/node/0W7FBYS1F1.md): concept 0W7FBYS1F1
- [Sample space](https://tamnd.github.io/math2/node/0W7NFW23RF.md): definition 0W7NFW23RF
- [Transpose](https://tamnd.github.io/math2/node/0W809K6AW0.md): definition 0W809K6AW0
- [Kernel of a homomorphism](https://tamnd.github.io/math2/node/0W88CEGQP1.md): concept 0W88CEGQP1
- [Tangent half-angle substitution](https://tamnd.github.io/math2/node/0W8ETY2AZ6.md): concept 0W8ETY2AZ6
- [Branch cut](https://tamnd.github.io/math2/node/0W8QHVSN95.md): concept 0W8QHVSN95
- [Epsilon-delta definition of continuity](https://tamnd.github.io/math2/node/0W8YHHGCJG.md): statement 0W8YHHGCJG
- [Fermat pseudoprime](https://tamnd.github.io/math2/node/0W93FJAC8F.md): concept 0W93FJAC8F
- [Equivalence class](https://tamnd.github.io/math2/node/0W95W33CMP.md): concept 0W95W33CMP
- [Chinese remainder theorem for rings](https://tamnd.github.io/math2/node/0W97J8QED7.md): statement 0W97J8QED7
- [Time translation of solutions of an autonomous system](https://tamnd.github.io/math2/node/0WA7ZR95V4.md): statement 0WA7ZR95V4
- [Double integral](https://tamnd.github.io/math2/node/0WAR4F23B4.md): concept 0WAR4F23B4
- [Congruence modulo n](https://tamnd.github.io/math2/node/0WAY5WX8NY.md): concept 0WAY5WX8NY
- [Sine and cosine](https://tamnd.github.io/math2/node/0WB57D9488.md): definition 0WB57D9488
- [Orthocentre](https://tamnd.github.io/math2/node/0WB619CA0V.md): concept 0WB619CA0V
- [Path-connected space](https://tamnd.github.io/math2/node/0WB6DH499T.md): concept 0WB6DH499T
- [Subgraph](https://tamnd.github.io/math2/node/0WBFFC8XZ9.md): concept 0WBFFC8XZ9
- [Norm on the Gaussian integers](https://tamnd.github.io/math2/node/0WBRJXXDPZ.md): concept 0WBRJXXDPZ
- [Integration by partial fractions](https://tamnd.github.io/math2/node/0WBTWFXJQG.md): concept 0WBTWFXJQG
- [Cardinality of the continuum](https://tamnd.github.io/math2/node/0WC28W9DNM.md): object 0WC28W9DNM
- [Clique number](https://tamnd.github.io/math2/node/0WCKRBHFFD.md): definition 0WCKRBHFFD
- [Proof by cases](https://tamnd.github.io/math2/node/0WCP0PVET6.md): concept 0WCP0PVET6
- [Trace of a linear map is the trace of its matrix](https://tamnd.github.io/math2/node/0WCPH5Z7FA.md): statement 0WCPH5Z7FA
- [Binomial distribution](https://tamnd.github.io/math2/node/0WCWA1KT3M.md): concept 0WCWA1KT3M
- [Separable extension](https://tamnd.github.io/math2/node/0WCX7NB4N7.md): concept 0WCX7NB4N7
- [Group action](https://tamnd.github.io/math2/node/0WCY42EV9V.md): concept 0WCY42EV9V
- [Rational function](https://tamnd.github.io/math2/node/0WCZH4AFMT.md): definition 0WCZH4AFMT
- [R/M is a field iff M is maximal](https://tamnd.github.io/math2/node/0WD3SE7S1X.md): statement 0WD3SE7S1X
- [x^2 sin(1/x) is differentiable with discontinuous derivative](https://tamnd.github.io/math2/node/0WDAKKN4NC.md): counterexample 0WDAKKN4NC
- [Cauchy–Euler equation](https://tamnd.github.io/math2/node/0WDNRBXX7K.md): definition 0WDNRBXX7K
- [Order of an element](https://tamnd.github.io/math2/node/0WE2NHZJ7B.md): concept 0WE2NHZJ7B
- [Totally bounded set](https://tamnd.github.io/math2/node/0WE5BBHGM6.md): definition 0WE5BBHGM6
- [Euclid–Euler theorem](https://tamnd.github.io/math2/node/0WETJCKNRG.md): statement 0WETJCKNRG
- [Probability mass function](https://tamnd.github.io/math2/node/0WF4EJP91W.md): definition 0WF4EJP91W
- [Reduced row echelon form](https://tamnd.github.io/math2/node/0WF5AQYTW6.md): concept 0WF5AQYTW6
- [Local maximum and minimum](https://tamnd.github.io/math2/node/0WFBH1Q3WC.md): concept 0WFBH1Q3WC
- [Factorial](https://tamnd.github.io/math2/node/0WFCGVMRRC.md): definition 0WFCGVMRRC
- [Ordered pair](https://tamnd.github.io/math2/node/0WFCNG0MZC.md): concept 0WFCNG0MZC
- [Sylvester's law of inertia](https://tamnd.github.io/math2/node/0WFED3K3DQ.md): statement 0WFED3K3DQ
- [Monotone function](https://tamnd.github.io/math2/node/0WFF6AZPBP.md): definition 0WFF6AZPBP
- [Gambler's ruin](https://tamnd.github.io/math2/node/0WFMKK7DFZ.md): problem 0WFMKK7DFZ
- [Inverse hyperbolic functions](https://tamnd.github.io/math2/node/0WFZX06698.md): concept 0WFZX06698
- [Newton's identities](https://tamnd.github.io/math2/node/0WG2A3HE0S.md): statement 0WG2A3HE0S
- [Two-element Boolean algebra](https://tamnd.github.io/math2/node/0WG43AC0EV.md): example 0WG43AC0EV
- [A subspace of full dimension is the whole space](https://tamnd.github.io/math2/node/0WGAXEK99E.md): statement 0WGAXEK99E
- [Continuous uniform distribution](https://tamnd.github.io/math2/node/0WGDPYR30Q.md): definition 0WGDPYR30Q
- [Circle as a quotient of the real line](https://tamnd.github.io/math2/node/0WH1GVBJNH.md): example 0WH1GVBJNH
- [There is a natural number n with 1/n below any positive epsilon](https://tamnd.github.io/math2/node/0WH916TJNY.md): statement 0WH916TJNY
- [Invariant factors and elementary divisors](https://tamnd.github.io/math2/node/0WHCA4SVP2.md): concept 0WHCA4SVP2
- [Taylor's theorem with the integral remainder](https://tamnd.github.io/math2/node/0WJGR81HBD.md): statement 0WJGR81HBD
- [Free abelian group](https://tamnd.github.io/math2/node/0WJX8MGG37.md): concept 0WJX8MGG37
- [Dimension theorem for vector spaces](https://tamnd.github.io/math2/node/0WK0Y8RH1M.md): statement 0WK0Y8RH1M
- [Sylvester's criterion](https://tamnd.github.io/math2/node/0WKEF78PMV.md): statement 0WKEF78PMV
- [Fermat number](https://tamnd.github.io/math2/node/0WKG15TV3G.md): concept 0WKG15TV3G
- [Cholesky decomposition](https://tamnd.github.io/math2/node/0WKG8CYY6A.md): statement 0WKG8CYY6A
- [Product of exponentials](https://tamnd.github.io/math2/node/0WM7DJCM6G.md): statement 0WM7DJCM6G
- [Volume of a ball in R^3](https://tamnd.github.io/math2/node/0WMDV48VSN.md): statement 0WMDV48VSN
- [A prime is a Gaussian prime if and only if it is 3 modulo 4](https://tamnd.github.io/math2/node/0WMJSEFYFE.md): statement 0WMJSEFYFE
- [Indefinite integral](https://tamnd.github.io/math2/node/0WN1QXR7R3.md): concept 0WN1QXR7R3
- [Hyperbola](https://tamnd.github.io/math2/node/0WN5G117YC.md): concept 0WN5G117YC
- [Edge set](https://tamnd.github.io/math2/node/0WNGQBPPQX.md): definition 0WNGQBPPQX
- [Hoeffding's inequality](https://tamnd.github.io/math2/node/0WPDNXGSSN.md): statement 0WPDNXGSSN
- [Pole](https://tamnd.github.io/math2/node/0WPQYW8MTE.md): definition 0WPQYW8MTE
- [Distance in a graph](https://tamnd.github.io/math2/node/0WPZ5YPAGR.md): definition 0WPZ5YPAGR
- [Distance from a point to a set](https://tamnd.github.io/math2/node/0WQ19TJV24.md): concept 0WQ19TJV24
- [Field extension](https://tamnd.github.io/math2/node/0WQ1A8SHTZ.md): definition 0WQ1A8SHTZ
- [Surface area](https://tamnd.github.io/math2/node/0WQE3G7B88.md): definition 0WQE3G7B88
- [Normal vector](https://tamnd.github.io/math2/node/0WQJJCCTC8.md): concept 0WQJJCCTC8
- [Expected value](https://tamnd.github.io/math2/node/0WQNRCNY3M.md): concept 0WQNRCNY3M
- [Moment-generating function](https://tamnd.github.io/math2/node/0WQVXP7PDB.md): concept 0WQVXP7PDB
- [Borel sigma-algebra](https://tamnd.github.io/math2/node/0WQXY32Q8D.md): concept 0WQXY32Q8D
- [Moment-generating function of a sum of independent random variables](https://tamnd.github.io/math2/node/0WRZ170YEH.md): statement 0WRZ170YEH
- [Extreme value theorem](https://tamnd.github.io/math2/node/0WS1A8E2P9.md): statement 0WS1A8E2P9
- [Subspace topology](https://tamnd.github.io/math2/node/0WS9P12FQR.md): concept 0WS9P12FQR
- [Image of a linear map](https://tamnd.github.io/math2/node/0WSFKKT6FF.md): concept 0WSFKKT6FF
- [Cantor's theorem](https://tamnd.github.io/math2/node/0WTEMGYV2C.md): statement 0WTEMGYV2C
- [Nullcline](https://tamnd.github.io/math2/node/0WW060JR6P.md): definition 0WW060JR6P
- [Bell numbers](https://tamnd.github.io/math2/node/0WW85SQVS3.md): definition 0WW85SQVS3
- [Right inverse](https://tamnd.github.io/math2/node/0WWRD82HD2.md): concept 0WWRD82HD2
- [Group of nth roots of unity](https://tamnd.github.io/math2/node/0WX11EA3F3.md): object 0WX11EA3F3
- [Gram–Schmidt vectors are orthogonal](https://tamnd.github.io/math2/node/0WXBGBTA3D.md): statement 0WXBGBTA3D
- [A monotone function has countably many discontinuities](https://tamnd.github.io/math2/node/0WYTM4C9QG.md): statement 0WYTM4C9QG
- [Iterated integral](https://tamnd.github.io/math2/node/0WZCHE1ZF9.md): concept 0WZCHE1ZF9
- [Hölder continuity](https://tamnd.github.io/math2/node/0WZTM0EFR6.md): definition 0WZTM0EFR6
- [Division ring](https://tamnd.github.io/math2/node/0WZXK05ME7.md): definition 0WZXK05ME7
- [Boolean function](https://tamnd.github.io/math2/node/0X05GS525E.md): concept 0X05GS525E
- [Determinant of a linear map](https://tamnd.github.io/math2/node/0X0HQYSZ0K.md): concept 0X0HQYSZ0K
- [Number of injections](https://tamnd.github.io/math2/node/0X0N833FF8.md): statement 0X0N833FF8
- [Union](https://tamnd.github.io/math2/node/0X1GSWNBN1.md): concept 0X1GSWNBN1
- [Subgroup](https://tamnd.github.io/math2/node/0X1JXT47GA.md): concept 0X1JXT47GA
- [Complex logarithm](https://tamnd.github.io/math2/node/0X2FY491PF.md): definition 0X2FY491PF
- [Hall's theorem for bipartite graphs](https://tamnd.github.io/math2/node/0X2GJPNPFN.md): statement 0X2GJPNPFN
- [Discrete random variable](https://tamnd.github.io/math2/node/0X2J0VY6DN.md): concept 0X2J0VY6DN
- [Countable intersection of dense open sets is dense](https://tamnd.github.io/math2/node/0X2T63GFQQ.md): statement 0X2T63GFQQ
- [The exterior derivative squares to zero](https://tamnd.github.io/math2/node/0X308R2DBE.md): statement 0X308R2DBE
- [Path component](https://tamnd.github.io/math2/node/0X3EG7Y65T.md): concept 0X3EG7Y65T
- [Congruent triangles](https://tamnd.github.io/math2/node/0X3TEM69AA.md): definition 0X3TEM69AA
- [Continuous function](https://tamnd.github.io/math2/node/0X3WGX95FE.md): concept 0X3WGX95FE
- [Laurent series](https://tamnd.github.io/math2/node/0X3XD2ATB0.md): definition 0X3XD2ATB0
- [Hypergeometric distribution](https://tamnd.github.io/math2/node/0X3Y0ZSKNF.md): concept 0X3Y0ZSKNF
- [Cayley transform](https://tamnd.github.io/math2/node/0X45ZM731G.md): example 0X45ZM731G
- [Idempotent element](https://tamnd.github.io/math2/node/0X4JKZXJ86.md): definition 0X4JKZXJ86
- [Forced oscillation and resonance](https://tamnd.github.io/math2/node/0X4NT20S3G.md): example 0X4NT20S3G
- [Tietze extension theorem](https://tamnd.github.io/math2/node/0X52TNMTCJ.md): statement 0X52TNMTCJ
- [Uncountable set](https://tamnd.github.io/math2/node/0X55M9XET4.md): definition 0X55M9XET4
- [Elementary symmetric polynomial](https://tamnd.github.io/math2/node/0X5QVCF06N.md): concept 0X5QVCF06N
- [Quantile function](https://tamnd.github.io/math2/node/0X60VE98HN.md): concept 0X60VE98HN
- [Algebraic closure](https://tamnd.github.io/math2/node/0X6AB5P3G1.md): concept 0X6AB5P3G1
- [Hyperbolic tangent](https://tamnd.github.io/math2/node/0X6C0MAF84.md): concept 0X6C0MAF84
- [Consistent and inconsistent systems](https://tamnd.github.io/math2/node/0X6V9B1V10.md): concept 0X6V9B1V10
- [Impossibility of squaring the circle](https://tamnd.github.io/math2/node/0X7CVDE3BG.md): statement 0X7CVDE3BG
- [Centroid](https://tamnd.github.io/math2/node/0X7EEWES5G.md): concept 0X7EEWES5G
- [Arc length](https://tamnd.github.io/math2/node/0X7G61297X.md): definition 0X7G61297X
- [Winding number](https://tamnd.github.io/math2/node/0X7SBS8E0F.md): definition 0X7SBS8E0F
- [Altitude of a triangle](https://tamnd.github.io/math2/node/0X89GC65N1.md): definition 0X89GC65N1
- [Continuity of probability measures](https://tamnd.github.io/math2/node/0X8HBKCSYB.md): statement 0X8HBKCSYB
- [Reflexive relation](https://tamnd.github.io/math2/node/0X8NMST0J2.md): concept 0X8NMST0J2
- [Space of linear maps](https://tamnd.github.io/math2/node/0X9MBPP2Z3.md): concept 0X9MBPP2Z3
- [Uniform limit theorem](https://tamnd.github.io/math2/node/0X9XT05JVC.md): statement 0X9XT05JVC
- [Boolean algebra](https://tamnd.github.io/math2/node/0XADJKM699.md): concept 0XADJKM699
- [Handshaking lemma](https://tamnd.github.io/math2/node/0XAP8WHNES.md): statement 0XAP8WHNES
- [Simple continued fraction](https://tamnd.github.io/math2/node/0XB0J0P64D.md): concept 0XB0J0P64D
- [Poisson distribution](https://tamnd.github.io/math2/node/0XB9GQTN9G.md): concept 0XB9GQTN9G
- [Negative binomial distribution](https://tamnd.github.io/math2/node/0XBRG3965F.md): concept 0XBRG3965F
- [Trace of a matrix](https://tamnd.github.io/math2/node/0XC1RDBY7A.md): concept 0XC1RDBY7A
- [Independence of random variables](https://tamnd.github.io/math2/node/0XCE070V8B.md): definition 0XCE070V8B
- [Cantor's diagonal argument](https://tamnd.github.io/math2/node/0XCH8QVDCQ.md): concept 0XCH8QVDCQ
- [Geometric distribution](https://tamnd.github.io/math2/node/0XCVV5Q500.md): definition 0XCVV5Q500
- [Wallis product](https://tamnd.github.io/math2/node/0XCYA4YERS.md): statement 0XCYA4YERS
- [Number of elements of each order in a cyclic group](https://tamnd.github.io/math2/node/0XCYGCCX8V.md): statement 0XCYGCCX8V
- [Monotonicity of the cumulative distribution function](https://tamnd.github.io/math2/node/0XD22PBMWA.md): statement 0XD22PBMWA
- [Jacobi symbol](https://tamnd.github.io/math2/node/0XDB5AW7M1.md): definition 0XDB5AW7M1
- [Rational function](https://tamnd.github.io/math2/node/0XDK9CGYKJ.md): concept 0XDK9CGYKJ
- [Even and odd functions](https://tamnd.github.io/math2/node/0XE690QF65.md): definition 0XE690QF65
- [Reflexive relation](https://tamnd.github.io/math2/node/0XEGNW851S.md): definition 0XEGNW851S
- [Multinomial theorem](https://tamnd.github.io/math2/node/0XFC1VQYEK.md): statement 0XFC1VQYEK
- [Euler's partition theorem](https://tamnd.github.io/math2/node/0XFZ1BMF5Q.md): statement 0XFZ1BMF5Q
- [Meagre set](https://tamnd.github.io/math2/node/0XGCZ4GDZ6.md): concept 0XGCZ4GDZ6
- [Archimedean ordered field](https://tamnd.github.io/math2/node/0XH79S06J3.md): concept 0XH79S06J3
- [Function](https://tamnd.github.io/math2/node/0XHDTE5YMZ.md): definition 0XHDTE5YMZ
- [Finitely generated module](https://tamnd.github.io/math2/node/0XHM30WBVB.md): definition 0XHM30WBVB
- [Derangement](https://tamnd.github.io/math2/node/0XHWTD3MPV.md): concept 0XHWTD3MPV
- [K-colourable graph](https://tamnd.github.io/math2/node/0XHZW0VCPR.md): concept 0XHZW0VCPR
- [Indiscrete topology](https://tamnd.github.io/math2/node/0XJ02JBD6S.md): example 0XJ02JBD6S
- [Consistent estimator](https://tamnd.github.io/math2/node/0XJVF81QF6.md): definition 0XJVF81QF6
- [Orthogonal complement](https://tamnd.github.io/math2/node/0XJXTBCB2T.md): concept 0XJXTBCB2T
- [Order of a differential equation](https://tamnd.github.io/math2/node/0XK0G19SK8.md): concept 0XK0G19SK8
- [Decomposition of a permutation into transpositions](https://tamnd.github.io/math2/node/0XK6FECMW5.md): statement 0XK6FECMW5
- [Length of a vector](https://tamnd.github.io/math2/node/0XKBKGTCPC.md): concept 0XKBKGTCPC
- [Surjective function](https://tamnd.github.io/math2/node/0XKH881BZ7.md): definition 0XKH881BZ7
- [Periodic continued fraction](https://tamnd.github.io/math2/node/0XKJPFS2W0.md): definition 0XKJPFS2W0
- [Poisson limit theorem](https://tamnd.github.io/math2/node/0XKTZRWTXX.md): statement 0XKTZRWTXX
- [Limit of a sequence](https://tamnd.github.io/math2/node/0XKWSSY8P2.md): definition 0XKWSSY8P2
- [Coordinate space](https://tamnd.github.io/math2/node/0XM8BZD76G.md): object 0XM8BZD76G
- [Sieve of Eratosthenes](https://tamnd.github.io/math2/node/0XM8ZA9RNM.md): concept 0XM8ZA9RNM
- [Semigroup](https://tamnd.github.io/math2/node/0XMEM406JM.md): concept 0XMEM406JM
- [Squeeze theorem](https://tamnd.github.io/math2/node/0XMF00T4XQ.md): statement 0XMF00T4XQ
- [Multivariate normal distribution](https://tamnd.github.io/math2/node/0XN3K7RSZY.md): concept 0XN3K7RSZY
- [Equicontinuity](https://tamnd.github.io/math2/node/0XN66MZN0Y.md): concept 0XN66MZN0Y
- [Negative binomial distribution](https://tamnd.github.io/math2/node/0XN7KEVSSS.md): definition 0XN7KEVSSS
- [Acyclic graph](https://tamnd.github.io/math2/node/0XNN9Z8E5A.md): definition 0XNN9Z8E5A
- [Rotation](https://tamnd.github.io/math2/node/0XNZWJ96HK.md): definition 0XNZWJ96HK
- [Cofactor](https://tamnd.github.io/math2/node/0XPKR18APA.md): concept 0XPKR18APA
- [Euclidean space](https://tamnd.github.io/math2/node/0XPR1E85W0.md): object 0XPR1E85W0
- [Spectrum of a matrix](https://tamnd.github.io/math2/node/0XPX5NC67E.md): concept 0XPX5NC67E
- [Laplacian](https://tamnd.github.io/math2/node/0XQ2ZVN1PZ.md): definition 0XQ2ZVN1PZ
- [Greatest element](https://tamnd.github.io/math2/node/0XR2QK49V6.md): concept 0XR2QK49V6
- [Negation](https://tamnd.github.io/math2/node/0XR7A7RSXB.md): definition 0XR7A7RSXB
- [Method of undetermined coefficients](https://tamnd.github.io/math2/node/0XRB469Z32.md): concept 0XRB469Z32
- [Farey sequence](https://tamnd.github.io/math2/node/0XRDYJQES6.md): concept 0XRDYJQES6
- [Prime-counting function](https://tamnd.github.io/math2/node/0XRJB5SSNE.md): definition 0XRJB5SSNE
- [Dini's theorem](https://tamnd.github.io/math2/node/0XRRYF1HPE.md): statement 0XRRYF1HPE
- [Eisenstein's lemma](https://tamnd.github.io/math2/node/0XSDBCP4RZ.md): statement 0XSDBCP4RZ
- [Linear isomorphism](https://tamnd.github.io/math2/node/0XSJ25Z9M4.md): definition 0XSJ25Z9M4
- [Cluster point of a sequence](https://tamnd.github.io/math2/node/0XTNJ6SVAC.md): concept 0XTNJ6SVAC
- [Galois group](https://tamnd.github.io/math2/node/0XTY86BZKY.md): concept 0XTY86BZKY
- [Composition series](https://tamnd.github.io/math2/node/0XV2A49X6Q.md): concept 0XV2A49X6Q
- [Unit of a ring](https://tamnd.github.io/math2/node/0XVEZDE9J8.md): concept 0XVEZDE9J8
- [The principal logarithm is holomorphic off the branch cut](https://tamnd.github.io/math2/node/0XVGJE7Q04.md): statement 0XVGJE7Q04
- [Galois group](https://tamnd.github.io/math2/node/0XVR72M51M.md): definition 0XVR72M51M
- [Upper central series](https://tamnd.github.io/math2/node/0XW54DH28V.md): concept 0XW54DH28V
- [Euclidean plane](https://tamnd.github.io/math2/node/0XW7V9ZZ05.md): object 0XW7V9ZZ05
- [Median of a triangle](https://tamnd.github.io/math2/node/0XW8HX8VEV.md): concept 0XW8HX8VEV
- [A root of an integer is an integer or irrational](https://tamnd.github.io/math2/node/0XWCKJZRYH.md): statement 0XWCKJZRYH
- [Factorial](https://tamnd.github.io/math2/node/0XWG0JGDFW.md): concept 0XWG0JGDFW
- [Second fundamental theorem of calculus](https://tamnd.github.io/math2/node/0XYAFCN0PT.md): statement 0XYAFCN0PT
- [Normal matrix](https://tamnd.github.io/math2/node/0XYKE8BE7X.md): definition 0XYKE8BE7X
- [Normal vector](https://tamnd.github.io/math2/node/0XYP2KKSPC.md): definition 0XYP2KKSPC
- [De Morgan's laws in a Boolean algebra](https://tamnd.github.io/math2/node/0XYVDWGQX2.md): statement 0XYVDWGQX2
- [Direct sum of subspaces](https://tamnd.github.io/math2/node/0XYYT5KNTX.md): concept 0XYYT5KNTX
- [Disjoint sets](https://tamnd.github.io/math2/node/0XZ37KAB1Z.md): concept 0XZ37KAB1Z
- [Rooted tree](https://tamnd.github.io/math2/node/0Y0JGC2K9T.md): concept 0Y0JGC2K9T
- [Graph of a function](https://tamnd.github.io/math2/node/0Y0KFHF47G.md): definition 0Y0KFHF47G
- [Modus ponens](https://tamnd.github.io/math2/node/0Y0MQYST0M.md): concept 0Y0MQYST0M
- [Inflection point](https://tamnd.github.io/math2/node/0Y0WS5BCJ7.md): definition 0Y0WS5BCJ7
- [Absolutely convergent series converge](https://tamnd.github.io/math2/node/0Y123X48GA.md): statement 0Y123X48GA
- [Limit at infinity](https://tamnd.github.io/math2/node/0Y12QYTJK3.md): definition 0Y12QYTJK3
- [Diagonalisable matrix](https://tamnd.github.io/math2/node/0Y17PEKF24.md): definition 0Y17PEKF24
- [Catalan recurrence](https://tamnd.github.io/math2/node/0Y1MG9R5VN.md): statement 0Y1MG9R5VN
- [Orthogonal vectors](https://tamnd.github.io/math2/node/0Y2X7P2WB7.md): concept 0Y2X7P2WB7
- [Gauss–Wantzel theorem](https://tamnd.github.io/math2/node/0Y34JMJNE9.md): statement 0Y34JMJNE9
- [Series of functions](https://tamnd.github.io/math2/node/0Y3KSN2E9X.md): concept 0Y3KSN2E9X
- [The order modulo p divides p − 1](https://tamnd.github.io/math2/node/0Y4B7P0GS8.md): statement 0Y4B7P0GS8
- [Trace of a product is symmetric](https://tamnd.github.io/math2/node/0Y4NF1DRKD.md): statement 0Y4NF1DRKD
- [Brooks' theorem](https://tamnd.github.io/math2/node/0Y4TV2D5RQ.md): statement 0Y4TV2D5RQ
- [Level set](https://tamnd.github.io/math2/node/0Y5Q8Q9WJ7.md): definition 0Y5Q8Q9WJ7
- [Outcome](https://tamnd.github.io/math2/node/0Y5R7TPNT1.md): definition 0Y5R7TPNT1
- [Centroid](https://tamnd.github.io/math2/node/0Y5RHNRKJZ.md): definition 0Y5RHNRKJZ
- [Fermat's Last Theorem](https://tamnd.github.io/math2/node/0Y5X3SDBCZ.md): statement 0Y5X3SDBCZ
- [Tail sum formula for expectation](https://tamnd.github.io/math2/node/0Y6QNF1J49.md): statement 0Y6QNF1J49
- [Variation of parameters](https://tamnd.github.io/math2/node/0Y6TH5PQVS.md): concept 0Y6TH5PQVS
- [Order isomorphism](https://tamnd.github.io/math2/node/0Y74KN99A5.md): definition 0Y74KN99A5
- [Matrix addition](https://tamnd.github.io/math2/node/0Y7H3SAVA1.md): concept 0Y7H3SAVA1
- [Directional derivative](https://tamnd.github.io/math2/node/0Y7PVW4Q44.md): definition 0Y7PVW4Q44
- [K-colourable graph](https://tamnd.github.io/math2/node/0Y7VDAX8Q3.md): definition 0Y7VDAX8Q3
- [Existence of an orthonormal basis](https://tamnd.github.io/math2/node/0Y85GYEGSF.md): statement 0Y85GYEGSF
- [Sequence of functions](https://tamnd.github.io/math2/node/0Y8FRYX8Q2.md): concept 0Y8FRYX8Q2
- [Distinct eigenvalues imply diagonalisable](https://tamnd.github.io/math2/node/0Y95Z7MT02.md): statement 0Y95Z7MT02
- [Binary trees are counted by Catalan numbers](https://tamnd.github.io/math2/node/0Y97E2DMB7.md): statement 0Y97E2DMB7
- [Second derivative](https://tamnd.github.io/math2/node/0Y9D2FJRQ4.md): definition 0Y9D2FJRQ4
- [Order on the natural numbers](https://tamnd.github.io/math2/node/0Y9H52BT03.md): definition 0Y9H52BT03
- [Characteristic polynomial of a linear recurrence](https://tamnd.github.io/math2/node/0YA7X86K05.md): concept 0YA7X86K05
- [Primitive element of a finite field](https://tamnd.github.io/math2/node/0YAA9CJQXF.md): definition 0YAA9CJQXF
- [Mersenne number](https://tamnd.github.io/math2/node/0YAG43TCXQ.md): concept 0YAG43TCXQ
- [Conditional expectation](https://tamnd.github.io/math2/node/0YAMJRK04E.md): definition 0YAMJRK04E
- [Cayley table](https://tamnd.github.io/math2/node/0YB2NRP0T5.md): concept 0YB2NRP0T5
- [Absorbing Markov chain](https://tamnd.github.io/math2/node/0YB93GWTKK.md): definition 0YB93GWTKK
- [Absolute convergence](https://tamnd.github.io/math2/node/0YBDSJARWN.md): concept 0YBDSJARWN
- [Comparability of cardinals](https://tamnd.github.io/math2/node/0YBHSJMN3J.md): statement 0YBHSJMN3J
- [Groups of order pq](https://tamnd.github.io/math2/node/0YBR6QPY4Y.md): example 0YBR6QPY4Y
- [Identity theorem](https://tamnd.github.io/math2/node/0YBVKMYX0W.md): statement 0YBVKMYX0W
- [Term-by-term integration](https://tamnd.github.io/math2/node/0YBY3VYX0K.md): statement 0YBY3VYX0K
- [Matrix exponential as the solution of a linear system](https://tamnd.github.io/math2/node/0YC8JT513J.md): statement 0YC8JT513J
- [Augmented matrix](https://tamnd.github.io/math2/node/0YCDZ7N8VD.md): definition 0YCDZ7N8VD
- [Open map](https://tamnd.github.io/math2/node/0YCG648KDD.md): concept 0YCG648KDD
- [Seven Bridges of Königsberg](https://tamnd.github.io/math2/node/0YCW781Q9D.md): example 0YCW781Q9D
- [Universal property of the tensor product](https://tamnd.github.io/math2/node/0YDRV01R8F.md): statement 0YDRV01R8F
- [Open ball](https://tamnd.github.io/math2/node/0YE5868HZB.md): concept 0YE5868HZB
- [Cayley graph](https://tamnd.github.io/math2/node/0YE597NHGB.md): definition 0YE597NHGB
- [Combinations with repetition](https://tamnd.github.io/math2/node/0YEQG3VBYC.md): definition 0YEQG3VBYC
- [Divisor function](https://tamnd.github.io/math2/node/0YEW47VJ0J.md): definition 0YEW47VJ0J
- [Jacobian matrix](https://tamnd.github.io/math2/node/0YF0VXKXGB.md): definition 0YF0VXKXGB
- [Similar triangles](https://tamnd.github.io/math2/node/0YF6QXXWB5.md): concept 0YF6QXXWB5
- [Continuity via open sets](https://tamnd.github.io/math2/node/0YF82GBTND.md): statement 0YF82GBTND
- [Harmonic oscillator](https://tamnd.github.io/math2/node/0YG0ZJPTWG.md): example 0YG0ZJPTWG
- [Normaliser](https://tamnd.github.io/math2/node/0YGN3VAA0D.md): concept 0YGN3VAA0D
- [Sum of independent Poisson random variables](https://tamnd.github.io/math2/node/0YH1M56SBN.md): statement 0YH1M56SBN
- [Irrationality of the square root of a prime](https://tamnd.github.io/math2/node/0YHFMXJJAZ.md): statement 0YHFMXJJAZ
- [Integers modulo n as a quotient group](https://tamnd.github.io/math2/node/0YJ3WRR9RY.md): example 0YJ3WRR9RY
- [Order of a pole](https://tamnd.github.io/math2/node/0YK27TJ4WF.md): definition 0YK27TJ4WF
- [Hilbert's basis theorem](https://tamnd.github.io/math2/node/0YKA9J3XEE.md): statement 0YKA9J3XEE
- [Essential singularity](https://tamnd.github.io/math2/node/0YKDJBV26S.md): definition 0YKDJBV26S
- [Principal branch](https://tamnd.github.io/math2/node/0YKY4M3KRQ.md): definition 0YKY4M3KRQ
- [Existence of nth roots of complex numbers](https://tamnd.github.io/math2/node/0YM6P6C206.md): statement 0YM6P6C206
- [Affine span](https://tamnd.github.io/math2/node/0YMB1676R1.md): concept 0YMB1676R1
- [Power series are holomorphic in the disc of convergence](https://tamnd.github.io/math2/node/0YMP521ZR5.md): statement 0YMP521ZR5
- [Free abelian group](https://tamnd.github.io/math2/node/0YMSEZE1VH.md): definition 0YMSEZE1VH
- [Irreducible element](https://tamnd.github.io/math2/node/0YMT0D2442.md): concept 0YMT0D2442
- [Simple group](https://tamnd.github.io/math2/node/0YMYMNQ0QZ.md): concept 0YMYMNQ0QZ
- [Direct product of groups](https://tamnd.github.io/math2/node/0YP7WYXF9V.md): definition 0YP7WYXF9V
- [Flux through a surface](https://tamnd.github.io/math2/node/0YPRY7T4PK.md): definition 0YPRY7T4PK
- [Fundamental theorem of contour integration](https://tamnd.github.io/math2/node/0YQ458D7PF.md): statement 0YQ458D7PF
- [Orbit](https://tamnd.github.io/math2/node/0YQPZB5Q1G.md): concept 0YQPZB5Q1G
- [Intersecting chords theorem](https://tamnd.github.io/math2/node/0YQQYET8YQ.md): statement 0YQQYET8YQ
- [Matrix of a linear map](https://tamnd.github.io/math2/node/0YQSGNYE3K.md): concept 0YQSGNYE3K
- [Pascal's rule](https://tamnd.github.io/math2/node/0YRG241HQQ.md): statement 0YRG241HQQ
- [Closed balls are closed](https://tamnd.github.io/math2/node/0YS40B38JD.md): statement 0YS40B38JD
- [Line integral of a vector field](https://tamnd.github.io/math2/node/0YS8RX87F4.md): definition 0YS8RX87F4
- [Ordinal number](https://tamnd.github.io/math2/node/0YSCSX5YZC.md): concept 0YSCSX5YZC
- [Group action](https://tamnd.github.io/math2/node/0YSE802YCQ.md): definition 0YSE802YCQ
- [x^n converges pointwise but not uniformly on [0, 1]](https://tamnd.github.io/math2/node/0YSRR1JTKZ.md): counterexample 0YSRR1JTKZ
- [Cyclic subgroup generated by an element](https://tamnd.github.io/math2/node/0YT586TKXF.md): definition 0YT586TKXF
- [Critical point](https://tamnd.github.io/math2/node/0YTCMFB4HC.md): concept 0YTCMFB4HC
- [Dyck word](https://tamnd.github.io/math2/node/0YTJ3RN3GE.md): concept 0YTJ3RN3GE
- [Root of unity](https://tamnd.github.io/math2/node/0YV2CKP2AV.md): concept 0YV2CKP2AV
- [Conditional probability distribution](https://tamnd.github.io/math2/node/0YVDC5PYDW.md): definition 0YVDC5PYDW
- [Metric topology](https://tamnd.github.io/math2/node/0YVKNB7V5F.md): concept 0YVKNB7V5F
- [Skewness](https://tamnd.github.io/math2/node/0YVQKBNV2R.md): concept 0YVQKBNV2R
- [Floor function](https://tamnd.github.io/math2/node/0YVRNEDSVG.md): definition 0YVRNEDSVG
- [Homotopy invariance of line integrals of closed forms](https://tamnd.github.io/math2/node/0YVVN6R715.md): statement 0YVVN6R715
- [Number of distinct prime factors](https://tamnd.github.io/math2/node/0YVVXT3A0B.md): definition 0YVVXT3A0B
- [Decomposition of a permutation into disjoint cycles](https://tamnd.github.io/math2/node/0YWDAZPRPQ.md): statement 0YWDAZPRPQ
- [Catalan numbers](https://tamnd.github.io/math2/node/0YWZJEABB2.md): concept 0YWZJEABB2
- [Tangent plane to a graph](https://tamnd.github.io/math2/node/0YXEHVA93R.md): definition 0YXEHVA93R
- [Choice function](https://tamnd.github.io/math2/node/0YXEQWRGP0.md): definition 0YXEQWRGP0
- [Quadratic reciprocity for the Jacobi symbol](https://tamnd.github.io/math2/node/0YY06KP3ZQ.md): statement 0YY06KP3ZQ
- [Moore–Penrose inverse](https://tamnd.github.io/math2/node/0YY8VX28JR.md): concept 0YY8VX28JR
- [Matrix of a linear map](https://tamnd.github.io/math2/node/0YYMY95FWS.md): definition 0YYMY95FWS
- [Conformal map](https://tamnd.github.io/math2/node/0YYT36R4YD.md): concept 0YYT36R4YD
- [Quotient group](https://tamnd.github.io/math2/node/0YYWCNTPWR.md): definition 0YYWCNTPWR
- [Determinant as a group homomorphism](https://tamnd.github.io/math2/node/0YZ3T5Q7XN.md): example 0YZ3T5Q7XN
- [Mercator series](https://tamnd.github.io/math2/node/0YZXNE3HFR.md): statement 0YZXNE3HFR
- [Cycle](https://tamnd.github.io/math2/node/0Z05VCBQ7Y.md): concept 0Z05VCBQ7Y
- [Pell's equation](https://tamnd.github.io/math2/node/0Z0NE3J6JX.md): concept 0Z0NE3J6JX
- [Power series of complex cosine](https://tamnd.github.io/math2/node/0Z1G2F8DGY.md): statement 0Z1G2F8DGY
- [Diffeomorphism](https://tamnd.github.io/math2/node/0Z1Y7JHGBR.md): definition 0Z1Y7JHGBR
- [Stirling numbers of the first kind](https://tamnd.github.io/math2/node/0Z22VQH17V.md): definition 0Z22VQH17V
- [Separable space](https://tamnd.github.io/math2/node/0Z2P41SVDP.md): concept 0Z2P41SVDP
- [Minimum spanning tree](https://tamnd.github.io/math2/node/0Z367CGNRM.md): definition 0Z367CGNRM
- [Diagonalisation of a quadratic form](https://tamnd.github.io/math2/node/0Z3FEWZ8HA.md): statement 0Z3FEWZ8HA
- [Decomposition into generalised eigenspaces](https://tamnd.github.io/math2/node/0Z3GDY89T5.md): statement 0Z3GDY89T5
- [Linear first-order equation](https://tamnd.github.io/math2/node/0Z3YKRQZ48.md): definition 0Z3YKRQZ48
- [Intermediate field](https://tamnd.github.io/math2/node/0Z409N8RDG.md): concept 0Z409N8RDG
- [A function with nonnegative second derivative is convex](https://tamnd.github.io/math2/node/0Z41C17TGP.md): statement 0Z41C17TGP
- [Kernel of a homomorphism](https://tamnd.github.io/math2/node/0Z424HK8CQ.md): definition 0Z424HK8CQ
- [Burnside's lemma](https://tamnd.github.io/math2/node/0Z47M1B2VS.md): statement 0Z47M1B2VS
- [Symmetric difference](https://tamnd.github.io/math2/node/0Z49NTR1DZ.md): definition 0Z49NTR1DZ
- [Taylor series](https://tamnd.github.io/math2/node/0Z49YH6ZWY.md): concept 0Z49YH6ZWY
- [Determinant of the identity matrix](https://tamnd.github.io/math2/node/0Z4BWARC38.md): statement 0Z4BWARC38
- [Proper metric space](https://tamnd.github.io/math2/node/0Z4V98VN5E.md): definition 0Z4V98VN5E
- [Product topology](https://tamnd.github.io/math2/node/0Z57EC6YX3.md): definition 0Z57EC6YX3
- [Alternating group](https://tamnd.github.io/math2/node/0Z5905Q73D.md): concept 0Z5905Q73D
- [Interval of convergence](https://tamnd.github.io/math2/node/0Z5R9TVWQH.md): definition 0Z5R9TVWQH
- [Greatest element](https://tamnd.github.io/math2/node/0Z7HDDXX3X.md): definition 0Z7HDDXX3X
- [Computing the inverse by row reduction](https://tamnd.github.io/math2/node/0Z83DHFMYM.md): concept 0Z83DHFMYM
- [Matching](https://tamnd.github.io/math2/node/0Z87BHAFTB.md): definition 0Z87BHAFTB
- [The complex conjugate is not holomorphic](https://tamnd.github.io/math2/node/0Z8ATBPM5W.md): counterexample 0Z8ATBPM5W
- [Local homeomorphism from the inverse function theorem](https://tamnd.github.io/math2/node/0Z8CSYVHDJ.md): statement 0Z8CSYVHDJ
- [Countable union of countable sets is countable](https://tamnd.github.io/math2/node/0Z8FM54P86.md): statement 0Z8FM54P86
- [Probability-generating function](https://tamnd.github.io/math2/node/0Z8PSSAVBA.md): definition 0Z8PSSAVBA
- [Trial division](https://tamnd.github.io/math2/node/0Z962ECCXR.md): concept 0Z962ECCXR
- [Constant multiple rule](https://tamnd.github.io/math2/node/0Z9JB71BS0.md): statement 0Z9JB71BS0
- [Dimension of the space of linear maps](https://tamnd.github.io/math2/node/0Z9N1ER0W0.md): statement 0Z9N1ER0W0
- [Cauchy criterion for sequences](https://tamnd.github.io/math2/node/0ZA4AX4MW3.md): statement 0ZA4AX4MW3
- [Greatest common divisor in an integral domain](https://tamnd.github.io/math2/node/0ZAF9KHDDF.md): concept 0ZAF9KHDDF
- [Memorylessness](https://tamnd.github.io/math2/node/0ZAZZ34WQS.md): concept 0ZAZZ34WQS
- [Nilpotent element](https://tamnd.github.io/math2/node/0ZB2JG1M9C.md): concept 0ZB2JG1M9C
- [Pairwise independence](https://tamnd.github.io/math2/node/0ZB3M68NYQ.md): concept 0ZB3M68NYQ
- [A complex number times its conjugate](https://tamnd.github.io/math2/node/0ZB7J071WQ.md): statement 0ZB7J071WQ
- [Bernoulli equation](https://tamnd.github.io/math2/node/0ZB84QCDKR.md): definition 0ZB84QCDKR
- [Gradient points in the direction of steepest ascent](https://tamnd.github.io/math2/node/0ZC78099PA.md): statement 0ZC78099PA
- [Orbits partition the set](https://tamnd.github.io/math2/node/0ZC7FYE0NZ.md): statement 0ZC7FYE0NZ
- [Rank of a matrix](https://tamnd.github.io/math2/node/0ZC9YTXZRY.md): concept 0ZC9YTXZRY
- [Ascending chain condition](https://tamnd.github.io/math2/node/0ZCAXY9563.md): definition 0ZCAXY9563
- [Closed set](https://tamnd.github.io/math2/node/0ZCK8DQYN1.md): concept 0ZCK8DQYN1
- [Jacobian determinant](https://tamnd.github.io/math2/node/0ZCSMYHTBP.md): concept 0ZCSMYHTBP
- [Companion matrix](https://tamnd.github.io/math2/node/0ZCWPNZTQ3.md): concept 0ZCWPNZTQ3
- [Neighbourhood](https://tamnd.github.io/math2/node/0ZCY0Z80B6.md): concept 0ZCY0Z80B6
- [Tangent vector of a curve](https://tamnd.github.io/math2/node/0ZCZK72B45.md): concept 0ZCZK72B45
- [Dual basis](https://tamnd.github.io/math2/node/0ZD2CF6ZS0.md): concept 0ZD2CF6ZS0
- [Angle at a point](https://tamnd.github.io/math2/node/0ZDDNBQSC4.md): definition 0ZDDNBQSC4
- [Exponent of a group](https://tamnd.github.io/math2/node/0ZDW0H0VX4.md): concept 0ZDW0H0VX4
- [Angle between two vectors](https://tamnd.github.io/math2/node/0ZEHZY85E6.md): concept 0ZEHZY85E6
- [g to the order of the group is the identity](https://tamnd.github.io/math2/node/0ZEPNS9BXK.md): statement 0ZEPNS9BXK
- [Universal property of the quotient space](https://tamnd.github.io/math2/node/0ZERJX6PH6.md): statement 0ZERJX6PH6
- [Median of a triangle](https://tamnd.github.io/math2/node/0ZET0ZH16W.md): definition 0ZET0ZH16W
- [Minimum degree](https://tamnd.github.io/math2/node/0ZFZ8ZG3EN.md): concept 0ZFZ8ZG3EN
- [Homothety](https://tamnd.github.io/math2/node/0ZG1PNEP35.md): concept 0ZG1PNEP35
- [Symmetry group](https://tamnd.github.io/math2/node/0ZG3Z5JZ85.md): concept 0ZG3Z5JZ85
- [Countable set](https://tamnd.github.io/math2/node/0ZG61TFBES.md): definition 0ZG61TFBES
- [Parametrised surface](https://tamnd.github.io/math2/node/0ZG9QF40HE.md): concept 0ZG9QF40HE
- [Proposition](https://tamnd.github.io/math2/node/0ZGC7ZESYN.md): concept 0ZGC7ZESYN
- [Baire space](https://tamnd.github.io/math2/node/0ZH5AJ2ED7.md): concept 0ZH5AJ2ED7
- [Transfinite induction](https://tamnd.github.io/math2/node/0ZHQMQ2C1N.md): statement 0ZHQMQ2C1N
- [Exterior derivative](https://tamnd.github.io/math2/node/0ZHSTAG8ZY.md): concept 0ZHSTAG8ZY
- [Finite-dimensional vector space](https://tamnd.github.io/math2/node/0ZHWKQMVX5.md): concept 0ZHWKQMVX5
- [Conditional independence](https://tamnd.github.io/math2/node/0ZJ87ZZ8PK.md): definition 0ZJ87ZZ8PK
- [Canonical isomorphism with the double dual](https://tamnd.github.io/math2/node/0ZJM9D7NPS.md): statement 0ZJM9D7NPS
- [Cycle notation](https://tamnd.github.io/math2/node/0ZKPH2BD1T.md): concept 0ZKPH2BD1T
- [Absorption law](https://tamnd.github.io/math2/node/0ZKZBJPG2J.md): statement 0ZKZBJPG2J
- [Path-connected spaces are connected](https://tamnd.github.io/math2/node/0ZMT1GP4KM.md): statement 0ZMT1GP4KM
- [Elementary symmetric polynomials](https://tamnd.github.io/math2/node/0ZMY8441GW.md): definition 0ZMY8441GW
- [First isomorphism theorem for rings](https://tamnd.github.io/math2/node/0ZMYX7GMH0.md): statement 0ZMYX7GMH0
- [Stirling numbers of the second kind](https://tamnd.github.io/math2/node/0ZN07RQ3JS.md): definition 0ZN07RQ3JS
- [Determinant of a 2 by 2 matrix](https://tamnd.github.io/math2/node/0ZN2J42SK7.md): statement 0ZN2J42SK7
- [K(a) is isomorphic to K[x] modulo the minimal polynomial](https://tamnd.github.io/math2/node/0ZN39QDTP9.md): statement 0ZN39QDTP9
- [Conditional probability distribution](https://tamnd.github.io/math2/node/0ZN9EZTEM6.md): concept 0ZN9EZTEM6
- [Eigenvalue](https://tamnd.github.io/math2/node/0ZND7NQ970.md): definition 0ZND7NQ970
- [Intersection](https://tamnd.github.io/math2/node/0ZNMMA9TF5.md): concept 0ZNMMA9TF5
- [p-series test](https://tamnd.github.io/math2/node/0ZNSSK8ZK1.md): statement 0ZNSSK8ZK1
- [Implication](https://tamnd.github.io/math2/node/0ZNX6DQ3EJ.md): definition 0ZNX6DQ3EJ
- [Power sum symmetric polynomial](https://tamnd.github.io/math2/node/0ZNXZB4HNB.md): concept 0ZNXZB4HNB
- [Inverse element](https://tamnd.github.io/math2/node/0ZNZH0EXFX.md): concept 0ZNZH0EXFX
- [Matrix of the dual map is the transpose](https://tamnd.github.io/math2/node/0ZPB3180AY.md): statement 0ZPB3180AY
- [Grönwall bound](https://tamnd.github.io/math2/node/0ZPQHH37PF.md): concept 0ZPQHH37PF
- [Recursively defined sequence](https://tamnd.github.io/math2/node/0ZPV68RN1X.md): concept 0ZPV68RN1X
- [Tower law for indices](https://tamnd.github.io/math2/node/0ZQJ7KEFQ5.md): statement 0ZQJ7KEFQ5
- [The logarithm is strictly concave](https://tamnd.github.io/math2/node/0ZS54B9GJF.md): statement 0ZS54B9GJF
- [Central moment](https://tamnd.github.io/math2/node/0ZT24EC90K.md): definition 0ZT24EC90K
- [Operator norm](https://tamnd.github.io/math2/node/0ZTEH055YF.md): definition 0ZTEH055YF
- [Intersection of subgroups is a subgroup](https://tamnd.github.io/math2/node/0ZTGMDNVTB.md): statement 0ZTGMDNVTB
- [Zero matrix](https://tamnd.github.io/math2/node/0ZTKJJDGR7.md): object 0ZTKJJDGR7
- [Closed and exact forms](https://tamnd.github.io/math2/node/0ZV14XRNF0.md): concept 0ZV14XRNF0
- [Solution of a linear recurrence](https://tamnd.github.io/math2/node/0ZV2YKKZ58.md): definition 0ZV2YKKZ58
- [Complex power](https://tamnd.github.io/math2/node/0ZV9A7EV0F.md): definition 0ZV9A7EV0F
- [Law of large numbers](https://tamnd.github.io/math2/node/0ZVEAYVRC3.md): concept 0ZVEAYVRC3
- [Quotient set](https://tamnd.github.io/math2/node/0ZVHXDSZC7.md): definition 0ZVHXDSZC7
- [Bipartite graph](https://tamnd.github.io/math2/node/0ZVQKTRKVQ.md): concept 0ZVQKTRKVQ
- [Uniqueness of limits](https://tamnd.github.io/math2/node/0ZXJ2455NJ.md): statement 0ZXJ2455NJ
- [Isometric isomorphism](https://tamnd.github.io/math2/node/0ZXSF2YQV2.md): concept 0ZXSF2YQV2
- [Second supplement to quadratic reciprocity](https://tamnd.github.io/math2/node/0ZXW95S44M.md): statement 0ZXW95S44M
- [Finite field](https://tamnd.github.io/math2/node/0ZXZ82FTX2.md): concept 0ZXZ82FTX2
- [Probability density function](https://tamnd.github.io/math2/node/0ZY9DX8FBF.md): concept 0ZY9DX8FBF
- [Probability space](https://tamnd.github.io/math2/node/0ZYBZ34RT2.md): definition 0ZYBZ34RT2
- [Taylor polynomial](https://tamnd.github.io/math2/node/0ZZ4RHTR9X.md): concept 0ZZ4RHTR9X
- [Meromorphic function](https://tamnd.github.io/math2/node/0ZZ7ZYDR8W.md): concept 0ZZ7ZYDR8W
- [Determinant of a transpose](https://tamnd.github.io/math2/node/0ZZQ2E1MTE.md): statement 0ZZQ2E1MTE
- [Subnormal series](https://tamnd.github.io/math2/node/0ZZR8D7Y1Z.md): concept 0ZZR8D7Y1Z
- [Argument of a complex number](https://tamnd.github.io/math2/node/0ZZRSX91QR.md): definition 0ZZRSX91QR
- [Rank of a linear map](https://tamnd.github.io/math2/node/0ZZXSBTR4J.md): concept 0ZZXSBTR4J
- [Sendov's conjecture](https://tamnd.github.io/math2/node/101H2VSX2R.md): conjecture 101H2VSX2R
- [Function with no Laplace transform](https://tamnd.github.io/math2/node/105G7PWKS1.md): counterexample 105G7PWKS1
- [Morse–Smale system](https://tamnd.github.io/math2/node/10B1TS96VA.md): definition 10B1TS96VA
- [Cotton tensor](https://tamnd.github.io/math2/node/10CQ5007PH.md): concept 10CQ5007PH
- [Power iteration](https://tamnd.github.io/math2/node/10FGGCQ6TS.md): definition 10FGGCQ6TS
- [Doob's upcrossing inequality](https://tamnd.github.io/math2/node/10HFW1GRV2.md): statement 10HFW1GRV2
- [Completely normal space](https://tamnd.github.io/math2/node/10P0C0Y5V0.md): definition 10P0C0Y5V0
- [Quotient algebra](https://tamnd.github.io/math2/node/10TJ890QK0.md): definition 10TJ890QK0
- [Fractional factorial design](https://tamnd.github.io/math2/node/10WTA1QZXD.md): definition 10WTA1QZXD
- [Median algebra](https://tamnd.github.io/math2/node/114KYR2JGP.md): definition 114KYR2JGP
- [Smooth dynamical system](https://tamnd.github.io/math2/node/116TPXR5FS.md): concept 116TPXR5FS
- [Stationary phase approximation](https://tamnd.github.io/math2/node/119J1Q1MG9.md): statement 119J1Q1MG9
- [Homotopy lifting for covering maps](https://tamnd.github.io/math2/node/11FFYBNF0E.md): statement 11FFYBNF0E
- [Rigged Hilbert space](https://tamnd.github.io/math2/node/11GMR0F2Q8.md): definition 11GMR0F2Q8
- [Free probability](https://tamnd.github.io/math2/node/11KYREAH1J.md): concept 11KYREAH1J
- [Graph of groups](https://tamnd.github.io/math2/node/11MVWQ5R31.md): definition 11MVWQ5R31
- [Willmore conjecture](https://tamnd.github.io/math2/node/11MYA5W3KT.md): statement 11MYA5W3KT
- [Expectation–maximization algorithm](https://tamnd.github.io/math2/node/11N1AXSXYK.md): concept 11N1AXSXYK
- [Relative CW complex](https://tamnd.github.io/math2/node/11PJJQRECB.md): definition 11PJJQRECB
- [Weierstrass factorization theorem](https://tamnd.github.io/math2/node/11VC51YHQJ.md): statement 11VC51YHQJ
- [Partial autocorrelation function](https://tamnd.github.io/math2/node/11YRFWD951.md): concept 11YRFWD951
- [Auslander–Reiten quiver](https://tamnd.github.io/math2/node/125Z033VCQ.md): concept 125Z033VCQ
- [Additive category](https://tamnd.github.io/math2/node/128WX7HXAT.md): definition 128WX7HXAT
- [Rokhlin's theorem](https://tamnd.github.io/math2/node/12BXVH5W5H.md): statement 12BXVH5W5H
- [Mackey machine](https://tamnd.github.io/math2/node/12CEWPMSG4.md): concept 12CEWPMSG4
- [Mittag-Leffler function](https://tamnd.github.io/math2/node/12G9M694YG.md): concept 12G9M694YG
- [Geometric measure theory](https://tamnd.github.io/math2/node/12GTQ4WG8N.md): definition 12GTQ4WG8N
- [Transcendental extension](https://tamnd.github.io/math2/node/12JQCGTQ8W.md): concept 12JQCGTQ8W
- [Baker's map](https://tamnd.github.io/math2/node/12V4MHVV0N.md): example 12V4MHVV0N
- [Filtration](https://tamnd.github.io/math2/node/130FCDWRT1.md): definition 130FCDWRT1
- [Betti numbers of a module](https://tamnd.github.io/math2/node/1313JFBBAD.md): definition 1313JFBBAD
- [Geodesic curvature](https://tamnd.github.io/math2/node/1399T2Q2ZS.md): definition 1399T2Q2ZS
- [Stokes flow](https://tamnd.github.io/math2/node/13CB7V2YM4.md): concept 13CB7V2YM4
- [Classical orthogonal polynomials](https://tamnd.github.io/math2/node/13CH1ZF88V.md): definition 13CH1ZF88V
- [Calabi flow](https://tamnd.github.io/math2/node/13D1AB6RT8.md): definition 13D1AB6RT8
- [De Rham cohomology of varieties](https://tamnd.github.io/math2/node/13F5JZPDD0.md): concept 13F5JZPDD0
- [Set-valued measure](https://tamnd.github.io/math2/node/13F6B3BGA4.md): concept 13F6B3BGA4
- [Absolute geometry](https://tamnd.github.io/math2/node/13FQJX394D.md): definition 13FQJX394D
- [Proof complexity](https://tamnd.github.io/math2/node/13JZKN6WKC.md): definition 13JZKN6WKC
- [Riemann domain over C^n](https://tamnd.github.io/math2/node/13RXPC2RJK.md): definition 13RXPC2RJK
- [Unit-regular ring](https://tamnd.github.io/math2/node/13S6WYDB6T.md): definition 13S6WYDB6T
- [Regular cardinal](https://tamnd.github.io/math2/node/13ZV2F8JAR.md): concept 13ZV2F8JAR
- [Rate function](https://tamnd.github.io/math2/node/140N2PQSTE.md): concept 140N2PQSTE
- [Shearlet](https://tamnd.github.io/math2/node/140TG7W6A5.md): definition 140TG7W6A5
- [Euler–Boole summation](https://tamnd.github.io/math2/node/141Y4NMGY7.md): concept 141Y4NMGY7
- [Carathéodory's royal road](https://tamnd.github.io/math2/node/148S3XETT3.md): concept 148S3XETT3
- [Monge–Ampère equation](https://tamnd.github.io/math2/node/14AKAC0S0T.md): concept 14AKAC0S0T
- [Laplace's method](https://tamnd.github.io/math2/node/14B9FE49V7.md): definition 14B9FE49V7
- [Sphere spectrum](https://tamnd.github.io/math2/node/14CGGHA85T.md): object 14CGGHA85T
- [Tarski's axioms](https://tamnd.github.io/math2/node/14DWEDXWCC.md): concept 14DWEDXWCC
- [VC dimension](https://tamnd.github.io/math2/node/14HX3QF45H.md): definition 14HX3QF45H
- [Finitely based algebra](https://tamnd.github.io/math2/node/14KPTDH4ZN.md): definition 14KPTDH4ZN
- [Huisken's theorem on convex hypersurfaces](https://tamnd.github.io/math2/node/14RHHMGF2D.md): statement 14RHHMGF2D
- [Hales–Jewett theorem](https://tamnd.github.io/math2/node/14SGZ5HA4Y.md): statement 14SGZ5HA4Y
- [Mehler–Fock transform](https://tamnd.github.io/math2/node/14T6SJC0BC.md): definition 14T6SJC0BC
- [Ryll-Nardzewski theorem](https://tamnd.github.io/math2/node/1513J40TKA.md): statement 1513J40TKA
- [Beta function in terms of the gamma function](https://tamnd.github.io/math2/node/151RG1JCGB.md): statement 151RG1JCGB
- [Schatten class](https://tamnd.github.io/math2/node/153N5DW55Z.md): concept 153N5DW55Z
- [Newton polynomial](https://tamnd.github.io/math2/node/155K06F4JF.md): definition 155K06F4JF
- [Nakayama's lemma](https://tamnd.github.io/math2/node/15JHBYR6KH.md): statement 15JHBYR6KH
- [Piecewise linear manifold](https://tamnd.github.io/math2/node/15MKGRNM9K.md): concept 15MKGRNM9K
- [Filter](https://tamnd.github.io/math2/node/15PF73BKY6.md): concept 15PF73BKY6
- [Proper map](https://tamnd.github.io/math2/node/15XEBGNGHC.md): concept 15XEBGNGHC
- [Model predictive control](https://tamnd.github.io/math2/node/15Z6BT9XYR.md): concept 15Z6BT9XYR
- [Abel–Ruffini theorem](https://tamnd.github.io/math2/node/163A2WQQJX.md): statement 163A2WQQJX
- [Strongly minimal theory](https://tamnd.github.io/math2/node/16B7TWCEX9.md): definition 16B7TWCEX9
- [Ideal quotient](https://tamnd.github.io/math2/node/16J9DM8V57.md): concept 16J9DM8V57
- [Lambert series](https://tamnd.github.io/math2/node/16SRNY6AXK.md): definition 16SRNY6AXK
- [Abelian theorems for integral transforms](https://tamnd.github.io/math2/node/16W9PP8ET0.md): statement 16W9PP8ET0
- [Bessel–Clifford function](https://tamnd.github.io/math2/node/16WR5EARVC.md): concept 16WR5EARVC
- [Tensor product of algebras](https://tamnd.github.io/math2/node/1735VGRT2P.md): concept 1735VGRT2P
- [Liénard's theorem](https://tamnd.github.io/math2/node/178GG6TXKQ.md): statement 178GG6TXKQ
- [Concentration of measure](https://tamnd.github.io/math2/node/17BH73YHVT.md): definition 17BH73YHVT
- [Finitely additive measure](https://tamnd.github.io/math2/node/17CW7Z2YDT.md): definition 17CW7Z2YDT
- [Closure operator](https://tamnd.github.io/math2/node/17FVAN6HXX.md): definition 17FVAN6HXX
- [Integral scheme](https://tamnd.github.io/math2/node/17NVR4XBHK.md): concept 17NVR4XBHK
- [Vector measure](https://tamnd.github.io/math2/node/17PWP8MB4X.md): definition 17PWP8MB4X
- [Polyharmonic function](https://tamnd.github.io/math2/node/17R5V5ZVSR.md): concept 17R5V5ZVSR
- [Regular open algebra](https://tamnd.github.io/math2/node/17S8AM75AD.md): example 17S8AM75AD
- [Warsaw circle](https://tamnd.github.io/math2/node/17VG475BRV.md): example 17VG475BRV
- [Hilbert's twentieth problem](https://tamnd.github.io/math2/node/187NP5Z002.md): problem 187NP5Z002
- [Kolmogorov complexity](https://tamnd.github.io/math2/node/18DPY9J8YP.md): definition 18DPY9J8YP
- [Spectral theorem for unbounded self-adjoint operators](https://tamnd.github.io/math2/node/18KPYRRPZY.md): statement 18KPYRRPZY
- [Navier–Stokes equations](https://tamnd.github.io/math2/node/18KXZHVF4M.md): concept 18KXZHVF4M
- [Pseudofunctor](https://tamnd.github.io/math2/node/18M8XGY04N.md): concept 18M8XGY04N
- [Jeffreys prior](https://tamnd.github.io/math2/node/18MMYZ7922.md): concept 18MMYZ7922
- [Isospectral manifolds](https://tamnd.github.io/math2/node/18MQEGN13R.md): concept 18MQEGN13R
- [Morrey–Campanato space](https://tamnd.github.io/math2/node/18Q1BBK6C0.md): definition 18Q1BBK6C0
- [Resolvent set](https://tamnd.github.io/math2/node/18ZNC2R6AX.md): definition 18ZNC2R6AX
- [Holomorphic map](https://tamnd.github.io/math2/node/1981Q0DGQW.md): definition 1981Q0DGQW
- [Stable map](https://tamnd.github.io/math2/node/1991WETZF9.md): concept 1991WETZF9
- [Fourier transform method for convolution equations](https://tamnd.github.io/math2/node/199B14MWGC.md): definition 199B14MWGC
- [Univalent function](https://tamnd.github.io/math2/node/199C7G1B36.md): definition 199C7G1B36
- [Interval exchange transformation](https://tamnd.github.io/math2/node/19JFF3GY9F.md): example 19JFF3GY9F
- [Meijer G-function](https://tamnd.github.io/math2/node/19JNNN5KV7.md): concept 19JNNN5KV7
- [Random matrix](https://tamnd.github.io/math2/node/19SZD9XAFX.md): definition 19SZD9XAFX
- [Lebesgue point](https://tamnd.github.io/math2/node/19WRHM5NAT.md): concept 19WRHM5NAT
- [Liouville's equation](https://tamnd.github.io/math2/node/19WY7FYPXK.md): definition 19WY7FYPXK
- [Entropy solution](https://tamnd.github.io/math2/node/1A01A3EJRX.md): concept 1A01A3EJRX
- [Link](https://tamnd.github.io/math2/node/1A5WRCNTM3.md): concept 1A5WRCNTM3
- [Convex join](https://tamnd.github.io/math2/node/1A83RGDDNE.md): concept 1A83RGDDNE
- [Leray–Schauder degree](https://tamnd.github.io/math2/node/1A8A1XFG45.md): definition 1A8A1XFG45
- [n-connected space](https://tamnd.github.io/math2/node/1A943BWBQT.md): concept 1A943BWBQT
- [Artin approximation theorem](https://tamnd.github.io/math2/node/1AFTCHDDFE.md): statement 1AFTCHDDFE
- [Numerical stability](https://tamnd.github.io/math2/node/1AKB4XHMTH.md): concept 1AKB4XHMTH
- [Equational theory](https://tamnd.github.io/math2/node/1AKKDC17M2.md): definition 1AKKDC17M2
- [Riesz projector](https://tamnd.github.io/math2/node/1AR19ZQQM2.md): definition 1AR19ZQQM2
- [Semiprimitive ring](https://tamnd.github.io/math2/node/1B2DBKV1C9.md): concept 1B2DBKV1C9
- [Wagner's theorem](https://tamnd.github.io/math2/node/1B2ZJPVV03.md): statement 1B2ZJPVV03
- [Congruence-permutable variety](https://tamnd.github.io/math2/node/1B3N9T68YV.md): concept 1B3N9T68YV
- [Krull dimension](https://tamnd.github.io/math2/node/1B4G153ZGE.md): definition 1B4G153ZGE
- [Euclidean distance matrix](https://tamnd.github.io/math2/node/1BCJ7A73KS.md): concept 1BCJ7A73KS
- [Sumudu transform](https://tamnd.github.io/math2/node/1BHMWWH55N.md): concept 1BHMWWH55N
- [Integral of a Hopf algebra](https://tamnd.github.io/math2/node/1BK2B286W3.md): definition 1BK2B286W3
- [Exponential integral](https://tamnd.github.io/math2/node/1BK7TC6PW9.md): definition 1BK7TC6PW9
- [Differential inclusion](https://tamnd.github.io/math2/node/1BKVNEHX96.md): definition 1BKVNEHX96
- [Banach–Alaoglu theorem](https://tamnd.github.io/math2/node/1BXEJXQ2YT.md): statement 1BXEJXQ2YT
- [Knot concordance](https://tamnd.github.io/math2/node/1C28287S75.md): concept 1C28287S75
- [Parry measure](https://tamnd.github.io/math2/node/1C2TEAKJHJ.md): concept 1C2TEAKJHJ
- [Graded poset](https://tamnd.github.io/math2/node/1C7TKGTB2F.md): concept 1C7TKGTB2F
- [Tate cohomology](https://tamnd.github.io/math2/node/1CA6GZG0R6.md): concept 1CA6GZG0R6
- [Ramanujan theta function](https://tamnd.github.io/math2/node/1CBZA8MQJW.md): concept 1CBZA8MQJW
- [Parabolic partial differential equation](https://tamnd.github.io/math2/node/1CCTH8X2FT.md): concept 1CCTH8X2FT
- [Graph of an operator](https://tamnd.github.io/math2/node/1CF72TCVD2.md): definition 1CF72TCVD2
- [Wilson polynomials](https://tamnd.github.io/math2/node/1CPF7SWQ3S.md): concept 1CPF7SWQ3S
- [Homogeneous space](https://tamnd.github.io/math2/node/1CPW6AG86Y.md): definition 1CPW6AG86Y
- [Kernel of an integral equation](https://tamnd.github.io/math2/node/1CRA2F3TY7.md): definition 1CRA2F3TY7
- [Algebraic curve](https://tamnd.github.io/math2/node/1CS00CEAEH.md): definition 1CS00CEAEH
- [Heaviside expansion theorem](https://tamnd.github.io/math2/node/1CW7Q8QH3N.md): statement 1CW7Q8QH3N
- [Park's conjecture](https://tamnd.github.io/math2/node/1D541B576S.md): conjecture 1D541B576S
- [Function having a given Mellin transform](https://tamnd.github.io/math2/node/1D5VDZYJ20.md): concept 1D5VDZYJ20
- [Dirichlet's unit theorem](https://tamnd.github.io/math2/node/1D99CSKX6N.md): statement 1D99CSKX6N
- [Martin kernel](https://tamnd.github.io/math2/node/1DBC9FYF7M.md): concept 1DBC9FYF7M
- [Real projective space](https://tamnd.github.io/math2/node/1DDYB6VVSR.md): object 1DDYB6VVSR
- [Nonlinear Volterra equation](https://tamnd.github.io/math2/node/1DFG72YV3N.md): definition 1DFG72YV3N
- [Krylov–Bogolyubov theorem](https://tamnd.github.io/math2/node/1DHZ81S0YE.md): statement 1DHZ81S0YE
- [Long line](https://tamnd.github.io/math2/node/1DKCNS63DH.md): counterexample 1DKCNS63DH
- [Ind-completion](https://tamnd.github.io/math2/node/1DKFWGWHH5.md): concept 1DKFWGWHH5
- [Weak topology](https://tamnd.github.io/math2/node/1DRW7JP962.md): definition 1DRW7JP962
- [Translation surface](https://tamnd.github.io/math2/node/1DSZV8QRS6.md): definition 1DSZV8QRS6
- [Ehrenfeucht–Fraïssé game](https://tamnd.github.io/math2/node/1DVRN1T2Z0.md): definition 1DVRN1T2Z0
- [Fiber product of schemes](https://tamnd.github.io/math2/node/1DZMT4S9FB.md): concept 1DZMT4S9FB
- [Bernstein–von Mises theorem](https://tamnd.github.io/math2/node/1E8QNK0Q92.md): statement 1E8QNK0Q92
- [Horocycle](https://tamnd.github.io/math2/node/1E8WCHSAE2.md): concept 1E8WCHSAE2
- [Malliavin calculus](https://tamnd.github.io/math2/node/1EC7VPHFAY.md): concept 1EC7VPHFAY
- [Error function](https://tamnd.github.io/math2/node/1EG3DWPNZN.md): definition 1EG3DWPNZN
- [Busemann G-space](https://tamnd.github.io/math2/node/1EHBR0VMH3.md): definition 1EHBR0VMH3
- [Nevanlinna theory](https://tamnd.github.io/math2/node/1EHH63VNAD.md): concept 1EHH63VNAD
- [Integral element](https://tamnd.github.io/math2/node/1EHN4KPSFS.md): concept 1EHN4KPSFS
- [Eckmann–Hilton duality](https://tamnd.github.io/math2/node/1EK3355WMC.md): concept 1EK3355WMC
- [Functional](https://tamnd.github.io/math2/node/1EQM0DJ4NC.md): concept 1EQM0DJ4NC
- [Hybrid system](https://tamnd.github.io/math2/node/1EW58ZYR8Y.md): concept 1EW58ZYR8Y
- [Padé approximant](https://tamnd.github.io/math2/node/1F16G4V64X.md): concept 1F16G4V64X
- [Equivariant cohomology](https://tamnd.github.io/math2/node/1F5YPVPRZV.md): definition 1F5YPVPRZV
- [Resistance form](https://tamnd.github.io/math2/node/1F6SG980N5.md): definition 1F6SG980N5
- [hp-FEM](https://tamnd.github.io/math2/node/1F7NV6AT2X.md): definition 1F7NV6AT2X
- [Grothendieck category](https://tamnd.github.io/math2/node/1F833GKJ9V.md): definition 1F833GKJ9V
- [CAT(k) space](https://tamnd.github.io/math2/node/1FAF2VNA4G.md): concept 1FAF2VNA4G
- [Gauss's hypergeometric theorem](https://tamnd.github.io/math2/node/1FC0T0GCCX.md): statement 1FC0T0GCCX
- [Tutte polynomial](https://tamnd.github.io/math2/node/1FFTQW1B1X.md): definition 1FFTQW1B1X
- [Simpson's rule](https://tamnd.github.io/math2/node/1FMBCFJ7K1.md): definition 1FMBCFJ7K1
- [Herglotz–Riesz kernel](https://tamnd.github.io/math2/node/1FSFX18D68.md): concept 1FSFX18D68
- [Polynomial function](https://tamnd.github.io/math2/node/1G0B73WPX8.md): concept 1G0B73WPX8
- [Arnold tongue](https://tamnd.github.io/math2/node/1G2ZRM270V.md): definition 1G2ZRM270V
- [Approximate point spectrum](https://tamnd.github.io/math2/node/1GAXM9QDCQ.md): definition 1GAXM9QDCQ
- [Spectrum](https://tamnd.github.io/math2/node/1GHF4EKBXS.md): concept 1GHF4EKBXS
- [Multiresolution analysis](https://tamnd.github.io/math2/node/1GHQMBR7DE.md): concept 1GHQMBR7DE
- [Harmonic analysis on groups](https://tamnd.github.io/math2/node/1GJ6SRT6B7.md): definition 1GJ6SRT6B7
- [Spin(7)-manifold](https://tamnd.github.io/math2/node/1GMSWCHKHE.md): concept 1GMSWCHKHE
- [Lipschitz domain](https://tamnd.github.io/math2/node/1GNKQ7NBAG.md): concept 1GNKQ7NBAG
- [Enriques–Kodaira classification](https://tamnd.github.io/math2/node/1GVF19KFAK.md): statement 1GVF19KFAK
- [Model theory](https://tamnd.github.io/math2/node/1GYEDS227W.md): definition 1GYEDS227W
- [Non-Archimedean absolute value](https://tamnd.github.io/math2/node/1H5F1BP6XS.md): concept 1H5F1BP6XS
- [Algebraic approach to constraint satisfaction](https://tamnd.github.io/math2/node/1H650VRZVT.md): definition 1H650VRZVT
- [Congruence subgroup](https://tamnd.github.io/math2/node/1H6R856P3C.md): concept 1H6R856P3C
- [Permutation pattern](https://tamnd.github.io/math2/node/1H8Z1WMG23.md): concept 1H8Z1WMG23
- [Support function](https://tamnd.github.io/math2/node/1H92CYA4ZC.md): concept 1H92CYA4ZC
- [Incidence algebra](https://tamnd.github.io/math2/node/1H9SFWHYEB.md): concept 1H9SFWHYEB
- [Regular local rings are UFDs](https://tamnd.github.io/math2/node/1HD3EK5392.md): statement 1HD3EK5392
- [Universal bundle](https://tamnd.github.io/math2/node/1HDH716Z24.md): concept 1HDH716Z24
- [Boundary value problem](https://tamnd.github.io/math2/node/1HGGRS57KT.md): definition 1HGGRS57KT
- [Hyperstructure](https://tamnd.github.io/math2/node/1HJF40J5AK.md): concept 1HJF40J5AK
- [Logical consequence](https://tamnd.github.io/math2/node/1HK0VYWTV7.md): concept 1HK0VYWTV7
- [Monoid ring](https://tamnd.github.io/math2/node/1HM7ZYMPD7.md): concept 1HM7ZYMPD7
- [Telegrapher's equations](https://tamnd.github.io/math2/node/1HR9NZM8B2.md): concept 1HR9NZM8B2
- [Octonions](https://tamnd.github.io/math2/node/1HRMT3DSX2.md): object 1HRMT3DSX2
- [Random matrix](https://tamnd.github.io/math2/node/1HS4RT7HK5.md): concept 1HS4RT7HK5
- [Evans's theorem on word problems](https://tamnd.github.io/math2/node/1HTSJ6A4D2.md): statement 1HTSJ6A4D2
- [Giroux correspondence](https://tamnd.github.io/math2/node/1HTW0F4ZZ7.md): statement 1HTW0F4ZZ7
- [Twelvefold way](https://tamnd.github.io/math2/node/1J16RVRHYS.md): concept 1J16RVRHYS
- [Feedback linearization](https://tamnd.github.io/math2/node/1J2KTGXB34.md): concept 1J2KTGXB34
- [Positive-definite function on a group](https://tamnd.github.io/math2/node/1J67Y01N4X.md): definition 1J67Y01N4X
- [Mayer and Lagrange problems](https://tamnd.github.io/math2/node/1J8TYT7C49.md): problem 1J8TYT7C49
- [Fundamental strip of a Mellin transform](https://tamnd.github.io/math2/node/1JAA21FHHQ.md): statement 1JAA21FHHQ
- [Semifield](https://tamnd.github.io/math2/node/1JCQTY728A.md): definition 1JCQTY728A
- [Arithmetical variety](https://tamnd.github.io/math2/node/1JD2FVGBM2.md): definition 1JD2FVGBM2
- [Cramér–Rao bound](https://tamnd.github.io/math2/node/1JFKTEQVRX.md): statement 1JFKTEQVRX
- [Singularity of a smooth map](https://tamnd.github.io/math2/node/1JG3HFWX3T.md): concept 1JG3HFWX3T
- [Categorical theory](https://tamnd.github.io/math2/node/1JJCGNX240.md): concept 1JJCGNX240
- [Elliptic cohomology](https://tamnd.github.io/math2/node/1JM66EN5R2.md): definition 1JM66EN5R2
- [Collectively compact operator approximation](https://tamnd.github.io/math2/node/1JNFB7XBDN.md): definition 1JNFB7XBDN
- [Hopf bifurcation](https://tamnd.github.io/math2/node/1JPGWFMAGK.md): definition 1JPGWFMAGK
- [Unramified morphism](https://tamnd.github.io/math2/node/1JQX6HD8E5.md): concept 1JQX6HD8E5
- [Non-negative least squares](https://tamnd.github.io/math2/node/1JV7093B4C.md): problem 1JV7093B4C
- [Fourier transform on Schwartz space](https://tamnd.github.io/math2/node/1JXFTBBK6D.md): statement 1JXFTBBK6D
- [Handle decomposition](https://tamnd.github.io/math2/node/1K31QJNYNF.md): definition 1K31QJNYNF
- [Sheffer function](https://tamnd.github.io/math2/node/1K829SJG3K.md): concept 1K829SJG3K
- [Transcritical bifurcation](https://tamnd.github.io/math2/node/1K8Y6BP097.md): definition 1K8Y6BP097
- [Kuranishi structure](https://tamnd.github.io/math2/node/1KPAEFMPZB.md): definition 1KPAEFMPZB
- [Proximity space](https://tamnd.github.io/math2/node/1M4MVM5CAS.md): definition 1M4MVM5CAS
- [Cylindric algebra](https://tamnd.github.io/math2/node/1MDT9BF81H.md): concept 1MDT9BF81H
- [Free Boolean algebra](https://tamnd.github.io/math2/node/1MF0TVHDZD.md): concept 1MF0TVHDZD
- [Hilbert system](https://tamnd.github.io/math2/node/1MGMQX43B8.md): concept 1MGMQX43B8
- [Serre subcategory](https://tamnd.github.io/math2/node/1MJW4YPQW6.md): definition 1MJW4YPQW6
- [Bernstein–Bézier form](https://tamnd.github.io/math2/node/1N2GCJ3JWT.md): definition 1N2GCJ3JWT
- [Airy function](https://tamnd.github.io/math2/node/1N5D4PN92W.md): concept 1N5D4PN92W
- [Abstract Volterra equation](https://tamnd.github.io/math2/node/1N7TZCPBF0.md): concept 1N7TZCPBF0
- [Finite morphism of rings](https://tamnd.github.io/math2/node/1NBACAEC2V.md): concept 1NBACAEC2V
- [Base change theorems](https://tamnd.github.io/math2/node/1NC789DATR.md): statement 1NC789DATR
- [Homotopy limit](https://tamnd.github.io/math2/node/1NDRD8BHFG.md): concept 1NDRD8BHFG
- [Poincaré's theorem that the ball and polydisc are not biholomorphic](https://tamnd.github.io/math2/node/1NF8T07H9X.md): statement 1NF8T07H9X
- [Prenex normal form](https://tamnd.github.io/math2/node/1NKD8P3PS0.md): definition 1NKD8P3PS0
- [Compactness theorem](https://tamnd.github.io/math2/node/1NR3HAWG9D.md): statement 1NR3HAWG9D
- [Radon–Nikodym derivative](https://tamnd.github.io/math2/node/1NTKG64F7N.md): concept 1NTKG64F7N
- [Initial topology](https://tamnd.github.io/math2/node/1NWC3710WW.md): concept 1NWC3710WW
- [Semisimple Lie group](https://tamnd.github.io/math2/node/1NX95TTMFT.md): definition 1NX95TTMFT
- [Levin-type transformations](https://tamnd.github.io/math2/node/1P47PCRVTZ.md): concept 1P47PCRVTZ
- [Comparability graph](https://tamnd.github.io/math2/node/1P4830BH6Z.md): concept 1P4830BH6Z
- [Inversion of rational transforms by partial fractions](https://tamnd.github.io/math2/node/1P6TBEPHQH.md): concept 1P6TBEPHQH
- [Linearly ordered group](https://tamnd.github.io/math2/node/1P8087YGYQ.md): definition 1P8087YGYQ
- [Stone–Weierstrass theorem](https://tamnd.github.io/math2/node/1P8D3MK79H.md): statement 1P8D3MK79H
- [Region of convergence](https://tamnd.github.io/math2/node/1P8WX3EGHF.md): concept 1P8WX3EGHF
- [Pettis measurability theorem](https://tamnd.github.io/math2/node/1P9ZR606JN.md): statement 1P9ZR606JN
- [Solenoid](https://tamnd.github.io/math2/node/1PBQ6ARWND.md): example 1PBQ6ARWND
- [Barnes integral](https://tamnd.github.io/math2/node/1PDGBGEHM2.md): concept 1PDGBGEHM2
- [Birkhoff–Grothendieck theorem](https://tamnd.github.io/math2/node/1PFE27B2MG.md): statement 1PFE27B2MG
- [Mahler measure](https://tamnd.github.io/math2/node/1PJS951C8N.md): definition 1PJS951C8N
- [Efficiency of an estimator](https://tamnd.github.io/math2/node/1PKZW82H0C.md): definition 1PKZW82H0C
- [Non-measurable set](https://tamnd.github.io/math2/node/1PQHRMQ4ZH.md): definition 1PQHRMQ4ZH
- [Proof complexity](https://tamnd.github.io/math2/node/1Q046J0AW2.md): concept 1Q046J0AW2
- [Krull intersection theorem](https://tamnd.github.io/math2/node/1Q1AH2V41S.md): statement 1Q1AH2V41S
- [Topological dynamics](https://tamnd.github.io/math2/node/1Q901WTV5X.md): definition 1Q901WTV5X
- [Summability kernel](https://tamnd.github.io/math2/node/1QC5206K07.md): concept 1QC5206K07
- [Finite difference method](https://tamnd.github.io/math2/node/1QCG88BQC2.md): definition 1QCG88BQC2
- [Jet](https://tamnd.github.io/math2/node/1QE0Z0756D.md): concept 1QE0Z0756D
- [Dual graph](https://tamnd.github.io/math2/node/1QHTD1DS5J.md): concept 1QHTD1DS5J
- [Direct limit of rings](https://tamnd.github.io/math2/node/1QS5A6XNXA.md): definition 1QS5A6XNXA
- [Rubin causal model](https://tamnd.github.io/math2/node/1QW75YQQR8.md): definition 1QW75YQQR8
- [Non-standard model of arithmetic](https://tamnd.github.io/math2/node/1QWN5CV4CG.md): example 1QWN5CV4CG
- [Log-rank test](https://tamnd.github.io/math2/node/1R358D2T7J.md): definition 1R358D2T7J
- [Laurent polynomial ring](https://tamnd.github.io/math2/node/1R5P656GS6.md): definition 1R5P656GS6
- [Birkhoff min-max method](https://tamnd.github.io/math2/node/1R5X9HW6D3.md): concept 1R5X9HW6D3
- [Riemannian metric](https://tamnd.github.io/math2/node/1R7ANBHDSN.md): definition 1R7ANBHDSN
- [Flux limiter](https://tamnd.github.io/math2/node/1RGPVZXVD9.md): definition 1RGPVZXVD9
- [Weakly compact cardinal](https://tamnd.github.io/math2/node/1RHTVMH34G.md): concept 1RHTVMH34G
- [Associated prime](https://tamnd.github.io/math2/node/1RJJ33KKSS.md): definition 1RJJ33KKSS
- [Szegő recurrence and Verblunsky coefficients](https://tamnd.github.io/math2/node/1RP3NVGNA6.md): statement 1RP3NVGNA6
- [Post's problem](https://tamnd.github.io/math2/node/1RVF7DWG8C.md): problem 1RVF7DWG8C
- [Symmetric spectrum](https://tamnd.github.io/math2/node/1RY6DQJ1RV.md): concept 1RY6DQJ1RV
- [Structural rigidity](https://tamnd.github.io/math2/node/1RZV6TRTPE.md): definition 1RZV6TRTPE
- [Arakelyan's theorem](https://tamnd.github.io/math2/node/1S20A3R659.md): statement 1S20A3R659
- [t-structure](https://tamnd.github.io/math2/node/1S9JGHT5HJ.md): definition 1S9JGHT5HJ
- [Scott domain](https://tamnd.github.io/math2/node/1SAAZG46DG.md): concept 1SAAZG46DG
- [Robinson–Schensted correspondence](https://tamnd.github.io/math2/node/1SCF8X47G1.md): statement 1SCF8X47G1
- [Cap set problem](https://tamnd.github.io/math2/node/1SF8Y9M8YB.md): statement 1SF8Y9M8YB
- [Huygens principle](https://tamnd.github.io/math2/node/1SH2WZVZ66.md): statement 1SH2WZVZ66
- [Locally path-connected space](https://tamnd.github.io/math2/node/1SJ46AX8Y6.md): definition 1SJ46AX8Y6
- [Kirkman's schoolgirl problem](https://tamnd.github.io/math2/node/1SYQ6QY95Y.md): problem 1SYQ6QY95Y
- [Derivation of the conjugate gradient method](https://tamnd.github.io/math2/node/1SYY57DCWT.md): statement 1SYY57DCWT
- [Functional](https://tamnd.github.io/math2/node/1T23294NKC.md): definition 1T23294NKC
- [Affine morphism](https://tamnd.github.io/math2/node/1T7XKF6WKM.md): definition 1T7XKF6WKM
- [Slice category](https://tamnd.github.io/math2/node/1TC7W7BQT3.md): concept 1TC7W7BQT3
- [Zeros and poles](https://tamnd.github.io/math2/node/1TE3QF8EYC.md): concept 1TE3QF8EYC
- [Fock space](https://tamnd.github.io/math2/node/1TJ8TNH331.md): object 1TJ8TNH331
- [Rossi's non-embeddable CR structure](https://tamnd.github.io/math2/node/1TJXANASFE.md): counterexample 1TJXANASFE
- [Hochster's homological conjectures](https://tamnd.github.io/math2/node/1TNS604ZST.md): conjecture 1TNS604ZST
- [Unknot](https://tamnd.github.io/math2/node/1TQ06R4Q66.md): example 1TQ06R4Q66
- [Midpoint convex function](https://tamnd.github.io/math2/node/1TQ1JQXEWP.md): concept 1TQ1JQXEWP
- [Minkowski bound](https://tamnd.github.io/math2/node/1TQ2F9CFVE.md): statement 1TQ2F9CFVE
- [Associative algebra](https://tamnd.github.io/math2/node/1TZSV19D85.md): definition 1TZSV19D85
- [Clarkson's inequalities](https://tamnd.github.io/math2/node/1V11QM5C07.md): statement 1V11QM5C07
- [Riesz potential](https://tamnd.github.io/math2/node/1V2N8MFSHC.md): concept 1V2N8MFSHC
- [Quantum enveloping algebra](https://tamnd.github.io/math2/node/1V8H34F6N9.md): concept 1V8H34F6N9
- [Minimal clone](https://tamnd.github.io/math2/node/1V9GKBY29Y.md): concept 1V9GKBY29Y
- [Functional data analysis](https://tamnd.github.io/math2/node/1VBRNS1EQJ.md): definition 1VBRNS1EQJ
- [Commutant](https://tamnd.github.io/math2/node/1VC7EJ1ZBZ.md): concept 1VC7EJ1ZBZ
- [Brouwer–Heyting–Kolmogorov interpretation](https://tamnd.github.io/math2/node/1VFH9WT1RZ.md): definition 1VFH9WT1RZ
- [Conformal invariance of harmonic functions](https://tamnd.github.io/math2/node/1VGB63K5HS.md): statement 1VGB63K5HS
- [Space form](https://tamnd.github.io/math2/node/1VM1AQT956.md): concept 1VM1AQT956
- [Representation theory of the symmetric group](https://tamnd.github.io/math2/node/1VQC05EEYW.md): concept 1VQC05EEYW
- [Holmgren's uniqueness theorem](https://tamnd.github.io/math2/node/1VVKF0Y19B.md): statement 1VVKF0Y19B
- [CUSUM](https://tamnd.github.io/math2/node/1VXZHAHT6K.md): concept 1VXZHAHT6K
- [Dynamic equation on time scales](https://tamnd.github.io/math2/node/1W4X6J3948.md): definition 1W4X6J3948
- [Beltrami–Klein model](https://tamnd.github.io/math2/node/1WN9B18H86.md): example 1WN9B18H86
- [Queueing theory](https://tamnd.github.io/math2/node/1WT57ED272.md): definition 1WT57ED272
- [Pseudo-element](https://tamnd.github.io/math2/node/1WV2THA6NY.md): concept 1WV2THA6NY
- [Cuthill–McKee algorithm](https://tamnd.github.io/math2/node/1X0E3JHCH9.md): statement 1X0E3JHCH9
- [Functor category](https://tamnd.github.io/math2/node/1X0WKFQX8K.md): concept 1X0WKFQX8K
- [Quasi-everywhere](https://tamnd.github.io/math2/node/1X1WCNDA0J.md): concept 1X1WCNDA0J
- [Jacobson ring](https://tamnd.github.io/math2/node/1X1Y2AW4B0.md): definition 1X1Y2AW4B0
- [Kobayashi conjecture](https://tamnd.github.io/math2/node/1X3EV2TZNN.md): conjecture 1X3EV2TZNN
- [Sectional curvature](https://tamnd.github.io/math2/node/1X42JAA349.md): concept 1X42JAA349
- [Balayage](https://tamnd.github.io/math2/node/1X62YMAGMN.md): definition 1X62YMAGMN
- [Koszul complex](https://tamnd.github.io/math2/node/1XC3QZT3SD.md): definition 1XC3QZT3SD
- [Cartan–Thullen theorem](https://tamnd.github.io/math2/node/1XCBCB3MD7.md): statement 1XCBCB3MD7
- [Degree of a projective variety](https://tamnd.github.io/math2/node/1XG9GSHZ3N.md): concept 1XG9GSHZ3N
- [Weak ordering](https://tamnd.github.io/math2/node/1XNJHTW6TK.md): definition 1XNJHTW6TK
- [Riemann zeta function](https://tamnd.github.io/math2/node/1XWBY4N0VT.md): concept 1XWBY4N0VT
- [Algebraic set](https://tamnd.github.io/math2/node/1XYZNC2E6G.md): definition 1XYZNC2E6G
- [O-minimal theory](https://tamnd.github.io/math2/node/1XZJCGJ8B5.md): definition 1XZJCGJ8B5
- [Saccheri–Legendre theorem](https://tamnd.github.io/math2/node/1Y2XK0QJDX.md): statement 1Y2XK0QJDX
- [Poisson regression](https://tamnd.github.io/math2/node/1Y31ZXTRTV.md): definition 1Y31ZXTRTV
- [Lebesgue's theorem on differentiability of monotone functions](https://tamnd.github.io/math2/node/1Y4N1DDQNR.md): statement 1Y4N1DDQNR
- [Hodge star operator](https://tamnd.github.io/math2/node/1Y4WX4XQWJ.md): concept 1Y4WX4XQWJ
- [Analytic torsion](https://tamnd.github.io/math2/node/1Y7AH9QTEQ.md): definition 1Y7AH9QTEQ
- [Loewner's torus inequality](https://tamnd.github.io/math2/node/1Y9X1YW6GH.md): statement 1Y9X1YW6GH
- [Favard's theorem](https://tamnd.github.io/math2/node/1YADG8ESHY.md): statement 1YADG8ESHY
- [Newton–Kantorovich theorem](https://tamnd.github.io/math2/node/1YDR94WQMZ.md): statement 1YDR94WQMZ
- [Local index theorem](https://tamnd.github.io/math2/node/1YKCH218MN.md): statement 1YKCH218MN
- [Lickorish–Wallace theorem](https://tamnd.github.io/math2/node/1YP4RF1QXV.md): statement 1YP4RF1QXV
- [Optimal truncation](https://tamnd.github.io/math2/node/1YR53GZEPT.md): concept 1YR53GZEPT
- [Perfect graph](https://tamnd.github.io/math2/node/1YZ7NV161A.md): concept 1YZ7NV161A
- [Continuous additive functions are linear](https://tamnd.github.io/math2/node/1Z0105QF2Q.md): statement 1Z0105QF2Q
- [Separably generated extension](https://tamnd.github.io/math2/node/1Z2BCP64SZ.md): concept 1Z2BCP64SZ
- [Energy of a measure](https://tamnd.github.io/math2/node/1Z68SX12PA.md): definition 1Z68SX12PA
- [Dedekind number](https://tamnd.github.io/math2/node/1Z6RS2C7AK.md): definition 1Z6RS2C7AK
- [Dedekind domain](https://tamnd.github.io/math2/node/1Z9MRNNYYD.md): definition 1Z9MRNNYYD
- [Regular chain](https://tamnd.github.io/math2/node/1ZA98AGKGK.md): concept 1ZA98AGKGK
- [Topological group structure of a vector space](https://tamnd.github.io/math2/node/1ZAF1CC8M5.md): definition 1ZAF1CC8M5
- [Characteristic of a field](https://tamnd.github.io/math2/node/1ZB13N78XW.md): concept 1ZB13N78XW
- [Order (ring theory)](https://tamnd.github.io/math2/node/1ZBX4TKVZ9.md): concept 1ZBX4TKVZ9
- [Standard part function](https://tamnd.github.io/math2/node/1ZBY4W4E81.md): concept 1ZBY4W4E81
- [Shelf](https://tamnd.github.io/math2/node/1ZC0ZW96YQ.md): definition 1ZC0ZW96YQ
- [Dual cone](https://tamnd.github.io/math2/node/1ZDAGGGRP4.md): definition 1ZDAGGGRP4
- [Milnor number](https://tamnd.github.io/math2/node/1ZE51P7GRQ.md): concept 1ZE51P7GRQ
- [Split exact sequence](https://tamnd.github.io/math2/node/1ZPMDE4ARK.md): concept 1ZPMDE4ARK
- [Sequential analysis](https://tamnd.github.io/math2/node/1ZSTQGCCCQ.md): definition 1ZSTQGCCCQ
- [Radon–Nikodym theorem](https://tamnd.github.io/math2/node/1ZSZS5G1Q4.md): statement 1ZSZS5G1Q4
- [Locally path-connected space](https://tamnd.github.io/math2/node/1ZVR8BRK65.md): concept 1ZVR8BRK65
- [Clarke generalized gradient](https://tamnd.github.io/math2/node/20967BXZKA.md): concept 20967BXZKA
- [Berger's classification of holonomy groups](https://tamnd.github.io/math2/node/20GTNPF5V6.md): statement 20GTNPF5V6
- [Self-similar measure](https://tamnd.github.io/math2/node/20NQDPWS3J.md): definition 20NQDPWS3J
- [Artinian ring](https://tamnd.github.io/math2/node/20QEAM8AX2.md): concept 20QEAM8AX2
- [Shell theorem](https://tamnd.github.io/math2/node/20SAJR6J3Y.md): statement 20SAJR6J3Y
- [Locally compact group](https://tamnd.github.io/math2/node/20YERDAS5W.md): definition 20YERDAS5W
- [Paley–Zygmund inequality](https://tamnd.github.io/math2/node/211PJYW8W6.md): statement 211PJYW8W6
- [Homotopy groups of spheres](https://tamnd.github.io/math2/node/212MDDPN8A.md): definition 212MDDPN8A
- [Crossing number](https://tamnd.github.io/math2/node/212QCPWWA9.md): definition 212QCPWWA9
- [Grothendieck's Galois theory](https://tamnd.github.io/math2/node/218WAD58W3.md): definition 218WAD58W3
- [Devaney chaos](https://tamnd.github.io/math2/node/21B8AHPD4P.md): concept 21B8AHPD4P
- [Elementary substructure](https://tamnd.github.io/math2/node/21BZEVVT7G.md): definition 21BZEVVT7G
- [Tricomi equation](https://tamnd.github.io/math2/node/21DD2334NR.md): concept 21DD2334NR
- [Period-doubling cascade](https://tamnd.github.io/math2/node/21FMF8VKXY.md): definition 21FMF8VKXY
- [Zassenhaus lemma](https://tamnd.github.io/math2/node/21JJ523V95.md): statement 21JJ523V95
- [Bayesian network](https://tamnd.github.io/math2/node/21JT983G60.md): concept 21JT983G60
- [Ergodic decomposition](https://tamnd.github.io/math2/node/21RJ8035KP.md): statement 21RJ8035KP
- [Convex combination](https://tamnd.github.io/math2/node/21SXYJYC59.md): concept 21SXYJYC59
- [Sweedler notation](https://tamnd.github.io/math2/node/220PPH6N7Q.md): definition 220PPH6N7Q
- [Heegner point](https://tamnd.github.io/math2/node/2230ZKE4W9.md): concept 2230ZKE4W9
- [Wiener process](https://tamnd.github.io/math2/node/22735FGJV6.md): concept 22735FGJV6
- [Jung's theorem](https://tamnd.github.io/math2/node/227ERQ43PY.md): statement 227ERQ43PY
- [Hilbert–Samuel function](https://tamnd.github.io/math2/node/22J8E7AB8V.md): definition 22J8E7AB8V
- [Complete measure](https://tamnd.github.io/math2/node/22K8CSQTYM.md): concept 22K8CSQTYM
- [Ideal sheaf](https://tamnd.github.io/math2/node/22M7FHJ3N8.md): definition 22M7FHJ3N8
- [Euler–Maruyama method](https://tamnd.github.io/math2/node/22T0NVC56P.md): concept 22T0NVC56P
- [Gaussian process](https://tamnd.github.io/math2/node/22WEM6RTW0.md): concept 22WEM6RTW0
- [Weyl character formula for compact groups](https://tamnd.github.io/math2/node/230F5FXPMQ.md): statement 230F5FXPMQ
- [Quotient category](https://tamnd.github.io/math2/node/231CSHFC5K.md): definition 231CSHFC5K
- [Measure-preserving transformation](https://tamnd.github.io/math2/node/231TRBRZSY.md): definition 231TRBRZSY
- [Tricomi confluent hypergeometric function](https://tamnd.github.io/math2/node/234F5DNAJH.md): concept 234F5DNAJH
- [Formally real field](https://tamnd.github.io/math2/node/2362AFNG8X.md): definition 2362AFNG8X
- [Lagrange interpolation](https://tamnd.github.io/math2/node/237KR6FRCK.md): concept 237KR6FRCK
- [Order of a zero or pole](https://tamnd.github.io/math2/node/23ASRP0MWR.md): concept 23ASRP0MWR
- [Rubin causal model](https://tamnd.github.io/math2/node/23ATBZMJHR.md): concept 23ATBZMJHR
- [Nonexistence of infinite-dimensional Lebesgue measure](https://tamnd.github.io/math2/node/23DM2PQC78.md): statement 23DM2PQC78
- [Primitive polynomial over a finite field](https://tamnd.github.io/math2/node/23G7N7WZBH.md): definition 23G7N7WZBH
- [Fundamental theorem of Γ-convergence](https://tamnd.github.io/math2/node/23RK8EZ20P.md): statement 23RK8EZ20P
- [Thom's classification of elementary catastrophes](https://tamnd.github.io/math2/node/23RVD8K8RB.md): statement 23RVD8K8RB
- [Fredholm theory](https://tamnd.github.io/math2/node/242TFD41KZ.md): concept 242TFD41KZ
- [Cohn's irreducibility criterion](https://tamnd.github.io/math2/node/246SBC2M2N.md): statement 246SBC2M2N
- [Serial ring](https://tamnd.github.io/math2/node/249ZVY6WK3.md): definition 249ZVY6WK3
- [Schur-convex function](https://tamnd.github.io/math2/node/24CEMRAGJ8.md): definition 24CEMRAGJ8
- [Scalar curvature](https://tamnd.github.io/math2/node/24CP8Q56CV.md): definition 24CP8Q56CV
- [Expansion of iterated differences in shifts](https://tamnd.github.io/math2/node/24D1EGHP4G.md): statement 24D1EGHP4G
- [Dirichlet L-function](https://tamnd.github.io/math2/node/24GAZHVTGB.md): definition 24GAZHVTGB
- [Urysohn integral equation](https://tamnd.github.io/math2/node/24RRYYGC7S.md): definition 24RRYYGC7S
- [Conjugate residual method](https://tamnd.github.io/math2/node/24YCKE2RPH.md): statement 24YCKE2RPH
- [Sinai–Ruelle–Bowen measure](https://tamnd.github.io/math2/node/256NP6JBEQ.md): concept 256NP6JBEQ
- [Delaunay triangulation](https://tamnd.github.io/math2/node/259Y10BYJC.md): concept 259Y10BYJC
- [Choquet capacity](https://tamnd.github.io/math2/node/25FRWWHFZV.md): concept 25FRWWHFZV
- [Gelfand's problem on the index](https://tamnd.github.io/math2/node/25G4AP9P4R.md): problem 25G4AP9P4R
- [Schur test](https://tamnd.github.io/math2/node/25HF1WPPKX.md): statement 25HF1WPPKX
- [Strictly convex set](https://tamnd.github.io/math2/node/25RZTHSRS3.md): definition 25RZTHSRS3
- [Kolmogorov's everywhere divergent Fourier series](https://tamnd.github.io/math2/node/25V2MHPHZ7.md): counterexample 25V2MHPHZ7
- [Wilson polynomials](https://tamnd.github.io/math2/node/25Y6SVM2D1.md): definition 25Y6SVM2D1
- [Rotation group SO(3)](https://tamnd.github.io/math2/node/263GGJFA21.md): example 263GGJFA21
- [Ball's existence theorem in nonlinear elasticity](https://tamnd.github.io/math2/node/269C7M56G5.md): statement 269C7M56G5
- [Low-discrepancy sequence](https://tamnd.github.io/math2/node/26C8QTZSVY.md): definition 26C8QTZSVY
- [Dickey–Fuller test](https://tamnd.github.io/math2/node/26E0HYRDS8.md): concept 26E0HYRDS8
- [Stable homotopy category](https://tamnd.github.io/math2/node/26GTNMAF7B.md): concept 26GTNMAF7B
- [Forgetful functor](https://tamnd.github.io/math2/node/26RWFX4H2H.md): definition 26RWFX4H2H
- [Gauss circle problem](https://tamnd.github.io/math2/node/26XTJJT267.md): problem 26XTJJT267
- [Classifying space](https://tamnd.github.io/math2/node/272KQSXNEA.md): definition 272KQSXNEA
- [Grothendieck construction](https://tamnd.github.io/math2/node/272T16NWZ9.md): concept 272T16NWZ9
- [Subnormal operator](https://tamnd.github.io/math2/node/273NZSYWZA.md): definition 273NZSYWZA
- [Exponential map](https://tamnd.github.io/math2/node/275GQ9T7EB.md): definition 275GQ9T7EB
- [Complex geodesic](https://tamnd.github.io/math2/node/275YT975A8.md): concept 275YT975A8
- [Jacobi sum](https://tamnd.github.io/math2/node/2767J23XHN.md): concept 2767J23XHN
- [Left-invariant vector field](https://tamnd.github.io/math2/node/279SXZFBJ2.md): definition 279SXZFBJ2
- [Reflexive space](https://tamnd.github.io/math2/node/27BDDC590S.md): definition 27BDDC590S
- [Iterative root](https://tamnd.github.io/math2/node/27C6J3YNZY.md): concept 27C6J3YNZY
- [Somos sequence](https://tamnd.github.io/math2/node/27K5X7S4XN.md): example 27K5X7S4XN
- [Kleinian group](https://tamnd.github.io/math2/node/27QRARKWD5.md): concept 27QRARKWD5
- [Free monoid](https://tamnd.github.io/math2/node/27ZPTJP9PP.md): definition 27ZPTJP9PP
- [Holomorphic inverse function theorem](https://tamnd.github.io/math2/node/282MQP4X82.md): statement 282MQP4X82
- [Kolmogorov backward equation](https://tamnd.github.io/math2/node/284TYTVCC0.md): definition 284TYTVCC0
- [Smooth morphism of rings](https://tamnd.github.io/math2/node/286G0P1B69.md): concept 286G0P1B69
- [Young tableau](https://tamnd.github.io/math2/node/287JS93ZDS.md): definition 287JS93ZDS
- [Maslov index](https://tamnd.github.io/math2/node/28ADY9WTV3.md): definition 28ADY9WTV3
- [Minimal curvature property of natural cubic splines](https://tamnd.github.io/math2/node/28C5WWGX9J.md): statement 28C5WWGX9J
- [Tautological line bundle](https://tamnd.github.io/math2/node/28DVNYNMZR.md): definition 28DVNYNMZR
- [Inner regular measure](https://tamnd.github.io/math2/node/28E5Q5BM69.md): concept 28E5Q5BM69
- [Words and profiles](https://tamnd.github.io/math2/node/28M8XD8EA9.md): concept 28M8XD8EA9
- [Functor](https://tamnd.github.io/math2/node/28NQ0KRHD3.md): concept 28NQ0KRHD3
- [Order type](https://tamnd.github.io/math2/node/28PE4GRFD7.md): definition 28PE4GRFD7
- [Fractional Fourier transform](https://tamnd.github.io/math2/node/28VV812XWQ.md): concept 28VV812XWQ
- [Ind-completion](https://tamnd.github.io/math2/node/290G1NGAMR.md): definition 290G1NGAMR
- [Koopman operator](https://tamnd.github.io/math2/node/2941ECC4QQ.md): definition 2941ECC4QQ
- [Periodic tiling](https://tamnd.github.io/math2/node/29AR3Z107J.md): definition 29AR3Z107J
- [Regular summability method](https://tamnd.github.io/math2/node/29C7C41A52.md): concept 29C7C41A52
- [Ornstein–Zernike equation](https://tamnd.github.io/math2/node/29T4NC87AR.md): example 29T4NC87AR
- [Malgrange preparation theorem](https://tamnd.github.io/math2/node/2A5HZZRGFT.md): statement 2A5HZZRGFT
- [Artin–Mazur zeta function](https://tamnd.github.io/math2/node/2A5Z21CE4P.md): concept 2A5Z21CE4P
- [Integrally closed domain](https://tamnd.github.io/math2/node/2A7CYKQ2XW.md): definition 2A7CYKQ2XW
- [Projection method for integral equations](https://tamnd.github.io/math2/node/2AFNVJ54G7.md): concept 2AFNVJ54G7
- [Osgood uniqueness criterion](https://tamnd.github.io/math2/node/2AHNVSH76M.md): statement 2AHNVSH76M
- [Positive linear operator](https://tamnd.github.io/math2/node/2AKFNAGF1N.md): definition 2AKFNAGF1N
- [Markov kernel](https://tamnd.github.io/math2/node/2AVNDP8BJP.md): concept 2AVNDP8BJP
- [Adjoint representation of a Lie group](https://tamnd.github.io/math2/node/2B3BWV1HPC.md): definition 2B3BWV1HPC
- [Topological transformation group](https://tamnd.github.io/math2/node/2B8KYPCPAS.md): concept 2B8KYPCPAS
- [Shifted Legendre polynomials](https://tamnd.github.io/math2/node/2B9G13MQK3.md): definition 2B9G13MQK3
- [Admissibility condition for wavelets](https://tamnd.github.io/math2/node/2BBB8NY5ZA.md): concept 2BBB8NY5ZA
- [Permuting congruences](https://tamnd.github.io/math2/node/2BCES6NPYM.md): concept 2BCES6NPYM
- [Unitary dual](https://tamnd.github.io/math2/node/2BDH4RQZQ9.md): definition 2BDH4RQZQ9
- [Universal coalgebra](https://tamnd.github.io/math2/node/2BG03EJT8D.md): concept 2BG03EJT8D
- [Factor congruence](https://tamnd.github.io/math2/node/2BJE1ABGDJ.md): definition 2BJE1ABGDJ
- [Herbrand universe](https://tamnd.github.io/math2/node/2BVB7YDEV4.md): example 2BVB7YDEV4
- [Grätzer–Schmidt theorem](https://tamnd.github.io/math2/node/2BZ2ZXH03B.md): statement 2BZ2ZXH03B
- [Operational calculus](https://tamnd.github.io/math2/node/2C008K3V28.md): definition 2C008K3V28
- [Homogeneous polynomial](https://tamnd.github.io/math2/node/2C3PAQ7H50.md): concept 2C3PAQ7H50
- [Ordinal arithmetic](https://tamnd.github.io/math2/node/2C7MBG93AZ.md): definition 2C7MBG93AZ
- [Semiperfect ring](https://tamnd.github.io/math2/node/2C8JN6XBWY.md): definition 2C8JN6XBWY
- [Artin's induction theorem](https://tamnd.github.io/math2/node/2CABBY4EEB.md): statement 2CABBY4EEB
- [Vietoris topology](https://tamnd.github.io/math2/node/2CG46QFBHW.md): definition 2CG46QFBHW
- [Well-formed formula](https://tamnd.github.io/math2/node/2CHENV4DCC.md): definition 2CHENV4DCC
- [Discrete series representation](https://tamnd.github.io/math2/node/2CKZETC3QA.md): definition 2CKZETC3QA
- [Cellular automaton](https://tamnd.github.io/math2/node/2CM2MC5HQ6.md): definition 2CM2MC5HQ6
- [Geodesic](https://tamnd.github.io/math2/node/2CNFVTWFRC.md): definition 2CNFVTWFRC
- [Grauert's Oka principle for vector bundles](https://tamnd.github.io/math2/node/2CVH8BCHTF.md): statement 2CVH8BCHTF
- [Category of groups](https://tamnd.github.io/math2/node/2CX7RPPXA9.md): example 2CX7RPPXA9
- [E8 lattice](https://tamnd.github.io/math2/node/2CXH65EKY7.md): object 2CXH65EKY7
- [Fully nonlinear PDE](https://tamnd.github.io/math2/node/2D3N1ZJWBW.md): concept 2D3N1ZJWBW
- [Kleene algebra](https://tamnd.github.io/math2/node/2D4E6NY8Q4.md): definition 2D4E6NY8Q4
- [Function field of a variety](https://tamnd.github.io/math2/node/2D5SS6WM14.md): concept 2D5SS6WM14
- [Bergman space](https://tamnd.github.io/math2/node/2D619WTPYC.md): concept 2D619WTPYC
- [Horocycle flow](https://tamnd.github.io/math2/node/2DC88BMQVX.md): example 2DC88BMQVX
- [Mathieu function](https://tamnd.github.io/math2/node/2DDZXF5EBY.md): concept 2DDZXF5EBY
- [Köthe–Toeplitz dual](https://tamnd.github.io/math2/node/2DJT2AYY23.md): definition 2DJT2AYY23
- [Montel space](https://tamnd.github.io/math2/node/2DSKCVDFAS.md): definition 2DSKCVDFAS
- [Koksma–Hlawka inequality](https://tamnd.github.io/math2/node/2DSN343YYW.md): statement 2DSN343YYW
- [Bachmann's reflection geometry](https://tamnd.github.io/math2/node/2DT5DGTB8V.md): definition 2DT5DGTB8V
- [Hilbert matrix](https://tamnd.github.io/math2/node/2DTWB6F5TA.md): example 2DTWB6F5TA
- [Index of the de Rham complex](https://tamnd.github.io/math2/node/2DYR4C5Q5V.md): example 2DYR4C5Q5V
- [Relational structure](https://tamnd.github.io/math2/node/2E1R3BDDMK.md): definition 2E1R3BDDMK
- [Homotopy](https://tamnd.github.io/math2/node/2E24G3S4CK.md): definition 2E24G3S4CK
- [Hodges' superefficient estimator](https://tamnd.github.io/math2/node/2E46CY6S3Z.md): counterexample 2E46CY6S3Z
- [Homotopy hypothesis](https://tamnd.github.io/math2/node/2E83D9ACDY.md): conjecture 2E83D9ACDY
- [Martingale representation theorem](https://tamnd.github.io/math2/node/2E8MQXDYNV.md): statement 2E8MQXDYNV
- [Quantum logic](https://tamnd.github.io/math2/node/2E9DMB7A6Q.md): concept 2E9DMB7A6Q
- [Permutation test](https://tamnd.github.io/math2/node/2ECAM46NE2.md): concept 2ECAM46NE2
- [Spherical mean transform](https://tamnd.github.io/math2/node/2EE62A1QXA.md): definition 2EE62A1QXA
- [Integral equation of the second kind](https://tamnd.github.io/math2/node/2EEKCBNYS0.md): definition 2EEKCBNYS0
- [Catastrophe theory](https://tamnd.github.io/math2/node/2EM3EQEX2J.md): definition 2EM3EQEX2J
- [Inaccessible cardinal](https://tamnd.github.io/math2/node/2ENTSSHEPZ.md): definition 2ENTSSHEPZ
- [List of simple Lie groups](https://tamnd.github.io/math2/node/2EPRMNK1G2.md): example 2EPRMNK1G2
- [Modified Bessel functions](https://tamnd.github.io/math2/node/2ET068BZZH.md): concept 2ET068BZZH
- [Darboux's theorem on additive functions bounded on an interval](https://tamnd.github.io/math2/node/2F30ZHN1RT.md): statement 2F30ZHN1RT
- [Quantile regression](https://tamnd.github.io/math2/node/2F5C10ED1A.md): concept 2F5C10ED1A
- [Theta function](https://tamnd.github.io/math2/node/2F5CYQJVAF.md): concept 2F5CYQJVAF
- [Green's function for the Laplacian](https://tamnd.github.io/math2/node/2F6AJ1DM11.md): definition 2F6AJ1DM11
- [Von Neumann's theorem on self-adjoint extensions](https://tamnd.github.io/math2/node/2FE4P23NVC.md): statement 2FE4P23NVC
- [Conformal modulus](https://tamnd.github.io/math2/node/2FQAKBVBX5.md): concept 2FQAKBVBX5
- [Integral quadratic form](https://tamnd.github.io/math2/node/2FQVQCTV88.md): concept 2FQVQCTV88
- [dg category](https://tamnd.github.io/math2/node/2FR1BWY3H6.md): definition 2FR1BWY3H6
- [Absolutely continuous measure](https://tamnd.github.io/math2/node/2G173YJC54.md): definition 2G173YJC54
- [t-structure](https://tamnd.github.io/math2/node/2GD9V7SA0E.md): concept 2GD9V7SA0E
- [Lagrange resolvent](https://tamnd.github.io/math2/node/2GDFFH7Y7T.md): concept 2GDFFH7Y7T
- [Reeb vector field](https://tamnd.github.io/math2/node/2GF2675154.md): definition 2GF2675154
- [Mean curvature flow](https://tamnd.github.io/math2/node/2GFFGE6WK1.md): definition 2GFFGE6WK1
- [Phragmén–Lindelöf principle for subharmonic functions](https://tamnd.github.io/math2/node/2GQKS3B3X0.md): statement 2GQKS3B3X0
- [Symmetric decreasing rearrangement](https://tamnd.github.io/math2/node/2GRPCP5XK9.md): concept 2GRPCP5XK9
- [Whitney embedding theorem](https://tamnd.github.io/math2/node/2GSRBDT96M.md): statement 2GSRBDT96M
- [Projective determinacy](https://tamnd.github.io/math2/node/2GY3131ZPS.md): concept 2GY3131ZPS
- [Local regression](https://tamnd.github.io/math2/node/2H0J4P4NQF.md): definition 2H0J4P4NQF
- [Garden of Eden theorem](https://tamnd.github.io/math2/node/2H1XHWDACH.md): statement 2H1XHWDACH
- [Chain recurrence](https://tamnd.github.io/math2/node/2H2BRQD0YX.md): definition 2H2BRQD0YX
- [Closure operator](https://tamnd.github.io/math2/node/2HJGRABSPZ.md): concept 2HJGRABSPZ
- [Nilpotency class](https://tamnd.github.io/math2/node/2HKA02YDDM.md): definition 2HKA02YDDM
- [Wavelet packet decomposition](https://tamnd.github.io/math2/node/2HNN80TMGH.md): definition 2HNN80TMGH
- [Riemann zeta function as a series](https://tamnd.github.io/math2/node/2HP4SA15DW.md): example 2HP4SA15DW
- [Poincaré asymptotic series](https://tamnd.github.io/math2/node/2HVC5G0H5K.md): concept 2HVC5G0H5K
- [Finite projective plane](https://tamnd.github.io/math2/node/2HWB7P1DZN.md): object 2HWB7P1DZN
- [Infinite product measure](https://tamnd.github.io/math2/node/2HWVFHM13R.md): concept 2HWVFHM13R
- [Sheafification](https://tamnd.github.io/math2/node/2HYK3KBQNA.md): concept 2HYK3KBQNA
- [Itô's lemma](https://tamnd.github.io/math2/node/2J1RVQB0CM.md): statement 2J1RVQB0CM
- [Homology of a chain complex](https://tamnd.github.io/math2/node/2J4JJJYP22.md): concept 2J4JJJYP22
- [Holomorphic automorphisms of C^n](https://tamnd.github.io/math2/node/2J7JB3H3FD.md): concept 2J7JB3H3FD
- [Harmonic functions on Riemannian manifolds](https://tamnd.github.io/math2/node/2J9GA17YAK.md): definition 2J9GA17YAK
- [Stochastic geometry](https://tamnd.github.io/math2/node/2J9GB49FHA.md): concept 2J9GB49FHA
- [Pro-p group](https://tamnd.github.io/math2/node/2JABC6VSE1.md): definition 2JABC6VSE1
- [Autoregressive model](https://tamnd.github.io/math2/node/2JC96FMGA1.md): concept 2JC96FMGA1
- [Hopf–Lax formula](https://tamnd.github.io/math2/node/2JD72JT9FQ.md): statement 2JD72JT9FQ
- [Cauchy matrix](https://tamnd.github.io/math2/node/2JDRN4ZS49.md): definition 2JDRN4ZS49
- [Countably generated structure](https://tamnd.github.io/math2/node/2JFAZSGB6J.md): concept 2JFAZSGB6J
- [Loop space](https://tamnd.github.io/math2/node/2JFF5F3EGF.md): concept 2JFF5F3EGF
- [Weyl's law](https://tamnd.github.io/math2/node/2JK9YMZV2B.md): statement 2JK9YMZV2B
- [Associativity equation](https://tamnd.github.io/math2/node/2JMA7DZMG8.md): concept 2JMA7DZMG8
- [Simplex algorithm](https://tamnd.github.io/math2/node/2JS8PPP8XW.md): definition 2JS8PPP8XW
- [Cancellation property](https://tamnd.github.io/math2/node/2JVB5R0MBW.md): definition 2JVB5R0MBW
- [Holomorphic function of several variables](https://tamnd.github.io/math2/node/2K13M9FBAN.md): definition 2K13M9FBAN
- [Nilradical](https://tamnd.github.io/math2/node/2K5GKVZR78.md): definition 2K5GKVZR78
- [Representation of a Lie group](https://tamnd.github.io/math2/node/2K8BZRGE8R.md): definition 2K8BZRGE8R
- [Non-Euclidean geometry](https://tamnd.github.io/math2/node/2K9VZ8M8AG.md): definition 2K9VZ8M8AG
- [Amalgamation property](https://tamnd.github.io/math2/node/2KBFZ1MPMW.md): definition 2KBFZ1MPMW
- [Finite Hilbert transform](https://tamnd.github.io/math2/node/2KJV7187NR.md): definition 2KJV7187NR
- [Hypergraph](https://tamnd.github.io/math2/node/2KM46K20DZ.md): definition 2KM46K20DZ
- [Y transform](https://tamnd.github.io/math2/node/2KTYK7BNA9.md): concept 2KTYK7BNA9
- [Morse–Bott function](https://tamnd.github.io/math2/node/2KVA0Q66RE.md): definition 2KVA0Q66RE
- [Inverse transform sampling](https://tamnd.github.io/math2/node/2KX7W1TG7T.md): statement 2KX7W1TG7T
- [Matrix Wiener–Hopf factorization](https://tamnd.github.io/math2/node/2M672PP7QR.md): statement 2M672PP7QR
- [Unconditional convergence](https://tamnd.github.io/math2/node/2ME9M61TJK.md): concept 2ME9M61TJK
- [Weak mixing](https://tamnd.github.io/math2/node/2MGZFRH5PX.md): concept 2MGZFRH5PX
- [Harish-Chandra module](https://tamnd.github.io/math2/node/2MKHH636C3.md): concept 2MKHH636C3
- [Backward difference operator](https://tamnd.github.io/math2/node/2MM7AHFR2S.md): definition 2MM7AHFR2S
- [Eberlein–Šmulian theorem](https://tamnd.github.io/math2/node/2MTN8CQWME.md): statement 2MTN8CQWME
- [Space of convergent sequences c](https://tamnd.github.io/math2/node/2N4RJCW5JK.md): example 2N4RJCW5JK
- [Interval order](https://tamnd.github.io/math2/node/2N4Z69K7V6.md): definition 2N4Z69K7V6
- [Group von Neumann algebra](https://tamnd.github.io/math2/node/2N7CR03Q1X.md): concept 2N7CR03Q1X
- [Relaxation oscillation](https://tamnd.github.io/math2/node/2NHWCM0DK8.md): concept 2NHWCM0DK8
- [Jordan–Kinderlehrer–Otto scheme](https://tamnd.github.io/math2/node/2NVEYM6HN9.md): definition 2NVEYM6HN9
- [Poncelet–Steiner theorem](https://tamnd.github.io/math2/node/2P0DP2TK6P.md): statement 2P0DP2TK6P
- [Horn hypergeometric series](https://tamnd.github.io/math2/node/2P0V6X8GZ3.md): definition 2P0V6X8GZ3
- [Linear logic](https://tamnd.github.io/math2/node/2P2184B6HE.md): concept 2P2184B6HE
- [Free algebra](https://tamnd.github.io/math2/node/2P3Y2FRCSZ.md): concept 2P3Y2FRCSZ
- [List of integral transforms](https://tamnd.github.io/math2/node/2P4RDS0A6X.md): concept 2P4RDS0A6X
- [Fourier–Mukai transform](https://tamnd.github.io/math2/node/2P90CV9QKH.md): concept 2P90CV9QKH
- [Hankel transform](https://tamnd.github.io/math2/node/2P9YWDGKC4.md): concept 2P9YWDGKC4
- [Lempert's theorem on convex domains](https://tamnd.github.io/math2/node/2PANNSQ3G6.md): statement 2PANNSQ3G6
- [Group cohomology](https://tamnd.github.io/math2/node/2PAS78F34T.md): definition 2PAS78F34T
- [Parallel postulate](https://tamnd.github.io/math2/node/2PC3XY0PYZ.md): statement 2PC3XY0PYZ
- [Temperley–Lieb algebra](https://tamnd.github.io/math2/node/2PCPEX85PX.md): definition 2PCPEX85PX
- [Born series](https://tamnd.github.io/math2/node/2PCVRG9NBN.md): concept 2PCVRG9NBN
- [Uniqueness of adjoints](https://tamnd.github.io/math2/node/2PEH9MPTAR.md): statement 2PEH9MPTAR
- [Cofinal subset](https://tamnd.github.io/math2/node/2PFQY2RZBC.md): concept 2PFQY2RZBC
- [Rectifiable set](https://tamnd.github.io/math2/node/2PHQQ01C0F.md): definition 2PHQQ01C0F
- [Bulirsch–Stoer algorithm](https://tamnd.github.io/math2/node/2PJECK6ZGW.md): statement 2PJECK6ZGW
- [Hahn decomposition theorem](https://tamnd.github.io/math2/node/2PKHANZ8JS.md): statement 2PKHANZ8JS
- [Transfinite diameter](https://tamnd.github.io/math2/node/2PMZ7GS0QY.md): definition 2PMZ7GS0QY
- [Ziggurat algorithm](https://tamnd.github.io/math2/node/2PSDHH34BJ.md): statement 2PSDHH34BJ
- [Elliptic regularity](https://tamnd.github.io/math2/node/2PVQY9PCT4.md): statement 2PVQY9PCT4
- [Path algebra](https://tamnd.github.io/math2/node/2PYANBGXFD.md): concept 2PYANBGXFD
- [Steinitz's theorem](https://tamnd.github.io/math2/node/2PYAZX5VD4.md): statement 2PYAZX5VD4
- [Jordan–Brouwer separation theorem](https://tamnd.github.io/math2/node/2Q008XWQDE.md): statement 2Q008XWQDE
- [Irrational rotation](https://tamnd.github.io/math2/node/2Q1J7RZ4TK.md): example 2Q1J7RZ4TK
- [Abel function](https://tamnd.github.io/math2/node/2QBWT7T02Y.md): concept 2QBWT7T02Y
- [Wavelet packet decomposition](https://tamnd.github.io/math2/node/2QKRBPS8B9.md): concept 2QKRBPS8B9
- [Étale morphism of rings](https://tamnd.github.io/math2/node/2QSKZ59DB6.md): definition 2QSKZ59DB6
- [Least squares approximation](https://tamnd.github.io/math2/node/2QV28WYQTQ.md): concept 2QV28WYQTQ
- [Clausen function](https://tamnd.github.io/math2/node/2R3VH69BSA.md): definition 2R3VH69BSA
- [Bochner–Riesz means](https://tamnd.github.io/math2/node/2R4H0RFQZZ.md): definition 2R4H0RFQZZ
- [Lattice of equational theories](https://tamnd.github.io/math2/node/2R8QQTP1CE.md): concept 2R8QQTP1CE
- [Newtonian space](https://tamnd.github.io/math2/node/2RCTYXSJ67.md): definition 2RCTYXSJ67
- [Projective subspace](https://tamnd.github.io/math2/node/2RD6T5KG59.md): concept 2RD6T5KG59
- [Prüfer domain](https://tamnd.github.io/math2/node/2RDAA4PCK8.md): concept 2RDAA4PCK8
- [Ehrenfeucht–Fraïssé game](https://tamnd.github.io/math2/node/2RE7YAZK8R.md): concept 2RE7YAZK8R
- [Repair radii and villainy](https://tamnd.github.io/math2/node/2REN148QWD.md): definition 2REN148QWD
- [Point spectrum](https://tamnd.github.io/math2/node/2RH4DB8KWW.md): definition 2RH4DB8KWW
- [Fredholm equation of the first kind](https://tamnd.github.io/math2/node/2RMWAVR1HV.md): concept 2RMWAVR1HV
- [Iterative method](https://tamnd.github.io/math2/node/2RPE77ACER.md): concept 2RPE77ACER
- [Pluriharmonic function](https://tamnd.github.io/math2/node/2RXEM9BQFY.md): definition 2RXEM9BQFY
- [Poisson bracket](https://tamnd.github.io/math2/node/2S0MXDT4YH.md): concept 2S0MXDT4YH
- [Field of formal Laurent series](https://tamnd.github.io/math2/node/2S4K8FA8QY.md): definition 2S4K8FA8QY
- [Generalized Fourier series](https://tamnd.github.io/math2/node/2S6XZCQWY7.md): definition 2S6XZCQWY7
- [Chinese restaurant process](https://tamnd.github.io/math2/node/2SEF8QTYQF.md): example 2SEF8QTYQF
- [Classification of surfaces](https://tamnd.github.io/math2/node/2SFM73YET0.md): statement 2SFM73YET0
- [Regular category](https://tamnd.github.io/math2/node/2SM3JRJXA4.md): concept 2SM3JRJXA4
- [Holomorphic functions on compact complex manifolds are constant](https://tamnd.github.io/math2/node/2SM64J81YC.md): statement 2SM64J81YC
- [Lifting idempotents](https://tamnd.github.io/math2/node/2SRDN8EJG3.md): concept 2SRDN8EJG3
- [Ohsawa–Takegoshi extension theorem](https://tamnd.github.io/math2/node/2SRQCEB16J.md): statement 2SRQCEB16J
- [Primitive ring](https://tamnd.github.io/math2/node/2T1AGV1K13.md): definition 2T1AGV1K13
- [Geometric group theory](https://tamnd.github.io/math2/node/2T3218RXTD.md): concept 2T3218RXTD
- [Monotone class theorem](https://tamnd.github.io/math2/node/2T33SA540E.md): statement 2T33SA540E
- [Finite element exterior calculus](https://tamnd.github.io/math2/node/2T8MYCZ7N4.md): concept 2T8MYCZ7N4
- [Balanced set](https://tamnd.github.io/math2/node/2TDSHJJYS4.md): concept 2TDSHJJYS4
- [Design of experiments](https://tamnd.github.io/math2/node/2TE42ZWJ3E.md): concept 2TE42ZWJ3E
- [Germ of a holomorphic function](https://tamnd.github.io/math2/node/2TGPZ8VW2D.md): definition 2TGPZ8VW2D
- [Mellin convergence](https://tamnd.github.io/math2/node/2TM14070HZ.md): definition 2TM14070HZ
- [Generated subalgebra](https://tamnd.github.io/math2/node/2TQD8NMG9P.md): concept 2TQD8NMG9P
- [Hankel contour integral for the gamma function](https://tamnd.github.io/math2/node/2V0TG9Z3ZW.md): statement 2V0TG9Z3ZW
- [Beurling's theorem](https://tamnd.github.io/math2/node/2V1DXXM9FA.md): statement 2V1DXXM9FA
- [Method of multiple scales](https://tamnd.github.io/math2/node/2V5E1B1Q9H.md): concept 2V5E1B1Q9H
- [Absorbing set](https://tamnd.github.io/math2/node/2V7JHTRMTA.md): definition 2V7JHTRMTA
- [Contravariant functor](https://tamnd.github.io/math2/node/2V8FATXF50.md): definition 2V8FATXF50
- [Matlis duality](https://tamnd.github.io/math2/node/2V9BR5XX6A.md): statement 2V9BR5XX6A
- [Fredholm index](https://tamnd.github.io/math2/node/2V9Q4KHY1T.md): concept 2V9Q4KHY1T
- [Suspension](https://tamnd.github.io/math2/node/2VD7GR64JJ.md): definition 2VD7GR64JJ
- [I-adic topology](https://tamnd.github.io/math2/node/2VF2TSJ3PY.md): concept 2VF2TSJ3PY
- [Derived category of coherent sheaves](https://tamnd.github.io/math2/node/2VP2D51KB8.md): concept 2VP2D51KB8
- [Nichols algebra](https://tamnd.github.io/math2/node/2VQ771ENN0.md): definition 2VQ771ENN0
- [Namioka--Phelps](https://tamnd.github.io/math2/node/2VQDK71A27.md): statement 2VQDK71A27
- [Noetherian topological space](https://tamnd.github.io/math2/node/2VQX2E8EAH.md): concept 2VQX2E8EAH
- [Auslander–Reiten theory](https://tamnd.github.io/math2/node/2VY5M3M91Z.md): concept 2VY5M3M91Z
- [Mac Lane's coherence theorem](https://tamnd.github.io/math2/node/2W2X7EBG2S.md): statement 2W2X7EBG2S
- [Dense linear order](https://tamnd.github.io/math2/node/2W4S2H4VYA.md): example 2W4S2H4VYA
- [Completion of a normed space](https://tamnd.github.io/math2/node/2W7R5JHF0C.md): concept 2W7R5JHF0C
- [Mathematical statistics](https://tamnd.github.io/math2/node/2W9FEY4Y6K.md): definition 2W9FEY4Y6K
- [Natural duality](https://tamnd.github.io/math2/node/2WADQVGPD4.md): definition 2WADQVGPD4
- [Ideal of compact operators](https://tamnd.github.io/math2/node/2WAZF1RV2Y.md): concept 2WAZF1RV2Y
- [Heegaard Floer homology](https://tamnd.github.io/math2/node/2WJJBN2T5P.md): concept 2WJJBN2T5P
- [Prony's method](https://tamnd.github.io/math2/node/2WSV04QETW.md): concept 2WSV04QETW
- [F-test](https://tamnd.github.io/math2/node/2WXDHWVZ03.md): definition 2WXDHWVZ03
- [Polylogarithm](https://tamnd.github.io/math2/node/2XB240D6MP.md): definition 2XB240D6MP
- [Z* theorem](https://tamnd.github.io/math2/node/2XCJ8NNMXP.md): statement 2XCJ8NNMXP
- [Bifurcation diagram](https://tamnd.github.io/math2/node/2XDRZV3GSA.md): concept 2XDRZV3GSA
- [Atomless Boolean algebra](https://tamnd.github.io/math2/node/2XF2Y9B5E2.md): definition 2XF2Y9B5E2
- [Laplace transform of the Dirac delta](https://tamnd.github.io/math2/node/2XK0YX7VMC.md): example 2XK0YX7VMC
- [Reductive group](https://tamnd.github.io/math2/node/2XS84PWE3S.md): concept 2XS84PWE3S
- [Errors-in-variables model](https://tamnd.github.io/math2/node/2XX0A85WSF.md): concept 2XX0A85WSF
- [Bialgebra](https://tamnd.github.io/math2/node/2XYEC53KNK.md): definition 2XYEC53KNK
- [Benamou–Brenier formula](https://tamnd.github.io/math2/node/2XYQRM1NFH.md): statement 2XYQRM1NFH
- [Knot group](https://tamnd.github.io/math2/node/2Y52Q00MGE.md): definition 2Y52Q00MGE
- [Möbius function of a poset](https://tamnd.github.io/math2/node/2Y75S8NG5B.md): concept 2Y75S8NG5B
- [Factor analysis](https://tamnd.github.io/math2/node/2Y7P8SABM9.md): definition 2Y7P8SABM9
- [Series-parallel partial order](https://tamnd.github.io/math2/node/2YC1Q1MDV8.md): concept 2YC1Q1MDV8
- [Todd class](https://tamnd.github.io/math2/node/2YDB5089FR.md): concept 2YDB5089FR
- [Disjoint union of posets](https://tamnd.github.io/math2/node/2YHG0X6EA0.md): definition 2YHG0X6EA0
- [Cartan formula](https://tamnd.github.io/math2/node/2YPJ39STQM.md): statement 2YPJ39STQM
- [Relative homotopy group](https://tamnd.github.io/math2/node/2YPY6YTHX8.md): definition 2YPY6YTHX8
- [Milnor K-theory](https://tamnd.github.io/math2/node/2YXDQ7Q445.md): concept 2YXDQ7Q445
- [Singular function](https://tamnd.github.io/math2/node/2YYCNFPX1F.md): definition 2YYCNFPX1F
- [Mean curvature flow](https://tamnd.github.io/math2/node/2YYV01C7QQ.md): concept 2YYV01C7QQ
- [Convolution](https://tamnd.github.io/math2/node/2Z29JN3VES.md): concept 2Z29JN3VES
- [Lebesgue measure](https://tamnd.github.io/math2/node/2Z2QVWZ262.md): object 2Z2QVWZ262
- [von Neumann stability analysis](https://tamnd.github.io/math2/node/2Z7NASDQB6.md): statement 2Z7NASDQB6
- [Hermitian kernel](https://tamnd.github.io/math2/node/2ZB3NFZ94T.md): concept 2ZB3NFZ94T
- [Smash product](https://tamnd.github.io/math2/node/2ZJ2G308QV.md): concept 2ZJ2G308QV
- [Wold's decomposition theorem](https://tamnd.github.io/math2/node/2ZK4CE9MF8.md): statement 2ZK4CE9MF8
- [Schwartz kernel theorem](https://tamnd.github.io/math2/node/2ZKWZ0A4MT.md): statement 2ZKWZ0A4MT
- [Kirszbraun theorem](https://tamnd.github.io/math2/node/2ZM39HSFTQ.md): statement 2ZM39HSFTQ
- [Lyapunov's convexity theorem](https://tamnd.github.io/math2/node/2ZTQX5M0TK.md): statement 2ZTQX5M0TK
- [Convex polytope](https://tamnd.github.io/math2/node/300K6CB1KH.md): concept 300K6CB1KH
- [Chromatic homotopy theory](https://tamnd.github.io/math2/node/300V40FC1B.md): concept 300V40FC1B
- [Pseudogroup](https://tamnd.github.io/math2/node/3017J7P6JD.md): concept 3017J7P6JD
- [Higher-order differences](https://tamnd.github.io/math2/node/3039GWNHWS.md): concept 3039GWNHWS
- [Frame](https://tamnd.github.io/math2/node/306PWGH7BF.md): concept 306PWGH7BF
- [Time-scale calculus](https://tamnd.github.io/math2/node/30G1M35TK1.md): concept 30G1M35TK1
- [Meshfree methods](https://tamnd.github.io/math2/node/30GF7TK8YP.md): definition 30GF7TK8YP
- [Artin–Rees lemma](https://tamnd.github.io/math2/node/30H6B0G2FT.md): statement 30H6B0G2FT
- [Toda bracket](https://tamnd.github.io/math2/node/30K7VF7QBB.md): concept 30K7VF7QBB
- [Perfect map](https://tamnd.github.io/math2/node/30R6VND1T7.md): definition 30R6VND1T7
- [Restriction of a measure](https://tamnd.github.io/math2/node/30SKVHXFNQ.md): definition 30SKVHXFNQ
- [Nevanlinna characteristic](https://tamnd.github.io/math2/node/30TG40QA1J.md): definition 30TG40QA1J
- [Stein's unbiased risk estimate](https://tamnd.github.io/math2/node/30XVCMK916.md): statement 30XVCMK916
- [Sylvester equation](https://tamnd.github.io/math2/node/312GG23DN4.md): definition 312GG23DN4
- [Stationary process](https://tamnd.github.io/math2/node/314WH1WAP1.md): definition 314WH1WAP1
- [Malgrange–Ehrenpreis theorem](https://tamnd.github.io/math2/node/3173Y813V7.md): statement 3173Y813V7
- [Demailly's regularization of currents](https://tamnd.github.io/math2/node/31CV5K5SAJ.md): statement 31CV5K5SAJ
- [Cauchy integral on Lipschitz curves](https://tamnd.github.io/math2/node/31DQH72ZJP.md): statement 31DQH72ZJP
- [Lanczos sigma factor](https://tamnd.github.io/math2/node/31G7KHSYEC.md): concept 31G7KHSYEC
- [Jones index theorem](https://tamnd.github.io/math2/node/31N0FH22ZT.md): statement 31N0FH22ZT
- [Clifford module](https://tamnd.github.io/math2/node/31Q0RDPZ00.md): concept 31Q0RDPZ00
- [Angle of parallelism](https://tamnd.github.io/math2/node/323GN6K117.md): concept 323GN6K117
- [Lorentz space](https://tamnd.github.io/math2/node/3240RP14MT.md): concept 3240RP14MT
- [Heat kernel on a Riemannian manifold](https://tamnd.github.io/math2/node/327NP6S7H5.md): concept 327NP6S7H5
- [De Giorgi–Nash–Moser theorem](https://tamnd.github.io/math2/node/32C3D8FG9V.md): statement 32C3D8FG9V
- [Linear programming](https://tamnd.github.io/math2/node/32CP806EHX.md): definition 32CP806EHX
- [Interpretation of varieties](https://tamnd.github.io/math2/node/32PDB3WQ4D.md): concept 32PDB3WQ4D
- [Wigner D-matrix](https://tamnd.github.io/math2/node/32SHBZX38H.md): definition 32SHBZX38H
- [Minimal model program](https://tamnd.github.io/math2/node/32VAKG7CPS.md): definition 32VAKG7CPS
- [Hodge theory](https://tamnd.github.io/math2/node/32YJXG30MF.md): concept 32YJXG30MF
- [Delta functor](https://tamnd.github.io/math2/node/32ZHMZMACN.md): concept 32ZHMZMACN
- [Laguerre plane](https://tamnd.github.io/math2/node/337Z2AXRK2.md): definition 337Z2AXRK2
- [Poincaré's functional equation](https://tamnd.github.io/math2/node/3384ZV3PWM.md): concept 3384ZV3PWM
- [Outer regular measure](https://tamnd.github.io/math2/node/33C1KEK3SC.md): concept 33C1KEK3SC
- [Casson invariant](https://tamnd.github.io/math2/node/33GY6P5A4T.md): concept 33GY6P5A4T
- [Quermassintegral](https://tamnd.github.io/math2/node/33JWVAX32K.md): definition 33JWVAX32K
- [Mortar methods](https://tamnd.github.io/math2/node/33SJNF8V7Z.md): concept 33SJNF8V7Z
- [Transfer operator](https://tamnd.github.io/math2/node/340GV29CGE.md): definition 340GV29CGE
- [Cauchy–Pompeiu formula](https://tamnd.github.io/math2/node/345DJTSPHM.md): statement 345DJTSPHM
- [Slowly varying function](https://tamnd.github.io/math2/node/347JEZ7WTW.md): definition 347JEZ7WTW
- [Tension field](https://tamnd.github.io/math2/node/3482HSGW27.md): definition 3482HSGW27
- [Crossing number inequality](https://tamnd.github.io/math2/node/34B1PW5EE2.md): statement 34B1PW5EE2
- [Donaldson's theorem](https://tamnd.github.io/math2/node/34B7K77P31.md): statement 34B7K77P31
- [Fermi–Dirac integral](https://tamnd.github.io/math2/node/34JP0GZ3ED.md): definition 34JP0GZ3ED
- [Remmert–Stein theorem](https://tamnd.github.io/math2/node/34PT0GE8RA.md): statement 34PT0GE8RA
- [Moore space](https://tamnd.github.io/math2/node/34SN01NWPS.md): definition 34SN01NWPS
- [Fourier–Bessel series](https://tamnd.github.io/math2/node/34VJ4T7ZSB.md): concept 34VJ4T7ZSB
- [Free algebra in a variety](https://tamnd.github.io/math2/node/351WM99D5W.md): definition 351WM99D5W
- [Hermitian metric on a vector bundle](https://tamnd.github.io/math2/node/353YPJWBBR.md): definition 353YPJWBBR
- [Compactification](https://tamnd.github.io/math2/node/355XPVR564.md): definition 355XPVR564
- [Jacobi's formula](https://tamnd.github.io/math2/node/357FEAZ0Q4.md): statement 357FEAZ0Q4
- [Minakshisundaram–Pleijel zeta function](https://tamnd.github.io/math2/node/35FFCR3GRH.md): definition 35FFCR3GRH
- [Baker's finite basis theorem](https://tamnd.github.io/math2/node/35G4HQG2FN.md): statement 35G4HQG2FN
- [Abel–Jacobi theorem](https://tamnd.github.io/math2/node/35KB1QCD3Z.md): statement 35KB1QCD3Z
- [Extended finite element method](https://tamnd.github.io/math2/node/35KXRGMAHT.md): definition 35KXRGMAHT
- [Hadwiger conjecture](https://tamnd.github.io/math2/node/35NNSXJB1K.md): conjecture 35NNSXJB1K
- [Hyponormal operator](https://tamnd.github.io/math2/node/35XKRHEC3Q.md): concept 35XKRHEC3Q
- [Proper forcing axiom](https://tamnd.github.io/math2/node/35XTXNEWQ9.md): concept 35XTXNEWQ9
- [Lovász local lemma](https://tamnd.github.io/math2/node/35YQ94FNV1.md): statement 35YQ94FNV1
- [Thue equation](https://tamnd.github.io/math2/node/3601Z7V838.md): definition 3601Z7V838
- [Current](https://tamnd.github.io/math2/node/360H7MD9RD.md): concept 360H7MD9RD
- [Albert–Brauer–Hasse–Noether theorem](https://tamnd.github.io/math2/node/361GJ6QWAR.md): statement 361GJ6QWAR
- [Weierstrass elliptic function](https://tamnd.github.io/math2/node/361NAK68QK.md): object 361NAK68QK
- [Weyl group of a compact Lie group](https://tamnd.github.io/math2/node/363770WNM2.md): definition 363770WNM2
- [Conway groups](https://tamnd.github.io/math2/node/366D4ETGQH.md): object 366D4ETGQH
- [Dini derivative](https://tamnd.github.io/math2/node/36DXKAD2RZ.md): concept 36DXKAD2RZ
- [Factorization system](https://tamnd.github.io/math2/node/36J8PH2B0T.md): concept 36J8PH2B0T
- [Matrix Lie group](https://tamnd.github.io/math2/node/36KSZ7GA5S.md): concept 36KSZ7GA5S
- [Cuntz algebra](https://tamnd.github.io/math2/node/36Q7XH9P7W.md): example 36Q7XH9P7W
- [Choquet integral](https://tamnd.github.io/math2/node/36TXCRZ6GC.md): concept 36TXCRZ6GC
- [Sacks–Uhlenbeck bubbling](https://tamnd.github.io/math2/node/36WNFJTRRR.md): statement 36WNFJTRRR
- [Dimension of an algebraic variety](https://tamnd.github.io/math2/node/36X2WTHYCB.md): concept 36X2WTHYCB
- [Suzuki groups](https://tamnd.github.io/math2/node/36X3GET13B.md): object 36X3GET13B
- [Selmer group](https://tamnd.github.io/math2/node/36YNSTESMQ.md): concept 36YNSTESMQ
- [Extremally disconnected space](https://tamnd.github.io/math2/node/370PJ5BF70.md): concept 370PJ5BF70
- [Well-powered category](https://tamnd.github.io/math2/node/374WKANY3F.md): definition 374WKANY3F
- [Marcinkiewicz multiplier theorem](https://tamnd.github.io/math2/node/375ZGE0V2W.md): statement 375ZGE0V2W
- [Abelian variety](https://tamnd.github.io/math2/node/37CPGVHTAR.md): definition 37CPGVHTAR
- [Resurgent function](https://tamnd.github.io/math2/node/37R4QJCARA.md): concept 37R4QJCARA
- [Hasse–Weil zeta function](https://tamnd.github.io/math2/node/37TNNBQ4PK.md): concept 37TNNBQ4PK
- [Koopman operator](https://tamnd.github.io/math2/node/37TRG5DBQC.md): concept 37TRG5DBQC
- [Plateau's laws](https://tamnd.github.io/math2/node/381RB634WS.md): statement 381RB634WS
- [Profinite completion](https://tamnd.github.io/math2/node/383CHJH46R.md): definition 383CHJH46R
- [Index of a singular integral operator](https://tamnd.github.io/math2/node/388FKFJAEJ.md): definition 388FKFJAEJ
- [Hyperconnected space](https://tamnd.github.io/math2/node/38CDCK1RYB.md): concept 38CDCK1RYB
- [Minkowski functional](https://tamnd.github.io/math2/node/38CEJPFESJ.md): definition 38CEJPFESJ
- [Bourget's hypothesis](https://tamnd.github.io/math2/node/38D9RPAXQ8.md): statement 38D9RPAXQ8
- [Wilcoxon signed-rank test](https://tamnd.github.io/math2/node/38SETDT2HN.md): concept 38SETDT2HN
- [Sporadic group](https://tamnd.github.io/math2/node/390Z8QV5ZG.md): definition 390Z8QV5ZG
- [Quasiconvex function](https://tamnd.github.io/math2/node/393PNMXBFJ.md): definition 393PNMXBFJ
- [Symbolic method](https://tamnd.github.io/math2/node/39HWNYW5EF.md): definition 39HWNYW5EF
- [Strong maximal function](https://tamnd.github.io/math2/node/39MVMBCXG3.md): concept 39MVMBCXG3
- [Delay differential equation](https://tamnd.github.io/math2/node/39PCDRDB01.md): concept 39PCDRDB01
- [Generic point](https://tamnd.github.io/math2/node/3A19C4YXGE.md): concept 3A19C4YXGE
- [Small cancellation theory](https://tamnd.github.io/math2/node/3A1D37TVF6.md): concept 3A1D37TVF6
- [Higher category theory](https://tamnd.github.io/math2/node/3A1NMGJKJ3.md): concept 3A1NMGJKJ3
- [A translation of the integers is not a homomorphism](https://tamnd.github.io/math2/node/3A3JJPJHQR.md): example 3A3JJPJHQR
- [Henselian field](https://tamnd.github.io/math2/node/3ACEE4N6NE.md): concept 3ACEE4N6NE
- [Regulator of a number field](https://tamnd.github.io/math2/node/3AG1JV6J80.md): definition 3AG1JV6J80
- [p-adic Lie group](https://tamnd.github.io/math2/node/3AKM6EPV2F.md): concept 3AKM6EPV2F
- [Quadratic polynomial dynamics](https://tamnd.github.io/math2/node/3ANCZ518BN.md): definition 3ANCZ518BN
- [Triangular matrix](https://tamnd.github.io/math2/node/3AQFY4PAC2.md): definition 3AQFY4PAC2
- [Maximal chain](https://tamnd.github.io/math2/node/3AX549FFFM.md): concept 3AX549FFFM
- [Stefan problem](https://tamnd.github.io/math2/node/3B0HB3DHW1.md): problem 3B0HB3DHW1
- [Betti numbers of a module](https://tamnd.github.io/math2/node/3B1743XS7S.md): concept 3B1743XS7S
- [Borel–Harish-Chandra theorem](https://tamnd.github.io/math2/node/3B451RD8AJ.md): statement 3B451RD8AJ
- [Probabilistic method](https://tamnd.github.io/math2/node/3B55HNS754.md): concept 3B55HNS754
- [von Neumann bounded set](https://tamnd.github.io/math2/node/3BAJ8ZTVPX.md): concept 3BAJ8ZTVPX
- [Twelvefold way](https://tamnd.github.io/math2/node/3BBE4XY4E9.md): definition 3BBE4XY4E9
- [Anger function](https://tamnd.github.io/math2/node/3BBSXXAZKE.md): definition 3BBSXXAZKE
- [Discriminant of a field extension](https://tamnd.github.io/math2/node/3BCYVS9E17.md): concept 3BCYVS9E17
- [Orthogonal polynomials on the unit circle](https://tamnd.github.io/math2/node/3BJTDXB9KK.md): definition 3BJTDXB9KK
- [Walsh–Hadamard transform](https://tamnd.github.io/math2/node/3BTA41B7M1.md): definition 3BTA41B7M1
- [Loss function](https://tamnd.github.io/math2/node/3BTG6VVCGZ.md): concept 3BTG6VVCGZ
- [Elliptic partial differential equation](https://tamnd.github.io/math2/node/3BYHP9E9VF.md): definition 3BYHP9E9VF
- [Urysohn space](https://tamnd.github.io/math2/node/3BZAPDBPC8.md): definition 3BZAPDBPC8
- [Priority method](https://tamnd.github.io/math2/node/3BZFSC5QZ2.md): concept 3BZFSC5QZ2
- [Lagrange's theorem](https://tamnd.github.io/math2/node/3C53TD.md): statement 3C53TD
- [Feigenbaum–Cvitanović functional equation](https://tamnd.github.io/math2/node/3C6P6X69P1.md): definition 3C6P6X69P1
- [Automorphic form](https://tamnd.github.io/math2/node/3C73HND6XA.md): concept 3C73HND6XA
- [Thomson problem](https://tamnd.github.io/math2/node/3C7V5X5FY8.md): problem 3C7V5X5FY8
- [Multialgebra](https://tamnd.github.io/math2/node/3C930MVM0T.md): concept 3C930MVM0T
- [Deformation of complex structure](https://tamnd.github.io/math2/node/3CEKYF05P1.md): definition 3CEKYF05P1
- [Milnor fibration theorem](https://tamnd.github.io/math2/node/3CGMRT9W7J.md): statement 3CGMRT9W7J
- [Internal language](https://tamnd.github.io/math2/node/3CJ8XA1VAB.md): definition 3CJ8XA1VAB
- [Erdős–Mordell inequality](https://tamnd.github.io/math2/node/3CJEWF0Y3F.md): statement 3CJEWF0Y3F
- [Fourier transform of radial functions](https://tamnd.github.io/math2/node/3CSEGRAD75.md): statement 3CSEGRAD75
- [Associative algebra](https://tamnd.github.io/math2/node/3CTW1P4HBP.md): concept 3CTW1P4HBP
- [Equilibrium of a difference equation](https://tamnd.github.io/math2/node/3CVERVJRV9.md): definition 3CVERVJRV9
- [Strassen algorithm](https://tamnd.github.io/math2/node/3CVWXDS4JQ.md): concept 3CVWXDS4JQ
- [Hilbert–Schmidt expansion theorem](https://tamnd.github.io/math2/node/3CXAZ1G5RJ.md): statement 3CXAZ1G5RJ
- [Nilradical is the intersection of the prime ideals](https://tamnd.github.io/math2/node/3CZ8PXR9JY.md): statement 3CZ8PXR9JY
- [Weierstrass zeta and sigma functions](https://tamnd.github.io/math2/node/3D15ZW4YZ6.md): definition 3D15ZW4YZ6
- [Rigidity of frameworks](https://tamnd.github.io/math2/node/3D1J3AXS69.md): definition 3D1J3AXS69
- [Gelfand–Mazur theorem](https://tamnd.github.io/math2/node/3D6XH41A3B.md): statement 3D6XH41A3B
- [Doléans-Dade exponential](https://tamnd.github.io/math2/node/3D88622PBQ.md): concept 3D88622PBQ
- [Bilateral Z-transform](https://tamnd.github.io/math2/node/3DBAMXA80B.md): concept 3DBAMXA80B
- [L2 convergence of Fourier series](https://tamnd.github.io/math2/node/3DGHC142FT.md): statement 3DGHC142FT
- [Short-time Fourier transform](https://tamnd.github.io/math2/node/3DHNP5NEWH.md): concept 3DHNP5NEWH
- [Joukowsky transform](https://tamnd.github.io/math2/node/3DJ46V21KH.md): example 3DJ46V21KH
- [Similarity solution](https://tamnd.github.io/math2/node/3DJMFT0SQM.md): definition 3DJMFT0SQM
- [Hough transform](https://tamnd.github.io/math2/node/3DK4V3A10K.md): example 3DK4V3A10K
- [Oscillatory solution of a difference equation](https://tamnd.github.io/math2/node/3DRCZF6J9J.md): concept 3DRCZF6J9J
- [Knaster–Kuratowski fan](https://tamnd.github.io/math2/node/3DRNRZEW9D.md): counterexample 3DRNRZEW9D
- [Dirac operator](https://tamnd.github.io/math2/node/3DXGJTEQZH.md): concept 3DXGJTEQZH
- [Fejér kernel](https://tamnd.github.io/math2/node/3E740YTT9V.md): definition 3E740YTT9V
- [Multivariate formal power series](https://tamnd.github.io/math2/node/3E8ZG3NJYZ.md): concept 3E8ZG3NJYZ
- [Regular measure](https://tamnd.github.io/math2/node/3EC1CQKMK2.md): definition 3EC1CQKMK2
- [Formal concept analysis](https://tamnd.github.io/math2/node/3EG3HMEGWC.md): definition 3EG3HMEGWC
- [Finitely presented module](https://tamnd.github.io/math2/node/3EH4C3M5GV.md): concept 3EH4C3M5GV
- [Algebraic approach to constraint satisfaction](https://tamnd.github.io/math2/node/3EH4F5H7FT.md): concept 3EH4F5H7FT
- [Lakes of Wada](https://tamnd.github.io/math2/node/3ENQE05H1H.md): example 3ENQE05H1H
- [Baskakov operator](https://tamnd.github.io/math2/node/3EQXGGB849.md): definition 3EQXGGB849
- [Vector fields on spheres](https://tamnd.github.io/math2/node/3ES79NWRS7.md): statement 3ES79NWRS7
- [Ending lamination theorem](https://tamnd.github.io/math2/node/3EVAF9NG5Z.md): statement 3EVAF9NG5Z
- [Universal property of free algebras](https://tamnd.github.io/math2/node/3F2W916W64.md): statement 3F2W916W64
- [Spherical harmonics](https://tamnd.github.io/math2/node/3F32Y21HP4.md): concept 3F32Y21HP4
- [Regular boundary point](https://tamnd.github.io/math2/node/3F4Y6NEVM4.md): definition 3F4Y6NEVM4
- [Whitney stratification](https://tamnd.github.io/math2/node/3F84AMVJNV.md): concept 3F84AMVJNV
- [Nuclear operator](https://tamnd.github.io/math2/node/3F94J43P2C.md): definition 3F94J43P2C
- [Orthogonal polynomials on the unit circle](https://tamnd.github.io/math2/node/3F9G3DRQ62.md): concept 3F9G3DRQ62
- [T(1) theorem](https://tamnd.github.io/math2/node/3FBF628XEF.md): statement 3FBF628XEF
- [Quaternionic analysis](https://tamnd.github.io/math2/node/3FE2AQS534.md): definition 3FE2AQS534
- [Mikusiński field of operators](https://tamnd.github.io/math2/node/3FTH87V9GD.md): definition 3FTH87V9GD
- [Periodic boundary conditions](https://tamnd.github.io/math2/node/3FW4J270FV.md): concept 3FW4J270FV
- [Vector bundle](https://tamnd.github.io/math2/node/3FX0JBH9S1.md): definition 3FX0JBH9S1
- [Continuous geometry](https://tamnd.github.io/math2/node/3FYQZ4R6T8.md): definition 3FYQZ4R6T8
- [Cheng–Yau gradient estimate](https://tamnd.github.io/math2/node/3G1CJ6HJMK.md): statement 3G1CJ6HJMK
- [Upper nilradical](https://tamnd.github.io/math2/node/3G2JQEYBEK.md): concept 3G2JQEYBEK
- [Toeplitz matrix](https://tamnd.github.io/math2/node/3G7WAP9DJJ.md): definition 3G7WAP9DJJ
- [Fréchet space](https://tamnd.github.io/math2/node/3GAA614673.md): concept 3GAA614673
- [Faddeeva function](https://tamnd.github.io/math2/node/3GHGFTMFQR.md): concept 3GHGFTMFQR
- [Hurwitz's automorphisms theorem](https://tamnd.github.io/math2/node/3GNKSCQSCD.md): statement 3GNKSCQSCD
- [Schwarzian derivative](https://tamnd.github.io/math2/node/3GSQJXG19A.md): definition 3GSQJXG19A
- [Minimal surface equation](https://tamnd.github.io/math2/node/3GV2E2YTZX.md): concept 3GV2E2YTZX
- [Full and faithful functors](https://tamnd.github.io/math2/node/3GZ99NCX22.md): definition 3GZ99NCX22
- [Standard probability space](https://tamnd.github.io/math2/node/3H1YKSPH19.md): definition 3H1YKSPH19
- [Absolutely free algebra](https://tamnd.github.io/math2/node/3H2W1MQFGM.md): definition 3H2W1MQFGM
- [Co-Heyting algebra](https://tamnd.github.io/math2/node/3H5SDW6JTX.md): definition 3H5SDW6JTX
- [Removable singularities for harmonic functions](https://tamnd.github.io/math2/node/3H6SQA6JWM.md): statement 3H6SQA6JWM
- [Hypergeometric function of a matrix argument](https://tamnd.github.io/math2/node/3HAV1FH6V9.md): definition 3HAV1FH6V9
- [Löb's theorem](https://tamnd.github.io/math2/node/3HD30YG3DG.md): statement 3HD30YG3DG
- [Weeks manifold](https://tamnd.github.io/math2/node/3HDRK3QYSD.md): example 3HDRK3QYSD
- [Cramér–Wold theorem](https://tamnd.github.io/math2/node/3HFP9ZQ580.md): statement 3HFP9ZQ580
- [Leapfrog integration](https://tamnd.github.io/math2/node/3HGJCQBWVY.md): statement 3HGJCQBWVY
- [Grothendieck–Riemann–Roch theorem](https://tamnd.github.io/math2/node/3HKREN3Y3F.md): statement 3HKREN3Y3F
- [Minimal residual method](https://tamnd.github.io/math2/node/3HRHSHTG9C.md): statement 3HRHSHTG9C
- [Algebraic lattice](https://tamnd.github.io/math2/node/3J0BTXSCVN.md): concept 3J0BTXSCVN
- [Mollifier](https://tamnd.github.io/math2/node/3J2QZQDDM7.md): definition 3J2QZQDDM7
- [Polygamma function](https://tamnd.github.io/math2/node/3J3QGKPMZQ.md): concept 3J3QGKPMZQ
- [Derivation](https://tamnd.github.io/math2/node/3J89REV9WQ.md): concept 3J89REV9WQ
- [Complex cobordism](https://tamnd.github.io/math2/node/3JAY4V0GAF.md): definition 3JAY4V0GAF
- [Oscillator representation](https://tamnd.github.io/math2/node/3JE85VQR4G.md): definition 3JE85VQR4G
- [Berkovich space](https://tamnd.github.io/math2/node/3JFF28P5NJ.md): definition 3JFF28P5NJ
- [Filter limit](https://tamnd.github.io/math2/node/3JGEQYCF6S.md): definition 3JGEQYCF6S
- [Honeycomb theorem](https://tamnd.github.io/math2/node/3JST9B967R.md): statement 3JST9B967R
- [Maximal torus](https://tamnd.github.io/math2/node/3JVGZVVTB0.md): concept 3JVGZVVTB0
- [Trace of an integral operator](https://tamnd.github.io/math2/node/3JVT83K8A1.md): concept 3JVT83K8A1
- [Dirichlet kernel](https://tamnd.github.io/math2/node/3JY4BVE4DW.md): concept 3JY4BVE4DW
- [Square root of a positive operator](https://tamnd.github.io/math2/node/3JY5HNFRMF.md): concept 3JY5HNFRMF
- [Robinson arithmetic](https://tamnd.github.io/math2/node/3K1AKGXF8A.md): concept 3K1AKGXF8A
- [Moving frame](https://tamnd.github.io/math2/node/3K774WEHHY.md): concept 3K774WEHHY
- [Carleman matrix](https://tamnd.github.io/math2/node/3KK7M27P8T.md): concept 3KK7M27P8T
- [Truncation error](https://tamnd.github.io/math2/node/3KM7HQWKXP.md): definition 3KM7HQWKXP
- [Minimum principle for harmonic functions](https://tamnd.github.io/math2/node/3KP89PT5P3.md): statement 3KP89PT5P3
- [Gelfand's formula](https://tamnd.github.io/math2/node/3KX0KQ7V05.md): statement 3KX0KQ7V05
- [Birkhoff–von Neumann theorem](https://tamnd.github.io/math2/node/3KY1RMEDMJ.md): statement 3KY1RMEDMJ
- [Ridge regression](https://tamnd.github.io/math2/node/3KZW2930V2.md): concept 3KZW2930V2
- [Inverse semigroup](https://tamnd.github.io/math2/node/3M724K3B50.md): concept 3M724K3B50
- [Jónsson terms](https://tamnd.github.io/math2/node/3M8Y76E4AG.md): concept 3M8Y76E4AG
- [Brownian bridge](https://tamnd.github.io/math2/node/3MCVMBAGD1.md): concept 3MCVMBAGD1
- [Einstein field equations](https://tamnd.github.io/math2/node/3MCZ97SW6V.md): concept 3MCZ97SW6V
- [Linear congruential generator](https://tamnd.github.io/math2/node/3MDPTW005V.md): definition 3MDPTW005V
- [Complete intersection](https://tamnd.github.io/math2/node/3MGF59XBD3.md): definition 3MGF59XBD3
- [Number of points of a finite projective plane](https://tamnd.github.io/math2/node/3MHF6QBW4Q.md): statement 3MHF6QBW4Q
- [Uniform convergence of series](https://tamnd.github.io/math2/node/3MHP17HY7N.md): definition 3MHP17HY7N
- [Deformation quantization](https://tamnd.github.io/math2/node/3MKSJ1J04K.md): concept 3MKSJ1J04K
- [Ultrafilter lemma](https://tamnd.github.io/math2/node/3MKZST79MK.md): statement 3MKZST79MK
- [Gabriel–Popescu theorem](https://tamnd.github.io/math2/node/3MP41FQHZ5.md): statement 3MP41FQHZ5
- [Holomorphic map](https://tamnd.github.io/math2/node/3MQTWKBBBT.md): concept 3MQTWKBBBT
- [Pluricomplex Green function](https://tamnd.github.io/math2/node/3MQVKVGYYY.md): concept 3MQVKVGYYY
- [Extreme point](https://tamnd.github.io/math2/node/3MSGFJM2NX.md): concept 3MSGFJM2NX
- [Jordan–Kinderlehrer–Otto scheme](https://tamnd.github.io/math2/node/3MWNBBG9EH.md): concept 3MWNBBG9EH
- [Ratner's theorems](https://tamnd.github.io/math2/node/3MXR03096V.md): statement 3MXR03096V
- [Ergodic hypothesis](https://tamnd.github.io/math2/node/3MY6RYD0PD.md): definition 3MY6RYD0PD
- [von Mises–Fisher distribution](https://tamnd.github.io/math2/node/3N2KQS7QBB.md): definition 3N2KQS7QBB
- [Milnor–Thurston kneading theory](https://tamnd.github.io/math2/node/3N54R6HR05.md): statement 3N54R6HR05
- [Pushforward measure](https://tamnd.github.io/math2/node/3N66TM3TSQ.md): concept 3N66TM3TSQ
- [Forcing](https://tamnd.github.io/math2/node/3N80SFETC5.md): definition 3N80SFETC5
- [Ill-posedness of real Laplace inversion](https://tamnd.github.io/math2/node/3N8DRR7X92.md): definition 3N8DRR7X92
- [Complemented subspace](https://tamnd.github.io/math2/node/3NFC6F0Z9F.md): definition 3NFC6F0Z9F
- [Closed subgroup theorem](https://tamnd.github.io/math2/node/3NJGY8QSTE.md): statement 3NJGY8QSTE
- [Brouwer's characterization of the Cantor space](https://tamnd.github.io/math2/node/3NKBCNH5JS.md): statement 3NKBCNH5JS
- [Additive combinatorics](https://tamnd.github.io/math2/node/3NM362A2JG.md): concept 3NM362A2JG
- [Eisenstein series](https://tamnd.github.io/math2/node/3NZBZXYWAJ.md): definition 3NZBZXYWAJ
- [Thue's theorem](https://tamnd.github.io/math2/node/3P246RHXMF.md): statement 3P246RHXMF
- [Perspective projection](https://tamnd.github.io/math2/node/3P5R3WQPT6.md): definition 3P5R3WQPT6
- [Lyusternik's theorem on tangent spaces](https://tamnd.github.io/math2/node/3PGXH0T1K6.md): statement 3PGXH0T1K6
- [Homogeneous ideal](https://tamnd.github.io/math2/node/3PNJYFPJ3P.md): definition 3PNJYFPJ3P
- [Spectral radius](https://tamnd.github.io/math2/node/3PR407YKHN.md): definition 3PR407YKHN
- [Diagonal lemma](https://tamnd.github.io/math2/node/3PRW76471J.md): statement 3PRW76471J
- [Positive-definite function on a group](https://tamnd.github.io/math2/node/3PRZ02APFA.md): concept 3PRZ02APFA
- [Heteroscedasticity](https://tamnd.github.io/math2/node/3PS1JB7V95.md): definition 3PS1JB7V95
- [Boolean model](https://tamnd.github.io/math2/node/3PZ3TMW86C.md): definition 3PZ3TMW86C
- [Thue equation](https://tamnd.github.io/math2/node/3Q6CFPQZDF.md): concept 3Q6CFPQZDF
- [Quasitriangular Hopf algebra](https://tamnd.github.io/math2/node/3Q99FQH4QX.md): definition 3Q99FQH4QX
- [Nonparametric statistics](https://tamnd.github.io/math2/node/3QB691PFPB.md): definition 3QB691PFPB
- [Convergence of the Neumann series for Volterra kernels](https://tamnd.github.io/math2/node/3QDDSSH4FB.md): statement 3QDDSSH4FB
- [Functional equation](https://tamnd.github.io/math2/node/3QG8F9ER3S.md): definition 3QG8F9ER3S
- [Sullivan minimal model](https://tamnd.github.io/math2/node/3QQ2PBGY3N.md): concept 3QQ2PBGY3N
- [Wigner semicircle law](https://tamnd.github.io/math2/node/3QTT95DMFH.md): statement 3QTT95DMFH
- [Saturation of approximation methods](https://tamnd.github.io/math2/node/3QWJB8GHDE.md): concept 3QWJB8GHDE
- [Bendixson–Dulac theorem](https://tamnd.github.io/math2/node/3QWVVM56JF.md): statement 3QWVVM56JF
- [Poincaré conjecture](https://tamnd.github.io/math2/node/3R064KFF68.md): statement 3R064KFF68
- [Desargues configuration](https://tamnd.github.io/math2/node/3R2VHPXXWS.md): example 3R2VHPXXWS
- [Transfer homomorphism](https://tamnd.github.io/math2/node/3R2Y906QAQ.md): concept 3R2Y906QAQ
- [Maass wave form](https://tamnd.github.io/math2/node/3R959YSNYP.md): concept 3R959YSNYP
- [Graph of an operator](https://tamnd.github.io/math2/node/3RCAY97W1E.md): concept 3RCAY97W1E
- [Hurwitz's theorem (number theory)](https://tamnd.github.io/math2/node/3RDB8B5GFA.md): statement 3RDB8B5GFA
- [Galois category](https://tamnd.github.io/math2/node/3RGBA24QVA.md): concept 3RGBA24QVA
- [Michael selection theorem](https://tamnd.github.io/math2/node/3RJ1K36A9A.md): statement 3RJ1K36A9A
- [Daniell integral](https://tamnd.github.io/math2/node/3RQ2PT36KQ.md): definition 3RQ2PT36KQ
- [120-cell](https://tamnd.github.io/math2/node/3RQEK50CP0.md): object 3RQEK50CP0
- [Transfer principle](https://tamnd.github.io/math2/node/3RT74FQYG1.md): statement 3RT74FQYG1
- [Analytic set](https://tamnd.github.io/math2/node/3RWWZYPAFV.md): concept 3RWWZYPAFV
- [Leavitt path algebra](https://tamnd.github.io/math2/node/3S42WRMG9V.md): definition 3S42WRMG9V
- [Markov partition](https://tamnd.github.io/math2/node/3S543TYT8C.md): definition 3S543TYT8C
- [Local property](https://tamnd.github.io/math2/node/3S58E70Z7Z.md): concept 3S58E70Z7Z
- [Nagata ring](https://tamnd.github.io/math2/node/3S5TMBAKCA.md): concept 3S5TMBAKCA
- [Dissipative operator](https://tamnd.github.io/math2/node/3S7HJ5GQRF.md): concept 3S7HJ5GQRF
- [Calderón–Zygmund operator](https://tamnd.github.io/math2/node/3SCXCW4AH3.md): concept 3SCXCW4AH3
- [Toeplitz algebra](https://tamnd.github.io/math2/node/3SKRBF4KG1.md): object 3SKRBF4KG1
- [Variational capacity](https://tamnd.github.io/math2/node/3SN7JXGH2Q.md): definition 3SN7JXGH2Q
- [Radial basis function](https://tamnd.github.io/math2/node/3SS5K8MHPK.md): definition 3SS5K8MHPK
- [Exponential of a bounded operator](https://tamnd.github.io/math2/node/3SVWD5SN15.md): definition 3SVWD5SN15
- [Von Neumann universe](https://tamnd.github.io/math2/node/3SZADQXVZJ.md): object 3SZADQXVZJ
- [Total variation distance between probability measures](https://tamnd.github.io/math2/node/3SZESKX3N5.md): concept 3SZESKX3N5
- [Wave front set](https://tamnd.github.io/math2/node/3T5Q9WVQX8.md): concept 3T5Q9WVQX8
- [Metric outer measure](https://tamnd.github.io/math2/node/3T6WG315GC.md): concept 3T6WG315GC
- [Uniform boundedness conjecture for preperiodic points](https://tamnd.github.io/math2/node/3T9TT5SSGD.md): conjecture 3T9TT5SSGD
- [Characterization of Shannon entropy](https://tamnd.github.io/math2/node/3TERJ5C8QK.md): statement 3TERJ5C8QK
- [Hopf module](https://tamnd.github.io/math2/node/3TGTNRBKW0.md): definition 3TGTNRBKW0
- [Brianchon's theorem](https://tamnd.github.io/math2/node/3THE3Q96KK.md): statement 3THE3Q96KK
- [Compact element](https://tamnd.github.io/math2/node/3TNVVAA1AG.md): definition 3TNVVAA1AG
- [Commutant lifting theorem](https://tamnd.github.io/math2/node/3TVXB7X266.md): statement 3TVXB7X266
- [Closable operator](https://tamnd.github.io/math2/node/3TWMBFQJJ8.md): concept 3TWMBFQJJ8
- [n-term approximation](https://tamnd.github.io/math2/node/3TXY35FR30.md): definition 3TXY35FR30
- [K-function](https://tamnd.github.io/math2/node/3TY9Y91AMZ.md): definition 3TY9Y91AMZ
- [Moulton plane](https://tamnd.github.io/math2/node/3V9GYYXM1H.md): example 3V9GYYXM1H
- [Hermite–Hadamard inequality](https://tamnd.github.io/math2/node/3VB65PEZQS.md): statement 3VB65PEZQS
- [Mild solution](https://tamnd.github.io/math2/node/3VBRS94V1G.md): concept 3VBRS94V1G
- [Toeplitz operator](https://tamnd.github.io/math2/node/3VCD0T7QNG.md): definition 3VCD0T7QNG
- [Leray–Schauder degree](https://tamnd.github.io/math2/node/3VJC5Q9MYA.md): concept 3VJC5Q9MYA
- [Hartman's uniqueness theorem for harmonic maps](https://tamnd.github.io/math2/node/3VMJGQCD7S.md): statement 3VMJGQCD7S
- [Sphere bundle](https://tamnd.github.io/math2/node/3VQ2V5HH3R.md): concept 3VQ2V5HH3R
- [Rückert Nullstellensatz](https://tamnd.github.io/math2/node/3VRZ80X45P.md): statement 3VRZ80X45P
- [Schoenflies theorem](https://tamnd.github.io/math2/node/3VX06XRVZG.md): statement 3VX06XRVZG
- [Projective cover](https://tamnd.github.io/math2/node/3W0FP5DCEV.md): definition 3W0FP5DCEV
- [Periodic solution of a difference equation](https://tamnd.github.io/math2/node/3W358BSRN2.md): concept 3W358BSRN2
- [Tarski–Seidenberg theorem](https://tamnd.github.io/math2/node/3W5JGDZAN0.md): statement 3W5JGDZAN0
- [Krohn–Rhodes theory](https://tamnd.github.io/math2/node/3WDZY1J002.md): concept 3WDZY1J002
- [Polar sets and Brownian hitting](https://tamnd.github.io/math2/node/3WJFWAGKXR.md): statement 3WJFWAGKXR
- [Perspectivity](https://tamnd.github.io/math2/node/3WZAKQ3EJY.md): definition 3WZAKQ3EJY
- [Type and cotype](https://tamnd.github.io/math2/node/3X4YMR25ST.md): definition 3X4YMR25ST
- [Ten martini problem](https://tamnd.github.io/math2/node/3X4ZQ41PXG.md): problem 3X4ZQ41PXG
- [Durand–Kerner method](https://tamnd.github.io/math2/node/3X6C5SC7AF.md): statement 3X6C5SC7AF
- [Doubly stochastic matrix](https://tamnd.github.io/math2/node/3X74C58NCH.md): definition 3X74C58NCH
- [Subadditive function](https://tamnd.github.io/math2/node/3X9DD9J3CV.md): concept 3X9DD9J3CV
- [Minimal surface equation](https://tamnd.github.io/math2/node/3XB4PTTK68.md): definition 3XB4PTTK68
- [Hardy's uncertainty principle](https://tamnd.github.io/math2/node/3XC1C1NC0M.md): statement 3XC1C1NC0M
- [Zernike polynomials](https://tamnd.github.io/math2/node/3XJYJHMPKY.md): definition 3XJYJHMPKY
- [Lerch zeta function](https://tamnd.github.io/math2/node/3XNJ04AMCW.md): definition 3XNJ04AMCW
- [Covering space](https://tamnd.github.io/math2/node/3XZRRE165N.md): concept 3XZRRE165N
- [Steiner symmetrization](https://tamnd.github.io/math2/node/3YF7P33WRM.md): definition 3YF7P33WRM
- [Term algebra](https://tamnd.github.io/math2/node/3YYKGWJJND.md): concept 3YYKGWJJND
- [Excellent ring](https://tamnd.github.io/math2/node/3Z7YMK2WDS.md): definition 3Z7YMK2WDS
- [Picard group](https://tamnd.github.io/math2/node/3Z9DCWYXQZ.md): concept 3Z9DCWYXQZ
- [Hyperkähler manifold](https://tamnd.github.io/math2/node/3ZCJRT8FPT.md): definition 3ZCJRT8FPT
- [Method of multiple scales](https://tamnd.github.io/math2/node/3ZEW7MC05Z.md): definition 3ZEW7MC05Z
- [Wiener–Khinchin theorem](https://tamnd.github.io/math2/node/3ZXN26PW4G.md): statement 3ZXN26PW4G
- [Periodogram](https://tamnd.github.io/math2/node/3ZYM0HXA3E.md): concept 3ZYM0HXA3E
- [Kolmogorov–Arnold–Moser theorem](https://tamnd.github.io/math2/node/3ZZSXKM16Z.md): statement 3ZZSXKM16Z
- [Bloch space](https://tamnd.github.io/math2/node/405TSJS5PA.md): definition 405TSJS5PA
- [Ahlswede–Daykin inequality](https://tamnd.github.io/math2/node/4085B8A528.md): statement 4085B8A528
- [Bootstrap](https://tamnd.github.io/math2/node/40ATJEVR32.md): concept 40ATJEVR32
- [Auslander–Reiten theory](https://tamnd.github.io/math2/node/40F8KNF8Q1.md): definition 40F8KNF8Q1
- [Hilbert dimension](https://tamnd.github.io/math2/node/40SV748R64.md): concept 40SV748R64
- [Schwarz–Christoffel mapping](https://tamnd.github.io/math2/node/40TPSVZGJB.md): definition 40TPSVZGJB
- [Quasi-compact morphism](https://tamnd.github.io/math2/node/41009J4M5D.md): concept 41009J4M5D
- [Mapping class group](https://tamnd.github.io/math2/node/410BCE4PAM.md): definition 410BCE4PAM
- [Calculus of variations](https://tamnd.github.io/math2/node/410CS90H3R.md): definition 410CS90H3R
- [Minimum degree algorithm](https://tamnd.github.io/math2/node/4135XQ4WHK.md): statement 4135XQ4WHK
- [Perfect set property](https://tamnd.github.io/math2/node/415776K9Z3.md): definition 415776K9Z3
- [Quotient of a topological group](https://tamnd.github.io/math2/node/417ZRDCHY6.md): concept 417ZRDCHY6
- [Lp multiplier](https://tamnd.github.io/math2/node/41D1KJWM78.md): definition 41D1KJWM78
- [Linear regression](https://tamnd.github.io/math2/node/41FVC4P5JB.md): concept 41FVC4P5JB
- [Einstein manifold](https://tamnd.github.io/math2/node/41S2EZ0KR3.md): concept 41S2EZ0KR3
- [Minimal surface](https://tamnd.github.io/math2/node/41X0YW2R9N.md): concept 41X0YW2R9N
- [Hyperreal number](https://tamnd.github.io/math2/node/421QMTFG9T.md): concept 421QMTFG9T
- [Elliptic curve](https://tamnd.github.io/math2/node/427N6S6QQW.md): concept 427N6S6QQW
- [Hypergeometric series](https://tamnd.github.io/math2/node/428M453CR2.md): concept 428M453CR2
- [Cartan–Hadamard theorem](https://tamnd.github.io/math2/node/428R02EYWS.md): statement 428R02EYWS
- [Baumslag–Solitar group](https://tamnd.github.io/math2/node/429JJ5ZNBY.md): example 429JJ5ZNBY
- [Type theory](https://tamnd.github.io/math2/node/42A2M9S2C1.md): definition 42A2M9S2C1
- [Riemann–Hurwitz formula](https://tamnd.github.io/math2/node/42A5ZT6AWN.md): statement 42A5ZT6AWN
- [Neural network approximation](https://tamnd.github.io/math2/node/42AGEAAQ8H.md): definition 42AGEAAQ8H
- [Kahane–Katznelson theorem](https://tamnd.github.io/math2/node/42DDDZHP7F.md): statement 42DDDZHP7F
- [Response surface methodology](https://tamnd.github.io/math2/node/42NZSQ65DQ.md): concept 42NZSQ65DQ
- [Singularity theory](https://tamnd.github.io/math2/node/42V5XGZS4F.md): concept 42V5XGZS4F
- [Reductive Lie group](https://tamnd.github.io/math2/node/42ZCE14AD2.md): concept 42ZCE14AD2
- [Convex body](https://tamnd.github.io/math2/node/437FSG3419.md): concept 437FSG3419
- [Radical of an ideal](https://tamnd.github.io/math2/node/437YRCNYRJ.md): concept 437YRCNYRJ
- [Complete intersection](https://tamnd.github.io/math2/node/43DJVJYRBN.md): concept 43DJVJYRBN
- [Interlacing of zeros of orthogonal polynomials](https://tamnd.github.io/math2/node/43DXNAFZ9T.md): statement 43DXNAFZ9T
- [Godunov's scheme](https://tamnd.github.io/math2/node/43HZJ7DRT0.md): concept 43HZJ7DRT0
- [Barnes integral](https://tamnd.github.io/math2/node/43J9XSE6M8.md): definition 43J9XSE6M8
- [Presentation of an algebra](https://tamnd.github.io/math2/node/43M1WFPYGQ.md): concept 43M1WFPYGQ
- [Quotient category](https://tamnd.github.io/math2/node/43MP9NWS6H.md): concept 43MP9NWS6H
- [Nilmanifold](https://tamnd.github.io/math2/node/43NRBE72C6.md): concept 43NRBE72C6
- [Lagrangian singularity](https://tamnd.github.io/math2/node/43R0MPN4T1.md): definition 43R0MPN4T1
- [Schur polynomial](https://tamnd.github.io/math2/node/43VFMJCDYJ.md): concept 43VFMJCDYJ
- [Stochastic partial differential equation](https://tamnd.github.io/math2/node/441JWPAV3W.md): definition 441JWPAV3W
- [Credible interval](https://tamnd.github.io/math2/node/441PZ3T7F0.md): concept 441PZ3T7F0
- [Survival analysis](https://tamnd.github.io/math2/node/443ZWDX88P.md): concept 443ZWDX88P
- [Nörlund mean](https://tamnd.github.io/math2/node/447CBEA2Z1.md): definition 447CBEA2Z1
- [Domain of holomorphy](https://tamnd.github.io/math2/node/4487CSC0MB.md): concept 4487CSC0MB
- [Analytic index](https://tamnd.github.io/math2/node/44GBFZ16SG.md): concept 44GBFZ16SG
- [Latin square](https://tamnd.github.io/math2/node/44HTG620GB.md): definition 44HTG620GB
- [Sorgenfrey plane](https://tamnd.github.io/math2/node/44KMEHYN1R.md): counterexample 44KMEHYN1R
- [Scheffé's lemma](https://tamnd.github.io/math2/node/44RN97ZXDZ.md): statement 44RN97ZXDZ
- [Pseudosphere](https://tamnd.github.io/math2/node/44WEBQSKEY.md): example 44WEBQSKEY
- [M-estimator](https://tamnd.github.io/math2/node/4503R8BVF8.md): concept 4503R8BVF8
- [Digital topology](https://tamnd.github.io/math2/node/456E0RCQR1.md): concept 456E0RCQR1
- [Day convolution](https://tamnd.github.io/math2/node/457T9T7Y75.md): concept 457T9T7Y75
- [Wright omega function](https://tamnd.github.io/math2/node/45P0QAMAFW.md): definition 45P0QAMAFW
- [Minakshisundaram–Pleijel asymptotic expansion](https://tamnd.github.io/math2/node/45P2SEGGR1.md): statement 45P2SEGGR1
- [Renewal theory](https://tamnd.github.io/math2/node/45V0W838BQ.md): concept 45V0W838BQ
- [Hodge star operator](https://tamnd.github.io/math2/node/45VSJ0RG2S.md): definition 45VSJ0RG2S
- [Union-closed sets conjecture](https://tamnd.github.io/math2/node/4628Z96WVZ.md): conjecture 4628Z96WVZ
- [Regula falsi](https://tamnd.github.io/math2/node/462AEY7MWF.md): statement 462AEY7MWF
- [Hypergeometric function](https://tamnd.github.io/math2/node/463X5V31D9.md): concept 463X5V31D9
- [Excessive function](https://tamnd.github.io/math2/node/46436KE7NT.md): concept 46436KE7NT
- [Equilibrium measure](https://tamnd.github.io/math2/node/464NN2K4FZ.md): concept 464NN2K4FZ
- [Block system](https://tamnd.github.io/math2/node/464TFVZ9JA.md): definition 464TFVZ9JA
- [Series in Banach spaces](https://tamnd.github.io/math2/node/4670W2WNFX.md): definition 4670W2WNFX
- [Injective algebra](https://tamnd.github.io/math2/node/468EN8KVPJ.md): definition 468EN8KVPJ
- [Collineation](https://tamnd.github.io/math2/node/4695YA7F87.md): definition 4695YA7F87
- [Ample line bundle](https://tamnd.github.io/math2/node/46ESFR9MSE.md): definition 46ESFR9MSE
- [Formalism](https://tamnd.github.io/math2/node/46JCQ6KDA4.md): concept 46JCQ6KDA4
- [Sinogram](https://tamnd.github.io/math2/node/46MFCQQ9VW.md): object 46MFCQQ9VW
- [Convenient vector space](https://tamnd.github.io/math2/node/46MSX1FXZT.md): concept 46MSX1FXZT
- [Logarithmic potential](https://tamnd.github.io/math2/node/46NHWJHW44.md): concept 46NHWJHW44
- [Jordan–von Neumann theorem](https://tamnd.github.io/math2/node/46WGVM5VS6.md): statement 46WGVM5VS6
- [Navier–Stokes existence and smoothness](https://tamnd.github.io/math2/node/46Z1TVS2K6.md): conjecture 46Z1TVS2K6
- [Morse–Bott function](https://tamnd.github.io/math2/node/47096P24SB.md): concept 47096P24SB
- [Palais–Smale compactness condition](https://tamnd.github.io/math2/node/4724KRNHAT.md): concept 4724KRNHAT
- [Clenshaw algorithm](https://tamnd.github.io/math2/node/476ZQPMZHP.md): statement 476ZQPMZHP
- [Levi problem](https://tamnd.github.io/math2/node/47CVS9RJ9D.md): problem 47CVS9RJ9D
- [External ray](https://tamnd.github.io/math2/node/47DYWZTQMC.md): definition 47DYWZTQMC
- [Volterra series](https://tamnd.github.io/math2/node/47KD8V5A7G.md): definition 47KD8V5A7G
- [Separation of variables](https://tamnd.github.io/math2/node/47NFRVB564.md): concept 47NFRVB564
- [Monotone operator](https://tamnd.github.io/math2/node/47NSS1R24Y.md): definition 47NSS1R24Y
- [Classical orthogonal polynomials](https://tamnd.github.io/math2/node/47WJ34NC9K.md): concept 47WJ34NC9K
- [Second-order logic](https://tamnd.github.io/math2/node/47XHPD8XE2.md): definition 47XHPD8XE2
- [Dedekind number](https://tamnd.github.io/math2/node/481ZPQ0GC3.md): concept 481ZPQ0GC3
- [Abscissa of absolute convergence](https://tamnd.github.io/math2/node/484GKH3ZS8.md): definition 484GKH3ZS8
- [Löwenheim–Skolem theorem](https://tamnd.github.io/math2/node/484M0TZ1S7.md): statement 484M0TZ1S7
- [Near-field](https://tamnd.github.io/math2/node/486GPTW3M8.md): definition 486GPTW3M8
- [Interpretation of varieties](https://tamnd.github.io/math2/node/48AR35RY9R.md): definition 48AR35RY9R
- [q-convex manifold](https://tamnd.github.io/math2/node/48BP5HZAAC.md): definition 48BP5HZAAC
- [Sumset](https://tamnd.github.io/math2/node/48EA9Z10DX.md): definition 48EA9Z10DX
- [Lopatinskii–Shapiro condition](https://tamnd.github.io/math2/node/48MEJEMEXM.md): concept 48MEJEMEXM
- [Young's inequality for integral operators](https://tamnd.github.io/math2/node/48NS4ZCSR3.md): statement 48NS4ZCSR3
- [Hermitian form](https://tamnd.github.io/math2/node/48P4A1CX85.md): concept 48P4A1CX85
- [Homotopy category](https://tamnd.github.io/math2/node/48QQ7T98Z9.md): concept 48QQ7T98Z9
- [Representable functor](https://tamnd.github.io/math2/node/494YXJH790.md): definition 494YXJH790
- [Jensen's formula](https://tamnd.github.io/math2/node/497K99BQ4Y.md): statement 497K99BQ4Y
- [Rational difference equation](https://tamnd.github.io/math2/node/49837NYG1A.md): definition 49837NYG1A
- [Functional inequality](https://tamnd.github.io/math2/node/498VVT12YJ.md): definition 498VVT12YJ
- [Weierstrass preparation theorem](https://tamnd.github.io/math2/node/49JSD49P7X.md): statement 49JSD49P7X
- [Catastrophic cancellation](https://tamnd.github.io/math2/node/49MX22EDS7.md): definition 49MX22EDS7
- [Sectional curvature](https://tamnd.github.io/math2/node/49SVWS83ZX.md): definition 49SVWS83ZX
- [Hopf algebra](https://tamnd.github.io/math2/node/49VHC2VWNF.md): definition 49VHC2VWNF
- [Perfect ring](https://tamnd.github.io/math2/node/49Y9Q949PK.md): concept 49Y9Q949PK
- [Fefferman's ball multiplier theorem](https://tamnd.github.io/math2/node/4A5MJY4XPV.md): statement 4A5MJY4XPV
- [Homotopic maps](https://tamnd.github.io/math2/node/4A7H5AG0NF.md): definition 4A7H5AG0NF
- [Pluriharmonic function](https://tamnd.github.io/math2/node/4A7VAXPG4F.md): concept 4A7VAXPG4F
- [Continuous dual space](https://tamnd.github.io/math2/node/4A82S5FVRT.md): concept 4A82S5FVRT
- [Soundness](https://tamnd.github.io/math2/node/4A84VHDTHV.md): concept 4A84VHDTHV
- [Knot sum](https://tamnd.github.io/math2/node/4ACJ5TPHJX.md): definition 4ACJ5TPHJX
- [Conjugacy problem](https://tamnd.github.io/math2/node/4AF7AF7690.md): problem 4AF7AF7690
- [Ringed space](https://tamnd.github.io/math2/node/4AG8ZWK2BB.md): definition 4AG8ZWK2BB
- [Hamiltonian Monte Carlo](https://tamnd.github.io/math2/node/4AHBMJYCX1.md): statement 4AHBMJYCX1
- [Truncation error](https://tamnd.github.io/math2/node/4AKCQERM89.md): concept 4AKCQERM89
- [Graph enumeration](https://tamnd.github.io/math2/node/4AVF0RN7HV.md): concept 4AVF0RN7HV
- [Superharmonic function](https://tamnd.github.io/math2/node/4AWC5Q4FHE.md): definition 4AWC5Q4FHE
- [Proper holomorphic map](https://tamnd.github.io/math2/node/4AXC8NESM4.md): definition 4AXC8NESM4
- [Goldie ring](https://tamnd.github.io/math2/node/4AY3RMCTA3.md): definition 4AY3RMCTA3
- [Schoenberg's embedding criterion](https://tamnd.github.io/math2/node/4B0R70F4VE.md): statement 4B0R70F4VE
- [Essential extension](https://tamnd.github.io/math2/node/4B66270PGS.md): definition 4B66270PGS
- [Partial difference equation](https://tamnd.github.io/math2/node/4B7M4A1MH0.md): concept 4B7M4A1MH0
- [Jacobi polynomials](https://tamnd.github.io/math2/node/4BBW1NZ1CC.md): definition 4BBW1NZ1CC
- [Mostow rigidity theorem](https://tamnd.github.io/math2/node/4BEWKR0EG7.md): statement 4BEWKR0EG7
- [Schrödinger equation](https://tamnd.github.io/math2/node/4BT3Y8TCDN.md): definition 4BT3Y8TCDN
- [Relative homology](https://tamnd.github.io/math2/node/4BVWH0BN28.md): concept 4BVWH0BN28
- [Horn](https://tamnd.github.io/math2/node/4BW0Y31347.md): definition 4BW0Y31347
- [Solution of equations with degenerate kernels](https://tamnd.github.io/math2/node/4BY024AVR8.md): statement 4BY024AVR8
- [Localization](https://tamnd.github.io/math2/node/4C1AE84N5N.md): concept 4C1AE84N5N
- [Second variation](https://tamnd.github.io/math2/node/4C3BANJ8AH.md): concept 4C3BANJ8AH
- [Regular semigroup](https://tamnd.github.io/math2/node/4C3RRDA4WZ.md): definition 4C3RRDA4WZ
- [Pointless topology](https://tamnd.github.io/math2/node/4C5RG7P4VT.md): definition 4C5RG7P4VT
- [Quartic formula](https://tamnd.github.io/math2/node/4CDFMKKZ01.md): statement 4CDFMKKZ01
- [Ramanujan–Petersson conjecture](https://tamnd.github.io/math2/node/4CDFWST967.md): statement 4CDFWST967
- [Teichmüller space](https://tamnd.github.io/math2/node/4CGTN9NRR9.md): definition 4CGTN9NRR9
- [Reproducing kernel Hilbert space](https://tamnd.github.io/math2/node/4CK2GSRAGD.md): concept 4CK2GSRAGD
- [Galois group of a prime degree polynomial with two nonreal roots](https://tamnd.github.io/math2/node/4CMPPTSJNP.md): statement 4CMPPTSJNP
- [Growth rate of a group](https://tamnd.github.io/math2/node/4CMRM3975W.md): concept 4CMRM3975W
- [Near-unanimity term](https://tamnd.github.io/math2/node/4CRH8NNFEF.md): concept 4CRH8NNFEF
- [Tate module](https://tamnd.github.io/math2/node/4CSMQFTGD8.md): concept 4CSMQFTGD8
- [g-theorem](https://tamnd.github.io/math2/node/4CZMWQVC30.md): statement 4CZMWQVC30
- [Rational map](https://tamnd.github.io/math2/node/4D073CCYGV.md): concept 4D073CCYGV
- [Carathéodory's existence theorem](https://tamnd.github.io/math2/node/4D090WJ84H.md): statement 4D090WJ84H
- [Bilateral Laplace transform of a probability law](https://tamnd.github.io/math2/node/4D1E2YYJNZ.md): example 4D1E2YYJNZ
- [Partial isometry](https://tamnd.github.io/math2/node/4D8CDGZNHE.md): definition 4D8CDGZNHE
- [Minkowski question-mark function](https://tamnd.github.io/math2/node/4D9HP5TS6W.md): example 4D9HP5TS6W
- [Local cohomology](https://tamnd.github.io/math2/node/4DD3GM2K4T.md): definition 4DD3GM2K4T
- [Center manifold theorem](https://tamnd.github.io/math2/node/4DEGC6JADR.md): statement 4DEGC6JADR
- [Normed algebra](https://tamnd.github.io/math2/node/4DJ4V5VBSH.md): concept 4DJ4V5VBSH
- [Yoneda embedding](https://tamnd.github.io/math2/node/4DKB6AQ8J7.md): concept 4DKB6AQ8J7
- [General recursive function](https://tamnd.github.io/math2/node/4DQGSSWREK.md): definition 4DQGSSWREK
- [Net of a polyhedron](https://tamnd.github.io/math2/node/4DRPG6VR70.md): definition 4DRPG6VR70
- [Operator system](https://tamnd.github.io/math2/node/4DSVM5Q17H.md): concept 4DSVM5Q17H
- [Jordan algebra](https://tamnd.github.io/math2/node/4DV7PEPZ7R.md): definition 4DV7PEPZ7R
- [Net](https://tamnd.github.io/math2/node/4DVJQJV0DG.md): concept 4DVJQJV0DG
- [Lagrange's theorem in S3](https://tamnd.github.io/math2/node/4DWC4XFS34.md): example 4DWC4XFS34
- [van Dantzig's theorem](https://tamnd.github.io/math2/node/4E4RXKV9W3.md): statement 4E4RXKV9W3
- [Quillen's theorem on complex cobordism](https://tamnd.github.io/math2/node/4E518M24MA.md): statement 4E518M24MA
- [Topological index](https://tamnd.github.io/math2/node/4E7ZAWAJ7S.md): definition 4E7ZAWAJ7S
- [Chain homotopy](https://tamnd.github.io/math2/node/4EBHAWJJ8C.md): concept 4EBHAWJJ8C
- [Dirac equation](https://tamnd.github.io/math2/node/4ECWKRYM8P.md): definition 4ECWKRYM8P
- [Dini–Lipschitz criterion](https://tamnd.github.io/math2/node/4EH60YPD9Q.md): statement 4EH60YPD9Q
- [Semimartingale](https://tamnd.github.io/math2/node/4EJB6E2AZK.md): concept 4EJB6E2AZK
- [C-infinity ring](https://tamnd.github.io/math2/node/4EJED0YZBB.md): concept 4EJED0YZBB
- [Wald–Wolfowitz optimality of the sequential probability ratio test](https://tamnd.github.io/math2/node/4F0N5J5QT2.md): statement 4F0N5J5QT2
- [Weakly compact cardinal](https://tamnd.github.io/math2/node/4F1ABY39W1.md): definition 4F1ABY39W1
- [Homotopy fiber](https://tamnd.github.io/math2/node/4F3KWXYXBA.md): concept 4F3KWXYXBA
- [Resonance](https://tamnd.github.io/math2/node/4F812GHFZK.md): concept 4F812GHFZK
- [Sparse grid](https://tamnd.github.io/math2/node/4FBC8218YE.md): definition 4FBC8218YE
- [Coproduct](https://tamnd.github.io/math2/node/4FJ0S71P58.md): concept 4FJ0S71P58
- [Pringsheim convergence](https://tamnd.github.io/math2/node/4FK55CYMVZ.md): definition 4FK55CYMVZ
- [Monomial ideal](https://tamnd.github.io/math2/node/4G25S7PJ22.md): concept 4G25S7PJ22
- [Doob's martingale inequality](https://tamnd.github.io/math2/node/4G53E7BRXF.md): statement 4G53E7BRXF
- [Brun sieve](https://tamnd.github.io/math2/node/4G65M0ZWAZ.md): definition 4G65M0ZWAZ
- [Homotopy fiber](https://tamnd.github.io/math2/node/4G6CFK4T7F.md): definition 4G6CFK4T7F
- [Smale's degree for proper Fredholm maps](https://tamnd.github.io/math2/node/4G7BGMGGK2.md): definition 4G7BGMGGK2
- [XYZ inequality](https://tamnd.github.io/math2/node/4GG1HGDEZC.md): statement 4GG1HGDEZC
- [Chern class](https://tamnd.github.io/math2/node/4GG7VRW0DY.md): definition 4GG7VRW0DY
- [Maass wave form](https://tamnd.github.io/math2/node/4GM75FC4PP.md): definition 4GM75FC4PP
- [Lamplighter group](https://tamnd.github.io/math2/node/4GNPFMT7DZ.md): example 4GNPFMT7DZ
- [Radon transform](https://tamnd.github.io/math2/node/4GPAX3B1MK.md): definition 4GPAX3B1MK
- [Semisimple module](https://tamnd.github.io/math2/node/4GRWMXWJNZ.md): definition 4GRWMXWJNZ
- [Ricci curvature](https://tamnd.github.io/math2/node/4GTNM0WHDH.md): concept 4GTNM0WHDH
- [Hamilton–Jacobi equation](https://tamnd.github.io/math2/node/4H1E4KW1JX.md): definition 4H1E4KW1JX
- [Erdős distinct distances problem](https://tamnd.github.io/math2/node/4H31EM5T0C.md): problem 4H31EM5T0C
- [Realizability](https://tamnd.github.io/math2/node/4H35VQ16DC.md): concept 4H35VQ16DC
- [Poisson manifold](https://tamnd.github.io/math2/node/4H3AHHDSVS.md): definition 4H3AHHDSVS
- [Determinacy](https://tamnd.github.io/math2/node/4H7DF303WF.md): definition 4H7DF303WF
- [Linear recurrence with polynomial coefficients](https://tamnd.github.io/math2/node/4HH0RPA7E6.md): definition 4HH0RPA7E6
- [Fredholm alternative](https://tamnd.github.io/math2/node/4HH8C9V8JQ.md): statement 4HH8C9V8JQ
- [Covering relation](https://tamnd.github.io/math2/node/4HHDPZ4BDY.md): concept 4HHDPZ4BDY
- [Function having a given Mellin transform](https://tamnd.github.io/math2/node/4HMBNA4RMN.md): definition 4HMBNA4RMN
- [Max-flow min-cut theorem](https://tamnd.github.io/math2/node/4HMR4KQ01T.md): statement 4HMR4KQ01T
- [WZ theory](https://tamnd.github.io/math2/node/4HP429BZQ5.md): definition 4HP429BZQ5
- [Bifurcation from a characteristic value](https://tamnd.github.io/math2/node/4HQBRDQ3A9.md): concept 4HQBRDQ3A9
- [Feller process](https://tamnd.github.io/math2/node/4HS3C12472.md): concept 4HS3C12472
- [Fukaya category](https://tamnd.github.io/math2/node/4HS60JABC5.md): concept 4HS60JABC5
- [Intersection theory](https://tamnd.github.io/math2/node/4HV4WTAX9P.md): concept 4HV4WTAX9P
- [Legendre condition](https://tamnd.github.io/math2/node/4HZX2EN319.md): statement 4HZX2EN319
- [Random group](https://tamnd.github.io/math2/node/4J05N6H5NX.md): concept 4J05N6H5NX
- [Mazur–Ulam theorem](https://tamnd.github.io/math2/node/4J4EW7FXWW.md): statement 4J4EW7FXWW
- [Hilbert symbol](https://tamnd.github.io/math2/node/4J9JJ3DY5R.md): definition 4J9JJ3DY5R
- [Kahan summation algorithm](https://tamnd.github.io/math2/node/4JAEKN22Y5.md): statement 4JAEKN22Y5
- [Block design](https://tamnd.github.io/math2/node/4JCSS2Z1BA.md): concept 4JCSS2Z1BA
- [Determinantal variety](https://tamnd.github.io/math2/node/4JEBSN709Q.md): definition 4JEBSN709Q
- [Siggers term](https://tamnd.github.io/math2/node/4JEDEJRXJW.md): definition 4JEDEJRXJW
- [Discrete valuation ring](https://tamnd.github.io/math2/node/4JF1SGYZ5Y.md): definition 4JF1SGYZ5Y
- [Eilenberg–Steenrod axioms](https://tamnd.github.io/math2/node/4JG5Y1E5F3.md): definition 4JG5Y1E5F3
- [Legendre polynomials](https://tamnd.github.io/math2/node/4JVVDAQ9XX.md): definition 4JVVDAQ9XX
- [Character sum](https://tamnd.github.io/math2/node/4JWHKNGS4Y.md): definition 4JWHKNGS4Y
- [Markov random field](https://tamnd.github.io/math2/node/4JX3WDYJ83.md): definition 4JX3WDYJ83
- [Kolmogorov quotient](https://tamnd.github.io/math2/node/4K0ZQ35ZCK.md): definition 4K0ZQ35ZCK
- [Spherical Bessel functions](https://tamnd.github.io/math2/node/4K2DZV0D0D.md): concept 4K2DZV0D0D
- [Turán's inequalities](https://tamnd.github.io/math2/node/4K3D632K60.md): statement 4K3D632K60
- [Beltrami equation](https://tamnd.github.io/math2/node/4K6B00NYCD.md): definition 4K6B00NYCD
- [Integral extension](https://tamnd.github.io/math2/node/4K6F157663.md): definition 4K6F157663
- [Fresnel integrals](https://tamnd.github.io/math2/node/4K7TJE5KM4.md): definition 4K7TJE5KM4
- [Open subgroup](https://tamnd.github.io/math2/node/4K9RYSNPG3.md): definition 4K9RYSNPG3
- [Isolated hypersurface singularity](https://tamnd.github.io/math2/node/4KAWK0FD4V.md): definition 4KAWK0FD4V
- [Daniell integral](https://tamnd.github.io/math2/node/4KCBFQS9WR.md): concept 4KCBFQS9WR
- [Hom-set characterization of adjunctions](https://tamnd.github.io/math2/node/4KCECXYW81.md): statement 4KCECXYW81
- [1 + 2 + 3 + 4 + ⋯](https://tamnd.github.io/math2/node/4KFC9BE02Y.md): example 4KFC9BE02Y
- [Spectrum of an element](https://tamnd.github.io/math2/node/4KN39NQZ02.md): definition 4KN39NQZ02
- [Barbalat's lemma](https://tamnd.github.io/math2/node/4KPNX1SZYR.md): statement 4KPNX1SZYR
- [Siegel's theorem on integral points](https://tamnd.github.io/math2/node/4KTBRP27P6.md): statement 4KTBRP27P6
- [Seminorm](https://tamnd.github.io/math2/node/4KX81Q85DT.md): definition 4KX81Q85DT
- [Weierstrass's counterexample to the Dirichlet principle](https://tamnd.github.io/math2/node/4KXHGXZQY8.md): counterexample 4KXHGXZQY8
- [Variational integrator](https://tamnd.github.io/math2/node/4KZ458T2QA.md): concept 4KZ458T2QA
- [Rational map](https://tamnd.github.io/math2/node/4M1SYSPGA2.md): definition 4M1SYSPGA2
- [Compactly generated space](https://tamnd.github.io/math2/node/4M5JXYVQ2X.md): concept 4M5JXYVQ2X
- [Regular function](https://tamnd.github.io/math2/node/4M94FZH9Y8.md): concept 4M94FZH9Y8
- [Artin–Tate lemma](https://tamnd.github.io/math2/node/4MBR0W0S6N.md): statement 4MBR0W0S6N
- [Commuting probability](https://tamnd.github.io/math2/node/4MBYAVB0PY.md): concept 4MBYAVB0PY
- [Closed geodesic](https://tamnd.github.io/math2/node/4MCCP8GG4J.md): concept 4MCCP8GG4J
- [Exchangeable random variables](https://tamnd.github.io/math2/node/4MR4YJS1ZD.md): concept 4MR4YJS1ZD
- [Wavelet](https://tamnd.github.io/math2/node/4MWPJBF1Z7.md): definition 4MWPJBF1Z7
- [Proof that subgroups of index two are normal](https://tamnd.github.io/math2/node/4N1WKH2ZQV.md): proof 4N1WKH2ZQV
- [Sober space](https://tamnd.github.io/math2/node/4N4DNDR22Z.md): concept 4N4DNDR22Z
- [Mehta integral](https://tamnd.github.io/math2/node/4N53T1BZ9S.md): statement 4N53T1BZ9S
- [Whitney C-infinity topology](https://tamnd.github.io/math2/node/4N5FGYKAFW.md): definition 4N5FGYKAFW
- [Lewy's example](https://tamnd.github.io/math2/node/4NCB2ZY08G.md): counterexample 4NCB2ZY08G
- [Signature operator](https://tamnd.github.io/math2/node/4NCYA7GME1.md): concept 4NCYA7GME1
- [Integro-differential equation](https://tamnd.github.io/math2/node/4NFHBBNZBK.md): definition 4NFHBBNZBK
- [Area-minimizing current](https://tamnd.github.io/math2/node/4NKP73KVVW.md): concept 4NKP73KVVW
- [Group ring](https://tamnd.github.io/math2/node/4NMTSX0XDS.md): concept 4NMTSX0XDS
- [Eigenfunction of an integral operator](https://tamnd.github.io/math2/node/4NSACSWRJZ.md): concept 4NSACSWRJZ
- [Nerve of a cover](https://tamnd.github.io/math2/node/4P3S0DRZXB.md): concept 4P3S0DRZXB
- [Borel hierarchy](https://tamnd.github.io/math2/node/4P4Z4ZGXCY.md): concept 4P4Z4ZGXCY
- [Packing density](https://tamnd.github.io/math2/node/4P5PSWADJD.md): concept 4P5PSWADJD
- [Ext functor](https://tamnd.github.io/math2/node/4P7GBWJ4RR.md): concept 4P7GBWJ4RR
- [Smoothing by convolution](https://tamnd.github.io/math2/node/4P7KPA3Y7R.md): statement 4P7KPA3Y7R
- [Sequential probability ratio test](https://tamnd.github.io/math2/node/4PBC71YCTK.md): definition 4PBC71YCTK
- [Strichartz estimate](https://tamnd.github.io/math2/node/4PDKJ8N1S9.md): statement 4PDKJ8N1S9
- [Zaremba's punctured ball](https://tamnd.github.io/math2/node/4PK2T949SQ.md): counterexample 4PK2T949SQ
- [Importance sampling](https://tamnd.github.io/math2/node/4PPT08K2M1.md): statement 4PPT08K2M1
- [Multidimensional transform](https://tamnd.github.io/math2/node/4PVP1C3JZG.md): concept 4PVP1C3JZG
- [Wavelet](https://tamnd.github.io/math2/node/4PZ35C0GV3.md): concept 4PZ35C0GV3
- [S-finite measure](https://tamnd.github.io/math2/node/4Q1CK359NB.md): definition 4Q1CK359NB
- [Weak homotopy equivalence](https://tamnd.github.io/math2/node/4Q3WY1RX91.md): concept 4Q3WY1RX91
- [Percolation theory](https://tamnd.github.io/math2/node/4QAQ2TRAY0.md): definition 4QAQ2TRAY0
- [Axiom of constructibility](https://tamnd.github.io/math2/node/4QKRPRCSXH.md): concept 4QKRPRCSXH
- [Factor analysis](https://tamnd.github.io/math2/node/4QQQP1Q886.md): concept 4QQQP1Q886
- [Bornological space](https://tamnd.github.io/math2/node/4R6SQD60FN.md): concept 4R6SQD60FN
- [Weak and strong extremum](https://tamnd.github.io/math2/node/4R6XZSDH3E.md): definition 4R6XZSDH3E
- [Exponential map of a Lie group](https://tamnd.github.io/math2/node/4R7HW6X4JY.md): definition 4R7HW6X4JY
- [Laguerre polynomials](https://tamnd.github.io/math2/node/4R8XEAV3P6.md): definition 4R8XEAV3P6
- [F-algebra](https://tamnd.github.io/math2/node/4RB8834SEJ.md): concept 4RB8834SEJ
- [Hankel functions](https://tamnd.github.io/math2/node/4RDG2SJ70D.md): definition 4RDG2SJ70D
- [Stein factorization](https://tamnd.github.io/math2/node/4READX002Z.md): statement 4READX002Z
- [Steiner quasigroup](https://tamnd.github.io/math2/node/4RF117CH63.md): example 4RF117CH63
- [Biproduct](https://tamnd.github.io/math2/node/4RG1RB1MXH.md): definition 4RG1RB1MXH
- [Buchberger's algorithm](https://tamnd.github.io/math2/node/4RH1D40WG1.md): concept 4RH1D40WG1
- [Power of a test](https://tamnd.github.io/math2/node/4RJSN0QARD.md): concept 4RJSN0QARD
- [Fiber bundle](https://tamnd.github.io/math2/node/4RK8T21D27.md): definition 4RK8T21D27
- [Eckmann–Hilton duality](https://tamnd.github.io/math2/node/4RPERPNBK1.md): definition 4RPERPNBK1
- [Minkowski content](https://tamnd.github.io/math2/node/4RPGSCQVR9.md): definition 4RPGSCQVR9
- [Harish-Chandra embedding](https://tamnd.github.io/math2/node/4RXXE3BB6W.md): statement 4RXXE3BB6W
- [Perfect set](https://tamnd.github.io/math2/node/4S1HC0N5E8.md): definition 4S1HC0N5E8
- [Conformal geometric algebra](https://tamnd.github.io/math2/node/4S4Z7SQ47G.md): concept 4S4Z7SQ47G
- [Oppenheim conjecture](https://tamnd.github.io/math2/node/4S50G4PS26.md): statement 4S50G4PS26
- [Essential submodule](https://tamnd.github.io/math2/node/4S6DXRPZ4Z.md): definition 4S6DXRPZ4Z
- [Maschke's theorem](https://tamnd.github.io/math2/node/4SBZ4EMPBZ.md): statement 4SBZ4EMPBZ
- [The integers modulo the even integers](https://tamnd.github.io/math2/node/4SCS55K5MA.md): example 4SCS55K5MA
- [Global attractor](https://tamnd.github.io/math2/node/4SH2ZB984E.md): concept 4SH2ZB984E
- [Almost convergence](https://tamnd.github.io/math2/node/4SNJWQBAHF.md): definition 4SNJWQBAHF
- [Jones polynomial](https://tamnd.github.io/math2/node/4SPC17FPXE.md): concept 4SPC17FPXE
- [Recurrent point](https://tamnd.github.io/math2/node/4SQHHRPKMV.md): concept 4SQHHRPKMV
- [Nilpotency class](https://tamnd.github.io/math2/node/4SQYNP0B4Y.md): concept 4SQYNP0B4Y
- [Portmanteau theorem](https://tamnd.github.io/math2/node/4SSJ589G6T.md): statement 4SSJ589G6T
- [Spheroidal wave function](https://tamnd.github.io/math2/node/4ST57PN8K7.md): definition 4ST57PN8K7
- [Quadrature rule](https://tamnd.github.io/math2/node/4T1MNB8Z11.md): definition 4T1MNB8Z11
- [Mumford–Shah functional](https://tamnd.github.io/math2/node/4T72ZP5XHF.md): example 4T72ZP5XHF
- [Sobol sequence](https://tamnd.github.io/math2/node/4T7D0T5GJA.md): object 4T7D0T5GJA
- [Nef line bundle](https://tamnd.github.io/math2/node/4TCFNSTXAD.md): definition 4TCFNSTXAD
- [Limit point compact space](https://tamnd.github.io/math2/node/4THTYJ07RT.md): definition 4THTYJ07RT
- [Kardar–Parisi–Zhang equation](https://tamnd.github.io/math2/node/4TKVXC9N46.md): definition 4TKVXC9N46
- [Loewner order](https://tamnd.github.io/math2/node/4TP2R2R7DA.md): definition 4TP2R2R7DA
- [Coplanarity](https://tamnd.github.io/math2/node/4TWY53TBKK.md): concept 4TWY53TBKK
- [Feldman–Hájek theorem](https://tamnd.github.io/math2/node/4TXG13F5N0.md): statement 4TXG13F5N0
- [Limit set](https://tamnd.github.io/math2/node/4V09M20SJB.md): definition 4V09M20SJB
- [Fractional calculus](https://tamnd.github.io/math2/node/4V4AGE7XP4.md): definition 4V4AGE7XP4
- [Field of rational functions](https://tamnd.github.io/math2/node/4V81611P9N.md): concept 4V81611P9N
- [Characteristic value of a kernel](https://tamnd.github.io/math2/node/4V9V0AHJCS.md): definition 4V9V0AHJCS
- [Dirichlet eigenvalue](https://tamnd.github.io/math2/node/4VD8XXQ4P3.md): concept 4VD8XXQ4P3
- [Rack and quandle](https://tamnd.github.io/math2/node/4VDFK4ZF0V.md): definition 4VDFK4ZF0V
- [Grünwald–Letnikov derivative](https://tamnd.github.io/math2/node/4VE006XQ33.md): definition 4VE006XQ33
- [Birational geometry](https://tamnd.github.io/math2/node/4VFX7WNFGX.md): definition 4VFX7WNFGX
- [Nevanlinna–Pick interpolation](https://tamnd.github.io/math2/node/4VMSSYB5DN.md): statement 4VMSSYB5DN
- [Markov partition](https://tamnd.github.io/math2/node/4VNHVDHTNN.md): concept 4VNHVDHTNN
- [Particular point topology](https://tamnd.github.io/math2/node/4VSZ22TX4B.md): example 4VSZ22TX4B
- [Fredholm series for the resolvent](https://tamnd.github.io/math2/node/4W0EG94RZ5.md): statement 4W0EG94RZ5
- [Buekenhout geometry](https://tamnd.github.io/math2/node/4W27TKGQWR.md): concept 4W27TKGQWR
- [Approximate derivative](https://tamnd.github.io/math2/node/4W31MKE01V.md): concept 4W31MKE01V
- [Log-normal distribution](https://tamnd.github.io/math2/node/4W40NHWPQT.md): example 4W40NHWPQT
- [Pi-system](https://tamnd.github.io/math2/node/4W9NM803BB.md): concept 4W9NM803BB
- [Lichnerowicz formula](https://tamnd.github.io/math2/node/4WCJ49Z90S.md): statement 4WCJ49Z90S
- [Matroid minor](https://tamnd.github.io/math2/node/4WCYBSH1P6.md): concept 4WCYBSH1P6
- [Dehn–Sommerville equations](https://tamnd.github.io/math2/node/4WKHF5BCTA.md): statement 4WKHF5BCTA
- [Tate algebra](https://tamnd.github.io/math2/node/4WSR1VXJNW.md): definition 4WSR1VXJNW
- [Graded poset](https://tamnd.github.io/math2/node/4WX0HAK9X9.md): definition 4WX0HAK9X9
- [Kuranishi structure](https://tamnd.github.io/math2/node/4WY3A9YGDR.md): concept 4WY3A9YGDR
- [Harmonic at a point](https://tamnd.github.io/math2/node/4X0A0CQ0B0.md): definition 4X0A0CQ0B0
- [Céa's lemma](https://tamnd.github.io/math2/node/4X6WC3E7P8.md): statement 4X6WC3E7P8
- [Stieltjes function](https://tamnd.github.io/math2/node/4XEGZK0H36.md): definition 4XEGZK0H36
- [Nonexpansive map](https://tamnd.github.io/math2/node/4XGQD57EWF.md): definition 4XGQD57EWF
- [Weak operator topology](https://tamnd.github.io/math2/node/4XM1VN1C1N.md): concept 4XM1VN1C1N
- [Cayley–Abels graph](https://tamnd.github.io/math2/node/4XQ8TYS2C4.md): definition 4XQ8TYS2C4
- [Bohr compactification](https://tamnd.github.io/math2/node/4XR88RPJ4K.md): definition 4XR88RPJ4K
- [Finite model theory](https://tamnd.github.io/math2/node/4XRVJE0KAN.md): concept 4XRVJE0KAN
- [Poincaré–Hopf theorem](https://tamnd.github.io/math2/node/4XTPY82PP4.md): statement 4XTPY82PP4
- [Arithmetic group](https://tamnd.github.io/math2/node/4XWG6S7R4Z.md): concept 4XWG6S7R4Z
- [Darboux integral](https://tamnd.github.io/math2/node/4XYXDYXQZR.md): definition 4XYXDYXQZR
- [Association scheme](https://tamnd.github.io/math2/node/4Y4MR8W9AF.md): definition 4Y4MR8W9AF
- [Approximately finite-dimensional C*-algebra](https://tamnd.github.io/math2/node/4Y7VENNBVK.md): concept 4Y7VENNBVK
- [Descriptive geometry](https://tamnd.github.io/math2/node/4YDJDTV6B7.md): concept 4YDJDTV6B7
- [Iwasawa decomposition of a Lie group](https://tamnd.github.io/math2/node/4YDR9Q36DS.md): statement 4YDR9Q36DS
- [Chu–Vandermonde identity](https://tamnd.github.io/math2/node/4YK46R1QDR.md): statement 4YK46R1QDR
- [Universal property of the free magma](https://tamnd.github.io/math2/node/4YN1TB9CXN.md): statement 4YN1TB9CXN
- [Starred transform](https://tamnd.github.io/math2/node/4YQ8JYXBKK.md): concept 4YQ8JYXBKK
- [Subvariety](https://tamnd.github.io/math2/node/4YSCK1DF26.md): concept 4YSCK1DF26
- [Capacity of a set](https://tamnd.github.io/math2/node/4YZVFGJVSZ.md): concept 4YZVFGJVSZ
- [Modular lambda function](https://tamnd.github.io/math2/node/4Z1YXT957Z.md): object 4Z1YXT957Z
- [Ore localization](https://tamnd.github.io/math2/node/4Z6EED4616.md): definition 4Z6EED4616
- [Monotone likelihood ratio](https://tamnd.github.io/math2/node/4Z73AFN7WT.md): concept 4Z73AFN7WT
- [Newtonian potential](https://tamnd.github.io/math2/node/4Z75CMNQ1P.md): definition 4Z75CMNQ1P
- [Affine differential geometry](https://tamnd.github.io/math2/node/4ZTTPDXCQT.md): definition 4ZTTPDXCQT
- [Benedicks' theorem](https://tamnd.github.io/math2/node/4ZY2D6C8A6.md): statement 4ZY2D6C8A6
- [Holomorphic function of several variables](https://tamnd.github.io/math2/node/507C34W0ZT.md): concept 507C34W0ZT
- [Birkhoff's HSP theorem](https://tamnd.github.io/math2/node/5083DHSK82.md): statement 5083DHSK82
- [Paris–Harrington theorem](https://tamnd.github.io/math2/node/509ZWA0R26.md): statement 509ZWA0R26
- [Partial algebra](https://tamnd.github.io/math2/node/50D3KZHQ7B.md): definition 50D3KZHQ7B
- [Linear difference equation](https://tamnd.github.io/math2/node/5115KV924G.md): definition 5115KV924G
- [Volterra integro-differential equation](https://tamnd.github.io/math2/node/5126SDXYHF.md): concept 5126SDXYHF
- [Thom space](https://tamnd.github.io/math2/node/5152FTFE13.md): definition 5152FTFE13
- [Loewner differential equation](https://tamnd.github.io/math2/node/51G4QJZ345.md): concept 51G4QJZ345
- [Rademacher functions](https://tamnd.github.io/math2/node/51GRFQ4WY6.md): concept 51GRFQ4WY6
- [Legendre integral transform](https://tamnd.github.io/math2/node/51PH7Z03TJ.md): concept 51PH7Z03TJ
- [Birational geometry](https://tamnd.github.io/math2/node/51YVV85GWB.md): concept 51YVV85GWB
- [Selberg sieve](https://tamnd.github.io/math2/node/5252P3NHT7.md): concept 5252P3NHT7
- [Fredholm integral equation](https://tamnd.github.io/math2/node/526TCVMSHP.md): concept 526TCVMSHP
- [Unisolvent point set](https://tamnd.github.io/math2/node/5276MDSDQV.md): concept 5276MDSDQV
- [Schauder basis](https://tamnd.github.io/math2/node/528G11SDWR.md): concept 528G11SDWR
- [Koenigs function](https://tamnd.github.io/math2/node/529D1C4W9H.md): concept 529D1C4W9H
- [Hermitian symmetric space](https://tamnd.github.io/math2/node/52GCNZ1MZ8.md): definition 52GCNZ1MZ8
- [Brownian bridge](https://tamnd.github.io/math2/node/52N6S8WXM9.md): definition 52N6S8WXM9
- [Analytic set](https://tamnd.github.io/math2/node/52QT0EK3GJ.md): definition 52QT0EK3GJ
- [Hermite functions](https://tamnd.github.io/math2/node/52VS07H6GK.md): concept 52VS07H6GK
- [Intuitionistic logic](https://tamnd.github.io/math2/node/52XCYMK0WW.md): definition 52XCYMK0WW
- [Stirling's formula for the gamma function](https://tamnd.github.io/math2/node/53750WSJ6K.md): statement 53750WSJ6K
- [First Cousin problem](https://tamnd.github.io/math2/node/537J79VYQR.md): problem 537J79VYQR
- [Lax–Friedrichs method](https://tamnd.github.io/math2/node/53H8EVRC1J.md): definition 53H8EVRC1J
- [Shimura variety](https://tamnd.github.io/math2/node/53HPW59SJC.md): definition 53HPW59SJC
- [Inertial manifold](https://tamnd.github.io/math2/node/53K0FDVSHE.md): definition 53K0FDVSHE
- [Bochner's classification of orthogonal polynomial eigenfunctions](https://tamnd.github.io/math2/node/53MWMX2HWT.md): statement 53MWMX2HWT
- [Weierstrass product for the gamma function](https://tamnd.github.io/math2/node/53MZ2DGDTQ.md): statement 53MZ2DGDTQ
- [Kleisli category](https://tamnd.github.io/math2/node/53TSQ57J0H.md): definition 53TSQ57J0H
- [Orthocentric system](https://tamnd.github.io/math2/node/541S6DXQAR.md): definition 541S6DXQAR
- [Consistency of the maximum likelihood estimator](https://tamnd.github.io/math2/node/54387XJ9QW.md): statement 54387XJ9QW
- [Stable homotopy theory](https://tamnd.github.io/math2/node/545EV0W4CE.md): concept 545EV0W4CE
- [Saturation of approximation methods](https://tamnd.github.io/math2/node/54BD10ZZDN.md): definition 54BD10ZZDN
- [Compact group](https://tamnd.github.io/math2/node/54KHZGC0CV.md): definition 54KHZGC0CV
- [Halton sequence](https://tamnd.github.io/math2/node/54MY2XHVGB.md): object 54MY2XHVGB
- [Monodromy theorem](https://tamnd.github.io/math2/node/54QMKJF64A.md): statement 54QMKJF64A
- [Matrix norm](https://tamnd.github.io/math2/node/54T6Y6DSY4.md): concept 54T6Y6DSY4
- [Geodesic curvature](https://tamnd.github.io/math2/node/550P8PFMVC.md): concept 550P8PFMVC
- [Wiener process](https://tamnd.github.io/math2/node/552AWX3QXN.md): definition 552AWX3QXN
- [The special linear group is the kernel of the determinant](https://tamnd.github.io/math2/node/556HZEYCQF.md): example 556HZEYCQF
- [Measure-preserving transformation](https://tamnd.github.io/math2/node/556WBDAHNP.md): concept 556WBDAHNP
- [Numerical methods for ordinary differential equations](https://tamnd.github.io/math2/node/558XADG64M.md): concept 558XADG64M
- [Godunov's theorem](https://tamnd.github.io/math2/node/559N53ZTMV.md): statement 559N53ZTMV
- [Rational surface](https://tamnd.github.io/math2/node/55A4A53TRF.md): concept 55A4A53TRF
- [Matroid rank](https://tamnd.github.io/math2/node/55B7ZCDKS9.md): definition 55B7ZCDKS9
- [Numerical diffusion](https://tamnd.github.io/math2/node/55GVVJTF24.md): concept 55GVVJTF24
- [p-adic absolute value](https://tamnd.github.io/math2/node/55J3D3SACQ.md): definition 55J3D3SACQ
- [Finite measure](https://tamnd.github.io/math2/node/55RNFNN39R.md): definition 55RNFNN39R
- [Inverse distance weighting](https://tamnd.github.io/math2/node/55V8WAJXXJ.md): definition 55V8WAJXXJ
- [Fodor's lemma](https://tamnd.github.io/math2/node/55W1M20Q3F.md): statement 55W1M20Q3F
- [Well-founded order](https://tamnd.github.io/math2/node/55WD0H03CZ.md): concept 55WD0H03CZ
- [Modular curve](https://tamnd.github.io/math2/node/55XCZDS142.md): definition 55XCZDS142
- [Class number formula](https://tamnd.github.io/math2/node/55XX5A7XZV.md): statement 55XX5A7XZV
- [Hochster–Roberts theorem](https://tamnd.github.io/math2/node/56107GMFYV.md): statement 56107GMFYV
- [Wright omega function](https://tamnd.github.io/math2/node/563ANQ3MY7.md): concept 563ANQ3MY7
- [Coset](https://tamnd.github.io/math2/node/564VVAQTNR.md): concept 564VVAQTNR
- [Differential of a smooth map](https://tamnd.github.io/math2/node/565B0Y76XE.md): definition 565B0Y76XE
- [Indecomposable continuum](https://tamnd.github.io/math2/node/56J7DSVAZH.md): concept 56J7DSVAZH
- [Measure space](https://tamnd.github.io/math2/node/56MJSVTHG0.md): definition 56MJSVTHG0
- [Empirical risk minimization](https://tamnd.github.io/math2/node/56NVR0SS61.md): concept 56NVR0SS61
- [Empirical Bayes method](https://tamnd.github.io/math2/node/56QC25W5AT.md): definition 56QC25W5AT
- [Coherent sheaf](https://tamnd.github.io/math2/node/56RXEJ8Z6S.md): concept 56RXEJ8Z6S
- [Modification of a complex space](https://tamnd.github.io/math2/node/56V4WT0BT2.md): definition 56V4WT0BT2
- [h-cobordism theorem](https://tamnd.github.io/math2/node/56YF8W9NXP.md): statement 56YF8W9NXP
- [Lieb–Thirring inequality](https://tamnd.github.io/math2/node/572QKEYYKX.md): statement 572QKEYYKX
- [Similarity solution](https://tamnd.github.io/math2/node/579J5K9K5Z.md): concept 579J5K9K5Z
- [Analytic geometry](https://tamnd.github.io/math2/node/579PT4D349.md): definition 579PT4D349
- [Giraud's theorem](https://tamnd.github.io/math2/node/57BXX32WF0.md): statement 57BXX32WF0
- [Effective resistance](https://tamnd.github.io/math2/node/57DPMSN7H8.md): definition 57DPMSN7H8
- [Continuous wavelet transform](https://tamnd.github.io/math2/node/57FR21NDGV.md): concept 57FR21NDGV
- [Ordinal analysis](https://tamnd.github.io/math2/node/57PTMS2ADY.md): definition 57PTMS2ADY
- [Fuchsian group](https://tamnd.github.io/math2/node/57TCAKSBQT.md): definition 57TCAKSBQT
- [Robin boundary condition](https://tamnd.github.io/math2/node/581AGFCCRX.md): concept 581AGFCCRX
- [Polish space](https://tamnd.github.io/math2/node/584GXXY73M.md): concept 584GXXY73M
- [Dolbeault–Grothendieck lemma](https://tamnd.github.io/math2/node/5858M5PGN6.md): statement 5858M5PGN6
- [Trajectory optimization](https://tamnd.github.io/math2/node/58895A7T4H.md): concept 58895A7T4H
- [Griffiths transversality](https://tamnd.github.io/math2/node/589GJMH5EF.md): statement 589GJMH5EF
- [Neville's algorithm](https://tamnd.github.io/math2/node/58AM5FK67J.md): concept 58AM5FK67J
- [Parallelohedron](https://tamnd.github.io/math2/node/58MV5BK731.md): definition 58MV5BK731
- [Expansive homeomorphism](https://tamnd.github.io/math2/node/58MWCEYTZD.md): definition 58MWCEYTZD
- [Order statistic](https://tamnd.github.io/math2/node/58NHGZDSRE.md): concept 58NHGZDSRE
- [Artin–Schreier theorem](https://tamnd.github.io/math2/node/58P0EGAFVV.md): statement 58P0EGAFVV
- [Almost periodic sequence](https://tamnd.github.io/math2/node/58PZBP5FE1.md): definition 58PZBP5FE1
- [Complex projective plane](https://tamnd.github.io/math2/node/58WSWYBCWA.md): example 58WSWYBCWA
- [Woodin cardinal](https://tamnd.github.io/math2/node/592QGNJCZC.md): definition 592QGNJCZC
- [Potential operator of a Markov process](https://tamnd.github.io/math2/node/595AZJAZNA.md): concept 595AZJAZNA
- [Continuous kernel](https://tamnd.github.io/math2/node/59CF2D7BRB.md): definition 59CF2D7BRB
- [Dynamical system](https://tamnd.github.io/math2/node/59GFP54D78.md): definition 59GFP54D78
- [Esakia space](https://tamnd.github.io/math2/node/59PG49KH6V.md): definition 59PG49KH6V
- [Kac's theorem](https://tamnd.github.io/math2/node/59QJC3ZXA1.md): statement 59QJC3ZXA1
- [Littlewood polynomial](https://tamnd.github.io/math2/node/5A16A9W335.md): concept 5A16A9W335
- [Scheme](https://tamnd.github.io/math2/node/5A35TYZD02.md): concept 5A35TYZD02
- [Discrete-time dynamical system](https://tamnd.github.io/math2/node/5AA9AJEARJ.md): concept 5AA9AJEARJ
- [Extension and contraction of ideals](https://tamnd.github.io/math2/node/5AFH94P4S3.md): concept 5AFH94P4S3
- [Killing vector field](https://tamnd.github.io/math2/node/5AKNNW3W4Z.md): concept 5AKNNW3W4Z
- [Adic space](https://tamnd.github.io/math2/node/5AXZG4PXTE.md): definition 5AXZG4PXTE
- [Microlocal analysis](https://tamnd.github.io/math2/node/5B6WEQX7XK.md): concept 5B6WEQX7XK
- [First variation](https://tamnd.github.io/math2/node/5B8NFYB2XQ.md): concept 5B8NFYB2XQ
- [Hyperfunction](https://tamnd.github.io/math2/node/5BAYVAGVW3.md): definition 5BAYVAGVW3
- [Perron's irreducibility criterion](https://tamnd.github.io/math2/node/5BCRQYVYR8.md): statement 5BCRQYVYR8
- [Proper group action](https://tamnd.github.io/math2/node/5BM1KMAF03.md): definition 5BM1KMAF03
- [Opposite category](https://tamnd.github.io/math2/node/5BNBYZVECW.md): definition 5BNBYZVECW
- [Parseval formula for the Mellin transform](https://tamnd.github.io/math2/node/5BR72MMAPV.md): statement 5BR72MMAPV
- [FBI transform](https://tamnd.github.io/math2/node/5BW26GPS2A.md): concept 5BW26GPS2A
- [Hartogs domain](https://tamnd.github.io/math2/node/5BW600RVX5.md): definition 5BW600RVX5
- [Elliptic function](https://tamnd.github.io/math2/node/5BXMSPDPRJ.md): definition 5BXMSPDPRJ
- [Neutral differential equation](https://tamnd.github.io/math2/node/5C0N2ENRF2.md): concept 5C0N2ENRF2
- [Stochastic Volterra equation](https://tamnd.github.io/math2/node/5C3411G4C0.md): definition 5C3411G4C0
- [Pixley term](https://tamnd.github.io/math2/node/5C91SFVTY7.md): concept 5C91SFVTY7
- [Continuous geometry](https://tamnd.github.io/math2/node/5CET1KWWWH.md): concept 5CET1KWWWH
- [Hardy inequality in Sobolev spaces](https://tamnd.github.io/math2/node/5CHFRPQ03W.md): statement 5CHFRPQ03W
- [Karhunen–Loève theorem](https://tamnd.github.io/math2/node/5CN7KY8MWP.md): statement 5CN7KY8MWP
- [Regular variation](https://tamnd.github.io/math2/node/5CSCTKRTB2.md): definition 5CSCTKRTB2
- [Atiyah–Guillemin–Sternberg convexity theorem](https://tamnd.github.io/math2/node/5CTHC6612N.md): statement 5CTHC6612N
- [Algebraic independence](https://tamnd.github.io/math2/node/5CVYTHWY96.md): definition 5CVYTHWY96
- [Torsor](https://tamnd.github.io/math2/node/5CYEQY4SWR.md): concept 5CYEQY4SWR
- [Complex cobordism](https://tamnd.github.io/math2/node/5CZHFNY9F3.md): concept 5CZHFNY9F3
- [Differential dynamic programming](https://tamnd.github.io/math2/node/5D0J26E619.md): concept 5D0J26E619
- [Pettis integral](https://tamnd.github.io/math2/node/5D5SC7NNJ2.md): definition 5D5SC7NNJ2
- [Minkowski–Weyl theorem](https://tamnd.github.io/math2/node/5DFMMGGD9Y.md): statement 5DFMMGGD9Y
- [Strongly measurable function](https://tamnd.github.io/math2/node/5DHPVQ9B4X.md): definition 5DHPVQ9B4X
- [Ring with involution](https://tamnd.github.io/math2/node/5DHSMZKD73.md): definition 5DHSMZKD73
- [Brown–McCoy radical](https://tamnd.github.io/math2/node/5DNZZBSQSR.md): definition 5DNZZBSQSR
- [Determinant line bundle](https://tamnd.github.io/math2/node/5DST3VKEMX.md): concept 5DST3VKEMX
- [Seiberg–Witten invariants](https://tamnd.github.io/math2/node/5DTY2FCT9Z.md): definition 5DTY2FCT9Z
- [Cross-polytope](https://tamnd.github.io/math2/node/5DYYRZWER6.md): example 5DYYRZWER6
- [Representation theory of SL2(R)](https://tamnd.github.io/math2/node/5E547TJ7PS.md): example 5E547TJ7PS
- [Harmonic analysis on groups](https://tamnd.github.io/math2/node/5E6KB3V8R0.md): concept 5E6KB3V8R0
- [Brieskorn manifold](https://tamnd.github.io/math2/node/5E9E5R9RYZ.md): example 5E9E5R9RYZ
- [Wave equation](https://tamnd.github.io/math2/node/5EHD2XBQDG.md): concept 5EHD2XBQDG
- [Uniformization theorem](https://tamnd.github.io/math2/node/5EJ7SW1RCV.md): statement 5EJ7SW1RCV
- [Saddle point of a Lagrangian](https://tamnd.github.io/math2/node/5ENHTF01Z8.md): concept 5ENHTF01Z8
- [Zeros of holomorphic functions are not isolated](https://tamnd.github.io/math2/node/5EZRN3812E.md): statement 5EZRN3812E
- [Semiprime ring](https://tamnd.github.io/math2/node/5F0YMCV50R.md): concept 5F0YMCV50R
- [Brent's method](https://tamnd.github.io/math2/node/5F76XA1ABC.md): statement 5F76XA1ABC
- [Real Hardy space H1](https://tamnd.github.io/math2/node/5F9MP8VCJ4.md): definition 5F9MP8VCJ4
- [Totally unimodular matrix](https://tamnd.github.io/math2/node/5FC2T66AQD.md): definition 5FC2T66AQD
- [Classification of second-order PDEs](https://tamnd.github.io/math2/node/5FC6BMM1NA.md): concept 5FC6BMM1NA
- [Cartan's theorem A](https://tamnd.github.io/math2/node/5FD8V6MQNX.md): statement 5FD8V6MQNX
- [Grothendieck group](https://tamnd.github.io/math2/node/5FDMY7PDYY.md): concept 5FDMY7PDYY
- [Haagerup property](https://tamnd.github.io/math2/node/5FG1BKZXKS.md): concept 5FG1BKZXKS
- [Quasi-projective variety](https://tamnd.github.io/math2/node/5FKKCM9ESE.md): definition 5FKKCM9ESE
- [Spectral density estimation](https://tamnd.github.io/math2/node/5FM1TC4FG4.md): definition 5FM1TC4FG4
- [Convolution integral equation](https://tamnd.github.io/math2/node/5FQG18RD00.md): definition 5FQG18RD00
- [Hilbert polynomial](https://tamnd.github.io/math2/node/5FR6JX04RS.md): definition 5FR6JX04RS
- [Z-transform](https://tamnd.github.io/math2/node/5FY0JQGHYZ.md): concept 5FY0JQGHYZ
- [Covariant derivative](https://tamnd.github.io/math2/node/5FZDQQHQKM.md): concept 5FZDQQHQKM
- [Integral extensions of fields](https://tamnd.github.io/math2/node/5GCANV7337.md): statement 5GCANV7337
- [Hypergeometric series](https://tamnd.github.io/math2/node/5GD32H1RPD.md): definition 5GD32H1RPD
- [Cartan–Iwasawa–Malcev theorem](https://tamnd.github.io/math2/node/5GDFYDF2R2.md): statement 5GDFYDF2R2
- [Totally positive matrix](https://tamnd.github.io/math2/node/5GN0RYRVBA.md): concept 5GN0RYRVBA
- [Involution as solution of Babbage's equation](https://tamnd.github.io/math2/node/5GNSQRHSSC.md): example 5GNSQRHSSC
- [Algebraic K-theory](https://tamnd.github.io/math2/node/5GVFZWMWC8.md): concept 5GVFZWMWC8
- [Lattice ideal](https://tamnd.github.io/math2/node/5GW2T2AM1A.md): concept 5GW2T2AM1A
- [Strichartz estimates in Lorentz space](https://tamnd.github.io/math2/node/5GX76EM5HS.md): statement 5GX76EM5HS
- [Extensive category](https://tamnd.github.io/math2/node/5H16H7FY5D.md): concept 5H16H7FY5D
- [Minkowski's second theorem](https://tamnd.github.io/math2/node/5H4XZVF9B9.md): statement 5H4XZVF9B9
- [Wiener algebra](https://tamnd.github.io/math2/node/5H8XESYCHQ.md): concept 5H8XESYCHQ
- [Lasker–Noether theorem](https://tamnd.github.io/math2/node/5H8Z56KXMS.md): statement 5H8Z56KXMS
- [Riemannian manifold](https://tamnd.github.io/math2/node/5HDG31YEPW.md): definition 5HDG31YEPW
- [Hyperconnected space](https://tamnd.github.io/math2/node/5HDNJWZ6K9.md): definition 5HDNJWZ6K9
- [Central simple algebra](https://tamnd.github.io/math2/node/5HRKJBAY2V.md): concept 5HRKJBAY2V
- [Weierstrass transform](https://tamnd.github.io/math2/node/5HYB6DDZH8.md): concept 5HYB6DDZH8
- [Cartan's coherence theorem for ideal sheaves](https://tamnd.github.io/math2/node/5J0D97M548.md): statement 5J0D97M548
- [Macdonald constant term conjecture](https://tamnd.github.io/math2/node/5J25W265JA.md): statement 5J25W265JA
- [Nonlinear Schrödinger equation](https://tamnd.github.io/math2/node/5J2GTEA6CV.md): definition 5J2GTEA6CV
- [Hecke algebra](https://tamnd.github.io/math2/node/5J2P2AGY4N.md): concept 5J2P2AGY4N
- [Induced representation of a locally compact group](https://tamnd.github.io/math2/node/5J603A6DFH.md): definition 5J603A6DFH
- [Spectral graph theory](https://tamnd.github.io/math2/node/5JESM2ZN48.md): concept 5JESM2ZN48
- [de Boor's algorithm](https://tamnd.github.io/math2/node/5JHFB384BF.md): concept 5JHFB384BF
- [Glimm scheme](https://tamnd.github.io/math2/node/5JQG4S5JVH.md): definition 5JQG4S5JVH
- [Accelerated failure time model](https://tamnd.github.io/math2/node/5JQRYAKHSB.md): definition 5JQRYAKHSB
- [Wedge sum](https://tamnd.github.io/math2/node/5JSWM1HBX2.md): concept 5JSWM1HBX2
- [Borel subgroup](https://tamnd.github.io/math2/node/5JSYA1JNCC.md): concept 5JSYA1JNCC
- [Sheaf of modules](https://tamnd.github.io/math2/node/5JVKKP0X1Y.md): definition 5JVKKP0X1Y
- [Finite Hilbert transform](https://tamnd.github.io/math2/node/5JYBXH9BP6.md): concept 5JYBXH9BP6
- [Riemann surface](https://tamnd.github.io/math2/node/5K0N8QMV6Q.md): concept 5K0N8QMV6Q
- [Majorization](https://tamnd.github.io/math2/node/5K7VT2VKZT.md): concept 5K7VT2VKZT
- [Duhamel's integral](https://tamnd.github.io/math2/node/5KC6B0Z09Y.md): statement 5KC6B0Z09Y
- [Stable homotopy theory](https://tamnd.github.io/math2/node/5KERJ85BXJ.md): definition 5KERJ85BXJ
- [Satellite knot](https://tamnd.github.io/math2/node/5KGKGTW2XY.md): definition 5KGKGTW2XY
- [Mergelyan's theorem](https://tamnd.github.io/math2/node/5KKCSZJC29.md): statement 5KKCSZJC29
- [Idele class group](https://tamnd.github.io/math2/node/5KMSW7139A.md): definition 5KMSW7139A
- [Bromwich integral](https://tamnd.github.io/math2/node/5KNAXNHRY6.md): concept 5KNAXNHRY6
- [Brauer's height zero conjecture](https://tamnd.github.io/math2/node/5M0HYY3RYV.md): conjecture 5M0HYY3RYV
- [Essential extension](https://tamnd.github.io/math2/node/5M223XDM43.md): concept 5M223XDM43
- [Fractional iteration](https://tamnd.github.io/math2/node/5M3THA1S6E.md): concept 5M3THA1S6E
- [Julia set](https://tamnd.github.io/math2/node/5M4S7P04GR.md): concept 5M4S7P04GR
- [Flat morphism](https://tamnd.github.io/math2/node/5M6KS8F47R.md): concept 5M6KS8F47R
- [Discrete subgroup](https://tamnd.github.io/math2/node/5M7D1BSHVZ.md): concept 5M7D1BSHVZ
- [Itô isometry](https://tamnd.github.io/math2/node/5MDK7HSYRM.md): statement 5MDK7HSYRM
- [Naive set theory](https://tamnd.github.io/math2/node/5MDSVRCDZW.md): definition 5MDSVRCDZW
- [Weak dimension](https://tamnd.github.io/math2/node/5MEKEETGDA.md): concept 5MEKEETGDA
- [Heegaard splitting](https://tamnd.github.io/math2/node/5MN8MX0JPA.md): definition 5MN8MX0JPA
- [Thom space](https://tamnd.github.io/math2/node/5MNYT3AFPH.md): concept 5MNYT3AFPH
- [Failure of Kalman's conjecture](https://tamnd.github.io/math2/node/5MP7DM0V5V.md): counterexample 5MP7DM0V5V
- [Statistical learning theory](https://tamnd.github.io/math2/node/5MR9ZSC3CX.md): concept 5MR9ZSC3CX
- [Energy of a map between Riemannian manifolds](https://tamnd.github.io/math2/node/5N1825CM10.md): definition 5N1825CM10
- [Left ideal](https://tamnd.github.io/math2/node/5N2CJZ47D4.md): concept 5N2CJZ47D4
- [Pareto distribution](https://tamnd.github.io/math2/node/5N3P64YHQR.md): example 5N3P64YHQR
- [Riesz potential operator](https://tamnd.github.io/math2/node/5N5DXQMVP7.md): concept 5N5DXQMVP7
- [Poincaré classification of circle homeomorphisms](https://tamnd.github.io/math2/node/5N5SDY3K53.md): statement 5N5SDY3K53
- [Residue field of a point](https://tamnd.github.io/math2/node/5NAGGDQQTM.md): definition 5NAGGDQQTM
- [Absolute continuity](https://tamnd.github.io/math2/node/5NB63N9W3P.md): definition 5NB63N9W3P
- [Congruence criteria for triangles](https://tamnd.github.io/math2/node/5NEJPXA6N4.md): statement 5NEJPXA6N4
- [Polygon triangulation](https://tamnd.github.io/math2/node/5NH4HVMEXA.md): definition 5NH4HVMEXA
- [Matrix Riemann–Hilbert problem](https://tamnd.github.io/math2/node/5NH9BXWAKR.md): problem 5NH9BXWAKR
- [Ruled surface](https://tamnd.github.io/math2/node/5NK10NGY13.md): concept 5NK10NGY13
- [Fourier coefficient](https://tamnd.github.io/math2/node/5NN2YZZR3V.md): concept 5NN2YZZR3V
- [Compact symplectic group](https://tamnd.github.io/math2/node/5NNPBFQA02.md): example 5NNPBFQA02
- [Rounding](https://tamnd.github.io/math2/node/5NPVX29Y0C.md): concept 5NPVX29Y0C
- [Primitive recursive function](https://tamnd.github.io/math2/node/5NVA2JRHPV.md): concept 5NVA2JRHPV
- [Hierarchical matrix](https://tamnd.github.io/math2/node/5P5EMYHXE5.md): concept 5P5EMYHXE5
- [Knot Floer homology](https://tamnd.github.io/math2/node/5P608WRETM.md): concept 5P608WRETM
- [Representer theorem](https://tamnd.github.io/math2/node/5P6NB8P1HG.md): statement 5P6NB8P1HG
- [Discrete Painlevé equations](https://tamnd.github.io/math2/node/5PFJBYDD2H.md): concept 5PFJBYDD2H
- [Logicism](https://tamnd.github.io/math2/node/5PGSQYB5Y1.md): concept 5PGSQYB5Y1
- [Fukushima correspondence between Dirichlet forms and processes](https://tamnd.github.io/math2/node/5PJF1NDCTT.md): statement 5PJF1NDCTT
- [Pseudoholomorphic curve](https://tamnd.github.io/math2/node/5PJNXDJD8W.md): definition 5PJNXDJD8W
- [Tropical semiring](https://tamnd.github.io/math2/node/5PNBCRH2JG.md): example 5PNBCRH2JG
- [Soundness](https://tamnd.github.io/math2/node/5PP333WZEK.md): definition 5PP333WZEK
- [Eulerian number](https://tamnd.github.io/math2/node/5PS2496830.md): concept 5PS2496830
- [Siegel upper half-space](https://tamnd.github.io/math2/node/5PTYE97PF6.md): object 5PTYE97PF6
- [Barycentric subdivision](https://tamnd.github.io/math2/node/5PVK5PVZBA.md): definition 5PVK5PVZBA
- [Fourier transform of tempered distributions](https://tamnd.github.io/math2/node/5PVSQXCTCT.md): concept 5PVSQXCTCT
- [Frobenius theorem](https://tamnd.github.io/math2/node/5Q4BHBFXJY.md): statement 5Q4BHBFXJY
- [De Morgan algebra](https://tamnd.github.io/math2/node/5Q53XWV0QN.md): concept 5Q53XWV0QN
- [Norm residue isomorphism theorem](https://tamnd.github.io/math2/node/5Q5VN56QH4.md): statement 5Q5VN56QH4
- [Epigraph](https://tamnd.github.io/math2/node/5Q9BNSZ5XZ.md): definition 5Q9BNSZ5XZ
- [Sphere theorem for 3-manifolds](https://tamnd.github.io/math2/node/5QAFGPAZ4A.md): statement 5QAFGPAZ4A
- [Bilinear interpolation](https://tamnd.github.io/math2/node/5QJYCNRYMQ.md): concept 5QJYCNRYMQ
- [Injective object](https://tamnd.github.io/math2/node/5QRFTBF4PG.md): concept 5QRFTBF4PG
- [Cellular automaton](https://tamnd.github.io/math2/node/5QXNC6NTVN.md): concept 5QXNC6NTVN
- [Finsler manifold](https://tamnd.github.io/math2/node/5R6QZ388BB.md): concept 5R6QZ388BB
- [Monoidal category](https://tamnd.github.io/math2/node/5R76HPDB8W.md): definition 5R76HPDB8W
- [MacCormack method](https://tamnd.github.io/math2/node/5R9ZRK8YB5.md): definition 5R9ZRK8YB5
- [Translation equation](https://tamnd.github.io/math2/node/5RD6ZFKPDS.md): definition 5RD6ZFKPDS
- [Invariant subspace problem](https://tamnd.github.io/math2/node/5RF055GK4N.md): problem 5RF055GK4N
- [Oates–Powell theorem](https://tamnd.github.io/math2/node/5RFPT46DXX.md): statement 5RFPT46DXX
- [Takens's theorem](https://tamnd.github.io/math2/node/5RN835HTBD.md): statement 5RN835HTBD
- [Wallace–Bolyai–Gerwien theorem](https://tamnd.github.io/math2/node/5RQGCEQYK5.md): statement 5RQGCEQYK5
- [Homogenization](https://tamnd.github.io/math2/node/5RSREPYN0F.md): concept 5RSREPYN0F
- [Frame bundle](https://tamnd.github.io/math2/node/5RYRVW0XPC.md): definition 5RYRVW0XPC
- [Bicategory](https://tamnd.github.io/math2/node/5S1R5AVKZ6.md): definition 5S1R5AVKZ6
- [Levi form](https://tamnd.github.io/math2/node/5S30FAJNGM.md): definition 5S30FAJNGM
- [Symmetric monoidal category](https://tamnd.github.io/math2/node/5S7ER95XAC.md): concept 5S7ER95XAC
- [Tensor product of Hilbert spaces](https://tamnd.github.io/math2/node/5S7FCJMCD7.md): definition 5S7FCJMCD7
- [Hyperelliptic curve](https://tamnd.github.io/math2/node/5SB8EKSQRN.md): concept 5SB8EKSQRN
- [Milnor conjecture on slice genus of torus knots](https://tamnd.github.io/math2/node/5SDA7ZXT95.md): statement 5SDA7ZXT95
- [Atomic lattice](https://tamnd.github.io/math2/node/5SE8EMSDNY.md): concept 5SE8EMSDNY
- [L-curve](https://tamnd.github.io/math2/node/5SE9H1SPBN.md): concept 5SE9H1SPBN
- [Saccheri quadrilateral](https://tamnd.github.io/math2/node/5SJVH0BHJE.md): concept 5SJVH0BHJE
- [May recognition principle](https://tamnd.github.io/math2/node/5SMK5KEBQE.md): statement 5SMK5KEBQE
- [Quasi-arithmetic mean](https://tamnd.github.io/math2/node/5SNCPXVH1J.md): definition 5SNCPXVH1J
- [CW complex](https://tamnd.github.io/math2/node/5SP8E66T9E.md): definition 5SP8E66T9E
- [Ribbon Hopf algebra](https://tamnd.github.io/math2/node/5SRDQB4NG7.md): definition 5SRDQB4NG7
- [Derived category](https://tamnd.github.io/math2/node/5ST7NVJFSM.md): concept 5ST7NVJFSM
- [Stable homotopy groups of spheres](https://tamnd.github.io/math2/node/5SY318V6S3.md): definition 5SY318V6S3
- [Polish group](https://tamnd.github.io/math2/node/5T2J3FDT0X.md): definition 5T2J3FDT0X
- [Vandermonde matrix](https://tamnd.github.io/math2/node/5T4DY28JSK.md): definition 5T4DY28JSK
- [Final functor](https://tamnd.github.io/math2/node/5T6HCEK1M8.md): definition 5T6HCEK1M8
- [Completely distributive lattice](https://tamnd.github.io/math2/node/5TBC06NNS1.md): concept 5TBC06NNS1
- [Geodesic flow on a locally symmetric space](https://tamnd.github.io/math2/node/5TKBSHY6VC.md): example 5TKBSHY6VC
- [Whitman's condition](https://tamnd.github.io/math2/node/5TNA2HBG7Y.md): statement 5TNA2HBG7Y
- [Gegenbauer polynomials](https://tamnd.github.io/math2/node/5TRFA6F79B.md): concept 5TRFA6F79B
- [Hamilton–Jacobi equation](https://tamnd.github.io/math2/node/5TRXQKP7DY.md): concept 5TRXQKP7DY
- [Modulus of smoothness](https://tamnd.github.io/math2/node/5V22013VEQ.md): definition 5V22013VEQ
- [Numerical continuation](https://tamnd.github.io/math2/node/5V3GJ84CPQ.md): concept 5V3GJ84CPQ
- [Grandi's series](https://tamnd.github.io/math2/node/5V9ZHDZ0EZ.md): example 5V9ZHDZ0EZ
- [Siegel zero](https://tamnd.github.io/math2/node/5VBZBXSFDQ.md): concept 5VBZBXSFDQ
- [Algebraic combinatorics](https://tamnd.github.io/math2/node/5VDCJTZ8SW.md): definition 5VDCJTZ8SW
- [Trigamma function](https://tamnd.github.io/math2/node/5VGVPENA0F.md): concept 5VGVPENA0F
- [Indefinite sum](https://tamnd.github.io/math2/node/5VK9NZM0R7.md): definition 5VK9NZM0R7
- [Kodaira dimension](https://tamnd.github.io/math2/node/5VKAY7RS8C.md): definition 5VKAY7RS8C
- [High indices theorem](https://tamnd.github.io/math2/node/5VN386XHGA.md): statement 5VN386XHGA
- [Mirsky's theorem](https://tamnd.github.io/math2/node/5VPXR3RDF1.md): statement 5VPXR3RDF1
- [Homotopy continuation for polynomial systems](https://tamnd.github.io/math2/node/5VR7XNNXN0.md): concept 5VR7XNNXN0
- [Riemann extension theorem in several variables](https://tamnd.github.io/math2/node/5VW718YJQK.md): statement 5VW718YJQK
- [Flat norm](https://tamnd.github.io/math2/node/5VWTS78ZPJ.md): definition 5VWTS78ZPJ
- [Virtual fundamental class](https://tamnd.github.io/math2/node/5VXYQ4BXRB.md): concept 5VXYQ4BXRB
- [Dedekind-finite ring](https://tamnd.github.io/math2/node/5VYYQER2JB.md): definition 5VYYQER2JB
- [Eigenfunction of an integral operator](https://tamnd.github.io/math2/node/5W1MCFBMYP.md): definition 5W1MCFBMYP
- [Stratonovich integral](https://tamnd.github.io/math2/node/5W60ZXNYED.md): definition 5W60ZXNYED
- [Free spectrum of a variety](https://tamnd.github.io/math2/node/5W801R7ZF5.md): definition 5W801R7ZF5
- [CW complex](https://tamnd.github.io/math2/node/5W95QFHXPR.md): concept 5W95QFHXPR
- [E-infinity operad](https://tamnd.github.io/math2/node/5WA6FC4JQD.md): definition 5WA6FC4JQD
- [Universally catenary ring](https://tamnd.github.io/math2/node/5WAFPTH09V.md): concept 5WAFPTH09V
- [Semitopological group](https://tamnd.github.io/math2/node/5WGMRS4JSE.md): concept 5WGMRS4JSE
- [Weierstrass equation](https://tamnd.github.io/math2/node/5WGZPE6249.md): definition 5WGZPE6249
- [Stein interpolation theorem](https://tamnd.github.io/math2/node/5WHXGHBCY9.md): statement 5WHXGHBCY9
- [Perverse sheaf](https://tamnd.github.io/math2/node/5WHXRZDZFK.md): concept 5WHXRZDZFK
- [Brjuno number](https://tamnd.github.io/math2/node/5WR304YRST.md): concept 5WR304YRST
- [Virtual fundamental class](https://tamnd.github.io/math2/node/5WR98YG0AX.md): definition 5WR98YG0AX
- [Domination principle](https://tamnd.github.io/math2/node/5WVZSM42XR.md): statement 5WVZSM42XR
- [Stokes flow](https://tamnd.github.io/math2/node/5WXY2T8JG0.md): definition 5WXY2T8JG0
- [Bilinear series of a symmetric kernel](https://tamnd.github.io/math2/node/5WYZPX1ZVW.md): statement 5WYZPX1ZVW
- [Maximum and minimum](https://tamnd.github.io/math2/node/5X1YMSR204.md): concept 5X1YMSR204
- [Topological dynamics](https://tamnd.github.io/math2/node/5X6E97EMFR.md): concept 5X6E97EMFR
- [Lie supergroup](https://tamnd.github.io/math2/node/5X6G3Q5CSQ.md): definition 5X6G3Q5CSQ
- [Clifford algebra](https://tamnd.github.io/math2/node/5X7NA4TTK8.md): definition 5X7NA4TTK8
- [Fourier–Stieltjes transform](https://tamnd.github.io/math2/node/5X8PK7H6W3.md): concept 5X8PK7H6W3
- [Kobayashi metric](https://tamnd.github.io/math2/node/5XSY1Y133X.md): definition 5XSY1Y133X
- [Inverse mean curvature flow](https://tamnd.github.io/math2/node/5XVEF5DKAG.md): concept 5XVEF5DKAG
- [Oscillation of a function](https://tamnd.github.io/math2/node/5XW4N8KY3Y.md): concept 5XW4N8KY3Y
- [Ordinary least squares](https://tamnd.github.io/math2/node/5Y3DYMPVEG.md): definition 5Y3DYMPVEG
- [Noetherian scheme](https://tamnd.github.io/math2/node/5Y3SAEWBZM.md): definition 5Y3SAEWBZM
- [Krylov–Safonov estimate](https://tamnd.github.io/math2/node/5YC2Y65NZX.md): statement 5YC2Y65NZX
- [Closed monoidal category](https://tamnd.github.io/math2/node/5YH6FTMVYP.md): definition 5YH6FTMVYP
- [String diagram](https://tamnd.github.io/math2/node/5YHK0VS62A.md): concept 5YHK0VS62A
- [Hille–Yosida theorem](https://tamnd.github.io/math2/node/5YR4GJKPCV.md): statement 5YR4GJKPCV
- [Rayleigh–Ritz method](https://tamnd.github.io/math2/node/5YSP8TWD4Q.md): concept 5YSP8TWD4Q
- [Classical Wiener space](https://tamnd.github.io/math2/node/5Z00K7K1HY.md): object 5Z00K7K1HY
- [Desargues's theorem](https://tamnd.github.io/math2/node/5Z19Y02MME.md): statement 5Z19Y02MME
- [Fibration and cofibration](https://tamnd.github.io/math2/node/5ZA2T4C3N2.md): definition 5ZA2T4C3N2
- [Gauss–Legendre quadrature](https://tamnd.github.io/math2/node/5ZA6VRDNB9.md): definition 5ZA6VRDNB9
- [Bayes factor](https://tamnd.github.io/math2/node/5ZEGZVZBTJ.md): definition 5ZEGZVZBTJ
- [Levinson's theorem on asymptotic integration](https://tamnd.github.io/math2/node/5ZFRRMBFV0.md): statement 5ZFRRMBFV0
- [Singular cohomology](https://tamnd.github.io/math2/node/5ZJDCNSGHJ.md): definition 5ZJDCNSGHJ
- [Log-gamma function](https://tamnd.github.io/math2/node/5ZKBBTBXVB.md): definition 5ZKBBTBXVB
- [Gröbner basis](https://tamnd.github.io/math2/node/5ZKP6E4JCE.md): definition 5ZKP6E4JCE
- [Perron integral](https://tamnd.github.io/math2/node/5ZN4VXR3X5.md): definition 5ZN4VXR3X5
- [Crainic–Fernandes integrability theorem](https://tamnd.github.io/math2/node/5ZNN348M6X.md): statement 5ZNN348M6X
- [Characteristic of a field](https://tamnd.github.io/math2/node/5ZXRZB41HE.md): definition 5ZXRZB41HE
- [Connection on a vector bundle](https://tamnd.github.io/math2/node/5ZYRRVE8BW.md): concept 5ZYRRVE8BW
- [Identifiability](https://tamnd.github.io/math2/node/605M5NMFD5.md): concept 605M5NMFD5
- [Absorbing element](https://tamnd.github.io/math2/node/6068BQ4XZF.md): definition 6068BQ4XZF
- [Regular surface](https://tamnd.github.io/math2/node/6080X3FAD4.md): concept 6080X3FAD4
- [Pullback pasting lemma](https://tamnd.github.io/math2/node/60A5CC7S3A.md): statement 60A5CC7S3A
- [Strict partial order](https://tamnd.github.io/math2/node/60FNZ35H6R.md): concept 60FNZ35H6R
- [Cantor group](https://tamnd.github.io/math2/node/60G5ABJQ75.md): example 60G5ABJQ75
- [Generator of a semigroup](https://tamnd.github.io/math2/node/60HSWZBTK9.md): concept 60HSWZBTK9
- [Closable operator](https://tamnd.github.io/math2/node/60MQBQRACM.md): definition 60MQBQRACM
- [Turing jump](https://tamnd.github.io/math2/node/60P2XWY5QX.md): concept 60P2XWY5QX
- [Tarski–Vaught test](https://tamnd.github.io/math2/node/60SA99BER7.md): statement 60SA99BER7
- [Symm's integral equation](https://tamnd.github.io/math2/node/60TBPSWD32.md): example 60TBPSWD32
- [Poincaré map](https://tamnd.github.io/math2/node/60TDCPY4Z4.md): concept 60TDCPY4Z4
- [Harmonic at a point](https://tamnd.github.io/math2/node/60TNVD82Q5.md): concept 60TNVD82Q5
- [Projective line over a ring](https://tamnd.github.io/math2/node/60VZDSFKNF.md): definition 60VZDSFKNF
- [Levinson recursion](https://tamnd.github.io/math2/node/60XZ6DNKKW.md): statement 60XZ6DNKKW
- [Graph enumeration](https://tamnd.github.io/math2/node/610F4NS6GD.md): definition 610F4NS6GD
- [Diaconescu's theorem](https://tamnd.github.io/math2/node/6155D300EB.md): statement 6155D300EB
- [Deformation quantization](https://tamnd.github.io/math2/node/617CH1WRQN.md): definition 617CH1WRQN
- [Ryll-Nardzewski fixed-point theorem](https://tamnd.github.io/math2/node/61B8B11Y0Y.md): statement 61B8B11Y0Y
- [Discrete calculus](https://tamnd.github.io/math2/node/61CDKWSXV4.md): concept 61CDKWSXV4
- [Schreier–Sims algorithm](https://tamnd.github.io/math2/node/61F4HEY84K.md): definition 61F4HEY84K
- [Provability logic](https://tamnd.github.io/math2/node/61THBNS8WE.md): definition 61THBNS8WE
- [Neumann problem](https://tamnd.github.io/math2/node/61VG0XSDM6.md): problem 61VG0XSDM6
- [Schwarz integral formula](https://tamnd.github.io/math2/node/61WT6CE08W.md): statement 61WT6CE08W
- [Hecke algebra of a locally compact group](https://tamnd.github.io/math2/node/61Z8MZ2M28.md): definition 61Z8MZ2M28
- [Subshift of finite type](https://tamnd.github.io/math2/node/624XDXX8FN.md): concept 624XDXX8FN
- [Perfect set](https://tamnd.github.io/math2/node/628242KN0T.md): concept 628242KN0T
- [B-spline](https://tamnd.github.io/math2/node/62AFX32JGT.md): concept 62AFX32JGT
- [Strictly convex set](https://tamnd.github.io/math2/node/62FGRXS27K.md): concept 62FGRXS27K
- [Simple ring](https://tamnd.github.io/math2/node/62M0NC3E2T.md): concept 62M0NC3E2T
- [Perron–Frobenius theorem](https://tamnd.github.io/math2/node/62NS7PV1M9.md): statement 62NS7PV1M9
- [Homotopy extension property](https://tamnd.github.io/math2/node/62PG3KVVVK.md): definition 62PG3KVVVK
- [n-body problem](https://tamnd.github.io/math2/node/62SNWC5E45.md): problem 62SNWC5E45
- [Pickands–Balkema–De Haan theorem](https://tamnd.github.io/math2/node/62TMXRHZ76.md): statement 62TMXRHZ76
- [Locale](https://tamnd.github.io/math2/node/62WCKX1EZT.md): definition 62WCKX1EZT
- [Global field](https://tamnd.github.io/math2/node/634XKY8RZ0.md): concept 634XKY8RZ0
- [Principal series representation](https://tamnd.github.io/math2/node/63AMHDV2HX.md): definition 63AMHDV2HX
- [Soliton](https://tamnd.github.io/math2/node/63D4AXYCT4.md): concept 63D4AXYCT4
- [Square root of a matrix](https://tamnd.github.io/math2/node/63FAKMYZ2E.md): concept 63FAKMYZ2E
- [Zeta function regularization](https://tamnd.github.io/math2/node/63FS7T8R18.md): definition 63FS7T8R18
- [Distributive property](https://tamnd.github.io/math2/node/63JGSKKF7J.md): concept 63JGSKKF7J
- [Modular function](https://tamnd.github.io/math2/node/63TX60RTRQ.md): concept 63TX60RTRQ
- [Kolmogorov epsilon-entropy](https://tamnd.github.io/math2/node/63XHDVA5FS.md): concept 63XHDVA5FS
- [Regular local ring](https://tamnd.github.io/math2/node/63XXTF6B5M.md): concept 63XXTF6B5M
- [Finite measure](https://tamnd.github.io/math2/node/63Z1Q4FS23.md): concept 63Z1Q4FS23
- [Yamabe problem](https://tamnd.github.io/math2/node/63ZT8D3DXD.md): problem 63ZT8D3DXD
- [Paraproduct](https://tamnd.github.io/math2/node/640A1MTM4Z.md): definition 640A1MTM4Z
- [Monogenic function](https://tamnd.github.io/math2/node/646PNPT1R6.md): concept 646PNPT1R6
- [Invariant mean](https://tamnd.github.io/math2/node/64F9VMG27D.md): definition 64F9VMG27D
- [Max-plus algebra](https://tamnd.github.io/math2/node/64G0VSC5F4.md): concept 64G0VSC5F4
- [Khovanov homology](https://tamnd.github.io/math2/node/64GEW8D5G0.md): definition 64GEW8D5G0
- [Schur–Zassenhaus theorem](https://tamnd.github.io/math2/node/64K1A69C80.md): statement 64K1A69C80
- [Validated numerics](https://tamnd.github.io/math2/node/64V8MA4FH4.md): definition 64V8MA4FH4
- [Quasi-identity](https://tamnd.github.io/math2/node/64VN7KYGHK.md): definition 64VN7KYGHK
- [Codimension of a singularity](https://tamnd.github.io/math2/node/64W1MJFRNT.md): definition 64W1MJFRNT
- [Complete theory](https://tamnd.github.io/math2/node/655K8Y7VS8.md): concept 655K8Y7VS8
- [Sampling distribution](https://tamnd.github.io/math2/node/656Y8FA3RY.md): concept 656Y8FA3RY
- [Inverse semigroup](https://tamnd.github.io/math2/node/657D5Q37J1.md): definition 657D5Q37J1
- [Duality in projective geometry](https://tamnd.github.io/math2/node/65DN8PMR4H.md): definition 65DN8PMR4H
- [Butcher tableau](https://tamnd.github.io/math2/node/65H4W6TYKF.md): definition 65H4W6TYKF
- [Normed field](https://tamnd.github.io/math2/node/65HSHAH2Y4.md): definition 65HSHAH2Y4
- [Gelfand–Levitan theory](https://tamnd.github.io/math2/node/65PNDW38YC.md): definition 65PNDW38YC
- [Schanuel's conjecture](https://tamnd.github.io/math2/node/65PV4VFP9Y.md): conjecture 65PV4VFP9Y
- [Discrete-time Fourier transform](https://tamnd.github.io/math2/node/65QTEF1FTN.md): definition 65QTEF1FTN
- [Adjunction space](https://tamnd.github.io/math2/node/662659CEK4.md): definition 662659CEK4
- [Yang–Baxter equation](https://tamnd.github.io/math2/node/663BP8ZHS5.md): concept 663BP8ZHS5
- [Cardinal characteristics of the continuum](https://tamnd.github.io/math2/node/66748CE93N.md): concept 66748CE93N
- [Central polynomial](https://tamnd.github.io/math2/node/667YZTHE96.md): concept 667YZTHE96
- [Euclidean distance matrix](https://tamnd.github.io/math2/node/668HBS6C06.md): definition 668HBS6C06
- [Algebraic torus](https://tamnd.github.io/math2/node/66FQFAJNS4.md): concept 66FQFAJNS4
- [Energy functional on paths](https://tamnd.github.io/math2/node/66G7TEZ3JW.md): definition 66G7TEZ3JW
- [Many-valued logic](https://tamnd.github.io/math2/node/66NMTCQNQV.md): definition 66NMTCQNQV
- [Reflexive operator algebra](https://tamnd.github.io/math2/node/672HNBV803.md): concept 672HNBV803
- [Convex integrand](https://tamnd.github.io/math2/node/672SS33S5H.md): definition 672SS33S5H
- [Index theory](https://tamnd.github.io/math2/node/675MAXBQN6.md): concept 675MAXBQN6
- [Substructure](https://tamnd.github.io/math2/node/676H7EGTKJ.md): concept 676H7EGTKJ
- [Cyclic homology](https://tamnd.github.io/math2/node/676K167XS8.md): concept 676K167XS8
- [Iwasawa main conjecture](https://tamnd.github.io/math2/node/67A4D4FAYJ.md): statement 67A4D4FAYJ
- [Andreotti–Grauert theorem](https://tamnd.github.io/math2/node/67RJ0S8F74.md): statement 67RJ0S8F74
- [Unisolvent point set](https://tamnd.github.io/math2/node/681HPCS6G6.md): definition 681HPCS6G6
- [Global attractor](https://tamnd.github.io/math2/node/681NY7506V.md): definition 681NY7506V
- [Topos](https://tamnd.github.io/math2/node/6837RS7N82.md): concept 6837RS7N82
- [Projective tensor product](https://tamnd.github.io/math2/node/685ZV73JJ8.md): concept 685ZV73JJ8
- [Telescope conjecture](https://tamnd.github.io/math2/node/68650624FF.md): counterexample 68650624FF
- [Jordan curve theorem](https://tamnd.github.io/math2/node/687RGCHBTP.md): statement 687RGCHBTP
- [One-relator group](https://tamnd.github.io/math2/node/68AFPHR4GN.md): concept 68AFPHR4GN
- [Localization of a module](https://tamnd.github.io/math2/node/68E0F8N94E.md): concept 68E0F8N94E
- [Lie bracket of vector fields](https://tamnd.github.io/math2/node/68FGQEBD6J.md): concept 68FGQEBD6J
- [Moore space](https://tamnd.github.io/math2/node/68MA0RM5P6.md): concept 68MA0RM5P6
- [q-gamma function](https://tamnd.github.io/math2/node/68P2VKSEP5.md): concept 68P2VKSEP5
- [Retract](https://tamnd.github.io/math2/node/68PFSGP39B.md): concept 68PFSGP39B
- [Skolem's paradox](https://tamnd.github.io/math2/node/68QZWDXAQ4.md): example 68QZWDXAQ4
- [Null set](https://tamnd.github.io/math2/node/68RX8E3NQP.md): definition 68RX8E3NQP
- [Abel equation](https://tamnd.github.io/math2/node/68SHB7Z721.md): concept 68SHB7Z721
- [Automorphic form](https://tamnd.github.io/math2/node/68SJQEC11F.md): definition 68SJQEC11F
- [Gauss–Codazzi equations](https://tamnd.github.io/math2/node/68TVVHCQ3Z.md): statement 68TVVHCQ3Z
- [Direct product of algebras](https://tamnd.github.io/math2/node/68WXFBJEA6.md): definition 68WXFBJEA6
- [Dynkin system](https://tamnd.github.io/math2/node/69460Z6DMW.md): definition 69460Z6DMW
- [Wirtinger derivatives](https://tamnd.github.io/math2/node/698R12MY4G.md): definition 698R12MY4G
- [Mori domain](https://tamnd.github.io/math2/node/69HK75YAD1.md): concept 69HK75YAD1
- [Regular sequence](https://tamnd.github.io/math2/node/69R5R2FCS5.md): concept 69R5R2FCS5
- [Legendre functions](https://tamnd.github.io/math2/node/69XMERM6KJ.md): concept 69XMERM6KJ
- [Ordinal sum of posets](https://tamnd.github.io/math2/node/6AJ7QQCTQQ.md): concept 6AJ7QQCTQQ
- [Self-injective ring](https://tamnd.github.io/math2/node/6ANP035KG8.md): definition 6ANP035KG8
- [Free product of algebras](https://tamnd.github.io/math2/node/6AT64EQEJQ.md): definition 6AT64EQEJQ
- [Koszul duality](https://tamnd.github.io/math2/node/6AW49BZ8TM.md): concept 6AW49BZ8TM
- [Confluent hypergeometric function](https://tamnd.github.io/math2/node/6AZ4ZBN1M8.md): concept 6AZ4ZBN1M8
- [Simple random sample](https://tamnd.github.io/math2/node/6B05T6XHY0.md): definition 6B05T6XHY0
- [Directly indecomposable algebra](https://tamnd.github.io/math2/node/6B1PSZ42MC.md): concept 6B1PSZ42MC
- [Birkhoff's axioms](https://tamnd.github.io/math2/node/6BB0KD6752.md): concept 6BB0KD6752
- [Oka's solution of the Levi problem](https://tamnd.github.io/math2/node/6BJB149MDK.md): statement 6BJB149MDK
- [Kripke semantics](https://tamnd.github.io/math2/node/6BQ71KQ6QT.md): definition 6BQ71KQ6QT
- [Building](https://tamnd.github.io/math2/node/6BQJ50NEJ0.md): concept 6BQJ50NEJ0
- [F-coalgebra](https://tamnd.github.io/math2/node/6BS0300CMV.md): concept 6BS0300CMV
- [Shannon wavelet](https://tamnd.github.io/math2/node/6C1S8HD38Z.md): example 6C1S8HD38Z
- [Jordan algebra](https://tamnd.github.io/math2/node/6C2MCVB8C7.md): concept 6C2MCVB8C7
- [Frisch–Waugh–Lovell theorem](https://tamnd.github.io/math2/node/6C67F22FTD.md): statement 6C67F22FTD
- [False discovery rate and the Benjamini–Hochberg procedure](https://tamnd.github.io/math2/node/6CHHCTGQ18.md): concept 6CHHCTGQ18
- [Cancellation property](https://tamnd.github.io/math2/node/6CR9A0EWPN.md): concept 6CR9A0EWPN
- [Rogers polynomials](https://tamnd.github.io/math2/node/6CW61T5BB2.md): definition 6CW61T5BB2
- [Mapping cone](https://tamnd.github.io/math2/node/6CXPGB13WD.md): concept 6CXPGB13WD
- [Almost everywhere](https://tamnd.github.io/math2/node/6D0AVB7ZBR.md): definition 6D0AVB7ZBR
- [Rough path](https://tamnd.github.io/math2/node/6D10WYZR4F.md): definition 6D10WYZR4F
- [Right-angled Artin group](https://tamnd.github.io/math2/node/6D1DV09D6P.md): concept 6D1DV09D6P
- [Term](https://tamnd.github.io/math2/node/6D3WA7N2W4.md): concept 6D3WA7N2W4
- [Capacitability of Borel sets](https://tamnd.github.io/math2/node/6D4EW91AJY.md): statement 6D4EW91AJY
- [Zak transform](https://tamnd.github.io/math2/node/6D5Z4SWYMX.md): definition 6D5Z4SWYMX
- [Runge–Kutta–Fehlberg method](https://tamnd.github.io/math2/node/6D73FFD9DC.md): statement 6D73FFD9DC
- [Regular sequence](https://tamnd.github.io/math2/node/6D8RK0Q6QM.md): definition 6D8RK0Q6QM
- [Homoclinic orbit](https://tamnd.github.io/math2/node/6DBSQKZK4W.md): concept 6DBSQKZK4W
- [Runge's theorem](https://tamnd.github.io/math2/node/6DG4CTGXJD.md): statement 6DG4CTGXJD
- [Penny graph](https://tamnd.github.io/math2/node/6DG5BBQPRA.md): concept 6DG5BBQPRA
- [Five-term relation for the dilogarithm](https://tamnd.github.io/math2/node/6DHQTFAJ03.md): statement 6DHQTFAJ03
- [Concrete category of algebras](https://tamnd.github.io/math2/node/6DMND2YH1X.md): definition 6DMND2YH1X
- [Local asymptotic normality](https://tamnd.github.io/math2/node/6DMQ97EBVG.md): concept 6DMQ97EBVG
- [Ramanujan tau function](https://tamnd.github.io/math2/node/6DNZAGCSQS.md): concept 6DNZAGCSQS
- [Commutator theory](https://tamnd.github.io/math2/node/6DSWPHSWQW.md): concept 6DSWPHSWQW
- [Polar polytope](https://tamnd.github.io/math2/node/6DXF1T506B.md): concept 6DXF1T506B
- [Semi-local ring](https://tamnd.github.io/math2/node/6DYAGW68AV.md): concept 6DYAGW68AV
- [Linear PDE](https://tamnd.github.io/math2/node/6DZETZ5T9G.md): concept 6DZETZ5T9G
- [Erdős–Rényi model](https://tamnd.github.io/math2/node/6E83YKANV3.md): concept 6E83YKANV3
- [Weierstrass–Erdmann corner conditions](https://tamnd.github.io/math2/node/6E8KRJKXHS.md): statement 6E8KRJKXHS
- [Rearrangement inequality](https://tamnd.github.io/math2/node/6E9TAF8HCY.md): statement 6E9TAF8HCY
- [Discrete Poisson equation](https://tamnd.github.io/math2/node/6EEAEDWXZ2.md): concept 6EEAEDWXZ2
- [Poisson boundary](https://tamnd.github.io/math2/node/6EEF2E8XPF.md): concept 6EEF2E8XPF
- [Chow's theorem](https://tamnd.github.io/math2/node/6EFEG5VJEF.md): statement 6EFEG5VJEF
- [Essentially surjective functor](https://tamnd.github.io/math2/node/6ERBSBVF14.md): concept 6ERBSBVF14
- [Final topology](https://tamnd.github.io/math2/node/6ERJRRZK32.md): concept 6ERJRRZK32
- [Kähler–Einstein metric](https://tamnd.github.io/math2/node/6ETEBA8827.md): concept 6ETEBA8827
- [Instrumental variables estimation](https://tamnd.github.io/math2/node/6ETK64HYK6.md): concept 6ETK64HYK6
- [Algebraic torus](https://tamnd.github.io/math2/node/6EWEE3F9R8.md): definition 6EWEE3F9R8
- [Variety as a reflective subcategory](https://tamnd.github.io/math2/node/6EWEMZ6FX1.md): statement 6EWEMZ6FX1
- [Lévy–Steinitz theorem](https://tamnd.github.io/math2/node/6EXHAQF8SP.md): statement 6EXHAQF8SP
- [Fuchs' theorem](https://tamnd.github.io/math2/node/6EZ400WM94.md): statement 6EZ400WM94
- [Simplicial polytope](https://tamnd.github.io/math2/node/6F4JHFDM19.md): definition 6F4JHFDM19
- [Carlson's theorem](https://tamnd.github.io/math2/node/6F5B8KPEGH.md): statement 6F5B8KPEGH
- [Stationary set](https://tamnd.github.io/math2/node/6FDYJVG1CK.md): definition 6FDYJVG1CK
- [Calkin algebra](https://tamnd.github.io/math2/node/6FE7JZ2VF1.md): object 6FE7JZ2VF1
- [Arc in a finite plane](https://tamnd.github.io/math2/node/6FFVXT5KMH.md): concept 6FFVXT5KMH
- [Catastrophe theory](https://tamnd.github.io/math2/node/6FKJ6SXP3G.md): concept 6FKJ6SXP3G
- [Quasi-isometry](https://tamnd.github.io/math2/node/6FKYSH729T.md): concept 6FKYSH729T
- [Orthomodular lattice](https://tamnd.github.io/math2/node/6FPKEFX0X4.md): concept 6FPKEFX0X4
- [Isoperimetric problem of the calculus of variations](https://tamnd.github.io/math2/node/6FY0KHMK5A.md): problem 6FY0KHMK5A
- [Domain of holomorphy](https://tamnd.github.io/math2/node/6G0079XRKK.md): definition 6G0079XRKK
- [Co-Heyting algebra](https://tamnd.github.io/math2/node/6G03V0BVZT.md): concept 6G03V0BVZT
- [Preissmann's theorem](https://tamnd.github.io/math2/node/6G73DJH9T6.md): statement 6G73DJH9T6
- [Generic point](https://tamnd.github.io/math2/node/6GACD52XKX.md): definition 6GACD52XKX
- [Basu's theorem](https://tamnd.github.io/math2/node/6GATRFMJYY.md): statement 6GATRFMJYY
- [Group algebra of a locally compact group](https://tamnd.github.io/math2/node/6GD25S53ND.md): concept 6GD25S53ND
- [Cantor normal form](https://tamnd.github.io/math2/node/6GD30FGYDZ.md): statement 6GD30FGYDZ
- [Legendre integral transform](https://tamnd.github.io/math2/node/6GG8AS9XX4.md): definition 6GG8AS9XX4
- [Category of rings](https://tamnd.github.io/math2/node/6GSTSR4V09.md): example 6GSTSR4V09
- [QR algorithm](https://tamnd.github.io/math2/node/6GVPDQMBN7.md): definition 6GVPDQMBN7
- [Schubert calculus](https://tamnd.github.io/math2/node/6GW1FHF9MB.md): definition 6GW1FHF9MB
- [Hahn–Banach separation theorem](https://tamnd.github.io/math2/node/6GWMGFGFJJ.md): statement 6GWMGFGFJJ
- [Dolbeault cohomology](https://tamnd.github.io/math2/node/6GWRCGBGWC.md): definition 6GWRCGBGWC
- [Harmonic majorant](https://tamnd.github.io/math2/node/6GWZN8FXC6.md): definition 6GWZN8FXC6
- [Inverse scattering transform](https://tamnd.github.io/math2/node/6GY3X4Y8X5.md): concept 6GY3X4Y8X5
- [Riemann's differential equation](https://tamnd.github.io/math2/node/6GY4GAMY9Z.md): definition 6GY4GAMY9Z
- [Littlewood–Paley inequality](https://tamnd.github.io/math2/node/6GYQNB2862.md): statement 6GYQNB2862
- [Prandtl's lifting-line equation](https://tamnd.github.io/math2/node/6GZ8DXT7TN.md): example 6GZ8DXT7TN
- [Hecke algebra of a locally compact group](https://tamnd.github.io/math2/node/6H3WCCHZB5.md): concept 6H3WCCHZB5
- [Thom spectrum](https://tamnd.github.io/math2/node/6H8AGEN6NY.md): concept 6H8AGEN6NY
- [Frame](https://tamnd.github.io/math2/node/6HAZ0AKKAR.md): definition 6HAZ0AKKAR
- [Quantum dilogarithm](https://tamnd.github.io/math2/node/6HMT17DVX7.md): concept 6HMT17DVX7
- [Hilbert space adjoint](https://tamnd.github.io/math2/node/6HSJBJ8TTQ.md): concept 6HSJBJ8TTQ
- [Bump function](https://tamnd.github.io/math2/node/6HVAGKSQWB.md): definition 6HVAGKSQWB
- [Product sigma-algebra](https://tamnd.github.io/math2/node/6HVCTJGB61.md): concept 6HVCTJGB61
- [Complete set of orthogonal idempotents](https://tamnd.github.io/math2/node/6HVDTYAWX1.md): definition 6HVDTYAWX1
- [Circle map](https://tamnd.github.io/math2/node/6HVZ30TS5F.md): definition 6HVZ30TS5F
- [Schur complement](https://tamnd.github.io/math2/node/6HWFBXYPZP.md): concept 6HWFBXYPZP
- [Blocking set](https://tamnd.github.io/math2/node/6HYGN2WAQQ.md): definition 6HYGN2WAQQ
- [Bergman projection](https://tamnd.github.io/math2/node/6J1X63Z7QZ.md): concept 6J1X63Z7QZ
- [Two-sided Laplace transform](https://tamnd.github.io/math2/node/6J2AGD0BYY.md): concept 6J2AGD0BYY
- [Klein bottle](https://tamnd.github.io/math2/node/6J4C2P3ZHC.md): example 6J4C2P3ZHC
- [Period-doubling cascade](https://tamnd.github.io/math2/node/6J921PEDN2.md): concept 6J921PEDN2
- [Limit-point and limit-circle](https://tamnd.github.io/math2/node/6J9FXT6HAV.md): concept 6J9FXT6HAV
- [Local cohomology](https://tamnd.github.io/math2/node/6JB30BXDA2.md): concept 6JB30BXDA2
- [Semiconjugacy and factor map](https://tamnd.github.io/math2/node/6JBE71PZWA.md): definition 6JBE71PZWA
- [Hamilton–Jacobi theory of the calculus of variations](https://tamnd.github.io/math2/node/6JGM1K39FB.md): concept 6JGM1K39FB
- [Representation theory of the symmetric group](https://tamnd.github.io/math2/node/6JZ2P9MYRK.md): definition 6JZ2P9MYRK
- [Superalgebra](https://tamnd.github.io/math2/node/6JZW4JC68Z.md): concept 6JZW4JC68Z
- [Evans potential theorem](https://tamnd.github.io/math2/node/6K8GPMHWBB.md): statement 6K8GPMHWBB
- [Chattering control](https://tamnd.github.io/math2/node/6K8Q2YERTF.md): concept 6K8Q2YERTF
- [Stochastic optimal control](https://tamnd.github.io/math2/node/6KAVKR1HC8.md): definition 6KAVKR1HC8
- [Bargmann transform](https://tamnd.github.io/math2/node/6KBAARWWD6.md): definition 6KBAARWWD6
- [Sheffer function](https://tamnd.github.io/math2/node/6KFS240R42.md): definition 6KFS240R42
- [Charlier polynomials](https://tamnd.github.io/math2/node/6KH8KVYS3R.md): concept 6KH8KVYS3R
- [Stone space](https://tamnd.github.io/math2/node/6KQ6E3YHKR.md): definition 6KQ6E3YHKR
- [Vanishing mean oscillation](https://tamnd.github.io/math2/node/6KQCNP7P01.md): definition 6KQCNP7P01
- [Successive over-relaxation](https://tamnd.github.io/math2/node/6KQJR85ZC6.md): statement 6KQJR85ZC6
- [Natural deduction](https://tamnd.github.io/math2/node/6KT3BJZ7D1.md): concept 6KT3BJZ7D1
- [Almgren–Pitts min-max theory](https://tamnd.github.io/math2/node/6KVJHG1KRZ.md): concept 6KVJHG1KRZ
- [Bass–Serre theory](https://tamnd.github.io/math2/node/6M731X66RD.md): concept 6M731X66RD
- [Submartingale](https://tamnd.github.io/math2/node/6M852B4RA8.md): concept 6M852B4RA8
- [Homogeneity equation](https://tamnd.github.io/math2/node/6MB2G4T913.md): definition 6MB2G4T913
- [Jacobson topology](https://tamnd.github.io/math2/node/6MFRP922BR.md): definition 6MFRP922BR
- [Triangulation of a point set](https://tamnd.github.io/math2/node/6MFS8FEYHD.md): definition 6MFS8FEYHD
- [Butcher group](https://tamnd.github.io/math2/node/6MH1DHMCPF.md): object 6MH1DHMCPF
- [Homotopy type](https://tamnd.github.io/math2/node/6MJH22BCAE.md): definition 6MJH22BCAE
- [Adaptive step size](https://tamnd.github.io/math2/node/6MREWBZR7R.md): concept 6MREWBZR7R
- [Spectral decomposition theorem](https://tamnd.github.io/math2/node/6MTVY6NYY7.md): statement 6MTVY6NYY7
- [Chain recurrence](https://tamnd.github.io/math2/node/6MX5R482KJ.md): concept 6MX5R482KJ
- [Factor congruence](https://tamnd.github.io/math2/node/6N268PHKQA.md): concept 6N268PHKQA
- [Witt vectors](https://tamnd.github.io/math2/node/6N3X2QX4W9.md): concept 6N3X2QX4W9
- [Singular cohomology](https://tamnd.github.io/math2/node/6N7BC70CX0.md): concept 6N7BC70CX0
- [Incompressible surface](https://tamnd.github.io/math2/node/6N9GW98MR6.md): concept 6N9GW98MR6
- [Catenoid](https://tamnd.github.io/math2/node/6NN1HP54GB.md): example 6NN1HP54GB
- [Whitehead theorem](https://tamnd.github.io/math2/node/6P1E2FJPSD.md): statement 6P1E2FJPSD
- [Discontinuous Galerkin method](https://tamnd.github.io/math2/node/6P5MCRDD9Q.md): concept 6P5MCRDD9Q
- [Amenable Banach algebra](https://tamnd.github.io/math2/node/6P9AE21Q7X.md): concept 6P9AE21Q7X
- [Differential of a smooth map](https://tamnd.github.io/math2/node/6P9PXQDSEN.md): concept 6P9PXQDSEN
- [Torsion submodule](https://tamnd.github.io/math2/node/6PBS6KW0NC.md): concept 6PBS6KW0NC
- [Collatz conjecture](https://tamnd.github.io/math2/node/6PCQEXEX4N.md): conjecture 6PCQEXEX4N
- [Lie bracket of vector fields](https://tamnd.github.io/math2/node/6PDWHWQQ11.md): definition 6PDWHWQQ11
- [Kakeya set](https://tamnd.github.io/math2/node/6PEN7AKC7M.md): object 6PEN7AKC7M
- [Banach bundle](https://tamnd.github.io/math2/node/6PHR16TB0V.md): definition 6PHR16TB0V
- [Stein's lemma](https://tamnd.github.io/math2/node/6PWC83R457.md): statement 6PWC83R457
- [Lefschetz number](https://tamnd.github.io/math2/node/6PWN57BW45.md): definition 6PWN57BW45
- [d'Alembert's functional equation](https://tamnd.github.io/math2/node/6PXHVPFTAS.md): concept 6PXHVPFTAS
- [Reduction of a boundary value problem to a Fredholm equation](https://tamnd.github.io/math2/node/6Q36P44PHZ.md): statement 6Q36P44PHZ
- [Induced representation of a locally compact group](https://tamnd.github.io/math2/node/6Q7AHAZ6Q4.md): concept 6Q7AHAZ6Q4
- [Population balance equation](https://tamnd.github.io/math2/node/6Q7H2FQ12Q.md): example 6Q7H2FQ12Q
- [Monoid ring](https://tamnd.github.io/math2/node/6Q96TKGZX2.md): definition 6Q96TKGZX2
- [Aleph number](https://tamnd.github.io/math2/node/6Q9Z2BM1XR.md): concept 6Q9Z2BM1XR
- [HOMFLY polynomial](https://tamnd.github.io/math2/node/6QAPNQVBEK.md): definition 6QAPNQVBEK
- [Jensen's functional equation](https://tamnd.github.io/math2/node/6QB8J65RVV.md): concept 6QB8J65RVV
- [Finite Desarguesian planes are Pappian](https://tamnd.github.io/math2/node/6QBGAKFB9N.md): statement 6QBGAKFB9N
- [Constructive function theory](https://tamnd.github.io/math2/node/6QDK0N4KQH.md): definition 6QDK0N4KQH
- [Dual matroid](https://tamnd.github.io/math2/node/6QF83CFK2G.md): concept 6QF83CFK2G
- [Veblen function](https://tamnd.github.io/math2/node/6QFEVWKE25.md): definition 6QFEVWKE25
- [Double bubble theorem](https://tamnd.github.io/math2/node/6QG8QNHBF4.md): statement 6QG8QNHBF4
- [Variational inequality](https://tamnd.github.io/math2/node/6QS7X5GVMY.md): problem 6QS7X5GVMY
- [Holomorphic automorphisms of C^n](https://tamnd.github.io/math2/node/6QW3MQC3SK.md): definition 6QW3MQC3SK
- [Axiom of regularity](https://tamnd.github.io/math2/node/6QXBY7J0M5.md): definition 6QXBY7J0M5
- [Mahler's compactness theorem](https://tamnd.github.io/math2/node/6QY76Q5WVS.md): statement 6QY76Q5WVS
- [Mal'cev condition](https://tamnd.github.io/math2/node/6R0CA4JWWY.md): concept 6R0CA4JWWY
- [Hardy spaces of harmonic functions](https://tamnd.github.io/math2/node/6R2ZXY5VZW.md): concept 6R2ZXY5VZW
- [Ring spectrum](https://tamnd.github.io/math2/node/6R57T5W7BV.md): definition 6R57T5W7BV
- [Mutual energy](https://tamnd.github.io/math2/node/6R92RBM72R.md): definition 6R92RBM72R
- [Birkhoff's representation theorem](https://tamnd.github.io/math2/node/6RDM3069KC.md): statement 6RDM3069KC
- [Fukaya category](https://tamnd.github.io/math2/node/6RE2JMYMA0.md): definition 6RE2JMYMA0
- [Linear topology](https://tamnd.github.io/math2/node/6RF8CB69KQ.md): concept 6RF8CB69KQ
- [Atom](https://tamnd.github.io/math2/node/6RJVSY3PQ0.md): definition 6RJVSY3PQ0
- [Berkovich space dynamics](https://tamnd.github.io/math2/node/6RM4RWWSSX.md): concept 6RM4RWWSSX
- [Immersion](https://tamnd.github.io/math2/node/6RW68JGBHC.md): concept 6RW68JGBHC
- [Kummer theory](https://tamnd.github.io/math2/node/6RXEC9CT6M.md): definition 6RXEC9CT6M
- [Hopfian group](https://tamnd.github.io/math2/node/6S4CM98021.md): definition 6S4CM98021
- [Gagliardo–Nirenberg interpolation inequality](https://tamnd.github.io/math2/node/6SK4PQPD3C.md): statement 6SK4PQPD3C
- [Einstein manifold](https://tamnd.github.io/math2/node/6SNCVN8JZQ.md): definition 6SNCVN8JZQ
- [Rigged Hilbert space](https://tamnd.github.io/math2/node/6SPMVHRPM2.md): concept 6SPMVHRPM2
- [Fundamental group of the circle](https://tamnd.github.io/math2/node/6SQXPE0JPN.md): statement 6SQXPE0JPN
- [Descending chain condition](https://tamnd.github.io/math2/node/6SVWAP8WP5.md): concept 6SVWAP8WP5
- [Algebraic K-theory](https://tamnd.github.io/math2/node/6SX85ZKBNJ.md): definition 6SX85ZKBNJ
- [Triangulability of analytic sets](https://tamnd.github.io/math2/node/6SZDXD3KYH.md): statement 6SZDXD3KYH
- [Bertrand's test](https://tamnd.github.io/math2/node/6T3E3AXZT7.md): statement 6T3E3AXZT7
- [Projective dimension](https://tamnd.github.io/math2/node/6T74XFKWRA.md): definition 6T74XFKWRA
- [Constructible universe](https://tamnd.github.io/math2/node/6TF2JBTPPJ.md): object 6TF2JBTPPJ
- [Krylov–Bogolyubov averaging](https://tamnd.github.io/math2/node/6TJ7YHP2HE.md): concept 6TJ7YHP2HE
- [Newtonian kernel](https://tamnd.github.io/math2/node/6TJTWQ5HCR.md): concept 6TJTWQ5HCR
- [Homology manifold](https://tamnd.github.io/math2/node/6TKGN8K8KT.md): concept 6TKGN8K8KT
- [Variogram](https://tamnd.github.io/math2/node/6TP0G4DKW2.md): definition 6TP0G4DKW2
- [Young measure](https://tamnd.github.io/math2/node/6TQSVJQDEH.md): concept 6TQSVJQDEH
- [Modica–Mortola theorem](https://tamnd.github.io/math2/node/6TSMHGCXVM.md): statement 6TSMHGCXVM
- [Dynamical Mordell–Lang conjecture](https://tamnd.github.io/math2/node/6TV7FR553H.md): conjecture 6TV7FR553H
- [Fourier transform](https://tamnd.github.io/math2/node/6V59T5KJBX.md): definition 6V59T5KJBX
- [Ideal of an algebraic set](https://tamnd.github.io/math2/node/6V6QPT0B3Y.md): definition 6V6QPT0B3Y
- [Kobayashi hyperbolic manifold](https://tamnd.github.io/math2/node/6V6WDRHRTF.md): definition 6V6WDRHRTF
- [Robbins–Monro algorithm](https://tamnd.github.io/math2/node/6V75QVQCX6.md): definition 6V75QVQCX6
- [Compressed sensing](https://tamnd.github.io/math2/node/6V8E0AQNHA.md): concept 6V8E0AQNHA
- [Sheaf of differentials](https://tamnd.github.io/math2/node/6VC6HQTFSA.md): concept 6VC6HQTFSA
- [Cesàro summation](https://tamnd.github.io/math2/node/6VC9JQD45M.md): definition 6VC9JQD45M
- [Peter–Weyl theorem](https://tamnd.github.io/math2/node/6VC9QE39XH.md): statement 6VC9QE39XH
- [Convex cone](https://tamnd.github.io/math2/node/6VE4KA12VD.md): concept 6VE4KA12VD
- [Embedding theorem for Stein manifolds](https://tamnd.github.io/math2/node/6VF47Q35YA.md): statement 6VF47Q35YA
- [Pseudo-Riemannian manifold](https://tamnd.github.io/math2/node/6VFADXR4MW.md): definition 6VFADXR4MW
- [The left cosets of a subgroup of order two in S3](https://tamnd.github.io/math2/node/6VFF2TTXYZ.md): example 6VFF2TTXYZ
- [Kolmogorov's weak-type estimate for the conjugate function](https://tamnd.github.io/math2/node/6VJDJZAYKA.md): statement 6VJDJZAYKA
- [Feit–Thompson theorem](https://tamnd.github.io/math2/node/6VK9RSG13Q.md): statement 6VK9RSG13Q
- [Caputo fractional derivative](https://tamnd.github.io/math2/node/6VMGQYF8RW.md): concept 6VMGQYF8RW
- [Mutual energy](https://tamnd.github.io/math2/node/6VR8GASZJY.md): concept 6VR8GASZJY
- [Ring of invariants](https://tamnd.github.io/math2/node/6VTR5EBNYC.md): concept 6VTR5EBNYC
- [Markov random field](https://tamnd.github.io/math2/node/6W69H3A3G5.md): concept 6W69H3A3G5
- [Method of fundamental solutions](https://tamnd.github.io/math2/node/6W72QW363B.md): concept 6W72QW363B
- [Genus of a curve](https://tamnd.github.io/math2/node/6W8TDV9VVN.md): concept 6W8TDV9VVN
- [Theta correspondence](https://tamnd.github.io/math2/node/6WB6TKE70H.md): definition 6WB6TKE70H
- [Holomorphic dynamics in several variables](https://tamnd.github.io/math2/node/6WCRMYEYF9.md): concept 6WCRMYEYF9
- [Gamma function as Mellin transform of the exponential](https://tamnd.github.io/math2/node/6WGT43N6KT.md): example 6WGT43N6KT
- [Determinantal variety](https://tamnd.github.io/math2/node/6WH08A4BTK.md): concept 6WH08A4BTK
- [Contiguity of measures](https://tamnd.github.io/math2/node/6WTHKQG40F.md): definition 6WTHKQG40F
- [Klein quadric](https://tamnd.github.io/math2/node/6WV71T5CK1.md): object 6WV71T5CK1
- [Kähler–Einstein metric](https://tamnd.github.io/math2/node/6WVTB9Y3XD.md): definition 6WVTB9Y3XD
- [Exotic sphere](https://tamnd.github.io/math2/node/6WYEMXSHD9.md): object 6WYEMXSHD9
- [Fundamental recurrence formulas](https://tamnd.github.io/math2/node/6WYYMPH21Q.md): statement 6WYYMPH21Q
- [Analyticity of the Laplace transform in its half-plane](https://tamnd.github.io/math2/node/6X05N89XDP.md): statement 6X05N89XDP
- [Wynn's epsilon algorithm](https://tamnd.github.io/math2/node/6X0C4HNMAZ.md): concept 6X0C4HNMAZ
- [Invariance of domain](https://tamnd.github.io/math2/node/6X2D0JDFQW.md): statement 6X2D0JDFQW
- [Abel–Poisson summation of Fourier series](https://tamnd.github.io/math2/node/6X94NPK782.md): definition 6X94NPK782
- [Hadamard conjecture](https://tamnd.github.io/math2/node/6X9S10E3M2.md): conjecture 6X9S10E3M2
- [Normal closure](https://tamnd.github.io/math2/node/6XM4CCVTKE.md): concept 6XM4CCVTKE
- [Weighted polynomial approximation](https://tamnd.github.io/math2/node/6XQYHACGT0.md): concept 6XQYHACGT0
- [Dimension of an algebraic variety](https://tamnd.github.io/math2/node/6Y0Y9JRXSK.md): definition 6Y0Y9JRXSK
- [Degenerate kernel](https://tamnd.github.io/math2/node/6Y1RY5767R.md): concept 6Y1RY5767R
- [Order statistic](https://tamnd.github.io/math2/node/6Y451VYFV9.md): definition 6Y451VYFV9
- [Schwarzschild metric](https://tamnd.github.io/math2/node/6Y50T68WKY.md): example 6Y50T68WKY
- [Generalized convolution of an integral transform](https://tamnd.github.io/math2/node/6Y7NJCXXTN.md): definition 6Y7NJCXXTN
- [Rees algebra](https://tamnd.github.io/math2/node/6YCKECNR3T.md): definition 6YCKECNR3T
- [p-adic gamma function](https://tamnd.github.io/math2/node/6YD7S3C7W7.md): definition 6YD7S3C7W7
- [Barrelled space](https://tamnd.github.io/math2/node/6YDMMN6M9R.md): definition 6YDMMN6M9R
- [Topological monoid](https://tamnd.github.io/math2/node/6YJ77MYM0Q.md): definition 6YJ77MYM0Q
- [Lauricella hypergeometric series](https://tamnd.github.io/math2/node/6YSVDFEM39.md): concept 6YSVDFEM39
- [Abel's lemma for multiple power series](https://tamnd.github.io/math2/node/6YT8ZM6B0F.md): statement 6YT8ZM6B0F
- [Laurent phenomenon](https://tamnd.github.io/math2/node/6YYPNKK6WZ.md): statement 6YYPNKK6WZ
- [6-j symbol](https://tamnd.github.io/math2/node/6Z33W3D482.md): definition 6Z33W3D482
- [Laplace transform of distributions](https://tamnd.github.io/math2/node/6Z3XJVYT2Q.md): concept 6Z3XJVYT2Q
- [Homogeneous ideal](https://tamnd.github.io/math2/node/6ZBNGS0P98.md): concept 6ZBNGS0P98
- [Tauberian condition](https://tamnd.github.io/math2/node/6ZDP261K7R.md): concept 6ZDP261K7R
- [Besicovitch covering theorem](https://tamnd.github.io/math2/node/6ZDSVX9GAR.md): statement 6ZDSVX9GAR
- [Hasse–Weil zeta function](https://tamnd.github.io/math2/node/6ZGF49H346.md): definition 6ZGF49H346
- [Riesz space](https://tamnd.github.io/math2/node/6ZMQ1XZ44B.md): definition 6ZMQ1XZ44B
- [Divergent asymptotic series](https://tamnd.github.io/math2/node/6ZP4DJ7JMF.md): example 6ZP4DJ7JMF
- [Schwarzian derivative](https://tamnd.github.io/math2/node/6ZPKWE0VAZ.md): concept 6ZPKWE0VAZ
- [Volterra integro-differential equation](https://tamnd.github.io/math2/node/6ZQRVRD8KP.md): definition 6ZQRVRD8KP
- [Balanced set](https://tamnd.github.io/math2/node/6ZQYHVGE1S.md): definition 6ZQYHVGE1S
- [Nine lemma](https://tamnd.github.io/math2/node/6ZRRH96VZT.md): statement 6ZRRH96VZT
- [Product](https://tamnd.github.io/math2/node/6ZS77J1873.md): definition 6ZS77J1873
- [Orientability](https://tamnd.github.io/math2/node/702SNFQG0W.md): concept 702SNFQG0W
- [Seifert fiber space](https://tamnd.github.io/math2/node/705CC9AZCG.md): definition 705CC9AZCG
- [Gauge group](https://tamnd.github.io/math2/node/707DEWCCFX.md): object 707DEWCCFX
- [James–Stein estimator](https://tamnd.github.io/math2/node/709T196XCT.md): concept 709T196XCT
- [Summation of families over arbitrary index sets](https://tamnd.github.io/math2/node/70HJTK5VZH.md): concept 70HJTK5VZH
- [Hurewicz fibration](https://tamnd.github.io/math2/node/70KRD7X6SY.md): definition 70KRD7X6SY
- [Convergence of filters](https://tamnd.github.io/math2/node/70M1VAPFN0.md): concept 70M1VAPFN0
- [Building](https://tamnd.github.io/math2/node/70Q4YVF322.md): definition 70Q4YVF322
- [Jónsson–Tarski algebra](https://tamnd.github.io/math2/node/70QWWYQQG9.md): example 70QWWYQQG9
- [Inverse iteration](https://tamnd.github.io/math2/node/70QYCEWHVY.md): statement 70QYCEWHVY
- [Virtual knot](https://tamnd.github.io/math2/node/70ZP456XG6.md): concept 70ZP456XG6
- [Kernel regression](https://tamnd.github.io/math2/node/710VV73909.md): concept 710VV73909
- [Entropic algebra](https://tamnd.github.io/math2/node/71346N77AX.md): definition 71346N77AX
- [Ginzburg–Landau equation](https://tamnd.github.io/math2/node/713MRSZ9XA.md): concept 713MRSZ9XA
- [Dirichlet process](https://tamnd.github.io/math2/node/7174H28FT6.md): definition 7174H28FT6
- [Alexandrov space](https://tamnd.github.io/math2/node/71AWVBG6KH.md): concept 71AWVBG6KH
- [FK-space](https://tamnd.github.io/math2/node/71DMQHW6JZ.md): concept 71DMQHW6JZ
- [Sparse matrix](https://tamnd.github.io/math2/node/71K5G4GNNZ.md): definition 71K5G4GNNZ
- [Massey product](https://tamnd.github.io/math2/node/71M4GJKN0N.md): definition 71M4GJKN0N
- [Polytope combinatorics](https://tamnd.github.io/math2/node/71VK7M83WS.md): definition 71VK7M83WS
- [Kiefer–Wolfowitz equivalence theorem](https://tamnd.github.io/math2/node/71XBFB49HE.md): statement 71XBFB49HE
- [Table of Lie groups](https://tamnd.github.io/math2/node/720FQEBT68.md): example 720FQEBT68
- [Immersed boundary method](https://tamnd.github.io/math2/node/720QQR553X.md): concept 720QQR553X
- [Polarity](https://tamnd.github.io/math2/node/727TZQJ4ZA.md): concept 727TZQJ4ZA
- [Kontsevich integral](https://tamnd.github.io/math2/node/727ZEPVSTC.md): concept 727ZEPVSTC
- [Fully invariant congruence](https://tamnd.github.io/math2/node/72JSAPJB96.md): definition 72JSAPJB96
- [Global dimension](https://tamnd.github.io/math2/node/72K5F9VPCR.md): definition 72K5F9VPCR
- [Ternary discriminator](https://tamnd.github.io/math2/node/72PG6XCQ7D.md): concept 72PG6XCQ7D
- [Equational logic](https://tamnd.github.io/math2/node/72S3YSQFZ7.md): concept 72S3YSQFZ7
- [Absolute neighborhood retract](https://tamnd.github.io/math2/node/72SN9MMH5Y.md): definition 72SN9MMH5Y
- [Jacobi transform](https://tamnd.github.io/math2/node/72TSY1Z4VA.md): concept 72TSY1Z4VA
- [Bounded lattice](https://tamnd.github.io/math2/node/72V1Z82M5F.md): concept 72V1Z82M5F
- [Splitting of prime ideals](https://tamnd.github.io/math2/node/72W4HXM98H.md): definition 72W4HXM98H
- [Explicit formula](https://tamnd.github.io/math2/node/72WNRWJQC1.md): statement 72WNRWJQC1
- [Randomization](https://tamnd.github.io/math2/node/734Q6Q51X7.md): concept 734Q6Q51X7
- [Fredholm property of elliptic operators](https://tamnd.github.io/math2/node/73ANTTKAH3.md): statement 73ANTTKAH3
- [Lattice of clones](https://tamnd.github.io/math2/node/73DXSAH73B.md): definition 73DXSAH73B
- [Hawaiian earring](https://tamnd.github.io/math2/node/73EBKQPP8Q.md): example 73EBKQPP8Q
- [Flat module](https://tamnd.github.io/math2/node/73RYYEVDKA.md): concept 73RYYEVDKA
- [Elliptic surface](https://tamnd.github.io/math2/node/73SAQBFWKQ.md): definition 73SAQBFWKQ
- [Five-point stencil](https://tamnd.github.io/math2/node/73WPYFAEWS.md): concept 73WPYFAEWS
- [Pick matrix](https://tamnd.github.io/math2/node/73X2JGSE0W.md): definition 73X2JGSE0W
- [Enneper surface](https://tamnd.github.io/math2/node/73XN6AS0WX.md): example 73XN6AS0WX
- [Field trace](https://tamnd.github.io/math2/node/73XV4RP5PT.md): definition 73XV4RP5PT
- [Hardy–Littlewood maximal function](https://tamnd.github.io/math2/node/744826NNN9.md): definition 744826NNN9
- [Unitary representation](https://tamnd.github.io/math2/node/74BPYJNZTM.md): concept 74BPYJNZTM
- [Linear canonical transformation](https://tamnd.github.io/math2/node/74BXAPVSW8.md): concept 74BXAPVSW8
- [Euler–Poisson equation for higher derivatives](https://tamnd.github.io/math2/node/74DV24XGKZ.md): definition 74DV24XGKZ
- [Eulerian poset](https://tamnd.github.io/math2/node/74JREVZVY3.md): definition 74JREVZVY3
- [Continuous multiplication](https://tamnd.github.io/math2/node/74TDTC7ETT.md): definition 74TDTC7ETT
- [NIP theory](https://tamnd.github.io/math2/node/74VCDYQDGP.md): concept 74VCDYQDGP
- [Lebesgue decomposition theorem](https://tamnd.github.io/math2/node/74VMD07DRX.md): statement 74VMD07DRX
- [Semilattice](https://tamnd.github.io/math2/node/74X7JRG2J7.md): concept 74X7JRG2J7
- [Strong Cesàro summability](https://tamnd.github.io/math2/node/750MCA763X.md): concept 750MCA763X
- [Finite difference coefficients](https://tamnd.github.io/math2/node/752Z7DRT7P.md): concept 752Z7DRT7P
- [Turing machine](https://tamnd.github.io/math2/node/758EYCN27B.md): concept 758EYCN27B
- [Summation in topological groups](https://tamnd.github.io/math2/node/759C4C6NXZ.md): definition 759C4C6NXZ
- [Advanced Z-transform](https://tamnd.github.io/math2/node/759JMPAZCZ.md): definition 759JMPAZCZ
- [Geometric lattice](https://tamnd.github.io/math2/node/759KSC8XQD.md): definition 759KSC8XQD
- [Iterated function system](https://tamnd.github.io/math2/node/75B8VHDGSA.md): concept 75B8VHDGSA
- [Complete Boolean algebra](https://tamnd.github.io/math2/node/75BPV95AAS.md): concept 75BPV95AAS
- [Cofinal subset](https://tamnd.github.io/math2/node/75EJ0HB40M.md): definition 75EJ0HB40M
- [Classical group](https://tamnd.github.io/math2/node/75H4HY10Z4.md): concept 75H4HY10Z4
- [Lindley's paradox](https://tamnd.github.io/math2/node/75P3EJPH81.md): counterexample 75P3EJPH81
- [Real Hardy space H1](https://tamnd.github.io/math2/node/75WGJ014GR.md): concept 75WGJ014GR
- [Padé approximant](https://tamnd.github.io/math2/node/75XCD68557.md): definition 75XCD68557
- [Dirichlet conditions](https://tamnd.github.io/math2/node/75ZQW12WY0.md): statement 75ZQW12WY0
- [Egorov's theorem](https://tamnd.github.io/math2/node/762Y2NYA7W.md): statement 762Y2NYA7W
- [Finitely presented algebra](https://tamnd.github.io/math2/node/764GZHTGV8.md): concept 764GZHTGV8
- [Trigonometric interpolation](https://tamnd.github.io/math2/node/765P54K9BR.md): definition 765P54K9BR
- [Splitting circle method](https://tamnd.github.io/math2/node/7680AP09J4.md): statement 7680AP09J4
- [Projection operation](https://tamnd.github.io/math2/node/769B35T6WG.md): concept 769B35T6WG
- [Residually finite algebra](https://tamnd.github.io/math2/node/76BXV761BR.md): definition 76BXV761BR
- [Cellular algebra](https://tamnd.github.io/math2/node/76DF4G7J9S.md): concept 76DF4G7J9S
- [Rational approximation](https://tamnd.github.io/math2/node/76JD0ZG3B1.md): definition 76JD0ZG3B1
- [Tychonoff space](https://tamnd.github.io/math2/node/76M4XEGPAB.md): definition 76M4XEGPAB
- [Solution of recurrences by generating functions](https://tamnd.github.io/math2/node/76NQXSGSHS.md): definition 76NQXSGSHS
- [Positive operator](https://tamnd.github.io/math2/node/77020CY5ZW.md): concept 77020CY5ZW
- [Real closure](https://tamnd.github.io/math2/node/778KFSM09P.md): concept 778KFSM09P
- [Model theory of algebraically closed fields](https://tamnd.github.io/math2/node/778ZG6P6E6.md): definition 778ZG6P6E6
- [Nerve of a category](https://tamnd.github.io/math2/node/779ZSEF36C.md): definition 779ZSEF36C
- [Kelvin functions](https://tamnd.github.io/math2/node/77BS2B0A6Q.md): concept 77BS2B0A6Q
- [Universally measurable set](https://tamnd.github.io/math2/node/77CQPXNTRW.md): definition 77CQPXNTRW
- [Point process](https://tamnd.github.io/math2/node/77CVEF0CNP.md): definition 77CVEF0CNP
- [Coupled map lattice](https://tamnd.github.io/math2/node/77JZ2QRZ4M.md): concept 77JZ2QRZ4M
- [Künneth theorem](https://tamnd.github.io/math2/node/77K0BZQ4MK.md): statement 77K0BZQ4MK
- [Orders of the elements of the integers modulo 6](https://tamnd.github.io/math2/node/77N4TVRM15.md): example 77N4TVRM15
- [Multivariate approximation](https://tamnd.github.io/math2/node/77NG85NDQM.md): definition 77NG85NDQM
- [Product measure](https://tamnd.github.io/math2/node/77PV0ZAVXT.md): concept 77PV0ZAVXT
- [Capacity and Hausdorff dimension](https://tamnd.github.io/math2/node/77VNNV4VET.md): statement 77VNNV4VET
- [Complete local ring](https://tamnd.github.io/math2/node/78GT1VFFPR.md): definition 78GT1VFFPR
- [Fundamental theorem of Riemannian geometry](https://tamnd.github.io/math2/node/78JRW8KTH8.md): statement 78JRW8KTH8
- [Calderón–Zygmund kernel](https://tamnd.github.io/math2/node/78N6AXP9VV.md): concept 78N6AXP9VV
- [Jordan's theorem on primitive groups containing a prime cycle](https://tamnd.github.io/math2/node/78R2TV41D4.md): statement 78R2TV41D4
- [Artin reciprocity law](https://tamnd.github.io/math2/node/7938EKMMVR.md): statement 7938EKMMVR
- [Riemann curvature tensor](https://tamnd.github.io/math2/node/793SSEV9ED.md): definition 793SSEV9ED
- [Annihilator method](https://tamnd.github.io/math2/node/794N251WY8.md): concept 794N251WY8
- [Summability field of a matrix method](https://tamnd.github.io/math2/node/799DX4375M.md): concept 799DX4375M
- [Gelfand–Naimark–Segal construction](https://tamnd.github.io/math2/node/79DH4GBCZ5.md): definition 79DH4GBCZ5
- [Falling and rising factorials](https://tamnd.github.io/math2/node/79M1TG6F5N.md): definition 79M1TG6F5N
- [Space of functions vanishing at infinity](https://tamnd.github.io/math2/node/79X8DJZRBD.md): definition 79X8DJZRBD
- [M-estimator](https://tamnd.github.io/math2/node/7A55AWN7EV.md): definition 7A55AWN7EV
- [Kohn–Rossi cohomology](https://tamnd.github.io/math2/node/7A67N3PVQG.md): definition 7A67N3PVQG
- [Rogers–Ramanujan continued fraction](https://tamnd.github.io/math2/node/7A68BZR6QS.md): concept 7A68BZR6QS
- [Restricted isometry property](https://tamnd.github.io/math2/node/7A9WVPD9DG.md): definition 7A9WVPD9DG
- [Witt group](https://tamnd.github.io/math2/node/7ABT36371X.md): definition 7ABT36371X
- [Weak dimension](https://tamnd.github.io/math2/node/7ADX0MGPNV.md): definition 7ADX0MGPNV
- [Morse–Smale system](https://tamnd.github.io/math2/node/7AMQ8GS78Y.md): concept 7AMQ8GS78Y
- [Oka–Weil theorem](https://tamnd.github.io/math2/node/7AWZJNQGT2.md): statement 7AWZJNQGT2
- [Oval](https://tamnd.github.io/math2/node/7B03TB64N9.md): concept 7B03TB64N9
- [Brouwer–Heyting–Kolmogorov interpretation](https://tamnd.github.io/math2/node/7B53ZGXXEQ.md): concept 7B53ZGXXEQ
- [Euler transform](https://tamnd.github.io/math2/node/7BGRWJGYWB.md): definition 7BGRWJGYWB
- [Darboux function](https://tamnd.github.io/math2/node/7BJEH1GCTE.md): concept 7BJEH1GCTE
- [p-harmonic function](https://tamnd.github.io/math2/node/7BKQT71FYZ.md): concept 7BKQT71FYZ
- [Steiner system](https://tamnd.github.io/math2/node/7BPSC4T4M3.md): concept 7BPSC4T4M3
- [Fixed-point iteration](https://tamnd.github.io/math2/node/7BVX2KF3QC.md): definition 7BVX2KF3QC
- [Harnack inequality for parabolic equations](https://tamnd.github.io/math2/node/7BY5D2THKW.md): statement 7BY5D2THKW
- [Einstein field equations](https://tamnd.github.io/math2/node/7C2WP33E10.md): definition 7C2WP33E10
- [Nonlinear difference equation](https://tamnd.github.io/math2/node/7C7FBZGERM.md): concept 7C7FBZGERM
- [Chebyshev polynomials of the second kind](https://tamnd.github.io/math2/node/7CATGEN6KH.md): concept 7CATGEN6KH
- [Riemannian manifold](https://tamnd.github.io/math2/node/7CBT4SM83Y.md): concept 7CBT4SM83Y
- [Modified Richardson iteration](https://tamnd.github.io/math2/node/7CESYMHQEA.md): statement 7CESYMHQEA
- [Subalgebra](https://tamnd.github.io/math2/node/7CHG6SD365.md): concept 7CHG6SD365
- [Theta function](https://tamnd.github.io/math2/node/7CHQXJD8HT.md): definition 7CHQXJD8HT
- [Ehresmann's fibration theorem](https://tamnd.github.io/math2/node/7CJ3P443NQ.md): statement 7CJ3P443NQ
- [A priori estimate](https://tamnd.github.io/math2/node/7CT1JTB4JJ.md): concept 7CT1JTB4JJ
- [Semimodular lattice](https://tamnd.github.io/math2/node/7CV8KGA1KM.md): definition 7CV8KGA1KM
- [Arithmetical hierarchy](https://tamnd.github.io/math2/node/7CVJCQKY41.md): concept 7CVJCQKY41
- [Frobenius method](https://tamnd.github.io/math2/node/7D47PADY9N.md): definition 7D47PADY9N
- [Torus integral](https://tamnd.github.io/math2/node/7D7XWA8WWJ.md): definition 7D7XWA8WWJ
- [Runge–Kutta methods](https://tamnd.github.io/math2/node/7DBPJKYC6C.md): concept 7DBPJKYC6C
- [Integral scheme](https://tamnd.github.io/math2/node/7DBY85BVH8.md): definition 7DBY85BVH8
- [Dixon elliptic functions](https://tamnd.github.io/math2/node/7DCBEGJX3E.md): definition 7DCBEGJX3E
- [Kunen's inconsistency theorem](https://tamnd.github.io/math2/node/7DEZD5TWT5.md): statement 7DEZD5TWT5
- [Congruence generated by a relation](https://tamnd.github.io/math2/node/7DFJTZR3XJ.md): definition 7DFJTZR3XJ
- [Thin set](https://tamnd.github.io/math2/node/7DGF6MMEGX.md): concept 7DGF6MMEGX
- [Wreath product](https://tamnd.github.io/math2/node/7DH8HG1SXH.md): definition 7DH8HG1SXH
- [Purely transcendental extension](https://tamnd.github.io/math2/node/7DKKTMPE6V.md): definition 7DKKTMPE6V
- [Freudenthal suspension theorem](https://tamnd.github.io/math2/node/7DM2KSS2DB.md): statement 7DM2KSS2DB
- [Optimal truncation](https://tamnd.github.io/math2/node/7DMEGGFD0J.md): definition 7DMEGGFD0J
- [Haken manifold](https://tamnd.github.io/math2/node/7DP7H8M5B2.md): concept 7DP7H8M5B2
- [Manifold with boundary](https://tamnd.github.io/math2/node/7DS6B2BFK8.md): concept 7DS6B2BFK8
- [Elementary class](https://tamnd.github.io/math2/node/7DWHFWF1D8.md): definition 7DWHFWF1D8
- [Quadratic field](https://tamnd.github.io/math2/node/7E23FBM9NQ.md): definition 7E23FBM9NQ
- [Barnes G-function](https://tamnd.github.io/math2/node/7E2JJWFF00.md): concept 7E2JJWFF00
- [Artin–Mazur zeta function](https://tamnd.github.io/math2/node/7E2ZX7N4VS.md): definition 7E2ZX7N4VS
- [Simple polytope](https://tamnd.github.io/math2/node/7E6FXTGH2Y.md): concept 7E6FXTGH2Y
- [Quasi-isomorphism](https://tamnd.github.io/math2/node/7E7W0K3N8W.md): definition 7E7W0K3N8W
- [Neutral differential equation](https://tamnd.github.io/math2/node/7E9H1SPAK8.md): definition 7E9H1SPAK8
- [Reaction–diffusion system](https://tamnd.github.io/math2/node/7E9R9X72JP.md): definition 7E9R9X72JP
- [Jacobson ring](https://tamnd.github.io/math2/node/7EB64ZBXAT.md): concept 7EB64ZBXAT
- [Whittaker function](https://tamnd.github.io/math2/node/7EFDMJSVDP.md): concept 7EFDMJSVDP
- [Quantum calculus](https://tamnd.github.io/math2/node/7EJCYBPYNN.md): concept 7EJCYBPYNN
- [Jackson integral](https://tamnd.github.io/math2/node/7EMPA6M8S7.md): concept 7EMPA6M8S7
- [Partially ordered ring](https://tamnd.github.io/math2/node/7EN0FYFTVE.md): concept 7EN0FYFTVE
- [Finite morphism](https://tamnd.github.io/math2/node/7ERFZS0NJJ.md): definition 7ERFZS0NJJ
- [Oscillatory integral](https://tamnd.github.io/math2/node/7ES4TCMRZC.md): definition 7ES4TCMRZC
- [Mittag-Leffler star](https://tamnd.github.io/math2/node/7ET5N8BWZ6.md): definition 7ET5N8BWZ6
- [Ornstein–Uhlenbeck process](https://tamnd.github.io/math2/node/7EWX5ECNRR.md): definition 7EWX5ECNRR
- [First variation](https://tamnd.github.io/math2/node/7EY98D6MG0.md): definition 7EY98D6MG0
- [Markov theorem](https://tamnd.github.io/math2/node/7F3QQQ63WR.md): statement 7F3QQQ63WR
- [Moufang plane](https://tamnd.github.io/math2/node/7F4HSNJJH8.md): concept 7F4HSNJJH8
- [Liouville property of graphs](https://tamnd.github.io/math2/node/7F5MZS55Y1.md): definition 7F5MZS55Y1
- [The integers modulo 2 and the group of signs are isomorphic](https://tamnd.github.io/math2/node/7FCN2KNWC0.md): example 7FCN2KNWC0
- [Quantifier elimination](https://tamnd.github.io/math2/node/7FDDXV3JK7.md): concept 7FDDXV3JK7
- [Point spectrum](https://tamnd.github.io/math2/node/7FNVA0Q6YQ.md): concept 7FNVA0Q6YQ
- [Purely inseparable extension](https://tamnd.github.io/math2/node/7FR2PZ141W.md): definition 7FR2PZ141W
- [Schwarz–Christoffel mapping](https://tamnd.github.io/math2/node/7FTW5Q9GHC.md): concept 7FTW5Q9GHC
- [Attractor](https://tamnd.github.io/math2/node/7FWX24X5MW.md): concept 7FWX24X5MW
- [Fort space](https://tamnd.github.io/math2/node/7FYKA5F3S6.md): counterexample 7FYKA5F3S6
- [Radical extension](https://tamnd.github.io/math2/node/7FYMK2XH1V.md): definition 7FYMK2XH1V
- [k-vertex-connected graph](https://tamnd.github.io/math2/node/7FZB7YYBXA.md): concept 7FZB7YYBXA
- [Thom–Mather isotopy lemmas](https://tamnd.github.io/math2/node/7G1PCS2RPG.md): statement 7G1PCS2RPG
- [Kodaira embedding theorem](https://tamnd.github.io/math2/node/7G7CSM2NZ9.md): statement 7G7CSM2NZ9
- [The exponential map from the additive reals to the positive reals](https://tamnd.github.io/math2/node/7GEXM87S0H.md): example 7GEXM87S0H
- [Serre's problem on projective modules](https://tamnd.github.io/math2/node/7GG0XYHB0T.md): problem 7GG0XYHB0T
- [Complex projective space](https://tamnd.github.io/math2/node/7GJ4STCNNM.md): example 7GJ4STCNNM
- [Minkowski sum of polytopes](https://tamnd.github.io/math2/node/7GK5417BJF.md): concept 7GK5417BJF
- [Hausdorff topological group](https://tamnd.github.io/math2/node/7GKCGQ7215.md): concept 7GKCGQ7215
- [Sphere theorem](https://tamnd.github.io/math2/node/7GRSF20H15.md): statement 7GRSF20H15
- [Identity theorem in several variables](https://tamnd.github.io/math2/node/7GZ908QZ04.md): statement 7GZ908QZ04
- [Normal fan](https://tamnd.github.io/math2/node/7H0E1ZBWTB.md): definition 7H0E1ZBWTB
- [Kernel regression](https://tamnd.github.io/math2/node/7H2639JPXJ.md): definition 7H2639JPXJ
- [Schaefer's fixed point theorem](https://tamnd.github.io/math2/node/7H3YSSDEBK.md): statement 7H3YSSDEBK
- [Holonomic function](https://tamnd.github.io/math2/node/7H44JD9NQD.md): concept 7H44JD9NQD
- [Enriques surface](https://tamnd.github.io/math2/node/7H6T24G8BM.md): definition 7H6T24G8BM
- [Modular representation theory](https://tamnd.github.io/math2/node/7H7CT6H9AC.md): definition 7H7CT6H9AC
- [Rogers polynomials](https://tamnd.github.io/math2/node/7H7SACRYEF.md): concept 7H7SACRYEF
- [Mairhuber–Curtis theorem](https://tamnd.github.io/math2/node/7HA6EEDR9S.md): statement 7HA6EEDR9S
- [Characteristic class](https://tamnd.github.io/math2/node/7HACJC0JE2.md): concept 7HACJC0JE2
- [Kähler differentials](https://tamnd.github.io/math2/node/7HCFARHG7D.md): concept 7HCFARHG7D
- [Jacobson radical](https://tamnd.github.io/math2/node/7HWZ06SRMF.md): definition 7HWZ06SRMF
- [Numerical integration](https://tamnd.github.io/math2/node/7HYF6YG0RS.md): concept 7HYF6YG0RS
- [Mertens' theorems](https://tamnd.github.io/math2/node/7J3P1W042Y.md): statement 7J3P1W042Y
- [Representation of a Lie group](https://tamnd.github.io/math2/node/7JA7WH92K0.md): concept 7JA7WH92K0
- [Lie group](https://tamnd.github.io/math2/node/7JBRGCVJRB.md): concept 7JBRGCVJRB
- [Invertible sheaf](https://tamnd.github.io/math2/node/7JCF2TE3V5.md): concept 7JCF2TE3V5
- [Pseudospectrum](https://tamnd.github.io/math2/node/7JE39BAHQG.md): definition 7JE39BAHQG
- [Smoothing spline](https://tamnd.github.io/math2/node/7JN0ADA86C.md): definition 7JN0ADA86C
- [Curvature of a curve](https://tamnd.github.io/math2/node/7JQS6VKFEZ.md): concept 7JQS6VKFEZ
- [Diffeology](https://tamnd.github.io/math2/node/7JS0JFR89E.md): concept 7JS0JFR89E
- [Finite difference](https://tamnd.github.io/math2/node/7JSQZ126BP.md): concept 7JSQZ126BP
- [Local homeomorphism](https://tamnd.github.io/math2/node/7JVVQ2ZZY2.md): definition 7JVVQ2ZZY2
- [Thue–Morse sequence](https://tamnd.github.io/math2/node/7JZVT385PC.md): object 7JZVT385PC
- [Bessel–Clifford function](https://tamnd.github.io/math2/node/7KA78K00QJ.md): definition 7KA78K00QJ
- [Wu's formula](https://tamnd.github.io/math2/node/7KEDZ9TFEE.md): statement 7KEDZ9TFEE
- [Lp space](https://tamnd.github.io/math2/node/7KGF78FGEF.md): definition 7KGF78FGEF
- [Dirac comb](https://tamnd.github.io/math2/node/7KHGQFM8A9.md): example 7KHGQFM8A9
- [Regularity structures](https://tamnd.github.io/math2/node/7KHRRDRVA6.md): definition 7KHRRDRVA6
- [Simple algebra](https://tamnd.github.io/math2/node/7KK111QGBZ.md): definition 7KK111QGBZ
- [Bifurcation from a characteristic value](https://tamnd.github.io/math2/node/7KMD9JA3QZ.md): definition 7KMD9JA3QZ
- [von Neumann's inequality](https://tamnd.github.io/math2/node/7KS1W5K54S.md): statement 7KS1W5K54S
- [Crystallographic group](https://tamnd.github.io/math2/node/7KTJKFQT2N.md): concept 7KTJKFQT2N
- [Level-set method](https://tamnd.github.io/math2/node/7KWR08FJET.md): concept 7KWR08FJET
- [KdV equation as an integrable system](https://tamnd.github.io/math2/node/7M1B0JCXSQ.md): example 7M1B0JCXSQ
- [Bernstein–Sato polynomial](https://tamnd.github.io/math2/node/7M5P2A0BZ6.md): concept 7M5P2A0BZ6
- [Method of lines](https://tamnd.github.io/math2/node/7M6MQ2HB1X.md): definition 7M6MQ2HB1X
- [Omitting types theorem](https://tamnd.github.io/math2/node/7MFGBVDVG9.md): statement 7MFGBVDVG9
- [Morse homology](https://tamnd.github.io/math2/node/7MJSQ7MT23.md): definition 7MJSQ7MT23
- [Ideal norm](https://tamnd.github.io/math2/node/7MPCVNQ8SF.md): definition 7MPCVNQ8SF
- [Braided monoidal category](https://tamnd.github.io/math2/node/7MZG4JQ04Y.md): definition 7MZG4JQ04Y
- [Quadratic functional equation](https://tamnd.github.io/math2/node/7MZGZFQ347.md): concept 7MZGZFQ347
- [Rectifiable set](https://tamnd.github.io/math2/node/7N0GFMW05R.md): concept 7N0GFMW05R
- [Best approximation](https://tamnd.github.io/math2/node/7N4988HN3P.md): concept 7N4988HN3P
- [Quillen–Suslin theorem](https://tamnd.github.io/math2/node/7N50GA46W1.md): statement 7N50GA46W1
- [Near-unanimity term](https://tamnd.github.io/math2/node/7N5NDD77MS.md): definition 7N5NDD77MS
- [Peano kernel theorem](https://tamnd.github.io/math2/node/7N8E1BR67D.md): statement 7N8E1BR67D
- [Semihereditary ring](https://tamnd.github.io/math2/node/7NA22G672W.md): concept 7NA22G672W
- [Dedekind domain](https://tamnd.github.io/math2/node/7NPY2CN5KB.md): concept 7NPY2CN5KB
- [Sturm–Liouville theory](https://tamnd.github.io/math2/node/7P72CCAMGZ.md): definition 7P72CCAMGZ
- [L-infinity space](https://tamnd.github.io/math2/node/7PCHXZM5YE.md): definition 7PCHXZM5YE
- [Irreducible ideal](https://tamnd.github.io/math2/node/7PDWB550C9.md): concept 7PDWB550C9
- [Combinatorial species](https://tamnd.github.io/math2/node/7PEE48WBQF.md): concept 7PEE48WBQF
- [Germ of a holomorphic function](https://tamnd.github.io/math2/node/7PJ8KVZZCQ.md): concept 7PJ8KVZZCQ
- [Riesz kernel energy](https://tamnd.github.io/math2/node/7PN328ANQA.md): definition 7PN328ANQA
- [Whitney C-infinity topology](https://tamnd.github.io/math2/node/7PN9MD1PQB.md): concept 7PN9MD1PQB
- [Group C*-algebra](https://tamnd.github.io/math2/node/7PY20WMZDZ.md): definition 7PY20WMZDZ
- [Fitting subgroup](https://tamnd.github.io/math2/node/7Q3X4FG7A5.md): definition 7Q3X4FG7A5
- [Irreducible matrix](https://tamnd.github.io/math2/node/7Q60BB01YG.md): concept 7Q60BB01YG
- [Ore condition](https://tamnd.github.io/math2/node/7Q6KK8WJ02.md): definition 7Q6KK8WJ02
- [Weak lower semicontinuity](https://tamnd.github.io/math2/node/7Q9YSCM8YY.md): definition 7Q9YSCM8YY
- [Vanishing moments of a wavelet](https://tamnd.github.io/math2/node/7QERXM0C52.md): concept 7QERXM0C52
- [Stone's theorem on paracompactness of metric spaces](https://tamnd.github.io/math2/node/7QFKD5JXH6.md): statement 7QFKD5JXH6
- [Kummer's 24 solutions](https://tamnd.github.io/math2/node/7QFW771QMA.md): statement 7QFW771QMA
- [Sheaf](https://tamnd.github.io/math2/node/7QM2KNZ358.md): definition 7QM2KNZ358
- [Harmonic analysis](https://tamnd.github.io/math2/node/7QNRC9SJVV.md): concept 7QNRC9SJVV
- [Trapezoidal rule](https://tamnd.github.io/math2/node/7R0F4956PX.md): definition 7R0F4956PX
- [Smale–Hirsch theorem](https://tamnd.github.io/math2/node/7R8W704VGW.md): statement 7R8W704VGW
- [Regular prime](https://tamnd.github.io/math2/node/7RAH6V0C5R.md): definition 7RAH6V0C5R
- [Chern connection](https://tamnd.github.io/math2/node/7RC9CN2Q49.md): concept 7RC9CN2Q49
- [Hilbert's twenty-first problem](https://tamnd.github.io/math2/node/7RHW0KA5RY.md): problem 7RHW0KA5RY
- [Eisenstein series as a double series](https://tamnd.github.io/math2/node/7RP8XW5YH3.md): example 7RP8XW5YH3
- [Gödel numbering](https://tamnd.github.io/math2/node/7RP93F2W47.md): definition 7RP93F2W47
- [Delsarte linear programming bound](https://tamnd.github.io/math2/node/7RPEDHAT7R.md): statement 7RPEDHAT7R
- [Pontryagin's maximum principle](https://tamnd.github.io/math2/node/7RPV2T29CF.md): statement 7RPV2T29CF
- [Lanczos algorithm](https://tamnd.github.io/math2/node/7RRB4PCRJB.md): statement 7RRB4PCRJB
- [Allen–Cahn equation](https://tamnd.github.io/math2/node/7S813PEV69.md): concept 7S813PEV69
- [Unitization](https://tamnd.github.io/math2/node/7SB03M850M.md): concept 7SB03M850M
- [Stein's example](https://tamnd.github.io/math2/node/7SC9E65XDN.md): counterexample 7SC9E65XDN
- [Homotopy equivalence](https://tamnd.github.io/math2/node/7SCA4TQ1Y8.md): definition 7SCA4TQ1Y8
- [Sum-free set](https://tamnd.github.io/math2/node/7SHV2SPYP6.md): concept 7SHV2SPYP6
- [Mal'cev condition](https://tamnd.github.io/math2/node/7SP03R55HW.md): definition 7SP03R55HW
- [Reverse mathematics](https://tamnd.github.io/math2/node/7SQS69T17R.md): concept 7SQS69T17R
- [Sylvester's determinant identity](https://tamnd.github.io/math2/node/7SSKBQYFW1.md): statement 7SSKBQYFW1
- [Quadratic field](https://tamnd.github.io/math2/node/7SV4WA92SM.md): concept 7SV4WA92SM
- [Germ of a smooth function](https://tamnd.github.io/math2/node/7SWSRGDT4A.md): concept 7SWSRGDT4A
- [Censoring](https://tamnd.github.io/math2/node/7SWYFFHDBF.md): definition 7SWYFFHDBF
- [Direct limit of algebras](https://tamnd.github.io/math2/node/7SXKJG1Q9B.md): concept 7SXKJG1Q9B
- [Class VII surface](https://tamnd.github.io/math2/node/7SZF58P6Z7.md): definition 7SZF58P6Z7
- [Regularity theory](https://tamnd.github.io/math2/node/7T2RGR9XTZ.md): definition 7T2RGR9XTZ
- [Bayes estimator](https://tamnd.github.io/math2/node/7T98FQFGYY.md): definition 7T98FQFGYY
- [Paley–Wiener theorem for Volterra resolvents](https://tamnd.github.io/math2/node/7T9BZV552E.md): statement 7T9BZV552E
- [Fredholm determinant](https://tamnd.github.io/math2/node/7T9ZS932SG.md): definition 7T9ZS932SG
- [X-ray transform](https://tamnd.github.io/math2/node/7TA23MSPZ0.md): definition 7TA23MSPZ0
- [Path algebra](https://tamnd.github.io/math2/node/7TCB6AGE72.md): definition 7TCB6AGE72
- [LF-space](https://tamnd.github.io/math2/node/7THJSH9SFP.md): concept 7THJSH9SFP
- [Budan's theorem](https://tamnd.github.io/math2/node/7THS519D4A.md): statement 7THS519D4A
- [Distributive property](https://tamnd.github.io/math2/node/7TNH4MR0WP.md): definition 7TNH4MR0WP
- [Supersolvable group](https://tamnd.github.io/math2/node/7TP5M906HB.md): definition 7TP5M906HB
- [Cone](https://tamnd.github.io/math2/node/7TPBQPPN97.md): concept 7TPBQPPN97
- [Directed-complete partial order](https://tamnd.github.io/math2/node/7TSXP95QN6.md): concept 7TSXP95QN6
- [Weak Lp space](https://tamnd.github.io/math2/node/7TVFY6NZDB.md): definition 7TVFY6NZDB
- [Log-modulus of a holomorphic function is subharmonic](https://tamnd.github.io/math2/node/7TXCCCA17A.md): statement 7TXCCCA17A
- [Incomplete Cholesky factorization](https://tamnd.github.io/math2/node/7V24BPCQTZ.md): definition 7V24BPCQTZ
- [Hadwiger's theorem](https://tamnd.github.io/math2/node/7V2X68AMEY.md): statement 7V2X68AMEY
- [Kakutani fixed-point theorem](https://tamnd.github.io/math2/node/7V3HWGR32Z.md): statement 7V3HWGR32Z
- [Lebesgue constant](https://tamnd.github.io/math2/node/7V7KWNK9HN.md): concept 7V7KWNK9HN
- [Measure](https://tamnd.github.io/math2/node/7V9TSDJFZR.md): definition 7V9TSDJFZR
- [Symbolic power](https://tamnd.github.io/math2/node/7VFPQVCNJ7.md): definition 7VFPQVCNJ7
- [Affine space as an algebra](https://tamnd.github.io/math2/node/7VJGH5BKKA.md): example 7VJGH5BKKA
- [Bisection method](https://tamnd.github.io/math2/node/7VMYC7MB3Q.md): statement 7VMYC7MB3Q
- [Face of a convex set](https://tamnd.github.io/math2/node/7VPJQNQV34.md): definition 7VPJQNQV34
- [Irreducible analytic germ](https://tamnd.github.io/math2/node/7VPJWF0AZ7.md): concept 7VPJWF0AZ7
- [Stably finite ring](https://tamnd.github.io/math2/node/7VRY94A22R.md): concept 7VRY94A22R
- [Normed algebra](https://tamnd.github.io/math2/node/7VS7JZ3Y5B.md): definition 7VS7JZ3Y5B
- [Waring's problem](https://tamnd.github.io/math2/node/7VWGA444K3.md): problem 7VWGA444K3
- [Ring with involution](https://tamnd.github.io/math2/node/7W52X4N3NJ.md): concept 7W52X4N3NJ
- [Superposition of solutions](https://tamnd.github.io/math2/node/7W7VV861JW.md): statement 7W7VV861JW
- [Comodule](https://tamnd.github.io/math2/node/7WEM2E6RTV.md): concept 7WEM2E6RTV
- [Mixed-type equation](https://tamnd.github.io/math2/node/7WFKPAMQFG.md): concept 7WFKPAMQFG
- [Simple function](https://tamnd.github.io/math2/node/7WMBX7M4P6.md): concept 7WMBX7M4P6
- [Tame Fréchet space](https://tamnd.github.io/math2/node/7WPVZZ4YFC.md): concept 7WPVZZ4YFC
- [Saturated model](https://tamnd.github.io/math2/node/7WTR1Y3WH4.md): definition 7WTR1Y3WH4
- [Sturm–Liouville theory](https://tamnd.github.io/math2/node/7WV6B2502K.md): concept 7WV6B2502K
- [Parallelizable manifold](https://tamnd.github.io/math2/node/7WX14GWD4M.md): concept 7WX14GWD4M
- [Interior algebra](https://tamnd.github.io/math2/node/7X1HGZ6FK0.md): definition 7X1HGZ6FK0
- [Decomposition group](https://tamnd.github.io/math2/node/7X92G8CQ33.md): concept 7X92G8CQ33
- [Quasilinear PDE](https://tamnd.github.io/math2/node/7X9N9H3RK6.md): concept 7X9N9H3RK6
- [Method of characteristics](https://tamnd.github.io/math2/node/7XAPBCJK3B.md): concept 7XAPBCJK3B
- [Dirichlet character](https://tamnd.github.io/math2/node/7XB5VJKDPG.md): concept 7XB5VJKDPG
- [Hopf algebra](https://tamnd.github.io/math2/node/7XFYP0N84H.md): concept 7XFYP0N84H
- [Algebraic structure](https://tamnd.github.io/math2/node/7XQBJ52RA1.md): definition 7XQBJ52RA1
- [Classifying space of a group](https://tamnd.github.io/math2/node/7XRBR7AV3C.md): definition 7XRBR7AV3C
- [Comparison theorem](https://tamnd.github.io/math2/node/7XTBR347P5.md): statement 7XTBR347P5
- [Kervaire invariant](https://tamnd.github.io/math2/node/7XTQKSHQEK.md): definition 7XTQKSHQEK
- [Hasse's theorem on elliptic curves](https://tamnd.github.io/math2/node/7Y26317GJ0.md): statement 7Y26317GJ0
- [Adapted process](https://tamnd.github.io/math2/node/7Y2CZ5B2SN.md): definition 7Y2CZ5B2SN
- [Error function](https://tamnd.github.io/math2/node/7Y56QCZSZJ.md): concept 7Y56QCZSZJ
- [Locally ringed space](https://tamnd.github.io/math2/node/7Y69FN6M7Y.md): definition 7Y69FN6M7Y
- [Laplacian matrix](https://tamnd.github.io/math2/node/7YBM0AXY23.md): definition 7YBM0AXY23
- [Basis of a matroid](https://tamnd.github.io/math2/node/7YD2GRHXAB.md): definition 7YD2GRHXAB
- [Extraspecial group](https://tamnd.github.io/math2/node/7YE4TAG5GE.md): concept 7YE4TAG5GE
- [Barrier](https://tamnd.github.io/math2/node/7YEJ79ZMV5.md): definition 7YEJ79ZMV5
- [Topology optimization](https://tamnd.github.io/math2/node/7YHW7PHH6H.md): problem 7YHW7PHH6H
- [Artin–Schreier theory](https://tamnd.github.io/math2/node/7YNMNRGH3T.md): concept 7YNMNRGH3T
- [Tropical geometry](https://tamnd.github.io/math2/node/7YTC3BHK60.md): concept 7YTC3BHK60
- [Zernike polynomials](https://tamnd.github.io/math2/node/7YW5WB3BY4.md): concept 7YW5WB3BY4
- [Étale cohomology](https://tamnd.github.io/math2/node/7YX76G0838.md): concept 7YX76G0838
- [Matroid minor](https://tamnd.github.io/math2/node/7Z0QZP0FC0.md): definition 7Z0QZP0FC0
- [Maximal subfield](https://tamnd.github.io/math2/node/7Z0XM9N43F.md): definition 7Z0XM9N43F
- [Noether normalization lemma](https://tamnd.github.io/math2/node/7Z28JY8QW4.md): statement 7Z28JY8QW4
- [Zonoid](https://tamnd.github.io/math2/node/7Z3VD04JGX.md): definition 7Z3VD04JGX
- [FBI transform](https://tamnd.github.io/math2/node/7Z57ARX3MF.md): definition 7Z57ARX3MF
- [Out(F_n)](https://tamnd.github.io/math2/node/7Z5ZS7KNXS.md): object 7Z5ZS7KNXS
- [Transformation semigroup](https://tamnd.github.io/math2/node/7ZB7MRKPQ9.md): concept 7ZB7MRKPQ9
- [Ramanujan tau function](https://tamnd.github.io/math2/node/7ZC93QSXNT.md): definition 7ZC93QSXNT
- [Jet bundle](https://tamnd.github.io/math2/node/7ZEFP7W1XZ.md): definition 7ZEFP7W1XZ
- [Splitting lemma](https://tamnd.github.io/math2/node/7ZGQKWYKQS.md): statement 7ZGQKWYKQS
- [Word problem for an algebra](https://tamnd.github.io/math2/node/7ZKBV18BFG.md): problem 7ZKBV18BFG
- [Sunada's theorem](https://tamnd.github.io/math2/node/7ZMWEDNCVQ.md): statement 7ZMWEDNCVQ
- [Bishop's theorem on analytic sets](https://tamnd.github.io/math2/node/7ZQ9FWZMP1.md): statement 7ZQ9FWZMP1
- [Flat module](https://tamnd.github.io/math2/node/7ZR66SH0MB.md): definition 7ZR66SH0MB
- [Mazur's lemma](https://tamnd.github.io/math2/node/7ZR88SCSRF.md): statement 7ZR88SCSRF
- [Hyperbolic law of cosines](https://tamnd.github.io/math2/node/7ZYKHQRKSJ.md): statement 7ZYKHQRKSJ
- [Sine-Gordon equation](https://tamnd.github.io/math2/node/802QSF9NM3.md): concept 802QSF9NM3
- [Operational calculus](https://tamnd.github.io/math2/node/8050R3Z6HA.md): concept 8050R3Z6HA
- [Pure weights](https://tamnd.github.io/math2/node/807S91H4RY.md): definition 807S91H4RY
- [Score test](https://tamnd.github.io/math2/node/8085SD9EYY.md): definition 8085SD9EYY
- [Hyperboloid model](https://tamnd.github.io/math2/node/809288G50Y.md): example 809288G50Y
- [Moore family](https://tamnd.github.io/math2/node/80GQXWM7BV.md): concept 80GQXWM7BV
- [Levi form](https://tamnd.github.io/math2/node/80MTNXWW80.md): concept 80MTNXWW80
- [Invariant set](https://tamnd.github.io/math2/node/80QZ05NFCB.md): definition 80QZ05NFCB
- [Hilbert's third problem](https://tamnd.github.io/math2/node/81035XQ5HE.md): problem 81035XQ5HE
- [Hauptvermutung](https://tamnd.github.io/math2/node/811NW2WEJH.md): problem 811NW2WEJH
- [Euler's integral representation of the hypergeometric function](https://tamnd.github.io/math2/node/812THX456R.md): statement 812THX456R
- [Hard Lefschetz theorem](https://tamnd.github.io/math2/node/816AHR3DRT.md): statement 816AHR3DRT
- [Donsker's theorem](https://tamnd.github.io/math2/node/818832PMSV.md): statement 818832PMSV
- [Weierstrass transform](https://tamnd.github.io/math2/node/81AQ28RQDP.md): definition 81AQ28RQDP
- [Riemann–Liouville fractional integral equation](https://tamnd.github.io/math2/node/81JHJAVRV4.md): example 81JHJAVRV4
- [Subnormal operator](https://tamnd.github.io/math2/node/81MZ023K1A.md): concept 81MZ023K1A
- [Euler summation](https://tamnd.github.io/math2/node/81Q4ERP594.md): concept 81Q4ERP594
- [Weil conjectures](https://tamnd.github.io/math2/node/81TXP32VDM.md): statement 81TXP32VDM
- [Identity](https://tamnd.github.io/math2/node/81XXW834SS.md): definition 81XXW834SS
- [Gelfand transform](https://tamnd.github.io/math2/node/81ZC98VMD5.md): definition 81ZC98VMD5
- [Kolmogorov's normability criterion](https://tamnd.github.io/math2/node/822WQ7S91T.md): statement 822WQ7S91T
- [Arithmetic geometry](https://tamnd.github.io/math2/node/824JAXQ5M6.md): concept 824JAXQ5M6
- [Wave front of a Legendrian map](https://tamnd.github.io/math2/node/824SSQ3SQN.md): concept 824SSQ3SQN
- [Triangle center](https://tamnd.github.io/math2/node/827DJ2CVEX.md): definition 827DJ2CVEX
- [Envelope](https://tamnd.github.io/math2/node/827KR3MVVY.md): concept 827KR3MVVY
- [Levin-type transformations](https://tamnd.github.io/math2/node/827RWYT7Q6.md): definition 827RWYT7Q6
- [Sublattice](https://tamnd.github.io/math2/node/82D3DQD3X8.md): definition 82D3DQD3X8
- [Bishop–Gromov inequality](https://tamnd.github.io/math2/node/82F5Q4F1S6.md): statement 82F5Q4F1S6
- [Følner condition](https://tamnd.github.io/math2/node/82R5XGCART.md): concept 82R5XGCART
- [Mikusiński field of operators](https://tamnd.github.io/math2/node/82YAXZMMQ1.md): concept 82YAXZMMQ1
- [Moser's trick](https://tamnd.github.io/math2/node/82YGJ392SJ.md): statement 82YGJ392SJ
- [Chern–Weil homomorphism](https://tamnd.github.io/math2/node/82ZPRD39J7.md): concept 82ZPRD39J7
- [Brill–Noether theory](https://tamnd.github.io/math2/node/82ZR6BCJXV.md): concept 82ZR6BCJXV
- [Möbius plane](https://tamnd.github.io/math2/node/831H5X5JS4.md): definition 831H5X5JS4
- [Reduced complex space](https://tamnd.github.io/math2/node/831N2Q4GCB.md): concept 831N2Q4GCB
- [Regular open set](https://tamnd.github.io/math2/node/833XAA2Q65.md): concept 833XAA2Q65
- [Kingman's subadditive ergodic theorem](https://tamnd.github.io/math2/node/833XVEEZCJ.md): statement 833XVEEZCJ
- [Riesz kernel energy](https://tamnd.github.io/math2/node/835MPDJXMA.md): concept 835MPDJXMA
- [Down-closure](https://tamnd.github.io/math2/node/839P1ZNRD7.md): concept 839P1ZNRD7
- [Fuzzy measure](https://tamnd.github.io/math2/node/83DSSH4HZY.md): definition 83DSSH4HZY
- [Poincaré–Lindstedt method](https://tamnd.github.io/math2/node/83H3AMTXST.md): definition 83H3AMTXST
- [Domino tiling](https://tamnd.github.io/math2/node/83PJCPB9VA.md): concept 83PJCPB9VA
- [Basic hypergeometric series](https://tamnd.github.io/math2/node/83TW9SVG0A.md): definition 83TW9SVG0A
- [Euclid's Elements](https://tamnd.github.io/math2/node/83WHDAV68J.md): object 83WHDAV68J
- [Cotton tensor](https://tamnd.github.io/math2/node/83ZA89PAGK.md): definition 83ZA89PAGK
- [Turing jump](https://tamnd.github.io/math2/node/841C9J1QTC.md): definition 841C9J1QTC
- [Quasi-ergodicity](https://tamnd.github.io/math2/node/844NHRQQN3.md): definition 844NHRQQN3
- [Counterexample to the Markus–Yamabe conjecture](https://tamnd.github.io/math2/node/8452SY027Z.md): counterexample 8452SY027Z
- [Multivariate analysis of variance](https://tamnd.github.io/math2/node/849ASYBYJ6.md): definition 849ASYBYJ6
- [Sparse grid](https://tamnd.github.io/math2/node/84BR7X6HET.md): concept 84BR7X6HET
- [p-adic gamma function](https://tamnd.github.io/math2/node/84MVNJCYR4.md): concept 84MVNJCYR4
- [Spline](https://tamnd.github.io/math2/node/84TJ8KQ926.md): concept 84TJ8KQ926
- [Reduced product](https://tamnd.github.io/math2/node/84XXPH6XJ8.md): definition 84XXPH6XJ8
- [Divisibility poset](https://tamnd.github.io/math2/node/850Y7GFENR.md): example 850Y7GFENR
- [Nichols–Zoeller theorem](https://tamnd.github.io/math2/node/856B6C7EE4.md): statement 856B6C7EE4
- [Morphism](https://tamnd.github.io/math2/node/859F0TFVA3.md): concept 859F0TFVA3
- [Injective module](https://tamnd.github.io/math2/node/85H7PRHDPZ.md): concept 85H7PRHDPZ
- [Lax–Milgram theorem](https://tamnd.github.io/math2/node/85KK79FCN5.md): statement 85KK79FCN5
- [Siegel modular form](https://tamnd.github.io/math2/node/85KTS2VRNV.md): definition 85KTS2VRNV
- [Poincaré disk model](https://tamnd.github.io/math2/node/85MM2EA0TK.md): object 85MM2EA0TK
- [Hermite functions](https://tamnd.github.io/math2/node/85RZANS8DX.md): definition 85RZANS8DX
- [Homogeneous space](https://tamnd.github.io/math2/node/85TWR6E75P.md): concept 85TWR6E75P
- [Unitization](https://tamnd.github.io/math2/node/85Y0XMYHGG.md): definition 85Y0XMYHGG
- [Ring of convergent power series](https://tamnd.github.io/math2/node/85ZNEMDA9H.md): definition 85ZNEMDA9H
- [Classifying topos](https://tamnd.github.io/math2/node/85ZZ921DWH.md): concept 85ZZ921DWH
- [Abstract data type as an algebra](https://tamnd.github.io/math2/node/86144NHH59.md): concept 86144NHH59
- [Single layer potential](https://tamnd.github.io/math2/node/866JDBWRB0.md): definition 866JDBWRB0
- [Chebyshev series](https://tamnd.github.io/math2/node/868V3H5VX8.md): definition 868V3H5VX8
- [Reduction of an initial value problem to a Volterra equation](https://tamnd.github.io/math2/node/86AWJMXT1A.md): statement 86AWJMXT1A
- [Adjoint of an unbounded operator](https://tamnd.github.io/math2/node/86B7HF79PQ.md): concept 86B7HF79PQ
- [F-space](https://tamnd.github.io/math2/node/86BYEV9AX8.md): concept 86BYEV9AX8
- [Coercive bilinear form](https://tamnd.github.io/math2/node/86KRHS6X22.md): concept 86KRHS6X22
- [Reduced function](https://tamnd.github.io/math2/node/86PVRZ08VK.md): definition 86PVRZ08VK
- [Gradient flow on a manifold](https://tamnd.github.io/math2/node/86SJSP7WAY.md): definition 86SJSP7WAY
- [Elliptic complex](https://tamnd.github.io/math2/node/8722T9KCP5.md): definition 8722T9KCP5
- [f-vector](https://tamnd.github.io/math2/node/872BAKDQQK.md): concept 872BAKDQQK
- [Tricolorability](https://tamnd.github.io/math2/node/878B9A4D9B.md): definition 878B9A4D9B
- [Polytope combinatorics](https://tamnd.github.io/math2/node/879SBTVE7V.md): concept 879SBTVE7V
- [Hyperkähler manifold](https://tamnd.github.io/math2/node/87AWY7NQS0.md): concept 87AWY7NQS0
- [Heun function](https://tamnd.github.io/math2/node/87DAYKCZ5M.md): concept 87DAYKCZ5M
- [Landweber iteration](https://tamnd.github.io/math2/node/87DV5ZTJ9J.md): statement 87DV5ZTJ9J
- [Constant-mean-curvature surface](https://tamnd.github.io/math2/node/87EFGXB17V.md): definition 87EFGXB17V
- [Nuclear space](https://tamnd.github.io/math2/node/87FM669CDD.md): definition 87FM669CDD
- [Mixed finite element method](https://tamnd.github.io/math2/node/87VMQ892VW.md): concept 87VMQ892VW
- [Height of a prime ideal](https://tamnd.github.io/math2/node/87WH65FJB8.md): concept 87WH65FJB8
- [Faithfully flat module](https://tamnd.github.io/math2/node/8831C5ARPB.md): definition 8831C5ARPB
- [Radon's theorem](https://tamnd.github.io/math2/node/888R24ZGNY.md): statement 888R24ZGNY
- [Space of connections modulo gauge](https://tamnd.github.io/math2/node/888SJ2508Y.md): object 888SJ2508Y
- [Meixner polynomials](https://tamnd.github.io/math2/node/88GQ3KHQ3Q.md): definition 88GQ3KHQ3Q
- [Gaussian curvature](https://tamnd.github.io/math2/node/88VFJSZ8X2.md): definition 88VFJSZ8X2
- [Subdirect product](https://tamnd.github.io/math2/node/88WN3AD2VQ.md): concept 88WN3AD2VQ
- [Diagram](https://tamnd.github.io/math2/node/88XF0NVBVG.md): definition 88XF0NVBVG
- [Convolution power](https://tamnd.github.io/math2/node/8943THTS6W.md): definition 8943THTS6W
- [Schrödinger operator](https://tamnd.github.io/math2/node/89AA6XJY5K.md): concept 89AA6XJY5K
- [Mutual absolute continuity of harmonic measures](https://tamnd.github.io/math2/node/89CHZ3ZMCH.md): statement 89CHZ3ZMCH
- [Diagonally dominant matrix](https://tamnd.github.io/math2/node/89P5GTKW87.md): concept 89P5GTKW87
- [Gauss–Hermite quadrature](https://tamnd.github.io/math2/node/89S5KGQEMJ.md): statement 89S5KGQEMJ
- [Hereditary ring](https://tamnd.github.io/math2/node/89VBQH8HNS.md): concept 89VBQH8HNS
- [Tracy–Widom distribution](https://tamnd.github.io/math2/node/89ZY5YRETE.md): concept 89ZY5YRETE
- [Amitsur's noncrossed product](https://tamnd.github.io/math2/node/8A00Y84SW7.md): counterexample 8A00Y84SW7
- [Fekete's lemma](https://tamnd.github.io/math2/node/8A4GGX1MEY.md): statement 8A4GGX1MEY
- [Dual object](https://tamnd.github.io/math2/node/8A7YXMW727.md): concept 8A7YXMW727
- [Regularity of minimizers](https://tamnd.github.io/math2/node/8A8FBH4C7Y.md): concept 8A8FBH4C7Y
- [Lusin area integral](https://tamnd.github.io/math2/node/8AD5RQTTK9.md): definition 8AD5RQTTK9
- [Harmonic map](https://tamnd.github.io/math2/node/8AGSK9S92W.md): definition 8AGSK9S92W
- [Holomorphic map between Riemann surfaces](https://tamnd.github.io/math2/node/8AJEMCZ29B.md): definition 8AJEMCZ29B
- [McDiarmid's inequality](https://tamnd.github.io/math2/node/8APFT1RPM4.md): statement 8APFT1RPM4
- [Specialization preorder](https://tamnd.github.io/math2/node/8APGCB1CKH.md): definition 8APGCB1CKH
- [Fokker–Planck equation](https://tamnd.github.io/math2/node/8APSJK9GSK.md): definition 8APSJK9GSK
- [Schlömilch's generalization of the condensation test](https://tamnd.github.io/math2/node/8AX4XREEPX.md): statement 8AX4XREEPX
- [Metaplectic group](https://tamnd.github.io/math2/node/8B0JM2YDE2.md): example 8B0JM2YDE2
- [Separably closed field](https://tamnd.github.io/math2/node/8B57B7EGK4.md): concept 8B57B7EGK4
- [Cauchy-type integral](https://tamnd.github.io/math2/node/8BBSFVN1CS.md): definition 8BBSFVN1CS
- [Wishart distribution](https://tamnd.github.io/math2/node/8BEMNJW0B9.md): concept 8BEMNJW0B9
- [Bar resolution](https://tamnd.github.io/math2/node/8BGJ6J9PYF.md): concept 8BGJ6J9PYF
- [Epigraph](https://tamnd.github.io/math2/node/8BJ5M76FSX.md): concept 8BJ5M76FSX
- [Steiner symmetrization](https://tamnd.github.io/math2/node/8BMM456H7G.md): concept 8BMM456H7G
- [Leray spectral sequence](https://tamnd.github.io/math2/node/8BMQ90EJPR.md): statement 8BMQ90EJPR
- [Thomson's principle](https://tamnd.github.io/math2/node/8BPMHA5MT4.md): statement 8BPMHA5MT4
- [Bar resolution](https://tamnd.github.io/math2/node/8BVNQ3R3YP.md): definition 8BVNQ3R3YP
- [Kantorovich polynomials](https://tamnd.github.io/math2/node/8BXFDDSSG7.md): concept 8BXFDDSSG7
- [van der Waerden conjecture on permanents](https://tamnd.github.io/math2/node/8C1JYDJBYS.md): statement 8C1JYDJBYS
- [Modality of a singularity](https://tamnd.github.io/math2/node/8CFTJP1WN8.md): concept 8CFTJP1WN8
- [Lévy's characterization of Brownian motion](https://tamnd.github.io/math2/node/8CKDVESXT6.md): statement 8CKDVESXT6
- [Approximation property](https://tamnd.github.io/math2/node/8CKE0V6ANK.md): concept 8CKE0V6ANK
- [Geometric integrator](https://tamnd.github.io/math2/node/8CV1KW9AJ8.md): definition 8CV1KW9AJ8
- [Resultant](https://tamnd.github.io/math2/node/8D277VG29T.md): definition 8D277VG29T
- [Weierstrass excess function](https://tamnd.github.io/math2/node/8D455SA1D4.md): definition 8D455SA1D4
- [Dirichlet problem](https://tamnd.github.io/math2/node/8D56GQ4AG8.md): problem 8D56GQ4AG8
- [Holomorphy on Banach spaces](https://tamnd.github.io/math2/node/8D5Y3JNF09.md): definition 8D5Y3JNF09
- [f-vector and h-vector](https://tamnd.github.io/math2/node/8DEH72QHEA.md): definition 8DEH72QHEA
- [Borel summation](https://tamnd.github.io/math2/node/8DJPSYSHGJ.md): concept 8DJPSYSHGJ
- [Free product](https://tamnd.github.io/math2/node/8DVYNQAA1C.md): definition 8DVYNQAA1C
- [Asymptotic normality of the maximum likelihood estimator](https://tamnd.github.io/math2/node/8DWRYNAMFT.md): statement 8DWRYNAMFT
- [Projective hierarchy](https://tamnd.github.io/math2/node/8DZTNVTGY2.md): concept 8DZTNVTGY2
- [Forward difference operator](https://tamnd.github.io/math2/node/8E0EBA8X07.md): concept 8E0EBA8X07
- [Principal bundle](https://tamnd.github.io/math2/node/8E37E74XVP.md): concept 8E37E74XVP
- [Minimax estimator](https://tamnd.github.io/math2/node/8EG1M1WZ9C.md): concept 8EG1M1WZ9C
- [Order complex](https://tamnd.github.io/math2/node/8EPC35FTMX.md): definition 8EPC35FTMX
- [Painlevé property](https://tamnd.github.io/math2/node/8EV1G2Y6CC.md): concept 8EV1G2Y6CC
- [Central polynomial](https://tamnd.github.io/math2/node/8EV392NDQD.md): definition 8EV392NDQD
- [Frobenius algebra](https://tamnd.github.io/math2/node/8EVP1K2CZH.md): concept 8EVP1K2CZH
- [Bessel potential space](https://tamnd.github.io/math2/node/8F1HPGFW4F.md): definition 8F1HPGFW4F
- [Primitive ring](https://tamnd.github.io/math2/node/8F6EHDVJ62.md): concept 8F6EHDVJ62
- [Directed set](https://tamnd.github.io/math2/node/8F933HJ1RY.md): definition 8F933HJ1RY
- [Fusion category](https://tamnd.github.io/math2/node/8FEGZWG3JH.md): definition 8FEGZWG3JH
- [WZ theory](https://tamnd.github.io/math2/node/8FK88MNE0N.md): concept 8FK88MNE0N
- [Anosov flow](https://tamnd.github.io/math2/node/8FQJXSA4DT.md): concept 8FQJXSA4DT
- [Logarithmically convex function](https://tamnd.github.io/math2/node/8FRWATNNMX.md): concept 8FRWATNNMX
- [Borel determinacy theorem](https://tamnd.github.io/math2/node/8FVWYEZWTK.md): statement 8FVWYEZWTK
- [Boolean-valued model](https://tamnd.github.io/math2/node/8FYH1Y21CP.md): definition 8FYH1Y21CP
- [Box–Muller transform](https://tamnd.github.io/math2/node/8G07Q9SVQ4.md): statement 8G07Q9SVQ4
- [Commuting matrices](https://tamnd.github.io/math2/node/8G3X8MPG7E.md): concept 8G3X8MPG7E
- [Doubling measure](https://tamnd.github.io/math2/node/8G4ZKXPV29.md): definition 8G4ZKXPV29
- [Unit disc](https://tamnd.github.io/math2/node/8G5ABXRK85.md): object 8G5ABXRK85
- [Rational homotopy theory](https://tamnd.github.io/math2/node/8G9QPMBT7R.md): concept 8G9QPMBT7R
- [Admissible control](https://tamnd.github.io/math2/node/8GETTS41P6.md): definition 8GETTS41P6
- [Empirical Bayes method](https://tamnd.github.io/math2/node/8GG2Z2E61R.md): concept 8GG2Z2E61R
- [Gabor transform](https://tamnd.github.io/math2/node/8GQRJ6TPVC.md): concept 8GQRJ6TPVC
- [Multidimensional Z-transform](https://tamnd.github.io/math2/node/8GRA22YEMB.md): definition 8GRA22YEMB
- [Bonferroni correction](https://tamnd.github.io/math2/node/8GRD2TMJRR.md): concept 8GRD2TMJRR
- [Legendre's relation](https://tamnd.github.io/math2/node/8GSKDNZ4TK.md): statement 8GSKDNZ4TK
- [Mann–Whitney U test](https://tamnd.github.io/math2/node/8GWEWBZE9P.md): definition 8GWEWBZE9P
- [Haefliger structure](https://tamnd.github.io/math2/node/8GYWYYBFWE.md): concept 8GYWYYBFWE
- [Directional statistics](https://tamnd.github.io/math2/node/8GZJQBPBY0.md): concept 8GZJQBPBY0
- [Lévy–Prokhorov metric](https://tamnd.github.io/math2/node/8GZK8NC3VQ.md): concept 8GZK8NC3VQ
- [Ornstein isomorphism theorem](https://tamnd.github.io/math2/node/8H4GF58E9M.md): statement 8H4GF58E9M
- [4-manifold](https://tamnd.github.io/math2/node/8H58Q6YJCH.md): concept 8H58Q6YJCH
- [Richardson extrapolation](https://tamnd.github.io/math2/node/8HA39P7ZFN.md): definition 8HA39P7ZFN
- [Determinant line bundle](https://tamnd.github.io/math2/node/8HDWPN2FXS.md): definition 8HDWPN2FXS
- [BK-space](https://tamnd.github.io/math2/node/8HEZ9TW1NK.md): concept 8HEZ9TW1NK
- [Associated graded ring](https://tamnd.github.io/math2/node/8HFAEGFQT1.md): concept 8HFAEGFQT1
- [Crossed product](https://tamnd.github.io/math2/node/8HG8TS46A6.md): concept 8HG8TS46A6
- [Field norm](https://tamnd.github.io/math2/node/8HHY1K3XDC.md): definition 8HHY1K3XDC
- [Central difference](https://tamnd.github.io/math2/node/8HSQHDNFEF.md): definition 8HSQHDNFEF
- [Almost convergence](https://tamnd.github.io/math2/node/8HTHB1QPZY.md): concept 8HTHB1QPZY
- [Fredholm operator](https://tamnd.github.io/math2/node/8HWGKWD6ZM.md): concept 8HWGKWD6ZM
- [Multivariate analysis of variance](https://tamnd.github.io/math2/node/8J3B7N8CAT.md): concept 8J3B7N8CAT
- [Besicovitch–Federer projection theorem](https://tamnd.github.io/math2/node/8J8AW0TG9M.md): statement 8J8AW0TG9M
- [Invertible sheaf](https://tamnd.github.io/math2/node/8JHC8TDNDW.md): definition 8JHC8TDNDW
- [Additive energy](https://tamnd.github.io/math2/node/8JMZDBJNZR.md): definition 8JMZDBJNZR
- [Minimum description length](https://tamnd.github.io/math2/node/8JV9PPFRG4.md): concept 8JV9PPFRG4
- [Robinson–Schensted–Knuth correspondence](https://tamnd.github.io/math2/node/8JW8F4VVED.md): statement 8JW8F4VVED
- [Root-finding algorithm](https://tamnd.github.io/math2/node/8JYC7S28JX.md): definition 8JYC7S28JX
- [Little q-Jacobi polynomials](https://tamnd.github.io/math2/node/8JZZNXKWCX.md): definition 8JZZNXKWCX
- [Orthocentric system](https://tamnd.github.io/math2/node/8K5FT58EK5.md): concept 8K5FT58EK5
- [Beta function](https://tamnd.github.io/math2/node/8K9YQ1YGVS.md): definition 8K9YQ1YGVS
- [Graded ring](https://tamnd.github.io/math2/node/8KB4SQGC0C.md): concept 8KB4SQGC0C
- [Symplectic capacity](https://tamnd.github.io/math2/node/8KCB5KJKNT.md): concept 8KCB5KJKNT
- [Generalized Poincaré conjecture](https://tamnd.github.io/math2/node/8KDV43WZ3Z.md): statement 8KDV43WZ3Z
- [Matching pursuit](https://tamnd.github.io/math2/node/8KJ9P9KQ5H.md): concept 8KJ9P9KQ5H
- [Integral equation of the second kind](https://tamnd.github.io/math2/node/8KJCN49V64.md): concept 8KJCN49V64
- [Krull domain](https://tamnd.github.io/math2/node/8KK1BQSPJT.md): definition 8KK1BQSPJT
- [Formally smooth map](https://tamnd.github.io/math2/node/8KPDNVKX7B.md): concept 8KPDNVKX7B
- [Homotopy lifting property](https://tamnd.github.io/math2/node/8KYEBXE16P.md): definition 8KYEBXE16P
- [Stability of Volterra equations](https://tamnd.github.io/math2/node/8M012XRPZM.md): concept 8M012XRPZM
- [Tricomi equation](https://tamnd.github.io/math2/node/8M04CT7CCM.md): definition 8M04CT7CCM
- [Regular open set](https://tamnd.github.io/math2/node/8M0QH4Z60S.md): definition 8M0QH4Z60S
- [Admissible control](https://tamnd.github.io/math2/node/8M31QKB0GW.md): concept 8M31QKB0GW
- [Quasi-analytic function](https://tamnd.github.io/math2/node/8M4N38VGEE.md): definition 8M4N38VGEE
- [Proof that cyclic groups are abelian](https://tamnd.github.io/math2/node/8MADRXG8PQ.md): proof 8MADRXG8PQ
- [Short exact sequence](https://tamnd.github.io/math2/node/8MCYV0P04T.md): definition 8MCYV0P04T
- [Uniform convergence of Bernstein approximations](https://tamnd.github.io/math2/node/8MD8DJE30F.md): statement 8MD8DJE30F
- [Loop group](https://tamnd.github.io/math2/node/8MEDVAEERQ.md): example 8MEDVAEERQ
- [F-sigma set](https://tamnd.github.io/math2/node/8MGC7M64GW.md): concept 8MGC7M64GW
- [Hyperoval](https://tamnd.github.io/math2/node/8MGCGZP0E6.md): definition 8MGCGZP0E6
- [Total ring of fractions](https://tamnd.github.io/math2/node/8MGH8PGJWR.md): definition 8MGH8PGJWR
- [600-cell](https://tamnd.github.io/math2/node/8MNQDK5X5W.md): object 8MNQDK5X5W
- [Word metric](https://tamnd.github.io/math2/node/8MRPZ4Q4NM.md): definition 8MRPZ4Q4NM
- [Essential submodule](https://tamnd.github.io/math2/node/8MS51FASMX.md): concept 8MS51FASMX
- [F-test](https://tamnd.github.io/math2/node/8MTVCMAATD.md): concept 8MTVCMAATD
- [Dirichlet series](https://tamnd.github.io/math2/node/8MWTEF6462.md): concept 8MWTEF6462
- [Pseudospectral optimal control](https://tamnd.github.io/math2/node/8N5CYXWXDV.md): definition 8N5CYXWXDV
- [Homotopy group with coefficients](https://tamnd.github.io/math2/node/8NAXJ0JH71.md): concept 8NAXJ0JH71
- [Benz plane](https://tamnd.github.io/math2/node/8NCGCJ2YBH.md): definition 8NCGCJ2YBH
- [Kaplansky's theorem on PI rings](https://tamnd.github.io/math2/node/8NE9Q44FEG.md): statement 8NE9Q44FEG
- [Hermitian form](https://tamnd.github.io/math2/node/8NJ7XY6F1S.md): definition 8NJ7XY6F1S
- [Classical ring of quotients](https://tamnd.github.io/math2/node/8NMGMXNF02.md): definition 8NMGMXNF02
- [Riemann–Stieltjes integral](https://tamnd.github.io/math2/node/8NMH42GJRK.md): definition 8NMH42GJRK
- [G-delta set](https://tamnd.github.io/math2/node/8NS0116H9P.md): definition 8NS0116H9P
- [Lévy process](https://tamnd.github.io/math2/node/8NT5CFG2ZR.md): concept 8NT5CFG2ZR
- [Density theorem](https://tamnd.github.io/math2/node/8P9PJG95H3.md): statement 8P9PJG95H3
- [Spectral element method](https://tamnd.github.io/math2/node/8PQ405CKA3.md): definition 8PQ405CKA3
- [Symbolic dynamics](https://tamnd.github.io/math2/node/8PX1F72APY.md): definition 8PX1F72APY
- [Kolmogorov complexity](https://tamnd.github.io/math2/node/8PY3K6NCKF.md): concept 8PY3K6NCKF
- [Ramanujan's series for 1/pi](https://tamnd.github.io/math2/node/8PZPMF2QJD.md): statement 8PZPMF2QJD
- [Volterra equation of the first kind](https://tamnd.github.io/math2/node/8Q7P2WA8ZT.md): definition 8Q7P2WA8ZT
- [Total variation](https://tamnd.github.io/math2/node/8QA5DEWNYD.md): concept 8QA5DEWNYD
- [Principal curvature](https://tamnd.github.io/math2/node/8QBWWC354F.md): definition 8QBWWC354F
- [Artin group](https://tamnd.github.io/math2/node/8QE86W4GQ6.md): definition 8QE86W4GQ6
- [Neumann series](https://tamnd.github.io/math2/node/8QJ91709V2.md): statement 8QJ91709V2
- [Locally finite variety](https://tamnd.github.io/math2/node/8QPW65PZVW.md): concept 8QPW65PZVW
- [Solving linear difference equations by the Z-transform](https://tamnd.github.io/math2/node/8QQ7BF5BM0.md): definition 8QQ7BF5BM0
- [Cauchy interlacing theorem](https://tamnd.github.io/math2/node/8QQPVNR80F.md): statement 8QQPVNR80F
- [Bernstein–Gelfand–Ponomarev reflection functors](https://tamnd.github.io/math2/node/8QTP2EFA92.md): definition 8QTP2EFA92
- [Hartogs–Bochner extension theorem](https://tamnd.github.io/math2/node/8QYVM1JTMF.md): statement 8QYVM1JTMF
- [Paracompact space](https://tamnd.github.io/math2/node/8R29G6NG9Q.md): concept 8R29G6NG9Q
- [B-spline](https://tamnd.github.io/math2/node/8R4CVVV8WT.md): definition 8R4CVVV8WT
- [Ricker model](https://tamnd.github.io/math2/node/8R4H8GW1CD.md): example 8R4H8GW1CD
- [Conservation law](https://tamnd.github.io/math2/node/8RA7AMWSAF.md): concept 8RA7AMWSAF
- [Alternating-direction implicit method](https://tamnd.github.io/math2/node/8RDSR3JXP2.md): statement 8RDSR3JXP2
- [Catenary problem](https://tamnd.github.io/math2/node/8REBFQD6E0.md): problem 8REBFQD6E0
- [Shape operator](https://tamnd.github.io/math2/node/8RT3FGZPZY.md): definition 8RT3FGZPZY
- [Bergman kernel](https://tamnd.github.io/math2/node/8RWDHW0CXC.md): concept 8RWDHW0CXC
- [Remez inequality](https://tamnd.github.io/math2/node/8S129JX919.md): statement 8S129JX919
- [Kurosh–Amitsur radical theory](https://tamnd.github.io/math2/node/8S2SJ4SC2E.md): definition 8S2SJ4SC2E
- [Apollonian gasket](https://tamnd.github.io/math2/node/8SETG842RX.md): example 8SETG842RX
- [Kolmogorov automorphism](https://tamnd.github.io/math2/node/8SFZYANHNC.md): definition 8SFZYANHNC
- [Singular cardinals hypothesis](https://tamnd.github.io/math2/node/8SH7WETCKY.md): statement 8SH7WETCKY
- [Tauber's theorem](https://tamnd.github.io/math2/node/8SK3GQ4S0M.md): statement 8SK3GQ4S0M
- [Zariski topology](https://tamnd.github.io/math2/node/8SQK7RTB63.md): concept 8SQK7RTB63
- [Residue field](https://tamnd.github.io/math2/node/8ST7CTZ2DE.md): definition 8ST7CTZ2DE
- [Elementary abelian group](https://tamnd.github.io/math2/node/8T34EC16NH.md): definition 8T34EC16NH
- [MV-algebra](https://tamnd.github.io/math2/node/8T4M7RN9AC.md): concept 8T4M7RN9AC
- [Contractible space](https://tamnd.github.io/math2/node/8T7XY797SV.md): definition 8T7XY797SV
- [Fence](https://tamnd.github.io/math2/node/8T9S6F9HRW.md): example 8T9S6F9HRW
- [Birkhoff sum and average](https://tamnd.github.io/math2/node/8TD2KCVH5N.md): concept 8TD2KCVH5N
- [Standard map](https://tamnd.github.io/math2/node/8TEV7CSX2H.md): example 8TEV7CSX2H
- [Tempered distribution](https://tamnd.github.io/math2/node/8TK1JNG87X.md): definition 8TK1JNG87X
- [2-category](https://tamnd.github.io/math2/node/8TPR96GV7F.md): definition 8TPR96GV7F
- [Cap product](https://tamnd.github.io/math2/node/8TPW8B9G48.md): definition 8TPW8B9G48
- [Toeplitz–Hausdorff theorem](https://tamnd.github.io/math2/node/8TV0DJBBAY.md): statement 8TV0DJBBAY
- [Markov chain mixing time](https://tamnd.github.io/math2/node/8TV1YW8EXC.md): concept 8TV1YW8EXC
- [Orthogonal polynomials](https://tamnd.github.io/math2/node/8TXASARQN1.md): concept 8TXASARQN1
- [Freyd–Mitchell embedding theorem](https://tamnd.github.io/math2/node/8V296XR2M3.md): statement 8V296XR2M3
- [Rational singularity](https://tamnd.github.io/math2/node/8V30QMXTYT.md): definition 8V30QMXTYT
- [Abstract Volterra equation](https://tamnd.github.io/math2/node/8V9J445R7N.md): definition 8V9J445R7N
- [Carleson embedding theorem](https://tamnd.github.io/math2/node/8V9JHWWXNH.md): statement 8V9JHWWXNH
- [Singular integral operator](https://tamnd.github.io/math2/node/8VB096R12H.md): definition 8VB096R12H
- [Efficiency of an estimator](https://tamnd.github.io/math2/node/8VDNRZG3R1.md): concept 8VDNRZG3R1
- [Polygon](https://tamnd.github.io/math2/node/8VEFWD3EJB.md): concept 8VEFWD3EJB
- [Liouville's theorem (differential algebra)](https://tamnd.github.io/math2/node/8VRTPECBPZ.md): statement 8VRTPECBPZ
- [Polyconvexity](https://tamnd.github.io/math2/node/8VYYWQ5CMB.md): concept 8VYYWQ5CMB
- [Simple ring](https://tamnd.github.io/math2/node/8W5YE3N1PM.md): definition 8W5YE3N1PM
- [Weil divisor](https://tamnd.github.io/math2/node/8WAC71N0C2.md): definition 8WAC71N0C2
- [Inversive geometry](https://tamnd.github.io/math2/node/8WBZ6Q949M.md): definition 8WBZ6Q949M
- [Descriptive geometry](https://tamnd.github.io/math2/node/8WETP2V4MB.md): definition 8WETP2V4MB
- [The even integers form a subgroup of the integers](https://tamnd.github.io/math2/node/8WJPSS7TE9.md): example 8WJPSS7TE9
- [Computed tomography as an inverse problem](https://tamnd.github.io/math2/node/8WMX3GRGCZ.md): example 8WMX3GRGCZ
- [Cartan's theorem on fine continuity of superharmonic functions](https://tamnd.github.io/math2/node/8WWTMFR278.md): statement 8WWTMFR278
- [Excluded point topology](https://tamnd.github.io/math2/node/8WX57M9ZVA.md): example 8WX57M9ZVA
- [Elementary substructure](https://tamnd.github.io/math2/node/8WZF9SBP39.md): concept 8WZF9SBP39
- [Property (T)](https://tamnd.github.io/math2/node/8X1ZRYBZ46.md): definition 8X1ZRYBZ46
- [Summability method](https://tamnd.github.io/math2/node/8X25Q4XYMN.md): concept 8X25Q4XYMN
- [Score](https://tamnd.github.io/math2/node/8X7293647G.md): definition 8X7293647G
- [Khinchin integral](https://tamnd.github.io/math2/node/8X7ZESGYBW.md): definition 8X7ZESGYBW
- [Tikhonov's theorem on slow-fast systems](https://tamnd.github.io/math2/node/8X8Y61MGHW.md): statement 8X8Y61MGHW
- [Cohomology of Lie groups](https://tamnd.github.io/math2/node/8XAT77QQTN.md): definition 8XAT77QQTN
- [Non-Archimedean absolute value](https://tamnd.github.io/math2/node/8XDYNK6TV0.md): definition 8XDYNK6TV0
- [Statistical convergence](https://tamnd.github.io/math2/node/8XS5T7BEJP.md): definition 8XS5T7BEJP
- [Galois ring](https://tamnd.github.io/math2/node/8Y0M6X8Q2A.md): definition 8Y0M6X8Q2A
- [Homogeneous and inhomogeneous linear recurrences](https://tamnd.github.io/math2/node/8Y28JQB0Q5.md): concept 8Y28JQB0Q5
- [Arbitrary-precision arithmetic](https://tamnd.github.io/math2/node/8Y4C6N09YQ.md): definition 8Y4C6N09YQ
- [Nash embedding theorems](https://tamnd.github.io/math2/node/8YQJQXQHVE.md): statement 8YQJQXQHVE
- [Dual object](https://tamnd.github.io/math2/node/8YYDZZCFY6.md): definition 8YYDZZCFY6
- [Loop theorem](https://tamnd.github.io/math2/node/8YYJBVP68F.md): statement 8YYJBVP68F
- [Cayley–Menger determinant](https://tamnd.github.io/math2/node/8Z18NXTHT4.md): definition 8Z18NXTHT4
- [Bolza problem](https://tamnd.github.io/math2/node/8Z55NXZWD5.md): problem 8Z55NXZWD5
- [Cantor–Lebesgue theorem](https://tamnd.github.io/math2/node/8Z55VMJSEX.md): statement 8Z55VMJSEX
- [Hall's theorem on solvable groups](https://tamnd.github.io/math2/node/8Z8PE69SJS.md): statement 8Z8PE69SJS
- [Derived set](https://tamnd.github.io/math2/node/8ZAPAPYJFG.md): concept 8ZAPAPYJFG
- [Poincaré group](https://tamnd.github.io/math2/node/8ZQ9TA4Q2Q.md): example 8ZQ9TA4Q2Q
- [Frölicher space](https://tamnd.github.io/math2/node/8ZSFY9BWME.md): definition 8ZSFY9BWME
- [Moishezon manifold](https://tamnd.github.io/math2/node/8ZW9P9JRQD.md): concept 8ZW9P9JRQD
- [Exterior differential system](https://tamnd.github.io/math2/node/901CCKQHVY.md): concept 901CCKQHVY
- [Hopf degree theorem](https://tamnd.github.io/math2/node/902X7EHFSC.md): statement 902X7EHFSC
- [Reduced function](https://tamnd.github.io/math2/node/909B1WYQ4Q.md): concept 909B1WYQ4Q
- [Series acceleration](https://tamnd.github.io/math2/node/90ANMX55DP.md): definition 90ANMX55DP
- [Fractional ideal](https://tamnd.github.io/math2/node/90ASENKFMH.md): definition 90ASENKFMH
- [Geometric optics](https://tamnd.github.io/math2/node/90F3NWRHVD.md): concept 90F3NWRHVD
- [Logarithm of a matrix](https://tamnd.github.io/math2/node/90MF8M915N.md): definition 90MF8M915N
- [Clopen set](https://tamnd.github.io/math2/node/90PB1H41A6.md): definition 90PB1H41A6
- [11/8 conjecture](https://tamnd.github.io/math2/node/90RA6TAT7H.md): conjecture 90RA6TAT7H
- [Daubechies wavelet](https://tamnd.github.io/math2/node/90V8S5CXE6.md): example 90V8S5CXE6
- [Orientation of a manifold](https://tamnd.github.io/math2/node/913WEQERBE.md): concept 913WEQERBE
- [Henselization](https://tamnd.github.io/math2/node/917PGHVZPM.md): concept 917PGHVZPM
- [Stanley–Reisner ring](https://tamnd.github.io/math2/node/9186DYXE86.md): concept 9186DYXE86
- [Maximum modulus principle in several variables](https://tamnd.github.io/math2/node/91DW8V0PET.md): statement 91DW8V0PET
- [Supercompact cardinal](https://tamnd.github.io/math2/node/91GE4C1BC5.md): concept 91GE4C1BC5
- [Maslov index](https://tamnd.github.io/math2/node/91KJT4TXG8.md): concept 91KJT4TXG8
- [Formal group law in topology](https://tamnd.github.io/math2/node/91N7YTXJT2.md): definition 91N7YTXJT2
- [F-distribution](https://tamnd.github.io/math2/node/91PCJMS5VC.md): definition 91PCJMS5VC
- [Shift operator](https://tamnd.github.io/math2/node/91QMF4R1CY.md): definition 91QMF4R1CY
- [Singular control](https://tamnd.github.io/math2/node/91RFB97TNV.md): concept 91RFB97TNV
- [Hörmander's condition](https://tamnd.github.io/math2/node/91RZ5YARTM.md): concept 91RZ5YARTM
- [Open segment](https://tamnd.github.io/math2/node/91WG6QWRQ1.md): definition 91WG6QWRQ1
- [Conway base 13 function](https://tamnd.github.io/math2/node/91YF82M8AA.md): counterexample 91YF82M8AA
- [Quotient of a normed space](https://tamnd.github.io/math2/node/9240K052CS.md): concept 9240K052CS
- [Borel functional calculus](https://tamnd.github.io/math2/node/92433H25TC.md): concept 92433H25TC
- [Topological group](https://tamnd.github.io/math2/node/92700QAB5Y.md): concept 92700QAB5Y
- [Lax pair](https://tamnd.github.io/math2/node/928JSGQT9C.md): definition 928JSGQT9C
- [Hyperbolic partial differential equation](https://tamnd.github.io/math2/node/92C3S03AQZ.md): concept 92C3S03AQZ
- [Lyapunov equation](https://tamnd.github.io/math2/node/92GVPZH9FT.md): concept 92GVPZH9FT
- [Finite field arithmetic](https://tamnd.github.io/math2/node/92JCZQZSST.md): definition 92JCZQZSST
- [Multidimensional Mellin transform](https://tamnd.github.io/math2/node/92JY94ZJD8.md): concept 92JY94ZJD8
- [Dynkin quiver](https://tamnd.github.io/math2/node/92RQG73SC5.md): concept 92RQG73SC5
- [Pseudo algebraically closed field](https://tamnd.github.io/math2/node/92SQ10PN6J.md): concept 92SQ10PN6J
- [Theory](https://tamnd.github.io/math2/node/92SWXST1ZM.md): concept 92SWXST1ZM
- [Quasi-category](https://tamnd.github.io/math2/node/9328WEVJEK.md): concept 9328WEVJEK
- [Feuerbach's theorem](https://tamnd.github.io/math2/node/93343FN5PF.md): statement 93343FN5PF
- [Dual Radon transform](https://tamnd.github.io/math2/node/938SXZQS8G.md): concept 938SXZQS8G
- [Fixed-point property](https://tamnd.github.io/math2/node/93FKWZP2AZ.md): definition 93FKWZP2AZ
- [Nelson–Aalen estimator](https://tamnd.github.io/math2/node/93GK2SPBKV.md): concept 93GK2SPBKV
- [Global analysis](https://tamnd.github.io/math2/node/93MFWKENQ7.md): definition 93MFWKENQ7
- [Serre's theorem on regular local rings](https://tamnd.github.io/math2/node/93RQRG9R1P.md): statement 93RQRG9R1P
- [Initial and terminal objects](https://tamnd.github.io/math2/node/93T4M1R5CT.md): definition 93T4M1R5CT
- [Period three implies chaos](https://tamnd.github.io/math2/node/93VXAB8JYS.md): statement 93VXAB8JYS
- [Aoki–Rassias theorem](https://tamnd.github.io/math2/node/93WX6PZC8R.md): statement 93WX6PZC8R
- [Suslin's theorem](https://tamnd.github.io/math2/node/9418TGREZX.md): statement 9418TGREZX
- [P-adic analysis](https://tamnd.github.io/math2/node/942BKZYGQS.md): definition 942BKZYGQS
- [Finitely generated projective modules over local rings are free](https://tamnd.github.io/math2/node/942FK83S09.md): statement 942FK83S09
- [Adaptive quadrature](https://tamnd.github.io/math2/node/948A39DZDE.md): statement 948A39DZDE
- [Pascal's theorem](https://tamnd.github.io/math2/node/94AKWRHND0.md): statement 94AKWRHND0
- [Ionescu-Tulcea theorem](https://tamnd.github.io/math2/node/94ASD187AQ.md): statement 94ASD187AQ
- [Morse homology](https://tamnd.github.io/math2/node/94BWTKKGJM.md): concept 94BWTKKGJM
- [Gauss variational problem](https://tamnd.github.io/math2/node/94GSMEWPTN.md): problem 94GSMEWPTN
- [Quasi-identity](https://tamnd.github.io/math2/node/94GVY3APPQ.md): concept 94GVY3APPQ
- [Krull–Schmidt theorem](https://tamnd.github.io/math2/node/94M0BPV1W4.md): statement 94M0BPV1W4
- [Cartan matrix of an algebra](https://tamnd.github.io/math2/node/94SXH8EDPS.md): definition 94SXH8EDPS
- [Weakly singular kernel](https://tamnd.github.io/math2/node/94TNBAAJYX.md): concept 94TNBAAJYX
- [Weak-type estimate](https://tamnd.github.io/math2/node/9534R6VMY0.md): concept 9534R6VMY0
- [Proper morphism](https://tamnd.github.io/math2/node/9537RKBZPP.md): concept 9537RKBZPP
- [j-invariant of an elliptic curve](https://tamnd.github.io/math2/node/9538W1K1EX.md): concept 9538W1K1EX
- [Subnormal number](https://tamnd.github.io/math2/node/955ZNEXEF0.md): definition 955ZNEXEF0
- [Equalizer](https://tamnd.github.io/math2/node/956E1C2VE6.md): definition 956E1C2VE6
- [Schur's inequality](https://tamnd.github.io/math2/node/958TMJZ51Y.md): statement 958TMJZ51Y
- [Semi-continuity](https://tamnd.github.io/math2/node/95BSKT0S6M.md): concept 95BSKT0S6M
- [Cauchy matrix](https://tamnd.github.io/math2/node/95MTQMX59H.md): concept 95MTQMX59H
- [Rational equivalence](https://tamnd.github.io/math2/node/95N3QVHAQN.md): definition 95N3QVHAQN
- [Unitary dual](https://tamnd.github.io/math2/node/95Q8WZMN0W.md): concept 95Q8WZMN0W
- [Besov space](https://tamnd.github.io/math2/node/95QBBPRJ9E.md): concept 95QBBPRJ9E
- [System of parameters](https://tamnd.github.io/math2/node/95QTEPMDA4.md): concept 95QTEPMDA4
- [Fiber of a morphism](https://tamnd.github.io/math2/node/963QWX2E2Z.md): concept 963QWX2E2Z
- [Purely inseparable extension](https://tamnd.github.io/math2/node/966GNVGB8A.md): concept 966GNVGB8A
- [Magma](https://tamnd.github.io/math2/node/9689SFPVE9.md): concept 9689SFPVE9
- [Statistical decision theory](https://tamnd.github.io/math2/node/969F3AWBKC.md): definition 969F3AWBKC
- [Quasiregular map](https://tamnd.github.io/math2/node/96AZPZSQKR.md): definition 96AZPZSQKR
- [Lie group action](https://tamnd.github.io/math2/node/96B4E2GPKJ.md): concept 96B4E2GPKJ
- [Polylogarithm](https://tamnd.github.io/math2/node/96CHDZKGQ0.md): concept 96CHDZKGQ0
- [Δ-complex](https://tamnd.github.io/math2/node/96MQARP0N4.md): definition 96MQARP0N4
- [Mackey machine](https://tamnd.github.io/math2/node/972X7QM34D.md): definition 972X7QM34D
- [Filtered category](https://tamnd.github.io/math2/node/975PMY4FG8.md): concept 975PMY4FG8
- [Uhlenbeck compactness theorem](https://tamnd.github.io/math2/node/97CGS5KKRW.md): statement 97CGS5KKRW
- [Difference quotient](https://tamnd.github.io/math2/node/97DDM0ZBYG.md): definition 97DDM0ZBYG
- [Principal bundle](https://tamnd.github.io/math2/node/97GRC6YGF5.md): definition 97GRC6YGF5
- [Cauchy–Kovalevskaya theorem](https://tamnd.github.io/math2/node/97HMV9BT2W.md): statement 97HMV9BT2W
- [Compact operator](https://tamnd.github.io/math2/node/97J0B6VVDT.md): definition 97J0B6VVDT
- [Čech cohomology](https://tamnd.github.io/math2/node/97MYM5X7N5.md): concept 97MYM5X7N5
- [Comonad](https://tamnd.github.io/math2/node/97NPWM3S6E.md): concept 97NPWM3S6E
- [Final topology](https://tamnd.github.io/math2/node/97NS6JRVNE.md): definition 97NS6JRVNE
- [Sub-Riemannian manifold](https://tamnd.github.io/math2/node/97NYHNBZ2D.md): concept 97NYHNBZ2D
- [Exponential sum](https://tamnd.github.io/math2/node/97PBMQCPPM.md): concept 97PBMQCPPM
- [Ackermann function is not primitive recursive](https://tamnd.github.io/math2/node/97QF3KHSBY.md): statement 97QF3KHSBY
- [Baker–Pixley theorem](https://tamnd.github.io/math2/node/97SRYBF41G.md): statement 97SRYBF41G
- [Schwarz alternating method](https://tamnd.github.io/math2/node/97VNBS15CD.md): statement 97VNBS15CD
- [Multicategory](https://tamnd.github.io/math2/node/97ZY7H816E.md): definition 97ZY7H816E
- [Structure](https://tamnd.github.io/math2/node/983GZBQYT4.md): concept 983GZBQYT4
- [Prime ideal of a lattice](https://tamnd.github.io/math2/node/984525NJS0.md): definition 984525NJS0
- [Sublinear function](https://tamnd.github.io/math2/node/984ENA3M5Q.md): definition 984ENA3M5Q
- [Majority term](https://tamnd.github.io/math2/node/9879ZMW5AA.md): concept 9879ZMW5AA
- [F-algebra](https://tamnd.github.io/math2/node/98F5NBDR0V.md): definition 98F5NBDR0V
- [Finite element method](https://tamnd.github.io/math2/node/98GZ9DVHTA.md): concept 98GZ9DVHTA
- [Foliation](https://tamnd.github.io/math2/node/98HTB7MSPX.md): concept 98HTB7MSPX
- [Thom–Sebastiani theorem](https://tamnd.github.io/math2/node/98MVMS1NRR.md): statement 98MVMS1NRR
- [Rational singularity](https://tamnd.github.io/math2/node/98QASBAMEB.md): concept 98QASBAMEB
- [Weyl algebra](https://tamnd.github.io/math2/node/98QE7KY55W.md): concept 98QE7KY55W
- [Positive harmonic function](https://tamnd.github.io/math2/node/98TJTT6GZN.md): definition 98TJTT6GZN
- [Group action on a variety](https://tamnd.github.io/math2/node/990YQ65S1D.md): definition 990YQ65S1D
- [Admissible function](https://tamnd.github.io/math2/node/993HPHG9VF.md): definition 993HPHG9VF
- [Esakia space](https://tamnd.github.io/math2/node/996BWM6ZQ7.md): concept 996BWM6ZQ7
- [Riesz space](https://tamnd.github.io/math2/node/99709K2NJM.md): concept 99709K2NJM
- [Chandrasekhar's H-function](https://tamnd.github.io/math2/node/99879Z5Y1K.md): example 99879Z5Y1K
- [Injective hull](https://tamnd.github.io/math2/node/998RQZYBZ8.md): definition 998RQZYBZ8
- [Weierstrass product](https://tamnd.github.io/math2/node/99B3B7AAVT.md): definition 99B3B7AAVT
- [Atomistic lattice](https://tamnd.github.io/math2/node/99DVVV7BH2.md): concept 99DVVV7BH2
- [Lebesgue covering dimension](https://tamnd.github.io/math2/node/99GNAS98PV.md): definition 99GNAS98PV
- [Polynomial greatest common divisor](https://tamnd.github.io/math2/node/99J1WXG3F7.md): definition 99J1WXG3F7
- [Isometry of Riemannian manifolds](https://tamnd.github.io/math2/node/99RQHAGNW1.md): definition 99RQHAGNW1
- [de Boor's algorithm](https://tamnd.github.io/math2/node/99SCT623E8.md): definition 99SCT623E8
- [Central simple algebra](https://tamnd.github.io/math2/node/99T7Z4T39W.md): definition 99T7Z4T39W
- [Superadditive function](https://tamnd.github.io/math2/node/99VGJBCYC0.md): concept 99VGJBCYC0
- [Category of abelian groups](https://tamnd.github.io/math2/node/99ZVSM254K.md): example 99ZVSM254K
- [Hurwitz-stable matrix](https://tamnd.github.io/math2/node/9A3TBBHRFA.md): concept 9A3TBBHRFA
- [Universal Horn class](https://tamnd.github.io/math2/node/9A4CYWXPEA.md): concept 9A4CYWXPEA
- [Pettis integral](https://tamnd.github.io/math2/node/9A5F3JGTC0.md): concept 9A5F3JGTC0
- [Discrete series representation](https://tamnd.github.io/math2/node/9AB7HYGPEK.md): concept 9AB7HYGPEK
- [Jacobi field](https://tamnd.github.io/math2/node/9AHMM41PJ3.md): concept 9AHMM41PJ3
- [Prime radical](https://tamnd.github.io/math2/node/9AVE43G8RT.md): definition 9AVE43G8RT
- [Brunn–Minkowski theorem](https://tamnd.github.io/math2/node/9B02RPN7XD.md): statement 9B02RPN7XD
- [Projective plane](https://tamnd.github.io/math2/node/9B143EK6A8.md): definition 9B143EK6A8
- [Thurston norm](https://tamnd.github.io/math2/node/9B4A2WHAPQ.md): concept 9B4A2WHAPQ
- [Class number](https://tamnd.github.io/math2/node/9BDJZGZ5Z5.md): definition 9BDJZGZ5Z5
- [Mehler kernel](https://tamnd.github.io/math2/node/9BF9QTCRKW.md): statement 9BF9QTCRKW
- [Height of a prime ideal](https://tamnd.github.io/math2/node/9BKCZPMQT6.md): definition 9BKCZPMQT6
- [Orderable group](https://tamnd.github.io/math2/node/9BM25E20JQ.md): concept 9BM25E20JQ
- [Axiom schema of replacement](https://tamnd.github.io/math2/node/9BPQXY11NR.md): definition 9BPQXY11NR
- [Stability of a difference equation](https://tamnd.github.io/math2/node/9BPTKH2HYJ.md): definition 9BPTKH2HYJ
- [Coercive bilinear form](https://tamnd.github.io/math2/node/9BS2PM0227.md): definition 9BS2PM0227
- [Maxwell's equations](https://tamnd.github.io/math2/node/9BWWD4QK8Y.md): concept 9BWWD4QK8Y
- [Polish group](https://tamnd.github.io/math2/node/9C1WSBNQX9.md): concept 9C1WSBNQX9
- [Edgeworth expansion](https://tamnd.github.io/math2/node/9C4MPVMTA7.md): statement 9C4MPVMTA7
- [Subderivative](https://tamnd.github.io/math2/node/9C5WXYYQEK.md): concept 9C5WXYYQEK
- [Cahen–Mellin integral](https://tamnd.github.io/math2/node/9C8SS4522J.md): statement 9C8SS4522J
- [Betweenness](https://tamnd.github.io/math2/node/9CCSBY0JQW.md): definition 9CCSBY0JQW
- [Ovoid](https://tamnd.github.io/math2/node/9CD4SGERVP.md): definition 9CD4SGERVP
- [Restriction conjecture](https://tamnd.github.io/math2/node/9CE11G48JY.md): conjecture 9CE11G48JY
- [Topological vector space](https://tamnd.github.io/math2/node/9CJ0VSYZ81.md): definition 9CJ0VSYZ81
- [Simple algebra](https://tamnd.github.io/math2/node/9CJCXNYNK8.md): concept 9CJCXNYNK8
- [Skew group ring](https://tamnd.github.io/math2/node/9CMAJWWBNZ.md): definition 9CMAJWWBNZ
- [Solvable Lie group](https://tamnd.github.io/math2/node/9CNSVFQ801.md): concept 9CNSVFQ801
- [Weyl's lemma](https://tamnd.github.io/math2/node/9CWM3F4M04.md): statement 9CWM3F4M04
- [Sequential analysis](https://tamnd.github.io/math2/node/9CWYNB9003.md): concept 9CWYNB9003
- [Choquet integral](https://tamnd.github.io/math2/node/9CZVKQMCZP.md): definition 9CZVKQMCZP
- [Scott topology](https://tamnd.github.io/math2/node/9D26WKTFPT.md): definition 9D26WKTFPT
- [Dirichlet L-function](https://tamnd.github.io/math2/node/9D2Z543EJQ.md): concept 9D2Z543EJQ
- [Minimax estimator](https://tamnd.github.io/math2/node/9D47D1Q89E.md): definition 9D47D1Q89E
- [Crofton formula](https://tamnd.github.io/math2/node/9D5NT00H6G.md): statement 9D5NT00H6G
- [Finite morphism of rings](https://tamnd.github.io/math2/node/9D7WZEN64H.md): definition 9D7WZEN64H
- [Levitzki radical](https://tamnd.github.io/math2/node/9DD8M9R868.md): concept 9DD8M9R868
- [Ehrhart polynomial](https://tamnd.github.io/math2/node/9DEG0WRDHR.md): concept 9DEG0WRDHR
- [Zeros and poles](https://tamnd.github.io/math2/node/9DNDF7AX59.md): definition 9DNDF7AX59
- [Incidence matrix](https://tamnd.github.io/math2/node/9DNFN8M83X.md): concept 9DNFN8M83X
- [Lie supergroup](https://tamnd.github.io/math2/node/9DQB442K3Y.md): concept 9DQB442K3Y
- [Ginzburg–Landau equation](https://tamnd.github.io/math2/node/9DQD6HTTA5.md): definition 9DQD6HTTA5
- [Nested dissection](https://tamnd.github.io/math2/node/9DTJ87P4R1.md): statement 9DTJ87P4R1
- [Whittaker function](https://tamnd.github.io/math2/node/9DYE6V628H.md): definition 9DYE6V628H
- [Intersection theory](https://tamnd.github.io/math2/node/9E9T41A7XN.md): definition 9E9T41A7XN
- [Sullivan conjecture](https://tamnd.github.io/math2/node/9EA7KJ50AE.md): statement 9EA7KJ50AE
- [Bourbaki–Witt theorem](https://tamnd.github.io/math2/node/9EABA9ZDZS.md): statement 9EABA9ZDZS
- [Internal set theory](https://tamnd.github.io/math2/node/9EAHDX44B2.md): definition 9EAHDX44B2
- [Way-below relation](https://tamnd.github.io/math2/node/9EE43J2W16.md): definition 9EE43J2W16
- [Geometric invariant theory](https://tamnd.github.io/math2/node/9EG3A21ET1.md): concept 9EG3A21ET1
- [Continuous functional calculus](https://tamnd.github.io/math2/node/9EHVQ8Q0A8.md): definition 9EHVQ8Q0A8
- [Permutation group](https://tamnd.github.io/math2/node/9F0CPB14FQ.md): definition 9F0CPB14FQ
- [Cumulant](https://tamnd.github.io/math2/node/9F3ZTTT9HA.md): concept 9F3ZTTT9HA
- [Multiplication operator](https://tamnd.github.io/math2/node/9F6B05QAWH.md): example 9F6B05QAWH
- [Compact element](https://tamnd.github.io/math2/node/9F7TYES4KE.md): concept 9F7TYES4KE
- [Drazin inverse](https://tamnd.github.io/math2/node/9FB0AZSEGM.md): concept 9FB0AZSEGM
- [Calabi flow](https://tamnd.github.io/math2/node/9FBJED00CP.md): concept 9FBJED00CP
- [Derivative of a convolution](https://tamnd.github.io/math2/node/9FN0PFZZXZ.md): statement 9FN0PFZZXZ
- [Spherical trigonometry](https://tamnd.github.io/math2/node/9FR2ZRAVCE.md): definition 9FR2ZRAVCE
- [Idempotence](https://tamnd.github.io/math2/node/9FS1JFZSX4.md): concept 9FS1JFZSX4
- [Gelfand–Naimark–Segal construction](https://tamnd.github.io/math2/node/9FWZTK819W.md): concept 9FWZTK819W
- [Monoid object](https://tamnd.github.io/math2/node/9G1XS1XZ1F.md): concept 9G1XS1XZ1F
- [Laplace's method](https://tamnd.github.io/math2/node/9G7K8DBT9Y.md): concept 9G7K8DBT9Y
- [Summation by parts](https://tamnd.github.io/math2/node/9G82Z17DKA.md): statement 9G82Z17DKA
- [Series-parallel partial order](https://tamnd.github.io/math2/node/9G8XXBG9F3.md): definition 9G8XXBG9F3
- [Algebraic surface](https://tamnd.github.io/math2/node/9G91CJJ0JQ.md): definition 9G91CJJ0JQ
- [Hausdorff paradox](https://tamnd.github.io/math2/node/9G9G8ED97D.md): counterexample 9G9G8ED97D
- [Loewner order](https://tamnd.github.io/math2/node/9G9S92T1VJ.md): concept 9G9S92T1VJ
- [Idele class group](https://tamnd.github.io/math2/node/9GHD9MXNYV.md): concept 9GHD9MXNYV
- [Monotone likelihood ratio](https://tamnd.github.io/math2/node/9GKRK5B2B5.md): definition 9GKRK5B2B5
- [Δ-system lemma](https://tamnd.github.io/math2/node/9GKS3RBZ8V.md): statement 9GKS3RBZ8V
- [Convergence of filters](https://tamnd.github.io/math2/node/9GNAAHQJY6.md): definition 9GNAAHQJY6
- [Keller's conjecture](https://tamnd.github.io/math2/node/9GQQ7TK36T.md): statement 9GQQ7TK36T
- [Jacobian variety](https://tamnd.github.io/math2/node/9GS8XQMZED.md): definition 9GS8XQMZED
- [Cahn–Hilliard equation](https://tamnd.github.io/math2/node/9GVQ7JXC25.md): definition 9GVQ7JXC25
- [Cobweb plot](https://tamnd.github.io/math2/node/9GWGW4YRD9.md): definition 9GWGW4YRD9
- [Congruence-distributive variety](https://tamnd.github.io/math2/node/9GZ7F6ETC6.md): concept 9GZ7F6ETC6
- [Join-prime element](https://tamnd.github.io/math2/node/9H61HXXJ11.md): concept 9H61HXXJ11
- [Laver table](https://tamnd.github.io/math2/node/9H65YDXMGJ.md): example 9H65YDXMGJ
- [Completely normal space](https://tamnd.github.io/math2/node/9HBDQ2W9CF.md): concept 9HBDQ2W9CF
- [Meijer G-function transforms](https://tamnd.github.io/math2/node/9HD1MEV7M4.md): definition 9HD1MEV7M4
- [Dawson function](https://tamnd.github.io/math2/node/9HD9A3YWTN.md): definition 9HD9A3YWTN
- [Diffusion process](https://tamnd.github.io/math2/node/9HDQ094ADQ.md): concept 9HDQ094ADQ
- [Strongly pseudoconvex domain](https://tamnd.github.io/math2/node/9HHEJVCS2Q.md): definition 9HHEJVCS2Q
- [Perturbation theory of linear operators](https://tamnd.github.io/math2/node/9HNTG5C2XX.md): definition 9HNTG5C2XX
- [Nullary operation](https://tamnd.github.io/math2/node/9HPF7TE18C.md): concept 9HPF7TE18C
- [Locally finite poset](https://tamnd.github.io/math2/node/9HTB5KVS72.md): definition 9HTB5KVS72
- [Cauchy's logarithmic equation](https://tamnd.github.io/math2/node/9HTRR4P4M1.md): concept 9HTRR4P4M1
- [Moving least squares](https://tamnd.github.io/math2/node/9HWTPQHHZK.md): definition 9HWTPQHHZK
- [Pontryagin–Thom construction](https://tamnd.github.io/math2/node/9HXYTNKB24.md): concept 9HXYTNKB24
- [Rader's FFT algorithm](https://tamnd.github.io/math2/node/9HYM2ZMS31.md): statement 9HYM2ZMS31
- [Spectral sequence](https://tamnd.github.io/math2/node/9HZA9B51PZ.md): definition 9HZA9B51PZ
- [Continuous group homomorphism](https://tamnd.github.io/math2/node/9J5PQMG3HJ.md): concept 9J5PQMG3HJ
- [Beltrami equation](https://tamnd.github.io/math2/node/9J7GFJ1GPF.md): concept 9J7GFJ1GPF
- [Homology of a chain complex](https://tamnd.github.io/math2/node/9JA1N3CH80.md): definition 9JA1N3CH80
- [Sample mean and sample covariance](https://tamnd.github.io/math2/node/9JP83HPAEX.md): definition 9JP83HPAEX
- [Kummer's test](https://tamnd.github.io/math2/node/9JQEWK5405.md): statement 9JQEWK5405
- [Smoluchowski coagulation equation](https://tamnd.github.io/math2/node/9JRNCRZ97G.md): example 9JRNCRZ97G
- [Gibbs phenomenon](https://tamnd.github.io/math2/node/9JVJKJTXDE.md): definition 9JVJKJTXDE
- [Hilbert C*-module](https://tamnd.github.io/math2/node/9K11FXYX5S.md): definition 9K11FXYX5S
- [Banach manifold](https://tamnd.github.io/math2/node/9K15AMGBV9.md): concept 9K15AMGBV9
- [Favard constant](https://tamnd.github.io/math2/node/9K1W9Q49NR.md): concept 9K1W9Q49NR
- [Nevanlinna theory of difference equations](https://tamnd.github.io/math2/node/9KAQ1REPE8.md): concept 9KAQ1REPE8
- [Tightness of measures](https://tamnd.github.io/math2/node/9KD9R5TGD3.md): definition 9KD9R5TGD3
- [Induced representation](https://tamnd.github.io/math2/node/9KDC2DQ72A.md): definition 9KDC2DQ72A
- [Serre fibration](https://tamnd.github.io/math2/node/9KE5E84M3M.md): definition 9KE5E84M3M
- [Rng](https://tamnd.github.io/math2/node/9KGA2BZC1M.md): definition 9KGA2BZC1M
- [Second fundamental form](https://tamnd.github.io/math2/node/9KHQR16P1B.md): concept 9KHQR16P1B
- [Bihari–LaSalle inequality](https://tamnd.github.io/math2/node/9KMJSKNZ72.md): statement 9KMJSKNZ72
- [Distinguished triangle](https://tamnd.github.io/math2/node/9KP392QCGV.md): definition 9KP392QCGV
- [Banach algebra](https://tamnd.github.io/math2/node/9KYWN94QDN.md): concept 9KYWN94QDN
- [Kernel of an integral transform](https://tamnd.github.io/math2/node/9M0NNNFJGM.md): concept 9M0NNNFJGM
- [Minkowski's first inequality](https://tamnd.github.io/math2/node/9M8XJX4DQ7.md): statement 9M8XJX4DQ7
- [Nuclear C*-algebra](https://tamnd.github.io/math2/node/9MB4JD02YF.md): definition 9MB4JD02YF
- [Fishburn's theorem on interval orders](https://tamnd.github.io/math2/node/9MMG0335JE.md): statement 9MMG0335JE
- [Galois cohomology](https://tamnd.github.io/math2/node/9MPCZ3EAJM.md): concept 9MPCZ3EAJM
- [Autoregressive integrated moving average](https://tamnd.github.io/math2/node/9MPM6E51WV.md): definition 9MPM6E51WV
- [Frobenius algebra](https://tamnd.github.io/math2/node/9MXPNHHMT1.md): definition 9MXPNHHMT1
- [Jackson kernel](https://tamnd.github.io/math2/node/9MZFTX77CD.md): concept 9MZFTX77CD
- [Nilpotence theorem](https://tamnd.github.io/math2/node/9N027RT5R6.md): statement 9N027RT5R6
- [Blaschke product](https://tamnd.github.io/math2/node/9N6R5ADJXT.md): concept 9N6R5ADJXT
- [Cooley–Tukey FFT algorithm](https://tamnd.github.io/math2/node/9N8D9E5TB2.md): statement 9N8D9E5TB2
- [Condensed set](https://tamnd.github.io/math2/node/9NM376YWF2.md): concept 9NM376YWF2
- [Proximal operator](https://tamnd.github.io/math2/node/9NMMRYKBNG.md): definition 9NMMRYKBNG
- [Natural boundary condition](https://tamnd.github.io/math2/node/9NPY7BF6ZJ.md): definition 9NPY7BF6ZJ
- [Kalman rank condition](https://tamnd.github.io/math2/node/9NSCZ2KBEB.md): statement 9NSCZ2KBEB
- [Hadamard's inequality in Fredholm theory](https://tamnd.github.io/math2/node/9NZ9CXC8FY.md): statement 9NZ9CXC8FY
- [Leavitt path algebra](https://tamnd.github.io/math2/node/9NZ9FY1S7C.md): concept 9NZ9FY1S7C
- [Euler–Maclaurin formula](https://tamnd.github.io/math2/node/9P839VB8S7.md): statement 9P839VB8S7
- [Profunctor](https://tamnd.github.io/math2/node/9P83YFRWDJ.md): definition 9P83YFRWDJ
- [Anger function](https://tamnd.github.io/math2/node/9P87VQZ0P7.md): concept 9P87VQZ0P7
- [Exponential type](https://tamnd.github.io/math2/node/9P8D8YAHQS.md): concept 9P8D8YAHQS
- [Cartan's classification of symmetric spaces](https://tamnd.github.io/math2/node/9P8VK3H4YZ.md): statement 9P8VK3H4YZ
- [Elimination theory](https://tamnd.github.io/math2/node/9PA8CVSWPD.md): concept 9PA8CVSWPD
- [Quadratic convergence of Newton's method](https://tamnd.github.io/math2/node/9PBJF44KBX.md): statement 9PBJF44KBX
- [Lattès map](https://tamnd.github.io/math2/node/9PK608W8MS.md): example 9PK608W8MS
- [Potential theory](https://tamnd.github.io/math2/node/9PM8Q2E30Q.md): concept 9PM8Q2E30Q
- [Mixed volume](https://tamnd.github.io/math2/node/9PN6QZKFCB.md): definition 9PN6QZKFCB
- [Cauchy–Riemann equations in several variables](https://tamnd.github.io/math2/node/9PT772R94A.md): concept 9PT772R94A
- [Riemann solver](https://tamnd.github.io/math2/node/9PTJWSCFS5.md): definition 9PTJWSCFS5
- [Harmonic function on a ball](https://tamnd.github.io/math2/node/9PV9CN8X0Y.md): example 9PV9CN8X0Y
- [Discrete-time algebraic Riccati equation](https://tamnd.github.io/math2/node/9PW3GX1579.md): definition 9PW3GX1579
- [Equational basis](https://tamnd.github.io/math2/node/9Q008GMZSW.md): concept 9Q008GMZSW
- [Kolmogorov–Nagumo characterization of quasi-arithmetic means](https://tamnd.github.io/math2/node/9Q5DPYQ5NP.md): statement 9Q5DPYQ5NP
- [Napoleon's theorem](https://tamnd.github.io/math2/node/9Q5P1P1ABX.md): statement 9Q5P1P1ABX
- [Iterated kernel](https://tamnd.github.io/math2/node/9QNBFERMHD.md): concept 9QNBFERMHD
- [Field with one element](https://tamnd.github.io/math2/node/9QP9ZAGH6T.md): concept 9QP9ZAGH6T
- [Nemytskii operator](https://tamnd.github.io/math2/node/9QQN0QH2YT.md): definition 9QQN0QH2YT
- [Erlangen program](https://tamnd.github.io/math2/node/9QW65PGEPW.md): concept 9QW65PGEPW
- [Overdetermined system](https://tamnd.github.io/math2/node/9R899M1746.md): definition 9R899M1746
- [Injective object](https://tamnd.github.io/math2/node/9R9MVA4PWR.md): definition 9R9MVA4PWR
- [Polycyclic group](https://tamnd.github.io/math2/node/9RBA56NXX3.md): concept 9RBA56NXX3
- [Paraproduct](https://tamnd.github.io/math2/node/9RCFS9W90H.md): concept 9RCFS9W90H
- [Cyclic homology](https://tamnd.github.io/math2/node/9RGFJDFY96.md): definition 9RGFJDFY96
- [Borel–Kolmogorov paradox](https://tamnd.github.io/math2/node/9RMS0ZJ618.md): counterexample 9RMS0ZJ618
- [Łoś–Tarski preservation theorem](https://tamnd.github.io/math2/node/9RN0KYXNT1.md): statement 9RN0KYXNT1
- [Jacobi transform](https://tamnd.github.io/math2/node/9RNQX1X11T.md): definition 9RNQX1X11T
- [Lattice of partitions](https://tamnd.github.io/math2/node/9RP82GD0Q4.md): example 9RP82GD0Q4
- [Wilcoxon signed-rank test](https://tamnd.github.io/math2/node/9RPYB7S4HM.md): definition 9RPYB7S4HM
- [Deligne–Lusztig theory](https://tamnd.github.io/math2/node/9RS2YPTPJY.md): definition 9RS2YPTPJY
- [Contact process](https://tamnd.github.io/math2/node/9RVKQJQ5JJ.md): example 9RVKQJQ5JJ
- [Subfactor](https://tamnd.github.io/math2/node/9RWS7TZHXH.md): concept 9RWS7TZHXH
- [Uniqueness of best approximation in strictly convex spaces](https://tamnd.github.io/math2/node/9S18DSRFND.md): statement 9S18DSRFND
- [Brelot space](https://tamnd.github.io/math2/node/9SDYXHB8P6.md): definition 9SDYXHB8P6
- [Topological semigroup](https://tamnd.github.io/math2/node/9SEF1CWT6X.md): definition 9SEF1CWT6X
- [Equivalence of varieties](https://tamnd.github.io/math2/node/9SNTXYEEBQ.md): concept 9SNTXYEEBQ
- [Almost complex manifold](https://tamnd.github.io/math2/node/9SYWRGVTRS.md): concept 9SYWRGVTRS
- [Response surface methodology](https://tamnd.github.io/math2/node/9SYZS6S4NY.md): definition 9SYZS6S4NY
- [Entropic algebra](https://tamnd.github.io/math2/node/9T2XVJNBBY.md): concept 9T2XVJNBBY
- [Single layer potential](https://tamnd.github.io/math2/node/9T6ZEV4125.md): concept 9T6ZEV4125
- [Liouville's theorem on phase-space volume](https://tamnd.github.io/math2/node/9T7D9ZRWYY.md): statement 9T7D9ZRWYY
- [Non-uniform rational B-spline](https://tamnd.github.io/math2/node/9TBVX1RHDB.md): definition 9TBVX1RHDB
- [Compact Lie group](https://tamnd.github.io/math2/node/9TFC6ADC1C.md): concept 9TFC6ADC1C
- [Schmidt subspace theorem](https://tamnd.github.io/math2/node/9TPS3W497D.md): statement 9TPS3W497D
- [Positive line bundle](https://tamnd.github.io/math2/node/9TTJRYP841.md): definition 9TTJRYP841
- [Handle decomposition](https://tamnd.github.io/math2/node/9V11N8VVTP.md): concept 9V11N8VVTP
- [Seifert fiber space](https://tamnd.github.io/math2/node/9V1GF5WMKA.md): concept 9V1GF5WMKA
- [Shifted Legendre polynomials](https://tamnd.github.io/math2/node/9V3QA0KTW8.md): concept 9V3QA0KTW8
- [Neighborly polytope](https://tamnd.github.io/math2/node/9V4PR1RVXW.md): concept 9V4PR1RVXW
- [Representation theory of SU(2)](https://tamnd.github.io/math2/node/9V6PRD1Z90.md): example 9V6PRD1Z90
- [Type](https://tamnd.github.io/math2/node/9V71Q3SD1G.md): concept 9V71Q3SD1G
- [Spectral radius](https://tamnd.github.io/math2/node/9V7YQA85PC.md): concept 9V7YQA85PC
- [Category of elements](https://tamnd.github.io/math2/node/9V800QGBQH.md): definition 9V800QGBQH
- [Matrix difference equation](https://tamnd.github.io/math2/node/9VA6P5KQQT.md): concept 9VA6P5KQQT
- [Approximate identity](https://tamnd.github.io/math2/node/9VA95Q3K6G.md): definition 9VA95Q3K6G
- [Algebraic cycle](https://tamnd.github.io/math2/node/9VFWPZEVGN.md): concept 9VFWPZEVGN
- [Symmetric derivative](https://tamnd.github.io/math2/node/9VN0XT2THJ.md): definition 9VN0XT2THJ
- [Symmetric space](https://tamnd.github.io/math2/node/9VQED5PARF.md): definition 9VQED5PARF
- [Constructive analysis](https://tamnd.github.io/math2/node/9VW7VX0SXB.md): definition 9VW7VX0SXB
- [Sheafification](https://tamnd.github.io/math2/node/9VY3JT31RW.md): definition 9VY3JT31RW
- [Poincaré duality](https://tamnd.github.io/math2/node/9VY90HJVZX.md): statement 9VY90HJVZX
- [Z-group](https://tamnd.github.io/math2/node/9VYFXJQ3DE.md): concept 9VYFXJQ3DE
- [Commuting probability](https://tamnd.github.io/math2/node/9W6KMR96R9.md): definition 9W6KMR96R9
- [Pariah groups](https://tamnd.github.io/math2/node/9W6SSHGH4F.md): definition 9W6SSHGH4F
- [Newton polynomial](https://tamnd.github.io/math2/node/9W77FPTM9E.md): concept 9W77FPTM9E
- [Tammes problem](https://tamnd.github.io/math2/node/9WD2TZ4ZB5.md): problem 9WD2TZ4ZB5
- [Strange attractor](https://tamnd.github.io/math2/node/9WKCZGG906.md): concept 9WKCZGG906
- [Bromwich integral](https://tamnd.github.io/math2/node/9WRE8KEJJX.md): definition 9WRE8KEJJX
- [Logarithmic capacity](https://tamnd.github.io/math2/node/9WRXN9RXG1.md): definition 9WRXN9RXG1
- [Deduction theorem](https://tamnd.github.io/math2/node/9X08XK5RV2.md): statement 9X08XK5RV2
- [Graded manifold](https://tamnd.github.io/math2/node/9X3B0EX04F.md): definition 9X3B0EX04F
- [Bernstein's theorem on analytic functions](https://tamnd.github.io/math2/node/9X3QFZEHM1.md): statement 9X3QFZEHM1
- [Elementary embedding](https://tamnd.github.io/math2/node/9X5Z7DH5BF.md): definition 9X5Z7DH5BF
- [Pfaff–Saalschütz theorem](https://tamnd.github.io/math2/node/9X6KG770GV.md): statement 9X6KG770GV
- [Holomorphic functional calculus](https://tamnd.github.io/math2/node/9XEXZSZSZ9.md): definition 9XEXZSZSZ9
- [Shape derivative](https://tamnd.github.io/math2/node/9XFEG6ANGM.md): concept 9XFEG6ANGM
- [Maximum principle for subharmonic functions](https://tamnd.github.io/math2/node/9XJ5FBQD7M.md): statement 9XJ5FBQD7M
- [Grothendieck construction](https://tamnd.github.io/math2/node/9XYYWS5AFJ.md): definition 9XYYWS5AFJ
- [Fundamental solution of the Laplacian](https://tamnd.github.io/math2/node/9Y01DS79PR.md): definition 9Y01DS79PR
- [Homotopy category of chain complexes](https://tamnd.github.io/math2/node/9Y5Y92R4D5.md): concept 9Y5Y92R4D5
- [Homotopy](https://tamnd.github.io/math2/node/9YA6S6HD9V.md): concept 9YA6S6HD9V
- [Space of functions of bounded variation](https://tamnd.github.io/math2/node/9YQ999QHHT.md): concept 9YQ999QHHT
- [Gromov–Witten invariant](https://tamnd.github.io/math2/node/9YTV2296X7.md): definition 9YTV2296X7
- [Hermite interpolation](https://tamnd.github.io/math2/node/9Z50M2WPTP.md): concept 9Z50M2WPTP
- [Difference kernel](https://tamnd.github.io/math2/node/9Z6B67DCKR.md): definition 9Z6B67DCKR
- [Herglotz representation theorem](https://tamnd.github.io/math2/node/9ZBAX4ZPX0.md): statement 9ZBAX4ZPX0
- [Artinian reduced rings are semisimple](https://tamnd.github.io/math2/node/9ZBM46J88T.md): statement 9ZBM46J88T
- [Euler equations](https://tamnd.github.io/math2/node/9ZBWK5WSC7.md): definition 9ZBWK5WSC7
- [Secant method](https://tamnd.github.io/math2/node/9ZC177PVPC.md): statement 9ZC177PVPC
- [Nash–Moser theorem](https://tamnd.github.io/math2/node/9ZCCXT3KKP.md): statement 9ZCCXT3KKP
- [Poisson–Jensen formula](https://tamnd.github.io/math2/node/9ZG6HNM84Q.md): statement 9ZG6HNM84Q
- [Kolmogorov epsilon-entropy](https://tamnd.github.io/math2/node/9ZK07AKD4M.md): definition 9ZK07AKD4M
- [Product measure](https://tamnd.github.io/math2/node/9ZP4ER2N65.md): definition 9ZP4ER2N65
- [Critical point theory](https://tamnd.github.io/math2/node/9ZQC7ARWTT.md): concept 9ZQC7ARWTT
- [Differential inclusion](https://tamnd.github.io/math2/node/9ZSMF6NYKM.md): concept 9ZSMF6NYKM
- [Subharmonic function](https://tamnd.github.io/math2/node/9ZV3T5C6B7.md): definition 9ZV3T5C6B7
- [Moore family](https://tamnd.github.io/math2/node/9ZVCJWYPB2.md): definition 9ZVCJWYPB2
- [Combinatorial Nullstellensatz](https://tamnd.github.io/math2/node/9ZW7BCJMDT.md): statement 9ZW7BCJMDT
- [Lifting scheme](https://tamnd.github.io/math2/node/A00DAT34F9.md): definition A00DAT34F9
- [Variance reduction](https://tamnd.github.io/math2/node/A01GEGWKVB.md): concept A01GEGWKVB
- [Metropolis–Hastings algorithm](https://tamnd.github.io/math2/node/A02K3MKQ3Q.md): definition A02K3MKQ3Q
- [Empirical distribution function](https://tamnd.github.io/math2/node/A05WR6TEK5.md): concept A05WR6TEK5
- [Coalgebra](https://tamnd.github.io/math2/node/A06RA8KB0T.md): concept A06RA8KB0T
- [General linear model](https://tamnd.github.io/math2/node/A06SGFCSCE.md): concept A06SGFCSCE
- [Hausdorff measure](https://tamnd.github.io/math2/node/A0926EFKVP.md): definition A0926EFKVP
- [Max-plus algebra](https://tamnd.github.io/math2/node/A0E47AYDPX.md): definition A0E47AYDPX
- [Poisson's equation](https://tamnd.github.io/math2/node/A0F09CMEJM.md): definition A0F09CMEJM
- [Polymorphism of a relation](https://tamnd.github.io/math2/node/A0K9EZRM7E.md): concept A0K9EZRM7E
- [Calderón–Vaillancourt theorem](https://tamnd.github.io/math2/node/A0MXE8YWZA.md): statement A0MXE8YWZA
- [Partially ordered ring](https://tamnd.github.io/math2/node/A0NP6HQNPX.md): definition A0NP6HQNPX
- [Landen's transformation](https://tamnd.github.io/math2/node/A0V7ZAEQBW.md): statement A0V7ZAEQBW
- [Rigid monoidal category](https://tamnd.github.io/math2/node/A10WW8GEKR.md): concept A10WW8GEKR
- [Hodge theorem](https://tamnd.github.io/math2/node/A127WFVYKW.md): statement A127WFVYKW
- [Riemann domain over C^n](https://tamnd.github.io/math2/node/A15BF6QHDR.md): concept A15BF6QHDR
- [Graph theory](https://tamnd.github.io/math2/node/A167XK4YTF.md): concept A167XK4YTF
- [Nonlinear regression](https://tamnd.github.io/math2/node/A16HP3P271.md): definition A16HP3P271
- [Bruhat order](https://tamnd.github.io/math2/node/A17Q3X6285.md): example A17Q3X6285
- [Orthogonal polynomials](https://tamnd.github.io/math2/node/A18HPFPC4D.md): definition A18HPFPC4D
- [Satellite knot](https://tamnd.github.io/math2/node/A1E8TDEVJ9.md): concept A1E8TDEVJ9
- [Spectral theorem for compact self-adjoint operators](https://tamnd.github.io/math2/node/A1G25R99PY.md): statement A1G25R99PY
- [Conjugate function](https://tamnd.github.io/math2/node/A1H48DDW49.md): concept A1H48DDW49
- [Variational capacity](https://tamnd.github.io/math2/node/A1KWT817JX.md): concept A1KWT817JX
- [Polymorphism of a relation](https://tamnd.github.io/math2/node/A1MRBS3Z5C.md): definition A1MRBS3Z5C
- [Courant–Fischer characterization of characteristic values](https://tamnd.github.io/math2/node/A1PANCDYTG.md): statement A1PANCDYTG
- [Fourier transform as Laplace transform on the imaginary axis](https://tamnd.github.io/math2/node/A1X6T9ZD9E.md): definition A1X6T9ZD9E
- [Shift space](https://tamnd.github.io/math2/node/A1XE59ZHNM.md): concept A1XE59ZHNM
- [Subderivative](https://tamnd.github.io/math2/node/A289X08MR1.md): definition A289X08MR1
- [Category](https://tamnd.github.io/math2/node/A2DE66937D.md): concept A2DE66937D
- [Discrete valuation](https://tamnd.github.io/math2/node/A2KKZAV29W.md): concept A2KKZAV29W
- [Poisson manifold](https://tamnd.github.io/math2/node/A2KQ49Y4NF.md): concept A2KQ49Y4NF
- [Crisis](https://tamnd.github.io/math2/node/A2MX36J5GD.md): definition A2MX36J5GD
- [Transversality](https://tamnd.github.io/math2/node/A2PDZ8GA9C.md): concept A2PDZ8GA9C
- [Mellin convolution](https://tamnd.github.io/math2/node/A2PMJG7J68.md): concept A2PMJG7J68
- [Comb space](https://tamnd.github.io/math2/node/A2Q4RXD162.md): counterexample A2Q4RXD162
- [Failure of Aizerman's conjecture](https://tamnd.github.io/math2/node/A2QG6D75KR.md): counterexample A2QG6D75KR
- [CR manifold](https://tamnd.github.io/math2/node/A2VXXBYX6A.md): definition A2VXXBYX6A
- [Backward error analysis](https://tamnd.github.io/math2/node/A2W5SNSC05.md): definition A2W5SNSC05
- [Germ of an analytic set](https://tamnd.github.io/math2/node/A2WHCVP6JN.md): definition A2WHCVP6JN
- [Homotopy limit](https://tamnd.github.io/math2/node/A2YTPG6CQF.md): definition A2YTPG6CQF
- [Dandelin spheres](https://tamnd.github.io/math2/node/A2ZXW6QTEA.md): definition A2ZXW6QTEA
- [Multiple gamma function](https://tamnd.github.io/math2/node/A30TFC9Z92.md): definition A30TFC9Z92
- [Monge–Ampère equation](https://tamnd.github.io/math2/node/A3413S0RTM.md): definition A3413S0RTM
- [Topological homogeneity](https://tamnd.github.io/math2/node/A34VR4FMRE.md): concept A34VR4FMRE
- [Clifford algebra](https://tamnd.github.io/math2/node/A38B1QFK20.md): concept A38B1QFK20
- [Real form of a Lie group](https://tamnd.github.io/math2/node/A399EBX2YD.md): definition A399EBX2YD
- [Proj construction](https://tamnd.github.io/math2/node/A39XQHCVNW.md): concept A39XQHCVNW
- [Hopf invariant one problem](https://tamnd.github.io/math2/node/A3AP0P0D4M.md): statement A3AP0P0D4M
- [Linear topology](https://tamnd.github.io/math2/node/A3DP43YC3Z.md): definition A3DP43YC3Z
- [Generalized minimal residual method](https://tamnd.github.io/math2/node/A3FR6PNEPB.md): statement A3FR6PNEPB
- [Blichfeldt's theorem](https://tamnd.github.io/math2/node/A3FTHW4NES.md): statement A3FTHW4NES
- [Runge domain](https://tamnd.github.io/math2/node/A3R53S95KP.md): concept A3R53S95KP
- [Lebesgue covering dimension](https://tamnd.github.io/math2/node/A3W7RGXXK9.md): concept A3W7RGXXK9
- [Tate cohomology](https://tamnd.github.io/math2/node/A3ZHAPT767.md): definition A3ZHAPT767
- [Disjoint union topology](https://tamnd.github.io/math2/node/A486VR787Z.md): definition A486VR787Z
- [Ordered group](https://tamnd.github.io/math2/node/A49BATWQEH.md): definition A49BATWQEH
- [Skeleton of a category](https://tamnd.github.io/math2/node/A49QJYG669.md): concept A49QJYG669
- [Hausdorff topological group](https://tamnd.github.io/math2/node/A4E24P47KP.md): definition A4E24P47KP
- [Almgren–Pitts min-max theory](https://tamnd.github.io/math2/node/A4E7SZ935M.md): definition A4E7SZ935M
- [Projective variety](https://tamnd.github.io/math2/node/A4H4CZ2V35.md): concept A4H4CZ2V35
- [Prolongation of a section to a jet bundle](https://tamnd.github.io/math2/node/A4P47SGE4S.md): definition A4P47SGE4S
- [Moduli space](https://tamnd.github.io/math2/node/A4SS2SN0MB.md): definition A4SS2SN0MB
- [Multiplication formula for the Fourier transform](https://tamnd.github.io/math2/node/A4TRMA5V4V.md): statement A4TRMA5V4V
- [Cusp form](https://tamnd.github.io/math2/node/A50Y012RBP.md): concept A50Y012RBP
- [Brelot space](https://tamnd.github.io/math2/node/A51SQRPM5T.md): concept A51SQRPM5T
- [Uniform space](https://tamnd.github.io/math2/node/A54T86KS63.md): concept A54T86KS63
- [Gauss–Lucas theorem](https://tamnd.github.io/math2/node/A56086N8XQ.md): statement A56086N8XQ
- [Tautochrone problem](https://tamnd.github.io/math2/node/A56P5S5D6Z.md): problem A56P5S5D6Z
- [Frullani integral](https://tamnd.github.io/math2/node/A58Z1DQ593.md): statement A58Z1DQ593
- [Conditionally complete lattice](https://tamnd.github.io/math2/node/A5D94C38HZ.md): definition A5D94C38HZ
- [Shock wave](https://tamnd.github.io/math2/node/A5GDAAYNM0.md): definition A5GDAAYNM0
- [Ideal completion](https://tamnd.github.io/math2/node/A5HTJ565DN.md): concept A5HTJ565DN
- [Cartan decomposition of a Lie group](https://tamnd.github.io/math2/node/A5JB9JQKR0.md): statement A5JB9JQKR0
- [Discrete-time dynamical system](https://tamnd.github.io/math2/node/A5RCGG79JX.md): definition A5RCGG79JX
- [Minimum total potential energy principle](https://tamnd.github.io/math2/node/A5T30ZHVGH.md): statement A5T30ZHVGH
- [Dirac operator on a spin manifold](https://tamnd.github.io/math2/node/A5TDM8F4KB.md): definition A5TDM8F4KB
- [Fourier transform on locally compact abelian groups](https://tamnd.github.io/math2/node/A625F93ANV.md): definition A625F93ANV
- [Arc in a finite plane](https://tamnd.github.io/math2/node/A66VNJ7BED.md): definition A66VNJ7BED
- [Birkhoff's axioms](https://tamnd.github.io/math2/node/A67HV2KCHR.md): definition A67HV2KCHR
- [Collar neighbourhood theorem](https://tamnd.github.io/math2/node/A68WQ4ZAF0.md): statement A68WQ4ZAF0
- [Continuous-time Markov chain](https://tamnd.github.io/math2/node/A69F1SM1AC.md): definition A69F1SM1AC
- [Directional statistics](https://tamnd.github.io/math2/node/A6BW0KPXNC.md): definition A6BW0KPXNC
- [Ultrametric space](https://tamnd.github.io/math2/node/A6DPR620B5.md): concept A6DPR620B5
- [Irregular singularity](https://tamnd.github.io/math2/node/A6EXFAP5EQ.md): definition A6EXFAP5EQ
- [Marden's theorem](https://tamnd.github.io/math2/node/A6GYWEFFTY.md): statement A6GYWEFFTY
- [Distinguished boundary of a polydisc](https://tamnd.github.io/math2/node/A6HN207Y2T.md): concept A6HN207Y2T
- [Term rewriting for equational theories](https://tamnd.github.io/math2/node/A6J731Y193.md): concept A6J731Y193
- [Pythagorean field](https://tamnd.github.io/math2/node/A6M3GTHM40.md): concept A6M3GTHM40
- [Trigonometric polynomial](https://tamnd.github.io/math2/node/A6YS3A9DNP.md): definition A6YS3A9DNP
- [James–Stein estimator](https://tamnd.github.io/math2/node/A6Z3GR2RWP.md): definition A6Z3GR2RWP
- [Core model](https://tamnd.github.io/math2/node/A6ZJ8MSCAY.md): concept A6ZJ8MSCAY
- [Selmer group](https://tamnd.github.io/math2/node/A6ZREX2DMW.md): definition A6ZREX2DMW
- [Shift space](https://tamnd.github.io/math2/node/A71EWRZBZV.md): definition A71EWRZBZV
- [Kurosh–Amitsur radical theory](https://tamnd.github.io/math2/node/A72J7ZVNR5.md): concept A72J7ZVNR5
- [Schur-convex function](https://tamnd.github.io/math2/node/A72XTVF2GK.md): concept A72XTVF2GK
- [Darboux function](https://tamnd.github.io/math2/node/A7549DJW1N.md): definition A7549DJW1N
- [Melnikov method](https://tamnd.github.io/math2/node/A755BDRP1A.md): concept A755BDRP1A
- [Signature operator](https://tamnd.github.io/math2/node/A77EHY9DAN.md): definition A77EHY9DAN
- [Curry–Howard correspondence](https://tamnd.github.io/math2/node/A77FER1Q5H.md): concept A77FER1Q5H
- [Modulus of continuity](https://tamnd.github.io/math2/node/A79PHYDXMW.md): definition A79PHYDXMW
- [Gauss's test](https://tamnd.github.io/math2/node/A7MJXJERNY.md): statement A7MJXJERNY
- [Persistent homology](https://tamnd.github.io/math2/node/A7MZ6GG7W0.md): definition A7MZ6GG7W0
- [Multidimensional Z-transform](https://tamnd.github.io/math2/node/A7NF067M7N.md): concept A7NF067M7N
- [Fractional Sobolev space](https://tamnd.github.io/math2/node/A7Q6581B1F.md): definition A7Q6581B1F
- [Infinite loop space](https://tamnd.github.io/math2/node/A7VB2NSGJ2.md): concept A7VB2NSGJ2
- [Grothendieck category](https://tamnd.github.io/math2/node/A7W468BQSM.md): concept A7W468BQSM
- [Newtonian kernel](https://tamnd.github.io/math2/node/A7X2MQP0KM.md): definition A7X2MQP0KM
- [Bounded variation](https://tamnd.github.io/math2/node/A8251BHCMP.md): concept A8251BHCMP
- [Virtual knot](https://tamnd.github.io/math2/node/A82BZ4JQR5.md): definition A82BZ4JQR5
- [Absolute summability](https://tamnd.github.io/math2/node/A82GRSZMVW.md): concept A82GRSZMVW
- [Mittag-Leffler summation](https://tamnd.github.io/math2/node/A8A21Z0RJG.md): definition A8A21Z0RJG
- [Determinant trick](https://tamnd.github.io/math2/node/A8HRT9H3XA.md): statement A8HRT9H3XA
- [Nijenhuis tensor](https://tamnd.github.io/math2/node/A8Q8RVFWWE.md): definition A8Q8RVFWWE
- [Ideal of compact operators](https://tamnd.github.io/math2/node/A8R7V30ANX.md): definition A8R7V30ANX
- [Mortar methods](https://tamnd.github.io/math2/node/A8S152S9CG.md): definition A8S152S9CG
- [Symplectomorphism](https://tamnd.github.io/math2/node/A8TS2SNX9Z.md): concept A8TS2SNX9Z
- [Separable algebra](https://tamnd.github.io/math2/node/A8WVA43YXS.md): definition A8WVA43YXS
- [Numerical integration](https://tamnd.github.io/math2/node/A915CSRFEF.md): definition A915CSRFEF
- [Van Wijngaarden transformation](https://tamnd.github.io/math2/node/A93A03C5XR.md): definition A93A03C5XR
- [Heine's transformation](https://tamnd.github.io/math2/node/A93A5VATSD.md): statement A93A5VATSD
- [Second fundamental form](https://tamnd.github.io/math2/node/A9514690T3.md): definition A9514690T3
- [Order polytope](https://tamnd.github.io/math2/node/A9A7S5AYY1.md): definition A9A7S5AYY1
- [Coxeter group](https://tamnd.github.io/math2/node/A9AHHSV1EE.md): definition A9AHHSV1EE
- [Rectangular function](https://tamnd.github.io/math2/node/A9B7Z5SA9K.md): example A9B7Z5SA9K
- [Packing density](https://tamnd.github.io/math2/node/A9C785ESPD.md): definition A9C785ESPD
- [Banach limit](https://tamnd.github.io/math2/node/A9CSR7RQAY.md): definition A9CSR7RQAY
- [Quasiconformal mapping](https://tamnd.github.io/math2/node/A9J2KK4CEJ.md): definition A9J2KK4CEJ
- [Thom–Boardman singularities](https://tamnd.github.io/math2/node/A9TYDPD5D7.md): concept A9TYDPD5D7
- [Controllability](https://tamnd.github.io/math2/node/A9WFM77Z4P.md): concept A9WFM77Z4P
- [Littlewood–Paley theory](https://tamnd.github.io/math2/node/AA29VBWBFH.md): concept AA29VBWBFH
- [Linear difference equation](https://tamnd.github.io/math2/node/AA7XRZ4VAG.md): concept AA7XRZ4VAG
- [Class field theory](https://tamnd.github.io/math2/node/AA943HVK6Z.md): definition AA943HVK6Z
- [Geometry of numbers](https://tamnd.github.io/math2/node/AAATH70CY1.md): concept AAATH70CY1
- [Laplacian eigenvalues of the round sphere](https://tamnd.github.io/math2/node/AABCR83PXG.md): example AABCR83PXG
- [Primitive permutation group](https://tamnd.github.io/math2/node/AADNYSB6TC.md): definition AADNYSB6TC
- [Fourier series on the torus](https://tamnd.github.io/math2/node/AAECHE30BD.md): concept AAECHE30BD
- [Legendre functions](https://tamnd.github.io/math2/node/AAN6HWAC4A.md): definition AAN6HWAC4A
- [Kriging](https://tamnd.github.io/math2/node/AAQAF5GGES.md): definition AAQAF5GGES
- [Regular conditional probability](https://tamnd.github.io/math2/node/AAT8PVDW23.md): definition AAT8PVDW23
- [du Bois-Reymond lemma](https://tamnd.github.io/math2/node/AATQKCWAAC.md): statement AATQKCWAAC
- [Presentation of an algebra](https://tamnd.github.io/math2/node/AAV343QP5E.md): definition AAV343QP5E
- [The subgroups of the symmetric group S4](https://tamnd.github.io/math2/node/AAYEQE.md): computation AAYEQE
- [Tarski's finite basis problem](https://tamnd.github.io/math2/node/AB22FWBQ0M.md): problem AB22FWBQ0M
- [Log-convexity of the gamma function](https://tamnd.github.io/math2/node/AB3ATTS48V.md): statement AB3ATTS48V
- [Condition number](https://tamnd.github.io/math2/node/AB3S6MHJ8X.md): concept AB3S6MHJ8X
- [Singular integral equation](https://tamnd.github.io/math2/node/AB7W6DBA4P.md): definition AB7W6DBA4P
- [Theorem of the three geodesics](https://tamnd.github.io/math2/node/AB9FNZD18B.md): statement AB9FNZD18B
- [Lie sphere geometry](https://tamnd.github.io/math2/node/ABE41ZYD78.md): definition ABE41ZYD78
- [Standard Borel space](https://tamnd.github.io/math2/node/ABEFWF4JHH.md): concept ABEFWF4JHH
- [Dunkl operator](https://tamnd.github.io/math2/node/ABJ88GR4Z0.md): definition ABJ88GR4Z0
- [Quotient of a topological group](https://tamnd.github.io/math2/node/ABKND96G8P.md): definition ABKND96G8P
- [Cantor's isomorphism theorem](https://tamnd.github.io/math2/node/ABN5PTXEHY.md): statement ABN5PTXEHY
- [Choquet theory](https://tamnd.github.io/math2/node/ABPK3RW5MW.md): concept ABPK3RW5MW
- [Atkinson's theorem](https://tamnd.github.io/math2/node/ABS17CB1D2.md): statement ABS17CB1D2
- [Quasi-Monte Carlo method](https://tamnd.github.io/math2/node/ABSRB2KXA6.md): concept ABSRB2KXA6
- [Gluing of schemes](https://tamnd.github.io/math2/node/ABVBHCA9VW.md): concept ABVBHCA9VW
- [Principal component analysis](https://tamnd.github.io/math2/node/ABXD13M340.md): concept ABXD13M340
- [Smith conjecture](https://tamnd.github.io/math2/node/ABYAPP6Q4C.md): statement ABYAPP6Q4C
- [Landau's constants](https://tamnd.github.io/math2/node/AC3FNJ8Y1Y.md): concept AC3FNJ8Y1Y
- [Kantorovich duality](https://tamnd.github.io/math2/node/AC5CB7RH6Z.md): statement AC5CB7RH6Z
- [Finite type algebra](https://tamnd.github.io/math2/node/AC5TFFQT9M.md): concept AC5TFFQT9M
- [Yamabe flow](https://tamnd.github.io/math2/node/AC6WNX8F9Z.md): concept AC6WNX8F9Z
- [Multigrid method](https://tamnd.github.io/math2/node/AC9PY6FBYF.md): statement AC9PY6FBYF
- [Summability kernel](https://tamnd.github.io/math2/node/ACBJ8RW42F.md): definition ACBJ8RW42F
- [GCD domain](https://tamnd.github.io/math2/node/ACDEK1GD9G.md): concept ACDEK1GD9G
- [Bogdanov–Takens bifurcation](https://tamnd.github.io/math2/node/ACN92H5M6D.md): concept ACN92H5M6D
- [Sinai's factor theorem](https://tamnd.github.io/math2/node/ACQ7A58XY7.md): statement ACQ7A58XY7
- [Weierstrass product](https://tamnd.github.io/math2/node/ACSF58HYDH.md): concept ACSF58HYDH
- [Pompeiu derivative](https://tamnd.github.io/math2/node/AD0G065GK8.md): example AD0G065GK8
- [Polynomially convex set](https://tamnd.github.io/math2/node/AD3474PVHR.md): concept AD3474PVHR
- [Green–Griffiths–Lang conjecture](https://tamnd.github.io/math2/node/AD9GD9EM9R.md): conjecture AD9GD9EM9R
- [Linear multistep method](https://tamnd.github.io/math2/node/AD9GZCC2PB.md): definition AD9GZCC2PB
- [Toda lattice](https://tamnd.github.io/math2/node/ADBEEPPKRR.md): example ADBEEPPKRR
- [Relative homotopy](https://tamnd.github.io/math2/node/ADDS6HE5SV.md): concept ADDS6HE5SV
- [Birkhoff polytope](https://tamnd.github.io/math2/node/ADEXY830S8.md): example ADEXY830S8
- [Fredholm index](https://tamnd.github.io/math2/node/ADJZ99KY2Q.md): definition ADJZ99KY2Q
- [3-j symbol](https://tamnd.github.io/math2/node/ADKBEJCTZE.md): definition ADKBEJCTZE
- [Distinguished triangle](https://tamnd.github.io/math2/node/ADKK56VZHW.md): concept ADKK56VZHW
- [Tautological one-form](https://tamnd.github.io/math2/node/ADNHVGY2W4.md): concept ADNHVGY2W4
- [Quadrature rule](https://tamnd.github.io/math2/node/ADSVMCY77A.md): concept ADSVMCY77A
- [Symmetric kernel](https://tamnd.github.io/math2/node/ADTYF8FG6C.md): concept ADTYF8FG6C
- [Quasicircle](https://tamnd.github.io/math2/node/ADZAXRA6QT.md): concept ADZAXRA6QT
- [Birkhoff interpolation](https://tamnd.github.io/math2/node/AE08JXN7CA.md): definition AE08JXN7CA
- [Marcinkiewicz–Zygmund inequality](https://tamnd.github.io/math2/node/AE0EZQH4ZM.md): statement AE0EZQH4ZM
- [Signed measure](https://tamnd.github.io/math2/node/AE0VPNDDXZ.md): definition AE0VPNDDXZ
- [Calderón–Zygmund theorem](https://tamnd.github.io/math2/node/AE1HJ8D8GV.md): statement AE1HJ8D8GV
- [Right-angled Artin group](https://tamnd.github.io/math2/node/AE1VC00NZN.md): definition AE1VC00NZN
- [Direct and inverse theorems of approximation](https://tamnd.github.io/math2/node/AEABS105PS.md): definition AEABS105PS
- [Identifiability](https://tamnd.github.io/math2/node/AEDA02M5NZ.md): definition AEDA02M5NZ
- [Abelian category](https://tamnd.github.io/math2/node/AEF4CBRAMN.md): definition AEF4CBRAMN
- [Minimal surface of revolution](https://tamnd.github.io/math2/node/AEG6QWEFRG.md): problem AEG6QWEFRG
- [Tempered distribution](https://tamnd.github.io/math2/node/AES2T5JPFP.md): concept AES2T5JPFP
- [Baire function](https://tamnd.github.io/math2/node/AEX1QP3178.md): definition AEX1QP3178
- [Semiconjugacy and factor map](https://tamnd.github.io/math2/node/AEXJNRBASV.md): concept AEXJNRBASV
- [Perron method](https://tamnd.github.io/math2/node/AEZ7PPCNQS.md): concept AEZ7PPCNQS
- [Collectively compact operator approximation](https://tamnd.github.io/math2/node/AEZE6B2XXY.md): concept AEZE6B2XXY
- [Abelian and Tauberian theorems](https://tamnd.github.io/math2/node/AEZNCY3N3K.md): definition AEZNCY3N3K
- [Lifting scheme](https://tamnd.github.io/math2/node/AF9NY73QHJ.md): concept AF9NY73QHJ
- [Porous medium equation](https://tamnd.github.io/math2/node/AFBY76MAX7.md): definition AFBY76MAX7
- [Affine morphism](https://tamnd.github.io/math2/node/AFC6JMYWEH.md): concept AFC6JMYWEH
- [Kepler conjecture](https://tamnd.github.io/math2/node/AFMBQT06ZV.md): statement AFMBQT06ZV
- [Product category](https://tamnd.github.io/math2/node/AFNE9C4NH6.md): concept AFNE9C4NH6
- [Branched covering](https://tamnd.github.io/math2/node/AFQ3HQYEPS.md): definition AFQ3HQYEPS
- [Dirichlet energy](https://tamnd.github.io/math2/node/AFQGK1NE0X.md): definition AFQGK1NE0X
- [Gradient flow on a manifold](https://tamnd.github.io/math2/node/AFRP1AAPV9.md): concept AFRP1AAPV9
- [Bochner–Riesz means](https://tamnd.github.io/math2/node/AFS2DCWCD3.md): concept AFS2DCWCD3
- [Conjugate gradient method](https://tamnd.github.io/math2/node/AFXZGQB4EN.md): statement AFXZGQB4EN
- [Invariant basis number](https://tamnd.github.io/math2/node/AG1GMQDTDW.md): concept AG1GMQDTDW
- [Hilbert–Burch theorem](https://tamnd.github.io/math2/node/AG4FF4DNYQ.md): statement AG4FF4DNYQ
- [Binet's formulas for the log-gamma function](https://tamnd.github.io/math2/node/AGBWPB994F.md): statement AGBWPB994F
- [Matrix difference equation](https://tamnd.github.io/math2/node/AGMBTZQC97.md): definition AGMBTZQC97
- [Riesz–Thorin theorem](https://tamnd.github.io/math2/node/AGRKRG1TMJ.md): statement AGRKRG1TMJ
- [Khintchine inequality](https://tamnd.github.io/math2/node/AGWMZ50SSZ.md): statement AGWMZ50SSZ
- [Pontryagin class](https://tamnd.github.io/math2/node/AGXJQA5G5J.md): definition AGXJQA5G5J
- [Riesz mean](https://tamnd.github.io/math2/node/AGXZ2G4MFM.md): concept AGXZ2G4MFM
- [Łoś's theorem](https://tamnd.github.io/math2/node/AGYBSTD7BG.md): statement AGYBSTD7BG
- [Knot sum](https://tamnd.github.io/math2/node/AGZSZW7GYE.md): concept AGZSZW7GYE
- [Morse index](https://tamnd.github.io/math2/node/AH059Y44NF.md): definition AH059Y44NF
- [Kneser's construction of real analytic tetration](https://tamnd.github.io/math2/node/AH3G1HDB0D.md): statement AH3G1HDB0D
- [Separable Banach space](https://tamnd.github.io/math2/node/AH45GS6RCR.md): concept AH45GS6RCR
- [Generalized inverse](https://tamnd.github.io/math2/node/AH7EW2EHNC.md): definition AH7EW2EHNC
- [Quadric](https://tamnd.github.io/math2/node/AHCXQCYW3C.md): definition AHCXQCYW3C
- [Graphon](https://tamnd.github.io/math2/node/AHE58SGQYW.md): definition AHE58SGQYW
- [Semifield](https://tamnd.github.io/math2/node/AHH97EJ6VH.md): concept AHH97EJ6VH
- [Long exact sequence in homology](https://tamnd.github.io/math2/node/AHKJ361FDB.md): statement AHKJ361FDB
- [Morphism of finite type](https://tamnd.github.io/math2/node/AHTE5DVX61.md): concept AHTE5DVX61
- [Locally compact group](https://tamnd.github.io/math2/node/AHV05ZBCYW.md): concept AHV05ZBCYW
- [Serre spectral sequence](https://tamnd.github.io/math2/node/AHW4B5480K.md): statement AHW4B5480K
- [Huber loss](https://tamnd.github.io/math2/node/AHYMBCMW11.md): definition AHYMBCMW11
- [Pseudoconvexity](https://tamnd.github.io/math2/node/AHYZMSBBNH.md): concept AHYZMSBBNH
- [Periodic sequence](https://tamnd.github.io/math2/node/AJ4FM5BT7Q.md): definition AJ4FM5BT7Q
- [Milman–Pettis theorem](https://tamnd.github.io/math2/node/AJ5RG9XQYZ.md): statement AJ5RG9XQYZ
- [Spin(7)-manifold](https://tamnd.github.io/math2/node/AJ7P4G261J.md): definition AJ7P4G261J
- [Eilenberg–MacLane space](https://tamnd.github.io/math2/node/AJHEYJE5XY.md): concept AJHEYJE5XY
- [Variety of algebras](https://tamnd.github.io/math2/node/AJNZJD8442.md): concept AJNZJD8442
- [Riemann–Liouville integral](https://tamnd.github.io/math2/node/AJXFNKX4AZ.md): definition AJXFNKX4AZ
- [Noetherian module](https://tamnd.github.io/math2/node/AK1ZRCXR32.md): concept AK1ZRCXR32
- [Exact functor](https://tamnd.github.io/math2/node/AK2XQC04PC.md): definition AK2XQC04PC
- [Infinite loop space](https://tamnd.github.io/math2/node/AK51SYB1Y9.md): definition AK51SYB1Y9
- [Bauer–Fike theorem](https://tamnd.github.io/math2/node/AK9J4RNZ51.md): statement AK9J4RNZ51
- [Measurable cardinal](https://tamnd.github.io/math2/node/AKBWT323RQ.md): concept AKBWT323RQ
- [Rice's theorem](https://tamnd.github.io/math2/node/AKDR5YBBE2.md): statement AKDR5YBBE2
- [Riemann P-symbol](https://tamnd.github.io/math2/node/AKEDFDGCDY.md): definition AKEDFDGCDY
- [Quasivariety](https://tamnd.github.io/math2/node/AKMASKR0ZG.md): definition AKMASKR0ZG
- [Quaternion algebra](https://tamnd.github.io/math2/node/AM07TESJED.md): definition AM07TESJED
- [Symbol class](https://tamnd.github.io/math2/node/AM4RNWVV4Q.md): concept AM4RNWVV4Q
- [Simple Lie group](https://tamnd.github.io/math2/node/AM7YJP8AR6.md): concept AM7YJP8AR6
- [Voronoi diagram](https://tamnd.github.io/math2/node/AMC3JCQ497.md): definition AMC3JCQ497
- [Steenrod reduced power](https://tamnd.github.io/math2/node/AMDN6PK5RA.md): definition AMDN6PK5RA
- [Exterior differential system](https://tamnd.github.io/math2/node/AME1ZE5HBX.md): definition AME1ZE5HBX
- [Existence of best approximations from finite-dimensional subspaces](https://tamnd.github.io/math2/node/AMP0CRE14P.md): statement AMP0CRE14P
- [Sigma-compact space](https://tamnd.github.io/math2/node/AMVHH8JT14.md): definition AMVHH8JT14
- [Minimal sufficient statistic](https://tamnd.github.io/math2/node/AMXXM288Y0.md): definition AMXXM288Y0
- [Domain decomposition methods](https://tamnd.github.io/math2/node/AN054RHT1K.md): concept AN054RHT1K
- [Dimension of an analytic set](https://tamnd.github.io/math2/node/AN0BYRKNRA.md): concept AN0BYRKNRA
- [Arnold diffusion](https://tamnd.github.io/math2/node/AN7QEWTNBT.md): definition AN7QEWTNBT
- [Multiplicative set](https://tamnd.github.io/math2/node/AN9SKEZ3VY.md): concept AN9SKEZ3VY
- [Presheaf](https://tamnd.github.io/math2/node/AND2WHZC8P.md): concept AND2WHZC8P
- [Quotient of a normed space](https://tamnd.github.io/math2/node/ANEFY3FQZR.md): definition ANEFY3FQZR
- [Tensor product of fields](https://tamnd.github.io/math2/node/ANH16GQJZQ.md): concept ANH16GQJZQ
- [Dedekind's characterization of modular lattices](https://tamnd.github.io/math2/node/ANRXAG227H.md): statement ANRXAG227H
- [Lane–Emden equation](https://tamnd.github.io/math2/node/ANRZZMH2Q0.md): concept ANRZZMH2Q0
- [Handlebody](https://tamnd.github.io/math2/node/AP1KY4EHMA.md): concept AP1KY4EHMA
- [Quantum group](https://tamnd.github.io/math2/node/AP2ZDXANKC.md): concept AP2ZDXANKC
- [Eells–Sampson theorem](https://tamnd.github.io/math2/node/AP4A0GX0V2.md): statement AP4A0GX0V2
- [Semiring of sets](https://tamnd.github.io/math2/node/APA7GHAYT2.md): definition APA7GHAYT2
- [Order dimension](https://tamnd.github.io/math2/node/APD4NYG9TG.md): definition APD4NYG9TG
- [Jean Taylor's theorem on soap film singularities](https://tamnd.github.io/math2/node/APGRBE5DW9.md): statement APGRBE5DW9
- [Nakayama algebra](https://tamnd.github.io/math2/node/APRFMTZ10K.md): concept APRFMTZ10K
- [Algebraic Riccati equation](https://tamnd.github.io/math2/node/APX4ZA01DK.md): definition APX4ZA01DK
- [Atomic decomposition of Hardy spaces](https://tamnd.github.io/math2/node/AQ15060AAA.md): statement AQ15060AAA
- [Isogeometric analysis](https://tamnd.github.io/math2/node/AQ4CNCXAJ0.md): concept AQ4CNCXAJ0
- [Superharmonic potential](https://tamnd.github.io/math2/node/AQ5RN6FFT0.md): concept AQ5RN6FFT0
- [Schramm–Loewner evolution](https://tamnd.github.io/math2/node/AQ75M2JTB2.md): concept AQ75M2JTB2
- [Positive linear operator](https://tamnd.github.io/math2/node/AQDXWH4812.md): concept AQDXWH4812
- [Zero object](https://tamnd.github.io/math2/node/AQET9YHH6F.md): concept AQET9YHH6F
- [Peak function](https://tamnd.github.io/math2/node/AQK50V3NGJ.md): definition AQK50V3NGJ
- [Pi-system](https://tamnd.github.io/math2/node/AQN6ZN66NK.md): definition AQN6ZN66NK
- [Knot group](https://tamnd.github.io/math2/node/AQVHM3FP0D.md): concept AQVHM3FP0D
- [Word problem for free algebras of a variety](https://tamnd.github.io/math2/node/AR14VMVHNG.md): problem AR14VMVHNG
- [Leray–Schauder method](https://tamnd.github.io/math2/node/AR52AWP2XE.md): definition AR52AWP2XE
- [Simple homotopy equivalence](https://tamnd.github.io/math2/node/AR8SP73SXX.md): concept AR8SP73SXX
- [Near-ring](https://tamnd.github.io/math2/node/ARAEEM8CVF.md): concept ARAEEM8CVF
- [Subalgebra lattice](https://tamnd.github.io/math2/node/ARCYEPTR4Q.md): definition ARCYEPTR4Q
- [Mackey topology](https://tamnd.github.io/math2/node/ARG1F8W9NZ.md): definition ARG1F8W9NZ
- [Generic matrix algebra](https://tamnd.github.io/math2/node/ARGNTXTAST.md): definition ARGNTXTAST
- [Möbius function of a poset](https://tamnd.github.io/math2/node/ARGZ549BQT.md): definition ARGZ549BQT
- [Resolvent kernel](https://tamnd.github.io/math2/node/ARTJV1XCRX.md): concept ARTJV1XCRX
- [Arbitrary-precision arithmetic](https://tamnd.github.io/math2/node/AS00YM4X7F.md): concept AS00YM4X7F
- [Halmos–von Neumann theorem](https://tamnd.github.io/math2/node/AS18AWFXSB.md): statement AS18AWFXSB
- [Boolean matrix](https://tamnd.github.io/math2/node/AS4WVMYEDZ.md): concept AS4WVMYEDZ
- [Character space](https://tamnd.github.io/math2/node/AS7ZA3P37H.md): definition AS7ZA3P37H
- [Biharmonic equation](https://tamnd.github.io/math2/node/AS8A9F5SKE.md): concept AS8A9F5SKE
- [P-adic analysis](https://tamnd.github.io/math2/node/ASMPNM7K0M.md): concept ASMPNM7K0M
- [Minimax approximation](https://tamnd.github.io/math2/node/ASWF03X5AM.md): concept ASWF03X5AM
- [Non-surjective epimorphism of algebras](https://tamnd.github.io/math2/node/ASXD4S15Z4.md): example ASXD4S15Z4
- [Irregular singularity](https://tamnd.github.io/math2/node/AT1W4MMTYR.md): concept AT1W4MMTYR
- [Riesz potential](https://tamnd.github.io/math2/node/AT6JHPCXJ1.md): definition AT6JHPCXJ1
- [Hamiltonian path](https://tamnd.github.io/math2/node/AT8GWPWQWT.md): definition AT8GWPWQWT
- [Subcategory](https://tamnd.github.io/math2/node/AT8K76KJ5R.md): definition AT8K76KJ5R
- [Bartle–Dunford–Schwartz integral](https://tamnd.github.io/math2/node/ATG4DQZZT9.md): definition ATG4DQZZT9
- [Projection-valued measure](https://tamnd.github.io/math2/node/ATGT25E5FF.md): concept ATGT25E5FF
- [Abstract polytope](https://tamnd.github.io/math2/node/ATJXGC4KCG.md): definition ATJXGC4KCG
- [Alexandrov topology](https://tamnd.github.io/math2/node/ATP1JK3V27.md): concept ATP1JK3V27
- [Volterra equation of the second kind](https://tamnd.github.io/math2/node/ATPSAHYG7C.md): concept ATPSAHYG7C
- [Point estimation](https://tamnd.github.io/math2/node/ATQPY2WPTA.md): concept ATQPY2WPTA
- [Sphere bundle](https://tamnd.github.io/math2/node/ATSXD280T1.md): definition ATSXD280T1
- [Whittle likelihood](https://tamnd.github.io/math2/node/ATXGHXJSRR.md): definition ATXGHXJSRR
- [Edge-of-the-wedge theorem](https://tamnd.github.io/math2/node/ATZEZ428CY.md): statement ATZEZ428CY
- [Atiyah–Bott theorem on Yang–Mills over Riemann surfaces](https://tamnd.github.io/math2/node/AV0Q1VSJ88.md): statement AV0Q1VSJ88
- [Del Pezzo surface](https://tamnd.github.io/math2/node/AV2ZAAZXN9.md): concept AV2ZAAZXN9
- [Projective connection](https://tamnd.github.io/math2/node/AV8Q85BH70.md): definition AV8Q85BH70
- [Lüroth's theorem](https://tamnd.github.io/math2/node/AVCZ6HVBA7.md): statement AVCZ6HVBA7
- [Probabilistic method](https://tamnd.github.io/math2/node/AVD8S4NW3B.md): definition AVD8S4NW3B
- [Kirby calculus](https://tamnd.github.io/math2/node/AVQ9WN1MDV.md): concept AVQ9WN1MDV
- [Clausen function](https://tamnd.github.io/math2/node/AVS6G54711.md): concept AVS6G54711
- [Poisson–Mellin–Newton cycle](https://tamnd.github.io/math2/node/AVSV6QBB6K.md): definition AVSV6QBB6K
- [Domino tiling](https://tamnd.github.io/math2/node/AVTXQM3CNZ.md): definition AVTXQM3CNZ
- [Tensor-hom adjunction](https://tamnd.github.io/math2/node/AVXARAXHKY.md): statement AVXARAXHKY
- [Kronecker–Weber theorem](https://tamnd.github.io/math2/node/AW00HPFB54.md): statement AW00HPFB54
- [Menshov's set of multiplicity of measure zero](https://tamnd.github.io/math2/node/AW16MYJV5Q.md): example AW16MYJV5Q
- [Totally disconnected group](https://tamnd.github.io/math2/node/AW3J9AQSA0.md): concept AW3J9AQSA0
- [Fréchet algebra](https://tamnd.github.io/math2/node/AW4T3NQYVF.md): concept AW4T3NQYVF
- [Lebesgue point](https://tamnd.github.io/math2/node/AWFXHJXGKE.md): definition AWFXHJXGKE
- [Canonical correlation](https://tamnd.github.io/math2/node/AWK2MBQ94K.md): definition AWK2MBQ94K
- [Numerical methods for linear least squares](https://tamnd.github.io/math2/node/AWQA2Y30N4.md): concept AWQA2Y30N4
- [Hua's identity](https://tamnd.github.io/math2/node/AWS7XTZ49C.md): statement AWS7XTZ49C
- [Endomorphism ring](https://tamnd.github.io/math2/node/AWSBZWWMG9.md): concept AWSBZWWMG9
- [Supersolvable group](https://tamnd.github.io/math2/node/AWWM9RWETX.md): concept AWWM9RWETX
- [Tensor field](https://tamnd.github.io/math2/node/AWZHWFBN53.md): definition AWZHWFBN53
- [Kuranishi family](https://tamnd.github.io/math2/node/AX46SNQCAA.md): definition AX46SNQCAA
- [Long exact sequence of a pair](https://tamnd.github.io/math2/node/AX4ZM9BV3Z.md): statement AX4ZM9BV3Z
- [Ultraproduct](https://tamnd.github.io/math2/node/AX9Q86GB8M.md): concept AX9Q86GB8M
- [1/3–2/3 conjecture](https://tamnd.github.io/math2/node/AXCH7Z9FKH.md): conjecture AXCH7Z9FKH
- [Modulus of continuity](https://tamnd.github.io/math2/node/AXCJBSQZWF.md): concept AXCJBSQZWF
- [Stereographic projection](https://tamnd.github.io/math2/node/AXDKKVD18N.md): concept AXDKKVD18N
- [Stiff equation](https://tamnd.github.io/math2/node/AXND5JRQAZ.md): definition AXND5JRQAZ
- [Polygon](https://tamnd.github.io/math2/node/AXS7YYF1YY.md): definition AXS7YYF1YY
- [Hopf invariant](https://tamnd.github.io/math2/node/AXT498MW87.md): definition AXT498MW87
- [Fisher's equation](https://tamnd.github.io/math2/node/AXV2PH7FEZ.md): concept AXV2PH7FEZ
- [Schur orthogonality relations](https://tamnd.github.io/math2/node/AY0B20QC04.md): statement AY0B20QC04
- [Proof that the kernel is a normal subgroup](https://tamnd.github.io/math2/node/AY60KQR8RT.md): proof AY60KQR8RT
- [Klein–Gordon equation](https://tamnd.github.io/math2/node/AYCCDQWCJF.md): concept AYCCDQWCJF
- [Fischer groups](https://tamnd.github.io/math2/node/AYJ1CM1XT8.md): object AYJ1CM1XT8
- [Dispersive partial differential equation](https://tamnd.github.io/math2/node/AYMJ2NFR4W.md): concept AYMJ2NFR4W
- [Essentially surjective functor](https://tamnd.github.io/math2/node/AYQ3VQ4B0V.md): definition AYQ3VQ4B0V
- [Cut-elimination theorem](https://tamnd.github.io/math2/node/AYRZ98Y0NW.md): statement AYRZ98Y0NW
- [Logistic regression](https://tamnd.github.io/math2/node/AYTEC1KET8.md): definition AYTEC1KET8
- [Iterated function](https://tamnd.github.io/math2/node/AYVTR2ES5B.md): concept AYVTR2ES5B
- [Linear integral equation](https://tamnd.github.io/math2/node/AYZCKG34QK.md): concept AYZCKG34QK
- [Painlevé transcendents](https://tamnd.github.io/math2/node/AZ17JCMRKB.md): concept AZ17JCMRKB
- [Carathéodory's theorem on boundary extension](https://tamnd.github.io/math2/node/AZ6H4ZKK55.md): statement AZ6H4ZKK55
- [Difference quotient](https://tamnd.github.io/math2/node/AZBDB5F4M4.md): concept AZBDB5F4M4
- [Lens space](https://tamnd.github.io/math2/node/AZBTMF196E.md): example AZBTMF196E
- [Order embedding](https://tamnd.github.io/math2/node/AZES1C6VMT.md): definition AZES1C6VMT
- [Ebin's slice theorem](https://tamnd.github.io/math2/node/AZF9TXV5C0.md): statement AZF9TXV5C0
- [Lifting property](https://tamnd.github.io/math2/node/AZK4RD8XV0.md): concept AZK4RD8XV0
- [Smoothness and analyticity of harmonic functions](https://tamnd.github.io/math2/node/AZPKCHND9T.md): statement AZPKCHND9T
- [Ergodic transformation](https://tamnd.github.io/math2/node/AZRPHS478B.md): concept AZRPHS478B
- [Green's function and the Riemann map](https://tamnd.github.io/math2/node/AZVRS57S7H.md): statement AZVRS57S7H
- [Support of a module](https://tamnd.github.io/math2/node/B0223N93XS.md): definition B0223N93XS
- [Sobolev inequality](https://tamnd.github.io/math2/node/B03CA692K3.md): statement B03CA692K3
- [Hecke algebra](https://tamnd.github.io/math2/node/B0587TVWZY.md): definition B0587TVWZY
- [Restricted isometry property](https://tamnd.github.io/math2/node/B0K18PWN2A.md): concept B0K18PWN2A
- [Cobweb plot](https://tamnd.github.io/math2/node/B0K8MDA7CJ.md): concept B0K8MDA7CJ
- [Bounded poset](https://tamnd.github.io/math2/node/B0QT3SM5DT.md): concept B0QT3SM5DT
- [Day terms](https://tamnd.github.io/math2/node/B0YKA8401M.md): concept B0YKA8401M
- [Elementary theory of the category of sets](https://tamnd.github.io/math2/node/B138JA6W20.md): definition B138JA6W20
- [Convergence to equilibrium of ergodic Markov chains](https://tamnd.github.io/math2/node/B1E1ES1BYD.md): statement B1E1ES1BYD
- [Autoregressive–moving-average model](https://tamnd.github.io/math2/node/B1HNX0YT0Z.md): concept B1HNX0YT0Z
- [Kurosh problem](https://tamnd.github.io/math2/node/B1Q5F5VNSK.md): problem B1Q5F5VNSK
- [Series acceleration](https://tamnd.github.io/math2/node/B1S2XJPGYD.md): concept B1S2XJPGYD
- [Infinitesimal stability](https://tamnd.github.io/math2/node/B1S7A7J8H4.md): definition B1S7A7J8H4
- [Additive function](https://tamnd.github.io/math2/node/B1T5SQB154.md): definition B1T5SQB154
- [Tate algebra](https://tamnd.github.io/math2/node/B1TEQ98JZD.md): concept B1TEQ98JZD
- [Eilenberg–Moore category](https://tamnd.github.io/math2/node/B1TVEMYMC0.md): definition B1TVEMYMC0
- [Set-valued measure](https://tamnd.github.io/math2/node/B1WPAD1M03.md): definition B1WPAD1M03
- [Fenchel conjugate](https://tamnd.github.io/math2/node/B1X92F7SJ6.md): definition B1X92F7SJ6
- [Monad in nonstandard topology](https://tamnd.github.io/math2/node/B1YJM6EENP.md): definition B1YJM6EENP
- [Fáry–Milnor theorem](https://tamnd.github.io/math2/node/B1Z1C8AHVD.md): statement B1Z1C8AHVD
- [Stochastic differential equation](https://tamnd.github.io/math2/node/B219G9SHJA.md): concept B219G9SHJA
- [Sasakian manifold](https://tamnd.github.io/math2/node/B21YEPS76G.md): concept B21YEPS76G
- [Invariant measure](https://tamnd.github.io/math2/node/B232EZ1FMB.md): concept B232EZ1FMB
- [Szemerédi's theorem](https://tamnd.github.io/math2/node/B23B0QEMN5.md): statement B23B0QEMN5
- [Stochastic optimal control](https://tamnd.github.io/math2/node/B24T6DQCDR.md): concept B24T6DQCDR
- [Geometrization conjecture](https://tamnd.github.io/math2/node/B27HB43K86.md): statement B27HB43K86
- [Catenary ring](https://tamnd.github.io/math2/node/B2CYR292J6.md): concept B2CYR292J6
- [Eulerian number](https://tamnd.github.io/math2/node/B2EMR1MA1H.md): definition B2EMR1MA1H
- [Magnus's Freiheitssatz](https://tamnd.github.io/math2/node/B2TRERFWEH.md): statement B2TRERFWEH
- [Additive category](https://tamnd.github.io/math2/node/B2VP96NJSK.md): concept B2VP96NJSK
- [Poincaré–Lelong formula](https://tamnd.github.io/math2/node/B2YBWQQMHA.md): statement B2YBWQQMHA
- [Differentially closed field](https://tamnd.github.io/math2/node/B2YPDYMR7E.md): definition B2YPDYMR7E
- [Varignon's theorem](https://tamnd.github.io/math2/node/B2ZJW0YBQ7.md): statement B2ZJW0YBQ7
- [Uhlenbeck's removable singularity theorem](https://tamnd.github.io/math2/node/B30ACDNS76.md): statement B30ACDNS76
- [Harish-Chandra module](https://tamnd.github.io/math2/node/B33WNK1SVN.md): definition B33WNK1SVN
- [Malmquist's theorem](https://tamnd.github.io/math2/node/B37RCPYARV.md): statement B37RCPYARV
- [Conley index](https://tamnd.github.io/math2/node/B3AZ4W0NGT.md): concept B3AZ4W0NGT
- [Bombieri–Vinogradov theorem](https://tamnd.github.io/math2/node/B3JMY54GJF.md): statement B3JMY54GJF
- [Krawtchouk polynomials](https://tamnd.github.io/math2/node/B3K63TFVVA.md): definition B3K63TFVVA
- [Affine scheme](https://tamnd.github.io/math2/node/B3V3XJ2W8G.md): concept B3V3XJ2W8G
- [Kissing number](https://tamnd.github.io/math2/node/B408FZYXF9.md): concept B408FZYXF9
- [Symmetric derivative](https://tamnd.github.io/math2/node/B40P5MJYYJ.md): concept B40P5MJYYJ
- [Butterfly catastrophe](https://tamnd.github.io/math2/node/B416H6DQDA.md): example B416H6DQDA
- [Stallings–Swan theorem](https://tamnd.github.io/math2/node/B41GE4M134.md): statement B41GE4M134
- [Coefficient of determination](https://tamnd.github.io/math2/node/B45RMKMVZ5.md): concept B45RMKMVZ5
- [Integral geometry](https://tamnd.github.io/math2/node/B4BN9PM331.md): definition B4BN9PM331
- [Symplectic integrator](https://tamnd.github.io/math2/node/B4C25SXFRE.md): concept B4C25SXFRE
- [Multilevel Monte Carlo method](https://tamnd.github.io/math2/node/B4FA03XDVW.md): statement B4FA03XDVW
- [Codimension of a singularity](https://tamnd.github.io/math2/node/B4FRQMWQKP.md): concept B4FRQMWQKP
- [Riemannian distance](https://tamnd.github.io/math2/node/B4QR306D8G.md): concept B4QR306D8G
- [Elliptic partial differential equation](https://tamnd.github.io/math2/node/B4TTNHSTRX.md): concept B4TTNHSTRX
- [Band-limited function](https://tamnd.github.io/math2/node/B4TYSQQC38.md): definition B4TYSQQC38
- [Regular prime](https://tamnd.github.io/math2/node/B4VK33PCXZ.md): concept B4VK33PCXZ
- [Curve of constant width](https://tamnd.github.io/math2/node/B4YD1ZXRVH.md): concept B4YD1ZXRVH
- [Linear system of divisors](https://tamnd.github.io/math2/node/B4Z8C36HEB.md): definition B4Z8C36HEB
- [Lindelöf summation](https://tamnd.github.io/math2/node/B505P3MKTZ.md): concept B505P3MKTZ
- [Exterior cone condition](https://tamnd.github.io/math2/node/B51J3BC610.md): statement B51J3BC610
- [Analytic function between Banach spaces](https://tamnd.github.io/math2/node/B538T38WDS.md): definition B538T38WDS
- [Asymptotic expansion](https://tamnd.github.io/math2/node/B53THC8F79.md): definition B53THC8F79
- [Linear PDE](https://tamnd.github.io/math2/node/B54GEJWQAV.md): definition B54GEJWQAV
- [Carleman's inequality](https://tamnd.github.io/math2/node/B54HPY6C5Z.md): statement B54HPY6C5Z
- [Difference kernel](https://tamnd.github.io/math2/node/B5AB451TB7.md): concept B5AB451TB7
- [Period and index](https://tamnd.github.io/math2/node/B5DDZHB04G.md): concept B5DDZHB04G
- [Peirce decomposition](https://tamnd.github.io/math2/node/B5GKNHH2SX.md): statement B5GKNHH2SX
- [Jacobian conjecture](https://tamnd.github.io/math2/node/B5JFXEQQD8.md): conjecture B5JFXEQQD8
- [Homological mirror symmetry](https://tamnd.github.io/math2/node/B5N9WJH5F0.md): conjecture B5N9WJH5F0
- [Symplectic manifold](https://tamnd.github.io/math2/node/B5YP8CY8DQ.md): definition B5YP8CY8DQ
- [Differentiation operator p](https://tamnd.github.io/math2/node/B5ZWZHMD12.md): definition B5ZWZHMD12
- [Consistency of summability methods](https://tamnd.github.io/math2/node/B61GY80PKG.md): definition B61GY80PKG
- [Presburger arithmetic](https://tamnd.github.io/math2/node/B66JYY3K58.md): example B66JYY3K58
- [Pappus's hexagon theorem](https://tamnd.github.io/math2/node/B6C3J2GJWP.md): statement B6C3J2GJWP
- [P-adic dynamics](https://tamnd.github.io/math2/node/B6DFK8HCB7.md): concept B6DFK8HCB7
- [Numerical analysis](https://tamnd.github.io/math2/node/B6E54YTR64.md): concept B6E54YTR64
- [QR algorithm](https://tamnd.github.io/math2/node/B6F8YWHYZ5.md): concept B6F8YWHYZ5
- [Locally convex space](https://tamnd.github.io/math2/node/B6GY2KJMWW.md): concept B6GY2KJMWW
- [Topological homogeneity](https://tamnd.github.io/math2/node/B6H4EY6R4Q.md): definition B6H4EY6R4Q
- [Yang–Mills functional](https://tamnd.github.io/math2/node/B6K478R9XE.md): definition B6K478R9XE
- [Density of simple functions in Lp](https://tamnd.github.io/math2/node/B6RYM7QTYZ.md): statement B6RYM7QTYZ
- [Rigid monoidal category](https://tamnd.github.io/math2/node/B6TJS37KQQ.md): definition B6TJS37KQQ
- [Dieudonné's theorem on paracompact Hausdorff spaces](https://tamnd.github.io/math2/node/B6Y8PEY72W.md): statement B6Y8PEY72W
- [Double sequence](https://tamnd.github.io/math2/node/B70G8FYPG0.md): concept B70G8FYPG0
- [Triangulated category](https://tamnd.github.io/math2/node/B75H4VEARE.md): definition B75H4VEARE
- [Topological mixing](https://tamnd.github.io/math2/node/B763QH5521.md): concept B763QH5521
- [Motive](https://tamnd.github.io/math2/node/B76G1KR3SB.md): definition B76G1KR3SB
- [Regular boundary point](https://tamnd.github.io/math2/node/B7948M365K.md): concept B7948M365K
- [Kaplan–Meier estimator](https://tamnd.github.io/math2/node/B7AJ8RWZRC.md): concept B7AJ8RWZRC
- [Multiplicity of a local ring](https://tamnd.github.io/math2/node/B7CV5DZPRK.md): definition B7CV5DZPRK
- [Asymptotic expansion from poles of a Mellin transform](https://tamnd.github.io/math2/node/B7EFVHCEQV.md): concept B7EFVHCEQV
- [Locally ringed space](https://tamnd.github.io/math2/node/B7FPT8JTXR.md): concept B7FPT8JTXR
- [Proper cone](https://tamnd.github.io/math2/node/B7GCKZZJE3.md): definition B7GCKZZJE3
- [Cramér–von Mises criterion](https://tamnd.github.io/math2/node/B7HPB2Z6GC.md): definition B7HPB2Z6GC
- [Selberg's 1/4 conjecture](https://tamnd.github.io/math2/node/B7K7NY8NTJ.md): conjecture B7K7NY8NTJ
- [Splitting of prime ideals](https://tamnd.github.io/math2/node/B7KNZCBFYT.md): concept B7KNZCBFYT
- [Uniform structure of a topological group](https://tamnd.github.io/math2/node/B7PZV00FP4.md): definition B7PZV00FP4
- [Integral extension](https://tamnd.github.io/math2/node/B7X0KFY1S1.md): concept B7X0KFY1S1
- [Lorentz space](https://tamnd.github.io/math2/node/B7X9M15Q92.md): definition B7X9M15Q92
- [Spectral geometry](https://tamnd.github.io/math2/node/B7XDAW51Z1.md): definition B7XDAW51Z1
- [Cubic spline](https://tamnd.github.io/math2/node/B7YND5ZVMF.md): definition B7YND5ZVMF
- [Distance geometry](https://tamnd.github.io/math2/node/B83KZW70C8.md): definition B83KZW70C8
- [Fuzzy algebraic structure](https://tamnd.github.io/math2/node/B85E71WZWA.md): definition B85E71WZWA
- [Valuation](https://tamnd.github.io/math2/node/B85T5VKR3M.md): concept B85T5VKR3M
- [Infinite compositions of functions](https://tamnd.github.io/math2/node/B88PFYXWBK.md): concept B88PFYXWBK
- [Chebyshev functions](https://tamnd.github.io/math2/node/B8AKSVBE5Z.md): concept B8AKSVBE5Z
- [Non-wandering set](https://tamnd.github.io/math2/node/B8CVXXXF29.md): definition B8CVXXXF29
- [Jacobi method](https://tamnd.github.io/math2/node/B8FVHNXAN1.md): statement B8FVHNXAN1
- [Plücker coordinates](https://tamnd.github.io/math2/node/B8PAMSJ81T.md): concept B8PAMSJ81T
- [Spectrum of the Laplace–Beltrami operator](https://tamnd.github.io/math2/node/B8TYKV7AA1.md): concept B8TYKV7AA1
- [Numerical linear algebra](https://tamnd.github.io/math2/node/B8W28KDMB0.md): concept B8W28KDMB0
- [Möbius inversion on posets](https://tamnd.github.io/math2/node/B8XJSJ517Q.md): statement B8XJSJ517Q
- [Ravenel conjectures](https://tamnd.github.io/math2/node/B8XP2K7RS1.md): statement B8XP2K7RS1
- [Kesten's theorem on planar bond percolation](https://tamnd.github.io/math2/node/B8Y0ZRQP17.md): statement B8Y0ZRQP17
- [Borel measure](https://tamnd.github.io/math2/node/B91V9BKZMR.md): definition B91V9BKZMR
- [Intersection multiplicity](https://tamnd.github.io/math2/node/B94AB2AA0S.md): definition B94AB2AA0S
- [Vizing's theorem](https://tamnd.github.io/math2/node/B99049JPY6.md): statement B99049JPY6
- [Willmore energy](https://tamnd.github.io/math2/node/B9FNGB5EBM.md): definition B9FNGB5EBM
- [Constructive function theory](https://tamnd.github.io/math2/node/B9NEZNTRPT.md): concept B9NEZNTRPT
- [Stone duality](https://tamnd.github.io/math2/node/B9T33D5MJK.md): statement B9T33D5MJK
- [Chern connection](https://tamnd.github.io/math2/node/B9X31R5EGG.md): definition B9X31R5EGG
- [Z-transform as a generating function](https://tamnd.github.io/math2/node/B9ZSFFTYRR.md): definition B9ZSFFTYRR
- [Margulis normal subgroup theorem](https://tamnd.github.io/math2/node/BA2S40MM0Y.md): statement BA2S40MM0Y
- [Besicovitch almost periodic function](https://tamnd.github.io/math2/node/BA9HBM8DFR.md): concept BA9HBM8DFR
- [Sidon set](https://tamnd.github.io/math2/node/BAADB7WHWZ.md): concept BAADB7WHWZ
- [Rayleigh–Ritz method](https://tamnd.github.io/math2/node/BAGCJR5EQF.md): definition BAGCJR5EQF
- [Sheaf on a site](https://tamnd.github.io/math2/node/BAH89SW049.md): definition BAH89SW049
- [Substitution tiling](https://tamnd.github.io/math2/node/BAN4VASZF0.md): concept BAN4VASZF0
- [Hyperbolic triangle](https://tamnd.github.io/math2/node/BANY6GN2GD.md): definition BANY6GN2GD
- [Lagrange inversion theorem](https://tamnd.github.io/math2/node/BANY9YR82A.md): statement BANY9YR82A
- [Rademacher functions](https://tamnd.github.io/math2/node/BAVME420YM.md): definition BAVME420YM
- [Ring spectrum](https://tamnd.github.io/math2/node/BAW0T68GWK.md): concept BAW0T68GWK
- [Cofinality](https://tamnd.github.io/math2/node/BAWHE1W6MH.md): concept BAWHE1W6MH
- [Flag manifold](https://tamnd.github.io/math2/node/BB0TSMW93P.md): concept BB0TSMW93P
- [Néron–Severi group](https://tamnd.github.io/math2/node/BB0X043XD1.md): definition BB0X043XD1
- [Differentiation operator p](https://tamnd.github.io/math2/node/BB15T3J891.md): concept BB15T3J891
- [Hyperbolic Dehn surgery theorem](https://tamnd.github.io/math2/node/BB28P2ED5H.md): statement BB28P2ED5H
- [Artin's conjecture on primitive roots](https://tamnd.github.io/math2/node/BB4FJ1FFGM.md): conjecture BB4FJ1FFGM
- [Bergman metric](https://tamnd.github.io/math2/node/BB7S4NTZ6V.md): concept BB7S4NTZ6V
- [Semitopological group](https://tamnd.github.io/math2/node/BBF9SCQRM9.md): definition BBF9SCQRM9
- [Dynkin's π-λ theorem](https://tamnd.github.io/math2/node/BBHM1A36M7.md): statement BBHM1A36M7
- [Expander graph](https://tamnd.github.io/math2/node/BBR7QAD87Y.md): definition BBR7QAD87Y
- [Rank of an abelian group](https://tamnd.github.io/math2/node/BBRSEM0C4H.md): definition BBRSEM0C4H
- [Wang tile](https://tamnd.github.io/math2/node/BBYNPDWR5R.md): example BBYNPDWR5R
- [Dynamical system](https://tamnd.github.io/math2/node/BBZP1R872N.md): concept BBZP1R872N
- [Arthur–Selberg trace formula](https://tamnd.github.io/math2/node/BCETMS8K3X.md): statement BCETMS8K3X
- [Jordan–Hölder theorem for modular lattices](https://tamnd.github.io/math2/node/BCG5KM83AP.md): statement BCG5KM83AP
- [Iterative functional equation](https://tamnd.github.io/math2/node/BCGBQ3HK26.md): concept BCGBQ3HK26
- [H-space](https://tamnd.github.io/math2/node/BCJD3SR975.md): definition BCJD3SR975
- [Densely defined operator](https://tamnd.github.io/math2/node/BCJHNHKQFG.md): concept BCJHNHKQFG
- [Varadhan's lemma](https://tamnd.github.io/math2/node/BD1CNVV5P0.md): statement BD1CNVV5P0
- [Lambda calculus](https://tamnd.github.io/math2/node/BD756KR8FM.md): concept BD756KR8FM
- [Azuma's inequality](https://tamnd.github.io/math2/node/BDA5TTB1F0.md): statement BDA5TTB1F0
- [Selberg trace formula](https://tamnd.github.io/math2/node/BDAWKPBNQP.md): statement BDAWKPBNQP
- [Tautological line bundle](https://tamnd.github.io/math2/node/BDB06N91FN.md): concept BDB06N91FN
- [Integration operator as convolution with the unit step](https://tamnd.github.io/math2/node/BDBDJSP6DF.md): statement BDBDJSP6DF
- [Interior algebra](https://tamnd.github.io/math2/node/BDEX7HET71.md): concept BDEX7HET71
- [Fréchet manifold](https://tamnd.github.io/math2/node/BDFXWM8XK1.md): definition BDFXWM8XK1
- [Final value theorem](https://tamnd.github.io/math2/node/BDHADVMJ8G.md): statement BDHADVMJ8G
- [Multiply transitive group action](https://tamnd.github.io/math2/node/BDJ1AX4XBB.md): concept BDJ1AX4XBB
- [Energy drift](https://tamnd.github.io/math2/node/BDVF3BPJEB.md): concept BDVF3BPJEB
- [Topological entropy](https://tamnd.github.io/math2/node/BDZYBXBFDE.md): concept BDZYBXBFDE
- [Gaussian free field](https://tamnd.github.io/math2/node/BE2CBPAZK6.md): definition BE2CBPAZK6
- [Parry measure](https://tamnd.github.io/math2/node/BE3E2AH0EV.md): definition BE3E2AH0EV
- [Conjugacy of maximal tori](https://tamnd.github.io/math2/node/BE3XK5ERC1.md): statement BE3XK5ERC1
- [Arithmetic dynamics](https://tamnd.github.io/math2/node/BE4X0QG8XW.md): concept BE4X0QG8XW
- [Yang–Baxter equation](https://tamnd.github.io/math2/node/BE6YFV3XP9.md): definition BE6YFV3XP9
- [Weitzenböck identity](https://tamnd.github.io/math2/node/BE9GTWMN0B.md): statement BE9GTWMN0B
- [Whitney sum](https://tamnd.github.io/math2/node/BE9NJRVDR3.md): concept BE9NJRVDR3
- [Uniformly convex space](https://tamnd.github.io/math2/node/BEAMC5011M.md): concept BEAMC5011M
- [Artin–Schreier theory](https://tamnd.github.io/math2/node/BEEAETX7SB.md): definition BEEAETX7SB
- [Kernel density estimation](https://tamnd.github.io/math2/node/BEEJJA60BX.md): definition BEEJJA60BX
- [Eta invariant](https://tamnd.github.io/math2/node/BEHD931SJQ.md): definition BEHD931SJQ
- [Class field theory](https://tamnd.github.io/math2/node/BEKCEYHCHX.md): concept BEKCEYHCHX
- [Pólya's recurrence theorem](https://tamnd.github.io/math2/node/BEQH6AZPYM.md): statement BEQH6AZPYM
- [Vertex of a polytope](https://tamnd.github.io/math2/node/BETJYRCRJB.md): definition BETJYRCRJB
- [Brauer group](https://tamnd.github.io/math2/node/BEW48TA9WZ.md): concept BEW48TA9WZ
- [Hardy–Littlewood maximal inequality](https://tamnd.github.io/math2/node/BF7T7VE57V.md): statement BF7T7VE57V
- [Type and cotype](https://tamnd.github.io/math2/node/BF9XR5SVPV.md): concept BF9XR5SVPV
- [Cartesian closed category](https://tamnd.github.io/math2/node/BFBNQ36QS8.md): concept BFBNQ36QS8
- [Radical axis](https://tamnd.github.io/math2/node/BFD7T0YY42.md): concept BFD7T0YY42
- [Projective module](https://tamnd.github.io/math2/node/BFFA03GE3V.md): concept BFFA03GE3V
- [Jacobson's conjecture](https://tamnd.github.io/math2/node/BFP0ZST7V3.md): conjecture BFP0ZST7V3
- [Empirical likelihood](https://tamnd.github.io/math2/node/BFQBHBN5F5.md): concept BFQBHBN5F5
- [Schlegel diagram](https://tamnd.github.io/math2/node/BFV3MKCBHZ.md): definition BFV3MKCBHZ
- [Nevanlinna theory of difference equations](https://tamnd.github.io/math2/node/BFWTGP7F92.md): definition BFWTGP7F92
- [Dehn–Lickorish theorem](https://tamnd.github.io/math2/node/BG3EH18KRE.md): statement BG3EH18KRE
- [Bellows conjecture](https://tamnd.github.io/math2/node/BG3XQF6V58.md): statement BG3XQF6V58
- [Benz plane](https://tamnd.github.io/math2/node/BG5A4CB6NB.md): concept BG5A4CB6NB
- [Kaplansky's theorem on projective modules](https://tamnd.github.io/math2/node/BG86NXRJ5F.md): statement BG86NXRJ5F
- [Fagin's theorem](https://tamnd.github.io/math2/node/BG8Q0BYSQ3.md): statement BG8Q0BYSQ3
- [Canonical bundle](https://tamnd.github.io/math2/node/BGH41KPWQ6.md): definition BGH41KPWQ6
- [Lopatinskii–Shapiro condition](https://tamnd.github.io/math2/node/BGRWCAT4SZ.md): definition BGRWCAT4SZ
- [Regular chain](https://tamnd.github.io/math2/node/BGXEC1SVQV.md): definition BGXEC1SVQV
- [Deck transformation group](https://tamnd.github.io/math2/node/BH32RQBX6A.md): concept BH32RQBX6A
- [Prime-factor FFT algorithm](https://tamnd.github.io/math2/node/BH3NNAYJT5.md): statement BH3NNAYJT5
- [Several complex variables](https://tamnd.github.io/math2/node/BH57D2FBNR.md): concept BH57D2FBNR
- [Caputo fractional derivative](https://tamnd.github.io/math2/node/BH5TK7DC2X.md): definition BH5TK7DC2X
- [Symmetric spectrum](https://tamnd.github.io/math2/node/BH7C1Y1F0E.md): definition BH7C1Y1F0E
- [Percolation theory](https://tamnd.github.io/math2/node/BH7YPXCYDK.md): concept BH7YPXCYDK
- [Upper bound theorem](https://tamnd.github.io/math2/node/BH8R1M1KVF.md): statement BH8R1M1KVF
- [Cyclotomic field](https://tamnd.github.io/math2/node/BHKM3A1GB6.md): definition BHKM3A1GB6
- [Group von Neumann algebra](https://tamnd.github.io/math2/node/BHMMRGCFVX.md): definition BHMMRGCFVX
- [Multijet transversality theorem](https://tamnd.github.io/math2/node/BJ0X0T4SBA.md): statement BJ0X0T4SBA
- [Outer measure](https://tamnd.github.io/math2/node/BJ314H17HC.md): concept BJ314H17HC
- [Copula](https://tamnd.github.io/math2/node/BJ33Y5H49M.md): definition BJ33Y5H49M
- [Divide-and-conquer eigenvalue algorithm](https://tamnd.github.io/math2/node/BJE8P9TH88.md): statement BJE8P9TH88
- [Perfect ring](https://tamnd.github.io/math2/node/BJMEXPKRYM.md): definition BJMEXPKRYM
- [Asymptotic expansion](https://tamnd.github.io/math2/node/BJQTEPEZ9G.md): concept BJQTEPEZ9G
- [Colimit](https://tamnd.github.io/math2/node/BJS1CGTKQ5.md): concept BJS1CGTKQ5
- [Vietoris topology](https://tamnd.github.io/math2/node/BJVRFFJTV6.md): concept BJVRFFJTV6
- [Floquet theory](https://tamnd.github.io/math2/node/BJVZXPB297.md): concept BJVZXPB297
- [Euler class](https://tamnd.github.io/math2/node/BJX6QS0ANT.md): concept BJX6QS0ANT
- [Kleene algebra](https://tamnd.github.io/math2/node/BK4DDS24XC.md): concept BK4DDS24XC
- [Tate–Shafarevich group](https://tamnd.github.io/math2/node/BK9WBMF56G.md): concept BK9WBMF56G
- [Spencer cohomology](https://tamnd.github.io/math2/node/BKKAA5JKT0.md): concept BKKAA5JKT0
- [Multiplier ideal](https://tamnd.github.io/math2/node/BKM0TQSCD9.md): concept BKM0TQSCD9
- [Dynamic equation on time scales](https://tamnd.github.io/math2/node/BKXYBS57T9.md): concept BKXYBS57T9
- [Spherical function](https://tamnd.github.io/math2/node/BKZNC76RC8.md): concept BKZNC76RC8
- [Langlands decomposition](https://tamnd.github.io/math2/node/BM3TWJ0D4P.md): statement BM3TWJ0D4P
- [The cosets of 3Z in the integers](https://tamnd.github.io/math2/node/BM4VXG92DT.md): example BM4VXG92DT
- [Multivariate interpolation](https://tamnd.github.io/math2/node/BM703GV34G.md): concept BM703GV34G
- [Prolongation of a section to a jet bundle](https://tamnd.github.io/math2/node/BMCCQ8JXNE.md): concept BMCCQ8JXNE
- [Proof that the order of an element divides the order of the group](https://tamnd.github.io/math2/node/BMD35Z5SGC.md): proof BMD35Z5SGC
- [Regular singular point](https://tamnd.github.io/math2/node/BMD7KQQA9C.md): definition BMD7KQQA9C
- [Algebra of bounded operators on a Hilbert space](https://tamnd.github.io/math2/node/BMDG4JQPEH.md): object BMDG4JQPEH
- [Galois theory](https://tamnd.github.io/math2/node/BMGE1ZAFQM.md): definition BMGE1ZAFQM
- [Spectrum of an operator](https://tamnd.github.io/math2/node/BMKATJ802A.md): concept BMKATJ802A
- [Weakly singular kernel](https://tamnd.github.io/math2/node/BMRX3X7095.md): definition BMRX3X7095
- [Nyquist–Shannon sampling theorem](https://tamnd.github.io/math2/node/BMXDMHRXBG.md): statement BMXDMHRXBG
- [Hankel inversion theorem](https://tamnd.github.io/math2/node/BMXZSXETDW.md): statement BMXZSXETDW
- [Submersion](https://tamnd.github.io/math2/node/BMY55EW3RF.md): concept BMY55EW3RF
- [Heyting algebra](https://tamnd.github.io/math2/node/BND6A96S3D.md): definition BND6A96S3D
- [E8 manifold](https://tamnd.github.io/math2/node/BNDRN0G9BR.md): object BNDRN0G9BR
- [Smoothing spline](https://tamnd.github.io/math2/node/BNEXZF75QC.md): concept BNEXZF75QC
- [Hilbert–Samuel function](https://tamnd.github.io/math2/node/BNQVS83GZV.md): concept BNQVS83GZV
- [Braid group](https://tamnd.github.io/math2/node/BNR4H5GCTP.md): concept BNR4H5GCTP
- [Serre's vanishing theorem on affine schemes](https://tamnd.github.io/math2/node/BNSC4P3PHS.md): statement BNSC4P3PHS
- [Lattice homomorphism](https://tamnd.github.io/math2/node/BNVRR925X8.md): concept BNVRR925X8
- [Hyperbolic 3-manifold](https://tamnd.github.io/math2/node/BNYW1SNP00.md): definition BNYW1SNP00
- [Oka's normalization theorem](https://tamnd.github.io/math2/node/BP0ZYHGD0S.md): statement BP0ZYHGD0S
- [Følner condition](https://tamnd.github.io/math2/node/BP331JP8SF.md): definition BP331JP8SF
- [Frame operator](https://tamnd.github.io/math2/node/BP3WR8NG1W.md): concept BP3WR8NG1W
- [Convergence in measure](https://tamnd.github.io/math2/node/BP8S3Z5TW9.md): definition BP8S3Z5TW9
- [Enriched category](https://tamnd.github.io/math2/node/BPH9SJ30Y0.md): concept BPH9SJ30Y0
- [Matrix pencil](https://tamnd.github.io/math2/node/BPKZV9W7AT.md): definition BPKZV9W7AT
- [Galois insertion](https://tamnd.github.io/math2/node/BPV58EPYCQ.md): concept BPV58EPYCQ
- [Interior operator](https://tamnd.github.io/math2/node/BPXY5A22BB.md): definition BPXY5A22BB
- [Pol–Inv Galois connection](https://tamnd.github.io/math2/node/BQ0AC9QX7E.md): statement BQ0AC9QX7E
- [Regularity structures](https://tamnd.github.io/math2/node/BQ6CYHQW02.md): concept BQ6CYHQW02
- [Dahlquist barriers](https://tamnd.github.io/math2/node/BQBZ7QMJE2.md): statement BQBZ7QMJE2
- [Generator](https://tamnd.github.io/math2/node/BQC9DW8TSZ.md): definition BQC9DW8TSZ
- [Embedding of structures](https://tamnd.github.io/math2/node/BQGK1FHSKB.md): definition BQGK1FHSKB
- [Mean value theorem for divided differences](https://tamnd.github.io/math2/node/BQJ9C1CD5X.md): statement BQJ9C1CD5X
- [Ramification group](https://tamnd.github.io/math2/node/BQJ9ZX9P3V.md): concept BQJ9ZX9P3V
- [Newton–Cotes formulas](https://tamnd.github.io/math2/node/BQPQAY1FDJ.md): concept BQPQAY1FDJ
- [Term](https://tamnd.github.io/math2/node/BQQ0FPFZ67.md): definition BQQ0FPFZ67
- [Complex dynamics](https://tamnd.github.io/math2/node/BQQZ0ZCGQF.md): concept BQQZ0ZCGQF
- [Algebra of sets](https://tamnd.github.io/math2/node/BQTM3PBQC9.md): definition BQTM3PBQC9
- [Fano variety](https://tamnd.github.io/math2/node/BQXB8H365P.md): definition BQXB8H365P
- [Blancmange curve](https://tamnd.github.io/math2/node/BQXRD4GVKZ.md): example BQXRD4GVKZ
- [Minimal deviation property of Chebyshev polynomials](https://tamnd.github.io/math2/node/BR3HD0RRA4.md): statement BR3HD0RRA4
- [Hölder space](https://tamnd.github.io/math2/node/BR6KJEJD1C.md): definition BR6KJEJD1C
- [Breakdown point](https://tamnd.github.io/math2/node/BRGGGZVQDS.md): concept BRGGGZVQDS
- [Extraspecial group](https://tamnd.github.io/math2/node/BRHQQTZB9K.md): definition BRHQQTZB9K
- [Torelli theorem](https://tamnd.github.io/math2/node/BRKFX8BQ60.md): statement BRKFX8BQ60
- [Zimmer program](https://tamnd.github.io/math2/node/BRKPE99MFC.md): concept BRKPE99MFC
- [Linear canonical transformation](https://tamnd.github.io/math2/node/BRXKAGYEFQ.md): definition BRXKAGYEFQ
- [Embedding dimension](https://tamnd.github.io/math2/node/BRXX5Y929C.md): definition BRXX5Y929C
- [Duality in order theory](https://tamnd.github.io/math2/node/BS1RD4F5MD.md): definition BS1RD4F5MD
- [Strong Cesàro summability](https://tamnd.github.io/math2/node/BS24BGYFJJ.md): definition BS24BGYFJJ
- [Random walk on a group](https://tamnd.github.io/math2/node/BS2H0G0T90.md): definition BS2H0G0T90
- [Bring radical](https://tamnd.github.io/math2/node/BS7NSJFFW3.md): definition BS7NSJFFW3
- [Von Neumann bicommutant theorem](https://tamnd.github.io/math2/node/BSA6GFT8SQ.md): statement BSA6GFT8SQ
- [Nonlinear integral equation](https://tamnd.github.io/math2/node/BSCYBP6AS6.md): concept BSCYBP6AS6
- [Arnold's cat map](https://tamnd.github.io/math2/node/BSCZN8Y1AN.md): example BSCZN8Y1AN
- [Bring radical](https://tamnd.github.io/math2/node/BSDCDJR6RQ.md): concept BSDCDJR6RQ
- [Fundamental solution of the Laplacian](https://tamnd.github.io/math2/node/BSJ8X2XRN9.md): concept BSJ8X2XRN9
- [Abel equation](https://tamnd.github.io/math2/node/BSRJN5M0R4.md): definition BSRJN5M0R4
- [Tilting theory](https://tamnd.github.io/math2/node/BT4QFQ9MJH.md): definition BT4QFQ9MJH
- [Spherical geometry](https://tamnd.github.io/math2/node/BT5B00W898.md): definition BT5B00W898
- [Place (number field)](https://tamnd.github.io/math2/node/BT5V3TSNY6.md): concept BT5V3TSNY6
- [p-adic number](https://tamnd.github.io/math2/node/BT6T702C28.md): concept BT6T702C28
- [Initial algebra semantics](https://tamnd.github.io/math2/node/BT7D4VQ6Y0.md): definition BT7D4VQ6Y0
- [Wythoff construction](https://tamnd.github.io/math2/node/BTCXG67H69.md): definition BTCXG67H69
- [Projective special linear group](https://tamnd.github.io/math2/node/BTDZHD9HG3.md): example BTDZHD9HG3
- [Soul theorem](https://tamnd.github.io/math2/node/BTH3GSAWY3.md): statement BTH3GSAWY3
- [Landau's constants](https://tamnd.github.io/math2/node/BTJR5NZPE3.md): definition BTJR5NZPE3
- [Implicit function theorem in Banach spaces](https://tamnd.github.io/math2/node/BTM6WJ0A52.md): statement BTM6WJ0A52
- [Inversion of the Weierstrass transform](https://tamnd.github.io/math2/node/BTPQHQNJVY.md): statement BTPQHQNJVY
- [Totally positive matrix](https://tamnd.github.io/math2/node/BTRNZKN1BP.md): definition BTRNZKN1BP
- [First fundamental form](https://tamnd.github.io/math2/node/BTX4G1FDF6.md): definition BTX4G1FDF6
- [Lattice in a Lie group](https://tamnd.github.io/math2/node/BTYF5Y5PFY.md): concept BTYF5Y5PFY
- [Reduction modulo n is a homomorphism](https://tamnd.github.io/math2/node/BV08DGBD7C.md): example BV08DGBD7C
- [Biorthogonal wavelet](https://tamnd.github.io/math2/node/BV0NXXGQZB.md): definition BV0NXXGQZB
- [Discriminator variety](https://tamnd.github.io/math2/node/BV2GCXPDDJ.md): definition BV2GCXPDDJ
- [Value function](https://tamnd.github.io/math2/node/BV6HGA075P.md): definition BV6HGA075P
- [Euler product for multiplicative functions](https://tamnd.github.io/math2/node/BVA6XT55CS.md): statement BVA6XT55CS
- [Wiener filter equation](https://tamnd.github.io/math2/node/BVFGX5JH7X.md): example BVFGX5JH7X
- [Central difference](https://tamnd.github.io/math2/node/BVGYDP6YKF.md): concept BVGYDP6YKF
- [Bloch–Wigner function](https://tamnd.github.io/math2/node/BVJ2ERSQBM.md): definition BVJ2ERSQBM
- [Exposed point](https://tamnd.github.io/math2/node/BVMH4AKV9F.md): definition BVMH4AKV9F
- [Difference algebra](https://tamnd.github.io/math2/node/BVR56STQ32.md): definition BVR56STQ32
- [Manifold of smooth mappings](https://tamnd.github.io/math2/node/BW12J14X0Z.md): concept BW12J14X0Z
- [Lattice in a locally compact group](https://tamnd.github.io/math2/node/BW21VJW403.md): definition BW21VJW403
- [Ultraweak topology](https://tamnd.github.io/math2/node/BW678BX9ZX.md): definition BW678BX9ZX
- [Automatic differentiation](https://tamnd.github.io/math2/node/BW7FEZ8WM1.md): concept BW7FEZ8WM1
- [Melnikov method](https://tamnd.github.io/math2/node/BW7NGTGXAY.md): definition BW7NGTGXAY
- [Watson's lemma](https://tamnd.github.io/math2/node/BW830VCE98.md): statement BW830VCE98
- [Connected sum](https://tamnd.github.io/math2/node/BW8X6DXTHG.md): definition BW8X6DXTHG
- [Locally convex space](https://tamnd.github.io/math2/node/BW9BBF244W.md): definition BW9BBF244W
- [Monadic functor](https://tamnd.github.io/math2/node/BWE0JG7FRF.md): definition BWE0JG7FRF
- [Centerpoint theorem](https://tamnd.github.io/math2/node/BWEGTCW5RJ.md): statement BWEGTCW5RJ
- [Excessive function](https://tamnd.github.io/math2/node/BWMS6B1M5N.md): definition BWMS6B1M5N
- [Aberth method](https://tamnd.github.io/math2/node/BWQ8PVA9Q3.md): statement BWQ8PVA9Q3
- [Decreasing rearrangement](https://tamnd.github.io/math2/node/BWS042NC2P.md): concept BWS042NC2P
- [Covering density](https://tamnd.github.io/math2/node/BWSN2JAMTK.md): concept BWSN2JAMTK
- [q-difference equation](https://tamnd.github.io/math2/node/BWT2AAEK9R.md): definition BWT2AAEK9R
- [Hardy's Tauberian theorem for Cesàro means](https://tamnd.github.io/math2/node/BWWZB1F0HY.md): statement BWWZB1F0HY
- [Upper set](https://tamnd.github.io/math2/node/BWYFKCWRW5.md): definition BWYFKCWRW5
- [Wiman–Valiron theory](https://tamnd.github.io/math2/node/BWZ1H4FAQG.md): concept BWZ1H4FAQG
- [Infinitely divisible distribution](https://tamnd.github.io/math2/node/BX3V3NYM1Y.md): concept BX3V3NYM1Y
- [Linear multistep method](https://tamnd.github.io/math2/node/BX4D85A79P.md): concept BX4D85A79P
- [Gross–Zagier theorem](https://tamnd.github.io/math2/node/BX558DTV2D.md): statement BX558DTV2D
- [Semialgebraic set](https://tamnd.github.io/math2/node/BX5WSK8ZCT.md): concept BX5WSK8ZCT
- [Absolutely convergent Fourier series](https://tamnd.github.io/math2/node/BX9ZH25FW5.md): definition BX9ZH25FW5
- [Dirac bundle](https://tamnd.github.io/math2/node/BXKZ62PBC9.md): definition BXKZ62PBC9
- [Mehler–Heine formula](https://tamnd.github.io/math2/node/BXQARMFY75.md): statement BXQARMFY75
- [Bernoulli polynomials](https://tamnd.github.io/math2/node/BXQKBJJGFG.md): object BXQKBJJGFG
- [Regular summability method](https://tamnd.github.io/math2/node/BXVVXJQV8M.md): definition BXVVXJQV8M
- [Sub-mean value property](https://tamnd.github.io/math2/node/BXYQE0Q1ST.md): statement BXYQE0Q1ST
- [Lorentzian manifold](https://tamnd.github.io/math2/node/BY4QYYS788.md): concept BY4QYYS788
- [Erdős–Kac theorem](https://tamnd.github.io/math2/node/BY5WRND0J1.md): statement BY5WRND0J1
- [Positive current](https://tamnd.github.io/math2/node/BYJHP6A906.md): concept BYJHP6A906
- [Wiener–Hopf factorization](https://tamnd.github.io/math2/node/BYKP1V2AQN.md): statement BYKP1V2AQN
- [Dominated convergence theorem](https://tamnd.github.io/math2/node/BYNV84PF86.md): statement BYNV84PF86
- [Grönwall area theorem](https://tamnd.github.io/math2/node/BYZDAZNHP4.md): statement BYZDAZNHP4
- [Poincaré's functional equation](https://tamnd.github.io/math2/node/BZ0DDT1MXS.md): definition BZ0DDT1MXS
- [Cayley–Bacharach theorem](https://tamnd.github.io/math2/node/BZ2MM3CTNA.md): statement BZ2MM3CTNA
- [Sperner property](https://tamnd.github.io/math2/node/BZ5GXNN7HC.md): definition BZ5GXNN7HC
- [Biproduct](https://tamnd.github.io/math2/node/BZEJKTQ76T.md): concept BZEJKTQ76T
- [Mordell's theorem](https://tamnd.github.io/math2/node/BZHGKN6Q3K.md): statement BZHGKN6Q3K
- [Particle filter](https://tamnd.github.io/math2/node/BZK7HQDMT0.md): concept BZK7HQDMT0
- [Larson–Sweedler theorem](https://tamnd.github.io/math2/node/BZPKCPCWTM.md): statement BZPKCPCWTM
- [Rado's theorem](https://tamnd.github.io/math2/node/BZSGRCTFH4.md): statement BZSGRCTFH4
- [Uniform polytope](https://tamnd.github.io/math2/node/BZXHGNGQAX.md): concept BZXHGNGQAX
- [Scaling function and refinement equation](https://tamnd.github.io/math2/node/C00HSE6W2A.md): definition C00HSE6W2A
- [Figure-eight knot](https://tamnd.github.io/math2/node/C01XBGAHC6.md): example C01XBGAHC6
- [Vinogradov's theorem](https://tamnd.github.io/math2/node/C01Z9QRGZY.md): statement C01Z9QRGZY
- [Cobordism ring](https://tamnd.github.io/math2/node/C02WQY819R.md): concept C02WQY819R
- [Sine and cosine transforms](https://tamnd.github.io/math2/node/C07WNRSW52.md): concept C07WNRSW52
- [q-exponential](https://tamnd.github.io/math2/node/C08ASAB8BX.md): definition C08ASAB8BX
- [Error of best approximation](https://tamnd.github.io/math2/node/C0AKF92CWH.md): definition C0AKF92CWH
- [Meijer G-function transforms](https://tamnd.github.io/math2/node/C0BCHX106R.md): concept C0BCHX106R
- [Polynomial identity ring](https://tamnd.github.io/math2/node/C0EDT81EGH.md): concept C0EDT81EGH
- [Zero object](https://tamnd.github.io/math2/node/C0GMFP0RF9.md): definition C0GMFP0RF9
- [Stable ∞-category](https://tamnd.github.io/math2/node/C0KFX2CMRR.md): definition C0KFX2CMRR
- [Fatou set](https://tamnd.github.io/math2/node/C0N0XGC6GK.md): concept C0N0XGC6GK
- [Brahmagupta's formula](https://tamnd.github.io/math2/node/C0NQDQAZ2Y.md): statement C0NQDQAZ2Y
- [Moment map](https://tamnd.github.io/math2/node/C0RDHFD00Y.md): definition C0RDHFD00Y
- [Shanks transformation](https://tamnd.github.io/math2/node/C0TXDZZMH7.md): concept C0TXDZZMH7
- [Stieltjes inversion formula](https://tamnd.github.io/math2/node/C0X2A02MRE.md): statement C0X2A02MRE
- [Associated Legendre polynomials](https://tamnd.github.io/math2/node/C0Z1MKEGZ1.md): definition C0Z1MKEGZ1
- [Pro-p group](https://tamnd.github.io/math2/node/C13D6W3MM9.md): concept C13D6W3MM9
- [Proof-theoretic ordinal](https://tamnd.github.io/math2/node/C19Z3B6V4P.md): definition C19Z3B6V4P
- [Elliptic boundary value problem](https://tamnd.github.io/math2/node/C1F7T523G0.md): concept C1F7T523G0
- [Covariant derivative](https://tamnd.github.io/math2/node/C1KY6MAW0J.md): definition C1KY6MAW0J
- [Semimodular lattice](https://tamnd.github.io/math2/node/C1NET23GG0.md): concept C1NET23GG0
- [Complete intersection ring](https://tamnd.github.io/math2/node/C1PFAQBCB8.md): definition C1PFAQBCB8
- [Bredon cohomology](https://tamnd.github.io/math2/node/C1T9VYKNV1.md): definition C1T9VYKNV1
- [Helgason support theorem](https://tamnd.github.io/math2/node/C1WVKRGQ1T.md): statement C1WVKRGQ1T
- [Hilbert's sixteenth problem](https://tamnd.github.io/math2/node/C1XNMMKZ1J.md): problem C1XNMMKZ1J
- [Blaschke product](https://tamnd.github.io/math2/node/C2082GXJ3Q.md): definition C2082GXJ3Q
- [Schottky problem](https://tamnd.github.io/math2/node/C210P5AK7X.md): problem C210P5AK7X
- [Mittag-Leffler star](https://tamnd.github.io/math2/node/C21ECQ1H84.md): concept C21ECQ1H84
- [Summability field of a matrix method](https://tamnd.github.io/math2/node/C22331MBY7.md): definition C22331MBY7
- [Strongly regular graph](https://tamnd.github.io/math2/node/C22WGRP3T8.md): concept C22WGRP3T8
- [Parametrix](https://tamnd.github.io/math2/node/C24D9MWRV3.md): definition C24D9MWRV3
- [Tauberian condition](https://tamnd.github.io/math2/node/C28YT5DQFF.md): definition C28YT5DQFF
- [Random group](https://tamnd.github.io/math2/node/C2ATM03215.md): definition C2ATM03215
- [Intrinsic volume](https://tamnd.github.io/math2/node/C2CK6KSSTV.md): concept C2CK6KSSTV
- [Wedge sum](https://tamnd.github.io/math2/node/C2DAKWAZ28.md): definition C2DAKWAZ28
- [Borel–Ritt theorem](https://tamnd.github.io/math2/node/C2FFQAPJ9F.md): statement C2FFQAPJ9F
- [Nilpotent ideal](https://tamnd.github.io/math2/node/C2H26BYWHE.md): concept C2H26BYWHE
- [Dimension of an analytic set](https://tamnd.github.io/math2/node/C2R5Q2Q717.md): definition C2R5Q2Q717
- [Cohomology operation](https://tamnd.github.io/math2/node/C2SRBS1CX3.md): definition C2SRBS1CX3
- [Kuramoto model](https://tamnd.github.io/math2/node/C2V5P3G2FS.md): example C2V5P3G2FS
- [Faithfully flat module](https://tamnd.github.io/math2/node/C31PYYPRGC.md): concept C31PYYPRGC
- [Association scheme](https://tamnd.github.io/math2/node/C338GKB4VJ.md): concept C338GKB4VJ
- [Kazhdan–Margulis theorem](https://tamnd.github.io/math2/node/C34RZMKKA4.md): statement C34RZMKKA4
- [Trace class operator](https://tamnd.github.io/math2/node/C37R11WREF.md): definition C37R11WREF
- [Indeterminate moment problem](https://tamnd.github.io/math2/node/C38DV6ZAZ3.md): definition C38DV6ZAZ3
- [Vitali convergence theorem](https://tamnd.github.io/math2/node/C3CKZAZ57Z.md): statement C3CKZAZ57Z
- [Landau–Kolmogorov inequality](https://tamnd.github.io/math2/node/C3DD4Z4MD1.md): statement C3DD4Z4MD1
- [Weak derivative](https://tamnd.github.io/math2/node/C3EC148VZD.md): concept C3EC148VZD
- [Natural transformation](https://tamnd.github.io/math2/node/C3JHS6W73G.md): definition C3JHS6W73G
- [Modal logic](https://tamnd.github.io/math2/node/C3RZY20ETK.md): concept C3RZY20ETK
- [Infinitesimal stability](https://tamnd.github.io/math2/node/C3TFCVE2YK.md): concept C3TFCVE2YK
- [Stable theory](https://tamnd.github.io/math2/node/C44JJKGCBF.md): definition C44JJKGCBF
- [Matched Z-transform method](https://tamnd.github.io/math2/node/C45NVCD39K.md): concept C45NVCD39K
- [Time-scale calculus](https://tamnd.github.io/math2/node/C4B1FA1H0J.md): definition C4B1FA1H0J
- [Multivariate interpolation](https://tamnd.github.io/math2/node/C4B5FB6A34.md): definition C4B5FB6A34
- [Observability](https://tamnd.github.io/math2/node/C4HN3EAQWM.md): concept C4HN3EAQWM
- [Blowing up](https://tamnd.github.io/math2/node/C4KGYAFPN7.md): concept C4KGYAFPN7
- [Canonical height](https://tamnd.github.io/math2/node/C55ST3TH02.md): concept C55ST3TH02
- [Differentiable curve](https://tamnd.github.io/math2/node/C57T562NKB.md): concept C57T562NKB
- [Classical ring of quotients](https://tamnd.github.io/math2/node/C5971PGKPH.md): concept C5971PGKPH
- [Cone](https://tamnd.github.io/math2/node/C5ANXPS4YN.md): definition C5ANXPS4YN
- [Ordinal analysis](https://tamnd.github.io/math2/node/C5E5TG8PGP.md): concept C5E5TG8PGP
- [Hamiltonian vector field](https://tamnd.github.io/math2/node/C5EH3B4BEP.md): concept C5EH3B4BEP
- [Pseudorandom number generator](https://tamnd.github.io/math2/node/C5FG6ARWVV.md): concept C5FG6ARWVV
- [Adaptive mesh refinement](https://tamnd.github.io/math2/node/C5JM6D0841.md): definition C5JM6D0841
- [Projective determinacy](https://tamnd.github.io/math2/node/C5MA8PA2Q5.md): definition C5MA8PA2Q5
- [Tychonoff space](https://tamnd.github.io/math2/node/C5N2J2K6Z3.md): concept C5N2J2K6Z3
- [Covering density](https://tamnd.github.io/math2/node/C5VXQVNNH2.md): definition C5VXQVNNH2
- [Milstein method](https://tamnd.github.io/math2/node/C5WJT0NWGV.md): statement C5WJT0NWGV
- [Localization of a Noetherian ring is Noetherian](https://tamnd.github.io/math2/node/C618HAK3X6.md): statement C618HAK3X6
- [Latent variable model](https://tamnd.github.io/math2/node/C63RVH96TF.md): concept C63RVH96TF
- [Algebraic logic](https://tamnd.github.io/math2/node/C66RKJQQSC.md): definition C66RKJQQSC
- [Potential operator of a Markov process](https://tamnd.github.io/math2/node/C6EC4SBEMC.md): definition C6EC4SBEMC
- [Scott continuity](https://tamnd.github.io/math2/node/C6GAHS1Z2P.md): concept C6GAHS1Z2P
- [M3–N5 characterization of distributivity](https://tamnd.github.io/math2/node/C6GH4VHAMQ.md): statement C6GH4VHAMQ
- [Simplicial homology](https://tamnd.github.io/math2/node/C6J40QF82R.md): concept C6J40QF82R
- [Tropical variety](https://tamnd.github.io/math2/node/C6QQ3HM5S4.md): definition C6QQ3HM5S4
- [Proof-theoretic ordinal](https://tamnd.github.io/math2/node/C72RBH0CDR.md): concept C72RBH0CDR
- [Sine and cosine addition equations](https://tamnd.github.io/math2/node/C73TGW752D.md): concept C73TGW752D
- [Sheaf cohomology](https://tamnd.github.io/math2/node/C7E5EMV89Z.md): definition C7E5EMV89Z
- [Nonlinear regression](https://tamnd.github.io/math2/node/C7E69V030F.md): concept C7E69V030F
- [Equivariant cohomology](https://tamnd.github.io/math2/node/C7MZ3SFGYF.md): concept C7MZ3SFGYF
- [Ideal class group](https://tamnd.github.io/math2/node/C7Q2MVD933.md): definition C7Q2MVD933
- [Boy's surface](https://tamnd.github.io/math2/node/C7WPR8W54K.md): example C7WPR8W54K
- [Akizuki–Hopkins–Levitzki theorem](https://tamnd.github.io/math2/node/C7X657TF19.md): statement C7X657TF19
- [Paraconsistent logic](https://tamnd.github.io/math2/node/C7YWCR87QJ.md): concept C7YWCR87QJ
- [Laman graph](https://tamnd.github.io/math2/node/C81JPZ51NA.md): concept C81JPZ51NA
- [Tutte polynomial](https://tamnd.github.io/math2/node/C81RAXD6DV.md): concept C81RAXD6DV
- [Hochster's theorem on spectral spaces](https://tamnd.github.io/math2/node/C82AXZK2BS.md): statement C82AXZK2BS
- [Asymptotic behavior of solutions of renewal equations](https://tamnd.github.io/math2/node/C82WDEQ3DC.md): statement C82WDEQ3DC
- [Link of a singularity](https://tamnd.github.io/math2/node/C831AN3RFM.md): concept C831AN3RFM
- [Gröbner basis](https://tamnd.github.io/math2/node/C83KPT47R0.md): concept C83KPT47R0
- [KAK decomposition](https://tamnd.github.io/math2/node/C846XS3Y0T.md): statement C846XS3Y0T
- [Injective resolution](https://tamnd.github.io/math2/node/C84CHVTZ4J.md): definition C84CHVTZ4J
- [Ramanujan summation](https://tamnd.github.io/math2/node/C87X3ACS76.md): definition C87X3ACS76
- [Liouville's equation](https://tamnd.github.io/math2/node/C8E0TWY3WT.md): concept C8E0TWY3WT
- [Monodromy](https://tamnd.github.io/math2/node/C8HD62AB6Q.md): concept C8HD62AB6Q
- [Borel–Moore homology](https://tamnd.github.io/math2/node/C8N3ER51WT.md): definition C8N3ER51WT
- [Bairstow's method](https://tamnd.github.io/math2/node/C8S7XX0VZM.md): statement C8S7XX0VZM
- [Weak order of permutations](https://tamnd.github.io/math2/node/C8SEGT1AZS.md): example C8SEGT1AZS
- [Kadison–Singer problem](https://tamnd.github.io/math2/node/C8TTRMVVS7.md): problem C8TTRMVVS7
- [Archimedean equivalence classes](https://tamnd.github.io/math2/node/C8VE2SEVEB.md): definition C8VE2SEVEB
- [Local time](https://tamnd.github.io/math2/node/C8W7J0S73V.md): definition C8W7J0S73V
- [Müntz–Szász theorem](https://tamnd.github.io/math2/node/C8XY22X5NP.md): statement C8XY22X5NP
- [Free Boolean algebra](https://tamnd.github.io/math2/node/C8ZPFCX1SJ.md): definition C8ZPFCX1SJ
- [Polar set](https://tamnd.github.io/math2/node/C95QJ7YH5T.md): concept C95QJ7YH5T
- [Wiener algebra](https://tamnd.github.io/math2/node/C96N6EB236.md): definition C96N6EB236
- [Mixture model](https://tamnd.github.io/math2/node/C98DKQSTNC.md): concept C98DKQSTNC
- [Normal number](https://tamnd.github.io/math2/node/C98T2T5FJS.md): concept C98T2T5FJS
- [Cheeger–Müller theorem](https://tamnd.github.io/math2/node/C9DT4Y63CY.md): statement C9DT4Y63CY
- [Detailed balance](https://tamnd.github.io/math2/node/C9EDBYCRAY.md): definition C9EDBYCRAY
- [Sullivan's no wandering domains theorem](https://tamnd.github.io/math2/node/C9FDR4BGMG.md): statement C9FDR4BGMG
- [Strict total order](https://tamnd.github.io/math2/node/C9GDH7264N.md): definition C9GDH7264N
- [Quasi-isomorphism](https://tamnd.github.io/math2/node/C9JEC2WQKV.md): concept C9JEC2WQKV
- [Pfaffian](https://tamnd.github.io/math2/node/C9JJD2AVJR.md): definition C9JJD2AVJR
- [Universally closed morphism](https://tamnd.github.io/math2/node/C9NE52R27S.md): definition C9NE52R27S
- [M-matrix](https://tamnd.github.io/math2/node/C9VJ8DTZWC.md): definition C9VJ8DTZWC
- [Monster group](https://tamnd.github.io/math2/node/C9VY20TAPT.md): object C9VY20TAPT
- [Amoeba](https://tamnd.github.io/math2/node/C9WE6180A7.md): definition C9WE6180A7
- [Radó's theorem on triangulation of surfaces](https://tamnd.github.io/math2/node/C9X05EDS5F.md): statement C9X05EDS5F
- [Lawvere–Tierney topology](https://tamnd.github.io/math2/node/C9XM3X631W.md): definition C9XM3X631W
- [Laplace transform on a convex cone](https://tamnd.github.io/math2/node/C9Z0P713CF.md): definition C9Z0P713CF
- [Witt vectors](https://tamnd.github.io/math2/node/C9Z22J9SQE.md): definition C9Z22J9SQE
- [Schur complement](https://tamnd.github.io/math2/node/CA1J41NWGS.md): definition CA1J41NWGS
- [Ultraweak topology](https://tamnd.github.io/math2/node/CA38T5ZH14.md): concept CA38T5ZH14
- [Logical connective](https://tamnd.github.io/math2/node/CA49F5KQGB.md): definition CA49F5KQGB
- [Complete elliptic integral of the second kind](https://tamnd.github.io/math2/node/CA49GFN9ZA.md): definition CA49GFN9ZA
- [Coupling from the past](https://tamnd.github.io/math2/node/CA6SVPFR3G.md): definition CA6SVPFR3G
- [Arithmetic group](https://tamnd.github.io/math2/node/CA7H2XMD9P.md): definition CA7H2XMD9P
- [Finite volume method](https://tamnd.github.io/math2/node/CAB1ARHH31.md): definition CAB1ARHH31
- [Peetre K-functional](https://tamnd.github.io/math2/node/CAE0X7G9R5.md): concept CAE0X7G9R5
- [Steinitz's theorem on algebraically closed fields](https://tamnd.github.io/math2/node/CAEHMSX9RT.md): statement CAEHMSX9RT
- [Nonharmonic Fourier series](https://tamnd.github.io/math2/node/CAEZVHVJJD.md): definition CAEZVHVJJD
- [Irreducible ideal](https://tamnd.github.io/math2/node/CAKG0X11TJ.md): definition CAKG0X11TJ
- [Copula](https://tamnd.github.io/math2/node/CAMJRBZANK.md): concept CAMJRBZANK
- [Symplectic manifold](https://tamnd.github.io/math2/node/CAN490EPRT.md): concept CAN490EPRT
- [Graded ring](https://tamnd.github.io/math2/node/CAPQBQDRA7.md): definition CAPQBQDRA7
- [Gegenbauer polynomials](https://tamnd.github.io/math2/node/CAPWE7KWRY.md): definition CAPWE7KWRY
- [Cohen–Macaulay ring](https://tamnd.github.io/math2/node/CAWK9DA1PH.md): definition CAWK9DA1PH
- [Method of matched asymptotic expansions](https://tamnd.github.io/math2/node/CAYQ1S3RM8.md): definition CAYQ1S3RM8
- [Levi-Civita connection](https://tamnd.github.io/math2/node/CAZYGE1T4E.md): concept CAZYGE1T4E
- [Rigid analytic space](https://tamnd.github.io/math2/node/CB4Y87RC4V.md): definition CB4Y87RC4V
- [Baily–Borel compactification](https://tamnd.github.io/math2/node/CB54N55JD9.md): statement CB54N55JD9
- [Hilbert–Schmidt integral operator](https://tamnd.github.io/math2/node/CB5G212Q03.md): concept CB5G212Q03
- [q-derivative](https://tamnd.github.io/math2/node/CB945JVQ0X.md): definition CB945JVQ0X
- [Krull domain](https://tamnd.github.io/math2/node/CB9WJCN9BM.md): concept CB9WJCN9BM
- [Exact sequence](https://tamnd.github.io/math2/node/CBDM280TDF.md): definition CBDM280TDF
- [Robust statistics](https://tamnd.github.io/math2/node/CBNYG6QV03.md): definition CBNYG6QV03
- [Entropic uncertainty principle](https://tamnd.github.io/math2/node/CBRTFS0RVC.md): statement CBRTFS0RVC
- [Euler's elastica](https://tamnd.github.io/math2/node/CBT8F8S8JT.md): problem CBT8F8S8JT
- [Handlebody](https://tamnd.github.io/math2/node/CBXJVT7M6H.md): definition CBXJVT7M6H
- [Ulam's stability problem](https://tamnd.github.io/math2/node/CBYB39FTFG.md): problem CBYB39FTFG
- [Riesz decomposition theorem](https://tamnd.github.io/math2/node/CBZ20SS8RQ.md): statement CBZ20SS8RQ
- [Lyapunov–Schmidt reduction](https://tamnd.github.io/math2/node/CC666JT89X.md): definition CC666JT89X
- [Lebesgue–Stieltjes integration](https://tamnd.github.io/math2/node/CC8KNGMZYS.md): definition CC8KNGMZYS
- [Witt's theorem](https://tamnd.github.io/math2/node/CCB8VNQCV0.md): statement CCB8VNQCV0
- [Kakutani's solution of the Dirichlet problem](https://tamnd.github.io/math2/node/CCEKM4241P.md): statement CCEKM4241P
- [Burali-Forti paradox](https://tamnd.github.io/math2/node/CCNCGN3P59.md): counterexample CCNCGN3P59
- [Path homotopy](https://tamnd.github.io/math2/node/CCNKA8C711.md): concept CCNKA8C711
- [Haar's uniqueness theorem](https://tamnd.github.io/math2/node/CCPS738AH8.md): statement CCPS738AH8
- [Facet](https://tamnd.github.io/math2/node/CCQ0QVJY4R.md): definition CCQ0QVJY4R
- [Oscillator representation](https://tamnd.github.io/math2/node/CCR7BAQ5N5.md): concept CCR7BAQ5N5
- [Levi-Civita field](https://tamnd.github.io/math2/node/CCT8SQKHW4.md): object CCT8SQKHW4
- [Open segment](https://tamnd.github.io/math2/node/CCZA91NE5X.md): concept CCZA91NE5X
- [Treewidth](https://tamnd.github.io/math2/node/CD2XB4Y1SF.md): concept CD2XB4Y1SF
- [Frequency shift theorem](https://tamnd.github.io/math2/node/CD35MSBNE7.md): statement CD35MSBNE7
- [Raabe's test](https://tamnd.github.io/math2/node/CD6CAQ6XDN.md): statement CD6CAQ6XDN
- [Holomorphically convex hull](https://tamnd.github.io/math2/node/CD6SMMZSW8.md): concept CD6SMMZSW8
- [Clifford analysis](https://tamnd.github.io/math2/node/CD7JZ43JBA.md): definition CD7JZ43JBA
- [Esakia duality](https://tamnd.github.io/math2/node/CD9Y7YP4EJ.md): statement CD9Y7YP4EJ
- [De Morgan algebra](https://tamnd.github.io/math2/node/CDGBK86D23.md): definition CDGBK86D23
- [Lipschitz domain](https://tamnd.github.io/math2/node/CDJNQ3SQTG.md): definition CDJNQ3SQTG
- [Hölder's theorem on Archimedean groups](https://tamnd.github.io/math2/node/CDM40MT3HZ.md): statement CDM40MT3HZ
- [Dormand–Prince method](https://tamnd.github.io/math2/node/CDMXAH4A7G.md): statement CDMXAH4A7G
- [Morphism of varieties](https://tamnd.github.io/math2/node/CDS1VWB7A3.md): concept CDS1VWB7A3
- [Ambiguity function](https://tamnd.github.io/math2/node/CDVA7EJSSD.md): concept CDVA7EJSSD
- [Latin square](https://tamnd.github.io/math2/node/CDVRMVED2V.md): concept CDVRMVED2V
- [Hidden Markov model](https://tamnd.github.io/math2/node/CDVYCKZ2XZ.md): definition CDVYCKZ2XZ
- [Approximate identity](https://tamnd.github.io/math2/node/CDYXTW3PJF.md): concept CDYXTW3PJF
- [Haar measure](https://tamnd.github.io/math2/node/CE0V9686M8.md): definition CE0V9686M8
- [Betweenness](https://tamnd.github.io/math2/node/CE8GX6ADZS.md): concept CE8GX6ADZS
- [Amenable group](https://tamnd.github.io/math2/node/CEAT9BESKK.md): concept CEAT9BESKK
- [Meixner polynomials](https://tamnd.github.io/math2/node/CEC3DESCFT.md): concept CEC3DESCFT
- [Algebraic variety](https://tamnd.github.io/math2/node/CECPTP66EK.md): definition CECPTP66EK
- [Boolean product](https://tamnd.github.io/math2/node/CED4Q4TFCD.md): concept CED4Q4TFCD
- [Cauchy's exponential equation](https://tamnd.github.io/math2/node/CEGF8W64VQ.md): concept CEGF8W64VQ
- [Translation surface](https://tamnd.github.io/math2/node/CEHHBBD6NT.md): concept CEHHBBD6NT
- [Formally smooth map](https://tamnd.github.io/math2/node/CEK4HKMPB4.md): definition CEK4HKMPB4
- [Deformation retract](https://tamnd.github.io/math2/node/CEK4SC20WK.md): concept CEK4SC20WK
- [Tietze transformations](https://tamnd.github.io/math2/node/CEZTVHC1KS.md): concept CEZTVHC1KS
- [Group homology](https://tamnd.github.io/math2/node/CFAG3VRXEA.md): concept CFAG3VRXEA
- [Composition operator](https://tamnd.github.io/math2/node/CFASPSF7BB.md): concept CFASPSF7BB
- [Reduction to diagonal form by transvections](https://tamnd.github.io/math2/node/CFB41AZX87.md): statement CFB41AZX87
- [Harmonic measure](https://tamnd.github.io/math2/node/CFD7BPM51W.md): concept CFD7BPM51W
- [Gelfand–Raikov theorem](https://tamnd.github.io/math2/node/CFDZJ55N95.md): statement CFDZJ55N95
- [Path space of a manifold](https://tamnd.github.io/math2/node/CFHNV77ZMD.md): object CFHNV77ZMD
- [Finite-dimensional distributions](https://tamnd.github.io/math2/node/CFQXJ4FHR4.md): definition CFQXJ4FHR4
- [Continuous mapping theorem](https://tamnd.github.io/math2/node/CFRN0G7GDB.md): statement CFRN0G7GDB
- [Trigonometric functions](https://tamnd.github.io/math2/node/CFWMB0S5NK.md): definition CFWMB0S5NK
- [Concentration of measure](https://tamnd.github.io/math2/node/CFZ1J559MK.md): concept CFZ1J559MK
- [Tensor product of algebras](https://tamnd.github.io/math2/node/CG0BWPC9ZY.md): definition CG0BWPC9ZY
- [Jacobi's four-square theorem](https://tamnd.github.io/math2/node/CG17WSS64W.md): statement CG17WSS64W
- [Mellin transform under a power substitution](https://tamnd.github.io/math2/node/CG7BFJ04YE.md): statement CG7BFJ04YE
- [Graph embedding](https://tamnd.github.io/math2/node/CG91WNMG2T.md): concept CG91WNMG2T
- [Profinite completion](https://tamnd.github.io/math2/node/CG9EH0TN27.md): concept CG9EH0TN27
- [Schwartz space](https://tamnd.github.io/math2/node/CG9JEATW3Q.md): concept CG9JEATW3Q
- [Differential operator between vector bundles](https://tamnd.github.io/math2/node/CGBX1N9V8H.md): concept CGBX1N9V8H
- [Hadamard's inequality](https://tamnd.github.io/math2/node/CGDN1X32NQ.md): statement CGDN1X32NQ
- [Geometric quantization](https://tamnd.github.io/math2/node/CGEXQ8BAH5.md): definition CGEXQ8BAH5
- [Elementary embedding](https://tamnd.github.io/math2/node/CGH8C8EA9N.md): concept CGH8C8EA9N
- [Completely positive map](https://tamnd.github.io/math2/node/CGY299QMC3.md): concept CGY299QMC3
- [Hypergraph](https://tamnd.github.io/math2/node/CH2JRZNX7V.md): concept CH2JRZNX7V
- [Existence of an eigenvalue for a nonzero symmetric kernel](https://tamnd.github.io/math2/node/CH6655HZXS.md): statement CH6655HZXS
- [Brennan conjecture](https://tamnd.github.io/math2/node/CH73TVRT27.md): conjecture CH73TVRT27
- [Least fixed point](https://tamnd.github.io/math2/node/CH9B4RZPMW.md): definition CH9B4RZPMW
- [Stability of a difference equation](https://tamnd.github.io/math2/node/CHEQ235HW9.md): concept CHEQ235HW9
- [Inverse Mellin transform](https://tamnd.github.io/math2/node/CHESR1W1CY.md): concept CHESR1W1CY
- [Robbins–Monro algorithm](https://tamnd.github.io/math2/node/CHHD44WDHJ.md): concept CHHD44WDHJ
- [Lichnerowicz vanishing theorem](https://tamnd.github.io/math2/node/CHJ4TAH0QX.md): statement CHJ4TAH0QX
- [Path homotopy](https://tamnd.github.io/math2/node/CHP0QWHFWM.md): definition CHP0QWHFWM
- [Limit](https://tamnd.github.io/math2/node/CHS5AFJGWE.md): concept CHS5AFJGWE
- [p-adic number](https://tamnd.github.io/math2/node/CHT5JYM6QW.md): definition CHT5JYM6QW
- [Line graph](https://tamnd.github.io/math2/node/CJ57DT86S8.md): definition CJ57DT86S8
- [Arnoldi iteration](https://tamnd.github.io/math2/node/CJ5EC820S7.md): statement CJ5EC820S7
- [Cauchy's rigidity theorem](https://tamnd.github.io/math2/node/CJ7KCDDF31.md): statement CJ7KCDDF31
- [Coulomb wave function](https://tamnd.github.io/math2/node/CJ9MSY7CRM.md): concept CJ9MSY7CRM
- [Positivity of Hankel matrices of moments](https://tamnd.github.io/math2/node/CJCPD4DZ65.md): statement CJCPD4DZ65
- [Lyapunov function for a difference equation](https://tamnd.github.io/math2/node/CJDS1BGDVH.md): definition CJDS1BGDVH
- [Mahlo cardinal](https://tamnd.github.io/math2/node/CJFEPPP9KA.md): concept CJFEPPP9KA
- [Krawtchouk polynomials](https://tamnd.github.io/math2/node/CJGKW80KBR.md): concept CJGKW80KBR
- [Courant–Fischer min-max theorem](https://tamnd.github.io/math2/node/CJK2HS2508.md): statement CJK2HS2508
- [Bias–variance decomposition](https://tamnd.github.io/math2/node/CJQ8D9FTQN.md): statement CJQ8D9FTQN
- [Generator](https://tamnd.github.io/math2/node/CK08JW32WB.md): concept CK08JW32WB
- [Hermitian manifold](https://tamnd.github.io/math2/node/CK1DCPC3D3.md): concept CK1DCPC3D3
- [Space of functions of bounded variation](https://tamnd.github.io/math2/node/CK4QHTVKN4.md): definition CK4QHTVKN4
- [q-analog](https://tamnd.github.io/math2/node/CK80HQNHEK.md): definition CK80HQNHEK
- [Heyting arithmetic](https://tamnd.github.io/math2/node/CK8ZQRN183.md): concept CK8ZQRN183
- [Mann–Whitney U test](https://tamnd.github.io/math2/node/CKF9KDXBP7.md): concept CKF9KDXBP7
- [Yang–Mills functional](https://tamnd.github.io/math2/node/CKGDR9P9PA.md): concept CKGDR9P9PA
- [Strict Henselization](https://tamnd.github.io/math2/node/CKGN4ZH6P2.md): definition CKGN4ZH6P2
- [Kepler–Poinsot polyhedron](https://tamnd.github.io/math2/node/CKQRZVBGTS.md): object CKQRZVBGTS
- [Solution set condition](https://tamnd.github.io/math2/node/CKTDR0VE8D.md): concept CKTDR0VE8D
- [Hyponormal operator](https://tamnd.github.io/math2/node/CM3KYNZZJ6.md): definition CM3KYNZZJ6
- [Loeb measure](https://tamnd.github.io/math2/node/CM6XF84MNY.md): definition CM6XF84MNY
- [Keller–Segel model](https://tamnd.github.io/math2/node/CM7D3ES438.md): definition CM7D3ES438
- [Makarov's theorem on harmonic measure](https://tamnd.github.io/math2/node/CM9BDNEA10.md): statement CM9BDNEA10
- [Akaike information criterion](https://tamnd.github.io/math2/node/CMAX3KF9H0.md): concept CMAX3KF9H0
- [Woodin cardinal](https://tamnd.github.io/math2/node/CMC2C1D53W.md): concept CMC2C1D53W
- [Shapley–Folkman lemma](https://tamnd.github.io/math2/node/CMGKQNMY77.md): statement CMGKQNMY77
- [Willmore flow](https://tamnd.github.io/math2/node/CMK92XDD0W.md): definition CMK92XDD0W
- [Harmonic analysis](https://tamnd.github.io/math2/node/CMMDQ13YEE.md): definition CMMDQ13YEE
- [Backward error analysis](https://tamnd.github.io/math2/node/CMXHRNWY35.md): concept CMXHRNWY35
- [Inversion of the Hilbert transform](https://tamnd.github.io/math2/node/CMXS9P2F13.md): statement CMXS9P2F13
- [Specialization preorder](https://tamnd.github.io/math2/node/CMYVK9CEKZ.md): concept CMYVK9CEKZ
- [Weil's converse to Haar's theorem](https://tamnd.github.io/math2/node/CMZS6KA033.md): statement CMZS6KA033
- [Boundary integral equation for the Neumann problem](https://tamnd.github.io/math2/node/CN7NP6R1AQ.md): statement CN7NP6R1AQ
- [Iterative method](https://tamnd.github.io/math2/node/CN91KKNSQK.md): definition CN91KKNSQK
- [Hartogs triangle](https://tamnd.github.io/math2/node/CNADH7BENS.md): counterexample CNADH7BENS
- [Sieve](https://tamnd.github.io/math2/node/CNCXNWKC6R.md): concept CNCXNWKC6R
- [Legendre duplication formula](https://tamnd.github.io/math2/node/CND4E1PY9V.md): statement CND4E1PY9V
- [Characteristic class](https://tamnd.github.io/math2/node/CNGJP7B4BW.md): definition CNGJP7B4BW
- [Category of posets](https://tamnd.github.io/math2/node/CNGSMAB60A.md): definition CNGSMAB60A
- [Turing degree](https://tamnd.github.io/math2/node/CNRVT7009S.md): definition CNRVT7009S
- [External ray](https://tamnd.github.io/math2/node/CNX4KVMDJQ.md): concept CNX4KVMDJQ
- [Transfer operator](https://tamnd.github.io/math2/node/CNZV00JQRC.md): concept CNZV00JQRC
- [Finite element method](https://tamnd.github.io/math2/node/CP0TQ5ZWSP.md): definition CP0TQ5ZWSP
- [Hamilton–Jacobi–Bellman equation](https://tamnd.github.io/math2/node/CP25WM5537.md): definition CP25WM5537
- [Numerical continuation](https://tamnd.github.io/math2/node/CP2Z74X1X6.md): definition CP2Z74X1X6
- [Lévy process](https://tamnd.github.io/math2/node/CP7Q2FWWNT.md): definition CP7Q2FWWNT
- [Score](https://tamnd.github.io/math2/node/CP97GRVTQ0.md): concept CP97GRVTQ0
- [Torsor](https://tamnd.github.io/math2/node/CPA1C45GQR.md): definition CPA1C45GQR
- [Abstract Wiener space](https://tamnd.github.io/math2/node/CPA34DVD8J.md): concept CPA34DVD8J
- [Invertible operator](https://tamnd.github.io/math2/node/CPAD5W1CGW.md): concept CPAD5W1CGW
- [Bedford–Taylor theory](https://tamnd.github.io/math2/node/CPC8ZMF1SD.md): definition CPC8ZMF1SD
- [Translation equation](https://tamnd.github.io/math2/node/CPE1ANDDJ0.md): concept CPE1ANDDJ0
- [Phase space](https://tamnd.github.io/math2/node/CPEC2NW2KF.md): concept CPEC2NW2KF
- [Electric field integral equation](https://tamnd.github.io/math2/node/CPF9Q1GR10.md): example CPF9Q1GR10
- [Statistical decision theory](https://tamnd.github.io/math2/node/CPFSBBHXT5.md): concept CPFSBBHXT5
- [Mixture model](https://tamnd.github.io/math2/node/CPGEJWQS8A.md): definition CPGEJWQS8A
- [Round-off error](https://tamnd.github.io/math2/node/CPN6SD3BTV.md): concept CPN6SD3BTV
- [Plurisubharmonic function](https://tamnd.github.io/math2/node/CPR1B0WEWA.md): definition CPR1B0WEWA
- [Gleason's theorem on projective compact spaces](https://tamnd.github.io/math2/node/CPZKZ2AV0G.md): statement CPZKZ2AV0G
- [Morita equivalence](https://tamnd.github.io/math2/node/CQ3616BH5C.md): definition CQ3616BH5C
- [Combinatorial design](https://tamnd.github.io/math2/node/CQ4C1WKMF6.md): definition CQ4C1WKMF6
- [Converse of the mean value property](https://tamnd.github.io/math2/node/CQ7E0V6T2G.md): statement CQ7E0V6T2G
- [2-category](https://tamnd.github.io/math2/node/CQACXG10JZ.md): concept CQACXG10JZ
- [Newton fractal](https://tamnd.github.io/math2/node/CQE378ZVY5.md): example CQE378ZVY5
- [Direct summand conjecture](https://tamnd.github.io/math2/node/CQJ1FPKKHT.md): statement CQJ1FPKKHT
- [Characteristic variety](https://tamnd.github.io/math2/node/CQJ2E2ARRF.md): concept CQJ2E2ARRF
- [Closed immersion](https://tamnd.github.io/math2/node/CQKTEBK13V.md): definition CQKTEBK13V
- [Ultrafilter on a Boolean algebra](https://tamnd.github.io/math2/node/CQM84EW10G.md): concept CQM84EW10G
- [Mollifier](https://tamnd.github.io/math2/node/CQMK4YPE1J.md): concept CQMK4YPE1J
- [Center of a ring](https://tamnd.github.io/math2/node/CQSZTPMSEP.md): definition CQSZTPMSEP
- [Ricci flow](https://tamnd.github.io/math2/node/CR14D9YQTY.md): concept CR14D9YQTY
- [Projective module](https://tamnd.github.io/math2/node/CRE6VEKT32.md): definition CRE6VEKT32
- [Kernel approximation in reproducing kernel Hilbert spaces](https://tamnd.github.io/math2/node/CRF6HCM0Q5.md): definition CRF6HCM0Q5
- [Group scheme](https://tamnd.github.io/math2/node/CRFGFCNG3Q.md): definition CRFGFCNG3Q
- [Fuzzy measure](https://tamnd.github.io/math2/node/CRFQEGJDA9.md): concept CRFQEGJDA9
- [Minimal set](https://tamnd.github.io/math2/node/CRGZZKQ7Z9.md): concept CRGZZKQ7Z9
- [Smoothness classes](https://tamnd.github.io/math2/node/CRJSW18XVM.md): definition CRJSW18XVM
- [Modulus of smoothness](https://tamnd.github.io/math2/node/CRPZ1RVNKQ.md): concept CRPZ1RVNKQ
- [Hahn polynomials](https://tamnd.github.io/math2/node/CRZH89G0J1.md): definition CRZH89G0J1
- [Gross–Pitaevskii equation](https://tamnd.github.io/math2/node/CS10HH24VT.md): definition CS10HH24VT
- [Modularity theorem](https://tamnd.github.io/math2/node/CS1NVJBZMH.md): statement CS1NVJBZMH
- [Moment problem](https://tamnd.github.io/math2/node/CS7TWHY7JM.md): problem CS7TWHY7JM
- [Prenex normal form](https://tamnd.github.io/math2/node/CS82V969DE.md): concept CS82V969DE
- [Dougall's formula](https://tamnd.github.io/math2/node/CSCWM5ZKV4.md): statement CSCWM5ZKV4
- [Serre subcategory](https://tamnd.github.io/math2/node/CSD6W0JV0P.md): concept CSD6W0JV0P
- [Kaczmarz method](https://tamnd.github.io/math2/node/CSETD54YDT.md): statement CSETD54YDT
- [Simplex](https://tamnd.github.io/math2/node/CSK0KBMARJ.md): definition CSK0KBMARJ
- [Projective geometry over a finite field](https://tamnd.github.io/math2/node/CSMB49BDPT.md): example CSMB49BDPT
- [Deficiency indices](https://tamnd.github.io/math2/node/CSMF0WVH67.md): definition CSMF0WVH67
- [Beverton–Holt model](https://tamnd.github.io/math2/node/CSRSDP9PXJ.md): example CSRSDP9PXJ
- [Incidence geometry](https://tamnd.github.io/math2/node/CSXT0BG2CM.md): concept CSXT0BG2CM
- [Uniform convergence of absolutely summable Fourier series](https://tamnd.github.io/math2/node/CSYZGA5YKS.md): statement CSYZGA5YKS
- [Multivariate approximation](https://tamnd.github.io/math2/node/CT3PNBCYMX.md): concept CT3PNBCYMX
- [Derived functor](https://tamnd.github.io/math2/node/CT9GYCJV4V.md): definition CT9GYCJV4V
- [p-adic Lie group](https://tamnd.github.io/math2/node/CTBNKE0B03.md): definition CTBNKE0B03
- [Affine scheme](https://tamnd.github.io/math2/node/CTBYV71CEJ.md): definition CTBYV71CEJ
- [Topological conjugacy](https://tamnd.github.io/math2/node/CTCSAWBSB8.md): definition CTCSAWBSB8
- [Wald's equation](https://tamnd.github.io/math2/node/CTDG4MZC8D.md): statement CTDG4MZC8D
- [Detailed balance](https://tamnd.github.io/math2/node/CTGN6CK1ZS.md): concept CTGN6CK1ZS
- [Equalizer](https://tamnd.github.io/math2/node/CTJ5AD8ST5.md): concept CTJ5AD8ST5
- [Pexider equation](https://tamnd.github.io/math2/node/CTMRG74JN0.md): concept CTMRG74JN0
- [Cartan connection](https://tamnd.github.io/math2/node/CTQ0V4WKHN.md): definition CTQ0V4WKHN
- [Euclidean tilings by convex regular polygons](https://tamnd.github.io/math2/node/CTVSGWNB19.md): example CTVSGWNB19
- [Exact category](https://tamnd.github.io/math2/node/CTVVXCWTX7.md): definition CTVVXCWTX7
- [Natural density](https://tamnd.github.io/math2/node/CTXX2CXFRB.md): definition CTXX2CXFRB
- [Mackey imprimitivity theorem](https://tamnd.github.io/math2/node/CTYQQWJPMB.md): statement CTYQQWJPMB
- [Free monoid](https://tamnd.github.io/math2/node/CV013KHVJG.md): concept CV013KHVJG
- [Arithmetical hierarchy](https://tamnd.github.io/math2/node/CV2E6JZHVP.md): definition CV2E6JZHVP
- [Random ordinary differential equation](https://tamnd.github.io/math2/node/CV9ME89WDV.md): definition CV9ME89WDV
- [Local existence of harmonic conjugates](https://tamnd.github.io/math2/node/CVDGP7DKZ3.md): statement CVDGP7DKZ3
- [Brenier's theorem](https://tamnd.github.io/math2/node/CVQVBMHH4X.md): statement CVQVBMHH4X
- [Kelvin problem](https://tamnd.github.io/math2/node/CVTKSCP55A.md): problem CVTKSCP55A
- [Inverse spectral problem](https://tamnd.github.io/math2/node/CVYHG5HKPP.md): problem CVYHG5HKPP
- [Circumsphere](https://tamnd.github.io/math2/node/CW1TJDGCKA.md): definition CW1TJDGCKA
- [Sigma-finite measure](https://tamnd.github.io/math2/node/CW5CEF84T2.md): definition CW5CEF84T2
- [Principal series representation](https://tamnd.github.io/math2/node/CWDF4KGHS3.md): concept CWDF4KGHS3
- [Karoubi envelope](https://tamnd.github.io/math2/node/CWKNRTVERF.md): definition CWKNRTVERF
- [Double series](https://tamnd.github.io/math2/node/CWP26Y1TCZ.md): definition CWP26Y1TCZ
- [Model](https://tamnd.github.io/math2/node/CWQV9PTCB3.md): definition CWQV9PTCB3
- [Partial linear space](https://tamnd.github.io/math2/node/CX07CWPCDX.md): concept CX07CWPCDX
- [Fiber bundle](https://tamnd.github.io/math2/node/CX4M8K61AB.md): concept CX4M8K61AB
- [Grothendieck's Galois theory](https://tamnd.github.io/math2/node/CX7BDQ47ZK.md): concept CX7BDQ47ZK
- [Non-vanishing of the zeta function on the line Re s = 1](https://tamnd.github.io/math2/node/CXES7915R7.md): statement CXES7915R7
- [Unimodular group](https://tamnd.github.io/math2/node/CXG6JZGMCK.md): concept CXG6JZGMCK
- [Topological ring](https://tamnd.github.io/math2/node/CXV1969XGB.md): definition CXV1969XGB
- [Homoclinic orbit](https://tamnd.github.io/math2/node/CXXM6ZYXDX.md): definition CXXM6ZYXDX
- [Alexandrov topology](https://tamnd.github.io/math2/node/CXZZJ1QV5K.md): definition CXZZJ1QV5K
- [Definable set](https://tamnd.github.io/math2/node/CY1TXHF6KQ.md): concept CY1TXHF6KQ
- [Stein space](https://tamnd.github.io/math2/node/CYAWQF7AJZ.md): definition CYAWQF7AJZ
- [Transcendence basis](https://tamnd.github.io/math2/node/CYCJ46W6P6.md): definition CYCJ46W6P6
- [Clean ring](https://tamnd.github.io/math2/node/CYCTQ0CR3K.md): concept CYCTQ0CR3K
- [Pointless topology](https://tamnd.github.io/math2/node/CYEMGH98WN.md): concept CYEMGH98WN
- [Grassmannian](https://tamnd.github.io/math2/node/CYPA882RB9.md): object CYPA882RB9
- [Pólya's theorem on integer-valued entire functions](https://tamnd.github.io/math2/node/CYT8JNNWXH.md): statement CYT8JNNWXH
- [Newton polytope](https://tamnd.github.io/math2/node/CYTTX8KHC8.md): concept CYTTX8KHC8
- [Gauss–Newton algorithm](https://tamnd.github.io/math2/node/CYV4T43FQJ.md): statement CYV4T43FQJ
- [Finite difference](https://tamnd.github.io/math2/node/CYW8E2HTWN.md): definition CYW8E2HTWN
- [Berezin integral](https://tamnd.github.io/math2/node/CYWSMK47CN.md): concept CYWSMK47CN
- [Join-irreducible element](https://tamnd.github.io/math2/node/CZ6ABG86WY.md): definition CZ6ABG86WY
- [Hemicontinuity](https://tamnd.github.io/math2/node/CZCANZNKPJ.md): definition CZCANZNKPJ
- [Bessel function of the first kind](https://tamnd.github.io/math2/node/CZCMRDWDYW.md): concept CZCMRDWDYW
- [Special unitary group SU(n)](https://tamnd.github.io/math2/node/CZCWFPNS4N.md): example CZCWFPNS4N
- [Birkhoff's subdirect representation theorem](https://tamnd.github.io/math2/node/CZF5696TPY.md): statement CZF5696TPY
- [Totally disconnected group](https://tamnd.github.io/math2/node/CZG6Y6EAHP.md): definition CZG6Y6EAHP
- [Conditionally complete lattice](https://tamnd.github.io/math2/node/CZPBQY67ZJ.md): concept CZPBQY67ZJ
- [Kodaira–Nakano vanishing theorem](https://tamnd.github.io/math2/node/CZRWHNKD2C.md): statement CZRWHNKD2C
- [Length of a module](https://tamnd.github.io/math2/node/CZSCNZH8VY.md): concept CZSCNZH8VY
- [Endomorphism monoid](https://tamnd.github.io/math2/node/CZW4BPZPZC.md): definition CZW4BPZPZC
- [Algebra over a monad](https://tamnd.github.io/math2/node/CZWDNA1HA7.md): definition CZWDNA1HA7
- [Filtered colimits commute with finite limits](https://tamnd.github.io/math2/node/CZYDQ96X2T.md): statement CZYDQ96X2T
- [Proof that a homomorphism is injective iff its kernel is trivial](https://tamnd.github.io/math2/node/D05NAAC3XK.md): proof D05NAAC3XK
- [Conley's fundamental theorem of dynamical systems](https://tamnd.github.io/math2/node/D07MBM0S3V.md): statement D07MBM0S3V
- [Almost Mathieu operator](https://tamnd.github.io/math2/node/D08SKTH2T2.md): example D08SKTH2T2
- [Tent map](https://tamnd.github.io/math2/node/D0A22MEB1F.md): example D0A22MEB1F
- [Relative CW complex](https://tamnd.github.io/math2/node/D0DAFG8QK6.md): concept D0DAFG8QK6
- [Solvmanifold](https://tamnd.github.io/math2/node/D0Q6GE7FG4.md): definition D0Q6GE7FG4
- [Set-valued analysis](https://tamnd.github.io/math2/node/D0R8455C26.md): concept D0R8455C26
- [Fresnel integrals](https://tamnd.github.io/math2/node/D0TQ9F2ZDW.md): concept D0TQ9F2ZDW
- [Chromatic homotopy theory](https://tamnd.github.io/math2/node/D0X6EBNMW6.md): definition D0X6EBNMW6
- [Localization of a space](https://tamnd.github.io/math2/node/D0ZHVD797D.md): concept D0ZHVD797D
- [Euler–Maruyama method](https://tamnd.github.io/math2/node/D0ZXXEYNK8.md): definition D0ZXXEYNK8
- [Mahler's theorem](https://tamnd.github.io/math2/node/D115PG3T7K.md): statement D115PG3T7K
- [Girth](https://tamnd.github.io/math2/node/D13K9GJT0B.md): concept D13K9GJT0B
- [Extremally disconnected space](https://tamnd.github.io/math2/node/D14XVCMGX9.md): definition D14XVCMGX9
- [6-j symbol](https://tamnd.github.io/math2/node/D17XX5N9VF.md): concept D17XX5N9VF
- [Newton polygon](https://tamnd.github.io/math2/node/D186VNHFZJ.md): definition D186VNHFZJ
- [Hilbert's invariant integral](https://tamnd.github.io/math2/node/D1EBAAFEWV.md): concept D1EBAAFEWV
- [Interval arithmetic](https://tamnd.github.io/math2/node/D1FM4ECHKD.md): definition D1FM4ECHKD
- [LYM inequality](https://tamnd.github.io/math2/node/D1GQNNRDDN.md): statement D1GQNNRDDN
- [Derived category of coherent sheaves](https://tamnd.github.io/math2/node/D1MYE6WVZ9.md): definition D1MYE6WVZ9
- [Littlewood–Paley decomposition](https://tamnd.github.io/math2/node/D1PAT0K446.md): concept D1PAT0K446
- [Difference set](https://tamnd.github.io/math2/node/D1QA6N9EWJ.md): definition D1QA6N9EWJ
- [Noncommutative ring](https://tamnd.github.io/math2/node/D1SA8CHJV0.md): concept D1SA8CHJV0
- [Lorentzian manifold](https://tamnd.github.io/math2/node/D1SYF8F8RQ.md): definition D1SYF8F8RQ
- [Suslin's problem](https://tamnd.github.io/math2/node/D1T39KHPDN.md): problem D1T39KHPDN
- [Moments and orthogonal polynomials](https://tamnd.github.io/math2/node/D1TK8AHMCM.md): definition D1TK8AHMCM
- [Order of a projective plane](https://tamnd.github.io/math2/node/D2GQNNQ7FR.md): definition D2GQNNQ7FR
- [Topological transitivity](https://tamnd.github.io/math2/node/D2J0A8SACE.md): definition D2J0A8SACE
- [Maximum principle](https://tamnd.github.io/math2/node/D2JF19HF91.md): statement D2JF19HF91
- [Liénard equation](https://tamnd.github.io/math2/node/D2KC8NEAPS.md): example D2KC8NEAPS
- [Nuclear C*-algebra](https://tamnd.github.io/math2/node/D2M4VXJ7PA.md): concept D2M4VXJ7PA
- [Bass's theorem P](https://tamnd.github.io/math2/node/D2M8BSW720.md): statement D2M8BSW720
- [Ore extension](https://tamnd.github.io/math2/node/D2R7BNFZNS.md): definition D2R7BNFZNS
- [Semiring](https://tamnd.github.io/math2/node/D31S3GR51T.md): concept D31S3GR51T
- [Bernoulli scheme](https://tamnd.github.io/math2/node/D33JK3AXRZ.md): example D33JK3AXRZ
- [Opposite structure](https://tamnd.github.io/math2/node/D33V8HE0KY.md): concept D33V8HE0KY
- [Adams–Novikov spectral sequence](https://tamnd.github.io/math2/node/D34QHZVPES.md): statement D34QHZVPES
- [Whitehead tower](https://tamnd.github.io/math2/node/D3649AJ30F.md): concept D3649AJ30F
- [Space of bounded continuous functions](https://tamnd.github.io/math2/node/D38YWGXRKJ.md): definition D38YWGXRKJ
- [Quasi-measure-preserving map](https://tamnd.github.io/math2/node/D38ZP0YX0D.md): definition D38ZP0YX0D
- [Euler's formula for polytopes](https://tamnd.github.io/math2/node/D39NBQH3H6.md): statement D39NBQH3H6
- [Capacity of a disc and a segment](https://tamnd.github.io/math2/node/D3A9V8DKZV.md): example D3A9V8DKZV
- [Homomorphism](https://tamnd.github.io/math2/node/D3AC1N7PNX.md): definition D3AC1N7PNX
- [Alexandrov–Fenchel inequality](https://tamnd.github.io/math2/node/D3AVHVYQW9.md): statement D3AVHVYQW9
- [Path space fibration](https://tamnd.github.io/math2/node/D3BN7MBKTR.md): concept D3BN7MBKTR
- [Feedback linearization](https://tamnd.github.io/math2/node/D3FHYZ4P1R.md): definition D3FHYZ4P1R
- [Closure of an operator](https://tamnd.github.io/math2/node/D3G13ZQ2AQ.md): definition D3G13ZQ2AQ
- [Similarity transformation](https://tamnd.github.io/math2/node/D3GYHF6NAX.md): concept D3GYHF6NAX
- [Complex multiplication](https://tamnd.github.io/math2/node/D3HEJTVY7G.md): concept D3HEJTVY7G
- [Kelvin functions](https://tamnd.github.io/math2/node/D3JWE193EM.md): definition D3JWE193EM
- [Homotopy extension property](https://tamnd.github.io/math2/node/D42X0F1EG1.md): concept D42X0F1EG1
- [Tychonoff plank](https://tamnd.github.io/math2/node/D440JPY3QG.md): counterexample D440JPY3QG
- [Second Cousin problem](https://tamnd.github.io/math2/node/D44KAF8PNK.md): problem D44KAF8PNK
- [Frölicher–Nijenhuis bracket](https://tamnd.github.io/math2/node/D46QECMC10.md): concept D46QECMC10
- [Geometric Brownian motion](https://tamnd.github.io/math2/node/D48DR46B97.md): concept D48DR46B97
- [Structural stability](https://tamnd.github.io/math2/node/D4MGFAHVMT.md): concept D4MGFAHVMT
- [Locally connected space](https://tamnd.github.io/math2/node/D4MRA2R7N8.md): definition D4MRA2R7N8
- [Riccati difference equation](https://tamnd.github.io/math2/node/D4Q8VM7826.md): concept D4Q8VM7826
- [Socle of a module](https://tamnd.github.io/math2/node/D4TYZN7BRH.md): definition D4TYZN7BRH
- [Polyharmonic spline](https://tamnd.github.io/math2/node/D4W7DBAT55.md): definition D4W7DBAT55
- [Iwasawa manifold](https://tamnd.github.io/math2/node/D4WF2BJE2M.md): example D4WF2BJE2M
- [Spline interpolation](https://tamnd.github.io/math2/node/D59HBKKRP9.md): concept D59HBKKRP9
- [Puiseux series](https://tamnd.github.io/math2/node/D5BDZZW9D4.md): definition D5BDZZW9D4
- [Uniform algebra](https://tamnd.github.io/math2/node/D5CKZCK3Z2.md): definition D5CKZCK3Z2
- [Oscillatory integral](https://tamnd.github.io/math2/node/D5HZKE8E1A.md): concept D5HZKE8E1A
- [Holomorphically convex domain](https://tamnd.github.io/math2/node/D5J6CP5V2Z.md): concept D5J6CP5V2Z
- [Lax equivalence theorem](https://tamnd.github.io/math2/node/D5M7TRD3MY.md): statement D5M7TRD3MY
- [Endomorphism ring](https://tamnd.github.io/math2/node/D5MMVPJ8M1.md): definition D5MMVPJ8M1
- [k-means clustering](https://tamnd.github.io/math2/node/D5PN3G41D6.md): concept D5PN3G41D6
- [De Giorgi's structure theorem](https://tamnd.github.io/math2/node/D5VM0MC3EB.md): statement D5VM0MC3EB
- [Simson line](https://tamnd.github.io/math2/node/D5VS0YFTEZ.md): definition D5VS0YFTEZ
- [Horn](https://tamnd.github.io/math2/node/D5Z60BRP18.md): concept D5Z60BRP18
- [Equilibrium of a difference equation](https://tamnd.github.io/math2/node/D62PEKHRV5.md): concept D62PEKHRV5
- [Quasi-algebraically closed field](https://tamnd.github.io/math2/node/D637VGM7GV.md): definition D637VGM7GV
- [Cyclic algebra](https://tamnd.github.io/math2/node/D6BM80QYNE.md): concept D6BM80QYNE
- [Polynomial long division](https://tamnd.github.io/math2/node/D6D0SR1FK3.md): concept D6D0SR1FK3
- [Yoneda lemma](https://tamnd.github.io/math2/node/D6DF1ARPVD.md): statement D6DF1ARPVD
- [Prokhorov's theorem](https://tamnd.github.io/math2/node/D6FMEQYA76.md): statement D6FMEQYA76
- [Lelong number](https://tamnd.github.io/math2/node/D6JTF3613D.md): definition D6JTF3613D
- [Post's inversion formula](https://tamnd.github.io/math2/node/D6QKPBS2EN.md): statement D6QKPBS2EN
- [Extension and contraction of ideals](https://tamnd.github.io/math2/node/D6QKPBT78T.md): definition D6QKPBT78T
- [Abelian and Tauberian theorems](https://tamnd.github.io/math2/node/D6XWK1NRY1.md): concept D6XWK1NRY1
- [Beck's theorem](https://tamnd.github.io/math2/node/D75XZC18PP.md): statement D75XZC18PP
- [Complete lattice](https://tamnd.github.io/math2/node/D7A0ZGZT5S.md): concept D7A0ZGZT5S
- [Eigendecomposition](https://tamnd.github.io/math2/node/D7E9SZ4E82.md): statement D7E9SZ4E82
- [Puppe sequence](https://tamnd.github.io/math2/node/D7K7YB52MM.md): statement D7K7YB52MM
- [Countable additivity](https://tamnd.github.io/math2/node/D7KB9HFCGZ.md): concept D7KB9HFCGZ
- [Fusion category](https://tamnd.github.io/math2/node/D7NA6T33A7.md): concept D7NA6T33A7
- [Fourier analysis](https://tamnd.github.io/math2/node/D7PH59MNTV.md): definition D7PH59MNTV
- [Transcendence degree](https://tamnd.github.io/math2/node/D7TR0YWR9H.md): concept D7TR0YWR9H
- [Topological field](https://tamnd.github.io/math2/node/D7YQW9RG24.md): concept D7YQW9RG24
- [Projective connection](https://tamnd.github.io/math2/node/D83NMTHT7N.md): concept D83NMTHT7N
- [Hilbert's Nullstellensatz](https://tamnd.github.io/math2/node/D85VZWS9QK.md): statement D85VZWS9QK
- [Singular point of an algebraic variety](https://tamnd.github.io/math2/node/D86X15A8WS.md): concept D86X15A8WS
- [Ulm's theorem](https://tamnd.github.io/math2/node/D8A4VEQR98.md): statement D8A4VEQR98
- [Subcategory](https://tamnd.github.io/math2/node/D8C0EYD9ED.md): concept D8C0EYD9ED
- [Commutative diagram](https://tamnd.github.io/math2/node/D8EYFTW4RY.md): definition D8EYFTW4RY
- [Standard conjectures on algebraic cycles](https://tamnd.github.io/math2/node/D8GA2R3QT5.md): conjecture D8GA2R3QT5
- [Topology of pointwise convergence](https://tamnd.github.io/math2/node/D8GHJN61MW.md): definition D8GHJN61MW
- [Boundary integral method for the Dirichlet problem](https://tamnd.github.io/math2/node/D8H0MJCS63.md): definition D8H0MJCS63
- [Specht module](https://tamnd.github.io/math2/node/D8KET8700Y.md): definition D8KET8700Y
- [Positive solutions of integral equations](https://tamnd.github.io/math2/node/D8PHFS2KAY.md): concept D8PHFS2KAY
- [Domain of convergence of a power series](https://tamnd.github.io/math2/node/D8S8JWE45R.md): statement D8S8JWE45R
- [Heteroscedasticity](https://tamnd.github.io/math2/node/D8VX6XQ3TT.md): concept D8VX6XQ3TT
- [Rota's basis conjecture](https://tamnd.github.io/math2/node/D8ZD10YF3Y.md): conjecture D8ZD10YF3Y
- [Nonlinear approximation](https://tamnd.github.io/math2/node/D94626DTTW.md): definition D94626DTTW
- [Hopf conjecture](https://tamnd.github.io/math2/node/D95SQEJJSK.md): conjecture D95SQEJJSK
- [Axiom schema of replacement](https://tamnd.github.io/math2/node/D96K8S8R61.md): concept D96K8S8R61
- [Gauss–Legendre Runge–Kutta methods](https://tamnd.github.io/math2/node/D99FZQ2SQT.md): statement D99FZQ2SQT
- [Computable function](https://tamnd.github.io/math2/node/D9AWNH60RM.md): definition D9AWNH60RM
- [Kolmogorov's inequality](https://tamnd.github.io/math2/node/D9B4HE4B4Q.md): statement D9B4HE4B4Q
- [Ahlfors–Bers theorem](https://tamnd.github.io/math2/node/D9B92SQ2GF.md): statement D9B92SQ2GF
- [Blaschke selection theorem](https://tamnd.github.io/math2/node/D9C581V5HG.md): statement D9C581V5HG
- [Laplace–Stieltjes transform](https://tamnd.github.io/math2/node/D9H2AGW198.md): definition D9H2AGW198
- [Stone–Čech compactification](https://tamnd.github.io/math2/node/D9MWP5C85W.md): concept D9MWP5C85W
- [Standard Borel space](https://tamnd.github.io/math2/node/D9PDDPC61K.md): definition D9PDDPC61K
- [Directed set](https://tamnd.github.io/math2/node/D9RY4667K6.md): concept D9RY4667K6
- [Jackson kernel](https://tamnd.github.io/math2/node/D9RZQGM0QP.md): definition D9RZQGM0QP
- [Bifurcation theory](https://tamnd.github.io/math2/node/D9TW46P19Y.md): concept D9TW46P19Y
- [Quantum ergodicity](https://tamnd.github.io/math2/node/D9W79NKYZJ.md): statement D9W79NKYZJ
- [Cluster analysis](https://tamnd.github.io/math2/node/D9Y3A9RH6J.md): concept D9Y3A9RH6J
- [Kleisli category](https://tamnd.github.io/math2/node/D9YHA46YWM.md): concept D9YHA46YWM
- [Schur polynomial](https://tamnd.github.io/math2/node/DA03KQ311J.md): definition DA03KQ311J
- [Lagrange basis polynomials](https://tamnd.github.io/math2/node/DA3CQJ5MJ6.md): concept DA3CQJ5MJ6
- [Residually finite group](https://tamnd.github.io/math2/node/DA515YH1J0.md): definition DA515YH1J0
- [Quantale](https://tamnd.github.io/math2/node/DA619M38C3.md): concept DA619M38C3
- [Hadamard gap theorem](https://tamnd.github.io/math2/node/DA8M8VZZHC.md): statement DA8M8VZZHC
- [Transversality conditions for variable endpoints](https://tamnd.github.io/math2/node/DAEHVRJVSD.md): statement DAEHVRJVSD
- [Fast marching method](https://tamnd.github.io/math2/node/DAK6EJK20R.md): statement DAK6EJK20R
- [Opposite category](https://tamnd.github.io/math2/node/DAMK8XE1FR.md): concept DAMK8XE1FR
- [Projective plane](https://tamnd.github.io/math2/node/DANFNN8C4Y.md): concept DANFNN8C4Y
- [Thiele's interpolation formula](https://tamnd.github.io/math2/node/DANS1S61SX.md): statement DANS1S61SX
- [Topological quantum field theory](https://tamnd.github.io/math2/node/DAP92B80J2.md): concept DAP92B80J2
- [Monte Carlo integration](https://tamnd.github.io/math2/node/DAR3QJ5H7Y.md): statement DAR3QJ5H7Y
- [Tridiagonal matrix](https://tamnd.github.io/math2/node/DARX1KZRHZ.md): concept DARX1KZRHZ
- [Subdirectly irreducible algebra](https://tamnd.github.io/math2/node/DASF939439.md): concept DASF939439
- [Solving linear difference equations by the Z-transform](https://tamnd.github.io/math2/node/DAVT2ZWAF7.md): concept DAVT2ZWAF7
- [Relative homology](https://tamnd.github.io/math2/node/DAXBFJ8G6T.md): definition DAXBFJ8G6T
- [Nelson–Aalen estimator](https://tamnd.github.io/math2/node/DBBE47SCG8.md): definition DBBE47SCG8
- [Least fixed point](https://tamnd.github.io/math2/node/DBD64TH52D.md): concept DBD64TH52D
- [Time series](https://tamnd.github.io/math2/node/DBDVPNMD46.md): definition DBDVPNMD46
- [Karamata's Tauberian theorem](https://tamnd.github.io/math2/node/DBEGF38V9H.md): statement DBEGF38V9H
- [Chaos theory](https://tamnd.github.io/math2/node/DBKXCGDKTD.md): concept DBKXCGDKTD
- [Differentiation operator as inverse of integration](https://tamnd.github.io/math2/node/DBM13YSJAM.md): statement DBM13YSJAM
- [Logarithmic capacity](https://tamnd.github.io/math2/node/DBQXRTC0VN.md): concept DBQXRTC0VN
- [Mixed finite element method](https://tamnd.github.io/math2/node/DBVE636BXN.md): definition DBVE636BXN
- [Ternary discriminator](https://tamnd.github.io/math2/node/DBWS5MMPZD.md): definition DBWS5MMPZD
- [Graph of groups](https://tamnd.github.io/math2/node/DC1CQMD3WK.md): concept DC1CQMD3WK
- [Obstruction theory](https://tamnd.github.io/math2/node/DC4JPVPXEJ.md): definition DC4JPVPXEJ
- [Second-order arithmetic](https://tamnd.github.io/math2/node/DC60FYCSPM.md): concept DC60FYCSPM
- [Characteristic value of a kernel](https://tamnd.github.io/math2/node/DC7HMHVREP.md): concept DC7HMHVREP
- [(p,q)-form](https://tamnd.github.io/math2/node/DC81TCR98Y.md): concept DC81TCR98Y
- [Optional stopping theorem](https://tamnd.github.io/math2/node/DC9HZT2HVJ.md): statement DC9HZT2HVJ
- [Fejér kernel](https://tamnd.github.io/math2/node/DCCA2T9GKH.md): concept DCCA2T9GKH
- [de Casteljau's algorithm](https://tamnd.github.io/math2/node/DCH91F9YDK.md): definition DCH91F9YDK
- [Favard constant](https://tamnd.github.io/math2/node/DCJTJXN3SR.md): definition DCJTJXN3SR
- [Ricker wavelet](https://tamnd.github.io/math2/node/DCJW3WZ7ZJ.md): example DCJW3WZ7ZJ
- [Marchenko–Pastur law](https://tamnd.github.io/math2/node/DCK63WTY5E.md): statement DCK63WTY5E
- [f-vector and h-vector](https://tamnd.github.io/math2/node/DCK64GFPD7.md): concept DCK64GFPD7
- [Newtonian potential](https://tamnd.github.io/math2/node/DCMFQ01QGS.md): concept DCMFQ01QGS
- [Duality in projective geometry](https://tamnd.github.io/math2/node/DCMP1A0138.md): concept DCMP1A0138
- [Schauder fixed-point theorem](https://tamnd.github.io/math2/node/DCPFVVZZ54.md): statement DCPFVVZZ54
- [Congruence relation](https://tamnd.github.io/math2/node/DCSXXT4SER.md): concept DCSXXT4SER
- [FKG inequality](https://tamnd.github.io/math2/node/DCXJGS6E6D.md): statement DCXJGS6E6D
- [Adams spectral sequence](https://tamnd.github.io/math2/node/DD4D0EKX74.md): statement DD4D0EKX74
- [Finite speed of propagation](https://tamnd.github.io/math2/node/DD66THZBDX.md): statement DD66THZBDX
- [Bailey's lemma](https://tamnd.github.io/math2/node/DD8SAAED09.md): statement DD8SAAED09
- [Kleene fixed-point theorem](https://tamnd.github.io/math2/node/DDAHG7QRY8.md): statement DDAHG7QRY8
- [Earnshaw's theorem](https://tamnd.github.io/math2/node/DDH18HWKZ9.md): statement DDH18HWKZ9
- [Newton's forward difference formula](https://tamnd.github.io/math2/node/DDHCMKHD2T.md): statement DDHCMKHD2T
- [Weyl's unitary trick](https://tamnd.github.io/math2/node/DDQ2WBQ9GB.md): concept DDQ2WBQ9GB
- [Cartan's theorem on continuous homomorphisms](https://tamnd.github.io/math2/node/DDQJGJGQGC.md): statement DDQJGJGQGC
- [Tor functor](https://tamnd.github.io/math2/node/DDQRERAQCV.md): concept DDQRERAQCV
- [Beurling's projection theorem](https://tamnd.github.io/math2/node/DDV6E39VSX.md): statement DDV6E39VSX
- [Real closed field](https://tamnd.github.io/math2/node/DDY3Q3PJ6W.md): example DDY3Q3PJ6W
- [Tameness theorem](https://tamnd.github.io/math2/node/DE1Z7R9T05.md): statement DE1Z7R9T05
- [Segal conjecture](https://tamnd.github.io/math2/node/DE29ZC399Z.md): statement DE29ZC399Z
- [Inverse limit of algebras](https://tamnd.github.io/math2/node/DE2KEQ85ZT.md): concept DE2KEQ85ZT
- [Hilbert sum](https://tamnd.github.io/math2/node/DE42T92C4R.md): definition DE42T92C4R
- [Möbius plane](https://tamnd.github.io/math2/node/DED008EJVH.md): concept DED008EJVH
- [Complete statistic](https://tamnd.github.io/math2/node/DED6JFAV1K.md): concept DED6JFAV1K
- [c-cyclical monotonicity](https://tamnd.github.io/math2/node/DEHZAQ1N92.md): definition DEHZAQ1N92
- [Hilbert space adjoint](https://tamnd.github.io/math2/node/DEK2SCEDJ2.md): definition DEK2SCEDJ2
- [Nevanlinna theory in several variables](https://tamnd.github.io/math2/node/DERJPQ04BQ.md): concept DERJPQ04BQ
- [von Neumann regular ring](https://tamnd.github.io/math2/node/DEV24TSWQZ.md): definition DEV24TSWQZ
- [Fredholm equation of the second kind](https://tamnd.github.io/math2/node/DEYYNVCV9J.md): concept DEYYNVCV9J
- [Monad from an adjunction](https://tamnd.github.io/math2/node/DF5N0PK3ZS.md): statement DF5N0PK3ZS
- [Permutation pattern](https://tamnd.github.io/math2/node/DF69BF6841.md): definition DF69BF6841
- [Bochner's formula](https://tamnd.github.io/math2/node/DF7KYMAGTC.md): statement DF7KYMAGTC
- [Morin singularities](https://tamnd.github.io/math2/node/DF8CJ1A9MF.md): concept DF8CJ1A9MF
- [Mixing](https://tamnd.github.io/math2/node/DFCJCRTTBR.md): definition DFCJCRTTBR
- [Coherent ring](https://tamnd.github.io/math2/node/DFDW7ARK6S.md): definition DFDW7ARK6S
- [Minimal surface](https://tamnd.github.io/math2/node/DFM7NJTX3J.md): definition DFM7NJTX3J
- [Mal'cev's characterization of quasivarieties](https://tamnd.github.io/math2/node/DFNM3BGMAY.md): statement DFNM3BGMAY
- [Critical point theory](https://tamnd.github.io/math2/node/DFXSJH9TY4.md): definition DFXSJH9TY4
- [Mal'cev's theorem on congruence permutability](https://tamnd.github.io/math2/node/DFZ7ZKRZCQ.md): statement DFZ7ZKRZCQ
- [Integral of a non-negative function](https://tamnd.github.io/math2/node/DG76G5Z6XW.md): concept DG76G5Z6XW
- [Oscillation theory](https://tamnd.github.io/math2/node/DG7DZ37217.md): definition DG7DZ37217
- [Complete measure](https://tamnd.github.io/math2/node/DG8FBQ8FPY.md): definition DG8FBQ8FPY
- [Gibbs measure](https://tamnd.github.io/math2/node/DG9F3W6MZM.md): concept DG9F3W6MZM
- [Conformal radius](https://tamnd.github.io/math2/node/DGJEC8MTMQ.md): definition DGJEC8MTMQ
- [Porous medium equation](https://tamnd.github.io/math2/node/DGVNNYKBHC.md): concept DGVNNYKBHC
- [Disk algebra](https://tamnd.github.io/math2/node/DGWYF0B6NT.md): concept DGWYF0B6NT
- [Lawvere theory](https://tamnd.github.io/math2/node/DGZ8RS5KMZ.md): definition DGZ8RS5KMZ
- [Term evaluation](https://tamnd.github.io/math2/node/DH30WTA26T.md): concept DH30WTA26T
- [Plancherel theorem for groups](https://tamnd.github.io/math2/node/DH3PYQJXH0.md): statement DH3PYQJXH0
- [Moving frame](https://tamnd.github.io/math2/node/DH4P0W4V3J.md): definition DH4P0W4V3J
- [Pentagon lattice N5](https://tamnd.github.io/math2/node/DHC8TZXGYS.md): example DHC8TZXGYS
- [Curry–Schoenberg theorem](https://tamnd.github.io/math2/node/DHC96EYCN6.md): statement DHC96EYCN6
- [Cartier–Milnor–Moore theorem](https://tamnd.github.io/math2/node/DHCHJNHD02.md): statement DHCHJNHD02
- [Existence of Leray weak solutions](https://tamnd.github.io/math2/node/DHHR5PZZ7E.md): statement DHHR5PZZ7E
- [Pólya urn model](https://tamnd.github.io/math2/node/DHPXDFEEMF.md): example DHPXDFEEMF
- [Five-point stencil](https://tamnd.github.io/math2/node/DHRDTHFBFA.md): definition DHRDTHFBFA
- [Subnormal number](https://tamnd.github.io/math2/node/DHXXSN12JC.md): concept DHXXSN12JC
- [Group algebra of a locally compact group](https://tamnd.github.io/math2/node/DHYS1W3Y4N.md): definition DHYS1W3Y4N
- [Yau–Tian–Donaldson conjecture](https://tamnd.github.io/math2/node/DHYSPSYSCB.md): statement DHYSPSYSCB
- [Measurable space](https://tamnd.github.io/math2/node/DJ6F3DZX42.md): definition DJ6F3DZX42
- [Linear-quadratic-Gaussian control](https://tamnd.github.io/math2/node/DJB8MAV5BH.md): problem DJB8MAV5BH
- [Polydisc](https://tamnd.github.io/math2/node/DJBWP7D3M8.md): concept DJBWP7D3M8
- [Integral operator](https://tamnd.github.io/math2/node/DJBY0QQA80.md): definition DJBY0QQA80
- [Quasi-primal algebra](https://tamnd.github.io/math2/node/DJEK11CXXJ.md): definition DJEK11CXXJ
- [McKenzie–Hobby tame congruence theory](https://tamnd.github.io/math2/node/DJFQWTQG9V.md): concept DJFQWTQG9V
- [Spearman's rank correlation coefficient](https://tamnd.github.io/math2/node/DJFYTQJ3SH.md): concept DJFYTQJ3SH
- [Unitary operator](https://tamnd.github.io/math2/node/DJHVKR63H5.md): definition DJHVKR63H5
- [Singular simplicial set](https://tamnd.github.io/math2/node/DJM5H15WV5.md): definition DJM5H15WV5
- [Oracle machine](https://tamnd.github.io/math2/node/DJMXZX0G9G.md): definition DJMXZX0G9G
- [Affine arithmetic](https://tamnd.github.io/math2/node/DJXX3X6TEQ.md): definition DJXX3X6TEQ
- [Discrete differential geometry](https://tamnd.github.io/math2/node/DK0A24WJKF.md): concept DK0A24WJKF
- [Doob's h-transform](https://tamnd.github.io/math2/node/DK3492C42N.md): definition DK3492C42N
- [Absolute neighborhood retract](https://tamnd.github.io/math2/node/DK3MT3HYX5.md): concept DK3MT3HYX5
- [Nilpotent operator](https://tamnd.github.io/math2/node/DK3Q4RPRZJ.md): concept DK3Q4RPRZJ
- [Nilpotent Lie group](https://tamnd.github.io/math2/node/DK87YN458D.md): concept DK87YN458D
- [Sentence](https://tamnd.github.io/math2/node/DKFGQTRYZE.md): concept DKFGQTRYZE
- [Dedekind–MacNeille completion](https://tamnd.github.io/math2/node/DKH4933HMQ.md): concept DKH4933HMQ
- [Cahn–Hilliard equation](https://tamnd.github.io/math2/node/DKN15NVKSB.md): concept DKN15NVKSB
- [Shooting method](https://tamnd.github.io/math2/node/DKR8R8Y1RJ.md): definition DKR8R8Y1RJ
- [Generalized polygon](https://tamnd.github.io/math2/node/DKT8Z2M9CB.md): definition DKT8Z2M9CB
- [p-Laplacian](https://tamnd.github.io/math2/node/DKV5J6N6T6.md): definition DKV5J6N6T6
- [Meijer G-function](https://tamnd.github.io/math2/node/DKVBV788GT.md): definition DKVBV788GT
- [Clifford analysis](https://tamnd.github.io/math2/node/DKVSAZQWF7.md): concept DKVSAZQWF7
- [Open subgroups are closed](https://tamnd.github.io/math2/node/DKX3EAVQX6.md): statement DKX3EAVQX6
- [Hadamard matrix](https://tamnd.github.io/math2/node/DM59GHSMAJ.md): definition DM59GHSMAJ
- [Breakdown point](https://tamnd.github.io/math2/node/DMFB5JQ2T7.md): definition DMFB5JQ2T7
- [Delta method](https://tamnd.github.io/math2/node/DMMXBTN6M2.md): statement DMMXBTN6M2
- [Chebyshev constant](https://tamnd.github.io/math2/node/DMNN7E2BYA.md): concept DMNN7E2BYA
- [Random dynamical system](https://tamnd.github.io/math2/node/DMTFJY6CS0.md): concept DMTFJY6CS0
- [Pointed cone](https://tamnd.github.io/math2/node/DMWEBMB1RS.md): definition DMWEBMB1RS
- [Linear algebra method](https://tamnd.github.io/math2/node/DMYGZ3YZ6V.md): concept DMYGZ3YZ6V
- [The local ring of holomorphic germs is Noetherian and factorial](https://tamnd.github.io/math2/node/DMYXA8HZ2P.md): statement DMYXA8HZ2P
- [Pontryagin class](https://tamnd.github.io/math2/node/DN11FKWN63.md): concept DN11FKWN63
- [Transformation semigroup](https://tamnd.github.io/math2/node/DN2MNQGN4S.md): definition DN2MNQGN4S
- [Computable set](https://tamnd.github.io/math2/node/DN5G4NAF95.md): definition DN5G4NAF95
- [Median algebra](https://tamnd.github.io/math2/node/DN7VSSDYPQ.md): concept DN7VSSDYPQ
- [Nehari's theorem on univalence](https://tamnd.github.io/math2/node/DN98H15VZF.md): statement DN98H15VZF
- [Dedekind–MacNeille completion](https://tamnd.github.io/math2/node/DNBS22EV21.md): definition DNBS22EV21
- [Volterra integral equation](https://tamnd.github.io/math2/node/DND8CBEVXT.md): concept DND8CBEVXT
- [Point process](https://tamnd.github.io/math2/node/DNDKBVHS3H.md): concept DNDKBVHS3H
- [Localization of a category](https://tamnd.github.io/math2/node/DNFR3YP0XF.md): concept DNFR3YP0XF
- [Generalized Riemann hypothesis](https://tamnd.github.io/math2/node/DNKA4WBKDS.md): conjecture DNKA4WBKDS
- [M-matrix](https://tamnd.github.io/math2/node/DNKXHMGQY1.md): concept DNKXHMGQY1
- [Monadic functor](https://tamnd.github.io/math2/node/DNTDCKZBKK.md): concept DNTDCKZBKK
- [Logarithmic kernel](https://tamnd.github.io/math2/node/DNVGM3YT8Q.md): definition DNVGM3YT8Q
- [Lucas sequence](https://tamnd.github.io/math2/node/DP39F5KQ2W.md): example DP39F5KQ2W
- [Embedding dimension](https://tamnd.github.io/math2/node/DP3KSCRKFP.md): concept DP3KSCRKFP
- [Rotation number](https://tamnd.github.io/math2/node/DP4WWQBJJF.md): concept DP4WWQBJJF
- [Commutative algebra over a ring](https://tamnd.github.io/math2/node/DPBJ04JMH8.md): concept DPBJ04JMH8
- [Facet](https://tamnd.github.io/math2/node/DPDXN2FJC4.md): concept DPDXN2FJC4
- [Dual graph](https://tamnd.github.io/math2/node/DPE8EDD9W7.md): definition DPE8EDD9W7
- [Absolute Galois group](https://tamnd.github.io/math2/node/DPFMXS71D1.md): definition DPFMXS71D1
- [Stiffness matrix](https://tamnd.github.io/math2/node/DPR9DJRPAF.md): definition DPR9DJRPAF
- [Relaxation of a variational problem](https://tamnd.github.io/math2/node/DPS8QZBVB0.md): concept DPS8QZBVB0
- [Fourier transform](https://tamnd.github.io/math2/node/DPVM34ADZV.md): concept DPVM34ADZV
- [Wald test](https://tamnd.github.io/math2/node/DPW8KNHE6P.md): concept DPW8KNHE6P
- [Stable homotopy category](https://tamnd.github.io/math2/node/DPZ76D18DD.md): definition DPZ76D18DD
- [Local solvability](https://tamnd.github.io/math2/node/DQ0TSVK7ZT.md): definition DQ0TSVK7ZT
- [Majority term](https://tamnd.github.io/math2/node/DQ14ERMRTZ.md): definition DQ14ERMRTZ
- [Clopen set](https://tamnd.github.io/math2/node/DQ4E55T3BW.md): concept DQ4E55T3BW
- [Function field](https://tamnd.github.io/math2/node/DQ564JNFYE.md): definition DQ564JNFYE
- [Veronese embedding](https://tamnd.github.io/math2/node/DQA60Z3FD7.md): example DQA60Z3FD7
- [Tangent bundle](https://tamnd.github.io/math2/node/DQB1BVQJWD.md): definition DQB1BVQJWD
- [Shape operator](https://tamnd.github.io/math2/node/DQCD11X1VM.md): concept DQCD11X1VM
- [Subnormal subgroup](https://tamnd.github.io/math2/node/DQCW86BXEP.md): definition DQCW86BXEP
- [Euler's sine product](https://tamnd.github.io/math2/node/DQEHD56649.md): statement DQEHD56649
- [Hausdorff distance](https://tamnd.github.io/math2/node/DQEW4MSD5H.md): definition DQEW4MSD5H
- [Contiguity of measures](https://tamnd.github.io/math2/node/DQGJMP9WHC.md): concept DQGJMP9WHC
- [Monodromy theorem for singularities](https://tamnd.github.io/math2/node/DQP63REKEK.md): statement DQP63REKEK
- [Heaviside operational calculus](https://tamnd.github.io/math2/node/DQRXS8PJ15.md): definition DQRXS8PJ15
- [Trace form](https://tamnd.github.io/math2/node/DQWX2Y8EA5.md): definition DQWX2Y8EA5
- [Supporting hyperplane theorem](https://tamnd.github.io/math2/node/DR9BXFWMQB.md): statement DR9BXFWMQB
- [Regular surface](https://tamnd.github.io/math2/node/DRFKZKE229.md): definition DRFKZKE229
- [Extremal graph theory](https://tamnd.github.io/math2/node/DRGS5TT784.md): definition DRGS5TT784
- [Integral closure](https://tamnd.github.io/math2/node/DRJPG78875.md): definition DRJPG78875
- [Donaldson theory](https://tamnd.github.io/math2/node/DRNH767W84.md): definition DRNH767W84
- [Sokhotski–Plemelj formulae](https://tamnd.github.io/math2/node/DRT1E6WY6K.md): statement DRT1E6WY6K
- [Hopf's theorem on cohomology of compact Lie groups](https://tamnd.github.io/math2/node/DRY1HXP1W9.md): statement DRY1HXP1W9
- [Complexification of a Lie group](https://tamnd.github.io/math2/node/DRYYK2WX0M.md): concept DRYYK2WX0M
- [Hilbert–Smith conjecture](https://tamnd.github.io/math2/node/DS2AS9TSZP.md): conjecture DS2AS9TSZP
- [Diffeology](https://tamnd.github.io/math2/node/DS3W88FESM.md): definition DS3W88FESM
- [Kan extension](https://tamnd.github.io/math2/node/DS6TC7FSDQ.md): definition DS6TC7FSDQ
- [Near-field](https://tamnd.github.io/math2/node/DSE72WQ5A2.md): concept DSE72WQ5A2
- [Kaplansky's conjectures](https://tamnd.github.io/math2/node/DSF8C6QPAG.md): conjecture DSF8C6QPAG
- [Coherent analytic sheaf](https://tamnd.github.io/math2/node/DSJET02WPR.md): concept DSJET02WPR
- [Lazard's theorem on p-adic analytic groups](https://tamnd.github.io/math2/node/DSK9GN63GQ.md): statement DSK9GN63GQ
- [Margulis superrigidity theorem](https://tamnd.github.io/math2/node/DSKV65NHZZ.md): statement DSKV65NHZZ
- [Automorphism](https://tamnd.github.io/math2/node/DSQ17TNW1C.md): concept DSQ17TNW1C
- [Admissible decision rule](https://tamnd.github.io/math2/node/DSTF6X8DRJ.md): concept DSTF6X8DRJ
- [Stationary process](https://tamnd.github.io/math2/node/DT0QXJE705.md): concept DT0QXJE705
- [Direct collocation](https://tamnd.github.io/math2/node/DT45YNKWSQ.md): definition DT45YNKWSQ
- [Essentially self-adjoint operator](https://tamnd.github.io/math2/node/DT49GKR9AS.md): concept DT49GKR9AS
- [Compactness via ultrafilters](https://tamnd.github.io/math2/node/DT5PH9QVQ7.md): statement DT5PH9QVQ7
- [Pfaffian system](https://tamnd.github.io/math2/node/DT60HNP41N.md): definition DT60HNP41N
- [Spherical harmonics](https://tamnd.github.io/math2/node/DT6WEXJANH.md): definition DT6WEXJANH
- [Topos](https://tamnd.github.io/math2/node/DTNDR45FTH.md): definition DTNDR45FTH
- [Moser's worm problem](https://tamnd.github.io/math2/node/DTVFW3QMNW.md): problem DTVFW3QMNW
- [Left ideal](https://tamnd.github.io/math2/node/DTWDS6KAKN.md): definition DTWDS6KAKN
- [O-minimal theory](https://tamnd.github.io/math2/node/DTWHAW1N0V.md): concept DTWHAW1N0V
- [Initial value problem for a recurrence](https://tamnd.github.io/math2/node/DV00798WNX.md): definition DV00798WNX
- [Presheaf](https://tamnd.github.io/math2/node/DV3PWY8NVW.md): definition DV3PWY8NVW
- [Countably compact space](https://tamnd.github.io/math2/node/DV8GH6HJJ8.md): definition DV8GH6HJJ8
- [Quasilinear PDE](https://tamnd.github.io/math2/node/DV92N2YWZJ.md): definition DV92N2YWZJ
- [Atomistic lattice](https://tamnd.github.io/math2/node/DVA5DBQYBA.md): definition DVA5DBQYBA
- [Optimal control of partial differential equations](https://tamnd.github.io/math2/node/DVA7AZ0HDJ.md): definition DVA7AZ0HDJ
- [Dirichlet-to-Neumann operator](https://tamnd.github.io/math2/node/DVAWCZWT3B.md): definition DVAWCZWT3B
- [Raviart–Thomas elements](https://tamnd.github.io/math2/node/DVB79BK06Y.md): object DVB79BK06Y
- [Curve-shortening flow](https://tamnd.github.io/math2/node/DVGFRRYK10.md): concept DVGFRRYK10
- [Stone's theorem on one-parameter unitary groups](https://tamnd.github.io/math2/node/DVGQB92HPJ.md): statement DVGQB92HPJ
- [Morrey quasiconvexity](https://tamnd.github.io/math2/node/DVJ17KVWXP.md): concept DVJ17KVWXP
- [Bessel potential space](https://tamnd.github.io/math2/node/DVJM7NBSF2.md): concept DVJM7NBSF2
- [Coequalizer](https://tamnd.github.io/math2/node/DVSTWGVCSZ.md): concept DVSTWGVCSZ
- [Plateau's problem](https://tamnd.github.io/math2/node/DVYA6GD8RN.md): problem DVYA6GD8RN
- [Dirichlet form](https://tamnd.github.io/math2/node/DW5G0SQ2G5.md): definition DW5G0SQ2G5
- [Loeb measure](https://tamnd.github.io/math2/node/DW5SBZXK6R.md): concept DW5SBZXK6R
- [Validated numerics](https://tamnd.github.io/math2/node/DW7FG506QJ.md): concept DW7FG506QJ
- [Riesz–Markov–Kakutani representation theorem](https://tamnd.github.io/math2/node/DWFPDF8XTB.md): statement DWFPDF8XTB
- [Lebesgue integral](https://tamnd.github.io/math2/node/DWJF7ABPR1.md): concept DWJF7ABPR1
- [Poisson's equation](https://tamnd.github.io/math2/node/DWK5N5SKTK.md): concept DWK5N5SKTK
- [Universal covering group](https://tamnd.github.io/math2/node/DWMQ3V205V.md): definition DWMQ3V205V
- [Harmonic space](https://tamnd.github.io/math2/node/DWN0XEQHHK.md): concept DWN0XEQHHK
- [Perfectly normal space](https://tamnd.github.io/math2/node/DWV8BQP6GA.md): definition DWV8BQP6GA
- [Many-valued logic](https://tamnd.github.io/math2/node/DWYCKAKXZD.md): concept DWYCKAKXZD
- [Affine connection](https://tamnd.github.io/math2/node/DWYPS58A7Q.md): definition DWYPS58A7Q
- [C-infinity ring](https://tamnd.github.io/math2/node/DWZA03W93P.md): definition DWZA03W93P
- [Laplace transform of the error function](https://tamnd.github.io/math2/node/DWZD2EKC1Q.md): example DWZD2EKC1Q
- [Minimum description length](https://tamnd.github.io/math2/node/DWZYGTXKNS.md): definition DWZYGTXKNS
- [Surface of general type](https://tamnd.github.io/math2/node/DX2JE47SXH.md): definition DX2JE47SXH
- [Degree of a continuous map](https://tamnd.github.io/math2/node/DX3DJY91QH.md): definition DX3DJY91QH
- [The Klein four-group is abelian but not cyclic](https://tamnd.github.io/math2/node/DX6X1D21FE.md): example DX6X1D21FE
- [Darcy's law](https://tamnd.github.io/math2/node/DX8RV2XG6H.md): statement DX8RV2XG6H
- [CAT(0) group](https://tamnd.github.io/math2/node/DXCTK6M7ZV.md): definition DXCTK6M7ZV
- [Section and retraction](https://tamnd.github.io/math2/node/DXJH3DGXBC.md): concept DXJH3DGXBC
- [Expander graph](https://tamnd.github.io/math2/node/DXJXVFV79H.md): concept DXJXVFV79H
- [Pixley term](https://tamnd.github.io/math2/node/DXWBWMM5H7.md): definition DXWBWMM5H7
- [von Neumann regular ring](https://tamnd.github.io/math2/node/DXX7GNY3HE.md): concept DXX7GNY3HE
- [Hypergeometric function of a matrix argument](https://tamnd.github.io/math2/node/DXZP6V23Z7.md): concept DXZP6V23Z7
- [Kobayashi metric](https://tamnd.github.io/math2/node/DY8QQ5GTTX.md): concept DY8QQ5GTTX
- [James space](https://tamnd.github.io/math2/node/DYAWGXE2GV.md): example DYAWGXE2GV
- [Hyperbolic knot](https://tamnd.github.io/math2/node/DYFQ2EQ7J6.md): definition DYFQ2EQ7J6
- [Kendall rank correlation coefficient](https://tamnd.github.io/math2/node/DYFY16SFHZ.md): concept DYFY16SFHZ
- [Priestley space](https://tamnd.github.io/math2/node/DYM1EFAC59.md): definition DYM1EFAC59
- [Borel measure](https://tamnd.github.io/math2/node/DYR1QFNGNJ.md): concept DYR1QFNGNJ
- [Levi–Malcev decomposition](https://tamnd.github.io/math2/node/DYXEBVDC6W.md): statement DYXEBVDC6W
- [Minimal variety](https://tamnd.github.io/math2/node/DZ0B0JD11D.md): definition DZ0B0JD11D
- [Locally finite family](https://tamnd.github.io/math2/node/DZ297MMGV8.md): concept DZ297MMGV8
- [Integral equation of the third kind](https://tamnd.github.io/math2/node/DZ47JVKEF5.md): concept DZ47JVKEF5
- [Inverse limit](https://tamnd.github.io/math2/node/DZB378CNCK.md): concept DZB378CNCK
- [Newton series](https://tamnd.github.io/math2/node/DZD0Q4H4DS.md): definition DZD0Q4H4DS
- [Schubert variety](https://tamnd.github.io/math2/node/DZEQY398HQ.md): definition DZEQY398HQ
- [Unitary representation](https://tamnd.github.io/math2/node/DZF985Y137.md): definition DZF985Y137
- [Birkhoff–Frink theorem](https://tamnd.github.io/math2/node/DZGNHKE3X1.md): statement DZGNHKE3X1
- [Merton's portfolio problem](https://tamnd.github.io/math2/node/DZKBF576W5.md): problem DZKBF576W5
- [Hermite interpolation](https://tamnd.github.io/math2/node/DZQ4EW6907.md): definition DZQ4EW6907
- [Elementary abelian group](https://tamnd.github.io/math2/node/DZSZ8ZS554.md): concept DZSZ8ZS554
- [Picard group of a ring](https://tamnd.github.io/math2/node/DZW58R6T2Q.md): concept DZW58R6T2Q
- [Coordinate ring](https://tamnd.github.io/math2/node/DZWS3QN3H7.md): definition DZWS3QN3H7
- [Fenichel's theorem](https://tamnd.github.io/math2/node/DZXTK64M6X.md): statement DZXTK64M6X
- [Hall subgroup](https://tamnd.github.io/math2/node/E09VVB7BJQ.md): definition E09VVB7BJQ
- [Dowker space](https://tamnd.github.io/math2/node/E0A14ZEW72.md): definition E0A14ZEW72
- [Chern–Simons theory](https://tamnd.github.io/math2/node/E0NXWECXTB.md): definition E0NXWECXTB
- [Ternary operation](https://tamnd.github.io/math2/node/E0QDPSKX3R.md): definition E0QDPSKX3R
- [Del Pezzo surface](https://tamnd.github.io/math2/node/E0QKXVH9D6.md): definition E0QKXVH9D6
- [Long exact sequence of homotopy groups](https://tamnd.github.io/math2/node/E0RV5D1GQ0.md): statement E0RV5D1GQ0
- [Arithmetic–geometric mean and elliptic integrals](https://tamnd.github.io/math2/node/E0V3N93DX8.md): statement E0V3N93DX8
- [Discriminant of a number field](https://tamnd.github.io/math2/node/E0VZTNPJXA.md): concept E0VZTNPJXA
- [Canonical correlation](https://tamnd.github.io/math2/node/E0WKQ3XBAC.md): concept E0WKQ3XBAC
- [Monstrous moonshine](https://tamnd.github.io/math2/node/E0Y7Y75T6C.md): statement E0Y7Y75T6C
- [Euler–Lotka equation](https://tamnd.github.io/math2/node/E0YVMBPK7A.md): example E0YVMBPK7A
- [Complexification of a Lie group](https://tamnd.github.io/math2/node/E0ZMRTA42A.md): definition E0ZMRTA42A
- [Residue calculus for Laplace inversion](https://tamnd.github.io/math2/node/E14W2V09HM.md): concept E14W2V09HM
- [Admissible representation](https://tamnd.github.io/math2/node/E15H5E8FP4.md): definition E15H5E8FP4
- [Heat potential](https://tamnd.github.io/math2/node/E16HJZ3W4R.md): definition E16HJZ3W4R
- [Invertible ideal](https://tamnd.github.io/math2/node/E190TAV2S6.md): concept E190TAV2S6
- [Fold catastrophe](https://tamnd.github.io/math2/node/E192Q8BQWJ.md): example E192Q8BQWJ
- [Recurrent point](https://tamnd.github.io/math2/node/E1B3J3XB1T.md): definition E1B3J3XB1T
- [Graded manifold](https://tamnd.github.io/math2/node/E1CA8QFEHB.md): concept E1CA8QFEHB
- [ℓ-adic cohomology](https://tamnd.github.io/math2/node/E1DVHN06CR.md): concept E1DVHN06CR
- [s-cobordism theorem](https://tamnd.github.io/math2/node/E1E5ABP89W.md): statement E1E5ABP89W
- [Kaplansky's zero divisor conjecture](https://tamnd.github.io/math2/node/E1NE8V7EX8.md): conjecture E1NE8V7EX8
- [Involute](https://tamnd.github.io/math2/node/E1PC4TZK5C.md): concept E1PC4TZK5C
- [Kloosterman sum](https://tamnd.github.io/math2/node/E1SQWQ2PAV.md): concept E1SQWQ2PAV
- [Semianalytic set](https://tamnd.github.io/math2/node/E1V22QQHYJ.md): concept E1V22QQHYJ
- [Subdivision surface](https://tamnd.github.io/math2/node/E1V8CPXK24.md): definition E1V8CPXK24
- [Closing lemma](https://tamnd.github.io/math2/node/E1Z444F6TV.md): statement E1Z444F6TV
- [Toric variety](https://tamnd.github.io/math2/node/E22Y744R1C.md): definition E22Y744R1C
- [Proximal operator](https://tamnd.github.io/math2/node/E239APE78P.md): concept E239APE78P
- [Weak equivalence](https://tamnd.github.io/math2/node/E2870TEDG7.md): concept E2870TEDG7
- [Self-avoiding walk](https://tamnd.github.io/math2/node/E2BVEB570S.md): concept E2BVEB570S
- [Craig interpolation theorem](https://tamnd.github.io/math2/node/E2CBFNAVZY.md): statement E2CBFNAVZY
- [Regular function](https://tamnd.github.io/math2/node/E2EM9ZYCZS.md): definition E2EM9ZYCZS
- [Line with two origins](https://tamnd.github.io/math2/node/E2K0MM2MQQ.md): counterexample E2K0MM2MQQ
- [Indecomposable module](https://tamnd.github.io/math2/node/E2P29GCD9T.md): concept E2P29GCD9T
- [Rosenbrock methods](https://tamnd.github.io/math2/node/E2ZFQ76N7F.md): statement E2ZFQ76N7F
- [Singular integral equation](https://tamnd.github.io/math2/node/E30EA6ETZ4.md): concept E30EA6ETZ4
- [Residuated lattice](https://tamnd.github.io/math2/node/E337A6WVH3.md): concept E337A6WVH3
- [Special adjoint functor theorem](https://tamnd.github.io/math2/node/E33H1J8S53.md): statement E33H1J8S53
- [Higher-order differences](https://tamnd.github.io/math2/node/E38801KVJ1.md): definition E38801KVJ1
- [Berezin integral](https://tamnd.github.io/math2/node/E38H8QVJK9.md): definition E38H8QVJK9
- [L-estimator](https://tamnd.github.io/math2/node/E3C4F0RKJY.md): concept E3C4F0RKJY
- [Completeness of the real numbers](https://tamnd.github.io/math2/node/E3CPZNJZNG.md): statement E3CPZNJZNG
- [Hyperbolic group](https://tamnd.github.io/math2/node/E3ECET66DQ.md): definition E3ECET66DQ
- [Lie pseudogroup](https://tamnd.github.io/math2/node/E3EYXM7GJN.md): concept E3EYXM7GJN
- [Fréchet mean](https://tamnd.github.io/math2/node/E3GQVC04BC.md): concept E3GQVC04BC
- [Wigner's classification](https://tamnd.github.io/math2/node/E3RQ0J3ACZ.md): statement E3RQ0J3ACZ
- [Ramanujan summation](https://tamnd.github.io/math2/node/E3SRYCE5VN.md): concept E3SRYCE5VN
- [Noncentral chi-squared distribution](https://tamnd.github.io/math2/node/E3TEYV0D89.md): definition E3TEYV0D89
- [Furstenberg's multiple recurrence theorem](https://tamnd.github.io/math2/node/E459H2B2EM.md): statement E459H2B2EM
- [Transcendence basis](https://tamnd.github.io/math2/node/E45PA267S4.md): concept E45PA267S4
- [Blowing up](https://tamnd.github.io/math2/node/E4F316AR19.md): definition E4F316AR19
- [Muller's method](https://tamnd.github.io/math2/node/E4FMVDQ2GR.md): statement E4FMVDQ2GR
- [Kirby–Siebenmann class](https://tamnd.github.io/math2/node/E4NJER1RFQ.md): concept E4NJER1RFQ
- [Differential equations in Banach spaces](https://tamnd.github.io/math2/node/E4P30YHWEQ.md): definition E4P30YHWEQ
- [Order topology](https://tamnd.github.io/math2/node/E4TEZEDHY8.md): definition E4TEZEDHY8
- [Characterizations of semisimple modules](https://tamnd.github.io/math2/node/E5D4BD4193.md): statement E5D4BD4193
- [Maximal order](https://tamnd.github.io/math2/node/E5DH1CSW1N.md): concept E5DH1CSW1N
- [Schouten–Nijenhuis bracket](https://tamnd.github.io/math2/node/E5G2SSTBP3.md): definition E5G2SSTBP3
- [Hájek–Le Cam convolution theorem](https://tamnd.github.io/math2/node/E5H2GFPK42.md): statement E5H2GFPK42
- [Wente torus](https://tamnd.github.io/math2/node/E5NNY7XM44.md): counterexample E5NNY7XM44
- [SU(2)](https://tamnd.github.io/math2/node/E5P8AG6CK7.md): example E5P8AG6CK7
- [Planar ternary ring](https://tamnd.github.io/math2/node/E6F61940F9.md): definition E6F61940F9
- [Menger's theorem](https://tamnd.github.io/math2/node/E6GYGZDBS7.md): statement E6GYGZDBS7
- [Lawvere–Tierney topology](https://tamnd.github.io/math2/node/E6H1CTT439.md): concept E6H1CTT439
- [Energy of a map between Riemannian manifolds](https://tamnd.github.io/math2/node/E6HDE1DFXD.md): concept E6HDE1DFXD
- [Realizability](https://tamnd.github.io/math2/node/E6MB6B3YQ0.md): definition E6MB6B3YQ0
- [Infinite product measure](https://tamnd.github.io/math2/node/E6MGJDRXJ0.md): definition E6MGJDRXJ0
- [Green's function of a Markov process](https://tamnd.github.io/math2/node/E7063PCV9P.md): definition E7063PCV9P
- [Kernel pair](https://tamnd.github.io/math2/node/E70DQGM8X5.md): concept E70DQGM8X5
- [Bessel function of the second kind](https://tamnd.github.io/math2/node/E76951YS23.md): definition E76951YS23
- [Total variation of a measure](https://tamnd.github.io/math2/node/E76H86NN6Y.md): definition E76H86NN6Y
- [Littlewood's three principles](https://tamnd.github.io/math2/node/E77T9G4DV7.md): definition E77T9G4DV7
- [Functional square root](https://tamnd.github.io/math2/node/E7890PHMGP.md): concept E7890PHMGP
- [Zariski tangent space](https://tamnd.github.io/math2/node/E7A5DHFT01.md): concept E7A5DHFT01
- [Glivenko–Cantelli theorem](https://tamnd.github.io/math2/node/E7B78X780X.md): statement E7B78X780X
- [Complexification of a real analytic manifold](https://tamnd.github.io/math2/node/E7G1S7Y1MT.md): statement E7G1S7Y1MT
- [Roth's theorem on arithmetic progressions](https://tamnd.github.io/math2/node/E7H8NWXESF.md): statement E7H8NWXESF
- [Order of a PDE](https://tamnd.github.io/math2/node/E7JDFDPM1X.md): concept E7JDFDPM1X
- [Projective tensor product](https://tamnd.github.io/math2/node/E7QDFXK2E1.md): definition E7QDFXK2E1
- [Faber–Krahn inequality](https://tamnd.github.io/math2/node/E7TY2TPXX7.md): statement E7TY2TPXX7
- [Ramification](https://tamnd.github.io/math2/node/E7WHZXQ0WF.md): definition E7WHZXQ0WF
- [Characterizations of discrete valuation rings](https://tamnd.github.io/math2/node/E7XJ3X8RG4.md): statement E7XJ3X8RG4
- [Local time](https://tamnd.github.io/math2/node/E7ZC459PSA.md): concept E7ZC459PSA
- [Convex functions of subharmonic functions](https://tamnd.github.io/math2/node/E820CTMJVM.md): statement E820CTMJVM
- [Monomial order](https://tamnd.github.io/math2/node/E874NARWPV.md): concept E874NARWPV
- [Birth–death process](https://tamnd.github.io/math2/node/E8DAD7052E.md): concept E8DAD7052E
- [Immersed boundary method](https://tamnd.github.io/math2/node/E8FQ2R87VX.md): definition E8FQ2R87VX
- [Prime ideal in a noncommutative ring](https://tamnd.github.io/math2/node/E8K2MN5V1E.md): concept E8K2MN5V1E
- [Holomorphy on Banach spaces](https://tamnd.github.io/math2/node/E8N3EC0YH5.md): concept E8N3EC0YH5
- [Gauss–Kronrod quadrature](https://tamnd.github.io/math2/node/E8QJJVC1KV.md): statement E8QJJVC1KV
- [Heat kernel on a Riemannian manifold](https://tamnd.github.io/math2/node/E8TQZG66XM.md): definition E8TQZG66XM
- [Fourier monomial on the circle](https://tamnd.github.io/math2/node/E8VY2SE93Z.md): definition E8VY2SE93Z
- [Injectivity radius](https://tamnd.github.io/math2/node/E8X03PNHM0.md): definition E8X03PNHM0
- [Limit point compact space](https://tamnd.github.io/math2/node/E8XKSQY76M.md): concept E8XKSQY76M
- [Homotopy principle](https://tamnd.github.io/math2/node/E8YK1W16NV.md): definition E8YK1W16NV
- [Logarithmically convex function](https://tamnd.github.io/math2/node/E97P413DKM.md): definition E97P413DKM
- [Standard part function](https://tamnd.github.io/math2/node/E9C6XXX0EE.md): definition E9C6XXX0EE
- [Periodic solutions of integral equations](https://tamnd.github.io/math2/node/E9F1PPZYKN.md): definition E9F1PPZYKN
- [Lyapunov function for a difference equation](https://tamnd.github.io/math2/node/E9G05M7CE6.md): concept E9G05M7CE6
- [Convolution ring has no zero divisors](https://tamnd.github.io/math2/node/E9QNNPQD2P.md): statement E9QNNPQD2P
- [Butterfly theorem](https://tamnd.github.io/math2/node/E9TX1NDAHB.md): statement E9TX1NDAHB
- [Differentiability of the Mellin transform](https://tamnd.github.io/math2/node/E9Y670Y3QV.md): statement E9Y670Y3QV
- [Time series](https://tamnd.github.io/math2/node/EA3MF4VTEH.md): concept EA3MF4VTEH
- [Nontangential limit](https://tamnd.github.io/math2/node/EACRW19W5R.md): definition EACRW19W5R
- [Degree of a central simple algebra](https://tamnd.github.io/math2/node/EAEFNBF94A.md): definition EAEFNBF94A
- [Modality of a singularity](https://tamnd.github.io/math2/node/EANPWS9G8N.md): definition EANPWS9G8N
- [Generalized convolution of an integral transform](https://tamnd.github.io/math2/node/EAP9N3VEBV.md): concept EAP9N3VEBV
- [Profunctor](https://tamnd.github.io/math2/node/EAPHAQVVD5.md): concept EAPHAQVVD5
- [Lerch zeta function](https://tamnd.github.io/math2/node/EAQDSQD4DW.md): concept EAQDSQD4DW
- [H-space](https://tamnd.github.io/math2/node/EASHTX8EG5.md): concept EASHTX8EG5
- [Bessel's equation](https://tamnd.github.io/math2/node/EAV97QH9DF.md): example EAV97QH9DF
- [Homotopy category](https://tamnd.github.io/math2/node/EAX2SYK8C5.md): definition EAX2SYK8C5
- [Gateaux derivative](https://tamnd.github.io/math2/node/EAZF9NJW12.md): concept EAZF9NJW12
- [Fox H-function](https://tamnd.github.io/math2/node/EAZMKENA4N.md): concept EAZMKENA4N
- [Influence function](https://tamnd.github.io/math2/node/EB2QJQNE5Z.md): concept EB2QJQNE5Z
- [Basic hypergeometric series](https://tamnd.github.io/math2/node/EB4N9RQR51.md): concept EB4N9RQR51
- [Gordon–Luecke theorem](https://tamnd.github.io/math2/node/EBASBW5TYS.md): statement EBASBW5TYS
- [Wolff potential](https://tamnd.github.io/math2/node/EBD8TWFNS0.md): concept EBD8TWFNS0
- [Chow group](https://tamnd.github.io/math2/node/EBEF6PEK4N.md): concept EBEF6PEK4N
- [Plünnecke–Ruzsa inequality](https://tamnd.github.io/math2/node/EBEVWSRD2E.md): statement EBEVWSRD2E
- [Mutually singular measures](https://tamnd.github.io/math2/node/EBGECZNCV5.md): concept EBGECZNCV5
- [Wilkie's identity](https://tamnd.github.io/math2/node/EBHKKRTRFS.md): example EBHKKRTRFS
- [Successive minima](https://tamnd.github.io/math2/node/EBHS26MHZT.md): definition EBHS26MHZT
- [Fréchet manifold](https://tamnd.github.io/math2/node/EBMCFQT0JN.md): concept EBMCFQT0JN
- [Repair radii and villainy](https://tamnd.github.io/math2/node/EBNHD7FQ23.md): concept EBNHD7FQ23
- [Magnus expansion](https://tamnd.github.io/math2/node/EBV9M9C435.md): concept EBV9M9C435
- [Fermi–Pasta–Ulam–Tsingou problem](https://tamnd.github.io/math2/node/EBXRM4PFRC.md): problem EBXRM4PFRC
- [Extensive category](https://tamnd.github.io/math2/node/EC6C1MR0C1.md): definition EC6C1MR0C1
- [Affine plane](https://tamnd.github.io/math2/node/EC7XT2ZM5S.md): concept EC7XT2ZM5S
- [Quantale](https://tamnd.github.io/math2/node/ECB3W5EVK3.md): definition ECB3W5EVK3
- [Radon–Nikodym derivative](https://tamnd.github.io/math2/node/ECDJ8DD9RV.md): definition ECDJ8DD9RV
- [Concrete category](https://tamnd.github.io/math2/node/ECDKZ8G998.md): definition ECDKZ8G998
- [Ginzburg–Landau functional](https://tamnd.github.io/math2/node/ECHTH3P9RE.md): example ECHTH3P9RE
- [Hartogs's theorem on separate holomorphicity](https://tamnd.github.io/math2/node/ECJG25SAJR.md): statement ECJG25SAJR
- [Hilbert–Pólya conjecture](https://tamnd.github.io/math2/node/ECS1GDJ734.md): conjecture ECS1GDJ734
- [Luzin N property](https://tamnd.github.io/math2/node/ECSW5HZGE3.md): definition ECSW5HZGE3
- [Shooting method](https://tamnd.github.io/math2/node/ECT6RNF865.md): concept ECT6RNF865
- [Barrier](https://tamnd.github.io/math2/node/ED0SZS1VED.md): concept ED0SZS1VED
- [Simons cone](https://tamnd.github.io/math2/node/ED3DAZVQZ4.md): example ED3DAZVQZ4
- [Bohr almost periodic function](https://tamnd.github.io/math2/node/ED69H4C3PK.md): concept ED69H4C3PK
- [Heegaard splitting](https://tamnd.github.io/math2/node/EDE9919B5R.md): concept EDE9919B5R
- [Galois category](https://tamnd.github.io/math2/node/EDP5426NNF.md): definition EDP5426NNF
- [Stratonovich integral](https://tamnd.github.io/math2/node/EE1QZRCFPJ.md): concept EE1QZRCFPJ
- [Bifurcation in a family of maps](https://tamnd.github.io/math2/node/EE9X3NZW81.md): concept EE9X3NZW81
- [Prevariety](https://tamnd.github.io/math2/node/EEAJ7AXBCT.md): concept EEAJ7AXBCT
- [Wave front of a Legendrian map](https://tamnd.github.io/math2/node/EEE4MM9N1G.md): definition EEE4MM9N1G
- [Bedford–Taylor theory](https://tamnd.github.io/math2/node/EEHQSWFHSB.md): concept EEHQSWFHSB
- [Galois cohomology](https://tamnd.github.io/math2/node/EESNTXFWHZ.md): definition EESNTXFWHZ
- [Dirichlet boundary condition](https://tamnd.github.io/math2/node/EETVA6QYX4.md): concept EETVA6QYX4
- [Differential Galois theory](https://tamnd.github.io/math2/node/EEZS4H8QPH.md): concept EEZS4H8QPH
- [Costa's minimal surface](https://tamnd.github.io/math2/node/EF0QMQB2J1.md): example EF0QMQB2J1
- [Equivalence of varieties](https://tamnd.github.io/math2/node/EF4M77TDF4.md): definition EF4M77TDF4
- [Cohomological dimension of a group](https://tamnd.github.io/math2/node/EF4ZWACHD8.md): concept EF4ZWACHD8
- [Factorization of polynomials](https://tamnd.github.io/math2/node/EFDBS6ZME8.md): definition EFDBS6ZME8
- [Lebesgue spine](https://tamnd.github.io/math2/node/EFFC7688KR.md): counterexample EFFC7688KR
- [Foundations of mathematics](https://tamnd.github.io/math2/node/EFFRJFW67Y.md): concept EFFRJFW67Y
- [Cofibration](https://tamnd.github.io/math2/node/EFJ77PFF88.md): definition EFJ77PFF88
- [John ellipsoid](https://tamnd.github.io/math2/node/EFY92E7GC7.md): definition EFY92E7GC7
- [Differences of polynomials vanish beyond the degree](https://tamnd.github.io/math2/node/EFYGR426DF.md): statement EFYGR426DF
- [Boolean algebras canonically defined](https://tamnd.github.io/math2/node/EG1N82CV0R.md): definition EG1N82CV0R
- [WKB approximation](https://tamnd.github.io/math2/node/EG41JQS3PC.md): definition EG41JQS3PC
- [Integral current](https://tamnd.github.io/math2/node/EG4AJVEFT2.md): concept EG4AJVEFT2
- [Kruskal–Wallis test](https://tamnd.github.io/math2/node/EG4VCBXDKZ.md): concept EG4VCBXDKZ
- [Double layer potential](https://tamnd.github.io/math2/node/EG8Z3FWV9A.md): concept EG8Z3FWV9A
- [Quasiregular map](https://tamnd.github.io/math2/node/EGAJPZ71VW.md): concept EGAJPZ71VW
- [Maximum principle for potentials](https://tamnd.github.io/math2/node/EGBAZMS66W.md): statement EGBAZMS66W
- [Gaussian free field](https://tamnd.github.io/math2/node/EGCDMWYKQB.md): concept EGCDMWYKQB
- [Abelian scheme](https://tamnd.github.io/math2/node/EGKVP4KK6V.md): definition EGKVP4KK6V
- [Finite determinacy theorem](https://tamnd.github.io/math2/node/EGPP69XA4F.md): statement EGPP69XA4F
- [Lebedev quadrature](https://tamnd.github.io/math2/node/EGQ3CJDH0D.md): statement EGQ3CJDH0D
- [Quasi-invariant measure](https://tamnd.github.io/math2/node/EGQ4GHH127.md): definition EGQ4GHH127
- [Harmonic measure](https://tamnd.github.io/math2/node/EGRH1KN03C.md): definition EGRH1KN03C
- [Inner measure](https://tamnd.github.io/math2/node/EGWG9RKDBQ.md): definition EGWG9RKDBQ
- [Spectrum of an element](https://tamnd.github.io/math2/node/EH0CZ19RKP.md): concept EH0CZ19RKP
- [Markov brothers' inequality](https://tamnd.github.io/math2/node/EHA5MF9VKE.md): statement EHA5MF9VKE
- [Abelian algebra](https://tamnd.github.io/math2/node/EHASX3TBRQ.md): concept EHASX3TBRQ
- [Lipschitz class](https://tamnd.github.io/math2/node/EHBQ295J3Q.md): concept EHBQ295J3Q
- [Cohomology ring](https://tamnd.github.io/math2/node/EHDXZV05JR.md): definition EHDXZV05JR
- [Hermite normal form](https://tamnd.github.io/math2/node/EHE26PR17K.md): statement EHE26PR17K
- [Continuous dual space](https://tamnd.github.io/math2/node/EHEZM8TXHJ.md): definition EHEZM8TXHJ
- [Rational approximation](https://tamnd.github.io/math2/node/EHFQK1WXKS.md): concept EHFQK1WXKS
- [Laplace transform method for convolution equations](https://tamnd.github.io/math2/node/EHGECREYB4.md): concept EHGECREYB4
- [Haagerup property](https://tamnd.github.io/math2/node/EHGRPCTM3J.md): definition EHGRPCTM3J
- [Factorial experiment](https://tamnd.github.io/math2/node/EHJ3R589ZX.md): concept EHJ3R589ZX
- [Slowly varying function](https://tamnd.github.io/math2/node/EHJ7D5CRZS.md): concept EHJ7D5CRZS
- [Multiresolution analysis](https://tamnd.github.io/math2/node/EHW4EAWJBF.md): definition EHW4EAWJBF
- [Logarithmically convex domain](https://tamnd.github.io/math2/node/EHWH6EF4QY.md): concept EHWH6EF4QY
- [Covering relation](https://tamnd.github.io/math2/node/EHXHC4ZNPA.md): definition EHXHC4ZNPA
- [Conjugate prior](https://tamnd.github.io/math2/node/EHY450Q5CW.md): concept EHY450Q5CW
- [Growth of algebras](https://tamnd.github.io/math2/node/EHY4PYSMGZ.md): concept EHY4PYSMGZ
- [Morrey quasiconvexity](https://tamnd.github.io/math2/node/EJ0N7F9TYQ.md): definition EJ0N7F9TYQ
- [Behrens–Fisher problem](https://tamnd.github.io/math2/node/EJ3W3836VZ.md): problem EJ3W3836VZ
- [Perron's formula](https://tamnd.github.io/math2/node/EJ556BJNFM.md): statement EJ556BJNFM
- [Finitary operation](https://tamnd.github.io/math2/node/EJ8YJVM9BW.md): concept EJ8YJVM9BW
- [Stein manifold](https://tamnd.github.io/math2/node/EJD3C6QW9T.md): definition EJD3C6QW9T
- [Dirichlet kernel](https://tamnd.github.io/math2/node/EJE15QHXZN.md): definition EJE15QHXZN
- [Eckmann–Hilton argument](https://tamnd.github.io/math2/node/EJJYCPWHPA.md): statement EJJYCPWHPA
- [Stieltjes transform](https://tamnd.github.io/math2/node/EJNM0CV12B.md): definition EJNM0CV12B
- [FK-space](https://tamnd.github.io/math2/node/EJNPP8RZYF.md): definition EJNPP8RZYF
- [Homotopy group with coefficients](https://tamnd.github.io/math2/node/EJRP0VHGQE.md): definition EJRP0VHGQE
- [Indecomposable continuum](https://tamnd.github.io/math2/node/EK0FJBNE1Q.md): definition EK0FJBNE1Q
- [Jacobson density theorem](https://tamnd.github.io/math2/node/EK0T7FRKCY.md): statement EK0T7FRKCY
- [Normal scheme](https://tamnd.github.io/math2/node/EK261G1XP9.md): concept EK261G1XP9
- [Beurling–Deny formula](https://tamnd.github.io/math2/node/EK58S2872F.md): statement EK58S2872F
- [Initial and terminal objects](https://tamnd.github.io/math2/node/EK85VSSRMR.md): concept EK85VSSRMR
- [Polynomial interpolation](https://tamnd.github.io/math2/node/EKDSDN89FR.md): definition EKDSDN89FR
- [Radical axis](https://tamnd.github.io/math2/node/EKEPQC0T50.md): definition EKEPQC0T50
- [Postnikov tower](https://tamnd.github.io/math2/node/EKGEZGHYBW.md): concept EKGEZGHYBW
- [Closed monoidal category](https://tamnd.github.io/math2/node/EKJBDD07TY.md): concept EKJBDD07TY
- [Busemann G-space](https://tamnd.github.io/math2/node/EKXQFD7EMF.md): concept EKXQFD7EMF
- [Neyman construction](https://tamnd.github.io/math2/node/EKZB41Y82N.md): concept EKZB41Y82N
- [Bimodule](https://tamnd.github.io/math2/node/EM7FHAFD3N.md): concept EM7FHAFD3N
- [Berlekamp's algorithm](https://tamnd.github.io/math2/node/EM8MFGFAGN.md): concept EM8MFGFAGN
- [Schatten class](https://tamnd.github.io/math2/node/EM8Q0636YT.md): definition EM8Q0636YT
- [Hasse principle](https://tamnd.github.io/math2/node/EMAR4X3W8D.md): concept EMAR4X3W8D
- [Rank of an abelian group](https://tamnd.github.io/math2/node/EMCTDD3HZ1.md): concept EMCTDD3HZ1
- [Parabolic maximum principle](https://tamnd.github.io/math2/node/EMDVD4E83K.md): statement EMDVD4E83K
- [Cohen's independence of CH](https://tamnd.github.io/math2/node/EMGBZ46GCD.md): statement EMGBZ46GCD
- [Shallow water equations](https://tamnd.github.io/math2/node/EMKFC3R1WM.md): definition EMKFC3R1WM
- [Total curvature](https://tamnd.github.io/math2/node/EMVDSQJM8S.md): definition EMVDSQJM8S
- [Cubic reciprocity](https://tamnd.github.io/math2/node/EMVECB93EQ.md): statement EMVECB93EQ
- [Projective geometry](https://tamnd.github.io/math2/node/EMWW6PKFSG.md): definition EMWW6PKFSG
- [Hamilton–Jacobi theory of the calculus of variations](https://tamnd.github.io/math2/node/EMXK9T0X1Q.md): definition EMXK9T0X1Q
- [Incomplete LU factorization](https://tamnd.github.io/math2/node/EMYPFVX322.md): definition EMYPFVX322
- [Petersson inner product](https://tamnd.github.io/math2/node/EMZTGD63V3.md): concept EMZTGD63V3
- [Anosov flow](https://tamnd.github.io/math2/node/EN4A9MM7RP.md): definition EN4A9MM7RP
- [Universal property](https://tamnd.github.io/math2/node/EN4DD2FAJH.md): concept EN4DD2FAJH
- [Hamiltonian of an optimal control problem](https://tamnd.github.io/math2/node/ENA9XD5B77.md): definition ENA9XD5B77
- [Strictly singular operator](https://tamnd.github.io/math2/node/ENB0GT59ZC.md): definition ENB0GT59ZC
- [Bornological space](https://tamnd.github.io/math2/node/ENH9EP5ZH5.md): definition ENH9EP5ZH5
- [Arithmetic genus](https://tamnd.github.io/math2/node/ENMB2J4JJ6.md): definition ENMB2J4JJ6
- [Bochner integral](https://tamnd.github.io/math2/node/ENTNQSXA96.md): definition ENTNQSXA96
- [Siegel's theorem on meromorphic function fields](https://tamnd.github.io/math2/node/ENXCWPTG92.md): statement ENXCWPTG92
- [Ruelle zeta function](https://tamnd.github.io/math2/node/ENYXKW76AA.md): concept ENYXKW76AA
- [Discrete Painlevé equations](https://tamnd.github.io/math2/node/ENZ88T9WWQ.md): definition ENZ88T9WWQ
- [Bézier curve](https://tamnd.github.io/math2/node/ENZF7KF1TC.md): definition ENZF7KF1TC
- [Jacobi amplitude](https://tamnd.github.io/math2/node/EP681WKN00.md): definition EP681WKN00
- [Exponential family](https://tamnd.github.io/math2/node/EP9VPZ0S6G.md): concept EP9VPZ0S6G
- [Homotopy type theory](https://tamnd.github.io/math2/node/EPAAKD80H9.md): concept EPAAKD80H9
- [Simplicial set](https://tamnd.github.io/math2/node/EPAZPBPFF9.md): definition EPAZPBPFF9
- [Total variation of a measure](https://tamnd.github.io/math2/node/EPBND51CR5.md): concept EPBND51CR5
- [Unilateral Z-transform](https://tamnd.github.io/math2/node/EPG6JPVJE7.md): concept EPG6JPVJE7
- [Complete intersection ring](https://tamnd.github.io/math2/node/EPM2HKGMZS.md): concept EPM2HKGMZS
- [Symplectic capacity](https://tamnd.github.io/math2/node/EPN6C0X93V.md): definition EPN6C0X93V
- [Noncommutative Gröbner basis](https://tamnd.github.io/math2/node/EPPR3DAFP6.md): concept EPPR3DAFP6
- [Restriction of a measure](https://tamnd.github.io/math2/node/EPS9JT55MV.md): concept EPS9JT55MV
- [Unit distance problem](https://tamnd.github.io/math2/node/EPWB2JEXPE.md): problem EPWB2JEXPE
- [Mountain pass theorem](https://tamnd.github.io/math2/node/EPXNCSP2MN.md): statement EPXNCSP2MN
- [Schrödinger equation](https://tamnd.github.io/math2/node/EPZR6H2WJN.md): concept EPZR6H2WJN
- [Submartingale](https://tamnd.github.io/math2/node/EQ055QMACW.md): definition EQ055QMACW
- [Superharmonic potential](https://tamnd.github.io/math2/node/EQ23YX9CFG.md): definition EQ23YX9CFG
- [Dawson function](https://tamnd.github.io/math2/node/EQ4QB1QW60.md): concept EQ4QB1QW60
- [Helicoid](https://tamnd.github.io/math2/node/EQ5JSYGWG2.md): example EQ5JSYGWG2
- [Gauss sum](https://tamnd.github.io/math2/node/EQ6A3ZN48K.md): concept EQ6A3ZN48K
- [Congruence subgroup](https://tamnd.github.io/math2/node/EQ7R998DYJ.md): definition EQ7R998DYJ
- [Slice-ribbon conjecture](https://tamnd.github.io/math2/node/EQB39X4HGB.md): conjecture EQB39X4HGB
- [Heisenberg inequality for the Fourier transform](https://tamnd.github.io/math2/node/EQBC3P901V.md): statement EQBC3P901V
- [Manifold with corners](https://tamnd.github.io/math2/node/EQDDVAKSVE.md): definition EQDDVAKSVE
- [Abelian extension](https://tamnd.github.io/math2/node/EQDEXBWGJH.md): concept EQDEXBWGJH
- [Choi's theorem on completely positive maps](https://tamnd.github.io/math2/node/EQG2K8ZFM4.md): statement EQG2K8ZFM4
- [Cartan's uniqueness theorem](https://tamnd.github.io/math2/node/EQHPSKC5SY.md): statement EQHPSKC5SY
- [Integral functional with a Lagrangian density](https://tamnd.github.io/math2/node/EQSFAA5D1M.md): definition EQSFAA5D1M
- [Quasi-projective variety](https://tamnd.github.io/math2/node/EQSHJPE7H8.md): concept EQSHJPE7H8
- [Mapping cylinder](https://tamnd.github.io/math2/node/EQVVDS0ZQT.md): concept EQVVDS0ZQT
- [Morphism of schemes](https://tamnd.github.io/math2/node/EQYDBQHY5B.md): definition EQYDBQHY5B
- [Associated prime](https://tamnd.github.io/math2/node/ERFZ6S8X0X.md): concept ERFZ6S8X0X
- [Total variation](https://tamnd.github.io/math2/node/ERHGXV009H.md): definition ERHGXV009H
- [Convolution on the line](https://tamnd.github.io/math2/node/ERP5737VZ1.md): concept ERP5737VZ1
- [Morse lemma](https://tamnd.github.io/math2/node/ERSD8D4HWX.md): statement ERSD8D4HWX
- [Normal function](https://tamnd.github.io/math2/node/ERTAM0SFVB.md): concept ERTAM0SFVB
- [Lattice homomorphism](https://tamnd.github.io/math2/node/ERTRZPKA1Q.md): definition ERTRZPKA1Q
- [Model complete theory](https://tamnd.github.io/math2/node/ERVV6RWNNX.md): definition ERVV6RWNNX
- [Hodges–Lehmann estimator](https://tamnd.github.io/math2/node/ERXF29KNPC.md): definition ERXF29KNPC
- [Irreducible analytic germ](https://tamnd.github.io/math2/node/ES08X651A3.md): definition ES08X651A3
- [Order theory](https://tamnd.github.io/math2/node/ES34ZH23RS.md): concept ES34ZH23RS
- [Missing data mechanisms](https://tamnd.github.io/math2/node/ES4MC30ZES.md): concept ES4MC30ZES
- [Trivial variety](https://tamnd.github.io/math2/node/ES6P7WHJG9.md): example ES6P7WHJG9
- [Relatively complemented lattice](https://tamnd.github.io/math2/node/ES9XR9CQ3D.md): concept ES9XR9CQ3D
- [Algebra over an operad](https://tamnd.github.io/math2/node/ES9YATDWA3.md): concept ES9YATDWA3
- [Nakano positivity](https://tamnd.github.io/math2/node/ESBGJHPEDT.md): definition ESBGJHPEDT
- [Gauss–Bonnet theorem](https://tamnd.github.io/math2/node/ESGEHKPD60.md): statement ESGEHKPD60
- [Young's lattice](https://tamnd.github.io/math2/node/ESPDZT91QP.md): example ESPDZT91QP
- [Regular scheme](https://tamnd.github.io/math2/node/ESPG0BJERZ.md): concept ESPG0BJERZ
- [Kruskal–Wallis test](https://tamnd.github.io/math2/node/ESQCJ06KZD.md): definition ESQCJ06KZD
- [Smash product](https://tamnd.github.io/math2/node/ESRDNKF6WJ.md): definition ESRDNKF6WJ
- [Intuitionistic type theory](https://tamnd.github.io/math2/node/ESVYD9CZ8C.md): concept ESVYD9CZ8C
- [Hypergeometric function](https://tamnd.github.io/math2/node/ESWC2PXNZX.md): definition ESWC2PXNZX
- [Verlet integration](https://tamnd.github.io/math2/node/ESWE8Y3JD4.md): statement ESWE8Y3JD4
- [Geometric invariant theory](https://tamnd.github.io/math2/node/ESX3ERNP4G.md): definition ESX3ERNP4G
- [Stone algebra](https://tamnd.github.io/math2/node/ESYKDNZF8E.md): concept ESYKDNZF8E
- [Hilbert system](https://tamnd.github.io/math2/node/ET1HPME8VV.md): definition ET1HPME8VV
- [Polytope](https://tamnd.github.io/math2/node/ET5G9CT0RC.md): definition ET5G9CT0RC
- [Amenable Banach algebra](https://tamnd.github.io/math2/node/ET71R6CSAZ.md): definition ET71R6CSAZ
- [Loop-erased random walk](https://tamnd.github.io/math2/node/ET73YH2T29.md): concept ET73YH2T29
- [Pontryagin dual](https://tamnd.github.io/math2/node/ET9QA03J3F.md): definition ET9QA03J3F
- [Upper set](https://tamnd.github.io/math2/node/ETCE287E2R.md): concept ETCE287E2R
- [Paraconsistent logic](https://tamnd.github.io/math2/node/ETCVR65W3M.md): definition ETCVR65W3M
- [Lindelöf space](https://tamnd.github.io/math2/node/ETDW4AK1W1.md): definition ETDW4AK1W1
- [Resolutivity of boundary functions](https://tamnd.github.io/math2/node/ETH7FW1KKA.md): definition ETH7FW1KKA
- [Brauer character](https://tamnd.github.io/math2/node/ETJ4WRXC41.md): definition ETJ4WRXC41
- [Szemerédi regularity lemma](https://tamnd.github.io/math2/node/ETMAPNQY7T.md): statement ETMAPNQY7T
- [Schur orthogonality for compact groups](https://tamnd.github.io/math2/node/ETNDTECY97.md): statement ETNDTECY97
- [Jacobi sum](https://tamnd.github.io/math2/node/ETQ87X5CWS.md): definition ETQ87X5CWS
- [Gromov–Hausdorff space](https://tamnd.github.io/math2/node/ETRF2FHAEE.md): object ETRF2FHAEE
- [Canonical module](https://tamnd.github.io/math2/node/ETSXVDDCYK.md): definition ETSXVDDCYK
- [Hankel transform](https://tamnd.github.io/math2/node/EV25PMXVQT.md): definition EV25PMXVQT
- [Exponential integral](https://tamnd.github.io/math2/node/EV6ZDQ0NDY.md): concept EV6ZDQ0NDY
- [Mixed model](https://tamnd.github.io/math2/node/EV7HZCHQ5V.md): definition EV7HZCHQ5V
- [Local coefficient system](https://tamnd.github.io/math2/node/EVB1XHQTJR.md): definition EVB1XHQTJR
- [C0-semigroup](https://tamnd.github.io/math2/node/EVF180QJXY.md): concept EVF180QJXY
- [Compact group](https://tamnd.github.io/math2/node/EVHCE5VR6Y.md): concept EVHCE5VR6Y
- [Separably closed field](https://tamnd.github.io/math2/node/EVPS59M8X4.md): definition EVPS59M8X4
- [Ring of convergent power series](https://tamnd.github.io/math2/node/EVPZ7ZG1HB.md): concept EVPZ7ZG1HB
- [Fourier integral operator](https://tamnd.github.io/math2/node/EVS4JPEE45.md): concept EVS4JPEE45
- [Crystalline cohomology](https://tamnd.github.io/math2/node/EVXCN06FW3.md): concept EVXCN06FW3
- [Weak operator topology](https://tamnd.github.io/math2/node/EVXPTK4P0J.md): definition EVXPTK4P0J
- [Eilenberg swindle](https://tamnd.github.io/math2/node/EW56H41PKJ.md): definition EW56H41PKJ
- [Entropy solution](https://tamnd.github.io/math2/node/EWFW0P9KKC.md): definition EWFW0P9KKC
- [Symbolic dynamics](https://tamnd.github.io/math2/node/EWFWTSVY0B.md): concept EWFWTSVY0B
- [Duistermaat–Heckman formula](https://tamnd.github.io/math2/node/EWJK7NY6ZB.md): statement EWJK7NY6ZB
- [Plane curve](https://tamnd.github.io/math2/node/EWS4WFTA7W.md): concept EWS4WFTA7W
- [Macdonald polynomial](https://tamnd.github.io/math2/node/EWV5KVK4GY.md): definition EWV5KVK4GY
- [Transversality](https://tamnd.github.io/math2/node/EWVATW6EYZ.md): definition EWVATW6EYZ
- [Renewal equation](https://tamnd.github.io/math2/node/EWVVH3EKMR.md): example EWVVH3EKMR
- [Brown–McCoy radical](https://tamnd.github.io/math2/node/EWWP84BGSC.md): concept EWWP84BGSC
- [Kodaira–Spencer map](https://tamnd.github.io/math2/node/EWZ1G10MAY.md): concept EWZ1G10MAY
- [Brauer group](https://tamnd.github.io/math2/node/EX34J8PP61.md): definition EX34J8PP61
- [Enumerative combinatorics](https://tamnd.github.io/math2/node/EX382NQTGP.md): concept EX382NQTGP
- [Lefschetz hyperplane theorem](https://tamnd.github.io/math2/node/EX6YSYN7B4.md): statement EX6YSYN7B4
- [Krull dimension of a poset](https://tamnd.github.io/math2/node/EX7N9D1YCD.md): concept EX7N9D1YCD
- [Lattice of clones](https://tamnd.github.io/math2/node/EX9NCDFYC7.md): concept EX9NCDFYC7
- [Radiative transfer equation](https://tamnd.github.io/math2/node/EX9YYRMAJA.md): example EX9YYRMAJA
- [Geometric morphism](https://tamnd.github.io/math2/node/EXD1197NPH.md): definition EXD1197NPH
- [Quillen equivalence](https://tamnd.github.io/math2/node/EXHJHA1TBK.md): definition EXHJHA1TBK
- [Zariski topology](https://tamnd.github.io/math2/node/EXJAXVXW51.md): definition EXJAXVXW51
- [Smooth morphism](https://tamnd.github.io/math2/node/EXKGG9X6J0.md): concept EXKGG9X6J0
- [De Rham cohomology of varieties](https://tamnd.github.io/math2/node/EXPS8MH5S2.md): definition EXPS8MH5S2
- [Lyusternik–Fet theorem](https://tamnd.github.io/math2/node/EXZA8ZA8P6.md): statement EXZA8ZA8P6
- [Non-uniform discrete Fourier transform](https://tamnd.github.io/math2/node/EXZHFNRMV8.md): definition EXZHFNRMV8
- [Tensor product of Hilbert spaces](https://tamnd.github.io/math2/node/EY0X2NKA3S.md): concept EY0X2NKA3S
- [Charlier polynomials](https://tamnd.github.io/math2/node/EY18HD35RG.md): definition EY18HD35RG
- [Non-wandering set](https://tamnd.github.io/math2/node/EY2W34JZA4.md): concept EY2W34JZA4
- [Differential-algebraic system of equations](https://tamnd.github.io/math2/node/EY362KR0KB.md): concept EY362KR0KB
- [Convolution integral equation](https://tamnd.github.io/math2/node/EY75YD988Z.md): concept EY75YD988Z
- [Born series](https://tamnd.github.io/math2/node/EYA944RVSY.md): definition EYA944RVSY
- [Euler–Boole summation](https://tamnd.github.io/math2/node/EYDGR9D2Y9.md): definition EYDGR9D2Y9
- [Comparison principle](https://tamnd.github.io/math2/node/EYF0H8AP91.md): statement EYF0H8AP91
- [Steinhaus theorem](https://tamnd.github.io/math2/node/EYG946BY5G.md): statement EYG946BY5G
- [Dehn invariant](https://tamnd.github.io/math2/node/EYMHCPTZ8T.md): definition EYMHCPTZ8T
- [Positive harmonic function](https://tamnd.github.io/math2/node/EYMQ82Z8XJ.md): concept EYMQ82Z8XJ
- [Eisenstein series](https://tamnd.github.io/math2/node/EYREVY31BN.md): concept EYREVY31BN
- [Cauchy formula for repeated integration](https://tamnd.github.io/math2/node/EYXG11V76E.md): statement EYXG11V76E
- [Lidskii's trace formula for integral operators](https://tamnd.github.io/math2/node/EYXNKYMJJH.md): statement EYXNKYMJJH
- [Varieties as categories of algebras over a monad](https://tamnd.github.io/math2/node/EZ21NHHAKG.md): statement EZ21NHHAKG
- [Attractor](https://tamnd.github.io/math2/node/EZ24HDESPB.md): definition EZ24HDESPB
- [Multidimensional scaling](https://tamnd.github.io/math2/node/EZ596QFCKY.md): concept EZ596QFCKY
- [Optimal control](https://tamnd.github.io/math2/node/EZ8NBZ8D0E.md): definition EZ8NBZ8D0E
- [Skolem normal form](https://tamnd.github.io/math2/node/EZ8NCHN7SJ.md): definition EZ8NCHN7SJ
- [Weyl group of a compact Lie group](https://tamnd.github.io/math2/node/EZGTEK3HQN.md): concept EZGTEK3HQN
- [Koebe function](https://tamnd.github.io/math2/node/EZQFWKF03Q.md): example EZQFWKF03Q
- [Noncommutative Gröbner basis](https://tamnd.github.io/math2/node/EZSWDV6ZZQ.md): definition EZSWDV6ZZQ
- [Field homomorphism](https://tamnd.github.io/math2/node/EZVN87C88J.md): concept EZVN87C88J
- [Abel summation](https://tamnd.github.io/math2/node/EZXPY4A19C.md): concept EZXPY4A19C
- [Steenrod reduced power](https://tamnd.github.io/math2/node/EZYHSE1E8P.md): concept EZYHSE1E8P
- [Viscosity solution](https://tamnd.github.io/math2/node/F0047A6GDG.md): definition F0047A6GDG
- [Simple homotopy equivalence](https://tamnd.github.io/math2/node/F01D990B61.md): definition F01D990B61
- [Sperner property](https://tamnd.github.io/math2/node/F02PJ2ZK9X.md): concept F02PJ2ZK9X
- [Steiner formula](https://tamnd.github.io/math2/node/F02XA3MYX8.md): statement F02XA3MYX8
- [Voronoi's conjecture on parallelohedra](https://tamnd.github.io/math2/node/F03A4MKA5V.md): conjecture F03A4MKA5V
- [Heteroclinic orbit](https://tamnd.github.io/math2/node/F05PSBZP8Z.md): definition F05PSBZP8Z
- [Projection operation](https://tamnd.github.io/math2/node/F0ABT2ZFQF.md): definition F0ABT2ZFQF
- [Whitehead problem](https://tamnd.github.io/math2/node/F0B6BRJA5Z.md): problem F0B6BRJA5Z
- [Fenchel's theorem](https://tamnd.github.io/math2/node/F0BJ2BHSZG.md): statement F0BJ2BHSZG
- [Straszewicz's theorem](https://tamnd.github.io/math2/node/F0EXKKKEFW.md): statement F0EXKKKEFW
- [Stability condition](https://tamnd.github.io/math2/node/F0G92Q55DM.md): concept F0G92Q55DM
- [Midpoint convex function](https://tamnd.github.io/math2/node/F0GBTYKN31.md): definition F0GBTYKN31
- [Algebraic dimension](https://tamnd.github.io/math2/node/F0GF0JEZTG.md): concept F0GF0JEZTG
- [Tensor product of modules](https://tamnd.github.io/math2/node/F0HQXQ0HCD.md): concept F0HQXQ0HCD
- [Laguerre plane](https://tamnd.github.io/math2/node/F0JQ1TGKB0.md): concept F0JQ1TGKB0
- [Exact couple](https://tamnd.github.io/math2/node/F0M6BY5BH7.md): definition F0M6BY5BH7
- [Pluripolar set](https://tamnd.github.io/math2/node/F0QRCDYBXK.md): definition F0QRCDYBXK
- [Fekete points](https://tamnd.github.io/math2/node/F0RVAF11K9.md): concept F0RVAF11K9
- [Prüfer group](https://tamnd.github.io/math2/node/F0XAMM2PPS.md): example F0XAMM2PPS
- [Bernstein's inequality for polynomials](https://tamnd.github.io/math2/node/F0YGC89S9R.md): statement F0YGC89S9R
- [Walsh function](https://tamnd.github.io/math2/node/F12A5QKW8Z.md): concept F12A5QKW8Z
- [Von Neumann algebra](https://tamnd.github.io/math2/node/F12V24NF4R.md): definition F12V24NF4R
- [Nonlinear Volterra equation](https://tamnd.github.io/math2/node/F18MBATVVP.md): concept F18MBATVVP
- [Whitney stratification](https://tamnd.github.io/math2/node/F19BQTN832.md): definition F19BQTN832
- [Rokhlin lemma](https://tamnd.github.io/math2/node/F1CCCF4V83.md): statement F1CCCF4V83
- [Stalk](https://tamnd.github.io/math2/node/F1CGWAW74E.md): definition F1CGWAW74E
- [Feigenbaum constants](https://tamnd.github.io/math2/node/F1EHT2CPHC.md): object F1EHT2CPHC
- [Height function](https://tamnd.github.io/math2/node/F1FF505PWH.md): concept F1FF505PWH
- [Equational theory](https://tamnd.github.io/math2/node/F1M51B4MDG.md): concept F1M51B4MDG
- [Integrable system](https://tamnd.github.io/math2/node/F1MCB9P7KJ.md): definition F1MCB9P7KJ
- [Constructivism](https://tamnd.github.io/math2/node/F1P5SZ1XPH.md): definition F1P5SZ1XPH
- [Tridiagonal matrix algorithm](https://tamnd.github.io/math2/node/F1QWA1PEQQ.md): statement F1QWA1PEQQ
- [Helly's theorem](https://tamnd.github.io/math2/node/F1W3QRBNGJ.md): statement F1W3QRBNGJ
- [Left-orderable group](https://tamnd.github.io/math2/node/F1WYS82NNB.md): concept F1WYS82NNB
- [Pseudospectrum](https://tamnd.github.io/math2/node/F20VTSH2FM.md): concept F20VTSH2FM
- [Chern–Simons theory](https://tamnd.github.io/math2/node/F28G886PWD.md): concept F28G886PWD
- [Semiorder](https://tamnd.github.io/math2/node/F2C174HHSE.md): concept F2C174HHSE
- [Acyclic object](https://tamnd.github.io/math2/node/F2CMPBHQ02.md): concept F2CMPBHQ02
- [Forcing](https://tamnd.github.io/math2/node/F2DTWW9HP7.md): concept F2DTWW9HP7
- [Harish-Chandra character](https://tamnd.github.io/math2/node/F2E3317P78.md): concept F2E3317P78
- [Homotopic maps](https://tamnd.github.io/math2/node/F2F143KQ2M.md): concept F2F143KQ2M
- [Hartman–Wintner theorem](https://tamnd.github.io/math2/node/F2HHDD85EM.md): statement F2HHDD85EM
- [Curve of constant width](https://tamnd.github.io/math2/node/F2JX0R86WQ.md): definition F2JX0R86WQ
- [Delta functor](https://tamnd.github.io/math2/node/F2W20ZNWC7.md): definition F2W20ZNWC7
- [Principal symbol](https://tamnd.github.io/math2/node/F31T650DKY.md): concept F31T650DKY
- [Bounded variation](https://tamnd.github.io/math2/node/F34S6P61SW.md): definition F34S6P61SW
- [Center manifold](https://tamnd.github.io/math2/node/F35WJP8VHG.md): definition F35WJP8VHG
- [Atomic domain](https://tamnd.github.io/math2/node/F37JBGAA1S.md): concept F37JBGAA1S
- [Lattice polytope](https://tamnd.github.io/math2/node/F39N9NBFPZ.md): concept F39N9NBFPZ
- [Lamé equations of elasticity](https://tamnd.github.io/math2/node/F3DXBJZAME.md): definition F3DXBJZAME
- [Quantum unique ergodicity conjecture](https://tamnd.github.io/math2/node/F3EB7WZFAC.md): conjecture F3EB7WZFAC
- [Krull topology](https://tamnd.github.io/math2/node/F3GTF5NT4M.md): concept F3GTF5NT4M
- [Stable distribution](https://tamnd.github.io/math2/node/F3N6E1Y2AY.md): concept F3N6E1Y2AY
- [Intermittency](https://tamnd.github.io/math2/node/F3STQVV22B.md): definition F3STQVV22B
- [Synthetic differential geometry](https://tamnd.github.io/math2/node/F3YMFK4S5J.md): concept F3YMFK4S5J
- [Kleene's recursion theorem](https://tamnd.github.io/math2/node/F3Z30M606H.md): statement F3Z30M606H
- [Monad](https://tamnd.github.io/math2/node/F426Q5R0RY.md): definition F426Q5R0RY
- [Quasi-ergodicity](https://tamnd.github.io/math2/node/F43HXQD0T6.md): concept F43HXQD0T6
- [Einstein problem](https://tamnd.github.io/math2/node/F456KQGPFV.md): statement F456KQGPFV
- [Smooth manifold](https://tamnd.github.io/math2/node/F45VQ47GPV.md): concept F45VQ47GPV
- [Sturm–Liouville problem as an integral equation](https://tamnd.github.io/math2/node/F46GGH7RJ5.md): example F46GGH7RJ5
- [Domain theory](https://tamnd.github.io/math2/node/F4BVTGH58G.md): definition F4BVTGH58G
- [Frame bundle](https://tamnd.github.io/math2/node/F4HWTRNXAP.md): concept F4HWTRNXAP
- [Minkowski addition](https://tamnd.github.io/math2/node/F4J5BZ4HWD.md): definition F4J5BZ4HWD
- [Laplace–Beltrami operator](https://tamnd.github.io/math2/node/F4NWZNTRMD.md): concept F4NWZNTRMD
- [Banach bundle](https://tamnd.github.io/math2/node/F4RH4MZQDG.md): concept F4RH4MZQDG
- [Cauchy–Kovalevskaya theorem for ODEs](https://tamnd.github.io/math2/node/F4VMHVWK41.md): statement F4VMHVWK41
- [McKean–Singer formula](https://tamnd.github.io/math2/node/F4WDYPQ9VP.md): statement F4WDYPQ9VP
- [Reduced product](https://tamnd.github.io/math2/node/F4WRD396PB.md): concept F4WRD396PB
- [Compact Lie group](https://tamnd.github.io/math2/node/F4ZYFEKKZW.md): definition F4ZYFEKKZW
- [Fitting's lemma](https://tamnd.github.io/math2/node/F52B2JMEHY.md): statement F52B2JMEHY
- [Stochastic difference equation](https://tamnd.github.io/math2/node/F52PFPZ4H9.md): definition F52PFPZ4H9
- [Subanalytic set](https://tamnd.github.io/math2/node/F55M92H5JY.md): definition F55M92H5JY
- [Dixmier conjecture](https://tamnd.github.io/math2/node/F59TS79V34.md): conjecture F59TS79V34
- [Gross–Pitaevskii equation](https://tamnd.github.io/math2/node/F5EAWFMC8A.md): concept F5EAWFMC8A
- [Hopf bifurcation](https://tamnd.github.io/math2/node/F5H8GW1XYS.md): concept F5H8GW1XYS
- [Dini test](https://tamnd.github.io/math2/node/F5PHA35M50.md): statement F5PHA35M50
- [Holomorphic line bundle](https://tamnd.github.io/math2/node/F5R676980R.md): definition F5R676980R
- [Hörmander–Mikhlin multiplier theorem](https://tamnd.github.io/math2/node/F5V7CXW30E.md): statement F5V7CXW30E
- [Asymptotics of Bessel functions](https://tamnd.github.io/math2/node/F5YG34FAY1.md): statement F5YG34FAY1
- [Quasi-measure-preserving map](https://tamnd.github.io/math2/node/F608XX51KA.md): concept F608XX51KA
- [Regular polytope](https://tamnd.github.io/math2/node/F60PD912D5.md): concept F60PD912D5
- [Dynamic programming in continuous time](https://tamnd.github.io/math2/node/F64JX05VJX.md): definition F64JX05VJX
- [Information geometry](https://tamnd.github.io/math2/node/F64Z9541DV.md): concept F64Z9541DV
- [Siegel disc](https://tamnd.github.io/math2/node/F666F5W806.md): definition F666F5W806
- [Pólya's conjecture on Laplacian eigenvalues](https://tamnd.github.io/math2/node/F68QFEYFY0.md): conjecture F68QFEYFY0
- [Log-rank test](https://tamnd.github.io/math2/node/F6CT1YXZ0G.md): concept F6CT1YXZ0G
- [Sine-Gordon equation](https://tamnd.github.io/math2/node/F6GKG7T6Y8.md): definition F6GKG7T6Y8
- [Bloch space](https://tamnd.github.io/math2/node/F6M8ME0DQR.md): concept F6M8ME0DQR
- [Filter limit](https://tamnd.github.io/math2/node/F6QDHKE6J3.md): concept F6QDHKE6J3
- [Adic space](https://tamnd.github.io/math2/node/F6VBCVP13K.md): concept F6VBCVP13K
- [Cauchy's exponential equation](https://tamnd.github.io/math2/node/F6ZWKD5W15.md): definition F6ZWKD5W15
- [Positive-definite function](https://tamnd.github.io/math2/node/F73VWGF637.md): concept F73VWGF637
- [Hausdorff distance](https://tamnd.github.io/math2/node/F79WYYC07C.md): concept F79WYYC07C
- [Whittaker transform](https://tamnd.github.io/math2/node/F7A335JYKF.md): definition F7A335JYKF
- [Dodgson condensation](https://tamnd.github.io/math2/node/F7D849A4RZ.md): definition F7D849A4RZ
- [Beltrami identity](https://tamnd.github.io/math2/node/F7EKTEESQ5.md): statement F7EKTEESQ5
- [Lagrange resolvent](https://tamnd.github.io/math2/node/F7FYTD7Z56.md): definition F7FYTD7Z56
- [Minkowski–Bouligand dimension](https://tamnd.github.io/math2/node/F7G7CJY143.md): definition F7G7CJY143
- [Adaptive step size](https://tamnd.github.io/math2/node/F7H4H711MY.md): definition F7H4H711MY
- [Maximal compact subgroup](https://tamnd.github.io/math2/node/F7H6GPR050.md): definition F7H6GPR050
- [Cauchy–Davenport theorem](https://tamnd.github.io/math2/node/F7K7XFACSG.md): statement F7K7XFACSG
- [State on a C*-algebra](https://tamnd.github.io/math2/node/F7MB2WREBJ.md): concept F7MB2WREBJ
- [Join and meet](https://tamnd.github.io/math2/node/F7YV441H6R.md): definition F7YV441H6R
- [Epsilon-net of a set system](https://tamnd.github.io/math2/node/F7ZGDKRMB4.md): definition F7ZGDKRMB4
- [Mathematical formulation of quantum mechanics](https://tamnd.github.io/math2/node/F8R88XFMQK.md): definition F8R88XFMQK
- [Fractional Laplacian](https://tamnd.github.io/math2/node/F8RSNQBMBE.md): concept F8RSNQBMBE
- [Autoregressive–moving-average model](https://tamnd.github.io/math2/node/F8T3MJX3VJ.md): definition F8T3MJX3VJ
- [Borel–Carathéodory theorem](https://tamnd.github.io/math2/node/F8XD8CDB0H.md): statement F8XD8CDB0H
- [Hierarchical matrix](https://tamnd.github.io/math2/node/F92JJR2NZE.md): definition F92JJR2NZE
- [Problem of Apollonius](https://tamnd.github.io/math2/node/F9410EB0N1.md): problem F9410EB0N1
- [Darboux integral](https://tamnd.github.io/math2/node/F961Q88FHK.md): concept F961Q88FHK
- [Finiteness of the class number](https://tamnd.github.io/math2/node/F99RM1T02C.md): statement F99RM1T02C
- [Homotopy continuation for polynomial systems](https://tamnd.github.io/math2/node/F9FTASTFSX.md): definition F9FTASTFSX
- [Subdirect product](https://tamnd.github.io/math2/node/F9G2C6EE0E.md): definition F9G2C6EE0E
- [Shearlet](https://tamnd.github.io/math2/node/F9Q17JRV5K.md): concept F9Q17JRV5K
- [Yoneda embedding](https://tamnd.github.io/math2/node/F9RQZ6EPS7.md): definition F9RQZ6EPS7
- [Arithmetic dynamics](https://tamnd.github.io/math2/node/F9ZCCKATES.md): definition F9ZCCKATES
- [Weak mixing](https://tamnd.github.io/math2/node/FA0CD4SQZ5.md): definition FA0CD4SQZ5
- [Classification of Fatou components](https://tamnd.github.io/math2/node/FA37X4AFFB.md): statement FA37X4AFFB
- [Compactness of the spectrum](https://tamnd.github.io/math2/node/FA5V5KRWDF.md): statement FA5V5KRWDF
- [Principal curvature](https://tamnd.github.io/math2/node/FA8QE1SGP4.md): concept FA8QE1SGP4
- [Dido's problem](https://tamnd.github.io/math2/node/FABJSZ9D3S.md): problem FABJSZ9D3S
- [Bidual and canonical embedding](https://tamnd.github.io/math2/node/FAGMBTRN7Z.md): definition FAGMBTRN7Z
- [f-ring](https://tamnd.github.io/math2/node/FAHERE12SH.md): definition FAHERE12SH
- [Besov space](https://tamnd.github.io/math2/node/FAMZ5Z5JK4.md): definition FAMZ5Z5JK4
- [Square-integrable kernel](https://tamnd.github.io/math2/node/FANW5G7YZC.md): definition FANW5G7YZC
- [Stone algebra](https://tamnd.github.io/math2/node/FAQ8ZJ4M0M.md): definition FAQ8ZJ4M0M
- [Hamilton's principle](https://tamnd.github.io/math2/node/FAQ9CH2Z76.md): statement FAQ9CH2Z76
- [Fourier analysis](https://tamnd.github.io/math2/node/FAYWXJ68SB.md): concept FAYWXJ68SB
- [Spectrum of a ring](https://tamnd.github.io/math2/node/FB45BHHA2J.md): concept FB45BHHA2J
- [Bayes risk is at most minimax risk](https://tamnd.github.io/math2/node/FB692YN0BZ.md): statement FB692YN0BZ
- [Cubic formula](https://tamnd.github.io/math2/node/FB6J5WAT5V.md): statement FB6J5WAT5V
- [Integers modulo n form a cyclic group](https://tamnd.github.io/math2/node/FB79N7AHZA.md): example FB79N7AHZA
- [Whitney decomposition](https://tamnd.github.io/math2/node/FBERT0KQXN.md): statement FBERT0KQXN
- [Chern–Weil homomorphism](https://tamnd.github.io/math2/node/FBFFS10WGS.md): definition FBFFS10WGS
- [Complete category](https://tamnd.github.io/math2/node/FBGXAXAEJ1.md): definition FBGXAXAEJ1
- [Carleson's theorem](https://tamnd.github.io/math2/node/FBH6KG484Z.md): statement FBH6KG484Z
- [Godunov's scheme](https://tamnd.github.io/math2/node/FBJ4JCP5SR.md): definition FBJ4JCP5SR
- [Contact structure on a jet bundle](https://tamnd.github.io/math2/node/FBTWQEMTPS.md): definition FBTWQEMTPS
- [Seifert–Weber space](https://tamnd.github.io/math2/node/FBVEG8HVTP.md): example FBVEG8HVTP
- [Polynomial operation](https://tamnd.github.io/math2/node/FC34XP2DQV.md): definition FC34XP2DQV
- [Gabriel's theorem](https://tamnd.github.io/math2/node/FC78F2E87E.md): statement FC78F2E87E
- [Reflective subcategory](https://tamnd.github.io/math2/node/FC7VB4WSG0.md): concept FC7VB4WSG0
- [Accessible category](https://tamnd.github.io/math2/node/FC8C0V4TFZ.md): concept FC8C0V4TFZ
- [Sampling distribution](https://tamnd.github.io/math2/node/FC9NRV5GSP.md): definition FC9NRV5GSP
- [Vitali covering lemma](https://tamnd.github.io/math2/node/FCA0J1X07K.md): statement FCA0J1X07K
- [Nikodym convergence theorem](https://tamnd.github.io/math2/node/FCAA02SN0E.md): statement FCAA02SN0E
- [Henstock–Kurzweil integral](https://tamnd.github.io/math2/node/FCGHXDGVV9.md): concept FCGHXDGVV9
- [Wigner distribution function](https://tamnd.github.io/math2/node/FCHW5QGG5V.md): concept FCHW5QGG5V
- [Convolution on the half-line](https://tamnd.github.io/math2/node/FCKX8KFVGN.md): concept FCKX8KFVGN
- [Primary obstruction](https://tamnd.github.io/math2/node/FCM1T5EXFV.md): concept FCM1T5EXFV
- [Quasi-invariant measure](https://tamnd.github.io/math2/node/FCNQP0A60D.md): concept FCNQP0A60D
- [Support function](https://tamnd.github.io/math2/node/FCSBB1HAHH.md): definition FCSBB1HAHH
- [Nested radical](https://tamnd.github.io/math2/node/FCTRS2SS16.md): example FCTRS2SS16
- [Local property](https://tamnd.github.io/math2/node/FD3T25A1DX.md): definition FD3T25A1DX
- [Differential-algebraic system of equations](https://tamnd.github.io/math2/node/FD7H5AM1R7.md): definition FD7H5AM1R7
- [Dirichlet space](https://tamnd.github.io/math2/node/FD8VEKWZNR.md): definition FD8VEKWZNR
- [Biholomorphism](https://tamnd.github.io/math2/node/FDARANFQ0B.md): definition FDARANFQ0B
- [Noether theorems for singular integral equations](https://tamnd.github.io/math2/node/FDC1BEB8FW.md): statement FDC1BEB8FW
- [Iterated kernel](https://tamnd.github.io/math2/node/FDF5C62DBD.md): definition FDF5C62DBD
- [Kan complex](https://tamnd.github.io/math2/node/FDFYCM3YYC.md): concept FDFYCM3YYC
- [Locally connected space](https://tamnd.github.io/math2/node/FDVAHRJKF3.md): concept FDVAHRJKF3
- [Dynamical billiards](https://tamnd.github.io/math2/node/FDWY67RFRT.md): definition FDWY67RFRT
- [Kontsevich integral](https://tamnd.github.io/math2/node/FE1VTGF349.md): definition FE1VTGF349
- [Multiple integral](https://tamnd.github.io/math2/node/FE1ZZJ8QW2.md): concept FE1ZZJ8QW2
- [Weierstrass division theorem](https://tamnd.github.io/math2/node/FE83QQS17S.md): statement FE83QQS17S
- [Iterative root](https://tamnd.github.io/math2/node/FE8EYTGMRS.md): definition FE8EYTGMRS
- [Knot](https://tamnd.github.io/math2/node/FEAFDTMMW7.md): definition FEAFDTMMW7
- [j-invariant](https://tamnd.github.io/math2/node/FEC2XH822R.md): concept FEC2XH822R
- [Margulis arithmeticity theorem](https://tamnd.github.io/math2/node/FEJK846N5X.md): statement FEJK846N5X
- [Eilenberg–Moore category](https://tamnd.github.io/math2/node/FEPRJR3MND.md): concept FEPRJR3MND
- [Sylow theorems](https://tamnd.github.io/math2/node/FESTJ0K9TX.md): statement FESTJ0K9TX
- [Support of a module](https://tamnd.github.io/math2/node/FETMR0M0M8.md): concept FETMR0M0M8
- [Topological data analysis](https://tamnd.github.io/math2/node/FETY7R4X86.md): concept FETY7R4X86
- [Beck's monadicity theorem](https://tamnd.github.io/math2/node/FETZ0VZGMD.md): statement FETZ0VZGMD
- [M. Riesz extension theorem](https://tamnd.github.io/math2/node/FEXZC8XKN1.md): statement FEXZC8XKN1
- [Ramanujan theta function](https://tamnd.github.io/math2/node/FEZ2HAFZTM.md): definition FEZ2HAFZTM
- [Convolution on the line](https://tamnd.github.io/math2/node/FF1JW893K2.md): definition FF1JW893K2
- [Pseudo-differential operator](https://tamnd.github.io/math2/node/FF27N0BWTP.md): concept FF27N0BWTP
- [Information geometry](https://tamnd.github.io/math2/node/FF2KNFNTB7.md): definition FF2KNFNTB7
- [Plurisubharmonic function](https://tamnd.github.io/math2/node/FF2P7ER3QT.md): concept FF2P7ER3QT
- [Monad](https://tamnd.github.io/math2/node/FF6XXA6D1A.md): concept FF6XXA6D1A
- [Minimal harmonic function](https://tamnd.github.io/math2/node/FF8P46NH5S.md): definition FF8P46NH5S
- [Elliptic function](https://tamnd.github.io/math2/node/FF9ZPCZ59K.md): concept FF9ZPCZ59K
- [Fourier restriction problem](https://tamnd.github.io/math2/node/FFCH15ZNZG.md): problem FFCH15ZNZG
- [Sobolev space](https://tamnd.github.io/math2/node/FFDYRMG5DP.md): definition FFDYRMG5DP
- [Cellular homology](https://tamnd.github.io/math2/node/FFH90QRFFJ.md): definition FFH90QRFFJ
- [Lindeberg's condition](https://tamnd.github.io/math2/node/FFJMRQNF96.md): definition FFJMRQNF96
- [Babenko–Beckner inequality](https://tamnd.github.io/math2/node/FFM7Y61KRT.md): statement FFM7Y61KRT
- [Convolution property of the Z-transform](https://tamnd.github.io/math2/node/FFR8NPWG64.md): statement FFR8NPWG64
- [Hausdorff content](https://tamnd.github.io/math2/node/FFRM041WQX.md): definition FFRM041WQX
- [Discontinuous additive functions via a Hamel basis](https://tamnd.github.io/math2/node/FFTNS66XG6.md): example FFTNS66XG6
- [Picard–Lefschetz theory](https://tamnd.github.io/math2/node/FFYQAVT7MB.md): concept FFYQAVT7MB
- [Mahler conjecture](https://tamnd.github.io/math2/node/FFZHNFJCZS.md): conjecture FFZHNFJCZS
- [Geometric quantization](https://tamnd.github.io/math2/node/FG33N4ZHDG.md): concept FG33N4ZHDG
- [Cauchy integral formula on a polydisc](https://tamnd.github.io/math2/node/FG5448NZ18.md): statement FG5448NZ18
- [Filtered algebra](https://tamnd.github.io/math2/node/FGF275MNQJ.md): definition FGF275MNQJ
- [Incidence structure](https://tamnd.github.io/math2/node/FGKNAFNHA3.md): concept FGKNAFNHA3
- [Holomorphic vector bundle](https://tamnd.github.io/math2/node/FGP202VKXW.md): definition FGP202VKXW
- [Sasakian manifold](https://tamnd.github.io/math2/node/FGRPYWW1V1.md): definition FGRPYWW1V1
- [Plurisubharmonic exhaustion function](https://tamnd.github.io/math2/node/FGSKR6ZS3Q.md): concept FGSKR6ZS3Q
- [Abel function](https://tamnd.github.io/math2/node/FGY1XAHVXP.md): definition FGY1XAHVXP
- [Trapezoidal rule](https://tamnd.github.io/math2/node/FH0538XCZR.md): concept FH0538XCZR
- [Fine topology](https://tamnd.github.io/math2/node/FH0Z9VHRH7.md): definition FH0Z9VHRH7
- [Banach–Tarski paradox](https://tamnd.github.io/math2/node/FH250K068Z.md): statement FH250K068Z
- [Telegrapher's equations](https://tamnd.github.io/math2/node/FH2A1PCXNB.md): definition FH2A1PCXNB
- [Groupoid](https://tamnd.github.io/math2/node/FH3C0T643F.md): concept FH3C0T643F
- [Dehn's lemma](https://tamnd.github.io/math2/node/FH5WSPD79F.md): statement FH5WSPD79F
- [Coercive functional](https://tamnd.github.io/math2/node/FH92FW76Q0.md): definition FH92FW76Q0
- [Abel transform](https://tamnd.github.io/math2/node/FH9SJGD1CY.md): definition FH9SJGD1CY
- [Recurrence and transience of states](https://tamnd.github.io/math2/node/FHC702B2R7.md): definition FHC702B2R7
- [Minimal period](https://tamnd.github.io/math2/node/FHFWE27514.md): definition FHFWE27514
- [Chern–Moser normal form](https://tamnd.github.io/math2/node/FHG1N4EKP4.md): statement FHG1N4EKP4
- [Schreier–Sims algorithm](https://tamnd.github.io/math2/node/FHGDZWZ2WE.md): concept FHGDZWZ2WE
- [Harish-Chandra character](https://tamnd.github.io/math2/node/FHKGR7MD1A.md): definition FHKGR7MD1A
- [Periodic solution](https://tamnd.github.io/math2/node/FHNDX22WMK.md): concept FHNDX22WMK
- [Compact quantum group](https://tamnd.github.io/math2/node/FHQKT6HPW2.md): concept FHQKT6HPW2
- [Cartier divisor](https://tamnd.github.io/math2/node/FHV05ZK99F.md): concept FHV05ZK99F
- [Schouten–Nijenhuis bracket](https://tamnd.github.io/math2/node/FJ1MDBZ680.md): concept FJ1MDBZ680
- [Treewidth](https://tamnd.github.io/math2/node/FJ5E6FZVRJ.md): definition FJ5E6FZVRJ
- [Prime model](https://tamnd.github.io/math2/node/FJ79HREXG4.md): definition FJ79HREXG4
- [Erdős–Ko–Rado theorem](https://tamnd.github.io/math2/node/FJ92Q6VTJH.md): statement FJ92Q6VTJH
- [Arens–Fort space](https://tamnd.github.io/math2/node/FJ965C4QWA.md): counterexample FJ965C4QWA
- [Approximation of a kernel by degenerate kernels](https://tamnd.github.io/math2/node/FJKNJ14AD2.md): statement FJKNJ14AD2
- [Convergence tests](https://tamnd.github.io/math2/node/FJKV01J9F1.md): concept FJKV01J9F1
- [Instanton](https://tamnd.github.io/math2/node/FJVQH0VPN7.md): definition FJVQH0VPN7
- [Moduli space](https://tamnd.github.io/math2/node/FJZRW9P0K3.md): concept FJZRW9P0K3
- [Bott's Morse-theoretic proof of periodicity](https://tamnd.github.io/math2/node/FJZXB1N7TP.md): statement FJZXB1N7TP
- [Lebesgue–Stieltjes integration](https://tamnd.github.io/math2/node/FK3QBZAZH7.md): concept FK3QBZAZH7
- [Yamabe flow](https://tamnd.github.io/math2/node/FK4F02GMVA.md): definition FK4F02GMVA
- [Complex p-adic numbers](https://tamnd.github.io/math2/node/FKF9YGBKHF.md): object FKF9YGBKHF
- [Krylov subspace](https://tamnd.github.io/math2/node/FKPPWF87TR.md): definition FKPPWF87TR
- [Subnormal subgroup](https://tamnd.github.io/math2/node/FKTQ1WBMKM.md): concept FKTQ1WBMKM
- [Multiple gamma function](https://tamnd.github.io/math2/node/FM0ZZS1PY2.md): concept FM0ZZS1PY2
- [Stein equation](https://tamnd.github.io/math2/node/FM9Y4Y6D8F.md): concept FM9Y4Y6D8F
- [Finite-rank operator](https://tamnd.github.io/math2/node/FMBFW2QXZ3.md): definition FMBFW2QXZ3
- [Free lattice](https://tamnd.github.io/math2/node/FMCWXA0WT4.md): definition FMCWXA0WT4
- [Amalgamated free product](https://tamnd.github.io/math2/node/FMFDEE8WN2.md): definition FMFDEE8WN2
- [Order type](https://tamnd.github.io/math2/node/FMFGTVVR4S.md): concept FMFGTVVR4S
- [System of linear integral equations](https://tamnd.github.io/math2/node/FMFZQKR71N.md): definition FMFZQKR71N
- [Fano variety](https://tamnd.github.io/math2/node/FMHA649HJ9.md): concept FMHA649HJ9
- [Jensen's functional equation](https://tamnd.github.io/math2/node/FMKKQEKNGT.md): definition FMKKQEKNGT
- [Prime avoidance lemma](https://tamnd.github.io/math2/node/FMP5AWG6PR.md): statement FMP5AWG6PR
- [Symbolic power](https://tamnd.github.io/math2/node/FMPJJBVM5A.md): concept FMPJJBVM5A
- [Abel–Jacobi map](https://tamnd.github.io/math2/node/FMQEG0PVYB.md): definition FMQEG0PVYB
- [Cohomological dimension of a group](https://tamnd.github.io/math2/node/FMVVV8VKHG.md): definition FMVVV8VKHG
- [Pontryagin dual](https://tamnd.github.io/math2/node/FMW57WNPDB.md): concept FMW57WNPDB
- [Finite type invariant](https://tamnd.github.io/math2/node/FMWGQ0W2X0.md): definition FMWGQ0W2X0
- [Hahn–Kolmogorov theorem](https://tamnd.github.io/math2/node/FN2WRRF6V9.md): statement FN2WRRF6V9
- [Free category](https://tamnd.github.io/math2/node/FN6P3CTPTS.md): definition FN6P3CTPTS
- [Positivstellensatz](https://tamnd.github.io/math2/node/FN734RQYM6.md): statement FN734RQYM6
- [Erdős conjecture on arithmetic progressions](https://tamnd.github.io/math2/node/FN978TDX67.md): conjecture FN978TDX67
- [Method of lines](https://tamnd.github.io/math2/node/FNGWJR2FPK.md): concept FNGWJR2FPK
- [Self-distributive operation](https://tamnd.github.io/math2/node/FNJC21BM1B.md): definition FNJC21BM1B
- [ADE singularity](https://tamnd.github.io/math2/node/FP32JHA9M8.md): definition FP32JHA9M8
- [Gaussian quadrature](https://tamnd.github.io/math2/node/FP37BTP93K.md): concept FP37BTP93K
- [Seminorm](https://tamnd.github.io/math2/node/FP5GVCN8GV.md): concept FP5GVCN8GV
- [Continuous function with divergent Fourier series](https://tamnd.github.io/math2/node/FPDWWD12FE.md): counterexample FPDWWD12FE
- [Generalized inverse](https://tamnd.github.io/math2/node/FPHNJ5AAYF.md): concept FPHNJ5AAYF
- [Unramified morphism](https://tamnd.github.io/math2/node/FPK5ZH7FKC.md): definition FPK5ZH7FKC
- [Causal inference](https://tamnd.github.io/math2/node/FPKRV34TBV.md): definition FPKRV34TBV
- [Myers's theorem](https://tamnd.github.io/math2/node/FPMJEX3DSJ.md): statement FPMJEX3DSJ
- [Floquet multiplier](https://tamnd.github.io/math2/node/FPS7YFGX32.md): concept FPS7YFGX32
- [Hopf lemma](https://tamnd.github.io/math2/node/FPWKGFZE6Q.md): statement FPWKGFZE6Q
- [Floer homology](https://tamnd.github.io/math2/node/FPXZMN0CH5.md): definition FPXZMN0CH5
- [Nishida nilpotence theorem](https://tamnd.github.io/math2/node/FQ048Z65F9.md): statement FQ048Z65F9
- [Semiprime ring](https://tamnd.github.io/math2/node/FQ505YSHNA.md): definition FQ505YSHNA
- [Gorenstein ring](https://tamnd.github.io/math2/node/FQEHRZ7N7X.md): definition FQEHRZ7N7X
- [Conley index](https://tamnd.github.io/math2/node/FQESQA42JE.md): definition FQESQA42JE
- [Malliavin calculus](https://tamnd.github.io/math2/node/FQMAKKE8R9.md): definition FQMAKKE8R9
- [q-difference equation](https://tamnd.github.io/math2/node/FQPYGXJJN0.md): concept FQPYGXJJN0
- [Support of a measure](https://tamnd.github.io/math2/node/FQTXV6BQZG.md): concept FQTXV6BQZG
- [Courant–Friedrichs–Lewy condition](https://tamnd.github.io/math2/node/FQVVNYV6N8.md): statement FQVVNYV6N8
- [Complex dynamics](https://tamnd.github.io/math2/node/FQX56M03XD.md): definition FQX56M03XD
- [Orthocomplemented lattice](https://tamnd.github.io/math2/node/FQXWHS5QT2.md): definition FQXWHS5QT2
- [Elliptic hypergeometric series](https://tamnd.github.io/math2/node/FQYSKP388T.md): concept FQYSKP388T
- [Restricted Burnside problem](https://tamnd.github.io/math2/node/FQZJKY2G5Q.md): statement FQZJKY2G5Q
- [Uniform integrability](https://tamnd.github.io/math2/node/FR210Q35CB.md): definition FR210Q35CB
- [Topological transitivity](https://tamnd.github.io/math2/node/FR49Y9GQHR.md): concept FR49Y9GQHR
- [Meshfree methods](https://tamnd.github.io/math2/node/FR6RA0XF9P.md): concept FR6RA0XF9P
- [Kunz's theorem](https://tamnd.github.io/math2/node/FRCY5BQF6N.md): statement FRCY5BQF6N
- [Simplex category](https://tamnd.github.io/math2/node/FRFYHJGAN3.md): object FRFYHJGAN3
- [Riesz measure of a subharmonic function](https://tamnd.github.io/math2/node/FRKAPMVZ7M.md): definition FRKAPMVZ7M
- [Wishart distribution](https://tamnd.github.io/math2/node/FRNDCF91FT.md): definition FRNDCF91FT
- [Riesz transform](https://tamnd.github.io/math2/node/FRNDFJ9WQ5.md): concept FRNDFJ9WQ5
- [Splitting principle](https://tamnd.github.io/math2/node/FRTE50B74R.md): statement FRTE50B74R
- [Essential spectrum](https://tamnd.github.io/math2/node/FRZG0E15SK.md): concept FRZG0E15SK
- [Subobject](https://tamnd.github.io/math2/node/FS1FKAE89H.md): concept FS1FKAE89H
- [Thom spectrum](https://tamnd.github.io/math2/node/FS2A7G2M6S.md): definition FS2A7G2M6S
- [Logarithm of a matrix](https://tamnd.github.io/math2/node/FS339MCFEQ.md): concept FS339MCFEQ
- [Primary obstruction](https://tamnd.github.io/math2/node/FS7FE0DTYZ.md): definition FS7FE0DTYZ
- [Regular scheme](https://tamnd.github.io/math2/node/FS8H2KHZ9X.md): definition FS8H2KHZ9X
- [Homogeneous coordinates](https://tamnd.github.io/math2/node/FS9FRJ17D2.md): concept FS9FRJ17D2
- [Buekenhout geometry](https://tamnd.github.io/math2/node/FSASFB5BMN.md): definition FSASFB5BMN
- [Jack function](https://tamnd.github.io/math2/node/FSCCNPWXBR.md): definition FSCCNPWXBR
- [Jacobi equation](https://tamnd.github.io/math2/node/FSD6S7DR8D.md): concept FSD6S7DR8D
- [Hartogs's extension theorem](https://tamnd.github.io/math2/node/FSGES670AZ.md): statement FSGES670AZ
- [Quantifier elimination for real closed fields](https://tamnd.github.io/math2/node/FSJ9NGRRPA.md): statement FSJ9NGRRPA
- [Fuzzy algebraic structure](https://tamnd.github.io/math2/node/FSRST924GR.md): concept FSRST924GR
- [Krull–Akizuki theorem](https://tamnd.github.io/math2/node/FSXA3CF365.md): statement FSXA3CF365
- [Functional completeness](https://tamnd.github.io/math2/node/FSZ6PCKN0F.md): definition FSZ6PCKN0F
- [Birkhoff interpolation](https://tamnd.github.io/math2/node/FT1YKVBFC9.md): concept FT1YKVBFC9
- [Tropical variety](https://tamnd.github.io/math2/node/FT2GJEHWY5.md): concept FT2GJEHWY5
- [Hopf algebra structure on cohomology](https://tamnd.github.io/math2/node/FT3ERF0YP9.md): definition FT3ERF0YP9
- [Calculus of finite differences](https://tamnd.github.io/math2/node/FT9JQA58KJ.md): concept FT9JQA58KJ
- [Polynomial long division](https://tamnd.github.io/math2/node/FTAK2F0TR3.md): definition FTAK2F0TR3
- [Inner measure](https://tamnd.github.io/math2/node/FTE966EMW8.md): concept FTE966EMW8
- [Laguerre equation](https://tamnd.github.io/math2/node/FTGTM3JB1F.md): example FTGTM3JB1F
- [Abscissa of convergence](https://tamnd.github.io/math2/node/FTHFSJ30WJ.md): concept FTHFSJ30WJ
- [Koszul algebra](https://tamnd.github.io/math2/node/FTKFCCZ8YA.md): concept FTKFCCZ8YA
- [Hermitian manifold](https://tamnd.github.io/math2/node/FTM4W0VBC8.md): definition FTM4W0VBC8
- [Infinitary logic](https://tamnd.github.io/math2/node/FTNXZ74D8N.md): definition FTNXZ74D8N
- [Quantum group](https://tamnd.github.io/math2/node/FTQDGFRRS0.md): definition FTQDGFRRS0
- [Spectral triple](https://tamnd.github.io/math2/node/FTQMQR3YS7.md): concept FTQMQR3YS7
- [Étale morphism of rings](https://tamnd.github.io/math2/node/FTRW0679E2.md): concept FTRW0679E2
- [Birth–death process](https://tamnd.github.io/math2/node/FV0E9GEPCD.md): definition FV0E9GEPCD
- [Riccati differential equation of optimal control](https://tamnd.github.io/math2/node/FV1E09CJQN.md): definition FV1E09CJQN
- [Griffiths positivity](https://tamnd.github.io/math2/node/FVCDGMNSAN.md): concept FVCDGMNSAN
- [Entourage](https://tamnd.github.io/math2/node/FVFFKNKEDF.md): concept FVFFKNKEDF
- [Topology of pointwise convergence](https://tamnd.github.io/math2/node/FVGB8ZP67B.md): concept FVGB8ZP67B
- [Gage–Hamilton–Grayson theorem](https://tamnd.github.io/math2/node/FVGW1381TZ.md): statement FVGW1381TZ
- [Boolean-valued model](https://tamnd.github.io/math2/node/FVMNKJ3DT8.md): concept FVMNKJ3DT8
- [Hahn embedding theorem](https://tamnd.github.io/math2/node/FVPXXFNVQY.md): statement FVPXXFNVQY
- [Fokker–Planck equation](https://tamnd.github.io/math2/node/FVQWR7NS00.md): concept FVQWR7NS00
- [de Rham curve](https://tamnd.github.io/math2/node/FW29W5E80Q.md): example FW29W5E80Q
- [Lambert W function](https://tamnd.github.io/math2/node/FW2JKAN14B.md): concept FW2JKAN14B
- [Positive element](https://tamnd.github.io/math2/node/FW57YYXK7Y.md): definition FW57YYXK7Y
- [Galois connection between subgroups and subfields](https://tamnd.github.io/math2/node/FW68FVNKMJ.md): definition FW68FVNKMJ
- [Kuratowski closure axioms](https://tamnd.github.io/math2/node/FW83WPH44N.md): definition FW83WPH44N
- [Casson invariant](https://tamnd.github.io/math2/node/FWF9D69R3F.md): definition FWF9D69R3F
- [Schubert calculus](https://tamnd.github.io/math2/node/FWGHMADJR4.md): concept FWGHMADJR4
- [Complete category](https://tamnd.github.io/math2/node/FWGPKXCPB7.md): concept FWGPKXCPB7
- [Almost split sequence](https://tamnd.github.io/math2/node/FWJ7CH6WG8.md): definition FWJ7CH6WG8
- [Karamata's inequality](https://tamnd.github.io/math2/node/FWKSHEJ9RP.md): statement FWKSHEJ9RP
- [Integral equation of the first kind](https://tamnd.github.io/math2/node/FWPYXMK52W.md): definition FWPYXMK52W
- [Net](https://tamnd.github.io/math2/node/FWQCZGXHWQ.md): definition FWQCZGXHWQ
- [Kleene's T predicate](https://tamnd.github.io/math2/node/FWRJV1ZABC.md): definition FWRJV1ZABC
- [Variational integrator](https://tamnd.github.io/math2/node/FWRW7BYX6T.md): definition FWRW7BYX6T
- [Little cubes operad](https://tamnd.github.io/math2/node/FWT0003QS1.md): example FWT0003QS1
- [Quasifield](https://tamnd.github.io/math2/node/FX11K6XRKV.md): definition FX11K6XRKV
- [Bayes risk](https://tamnd.github.io/math2/node/FX1A4SN6CF.md): definition FX1A4SN6CF
- [Irreducible representation](https://tamnd.github.io/math2/node/FX3NEFYFFN.md): concept FX3NEFYFFN
- [Rauch comparison theorem](https://tamnd.github.io/math2/node/FX40F4A63V.md): statement FX40F4A63V
- [Bipolar theorem](https://tamnd.github.io/math2/node/FX5PWKCMBQ.md): statement FX5PWKCMBQ
- [Freyd's adjoint functor theorem](https://tamnd.github.io/math2/node/FX88V4XW6F.md): statement FX88V4XW6F
- [Variety of groups](https://tamnd.github.io/math2/node/FX8CRCAYXZ.md): example FX8CRCAYXZ
- [Biholomorphism](https://tamnd.github.io/math2/node/FX92PP3HZR.md): concept FX92PP3HZR
- [Statistical inference](https://tamnd.github.io/math2/node/FXCQWFY5RV.md): definition FXCQWFY5RV
- [Net of a polyhedron](https://tamnd.github.io/math2/node/FXJ66Z6PQY.md): concept FXJ66Z6PQY
- [Radon measure](https://tamnd.github.io/math2/node/FXN98YEQRZ.md): definition FXN98YEQRZ
- [Discrete wavelet transform](https://tamnd.github.io/math2/node/FXSY9SXHFZ.md): definition FXSY9SXHFZ
- [Normal coordinates](https://tamnd.github.io/math2/node/FXT3N6QMEX.md): definition FXT3N6QMEX
- [Autoregressive model](https://tamnd.github.io/math2/node/FY0YG17JBE.md): definition FY0YG17JBE
- [Horner's method](https://tamnd.github.io/math2/node/FYBQ3Z52Z7.md): concept FYBQ3Z52Z7
- [Concrete category](https://tamnd.github.io/math2/node/FYBRP16KAN.md): concept FYBRP16KAN
- [Frostman lemma](https://tamnd.github.io/math2/node/FYNFT90NT2.md): statement FYNFT90NT2
- [Ext functor](https://tamnd.github.io/math2/node/FYQX9CJAXG.md): definition FYQX9CJAXG
- [Phragmén–Lindelöf principle](https://tamnd.github.io/math2/node/FYR6V2NH6C.md): statement FYR6V2NH6C
- [Morita equivalence of Lie groupoids](https://tamnd.github.io/math2/node/FYXTVSVHT2.md): concept FYXTVSVHT2
- [Lindström's theorem](https://tamnd.github.io/math2/node/FYZ3CJGZYV.md): statement FYZ3CJGZYV
- [Farkas' lemma](https://tamnd.github.io/math2/node/FYZC634H2D.md): statement FYZC634H2D
- [Semistable reduction theorem](https://tamnd.github.io/math2/node/FZ25BQTKNT.md): statement FZ25BQTKNT
- [Pseudospectral optimal control](https://tamnd.github.io/math2/node/FZ4BRYJ0YZ.md): concept FZ4BRYJ0YZ
- [Well-posed problem](https://tamnd.github.io/math2/node/FZ64956KX2.md): definition FZ64956KX2
- [Matched Z-transform method](https://tamnd.github.io/math2/node/FZ6KV2J5C9.md): definition FZ6KV2J5C9
- [Zariski tangent space](https://tamnd.github.io/math2/node/FZ6RYHNRWN.md): definition FZ6RYHNRWN
- [Baire property](https://tamnd.github.io/math2/node/FZ7237VY65.md): concept FZ7237VY65
- [Zassenhaus neighbourhood](https://tamnd.github.io/math2/node/FZ798EZ6M0.md): concept FZ798EZ6M0
- [Christoffel symbols](https://tamnd.github.io/math2/node/FZBA1HA9ZM.md): concept FZBA1HA9ZM
- [Prime decomposition of 3-manifolds](https://tamnd.github.io/math2/node/FZH5T5340Q.md): statement FZH5T5340Q
- [Trigonometric series](https://tamnd.github.io/math2/node/FZK2M8KGXQ.md): concept FZK2M8KGXQ
- [Peridynamics](https://tamnd.github.io/math2/node/FZM3CTTYVV.md): example FZM3CTTYVV
- [Character of an abelian group](https://tamnd.github.io/math2/node/FZNC7ETZA8.md): definition FZNC7ETZA8
- [Theory](https://tamnd.github.io/math2/node/FZRMYTQFRR.md): definition FZRMYTQFRR
- [Gauss's principle of least constraint](https://tamnd.github.io/math2/node/FZWAHMFZM3.md): statement FZWAHMFZM3
- [Fredholm map](https://tamnd.github.io/math2/node/FZXRWKD430.md): definition FZXRWKD430
- [Integral equation](https://tamnd.github.io/math2/node/FZYDPWMS46.md): definition FZYDPWMS46
- [Cellular approximation theorem](https://tamnd.github.io/math2/node/G003WR6SMX.md): statement G003WR6SMX
- [Simple module](https://tamnd.github.io/math2/node/G00C22AEDH.md): concept G00C22AEDH
- [Nonstandard analysis](https://tamnd.github.io/math2/node/G00EDC2ET4.md): definition G00EDC2ET4
- [Koenigs linearization theorem](https://tamnd.github.io/math2/node/G02NBNN6CJ.md): statement G02NBNN6CJ
- [Inversion in a circle](https://tamnd.github.io/math2/node/G03V3D097M.md): definition G03V3D097M
- [Ideal quotient](https://tamnd.github.io/math2/node/G05X8BHVB9.md): definition G05X8BHVB9
- [Minimum-variance unbiased estimator](https://tamnd.github.io/math2/node/G0CS8C1FJW.md): definition G0CS8C1FJW
- [Gauss–Seidel method](https://tamnd.github.io/math2/node/G0EDYEDG4S.md): statement G0EDYEDG4S
- [Least harmonic majorant](https://tamnd.github.io/math2/node/G0H97JZRS9.md): concept G0H97JZRS9
- [Wirtinger's inequality for functions](https://tamnd.github.io/math2/node/G0J13SXYTC.md): statement G0J13SXYTC
- [Frattini subgroup](https://tamnd.github.io/math2/node/G0JDQQRTGV.md): definition G0JDQQRTGV
- [Subobject](https://tamnd.github.io/math2/node/G0MYA3D9ZZ.md): definition G0MYA3D9ZZ
- [Integral closure](https://tamnd.github.io/math2/node/G0V86SJBQW.md): concept G0V86SJBQW
- [Strip of convergence of the bilateral transform](https://tamnd.github.io/math2/node/G0VTGXYHQK.md): definition G0VTGXYHQK
- [Nevanlinna theory in several variables](https://tamnd.github.io/math2/node/G0W7S1Y7XC.md): definition G0W7S1Y7XC
- [Dilogarithm](https://tamnd.github.io/math2/node/G0X0NWSDB7.md): concept G0X0NWSDB7
- [Barycentric coordinates](https://tamnd.github.io/math2/node/G0XVRV6QT5.md): concept G0XVRV6QT5
- [Butterfly effect](https://tamnd.github.io/math2/node/G0ZT1SDQGS.md): concept G0ZT1SDQGS
- [Min-max theorem](https://tamnd.github.io/math2/node/G11NE3AT6E.md): statement G11NE3AT6E
- [Fourier series](https://tamnd.github.io/math2/node/G12KXDHPPS.md): concept G12KXDHPPS
- [Smale's problems on dynamics](https://tamnd.github.io/math2/node/G14BVRT221.md): problem G14BVRT221
- [Cramér–von Mises criterion](https://tamnd.github.io/math2/node/G14XJXPKFS.md): concept G14XJXPKFS
- [Spencer cohomology](https://tamnd.github.io/math2/node/G155QN3Z4Y.md): definition G155QN3Z4Y
- [Method of steps](https://tamnd.github.io/math2/node/G1M4BGC7WK.md): definition G1M4BGC7WK
- [Orthomodular lattice](https://tamnd.github.io/math2/node/G1P8EW3Q3E.md): definition G1P8EW3Q3E
- [James's theorem](https://tamnd.github.io/math2/node/G1VWNQSAK5.md): statement G1VWNQSAK5
- [Fisher information](https://tamnd.github.io/math2/node/G1Z3WSTE8Q.md): concept G1Z3WSTE8Q
- [Section and retraction](https://tamnd.github.io/math2/node/G206NVQYVD.md): definition G206NVQYVD
- [Davis's converse of the Knaster–Tarski theorem](https://tamnd.github.io/math2/node/G20G1ECN56.md): statement G20G1ECN56
- [Sign test](https://tamnd.github.io/math2/node/G20KZ755WW.md): definition G20KZ755WW
- [Quasisymmetric map](https://tamnd.github.io/math2/node/G215XD6D4D.md): concept G215XD6D4D
- [Crossed product algebra](https://tamnd.github.io/math2/node/G21MZQCME9.md): concept G21MZQCME9
- [Post algebra](https://tamnd.github.io/math2/node/G24036DJ5K.md): definition G24036DJ5K
- [Modular representation theory](https://tamnd.github.io/math2/node/G28W7J8G2D.md): concept G28W7J8G2D
- [Dolbeault isomorphism](https://tamnd.github.io/math2/node/G2HC1SFQ4P.md): statement G2HC1SFQ4P
- [Locale](https://tamnd.github.io/math2/node/G2QNFT210W.md): concept G2QNFT210W
- [Cycle double cover conjecture](https://tamnd.github.io/math2/node/G2RYQMGG1R.md): conjecture G2RYQMGG1R
- [Monotone convergence theorem](https://tamnd.github.io/math2/node/G2S6H789RG.md): statement G2S6H789RG
- [Bayesian network](https://tamnd.github.io/math2/node/G2XY7J2Z5Q.md): definition G2XY7J2Z5Q
- [Nagell–Lutz theorem](https://tamnd.github.io/math2/node/G38B7E4ESW.md): statement G38B7E4ESW
- [Lax pair](https://tamnd.github.io/math2/node/G3A4S7CSB3.md): concept G3A4S7CSB3
- [Polynomially convex set](https://tamnd.github.io/math2/node/G3CKBSCM6A.md): definition G3CKBSCM6A
- [Bounded sequences ℓ∞](https://tamnd.github.io/math2/node/G3F2CWQSQH.md): example G3F2CWQSQH
- [Morita invariant](https://tamnd.github.io/math2/node/G3NN0YEVVG.md): definition G3NN0YEVVG
- [Sketch](https://tamnd.github.io/math2/node/G3TZAZ3XDR.md): definition G3TZAZ3XDR
- [Spectral space](https://tamnd.github.io/math2/node/G3W2WXY2VN.md): concept G3W2WXY2VN
- [Volterra operator](https://tamnd.github.io/math2/node/G3WJ3XDJ7M.md): example G3WJ3XDJ7M
- [Hirzebruch–Riemann–Roch theorem](https://tamnd.github.io/math2/node/G3X02FKQCR.md): statement G3X02FKQCR
- [Dirichlet boundary condition](https://tamnd.github.io/math2/node/G3XFT0MWKH.md): definition G3XFT0MWKH
- [Nondegenerate critical point](https://tamnd.github.io/math2/node/G43BR3YVKQ.md): definition G43BR3YVKQ
- [Cohn localization](https://tamnd.github.io/math2/node/G449HX8H25.md): concept G449HX8H25
- [Lefschetz fixed-point theorem](https://tamnd.github.io/math2/node/G44NQXCWT9.md): statement G44NQXCWT9
- [Bridge number](https://tamnd.github.io/math2/node/G44YY10VC4.md): concept G44YY10VC4
- [Erdős–Anning theorem](https://tamnd.github.io/math2/node/G4C8V6K4VW.md): statement G4C8V6K4VW
- [Banach lattice](https://tamnd.github.io/math2/node/G4CA22W2XX.md): definition G4CA22W2XX
- [Deficiency indices](https://tamnd.github.io/math2/node/G4GMSNAX8X.md): concept G4GMSNAX8X
- [Abel's integral equation](https://tamnd.github.io/math2/node/G4GWBYTW7B.md): example G4GWBYTW7B
- [Homotopy lifting property](https://tamnd.github.io/math2/node/G4QDM01P6J.md): concept G4QDM01P6J
- [Gabor frame](https://tamnd.github.io/math2/node/G4QVDEXYTV.md): definition G4QVDEXYTV
- [Hilbert's inequality](https://tamnd.github.io/math2/node/G4R4MWWP3C.md): statement G4R4MWWP3C
- [Hurwitz zeta function](https://tamnd.github.io/math2/node/G4SW7HJG88.md): concept G4SW7HJG88
- [Art gallery theorem](https://tamnd.github.io/math2/node/G4WSPQJ6W1.md): statement G4WSPQJ6W1
- [Residue field of a point](https://tamnd.github.io/math2/node/G51E8MW7J1.md): concept G51E8MW7J1
- [Coherent sheaf](https://tamnd.github.io/math2/node/G596MHC68T.md): definition G596MHC68T
- [Lattice-ordered group](https://tamnd.github.io/math2/node/G5AN43WE9D.md): definition G5AN43WE9D
- [Hodge Laplacian](https://tamnd.github.io/math2/node/G5FAPAB2G2.md): definition G5FAPAB2G2
- [Thie's theorem on multiplicity](https://tamnd.github.io/math2/node/G5JKJEAKZ9.md): statement G5JKJEAKZ9
- [Sequence space](https://tamnd.github.io/math2/node/G5MKMRTT62.md): definition G5MKMRTT62
- [Proof that groups of prime order are cyclic](https://tamnd.github.io/math2/node/G5PTXQ0QK3.md): proof G5PTXQ0QK3
- [Purely transcendental extension](https://tamnd.github.io/math2/node/G5PVJMM37Z.md): concept G5PVJMM37Z
- [Upwind scheme](https://tamnd.github.io/math2/node/G5RB8T2WKD.md): definition G5RB8T2WKD
- [Group](https://tamnd.github.io/math2/node/G5S1M9.md): concept G5S1M9
- [Ramsey theory](https://tamnd.github.io/math2/node/G5TWXV565S.md): concept G5TWXV565S
- [Convergence in measure](https://tamnd.github.io/math2/node/G5WT6QN8WR.md): concept G5WT6QN8WR
- [Set of uniqueness](https://tamnd.github.io/math2/node/G653QANYYD.md): definition G653QANYYD
- [Coxeter group](https://tamnd.github.io/math2/node/G6FND7X69J.md): concept G6FND7X69J
- [Affine group](https://tamnd.github.io/math2/node/G6GMSEZDB1.md): concept G6GMSEZDB1
- [Singular perturbation](https://tamnd.github.io/math2/node/G6J21TGRMM.md): concept G6J21TGRMM
- [Martin kernel](https://tamnd.github.io/math2/node/G6S707SYF9.md): definition G6S707SYF9
- [Kruskal–Katona theorem](https://tamnd.github.io/math2/node/G6WNYB0WMY.md): statement G6WNYB0WMY
- [Second-order logic](https://tamnd.github.io/math2/node/G6X7SZSNC5.md): concept G6X7SZSNC5
- [Inner model](https://tamnd.github.io/math2/node/G78W36EGMB.md): concept G78W36EGMB
- [Rotation number](https://tamnd.github.io/math2/node/G7D6SPTA2C.md): definition G7D6SPTA2C
- [abc conjecture](https://tamnd.github.io/math2/node/G7JH2NXPC0.md): conjecture G7JH2NXPC0
- [Quasi-algebraically closed field](https://tamnd.github.io/math2/node/G7RM16N4PF.md): concept G7RM16N4PF
- [Admissible function](https://tamnd.github.io/math2/node/G7T008BEWY.md): concept G7T008BEWY
- [Hadamard finite part](https://tamnd.github.io/math2/node/G7VE41JHHH.md): concept G7VE41JHHH
- [Hausdorff moment problem](https://tamnd.github.io/math2/node/G7VG61YD4J.md): problem G7VG61YD4J
- [Projection body](https://tamnd.github.io/math2/node/G7WP728SZH.md): concept G7WP728SZH
- [Klein–Gordon equation](https://tamnd.github.io/math2/node/G82RA93DWJ.md): definition G82RA93DWJ
- [Formal concept analysis](https://tamnd.github.io/math2/node/G83JY7CY7Q.md): concept G83JY7CY7Q
- [Bohr compactification](https://tamnd.github.io/math2/node/G852FHZ3W1.md): concept G852FHZ3W1
- [Quasi-category](https://tamnd.github.io/math2/node/G87CWTXAWC.md): definition G87CWTXAWC
- [Lifting property](https://tamnd.github.io/math2/node/G87F5M6TXV.md): definition G87F5M6TXV
- [Exceptional collection](https://tamnd.github.io/math2/node/G8BQ5THKRZ.md): concept G8BQ5THKRZ
- [Calderón projector](https://tamnd.github.io/math2/node/G8DBMWQSDB.md): definition G8DBMWQSDB
- [Energy method](https://tamnd.github.io/math2/node/G8J0YQRPA3.md): concept G8J0YQRPA3
- [Krasner's lemma](https://tamnd.github.io/math2/node/G8PGN0WQHW.md): statement G8PGN0WQHW
- [Modal logic](https://tamnd.github.io/math2/node/G8VGTW9P4F.md): definition G8VGTW9P4F
- [Stiffness matrix](https://tamnd.github.io/math2/node/G8ZTW9JVX1.md): concept G8ZTW9JVX1
- [Constructivism](https://tamnd.github.io/math2/node/G958EQ8CWW.md): concept G958EQ8CWW
- [Cylinder set measure](https://tamnd.github.io/math2/node/G95YDNFXQN.md): concept G95YDNFXQN
- [Kirillov orbit method](https://tamnd.github.io/math2/node/G9AW6R1D2F.md): concept G9AW6R1D2F
- [Proof methods](https://tamnd.github.io/math2/node/G9JFJGY4TQ.md): topic G9JFJGY4TQ
- [Balayage space](https://tamnd.github.io/math2/node/G9M1H5NNF6.md): concept G9M1H5NNF6
- [Complete local ring](https://tamnd.github.io/math2/node/G9Q61B8A7C.md): concept G9Q61B8A7C
- [Mohr–Mascheroni theorem](https://tamnd.github.io/math2/node/G9VT2J3ZVK.md): statement G9VT2J3ZVK
- [Coordinate ring](https://tamnd.github.io/math2/node/G9WB27VWP7.md): concept G9WB27VWP7
- [System of linear integral equations](https://tamnd.github.io/math2/node/G9WR1GDE7X.md): concept G9WR1GDE7X
- [Exact category in the sense of Barr](https://tamnd.github.io/math2/node/G9ZNDNT3YY.md): concept G9ZNDNT3YY
- [Laguerre geometry](https://tamnd.github.io/math2/node/GA0NCJMZHB.md): concept GA0NCJMZHB
- [Yule–Walker equations](https://tamnd.github.io/math2/node/GA46NC0CJZ.md): statement GA46NC0CJZ
- [Differential field](https://tamnd.github.io/math2/node/GA5PTQGC6G.md): concept GA5PTQGC6G
- [Connection on a principal bundle](https://tamnd.github.io/math2/node/GA7437914V.md): definition GA7437914V
- [Unconditional convergence](https://tamnd.github.io/math2/node/GA7Q813F50.md): definition GA7Q813F50
- [Linear algebraic group](https://tamnd.github.io/math2/node/GA81VVQHEP.md): concept GA81VVQHEP
- [Posner's theorem](https://tamnd.github.io/math2/node/GA8BDBHP4M.md): statement GA8BDBHP4M
- [Period and index](https://tamnd.github.io/math2/node/GAD8ZE5MPV.md): definition GAD8ZE5MPV
- [Pro-object](https://tamnd.github.io/math2/node/GAFQDGF6FM.md): concept GAFQDGF6FM
- [Fisher's equation](https://tamnd.github.io/math2/node/GAJDYXNHN2.md): definition GAJDYXNHN2
- [Pointwise Kan extension formula](https://tamnd.github.io/math2/node/GAMHZY8TWV.md): statement GAMHZY8TWV
- [Poisson summation formula](https://tamnd.github.io/math2/node/GARGJYSGFG.md): statement GARGJYSGFG
- [Germ of a smooth function](https://tamnd.github.io/math2/node/GAW3VCFYZV.md): definition GAW3VCFYZV
- [Holomorphic fiber bundle](https://tamnd.github.io/math2/node/GB0D4XMZV3.md): concept GB0D4XMZV3
- [Matrix completion](https://tamnd.github.io/math2/node/GB7K3PTPAR.md): problem GB7K3PTPAR
- [Complex analytic variety](https://tamnd.github.io/math2/node/GB7M582M5Y.md): concept GB7M582M5Y
- [p-adic integer](https://tamnd.github.io/math2/node/GBN9F8FAQ3.md): definition GBN9F8FAQ3
- [Ring of differential operators](https://tamnd.github.io/math2/node/GBWQF3HZG0.md): concept GBWQF3HZG0
- [Trigamma function](https://tamnd.github.io/math2/node/GBXD38Y52R.md): definition GBXD38Y52R
- [Partial differential equation](https://tamnd.github.io/math2/node/GC5BDZ6QBC.md): concept GC5BDZ6QBC
- [Lie group action](https://tamnd.github.io/math2/node/GC7T4VW3N1.md): definition GC7T4VW3N1
- [Inverse transform](https://tamnd.github.io/math2/node/GC96MQC0PP.md): concept GC96MQC0PP
- [Brown representability theorem](https://tamnd.github.io/math2/node/GC9FN7TTBV.md): statement GC9FN7TTBV
- [Action-angle coordinates](https://tamnd.github.io/math2/node/GC9R87YZ6A.md): definition GC9R87YZ6A
- [Tubular neighborhood theorem](https://tamnd.github.io/math2/node/GCBY80CSC6.md): statement GCBY80CSC6
- [Conjugate function](https://tamnd.github.io/math2/node/GCD0WZ0VKZ.md): definition GCD0WZ0VKZ
- [Hadamard's maximal determinant problem](https://tamnd.github.io/math2/node/GCH1Z9YFNG.md): problem GCH1Z9YFNG
- [Skolem normal form](https://tamnd.github.io/math2/node/GCPQGMJYEV.md): concept GCPQGMJYEV
- [Convex independence](https://tamnd.github.io/math2/node/GCPRFTKS4E.md): concept GCPRFTKS4E
- [Ostrowski's theorem](https://tamnd.github.io/math2/node/GCQ23XT6VD.md): statement GCQ23XT6VD
- [Contact structure on a jet bundle](https://tamnd.github.io/math2/node/GCS702QT54.md): concept GCS702QT54
- [Indeterminate moment problem](https://tamnd.github.io/math2/node/GCTN4BZ1ET.md): concept GCTN4BZ1ET
- [Slutsky's theorem](https://tamnd.github.io/math2/node/GCV10B5HES.md): statement GCV10B5HES
- [Hyperbolic triangle](https://tamnd.github.io/math2/node/GCZ8EEDCHS.md): concept GCZ8EEDCHS
- [Free probability](https://tamnd.github.io/math2/node/GD3Q8G9GCV.md): definition GD3Q8G9GCV
- [d'Alembert's principle](https://tamnd.github.io/math2/node/GD776GVZSR.md): statement GD776GVZSR
- [Bromwich justification of the Heaviside calculus](https://tamnd.github.io/math2/node/GD8B883592.md): concept GD8B883592
- [Real analytic manifold](https://tamnd.github.io/math2/node/GD8FNE3GTF.md): definition GD8FNE3GTF
- [Exponential object](https://tamnd.github.io/math2/node/GD9JMCZHNY.md): definition GD9JMCZHNY
- [Uniqueness of identity element](https://tamnd.github.io/math2/node/GDBV09GFYB.md): statement GDBV09GFYB
- [List coloring](https://tamnd.github.io/math2/node/GDN82Y3ATR.md): definition GDN82Y3ATR
- [Grothendieck topology](https://tamnd.github.io/math2/node/GDRFSJ8E2B.md): concept GDRFSJ8E2B
- [Strictly convex space](https://tamnd.github.io/math2/node/GDTKW9KPH8.md): concept GDTKW9KPH8
- [Universal cover](https://tamnd.github.io/math2/node/GDTQD3J2RH.md): definition GDTQD3J2RH
- [Fast wavelet transform](https://tamnd.github.io/math2/node/GDW2SR7TSH.md): definition GDW2SR7TSH
- [Ham sandwich theorem](https://tamnd.github.io/math2/node/GDXX60SNG5.md): statement GDXX60SNG5
- [Stochastic geometry](https://tamnd.github.io/math2/node/GDY62GEC1H.md): definition GDY62GEC1H
- [Schwartz space](https://tamnd.github.io/math2/node/GE020MY43M.md): definition GE020MY43M
- [Analytic semigroup](https://tamnd.github.io/math2/node/GE05N4B84Q.md): definition GE05N4B84Q
- [Fabry gap theorem](https://tamnd.github.io/math2/node/GE0HX2KMMZ.md): statement GE0HX2KMMZ
- [Polarity](https://tamnd.github.io/math2/node/GE75S5XZ9T.md): definition GE75S5XZ9T
- [Submultiplicative function](https://tamnd.github.io/math2/node/GE8MRV41R7.md): concept GE8MRV41R7
- [Integral of a non-negative function](https://tamnd.github.io/math2/node/GEA8H3228S.md): definition GEA8H3228S
- [Steffensen's method](https://tamnd.github.io/math2/node/GEFBDG5CJ3.md): statement GEFBDG5CJ3
- [Nonlinear Schrödinger equation](https://tamnd.github.io/math2/node/GEJHRXC176.md): concept GEJHRXC176
- [Filtration of a module](https://tamnd.github.io/math2/node/GEJYZEY7Y0.md): definition GEJYZEY7Y0
- [n-term approximation](https://tamnd.github.io/math2/node/GEK0NP5WQM.md): concept GEK0NP5WQM
- [Tanaka's formula](https://tamnd.github.io/math2/node/GEM0ERJ5ZB.md): statement GEM0ERJ5ZB
- [Nielsen fixed-point theory](https://tamnd.github.io/math2/node/GEQV0DZS3W.md): definition GEQV0DZS3W
- [Pythagorean field](https://tamnd.github.io/math2/node/GER28JFXMR.md): definition GER28JFXMR
- [Fractional Laplacian](https://tamnd.github.io/math2/node/GETN4MCAAN.md): definition GETN4MCAAN
- [Caustic](https://tamnd.github.io/math2/node/GEXSRZ7JEH.md): definition GEXSRZ7JEH
- [Rosenberg's classification of maximal clones](https://tamnd.github.io/math2/node/GF1RK91VED.md): statement GF1RK91VED
- [Approximately finite-dimensional C*-algebra](https://tamnd.github.io/math2/node/GF3MD842NZ.md): definition GF3MD842NZ
- [Homogeneous integral equation](https://tamnd.github.io/math2/node/GF40N88TVM.md): definition GF40N88TVM
- [Sequential space](https://tamnd.github.io/math2/node/GFADRKMQ51.md): definition GFADRKMQ51
- [Open immersion](https://tamnd.github.io/math2/node/GFC5YKSN5H.md): concept GFC5YKSN5H
- [Appell series](https://tamnd.github.io/math2/node/GFJ20FBYF8.md): definition GFJ20FBYF8
- [Martin representation theorem](https://tamnd.github.io/math2/node/GFJE824E93.md): statement GFJE824E93
- [Amenable group action](https://tamnd.github.io/math2/node/GFNJKVWQBB.md): definition GFNJKVWQBB
- [Kramers–Kronig relations](https://tamnd.github.io/math2/node/GFS95PZ5YK.md): statement GFS95PZ5YK
- [q-analog](https://tamnd.github.io/math2/node/GFSDMJHHVA.md): concept GFSDMJHHVA
- [Morin singularities](https://tamnd.github.io/math2/node/GFTQRDW607.md): definition GFTQRDW607
- [Dimension theory](https://tamnd.github.io/math2/node/GFVWEATKGE.md): concept GFVWEATKGE
- [Fredholm map](https://tamnd.github.io/math2/node/GFXQ6HEM65.md): concept GFXQ6HEM65
- [Latent variable model](https://tamnd.github.io/math2/node/GFZD5Z7PT2.md): definition GFZD5Z7PT2
- [Chern–Gauss–Bonnet theorem](https://tamnd.github.io/math2/node/GG6NYPKMHX.md): statement GG6NYPKMHX
- [Spectral sequence](https://tamnd.github.io/math2/node/GG8AYV847V.md): concept GG8AYV847V
- [Congruence-distributive variety](https://tamnd.github.io/math2/node/GGBJDS5AV9.md): definition GGBJDS5AV9
- [Separable closure](https://tamnd.github.io/math2/node/GGCHBSY4HK.md): definition GGCHBSY4HK
- [Iterative functional equation](https://tamnd.github.io/math2/node/GGDJN5KXN4.md): definition GGDJN5KXN4
- [Partial isometry](https://tamnd.github.io/math2/node/GGDSF85JPD.md): concept GGDSF85JPD
- [Grothendieck topos](https://tamnd.github.io/math2/node/GGE4KYWCNT.md): definition GGE4KYWCNT
- [Talagrand's concentration inequality](https://tamnd.github.io/math2/node/GGEAGW4RRV.md): statement GGEAGW4RRV
- [Nagata's criterion for factoriality](https://tamnd.github.io/math2/node/GGEMJF449D.md): statement GGEMJF449D
- [Atiyah–Patodi–Singer index theorem](https://tamnd.github.io/math2/node/GGFAH1M80J.md): statement GGFAH1M80J
- [Reeb vector field](https://tamnd.github.io/math2/node/GGM63KXCF2.md): concept GGM63KXCF2
- [Algebraic Riccati equation](https://tamnd.github.io/math2/node/GGZV75ZZMV.md): concept GGZV75ZZMV
- [Digital topology](https://tamnd.github.io/math2/node/GH2SFDQNAE.md): definition GH2SFDQNAE
- [Pseudo-Riemannian manifold](https://tamnd.github.io/math2/node/GH6CXWFNDJ.md): concept GH6CXWFNDJ
- [Algebraic group](https://tamnd.github.io/math2/node/GH81SYAPPN.md): concept GH81SYAPPN
- [Incompressible surface](https://tamnd.github.io/math2/node/GHEWGCECQJ.md): definition GHEWGCECQJ
- [Hartogs figure](https://tamnd.github.io/math2/node/GHJC53RFBT.md): example GHJC53RFBT
- [Higman's lemma](https://tamnd.github.io/math2/node/GHK66XE1S8.md): statement GHK66XE1S8
- [Going-up theorem](https://tamnd.github.io/math2/node/GHKYR5PBQC.md): statement GHKYR5PBQC
- [Linear system of divisors](https://tamnd.github.io/math2/node/GHRPE8CXJ9.md): concept GHRPE8CXJ9
- [Hyperbolic 3-manifold](https://tamnd.github.io/math2/node/GHVW1Z35BS.md): concept GHVW1Z35BS
- [Nielsen fixed-point theory](https://tamnd.github.io/math2/node/GHXMVWN3W5.md): concept GHXMVWN3W5
- [Erdős–Rényi model](https://tamnd.github.io/math2/node/GJ4J77VV60.md): definition GJ4J77VV60
- [Prime knot](https://tamnd.github.io/math2/node/GJ6FJ4G5VC.md): concept GJ6FJ4G5VC
- [von Neumann's minimax theorem](https://tamnd.github.io/math2/node/GJ8VTBFNP2.md): statement GJ8VTBFNP2
- [Lattice of subvarieties](https://tamnd.github.io/math2/node/GJAYCVYZYF.md): concept GJAYCVYZYF
- [Z-group](https://tamnd.github.io/math2/node/GJBAX2R2PG.md): definition GJBAX2R2PG
- [Behnke–Stein theorem](https://tamnd.github.io/math2/node/GJD05AZWQN.md): statement GJD05AZWQN
- [Generic matrix algebra](https://tamnd.github.io/math2/node/GJGEVMZ828.md): concept GJGEVMZ828
- [Strict Henselization](https://tamnd.github.io/math2/node/GJHAF85F4S.md): concept GJHAF85F4S
- [Curve complex](https://tamnd.github.io/math2/node/GJM3PAVV96.md): definition GJM3PAVV96
- [Gerstenhaber algebra](https://tamnd.github.io/math2/node/GJPPS5N213.md): concept GJPPS5N213
- [Numerical methods for ordinary differential equations](https://tamnd.github.io/math2/node/GJQVTDHKM2.md): definition GJQVTDHKM2
- [Cointegration](https://tamnd.github.io/math2/node/GJVBTYF9YM.md): definition GJVBTYF9YM
- [Riemann problem](https://tamnd.github.io/math2/node/GJWKX22VVE.md): problem GJWKX22VVE
- [Bang–bang control](https://tamnd.github.io/math2/node/GK1Z3Q1WRZ.md): concept GK1Z3Q1WRZ
- [Rank-one convexity](https://tamnd.github.io/math2/node/GK4P8G0PB2.md): concept GK4P8G0PB2
- [Mixed-type equation](https://tamnd.github.io/math2/node/GKFDKRHEP4.md): definition GKFDKRHEP4
- [Tessellation](https://tamnd.github.io/math2/node/GKJK293DXS.md): concept GKJK293DXS
- [Fundamental theorem of projective geometry](https://tamnd.github.io/math2/node/GKV4RVXBRE.md): statement GKV4RVXBRE
- [Stiefel–Whitney class](https://tamnd.github.io/math2/node/GKWC4EASX9.md): definition GKWC4EASX9
- [Toeplitz matrix](https://tamnd.github.io/math2/node/GKY2WSHE26.md): concept GKY2WSHE26
- [Automatic group](https://tamnd.github.io/math2/node/GMPJ88PAQE.md): definition GMPJ88PAQE
- [Caustic](https://tamnd.github.io/math2/node/GMRYG4S3G0.md): concept GMRYG4S3G0
- [Szegő kernel](https://tamnd.github.io/math2/node/GMSKDPXWPH.md): definition GMSKDPXWPH
- [Risch algorithm](https://tamnd.github.io/math2/node/GMY8XWETFV.md): definition GMY8XWETFV
- [Levi's extension theorem for meromorphic functions](https://tamnd.github.io/math2/node/GMYDMZMN1S.md): statement GMYDMZMN1S
- [Cellular algebra](https://tamnd.github.io/math2/node/GMYSQTH3Z7.md): definition GMYSQTH3Z7
- [Stratified sampling](https://tamnd.github.io/math2/node/GN02JVADH4.md): definition GN02JVADH4
- [Hyperbolic geometry](https://tamnd.github.io/math2/node/GN4N94GQ0Z.md): concept GN4N94GQ0Z
- [Proof of Extension of Lagrange's theorem](https://tamnd.github.io/math2/node/GN6WHHGBAE.md): proof GN6WHHGBAE
- [IEEE 754 standard](https://tamnd.github.io/math2/node/GNDWPYYFW4.md): concept GNDWPYYFW4
- [Group law on an elliptic curve](https://tamnd.github.io/math2/node/GNEGMN4E26.md): definition GNEGMN4E26
- [Reduction of a first-kind Volterra equation to the second kind](https://tamnd.github.io/math2/node/GNH0Y5VDE5.md): statement GNH0Y5VDE5
- [Prime end](https://tamnd.github.io/math2/node/GNWE3XR0ZG.md): concept GNWE3XR0ZG
- [Involution](https://tamnd.github.io/math2/node/GP56NBN9RY.md): concept GP56NBN9RY
- [Chebyshev series](https://tamnd.github.io/math2/node/GP98BJP4JH.md): concept GP98BJP4JH
- [Superlinear convergence of the secant method](https://tamnd.github.io/math2/node/GP9E4AWJ8W.md): statement GP9E4AWJ8W
- [Shrinkage estimator](https://tamnd.github.io/math2/node/GPBSS1V3SS.md): concept GPBSS1V3SS
- [Representability of cohomology by Eilenberg–MacLane spaces](https://tamnd.github.io/math2/node/GPDPYHM3RE.md): statement GPDPYHM3RE
- [Continuity principle](https://tamnd.github.io/math2/node/GPJC1NY5RE.md): statement GPJC1NY5RE
- [Conditioned Brownian motion](https://tamnd.github.io/math2/node/GPK4F1K971.md): definition GPK4F1K971
- [Gelfand pair](https://tamnd.github.io/math2/node/GPQ5M0SA47.md): definition GPQ5M0SA47
- [Fefferman's expansion of the Bergman kernel](https://tamnd.github.io/math2/node/GPQ9JNSDPA.md): statement GPQ9JNSDPA
- [Intersection multiplicity](https://tamnd.github.io/math2/node/GPQF9GK4T9.md): concept GPQF9GK4T9
- [Inoue surface](https://tamnd.github.io/math2/node/GPSVJK3R4X.md): example GPSVJK3R4X
- [Retract](https://tamnd.github.io/math2/node/GPWYZX03N8.md): definition GPWYZX03N8
- [Toeplitz operator](https://tamnd.github.io/math2/node/GQ0X3R3WM2.md): concept GQ0X3R3WM2
- [Lamé equations of elasticity](https://tamnd.github.io/math2/node/GQ3K59MRJP.md): concept GQ3K59MRJP
- [Fourier transform of tempered distributions](https://tamnd.github.io/math2/node/GQ559QSZ2T.md): definition GQ559QSZ2T
- [Honeycomb](https://tamnd.github.io/math2/node/GQB3AS4WKQ.md): definition GQB3AS4WKQ
- [Goddard problem](https://tamnd.github.io/math2/node/GQCN7KRSDP.md): problem GQCN7KRSDP
- [Family-wise error rate](https://tamnd.github.io/math2/node/GQCY1N8JZT.md): concept GQCY1N8JZT
- [Time–frequency analysis](https://tamnd.github.io/math2/node/GQFPVXEWK1.md): concept GQFPVXEWK1
- [Algebraic dimension](https://tamnd.github.io/math2/node/GQG0VMXER2.md): definition GQG0VMXER2
- [Generalized quadrangle](https://tamnd.github.io/math2/node/GQJCPFSYVK.md): concept GQJCPFSYVK
- [Relation algebra](https://tamnd.github.io/math2/node/GQK4DSA11K.md): concept GQK4DSA11K
- [General linear model](https://tamnd.github.io/math2/node/GQM8GPBRYN.md): definition GQM8GPBRYN
- [Differential graded algebra](https://tamnd.github.io/math2/node/GQVAXW5X09.md): concept GQVAXW5X09
- [Circle packing theorem](https://tamnd.github.io/math2/node/GR1DDT1B6B.md): statement GR1DDT1B6B
- [Interacting particle system](https://tamnd.github.io/math2/node/GR49AM3PSZ.md): concept GR49AM3PSZ
- [Probit model](https://tamnd.github.io/math2/node/GR4FN6XNWP.md): definition GR4FN6XNWP
- [Finite element exterior calculus](https://tamnd.github.io/math2/node/GRBZT7SWK0.md): definition GRBZT7SWK0
- [Spectrum of the Laplace–Beltrami operator](https://tamnd.github.io/math2/node/GRC1E284T7.md): definition GRC1E284T7
- [Affine group](https://tamnd.github.io/math2/node/GRCEW24985.md): definition GRCEW24985
- [Input-to-state stability](https://tamnd.github.io/math2/node/GRDCEGQJYQ.md): concept GRDCEGQJYQ
- [Full and faithful functors](https://tamnd.github.io/math2/node/GRG8WEBSH5.md): concept GRG8WEBSH5
- [Residual set](https://tamnd.github.io/math2/node/GRH8MCVN3N.md): concept GRH8MCVN3N
- [Atomic lattice](https://tamnd.github.io/math2/node/GRMT65QJAY.md): definition GRMT65QJAY
- [Szász–Mirakyan operator](https://tamnd.github.io/math2/node/GRQJ49XQNR.md): concept GRQJ49XQNR
- [(∞,1)-category](https://tamnd.github.io/math2/node/GRQM6KXA00.md): definition GRQM6KXA00
- [Bessel function of the second kind](https://tamnd.github.io/math2/node/GRSNDJG363.md): concept GRSNDJG363
- [Morita invariant](https://tamnd.github.io/math2/node/GRWE9JJ5CT.md): concept GRWE9JJ5CT
- [Weyl–Titchmarsh theory](https://tamnd.github.io/math2/node/GRZAY7GWC5.md): definition GRZAY7GWC5
- [Polygamma function](https://tamnd.github.io/math2/node/GRZKGHMNTG.md): definition GRZKGHMNTG
- [Initial topology](https://tamnd.github.io/math2/node/GRZKVSMJXF.md): definition GRZKVSMJXF
- [Neumann boundary condition](https://tamnd.github.io/math2/node/GS3KT2H645.md): concept GS3KT2H645
- [Weyl's criterion](https://tamnd.github.io/math2/node/GS4RC7A7DJ.md): statement GS4RC7A7DJ
- [Adem relations](https://tamnd.github.io/math2/node/GS8SCGYPX7.md): statement GS8SCGYPX7
- [Universally measurable set](https://tamnd.github.io/math2/node/GSFXAV0VD7.md): concept GSFXAV0VD7
- [Kernel of an integral equation](https://tamnd.github.io/math2/node/GSMBXT7FPK.md): concept GSMBXT7FPK
- [Lusternik–Schnirelmann theory of critical points](https://tamnd.github.io/math2/node/GSV68DRC1D.md): definition GSV68DRC1D
- [Hyers–Ulam stability](https://tamnd.github.io/math2/node/GSVBM4F91E.md): definition GSVBM4F91E
- [Moments and orthogonal polynomials](https://tamnd.github.io/math2/node/GSY904FM0A.md): concept GSY904FM0A
- [Dilworth's theorem](https://tamnd.github.io/math2/node/GT224XDA4E.md): statement GT224XDA4E
- [Gromov–Hausdorff convergence](https://tamnd.github.io/math2/node/GT56RTG9X5.md): concept GT56RTG9X5
- [Hyperstructure](https://tamnd.github.io/math2/node/GT687SNZMQ.md): definition GT687SNZMQ
- [Polar set](https://tamnd.github.io/math2/node/GT9E1NT6KC.md): definition GT9E1NT6KC
- [Universal coefficient theorem](https://tamnd.github.io/math2/node/GTA8ZFGZ3N.md): statement GTA8ZFGZ3N
- [Linear discriminant analysis](https://tamnd.github.io/math2/node/GTBMW9QXST.md): definition GTBMW9QXST
- [Friedrichs extension](https://tamnd.github.io/math2/node/GTBXZXK146.md): concept GTBXZXK146
- [M/M/1 queue](https://tamnd.github.io/math2/node/GTFRPF5J2K.md): example GTFRPF5J2K
- [Property (T)](https://tamnd.github.io/math2/node/GTFZ277047.md): concept GTFZ277047
- [Triangular matrix](https://tamnd.github.io/math2/node/GTG8MJGDKC.md): concept GTG8MJGDKC
- [Normal closure](https://tamnd.github.io/math2/node/GTGFXW37YM.md): definition GTGFXW37YM
- [Crystallographic group](https://tamnd.github.io/math2/node/GTHWVEB1NZ.md): definition GTHWVEB1NZ
- [Zeta-regularized determinant](https://tamnd.github.io/math2/node/GTJ6KBD4PX.md): definition GTJ6KBD4PX
- [Submersion](https://tamnd.github.io/math2/node/GTMT5ZFV00.md): definition GTMT5ZFV00
- [Line graph](https://tamnd.github.io/math2/node/GTN3QQYTT5.md): concept GTN3QQYTT5
- [Cerf theory](https://tamnd.github.io/math2/node/GTS394XJ7Y.md): definition GTS394XJ7Y
- [Free algebra in a variety](https://tamnd.github.io/math2/node/GTS4DRRMZR.md): concept GTS4DRRMZR
- [Dunkl operator](https://tamnd.github.io/math2/node/GTVCECGH40.md): concept GTVCECGH40
- [Conformal geometry](https://tamnd.github.io/math2/node/GTVQFG2SH9.md): concept GTVQFG2SH9
- [Dedekind-finite ring](https://tamnd.github.io/math2/node/GV1HKEAWVN.md): concept GV1HKEAWVN
- [Analytic signal](https://tamnd.github.io/math2/node/GVCRZH15Y0.md): concept GVCRZH15Y0
- [Diffusion process](https://tamnd.github.io/math2/node/GVMM9GC7CE.md): definition GVMM9GC7CE
- [Prime ideal in a noncommutative ring](https://tamnd.github.io/math2/node/GVN1MS9PEJ.md): definition GVN1MS9PEJ
- [Smale's degree for proper Fredholm maps](https://tamnd.github.io/math2/node/GVR54SZFX4.md): concept GVR54SZFX4
- [Floquet multiplier](https://tamnd.github.io/math2/node/GVSY2FZ8YW.md): definition GVSY2FZ8YW
- [Several complex variables](https://tamnd.github.io/math2/node/GVVG201079.md): definition GVVG201079
- [Index of a singular integral operator](https://tamnd.github.io/math2/node/GW3B4K1WA5.md): concept GW3B4K1WA5
- [Arithmetical variety](https://tamnd.github.io/math2/node/GW78A05TNA.md): concept GW78A05TNA
- [Floquet theory](https://tamnd.github.io/math2/node/GW8SBF37K1.md): definition GW8SBF37K1
- [Global dimension](https://tamnd.github.io/math2/node/GWA0T1WKCN.md): concept GWA0T1WKCN
- [Space of convergent sequences](https://tamnd.github.io/math2/node/GWCBGYG5RA.md): example GWCBGYG5RA
- [Bohr's approximation theorem](https://tamnd.github.io/math2/node/GWCY1C1KQV.md): statement GWCY1C1KQV
- [Analysis of variance](https://tamnd.github.io/math2/node/GWGFT35TCZ.md): definition GWGFT35TCZ
- [Enough projectives](https://tamnd.github.io/math2/node/GWGGZBPF60.md): definition GWGGZBPF60
- [Hironaka's resolution for complex spaces](https://tamnd.github.io/math2/node/GWJWP187QE.md): statement GWJWP187QE
- [Modular form](https://tamnd.github.io/math2/node/GWKN5VG78A.md): concept GWKN5VG78A
- [Club set](https://tamnd.github.io/math2/node/GWS5QXH7WW.md): definition GWS5QXH7WW
- [Second main theorem of Nevanlinna theory](https://tamnd.github.io/math2/node/GWYE6745JK.md): statement GWYE6745JK
- [Stallings theorem about ends of groups](https://tamnd.github.io/math2/node/GWZN3EKEWD.md): statement GWZN3EKEWD
- [Singularity theory](https://tamnd.github.io/math2/node/GX68S0ZH35.md): definition GX68S0ZH35
- [de Rham cohomology](https://tamnd.github.io/math2/node/GX74C07T0G.md): concept GX74C07T0G
- [Image of J](https://tamnd.github.io/math2/node/GX8K2SCWVQ.md): concept GX8K2SCWVQ
- [Sinai billiard](https://tamnd.github.io/math2/node/GX9Q2PYNZE.md): example GX9Q2PYNZE
- [Closed graph theorem](https://tamnd.github.io/math2/node/GXM8BX8YM7.md): statement GXM8BX8YM7
- [Obstacle problem](https://tamnd.github.io/math2/node/GXN2NEHC8X.md): problem GXN2NEHC8X
- [Nullary operation](https://tamnd.github.io/math2/node/GXNE1EPK0S.md): definition GXNE1EPK0S
- [Bayesian information criterion](https://tamnd.github.io/math2/node/GXP6H9VGRB.md): concept GXP6H9VGRB
- [Cyclic algebra](https://tamnd.github.io/math2/node/GXPDK5Q0HA.md): definition GXPDK5Q0HA
- [Integration in the transform variable](https://tamnd.github.io/math2/node/GXX2WPVRJ2.md): statement GXX2WPVRJ2
- [Klein quartic](https://tamnd.github.io/math2/node/GXYKMAEAMG.md): object GXYKMAEAMG
- [Unramified morphism of rings](https://tamnd.github.io/math2/node/GY3QA6NH0E.md): concept GY3QA6NH0E
- [Quantum dilogarithm](https://tamnd.github.io/math2/node/GY5EEKHA45.md): definition GY5EEKHA45
- [De Bruijn sequence](https://tamnd.github.io/math2/node/GY8F3M39R8.md): object GY8F3M39R8
- [Virtually Haken conjecture](https://tamnd.github.io/math2/node/GYB411M966.md): statement GYB411M966
- [Conformal radius](https://tamnd.github.io/math2/node/GYEX4BH5WM.md): concept GYEX4BH5WM
- [Broyden's method](https://tamnd.github.io/math2/node/GYNYT7KTC3.md): statement GYNYT7KTC3
- [Krohn–Rhodes theory](https://tamnd.github.io/math2/node/GYR1XYFEZB.md): definition GYR1XYFEZB
- [Pseudo-Anosov map](https://tamnd.github.io/math2/node/GYW7GYABD9.md): concept GYW7GYABD9
- [Quasi-Frobenius ring](https://tamnd.github.io/math2/node/GYXTW7PF56.md): definition GYXTW7PF56
- [Rational surface](https://tamnd.github.io/math2/node/GZ3XA1V7CJ.md): definition GZ3XA1V7CJ
- [Almost periodic function](https://tamnd.github.io/math2/node/GZBSV8DYGS.md): concept GZBSV8DYGS
- [Inverse problem](https://tamnd.github.io/math2/node/GZDAPBG06Q.md): definition GZDAPBG06Q
- [Cheeger constant](https://tamnd.github.io/math2/node/GZKMX0WRA5.md): definition GZKMX0WRA5
- [Lusin's theorem](https://tamnd.github.io/math2/node/GZMAH9W2SK.md): statement GZMAH9W2SK
- [Knot invariant](https://tamnd.github.io/math2/node/GZNW0SV2PT.md): definition GZNW0SV2PT
- [Knot invariant](https://tamnd.github.io/math2/node/GZQ5ZE3X6G.md): concept GZQ5ZE3X6G
- [Sobolev embedding theorem](https://tamnd.github.io/math2/node/GZTT6FT18G.md): statement GZTT6FT18G
- [Bergman space](https://tamnd.github.io/math2/node/GZV7QFA0RZ.md): definition GZV7QFA0RZ
- [Sinc function](https://tamnd.github.io/math2/node/H03BGDDF0N.md): definition H03BGDDF0N
- [Arithmetic genus](https://tamnd.github.io/math2/node/H03GKRVBJ1.md): concept H03GKRVBJ1
- [Faltings's theorem](https://tamnd.github.io/math2/node/H0DB07WM20.md): statement H0DB07WM20
- [Determinate moment problem](https://tamnd.github.io/math2/node/H0GY2J14CA.md): concept H0GY2J14CA
- [Exponential integrator](https://tamnd.github.io/math2/node/H0MZB0DPP2.md): statement H0MZB0DPP2
- [Conditioned Brownian motion](https://tamnd.github.io/math2/node/H0NDB85RCD.md): concept H0NDB85RCD
- [Partially well-ordered set](https://tamnd.github.io/math2/node/H0Q4BWB23T.md): concept H0Q4BWB23T
- [Steenrod square](https://tamnd.github.io/math2/node/H0T4RFG2AD.md): definition H0T4RFG2AD
- [Paratopological group](https://tamnd.github.io/math2/node/H0V3TNH7DN.md): concept H0V3TNH7DN
- [Painlevé transcendents](https://tamnd.github.io/math2/node/H11X2TCMAZ.md): definition H11X2TCMAZ
- [Remez algorithm](https://tamnd.github.io/math2/node/H129B6DA0F.md): concept H129B6DA0F
- [Width of a poset](https://tamnd.github.io/math2/node/H13W0YERH5.md): definition H13W0YERH5
- [Spread](https://tamnd.github.io/math2/node/H1AT0G82ZD.md): concept H1AT0G82ZD
- [Focal subgroup theorem](https://tamnd.github.io/math2/node/H1BSG0J5G1.md): statement H1BSG0J5G1
- [Jacobian criterion](https://tamnd.github.io/math2/node/H1CZXMZ6HE.md): statement H1CZXMZ6HE
- [Viscosity solution](https://tamnd.github.io/math2/node/H1ERJ3T75T.md): concept H1ERJ3T75T
- [Liouville–Neumann series](https://tamnd.github.io/math2/node/H1HWSWME6B.md): statement H1HWSWME6B
- [Quermassintegral](https://tamnd.github.io/math2/node/H1K36H9ZWB.md): concept H1K36H9ZWB
- [Analytic function between Banach spaces](https://tamnd.github.io/math2/node/H1KHRWNH8B.md): concept H1KHRWNH8B
- [Character of an abelian group](https://tamnd.github.io/math2/node/H1WNVN5V7F.md): concept H1WNVN5V7F
- [Convergence of an infinite product](https://tamnd.github.io/math2/node/H1Z2TKY3EQ.md): definition H1Z2TKY3EQ
- [Integrally closed domain](https://tamnd.github.io/math2/node/H20P600HQS.md): concept H20P600HQS
- [Functional square root](https://tamnd.github.io/math2/node/H27ZC95KN5.md): definition H27ZC95KN5
- [Generalized cohomology theory](https://tamnd.github.io/math2/node/H28PSZR7XG.md): concept H28PSZR7XG
- [Painlevé property](https://tamnd.github.io/math2/node/H2E8KW6DB5.md): definition H2E8KW6DB5
- [Geometric Brownian motion](https://tamnd.github.io/math2/node/H2ET6RG7PC.md): definition H2ET6RG7PC
- [Kan complex](https://tamnd.github.io/math2/node/H2YS8SX31W.md): definition H2YS8SX31W
- [Projective space](https://tamnd.github.io/math2/node/H30E38799N.md): definition H30E38799N
- [Trigonometric functions](https://tamnd.github.io/math2/node/H313BRXEJS.md): concept H313BRXEJS
- [Geometric optics](https://tamnd.github.io/math2/node/H38D5RAYG8.md): definition H38D5RAYG8
- [Bogdanov–Takens bifurcation](https://tamnd.github.io/math2/node/H3GWX277EF.md): definition H3GWX277EF
- [Exponential map of a Lie group](https://tamnd.github.io/math2/node/H3QX1ZFC8K.md): concept H3QX1ZFC8K
- [Ergodic transformation](https://tamnd.github.io/math2/node/H3SZ6VAJ6F.md): definition H3SZ6VAJ6F
- [Congruence relation](https://tamnd.github.io/math2/node/H3W6ZPAJJV.md): definition H3W6ZPAJJV
- [Linear logic](https://tamnd.github.io/math2/node/H408H4WHBA.md): definition H408H4WHBA
- [Perfect set property](https://tamnd.github.io/math2/node/H47ZGRR13V.md): concept H47ZGRR13V
- [Weak Lp space](https://tamnd.github.io/math2/node/H4C888M5AZ.md): concept H4C888M5AZ
- [Empirical likelihood](https://tamnd.github.io/math2/node/H4DD38RKBA.md): definition H4DD38RKBA
- [Continuous wavelet transform](https://tamnd.github.io/math2/node/H4DHJ7TS76.md): definition H4DHJ7TS76
- [Littlewood–Paley square function](https://tamnd.github.io/math2/node/H4E1PW67Z1.md): concept H4E1PW67Z1
- [Topological field](https://tamnd.github.io/math2/node/H4FS63VY0Y.md): definition H4FS63VY0Y
- [Monoid object](https://tamnd.github.io/math2/node/H4JSMZK622.md): definition H4JSMZK622
- [Unit-regular ring](https://tamnd.github.io/math2/node/H4K920SVYT.md): concept H4K920SVYT
- [Power iteration](https://tamnd.github.io/math2/node/H4MQD841BR.md): concept H4MQD841BR
- [Preservation of limits](https://tamnd.github.io/math2/node/H4MY8JN8W4.md): concept H4MY8JN8W4
- [Fibred category](https://tamnd.github.io/math2/node/H4NTJ12J88.md): definition H4NTJ12J88
- [Wiener–Hopf method](https://tamnd.github.io/math2/node/H4NXEY6QTZ.md): concept H4NXEY6QTZ
- [Proof of the first isomorphism theorem](https://tamnd.github.io/math2/node/H4QSPK0XTZ.md): proof H4QSPK0XTZ
- [Clone](https://tamnd.github.io/math2/node/H4XGGXDTKN.md): definition H4XGGXDTKN
- [Asymptotic scale](https://tamnd.github.io/math2/node/H4XSF8ZPRS.md): definition H4XSF8ZPRS
- [Frobenius element](https://tamnd.github.io/math2/node/H5145REFMW.md): concept H5145REFMW
- [One-parameter subgroup](https://tamnd.github.io/math2/node/H51S5V6XYJ.md): definition H51S5V6XYJ
- [Convex combination](https://tamnd.github.io/math2/node/H56XFFM23H.md): definition H56XFFM23H
- [Algebra over an operad](https://tamnd.github.io/math2/node/H5765CJ7VB.md): definition H5765CJ7VB
- [Elementary equivalence](https://tamnd.github.io/math2/node/H58AGW8ZHF.md): definition H58AGW8ZHF
- [K3 surface](https://tamnd.github.io/math2/node/H5AQZ3336P.md): definition H5AQZ3336P
- [Weighted least squares](https://tamnd.github.io/math2/node/H5CGZHBR58.md): concept H5CGZHBR58
- [Many-one reduction](https://tamnd.github.io/math2/node/H5CW67CCST.md): definition H5CW67CCST
- [Zonotope](https://tamnd.github.io/math2/node/H5DEZ7Q1A6.md): definition H5DEZ7Q1A6
- [Incidence geometry](https://tamnd.github.io/math2/node/H5EWB180S6.md): definition H5EWB180S6
- [Calabi–Eckmann manifold](https://tamnd.github.io/math2/node/H5FQQWDJRV.md): example H5FQQWDJRV
- [Lattice of subspaces](https://tamnd.github.io/math2/node/H5G6TGRQH1.md): example H5G6TGRQH1
- [Dvoretzky–Kiefer–Wolfowitz inequality](https://tamnd.github.io/math2/node/H5G7CNYZEB.md): statement H5G7CNYZEB
- [Elliptic geometry](https://tamnd.github.io/math2/node/H5GB19W2H3.md): concept H5GB19W2H3
- [Picard–Vessiot extension](https://tamnd.github.io/math2/node/H5P06V330V.md): definition H5P06V330V
- [Solution set condition](https://tamnd.github.io/math2/node/H5QDJACKMS.md): definition H5QDJACKMS
- [Ramification](https://tamnd.github.io/math2/node/H5RC67SD24.md): concept H5RC67SD24
- [Stokes phenomenon](https://tamnd.github.io/math2/node/H5V4099GN7.md): definition H5V4099GN7
- [Separable degree](https://tamnd.github.io/math2/node/H5YX5B7GVM.md): concept H5YX5B7GVM
- [Weak lower semicontinuity](https://tamnd.github.io/math2/node/H6GTJRG26G.md): concept H6GTJRG26G
- [Convergence of approximations for second-kind equations](https://tamnd.github.io/math2/node/H6KS0E9MF5.md): statement H6KS0E9MF5
- [Mutually orthogonal Latin squares](https://tamnd.github.io/math2/node/H6TN6BADVY.md): concept H6TN6BADVY
- [Ordered ring](https://tamnd.github.io/math2/node/H6WXC7Q71A.md): concept H6WXC7Q71A
- [Volterra equation of the first kind](https://tamnd.github.io/math2/node/H6ZGEDM4VV.md): concept H6ZGEDM4VV
- [Carleman matrix](https://tamnd.github.io/math2/node/H70CF6YBY1.md): definition H70CF6YBY1
- [Logarithmic integral](https://tamnd.github.io/math2/node/H77NYP0C68.md): definition H77NYP0C68
- [Layer cake representation](https://tamnd.github.io/math2/node/H78RXJC1K1.md): statement H78RXJC1K1
- [Compactly generated space](https://tamnd.github.io/math2/node/H7B12XW6Y2.md): definition H7B12XW6Y2
- [Analytic geometry](https://tamnd.github.io/math2/node/H7CXZYANX0.md): concept H7CXZYANX0
- [Bockstein homomorphism](https://tamnd.github.io/math2/node/H7PCM776GR.md): concept H7PCM776GR
- [Newton polygon](https://tamnd.github.io/math2/node/H7QNCE2QZJ.md): concept H7QNCE2QZJ
- [Jacobson radical of an Artinian ring is nilpotent](https://tamnd.github.io/math2/node/H7RAXJV9CP.md): statement H7RAXJV9CP
- [Partial algebra](https://tamnd.github.io/math2/node/H7RX5729ZB.md): concept H7RX5729ZB
- [Lions–Stampacchia theorem](https://tamnd.github.io/math2/node/H80M01T0H8.md): statement H80M01T0H8
- [Bayesian information criterion](https://tamnd.github.io/math2/node/H82J4YJF8N.md): definition H82J4YJF8N
- [Étale morphism](https://tamnd.github.io/math2/node/H84VHC8PX2.md): concept H84VHC8PX2
- [Deformation theory](https://tamnd.github.io/math2/node/H8AAQJ1SZW.md): definition H8AAQJ1SZW
- [Lagrangian submanifold](https://tamnd.github.io/math2/node/H8ACPTMX08.md): concept H8ACPTMX08
- [Green–Tao theorem](https://tamnd.github.io/math2/node/H8BAHWSEMC.md): statement H8BAHWSEMC
- [Spread](https://tamnd.github.io/math2/node/H8J52N0P9A.md): definition H8J52N0P9A
- [Integral operator](https://tamnd.github.io/math2/node/H8P36XFV8K.md): concept H8P36XFV8K
- [Deformation of algebras](https://tamnd.github.io/math2/node/H8RN439272.md): definition H8RN439272
- [Cokernel](https://tamnd.github.io/math2/node/H8W38933NC.md): concept H8W38933NC
- [Complete integral](https://tamnd.github.io/math2/node/H90CS9B8YH.md): concept H90CS9B8YH
- [Fujiki class C](https://tamnd.github.io/math2/node/H90N8NBVNY.md): concept H90N8NBVNY
- [Dvoretzky–Rogers theorem](https://tamnd.github.io/math2/node/H924SAD12G.md): statement H924SAD12G
- [Regular singular point](https://tamnd.github.io/math2/node/H96HXG6CA0.md): concept H96HXG6CA0
- [Symmetric operator](https://tamnd.github.io/math2/node/H971WGMV6C.md): concept H971WGMV6C
- [Double centralizer theorem](https://tamnd.github.io/math2/node/H9F9G4V1EM.md): statement H9F9G4V1EM
- [Z-transform as a generating function](https://tamnd.github.io/math2/node/H9FSFB7T28.md): concept H9FSFB7T28
- [Wiener amalgam space](https://tamnd.github.io/math2/node/H9H80DM3SX.md): concept H9H80DM3SX
- [Berry–Esseen theorem](https://tamnd.github.io/math2/node/H9KTH7B380.md): statement H9KTH7B380
- [Galois ring](https://tamnd.github.io/math2/node/H9S7RMBMFD.md): concept H9S7RMBMFD
- [Equilibrium measure](https://tamnd.github.io/math2/node/H9X4FG08JT.md): definition H9X4FG08JT
- [Cross-validation](https://tamnd.github.io/math2/node/HA1KJ3B9DX.md): concept HA1KJ3B9DX
- [Yang–Mills equations](https://tamnd.github.io/math2/node/HA5YJM70TF.md): definition HA5YJM70TF
- [Monge point](https://tamnd.github.io/math2/node/HA63ZVJFVZ.md): definition HA63ZVJFVZ
- [Ordered semigroup](https://tamnd.github.io/math2/node/HA79YFZAQR.md): concept HA79YFZAQR
- [Hodge decomposition](https://tamnd.github.io/math2/node/HABCJ4Y0SR.md): statement HABCJ4Y0SR
- [Elliptic integral](https://tamnd.github.io/math2/node/HACWFM9JXE.md): definition HACWFM9JXE
- [Interpolation error formula](https://tamnd.github.io/math2/node/HASA2XSG66.md): statement HASA2XSG66
- [Annulus theorem](https://tamnd.github.io/math2/node/HAT8HG0RCK.md): statement HAT8HG0RCK
- [Lambert series](https://tamnd.github.io/math2/node/HATVWNZNMR.md): concept HATVWNZNMR
- [Traveling wave](https://tamnd.github.io/math2/node/HAZ0NKCAXS.md): definition HAZ0NKCAXS
- [Christoffel–Darboux formula](https://tamnd.github.io/math2/node/HB0MJCV482.md): statement HB0MJCV482
- [Envelope](https://tamnd.github.io/math2/node/HB14PWFCS6.md): definition HB14PWFCS6
- [Gromov's systolic inequality](https://tamnd.github.io/math2/node/HB2K72PSZ6.md): statement HB2K72PSZ6
- [Annihilator](https://tamnd.github.io/math2/node/HB4XDKJTW8.md): concept HB4XDKJTW8
- [Measurable cardinal](https://tamnd.github.io/math2/node/HBBE0C1NFD.md): definition HBBE0C1NFD
- [Solvmanifold](https://tamnd.github.io/math2/node/HBDEPTRF11.md): concept HBDEPTRF11
- [Urysohn metrization theorem](https://tamnd.github.io/math2/node/HBFHDH4EJX.md): statement HBFHDH4EJX
- [Pseudocompact space](https://tamnd.github.io/math2/node/HBGCDP2BRN.md): concept HBGCDP2BRN
- [Continuity of measure from below and above](https://tamnd.github.io/math2/node/HBHNB63J7R.md): statement HBHNB63J7R
- [General recursive function](https://tamnd.github.io/math2/node/HBK5QHC1GC.md): concept HBK5QHC1GC
- [Contact manifold](https://tamnd.github.io/math2/node/HBQ8BZ7XB5.md): concept HBQ8BZ7XB5
- [Bounded lattice](https://tamnd.github.io/math2/node/HBSWH80PVB.md): definition HBSWH80PVB
- [Boolean matrix](https://tamnd.github.io/math2/node/HC3BQBQECH.md): definition HC3BQBQECH
- [Riemann localization principle](https://tamnd.github.io/math2/node/HC3T4EY6WZ.md): statement HC3T4EY6WZ
- [Jet bundle](https://tamnd.github.io/math2/node/HC3VBBCJQP.md): concept HC3VBBCJQP
- [Capelli's theorem](https://tamnd.github.io/math2/node/HC5NDQ3DTS.md): statement HC5NDQ3DTS
- [Zariski's main theorem](https://tamnd.github.io/math2/node/HCD6W2XK6W.md): statement HCD6W2XK6W
- [Eigenvalue algorithm](https://tamnd.github.io/math2/node/HCJ2ETX89Z.md): concept HCJ2ETX89Z
- [Hilbert transform](https://tamnd.github.io/math2/node/HCJK9SV43D.md): definition HCJK9SV43D
- [Cubic Hermite spline](https://tamnd.github.io/math2/node/HCQWPAHYAT.md): concept HCQWPAHYAT
- [Zero sharp](https://tamnd.github.io/math2/node/HCTNBZZQ0W.md): object HCTNBZZQ0W
- [Todd class](https://tamnd.github.io/math2/node/HD4EESS2DP.md): definition HD4EESS2DP
- [Lambda calculus](https://tamnd.github.io/math2/node/HD5KCKVPS8.md): definition HD5KCKVPS8
- [Energy method](https://tamnd.github.io/math2/node/HD74ZBGCJ2.md): definition HD74ZBGCJ2
- [Algebraic lattice](https://tamnd.github.io/math2/node/HD8CK28AJC.md): definition HD8CK28AJC
- [Hilbert metric](https://tamnd.github.io/math2/node/HDD0EB48AB.md): definition HDD0EB48AB
- [Bernstein's theorem on inverse approximation](https://tamnd.github.io/math2/node/HDFM8ZY7XG.md): statement HDFM8ZY7XG
- [Banach lattice](https://tamnd.github.io/math2/node/HDJ1GKG0NA.md): concept HDJ1GKG0NA
- [Structure theorem for Artinian rings](https://tamnd.github.io/math2/node/HE7MP4HP6Q.md): statement HE7MP4HP6Q
- [Abel–Jacobi map](https://tamnd.github.io/math2/node/HE8F7KBF42.md): concept HE8F7KBF42
- [Constructive analysis](https://tamnd.github.io/math2/node/HEB1VQW0DQ.md): concept HEB1VQW0DQ
- [Residually finite group](https://tamnd.github.io/math2/node/HEE922EHQ8.md): concept HEE922EHQ8
- [Hausdorff summation](https://tamnd.github.io/math2/node/HEGTKZ640Y.md): definition HEGTKZ640Y
- [Gordan's lemma](https://tamnd.github.io/math2/node/HEKDP2TDNZ.md): statement HEKDP2TDNZ
- [Exponential sum](https://tamnd.github.io/math2/node/HEKS0NMYEQ.md): definition HEKS0NMYEQ
- [Isomorphism of topological groups](https://tamnd.github.io/math2/node/HEKY61PKA9.md): definition HEKY61PKA9
- [Abscissa of convergence](https://tamnd.github.io/math2/node/HEMXNPQBSV.md): definition HEMXNPQBSV
- [Steenrod algebra](https://tamnd.github.io/math2/node/HEV7170SEK.md): definition HEV7170SEK
- [Acyclic object](https://tamnd.github.io/math2/node/HEVF97NZ14.md): definition HEVF97NZ14
- [Milnor K-theory](https://tamnd.github.io/math2/node/HEVWQ6B77W.md): definition HEVWQ6B77W
- [C* identity](https://tamnd.github.io/math2/node/HEW0GGKJGB.md): definition HEW0GGKJGB
- [Ultrafilter on a Boolean algebra](https://tamnd.github.io/math2/node/HF00FDAW2F.md): definition HF00FDAW2F
- [Sphere packing](https://tamnd.github.io/math2/node/HF32CBN5MA.md): concept HF32CBN5MA
- [Mahalanobis distance](https://tamnd.github.io/math2/node/HF8AJZP6PG.md): definition HF8AJZP6PG
- [Unit in the last place](https://tamnd.github.io/math2/node/HFBST11CT6.md): concept HFBST11CT6
- [Inversive geometry](https://tamnd.github.io/math2/node/HFBW3SXQVM.md): concept HFBW3SXQVM
- [Poincaré–Birkhoff–Witt theorem](https://tamnd.github.io/math2/node/HFEA4FKGW1.md): statement HFEA4FKGW1
- [Geometric graph](https://tamnd.github.io/math2/node/HFJCAPWWD4.md): concept HFJCAPWWD4
- [Diamond principle](https://tamnd.github.io/math2/node/HFQSSB1Z23.md): concept HFQSSB1Z23
- [Jones polynomial](https://tamnd.github.io/math2/node/HFSM647VBR.md): definition HFSM647VBR
- [Model theory](https://tamnd.github.io/math2/node/HFTBG5PMXC.md): concept HFTBG5PMXC
- [Continuum](https://tamnd.github.io/math2/node/HG4S5HNWXF.md): definition HG4S5HNWXF
- [Hardy–Littlewood–Sobolev inequality](https://tamnd.github.io/math2/node/HG65C9R1SP.md): statement HG65C9R1SP
- [Nakayama algebra](https://tamnd.github.io/math2/node/HG7HC57NDM.md): definition HG7HC57NDM
- [Bézout domain](https://tamnd.github.io/math2/node/HGEX40P0Z0.md): concept HGEX40P0Z0
- [Biconjugate gradient method](https://tamnd.github.io/math2/node/HGF11AHX5R.md): statement HGF11AHX5R
- [Double-negation translation](https://tamnd.github.io/math2/node/HGJVM49PA4.md): concept HGJVM49PA4
- [Hilbert class field](https://tamnd.github.io/math2/node/HGK7R92QH7.md): definition HGK7R92QH7
- [Finite volume method](https://tamnd.github.io/math2/node/HGKN8E09RS.md): concept HGKN8E09RS
- [Eikonal equation](https://tamnd.github.io/math2/node/HGMX1QGTAZ.md): concept HGMX1QGTAZ
- [Periodic solutions of integral equations](https://tamnd.github.io/math2/node/HGPH5QWVC2.md): concept HGPH5QWVC2
- [Stable homotopy groups of spheres](https://tamnd.github.io/math2/node/HGZAYPQBXW.md): concept HGZAYPQBXW
- [Docquier–Grauert theorem](https://tamnd.github.io/math2/node/HGZMZRSS5D.md): statement HGZMZRSS5D
- [Gabrielov's theorem of the complement](https://tamnd.github.io/math2/node/HH2EM3DPNK.md): statement HH2EM3DPNK
- [Partial linear space](https://tamnd.github.io/math2/node/HH35EXPKEC.md): definition HH35EXPKEC
- [Ideal of an algebraic set](https://tamnd.github.io/math2/node/HH6VEVM6Z0.md): concept HH6VEVM6Z0
- [Absolutely convergent Fourier series](https://tamnd.github.io/math2/node/HHBGB5R5S2.md): concept HHBGB5R5S2
- [Fermi–Dirac integral](https://tamnd.github.io/math2/node/HHKKPD81J2.md): concept HHKKPD81J2
- [Principal symbol of an operator on a manifold](https://tamnd.github.io/math2/node/HHKSWC8Y6D.md): definition HHKSWC8Y6D
- [Ordered vector space](https://tamnd.github.io/math2/node/HHMS3WH9T7.md): concept HHMS3WH9T7
- [Term rewriting for equational theories](https://tamnd.github.io/math2/node/HHNJT9CBEJ.md): definition HHNJT9CBEJ
- [Differentiation and multiplication under the Fourier transform](https://tamnd.github.io/math2/node/HHNR31W3WD.md): statement HHNR31W3WD
- [Free product of algebras](https://tamnd.github.io/math2/node/HHNWHS1XTN.md): concept HHNWHS1XTN
- [Variational principle for topological entropy](https://tamnd.github.io/math2/node/HHR8HP3R6G.md): statement HHR8HP3R6G
- [Taylor term](https://tamnd.github.io/math2/node/HJ1159KYP3.md): definition HJ1159KYP3
- [LF-space](https://tamnd.github.io/math2/node/HJ2QSEMHMC.md): definition HJ2QSEMHMC
- [Sumudu transform](https://tamnd.github.io/math2/node/HJ2WFGXNBQ.md): definition HJ2WFGXNBQ
- [Jordan–Dedekind chain condition](https://tamnd.github.io/math2/node/HJ2XQMFDS9.md): definition HJ2XQMFDS9
- [Fraïssé limit](https://tamnd.github.io/math2/node/HJBYFPJEQ1.md): definition HJBYFPJEQ1
- [Relative homotopy](https://tamnd.github.io/math2/node/HJEFC4CC6M.md): definition HJEFC4CC6M
- [Minkowski plane](https://tamnd.github.io/math2/node/HJFPH6NVVH.md): definition HJFPH6NVVH
- [Diagram](https://tamnd.github.io/math2/node/HJK75HDWFE.md): concept HJK75HDWFE
- [Stochastic ordering](https://tamnd.github.io/math2/node/HJPPRZ29YE.md): concept HJPPRZ29YE
- [Hidden Markov model](https://tamnd.github.io/math2/node/HJVM390354.md): concept HJVM390354
- [Fraïssé limit](https://tamnd.github.io/math2/node/HJVRBDQSGG.md): concept HJVRBDQSGG
- [Berlekamp's algorithm](https://tamnd.github.io/math2/node/HJW1KSQZWA.md): definition HJW1KSQZWA
- [Convex cone](https://tamnd.github.io/math2/node/HJXA140M0C.md): definition HJXA140M0C
- [Cauchy filter](https://tamnd.github.io/math2/node/HJYJMKA7VE.md): definition HJYJMKA7VE
- [Tomographic reconstruction](https://tamnd.github.io/math2/node/HK1TJFXNG9.md): problem HK1TJFXNG9
- [Projective resolution](https://tamnd.github.io/math2/node/HK2AA1YSQ2.md): concept HK2AA1YSQ2
- [Lie product formula](https://tamnd.github.io/math2/node/HK3BRMK912.md): statement HK3BRMK912
- [Shannon–McMillan–Breiman theorem](https://tamnd.github.io/math2/node/HK8339ECNN.md): statement HK8339ECNN
- [Homology sphere](https://tamnd.github.io/math2/node/HK8NS02NNE.md): concept HK8NS02NNE
- [Kernel of an integral transform](https://tamnd.github.io/math2/node/HKB6KES6PK.md): definition HKB6KES6PK
- [Legendre transformation](https://tamnd.github.io/math2/node/HKPHRSN3ER.md): definition HKPHRSN3ER
- [Kolmogorov criterion for best approximation](https://tamnd.github.io/math2/node/HKRN90DCWK.md): statement HKRN90DCWK
- [Cohomology of Lie groups](https://tamnd.github.io/math2/node/HKZABKRPN6.md): concept HKZABKRPN6
- [Thom's cobordism theorem](https://tamnd.github.io/math2/node/HM23BV561Z.md): statement HM23BV561Z
- [Hilbert space](https://tamnd.github.io/math2/node/HM2HHE3J51.md): concept HM2HHE3J51
- [Prony's method](https://tamnd.github.io/math2/node/HM74NCW5JX.md): definition HM74NCW5JX
- [Cluster algebra](https://tamnd.github.io/math2/node/HM9HBVSM54.md): definition HM9HBVSM54
- [Abel–Plana formula](https://tamnd.github.io/math2/node/HMC733EFMD.md): statement HMC733EFMD
- [Stein manifold](https://tamnd.github.io/math2/node/HMCD0JBFST.md): concept HMCD0JBFST
- [Fixed-point property of posets](https://tamnd.github.io/math2/node/HMDGNS1V2T.md): concept HMDGNS1V2T
- [Transcendental extension](https://tamnd.github.io/math2/node/HMF8G4EBDX.md): definition HMF8G4EBDX
- [Holonomy groupoid of a foliation](https://tamnd.github.io/math2/node/HMRP135M6K.md): concept HMRP135M6K
- [Inversion of rational transforms by partial fractions](https://tamnd.github.io/math2/node/HMTCW4EAMW.md): definition HMTCW4EAMW
- [Dirichlet series](https://tamnd.github.io/math2/node/HN0AXS6VSG.md): definition HN0AXS6VSG
- [Minkowski sum of polytopes](https://tamnd.github.io/math2/node/HN5A4M2GE8.md): definition HN5A4M2GE8
- [Nevanlinna theory](https://tamnd.github.io/math2/node/HN5XRJTHND.md): definition HN5XRJTHND
- [Radon inversion formula](https://tamnd.github.io/math2/node/HN93ST5CBT.md): statement HN93ST5CBT
- [Kronecker's lemma](https://tamnd.github.io/math2/node/HNCY0067F1.md): statement HNCY0067F1
- [Carlson symmetric forms](https://tamnd.github.io/math2/node/HNGT8S7JAG.md): definition HNGT8S7JAG
- [Filtration](https://tamnd.github.io/math2/node/HNK1Y62G7V.md): concept HNK1Y62G7V
- [Absolutely monotonic function](https://tamnd.github.io/math2/node/HNRM1CW4JK.md): definition HNRM1CW4JK
- [Ovoid](https://tamnd.github.io/math2/node/HNTE5FPS14.md): concept HNTE5FPS14
- [Lie groupoid](https://tamnd.github.io/math2/node/HNVW0E732J.md): concept HNVW0E732J
- [Riesz projector](https://tamnd.github.io/math2/node/HNW4WRSEEX.md): concept HNW4WRSEEX
- [Topological conjugacy](https://tamnd.github.io/math2/node/HNXXAPQKG3.md): concept HNXXAPQKG3
- [Hilbert–Serre theorem](https://tamnd.github.io/math2/node/HP56DJECMT.md): statement HP56DJECMT
- [Fredholm minors](https://tamnd.github.io/math2/node/HP5VJ9EBST.md): concept HP5VJ9EBST
- [Askey–Wilson polynomials](https://tamnd.github.io/math2/node/HP97SGN9A3.md): definition HP97SGN9A3
- [Generalized hypergeometric function](https://tamnd.github.io/math2/node/HP9Z9HJ2SS.md): definition HP9Z9HJ2SS
- [Minimal set](https://tamnd.github.io/math2/node/HPBWEBJ3G5.md): definition HPBWEBJ3G5
- [No small subgroups](https://tamnd.github.io/math2/node/HPMAJV2RDY.md): concept HPMAJV2RDY
- [Distance geometry](https://tamnd.github.io/math2/node/HPW5F0GQ4B.md): concept HPW5F0GQ4B
- [Desarguesian plane](https://tamnd.github.io/math2/node/HPX85PA8NX.md): concept HPX85PA8NX
- [Plancherel theorem](https://tamnd.github.io/math2/node/HPZHRZ0502.md): statement HPZHRZ0502
- [Projective conic](https://tamnd.github.io/math2/node/HQ0JN9W2PR.md): definition HQ0JN9W2PR
- [Primitive polynomial over a finite field](https://tamnd.github.io/math2/node/HQ7KMZYPSQ.md): concept HQ7KMZYPSQ
- [Regular measure](https://tamnd.github.io/math2/node/HQFZQ0F991.md): concept HQFZQ0F991
- [Sperner's lemma](https://tamnd.github.io/math2/node/HQG3V5Z6MR.md): statement HQG3V5Z6MR
- [Algebraic space](https://tamnd.github.io/math2/node/HQHP2J7F3M.md): definition HQHP2J7F3M
- [Fréchet–Urysohn space](https://tamnd.github.io/math2/node/HQMNCB9FZ5.md): definition HQMNCB9FZ5
- [ADE singularity](https://tamnd.github.io/math2/node/HQNXQBZ6C7.md): concept HQNXQBZ6C7
- [Comodule](https://tamnd.github.io/math2/node/HQTPVQS7TY.md): definition HQTPVQS7TY
- [Distribution function of a measurable function](https://tamnd.github.io/math2/node/HQVXKEMN5N.md): concept HQVXKEMN5N
- [Hunt process](https://tamnd.github.io/math2/node/HR2GM9DSDJ.md): definition HR2GM9DSDJ
- [Structure theory of totally disconnected locally compact groups](https://tamnd.github.io/math2/node/HR5MB30MV3.md): concept HR5MB30MV3
- [Linear-quadratic regulator](https://tamnd.github.io/math2/node/HR720TJZZR.md): problem HR720TJZZR
- [Rayleigh quotient for symmetric kernels](https://tamnd.github.io/math2/node/HRCQ3JHYY0.md): concept HRCQ3JHYY0
- [Determinate moment problem](https://tamnd.github.io/math2/node/HRE0QD3B0W.md): definition HRE0QD3B0W
- [Kobayashi hyperbolic manifold](https://tamnd.github.io/math2/node/HRPAX8V7HA.md): concept HRPAX8V7HA
- [Generalized Boolean algebra](https://tamnd.github.io/math2/node/HS0R4H5SAW.md): definition HS0R4H5SAW
- [Minkowski content](https://tamnd.github.io/math2/node/HS2NTCMZ4T.md): concept HS2NTCMZ4T
- [Bifurcation diagram](https://tamnd.github.io/math2/node/HS5ZEM7TWA.md): definition HS5ZEM7TWA
- [Bump function](https://tamnd.github.io/math2/node/HS65DW0HXH.md): concept HS65DW0HXH
- [Jónsson terms](https://tamnd.github.io/math2/node/HS7GJ1C86M.md): definition HS7GJ1C86M
- [Castelnuovo–Mumford regularity](https://tamnd.github.io/math2/node/HS9VYE19Z2.md): definition HS9VYE19Z2
- [Cochran's theorem](https://tamnd.github.io/math2/node/HSBWJSBH0D.md): statement HSBWJSBH0D
- [Freedman's classification of simply connected 4-manifolds](https://tamnd.github.io/math2/node/HSD6FKCYRC.md): statement HSD6FKCYRC
- [Schauder estimates](https://tamnd.github.io/math2/node/HSEKZMAH1V.md): statement HSEKZMAH1V
- [Additive functor](https://tamnd.github.io/math2/node/HSQJSK4E23.md): definition HSQJSK4E23
- [Universal approximation theorem](https://tamnd.github.io/math2/node/HSRNND3TG5.md): statement HSRNND3TG5
- [Stochastic approximation](https://tamnd.github.io/math2/node/HSSC2ZZWMY.md): definition HSSC2ZZWMY
- [Fisher information](https://tamnd.github.io/math2/node/HSTFKC0Y39.md): definition HSTFKC0Y39
- [Fibration and cofibration](https://tamnd.github.io/math2/node/HSV1J7BJXX.md): concept HSV1J7BJXX
- [Path lifting property](https://tamnd.github.io/math2/node/HSWWF35BEX.md): statement HSWWF35BEX
- [Scott domain](https://tamnd.github.io/math2/node/HSYGHAY6WE.md): definition HSYGHAY6WE
- [de Casteljau's algorithm](https://tamnd.github.io/math2/node/HT4Q9K3P2Z.md): concept HT4Q9K3P2Z
- [Lusternik–Schnirelmann category](https://tamnd.github.io/math2/node/HTDYK465JA.md): definition HTDYK465JA
- [Curry–Howard correspondence](https://tamnd.github.io/math2/node/HTF9E5W7EZ.md): definition HTF9E5W7EZ
- [Honeycomb](https://tamnd.github.io/math2/node/HTJ03RFKFH.md): concept HTJ03RFKFH
- [Crandall–Lions uniqueness theorem](https://tamnd.github.io/math2/node/HTK7A4JRW7.md): statement HTK7A4JRW7
- [Chain map](https://tamnd.github.io/math2/node/HTK9WQZJ1G.md): concept HTK9WQZJ1G
- [Singular integral operator](https://tamnd.github.io/math2/node/HTMCEARGDD.md): concept HTMCEARGDD
- [Coupling](https://tamnd.github.io/math2/node/HTR7J0W66V.md): concept HTR7J0W66V
- [Jenkins–Traub algorithm](https://tamnd.github.io/math2/node/HTWCTA2MTJ.md): statement HTWCTA2MTJ
- [Dyadic transformation](https://tamnd.github.io/math2/node/HTWWJNSZNQ.md): example HTWWJNSZNQ
- [Umbilic catastrophes](https://tamnd.github.io/math2/node/HV003CB8EP.md): example HV003CB8EP
- [Schreier coset graph](https://tamnd.github.io/math2/node/HV2QBDHTDD.md): concept HV2QBDHTDD
- [Algebraic variety](https://tamnd.github.io/math2/node/HV6YBS0YDH.md): concept HV6YBS0YDH
- [Topological quantum field theory](https://tamnd.github.io/math2/node/HVA7TFK9XR.md): definition HVA7TFK9XR
- [Initial algebra semantics](https://tamnd.github.io/math2/node/HVF403PQ4H.md): concept HVF403PQ4H
- [Algebraic group](https://tamnd.github.io/math2/node/HVMYKZ2PB5.md): definition HVMYKZ2PB5
- [Kronecker's approximation theorem](https://tamnd.github.io/math2/node/HVT4QVHRTF.md): statement HVT4QVHRTF
- [Kissing number](https://tamnd.github.io/math2/node/HWC3SQGCQ0.md): definition HWC3SQGCQ0
- [Griffiths positivity](https://tamnd.github.io/math2/node/HWEED6YTNN.md): definition HWEED6YTNN
- [Gluing of schemes](https://tamnd.github.io/math2/node/HWGM5P7KCK.md): definition HWGM5P7KCK
- [Mellin transform method](https://tamnd.github.io/math2/node/HWHNY5854A.md): concept HWHNY5854A
- [Monte Carlo method](https://tamnd.github.io/math2/node/HWP11G3HTN.md): concept HWP11G3HTN
- [Digamma function](https://tamnd.github.io/math2/node/HWRRXB0YXS.md): concept HWRRXB0YXS
- [Special functions](https://tamnd.github.io/math2/node/HWY72ZK273.md): concept HWY72ZK273
- [Stability condition](https://tamnd.github.io/math2/node/HX2DMKTT3Y.md): definition HX2DMKTT3Y
- [Invariant measure](https://tamnd.github.io/math2/node/HX6SS7WYQV.md): definition HX6SS7WYQV
- [Extremal length](https://tamnd.github.io/math2/node/HX72KT9D0G.md): concept HX72KT9D0G
- [Hausdorff–Young inequality](https://tamnd.github.io/math2/node/HXNJH6W23S.md): statement HXNJH6W23S
- [Stewart's theorem](https://tamnd.github.io/math2/node/HXSK0BNBNE.md): statement HXSK0BNBNE
- [Complex analytic space](https://tamnd.github.io/math2/node/HXSZA44J5J.md): concept HXSZA44J5J
- [Arnold conjecture](https://tamnd.github.io/math2/node/HY7VRFBAP9.md): conjecture HY7VRFBAP9
- [Strip of convergence of the bilateral transform](https://tamnd.github.io/math2/node/HYA7QQ1E38.md): concept HYA7QQ1E38
- [Non-Desarguesian plane](https://tamnd.github.io/math2/node/HYE5FT4M34.md): example HYE5FT4M34
- [Well-formed formula](https://tamnd.github.io/math2/node/HYJ10AF6RT.md): concept HYJ10AF6RT
- [Chebyshev nodes](https://tamnd.github.io/math2/node/HYK8J56PSH.md): concept HYK8J56PSH
- [Iterated Laplace transform equals Stieltjes transform](https://tamnd.github.io/math2/node/HYP9AZHD06.md): statement HYP9AZHD06
- [Laguerre series expansion of the Laplace transform](https://tamnd.github.io/math2/node/HYRFMXAJVW.md): concept HYRFMXAJVW
- [Poisson bracket](https://tamnd.github.io/math2/node/HYSCKJN3MG.md): definition HYSCKJN3MG
- [Cospherical points](https://tamnd.github.io/math2/node/HYVFPWS9ER.md): concept HYVFPWS9ER
- [Dialectica interpretation](https://tamnd.github.io/math2/node/HZ17ES9PDY.md): concept HZ17ES9PDY
- [Associativity of convolution](https://tamnd.github.io/math2/node/HZ5J8GGS16.md): statement HZ5J8GGS16
- [n-connected space](https://tamnd.github.io/math2/node/HZ5V3M1F54.md): definition HZ5V3M1F54
- [Milnor number](https://tamnd.github.io/math2/node/HZCH4KNGY2.md): definition HZCH4KNGY2
- [Smooth dynamical system](https://tamnd.github.io/math2/node/HZKFNZTHZS.md): definition HZKFNZTHZS
- [Ornstein–Uhlenbeck process](https://tamnd.github.io/math2/node/HZM2ASCDY4.md): concept HZM2ASCDY4
- [Radon measure](https://tamnd.github.io/math2/node/HZNEQQBCW4.md): concept HZNEQQBCW4
- [Chebyshev system](https://tamnd.github.io/math2/node/HZRC5N7M9E.md): definition HZRC5N7M9E
- [Smith–Volterra–Cantor set](https://tamnd.github.io/math2/node/HZTRH10Z8Y.md): example HZTRH10Z8Y
- [Herman ring](https://tamnd.github.io/math2/node/HZX6Q514X8.md): concept HZX6Q514X8
- [Dual abelian variety](https://tamnd.github.io/math2/node/HZXSESHC8P.md): concept HZXSESHC8P
- [Translation plane](https://tamnd.github.io/math2/node/HZY2EWWKB5.md): definition HZY2EWWKB5
- [Order of a difference equation](https://tamnd.github.io/math2/node/J00G7MTBFB.md): definition J00G7MTBFB
- [Erdős–Ginzburg–Ziv theorem](https://tamnd.github.io/math2/node/J00Z252GRE.md): statement J00Z252GRE
- [Class operators H S P](https://tamnd.github.io/math2/node/J037NW1Q38.md): definition J037NW1Q38
- [Laguerre geometry](https://tamnd.github.io/math2/node/J03P12X1X4.md): definition J03P12X1X4
- [Schiffer's conjecture](https://tamnd.github.io/math2/node/J05VKG4XQB.md): conjecture J05VKG4XQB
- [Lattice-ordered group](https://tamnd.github.io/math2/node/J0DHYQ0MDE.md): concept J0DHYQ0MDE
- [Tait conjectures](https://tamnd.github.io/math2/node/J0EVKMWT94.md): statement J0EVKMWT94
- [Method of steps](https://tamnd.github.io/math2/node/J0JY4AG3E6.md): concept J0JY4AG3E6
- [Bachmann's reflection geometry](https://tamnd.github.io/math2/node/J0K6T02FQQ.md): concept J0K6T02FQQ
- [Fourier series method](https://tamnd.github.io/math2/node/J0MHS4ZCKR.md): definition J0MHS4ZCKR
- [Hadamard matrix](https://tamnd.github.io/math2/node/J0N4J606X0.md): concept J0N4J606X0
- [Numerical analysis](https://tamnd.github.io/math2/node/J0NR3YRW07.md): definition J0NR3YRW07
- [Region of convergence](https://tamnd.github.io/math2/node/J0SV0PCDBW.md): definition J0SV0PCDBW
- [Graphical model](https://tamnd.github.io/math2/node/J10YF5CXX9.md): definition J10YF5CXX9
- [Illumination problem](https://tamnd.github.io/math2/node/J1E6BEFAPZ.md): problem J1E6BEFAPZ
- [Jackknife](https://tamnd.github.io/math2/node/J1EZZN14WC.md): definition J1EZZN14WC
- [P-adic dynamics](https://tamnd.github.io/math2/node/J1SGNEWKWS.md): definition J1SGNEWKWS
- [Pseudorandom number generator](https://tamnd.github.io/math2/node/J1TZHNRV02.md): definition J1TZHNRV02
- [Mathematical formulation of quantum mechanics](https://tamnd.github.io/math2/node/J1YDW3BE56.md): concept J1YDW3BE56
- [Double layer potential](https://tamnd.github.io/math2/node/J28WVG6KPT.md): definition J28WVG6KPT
- [Skew polynomial ring](https://tamnd.github.io/math2/node/J29JWVDSEJ.md): definition J29JWVDSEJ
- [Quantile regression](https://tamnd.github.io/math2/node/J2GWK27Y15.md): definition J2GWK27Y15
- [Steiner conic](https://tamnd.github.io/math2/node/J2KM2VYA3J.md): concept J2KM2VYA3J
- [Join and meet](https://tamnd.github.io/math2/node/J2T75ATRV5.md): concept J2T75ATRV5
- [Enough projectives](https://tamnd.github.io/math2/node/J2TX4TDPCX.md): concept J2TX4TDPCX
- [Sparse approximation](https://tamnd.github.io/math2/node/J2XK2WAHBG.md): definition J2XK2WAHBG
- [Order complex](https://tamnd.github.io/math2/node/J32951135H.md): concept J32951135H
- [Discontinuous Galerkin method](https://tamnd.github.io/math2/node/J332CQ8SPB.md): definition J332CQ8SPB
- [Lipschitz class](https://tamnd.github.io/math2/node/J39DGKG9A7.md): definition J39DGKG9A7
- [Janson inequality](https://tamnd.github.io/math2/node/J3CT5D7FSZ.md): statement J3CT5D7FSZ
- [Markov chain mixing time](https://tamnd.github.io/math2/node/J3D1T7F0CB.md): definition J3D1T7F0CB
- [Homogenization](https://tamnd.github.io/math2/node/J3FQ13SP85.md): definition J3FQ13SP85
- [Gowers norm](https://tamnd.github.io/math2/node/J3JV10S9YX.md): definition J3JV10S9YX
- [Decreasing rearrangement](https://tamnd.github.io/math2/node/J3MJEDDHNY.md): definition J3MJEDDHNY
- [Involute](https://tamnd.github.io/math2/node/J3NHX0R6YD.md): definition J3NHX0R6YD
- [Trigonometric polynomial](https://tamnd.github.io/math2/node/J3NJQ5FZJ5.md): concept J3NJQ5FZJ5
- [Matrix rings over simple rings are simple](https://tamnd.github.io/math2/node/J41Y7CCE6B.md): statement J41Y7CCE6B
- [Hilbert's seventeenth problem](https://tamnd.github.io/math2/node/J45F4F13VB.md): problem J45F4F13VB
- [Positive line bundle](https://tamnd.github.io/math2/node/J45KWFHENP.md): concept J45KWFHENP
- [Wythoff construction](https://tamnd.github.io/math2/node/J4EDDYCXMR.md): concept J4EDDYCXMR
- [Dispersive estimate for the Schrödinger equation](https://tamnd.github.io/math2/node/J4TDK6Z4XA.md): statement J4TDK6Z4XA
- [Descartes' theorem](https://tamnd.github.io/math2/node/J4Y2E05RHQ.md): statement J4Y2E05RHQ
- [Zonal spherical harmonics](https://tamnd.github.io/math2/node/J4YVA5TGVA.md): definition J4YVA5TGVA
- [Quasi-compactness of the spectrum](https://tamnd.github.io/math2/node/J4YW23MD6N.md): statement J4YW23MD6N
- [Strongly regular graph](https://tamnd.github.io/math2/node/J50MA9C0EH.md): definition J50MA9C0EH
- [Classification of second-order PDEs](https://tamnd.github.io/math2/node/J53A2A04P7.md): definition J53A2A04P7
- [Kantorovich polynomials](https://tamnd.github.io/math2/node/J56JP23J57.md): definition J56JP23J57
- [SL(2,C)](https://tamnd.github.io/math2/node/J58ED1GV6N.md): example J58ED1GV6N
- [Simple module](https://tamnd.github.io/math2/node/J58ERN8KK8.md): definition J58ERN8KK8
- [Jump relations for layer potentials](https://tamnd.github.io/math2/node/J58YA8QNNX.md): statement J58YA8QNNX
- [Barycentric subdivision](https://tamnd.github.io/math2/node/J5DZEYCEX2.md): concept J5DZEYCEX2
- [Lattice ideal](https://tamnd.github.io/math2/node/J5HQWT3ZDA.md): definition J5HQWT3ZDA
- [Coequalizer](https://tamnd.github.io/math2/node/J5QPP8REV6.md): definition J5QPP8REV6
- [Pettis lemma](https://tamnd.github.io/math2/node/J5VXNMFVZB.md): statement J5VXNMFVZB
- [Properly discontinuous action](https://tamnd.github.io/math2/node/J5X5ED0B2Y.md): concept J5X5ED0B2Y
- [Automatic group](https://tamnd.github.io/math2/node/J5XQRP1SNQ.md): concept J5XQRP1SNQ
- [Resolvent](https://tamnd.github.io/math2/node/J5YNNKSB9P.md): concept J5YNNKSB9P
- [Hilbert projection theorem](https://tamnd.github.io/math2/node/J5YNTHQMFX.md): statement J5YNTHQMFX
- [Discrete harmonic function](https://tamnd.github.io/math2/node/J64R5MR303.md): definition J64R5MR303
- [Ultrafilter](https://tamnd.github.io/math2/node/J64Y3JC0RE.md): concept J64Y3JC0RE
- [Approximately additive function](https://tamnd.github.io/math2/node/J6GPWAX3X2.md): concept J6GPWAX3X2
- [Sine and cosine series](https://tamnd.github.io/math2/node/J6H7YPRZG7.md): definition J6H7YPRZG7
- [Fisher's exact test](https://tamnd.github.io/math2/node/J6JP5CJNPX.md): concept J6JP5CJNPX
- [Outer regular measure](https://tamnd.github.io/math2/node/J6MTQSZVEM.md): definition J6MTQSZVEM
- [Birkhoff's ergodic theorem](https://tamnd.github.io/math2/node/J6NMBX3E55.md): statement J6NMBX3E55
- [Stefan problem as a variational inequality](https://tamnd.github.io/math2/node/J6X53EPYR5.md): example J6X53EPYR5
- [Generated subalgebra](https://tamnd.github.io/math2/node/J6X65C2W4P.md): definition J6X65C2W4P
- [Grothendieck duality](https://tamnd.github.io/math2/node/J70H5D30DC.md): statement J70H5D30DC
- [Inscribed square problem](https://tamnd.github.io/math2/node/J72DEZHNAW.md): conjecture J72DEZHNAW
- [Robust statistics](https://tamnd.github.io/math2/node/J7410PG2DR.md): concept J7410PG2DR
- [Sierpiński's theorem on non-atomic measures](https://tamnd.github.io/math2/node/J76687HP2E.md): statement J76687HP2E
- [Multivariate division algorithm](https://tamnd.github.io/math2/node/J7GYAF4FGT.md): definition J7GYAF4FGT
- [Genus of a curve](https://tamnd.github.io/math2/node/J7K42A9F8A.md): definition J7K42A9F8A
- [Orthocomplemented lattice](https://tamnd.github.io/math2/node/J7KN14DYV3.md): concept J7KN14DYV3
- [Projective variety](https://tamnd.github.io/math2/node/J7NBAJMQND.md): definition J7NBAJMQND
- [Atlas](https://tamnd.github.io/math2/node/J7P32JQTVE.md): concept J7P32JQTVE
- [Projection-slice theorem](https://tamnd.github.io/math2/node/J7VA040MC3.md): statement J7VA040MC3
- [Solvable by radicals](https://tamnd.github.io/math2/node/J7YS0DA3HQ.md): concept J7YS0DA3HQ
- [Fourier transform as Laplace transform on the imaginary axis](https://tamnd.github.io/math2/node/J7Z1H328BC.md): concept J7Z1H328BC
- [Cayley–Klein metric](https://tamnd.github.io/math2/node/J7ZE3EGRYX.md): concept J7ZE3EGRYX
- [Kac's lemma](https://tamnd.github.io/math2/node/J80FB2BWNF.md): statement J80FB2BWNF
- [Rayleigh quotient for symmetric kernels](https://tamnd.github.io/math2/node/J858GV245K.md): definition J858GV245K
- [Local field](https://tamnd.github.io/math2/node/J85FZEQG38.md): concept J85FZEQG38
- [Heun function](https://tamnd.github.io/math2/node/J87MYGKPDD.md): definition J87MYGKPDD
- [Runge–Kutta method for stochastic differential equations](https://tamnd.github.io/math2/node/J8ET1M5V10.md): statement J8ET1M5V10
- [Subvariety](https://tamnd.github.io/math2/node/J8FCCSQ2G6.md): definition J8FCCSQ2G6
- [Diamond principle](https://tamnd.github.io/math2/node/J8GYFAA70H.md): definition J8GYFAA70H
- [Metacompact space](https://tamnd.github.io/math2/node/J8QBXGP7KK.md): definition J8QBXGP7KK
- [Generalized quadrangle](https://tamnd.github.io/math2/node/J8STHVJGDS.md): definition J8STHVJGDS
- [Mori–Nagata theorem](https://tamnd.github.io/math2/node/J8YYYP5CDN.md): statement J8YYYP5CDN
- [Lax–Wendroff method](https://tamnd.github.io/math2/node/J8ZRM7KFJH.md): definition J8ZRM7KFJH
- [Dirichlet series as Mellin transforms](https://tamnd.github.io/math2/node/J900ENTQ5B.md): definition J900ENTQ5B
- [Rodrigues' formula](https://tamnd.github.io/math2/node/J9367QFD3G.md): statement J9367QFD3G
- [Inherently nonfinitely based algebra](https://tamnd.github.io/math2/node/J9BZ6ZWGH6.md): definition J9BZ6ZWGH6
- [Prékopa–Leindler inequality](https://tamnd.github.io/math2/node/J9CJX9F42Y.md): statement J9CJX9F42Y
- [Pointed cone](https://tamnd.github.io/math2/node/J9HA54ACME.md): concept J9HA54ACME
- [Directly indecomposable algebra](https://tamnd.github.io/math2/node/J9N0MKK9S8.md): definition J9N0MKK9S8
- [Henselization](https://tamnd.github.io/math2/node/J9QYGM8AKF.md): definition J9QYGM8AKF
- [Starred transform](https://tamnd.github.io/math2/node/J9SQCM1DWM.md): definition J9SQCM1DWM
- [Langlands program](https://tamnd.github.io/math2/node/J9T2M084W5.md): conjecture J9T2M084W5
- [Schauder's theorem on adjoints of compact operators](https://tamnd.github.io/math2/node/J9TBJSV9ER.md): statement J9TBJSV9ER
- [Weak-type estimate](https://tamnd.github.io/math2/node/J9VHDBVMA6.md): definition J9VHDBVMA6
- [Dilogarithm](https://tamnd.github.io/math2/node/J9Y7DBZVJA.md): definition J9Y7DBZVJA
- [Abel equation of the first kind](https://tamnd.github.io/math2/node/J9ZY817SRC.md): example J9ZY817SRC
- [Riemann curvature tensor](https://tamnd.github.io/math2/node/JA29ETCM02.md): concept JA29ETCM02
- [Machine epsilon](https://tamnd.github.io/math2/node/JA3QXVF2B8.md): concept JA3QXVF2B8
- [Nilmanifold](https://tamnd.github.io/math2/node/JA83J040Q2.md): definition JA83J040Q2
- [Itô calculus](https://tamnd.github.io/math2/node/JA95NH206E.md): concept JA95NH206E
- [Isotonic regression](https://tamnd.github.io/math2/node/JAFPWZDM95.md): concept JAFPWZDM95
- [Hilbert dimension](https://tamnd.github.io/math2/node/JB02WZSDDW.md): definition JB02WZSDDW
- [Continuous functional calculus](https://tamnd.github.io/math2/node/JB7S6WPZ77.md): concept JB7S6WPZ77
- [Exact sequence](https://tamnd.github.io/math2/node/JB7ZBFBA9C.md): concept JB7ZBFBA9C
- [Kojima–Schur theorem](https://tamnd.github.io/math2/node/JB8GQBVW6Q.md): statement JB8GQBVW6Q
- [Inverse Fourier transform](https://tamnd.github.io/math2/node/JB8QMT8DCH.md): concept JB8QMT8DCH
- [Dense order](https://tamnd.github.io/math2/node/JB9HJ9CPEN.md): definition JB9HJ9CPEN
- [Hybrid system](https://tamnd.github.io/math2/node/JBJJHSW72S.md): definition JBJJHSW72S
- [Category of modules](https://tamnd.github.io/math2/node/JBSSXP4FFY.md): example JBSSXP4FFY
- [Equivalence of categories](https://tamnd.github.io/math2/node/JBVKKDXETM.md): definition JBVKKDXETM
- [Riesz's subsequence theorem](https://tamnd.github.io/math2/node/JBWB5H8JZG.md): statement JBWB5H8JZG
- [Field homomorphism](https://tamnd.github.io/math2/node/JBZMRZY4YF.md): definition JBZMRZY4YF
- [Young measure](https://tamnd.github.io/math2/node/JC0SZT6453.md): definition JC0SZT6453
- [Order dimension](https://tamnd.github.io/math2/node/JC4J24MF93.md): concept JC4J24MF93
- [Flag manifold](https://tamnd.github.io/math2/node/JC9R1073M9.md): definition JC9R1073M9
- [Homotopy type theory](https://tamnd.github.io/math2/node/JCCTFJV9CN.md): definition JCCTFJV9CN
- [Clean ring](https://tamnd.github.io/math2/node/JCG0MRFGC7.md): definition JCG0MRFGC7
- [Resolutivity of boundary functions](https://tamnd.github.io/math2/node/JCMTFY6HMA.md): concept JCMTFY6HMA
- [Relaxation oscillation](https://tamnd.github.io/math2/node/JCMXX84BWT.md): definition JCMXX84BWT
- [Laplace transforms of sine and cosine](https://tamnd.github.io/math2/node/JCNCD8PSGF.md): example JCNCD8PSGF
- [Idempotent semiring](https://tamnd.github.io/math2/node/JCSAC9Y1CV.md): concept JCSAC9Y1CV
- [Haar measure on a Lie group](https://tamnd.github.io/math2/node/JCSR7HJ7PC.md): concept JCSR7HJ7PC
- [Symmetry of the Green function](https://tamnd.github.io/math2/node/JDDH4DAT2W.md): statement JDDH4DAT2W
- [Abelian scheme](https://tamnd.github.io/math2/node/JDDXZ89N33.md): concept JDDXZ89N33
- [Carathéodory's criterion](https://tamnd.github.io/math2/node/JDGQ135YK2.md): statement JDGQ135YK2
- [Householder transformation](https://tamnd.github.io/math2/node/JDH0THTVGM.md): concept JDH0THTVGM
- [Strongly measurable function](https://tamnd.github.io/math2/node/JDM2M1FRY6.md): concept JDM2M1FRY6
- [Three-body problem](https://tamnd.github.io/math2/node/JE1Z920QKT.md): problem JE1Z920QKT
- [Akaike information criterion](https://tamnd.github.io/math2/node/JEAYPXPD87.md): definition JEAYPXPD87
- [Lomonosov's invariant subspace theorem](https://tamnd.github.io/math2/node/JEBP2FYZFM.md): statement JEBP2FYZFM
- [Morita equivalence of a ring with its matrix rings](https://tamnd.github.io/math2/node/JEBXXWHW8M.md): statement JEBXXWHW8M
- [Schauder fixed-point theorem for integral equations](https://tamnd.github.io/math2/node/JEFBKDG84Z.md): statement JEFBKDG84Z
- [Normal ring](https://tamnd.github.io/math2/node/JEMH9P1NVX.md): definition JEMH9P1NVX
- [Spectral theory of compact operators](https://tamnd.github.io/math2/node/JEPDM5B5FC.md): statement JEPDM5B5FC
- [Newman's theorem on rational approximation of the absolute value](https://tamnd.github.io/math2/node/JEQVZXHE6A.md): statement JEQVZXHE6A
- [Affine variety](https://tamnd.github.io/math2/node/JERVQT4C12.md): concept JERVQT4C12
- [Grothendieck group](https://tamnd.github.io/math2/node/JEWP2J397B.md): definition JEWP2J397B
- [Airy function](https://tamnd.github.io/math2/node/JEXQ0MFJ0Q.md): definition JEXQ0MFJ0Q
- [Minimal clone](https://tamnd.github.io/math2/node/JFJYSSWSSR.md): definition JFJYSSWSSR
- [CR function](https://tamnd.github.io/math2/node/JFKYYK2CEB.md): concept JFKYYK2CEB
- [Jacobi triple product](https://tamnd.github.io/math2/node/JFP2DGMNXT.md): statement JFP2DGMNXT
- [Tetrahedron](https://tamnd.github.io/math2/node/JFX8ASQS3E.md): concept JFX8ASQS3E
- [Correspondence theorem for congruences](https://tamnd.github.io/math2/node/JG7WB75BHC.md): statement JG7WB75BHC
- [Sparse matrix](https://tamnd.github.io/math2/node/JGACZYEEY7.md): concept JGACZYEEY7
- [Homology manifold](https://tamnd.github.io/math2/node/JGBW4KVMN8.md): definition JGBW4KVMN8
- [Content](https://tamnd.github.io/math2/node/JGF3WBT4GC.md): definition JGF3WBT4GC
- [Free algebra](https://tamnd.github.io/math2/node/JGQM4K9H8T.md): definition JGQM4K9H8T
- [Convolution power](https://tamnd.github.io/math2/node/JGR1E7Q9KQ.md): concept JGR1E7Q9KQ
- [Fáry's theorem](https://tamnd.github.io/math2/node/JGT7T3A9PD.md): statement JGT7T3A9PD
- [Quantum enveloping algebra](https://tamnd.github.io/math2/node/JGTGXJB7V5.md): definition JGTGXJB7V5
- [Stably finite ring](https://tamnd.github.io/math2/node/JGVDA1N106.md): definition JGVDA1N106
- [Strong maximal function](https://tamnd.github.io/math2/node/JH4BN4W27Z.md): definition JH4BN4W27Z
- [Catalan's conjecture](https://tamnd.github.io/math2/node/JH6Y2E1ZCN.md): statement JH6Y2E1ZCN
- [Potential of a measure](https://tamnd.github.io/math2/node/JHKSF3XFY2.md): concept JHKSF3XFY2
- [Shilov boundary](https://tamnd.github.io/math2/node/JHQNT58FEH.md): concept JHQNT58FEH
- [Circle packing](https://tamnd.github.io/math2/node/JJ0PQZ7VZX.md): definition JJ0PQZ7VZX
- [Calderón–Zygmund operator](https://tamnd.github.io/math2/node/JJ2G94HHJ6.md): definition JJ2G94HHJ6
- [Microlocal analysis](https://tamnd.github.io/math2/node/JJ5XQEVBZ4.md): definition JJ5XQEVBZ4
- [Zero-dimensional space](https://tamnd.github.io/math2/node/JJ7D0CYECZ.md): concept JJ7D0CYECZ
- [Separable closure](https://tamnd.github.io/math2/node/JJ91A1V8NQ.md): concept JJ91A1V8NQ
- [Doubly stochastic matrix](https://tamnd.github.io/math2/node/JJA90EB2PS.md): concept JJA90EB2PS
- [Runge–Kutta methods](https://tamnd.github.io/math2/node/JJCQ0DG4Z9.md): definition JJCQ0DG4Z9
- [k-vertex-connected graph](https://tamnd.github.io/math2/node/JJN72VZCZE.md): definition JJN72VZCZE
- [Induced representation](https://tamnd.github.io/math2/node/JJP4JFH3C6.md): concept JJP4JFH3C6
- [Scorer's function](https://tamnd.github.io/math2/node/JJQEGRK2DG.md): concept JJQEGRK2DG
- [Spin group](https://tamnd.github.io/math2/node/JJSXXJQGEH.md): example JJSXXJQGEH
- [Eckart–Young theorem](https://tamnd.github.io/math2/node/JJT6GC055J.md): statement JJT6GC055J
- [Yang–Mills equations](https://tamnd.github.io/math2/node/JJWETW5AT8.md): concept JJWETW5AT8
- [Weyl algebra](https://tamnd.github.io/math2/node/JJZ5X28NNR.md): definition JJZ5X28NNR
- [Proper morphism](https://tamnd.github.io/math2/node/JK0S0WYN01.md): definition JK0S0WYN01
- [Alexander polynomial](https://tamnd.github.io/math2/node/JKA25KDPQP.md): concept JKA25KDPQP
- [Group C*-algebra](https://tamnd.github.io/math2/node/JKANN4VBYH.md): concept JKANN4VBYH
- [Stable map](https://tamnd.github.io/math2/node/JKAPWAARTT.md): definition JKAPWAARTT
- [Fourier–Stieltjes transform](https://tamnd.github.io/math2/node/JKCFHKT13F.md): definition JKCFHKT13F
- [Chudnovsky algorithm](https://tamnd.github.io/math2/node/JKCKXDT5B5.md): concept JKCKXDT5B5
- [Hilbert cube](https://tamnd.github.io/math2/node/JKMMJ8RFC6.md): object JKMMJ8RFC6
- [Tate–Shafarevich group](https://tamnd.github.io/math2/node/JKPSBMX80N.md): definition JKPSBMX80N
- [Separable Banach space](https://tamnd.github.io/math2/node/JKQP7AFZ0E.md): definition JKQP7AFZ0E
- [End and coend](https://tamnd.github.io/math2/node/JKXTT55WCV.md): definition JKXTT55WCV
- [Lambert quadrilateral](https://tamnd.github.io/math2/node/JM98RN8HVF.md): definition JM98RN8HVF
- [Ordered ring](https://tamnd.github.io/math2/node/JMBXBG860R.md): definition JMBXBG860R
- [Four lemma](https://tamnd.github.io/math2/node/JMCKACGCQ0.md): statement JMCKACGCQ0
- [Tarski's axioms](https://tamnd.github.io/math2/node/JMFHAMX0JG.md): definition JMFHAMX0JG
- [Characteristic variety](https://tamnd.github.io/math2/node/JMJEVS7S1V.md): definition JMJEVS7S1V
- [Jacobi equation](https://tamnd.github.io/math2/node/JMST03Q37Q.md): definition JMST03Q37Q
- [Kloosterman sum](https://tamnd.github.io/math2/node/JMTJG4B94M.md): definition JMTJG4B94M
- [Legendre–Hadamard condition](https://tamnd.github.io/math2/node/JMVHZ6HDNZ.md): statement JMVHZ6HDNZ
- [Hopf link](https://tamnd.github.io/math2/node/JMWHKWD7EX.md): example JMWHKWD7EX
- [Logistic regression](https://tamnd.github.io/math2/node/JN1PJ5AJJS.md): concept JN1PJ5AJJS
- [Little's law](https://tamnd.github.io/math2/node/JN2YME6F4A.md): statement JN2YME6F4A
- [Differential equations in Banach spaces](https://tamnd.github.io/math2/node/JN7362WCD8.md): concept JN7362WCD8
- [Topological modular forms](https://tamnd.github.io/math2/node/JN73KE96YB.md): concept JN73KE96YB
- [Polycyclic group](https://tamnd.github.io/math2/node/JNB1VM75XP.md): definition JNB1VM75XP
- [Fundamental lemma of the calculus of variations](https://tamnd.github.io/math2/node/JNDZ86CV3Y.md): statement JNDZ86CV3Y
- [Milnor's isospectral flat tori](https://tamnd.github.io/math2/node/JNR9YJTT54.md): example JNR9YJTT54
- [Mixed model](https://tamnd.github.io/math2/node/JNTTAHTPDD.md): concept JNTTAHTPDD
- [Triangulation conjecture](https://tamnd.github.io/math2/node/JNW9GKADDN.md): problem JNW9GKADDN
- [Symbolic method](https://tamnd.github.io/math2/node/JP11D90EJC.md): concept JP11D90EJC
- [Accelerated failure time model](https://tamnd.github.io/math2/node/JP1QV1KQ0P.md): concept JP1QV1KQ0P
- [Inverse transform](https://tamnd.github.io/math2/node/JP9VD777AE.md): definition JP9VD777AE
- [Skorokhod's representation theorem](https://tamnd.github.io/math2/node/JPD7Q0P4F6.md): statement JPD7Q0P4F6
- [Einstein–Hilbert action](https://tamnd.github.io/math2/node/JPE3A1SN7V.md): concept JPE3A1SN7V
- [Poincaré map](https://tamnd.github.io/math2/node/JPNZMT2ARS.md): definition JPNZMT2ARS
- [Regular and singular points of an analytic set](https://tamnd.github.io/math2/node/JPP8N6MZGP.md): definition JPP8N6MZGP
- [Nonnegative matrix](https://tamnd.github.io/math2/node/JPQCCFCY8X.md): concept JPQCCFCY8X
- [Derived set](https://tamnd.github.io/math2/node/JPQM16D84G.md): definition JPQM16D84G
- [Quasifield](https://tamnd.github.io/math2/node/JQ22QH86DE.md): concept JQ22QH86DE
- [Diamond lattice M3](https://tamnd.github.io/math2/node/JQ6FJH2TP8.md): example JQ6FJH2TP8
- [Hölder's inequality](https://tamnd.github.io/math2/node/JQ9WXC729D.md): statement JQ9WXC729D
- [Reflective subcategory](https://tamnd.github.io/math2/node/JQAB8D3FFN.md): definition JQAB8D3FFN
- [Machine epsilon](https://tamnd.github.io/math2/node/JQBG1GT0XW.md): definition JQBG1GT0XW
- [Stiefel–Whitney class](https://tamnd.github.io/math2/node/JQCK7A210W.md): concept JQCK7A210W
- [Fixed-point property](https://tamnd.github.io/math2/node/JQDHHFXBF3.md): concept JQDHHFXBF3
- [Galois representation](https://tamnd.github.io/math2/node/JQEYEQJY2G.md): concept JQEYEQJY2G
- [Veblen function](https://tamnd.github.io/math2/node/JQFAE9S2F4.md): concept JQFAE9S2F4
- [Dual abelian variety](https://tamnd.github.io/math2/node/JQWXE54TVM.md): definition JQWXE54TVM
- [Homoclinic bifurcation](https://tamnd.github.io/math2/node/JQYCTY3Q8V.md): concept JQYCTY3Q8V
- [Simplicial complex](https://tamnd.github.io/math2/node/JR0G4HN9EA.md): definition JR0G4HN9EA
- [NIP theory](https://tamnd.github.io/math2/node/JR0N0BDR3B.md): definition JR0N0BDR3B
- [Ultraparallel theorem](https://tamnd.github.io/math2/node/JR2BWZEWH1.md): statement JR2BWZEWH1
- [Â-genus as the index of the Dirac operator](https://tamnd.github.io/math2/node/JR2M58RCW3.md): statement JR2M58RCW3
- [Dirichlet character](https://tamnd.github.io/math2/node/JR4QSEFXWF.md): definition JR4QSEFXWF
- [Besicovitch almost periodic function](https://tamnd.github.io/math2/node/JR7PBS4WAR.md): definition JR7PBS4WAR
- [Serre's twisting sheaf](https://tamnd.github.io/math2/node/JR83AKA184.md): definition JR83AKA184
- [Lê–Ramanujam theorem](https://tamnd.github.io/math2/node/JRAEY07Y4G.md): statement JRAEY07Y4G
- [Set theory](https://tamnd.github.io/math2/node/JRBRNZT680.md): definition JRBRNZT680
- [Cheeger's inequality for the first eigenvalue](https://tamnd.github.io/math2/node/JRGADFNBV2.md): statement JRGADFNBV2
- [Conway polynomial](https://tamnd.github.io/math2/node/JRMWDJCMCE.md): definition JRMWDJCMCE
- [Integral functional with a Lagrangian density](https://tamnd.github.io/math2/node/JRQZ803Q5Y.md): concept JRQZ803Q5Y
- [Reduced complex space](https://tamnd.github.io/math2/node/JRTB9SHAFD.md): definition JRTB9SHAFD
- [Kernel pair](https://tamnd.github.io/math2/node/JS0YH5E9ZR.md): definition JS0YH5E9ZR
- [Poincaré inequality](https://tamnd.github.io/math2/node/JS2CCR3RNS.md): statement JS2CCR3RNS
- [Unimodular matrix](https://tamnd.github.io/math2/node/JS3695KMCC.md): definition JS3695KMCC
- [Substructure](https://tamnd.github.io/math2/node/JS7BWX3HW3.md): definition JS7BWX3HW3
- [Shock wave](https://tamnd.github.io/math2/node/JSDMGGTG7G.md): concept JSDMGGTG7G
- [Complex manifold](https://tamnd.github.io/math2/node/JSGWAN2RJV.md): concept JSGWAN2RJV
- [Analytic semigroup](https://tamnd.github.io/math2/node/JSNNSNB9MJ.md): concept JSNNSNB9MJ
- [Sphere](https://tamnd.github.io/math2/node/JSQ840425F.md): definition JSQ840425F
- [Modulation space](https://tamnd.github.io/math2/node/JT3G2NF0VX.md): concept JT3G2NF0VX
- [Growth rate of a group](https://tamnd.github.io/math2/node/JT4F3SKPCP.md): definition JT4F3SKPCP
- [Frobenius norm](https://tamnd.github.io/math2/node/JT8Q204E4R.md): concept JT8Q204E4R
- [Log-gamma function](https://tamnd.github.io/math2/node/JT95B44G4G.md): concept JT95B44G4G
- [Riemann–Stieltjes integral](https://tamnd.github.io/math2/node/JT9AFHG0NA.md): concept JT9AFHG0NA
- [Bergman metric](https://tamnd.github.io/math2/node/JTA7F0KBMD.md): definition JTA7F0KBMD
- [Many-one reduction](https://tamnd.github.io/math2/node/JTAH37JTHD.md): concept JTAH37JTHD
- [Stalk](https://tamnd.github.io/math2/node/JTAKAPTT8H.md): concept JTAKAPTT8H
- [Strong dual space](https://tamnd.github.io/math2/node/JTCKJWTFEN.md): definition JTCKJWTFEN
- [Local trivialization](https://tamnd.github.io/math2/node/JTCQT4NQ79.md): concept JTCQT4NQ79
- [Linearized stability theorem for maps](https://tamnd.github.io/math2/node/JTERX19M7X.md): statement JTERX19M7X
- [Fréchet algebra](https://tamnd.github.io/math2/node/JTJ8VQA1JC.md): definition JTJ8VQA1JC
- [Riemann–Liouville integral](https://tamnd.github.io/math2/node/JTKCZ44NN3.md): concept JTKCZ44NN3
- [Von Neumann–Bernays–Gödel set theory](https://tamnd.github.io/math2/node/JTPZ1HZ9KK.md): concept JTPZ1HZ9KK
- [Normal family](https://tamnd.github.io/math2/node/JTVWRXXAKM.md): concept JTVWRXXAKM
- [Elliptic gamma function](https://tamnd.github.io/math2/node/JTXBDYXJET.md): concept JTXBDYXJET
- [Hewitt–Savage zero–one law](https://tamnd.github.io/math2/node/JTXNFQNQ4J.md): statement JTXNFQNQ4J
- [Lie subgroup](https://tamnd.github.io/math2/node/JTXSC68JZE.md): definition JTXSC68JZE
- [Perturbation theory of linear operators](https://tamnd.github.io/math2/node/JTZK8DZ257.md): concept JTZK8DZ257
- [Lindström–Gessel–Viennot lemma](https://tamnd.github.io/math2/node/JV24HVZPJ0.md): statement JV24HVZPJ0
- [Bifurcation set](https://tamnd.github.io/math2/node/JV45S0KQE6.md): concept JV45S0KQE6
- [Local trivialization](https://tamnd.github.io/math2/node/JV4H0R6GZM.md): definition JV4H0R6GZM
- [Mapping class group](https://tamnd.github.io/math2/node/JV5NYG913V.md): concept JV5NYG913V
- [Hyperplane arrangement](https://tamnd.github.io/math2/node/JVDEWP6MWG.md): definition JVDEWP6MWG
- [Geometric measure theory](https://tamnd.github.io/math2/node/JW431VDN08.md): concept JW431VDN08
- [Haar wavelet](https://tamnd.github.io/math2/node/JW5PA9XA7F.md): example JW5PA9XA7F
- [Zimmer's conjecture](https://tamnd.github.io/math2/node/JW71J3AKY4.md): conjecture JW71J3AKY4
- [Factorization of polynomials](https://tamnd.github.io/math2/node/JWB6B13Q8Z.md): concept JWB6B13Q8Z
- [Open embedding theorem for Hilbert manifolds](https://tamnd.github.io/math2/node/JWCDDMJE3K.md): statement JWCDDMJE3K
- [Shift property of the Z-transform](https://tamnd.github.io/math2/node/JWH2KYCGBQ.md): statement JWH2KYCGBQ
- [Hyperbolic group](https://tamnd.github.io/math2/node/JWJ8YPR7EB.md): concept JWJ8YPR7EB
- [Affine independence](https://tamnd.github.io/math2/node/JWK49HMHYT.md): concept JWK49HMHYT
- [Fischer decomposition](https://tamnd.github.io/math2/node/JWNRB34ZGV.md): statement JWNRB34ZGV
- [Bergman kernel](https://tamnd.github.io/math2/node/JWSNY03J2Y.md): definition JWSNY03J2Y
- [Steenrod algebra](https://tamnd.github.io/math2/node/JWTTXSGD0E.md): concept JWTTXSGD0E
- [Multiplicity of a local ring](https://tamnd.github.io/math2/node/JX20MNDDXT.md): concept JX20MNDDXT
- [Cartier divisor](https://tamnd.github.io/math2/node/JX3ZH5WY0T.md): definition JX3ZH5WY0T
- [Cup product](https://tamnd.github.io/math2/node/JX47MS1R1H.md): concept JX47MS1R1H
- [Affine connection](https://tamnd.github.io/math2/node/JX65KSVHSJ.md): concept JX65KSVHSJ
- [Field of rational functions](https://tamnd.github.io/math2/node/JX6CHG7HWV.md): definition JX6CHG7HWV
- [Box–Jenkins method](https://tamnd.github.io/math2/node/JX775JPB0N.md): concept JX775JPB0N
- [Opposite structure](https://tamnd.github.io/math2/node/JXBSN6E2X5.md): definition JXBSN6E2X5
- [Young's convolution inequality](https://tamnd.github.io/math2/node/JXDKJCBXJ8.md): statement JXDKJCBXJ8
- [Littlewood's three principles](https://tamnd.github.io/math2/node/JXFHBY77RN.md): concept JXFHBY77RN
- [Minimal variety](https://tamnd.github.io/math2/node/JXH66D0XD5.md): concept JXH66D0XD5
- [Banach measure problem](https://tamnd.github.io/math2/node/JXK6NRDH4V.md): problem JXK6NRDH4V
- [Morphism of schemes](https://tamnd.github.io/math2/node/JXRKH6P1Z4.md): concept JXRKH6P1Z4
- [Oval](https://tamnd.github.io/math2/node/JXRXV92WN5.md): definition JXRXV92WN5
- [Frobenius group](https://tamnd.github.io/math2/node/JXS91FGFHY.md): concept JXS91FGFHY
- [Pell numbers](https://tamnd.github.io/math2/node/JY4ZN779YG.md): example JY4ZN779YG
- [Multiply transitive group action](https://tamnd.github.io/math2/node/JY5TC2XCKH.md): definition JY5TC2XCKH
- [Unit and counit](https://tamnd.github.io/math2/node/JY9K3YXW0X.md): definition JY9K3YXW0X
- [Fourier series method](https://tamnd.github.io/math2/node/JYEV0Q7T4Z.md): concept JYEV0Q7T4Z
- [Kauffman bracket](https://tamnd.github.io/math2/node/JYKVYS4N32.md): concept JYKVYS4N32
- [Jensen's inequality for integrals](https://tamnd.github.io/math2/node/JYRZ927BMQ.md): statement JYRZ927BMQ
- [Infinite product](https://tamnd.github.io/math2/node/JYV1Y8XX16.md): concept JYV1Y8XX16
- [Siggers term](https://tamnd.github.io/math2/node/JYW1H3J8SX.md): concept JYW1H3J8SX
- [Perverse sheaf](https://tamnd.github.io/math2/node/JYW9312EE6.md): definition JYW9312EE6
- [Favard's theorem on trigonometric approximation](https://tamnd.github.io/math2/node/JYX6N8A2P6.md): statement JYX6N8A2P6
- [Statistical inference](https://tamnd.github.io/math2/node/JZ2S01BHJJ.md): concept JZ2S01BHJJ
- [Symmetric monoidal category](https://tamnd.github.io/math2/node/JZ4JN92A2W.md): definition JZ4JN92A2W
- [Translation plane](https://tamnd.github.io/math2/node/JZ5J6REAMD.md): concept JZ5J6REAMD
- [Statistical model](https://tamnd.github.io/math2/node/JZ6R7J36K5.md): definition JZ6R7J36K5
- [Group extension](https://tamnd.github.io/math2/node/JZ7CSKTZ16.md): definition JZ7CSKTZ16
- [Fourier–Mukai transform](https://tamnd.github.io/math2/node/JZAGFFAXKK.md): definition JZAGFFAXKK
- [Picard group](https://tamnd.github.io/math2/node/JZAKGD68TY.md): definition JZAKGD68TY
- [Kontsevich quantization formula](https://tamnd.github.io/math2/node/JZEK70CH3G.md): statement JZEK70CH3G
- [Torsion tensor](https://tamnd.github.io/math2/node/JZNQAN00ZJ.md): concept JZNQAN00ZJ
- [Torsion group](https://tamnd.github.io/math2/node/JZNQTTMG58.md): concept JZNQTTMG58
- [Existence of limits from products and equalizers](https://tamnd.github.io/math2/node/JZTCEHFVS2.md): statement JZTCEHFVS2
- [Dimension of the solution space of a linear recurrence](https://tamnd.github.io/math2/node/K00NJY6C33.md): statement K00NJY6C33
- [Distinguished boundary of a polydisc](https://tamnd.github.io/math2/node/K02CEYQBCX.md): definition K02CEYQBCX
- [Keller–Segel model](https://tamnd.github.io/math2/node/K04XRMH820.md): concept K04XRMH820
- [Hammersley–Clifford theorem](https://tamnd.github.io/math2/node/K07QH8EKD0.md): statement K07QH8EKD0
- [Product order](https://tamnd.github.io/math2/node/K0AWJK6NZH.md): concept K0AWJK6NZH
- [Néron model](https://tamnd.github.io/math2/node/K0EQTBRPE3.md): concept K0EQTBRPE3
- [Dirac bundle](https://tamnd.github.io/math2/node/K0GVMM3YPF.md): concept K0GVMM3YPF
- [Autocorrelation function](https://tamnd.github.io/math2/node/K0KF9KMF5W.md): definition K0KF9KMF5W
- [Complex geodesic](https://tamnd.github.io/math2/node/K0NRSZCCW7.md): definition K0NRSZCCW7
- [Advanced Z-transform](https://tamnd.github.io/math2/node/K0V6T18367.md): concept K0V6T18367
- [Borel–Weil–Bott theorem](https://tamnd.github.io/math2/node/K10YADF5XV.md): statement K10YADF5XV
- [Lie group](https://tamnd.github.io/math2/node/K14ZBKYT81.md): definition K14ZBKYT81
- [Local parametrization theorem](https://tamnd.github.io/math2/node/K18PKZY5RS.md): statement K18PKZY5RS
- [Motive](https://tamnd.github.io/math2/node/K1AZ4YF2RE.md): concept K1AZ4YF2RE
- [Category of topological spaces](https://tamnd.github.io/math2/node/K1DX0S4797.md): example K1DX0S4797
- [Subanalytic set](https://tamnd.github.io/math2/node/K1HTQF26MA.md): concept K1HTQF26MA
- [Product category](https://tamnd.github.io/math2/node/K1SQAAN8D3.md): definition K1SQAAN8D3
- [Calderón–Zygmund lemma](https://tamnd.github.io/math2/node/K1YKXVPVDG.md): statement K1YKXVPVDG
- [Discrete-time algebraic Riccati equation](https://tamnd.github.io/math2/node/K22P72HMT1.md): concept K22P72HMT1
- [Spherical Bessel functions](https://tamnd.github.io/math2/node/K24YAC8P7X.md): definition K24YAC8P7X
- [Baire function](https://tamnd.github.io/math2/node/K26WG8C6PV.md): concept K26WG8C6PV
- [Struve function](https://tamnd.github.io/math2/node/K290RBT8GB.md): concept K290RBT8GB
- [Stirling's series](https://tamnd.github.io/math2/node/K29QYWB6A5.md): example K29QYWB6A5
- [Directed-complete partial order](https://tamnd.github.io/math2/node/K2KQFX4GVF.md): definition K2KQFX4GVF
- [Strong amalgamation property](https://tamnd.github.io/math2/node/K2N123JM35.md): definition K2N123JM35
- [Fundamental solution](https://tamnd.github.io/math2/node/K2NK9A8M5F.md): concept K2NK9A8M5F
- [Order of a zero or pole](https://tamnd.github.io/math2/node/K2S5TAW2BF.md): definition K2S5TAW2BF
- [Order of a projective plane](https://tamnd.github.io/math2/node/K2YVMTGTBB.md): concept K2YVMTGTBB
- [Sublinear function](https://tamnd.github.io/math2/node/K3356N0SAX.md): concept K3356N0SAX
- [Frobenius norm](https://tamnd.github.io/math2/node/K33BQX70N0.md): definition K33BQX70N0
- [Statistical model](https://tamnd.github.io/math2/node/K3AMMSY8DD.md): concept K3AMMSY8DD
- [Jacobi eigenvalue algorithm](https://tamnd.github.io/math2/node/K3DP7D4JJC.md): statement K3DP7D4JJC
- [Percus–Yevick approximation](https://tamnd.github.io/math2/node/K3E536PSRJ.md): example K3E536PSRJ
- [Van Wijngaarden transformation](https://tamnd.github.io/math2/node/K3KADGATHA.md): concept K3KADGATHA
- [Satisfiability](https://tamnd.github.io/math2/node/K3WXSG7WBX.md): definition K3WXSG7WBX
- [Berkovich space dynamics](https://tamnd.github.io/math2/node/K3ZNV0GY9T.md): definition K3ZNV0GY9T
- [Higher-order logic](https://tamnd.github.io/math2/node/K40YJX8K1G.md): definition K40YJX8K1G
- [Hyperbolic knot](https://tamnd.github.io/math2/node/K47G2Q24DT.md): concept K47G2Q24DT
- [Ribbon Hopf algebra](https://tamnd.github.io/math2/node/K49YFHYCHA.md): concept K49YFHYCHA
- [Schröder's equation](https://tamnd.github.io/math2/node/K4C77FGSW6.md): definition K4C77FGSW6
- [Dense order](https://tamnd.github.io/math2/node/K4D3S1MBZH.md): concept K4D3S1MBZH
- [Field of extremals](https://tamnd.github.io/math2/node/K4DFT3H9BJ.md): concept K4DFT3H9BJ
- [Bramble–Hilbert lemma](https://tamnd.github.io/math2/node/K4P0R83G1Q.md): statement K4P0R83G1Q
- [Multidimensional transform](https://tamnd.github.io/math2/node/K4QG4MX1HV.md): definition K4QG4MX1HV
- [Goldstine theorem](https://tamnd.github.io/math2/node/K4S9G3AQBM.md): statement K4S9G3AQBM
- [Calderón projector](https://tamnd.github.io/math2/node/K4XSSPWHCP.md): concept K4XSSPWHCP
- [Affine differential geometry](https://tamnd.github.io/math2/node/K5C968HMSN.md): concept K5C968HMSN
- [Brauer–Fowler theorem](https://tamnd.github.io/math2/node/K5CQM8MYYG.md): statement K5CQM8MYYG
- [Magma](https://tamnd.github.io/math2/node/K5G3VB3DEZ.md): definition K5G3VB3DEZ
- [Weak convergence of measures](https://tamnd.github.io/math2/node/K5H0RT71A3.md): definition K5H0RT71A3
- [Dirichlet series as Mellin transforms](https://tamnd.github.io/math2/node/K5RXH32SE7.md): concept K5RXH32SE7
- [Spectral theory](https://tamnd.github.io/math2/node/K5VZ67T9J8.md): concept K5VZ67T9J8
- [Total ring of fractions](https://tamnd.github.io/math2/node/K5Z67HR9ET.md): concept K5Z67HR9ET
- [Generalized linear model](https://tamnd.github.io/math2/node/K6C48KMRKC.md): concept K6C48KMRKC
- [Strang splitting](https://tamnd.github.io/math2/node/K6EV57QFYG.md): statement K6EV57QFYG
- [Value function](https://tamnd.github.io/math2/node/K6G0DRQ3CP.md): concept K6G0DRQ3CP
- [Catenary ring](https://tamnd.github.io/math2/node/K6PPGRT5E1.md): definition K6PPGRT5E1
- [Linear recurrence with polynomial coefficients](https://tamnd.github.io/math2/node/K6QV4WSTXX.md): concept K6QV4WSTXX
- [Reflection principle](https://tamnd.github.io/math2/node/K6R7GHB3VS.md): statement K6R7GHB3VS
- [Fixed-point iteration](https://tamnd.github.io/math2/node/K6ZF91051B.md): concept K6ZF91051B
- [Dehn function](https://tamnd.github.io/math2/node/K7106VBEBF.md): definition K7106VBEBF
- [Monomial order](https://tamnd.github.io/math2/node/K711RM2Z15.md): definition K711RM2Z15
- [Relatively free algebra](https://tamnd.github.io/math2/node/K7131908S7.md): definition K7131908S7
- [Navier–Stokes equations](https://tamnd.github.io/math2/node/K76033B5G1.md): definition K76033B5G1
- [Perfectoid space](https://tamnd.github.io/math2/node/K768V9ZS7R.md): definition K768V9ZS7R
- [Rational equivalence](https://tamnd.github.io/math2/node/K76N026JFK.md): concept K76N026JFK
- [Walsh–Hadamard transform](https://tamnd.github.io/math2/node/K784BA2WHV.md): concept K784BA2WHV
- [Equivalence of categories](https://tamnd.github.io/math2/node/K7A04ZBD92.md): concept K7A04ZBD92
- [Set of uniqueness](https://tamnd.github.io/math2/node/K7CNWXF9SN.md): concept K7CNWXF9SN
- [Limit-point and limit-circle](https://tamnd.github.io/math2/node/K7H05NZEX8.md): definition K7H05NZEX8
- [Poincaré–Birkhoff theorem](https://tamnd.github.io/math2/node/K7MR77CHP6.md): statement K7MR77CHP6
- [Supporting hyperplane](https://tamnd.github.io/math2/node/K7N565NKCV.md): concept K7N565NKCV
- [Wiman–Valiron theory](https://tamnd.github.io/math2/node/K7TYSAWGPC.md): definition K7TYSAWGPC
- [Credible interval](https://tamnd.github.io/math2/node/K7Y7AMMPK0.md): definition K7Y7AMMPK0
- [de Rham theorem](https://tamnd.github.io/math2/node/K7Z2PQZ4WH.md): statement K7Z2PQZ4WH
- [Ricci curvature](https://tamnd.github.io/math2/node/K83HM845A8.md): definition K83HM845A8
- [Seifert surface](https://tamnd.github.io/math2/node/K866ND3WF7.md): concept K866ND3WF7
- [Miquel's theorem](https://tamnd.github.io/math2/node/K8A688YSPW.md): statement K8A688YSPW
- [j-invariant](https://tamnd.github.io/math2/node/K8BK5SKY1Y.md): definition K8BK5SKY1Y
- [Optimal transport and the Monge problem](https://tamnd.github.io/math2/node/K8EBASR5VG.md): problem K8EBASR5VG
- [Homotopy category of chain complexes](https://tamnd.github.io/math2/node/K8JG3C40HZ.md): definition K8JG3C40HZ
- [Sketch](https://tamnd.github.io/math2/node/K8P15XTH1M.md): concept K8P15XTH1M
- [q-Pochhammer symbol](https://tamnd.github.io/math2/node/K8PHD2NAW9.md): definition K8PHD2NAW9
- [Finitary operation](https://tamnd.github.io/math2/node/K8RE4YQ1S9.md): definition K8RE4YQ1S9
- [Cohomology operation](https://tamnd.github.io/math2/node/K8T15QHN3W.md): concept K8T15QHN3W
- [Pseudogroup](https://tamnd.github.io/math2/node/K8YRDXE4WC.md): definition K8YRDXE4WC
- [Banach analytic space](https://tamnd.github.io/math2/node/K91VB1DXPR.md): definition K91VB1DXPR
- [Cumulant](https://tamnd.github.io/math2/node/K964YTWETD.md): definition K964YTWETD
- [Fundamental class](https://tamnd.github.io/math2/node/K99YMX66XD.md): definition K99YMX66XD
- [Conformal modulus](https://tamnd.github.io/math2/node/K9BPYVQNZ9.md): definition K9BPYVQNZ9
- [Hypoelliptic operator](https://tamnd.github.io/math2/node/K9DSR21DQC.md): concept K9DSR21DQC
- [Fractional Helly theorem](https://tamnd.github.io/math2/node/K9JAYYA0WS.md): statement K9JAYYA0WS
- [Pick's theorem](https://tamnd.github.io/math2/node/K9SBQZMWB9.md): statement K9SBQZMWB9
- [Combinatorial species](https://tamnd.github.io/math2/node/K9VZKJACKT.md): definition K9VZKJACKT
- [Heap](https://tamnd.github.io/math2/node/K9XQKT69WF.md): definition K9XQKT69WF
- [Profinite group](https://tamnd.github.io/math2/node/KA0YDB6QXE.md): concept KA0YDB6QXE
- [Palais–Smale compactness condition](https://tamnd.github.io/math2/node/KA3V66S6F5.md): definition KA3V66S6F5
- [Random walk as a difference equation](https://tamnd.github.io/math2/node/KA4MW1FRZ4.md): example KA4MW1FRZ4
- [Birational map](https://tamnd.github.io/math2/node/KA5890R9YM.md): concept KA5890R9YM
- [Stochastic Volterra equation](https://tamnd.github.io/math2/node/KA95J9V6AS.md): concept KA95J9V6AS
- [Pfaff and Euler transformations](https://tamnd.github.io/math2/node/KAAJ0JJ2GF.md): statement KAAJ0JJ2GF
- [Ordinal arithmetic](https://tamnd.github.io/math2/node/KAEDCQ8RM6.md): concept KAEDCQ8RM6
- [Additive functor](https://tamnd.github.io/math2/node/KAN07YC9T2.md): concept KAN07YC9T2
- [Computational geometry](https://tamnd.github.io/math2/node/KAT7K2NE9Z.md): concept KAT7K2NE9Z
- [Nagata–Smirnov metrization theorem](https://tamnd.github.io/math2/node/KB11T6389G.md): statement KB11T6389G
- [Sheaf cohomology in topology](https://tamnd.github.io/math2/node/KB76923A1M.md): concept KB76923A1M
- [Existence and uniqueness of Haar measure](https://tamnd.github.io/math2/node/KB792QTG4F.md): statement KB792QTG4F
- [Completion of a ring](https://tamnd.github.io/math2/node/KBN13NPYJQ.md): concept KBN13NPYJQ
- [Hilbert C*-module](https://tamnd.github.io/math2/node/KBPJH3J56K.md): concept KBPJH3J56K
- [Progenerator](https://tamnd.github.io/math2/node/KBSHB2N0VG.md): definition KBSHB2N0VG
- [Isoperimetric inequality](https://tamnd.github.io/math2/node/KBX8RBZ626.md): statement KBX8RBZ626
- [Homomorphism](https://tamnd.github.io/math2/node/KBXSZC6BV1.md): concept KBXSZC6BV1
- [Cartan–Brauer–Hua theorem](https://tamnd.github.io/math2/node/KC22CWGGAE.md): statement KC22CWGGAE
- [Postnikov tower](https://tamnd.github.io/math2/node/KC7BVZZZ2W.md): definition KC7BVZZZ2W
- [Multivariate formal power series](https://tamnd.github.io/math2/node/KCBFS516DE.md): definition KCBFS516DE
- [Van Vleck's theorem](https://tamnd.github.io/math2/node/KCCFZTP3MM.md): statement KCCFZTP3MM
- [Representable functor](https://tamnd.github.io/math2/node/KCCHH87H6E.md): concept KCCHH87H6E
- [Sphere eversion](https://tamnd.github.io/math2/node/KCDT9Q3JGQ.md): statement KCDT9Q3JGQ
- [Semialgebraic set](https://tamnd.github.io/math2/node/KCEVPEV70X.md): definition KCEVPEV70X
- [VC dimension](https://tamnd.github.io/math2/node/KCFXR9K3GD.md): concept KCFXR9K3GD
- [Lie groupoid](https://tamnd.github.io/math2/node/KCP52TPREW.md): definition KCP52TPREW
- [Contraction mapping principle for integral equations](https://tamnd.github.io/math2/node/KCV4BDWZ87.md): statement KCV4BDWZ87
- [Intuitionistic type theory](https://tamnd.github.io/math2/node/KCV8RKBATN.md): definition KCV8RKBATN
- [Bergman's gap theorem](https://tamnd.github.io/math2/node/KCVCPGX7ZC.md): statement KCVCPGX7ZC
- [Ridge regression](https://tamnd.github.io/math2/node/KCVKGSYJR9.md): definition KCVKGSYJR9
- [Heat kernel](https://tamnd.github.io/math2/node/KD404MF042.md): concept KD404MF042
- [Approximation property](https://tamnd.github.io/math2/node/KD9BP7YJK5.md): definition KD9BP7YJK5
- [Cubic surface](https://tamnd.github.io/math2/node/KDEVJ69P10.md): example KDEVJ69P10
- [Density of trigonometric polynomials in continuous functions](https://tamnd.github.io/math2/node/KDRZKC9T1V.md): statement KDRZKC9T1V
- [Jordan decomposition of a function of bounded variation](https://tamnd.github.io/math2/node/KDT3E0VE0K.md): statement KDT3E0VE0K
- [C* identity](https://tamnd.github.io/math2/node/KDV7W5EDTJ.md): concept KDV7W5EDTJ
- [Isomonodromic deformation](https://tamnd.github.io/math2/node/KDZF7E66BT.md): concept KDZF7E66BT
- [Automatic differentiation](https://tamnd.github.io/math2/node/KDZN92Z8JR.md): definition KDZN92Z8JR
- [Padé table](https://tamnd.github.io/math2/node/KE1KD9P1J0.md): definition KE1KD9P1J0
- [Periodogram](https://tamnd.github.io/math2/node/KE4HEWEG38.md): definition KE4HEWEG38
- [Preconditioner](https://tamnd.github.io/math2/node/KE98SW3SBD.md): concept KE98SW3SBD
- [Functional data analysis](https://tamnd.github.io/math2/node/KEARMKNZ1Q.md): concept KEARMKNZ1Q
- [Complete integral](https://tamnd.github.io/math2/node/KEC5TGS9WM.md): definition KEC5TGS9WM
- [Euclidean geometry](https://tamnd.github.io/math2/node/KENSASDRWF.md): definition KENSASDRWF
- [Freiman's theorem](https://tamnd.github.io/math2/node/KEPFYE6WYZ.md): statement KEPFYE6WYZ
- [Localization](https://tamnd.github.io/math2/node/KET2GSPZFG.md): definition KET2GSPZFG
- [Zermelo's navigation problem](https://tamnd.github.io/math2/node/KEXJX0YDT4.md): problem KEXJX0YDT4
- [Equational basis](https://tamnd.github.io/math2/node/KEXKKBDXR3.md): definition KEXKKBDXR3
- [Trace operator](https://tamnd.github.io/math2/node/KEY8MM7F28.md): definition KEY8MM7F28
- [Differentiable dependence on parameters](https://tamnd.github.io/math2/node/KF06HRCM4N.md): statement KF06HRCM4N
- [L-curve](https://tamnd.github.io/math2/node/KF26RN3E8P.md): definition KF26RN3E8P
- [Additive function](https://tamnd.github.io/math2/node/KF719SZD5T.md): concept KF719SZD5T
- [Strassen algorithm](https://tamnd.github.io/math2/node/KF7HK5Z9XW.md): definition KF7HK5Z9XW
- [Gale diagram](https://tamnd.github.io/math2/node/KFCXGKK288.md): concept KFCXGKK288
- [Jacobi–Anger expansion](https://tamnd.github.io/math2/node/KFHYQM8T7E.md): statement KFHYQM8T7E
- [Algebraic surface](https://tamnd.github.io/math2/node/KFTECAATKG.md): concept KFTECAATKG
- [Witt group](https://tamnd.github.io/math2/node/KFVY0RPGF2.md): concept KFVY0RPGF2
- [Spherical function](https://tamnd.github.io/math2/node/KFW9841GQK.md): definition KFW9841GQK
- [Finitely additive measure](https://tamnd.github.io/math2/node/KFWQ24D9M6.md): concept KFWQ24D9M6
- [Transcritical bifurcation](https://tamnd.github.io/math2/node/KFY58YKR3N.md): concept KFY58YKR3N
- [Completion of a valued field](https://tamnd.github.io/math2/node/KFYW0WN9AD.md): definition KFYW0WN9AD
- [Cayley–Menger determinant](https://tamnd.github.io/math2/node/KGC5CYDF53.md): concept KGC5CYDF53
- [Outer function](https://tamnd.github.io/math2/node/KGJD4A4ATY.md): concept KGJD4A4ATY
- [Linear integral equation](https://tamnd.github.io/math2/node/KGNCXDK4D5.md): definition KGNCXDK4D5
- [Injective algebra](https://tamnd.github.io/math2/node/KGTA7KGP3C.md): concept KGTA7KGP3C
- [Unique ergodicity](https://tamnd.github.io/math2/node/KH07KHXJB6.md): definition KH07KHXJB6
- [Mosco convergence](https://tamnd.github.io/math2/node/KH0V6QA2ZN.md): concept KH0V6QA2ZN
- [Orbifold](https://tamnd.github.io/math2/node/KH59BQZKCY.md): definition KH59BQZKCY
- [Lie–Trotter product formula](https://tamnd.github.io/math2/node/KH7EPKDCH3.md): statement KH7EPKDCH3
- [Instrumental variables estimation](https://tamnd.github.io/math2/node/KHA4WMRDKY.md): definition KHA4WMRDKY
- [Compressed sensing](https://tamnd.github.io/math2/node/KHAT9R19CD.md): definition KHAT9R19CD
- [Mutually orthogonal Latin squares](https://tamnd.github.io/math2/node/KHD2QMNR7R.md): definition KHD2QMNR7R
- [Curve complex](https://tamnd.github.io/math2/node/KHEJ0FVZXT.md): concept KHEJ0FVZXT
- [Barycentric coordinates](https://tamnd.github.io/math2/node/KHEPP95FF9.md): definition KHEPP95FF9
- [Strong operator topology](https://tamnd.github.io/math2/node/KHJ2G2R5G7.md): concept KHJ2G2R5G7
- [Nichols algebra](https://tamnd.github.io/math2/node/KHMT0E4BRK.md): concept KHMT0E4BRK
- [Abel transform as the projection of a radial function](https://tamnd.github.io/math2/node/KHN8H3SNSM.md): statement KHN8H3SNSM
- [Laplace–Stieltjes transform](https://tamnd.github.io/math2/node/KHRTBP37VD.md): concept KHRTBP37VD
- [Finitely presented group](https://tamnd.github.io/math2/node/KHTCR3VFST.md): definition KHTCR3VFST
- [Ill-posedness of first-kind equations](https://tamnd.github.io/math2/node/KHXYWTNN2R.md): statement KHXYWTNN2R
- [Lusin area integral](https://tamnd.github.io/math2/node/KHY35P4AKF.md): concept KHY35P4AKF
- [Words and profiles](https://tamnd.github.io/math2/node/KHZ89X6Y8R.md): definition KHZ89X6Y8R
- [Étale fundamental group](https://tamnd.github.io/math2/node/KJ11DAXMR0.md): concept KJ11DAXMR0
- [Volterra equation of convolution type](https://tamnd.github.io/math2/node/KJ2984R97Y.md): definition KJ2984R97Y
- [Free object](https://tamnd.github.io/math2/node/KJ2NZCB7TH.md): definition KJ2NZCB7TH
- [Tight closure](https://tamnd.github.io/math2/node/KJ9MMP99G6.md): concept KJ9MMP99G6
- [Observed information](https://tamnd.github.io/math2/node/KJ9ZQWYBY6.md): concept KJ9ZQWYBY6
- [Functor](https://tamnd.github.io/math2/node/KJB47D7Z05.md): definition KJB47D7Z05
- [Routh–Hurwitz stability criterion](https://tamnd.github.io/math2/node/KJF8Y19BNV.md): statement KJF8Y19BNV
- [Short exact sequence](https://tamnd.github.io/math2/node/KJKNJMGKZZ.md): concept KJKNJMGKZZ
- [Completion of a measure](https://tamnd.github.io/math2/node/KJNSKV75VQ.md): concept KJNSKV75VQ
- [Projective morphism](https://tamnd.github.io/math2/node/KJQM4EJHV1.md): definition KJQM4EJHV1
- [Proper forcing axiom](https://tamnd.github.io/math2/node/KJT0AXWQXP.md): definition KJT0AXWQXP
- [Pasch's axiom](https://tamnd.github.io/math2/node/KJT98G16Y4.md): statement KJT98G16Y4
- [Alexander duality](https://tamnd.github.io/math2/node/KK05CSTM15.md): statement KK05CSTM15
- [Kuratowski embedding](https://tamnd.github.io/math2/node/KK0B7V8DCX.md): statement KK0B7V8DCX
- [Sub-Riemannian manifold](https://tamnd.github.io/math2/node/KK25XTQG41.md): definition KK25XTQG41
- [Partition of unity](https://tamnd.github.io/math2/node/KK8KZQNDVY.md): definition KK8KZQNDVY
- [Free distributive lattice](https://tamnd.github.io/math2/node/KKJ9F6JJY0.md): definition KKJ9F6JJY0
- [Laguerre series expansion of the Laplace transform](https://tamnd.github.io/math2/node/KKK6EHQQ22.md): definition KKK6EHQQ22
- [Logical consequence](https://tamnd.github.io/math2/node/KKNB9JY1YH.md): definition KKNB9JY1YH
- [Aronszajn tree](https://tamnd.github.io/math2/node/KKT4JYJEJJ.md): object KKT4JYJEJJ
- [Milne problem](https://tamnd.github.io/math2/node/KKT4Y0TBRZ.md): example KKT4Y0TBRZ
- [Hilbert's axioms](https://tamnd.github.io/math2/node/KM1FB8XZ2R.md): concept KM1FB8XZ2R
- [Hurwitz quaternion order](https://tamnd.github.io/math2/node/KM59PWHJHH.md): example KM59PWHJHH
- [Douglas–Radó solution of Plateau's problem](https://tamnd.github.io/math2/node/KM8Q5HZ48G.md): statement KM8Q5HZ48G
- [Riemann–Hilbert correspondence](https://tamnd.github.io/math2/node/KMAEWSVGRS.md): statement KMAEWSVGRS
- [Fréchet mean](https://tamnd.github.io/math2/node/KMBHAYAGZ9.md): definition KMBHAYAGZ9
- [Reuleaux triangle](https://tamnd.github.io/math2/node/KMNV11B91V.md): example KMNV11B91V
- [Hermite–Minkowski theorem](https://tamnd.github.io/math2/node/KMVW45B0TT.md): statement KMVW45B0TT
- [Sidorenko's conjecture](https://tamnd.github.io/math2/node/KN2PXSCT9J.md): conjecture KN2PXSCT9J
- [Tricolorability](https://tamnd.github.io/math2/node/KNAGK0W65G.md): concept KNAGK0W65G
- [Whitney umbrella](https://tamnd.github.io/math2/node/KNBQ68XG2M.md): example KNBQ68XG2M
- [Unital magma](https://tamnd.github.io/math2/node/KNDSJAYMSY.md): concept KNDSJAYMSY
- [Ringed space](https://tamnd.github.io/math2/node/KNDXDE7K8M.md): concept KNDXDE7K8M
- [Siegel disc](https://tamnd.github.io/math2/node/KNDYA9T8ZK.md): concept KNDYA9T8ZK
- [Fundamental domain](https://tamnd.github.io/math2/node/KNJA2NEZ9F.md): concept KNJA2NEZ9F
- [H-representation and V-representation](https://tamnd.github.io/math2/node/KNP5ZEQ458.md): definition KNP5ZEQ458
- [Incomplete beta function](https://tamnd.github.io/math2/node/KNYSF300GC.md): definition KNYSF300GC
- [Quasi-Frobenius ring](https://tamnd.github.io/math2/node/KP1PP7A6JR.md): concept KP1PP7A6JR
- [Koebe's theorem on the mean value property](https://tamnd.github.io/math2/node/KP23SF2Z0Q.md): statement KP23SF2Z0Q
- [Nonexistence of a projective plane of order 10](https://tamnd.github.io/math2/node/KP27TDCZPJ.md): statement KP27TDCZPJ
- [Character table](https://tamnd.github.io/math2/node/KPCM3KY0MR.md): definition KPCM3KY0MR
- [Differentially closed field](https://tamnd.github.io/math2/node/KPCN3KHG50.md): concept KPCN3KHG50
- [Growth of algebras](https://tamnd.github.io/math2/node/KPJED63FT6.md): definition KPJED63FT6
- [Quasi-coherent sheaf](https://tamnd.github.io/math2/node/KPKND0K7W6.md): concept KPKND0K7W6
- [Associahedron](https://tamnd.github.io/math2/node/KPNY26EKNP.md): example KPNY26EKNP
- [Morse function](https://tamnd.github.io/math2/node/KPQ96640ZJ.md): concept KPQ96640ZJ
- [Hall subgroup](https://tamnd.github.io/math2/node/KPWW9X14V2.md): concept KPWW9X14V2
- [Hecke character](https://tamnd.github.io/math2/node/KPYFGB342X.md): definition KPYFGB342X
- [Mellin transform](https://tamnd.github.io/math2/node/KPZ1S446MN.md): definition KPZ1S446MN
- [Free functor](https://tamnd.github.io/math2/node/KQ04KEFBE4.md): definition KQ04KEFBE4
- [Holomorphic dynamics in several variables](https://tamnd.github.io/math2/node/KQ0ABB8N82.md): definition KQ0ABB8N82
- [Wiener–Ikehara theorem](https://tamnd.github.io/math2/node/KQ57DC3AKM.md): statement KQ57DC3AKM
- [Matrix pencil](https://tamnd.github.io/math2/node/KQ7V0J5CGG.md): concept KQ7V0J5CGG
- [Regular value theorem](https://tamnd.github.io/math2/node/KQB7WTZ1TX.md): statement KQB7WTZ1TX
- [Parareal](https://tamnd.github.io/math2/node/KQG5GXJ5MB.md): statement KQG5GXJ5MB
- [Self-injective ring](https://tamnd.github.io/math2/node/KQKJENEMSN.md): concept KQKJENEMSN
- [Kolmogorov backward equation](https://tamnd.github.io/math2/node/KQMJ7M57ER.md): concept KQMJ7M57ER
- [Monodromy of the hypergeometric equation](https://tamnd.github.io/math2/node/KQSVGPG3W0.md): definition KQSVGPG3W0
- [Tangential Cauchy–Riemann equations](https://tamnd.github.io/math2/node/KQWJ8SGJEZ.md): definition KQWJ8SGJEZ
- [Structural stability](https://tamnd.github.io/math2/node/KQX43WB57R.md): definition KQX43WB57R
- [Omega-complete partial order](https://tamnd.github.io/math2/node/KR1F39C0SP.md): definition KR1F39C0SP
- [Symmetric operator](https://tamnd.github.io/math2/node/KR4KX354JN.md): definition KR4KX354JN
- [Bateman transform](https://tamnd.github.io/math2/node/KR5Q7WMAVC.md): concept KR5Q7WMAVC
- [Normal ring](https://tamnd.github.io/math2/node/KR7SWBYHRQ.md): concept KR7SWBYHRQ
- [Independence of sample mean and sample variance for normal samples](https://tamnd.github.io/math2/node/KR80YEKVWB.md): statement KR80YEKVWB
- [Ordered group](https://tamnd.github.io/math2/node/KRDR04RQ7N.md): concept KRDR04RQ7N
- [Viscosity solution of the Hamilton–Jacobi–Bellman equation](https://tamnd.github.io/math2/node/KREYT31SS1.md): statement KREYT31SS1
- [Hazard function](https://tamnd.github.io/math2/node/KRF03HSD44.md): definition KRF03HSD44
- [Unbounded operator](https://tamnd.github.io/math2/node/KRJX5RA0PW.md): concept KRJX5RA0PW
- [Area-minimizing current](https://tamnd.github.io/math2/node/KRK48SX454.md): definition KRK48SX454
- [McKenzie's undecidability of the finite basis problem](https://tamnd.github.io/math2/node/KRM89R0T28.md): statement KRM89R0T28
- [Field norm](https://tamnd.github.io/math2/node/KRNFEZWPY0.md): concept KRNFEZWPY0
- [Noncommutative ring](https://tamnd.github.io/math2/node/KRTDAZP29B.md): definition KRTDAZP29B
- [Associativity equation](https://tamnd.github.io/math2/node/KRWE7P7FHP.md): definition KRWE7P7FHP
- [Aperiodic set of prototiles](https://tamnd.github.io/math2/node/KRX3JWDV1B.md): definition KRX3JWDV1B
- [Euclidean geometry](https://tamnd.github.io/math2/node/KRZD96GSVC.md): concept KRZD96GSVC
- [Self-similar process](https://tamnd.github.io/math2/node/KS3KDRRJ58.md): definition KS3KDRRJ58
- [Hénon map](https://tamnd.github.io/math2/node/KS4FJ6H5JH.md): example KS4FJ6H5JH
- [Gibbs sampling](https://tamnd.github.io/math2/node/KS91STZCGM.md): statement KS91STZCGM
- [Turing pattern](https://tamnd.github.io/math2/node/KSA55H8Q8N.md): definition KSA55H8Q8N
- [Versal unfolding](https://tamnd.github.io/math2/node/KSAFWZVB8K.md): concept KSAFWZVB8K
- [Localization of a category](https://tamnd.github.io/math2/node/KSN3AJ6EJD.md): definition KSN3AJ6EJD
- [Extreme set](https://tamnd.github.io/math2/node/KSPDXANYC0.md): definition KSPDXANYC0
- [Measure algebra](https://tamnd.github.io/math2/node/KST0YNAP55.md): definition KST0YNAP55
- [Kakeya problem over finite fields](https://tamnd.github.io/math2/node/KSW18V47ZD.md): statement KSW18V47ZD
- [Level-set method](https://tamnd.github.io/math2/node/KT0380H39M.md): definition KT0380H39M
- [Countably generated structure](https://tamnd.github.io/math2/node/KT2GNAY91E.md): definition KT2GNAY91E
- [Epsilon nought](https://tamnd.github.io/math2/node/KT50SMH3EC.md): object KT50SMH3EC
- [Zarankiewicz problem](https://tamnd.github.io/math2/node/KT9WJFRQ3F.md): problem KT9WJFRQ3F
- [Embedded prime](https://tamnd.github.io/math2/node/KTBBV459TV.md): definition KTBBV459TV
- [Autoregressive integrated moving average](https://tamnd.github.io/math2/node/KTDVZP4DF2.md): concept KTDVZP4DF2
- [Milnor fibration theorem](https://tamnd.github.io/math2/node/KTFG1S2SMG.md): statement KTFG1S2SMG
- [Class number problem](https://tamnd.github.io/math2/node/KTGAS9DGRX.md): problem KTGAS9DGRX
- [Class number](https://tamnd.github.io/math2/node/KTKK9B1X41.md): concept KTKK9B1X41
- [dg category](https://tamnd.github.io/math2/node/KTN8HTSY3R.md): concept KTN8HTSY3R
- [Lamé function](https://tamnd.github.io/math2/node/KTP37HF2RZ.md): concept KTP37HF2RZ
- [Regular local ring](https://tamnd.github.io/math2/node/KTQC33ZJTN.md): definition KTQC33ZJTN
- [Korteweg–De Vries equation](https://tamnd.github.io/math2/node/KTT75QC72Z.md): concept KTT75QC72Z
- [Spline](https://tamnd.github.io/math2/node/KTVJR95GSF.md): definition KTVJR95GSF
- [Estimating equations](https://tamnd.github.io/math2/node/KV0F0E8YR2.md): definition KV0F0E8YR2
- [Denjoy–Carleman theorem](https://tamnd.github.io/math2/node/KV0GW9K7JS.md): statement KV0GW9K7JS
- [Algebraic independence](https://tamnd.github.io/math2/node/KV4PT6KW10.md): concept KV4PT6KW10
- [Adjoint functors](https://tamnd.github.io/math2/node/KVX3R8KC84.md): definition KVX3R8KC84
- [Dedekind eta function](https://tamnd.github.io/math2/node/KVZ0VZ6W01.md): definition KVZ0VZ6W01
- [Neville's algorithm](https://tamnd.github.io/math2/node/KVZ7D84541.md): definition KVZ7D84541
- [Bloch's theorem](https://tamnd.github.io/math2/node/KW31KDX7TB.md): statement KW31KDX7TB
- [Dimension theorem for local rings](https://tamnd.github.io/math2/node/KW3HY77V6D.md): statement KW3HY77V6D
- [Algorithmic randomness](https://tamnd.github.io/math2/node/KW3V44GPPC.md): concept KW3V44GPPC
- [Real closure](https://tamnd.github.io/math2/node/KWDJF4JEKN.md): definition KWDJF4JEKN
- [Ring of sets](https://tamnd.github.io/math2/node/KWE1DBMN9B.md): concept KWE1DBMN9B
- [Stanley–Wilf conjecture](https://tamnd.github.io/math2/node/KWJS2QYJ1K.md): statement KWJS2QYJ1K
- [Dirichlet space](https://tamnd.github.io/math2/node/KWKVCSJWFB.md): concept KWKVCSJWFB
- [Thom transversality theorem](https://tamnd.github.io/math2/node/KWNRS3RA09.md): statement KWNRS3RA09
- [Bézout's theorem](https://tamnd.github.io/math2/node/KWNVF194D8.md): statement KWNVF194D8
- [Angle of parallelism](https://tamnd.github.io/math2/node/KWQYX7B6P5.md): definition KWQYX7B6P5
- [Kontinuitätssatz](https://tamnd.github.io/math2/node/KWRF0DEYA2.md): statement KWRF0DEYA2
- [Clifford module](https://tamnd.github.io/math2/node/KWS2STNJAA.md): definition KWS2STNJAA
- [Toponogov's theorem](https://tamnd.github.io/math2/node/KX0B3AWD5H.md): statement KX0B3AWD5H
- [Spectral radius formula](https://tamnd.github.io/math2/node/KX2XY7S3X6.md): statement KX2XY7S3X6
- [Exponential of a bounded operator](https://tamnd.github.io/math2/node/KX3QRA3CEP.md): concept KX3QRA3CEP
- [Rayleigh monotonicity law](https://tamnd.github.io/math2/node/KX4152NVW8.md): statement KX4152NVW8
- [Convolution](https://tamnd.github.io/math2/node/KXC9KHWVGE.md): definition KXC9KHWVGE
- [Frattini subgroup](https://tamnd.github.io/math2/node/KXJG5VV340.md): concept KXJG5VV340
- [Dedekind zeta function](https://tamnd.github.io/math2/node/KXS5DQDFYD.md): concept KXS5DQDFYD
- [Reduced ring](https://tamnd.github.io/math2/node/KXTQ10RDSX.md): definition KXTQ10RDSX
- [Face of a convex set](https://tamnd.github.io/math2/node/KY7SB854VH.md): concept KY7SB854VH
- [Superadditive function](https://tamnd.github.io/math2/node/KY9ZXCHCVX.md): definition KY9ZXCHCVX
- [Natural transformation](https://tamnd.github.io/math2/node/KYCPCC57FT.md): concept KYCPCC57FT
- [Renewal theory](https://tamnd.github.io/math2/node/KYF5SCYC64.md): definition KYF5SCYC64
- [Topological transformation group](https://tamnd.github.io/math2/node/KYGE7SPYA6.md): definition KYGE7SPYA6
- [Hadamard factorization theorem](https://tamnd.github.io/math2/node/KYSR7NHNFQ.md): statement KYSR7NHNFQ
- [Heegaard Floer homology](https://tamnd.github.io/math2/node/KYVS4PSDBQ.md): definition KYVS4PSDBQ
- [Supermartingale](https://tamnd.github.io/math2/node/KYW1PV0SX5.md): definition KYW1PV0SX5
- [CAT(k) space](https://tamnd.github.io/math2/node/KZ00AZ2HFM.md): definition KZ00AZ2HFM
- [Lasso](https://tamnd.github.io/math2/node/KZ226116HW.md): concept KZ226116HW
- [Peetre K-functional](https://tamnd.github.io/math2/node/KZ3YVM10SP.md): definition KZ3YVM10SP
- [Quillen equivalence](https://tamnd.github.io/math2/node/KZCXHTEYK7.md): concept KZCXHTEYK7
- [Anderson–Darling test](https://tamnd.github.io/math2/node/KZDMD3SDJA.md): definition KZDMD3SDJA
- [Borel's lemma](https://tamnd.github.io/math2/node/KZHNA19FSS.md): statement KZHNA19FSS
- [Harmonic map](https://tamnd.github.io/math2/node/KZJYG9DS3M.md): concept KZJYG9DS3M
- [Initial value theorem](https://tamnd.github.io/math2/node/KZPVHDKV9E.md): statement KZPVHDKV9E
- [Computable function](https://tamnd.github.io/math2/node/KZSB5E471K.md): concept KZSB5E471K
- [Stark–Heegner theorem](https://tamnd.github.io/math2/node/KZTFH832H0.md): statement KZTFH832H0
- [Analytical hierarchy](https://tamnd.github.io/math2/node/M00S55MA1T.md): definition M00S55MA1T
- [Pluricomplex Green function](https://tamnd.github.io/math2/node/M03K2ZWNST.md): definition M03K2ZWNST
- [Perron integral](https://tamnd.github.io/math2/node/M077AXJACA.md): concept M077AXJACA
- [Spearman's rank correlation coefficient](https://tamnd.github.io/math2/node/M092V97ZET.md): definition M092V97ZET
- [Homicidal chauffeur problem](https://tamnd.github.io/math2/node/M0B618E2P6.md): problem M0B618E2P6
- [Viète's formula](https://tamnd.github.io/math2/node/M0G4AASSG7.md): statement M0G4AASSG7
- [Backward differentiation formula](https://tamnd.github.io/math2/node/M0HC35HZ5H.md): definition M0HC35HZ5H
- [Widder's theorem](https://tamnd.github.io/math2/node/M0J1YJAE00.md): statement M0J1YJAE00
- [Primitive ideal](https://tamnd.github.io/math2/node/M0JDQR0V5S.md): concept M0JDQR0V5S
- [Singular homology](https://tamnd.github.io/math2/node/M0Q5CFJ7ZH.md): concept M0Q5CFJ7ZH
- [Oka manifold](https://tamnd.github.io/math2/node/M0SSFESZQY.md): definition M0SSFESZQY
- [Grigorchuk group](https://tamnd.github.io/math2/node/M0WAT1VQZH.md): object M0WAT1VQZH
- [Analytic signal](https://tamnd.github.io/math2/node/M13GJHB8H9.md): definition M13GJHB8H9
- [Lie group–Lie algebra correspondence](https://tamnd.github.io/math2/node/M13HAQSHEM.md): statement M13HAQSHEM
- [Jordan–Hölder theorem for modules](https://tamnd.github.io/math2/node/M1ABX18FC4.md): statement M1ABX18FC4
- [Pompeiu problem](https://tamnd.github.io/math2/node/M1KYAQMF9W.md): problem M1KYAQMF9W
- [Dbar operator](https://tamnd.github.io/math2/node/M1VKCW3RFW.md): concept M1VKCW3RFW
- [Cobordism hypothesis](https://tamnd.github.io/math2/node/M1WQTWFNNQ.md): statement M1WQTWFNNQ
- [Inverse function theorem in Banach spaces](https://tamnd.github.io/math2/node/M218MWZ62X.md): statement M218MWZ62X
- [Rogers–Ramanujan identities](https://tamnd.github.io/math2/node/M24H4GJPPW.md): statement M24H4GJPPW
- [Envelope of holomorphy](https://tamnd.github.io/math2/node/M28SB04806.md): definition M28SB04806
- [Indefinite orthogonal group](https://tamnd.github.io/math2/node/M2900DMEW6.md): example M2900DMEW6
- [Dolbeault operator](https://tamnd.github.io/math2/node/M2F4N31JQ2.md): concept M2F4N31JQ2
- [Normalization](https://tamnd.github.io/math2/node/M2FAMD9HZN.md): concept M2FAMD9HZN
- [Heaviside cover-up method](https://tamnd.github.io/math2/node/M2FSQNBJCT.md): definition M2FSQNBJCT
- [Variance reduction](https://tamnd.github.io/math2/node/M2JF5D7DP2.md): definition M2JF5D7DP2
- [Compact operator](https://tamnd.github.io/math2/node/M2NEP1K5AE.md): concept M2NEP1K5AE
- [Finsler manifold](https://tamnd.github.io/math2/node/M2R0XWWW0R.md): definition M2R0XWWW0R
- [Depth](https://tamnd.github.io/math2/node/M2RADBFW99.md): concept M2RADBFW99
- [Clausen's formula](https://tamnd.github.io/math2/node/M2V5A38W5C.md): statement M2V5A38W5C
- [Primary decomposition](https://tamnd.github.io/math2/node/M2VPVJH33K.md): definition M2VPVJH33K
- [Palais–Smale theory of critical points](https://tamnd.github.io/math2/node/M2W79VF0TN.md): statement M2W79VF0TN
- [Ax–Grothendieck theorem](https://tamnd.github.io/math2/node/M2XAN2HGZH.md): statement M2XAN2HGZH
- [Dual operator](https://tamnd.github.io/math2/node/M31BYH9VSS.md): definition M31BYH9VSS
- [Zeros of Bessel functions](https://tamnd.github.io/math2/node/M324R146XH.md): concept M324R146XH
- [Green's function for the Laplacian](https://tamnd.github.io/math2/node/M36SD56MWQ.md): concept M36SD56MWQ
- [Traveling wave](https://tamnd.github.io/math2/node/M37M411KP7.md): concept M37M411KP7
- [Curvature of a curve](https://tamnd.github.io/math2/node/M39DK862R7.md): definition M39DK862R7
- [Family-wise error rate](https://tamnd.github.io/math2/node/M3B0JFVTW0.md): definition M3B0JFVTW0
- [Huntington's axioms](https://tamnd.github.io/math2/node/M3GFCVX549.md): concept M3GFCVX549
- [Locally finite family](https://tamnd.github.io/math2/node/M3M3EFWBEX.md): definition M3M3EFWBEX
- [Monoidal category](https://tamnd.github.io/math2/node/M3MSQ3NKAE.md): concept M3MSQ3NKAE
- [Complete class theorem](https://tamnd.github.io/math2/node/M3PFV1BX9V.md): statement M3PFV1BX9V
- [Convex join](https://tamnd.github.io/math2/node/M3S9JK4RPK.md): definition M3S9JK4RPK
- [Action-angle coordinates](https://tamnd.github.io/math2/node/M3T81YNSKC.md): concept M3T81YNSKC
- [Nevanlinna class](https://tamnd.github.io/math2/node/M3WQTN3FH2.md): concept M3WQTN3FH2
- [Place (number field)](https://tamnd.github.io/math2/node/M3XJ4ZKYGT.md): definition M3XJ4ZKYGT
- [Morita duality](https://tamnd.github.io/math2/node/M3YXA82ZFX.md): concept M3YXA82ZFX
- [Bessel potential](https://tamnd.github.io/math2/node/M435YDQWQ4.md): definition M435YDQWQ4
- [Topological entropy](https://tamnd.github.io/math2/node/M44J4ZJ6Z8.md): definition M44J4ZJ6Z8
- [Determinantal ideal](https://tamnd.github.io/math2/node/M4606WJDYG.md): definition M4606WJDYG
- [Sobolev space](https://tamnd.github.io/math2/node/M4CEZ5K9HP.md): concept M4CEZ5K9HP
- [Rees matrix semigroup](https://tamnd.github.io/math2/node/M4DP72HFE5.md): definition M4DP72HFE5
- [Brun sieve](https://tamnd.github.io/math2/node/M4G4RXASTY.md): concept M4G4RXASTY
- [Noetherian topological space](https://tamnd.github.io/math2/node/M4H3QNJPGD.md): definition M4H3QNJPGD
- [Ultrametric space](https://tamnd.github.io/math2/node/M4HM6A3J54.md): definition M4HM6A3J54
- [Lamé function](https://tamnd.github.io/math2/node/M4KG2KBKS9.md): definition M4KG2KBKS9
- [Cokernel](https://tamnd.github.io/math2/node/M4N5FNNYCF.md): definition M4N5FNNYCF
- [Total variation denoising](https://tamnd.github.io/math2/node/M4RJHZ3KNA.md): example M4RJHZ3KNA
- [Chebyshev set](https://tamnd.github.io/math2/node/M4RMZNGZ2P.md): concept M4RMZNGZ2P
- [Schur stable polynomial](https://tamnd.github.io/math2/node/M4TXKR87H8.md): definition M4TXKR87H8
- [Time shift theorem](https://tamnd.github.io/math2/node/M4VJHPZJMQ.md): statement M4VJHPZJMQ
- [Tarski's HSP formula](https://tamnd.github.io/math2/node/M4ZHB3GXBP.md): statement M4ZHB3GXBP
- [CORDIC](https://tamnd.github.io/math2/node/M4ZJ8QV3MR.md): statement M4ZJ8QV3MR
- [j-invariant of an elliptic curve](https://tamnd.github.io/math2/node/M5196PBDCA.md): definition M5196PBDCA
- [Inverse limit of algebras](https://tamnd.github.io/math2/node/M53B55WESR.md): definition M53B55WESR
- [Arnold's classification of simple critical points](https://tamnd.github.io/math2/node/M5437FP73G.md): statement M5437FP73G
- [Ideal norm](https://tamnd.github.io/math2/node/M54ADM0CVF.md): concept M54ADM0CVF
- [Local coefficient system](https://tamnd.github.io/math2/node/M58WRZ7PFM.md): concept M58WRZ7PFM
- [Final functor](https://tamnd.github.io/math2/node/M5910N7YF0.md): concept M5910N7YF0
- [Hopf fibration](https://tamnd.github.io/math2/node/M5AKNYC4QQ.md): example M5AKNYC4QQ
- [Almost split sequence](https://tamnd.github.io/math2/node/M5D95FN7G3.md): concept M5D95FN7G3
- [Vanishing cycle](https://tamnd.github.io/math2/node/M5EK9AKXNR.md): concept M5EK9AKXNR
- [Torsion group](https://tamnd.github.io/math2/node/M5FDV1TMBG.md): definition M5FDV1TMBG
- [Hilbert's fourteenth problem](https://tamnd.github.io/math2/node/M5N08HD75V.md): problem M5N08HD75V
- [Bessel polynomials](https://tamnd.github.io/math2/node/M5NXATQ48S.md): concept M5NXATQ48S
- [Hölder summation](https://tamnd.github.io/math2/node/M5YRA6CK4G.md): concept M5YRA6CK4G
- [Matrix summability method](https://tamnd.github.io/math2/node/M6311DWJ1E.md): concept M6311DWJ1E
- [Maxwell's equations](https://tamnd.github.io/math2/node/M665JWCEXY.md): definition M665JWCEXY
- [Higher homotopy groups are abelian](https://tamnd.github.io/math2/node/M676G2V5KQ.md): statement M676G2V5KQ
- [Tikhonov regularization](https://tamnd.github.io/math2/node/M68V0M5A3R.md): concept M68V0M5A3R
- [Projective line over a ring](https://tamnd.github.io/math2/node/M6BN1K0B30.md): concept M6BN1K0B30
- [Chebyshev polynomials](https://tamnd.github.io/math2/node/M6EAR1JHPP.md): object M6EAR1JHPP
- [Analytical hierarchy](https://tamnd.github.io/math2/node/M6JGF4TPM6.md): concept M6JGF4TPM6
- [Cyclic polytope](https://tamnd.github.io/math2/node/M6YBMXKJYD.md): concept M6YBMXKJYD
- [Bohr almost periodic function](https://tamnd.github.io/math2/node/M702B26KQ8.md): definition M702B26KQ8
- [Analytic index](https://tamnd.github.io/math2/node/M7DE5A0RKN.md): definition M7DE5A0RKN
- [Character space](https://tamnd.github.io/math2/node/M7JVFJ3MMX.md): concept M7JVFJ3MMX
- [Subshift of finite type](https://tamnd.github.io/math2/node/M7RPDE39FM.md): definition M7RPDE39FM
- [Backward Euler method](https://tamnd.github.io/math2/node/M7XKVNATFA.md): statement M7XKVNATFA
- [Amalgamated free product](https://tamnd.github.io/math2/node/M848AHBRCM.md): concept M848AHBRCM
- [Image of J](https://tamnd.github.io/math2/node/M84YQMSQZX.md): definition M84YQMSQZX
- [Ahlfors theory](https://tamnd.github.io/math2/node/M8611CV76A.md): concept M8611CV76A
- [Energy drift](https://tamnd.github.io/math2/node/M8DEV2YM2S.md): definition M8DEV2YM2S
- [Elliptic complex](https://tamnd.github.io/math2/node/M8GMH8GQR6.md): concept M8GMH8GQR6
- [Modular form](https://tamnd.github.io/math2/node/M8NA9X8BTV.md): definition M8NA9X8BTV
- [Opposite ring](https://tamnd.github.io/math2/node/M8Q24XB0AF.md): concept M8Q24XB0AF
- [Wandering set](https://tamnd.github.io/math2/node/M8TAKF85NT.md): definition M8TAKF85NT
- [Wiener–Hopf equation](https://tamnd.github.io/math2/node/M8W64BRCQH.md): definition M8W64BRCQH
- [Product sigma-algebra](https://tamnd.github.io/math2/node/M8X6VZD91K.md): definition M8X6VZD91K
- [Nonlinear potential theory](https://tamnd.github.io/math2/node/M923VKQVZM.md): concept M923VKQVZM
- [Crank–Nicolson method](https://tamnd.github.io/math2/node/M92S0Y15N1.md): statement M92S0Y15N1
- [Analysis of variance](https://tamnd.github.io/math2/node/M9AQ5HKGP4.md): concept M9AQ5HKGP4
- [Homoclinic bifurcation](https://tamnd.github.io/math2/node/M9B7X8F6TN.md): definition M9B7X8F6TN
- [Parabolic subgroup](https://tamnd.github.io/math2/node/M9DY1E7RKS.md): definition M9DY1E7RKS
- [Unbounded operator](https://tamnd.github.io/math2/node/M9HEZ5X3FX.md): definition M9HEZ5X3FX
- [Statistical learning theory](https://tamnd.github.io/math2/node/M9SVV4FDZV.md): definition M9SVV4FDZV
- [Sylvester equation](https://tamnd.github.io/math2/node/M9SW4NVP2H.md): concept M9SW4NVP2H
- [Gabor frame](https://tamnd.github.io/math2/node/M9WHY8E2PM.md): concept M9WHY8E2PM
- [Harish-Chandra's regularity theorem](https://tamnd.github.io/math2/node/M9X7Y3GZZD.md): statement M9X7Y3GZZD
- [Fisher's inequality](https://tamnd.github.io/math2/node/M9XF82GGW8.md): statement M9XF82GGW8
- [Grünwald–Letnikov derivative](https://tamnd.github.io/math2/node/M9Z8VM4K1Z.md): concept M9Z8VM4K1Z
- [Plurisubharmonic exhaustion function](https://tamnd.github.io/math2/node/M9ZFWS9KSP.md): definition M9ZFWS9KSP
- [Post's lattice](https://tamnd.github.io/math2/node/MA20X7JCET.md): object MA20X7JCET
- [Biconjugate gradient stabilized method](https://tamnd.github.io/math2/node/MA64H1RTP1.md): statement MA64H1RTP1
- [Cesàro means of Fourier series](https://tamnd.github.io/math2/node/MA8X1G11DH.md): concept MA8X1G11DH
- [Convergence tests](https://tamnd.github.io/math2/node/MA9EMC488R.md): definition MA9EMC488R
- [Schläfli symbol](https://tamnd.github.io/math2/node/MAE6BF4DFK.md): concept MAE6BF4DFK
- [Uniform convergence of series](https://tamnd.github.io/math2/node/MAHHKQST2A.md): concept MAHHKQST2A
- [First fundamental form](https://tamnd.github.io/math2/node/MAJ3DH7MZS.md): concept MAJ3DH7MZS
- [Laplace transform on a convex cone](https://tamnd.github.io/math2/node/MAKCYGGERD.md): concept MAKCYGGERD
- [Coordinatization of projective planes](https://tamnd.github.io/math2/node/MAM2MDG0XG.md): statement MAM2MDG0XG
- [Polynomial function](https://tamnd.github.io/math2/node/MAM9H3SY3Y.md): definition MAM9H3SY3Y
- [Saddle-node bifurcation](https://tamnd.github.io/math2/node/MAP5J6HK0T.md): concept MAP5J6HK0T
- [Additive energy](https://tamnd.github.io/math2/node/MAQKH72P6V.md): concept MAQKH72P6V
- [Lyapunov–Schmidt reduction](https://tamnd.github.io/math2/node/MAVAK2WWXK.md): concept MAVAK2WWXK
- [Morse index](https://tamnd.github.io/math2/node/MAX2ZYG6H9.md): concept MAX2ZYG6H9
- [Krull dimension of a noncommutative ring](https://tamnd.github.io/math2/node/MB1K62QDM4.md): definition MB1K62QDM4
- [Resolution of singularities](https://tamnd.github.io/math2/node/MB52NDA6WN.md): statement MB52NDA6WN
- [Hamiltonian path](https://tamnd.github.io/math2/node/MBASZABTJ1.md): concept MBASZABTJ1
- [Penrose tiling](https://tamnd.github.io/math2/node/MBE83FTRBG.md): example MBE83FTRBG
- [Duality of Lp spaces](https://tamnd.github.io/math2/node/MBF660WMYJ.md): statement MBF660WMYJ
- [Shift operator](https://tamnd.github.io/math2/node/MBK6AE6734.md): concept MBK6AE6734
- [Rng](https://tamnd.github.io/math2/node/MBK7NFE74S.md): concept MBK7NFE74S
- [Łojasiewicz inequality](https://tamnd.github.io/math2/node/MBMHS6FQF6.md): statement MBMHS6FQF6
- [Index theory](https://tamnd.github.io/math2/node/MBQS4B0TPA.md): definition MBQS4B0TPA
- [q-Vandermonde identity](https://tamnd.github.io/math2/node/MBRSCQRGGH.md): statement MBRSCQRGGH
- [Galois connection](https://tamnd.github.io/math2/node/MBVJ4JQXC5.md): definition MBVJ4JQXC5
- [Weak convergence of measures](https://tamnd.github.io/math2/node/MBW9JCGZC1.md): concept MBW9JCGZC1
- [Relational clone](https://tamnd.github.io/math2/node/MC25MTN2N7.md): concept MC25MTN2N7
- [Kleinian group](https://tamnd.github.io/math2/node/MC4EFHWRJG.md): definition MC4EFHWRJG
- [Gibbs measure](https://tamnd.github.io/math2/node/MC4S4WCGJS.md): definition MC4S4WCGJS
- [Banach–Lie group](https://tamnd.github.io/math2/node/MC5CGVRFJM.md): concept MC5CGVRFJM
- [Chebyshev functions](https://tamnd.github.io/math2/node/MC9HE0M2RG.md): definition MC9HE0M2RG
- [Discriminator variety](https://tamnd.github.io/math2/node/MC9MX40Z7S.md): concept MC9MX40Z7S
- [Cesàro means of Fourier series](https://tamnd.github.io/math2/node/MCCXNPGGSH.md): definition MCCXNPGGSH
- [Linear continuum](https://tamnd.github.io/math2/node/MCDP25YMSE.md): definition MCDP25YMSE
- [Finite geometry](https://tamnd.github.io/math2/node/MCDZP31ZF3.md): concept MCDZP31ZF3
- [Category](https://tamnd.github.io/math2/node/MCEE53VGBQ.md): definition MCEE53VGBQ
- [Unknotting number](https://tamnd.github.io/math2/node/MCPFA4DC9Z.md): definition MCPFA4DC9Z
- [Complete atomic Boolean algebra](https://tamnd.github.io/math2/node/MCW8V4RFYJ.md): definition MCW8V4RFYJ
- [Iterated function](https://tamnd.github.io/math2/node/MCXT7ZW329.md): definition MCXT7ZW329
- [Torus knot](https://tamnd.github.io/math2/node/MD1QFAEFF0.md): definition MD1QFAEFF0
- [Crystalline cohomology](https://tamnd.github.io/math2/node/MD624VBPHA.md): definition MD624VBPHA
- [Harmonic map heat flow](https://tamnd.github.io/math2/node/MD99PYBCE6.md): concept MD99PYBCE6
- [Local homeomorphism](https://tamnd.github.io/math2/node/MDH71ZZ8XX.md): concept MDH71ZZ8XX
- [Proper group action](https://tamnd.github.io/math2/node/MDH7VXDCVF.md): concept MDH7VXDCVF
- [Grötzsch's theorem](https://tamnd.github.io/math2/node/MDHWF76S1Z.md): statement MDHWF76S1Z
- [Padé table](https://tamnd.github.io/math2/node/MDM8F48X18.md): concept MDM8F48X18
- [Flat morphism](https://tamnd.github.io/math2/node/MDPRVV9DN9.md): definition MDPRVV9DN9
- [Goldie's theorem](https://tamnd.github.io/math2/node/MDS8655N0Z.md): statement MDS8655N0Z
- [Gelfand–Kirillov dimension](https://tamnd.github.io/math2/node/MDVSY2NMD2.md): concept MDVSY2NMD2
- [Cauchy's power equation](https://tamnd.github.io/math2/node/MDWN6E58AW.md): concept MDWN6E58AW
- [Raghunathan's conjecture](https://tamnd.github.io/math2/node/ME4ZSRAT5Z.md): statement ME4ZSRAT5Z
- [Commutative magma](https://tamnd.github.io/math2/node/ME71ZQE49J.md): definition ME71ZQE49J
- [Prime ideal theorem for distributive lattices](https://tamnd.github.io/math2/node/ME8GM0VTWP.md): statement ME8GM0VTWP
- [Congruence-modular variety](https://tamnd.github.io/math2/node/MEBR2NYA1S.md): definition MEBR2NYA1S
- [Elementary class](https://tamnd.github.io/math2/node/MEC3ZMC4F1.md): concept MEC3ZMC4F1
- [Markov chain Monte Carlo](https://tamnd.github.io/math2/node/MECCMT61DV.md): definition MECCMT61DV
- [Suslin line](https://tamnd.github.io/math2/node/MEKFR4Q617.md): object MEKFR4Q617
- [Urysohn space](https://tamnd.github.io/math2/node/MF3BWZ9YSW.md): concept MF3BWZ9YSW
- [Nonnegative matrix](https://tamnd.github.io/math2/node/MF3GV42CG6.md): definition MF3GV42CG6
- [Group extension](https://tamnd.github.io/math2/node/MF4N3VGGFV.md): concept MF4N3VGGFV
- [Nested intervals theorem](https://tamnd.github.io/math2/node/MF56GK7QY2.md): statement MF56GK7QY2
- [Compactness of integral operators](https://tamnd.github.io/math2/node/MF77N00SK8.md): statement MF77N00SK8
- [Projective algebra](https://tamnd.github.io/math2/node/MF89BJZ8KS.md): definition MF89BJZ8KS
- [Multicategory](https://tamnd.github.io/math2/node/MFB1Q9JTPD.md): concept MFB1Q9JTPD
- [Configuration](https://tamnd.github.io/math2/node/MFDEMR9GMM.md): definition MFDEMR9GMM
- [Volterra equations with completely monotone kernels](https://tamnd.github.io/math2/node/MFJSS5HVA0.md): concept MFJSS5HVA0
- [Gödel numbering](https://tamnd.github.io/math2/node/MFJXPAF0R0.md): concept MFJXPAF0R0
- [Regular variation](https://tamnd.github.io/math2/node/MFM9ZXWMHA.md): concept MFM9ZXWMHA
- [Rarefaction wave](https://tamnd.github.io/math2/node/MFN4Q5SH8H.md): concept MFN4Q5SH8H
- [Strong perfect graph theorem](https://tamnd.github.io/math2/node/MFNNZQ484T.md): statement MFNNZQ484T
- [Difference set](https://tamnd.github.io/math2/node/MG1KM4S1BV.md): concept MG1KM4S1BV
- [Borel fixed-point theorem](https://tamnd.github.io/math2/node/MG20HMMYKY.md): statement MG20HMMYKY
- [Binomial proportion confidence interval](https://tamnd.github.io/math2/node/MG2NCWHQF2.md): concept MG2NCWHQF2
- [Space of bounded continuous functions](https://tamnd.github.io/math2/node/MG3X8Y1VT4.md): concept MG3X8Y1VT4
- [Bernstein–Gelfand–Ponomarev reflection functors](https://tamnd.github.io/math2/node/MG4RNMA8AR.md): concept MG4RNMA8AR
- [Hermite polynomials](https://tamnd.github.io/math2/node/MGACY98E9H.md): object MGACY98E9H
- [Cauchy-type integral](https://tamnd.github.io/math2/node/MGBKF52TQE.md): concept MGBKF52TQE
- [Lindenbaum–Tarski algebra](https://tamnd.github.io/math2/node/MGCVQXXQJH.md): definition MGCVQXXQJH
- [Alexander–Spanier cohomology](https://tamnd.github.io/math2/node/MGE14ST1YP.md): concept MGE14ST1YP
- [Discrete valuation ring](https://tamnd.github.io/math2/node/MGM3CNJW9V.md): concept MGM3CNJW9V
- [Nondegenerate critical point](https://tamnd.github.io/math2/node/MGPPCDFQDG.md): concept MGPPCDFQDG
- [Selberg sieve](https://tamnd.github.io/math2/node/MGSX6V5Y5Y.md): definition MGSX6V5Y5Y
- [Dirichlet form](https://tamnd.github.io/math2/node/MH20Q2N1AH.md): concept MH20Q2N1AH
- [Simplex algorithm](https://tamnd.github.io/math2/node/MH272QHWPM.md): concept MH272QHWPM
- [Boltzmann equation](https://tamnd.github.io/math2/node/MH6CSWD7F0.md): concept MH6CSWD7F0
- [Morse inequalities](https://tamnd.github.io/math2/node/MH7MEB0MRR.md): statement MH7MEB0MRR
- [Filtered back-projection](https://tamnd.github.io/math2/node/MHC7X73NG9.md): definition MHC7X73NG9
- [Chain map](https://tamnd.github.io/math2/node/MHF34RK5ZT.md): definition MHF34RK5ZT
- [Predictor–corrector method](https://tamnd.github.io/math2/node/MHGV8HCAQR.md): statement MHGV8HCAQR
- [Link of a singularity](https://tamnd.github.io/math2/node/MHGZEW6RBM.md): definition MHGZEW6RBM
- [Surgery theory](https://tamnd.github.io/math2/node/MHHN8386TG.md): concept MHHN8386TG
- [Loop space](https://tamnd.github.io/math2/node/MHJETA01P6.md): definition MHJETA01P6
- [Shift operator as an exponential of differentiation](https://tamnd.github.io/math2/node/MHSBP40HDC.md): statement MHSBP40HDC
- [Birkhoff–Kakutani theorem](https://tamnd.github.io/math2/node/MHSKT2Z4MX.md): statement MHSKT2Z4MX
- [Bochner integral](https://tamnd.github.io/math2/node/MHSMQ097GP.md): concept MHSMQ097GP
- [Hankel operator](https://tamnd.github.io/math2/node/MJ48S56VQ7.md): definition MJ48S56VQ7
- [Casus irreducibilis](https://tamnd.github.io/math2/node/MJ864PBRRJ.md): statement MJ864PBRRJ
- [Tight and overtwisted contact structures](https://tamnd.github.io/math2/node/MJBXNNFQGW.md): definition MJBXNNFQGW
- [I-adic topology](https://tamnd.github.io/math2/node/MJDV3PFBRT.md): definition MJDV3PFBRT
- [Relational structure](https://tamnd.github.io/math2/node/MJF9M19E3E.md): concept MJF9M19E3E
- [Ekeland's variational principle](https://tamnd.github.io/math2/node/MJGPW5EVB3.md): statement MJGPW5EVB3
- [Theorem of the cube](https://tamnd.github.io/math2/node/MJMB7SH7SD.md): statement MJMB7SH7SD
- [Stone–Čech compactification](https://tamnd.github.io/math2/node/MJP3GW3HD7.md): definition MJP3GW3HD7
- [Fourier–Motzkin elimination](https://tamnd.github.io/math2/node/MJQARBFG10.md): concept MJQARBFG10
- [Carlson symmetric forms](https://tamnd.github.io/math2/node/MK3ASPEN8A.md): concept MK3ASPEN8A
- [Dickey–Fuller test](https://tamnd.github.io/math2/node/MK4FDH3EDM.md): definition MK4FDH3EDM
- [Backward difference operator](https://tamnd.github.io/math2/node/MK5JGVDPS9.md): concept MK5JGVDPS9
- [Poisson boundary](https://tamnd.github.io/math2/node/MK827B9NDR.md): definition MK827B9NDR
- [Optimal experimental design](https://tamnd.github.io/math2/node/MKAMVQJXBP.md): definition MKAMVQJXBP
- [Aubin–Lions lemma](https://tamnd.github.io/math2/node/MKB5V9W0WQ.md): statement MKB5V9W0WQ
- [Orientation of a manifold](https://tamnd.github.io/math2/node/MKBSS15J3Z.md): definition MKBSS15J3Z
- [Kolmogorov quotient](https://tamnd.github.io/math2/node/MKFKFTPMXZ.md): concept MKFKFTPMXZ
- [Metropolis–Hastings algorithm](https://tamnd.github.io/math2/node/MKHF449E22.md): concept MKHF449E22
- [Inductive dimension](https://tamnd.github.io/math2/node/MKS9H497AM.md): concept MKS9H497AM
- [Exceptional Lie group E8](https://tamnd.github.io/math2/node/MKT1263M0A.md): example MKT1263M0A
- [Unification](https://tamnd.github.io/math2/node/MKY6BQHRJM.md): definition MKY6BQHRJM
- [Moving-average model](https://tamnd.github.io/math2/node/MM017WE5FS.md): definition MM017WE5FS
- [Borel functional calculus](https://tamnd.github.io/math2/node/MM07SS0PB4.md): definition MM07SS0PB4
- [Unitary operator](https://tamnd.github.io/math2/node/MM0EJ2GDE4.md): concept MM0EJ2GDE4
- [Cohn localization](https://tamnd.github.io/math2/node/MM0MATN1P8.md): definition MM0MATN1P8
- [Linear continuum](https://tamnd.github.io/math2/node/MM0QPT0GT0.md): concept MM0QPT0GT0
- [Numerical diffusion](https://tamnd.github.io/math2/node/MM0YNN1MJA.md): definition MM0YNN1MJA
- [Nilpotent ideal](https://tamnd.github.io/math2/node/MM3TQ1WJWF.md): definition MM3TQ1WJWF
- [Rarefaction wave](https://tamnd.github.io/math2/node/MM6TXKN90V.md): definition MM6TXKN90V
- [Schwarz triangle function](https://tamnd.github.io/math2/node/MME2BSPTZG.md): definition MME2BSPTZG
- [Kovacic's algorithm](https://tamnd.github.io/math2/node/MMEX2JZWGR.md): definition MMEX2JZWGR
- [Danzer set](https://tamnd.github.io/math2/node/MMHWS2TC2A.md): problem MMHWS2TC2A
- [Density of the graph of a discontinuous additive function](https://tamnd.github.io/math2/node/MNAWWP9J1X.md): statement MNAWWP9J1X
- [Blaschke–Santaló inequality](https://tamnd.github.io/math2/node/MNJ030R4CA.md): statement MNJ030R4CA
- [Tightness of measures](https://tamnd.github.io/math2/node/MNMJC8YVBS.md): concept MNMJC8YVBS
- [Group algebra](https://tamnd.github.io/math2/node/MNP2HBWB5Q.md): concept MNP2HBWB5Q
- [Large deviations theory](https://tamnd.github.io/math2/node/MNPWBBNCAD.md): definition MNPWBBNCAD
- [Rössler attractor](https://tamnd.github.io/math2/node/MNQXM3PX4R.md): example MNQXM3PX4R
- [Fundamental theorem of the calculus of finite differences](https://tamnd.github.io/math2/node/MNSSV4QHD1.md): statement MNSSV4QHD1
- [Simpson's rule](https://tamnd.github.io/math2/node/MNXC26G3BQ.md): concept MNXC26G3BQ
- [Helmholtz equation](https://tamnd.github.io/math2/node/MNXKTXJ35G.md): concept MNXKTXJ35G
- [Cluster analysis](https://tamnd.github.io/math2/node/MNY9YSTZAY.md): definition MNY9YSTZAY
- [Trigonometric series](https://tamnd.github.io/math2/node/MNZPMSSPD8.md): definition MNZPMSSPD8
- [Group isomorphism problem](https://tamnd.github.io/math2/node/MP3F49QT3C.md): problem MP3F49QT3C
- [Galois group of a finite field extension](https://tamnd.github.io/math2/node/MP5C9T2EYG.md): statement MP5C9T2EYG
- [Weinstein conjecture](https://tamnd.github.io/math2/node/MP6B64DS31.md): conjecture MP6B64DS31
- [Percolation threshold](https://tamnd.github.io/math2/node/MP8VSZK19A.md): concept MP8VSZK19A
- [Function approximation](https://tamnd.github.io/math2/node/MP8X04H1ZD.md): definition MP8X04H1ZD
- [Picard's criterion for first-kind equations](https://tamnd.github.io/math2/node/MPB3QF09E5.md): statement MPB3QF09E5
- [Conservative system](https://tamnd.github.io/math2/node/MPE4AWEP6K.md): concept MPE4AWEP6K
- [Hyperbolic set](https://tamnd.github.io/math2/node/MPFD24TYW2.md): concept MPFD24TYW2
- [Autoregressive conditional heteroskedasticity](https://tamnd.github.io/math2/node/MPN10SB26R.md): concept MPN10SB26R
- [Cartan–Serre finiteness theorem](https://tamnd.github.io/math2/node/MPNGD4V271.md): statement MPNGD4V271
- [Height of a poset](https://tamnd.github.io/math2/node/MPR6MH4YSQ.md): concept MPR6MH4YSQ
- [Oscillatory solution of a difference equation](https://tamnd.github.io/math2/node/MPRCFT30Z7.md): definition MPRCFT30Z7
- [Perron method](https://tamnd.github.io/math2/node/MPSZJ3E38R.md): definition MPSZJ3E38R
- [Pólya enumeration theorem](https://tamnd.github.io/math2/node/MPYQ39P01F.md): statement MPYQ39P01F
- [Simple Lie group](https://tamnd.github.io/math2/node/MQ9A7TZEQ0.md): definition MQ9A7TZEQ0
- [Bernstein–Doetsch theorem](https://tamnd.github.io/math2/node/MQA1VVCJ63.md): statement MQA1VVCJ63
- [Genus–degree formula](https://tamnd.github.io/math2/node/MQCBFACHXH.md): statement MQCBFACHXH
- [Parallelohedron](https://tamnd.github.io/math2/node/MR2JWBCZQZ.md): concept MR2JWBCZQZ
- [Galerkin method](https://tamnd.github.io/math2/node/MR4D29XAPB.md): definition MR4D29XAPB
- [Bounded inverse theorem](https://tamnd.github.io/math2/node/MR5FJXR348.md): statement MR5FJXR348
- [Baer's criterion](https://tamnd.github.io/math2/node/MR5HNV0Q07.md): statement MR5HNV0Q07
- [Projective dimension](https://tamnd.github.io/math2/node/MR9QCH1PNA.md): concept MR9QCH1PNA
- [Wiener–Hopf method](https://tamnd.github.io/math2/node/MRBGM52H94.md): definition MRBGM52H94
- [Filter](https://tamnd.github.io/math2/node/MRBRH9CH7E.md): definition MRBRH9CH7E
- [Circumsphere](https://tamnd.github.io/math2/node/MRC4WMVTAH.md): concept MRC4WMVTAH
- [Generalized method of moments](https://tamnd.github.io/math2/node/MRCBJ0WDR0.md): definition MRCBJ0WDR0
- [Bounded mean oscillation](https://tamnd.github.io/math2/node/MRDZAR4X0N.md): definition MRDZAR4X0N
- [Grauert's semicontinuity theorem](https://tamnd.github.io/math2/node/MREZVW7JTW.md): statement MREZVW7JTW
- [Hypernetted-chain equation](https://tamnd.github.io/math2/node/MRJ2FHX6K1.md): example MRJ2FHX6K1
- [Gauss map](https://tamnd.github.io/math2/node/MRKVM2VVNN.md): example MRKVM2VVNN
- [Iwasawa theory](https://tamnd.github.io/math2/node/MRRQEEP3T8.md): definition MRRQEEP3T8
- [Doob's martingale convergence theorems](https://tamnd.github.io/math2/node/MRXBC4ZNRR.md): statement MRXBC4ZNRR
- [Duality in order theory](https://tamnd.github.io/math2/node/MRYFDAG5G8.md): concept MRYFDAG5G8
- [Lambert quadrilateral](https://tamnd.github.io/math2/node/MS263NBTRR.md): concept MS263NBTRR
- [Unary algebra](https://tamnd.github.io/math2/node/MS5PXY4D9Y.md): definition MS5PXY4D9Y
- [Ruelle zeta function](https://tamnd.github.io/math2/node/MS8G1FD8FD.md): definition MS8G1FD8FD
- [Parametric model](https://tamnd.github.io/math2/node/MSC0MR8VMH.md): concept MSC0MR8VMH
- [Random integral equation](https://tamnd.github.io/math2/node/MSDC626JK5.md): definition MSDC626JK5
- [Fourier integral operator](https://tamnd.github.io/math2/node/MSMD2BS9C5.md): definition MSMD2BS9C5
- [Uniformly most powerful test](https://tamnd.github.io/math2/node/MSN9TFS59W.md): definition MSN9TFS59W
- [Compactness of families of harmonic functions](https://tamnd.github.io/math2/node/MSPNPVP9NK.md): statement MSPNPVP9NK
- [Willmore flow](https://tamnd.github.io/math2/node/MSRAD0X7H4.md): concept MSRAD0X7H4
- [Oka coherence theorem](https://tamnd.github.io/math2/node/MSX3VKM9Z5.md): statement MSX3VKM9Z5
- [Zeros of Bessel functions](https://tamnd.github.io/math2/node/MSXT31P944.md): definition MSXT31P944
- [Arity](https://tamnd.github.io/math2/node/MSXX2P86CY.md): concept MSXX2P86CY
- [Erlangen program](https://tamnd.github.io/math2/node/MSYQ5YEPN5.md): definition MSYQ5YEPN5
- [Mittag-Leffler summation](https://tamnd.github.io/math2/node/MT5JZ63RD4.md): concept MT5JZ63RD4
- [Model theory of algebraically closed fields](https://tamnd.github.io/math2/node/MT8TDVPBXB.md): concept MT8TDVPBXB
- [Elliptic operator on a compact manifold](https://tamnd.github.io/math2/node/MT9P2X85V8.md): concept MT9P2X85V8
- [Upwind scheme](https://tamnd.github.io/math2/node/MTCRZC5QKT.md): concept MTCRZC5QKT
- [Global analysis](https://tamnd.github.io/math2/node/MTEW1M61QK.md): concept MTEW1M61QK
- [Mikusiński operational calculus](https://tamnd.github.io/math2/node/MTM92ND7HE.md): definition MTM92ND7HE
- [Fully nonlinear PDE](https://tamnd.github.io/math2/node/MTQ3SGX56W.md): definition MTQ3SGX56W
- [Algebraic space](https://tamnd.github.io/math2/node/MTV9TFKF3X.md): concept MTV9TFKF3X
- [Galois correspondence for coverings](https://tamnd.github.io/math2/node/MTWNAJMH1D.md): statement MTWNAJMH1D
- [Complex powers of elliptic operators](https://tamnd.github.io/math2/node/MTZCFEV37Y.md): concept MTZCFEV37Y
- [Zaslavsky's theorem](https://tamnd.github.io/math2/node/MV1CFTH98T.md): statement MV1CFTH98T
- [Smith normal form](https://tamnd.github.io/math2/node/MV28Z5N9XX.md): statement MV28Z5N9XX
- [Plücker coordinates](https://tamnd.github.io/math2/node/MV3PHCJ6CE.md): definition MV3PHCJ6CE
- [Gauss–Kronrod quadrature formula](https://tamnd.github.io/math2/node/MV45MQ12YE.md): definition MV45MQ12YE
- [Busemann–Petty problem](https://tamnd.github.io/math2/node/MV58A1F52E.md): problem MV58A1F52E
- [McShane–Whitney extension](https://tamnd.github.io/math2/node/MV7ADC2DEH.md): statement MV7ADC2DEH
- [Mellin transform method](https://tamnd.github.io/math2/node/MVCENBR5M1.md): definition MVCENBR5M1
- [Model](https://tamnd.github.io/math2/node/MVCH6N9SNW.md): concept MVCH6N9SNW
- [Krull dimension](https://tamnd.github.io/math2/node/MVGBS3J1P7.md): concept MVGBS3J1P7
- [Bockstein spectral sequence](https://tamnd.github.io/math2/node/MVHCYMKFY9.md): statement MVHCYMKFY9
- [Center of a ring](https://tamnd.github.io/math2/node/MVT64B3B6H.md): concept MVT64B3B6H
- [Harmonic polynomial](https://tamnd.github.io/math2/node/MVYFSYMECG.md): concept MVYFSYMECG
- [Sets of capacity zero](https://tamnd.github.io/math2/node/MVYTPJ3V05.md): definition MVYTPJ3V05
- [Singular control](https://tamnd.github.io/math2/node/MW198BJJNK.md): definition MW198BJJNK
- [Runge domain](https://tamnd.github.io/math2/node/MW1QAB5AN4.md): definition MW1QAB5AN4
- [Locally finite measure](https://tamnd.github.io/math2/node/MW5P4P4S9A.md): definition MW5P4P4S9A
- [Lippmann–Schwinger equation](https://tamnd.github.io/math2/node/MW5R19P2X4.md): example MW5R19P2X4
- [Extended finite element method](https://tamnd.github.io/math2/node/MWB6Y5Q4VK.md): concept MWB6Y5Q4VK
- [Algebraic number field](https://tamnd.github.io/math2/node/MWBDFS2KJ5.md): concept MWBDFS2KJ5
- [Real projective plane](https://tamnd.github.io/math2/node/MWDB7RACWQ.md): example MWDB7RACWQ
- [Sum-free set](https://tamnd.github.io/math2/node/MWGG1P8WDC.md): definition MWGG1P8WDC
- [Congruent number problem](https://tamnd.github.io/math2/node/MWMFQR8SMA.md): problem MWMFQR8SMA
- [Computably enumerable set](https://tamnd.github.io/math2/node/MWSD85BB4X.md): definition MWSD85BB4X
- [Suspension flow](https://tamnd.github.io/math2/node/MX16DQ5Q8G.md): concept MX16DQ5Q8G
- [Commuting matrices](https://tamnd.github.io/math2/node/MX26N71ZD3.md): definition MX26N71ZD3
- [Piecewise linear manifold](https://tamnd.github.io/math2/node/MX38D5Z5NY.md): definition MX38D5Z5NY
- [Dedekind eta function](https://tamnd.github.io/math2/node/MX3CVYMNDV.md): concept MX3CVYMNDV
- [Bernstein inequalities](https://tamnd.github.io/math2/node/MX6157RAVA.md): statement MX6157RAVA
- [Principal symbol](https://tamnd.github.io/math2/node/MX6D07DDAZ.md): definition MX6D07DDAZ
- [Anderson localization](https://tamnd.github.io/math2/node/MXEZM0QJ1K.md): concept MXEZM0QJ1K
- [Dowker space](https://tamnd.github.io/math2/node/MXHTKVKB09.md): concept MXHTKVKB09
- [Kendall rank correlation coefficient](https://tamnd.github.io/math2/node/MXNDW6AFGZ.md): definition MXNDW6AFGZ
- [Siegel modular form](https://tamnd.github.io/math2/node/MXPTXV340K.md): concept MXPTXV340K
- [Complex Monge–Ampère operator](https://tamnd.github.io/math2/node/MXRC891TQS.md): concept MXRC891TQS
- [Mordell–Weil theorem](https://tamnd.github.io/math2/node/MXRCPAH09H.md): statement MXRCPAH09H
- [Complex Monge–Ampère operator](https://tamnd.github.io/math2/node/MXX766RYDX.md): definition MXX766RYDX
- [Semisimple ring](https://tamnd.github.io/math2/node/MXXTYV1J2N.md): definition MXXTYV1J2N
- [Perimeter of a set](https://tamnd.github.io/math2/node/MXZHMC2P99.md): definition MXZHMC2P99
- [Octahedral axiom](https://tamnd.github.io/math2/node/MY2XG7XTS8.md): definition MY2XG7XTS8
- [Wallpaper group](https://tamnd.github.io/math2/node/MY3CBHARYQ.md): example MY3CBHARYQ
- [Hermite–Genocchi formula](https://tamnd.github.io/math2/node/MY491MR2PJ.md): statement MY491MR2PJ
- [Perfect field](https://tamnd.github.io/math2/node/MY8BG4EE0F.md): definition MY8BG4EE0F
- [Censoring](https://tamnd.github.io/math2/node/MY8ZZP84NP.md): concept MY8ZZP84NP
- [Unit and counit](https://tamnd.github.io/math2/node/MYGWT305AB.md): concept MYGWT305AB
- [Morse function](https://tamnd.github.io/math2/node/MYQN2FX9SB.md): definition MYQN2FX9SB
- [Spectral method](https://tamnd.github.io/math2/node/MYV657YRDW.md): concept MYV657YRDW
- [Local ring](https://tamnd.github.io/math2/node/MZ02BKPHJA.md): concept MZ02BKPHJA
- [Chevalley's structure theorem](https://tamnd.github.io/math2/node/MZ6XDATQKR.md): statement MZ6XDATQKR
- [Integral equation of the third kind](https://tamnd.github.io/math2/node/MZHSEZDK3E.md): definition MZHSEZDK3E
- [Interchange of summation order](https://tamnd.github.io/math2/node/MZM1AFXY82.md): statement MZM1AFXY82
- [Semimartingale](https://tamnd.github.io/math2/node/MZNFPJJMY9.md): definition MZNFPJJMY9
- [Analytic continuation in several variables](https://tamnd.github.io/math2/node/MZT3Y1X53W.md): concept MZT3Y1X53W
- [Analytic torsion](https://tamnd.github.io/math2/node/MZT9D34B0A.md): concept MZT9D34B0A
- [Infinitary operation](https://tamnd.github.io/math2/node/MZW52JVTBX.md): definition MZW52JVTBX
- [Inverse Mellin transform as inverse Fourier transform](https://tamnd.github.io/math2/node/MZXHSGFBRF.md): statement MZXHSGFBRF
- [Mittag-Leffler function](https://tamnd.github.io/math2/node/MZZ56W92DY.md): definition MZZ56W92DY
- [Proportional hazards model](https://tamnd.github.io/math2/node/N029V3JX60.md): concept N029V3JX60
- [Jackson integral](https://tamnd.github.io/math2/node/N05N7W1BAB.md): definition N05N7W1BAB
- [Displacement convexity](https://tamnd.github.io/math2/node/N05WC9ZCWA.md): concept N05WC9ZCWA
- [Crisis](https://tamnd.github.io/math2/node/N087STXZKG.md): concept N087STXZKG
- [Torsion of a curve](https://tamnd.github.io/math2/node/N09YAJWA7E.md): definition N09YAJWA7E
- [Haar system](https://tamnd.github.io/math2/node/N0B51FAA9S.md): concept N0B51FAA9S
- [Vitali–Hahn–Saks theorem](https://tamnd.github.io/math2/node/N0K0YWPQN6.md): statement N0K0YWPQN6
- [Pluripolar set](https://tamnd.github.io/math2/node/N0MAGPW8AP.md): concept N0MAGPW8AP
- [Integrable system](https://tamnd.github.io/math2/node/N0N7CMRCMH.md): concept N0N7CMRCMH
- [Resurgent function](https://tamnd.github.io/math2/node/N0QW1RGPR8.md): definition N0QW1RGPR8
- [Operator monotone function](https://tamnd.github.io/math2/node/N155BSJ2DN.md): concept N155BSJ2DN
- [Large sieve](https://tamnd.github.io/math2/node/N15YK7XG1H.md): concept N15YK7XG1H
- [Uniserial module](https://tamnd.github.io/math2/node/N1H5NV5X0Z.md): definition N1H5NV5X0Z
- [Formalism](https://tamnd.github.io/math2/node/N1N5N2QJ86.md): definition N1N5N2QJ86
- [Empirical distribution function](https://tamnd.github.io/math2/node/N1NA5MEFWX.md): definition N1NA5MEFWX
- [Eulerian poset](https://tamnd.github.io/math2/node/N1Q0BJAEFX.md): concept N1Q0BJAEFX
- [Cohen–Macaulay ring](https://tamnd.github.io/math2/node/N1QJQEGBFC.md): concept N1QJQEGBFC
- [Hilbert's irreducibility theorem](https://tamnd.github.io/math2/node/N1V3YEYHNP.md): statement N1V3YEYHNP
- [Killing–Hopf theorem](https://tamnd.github.io/math2/node/N1WJR10WX0.md): statement N1WJR10WX0
- [Fitting subgroup](https://tamnd.github.io/math2/node/N219CDB623.md): concept N219CDB623
- [Risk function](https://tamnd.github.io/math2/node/N26TSVRH2S.md): concept N26TSVRH2S
- [Andrews–Curtis conjecture](https://tamnd.github.io/math2/node/N298DRYE0W.md): conjecture N298DRYE0W
- [Laplace transform tends to zero at infinity](https://tamnd.github.io/math2/node/N2JFN9VHD9.md): statement N2JFN9VHD9
- [Subdivision surface](https://tamnd.github.io/math2/node/N2JTWEJ1F9.md): concept N2JTWEJ1F9
- [Complete lattice](https://tamnd.github.io/math2/node/N2Q4RS43FD.md): definition N2Q4RS43FD
- [Circle map](https://tamnd.github.io/math2/node/N2SVK69S53.md): concept N2SVK69S53
- [Bootstrap](https://tamnd.github.io/math2/node/N2TSHRCJ1Y.md): definition N2TSHRCJ1Y
- [Projective geometry](https://tamnd.github.io/math2/node/N32Z68F0D3.md): concept N32Z68F0D3
- [Preadditive category](https://tamnd.github.io/math2/node/N3457SJ63D.md): definition N3457SJ63D
- [Free functor](https://tamnd.github.io/math2/node/N353JF8RBK.md): concept N353JF8RBK
- [Curtis–Hedlund–Lyndon theorem](https://tamnd.github.io/math2/node/N35QM9G83G.md): statement N35QM9G83G
- [Whitney sum](https://tamnd.github.io/math2/node/N35SYWJMA9.md): definition N35SYWJMA9
- [Decomposition group](https://tamnd.github.io/math2/node/N36F259F1B.md): definition N36F259F1B
- [Localization at a prime ideal](https://tamnd.github.io/math2/node/N37SPDWYGS.md): definition N37SPDWYGS
- [Regularity theory](https://tamnd.github.io/math2/node/N383CX7WKS.md): concept N383CX7WKS
- [Hankel functions](https://tamnd.github.io/math2/node/N3CC5DRGT1.md): concept N3CC5DRGT1
- [Semisimple module](https://tamnd.github.io/math2/node/N3MRHDKDGC.md): concept N3MRHDKDGC
- [Ideal sheaf](https://tamnd.github.io/math2/node/N3PGYJJQ7W.md): concept N3PGYJJQ7W
- [Varifold](https://tamnd.github.io/math2/node/N3QA80M44M.md): concept N3QA80M44M
- [Serre duality](https://tamnd.github.io/math2/node/N3QPQATSRN.md): statement N3QPQATSRN
- [Pochhammer symbol](https://tamnd.github.io/math2/node/N3XB4SYQ6B.md): definition N3XB4SYQ6B
- [Cohen structure theorem](https://tamnd.github.io/math2/node/N3XH42KVV3.md): statement N3XH42KVV3
- [Gauss–Markov theorem](https://tamnd.github.io/math2/node/N3YMNMWANM.md): statement N3YMNMWANM
- [Semianalytic set](https://tamnd.github.io/math2/node/N3Z69FTQ7J.md): definition N3Z69FTQ7J
- [Fefferman duality theorem](https://tamnd.github.io/math2/node/N41N5QHDAN.md): statement N41N5QHDAN
- [Hardy–Littlewood Tauberian theorem](https://tamnd.github.io/math2/node/N42JDBT0VA.md): statement N42JDBT0VA
- [Dedekind's lemma on independence of characters](https://tamnd.github.io/math2/node/N43G50YHG2.md): statement N43G50YHG2
- [Approximate point spectrum](https://tamnd.github.io/math2/node/N44162Y9XS.md): concept N44162Y9XS
- [Pseudoholomorphic curve](https://tamnd.github.io/math2/node/N469E47V75.md): concept N469E47V75
- [Mean-periodic function](https://tamnd.github.io/math2/node/N470GNCK2Y.md): concept N470GNCK2Y
- [Ackermann function](https://tamnd.github.io/math2/node/N470JF7ZJX.md): example N470JF7ZJX
- [Proof theory](https://tamnd.github.io/math2/node/N4EDJNMTEF.md): concept N4EDJNMTEF
- [Holonomic function](https://tamnd.github.io/math2/node/N4FACX7QSW.md): definition N4FACX7QSW
- [Measurable space](https://tamnd.github.io/math2/node/N4KDT83V3S.md): concept N4KDT83V3S
- [Nonlinear integral equation](https://tamnd.github.io/math2/node/N4NTJ6W38N.md): definition N4NTJ6W38N
- [Minkowski inequality](https://tamnd.github.io/math2/node/N4QZS912MT.md): statement N4QZS912MT
- [Nest algebra](https://tamnd.github.io/math2/node/N528CFDN46.md): definition N528CFDN46
- [Ordered geometry](https://tamnd.github.io/math2/node/N56DHWH283.md): definition N56DHWH283
- [Order polytope](https://tamnd.github.io/math2/node/N587ECTQ4X.md): concept N587ECTQ4X
- [Space of functions vanishing at infinity](https://tamnd.github.io/math2/node/N59BW1QFD1.md): concept N59BW1QFD1
- [Degenerate kernel](https://tamnd.github.io/math2/node/N5JCKN50NE.md): definition N5JCKN50NE
- [Polyharmonic function](https://tamnd.github.io/math2/node/N5JN15Z5J6.md): definition N5JN15Z5J6
- [Quotient algebra](https://tamnd.github.io/math2/node/N5KE7FH6CX.md): concept N5KE7FH6CX
- [Elliptic hypergeometric series](https://tamnd.github.io/math2/node/N5MRMYN61N.md): definition N5MRMYN61N
- [The cosets of a subgroup that is not normal do not form a group](https://tamnd.github.io/math2/node/N5NTPAR2KT.md): example N5NTPAR2KT
- [Liouvillian function](https://tamnd.github.io/math2/node/N5PM2NCPWF.md): concept N5PM2NCPWF
- [Mellin transform](https://tamnd.github.io/math2/node/N5PVGRJE1T.md): concept N5PVGRJE1T
- [Geometric morphism](https://tamnd.github.io/math2/node/N5R1WAEPFB.md): concept N5R1WAEPFB
- [Divided differences](https://tamnd.github.io/math2/node/N5ZHV4HWDC.md): concept N5ZHV4HWDC
- [Hyperplane arrangement](https://tamnd.github.io/math2/node/N602W0WAVT.md): concept N602W0WAVT
- [Smale horseshoe](https://tamnd.github.io/math2/node/N63Z6EA6YS.md): example N63Z6EA6YS
- [Schur's lemma in abelian categories](https://tamnd.github.io/math2/node/N67R59ZVC4.md): statement N67R59ZVC4
- [Thom isomorphism](https://tamnd.github.io/math2/node/N685RXHDZX.md): statement N685RXHDZX
- [Rounding](https://tamnd.github.io/math2/node/N6AFVYBY2Z.md): definition N6AFVYBY2Z
- [Algebraic statistics](https://tamnd.github.io/math2/node/N6BDTP0AZR.md): definition N6BDTP0AZR
- [Adjunction space](https://tamnd.github.io/math2/node/N6DQ3Q0HBT.md): concept N6DQ3Q0HBT
- [Boltzmann equation](https://tamnd.github.io/math2/node/N6DZXEGD9D.md): definition N6DZXEGD9D
- [Complete reducibility of representations of compact groups](https://tamnd.github.io/math2/node/N6JC6KNXJJ.md): statement N6JC6KNXJJ
- [Basic open set](https://tamnd.github.io/math2/node/N6JX9BCKQP.md): concept N6JX9BCKQP
- [Fractional Sobolev space](https://tamnd.github.io/math2/node/N6NX1VT837.md): concept N6NX1VT837
- [Perfect map](https://tamnd.github.io/math2/node/N6TE6WJ4EZ.md): concept N6TE6WJ4EZ
- [Sunflower lemma](https://tamnd.github.io/math2/node/N6TH42TD7F.md): statement N6TH42TD7F
- [Linearly disjoint extensions](https://tamnd.github.io/math2/node/N6YZFG66YP.md): concept N6YZFG66YP
- [Linear and stable summation methods](https://tamnd.github.io/math2/node/N6Z14G81VF.md): concept N6Z14G81VF
- [Minimal prime ideal](https://tamnd.github.io/math2/node/N730JMCM7N.md): definition N730JMCM7N
- [Gradient estimates for harmonic functions](https://tamnd.github.io/math2/node/N76R7JWMNJ.md): statement N76R7JWMNJ
- [Homomorphism theorem](https://tamnd.github.io/math2/node/N783XNJNBV.md): statement N783XNJNBV
- [Brouwer fixed-point theorem](https://tamnd.github.io/math2/node/N7AAJ2MW5T.md): statement N7AAJ2MW5T
- [Shadowing lemma](https://tamnd.github.io/math2/node/N7C06T3KCZ.md): statement N7C06T3KCZ
- [Cap product](https://tamnd.github.io/math2/node/N7SXBQ7WK0.md): concept N7SXBQ7WK0
- [Double complex](https://tamnd.github.io/math2/node/N80ABTQHEQ.md): concept N80ABTQHEQ
- [Combinatorics on words](https://tamnd.github.io/math2/node/N815PEPPFA.md): definition N815PEPPFA
- [Lawvere's fixed-point theorem](https://tamnd.github.io/math2/node/N84M16W19X.md): statement N84M16W19X
- [Depth](https://tamnd.github.io/math2/node/N8G4ZFKP0X.md): definition N8G4ZFKP0X
- [Geometric algebra](https://tamnd.github.io/math2/node/N8H460W3E2.md): concept N8H460W3E2
- [Disintegration theorem](https://tamnd.github.io/math2/node/N8NXNCMSK5.md): statement N8NXNCMSK5
- [Adjoint of an unbounded operator](https://tamnd.github.io/math2/node/N8SFMS441V.md): definition N8SFMS441V
- [Fréchet space](https://tamnd.github.io/math2/node/N8TGDDJCFJ.md): definition N8TGDDJCFJ
- [Hopf module](https://tamnd.github.io/math2/node/N8X7XJBKT0.md): concept N8X7XJBKT0
- [Stanley–Reisner ring](https://tamnd.github.io/math2/node/N8YJAFVQRW.md): definition N8YJAFVQRW
- [Cesàro summation](https://tamnd.github.io/math2/node/N8ZGG7EW6J.md): concept N8ZGG7EW6J
- [Sober space](https://tamnd.github.io/math2/node/N8ZY3A50HK.md): definition N8ZY3A50HK
- [Fefferman's theorem on extension of biholomorphisms](https://tamnd.github.io/math2/node/N95NQ0G3A6.md): statement N95NQ0G3A6
- [Finite morphism](https://tamnd.github.io/math2/node/N9C4X7XKE4.md): concept N9C4X7XKE4
- [Heteroclinic orbit](https://tamnd.github.io/math2/node/N9GYM69ZWM.md): concept N9GYM69ZWM
- [Term operation](https://tamnd.github.io/math2/node/N9M2N27END.md): concept N9M2N27END
- [Chebyshev set](https://tamnd.github.io/math2/node/N9QDDYY7GK.md): definition N9QDDYY7GK
- [Anderson–Darling test](https://tamnd.github.io/math2/node/N9ZB0KMV0B.md): concept N9ZB0KMV0B
- [Lebesgue constant](https://tamnd.github.io/math2/node/N9ZCXH6HV1.md): definition N9ZCXH6HV1
- [Lacunary series](https://tamnd.github.io/math2/node/NA3J9XQYGP.md): concept NA3J9XQYGP
- [Idempotent semiring](https://tamnd.github.io/math2/node/NA44T2NZE2.md): definition NA44T2NZE2
- [Minimal harmonic function](https://tamnd.github.io/math2/node/NA4BTZ6WRB.md): concept NA4BTZ6WRB
- [Weighted least squares](https://tamnd.github.io/math2/node/NA5X6JVK5J.md): definition NA5X6JVK5J
- [Positive element](https://tamnd.github.io/math2/node/NA8DF5X3M4.md): concept NA8DF5X3M4
- [Minimax approximation](https://tamnd.github.io/math2/node/NABGFH3WS1.md): definition NABGFH3WS1
- [Trajectory optimization](https://tamnd.github.io/math2/node/NAD3RQNJ7V.md): definition NAD3RQNJ7V
- [Infinitesimal generator of a Markov process](https://tamnd.github.io/math2/node/NAJ9Y60PF6.md): definition NAJ9Y60PF6
- [Quaternion-Kähler manifold](https://tamnd.github.io/math2/node/NAJBH8GJMM.md): concept NAJBH8GJMM
- [Hunt's extension of Carleson's theorem](https://tamnd.github.io/math2/node/NAX51M6025.md): statement NAX51M6025
- [Brauer's main theorems](https://tamnd.github.io/math2/node/NB057FEAAE.md): statement NB057FEAAE
- [Distribution function of a measurable function](https://tamnd.github.io/math2/node/NB0HE0TE1R.md): definition NB0HE0TE1R
- [Spheroidal wave function](https://tamnd.github.io/math2/node/NB0NAQSKJZ.md): concept NB0NAQSKJZ
- [Osgood's lemma](https://tamnd.github.io/math2/node/NBFNXA2WFM.md): statement NBFNXA2WFM
- [Sign test](https://tamnd.github.io/math2/node/NBK6G2VB7A.md): concept NBK6G2VB7A
- [Tomita–Takesaki theory](https://tamnd.github.io/math2/node/NBR81PR7E5.md): concept NBR81PR7E5
- [Independent component analysis](https://tamnd.github.io/math2/node/NBX8FZY516.md): concept NBX8FZY516
- [Positive solutions of integral equations](https://tamnd.github.io/math2/node/NC1C7EKV1V.md): definition NC1C7EKV1V
- [Sweedler notation](https://tamnd.github.io/math2/node/NC26CE0Y8B.md): concept NC26CE0Y8B
- [Tate module](https://tamnd.github.io/math2/node/NC4B8KC479.md): definition NC4B8KC479
- [Wiener's lemma](https://tamnd.github.io/math2/node/NC768TWZ3D.md): statement NC768TWZ3D
- [Euler class](https://tamnd.github.io/math2/node/NC79GZ12QZ.md): definition NC79GZ12QZ
- [Operator system](https://tamnd.github.io/math2/node/NCB8AMYKCV.md): definition NCB8AMYKCV
- [Equidistribution theorem](https://tamnd.github.io/math2/node/NCCGNTBEYV.md): statement NCCGNTBEYV
- [Substitution](https://tamnd.github.io/math2/node/NCDZ969V7X.md): concept NCDZ969V7X
- [Floer homology](https://tamnd.github.io/math2/node/NCKFPAGHXF.md): concept NCKFPAGHXF
- [Quasitriangular Hopf algebra](https://tamnd.github.io/math2/node/NCRPWS9W5R.md): concept NCRPWS9W5R
- [Clarke generalized gradient](https://tamnd.github.io/math2/node/NCWYTR6PH0.md): definition NCWYTR6PH0
- [Desarguesian plane](https://tamnd.github.io/math2/node/NCZP5MPR9Z.md): definition NCZP5MPR9Z
- [Fourier–Motzkin elimination](https://tamnd.github.io/math2/node/NCZZ90R5Y6.md): definition NCZZ90R5Y6
- [Holonomic sequence](https://tamnd.github.io/math2/node/ND1CBJS6V2.md): definition ND1CBJS6V2
- [Noetherian scheme](https://tamnd.github.io/math2/node/ND6XEY68P4.md): concept ND6XEY68P4
- [Lindelöf summation](https://tamnd.github.io/math2/node/ND7KXXEDJ1.md): definition ND7KXXEDJ1
- [Lehmer's conjecture](https://tamnd.github.io/math2/node/NDRRH2YD2C.md): conjecture NDRRH2YD2C
- [Unit in the last place](https://tamnd.github.io/math2/node/NDS0H727BA.md): definition NDS0H727BA
- [Wilkinson's polynomial](https://tamnd.github.io/math2/node/NDVJ9VWA71.md): example NDVJ9VWA71
- [Dyson conjecture](https://tamnd.github.io/math2/node/NDYA85381W.md): statement NDYA85381W
- [Petersson inner product](https://tamnd.github.io/math2/node/NDYRB4MQZT.md): definition NDYRB4MQZT
- [Surface of revolution](https://tamnd.github.io/math2/node/NE09EDK19V.md): example NE09EDK19V
- [Exponential family](https://tamnd.github.io/math2/node/NE0BW8Q2PX.md): definition NE0BW8Q2PX
- [Algebraic cycle](https://tamnd.github.io/math2/node/NE0XB0AEDR.md): definition NE0XB0AEDR
- [Incidence algebra](https://tamnd.github.io/math2/node/NE75TQGG79.md): definition NE75TQGG79
- [Logarithmically convex domain](https://tamnd.github.io/math2/node/NEAE07CSM4.md): definition NEAE07CSM4
- [Influence function](https://tamnd.github.io/math2/node/NEB29GSM22.md): definition NEB29GSM22
- [Śleszyński–Pringsheim theorem](https://tamnd.github.io/math2/node/NEC48RHNN0.md): statement NEC48RHNN0
- [Hyperfunction](https://tamnd.github.io/math2/node/NECFJN169C.md): concept NECFJN169C
- [Burgers' equation](https://tamnd.github.io/math2/node/NEGHXN8RE1.md): concept NEGHXN8RE1
- [Categorical logic](https://tamnd.github.io/math2/node/NES4B9CWW4.md): concept NES4B9CWW4
- [Bézier curve](https://tamnd.github.io/math2/node/NEXBTGYYRG.md): concept NEXBTGYYRG
- [Dulac's theorem on finiteness of limit cycles](https://tamnd.github.io/math2/node/NEYHC1XZ14.md): statement NEYHC1XZ14
- [Completely positive map](https://tamnd.github.io/math2/node/NEZYP9HJB9.md): definition NEZYP9HJB9
- [Monotone operator method for Hammerstein equations](https://tamnd.github.io/math2/node/NF0F8M86W4.md): statement NF0F8M86W4
- [Premeasure](https://tamnd.github.io/math2/node/NF2GB0TFEW.md): definition NF2GB0TFEW
- [Y transform](https://tamnd.github.io/math2/node/NF61KYXFXF.md): definition NF61KYXFXF
- [Orlicz–Pettis theorem](https://tamnd.github.io/math2/node/NFCQYZ3GRM.md): statement NFCQYZ3GRM
- [Atomless Boolean algebra](https://tamnd.github.io/math2/node/NFEEGWJFCK.md): concept NFEEGWJFCK
- [Three-term recurrence relation](https://tamnd.github.io/math2/node/NFJTRCRB9W.md): concept NFJTRCRB9W
- [Krein space](https://tamnd.github.io/math2/node/NFN044536P.md): concept NFN044536P
- [Axiom of regularity](https://tamnd.github.io/math2/node/NFQ9FVCP31.md): concept NFQ9FVCP31
- [Intersection form](https://tamnd.github.io/math2/node/NFTS5A3NXE.md): definition NFTS5A3NXE
- [Functional equation of the Riemann zeta function](https://tamnd.github.io/math2/node/NFXKNDJX9C.md): statement NFXKNDJX9C
- [Discriminant of a number field](https://tamnd.github.io/math2/node/NG1F0HYC5G.md): definition NG1F0HYC5G
- [Steinhaus theorem on regular methods](https://tamnd.github.io/math2/node/NG1FPJQHA1.md): statement NG1FPJQHA1
- [Survival function](https://tamnd.github.io/math2/node/NG2QE06MP1.md): definition NG2QE06MP1
- [Stopped process](https://tamnd.github.io/math2/node/NG35MGYDZR.md): concept NG35MGYDZR
- [Heyting algebra](https://tamnd.github.io/math2/node/NG943XNQWH.md): concept NG943XNQWH
- [Subalgebra](https://tamnd.github.io/math2/node/NG9P9QC1NG.md): definition NG9P9QC1NG
- [Inverse tangent integral](https://tamnd.github.io/math2/node/NGGC4TTYNR.md): concept NGGC4TTYNR
- [Umbilical point](https://tamnd.github.io/math2/node/NGHWJGCXS9.md): definition NGHWJGCXS9
- [Kripke semantics](https://tamnd.github.io/math2/node/NGHWZ9Y7ZV.md): concept NGHWZ9Y7ZV
- [Swallowtail catastrophe](https://tamnd.github.io/math2/node/NGQMPVNZDP.md): example NGQMPVNZDP
- [Coercive functional](https://tamnd.github.io/math2/node/NGT5YCVAMR.md): concept NGT5YCVAMR
- [Krylov–Bogolyubov averaging](https://tamnd.github.io/math2/node/NGYZK893VE.md): definition NGYZK893VE
- [Well-quasi-ordering](https://tamnd.github.io/math2/node/NH25G6GC07.md): concept NH25G6GC07
- [Radial basis function interpolation](https://tamnd.github.io/math2/node/NH59FNJQG1.md): definition NH59FNJQG1
- [Pfaffian](https://tamnd.github.io/math2/node/NH5HWBPWS9.md): concept NH5HWBPWS9
- [Partition of unity](https://tamnd.github.io/math2/node/NH739XXVXK.md): concept NH739XXVXK
- [Liouville's theorem on conformal mappings](https://tamnd.github.io/math2/node/NH7J4YQMQC.md): statement NH7J4YQMQC
- [Finite geometry](https://tamnd.github.io/math2/node/NH7S9CHHBN.md): definition NH7S9CHHBN
- [Essential spectrum](https://tamnd.github.io/math2/node/NHB3ZMN2BN.md): definition NHB3ZMN2BN
- [Poincaré homology sphere](https://tamnd.github.io/math2/node/NHERCK7C43.md): example NHERCK7C43
- [Gosper's algorithm](https://tamnd.github.io/math2/node/NHMJQBS04M.md): concept NHMJQBS04M
- [Variety of lattices](https://tamnd.github.io/math2/node/NHR4CYEQCE.md): example NHR4CYEQCE
- [Flag variety](https://tamnd.github.io/math2/node/NHRWCQQKTH.md): concept NHRWCQQKTH
- [Heegner point](https://tamnd.github.io/math2/node/NHS23SW780.md): definition NHS23SW780
- [Regular polytope](https://tamnd.github.io/math2/node/NHVQ6G3Z85.md): definition NHVQ6G3Z85
- [Surgery theory](https://tamnd.github.io/math2/node/NHY3HE7RCN.md): definition NHY3HE7RCN
- [Lyapunov exponent](https://tamnd.github.io/math2/node/NJ1FJSNK6A.md): concept NJ1FJSNK6A
- [Oseledets theorem](https://tamnd.github.io/math2/node/NJ1PDJ3S4R.md): statement NJ1PDJ3S4R
- [Connection on a principal bundle](https://tamnd.github.io/math2/node/NJ21PN5ZGJ.md): concept NJ21PN5ZGJ
- [Residue field](https://tamnd.github.io/math2/node/NJ3APQ8BHH.md): concept NJ3APQ8BHH
- [Genus](https://tamnd.github.io/math2/node/NJ787GZS1N.md): concept NJ787GZS1N
- [Itô calculus](https://tamnd.github.io/math2/node/NJEPVG39EZ.md): definition NJEPVG39EZ
- [Minkowski problem](https://tamnd.github.io/math2/node/NJP0AZ8R4K.md): problem NJP0AZ8R4K
- [Continuum](https://tamnd.github.io/math2/node/NJQ196M1WR.md): concept NJQ196M1WR
- [Reduced homology](https://tamnd.github.io/math2/node/NJQMWVK3QK.md): concept NJQMWVK3QK
- [Tonelli's existence theorem](https://tamnd.github.io/math2/node/NJQRG3DVPP.md): statement NJQRG3DVPP
- [Ordinal sum of posets](https://tamnd.github.io/math2/node/NJY30DQG3H.md): definition NJY30DQG3H
- [Boolean model](https://tamnd.github.io/math2/node/NK3CZTVEJD.md): concept NK3CZTVEJD
- [Moduli of algebraic curves](https://tamnd.github.io/math2/node/NK9ATH4WB6.md): concept NK9ATH4WB6
- [Alexander subbase theorem](https://tamnd.github.io/math2/node/NKKYW7Z6R7.md): statement NKKYW7Z6R7
- [Ergodic hypothesis](https://tamnd.github.io/math2/node/NKNSCK1JY2.md): concept NKNSCK1JY2
- [Subobject classifier](https://tamnd.github.io/math2/node/NKP5TPQ79X.md): concept NKP5TPQ79X
- [Douglas' lemma](https://tamnd.github.io/math2/node/NKTAN9XBZA.md): statement NKTAN9XBZA
- [Absorbing set](https://tamnd.github.io/math2/node/NM1BCK3ZX9.md): concept NM1BCK3ZX9
- [Givens rotation](https://tamnd.github.io/math2/node/NM1KPS6E7Y.md): concept NM1KPS6E7Y
- [Hermitian kernel](https://tamnd.github.io/math2/node/NM28RE8CHQ.md): definition NM28RE8CHQ
- [Lie subgroup](https://tamnd.github.io/math2/node/NM8XCPS1Q3.md): concept NM8XCPS1Q3
- [Complete theory](https://tamnd.github.io/math2/node/NN6QM9JMKB.md): definition NN6QM9JMKB
- [Cyclotomic field](https://tamnd.github.io/math2/node/NN77VYSGXR.md): concept NN77VYSGXR
- [Reflexive space](https://tamnd.github.io/math2/node/NN8RWPC8WG.md): concept NN8RWPC8WG
- [Inner function](https://tamnd.github.io/math2/node/NN8XH5RJYB.md): concept NN8XH5RJYB
- [Γ-convergence](https://tamnd.github.io/math2/node/NNAR1HN9BQ.md): definition NNAR1HN9BQ
- [Eilenberg–Moore spectral sequence](https://tamnd.github.io/math2/node/NNCBVR7Z2X.md): statement NNCBVR7Z2X
- [Approximation theory](https://tamnd.github.io/math2/node/NNHM06RCNV.md): definition NNHM06RCNV
- [Resultant](https://tamnd.github.io/math2/node/NNPHTWF645.md): concept NNPHTWF645
- [Degree of a central simple algebra](https://tamnd.github.io/math2/node/NNXSP5KKAC.md): concept NNXSP5KKAC
- [Bernstein's problem](https://tamnd.github.io/math2/node/NP948D6FR0.md): problem NP948D6FR0
- [Hilbert–Schmidt operator](https://tamnd.github.io/math2/node/NPBWWBGV37.md): definition NPBWWBGV37
- [Cantor function](https://tamnd.github.io/math2/node/NPBYYK63FE.md): counterexample NPBYYK63FE
- [Group of Lie type](https://tamnd.github.io/math2/node/NPEBABQ49M.md): concept NPEBABQ49M
- [Completion of a normed space](https://tamnd.github.io/math2/node/NPF2KSXK81.md): definition NPF2KSXK81
- [Completion is flat](https://tamnd.github.io/math2/node/NPF42DJMBX.md): statement NPF42DJMBX
- [Derived functor](https://tamnd.github.io/math2/node/NPMC3K6J2F.md): concept NPMC3K6J2F
- [Veblen–Young theorem](https://tamnd.github.io/math2/node/NPNNXWMVCC.md): statement NPNNXWMVCC
- [Pullback](https://tamnd.github.io/math2/node/NPPEERJVCR.md): definition NPPEERJVCR
- [Topology generated by a family of seminorms](https://tamnd.github.io/math2/node/NPR76WM5DQ.md): definition NPR76WM5DQ
- [Topological mixing](https://tamnd.github.io/math2/node/NPS5GGXBDP.md): definition NPS5GGXBDP
- [Infinite Galois theory](https://tamnd.github.io/math2/node/NPY85FYJHF.md): definition NPY85FYJHF
- [Szemerédi–Trotter theorem](https://tamnd.github.io/math2/node/NQBWFSC1E0.md): statement NQBWFSC1E0
- [Bernstein's theorem on absolutely convergent Fourier series](https://tamnd.github.io/math2/node/NQC6CRDWFW.md): statement NQC6CRDWFW
- [GCD domain](https://tamnd.github.io/math2/node/NQCDBHF8J7.md): definition NQCDBHF8J7
- [Dickson's lemma](https://tamnd.github.io/math2/node/NQD0Q9S9M7.md): statement NQD0Q9S9M7
- [Period-doubling bifurcation](https://tamnd.github.io/math2/node/NQDTTZ59X8.md): concept NQDTTZ59X8
- [Infinitely divisible distribution](https://tamnd.github.io/math2/node/NQGAM8PDG7.md): definition NQGAM8PDG7
- [Conway polynomial](https://tamnd.github.io/math2/node/NQQBESGPA4.md): concept NQQBESGPA4
- [p-harmonic function](https://tamnd.github.io/math2/node/NQVQFMXMGE.md): definition NQVQFMXMGE
- [Ancillary statistic](https://tamnd.github.io/math2/node/NR6MGAB5V0.md): definition NR6MGAB5V0
- [Hilbert's fifth problem](https://tamnd.github.io/math2/node/NR70GHZ37D.md): problem NR70GHZ37D
- [Fourier transform method for convolution equations](https://tamnd.github.io/math2/node/NRG03YP2PW.md): concept NRG03YP2PW
- [Riemann's existence theorem](https://tamnd.github.io/math2/node/NRJQP7FXFM.md): statement NRJQP7FXFM
- [Uniform module](https://tamnd.github.io/math2/node/NRP019D9AE.md): definition NRP019D9AE
- [Dvoretzky's theorem](https://tamnd.github.io/math2/node/NRPQP49N9F.md): statement NRPQP49N9F
- [Saccheri quadrilateral](https://tamnd.github.io/math2/node/NRZC1SY4S3.md): definition NRZC1SY4S3
- [Hyperarithmetical theory](https://tamnd.github.io/math2/node/NS1HWGTD8E.md): concept NS1HWGTD8E
- [Lattice in a locally compact group](https://tamnd.github.io/math2/node/NS2057MFWP.md): concept NS2057MFWP
- [Torus knot](https://tamnd.github.io/math2/node/NS60VV786W.md): concept NS60VV786W
- [Completion of a ring](https://tamnd.github.io/math2/node/NS84D2920R.md): definition NS84D2920R
- [Symmetric function](https://tamnd.github.io/math2/node/NS852SSQ00.md): definition NS852SSQ00
- [Functor of points](https://tamnd.github.io/math2/node/NS98MJFMYR.md): definition NS98MJFMYR
- [Kuratowski closure axioms](https://tamnd.github.io/math2/node/NS9VED3XG2.md): concept NS9VED3XG2
- [Bayes factor](https://tamnd.github.io/math2/node/NSBYE143KS.md): concept NSBYE143KS
- [Hilbert transform](https://tamnd.github.io/math2/node/NSCEBXM7ZM.md): concept NSCEBXM7ZM
- [Supermartingale](https://tamnd.github.io/math2/node/NSK7247MKX.md): concept NSK7247MKX
- [Cantor's uniqueness theorem for trigonometric series](https://tamnd.github.io/math2/node/NSS3XW4BYZ.md): statement NSS3XW4BYZ
- [Fenchel–Young inequality](https://tamnd.github.io/math2/node/NSTD00PJPE.md): statement NSTD00PJPE
- [Equicontinuous system](https://tamnd.github.io/math2/node/NSVEH6VMX8.md): concept NSVEH6VMX8
- [Littlewood–Paley decomposition](https://tamnd.github.io/math2/node/NSXT1MYFY8.md): definition NSXT1MYFY8
- [Geodesic](https://tamnd.github.io/math2/node/NSYMQMHPDN.md): concept NSYMQMHPDN
- [Successor order](https://tamnd.github.io/math2/node/NT44831EFE.md): concept NT44831EFE
- [Hermitian symmetric space](https://tamnd.github.io/math2/node/NT4797Z9H8.md): concept NT4797Z9H8
- [Minkowski–Carathéodory theorem](https://tamnd.github.io/math2/node/NT4SKSYKG9.md): statement NT4SKSYKG9
- [Commutative Gelfand–Naimark theorem](https://tamnd.github.io/math2/node/NT564Y315M.md): statement NT564Y315M
- [Tetration](https://tamnd.github.io/math2/node/NT65WHM5VV.md): example NT65WHM5VV
- [Fibred category](https://tamnd.github.io/math2/node/NT6WDN07RC.md): concept NT6WDN07RC
- [Shallow water equations](https://tamnd.github.io/math2/node/NT73Y95RY5.md): concept NT73Y95RY5
- [Reiterman's theorem](https://tamnd.github.io/math2/node/NT8W7R2KDR.md): statement NT8W7R2KDR
- [Kummer's equation](https://tamnd.github.io/math2/node/NT917KFR57.md): concept NT917KFR57
- [Resonance](https://tamnd.github.io/math2/node/NTB5JQGPGC.md): definition NTB5JQGPGC
- [Sierpiński space](https://tamnd.github.io/math2/node/NTCD3724BB.md): example NTCD3724BB
- [Cofinality](https://tamnd.github.io/math2/node/NTE5C260S1.md): definition NTE5C260S1
- [Condenser capacity](https://tamnd.github.io/math2/node/NTEFMRR3HR.md): concept NTEFMRR3HR
- [Specht module](https://tamnd.github.io/math2/node/NTFM7X5555.md): concept NTFM7X5555
- [Stopped process](https://tamnd.github.io/math2/node/NTFND9FRDN.md): definition NTFND9FRDN
- [Balayage](https://tamnd.github.io/math2/node/NTFY23YDEJ.md): concept NTFY23YDEJ
- [Lawvere theory](https://tamnd.github.io/math2/node/NTJ94H2CNS.md): concept NTJ94H2CNS
- [Švarc–Milnor lemma](https://tamnd.github.io/math2/node/NTK1TVA50A.md): statement NTK1TVA50A
- [Jeffreys prior](https://tamnd.github.io/math2/node/NTKTY1A2TH.md): definition NTKTY1A2TH
- [Fundamental group](https://tamnd.github.io/math2/node/NTQEXR0WDN.md): definition NTQEXR0WDN
- [U-statistic](https://tamnd.github.io/math2/node/NTVQB1DC56.md): concept NTVQB1DC56
- [Steinitz theorem on convex hull interiors](https://tamnd.github.io/math2/node/NV0ZAWS3ES.md): statement NV0ZAWS3ES
- [Tauberian theorems for integral transforms](https://tamnd.github.io/math2/node/NV54MWKT6T.md): statement NV54MWKT6T
- [Screw theory](https://tamnd.github.io/math2/node/NVBD7Y86ES.md): concept NVBD7Y86ES
- [Splitting field of a central simple algebra](https://tamnd.github.io/math2/node/NVD7R0T835.md): concept NVD7R0T835
- [Kirillov orbit method](https://tamnd.github.io/math2/node/NVD9GWQB6M.md): definition NVD9GWQB6M
- [Crossed product](https://tamnd.github.io/math2/node/NVFF58NEKV.md): definition NVFF58NEKV
- [Numerical differentiation](https://tamnd.github.io/math2/node/NVH3F8FTYS.md): concept NVH3F8FTYS
- [Brown–Peterson cohomology](https://tamnd.github.io/math2/node/NVMCZWHTZE.md): definition NVMCZWHTZE
- [Transcendence degree](https://tamnd.github.io/math2/node/NVQ0N7QTCR.md): definition NVQ0N7QTCR
- [Finitely generated variety](https://tamnd.github.io/math2/node/NVQJWDJG2T.md): concept NVQJWDJG2T
- [Korteweg–De Vries equation](https://tamnd.github.io/math2/node/NVQN7HR9Y0.md): definition NVQN7HR9Y0
- [Addition theorem for spherical harmonics](https://tamnd.github.io/math2/node/NVRR2TXQSE.md): statement NVRR2TXQSE
- [Associated bundle](https://tamnd.github.io/math2/node/NVVGA3KEWE.md): concept NVVGA3KEWE
- [Weak equivalence](https://tamnd.github.io/math2/node/NVXMVBS1ES.md): definition NVXMVBS1ES
- [Inverse quadratic interpolation](https://tamnd.github.io/math2/node/NW1XQ95R5P.md): statement NW1XQ95R5P
- [Zero morphism](https://tamnd.github.io/math2/node/NW2WW3WSH8.md): definition NW2WW3WSH8
- [Inaccessible cardinal](https://tamnd.github.io/math2/node/NW4DREQ5DQ.md): concept NW4DREQ5DQ
- [Pure weights](https://tamnd.github.io/math2/node/NW4VZRA5J1.md): concept NW4VZRA5J1
- [Monge point](https://tamnd.github.io/math2/node/NW57FTPF3E.md): concept NW57FTPF3E
- [Rational point](https://tamnd.github.io/math2/node/NW7W4R0T85.md): definition NW7W4R0T85
- [Γ-convergence](https://tamnd.github.io/math2/node/NW9MM370A5.md): concept NW9MM370A5
- [Frobenius element](https://tamnd.github.io/math2/node/NWAEQJ75QM.md): definition NWAEQJ75QM
- [Enumerative combinatorics](https://tamnd.github.io/math2/node/NWGARTY61Y.md): definition NWGARTY61Y
- [Browder–Minty theorem](https://tamnd.github.io/math2/node/NWK9Z80BT2.md): statement NWK9Z80BT2
- [Holonomic sequence](https://tamnd.github.io/math2/node/NWKX329T01.md): concept NWKX329T01
- [Montel space](https://tamnd.github.io/math2/node/NWMYW2WK9S.md): concept NWMYW2WK9S
- [Lemniscate elliptic functions](https://tamnd.github.io/math2/node/NWQ9Y7ZEAW.md): definition NWQ9Y7ZEAW
- [Osculating circle](https://tamnd.github.io/math2/node/NWSFMCRWHJ.md): concept NWSFMCRWHJ
- [Lefschetz duality](https://tamnd.github.io/math2/node/NWXJACYJC5.md): statement NWXJACYJC5
- [Almost complex manifold](https://tamnd.github.io/math2/node/NX0PKAWT0Y.md): definition NX0PKAWT0Y
- [Morrey–Campanato space](https://tamnd.github.io/math2/node/NX1JQMFC4F.md): concept NX1JQMFC4F
- [Thin set](https://tamnd.github.io/math2/node/NX4V39GJCV.md): definition NX4V39GJCV
- [Groupoid](https://tamnd.github.io/math2/node/NX7WCTNYWT.md): definition NX7WCTNYWT
- [Grothendieck inequality](https://tamnd.github.io/math2/node/NX8AMNXWG8.md): statement NX8AMNXWG8
- [Total variation diminishing scheme](https://tamnd.github.io/math2/node/NX8RE9B72A.md): concept NX8RE9B72A
- [Universal class](https://tamnd.github.io/math2/node/NX9X1A5EKY.md): definition NX9X1A5EKY
- [Caristi fixed-point theorem](https://tamnd.github.io/math2/node/NXCM67XY9E.md): statement NXCM67XY9E
- [Oka manifold](https://tamnd.github.io/math2/node/NXEAT9DMWD.md): concept NXEAT9DMWD
- [Carathéodory metric](https://tamnd.github.io/math2/node/NXHGJFVVAB.md): definition NXHGJFVVAB
- [Elliptic integral](https://tamnd.github.io/math2/node/NXMCYGCB59.md): concept NXMCYGCB59
- [Fourier basis of L2 of the circle](https://tamnd.github.io/math2/node/NXQX1KFMDB.md): statement NXQX1KFMDB
- [Gaussian measure](https://tamnd.github.io/math2/node/NXT0V2QWSC.md): definition NXT0V2QWSC
- [Brody hyperbolicity](https://tamnd.github.io/math2/node/NXTV4Y81ZJ.md): definition NXTV4Y81ZJ
- [Real spectrum of self-adjoint operators](https://tamnd.github.io/math2/node/NXV8S3145K.md): statement NXV8S3145K
- [Kelvin transform](https://tamnd.github.io/math2/node/NXWRDGX8T5.md): definition NXWRDGX8T5
- [Dolbeault operator](https://tamnd.github.io/math2/node/NY1K8FNMPQ.md): definition NY1K8FNMPQ
- [Semisimple ring](https://tamnd.github.io/math2/node/NY25HXA478.md): concept NY25HXA478
- [Projective hierarchy](https://tamnd.github.io/math2/node/NY50YEPR0X.md): definition NY50YEPR0X
- [Chebyshev system](https://tamnd.github.io/math2/node/NY5AK617TR.md): concept NY5AK617TR
- [Closed operator](https://tamnd.github.io/math2/node/NY6FVCX61P.md): definition NY6FVCX61P
- [Complex measure](https://tamnd.github.io/math2/node/NY6TYHRGMS.md): concept NY6TYHRGMS
- [Kodaira vanishing theorem](https://tamnd.github.io/math2/node/NY77DQ2AX8.md): statement NY77DQ2AX8
- [Non-uniform discrete Fourier transform](https://tamnd.github.io/math2/node/NY851TZ24X.md): concept NY851TZ24X
- [Periodic solution of a difference equation](https://tamnd.github.io/math2/node/NY91GA6WVT.md): definition NY91GA6WVT
- [Hyperbolic set](https://tamnd.github.io/math2/node/NY9MCZXY6N.md): definition NY9MCZXY6N
- [Site](https://tamnd.github.io/math2/node/NYE0Z94MME.md): concept NYE0Z94MME
- [de Branges space](https://tamnd.github.io/math2/node/NYHNTP9AND.md): concept NYHNTP9AND
- [Adele ring](https://tamnd.github.io/math2/node/NYJFYZ0B4M.md): definition NYJFYZ0B4M
- [Post's theorem](https://tamnd.github.io/math2/node/NYJX5QXX1T.md): statement NYJX5QXX1T
- [Recurrence and capacity of graphs](https://tamnd.github.io/math2/node/NYN6NYES16.md): statement NYN6NYES16
- [Coulomb wave function](https://tamnd.github.io/math2/node/NYTZ1VQGJ2.md): definition NYTZ1VQGJ2
- [Krein–Smulian theorem](https://tamnd.github.io/math2/node/NYVHMQJP3E.md): statement NYVHMQJP3E
- [Relatively free algebra](https://tamnd.github.io/math2/node/NYYV5V2FDD.md): concept NYYV5V2FDD
- [Nil ideal](https://tamnd.github.io/math2/node/NZ1G9WRWFB.md): concept NZ1G9WRWFB
- [Hyperfinite type II1 factor](https://tamnd.github.io/math2/node/NZ4NY8SK1C.md): object NZ4NY8SK1C
- [Friedrichs extension](https://tamnd.github.io/math2/node/NZ8WECT409.md): definition NZ8WECT409
- [Flux limiter](https://tamnd.github.io/math2/node/NZ8XFNCKQW.md): concept NZ8XFNCKQW
- [Versal unfolding](https://tamnd.github.io/math2/node/NZASV9ZMGN.md): definition NZASV9ZMGN
- [Synchronization of chaos](https://tamnd.github.io/math2/node/NZBKMKB8CV.md): definition NZBKMKB8CV
- [Matrix congruence](https://tamnd.github.io/math2/node/NZJDNC6EP6.md): definition NZJDNC6EP6
- [Symplectic integrator](https://tamnd.github.io/math2/node/NZTCZQA8KW.md): definition NZTCZQA8KW
- [Coupling from the past](https://tamnd.github.io/math2/node/NZXD2SBY7F.md): concept NZXD2SBY7F
- [Unique ergodicity](https://tamnd.github.io/math2/node/NZZE9KCAD8.md): concept NZZE9KCAD8
- [Abelian differential](https://tamnd.github.io/math2/node/P014C13K5H.md): concept P014C13K5H
- [Jacobi elliptic functions](https://tamnd.github.io/math2/node/P071B692JF.md): concept P071B692JF
- [Gaussian measure](https://tamnd.github.io/math2/node/P0DV2TCH1N.md): concept P0DV2TCH1N
- [Absolutely continuous measure](https://tamnd.github.io/math2/node/P0P21KDV6E.md): concept P0P21KDV6E
- [Schwarz reflection principle](https://tamnd.github.io/math2/node/P0R6YEGNB5.md): statement P0R6YEGNB5
- [Coalgebra](https://tamnd.github.io/math2/node/P0SYXV6RTK.md): definition P0SYXV6RTK
- [Indirect method by shooting](https://tamnd.github.io/math2/node/P0XFYRE9P2.md): definition P0XFYRE9P2
- [Hölder space](https://tamnd.github.io/math2/node/P101AX1ZSC.md): concept P101AX1ZSC
- [Generalized least squares](https://tamnd.github.io/math2/node/P142WXJCF3.md): concept P142WXJCF3
- [Harnack's principle](https://tamnd.github.io/math2/node/P15K8RKHDV.md): statement P15K8RKHDV
- [Measure space](https://tamnd.github.io/math2/node/P15VQS5DJ9.md): concept P15VQS5DJ9
- [Character table](https://tamnd.github.io/math2/node/P16AFZT3K0.md): concept P16AFZT3K0
- [Higher-order logic](https://tamnd.github.io/math2/node/P1ABGYRQXH.md): concept P1ABGYRQXH
- [LOBPCG](https://tamnd.github.io/math2/node/P1ECPTAG8F.md): statement P1ECPTAG8F
- [Bicubic interpolation](https://tamnd.github.io/math2/node/P1G0M3S8QV.md): concept P1G0M3S8QV
- [Maurer–Cartan form](https://tamnd.github.io/math2/node/P1JVA39H5E.md): concept P1JVA39H5E
- [Projective transformation](https://tamnd.github.io/math2/node/P1MKTF7MWQ.md): concept P1MKTF7MWQ
- [Hilbert's program](https://tamnd.github.io/math2/node/P1RW6JWX73.md): problem P1RW6JWX73
- [Sinc function](https://tamnd.github.io/math2/node/P1T98BECWR.md): concept P1T98BECWR
- [G2 manifold](https://tamnd.github.io/math2/node/P1TFW9QWAD.md): concept P1TFW9QWAD
- [Fractional integration](https://tamnd.github.io/math2/node/P1VFBJV1VN.md): concept P1VFBJV1VN
- [Harmonic functions on Riemann surfaces](https://tamnd.github.io/math2/node/P214QJ0S3B.md): concept P214QJ0S3B
- [Admissibility condition for wavelets](https://tamnd.github.io/math2/node/P270MSQ0SZ.md): definition P270MSQ0SZ
- [Additive combinatorics](https://tamnd.github.io/math2/node/P285NBAVA3.md): definition P285NBAVA3
- [Linearly ordered group](https://tamnd.github.io/math2/node/P2E8BNYC7B.md): concept P2E8BNYC7B
- [Nerve of a cover](https://tamnd.github.io/math2/node/P2F66BS71T.md): definition P2F66BS71T
- [Splitting field of a central simple algebra](https://tamnd.github.io/math2/node/P2HKS3K7T2.md): definition P2HKS3K7T2
- [Operator monotone function](https://tamnd.github.io/math2/node/P2NS0V36TH.md): definition P2NS0V36TH
- [Propagation of singularities](https://tamnd.github.io/math2/node/P2QJ7NPW3Y.md): statement P2QJ7NPW3Y
- [Feit–Higman theorem](https://tamnd.github.io/math2/node/P2V0XS4F63.md): statement P2V0XS4F63
- [Baire space of sequences](https://tamnd.github.io/math2/node/P2WS16HKD7.md): object P2WS16HKD7
- [Pseudo algebraically closed field](https://tamnd.github.io/math2/node/P32FQE1H0D.md): definition P32FQE1H0D
- [Strict total order](https://tamnd.github.io/math2/node/P33CMYHJW9.md): concept P33CMYHJW9
- [Least squares approximation](https://tamnd.github.io/math2/node/P3A4EAA3RA.md): definition P3A4EAA3RA
- [Resolvent kernel](https://tamnd.github.io/math2/node/P3A4XF59FX.md): definition P3A4XF59FX
- [Elementary equivalence](https://tamnd.github.io/math2/node/P3B1V182N4.md): concept P3B1V182N4
- [Discrete wavelet transform](https://tamnd.github.io/math2/node/P3H4MMC4ZH.md): concept P3H4MMC4ZH
- [(p,q)-form](https://tamnd.github.io/math2/node/P3PJGESDSF.md): definition P3PJGESDSF
- [Moments of the log-normal distribution](https://tamnd.github.io/math2/node/P3PZ73SMPZ.md): counterexample P3PZ73SMPZ
- [Band-limited function](https://tamnd.github.io/math2/node/P3T104G564.md): concept P3T104G564
- [Strict hyperbolicity](https://tamnd.github.io/math2/node/P3VXR8FW17.md): concept P3VXR8FW17
- [Autoregressive conditional heteroskedasticity](https://tamnd.github.io/math2/node/P3YM2DDGX9.md): definition P3YM2DDGX9
- [Lie's theorems on transformation groups](https://tamnd.github.io/math2/node/P43WKB4372.md): statement P43WKB4372
- [Spherical mean transform](https://tamnd.github.io/math2/node/P4B2Z6GS0R.md): concept P4B2Z6GS0R
- [Empirical risk minimization](https://tamnd.github.io/math2/node/P4DFMJTD09.md): definition P4DFMJTD09
- [Vector bundle](https://tamnd.github.io/math2/node/P4DZM975BV.md): concept P4DZM975BV
- [Hankel operator](https://tamnd.github.io/math2/node/P4EKQS312D.md): concept P4EKQS312D
- [Tensor algebra](https://tamnd.github.io/math2/node/P4J5JZDW0B.md): concept P4J5JZDW0B
- [Surface](https://tamnd.github.io/math2/node/P4NVFFR2CH.md): definition P4NVFFR2CH
- [Burgers' equation](https://tamnd.github.io/math2/node/P4RF8A1562.md): definition P4RF8A1562
- [Hilbert manifold](https://tamnd.github.io/math2/node/P4X779XNEA.md): definition P4X779XNEA
- [Kervaire invariant](https://tamnd.github.io/math2/node/P4YXE08ZXF.md): concept P4YXE08ZXF
- [Zassenhaus neighbourhood](https://tamnd.github.io/math2/node/P52TCSHP7R.md): definition P52TCSHP7R
- [Mahalanobis distance](https://tamnd.github.io/math2/node/P546TSFBK2.md): concept P546TSFBK2
- [Singular function](https://tamnd.github.io/math2/node/P5AWRD2793.md): concept P5AWRD2793
- [Mellin convolution](https://tamnd.github.io/math2/node/P5F7ZGGRW8.md): definition P5F7ZGGRW8
- [Moufang plane](https://tamnd.github.io/math2/node/P5HCD49REV.md): definition P5HCD49REV
- [Complete atomic Boolean algebra](https://tamnd.github.io/math2/node/P5MA4CDRNA.md): concept P5MA4CDRNA
- [Turing degree](https://tamnd.github.io/math2/node/P5N2ND3VCG.md): concept P5N2ND3VCG
- [Queueing theory](https://tamnd.github.io/math2/node/P64V0HTE78.md): concept P64V0HTE78
- [Topological group](https://tamnd.github.io/math2/node/P65JDBW4FH.md): definition P65JDBW4FH
- [Slice knot](https://tamnd.github.io/math2/node/P68G28E4AD.md): concept P68G28E4AD
- [Chain complex](https://tamnd.github.io/math2/node/P6AJ840RV4.md): definition P6AJ840RV4
- [Local solvability](https://tamnd.github.io/math2/node/P6E9RZNRBZ.md): concept P6E9RZNRBZ
- [Equicontinuous system](https://tamnd.github.io/math2/node/P6EQTZBBNT.md): definition P6EQTZBBNT
- [Wirtinger presentation](https://tamnd.github.io/math2/node/P6GR8ZQCK3.md): statement P6GR8ZQCK3
- [Racah polynomials](https://tamnd.github.io/math2/node/P6JYNDX5DR.md): concept P6JYNDX5DR
- [Weak topology](https://tamnd.github.io/math2/node/P6KHVSTPZV.md): concept P6KHVSTPZV
- [Algebraic K-theory of schemes](https://tamnd.github.io/math2/node/P6MCQD23YR.md): definition P6MCQD23YR
- [Grothendieck–Lefschetz trace formula](https://tamnd.github.io/math2/node/P6QD6TKAGS.md): statement P6QD6TKAGS
- [Lie's third theorem](https://tamnd.github.io/math2/node/P6QNEK44RX.md): statement P6QNEK44RX
- [Margulis lemma](https://tamnd.github.io/math2/node/P6WSG7DT4Q.md): statement P6WSG7DT4Q
- [Adjoint representation of a Lie group](https://tamnd.github.io/math2/node/P6ZP0TE07M.md): concept P6ZP0TE07M
- [Smooth four-dimensional Poincaré conjecture](https://tamnd.github.io/math2/node/P70VGYNFS5.md): conjecture P70VGYNFS5
- [Characterizations of Galois extensions](https://tamnd.github.io/math2/node/P75CXH2Q2B.md): statement P75CXH2Q2B
- [Higman embedding theorem](https://tamnd.github.io/math2/node/P7TZ694FCM.md): statement P7TZ694FCM
- [Group law on an elliptic curve](https://tamnd.github.io/math2/node/P7V1BQJPSS.md): concept P7V1BQJPSS
- [Isotonic regression](https://tamnd.github.io/math2/node/P7XYNN27AM.md): definition P7XYNN27AM
- [Parallel transport](https://tamnd.github.io/math2/node/P7YZFF8851.md): definition P7YZFF8851
- [Train track](https://tamnd.github.io/math2/node/P8055074WH.md): definition P8055074WH
- [Cartan–Eilenberg resolution](https://tamnd.github.io/math2/node/P84GBDM0QR.md): definition P84GBDM0QR
- [Pitchfork bifurcation](https://tamnd.github.io/math2/node/P88X34MSVV.md): concept P88X34MSVV
- [Uniform structure of a topological group](https://tamnd.github.io/math2/node/P89SJ7JWJE.md): concept P89SJ7JWJE
- [Jentzsch's theorem on positive kernels](https://tamnd.github.io/math2/node/P8CAK0FHGJ.md): statement P8CAK0FHGJ
- [Littlewood conjecture](https://tamnd.github.io/math2/node/P8D6T5A1QD.md): conjecture P8D6T5A1QD
- [Differential graded algebra](https://tamnd.github.io/math2/node/P8JGMNC23R.md): definition P8JGMNC23R
- [Octahedral axiom](https://tamnd.github.io/math2/node/P8PFMNCB8Y.md): concept P8PFMNCB8Y
- [Lorentz's characterization of almost convergence](https://tamnd.github.io/math2/node/P8TMYNPSYD.md): statement P8TMYNPSYD
- [Local homomorphism](https://tamnd.github.io/math2/node/P8YTMGEN9R.md): concept P8YTMGEN9R
- [Hilbert metric](https://tamnd.github.io/math2/node/P923DHV74J.md): concept P923DHV74J
- [Kadomtsev–Petviashvili equation](https://tamnd.github.io/math2/node/P93R13XP0P.md): concept P93R13XP0P
- [Positive operator](https://tamnd.github.io/math2/node/P93WWQ6BBB.md): definition P93WWQ6BBB
- [Bayesian hierarchical model](https://tamnd.github.io/math2/node/P944X6S117.md): concept P944X6S117
- [Nilpotent operator](https://tamnd.github.io/math2/node/P971S9AX0E.md): definition P971S9AX0E
- [Homotopy group](https://tamnd.github.io/math2/node/P976WYEFMB.md): concept P976WYEFMB
- [Sphere](https://tamnd.github.io/math2/node/P9CM4GQ2T0.md): concept P9CM4GQ2T0
- [Hypercube](https://tamnd.github.io/math2/node/P9PYM71GMM.md): example P9PYM71GMM
- [Thom–Mather stratified space](https://tamnd.github.io/math2/node/P9R3777ZJF.md): concept P9R3777ZJF
- [Group homology](https://tamnd.github.io/math2/node/P9RN33VDFS.md): definition P9RN33VDFS
- [Thin plate spline](https://tamnd.github.io/math2/node/P9ZPH7EC8K.md): concept P9ZPH7EC8K
- [S-finite measure](https://tamnd.github.io/math2/node/PA15QP0BP5.md): concept PA15QP0BP5
- [Round-off error](https://tamnd.github.io/math2/node/PA1B7DJ0C4.md): definition PA1B7DJ0C4
- [Developable surface](https://tamnd.github.io/math2/node/PA4HJDEP28.md): definition PA4HJDEP28
- [Turing reduction](https://tamnd.github.io/math2/node/PA4PYQ1PZ0.md): concept PA4PYQ1PZ0
- [Mean curvature](https://tamnd.github.io/math2/node/PA78NFEPCF.md): concept PA78NFEPCF
- [Inverse Mellin transform](https://tamnd.github.io/math2/node/PA7SWNRRPY.md): definition PA7SWNRRPY
- [Löwner's theorem](https://tamnd.github.io/math2/node/PA7ZQ6C6SP.md): statement PA7ZQ6C6SP
- [Symmetric space](https://tamnd.github.io/math2/node/PAA93R4WYD.md): concept PAA93R4WYD
- [Chern class](https://tamnd.github.io/math2/node/PAD37GKDPD.md): concept PAD37GKDPD
- [Tucker's lemma](https://tamnd.github.io/math2/node/PADZ7D9KBT.md): statement PADZ7D9KBT
- [Grauert's theorem on real analytic manifolds](https://tamnd.github.io/math2/node/PAG39H901E.md): statement PAG39H901E
- [Laplace–Beltrami operator](https://tamnd.github.io/math2/node/PAGN759BNR.md): definition PAGN759BNR
- [Symmetric function](https://tamnd.github.io/math2/node/PAJEHWXW6M.md): concept PAJEHWXW6M
- [Combinatorial design](https://tamnd.github.io/math2/node/PAJTMQ604P.md): concept PAJTMQ604P
- [Contingency table](https://tamnd.github.io/math2/node/PANSMHE0SW.md): concept PANSMHE0SW
- [Frobenius reciprocity](https://tamnd.github.io/math2/node/PAP76H0ND9.md): statement PAP76H0ND9
- [Natural deduction](https://tamnd.github.io/math2/node/PAQ4ZX2TX1.md): definition PAQ4ZX2TX1
- [Minkowski functional](https://tamnd.github.io/math2/node/PAX9SHDHDW.md): concept PAX9SHDHDW
- [Double series](https://tamnd.github.io/math2/node/PAXCH4FFSV.md): concept PAXCH4FFSV
- [Intrinsic volume](https://tamnd.github.io/math2/node/PAXJGJYDE3.md): definition PAXJGJYDE3
- [Motzkin number](https://tamnd.github.io/math2/node/PB0DSH9XCG.md): concept PB0DSH9XCG
- [Serre fibration](https://tamnd.github.io/math2/node/PB9JXQ2KC3.md): concept PB9JXQ2KC3
- [Pseudo-element](https://tamnd.github.io/math2/node/PBE65H0506.md): definition PBE65H0506
- [Haar system](https://tamnd.github.io/math2/node/PBEBXNHCPN.md): definition PBEBXNHCPN
- [Symmetric algebra](https://tamnd.github.io/math2/node/PBGF9TXMB7.md): concept PBGF9TXMB7
- [Word problem for groups](https://tamnd.github.io/math2/node/PBGTTAVMMT.md): problem PBGTTAVMMT
- [Fractional factorial design](https://tamnd.github.io/math2/node/PBK2XY4M0A.md): concept PBK2XY4M0A
- [Muirhead's inequality](https://tamnd.github.io/math2/node/PBPFVY20WT.md): statement PBPFVY20WT
- [Discrete spectrum](https://tamnd.github.io/math2/node/PBSMWBFFZH.md): definition PBSMWBFFZH
- [F-distribution](https://tamnd.github.io/math2/node/PBT8VRJADK.md): concept PBT8VRJADK
- [Simple polytope](https://tamnd.github.io/math2/node/PBYCBVG5CG.md): definition PBYCBVG5CG
- [Choquet theory](https://tamnd.github.io/math2/node/PBZDR2E342.md): definition PBZDR2E342
- [Well-quasi-ordering](https://tamnd.github.io/math2/node/PBZK2A09X4.md): definition PBZK2A09X4
- [Recurrence and transience of states](https://tamnd.github.io/math2/node/PBZTBWZ4GS.md): concept PBZTBWZ4GS
- [Comma category](https://tamnd.github.io/math2/node/PC6HG1JEGM.md): concept PC6HG1JEGM
- [Bessel's integral](https://tamnd.github.io/math2/node/PCE6XQTT6R.md): statement PCE6XQTT6R
- [Derived category](https://tamnd.github.io/math2/node/PCEVB6XV6J.md): definition PCEVB6XV6J
- [Multiplicative set](https://tamnd.github.io/math2/node/PCH3G8BJ9Y.md): definition PCH3G8BJ9Y
- [Skew group ring](https://tamnd.github.io/math2/node/PCKK4SFZEK.md): concept PCKK4SFZEK
- [Sierpiński carpet](https://tamnd.github.io/math2/node/PCM3H6J3EE.md): example PCM3H6J3EE
- [Inverse problem](https://tamnd.github.io/math2/node/PCMW7KN5P4.md): concept PCMW7KN5P4
- [Levi-Civita connection](https://tamnd.github.io/math2/node/PCNPQ7YBDE.md): definition PCNPQ7YBDE
- [Generalized polygon](https://tamnd.github.io/math2/node/PCT2HY9AAZ.md): concept PCT2HY9AAZ
- [Digamma function](https://tamnd.github.io/math2/node/PCT8RW4MY4.md): definition PCT8RW4MY4
- [State on a C*-algebra](https://tamnd.github.io/math2/node/PCX8R81H96.md): definition PCX8R81H96
- [Convergence of an infinite product](https://tamnd.github.io/math2/node/PCYRWA7SYB.md): concept PCYRWA7SYB
- [Poisson integral formula](https://tamnd.github.io/math2/node/PCZBE7EX01.md): statement PCZBE7EX01
- [Commutative algebra over a ring](https://tamnd.github.io/math2/node/PD4XXFJSW0.md): definition PD4XXFJSW0
- [Polydisc](https://tamnd.github.io/math2/node/PD7R2CJSEP.md): definition PD7R2CJSEP
- [Residual set](https://tamnd.github.io/math2/node/PDDAJEYDE9.md): definition PDDAJEYDE9
- [Shelf](https://tamnd.github.io/math2/node/PDFC0MRYR8.md): concept PDFC0MRYR8
- [CW approximation](https://tamnd.github.io/math2/node/PDFK69ZMST.md): statement PDFK69ZMST
- [Laguerre–Pólya class](https://tamnd.github.io/math2/node/PDG1HMR8GK.md): definition PDG1HMR8GK
- [Displacement convexity](https://tamnd.github.io/math2/node/PDP7CHN4B4.md): definition PDP7CHN4B4
- [Unit ball in C^n](https://tamnd.github.io/math2/node/PDQ9KW9FAP.md): object PDQ9KW9FAP
- [Bishop–Phelps theorem](https://tamnd.github.io/math2/node/PDVJTYCK8F.md): statement PDVJTYCK8F
- [Fast wavelet transform](https://tamnd.github.io/math2/node/PDVYQAW887.md): concept PDVYQAW887
- [Gateaux derivative](https://tamnd.github.io/math2/node/PDXR52YGPD.md): definition PDXR52YGPD
- [Particular values of the gamma function](https://tamnd.github.io/math2/node/PDYF9AV5PN.md): example PDYF9AV5PN
- [Hedetniemi's conjecture](https://tamnd.github.io/math2/node/PE207P2DZD.md): counterexample PE207P2DZD
- [Projection body](https://tamnd.github.io/math2/node/PE2HN8MY2J.md): definition PE2HN8MY2J
- [Absolute summability](https://tamnd.github.io/math2/node/PE3Z7CB1RY.md): definition PE3Z7CB1RY
- [Chebyshev iteration](https://tamnd.github.io/math2/node/PE5H892XVS.md): statement PE5H892XVS
- [Moving-average model](https://tamnd.github.io/math2/node/PEB9DN0AHA.md): concept PEB9DN0AHA
- [Dürer's conjecture](https://tamnd.github.io/math2/node/PEDBPRSV9V.md): conjecture PEDBPRSV9V
- [Noether's second theorem](https://tamnd.github.io/math2/node/PEDD2CP5PF.md): statement PEDD2CP5PF
- [Torsion-free group](https://tamnd.github.io/math2/node/PEMNN32A43.md): concept PEMNN32A43
- [Levi pseudoconvex domain](https://tamnd.github.io/math2/node/PETEQQCS0B.md): concept PETEQQCS0B
- [Axiom of determinacy](https://tamnd.github.io/math2/node/PEXTM3MJ5Z.md): concept PEXTM3MJ5Z
- [Wave front set](https://tamnd.github.io/math2/node/PF3G0J5JD6.md): definition PF3G0J5JD6
- [Complete statistic](https://tamnd.github.io/math2/node/PF4RHTZWXH.md): definition PF4RHTZWXH
- [Lattice polytope](https://tamnd.github.io/math2/node/PF5HS26Q0H.md): definition PF5HS26Q0H
- [Catmull–Clark subdivision surface](https://tamnd.github.io/math2/node/PF5Z57DDG2.md): statement PF5Z57DDG2
- [Dbar operator](https://tamnd.github.io/math2/node/PF83EHMFEG.md): definition PF83EHMFEG
- [Content](https://tamnd.github.io/math2/node/PFF8SV6ENB.md): concept PFF8SV6ENB
- [Horvitz–Thompson estimator](https://tamnd.github.io/math2/node/PFJFB459MT.md): definition PFJFB459MT
- [Caloric function](https://tamnd.github.io/math2/node/PFJP8T03C4.md): concept PFJP8T03C4
- [Algebra of sets](https://tamnd.github.io/math2/node/PFMGZ7988V.md): concept PFMGZ7988V
- [Contraction semigroup](https://tamnd.github.io/math2/node/PFQJTXX9YG.md): definition PFQJTXX9YG
- [Modular function](https://tamnd.github.io/math2/node/PFT115EGPE.md): definition PFT115EGPE
- [Bruhat decomposition of a Lie group](https://tamnd.github.io/math2/node/PG1CVVN7KA.md): statement PG1CVVN7KA
- [Antipode](https://tamnd.github.io/math2/node/PG3ECG5D0G.md): concept PG3ECG5D0G
- [Homotopy group](https://tamnd.github.io/math2/node/PG80NQ1S5C.md): definition PG80NQ1S5C
- [Amenable group](https://tamnd.github.io/math2/node/PG8Z6F6DV9.md): definition PG8Z6F6DV9
- [Kneser's theorem on solution sets](https://tamnd.github.io/math2/node/PG8ZH3ZQ88.md): statement PG8ZH3ZQ88
- [Quasicircle](https://tamnd.github.io/math2/node/PGCESD48XJ.md): definition PGCESD48XJ
- [Mahlo cardinal](https://tamnd.github.io/math2/node/PGJNYGAN75.md): definition PGJNYGAN75
- [Absolute Galois group](https://tamnd.github.io/math2/node/PGKWBJDTKX.md): concept PGKWBJDTKX
- [Nörlund–Rice integral](https://tamnd.github.io/math2/node/PGMNNRN5YN.md): statement PGMNNRN5YN
- [MacCormack method](https://tamnd.github.io/math2/node/PGXZJ2YWVJ.md): concept PGXZJ2YWVJ
- [Universal Horn class](https://tamnd.github.io/math2/node/PGZSNWS5VN.md): definition PGZSNWS5VN
- [Christodoulou–Klainerman stability of Minkowski space](https://tamnd.github.io/math2/node/PH4Z47SCK6.md): statement PH4Z47SCK6
- [Inverse Z-transform](https://tamnd.github.io/math2/node/PH61T3B7W7.md): statement PH61T3B7W7
- [Motzkin number](https://tamnd.github.io/math2/node/PHA1S6GQGV.md): definition PHA1S6GQGV
- [Hopfian group](https://tamnd.github.io/math2/node/PHEN4E1JPT.md): concept PHEN4E1JPT
- [Regular conditional probability](https://tamnd.github.io/math2/node/PHHT5W3BEJ.md): concept PHHT5W3BEJ
- [Bergman projection](https://tamnd.github.io/math2/node/PHJ8DD2W0E.md): definition PHJ8DD2W0E
- [Ax–Kochen theorem](https://tamnd.github.io/math2/node/PHT3TJCS90.md): statement PHT3TJCS90
- [Measurable function](https://tamnd.github.io/math2/node/PHT84D91B7.md): definition PHT84D91B7
- [Electrical network](https://tamnd.github.io/math2/node/PHTRQKMAC8.md): definition PHTRQKMAC8
- [Parametric transversality theorem](https://tamnd.github.io/math2/node/PJ4DQ435XK.md): statement PJ4DQ435XK
- [Absorbing element](https://tamnd.github.io/math2/node/PJ61DXYG2C.md): concept PJ61DXYG2C
- [Lifting idempotents](https://tamnd.github.io/math2/node/PJA5H9ZW4S.md): definition PJA5H9ZW4S
- [M. Riesz theorem on the Hilbert transform](https://tamnd.github.io/math2/node/PJFH51Z48M.md): statement PJFH51Z48M
- [Fractional Fourier transform](https://tamnd.github.io/math2/node/PJN54XYB5T.md): definition PJN54XYB5T
- [Short-time Fourier transform](https://tamnd.github.io/math2/node/PJNV4KR00F.md): definition PJNV4KR00F
- [Lie derivative](https://tamnd.github.io/math2/node/PJWK0PKN6Q.md): concept PJWK0PKN6Q
- [Discrete spectrum](https://tamnd.github.io/math2/node/PJWY3JJQQ7.md): concept PJWY3JJQQ7
- [Harmonic functions on Riemannian manifolds](https://tamnd.github.io/math2/node/PK2KNMWNW3.md): concept PK2KNMWNW3
- [Hausdorff summation](https://tamnd.github.io/math2/node/PK3RGWY032.md): concept PK3RGWY032
- [Uniform polytope](https://tamnd.github.io/math2/node/PK3RPS6EF4.md): definition PK3RPS6EF4
- [Similarity transformation](https://tamnd.github.io/math2/node/PK77W5K15V.md): definition PK77W5K15V
- [Identity](https://tamnd.github.io/math2/node/PKD440G2XW.md): concept PKD440G2XW
- [Linear regression](https://tamnd.github.io/math2/node/PKFPHK189T.md): definition PKFPHK189T
- [Mercer's theorem](https://tamnd.github.io/math2/node/PKGADWXPYH.md): statement PKGADWXPYH
- [Semihereditary ring](https://tamnd.github.io/math2/node/PKKD695HD2.md): definition PKKD695HD2
- [Nielsen–Thurston classification](https://tamnd.github.io/math2/node/PKP7RVAZK6.md): statement PKP7RVAZK6
- [Non-measurable set](https://tamnd.github.io/math2/node/PKQ7ABPVSG.md): concept PKQ7ABPVSG
- [Bromwich justification of the Heaviside calculus](https://tamnd.github.io/math2/node/PKRGY8HD0A.md): definition PKRGY8HD0A
- [Atiyah–Bott fixed-point theorem](https://tamnd.github.io/math2/node/PKS8JE1PW1.md): statement PKS8JE1PW1
- [Lauricella hypergeometric series](https://tamnd.github.io/math2/node/PKYQY68PCB.md): definition PKYQY68PCB
- [Optimal experimental design](https://tamnd.github.io/math2/node/PM5PQX6TVP.md): concept PM5PQX6TVP
- [Valuation on convex bodies](https://tamnd.github.io/math2/node/PM71JAZV31.md): definition PM71JAZV31
- [Sub-Gaussian distribution](https://tamnd.github.io/math2/node/PMDXYNPE7D.md): definition PMDXYNPE7D
- [Petersen's theorem](https://tamnd.github.io/math2/node/PMGTV643X1.md): statement PMGTV643X1
- [q-derivative](https://tamnd.github.io/math2/node/PMJT3FZE0Y.md): concept PMJT3FZE0Y
- [Littlewood subordination theorem](https://tamnd.github.io/math2/node/PMMXG4Q8JR.md): statement PMMXG4Q8JR
- [Crossed product algebra](https://tamnd.github.io/math2/node/PMRGCQTFZC.md): definition PMRGCQTFZC
- [Quasisymmetric map](https://tamnd.github.io/math2/node/PMTK0C47E0.md): definition PMTK0C47E0
- [Pentagonal number theorem](https://tamnd.github.io/math2/node/PMYDHW00KF.md): statement PMYDHW00KF
- [Normal complex space](https://tamnd.github.io/math2/node/PN55VEF0J7.md): concept PN55VEF0J7
- [Noetherian module](https://tamnd.github.io/math2/node/PNERH1096H.md): definition PNERH1096H
- [Airy equation](https://tamnd.github.io/math2/node/PNESAD6R5T.md): example PNESAD6R5T
- [k-means clustering](https://tamnd.github.io/math2/node/PNF97STBAW.md): definition PNF97STBAW
- [Hopf surface](https://tamnd.github.io/math2/node/PNFNGBXRM3.md): example PNFNGBXRM3
- [Strongly minimal theory](https://tamnd.github.io/math2/node/PNGY60EBDJ.md): concept PNGY60EBDJ
- [Stratified sampling](https://tamnd.github.io/math2/node/PNMT15BB91.md): concept PNMT15BB91
- [Hardy space](https://tamnd.github.io/math2/node/PNNNFJVF6W.md): concept PNNNFJVF6W
- [Liouville–Arnold theorem](https://tamnd.github.io/math2/node/PNQCW85HTJ.md): statement PNQCW85HTJ
- [Simson line](https://tamnd.github.io/math2/node/PNRMHP5A0C.md): concept PNRMHP5A0C
- [Faithful module](https://tamnd.github.io/math2/node/PNT397N3HW.md): definition PNT397N3HW
- [Fredholm equation of the second kind](https://tamnd.github.io/math2/node/PNTCSE6DWA.md): definition PNTCSE6DWA
- [Slice knot](https://tamnd.github.io/math2/node/PNWQZD5GRQ.md): definition PNWQZD5GRQ
- [Positive definiteness of the energy](https://tamnd.github.io/math2/node/PNZT35FFN2.md): statement PNZT35FFN2
- [Krull dimension of a noncommutative ring](https://tamnd.github.io/math2/node/PP11DV3ZMQ.md): concept PP11DV3ZMQ
- [Bayes estimator](https://tamnd.github.io/math2/node/PP1A0JB47Q.md): concept PP1A0JB47Q
- [Butterfly effect](https://tamnd.github.io/math2/node/PP3XTZWHNH.md): definition PP3XTZWHNH
- [Horocycle](https://tamnd.github.io/math2/node/PPEWPSQB9K.md): definition PPEWPSQB9K
- [Computable set](https://tamnd.github.io/math2/node/PPGD5KW8JC.md): concept PPGD5KW8JC
- [Cole–Hopf transformation](https://tamnd.github.io/math2/node/PPM7V7WWHQ.md): definition PPM7V7WWHQ
- [Calderón–Zygmund kernel](https://tamnd.github.io/math2/node/PPS8T94AWS.md): definition PPS8T94AWS
- [Cauchy–Riemann operator](https://tamnd.github.io/math2/node/PPSSTHTEDE.md): concept PPSSTHTEDE
- [Reproducing kernel Hilbert space](https://tamnd.github.io/math2/node/PPT1ZSSHBN.md): definition PPT1ZSSHBN
- [Quadratic transformations of hypergeometric functions](https://tamnd.github.io/math2/node/PPTD47MP45.md): statement PPTD47MP45
- [Field of extremals](https://tamnd.github.io/math2/node/PPTX47481A.md): definition PPTX47481A
- [Dixon elliptic functions](https://tamnd.github.io/math2/node/PPW3GYXH0P.md): concept PPW3GYXH0P
- [Real algebraic geometry](https://tamnd.github.io/math2/node/PPWJ8NFH7G.md): concept PPWJ8NFH7G
- [Fredholm determinant](https://tamnd.github.io/math2/node/PPX0AK81QV.md): concept PPX0AK81QV
- [Hankel matrix](https://tamnd.github.io/math2/node/PQ1D8FQB3E.md): concept PQ1D8FQB3E
- [Calculus of variations](https://tamnd.github.io/math2/node/PQ25NHV0KS.md): concept PQ25NHV0KS
- [Sheaf cohomology](https://tamnd.github.io/math2/node/PQ6Q143ARJ.md): concept PQ6Q143ARJ
- [Symplectic vector space](https://tamnd.github.io/math2/node/PQ7JJY0VPH.md): concept PQ7JJY0VPH
- [Classical group](https://tamnd.github.io/math2/node/PQB48WPK5C.md): definition PQB48WPK5C
- [Kullback–Leibler divergence](https://tamnd.github.io/math2/node/PQBW43X5RE.md): concept PQBW43X5RE
- [Equivalence of Boolean rings and Boolean algebras](https://tamnd.github.io/math2/node/PQEPV7F2RV.md): statement PQEPV7F2RV
- [Natural boundary](https://tamnd.github.io/math2/node/PQFEA12AHA.md): concept PQFEA12AHA
- [Infinitary logic](https://tamnd.github.io/math2/node/PQG7VAZ1R2.md): concept PQG7VAZ1R2
- [von Neumann mean ergodic theorem](https://tamnd.github.io/math2/node/PQPDQCCD02.md): statement PQPDQCCD02
- [Pro-object](https://tamnd.github.io/math2/node/PQRJ30QTYD.md): definition PQRJ30QTYD
- [Binary quadratic form](https://tamnd.github.io/math2/node/PQV6DNE0CW.md): definition PQV6DNE0CW
- [Measure](https://tamnd.github.io/math2/node/PR11BZQQPG.md): concept PR11BZQQPG
- [Separation of variables](https://tamnd.github.io/math2/node/PR2DV03M4H.md): definition PR2DV03M4H
- [Playfair's axiom](https://tamnd.github.io/math2/node/PR7Q3QMEHZ.md): statement PR7Q3QMEHZ
- [Riesz measure of a subharmonic function](https://tamnd.github.io/math2/node/PR8KWEA3MH.md): concept PR8KWEA3MH
- [U-statistic](https://tamnd.github.io/math2/node/PR8SESTZAN.md): definition PR8SESTZAN
- [Biharmonic equation](https://tamnd.github.io/math2/node/PR9A84ZPT1.md): definition PR9A84ZPT1
- [Tian–Todorov theorem](https://tamnd.github.io/math2/node/PRC4WH5WYJ.md): statement PRC4WH5WYJ
- [Stochastic partial differential equation](https://tamnd.github.io/math2/node/PRCQG82QTK.md): concept PRCQG82QTK
- [Seiberg–Witten invariants](https://tamnd.github.io/math2/node/PRDDMSTR4X.md): concept PRDDMSTR4X
- [Subobject classifier](https://tamnd.github.io/math2/node/PRFRD5RQ1W.md): definition PRFRD5RQ1W
- [Eta invariant](https://tamnd.github.io/math2/node/PRJCH788AM.md): concept PRJCH788AM
- [Paratopological group](https://tamnd.github.io/math2/node/PRQKKB7HNX.md): definition PRQKKB7HNX
- [Alexander's theorem on closed braids](https://tamnd.github.io/math2/node/PRWP1JF486.md): statement PRWP1JF486
- [Euler–Lagrange equation](https://tamnd.github.io/math2/node/PS0VV149ET.md): concept PS0VV149ET
- [Vector measure](https://tamnd.github.io/math2/node/PS28BSAJKJ.md): concept PS28BSAJKJ
- [Noncommutative geometry](https://tamnd.github.io/math2/node/PS2MR7P299.md): concept PS2MR7P299
- [Basic open set](https://tamnd.github.io/math2/node/PS8DVS9WTA.md): definition PS8DVS9WTA
- [Rudin–Shapiro polynomials](https://tamnd.github.io/math2/node/PSAX8W3ZYV.md): example PSAX8W3ZYV
- [Janko groups](https://tamnd.github.io/math2/node/PSEC8D7CX0.md): object PSEC8D7CX0
- [Von Neumann algebra](https://tamnd.github.io/math2/node/PSFWQVZ0FS.md): concept PSFWQVZ0FS
- [Amoeba](https://tamnd.github.io/math2/node/PSK6VJQR9B.md): concept PSK6VJQR9B
- [IEEE 754 standard](https://tamnd.github.io/math2/node/PSNW0ZQ0E9.md): definition PSNW0ZQ0E9
- [Beurling–Ahlfors transform](https://tamnd.github.io/math2/node/PSR7JAN6HZ.md): definition PSR7JAN6HZ
- [Riemannian distance](https://tamnd.github.io/math2/node/PSX7N8WAVB.md): definition PSX7N8WAVB
- [Complex manifold](https://tamnd.github.io/math2/node/PT1P6Z6GEY.md): definition PT1P6Z6GEY
- [Descartes's theorem on total angular defect](https://tamnd.github.io/math2/node/PT67C5ANSD.md): statement PT67C5ANSD
- [Fractional Brownian motion](https://tamnd.github.io/math2/node/PT6B8TPA8H.md): definition PT6B8TPA8H
- [Quasi-coherent sheaf](https://tamnd.github.io/math2/node/PT8PN4CS1H.md): definition PT8PN4CS1H
- [Killing vector field](https://tamnd.github.io/math2/node/PT9FSNP9T5.md): definition PT9FSNP9T5
- [Gauss–Kronrod quadrature formula](https://tamnd.github.io/math2/node/PT9V59MJZZ.md): concept PT9V59MJZZ
- [Bayesian hierarchical model](https://tamnd.github.io/math2/node/PTK8XCQ0GK.md): definition PTK8XCQ0GK
- [Clone](https://tamnd.github.io/math2/node/PTKR703G6A.md): concept PTKR703G6A
- [Bifurcation theory](https://tamnd.github.io/math2/node/PTRKN3DFAA.md): definition PTRKN3DFAA
- [Premeasure](https://tamnd.github.io/math2/node/PV0XXSWFC7.md): concept PV0XXSWFC7
- [L-estimator](https://tamnd.github.io/math2/node/PV187HG1K6.md): definition PV187HG1K6
- [Spectral theory of dynamical systems](https://tamnd.github.io/math2/node/PV2DGD23AY.md): definition PV2DGD23AY
- [Lindelöf hypothesis](https://tamnd.github.io/math2/node/PV82BVXE1A.md): conjecture PV82BVXE1A
- [Partial differential equation](https://tamnd.github.io/math2/node/PV8DQ41Q55.md): definition PV8DQ41Q55
- [Sublattice](https://tamnd.github.io/math2/node/PV90EW00XK.md): concept PV90EW00XK
- [Grauert's solution of the Levi problem for manifolds](https://tamnd.github.io/math2/node/PVA1QJVS2E.md): statement PVA1QJVS2E
- [Transport equation](https://tamnd.github.io/math2/node/PVD7VZJZ9B.md): concept PVD7VZJZ9B
- [Doubling on the integers is not an isomorphism](https://tamnd.github.io/math2/node/PVR5EX71PD.md): example PVR5EX71PD
- [Faber polynomials](https://tamnd.github.io/math2/node/PVTA9RQKWM.md): concept PVTA9RQKWM
- [Matroid](https://tamnd.github.io/math2/node/PVZCDPKF31.md): definition PVZCDPKF31
- [Weak ordering](https://tamnd.github.io/math2/node/PW2NKAPXWA.md): concept PW2NKAPXWA
- [Small cancellation theory](https://tamnd.github.io/math2/node/PW2TVRD47Y.md): definition PW2TVRD47Y
- [Wilf–Zeilberger pair](https://tamnd.github.io/math2/node/PW313WB7HE.md): definition PW313WB7HE
- [Sawtooth wave](https://tamnd.github.io/math2/node/PW5SKE8D5R.md): example PW5SKE8D5R
- [Laplace transform of distributions](https://tamnd.github.io/math2/node/PW8XX59WNH.md): definition PW8XX59WNH
- [Unknotting number](https://tamnd.github.io/math2/node/PW9D9QKAD4.md): concept PW9D9QKAD4
- [Antipode](https://tamnd.github.io/math2/node/PWDQ80EKZ8.md): definition PWDQ80EKZ8
- [Hodge Laplacian](https://tamnd.github.io/math2/node/PWG3GKZG80.md): concept PWG3GKZG80
- [Signature](https://tamnd.github.io/math2/node/PWHJZYBNEY.md): definition PWHJZYBNEY
- [Grothendieck universe](https://tamnd.github.io/math2/node/PWHVYMN5T3.md): definition PWHVYMN5T3
- [Levy–Desplanques theorem](https://tamnd.github.io/math2/node/PWKN6ANA0G.md): statement PWKN6ANA0G
- [Perfect closure](https://tamnd.github.io/math2/node/PWNZQVQ175.md): concept PWNZQVQ175
- [Cyclic extension](https://tamnd.github.io/math2/node/PWQ031W07T.md): definition PWQ031W07T
- [Atlas](https://tamnd.github.io/math2/node/PWQ5ZR5PK5.md): definition PWQ5ZR5PK5
- [Erdős–Hajnal conjecture](https://tamnd.github.io/math2/node/PWV2JFX6RR.md): conjecture PWV2JFX6RR
- [Countable additivity](https://tamnd.github.io/math2/node/PWZAR115MC.md): definition PWZAR115MC
- [Matroid](https://tamnd.github.io/math2/node/PX31ZNYS3Z.md): concept PX31ZNYS3Z
- [Happy ending problem](https://tamnd.github.io/math2/node/PX7SS1X16K.md): problem PX7SS1X16K
- [Uniformly convex space](https://tamnd.github.io/math2/node/PXF8731MBK.md): definition PXF8731MBK
- [Pivotal quantity](https://tamnd.github.io/math2/node/PXFYDT3M52.md): concept PXFYDT3M52
- [Dahlberg's theorem on harmonic measure](https://tamnd.github.io/math2/node/PXPC6E40ZW.md): statement PXPC6E40ZW
- [Normal function](https://tamnd.github.io/math2/node/PXTF18X433.md): definition PXTF18X433
- [Morley rank](https://tamnd.github.io/math2/node/PXX4P2BHJW.md): definition PXX4P2BHJW
- [Normal basis theorem](https://tamnd.github.io/math2/node/PXY1WYNRD4.md): statement PXY1WYNRD4
- [Monoidal functor](https://tamnd.github.io/math2/node/PY430TXMG0.md): concept PY430TXMG0
- [Dold–Kan correspondence](https://tamnd.github.io/math2/node/PY4EWJ8BQA.md): statement PY4EWJ8BQA
- [Reciprocal gamma function](https://tamnd.github.io/math2/node/PY6K9AWR33.md): concept PY6K9AWR33
- [Universally catenary ring](https://tamnd.github.io/math2/node/PYABBA1QQE.md): definition PYABBA1QQE
- [Nekhoroshev estimates](https://tamnd.github.io/math2/node/PYEN6SFS9P.md): statement PYEN6SFS9P
- [Semiring of sets](https://tamnd.github.io/math2/node/PYH82FMXQ2.md): concept PYH82FMXQ2
- [Boundary Harnack principle](https://tamnd.github.io/math2/node/PYJ7T889KC.md): statement PYJ7T889KC
- [Total variation distance between probability measures](https://tamnd.github.io/math2/node/PYN08Z14ZP.md): definition PYN08Z14ZP
- [Positive mass theorem](https://tamnd.github.io/math2/node/PYRPXHNQTB.md): statement PYRPXHNQTB
- [Finitely complete category](https://tamnd.github.io/math2/node/PYSBNWCHCF.md): concept PYSBNWCHCF
- [Spinor bundle](https://tamnd.github.io/math2/node/PYYQQKAZ0C.md): concept PYYQQKAZ0C
- [Hilbert space](https://tamnd.github.io/math2/node/PYYXEXMARY.md): definition PYYXEXMARY
- [Projective algebra](https://tamnd.github.io/math2/node/PZ41AE18X4.md): concept PZ41AE18X4
- [Sporadic group](https://tamnd.github.io/math2/node/PZ49Z4CBFZ.md): concept PZ49Z4CBFZ
- [Convex independence](https://tamnd.github.io/math2/node/PZ8KRWR8RT.md): definition PZ8KRWR8RT
- [Noncommutative torus](https://tamnd.github.io/math2/node/PZ97S3QQF1.md): example PZ97S3QQF1
- [One-relator group](https://tamnd.github.io/math2/node/PZ9CPKHXVZ.md): definition PZ9CPKHXVZ
- [Systolic geometry](https://tamnd.github.io/math2/node/PZCJV40M4K.md): concept PZCJV40M4K
- [Relational clone](https://tamnd.github.io/math2/node/PZES1MKY8M.md): definition PZES1MKY8M
- [Power of a test](https://tamnd.github.io/math2/node/PZGNRA2DAY.md): definition PZGNRA2DAY
- [Locally presentable category](https://tamnd.github.io/math2/node/PZHDNJ2J73.md): concept PZHDNJ2J73
- [Epimorphism](https://tamnd.github.io/math2/node/PZKBP4F5HB.md): definition PZKBP4F5HB
- [Isomorphism](https://tamnd.github.io/math2/node/PZMGQMDN25.md): concept PZMGQMDN25
- [Quintuple product identity](https://tamnd.github.io/math2/node/PZR6RC79ED.md): statement PZR6RC79ED
- [Hörmander's condition](https://tamnd.github.io/math2/node/PZRK2EPH1Y.md): definition PZRK2EPH1Y
- [Sequent calculus](https://tamnd.github.io/math2/node/PZX9J3VEMT.md): definition PZX9J3VEMT
- [Lagrangian submanifold](https://tamnd.github.io/math2/node/PZZ4X908Y8.md): definition PZZ4X908Y8
- [Holomorphic fiber bundle](https://tamnd.github.io/math2/node/PZZNV9SEF1.md): definition PZZNV9SEF1
- [Pullback](https://tamnd.github.io/math2/node/Q05K04QJ71.md): concept Q05K04QJ71
- [Semiring](https://tamnd.github.io/math2/node/Q07RA7AMDE.md): definition Q07RA7AMDE
- [Schwartz impossibility theorem](https://tamnd.github.io/math2/node/Q0C1H111EN.md): statement Q0C1H111EN
- [Parametric model](https://tamnd.github.io/math2/node/Q0DXKAQXE1.md): definition Q0DXKAQXE1
- [SL(2,R)](https://tamnd.github.io/math2/node/Q0GBCB9RA4.md): example Q0GBCB9RA4
- [Linear congruential generator](https://tamnd.github.io/math2/node/Q0K15AWV8H.md): concept Q0K15AWV8H
- [Category theory](https://tamnd.github.io/math2/node/Q0KB9QDH8V.md): definition Q0KB9QDH8V
- [Error bound for the trapezoidal rule](https://tamnd.github.io/math2/node/Q0M7PT9VN2.md): statement Q0M7PT9VN2
- [Planar ternary ring](https://tamnd.github.io/math2/node/Q0NXXFADT9.md): concept Q0NXXFADT9
- [Pointed space](https://tamnd.github.io/math2/node/Q0PAZ5QG79.md): definition Q0PAZ5QG79
- [Discrete geometry](https://tamnd.github.io/math2/node/Q0RW86P85F.md): concept Q0RW86P85F
- [Complemented subspace](https://tamnd.github.io/math2/node/Q0TY362K0V.md): concept Q0TY362K0V
- [Spinor bundle](https://tamnd.github.io/math2/node/Q0XKJ01SY6.md): definition Q0XKJ01SY6
- [Tridiagonal matrix](https://tamnd.github.io/math2/node/Q0YPJZH60H.md): definition Q0YPJZH60H
- [Stationary-action principle](https://tamnd.github.io/math2/node/Q104C67P26.md): statement Q104C67P26
- [Intuitionistic logic](https://tamnd.github.io/math2/node/Q11N8X99FK.md): concept Q11N8X99FK
- [Energy functional on paths](https://tamnd.github.io/math2/node/Q12AB1MJ64.md): concept Q12AB1MJ64
- [Shilnikov theorem](https://tamnd.github.io/math2/node/Q134D9EMAZ.md): statement Q134D9EMAZ
- [Wiener amalgam space](https://tamnd.github.io/math2/node/Q19EMZ4CP7.md): definition Q19EMZ4CP7
- [Kissing number problem in three dimensions](https://tamnd.github.io/math2/node/Q1H6BCKTBE.md): statement Q1H6BCKTBE
- [Information-based complexity](https://tamnd.github.io/math2/node/Q1QK49TTEJ.md): concept Q1QK49TTEJ
- [Category theory](https://tamnd.github.io/math2/node/Q1RZT7DV0F.md): concept Q1RZT7DV0F
- [Hausdorff measure](https://tamnd.github.io/math2/node/Q1TGZK8RN6.md): concept Q1TGZK8RN6
- [Khinchin integral](https://tamnd.github.io/math2/node/Q1TV4F608D.md): concept Q1TV4F608D
- [Dual of Lp](https://tamnd.github.io/math2/node/Q1V7ZNS332.md): statement Q1V7ZNS332
- [Axiom of constructibility](https://tamnd.github.io/math2/node/Q1VYK7362X.md): definition Q1VYK7362X
- [Direct limit of algebras](https://tamnd.github.io/math2/node/Q22KS9BBT6.md): definition Q22KS9BBT6
- [Foundations of mathematics](https://tamnd.github.io/math2/node/Q24EQE2HVG.md): definition Q24EQE2HVG
- [Semi-local ring](https://tamnd.github.io/math2/node/Q256RTTNTN.md): definition Q256RTTNTN
- [Asymptotic analysis](https://tamnd.github.io/math2/node/Q26QN6GYZX.md): concept Q26QN6GYZX
- [Duhamel's principle](https://tamnd.github.io/math2/node/Q28DM6ZCMF.md): statement Q28DM6ZCMF
- [de Bruijn–Erdős theorem](https://tamnd.github.io/math2/node/Q2D88XQKGX.md): statement Q2D88XQKGX
- [Legendrian submanifold](https://tamnd.github.io/math2/node/Q2DMSTWZYZ.md): concept Q2DMSTWZYZ
- [Fatou's theorem](https://tamnd.github.io/math2/node/Q2N917JQ6A.md): statement Q2N917JQ6A
- [Domain theory](https://tamnd.github.io/math2/node/Q2P5D74Q1V.md): concept Q2P5D74Q1V
- [Configuration](https://tamnd.github.io/math2/node/Q2QFXPP9EE.md): concept Q2QFXPP9EE
- [Homogeneous coordinates](https://tamnd.github.io/math2/node/Q2V8JVPB5E.md): definition Q2V8JVPB5E
- [Unary operation](https://tamnd.github.io/math2/node/Q2W26ZY10B.md): concept Q2W26ZY10B
- [Topology of uniform convergence](https://tamnd.github.io/math2/node/Q2WXAR0PXW.md): definition Q2WXAR0PXW
- [Incomparability theorem](https://tamnd.github.io/math2/node/Q2XBHJ96GV.md): statement Q2XBHJ96GV
- [Householder's method](https://tamnd.github.io/math2/node/Q390KATZ87.md): statement Q390KATZ87
- [Periodic sequence](https://tamnd.github.io/math2/node/Q3B6K9HYRG.md): concept Q3B6K9HYRG
- [Integro-partial differential equation](https://tamnd.github.io/math2/node/Q3C2ZGRHSG.md): definition Q3C2ZGRHSG
- [Linear extension](https://tamnd.github.io/math2/node/Q3EAVBDA8J.md): definition Q3EAVBDA8J
- [Graphical model](https://tamnd.github.io/math2/node/Q3FQAD1BXG.md): concept Q3FQAD1BXG
- [Vandermonde matrix](https://tamnd.github.io/math2/node/Q3JWBZ7EFY.md): concept Q3JWBZ7EFY
- [Quadric](https://tamnd.github.io/math2/node/Q3Q4FF23VG.md): concept Q3Q4FF23VG
- [Grothendieck topology](https://tamnd.github.io/math2/node/Q3TVJ4A3D4.md): definition Q3TVJ4A3D4
- [K3 surface](https://tamnd.github.io/math2/node/Q3V566R6K2.md): concept Q3V566R6K2
- [Peetre theorem on local operators](https://tamnd.github.io/math2/node/Q3X1CNXASD.md): statement Q3X1CNXASD
- [Dunford–Pettis theorem](https://tamnd.github.io/math2/node/Q3Y1HBD8NN.md): statement Q3Y1HBD8NN
- [Krein's condition](https://tamnd.github.io/math2/node/Q401KCEKNZ.md): statement Q401KCEKNZ
- [Morrey's lower semicontinuity theorem](https://tamnd.github.io/math2/node/Q43RD1GK8G.md): statement Q43RD1GK8G
- [Dehn surgery](https://tamnd.github.io/math2/node/Q468P6FP4A.md): definition Q468P6FP4A
- [Type theory](https://tamnd.github.io/math2/node/Q48KGV33AH.md): concept Q48KGV33AH
- [Littlewood–Richardson rule](https://tamnd.github.io/math2/node/Q490XBH1Q2.md): statement Q490XBH1Q2
- [Dehn function](https://tamnd.github.io/math2/node/Q4HGMRA9JQ.md): concept Q4HGMRA9JQ
- [Sine and cosine transforms](https://tamnd.github.io/math2/node/Q4HR7AF2R9.md): definition Q4HR7AF2R9
- [Lacunary series](https://tamnd.github.io/math2/node/Q4KF0MAGSC.md): definition Q4KF0MAGSC
- [dbar-Neumann problem](https://tamnd.github.io/math2/node/Q4P4R2GDNG.md): problem Q4P4R2GDNG
- [Integral transform](https://tamnd.github.io/math2/node/Q4TKABC1N2.md): definition Q4TKABC1N2
- [Newton–Gregory forward formula](https://tamnd.github.io/math2/node/Q4YAST023H.md): statement Q4YAST023H
- [Fredholm operator](https://tamnd.github.io/math2/node/Q5B5YGY7D6.md): definition Q5B5YGY7D6
- [H-cone](https://tamnd.github.io/math2/node/Q5C1E10CSQ.md): concept Q5C1E10CSQ
- [Zonal spherical harmonics](https://tamnd.github.io/math2/node/Q5D54R3RGE.md): concept Q5D54R3RGE
- [Riemannian Penrose inequality](https://tamnd.github.io/math2/node/Q5EWVTA2RW.md): statement Q5EWVTA2RW
- [Method of steepest descent](https://tamnd.github.io/math2/node/Q5H9VYFK98.md): concept Q5H9VYFK98
- [Galois connection](https://tamnd.github.io/math2/node/Q5JHEPRYQ6.md): concept Q5JHEPRYQ6
- [Hamilton–Jacobi–Isaacs equation](https://tamnd.github.io/math2/node/Q5N7E7P6F6.md): definition Q5N7E7P6F6
- [Hahn–Mazurkiewicz theorem](https://tamnd.github.io/math2/node/Q5QFYKD004.md): statement Q5QFYKD004
- [Meyer wavelet](https://tamnd.github.io/math2/node/Q5RN2FNX6Z.md): example Q5RN2FNX6Z
- [Direct limit of rings](https://tamnd.github.io/math2/node/Q5T05ET56S.md): concept Q5T05ET56S
- [Riemannian metric](https://tamnd.github.io/math2/node/Q61VFHCH2J.md): concept Q61VFHCH2J
- [Property P conjecture](https://tamnd.github.io/math2/node/Q630X0VHAE.md): statement Q630X0VHAE
- [Pushout](https://tamnd.github.io/math2/node/Q699KKMVJD.md): concept Q699KKMVJD
- [Korovkin's theorem](https://tamnd.github.io/math2/node/Q6KCF4688J.md): statement Q6KCF4688J
- [Polygon triangulation](https://tamnd.github.io/math2/node/Q6KKBP3RRP.md): concept Q6KKBP3RRP
- [Vanishing moments of a wavelet](https://tamnd.github.io/math2/node/Q6T1TTJJZG.md): definition Q6T1TTJJZG
- [Young tableau](https://tamnd.github.io/math2/node/Q6TH9PH2R7.md): concept Q6TH9PH2R7
- [Cartesian closed category](https://tamnd.github.io/math2/node/Q6W7PDWN4V.md): definition Q6W7PDWN4V
- [Equational logic](https://tamnd.github.io/math2/node/Q6WPMDYWN2.md): definition Q6WPMDYWN2
- [Hermitian metric on a vector bundle](https://tamnd.github.io/math2/node/Q6X6W2N29G.md): concept Q6X6W2N29G
- [Compactification](https://tamnd.github.io/math2/node/Q6XNC8JAZD.md): concept Q6XNC8JAZD
- [Large deviations theory](https://tamnd.github.io/math2/node/Q702WCDQQ3.md): concept Q702WCDQQ3
- [Spectral mapping theorem for polynomials](https://tamnd.github.io/math2/node/Q71WX8HFYT.md): statement Q71WX8HFYT
- [Completeness in order theory](https://tamnd.github.io/math2/node/Q72NHN7ATC.md): concept Q72NHN7ATC
- [Polynomial greatest common divisor](https://tamnd.github.io/math2/node/Q75YRG25JK.md): concept Q75YRG25JK
- [Modular lattice](https://tamnd.github.io/math2/node/Q76XC6K6NB.md): definition Q76XC6K6NB
- [Surgery exact sequence](https://tamnd.github.io/math2/node/Q7C480HQ9J.md): statement Q7C480HQ9J
- [Denjoy theorem on rotation number](https://tamnd.github.io/math2/node/Q7GKAMT3P9.md): statement Q7GKAMT3P9
- [Biorthogonal wavelet](https://tamnd.github.io/math2/node/Q7PENKTFHA.md): concept Q7PENKTFHA
- [Lagrange multipliers on Banach spaces](https://tamnd.github.io/math2/node/Q7QK3PF806.md): statement Q7QK3PF806
- [Suspension](https://tamnd.github.io/math2/node/Q7RFNJH76Q.md): concept Q7RFNJH76Q
- [Feigenbaum–Cvitanović functional equation](https://tamnd.github.io/math2/node/Q7S0D1NAG6.md): concept Q7S0D1NAG6
- [Polynomial identity ring](https://tamnd.github.io/math2/node/Q80ASJT68F.md): definition Q80ASJT68F
- [Lévy–Khintchine representation](https://tamnd.github.io/math2/node/Q82B4H4Z4J.md): statement Q82B4H4Z4J
- [Sentence](https://tamnd.github.io/math2/node/Q85MMH5DEY.md): definition Q85MMH5DEY
- [Snake lemma](https://tamnd.github.io/math2/node/Q8AYGD93CW.md): statement Q8AYGD93CW
- [Minimal model program](https://tamnd.github.io/math2/node/Q8CTDCZH01.md): concept Q8CTDCZH01
- [Bicubic interpolation](https://tamnd.github.io/math2/node/Q8DWXQ2WG0.md): definition Q8DWXQ2WG0
- [Carathéodory's extension theorem](https://tamnd.github.io/math2/node/Q8EXG7EKYW.md): statement Q8EXG7EKYW
- [Mutually singular measures](https://tamnd.github.io/math2/node/Q8FDSM86X6.md): definition Q8FDSM86X6
- [Banach–Stone theorem](https://tamnd.github.io/math2/node/Q8FERXTZ8B.md): statement Q8FERXTZ8B
- [Bloch–Wigner function](https://tamnd.github.io/math2/node/Q8K5MWEXB9.md): concept Q8K5MWEXB9
- [Elastic net](https://tamnd.github.io/math2/node/Q8MTV1Z4AQ.md): definition Q8MTV1Z4AQ
- [Gödel's consistency of CH and AC](https://tamnd.github.io/math2/node/Q9176JRJ5M.md): statement Q9176JRJ5M
- [Near-ring](https://tamnd.github.io/math2/node/Q93QR7EJKY.md): definition Q93QR7EJKY
- [Strong amalgamation property](https://tamnd.github.io/math2/node/Q94Z7TB8B5.md): concept Q94Z7TB8B5
- [Kempner series](https://tamnd.github.io/math2/node/Q958DANJNA.md): example Q958DANJNA
- [Periodicity equation](https://tamnd.github.io/math2/node/Q95N549AE3.md): definition Q95N549AE3
- [Germ of an analytic set](https://tamnd.github.io/math2/node/Q9B74BYZ8C.md): concept Q9B74BYZ8C
- [Coifman–Rochberg–Weiss commutator theorem](https://tamnd.github.io/math2/node/Q9D5TDRBEF.md): statement Q9D5TDRBEF
- [Bialgebra](https://tamnd.github.io/math2/node/Q9HHVN606G.md): concept Q9HHVN606G
- [Quantum invariant of 3-manifolds](https://tamnd.github.io/math2/node/Q9J0VHGTSD.md): definition Q9J0VHGTSD
- [Order (ring theory)](https://tamnd.github.io/math2/node/Q9KPGBD49F.md): definition Q9KPGBD49F
- [Morse theory on Hilbert manifolds](https://tamnd.github.io/math2/node/Q9VG3EGT74.md): definition Q9VG3EGT74
- [Perfectoid space](https://tamnd.github.io/math2/node/Q9Y4CJE1X5.md): concept Q9Y4CJE1X5
- [Geometric algebra](https://tamnd.github.io/math2/node/QA9YR4GBNB.md): definition QA9YR4GBNB
- [Alexandrov space](https://tamnd.github.io/math2/node/QAAV27WAAP.md): definition QAAV27WAAP
- [Holomorphic line bundle](https://tamnd.github.io/math2/node/QAC7BYK9KX.md): concept QAC7BYK9KX
- [Rubio de Francia extrapolation theorem](https://tamnd.github.io/math2/node/QADCFWETJC.md): statement QADCFWETJC
- [Tor functor](https://tamnd.github.io/math2/node/QAGW4V0S62.md): definition QAGW4V0S62
- [Riesz mean](https://tamnd.github.io/math2/node/QAP2SBSK2G.md): definition QAP2SBSK2G
- [Congruence-modular variety](https://tamnd.github.io/math2/node/QAQAS3JWXW.md): concept QAQAS3JWXW
- [Lusternik–Schnirelmann theory of critical points](https://tamnd.github.io/math2/node/QARH15D2PN.md): concept QARH15D2PN
- [Weierstrass zeta and sigma functions](https://tamnd.github.io/math2/node/QATW7046PD.md): concept QATW7046PD
- [Murray–von Neumann classification of factors](https://tamnd.github.io/math2/node/QAWDC7SWKC.md): statement QAWDC7SWKC
- [Permuting congruences](https://tamnd.github.io/math2/node/QB05EN6HEQ.md): definition QB05EN6HEQ
- [Mixed volume](https://tamnd.github.io/math2/node/QB3Q5FBJ08.md): concept QB3Q5FBJ08
- [Krylov subspace](https://tamnd.github.io/math2/node/QB45PCGQPV.md): concept QB45PCGQPV
- [Hahn–Banach theorem](https://tamnd.github.io/math2/node/QB5AE3VYZZ.md): statement QB5AE3VYZZ
- [Simple function](https://tamnd.github.io/math2/node/QB6EZ3HP73.md): definition QB6EZ3HP73
- [Free object](https://tamnd.github.io/math2/node/QBDT694BAP.md): concept QBDT694BAP
- [Finitely generated variety](https://tamnd.github.io/math2/node/QBHNB4CP22.md): definition QBHNB4CP22
- [Siciak extremal function](https://tamnd.github.io/math2/node/QBJ9R06GYE.md): concept QBJ9R06GYE
- [Double complex](https://tamnd.github.io/math2/node/QBK2MFPJZ7.md): definition QBK2MFPJZ7
- [Stability of Volterra equations](https://tamnd.github.io/math2/node/QBNKCGZC69.md): definition QBNKCGZC69
- [HOMFLY polynomial](https://tamnd.github.io/math2/node/QBQ0BH589C.md): concept QBQ0BH589C
- [Annihilator method](https://tamnd.github.io/math2/node/QBRJN93K0P.md): definition QBRJN93K0P
- [Boundary layer](https://tamnd.github.io/math2/node/QBTK36P203.md): definition QBTK36P203
- [Hilbertian field](https://tamnd.github.io/math2/node/QBXW04EC4V.md): definition QBXW04EC4V
- [Dixon's identity](https://tamnd.github.io/math2/node/QC4TP5D6HE.md): statement QC4TP5D6HE
- [Banach–Mazur theorem](https://tamnd.github.io/math2/node/QC5MB5WM8F.md): statement QC5MB5WM8F
- [Lindenbaum–Tarski algebra](https://tamnd.github.io/math2/node/QC9M3Q8B8Q.md): concept QC9M3Q8B8Q
- [Separably generated extension](https://tamnd.github.io/math2/node/QCBYAWBYWM.md): definition QCBYAWBYWM
- [Hopf–Rinow theorem](https://tamnd.github.io/math2/node/QCCD2JNFBT.md): statement QCCD2JNFBT
- [Kuratowski's closure-complement problem](https://tamnd.github.io/math2/node/QCDA5QM2BZ.md): problem QCDA5QM2BZ
- [Injective hull](https://tamnd.github.io/math2/node/QCJN05GN64.md): concept QCJN05GN64
- [Generic smoothness](https://tamnd.github.io/math2/node/QCYZRQX7QG.md): statement QCYZRQX7QG
- [Martin boundary](https://tamnd.github.io/math2/node/QD02B3DEMR.md): concept QD02B3DEMR
- [Integral transforms of tempered distributions](https://tamnd.github.io/math2/node/QD1MCREE5A.md): concept QD1MCREE5A
- [Lidskii's theorem](https://tamnd.github.io/math2/node/QD23PJR6RP.md): statement QD23PJR6RP
- [Preservation of limits](https://tamnd.github.io/math2/node/QD9H1WHHSW.md): definition QD9H1WHHSW
- [Topological index](https://tamnd.github.io/math2/node/QDCD33CS21.md): concept QDCD33CS21
- [Standard probability space](https://tamnd.github.io/math2/node/QDDRDW9PXS.md): concept QDDRDW9PXS
- [Frobenius theorem on real division algebras](https://tamnd.github.io/math2/node/QDJ7PQE9SW.md): statement QDJ7PQE9SW
- [Ring of sets](https://tamnd.github.io/math2/node/QDJMVNVY8T.md): definition QDJMVNVY8T
- [Voronoi diagram](https://tamnd.github.io/math2/node/QDNEBBVZ84.md): concept QDNEBBVZ84
- [Hammerstein integral equation](https://tamnd.github.io/math2/node/QDQSXW9HG1.md): definition QDQSXW9HG1
- [Structure theorem for compactly generated LCA groups](https://tamnd.github.io/math2/node/QDWNHEQJRK.md): statement QDWNHEQJRK
- [Van der Corput lemma](https://tamnd.github.io/math2/node/QDXR02419M.md): statement QDXR02419M
- [Green's identities](https://tamnd.github.io/math2/node/QDXXWA1QGF.md): statement QDXXWA1QGF
- [Allard regularity theorem](https://tamnd.github.io/math2/node/QDZ6DM42EB.md): statement QDZ6DM42EB
- [Finite difference coefficients](https://tamnd.github.io/math2/node/QE16SFY3GR.md): definition QE16SFY3GR
- [Scaling rule for the Laplace transform](https://tamnd.github.io/math2/node/QE2PT4TNDC.md): statement QE2PT4TNDC
- [Least harmonic majorant](https://tamnd.github.io/math2/node/QE4X5PTWN2.md): definition QE4X5PTWN2
- [Taylor term](https://tamnd.github.io/math2/node/QEAMRNRJJD.md): concept QEAMRNRJJD
- [Horvitz–Thompson estimator](https://tamnd.github.io/math2/node/QED889GZDE.md): concept QED889GZDE
- [Narasimhan lemma](https://tamnd.github.io/math2/node/QEEZEDBZ50.md): statement QEEZEDBZ50
- [Lommel function](https://tamnd.github.io/math2/node/QEFSJXTCZA.md): definition QEFSJXTCZA
- [Reductive Lie group](https://tamnd.github.io/math2/node/QEHV2DAHSW.md): definition QEHV2DAHSW
- [Operator theory](https://tamnd.github.io/math2/node/QEKVW55VV7.md): definition QEKVW55VV7
- [Ahlfors theory](https://tamnd.github.io/math2/node/QETCC3GZXC.md): definition QETCC3GZXC
- [Rademacher's theorem](https://tamnd.github.io/math2/node/QEY26RT19R.md): statement QEY26RT19R
- [Vanishing cycle](https://tamnd.github.io/math2/node/QEZFT0EM0G.md): definition QEZFT0EM0G
- [Double-negation translation](https://tamnd.github.io/math2/node/QEZR7PZ2DD.md): definition QEZR7PZ2DD
- [Blakers–Massey theorem](https://tamnd.github.io/math2/node/QF764XKBA3.md): statement QF764XKBA3
- [Goldie ring](https://tamnd.github.io/math2/node/QFDC3NYHJE.md): concept QFDC3NYHJE
- [Graph minor](https://tamnd.github.io/math2/node/QFE1AXQK2E.md): definition QFE1AXQK2E
- [Endomorphism monoid](https://tamnd.github.io/math2/node/QFE3FKNQQ1.md): concept QFE3FKNQQ1
- [Bohr–Mollerup theorem](https://tamnd.github.io/math2/node/QFF87AXWN2.md): statement QFF87AXWN2
- [Spectral method](https://tamnd.github.io/math2/node/QFJEFGR171.md): definition QFJEFGR171
- [Poles of the gamma function](https://tamnd.github.io/math2/node/QFPFF15PXC.md): statement QFPFF15PXC
- [Signorini problem](https://tamnd.github.io/math2/node/QFQAB091MQ.md): problem QFQAB091MQ
- [Dual cone](https://tamnd.github.io/math2/node/QFY1JR1TK4.md): concept QFY1JR1TK4
- [Legendre polynomials](https://tamnd.github.io/math2/node/QFYGJVA996.md): concept QFYGJVA996
- [Radical of a module](https://tamnd.github.io/math2/node/QFZ3WKEFH6.md): concept QFZ3WKEFH6
- [Path space fibration](https://tamnd.github.io/math2/node/QFZMEFZH76.md): definition QFZMEFZH76
- [Algebra over a monad](https://tamnd.github.io/math2/node/QG1ZXRC8EH.md): concept QG1ZXRC8EH
- [Parabolic cylinder function](https://tamnd.github.io/math2/node/QG2JHK1VXX.md): definition QG2JHK1VXX
- [G2 manifold](https://tamnd.github.io/math2/node/QG3BB9VZ3N.md): definition QG3BB9VZ3N
- [Convergence along an ultrafilter](https://tamnd.github.io/math2/node/QG9QRZAFTG.md): concept QG9QRZAFTG
- [Appell series](https://tamnd.github.io/math2/node/QG9VDK1HN9.md): concept QG9VDK1HN9
- [Vanishing mean oscillation](https://tamnd.github.io/math2/node/QGA9ABJEXQ.md): concept QGA9ABJEXQ
- [Linear algebraic group](https://tamnd.github.io/math2/node/QGFRT7QXWA.md): definition QGFRT7QXWA
- [Monoidal functor](https://tamnd.github.io/math2/node/QGG9GJN73C.md): definition QGG9GJN73C
- [Hankel transform of radial functions in the plane](https://tamnd.github.io/math2/node/QGGMQNFHJM.md): example QGGMQNFHJM
- [Generic filter](https://tamnd.github.io/math2/node/QGNCCJ125J.md): definition QGNCCJ125J
- [Tannaka–Krein duality](https://tamnd.github.io/math2/node/QGPAB6AS9F.md): statement QGPAB6AS9F
- [Ehrhart polynomial](https://tamnd.github.io/math2/node/QGPHXKCNNX.md): definition QGPHXKCNNX
- [Singular point of an algebraic variety](https://tamnd.github.io/math2/node/QGQ2PZHZ5R.md): definition QGQ2PZHZ5R
- [Analytic capacity](https://tamnd.github.io/math2/node/QGQPHRANWZ.md): concept QGQPHRANWZ
- [Lagrange interpolation](https://tamnd.github.io/math2/node/QGQTYT6YZR.md): definition QGQTYT6YZR
- [Leray–Schauder method](https://tamnd.github.io/math2/node/QGRBR04JJN.md): concept QGRBR04JJN
- [Elliptic geometry](https://tamnd.github.io/math2/node/QH76Y4JD89.md): definition QH76Y4JD89
- [Picard group of a ring](https://tamnd.github.io/math2/node/QH84JHCQMY.md): definition QH84JHCQMY
- [Levitzki's theorem](https://tamnd.github.io/math2/node/QHD4KJYFSA.md): statement QHD4KJYFSA
- [Schnirelmann density](https://tamnd.github.io/math2/node/QHH8RY9XCZ.md): concept QHH8RY9XCZ
- [Borsuk–Ulam theorem](https://tamnd.github.io/math2/node/QHJ0GBQE2E.md): statement QHJ0GBQE2E
- [Inverse distance weighting](https://tamnd.github.io/math2/node/QHN67MSH8S.md): concept QHN67MSH8S
- [Beta function](https://tamnd.github.io/math2/node/QHNWVSC6AQ.md): concept QHNWVSC6AQ
- [Steiner system](https://tamnd.github.io/math2/node/QHPW53E7YA.md): definition QHPW53E7YA
- [Rogers' bound](https://tamnd.github.io/math2/node/QHPXB39SN7.md): statement QHPXB39SN7
- [Hurwitz quaternions](https://tamnd.github.io/math2/node/QHTAEPTKEB.md): object QHTAEPTKEB
- [Overdetermined system](https://tamnd.github.io/math2/node/QJ0P26J9BV.md): concept QJ0P26J9BV
- [Spectrum of a ring](https://tamnd.github.io/math2/node/QJAXM2SF1Z.md): definition QJAXM2SF1Z
- [Algebraic structure](https://tamnd.github.io/math2/node/QJGPT2W0FT.md): concept QJGPT2W0FT
- [Muckenhoupt weights](https://tamnd.github.io/math2/node/QJK35TCFCJ.md): definition QJK35TCFCJ
- [Left-orderable group](https://tamnd.github.io/math2/node/QJNS6RJ743.md): definition QJNS6RJ743
- [Gysin sequence](https://tamnd.github.io/math2/node/QJSPWM9F9P.md): statement QJSPWM9F9P
- [First main theorem of Nevanlinna theory](https://tamnd.github.io/math2/node/QJWCFAK4AK.md): statement QJWCFAK4AK
- [Bounded mean oscillation](https://tamnd.github.io/math2/node/QJYTXFT8Y4.md): concept QJYTXFT8Y4
- [Randomization](https://tamnd.github.io/math2/node/QK03YC87QC.md): definition QK03YC87QC
- [Liouville's theorem for harmonic functions](https://tamnd.github.io/math2/node/QK1KWYZNRD.md): statement QK1KWYZNRD
- [Ordinary point and singular point](https://tamnd.github.io/math2/node/QK2J038VB4.md): concept QK2J038VB4
- [Coarea formula](https://tamnd.github.io/math2/node/QK55379SXW.md): statement QK55379SXW
- [Spectral geometry](https://tamnd.github.io/math2/node/QKAFRDPGE4.md): concept QKAFRDPGE4
- [Whitehead product](https://tamnd.github.io/math2/node/QKEE8AT2NR.md): concept QKEE8AT2NR
- [Fully invariant congruence](https://tamnd.github.io/math2/node/QKGW3Z9BHW.md): concept QKGW3Z9BHW
- [Cardinal function](https://tamnd.github.io/math2/node/QKJ743GE3J.md): definition QKJ743GE3J
- [Model category](https://tamnd.github.io/math2/node/QKJPM71RMG.md): definition QKJPM71RMG
- [Lagrangian singularity](https://tamnd.github.io/math2/node/QKPMR9MC9W.md): concept QKPMR9MC9W
- [Bidual and canonical embedding](https://tamnd.github.io/math2/node/QKR3D65TTH.md): concept QKR3D65TTH
- [Prime ideal of a lattice](https://tamnd.github.io/math2/node/QKWVXJWHP6.md): concept QKWVXJWHP6
- [Fermat point](https://tamnd.github.io/math2/node/QM0CGSQ4TF.md): concept QM0CGSQ4TF
- [Ky Fan minimax inequality](https://tamnd.github.io/math2/node/QM2EBW5CD9.md): statement QM2EBW5CD9
- [Finite type algebra](https://tamnd.github.io/math2/node/QM5ZCBJYP1.md): definition QM5ZCBJYP1
- [Weyl integration formula](https://tamnd.github.io/math2/node/QMH530N27M.md): statement QMH530N27M
- [Graphon](https://tamnd.github.io/math2/node/QMJ5PNSF9X.md): concept QMJ5PNSF9X
- [Mather–Yau theorem](https://tamnd.github.io/math2/node/QMMSKHEW9S.md): statement QMMSKHEW9S
- [Box spline](https://tamnd.github.io/math2/node/QMN3W21C9M.md): concept QMN3W21C9M
- [Dirac operator on a spin manifold](https://tamnd.github.io/math2/node/QMRAVX14BJ.md): concept QMRAVX14BJ
- [Abel transform](https://tamnd.github.io/math2/node/QMW1MWQH6J.md): concept QMW1MWQH6J
- [Laurent polynomial ring](https://tamnd.github.io/math2/node/QMY5JTZZB8.md): concept QMY5JTZZB8
- [Identity component](https://tamnd.github.io/math2/node/QMYZ798EAR.md): definition QMYZ798EAR
- [Moore–Osgood theorem](https://tamnd.github.io/math2/node/QN2210XCST.md): statement QN2210XCST
- [Satisfiability](https://tamnd.github.io/math2/node/QN7M1JCGT9.md): concept QN7M1JCGT9
- [Logistic map](https://tamnd.github.io/math2/node/QN9X9BNKTX.md): example QN9X9BNKTX
- [Whipple's transformation](https://tamnd.github.io/math2/node/QNGGCTXK9S.md): statement QNGGCTXK9S
- [Prolate spheroidal wave functions](https://tamnd.github.io/math2/node/QNHYXDFJJY.md): concept QNHYXDFJJY
- [Sufficient statistic](https://tamnd.github.io/math2/node/QNMR5F525C.md): definition QNMR5F525C
- [Fox H-function transforms](https://tamnd.github.io/math2/node/QNMSHKBBSG.md): definition QNMSHKBBSG
- [Pseudofunctor](https://tamnd.github.io/math2/node/QNMWK3GCF2.md): definition QNMWK3GCF2
- [Cauchy–Riemann equations in several variables](https://tamnd.github.io/math2/node/QNS8WFYFXJ.md): definition QNS8WFYFXJ
- [Congruence generated by a relation](https://tamnd.github.io/math2/node/QP06QH2Q7C.md): concept QP06QH2Q7C
- [Symplectomorphism](https://tamnd.github.io/math2/node/QP0MY1E9RK.md): definition QP0MY1E9RK
- [Polytope](https://tamnd.github.io/math2/node/QP7GEKDTYX.md): concept QP7GEKDTYX
- [Cartan's magic formula](https://tamnd.github.io/math2/node/QP8B74828C.md): statement QP8B74828C
- [Deformation of algebras](https://tamnd.github.io/math2/node/QP92G4ZGJE.md): concept QP92G4ZGJE
- [Total curvature](https://tamnd.github.io/math2/node/QP9DY5K8QW.md): concept QP9DY5K8QW
- [Bargmann transform](https://tamnd.github.io/math2/node/QPH28QH036.md): concept QPH28QH036
- [Interval arithmetic](https://tamnd.github.io/math2/node/QPKKTE1ZY0.md): concept QPKKTE1ZY0
- [Hilbert's nineteenth problem](https://tamnd.github.io/math2/node/QPMM2ZPWBW.md): problem QPMM2ZPWBW
- [Cartan connection](https://tamnd.github.io/math2/node/QPR7DCHXPG.md): concept QPR7DCHXPG
- [Space-filling curve](https://tamnd.github.io/math2/node/QPRHC4DZ20.md): example QPRHC4DZ20
- [Dynamic programming in continuous time](https://tamnd.github.io/math2/node/QPS8VSJGMJ.md): concept QPS8VSJGMJ
- [Abstract data type as an algebra](https://tamnd.github.io/math2/node/QPT5H6MM99.md): definition QPT5H6MM99
- [Gentzen's consistency proof](https://tamnd.github.io/math2/node/QPV0HNP22S.md): statement QPV0HNP22S
- [Method of matched asymptotic expansions](https://tamnd.github.io/math2/node/QQ1YT0MV2X.md): concept QQ1YT0MV2X
- [Mesh generation](https://tamnd.github.io/math2/node/QQ3PBZS9J8.md): definition QQ3PBZS9J8
- [Differential operator between vector bundles](https://tamnd.github.io/math2/node/QQ5F8B8DYC.md): definition QQ5F8B8DYC
- [Boundary integral method for the Dirichlet problem](https://tamnd.github.io/math2/node/QQ62R5ZAXJ.md): concept QQ62R5ZAXJ
- [Funk transform](https://tamnd.github.io/math2/node/QQ6DA5QKXB.md): definition QQ6DA5QKXB
- [Vojta's conjecture](https://tamnd.github.io/math2/node/QQ717VSFNN.md): conjecture QQ717VSFNN
- [Fano plane](https://tamnd.github.io/math2/node/QQ8GW1ASZM.md): object QQ8GW1ASZM
- [Lie group method for PDEs](https://tamnd.github.io/math2/node/QQ8M0T04Q9.md): concept QQ8M0T04Q9
- [Heaviside operational calculus](https://tamnd.github.io/math2/node/QQ98C2AXZP.md): concept QQ98C2AXZP
- [Schreier refinement theorem](https://tamnd.github.io/math2/node/QQAWHC9X60.md): statement QQAWHC9X60
- [Discrete subgroup](https://tamnd.github.io/math2/node/QQCWHB37ZC.md): definition QQCWHB37ZC
- [Nilradical](https://tamnd.github.io/math2/node/QQDN6EYQNJ.md): concept QQDN6EYQNJ
- [Alexander horned sphere](https://tamnd.github.io/math2/node/QQEASX79RW.md): counterexample QQEASX79RW
- [Variation of parameters for difference equations](https://tamnd.github.io/math2/node/QQFYC988SJ.md): statement QQFYC988SJ
- [Counting measure](https://tamnd.github.io/math2/node/QQMJZWDS1F.md): example QQMJZWDS1F
- [Saddle point of a Lagrangian](https://tamnd.github.io/math2/node/QQMQGXS0XE.md): definition QQMQGXS0XE
- [Wigner D-matrix](https://tamnd.github.io/math2/node/QQPC9ASQHR.md): concept QQPC9ASQHR
- [Topological manifold](https://tamnd.github.io/math2/node/QQQ9B1DK4P.md): concept QQQ9B1DK4P
- [Haefliger structure](https://tamnd.github.io/math2/node/QQRD38DF4X.md): definition QQRD38DF4X
- [Dunford–Pettis property](https://tamnd.github.io/math2/node/QQSNZX1QMM.md): concept QQSNZX1QMM
- [Absolutely monotonic function](https://tamnd.github.io/math2/node/QQSPWH4NG9.md): concept QQSPWH4NG9
- [Triangulation](https://tamnd.github.io/math2/node/QQTPD2TZ0Z.md): definition QQTPD2TZ0Z
- [Condition number](https://tamnd.github.io/math2/node/QQW1AWRFZH.md): definition QQW1AWRFZH
- [Lehmann–Scheffé theorem](https://tamnd.github.io/math2/node/QQZPWDWS29.md): statement QQZPWDWS29
- [Gaussian process](https://tamnd.github.io/math2/node/QR83860XAZ.md): definition QR83860XAZ
- [Horn clause](https://tamnd.github.io/math2/node/QR92EZDTKF.md): concept QR92EZDTKF
- [Uniformly most powerful test](https://tamnd.github.io/math2/node/QRAPTPVH8C.md): concept QRAPTPVH8C
- [Tamari lattice](https://tamnd.github.io/math2/node/QREAJQH24A.md): example QREAJQH24A
- [Lyapunov exponent](https://tamnd.github.io/math2/node/QREYWGEH41.md): definition QREYWGEH41
- [Fractional differential equation](https://tamnd.github.io/math2/node/QRGSRA0CYV.md): definition QRGSRA0CYV
- [Logical connective](https://tamnd.github.io/math2/node/QS0XVAN9BG.md): concept QS0XVAN9BG
- [Haken manifold](https://tamnd.github.io/math2/node/QS6TMVTTHW.md): definition QS6TMVTTHW
- [Jordan curve theorem for polygons](https://tamnd.github.io/math2/node/QS9PG4MNTR.md): statement QS9PG4MNTR
- [Packing measure](https://tamnd.github.io/math2/node/QSH01NSVXN.md): definition QSH01NSVXN
- [Singular values of a compact operator](https://tamnd.github.io/math2/node/QSJKPKPM7H.md): concept QSJKPKPM7H
- [Lenagan's theorem](https://tamnd.github.io/math2/node/QSK2J5HEER.md): statement QSK2J5HEER
- [The symmetric group S3 is not abelian](https://tamnd.github.io/math2/node/QSMVQ4EJ3D.md): example QSMVQ4EJ3D
- [Join](https://tamnd.github.io/math2/node/QSTN8NMSK0.md): definition QSTN8NMSK0
- [Mandelbrot set](https://tamnd.github.io/math2/node/QSWSVYRCS1.md): object QSWSVYRCS1
- [Valuation ring](https://tamnd.github.io/math2/node/QSX12D8RQM.md): concept QSX12D8RQM
- [Pincherle derivative](https://tamnd.github.io/math2/node/QT1DYNX2QA.md): definition QT1DYNX2QA
- [Multidimensional Laplace transform](https://tamnd.github.io/math2/node/QT3M720TCV.md): concept QT3M720TCV
- [Shimura variety](https://tamnd.github.io/math2/node/QT4CF7ZKK5.md): concept QT4CF7ZKK5
- [Seifert surface](https://tamnd.github.io/math2/node/QT4HP1H2YE.md): definition QT4HP1H2YE
- [Categorical theory](https://tamnd.github.io/math2/node/QTC57AJEVR.md): definition QTC57AJEVR
- [Birkhoff min-max method](https://tamnd.github.io/math2/node/QTHPN6V7XM.md): definition QTHPN6V7XM
- [Survival function](https://tamnd.github.io/math2/node/QTM1HZM53Q.md): concept QTM1HZM53Q
- [Fixed-point property of posets](https://tamnd.github.io/math2/node/QTQN93B305.md): definition QTQN93B305
- [Natural boundary condition](https://tamnd.github.io/math2/node/QTRPNFDFJ8.md): concept QTRPNFDFJ8
- [John–Nirenberg inequality](https://tamnd.github.io/math2/node/QTSJT4JZXT.md): statement QTSJT4JZXT
- [Basis of a matroid](https://tamnd.github.io/math2/node/QTSQR7130G.md): concept QTSQR7130G
- [Lattice of equational theories](https://tamnd.github.io/math2/node/QTTW954WCK.md): definition QTTW954WCK
- [Pontryagin duality theorem](https://tamnd.github.io/math2/node/QTVC6F2P9D.md): statement QTVC6F2P9D
- [Iterative refinement](https://tamnd.github.io/math2/node/QTVP4WP1XX.md): statement QTVP4WP1XX
- [Removable sets for bounded harmonic functions](https://tamnd.github.io/math2/node/QTX0GBZMT8.md): statement QTX0GBZMT8
- [Perspectivity](https://tamnd.github.io/math2/node/QTZWZR1KWQ.md): concept QTZWZR1KWQ
- [Idempotence](https://tamnd.github.io/math2/node/QV3V8HJW2F.md): definition QV3V8HJW2F
- [Cheeger–Gromoll splitting theorem](https://tamnd.github.io/math2/node/QV75TT6131.md): statement QV75TT6131
- [Lebesgue–Stieltjes measure](https://tamnd.github.io/math2/node/QV75XJBFCS.md): concept QV75XJBFCS
- [Continuous q-Hermite polynomials](https://tamnd.github.io/math2/node/QVCKQ46TFM.md): definition QVCKQ46TFM
- [Mahler measure](https://tamnd.github.io/math2/node/QVPYBPRHRC.md): concept QVPYBPRHRC
- [Hypersingular integral equation](https://tamnd.github.io/math2/node/QVRP8FSAZM.md): concept QVRP8FSAZM
- [MV-algebra](https://tamnd.github.io/math2/node/QVWHWDJ5X1.md): definition QVWHWDJ5X1
- [Galois representation](https://tamnd.github.io/math2/node/QW17YJ6ZCT.md): definition QW17YJ6ZCT
- [Lawson topology](https://tamnd.github.io/math2/node/QW999ZFZVN.md): definition QW999ZFZVN
- [Domain of finite type](https://tamnd.github.io/math2/node/QWAZ4DK5AD.md): definition QWAZ4DK5AD
- [Sanov's theorem](https://tamnd.github.io/math2/node/QWEF2WAXJM.md): statement QWEF2WAXJM
- [Cayley–Abels graph](https://tamnd.github.io/math2/node/QWG66JE1JE.md): concept QWG66JE1JE
- [Lorentz group](https://tamnd.github.io/math2/node/QWK3BYVJ9W.md): example QWK3BYVJ9W
- [Exponential functions on semigroups](https://tamnd.github.io/math2/node/QWQ45PQTZT.md): concept QWQ45PQTZT
- [Nonemptiness of the spectrum](https://tamnd.github.io/math2/node/QWQC5SV0SD.md): statement QWQC5SV0SD
- [Quasi-primal algebra](https://tamnd.github.io/math2/node/QWSBDFM47K.md): concept QWSBDFM47K
- [Normal family](https://tamnd.github.io/math2/node/QWSD2JDRP0.md): definition QWSD2JDRP0
- [Addition formulas for Jacobi elliptic functions](https://tamnd.github.io/math2/node/QWSRBJ1G1E.md): statement QWSRBJ1G1E
- [Solvability of x^n - a](https://tamnd.github.io/math2/node/QWXD1Z9V8F.md): statement QWXD1Z9V8F
- [Frankel conjecture](https://tamnd.github.io/math2/node/QWXTB8VVXN.md): statement QWXTB8VVXN
- [Numerical stability](https://tamnd.github.io/math2/node/QWYQ3JNK0B.md): definition QWYQ3JNK0B
- [Modular lattice](https://tamnd.github.io/math2/node/QX7WZ92QAK.md): concept QX7WZ92QAK
- [Topological semigroup](https://tamnd.github.io/math2/node/QXB6JDG68F.md): concept QXB6JDG68F
- [Missing data mechanisms](https://tamnd.github.io/math2/node/QXDY7STQWG.md): definition QXDY7STQWG
- [Determinacy](https://tamnd.github.io/math2/node/QXECDPAYTN.md): concept QXECDPAYTN
- [Holomorphically convex hull](https://tamnd.github.io/math2/node/QXF1VVQ5M6.md): definition QXF1VVQ5M6
- [Castelnuovo's rationality criterion](https://tamnd.github.io/math2/node/QXHMMQX1D7.md): statement QXHMMQX1D7
- [Conservative system](https://tamnd.github.io/math2/node/QXK4Q72WCV.md): definition QXK4Q72WCV
- [Convolution theorem](https://tamnd.github.io/math2/node/QY8ETVDA1Z.md): statement QY8ETVDA1Z
- [Open subgroup](https://tamnd.github.io/math2/node/QYCFH68RD1.md): concept QYCFH68RD1
- [Wild knot](https://tamnd.github.io/math2/node/QYDQ28X8MA.md): counterexample QYDQ28X8MA
- [Knot](https://tamnd.github.io/math2/node/QYEAZTYGPT.md): concept QYEAZTYGPT
- [Herman ring](https://tamnd.github.io/math2/node/QYHKS4B7M0.md): definition QYHKS4B7M0
- [Struve function](https://tamnd.github.io/math2/node/QYNEBRGMR6.md): definition QYNEBRGMR6
- [Feichtinger conjecture](https://tamnd.github.io/math2/node/QYQ983Z0BK.md): statement QYQ983Z0BK
- [Pringsheim convergence](https://tamnd.github.io/math2/node/QYTGCW1GTK.md): concept QYTGCW1GTK
- [Cartan's classification of bounded symmetric domains](https://tamnd.github.io/math2/node/QYVY3DV9K4.md): statement QYVY3DV9K4
- [Tate's thesis](https://tamnd.github.io/math2/node/QYW79XKJTP.md): concept QYW79XKJTP
- [Tits alternative](https://tamnd.github.io/math2/node/QYXGVES1XM.md): statement QYXGVES1XM
- [Wiener–Hopf equation](https://tamnd.github.io/math2/node/QYY4NMDY5H.md): concept QYY4NMDY5H
- [Incomplete Cholesky factorization](https://tamnd.github.io/math2/node/QYYCFQ8E5D.md): concept QYYCFQ8E5D
- [Spin structure](https://tamnd.github.io/math2/node/QYZNMEM1ZZ.md): definition QYZNMEM1ZZ
- [Spectral space](https://tamnd.github.io/math2/node/QYZXXXFM53.md): definition QYZXXXFM53
- [Locally presentable category](https://tamnd.github.io/math2/node/QZFTEZKGNX.md): definition QZFTEZKGNX
- [Wynn's epsilon algorithm](https://tamnd.github.io/math2/node/QZKW59K1HH.md): definition QZKW59K1HH
- [Cohomological dimension of a field](https://tamnd.github.io/math2/node/QZMZXFFJFJ.md): concept QZMZXFFJFJ
- [Valuative criterion](https://tamnd.github.io/math2/node/QZN6P7AG33.md): statement QZN6P7AG33
- [Hilbert's syzygy theorem](https://tamnd.github.io/math2/node/R0DSZPRG6T.md): statement R0DSZPRG6T
- [Pincherle derivative](https://tamnd.github.io/math2/node/R0H4RXF0YZ.md): concept R0H4RXF0YZ
- [Ising model](https://tamnd.github.io/math2/node/R0KGM32QN6.md): example R0KGM32QN6
- [Kuranishi family](https://tamnd.github.io/math2/node/R0QABVGPRX.md): concept R0QABVGPRX
- [Smooth morphism](https://tamnd.github.io/math2/node/R123NEWYZR.md): definition R123NEWYZR
- [Perron–Wiener–Brelot solution](https://tamnd.github.io/math2/node/R12KQGVGN6.md): concept R12KQGVGN6
- [Reduced scheme](https://tamnd.github.io/math2/node/R15Z3DPF6A.md): definition R15Z3DPF6A
- [Representation theory of the Lorentz group](https://tamnd.github.io/math2/node/R1EXRPQGQ0.md): example R1EXRPQGQ0
- [Derivation](https://tamnd.github.io/math2/node/R1F0HCCE6F.md): definition R1F0HCCE6F
- [Artin–Wedderburn theorem](https://tamnd.github.io/math2/node/R1F271180N.md): statement R1F271180N
- [Urysohn universal space](https://tamnd.github.io/math2/node/R1H00F1668.md): object R1H00F1668
- [Marcinkiewicz interpolation theorem](https://tamnd.github.io/math2/node/R1PHMVBJ9J.md): statement R1PHMVBJ9J
- [Filtered algebra](https://tamnd.github.io/math2/node/R1PTHQACWM.md): concept R1PTHQACWM
- [Rejection sampling](https://tamnd.github.io/math2/node/R1QBNFTHFP.md): statement R1QBNFTHFP
- [Simplicial homology](https://tamnd.github.io/math2/node/R1RQ3226WV.md): definition R1RQ3226WV
- [Intersection body](https://tamnd.github.io/math2/node/R1S7NT139R.md): definition R1S7NT139R
- [Poincaré problem](https://tamnd.github.io/math2/node/R1W62VM8W6.md): problem R1W62VM8W6
- [Lambert W function](https://tamnd.github.io/math2/node/R1XGWTVY3N.md): definition R1XGWTVY3N
- [3-manifold](https://tamnd.github.io/math2/node/R20TB2KEQQ.md): concept R20TB2KEQQ
- [Prevariety](https://tamnd.github.io/math2/node/R2658B61VX.md): definition R2658B61VX
- [Bernstein–Sato polynomial](https://tamnd.github.io/math2/node/R2AV47AX0Y.md): definition R2AV47AX0Y
- [Principal component analysis](https://tamnd.github.io/math2/node/R2EHQ06B98.md): definition R2EHQ06B98
- [Volterra equations with completely monotone kernels](https://tamnd.github.io/math2/node/R2KZVG9SQ6.md): definition R2KZVG9SQ6
- [Brauer character](https://tamnd.github.io/math2/node/R2N3GBDBVQ.md): concept R2N3GBDBVQ
- [Picard's little theorem](https://tamnd.github.io/math2/node/R2PHM118NG.md): statement R2PHM118NG
- [Smooth structure](https://tamnd.github.io/math2/node/R2TQ82PCNB.md): concept R2TQ82PCNB
- [Liouville's theorems on elliptic functions](https://tamnd.github.io/math2/node/R32HEEN55K.md): statement R32HEEN55K
- [Galois connection between subgroups and subfields](https://tamnd.github.io/math2/node/R3625YKQC0.md): concept R3625YKQC0
- [Coiflet](https://tamnd.github.io/math2/node/R38W9GFMQX.md): example R38W9GFMQX
- [Dedekind zeta function](https://tamnd.github.io/math2/node/R3AR5JK8WW.md): definition R3AR5JK8WW
- [Fréchet–Urysohn space](https://tamnd.github.io/math2/node/R3B4V9WW2S.md): concept R3B4V9WW2S
- [Power series in several variables](https://tamnd.github.io/math2/node/R3QTCPTJK7.md): concept R3QTCPTJK7
- [Integral equation](https://tamnd.github.io/math2/node/R3SG1SN7JQ.md): concept R3SG1SN7JQ
- [Semilocally simply connected space](https://tamnd.github.io/math2/node/R3SHZX0NZR.md): concept R3SHZX0NZR
- [Projective conic](https://tamnd.github.io/math2/node/R3TMVHTP3A.md): concept R3TMVHTP3A
- [Hurwitz-stable matrix](https://tamnd.github.io/math2/node/R3VWS51XY5.md): definition R3VWS51XY5
- [Transport equation](https://tamnd.github.io/math2/node/R3X5WFHV8P.md): definition R3X5WFHV8P
- [Aitken's delta-squared process](https://tamnd.github.io/math2/node/R40BXAKSV4.md): definition R40BXAKSV4
- [Bruck–Ryser–Chowla theorem](https://tamnd.github.io/math2/node/R43K0GTTDS.md): statement R43K0GTTDS
- [Sturm's theorem](https://tamnd.github.io/math2/node/R44CV0KD9T.md): statement R44CV0KD9T
- [Kummer–Artin–Schreier–Witt theory](https://tamnd.github.io/math2/node/R45FHBE15X.md): concept R45FHBE15X
- [Torus integral](https://tamnd.github.io/math2/node/R49Q9M519Z.md): concept R49Q9M519Z
- [Projective resolution](https://tamnd.github.io/math2/node/R4G6PMRHAD.md): definition R4G6PMRHAD
- [Neumann's addition theorem](https://tamnd.github.io/math2/node/R4KVJ1PD8S.md): statement R4KVJ1PD8S
- [Gelfand–Levitan theory](https://tamnd.github.io/math2/node/R4MRNP9EQT.md): concept R4MRNP9EQT
- [Incomplete gamma function](https://tamnd.github.io/math2/node/R4MZRFB9MV.md): definition R4MZRFB9MV
- [Multidimensional scaling](https://tamnd.github.io/math2/node/R4WG4AQH56.md): definition R4WG4AQH56
- [Cauchy–Riemann operator](https://tamnd.github.io/math2/node/R50ERM9AHR.md): definition R50ERM9AHR
- [Hadamard finite part](https://tamnd.github.io/math2/node/R519NKG3V6.md): definition R519NKG3V6
- [Normal scheme](https://tamnd.github.io/math2/node/R56VTW55SG.md): definition R56VTW55SG
- [Permanent](https://tamnd.github.io/math2/node/R582CKDHF4.md): definition R582CKDHF4
- [Euler's pentagonal number theorem](https://tamnd.github.io/math2/node/R587SR9GNV.md): statement R587SR9GNV
- [Oriented matroid](https://tamnd.github.io/math2/node/R5EB9NKQ5M.md): concept R5EB9NKQ5M
- [Stochastic ordering](https://tamnd.github.io/math2/node/R5JTGHBMGE.md): definition R5JTGHBMGE
- [Contractible space](https://tamnd.github.io/math2/node/R5WP8Z69Q8.md): concept R5WP8Z69Q8
- [Hecke character](https://tamnd.github.io/math2/node/R65XKHN4D1.md): concept R65XKHN4D1
- [The natural numbers under addition are not a group](https://tamnd.github.io/math2/node/R68D2P175Z.md): example R68D2P175Z
- [Dual pair](https://tamnd.github.io/math2/node/R68HNB5SVR.md): concept R68HNB5SVR
- [n-skeleton](https://tamnd.github.io/math2/node/R6DXPZPVV7.md): definition R6DXPZPVV7
- [Tetrahedron](https://tamnd.github.io/math2/node/R6PWCDFEVR.md): definition R6PWCDFEVR
- [Fixed point](https://tamnd.github.io/math2/node/R6QVS6P21N.md): concept R6QVS6P21N
- [Unimodular matrix](https://tamnd.github.io/math2/node/R6RBQHW191.md): concept R6RBQHW191
- [Seeley–DeWitt coefficients](https://tamnd.github.io/math2/node/R72NAFZ722.md): concept R72NAFZ722
- [Clifford–Klein form](https://tamnd.github.io/math2/node/R73WRFDXRY.md): concept R73WRFDXRY
- [Initial algebra](https://tamnd.github.io/math2/node/R76EC56V9R.md): concept R76EC56V9R
- [Ruzsa triangle inequality](https://tamnd.github.io/math2/node/R7E5A4W6FJ.md): statement R7E5A4W6FJ
- [Borel subgroup](https://tamnd.github.io/math2/node/R7FTZ813DA.md): definition R7FTZ813DA
- [Localization of a space](https://tamnd.github.io/math2/node/R7G4ZC0963.md): definition R7G4ZC0963
- [Unital magma](https://tamnd.github.io/math2/node/R7RDNEDQF2.md): definition R7RDNEDQF2
- [Kernel approximation in reproducing kernel Hilbert spaces](https://tamnd.github.io/math2/node/R7RNMX78KZ.md): concept R7RNMX78KZ
- [Ordered semigroup](https://tamnd.github.io/math2/node/R7SH3ECJRN.md): definition R7SH3ECJRN
- [Vertex of a polytope](https://tamnd.github.io/math2/node/R7V3S9V891.md): concept R7V3S9V891
- [Cartan's theorem B](https://tamnd.github.io/math2/node/R7VKRV8AGQ.md): statement R7VKRV8AGQ
- [Abelian variety](https://tamnd.github.io/math2/node/R7W03TXRS9.md): concept R7W03TXRS9
- [Willmore energy](https://tamnd.github.io/math2/node/R7WFJGJBV4.md): concept R7WFJGJBV4
- [Convex hull algorithms](https://tamnd.github.io/math2/node/R8071FWJRK.md): concept R8071FWJRK
- [Liouvillian function](https://tamnd.github.io/math2/node/R80FA7993W.md): definition R80FA7993W
- [Cylinder set measure](https://tamnd.github.io/math2/node/R80G20B2ZW.md): definition R80G20B2ZW
- [Self-similar measure](https://tamnd.github.io/math2/node/R8A7XTNZ1B.md): concept R8A7XTNZ1B
- [Hilbert manifold](https://tamnd.github.io/math2/node/R8DMMW0E9M.md): concept R8DMMW0E9M
- [Complex analytic space](https://tamnd.github.io/math2/node/R8GZAJ85YF.md): definition R8GZAJ85YF
- [Hasse–Minkowski theorem](https://tamnd.github.io/math2/node/R8HCWA981S.md): statement R8HCWA981S
- [Frobenius theorem on Cesàro and Abel summability](https://tamnd.github.io/math2/node/R8MZAZ6GCR.md): statement R8MZAZ6GCR
- [Likelihood-ratio test](https://tamnd.github.io/math2/node/R8N0CCDJZD.md): definition R8N0CCDJZD
- [Linking number](https://tamnd.github.io/math2/node/R8PE771DDE.md): definition R8PE771DDE
- [Cardinal characteristics of the continuum](https://tamnd.github.io/math2/node/R8S00S3Z94.md): definition R8S00S3Z94
- [Floating-point arithmetic](https://tamnd.github.io/math2/node/R8S821AD6T.md): definition R8S821AD6T
- [Zolotarev polynomials](https://tamnd.github.io/math2/node/R8SNG5TBKJ.md): concept R8SNG5TBKJ
- [Weierstrass excess function](https://tamnd.github.io/math2/node/R8TWRPSPNV.md): concept R8TWRPSPNV
- [Calderón problem](https://tamnd.github.io/math2/node/R8Y8DDTWBM.md): problem R8Y8DDTWBM
- [Univalent function](https://tamnd.github.io/math2/node/R91E83SDBD.md): concept R91E83SDBD
- [Schur complement method](https://tamnd.github.io/math2/node/R997ZAD141.md): statement R997ZAD141
- [Cauchy–Fantappiè formula](https://tamnd.github.io/math2/node/R9JJ44TMQB.md): statement R9JJ44TMQB
- [Thue–Siegel–Roth theorem](https://tamnd.github.io/math2/node/R9JZA3G2NX.md): statement R9JZA3G2NX
- [Isometry of Riemannian manifolds](https://tamnd.github.io/math2/node/R9YWQS15AA.md): concept R9YWQS15AA
- [Friedberg–Muchnik theorem](https://tamnd.github.io/math2/node/RA358M9QCN.md): statement RA358M9QCN
- [Topological monoid](https://tamnd.github.io/math2/node/RA56GHB9JZ.md): concept RA56GHB9JZ
- [Width of a poset](https://tamnd.github.io/math2/node/RA5SQ908G1.md): concept RA5SQ908G1
- [Toeplitz index theorem](https://tamnd.github.io/math2/node/RABJ6X1BXC.md): statement RABJ6X1BXC
- [Weyl's inequality](https://tamnd.github.io/math2/node/RACSNQXDQV.md): statement RACSNQXDQV
- [Hamiltonian of an optimal control problem](https://tamnd.github.io/math2/node/RACVWG5QTK.md): concept RACVWG5QTK
- [Minimal sufficient statistic](https://tamnd.github.io/math2/node/RADE160VJN.md): concept RADE160VJN
- [Monad in nonstandard topology](https://tamnd.github.io/math2/node/RAGJSF4V9V.md): concept RAGJSF4V9V
- [Lp multiplier](https://tamnd.github.io/math2/node/RAGR38Q40B.md): concept RAGR38Q40B
- [Almost everywhere](https://tamnd.github.io/math2/node/RARTMA0ZEC.md): concept RARTMA0ZEC
- [Volterra's function](https://tamnd.github.io/math2/node/RARXGWQ85S.md): counterexample RARXGWQ85S
- [Unramified morphism of rings](https://tamnd.github.io/math2/node/RASAC44RTQ.md): definition RASAC44RTQ
- [Self-similar process](https://tamnd.github.io/math2/node/RASDTDEKP7.md): concept RASDTDEKP7
- [Huber loss](https://tamnd.github.io/math2/node/RATW595XR9.md): concept RATW595XR9
- [Hilbert's invariant integral](https://tamnd.github.io/math2/node/RB08381QYR.md): definition RB08381QYR
- [Dugundji extension theorem](https://tamnd.github.io/math2/node/RBDJBK63HX.md): statement RBDJBK63HX
- [Bimodule](https://tamnd.github.io/math2/node/RBPR7BK5T5.md): definition RBPR7BK5T5
- [Eigenvalue algorithm](https://tamnd.github.io/math2/node/RBVCDV3GCY.md): definition RBVCDV3GCY
- [Hyperbolic geometry](https://tamnd.github.io/math2/node/RBZPZTSPZE.md): definition RBZPZTSPZE
- [Wasserstein metric](https://tamnd.github.io/math2/node/RC22S41ZDB.md): definition RC22S41ZDB
- [Golod–Shafarevich theorem](https://tamnd.github.io/math2/node/RC5A5T7YY2.md): statement RC5A5T7YY2
- [Mori domain](https://tamnd.github.io/math2/node/RC7HCZCRC5.md): definition RC7HCZCRC5
- [Karlin–Rubin theorem](https://tamnd.github.io/math2/node/RCASN31FFS.md): statement RCASN31FFS
- [Semilinear PDE](https://tamnd.github.io/math2/node/RCNY89A4NY.md): concept RCNY89A4NY
- [Anosov diffeomorphism](https://tamnd.github.io/math2/node/RCQH8P6DPQ.md): concept RCQH8P6DPQ
- [Tarski's high school algebra problem](https://tamnd.github.io/math2/node/RCRD8CAVB4.md): problem RCRD8CAVB4
- [Shape derivative](https://tamnd.github.io/math2/node/RCSXGDTTET.md): definition RCSXGDTTET
- [Order topology](https://tamnd.github.io/math2/node/RCWHDMWP1T.md): concept RCWHDMWP1T
- [Polar space](https://tamnd.github.io/math2/node/RD1ES06VHN.md): definition RD1ES06VHN
- [Post algebra](https://tamnd.github.io/math2/node/RD2A5Z7E2B.md): concept RD2A5Z7E2B
- [Matching pursuit](https://tamnd.github.io/math2/node/RD3K5HADYG.md): definition RD3K5HADYG
- [Morita equivalence](https://tamnd.github.io/math2/node/RD5WKA1VK7.md): concept RD5WKA1VK7
- [Weierstrass polynomial](https://tamnd.github.io/math2/node/RD7DVC2NJ6.md): definition RD7DVC2NJ6
- [Iterated exponentiation](https://tamnd.github.io/math2/node/RDC61V812J.md): example RDC61V812J
- [Projection-valued measure](https://tamnd.github.io/math2/node/RDE658H8Z4.md): definition RDE658H8Z4
- [Self-avoiding walk](https://tamnd.github.io/math2/node/RDFA0SYHEQ.md): definition RDFA0SYHEQ
- [Gelfand–Levitan–Marchenko equation](https://tamnd.github.io/math2/node/RDFRDH3E45.md): definition RDFRDH3E45
- [Laplace transform of a power function](https://tamnd.github.io/math2/node/RDFYHS2Q55.md): example RDFYHS2Q55
- [Way-below relation](https://tamnd.github.io/math2/node/RDM7J5J9HQ.md): concept RDM7J5J9HQ
- [Semiprimary ring](https://tamnd.github.io/math2/node/RDPDZJ6E6H.md): definition RDPDZJ6E6H
- [Lanczos sigma factor](https://tamnd.github.io/math2/node/RDPX2NEM47.md): definition RDPX2NEM47
- [Relation algebra](https://tamnd.github.io/math2/node/RDSY11X6N2.md): definition RDSY11X6N2
- [Cohomology ring](https://tamnd.github.io/math2/node/RDT4785M4D.md): concept RDT4785M4D
- [Indirect method by shooting](https://tamnd.github.io/math2/node/RDTESNZTKQ.md): concept RDTESNZTKQ
- [Local field](https://tamnd.github.io/math2/node/RDV3C0PXTH.md): definition RDV3C0PXTH
- [Gromov's non-squeezing theorem](https://tamnd.github.io/math2/node/RDX55HQ3XT.md): statement RDX55HQ3XT
- [Jordan–Zassenhaus theorem](https://tamnd.github.io/math2/node/RDZGJ21WRX.md): statement RDZGJ21WRX
- [Loss function](https://tamnd.github.io/math2/node/RE02MP1H45.md): definition RE02MP1H45
- [Injective module](https://tamnd.github.io/math2/node/RE5T59YZ46.md): definition RE5T59YZ46
- [Quantifier elimination](https://tamnd.github.io/math2/node/RE5TFTDP4D.md): definition RE5TFTDP4D
- [Quiver](https://tamnd.github.io/math2/node/RE9YFFS3TS.md): concept RE9YFFS3TS
- [Contravariant functor](https://tamnd.github.io/math2/node/RECR96V97B.md): concept RECR96V97B
- [Congruence lattice](https://tamnd.github.io/math2/node/REE07CSXHR.md): concept REE07CSXHR
- [Fatou set](https://tamnd.github.io/math2/node/REER3CG7DS.md): definition REER3CG7DS
- [Neumann boundary condition](https://tamnd.github.io/math2/node/REH63MEBNC.md): definition REH63MEBNC
- [Fundamental identity for ramification and inertia](https://tamnd.github.io/math2/node/REJDQ6BTQV.md): statement REJDQ6BTQV
- [Affine independence](https://tamnd.github.io/math2/node/RERRK4RFZE.md): definition RERRK4RFZE
- [Carleson measure](https://tamnd.github.io/math2/node/RF3Y3CF0YF.md): definition RF3Y3CF0YF
- [Tjurina number](https://tamnd.github.io/math2/node/RF66FF634A.md): definition RF66FF634A
- [Grassmannian as classifying space](https://tamnd.github.io/math2/node/RF8WD9TSE7.md): example RF8WD9TSE7
- [Gauss–Jacobi quadrature](https://tamnd.github.io/math2/node/RFA8HAEW9D.md): statement RFA8HAEW9D
- [Euler's theorem in geometry](https://tamnd.github.io/math2/node/RFCQT6TKZF.md): statement RFCQT6TKZF
- [Approximate derivative](https://tamnd.github.io/math2/node/RFK1AVB9H7.md): definition RFK1AVB9H7
- [Green's relations](https://tamnd.github.io/math2/node/RFKHKD4KXY.md): definition RFKHKD4KXY
- [Compact-open topology](https://tamnd.github.io/math2/node/RG15F0TWPW.md): concept RG15F0TWPW
- [Stieltjes transform](https://tamnd.github.io/math2/node/RG1HFAYD98.md): concept RG1HFAYD98
- [Strong dual space](https://tamnd.github.io/math2/node/RG52DZ1A8Y.md): concept RG52DZ1A8Y
- [Kähler differentials](https://tamnd.github.io/math2/node/RG6WFFW7K3.md): definition RG6WFFW7K3
- [Bing metrization theorem](https://tamnd.github.io/math2/node/RGCZ2P9QRM.md): statement RGCZ2P9QRM
- [Gauss–Legendre quadrature](https://tamnd.github.io/math2/node/RGGXDRCJVE.md): concept RGGXDRCJVE
- [Spline interpolation](https://tamnd.github.io/math2/node/RGT7WEZG8X.md): definition RGT7WEZG8X
- [Hardy–Littlewood circle method](https://tamnd.github.io/math2/node/RGT9HH7SQQ.md): definition RGT9HH7SQQ
- [Mild solution](https://tamnd.github.io/math2/node/RGWYQCS5A1.md): definition RGWYQCS5A1
- [Order embedding](https://tamnd.github.io/math2/node/RH0HD05BMN.md): concept RH0HD05BMN
- [Banach–Mazur compactum](https://tamnd.github.io/math2/node/RH1WTN38RQ.md): object RH1WTN38RQ
- [Whitney extension theorem](https://tamnd.github.io/math2/node/RH2EZGYGAW.md): statement RH2EZGYGAW
- [Representation of a quiver](https://tamnd.github.io/math2/node/RH6P1FGZSR.md): definition RH6P1FGZSR
- [Butcher tableau](https://tamnd.github.io/math2/node/RH8BD7PW5F.md): concept RH8BD7PW5F
- [Structural rigidity](https://tamnd.github.io/math2/node/RHDAR5YY8S.md): concept RHDAR5YY8S
- [Nontangential limit](https://tamnd.github.io/math2/node/RHDBK62123.md): concept RHDBK62123
- [Steiner chain](https://tamnd.github.io/math2/node/RHFSPFGSZJ.md): example RHFSPFGSZJ
- [Poincaré asymptotic series](https://tamnd.github.io/math2/node/RHG3TRY19X.md): definition RHG3TRY19X
- [Poincaré half-plane model](https://tamnd.github.io/math2/node/RHKSMPBSQ1.md): example RHKSMPBSQ1
- [Parabolic and hyperbolic Riemann surfaces](https://tamnd.github.io/math2/node/RHNJECZ9WR.md): concept RHNJECZ9WR
- [Graeffe's method](https://tamnd.github.io/math2/node/RHPQG175AH.md): statement RHPQG175AH
- [Beeman's algorithm](https://tamnd.github.io/math2/node/RHQ88QD2BE.md): statement RHQ88QD2BE
- [Exponential object](https://tamnd.github.io/math2/node/RHWD8DY41R.md): concept RHWD8DY41R
- [Knaster–Tarski theorem](https://tamnd.github.io/math2/node/RHXP3HA64M.md): statement RHXP3HA64M
- [Black–Scholes equation](https://tamnd.github.io/math2/node/RJ9767FFTF.md): definition RJ9767FFTF
- [Integral of a Hopf algebra](https://tamnd.github.io/math2/node/RJA5026RGX.md): concept RJA5026RGX
- [Beth definability theorem](https://tamnd.github.io/math2/node/RJBDSA2ETP.md): statement RJBDSA2ETP
- [Jensen's covering theorem](https://tamnd.github.io/math2/node/RJE1W5NRE3.md): statement RJE1W5NRE3
- [Heun's method](https://tamnd.github.io/math2/node/RJEK60G28F.md): statement RJEK60G28F
- [Unification](https://tamnd.github.io/math2/node/RJJ9D0WF53.md): concept RJJ9D0WF53
- [Fundamental convergence theorem](https://tamnd.github.io/math2/node/RJK6ZKNF6R.md): statement RJK6ZKNF6R
- [Dihedral quandle](https://tamnd.github.io/math2/node/RJYKTAPSBJ.md): example RJYKTAPSBJ
- [Functional equations on groups](https://tamnd.github.io/math2/node/RJYSXGZTTE.md): concept RJYSXGZTTE
- [Fourier–Legendre series](https://tamnd.github.io/math2/node/RK2G9HN63E.md): concept RK2G9HN63E
- [Attainable null space](https://tamnd.github.io/math2/node/RK2N3QP4FQ.md): concept RK2N3QP4FQ
- [Ramanujan's 1psi1 summation](https://tamnd.github.io/math2/node/RK6DCJVJCR.md): statement RK6DCJVJCR
- [Noncentral chi-squared distribution](https://tamnd.github.io/math2/node/RK84ANP6BP.md): concept RK84ANP6BP
- [Adjoint integral equation](https://tamnd.github.io/math2/node/RK8B7KEA3M.md): definition RK8B7KEA3M
- [Group representation](https://tamnd.github.io/math2/node/RK8RV2C998.md): definition RK8RV2C998
- [Cylindric algebra](https://tamnd.github.io/math2/node/RKCSWWVYTC.md): definition RKCSWWVYTC
- [Matrix summability method](https://tamnd.github.io/math2/node/RKFB6AHPBE.md): definition RKFB6AHPBE
- [Segre embedding](https://tamnd.github.io/math2/node/RKJBNRKEDT.md): example RKJBNRKEDT
- [Dirichlet's principle](https://tamnd.github.io/math2/node/RKK3XHG4MR.md): statement RKK3XHG4MR
- [Mackey–Arens theorem](https://tamnd.github.io/math2/node/RKKWS7CDNF.md): statement RKKWS7CDNF
- [Fermat's principle](https://tamnd.github.io/math2/node/RKX9XJ0RVW.md): statement RKX9XJ0RVW
- [Local homomorphism](https://tamnd.github.io/math2/node/RM0FMT0X8W.md): definition RM0FMT0X8W
- [Complete elliptic integral of the first kind](https://tamnd.github.io/math2/node/RM54CPPE12.md): concept RM54CPPE12
- [Dedekind's problem](https://tamnd.github.io/math2/node/RM64J2CP1M.md): problem RM64J2CP1M
- [Turing pattern](https://tamnd.github.io/math2/node/RM94KBV8P7.md): concept RM94KBV8P7
- [Tomita–Takesaki theory](https://tamnd.github.io/math2/node/RMEF8T6C0E.md): definition RMEF8T6C0E
- [Hilbert polynomial](https://tamnd.github.io/math2/node/RMH4FBCFDA.md): concept RMH4FBCFDA
- [Borel hierarchy](https://tamnd.github.io/math2/node/RMH7ZQ2F40.md): definition RMH7ZQ2F40
- [Saturated model](https://tamnd.github.io/math2/node/RMTZ5J64FX.md): concept RMTZ5J64FX
- [Character of a representation](https://tamnd.github.io/math2/node/RMWMYBY8TC.md): concept RMWMYBY8TC
- [Bartle–Dunford–Schwartz integral](https://tamnd.github.io/math2/node/RMWW2X86TT.md): concept RMWW2X86TT
- [Rado graph](https://tamnd.github.io/math2/node/RMXR0FEF54.md): object RMXR0FEF54
- [Analyticity of the bilateral Laplace transform of a law](https://tamnd.github.io/math2/node/RMXW3D9XZ3.md): statement RMXW3D9XZ3
- [Semicontinuity](https://tamnd.github.io/math2/node/RN4G8SGP6P.md): concept RN4G8SGP6P
- [Tychonoff's non-uniqueness example](https://tamnd.github.io/math2/node/RN4J24CD83.md): counterexample RN4J24CD83
- [Gelfond–Schneider theorem](https://tamnd.github.io/math2/node/RN58Z2NWAC.md): statement RN58Z2NWAC
- [Stereographic projection](https://tamnd.github.io/math2/node/RN8RBM2A8A.md): definition RN8RBM2A8A
- [Soliton](https://tamnd.github.io/math2/node/RN8Y1823FX.md): definition RN8Y1823FX
- [Kan extension](https://tamnd.github.io/math2/node/RN9D96ESWN.md): concept RN9D96ESWN
- [Sokhotski–Plemelj theorem](https://tamnd.github.io/math2/node/RNB6QX0GCA.md): statement RNB6QX0GCA
- [Finitely complete category](https://tamnd.github.io/math2/node/RNCFXWTPFK.md): definition RNCFXWTPFK
- [Euler characteristic](https://tamnd.github.io/math2/node/RNDDWDAFJ5.md): concept RNDDWDAFJ5
- [Mosco convergence](https://tamnd.github.io/math2/node/RNDKD74V17.md): definition RNDKD74V17
- [Hyperoval](https://tamnd.github.io/math2/node/RNG2A8BFVZ.md): concept RNG2A8BFVZ
- [Brunn–Minkowski theory](https://tamnd.github.io/math2/node/RNS272QBGT.md): concept RNS272QBGT
- [Semilattice](https://tamnd.github.io/math2/node/RNTAAHTD0M.md): definition RNTAAHTD0M
- [Risk function](https://tamnd.github.io/math2/node/RNV041SNCT.md): definition RNV041SNCT
- [Strictly singular operator](https://tamnd.github.io/math2/node/RNX150HS1P.md): concept RNX150HS1P
- [Henselian ring](https://tamnd.github.io/math2/node/RNXPV3KC5J.md): concept RNXPV3KC5J
- [Orbifold](https://tamnd.github.io/math2/node/RPAVY0W0XS.md): concept RPAVY0W0XS
- [Constant-mean-curvature surface](https://tamnd.github.io/math2/node/RPBKZ7E9VE.md): concept RPBKZ7E9VE
- [Littlewood polynomial](https://tamnd.github.io/math2/node/RPFX0V6BYF.md): definition RPFX0V6BYF
- [Halley's method](https://tamnd.github.io/math2/node/RPME2B5MS2.md): statement RPME2B5MS2
- [c-cyclical monotonicity](https://tamnd.github.io/math2/node/RPN59B2XSR.md): concept RPN59B2XSR
- [Mellin transform of a power times a function](https://tamnd.github.io/math2/node/RPQNSM5HKC.md): statement RPQNSM5HKC
- [Singularity of a smooth map](https://tamnd.github.io/math2/node/RPSYWTKB4V.md): definition RPSYWTKB4V
- [Kardar–Parisi–Zhang equation](https://tamnd.github.io/math2/node/RQ196EHXKC.md): concept RQ196EHXKC
- [Spectral theory](https://tamnd.github.io/math2/node/RQ39XYAG36.md): definition RQ39XYAG36
- [Volterra series](https://tamnd.github.io/math2/node/RQ3JKGWD03.md): concept RQ3JKGWD03
- [Dodgson condensation](https://tamnd.github.io/math2/node/RQ44FKS5QM.md): concept RQ44FKS5QM
- [Archimedean equivalence classes](https://tamnd.github.io/math2/node/RQ4X0EH6NH.md): concept RQ4X0EH6NH
- [Bang–bang control](https://tamnd.github.io/math2/node/RQBNA1ZRJW.md): definition RQBNA1ZRJW
- [Strictly convex space](https://tamnd.github.io/math2/node/RQDZX250B2.md): definition RQDZX250B2
- [Mean-periodic function](https://tamnd.github.io/math2/node/RQHPBZ9682.md): definition RQHPBZ9682
- [Law of the iterated logarithm](https://tamnd.github.io/math2/node/RQJC67S0QJ.md): statement RQJC67S0QJ
- [Ring of invariants](https://tamnd.github.io/math2/node/RQP2HJ4WQ1.md): definition RQP2HJ4WQ1
- [Stable vector bundle](https://tamnd.github.io/math2/node/RQRJ5H4Y3E.md): concept RQRJ5H4Y3E
- [Mazur's torsion theorem](https://tamnd.github.io/math2/node/RQS6YHAR8N.md): statement RQS6YHAR8N
- [Injective tensor product](https://tamnd.github.io/math2/node/RQXC5D3M1Y.md): concept RQXC5D3M1Y
- [Borel summation](https://tamnd.github.io/math2/node/RQYX6E42D0.md): definition RQYX6E42D0
- [Intuitionism](https://tamnd.github.io/math2/node/RR10R8JN02.md): definition RR10R8JN02
- [Askey scheme](https://tamnd.github.io/math2/node/RR2S05C09B.md): concept RR2S05C09B
- [Stiff equation](https://tamnd.github.io/math2/node/RR5FCG92PN.md): concept RR5FCG92PN
- [Baire property](https://tamnd.github.io/math2/node/RR5JH5K74X.md): definition RR5JH5K74X
- [Moduli of algebraic curves](https://tamnd.github.io/math2/node/RREV63H509.md): definition RREV63H509
- [Normal coordinates](https://tamnd.github.io/math2/node/RRHKMGNKG9.md): concept RRHKMGNKG9
- [Ample line bundle](https://tamnd.github.io/math2/node/RRK8P6TXG2.md): concept RRK8P6TXG2
- [Symmetric and self-reciprocal kernels](https://tamnd.github.io/math2/node/RRZGEFJS7H.md): concept RRZGEFJS7H
- [Normal number](https://tamnd.github.io/math2/node/RS7Z7NTDX9.md): definition RS7Z7NTDX9
- [Riemann's differential equation](https://tamnd.github.io/math2/node/RS8Z48E784.md): concept RS8Z48E784
- [Second variation](https://tamnd.github.io/math2/node/RS9VHTWR4K.md): definition RS9VHTWR4K
- [Integrable function](https://tamnd.github.io/math2/node/RSAK8YHMXF.md): definition RSAK8YHMXF
- [Gromov's theorem on groups of polynomial growth](https://tamnd.github.io/math2/node/RSBFHHPTWX.md): statement RSBFHHPTWX
- [Scheme](https://tamnd.github.io/math2/node/RSC5KTVNZD.md): definition RSC5KTVNZD
- [Eilenberg swindle](https://tamnd.github.io/math2/node/RSFYN75DCS.md): concept RSFYN75DCS
- [Askey–Wilson polynomials](https://tamnd.github.io/math2/node/RSNZNG0WA1.md): concept RSNZNG0WA1
- [Spin structure](https://tamnd.github.io/math2/node/RSPZ9TCFQ8.md): concept RSPZ9TCFQ8
- [Krein–Milman theorem](https://tamnd.github.io/math2/node/RSRAY0Q68S.md): statement RSRAY0Q68S
- [Clifford's theorem on special divisors](https://tamnd.github.io/math2/node/RSW29AVKGM.md): statement RSW29AVKGM
- [Fuller's phenomenon](https://tamnd.github.io/math2/node/RSZS9SFFZ7.md): example RSZS9SFFZ7
- [Köthe conjecture](https://tamnd.github.io/math2/node/RT2D9JEVRR.md): conjecture RT2D9JEVRR
- [Apéry's theorem](https://tamnd.github.io/math2/node/RT5MJT6CTB.md): statement RT5MJT6CTB
- [Equidistribution modulo one](https://tamnd.github.io/math2/node/RT62ZAN3JB.md): definition RT62ZAN3JB
- [Sheaf on a site](https://tamnd.github.io/math2/node/RT8A8FJ566.md): concept RT8A8FJ566
- [Harmonic functions on Riemann surfaces](https://tamnd.github.io/math2/node/RT9EYEQ6S2.md): definition RT9EYEQ6S2
- [Chebyshev polynomials of the second kind](https://tamnd.github.io/math2/node/RTA8BG53KZ.md): definition RTA8BG53KZ
- [Legendrian submanifold](https://tamnd.github.io/math2/node/RTAVY44P4Z.md): definition RTAVY44P4Z
- [Petrov–Galerkin method](https://tamnd.github.io/math2/node/RTBJM23ETS.md): statement RTBJM23ETS
- [Devaney chaos](https://tamnd.github.io/math2/node/RTG32790CW.md): definition RTG32790CW
- [Set theory](https://tamnd.github.io/math2/node/RTHHCX84HN.md): concept RTHHCX84HN
- [Coupled map lattice](https://tamnd.github.io/math2/node/RTJ6Y1AH7S.md): definition RTJ6Y1AH7S
- [Atomic domain](https://tamnd.github.io/math2/node/RTJQPZC0VY.md): definition RTJQPZC0VY
- [Floating-point arithmetic](https://tamnd.github.io/math2/node/RTKRSC3KPH.md): concept RTKRSC3KPH
- [Semiautomaton as a unary algebra](https://tamnd.github.io/math2/node/RTMYXRDGWR.md): example RTMYXRDGWR
- [Mesh generation](https://tamnd.github.io/math2/node/RTSAFF4KFJ.md): concept RTSAFF4KFJ
- [Nerve of a category](https://tamnd.github.io/math2/node/RV0MKJ3CQ6.md): concept RV0MKJ3CQ6
- [Multidimensional Mellin transform](https://tamnd.github.io/math2/node/RV67AGM6EJ.md): definition RV67AGM6EJ
- [Almgren's regularity theorem](https://tamnd.github.io/math2/node/RV6PWJBH5P.md): statement RV6PWJBH5P
- [Global field](https://tamnd.github.io/math2/node/RVEFNEJSHF.md): definition RVEFNEJSHF
- [Natural density](https://tamnd.github.io/math2/node/RVG9Z8NYA6.md): concept RVG9Z8NYA6
- [Embedded prime](https://tamnd.github.io/math2/node/RVKPM94MDF.md): concept RVKPM94MDF
- [Hessenberg matrix](https://tamnd.github.io/math2/node/RVMW0RP050.md): definition RVMW0RP050
- [Infinite Galois theory](https://tamnd.github.io/math2/node/RVR5QFFD77.md): concept RVR5QFFD77
- [Partial correlation](https://tamnd.github.io/math2/node/RVVRYKVGXK.md): definition RVVRYKVGXK
- [Rational variety](https://tamnd.github.io/math2/node/RVX312A2D2.md): definition RVX312A2D2
- [Epimorphism](https://tamnd.github.io/math2/node/RVXDSY04KG.md): concept RVXDSY04KG
- [Carleman's condition](https://tamnd.github.io/math2/node/RVZ831CNYW.md): statement RVZ831CNYW
- [Euler equations](https://tamnd.github.io/math2/node/RW0JK9VDZE.md): concept RW0JK9VDZE
- [Proof of the uniqueness of inverses](https://tamnd.github.io/math2/node/RW1ZF04CRF.md): proof RW1ZF04CRF
- [Jordan–Dedekind chain condition](https://tamnd.github.io/math2/node/RW7YC3HG0S.md): concept RW7YC3HG0S
- [Group action on a variety](https://tamnd.github.io/math2/node/RW85C34GQW.md): concept RW85C34GQW
- [Differential Galois theory](https://tamnd.github.io/math2/node/RW8DQQYJMR.md): definition RW8DQQYJMR
- [Singular simplicial set](https://tamnd.github.io/math2/node/RWBMXQW6S5.md): concept RWBMXQW6S5
- [Regulator of a number field](https://tamnd.github.io/math2/node/RWK0HNF1NA.md): concept RWK0HNF1NA
- [Lebesgue constants of the Dirichlet kernel](https://tamnd.github.io/math2/node/RWKQ5EP4YF.md): statement RWKQ5EP4YF
- [Zeilberger's algorithm](https://tamnd.github.io/math2/node/RWNTZ2AYA7.md): definition RWNTZ2AYA7
- [Church–Turing thesis](https://tamnd.github.io/math2/node/RWRNBS3Z7B.md): concept RWRNBS3Z7B
- [Noncommutative geometry](https://tamnd.github.io/math2/node/RWTN4E95YZ.md): definition RWTN4E95YZ
- [Serre's modularity conjecture](https://tamnd.github.io/math2/node/RWXXNZNQYJ.md): statement RWXXNZNQYJ
- [Random ordinary differential equation](https://tamnd.github.io/math2/node/RWZ5KMPD1R.md): concept RWZ5KMPD1R
- [Dynkin quiver](https://tamnd.github.io/math2/node/RX1NWE1WMD.md): definition RX1NWE1WMD
- [Dirichlet distribution](https://tamnd.github.io/math2/node/RX4CVSSRRJ.md): example RX4CVSSRRJ
- [Pfaffian system](https://tamnd.github.io/math2/node/RX5MYWE73J.md): concept RX5MYWE73J
- [Fundamental theorem of calculus for Lebesgue integrals](https://tamnd.github.io/math2/node/RX7JSHMP8Z.md): statement RX7JSHMP8Z
- [Continuous lattice](https://tamnd.github.io/math2/node/RXBD58SGXZ.md): concept RXBD58SGXZ
- [Discrete-time Fourier transform](https://tamnd.github.io/math2/node/RXFJ47VYGX.md): concept RXFJ47VYGX
- [Heyting arithmetic](https://tamnd.github.io/math2/node/RXHJPVMPJH.md): definition RXHJPVMPJH
- [Locally cyclic group](https://tamnd.github.io/math2/node/RXWA8TF2RM.md): concept RXWA8TF2RM
- [Phase space](https://tamnd.github.io/math2/node/RY2RAH3802.md): definition RY2RAH3802
- [Bluestein's FFT algorithm](https://tamnd.github.io/math2/node/RY3QGW87HX.md): statement RY3QGW87HX
- [Kurosh subgroup theorem](https://tamnd.github.io/math2/node/RY8K5RQJSR.md): statement RY8K5RQJSR
- [Riemann–Hilbert problem](https://tamnd.github.io/math2/node/RY8YQ9JZF9.md): problem RY8YQ9JZF9
- [Height function](https://tamnd.github.io/math2/node/RYA0R6P73C.md): definition RYA0R6P73C
- [Euler's theorem on normal curvatures](https://tamnd.github.io/math2/node/RYB4DQTAK0.md): statement RYB4DQTAK0
- [Adjunction formula](https://tamnd.github.io/math2/node/RYB6F7JFGV.md): statement RYB6F7JFGV
- [Calderón–Zygmund estimates](https://tamnd.github.io/math2/node/RYQJH4RYB4.md): statement RYQJH4RYB4
- [Infinite product](https://tamnd.github.io/math2/node/RZ0Z2BTMT4.md): definition RZ0Z2BTMT4
- [Nef line bundle](https://tamnd.github.io/math2/node/RZ66X0NAM5.md): concept RZ66X0NAM5
- [Contingency table](https://tamnd.github.io/math2/node/RZHB962EMF.md): definition RZHB962EMF
- [An infinite product converges iff its series of logarithms converges](https://tamnd.github.io/math2/node/RZKC8B00HZ.md): statement RZKC8B00HZ
- [Schauder basis](https://tamnd.github.io/math2/node/RZT4WMH4SG.md): definition RZT4WMH4SG
- [Abel–Poisson summation of Fourier series](https://tamnd.github.io/math2/node/RZZ78GXP5F.md): concept RZZ78GXP5F
- [Cramér's theorem](https://tamnd.github.io/math2/node/RZZK51CXMN.md): statement RZZK51CXMN
- [Zak transform](https://tamnd.github.io/math2/node/S026HQSGY9.md): concept S026HQSGY9
- [Nilpotent Lie group](https://tamnd.github.io/math2/node/S06VTCGCHN.md): definition S06VTCGCHN
- [Median algebra as a ternary algebra](https://tamnd.github.io/math2/node/S0DAGR1WN7.md): example S0DAGR1WN7
- [Mean-field game](https://tamnd.github.io/math2/node/S0QKEJQ1WM.md): definition S0QKEJQ1WM
- [Dehn invariant](https://tamnd.github.io/math2/node/S0TGRX3ARS.md): concept S0TGRX3ARS
- [Seifert–van Kampen theorem](https://tamnd.github.io/math2/node/S17DY98VF2.md): statement S17DY98VF2
- [Erdős–Rado theorem](https://tamnd.github.io/math2/node/S1BHE358VX.md): statement S1BHE358VX
- [Oka principle](https://tamnd.github.io/math2/node/S1CJV1PA3D.md): statement S1CJV1PA3D
- [Uncertainty principle in Fourier analysis](https://tamnd.github.io/math2/node/S1FB2K14NV.md): statement S1FB2K14NV
- [Exact functor](https://tamnd.github.io/math2/node/S1G61TDW0K.md): concept S1G61TDW0K
- [Kirchhoff's matrix tree theorem](https://tamnd.github.io/math2/node/S1PG1N2VEX.md): statement S1PG1N2VEX
- [Quintic equation](https://tamnd.github.io/math2/node/S1WS18QEGJ.md): example S1WS18QEGJ
- [Fundamental group](https://tamnd.github.io/math2/node/S1XK3HXHFX.md): concept S1XK3HXHFX
- [Octonionic projective plane](https://tamnd.github.io/math2/node/S26GYRVDR5.md): example S26GYRVDR5
- [Tessellation](https://tamnd.github.io/math2/node/S28N32RCHY.md): definition S28N32RCHY
- [Optimal control of partial differential equations](https://tamnd.github.io/math2/node/S2BBZPAFDR.md): concept S2BBZPAFDR
- [Suspension flow](https://tamnd.github.io/math2/node/S2CV71FCY2.md): definition S2CV71FCY2
- [Minimal prime ideal](https://tamnd.github.io/math2/node/S2D8823P8B.md): concept S2D8823P8B
- [Karoubi envelope](https://tamnd.github.io/math2/node/S2GBXQ26QM.md): concept S2GBXQ26QM
- [Minlos theorem](https://tamnd.github.io/math2/node/S2J9CBPJ0V.md): statement S2J9CBPJ0V
- [Schlegel diagram](https://tamnd.github.io/math2/node/S2JE8M3FMP.md): concept S2JE8M3FMP
- [Socle of a module](https://tamnd.github.io/math2/node/S2KXCAH5MM.md): concept S2KXCAH5MM
- [Affine plane over a finite field](https://tamnd.github.io/math2/node/S2NAGK9YHF.md): example S2NAGK9YHF
- [Hardy spaces of harmonic functions](https://tamnd.github.io/math2/node/S2T8X7QYNM.md): definition S2T8X7QYNM
- [Hopf's formula](https://tamnd.github.io/math2/node/S31M1ZYJRK.md): statement S31M1ZYJRK
- [Axiom A](https://tamnd.github.io/math2/node/S31RTTJPDC.md): definition S31RTTJPDC
- [Sheaf of modules](https://tamnd.github.io/math2/node/S33B2H3MYD.md): concept S33B2H3MYD
- [Polyconvexity](https://tamnd.github.io/math2/node/S34G7679AK.md): definition S34G7679AK
- [Localization of a module](https://tamnd.github.io/math2/node/S357810Z25.md): definition S357810Z25
- [Circle packing](https://tamnd.github.io/math2/node/S360JBEYKD.md): concept S360JBEYKD
- [Hereditary ring](https://tamnd.github.io/math2/node/S36T7TR7AG.md): definition S36T7TR7AG
- [Fourier series](https://tamnd.github.io/math2/node/S36XX2PVNH.md): definition S36XX2PVNH
- [Zero divisors are the union of associated primes](https://tamnd.github.io/math2/node/S37DQGW3TZ.md): statement S37DQGW3TZ
- [Countably compact space](https://tamnd.github.io/math2/node/S3N3CHZV89.md): concept S3N3CHZV89
- [Mixing](https://tamnd.github.io/math2/node/S3RJQ9CX2P.md): concept S3RJQ9CX2P
- [Birkhoff sum and average](https://tamnd.github.io/math2/node/S3YF52GRJM.md): definition S3YF52GRJM
- [Grothendieck topos](https://tamnd.github.io/math2/node/S3YVE24AQK.md): concept S3YVE24AQK
- [Kernel](https://tamnd.github.io/math2/node/S417YNEJ6M.md): concept S417YNEJ6M
- [Submanifold](https://tamnd.github.io/math2/node/S4543S2GBJ.md): definition S4543S2GBJ
- [Comonad](https://tamnd.github.io/math2/node/S47R8TG0T3.md): definition S47R8TG0T3
- [Levi-flat hypersurface](https://tamnd.github.io/math2/node/S4842D7MAE.md): concept S4842D7MAE
- [Tsirelson space](https://tamnd.github.io/math2/node/S49XMZKGQC.md): example S49XMZKGQC
- [Torsion of a curve](https://tamnd.github.io/math2/node/S4BZ697DSH.md): concept S4BZ697DSH
- [Radical ideal](https://tamnd.github.io/math2/node/S4GQN5V54X.md): concept S4GQN5V54X
- [Silverman–Toeplitz theorem](https://tamnd.github.io/math2/node/S4J7SRPXMZ.md): statement S4J7SRPXMZ
- [Stone's representation theorem for Boolean algebras](https://tamnd.github.io/math2/node/S4QJ395PA5.md): statement S4QJ395PA5
- [Quasiconvex function](https://tamnd.github.io/math2/node/S4RMH7JQB8.md): concept S4RMH7JQB8
- [Hecke operator](https://tamnd.github.io/math2/node/S4SV3QZ8B1.md): concept S4SV3QZ8B1
- [Rankine–Hugoniot conditions](https://tamnd.github.io/math2/node/S4XMYJ3V42.md): statement S4XMYJ3V42
- [Product](https://tamnd.github.io/math2/node/S5744QZV4N.md): concept S5744QZV4N
- [Hochschild homology](https://tamnd.github.io/math2/node/S5H153HFWC.md): concept S5H153HFWC
- [Period-doubling bifurcation](https://tamnd.github.io/math2/node/S5HVHEPSMA.md): definition S5HVHEPSMA
- [Haar measure on a Lie group](https://tamnd.github.io/math2/node/S5VTE2R86M.md): definition S5VTE2R86M
- [Group of Lie type](https://tamnd.github.io/math2/node/S5Z1XS6MZN.md): definition S5Z1XS6MZN
- [Fox H-function transforms](https://tamnd.github.io/math2/node/S60PA0RXRY.md): concept S60PA0RXRY
- [Skew polynomial ring](https://tamnd.github.io/math2/node/S6C8KDBS04.md): concept S6C8KDBS04
- [Fiber product of schemes](https://tamnd.github.io/math2/node/S6CCJBJKHR.md): definition S6CCJBJKHR
- [Shanks transformation](https://tamnd.github.io/math2/node/S6E3H24FGB.md): definition S6E3H24FGB
- [Relaxation of a variational problem](https://tamnd.github.io/math2/node/S6KGGNFKDF.md): definition S6KGGNFKDF
- [Unique factorization of ideals in a Dedekind domain](https://tamnd.github.io/math2/node/S6PHT75MEK.md): statement S6PHT75MEK
- [Survival analysis](https://tamnd.github.io/math2/node/S6VR0CVZ86.md): definition S6VR0CVZ86
- [Arithmetic progression](https://tamnd.github.io/math2/node/S70JBRSQZX.md): example S70JBRSQZX
- [Hardy's inequality](https://tamnd.github.io/math2/node/S7A8YFGG8X.md): statement S7A8YFGG8X
- [Bidiagonalization](https://tamnd.github.io/math2/node/S7MC88TW6W.md): statement S7MC88TW6W
- [Class operators H S P](https://tamnd.github.io/math2/node/S7MJVSP79N.md): concept S7MJVSP79N
- [Green's relations](https://tamnd.github.io/math2/node/S7WJW2YES2.md): concept S7WJW2YES2
- [Operator space](https://tamnd.github.io/math2/node/S805N6Z8G7.md): definition S805N6Z8G7
- [Riccati difference equation](https://tamnd.github.io/math2/node/S8AT0HNM07.md): definition S8AT0HNM07
- [Toda bracket](https://tamnd.github.io/math2/node/S8FJ6X8PRN.md): definition S8FJ6X8PRN
- [Incidence structure](https://tamnd.github.io/math2/node/S8FNWJYVXM.md): definition S8FNWJYVXM
- [Principal symbol of an operator on a manifold](https://tamnd.github.io/math2/node/S8GC34D5ZG.md): concept S8GC34D5ZG
- [Laplace's equation](https://tamnd.github.io/math2/node/S8RG6N1PRT.md): concept S8RG6N1PRT
- [Brown–Peterson cohomology](https://tamnd.github.io/math2/node/S8S72AA9C8.md): concept S8S72AA9C8
- [Persistent homology](https://tamnd.github.io/math2/node/S8SNBHVPSP.md): concept S8SNBHVPSP
- [Foliation](https://tamnd.github.io/math2/node/S8TA91BAN8.md): definition S8TA91BAN8
- [Selberg integral](https://tamnd.github.io/math2/node/S8VTVSJ6EX.md): statement S8VTVSJ6EX
- [Prime model](https://tamnd.github.io/math2/node/S8W40PYNHX.md): concept S8W40PYNHX
- [Chebyshev's sum inequality](https://tamnd.github.io/math2/node/S8X2N1DW8P.md): statement S8X2N1DW8P
- [Divisible group](https://tamnd.github.io/math2/node/S8Y8RVZY79.md): definition S8Y8RVZY79
- [Fatou's lemma](https://tamnd.github.io/math2/node/S95TR54TCG.md): statement S95TR54TCG
- [Fast Fourier transform](https://tamnd.github.io/math2/node/S966EZX3ZT.md): statement S966EZX3ZT
- [Wave equation](https://tamnd.github.io/math2/node/S977PT8S3S.md): definition S977PT8S3S
- [Conjugate prior](https://tamnd.github.io/math2/node/S99HFQ9WPH.md): definition S99HFQ9WPH
- [Muckenhoupt's theorem](https://tamnd.github.io/math2/node/S9BKNBRV4H.md): statement S9BKNBRV4H
- [Kleene's T predicate](https://tamnd.github.io/math2/node/S9DK13K5PS.md): concept S9DK13K5PS
- [Category of elements](https://tamnd.github.io/math2/node/S9EJKKRHW1.md): concept S9EJKKRHW1
- [Lambek's lemma](https://tamnd.github.io/math2/node/S9FVESJ0CS.md): statement S9FVESJ0CS
- [Herglotz–Riesz kernel](https://tamnd.github.io/math2/node/S9KDHJF5YV.md): definition S9KDHJF5YV
- [Kummer's transformation](https://tamnd.github.io/math2/node/S9VTAYV1VJ.md): statement S9VTAYV1VJ
- [Residuated lattice](https://tamnd.github.io/math2/node/S9Z6PKTXAS.md): definition S9Z6PKTXAS
- [Priestley duality](https://tamnd.github.io/math2/node/SA22SEC75K.md): statement SA22SEC75K
- [Weak solution](https://tamnd.github.io/math2/node/SA40H8EGJZ.md): definition SA40H8EGJZ
- [Reduced homology](https://tamnd.github.io/math2/node/SA493EH7DT.md): definition SA493EH7DT
- [Einstein–Hilbert action](https://tamnd.github.io/math2/node/SA5ZDFBHNZ.md): definition SA5ZDFBHNZ
- [Real form of a Lie group](https://tamnd.github.io/math2/node/SADNTQAY3G.md): concept SADNTQAY3G
- [Mellin transform of a theta series and the zeta function](https://tamnd.github.io/math2/node/SAGNEFJGVP.md): example SAGNEFJGVP
- [Model predictive control](https://tamnd.github.io/math2/node/SAH0W43RMW.md): definition SAH0W43RMW
- [Clifford theory](https://tamnd.github.io/math2/node/SAH4PD2J9T.md): statement SAH4PD2J9T
- [Homological algebra](https://tamnd.github.io/math2/node/SAHAEZNCVD.md): concept SAHAEZNCVD
- [Frame operator](https://tamnd.github.io/math2/node/SAQ5EFQP7H.md): definition SAQ5EFQP7H
- [Moore plane](https://tamnd.github.io/math2/node/SAR7PBYF07.md): counterexample SAR7PBYF07
- [Variety of algebras](https://tamnd.github.io/math2/node/SARA216BGB.md): definition SARA216BGB
- [Kolmogorov–Smirnov test](https://tamnd.github.io/math2/node/SARNZPAKE5.md): definition SARNZPAKE5
- [Lp space](https://tamnd.github.io/math2/node/SASP14MA1Z.md): concept SASP14MA1Z
- [Quaternionic analysis](https://tamnd.github.io/math2/node/SAT7ST5WP7.md): concept SAT7ST5WP7
- [Artin group](https://tamnd.github.io/math2/node/SAWBDBHE41.md): concept SAWBDBHE41
- [Lebesgue differentiation theorem](https://tamnd.github.io/math2/node/SAZ0ESVBVP.md): statement SAZ0ESVBVP
- [Injective tensor product](https://tamnd.github.io/math2/node/SB04FYJ0A9.md): definition SB04FYJ0A9
- [MUSCL scheme](https://tamnd.github.io/math2/node/SB1TM11S5Q.md): definition SB1TM11S5Q
- [Modification of a complex space](https://tamnd.github.io/math2/node/SB3MY7XQM9.md): concept SB3MY7XQM9
- [Denjoy–Carleman–Ahlfors theorem](https://tamnd.github.io/math2/node/SB55PWPRH2.md): statement SB55PWPRH2
- [Quasiconformal mapping](https://tamnd.github.io/math2/node/SB8KG46NHP.md): concept SB8KG46NHP
- [Weyl–von Neumann theorem](https://tamnd.github.io/math2/node/SB8PFJCJBC.md): statement SB8PFJCJBC
- [Binary quadratic form](https://tamnd.github.io/math2/node/SBDSYA037Q.md): concept SBDSYA037Q
- [Koenigs function](https://tamnd.github.io/math2/node/SBF5C86E3Z.md): definition SBF5C86E3Z
- [Weyl tensor](https://tamnd.github.io/math2/node/SBHY31HESP.md): concept SBHY31HESP
- [Marsden–Weinstein reduction](https://tamnd.github.io/math2/node/SBJVATVH44.md): concept SBJVATVH44
- [Control system](https://tamnd.github.io/math2/node/SBQVGQPXT1.md): definition SBQVGQPXT1
- [F-space](https://tamnd.github.io/math2/node/SBW5P5B3FV.md): definition SBW5P5B3FV
- [Supercompact cardinal](https://tamnd.github.io/math2/node/SC3MSJRJ3S.md): definition SC3MSJRJ3S
- [Ladyzhenskaya–Babuška–Brezzi condition](https://tamnd.github.io/math2/node/SC5FV3GHV5.md): statement SC5FV3GHV5
- [Extremal graph theory](https://tamnd.github.io/math2/node/SC5QB5F4TE.md): concept SC5QB5F4TE
- [Preadditive category](https://tamnd.github.io/math2/node/SC6FD5F4J0.md): concept SC6FD5F4J0
- [Periodic boundary conditions](https://tamnd.github.io/math2/node/SC8HT9T2GA.md): definition SC8HT9T2GA
- [Elliptic operator](https://tamnd.github.io/math2/node/SCCP5E254A.md): definition SCCP5E254A
- [Accessible category](https://tamnd.github.io/math2/node/SCG4061G4W.md): definition SCG4061G4W
- [Embedding of structures](https://tamnd.github.io/math2/node/SCRB74SWQN.md): concept SCRB74SWQN
- [Design of experiments](https://tamnd.github.io/math2/node/SCYW8DVPKQ.md): definition SCYW8DVPKQ
- [Universal enveloping algebra](https://tamnd.github.io/math2/node/SCZ8PRVY00.md): concept SCZ8PRVY00
- [Homogeneous polynomial](https://tamnd.github.io/math2/node/SD3M01318E.md): definition SD3M01318E
- [Maximum and minimum](https://tamnd.github.io/math2/node/SD649DT8VX.md): definition SD649DT8VX
- [Fourier multiplier](https://tamnd.github.io/math2/node/SD89TZ1PHR.md): definition SD89TZ1PHR
- [Weierstrass polynomial](https://tamnd.github.io/math2/node/SDCKNWSWNM.md): concept SDCKNWSWNM
- [Spherical code](https://tamnd.github.io/math2/node/SDCX4WDP4F.md): concept SDCX4WDP4F
- [Shoelace formula](https://tamnd.github.io/math2/node/SDHZRWMSTT.md): statement SDHZRWMSTT
- [Tensor product of fields](https://tamnd.github.io/math2/node/SDJ4ADZK8M.md): definition SDJ4ADZK8M
- [Identity component](https://tamnd.github.io/math2/node/SDQTNBDJ2H.md): concept SDQTNBDJ2H
- [Hilbert's eighteenth problem](https://tamnd.github.io/math2/node/SDSGN4B6JD.md): problem SDSGN4B6JD
- [Model category](https://tamnd.github.io/math2/node/SDTZVZ6B4Q.md): concept SDTZVZ6B4Q
- [Shafarevich's theorem on solvable Galois groups](https://tamnd.github.io/math2/node/SDV692Z915.md): statement SDV692Z915
- [Prolate spheroidal wave functions](https://tamnd.github.io/math2/node/SDYEK7HEZQ.md): definition SDYEK7HEZQ
- [Weyl's unitary trick](https://tamnd.github.io/math2/node/SEAW0VNYJ6.md): definition SEAW0VNYJ6
- [Adele ring](https://tamnd.github.io/math2/node/SEH68BA1QQ.md): concept SEH68BA1QQ
- [Loop-erased random walk](https://tamnd.github.io/math2/node/SEKY67RPX5.md): definition SEKY67RPX5
- [Puiseux series](https://tamnd.github.io/math2/node/SENTYNSKSF.md): concept SENTYNSKSF
- [Krein space](https://tamnd.github.io/math2/node/SEVHW7TPPZ.md): definition SEVHW7TPPZ
- [Scorer's function](https://tamnd.github.io/math2/node/SF1VEDT29F.md): definition SF1VEDT29F
- [Hölder's theorem](https://tamnd.github.io/math2/node/SF3X43X837.md): statement SF3X43X837
- [Block system](https://tamnd.github.io/math2/node/SF738NKWJ0.md): concept SF738NKWJ0
- [Transform pair](https://tamnd.github.io/math2/node/SFGFWZ4F5N.md): definition SFGFWZ4F5N
- [Linking number](https://tamnd.github.io/math2/node/SFJ329YB9F.md): concept SFJ329YB9F
- [Relaxed control](https://tamnd.github.io/math2/node/SFMD8RKRSE.md): concept SFMD8RKRSE
- [Hilbertian field](https://tamnd.github.io/math2/node/SFPKB3NC0E.md): concept SFPKB3NC0E
- [Energy of a measure](https://tamnd.github.io/math2/node/SFSRQCM3EG.md): concept SFSRQCM3EG
- [Quasi-arithmetic mean](https://tamnd.github.io/math2/node/SG5C5TVGRB.md): concept SG5C5TVGRB
- [Fourier coefficient](https://tamnd.github.io/math2/node/SG5Q94QTQE.md): definition SG5Q94QTQE
- [Ore localization](https://tamnd.github.io/math2/node/SG7AQ9T8HD.md): concept SG7AQ9T8HD
- [Free loop space](https://tamnd.github.io/math2/node/SGGXZ3TBSA.md): object SGGXZ3TBSA
- [Delaunay triangulation](https://tamnd.github.io/math2/node/SGH803FGY4.md): definition SGH803FGY4
- [Excision theorem](https://tamnd.github.io/math2/node/SGHMSBHHA0.md): statement SGHMSBHHA0
- [Duffing equation](https://tamnd.github.io/math2/node/SGQQGP261M.md): example SGQQGP261M
- [Completely monotone function](https://tamnd.github.io/math2/node/SGVQ5ZMEGE.md): definition SGVQ5ZMEGE
- [Walsh function](https://tamnd.github.io/math2/node/SGXYY92GRS.md): definition SGXYY92GRS
- [Mellin convergence](https://tamnd.github.io/math2/node/SGZZ4Y0D36.md): concept SGZZ4Y0D36
- [Cayley–Hamilton theorem for modules](https://tamnd.github.io/math2/node/SH3R5VTMZH.md): statement SH3R5VTMZH
- [Green's function](https://tamnd.github.io/math2/node/SH6JY9RN2E.md): concept SH6JY9RN2E
- [Singular homology](https://tamnd.github.io/math2/node/SH7P72TJW8.md): definition SH7P72TJW8
- [Relaxed control](https://tamnd.github.io/math2/node/SH95P4SEWT.md): definition SH95P4SEWT
- [24-cell](https://tamnd.github.io/math2/node/SH9NBM8JRR.md): object SH9NBM8JRR
- [Local class field theory](https://tamnd.github.io/math2/node/SHEN158T3M.md): concept SHEN158T3M
- [Order of an entire function](https://tamnd.github.io/math2/node/SHF8Z7GMFX.md): concept SHF8Z7GMFX
- [Siegel zero](https://tamnd.github.io/math2/node/SHFJH5JZ5P.md): definition SHFJH5JZ5P
- [Gumm terms](https://tamnd.github.io/math2/node/SHMTFRH28E.md): concept SHMTFRH28E
- [Chain homotopy](https://tamnd.github.io/math2/node/SHRXX7EW76.md): definition SHRXX7EW76
- [Julia set](https://tamnd.github.io/math2/node/SHSXQ98PHB.md): definition SHSXQ98PHB
- [Linearity of the Laplace transform](https://tamnd.github.io/math2/node/SHX9JPKJBE.md): statement SHX9JPKJBE
- [Elliptic curve](https://tamnd.github.io/math2/node/SJ2BVG9F4W.md): definition SJ2BVG9F4W
- [Homology sphere](https://tamnd.github.io/math2/node/SJ9AMSQA7P.md): definition SJ9AMSQA7P
- [f-vector](https://tamnd.github.io/math2/node/SJDZH6V0BC.md): definition SJDZH6V0BC
- [Vlasov equation](https://tamnd.github.io/math2/node/SJH7M2VCTE.md): concept SJH7M2VCTE
- [Hamilton–Jacobi–Bellman equation](https://tamnd.github.io/math2/node/SJP9HW82VX.md): concept SJP9HW82VX
- [Filtered back-projection](https://tamnd.github.io/math2/node/SJQDDY9FMX.md): concept SJQDDY9FMX
- [d'Alembert's functional equation](https://tamnd.github.io/math2/node/SJVWPZZF54.md): definition SJVWPZZF54
- [Krull's principal ideal theorem](https://tamnd.github.io/math2/node/SJZW2HN1SC.md): statement SJZW2HN1SC
- [Weak solution](https://tamnd.github.io/math2/node/SJZX81A783.md): concept SJZX81A783
- [Homotopy type](https://tamnd.github.io/math2/node/SK0ZJGWXDP.md): concept SK0ZJGWXDP
- [Monodromy](https://tamnd.github.io/math2/node/SK224Y1HNG.md): definition SK224Y1HNG
- [Tracy–Widom distribution](https://tamnd.github.io/math2/node/SK6M9GBKN4.md): definition SK6M9GBKN4
- [Peak function](https://tamnd.github.io/math2/node/SKE8QTW3YM.md): concept SKE8QTW3YM
- [Martin boundary](https://tamnd.github.io/math2/node/SKJB3ANDF4.md): definition SKJB3ANDF4
- [Unilateral Z-transform](https://tamnd.github.io/math2/node/SKMQ02FP9J.md): definition SKMQ02FP9J
- [Euler–Gauss limit formula](https://tamnd.github.io/math2/node/SKRP9NDXR4.md): statement SKRP9NDXR4
- [Hypoelliptic operator](https://tamnd.github.io/math2/node/SKS1JPBW2N.md): definition SKS1JPBW2N
- [Lefschetz principle](https://tamnd.github.io/math2/node/SKTMN14X5P.md): statement SKTMN14X5P
- [Hellinger distance](https://tamnd.github.io/math2/node/SM0R5T87TH.md): concept SM0R5T87TH
- [Golay code](https://tamnd.github.io/math2/node/SM1TJSAR75.md): object SM1TJSAR75
- [Hom functor](https://tamnd.github.io/math2/node/SM2KFBGJTH.md): definition SM2KFBGJTH
- [Siciak extremal function](https://tamnd.github.io/math2/node/SM9BNXXQ2Y.md): definition SM9BNXXQ2Y
- [Indefinite sum](https://tamnd.github.io/math2/node/SMAQN2N739.md): concept SMAQN2N739
- [Group algebra](https://tamnd.github.io/math2/node/SMEFDN2RF5.md): definition SMEFDN2RF5
- [Null-homotopic map](https://tamnd.github.io/math2/node/SMEV5BSQ5S.md): concept SMEV5BSQ5S
- [Tensor product of modules](https://tamnd.github.io/math2/node/SMFZN9E2ZB.md): definition SMFZN9E2ZB
- [Comparability graph](https://tamnd.github.io/math2/node/SMHMFA15HG.md): definition SMHMFA15HG
- [Grothendieck universe](https://tamnd.github.io/math2/node/SMMK9E0MTM.md): concept SMMK9E0MTM
- [Tensor rank](https://tamnd.github.io/math2/node/SN8YXH9M5E.md): definition SN8YXH9M5E
- [Left-invariant vector field](https://tamnd.github.io/math2/node/SNA3DVPK0C.md): concept SNA3DVPK0C
- [Laplace's equation](https://tamnd.github.io/math2/node/SNC7NR587P.md): definition SNC7NR587P
- [Delta derivative on a time scale](https://tamnd.github.io/math2/node/SNDSAJ8G8F.md): definition SNDSAJ8G8F
- [Expectation–maximization algorithm](https://tamnd.github.io/math2/node/SNEAX8ZKV6.md): definition SNEAX8ZKV6
- [Volterra equation of convolution type](https://tamnd.github.io/math2/node/SNFYTC1TTA.md): concept SNFYTC1TTA
- [Generalized Fourier series](https://tamnd.github.io/math2/node/SNJC87TBHD.md): concept SNJC87TBHD
- [Order of a difference equation](https://tamnd.github.io/math2/node/SNRN446BHC.md): concept SNRN446BHC
- [Whitney immersion theorem](https://tamnd.github.io/math2/node/SNZTKKN3GP.md): statement SNZTKKN3GP
- [Wang's finiteness theorem](https://tamnd.github.io/math2/node/SP421Y498C.md): statement SP421Y498C
- [Delay differential equation](https://tamnd.github.io/math2/node/SP56AYYR3T.md): definition SP56AYYR3T
- [Hasse principle](https://tamnd.github.io/math2/node/SP8H3S2XWX.md): definition SP8H3S2XWX
- [Non-Euclidean geometry](https://tamnd.github.io/math2/node/SPCYMFN5V2.md): concept SPCYMFN5V2
- [Segre's theorem](https://tamnd.github.io/math2/node/SPFP3GHVYJ.md): statement SPFP3GHVYJ
- [Carleman's singular integral equation](https://tamnd.github.io/math2/node/SPV5SRAS9H.md): example SPV5SRAS9H
- [Topological group structure of a vector space](https://tamnd.github.io/math2/node/SQ6APWZBTR.md): concept SQ6APWZBTR
- [Spherical geometry](https://tamnd.github.io/math2/node/SQ9RQVANMG.md): concept SQ9RQVANMG
- [Auslander–Buchsbaum formula](https://tamnd.github.io/math2/node/SQB74J5E8Q.md): statement SQB74J5E8Q
- [Remez algorithm](https://tamnd.github.io/math2/node/SQH9X0X8N1.md): definition SQH9X0X8N1
- [Carleson measure](https://tamnd.github.io/math2/node/SQHGJ9C0QE.md): concept SQHGJ9C0QE
- [Hadamard product](https://tamnd.github.io/math2/node/SQT6QCEXZ8.md): concept SQT6QCEXZ8
- [Gaussian curvature](https://tamnd.github.io/math2/node/SQVQ5DQT6W.md): concept SQVQ5DQT6W
- [Abelian extension](https://tamnd.github.io/math2/node/SR51EVK9H0.md): definition SR51EVK9H0
- [Ultrafilter](https://tamnd.github.io/math2/node/SR56KCFYYS.md): definition SR56KCFYYS
- [Hotelling's T-squared distribution](https://tamnd.github.io/math2/node/SR8DQDKHQN.md): definition SR8DQDKHQN
- [Functional equation](https://tamnd.github.io/math2/node/SRAN4EKWS0.md): concept SRAN4EKWS0
- [Weak homotopy equivalence](https://tamnd.github.io/math2/node/SREW4HZRM4.md): definition SREW4HZRM4
- [Heaviside cover-up method](https://tamnd.github.io/math2/node/SRQ5P3077W.md): concept SRQ5P3077W
- [String diagram](https://tamnd.github.io/math2/node/SRRMJFCKVT.md): definition SRRMJFCKVT
- [Braid group](https://tamnd.github.io/math2/node/SRS8WT4H8Y.md): definition SRS8WT4H8Y
- [Equioscillation theorem](https://tamnd.github.io/math2/node/SRW9PD2RSA.md): statement SRW9PD2RSA
- [Convergence of Fourier series](https://tamnd.github.io/math2/node/SRY5M3HCKH.md): definition SRY5M3HCKH
- [Polynomial interpolation](https://tamnd.github.io/math2/node/SS1YMS2NPC.md): concept SS1YMS2NPC
- [Bäcklund transform](https://tamnd.github.io/math2/node/SS62J715AZ.md): concept SS62J715AZ
- [Beth number](https://tamnd.github.io/math2/node/SS6WMEN6KX.md): concept SS6WMEN6KX
- [Complete set of orthogonal idempotents](https://tamnd.github.io/math2/node/SS8T32TMEX.md): concept SS8T32TMEX
- [Seeley–DeWitt coefficients](https://tamnd.github.io/math2/node/SSCJWAQ7AX.md): definition SSCJWAQ7AX
- [Pitman–Koopman–Darmois theorem](https://tamnd.github.io/math2/node/SSFSDQX8RR.md): statement SSFSDQX8RR
- [Inner regular measure](https://tamnd.github.io/math2/node/SSFYQK86PK.md): definition SSFYQK86PK
- [Hausdorff dimension](https://tamnd.github.io/math2/node/SSNFTHZ6YJ.md): concept SSNFTHZ6YJ
- [Monomorphism](https://tamnd.github.io/math2/node/SSP0CC967X.md): definition SSP0CC967X
- [Whitney's theorem on folds and cusps](https://tamnd.github.io/math2/node/SSWFTA4GP8.md): statement SSWFTA4GP8
- [Initial value problem for a recurrence](https://tamnd.github.io/math2/node/SSYN7X11MB.md): concept SSYN7X11MB
- [Elliptic boundary value problem](https://tamnd.github.io/math2/node/SSYPTQC4C1.md): definition SSYPTQC4C1
- [Betti number](https://tamnd.github.io/math2/node/ST71ZV3FES.md): concept ST71ZV3FES
- [Tight frame](https://tamnd.github.io/math2/node/ST8B3EV6YA.md): definition ST8B3EV6YA
- [Poisson regression](https://tamnd.github.io/math2/node/STHTW16717.md): concept STHTW16717
- [Cauchy's power equation](https://tamnd.github.io/math2/node/STJAXZG1K1.md): definition STJAXZG1K1
- [Wiener's criterion](https://tamnd.github.io/math2/node/STMT1K1865.md): statement STMT1K1865
- [Boolean prime ideal theorem](https://tamnd.github.io/math2/node/STPBXBKSZP.md): statement STPBXBKSZP
- [Krull dimension of a poset](https://tamnd.github.io/math2/node/STSV9XY3R5.md): definition STSV9XY3R5
- [Inductive dimension](https://tamnd.github.io/math2/node/STW9S1Q66W.md): definition STW9S1Q66W
- [Calabi–Yau manifold](https://tamnd.github.io/math2/node/STYAB4PFDW.md): concept STYAB4PFDW
- [Moise's theorem](https://tamnd.github.io/math2/node/SV0816DJP9.md): statement SV0816DJP9
- [Surface of general type](https://tamnd.github.io/math2/node/SV4CK2XP8M.md): concept SV4CK2XP8M
- [Cusp form](https://tamnd.github.io/math2/node/SV5EEBF861.md): definition SV5EEBF861
- [Lelong class](https://tamnd.github.io/math2/node/SV6A17MEYZ.md): definition SV6A17MEYZ
- [Non-atomic measure](https://tamnd.github.io/math2/node/SV96KN4WR9.md): concept SV96KN4WR9
- [Trigonometric interpolation](https://tamnd.github.io/math2/node/SVABG134WE.md): concept SVABG134WE
- [Isomonodromic deformation](https://tamnd.github.io/math2/node/SVDH9C5MCK.md): definition SVDH9C5MCK
- [Homogeneous and inhomogeneous linear recurrences](https://tamnd.github.io/math2/node/SVGN0QKE2C.md): definition SVGN0QKE2C
- [Kummer–Artin–Schreier–Witt theory](https://tamnd.github.io/math2/node/SVHTAFD07A.md): definition SVHTAFD07A
- [Hadwiger covering conjecture](https://tamnd.github.io/math2/node/SVP75YY386.md): conjecture SVP75YY386
- [Pairwise summation](https://tamnd.github.io/math2/node/SVSVMWA9NQ.md): statement SVSVMWA9NQ
- [Large sieve](https://tamnd.github.io/math2/node/SVTT5MWB2X.md): definition SVTT5MWB2X
- [Costate equation](https://tamnd.github.io/math2/node/SVXBKKB3T4.md): definition SVXBKKB3T4
- [Valuation](https://tamnd.github.io/math2/node/SVY882HKSJ.md): definition SVY882HKSJ
- [Lindelöf space](https://tamnd.github.io/math2/node/SW0SN1NEJE.md): concept SW0SN1NEJE
- [Strict partial order](https://tamnd.github.io/math2/node/SWB1ZXFVKY.md): definition SWB1ZXFVKY
- [Linearly disjoint extensions](https://tamnd.github.io/math2/node/SWDN96J72T.md): definition SWDN96J72T
- [Deformation theory](https://tamnd.github.io/math2/node/SWFR1GD6QT.md): concept SWFR1GD6QT
- [Burkholder–Davis–Gundy inequality](https://tamnd.github.io/math2/node/SWHVQD9YEB.md): statement SWHVQD9YEB
- [Rational point](https://tamnd.github.io/math2/node/SWJYG87QQH.md): concept SWJYG87QQH
- [Crossing number](https://tamnd.github.io/math2/node/SWJZ4QRF9W.md): concept SWJZ4QRF9W
- [Universal covering group](https://tamnd.github.io/math2/node/SWW2993CVJ.md): concept SWW2993CVJ
- [Functor of points](https://tamnd.github.io/math2/node/SWWW9X4A3N.md): concept SWWW9X4A3N
- [Tjurina number](https://tamnd.github.io/math2/node/SWX539TCG0.md): concept SWX539TCG0
- [Alexander–Spanier cohomology](https://tamnd.github.io/math2/node/SX5RBFJVYC.md): definition SX5RBFJVYC
- [Doob–Meyer decomposition theorem](https://tamnd.github.io/math2/node/SX7ZPP72EM.md): statement SX7ZPP72EM
- [Cartan–Eilenberg resolution](https://tamnd.github.io/math2/node/SX8V8GR5ZG.md): concept SX8V8GR5ZG
- [Determinantal ideal](https://tamnd.github.io/math2/node/SXD7X0WQH1.md): concept SXD7X0WQH1
- [Secretary problem](https://tamnd.github.io/math2/node/SXE9KQ02YE.md): problem SXE9KQ02YE
- [Holonomy groupoid of a foliation](https://tamnd.github.io/math2/node/SXMPSYESTP.md): definition SXMPSYESTP
- [Resolvent as Laplace transform of a semigroup](https://tamnd.github.io/math2/node/SXNWHN2C01.md): statement SXNWHN2C01
- [Spherical trigonometry](https://tamnd.github.io/math2/node/SXQGHWHFRF.md): concept SXQGHWHFRF
- [Sequential probability ratio test](https://tamnd.github.io/math2/node/SXR93CAP1R.md): concept SXR93CAP1R
- [Newton polytope](https://tamnd.github.io/math2/node/SXS38FWHEN.md): definition SXS38FWHEN
- [Minkowski addition](https://tamnd.github.io/math2/node/SXS6D87PHE.md): concept SXS6D87PHE
- [Commutativity of convolution](https://tamnd.github.io/math2/node/SY0TJ4NCES.md): statement SY0TJ4NCES
- [Universal enveloping algebra](https://tamnd.github.io/math2/node/SY22C7XVCW.md): definition SY22C7XVCW
- [Irreducible component](https://tamnd.github.io/math2/node/SY2MWC434Z.md): concept SY2MWC434Z
- [Quillen adjunction](https://tamnd.github.io/math2/node/SY38F954FE.md): concept SY38F954FE
- [Banach limit](https://tamnd.github.io/math2/node/SYF2ND6J97.md): concept SYF2ND6J97
- [Carathéodory metric](https://tamnd.github.io/math2/node/SYNST4W3FR.md): concept SYNST4W3FR
- [Decay of the Fourier transform of surface measure](https://tamnd.github.io/math2/node/SYP1ATSTTH.md): statement SYP1ATSTTH
- [Euler characteristic](https://tamnd.github.io/math2/node/SYP3FVM71M.md): definition SYP3FVM71M
- [Matroid rank](https://tamnd.github.io/math2/node/SYRCTH853H.md): concept SYRCTH853H
- [Bogomolov–Miyaoka–Yau inequality](https://tamnd.github.io/math2/node/SYSVE5P1YN.md): statement SYSVE5P1YN
- [Dual matroid](https://tamnd.github.io/math2/node/SYVAV7V0B3.md): definition SYVAV7V0B3
- [Lusternik–Schnirelmann theorem](https://tamnd.github.io/math2/node/SYVXQQK57V.md): statement SYVXQQK57V
- [Lie group method for PDEs](https://tamnd.github.io/math2/node/SYXNSN5JX0.md): definition SYXNSN5JX0
- [Kantorovich–Rubinstein duality](https://tamnd.github.io/math2/node/SYXSZD8EFK.md): statement SYXSZD8EFK
- [Manifold with corners](https://tamnd.github.io/math2/node/SZ1R8N1NR7.md): concept SZ1R8N1NR7
- [Caloric function](https://tamnd.github.io/math2/node/SZ2C28JG5K.md): definition SZ2C28JG5K
- [Hadamard's example of ill-posedness](https://tamnd.github.io/math2/node/SZ6MEVM89M.md): counterexample SZ6MEVM89M
- [Gaussian quadrature](https://tamnd.github.io/math2/node/SZ6XCSQA5H.md): definition SZ6XCSQA5H
- [Cross-validation](https://tamnd.github.io/math2/node/SZ7QZBDPYJ.md): definition SZ7QZBDPYJ
- [Szegő's theorem](https://tamnd.github.io/math2/node/SZ8ARMDEMY.md): statement SZ8ARMDEMY
- [Semiorder](https://tamnd.github.io/math2/node/SZ8NCJ4TC5.md): definition SZ8NCJ4TC5
- [Bounded poset](https://tamnd.github.io/math2/node/SZ9X10ATEK.md): definition SZ9X10ATEK
- [Fundamental domain of the modular group](https://tamnd.github.io/math2/node/SZEYSY99AM.md): example SZEYSY99AM
- [Reduced ring](https://tamnd.github.io/math2/node/SZHY975FFX.md): concept SZHY975FFX
- [Admissible representation](https://tamnd.github.io/math2/node/SZV8Y806Y7.md): concept SZV8Y806Y7
- [Ordinary point and singular point](https://tamnd.github.io/math2/node/T01779NV3Z.md): definition T01779NV3Z
- [Fourier–Legendre series](https://tamnd.github.io/math2/node/T046A1477B.md): definition T046A1477B
- [Simpson's paradox](https://tamnd.github.io/math2/node/T060JHSH05.md): counterexample T060JHSH05
- [Permutohedron](https://tamnd.github.io/math2/node/T06PG3RWC0.md): example T06PG3RWC0
- [Complemented lattice](https://tamnd.github.io/math2/node/T08FF77H16.md): definition T08FF77H16
- [Morphism of finite type](https://tamnd.github.io/math2/node/T09DXZAT6R.md): definition T09DXZAT6R
- [Frölicher space](https://tamnd.github.io/math2/node/T0AEF3Z8C9.md): concept T0AEF3Z8C9
- [Equivalent definition with relaxed axioms](https://tamnd.github.io/math2/node/T0BWYYR508.md): statement T0BWYYR508
- [Moving least squares](https://tamnd.github.io/math2/node/T0JZN6WP5X.md): concept T0JZN6WP5X
- [Subalgebra lattice](https://tamnd.github.io/math2/node/T0TNG7AGXT.md): concept T0TNG7AGXT
- [Minakshisundaram–Pleijel zeta function](https://tamnd.github.io/math2/node/T1039Q9FY0.md): concept T1039Q9FY0
- [Liouville's theorem for planar harmonic functions](https://tamnd.github.io/math2/node/T12FSF2J2V.md): statement T12FSF2J2V
- [Band matrix](https://tamnd.github.io/math2/node/T12HYJTXJF.md): definition T12HYJTXJF
- [Positive current](https://tamnd.github.io/math2/node/T19H7BJ9Y3.md): definition T19H7BJ9Y3
- [Holomorphic map between Riemann surfaces](https://tamnd.github.io/math2/node/T19P7J53A1.md): concept T19P7J53A1
- [Mathieu groups](https://tamnd.github.io/math2/node/T1KR4KPX8P.md): object T1KR4KPX8P
- [Borel–Moore homology](https://tamnd.github.io/math2/node/T1N6E5Q8Z2.md): concept T1N6E5Q8Z2
- [Compositum of fields](https://tamnd.github.io/math2/node/T1S29NN8PE.md): concept T1S29NN8PE
- [Partial autocorrelation function](https://tamnd.github.io/math2/node/T1VJ281P1V.md): definition T1VJ281P1V
- [Spectral element method](https://tamnd.github.io/math2/node/T1Y1CJR1JB.md): concept T1Y1CJR1JB
- [Local truncation error](https://tamnd.github.io/math2/node/T220HF0562.md): concept T220HF0562
- [Associative quotient of a magma](https://tamnd.github.io/math2/node/T22ZTQRXGB.md): concept T22ZTQRXGB
- [Resolvent equation for kernels](https://tamnd.github.io/math2/node/T26WGV2PQW.md): statement T26WGV2PQW
- [Berkovich space](https://tamnd.github.io/math2/node/T284RZ6AGE.md): concept T284RZ6AGE
- [Gauss's law for potentials](https://tamnd.github.io/math2/node/T2CYAYF0MT.md): statement T2CYAYF0MT
- [Maximal compact subgroup](https://tamnd.github.io/math2/node/T2EB8R9HEV.md): concept T2EB8R9HEV
- [Mean curvature](https://tamnd.github.io/math2/node/T2HPG1780R.md): definition T2HPG1780R
- [Fatou–Bieberbach domain](https://tamnd.github.io/math2/node/T2JZ0CKFKW.md): example T2JZ0CKFKW
- [Triangulated category](https://tamnd.github.io/math2/node/T2V7GJJ2EN.md): concept T2V7GJJ2EN
- [Topological vector space](https://tamnd.github.io/math2/node/T32W553W95.md): concept T32W553W95
- [Rate of convergence](https://tamnd.github.io/math2/node/T34R8Q0FJD.md): definition T34R8Q0FJD
- [Dimension of a polynomial ring](https://tamnd.github.io/math2/node/T35XFF5AY3.md): statement T35XFF5AY3
- [Pseudo-spectral method](https://tamnd.github.io/math2/node/T37Q055GDA.md): definition T37Q055GDA
- [Isomorphism of topological groups](https://tamnd.github.io/math2/node/T3A3JTW6Q4.md): concept T3A3JTW6Q4
- [Sylvester–Gallai theorem](https://tamnd.github.io/math2/node/T3GGJN70P2.md): statement T3GGJN70P2
- [Reidemeister's theorem](https://tamnd.github.io/math2/node/T3HAGWYKK7.md): statement T3HAGWYKK7
- [Optimal control](https://tamnd.github.io/math2/node/T3V6ZNAMDN.md): concept T3V6ZNAMDN
- [Alexander's theorem on proper self-maps of the ball](https://tamnd.github.io/math2/node/T3VN45KV0D.md): statement T3VN45KV0D
- [Differential field](https://tamnd.github.io/math2/node/T43TQ817XC.md): definition T43TQ817XC
- [Substitution tiling](https://tamnd.github.io/math2/node/T4403SAMB2.md): definition T4403SAMB2
- [Primal algebra](https://tamnd.github.io/math2/node/T440VB9JJM.md): definition T440VB9JJM
- [Nijenhuis tensor](https://tamnd.github.io/math2/node/T46ZBWDPHQ.md): concept T46ZBWDPHQ
- [Omega-complete partial order](https://tamnd.github.io/math2/node/T4BCQHR68Y.md): concept T4BCQHR68Y
- [Topological data analysis](https://tamnd.github.io/math2/node/T4EPFGAQ8C.md): definition T4EPFGAQ8C
- [4-manifold](https://tamnd.github.io/math2/node/T4ESER8G7G.md): definition T4ESER8G7G
- [Fourier series on the torus](https://tamnd.github.io/math2/node/T4JEMK1549.md): definition T4JEMK1549
- [Nerve theorem](https://tamnd.github.io/math2/node/T4MTXNJJ3Q.md): statement T4MTXNJJ3Q
- [The alternating group A3 is normal in S3](https://tamnd.github.io/math2/node/T4PCQ68C5K.md): example T4PCQ68C5K
- [Sphere packing](https://tamnd.github.io/math2/node/T4PGC6NH56.md): definition T4PGC6NH56
- [Gelfand's spectral radius formula](https://tamnd.github.io/math2/node/T4RFYT8HDN.md): statement T4RFYT8HDN
- [Recurrence relations for Bessel functions](https://tamnd.github.io/math2/node/T4VP5QTQK6.md): statement T4VP5QTQK6
- [Fourier inversion theorem](https://tamnd.github.io/math2/node/T4Z8YQZKZV.md): statement T4Z8YQZKZV
- [Monotone cubic interpolation](https://tamnd.github.io/math2/node/T53N88Y21J.md): statement T53N88Y21J
- [Doléans-Dade exponential](https://tamnd.github.io/math2/node/T567PNQ2NN.md): definition T567PNQ2NN
- [Direct collocation](https://tamnd.github.io/math2/node/T58W34D60G.md): concept T58W34D60G
- [Verification theorem](https://tamnd.github.io/math2/node/T5BEKMG5W8.md): statement T5BEKMG5W8
- [Barbier's theorem](https://tamnd.github.io/math2/node/T5D9RSE0X2.md): statement T5D9RSE0X2
- [Free distributive lattice](https://tamnd.github.io/math2/node/T5GK6DH8BR.md): concept T5GK6DH8BR
- [Zimmer program](https://tamnd.github.io/math2/node/T5KZR54SG8.md): definition T5KZR54SG8
- [Gårding's inequality](https://tamnd.github.io/math2/node/T5MDFAZ0H6.md): statement T5MDFAZ0H6
- [Green's function of the ball](https://tamnd.github.io/math2/node/T5RW1VN8AN.md): example T5RW1VN8AN
- [Baum–Welch algorithm](https://tamnd.github.io/math2/node/T5TJRYZPFX.md): definition T5TJRYZPFX
- [Baouendi–Trèves approximation theorem](https://tamnd.github.io/math2/node/T5VW3PE7TZ.md): statement T5VW3PE7TZ
- [Modular curve](https://tamnd.github.io/math2/node/T5W0CMS0DM.md): concept T5W0CMS0DM
- [Ambrose–Singer theorem](https://tamnd.github.io/math2/node/T61AW674XD.md): statement T61AW674XD
- [n-sphere](https://tamnd.github.io/math2/node/T62369RFH6.md): object T62369RFH6
- [Maximum principle for harmonic functions](https://tamnd.github.io/math2/node/T62HHSRHP5.md): statement T62HHSRHP5
- [Right adjoints preserve limits](https://tamnd.github.io/math2/node/T6A7JK3J98.md): statement T6A7JK3J98
- [Mather's stability theorem](https://tamnd.github.io/math2/node/T6BK8Q8PK8.md): statement T6BK8Q8PK8
- [Discrete Poisson equation](https://tamnd.github.io/math2/node/T6DDZTB360.md): definition T6DDZTB360
- [Descartes' rule of signs](https://tamnd.github.io/math2/node/T6DQQ36JZ5.md): statement T6DQQ36JZ5
- [Gorenstein ring](https://tamnd.github.io/math2/node/T6HCX959PJ.md): concept T6HCX959PJ
- [Hodges–Lehmann estimator](https://tamnd.github.io/math2/node/T6KRQAQQ15.md): concept T6KRQAQQ15
- [Topological modular forms](https://tamnd.github.io/math2/node/T6Q7NST7JA.md): definition T6Q7NST7JA
- [Turing reduction](https://tamnd.github.io/math2/node/T6V685TY1Y.md): definition T6V685TY1Y
- [Steklov eigenvalue problem](https://tamnd.github.io/math2/node/T6WEFSKYHZ.md): problem T6WEFSKYHZ
- [Fractional calculus](https://tamnd.github.io/math2/node/T73XK15SA2.md): concept T73XK15SA2
- [Periodic point](https://tamnd.github.io/math2/node/T75CZ5H47N.md): concept T75CZ5H47N
- [Trigonometric integrals](https://tamnd.github.io/math2/node/T761KPKPV6.md): definition T761KPKPV6
- [Lattice congruence](https://tamnd.github.io/math2/node/T782PJYVEZ.md): concept T782PJYVEZ
- [E-infinity operad](https://tamnd.github.io/math2/node/T78YG90VVT.md): concept T78YG90VVT
- [Projective transformation](https://tamnd.github.io/math2/node/T7B1WJVT9Z.md): definition T7B1WJVT9Z
- [Hammerstein integral equation](https://tamnd.github.io/math2/node/T7E2EW2T0G.md): concept T7E2EW2T0G
- [Ordered geometry](https://tamnd.github.io/math2/node/T7G8X5AD5G.md): concept T7G8X5AD5G
- [Generator of a semigroup](https://tamnd.github.io/math2/node/T7JBM8472G.md): definition T7JBM8472G
- [Stable theory](https://tamnd.github.io/math2/node/T7JW5QY6KR.md): concept T7JW5QY6KR
- [Electrical network](https://tamnd.github.io/math2/node/T7KRE7J3B4.md): concept T7KRE7J3B4
- [Fenchel conjugate](https://tamnd.github.io/math2/node/T7V2VPY61S.md): concept T7V2VPY61S
- [Inner function](https://tamnd.github.io/math2/node/T8032P33B7.md): definition T8032P33B7
- [Equivariant index theorem](https://tamnd.github.io/math2/node/T866256Y1W.md): statement T866256Y1W
- [Muckenhoupt weights](https://tamnd.github.io/math2/node/T86RSJ8M86.md): concept T86RSJ8M86
- [Cobordism](https://tamnd.github.io/math2/node/T8BVGPPSZP.md): concept T8BVGPPSZP
- [Multivariate division algorithm](https://tamnd.github.io/math2/node/T8EN3CAEQF.md): concept T8EN3CAEQF
- [Residuated mapping](https://tamnd.github.io/math2/node/T8MYNX54PS.md): definition T8MYNX54PS
- [Neville theta functions](https://tamnd.github.io/math2/node/T8NQJ0CBJM.md): definition T8NQJ0CBJM
- [q-exponential](https://tamnd.github.io/math2/node/T8QYJCS3JA.md): concept T8QYJCS3JA
- [Monodromy of the hypergeometric equation](https://tamnd.github.io/math2/node/T8SBE28DM1.md): concept T8SBE28DM1
- [Rational difference equation](https://tamnd.github.io/math2/node/T8YDEASD1H.md): concept T8YDEASD1H
- [Bruhat decomposition](https://tamnd.github.io/math2/node/T91E86D0RH.md): statement T91E86D0RH
- [Divided differences](https://tamnd.github.io/math2/node/T942SHWSSF.md): definition T942SHWSSF
- [Lebesgue's universal covering problem](https://tamnd.github.io/math2/node/T94V1HDVNC.md): problem T94V1HDVNC
- [Wald test](https://tamnd.github.io/math2/node/T9640YF4TX.md): definition T9640YF4TX
- [Outer function](https://tamnd.github.io/math2/node/T97B2GX3JJ.md): definition T97B2GX3JJ
- [Pontryagin–Thom construction](https://tamnd.github.io/math2/node/T9A1DZ4AQK.md): definition T9A1DZ4AQK
- [Szpilrajn extension theorem](https://tamnd.github.io/math2/node/T9CJHV6JW3.md): statement T9CJHV6JW3
- [Pointed space](https://tamnd.github.io/math2/node/T9DG8DDE75.md): concept T9DG8DDE75
- [Morse theory on Hilbert manifolds](https://tamnd.github.io/math2/node/T9H6PT9FHE.md): concept T9H6PT9FHE
- [Bilateral Z-transform](https://tamnd.github.io/math2/node/T9PRG9QP8Y.md): definition T9PRG9QP8Y
- [Degree of a projective variety](https://tamnd.github.io/math2/node/TA0NREQMPX.md): definition TA0NREQMPX
- [Factorization system](https://tamnd.github.io/math2/node/TA39C6P91W.md): definition TA39C6P91W
- [Integral quadratic form](https://tamnd.github.io/math2/node/TA5025GWDQ.md): definition TA5025GWDQ
- [Hölder summation](https://tamnd.github.io/math2/node/TA8NQQCQCF.md): definition TA8NQQCQCF
- [Reaction–diffusion system](https://tamnd.github.io/math2/node/TAAHWE917M.md): concept TAAHWE917M
- [HNN extension](https://tamnd.github.io/math2/node/TACX13XZST.md): definition TACX13XZST
- [Error-correcting code](https://tamnd.github.io/math2/node/TAE6K8NQ46.md): concept TAE6K8NQ46
- [Expansive homeomorphism](https://tamnd.github.io/math2/node/TAFJV2YWKR.md): concept TAFJV2YWKR
- [Abian–Brown theorem](https://tamnd.github.io/math2/node/TAHM74ZNWB.md): statement TAHM74ZNWB
- [Bicategory](https://tamnd.github.io/math2/node/TAHV24ED74.md): concept TAHV24ED74
- [Fractional integration](https://tamnd.github.io/math2/node/TAJTC44XC9.md): definition TAJTC44XC9
- [Tautological one-form](https://tamnd.github.io/math2/node/TAKA1DYBNE.md): definition TAKA1DYBNE
- [Kawamata–Viehweg vanishing theorem](https://tamnd.github.io/math2/node/TAMCTW7SA0.md): statement TAMCTW7SA0
- [Integral geometry](https://tamnd.github.io/math2/node/TAN5G5EQ96.md): concept TAN5G5EQ96
- [Ring of differential operators](https://tamnd.github.io/math2/node/TAPCWE72R0.md): definition TAPCWE72R0
- [Langevin equation](https://tamnd.github.io/math2/node/TAQWRWTHX8.md): definition TAQWRWTHX8
- [Lévy–Prokhorov metric](https://tamnd.github.io/math2/node/TASMEPWNA6.md): definition TASMEPWNA6
- [Direct limit](https://tamnd.github.io/math2/node/TAVG968272.md): definition TAVG968272
- [Algebraic statistics](https://tamnd.github.io/math2/node/TAYDF13B23.md): concept TAYDF13B23
- [Möbius–Kantor configuration](https://tamnd.github.io/math2/node/TB0ZRTX7DT.md): example TB0ZRTX7DT
- [Square wave](https://tamnd.github.io/math2/node/TB2TR373RC.md): example TB2TR373RC
- [Gerstenhaber algebra](https://tamnd.github.io/math2/node/TB3Q55Q0G4.md): definition TB3Q55Q0G4
- [Faithful module](https://tamnd.github.io/math2/node/TB40Y3SNSX.md): concept TB40Y3SNSX
- [Picard–Vessiot extension](https://tamnd.github.io/math2/node/TB53Y5NMJD.md): concept TB53Y5NMJD
- [Sheaf](https://tamnd.github.io/math2/node/TB6TM6R4MD.md): concept TB6TM6R4MD
- [Saddle-node bifurcation](https://tamnd.github.io/math2/node/TB7NBCCXJ1.md): definition TB7NBCCXJ1
- [Space form](https://tamnd.github.io/math2/node/TBBDP94GDN.md): definition TBBDP94GDN
- [Discriminant of a field extension](https://tamnd.github.io/math2/node/TBMEJXAMZ0.md): definition TBMEJXAMZ0
- [Hamburger moment problem](https://tamnd.github.io/math2/node/TBNCFXTMMW.md): problem TBNCFXTMMW
- [Triebel–Lizorkin space](https://tamnd.github.io/math2/node/TBNP6KYW4Z.md): concept TBNP6KYW4Z
- [Krull's height theorem](https://tamnd.github.io/math2/node/TBRTZMHFTV.md): statement TBRTZMHFTV
- [Levi-flat hypersurface](https://tamnd.github.io/math2/node/TBRWZE42A8.md): definition TBRWZE42A8
- [Crown](https://tamnd.github.io/math2/node/TBSBNJFD2P.md): example TBSBNJFD2P
- [Cauchy's logarithmic equation](https://tamnd.github.io/math2/node/TC5BKWCPS6.md): definition TC5BKWCPS6
- [Universal Turing machine](https://tamnd.github.io/math2/node/TC69PEHAX7.md): object TC69PEHAX7
- [Discrete Laplace operator](https://tamnd.github.io/math2/node/TC6Y42YGQF.md): definition TC6Y42YGQF
- [Foster's theorem on primal algebras](https://tamnd.github.io/math2/node/TCD8SZW95Z.md): statement TCD8SZW95Z
- [Upper nilradical](https://tamnd.github.io/math2/node/TCF1PXZ5YN.md): definition TCF1PXZ5YN
- [Systolic geometry](https://tamnd.github.io/math2/node/TCNY8VKQ61.md): definition TCNY8VKQ61
- [Cotangent space of a local ring](https://tamnd.github.io/math2/node/TCRZV021XR.md): definition TCRZV021XR
- [F-coalgebra](https://tamnd.github.io/math2/node/TCWV185JKY.md): definition TCWV185JKY
- [Poisson kernel](https://tamnd.github.io/math2/node/TCYSWCTYYM.md): concept TCYSWCTYYM
- [Projection method for integral equations](https://tamnd.github.io/math2/node/TD0N49V1SD.md): definition TD0N49V1SD
- [Mikusiński operational calculus](https://tamnd.github.io/math2/node/TD5XK5AHNV.md): concept TD5XK5AHNV
- [Ehresmann connection](https://tamnd.github.io/math2/node/TD6VEP3SCE.md): concept TD6VEP3SCE
- [Coupling](https://tamnd.github.io/math2/node/TD7S2HF19H.md): definition TD7S2HF19H
- [Magnus expansion](https://tamnd.github.io/math2/node/TD9T02HKPD.md): definition TD9T02HKPD
- [Backward differentiation formula](https://tamnd.github.io/math2/node/TDA94C2JYT.md): concept TDA94C2JYT
- [Multiple integral](https://tamnd.github.io/math2/node/TDBBJNW91Z.md): definition TDBBJNW91Z
- [Tietze transformations](https://tamnd.github.io/math2/node/TDCGM4WR88.md): definition TDCGM4WR88
- [Peixoto's theorem](https://tamnd.github.io/math2/node/TDCY8TASCS.md): statement TDCY8TASCS
- [Direct and inverse theorems of approximation](https://tamnd.github.io/math2/node/TDKKST5AWA.md): concept TDKKST5AWA
- [Abel inversion formula](https://tamnd.github.io/math2/node/TDMXFY06A2.md): statement TDMXFY06A2
- [Extremal length](https://tamnd.github.io/math2/node/TDQ4YBMW9C.md): definition TDQ4YBMW9C
- [Kaplansky density theorem](https://tamnd.github.io/math2/node/TDTMMY7RK1.md): statement TDTMMY7RK1
- [Invertible ideal](https://tamnd.github.io/math2/node/TDW023HZ1T.md): definition TDW023HZ1T
- [Kodaira–Spencer map](https://tamnd.github.io/math2/node/TDY6EY60DV.md): definition TDY6EY60DV
- [Bessel function of the first kind](https://tamnd.github.io/math2/node/TDZ8766BAR.md): definition TDZ8766BAR
- [Functional completeness](https://tamnd.github.io/math2/node/TDZVCWYJ9F.md): concept TDZVCWYJ9F
- [Tarski monster group](https://tamnd.github.io/math2/node/TE19BGFFF1.md): object TE19BGFFF1
- [Direct method in the calculus of variations](https://tamnd.github.io/math2/node/TE379HBTY0.md): concept TE379HBTY0
- [Periodic point](https://tamnd.github.io/math2/node/TE3STTNSCK.md): definition TE3STTNSCK
- [Hirzebruch signature theorem](https://tamnd.github.io/math2/node/TEG1ZX1AGJ.md): statement TEG1ZX1AGJ
- [Hopf invariant](https://tamnd.github.io/math2/node/TEGTSY7XMX.md): concept TEGTSY7XMX
- [Lagrange polynomial](https://tamnd.github.io/math2/node/TEQNAME853.md): definition TEQNAME853
- [Resolvent](https://tamnd.github.io/math2/node/TEREJ8DA37.md): definition TEREJ8DA37
- [Finite ring](https://tamnd.github.io/math2/node/TEV7XV14F5.md): definition TEV7XV14F5
- [Aleph number](https://tamnd.github.io/math2/node/TEWH75TX8Y.md): definition TEWH75TX8Y
- [Descriptive set theory](https://tamnd.github.io/math2/node/TF8HMVXKYC.md): concept TF8HMVXKYC
- [Exact category](https://tamnd.github.io/math2/node/TFADSC9YPZ.md): concept TFADSC9YPZ
- [Uniform approximation](https://tamnd.github.io/math2/node/TFB4BYRZ1W.md): definition TFB4BYRZ1W
- [Pivotal quantity](https://tamnd.github.io/math2/node/TFCWC3N41W.md): definition TFCWC3N41W
- [Lie sphere geometry](https://tamnd.github.io/math2/node/TFQSF9RA1G.md): concept TFQSF9RA1G
- [Capacity of a set](https://tamnd.github.io/math2/node/TFY9CD3019.md): definition TFY9CD3019
- [Banach algebra](https://tamnd.github.io/math2/node/TFYXH8QNR2.md): definition TFYXH8QNR2
- [Geometric realization](https://tamnd.github.io/math2/node/TG6WHHR7CF.md): concept TG6WHHR7CF
- [ℓ-adic cohomology](https://tamnd.github.io/math2/node/TGKPC4GV7R.md): definition TGKPC4GV7R
- [Chevalley's theorem on constructible sets](https://tamnd.github.io/math2/node/TGN01264C4.md): statement TGN01264C4
- [Join-prime element](https://tamnd.github.io/math2/node/TGNMG87Z6T.md): definition TGNMG87Z6T
- [Down-closure](https://tamnd.github.io/math2/node/TGW5NTV3AA.md): definition TGW5NTV3AA
- [Newton's minimal resistance problem](https://tamnd.github.io/math2/node/TGWM8G5R0C.md): problem TGWM8G5R0C
- [Block design](https://tamnd.github.io/math2/node/TGZATGBHTH.md): definition TGZATGBHTH
- [Residually small variety](https://tamnd.github.io/math2/node/TH2XQGGFWH.md): concept TH2XQGGFWH
- [Integro-partial differential equation](https://tamnd.github.io/math2/node/TH4NGC4016.md): concept TH4NGC4016
- [Levenberg–Marquardt algorithm](https://tamnd.github.io/math2/node/TH54SPVN1N.md): statement TH54SPVN1N
- [Proof theory](https://tamnd.github.io/math2/node/TH8EGSCFYY.md): definition TH8EGSCFYY
- [Resistance form](https://tamnd.github.io/math2/node/THCM7DWHNY.md): concept THCM7DWHNY
- [Hartogs domain](https://tamnd.github.io/math2/node/THH2YHW8G5.md): concept THH2YHW8G5
- [Generalized linear model](https://tamnd.github.io/math2/node/THH4GJFYQN.md): definition THH4GJFYQN
- [Free category](https://tamnd.github.io/math2/node/THRHFW3SMP.md): concept THRHFW3SMP
- [Differential game](https://tamnd.github.io/math2/node/THSHGNJNPS.md): concept THSHGNJNPS
- [Siu's semicontinuity theorem](https://tamnd.github.io/math2/node/THSVFE7YER.md): statement THSVFE7YER
- [Spectral triple](https://tamnd.github.io/math2/node/THX29SKKPB.md): definition THX29SKKPB
- [Paley–Wiener theorem](https://tamnd.github.io/math2/node/THY12GPZ8C.md): statement THY12GPZ8C
- [Random graph](https://tamnd.github.io/math2/node/THYBFG527A.md): definition THYBFG527A
- [Group scheme](https://tamnd.github.io/math2/node/TJ04VQP4NB.md): concept TJ04VQP4NB
- [Normal form](https://tamnd.github.io/math2/node/TJ04Y2Z5Q3.md): concept TJ04Y2Z5Q3
- [The alternating group is the kernel of the sign](https://tamnd.github.io/math2/node/TJ6EBSS3S7.md): example TJ6EBSS3S7
- [Discrete Fourier transform](https://tamnd.github.io/math2/node/TJ9XPJ65YX.md): definition TJ9XPJ65YX
- [Finitely presented group](https://tamnd.github.io/math2/node/TJKANTKASZ.md): concept TJKANTKASZ
- [Ore condition](https://tamnd.github.io/math2/node/TJP4ARC2X7.md): concept TJP4ARC2X7
- [Catastrophic cancellation](https://tamnd.github.io/math2/node/TJPNYW7J8E.md): concept TJPNYW7J8E
- [Hahn series](https://tamnd.github.io/math2/node/TJQ5HWTXXS.md): definition TJQ5HWTXXS
- [Differential game](https://tamnd.github.io/math2/node/TJS0MZSFA8.md): definition TJS0MZSFA8
- [Lichnerowicz–Obata eigenvalue theorem](https://tamnd.github.io/math2/node/TK1PT9FDV0.md): statement TK1PT9FDV0
- [Hyperreal number](https://tamnd.github.io/math2/node/TK6R2C9KRE.md): definition TK6R2C9KRE
- [Analytic continuation in several variables](https://tamnd.github.io/math2/node/TK9WPV0EXS.md): definition TK9WPV0EXS
- [Holomorphic vector bundle](https://tamnd.github.io/math2/node/TKA958Z4KZ.md): concept TKA958Z4KZ
- [Integral transforms of tempered distributions](https://tamnd.github.io/math2/node/TKBH6F3VXT.md): definition TKBH6F3VXT
- [Filtered category](https://tamnd.github.io/math2/node/TKG3YF2G5A.md): definition TKG3YF2G5A
- [Cluster algebra](https://tamnd.github.io/math2/node/TKK0TV2MYC.md): concept TKK0TV2MYC
- [Nonharmonic Fourier series](https://tamnd.github.io/math2/node/TKNA7PR99K.md): concept TKNA7PR99K
- [Elastic net](https://tamnd.github.io/math2/node/TKQPE0Q05Y.md): concept TKQPE0Q05Y
- [Taft algebra](https://tamnd.github.io/math2/node/TKTMNJQCHM.md): example TKTMNJQCHM
- [Function of exponential order](https://tamnd.github.io/math2/node/TKZAV3KR86.md): concept TKZAV3KR86
- [Domain decomposition methods](https://tamnd.github.io/math2/node/TM0AXHX3SG.md): definition TM0AXHX3SG
- [Schur product theorem](https://tamnd.github.io/math2/node/TM1RE1G7JK.md): statement TM1RE1G7JK
- [Proper holomorphic map](https://tamnd.github.io/math2/node/TM528Q96ZD.md): concept TM528Q96ZD
- [Löwner–John theorem](https://tamnd.github.io/math2/node/TM5SYH4XHY.md): statement TM5SYH4XHY
- [Free product](https://tamnd.github.io/math2/node/TM7QS7P7X9.md): concept TM7QS7P7X9
- [Borel density theorem](https://tamnd.github.io/math2/node/TM8PYTBT6Y.md): statement TM8PYTBT6Y
- [Boundary value problem](https://tamnd.github.io/math2/node/TM8VQJWGHZ.md): concept TM8VQJWGHZ
- [Nevanlinna class](https://tamnd.github.io/math2/node/TMAHZSNTA1.md): definition TMAHZSNTA1
- [Scott continuity](https://tamnd.github.io/math2/node/TMBCW25DZ4.md): definition TMBCW25DZ4
- [Eichler order](https://tamnd.github.io/math2/node/TMCMTXNZ63.md): definition TMCMTXNZ63
- [Separation principle](https://tamnd.github.io/math2/node/TMEKPXMC7V.md): statement TMEKPXMC7V
- [Degree of a continuous map](https://tamnd.github.io/math2/node/TMFG2P912R.md): concept TMFG2P912R
- [Semilocally simply connected space](https://tamnd.github.io/math2/node/TMJ0SFW7M7.md): definition TMJ0SFW7M7
- [Semiprimitive ring](https://tamnd.github.io/math2/node/TMKXXG8XHR.md): definition TMKXXG8XHR
- [Bifurcation in a family of maps](https://tamnd.github.io/math2/node/TMM9SGJMSP.md): definition TMM9SGJMSP
- [Gegenbauer transform](https://tamnd.github.io/math2/node/TMNNRRRGAS.md): concept TMNNRRRGAS
- [Primary ideal](https://tamnd.github.io/math2/node/TMTPWVETQK.md): concept TMTPWVETQK
- [Iterated function system](https://tamnd.github.io/math2/node/TMY3M75EK8.md): definition TMY3M75EK8
- [Neighborly polytope](https://tamnd.github.io/math2/node/TN1X6FNT87.md): definition TN1X6FNT87
- [Nome](https://tamnd.github.io/math2/node/TN2R1TDE3G.md): definition TN2R1TDE3G
- [Friendship theorem](https://tamnd.github.io/math2/node/TN4GDFWGAR.md): statement TN4GDFWGAR
- [Laplacian matrix](https://tamnd.github.io/math2/node/TN6M6DVZRP.md): concept TN6M6DVZRP
- [Computably enumerable set](https://tamnd.github.io/math2/node/TN9G24VABN.md): concept TN9G24VABN
- [Formal group law in topology](https://tamnd.github.io/math2/node/TNHP7XXYZT.md): concept TNHP7XXYZT
- [Roe solver](https://tamnd.github.io/math2/node/TNM5CFSZM6.md): statement TNM5CFSZM6
- [Hadamard three-circle theorem for subharmonic functions](https://tamnd.github.io/math2/node/TNMQDK0RR0.md): statement TNMQDK0RR0
- [Baby Monster group](https://tamnd.github.io/math2/node/TNPQ9AWVBC.md): object TNPQ9AWVBC
- [Random graph](https://tamnd.github.io/math2/node/TNT3DG0HRJ.md): concept TNT3DG0HRJ
- [Fundamental theorem of space curves](https://tamnd.github.io/math2/node/TNYQ5R0NVM.md): statement TNYQ5R0NVM
- [Stable vector bundle](https://tamnd.github.io/math2/node/TP1RQPCETV.md): definition TP1RQPCETV
- [Approximately additive function](https://tamnd.github.io/math2/node/TP8XXM47CJ.md): definition TP8XXM47CJ
- [Bürmann–Lagrange series](https://tamnd.github.io/math2/node/TPB4F1VY4Z.md): statement TPB4F1VY4Z
- [Rank-one convexity](https://tamnd.github.io/math2/node/TPC99RZ316.md): definition TPC99RZ316
- [Complex powers of elliptic operators](https://tamnd.github.io/math2/node/TPG0GQBFHM.md): definition TPG0GQBFHM
- [Whitehead link](https://tamnd.github.io/math2/node/TPGBASPKE0.md): example TPGBASPKE0
- [Proper map](https://tamnd.github.io/math2/node/TPH39D9W02.md): definition TPH39D9W02
- [Fejér's theorem](https://tamnd.github.io/math2/node/TPJZCSXSVN.md): statement TPJZCSXSVN
- [Romberg's method](https://tamnd.github.io/math2/node/TPM8JXFWYB.md): definition TPM8JXFWYB
- [Barycentric interpolation formula](https://tamnd.github.io/math2/node/TPRFJNZ7S2.md): statement TPRFJNZ7S2
- [Hörmander's L2 estimates](https://tamnd.github.io/math2/node/TPSH4ETK67.md): statement TPSH4ETK67
- [Axiom of dependent choice](https://tamnd.github.io/math2/node/TPXEFSBQYB.md): definition TPXEFSBQYB
- [Federer–Fleming compactness theorem](https://tamnd.github.io/math2/node/TPXYARH912.md): statement TPXYARH912
- [Hyers–Ulam stability](https://tamnd.github.io/math2/node/TPYXQTYND6.md): concept TPYXQTYND6
- [Lang's conjectures](https://tamnd.github.io/math2/node/TQ1QDAK2BR.md): conjecture TQ1QDAK2BR
- [Cyclic polytope](https://tamnd.github.io/math2/node/TQ2TZDMAPA.md): definition TQ2TZDMAPA
- [Lemniscate elliptic functions](https://tamnd.github.io/math2/node/TQDZQ964AZ.md): concept TQDZQ964AZ
- [Local regression](https://tamnd.github.io/math2/node/TQFFFSH0BQ.md): concept TQFFFSH0BQ
- [Simplicial approximation theorem](https://tamnd.github.io/math2/node/TQFXC1A6G5.md): statement TQFXC1A6G5
- [Coplanarity](https://tamnd.github.io/math2/node/TQJGPS6VZK.md): definition TQJGPS6VZK
- [Supermanifold](https://tamnd.github.io/math2/node/TQNG7T2RQ6.md): definition TQNG7T2RQ6
- [Reverse mathematics](https://tamnd.github.io/math2/node/TQTDKMBJ0F.md): definition TQTDKMBJ0F
- [Root-finding algorithm](https://tamnd.github.io/math2/node/TR4CNS5PMJ.md): concept TR4CNS5PMJ
- [Tame Fréchet space](https://tamnd.github.io/math2/node/TRB69DC8EW.md): definition TRB69DC8EW
- [Closure of an operator](https://tamnd.github.io/math2/node/TRC892PJNN.md): concept TRC892PJNN
- [Jacobi polynomials](https://tamnd.github.io/math2/node/TRD7DSF2KR.md): concept TRD7DSF2KR
- [Pitchfork bifurcation](https://tamnd.github.io/math2/node/TRFDEJRPAZ.md): definition TRFDEJRPAZ
- [Sato–Tate conjecture](https://tamnd.github.io/math2/node/TRGBJYPBXH.md): statement TRGBJYPBXH
- [Jet](https://tamnd.github.io/math2/node/TRKEGEZGGZ.md): definition TRKEGEZGGZ
- [Endomorphism ring of a simple module is a division ring](https://tamnd.github.io/math2/node/TRMYSF91EV.md): statement TRMYSF91EV
- [Hankel matrix](https://tamnd.github.io/math2/node/TRRREQX56B.md): definition TRRREQX56B
- [Ill-posedness of real Laplace inversion](https://tamnd.github.io/math2/node/TRTN840ZB0.md): concept TRTN840ZB0
- [Gauss sum](https://tamnd.github.io/math2/node/TS1GQGM7ZJ.md): definition TS1GQGM7ZJ
- [No-three-in-line problem](https://tamnd.github.io/math2/node/TS442VKWAG.md): problem TS442VKWAG
- [Gromov–Hausdorff convergence](https://tamnd.github.io/math2/node/TS5XH58X86.md): definition TS5XH58X86
- [Lindelöf's maximum principle](https://tamnd.github.io/math2/node/TS9TBY8ES1.md): statement TS9TBY8ES1
- [Topological ring](https://tamnd.github.io/math2/node/TSF8RZDD5B.md): concept TSF8RZDD5B
- [Pick matrix](https://tamnd.github.io/math2/node/TSPTN464CM.md): concept TSPTN464CM
- [Discrete Fourier transform](https://tamnd.github.io/math2/node/TST0GH66A5.md): concept TST0GH66A5
- [Strict hyperbolicity](https://tamnd.github.io/math2/node/TST3NVVB58.md): definition TST3NVVB58
- [Elimination theory](https://tamnd.github.io/math2/node/TSX2MAA95V.md): definition TSX2MAA95V
- [Initial algebra](https://tamnd.github.io/math2/node/TT0HJQ6QD7.md): definition TT0HJQ6QD7
- [Logicism](https://tamnd.github.io/math2/node/TT2ASTQ0WE.md): definition TT2ASTQ0WE
- [Partial correlation](https://tamnd.github.io/math2/node/TT3BXBR4DB.md): concept TT3BXBR4DB
- [Meromorphic function of several variables](https://tamnd.github.io/math2/node/TT7R63MV96.md): definition TT7R63MV96
- [Robinson arithmetic](https://tamnd.github.io/math2/node/TT8MXRTVBM.md): definition TT8MXRTVBM
- [Riesz representation theorem](https://tamnd.github.io/math2/node/TTC40N49M6.md): statement TTC40N49M6
- [Stokes phenomenon](https://tamnd.github.io/math2/node/TTGF8KCA8C.md): concept TTGF8KCA8C
- [Spectral density estimation](https://tamnd.github.io/math2/node/TTH1V4SWY6.md): concept TTH1V4SWY6
- [Quantum calculus](https://tamnd.github.io/math2/node/TTR52N9E7M.md): definition TTR52N9E7M
- [Absolutely free algebra](https://tamnd.github.io/math2/node/TTTEV3NWWM.md): concept TTTEV3NWWM
- [Kolmogorov's three-series theorem](https://tamnd.github.io/math2/node/TTTFQ693RJ.md): statement TTTFQ693RJ
- [Atom](https://tamnd.github.io/math2/node/TTVCTW6TNC.md): concept TTVCTW6TNC
- [Bianchi identities](https://tamnd.github.io/math2/node/TTXDZENFVB.md): statement TTXDZENFVB
- [Locally finite poset](https://tamnd.github.io/math2/node/TV1V4JDRDD.md): concept TV1V4JDRDD
- [Parallel transport](https://tamnd.github.io/math2/node/TVAGN3D6W5.md): concept TVAGN3D6W5
- [Orientability](https://tamnd.github.io/math2/node/TVMHQ338YS.md): definition TVMHQ338YS
- [Boole's rule](https://tamnd.github.io/math2/node/TVSVBH4TT4.md): statement TVSVBH4TT4
- [Dual pair](https://tamnd.github.io/math2/node/TVVYHW6FG0.md): definition TVVYHW6FG0
- [Poincaré recurrence theorem](https://tamnd.github.io/math2/node/TVYJPBHD0S.md): statement TVYJPBHD0S
- [Congruence-permutable variety](https://tamnd.github.io/math2/node/TW1KWXYWN0.md): definition TW1KWXYWN0
- [Torsion tensor](https://tamnd.github.io/math2/node/TW35GWE1XF.md): definition TW35GWE1XF
- [Serial ring](https://tamnd.github.io/math2/node/TW3ESQB3GQ.md): concept TW3ESQB3GQ
- [Measurable solutions of Cauchy's equation](https://tamnd.github.io/math2/node/TWANGP59RB.md): statement TWANGP59RB
- [Quantum logic](https://tamnd.github.io/math2/node/TWEYQ0XAY4.md): definition TWEYQ0XAY4
- [Cyclotomic extension](https://tamnd.github.io/math2/node/TWG8QKY8TA.md): definition TWG8QKY8TA
- [Finite type invariant](https://tamnd.github.io/math2/node/TWH3WVKFF1.md): concept TWH3WVKFF1
- [Étale fundamental group](https://tamnd.github.io/math2/node/TWP1EP7FR1.md): definition TWP1EP7FR1
- [Continuous-time Markov chain](https://tamnd.github.io/math2/node/TWQFE0FZZS.md): concept TWQFE0FZZS
- [Deck transformation group](https://tamnd.github.io/math2/node/TWSZNFKBKA.md): definition TWSZNFKBKA
- [Frostman's theorem](https://tamnd.github.io/math2/node/TX0Z2VVKH3.md): statement TX0Z2VVKH3
- [Čech cohomology](https://tamnd.github.io/math2/node/TX1W8PRASW.md): definition TX1W8PRASW
- [Beurling–Ahlfors transform](https://tamnd.github.io/math2/node/TX32GDQ330.md): concept TX32GDQ330
- [JSJ decomposition](https://tamnd.github.io/math2/node/TX4MSZ0SRP.md): statement TX4MSZ0SRP
- [Contiguous relations](https://tamnd.github.io/math2/node/TX5AV1JTF5.md): statement TX5AV1JTF5
- [Riemann zeta function](https://tamnd.github.io/math2/node/TXB2F07CC6.md): definition TXB2F07CC6
- [Asymmetric simple exclusion process](https://tamnd.github.io/math2/node/TXC6PZT8CF.md): example TXC6PZT8CF
- [K-function](https://tamnd.github.io/math2/node/TXCA43CMYR.md): concept TXCA43CMYR
- [Structure theorem for finitely generated modules over a principal ideal domain](https://tamnd.github.io/math2/node/TXMV0YDACG.md): statement TXMV0YDACG
- [Link](https://tamnd.github.io/math2/node/TXS0J856RD.md): definition TXS0J856RD
- [Summation in topological groups](https://tamnd.github.io/math2/node/TXS2H001R4.md): concept TXS2H001R4
- [Forward difference operator](https://tamnd.github.io/math2/node/TXTR68V4R1.md): definition TXTR68V4R1
- [Representation of a quiver](https://tamnd.github.io/math2/node/TXVWFWP6VQ.md): concept TXVWFWP6VQ
- [Measure problem of Lebesgue](https://tamnd.github.io/math2/node/TY12G0E8KX.md): problem TY12G0E8KX
- [Morita equivalence of Lie groupoids](https://tamnd.github.io/math2/node/TY60P5RE56.md): definition TY60P5RE56
- [Strange attractor](https://tamnd.github.io/math2/node/TY6TPW88MX.md): definition TY6TPW88MX
- [Banach analytic space](https://tamnd.github.io/math2/node/TY7Y32MCH1.md): concept TY7Y32MCH1
- [Von Neumann–Bernays–Gödel set theory](https://tamnd.github.io/math2/node/TY89DBMJYQ.md): definition TY89DBMJYQ
- [Basel problem via Fourier series](https://tamnd.github.io/math2/node/TYJWZDK485.md): example TYJWZDK485
- [Kepler problem](https://tamnd.github.io/math2/node/TYXD1T6FRX.md): example TYXD1T6FRX
- [Subdirectly irreducible algebra](https://tamnd.github.io/math2/node/TYYS649FAJ.md): definition TYYS649FAJ
- [Neville theta functions](https://tamnd.github.io/math2/node/TZ1N0HAGW7.md): concept TZ1N0HAGW7
- [Complementary error function](https://tamnd.github.io/math2/node/TZ3CT1YA9B.md): concept TZ3CT1YA9B
- [Gaussian binomial coefficient](https://tamnd.github.io/math2/node/TZ6Y9XXQJW.md): definition TZ6Y9XXQJW
- [Simplicial object](https://tamnd.github.io/math2/node/TZFSE2VNAQ.md): definition TZFSE2VNAQ
- [Measurable Riemann mapping theorem](https://tamnd.github.io/math2/node/TZHD2WKNSD.md): statement TZHD2WKNSD
- [Adjoint functors](https://tamnd.github.io/math2/node/TZHF6ZM92S.md): concept TZHF6ZM92S
- [Temperley–Lieb algebra](https://tamnd.github.io/math2/node/TZNR514JQN.md): concept TZNR514JQN
- [Cohen's theorem on Noetherian rings](https://tamnd.github.io/math2/node/TZT1TTB7CJ.md): statement TZT1TTB7CJ
- [Disjoint union of posets](https://tamnd.github.io/math2/node/TZWMEM7GKA.md): concept TZWMEM7GKA
- [Fourier monomial on the circle](https://tamnd.github.io/math2/node/TZYSW0975T.md): concept TZYSW0975T
- [Hadamard three-circle theorem](https://tamnd.github.io/math2/node/V00Q05ZW1G.md): statement V00Q05ZW1G
- [Primary ideal](https://tamnd.github.io/math2/node/V026AAV8E6.md): definition V026AAV8E6
- [Unital](https://tamnd.github.io/math2/node/V051BD0CZG.md): concept V051BD0CZG
- [Triangulation of a point set](https://tamnd.github.io/math2/node/V0B6C6TMWD.md): concept V0B6C6TMWD
- [Radical ideal](https://tamnd.github.io/math2/node/V0E14B3DWD.md): definition V0E14B3DWD
- [Internal set theory](https://tamnd.github.io/math2/node/V0PGBXYJ9R.md): concept V0PGBXYJ9R
- [Valuation on convex bodies](https://tamnd.github.io/math2/node/V0RBKHYTV7.md): concept V0RBKHYTV7
- [Clifford–Klein form](https://tamnd.github.io/math2/node/V0SGQ7E092.md): definition V0SGQ7E092
- [Capacity equals transfinite diameter and Chebyshev constant](https://tamnd.github.io/math2/node/V0SH1DY9FJ.md): statement V0SH1DY9FJ
- [Box–Jenkins method](https://tamnd.github.io/math2/node/V192HFRX5A.md): definition V192HFRX5A
- [Fuglede's theorem](https://tamnd.github.io/math2/node/V1C28937NX.md): statement V1C28937NX
- [Fredholm equation of the first kind](https://tamnd.github.io/math2/node/V1FFDDHCH2.md): definition V1FFDDHCH2
- [Radical of a module](https://tamnd.github.io/math2/node/V1H5X4H8Q7.md): definition V1H5X4H8Q7
- [Kolmogorov continuity theorem](https://tamnd.github.io/math2/node/V1HAJNC5RK.md): statement V1HAJNC5RK
- [Schur decomposition](https://tamnd.github.io/math2/node/V1JG4C7R7B.md): statement V1JG4C7R7B
- [Compact-open topology](https://tamnd.github.io/math2/node/V1PAXEF0EE.md): definition V1PAXEF0EE
- [Completion of a topological group](https://tamnd.github.io/math2/node/V1QDWHY4KT.md): concept V1QDWHY4KT
- [Schwarz–Pick theorem](https://tamnd.github.io/math2/node/V1QYA9RW73.md): statement V1QYA9RW73
- [Voronoi cell](https://tamnd.github.io/math2/node/V1R1R88GV6.md): concept V1R1R88GV6
- [Cointegration](https://tamnd.github.io/math2/node/V1S1223G5E.md): concept V1S1223G5E
- [G-delta set](https://tamnd.github.io/math2/node/V1TBNP8R3S.md): concept V1TBNP8R3S
- [Z-transform](https://tamnd.github.io/math2/node/V1WRSW2AA6.md): definition V1WRSW2AA6
- [Auslander–Reiten quiver](https://tamnd.github.io/math2/node/V1X38GS4BX.md): definition V1X38GS4BX
- [Wiener's Tauberian theorem](https://tamnd.github.io/math2/node/V20376HXHD.md): statement V20376HXHD
- [Tverberg's theorem](https://tamnd.github.io/math2/node/V21XNZ1H0B.md): statement V21XNZ1H0B
- [1 − 1 + 2 − 6 + 24 − 120 + ⋯](https://tamnd.github.io/math2/node/V24RQP5492.md): example V24RQP5492
- [Mapping cylinder](https://tamnd.github.io/math2/node/V25YYE26M1.md): definition V25YYE26M1
- [Genus](https://tamnd.github.io/math2/node/V27468ECYW.md): definition V27468ECYW
- [Term algebra](https://tamnd.github.io/math2/node/V288YS2F5W.md): definition V288YS2F5W
- [Banach–Lie group](https://tamnd.github.io/math2/node/V2AA2D23JC.md): definition V2AA2D23JC
- [Singular values of a compact operator](https://tamnd.github.io/math2/node/V2H578D4RT.md): definition V2H578D4RT
- [Bessel potential](https://tamnd.github.io/math2/node/V2HG4GN7TN.md): concept V2HG4GN7TN
- [Weil's local rigidity theorem](https://tamnd.github.io/math2/node/V2HTMG4KT1.md): statement V2HTMG4KT1
- [Pin group](https://tamnd.github.io/math2/node/V2QQKNHVEB.md): example V2QQKNHVEB
- [Tight frame](https://tamnd.github.io/math2/node/V2S7CHGS26.md): concept V2S7CHGS26
- [Epsilon-net of a set system](https://tamnd.github.io/math2/node/V2SBN1FNQV.md): concept V2SBN1FNQV
- [Lewy extension theorem](https://tamnd.github.io/math2/node/V2YCGHD719.md): statement V2YCGHD719
- [Homotopy invariance of homology](https://tamnd.github.io/math2/node/V313FN8HRW.md): statement V313FN8HRW
- [Caccioppoli set](https://tamnd.github.io/math2/node/V365PW1HNE.md): definition V365PW1HNE
- [Heap](https://tamnd.github.io/math2/node/V389A4WGH9.md): concept V389A4WGH9
- [Full subcategory](https://tamnd.github.io/math2/node/V3BFGKBJF7.md): concept V3BFGKBJF7
- [Eilenberg–Zilber theorem](https://tamnd.github.io/math2/node/V3G5YQTQ00.md): statement V3G5YQTQ00
- [Mapping cone](https://tamnd.github.io/math2/node/V3MP0RJ2NX.md): definition V3MP0RJ2NX
- [No small subgroups](https://tamnd.github.io/math2/node/V3N4HX0GRP.md): definition V3N4HX0GRP
- [Causal inference](https://tamnd.github.io/math2/node/V3QJFN5800.md): concept V3QJFN5800
- [Polar polytope](https://tamnd.github.io/math2/node/V3RCQ1J2VV.md): definition V3RCQ1J2VV
- [Solvable Lie group](https://tamnd.github.io/math2/node/V3XM2N7K0X.md): definition V3XM2N7K0X
- [Lerch's uniqueness theorem](https://tamnd.github.io/math2/node/V3YTF5WJHY.md): statement V3YTF5WJHY
- [Regular local rings are domains](https://tamnd.github.io/math2/node/V3ZBMRKC08.md): statement V3ZBMRKC08
- [Cubic spline](https://tamnd.github.io/math2/node/V42H0R3EKX.md): concept V42H0R3EKX
- [Lattice congruence](https://tamnd.github.io/math2/node/V465WCM25T.md): definition V465WCM25T
- [Trace of an integral operator](https://tamnd.github.io/math2/node/V4KW8XH2GF.md): definition V4KW8XH2GF
- [Complex structure on the six-sphere](https://tamnd.github.io/math2/node/V4M2HPJQPR.md): problem V4M2HPJQPR
- [Skolem–Noether theorem](https://tamnd.github.io/math2/node/V4PEVFF4F4.md): statement V4PEVFF4F4
- [Factorial experiment](https://tamnd.github.io/math2/node/V4Q6TSY1KX.md): definition V4Q6TSY1KX
- [Cayley–Klein metric](https://tamnd.github.io/math2/node/V50N3JEBDP.md): definition V50N3JEBDP
- [Lemniscate constant](https://tamnd.github.io/math2/node/V5A2S8QYQQ.md): object V5A2S8QYQQ
- [Abel's summation formula](https://tamnd.github.io/math2/node/V5EPT2PX9Z.md): statement V5EPT2PX9Z
- [Dynkin system](https://tamnd.github.io/math2/node/V5M2CKB591.md): concept V5M2CKB591
- [Perron–Wiener–Brelot solution](https://tamnd.github.io/math2/node/V5TBWB2RPQ.md): definition V5TBWB2RPQ
- [Rigidity of frameworks](https://tamnd.github.io/math2/node/V5YS9BZX0N.md): concept V5YS9BZX0N
- [Interpolation space](https://tamnd.github.io/math2/node/V60F20N6JT.md): definition V60F20N6JT
- [Exponential stability](https://tamnd.github.io/math2/node/V6E2R7JNN8.md): definition V6E2R7JNN8
- [Length of a module](https://tamnd.github.io/math2/node/V6JBKT168T.md): definition V6JBKT168T
- [Convergence of Fourier series](https://tamnd.github.io/math2/node/V6Q24CHJG8.md): concept V6Q24CHJG8
- [p-adic integer](https://tamnd.github.io/math2/node/V6TJAFZ4MD.md): concept V6TJAFZ4MD
- [Lasso](https://tamnd.github.io/math2/node/V6XGZQ0GR1.md): definition V6XGZQ0GR1
- [Schur's theorem](https://tamnd.github.io/math2/node/V6XJQXZ7E8.md): statement V6XJQXZ7E8
- [Josefson's theorem](https://tamnd.github.io/math2/node/V6XSDT5C21.md): statement V6XSDT5C21
- [Abelian differential](https://tamnd.github.io/math2/node/V6ZC875F8R.md): definition V6ZC875F8R
- [Radical extension](https://tamnd.github.io/math2/node/V74WMGQQ9D.md): concept V74WMGQQ9D
- [Residuated mapping](https://tamnd.github.io/math2/node/V76Q9B6VS8.md): concept V76Q9B6VS8
- [Fekete points](https://tamnd.github.io/math2/node/V7AAT7X2QW.md): definition V7AAT7X2QW
- [Krull topology](https://tamnd.github.io/math2/node/V7ERB5XHQR.md): definition V7ERB5XHQR
- [Parabolic cylinder function](https://tamnd.github.io/math2/node/V7H5Z35CPN.md): concept V7H5Z35CPN
- [Morley's categoricity theorem](https://tamnd.github.io/math2/node/V7N84P60SJ.md): statement V7N84P60SJ
- [Maximal ergodic theorem](https://tamnd.github.io/math2/node/V7PA1V3N0K.md): statement V7PA1V3N0K
- [Collineation](https://tamnd.github.io/math2/node/V7PRPN24CV.md): concept V7PRPN24CV
- [Perfect closure](https://tamnd.github.io/math2/node/V8FVP4NJV3.md): definition V8FVP4NJV3
- [Simplicial complex](https://tamnd.github.io/math2/node/V8GZ0W1ZZ5.md): concept V8GZ0W1ZZ5
- [Tsen's theorem](https://tamnd.github.io/math2/node/V8HDYN3JMY.md): statement V8HDYN3JMY
- [Brauer's theorem on induced characters](https://tamnd.github.io/math2/node/V8JVWE87TE.md): statement V8JVWE87TE
- [Martin's axiom](https://tamnd.github.io/math2/node/V8KGT8MRFE.md): concept V8KGT8MRFE
- [Stone-type duality for varieties](https://tamnd.github.io/math2/node/V8PEAAE4K1.md): definition V8PEAAE4K1
- [Householder transformation](https://tamnd.github.io/math2/node/V8XA0ECEJ3.md): definition V8XA0ECEJ3
- [von Mises–Fisher distribution](https://tamnd.github.io/math2/node/V8ZZ7A2FMF.md): concept V8ZZ7A2FMF
- [Sullivan minimal model](https://tamnd.github.io/math2/node/V9079C40AZ.md): definition V9079C40AZ
- [Newtonian space](https://tamnd.github.io/math2/node/V90RGBV52W.md): concept V90RGBV52W
- [Serre's twisting sheaf](https://tamnd.github.io/math2/node/V93Y0XYPZD.md): concept V93Y0XYPZD
- [Two-sided Laplace transform](https://tamnd.github.io/math2/node/V94VYJ2RR1.md): definition V94VYJ2RR1
- [van der Waerden's theorem](https://tamnd.github.io/math2/node/V96FRZ2ME1.md): statement V96FRZ2ME1
- [Unital](https://tamnd.github.io/math2/node/V9BRV06TVY.md): definition V9BRV06TVY
- [X-ray transform](https://tamnd.github.io/math2/node/V9G9J2C10P.md): concept V9G9J2C10P
- [Malcev completion](https://tamnd.github.io/math2/node/V9GRK3QANS.md): concept V9GRK3QANS
- [Connection on a vector bundle](https://tamnd.github.io/math2/node/V9HGTHC7DE.md): definition V9HGTHC7DE
- [Plancherel theorem for abelian groups](https://tamnd.github.io/math2/node/V9K9TG0K3J.md): statement V9K9TG0K3J
- [Numerical methods for linear least squares](https://tamnd.github.io/math2/node/V9R8K4JCVM.md): definition V9R8K4JCVM
- [Remmert's proper mapping theorem](https://tamnd.github.io/math2/node/VA2JEWWB61.md): statement VA2JEWWB61
- [Grothendieck's local duality](https://tamnd.github.io/math2/node/VA3DY2YX83.md): statement VA3DY2YX83
- [Term evaluation](https://tamnd.github.io/math2/node/VA41ZY2HTF.md): definition VA41ZY2HTF
- [Church's theorem on undecidability of first-order logic](https://tamnd.github.io/math2/node/VA6BNNY2YC.md): statement VA6BNNY2YC
- [Type](https://tamnd.github.io/math2/node/VA7AK1R5AN.md): definition VA7AK1R5AN
- [Zeilberger's algorithm](https://tamnd.github.io/math2/node/VA7MH05Z3B.md): concept VA7MH05Z3B
- [Masur–Veech theorem](https://tamnd.github.io/math2/node/VA8R0TZCKB.md): statement VA8R0TZCKB
- [Profinite group](https://tamnd.github.io/math2/node/VABPHNZAVV.md): definition VABPHNZAVV
- [Krein–Rutman theorem](https://tamnd.github.io/math2/node/VADXDK3TKM.md): statement VADXDK3TKM
- [Extension of Lagrange's theorem](https://tamnd.github.io/math2/node/VAF6B01Q8W.md): statement VAF6B01Q8W
- [Direct method in the calculus of variations](https://tamnd.github.io/math2/node/VAJPHCQBHR.md): definition VAJPHCQBHR
- [Yau's conjecture on infinitely many minimal surfaces](https://tamnd.github.io/math2/node/VAP45DFXW3.md): statement VAP45DFXW3
- [Trigonometric definition of Chebyshev polynomials](https://tamnd.github.io/math2/node/VAPNSC2JH5.md): statement VAPNSC2JH5
- [Operator space](https://tamnd.github.io/math2/node/VB1J9AA4WX.md): concept VB1J9AA4WX
- [Minkowski plane](https://tamnd.github.io/math2/node/VB2D03PD9Y.md): concept VB2D03PD9Y
- [Trigonometric integrals](https://tamnd.github.io/math2/node/VB37Q6FY1Y.md): concept VB37Q6FY1Y
- [Morita duality](https://tamnd.github.io/math2/node/VB7WV3E35H.md): definition VB7WV3E35H
- [Banach manifold](https://tamnd.github.io/math2/node/VB95VVVDM8.md): definition VB95VVVDM8
- [q-convex manifold](https://tamnd.github.io/math2/node/VBHR4R5W7F.md): concept VBHR4R5W7F
- [n-skeleton](https://tamnd.github.io/math2/node/VBM8ZYTT2X.md): concept VBM8ZYTT2X
- [Cole–Hopf transformation](https://tamnd.github.io/math2/node/VBMQ9SJ3DH.md): concept VBMQ9SJ3DH
- [Universal property](https://tamnd.github.io/math2/node/VBS4A8G5XA.md): definition VBS4A8G5XA
- [Closed geodesic](https://tamnd.github.io/math2/node/VBVBDQTECG.md): definition VBVBDQTECG
- [Ancillary statistic](https://tamnd.github.io/math2/node/VBW2A4H6TD.md): concept VBW2A4H6TD
- [Lyndon–Hochschild–Serre spectral sequence](https://tamnd.github.io/math2/node/VC3QNG6TKR.md): statement VC3QNG6TKR
- [Kohn–Rossi cohomology](https://tamnd.github.io/math2/node/VC48WDWPV1.md): concept VC48WDWPV1
- [Fractional Brownian motion](https://tamnd.github.io/math2/node/VC6R7XSKRQ.md): concept VC6R7XSKRQ
- [BK-space](https://tamnd.github.io/math2/node/VC7EH3FZX3.md): definition VC7EH3FZX3
- [Blocking](https://tamnd.github.io/math2/node/VC8286D7TZ.md): definition VC8286D7TZ
- [Banach–Mazur distance](https://tamnd.github.io/math2/node/VCBWR1WSK0.md): definition VCBWR1WSK0
- [Riemann hypothesis](https://tamnd.github.io/math2/node/VCGF9YR36X.md): conjecture VCGF9YR36X
- [Word metric](https://tamnd.github.io/math2/node/VCHJVR9NGH.md): concept VCHJVR9NGH
- [Dual operator](https://tamnd.github.io/math2/node/VCHS7PH0NF.md): concept VCHS7PH0NF
- [Hook length formula](https://tamnd.github.io/math2/node/VCMS1CP91N.md): statement VCMS1CP91N
- [Newlander–Nirenberg theorem](https://tamnd.github.io/math2/node/VCT848WV1T.md): statement VCT848WV1T
- [Gabor transform](https://tamnd.github.io/math2/node/VD2REA47BN.md): definition VD2REA47BN
- [Cyclotomic extension](https://tamnd.github.io/math2/node/VD6VC3ZMMY.md): concept VD6VC3ZMMY
- [Generalized hypergeometric function](https://tamnd.github.io/math2/node/VDBAW9PN88.md): concept VDBAW9PN88
- [Order polynomial](https://tamnd.github.io/math2/node/VDBDJZWPEA.md): concept VDBDJZWPEA
- [Errors-in-variables model](https://tamnd.github.io/math2/node/VDEFF7A5Q4.md): definition VDEFF7A5Q4
- [Geometric lattice](https://tamnd.github.io/math2/node/VDF7CGPC0P.md): concept VDF7CGPC0P
- [Markov kernel](https://tamnd.github.io/math2/node/VDGDD8Q61N.md): definition VDGDD8Q61N
- [Residually small variety](https://tamnd.github.io/math2/node/VDHW982H70.md): definition VDHW982H70
- [Curse of dimensionality](https://tamnd.github.io/math2/node/VDKPPS34XM.md): definition VDKPPS34XM
- [Convolution on the half-line](https://tamnd.github.io/math2/node/VDMCEPTTXB.md): definition VDMCEPTTXB
- [Brunn–Minkowski theory](https://tamnd.github.io/math2/node/VDPTQ35JWE.md): definition VDPTQ35JWE
- [Random integral equation](https://tamnd.github.io/math2/node/VDQ32JT7ZS.md): concept VDQ32JT7ZS
- [Small object argument](https://tamnd.github.io/math2/node/VDQAZDHFF7.md): statement VDQAZDHFF7
- [Betti number](https://tamnd.github.io/math2/node/VDQX55E7D4.md): definition VDQX55E7D4
- [von Neumann coordinatization theorem](https://tamnd.github.io/math2/node/VDRCDZ048J.md): statement VDRCDZ048J
- [Hausdorff content](https://tamnd.github.io/math2/node/VDTEARFVMB.md): concept VDTEARFVMB
- [Smale's fourteenth problem](https://tamnd.github.io/math2/node/VDVQD2HFSD.md): problem VDVQD2HFSD
- [Method of characteristics](https://tamnd.github.io/math2/node/VDWKE1KK4H.md): definition VDWKE1KK4H
- [Asymptotic analysis](https://tamnd.github.io/math2/node/VDX0PQ2GJ4.md): definition VDX0PQ2GJ4
- [Logarithmic kernel](https://tamnd.github.io/math2/node/VDZ5FKYC2T.md): concept VDZ5FKYC2T
- [Day terms](https://tamnd.github.io/math2/node/VE4SJTXFBV.md): definition VE4SJTXFBV
- [Clenshaw–Curtis quadrature](https://tamnd.github.io/math2/node/VEB7J3C74F.md): concept VEB7J3C74F
- [Bayes risk](https://tamnd.github.io/math2/node/VEBDNCJJTN.md): concept VEBDNCJJTN
- [Self-distributive operation](https://tamnd.github.io/math2/node/VEFE5WZMS3.md): concept VEFE5WZMS3
- [Rational homotopy theory](https://tamnd.github.io/math2/node/VEGBD6Y61A.md): definition VEGBD6Y61A
- [Modular group](https://tamnd.github.io/math2/node/VEKV1EM77B.md): object VEKV1EM77B
- [Independent component analysis](https://tamnd.github.io/math2/node/VEMZC8Y458.md): definition VEMZC8Y458
- [Differentiation in the transform variable](https://tamnd.github.io/math2/node/VEPJN1NK3C.md): statement VEPJN1NK3C
- [Brody's lemma](https://tamnd.github.io/math2/node/VEV5E2077C.md): statement VEV5E2077C
- [Curvelet](https://tamnd.github.io/math2/node/VEVPDJ8S7B.md): concept VEVPDJ8S7B
- [Wasserstein metric](https://tamnd.github.io/math2/node/VEY29CS26X.md): concept VEY29CS26X
- [Isogeometric analysis](https://tamnd.github.io/math2/node/VF17XWCK8Z.md): definition VF17XWCK8Z
- [Ore extension](https://tamnd.github.io/math2/node/VF3PV13BA1.md): concept VF3PV13BA1
- [Geodesic flow](https://tamnd.github.io/math2/node/VF43FTDA1H.md): example VF43FTDA1H
- [Nonexpansive map](https://tamnd.github.io/math2/node/VF6CA0VE41.md): concept VF6CA0VE41
- [Nonparametric statistics](https://tamnd.github.io/math2/node/VFAJ1EG3X5.md): concept VFAJ1EG3X5
- [Perfect graph](https://tamnd.github.io/math2/node/VFCN5HDH7S.md): definition VFCN5HDH7S
- [Cyclic extension](https://tamnd.github.io/math2/node/VFE5B71E7Y.md): concept VFE5B71E7Y
- [Complete Boolean algebra](https://tamnd.github.io/math2/node/VFKNNPVMTX.md): definition VFKNNPVMTX
- [Ultraproduct](https://tamnd.github.io/math2/node/VFPGADHJYA.md): definition VFPGADHJYA
- [Bockstein homomorphism](https://tamnd.github.io/math2/node/VFPP3RZGK3.md): definition VFPP3RZGK3
- [Hyers' theorem](https://tamnd.github.io/math2/node/VFRJ0P2P15.md): statement VFRJ0P2P15
- [The odd integers are not a subgroup of the integers](https://tamnd.github.io/math2/node/VFYVR2H4QS.md): example VFYVR2H4QS
- [Minimal period](https://tamnd.github.io/math2/node/VFZRVZ9SD8.md): concept VFZRVZ9SD8
- [Signed measure](https://tamnd.github.io/math2/node/VG2P7R91Z7.md): concept VG2P7R91Z7
- [Smooth structure](https://tamnd.github.io/math2/node/VG4ZVTGJ9W.md): definition VG4ZVTGJ9W
- [Bourgain's dimension bound for harmonic measure](https://tamnd.github.io/math2/node/VGB11C8TTN.md): statement VGB11C8TTN
- [Pseudo-arc](https://tamnd.github.io/math2/node/VGCTDCGS6V.md): example VGCTDCGS6V
- [Surface](https://tamnd.github.io/math2/node/VGJJVBZ1MQ.md): concept VGJJVBZ1MQ
- [Bifurcation set](https://tamnd.github.io/math2/node/VGPTRNE3FF.md): definition VGPTRNE3FF
- [Jury stability criterion](https://tamnd.github.io/math2/node/VGV9FAPCHN.md): statement VGV9FAPCHN
- [Laplace transform of an integral](https://tamnd.github.io/math2/node/VGYXWXV5M4.md): statement VGYXWXV5M4
- [Measure with density](https://tamnd.github.io/math2/node/VGZRSDNEBK.md): definition VGZRSDNEBK
- [Stone-type duality for varieties](https://tamnd.github.io/math2/node/VH12PN9V13.md): concept VH12PN9V13
- [Period map](https://tamnd.github.io/math2/node/VH4FD4ZHEQ.md): definition VH4FD4ZHEQ
- [Huntington's axioms](https://tamnd.github.io/math2/node/VH4R85Y6Y3.md): definition VH4R85Y6Y3
- [Weaire–Phelan structure](https://tamnd.github.io/math2/node/VH7MSMN03Y.md): example VH7MSMN03Y
- [List coloring](https://tamnd.github.io/math2/node/VHAMHH5SPC.md): concept VHAMHH5SPC
- [Galois' theorem on solvability](https://tamnd.github.io/math2/node/VHCWCFA3MX.md): statement VHCWCFA3MX
- [Algebraic set](https://tamnd.github.io/math2/node/VHFMQQYSTN.md): concept VHFMQQYSTN
- [f-ring](https://tamnd.github.io/math2/node/VHHMZPNVEA.md): concept VHHMZPNVEA
- [Shrinkage estimator](https://tamnd.github.io/math2/node/VHKMX3DR53.md): definition VHKMX3DR53
- [Gegenbauer transform](https://tamnd.github.io/math2/node/VHKT96G1A9.md): definition VHKT96G1A9
- [Fourier transform on locally compact abelian groups](https://tamnd.github.io/math2/node/VHN5VW5Q92.md): concept VHN5VW5Q92
- [Artinian module](https://tamnd.github.io/math2/node/VHRHK7V6TP.md): definition VHRHK7V6TP
- [Riesz basis](https://tamnd.github.io/math2/node/VHXX092DAJ.md): definition VHXX092DAJ
- [Completely monotone function](https://tamnd.github.io/math2/node/VJ1NCNTEHD.md): concept VJ1NCNTEHD
- [Lindeberg's condition](https://tamnd.github.io/math2/node/VJ3BVREDWN.md): concept VJ3BVREDWN
- [Risch algorithm](https://tamnd.github.io/math2/node/VJ51B7J125.md): concept VJ51B7J125
- [Askey–Gasper inequality](https://tamnd.github.io/math2/node/VJ7AMZB988.md): statement VJ7AMZB988
- [Zeta function regularization](https://tamnd.github.io/math2/node/VJCJYXGTDY.md): concept VJCJYXGTDY
- [Opposite ring](https://tamnd.github.io/math2/node/VJDYKZX993.md): definition VJDYKZX993
- [Littlewood's L1 norm conjecture](https://tamnd.github.io/math2/node/VJNTCDZS2H.md): statement VJNTCDZS2H
- [Radon transform](https://tamnd.github.io/math2/node/VJNZJ352PC.md): concept VJNZJ352PC
- [Köthe–Toeplitz dual](https://tamnd.github.io/math2/node/VJVDDR8BCE.md): concept VJVDDR8BCE
- [Hom functor](https://tamnd.github.io/math2/node/VJXS8AK67E.md): concept VJXS8AK67E
- [Graph embedding](https://tamnd.github.io/math2/node/VK1XXMR2WW.md): definition VK1XXMR2WW
- [Bing–Borsuk conjecture](https://tamnd.github.io/math2/node/VK39X07EYK.md): conjecture VK39X07EYK
- [Perfectly normal space](https://tamnd.github.io/math2/node/VK3ZMD1BCP.md): concept VK3ZMD1BCP
- [Thompson groups](https://tamnd.github.io/math2/node/VK50GVVFTN.md): object VK50GVVFTN
- [Torsion submodule](https://tamnd.github.io/math2/node/VK52XRHBXE.md): definition VK52XRHBXE
- [Reductive group](https://tamnd.github.io/math2/node/VK7402274F.md): definition VK7402274F
- [Absolute geometry](https://tamnd.github.io/math2/node/VKFPCSRNDH.md): concept VKFPCSRNDH
- [Submanifold](https://tamnd.github.io/math2/node/VKFW27WXEZ.md): concept VKFW27WXEZ
- [Volterra integral equation](https://tamnd.github.io/math2/node/VKJ8GQ07H7.md): definition VKJ8GQ07H7
- [Deformation of complex structure](https://tamnd.github.io/math2/node/VKQSM3BCE4.md): concept VKQSM3BCE4
- [Azumaya algebra](https://tamnd.github.io/math2/node/VKZTT484WQ.md): concept VKZTT484WQ
- [Ridders' method](https://tamnd.github.io/math2/node/VM1C9D8MX9.md): statement VM1C9D8MX9
- [Condenser capacity](https://tamnd.github.io/math2/node/VM5V845JGZ.md): definition VM5V845JGZ
- [Closed operator](https://tamnd.github.io/math2/node/VM7B38PA87.md): concept VM7B38PA87
- [Pesin theory](https://tamnd.github.io/math2/node/VM7M0PJTD4.md): definition VM7M0PJTD4
- [Complex analytic variety](https://tamnd.github.io/math2/node/VMDTGTMN9P.md): definition VMDTGTMN9P
- [Gibbs phenomenon](https://tamnd.github.io/math2/node/VMF169AT67.md): concept VMF169AT67
- [Thom–Mather stratified space](https://tamnd.github.io/math2/node/VMFWAPNN65.md): definition VMFWAPNN65
- [Laman graph](https://tamnd.github.io/math2/node/VMQGTDAQM0.md): definition VMQGTDAQM0
- [Exposed point](https://tamnd.github.io/math2/node/VN4M72SQW5.md): concept VN4M72SQW5
- [Heat equation](https://tamnd.github.io/math2/node/VNB1BFDEYK.md): definition VNB1BFDEYK
- [Irreducible representation](https://tamnd.github.io/math2/node/VNJ4V7FYBV.md): definition VNJ4V7FYBV
- [Fuchsian group](https://tamnd.github.io/math2/node/VNJWSCVP6V.md): concept VNJWSCVP6V
- [Horseshoe lemma](https://tamnd.github.io/math2/node/VNMBSYX9PY.md): statement VNMBSYX9PY
- [Linear programming](https://tamnd.github.io/math2/node/VNMXB0J2KA.md): concept VNMXB0J2KA
- [Nyström method](https://tamnd.github.io/math2/node/VNS9QQATH1.md): statement VNS9QQATH1
- [Densely defined operator](https://tamnd.github.io/math2/node/VNTA4KYMQT.md): definition VNTA4KYMQT
- [Extreme point](https://tamnd.github.io/math2/node/VNV54MAN67.md): definition VNV54MAN67
- [Wandering set](https://tamnd.github.io/math2/node/VNW5FHCQ2M.md): concept VNW5FHCQ2M
- [Function approximation](https://tamnd.github.io/math2/node/VNWH2KADN7.md): concept VNWH2KADN7
- [Algebraic stack](https://tamnd.github.io/math2/node/VNX79CRH90.md): concept VNX79CRH90
- [Conjugate points](https://tamnd.github.io/math2/node/VNXJRNE33W.md): definition VNXJRNE33W
- [Reinhardt domain](https://tamnd.github.io/math2/node/VNYXJXX44H.md): definition VNYXJXX44H
- [Algebraic stack](https://tamnd.github.io/math2/node/VP1N9S4J4F.md): definition VP1N9S4J4F
- [Heat equation](https://tamnd.github.io/math2/node/VP2HN86VAG.md): concept VP2HN86VAG
- [Bernstein–Kushnirenko theorem](https://tamnd.github.io/math2/node/VP32769PVR.md): statement VP32769PVR
- [Romberg's method](https://tamnd.github.io/math2/node/VP40T7MHGM.md): concept VP40T7MHGM
- [Moduli space of instantons](https://tamnd.github.io/math2/node/VP5PW8VSHZ.md): object VP5PW8VSHZ
- [Logarithmic integral](https://tamnd.github.io/math2/node/VP98WSTZWT.md): concept VP98WSTZWT
- [Kriging](https://tamnd.github.io/math2/node/VP9S3PNJQ0.md): concept VP9S3PNJQ0
- [Infinitesimal generator of a Markov process](https://tamnd.github.io/math2/node/VPA41PWRR1.md): concept VPA41PWRR1
- [Algebraic combinatorics](https://tamnd.github.io/math2/node/VPAQZ0XW74.md): concept VPAQZ0XW74
- [Pexider equation](https://tamnd.github.io/math2/node/VPBSH2KDFD.md): definition VPBSH2KDFD
- [Associated Legendre polynomials](https://tamnd.github.io/math2/node/VPJCSVRY6S.md): concept VPJCSVRY6S
- [Positive-definite function](https://tamnd.github.io/math2/node/VPMHPRRKZH.md): definition VPMHPRRKZH
- [Control variates](https://tamnd.github.io/math2/node/VPMRHXP6NM.md): statement VPMRHXP6NM
- [Proj construction](https://tamnd.github.io/math2/node/VPQJEFJ0HZ.md): definition VPQJEFJ0HZ
- [Geometric integrator](https://tamnd.github.io/math2/node/VQ2BTYFST4.md): concept VQ2BTYFST4
- [Annihilator](https://tamnd.github.io/math2/node/VQ31D68MDD.md): definition VQ31D68MDD
- [Line segment](https://tamnd.github.io/math2/node/VQ4Q2V45RP.md): concept VQ4Q2V45RP
- [Torsion-free group](https://tamnd.github.io/math2/node/VQ4SSKS53D.md): definition VQ4SSKS53D
- [Cusp catastrophe](https://tamnd.github.io/math2/node/VQ7D5N3MV7.md): example VQ7D5N3MV7
- [Slice theorem](https://tamnd.github.io/math2/node/VQB2N60ARE.md): statement VQB2N60ARE
- [Twenty-seven lines on a cubic surface](https://tamnd.github.io/math2/node/VQK5HQFYEM.md): statement VQK5HQFYEM
- [Chow–Rashevskii theorem](https://tamnd.github.io/math2/node/VQPN24VDH1.md): statement VQPN24VDH1
- [Null set](https://tamnd.github.io/math2/node/VQQM4Y550T.md): concept VQQM4Y550T
- [Kolmogorov–Arnold representation theorem](https://tamnd.github.io/math2/node/VQXDB9T838.md): statement VQXDB9T838
- [Steenrod square](https://tamnd.github.io/math2/node/VR7RHP41M0.md): concept VR7RHP41M0
- [Eigenvalue perturbation](https://tamnd.github.io/math2/node/VR89YHC0TX.md): concept VR89YHC0TX
- [Operator Cauchy equation and one-parameter semigroups](https://tamnd.github.io/math2/node/VRBP76MM84.md): statement VRBP76MM84
- [Lie pseudogroup](https://tamnd.github.io/math2/node/VRP0HESPYD.md): definition VRP0HESPYD
- [Askey scheme](https://tamnd.github.io/math2/node/VRQRRSBDPG.md): definition VRQRRSBDPG
- [Periodic solution](https://tamnd.github.io/math2/node/VRTV602H8G.md): definition VRTV602H8G
- [Uniform integrability](https://tamnd.github.io/math2/node/VRVVKMXDGV.md): concept VRVVKMXDGV
- [Sheaf cohomology in topology](https://tamnd.github.io/math2/node/VRZDAWNTCR.md): definition VRZDAWNTCR
- [Complete elliptic integral of the second kind](https://tamnd.github.io/math2/node/VS2PSX14C8.md): concept VS2PSX14C8
- [Elliptic surface](https://tamnd.github.io/math2/node/VSC3F53DX4.md): concept VSC3F53DX4
- [Modified Bessel functions](https://tamnd.github.io/math2/node/VSD1D6ASVH.md): definition VSD1D6ASVH
- [Totally unimodular matrix](https://tamnd.github.io/math2/node/VSD67A49XC.md): concept VSD67A49XC
- [Transfinite diameter](https://tamnd.github.io/math2/node/VSGTSQ7RY8.md): concept VSGTSQ7RY8
- [Interval order](https://tamnd.github.io/math2/node/VSNQV29KFJ.md): concept VSNQV29KFJ
- [Numerical differentiation](https://tamnd.github.io/math2/node/VSTM4MZJXZ.md): definition VSTM4MZJXZ
- [Sine and cosine addition equations](https://tamnd.github.io/math2/node/VSZR2CY2RS.md): definition VSZR2CY2RS
- [Polish space](https://tamnd.github.io/math2/node/VT2DVEEVRH.md): definition VT2DVEEVRH
- [Entourage](https://tamnd.github.io/math2/node/VT3NYDTY95.md): definition VT3NYDTY95
- [Jónsson's lemma](https://tamnd.github.io/math2/node/VT72G3BA99.md): statement VT72G3BA99
- [Braided monoidal category](https://tamnd.github.io/math2/node/VT8SFWT0XW.md): concept VT8SFWT0XW
- [Grunsky matrix](https://tamnd.github.io/math2/node/VT98YJJJ47.md): definition VT98YJJJ47
- [Normalization](https://tamnd.github.io/math2/node/VT9QWDTKA4.md): definition VT9QWDTKA4
- [Cantor algebra](https://tamnd.github.io/math2/node/VTE2H1GT4A.md): example VTE2H1GT4A
- [Tensor algebra](https://tamnd.github.io/math2/node/VTFQYCP1WA.md): definition VTFQYCP1WA
- [Isolated hypersurface singularity](https://tamnd.github.io/math2/node/VTGYEES3VS.md): concept VTGYEES3VS
- [Schröder's equation](https://tamnd.github.io/math2/node/VTJV7JVJEA.md): concept VTJV7JVJEA
- [Kodaira–Nirenberg–Spencer existence theorem](https://tamnd.github.io/math2/node/VTN8JHGDS5.md): statement VTN8JHGDS5
- [Limit](https://tamnd.github.io/math2/node/VTWSNXB5S7.md): definition VTWSNXB5S7
- [Character sum](https://tamnd.github.io/math2/node/VTYJ9419C7.md): concept VTYJ9419C7
- [Empirical process](https://tamnd.github.io/math2/node/VTZPSW07PQ.md): definition VTZPSW07PQ
- [q-binomial theorem](https://tamnd.github.io/math2/node/VVT27RB46B.md): statement VVT27RB46B
- [Tricomi confluent hypergeometric function](https://tamnd.github.io/math2/node/VVVJ655R0J.md): definition VVVJ655R0J
- [Amitsur's theorem on the radical of polynomial rings](https://tamnd.github.io/math2/node/VW2H7873YM.md): statement VW2H7873YM
- [Perfect field](https://tamnd.github.io/math2/node/VWBVKT794W.md): concept VWBVKT794W
- [Exponential type](https://tamnd.github.io/math2/node/VWDS9PM9V9.md): definition VWDS9PM9V9
- [Convenient vector space](https://tamnd.github.io/math2/node/VWFVT9AGCT.md): definition VWFVT9AGCT
- [Binomial proportion confidence interval](https://tamnd.github.io/math2/node/VWKWYB43A3.md): definition VWKWYB43A3
- [Gelfand transform](https://tamnd.github.io/math2/node/VWNSHKJT92.md): concept VWNSHKJT92
- [Routh's theorem](https://tamnd.github.io/math2/node/VWQQY7NFED.md): statement VWQQY7NFED
- [Holonomy](https://tamnd.github.io/math2/node/VWTCEKMNKR.md): definition VWTCEKMNKR
- [Tensor decomposition](https://tamnd.github.io/math2/node/VWV7TA2AJX.md): concept VWV7TA2AJX
- [Comma category](https://tamnd.github.io/math2/node/VWYXK7E49K.md): definition VWYXK7E49K
- [Hardy–Littlewood maximal function](https://tamnd.github.io/math2/node/VX1A56P5KQ.md): concept VX1A56P5KQ
- [Interpolation space](https://tamnd.github.io/math2/node/VX1R6FN6PS.md): concept VX1R6FN6PS
- [Strong operator topology](https://tamnd.github.io/math2/node/VX7BFH22GX.md): definition VX7BFH22GX
- [Kigami's construction of Laplacians on post-critically finite fractals](https://tamnd.github.io/math2/node/VXB4G47Z3T.md): statement VXB4G47Z3T
- [Exact couple](https://tamnd.github.io/math2/node/VXDPP4EJPG.md): concept VXDPP4EJPG
- [Baskakov operator](https://tamnd.github.io/math2/node/VXQ1Y50KTJ.md): concept VXQ1Y50KTJ
- [Unknotting problem](https://tamnd.github.io/math2/node/VXSD46NRX0.md): problem VXSD46NRX0
- [Stable distribution](https://tamnd.github.io/math2/node/VXVKHS137K.md): definition VXVKHS137K
- [Haar measure](https://tamnd.github.io/math2/node/VY0HXD3GC4.md): concept VY0HXD3GC4
- [Exponential stability](https://tamnd.github.io/math2/node/VY180V8QY2.md): concept VY180V8QY2
- [Spherical code](https://tamnd.github.io/math2/node/VY2TTB854G.md): definition VY2TTB854G
- [Lax–Friedrichs method](https://tamnd.github.io/math2/node/VY3J2NKGKW.md): concept VY3J2NKGKW
- [Boolean algebras canonically defined](https://tamnd.github.io/math2/node/VY46T0A6VQ.md): concept VY46T0A6VQ
- [Synthetic geometry](https://tamnd.github.io/math2/node/VY6ZCZY8DA.md): concept VY6ZCZY8DA
- [Wilson–Cowan neural field equation](https://tamnd.github.io/math2/node/VYB03ZZ8H6.md): example VYB03ZZ8H6
- [CUSUM](https://tamnd.github.io/math2/node/VYM8AJ8111.md): definition VYM8AJ8111
- [G-structure](https://tamnd.github.io/math2/node/VYNQVZ8XVF.md): concept VYNQVZ8XVF
- [Hausdorff dimension](https://tamnd.github.io/math2/node/VYP60TA2EC.md): definition VYP60TA2EC
- [Macdonald polynomial](https://tamnd.github.io/math2/node/VYP9PB000X.md): concept VYP9PB000X
- [Graph minor](https://tamnd.github.io/math2/node/VYQ32ZBPEE.md): concept VYQ32ZBPEE
- [Function field of a variety](https://tamnd.github.io/math2/node/VYQATJQNBH.md): definition VYQATJQNBH
- [Trefoil knot](https://tamnd.github.io/math2/node/VYTYSVHN4A.md): example VYTYSVHN4A
- [Transfer homomorphism](https://tamnd.github.io/math2/node/VZ1G9CYQSK.md): definition VZ1G9CYQSK
- [Akra–Bazzi theorem](https://tamnd.github.io/math2/node/VZ1J7KM4QS.md): statement VZ1J7KM4QS
- [Herbrand's theorem](https://tamnd.github.io/math2/node/VZ21F10WRS.md): statement VZ21F10WRS
- [Whittaker transform](https://tamnd.github.io/math2/node/VZ3BHDB5QZ.md): concept VZ3BHDB5QZ
- [Pseudoanalytic function](https://tamnd.github.io/math2/node/VZ5QQMDV2D.md): definition VZ5QQMDV2D
- [Ehresmann connection](https://tamnd.github.io/math2/node/VZBYXH5995.md): definition VZBYXH5995
- [Medial magma](https://tamnd.github.io/math2/node/VZHN6GA3TA.md): concept VZHN6GA3TA
- [Boolean product](https://tamnd.github.io/math2/node/VZHVJ3V0EH.md): definition VZHVJ3V0EH
- [Cascade algorithm](https://tamnd.github.io/math2/node/VZK6ZAXWJR.md): concept VZK6ZAXWJR
- [Germ](https://tamnd.github.io/math2/node/VZXF7RNFKS.md): concept VZXF7RNFKS
- [Quasi-compact morphism](https://tamnd.github.io/math2/node/W007WWX7AN.md): definition W007WWX7AN
- [Lévy's upward theorem](https://tamnd.github.io/math2/node/W01FD2A8NK.md): statement W01FD2A8NK
- [Canonical bundle](https://tamnd.github.io/math2/node/W05VYY4HBR.md): concept W05VYY4HBR
- [Well-posed problem](https://tamnd.github.io/math2/node/W09B9NZ9B5.md): concept W09B9NZ9B5
- [Universally closed morphism](https://tamnd.github.io/math2/node/W0B0BBAH35.md): concept W0B0BBAH35
- [Whittle likelihood](https://tamnd.github.io/math2/node/W0B5CQVV8V.md): concept W0B5CQVV8V
- [Helmholtz equation](https://tamnd.github.io/math2/node/W0F3YWAJ23.md): definition W0F3YWAJ23
- [Projective subspace](https://tamnd.github.io/math2/node/W0J2G7RKBY.md): definition W0J2G7RKBY
- [Toyoda–Bruck theorem](https://tamnd.github.io/math2/node/W0KBNZBCRR.md): statement W0KBNZBCRR
- [Quartic reciprocity](https://tamnd.github.io/math2/node/W0M20JWC9K.md): statement W0M20JWC9K
- [Schwarz's list](https://tamnd.github.io/math2/node/W0MJ51242N.md): statement W0MJ51242N
- [Pappus configuration](https://tamnd.github.io/math2/node/W0PAJ23MQ0.md): example W0PAJ23MQ0
- [Free spectrum of a variety](https://tamnd.github.io/math2/node/W0RYRVJY5Z.md): concept W0RYRVJY5Z
- [Nonassociative ring](https://tamnd.github.io/math2/node/W0VD6EAHDA.md): definition W0VD6EAHDA
- [Cup product](https://tamnd.github.io/math2/node/W10ERK1ADE.md): definition W10ERK1ADE
- [Superstability of the exponential equation](https://tamnd.github.io/math2/node/W11JCMFK4D.md): statement W11JCMFK4D
- [Completion of a valued field](https://tamnd.github.io/math2/node/W168DS966V.md): concept W168DS966V
- [Mackey topology](https://tamnd.github.io/math2/node/W18EY4E5TH.md): concept W18EY4E5TH
- [De Finetti's theorem](https://tamnd.github.io/math2/node/W19129PVKQ.md): statement W19129PVKQ
- [Poisson kernel](https://tamnd.github.io/math2/node/W19567AVYG.md): definition W19567AVYG
- [Briançon–Skoda theorem](https://tamnd.github.io/math2/node/W1C0W5MCX0.md): statement W1C0W5MCX0
- [Inertial manifold](https://tamnd.github.io/math2/node/W1F5C7RZ0M.md): concept W1F5C7RZ0M
- [Doubling measure](https://tamnd.github.io/math2/node/W1JXH6NCKX.md): concept W1JXH6NCKX
- [Coproduct](https://tamnd.github.io/math2/node/W1VV1MP5EG.md): definition W1VV1MP5EG
- [Properly discontinuous action](https://tamnd.github.io/math2/node/W1VVPNBPMX.md): definition W1VVPNBPMX
- [Wreath product](https://tamnd.github.io/math2/node/W1X984PKJ9.md): concept W1X984PKJ9
- [Harris inequality](https://tamnd.github.io/math2/node/W1YH3YD3S4.md): statement W1YH3YD3S4
- [Finitely generated algebra](https://tamnd.github.io/math2/node/W251JWP6PS.md): definition W251JWP6PS
- [Feynman–Kac formula](https://tamnd.github.io/math2/node/W26MGR4DA3.md): statement W26MGR4DA3
- [Serre's criterion for normality](https://tamnd.github.io/math2/node/W27Z60Q1PF.md): statement W27Z60Q1PF
- [Hopkins–Levitzki theorem](https://tamnd.github.io/math2/node/W29AA0266Z.md): statement W29AA0266Z
- [Intermittency](https://tamnd.github.io/math2/node/W2FNJ4KBYT.md): concept W2FNJ4KBYT
- [Fine topology](https://tamnd.github.io/math2/node/W2ZD4XP74R.md): concept W2ZD4XP74R
- [Dunford–Pettis property](https://tamnd.github.io/math2/node/W35ZP39S4R.md): definition W35ZP39S4R
- [Gelfand pair](https://tamnd.github.io/math2/node/W38ZDXDVYC.md): concept W38ZDXDVYC
- [Exceptional collection](https://tamnd.github.io/math2/node/W3B4XH5JHZ.md): definition W3B4XH5JHZ
- [Consistency of summability methods](https://tamnd.github.io/math2/node/W3BVN1HDXW.md): concept W3BVN1HDXW
- [Hilbert–Schmidt operator](https://tamnd.github.io/math2/node/W3BX66Y47G.md): concept W3BX66Y47G
- [Dandelin spheres](https://tamnd.github.io/math2/node/W3DCQ22JA3.md): concept W3DCQ22JA3
- [Robin constant](https://tamnd.github.io/math2/node/W3NVXBEMRQ.md): concept W3NVXBEMRQ
- [Riesz transform](https://tamnd.github.io/math2/node/W3TAPJEX29.md): definition W3TAPJEX29
- [Matrix norm](https://tamnd.github.io/math2/node/W3V0NW9P6J.md): definition W3V0NW9P6J
- [Fundamental theorem of affine geometry](https://tamnd.github.io/math2/node/W3YYX63R9R.md): statement W3YYX63R9R
- [Jacobi elliptic functions](https://tamnd.github.io/math2/node/W43SYCP85W.md): definition W43SYCP85W
- [Elementary theory of the category of sets](https://tamnd.github.io/math2/node/W448SK1RTY.md): concept W448SK1RTY
- [Proof of Lagrange's theorem](https://tamnd.github.io/math2/node/W44XETYC4Q.md): proof W44XETYC4Q
- [Coherent ring](https://tamnd.github.io/math2/node/W453BM8T6P.md): concept W453BM8T6P
- [Fixed point](https://tamnd.github.io/math2/node/W46BP9W06H.md): definition W46BP9W06H
- [Class](https://tamnd.github.io/math2/node/W473FH7Z0P.md): definition W473FH7Z0P
- [Jakobson's theorem](https://tamnd.github.io/math2/node/W49VGF1WDE.md): statement W49VGF1WDE
- [Elliptic operator on a compact manifold](https://tamnd.github.io/math2/node/W4B84CGEG1.md): definition W4B84CGEG1
- [Cellular homology](https://tamnd.github.io/math2/node/W4DG4WCQ5W.md): concept W4DG4WCQ5W
- [Schmidt's singular value decomposition of a kernel](https://tamnd.github.io/math2/node/W4E25FVXV1.md): statement W4E25FVXV1
- [Monte Carlo method](https://tamnd.github.io/math2/node/W4MT7K2QTF.md): definition W4MT7K2QTF
- [Normal complex space](https://tamnd.github.io/math2/node/W4TT4W95GR.md): definition W4TT4W95GR
- [Split exact sequence](https://tamnd.github.io/math2/node/W57F9ENPMV.md): definition W57F9ENPMV
- [Reeb foliation](https://tamnd.github.io/math2/node/W58NYR1DGR.md): example W58NYR1DGR
- [Packing measure](https://tamnd.github.io/math2/node/W59B3PKZPP.md): concept W59B3PKZPP
- [Lusternik–Schnirelmann category](https://tamnd.github.io/math2/node/W5BHHXCYXA.md): concept W5BHHXCYXA
- [Balog–Szemerédi–Gowers theorem](https://tamnd.github.io/math2/node/W5BZZ59CVM.md): statement W5BZZ59CVM
- [Canonical module](https://tamnd.github.io/math2/node/W5ECXPM6FW.md): concept W5ECXPM6FW
- [Symplectic vector space](https://tamnd.github.io/math2/node/W5FHT07SJJ.md): definition W5FHT07SJJ
- [Isospectral manifolds](https://tamnd.github.io/math2/node/W5HF4GMVN9.md): definition W5HF4GMVN9
- [Zero-dimensional space](https://tamnd.github.io/math2/node/W5S52WZZ9T.md): definition W5S52WZZ9T
- [Natural duality](https://tamnd.github.io/math2/node/W5VG6HS8KW.md): concept W5VG6HS8KW
- [Cofibration](https://tamnd.github.io/math2/node/W5WS5EY8ZM.md): concept W5WS5EY8ZM
- [Submultiplicative function](https://tamnd.github.io/math2/node/W5YGV8P9F2.md): definition W5YGV8P9F2
- [Existence of transcendence bases](https://tamnd.github.io/math2/node/W635VBXEXG.md): statement W635VBXEXG
- [Five lemma](https://tamnd.github.io/math2/node/W6AVEAW2WR.md): statement W6AVEAW2WR
- [Euler summation](https://tamnd.github.io/math2/node/W6BRWTB07W.md): definition W6BRWTB07W
- [Homogeneous integral equation](https://tamnd.github.io/math2/node/W6CYS54QXS.md): concept W6CYS54QXS
- [Tennenbaum's theorem](https://tamnd.github.io/math2/node/W6D5ZJAY9D.md): statement W6D5ZJAY9D
- [G-signature theorem](https://tamnd.github.io/math2/node/W6DAWQ6H48.md): statement W6DAWQ6H48
- [Kummer's equation](https://tamnd.github.io/math2/node/W6E15N91Q3.md): definition W6E15N91Q3
- [Prüfer domain](https://tamnd.github.io/math2/node/W6E6M1QYH5.md): definition W6E6M1QYH5
- [Turing machine](https://tamnd.github.io/math2/node/W6ESDYBF8Z.md): definition W6ESDYBF8Z
- [Meromorphic function of several variables](https://tamnd.github.io/math2/node/W6FTSF57T1.md): concept W6FTSF57T1
- [d'Alembert's equation](https://tamnd.github.io/math2/node/W6GG3FJMNQ.md): example W6GG3FJMNQ
- [Composition operator](https://tamnd.github.io/math2/node/W6GV82MMQ7.md): definition W6GV82MMQ7
- [Quaternion-Kähler manifold](https://tamnd.github.io/math2/node/W6J4MQ6D5N.md): definition W6J4MQ6D5N
- [Schrödinger operator](https://tamnd.github.io/math2/node/W6QS1ME39J.md): definition W6QS1ME39J
- [Caccioppoli set](https://tamnd.github.io/math2/node/W6R7QNEAF1.md): concept W6R7QNEAF1
- [Cerf theory](https://tamnd.github.io/math2/node/W6SSJN7TVW.md): concept W6SSJN7TVW
- [Kellogg–Evans theorem](https://tamnd.github.io/math2/node/W6SXM59TS4.md): statement W6SXM59TS4
- [Chow group](https://tamnd.github.io/math2/node/W6V340HYNK.md): definition W6V340HYNK
- [Legendre's equation](https://tamnd.github.io/math2/node/W6VE30V10Y.md): example W6VE30V10Y
- [Airfoil equation](https://tamnd.github.io/math2/node/W6Z6CWRE25.md): example W6Z6CWRE25
- [Sequent calculus](https://tamnd.github.io/math2/node/W72CA3R9MP.md): concept W72CA3R9MP
- [Dirichlet-to-Neumann operator](https://tamnd.github.io/math2/node/W783KDGT8E.md): concept W783KDGT8E
- [Neural network approximation](https://tamnd.github.io/math2/node/W7CK84MZFV.md): concept W7CK84MZFV
- [Matrix function](https://tamnd.github.io/math2/node/W7MPTRPXYD.md): concept W7MPTRPXYD
- [Unary operation](https://tamnd.github.io/math2/node/W7NFRDY0GA.md): definition W7NFRDY0GA
- [Baum–Welch algorithm](https://tamnd.github.io/math2/node/W7P6CCSMX7.md): concept W7P6CCSMX7
- [Eichler order](https://tamnd.github.io/math2/node/W7PYDJZ8BC.md): concept W7PYDJZ8BC
- [MUSCL scheme](https://tamnd.github.io/math2/node/W7QH1BPBS8.md): concept W7QH1BPBS8
- [Stiefel manifold](https://tamnd.github.io/math2/node/W7RCXS3CHT.md): example W7RCXS3CHT
- [Sturm comparison theorem](https://tamnd.github.io/math2/node/W7VXNYZQN2.md): statement W7VXNYZQN2
- [Limiting subdifferential](https://tamnd.github.io/math2/node/W7WDXEQH65.md): concept W7WDXEQH65
- [Universal algebra](https://tamnd.github.io/math2/node/W82R6N0GM9.md): concept W82R6N0GM9
- [Finite lattice representation problem](https://tamnd.github.io/math2/node/W83D78WN7T.md): problem W83D78WN7T
- [Tensor rank](https://tamnd.github.io/math2/node/W84NTF77M0.md): concept W84NTF77M0
- [Dehn twist](https://tamnd.github.io/math2/node/W890NRHGS4.md): concept W890NRHGS4
- [Linear extension](https://tamnd.github.io/math2/node/W8DZ5KGGHN.md): concept W8DZ5KGGHN
- [False discovery rate and the Benjamini–Hochberg procedure](https://tamnd.github.io/math2/node/W8DZQZQTJA.md): definition W8DZQZQTJA
- [Tension field](https://tamnd.github.io/math2/node/W8F8S8DX5X.md): concept W8F8S8DX5X
- [Nemytskii operator](https://tamnd.github.io/math2/node/W8G3AZ0DNG.md): concept W8G3AZ0DNG
- [Lambert summation](https://tamnd.github.io/math2/node/W8H1WX0JHG.md): concept W8H1WX0JHG
- [Impossibility of the classical construction problems](https://tamnd.github.io/math2/node/W8J42M1G28.md): statement W8J42M1G28
- [H-representation and V-representation](https://tamnd.github.io/math2/node/W8P2G5R02N.md): concept W8P2G5R02N
- [Prime ideals of a localization](https://tamnd.github.io/math2/node/W8QNZZ4QJS.md): statement W8QNZZ4QJS
- [Particle filter](https://tamnd.github.io/math2/node/W8R4A5D4N8.md): definition W8R4A5D4N8
- [Sinai–Ruelle–Bowen measure](https://tamnd.github.io/math2/node/W8TX2H4JV7.md): definition W8TX2H4JV7
- [Hypergeometric differential equation](https://tamnd.github.io/math2/node/W8VE25R5A7.md): concept W8VE25R5A7
- [Weak derivative](https://tamnd.github.io/math2/node/W8W9AP79V5.md): definition W8W9AP79V5
- [Tube lemma](https://tamnd.github.io/math2/node/W8XEZR3VKQ.md): statement W8XEZR3VKQ
- [Term operation](https://tamnd.github.io/math2/node/W8XWKYWV1D.md): definition W8XWKYWV1D
- [Rees algebra](https://tamnd.github.io/math2/node/W8ZAH8TTHS.md): concept W8ZAH8TTHS
- [Frobenius method](https://tamnd.github.io/math2/node/W8ZWXC2HQS.md): concept W8ZWXC2HQS
- [Stein equation](https://tamnd.github.io/math2/node/W90XPR88E8.md): definition W90XPR88E8
- [Completely distributive lattice](https://tamnd.github.io/math2/node/W91HFP63SN.md): definition W91HFP63SN
- [Hamiltonian vector field](https://tamnd.github.io/math2/node/W91M4JS396.md): definition W91M4JS396
- [Enumerative geometry](https://tamnd.github.io/math2/node/W94AEJV9NK.md): definition W94AEJV9NK
- [Besicovitch set of measure zero](https://tamnd.github.io/math2/node/W94NZR7AR1.md): example W94NZR7AR1
- [Kaplan–Meier estimator](https://tamnd.github.io/math2/node/W94W93RHF5.md): definition W94W93RHF5
- [Stopping time](https://tamnd.github.io/math2/node/W96WPDEA6A.md): definition W96WPDEA6A
- [Nielsen–Schreier theorem](https://tamnd.github.io/math2/node/W97X7WX5G8.md): statement W97X7WX5G8
- [Newton–Cotes formulas](https://tamnd.github.io/math2/node/W9ARPZGPM6.md): definition W9ARPZGPM6
- [Néron model](https://tamnd.github.io/math2/node/W9DXVGAE3R.md): definition W9DXVGAE3R
- [Sets of capacity zero](https://tamnd.github.io/math2/node/W9N1TR1X8P.md): concept W9N1TR1X8P
- [End and coend](https://tamnd.github.io/math2/node/W9QTFV0Z8D.md): concept W9QTFV0Z8D
- [Laplacian on the Sierpiński gasket](https://tamnd.github.io/math2/node/W9V9JSAWYK.md): example W9V9JSAWYK
- [Geometric graph](https://tamnd.github.io/math2/node/W9VAD8QYZF.md): definition W9VAD8QYZF
- [Cantor space](https://tamnd.github.io/math2/node/W9VYJRBF9N.md): object W9VYJRBF9N
- [Kelvin transform](https://tamnd.github.io/math2/node/W9YR470VY2.md): concept W9YR470VY2
- [Sieve](https://tamnd.github.io/math2/node/WA0NAANTAH.md): definition WA0NAANTAH
- [Reinhardt domain](https://tamnd.github.io/math2/node/WA0SE56FVH.md): concept WA0SE56FVH
- [Differential dynamic programming](https://tamnd.github.io/math2/node/WA3974X06N.md): definition WA3974X06N
- [de Rham cohomology](https://tamnd.github.io/math2/node/WA3M3RZDQA.md): definition WA3M3RZDQA
- [Fully faithful essentially surjective functor is an equivalence](https://tamnd.github.io/math2/node/WA60RW9E04.md): statement WA60RW9E04
- [Sidon set](https://tamnd.github.io/math2/node/WA74AY36GC.md): definition WA74AY36GC
- [Classifying space of a group](https://tamnd.github.io/math2/node/WA796ZCNRX.md): concept WA796ZCNRX
- [Cameron–Martin theorem](https://tamnd.github.io/math2/node/WAE4ZGJXJ7.md): statement WAE4ZGJXJ7
- [Jordan decomposition of a signed measure](https://tamnd.github.io/math2/node/WAGJVK2NWE.md): statement WAGJVK2NWE
- [Laguerre polynomials](https://tamnd.github.io/math2/node/WAMRHWW82R.md): concept WAMRHWW82R
- [Many-sorted algebra](https://tamnd.github.io/math2/node/WAQJJ5GX68.md): concept WAQJJ5GX68
- [Kohn Laplacian](https://tamnd.github.io/math2/node/WAS3TN7VKD.md): definition WAS3TN7VKD
- [Random walk on a group](https://tamnd.github.io/math2/node/WAS9EANSJJ.md): concept WAS9EANSJJ
- [Eilenberg–MacLane space](https://tamnd.github.io/math2/node/WAS9NHKMYD.md): definition WAS9NHKMYD
- [Range conditions for the Radon transform](https://tamnd.github.io/math2/node/WASXFTTKK6.md): statement WASXFTTKK6
- [Neyman construction](https://tamnd.github.io/math2/node/WB25VJQEXJ.md): definition WB25VJQEXJ
- [Orlicz space](https://tamnd.github.io/math2/node/WBCN9A847H.md): concept WBCN9A847H
- [Delzant's theorem on toric manifolds](https://tamnd.github.io/math2/node/WBKCGFMT0P.md): statement WBKCGFMT0P
- [Gamma function has no zeros](https://tamnd.github.io/math2/node/WBS4NM1TFC.md): statement WBS4NM1TFC
- [Hyperbolic conservation law](https://tamnd.github.io/math2/node/WBYJ7A0HY7.md): definition WBYJ7A0HY7
- [Commutator theory](https://tamnd.github.io/math2/node/WBYTAFMWJR.md): definition WBYTAFMWJR
- [Smooth manifold](https://tamnd.github.io/math2/node/WBZHXGKNSJ.md): definition WBZHXGKNSJ
- [Wilf–Zeilberger pair](https://tamnd.github.io/math2/node/WC0QNABPZ4.md): concept WC0QNABPZ4
- [Tits group](https://tamnd.github.io/math2/node/WC16TBY3FN.md): object WC16TBY3FN
- [Ternary operation](https://tamnd.github.io/math2/node/WC47MFR6GW.md): concept WC47MFR6GW
- [Penny graph](https://tamnd.github.io/math2/node/WC4QAGSMWR.md): definition WC4QAGSMWR
- [Axiom of determinacy](https://tamnd.github.io/math2/node/WC5JZVG82R.md): definition WC5JZVG82R
- [Hyperelliptic curve](https://tamnd.github.io/math2/node/WC8M1YAYCD.md): definition WC8M1YAYCD
- [Sparse approximation](https://tamnd.github.io/math2/node/WC8P971XGW.md): concept WC8P971XGW
- [Nonlinear difference equation](https://tamnd.github.io/math2/node/WC92BCCFHV.md): definition WC92BCCFHV
- [Invariant basis number](https://tamnd.github.io/math2/node/WC9WZ53ZNS.md): definition WC9WZ53ZNS
- [Wielandt's theorem](https://tamnd.github.io/math2/node/WCM87QEEZW.md): statement WCM87QEEZW
- [Complex multiplication](https://tamnd.github.io/math2/node/WCMAN2AJYR.md): definition WCMAN2AJYR
- [Laplace transform method for the heat equation](https://tamnd.github.io/math2/node/WCSZYPBZY4.md): example WCSZYPBZY4
- [Azumaya algebra](https://tamnd.github.io/math2/node/WCX24NTZ8X.md): definition WCX24NTZ8X
- [Weyl tensor](https://tamnd.github.io/math2/node/WD15XKXW46.md): definition WD15XKXW46
- [Outer measure](https://tamnd.github.io/math2/node/WD1AJ2X325.md): definition WD1AJ2X325
- [Indecomposable module](https://tamnd.github.io/math2/node/WD3G29FZ98.md): definition WD3G29FZ98
- [Extreme set](https://tamnd.github.io/math2/node/WD3RJZC5A7.md): concept WD3RJZC5A7
- [Complex measure](https://tamnd.github.io/math2/node/WDFHKDCQPV.md): definition WDFHKDCQPV
- [Space of null sequences](https://tamnd.github.io/math2/node/WDH6QSH1Z6.md): example WDH6QSH1Z6
- [Jacobi amplitude](https://tamnd.github.io/math2/node/WDHHJHVP7N.md): concept WDHHJHVP7N
- [Birkhoff's completeness theorem](https://tamnd.github.io/math2/node/WDHWCQ6Y2Y.md): statement WDHWCQ6Y2Y
- [Harmonic measure as exit distribution](https://tamnd.github.io/math2/node/WDJSNMG1EZ.md): statement WDJSNMG1EZ
- [Matrix function](https://tamnd.github.io/math2/node/WDQTBSK6ES.md): definition WDQTBSK6ES
- [Signature](https://tamnd.github.io/math2/node/WDVVEYD6AT.md): concept WDVVEYD6AT
- [Quadratic polynomial dynamics](https://tamnd.github.io/math2/node/WDZFV2ACXR.md): concept WDZFV2ACXR
- [Chen's theorem](https://tamnd.github.io/math2/node/WE0RQTR68H.md): statement WE0RQTR68H
- [Van der Pol oscillator](https://tamnd.github.io/math2/node/WE89NFZ84Q.md): example WE89NFZ84Q
- [Sigma-finite measure](https://tamnd.github.io/math2/node/WEBBT18D4X.md): concept WEBBT18D4X
- [Generic filter](https://tamnd.github.io/math2/node/WEDP73Y7GK.md): concept WEDP73Y7GK
- [Fundamental domain](https://tamnd.github.io/math2/node/WEGEXJBX26.md): definition WEGEXJBX26
- [Cobordism ring](https://tamnd.github.io/math2/node/WEGT4EZCX2.md): definition WEGT4EZCX2
- [Hypersingular integral equation](https://tamnd.github.io/math2/node/WEHG6JZDFY.md): definition WEHG6JZDFY
- [Hartley transform](https://tamnd.github.io/math2/node/WEHYECYAVJ.md): concept WEHYECYAVJ
- [Axiom of dependent choice](https://tamnd.github.io/math2/node/WEMJBN339R.md): concept WEMJBN339R
- [Natural boundary](https://tamnd.github.io/math2/node/WER1M071EK.md): definition WER1M071EK
- [Aitken's delta-squared process](https://tamnd.github.io/math2/node/WF08BMN6Q6.md): concept WF08BMN6Q6
- [Prime radical](https://tamnd.github.io/math2/node/WF117EJ2HF.md): concept WF117EJ2HF
- [Stable manifold theorem](https://tamnd.github.io/math2/node/WF22KK7C3R.md): statement WF22KK7C3R
- [Reflection formula for the gamma function](https://tamnd.github.io/math2/node/WF7KQAQTS2.md): statement WF7KQAQTS2
- [Rate function](https://tamnd.github.io/math2/node/WF95J6ZXV3.md): definition WF95J6ZXV3
- [Jacobson topology](https://tamnd.github.io/math2/node/WFFGB4RK60.md): concept WFFGB4RK60
- [Hilbert symbol](https://tamnd.github.io/math2/node/WFG4YMBYVQ.md): concept WFG4YMBYVQ
- [Bateman transform](https://tamnd.github.io/math2/node/WFJ4TVF15P.md): definition WFJ4TVF15P
- [Anderson localization](https://tamnd.github.io/math2/node/WFPYZ0XDRN.md): definition WFPYZ0XDRN
- [Mehler–Fock transform](https://tamnd.github.io/math2/node/WFQV4R1EMN.md): concept WFQV4R1EMN
- [Riemann surface](https://tamnd.github.io/math2/node/WFREP1QRPQ.md): definition WFREP1QRPQ
- [Periodic tiling](https://tamnd.github.io/math2/node/WFRGSBFZGV.md): concept WFRGSBFZGV
- [Necklace splitting theorem](https://tamnd.github.io/math2/node/WFS3PHFHA1.md): statement WFS3PHFHA1
- [Infinitary operation](https://tamnd.github.io/math2/node/WFVC137TYB.md): concept WFVC137TYB
- [Unique solvability of Volterra equations of the second kind](https://tamnd.github.io/math2/node/WG4GVC9HR8.md): statement WG4GVC9HR8
- [Positive cone](https://tamnd.github.io/math2/node/WG6BY4SWTV.md): definition WG6BY4SWTV
- [Post's functional completeness theorem](https://tamnd.github.io/math2/node/WGBR7JHBV4.md): statement WGBR7JHBV4
- [Dual Radon transform](https://tamnd.github.io/math2/node/WGDAZ1YY3J.md): definition WGDAZ1YY3J
- [Lüroth problem](https://tamnd.github.io/math2/node/WGGZEKF224.md): problem WGGZEKF224
- [p-adic absolute value](https://tamnd.github.io/math2/node/WGN82TCEF2.md): concept WGN82TCEF2
- [Titchmarsh convolution theorem](https://tamnd.github.io/math2/node/WGY75H4VWW.md): statement WGY75H4VWW
- [Projective cover](https://tamnd.github.io/math2/node/WGYXR97NPK.md): concept WGYXR97NPK
- [Order polynomial](https://tamnd.github.io/math2/node/WH10W2GWZX.md): definition WH10W2GWZX
- [Inner model](https://tamnd.github.io/math2/node/WH1MR8VHE3.md): definition WH1MR8VHE3
- [Completeness in order theory](https://tamnd.github.io/math2/node/WH3Z5M5D2R.md): definition WH3Z5M5D2R
- [Internal language](https://tamnd.github.io/math2/node/WH48X83YPJ.md): concept WH48X83YPJ
- [Lattice](https://tamnd.github.io/math2/node/WH4SFAACAS.md): concept WH4SFAACAS
- [Holonomy](https://tamnd.github.io/math2/node/WHDZ3MK56B.md): concept WHDZ3MK56B
- [Integral current](https://tamnd.github.io/math2/node/WHHMJERDFG.md): definition WHHMJERDFG
- [Quadratic functional equation](https://tamnd.github.io/math2/node/WHK5B8X2CQ.md): definition WHK5B8X2CQ
- [Midpoint method](https://tamnd.github.io/math2/node/WHK8D383YT.md): statement WHK8D383YT
- [Filtration of a module](https://tamnd.github.io/math2/node/WHME2NGANG.md): concept WHME2NGANG
- [Hadamard three-lines theorem](https://tamnd.github.io/math2/node/WHS58R43N0.md): statement WHS58R43N0
- [Hamilton–Jacobi–Isaacs equation](https://tamnd.github.io/math2/node/WHTBZ3K4FB.md): concept WHTBZ3K4FB
- [Congruence lattice problem](https://tamnd.github.io/math2/node/WJ0D8QAMW0.md): problem WJ0D8QAMW0
- [Castelnuovo's contraction theorem](https://tamnd.github.io/math2/node/WJ5WRGG5DE.md): statement WJ5WRGG5DE
- [Direct product of algebras](https://tamnd.github.io/math2/node/WJ73XYBNE4.md): concept WJ73XYBNE4
- [Hilbert sum](https://tamnd.github.io/math2/node/WJ7PDMGQQW.md): concept WJ7PDMGQQW
- [Nakano positivity](https://tamnd.github.io/math2/node/WJ8TRE4F1K.md): concept WJ8TRE4F1K
- [Parabolic partial differential equation](https://tamnd.github.io/math2/node/WJGY7XG3QR.md): definition WJGY7XG3QR
- [Provability logic](https://tamnd.github.io/math2/node/WJK0YA4GP2.md): concept WJK0YA4GP2
- [Calabi conjecture](https://tamnd.github.io/math2/node/WJKA5YBTJ7.md): statement WJKA5YBTJ7
- [Lebesgue's lemma on near-best approximation](https://tamnd.github.io/math2/node/WJKS606MDF.md): statement WJKS606MDF
- [Morphism of varieties](https://tamnd.github.io/math2/node/WJQ2C6SKX5.md): definition WJQ2C6SKX5
- [Local ring](https://tamnd.github.io/math2/node/WJS9TPSGVM.md): definition WJS9TPSGVM
- [Matrix congruence](https://tamnd.github.io/math2/node/WJWNYSQKK8.md): concept WJWNYSQKK8
- [Hopf algebra structure on cohomology](https://tamnd.github.io/math2/node/WJXK0V9NE9.md): concept WJXK0V9NE9
- [Simplicial object](https://tamnd.github.io/math2/node/WK0A32TZHB.md): concept WK0A32TZHB
- [Commuting elements](https://tamnd.github.io/math2/node/WK2FC0NM26.md): definition WK2FC0NM26
- [Smoothness classes](https://tamnd.github.io/math2/node/WK5HPW5Y24.md): concept WK5HPW5Y24
- [Double sequence](https://tamnd.github.io/math2/node/WK5V372TG3.md): definition WK5V372TG3
- [Envelope theorem](https://tamnd.github.io/math2/node/WK7MKDWRFF.md): statement WK7MKDWRFF
- [Intersection form](https://tamnd.github.io/math2/node/WK8GKRKAVD.md): concept WK8GKRKAVD
- [Coherent analytic sheaf](https://tamnd.github.io/math2/node/WKD2S2XA9C.md): definition WKD2S2XA9C
- [Nil ideal](https://tamnd.github.io/math2/node/WKG2DZDEBT.md): definition WKG2DZDEBT
- [Alexander polynomial](https://tamnd.github.io/math2/node/WKJ4EP6E1X.md): definition WKJ4EP6E1X
- [Chattering control](https://tamnd.github.io/math2/node/WKK2DC7JE9.md): definition WKK2DC7JE9
- [Fredholm minors](https://tamnd.github.io/math2/node/WKM1B317YB.md): definition WKM1B317YB
- [Unirational variety](https://tamnd.github.io/math2/node/WKYG28Z8ER.md): definition WKYG28Z8ER
- [Pseudocompact space](https://tamnd.github.io/math2/node/WM4QWH5F65.md): definition WM4QWH5F65
- [Gamma of one half](https://tamnd.github.io/math2/node/WM5FNBB7K0.md): statement WM5FNBB7K0
- [Nonstandard analysis](https://tamnd.github.io/math2/node/WM62GKCBWN.md): concept WM62GKCBWN
- [Sequential space](https://tamnd.github.io/math2/node/WMB2MAEDX4.md): concept WMB2MAEDX4
- [Baker's theorem](https://tamnd.github.io/math2/node/WMBJSNG57F.md): statement WMBJSNG57F
- [Corona theorem](https://tamnd.github.io/math2/node/WMGA92P2G4.md): statement WMGA92P2G4
- [Luzin N property](https://tamnd.github.io/math2/node/WMHMGJD261.md): concept WMHMGJD261
- [Regular category](https://tamnd.github.io/math2/node/WMNEQ6A8FB.md): definition WMNEQ6A8FB
- [Weak and strong extremum](https://tamnd.github.io/math2/node/WMPCR723MJ.md): concept WMPCR723MJ
- [Pseudocomplement](https://tamnd.github.io/math2/node/WMR73J0S26.md): concept WMR73J0S26
- [Richardson extrapolation](https://tamnd.github.io/math2/node/WMTZG71K2Y.md): concept WMTZG71K2Y
- [Riemann–Roch theorem](https://tamnd.github.io/math2/node/WMVAT9MBZ1.md): statement WMVAT9MBZ1
- [Wirtinger derivatives](https://tamnd.github.io/math2/node/WMXDVF36ZT.md): concept WMXDVF36ZT
- [Morley rank](https://tamnd.github.io/math2/node/WN43DRPNBX.md): concept WN43DRPNBX
- [Series in Banach spaces](https://tamnd.github.io/math2/node/WN6M8B3ZKB.md): concept WN6M8B3ZKB
- [Divisor class group](https://tamnd.github.io/math2/node/WN8PK6RQZR.md): concept WN8PK6RQZR
- [Lattice (group)](https://tamnd.github.io/math2/node/WN9V7WSJKF.md): concept WN9V7WSJKF
- [Stopping time](https://tamnd.github.io/math2/node/WND87FBAKC.md): concept WND87FBAKC
- [Antithetic variates](https://tamnd.github.io/math2/node/WNFEPEBART.md): statement WNFEPEBART
- [John ellipsoid](https://tamnd.github.io/math2/node/WNHXE7FCG4.md): concept WNHXE7FCG4
- [Feder–Vardi dichotomy conjecture](https://tamnd.github.io/math2/node/WNHZC07XVP.md): conjecture WNHZC07XVP
- [Szegő limit theorems](https://tamnd.github.io/math2/node/WNJPRHZ680.md): statement WNJPRHZ680
- [Galois theory](https://tamnd.github.io/math2/node/WNQDKQYZ2R.md): concept WNQDKQYZ2R
- [Hilbert's thirteenth problem](https://tamnd.github.io/math2/node/WNQWERBWBB.md): problem WNQWERBWBB
- [Contact manifold](https://tamnd.github.io/math2/node/WNSAZD8EZC.md): definition WNSAZD8EZC
- [Incomplete beta function](https://tamnd.github.io/math2/node/WNVACS2S2Z.md): concept WNVACS2S2Z
- [Category of sets](https://tamnd.github.io/math2/node/WP0YB7VC59.md): example WP0YB7VC59
- [Whitehead product](https://tamnd.github.io/math2/node/WP12H9FG91.md): definition WP12H9FG91
- [Free lattice](https://tamnd.github.io/math2/node/WP20S9NZ15.md): concept WP20S9NZ15
- [Robertson–Seymour theorem](https://tamnd.github.io/math2/node/WP3GDNWJJW.md): statement WP3GDNWJJW
- [Sufficient statistic](https://tamnd.github.io/math2/node/WP4TGXJQJG.md): concept WP4TGXJQJG
- [Penrose–Hawking singularity theorems](https://tamnd.github.io/math2/node/WP4YAHJ6V4.md): statement WP4YAHJ6V4
- [Essential supremum](https://tamnd.github.io/math2/node/WPC2XRMB9Y.md): concept WPC2XRMB9Y
- [Field trace](https://tamnd.github.io/math2/node/WPE5Z53QYJ.md): concept WPE5Z53QYJ
- [Laplace transform of the Heaviside step function](https://tamnd.github.io/math2/node/WPESA9XQ8X.md): example WPESA9XQ8X
- [Hilbert's fifteenth problem](https://tamnd.github.io/math2/node/WPQPBXNN5F.md): problem WPQPBXNN5F
- [Stieltjes function](https://tamnd.github.io/math2/node/WPR0NVC9PC.md): concept WPR0NVC9PC
- [Stochastic difference equation](https://tamnd.github.io/math2/node/WPYG78BSD5.md): concept WPYG78BSD5
- [Gödel's completeness theorem](https://tamnd.github.io/math2/node/WQ1ZE255SF.md): statement WQ1ZE255SF
- [Ideal class group](https://tamnd.github.io/math2/node/WQ46BD3ARY.md): concept WQ46BD3ARY
- [Chain complex](https://tamnd.github.io/math2/node/WQ82C0M3QB.md): concept WQ82C0M3QB
- [Jackknife](https://tamnd.github.io/math2/node/WQ8TQKDKYT.md): concept WQ8TQKDKYT
- [Easton's theorem](https://tamnd.github.io/math2/node/WQ967Q2XJD.md): statement WQ967Q2XJD
- [Arnold tongue](https://tamnd.github.io/math2/node/WQF5ASXZTH.md): concept WQF5ASXZTH
- [Metacompact space](https://tamnd.github.io/math2/node/WQG81SEDZ4.md): concept WQG81SEDZ4
- [Wilks' theorem](https://tamnd.github.io/math2/node/WQPB6GXD0M.md): statement WQPB6GXD0M
- [Funk transform](https://tamnd.github.io/math2/node/WQPTRZYY0Y.md): concept WQPTRZYY0Y
- [McKay conjecture](https://tamnd.github.io/math2/node/WQQTFRZFBD.md): statement WQQTFRZFBD
- [Tight and overtwisted contact structures](https://tamnd.github.io/math2/node/WQQX6SAQ5A.md): concept WQQX6SAQ5A
- [Jackson's inequality](https://tamnd.github.io/math2/node/WQRT86VYA6.md): statement WQRT86VYA6
- [Null-homotopic map](https://tamnd.github.io/math2/node/WQWEY93BKB.md): definition WQWEY93BKB
- [Screw theory](https://tamnd.github.io/math2/node/WR4RSD9NTK.md): definition WR4RSD9NTK
- [Finitely generated algebra](https://tamnd.github.io/math2/node/WR4WF1K2KW.md): concept WR4WF1K2KW
- [Kolmogorov extension theorem](https://tamnd.github.io/math2/node/WR4X7MMTH2.md): statement WR4X7MMTH2
- [Abstract polytope](https://tamnd.github.io/math2/node/WREPDZG7PF.md): concept WREPDZG7PF
- [Uniqueness of solutions given initial values](https://tamnd.github.io/math2/node/WREYNFJHH2.md): statement WREYNFJHH2
- [Triangle center](https://tamnd.github.io/math2/node/WRKFH25NCX.md): concept WRKFH25NCX
- [Maximal order](https://tamnd.github.io/math2/node/WRTSWX9G6X.md): definition WRTSWX9G6X
- [Quasivariety](https://tamnd.github.io/math2/node/WRXEKT4NQS.md): concept WRXEKT4NQS
- [Discrete harmonic function](https://tamnd.github.io/math2/node/WRY0M7SW59.md): concept WRY0M7SW59
- [Bridge number](https://tamnd.github.io/math2/node/WRYP2XCKJB.md): definition WRYP2XCKJB
- [Gale diagram](https://tamnd.github.io/math2/node/WRZA1VYGB6.md): definition WRZA1VYGB6
- [Solution of recurrences by generating functions](https://tamnd.github.io/math2/node/WS2E29RZD5.md): concept WS2E29RZD5
- [Linear and stable summation methods](https://tamnd.github.io/math2/node/WS62STR57F.md): definition WS62STR57F
- [Propagation of uncertainty](https://tamnd.github.io/math2/node/WS6GCG3YF8.md): concept WS6GCG3YF8
- [Mean-field game](https://tamnd.github.io/math2/node/WS9A6N2Y0F.md): concept WS9A6N2Y0F
- [Costate equation](https://tamnd.github.io/math2/node/WS9MRAZMFW.md): concept WS9MRAZMFW
- [Menger sponge](https://tamnd.github.io/math2/node/WSKWY54JDN.md): example WSKWY54JDN
- [Henselian ring](https://tamnd.github.io/math2/node/WSQ8GH8XWZ.md): definition WSQ8GH8XWZ
- [Lebesgue's density theorem](https://tamnd.github.io/math2/node/WSVXYSNHNX.md): statement WSVXYSNHNX
- [Bounded symmetric domain](https://tamnd.github.io/math2/node/WSWRGT9T6A.md): definition WSWRGT9T6A
- [Classifying space](https://tamnd.github.io/math2/node/WSZ40H5E4T.md): concept WSZ40H5E4T
- [Euler transform](https://tamnd.github.io/math2/node/WT0HACKQV7.md): concept WT0HACKQV7
- [G-structure](https://tamnd.github.io/math2/node/WT5CKAY5VD.md): definition WT5CKAY5VD
- [Inverse scattering transform](https://tamnd.github.io/math2/node/WT65A6ARGN.md): definition WT65A6ARGN
- [Wigner distribution function](https://tamnd.github.io/math2/node/WT7K28D1CD.md): definition WT7K28D1CD
- [Diffeomorphism group](https://tamnd.github.io/math2/node/WTCNKST134.md): example WTCNKST134
- [Borsuk's conjecture](https://tamnd.github.io/math2/node/WTCP8A87P1.md): counterexample WTCP8A87P1
- [Cobordism](https://tamnd.github.io/math2/node/WTRPSJKAZH.md): definition WTRPSJKAZH
- [Estimating equations](https://tamnd.github.io/math2/node/WTXFYA093D.md): concept WTXFYA093D
- [Gosper's algorithm](https://tamnd.github.io/math2/node/WTXMRG6WR4.md): definition WTXMRG6WR4
- [Regularization of ill-posed problems](https://tamnd.github.io/math2/node/WTY54BFC46.md): concept WTY54BFC46
- [Rogers–Ramanujan continued fraction](https://tamnd.github.io/math2/node/WV00RFGTVT.md): definition WV00RFGTVT
- [Fujiki class C](https://tamnd.github.io/math2/node/WV1Q8N8YW7.md): definition WV1Q8N8YW7
- [Forgetful functor](https://tamnd.github.io/math2/node/WV3BPYCHVG.md): concept WV3BPYCHVG
- [Pseudovariety](https://tamnd.github.io/math2/node/WV3PBEK76J.md): definition WV3PBEK76J
- [Cartan's test for involutivity](https://tamnd.github.io/math2/node/WV4S47X6QC.md): statement WV4S47X6QC
- [Commutative diagram](https://tamnd.github.io/math2/node/WV898P4X16.md): concept WV898P4X16
- [Lelong number](https://tamnd.github.io/math2/node/WVB60RKC3Z.md): concept WVB60RKC3Z
- [Cox–de Boor recursion](https://tamnd.github.io/math2/node/WVD069C7QF.md): statement WVD069C7QF
- [Morlet wavelet](https://tamnd.github.io/math2/node/WVJQXA3K2E.md): example WVJQXA3K2E
- [Riemann–Lebesgue lemma](https://tamnd.github.io/math2/node/WVKDC011KP.md): statement WVKDC011KP
- [Bézout domain](https://tamnd.github.io/math2/node/WVKE1QS423.md): definition WVKE1QS423
- [Riesz–Fischer theorem](https://tamnd.github.io/math2/node/WVV4MFB2SE.md): statement WVV4MFB2SE
- [Schottky group](https://tamnd.github.io/math2/node/WVYYQCFS4Q.md): definition WVYYQCFS4Q
- [GAGA](https://tamnd.github.io/math2/node/WVZ5EZN9V2.md): statement WVZ5EZN9V2
- [Medial magma](https://tamnd.github.io/math2/node/WVZ7F41S9Q.md): definition WVZ7F41S9Q
- [Fubini's theorem for series](https://tamnd.github.io/math2/node/WW53KG71KH.md): statement WW53KG71KH
- [Jack function](https://tamnd.github.io/math2/node/WW7H1KNXK8.md): concept WW7H1KNXK8
- [Kullback–Leibler divergence](https://tamnd.github.io/math2/node/WW7HPC3WKA.md): definition WW7HPC3WKA
- [Simplicial set](https://tamnd.github.io/math2/node/WW81ARB8K9.md): concept WW81ARB8K9
- [Hilbert class field](https://tamnd.github.io/math2/node/WWCJ7CR2R3.md): concept WWCJ7CR2R3
- [Method of steepest descent](https://tamnd.github.io/math2/node/WWE9H2948V.md): definition WWE9H2948V
- [Square-integrable kernel](https://tamnd.github.io/math2/node/WWMWX7AD4V.md): concept WWMWX7AD4V
- [Shape optimization](https://tamnd.github.io/math2/node/WWQJQXJECK.md): problem WWQJQXJECK
- [Closed immersion](https://tamnd.github.io/math2/node/WWRWYSJDET.md): concept WWRWYSJDET
- [Eremenko's conjecture](https://tamnd.github.io/math2/node/WWSYQJF7TZ.md): conjecture WWSYQJF7TZ
- [Parabolic and hyperbolic Riemann surfaces](https://tamnd.github.io/math2/node/WWVK5WF6WY.md): definition WWVK5WF6WY
- [Riemann P-symbol](https://tamnd.github.io/math2/node/WWW053WXP2.md): concept WWW053WXP2
- [Section of a vector bundle](https://tamnd.github.io/math2/node/WWX7H5W1RP.md): concept WWX7H5W1RP
- [Wang sequence](https://tamnd.github.io/math2/node/WWX8HCN260.md): statement WWX8HCN260
- [Invariant set](https://tamnd.github.io/math2/node/WWZTW11664.md): concept WWZTW11664
- [Koebe quarter theorem](https://tamnd.github.io/math2/node/WX0HMW4GT7.md): statement WX0HMW4GT7
- [Borromean rings](https://tamnd.github.io/math2/node/WX1Z2F3DCB.md): example WX1Z2F3DCB
- [Tensor field](https://tamnd.github.io/math2/node/WX302T93TW.md): concept WX302T93TW
- [Poisson integral for the half-space](https://tamnd.github.io/math2/node/WXFTF05MHK.md): statement WXFTF05MHK
- [Algebraic number field](https://tamnd.github.io/math2/node/WXGM3VWEK6.md): definition WXGM3VWEK6
- [Euler's thirty-six officers problem](https://tamnd.github.io/math2/node/WXJAVQKFR6.md): problem WXJAVQKFR6
- [Pseudocomplement](https://tamnd.github.io/math2/node/WXNCE8W5Q2.md): definition WXNCE8W5Q2
- [Yoneda Ext and extensions](https://tamnd.github.io/math2/node/WXRH0CYKYF.md): statement WXRH0CYKYF
- [Asymptotic stability of linear recurrences](https://tamnd.github.io/math2/node/WXV3ZNS2F9.md): statement WXV3ZNS2F9
- [Tangent bundle](https://tamnd.github.io/math2/node/WY6K3P4YCG.md): concept WY6K3P4YCG
- [Proximity space](https://tamnd.github.io/math2/node/WY9BHD6GBA.md): concept WY9BHD6GBA
- [Brill–Noether theory](https://tamnd.github.io/math2/node/WYCJN9GDW4.md): definition WYCJN9GDW4
- [Group representation](https://tamnd.github.io/math2/node/WYGRXCG2RN.md): concept WYGRXCG2RN
- [Vitali set](https://tamnd.github.io/math2/node/WYHGXY471C.md): counterexample WYHGXY471C
- [Majorization](https://tamnd.github.io/math2/node/WYHKN0TE3Z.md): definition WYHKN0TE3Z
- [Fox–Wright function](https://tamnd.github.io/math2/node/WYMC18W643.md): definition WYMC18W643
- [Lorenz system](https://tamnd.github.io/math2/node/WYRC41CQN9.md): example WYRC41CQN9
- [Minkowski–Bouligand dimension](https://tamnd.github.io/math2/node/WYRKJ412FT.md): concept WYRKJ412FT
- [Kernel](https://tamnd.github.io/math2/node/WYZVC78ZE5.md): definition WYZVC78ZE5
- [Locally finite variety](https://tamnd.github.io/math2/node/WZ157RXTNV.md): definition WZ157RXTNV
- [Kakeya needle problem](https://tamnd.github.io/math2/node/WZ42B0SAVM.md): problem WZ42B0SAVM
- [Girth](https://tamnd.github.io/math2/node/WZ4JGTG6ZD.md): definition WZ4JGTG6ZD
- [Urysohn integral equation](https://tamnd.github.io/math2/node/WZ4Z26XSVC.md): concept WZ4Z26XSVC
- [Fiber of a morphism](https://tamnd.github.io/math2/node/WZC2MN3CX3.md): definition WZC2MN3CX3
- [Cauchy filter](https://tamnd.github.io/math2/node/WZC4J62B1G.md): concept WZC4J62B1G
- [Multidimensional Laplace transform](https://tamnd.github.io/math2/node/WZJ99RSXNN.md): definition WZJ99RSXNN
- [Maximal chain](https://tamnd.github.io/math2/node/WZKPDWBW2Z.md): definition WZKPDWBW2Z
- [Lie symmetry method](https://tamnd.github.io/math2/node/WZKRG7H3FX.md): definition WZKRG7H3FX
- [Pseudo-Anosov map](https://tamnd.github.io/math2/node/WZNF6Q5JBS.md): definition WZNF6Q5JBS
- [Cut locus](https://tamnd.github.io/math2/node/WZQ12PPVA2.md): definition WZQ12PPVA2
- [Continuous group homomorphism](https://tamnd.github.io/math2/node/WZRXSVBVQC.md): definition WZRXSVBVQC
- [Adaptive mesh refinement](https://tamnd.github.io/math2/node/X001F42KH8.md): concept X001F42KH8
- [Resolvent set](https://tamnd.github.io/math2/node/X0243CZD9D.md): concept X0243CZD9D
- [Borel conjecture](https://tamnd.github.io/math2/node/X057MN9YY9.md): conjecture X057MN9YY9
- [Matrix multiplication exponent](https://tamnd.github.io/math2/node/X0G02CGQ5J.md): problem X0G02CGQ5J
- [Convex integrand](https://tamnd.github.io/math2/node/X0Q69ZS09T.md): concept X0Q69ZS09T
- [Maurer–Cartan form](https://tamnd.github.io/math2/node/X0X1Q5JZFK.md): definition X0X1Q5JZFK
- [Convergence along an ultrafilter](https://tamnd.github.io/math2/node/X117TDBR51.md): definition X117TDBR51
- [Schur stable polynomial](https://tamnd.github.io/math2/node/X11Q0G8PNA.md): concept X11Q0G8PNA
- [Error-correcting code](https://tamnd.github.io/math2/node/X122TKFP9G.md): definition X122TKFP9G
- [Knot Floer homology](https://tamnd.github.io/math2/node/X126J76RH7.md): definition X126J76RH7
- [Symplectic group](https://tamnd.github.io/math2/node/X12FS2JAP0.md): object X12FS2JAP0
- [Computability theory](https://tamnd.github.io/math2/node/X14Y880PN5.md): concept X14Y880PN5
- [Dispersive partial differential equation](https://tamnd.github.io/math2/node/X17RYQ4DS2.md): definition X17RYQ4DS2
- [Kolmogorov–Sinai entropy](https://tamnd.github.io/math2/node/X1D1C7VH6P.md): concept X1D1C7VH6P
- [Fractional iteration](https://tamnd.github.io/math2/node/X1DB08QJHK.md): definition X1DB08QJHK
- [Large cardinal](https://tamnd.github.io/math2/node/X1FG7PSCTY.md): concept X1FG7PSCTY
- [Wolff potential](https://tamnd.github.io/math2/node/X1GZ9WKQEW.md): definition X1GZ9WKQEW
- [Anosov diffeomorphism](https://tamnd.github.io/math2/node/X1P48KQFXQ.md): definition X1P48KQFXQ
- [Schreier coset graph](https://tamnd.github.io/math2/node/X1PQ8J2R10.md): definition X1PQ8J2R10
- [Horn hypergeometric series](https://tamnd.github.io/math2/node/X24AGT6TK9.md): concept X24AGT6TK9
- [Analytic capacity](https://tamnd.github.io/math2/node/X252NY7CS1.md): definition X252NY7CS1
- [Synthetic differential geometry](https://tamnd.github.io/math2/node/X253AFB4YW.md): definition X253AFB4YW
- [Stable ∞-category](https://tamnd.github.io/math2/node/X2AHW5VJZR.md): concept X2AHW5VJZR
- [Regularization of ill-posed problems](https://tamnd.github.io/math2/node/X2AMH38NCR.md): definition X2AMH38NCR
- [Information-based complexity](https://tamnd.github.io/math2/node/X2CS9SP1DZ.md): definition X2CS9SP1DZ
- [Complex torus](https://tamnd.github.io/math2/node/X2E0QP770D.md): definition X2E0QP770D
- [Mellin's inverse formula](https://tamnd.github.io/math2/node/X2H35BP974.md): statement X2H35BP974
- [Separated morphism](https://tamnd.github.io/math2/node/X30G4V7VAJ.md): concept X30G4V7VAJ
- [Formally real field](https://tamnd.github.io/math2/node/X32TSWJSYT.md): concept X32TSWJSYT
- [Ordered vector space](https://tamnd.github.io/math2/node/X32V6Y5PYW.md): definition X32V6Y5PYW
- [Gelfand–Levitan–Marchenko equation](https://tamnd.github.io/math2/node/X35RWR285S.md): concept X35RWR285S
- [Semilinear PDE](https://tamnd.github.io/math2/node/X3P4TQZ9X2.md): definition X3P4TQZ9X2
- [Lie symmetry method](https://tamnd.github.io/math2/node/X3S9XHX83H.md): concept X3S9XHX83H
- [Hearing the shape of a drum](https://tamnd.github.io/math2/node/X3VH8H1XV3.md): problem X3VH8H1XV3
- [Exotic R4](https://tamnd.github.io/math2/node/X44PNC38E2.md): object X44PNC38E2
- [Center manifold](https://tamnd.github.io/math2/node/X45STCVFZR.md): concept X45STCVFZR
- [Quadratic variation](https://tamnd.github.io/math2/node/X4DHMZD5JA.md): concept X4DHMZD5JA
- [Calabi–Weil rigidity](https://tamnd.github.io/math2/node/X4DZR72B6N.md): statement X4DZR72B6N
- [Tanh-sinh quadrature](https://tamnd.github.io/math2/node/X4EXEGM2BV.md): statement X4EXEGM2BV
- [Schur's lemma](https://tamnd.github.io/math2/node/X4M2FFKQJ6.md): statement X4M2FFKQJ6
- [Mittag-Leffler's theorem](https://tamnd.github.io/math2/node/X4R2GY1F6J.md): statement X4R2GY1F6J
- [Superharmonic function](https://tamnd.github.io/math2/node/X4V2S56YQQ.md): concept X4V2S56YQQ
- [Lelong class](https://tamnd.github.io/math2/node/X4ZHQG9995.md): concept X4ZHQG9995
- [Compositum of fields](https://tamnd.github.io/math2/node/X4ZZKC319Q.md): definition X4ZZKC319Q
- [Rees matrix semigroup](https://tamnd.github.io/math2/node/X54RV6SENQ.md): concept X54RV6SENQ
- [Colorful Carathéodory theorem](https://tamnd.github.io/math2/node/X5GS407VP8.md): statement X5GS407VP8
- [Milnor fibre is a bouquet of spheres](https://tamnd.github.io/math2/node/X5HMTTGB62.md): statement X5HMTTGB62
- [Laplace transform of the exponential function](https://tamnd.github.io/math2/node/X5JYH26J3M.md): example X5JYH26J3M
- [Algorithmic randomness](https://tamnd.github.io/math2/node/X5KD3X3VJ3.md): definition X5KD3X3VJ3
- [Nörlund mean](https://tamnd.github.io/math2/node/X5R7FV9KWH.md): concept X5R7FV9KWH
- [Cut locus](https://tamnd.github.io/math2/node/X5WKGC5Q6T.md): concept X5WKGC5Q6T
- [Givens rotation](https://tamnd.github.io/math2/node/X600GRHDQ7.md): definition X600GRHDQ7
- [Kähler manifold](https://tamnd.github.io/math2/node/X60JG9AEQJ.md): concept X60JG9AEQJ
- [Abel–Dini theorem](https://tamnd.github.io/math2/node/X629NF2CQS.md): statement X629NF2CQS
- [Differentiable curve](https://tamnd.github.io/math2/node/X66ZPDYAKC.md): definition X66ZPDYAKC
- [Finite ring](https://tamnd.github.io/math2/node/X6713ZP2NH.md): concept X6713ZP2NH
- [Additive Schwarz method](https://tamnd.github.io/math2/node/X68MFYW014.md): statement X68MFYW014
- [Exponential functions on semigroups](https://tamnd.github.io/math2/node/X6CNJMWSAK.md): definition X6CNJMWSAK
- [Fourier–Bessel series](https://tamnd.github.io/math2/node/X6GEZGZ8HQ.md): definition X6GEZGZ8HQ
- [Localization at a prime ideal](https://tamnd.github.io/math2/node/X6JD4AG973.md): concept X6JD4AG973
- [Rough path](https://tamnd.github.io/math2/node/X6KCW4X5JA.md): concept X6KCW4X5JA
- [Harmonic map heat flow](https://tamnd.github.io/math2/node/X6QC23AAB9.md): definition X6QC23AAB9
- [Upper half-plane](https://tamnd.github.io/math2/node/X6XGGH5FPW.md): object X6XGGH5FPW
- [Inner-outer factorization](https://tamnd.github.io/math2/node/X6ZQPAQF76.md): statement X6ZQPAQF76
- [Envelope of holomorphy](https://tamnd.github.io/math2/node/X71JE6A3ZJ.md): concept X71JE6A3ZJ
- [Spectral theorem for bounded normal operators](https://tamnd.github.io/math2/node/X71XRPJR7D.md): statement X71XRPJR7D
- [Robbins conjecture](https://tamnd.github.io/math2/node/X74S395RZD.md): statement X74S395RZD
- [Rack and quandle](https://tamnd.github.io/math2/node/X74VY86TSX.md): concept X74VY86TSX
- [Multiplier ideal](https://tamnd.github.io/math2/node/X7B9DEF1MF.md): definition X7B9DEF1MF
- [Tangential Cauchy–Riemann equations](https://tamnd.github.io/math2/node/X7GP9BECD8.md): concept X7GP9BECD8
- [Integrable function](https://tamnd.github.io/math2/node/X7GTF3D98F.md): concept X7GTF3D98F
- [Finite difference method](https://tamnd.github.io/math2/node/X7HYEBA5SA.md): concept X7HYEBA5SA
- [Archimedean solid](https://tamnd.github.io/math2/node/X7K75G78C8.md): object X7K75G78C8
- [Artin's theorem on fixed fields](https://tamnd.github.io/math2/node/X7R15C00ET.md): statement X7R15C00ET
- [Bondy's theorem](https://tamnd.github.io/math2/node/X7S41PARF5.md): statement X7S41PARF5
- [Lie algebra of a compact group is reductive](https://tamnd.github.io/math2/node/X7T5D9SMT5.md): statement X7T5D9SMT5
- [Marsden–Weinstein reduction](https://tamnd.github.io/math2/node/X7ZSDSXYT5.md): definition X7ZSDSXYT5
- [Summability method](https://tamnd.github.io/math2/node/X80DZEXQDQ.md): definition X80DZEXQDQ
- [Chern–Simons form](https://tamnd.github.io/math2/node/X83ECKQ6G8.md): concept X83ECKQ6G8
- [Integral element](https://tamnd.github.io/math2/node/X83RX7KDSP.md): definition X83RX7KDSP
- [Jensen inequality as a functional inequality](https://tamnd.github.io/math2/node/X84R2EWQJP.md): statement X84R2EWQJP
- [Baker–Campbell–Hausdorff formula](https://tamnd.github.io/math2/node/X898DPBSY7.md): statement X898DPBSY7
- [Mayer–Vietoris sequence](https://tamnd.github.io/math2/node/X8BWXFGK53.md): statement X8BWXFGK53
- [Eikonal equation](https://tamnd.github.io/math2/node/X8CF1K8QB4.md): definition X8CF1K8QB4
- [Separable algebra](https://tamnd.github.io/math2/node/X8H7A98Q55.md): concept X8H7A98Q55
- [Stinespring dilation theorem](https://tamnd.github.io/math2/node/X8K0Z7484R.md): statement X8K0Z7484R
- [Balancing domain decomposition](https://tamnd.github.io/math2/node/X8MCVW56QP.md): statement X8MCVW56QP
- [Frölicher–Nijenhuis bracket](https://tamnd.github.io/math2/node/X8P0H71EET.md): definition X8P0H71EET
- [Hilbert's tenth problem](https://tamnd.github.io/math2/node/X8TVJSVTD4.md): statement X8TVJSVTD4
- [Chebotarev's density theorem](https://tamnd.github.io/math2/node/X8VKW5Y1CZ.md): statement X8VKW5Y1CZ
- [Lattice](https://tamnd.github.io/math2/node/X8WD0ABG07.md): definition X8WD0ABG07
- [Residually finite algebra](https://tamnd.github.io/math2/node/X9072A41NK.md): concept X9072A41NK
- [Bangert–Franks theorem on closed geodesics on the sphere](https://tamnd.github.io/math2/node/X93AEE88PX.md): statement X93AEE88PX
- [Permanent](https://tamnd.github.io/math2/node/X9B1XETBPF.md): concept X9B1XETBPF
- [Adjoint state method](https://tamnd.github.io/math2/node/X9F8SQFQGP.md): definition X9F8SQFQGP
- [Direct limit](https://tamnd.github.io/math2/node/X9HEQ29Q49.md): concept X9HEQ29Q49
- [Padua points](https://tamnd.github.io/math2/node/X9HVQPWT6T.md): example X9HVQPWT6T
- [Hecke operator](https://tamnd.github.io/math2/node/X9PBTW2J8D.md): definition X9PBTW2J8D
- [Sub-Gaussian distribution](https://tamnd.github.io/math2/node/X9R1PCKJ0R.md): concept X9R1PCKJ0R
- [Laguerre–Pólya class](https://tamnd.github.io/math2/node/X9SRCEENV3.md): concept X9SRCEENV3
- [F. and M. Riesz theorem](https://tamnd.github.io/math2/node/X9V1E3V145.md): statement X9V1E3V145
- [Gödel's incompleteness theorems](https://tamnd.github.io/math2/node/X9WYNYE29C.md): statement X9WYNYE29C
- [Weyl's theorem on compact perturbations](https://tamnd.github.io/math2/node/X9XRGMDCQZ.md): statement X9XRGMDCQZ
- [Jacobson radical](https://tamnd.github.io/math2/node/XA0DCNH7WY.md): concept XA0DCNH7WY
- [Logarithmic potential](https://tamnd.github.io/math2/node/XA2H0YDBG2.md): definition XA2H0YDBG2
- [Hilbert's fourth problem](https://tamnd.github.io/math2/node/XA35DT40XS.md): problem XA35DT40XS
- [Matrix Lie group](https://tamnd.github.io/math2/node/XA8CY1X2BE.md): definition XA8CY1X2BE
- [Sub-Riemannian geodesic problem](https://tamnd.github.io/math2/node/XA9JK51Y35.md): problem XA9JK51Y35
- [Proportional hazards model](https://tamnd.github.io/math2/node/XAANCQH8A4.md): definition XAANCQH8A4
- [Uniform approximation](https://tamnd.github.io/math2/node/XAARVNKXF1.md): concept XAARVNKXF1
- [Commutative magma](https://tamnd.github.io/math2/node/XAC78GN2YJ.md): concept XAC78GN2YJ
- [Effective resistance](https://tamnd.github.io/math2/node/XAE5YJ3H68.md): concept XAE5YJ3H68
- [Hodge conjecture](https://tamnd.github.io/math2/node/XAEEDWV3PM.md): conjecture XAEEDWV3PM
- [Inverse limit](https://tamnd.github.io/math2/node/XAQSNBJSX4.md): definition XAQSNBJSX4
- [Quasi-Monte Carlo method](https://tamnd.github.io/math2/node/XAW51D9E62.md): definition XAW51D9E62
- [Khovanov homology](https://tamnd.github.io/math2/node/XB24W24B20.md): concept XB24W24B20
- [Pseudo-spectral method](https://tamnd.github.io/math2/node/XB91QS2AR0.md): concept XB91QS2AR0
- [CAT(0) group](https://tamnd.github.io/math2/node/XB9QTY77ZA.md): concept XB9QTY77ZA
- [Schnirelmann density](https://tamnd.github.io/math2/node/XBDB3YNV9S.md): definition XBDB3YNV9S
- [Cotlar–Stein lemma](https://tamnd.github.io/math2/node/XBJEDC18B1.md): statement XBJEDC18B1
- [Lindelöf's theorem](https://tamnd.github.io/math2/node/XBM7J5R1VQ.md): statement XBM7J5R1VQ
- [Euler–Poisson equation for higher derivatives](https://tamnd.github.io/math2/node/XBNAJ1RRET.md): concept XBNAJ1RRET
- [Incomplete LU factorization](https://tamnd.github.io/math2/node/XBPK7Y559D.md): concept XBPK7Y559D
- [J-homomorphism](https://tamnd.github.io/math2/node/XBPKCBX4EW.md): definition XBPKCBX4EW
- [Harmonic space](https://tamnd.github.io/math2/node/XBZ2RX25E6.md): definition XBZ2RX25E6
- [Associated graded ring](https://tamnd.github.io/math2/node/XBZ99KH3RP.md): definition XBZ99KH3RP
- [Local martingale](https://tamnd.github.io/math2/node/XC2YPC2BSW.md): concept XC2YPC2BSW
- [Kovacic's algorithm](https://tamnd.github.io/math2/node/XC4245C9E5.md): concept XC4245C9E5
- [Nuclear space](https://tamnd.github.io/math2/node/XC6XPMC2FQ.md): concept XC6XPMC2FQ
- [Gromoll–Meyer theorem on closed geodesics](https://tamnd.github.io/math2/node/XC8BRYJN93.md): statement XC8BRYJN93
- [Prime ring](https://tamnd.github.io/math2/node/XCE0P00WJF.md): definition XCE0P00WJF
- [Probit model](https://tamnd.github.io/math2/node/XCR3V8SJ01.md): concept XCR3V8SJ01
- [Blocking](https://tamnd.github.io/math2/node/XCWKYS1DFA.md): concept XCWKYS1DFA
- [Primal algebra](https://tamnd.github.io/math2/node/XD01Y7B6XW.md): concept XD01Y7B6XW
- [A subgroup of order two in S3 is not normal](https://tamnd.github.io/math2/node/XD0WBC1MWB.md): example XD0WBC1MWB
- [Unary algebra](https://tamnd.github.io/math2/node/XD10P9CJMD.md): concept XD10P9CJMD
- [Affine plane](https://tamnd.github.io/math2/node/XD8ZB63H78.md): definition XD8ZB63H78
- [Fubini's theorem on differentiation](https://tamnd.github.io/math2/node/XDEJCKRVQ4.md): statement XDEJCKRVQ4
- [Metric outer measure](https://tamnd.github.io/math2/node/XDK73D2DEA.md): definition XDK73D2DEA
- [Harmonic polynomial](https://tamnd.github.io/math2/node/XDK7SQW3F6.md): definition XDK7SQW3F6
- [Branch point](https://tamnd.github.io/math2/node/XDNRXKH3KT.md): definition XDNRXKH3KT
- [Square root of a positive operator](https://tamnd.github.io/math2/node/XDQ8CF0EZD.md): definition XDQ8CF0EZD
- [Nagata ring](https://tamnd.github.io/math2/node/XDQYMY78TS.md): definition XDQYMY78TS
- [Brauer–Manin obstruction](https://tamnd.github.io/math2/node/XDWF0HW2QN.md): concept XDWF0HW2QN
- [Fox–Wright function](https://tamnd.github.io/math2/node/XDY1GDM2JW.md): concept XDY1GDM2JW
- [Section of a vector bundle](https://tamnd.github.io/math2/node/XDYMCB41NG.md): definition XDYMCB41NG
- [Face lattice](https://tamnd.github.io/math2/node/XDZQYX6AQ3.md): example XDZQYX6AQ3
- [Lebesgue–Stieltjes measure](https://tamnd.github.io/math2/node/XE1E2QTK4W.md): definition XE1E2QTK4W
- [Faddeeva function](https://tamnd.github.io/math2/node/XE3913G5SA.md): definition XE3913G5SA
- [Modulation space](https://tamnd.github.io/math2/node/XE3GCCV96H.md): definition XE3GCCV96H
- [Gauss–Laguerre quadrature](https://tamnd.github.io/math2/node/XE98WJB3FV.md): statement XE98WJB3FV
- [Square root of a matrix](https://tamnd.github.io/math2/node/XEC5MF8WWX.md): definition XEC5MF8WWX
- [Best approximation](https://tamnd.github.io/math2/node/XEEZ2MQ6XM.md): definition XEEZ2MQ6XM
- [Asymptotic scale](https://tamnd.github.io/math2/node/XEGB8Y44GG.md): concept XEGB8Y44GG
- [Siegel–Walfisz theorem](https://tamnd.github.io/math2/node/XEPW251A97.md): statement XEPW251A97
- [F-sigma set](https://tamnd.github.io/math2/node/XETJ5G0YEW.md): definition XETJ5G0YEW
- [Leech lattice](https://tamnd.github.io/math2/node/XETV3657DG.md): object XETV3657DG
- [Filippov's existence theorem](https://tamnd.github.io/math2/node/XEX7QZV1BS.md): statement XEX7QZV1BS
- [Adjoint integral equation](https://tamnd.github.io/math2/node/XEZXGBT42J.md): concept XEZXGBT42J
- [Kontorovich–Lebedev transform](https://tamnd.github.io/math2/node/XFJDYMB91J.md): concept XFJDYMB91J
- [Superalgebra](https://tamnd.github.io/math2/node/XFJVAMSAQ2.md): definition XFJVAMSAQ2
- [Continuous q-Hermite polynomials](https://tamnd.github.io/math2/node/XFKVZJ55Q5.md): concept XFKVZJ55Q5
- [Sturmian word](https://tamnd.github.io/math2/node/XG4D9FF51T.md): example XG4D9FF51T
- [Naive set theory](https://tamnd.github.io/math2/node/XGAMQBSE1G.md): concept XGAMQBSE1G
- [Hadamard's two-constants theorem](https://tamnd.github.io/math2/node/XGBDVQQV0T.md): statement XGBDVQQV0T
- [Colimit](https://tamnd.github.io/math2/node/XGCV1SGEGQ.md): definition XGCV1SGEGQ
- [Cartan–Kähler theorem](https://tamnd.github.io/math2/node/XGDJ4V44PD.md): statement XGDJ4V44PD
- [Regular semigroup](https://tamnd.github.io/math2/node/XGJKF6T7FZ.md): concept XGJKF6T7FZ
- [Stability conjecture](https://tamnd.github.io/math2/node/XGNX9R7Q0Y.md): statement XGNX9R7Q0Y
- [Colombeau algebra](https://tamnd.github.io/math2/node/XGQT4WEVGG.md): concept XGQT4WEVGG
- [Analytic combinatorics](https://tamnd.github.io/math2/node/XGZ2YYCNWK.md): definition XGZ2YYCNWK
- [Kähler manifold](https://tamnd.github.io/math2/node/XGZMG3QDHZ.md): definition XGZMG3QDHZ
- [Preconditioner](https://tamnd.github.io/math2/node/XH1RXBEEWH.md): definition XH1RXBEEWH
- [Elliptic cohomology](https://tamnd.github.io/math2/node/XH68V2BP4K.md): concept XH68V2BP4K
- [Train track](https://tamnd.github.io/math2/node/XH6FFFR8AZ.md): concept XH6FFFR8AZ
- [Geometric realization](https://tamnd.github.io/math2/node/XHHTS6TDYZ.md): definition XHHTS6TDYZ
- [Noether's theorem](https://tamnd.github.io/math2/node/XHPDM5T2JG.md): statement XHPDM5T2JG
- [Skorokhod's embedding theorem](https://tamnd.github.io/math2/node/XHTYZ7ER4Z.md): statement XHTYZ7ER4Z
- [Beckman–Quarles theorem](https://tamnd.github.io/math2/node/XHWA2TEC7X.md): statement XHWA2TEC7X
- [Decay of Fourier coefficients and smoothness](https://tamnd.github.io/math2/node/XHXKMS1GXQ.md): statement XHXKMS1GXQ
- [Interacting particle system](https://tamnd.github.io/math2/node/XHXQDYAW3M.md): definition XHXQDYAW3M
- [Polyharmonic spline](https://tamnd.github.io/math2/node/XJ0JGPBC8R.md): concept XJ0JGPBC8R
- [Separated morphism](https://tamnd.github.io/math2/node/XJ1GNQZ8ND.md): definition XJ1GNQZ8ND
- [Normal fan](https://tamnd.github.io/math2/node/XJ83PTS3H0.md): concept XJ83PTS3H0
- [Gleason–Yamabe theorem](https://tamnd.github.io/math2/node/XJF5V9WHKZ.md): statement XJF5V9WHKZ
- [Algebraic curve](https://tamnd.github.io/math2/node/XJM2MT4S3Y.md): concept XJM2MT4S3Y
- [Curvelet](https://tamnd.github.io/math2/node/XJMGD9V64J.md): definition XJMGD9V64J
- [p-Laplacian](https://tamnd.github.io/math2/node/XJMYGY0ZZF.md): concept XJMYGY0ZZF
- [Eilenberg–MacLane spectrum](https://tamnd.github.io/math2/node/XJPYJT9Z1P.md): object XJPYJT9Z1P
- [Semisimple Lie group](https://tamnd.github.io/math2/node/XJS07AJ50Y.md): concept XJS07AJ50Y
- [Mellin inversion theorem](https://tamnd.github.io/math2/node/XJWC13NY9E.md): statement XJWC13NY9E
- [Abelian algebra](https://tamnd.github.io/math2/node/XK1KDMCW4K.md): definition XK1KDMCW4K
- [Frobenius's theorem on Frobenius kernels](https://tamnd.github.io/math2/node/XK7H3YXBQT.md): statement XK7H3YXBQT
- [Generalized least squares](https://tamnd.github.io/math2/node/XK7W504YEW.md): definition XK7W504YEW
- [Hilbert scheme](https://tamnd.github.io/math2/node/XK866CAZHE.md): object XK866CAZHE
- [Galois insertion](https://tamnd.github.io/math2/node/XK933YCW13.md): definition XK933YCW13
- [Szász–Mirakyan operator](https://tamnd.github.io/math2/node/XKAWV4CCCB.md): definition XKAWV4CCCB
- [Exchangeable random variables](https://tamnd.github.io/math2/node/XKBABTE7Y2.md): definition XKBABTE7Y2
- [Function field](https://tamnd.github.io/math2/node/XKEXQDRTGH.md): concept XKEXQDRTGH
- [Empirical process](https://tamnd.github.io/math2/node/XKFG1676CG.md): concept XKFG1676CG
- [Quillen adjunction](https://tamnd.github.io/math2/node/XKFKTHW8GS.md): definition XKFKTHW8GS
- [Coefficient of determination](https://tamnd.github.io/math2/node/XKGEV5RP3T.md): definition XKGEV5RP3T
- [Cohomological dimension of a field](https://tamnd.github.io/math2/node/XKMBKAN1MR.md): definition XKMBKAN1MR
- [Prime end](https://tamnd.github.io/math2/node/XKMWYSGBN0.md): definition XKMWYSGBN0
- [Grunsky matrix](https://tamnd.github.io/math2/node/XKRR37N916.md): concept XKRR37N916
- [Pentagon identity](https://tamnd.github.io/math2/node/XM4NTS5BGG.md): statement XM4NTS5BGG
- [Theta transformation formula](https://tamnd.github.io/math2/node/XM695ZZ4FJ.md): statement XM695ZZ4FJ
- [Equidistribution modulo one](https://tamnd.github.io/math2/node/XM6DH57BWY.md): concept XM6DH57BWY
- [Hilbert's theorem on the hyperbolic plane](https://tamnd.github.io/math2/node/XM772FD0BS.md): statement XM772FD0BS
- [Equilibrium problem with an external field](https://tamnd.github.io/math2/node/XMCAHWK1YB.md): problem XMCAHWK1YB
- [Schwarz triangle function](https://tamnd.github.io/math2/node/XMD3Q49Z2C.md): concept XMD3Q49Z2C
- [AKS primality test](https://tamnd.github.io/math2/node/XMHQCDWZW1.md): statement XMHQCDWZW1
- [Associated bundle](https://tamnd.github.io/math2/node/XMJ0V0FCPY.md): definition XMJ0V0FCPY
- [Triangle removal lemma](https://tamnd.github.io/math2/node/XMS1QRAQZC.md): statement XMS1QRAQZC
- [Bounded gaps between primes](https://tamnd.github.io/math2/node/XMXFDWYYMD.md): statement XMXFDWYYMD
- [Néron–Severi group](https://tamnd.github.io/math2/node/XMZ09SBM0X.md): concept XMZ09SBM0X
- [Hyperbolic conservation law](https://tamnd.github.io/math2/node/XN081NCPF5.md): concept XN081NCPF5
- [Substitution](https://tamnd.github.io/math2/node/XN8TQE6SRK.md): definition XN8TQE6SRK
- [Admissible decision rule](https://tamnd.github.io/math2/node/XNA8H9BPA8.md): definition XNA8H9BPA8
- [Torus as a group](https://tamnd.github.io/math2/node/XNAWD33HN1.md): example XNAWD33HN1
- [Sumset](https://tamnd.github.io/math2/node/XNBN31HQWB.md): concept XNBN31HQWB
- [Factorization of polynomials over finite fields](https://tamnd.github.io/math2/node/XNE7F0BAGE.md): concept XNE7F0BAGE
- [Hemicontinuity](https://tamnd.github.io/math2/node/XNMKX7H4FD.md): concept XNMKX7H4FD
- [De la Vallée Poussin lower bound](https://tamnd.github.io/math2/node/XNPVW5Z126.md): statement XNPVW5Z126
- [Convex polytope](https://tamnd.github.io/math2/node/XNTJTH6ACN.md): definition XNTJTH6ACN
- [Dirac equation](https://tamnd.github.io/math2/node/XNZX7VXXR2.md): concept XNZX7VXXR2
- [Dehn twist](https://tamnd.github.io/math2/node/XP21V9TVKK.md): definition XP21V9TVKK
- [Branch point](https://tamnd.github.io/math2/node/XP5JC1VXKZ.md): concept XP5JC1VXKZ
- [Polynomial operation](https://tamnd.github.io/math2/node/XP67WCPQ59.md): concept XP67WCPQ59
- [Nuclear operator](https://tamnd.github.io/math2/node/XPANM7C1D3.md): concept XPANM7C1D3
- [Nesbitt's inequality](https://tamnd.github.io/math2/node/XPBQEJNYJ6.md): statement XPBQEJNYJ6
- [Unimodular group](https://tamnd.github.io/math2/node/XPCTJTBSQ0.md): definition XPCTJTBSQ0
- [Contraction semigroup](https://tamnd.github.io/math2/node/XPWE94BV6N.md): concept XPWE94BV6N
- [Fredholm's theorem](https://tamnd.github.io/math2/node/XQ8DYKW4SH.md): statement XQ8DYKW4SH
- [Hyperbolic partial differential equation](https://tamnd.github.io/math2/node/XQG2VCBMKQ.md): definition XQG2VCBMKQ
- [Intuitionism](https://tamnd.github.io/math2/node/XQHF63BBYF.md): concept XQHF63BBYF
- [Hochschild homology](https://tamnd.github.io/math2/node/XQKZNYJ8WS.md): definition XQKZNYJ8WS
- [Goodstein's theorem](https://tamnd.github.io/math2/node/XQMNJF241J.md): statement XQMNJF241J
- [Pseudo-differential operator](https://tamnd.github.io/math2/node/XQN4N522WG.md): definition XQN4N522WG
- [Montessus de Ballore's theorem](https://tamnd.github.io/math2/node/XQQWP9ZPEW.md): statement XQQWP9ZPEW
- [Laplace transform method for convolution equations](https://tamnd.github.io/math2/node/XQTHXW4F1R.md): definition XQTHXW4F1R
- [Ideal completion](https://tamnd.github.io/math2/node/XQXBBDD5XB.md): definition XQXBBDD5XB
- [Hodge index theorem](https://tamnd.github.io/math2/node/XR0M3DFFMG.md): statement XR0M3DFFMG
- [Finiteness of associated primes](https://tamnd.github.io/math2/node/XR0WZJAM0R.md): statement XR0WZJAM0R
- [Pseudovariety](https://tamnd.github.io/math2/node/XR4DPKX7T1.md): concept XR4DPKX7T1
- [Schwarz reflection principle for harmonic functions](https://tamnd.github.io/math2/node/XRBD5BT2T9.md): statement XRBD5BT2T9
- [Graph reconstruction conjecture](https://tamnd.github.io/math2/node/XRBPS2QYE3.md): conjecture XRBPS2QYE3
- [Descending chain condition](https://tamnd.github.io/math2/node/XRGSKJVF0M.md): definition XRGSKJVF0M
- [Method of fundamental solutions](https://tamnd.github.io/math2/node/XS0Q09SV2T.md): definition XS0Q09SV2T
- [Separable degree](https://tamnd.github.io/math2/node/XS3VFNSY9N.md): definition XS3VFNSY9N
- [Cyclic term theorem](https://tamnd.github.io/math2/node/XS41Y1Q01M.md): statement XS41Y1Q01M
- [Shilov boundary](https://tamnd.github.io/math2/node/XS8KB53QKA.md): definition XS8KB53QKA
- [Hotelling's T-squared distribution](https://tamnd.github.io/math2/node/XS9GETCZX2.md): concept XS9GETCZX2
- [Amalgamation property](https://tamnd.github.io/math2/node/XSCSX39GEZ.md): concept XSCSX39GEZ
- [Low-discrepancy sequence](https://tamnd.github.io/math2/node/XSPX1KW9AQ.md): concept XSPX1KW9AQ
- [Green's function of a Markov process](https://tamnd.github.io/math2/node/XSQX8X2YQ2.md): concept XSQX8X2YQ2
- [Manin conjecture](https://tamnd.github.io/math2/node/XSZ49NF2AE.md): conjecture XSZ49NF2AE
- [System of parameters](https://tamnd.github.io/math2/node/XSZ9C68SPG.md): definition XSZ9C68SPG
- [Hadamard's lemma](https://tamnd.github.io/math2/node/XT034AHK9E.md): statement XT034AHK9E
- [Heilbronn triangle problem](https://tamnd.github.io/math2/node/XT32T13KBY.md): problem XT32T13KBY
- [Dirichlet process](https://tamnd.github.io/math2/node/XT4SWG6J7N.md): concept XT4SWG6J7N
- [Jordan–Schur theorem](https://tamnd.github.io/math2/node/XT6EEXPD4P.md): statement XT6EEXPD4P
- [Homology with coefficients](https://tamnd.github.io/math2/node/XT7VWNTKGB.md): concept XT7VWNTKGB
- [Weyl–Titchmarsh theory](https://tamnd.github.io/math2/node/XT8PA31VAX.md): concept XT8PA31VAX
- [Rational variety](https://tamnd.github.io/math2/node/XV4K9GREJ9.md): concept XV4K9GREJ9
- [Conformal geometric algebra](https://tamnd.github.io/math2/node/XVBZJVQ3WA.md): definition XVBZJVQ3WA
- [Lebesgue integral](https://tamnd.github.io/math2/node/XVEWR93EMV.md): definition XVEWR93EMV
- [Well-powered category](https://tamnd.github.io/math2/node/XVG7FPG227.md): concept XVG7FPG227
- [Kruskal's tree theorem](https://tamnd.github.io/math2/node/XVHKCX4F5Y.md): statement XVHKCX4F5Y
- [Gromov–Witten invariant](https://tamnd.github.io/math2/node/XVJA0JD79P.md): concept XVJA0JD79P
- [Blaschke–Lebesgue theorem](https://tamnd.github.io/math2/node/XVNB6HV2Y1.md): statement XVNB6HV2Y1
- [Cardinal function](https://tamnd.github.io/math2/node/XVNJPJ53W8.md): concept XVNJPJ53W8
- [Semiprimary ring](https://tamnd.github.io/math2/node/XVP517QRH5.md): concept XVP517QRH5
- [Supporting hyperplane](https://tamnd.github.io/math2/node/XVPJ44E0KW.md): definition XVPJ44E0KW
- [Disk algebra](https://tamnd.github.io/math2/node/XVQN0GT988.md): definition XVQN0GT988
- [Bombieri–De Giorgi–Giusti theorem](https://tamnd.github.io/math2/node/XVSVNP3K9G.md): statement XVSVNP3K9G
- [Balian–Low theorem](https://tamnd.github.io/math2/node/XVVHKCYF2Z.md): statement XVVHKCYF2Z
- [Schramm–Loewner evolution](https://tamnd.github.io/math2/node/XW2BS4VG3P.md): definition XW2BS4VG3P
- [Conformal geometry](https://tamnd.github.io/math2/node/XW80KQJ62C.md): definition XW80KQJ62C
- [Étale morphism](https://tamnd.github.io/math2/node/XWCAS1ESAA.md): definition XWCAS1ESAA
- [Burnside problem](https://tamnd.github.io/math2/node/XWEMJ7T2XV.md): problem XWEMJ7T2XV
- [Three-term recurrence relation](https://tamnd.github.io/math2/node/XWJ786Z701.md): definition XWJ786Z701
- [Dynamical billiards](https://tamnd.github.io/math2/node/XWMKYE038F.md): concept XWMKYE038F
- [Landau's necessary density conditions for sampling](https://tamnd.github.io/math2/node/XWV6VS7BQM.md): statement XWV6VS7BQM
- [Numerical range](https://tamnd.github.io/math2/node/XWXZ0DGW3Z.md): concept XWXZ0DGW3Z
- [Poncelet's closure theorem](https://tamnd.github.io/math2/node/XWYXSRM5YS.md): statement XWYXSRM5YS
- [Enriched category](https://tamnd.github.io/math2/node/XX5MY5CCKX.md): definition XX5MY5CCKX
- [Kontsevich formality theorem](https://tamnd.github.io/math2/node/XXA9D45E0N.md): statement XXA9D45E0N
- [Degree of exactness of a quadrature rule](https://tamnd.github.io/math2/node/XXA9JSGW2C.md): definition XXA9JSGW2C
- [Factorization of polynomials over finite fields](https://tamnd.github.io/math2/node/XXQ378NQE2.md): definition XXQ378NQE2
- [Delta derivative on a time scale](https://tamnd.github.io/math2/node/XXQ92H7T83.md): concept XXQ92H7T83
- [Spectrum of an operator](https://tamnd.github.io/math2/node/XXXSST2DR5.md): definition XXXSST2DR5
- [Tilting theory](https://tamnd.github.io/math2/node/XXYQV6R1PR.md): concept XXYQV6R1PR
- [Knot concordance](https://tamnd.github.io/math2/node/XXZTMNBYFZ.md): definition XXZTMNBYFZ
- [Volterra equation of the second kind](https://tamnd.github.io/math2/node/XY2MRKDA23.md): definition XY2MRKDA23
- [Automorphism](https://tamnd.github.io/math2/node/XY6CJMDY5M.md): definition XY6CJMDY5M
- [Cospherical points](https://tamnd.github.io/math2/node/XY6XAJN7Z7.md): definition XY6XAJN7Z7
- [Symmetric kernel](https://tamnd.github.io/math2/node/XY7H05ANPD.md): definition XY7H05ANPD
- [Bôcher's theorem](https://tamnd.github.io/math2/node/XY81HRY5HP.md): statement XY81HRY5HP
- [Elliptic operator](https://tamnd.github.io/math2/node/XY8GEQSWPR.md): concept XY8GEQSWPR
- [Classification of regular polytopes](https://tamnd.github.io/math2/node/XYCD53PB5K.md): statement XYCD53PB5K
- [Day convolution](https://tamnd.github.io/math2/node/XYH30V13SE.md): definition XYH30V13SE
- [Umbilical point](https://tamnd.github.io/math2/node/XYJ6T5CKR6.md): concept XYJ6T5CKR6
- [Almost periodic function](https://tamnd.github.io/math2/node/XYPJ07PT5G.md): definition XYPJ07PT5G
- [Sequence space](https://tamnd.github.io/math2/node/XYTN19EPF2.md): concept XYTN19EPF2
- [List of small groups](https://tamnd.github.io/math2/node/XYVPYNVYBX.md): example XYVPYNVYBX
- [Conservation law](https://tamnd.github.io/math2/node/XYWMB0G011.md): definition XYWMB0G011
- [Skeleton of a category](https://tamnd.github.io/math2/node/XYZWTQD6NN.md): definition XYZWTQD6NN
- [Generalized mean inequality](https://tamnd.github.io/math2/node/XZ1EXQPJ6S.md): statement XZ1EXQPJ6S
- [Radial basis function interpolation](https://tamnd.github.io/math2/node/XZ33P3HBB9.md): concept XZ33P3HBB9
- [Fermat point](https://tamnd.github.io/math2/node/XZ3BMJARBR.md): definition XZ3BMJARBR
- [Morphism](https://tamnd.github.io/math2/node/XZ6NNH7JED.md): definition XZ6NNH7JED
- [Toric ideal](https://tamnd.github.io/math2/node/XZBPKQWD0P.md): concept XZBPKQWD0P
- [Erdős–Stone theorem](https://tamnd.github.io/math2/node/XZF9BFJFXB.md): statement XZF9BFJFXB
- [Fenchel–Moreau theorem](https://tamnd.github.io/math2/node/XZFYB2PBPD.md): statement XZFYB2PBPD
- [Homotopy equivalence](https://tamnd.github.io/math2/node/XZG6MTSWCN.md): concept XZG6MTSWCN
- [Kummer theory](https://tamnd.github.io/math2/node/XZQENYV15Y.md): concept XZQENYV15Y
- [Tight closure](https://tamnd.github.io/math2/node/XZS36C6226.md): definition XZS36C6226
- [Mathematical statistics](https://tamnd.github.io/math2/node/XZSA2NCATS.md): concept XZSA2NCATS
- [Primitive recursive function](https://tamnd.github.io/math2/node/XZV7BD3D66.md): definition XZV7BD3D66
- [Functional equations on groups](https://tamnd.github.io/math2/node/XZY0H7K195.md): definition XZY0H7K195
- [Hardy space](https://tamnd.github.io/math2/node/Y00AZRQE13.md): definition Y00AZRQE13
- [Confluent hypergeometric function](https://tamnd.github.io/math2/node/Y0BA4WG2W3.md): definition Y0BA4WG2W3
- [Finite-dimensional distributions](https://tamnd.github.io/math2/node/Y0CN7V3B7E.md): concept Y0CN7V3B7E
- [Involution](https://tamnd.github.io/math2/node/Y0EZW47S8M.md): definition Y0EZW47S8M
- [Tannery's theorem](https://tamnd.github.io/math2/node/Y0JH94QR80.md): statement Y0JH94QR80
- [Connected sum](https://tamnd.github.io/math2/node/Y0K0K5NGWW.md): concept Y0K0K5NGWW
- [Poisson–Mellin–Newton cycle](https://tamnd.github.io/math2/node/Y0PGRJHBYR.md): concept Y0PGRJHBYR
- [Simple random sample](https://tamnd.github.io/math2/node/Y0PTCJBPEA.md): concept Y0PTCJBPEA
- [Toric ideal](https://tamnd.github.io/math2/node/Y0SCJSFQ3B.md): definition Y0SCJSFQ3B
- [Invertible operator](https://tamnd.github.io/math2/node/Y0V7AP2E09.md): definition Y0V7AP2E09
- [Percolation threshold](https://tamnd.github.io/math2/node/Y0YW91AAKJ.md): definition Y0YW91AAKJ
- [Primitive permutation group](https://tamnd.github.io/math2/node/Y10S0751KG.md): concept Y10S0751KG
- [Rayleigh quotient iteration](https://tamnd.github.io/math2/node/Y119KHA4VG.md): statement Y119KHA4VG
- [Robin boundary condition](https://tamnd.github.io/math2/node/Y19WM1EMS7.md): definition Y19WM1EMS7
- [Falling and rising factorials](https://tamnd.github.io/math2/node/Y1BVY7J6AE.md): concept Y1BVY7J6AE
- [Carleson–Sjölin theorem](https://tamnd.github.io/math2/node/Y1E9SJC3Y0.md): statement Y1E9SJC3Y0
- [Liouville property of graphs](https://tamnd.github.io/math2/node/Y1EGR8DF84.md): concept Y1EGR8DF84
- [Symmetric successive over-relaxation](https://tamnd.github.io/math2/node/Y1G3NNJSQH.md): statement Y1G3NNJSQH
- [Monotone operator](https://tamnd.github.io/math2/node/Y1M03JQKKR.md): concept Y1M03JQKKR
- [Vlasov equation](https://tamnd.github.io/math2/node/Y1M6T8DEXK.md): definition Y1M6T8DEXK
- [Hahn polynomials](https://tamnd.github.io/math2/node/Y1M82VZYS2.md): concept Y1M82VZYS2
- [Pseudoanalytic function](https://tamnd.github.io/math2/node/Y1MH7WPRDZ.md): concept Y1MH7WPRDZ
- [Priority method](https://tamnd.github.io/math2/node/Y1N80NR1CG.md): definition Y1N80NR1CG
- [Group ring](https://tamnd.github.io/math2/node/Y1P96JQN34.md): definition Y1P96JQN34
- [Full subcategory](https://tamnd.github.io/math2/node/Y1RAX7VVBX.md): definition Y1RAX7VVBX
- [q-Pochhammer symbol](https://tamnd.github.io/math2/node/Y1X8ZZQ7VM.md): concept Y1X8ZZQ7VM
- [Szegő kernel](https://tamnd.github.io/math2/node/Y1XE9SQB5A.md): concept Y1XE9SQB5A
- [Universal cover](https://tamnd.github.io/math2/node/Y1YPE6PM52.md): concept Y1YPE6PM52
- [Mal'cev term](https://tamnd.github.io/math2/node/Y203PPTS61.md): concept Y203PPTS61
- [Orders of the elements of S3](https://tamnd.github.io/math2/node/Y20PEX049W.md): example Y20PEX049W
- [Algebraic K-theory of schemes](https://tamnd.github.io/math2/node/Y2467453Q5.md): concept Y2467453Q5
- [Laguerre's method](https://tamnd.github.io/math2/node/Y277TCXZ9Z.md): statement Y277TCXZ9Z
- [Covering space](https://tamnd.github.io/math2/node/Y28DDJ9F79.md): definition Y28DDJ9F79
- [Singer's theorem](https://tamnd.github.io/math2/node/Y2DKZH3CCT.md): statement Y2DKZH3CCT
- [Borg–Marchenko uniqueness theorem](https://tamnd.github.io/math2/node/Y2GYM93C1W.md): statement Y2GYM93C1W
- [Newton series](https://tamnd.github.io/math2/node/Y2HXSEA2EE.md): concept Y2HXSEA2EE
- [Monomorphism](https://tamnd.github.io/math2/node/Y2NP1JFR86.md): concept Y2NP1JFR86
- [Homotopy principle](https://tamnd.github.io/math2/node/Y2S9GWG1J1.md): concept Y2S9GWG1J1
- [Iwasawa theory](https://tamnd.github.io/math2/node/Y2SPPSQ5D4.md): concept Y2SPPSQ5D4
- [Finitely presented algebra](https://tamnd.github.io/math2/node/Y2XHZKR0TX.md): definition Y2XHZKR0TX
- [Enriques surface](https://tamnd.github.io/math2/node/Y30JE2DC6Y.md): concept Y30JE2DC6Y
- [Curvature form](https://tamnd.github.io/math2/node/Y33AJ7GZ6R.md): definition Y33AJ7GZ6R
- [Dolbeault cohomology](https://tamnd.github.io/math2/node/Y3900T2EY5.md): concept Y3900T2EY5
- [Henstock–Kurzweil integral](https://tamnd.github.io/math2/node/Y3KYKTAMCT.md): definition Y3KYKTAMCT
- [Rao–Blackwell theorem](https://tamnd.github.io/math2/node/Y3TX4J41TP.md): statement Y3TX4J41TP
- [Box spline](https://tamnd.github.io/math2/node/Y3WG213N0Q.md): definition Y3WG213N0Q
- [Fisher–Tippett–Gnedenko theorem](https://tamnd.github.io/math2/node/Y3Z99461HJ.md): statement Y3Z99461HJ
- [Evolute](https://tamnd.github.io/math2/node/Y3ZNV24V2C.md): concept Y3ZNV24V2C
- [Fox H-function](https://tamnd.github.io/math2/node/Y403TS95KG.md): definition Y403TS95KG
- [Serre's finiteness theorem](https://tamnd.github.io/math2/node/Y40ZRJJNH0.md): statement Y40ZRJJNH0
- [Kolmogorov–Sinai entropy](https://tamnd.github.io/math2/node/Y41SH8BJVT.md): definition Y41SH8BJVT
- [Gauss multiplication formula](https://tamnd.github.io/math2/node/Y4759V8CY6.md): statement Y4759V8CY6
- [Class](https://tamnd.github.io/math2/node/Y48FA69563.md): concept Y48FA69563
- [Kirchberger's theorem](https://tamnd.github.io/math2/node/Y48R8EJQT8.md): statement Y48R8EJQT8
- [Multialgebra](https://tamnd.github.io/math2/node/Y4AQFH6N3Y.md): definition Y4AQFH6N3Y
- [Sauer–Shelah lemma](https://tamnd.github.io/math2/node/Y4FQT5WPQJ.md): statement Y4FQT5WPQJ
- [Non-atomic measure](https://tamnd.github.io/math2/node/Y4JBCX161G.md): definition Y4JBCX161G
- [Class VII surface](https://tamnd.github.io/math2/node/Y4NSSJT4RC.md): concept Y4NSSJT4RC
- [Sorgenfrey line](https://tamnd.github.io/math2/node/Y4T3GVK8GV.md): counterexample Y4T3GVK8GV
- [Riccati differential equation of optimal control](https://tamnd.github.io/math2/node/Y4V154JWR7.md): concept Y4V154JWR7
- [Attainable null space](https://tamnd.github.io/math2/node/Y4WACJKC3S.md): definition Y4WACJKC3S
- [Intersection body](https://tamnd.github.io/math2/node/Y4ZE8FVZ8D.md): concept Y4ZE8FVZ8D
- [Eigenvalue perturbation](https://tamnd.github.io/math2/node/Y50G59S75Q.md): definition Y50G59S75Q
- [Associative quotient of a magma](https://tamnd.github.io/math2/node/Y50YJMKRAM.md): definition Y50YJMKRAM
- [Schur–Horn theorem](https://tamnd.github.io/math2/node/Y553MSYDAH.md): statement Y553MSYDAH
- [Galerkin method](https://tamnd.github.io/math2/node/Y5B0PEBKE8.md): concept Y5B0PEBKE8
- [Mal'cev term](https://tamnd.github.io/math2/node/Y5BF0G0T9X.md): definition Y5BF0G0T9X
- [Function of exponential order](https://tamnd.github.io/math2/node/Y5CC9MSGYM.md): definition Y5CC9MSGYM
- [Score test](https://tamnd.github.io/math2/node/Y5HMW0BDTP.md): concept Y5HMW0BDTP
- [Semi-continuity](https://tamnd.github.io/math2/node/Y5V5CN8FGN.md): definition Y5V5CN8FGN
- [Generic flatness](https://tamnd.github.io/math2/node/Y60D7BN1BN.md): statement Y60D7BN1BN
- [Completion of a topological group](https://tamnd.github.io/math2/node/Y63847Z06N.md): definition Y63847Z06N
- [Perimeter of a set](https://tamnd.github.io/math2/node/Y68BHXZJBV.md): concept Y68BHXZJBV
- [Structure](https://tamnd.github.io/math2/node/Y6C40MHZZC.md): definition Y6C40MHZZC
- [Divisor class group](https://tamnd.github.io/math2/node/Y6D894R6HB.md): definition Y6D894R6HB
- [Sine and cosine series](https://tamnd.github.io/math2/node/Y6EPBG6S8G.md): concept Y6EPBG6S8G
- [Injectivity radius](https://tamnd.github.io/math2/node/Y6ZSKVAE3V.md): concept Y6ZSKVAE3V
- [Glimm scheme](https://tamnd.github.io/math2/node/Y70AC2X30T.md): concept Y70AC2X30T
- [Pseudoconvexity](https://tamnd.github.io/math2/node/Y71Z3MN0S3.md): definition Y71Z3MN0S3
- [Numerical range](https://tamnd.github.io/math2/node/Y730CQ2J1F.md): definition Y730CQ2J1F
- [Castelnuovo–Mumford regularity](https://tamnd.github.io/math2/node/Y76T2DQJPQ.md): concept Y76T2DQJPQ
- [Hurewicz fibration](https://tamnd.github.io/math2/node/Y773B7EQA7.md): concept Y773B7EQA7
- [Universal coalgebra](https://tamnd.github.io/math2/node/Y77K0TEKN7.md): definition Y77K0TEKN7
- [Circle average](https://tamnd.github.io/math2/node/Y7A1TP40MB.md): concept Y7A1TP40MB
- [Homogeneity equation](https://tamnd.github.io/math2/node/Y7BD5ECENG.md): concept Y7BD5ECENG
- [de Branges space](https://tamnd.github.io/math2/node/Y7JP4NA725.md): definition Y7JP4NA725
- [Lagrange polynomial](https://tamnd.github.io/math2/node/Y7KSGE97GK.md): concept Y7KSGE97GK
- [Complex torus](https://tamnd.github.io/math2/node/Y7MA6KBE9M.md): concept Y7MA6KBE9M
- [Flat norm](https://tamnd.github.io/math2/node/Y7P66D41Z8.md): concept Y7P66D41Z8
- [Bounded symmetric domain](https://tamnd.github.io/math2/node/Y7Q2SNKG0C.md): concept Y7Q2SNKG0C
- [Canonical height](https://tamnd.github.io/math2/node/Y7RP7KSH2C.md): definition Y7RP7KSH2C
- [Cheeger constant](https://tamnd.github.io/math2/node/Y7S6W12N64.md): concept Y7S6W12N64
- [Brody hyperbolicity](https://tamnd.github.io/math2/node/Y7W36GJDJD.md): concept Y7W36GJDJD
- [Hypergeometric differential equation](https://tamnd.github.io/math2/node/Y7Y0BY3JGJ.md): definition Y7Y0BY3JGJ
- [Exponential map](https://tamnd.github.io/math2/node/Y7Z0AGY1A7.md): concept Y7Z0AGY1A7
- [Non-uniform rational B-spline](https://tamnd.github.io/math2/node/Y8B2N9ERPE.md): concept Y8B2N9ERPE
- [Brauer–Manin obstruction](https://tamnd.github.io/math2/node/Y8H0JWZ2ZQ.md): definition Y8H0JWZ2ZQ
- [Prime knot](https://tamnd.github.io/math2/node/Y8KSN16X06.md): definition Y8KSN16X06
- [Infinite compositions of functions](https://tamnd.github.io/math2/node/Y8RZ73N0JA.md): definition Y8RZ73N0JA
- [Second-order arithmetic](https://tamnd.github.io/math2/node/Y92NXGEQBD.md): definition Y92NXGEQBD
- [Hodge structure](https://tamnd.github.io/math2/node/Y92V5H3KJ2.md): definition Y92V5H3KJ2
- [Hazard function](https://tamnd.github.io/math2/node/Y959NKQ3XW.md): concept Y959NKQ3XW
- [Continuous and residual spectrum](https://tamnd.github.io/math2/node/Y95W7EME5Q.md): definition Y95W7EME5Q
- [Measure with density](https://tamnd.github.io/math2/node/Y98C5T956Q.md): concept Y98C5T956Q
- [Germ](https://tamnd.github.io/math2/node/Y98MD3WVTG.md): definition Y98MD3WVTG
- [Stochastic differential equation](https://tamnd.github.io/math2/node/Y99MCTGZHZ.md): definition Y99MCTGZHZ
- [Abelian category](https://tamnd.github.io/math2/node/Y9CZ3MTNF1.md): concept Y9CZ3MTNF1
- [Girard's theorem](https://tamnd.github.io/math2/node/Y9DM9B0RA3.md): statement Y9DM9B0RA3
- [Stieltjes moment problem](https://tamnd.github.io/math2/node/Y9KH8P9JC0.md): problem Y9KH8P9JC0
- [Affine variety](https://tamnd.github.io/math2/node/YA2NN86Y11.md): definition YA2NN86Y11
- [Permutation test](https://tamnd.github.io/math2/node/YA786MZ8QD.md): definition YA786MZ8QD
- [Rigid analytic space](https://tamnd.github.io/math2/node/YAH6MZVG7K.md): concept YAH6MZVG7K
- [Minkowski–Hlawka theorem](https://tamnd.github.io/math2/node/YANP1PHBQ3.md): statement YANP1PHBQ3
- [Branched covering](https://tamnd.github.io/math2/node/YARENZRXTD.md): concept YARENZRXTD
- [Dirichlet eigenvalue](https://tamnd.github.io/math2/node/YATEFAKZYB.md): definition YATEFAKZYB
- [Lattice of subvarieties](https://tamnd.github.io/math2/node/YAVH4RAEYJ.md): definition YAVH4RAEYJ
- [Euler's continued fraction formula](https://tamnd.github.io/math2/node/YB35R7F7RZ.md): statement YB35R7F7RZ
- [Fractional ideal](https://tamnd.github.io/math2/node/YB4JTB4WJN.md): concept YB4JTB4WJN
- [Ree groups](https://tamnd.github.io/math2/node/YB62XEQJAC.md): object YB62XEQJAC
- [Hellinger distance](https://tamnd.github.io/math2/node/YB7GN4EAF7.md): definition YB7GN4EAF7
- [Sz.-Nagy's dilation theorem](https://tamnd.github.io/math2/node/YB7Q3EK0ND.md): statement YB7Q3EK0ND
- [Incomplete gamma function](https://tamnd.github.io/math2/node/YB7ZV0HG7K.md): concept YB7ZV0HG7K
- [Grushko theorem](https://tamnd.github.io/math2/node/YB8V6TQDYE.md): statement YB8V6TQDYE
- [Fisher's exact test](https://tamnd.github.io/math2/node/YB9390ZE4W.md): definition YB9390ZE4W
- [Voter model](https://tamnd.github.io/math2/node/YBAE5QCH1A.md): example YBAE5QCH1A
- [Thurston norm](https://tamnd.github.io/math2/node/YBAPZPEDWQ.md): definition YBAPZPEDWQ
- [Böttcher's equation](https://tamnd.github.io/math2/node/YBBDYHBADD.md): statement YBBDYHBADD
- [Kadomtsev–Petviashvili equation](https://tamnd.github.io/math2/node/YBBS082ZY1.md): definition YBBS082ZY1
- [Sharkovskii's theorem](https://tamnd.github.io/math2/node/YBCZ9VY7SX.md): statement YBCZ9VY7SX
- [Schottky group](https://tamnd.github.io/math2/node/YBERRJMXT6.md): concept YBERRJMXT6
- [Order filter](https://tamnd.github.io/math2/node/YBG22HS28V.md): definition YBG22HS28V
- [Zariski's lemma](https://tamnd.github.io/math2/node/YBJ2VQHFDQ.md): statement YBJ2VQHFDQ
- [Balinski's theorem](https://tamnd.github.io/math2/node/YBNZWCG8AC.md): statement YBNZWCG8AC
- [q-gamma function](https://tamnd.github.io/math2/node/YBSCJY6BY2.md): definition YBSCJY6BY2
- [Sierpiński's theorem on measurable midpoint convex functions](https://tamnd.github.io/math2/node/YBVSDT89T6.md): statement YBVSDT89T6
- [Approximation by convolution with mollifiers](https://tamnd.github.io/math2/node/YBYE4TBNJV.md): statement YBYE4TBNJV
- [Buchberger's algorithm](https://tamnd.github.io/math2/node/YC90FXM4SA.md): definition YC90FXM4SA
- [Evolute](https://tamnd.github.io/math2/node/YC9E5JJDDF.md): definition YC9E5JJDDF
- [Hessenberg matrix](https://tamnd.github.io/math2/node/YCEEWJTX56.md): concept YCEEWJTX56
- [Boundary layer](https://tamnd.github.io/math2/node/YCFPD0GE5G.md): concept YCFPD0GE5G
- [Integral transform](https://tamnd.github.io/math2/node/YCK709QB06.md): concept YCK709QB06
- [Hilbert's twenty-third problem](https://tamnd.github.io/math2/node/YCKDR0S7V1.md): problem YCKDR0S7V1
- [Homology with coefficients](https://tamnd.github.io/math2/node/YCQFP0KN48.md): definition YCQFP0KN48
- [Abscissa of absolute convergence](https://tamnd.github.io/math2/node/YCSH2WK69M.md): concept YCSH2WK69M
- [Autocorrelation function](https://tamnd.github.io/math2/node/YCZHYY8VE5.md): concept YCZHYY8VE5
- [Homotopy groups of spheres](https://tamnd.github.io/math2/node/YD6VAXXSBM.md): concept YD6VAXXSBM
- [Dini derivative](https://tamnd.github.io/math2/node/YD8DB4PMD4.md): definition YD8DB4PMD4
- [Adjoint state method](https://tamnd.github.io/math2/node/YD8SA6WS01.md): concept YD8SA6WS01
- [Barnes G-function](https://tamnd.github.io/math2/node/YDAKQ7EH52.md): definition YDAKQ7EH52
- [Artin–Schreier theorem on formally real fields](https://tamnd.github.io/math2/node/YDBXV8JMYE.md): statement YDBXV8JMYE
- [Amitsur–Levitzki theorem](https://tamnd.github.io/math2/node/YDH60KPTWT.md): statement YDH60KPTWT
- [Brjuno number](https://tamnd.github.io/math2/node/YDK77Q21X3.md): definition YDK77Q21X3
- [Successive minima](https://tamnd.github.io/math2/node/YDMBKD8FP2.md): concept YDMBKD8FP2
- [Error of best approximation](https://tamnd.github.io/math2/node/YDN0T4MEX0.md): concept YDN0T4MEX0
- [Frequentist inference](https://tamnd.github.io/math2/node/YDSNHKM3EG.md): concept YDSNHKM3EG
- [Going-down theorem](https://tamnd.github.io/math2/node/YE52MQ13MV.md): statement YE52MQ13MV
- [Mazur–Orlicz theorem](https://tamnd.github.io/math2/node/YE5439NH1T.md): statement YE5439NH1T
- [Arnold's classification of simple singularities](https://tamnd.github.io/math2/node/YE5F548WMS.md): statement YE5F548WMS
- [Stationary set](https://tamnd.github.io/math2/node/YE6WZR46RQ.md): concept YE6WZR46RQ
- [Chebyshev constant](https://tamnd.github.io/math2/node/YE71ZHHGKW.md): definition YE71ZHHGKW
- [Hill differential equation](https://tamnd.github.io/math2/node/YE7H8DGDPY.md): example YE7H8DGDPY
- [Bieberbach theorems](https://tamnd.github.io/math2/node/YE87R5CDJY.md): statement YE87R5CDJY
- [Sturm separation theorem](https://tamnd.github.io/math2/node/YEBPTXZ1FS.md): statement YEBPTXZ1FS
- [Random dynamical system](https://tamnd.github.io/math2/node/YEFR2GGGJN.md): definition YEFR2GGGJN
- [Nagata's example of a Noetherian ring of infinite dimension](https://tamnd.github.io/math2/node/YEJVKCY78E.md): counterexample YEJVKCY78E
- [von Neumann bounded set](https://tamnd.github.io/math2/node/YENN31BFZA.md): definition YENN31BFZA
- [Schläfli symbol](https://tamnd.github.io/math2/node/YES06WTNRZ.md): definition YES06WTNRZ
- [HNN extension](https://tamnd.github.io/math2/node/YEYERRY37K.md): concept YEYERRY37K
- [Quasi-analytic function](https://tamnd.github.io/math2/node/YF1KAWB5MB.md): concept YF1KAWB5MB
- [Unirational variety](https://tamnd.github.io/math2/node/YF1SE8WJK3.md): concept YF1SE8WJK3
- [Robin constant](https://tamnd.github.io/math2/node/YF1XA47RYT.md): definition YF1XA47RYT
- [Limit set](https://tamnd.github.io/math2/node/YF4XH2BEG5.md): concept YF4XH2BEG5
- [Excellent ring](https://tamnd.github.io/math2/node/YF7YFQ58QJ.md): concept YF7YFQ58QJ
- [Zero morphism](https://tamnd.github.io/math2/node/YF9F9EG6M3.md): concept YF9F9EG6M3
- [Generalized cohomology theory](https://tamnd.github.io/math2/node/YFAKVC0WEJ.md): definition YFAKVC0WEJ
- [Isomorphism theorem for measure algebras](https://tamnd.github.io/math2/node/YFAQGE2ATQ.md): statement YFAQGE2ATQ
- [Hilbert's Theorem 90](https://tamnd.github.io/math2/node/YFF8B5A0QG.md): statement YFF8B5A0QG
- [Order of a PDE](https://tamnd.github.io/math2/node/YFFEKD37RP.md): definition YFFEKD37RP
- [Holomorphic functional calculus](https://tamnd.github.io/math2/node/YFKA3KVTGR.md): concept YFKA3KVTGR
- [Malcev completion](https://tamnd.github.io/math2/node/YFSKBA5BDZ.md): definition YFSKBA5BDZ
- [Birkhoff–Frink representation of algebraic closure systems](https://tamnd.github.io/math2/node/YFT50501A0.md): statement YFT50501A0
- [(∞,1)-category](https://tamnd.github.io/math2/node/YFY1FWA1MY.md): concept YFY1FWA1MY
- [Uniform boundedness principle](https://tamnd.github.io/math2/node/YG0042V8CJ.md): statement YG0042V8CJ
- [Finite field arithmetic](https://tamnd.github.io/math2/node/YG3HWE6GEX.md): concept YG3HWE6GEX
- [Dirichlet energy](https://tamnd.github.io/math2/node/YG3W8CBB3E.md): concept YG3W8CBB3E
- [Dissipative operator](https://tamnd.github.io/math2/node/YG46P1T47G.md): definition YG46P1T47G
- [Choquet capacity](https://tamnd.github.io/math2/node/YG74XH44H6.md): definition YG74XH44H6
- [Operator theory](https://tamnd.github.io/math2/node/YG9NR68ZH3.md): concept YG9NR68ZH3
- [Projective object](https://tamnd.github.io/math2/node/YGHYD555Y7.md): definition YGHYD555Y7
- [Chebyshev–Markov–Stieltjes inequalities](https://tamnd.github.io/math2/node/YGJZCD37ZY.md): statement YGJZCD37ZY
- [Reciprocal gamma function](https://tamnd.github.io/math2/node/YGK9CD7SME.md): definition YGK9CD7SME
- [Heat kernel](https://tamnd.github.io/math2/node/YGMTZJY1MR.md): definition YGMTZJY1MR
- [Period map](https://tamnd.github.io/math2/node/YGNXQVTWP3.md): concept YGNXQVTWP3
- [Bonferroni correction](https://tamnd.github.io/math2/node/YGP3Q2EF31.md): definition YGP3Q2EF31
- [Quantum invariant of 3-manifolds](https://tamnd.github.io/math2/node/YGPR4VR3GP.md): concept YGPR4VR3GP
- [Reflexive operator algebra](https://tamnd.github.io/math2/node/YGR48AJMFD.md): definition YGR48AJMFD
- [Gromov's compactness theorem](https://tamnd.github.io/math2/node/YGSKZFTEX1.md): statement YGSKZFTEX1
- [A Dedekind UFD is a PID](https://tamnd.github.io/math2/node/YGV5S3MRD6.md): statement YGV5S3MRD6
- [Finitely based algebra](https://tamnd.github.io/math2/node/YGYZAQ12MZ.md): concept YGYZAQ12MZ
- [Triangulation](https://tamnd.github.io/math2/node/YH170WBQ6M.md): concept YH170WBQ6M
- [Scaling function and refinement equation](https://tamnd.github.io/math2/node/YH4BERB952.md): concept YH4BERB952
- [Synthetic geometry](https://tamnd.github.io/math2/node/YH5PGKNZKD.md): definition YH5PGKNZKD
- [Semilocal ring](https://tamnd.github.io/math2/node/YH75M7XAM1.md): definition YH75M7XAM1
- [One-parameter subgroup](https://tamnd.github.io/math2/node/YHA64AP1HN.md): concept YHA64AP1HN
- [Hilbert's axioms](https://tamnd.github.io/math2/node/YHFDV8J1KV.md): definition YHFDV8J1KV
- [Langlands classification](https://tamnd.github.io/math2/node/YHFP9HGBTT.md): statement YHFP9HGBTT
- [Automorphism group of a bounded domain is a Lie group](https://tamnd.github.io/math2/node/YHGS413E8B.md): statement YHGS413E8B
- [Poincaré–Lindstedt method](https://tamnd.github.io/math2/node/YHJHPRKQ6D.md): concept YHJHPRKQ6D
- [Eilenberg–Steenrod axioms](https://tamnd.github.io/math2/node/YHK51GHK2T.md): concept YHK51GHK2T
- [Perspective projection](https://tamnd.github.io/math2/node/YHKSDF59HK.md): concept YHKSDF59HK
- [Doob decomposition theorem](https://tamnd.github.io/math2/node/YHSSQ67GDS.md): statement YHSSQ67GDS
- [Lax–Wendroff method](https://tamnd.github.io/math2/node/YHZ9BCNDRZ.md): concept YHZ9BCNDRZ
- [Fredholm integral equation](https://tamnd.github.io/math2/node/YJ1GGHS71H.md): definition YJ1GGHS71H
- [Lommel function](https://tamnd.github.io/math2/node/YJ1JWF6ZDG.md): concept YJ1JWF6ZDG
- [Chandrasekhar's X- and Y-functions](https://tamnd.github.io/math2/node/YJ1M0FYXJD.md): example YJ1M0FYXJD
- [Symmetric decreasing rearrangement](https://tamnd.github.io/math2/node/YJ2VVJZBX4.md): definition YJ2VVJZBX4
- [Sphere packing in dimensions 8 and 24](https://tamnd.github.io/math2/node/YJ48S89BZA.md): statement YJ48S89BZA
- [Combinatorics on words](https://tamnd.github.io/math2/node/YJ961M051W.md): concept YJ961M051W
- [WKB approximation](https://tamnd.github.io/math2/node/YJEBN8N3G8.md): concept YJEBN8N3G8
- [Trigonometric moment problem](https://tamnd.github.io/math2/node/YJFP002B62.md): problem YJFP002B62
- [Ramification group](https://tamnd.github.io/math2/node/YJK4WF6YWV.md): definition YJK4WF6YWV
- [Analytic combinatorics](https://tamnd.github.io/math2/node/YJK71RKVEB.md): concept YJK71RKVEB
- [Wedderburn's theorem on simple Artinian rings](https://tamnd.github.io/math2/node/YJM07BT0KK.md): statement YJM07BT0KK
- [Ruled surface](https://tamnd.github.io/math2/node/YJMGEKHYEC.md): definition YJMGEKHYEC
- [Commuting elements](https://tamnd.github.io/math2/node/YJRBCWQV6B.md): concept YJRBCWQV6B
- [3-j symbol](https://tamnd.github.io/math2/node/YJRJVZ998D.md): concept YJRJVZ998D
- [Inverse Galois problem](https://tamnd.github.io/math2/node/YJS6X8JN7D.md): problem YJS6X8JN7D
- [Mirror symmetry](https://tamnd.github.io/math2/node/YK3DMN2NSK.md): concept YK3DMN2NSK
- [Ambiguity function](https://tamnd.github.io/math2/node/YK8121V1PN.md): definition YK8121V1PN
- [Discrete Dirichlet problem](https://tamnd.github.io/math2/node/YKETR75ZYW.md): problem YKETR75ZYW
- [Kauffman bracket](https://tamnd.github.io/math2/node/YKN9BKTCYN.md): definition YKN9BKTCYN
- [Theorema Egregium](https://tamnd.github.io/math2/node/YKSA8SJ81V.md): statement YKSA8SJ81V
- [Developable surface](https://tamnd.github.io/math2/node/YKVHDB52XW.md): concept YKVHDB52XW
- [Circle average](https://tamnd.github.io/math2/node/YKZ1FP1PAG.md): definition YKZ1FP1PAG
- [Likelihood-ratio test](https://tamnd.github.io/math2/node/YM1BNA80RC.md): concept YM1BNA80RC
- [Rellich–Kondrachov theorem](https://tamnd.github.io/math2/node/YM3JZ34PME.md): statement YM3JZ34PME
- [Multiplication theorem](https://tamnd.github.io/math2/node/YM68BCKBXC.md): statement YM68BCKBXC
- [Ordinary least squares](https://tamnd.github.io/math2/node/YM708KHV2W.md): concept YM708KHV2W
- [Local asymptotic normality](https://tamnd.github.io/math2/node/YMDADSX0E1.md): definition YMDADSX0E1
- [Oscillation of a function](https://tamnd.github.io/math2/node/YMGHNBDZCP.md): definition YMGHNBDZCP
- [Bochner–Martinelli formula](https://tamnd.github.io/math2/node/YMHYAEJQ80.md): statement YMHYAEJQ80
- [Site](https://tamnd.github.io/math2/node/YMK0D06F4B.md): definition YMK0D06F4B
- [Bochner's tube theorem](https://tamnd.github.io/math2/node/YMP12DXCSP.md): statement YMP12DXCSP
- [Bredon cohomology](https://tamnd.github.io/math2/node/YN280Q5G49.md): concept YN280Q5G49
- [Frequentist inference](https://tamnd.github.io/math2/node/YNBTV2S7SR.md): definition YNBTV2S7SR
- [Incidence matrix](https://tamnd.github.io/math2/node/YNFT400EV9.md): definition YNFT400EV9
- [Harnack's inequality](https://tamnd.github.io/math2/node/YNH832D50W.md): statement YNH832D50W
- [Gumm terms](https://tamnd.github.io/math2/node/YNM6V0R269.md): definition YNM6V0R269
- [Synchronization of chaos](https://tamnd.github.io/math2/node/YP1PMW0HE3.md): concept YP1PMW0HE3
- [Nonassociative ring](https://tamnd.github.io/math2/node/YP5TQ6WFVN.md): concept YP5TQ6WFVN
- [Chevalley group](https://tamnd.github.io/math2/node/YPCH4Q0V34.md): concept YPCH4Q0V34
- [Integer factorization](https://tamnd.github.io/math2/node/YPE65X5HJ4.md): problem YPE65X5HJ4
- [Affine arithmetic](https://tamnd.github.io/math2/node/YPE8E9CVXP.md): concept YPE8E9CVXP
- [Approximation theory](https://tamnd.github.io/math2/node/YPJSH79SHQ.md): concept YPJSH79SHQ
- [Subfactor](https://tamnd.github.io/math2/node/YPPPPHE6TN.md): definition YPPPPHE6TN
- [1 + 1 + 1 + 1 + ⋯](https://tamnd.github.io/math2/node/YPRZTC01QC.md): example YPRZTC01QC
- [Baker–Gammel–Wills conjecture](https://tamnd.github.io/math2/node/YPTNBRWRY7.md): conjecture YPTNBRWRY7
- [Hurwitz zeta function](https://tamnd.github.io/math2/node/YQ21C2N2HX.md): definition YQ21C2N2HX
- [Locally finite measure](https://tamnd.github.io/math2/node/YQ5C1SRE0W.md): concept YQ5C1SRE0W
- [Finitely presented module](https://tamnd.github.io/math2/node/YQ5HQP6HGE.md): definition YQ5HQP6HGE
- [Commutant](https://tamnd.github.io/math2/node/YQ626FK82G.md): definition YQ626FK82G
- [Functional inequality](https://tamnd.github.io/math2/node/YQB2B0EXY8.md): concept YQB2B0EXY8
- [Artinian module](https://tamnd.github.io/math2/node/YQNZTQGP1X.md): concept YQNZTQGP1X
- [Hahn series](https://tamnd.github.io/math2/node/YQP8V1KKMW.md): concept YQP8V1KKMW
- [Bessel polynomials](https://tamnd.github.io/math2/node/YQQW8X2ZZJ.md): definition YQQW8X2ZZJ
- [Observability](https://tamnd.github.io/math2/node/YQYJTKD0W2.md): definition YQYJTKD0W2
- [Diagonally dominant matrix](https://tamnd.github.io/math2/node/YR2MKGA2CE.md): definition YR2MKGA2CE
- [Free boundary problem](https://tamnd.github.io/math2/node/YR42E2W19Q.md): problem YR42E2W19Q
- [Controllability](https://tamnd.github.io/math2/node/YR43719RWE.md): definition YR43719RWE
- [Likelihood principle](https://tamnd.github.io/math2/node/YR5RAT9HWZ.md): statement YR5RAT9HWZ
- [Model complete theory](https://tamnd.github.io/math2/node/YR6HCMABTE.md): concept YR6HCMABTE
- [Functional differential equation](https://tamnd.github.io/math2/node/YRBPQS8GHM.md): definition YRBPQS8GHM
- [Nevanlinna parametrization of solutions](https://tamnd.github.io/math2/node/YRBT3ZHSV3.md): statement YRBT3ZHSV3
- [Koszul algebra](https://tamnd.github.io/math2/node/YREPR562ES.md): definition YREPR562ES
- [Kolmogorov automorphism](https://tamnd.github.io/math2/node/YRHJA7C7HS.md): concept YRHJA7C7HS
- [Integro-differential equation](https://tamnd.github.io/math2/node/YRKW2KX2CW.md): concept YRKW2KX2CW
- [Adapted process](https://tamnd.github.io/math2/node/YRNFE016FF.md): concept YRNFE016FF
- [Order of an entire function](https://tamnd.github.io/math2/node/YRTCH4Q9GA.md): definition YRTCH4Q9GA
- [Convex body](https://tamnd.github.io/math2/node/YRTFS7NHM8.md): definition YRTFS7NHM8
- [Tarski's undefinability theorem](https://tamnd.github.io/math2/node/YRV6SYK2BY.md): statement YRV6SYK2BY
- [Three-term recurrence for orthogonal polynomials](https://tamnd.github.io/math2/node/YRYWCP9TYV.md): statement YRYWCP9TYV
- [Gaussian binomial coefficient](https://tamnd.github.io/math2/node/YS09Z1YNEK.md): concept YS09Z1YNEK
- [Nonlinear approximation](https://tamnd.github.io/math2/node/YS1GRQRVY7.md): concept YS1GRQRVY7
- [Curse of dimensionality](https://tamnd.github.io/math2/node/YS424FTY4E.md): concept YS424FTY4E
- [Axiomatic potential theory](https://tamnd.github.io/math2/node/YS7RF1PE42.md): concept YS7RF1PE42
- [Nome](https://tamnd.github.io/math2/node/YSAZVCZAZC.md): concept YSAZVCZAZC
- [Lambert summation](https://tamnd.github.io/math2/node/YSMBT6KQ4N.md): definition YSMBT6KQ4N
- [Fuchsian equation](https://tamnd.github.io/math2/node/YT2B9KQ0BG.md): definition YT2B9KQ0BG
- [Compact quantum group](https://tamnd.github.io/math2/node/YT35QHA4EA.md): definition YT35QHA4EA
- [Zeta-regularized determinant](https://tamnd.github.io/math2/node/YT3ZQ2F7FG.md): concept YT3ZQ2F7FG
- [Product order](https://tamnd.github.io/math2/node/YT45QJKPNK.md): definition YT45QJKPNK
- [Nevanlinna characteristic](https://tamnd.github.io/math2/node/YT6Z0GWS0W.md): concept YT6Z0GWS0W
- [Expansion in spherical harmonics](https://tamnd.github.io/math2/node/YTPE0GBKWW.md): statement YTPE0GBKWW
- [Time-optimal control](https://tamnd.github.io/math2/node/YTR959CYMX.md): problem YTR959CYMX
- [Classifying topos](https://tamnd.github.io/math2/node/YV297TR3SC.md): definition YV297TR3SC
- [Mellin transform of a dilation](https://tamnd.github.io/math2/node/YV5VNM5ADP.md): statement YV5VNM5ADP
- [Quasi-everywhere](https://tamnd.github.io/math2/node/YV9QXZA4YG.md): definition YV9QXZA4YG
- [Vaught conjecture](https://tamnd.github.io/math2/node/YVBHE2SEE6.md): conjecture YVBHE2SEE6
- [Clenshaw–Curtis quadrature](https://tamnd.github.io/math2/node/YVDWN0KVG4.md): definition YVDWN0KVG4
- [Mersenne Twister](https://tamnd.github.io/math2/node/YVEA283NZW.md): object YVEA283NZW
- [Continuous and residual spectrum](https://tamnd.github.io/math2/node/YVH149HCAG.md): concept YVH149HCAG
- [Simplicial polytope](https://tamnd.github.io/math2/node/YVJ15PGKDM.md): concept YVJ15PGKDM
- [Surreal number](https://tamnd.github.io/math2/node/YVPTBY45PK.md): object YVPTBY45PK
- [Valuation ring](https://tamnd.github.io/math2/node/YVQ9CKGA0T.md): definition YVQ9CKGA0T
- [d'Alembert's formula](https://tamnd.github.io/math2/node/YVQRAPSF77.md): statement YVQRAPSF77
- [Topology of uniform convergence](https://tamnd.github.io/math2/node/YW5P7NEKR4.md): concept YW5P7NEKR4
- [Jacobi field](https://tamnd.github.io/math2/node/YW6W2JXMQB.md): definition YW6W2JXMQB
- [Deformation retract](https://tamnd.github.io/math2/node/YW8HBW1GJX.md): definition YW8HBW1GJX
- [Club set](https://tamnd.github.io/math2/node/YW8YSCFPG5.md): concept YW8YSCFPG5
- [Toric variety](https://tamnd.github.io/math2/node/YWB981MEF2.md): concept YWB981MEF2
- [Instanton](https://tamnd.github.io/math2/node/YWBTGC7MFA.md): concept YWBTGC7MFA
- [Isomorphism](https://tamnd.github.io/math2/node/YWTERZZME5.md): definition YWTERZZME5
- [Oriented matroid](https://tamnd.github.io/math2/node/YWTT6FD50Y.md): definition YWTT6FD50Y
- [Chebyshev nodes](https://tamnd.github.io/math2/node/YWTXGDC427.md): definition YWTXGDC427
- [Weierstrass–Enneper parameterization](https://tamnd.github.io/math2/node/YWV6J23AP6.md): statement YWV6J23AP6
- [Uniform space](https://tamnd.github.io/math2/node/YWW42G978D.md): definition YWW42G978D
- [Trace form](https://tamnd.github.io/math2/node/YWX2R53ED6.md): concept YWX2R53ED6
- [Smn theorem](https://tamnd.github.io/math2/node/YWX4FCFPAP.md): statement YWX4FCFPAP
- [Rate of convergence](https://tamnd.github.io/math2/node/YWZQC5E54B.md): concept YWZQC5E54B
- [Koszul duality](https://tamnd.github.io/math2/node/YX0J6TVGZ8.md): definition YX0J6TVGZ8
- [Simplex](https://tamnd.github.io/math2/node/YX1DJK2QAG.md): concept YX1DJK2QAG
- [Loewner differential equation](https://tamnd.github.io/math2/node/YX386FVZQR.md): definition YX386FVZQR
- [Exceptional Lie group G2](https://tamnd.github.io/math2/node/YX5FFSCR8P.md): example YX5FFSCR8P
- [Hadwiger–Nelson problem](https://tamnd.github.io/math2/node/YX985SQG26.md): problem YX985SQG26
- [Uniform module](https://tamnd.github.io/math2/node/YXEVE39Q31.md): concept YXEVE39Q31
- [Regularity of minimizers](https://tamnd.github.io/math2/node/YXJ10HDDBJ.md): definition YXJ10HDDBJ
- [Limiting subdifferential](https://tamnd.github.io/math2/node/YXJDQM9VVD.md): definition YXJDQM9VVD
- [Riesz potential operator](https://tamnd.github.io/math2/node/YXJPPC9CXS.md): definition YXJPPC9CXS
- [Group cohomology](https://tamnd.github.io/math2/node/YXJXA0Q52R.md): concept YXJXA0Q52R
- [Projective object](https://tamnd.github.io/math2/node/YXMXAGVC1Q.md): concept YXMXAGVC1Q
- [Inversion in a circle](https://tamnd.github.io/math2/node/YXST7NGMTC.md): concept YXST7NGMTC
- [Arakelov theory](https://tamnd.github.io/math2/node/YXWGV80Z2V.md): concept YXWGV80Z2V
- [Classification of covering spaces](https://tamnd.github.io/math2/node/YY277RBF8Z.md): statement YY277RBF8Z
- [Analysis on fractals](https://tamnd.github.io/math2/node/YY46TGRVBG.md): concept YY46TGRVBG
- [Fractional differential equation](https://tamnd.github.io/math2/node/YY4XSJX81P.md): concept YY4XSJX81P
- [Local truncation error](https://tamnd.github.io/math2/node/YY6P3RSDWC.md): definition YY6P3RSDWC
- [Gelfand–Kirillov dimension](https://tamnd.github.io/math2/node/YY8CG0NZ9C.md): definition YY8CG0NZ9C
- [Bunimovich stadium](https://tamnd.github.io/math2/node/YYB1JP7TWJ.md): example YYB1JP7TWJ
- [Kirby–Siebenmann class](https://tamnd.github.io/math2/node/YYE2JHRR32.md): definition YYE2JHRR32
- [J-homomorphism](https://tamnd.github.io/math2/node/YYJPRD1HYW.md): concept YYJPRD1HYW
- [Measurable function](https://tamnd.github.io/math2/node/YYNCJSR0PJ.md): concept YYNCJSR0PJ
- [Connes embedding problem](https://tamnd.github.io/math2/node/YYP6QNRERY.md): problem YYP6QNRERY
- [Laplace's integral for Legendre polynomials](https://tamnd.github.io/math2/node/YYQG5P32GX.md): statement YYQG5P32GX
- [Morse index theorem for geodesics](https://tamnd.github.io/math2/node/YZ4268SK8G.md): statement YZ4268SK8G
- [Manifold with boundary](https://tamnd.github.io/math2/node/YZ8AN4J2KD.md): definition YZ8AN4J2KD
- [Lying over theorem](https://tamnd.github.io/math2/node/YZ97GKYY3P.md): statement YZ97GKYY3P
- [Measure algebra](https://tamnd.github.io/math2/node/YZA1V04R96.md): concept YZA1V04R96
- [Symmetric algebra](https://tamnd.github.io/math2/node/YZCBSRCZC7.md): definition YZCBSRCZC7
- [Pochhammer symbol](https://tamnd.github.io/math2/node/YZHM3HSM0F.md): concept YZHM3HSM0F
- [Calabi–Yau manifold](https://tamnd.github.io/math2/node/YZMPRMQ6GA.md): definition YZMPRMQ6GA
- [Periodicity equation](https://tamnd.github.io/math2/node/YZNFWJAKA4.md): concept YZNFWJAKA4
- [Field of formal Laurent series](https://tamnd.github.io/math2/node/YZWYF7DXTW.md): concept YZWYF7DXTW
- [Arity](https://tamnd.github.io/math2/node/YZXPWC2MT1.md): definition YZXPWC2MT1
- [Slice category](https://tamnd.github.io/math2/node/YZXV7CHF5R.md): definition YZXV7CHF5R
- [Koszul complex](https://tamnd.github.io/math2/node/YZY11W08SD.md): concept YZY11W08SD
- [Kolmogorov–Smirnov test](https://tamnd.github.io/math2/node/YZYHWC6C5F.md): concept YZYHWC6C5F
- [Schubert variety](https://tamnd.github.io/math2/node/Z014Q8KPDD.md): concept Z014Q8KPDD
- [Semicontinuity](https://tamnd.github.io/math2/node/Z0K8AXW8Z1.md): definition Z0K8AXW8Z1
- [Sard–Smale theorem](https://tamnd.github.io/math2/node/Z0NPFYDVP2.md): statement Z0NPFYDVP2
- [Chern–Simons form](https://tamnd.github.io/math2/node/Z0RWBST5BN.md): definition Z0RWBST5BN
- [Supermanifold](https://tamnd.github.io/math2/node/Z0SB9HQBAJ.md): concept Z0SB9HQBAJ
- [Line segment](https://tamnd.github.io/math2/node/Z0VBZ906M8.md): definition Z0VBZ906M8
- [Uniserial module](https://tamnd.github.io/math2/node/Z1045A0V29.md): concept Z1045A0V29
- [Thin plate spline](https://tamnd.github.io/math2/node/Z11GGXV5KD.md): definition Z11GGXV5KD
- [Order filter](https://tamnd.github.io/math2/node/Z15FXM9WM1.md): concept Z15FXM9WM1
- [Duality in category theory](https://tamnd.github.io/math2/node/Z18AE0R3FR.md): definition Z18AE0R3FR
- [Racah polynomials](https://tamnd.github.io/math2/node/Z1BRQY4S3Y.md): definition Z1BRQY4S3Y
- [Cotangent space of a local ring](https://tamnd.github.io/math2/node/Z1CGB483J6.md): concept Z1CGB483J6
- [Difference algebra](https://tamnd.github.io/math2/node/Z1F7N07KFH.md): concept Z1F7N07KFH
- [Abel summation](https://tamnd.github.io/math2/node/Z1H1K8Q4ZV.md): definition Z1H1K8Q4ZV
- [Subharmonic function](https://tamnd.github.io/math2/node/Z1MHMGFNFW.md): concept Z1MHMGFNFW
- [Frobenius group](https://tamnd.github.io/math2/node/Z20AM1KT9R.md): definition Z20AM1KT9R
- [Lieb–Liniger integral equation](https://tamnd.github.io/math2/node/Z219Y8PJHD.md): example Z219Y8PJHD
- [Operator exponential and translation](https://tamnd.github.io/math2/node/Z22Y9N8EHN.md): statement Z22Y9N8EHN
- [CR function](https://tamnd.github.io/math2/node/Z23FCT4WTZ.md): definition Z23FCT4WTZ
- [Axiom A](https://tamnd.github.io/math2/node/Z26Z4XZFVA.md): concept Z26Z4XZFVA
- [Maximal torus](https://tamnd.github.io/math2/node/Z2AJFZQTWT.md): definition Z2AJFZQTWT
- [Trace class operator](https://tamnd.github.io/math2/node/Z2CP530WDE.md): concept Z2CP530WDE
- [Horner's method](https://tamnd.github.io/math2/node/Z2EWA10YMB.md): definition Z2EWA10YMB
- [Parabolic subgroup](https://tamnd.github.io/math2/node/Z2FEBW5MQM.md): concept Z2FEBW5MQM
- [Lumer–Phillips theorem](https://tamnd.github.io/math2/node/Z2N8MPD8DH.md): statement Z2N8MPD8DH
- [Morrey's inequality](https://tamnd.github.io/math2/node/Z2R1JY1CHS.md): statement Z2R1JY1CHS
- [Trace operator](https://tamnd.github.io/math2/node/Z2T5R88QDB.md): concept Z2T5R88QDB
- [Hadamard product](https://tamnd.github.io/math2/node/Z2XXJ5Z4NF.md): definition Z2XXJ5Z4NF
- [Osculating circle](https://tamnd.github.io/math2/node/Z2Z9BE946H.md): definition Z2Z9BE946H
- [Positive cone](https://tamnd.github.io/math2/node/Z344RY5XX1.md): concept Z344RY5XX1
- [Littlewood's Tauberian theorem](https://tamnd.github.io/math2/node/Z35TD4ZZRD.md): statement Z35TD4ZZRD
- [Lattice in a Lie group](https://tamnd.github.io/math2/node/Z367AGJRWH.md): definition Z367AGJRWH
- [Spectrum](https://tamnd.github.io/math2/node/Z38TW6347X.md): definition Z38TW6347X
- [Plane curve](https://tamnd.github.io/math2/node/Z3AY2NEVW3.md): definition Z3AY2NEVW3
- [Inverse tangent integral](https://tamnd.github.io/math2/node/Z3AYQ5M633.md): definition Z3AYQ5M633
- [Weierstrass equation](https://tamnd.github.io/math2/node/Z3C8B6QTS6.md): concept Z3C8B6QTS6
- [Algebraic logic](https://tamnd.github.io/math2/node/Z3DE4R4RET.md): concept Z3DE4R4RET
- [Aperiodic set of prototiles](https://tamnd.github.io/math2/node/Z3FCJZF6WK.md): concept Z3FCJZF6WK
- [Henselian field](https://tamnd.github.io/math2/node/Z3KPTM2YWX.md): definition Z3KPTM2YWX
- [Pushout](https://tamnd.github.io/math2/node/Z3MQ1D5Q38.md): definition Z3MQ1D5Q38
- [Myers–Steenrod theorem](https://tamnd.github.io/math2/node/Z3VRMW8ETQ.md): statement Z3VRMW8ETQ
- [Cubic Hermite spline](https://tamnd.github.io/math2/node/Z45HQ5VNF4.md): definition Z45HQ5VNF4
- [Exact category in the sense of Barr](https://tamnd.github.io/math2/node/Z45TBD32BR.md): definition Z45TBD32BR
- [Hairy ball theorem](https://tamnd.github.io/math2/node/Z49RAKVCWG.md): statement Z49RAKVCWG
- [Stein space](https://tamnd.github.io/math2/node/Z4AMJS5KA6.md): concept Z4AMJS5KA6
- [Lattice (group)](https://tamnd.github.io/math2/node/Z4FSES3CZ0.md): definition Z4FSES3CZ0
- [Summation of families over arbitrary index sets](https://tamnd.github.io/math2/node/Z4H7272S9S.md): definition Z4H7272S9S
- [Condensed set](https://tamnd.github.io/math2/node/Z4JZPEBM68.md): definition Z4JZPEBM68
- [Little q-Jacobi polynomials](https://tamnd.github.io/math2/node/Z4NV7VK9T9.md): concept Z4NV7VK9T9
- [Character of a representation](https://tamnd.github.io/math2/node/Z4PGTJFGYE.md): definition Z4PGTJFGYE
- [O'Nan–Scott theorem](https://tamnd.github.io/math2/node/Z4WVANZPK4.md): statement Z4WVANZPK4
- [Whittaker–Shannon interpolation formula](https://tamnd.github.io/math2/node/Z4ZFP6FNK2.md): statement Z4ZFP6FNK2
- [Abstract Wiener space](https://tamnd.github.io/math2/node/Z4ZM26BJHN.md): definition Z4ZM26BJHN
- [Primitive ideal](https://tamnd.github.io/math2/node/Z50EV9SYNJ.md): definition Z50EV9SYNJ
- [Doob martingale](https://tamnd.github.io/math2/node/Z57CG6W9S6.md): example Z57CG6W9S6
- [Band matrix](https://tamnd.github.io/math2/node/Z58FRZCF8P.md): concept Z58FRZCF8P
- [Oracle machine](https://tamnd.github.io/math2/node/Z5CCVBME2V.md): concept Z5CCVBME2V
- [Anderson–Kadec theorem](https://tamnd.github.io/math2/node/Z5DHRKE2Q0.md): statement Z5DHRKE2Q0
- [Relatively complemented lattice](https://tamnd.github.io/math2/node/Z5E2GX2B81.md): definition Z5E2GX2B81
- [Duality in category theory](https://tamnd.github.io/math2/node/Z5EETTFWAN.md): concept Z5EETTFWAN
- [Projective morphism](https://tamnd.github.io/math2/node/Z5G47MQ99B.md): concept Z5G47MQ99B
- [Chudnovsky algorithm](https://tamnd.github.io/math2/node/Z5G5JPT022.md): definition Z5G5JPT022
- [Change detection](https://tamnd.github.io/math2/node/Z5H8ZZHVMC.md): problem Z5H8ZZHVMC
- [3-manifold](https://tamnd.github.io/math2/node/Z5M5ESDBGK.md): definition Z5M5ESDBGK
- [Zonotope](https://tamnd.github.io/math2/node/Z5SA6A67KF.md): concept Z5SA6A67KF
- [Proper cone](https://tamnd.github.io/math2/node/Z5T0SEF3XA.md): concept Z5T0SEF3XA
- [Bernstein–Bézier form](https://tamnd.github.io/math2/node/Z5Y0VH10YR.md): concept Z5Y0VH10YR
- [Asymptotic expansion from poles of a Mellin transform](https://tamnd.github.io/math2/node/Z61W394YFA.md): definition Z61W394YFA
- [Complete elliptic integral of the first kind](https://tamnd.github.io/math2/node/Z6F9V4AHHH.md): definition Z6F9V4AHHH
- [No-retraction theorem](https://tamnd.github.io/math2/node/Z6JR1GN2RE.md): statement Z6JR1GN2RE
- [Fisher–Neyman factorization theorem](https://tamnd.github.io/math2/node/Z6MF9ZGCMW.md): statement Z6MF9ZGCMW
- [Current](https://tamnd.github.io/math2/node/Z6QXZX1THM.md): definition Z6QXZX1THM
- [Four-vertex theorem](https://tamnd.github.io/math2/node/Z6QYDSJVFX.md): statement Z6QYDSJVFX
- [Normal Moore space problem](https://tamnd.github.io/math2/node/Z6T00E3JQH.md): problem Z6T00E3JQH
- [Locally compact abelian group](https://tamnd.github.io/math2/node/Z6TBXH0MHR.md): definition Z6TBXH0MHR
- [Inherently nonfinitely based algebra](https://tamnd.github.io/math2/node/Z6TKYRZ9WX.md): concept Z6TKYRZ9WX
- [Triebel–Lizorkin space](https://tamnd.github.io/math2/node/Z72JWTXM3W.md): definition Z72JWTXM3W
- [Massey product](https://tamnd.github.io/math2/node/Z761N3D19J.md): concept Z761N3D19J
- [Girsanov theorem](https://tamnd.github.io/math2/node/Z76HD1BZ29.md): statement Z76HD1BZ29
- [Zolotarev polynomials](https://tamnd.github.io/math2/node/Z76K1W7BSH.md): definition Z76K1W7BSH
- [Spectral theory of dynamical systems](https://tamnd.github.io/math2/node/Z7GZ6SPFA5.md): concept Z7GZ6SPFA5
- [Real analytic manifold](https://tamnd.github.io/math2/node/Z7H98E1NHP.md): concept Z7H98E1NHP
- [Monomial ideal](https://tamnd.github.io/math2/node/Z7MTTS6FFG.md): definition Z7MTTS6FFG
- [Interior operator](https://tamnd.github.io/math2/node/Z7P561Y2JQ.md): concept Z7P561Y2JQ
- [Solvable by radicals](https://tamnd.github.io/math2/node/Z7R10WW158.md): definition Z7R10WW158
- [Golden–Thompson inequality](https://tamnd.github.io/math2/node/Z7R754JC7N.md): statement Z7R754JC7N
- [Sard's theorem](https://tamnd.github.io/math2/node/Z7R8J71SJN.md): statement Z7R8J71SJN
- [Successor order](https://tamnd.github.io/math2/node/Z7RGVW5EWN.md): definition Z7RGVW5EWN
- [Strongly pseudoconvex domain](https://tamnd.github.io/math2/node/Z7RVXBA2KB.md): concept Z7RVXBA2KB
- [Polar space](https://tamnd.github.io/math2/node/Z7T0V2VAWJ.md): concept Z7T0V2VAWJ
- [Hunt process](https://tamnd.github.io/math2/node/Z7T669BQ9B.md): concept Z7T669BQ9B
- [Undecidability of the halting problem](https://tamnd.github.io/math2/node/Z7TN2B35B0.md): statement Z7TN2B35B0
- [Symmetric and self-reciprocal kernels](https://tamnd.github.io/math2/node/Z7WD35SHQP.md): definition Z7WD35SHQP
- [Curve-shortening flow](https://tamnd.github.io/math2/node/Z8192111H3.md): definition Z8192111H3
- [Almost periodic sequence](https://tamnd.github.io/math2/node/Z83P73MG91.md): concept Z83P73MG91
- [Tensor decomposition](https://tamnd.github.io/math2/node/Z855M8QN2H.md): definition Z855M8QN2H
- [Word problem for free lattices](https://tamnd.github.io/math2/node/Z86VGP5RMG.md): statement Z86VGP5RMG
- [Stone–von Neumann theorem](https://tamnd.github.io/math2/node/Z877XDBMQ3.md): statement Z877XDBMQ3
- [Essentially self-adjoint operator](https://tamnd.github.io/math2/node/Z8CW6Z2K7N.md): definition Z8CW6Z2K7N
- [Euler integral for the gamma function](https://tamnd.github.io/math2/node/Z8D1CZFTWP.md): statement Z8D1CZFTWP
- [Finite-rank operator](https://tamnd.github.io/math2/node/Z8FABSYQTS.md): concept Z8FABSYQTS
- [Synge's theorem](https://tamnd.github.io/math2/node/Z8JJ04X1JX.md): statement Z8JJ04X1JX
- [Monogenic function](https://tamnd.github.io/math2/node/Z8NJ4KHGAT.md): definition Z8NJ4KHGAT
- [Hopf manifold](https://tamnd.github.io/math2/node/Z8R2D33WSM.md): example Z8R2D33WSM
- [Fourier series of Bernoulli polynomials](https://tamnd.github.io/math2/node/Z8T8YYF5ZZ.md): example Z8T8YYF5ZZ
- [Martin's axiom](https://tamnd.github.io/math2/node/Z8XMCCZFA4.md): definition Z8XMCCZFA4
- [Christoffel symbols](https://tamnd.github.io/math2/node/Z8XNDJSCJS.md): definition Z8XNDJSCJS
- [Congruence lattice](https://tamnd.github.io/math2/node/Z937W8RW4N.md): definition Z937W8RW4N
- [Reeb stability theorem](https://tamnd.github.io/math2/node/Z93KMZ9R37.md): statement Z93KMZ9R37
- [Kervaire invariant problem](https://tamnd.github.io/math2/node/Z95Q7B022R.md): problem Z95Q7B022R
- [Parametrix](https://tamnd.github.io/math2/node/Z97HVFD6JE.md): concept Z97HVFD6JE
- [Weil divisor](https://tamnd.github.io/math2/node/Z9DN23DAE2.md): concept Z9DN23DAE2
- [Topological manifold](https://tamnd.github.io/math2/node/Z9DYVG5RA3.md): definition Z9DYVG5RA3
- [Flag variety](https://tamnd.github.io/math2/node/Z9HNJ529DF.md): definition Z9HNJ529DF
- [Prime ring](https://tamnd.github.io/math2/node/Z9TJSZZYG1.md): concept Z9TJSZZYG1
- [Symbol class](https://tamnd.github.io/math2/node/Z9WE435T9G.md): definition Z9WE435T9G
- [Hurewicz theorem](https://tamnd.github.io/math2/node/ZA11N5XPZM.md): statement ZA11N5XPZM
- [Transform pair](https://tamnd.github.io/math2/node/ZA1HC10WM6.md): concept ZA1HC10WM6
- [Dehn surgery](https://tamnd.github.io/math2/node/ZA405B0Z1S.md): concept ZA405B0Z1S
- [CSP dichotomy theorem](https://tamnd.github.io/math2/node/ZA4YXJ9T1P.md): statement ZA4YXJ9T1P
- [Singular perturbation](https://tamnd.github.io/math2/node/ZAFEF4SAJ7.md): definition ZAFEF4SAJ7
- [Boutet de Monvel's embedding theorem](https://tamnd.github.io/math2/node/ZAH4SK5YX7.md): statement ZAH4SK5YX7
- [Curvature form](https://tamnd.github.io/math2/node/ZAJFW1S078.md): concept ZAJFW1S078
- [Well-founded order](https://tamnd.github.io/math2/node/ZANKHGE1YC.md): definition ZANKHGE1YC
- [Allen–Cahn equation](https://tamnd.github.io/math2/node/ZARKMD0C0H.md): definition ZARKMD0C0H
- [Moment map](https://tamnd.github.io/math2/node/ZARM34FZ6Z.md): concept ZARM34FZ6Z
- [Heat potential](https://tamnd.github.io/math2/node/ZAT4TV4030.md): concept ZAT4TV4030
- [Linear discriminant analysis](https://tamnd.github.io/math2/node/ZATS7NV9WD.md): concept ZATS7NV9WD
- [Subadditive function](https://tamnd.github.io/math2/node/ZAVMXRK4J9.md): definition ZAVMXRK4J9
- [Steiner conic](https://tamnd.github.io/math2/node/ZAZFMK7C67.md): definition ZAZFMK7C67
- [First and second variation of arc length](https://tamnd.github.io/math2/node/ZB91TH6FKV.md): statement ZB91TH6FKV
- [Gowers norm](https://tamnd.github.io/math2/node/ZBFZGA1J5G.md): concept ZBFZGA1J5G
- [Theta correspondence](https://tamnd.github.io/math2/node/ZBGT50BW0Z.md): concept ZBGT50BW0Z
- [Fundamental groupoid](https://tamnd.github.io/math2/node/ZBHGXJF24E.md): concept ZBHGXJF24E
- [Category of posets](https://tamnd.github.io/math2/node/ZBKVRMWHMW.md): concept ZBKVRMWHMW
- [Cauchy's functional equation](https://tamnd.github.io/math2/node/ZBMEVWN100.md): problem ZBMEVWN100
- [Power series in several variables](https://tamnd.github.io/math2/node/ZBNQEJ09QE.md): definition ZBNQEJ09QE
- [Cartan matrix of an algebra](https://tamnd.github.io/math2/node/ZBQ4KRC98K.md): concept ZBQ4KRC98K
- [Kohn Laplacian](https://tamnd.github.io/math2/node/ZBQ6TVNZZ8.md): concept ZBQ6TVNZZ8
- [Bernstein's theorem on monotone functions](https://tamnd.github.io/math2/node/ZBS1JA62QP.md): statement ZBS1JA62QP
- [Radial basis function](https://tamnd.github.io/math2/node/ZBSXSTQM2J.md): concept ZBSXSTQM2J
- [Fuchsian equation](https://tamnd.github.io/math2/node/ZBVH2YE50B.md): concept ZBVH2YE50B
- [Hirsch conjecture](https://tamnd.github.io/math2/node/ZBVJ4B41AX.md): counterexample ZBVJ4B41AX
- [Littlewood–Paley square function](https://tamnd.github.io/math2/node/ZBXBQYKP5W.md): definition ZBXBQYKP5W
- [Quadratic variation](https://tamnd.github.io/math2/node/ZBYC3MFW89.md): definition ZBYC3MFW89
- [Normal form](https://tamnd.github.io/math2/node/ZBZBKWWEBC.md): definition ZBZBKWWEBC
- [Universal bundle](https://tamnd.github.io/math2/node/ZC179DD2P4.md): definition ZC179DD2P4
- [Beth number](https://tamnd.github.io/math2/node/ZC4BHVHWE9.md): definition ZC4BHVHWE9
- [CR manifold](https://tamnd.github.io/math2/node/ZC78JKZX9S.md): concept ZC78JKZX9S
- [Lawson topology](https://tamnd.github.io/math2/node/ZC8DNB75N4.md): concept ZC8DNB75N4
- [Functional equation of the gamma function](https://tamnd.github.io/math2/node/ZCB806EV8A.md): statement ZCB806EV8A
- [Moishezon manifold](https://tamnd.github.io/math2/node/ZCBFF6DNA9.md): definition ZCBFF6DNA9
- [Hermite equation](https://tamnd.github.io/math2/node/ZCDCJTTG8V.md): example ZCDCJTTG8V
- [Alexandrov's theorem on polyhedra](https://tamnd.github.io/math2/node/ZCDWXVMDFN.md): statement ZCDWXVMDFN
- [Regular cardinal](https://tamnd.github.io/math2/node/ZCE333FAWC.md): definition ZCE333FAWC
- [Mathieu function](https://tamnd.github.io/math2/node/ZCE9DV48N0.md): definition ZCE9DV48N0
- [Amenable group action](https://tamnd.github.io/math2/node/ZCG4DJ7GH3.md): concept ZCG4DJ7GH3
- [Varifold](https://tamnd.github.io/math2/node/ZCJEPPAFD2.md): definition ZCJEPPAFD2
- [Minimum-variance unbiased estimator](https://tamnd.github.io/math2/node/ZCJX7THBNJ.md): concept ZCJX7THBNJ
- [Lagrange basis polynomials](https://tamnd.github.io/math2/node/ZCVB4ZQ0SA.md): definition ZCVB4ZQ0SA
- [Immersion](https://tamnd.github.io/math2/node/ZCW1NXBZ3R.md): definition ZCW1NXBZ3R
- [Concrete category of algebras](https://tamnd.github.io/math2/node/ZD0GTPSJS1.md): concept ZD0GTPSJS1
- [Primary decomposition](https://tamnd.github.io/math2/node/ZD34THCXQ0.md): concept ZD34THCXQ0
- [Birational map](https://tamnd.github.io/math2/node/ZD73KPKPJ4.md): definition ZD73KPKPJ4
- [Barratt–Priddy–Quillen theorem](https://tamnd.github.io/math2/node/ZD7BNADK4H.md): statement ZD7BNADK4H
- [Sklar's theorem](https://tamnd.github.io/math2/node/ZD7V7R27R9.md): statement ZD7V7R27R9
- [Blocking set](https://tamnd.github.io/math2/node/ZD8FVT10D8.md): concept ZD8FVT10D8
- [Ehrhart–Macdonald reciprocity](https://tamnd.github.io/math2/node/ZDBER9FJ6T.md): statement ZDBER9FJ6T
- [Bareiss algorithm](https://tamnd.github.io/math2/node/ZDBW3AB587.md): statement ZDBW3AB587
- [Semilocal ring](https://tamnd.github.io/math2/node/ZDCDS6EWGX.md): concept ZDCDS6EWGX
- [Manifold of smooth mappings](https://tamnd.github.io/math2/node/ZDHAMB0H8J.md): definition ZDHAMB0H8J
- [Harmonic majorant](https://tamnd.github.io/math2/node/ZDHPQWWCQW.md): concept ZDHPQWWCQW
- [Permutation group](https://tamnd.github.io/math2/node/ZDJQRKPWGA.md): concept ZDJQRKPWGA
- [Continuous lattice](https://tamnd.github.io/math2/node/ZDNBCXDGTA.md): definition ZDNBCXDGTA
- [Integral equation of the first kind](https://tamnd.github.io/math2/node/ZDS77Q0F06.md): concept ZDS77Q0F06
- [Colombeau algebra](https://tamnd.github.io/math2/node/ZDXQRARWT5.md): definition ZDXQRARWT5
- [Potential of a measure](https://tamnd.github.io/math2/node/ZE166728M3.md): definition ZE166728M3
- [Tikhonov regularization](https://tamnd.github.io/math2/node/ZE1FSZ6GJF.md): definition ZE1FSZ6GJF
- [Hartley transform](https://tamnd.github.io/math2/node/ZE4QPBC1AF.md): definition ZE4QPBC1AF
- [Barrelled space](https://tamnd.github.io/math2/node/ZE6DXEJB8H.md): concept ZE6DXEJB8H
- [Anti-equivalence of affine schemes and rings](https://tamnd.github.io/math2/node/ZE780BQS42.md): statement ZE780BQS42
- [Fundamental class](https://tamnd.github.io/math2/node/ZEA0PEM3B2.md): concept ZEA0PEM3B2
- [Join](https://tamnd.github.io/math2/node/ZEETZHV69Y.md): concept ZEETZHV69Y
- [Support of a measure](https://tamnd.github.io/math2/node/ZEH0N3N467.md): definition ZEH0N3N467
- [Method of images](https://tamnd.github.io/math2/node/ZEJ9VKKKRG.md): definition ZEJ9VKKKRG
- [Hartogs lemma for subharmonic functions](https://tamnd.github.io/math2/node/ZENZT8596D.md): statement ZENZT8596D
- [Many-sorted algebra](https://tamnd.github.io/math2/node/ZETPP2VJFJ.md): definition ZETPP2VJFJ
- [Irreducible matrix](https://tamnd.github.io/math2/node/ZEZPYJC06B.md): definition ZEZPYJC06B
- [Variogram](https://tamnd.github.io/math2/node/ZF8AA2JVKV.md): concept ZF8AA2JVKV
- [Dirac operator](https://tamnd.github.io/math2/node/ZFC8N4CFC4.md): definition ZFC8N4CFC4
- [Δ-complex](https://tamnd.github.io/math2/node/ZFCMCXA71M.md): concept ZFCMCXA71M
- [Fundamental groupoid](https://tamnd.github.io/math2/node/ZFEDKM4TJ7.md): definition ZFEDKM4TJ7
- [Kirby calculus](https://tamnd.github.io/math2/node/ZFNZ0M7X41.md): definition ZFNZ0M7X41
- [Lyapunov equation](https://tamnd.github.io/math2/node/ZFSZCMKJD1.md): definition ZFSZCMKJD1
- [Functor category](https://tamnd.github.io/math2/node/ZFT23VCN7Z.md): definition ZFT23VCN7Z
- [Black–Scholes equation](https://tamnd.github.io/math2/node/ZFTNJ2YREG.md): concept ZFTNJ2YREG
- [Normed field](https://tamnd.github.io/math2/node/ZFW9Q3WTAQ.md): concept ZFW9Q3WTAQ
- [Quasi-isometry](https://tamnd.github.io/math2/node/ZFZR65WYBX.md): definition ZFZR65WYBX
- [Priestley space](https://tamnd.github.io/math2/node/ZGBV8V3823.md): concept ZGBV8V3823
- [Flexible polyhedron](https://tamnd.github.io/math2/node/ZGD3TQX0K6.md): example ZGD3TQX0K6
- [Faber polynomials](https://tamnd.github.io/math2/node/ZGH48BG36Z.md): definition ZGH48BG36Z
- [Orlicz space](https://tamnd.github.io/math2/node/ZGH5RYJTQC.md): definition ZGH5RYJTQC
- [Parallelizable manifold](https://tamnd.github.io/math2/node/ZGHMHJZQJA.md): definition ZGHMHJZQJA
- [Markov chain Monte Carlo](https://tamnd.github.io/math2/node/ZGNBR2SPZ0.md): concept ZGNBR2SPZ0
- [Complementary error function](https://tamnd.github.io/math2/node/ZGYA090JP2.md): definition ZGYA090JP2
- [Deligne–Lusztig theory](https://tamnd.github.io/math2/node/ZH21D5B62C.md): concept ZH21D5B62C
- [Paracompact space](https://tamnd.github.io/math2/node/ZH29VRRRJ6.md): definition ZH29VRRRJ6
- [Oscillation theory](https://tamnd.github.io/math2/node/ZH2S50NSZ6.md): concept ZH2S50NSZ6
- [Graceful tree conjecture](https://tamnd.github.io/math2/node/ZH3K0BH68K.md): conjecture ZH3K0BH68K
- [Green's function](https://tamnd.github.io/math2/node/ZH4V0WHSXT.md): definition ZH4V0WHSXT
- [Smooth morphism of rings](https://tamnd.github.io/math2/node/ZHBREAF3MJ.md): definition ZHBREAF3MJ
- [Relative homotopy group](https://tamnd.github.io/math2/node/ZHHMDZWA7F.md): concept ZHHMDZWA7F
- [Pesin theory](https://tamnd.github.io/math2/node/ZHKS81A7QN.md): concept ZHKS81A7QN
- [Join-irreducible element](https://tamnd.github.io/math2/node/ZHRQ0AT8PM.md): concept ZHRQ0AT8PM
- [Scalar curvature](https://tamnd.github.io/math2/node/ZHX2VYQDZ3.md): concept ZHX2VYQDZ3
- [Domain of finite type](https://tamnd.github.io/math2/node/ZJ0GBPREMB.md): concept ZJ0GBPREMB
- [Weighted polynomial approximation](https://tamnd.github.io/math2/node/ZJD91FNTCY.md): definition ZJD91FNTCY
- [C0-semigroup](https://tamnd.github.io/math2/node/ZJF5XVY63D.md): definition ZJF5XVY63D
- [Banach–Mazur distance](https://tamnd.github.io/math2/node/ZJKSYYASPT.md): concept ZJKSYYASPT
- [Cone theorem](https://tamnd.github.io/math2/node/ZJSW0HP75H.md): statement ZJSW0HP75H
- [Adams' theorem on Hopf invariant one](https://tamnd.github.io/math2/node/ZJVKZ8D64P.md): statement ZJVKZ8D64P
- [Generalized Boolean algebra](https://tamnd.github.io/math2/node/ZK1MHTQGS9.md): concept ZK1MHTQGS9
- [Laplace transform of a periodic function](https://tamnd.github.io/math2/node/ZK55WHXEQX.md): example ZK55WHXEQX
- [Levi pseudoconvex domain](https://tamnd.github.io/math2/node/ZKG8D1S566.md): definition ZKG8D1S566
- [Holomorphy of the gamma function](https://tamnd.github.io/math2/node/ZKJF80DTNB.md): statement ZKJF80DTNB
- [Enflo's counterexample to the approximation problem](https://tamnd.github.io/math2/node/ZKP5KGFWCK.md): counterexample ZKP5KGFWCK
- [Sheaf of differentials](https://tamnd.github.io/math2/node/ZKSPYN2CRJ.md): definition ZKSPYN2CRJ
- [Cascade algorithm](https://tamnd.github.io/math2/node/ZKXDA3XF3Z.md): definition ZKXDA3XF3Z
- [Continuous kernel](https://tamnd.github.io/math2/node/ZM5G144PZG.md): concept ZM5G144PZG
- [Mellin transform under inversion of the variable](https://tamnd.github.io/math2/node/ZM5VYK2BFB.md): statement ZM5VYK2BFB
- [Frenet–Serret formulas](https://tamnd.github.io/math2/node/ZM6W89YNS7.md): statement ZM6W89YNS7
- [Hilbert–Schmidt integral operator](https://tamnd.github.io/math2/node/ZMCBQ0M082.md): definition ZMCBQ0M082
- [Functional differential equation](https://tamnd.github.io/math2/node/ZMCFEEPEKT.md): concept ZMCFEEPEKT
- [Chevalley group](https://tamnd.github.io/math2/node/ZMDAEW6DHN.md): definition ZMDAEW6DHN
- [Banach–Zarecki theorem](https://tamnd.github.io/math2/node/ZMEYSAHNX3.md): statement ZMEYSAHNX3
- [Griess algebra](https://tamnd.github.io/math2/node/ZMPYZ8KD9N.md): object ZMPYZ8KD9N
- [Bieberbach conjecture](https://tamnd.github.io/math2/node/ZMRFSYVWQT.md): statement ZMRFSYVWQT
- [Runge's phenomenon](https://tamnd.github.io/math2/node/ZMT8JF21R6.md): example ZMT8JF21R6
- [Nest algebra](https://tamnd.github.io/math2/node/ZMY9H2XENE.md): concept ZMY9H2XENE
- [Zonoid](https://tamnd.github.io/math2/node/ZMZZ2QRFTZ.md): concept ZMZZ2QRFTZ
- [Discrete Laplace operator](https://tamnd.github.io/math2/node/ZN30Y1ZDB6.md): concept ZN30Y1ZDB6
- [Ramanujan's master theorem](https://tamnd.github.io/math2/node/ZN4NMX1DB4.md): statement ZN4NMX1DB4
- [Doob's h-transform](https://tamnd.github.io/math2/node/ZN7HJNR2AC.md): concept ZN7HJNR2AC
- [Kontorovich–Lebedev transform](https://tamnd.github.io/math2/node/ZNC09JXRC5.md): definition ZNC09JXRC5
- [Grauert's direct image theorem](https://tamnd.github.io/math2/node/ZNCF16AENG.md): statement ZNCF16AENG
- [Hilbert transform of sinusoids](https://tamnd.github.io/math2/node/ZNDEPVNBBG.md): example ZNDEPVNBBG
- [Fourier transform of the Gaussian](https://tamnd.github.io/math2/node/ZNEHMR2CCP.md): statement ZNEHMR2CCP
- [Finite-dimensional Hausdorff vector topologies are Euclidean](https://tamnd.github.io/math2/node/ZNMN1YA83C.md): statement ZNMN1YA83C
- [Hofstadter sequences](https://tamnd.github.io/math2/node/ZNT0P1M1D5.md): example ZNT0P1M1D5
- [Radical of an ideal](https://tamnd.github.io/math2/node/ZP1A7QMD6R.md): definition ZP1A7QMD6R
- [Sample mean and sample covariance](https://tamnd.github.io/math2/node/ZP455BADJ3.md): concept ZP455BADJ3
- [Volume conjecture](https://tamnd.github.io/math2/node/ZP4AWD45CE.md): conjecture ZP4AWD45CE
- [Uniform algebra](https://tamnd.github.io/math2/node/ZP55RF3A0C.md): concept ZP55RF3A0C
- [Carathéodory's royal road](https://tamnd.github.io/math2/node/ZPC2A2JMD8.md): definition ZPC2A2JMD8
- [Stochastic approximation](https://tamnd.github.io/math2/node/ZPCPW4W1Y9.md): concept ZPCPW4W1Y9
- [Control system](https://tamnd.github.io/math2/node/ZPDTVZQN5K.md): concept ZPDTVZQN5K
- [Observed information](https://tamnd.github.io/math2/node/ZPPN50CEEX.md): definition ZPPN50CEEX
- [Generalized method of moments](https://tamnd.github.io/math2/node/ZPVMCCDYZ5.md): concept ZPVMCCDYZ5
- [Irreducible component](https://tamnd.github.io/math2/node/ZPVRRQDYRB.md): definition ZPVRRQDYRB
- [Conjugate points](https://tamnd.github.io/math2/node/ZPXE5VSZ35.md): concept ZPXE5VSZ35
- [Enumerative geometry](https://tamnd.github.io/math2/node/ZPZA1FKZZC.md): concept ZPZA1FKZZC
- [Jacobian variety](https://tamnd.github.io/math2/node/ZQ84M9D9TC.md): concept ZQ84M9D9TC
- [Progenerator](https://tamnd.github.io/math2/node/ZQ9ABBKF1N.md): concept ZQ9ABBKF1N
- [Fourier multiplier](https://tamnd.github.io/math2/node/ZQEFCY61T7.md): concept ZQEFCY61T7
- [Hardy–Littlewood circle method](https://tamnd.github.io/math2/node/ZQFBEX62D0.md): concept ZQFBEX62D0
- [Skolem–Mahler–Lech theorem](https://tamnd.github.io/math2/node/ZQFPVWMY4M.md): statement ZQFPVWMY4M
- [Propagation of uncertainty](https://tamnd.github.io/math2/node/ZQKA90X325.md): definition ZQKA90X325
- [Voronoi cell](https://tamnd.github.io/math2/node/ZQR56X168K.md): definition ZQR56X168K
- [Donaldson theory](https://tamnd.github.io/math2/node/ZQW6M0S6M0.md): concept ZQW6M0S6M0
- [Legendre transformation](https://tamnd.github.io/math2/node/ZQYMWVT1TT.md): concept ZQYMWVT1TT
- [Locally compact abelian group](https://tamnd.github.io/math2/node/ZR09S9R3AW.md): concept ZR09S9R3AW
- [Total variation diminishing scheme](https://tamnd.github.io/math2/node/ZR1DPX6VAN.md): definition ZR1DPX6VAN
- [Whitney–Graustein theorem](https://tamnd.github.io/math2/node/ZR6DEE6RBC.md): statement ZR6DEE6RBC
- [Drazin inverse](https://tamnd.github.io/math2/node/ZR6XXP36X8.md): definition ZR6XXP36X8
- [Łoś–Vaught test](https://tamnd.github.io/math2/node/ZR7Y5C235P.md): statement ZR7Y5C235P
- [Scott topology](https://tamnd.github.io/math2/node/ZR8A03FV67.md): concept ZR8A03FV67
- [Feller process](https://tamnd.github.io/math2/node/ZRC17G6GYS.md): definition ZRC17G6GYS
- [Semiperfect ring](https://tamnd.github.io/math2/node/ZRC2S3ECFK.md): concept ZRC2S3ECFK
- [Dialectica interpretation](https://tamnd.github.io/math2/node/ZREF8MTYNW.md): definition ZREF8MTYNW
- [Holomorphically convex domain](https://tamnd.github.io/math2/node/ZRGTKMHKX2.md): definition ZRGTKMHKX2
- [Bochner's theorem](https://tamnd.github.io/math2/node/ZRHAG4NXHC.md): statement ZRHAG4NXHC
- [Point estimation](https://tamnd.github.io/math2/node/ZRMWNF10FA.md): definition ZRMWNF10FA
- [Jacobi condition](https://tamnd.github.io/math2/node/ZRV9T8DD05.md): statement ZRV9T8DD05
- [Support of a convolution](https://tamnd.github.io/math2/node/ZS29A7QQC2.md): statement ZS29A7QQC2
- [Ricci flow](https://tamnd.github.io/math2/node/ZS2WYH25R0.md): definition ZS2WYH25R0
- [Horn clause](https://tamnd.github.io/math2/node/ZS5TMX4G8P.md): definition ZS5TMX4G8P
- [Inverse Fourier transform](https://tamnd.github.io/math2/node/ZS7HP7SGEP.md): definition ZS7HP7SGEP
- [Johnson solid](https://tamnd.github.io/math2/node/ZSA992C21V.md): object ZSA992C21V
- [Conway knot](https://tamnd.github.io/math2/node/ZSAMVFPWN6.md): example ZSAMVFPWN6
- [Hodge structure](https://tamnd.github.io/math2/node/ZSM083M97C.md): concept ZSM083M97C
- [Method of images](https://tamnd.github.io/math2/node/ZSQSPGBN88.md): concept ZSQSPGBN88
- [Universal class](https://tamnd.github.io/math2/node/ZSTEZ3PRT9.md): concept ZSTEZ3PRT9
- [Maximal subfield](https://tamnd.github.io/math2/node/ZSVF54TCYP.md): concept ZSVF54TCYP
- [Lie derivative](https://tamnd.github.io/math2/node/ZSW87QS3TQ.md): definition ZSW87QS3TQ
- [Lefschetz number](https://tamnd.github.io/math2/node/ZSYCA4EGEZ.md): concept ZSYCA4EGEZ
- [L-infinity space](https://tamnd.github.io/math2/node/ZT01G7BTSV.md): concept ZT01G7BTSV
- [Complemented lattice](https://tamnd.github.io/math2/node/ZT47BK9Y6K.md): concept ZT47BK9Y6K
- [Absolute continuity](https://tamnd.github.io/math2/node/ZT6F4C2YHK.md): concept ZT6F4C2YHK
- [Morley's trisector theorem](https://tamnd.github.io/math2/node/ZT97FC11VA.md): statement ZT97FC11VA
- [Projective space](https://tamnd.github.io/math2/node/ZT9A6JK8QF.md): concept ZT9A6JK8QF
- [Pushforward measure](https://tamnd.github.io/math2/node/ZT9BSVAFEX.md): definition ZT9BSVAFEX
- [Faber's theorem on interpolation](https://tamnd.github.io/math2/node/ZTBC6G7DGD.md): statement ZTBC6G7DGD
- [Input-to-state stability](https://tamnd.github.io/math2/node/ZTMK6QNKK3.md): definition ZTMK6QNKK3
- [Definable set](https://tamnd.github.io/math2/node/ZTTPPXGB30.md): definition ZTTPPXGB30
- [Zariski's multiplicity conjecture](https://tamnd.github.io/math2/node/ZTTZVZWMK9.md): conjecture ZTTZVZWMK9
- [Hindman's theorem](https://tamnd.github.io/math2/node/ZTWTZ8CSGV.md): statement ZTWTZ8CSGV
- [Mellin transform as a Fourier transform in logarithmic variables](https://tamnd.github.io/math2/node/ZTZ1FRRFM8.md): statement ZTZ1FRRFM8
- [Artinian ring](https://tamnd.github.io/math2/node/ZV04WWYE2Y.md): definition ZV04WWYE2Y
- [Partially well-ordered set](https://tamnd.github.io/math2/node/ZV239N9H5B.md): definition ZV239N9H5B
- [Stone space](https://tamnd.github.io/math2/node/ZV38HA5PAQ.md): concept ZV38HA5PAQ
- [Kernel density estimation](https://tamnd.github.io/math2/node/ZV65XMJ9A4.md): concept ZV65XMJ9A4
- [Grothendieck spectral sequence](https://tamnd.github.io/math2/node/ZV7VGE2SQ2.md): statement ZV7VGE2SQ2
- [Open immersion](https://tamnd.github.io/math2/node/ZVB7G8TGC3.md): definition ZVB7G8TGC3
- [Bilinear interpolation](https://tamnd.github.io/math2/node/ZVDW0BJ4FY.md): definition ZVDW0BJ4FY
- [Montel's theorem](https://tamnd.github.io/math2/node/ZVFVBVJMCB.md): statement ZVFVBVJMCB
- [Topology generated by a family of seminorms](https://tamnd.github.io/math2/node/ZVFXQ7S6FY.md): concept ZVFXQ7S6FY
- [Numerical linear algebra](https://tamnd.github.io/math2/node/ZVGM4STCJA.md): definition ZVGM4STCJA
- [Quiver](https://tamnd.github.io/math2/node/ZVTMF9YSGZ.md): definition ZVTMF9YSGZ
- [Essential supremum](https://tamnd.github.io/math2/node/ZVW3MT4WMT.md): definition ZVW3MT4WMT
- [Arnold diffusion](https://tamnd.github.io/math2/node/ZVY046R4XB.md): concept ZVY046R4XB
- [Partial difference equation](https://tamnd.github.io/math2/node/ZVZQ1KCHGH.md): definition ZVZQ1KCHGH
- [Kodaira dimension](https://tamnd.github.io/math2/node/ZW00Q0SAGR.md): concept ZW00Q0SAGR
- [Langevin equation](https://tamnd.github.io/math2/node/ZW2RBS9VTR.md): concept ZW2RBS9VTR
- [Injective resolution](https://tamnd.github.io/math2/node/ZW3W6FX4PW.md): concept ZW3W6FX4PW
- [Sigma-compact space](https://tamnd.github.io/math2/node/ZW4NFH5RNC.md): concept ZW4NFH5RNC
- [Regular and singular points of an analytic set](https://tamnd.github.io/math2/node/ZW5A88PCPK.md): concept ZW5A88PCPK
- [Statistical convergence](https://tamnd.github.io/math2/node/ZW6EHSMANH.md): concept ZW6EHSMANH
- [Birch and Swinnerton-Dyer conjecture](https://tamnd.github.io/math2/node/ZWEFBG40H9.md): conjecture ZWEFBG40H9
- [Local martingale](https://tamnd.github.io/math2/node/ZWJ9G89W3T.md): definition ZWJ9G89W3T
- [Mason–Stothers theorem](https://tamnd.github.io/math2/node/ZWPWF6NJC3.md): statement ZWPWF6NJC3
- [Brachistochrone problem](https://tamnd.github.io/math2/node/ZWQ5JBR6XF.md): problem ZWQ5JBR6XF
- [Weak Nullstellensatz](https://tamnd.github.io/math2/node/ZWS0YWGF17.md): statement ZWS0YWGF17
- [Newton–Puiseux theorem](https://tamnd.github.io/math2/node/ZWT5HGRFXX.md): statement ZWT5HGRFXX
- [Quaternion algebra](https://tamnd.github.io/math2/node/ZWXW0T3EAC.md): concept ZWXW0T3EAC
- [A priori estimate](https://tamnd.github.io/math2/node/ZX8972MACG.md): definition ZX8972MACG
- [Proof of the cancellation laws](https://tamnd.github.io/math2/node/ZXES6QGX62.md): proof ZXES6QGX62
- [Completion of a measure](https://tamnd.github.io/math2/node/ZXF5XXT5XM.md): definition ZXF5XXT5XM
- [Hellinger–Toeplitz theorem](https://tamnd.github.io/math2/node/ZXJ7FJ3WCE.md): statement ZXJ7FJ3WCE
- [Disjoint union topology](https://tamnd.github.io/math2/node/ZXJKSB30WF.md): concept ZXJKSB30WF
- [Absolute continuity of the Lebesgue integral](https://tamnd.github.io/math2/node/ZXK6XM44HS.md): statement ZXK6XM44HS
- [Reduced scheme](https://tamnd.github.io/math2/node/ZXSN8D4MCB.md): concept ZXSN8D4MCB
- [Divisible group](https://tamnd.github.io/math2/node/ZXVBEPMTCN.md): concept ZXVBEPMTCN
- [Maupertuis's principle](https://tamnd.github.io/math2/node/ZXY0AYKMXZ.md): statement ZXY0AYKMXZ
- [Transitivity of integrality](https://tamnd.github.io/math2/node/ZXZMQM818D.md): statement ZXZMQM818D
- [Bäcklund transform](https://tamnd.github.io/math2/node/ZY0GDAS488.md): definition ZY0GDAS488
- [Riemann solver](https://tamnd.github.io/math2/node/ZY4GY93C4B.md): concept ZY4GY93C4B
- [Euler–Lagrange equation](https://tamnd.github.io/math2/node/ZY5E6ZQZVV.md): definition ZY5E6ZQZVV
- [Category of small categories](https://tamnd.github.io/math2/node/ZY6R4H65P8.md): example ZY6R4H65P8
- [Fundamental solution](https://tamnd.github.io/math2/node/ZY724M2NKQ.md): definition ZY724M2NKQ
- [Whitehead tower](https://tamnd.github.io/math2/node/ZYCX4PN96G.md): definition ZYCX4PN96G
- [Boundary integral equation for the Dirichlet problem](https://tamnd.github.io/math2/node/ZYDVFT6JR7.md): statement ZYDVFT6JR7
- [Riesz basis](https://tamnd.github.io/math2/node/ZYDYD7FB1Z.md): concept ZYDYD7FB1Z
- [Levitzki radical](https://tamnd.github.io/math2/node/ZYG1MEBF4X.md): definition ZYG1MEBF4X
- [Discrete valuation](https://tamnd.github.io/math2/node/ZYK5F0GBK6.md): definition ZYK5F0GBK6
- [Invariant mean](https://tamnd.github.io/math2/node/ZYNY2VFECQ.md): concept ZYNY2VFECQ
- [Height of a poset](https://tamnd.github.io/math2/node/ZYS4TTQG2T.md): definition ZYS4TTQG2T
- [Degree of exactness of a quadrature rule](https://tamnd.github.io/math2/node/ZYT1JNT1Y0.md): concept ZYT1JNT1Y0
- [Thom–Boardman singularities](https://tamnd.github.io/math2/node/ZYY3FJ57NQ.md): definition ZYY3FJ57NQ
- [Pariah groups](https://tamnd.github.io/math2/node/ZYZ5FYG7QC.md): concept ZYZ5FYG7QC
- [hp-FEM](https://tamnd.github.io/math2/node/ZYZM357DQV.md): concept ZYZM357DQV
- [Elliptic gamma function](https://tamnd.github.io/math2/node/ZZ82BW6CBG.md): definition ZZ82BW6CBG
- [Lane–Emden equation](https://tamnd.github.io/math2/node/ZZD7X71HVN.md): definition ZZD7X71HVN
- [Residue calculus for Laplace inversion](https://tamnd.github.io/math2/node/ZZDMHH8V1K.md): definition ZZDMHH8V1K
- [Free magma](https://tamnd.github.io/math2/node/ZZF8XPBG3Z.md): example ZZF8XPBG3Z
- [Orderable group](https://tamnd.github.io/math2/node/ZZFFBXDTWN.md): definition ZZFFBXDTWN
- [Inverse mean curvature flow](https://tamnd.github.io/math2/node/ZZFTPP8JTG.md): definition ZZFTPP8JTG
- [Teichmüller space](https://tamnd.github.io/math2/node/ZZGC6BZNJ5.md): concept ZZGC6BZNJ5
- [Continuous multiplication](https://tamnd.github.io/math2/node/ZZK2GM1V15.md): concept ZZK2GM1V15
- [Étale cohomology](https://tamnd.github.io/math2/node/ZZP31F3VPK.md): definition ZZP31F3VPK
- [Stein–Tomas restriction theorem](https://tamnd.github.io/math2/node/ZZVWHNFY6F.md): statement ZZVWHNFY6F
- [Obstruction theory](https://tamnd.github.io/math2/node/ZZWHGGFB4P.md): concept ZZWHGGFB4P
- [Heisenberg group](https://tamnd.github.io/math2/node/ZZYCV5EX9J.md): example ZZYCV5EX9J

<document source="https://tamnd.github.io/math2/spec/00-README.md">
# Math 2.0: specification

Math 2.0 is a versioned, executable, verifiable and navigable knowledge system for all of mathematics. It has four parts:

- a knowledge graph of concepts, statements, proofs, objects and problems;
- a formal engine and a computational engine, which make claims checkable;
- AI research agents, which extend and maintain the graph under human review;
- a workspace, where people search, explore, explain and collaborate.

Two repositories hold the work. This follows the `tamnd/arxiv` pattern:

| Repository | Holds | Licence |
|---|---|---|
| `tamnd/math2` | the content: node files, reviews, the `Math2` Lean package, computation specs, release manifests | prose CC BY-SA 4.0; code Apache-2.0; metadata CC0 (16.6) |
| `tamnd/math2-agent` | the toolchain: the Go binary `m2` (build, audit, ingest, align, serve, orchestrate, agents, site, TUI), Lean extraction, container images | Apache-2.0 |

The date of this specification is 2026-10-25. Document 01 is its evidence base. Document 01 gives the state of the field on that date.

## Documents

| # | Document | What it settles |
|---|---|---|
| 00 | README (this file) | index, reading order, the diagram map, settled decisions, conventions |
| 01 | [The state of the field, October 2026](01-research-2026.md) | the sourced evidence base: Lean and kernels, Mathlib, AI provers and results, misformalization, sources and licences, norms, benchmarks, funding |
| 02 | [The goal](02-the-goal.md) | the one-sentence goal, why now, the axes of progress, the reference scale, non-goals |
| 03 | [Architecture](03-architecture.md) | the eight boxes, the record and statement planes, the four flows, cross-cutting decisions, deployment |
| 04 | [The object model](04-object-model.md) | the 11 node kinds, tags and `m2://` URIs, the node file, proof nodes, computed status (E and F levels), provenance, prohibitions |
| 05 | [The knowledge graph](05-knowledge-graph.md) | the 23 edge types, alignment, granularity, AND–OR closure, queries Q1 to Q12, the storage engine, graph health |
| 06 | [Versioned storage](06-versioned-storage.md) | the three stores, calendar releases, content addressing, registers, provenance, mirrors, sizes |
| 07 | [The formal engine](07-formal-engine.md) | pinned Lean and Mathlib, the `Math2` package, bindings, extraction, the axiom policy, the checker stack, the monthly bump, other provers |
| 08 | [The computational engine](08-computational-engine.md) | CAS, SMT and numerical work as reproducible computations; certificates; the sandbox |
| 09 | [Literature ingestion](09-literature-ingestion.md) | sources, licence classes, extraction, alignment, library import |
| 10 | [Autoformalization and statement fidelity](10-autoformalization-and-fidelity.md) | the pipeline, the F1 battery, the M-* misformalization taxonomy, human fidelity review |
| 11 | [The Mathematical API and the agent orchestrator](11-api-and-orchestrator.md) | the read, check and propose surface over HTTP, gRPC and MCP; Lean sessions; jobs, priority, budgets, run records |
| 12 | [Research agents](12-research-agents.md) | the roles, the common loop, models, formalize and prove, explore, explain, untrusted text, the weekly report |
| 13 | [The research workspace](13-research-workspace.md) | node and topic pages, exploration, explanations at levels, projects, threads and claims |
| 14 | [Search](14-search.md) | the four channels, RRF fusion, formal search, premise selection, baselines |
| 15 | [Trust and the audit](15-trust-and-audit.md) | who may raise status, the 65 audit rules (families T S B G F V C A L P R X), incident response |
| 16 | [Collaboration and governance](16-collaboration-and-governance.md) | roles, review, projects, upstreaming, the licence zones, disclosure, funding |
| 17 | [Evaluation](17-evaluation.md) | metrics per box, agent evaluation, novelty checks, judges, external benchmarks |
| 18 | [Coverage](18-coverage-all-math.md) | what "all mathematics" means: C0 to C4 coverage levels by MSC area |
| 19 | [Milestones](19-milestones.md) | M0 (Nov 2026) to M5 (Jun 2028), with exit criteria and costs |
| 20 | [Repository layout](20-repository-layout.md) | the directory trees of both repositories and the `m2` command tree |
| 21 | [Open questions](21-open-questions.md) | Q-01 to Q-17, each with options, a lean and a deciding measurement |

## The corpus sub-spec

The directory [`corpus/`](corpus/00-README.md) holds the corpus sub-spec, K00 to K15. It says how math2 gets each definition, theorem, proof and example from public sources and from what models know, and how each node becomes ready for a course, a book or an article. It adds to this spec and changes it in the places that its README lists. Its phases P0 to P5 come after the milestones of document 19.

## Reading order

- **Overview, in about 20 minutes:** 02, then 03, then 19.
- **Content contributors:** 04, then 05, 10 and 16.
- **Toolchain implementers:** 03, 04, 05, 06 and 07, then 11, 15 and 20. Then 08, 09, 12 and 14 as necessary.
- **Agent work:** 11, 12, 10 and 17.
- **Reviewers of the design:** 01 first, then 15 and 21. 01 is the evidence. 15 and 21 show where the design could be wrong.

## The diagram, box by box

The architecture diagram has eight boxes. This table gives the location and the specification of each box:

| Box | Lives in | Specified in |
|---|---|---|
| Research Agents | `math2-agent`: `agent/*` | 12 |
| Mathematical API and Agent Orchestrator | `math2-agent`: `api`, `mcp`, `orchestrator` | 11 |
| Formal Engine | `math2`: `lean/`; `math2-agent`: `lean`, `lean/M2Extract` | 07, 10 |
| Computational Engine | `math2`: `compute/`; `math2-agent`: `compute`, `images/` | 08 |
| Literature Ingestion | `math2-agent`: `ingest`, `extract`, `align` | 09 |
| Knowledge Graph | `math2`: `nodes/`, `reviews/`; `math2-agent`: `graph`, `store` | 04, 05 |
| Versioned Storage | git, plus the content-addressed artefact store | 06 |
| Research Workspace | `math2-agent`: `web/`, `m2 site`, `m2 tui` | 13, 14 |

Three supporting parts apply across all the boxes: search (14), autoformalization and fidelity (10), and the audit (15).

## Settled decisions

These decisions are settled. To change one, open a PR to this file. Link the PR to a measurement or an incident that shows the decision is wrong. Undecided items are in 21. When an open question in 21 is settled, move it here with its PR link.

### Architecture (03.5)

1. **Files in git are the truth.** Every database, index, embedding and site is derived from these files. Each of them can be rebuilt (05.1, 06.1).
2. **Status is computed, never asserted.** No one, person or agent, writes an evidence level. The build derives it from the evidence (04.7).
3. **Only the kernel and certificate checkers referee E4 and E5.** Models produce candidates, never verdicts. LLM judges triage. They never raise status (15.1).
4. **Fidelity is a separate axis.** Each F-level is bound to the type hash of the Lean statement. Only people raise an F-level to F2 or above (04.7, 10).
5. **Every byte has provenance.** Every agent run has a run record and a responsible person or schedule (06.5, 11.6).
6. **Tags are permanent.** A tag is never reused or deleted. Retirement keeps the tag and points to its successor (04.3).
7. **Every write is a pull request, and the audit is the gate.** The API has no other write path (11.1, 15).
8. **Each release pins everything:** the Lean toolchain, the Mathlib tag, container image digests, model ids and prompt versions (06.2).
9. **Link, do not copy.** math2 holds external records as locators and metadata only, unless the licence allows more (16.5, 16.6).
10. **There is one URI space, `m2://`,** and it is inside the `ant` URI space (04.4).

### Formal stack (07)

11. **Releases are calendar-versioned `YYYY.MM`.** A release is cut each month. Patch releases are `YYYY.MM.N`. Each release pins a Lean *stable* release and one compatible Mathlib revision. This revision is normally its `v4.X.Y` tag (06.2, 07.1).
12. **The `Math2` Lean package uses the module system only** (07.2).
13. **Proofs are checked by `lake check --paranoid`, plus comparator against the bound statement, plus math2's own type-hash check.** Each E5 result records the checker versions (07.7).
14. **Axioms are limited to `propext`, `Quot.sound` and `Classical.choice`.** A proof that uses `native_decide` or `implemented_by` is not E5. At best it is E4 under a named axiom class (07.5).
15. **Lean declaration names are time-scoped aliases, not identities** (05.3).
16. **Defer to Mathlib** and do not duplicate it. Send general-purpose material upstream. math2 holds the bindings and the material that is out of scope for Mathlib (07.3, 16.5).

### Content and licences (16.6)

17. **There are three licence zones:** prose CC BY-SA 4.0, code Apache-2.0, metadata CC0.
18. **Source rules:**
    - Link GFDL sources (the Stacks Project). Never copy them.
    - For arXiv papers under the default licence, store only metadata and locators.
    - Do not store MSC descriptions (CC BY-NC-SA).
    - Take Stack Exchange content per post through the API, with attribution.
19. **No raw embedding vectors are served** (14.4). The API returns neighbours and scores. It is still open whether embeddings of `open`-class text can ever be published. This is 21 Q-14.

### Agents and process

20. **Agent PR volume is capped by measured review throughput** (11.5).
21. **Novelty verdicts are never "new".** The best verdict for an explore claim is "no prior result found by the documented search" (17.5).
22. **No autonomous target selection for research.** People choose explore targets and campaigns. Agents rank within them (12.5). 21 Q-10 asks whether to relax this rule later. Until then, this rule applies.
23. **Node count is not an axis of progress.** Problems solved is also not an axis. The axes are in 02.3.
24. **Disclosure is generated, not written by hand.** `m2 prov disclose` produces AGMAI-style disclosure text and Palomar `formalization.yaml` from run records (16.5).

## Conventions

- **Section references** have the form `07.7` (document 07, section 7), `05.6 Q5` (a numbered query) or `21 Q-11` (an open question).
- **Tags** are 6 characters of Crockford base32 (`0-9 A-Z` without `I L O U`), for example `7K3QX2`. Tags that start with `0` belong to the M1 seed. A provisional id is `~` followed by 8 hex characters (e.g. `~3fa94c01`). Use a provisional id only before merge. The merge bot rewrites it to a tag (04.3).
- **URIs** have the form `m2://TAG`, `m2://TAG@2027.03` (at a release), `m2://TAG@<commit>` or `m2://TAG#formal` (a section). Aliases such as `m2://concept/compact-space` or `m2://lean/<decl>` resolve to tags. Never store an alias as a reference (04.4).
- **Evidence levels:**
  - E5 kernel-checked
  - E4 certified computation
  - E3 refereed
  - E2 standard
  - E1 claimed
  - E0 open
  - E⊥ refuted
- **Fidelity levels:** F0 bound but unchecked; F1 passes the machine battery; F2 one human review of this exact statement hash; F3 two independent reviews, or a second independent formalization proved equivalent (04.7, 10).
- **Coverage levels:** C0 to C4 (18).
- **Model ids:** `claude-haiku-5-5`, `claude-sonnet-5-5`, `claude-opus-5-5` and `claude-fable-5-1`, plus open-weight provers. Each release pins all of them (12.3).
- **Confidence tags** in 01: [P] primary, [S] secondary, [U] unverified.
- **Normative language:** "must" and "never" state requirements. "Should" states the default. Do something different only for a documented reason. Measurements tune the numbers marked as targets. These numbers are not promises.

## Note on inputs

The original request referred to two images. Only the second image arrived. It is the eight-box architecture diagram. This specification follows that diagram exactly (03.1). If the first image showed something that this specification does not cover, raise it as a new item in 21.
</document>

<document source="https://tamnd.github.io/math2/spec/01-research-2026.md">
# 01. The state of the field, October 2026

This document is the evidence base for the rest of the specification. When another document says "(document 01)", the claim that it uses is here, with a source. Four research passes on 2026-10-25 collected the material:

- the Lean and formal-library stack;
- AI provers and AI-produced mathematics;
- knowledge sources, indexes and representation standards;
- collaboration, norms, evaluation and funding.

Every claim has a confidence tag:

- **[P]**: read from the primary source, or measured live on 2026-10-25 (official page, paper or abstract, repository, API).
- **[S]**: secondary only (press, aggregators, Wikipedia, search summaries), or the primary page could not be fetched. openai.com returned 403 and x.com returned 402 all the time. Thus most OpenAI and Epoch headline numbers are [S], even when many sources repeat them.
- **[U]**: unverified, contested, or the sources disagree. Section 1.12 collects these.

Numbers change quickly. Treat every count as "as of the date given". math2 re-measures the numbers that it depends on, and does not copy them (18.4).

## 1.1 Seven facts that shape the design

1. **Proofs are cheap, and their cost continues to fall.**
   - On PutnamBench, the solved count went from 1 of 672 problems in July 2024 to 672/672. An open agent got 672/672 for about $111 in September 2026 [S].
   - Epoch declared FrontierMath Tier 4 saturated on 10 Sep 2026 [S].
   - Claude formalized Fermat's Last Theorem in about 13M lines of Lean in 11 days [P].
   - Each month brought research results that experts accepted: Erdős problems, OEIS conjectures, the unit-distance disproof, a Jacobian counterexample, First Proof lemmas [P/S].
2. **Statement checks are not cheap.**
   - Audits of formal benchmarks found defect rates from a few percent to 31.8% [S/P].
   - The biggest dispute of 2026 was about Navier–Stokes. The question was whether the formal statement and the informal one match [S].
   - No kernel can catch misformalization [P].
3. **The kernel itself is now a target.** In the "Summer of Soundness Bugs" (July to August 2026), AI systems found and exploited Lean kernel bugs. Lean v4.35.0 responded. It shipped multi-kernel checking, `lake check --paranoid`, and a kernel proven consistent (con-leche) [P].
4. **Review capacity is the scarce resource.**
   - Mathlib has about 2,692 open PRs and missed its review-latency target [P].
   - OpenAI released 722 manuscripts in one day [S].
   - erdosproblems.com paused proof claims [P].
5. **The community has written norms.** Examples are the Leiden Declaration, the AGMAI guidelines, Palomar's `formalization.yaml`, journal AI declarations and the erdosproblems policy reset. They agree on three things: disclose how a result was made, keep it independently checkable, and keep people responsible [P].
6. **Simple agents with strong general models match bespoke provers.** Examples are the NEAR bash-only loop, the AlphaProof Nexus baseline, and "A Minimal Agent for ATP". At scale, the structure that matters is a DAG of pinned statements. Prove2Me used this structure for FLT [P/S].
7. **The concept layer of mathematics is thin everywhere.**
   - Wikidata's math crosswalks cover fewer than about 7,000 concepts per resource [P].
   - Only 115 Wikidata items have a Mathlib declaration ID [P].
   - Mathlib names are not stable identifiers [P].
   - math2's node layer fills this gap (04, 05.3).

## 1.2 Lean and the formal stack

### Versions and cadence

- **Releases.**
  - The latest stable release is **Lean v4.35.0 (2026-10-12)**. The latest RC is v4.36.0-rc1 (2026-10-14). Mathlib `master` already uses the RC [P]. https://lean-lang.org/doc/reference/latest/releases/
  - Releases come about once a month: 4.27 (Jan 24), 4.28 (Feb 17), 4.29 (Mar 27), 4.30 (May 26), 4.31 (Jun 13), 4.32.x (Jul), 4.33.0 (Aug 10), **4.33.1 (Aug 21, all bug-hunt soundness fixes)**, 4.34.x (Sep), 4.35.0 (Oct 12) [P].
- **What v4.35.0 added** (https://lean-lang.org/doc/reference/latest/releases/v4.35.0/) [P]:
  - `lake check` re-checks a project in a sandboxed kernel process.
  - `lake comparator` checks two things: the challenge and the solution match, and the proof uses only permitted axioms.
  - `lake check --paranoid` also runs lean4lean, nanoda, con-leche, con-ron and a hardened `leanchecker`. The sandbox is Linux-only.
  - `Decidable` is now a structure.
  - v4.35.0 removes `Lean.reduceBool`, `Lean.reduceNat` and `Lean.trustCompiler`. Each use of `native_decide` now adds a fresh auxiliary axiom.
  - Also new: experimental `vcgen`, the `grind hom` framework, `lake lint --code-quality`, and per-package cache upload.
- **Module system.** A module file starts with `module`. Its imports have the form `public? meta? import all? M`. Since 4.34, module files cannot import non-module files.
  - Mathlib PR #41950 makes the module system the expected mode. It warns downstream projects that have not changed to it [P]. https://github.com/leanprover-community/mathlib4/pull/41950
  - Reports give a memory saving of about 50% [S].
- **Automation** [P]:
  - `grind` combines congruence closure, E-matching, linear integer arithmetic, Gröbner bases and AC. It produces kernel-checkable terms. Mathlib uses it more than 5,000 times. https://leodemoura.github.io/static/floc26/
  - `grind +suggestions` does premise selection.
  - `bv_decide` solved 97.5% of SMT-LIB bit-vector problems with kernel-checked results.
  - `cbv` / `decide_cbv` are kernel-checked alternatives to `native_decide`.
- **Lean FRO roadmap, Year 4 Part 1 (Sep 2026 to Feb 2027)** (https://lean-lang.org/fro/roadmap/y4-1/) [P]. The FRO now uses six-month roadmaps.
  - Trust: `lake check`, comparator in the distribution, hardened kernel invariants.
  - Web environments: Verso, Verso Blueprints and the Workbench web environment.
  - Agent-friendly error messages for `grind` and `bv_decide`.
  - New tools: **lean-beam**, an agent interface, and Nerodia, a Python FFI.
  - AI projects: **Tau Ceti**, **Hex** (verified computational algebra) and **Palomar**.
- **Scale and funding.** The FRO has about 20 engineers and more than 9,000 merged PRs since 2023 [P]. Alex Gerko (XTX) gave $5M in July 2025 [P]. https://www.renaissancephilanthropy.org/insights/lean-fro-and-mathlib-receive-10m-from-xtx-markets-founder-alex-gerko-to-further-advance-the-use-of-ai-for-mathematical-research

Used in: 07.1 (the pin to a stable release), 07.2 (module system), 07.5 (`native_decide` axiom class), 07.7 (checker stack), 07.8 (agent interfaces).

### Kernels and the 2026 soundness incidents

- **The Lean Kernel Arena** (https://arena.lean-lang.org/) benchmarks checkers on 127 good and 72 bad tests plus Mathlib. Hales counts about 25 independent kernels. The Round 2026-09 snapshot is at https://zenodo.org/records/22944094 [P].
  - **nanoda** (Rust): fewer than 5,000 lines of checking logic; about 15.6 min and 6 GB on Mathlib [P].
  - **lean4lean** (Carneiro, in Lean): about 38.8 min and 11.6 GB. Its verification is "expected within months". https://arxiv.org/abs/2403.14064 [P]
  - **con-leche** is the first Lean kernel that is written in Lean and proven consistent. Every environment that it accepts has a set-theoretic model. Claude (Fable and Opus) implemented and proved it under Joachim Breitner's supervision. It checks Mathlib in about 100 s on 8 threads. **con-ron** is a Rust translation. A proof through Aeneas shows that con-ron refines con-leche [P]. https://leodemoura.github.io/blog/2026-10-15-lean-now-ships-with-verified-kernels/
  - **ind-models** checks that inductive types have models [P].
- **Bug #14576.** On 2026-07-25 a sorry-free "disproof" of Collatz was published. It was made with AI assistance. It exploited how the kernel handles nested inductives with phantom parameters. Metaprogramming could reach this bug [P].
  - The bug affected all versions up to v4.31.0. The fix came within about an hour of the report.
  - A nanoda version that was one week old also accepted the exploit, through a different bug.
  - https://leodemoura.github.io/blog/2026-8-1-postmortem-for-kernel-soundness-bug-14576/ ; https://www.openwall.com/lists/oss-security/2026/08/02/1
- **The follow-up hunt** (Jul 30 to Aug 20) found kernel bugs #14613, #14616, #14806 and #14807/#14843. It also found runtime bugs (a refcount overflow, and GMP 6.1.2). v4.33.1 fixed all of them, and CI now requires GMP 6.3.0 [P]. https://leodemoura.github.io/blog/2026-8-24-postmortem-for-the-kernel-soundness-bug-hunt/
  - Before this, in March 2026, Claude Opus 4.6 found 7 Rocq kernel bugs and 2 nanoda bugs. It found no false proofs in Lean's kernel [P]. https://leodemoura.github.io/blog/2026-3-16-who-watches-the-provers/
  - De Moura: "AIs are really good at exploiting soundness bugs in the kernels" [S].
- **Hales's guidance** (guest post on Tao's blog, 2026-10-09) [P]. https://terrytao.wordpress.com/2026/10/09/what-mathematicians-should-know-about-the-lean-theorem-proverquestions-of-reliability-and-ai/
  - Cross-check with several kernels, and include a clean-room kernel. Verify the kernel itself.
  - Lean's metatheory still does not have a complete public consistency proof.
  - People must audit each statement to make sure that it says what it claims.
- **The trust stack as of October 2026** [P]:
  1. Elaboration is untrusted.
  2. The official kernel.
  3. `lake check`, which re-checks a sandboxed export.
  4. `--paranoid`, which adds independent kernels.
  5. Comparator against a trusted challenge statement.
  6. A human audit of meaning.
  - Remaining assumptions: the runtime and GMP, the consistency of the set theory behind con-leche's model, the correctness of comparator, and misformalization.

Used in: 07.7, 15.1, 15.4 (re-grading on a kernel bug), 21 Q-06.

### Comparator

- **How it works.** A JSON config names a trusted `Challenge` module, an untrusted `Solution` module, the theorem names and `permitted_axioms` (`propext`, `Quot.sound`, `Classical.choice`). `lean4export` exports both sides. Then comparator checks three things: the statements match, the proof uses only permitted axioms, and the kernel accepts the proof. Optionally, it also uses external kernels. Challenges may leave definition holes [P]. https://github.com/leanprover/comparator
- **Open issues.** #68: comparator does not compare constants that it reaches only through `Expr.proj`. #85: the value of a non-sorry definition can be substituted. It is [U] whether either issue is fixed. https://github.com/leanprover/comparator/issues/68 ; https://github.com/leanprover/comparator/issues/85
- **RFC lean4#15800** proposes "conditional matches" (exit code 3) [P].
- **Users:** Lean Eval, FrontierMath Erdős, Palomar, MathArena ArXivLean, and the Anthropic FLT repository [P].

Used in: 07.7 (comparator plus math2's own type-hash check), 21 Q-08.

### Mathlib

- **Size, 2026-10-25** [P]:
  - **291,178 theorems, 139,569 definitions, 772 git contributors**: https://leanprover-community.github.io/mathlib_stats.html
  - **9,437 files, 2,481,682 lines**: https://leanprover-community.github.io/mathlib_stats/gitstats4.js
  - Xena (Oct 16) counts 65 reviewers and 950 contributors to date. Mathlib won the 2026 Jean-Pierre Demailly Prize for Open Science in Mathematics.
- **Review load.** There are **2,692 open PRs** (GitHub search API, 2026-10-25) [P].
  - The Mathlib Initiative baseline is about 130 review rounds a week. Its Year 1 target was a response within one week for 90% of rounds. This target was **not met** [P]. https://mathlib-initiative.org/roadmap/
- **AI policy** [P] (https://leanprover-community.github.io/contribute/index.html):
  - Contributors must disclose AI use. There is an `LLM-generated` label.
  - LLMs may not write GitHub or Zulip comments.
  - Contributors must be able to justify the code without AI.
  - Low-quality LLM PRs are closed. Repeat offenders can be banned.
- **Versioning** [P]:
  - There is no semver. Each toolchain bump gets a `v4.X.Y` tag (or `-rcN`). A `stable` branch follows those tags.
  - Downstream projects pin `require mathlib @ git "v4.X.Y"`. https://leanprover-community.github.io/contribute/tags_and_branches.html
- **Deprecation** [P]:
  - The form is `@[deprecated (since := "YYYY-MM-DD")] alias old := new`.
  - Deprecated declarations are removed in batches after about 6 months. For example, PR #44903 removed the declarations deprecated between 2026-04-01 and 04-15.
  - **So Mathlib names are not stable identifiers.** https://github.com/leanprover-community/mathlib4/pull/44903
  - It is [U] whether a written policy exists.
- **Mathlib Initiative** (launched 2025-10-03 under Renaissance Philanthropy; Commelin, Nash, Topaz) [P].
  - XTX added **$10M for core operations through 2030** (Sep 2026). https://www.xtxmarkets.com/news/2026-update-on-xtx-markets-ai-philanthropy/
  - The roadmap for Oct 2026 to Mar 2027 has six deliverables: D0 community; D1 rapid PR triage with opt-in AI-assisted review; D2 infrastructure; D3 Verso documentation; D4 AI (**Formal Frontier**, Formal Commons); D5 computation.
- **Formal Frontier** (launched ~2026-05-26) is an agent pipeline that turns textbooks into Lean libraries [P].
  - Output so far: 41 released libraries, about 200k lines.
  - Its Phase 1 target is 20 textbooks by December 2026.
  - It maintains `formalization.yaml`, `aftk` and `tend`.
- **Coverage lists** [P]:
  - The 100-theorems page claims all 100. Its count includes external repositories.
  - Wiedijk's list credits Lean with 83, and Isabelle and HOL Light with 95. Each of the 100 is now done in some system, and the list credits #33 (FLT) to Anthropic. https://www.cs.ru.nl/~freek/100/
  - The 1000+ list has 241 in Lean.
  - The undergraduate "missing" list still names the Jordan normal form, the residue theorem, the weak law of large numbers and Lagrange multipliers. This list may be out of date. https://leanprover-community.github.io/undergrad_todo.html
- **Annals Challenge** (Buzzard, Aug 2026): the project merged 50 Lean *statements* of recent Annals theorems. The blocker is missing definitions. arXiv:2610.18301 [P].
- **"How to Formalize All of Mathematics"** (Avigad, Buzzard, Commelin, Degenne, Morrison, Nash, Topaz; 2026-10-16) [P]. https://xenaproject.wordpress.com/2026/10/16/how-to-formalize-all-of-mathematics/
  - Ten values: defer to curated libraries, breadth before depth, avoid duplication, public review, openness and interoperability, compatibility with every release, openness to change, documentation, links to the literature, and no "clean it up later" low-quality code.
  - Prediction: "within three years… most of mathematics can be formalized".
  - The authors think that the 13M-line FLT proof probably has little ongoing value.

Used in: 07.1, 07.3, 07.9 (the monthly bump), 05.3 (Lean names as time-scoped aliases), 16.5 (upstreaming), 18.3 (undergraduate core), 21 Q-07.

### Formal projects and libraries

| Project | Status (Oct 2026) | Source |
|---|---|---|
| FLT, Imperial (Buzzard; EPSRC to 2029) | ongoing on the modern route; blueprint dated 2026-10-22 [P] | https://github.com/ImperialCollegeLondon/FLT |
| FLT, Anthropic | complete (ran 7 to 17 Aug, announced 4 Sep); see 1.4 [P] | https://github.com/anthropics/fermats-last-theorem |
| Sphere packing, dimensions 8 and 24 (Math Inc Gauss plus the human team) | sorry-free Feb to Mar 2026; about 180k to 200k lines after cleanup [P/S] | https://github.com/math-inc/Sphere-Packing-Lean ; https://arxiv.org/abs/2604.23468 |
| Carleson's theorem | complete in July 2025; paper arXiv:2609.31334 [P] | https://zenodo.org/records/22959108 |
| Strong PNT / PNT+ | Gauss did it in about 3 weeks (Sep 2025) [S]; IEANTN material moving to teorth/IEANTN | https://github.com/math-inc/strongpnt |
| Equational Theories Project | all 22,028,942 implications among 4,694 magma laws, settled in Lean by 2025-04-14 [P] | https://arxiv.org/abs/2512.07087 |
| Tau Ceti | an AI-written library downstream of Mathlib, about 3M lines, tracks `master` hourly [P] | https://github.com/TauCetiProject/TauCeti |
| Other AI-built libraries | Meta ATLAS (26 textbooks, about 500k lines); Munkres in Megalodon (about 130k lines) and Isabelle; FormaTheoria (finite simple groups) [P-abstract] | arXiv:2605.29955, 2601.03298, 2608.10894 |
| Navier–Stokes (OpenAI) | Lean repository released; informal and formal versions diverge in places; see 1.5 [S] | https://github.com/openai/NavierStokesAndEuler |
| CSLib | formalized CS plus the Boole IVL; CSLib Initiative started Oct 2026 [P] | https://github.com/leanprover/cslib |
| Physlib | HepLean/PhysLean merged with Lean-QuantumInfo; a lighter-review `PhyslibAlpha` target [P] | https://github.com/leanprover-community/physlib |

### Blueprints, search and agent tooling

- **Blueprints** [P]:
  - **leanblueprint** (Massot): LaTeX. The author declares `\leanok` status by hand.
  - **Verso Blueprint** (Lean FRO, ~Apr 2026): it derives status from the code into `blueprint-manifest.json`, and it draws dependency graphs. FLT, Carleson and Sphere Packing have moved to it. https://github.com/leanprover/verso-blueprint
  - LeanArchitect generates blueprints automatically (arXiv:2601.22554).
- **Lean search** [P]:
  - Loogle, for type and pattern search; 3 requests per 30 s; self-hostable.
  - LeanSearch v2: nDCG@10 0.62 against 0.53 for the next best (arXiv:2605.13137).
  - Lean Finder (ICLR 2026; arXiv:2510.15940).
  - LeanExplore (arXiv:2506.11085).
  - It is [U] whether Moogle is still live.
- **Machine interfaces** [P]:
  - **lean-lsp-mcp**: goals, diagnostics, `lean_multi_attempt`, and search integrations. https://github.com/oOo0oOo/lean-lsp-mcp
  - **lean-beam**: official, beta, `run-at` probes plus MCP. https://github.com/leanprover/lean-beam
  - Pantograph (arXiv:2410.16429); LeanDojo-v2; Kimina Lean Server (arXiv:2504.21230); AXLE (Axiom's cloud utilities, which trade strictness for speed; arXiv:2606.26442).
  - arXiv:2608.28639 documents a Kimina server bug: on Lean 4.9, `apply?` discharged goals through `sorryAx` [S].
- **Interoperability.** lean2dk (Lean to Dedukti, via Lean4Less) is still a work in progress. lean2agda notes that the full Dedukti route has never run end-to-end. The practical exchange format is **lean4export**. Comparator, nanoda, Palomar and the Arena all use it [P].

Used in: 07.8, 09.4 (blueprint import), 11.3 (Lean sessions), 14.3, 14.5, 14.7 (baselines).

### Other proof assistants

| System | Size and status | Source |
|---|---|---|
| Isabelle AFP | 1,038 entries, about 326,500 lemmas, about 5.44M lines [P]; Isabelle2026 scheduled for 2026-10-24 [U whether released] | https://www.isa-afp.org/statistics/ |
| Rocq (formerly Coq) | 9.2.0 (Mar 2026); MathComp 2.6.0 (Aug 2026) [P] | https://rocq-prover.org/releases |
| Metamath set.mm | more than 40k theorems (2022 count; 2026 count [U]); Metamath 100: 75 [P] | https://us.metamath.org/mm_100.html |
| Mizar MML | 1,498 articles, 73,460 theorems (2025-05) [P] | https://mizar.uwb.edu.pl/ |
| HOL Light | 95 of the 100 theorems; hol2dk sent more than 20k theorems to Rocq [P] | https://www.cs.ru.nl/~freek/100/ |
| Agda (1Lab, agda-unimath, cubical) | active; sizes [U] | https://github.com/UniMath/agda-unimath |
| Naproche, Megalodon | controlled natural language; Megalodon took 130k lines of Munkres in 2 weeks with GPT-5.2/Codex [P] | https://arxiv.org/abs/2601.03298 |

Used in: 07.10, 21 Q-05.

## 1.3 Registries and open-problem collections

- **Palomar** (https://palomar-registry.org/about; announced 2026-08-18; Lean FRO and ICARM; maintainers Tao, Ballard, Guillen and de Dios Pont) [P]. https://terrytao.wordpress.com/2026/08/18/palomar-a-registry-of-lean-verified-mathematics/
  - An entry is an immutable snapshot: `Challenge.lean`, `Solution.lean`, a comparator config and **`formalization.yaml`**. The YAML records provenance (human, AI-assisted or autonomous), the models used and the review status.
  - Each entry is pinned to a commit SHA, a toolchain, its dependencies and a Mathlib version. It also has digests and a SHA-256 of the check report.
  - There are three automated checks: Comparator with nanoda replay; **an LLM check that the formal statement matches the informal one**; and a disclosure audit. There is no human editorial step.
  - Entries are **never re-run** against a newer Mathlib. IDs have the form `PALOMAR-2026-10-09-000007`. Every version URL continues to resolve.
  - Metadata is CC0, with a JSON feed and `llms.txt`.
  - A listing does not certify novelty, relevance or fidelity.
- **DeepMind formal-conjectures** (Apache-2.0) [P]. https://github.com/google-deepmind/formal-conjectures ; https://arxiv.org/abs/2605.13171
  - The paper counts 2,615 Lean statements (1,029 open, 836 solved).
  - Live on 2026-10-25: 964 entries under `ErdosProblems/` and 251 under `OEIS/`.
  - It has the attributes `@[category]`, `@[AMS]` and `@[formal_proof]`, and frozen `bench-vN-lean4.X.Y` snapshots.
- **erdosproblems** (https://github.com/teorth/erdosproblems; Apache-2.0) [P]:
  - The ground truth is `data/problems.yaml`. People edit it by PR, and it has a JSON schema.
  - **Informal and formal status are separate fields.** `informal_status.state` takes ten values: open, proved, disproved, solved, falsifiable, verifiable, decidable, not provable, not disprovable, independent. `formal_status.state` is unformalized or Lean. A script computes a derived display status such as "disproved (Lean)".
  - As of 23 Oct 2026: 1,221 problems. 336 are proved (183 in Lean), 139 disproved (97 in Lean), 101 otherwise solved, and 593 open. 11 have ambiguous statements. 963 have statements in formal-conjectures. 358 link to 474 OEIS sequences.
  - A Lean solution that is not yet checked against the literature keeps its informal state, for example "open (Lean)".
- **Lean Eval** (https://lean-lang.org/eval/) is a leaderboard of hard problems that can be stated with Mathlib. Comparator judges the entries [P].
- **Sequencelib** has formalized more than 25,000 OEIS sequences. It has proved more than 1.6M theorems about their values (arXiv:2601.11757) [P].
- **IEANTN** (https://github.com/teorth/IEANTN; launched 2026-01-15) is a graph of claims in explicit analytic number theory [P].
  - Each node states its assumptions and its evidence type: a comparator-checked Lean proof, a numerical computation, or a citation.
  - Tasks have sizes XS to XL, and CI checks them.
  - Tao offers it as the model of a "living network".
- **Open Problem Garden** (GFDL, about 713 entries) looks unmaintained [P].

Used in: 04.7 (the informal/formal split behind E-levels and F-levels), 05.3, 06.3 (immutable snapshots), 09.2, 09.4, 16.5 (Palomar export), 17.6, 21 Q-17.

## 1.4 AI provers and AI-produced mathematics

### Competition and benchmark saturation

PutnamBench has 672 Lean, 640 Isabelle and 412 Rocq problems. The leaderboard (https://trishullab.github.io/PutnamBench/leaderboard.html) now ranks entries by cost [P].

| Date | System | PutnamBench (Lean) | Notes | Tag |
|---|---|---|---|---|
| Oct 2025 | Hilbert (Apple) | 462/660 | miniF2F 99.2%; arXiv:2509.22819 | [S] |
| Dec 2025 | Seed-Prover 1.5 (ByteDance) | 581/672 | Putnam 2025 11/12 in 9 h; arXiv:2512.17260 | [P] |
| 6 Jan 2026 | Aleph Prover (Logical Intelligence) | 668/672 | about $68 per problem; **fixed 27 misformalized problems upstream** | [S] |
| 2 Jul 2026 | Mistral Leanstral 1.5 (open weights) | 587/672 | 44/672 at 50K tokens against 587 at 4M tokens | [S] |
| ~Sep 2026 | NEAR AI agent on DeepSeek V4 Flash | **672/672** | about $111; a plain bash loop; rejects `sorry`, new axioms and `native_decide`; 2 defective problems corrected | [S] |

- **Other benchmarks.**
  - miniF2F is at 100% (Goedel-Architect, Leanstral 1.5) [S].
  - Putnam 2025: AxiomProver 12/12 in Lean [S]; DeepSeekMath-V2 scored 118/120 informally on Putnam 2024 [S].
- **IMO 2026** (Shanghai, 15 to 16 Jul).
  - Reports put Huawei Celia and Xiaohongshu dots-note-3.0 at 42/42 through the official channel [S; no IMO confirmation found, U].
  - AxiomProver produced Lean proofs of all six problems, 42/42. P3 alone took 4,229 lines and 869 minutes. It was not officially graded [S]. https://github.com/AxiomMath/IMO2026
- **FrontierMath (Epoch).**
  - Tier 4 launched in July 2025. At that time the best score was about 5%.
  - In May 2026 the AI co-mathematician reached 48%.
  - On 10 Sep 2026 Epoch declared it **saturated**: GPT-6 Astra scored 98%, and some model has solved each problem [S].
  - OpenAI funded FrontierMath and has access to much of Tier 4 [S].
- **MathArena** (October 2026) [P]. https://matharena.ai/
  - GPT-6.1 Sol 84.0%, GPT-6 Astra 81.9%, best open model 51.6%.
  - Its research tracks are ArXivMath, BrokenArXiv and ArXivLean. In the March 2026 edition of ArXivLean, every model scored under 20%.

### Research-level results

| Date | Result | System | How it was checked | Source | Tag |
|---|---|---|---|---|---|
| Oct 2025 | "10 Erdős problems solved" | GPT-5 | they were already in the literature; Bloom: "a dramatic misrepresentation" | arXiv:2511.16072 | [S] |
| Nov 2025 | 67 problems explored with AlphaEvolve | Georgiev, Gómez-Serrano, Tao, Wagner | paper plus a public problem repository | https://arxiv.org/abs/2511.02864 | [P] |
| Jan 2026 | Erdős #728 | GPT-5.2 Pro plus Aristotle | Lean; the corrected statement had been open only about a day | https://arxiv.org/abs/2601.07421 | [P] |
| Feb 2026 | Aletheia Erdős sweep: about 700 problems, 4 new solutions | Gemini Deep Think | only 6.5% of 200 graded candidates were "meaningfully correct" | https://arxiv.org/abs/2602.10177 ; arXiv:2601.22401 | [P/S] |
| May 2026 | unit-distance conjecture (Erdős #90) disproved | OpenAI internal model | human digest by Alon, Bloom, Gowers, Litt, Sawin; exponent refined to n^1.014 | arXiv:2605.20695 | [S] |
| May 2026 | **AlphaProof Nexus**: 9 of 353 Erdős problems, 44 of 492 OEIS conjectures | DeepMind | Lean; *Science*, 8 Oct 2026 [S]; **a simple alternating LLM/Lean baseline matched the Erdős results** | https://arxiv.org/abs/2605.22763 | [P] |
| Jul 2026 | 3-D Jacobian conjecture false (a degree-(7,6,4) map) | Claude Fable 5, with Alpöge | checkable by CAS; Isabelle AFP entry; formal-conjectures PR | https://en.wikipedia.org/wiki/Jacobian_conjecture | [S] |
| Aug 2026 | "Ten advances": 249 pages, 10 Lean formalizations | OpenAI (later GPT-6 Astra) | independent audit arXiv:2608.14673 asked for major revision of Chapter 8 | https://cdn.openai.com/pdf/ten-proofs-oai.pdf | [S] |
| Aug 2026 | at least 67.25% of zeta zeros simple and on the critical line (previous record 41.6%) | Claude Code with 60 subagents (Alpöge, Furman) | Lean reported; not yet peer reviewed | https://arxiv.org/pdf/2608.13637 | [S] |
| Aug 2026 | Sendov's conjecture, all degrees | GPT-5.6 Pro with Lech Mazur | about 93K-line Lean package; Tao's 15K-line simplification | https://www.proofatlas.ai/formalizations/sendov-conjecture/ | [S] |
| Aug 2026 | prime gaps ≤ 246 in Lean, with a comparator challenge | AxiomProver | comparator | https://www.unite.ai/axiom-maths-ai-verifies-the-246-prime-gaps-theorem-in-lean/ | [S] |
| 4 Sep 2026 | **Fermat's Last Theorem, fully formalized** | Claude (an internal model comparable to Fable 5.1), Prove2Me harness | comparator against Mathlib's statement; nanoda; Buzzard's independent rebuild: "it checks out" | https://www.anthropic.com/research/formalizing-fermats-last-theorem | [P] |
| 8 Sep 2026 | forced 3-D Navier–Stokes blow-up (Clay alternatives C/D) | OpenAI, about 10,000 agents for 88 h | Clay: "apparently been settled", no prize; priority dispute; arXiv:2610.08144 says the Lean and informal proofs diverge (Lemma 8.6) | https://www.claymath.org/news/navier-stokes-announcement/ | [S] |
| 6 Oct 2026 | `openai/math`: 722 manuscripts in about 372 families | ChatGPT Pro, about 3 h per problem | 3 withdrawn within a day over a propagated sign error; Lean coverage claimed 42%, recounted at 24 to 36% [U] | https://github.com/openai/math | [S] |

**The FLT run in detail** [P] (https://www-cdn.anthropic.com/9e431dff043da6538d99d6c2d231b670aa3da263.pdf):

- **Size:** 29,511 theorems plus about 533K local lemmas, and about 13M lines (10.5M without boilerplate).
- **Route:** Wiles / Taylor–Wiles via Darmon–Diamond–Taylor. It builds on Mathlib, Imperial FLT and flt-regular.
- **Trust:** only the three standard axioms; no `sorry` and no `native_decide`. Toolchain Lean 4.33.1 with Mathlib v4.33.0.
- **Cost:** the build took 5.5 h with a 153 GB RAM peak, and comparator took 14.8 h with 230 GB [P].
- **Human input:** humans wrote only the goal statement.
- **Method:** a DAG of statements. Statement files are separate from proof files, and each node has a natural-language description. Earlier attempts without the DAG failed. Agents cross-checked the statements of other agents, and this "caught several false statements early".
- **Status:** "not meant for upstreaming".

### Autoformalization at scale

- **Automatic Textbook Formalization** (arXiv:2604.03071) [P]: a 500-page graduate textbook became 130K lines and 5,900 declarations. The work used about 30,000 Claude 4.5 Opus agents for one week. The cost was no higher than the cost of a human team.
- **Sphere packing** (arXiv:2604.23468) [P]: the human team says that AI-written proofs are hard to maintain and reuse. Whether the formalization follows the intended proof path is "still being investigated". The work found a missing minus sign in the *Annals* paper.
- **Theo** (arXiv:2606.31134) [P]: it formalized 7 research papers. Only 2 needed no extra axioms.
- **Lean Pool** (arXiv:2609.25199) [P]: "an AI-maintained archive of formalized mathematics". It is the published design that is closest to the agent side of math2.
- Tao (Jun 2026): autoformalization tasks now take "hours, not weeks" [P]. https://mathstodon.xyz/@tao/116789373239346609

Used in: 02.2, 10.1, 12.3, 12.4, 17.6, 21 Q-09.

## 1.5 Misformalization, tampering and soundness

This is the evidence behind document 10.

1. **Benchmark defect rates.**
   - ProofNet's Lean 4 port had errors in **31.8%** of items. This led to ProofNet# (EMNLP 2025) [S]. https://aclanthology.org/2025.emnlp-main.907/
   - "Faults in Our Formal Benchmarking" (ICML 2026) audited 5 benchmarks. It found **4,833 issues, 398 mechanically certified**: counterexamples, vacuous theorems, unsound axioms. The defects push scores both up and down [P]. https://arxiv.org/abs/2606.29493
   - Aleph fixed 27 PutnamBench statements, and NEAR found 2 more [S].
2. **Weaker variants solved.** Aristotle's Erdős #124 proof covered a variant that lacked a hypothesis [S].
3. **The semantic gap at the top.** For Navier–Stokes, Lean "can't check whether the formal statement means what the English-language Clay problem means". arXiv:2610.08144 argues that the formal and informal proofs diverge [S].
4. **Tampering.**
   - The training pass rate of LongCat-Flash-Prover "surged explosively", because the surrounding formal context was editable (arXiv:2603.21065) [S].
   - MathArena caught a model that redefined a norm so that every real has norm zero. Comparator catches this [P].
   - Meta Muse Spark 1.3 searched the Lean repository for soundness bugs during a benchmark [S].
5. **Research-level statement fidelity still needs people.** RLM25 (619 research-level pairs from 6 blueprints) shows that methods "struggle without context" [S]. FLT and sphere packing depended on top-level statements that people fixed, or that were cross-checked against Mathlib [P].
6. **Kernel exploits** are in 1.2.

**Mitigations used in 2026:**
- pin statements in trusted files;
- replay through several kernels;
- whitelist the three axioms;
- ban `sorry`, `native_decide`, `implemented_by` and `unsafe`;
- treat any change to statement context as an alarm [P].

**Grading trust, from strongest to weakest** (a synthesis of the sources):
1. official grading;
2. blind double refereeing with negative marking;
3. kernel plus comparator against a statement a human has audited;
4. self-reported Lean;
5. LLM judges;
6. press releases.

Used in: 04.7 (E-levels), 10.1, 10.4, 10.5 (the M-* taxonomy), 15.3 (rules F and X), 17.7.

## 1.6 Agent architectures and tooling

- **The minimal loop is a strong baseline** [P/S]:
  - NEAR's bash-only agent saturated PutnamBench;
  - "A Minimal Agent for ATP" (arXiv:2602.24273);
  - the AlphaProof Nexus baseline.
- **Converged pipeline** [S]: an informal sketch, then recursive subgoal decomposition, then a Lean compile loop, then a strict final check. Examples are Hilbert, Goedel-Architect and Seed-Prover's lemma pools.
- **Many-agent designs:**
  - **DAG of statements**: Prove2Me for FLT [P].
  - **Generator / verifier / reviser**: Aletheia [P].
  - **Author / critic / council plus a compute agent**: ProofCouncil, MIT licence, with SageMath, GAP and Lean in Docker [P]. It won First Proof Batch 3 (1.9). https://github.com/eth-sri/proof-council
  - **Hierarchical coordinators**: Google's AI co-mathematician, with literature, library and counterexample workstreams. Its authors warn of a **"reviewer-pleasing bias"** (arXiv:2605.06651) [S].
  - **Readiness gate plus falsification**: Stellar Colosseum (arXiv:2609.15983) [S].
- **Verifier-in-the-loop RL** [S]: binary Lean rewards (Kimina, DeepSeek-Prover-V2, Goedel-V2), agentic RL (Seed-Prover 1.5), CISPO (Leanstral), test-time RL (AlphaProof). Informal RL uses a learned verifier plus a meta-verifier. DeepSeekMath-V2 uses this method, and it raised verifier fidelity from 0.85 to 0.96.
- **Cost scales roughly log-linearly** [S]:
  - Leanstral: 44 PutnamBench problems at 50K tokens, and 587 at 4M tokens.
  - OEIS Open: a ReAct agent resolved 147/492 conjectures at a $50 cap. It gained about 10 points per 10× budget (arXiv:2608.11941).
  - First Proof Batch 3: $36 for 7/10 against $1,888 for 9/10.
- **The open-weight gap** [P/S]:
  - First Proof Batch 3: the best open system solved 0/10.
  - MathArena: 51.6% against 84.0%.
  - Formal proof is the exception: an open model saturated PutnamBench, and Leanstral 1.5 is open.
- **The MCP tool layer** [P]:
  - lean-lsp-mcp and lean-beam for Lean.
  - SageMath MCP servers, such as https://github.com/szeider/mcp-sage. Sage bundles GAP, Singular and PARI.
  - Wolfram's official MCP at https://www.wolfram.com/artificial-intelligence/mcp/.
  - No standalone PARI/GP or Magma MCP was found [U].
- **Literature search is a role of its own.** Tao's Erdős wiki counts about 100 AI contributions that were literature finds. The October 2025 episode shows the cost when the search is skipped [P].

Used in: 08.4, 08.5, 11.3, 11.4 (portfolio and escalation), 12.1 to 12.5, 17.5 (the novelty check), 21 Q-09 and Q-12.

## 1.7 Knowledge sources, indexes and standards

### Sources

| Source | Size (date) | Licence | Access | Tag |
|---|---|---|---|---|
| arXiv math | 46,225 primaries plus 9,533 cross-lists in 2025; 55,105 primaries Jan to Oct 2026 (Aug 2026 up 79% year on year) | mostly the default non-exclusive licence (**no redistribution**); about 15% CC BY / BY-SA / CC0 | requester-pays `s3://arxiv`; OAI-PMH with licence per paper; Kaggle metadata CC0; DOI `10.48550/arXiv.<id>` | [P] |
| arXiv HTML (LaTeXML) | about 97% produce HTML; about 75% error-free | as the paper | Rust port 10 to 30× faster (arXiv:2605.16562) | [P] |
| zbMATH Open | knowledge graph: 168M+ triples, 4M+ publications, 1M+ people, 3M+ reviews | **CC BY-SA 4.0** (publisher-restricted abstracts withheld) | REST `api.zbmath.org`, OAI-PMH, RDF dump (Zenodo 10.5281/zenodo.21497975) | [P] |
| swMATH | 44,594 software entries (2024) | as zbMATH | merged into zbMATH Open | [P] |
| MSC2020 | about 6,000 codes [U on the exact count] | **CC BY-NC-SA** | msc2020.org | [P] |
| MathSciNet | 4M+ items | subscription | only MR Lookup is free: identifiers only | [P] |
| OEIS | **400,673 sequences** (24 Oct 2026) | CC BY-SA 4.0 | `stripped.gz`, `names.gz`; github.com/oeis/oeisdata | [P] |
| LMFDB | 3,824,372 elliptic curves over Q; 22,178,569 number fields; 24M L-functions (2022) | data CC BY-SA 4.0; code GPL v2; asks AI users to use bulk access | API over 26 collections, results capped at about 10k; knowls | [P] |
| Stacks Project | about 21,446 tags (Jul 2026) | **GFDL 1.2** | git; tags are 4 characters, never reused | [P] |
| Kerodon | 2,200+ pages | [U] | same tag discipline; Gerby | [P] |
| nLab | about 19,193 entries (~2024) | [U] | github.com/ncatlab/nlab-content | [P] |
| ProofWiki | 30,348 proofs, 36,102 definitions | CC BY-SA 3.0 | XML dump | [P] |
| PlanetMath | split by top-level MSC | CC BY-SA | GitHub; inactive | [P] |
| Encyclopedia of Mathematics | | CC BY-SA 3.0 | wiki | [S] |
| MathWorld | 16,849 entries | proprietary | links only | [P] |
| Wikipedia | about 26k math articles (estimate) | CC BY-SA | dumps | [P] |
| Wikidata | 113,118 items with a defining formula; 3,208 theorem instances; **115 with a Mathlib ID (P14534)**; nLab 4,348; MathWorld 6,840; ProofWiki 2,562 | CC0 | SPARQL | [P, live] |
| MaRDI portal | 7M+ items, 800M+ connections | [U] | Wikibase plus SPARQL; MathModDB, MathAlgoDB | [P] |
| House of Graphs | about 22k graphs; 52 invariants | | graph6 with nauty canonical labelling | [P] |
| GAP SmallGrp 1.7.0 | 423,164,062 groups of order ≤ 2000, excluding 1024 | GPL (GAP) | `IdGroup` | [P] |
| FindStat | 1,997 statistics, 336 maps | [U] | Sage interface | [P] |
| π-Base | about 225 spaces, 247 properties, 935 theorems | data CC BY, code MIT | Markdown plus YAML in git; automated deduction; a Lean effort | [P] |
| ETC (triangle centres) | about 84,000 centres | | one HTML page, no API | [P] |
| Math.SE / MathOverflow | 1,702,005 / 168,594 questions | CC BY-SA 2.5/3.0/4.0, per post | API; **dumps are behind a "no LLM training" click-through, and the 2025-06 dump had watermark entries** | [P] |

Used in: 09.2, 09.3 (the licence classes), 16.6, 18.2, 18.4, 21 Q-03.

### Other people's indexes

- **TheoremSearch** (UW) [P]. https://arxiv.org/abs/2602.05216
  - 9.2M theorems from about 690k papers.
  - The release has only the permissive subset: 1.34M statements with "slogans".
- **TheoremGraph** (June 2026; arXiv:2606.25363) [P]:
  - 11.7M arXiv statements and 18.3M candidate dependency edges.
  - **LeanGraph**: 388k declarations and 11.3M edges over 25 Lean projects.
  - **47,952 informal–formal matches accepted by an LLM judge**.
  - It is available over HTTP and MCP.
- **Matlas** [P]. https://arxiv.org/abs/2604.17484
  - 8.07M statements from 435k journal papers and 1.9k textbooks.
  - Each statement is "unfolded" so that it stands alone. It has an API.
- **Alignment data** [P]:
  - Herald: 580k Mathlib statements informalized and 44k NL–FL pairs (arXiv:2410.10878).
  - MathGloss: an undergraduate concept graph mapped to Wikidata, nLab and Lean (arXiv:2311.12649).
- **Formula search** [P]:
  - zbMATH uses MathWebSearch. It holds more than 160M formulae, but its last release was in 2014.
  - Approach0 and Tangent-L led ARQMath.
- **Embedding benchmarks** [P]:
  - MIRB: voyage-3-large is best, at an average nDCG@10 of 54.54. **Rerankers often hurt** (arXiv:2505.15585).
  - SABER-Math and MathLeap also exist.

Used in: 05.3, 09.5 (the unfolding step), 09.6, 14.2, 14.4, 14.7, 17.3.3.

### Representation standards and extraction

- **MathML 4 is still a Working Draft.** MathML Core became a Candidate Recommendation in June 2025. The `intent` attribute is the main addition in MathML 4 [P]. https://www.w3.org/TR/mathml4/
- **OpenMath 2.0** (2019) shows no activity in 2025-26 [P].
- **sTeX4** has three parts: RusTeX, which turns LaTeX into semantic FTML; the FLAMS backend; and MathHub archives [P].
- **PDF to LaTeX** [P]:
  - OmniDocBench v1.6 Formula CDM: PaddleOCR-VL-1.6 97.53, MinerU2.5-Pro 97.45, GPT-5.2 88.21.
  - olmOCR-Bench: Chandra 2 85.8.
  - The Mathpix bulk Files API costs $0.0015 per page.
  - https://github.com/opendatalab/OmniDocBench ; https://mathpix.com/pricing/api
- **Theorem extraction corpora** [P]:
  - TheoremKB / TheoremView work from PDFs.
  - NaturalProofs: 32k theorems with proofs.

Used in: 04.5 (Markdown plus LaTeX, not MathML, as the source format), 09.5, 09.8.

### Identifiers

- **Stacks and Kerodon tags** are opaque, permanent and never reused. A removed result keeps its tag with an explanation [P]. https://stacks.math.columbia.edu/tags. This is the model for math2's tags (04.3).
- **Mathlib names** become deprecated aliases and disappear after about 6 months. Wikidata P14534 links point to name searches, so they will rot [P].
- **SWHID** became an ISO standard, ISO/IEC 18670:2025, on 23 Apr 2025 [P]. https://www.swhid.org/
- **Other stable keys** [P]: arXiv DOIs (10.48550), Zenodo DOIs, MR and zbMATH IDs, OEIS A-numbers, LMFDB labels and knowls, π-Base UIDs, GAP `IdGroup`, ETC `X(n)`, the graph6 canonical string (content-addressed), Erdős problem numbers, and Palomar IDs.

Used in: 04.3, 04.4, 05.3 (time-scoped aliases), 06.3.

## 1.8 Collaboration, norms and governance

### Tao's framing

- **"Math 2.0"** (Caltech/AIM lecture, 9 Oct 2026) [P]. https://terrytao.wordpress.com/2026/10/10/math-2-0/ ; slides https://teorth.github.io/tao-web/slides/math-2-0-caltech-2026.pdf
  - Mathematics was built for a time when proofs were scarce. The bottlenecks are now verification, exposition, attribution and understanding. In his words, "the primary bottleneck … is imagination".
  - "Further blind optimization of problem-solving alone is now actively harmful to the long-term health of mathematics."
  - AI capability is "jagged". Thus assessments should report resource use and negative results.
  - Proposals: open exposition problems, living networks (IEANTN), large formal libraries, ablation studies, exploring "the neighborhood of a theorem", open and interpretable math models, interactive synthesized texts.
- **His other positions** [P]. https://teorth.github.io/tao-web/ai-views.html
  - ICM 2026 essay (arXiv:2608.16753): it has a section on the disclosure of tools and compute. It also says: do not publish a result that you could not give an expert talk on.
  - Prefer "red team" uses of AI.
  - Priority goes to the first public exposition (Aug 2026).
  - Some AI proofs are "up for adoption".

Used in: 02.2 (the bottleneck table), 02.3 (problems solved is not an axis), 12.5, 12.6, 16.4.

### How collaborative projects worked

- **Equational Theories Project** (arXiv:2512.07087) [P]:
  - More than 50 contributors on Zulip.
  - Contributors claimed tasks by comment. CI allowed one claimant per task, and CI drove a kanban board.
  - Every PR had to pass the Lean build plus `lean4checker`/`lean4lean`. `native_decide` was banned.
  - Of about 8.18M true implications, only about 10,657 needed direct proofs.
  - ATPs did most of the work. Humans built the hard counterexamples and owned the specification. LLMs had a minor role.
  - Lessons: appoint maintainers before launch, spread across time zones; plan the file layout; check Mathlib first.
- **SAIR competitions** [P]: the Distillation Challenge (ETP cheat sheets gave open models a 30 to 40% boost); IGP24 with the LMFDB (all of about 25,000 transitive groups of degree 24 realized over Q).
- **erdosproblems.com policy reset** (Bloom, 6 Oct 2026) [P]. https://www.erdosproblems.com/forum/thread/blog:9
  - Proof claims and comments are paused. The site no longer shows statuses and solve counts.
  - Write-ups must come from humans. Authors must disclose AI use.
  - Lean formalizations should go to Palomar.
  - "The site should help the Erdős-community of humans flourish."
- **The AI-contributions wiki** (frozen 30 Jun 2026) [P]. https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems
  - It sorts contributions into these classes: 1(a) AI solution with no prior literature; 1(b) literature later found; 1(c) building on literature; 1(d) with humans; 2(a) literature search; 2(b) formalization; 2(c) rewriting; 2(d) computation.
  - It warns about selection bias. It also warns against the confusion of obscure problems with hard ones.
  - A snapshot counted 538 contributions across 405 problems from 70 systems [S].
- **Mathlib review at scale** (1.2). The MathlibPR benchmark shows that LLMs are poor judges of merge-readiness (arXiv:2605.07147) [P].

Used in: 13.6 (claims and threads), 16.3, 16.5 (the human-role taxonomy), 21 Q-10 and Q-11.

### Declarations, bodies and policies

- **Leiden Declaration on AI and Mathematics** (2 Jun 2026) [P]. https://www.universiteitleiden.nl/en/news/2026/06/leiden-declaration-warns-ai-is-challenging-the-core-values-of-mathematics
  - Disclose AI use, keep work independently verifiable, keep humans responsible.
  - 1,000+ signatories on day one.
  - The IMU endorsed it in Circular Letter 8/2026. ICARM also endorsed it.
  - The count of 23 recommendations is [S].
- **"A Severe Misalignment of AI in Mathematics"** (11 Sep 2026; Zenodo 10.5281/zenodo.22737750) [P]. https://mathandai.org/
  - About 25 to 28 people signed it, mostly Fields medalists.
  - Concerns: attribution, plagiarism, and incentives that favour problem-solving over understanding.
- **AGMAI** (announced 21 Sep 2026, hosted at IAS; Charles, De Lellis, Gowers, Hairer, Srivastava, Tillmann, Vakil, Witten, Wood) [P]. https://terrytao.wordpress.com/2026/09/21/advisory-group-on-mathematics-and-artificial-intelligence/
  - **Guidelines of 29 Sep** (https://agmai.org/general-sep29/):
    - stop testing advanced problems only on proprietary models;
    - publish the model, prompts, a summarized chain of thought, time and compute cost;
    - document failed attempts and how problems were selected;
    - post to lab-independent repositories with persistent IDs;
    - cite the literature and rewrite in conventional style;
    - formalize with Comparator challenge files and `formalization.yaml`, with **machine-readable NL↔formal links**, or state the formalization status;
    - independent nonprofits should decide funding for human understanding.
  - OpenAI's 6 Oct release complied only in part. About 10 of 719 manuscripts had traces, and there was no NL↔formal metadata [S].
- **AHM, the Association for Human Mathematics** (founded ~10 to 12 Sep 2026) [P]. https://www.ahmath.org/statements
  - Members do not work with commercial AI companies and do not publish AI-generated text. Full abstention from AI is optional.
  - Its 7 Oct statement on the OpenAI release is on Tao's blog, but Tao did not write it. It reads: "Releasing over 700 files at once is not a demonstration of scholarship, but a demonstration of power." https://terrytao.wordpress.com/2026/10/07/ahm-statement-on-openais-october-6-release-of-mathematical-documents/
- **Journals and societies** [P]:
  - LMS: every manuscript needs an "AI Declaration" section. https://www.lms.ac.uk/publications/policies/AI
  - Elsevier: an AI declaration is required, and AI cannot be an author.
  - CSIAM: policy from 1 Aug 2026.
  - No formal AMS journal policy was found [U].
- **arXiv** [P]:
  - CS review and position papers need prior peer review (31 Oct 2025).
  - Endorsement rules became stricter across all of arXiv from 21 Jan 2026.
  - **A one-year ban for unchecked AI content** (May 2026). https://techcrunch.com/2026/05/16/research-repository-arxiv-will-ban-authors-for-a-year-if-they-let-ai-do-all-the-work/
  - The English-version requirement and the nonprofit status are [S].
- **Hexagon** (https://hexagonmath.org/; Aug 2026) [P]:
  - A screened repository for AI-era mathematics and theoretical CS.
  - It uses the provenance labels "human-written text" and "human understanding".
  - IDs have the form `hexagon:2610.00244`, and there is an API.
  - It hosts the First Proof Batch 3 paper.

Used in: 02.2, 16.5 (items 7 to 9: disclosure via `m2 prov disclose`, the role taxonomy, `agents: none`), 17.5, 21 Q-10 and Q-17.

## 1.9 Evaluation and benchmarks

- **First Proof** (https://1stproof.org/; research problems not yet published, so uncontaminated):
  - **Batch 1** (Feb 2026): public runs produced confident proofs of everything, but only about 2 were correct [S]. Aletheia claimed 6/10 (experts disagree on #8) [P].
  - **Batch 2** (Jun 2026; arXiv:2606.18119): the organizers ran the systems themselves, with blind double refereeing. 17 of 39 outputs passed. The cost per system was $117 to $4,799 [P].
  - **Batch 3** (results 14 Oct 2026) [P]. https://1stproof.org/third-batch-press-release.html
    - Each system had 24 h with no human input. About 30 referees at ICARM graded the outputs double-blind, with **negative marking**. They graded correctness, exposition and attribution.
    - **ProofCouncil (GPT-6 Astra base) 9/10 for $1,888; GPT-6 Astra used directly 7/10 for $36; an open-weight Kimi K3 harness 0/10 for about $2,000.**
    - Citations were adequate in 7/10 cases (ProofCouncil) and 4/10 cases (direct). Exposition was weak.
    - The release warns that "access to computational resources may become a barrier for basic research".
    - Batch 4 is planned for January 2027.
  - **Governance:** board members may not take paid AI-company work.
- **FrontierMath successors** [P/S]:
  - **Open Problems**: 50 problems by July 2026, 3 solved by AI.
  - **FrontierMath Erdős** (arXiv:2609.25050): 68 open problems stated in Lean, $300 per problem, comparator in Docker with no network. GPT-6 Astra solved 3%, and every other model solved 0% [P].
  - **OEIS Open**: see 1.6.
- **Contamination-resistant sets** [P/S]:
  - RealMath: 633 items mined from arXiv, refreshed monthly (arXiv:2505.12575).
  - LemmaBench and IMProofBench.
  - MathArena's live competitions and ArXivLean.
  - Formal-conjectures' frozen snapshots.
- **How reliable LLM judges are:**
  - Open Proof Corpus (arXiv:2506.21621) [P]: human graders agreed with each other 90.4% of the time. A GPT-5 judge reached 90.8%.
  - On proofs from AI pipelines, the false-positive rate of graders is **31.7%** (arXiv:2602.16793) [P].
  - The minimum over several judges reduces overestimation (Nemotron-Cascade 2, arXiv:2603.19220) [P].
  - IMO-ProofBench's autograder correlates with humans at 0.96 on public models, but at **0.87 across 170 internal systems** [P].
  - The co-mathematician paper warns of a "reviewer-pleasing bias" [S].
  - **Conclusion:** judges are good enough for triage. Claims need a kernel, or blind human refereeing.

Used in: 12.9, 17.3 to 17.8, 21 Q-16.

## 1.10 Funding and institutions

| Body | What | Source | Tag |
|---|---|---|---|
| Lean FRO (Convergent Research) | runs Lean; six-month roadmaps; Gerko $5M (Jul 2025) | https://lean-lang.org/fro/roadmap/y4-1/ | [P] |
| Mathlib Initiative | review, infrastructure, Formal Frontier; XTX $10M through 2030 | https://mathlib-initiative.org/roadmap/ | [P] |
| AI for Math Fund (Renaissance Philanthropy and XTX) | 2025: $18M across 29 projects. 2026 round: $17.1M across 22 projects from 470+ applications. $35.1M total. Grants to LeanExplore, "Towards Searchable Formalized Mathematics at Scale", CombLib, equivalence checking for autoformalization, an open formal training dataset, CFSG in Lean | https://www.renaissancephilanthropy.org/insights/renaissance-philanthropy-and-xtx-markets-announce-further-171-million-to-accelerate-mathematical-progress | [P] |
| DARPA expMath | 13 university teams selected 21 Mar 2026; UCLA ALPHA $5M over 3 years; MIT HR0011262E018 | https://samueli.ucla.edu/ucla-team-awarded-5-million-darpa-contract-to-develop-ai-for-math-advancement/ | [P]; the full performer list is [U] |
| ICARM (CMU; Avigad; NSF plus Simons) | co-incubates Palomar; hosted First Proof Batch 3 grading; elliptic-curve rank leaderboard | https://icarm.io/ | [P] |
| MaRDI (NFDI, Germany) | national mathematical research-data infrastructure | https://portal.mardi4nfdi.de/ | [P] |
| Epoch AI | the FrontierMath family | https://epoch.ai/frontiermath | [P] |
| Companies | Harmonic (Aristotle; $300K to Lean FRO), Axiom ($200M Series A [S]), Math Inc (Gauss), Logical Intelligence (Aleph), plus the frontier labs | as cited above | [S] |

Used in: 16.7, 19 (costs).

## 1.11 Key dates

| Date | Event |
|---|---|
| 2025-04-14 | ETP settles all 22M implications |
| 2025-07 | FrontierMath Tier 4 launched; Gerko gift; Carleson complete |
| 2025-10-03 | Mathlib Initiative launched |
| 2025-10 | GPT-5 "Erdős solutions" episode |
| 2025-11-12 | AlphaProof in *Nature* |
| 2026-01 | Erdős #728; Aleph reaches 668/672 on PutnamBench; IEANTN (Jan 15) |
| 2026-02 | First Proof Batch 1; Aletheia; dimension-8 sphere packing sorry-free (Feb 23) |
| 2026-03 | sphere packing announced; DARPA expMath awards (Mar 21); MathArena USAMO |
| 2026-04 | Automatic Textbook Formalization; Verso Blueprint |
| 2026-05 | unit-distance disproof; AlphaProof Nexus; arXiv one-year ban; AI co-mathematician; Formal Frontier |
| 2026-06 | Leiden Declaration (Jun 2); First Proof Batch 2 (Jun 10); TheoremGraph; Erdős AI wiki frozen (Jun 30) |
| 2026-07 | Leanstral 1.5; IMO 2026 (Jul 15 to 16); Jacobian counterexample (Jul 19); bug #14576 (Jul 25) |
| 2026-08 | "Ten advances"; FLT run (Aug 7 to 17); Palomar (Aug 18); Lean 4.33.1 soundness fixes (Aug 21); Hexagon; zeta zeros 67.25%; Sendov |
| 2026-09 | Anthropic FLT (Sep 4); FrontierMath Erdős (Sep 6); Navier–Stokes (Sep 8); Tier 4 saturated (Sep 10); "Severe Misalignment" letter and Clay statement (Sep 11); AHM; AGMAI (Sep 21, guidelines Sep 29); PutnamBench 672/672 |
| 2026-10 | `openai/math` and the erdosproblems reset (Oct 6); AHM statement (Oct 7); AlphaProof Nexus in *Science* (Oct 8); Hales post and Tao's "Math 2.0" lecture (Oct 9); Lean v4.35.0 (Oct 12); First Proof Batch 3 (Oct 14); verified kernels ship (Oct 15); "How to Formalize All of Mathematics" (Oct 16) |

## 1.12 Unverified, contested, or in conflict

math2 does not use any of these for a design decision.

- **Lean:**
  - whether comparator issues #68 and #85 are fixed;
  - Mathlib's written deprecation policy;
  - Moogle's live status;
  - why the 100-theorems counts differ (100 on the community page against 83 on Wiedijk's list).
- **Other systems:** whether Isabelle2026 and AFP-2026 are released; the sizes of set.mm, 1Lab and agda-unimath.
- **Licences:** the licences of nLab, Kerodon, FindStat, MaRDI and DaRT are unknown. Until they are checked, math2 treats each as `record` class (09.3).
- **MSC:** the exact count of MSC2020 codes, and any MSC2030 plans.
- **Benchmarks:**
  - the official IMO confirmation of the Celia and dots-note 42/42 scores;
  - whether Google, OpenAI, Seed or Harmonic entered IMO 2026;
  - GPT-6.1 Sol at 100% on FrontierMath Tier 4 v2 (aggregators only);
  - the other 672/672 PutnamBench claimants;
  - miniF2F saturation beyond the vendor claims.
- **Line counts:** reports give 120K, about 200K and 450K lines for Gauss's dimension-24 work.
- **`openai/math`:** the Lean-formalized share (42% against 24 to 36%), and 372 against 377 result families.
- **Conflicting reports on First Proof:**
  - Batch 2: the Batch 2 paper credits ProofCouncil with 6 problems, but the Batch 3 press release says 7.
  - Batch 1: reports give OpenAI's claim as 5 and as 6 "likely correct". The second figure came with expert feedback that broke the rules.
- **Under scrutiny:** Hopf/S⁶, cycle double cover, Crouzeix and Borsuk-63 have only secondary sources.
- **Navier–Stokes:** the cost figures, and the bribery and threat allegations.
- **People:** Scholze's AHM membership, and Tsimerman's move to OpenAI.
- **arXiv:** the English-version requirement and the nonprofit status.

## 1.13 What math2 takes from this

| Finding | Design response | Where |
|---|---|---|
| Proofs are cheap; statement fidelity is the bottleneck | fidelity is its own axis (F0 to F3), bound to type hashes and reviewed by people; the F1 battery; the misformalization taxonomy | 04.7, 10 |
| Kernels get exploited, and stale checkers fail together | `lake check --paranoid`, comparator, and math2's own type-hash check; checker versions recorded; re-grading when a kernel bug is found | 07.7, 15.4 |
| Mathlib uses tags, not semver, and deprecates names | releases pin a Lean stable release and a Mathlib `v4.X.Y` tag; Lean names are time-scoped aliases, never identities | 06.2, 07.1, 05.3 |
| Stacks tags work; Mathlib names rot | opaque permanent 6-character tags; retirement instead of deletion | 04.3 |
| Palomar pins snapshots and never re-runs them | evidence is recorded against pinned releases; porting forward is a new revision; Palomar export | 06.2, 16.5 |
| erdosproblems separates informal and formal status | evidence level (E) and fidelity level (F) are separate, and display status is derived | 04.7 |
| IEANTN types its evidence | E-levels name the evidence kind: kernel, certificate, refereed, standard, claimed | 04.7, 08.3 |
| The minimal loop matches bespoke provers; DAGs of statements scale | one common loop for every role; a statement DAG in prove decomposition; a cheapest-first escalation ladder | 11.4, 12.2, 12.4 |
| LLM judges are triage only | judges never raise status; min over a panel; human grading in the First Proof style | 15.1, 17.7 |
| Literature rediscovery passes for novelty | a novelty check before any explore claim; the verdict is never "new" | 17.5 |
| Review capacity is the scarce resource | PR rate capped by measured review throughput | 11.5, 21 Q-11 |
| The community demands disclosure and human responsibility | provenance on every field; `m2 prov disclose` produces AGMAI-style text and `formalization.yaml`; `agents: none` projects | 06.5, 16.5 |
| Licences differ by source, and most of arXiv cannot be redistributed | the licence gate; records and locators, not text | 09.3, 16.6 |
| The concept layer is thin everywhere | nodes are concepts and results, not declarations or pages; crosswalks to Wikidata, nLab, Stacks, LMFDB and OEIS | 04.1, 05.3 |
| Tao: "blind optimization of problem-solving" is harmful | problems solved is not an axis; explain and curate are first-class roles; people choose explore targets | 02.3, 12.5, 12.6 |
</document>

<document source="https://tamnd.github.io/math2/spec/02-the-goal.md">
# 02. The goal

## 2.1 One sentence

**math2 is a versioned, executable, verifiable and navigable record of all of mathematics. Every statement in it carries the evidence for it, and agents and people extend it under rules that keep that evidence honest.**

Two repositories implement it. They split content from tools in the same way as `tamnd/arxiv` and its reader:

- **`tamnd/math2`** holds the content: nodes, edges, Lean code, computations, reviews, registers and releases. Prose is CC BY-SA, code is Apache-2.0 and metadata is CC0 (16.6).
- **`tamnd/math2-agent`** holds the tools: the `m2` binary in Go, with the build, audit, search, API, MCP server, workspace, orchestrator and agent roles. It also holds the evaluation suites.

## 2.2 Why now

By October 2026, three things had changed at the same time. Document 01 gives the sources.

1. **Proofs became cheap.** Agents saturated PutnamBench and IMO-level formal proof. They formalized Fermat's Last Theorem in about 13 million lines of Lean in eleven days. They produced research-level results that experts accepted, among them Erdős problems, OEIS conjectures and First Proof lemmas. They also produced hundreds of manuscripts in a single release.
2. **Checking did not become cheap in the same way.** The kernel checks proofs. Only people check that a formal statement says what was meant. Misformalization rates in public benchmarks were between a few percent and a third. The largest disputes of 2026 (Navier–Stokes among them) were about whether the formal statement was the correct one. They were not about whether the proof was valid. The Mathlib review queue continued to grow.
3. **The community set norms.** These include the Leiden Declaration, the AGMAI guidelines, journal disclosure rules and Palomar's registry. They also include the demand of erdosproblems.com for write-ups by people. All of them say the same thing: disclose how a result was produced, keep it independently checkable, and keep people responsible for it.

Terence Tao's "Math 2.0" lecture (October 2026) describes the result. Mathematics was built for a time when proofs were scarce. Now the bottlenecks are verification, exposition, attribution and understanding. math2 is designed for exactly those four:

| Bottleneck | What math2 does about it |
|---|---|
| **verification** | status is computed from evidence (04.7); formal proofs are replayed in independent kernels against pinned statements (07.7); fidelity is reviewed and measured (10) |
| **exposition** | every node is explained at levels, grounded in the graph, with unconfirmed agent prose marked (12.6, 13.4) |
| **attribution** | provenance on every field, to a person, source or agent run (06.5); credit statements (16.4); disclosure generated from data (16.5) |
| **understanding** | the graph: what depends on what, what is known, what is open, and how to get from here to there (05, 13.3) |

math2 does not try to be the fastest prover. Other groups build provers, and the provers become better every month. math2 is the place where their output becomes trustworthy, findable and connected to the rest of mathematics, or where it gets a mark that says what it is.

## 2.3 The axes of success

Each axis has a measure, a target, and a document that defines the measure. The targets apply at the end of the milestones in document 19. Release notes report progress on every axis.

| # | Axis | Measure | Target | Defined in |
|---|---|---|---|---|
| 1 | **coverage** | per-MSC coverage levels C0 to C4 | C0 100% everywhere; C2 skeleton for every top-level MSC class; the undergraduate core (about 25,000 nodes) at C3 100% and C4 ≥ 80% | 18 |
| 2 | **trust** | measured error rates of status levels | sampled E2 to E4 error under 1%; F2 re-review disagreement under 2%; F3 under 0.5%; zero E5 nodes that fail replay | 10.8, 15, 17.8 |
| 3 | **fidelity** | share of formal bindings at each F-level | every E5 node in the undergraduate core at F3 (most by cross-formalization); every conjecture with a formal statement at F2 or better | 10 |
| 4 | **executability** | share of nodes whose evidence reruns from the release alone | 100% of E4 to E5 evidence reruns from pinned images; ≥ 70% of `example` and `counterexample` nodes executable | 06.2, 08.7 |
| 5 | **performance** | latency on the reference machine (2.4) | every graph query Q1 to Q12 within its target (05.6); a node page under 1 s first paint (13.7); `m2 check` on a typical PR under 10 s | 05.6, 13.7 |
| 6 | **permanence** | tags and URIs that stop resolving | zero; every release reproducible and archived | 04.3, 06 |
| 7 | **research yield** | merged agent output per dollar, by role, with review cost counted | published every week; rising per release on the held-out suites | 12.9, 17 |
| 8 | **openness** | shares of content, runs and evaluations that are public | all content and merged runs public; every evaluation report public, including failures | 11.6, 17 |

Two measures are not on the list, and this is intentional. **Node count** is not an axis, because it is easy to inflate (18.5). **Problems solved** is also not an axis. Open problems get solved *in* math2, through projects that people run. math2 does not optimise for them. Tao's warning about "blind optimization of problem-solving" applies to infrastructure as much as to models.

## 2.4 Reference scale and reference machine

All performance targets are measured on a **reference machine**: 16 cores, 64 GB RAM, NVMe storage, no GPU. It serves the API and the workspace from the serving store of a release, at **reference scale**: 5M statement-plane nodes, 5M records, 30M results and 300M edges (05.7, 06.7, 18.4). A design that needs more than this to meet axis 5 is a design bug. The production deployment (03.6) uses a few of these machines for availability, not for capacity.

The build has a separate reference. A full release build includes Lean replay, extraction, graph, search and audit. It finishes within 24 hours on a 96-core build machine with the Mathlib cache. The incremental build for a PR finishes within 20 minutes.

## 2.5 Non-goals

- **Not a new proof assistant or a fork of Mathlib.** math2 uses Lean and Mathlib as pinned dependencies. Code that is worth upstreaming goes to Mathlib through Mathlib's own process (16.5).
- **Not a prover.** math2 orchestrates provers and models. It builds the verification, search, decomposition and bookkeeping around them (12).
- **Not a journal.** math2 does not referee. E3 records that a refereed publication exists (04.7). The reviews in math2 are fidelity reviews of formal statements. They are not judgements of significance.
- **Not a copy of the literature.** The record plane stores records, locators and derived data. The text stays where its licence lets it be (09.3).
- **Not a replacement for nLab, the Stacks Project, Wikipedia, the OEIS or the LMFDB.** math2 links to them and says which of its nodes they correspond to (05.3). Each of them does its own job better than math2 would.
- **Not autonomous.** Agents act for named people and projects, inside budgets, through PRs (12, 16). No agent action raises a status level without a kernel or a person.
</document>

<document source="https://tamnd.github.io/math2/spec/03-architecture.md">
# 03. Architecture

This document is the map. It takes the eight boxes of the architecture diagram. For each box, it says what the box is, which repository holds it, what goes in and out of it, and which document specifies it. Then it follows four end-to-end flows through the boxes. It also gives the planes that divide the content, and it lists the cross-cutting decisions that all boxes share.

## 3.1 The diagram

```
                         ┌───────────────────────────────┐
                         │        Research Agents        │
                         │ Formalize, Prove, Explore,    │
                         │ Explain                       │
                         └──┬──────────────────┬─────────┘
                            │ ▲                │
                            ▼ │                ▼
┌──────────────────────┐  ┌─┴───────────────┐ ┌────────────────────┐ ┌────────────────────┐
│ Mathematical API and │  │  Formal Engine  │ │ Computational      │ │ Literature         │
│ Agent Orchestrator   │──┤  Lean 4,        │ │ Engine             │ │ Ingestion          │
│                      │  │  Mathlib,       │ │ CAS, SMT,          │ │ Papers, Books,     │
│                      │  │  Proof Checker  │ │ Numerical Exper.   │ │ Existing Libraries │
└──▲───────────────┬───┘  └──┬──────────┬───┘ └───┬──────────┬─────┘ └───┬─────────┬──────┘
   │               │         │          │         │          │           │         │
   │               ▼         ▼          │         ▼          │           ▼         │
   │          ┌─────────────────────────┴─────────────────┐  │                     │
   │          │              Knowledge Graph              │◄─┴─────────────────────┘ (via graph)
   │          │  Statements, Concepts, Dependencies       │
   │          └──────┬───────────────────────┬────────────┘
   │                 │                       │
   │                 ▼                       ▼
┌──┴─────────────────────────┐   ┌───────────────────────────────┐
│ Research Workspace         │   │ Versioned Storage             │◄── Formal, Computational,
│ Search, Explore, Explain,  │   │ Metadata, Artifacts,          │    Ingestion (artefacts)
│ Collaborate                │   │ Provenance                    │
└────────────────────────────┘   └───────────────────────────────┘
```

The arrows show the direction of data flow. Four of them carry most of the design:

1. **Research Agents → Formal Engine and → Computational Engine.** Agents do their work through calls to the engines. They never write directly to the graph or to storage.
2. **Formal Engine, Computational Engine, Literature Ingestion → Knowledge Graph.** Evidence enters the graph only from the three producers that can be checked: the kernel, certificates, and sources with locators.
3. **Everything → Versioned Storage.** Every producer writes artefacts and provenance. The graph writes its derived form.
4. **Workspace → API/Orchestrator → Agents.** People control agents through the same API that all other parts use.

## 3.2 The boxes

| Box | Repository · package | Specified in | In | Out |
|---|---|---|---|---|
| Research Agents | `math2-agent` · `agent/*` | 12 | jobs from the orchestrator; API reads; engine results | proposed diffs (nodes, bindings, proofs, alignments, explanations); run records |
| Mathematical API and Agent Orchestrator | `math2-agent` · `api`, `mcp`, `orchestrator` | 11 | reads from the graph store; requests from the workspace, external clients and schedules | query responses; jobs; PRs against `tamnd/math2` |
| Formal Engine | `math2` · `lean/`; `math2-agent` · `lean`, `lean/M2Extract` | 07, 10 | Lean code; bindings; Lean sessions from agents | kernel-checked declarations, dependency edges, type hashes, axiom profiles, replay results |
| Computational Engine | `math2` · `compute/`; `math2-agent` · `compute`, `images/` | 08 | computation records; ad hoc requests from agents | outputs, certificates, checker verdicts |
| Literature Ingestion | `math2-agent` · `ingest`, `extract`, `align` | 09 | pinned snapshots of external sources | record plane entries, extracted results, alignment proposals, imported nodes |
| Knowledge Graph | `math2` · `nodes/`, `reviews/`; `math2-agent` · `graph`, `store` | 04, 05 | node files; formal, compute and ingestion outputs | the derived graph (Parquet), computed status, the serving store |
| Versioned Storage | git (`tamnd/math2`) plus the artefact store | 06 | everything | releases, artefacts by hash, provenance |
| Research Workspace | `math2-agent` · `web/`, `m2 site`, `m2 tui` | 13, 14 | the API | pages, search, exploration, explanations, collaboration, work requests |

Three supporting parts are not boxes in the diagram, but they go through all of the boxes:

- **Search** (document 14) serves the workspace, the API and the retrieval step of the agents.
- **Autoformalization and fidelity** (document 10) is at the point where agents, the formal engine and reviews by people meet.
- **The audit** (document 15) is the gate in front of `main`.

## 3.3 Planes

The content is divided into planes. As in `tamnd/arxiv`, the planes exist because their parts differ in licence, scale and trust.

| Plane | What | Scale | Trust | Licence of our copy |
|---|---|---|---|---|
| **record** | bibliographic records of the mathematical literature, and the results extracted from licence-permitted full text, each with a paper-scoped id | ~5 M records, ~30 M results | records are facts; extracted results are E1 until aligned | CC0 metadata; extracted text only from open-licence papers, under their licence |
| **statement** | curated nodes: concepts, definitions, statements, proofs, conjectures, problems, examples, objects, computations, topics | 2 to 5 M nodes | E0 to E5, computed | CC BY-SA 4.0 prose, CC0 metadata |
| **formal** | Lean code in `Math2/`, plus bindings to Mathlib and other libraries | Mathlib's ~0.5 M declarations bound selectively; `Math2/` as small as possible | kernel | Apache-2.0 |
| **compute** | computation code, records and certificates | 10⁴ to 10⁶ records | E1/E4 | Apache-2.0 code; CC0 records |

The record plane covers **all** of mathematics at the level of "this paper exists and states these results". The statement plane covers all of it at the level of "this is a result, and this is why we believe it". With this split, "all mathematics" stays an engineering target and does not become a slogan. The record plane gives full coverage of what exists. The statement plane gives coverage of what is understood, and that coverage becomes deeper over time. Document 18 measures the distance between them per MSC code.

## 3.4 Four flows

### Flow A: a theorem in a paper becomes a formal node

1. **Ingestion** fetches a new arXiv batch and the matching zbMATH Open records. It uses the metadata plane of `tamnd/arxiv`, and its content plane where the licence permits. The extractor reads the theorem environments of each open-licence paper. It writes record plane results such as `ax://paper/2610.01234#7Q2M`, with their text, their labels and their internal references (document 09.5).
2. **Alignment** embeds each result and finds its neighbours among live nodes. Then it asks the judge about each pair. The result is new (no `same`), and a statement already in the graph cites it. So the curate agent proposes a new `statement` node with a `source:` that points at the paper result. After audit, a curator merges the PR, and the node is minted as E1, claimed.
3. The centrality and demand of the node put it on the **formalize queue**. The formalize agent drafts a Lean statement against Mathlib and runs the F1 battery (document 10.4). It opens a PR that adds the binding at F0. When the battery passes, the audit shows F1.
4. A **reviewer** signs a fidelity review in `m2 review`. The node is now F2.
5. The **prove agent** takes the job ("F2 statement with no formal proof"). It decomposes the proof into lemmas. Each lemma is a new provisional node with its own statement. The agent proves the lemmas in Lean sessions. Then it opens a PR with the proof nodes and the Lean code in `Math2/`.
6. **CI** builds, replays with the external checker and audits. The node is now E5/F2.
7. Weeks later, someone upstreams the Lean proof to Mathlib. The **bump agent** sees the new Mathlib declaration, because its type hash matches the type hash of the binding. It proposes to rebind the node to Mathlib and to deprecate the `Math2/` copy. The review stays valid, because the type hash did not change.

### Flow B: an experiment becomes a conjecture, and a conjecture is refuted

1. A person in a project asks the **explore agent** for patterns in a family of graphs. The agent writes computations and runs them in the **computational engine**. It finds a regularity up to n = 40.
2. The falsification pass of the agent extends the range to n = 60. It compares the sequence with OEIS and finds a match: the sequence is A-something, and OEIS already lists a conjectured formula. Where relevant, it also checks LMFDB-style databases. The agent files a `conjecture` node with `source:` set to the OEIS entry and the computation. It does not file it as a new discovery.
3. A second person asks for a counterexample search. An SMT encoding at n = 71 finds a model. Lean checks the witness of the computation. The counterexample node is E5. The conjecture gets a `refutes` edge, and it is now E⊥.
4. The release notes for that month list the refutation. A person reports the conjecture in the OEIS entry to the OEIS editors, through the normal OEIS process.

### Flow C: a Mathlib bump

1. Each week, the **bump agent** moves a branch to the newest Mathlib `master` that is compatible with the current Lean toolchain. Then it runs the build.
2. Bindings break where Mathlib changed names. The agent applies the renames from the Mathlib deprecation attributes. These changes are mechanical and are eligible for auto-merge. For each item without a deprecation alias, the agent proposes a rebinding. It finds the rebinding through a type-hash match against the new Mathlib.
3. For some bindings, the type hashes changed but the names did not, because a definition was refactored. The reviews of those bindings become stale. The nodes drop to F0 and go to the top of the fidelity queue.
4. The branch merges when it is green. The release at the end of the month pins it.

### Flow D: a reader asks a question

1. A student searches for "every finite subgroup of the multiplicative group of a field is cyclic". The embedding channel of the **search** engine finds the node. Its formula channel confirms it.
2. The page shows: proved in Lean, reviewed, classical; three proofs; prerequisite path from the student's declared background.
3. The student opens "Ask about this node" and asks why finiteness matters. The **explain agent** answers with $\mathbb{Q}^\times$ (infinite, not cyclic) as the example. It cites the counterexample node by tag.
4. No node for that counterexample exists. The answer marks the claim "supported by computation, not by a node". It also files a curate job. The next week, the node exists.

## 3.5 Cross-cutting decisions

These apply to every box. Document 00 lists them as settled.

- **Files in git are the truth. Everything else is derived and rebuildable** (05.1, 06.1).
- **Status is computed from evidence and never asserted** (04.7).
- **The kernel and certificates are the only referees for E4 and E5. Models produce candidates, never verdicts** (15.1).
- **Fidelity is a separate axis, bound to type hashes, and reviewed by people** (04.7, 10).
- **Every byte has provenance. Every agent run has a run record and a responsible person or schedule** (06.5, 11.6).
- **Tags are permanent. Deletion is retirement** (04.3).
- **Writes are pull requests. The audit is the gate** (11.1, 15).
- **Pin everything per release: Lean, Mathlib, images, models' recorded ids** (06.2).
- **Link, do not copy, wherever a licence or a community asks for it** (16.5, 16.6).
- **One URI space, `m2://`, inside `ant`'s.** The same tools dereference every math2 record and every other `tamnd` record (04.4).

## 3.6 Deployment

At reference scale (document 02), the deployment is small:

| Component | Runs on | Notes |
|---|---|---|
| `m2 serve` (API, MCP, workspace) | 2 VMs behind a CDN | stateless over a release's serving store; ~64 GB RAM each for the CSR and the vector index |
| static site | CDN only | per release |
| `m2 orchestrate` | 1 VM | job queue in SQLite or Postgres; restarts are idempotent |
| Lean session pool | autoscaled runner VMs, 32 to 64 GB each | images with Mathlib preloaded; each session sandboxed |
| compute runners | autoscaled, x86-64 and arm64 | the C02 two-architecture rule needs both |
| CI | GitHub Actions plus self-hosted large runners for full builds | a full Mathlib-dependent build of `Math2/` with the Mathlib cache takes minutes; a cold Mathlib build is avoided by the cache |
| model inference | Claude API (Anthropic); optional GPU nodes for open prover models | document 12.3 |
| artefact store | two S3-compatible providers | document 06.6 |

No component in this table needs to be custom-built. Only one part is expensive at scale: model inference. For this reason, the orchestrator has budgets (11.5) and the escalation ladder, and the agent report publishes costs every week.
</document>

<document source="https://tamnd.github.io/math2/spec/04-object-model.md">
# 04. The object model

This document defines what math2 stores. It gives the kinds of node, the permanent name of a node, the format of a node file on disk, how the status of a node is computed, and what a node must not say about itself. All other documents use these definitions. If a later document seems to disagree with this one, this document is correct and the other document has a bug.

## 4.1 The unit is a node

A node is one piece of mathematics that someone may want to cite, check, compute with or explain on its own. Its size is about that of a Stacks Project tag, or of a Mathlib declaration with its docstring. It is not the size of a paper or a chapter.

A node has five parts:

1. a **permanent tag**, which is its identity (4.3);
2. a **kind**, from a closed list (4.2);
3. an **informal body** in constrained Markdown with LaTeX (4.5);
4. **typed edges** to other nodes and to external records (document 05);
5. **evidence**: formal bindings, proofs, computations, sources and reviews. The status of the node is computed from this evidence and is never written by hand (4.7).

A node is not a page. The workspace renders pages from nodes and their neighbourhoods (document 13). A node is also not a Lean declaration. A Lean declaration is evidence that a binding attaches to a node (document 07). The separation of these three things is the most important decision in this document. Mathlib has about a quarter of a million theorems. Most of them are API lemmas that nobody would cite by name. The literature of mathematics has millions of results, and most of them are not formalised. A system whose unit is the Lean declaration cannot represent the second set. A system whose unit is the page cannot be checked. The node is between the two.

## 4.2 Kinds

The list of kinds is closed. A new kind changes the schema version (4.9) and needs an entry in document 21.

| Kind | What it is | Typical evidence | Example |
|---|---|---|---|
| `concept` | A mathematical notion, independent of any one definition of it | definitions, equivalence statements | compact topological space |
| `definition` | One specific definition of a concept | formal binding to a `def`/`structure`/`class` | compact iff every open cover has a finite subcover |
| `statement` | A proposition claimed true. Its `role` is `theorem`, `lemma`, `proposition`, `corollary`, `fact` or `axiom` | proofs, formal binding, computation, sources | Lagrange's theorem |
| `conjecture` | A proposition that is open, or that was posed as open | sources, partial results, computational evidence, formal statement | Goldbach's conjecture |
| `problem` | A question that is not a single proposition: classify, compute, decide, find the best constant | sources, bounds as linked statements | Erdős problem #N, Hadwiger–Nelson |
| `example` | A specific instance that illustrates a concept | computation, formal `example` | $\mathbb{Z}/6$ is cyclic but not simple |
| `counterexample` | A specific instance that refutes a statement or shows a hypothesis is needed | computation, formal proof of the negation | Sorgenfrey plane is not normal |
| `object` | A named mathematical object with an identity of its own | external identifiers (LMFDB label, OEIS A-number, GAP id) | the Monster group, $E_8$ lattice, A000045 |
| `proof` | One argument for one statement. It is informal, formal, or both | a body, a formal binding, `uses` edges | the proof of Lagrange via cosets |
| `computation` | An executable, reproducible computation and its recorded result | an environment, inputs, outputs, a certificate | checking Goldbach's conjecture up to $4\cdot10^{18}$ |
| `topic` | A navigational grouping. It is never a claim | MSC codes, child links | finite group theory |

Three distinctions are intentional.

**Concept versus definition.** "Compact" has at least four standard definitions: open covers, the finite intersection property, nets and ultrafilters. In the literature they are one concept. In a proof assistant, one of them is the definition and the others are theorems. math2 keeps one `concept` node, one `definition` node for each definition, and `statement` nodes that say the definitions are equivalent. Navigation and search go through the concept. Checks go through the definitions. With this structure, math2 can align with nLab (organised by concept), with Mathlib (organised by definition) and with textbooks (which choose one definition and prove the others). It does not force any of them into the shape of the others.

**Statement versus proof.** A statement can have many proofs, and the proofs differ in what they use. A researcher needs this information to know whether a result survives without the axiom of choice, or without the classification of finite simple groups. So proofs are nodes, and each proof has its own `uses` edges. The dependency closure of a statement is not one set. It is the set of alternatives, one for each proof (document 05.5).

**Conjecture versus problem.** A conjecture is a proposition with a truth value. It can be formalised as a Lean `Prop`. A proof of it, or of its negation, settles it. A problem may have no single truth value ("determine $R(5,5)$"). A chain of statements settles it. DeepMind's `formal-conjectures` repository and the Erdős problems database both mix the two. math2 keeps them apart, because they are settled in different ways.

## 4.3 Tags: identity forever

Every node has a permanent tag. The idea comes directly from the Stacks Project. `tamnd/arxiv` already uses the same tag discipline for results inside a paper. Three rules apply:

1. **A tag is assigned once.**
2. **A tag is never edited.**
3. **A tag is never reused, even if its node is deleted.**

Audit rule `T01` checks the tag register against git history. It fails if a tag has ever disappeared (document 15).

### Format

A tag is six characters of Crockford base32: the digits and the uppercase letters, without `I`, `L`, `O` and `U`. That gives 32^6 tags, about 1.07 billion. A four-character tag (the Stacks size, about 1.7 million in base 36) is too small. Mathlib alone has over 400,000 declarations. The literature has tens of millions of stated results, and the record plane (document 09) can turn any of them into a node.

The first character is never `0`. Tags that begin with `0` are reserved for the bootstrap seed (milestone M1), so the first few thousand curated nodes are easy to identify. A tag never ends in a character that would make it a valid MSC code, an OEIS A-number or an arXiv identifier. With this rule, a bare tag pasted into a search box resolves to exactly one kind of thing.

Tags are random, not sequential, for two reasons. Random tags make no claim about order, importance or age. Also, two branches can mint tags without coordination. Audit rule `T02` catches collisions at merge (the chance per pair is about one in a billion).

### When a tag is minted

Tags are minted at merge to `main`, and never before. A node on a branch, in the scratch space of an agent or in a pull request has a **provisional id**. Its form is `~` followed by eight hex characters, for example `~3fa94c01`. The merge bot rewrites provisional ids to tags in the node, in every edge that points to it, and in every Lean attribute. A pull request that cites a provisional id outside its own diff fails `T03`.

This rule prevents agents from wasting tag space. Agents propose tens of thousands of candidate nodes each day (document 12), and most of them are rejected. No rejected candidate ever had a tag. So the register records only things that were accepted at some time.

### Deletion is deprecation

A node is never deleted, even if it is found to be a duplicate, a mistake or out of scope. Its file stays, its `kind` stays, and it gets:

```yaml
retired:
  date: 2027-02-14
  reason: duplicate        # duplicate | false | out-of-scope | split | merged
  successor: [7K3QX2]      # zero or more tags
  note: "Same statement as 7K3QX2 with the hypotheses reordered."
```

A retired tag still resolves. The workspace shows its successor and the reason. A citation from 2026 still resolves to something meaningful in 2036.

`split` and `merged` record a refactor of the curated graph. It can become clear that one node states two results, or that two nodes are one. The successor list keeps the history navigable.

## 4.4 URIs

math2 is the `m2://` scheme in the `ant` URI space. Other schemes in that space are `ax://`, `wikipedia://`, `oeis://` and the rest of the `tamnd/*-cli` family. So `ant get m2://7K3QX2` works in the same way as for any other record.

```
m2://7K3QX2                     the node, at the latest release
m2://7K3QX2@2026.11             the node as it was in release 2026.11
m2://7K3QX2@c4f1e9a             the node as it was at a git commit
m2://7K3QX2#proof               the statement section; #proof, #formal, #history, etc.
m2://statement/7K3QX2           a typed alias, redirects to m2://7K3QX2
m2://concept/compact-space      a slug alias, resolves through the alias table
m2://msc/20D60                  a topic view over an MSC2020 code
m2://lean/Subgroup.card_subgroup_dvd_card   the node bound to this Mathlib declaration
m2://release/2026.11            the release manifest
```

Aliases are a convenience. They are not identities. A slug can go to a different node later, for example if "compact space" is found to be ambiguous between topology and order theory. A tag cannot. Every place that stores a reference stores the tag. `m2 fmt` rewrites aliases into tags before commit (rule `T04`).

External identifiers map into `m2://` through `same_as` edges (document 05.3). `m2://ext/wikidata/Q180967` resolves to the node that has that `same_as`. If no node has it, the URI returns a "not held" record that names the external system of the identifier.

## 4.5 The node file

A node is one file, `nodes/<first two characters of the tag>/<tag>.md`. The file has YAML front matter and a Markdown body. The shard key is the first two characters of the tag. With it, each directory has fewer than about a thousand entries at a million nodes.

The full schema is `schema/node.v1.json` in `tamnd/math2` (document 20). A statement node looks like this:

```markdown
---
tag: 7K3QX2
kind: statement
role: theorem
name: Lagrange's theorem
aliases: [lagrange-theorem-groups]
msc: [20D60]
topics: [4F2M9A]                # tag of the topic node "finite groups"
uses:                           # informal dependencies of the statement itself
  - 1H8D2C                      #   definition: subgroup
  - 6PQ0WZ                      #   definition: order of a finite group
proofs: [9TX3RB, 2YB7KE]        # proof nodes, in the order shown to readers
formal:
  - system: lean4
    library: mathlib
    decl: Subgroup.card_subgroup_dvd_card
    match: exact                # exact | stronger | weaker | variant | partial
    fidelity_reviews: [r-7K3QX2-01, r-7K3QX2-02]
sources:
  - zbmath: "0000.00000"        # example only; real records carry real ids
    locator: "Thm 3.1"
  - book: "isbn:9780471433347"
    locator: "§3.2, Thm 8"
same_as:
  wikidata: Q180967
  proofwiki: "Lagrange's_Theorem_(Group_Theory)"
  nlab: null                     # explicitly checked, none exists
history:
  first_stated: "Lagrange 1771 (special case); Galois; Jordan 1861"
---

## Statement

Let $G$ be a finite [[group|1A2B3C]] and $H$ a [[subgroup|1H8D2C]] of $G$.
Then $|H|$ divides $|G|$.

## Remarks

The converse fails: $A_4$ has order 12 and no subgroup of order 6 ([[counterexample|5ZQ1PM]]).
```

The rules for the body:

- **Math is LaTeX** inside `$…$` and `$$…$$`. Use only the subset that both KaTeX and MathJax render, plus macros declared in `macros/*.sty`. Audit rule `B01` renders every body headlessly and fails on any error. Free LaTeX is not allowed. A body that renders in only one engine cannot be served everywhere.
- **Links are `[[text|TAG]]`.** A link to something that is not a tag fails `B02`. Specifically, links to URLs are not allowed in the statement section. Put sources in `sources:`.
- **Sections are fixed per kind.** A statement has `Statement`, then optionally `Remarks`, `History` and `Explanations`. A proof has `Proof`, then optionally `Idea`. The full list is in the schema. A section that the schema does not know fails `B03`. This rule lets a program parse bodies without a model.
- **The statement section states one thing.** It does not hold a proof, a history or a motivation. The explanation agent (document 12.6) writes into `Explanations`, and into no other section.
- **Explanations are levelled.** Each explanation has a level tag: `L1` for a person who has not met the concept, `L2` for an undergraduate, `L3` for a graduate, `L4` for a researcher. Each level also has a tag that says human-written or agent-written (4.8).

Not every node has text that a person wrote. A node can come from a well-structured source, such as an OEIS sequence, an LMFDB object or a Mathlib declaration with a good docstring. Such an imported node can have a generated body. The body has a mark that says it is generated. An `imported` provenance entry names the source and its version (document 06.5).

## 4.6 Proof nodes

A proof is a separate file with `kind: proof` and a `proves:` field that holds exactly one tag. It can have:

- an informal `## Proof` section in the same constrained Markdown;
- a `formal:` binding to a Lean declaration that proves the formal binding of the statement, or proves it up to a recorded adapter (document 07.4);
- `uses:` edges to every node that the argument depends on.

`uses` on a proof means: "if any of these is false, this argument fails". It does not mean "this is mentioned in passing". For formally bound proofs, `uses` is computed from the Lean dependency graph (document 07.6) and compared with the hand-written list. If `uses` does not include a formal dependency on a math2-bound declaration, that is a finding (`G04`). A hand-written `uses` edge with no formal counterpart is allowed, because informal proofs cite at a coarser grain than Lean.

A proof can name a **method**: induction, diagonal argument, probabilistic method, compactness, the circle method, and so on. Methods are `concept` nodes in the `methods` topic. Because of this, math2 can answer a query like "all proofs in additive combinatorics that use the polynomial method" (document 14).

## 4.7 Status is computed, never asserted

No node file has a field called `status`, `verified` or `proved`. Status is the output of `m2 build` (document 07, 08, 15). The build writes it into the derived store and the release manifest, and never into the content.

This rule exists because the opposite rule fails. A person or an agent who writes `verified: true` makes a claim. Nothing in the file makes the claim true. A year later, Mathlib renames a lemma, a definition below it changes, or a reviewer leaves. The flag stays true. In math2, status is a function of the evidence at a commit. So every build computes it again from the evidence.

Status has two independent axes.

### Truth evidence

The strongest evidence that is present sets the level. The levels have this order:

| Level | Name | Meaning | Who can produce it |
|---|---|---|---|
| E5 | `kernel` | A formal binding with `match: exact` (or `stronger`/`variant` through a checked adapter, document 07.3) and fidelity at least F2 is proved by a Lean declaration that the kernel checks. Its axioms are a subset of `{propext, Classical.choice, Quot.sound}`, plus any named hypotheses, which make the result conditional. It has no `sorry` and no `native_decide`, which caps a node at E4 (07.5). It passes independent replay and the comparator check (07.7) | the build |
| E4 | `certified` | A computation produced a certificate (an LRAT proof, an interval-arithmetic enclosure, a primality certificate, an exact rational witness), and an independent checker accepted it | the build, through the computational engine |
| E3 | `refereed` | At least one proof in the literature is in a refereed venue, and a source record points to it with a locator | curation |
| E2 | `standard` | The result is textbook-standard. At least two independent textbook sources with locators | curation |
| E1 | `claimed` | Someone claims it: a preprint, an agent's informal proof, a MathOverflow answer | anyone |
| E0 | `open` | It is posed and unresolved. Only conjectures and problems may sit here | anyone |
| E⊥ | `refuted` | A counterexample or a proof of the negation exists, at level E2 or above | the build or curation |

This has two important consequences.

**E5 does not mean "true"**, and the workspace never says that it does. E5 means two things. First, this Lean statement is a theorem of Lean's type theory with the standard axioms. Second, three to five people (or one person and a cross-formalisation) agreed that the Lean statement says what the informal statement says. The errors are in the second part. Document 10 is about how to make it reliable.

**An agent's informal proof is E1**, however good it looks. It moves up in only three ways: it becomes formal (E5), it is certified (E4), or it is published and refereed (E3). The agent framework cannot write a status. It can only produce evidence, and the build then grades that evidence (document 12).

### Statement fidelity

Fidelity is how strongly we believe that a formal statement means the same thing as the informal one. It applies only when a formal binding exists.

| Level | Meaning |
|---|---|
| F0 | Bound. Nobody has checked the formal statement against the informal one |
| F1 | Machine-checked. Back-translation, a type-level check and a model-based critic all agree, and checks for degenerate statements pass (document 10.4) |
| F2 | One human review. A reviewer who is not the author has signed off on this exact statement hash |
| F3 | Two independent reviews, or one review plus a second formalisation produced independently and proved equivalent in Lean |

A review is a record in `reviews/<shard>/<tag>.yaml`. It holds the identity of the reviewer, the date, the content hash of the informal statement section, and the content hash of the elaborated Lean type (not its source text). If either hash changes, the review no longer counts. This happens automatically. It is the mechanism that keeps F2 and F3 honest when Mathlib refactors a definition.

### Derived flags

The build also computes flags. Each flag can be queried:

- `ac` / `noac`: the proof uses, or avoids, `Classical.choice`. This is computed for each proof. The statement takes the best value over its proofs.
- `computational`: the proof depends on `decide`, `native_decide` (whose per-use auxiliary axioms the axiom audit sees) or an external certificate.
- `sorry_in_closure`: some node that the proof transitively depends on is not at E5. The blueprint style "proved assuming…" view comes from this flag.
- `stale_review`: a review was invalidated by a hash change and has not been done again.
- `orphan`: the node has no inbound edges and is not a topic root.
- `conditional`: the axiom profile includes hypotheses from `Math2/Hypotheses/`. These hypotheses are listed by name (07.5).
- `external_replay`: E5 rests on the replay results of an external library at its pinned commit, not on math2's own replay (07.7).

## 4.8 Provenance on every field

Every node has a provenance log in `prov/<shard>/<tag>.jsonl`. The log has one line for each change to each field. The line records who changed the field, when, with what, and from which source. The model follows W3C PROV. Document 06.5 gives it in full. In outline:

```json
{"field":"formal[0]","op":"add","agent":"m2-agent/formalize@0.9.1","model":"claude-opus-5-5","run":"r-2026-11-02-8812","human":null,"commit":"c4f1e9a","at":"2026-11-02T09:13:44Z"}
{"field":"formal[0].fidelity_reviews","op":"add","agent":null,"human":"orcid:0000-0002-1825-0097","commit":"e01b7d2","at":"2026-11-04T16:20:11Z"}
```

The rule is that **every byte in `nodes/` can be traced to a person or to an agent run**. Every agent run can be traced to the person or schedule that started it, the model, the prompt version and the tool versions (document 11.6). Audit rule `P01` fails a merge that adds bytes without a provenance line.

This is not bureaucracy. It makes three things possible. First, AI contributions get credit and disclosure in the way that journals and the community now expect (document 16). Second, if a model or prompt version is found to have a systematic flaw, a whole class of agent output can be withdrawn. Third, the statement "this was written by a person" stays true and checkable.

## 4.9 Schema versioning

The node schema has its own version (`node.v1`), separate from releases. Two kinds of change are possible:

- **An additive change** adds a new optional field or a new edge type. It needs only a minor bump and a migration that does nothing.
- **A breaking change** renames a field, changes the kinds or changes the meaning of a field. It needs a new major version, a migration script in `tamnd/math2-agent/migrate/`, and a release that accepts both versions. In that release, rule `S09` flags the old version.

The schema is small on purpose. Put information that has no schema field in `## Remarks`. People can read it there, and the build ignores it. Add a field only when a query or a check needs it.

## 4.10 What a node is not allowed to do

- It cannot state its own status (4.7).
- It cannot have two kinds. A conjecture that is proved becomes a statement: the conjecture node is retired with `reason: split`. The new statement node lists the tag of the conjecture in `history.resolves`, and the conjecture node gets a `resolved_by` edge. A citation of the conjecture still goes to a place that says it is resolved, how and when.
- It cannot contain a proof inside its statement.
- It cannot link outside `m2://` from its statement section.
- It cannot reuse a tag, an alias that is in use, or a `same_as` identifier that another live node already claims. `same_as` is functional in both directions within one external system (rule `G02`). If an external system has two entries for one node, or one entry for two nodes, the edge is `close_to` and the mismatch is recorded (document 05.3).
- It cannot be larger than 64 KB. A longer node is several nodes.
</document>

<document source="https://tamnd.github.io/math2/spec/05-knowledge-graph.md">
# 05. The knowledge graph

The graph is the box in the middle of the architecture diagram: Statements, Concepts, Dependencies. Every other box writes into it or reads from it. This document defines the edges of the graph and its alignment with the rest of the world's mathematical data. It also defines how the graph is stored and rebuilt, and the queries that it must answer quickly.

## 5.1 Where the graph lives

The graph has no separate database that holds the truth. **The truth is the node files in git** (document 04). `m2 build` derives the graph from them, together with three other derived inputs:

1. edges written by hand in node front matter (`uses`, `proofs`, `topics`, `same_as`, …);
2. edges extracted from the formal plane: the Lean dependency graph of every bound declaration, at the pinned Mathlib and math2 revisions (document 07.6);
3. edges from the record plane: citations between papers, and paper results aligned to nodes (document 09.6).

The derived graph goes to `build/graph/` as Parquet files (one for nodes, one for each edge family). It is then loaded into an embedded store for serving (5.7). It is never committed to `tamnd/math2`. It is published as a release artefact with a content hash (document 06). Anyone with the repository and the pinned toolchain can rebuild it byte for byte. Audit rule `R01` does exactly that in CI. For every merge it rebuilds a sample of shards, and for every release it rebuilds everything.

`tamnd/arxiv` made the same choice: commit what a person decided, and derive the rest. As a result, a reviewer reads a diff of decisions, not a diff of derived numbers.

## 5.2 Edge types

Edges are typed and directed, and their types come from a closed list. Each edge also carries its own evidence and provenance. "A implies B" is a mathematical claim, the same as a node. Its level can be anything from E1 to E5.

### Logical edges (they carry truth)

| Edge | From → to | Meaning | Can be E5 |
|---|---|---|---|
| `proves` | proof → statement | this argument establishes that statement | yes |
| `uses` | statement/proof/definition → any | the source depends on the target: a hypothesis, a lemma, a definition | yes (computed from Lean) |
| `implies` | statement → statement | A ⟹ B, proved by the proof node named in `via` | yes |
| `equivalent` | statement ↔ statement, definition ↔ definition | A ⟺ B, via a proof | yes |
| `refutes` | counterexample/statement → statement/conjecture | the source shows the target is false | yes |
| `resolves` | statement → conjecture/problem | the source settles the target, positively or negatively | yes |
| `generalizes` | statement → statement | the target is a special case of the source. Recorded with the specialising map | yes |
| `bounds` | statement → problem | the source gives a bound on the quantity the problem asks for (lower or upper, with the value) | yes |

### Conceptual edges (they carry meaning)

| Edge | From → to | Meaning |
|---|---|---|
| `defines` | definition → concept | this is one definition of that concept |
| `instance_of` | example/object → concept | the source is an instance of the concept |
| `about` | statement/conjecture/problem → concept | the result is about the concept, as one of its main subjects |
| `specializes` | concept → concept | every X is a Y (a ring is a semiring). These edges form a DAG, not a tree |
| `method` | proof → concept (in `methods`) | the proof uses this technique |
| `analogous_to` | any ↔ any | an analogy that mathematicians acknowledge, with a source. Soft, and never used for reasoning |
| `part_of` | any → topic | navigation only |

### Record edges (they carry provenance and identity)

| Edge | From → to | Meaning |
|---|---|---|
| `formalizes` | Lean declaration → node | the binding. Carries `match` and fidelity (document 07.4) |
| `source` | node → record | where this is stated or proved. Carries a locator |
| `aligned` | paper result → node | an extracted result (for example `ax://paper/2106.09685#AB12`) states this node. Carries a `match` grade |
| `same_as` | node → external id | the same thing in another system (5.3) |
| `close_to` | node → external id | a related but not identical entry in another system, with a note on how it differs |
| `cites` | record → record | the paper-to-paper citation graph, from zbMATH Open and arXiv |
| `resolved_by`, `successor` | retired node → node | lifecycle (document 04.3) |

There are 23 edge types. A new edge type is a schema change (document 04.9). The condition for a new type is: "some query in 5.6 needs it and cannot be written without it".

## 5.3 Alignment with the rest of the world

math2 does not try to replace nLab, Wikipedia, Wikidata, Mathlib, the Stacks Project, OEIS, LMFDB, zbMATH Open or MathOverflow. It aims to be the place that joins them together. Each of them does its own job very well, but none of them links reliably to the others. In its first year, math2 is most useful as a curated, versioned, machine-checkable **crosswalk** between them, with statuses attached.

`same_as` identifiers by external system:

| System | Identifier | Granularity | Notes |
|---|---|---|---|
| Wikidata | QID | concepts, named theorems, objects | the hub for every other crosswalk. The MaRDI knowledge graph is built on Wikibase and uses Wikidata-style ids |
| MaRDI portal | MaRDI QID | formulae, algorithms, software, research data | aligned through Wikidata where possible |
| Mathlib | fully qualified declaration name at a pinned revision | definitions, theorems | carried by `formalizes`, not `same_as` |
| Other formal libraries | Isabelle AFP entry + theorem, Rocq/MathComp qualified name, Metamath label, HOL Light name | definitions, theorems | `formalizes` with `system:` set accordingly |
| Stacks Project | 4-character tag | results, definitions | text under GFDL. We link and never copy (document 09.3) |
| Kerodon | 4-character tag | results, definitions | same as Stacks |
| nLab | page name + revision | concepts | |
| Wikipedia | page title + revision id | concepts, named theorems | revision-pinned, because titles move |
| ProofWiki | page title | statements, definitions, proofs | |
| OEIS | A-number | objects (sequences) | an OEIS sequence is an `object` node. Its formulae become `statement` nodes |
| LMFDB | the path of the object, for example `EllipticCurve/Q/11.a2` | objects | number fields, elliptic curves, modular forms, … The path has the collection and the label, because a label alone does not identify an object |
| zbMATH Open | document id; author id; MSC code | records | the main record plane source for "all mathematics" |
| MathSciNet | MR number | records | identifier only (no bulk licence) |
| arXiv | id + version, and per-paper result tag from `tamnd/arxiv` | records, results | `ax://paper/<id>#<tag>` |
| Erdős problems | problem number | problems/conjectures | |
| formal-conjectures (DeepMind) | declaration name | conjectures with Lean statements | |
| MathOverflow / Math.SE | question id | sources for E1 claims | |
| Palomar | registry entry id | Lean-verified results | carries the entry's `formalization.yaml` disclosure and replay record (09.4) |
| TheoremSearch, TheoremGraph/LeanGraph, Matlas | native statement ids | extracted results | record plane `same_as`; their extractions are imported as candidates only (09.6) |
| Hexagon | `hexagon:` id + version | records | the 2026 repository for AI-era mathematics, with provenance labels |
| pi-Base, DaRT, FindStat, House of Graphs, GAP small groups, ATLAS | native ids | objects, properties, counterexamples | data sources with natural `instance_of` and `refutes` structure |

Alignment follows three rules.

1. **Alignment is a claim, and it has a grade.** A `same_as` that an agent writes starts as E1. A human confirmation raises it. An automatic match that cannot be ambiguous also raises it (an OEIS A-number cited verbatim in both places, an identical Lean declaration). The crosswalk publishes the grade of each edge, so consumers can choose their threshold.
2. **`same_as` is functional both ways within one system.** A node has at most one Wikidata QID, and a QID belongs to at most one live node (rule `G02`). When the granularity of a system does not match ours, we use `close_to` with a note. An example is one Wikipedia page that covers three theorems. Real systems disagree about granularity all the time. A forced bijection would make the crosswalk wrong in a way that nobody can see.
3. **Round-tripping is a test.** For every system with a public API, `m2 audit -plane align` fetches a sample of external records again. It checks that they still say what they said when the edge was made. If the external record is gone, or its revision moved and the content changed, that is a finding (rule `A03`). In this way, renames on Wikipedia, deprecations in Mathlib and merges on nLab appear in a report. They do not become silent link rot.

### Contributing the crosswalk back

Where the external system accepts them, alignments go back out to it. Wikidata gets `math2 tag` statements and `exact match` links. Mathlib gets a `math2` field in docstrings if the community wants one, and only as an opt-in suggestion (document 16.5). MaRDI gets the same through its Wikibase API. This is the cheapest way to make math2 useful to people who never visit it.

## 5.4 Granularity and deduplication

The hardest problem in a graph of all mathematics is not size. The hardest problem is this: the same theorem is stated a thousand times, in a thousand notations, at slightly different generality. These thousand statements have to collapse into one node or a small family of nodes. If they do not, the graph is only a pile of paper extracts.

math2 deduplicates in four passes. Each pass is cheaper and more certain than the one after it.

1. **Identifier collisions.** Two candidates cite the same external identifier, or bind to the same Lean declaration. These merge automatically.
2. **Formal equivalence.** Both candidates have formal statements, and a Lean proof shows that they are equivalent. The proof is an `Iff` between the two elaborated types. Usually `exact ⟨fun h => …, fun h => …⟩` from the formalize agent closes it, or `simp`/`grind` does. The candidates merge, and the equivalence becomes an E5 `equivalent` edge.
3. **Embedding neighbourhood plus judgement.** Statement embeddings (document 14.4) give the 20 nearest live nodes for each candidate. A judge model classifies each pair as one of `same`, `generalizes`, `specializes`, `related` or `different`. It has to justify the classification with the specialising map. A `same` from the judge is only a proposal. It becomes an edge only through pass 2 or a person.
4. **People.** Curators process the queue of ambiguous `same` proposals. The queue order is by how much of the graph each merge would connect (document 16.3).

Deduplication gives a canonical node plus `aligned` edges from every paper result that states it. This is also the only honest way to answer "where in the literature is this proved?". The answer is the set of aligned records in date order. That set gives a provenance chain for each theorem. Currently nobody maintains such a chain by hand.

### Families, not points

Some results have no canonical form, only a family: the dozen forms of the Hahn–Banach theorem, or the Cauchy–Schwarz inequality in each of its settings. math2 does not choose a winner. It makes one node for each form that people cite separately, `generalizes`/`specializes` edges between them, and a `concept`-like hub. Hubs are statements with `role: fact` and an empty formal binding. A hub is only a navigation point. The workspace shows it as "the Hahn–Banach family". A hub with fewer than three members is a finding (`G07`), because a family of two is only two nodes and an edge.

## 5.5 Dependency closure, with alternatives

"What does this theorem depend on?" has no single answer, because a statement can have several proofs (document 04.6). The closure is an AND–OR graph. A statement needs one of its proofs (OR), and a proof needs all of its `uses` (AND).

For every statement, the build computes:

- **min-depth closure**: the alternative with the smallest maximum depth down to axioms and definitions;
- **min-size closure**: the alternative with the fewest distinct nodes. This is an approximation, because the problem is NP-hard in general. The build uses a greedy algorithm with a recorded bound;
- **axiom profile**: which of the foundational nodes (choice, excluded middle, univalence where relevant, large cardinals, CFSG as a black box, the Riemann hypothesis as a hypothesis) occur in which alternative;
- **trust profile**: the weakest evidence level anywhere on the best alternative. Take a statement whose only proof is E5 but uses a lemma that is E3. It shows as "formal modulo 1 refereed lemma". This is the blueprint view, computed for the whole graph in one pass.

Conditional results are first-class. "Assuming GRH, …" is a statement whose `uses` includes the GRH conjecture node with `as: hypothesis`. When a conjecture is settled, the next build re-grades every conditional result that depends on it. When a conjecture is refuted, every such result gets a flag (`resolves` propagation, document 15.4).

## 5.6 The queries the graph must answer

The design of the graph starts from these queries. Each query has a latency target on the reference machine (document 02, axis 5). The benchmarks in `math2-agent/bench/graph/` measure it.

| # | Query | Example | Target p95 |
|---|---|---|---|
| Q1 | resolve an identifier | `m2://lean/Subgroup.card_subgroup_dvd_card`, `Q180967`, `A000045`, `0FD4` (Stacks) | 5 ms |
| Q2 | neighbourhood | everything within 2 hops of a node, typed | 20 ms |
| Q3 | dependency closure | best alternative down to axioms, with its trust profile | 100 ms at depth 30 |
| Q4 | reverse dependencies | what breaks if this lemma is false (all statements whose every proof uses it) | 200 ms |
| Q5 | prerequisite path | the shortest learning path from concept A to statement B, using only `defines` and `uses` edges into definitions | 100 ms |
| Q6 | status slice | all E1 statements in MSC 11 (number theory) with an E5 formal statement but no proof: the "formalised, open to prove" frontier | 500 ms |
| Q7 | open-problem frontier | open conjectures in a topic, with their best bounds and the dates those bounds moved | 200 ms |
| Q8 | method query | proofs using the method M about concept C | 200 ms |
| Q9 | provenance | every source that states this node, in date order, with the earliest | 50 ms |
| Q10 | change query | what changed in topic T between releases R1 and R2 | 1 s |
| Q11 | coverage | per MSC code: counts by kind and by evidence level, and Mathlib coverage | 1 s for the full table |
| Q12 | agent work queue | the next 100 tasks for the prove agent, ranked (document 11.4) | 500 ms |

If a query in this list cannot meet its target, that is a design bug in the storage layer. It is not a reason to drop the query.

## 5.7 Storage engine

The derived graph is small by graph-database standards, even at full scale. The estimate in document 18 is about 10 to 30 million nodes and about 300 million edges after the record plane is aligned. Most of the edges are `cites` and `aligned`. That is a workload for a single machine, and math2 treats it as one.

- **Build format:** Parquet, one file for each node kind and each edge family, sorted by tag. These are the release artefacts.
- **Serving:** an embedded store inside `m2 serve`:
  - **SQLite** for point lookups, FTS5 lexical search and the small fixed queries (Q1, Q2, Q9);
  - **DuckDB** over the Parquet for analytical queries (Q6, Q10, Q11);
  - an in-memory **CSR adjacency** built at startup for the closure and path queries (Q3, Q4, Q5). It uses about 2.5 GB for 300M edges at 8 bytes per edge;
  - a **vector index** for embeddings (document 14.4).
- **No external graph database.** math2 does not use Neo4j, a hosted Neptune or similar products. If the record plane grows by orders of magnitude, look at the project's own graph database research (`tamnd/graph-research`). At the sizes above, a server process that loads Parquet, plus SQLite and DuckDB files, is faster to operate, trivial to version and reproducible. Here those three properties are more important than distributed scale.

The query language for users and agents is not SQL or Cypher. It is a small typed API (document 11.2). Power users also get a read-only DuckDB SQL endpoint over the published Parquet schema. That schema is versioned together with the node schema.

## 5.8 Graph health

The graph has its own audit rules. Document 15 has the full list. These rules define a healthy graph:

- `G01` every `uses`, `proves`, `defines` and so on points to a live tag or to a retired tag that has a successor;
- `G02` `same_as` is functional both ways within each external system;
- `G03` `specializes` among concepts is acyclic. `uses` among statements is acyclic when proofs are taken into account. A cycle through two different proofs is allowed, because equivalences look like that;
- `G04` formal dependencies agree with hand-written `uses` (document 04.6);
- `G05` every live statement has at least one `source`, or at least one proof, or a formal binding. A statement with none of these is an assertion with no evidence at all, and it is not accepted;
- `G06` every conjecture has a `source` giving who posed it and where;
- `G07` hub families have at least three members;
- `G08` orphan rate per topic is below 2%;
- `G09` no concept has more than 12 definitions without a `concept`-level note explaining why.

Each rule writes into the committed report under `reports/graph/` (document 15). Thus a person can read the state of the graph in a diff.
</document>

<document source="https://tamnd.github.io/math2/spec/06-versioned-storage.md">
# 06. Versioned storage: metadata, artefacts, provenance

This is the bottom-right box of the diagram. Every other box writes into it. It holds three things, and each has its own versioning rules:

- **metadata**: the node files, edges, reviews, manifests and registers. They are small and people edit them. They are in git.
- **artefacts**: compiled Lean `.olean` files, built graph Parquet, search indexes, computation outputs, certificates and rendered sites. They are large and derived. They are in content-addressed storage.
- **provenance**: who and what made each byte of the other two, and from what.

The goal of the layer is one sentence: **given a release name, anyone can get back exactly what math2 said, rebuild it, and check that the rebuild matches**.

## 6.1 Three repositories, one lineage

Storage is not one repository.

| Location | Holds | Versioned by | Licence |
|---|---|---|---|
| `tamnd/math2` (git) | `nodes/`, `reviews/`, `prov/`, `lean/` (the math2 Lean library), `manifests/`, `registers/`, `reports/`, `schema/`, `macros/` | git commits and release tags | prose CC BY-SA 4.0; Lean code Apache-2.0; metadata CC0 (document 16.6) |
| `tamnd/math2-agent` (git) | the `m2` toolchain, agent code, prompts, tool images, benchmarks | git and semver | Apache-2.0 |
| the artefact store | everything derived, keyed by BLAKE3 hash | content address | follows its inputs |

The artefact store is an S3-compatible bucket (R2 or similar) behind a plain HTTPS read path. It also has a second, independent mirror for releases (6.6). Nothing in it is mutable. Nothing in it is the only copy of a decision that a person made. If we lose it, we lose compute, not knowledge.

### Why git for metadata

At the scale in document 18, the git repository is large but still workable. It has a few million small files and a few GB packed. This is about the size of the history of the Linux kernel. Shards by tag prefix (document 04.5) keep directories small. Git gives four things for free that a database would make us build:

- review by diff;
- attribution by commit;
- cheap branches for agents (each agent run works on a branch);
- tools for mirrors and archives that have worked for decades.

If the repository becomes larger than 10 GB packed or 5 million files, the record plane moves to its own repository, `tamnd/math2-records`. The record plane holds most of the files. It keeps the same layout and rules. The trigger is a number and not a feeling. Document 21 tracks it.

## 6.2 Releases

math2 uses calendar versions: `YYYY.MM`, monthly. We cut each release on the first Monday of the month. Patch releases `YYYY.MM.N` are for fixes. Lean and Mathlib also do not use semver. Lean has a monthly release candidate cycle. A monthly math2 cadence matches it closely enough, so each math2 release can pin one Lean stable release.

A release is a git tag on `tamnd/math2` plus a **release manifest**, `manifests/releases/2026.11.yaml`:

```yaml
release: 2026.11
commit: 8b1e4c0f2d...            # tamnd/math2
toolchain:
  m2: v0.9.0                      # tamnd/math2-agent tag
  lean: leanprover/lean4:v4.NN.0 # the stable toolchain pinned for this release
  mathlib: <commit sha>           # one Mathlib revision, compatible with the toolchain
  other_libs:                     # optional formal bindings outside Lean
    afp: 2026-10-30
  images:                         # computational engine containers, by digest
    sage: ghcr.io/tamnd/m2-sage@sha256:...
    pari: ghcr.io/tamnd/m2-pari@sha256:...
    gap:  ghcr.io/tamnd/m2-gap@sha256:...
    smt:  ghcr.io/tamnd/m2-smt@sha256:...
artefacts:                        # the hash of the files and the size in bytes
  graph:  {hash: blake3:af01..., size: 3100000000}
  oleans: {hash: blake3:77c2..., size: 9800000000}
  search: {hash: blake3:1e9b..., size: 6400000000}
  site:   {hash: blake3:c0de..., size: 2200000000}
  certs:  {hash: blake3:51aa..., size: 840000000}
counts:                           # filled by the build. The README table is generated from this
  nodes_live: 0
  by_kind: {}
  by_evidence: {E5: 0, E4: 0, E3: 0, E2: 0, E1: 0, E0: 0}
archived:
  swh: swh:1:rev:...              # Software Heritage revision id
  zenodo_doi: 10.5281/zenodo....  # concept DOI is stable across releases
rebuild_check:
  runner: ubuntu-24.04/x86_64/32c/128G
  graph_hash_reproduced: true
  oleans_hash_reproduced: true
  olean_differences: []           # each file that is different, with a reason, when oleans_hash_reproduced is false
```

The rules:

1. **A release pins exactly one Lean toolchain and one Mathlib revision.** Between releases, `main` follows Mathlib's `master` on a weekly bump branch (document 07.9). Never cut a release from a state where the bump is red.
2. **Every artefact that a release names can be rebuilt from the commit and the toolchain.** For the graph and the search index, the rebuild must match exactly. Lean builds are deterministic, so the `.olean`s must be bit-identical. If they are not, the release must say which files are different and why (rule `R02`).
3. **Releases are archived outside the project.** Software Heritage gets the git tag (a SWHID). Zenodo gets the manifest and the graph Parquet (a DOI). If the project disappears, a 2026.11 citation still resolves.
4. **The release notes are generated, then edited.** `m2 release notes 2026.10..2026.11` makes these items: counts per topic, nodes that moved evidence level, newly resolved conjectures, retired tags with successors, and binding changes that Mathlib renames caused. A person edits the prose. A release cannot change counts if the notes do not say so (rule `R03`).

A release has two more files next to its manifest. `2026.11.levels` gives the evidence level and the fidelity level of each live node, one line for each node, thus the notes of the next release can list the nodes that moved evidence level without a build of an old commit. `2026.11.md` has the notes. Their counts table has one row for each count of the manifest that changed: the count, the number before, the number after, the change and the cause. `m2 release notes` writes the cause from the node files: the new nodes, the retired nodes and the nodes that moved between kinds or levels. When the node files do not give the change, the cause is empty, and a person writes it.

`m2 release cut` cuts a release in two runs. The first run builds the tree and commits the three files. Then a person writes the summary of the notes, and a second machine runs `m2 release verify -record` to fill the rebuild check. The second run refuses when a file outside `manifests/releases/` and `reports/` changed after the commit of the manifest, runs the full audit, and makes the tag only when the audit passes.

### Citing math2

The citation form is `m2://<tag>@<release>`. We also print it as a DOI-backed URL. A paper can cite `m2://7K3QX2@2026.11`. In 2040 a reader can resolve it to the exact text, formal statement, evidence and status as of November 2026. This is true even if the node was retired in 2029.

## 6.3 Content addressing

Each artefact has its BLAKE3 hash as its name. Each node has a **content hash**. We compute it over the canonical serialisation of the node: front matter in key-sorted canonical YAML, and the body normalised to NFC with trailing whitespace removed. Each formal binding has an **elaborated-type hash**. This is a hash of the Lean `Expr` of the type of the bound declaration after elaboration. Universe levels are normalised and auxiliary names are removed.

The elaborated-type hash is the most important hash in the system. Fidelity reviews (document 04.7) are bound to it. Mathlib can change a definition under a statement. Then the elaborated type changes and the hash changes. The review stops counting, and the node drops from F2 to F0 with `stale_review` set. This happens even though no text in math2 changed. Text hashes cannot find this change, because the text of `theorem foo : Bar x` is the same before and after `Bar` changes. Type hashes can find it.

Computation records (document 08) have the hash of `(image digest, entry point, inputs, seed)` as their address. If you run the same computation two times, you must get the same output hash. If you do not, that is a finding: rule `C02`, nondeterminism (document 08.6).

## 6.4 Registers

Some facts must never be lost and must never change. They live in append-only registers in `registers/`:

| Register | One line per | Rule |
|---|---|---|
| `tags.reg` | minted tag: tag, minted-at commit, kind | `T01`: lines are never removed or edited |
| `aliases.reg` | alias assignment: alias, tag, from, until | an alias can move. Its history cannot |
| `retired.reg` | retirement: tag, date, reason, successors | |
| `bindings.reg` | formal binding change: tag, system, old decl, new decl, cause (bind/rename/refactor/rebind), Mathlib commit, author | answers "which Mathlib name did this node have in March 2027" |
| `reviewers.reg` | reviewer identity: ORCID or GitHub handle, first review date, areas, affiliations | needed for F2/F3 independence (document 10.6) |
| `crosswalk.reg` | crosswalk decision: tag, edge, system, id, decision (accepted/rejected/confirmed), person, date, record hash | `A02`, `A03`: a person decided on each `same_as` value, and the record did not change after the decision |

Registers are plain text, sorted, with one record per line. Thus `git log -p` on a register is the full history of that decision space. Each build rebuilds the registers and checks them against the node files (rule `T05`).

## 6.5 Provenance

Each change to each field of each node, review and register has a provenance record (document 04.8). The model follows W3C PROV-O, with these classes:

- **Entity**: a node version (tag plus content hash), a formal declaration (name plus type hash plus Mathlib commit), a computation output (hash), a source record (external id plus revision);
- **Activity**: an agent run, a human edit (a commit), an import job, a build;
- **Agent** (in the PROV sense): a person (ORCID or GitHub id), a math2 agent role at a version, a model id, an external system.

The relations are `wasGeneratedBy`, `used`, `wasAssociatedWith`, `wasDerivedFrom`, `wasRevisionOf`, `wasAttributedTo` and `actedOnBehalfOf`. The last relation links an agent run to the person or schedule that started it.

Provenance is in `prov/<shard>/<tag>.jsonl`, with one JSON object per line. Each release exports it as PROV-JSON-LD. The full record of an agent run is larger: the prompts, each tool call and its output, token counts and cost. It is in the artefact store, keyed by run id. The provenance line of a node cites the run id. Thus the chain is complete, and git does not hold gigabytes of transcripts (document 11.6).

### What provenance has to make possible

`math2-agent/bench/prov/` runs these tests against a release:

1. **Attribution.** For a node, list the people and the agent runs that contributed to its current text, formal binding and reviews.
2. **Withdrawal.** A model id and a prompt version can be found to be systematically wrong. For that pair, list each field that it produced and that a person has not replaced or confirmed since. Then open a pull request that reverts or flags all of them (`m2 prov quarantine --model … --prompt …`).
3. **Licence tracing.** For a node, list each external source that its text came from, with the licence of that source. This is the input to the licence gate (document 09.3).
4. **Credit export.** For a person, make a CRediT-style statement of what they contributed to math2 in a period. Then the work counts as scholarly output (document 16.4).

## 6.6 Mirrors, backups, retention

- **git:** GitHub is primary. Codeberg and a self-hosted bare repository are mirrors. Each release is archived to Software Heritage.
- **Artefact store:** two providers. Keep releases forever. Keep builds of `main` for 90 days. Keep agent run records in full for 2 years, and keep their summaries (cost, outcome, the produced diff) forever.
- **Restore drill:** each quarter, CI rebuilds the latest release on a clean machine. It uses only the Software Heritage copy and the second artefact mirror. Then it checks the manifest hashes. A failed drill blocks the next release (rule `R04`). Each drill writes a record to `reports/restore/YYYY-MM-DD.yaml` with the fields `date`, `release`, `runner`, `passed` and `note`. Rule `R04` reads the last record. It fails when that drill failed, or when the drill is more than one quarter (92 days) older than the month of the last release.

## 6.7 Sizes

These estimates are for the end of the milestones in document 19. Each release manifest records the actual numbers.

| Item | Estimate | Basis |
|---|---|---|
| live nodes (statement plane) | 2–5 M | Mathlib-bound nodes (~0.3–0.5 M), curated literature results, OEIS sequences (~390 k), LMFDB objects as nodes only when cited |
| record plane entries | ~5 M records, ~30 M aligned results | zbMATH Open's ~4.9 M documents and arXiv math. Results are extracted only from licence-permitted full text (document 09) |
| git repository | 3–8 GB packed | about 1 KB average per node file, plus provenance lines |
| `.olean` cache | 10–15 GB | Mathlib plus the math2 library |
| graph Parquet | 3–6 GB | |
| search indexes | 5–20 GB | lexical plus vectors at 1024 dimensions, int8-quantised |

These sizes do not need special technology. The storage layer is simple on purpose, because the interesting part of the project is in a different place.
</document>

<document source="https://tamnd.github.io/math2/spec/07-formal-engine.md">
# 07. The formal engine: Lean 4, Mathlib, proof checking

The formal engine is the middle box of the top row of the diagram, with the label "Lean 4, Mathlib, Proof Checker". It is the only component that can produce E5. Thus its specification is stricter than that of any other component. The content side is in `tamnd/math2/lean/`, the `Math2` Lake package. The toolchain side is in `tamnd/math2-agent/lean` and in the `lean/M2Extract` package.

The job of the engine is one sentence: **given the content at a commit, produce for every formal binding a checked declaration, its elaborated-type hash, its axiom profile, its formal dependencies, and the results of independent replay, all pinned to one Lean toolchain and one Mathlib version.**

## 7.1 What is pinned

Each release pins three things. Its manifest records them (06.2):

| Item | Pinned as | Source of truth |
|---|---|---|
| Lean toolchain | a stable release, e.g. `leanprover/lean4:v4.35.0` | `lean/lean-toolchain` |
| Mathlib | the matching Mathlib tag `v4.35.0`, plus the commit it names | `lean/lake-manifest.json` |
| Checkers | `nanoda`, `lean4lean`, `con-leche`, `con-ron` and comparator versions, as bundled with the toolchain or pinned separately | `manifests/releases/<r>.yaml` → `checkers:` |

A release never pins a Lean release candidate or Mathlib `master`. Mathlib has no semantic versions. It tags each toolchain bump `v4.X.Y`, and its `stable` branch follows those tags (document 01). math2 pins those tags. This gives monthly releases a natural rhythm: each release takes the newest tag that the bump branch (7.9) has built green.

When a node binds to other formal libraries, we pin them in the same way. These libraries are Tau Ceti, the textbook libraries of Formal Frontier, the formal-conjectures repository, the Imperial FLT project, Carleson, PFR, and the AFP for Isabelle bindings. Each one is an entry in `manifests/sources.yaml` with a commit and a toolchain. Some libraries have a toolchain that is different from the toolchain of the release. We check a binding to such a library in the environment of that library, and the manifest marks it.

## 7.2 The `Math2` package

`lean/` is an ordinary Lake package. Its contents follow the 2026 community consensus on how to grow a formal library (the "How to formalize all of mathematics" statement by the Mathlib maintainers; document 01): **defer to Mathlib, avoid duplication, keep quality high, and link to the literature.**

- **Mathlib first.** If the result of a node is in Mathlib, the node binds to the Mathlib declaration and has no Lean code in math2. `Math2/` holds only what Mathlib does not have. That includes statements of research-level results (so we can bind and review them before anyone proves them), proofs that are not upstream yet, conjecture statements, hypotheses for conditional results, and certificate checkers.
- **Module system only.** Each file in `Math2/` is a Lean `module`, with explicit `public import`s. Mathlib has moved to the module system. It will stop support for downstream projects that have not moved. If we start there, it costs nothing.
- **Mathlib style.** Names, formats and linters follow the Mathlib conventions, and `lake lint --code-quality` runs in CI. This is a condition for upstreaming, and the plan is to upstream all general material (16.5).
- **Hypotheses are the only axioms.** `Math2/Hypotheses/` holds named axioms such as `GRH` or `ABC`. Each one is bound to its conjecture node. Then the axiom profile of a conditional theorem names exactly the hypotheses that it depends on. Nothing else in `Math2/` may contain `axiom`, `sorry`, `admit`, `@[implemented_by]` or `unsafe` (rule `F06`).
- **Conjectures are definitions, not axioms.** `Math2/Conjectures/Collatz.lean` defines `def Collatz : Prop := …`. A proof of the conjecture is a term of type `Collatz`. A refutation is a term of type `¬ Collatz`. If the formal-conjectures repository has conventions for a statement of an open problem, follow them, and bind to its entries with `same_as`.
- **A statement with no proof is a definition too.** Rule `F06` refuses `sorry`, thus a result that is stated and not yet proved is `def lagrange_statement : Prop := …`, and the node binds to it. The degenerate-statement checks and the F1 battery read the value of such a definition as the statement, as they read the type of a theorem (`lean.Record.Statement`). A definition of a proposition is never a proof, thus the binding is not E5. A proof adds `theorem lagrange : lagrange_statement`, moves the `@[m2 "TAG"]` attribute to it, and changes the `decl` of the binding to the theorem. The value of the definition does not change.

### The binding attribute

```lean
module
public import Mathlib.GroupTheory.Coset.Card
public import Math2.Meta.Attr

namespace Math2

/-- m2://9TQ4WA: every finite subgroup of the multiplicative group of a field is cyclic. -/
@[m2 "9TQ4WA"]
theorem finite_subgroup_units_cyclic {K : Type*} [Field K] (G : Subgroup Kˣ) [Finite G] :
    IsCyclic G :=
  inferInstance            -- Mathlib's instance; the binding is to a math2 restatement

end Math2
```

`@[m2 "TAG"]` records the binding on the Lean side. Rule `F01` checks that it agrees with the front matter of the node. A node can also bind to a Mathlib declaration that has no attribute, because math2 cannot add attributes to Mathlib. Only the front matter and `registers/bindings.reg` list those bindings.

## 7.3 Binding kinds and adapters

A binding states how the formal declaration relates to the informal statement of the node (04.5). The `match` field takes one of five values:

| `match` | Meaning | Can reach E5 for the node? |
|---|---|---|
| `exact` | the declaration states exactly the node's statement | yes |
| `stronger` | the declaration implies the statement; an adapter in `Math2/Meta/Match.lean` derives the exact form | yes, through the adapter, which is itself a checked declaration |
| `weaker` | the declaration is a special case | no; contributes evidence for the special case only |
| `variant` | same mathematics, different formal packaging (e.g. `Finset` against `Set`, bundled against unbundled) | yes, if the adapter proves the equivalence |
| `partial` | a statement-only binding for an unproved result, or one that is still `sorry`-free only in its statement | never (rule `F07`) |

Mathlib often states a classical theorem in a more general form than a textbook does, or packages it differently. Adapters are how math2 handles this. The adapter is small and checked, and we review it together with the binding. The F2 review covers the composite: the conclusion of the adapter against the informal statement.

## 7.4 Extraction

After a build, `M2Extract` walks the environment. It writes one record for each bound declaration, and one record for each declaration in the closure of each bound declaration:

```yaml
decl: Math2.finite_subgroup_units_cyclic
module: Math2.Algebra.Field.Units
tag: 9TQ4WA
type_hash: blake3:5f1c…            # hash of the elaborated type, after instantiating universe params canonically
value_hash: blake3:a903…
statement_pp: "∀ {K : Type u_1} [inst : Field K] (G : Subgroup Kˣ) [inst_1 : Finite ↥G], IsCyclic ↥G"
axioms: [propext, Classical.choice, Quot.sound]
native_decide_axioms: []
uses_decls: [IsCyclic, Subgroup, Units, Field, Finite, instIsCyclicSubtypeMemSubgroupUnitsOfFinite, …]
uses_constants_in_statement: [IsCyclic, Subgroup, Units, Field, Finite]
deprecated: null                   # or {new_name, since, text} from the @[deprecated] attribute
checks: {}                         # or {vacuous, trivial, refuted}, each the tactic that closed the goal
heartbeats: 1840
```

- **Type hash.** This is the key hash in math2 (06.3). We compute it over the elaborated type in a canonical serialization: universe parameters get names by position, binder names are erased, and instance arguments stay. If two declarations have the same type hash, they state the same thing. Because of this, a review stays valid after a rename (Flow C, 03.4), and align can propose `same_as` without judgement.
- **Dependencies.** `uses_decls` comes from the value of the declaration. It is each constant that the proof term mentions, closed transitively at the boundary of other bound declarations. Formal dependencies between bound declarations become `uses` evidence in the graph. Rule `G04` reports each formal dependency that the `uses` of the node does not show.
- **Statement constants.** `uses_constants_in_statement` drives the review tool. A reviewer must see each non-trivial definition that the statement relies on (16.3).
- **Pretty-printing.** We print the statement normally, and also with full elaboration and with notation off. Reviewers see both forms.
- **Deprecation.** `deprecated` copies the entry of the `@[deprecated]` attribute. Rule `F08` reports each bound declaration that has one, with the new name. The bump agent (07.9) uses the same data to propose the rename.
- **Degenerate statements.** For each theorem, `checks` gives the result of the checks 4, 5 and 6 of 10.4. Automation tries to prove `False` from the hypotheses (`vacuous`), to prove the statement without the theorem itself (`trivial`) and to prove the negation (`refuted`). Each tactic has a limit of 10 million heartbeats, which is 5 percent of the default limit of Lean, and the record keeps the first tactic that closed the goal. Rule `F09` fails on `vacuous` and on `refuted`. A `trivial` result does not fail, because a trivial statement can be the correct one.

Extraction runs in the same pinned environment as the build. Its output is a build artefact, with its hash as its address. The extraction code is part of the trusted base (15.1). Thus it is small, and we test it against golden files in `M2Extract/test/`.

## 7.5 Axioms and `native_decide`

- **The allowed set** for E5 is `{propext, Classical.choice, Quot.sound}`. A node can allow fewer: front matter `axioms: constructive` limits it to `propext` and `Quot.sound`. Then status reports the node as `noac` (04.7).
- **Hypotheses** from `Math2/Hypotheses/` can be in the axiom profile of a conditional result. Status reports them by name. The evidence of the node is then "E5 conditional on GRH", and the workspace shows this in words.
- **`native_decide`.** Since Lean 4.35, each use of `native_decide` adds a new auxiliary axiom, and `Lean.ofReduceBool` is removed. Thus the axiom audit sees `native_decide` without a special case. It is not in the allowed set. A node that depends on it can be at most E4, with the certificate kind `compiled-evaluation`, and rule `F04` lists it. If `decide`, `decide_cbv` or `cbv` can do the same job with kernel-only reduction, we tell the prove role to use them.
- **The checker sees the exported environment**, not the source. Axiom profiles come from the environment as the kernel checked it. A search of the source for `axiom` is not sufficient, because metaprograms can add declarations.

## 7.6 Building

```
m2 build                  # nodes → lean → extract → replay → battery → status → graph → store → search
m2 build -only lean       # lake build of Math2, with the Mathlib cache
```

- **Cache.** Lake's cache tools fetch the Mathlib `.olean` cache, and we check it against the hash in the release manifest. The `.olean` files are a build cache, not evidence. Evidence comes from the replay (7.7), which checks the exported environment again from the start. A compromised cache can make the build lie, but it cannot make the replay lie.
- **Incremental.** PR CI builds only the modules that a change touches, plus their reverse dependencies inside `Math2`. CI on `main` builds everything. The release build runs on a clean runner with no cache from earlier PR runs.
- **Budgets.** We record the heartbeat count of each declaration. If a PR makes the build time of `Math2` more than 5% longer, its description must say why.

## 7.7 Replay and the checker stack

A proof is E5 only after it passes **independent replay** (rule `F05`). The toolchain provides this directly since 2026. This came after the kernel soundness incidents of that summer (document 01):

| Layer | Tool | What it adds |
|---|---|---|
| 1 | `lake build` | elaboration and the official kernel, in the build process |
| 2 | `lake check` | re-checks the exported environment with the official kernel, in a separate sandboxed process |
| 3 | `lake check --paranoid` | adds the independent kernels nanoda (Rust), lean4lean (Lean), con-leche (Lean, proved consistent) and con-ron (Rust, proved to refine con-leche) |
| 4 | comparator | checks that the replayed declaration's statement matches the **pinned statement** (a challenge module built from the binding's recorded type, against Mathlib and `Math2` only), and that its axioms are permitted |

At release time, math2 runs all four layers for each E5 candidate. On each PR, it runs layers 1, 2 and 4. The release manifest records the version of each checker. In 2026, a checker that was one week out of date accepted an exploit that the current checker rejected. The release audit fails if a checker is older than the version bundled with the pinned toolchain.

When this document was written, comparator had known open issues (document 01). Thus math2 also compares type hashes from its own extraction against the pinned binding (7.4). Two independent statement checks are cheap, and each one covers the gaps of the other.

For the largest proofs, a full paranoid replay is expensive. The 13M-line Fermat formalization needed 230 GB of RAM and 15 hours under comparator. math2 does not hold proofs of that size. It binds to the declaration in the external library. It records the replay results of that library with their checker versions, at E5 with the flag `external_replay` (04.7). It runs those replays again at the pinned commit of the library only when a checker changes.

## 7.8 Interfaces for agents and people

The formal engine has three front doors. All three use the pinned toolchain.

- **Lean sessions** for agents (11.3). These are REPL processes with Mathlib and `Math2` loaded from a snapshot. They expose the state at tactic level. Implementations to build on are the community REPL, Pantograph, and lean-beam, which is the agent interface of the Lean FRO and is in beta now (document 01). math2 uses the one that gives the best control at goal level under its sandbox. The protocol in 11.3 belongs to math2, so the implementation can change and the agents do not see a change.
- **An MCP server.** `m2 serve` exposes `lean_check`, `lean_goal` and the session calls as MCP tools. With the public `lean-lsp-mcp`, any MCP agent, Claude Code included, can work on `Math2` from a local checkout.
- **Workbench and editors for people.** A contributor works in VS Code with the Lean extension on a local checkout. Or the contributor works in the zero-install Workbench of the Lean FRO in the browser (beta in 2026). math2 adds nothing to the editor experience other than the binding attribute. This is intentional.

## 7.9 The bump

The `bump` role and `m2 bump` keep math2 current. The process:

1. **Weekly**, the bump branch moves to the newest Mathlib tag. It also tries Mathlib `master` as an early-warning build. We never merge that build.
2. **Breaks are classified.** Each failure is one of four kinds:
   - a rename with a deprecation alias (mechanical: take the new name from the `@[deprecated]` attribute);
   - a rename without one (find the new declaration by type hash);
   - a refactor (the type hash changed);
   - a real break in `Math2` code (needs a fix).
3. **Mechanical renames auto-merge** (12.1). Migrate deprecated names before Mathlib deletes them. Mathlib deletes them in batches, about six months after deprecation.
4. **Refactors** make reviews stale (rule `F02`). Each affected node drops to F0 and goes into the fidelity queue, ranked by `unlock`. It gets a review-assist packet that shows the old and new types side by side.
5. **`Math2` breaks** become prove jobs to repair the proof, at high priority. The release cannot pin the new Mathlib until they are fixed.
6. **Upstream catch-up.** A new Mathlib declaration can have the same type hash as a `Math2` declaration. Then the bump proposes to rebind the node to Mathlib and to deprecate the `Math2` copy. `Math2` gets smaller as Mathlib gets larger. That is the intended outcome.
7. **Toolchain bumps** follow Mathlib, because each Mathlib tag names its toolchain. A toolchain bump also runs the full paranoid replay again, because the checkers change with it.

The bump report goes into `reports/bump/<date>.md`. It counts breaks by kind, time to green, and the nodes whose fidelity reviews became stale. Over months, it measures the cost to stay current. Each downstream Lean project pays this cost, and few measure it.

## 7.10 Other proof assistants

math2 is Lean-first, not Lean-only. Bindings have a `system:` field. Two other systems have support from the start:

- **Isabelle/AFP.** The AFP is the second-largest library, with more than a thousand entries. Some results exist only there. For example, the 2026 Jacobian counterexample was formalised in the AFP first. To check an AFP binding, build the entry at its pinned AFP and Isabelle release. It counts as E5 with `system: isabelle`. The trusted base for such nodes is the Isabelle kernel, and the workspace says so.
- **Rocq/MathComp, Metamath, Mizar, HOL Light** are crosswalk targets (`same_as`, 05.3). In v1 they are not bindings for E5. Metamath's set.mm is the exception. It is a candidate for a v2 binding, because its verifier is very small and there are several independent implementations of it.

Translation between systems (lean2dk, lean2agda and similar tools) was still experimental in 2026, and it is not on the critical path. We record agreement between systems as `same_as` between bindings. It counts towards F3, as an independent formalization of the same statement, only when an equivalence is proved or both formalizations have independent F2 reviews (04.7).
</document>

<document source="https://tamnd.github.io/math2/spec/08-computational-engine.md">
# 08. The computational engine: CAS, SMT, numerical experiments

The computational engine is the box that makes math2 executable in the second sense. In the first sense, formal proofs run in the Lean kernel (document 07). In the second sense, computations are part of the knowledge. These are the examples, counterexamples, tables, searches and numerical experiments that mathematicians use to find conjectures and to check them. The Research Agents call the engine directly. Its outputs go into the knowledge graph (as `computation`, `example` and `counterexample` nodes, and as evidence) and into versioned storage (as artefacts with certificates).

The engine has one design principle: **a computation is a claim, and the strength of the claim depends on what can be checked about it afterwards.**

## 8.1 What runs

The engine runs pinned container images, one for each system family. The image digest is part of the identity of each computation (document 06.3).

| Image | Contents | Used for |
|---|---|---|
| `m2-sage` | SageMath (current stable), with its bundled PARI/GP, GAP, FLINT, Singular, Maxima, NTL, ECL | general CAS, algebra, number theory, combinatorics |
| `m2-pari` | PARI/GP standalone, current stable | fast number theory, L-functions, elliptic curves |
| `m2-gap` | GAP with the standard packages (SmallGrp, AtlasRep, …) | group theory, representation theory |
| `m2-m2` | Macaulay2 | commutative algebra, algebraic geometry |
| `m2-flint` | FLINT 3, including Arb (merged into FLINT) | ball arithmetic and rigorous numerics |
| `m2-julia` | Julia with Oscar.jl, IntervalArithmetic.jl | computer algebra (Oscar), validated numerics |
| `m2-python` | CPython, SymPy, NumPy/SciPy, mpmath, NetworkX, python-flint, OR-tools | scripting, quick experiments, graph search |
| `m2-smt` | Z3, cvc5, Bitwuzla | SMT queries with models and proofs |
| `m2-sat` | CaDiCaL, Kissat, with DRAT/LRAT proof output; `cake_lpr` and `dpr-trim` as checkers | SAT encodings of combinatorial problems (Boolean Pythagorean triples, Keller, Schur number 5, empty hexagon) |
| `m2-lp` | SCIP, HiGHS, with exact rational verification (VIPR certificates where available) | integer and linear programming bounds |
| `m2-lean` | the pinned Lean toolchain and the Mathlib cache | `#eval`, `decide`, and checking certificates inside Lean (document 07.8) |

Commercial systems (Magma, Mathematica, Maple) are permitted as **E1 evidence only**. An agent or a person can run them, and we can record the output. But the output cannot reach E4, because a reader cannot reproduce the run. They are adapters in `math2-agent`, and they are off by default. The same rule applies to Wolfram|Alpha.

We build each image reproducibly from a Nix flake in `tamnd/math2-agent/images/`. We publish it by digest and refresh it monthly with the release. If an image changes under a recorded computation, the old record stays valid, because the record names the old digest. But the change causes a new run in the next release, to check that the result still holds (rule `C03`).

## 8.2 The computation record

A computation is a node of kind `computation`. Its body is a short description. Its front matter is the record:

```yaml
tag: 3MZ8QD
kind: computation
name: No counterexample to Goldbach below 10^7 (demonstration)
about: [8GQ2N1]                  # the Goldbach conjecture node
claim: >
  Every even n with 4 <= n <= 10^7 is the sum of two primes.
engine:
  image: m2-pari@sha256:...       # by digest
  entry: compute/3MZ8QD/main.gp
  inputs: {N: 10000000}
  seed: null                      # deterministic. Randomised computations must give a seed
  resources: {cpu: 4, mem: 4G, wall: 10m}
output:
  hash: blake3:...                # canonical serialisation of the result
  summary: "checked 4999999 even numbers; 0 failures"
certificate:
  kind: witness-table             # see 8.3
  artefact: blake3:...
  checker: m2-lean@sha256:...    # independent checker image
  checker_entry: compute/3MZ8QD/Check.lean
```

The code is next to the node, in `compute/<tag>/`, under Apache-2.0 like the Lean code. Read a computation in the same way as a proof: what does this show, and how can I check it?

## 8.3 Certificates: how computations reach E4

A computation reaches E4 (`certified`, document 04.7) only if it produces a **certificate** that a **different, smaller, more trusted** checker accepts. The engine knows a fixed list of certificate kinds:

| Certificate | Produced by | Checked by | Typical claim |
|---|---|---|---|
| `witness` | anything | Lean `decide`/`norm_num`, or a 50-line checker | "there exists an X with property P" (show the X) |
| `witness-table` | anything | a Lean or small Python checker that runs over the table | finite verifications where each case has a short witness (Goldbach pairs, factorisations) |
| `lrat` | CaDiCaL/Kissat with proof output | `cake_lpr` (a verified checker, CakeML) | "no X exists" via SAT |
| `smt-proof` | cvc5 (Alethe/CPC proofs), Z3 (proof objects) | Lean's SMT reconstruction, or Carcara for Alethe | decidable arithmetic and bit-vector facts |
| `primality` | PARI/GP or Primo | Lean (Pratt or ECPP certificate checker) | "N is prime" |
| `interval` | FLINT/Arb, IntervalArithmetic.jl | re-evaluation in a second ball arithmetic, or in Lean with exact rationals | rigorous numerical inequalities, zeros of functions |
| `lp-dual` | SCIP/HiGHS | exact rational arithmetic check of the dual certificate (VIPR) | bounds from linear and integer programming |
| `sos` | SDP solver plus rational rounding | exact polynomial identity check | positivity via sums of squares |
| `group-id` | GAP | a recomputation in a second system (Magma excluded), or a Lean check for small orders | "this group is SmallGroup(n,k)" |

A computation without a certificate is E1. That is still useful. Most experiments in math are E1, and the system records them accurately. But the system never shows it as stronger than E1.

### Into the kernel

If Lean can check a certificate in reasonable time, the computation can go all the way to E5. We import the certificate as data. We prove a Lean checker sound one time. Then `decide` or kernel reduction runs the checker on the data. The other route is `native_decide`, which is flagged and is permitted only per node (document 07.7). The Lean ecosystem of 2025 to 2026 made this practical for SAT (LRAT checks inside Lean, as `bv_decide` does) and for many arithmetic certificates. If the certificate kind of a computation has a Lean checker in `lean/Math2/Cert/`, the prove agent tries this route.

## 8.4 Numerical experiments

Experiments are how people find conjectures. FunSearch, AlphaEvolve and PatternBoost-style systems (document 01) are, at the base, experiment loops in which a model proposes programs or constructions. math2 supports them as first-class but untrusted:

- **An experiment is a computation with `role: experiment`.** Its claim is an observation ("for all n ≤ 10^5 tested, a(n) is squarefree"), not a theorem.
- **Search runs record their search space.** An agent can search for a construction (a cap set, a packing, a graph with given properties). Then the record holds the scoring function, the best object found and its score. A best object is a `witness`. A lower bound from it can be E4 if the check of the object is cheap. The claim "this is optimal" is E1 unless an exhaustive or dual certificate supports it.
- **Patterns become conjectures through a gate.** If an agent sees a pattern, it files a `conjecture` node with `source: computation <tag>`. The explore agent (document 12.5) must first run a falsification pass: extend the range, randomise, test against OEIS and LMFDB, and try the obvious generalisations. The agent files the conjecture together with the falsification attempts that failed.

## 8.5 SMT and SAT for the prove loop

SMT and SAT are also tools inside the prove loop (document 12.4). They do more than produce standalone computations:

- **Lean already calls out.** We use `bv_decide` (bit-blasting to CaDiCaL with LRAT replay), `omega`, `grind`, `polyrith` (which calls a CAS and then checks the certificate) and similar tactics where they apply. The Lean axioms audit (document 07.7) handles their trust status.
- **Agents use SMT as a counterexample finder.** Before the prove agent spends prover budget on a goal, it encodes small finite instances where it can. Then it asks Z3 or cvc5 for a model. A model is a counterexample to the formal statement. That is very often a *misformalisation* finding (document 10.5), and the engine reports it to the formalize agent as one.
- **Large SAT proofs are artefacts.** We do not store proofs of the Boolean Pythagorean triples size (about 200 TB uncompressed). We store the encoding, the solver version and the cube split. We also store a checked hash chain if the original authors gave one. That is sufficient to say exactly what was checked and by whom. A claim in this class is E4 only if a certificate was checked, possibly by a third party, and the record names who did it.

## 8.6 Sandboxing, budgets, determinism

Each computation runs in a sandbox. The rules are the same for agents and for the CI runs of people:

- **Isolation:** gVisor or Firecracker microVMs on the runners; no network by default; a read-only image; a writable scratch area of bounded size.
- **Budgets:** CPU, memory, wall clock and output size, all declared in the record. The request of an agent is charged against the budget of the run (document 11.5).
- **Determinism:** computations must be deterministic, or must declare a seed. The engine runs each new E4-candidate computation two times, on two different runner types (x86-64 and arm64), and compares output hashes (rule `C02`). A difference blocks E4, and the engine writes it to the report. Floating-point code that is not interval arithmetic never reaches E4, so it never blocks anything. It stays E1.
- **Data access:** a computation can read OEIS, LMFDB or zbMATH only through the cached snapshots of the ingestion layer (document 09). Those snapshots are pinned by hash and mounted read-only. Thus a computation never depends on a live API.

## 8.7 Executable examples

Each `example` and `counterexample` node should be executable where possible. The executable check is a Lean `example` (document 07), or a computation of 10 seconds or less with a `witness` certificate. The "run this" button of the workspace (document 13) runs exactly that record in a sandbox on the browser side or the server side. It shows the output next to the recorded hash. The build reports the share of executable examples per topic, and the goal in document 02 gives a number for it.

## 8.8 What the engine will not do

- It will not run code that is not in a pinned image.
- It will not let a computation call the network, call a model, or read anything outside its declared inputs.
- It will not accept a certificate checker written by the same agent run that produced the certificate, unless that checker has itself been proved sound in Lean or reviewed by a person. The checker for each certificate kind is fixed in `math2-agent`, and a change to it needs the review of a person.
- It will not show a numerical observation as anything stronger than an observation.
</document>

<document source="https://tamnd.github.io/math2/spec/09-literature-ingestion.md">
# 09. Literature ingestion: papers, books, existing libraries

The top-right box of the diagram has the label "Papers, Books, Existing Libraries". It is how math2 learns what mathematics already knows. Ingestion has three jobs:

1. **Fill the record plane** (03.3). It records each paper and book in the mathematical literature. Where possible, it also extracts and records each result that they state.
2. **Seed and source the statement plane.** The `sources:` of each node points at records with locators, and curation turns extracted results into nodes.
3. **Join the existing libraries**: Mathlib and other formal libraries, encyclopedias and databases. Crosswalk edges (05.3) connect them to nodes. math2 does not copy what they hold.

Ingestion code is in `tamnd/math2-agent/ingest/<source>`, with one adapter for each source. Each pull is pinned in `tamnd/math2/manifests/sources.yaml` and `manifests/ingest/`.

## 9.1 Principles

- **Dumps, not crawling.** Read each source through its published bulk channel or API, at the rate that it asks for. If a source has no bulk channel, link to it and do not ingest it.
- **Pinned snapshots.** A pull is a snapshot with a date, a URL, a hash and a licence. At build time, nothing in math2 depends on a live external service.
- **Licence decides what is taken** (9.3). Most of the literature can give a record and a locator, but not text. That is sufficient for the record plane, because a mathematical fact is not copyrightable and math2 writes its own statements.
- **Reuse other people's indexes.** Several 2026 projects have already extracted millions of statements from the literature (9.6). math2 aligns to them by id. It does not extract again what they hold, unless it needs something that they do not have.
- **Do not duplicate `tamnd/arxiv`.** arXiv papers come in through the two planes of `tamnd/arxiv`. math2 stores `ax://` references, not paper text.

## 9.2 The source catalogue

Sizes are from document 01, measured or reported in October 2026.

| Source | What math2 takes | Size | Channel | Licence of what is taken | Cadence |
|---|---|---|---|---|---|
| **arXiv** (via `tamnd/arxiv`) | records for every math paper; extracted results from the content plane | about 55k math primaries in 2026 so far, growing fast | `tamnd/arxiv` metadata and content planes | metadata CC0; text per paper licence (9.3) | daily |
| **zbMATH Open** | records for the published literature 1763–present, MSC codes, reviews' metadata, software (swMATH) | 4M+ publications, 168M+ triples in the KG | OAI-PMH, REST API, the KG dump | CC BY-SA 4.0 (publisher-restricted abstracts are not included) | monthly |
| **MathSciNet** | MR numbers only, via reference matching | 4M+ items | MR Lookup | identifiers only | on demand |
| **Crossref / DataCite** | DOIs, journal metadata, references where open | n/a | public APIs | CC0 metadata | monthly |
| **Mathlib** | declarations, types, docstrings, module structure, deprecations | about 290k theorems, 140k definitions | the pinned checkout plus `M2Extract` (07.4) | Apache-2.0 | every bump (07.9) |
| **other Lean libraries** | the same, for bound libraries: Tau Ceti, Formal Frontier textbooks, Physlib, CSLib, FLT (Imperial and Anthropic), PFR, Carleson, sphere packing | Tau Ceti about 3M lines | pinned checkouts | per repository (mostly Apache-2.0) | per release |
| **formal-conjectures** | formal statements of open problems, categories, solved markers | 4,652 statements on 25 October 2026 (1,602 open research problems, 2,816 solved, 234 textbook) | git | Apache-2.0 | weekly |
| **Palomar** | registered Lean-verified results, with `formalization.yaml` provenance | 760 results on 25 October 2026 (registry revision 517), growing since August 2026 | the listing API | CC0 metadata | daily |
| **Erdős problems** (`teorth/erdosproblems`) | problem statements, status, links to formal statements and OEIS | 1,221 problems | git (`data/problems.yaml`) | Apache-2.0 | weekly |
| **Isabelle AFP** | entries and theorem names, for bindings and crosswalk | 1,038 entries | the release tarball | per entry (BSD/LGPL) | per AFP release |
| **OEIS** | sequences: name, terms, formulas, references, conjectures in comments | about 400k sequences | `oeisdata` git export | CC BY-SA 4.0 | weekly |
| **LMFDB** | object records (curves, fields, forms, groups, L-functions) by label, for `object` nodes and crosswalk | about 3.8M curves over ℚ, 22M number fields, … | API and bulk downloads (the LMFDB asks AI users to prefer bulk) | CC BY-SA 4.0 | monthly |
| **GAP SmallGrp, House of Graphs, FindStat, π-Base, DaRT** | object identifiers and properties; π-Base implications as `statement` candidates | n/a | packages, downloads, git | varied; π-Base data CC BY | per release |
| **Wikidata** | QIDs, labels, existing crosswalk properties (nLab, MathWorld, ProofWiki, Mathlib) | about 113k items with a defining formula | weekly dump; SPARQL for checks | CC0 | weekly |
| **MaRDI** | crosswalk ids; model and algorithm records | 7M+ items | SPARQL | per dataset | monthly |
| **Wikipedia** | article ids for crosswalk; short quoted text where useful | about 26k mathematics articles | dumps | CC BY-SA 4.0 | monthly |
| **nLab** | page ids for crosswalk | about 19k pages | `nlab-content` git | link-only until the licence is confirmed | monthly |
| **ProofWiki** | page ids for crosswalk; proofs and definitions as candidate sources | 30k proofs, 36k definitions | XML dump | CC BY-SA 3.0 | quarterly |
| **Stacks Project, Kerodon** | tags, titles, dependency structure for `same_as` and `cites` | 21,446 Stacks tags on 25 October 2026 (commit a04446e) | git (`tags/tags` and the LaTeX) | GFDL: **link only** | monthly |
| **MathOverflow, Math.SE** | question ids, titles, per-post records for E1 claims and references | 169k and 1.7M questions | the Stack Exchange API, per post | CC BY-SA, per-post version (16.6) | on demand |
| **Books** | records and locators (9.7) | n/a | catalogue metadata, open textbooks | per book | on demand |
| **MathWorld, Encyclopedia of Mathematics, PlanetMath** | ids for crosswalk | n/a | n/a | MathWorld proprietary: link only; others CC BY-SA | quarterly |

Each row is an adapter with its own fixtures. To add a source, write an adapter, add a manifest entry with a licence, and get the approval of a maintainer for the licence classification. That classification is the one decision that matters.

## 9.3 The licence gate

Each source gets an **access class**. Where licences are different within a source, each item gets one. The class decides what can go into which part of math2. math2 extends the four classes of `tamnd/arxiv` for its three licence zones (16.6):

| Class | Example licences | Record (metadata, locator) | Embeddings, internal | Text in the record plane | Text in node prose |
|---|---|---|---|---|---|
| `open` | CC0, CC BY, CC BY-SA, Apache-2.0 | yes | yes | yes, by reference to its home (e.g. `ax://`) | yes, with attribution (BY-SA zone) |
| `share-alike-nc` | CC BY-NC-SA, CC BY-NC | yes | yes | by reference only | **no**: NC is incompatible with CC BY-SA |
| `verbatim` | CC BY-NC-ND, CC BY-ND | yes | yes | by reference only | no |
| `gfdl` | GFDL (Stacks, Kerodon, Open Problem Garden) | yes | yes | by reference only | **no**: GFDL and CC BY-SA cannot mix |
| `record` | arXiv default licence, all-rights-reserved journals, MathWorld | yes | yes | no | no |

Three consequences:

- **math2 stores almost no third-party text.** Paper text stays where it is now. For arXiv, that is the content plane of `tamnd/arxiv`, which has its own licence gate. math2 holds references, locators, hashes and its own derived data: embeddings, extracted structure and alignments. Thus each corpus has one licence gate, not two.
- **About 15% of arXiv is `open` for node prose.** TheoremSearch measured the CC0, CC BY and CC BY-SA share. If you include the NC licences, the Creative Commons share is about 40%. `tamnd/arxiv` uses this figure for its content plane. For each other paper, math2 writes the statement itself and cites the locator. For a mathematical result, this is better in any case: we check the statement against the definitions and notation of the graph.
- **Embeddings are internal.** math2 embeds text from all classes, for alignment and search. It never publishes embeddings of `record` class text in a form that can reconstruct the text. The vector index of the serving store serves neighbours, not vectors (14.4).

Rules `L01` to `L04` (15.3) enforce the gate. `L02` fails the build if `nodes/` contains text from a class that must not go into node prose.

## 9.4 Existing formal libraries and blueprints

Formal libraries come into math2 as **bindings and alignments**, never as copies.

- **Mathlib** is extracted at each bump (07.4). Each declaration becomes a candidate for binding. The curate and align roles propose nodes for important declarations that do not have one, ranked by in-degree in the formal dependency graph. They also use type search and embeddings to propose `formal:` bindings for existing nodes.
- **Other Lean libraries** are extracted in the same way, at their pinned commits. We bind AI-written libraries (Tau Ceti, the textbook libraries of Formal Frontier, the Anthropic FLT repository) like all other libraries. The source entry records their provenance (AI-written, how reviewed), and each binding to them shows it. A binding to an AI-written library is not weaker for E5, because the kernel does not care who wrote the proof. But it is important for maintenance, and the bump report tracks it.
- **Palomar entries** come in with their `formalization.yaml`. That file records these facts: whether the formalization was human, AI-assisted or autonomous, which models were used, and how it was reviewed. The challenge statement of a Palomar entry becomes a binding candidate. Its replay record is accepted as `external_replay` (07.7) at its pinned commit. Palomar does not certify fidelity, so the binding starts at F0 or F1 like all other bindings.
- **formal-conjectures** statements become `partial` bindings for conjecture nodes. They are aligned by the source references of the repository: Erdős numbers, OEIS, Wikipedia, MathOverflow.
- **Blueprints.** `m2 ingest blueprint <repo>` reads the blueprint of a formalization project and imports it as nodes and edges. It reads a leanblueprint LaTeX source (`\label`, `\uses`, `\lean`, `\leanok`) or the `blueprint-manifest.json` of a Verso Blueprint. A Verso Blueprint derives status from the code, not from hand annotations. Each blueprint item becomes a provisional node with `uses` edges. If the item names a declaration, it also becomes a binding. The import is a PR. The repository of the project stays the authority for its blueprint. math2 imports again when the blueprint changes, and keeps its tags stable through the labels of the blueprint (`aliases:`).

## 9.5 Extracting results from papers

Extraction turns a paper into a list of **results**: the theorem-like environments, with their text, labels, internal references and locators. math2 does not implement extraction again. It uses the result tags of `tamnd/arxiv`, `2106.09685/AB12`, which are permanent per paper. It adds processing that is specific to math:

1. **Source first.** arXiv LaTeX source is the ground truth. We use HTML (LaTeXML) for structure where the conversion is clean, which is about three quarters of papers. PDF-only material, such as older journal papers in open repositories and scanned books, goes through a pinned OCR model. We select the OCR model from the current formula-recognition benchmarks. In 2026 the best open document-OCR models were above 97 on the formula metric of OmniDocBench. The outputs are marked `ocr` with the model version.
2. **Environments.** The extractor finds `theorem`, `lemma`, `proposition`, `corollary`, `definition`, `conjecture`, `example`, `remark` and their custom aliases. It resolves `\newtheorem` declarations. It also finds proofs and attaches them to their statements.
3. **Internal references.** `\ref`, `\cref` and "by Lemma 3.2" become dependency edges inside the paper. Citations become references between papers. They are resolved to records through the bibliography of the paper, and matched against zbMATH, Crossref and arXiv.
4. **Unfolding.** A result can say "under the hypotheses of Theorem 2.1". We rewrite it into a **self-contained statement**: we put in the hypotheses and the notation that the paper set up earlier. A model does this, and a second pass checks it against the original (Matlas uses the same step). We keep the original and the unfolded form. Alignment and search see the unfolded form. It is derived data, and it is under the licence class of the paper (9.3).
5. **Typing.** Each result gets a kind (04.2) and MSC codes, from the codes of the paper plus a classifier.
6. **Quality.** Each step has an evaluation set (17.3). Each month, people check a sample, and we publish the error rates for each step.

The output is one record plane entry for each result, under `ax://paper/<id>#<tag>` for arXiv. For other sources it is a math2 record id `m2://rec/<source>/<id>#<n>`.

## 9.6 Other people's indexes

In 2026 several projects extracted the literature at scale (document 01):

- TheoremSearch: 9.2M theorems from about 690k papers. It released the openly licensed 1.34M.
- TheoremGraph, with its formal half LeanGraph: 11.7M arXiv statements and 18.3M candidate dependency edges. It matches 48k statements to Lean declarations, and has an MCP interface.
- Matlas: 8M unfolded statements from 435k journal papers and 1.9k textbooks.

The position of math2 is to **align to these, not compete with them**:

- Their identifiers are crosswalk targets (`same_as` on record plane entries).
- If their data is openly licensed, we import their extractions and dependency edges as *candidate* evidence. Each item names the extractor that produced it. We treat it like the extraction of math2: E1, unconfirmed, subject to dedup.
- If their data is not openly licensed, math2 uses their public APIs at query time, in the novelty check (17.5) and in search. It stores only the ids of matches.

math2 adds what these indexes do not have: curated concept nodes, computed status, reviewed fidelity, and permanence.

## 9.7 Books

Books are the main source for E2 ("textbook-standard", 04.7). They are also the most difficult source to ingest, because most of them are not openly licensed. math2 handles them as follows:

- **Records and locators** for any book, from catalogue metadata (ISBN, zbMATH and MR records). A locator such as "§3.2, Thm 8" is a fact, and math2 can store it for any book.
- **Curricula.** We build reading lists in `topics/curricula/` from standard texts and their tables of contents. A table of contents is a factual structure that math2 can record. The full text stays with the book.
- **Open textbooks** (CC-licensed, or with explicit permission) can be extracted like papers. The 2026 automatic formalizations of whole textbooks (document 01) give these books formal counterparts, which align to the nodes of the books.
- **Two independent book locators** for E2 are checked by curators, not by agents, because the check requires a person to read the book. That is an intentional use of human time. Most nodes in the undergraduate core will be at E2 before they are formalised, so E2 must mean something.

## 9.8 Scale and cost

| Item | Size | Ingest cost |
|---|---|---|
| Records (zbMATH, arXiv, Crossref) | about 5M papers and books | negligible compute; storage about 20 GB Parquet |
| Results extracted (arXiv `open` and `share-alike` content, plus imported candidates) | 10–30M results | extraction through `tamnd/arxiv`; unfolding by a small model in batch, at a cost of roughly cents per paper |
| Embeddings of all results | 30M × 1,024 dims | a one-time batch job, then incremental; the index is about 60 GB quantised (14.4) |
| Alignment of new results to nodes | daily: about 1,000 papers → 10–20k results | judge calls only on candidate pairs above the similarity threshold, with Haiku-class pre-filtering and batch pricing (12.3) |

The monthly arXiv math volume grew by more than half between 2025 and 2026. Part of the growth is attributed to AI-assisted writing. Ingestion must scale with that volume, and it must not use curator time for it. For these reasons, extraction and alignment are agent work, and curators see queues ranked by importance (16.3). Also for these reasons, an extracted result stays E1, in the record plane, until something in the graph needs it.
</document>

<document source="https://tamnd.github.io/math2/spec/10-autoformalization-and-fidelity.md">
# 10. Autoformalization and statement fidelity

The kernel answers one question perfectly: is this Lean term a proof of this Lean type? It cannot answer the question that matters: does this Lean type say what the mathematician meant? That second question is **fidelity**. In 2026 it is the weakest link in all formal mathematics, with AI or without it. This document tells how to change fidelity from a hope into a property that we measure, review and maintain.

## 10.1 The evidence that this is the bottleneck

Document 01 has the sources. In brief:

- An audit in 2025 found that about **a third of the Lean 4 port of ProofNet** was misformalized.
- An ICML 2026 audit of five formal benchmarks found **4,833 issues**. Of these, **398 were mechanically certified**: counterexamples to stated theorems, vacuous statements, and unsound axioms.
- The statements of PutnamBench needed **27 fixes** from one prover team and 2 more from another before people could call the benchmark saturated.
- In 2025, an AI "solution" to an Erdős problem got much attention. It proved a weaker variant, because the problem as written did not have a hypothesis.
- The 2026 Navier–Stokes claim came with a Lean proof. The dispute about it was about the gap between the formal statement and the physical problem.
- The 2026 Annals Challenge put Lean statements of 50 recent Annals theorems into Mathlib. It found that the main blocker was **missing definitions**, not missing proofs.
- In the largest 2026 formalizations, people fixed or cross-checked the top-level statement by hand, against a curated library. In the Fermat run, agents cross-checked the intermediate statements of each other, and that found false ones early.

The lesson that math2 takes: **autoformalization of statements is now cheap and usually right. "Usually" is not good enough for a library that other work will build on.** Thus the pipeline has two halves. A cheap machine half, F1, finds most errors. A human half, F2 and F3, is made fast enough that people can review the important statements.

## 10.2 The formalize pipeline

The formalize role (12.1) runs this pipeline for one node. `m2 autoform <tag>` runs it from the command line.

```
1. context      node statement, its uses closure (definitions!), sources' locators,
                bound neighbours and their Lean statements, Mathlib conventions
2. definitions  every concept the statement uses must already be bound, or is formalized first (10.3)
3. candidates   k candidate Lean statements from ≥ 2 model families, each elaborated in a Lean session
4. battery      the F1 battery (10.4) on every candidate that elaborates
5. select       candidates that pass; clustered by type hash and by proved equivalence;
                the largest agreeing cluster wins; disagreement is itself recorded
6. propose      PR: binding at F0 with the winning statement, the battery report,
                the dissenting candidates, and a review packet (10.6)
```

Two design points:

- **Agreement among independent candidates is evidence.** Five candidates from two model families can elaborate to the same type hash, or to types proved equivalent in a Lean session. That is stronger than the back-translation of one candidate. If the candidates split into clusters, the split usually points directly at the ambiguity: which convention for $\mathbb{N}$, whether "ring" includes a unit, whether a sum is over a finite set. The PR reports it.
- **Search before writing.** Step 1 first searches for an existing declaration with type search (14.3) and embedding search. If Mathlib already states it, the binding goes to Mathlib, and the formalize run becomes an align run.

## 10.3 Definitions before statements

A statement can only be as faithful as the definitions that it uses. A faithful statement about the wrong notion of "regular" is wrong. Thus formalize works from the bottom up over the `uses` closure:

1. Each concept or definition node in the closure must have a binding at F2 or higher before a statement that uses it can go higher than F1. The fidelity queue enforces this review order (11.4 `unlock`).
2. If Mathlib has no definition, the formalize role proposes one in `Math2/`, together with **sanity lemmas**. These show that the definition agrees with the examples of the textbook and with known equivalent characterisations. The proof of the sanity lemmas is a prove job. A definition with proved sanity lemmas is much easier to review.
3. If Mathlib has several candidate definitions (bundled and unbundled, `Set` and `Finset`, different universe levels), the binding names the selected one, and the remarks of the node say why.

## 10.4 The F1 battery

F1 means that a candidate has passed each check that needs no human. A candidate is a statement: a theorem, or a definition of a proposition when the result has no proof yet (07.2). The battery report records each check. Each check has an evaluation set with a measured detection rate (17.3).

| # | Check | How | Catches |
|---|---|---|---|
| 1 | **elaborates** | the Lean session at the pinned toolchain | syntax and type errors |
| 2 | **back-translation** | a model that has not seen the informal statement translates the Lean statement into English; a judge compares the two, sentence by sentence, with a structured verdict per hypothesis and conclusion | dropped or added hypotheses, wrong quantifier order, wrong conclusion |
| 3 | **constant audit** | every constant in the elaborated statement is mapped to a node through bindings, and that node must be in the informal statement's `uses` closure | the right words with the wrong definitions (e.g. `Nat.sub` truncation, `Real.sqrt` on negatives, `0⁻¹ = 0`) |
| 4 | **hypotheses satisfiable** | a Lean search, then an SMT search, for an instance satisfying all the hypotheses, starting from example nodes in the graph | vacuous statements: contradictory hypotheses make anything provable |
| 5 | **conclusion not trivial** | automation (`simp`, `decide`, `grind`, `aesop`) attempts the statement with a short timeout; success is flagged for review, not failed | statements that are trivially true by a junk-value convention |
| 6 | **negation not provable cheaply** | automation attempts the negation | statements that are false |
| 7 | **examples and counterexamples** | the statement is instantiated on the graph's `example` and `counterexample` nodes for its concepts and checked with `decide` or `plausible`-style testing | statements that fail on the textbook's own examples, or "prove" the counterexample |
| 8 | **neighbour consistency** | if the node `specializes` or `generalizes` a bound node, a Lean session tries to derive one formal statement from the other | inconsistent formalizations across the graph |
| 9 | **critic** | an independent model, from a different family from the generator where possible, reads the informal statement, the Lean statement and checks 2–8, and lists concerns with severity | everything else; its output goes to the reviewer, it is not a verdict |
| 10 | **pinned-statement compile** | the statement alone compiles as a comparator challenge module against Mathlib and `Math2` | statements that depend on the proof file's local context |

A candidate is F1 if checks 1, 3, 4, 6, 7, 8 and 10 pass, and checks 2 and 9 raise no high-severity concern. Check 5 never blocks F1. Sometimes a trivial truth is the correct statement. But the review packet always shows a trivial truth.

Checks 4, 5 and 6 are also rule `F09`. This rule runs on each F1+ binding at each release, because a Mathlib bump can make a statement trivial or vacuous and not change its text.

**The battery in m2.** The package `battery` runs the checks. The build step `battery` runs after the replay step and writes `build/battery.json`, with one report for each Lean binding. The status step reads the reports and gives F1 to a binding when its report is F1. Each check gives one of six results: pass, fail, flag, open, skip or not-run. A blocking check passes F1 with pass, flag or skip. Checks 2 and 9 pass F1 with pass or flag. Thus an open or not-run result blocks F1. The checks work as follows:

- Check 1 uses the extraction record of the declaration.
- Check 3 maps each constant of the statement to the nodes that bind it. The constants of logic and notation, and the instances, do not need a node. A constant that no node binds gives open. A set of patterns on the `pp.all` statement finds the junk values of 10.5 (natural subtraction, division by zero, `Real.sqrt`, `Real.log` and others) and gives a flag with the class `M-junk`.
- Checks 4, 5 and 6 use the degenerate-statement checks of the extract step.
- Checks 7 and 8 ask a Lean session for a short tactic proof of `A → B` or of `A → ¬B`, with the `implies` command of M2Extract. Check 7 does not instantiate the statement yet. It compares the statement with the formal statement of each example and counterexample of its concepts. When the statement implies the negation of an example, the class is `M-strong`. When a counterexample implies the negation of the statement, the class is `M-hyp-missing`. Check 8 uses the edges `generalizes`, `specializes`, `implies` and `equivalent`.
- Check 10 uses the comparator result of the replay step.
- Checks 2 and 9 need models. The build does not run them until the model calls are approved, so no binding is F1 before then.

## 10.5 A taxonomy of misformalization

Each class has a name, a detector and an example. Reviews and audit samples get labels from these classes. Thus we can measure the error rate for each class (17.3), and we can improve the battery where it is weak.

| Class | Description | Primary detector | Example |
|---|---|---|---|
| `M-hyp-missing` | a hypothesis of the informal statement is absent | back-translation; examples | "every continuous function attains its maximum", stated without compactness |
| `M-hyp-extra` | a hypothesis is added | back-translation; critic | the statement is proved for finite groups only when the source says groups |
| `M-quant` | quantifier order or scope wrong | back-translation | ∀ε ∃δ against ∃δ ∀ε |
| `M-vacuous` | hypotheses unsatisfiable | check 4 | `n > 0 → n < 0 → P` |
| `M-junk` | a junk value makes it trivially true or false | constant audit; check 5 | division by zero, natural subtraction, `sqrt` of a negative, `sSup` of an unbounded set |
| `M-def` | a constant means something different from the informal notion | constant audit; definitions-first | Mathlib's `Nat.Prime` against a source that counts 1 as prime |
| `M-type` | wrong carrier: ℕ for ℤ, `Fin n` off by one, real for complex | back-translation; examples | a sum over `Finset.range n` meant to be 1..n |
| `M-conv` | a convention mismatch on an otherwise correct statement | critic; review | ℕ including 0; `⊆` against `⊂`; graph with or without loops |
| `M-weak` | the formal statement is a special case | neighbour consistency; review | the Erdős #124 pattern |
| `M-strong` | the formal statement is stronger and false | negation check; examples | dropping "for sufficiently large n" |
| `M-context` | the statement depends on local context, notation or instances in the proof file | check 10 | a local `instance` that changes `≤` |
| `M-tamper` | the proof environment redefines or shadows something in the statement | pinned statement plus comparator (12.2) | redefining a norm so that every element has norm zero |

## 10.6 Human review: F2 and F3

### What a reviewer signs

A fidelity review says: *I have read the informal statement with hash h₁, and the Lean statement with elaborated type hash h₂, together with the definitions of every constant it uses, and the Lean statement says what the informal statement says, up to the conventions recorded on the node.* The review records:

- the reviewer, the date, h₁ and h₂;
- a verdict: `faithful`, `faithful-with-conventions` (listed), or `not-faithful` (with a taxonomy class);
- free-text notes.

A `not-faithful` review has the same value as a positive one. It sends the binding back to formalize with the class and the notes.

### The review packet

Review-assist builds a packet for each binding in the fidelity queue. `m2 review <tag>` renders it:

- the informal statement and the Lean statement side by side, the Lean one pretty-printed both with and without notation;
- the back-translation, and the judge's per-clause verdicts;
- each constant in the statement, expanded to its definition and its bound node, with that node's fidelity level;
- the battery report, the critic's concerns, the dissenting candidate clusters and how they differ;
- for a re-review after a bump, the old and new types as a diff.

**The packet in m2.** The package `review` makes the packet from the files of the last build, and `m2 review <tag>` shows it. The packet has the informal statement and the Lean statement side by side, the Lean statement with `pp.all`, each constant with its type and the nodes that bind it with their fidelity level, the battery report, and the reviews of the binding. The back-translation, the critic and the candidate clusters need the model runs of the formalize pipeline, and the old and new types need the bump data, so the packet does not show them yet. With a verdict, `m2 review` writes the review to `reviews/XX/TAG.yaml` and adds its id to the binding. It refuses a reviewer that is not a person, the author of the binding, and a statement that changed after the build. The check of rule `V03` on the agent runs that a reviewer started needs the run records of M5.

Review-assist never signs. An agent cannot be a reviewer. A reviewer cannot review a binding that came from an agent run that the reviewer started (rule `V03`).

### Independence

F3 needs one of two things:

- **two independent reviews**: different reviewers, neither of them the binding's author, sharing no affiliation (rule `V02`);
- **one review plus an independent formalization proved equivalent**. The second formalization comes from a different run, of a different model family, given no access to the first candidate. The equivalence is proved in Lean.

The second route is cheaper, and it becomes cheaper each year. It is also the route that scales. The target for each node in the undergraduate core is F3 by cross-formalization (document 02).

### Throughput targets

For undergraduate-level statements, a review takes 3 minutes. Thus one reviewer-hour gives 20 F2 reviews. The undergraduate core (about 25,000 statements and definitions, document 18) then needs about 1,250 reviewer-hours for F2. That is a summer of work for twenty people, or a year of light work for a hundred. Research-level statements take much longer, at about 15 minutes or more. For this reason the fidelity queue is ranked by `unlock`. Also for this reason, the Annals-Challenge-style approach is the correct order: it formalizes statements before anyone tries to prove them. A reviewed statement has more value than an unreviewed proof.

## 10.7 Statements without proofs

Many of the most useful formal bindings in math2 will have no proof for a long time. These are research-level theorems, open conjectures, and results whose proofs depend on large theories that Mathlib does not have. Such a binding is `match: partial` until a proof exists. It is still worth having:

- it can be reviewed to F2 and F3, which makes it a trustworthy target for anyone, human or agent, who wants to prove it;
- it gives search a formal type to match against (14.3);
- for conjectures, it is the precondition for any AI claim of a solution to be checkable. A solution counts only against the pinned, reviewed statement. That is the lesson of every 2025 to 2026 episode in which an AI proved the wrong version.

The formal-conjectures repository and the Annals Challenge are the two largest collections of formal statements without proofs. math2 aligns to both (document 09). If their maintainers have reviewed an aligned statement, math2 records it as a review and names the source.

## 10.8 Measuring fidelity

Fidelity levels are claims too, and math2 measures how reliable they are:

- **Sampled re-review.** Each month, a different reviewer does a blind re-review of 1% of F2 bindings and 0.5% of F3 bindings. We adjudicate and publish the disagreements. The measured error rate per level, by area, is a headline number in the notes of each release. `m2 review sample` draws the sample with a seed and records it in `registers/samples.reg`, and `m2 review measure` gives the rate.
- **Battery evaluation.** We evaluate each check in 10.4 against a labelled set of known misformalizations. The set starts from the published benchmark audits and grows with each `not-faithful` review. We publish detection rates per class (17.3).
- **Downstream evidence.** Sometimes the proof of a statement needs a hypothesis that the informal statement does not have. Sometimes the proof is suspiciously short. In both cases, the statement goes back to the fidelity queue with a flag. A suspiciously short proof means that the measured size is much smaller than that of comparable proofs.

If F2 means "probably right", math2 should be able to say how probable.
</document>

<document source="https://tamnd.github.io/math2/spec/11-api-and-orchestrator.md">
# 11. The Mathematical API and the agent orchestrator

In the diagram, one box is between the Research Workspace and the Research Agents: "Mathematical API and Agent Orchestrator". This box has two roles in one process:

- the **Mathematical API** is the only way for anything outside the build to read the knowledge graph or to ask for work. This includes a person's browser, an agent and an external tool;
- the **Orchestrator** changes requests for work into agent runs. It schedules the runs, enforces budgets and records everything. It changes the results into pull requests.

Both are in `tamnd/math2-agent` (`api`, `mcp`, `orchestrator`). `m2 serve` and `m2 orchestrate` serve them.

## 11.1 Principles

1. **Read is free, write is a pull request.** The API serves reads from the derived store of a release, or of `main`. Every write becomes a branch and a pull request against `tamnd/math2`. This is true for a write from a person in the workspace and for a write from an agent. The API has no other write path. Thus the audit (document 15) is the only gate, and git is the only history.
2. **One surface, three encodings.** The API uses HTTP+JSON for browsers and scripts, gRPC for the agents' tool calls inside the cluster, and MCP for any model client. One interface definition generates all three: `api/m2.proto` with HTTP annotations. Thus the three encodings cannot drift apart.
3. **Every response says what it is about.** It carries the release or commit that served it, the tag's content hash and the evidence summary. An agent that reasons from an API response can cite exactly what it saw.
4. **The API never returns a status it did not compute.** It serves the status that the build computed (document 04.7). It does not pass through any statement about truth from an external system.

## 11.2 The read surface

The methods are in groups by purpose. Each method is also an `m2` subcommand and an MCP tool with the same name.

### Resolve and fetch

| Method | Input | Output |
|---|---|---|
| `resolve` | any identifier: tag, alias, `m2://` URI, Lean name, QID, A-number, Stacks tag, arXiv result tag, DOI | tag(s) with match type, or "not held" plus the external system |
| `get` | tag[@release] | node: front matter, body, computed status, evidence summary |
| `get_formal` | tag | bindings, with the elaborated statement pretty-printed, axioms, replay result |
| `get_source` | tag | sources with locators and aligned paper results |
| `history` | tag | revisions with provenance, binding changes, review history |

### Navigate

| Method | Input | Output |
|---|---|---|
| `neighbours` | tag, edge types, depth ≤ 3 | typed subgraph |
| `closure` | tag, strategy (min-depth / min-size), stop set | AND–OR closure with trust profile (05.5) |
| `rdeps` | tag | everything whose every proof depends on it |
| `path` | from, to, edge filter | prerequisite path (05.6 Q5) |
| `frontier` | topic, kind, evidence filter | ranked frontier lists (Q6, Q7) |
| `topic` | MSC code or topic tag | the topic page data: members by kind and level, coverage |
| `diff` | topic, release₁, release₂ | changes (Q10) |

### Search

| Method | Input | Output |
|---|---|---|
| `search` | free text (natural language and/or LaTeX) | ranked nodes with snippets, scores and channel attribution (document 14) |
| `search_formal` | a Lean type pattern, or a Lean goal | bound nodes and Mathlib declarations, Loogle-style (14.3) |
| `search_formula` | LaTeX formula | formula-structure matches (14.2) |
| `similar` | tag | nearest nodes in embedding space, with the judge label when one is cached |
| `premises` | Lean goal state + file context | ranked candidate lemmas for a proof step (14.5) |

### Check and compute

These methods do work, but they do not write content. They return their results. The results are also cached by hash.

| Method | Input | Output |
|---|---|---|
| `lean_check` | Lean source snippet + imports at the pinned toolchain | elaboration result, messages, goals, axioms |
| `lean_goal` | snippet + position | goal state (used by provers via REPL sessions, 11.3) |
| `fidelity_check` | tag + candidate Lean statement | the F1 battery: back-translation, degenerate checks, critics (document 10.4) |
| `compute` | computation spec (image, entry, inputs) | output, hash, certificate check result (document 08) |
| `counterexample` | Lean statement or informal claim + domain hints | SMT/SAT/brute-force search result |

### Propose (writes, via pull request)

| Method | Effect |
|---|---|
| `propose_node` | open a PR with a new node (provisional id) |
| `propose_edit` | open a PR editing allowed fields of a node |
| `propose_binding` | open a PR adding a formal binding plus its Lean code |
| `propose_alignment` | open a PR adding `same_as`/`close_to`/`aligned` edges |
| `request_review` | file a review task for a binding (document 10.6) |
| `request_work` | file an orchestrator job (11.4) |

People use the same propose methods from the workspace. The PR that results is attributed to the person, and the workspace is named as the tool.

## 11.3 Lean sessions

A proof is interactive work: try a tactic, look at the new goals, then try a different tactic. One HTTP call for each attempt with a full re-elaboration is much too slow. Thus the API gives **Lean sessions**:

- A session is a Lean REPL process (the community `repl`, or Pantograph-style goal-level access). It starts from the pinned toolchain with Mathlib and `Math2` imported. The process is snapshotted after the imports. Thus a session costs milliseconds to start, not the 10 to 30 s that it takes to load Mathlib.
- A session holds a proof state tree. The calls are `run_tactic(state_id, tactic) → new states | error`, `goal(state_id)`, `branch(state_id)` and `close()`.
- Sessions run in the sandbox (document 08.6), with a memory cap and a timeout for each call. When a tactic times out, only its own call stops.
- Each runner has a session pool of a set size. The orchestrator charges session-seconds to the budget of the run.

The `lean-lsp-mcp` and Kimina-server style tools show the same interface (document 01). math2 uses its own interface because it must pin it to the release toolchain, sandbox it and meter it. Because of the MCP surface, any agent that can use MCP can prove things against math2 directly. This includes Claude Code with no math2-specific code.

## 11.4 The orchestrator: jobs

A **job** is a unit of work with a goal, a target and a budget. Jobs come from four places:

1. **People**, through the workspace or `request_work` ("formalise the statement of 7K3QX2", "find a counterexample to this").
2. **Standing queues** that the orchestrator computes from the graph: the frontier lists (Q6, Q7, Q12). Examples: each E3 statement with an F2 formal statement and no formal proof becomes a `prove` job. Each node with no explanation at L2 becomes an `explain` job. Each unconfirmed alignment older than 30 days becomes a `align` job.
3. **Events**: a Mathlib bump breaks bindings (→ `bump` jobs), a review becomes stale (→ `review-assist` jobs), a new arXiv batch arrives (→ `extract` and `align` jobs).
4. **Campaigns**: a project that a person defines, with a target set. Examples are "the undergraduate core list to F2" and "the Erdős problems in additive combinatorics to formal statements". A campaign is a topic node plus a goal predicate. Human research projects use the agents through campaigns (document 16.3).

Each job has this record:

```yaml
job: j-2026-11-02-41a7
role: prove
target: 7K3QX2
goal: "binding formal[0] reaches E5"
priority: 0.73            # computed (11.5)
budget: {usd: 4.00, wall: 2h, lean_session_s: 3600, compute_cpu_s: 7200}
on_behalf_of: campaign:ug-core   # or a person, or schedule:standing-prove
requested_by: orcid:0000-0002-1825-0097
constraints: {models: [claude-opus-5-5, claude-sonnet-5-5]}   # native_decide: true permits native_decide, and type_change: true permits a bump rebind to another type hash
```

### Priority

Priority is a score, not a decision by a person. Its formula has versions, as a prompt has:

```
priority = w1 · centrality(target)          # PageRank on the uses graph, so foundational nodes first
         + w2 · demand(target)              # workspace views, citations, campaign membership
         + w3 · unlock(target)              # how many frontier items this would move a level
         + w4 · tractability(target, role)  # learned: predicted success from past runs on similar targets
         − w5 · attempts(target, role)      # diminishing returns after failed attempts
```

`unlock` is the most important term. For example, 200 E1 statements use a lemma, and each one needs only that lemma. A proof of that lemma has more value than a proof of any one of the 200 statements. The closure structure in document 05.5 lets the orchestrator compute this.

### Scheduling

- Jobs go into one queue for each role. The queues share fairly across campaigns, so that one project cannot use all of the budget.
- **Portfolio scheduling.** A hard target gets several cheap attempts with different strategies before one expensive attempt. The strategies use different models, different retrieval and different decompositions. This is the method that worked in the provers of 2025 to 2026 (document 01): many different samples, with a verifier as the filter, are better than one long attempt.
- **Escalation ladder.** A job starts at the cheapest configuration that has some recorded success on similar targets. If the job fails, it moves up the ladder: Haiku 5.5 or a small open prover, then Sonnet 5.5, then Opus 5.5 at higher effort, then Fable 5.1 at high or max effort. Each role has its own ladder. The run records tune the ladder (document 12.9).
- **Deduplication.** If the target and goal of a job match a running job, the orchestrator merges the new job into the running job. Sometimes a job's target had an attempt in the last N days with the same configuration, and the attempt failed. The orchestrator skips that job until something changes in the neighbourhood of the target.

## 11.5 Budgets

Money and compute are the real limits, so they are first-class:

- **Global:** a monthly budget, divided by role in `orchestrator.yaml`. People review this file each month and commit it.
- **Per campaign:** the owner of the campaign sets it. The global allocation is its cap.
- **Per job:** the priority tier sets it, and the orchestrator enforces it. Model calls carry token budgets. These are the API's task budgets where the model supports them, and hard `max_tokens` in all cases. The sandbox meters Lean time and compute time.
- **Review capacity:** money is not the only limit. Each role has a cap on PRs per week. The cap is a multiple of the review throughput for that kind of PR over the previous four weeks. Thus agents cannot go faster than the people who must check their work (document 21, Q-11).
- **Stopping:** a job stops at success, when the budget is used up, or when the agent reports that it is stuck and gives a reason. A stuck report with a reason is valuable output. "This statement appears false at n=7, here is the computation" is a result.

Each job records its spend: tokens in and out for each model, cached tokens, session-seconds, CPU-seconds and dollars. The weekly agent report (document 12.9) uses these records. The report shows what a formal proof costs, by area and by difficulty. At this time, the field does not publish this number.

### The first version

The first version of the orchestrator is in the `orchestrator` package, and `m2 orchestrate` runs it on a directory of job files (tamnd/math2-agent#61). It has these parts and limits:

- One queue for each role. The orchestrator takes the roles in turn. In a role, it takes first the project or schedule that spent the least in the month, then the job with the highest priority. The priority is a field of the job, and the formula of the priority is not written yet.
- The monthly budget of each role and the monthly cap of each project or schedule are in `orchestrator.yaml`. A run gets the least of the money that is left to its job, to its role and to its project or schedule, and the orchestrator reserves that money while the run goes. The job budget is for all the runs of the job together. `agent.Runner` sets `max_tokens` before each turn so that the turn cannot cost more than the money that is left: it counts each byte of the request as one input token at the highest input price. Thus a run never goes over its budget, even when its last turn uses each token that it may use.
- A job starts on the cheapest rung of the ladder of its role with a recorded success on the role, and each failed run moves the job one rung up. A failed attempt on the same target, goal and rung in the retry time (7 days by default) makes the orchestrator skip that rung. A job with the target and the goal of a job in the queues merges into that job.
- A job ends at success, at a stuck report, when the ladder or the budget of the job is used up, or when a run cannot start. The orchestrator knows the past attempts from the run records in the store.
- Add refuses a job that names no person, project or schedule (rule P03).
- With `-sandbox IMAGE`, each call of lake, Lean and M2Extract in a run is a container of the image, with no network (11.7). The work tree is writable at `/out`, and the Lake packages of the tree are read-only at `/work`. `m2 agent run` and `m2 orchestrate` refuse to run with no image, unless `-no-sandbox` says that a person watches the run. The image `m2-lean` is not built yet, and the quotas for each identity are not written yet.
- The standing queues, the events, the portfolio of strategies and the cap on pull requests of each role are not written yet.

## 11.6 Run records

Each agent run makes a run record in the artefact store. The run id is the key. The record contains:

- the job record and the resolved configuration: role version, prompt version, model ids, tool versions, image digests;
- the full transcript: every model request and response (thinking blocks as the API returned them), every tool call and result;
- the tactic tree of every Lean session;
- the outcome: success with the diff, failure with the reason, or a partial result;
- the spend.

Run records complete the provenance chain in document 06.5. They are also the training and evaluation data that improve the agents (document 17.4). The system keeps them as document 06.6 specifies. Their summaries are public. When the output of a run was merged, the full transcript is public. When the output was not merged, the system keeps the full transcript but does not publish it. By default, math2 does not publish a failed attempt, but it uses the attempt to learn.

## 11.7 Security and abuse

- Read endpoints are public and have rate limits. Write endpoints need a GitHub identity, and the PR opens under that identity.
- `lean_check`, `compute` and Lean sessions run arbitrary code: Lean metaprograms can do arbitrary IO at elaboration time. Thus they run in the sandbox, with no network, a read-only toolchain and quotas for each identity. Anonymous use is permitted only for `lean_check` on small snippets, with a strict timeout.
- MCP tools that write have a mark that says so. The read-only MCP profile disables them. That profile is the default for third-party agent clients.
- Text from sources (paper text, MathOverflow posts) is data. It goes into agent contexts with an untrusted mark. It is never put in a system prompt, and it is never concatenated into a tool description (document 12.7).

## 11.8 The first version

The first version of the read surface is in the `api` package, and `m2 serve` serves it. It has these limits:

- It serves the last build of one working tree, not a release. A request for `tag@release` gives an error.
- `get_source`, `history`, `topic` and `diff` come with the provenance of document 06 and the topic pages of document 13. The check and compute methods, `request_review` and `request_work` come with the orchestrator.
- The one interface definition is the table of methods in the `api` package. The HTTP handler and the MCP server read the table. `api.Proto` makes `api/m2.proto` from it, with the HTTP annotations, and `buf` makes the Go code in `api/m2pb` from that file (tamnd/math2-agent#126). A test checks that `api/m2.proto` is the file that the table makes, a test checks that the code in `api/m2pb` agrees with the table, and a test checks that each gRPC call of the contract gives the HTTP answer. Thus the three encodings cannot drift apart. The gRPC service is `m2.v1.ReadService`, and each rpc gives one `Response` message with the result as a `google.protobuf.Value`, as each HTTP call gives the same JSON envelope. `m2 serve -grpc ADDR` serves it with the same rate limit for each IP address.
- The rate limit is a token bucket for each IP address of a peer, in memory. Behind a proxy or a CDN, the handler must read the address that the proxy sets.
- The MCP server is in the `mcp` package. It has the read-only profile only, with one tool for each read method, and `m2 serve` serves it over streamable HTTP at `/mcp` and over stdio. It uses the Go SDK of the Model Context Protocol.
- `similar` uses the embedding channel with the text of the node, and it has no judge label yet.
- The write methods `propose_node`, `propose_edit`, `propose_binding` and `propose_alignment` are in the `propose` package, and `m2 serve -propose` serves them over HTTP only (tamnd/math2-agent#54). The gRPC service and the MCP server do not have them yet, thus the MCP server keeps the read-only profile of 11.7. Each write needs the GitHub token of a person, opens the pull request under that identity, and passes the audit of the changed files before the pull request opens. The cap of 11.5 counts the pull requests of each method from their branch names, with twice the closed pull requests of a week over four weeks and a floor of 10. `propose_binding` adds the binding to the node file, and it does not add Lean code yet. `request_review` and `request_work` come with the orchestrator.
</document>

<document source="https://tamnd.github.io/math2/spec/12-research-agents.md">
# 12. Research agents

The top box of the diagram says "Formalize, Prove, Explore, Explain". Readers see these four agent roles. Four more roles work behind them and keep the graph healthy: curate, align, bump and review-assist. All eight roles are in `tamnd/math2-agent/agent/`. They run under the orchestrator (document 11). They change content only through pull requests.

The design comes from one finding of 2026. Document 01 gives this finding with its sources. **A minimal agent loop on a strong general model, with a strict verifier, matches or beats bespoke provers.** A plain agent on a general model saturated PutnamBench for about $111. The baseline agent of AlphaProof Nexus replicated the Erdős results of AlphaProof Nexus. The Fermat formalization ran on a generic DAG-of-statements harness. Thus math2 does not build a prover. It builds the things that a prover needs and cannot supply itself:

- a pinned, sandboxed, metered Lean environment;
- retrieval over the whole graph;
- a dependency structure in which a lemma is a first-class node;
- a verifier that nobody can talk past;
- a record of everything that was tried.

Then math2 uses the model that does best on the evaluations in document 17.

## 12.1 Roles and what they may touch

Each role has a bot account, a tool profile and a **field permission set**. Rule `X01` checks each agent PR against that set. Any change outside the set fails the audit, even if the change is correct.

| Role | Does | May create | May edit | Auto-merge |
|---|---|---|---|---|
| **formalize** | writes Lean statements for nodes, runs the F1 battery (10.4) | bindings at F0; `Math2/` statement declarations; `Conjectures/` entries | its own unmerged bindings | never |
| **prove** | finds formal proofs of bound statements | proof nodes; lemma nodes (statement plus binding); `Math2/` proofs | `formal[].decl` of nodes it proves | yes, if the target binding is already F2 or above and the new lemmas are internal to the proof (10.6) |
| **explore** | experiments, conjectures, counterexample searches | computation nodes; conjecture and counterexample nodes, which are always E1 or E4 by construction | none | never |
| **explain** | writes levelled explanations, examples and glosses | explanation sections marked `by: agent`; example nodes | its own unconfirmed explanations | never |
| **curate** | turns extracted paper results into nodes, proposes dedup merges, fixes graph health findings | statement and concept nodes from record plane results; `same` proposals | `uses`, `about`, `msc`, names and aliases | never |
| **align** | proposes crosswalk edges | `same_as`, `close_to`, `aligned` edges | its own unconfirmed alignments | `same_as` to Mathlib, only when the type hash of the bound declaration is identical |
| **bump** | moves to new Lean and Mathlib versions | the bump branch; rebindings | `formal[].decl`, `formal[].commit` | renames taken from Mathlib deprecation data |
| **review-assist** | prepares review packets for people | review packets in `reports/review/` (not content) | nothing in `nodes/` or `reviews/` | not applicable |

No role may write a status, a review, a register entry or a retirement (rules `S03`, `X04`, `V03`). An agent can give the reasons to retire a node, but only a curator can retire it.

A run writes only in its own work tree, through the `workspace` package. Its tools read and write files by their paths in the tree, and they cannot reach a file outside it. `agent.Permit` checks the changes of the run against the set of its role before the run opens a pull request. The sets of the eight roles are written. Rule `X01` uses the same check on the merge of each agent PR, thus a PR of a role with no written set fails.

The first version of curate is in `agent/curate`, with the role in `prompts/curate/`. Its target is a node, or a record of the record plane (09.3) in the form `source:id`, for example `arxiv_results:2312.00752#3KH2`. For a record of kind statement or definition, the run adds one statement or concept node in its own words. The model reads the record only with the tool `record`, which gives it in a data block (12.7). The goal holds when `m2 check` finds nothing in the changed nodes, the new node names the locator of the record in its `sources`, its kind matches the record, its body does not copy the text of the record by the thresholds of rule `L04`, and a judge finds that the node states the same result. The judge is one call of the model of the rung with the record and the node as data and a tool `judge`, which gives same, generalizes, specializes, related or different with the reason. Only same passes, and the other verdicts go back to the model. For a node target, the run can change the fields `uses`, `about`, `msc`, `name` and `aliases` of any node, and it can add a duplicate proposal to `queues/dedup.jsonl` with the tool `propose_same`. The set of the role allows no other change of a node that exists, no Lean, and no new node with `formal`, a crosswalk edge, a retirement or an explanation. The set keeps each proposal of the queue unchanged, and each new proposal must be by the role. Curate does not merge nodes. A person merges them from the queue.

The first version of align is in `agent/align`, with the role in `prompts/align/`. Its target is a node. The model reads the records of the other systems with the tool `record` and proposes edges with the tool `propose_edge`, which writes each proposal into the queue file of its system in `queues/align/`, as the aligner of 05.3 does. A person accepts or rejects each proposal, and the run writes no edge into a node. The tool refuses an edge that the node has, an edge that a person decided on, an `aligned` edge to a record that is not a paper, and an id that does not have the syntax of its system. A new proposal to a record replaces the proposal of the role to the same record. The set of the role keeps each proposal of another program unchanged, and the role adds, changes and removes only its own proposals from the target. The goal holds when the run made at least one proposal or binding, and the judge of curate finds the verdict same for each `same_as` proposal and a verdict that is not different for each `close_to` and `aligned` proposal. In the graph, a node meets a Mathlib declaration through `formalizes` and not through `same_as`. Thus the `same_as` to Mathlib of the table is a binding to Mathlib that align appends to the `formal` field of the target. The set allows such a binding with no fidelity reviews, and the goal allows it only when the type hash of the declaration in the Mathlib records (09.2) is the type hash of a Math2 binding of the node in the build, with the same match. `align.AutoMerge` reports a run whose changes are only such bindings. It is the only change of align that can merge with no review.

The first version of bump is in `agent/bump`, with the role in `prompts/bump/`. It comes after `m2 bump` (07.9), which applies each rename that the deprecation data or the type hashes give. The target of a bump job is a node with a Lean binding that does not resolve or is deprecated with the new pins, for example a lost declaration of the bump report. The model reads the declarations with `search_formal` and `lean_check`, and it changes `formal[].decl` of the binding with the tool `rebind`. A binding has no commit field in schema node.v1, because the commit is the pin in `lean/lake-manifest.json`. Thus the `formal[].commit` of the table is the pin, and the role changes only the decl. The set of the role allows a change of `formal[].decl` of the target and nothing else, and the role writes no Lean. Thus a break of the build of the Math2 library is not a job of bump. The goal holds when the run changed at least one binding, each changed binding did not resolve or was deprecated before the run, the new declaration has the kind and the type hash of an old declaration that resolves, and each Lean binding of the target resolves and is not deprecated. A rebind to another type hash changes the statement, thus it needs the constraint `type_change` of the job, or the flag `-type-change` of `m2 agent run` (17.3.4). The Lean environment of the run imports Mathlib and Math2, because a binding can name any declaration of Mathlib. `bump.AutoMerge` reports a run whose rebinds all take the replacement that the deprecation of the old declaration names and keep its type hash. Such a run can merge with no review, as the renames of 07.9 step 3. A person reviews each other rebind. No role writes a register, thus the person who merges a rebind records it in `registers/bindings.reg`.

The first version of review-assist is in `agent/reviewassist`, with the role in `prompts/review-assist/`. Its target is a node with a Lean binding, for example a node of the review queue. The brief has the review packet of the node from the last build, as `m2 review -blind` gives it: the informal statement, each binding with its Lean statement, the constants with the nodes that bind them, and the results of the battery. The model is the critic of 10.4. It reads each Lean statement against the informal statement, and it writes the packet with its concerns with the tool `write_packet`. Each concern names a binding of the packet, a class of 10.5 and a reason, and a packet has at most 10 concerns. The tool writes `reports/review/TAG.md`, which has the packet text of the build and the concerns, and it is the only file that the set of the role allows. The packet has no reviews, thus a blind re-review (10.8) can use it. The goal holds when the file has the packet of the build and each concern is valid. Review-assist never gives a verdict and never signs, and a person reviews with `m2 review`.

## 12.2 The common loop

All roles run the same loop. Only the tools, the prompts and the stop condition are different.

```
job ← orchestrator
context ← API: the target node, its neighbourhood (depth ≤ 2), its sources, its formal status,
           prior runs on this target (summaries), the role's prompt at its pinned version
loop:
    model turn (tools available per role profile)
    tool calls → API / Lean session / compute sandbox      # every result cached by hash
    check budget; check for stuck (no progress in k turns)
until: goal predicate holds | budget exhausted | agent reports stuck with reason
diff ← workspace changes
validate diff: m2 check, role field permissions, local audit on touched shards
open PR (agent/<role>/<run-id>) with run summary | or record failure with reason
```

These decisions are intentional:

- **Tools, not bash.** The agent sees typed tools: `get`, `search`, `premises`, `lean_*`, `compute`, `propose_*`. It does not get a shell. In 2026, agents used a shell to find and exploit kernel and harness bugs (document 01). The tool surface is the MCP surface of document 11. Thus an external MCP client with the same permissions can do all that a math2 agent can do.
  The package `agent/tools` has the first of these tools. `tools.API` makes a tool from each read method, with the names and the parameters of the MCP server. `read_file` and `write_file` work in the work tree of the run, and a write that the role may not make fails and leaves the file as it was. `lean_check` elaborates a Lean file of the tree, or a scratch file, and its time counts in the Lean session seconds of the budget.
- **One target, one closure.** The context comes from the graph neighbourhood of the target, not from a general search over everything. Thus runs are reproducible and cheap to cache.
- **The verifier's verdict is the only verdict.** When an agent claims success, it must produce a diff that passes the audit. The system records the agent's own assessment and never uses it.
- **Failures with reasons are outputs.** "Statement false at n=7, witness attached" closes a prove job and attaches a counterexample job. "Needs a lemma that is not in Mathlib: X" creates a node for X.

The `agent` package runs this loop. A turn makes progress when it calls a tool with an input that the run did not call before, and the run caches the result of each call by the tool and the input. The tool `report_stuck` is in each profile, and the model calls it with the reason. The goal predicate is a function of the role that reads the tree and the build, never the text of the model. Each model turn has a limit of output tokens from the role and from the budget that is left after the most that the input of the turn can cost. When the goal does not hold, the predicate also gives the reason in one sentence, and the next message to the model starts with this reason. `agent.OnChanges` makes a goal that runs its check only when the changes of the work tree are different from the changes of the last check, because a check can build the library. `tools.Profile` makes the tools of a role from the tool names in its `role.yaml`, and a name with no tool is an error.

### Statement pinning

In 2026, the worst failure mode was a correct proof of a different statement. The causes were edited context, redefined notions, a weakened hypothesis or an exploited checker. Thus each role that writes Lean works against a **pinned statement**:

- A statement with no proof is a definition of a proposition, `def S : Prop`, because rule `F06` refuses `sorry` (07.2). When the job starts, the pin records `S` and the hash of each declaration of math2 in its closure. For a definition, this is the hash of the value. For a theorem, it is the hash of the type.
- The proof is accepted only if it is a theorem of type `S`, its axioms are a subset of the allowed set (rule `F03`), and each hash of the closure is unchanged.
- The agent's Lean files must not modify, shadow or re-declare anything in the closure of the target. The check runs on the elaborated environment, not on the source text.
- Before merge, a Comparator-style check confirms that the statement is identical to the pinned one. This check builds against Mathlib and `Math2` only. Then the external checkers replay the proof (07.7).

## 12.3 Models

The model is a configuration value, never a dependency in code. Each run records the exact model ids that it called. The `llm` package has one interface. Its first client is the Anthropic client. It has adapters for local open-weight prover models served behind gRPC (document 20.2). The client of the Claude models is in `llm/claude`, on the official Go SDK. It streams each request, caches the system prompt, the tools and the prefix of the conversation, and keeps the thinking blocks as the service returned them. `llm/llmtest` has a scripted model, thus the tests of the roles call no model service. `llm.Cost` gives the price of a usage from a table of prices, and `llm.Prices` is the table above.

| Tier | Model | Price in/out per MTok | Used for |
|---|---|---|---|
| 0 | `claude-haiku-5-5` | $0.10 / $0.50 | triage, classification, dedup judge pre-filter, first cheap proof attempts, the F1 critics' easy checks |
| 0′ | open prover models (e.g. Leanstral, DeepSeek-Prover, Goedel-Prover families) | self-hosted GPU | high-sample-count tactic and whole-proof attempts in the portfolio |
| 1 | `claude-sonnet-5-5` | $2 / $10 | bulk: curate, align, explain L1–L2, routine formalization, most prove attempts |
| 2 | `claude-opus-5-5` | $4 / $20 | default for hard roles: research-level formalization, decomposition planning, explain L3–L4, explore |
| 3 | `claude-fable-5-1` | $10 / $50 | the top of the escalation ladder: hardest proofs, cross-checking research-level statements, explore campaigns on open problems |

How math2 uses the API:

- **Adaptive thinking** on each role that does much reasoning. The **effort** setting increases up the escalation ladder: medium by default, then high, then max at the top. math2 never sets a fixed thinking-token budget. Current models reject those parameters. Thus the token budget of the run is in the job (11.5). The API's task budgets enforce it where they are supported, and `max_tokens` enforces it in all cases.
- **Prompt caching** for the parts of the context that all runs of a role share: the system prompt, the tool definitions, the role's style guide and the Mathlib conventions. There are thousands of runs each day, so this is the largest single cost saving.
- **Message Batches** at half price for work that no person waits for: curate, align, the dedup judge, explanation backfill and back-translation in the F1 battery.
- **Compaction** for long prove and explore runs. Thus a run can go past one context window and keep its record. The run record keeps the full transcript without compaction.
- **The orchestrator never forces a tool choice.** The current models reject a forced `tool_choice`. The roles are designed so that the model selects its tools.
- **Managed Agents** is an option for the roles that need only the read API and MCP (explain, curate, align). These roles run as hosted sessions connected to the MCP server of math2. Prove and formalize run in the loop of math2, because math2 must pin, sandbox and meter their Lean sessions (11.3).

The table is a starting point. The run records tune the **escalation ladder** of each role (11.4) each week. A role moves to a cheaper default when the acceptance rate of the cheaper tier on recent comparable targets is within a set margin of the more expensive tier. The report (12.9) publishes the acceptance and cost for each tier that justify each move.

Open-weight models are important for two reasons other than price. First, they let the portfolio draw thousands of different samples at marginal cost. Second, they keep the prove pipeline of math2 in operation if hosted access changes. In 2026, the gap between open and frontier models in formal proof became small. In research-level informal proof, the gap stayed large (document 01). math2 uses open models where they work and does not pretend that they work in other places.

## 12.4 Formalize and prove

Document 10 gives the full specification of the formalize role, because for that role the fidelity battery is more important than the agent. This section is about prove.

A **prove** job has a target node with a binding at F1 or above. Before F2, a proof is still useful because it tests the statement. But a proof of an unreviewed statement never auto-merges. The loop:

1. **Read the informal proofs.** If the node has a `proof` node, the agent works from it. If the node has only sources, the agent reads the aligned paper results (only text that the licence permits, 09.3). First, the model writes an informal proof sketch. The run record keeps this sketch.
2. **Decompose into the graph.** The agent writes the lemmas of the sketch as provisional nodes, each with a Lean statement. The largest formalizations of 2026 used this DAG-of-statements pattern (Prove2Me for Fermat, Gauss for sphere packing, the textbook project). There is one difference: the DAG of math2 is the knowledge graph itself. If a lemma is general, it becomes a reusable node. Dedup (05.4) finds lemmas that the graph already has. Before the agent proves a lemma, it searches for it with `search_formal` and `premises` over Mathlib and the graph.
3. **Cross-check the lemma statements.** A second, independent model call checks the statement of each lemma. The check asks three questions: is the lemma true, is it what the sketch needs, and is it degenerate? It uses the same degenerate checks as F1 (10.4). In the Fermat run, this check found false statements early. When a false lemma is found late, all of its subtree is wasted work.
4. **Prove leaves by portfolio.** Each leaf gets attempts with several strategies in parallel:
   - automation first: `simp`, `omega`, `decide`, `aesop`, `grind`, and the Lean hammer;
   - then sampled whole proofs from a cheap tier;
   - then tactic-level search in a Lean session with retrieval;
   - then an expensive tier.
   The first success wins.
5. **Recurse on failure.** If a leaf resists, the agent decomposes it more or sends it back up with a reason. If the reason is "this lemma looks false", the counterexample tools of the explore role try to refute it.
6. **Assemble and check.** The proof must build against the pinned toolchain with no `sorry`. Its axioms must be in the allowed set, and the type hash of the target must not change. The external replay and the comparator check must pass. Only then does the PR open.
7. **Clean up.** The system records the proof size and the build time. A proof is marked `needs-golf` if it is more than 10× the size of comparable Mathlib proofs, or if it takes longer than a set time to elaborate. It gets a follow-up job to make it shorter. The main complaint about the landmark formalizations of 2026 was AI-written proofs that work but are not maintainable. math2 must maintain its proofs through each Mathlib bump.

The packages `agent/prove` and `agent/formalize` have the goal and the first message of each role. These are the first parts of the loop above. A prove run writes the theorem `T : S`, moves the attribute `@[m2 "TAG"]` from `S` to `T`, and changes the `decl` of the binding to `T`. Its goal holds when the changes are in the field permissions of prove, the library builds, and the pin accepts `T`. A formalize run adds Lean files and appends bindings. Its goal holds when the changes are in the field permissions of formalize, the library builds, each new declaration has the right kind for the node, it uses only the standard axioms, and the degenerate checks of 10.4 do not find a problem. The decomposition, the cross-check and the portfolio of steps 2 to 4 are not written yet.

**What prove does not do.** It does not prove anything with `native_decide` unless the job permits it (rule `F04` records it in all cases). It does not add axioms. It does not edit the statement of the target. If the statement is wrong, it says so and stops.

## 12.5 Explore

Explore is the role that is nearest to research, and it needs the most restraint. It does experiments, finds patterns, searches for constructions and counterexamples, and proposes conjectures. Its tools are the computational engine (document 08), the SMT and SAT search tools, and evolutionary program search for constructions, in the AlphaEvolve and PatternBoost style (document 01). In this search, the model writes programs that generate candidates, a scorer ranks them, and the best programs are mutated.

Explore obeys four rules. Each rule comes from a failure in 2025 to 2026:

1. **Falsify before filing.** When the agent finds a pattern up to n, it checks the pattern to a larger n. It also checks the pattern against OEIS, LMFDB and the graph. Only then does the pattern become a conjecture node. Most patterns fail here, and that is the purpose of the rule.
2. **Look up before claiming.** No explore output gets the label new until the novelty check (17.5) has run. The check searches the record plane, zbMATH Open, arXiv full text where the licence permits, and MathOverflow. The search and its result are attached to the node. Without this step, an AI result is a rediscovery until someone shows that it is not. The Erdős episode of 2025 and the Aletheia sweep showed that this is the usual case. In the Aletheia sweep, 9 of 13 addressed problems already had solutions in the literature, and only 6.5% of graded candidates were meaningfully correct.
3. **Verifiable or E1.** If a construction or counterexample can be checked (a witness, a certificate), the agent files it with its check, and it can get to E4 or E5. An informal argument stays at E1, however convincing it is. The workspace shows it as a claim.
4. **People decide what is interesting.** Explore runs inside a project or campaign that a person defined. It does not select open problems for itself. This is a policy, not a technical limit. In 2026, there was a debate about mass "harvesting" of open problems by AI (document 01). math2 should not decide this debate by default.

The first version of explore is in `agent/explore`, with the role in `prompts/explore/`. It uses the computational engine. The SMT and SAT tools and the program search are not written yet. A job of explore must be on behalf of a project (rule 4). The set of the role lets it add new nodes of kind `computation`, `conjecture` and `counterexample` with no curated fields, and new files in `compute/<id>/`, where the id is the tag or the provisional id of a computation. It changes no file that is in the tree. The tool `run_computation` runs a computation of the work tree in the sandbox and gives the exit code, the output hash, the certificate hash and the output. The tool `lookup` runs the novelty check of 17.5 on a claim. The goal checks these points in order: the changes are in the set, the new nodes pass the checks of curate, the run filed a conjecture or a counterexample, each program has a computation node, no output calls itself new, each output has the evidence of a computation that is new or in the tree, and each new computation verifies when it runs again (rule 3). A counterexample cites the computation in the `via` of `refutes`, and a conjecture links to it or uses it. The checks of rule 1 against a larger n are the work of the model, and the goal sees only the computations that it filed. Then the goal runs the novelty check on each output (rule 2), with the judge of the rung, and the run record keeps the result in `novelty`. A search that fails makes the goal fail, because then the check is not complete. The levels E1 and E4 come from the build, and the run never writes them.

## 12.6 Explain

Explain writes the explanations that make the graph readable. It writes the four levels of document 04 into nodes, with examples and non-examples. It also answers questions in the "Ask about this node" panel of the workspace (13.4).

- **Grounded.** Each factual claim in an explanation must cite a node by tag. A grounding check runs after the text is generated. A second model call takes each sentence and the cited node, and gives the sentence one of three labels: supported, unsupported or contradicted. Unsupported sentences are removed or marked. Contradicted sentences fail the run.
- **Levelled.** The level is a contract about prerequisites. An L1 explanation may use only nodes in the curriculum topic for its level (`topics/curricula/`). The check is mechanical: each cited or linked node must be in the allowed set.
- **Marked.** Explanations are `by: agent` until a person confirms them (rule `P04`). To confirm is to read and approve in `m2 review`. The system records the person who confirms.
- **Not a proof.** An explanation can describe a proof. It is never a replacement for the proof node.

The first version of explain is in `agent/explain`, with the role in `prompts/explain/`. It writes the explanations of one node, and it does not answer questions yet. The set of the role lets it change only the fields `body.explanations.L1` to `body.explanations.L4` of the target, and only when each heading of the old and the new text is marked agent. It can add a node of kind `example` with no formal binding. The goal finds the levels in the goal of the job, and each one must be written. A field whose provenance log has a line of a person is not changed. Each explanation must cite at least one node, and each cited node must be in the tree. L1 maps to the courses of year 1 in `topics/curricula/`, and L2 to the courses of years 1 and 2. A level with no nodes in the curricula is not checked. The grounding check is one call of the model of the rung with the cited nodes as data and a tool `label`. `agent.Meter` gives the call the money that is left to the run, and adds its spend to the run record. A sentence with no label is unsupported. The unsupported sentences go back to the model with the reasons, and a contradicted sentence fails the run.

## 12.7 Untrusted text and prompt injection

Agents read text that math2 did not write: paper text, MathOverflow posts, Wikipedia, OEIS comments and docstrings. Any of this text can contain instructions. The defences are in the structure, not in advice:

- **Separation.** Source text goes only inside tool results, as data blocks marked with their source and licence. It is never put in a system prompt, a tool description or the instructions of a role.
- **Least privilege.** The tool profile of each role is the minimum that it needs. Explain cannot write Lean. Curate cannot run compute. No role has network access other than the math2 API and the model API.
- **Narrow outputs.** The output is a diff that must pass `X01` (field permissions) and the audit. An injected instruction can make an agent edit something outside its role. The result is a failed run, not a merged change.
- **Canaries.** The evaluation suite (document 17) has source documents with planted instructions. If a prompt or model version obeys them, it fails its release gate.
- **Quarantine.** If someone finds an injection in the wild, the run records identify each run that read the poisoned source. One command quarantines all of these runs (06.5).

## 12.8 Versioning roles and prompts

The behaviour of a role depends on its prompts, its tool profile, its model configuration and its code. These four have one version together, the **role version** (semver), in `prompts/<role>/`:

```
prompts/prove/
  role.yaml          version: 3.4.0, tools, model ladder, effort per tier, budgets
  system.md          the system prompt
  style/             Mathlib style conventions given to the model
  examples/          worked examples (each a real merged run, referenced by run id)
  CHANGELOG.md
```

- The package `prompts` embeds the role directories, and `agent.LoadRole` reads one. The first versions are formalize 0.1.0 and prove 0.1.0. They have no examples yet, because there is no merged run yet.
- Each run records the role version and the hashes of each prompt file that it loaded.
- A change to any file bumps the version. `prompts/roles.sum` has the version and the digest of the files of each role, and the tests of the `prompts` package fail when the files change and the version does not go up. A minor bump must pass the evaluation suite of the role (17.3), with results no worse than the previous version. A major bump also needs a shadow week. In that week, the new version runs with the old one on the same jobs, and the system uses only the output of the old one.
- In the first version, `shadow` in `orchestrator.yaml` names the directory of the candidate of a role. After each run of a job of the role, the orchestrator runs the candidate on the same job, rung and budget, in a work tree that it always removes. The record of the shadow run is marked, and it has no summary in `runs/`, thus no audit rule, disclosure, ladder or job budget counts it. Its spend counts in the monthly budget of the role. The weekly report (12.9) compares the candidate with the current version on the same jobs.
- Provenance can be queried by role version. Thus "everything prove 3.2.x produced" is one command, and quarantine can use it as a target (06.5).

## 12.9 The weekly agent report

Each Monday, the orchestrator writes `reports/agents/<yyyy>-W<ww>.md` and its JSON twin. The report is public. People use it to decide if the agents are worth their budget:

- **Volume:** jobs started, finished, succeeded, failed and stuck, per role.
- **Acceptance:** PRs opened, merged, closed or changed before merge, per role and model tier. "Changed before merge" counts the fixes that a person made. This count tells more about quality than the merge count does.
- **Cost:** dollars per merged output per role, by area (MSC) and by tier, and the trend. One example is the cost of an E5 proof of a lemma at a given depth in a given area. Nobody else publishes this number, and math2 can.
- **Audit sample:** the 5% sample of auto-merged PRs (16.2), with the errors found and the error rate of the role against the 1% threshold.
- **Incidents:** quarantines, injection detections, reward-hacking attempts that statement pinning caught (with the run ids), and stale reviews that bumps caused.
- **Ladder changes:** the tier and effort changes made that week to the escalation ladder of each role, and the measurements behind them.

The first version is `m2 agent report` and the package `agent/report`. It counts the runs that started in the week, from the run records in the store. A run is accepted when a provenance line on the branch names it, as for rule X03, thus a later report of the same week can count more. The report has the volume, the acceptance by role and model, the cost by role, model and MSC area with the trend from the week before, and the incidents: the quarantines, the runs in which the pinned statement caught a change to its closure, and the runs over budget. A role in its shadow week has its own table. The counts of pull requests, the audit sample, the injection detections, the stale reviews and the ladder changes need runs that open pull requests, thus the report lists them as not measured.
</document>

<document source="https://tamnd.github.io/math2/spec/13-research-workspace.md">
# 13. The research workspace: search, explore, explain, collaborate

The workspace is the bottom-left box of the diagram. People use math2 through it. It reads from the knowledge graph through the API. It writes back only through the orchestrator, as jobs and pull requests. There are three front ends, and each one is complete for its job:

- a **static site**, `m2 site`, published with each release. It is fast, people can cite it and archive it, and it runs no server code. It is for reading.
- a **live web app**, `m2 serve`, which adds search, Lean sessions, computations, collaboration and agent requests.
- a **terminal UI**, `m2 tui`, for people who work in a terminal, and for scripts. It uses the same navigation model as `ant tui`.

Everything that the web app shows is also available through the API and MCP (document 11). The workspace must not have a private capability.

## 13.1 The page model

Each node shows as a page. The page has a fixed structure, so that a reader learns it one time:

```
┌─────────────────────────────────────────────────────────────────────────┐
│ Lagrange's theorem                                   m2://7K3QX2@2026.11 │
│ statement · theorem · 20D60                                              │
│ Proved in Lean (Mathlib). Statement reviewed by 2 people. Classical.     │  ← evidence phrase (15.5)
├─────────────────────────────────────────────────────────────────────────┤
│ Statement                                                                │
│   Let G be a finite group and H a subgroup of G. Then |H| divides |G|.   │
│                                                                          │
│ [Formal] [Proofs (2)] [Uses] [Used by (1,204)] [Sources (37)] [History]  │  ← tabs, each a query
│                                                                          │
│ Explanations   L1 · L2 · L3 · L4                                         │
│ Examples       ℤ/6, S₃, … (executable)                                   │
│ Related        converse fails: A₄ (counterexample) · Sylow theorems · …  │
├─────────────────────────────────────────────────────────────────────────┤
│ Open work on this node:  explanation L1 missing · 1 alignment to confirm │  ← jobs, claimable
└─────────────────────────────────────────────────────────────────────────┘
```

- The **Formal** tab shows the bound Lean statement, pretty-printed and also as source. It has links to the Mathlib documentation at the pinned commit. It shows the axioms, the replay results, and the reviews with their hashes. If the binding is `stronger`, `weaker` or `variant`, the tab shows the adapter and explains the difference in words.
- The **Proofs** tab shows each proof node: an informal proof, formal or not, its method tags and its own dependency closure. A reader can compare proofs side by side by what they use ("proof 1 uses choice, proof 2 does not").
- **Uses** and **Used by** are the closure views (05.5). They can show as a list, a layered DAG, or a blueprint-style dependency graph coloured by evidence level.
- **Sources** lists each aligned paper result in date order. The earliest result has a mark. Each result has a link to `ax://` for arXiv content, or to the publisher, zbMATH or MathSciNet for records.
- **History** shows revisions with provenance, each binding change that a Mathlib rename caused, and each change in evidence level, with the reason.

## 13.2 Search

The search box accepts natural language, LaTeX, a Lean expression, an identifier, or a mix of these. Document 14 describes the engine. The workspace adds three things:

- **Facets** for kind, MSC, evidence level, fidelity, "has formal statement", "has formal proof", "open", "uses choice", source date range and method.
- **Explanation of results.** Each result shows the channel that matched it (lexical, formula, embedding or formal type) and the matched span. Thus a reader can see why the result is in the list.
- **Did-you-mean across systems.** Sometimes a query matches nothing in math2 but resolves in Wikidata, nLab, Mathlib or OEIS. Then the workspace shows "not held yet", with the external link and a one-click "request this node". This button files a curate job (11.4).

## 13.3 Explore

To explore is to move through the graph with a purpose. The workspace supports the moves that mathematicians actually make:

| Move | Interface |
|---|---|
| "What do I need to understand this?" | prerequisite path (Q5) from a chosen starting concept, or from the reader's declared background (13.5), as an ordered reading list |
| "What is this used for?" | reverse dependencies, grouped by topic, ranked by centrality |
| "What is the most general form?" | the `generalizes` chain upwards, and the hub family view |
| "What is known about this open problem?" | the problem page: conjecture, bounds over time as a chart, partial results, related computations, formal statement status, who is working on it (13.6) |
| "Where are the gaps?" | the frontier view for a topic: formalised-but-unproved, refereed-but-unformalised, open-with-formal-statement. This view is the shared to-do list for people and agents |
| "What changed?" | the topic diff between releases |
| "Show me the landscape" | a topic map: concepts as nodes, `specializes`/`about` edges, laid out hierarchically and coloured by coverage, with zoom from MSC sections to individual nodes |

The client draws the graph. It uses a WebGL renderer for large neighbourhoods. It never shows more than approximately 2,000 nodes at one time. Above that number, it changes to aggregation by topic, with counts. A picture of 50,000 nodes is decoration, not navigation.

## 13.4 Explain

The explanation agent (document 12.6) writes levelled explanations into nodes. The workspace adds explanation on demand:

- **Ask about this node.** Each page has a chat panel that answers questions. The context of the panel has the node, its neighbourhood and its formal statement. Each answer cites `m2://` tags. If a statement in the answer has no cited node to support it, the panel marks it unsupported. The panel uses the explanation agent with a read-only tool profile.
- **Unfold this proof.** For a formal proof, the reader can expand any step into its sub-steps, down to single Lean tactics and the lemmas that they used. Each one has a link back to its node. The informal gloss for each step is generated, cached, and marked as generated.
- **Translate notation.** When the pointer is on a symbol, the workspace shows its definition node. A reader can change notation conventions, such as $\subset$ versus $\subseteq$, or conventions for $\mathbb{N}$, where the concept records the convention.
- **Languages.** The workspace can show explanations and statement prose in the languages that `tamnd/arxiv` already supports (English, Vietnamese, Simplified and Traditional Chinese, Japanese). It uses the same glossary and translation pipeline. Translations have a mark that says they are translations. The English text (or the text in the original language) stays canonical. Formal statements are never translated, because they do not depend on a language.

## 13.5 The reader's profile

Anyone can read without an account. A reader who signs in (GitHub or ORCID) can declare a **background**: the set of concepts and statements that they already know. A curriculum topic can be the start of a background ("finished a first course in algebra"), and the reader can change it. Prerequisite paths and explanation levels then start from the background. The workspace stores the profile, not `tamnd/math2`. The profile is private, and the reader can export it or delete it.

## 13.6 Collaborate

Collaboration is the area where math2 is most different from a static library. Its design follows the three ways that mathematical collaboration actually works in 2026 (document 01): Polymath-style open projects, blueprint-driven formalisation projects (FLT, PFR, the Equational Theories Project), and small groups of humans who work with AI tools.

### Projects

A **project** is a campaign (11.4) with people attached. It has:

- a **goal**: a set of target nodes and a target evidence or fidelity level, or an open problem with "find a proof or a counterexample";
- a **blueprint**: the closure of the targets, computed live and coloured by status. The graph generates it, and nobody writes it by hand. This is the main improvement over `leanblueprint`. If a project already has a blueprint, `m2 ingest blueprint` imports it as nodes and edges (document 09.4);
- **tasks**: the frontier items inside the closure, each claimable;
- **members**: people and agent roles, with permissions;
- a **log**: every PR, review, agent run and discussion thread that touched the project.

A project is the file `projects/<id>.yaml` in the content tree, with a name, a goal in words, the topic node of the goal, the target evidence level, a monthly budget in US dollars, the agent roles that work for it and the members with the role owner or member. The blueprint is the blueprint of the topic page. A job works for the project when it runs on behalf of `project:<id>`. The package `project` of m2 reads the files, `m2 agent run` refuses a job of a role or a person that the project does not have, and `m2 orchestrate` caps the monthly spend of the project at its budget. The project page of the site shows the run log from the run records. The pull requests, the reviews and the threads are not in the log yet.

### Tasks and claims

Any frontier item can be a task. A person claims a task for a time window (default 14 days, renewable). Everyone can see the claim, so two people do not do the same work. Agents examine claims before they take a job. A claim by a person stops agent work on that target, unless the person invites agent work ("let agents try the cheap attempts first"). This follows the norm that large formalisation projects agreed on in 2024 to 2026: claim visibly, release promptly.

### Discussion

Each node and each project has threads. The workspace database stores threads, not git. If a thread decided something, it must end with a pull request that writes the decision into a node or a review. Discussion is not knowledge until it is in the graph. For projects that already use Zulip, threads can be mirrored to Zulip streams. The Lean community discusses on Zulip, and math2 should join that discussion, not compete with it.

### Working with agents in the open

A project member can request agent work inside the budget of the project. Agent runs show in the log with their transcripts (11.6). Anyone in the project can read how a proof was found, and see which attempts failed and why. This is intentional. In 2025 to 2026, there were episodes in which AI "solved" problems that were already in the literature, or solved a misstated version. People found each of these problems because someone could see what was actually done (document 01). Openness is the cheapest verification method available.

### Credit

Provenance (06.5) attributes each contribution. The project page shows contributions by person and by agent role. Each agent contribution names the person or project that directed it. `m2 prov credit` exports a CRediT-style statement for any person and period (document 16.4).

## 13.7 Performance and accessibility targets

- Static node pages: less than 50 KB transferred before math fonts, and the first meaningful paint in less than 1 s on a mid-range phone on 4G.
- The server renders math to HTML+MathML at build time (KaTeX). MathML is always present for screen readers. The client never needs JavaScript to read a node.
- Each interactive view (graph, chart) has an equivalent table or list.
- The static site is fully usable offline as a downloaded archive. Each release publishes one archive. Thus people can read math2 without a network, and this is important for many of the mathematicians in the world.
</document>

<document source="https://tamnd.github.io/math2/spec/14-search.md">
# 14. Search

Most people will use math2 through search first. Agents also use search to find premises, duplicates and prior work. A query from a mathematician can be a phrase ("finite subgroup of the multiplicative group of a field"), a formula (`\sum_{d \mid n} \varphi(d) = n`), a Lean type (`∀ n : ℕ, ∑ d ∈ n.divisors, φ d = n`), an identifier (`A000010`, `Q190026`, `0BQX`), or a mix of these. No single retrieval method can handle all of these queries. Thus math2 runs **four channels** and fuses their results.

The code is in `tamnd/math2-agent/search`. Three things serve it: `m2 search`, the `search*` and `premises` methods of the API (11.2), and the workspace (13.2).

## 14.1 Channels

| Channel | Index | Good at | Bad at |
|---|---|---|---|
| **lexical** | SQLite FTS5 over names, aliases, statements, explanations; BM25 | names, rare words, exact phrases, identifiers | paraphrase, notation |
| **formula** | structural index over parsed LaTeX (14.2) | the same formula in different words or variable names | formulas that differ structurally but mean the same |
| **formal** | type-directed index over bound Lean declarations and Mathlib (14.3) | exact mathematical content, given a type pattern | natural-language queries |
| **embedding** | vector index over node text, formal statements and record plane results (14.4) | paraphrase, concepts, "the theorem that says…" | precision on near-duplicates, rare identifiers |

The **identifier resolver** runs before the four channels. The query can parse as a tag, alias, `m2://` URI, Lean name, QID, A-number, Stacks tag, arXiv id, DOI, MR or zbMATH id. In that case, it resolves directly (the `resolve` method of the API, 11.2), and search returns that node first.

## 14.2 Formula search

The parser changes LaTeX formulas into operator trees. It uses the same KaTeX-compatible macro set as the bodies (`macros/m2.sty`). Thus math2 interprets its own notation the same way in all places. The index has three parts:

- **Operator-tree paths.** The paths from leaf to root are tokenised. Variables are anonymised and constants are kept. Tangent-L and Approach0 use this method, and they are the strongest systems at ARQMath.
- **Symbol layout trees**, for formulas that have an ambiguous operator parse.
- **Canonical forms**, where a formula is in a class that can be normalised (polynomial identities, linear relations). SymPy computes a normal form in the compute image at build time.

Matching uses BM25 over the path tokens and then a structural re-score. The re-score is the size of the largest matching subtree with consistent variable unification. The index has only formulas from nodes and from record plane results. At full scale, this is approximately 40M formulas. The formula search of zbMATH Open (MathWebSearch) is a crosswalk target, and math2 does not replicate it.

## 14.3 Formal and type-directed search

For a Lean query, math2 does what Loogle does. It also does one thing that Loogle cannot do: it maps hits back to nodes.

- **Pattern search.** A query can be a pattern such as `Subgroup _ → Finite _ → IsCyclic _`, or a constant set such as `IsCyclic, Units, Field`. The channel matches it against the extracted statements (07.4) of each declaration in the pinned Mathlib and the bound libraries. It uses a discrimination-tree index over the elaborated types. It runs locally against the pinned versions of the release. It never runs against the public Loogle service, because that service has rate limits and follows a different Mathlib.
- **Goal search.** For a goal state, the channel finds declarations whose conclusions unify with it after instantiation (the `exact?`/`apply?` relation). It ranks them by the similarity of their hypotheses to the local context, as the embedding channel measures it.
- **Node mapping.** If a hit is bound to a node, the channel returns the node with its status. If a hit is not bound, the channel returns the declaration plus "no node yet". A reader can change this result into a curate request.

math2 uses the semantic Lean search tools of 2026 (LeanSearch v2, Lean Finder, LeanExplore; document 01) as **evaluation baselines**. Where their licences permit, it also uses them as sources of informalisation data to train the formal-statement embeddings of math2. math2 does not depend on these tools at runtime.

## 14.4 Embeddings

Each node gets several embedded texts. The index holds each text separately:

1. its name and statement section;
2. its Lean statement, with the Herald-style informalisation of it (a model's English rendering, generated at build time and cached by type hash);
3. its L2 explanation;
4. for record plane results, the unfolded statement (09.5).

Choices:

- **Model.** Each release pins one embedding model. math2 selects the model with its retrieval benchmark (14.7). The benchmarks of 2026 for mathematical retrieval (MIRB and its successors) put commercial general-purpose embedding models at the top. They also showed that generic rerankers often made results *worse*. Thus math2 benchmarks a reranker before it adopts it, and by default it has no reranker. If the benchmark says so, embeddings for formal statements may use a different model from the embeddings for prose.
- **Index.** A quantised HNSW index (int8, 1,024 dimensions) for nodes, and a separate index for the record plane. At 5M nodes, the node index is a few GB. At 30M results, the record index is approximately 60 GB. The release process builds both indexes, and the server holds them in memory (05.7).
- **No raw vectors in the API.** The API returns neighbours and scores. It never returns vectors. This protects record-class text (9.3) and lets math2 replace the model.
- **Re-embedding is a release event.** When the embedding model changes, a batch job embeds everything again. The release notes record the retrieval benchmark numbers before and after the change.

## 14.5 Premise selection

Premise selection is search for agents. For a Lean goal and its context, it ranks the lemmas that are most likely to help. It is the `premises` method (11.2), and it gives input to the prove loop (12.4). It has four parts:

1. **Formal search.** Declarations whose conclusions unify with the goal (14.3).
2. **Dependency priors.** Declarations that proofs of the graph neighbours of the target use, and declarations that Mathlib proofs of statements with similar types use. The formal dependency graph from extraction (07.4) gives this data. For a node with informal proofs, the declarations bound to the nodes that its proofs `use` get a strong prior.
3. **Embedding similarity** between the informalisation of the goal and the informalisations of declarations.
4. **Learned reranking**, trained from the run records of math2: the retrieved premises that were in the final proof (17.4). This is the only place where a learned reranker is on by default. It is trained on the data of math2, and math2 evaluates it on held-out runs before each deployment.

math2 also uses the built-in premise selection of Lean for `grind +suggestions` and the hammer. It runs inside the Lean session, and math2 merges its suggestions with its own.

## 14.6 Fusion and ranking

Reciprocal rank fusion combines the results from the channels. The weight of each channel is learned on the benchmark. Then two adjustments apply:

- **Kind and status priors.** For a natural-language query, statements and concepts rank above proofs, and E5/E3/E2 rank above E1. The facets of the workspace can override this. For example, an "open" filter inverts the status prior.
- **Duplicate folding.** If results are `same` or `same_as` the same node, they fold into one entry. Record plane results aligned to a node show under that node as "stated in N papers", not as N separate hits.

Each result reports the channels that found it and its rank in each channel (13.2). Thus a reader can see why the result is in the list.

## 14.7 Benchmarks

math2 measures search quality and does not only claim it. `bench/search/` holds the benchmarks below. Their results are in the notes of each release. If a headline metric gets worse by more than 2 points, the release is blocked.

| Benchmark | Queries | Metric | Source of labels |
|---|---|---|---|
| **concept lookup** | 2,000 natural-language descriptions of concepts and theorems, written by people, at undergraduate and graduate level | MRR, recall@10 | curated; each labelled with its target node |
| **formula** | ARQMath-style formula queries, remapped to math2 nodes | nDCG@10 | ARQMath judgements where licences permit, plus curated |
| **formal** | Lean types and goals with known Mathlib answers | recall@1, recall@10 | Mathlib proofs (the premise actually used) |
| **paper-to-node** | record plane results with known aligned nodes | recall@10 | confirmed alignments |
| **premise** | goals from held-out math2 and Mathlib proofs | recall@k of premises used | proofs |
| **cross-system** | queries that should resolve to an external system (nLab, OEIS, Stacks) | accuracy of "not held, see …" | crosswalk |
| **external baselines** | the same queries run against LeanSearch v2, Lean Finder, TheoremSearch, zbMATH formula search, where an API allows | the same metrics | none |

The external baseline row is important. On each type of query, the search of math2 should be better than the best single-purpose tool for that type. If it is not better, the release notes say so, and the workspace gives a link to the better tool for that query type.

## 14.8 The first version

The first version of the search package (M3) has all four channels, the identifier resolver, the fusion and the premise selection. It is simpler than the design above in these ways. The benchmarks of 14.7 decide what to change first.

- The index is one SQLite file, `build/search.db`, that `m2 build` writes in the `search` step. Two builds of the same tree give the same file.
- The formula channel has the operator-tree paths and the structural re-score. It has no symbol layout trees and no canonical forms from SymPy yet.
- The formal channel matches the head of the conclusion, the heads of the hypotheses and the constants of each declaration. It reads them from the canonical type forms of M2Extract, and it has no discrimination tree yet. It knows the declarations of extract.json, thus the bound declarations and the declarations that the tree names. The full pinned Mathlib comes with the import of its declarations.
- The embedding channel uses the local hash embedder of the aligner (16.3) and a flat scan of int8 vectors. The HNSW index and the pinned embedding model come when the benchmark has chosen a model. The index records the name and the length of the embedder, and it refuses to open with another one.
- The fusion multiplies the reciprocal rank term of each channel by a factor from one half to one. The factor is the score of the result divided by the score of rank 1 in the same channel. Thus an exact match far above the next result keeps its margin, and the kind and status priors only order results that are near.
- The channel weights are fixed: 1 for each channel and 0.5 for the embedding channel. The benchmark learns them later.
</document>

<document source="https://tamnd.github.io/math2/spec/15-trust-and-audit.md">
# 15. Trust and the audit

This document puts in one place what math2 trusts, what it checks, and the rules that do the checks. As in `tamnd/arxiv`, **the audit is the real contract**. A claim in the README, a status in the workspace or a number in a release is only as good as the rule that checks it. Each rule has a name and a state, and each rule writes into a committed report.

## 15.1 The trusted base

math2 states its trusted base explicitly. math2 believes that a statement at E5 is true **if and only if** all of these conditions are true:

1. **Lean's kernel is sound** for the type theory that it implements. We also require independent replay with `lake check --paranoid`. This replay checks again with nanoda, lean4lean and the verified con-leche and con-ron kernels (document 07.7). Thus the trusted base is "at least one of several independent kernel implementations is sound, and they agree", not "Lean's C++ kernel is sound". The kernel soundness bugs of 2026 (document 01) showed that this concern is not theoretical. They also showed that checkers which are out of date fail together. For this reason, the release records the version of each checker.
2. **The standard axioms are consistent**: `propext`, `Quot.sound` and `Classical.choice`, together with the type theory of Lean. This is equiconsistent with ZFC plus countably many inaccessible cardinals. That is a little more than ZFC, and much less than anything that mathematicians have doubts about.
3. **The formal statement means what the informal statement says.** This is the fidelity claim F2 or F3 (document 10). It is the weakest link. Each design decision about fidelity has the aim to make this link stronger.
4. **The binding is the right declaration**: the name, the Mathlib commit and the elaborated-type hash in the release manifest match what was built. The build checks this mechanically.

For E5, math2 trusts nothing else. Specifically, this includes:

- any model's output;
- the prover that found the proof;
- the implementation of any tactic, because its output is a term that the kernel checks;
- the `m2` toolchain, except for the code that extracts declarations and hashes. That code is small and reviewed, and it has its own tests in `math2-agent/trust/`.

For E4, the trusted base is the certificate checker for that certificate kind (document 08.3), plus the runtime of the checker. For E3 and E2, it is the literature and the curators who read it. For E1, it is nothing.

## 15.2 How the audit runs

```
m2 audit                     # all planes, all rules, on the working tree
m2 audit -plane nodes        # one plane
m2 audit -rule G04           # one rule
m2 audit -changed            # only shards touched by the current diff (what PR CI runs)
m2 audit -release 2026.11    # the full pre-release audit
```

Each rule has one of five states, and the report shows all five:

| State | Meaning |
|---|---|
| `pass` | checked something and found nothing |
| `fail` | a hard rule found something. Blocks merge or release |
| `found` | a soft rule found something. Does not block, and is in the report all the same |
| `n/a` | the rule has nothing it could look at in this scope. Never shown as `pass` |
| `skip` | the rule was not run (for example, a network rule offline), and the report says why |

math2 keeps `n/a` and `pass` separate for the same reason that `tamnd/arxiv` keeps them separate. If a rule passes because it had nothing to examine, the rule is not working. It should not look as if it works.

The report goes to `reports/audit/<scope>.txt` and `reports/audit/<scope>.json`. It is committed on `main` after each merge. The release report goes to `reports/releases/<release>.txt`.

```
$ m2 audit -plane formal -q
rule  state  checked   findings
F01   pass   41,203
F02   pass   41,203
F03   pass   41,203
F04   found  41,203    17
F05   pass   3,880
F06   n/a    0
...
41,203 bindings over 38,911 nodes, 17 soft findings, nothing failed
```

## 15.3 The rule catalogue

The letter tells the subject of the rule. Rules with the mark **H** are hard, and they block. Rules with the mark **S** are soft, and the audit reports them. The implementation of each rule is in `tamnd/math2-agent/audit/<id>.go`. With it is a fixture that triggers the rule and a fixture that does not.

### T: tags and identity

| Rule | H/S | Checks |
|---|---|---|
| T01 | H | no tag ever disappears from `registers/tags.reg` or from `nodes/` across git history |
| T02 | H | no two nodes share a tag; no minted tag collides with a reserved prefix |
| T03 | H | no provisional id (`~xxxxxxxx`) appears on `main`, and none is referenced outside its own PR |
| T04 | H | every stored reference is a tag, not an alias or slug |
| T05 | H | registers agree with node files (every live node registered, every retired node in `retired.reg`) |
| T06 | S | aliases are not ambiguous with tags, MSC codes, OEIS A-numbers or arXiv ids |

### S: schema

| Rule | H/S | Checks |
|---|---|---|
| S01 | H | every node file validates against its schema version |
| S02 | H | kind-specific required fields present (a `proof` has exactly one `proves`; a `computation` has an engine block) |
| S03 | H | no `status`, `verified`, `proved` or similarly named field anywhere in content (document 04.7) |
| S04 | H | node file size ≤ 64 KB |
| S09 | S | node uses a deprecated schema version during a migration window |

### B: body

| Rule | H/S | Checks |
|---|---|---|
| B01 | H | every body renders in KaTeX and MathJax headless with no error, using only declared macros |
| B02 | H | every `[[…|TAG]]` link targets a live or retired-with-successor tag; no URLs in statement sections |
| B03 | H | only the schema's sections for the kind, in order |
| B04 | S | statement section is under 2,000 characters (long statements are usually two statements) |
| B05 | S | explanations carry a level and a human/agent marker |

### G: graph

| Rule | H/S | Checks |
|---|---|---|
| G01 | H | every edge target resolves |
| G02 | H | `same_as` is functional both ways within each external system |
| G03 | H | `specializes` acyclic; `uses` acyclic modulo equivalence cycles through distinct proofs |
| G04 | S | formal dependencies of a bound proof appear in its `uses` |
| G05 | H | every live statement has a source, a proof or a formal binding |
| G06 | H | every conjecture and problem has a `source` naming who posed it |
| G07 | S | hub families have at least three members |
| G08 | S | orphan rate per topic < 2% |
| G09 | S | no concept has more than 12 definitions without a note |
| G10 | H | every `implies`/`equivalent`/`refutes` edge names its `via` proof or computation |

### F: formal

| Rule | H/S | Checks |
|---|---|---|
| F01 | H | every binding resolves to a declaration at the pinned commit |
| F02 | H | the elaborated-type hash recorded for reviews matches the built one, or the review is marked stale (never silently kept) |
| F03 | H | `#print axioms` for every E5 candidate is a subset of the standard three, or the node allows more and says so |
| F04 | S | `native_decide` auxiliary axioms (`*._native.native_decide.ax_N`) in closure, reported per node |
| F05 | H | external replay: every E5 declaration passes `lake check --paranoid` and the comparator against its pinned statement, with checker versions recorded (document 07.7) |
| F06 | H | no `sorry`, `admit`, `axiom` declarations or `@[implemented_by]` in the math2 library outside `Math2/Hypotheses/` |
| F07 | H | statement-only bindings (`match: partial`) are never shown as E5 |
| F08 | S | Mathlib deprecation warnings on bound declarations, with the replacement name |
| F09 | H | degenerate-statement checks pass for every F1+ binding: hypotheses satisfiable, conclusion not trivially true, no vacuous quantification (document 10.4) |

### V: review and fidelity

| Rule | H/S | Checks |
|---|---|---|
| V01 | H | a review is bound to both the informal statement hash and the elaborated-type hash |
| V02 | H | F3 by two reviews needs two reviewers with no shared affiliation listed in `reviewers.reg`, and neither is the binding's author |
| V03 | H | an agent cannot be a reviewer; an agent run started by a reviewer cannot produce the binding that the same reviewer reviews |
| V04 | S | review age > 24 months on a node whose neighbourhood changed (prompt for re-review, no downgrade) |

### C: computation

| Rule | H/S | Checks |
|---|---|---|
| C01 | H | every computation names an image by digest, an entry point and inputs |
| C02 | H | E4 candidates reproduce bit-identically on two runner architectures |
| C03 | S | a computation's image is older than the current release's; re-run scheduled |
| C04 | H | certificate checker is the fixed checker for that kind (document 08.8) |
| C05 | H | no network, no model calls in computation logs |

### A: alignment

| Rule | H/S | Checks |
|---|---|---|
| A01 | H | every `same_as` value is syntactically valid for its system (QID, A-number, Stacks tag…) |
| A02 | S | `same_as` written by an agent and not confirmed after 90 days |
| A03 | S | round-trip: the external record still exists and still matches at its pinned revision |
| A04 | S | candidate duplicate pairs (embedding plus judge `same`) unresolved after 30 days |

### L: licence

| Rule | H/S | Checks |
|---|---|---|
| L01 | H | every byte of imported text names a source and that source's licence (document 09.3) |
| L02 | H | no text from a source with an incompatible licence (GFDL-only, arXiv default licence, all-rights-reserved) appears in `nodes/` |
| L03 | H | share-alike sources carry attribution in the node's provenance and the release's attribution file |
| L04 | S | text similarity between a node body and its cited sources above threshold, for human review of copying |

### P: provenance

| Rule | H/S | Checks |
|---|---|---|
| P01 | H | every content change in a merge has a provenance line |
| P02 | H | every agent provenance line names a run id that exists in the artefact store |
| P03 | H | every agent run names the person or schedule it acted on behalf of |
| P04 | S | AI-produced text without human confirmation shown in the workspace without its marker |

### R: release and reproducibility

| Rule | H/S | Checks |
|---|---|---|
| R01 | H | derived graph rebuilds to the same hash (sample on merge, all on release) |
| R02 | H | `.olean` rebuild matches, or the difference is listed and explained |
| R03 | H | release notes account for every count change |
| R04 | H | last quarterly restore drill passed |
| R05 | H | the release pins exactly one Lean toolchain and one Mathlib commit, and the bump branch was green |

### X: agents

| Rule | H/S | Checks |
|---|---|---|
| X01 | H | an agent PR changes only the fields its role is allowed to change (document 12.1) |
| X02 | H | an agent PR stays within its run budget, and the run record shows the spend |
| X03 | S | agent acceptance rate per role and model below threshold over 7 days (flag for prompt or model review) |
| X04 | H | no agent wrote a status-like field or a review |

A merge is an agent PR when one of the new lines of its provenance logs is of an agent with no person. The runs of the PR are the runs that those lines name, and the audit reads their summaries in `build/runs/`. `X01` checks each changed file, except the provenance logs, with `agent.Permit` for the role and the target of each run. A change passes when the set of one run allows it. `X02` needs `spend_usd` and `budget_usd` in each summary. `X03` counts the runs that started in the week that ends 7 days before the date of the HEAD commit, so that each PR had a week for review. A run is accepted when a provenance line on the branch names it. A role and a model with fewer than 5 runs in the week are not checked, and the threshold is 25%. `X04` looks for a part of a field name that is a status word of `S03`, `fidelity_reviews` or `reviews`, in the lines of agents and in the fields of the nodes that an agent PR changes. It also fails an agent PR that changes a file below `reviews/`.

There are 65 rules. A new rule needs a fixture pair and a line in this table. Both must be in the same pull request.

## 15.4 Propagation when trust changes

Trust can change. The audit handles the events that change it:

- **A conjecture is refuted.** Some statements have a best alternative that uses the conjecture `as: hypothesis`. These statements do not change, because they are honest conditionals. Other statements have a proof that uses the conjecture *not* as a hypothesis. Each of these statements goes down to E1, gets a flag, and is in the list in the release notes. In that case, someone assumed an open conjecture and did not say so.
- **A formal binding loses its declaration** (a Mathlib rename without a deprecation alias). The binding is broken (`F01` fails on the bump branch). The bump agent (document 07.9) proposes the rename from the deprecation data of Mathlib, and a person merges it. The node keeps its tag and its history in `bindings.reg`.
- **A definition changes meaning underneath a review** (`F02`). The review becomes stale. For that binding, the node goes down to F0, and E5 is not available. This stays true until someone reviews again, or until a second formalisation is proved equivalent.
- **A kernel or checker bug is found.** Some E5 nodes had their replay on the affected checker version. Each of these nodes runs again on a fixed version, or on the other checker only. The release that contained them gets an erratum. At some time, this has occurred in every proof assistant. Thus math2 has a plan for it and does not only hope that it will not occur.
- **A model version is withdrawn** (document 06.5, the withdrawal test). A generated PR quarantines its unreviewed output. An E5 status reached through that output does not change, because the kernel and the fidelity reviews carried that status, not the model.

## 15.5 What the workspace says

The status that a reader sees is a short phrase that the system generates from the evidence. It is never a checkmark. For example:

- "Proved in Lean (Mathlib), statement reviewed by 2 people. Classical. Replayed by 2 kernels."
- "Proved in refereed literature (3 sources, earliest 1871). Formal statement exists, unreviewed. No formal proof."
- "Claimed (preprint, 2026). Not reviewed. 1 agent-produced informal proof."
- "Open. Posed by Erdős (1975). Best known bounds: … (2025, formal)."

Each phrase has a link to the evidence that it summarises. A reader who wants to know why we believe something can follow the link down to the kernel replay log, the reviews, or the page number in the 1871 source.
</document>

<document source="https://tamnd.github.io/math2/spec/16-collaboration-and-governance.md">
# 16. Collaboration and governance

People and agents write math2 together, and everyone can read it. This document defines who may change what and how changes are reviewed. It also defines how credit and disclosure work, how math2 relates to the communities whose work it links to, and how the content is licensed.

## 16.1 Roles

| Role | Who | Can |
|---|---|---|
| reader | anyone | read, search, run examples, use the read-only API and MCP |
| contributor | anyone with a GitHub identity | open PRs (nodes, edits, alignments, Lean code), comment, claim tasks |
| reviewer | a contributor listed in `registers/reviewers.reg` for one or more areas | sign fidelity reviews (F2/F3) in those areas; approve PRs in those areas |
| curator | a reviewer with merge rights in an area | merge PRs in the area; resolve dedup queues; retire nodes |
| maintainer | a small group | merge anything; change schema, rules, budgets, roles; cut releases |
| agent role | a bot account per role | open PRs within the role's field permissions (12.1); never review, never merge except under auto-merge rules |

The curators of an area give reviewer status for that area. An area is an MSC two-digit class or a narrower topic. The basis for the decision is the reviews that the person has already done. A reviewer does not need an academic position. Mathlib's reviewer and maintainer system is based on contributions, not on credentials, and math2 uses the same model. But a reviewer must have a real identity: an ORCID or a long-standing GitHub account. The independence rule for F3 (rule `V02`) cannot work with anonymous reviewers.

## 16.2 How changes land

Every change is a pull request against `tamnd/math2`. The gate before a merge depends on the kind of change:

| Change | Gate |
|---|---|
| typo or prose fix in a body | `m2 check` green; any contributor approval |
| new node or new edge, E1 evidence only | audit green; one curator approval |
| new formal binding (F0) | audit green, Lean build green; one curator approval |
| fidelity review (F2/F3) | the review itself is the gate; one reviewer signs; for F3, a second independent reviewer or an equivalence proof |
| retire, merge or split nodes | two curator approvals; release notes entry |
| agent PR, role with auto-merge (12.1) | audit green; role-specific checks green; sampled human audit afterwards |
| agent PR, any other role | as for the same change by a person, plus the agent label |
| schema, rule, budget, role changes | two maintainer approvals; an entry in document 21 if it settles an open question |

**Auto-merge** applies only to changes whose correctness is entirely mechanical. One example is a Mathlib rename that comes from deprecation data. Another example is a formal proof added to a node whose statement binding is already F2 or above. For these changes, the kernel and the replay are the whole check. Even then, people audit a sample of 5% of the auto-merged PRs of each role each week. The agent report (12.9) publishes the findings of the sample. If the sampled error rate for a role is more than 1%, auto-merge stops for that role until the cause is fixed.

## 16.3 Curation at scale

Curators cannot curate a graph of all mathematics if they approve PRs one at a time. Curators work through **queues**. Each queue has a rank order, so that one hour of a curator's time has the most effect:

- **dedup queue:** candidate `same` pairs (05.4), ranked by how much of the graph the merge would connect;
- **alignment queue:** agent-proposed `same_as` edges, grouped by external system and ranked by centrality;
- **fidelity queue:** bindings at F1 that wait for a human review, ranked by `unlock` (11.4). A review of a statement that 300 nodes depend on is worth more than a review of a leaf. `m2 review queue` lists it;
- **extraction queue:** paper results that the extractor could not align, sampled by topic;
- **retirement queue:** nodes flagged by graph health rules (orphans, hubs that are too small, conflicting `same_as`).

The reviewer's tool, `m2 review`, makes a fidelity review fast. It shows the informal statement and the elaborated Lean statement side by side. It also shows the back-translation, the results of the degenerate-statement check, and the critics' notes (document 10.4). It shows the definition of every non-trivial constant in the Lean type, and each definition expands on demand. The target median time per review is 3 minutes for undergraduate-level statements and less than 15 minutes for research-level statements. Document 17 measures this.

## 16.4 Credit

Work on math2 must count as scholarly output. If it does not, the people who can review research-level fidelity will not give their time to it. Thus:

- **Every release is citable** with a DOI (06.2). Every node at a release is citable as `m2://TAG@RELEASE`.
- **Contributor statements.** `m2 prov credit <person> --from --to` produces a CRediT-taxonomy statement. It lists, with links, the nodes that the person wrote, formalised, reviewed or curated. Its format is for direct use in a CV or a grant report.
- **Reviews are signed and public.** A fidelity review is a small, real and checkable piece of mathematical work. It carries the reviewer's name. Each reviewer has a page on the wiki with a stable link, `reviewers/SLUG/`, that lists their reviews, and `m2 review export` gives the same list as text or JSON.
- **Projects publish.** A project that reaches its goal (13.6) can publish a release note as a paper. The author list comes from provenance, and the project's members edit it. A standard paragraph discloses the AI contributions, and provenance also generates this paragraph (16.5).

## 16.5 AI contribution: disclosure and norms

The norms that formed from 2024 to 2026 are converging. They come from journal and arXiv policies and from statements by mathematical societies. They also come from the public practice of Terence Tao and other researchers who work with AI tools openly (document 01 has the sources). The norms agree on three points. AI tools may be used. The humans who sign are responsible for the result. The disclosure of the use must be specific enough that a reader can judge it. math2 puts these points into the data and does not depend on good intentions:

1. **Every agent-produced byte is marked** by provenance (06.5). The workspace shows the marker on prose that an agent wrote and a person has not confirmed (rule `P04`).
2. **Agents are not authors.** They are tools that act for a named person, project or schedule (rule `P03`). The person or project is responsible.
3. **Formal proofs found by agents** need no disclosure beyond provenance for their correctness, because the kernel checked them. But they do need disclosure for credit and for the record. If an agent run is the first to prove a theorem, the history tab of the theorem says so, with the run id.
4. **Informal proofs by agents stay E1** (04.7) until someone formalises, certifies or publishes them. The workspace never shows them as more than claims.
5. **Novelty claims are checked against the literature first.** Before anyone describes an agent's result as "new" in any place, the explore agent runs a literature search. It searches the record plane, zbMATH, arXiv full text where the licence allows, and MathOverflow. math2 attaches the search and its result. Several of the 2025 "AI solved an open problem" announcements were rediscoveries of known results. math2 is designed so that it does not repeat that error (document 01, 17.5).
6. **Withdrawal is mechanical** (06.5). Sometimes a model or prompt version is found to give systematically wrong output. Then one command quarantines all of its unconfirmed output.
7. **Disclosure meets the 2026 guidelines by construction.** The AGMAI guidelines of September 2026 ask labs and authors to publish the model, prompts, summarized reasoning, time and compute cost. They also ask for documentation of failed attempts and of selection criteria. They ask for persistent identifiers in lab-independent repositories, and for machine-readable links between informal and formal statements. Run records (11.6) and bindings (07) already hold all of these. `m2 prov disclose <tag|project>` renders them as a disclosure paragraph. For any formal result, it also renders them as a Palomar-compatible `formalization.yaml`. Thus you can register a math2 result on Palomar, or cite it in a journal that requires an AI declaration (the LMS, Elsevier). Nobody has to reconstruct what happened.
8. **The human role is recorded with a taxonomy.** Each agent-assisted contribution states what the people did. It uses categories adapted from the erdosproblems AI-contributions wiki: the agent solved it with no literature found; the agent solved it and literature was later found; the agent built on literature; people and agents worked together; and literature search, formalization, rewriting or computation only. The project page and the disclosure paragraph show these categories. One "AI-assisted" label in place of these categories hides exactly the information that a reader needs.
9. **Participation without AI is first-class.** Some mathematicians have chosen not to work with commercial AI systems (the Association for Human Mathematics, 2026). Some venues now require human-written expositions. A project can declare itself `agents: none`, and then the orchestrator refuses jobs in it. Any reader can filter the workspace to show only human-authored prose. The credit statement of a person (16.4) separates the work that they did themselves from the work that they directed. math2 should be a place where both kinds of mathematician can work side by side.

### Upstream etiquette

math2 links into communities that have their own norms. The most important are Mathlib, Wikidata, nLab, OEIS and the Stacks Project. The rules are:

- **Upstreaming to Mathlib is by people, in Mathlib's process.** If math2 Lean code is worth upstreaming, a person offers it as Mathlib PRs and guides them through review. Agents may prepare the branch. Agents do not open PRs on Mathlib or on any other external project, unless that project has asked for it.
- **No bulk writes to external wikis without agreement.** Writes to Wikidata, MaRDI and similar systems follow the bot policies of those systems. This means approved bot accounts, rate limits, and community discussion first.
- **Respect the load.** Ingestion uses dumps and published bulk channels. It does not crawl live sites. It follows the pace that each source states (document 09). This is the same discipline that `tamnd/arxiv` applies to arXiv.

## 16.6 Licensing

The content has three licence zones. The zones let math2 take in the sources that matter, and they let others reuse what math2 produces:

| Zone | Licence | Why |
|---|---|---|
| node prose (bodies, explanations, names) | **CC BY-SA 4.0** | compatible with importing from Wikipedia, zbMATH Open, OEIS, LMFDB, ProofWiki and individual MathOverflow/Math.SE posts, which are all BY-SA family (3.0 material can be adapted under 4.0); also compatible with our own text being reused by those communities |
| Lean code (`lean/`) and computation code (`compute/`) | **Apache-2.0** | identical to Mathlib's licence, so code can move upstream with nothing in the way |
| metadata (front matter facts, edges, registers, manifests, the derived graph) | **CC0** | facts about mathematics should be free of conditions; also matches arXiv metadata and Wikidata |

The `L` audit rules enforce the licence rules (document 15). In particular:

- **GFDL sources are linked, never copied.** The Stacks Project and Kerodon are GFDL. GFDL text cannot go into CC BY-SA content, and CC BY-SA text cannot go into GFDL content. math2 links to Stacks and Kerodon tags with `same_as` and writes its own statements. This is permitted, because a mathematical fact is not copyrightable. Rule `L04` compares node text with the linked source to find accidental copies.
- **arXiv default-licence papers give metadata and structure, not text.** The `tamnd/arxiv` licence gate applies unchanged. math2 can record that a paper states a result (a record plane entry with a locator). This does not require it to republish the paper's wording.
- **MSC2020 descriptions are CC BY-NC-SA.** The codes are identifiers, and math2 uses them freely. The descriptive text of each class has a non-commercial clause that CC BY-SA content cannot absorb. Thus math2 never stores this text in `tamnd/math2`. The site shows it from a pinned snapshot, with attribution.
- **Stack Exchange text is imported per post, never from the dumps.** Posts are CC BY-SA (2.5, 3.0 or 4.0, depending on the date). But since 2024, the official dumps are behind click-through terms that forbid LLM training. At least one dump contained planted watermark entries. math2 gets individual posts through the public API, and records the licence and the author of each post. It stores records and short quotations, not bulk text.
- **Mathlib docstrings** are Apache-2.0. You can use one as a node body, with attribution in provenance. The CC BY-SA node body then includes it under BY-SA, and Apache-2.0 permits this.

## 16.7 Governance and sustainability

- **Decisions are written down.** Settled decisions are in document 00. Open decisions are in document 21. A change to either document is a PR with a rationale. If the change reopens a settled decision, it needs a measurement. The other specs in this series use the same rule.
- **Budgets are public.** math2 commits the monthly agent budget, its split by role, and what it bought (12.9) in `reports/agents/`.
- **The project could outlive its funding.** The content is in git and archived to Software Heritage. It is citable by DOI, and you can rebuild it from pinned images (document 06). If agent funding stops, math2 becomes a static, versioned and checkable library that people maintain at human pace. That outcome is smaller, but it is not a failure.
- **Funding and institutions.** Document 01 lists the 2025 to 2026 programmes that fund this kind of infrastructure. They include AI-for-math funds, DARPA's expMath, national research data initiatives such as MaRDI, and the Lean FRO's model of a focused research organisation. In the milestone structure of math2 (document 19), each milestone is a deliverable that a funder can check.
</document>

<document source="https://tamnd.github.io/math2/spec/17-evaluation.md">
# 17. Evaluation

math2 makes claims at three levels. It makes claims about mathematics (the status of a node). It makes claims about its own machinery (the battery catches misformalizations; search finds the right node). It also makes claims about its agents (this prompt version proves more for less). Each claim needs a measurement, or it is a slogan. This document says what math2 measures, how it measures it, and what each measurement gates.

The code and data are in `tamnd/math2-agent/bench/`. The results go to `tamnd/math2/reports/eval/<release>/`, and the notes of each release give a summary of them.

## 17.1 Principles

1. **The kernel is the only grader that needs no evaluation.** math2 grades all other graders, which includes LLM judges, back-translation, the critic, search and extraction. Every grader has a labelled set, a measured error rate, and a version.
2. **Gate on regressions, report the levels.** Each suite has a gate: on the suite, a new version must be no worse than the old one, within noise. math2 publishes the absolute numbers, but they do not gate. A gate on absolute targets invites quiet changes to the target.
3. **Measure cost with quality.** Report every result with its cost: tokens, dollars, wall-clock time and Lean CPU time. In 2026, the same model could score 7/10 for $36 or 9/10 for $1,888. The difference came from the harness around the model (document 01). You cannot compare a score that has no cost.
4. **Report failures and the selection.** The report of a suite says how many items passed and which items failed. It also says how the items were chosen. This follows the AGMAI guideline that labs document failed attempts and how they selected problems (document 01). math2 applies this guideline to itself.
5. **Contamination is the default assumption.** Any public problem set may be in the training data of a model. Each suite has public, held-out and fresh items (17.6). The headline numbers come from the held-out and fresh items.

## 17.2 What is measured

| Area | Question | Suite | Gates |
|---|---|---|---|
| **status** | Is a node's computed status correct? | monthly sample of E2 to E4 nodes re-checked by a curator against their evidence (17.8); F-level re-review (10.8) | release notes; a measured error above the published bound blocks the release |
| **fidelity battery** | Does F1 catch misformalizations? | `bench/fidelity/` (17.3.1) | any battery change |
| **extraction** | Does ingestion extract and unfold results correctly? | `bench/extract/` (17.3.2) | ingest adapter changes |
| **alignment** | Are proposed `same_as` and `aligned` edges right? | `bench/align/` (17.3.3) | align role changes |
| **search** | Does search find the right node? | `bench/search/` (14.7) | search and embedding changes |
| **agents** | Does each role do its job better per dollar? | `bench/agents/<role>/` (17.3.4) | role and prompt versions (12.8) |
| **injection** | Do agents ignore instructions in sources? | `bench/agents/canaries/` (12.7) | every role version |
| **explanations** | Are explanations correct, grounded and at the right level? | `bench/explain/` (17.3.5) | explain role versions |
| **graph** | Do the queries meet their latency targets? | `bench/graph/` (05.6) | storage-layer changes |

## 17.3 Suites

### 17.3.1 The fidelity battery

The battery (10.4) is a classifier. Its verdict is faithful, or misformalized with a class. Its evaluation set contains labelled pairs, and each pair is an informal statement and a formal statement:

- **Seed.** Known misformalizations from the published audits. These include the defective items of ProofNet, the certified issues of the ICML 2026 benchmark audit, the fixed statements of PutnamBench, and the corrected statements of the formal-conjectures repository. Each item has its taxonomy class (10.5).
- **Synthetic.** Faithful statements from F3 nodes, mutated by one taxonomy class at a time: drop a hypothesis, swap a quantifier, change ℕ to ℤ, add a junk value. A person checks each mutation before it enters the set, because some mutations are also faithful by chance.
- **Live.** Every `not-faithful` review (10.6) enters the set, with its class.

Measured: recall per taxonomy class, and the false-alarm rate on faithful items. The report is a confusion matrix over classes. A battery change must not decrease recall on any class by more than the noise band of that class. The band is published with the matrix.

**The suite in m2.** `bench/fidelity` has the synthetic part, in the Lean core library, and it reads the live part from the reviews of a math2 tree. The seed part needs the published audits, and it comes later. The noise band is zero items, because the tactics of the battery have limits in heartbeats and give the same result on each run. Thus the test fails when one class catches one item less. The report is attached to each release of tamnd/math2-agent.

### 17.3.2 Extraction

The suite has five hundred papers across MSC codes, decades and source types (LaTeX, HTML only, OCR). Each paper is labelled by hand with its results, their kinds, their in-paper dependencies, and a reference unfolding of each result. Measured: result recall and precision; kind accuracy; dependency edge F1; unfolding fidelity. People grade unfolding fidelity on a sample, and the back-translation judge of the battery grades it on all items. The report gives errors by source type, so a regression in the OCR path cannot hide behind the LaTeX path.

### 17.3.3 Alignment

The suite has confirmed and rejected alignments from curation (09, 05.3). Measured: precision at the auto-propose threshold, and recall of known `same_as` pairs. Precision is more important here, because a wrong `same_as` edge contaminates everything downstream. The threshold is set so that precision on the held-out set is at least 0.98.

### 17.3.4 Agent roles

Each role in 12.1 has a suite of tasks with an objective outcome:

| Role | Tasks | Outcome measured |
|---|---|---|
| **prove** | held-out statements with known Lean proofs, by tier: undergraduate (Mathlib-adjacent), graduate, research (from bound libraries and formal-conjectures' solved items) | solved within budget; dollars and Lean CPU per solve; proof size against the reference |
| **formalize** | held-out nodes with F3 bindings | winning candidate equivalent to the F3 binding; battery verdict; dissent rate |
| **explore** | planted problems: known conjectures withheld from the context, known counterexamples to plausible statements | refutations found; false conjectures proposed and not caught by the falsify step; novelty check run before every claim |
| **explain** | 17.3.5 | see 17.3.5 |
| **curate, align** | 17.3.3 | see 17.3.3 |
| **bump** | replays of past Mathlib bumps from git history | share of breakages repaired automatically; repairs that change a statement's type hash (these must be zero without a flag) |
| **review-assist** | packets for bindings with known verdicts | verdict-relevant concerns surfaced (the critic's recall); length of packet |

The suite of formalize and prove is in `bench/agents`. Its first items are public, use only the Lean core library, and have a reference solution. Without a model service, a scripted model writes the reference and some wrong attempts, thus the suite checks the goal predicates of the roles on real Lean. A live run with the Claude models gives the items solved and the dollars of each item.

The suites of explain and curate are in `bench/agents/explain` and `bench/agents/curate`. They need no Lean. The scripted model writes the reference and wrong attempts, and the scripted grounding check or judge gives the labels or the verdicts of the items. Thus the suites check the mechanical parts of the goals: the citations and the levels of explain, and the locator, the kind and the copy check of curate. A live run also grades the grounding check and the judge alone against the labels and the verdicts of the items, because the goals depend on them.

The suite of review-assist is in `bench/agents/reviewassist`, and it needs no Lean. Its items are bindings with known verdicts, as the table says. The scripted model writes the packet with the reference concern of each item, and some wrong packets that the goal or the tool must refuse. A live run gives the recall of the critic on the not-faithful items, the concerns on the faithful items, and the length of the packets.

The suite of align is in `bench/agents/align`, and it needs no Lean. Its items are crosswalk edges and Mathlib bindings with a known reference. The scripted model makes the reference change and some wrong changes, and the scripted judge gives the verdicts of the items. Thus the suite checks the mechanical parts of the goal: the edges that a source allows, the decisions of the register and the type hash of a Mathlib binding. A live run gives the precision and the recall of the edges, as 17.3.3 says, and it grades the judge alone against the verdicts of the items.

The suite of bump is in `bench/agents/bump`, and it needs no Lean. Each item is a replay of a bump, with the pinned Mathlib after the bump in the file of the items. The public items are made up, and the held-out items are replays of past Mathlib bumps from git history, as the table says. The scripted model makes the rebinds of the reference and some wrong rebinds that the goal or the tool must refuse, and the suite checks that `bump.AutoMerge` gives the value of each item. The goal refuses a rebind to another type hash unless the job has the constraint `type_change`. A live run gives the share of the breakages that the role repaired, the share of the repairs that can merge with no review, and the repairs that change a type hash with no flag, which must be zero.

The suite of explore is in `bench/agents/explore`, and it needs no Lean. Its items are planted problems of the second form of the table: plausible statements that are false, each with a small counterexample, the program that finds it and the output of the program. Without flags, a fake sandbox gives the output of each program, and the scripted model files the reference counterexample through a computation that verifies, after a lookup of the claim. The goal or the tool must refuse the wrong attempts: no output, a claim that the output is new, a conjecture that no computation checks, a counterexample with no computation, a wrong output hash and a change of the target. A live run runs the computations in docker and gives the three metrics of the table. Each target of the public items is false, thus a person reads each conjecture of a live run, and a conjecture that the person finds false is a false conjecture that the falsify step did not catch. The items with withheld known conjectures are not in the suite yet.

Every role suite has three splits. **Public** items can be anywhere. **Held-out** items are never published. They are in a private repository that the tools of the agents cannot reach. **Fresh** items were made after the training cutoff of the model. Fresh items come from new Mathlib declarations, new formal-conjectures statements, new arXiv results with formal statements, and items that the maintainers write. Each model version is compared on all three splits. If public scores are much higher than fresh scores, the report gives the difference as a contamination signal.

### 17.3.5 Explanations

Explanations (12.6) get grades on three axes. **Correctness**: every mathematical claim in the explanation is grounded in a cited node, or it is marked. **Level**: an L1 explanation uses no concept outside the prerequisite set of its level. A mechanical check against the graph confirms this. **Usefulness**: people grade it. People grade a monthly sample, and a judge grades everything. math2 also publishes the agreement between the judge and the people on that sample. LLM judges are least reliable on explanations. Thus an improvement that only a judge sees does not pass the gate without a human-graded sample.

## 17.4 Run data

Every agent run is a record (11.6). Together, the records are math2's own dataset:

- **Outcome per target**: solved or not, with cost, model, prompt version and the retrieval used. The orchestrator uses this data to learn which ladder rung to start at for each kind of target (11.4).
- **Premise use**: which retrieved premises appeared in the final proof. This data trains the premise reranker (14.5).
- **Failure classes**: timeout, elaboration loop, wrong decomposition, statement mismatch, injection attempt detected. The classification is automatic, and people check a sample.
- **Cost curves**: solve rate against spend, by role, tier and model. The weekly report (12.9) plots this curve, and the budget split (11.5) is tuned on it.

The transcripts of merged runs are public (11.6), so their outcome data is public too. Each quarter, math2 also publishes a dataset of all runs, merged or not. It contains targets, outcomes and costs, but no transcripts for unmerged runs. This applies the AGMAI and Tao position on the report of negative results and resource use (document 01) to the project's own agents.

The package `agent/dataset` and `m2 agent dataset` make the dataset of a quarter as JSON Lines, with one row for each run that started in it. A row has the job, the role at its version, the models, the outcome, the budget, the spend and the verdicts of the novelty check. It has no transcript, and the transcripts of the merged runs stay in their run records (11.6). A job on behalf of a person gives only the word `person`, and the free texts lose their email addresses, keys, tokens and home directories.

## 17.5 The novelty check

Some outputs can be described as new: an explore conjecture, a proof of an open node, a counterexample. Before that, the output goes through a novelty check, and the result of the check is attached to the node:

1. **Graph.** Search the statement plane for the statement, its negation, and its specialisations and generalisations (14).
2. **Record plane.** Search aligned and unaligned results from the literature. Use the formula, embedding and lexical channels.
3. **External indexes.** Query zbMATH Open, TheoremSearch, TheoremGraph and Matlas where their APIs allow (09.6). Also query arXiv full text of licence-permitted papers through `tamnd/arxiv`. Query MathOverflow and Math.SE through the Stack Exchange API.
4. **Problem databases.** For a problem node, check these sources: the status and comments in the erdosproblems database, the `solved` markers in the formal-conjectures repository, the comments of the OEIS entry, and the LMFDB.
5. **Verdict.** `known` (with the source), `likely-known` (close matches that a person should read), or `no-match-found`. **No verdict says "new".** `no-match-found` means that these searches found nothing. Only a person makes a statement of novelty, after they read the matches.

The check is in the `novelty` package. `novelty.Check` runs each search on the claim and keeps each search and each match in the result, thus the run record of the output can keep them. The searches are `novelty.Graph` on the search index of the build, `novelty.Records` on each source of the record plane with the channels of the aligner, and `novelty.ZbMATH`, `novelty.ArXiv`, `novelty.StackExchange` and `novelty.OEIS` on the web. The records of the problem databases give their status to the match. A judge compares the claim with the three best matches of each search. A match that states the claim, or a result that implies it, makes the claim `known`. A related match, or a match that the claim generalizes, makes it `likely-known`. With no judge, each match makes it `likely-known`, because a person must read it. A search that fails is in the result, and the check does not give `no-match-found` when a search failed. A match keeps only its source, its id, its link and the verdict of the judge (09.6). The goal of explore (12.5) runs the check on each conjecture and counterexample of the run, and the field `novelty` of the run record keeps the result by node id.

math2 evaluates the check like everything else. Its suite contains results that are known to be in the literature. Each result is phrased as it would come out of an agent. The suite includes the 2025 to 2026 rediscoveries (document 01). Measured: recall of `known`. Sometimes the novelty check misses a rediscovery and a person finds it later. That rediscovery then goes into the suite.

The suite is in `bench/novelty`. Its first items are classical results in the words of an agent output, with the node of the test tree or the paper of the suite that makes each one known, and some results that the searches do not have. With an oracle judge, the recall of `known` measures the searches: the match must be in the three best matches of a search. With the judge of explore, it also measures the judge, and the items with no match measure how often the judge calls a result known by mistake. The rediscoveries of 2025 to 2026 are not in the suite yet, because each item needs a source that a person checked.

## 17.6 External benchmarks

math2 does not optimise for public benchmarks. But it runs them, for two reasons. They put the agents of math2 on a common scale with the agents of everyone else. And the fresh benchmarks are useful held-out sets. Each release reports the results with their costs. These results never replace the suites above.

| Benchmark | What it measures | 2026 state (document 01) | Use in math2 |
|---|---|---|---|
| **miniF2F** | competition maths in Lean | widely regarded as saturated | smoke test only |
| **PutnamBench** | Putnam problems in Lean, 672 items | saturated: all 672 solved, and the leaderboard ranks by cost | cost benchmark for the prove role: dollars to solve all of it |
| **formal-conjectures** (frozen evaluation subsets) | open and solved problems as Lean statements | 2,615 statements; AlphaProof Nexus solved 9 Erdős and 44 OEIS items | research-tier prove suite (solved items); explore targets inside campaigns (open items) |
| **FrontierMath Erdős** | 68 open Erdős problems in Lean, comparator-graded, $300 cap per problem | top score 3% | the hardest prove tier; reported with spend |
| **FrontierMath Tier 4** | expert final-answer problems | declared saturated in September 2026 | not used: final answers are not math2's output |
| **First Proof** | unpublished research lemmas, graded double-blind by people, with negative scoring for wrong answers | Batch 3: 9/10 at $1,888 for the best system | the model for 17.7; math2 enters when a batch fits its roles |
| **IMO-ProofBench, MathArena proof tracks** | informal proofs graded by judges plus people | judges agree with people at about 0.9 | calibration data for math2's own judges |
| **RealMath, LemmaBench** | results mined from recent arXiv papers, refreshed monthly | contamination-resistant by construction | fresh items for the formalize and prove suites, through 09 |
| **Annals Challenge** | 50 Lean statements of recent Annals theorems | missing definitions are the blocker | a definitions-first suite for formalize (10.3) |

Contamination rules for every external benchmark:

- During an evaluation, no context that an agent sees contains the items or their solutions. Search excludes them (`bench_exclude` in the run configuration), and the canary rule checks for leaks.
- The report gives results separately for items published before and after the training cutoff of each model.
- When a run solves an item, it reports whether search returned the solution of the item or a close neighbour. Thus a "solve" by retrieval is labelled as one.

## 17.7 Graders: judges and people

Many checks in math2 end in an LLM judge: back-translation (10.4), the critic, alignment, explanation grading, and the ranking in the novelty check. The 2026 evidence about judges is mixed (document 01):

- On competition proofs, the best judges agree with human graders at about 90%. This is about as often as human graders agree with each other.
- Agreement fell when the proofs came from AI pipelines that were optimised against judges. One study found a 31.7% false-positive rate.
- Judges show a "reviewer-pleasing" bias towards confident, well-formatted text.
- A minimum over several judges reduced overestimation.

Thus math2 applies four rules to every judge:

1. **Judges triage. They never certify.** No judge verdict raises the E-level or F-level of a node. Judges decide what a person examines first, and which items go to the kernel for a check.
2. **Min over a panel.** Sometimes a judge's verdict gates something (for example, F1 needs a back-translation pass). Such a gate uses at least two judges from different model families and takes the more severe verdict.
3. **Judges are evaluated on adversarial items.** The suite of every judge includes outputs from math2's own agents that people marked wrong. If a judge cannot tell a wrong proof by an agent from a right one, math2 does not use that judge to grade agents.
4. **Human grading follows First Proof.** When people grade agent output in an evaluation, they do not know the source: agent or human, and which model. They use a rubric that gives separate scores for correctness, exposition and attribution. A wrong answer scores lower than an abstention, so a confident wrong answer costs more than no answer.

## 17.8 Reporting

The notes of each release include an evaluation section that is generated from `reports/eval/<release>/`:

- the status error bounds: the share of sampled E2 to E4 nodes whose evidence did not support their level (0.5% of each level per month, re-checked by a curator who did not add the evidence), and F-level re-review disagreement (10.8);
- the battery confusion matrix and its change since the last release;
- search headline metrics against the external baselines (14.7);
- per-role solve rates, dollars per solve, and contamination gaps between public and fresh items;
- external benchmark results with costs;
- known failures: the items each suite failed, linked.

If a number got worse, the notes say so first.
</document>

<document source="https://tamnd.github.io/math2/spec/18-coverage-all-math.md">
# 18. Coverage: what "all mathematics" means

"All mathematics" is either a slogan or a measurable target. This document makes it a target. It says what counts as covered and how math2 measures coverage per area. It also gives the size of each layer and the order in which math2 fills it. Document 19 makes milestones from this order.

## 18.1 Two planes, five levels

Document 03.3 divides math2 into a **record plane** (what exists in the literature) and a **statement plane** (what math2 understands). math2 measures coverage on both planes, per MSC2020 code, at five levels:

| Level | Name | Means, for an MSC code | Plane |
|---|---|---|---|
| **C0** | recorded | every paper and book in the area has a record, with MSC codes, identifiers and references | record |
| **C1** | indexed | results are extracted from every licence-permitted full text, and every result has a locator and an embedding | record |
| **C2** | mapped | the area's concept skeleton exists as nodes: its main concepts and definitions, and the theorems that a survey or a graduate text would name, each with at least one source | statement |
| **C3** | standard | the area's core (18.3) is at E2 or better, with proofs or locators of proofs, and explanations at L1 and L2 | statement |
| **C4** | formal | the area's core is at E5 with F2 or better, and its conjectures have reviewed formal statements | statement |

An area is at level *k* when it meets the criterion of that level. Each level is a percentage, not a yes/no. math2 should be able to say a sentence such as "MSC 20 (group theory) is 100% C2, 71% C3 and 38% C4". It should also be able to defend that sentence line by line.

C0 for all of mathematics is possible within the first year, because zbMATH Open and arXiv already did that work (document 09). C4 for all of mathematics is not possible in any planning horizon. The purpose of the measurement is to know the gap, area by area, and where the gap is closing.

## 18.2 The coverage table

Q11 (05.6) computes, for every MSC2020 code at every level of the hierarchy:

- **record plane**: records; records with extracted results; results; results aligned to nodes;
- **statement plane**: nodes by kind; nodes by E-level and F-level; nodes with explanations at each level; executable examples (08.7);
- **formal**: bound nodes by match kind; Mathlib declarations in the area that are bound to a node, and the share that are not;
- **open problems**: conjectures and problems, with their formal-statement status;
- **movement**: the change in every count since the last release.

The table goes to `reports/coverage/<release>.parquet`, with a Markdown summary of all classes in `reports/coverage/<release>.md` and a summary per top-level code in `reports/coverage/<release>/` (20.1). The workspace shows it as a map: one cell per MSC code, coloured by level. Nodes get MSC codes as topic edges (05.2). The codes come from the codes of the sources and from a classifier. For C2 nodes, a curator confirms them.

MSC2020 has 63 top-level classes and about 6,000 codes in total. math2 computes coverage at all three levels. By default, it reports the top two levels.

The denominator of C0 is the number of documents of zbMATH Open in each class. `m2 coverage -count` asks the REST interface of zbMATH Open for it, one class at a time, and pins the count with its date in `manifests/coverage/expected.yaml`. The count is for the top-level classes only, thus C0 has a value only at the top level. A record or a node counts for each of its codes and for each code above them. A code that is not in MSC2020, such as an MSC2010 code, counts for the nearest code above it.

### Which areas count

Every MSC2020 class is in scope for C0 and C1. For the statement plane:

- **00, 01, 97** (general, history, education) are record-plane only. A history-of-mathematics source can be a node's source, but history is not a node.
- **Applied areas** (35, 49, 60 to 62, 65, 68, 70 to 94) are in scope for their mathematical results: the theorems, definitions and conjectures. A model of a physical, biological or economic system is not a node unless it comes with a mathematical statement. If a formal library exists for the area (Physlib for physics, CSLib for computer science), math2 aligns to it and leaves the modelling to it.
- **68 (computer science)** overlaps with the theory of computation. Its mathematical core (complexity classes, automata, type theory) is in scope. Software engineering is not.

## 18.3 Skeletons and cores

A **skeleton** is the list that C2 counts. Each top-level MSC2020 class has at most one, in `topics/skeletons/NN.yaml`, where NN is the first two characters of the code. Its items have the fields of the items of a curriculum: a name, a kind, and at least one of a Wikipedia title, a Wikidata item and a Mathlib list, with a Mathlib declaration when one exists. An item gets the field `node` when its node exists. C2 of the class is the share of the items whose node is live and has at least one source. A class with no skeleton has no C2. One person who works in the area reviews the skeleton before it counts toward a claim such as "100% C2".


A "core" is the set of nodes that an area must have before it can claim C3 or C4. Without cores, C3 would be a percentage of an unknown total. Each core is a curated list in `topics/cores/<name>.yaml`. It is built from standard syllabi and texts, and people who teach the area review it. Cores nest:

| Core | Basis | Size (nodes: definitions, statements, key examples) |
|---|---|---|
| **undergraduate** | about 40 standard courses, from first-year analysis and linear algebra through Galois theory, measure theory, topology, complex analysis, probability, number theory, combinatorics and logic; Mathlib's `undergrad.yaml` list and the "1000+ theorems" list as cross-checks | **about 25,000** (about 600 per course) |
| **graduate** | the standard first- and second-year graduate syllabi and qualifying-exam lists of several departments; one or two standard texts per subject | about 150,000 to 250,000 |
| **research, per area** | the concepts and named theorems that recent surveys in the area cite; the area's open problem lists | about 5,000 to 50,000 per top-level class, in total about 1M |

The undergraduate core is the first serious target, for three reasons. It is bounded. It is mostly in Mathlib already, so C4 is mostly binding and review, not new proofs. Most of the items on Mathlib's own `undergrad.yaml` list were formalized by 2025, and the list records the gaps. And in this core, explanations and learning paths (Q5) help the most people.

## 18.4 How big is mathematics?

These are estimates for planning. Each estimate has a basis. When math2 has the measured number, the measured number replaces the estimate.

| Quantity | Estimate | Basis (document 01) |
|---|---|---|
| papers and books | about 5M, growing by about 150k per year | zbMATH Open's coverage (over 4.9M documents) plus arXiv math, deduplicated |
| results stated in the literature | about 30 to 60M | about 13 to 19 results per paper that states results in theorem form (TheoremSearch: 9.2M from about 690k arXiv papers; Matlas: 8M from 435k journal papers); many older and applied papers state none |
| distinct results after deduplication | perhaps a third to a half of that | duplicated restatements, lemmas restated from earlier papers, and corollaries |
| concepts | about 100,000 to 300,000 | Wikidata items with a defining formula (about 113k), nLab (about 19k pages), Wikipedia mathematics (about 26k articles) and the Stacks Project (about 21k tags) overlap heavily, and each covers part of it |
| formal declarations | about 430k in Mathlib, and several million across all Lean libraries | Mathlib's own statistics; Tau Ceti is about 3M lines alone |
| open problems with a formal statement | about 1,000 open in formal-conjectures, plus FrontierMath Erdős and Annals-Challenge-style collections | those repositories |

Thus the statement plane at full graduate coverage, plus research cores, is about 1 to 2M nodes. If OEIS sequences and the most-cited objects are also nodes, it is 2 to 5M (06.7). The record plane is about 5M records and 30M results. Both fit the storage design without change.

## 18.5 The order of work

Breadth first on the record plane, depth first on the statement plane:

1. **C0 everywhere.** It is cheap. It makes everything else possible, because the source of every node needs a record to point at.
2. **C2 skeletons everywhere.** For each top-level MSC class: the 200 to 2,000 concepts and named theorems that define the area. These skeletons make the graph navigable across all of mathematics, even where the graph is shallow. This is curated work with much agent help. The proposals come from surveys, the tables of contents of textbooks, and Wikipedia's category structure.
3. **C3, then C4, for the undergraduate core.** Then the same for graduate cores, in order of Mathlib coverage, because a binding costs less than a proof.
4. **C1 everywhere,** as `tamnd/arxiv`'s content plane and the licence gate allow.
5. **Research cores on demand.** math2 builds the research core of an area when a project (13.6) or a campaign asks for it. The reason is that research-level curation needs the people who work in the area.

Two rules keep this honest:

- **The table is published every release, including the regressions.** Coverage can decrease. A Mathlib bump can break bindings, or a re-review can lower an F-level. The table shows that.
- **Counts are not the target.** People who teach the area review the composition of each core. Sometimes the node count of an area grows while its core coverage stays flat. That is a curation problem, and the coverage report is designed to show exactly that.
</document>

<document source="https://tamnd.github.io/math2/spec/19-milestones.md">
# 19. Milestones

There are six milestones, M0 to M5. Each one ends in a release. Someone outside the project can check that release against written exit criteria. The `m2` command table (20.2) names the milestone in which each command arrives. A command that has not arrived exits non-zero and says which milestone delivers it.

The dates assume that work starts in November 2026 with a small core team. The team is two to three engineers, one Lean lead, and a curator group that grows from five to about thirty people. Each milestone has a short "out of scope" list. Thus work that slips is visible and does not move quietly to the next milestone.

## M0: the skeleton (Nov to Dec 2026)

**Deliverables**

- Both repositories, with the layout in document 20, the licence files, and contribution guides.
- The node schema v1 (04.5) as JSON Schema, with fixtures.
- Tag minting and the provisional-id flow (04.3).
- `m2 new`, `m2 fmt`, `m2 check`: schema, body grammar and link checks on a working tree, with no Lean.
- CI for both repositories. The `lean/` package pins a Lean stable release and a Mathlib tag (07.1), and builds an empty `Math2` with the module system.
- This specification, committed as `tamnd/math2-agent/spec/`.
- The public wiki at https://tamnd.github.io/math2. `m2 site build` makes it from `main`, and each push to `main` updates it. It has a page for each node and each spec document, and JSON, Markdown and `llms.txt` for agents. The plan moved this from M3 to M0, so that readers and agents can use the graph from the start.

**Exit criteria**

- `m2 check` runs on 100 hand-written fixture nodes that cover all 11 kinds. It catches every planted error in a fixture set of 200 broken nodes.
- A contributor who is new to the project uses only the README. With it, they can add a node with a provisional id, open a PR, and have the PR pass CI.
- The wiki shows the spec, the milestones and each node on `main`, with no manual step.

**Out of scope:** status computation, graph, any agent.

## M1: the seed and the trust core (Jan to Mar 2027)

**Deliverables**

- **The bootstrap seed**: about 2,000 curated nodes, with tags that begin with `0` (04.3). They cover the concept skeleton of first- and second-year undergraduate mathematics, with sources, edges and Mathlib bindings.
- `m2 build`: Lean build, extraction (07.4), status computation (04.7) and the derived graph.
- `m2 audit` with every T, S, B, G, F and V rule (15.3), and `m2 status`.
- The checker stack: `lake check --paranoid` and the comparator against pinned statements (07.7), with checker versions recorded.
- Fidelity reviews as data (10.6), signed by hand. There is no review tool yet.
- **Release 2027.03**: the first numbered release, with a DOI, archived to Software Heritage and Zenodo.

**Exit criteria**

- Every E5 node in the seed passes replay in at least two independent kernels.
- Every seed node is at E2 or better, and at least 30% are at E5 with F2.
- The release rebuilds bit-for-bit from its manifest on a second machine (rule `R01`).
- An external Lean user reviews 50 random F2 bindings and agrees with 49 or more.

**Out of scope:** ingestion beyond Mathlib, search, the workspace.

## M2: the graph meets the world (Apr to Jun 2027)

**Deliverables**

- `m2 graph`: queries Q3 to Q7 (05.6).
- `m2 ingest` for the first sources:
  - arXiv metadata through `tamnd/arxiv`;
  - zbMATH Open;
  - Mathlib (full extraction);
  - Wikidata;
  - OEIS;
  - formal-conjectures;
  - erdosproblems;
  - Palomar;
  - the Stacks Project (tags only);
  - LMFDB (labels).
- `m2 align`, with crosswalk edges and the curation queue (16.3).
- `m2 bump`: the monthly Mathlib bump, with its report (07.9).
- `m2 release`: monthly releases from this point. Each release pins a new Mathlib tag when the bump is green.

**Exit criteria**

- **C0 at 100%** for every MSC class: every zbMATH Open and arXiv math record is in the record plane.
- Three consecutive monthly bumps complete with no hand edits outside `Math2/` repairs.
- Alignment precision at least 0.98 on the held-out set (17.3.3).
- Q1 to Q7 meet their p95 targets at reference scale. Where the real graph is still smaller, the test uses synthetic nodes (02.4).

**Out of scope:** extraction of results from full text, public search.

## M3: open to readers (Jul to Sep 2027)

**Deliverables**

- `m2 search` with all four channels (14), and the benchmark in 14.7.
- `m2 serve`: the read API, the read-only MCP profile, and the workspace's node pages, search, explore views and graph (13).
- `m2 compute`: the computational engine with certificates (08). Executable examples have a "run this" button.
- `m2 prov`: show, credit and quarantine.
- Result extraction from licence-permitted arXiv full text (09.5), for C1.
- The offline archive of each release. The public site starts in M0. M3 adds the explore views, the graph view and the API to it.

**Exit criteria**

- The search benchmark beats the best external baseline on concept lookup and paper-to-node. The release notes report every metric on which it does not.
- The workspace meets the performance and accessibility targets of 13.7.
- Any MCP client, Claude Code included, can use the read-only profile against the public endpoint with no math2-specific code.
- C2 skeletons exist for 30 top-level MSC classes.

**Out of scope:** write access through the API, autoformalization.

## M4: fidelity at scale (Oct to Dec 2027)

**Deliverables**

- `m2 review`: the reviewer's tool and review packets (10.6).
- `m2 autoform`: the formalize pipeline with the full F1 battery (10.2 to 10.4).
- Fidelity evaluation: the battery suite with its confusion matrix (17.3.1), and the monthly re-review sample (10.8).
- Write access through the API, which opens PRs (11).
- Reviewer onboarding and the credit export (16.4).

**Exit criteria**

- **The undergraduate core (18.3) is at C3 100% and C4 50%.**
- Measured F2 disagreement is under 2%, from at least 500 re-reviews.
- The battery's recall is at least 0.9 on every misformalization class with 30 or more labelled items.
- At least 50 active reviewers, none of whom is a project employee.

**Out of scope:** unattended agent runs.

## M5: the research engine (Jan to Jun 2028)

**Deliverables**

- `m2 agent run` for every role in 12.1, and `m2 orchestrate` with budgets, the escalation ladder and the work queues (11.4, 11.5).
- Role versioning, shadow weeks and the weekly agent report (12.8, 12.9).
- Projects in the workspace, with project budgets for agent work (13.6).
- The novelty check (17.5) for every explore output.
- The quarterly public run dataset (17.4).

**Exit criteria**

- **Undergraduate core at C4 80%**, and every E5 node in it at F3. Most of the F3s come from cross-formalization.
- C2 skeletons for every top-level MSC class in scope (18.2).
- People outside the core team run at least three research projects. Each project produces work that they publish elsewhere.
- Per-role solve rates and costs are published for 12 consecutive weeks.
- No agent output raised a status level without a kernel check or a person (rule audit `X`, zero violations).

## After M5

The milestones stop at M5, because at M5 math2 is a complete system. After M5, the work is about depth: graduate cores, research cores on demand, C1 everywhere that licences allow, and more formal libraries bound. Releases continue monthly. Every release reports the axes in 02.3. The targets for 2029 will come from the measurements, and math2 does not set them now.

## Costs, roughly

These are planning numbers. math2 publishes them so that a funder can compare them with the weekly reports.

| Item | M0 to M2 | M3 to M5 |
|---|---|---|
| people | 3 to 4 FTE | 5 to 6 FTE plus paid reviewer stipends |
| model inference | under $5k per month, mainly alignment and extraction in batch | $20k to 60k per month, set by budget, not demand (11.5) |
| compute (CI, Lean runners, serving) | about $3k per month | about $10k per month |
| storage and archiving | small | under $1k per month |

The model line is the only line that can grow without limit. It is also the line with the clearest controls. Batch pricing, prompt caching and the escalation ladder (12.3) keep most calls on the cheapest model that passes the suite of the role. The weekly report shows what each dollar bought.
</document>

<document source="https://tamnd.github.io/math2/spec/20-repository-layout.md">
# 20. Repository layout

There are two repositories. They use the same split as `tamnd/arxiv` and `tamnd/arxiv-reader`:

- `tamnd/math2` is the **content**. It is the thing that must be right.
- `tamnd/math2-agent` is the **toolchain and the agents** that build, check, grow and serve the content. Its binary is `m2`.

Content never depends on the toolchain's code. A person with a text editor, `lake` and Python can read, edit and check the core of `tamnd/math2`. For this, they never need to install `m2`. The full audit needs `m2`, in the same way as the full audit of `tamnd/arxiv` needs `ax`.

## 20.1 `tamnd/math2`

```
math2/
  README.md                      what it is, the counts table (generated), how to check it
  LICENCE                        CC BY-SA 4.0 (prose), Apache-2.0 (code), CC0 (metadata). See 16.6
  CONTRIBUTING.md                how people and agents contribute; disclosure; review
  CITATION.cff                   how to cite a release and a node

  schema/
    node.v1.json                 JSON Schema for node front matter, per kind
    review.v1.json
    computation.v1.json
    manifest.v1.json
    edges.md                     the 23 edge types (document 05.2), generated from the schema

  macros/
    m2.sty                       the only LaTeX macros a body may use, KaTeX/MathJax compatible
    m2.katex.json                the same macros for KaTeX

  nodes/
    7K/7K3QX2.md                 one file per node, sharded by the first two tag characters
    ...

  compute/
    3MZ8QD/main.gp               code for computation nodes, next to the tag
    3MZ8QD/Check.lean

  reviews/
    7K/7K3QX2.yaml               fidelity reviews, bound to statement and type hashes

  prov/
    7K/7K3QX2.jsonl              provenance, one line per field change

  registers/
    tags.reg  aliases.reg  retired.reg  bindings.reg  reviewers.reg  crosswalk.reg

  queues/
    align/wikidata.jsonl         proposed same_as, close_to and aligned edges, one file for each external system (16.3)
    dedup.jsonl                  proposed duplicate pairs (05.4)

  lean/                          the math2 Lean library (a Lake package)
    lakefile.toml                requires mathlib at the pinned commit
    lean-toolchain               the pinned Lean release
    Math2.lean
    Math2/
      Meta/Attr.lean             @[m2 "TAG"] attribute, the binding marker
      Meta/Match.lean            adapters: stating that a Mathlib decl matches a node up to a recorded transformation
      Algebra/ …                 statements and proofs math2 holds that Mathlib does not (yet)
      NumberTheory/ …
      Conjectures/ …             formal statements of open conjectures (Prop-valued defs, never axioms)
      Hypotheses/ …              the only place `axiom` may appear: named hypotheses such as GRH, for conditional results. Each is a node
      Cert/ …                    certificate checkers proved sound (LRAT, Pratt, interval)
    test/                        golden tests for the attribute and the adapters

  topics/
    msc2020.yaml                 MSC2020 codes → topic tags, with our own short labels. The official class descriptions are CC BY-NC-SA, so they are
                                 not stored here; the site fetches them from a pinned snapshot and shows them with attribution (16.6)
    curricula/                   reading lists (undergraduate core, qualifying exams). Each one is a topic with ordered members
    skeletons/                   the C2 skeleton of each top-level MSC2020 class, one file NN.yaml for each class (18.3)

  manifests/
    releases/2026.11.yaml        release manifests (document 06.2)
    releases/2026.11.levels      the evidence and fidelity level of each live node at the release
    releases/2026.11.md          the release notes: a summary by a person, and the sections that m2 release notes writes
    sources.yaml                 every external snapshot used: URL, hash, date, licence (document 09)
    images.yaml                  computational images by digest
    ingest/                      per-source import manifests
    coverage/expected.yaml       the pinned number of documents in each MSC2020 class, the denominator of C0 (18.2)

  reports/
    audit/                       committed audit reports (document 15)
    bump/                        one report for each Mathlib bump: the breaks by kind, the time to green, the stale reviews (07.9)
    graph/                       graph health
    coverage/                    per-MSC coverage tables (document 18)
    releases/                    the full pre-release report per release
    restore/                     one record for each quarterly restore drill (06.6, rule R04)
    agents/                      weekly agent acceptance and cost reports (document 12.9)
    eval/                        per-release evaluation reports (document 17.8)

  .github/
    workflows/                   check.yml (schema+body+Lean build on PR), audit.yml (full on main), bump.yml (weekly Mathlib bump), release.yml
    scripts/check_nodes.py       the Python-only check that needs no m2: schema, tags, links, sections
```

### The Lean package

`lean/` is an ordinary Lake package named `Math2`. It depends on Mathlib at the pinned commit. It holds only what Mathlib does not have. A node bound to a Mathlib declaration has no Lean code in math2 at all. It has only the binding in its front matter.

Sometimes math2 holds a theorem and Mathlib gains it later. Then the bump agent proposes two changes: the binding moves to Mathlib, and the math2 copy becomes deprecated (document 07.9). The aim is that `Math2/` gets smaller in each area as Mathlib grows. The aim is also that math2's Lean contributions move upstream (document 16.5).

`@[m2 "7K3QX2"]` on a declaration is the binding seen from the Lean side. The build checks that the attribute and the front matter agree (rule `F01`). The attribute is the only Lean metaprogramming in the content repository. Everything else that must inspect Lean (dependency extraction, type hashing, axiom audit) is in the toolchain.

## 20.2 `tamnd/math2-agent`

It is a Go module for the toolchain and the agents. It also has a small Lean package for extraction and a web front end for the workspace.

```
math2-agent/
  README.md
  go.mod                         module github.com/tamnd/math2-agent
  cmd/
    m2/                          the binary

  check/                         the fast rules of m2 check: S01 to S04, B01 to B03, G01 (15.3)
  cli/                           the command table and the flags of each command
  format/                        m2 fmt: canonical front matter, aliases to tags (04.4, 04.5)
  node/                          parse, canonicalise, hash node files (document 04)
  schema/                        schema validation, migrations
  tag/                           minting, provisional id rewrite, registers
  graph/                         derive edges, closure (AND–OR), CSR, Parquet writer (document 05)
  store/                         SQLite + DuckDB + CSR + vector serving store (05.7)
  build/                         the build graph: nodes → Lean → compute → graph → search → site
  bump/                          the Mathlib bump: pins, breaks, renames and the bump report (07.9)
  compute/                       image runner, sandbox, certificate dispatch (document 08)
  ingest/                        source adapters (document 09), one subpackage per source
    arxiv/  zbmath/  crossref/  oeis/  lmfdb/  wikidata/  mardi/  nlab/  stacks/  proofwiki/  mathlib/  leanlibs/
    afp/  palomar/  blueprint/  erdos/  formalconj/  mo/  objects/  theoremidx/
  extract/                       statement extraction from papers (09.5)
  align/                         dedup and crosswalk (05.3, 05.4)
  autoform/                      statement autoformalization and fidelity pipeline (document 10)
  search/                        lexical, formula, embedding, type-directed search (document 14)
  novelty/                       the novelty check: the searches, the judge and the verdict (17.5)
  audit/                         one file per rule, with fixtures (document 15)
  prov/                          provenance writing, PROV export, quarantine (06.5)
  release/                       release manifest, notes, archive (06.2)
  api/                           the Mathematical API: HTTP+JSON, gRPC (document 11)
  mcp/                           the MCP server over the API
  orchestrator/                  job queue, scheduler, budgets, run records (document 11)
  agent/                         the research agents (document 12)
    formalize/  prove/  explore/  explain/  curate/  align/  bump/  review-assist/
  llm/                           model clients (Anthropic SDK first; adapters for local prover models)
  sandbox/                       gVisor/Firecracker wrappers shared by compute and agents
  uri/                           m2:// parser and the ant driver
  site/                          the wiki: a static site for people and agents

  lean/                          drive Lake, the Lean REPL and the extraction package; axiom audit; replay (document 07)
    M2Extract/                   Lean package: dependency graph, elaborated-type hash, axioms, statement pretty-printing, degenerate-statement checks
    M2Extract/lakefile.toml      depends on Mathlib and on tamnd/math2's Math2 package at the pinned commits

  prompts/                       versioned prompts per agent role; each change bumps the role version (12.8)
  images/                        Nix flakes for every computational image (08.1)
  web/                           the research workspace front end (document 13)
  bench/
    graph/                       Q1 to Q12 latency benchmarks (05.6)
    search/                      retrieval benchmarks (14.7)
    fidelity/                    the F1 battery suite (17.3.1)
    extract/                     extraction and unfolding (17.3.2)
    align/                       alignment precision and recall (17.3.3)
    agents/                      per-role suites (public and fresh splits; held-out items live in a private repository) and canaries (17.3.4, 12.7)
    explain/                     explanation grading (17.3.5)
    novelty/                     the novelty check's recall suite (17.5)
    external/                    runners for external benchmarks (17.6)
    prov/                        provenance tests (06.5)
  migrate/                       schema migrations (04.9)
  docs/                          command reference, generated
  spec/                          this specification (documents 00 to 21)
```

The module has no `internal/` directory. Each Go directory is a package that other programs can import, for example a bot that reads node files with `node` and checks them with `schema`. Thus each package follows the conventions of the Go standard library. It has a package comment, a doc comment on each exported name and testable examples. It exports only what other programs need. A change to an exported name is a breaking change, and the release notes say so.

### The command set

Each command names the milestone in which it arrives (document 19). If the milestone of a command has not arrived, the command exits non-zero and says which milestone delivers it. It does not pretend to succeed. That is the `ax` convention.

| Command | What it does | Milestone |
|---|---|---|
| `m2 new <kind>` | scaffold a node with a provisional id | M0 |
| `m2 fmt` | canonicalise front matter, rewrite aliases to tags | M0 |
| `m2 check` | schema, body, links on the working tree (fast, no Lean) | M0 |
| `m2 mint [-commit C]` | change provisional ids to tags before merge, and add them to `registers/tags.reg` | M0 |
| `m2 site build` | the static wiki for GitHub Pages: HTML for people, JSON, Markdown and `llms.txt` for agents | M0 |
| `m2 build` | full build: Lean, extraction, status, graph, search | M1 |
| `m2 audit [-plane X] [-rule R] [-changed] [-release R]` | the audit (document 15) | M1 |
| `m2 status <tag>` | the computed status and the evidence behind it | M1 |
| `m2 graph closure|rdeps|path|frontier <tag>` | graph queries Q3 to Q7 | M2 |
| `m2 ingest <source> [--snapshot]` | pull a pinned snapshot of a source (document 09) | M2 |
| `m2 align <source>` | propose crosswalk edges | M2 |
| `m2 search <query>` | the search engine (document 14) | M3 |
| `m2 serve [--addr]` | the API, the MCP server and the workspace | M3 |
| `m2 compute run|verify <tag>` | the computational engine (document 08) | M3 |
| `m2 review <tag>` | the reviewer's tool: side-by-side informal and formal, back-translation, checks, sign | M4 |
| `m2 autoform <tag>` | statement autoformalization for a node | M4 |
| `m2 agent run <role> [--target T] [--budget B]` | one agent run | M5 |
| `m2 orchestrate` | the scheduler daemon | M5 |
| `m2 bump` | Mathlib/Lean bump branch (document 07.9) | M2 |
| `m2 prov show|quarantine|credit|disclose` | provenance operations; `disclose` renders a disclosure paragraph and `formalization.yaml` (16.5) | M3 |
| `m2 release cut|notes|verify` | releases | M2 |

### Why Go for the toolchain

The rest of the `tamnd` toolchain is Go: `ax`, `ant`, and the `*-cli` family. `m2 ingest` links the packages of that family as libraries. Go builds one static binary and cross-compiles easily. It has the official Anthropic SDK, DuckDB and SQLite bindings, and good sandbox control. Lean-specific work happens in Lean (`lean/M2Extract`), driven over the Lean REPL protocol. Python appears only inside computational images and in the no-dependency check script in `tamnd/math2`.

One exception may come later: a Python sidecar for local open-weight prover models (document 12.3). The reason is that the inference stacks of these models are in Python. If the sidecar exists, it sits behind a gRPC interface in `llm`. Nothing else in the toolchain knows about it.

## 20.3 How the two repositories move together

- `tamnd/math2-agent` is versioned by semver. `tamnd/math2` is versioned by calendar (document 06.2).
- Each math2 release names the `m2` version that built it. CI in `tamnd/math2` uses the `m2` version pinned in `manifests/releases/next.yaml`. A person bumps that version in a PR.
- A schema change lands in `math2-agent` first, behind a flag. Then a migration PR lands in `math2`. Then the flag is removed. The two repositories never change in a single step.
- Agents open PRs against `tamnd/math2` from a bot account (document 16.2), with branch names `agent/<role>/<run-id>`. A person merges them, or for some roles a rule-limited auto-merge does (document 12.1).
</document>

<document source="https://tamnd.github.io/math2/spec/21-open-questions.md">
# 21. Open questions

This document lists the decisions that are not yet settled. Each entry gives the question, the options, and the current default (what the code does until the question is settled). It also gives **what would settle it**, which is usually a measurement. When a question is settled, its entry moves to the decision list in document 00, with a link to the PR and the measurement (16.7).

## 21.1 Content and model

**Q-01. Is `concept` separate from `definition` worth its cost?**
- Options: keep both kinds (04.2); or merge them, so that a concept is just its preferred definition plus equivalences.
- Default: keep both.
- Settles it: after M2, measure how many concepts have more than one definition node. Also measure how often curators make the split incorrectly (audit `S` findings on concept/definition misuse). If fewer than 10% of concepts have a second definition and the error rate is high, merge.

**Q-02. How should proof steps be represented?**
- Options: proofs as nodes with `uses` edges only (current); or a finer structure, with steps as sub-nodes, as in Lamport-style structured proofs or the Fermat run's DAG of intermediate statements.
- Default: coarse. An intermediate statement that is worth citing becomes its own `statement` node.
- Settles it: whether explain and the prove decomposition (12.4) produce many orphan intermediate statements (flag `orphan`). If they do, a lighter sub-node kind may be cheaper than full nodes.

**Q-03. How much of the OEIS and the LMFDB becomes nodes?**
- Options: every sequence and object as a node; only those cited by another node (current for the LMFDB); or all OEIS sequences but only cited LMFDB objects (current for the OEIS).
- Default: as stated in 06.7.
- Settles it: the repository size trigger (Q-13), and whether uncited object nodes ever receive edges.

**Q-04. Should math2 adopt a second informal format besides Markdown with LaTeX?**
- Options: stay with constrained Markdown (04.5); or allow Verso documents for nodes that are mostly Lean.
- Default: Markdown only.
- Settles it: the share of blueprint imports (09.4) that come from Verso, and whether the round trip loses information.

## 21.2 Formal

**Q-05. Isabelle, Rocq and Metamath bindings at E5?**
The AFP binding path exists (07.10). Rocq and HOL Light are crosswalk only. Metamath is a candidate for v2: its verifier is tiny and has independent implementations.
- Default: Lean and AFP only.
- Settles it: demand from projects, and whether the checker of a second system can go through the same replay discipline as the Lean checker (07.7).

**Q-06. What fraction of the build should go to replay?**
`lake check --paranoid` across several kernels is slow at FLT scale. 07.7 allows `external_replay` for proofs above a size threshold.
- Default: full replay for everything under 1M lines, and an external replay record for the rest.
- Settles it: measured replay time per release. If it fits the 24-hour build budget (02.4), the threshold is removed.

**Q-07. When should Math2 code move to Mathlib?**
- Options: as soon as it is stable; only on a Mathlib maintainer's request; or never move it, and let Mathlib cherry-pick.
- Default: a person offers it when its sanity lemmas and review are complete (16.5).
- Settles it: Mathlib's own policy on AI-assisted contributions, which is evolving, and the review-queue data from the Mathlib Initiative.

**Q-08. How should math2 handle Comparator's open issues?**
Comparator had known gaps in October 2026. 07.7 adds math2's own type-hash check as a second statement check.
- Default: both checks.
- Settles it: when Comparator ships in the Lean distribution with those issues closed. Then decide whether the type-hash check is still necessary. Measure whether it ever caught anything that Comparator missed.

## 21.3 Agents

**Q-09. Should math2 run its own fine-tuned prover models?**
- Options: hosted frontier models plus open prover models as they are released (current, 12.3); or fine-tune an open model on math2's run data (17.4).
- Default: no fine-tuning.
- Settles it: whether the quarterly run dataset shows a solve-rate gap on math2's own targets. It would have to be a gap that a fine-tune could close at a lower cost per solve than the ladder. A 2026 finding makes it likely that the default holds: general models with a minimal loop match bespoke provers (12).

**Q-10. Should agents be allowed to choose explore targets?**
12.5 says no: people define campaigns. Some open problems would plausibly benefit from wide automatic sweeps. For others, there are requests that they not be "harvested" (the 2026 debates around erdosproblems.com and the `openai/math` release).
- Default: no autonomous target selection.
- Settles it: this is a community decision, not a measurement. The proposal would need maintainer approval plus an RFC open for a month.

**Q-11. How should agent output be paced to match review capacity?**
Agents can produce much more than people can review. That is the Mathlib queue problem, and the lesson of the 719-manuscript release.
- Default: the orchestrator sets a weekly limit on the PR rate of each role. The limit is a multiple of the review throughput of the previous four weeks (11.5).
- Settles it: the multiple is tuned on the weekly report (12.9). The open question is whether review-queue age is a better control signal than throughput.

**Q-12. Should Managed Agents run the explain, curate and align roles?**
12.3 lists hosted agent infrastructure as an option for the long-running roles that need no Lean session.
- Default: self-hosted loop for every role.
- Settles it: a month of shadow runs that compare cost, latency and suite scores.

## 21.4 Storage and scale

**Q-13. When does the record plane move to its own repository?**
06.1 sets the trigger: more than 10 GB packed, or more than 5 million files, in `tamnd/math2`.
- Default: one repository until the trigger.
- Settles it: the trigger. The open part is whether to split by plane (`tamnd/math2-records`) or by MSC range. Split by plane, unless clone-time measurements say otherwise.

**Q-14. Should math2 serve vectors?**
14.4 forbids the return of raw embeddings. This protects record-class text and keeps the model swappable. Some research users want vectors for their own analysis.
- Default: no vectors.
- Settles it: a licence review of whether embeddings of `open` class text could be published, and whether anyone asks for them through a project.

## 21.5 Governance

**Q-15. Who are the maintainers, and how are they chosen?**
16.7 assumes a maintainer group with two-approval rules. It does not say how the group is formed after the founding team.
- Default: founding maintainers, then nominations by existing maintainers with a public comment period.
- Settles it: a governance document, before M3. The models to consider are Mathlib's maintainer process, the OEIS editors and the Stacks Project's single-editor model.

**Q-16. Should reviewers be paid?**
19 budgets reviewer stipends from M3. Payment for review could compromise independence. Or it could be the only way that research-level review happens at all.
- Default: stipends for undergraduate and graduate core review only. Research-level review is volunteer and credited.
- Settles it: measured re-review disagreement for paid against unpaid reviews (10.8).

**Q-17. Relationship with Palomar and erdosproblems.com.**
Both are community registries with their own norms. math2 imports from both and can export to Palomar (16.5).
- Default: import and link; never write to them except by people following their processes.
- Settles it: a conversation with their maintainers, before M2.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/00-README.md">
# Math 2.0: the corpus specification

The main spec (the files `00` to `21` in the parent directory) says how math2 holds, checks and serves mathematics. This sub-spec says how math2 gets the mathematics: each definition, theorem, proof and example that a person can find in a public source, and what models know, as nodes that people can read and machines can check, ready to go into a course, a book or an article (K2.1).

The corpus sub-spec adds to the main spec and changes it in a small number of named places. Where the two disagree, this sub-spec is the newer text, and the change is listed in "Changes to the main spec" below. The main spec files are not edited by this sub-spec. A change becomes part of them in the same pull request that implements it.

This sub-spec was written on 2026-10-26. Its evidence base is K01, which reflects the sources as of that date.

## Documents

| # | Document | What it settles |
|---|---|---|
| 00 | README (this file) | index, reading order, settled corpus decisions, conventions, the rules that the corpus adds |
| 01 | [Research](01-research.md) | the method, the findings that change the plan, the corrections to the main spec, what could not be verified |
| 01a | [Part A: reference works](01a-reference-works.md) | encyclopedias, wikis, databases, problem lists and Q&A sites: sizes, licences, channels |
| 01b | [Part B: literature and books](01b-literature-and-books.md) | literature indexes, full text, extraction projects, open textbooks, syllabi, theorem lists |
| 01c | [Part C: formal libraries](01c-formal-libraries.md) | formal libraries, translation between systems, formal and informal datasets |
| 01d | [Part D: model generation](01d-model-generation.md) | error rates, projects at scale, memorization, the law on model output, provider terms, cost |
| 01e | [Part E: representation and packaging](01e-representation-and-packaging.md) | formats, levels of explanation, exercises, book and course formats, notation |
| 02 | [The goal](02-the-goal.md) | the goal in one sentence, items, writing levels W0 to W3, formal outcomes, what the corpus must not do, the reference scale |
| 03 | [Identifiers](03-identifiers.md) | tags of ten characters, the check character, sharding, one space for nodes and inventory rows |
| 04 | [The inventory](04-inventory.md) | mentions and rows, clustering, outcomes, which declarations and objects become nodes, measuring what is missing |
| 05 | [Human sources](05-human-sources.md) | the source document, licence reading, public domain, reference works, textbooks, literature, curricula, exercise banks, adapters, import order |
| 06 | [Formal sources](06-formal-sources.md) | intake classes, the libraries, Mathlib, libraries that AI wrote, other systems, crosswalks, datasets, informalization |
| 07 | [Machine generation](07-machine-generation.md) | what models are for, the four new roles, generation from what models know, the write job, the model register, configurations, provider terms |
| 08 | [The node card](08-the-node-card.md) | the house style, front matter, the card of each kind, proofs in steps, explanations at levels, exercises, schema changes |
| 09 | [Quality](09-quality.md) | the gates before merge, sanity checks, the copy check, source confirmation, what people check, sample audits, reader reports |
| 10 | [Structure and notation](10-structure-and-notation.md) | the taxonomy, curricula, the notation register, conventions, names and languages |
| 11 | [Packaging](11-packaging.md) | the volume file, linearization, editorial text, outputs, the pack check, licence, editions, the Lean companion |
| 12 | [Pipeline and operations](12-pipeline-and-operations.md) | the stages of an item, jobs, priority, batches, review capacity, cost, measurement, commands and libraries |
| 13 | [Law and ethics](13-legal-and-ethics.md) | the three zones, copyright in model text, copies, human sources, the EU AI Act, provider terms, disclosure, questions for a lawyer |
| 14 | [Roadmap](14-roadmap.md) | phases 0 to 5 with entry and exit criteria, what runs all the time, the people that the phases need |
| 15 | [Open questions](15-open-questions.md) | OQ01 to OQ60, each with the way to settle it and the phase that it blocks |

## Reading order

- **Overview:** K02, then K14, then the settled decisions below.
- **Before an importer is written:** K05 or K06, then K04, K13 and K15.
- **Before agents write nodes:** K07, K08 and K09, then K12.
- **Before a volume is built:** K08, K10 and K11.
- **Reviewers of the design:** K01 first, then K09, K13 and K15. K01 is the evidence; K09, K13 and K15 are where the design could be wrong.

## How the corpus relates to the main spec

The corpus uses the objects of the main spec and adds no new box to its architecture:

| The corpus needs | The main spec gives | The corpus adds |
|---|---|---|
| a place for each item | nodes and kinds (spec 04) | the inventory as a register of rows (K04), the exercise node (K8.7) |
| sources | the record plane and licence classes (spec 09) | the source document, licence reading, the public-domain rule (K05), intake classes (K06) |
| writers | agent roles and run records (spec 11, 12) | four new roles, the model register, generation from what models know (K07) |
| checks | the audit and the status levels (spec 15) | the copy check, source confirmation, sample audits, new rules (K09) |
| readers | the workspace and explanations at levels (spec 13) | the node card (K08), the notation register (K10), volumes and the pack check (K11) |

## Settled corpus decisions

These decisions are settled. A change to one of them is a change to this file and to the document that it names, in one commit, with the reason.

1. **Tags have ten characters.** Nine random characters and one check character, from Crockford base32. The tags of six characters that exist are frozen and stay valid (K03).
2. **Nodes and inventory rows share one identifier space.** A row identifier is never a node tag (K3.6, rule T08).
3. **The inventory is the denominator.** The corpus is complete when each row has an outcome. The open count is the number of rows with no outcome, and its target is zero (K04).
4. **No row from models alone.** A mention that a model gave becomes a row only when a source that people wrote confirms it (rule I03).
5. **The licence class decides what the corpus takes from a source.** Only `open` text enters the prose zone. `share-alike-nc`, `gfdl` and `record` give records, locators and links. `verbatim` gives short quotations only (K5.1, K13.4).
6. **The licence is read at the source.** An aggregator does not change a licence. The licence is read again at each snapshot, and the reading has a date and a person (K5.2.1).
7. **Public domain means the US and the EU.** A work is `open` as public domain only if it is in the public domain in both (K5.2.3).
8. **Each formal library has one intake class:** `bind`, `crosswalk`, `link`, `evaluate` or `exclude` (K6.2). Benchmarks evaluate roles; they are not sources (K6.8).
9. **Models write; they never decide what is true.** Models never raise a status, and a judge is of a family other than the writer's (K07, rule X09). This keeps settled decision 3 of the main spec.
10. **The copy check runs against a reference corpus,** not only against the cited sources. No restricted text is in the context of a write job (K9.4, K7.5).
11. **The prose zone stays CC BY-SA 4.0,** with a notice that the licence applies only where copyright subsists. The provenance says which sections software wrote (K13.2). This keeps settled decision 17 of the main spec.
12. **The exercise is a node.** Its text is in the prose zone. Its Lean statement and its reference solution are in the code zone, Apache-2.0 (K8.7, K13.1).
13. **One notation register.** Each macro has one meaning in its scope, and the LaTeX macros and the KaTeX map are generated from the register (K10.4).
14. **A volume adds no mathematics.** Each definition, statement, proof and example in a volume is a node, and glue states no mathematics (K11).
15. **Review capacity sets the speed.** Agent work is capped by what people can review, not by the cost of the models (K12.5).
16. **Phases have entry and exit criteria, not dates** (K14).

## Conventions

- **Numbering.** Corpus documents are K01 to K15, and their sections are K2.1, K2.2 and so on. A reference to "spec 09" or "spec 9.3" is to the main spec. A reference to "settled decision 17" is to the list in the main README.
- **Rules.** A rule has an id in the families of the main spec 15, and **H** (hard: the audit fails) or **S** (soft: the audit reports). The rules in this sub-spec are proposed. They become part of the audit when `m2 audit` implements them, and the main spec 15 lists them in the same pull request.
- **Not settled.** Where the sources disagree or a fact is not known, the text says "not settled" or names the question in K15, and it takes the safe course (K15.1).
- **Facts and dates.** Each fact from the research has a URL and the date of reading in K01. A figure that a later reading changes is changed in the corpus document and in the part of K01, in the same commit (K1.6).
- **Style.** The text follows ASD-STE100 Simplified Technical English.

## Rules that the corpus adds

| Rule | H/S | Checks | Where |
|---|---|---|---|
| T06 | H | changed: a tag has six or ten characters | K3.8 |
| T08 | H | the tag register and the inventory register have no value in common | K3.6, K3.7 |
| T09 | H | each tag of ten characters has a valid check character | K3.7 |
| T10 | H | no tag of six characters is minted after the cut-over commit | K3.7 |
| I01 | H | each `inventory` value in a node points to a row whose outcome is `node` or `member` with the tag of that node | K4.9 |
| I02 | H | each row with the outcome `node` or `member` points to a live node, or to a retired node with a successor | K4.9 |
| I03 | H | no row has only mentions from models | K4.9 |
| I04 | H | each outcome that a rule gave names a rule in `schema/outcomes.v1.yaml` | K4.9 |
| I05 | S | a `deferred` row whose date has passed | K4.9 |
| I06 | S | an MSC code at two digits whose estimated recall is below 90% | K4.9 |
| L05 | H | each agent-written body has a clean copy check result. Soft until the calibration is done | K9.9 |
| L06 | H | each item from a source with a licence for each item records the licence of the item | K13.9 |
| L07 | S | a node with a takedown report older than two working days that is not quarantined | K13.9 |
| L08 | H | each source document has the required fields, and each mention names a source | K5.14 |
| L09 | H | a source with a choice of licences has an `election` | K5.14 |
| L10 | H | a public-domain source of the `open` class is public domain in the US and the EU | K5.14 |
| L11 | H | a mention with text comes from a snapshot whose licence was read on or before the snapshot date | K5.14 |
| L12 | S | a source whose licence was last read more than twelve months ago | K5.14 |
| F10 | H | each mention from a library records the library, the pin, the declaration, the intake class and the licence of the item | K6.12 |
| F11 | H | no section holds text from a library of class `link`, `evaluate` or `exclude`, or from an item whose licence is not open | K6.12 |
| F12 | H | each library source document has `authored`, and each mention copies it | K6.12 |
| F13 | S | a binding to an `ai` or `mixed` declaration whose F2 review does not cover the definitions that its statement uses | K6.12 |
| P05 | H | each source that an agent added resolves to a record, and the record matches the claim | K9.9 |
| P06 | S | a source with `locator_unconfirmed` older than 180 days | K9.9 |
| X05 | H | an agent-written statement or proof has the results of the sanity checks in its run record | K9.9 |
| X06 | S | a stratum whose error rate is near its threshold | K9.9 |
| X07 | S | a theorem that a body names that resolves to no node and has no source | K9.9 |
| X08 | H | each field that an agent wrote names a model in the model register | K7.10 |
| X09 | H | each judge verdict is from a model of a family other than the writer's | K7.10 |
| N01 | H | each macro in a body has a register entry in scope, or the node defines it | K10.7 |
| N02 | H | two entries with the same glyph in overlapping scopes are the same macro, or list each other in `conflicts` | K10.7 |
| N03 | S | a curriculum puts a member before one of its prerequisites | K10.7 |
| N04 | S | a node uses a non-default option of a convention and does not say so | K10.7 |
| N05 | S | a concept with no `specializes` edge and no topic other than its MSC code | K10.7 |
| N06 | H | each macro has one `denotes`, and the macro files are the output of the register | K10.7 |
| N07 | S | a symbol in a `notation` field that no register entry has as `introduced_by` | K10.7 |

The pack check of a volume (PC1 to PC10, K11.6) is not an audit rule. It runs when a volume is built, and a volume that fails it is not published.

## Changes to the main spec

| Main spec | Change | Where |
|---|---|---|
| 00, 04.3, 04.4, 04.5, 05.1, 06.4, 06.7, 15.3, 20.1 | tags of ten characters, the shard key, the inventory register; rules T08 to T10 | K3.7 |
| 04.9 | the new edges, sections and fields of node.v1; the exercise kind in node.v2 | K8.9 |
| 1.7, 9.2, 9.3, 9.6, 16.6 | figures and licences of sources, including Kerodon as `record` | K1.4, K5.13 |
| 1.7, 9.4 | figures of formal libraries | K6.11 |
| 9.1 | adapters are `importer/<source>` | K5.11 |
| 12 | the roles `enumerate`, `write`, `informalize` and `exercise` | K7.3 |
| 15 | the rules above | this file |
| 16.6 | the zone of the Lean part of an exercise is the code zone | K13.1 |
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/01-research.md">
# K01. Research

The corpus spec rests on five research notes, written on 2026-10-26. They are parts A to E of this document, each in its own file. This file says how the research was done, what each part covers, which findings change the plan, which corrections they make to the main spec, and what the research could not verify.

The main spec has its own research document (spec 01). The parts here extend it for the corpus. They do not repeat it, except where a figure has changed or a statement in spec 01 is wrong.

## K1.1 Method

- **One date.** Each fact was read on 2026-10-26, unless the part gives another date. Each fact has its URL next to it.
- **Polite reading.** The research made single requests, with the User-Agent `m2/research (+https://github.com/tamnd/math2-agent)`. It did not crawl and it downloaded no dump. Dump sizes come from HEAD requests and directory listings. The largest samples were three days of arXiv OAI-PMH for mathematics (1,421 records), the mathematics list of the Open Textbook Library, four Mathlib YAML files and some tree-only listings of git repositories.
- **Measured and reported.** A count that the research made itself, for example from a Wikidata query or a repository tree, is marked **[measured]**. Other counts are as the source reports them.
- **Primary and secondary.** Part D marks each fact **[P]** (paper, official page, licence text, court docket), **[S]** (a law-firm note, a tracker, news) or **[snippet]** (seen only in a search excerpt). A spec section may quote a [snippet] fact only after a person reads the primary source.
- **UNVERIFIED** marks a fact that the research could not confirm, with the reason. These facts are in K1.5 and in K15.
- **No model calls.** The research called no model API, and it used no account or key.

## K1.2 The parts

| Part | File | Scope | Used by |
|---|---|---|---|
| A | `01a-reference-works.md` | encyclopedias, wikis and databases that people wrote: Wikipedia, Wikidata, nLab, ProofWiki, PlanetMath, the EoM, MathWorld, OEIS, DLMF, LMFDB, the subwikis, π-Base, the graph and knot databases, problem lists, MathOverflow and Math.SE | K05, K13 |
| B | `01b-literature-and-books.md` | literature indexes and full text, extraction projects, open textbooks, syllabi and theorem lists | K05, K10, K13 |
| C | `01c-formal-libraries.md` | formal libraries, translation between systems, formal and informal datasets, informalization | K06 |
| D | `01d-model-generation.md` | the error rates of model-written mathematics, projects at scale, memorization and copy detection, the law on model output, provider terms, cost | K07, K09, K12, K13 |
| E | `01e-representation-and-packaging.md` | representation formats, levels of explanation, exercises, book and course formats, notation | K08, K10, K11 |

## K1.3 Findings that change the plan

**Sources (parts A and B).**

1. **Most of the Wikidata "formulas" are primes.** Of the 113,118 items with a defining formula, 105,345 are prime numbers. Only 6,560 are other formulas. Wikidata is still the best crosswalk: 6,840 MathWorld ids, 4,348 nLab ids, 2,562 ProofWiki ids, 1,224 EoM ids, 438 Metamath ids and 115 Mathlib ids.
2. **Some large sources give no text.** The nLab (20,778 pages) has no licence. The originals of the EoM are © Springer. The DLMF is © NIST and not in the public domain. Kerodon states no licence, so it is `record` and not GFDL as the main spec says. These sources give mentions, locators and links (K5.3).
3. **Some large sources give text with attribution.** Wikipedia (28,196 mathematics articles), ProofWiki (47,254 content pages, a daily dump), PlanetMath (9,477 entries), OEIS (400,709 sequences), groupprops, π-Base, DaRT and Fungrim are `open`.
4. **Access is closing.** Wikimedia set API rate limits in 2026 and rewrote its robot policy. The LMFDB has a JavaScript gate and forbids `/api/` to robots. Stack Exchange put its dumps behind a click-through that excludes the training of language models. The importers use dumps and bulk downloads, and keep each limit (K5.10).
5. **New arXiv mathematics is more open than the stock.** In a 2026 sample, 36.9% of new papers were `open`, against about 15% for all papers.
6. **Open textbooks are fewer than they look.** About 48% of the mathematics books of the Open Textbook Library are `open`, and its licence data is sometimes stale. Some well-known free books (Hatcher, *Book of Proof*, Axler) are not open for the prose zone. Table A of K5.4 lists the books that are.
7. **Extraction projects mix licences.** The CC BY-SA release of TheoremSearch includes GFDL text from the Stacks Project. Its `-permissive` release is clean. Matlas states no licence.

**Formal libraries (part C).**

8. **Mathlib has three different counts.** 430,853 theorems and definitions (the statistics page), 483,112 declarations that a user can complete, and 797,294 constants. The corpus counts the second (K6.4).
9. **Translation between systems is weak.** No tool translates the large libraries of other systems into Lean. Model translation succeeds in a minority of cases (K6.6).
10. **Benchmarks have errors.** Several formal benchmarks have measured error rates in their statements, so the corpus uses them to evaluate roles, not as sources (K6.8).

**Model generation (part D).**

11. **Checks by models alone are not enough.** Model judges accept many wrong steps, strong models "prove" false statements, and between 11.4% and 56.8% of generated citations do not exist (part D, §1.6). The checks of K09 follow from these numbers.
12. **Models reproduce text.** In ordinary use, encyclopedia-style output had more than 27% of its characters in verbatim matches of 50 characters or more with public web text. The human median is 0%. So the copy check compares with a reference corpus (K9.4).
13. **The law protects the human part.** In the United States, text that a model wrote with no human control has no copyright. The output cases are not decided. K13 gives the decisions that follow.
14. **The cost of model text is small next to review.** One million nodes cost about $4,000 to $400,000 in model calls, by configuration, before the acceptance rate. Review capacity is the limit (K12.5).

**Representation and packaging (part E).**

15. **MathML 4 is still a working draft.** Its charter target of a Candidate Recommendation in May 2026 was missed. QTI 3.0 supports MathML 3 only. So the corpus stores LaTeX with its notation register, and the packager generates MathML (K10.4, K11).
16. **No system makes a textbook from a knowledge graph.** The packaging of K11 is new and has no prior evaluation.
17. **Exercise banks are mostly non-commercial.** The WeBWorK OPL, OpenStax and APEX are NC. Exercises are math2's own text, with links to the banks (K8.7).

## K1.4 Corrections to the main spec

The research corrects the main spec in these places. The detail is in the section named:

| Main spec | Correction | Detail |
|---|---|---|
| 1.7, 9.2 | Wikidata formulas, Wikipedia, nLab, ProofWiki, EoM, Erdős problems, LMFDB, zbMATH, MathWorld and OEIS figures and licences | K5.13 |
| 9.3, 16.6 | Kerodon is `record`, not GFDL | K5.13 |
| 9.3 | the `open` share of arXiv: two figures, the stock and the 2026 flow | K5.6 |
| 9.6 | TheoremSearch: 8.4M live; take the `-permissive` release | K5.7 |
| 1.7, 9.4 | Mathlib, AFP, Agda, Megalodon, Mizar, Metamath and Tau Ceti figures | K6.11 |
| 1.2 | FLT keeps both a leanblueprint and a Verso blueprint. Only verso-flt and verso-sphere-packing are in the Verso Blueprint catalogue, and Carleson waits for a toolchain bump. "Moved to it" is too strong | part E, §1.7 |
| 16.6 | the zone of the Lean statement and the reference solution of an exercise is not stated. It is the code zone, Apache-2.0 | K13.1 |
| 9.1 | adapters are `importer/<source>`, not `internal/ingest/<source>` | K5.11 |

## K1.5 What the research could not verify

Each item below is in K15 with the question that settles it.

- **Sites that blocked or failed:** the Encyclopedia of Mathematics (HTTP 502), EuDML, the zbMATH API terms, the HathiTrust terms, CORE, the Unpaywall terms, the Internet Archive terms, GroupNames, and the current page of LMFDB downloads.
- **Terms that could not be read:** the Stack Exchange Public Network Terms of 13 November 2025 (loaded by JavaScript), and the Semantic Scholar licence of abstracts.
- **Licences that sources leave open:** the nLab, the Manifold Atlas, FindStat, House of Graphs, KnotInfo, the knowls of the LMFDB, the zbMATH knowledge graph (CC BY or CC BY-SA), the 14 AIM titles, the US port of CC BY-SA 3.0, Coquelicot, the remaining 20 Formal Frontier repositories, Verso Blueprint, LaTeXML and MMT.
- **Counts that do not agree:** the Mizar articles (1,493 or 1,498), the Tau Ceti size, the Metamath counts, the TheoremSearch rows, the ProofWiki Main Page counters, the Mathlib `100.yaml` and Wiedijk's Lean count, and the 1000+ counts.
- **Facts seen only in search excerpts:** marked [snippet] in part D and "from search" in parts A and B. Before a spec section quotes one of them as a rule, a person reads the primary source.
- **The law outside the United States** on the copyright of model output (part D, §3b).

## K1.6 Facts age

The research is a snapshot. Counts grow, licences change, and terms change without notice (the OEIS licence page says so). So:

- Each source document (K5.2) holds its own date of reading, and rule L12 reports a licence that is older than twelve months.
- Before each phase of K14, a person reads again the licences and terms of the sources that the phase imports, and updates the parts of this document that changed.
- A figure in a corpus document that comes from this research names the part and the date. A figure that a later reading changes is changed in the corpus document and in the part, in the same commit.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/01a-reference-works.md">
# K01, part A. Encyclopedic, reference and database sources of mathematics written by people

Research for the math2 spec. This file extends `spec/01-research-2026.md` §1.7, `spec/09-literature-ingestion.md` §9.2 and §9.3, and `spec/16-collaboration-and-governance.md` §16.6. It does not repeat what those sections already say, except where a figure has changed or a statement in the spec is wrong.

- **Read date.** Every fact below was read on **2026-10-26**, unless another date is given. Each fact has its URL.
- **Fetches.** These were polite, single-page fetches. Where a fetch tool was used, the User-Agent was `m2/research (+https://github.com/tamnd/math2-agent)`.
- **Own counts.** Some counts are mine, not the source's. They are marked **[measured]**. They come from one of these:
  - a Wikidata SPARQL query (WDQS, 2026-10-26 08:25 to 08:47 UTC);
  - a MediaWiki `siteinfo` API call;
  - a tree-only listing of a git repository (`git clone --filter=blob:none --no-checkout`, then `git ls-tree`; no file contents were downloaded).
- **No dumps.** No dump was downloaded. Dump sizes come from HEAD requests or directory listings.
- **UNVERIFIED** marks a fact that I could not confirm. The reason is given each time.
- **Licence gate.** "What math2 can take" uses the classes in spec 9.3: `open`, `share-alike-nc`, `verbatim`, `gfdl` and `record`. The three practical outcomes are:
  - **text with attribution**: `open`, so text can go into node prose in the BY-SA zone;
  - **link only**: `gfdl`, or no grant of rights; math2 stores the locator and writes its own statement;
  - **record only**: `record`; metadata, identifiers and facts, but no text.

  Facts, identifiers and numerical data are not copyrightable, so math2 can take them as CC0 metadata (16.6) from any source. The one exception is database rights in the EU sui generis sense. Rows below say so where a source asserts this.

---

## A.1 Wikipedia (mathematics)

**What it holds and its size**

English Wikipedia:
- The WikiProject Mathematics assessment table counts **33,957 pages** in the project. 33,913 are assessed and 44 unassessed.
  https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Wikipedia_1.0/Assessment
  - The page gives no date for the counts. Its last edit was 28 Sep 2026 14:10 UTC.
- The table includes non-article pages: Category 2,302, Redirect 1,658, Template 1,012, Draft 172, and 503 others.
- The article classes are FA 32, FL 3, GA 231, B 1,446, C 4,459, Start 13,847, Stub 7,741 and List 437. They sum to **28,196 article pages** [measured from the table].
  - This replaces the "about 26k (estimate)" in spec 1.7.
- By importance: Top 208, High 955, Mid 4,363, Low 15,324.

Other languages. Each wiki counts in its own way, so the figures are not comparable:
- **German**: Portal:Mathematik says it is the central page for "20.732 mathematischen Artikeln". The figure is undated; the portal was last edited 7 May 2026. https://de.wikipedia.org/wiki/Portal:Mathematik
- **French**: Projet:Mathématiques has 12,192 project articles, of which 6,825 are evaluated (56%). It was "mis à jour 25 octobre 2026". https://fr.wikipedia.org/wiki/Projet:Mathématiques/Évaluation
- **Japanese**: Portal:数学 shows 8,865 articles. The page was last updated 1 Apr 2026, and the count is undated. https://ja.wikipedia.org/wiki/Portal:数学
- **Russian**: Проект:Математика lists 3,928 articles. This is the *project-tagged* set, not all mathematics articles. The figure is undated and "обновляется автоматически". https://ru.wikipedia.org/wiki/Проект:Математика
- **Chinese**: Wikipedia:数学专题 says "数学专题目前已有5,682篇条目". Its assessment tables sum to about 6,033. https://zh.wikipedia.org/wiki/Wikipedia:数学专题
  - The timestamp on the statistics equals the fetch time, so it is probably a live magic word, not a date of the count.

**Licence of the text**
- CC BY-SA 4.0. Text is also under the unversioned GFDL "unless otherwise noted". A page that includes text available *only* under CC BY-SA is not available under the GFDL. https://en.wikipedia.org/wiki/Wikipedia:Copyrights
- The dumps page says the same. It also says Wikidata structured data is CC0. https://dumps.wikimedia.org/legal.html

**Bulk channels and their terms**
- **XML dumps.**
  - In 2026 the `enwiki` dump directory has runs only on the 1st of each month: 20260401 to 20261001. https://dumps.wikimedia.org/enwiki/
    - Wikimedia used to run twice a month. Whether that changed formally is UNVERIFIED: I saw only the listing, not an announcement.
  - `enwiki-latest-pages-articles-multistream.xml.bz2` is 26,917,225,724 bytes, dated 08 Oct 2026. https://dumps.wikimedia.org/enwiki/latest/
- **HTML ("enterprise_html") dumps** on dumps.wikimedia.org. The last run listed is **20250320**. https://dumps.wikimedia.org/other/enterprise_html/runs/
  - The free public HTML snapshots look stopped, or moved to Wikimedia Enterprise. The reason is UNVERIFIED.
- **Wikimedia Enterprise.** https://enterprise.wikimedia.com/pricing/
  - Free account: Snapshot API with monthly updates, "up to 30/mo" requests and "up to 1,500/mo" snapshot chunks; On-demand API up to 50,000 requests a month.
  - Paid account: daily snapshots, the Realtime API, and bespoke pricing.
  - "Every response includes machine-readable license information."
- **API rate limits.** These are new in 2026 and "subject to experimentation and change". https://www.mediawiki.org/wiki/Wikimedia_APIs/Rate_limits (revision 8654799, edited 15 Oct 2026)
  - The limits are per minute, per user:

    | Client | Limit |
    |---|---|
    | unidentified (IP only) | **10** |
    | compliant User-Agent only | 200 |
    | authenticated, new user | 200 |
    | established editor | 2,000 |
    | bot flag, WMCS or known client | exempt |

  - Rollout: low limits from early March 2026, identified tiers from early April 2026. https://www.mail-archive.com/wikitech-l@lists.wikimedia.org/msg97202.html (2 Mar 2026; summary from search, not read in full)
- **Robot policy.** This rewrites the 2009 policy; the page was last edited 16 Mar 2026. https://wikitech.wikimedia.org/wiki/Robot_policy
  - Crawl only `/wiki/Article_name`, with fewer than 10 concurrent requests and fewer than 20 requests per second on average.
  - REST API: unauthenticated, concurrency 3 and fewer than 5 requests per second; authenticated, up to 10 per second.
  - Action API: unauthenticated, concurrency 1 and fewer than 5 requests per second.
  - "Consider dumps before making live requests."
  - There is no AI-specific clause.

**AI terms and 2024–2026 changes**
- **10 Nov 2025.** The Wikimedia Foundation asked AI developers to attribute the human contributors and to use Wikimedia Enterprise instead of scraping. It made no legal threat. https://wikimediafoundation.org/news/2025/11/10/in-the-ai-era-wikipedia-has-never-been-more-valuable/ (the post itself was not fetched; content from https://techcrunch.com/2025/11/10/wikipedia-urges-ai-companies-to-use-its-paid-api-and-stop-scraping/)
- **15 Jan 2026.** Enterprise announced Amazon, Meta, Microsoft, Mistral AI and Perplexity as partners, alongside Google and others. https://enterprise.wikimedia.com/blog/wikipedia-25-enterprise-partners/ (from a search summary)
- **No licence change.** The text stays CC BY-SA 4.0. What these deals buy is volume and an SLA, not rights.
- **Rogue OpenAI agents.** On 5 to 6 Oct 2026 the Foundation reported that rogue OpenAI agents made millions of API requests and that this may have contributed to the WDQS outage of 7 to 11 May 2026. https://www.theregister.com/ai-and-ml/2026/10/06/wikimedia-foundation-comes-forward-as-latest-openai-agent-assault-victim/5301400 ; https://thehackernews.com/2026/10/wikimedia-says-openai-agents-tried-to.html (from search summaries)
  - This is a likely driver of the stricter limits.
  - Wikipedia's own article on the Foundation states the dates in a confusing way.

**What math2 can take:** `open`.
- Short text with attribution can go into node prose, because CC BY-SA 4.0 equals the node prose zone. Each item needs the page and revision ID, and attribution through the history URL.
- Get articles from the monthly dump or the free Enterprise tier, never by crawling.
- An imported page may contain CC BY-SA-only text. That text cannot be used under the GFDL, but math2 does not need the GFDL.

---

## A.2 Wikidata

**What it holds** [measured, WDQS, 2026-10-26 ~08:25 UTC]
- **Defining formula (P2534): 113,118 items.** But **105,345 of these are instances of *prime number* (Q49008)**. Megaprimes, gigantic primes and Mersenne numbers add more.
  - Only **6,560** items with P2534 are *not* primes or prime-like. That is the real pool of mathematical formulas.
  - Only **5,492** P2534 items have an English Wikipedia article.
  - The top non-prime classes are formula 865, mathematical concept 458, theorem 436, function 95 and physical law 74.
  - **This changes the meaning of the "about 113k items with a defining formula" figure in spec 1.7 and 9.2.**
- **Theorem.** Instances of theorem (Q65943), counting subclasses: **3,208**. This matches spec 1.7.
- Crosswalk properties, as `P-id`: label, number of items:

  | Property | Label | Items |
  |---|---|---|
  | P2812 | MathWorld ID | 6,840 |
  | P4215 | nLab ID | 4,348 |
  | P8427 | KnotInfo ID | 2,978 |
  | P6781 | ProofWiki ID | 2,562 |
  | P8426 | Knot Atlas ID | 2,228 |
  | P7554 | Encyclopedia of Mathematics article ID | 1,224 |
  | P7726 | PlanetMath ID | 951 |
  | P12975 | Lexikon der Mathematik entry ID | 603 |
  | P829 | OEIS ID | 469 |
  | P12888 | Metamath statement ID | 438 |
  | P3285 | Mathematics Subject Classification ID | 408 |
  | P12934 | The House of Graphs ID | 213 |
  | P11497 | Digital Library of Mathematical Functions ID | 199 |
  | P14534 | Mathlib Declaration ID | 115 |
  | P12987 | LMFDB knowl ID | 109 |
  | P5610 | OpenMath ID | 54 |
  | P14106 | Erdős Problem number | 3 |

  Other relevant properties:

  | Property | Label | Items |
  |---|---|---|
  | P7263 | Prime Pages ID | 126,184 (the source of the prime items) |
  | P4252 | All-Russian Mathematical Portal ID | 5,090 |
  | P1482 | Stack Exchange tag | 7,215 (all sites) |
  | P894 | zbMATH Open document ID | 1,888 |
  | P889 | Mathematical Reviews ID | 141 |
  | P6830 | swMATH work ID | 125 |

  The query that found these matched property labels, so a property with an unusual label may be missing.
- **Gaps.** There is no Wikidata property for the Stacks Project, Kerodon, groupprops, Topospaces, π-Base, ISGCI, DaRT, FindStat, Fungrim or the Wolfram Functions Site. My label search found none.
  - P14106 (Erdős Problem number) is used on only **3** items, against 1,221 problems.

**Licence:** structured data CC0. Unstructured content, such as talk pages, is CC BY-SA 4.0. https://dumps.wikimedia.org/legal.html

**Bulk channels and their terms**
- **JSON dump.** `latest-all.json.bz2` is 103,488,697,311 bytes, dated 21 Oct 2026. There are also `.nt` and `.ttl` dumps. https://dumps.wikimedia.org/wikidatawiki/entities/
- **WDQS.** https://www.mediawiki.org/wiki/Wikidata_Query_Service/User_Manual
  - The hard timeout is 60 s.
  - One client, defined as User-Agent plus IP, gets 60 s of processing time per 60 s.
  - The limit is 5 parallel queries per IP and 30 error queries per minute. Clients over a limit get HTTP 429.
  - The manual says WDQS is "migrating to a new backend in 2026–2027" with new v2 endpoint URLs.
  - The details of the scholarly graph split were UNVERIFIED: the manual page does not describe them.
- The general API rate limits (A.1) apply.

**AI terms:** none specific; CC0. See A.1 for the May 2026 WDQS outage.

**What math2 can take:** `open`, CC0. All of it can go into the metadata zone.
- Use the weekly dump, not WDQS, for anything bulk.
- Treat P2534 as a source of formulas only after removing the prime-number items.

---

## A.3 nLab

**What it holds and its size**
- **20,778 pages** [measured]: the count of `content.md` files in `ncatlab/nlab-content` at the last commit, 2026-10-25T23:35:34Z ("Revision ID: 236426 … Page name: category theory").
  - The tree is not truncated: 73,134 entries in all. https://api.github.com/repos/ncatlab/nlab-content/git/trees/master?recursive=1
  - The repository has 208,017 commits. Each wiki edit is mirrored as a commit, in Markdown plus itex2MML. https://github.com/ncatlab/nlab-content
  - This replaces "about 19,193 entries (~2024)" in spec 1.7.

**Licence:** **none.**
- The HomePage says: "Using and distributing content obtained from the nLab is free and encouraged if you acknowledge the source" and "There is currently no consensus on a more formal license statement". It suggests checking the nLab homepages of individual contributors. https://ncatlab.org/nlab/show/HomePage
- The GitHub repository has no licence file (`license: null`). https://api.github.com/repos/ncatlab/nlab-content
- The history of the question is in a 2013 nForum thread, where the CC link was removed from the HomePage. https://nforum.ncatlab.org/discussion/3545/
  - In that thread Toby Bartels says a licence cannot be retroactive, and that copying nLab text into Wikipedia would be illegal even with a licence statement.
- There was no change in 2024 to 2026 that I could find.

**Bulk channel:** the git mirror, updated within minutes of each edit. No API terms are stated.

**AI terms:** none stated.

**What math2 can take:** **link only**, with records and identifiers.
- The "free … if you acknowledge the source" permission covers using and distributing the text. It grants no clear right to adapt and relicense under CC BY-SA, and it has not been agreed by the many authors. So the gate's `record`/link-only class still applies.
- Page names and the link graph are facts, so math2 can take them as CC0 metadata.
- To lift the restriction, math2 would need an explicit statement from the nLab steering committee. Even then, it could only cover authors who agree.

---

## A.4 ProofWiki

**What it holds and its size**
- Special:Statistics: **47,254 content pages**, 128,658 pages in all, 821,010 edits, 5,412 registered users, **12 active users** in the last 30 days. https://proofwiki.org/wiki/Special:Statistics
- The Main Page reports "30,348 Proofs" and "36,102 Definitions". https://proofwiki.org/wiki/Main_Page
  - These equal the figures in spec 1.7. Whether the Main Page counters are live is UNVERIFIED.

**Licence:** **CC BY-SA 3.0 and the GFDL** for all content except talk pages and user pages: "All content, except for that on Discusson pages and user home pages, is subject to the CC-by-SA 3.0 licence" and the GNU Free Documentation Licence. Talk pages are exempt from both. https://proofwiki.org/wiki/ProofWiki:Copyrights

**Bulk channel:** **a daily XML dump is live.**
- `https://proofwiki.org/xmldump/latest.xml.gz` returned HTTP 200, `content-length: 36599568` (about 36.6 MB gzipped), `last-modified: Mon, 26 Oct 2026 04:48:09 GMT`. This was a HEAD request only.
- The old URL `latest.xml` returns 301.
- Historical copies exist on the Internet Archive (WikiTeam 2017). https://archive.org/details/proofwikiorg_w-20170808-wikidump

**AI terms:** none stated. NaturalProofs (2021) and MathPile (2023) built training sets from the dump. https://arxiv.org/abs/2104.01112 ; https://arxiv.org/pdf/2312.17120

**What math2 can take:** `open`. Proofs and definitions can go into node prose with attribution.
- CC BY-SA 3.0 adaptations may be licensed under BY-SA 4.0, as spec 16.6 already says.
- Use the daily dump, not page crawls.
- Record the page title and revision ID for each item.

---

## A.5 PlanetMath

**What it holds and its size**
- **9,477 `.tex` entries** [measured]: tree-only clones of all 64 repositories in the `planetmath` GitHub organisation. Each repository is one top-level MSC class, plus `fem2016`. https://github.com/planetmath
  - For example: 03 Mathematical logic 906, 11 Number theory 1,071, 54 General topology 483.
- **The project is inactive.** The most recent push to any repository was 2024-02-19 (`16_Associative_rings_and_algebras`), and most last changed in March 2018.
  - The `planetmath.org` server's front page has `Last-Modified: Sat, 10 Mar 2018`.
  - Wikipedia's infobox says "Inactive". https://en.wikipedia.org/wiki/PlanetMath

**Licence:** **CC BY-SA 3.0 Unported**. The `LICENSE` file in each repository is the CC BY-SA 3.0 legal code. https://raw.githubusercontent.com/planetmath/11_Number_theory/master/LICENSE

**Bulk channel:** git. No API.

**AI terms:** none.

**What math2 can take:** `open`. Text can go in with attribution. Record the repository and the file path.

---

## A.6 Encyclopedia of Mathematics (encyclopediaofmath.org) and the Springer *Encyclopaedia of Mathematics*

**History**
- The original is the Soviet *Matematicheskaya entsiklopediya* (1977, ed. I. M. Vinogradov).
- The English edition, edited by Michiel Hazewinkel, came out from Kluwer in 10 volumes (1987–1994), with Supplements I to III (1997, 2000, 2002) and a CD-ROM (1998).
- Kluwer became part of Springer in 2003. "The 2002 version contains more than 8,000 entries".
- The free static online version redirected to the wiki from 29 Nov 2011. The wiki was "a collaboration between Springer and the European Mathematical Society". https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics
- An undated notice on the wiki's main page says the EoM "has moved from Springer Verlag to EMS Press". From search snippets, the snapshot is about mid-2020. https://encyclopediaofmath.org/wiki/Main_Page

**Licence: mixed, and mostly NOT open.**
- The main page says: "The original articles from the Encyclopaedia of Mathematics remain copyrighted to Springer but any new articles added and any changes made to existing articles within encyclopediaofmath.org will come under the Creative Commons Attribution Share-Alike License".
- The License Notice says new text is co-licensed under CC BY-SA 3.0 Unported and the GFDL. https://encyclopediaofmath.org/wiki/Encyclopedia_of_Mathematics:License_Notice
- Both statements were read through search-engine snippets. **The live site returned HTTP 502** on every attempt on 2026-10-26, between about 08:30 and 08:45 UTC, so the current wording is **UNVERIFIED**.
- Whether the move to EMS Press transferred or relicensed Springer's copyright is **UNVERIFIED**. I found no statement either way.
- A Creative Commons wiki case page from 2010 lists the Springer EoM as "copyright None" and "Open or Free Statement: no". https://wiki.creativecommons.org/wiki/Springer_-_Springer_Link_-_Encyclopaedia_of_Mathematics
- An EMS Magazine article (2025) by FIZ Karlsruhe staff says EoM runs the same wiki software as Wikipedia "albeit under a different license and editorial policies". zbMATH Open entity linking now links to EoM, nLab, MathWorld and Wikipedia. https://euromathsoc.org/magazine/articles/252 (from search summary)

**Size:** UNVERIFIED (the site was down). Wikidata has 1,224 items with an EoM ID (P7554) [measured].

**Bulk channel:** none known. There is a MediaWiki API (`api.php`); it was also 502 on the read date. No dump.

**AI terms:** none found.

**What math2 can take:** **record only, by default.**
- Most articles are derived from Springer-copyrighted originals. An edited article is a derivative of that text, so its BY-SA grant covers only the changes.
- Only an article that the page history shows to be wholly new on the wiki is `open`.
- Spec 1.7 and 9.2 list EoM as "CC BY-SA 3.0". **That is wrong for most of the text and should change.**

---

## A.7 Wolfram MathWorld

**Size:** **16,939 entries**, "Last Updated: Sun Oct 25 2026". https://mathworld.wolfram.com/ (Spec 1.7 says 16,849.)

**Licence and terms:** proprietary. https://www.wolfram.com/legal/terms/mathworld.html
- "these pages may not be copied, mirrored, redistributed, printed, or reproduced in bulk without permission."
- "Reproduction for commercial purposes is not permitted, nor is use of robots to create archival copies."
- Prefetching "may violate the above proscription against bulk downloading."
- "Limited use of material for academic research and teaching is permitted, providing conventional citation standards are followed" (quoted from the search snippet of the same page).
- The terms say nothing about AI.
- `robots.txt` says "Bulk downloading of these pages is not permitted for the purposes of making archival copies" and disallows only `/random.html`. It links a sitemap. https://mathworld.wolfram.com/robots.txt

**Bulk channel:** none. Wikidata P2812 holds 6,840 MathWorld IDs [measured], so the crosswalk does not need the site.

**What math2 can take:** **record only**: entry IDs from Wikidata, plus links. This confirms the spec.

---

## A.8 OEIS

**Size:** "Contains **400709** sequences". https://oeis.org/ (Spec 1.7 has 400,673 on 24 Oct.)

**Licence:** CC BY-SA 4.0. https://oeis.org/wiki/The_OEIS_End-User_License_Agreement (revision approved 24 Feb 2023)
- Attribution "should credit The Online Encyclopedia of Integer Sequences and provide a URL to the main page https://oeis.org/ or to a specific sequence".
- The EULA "may change … from time to time, without advance notice".

**Bulk channels**
- The download options are `stripped.gz`, `names.gz` and the git repository `oeis/oeisdata`, which has "incremental updates". https://oeis.org/wiki/Download
- `oeisdata` was last pushed 2026-10-26T07:03:53Z (commit "oeis data export"). GitHub reports its licence as "Other". https://api.github.com/repos/oeis/oeisdata
- `robots.txt` has `Crawl-Delay: 10`, disallows `/search`, `/edit` and `/history`, and fully blocks one named bot ("ShapBot"). https://oeis.org/robots.txt

**AI terms:**
- None in the EULA or the Download page.
- The Terms of Use text contains no clause on robots, automation, scraping, bulk use, machine learning or crawling. I checked by keyword search over https://oeis.org/OEISTermsOfUse.odt.txt.

**What math2 can take:** `open`, with text and attribution. This confirms the spec. Use `oeisdata` only.

---

## A.9 DLMF (NIST Digital Library of Mathematical Functions)

**Version:** 1.2.8, released 2026-09-15. https://dlmf.nist.gov/about/notices

**Licence:** **NIST-owned copyright; it is not public domain.**
- "All materials on this website are owned by NIST". NIST holds copyrights assigned by the DLMF authors.
- "Limited copying and internal distribution of the content of these pages is permitted for research and teaching."
- "Reproduction, copying, or distribution for any commercial purpose is strictly prohibited."
- "Bulk copying, reproduction, or redistribution in any form is not permitted."
- Excessive downloading may get the offending subnet blocked.
- There is nothing on AI.
- This is a surprise, because works of the US government are usually public domain. The DLMF authors were mostly not federal employees, and they assigned copyright to NIST.

**Bulk channel:** none. Wikidata P11497 holds 199 DLMF IDs [measured].

**What math2 can take:** **record only**.
- Equation numbers such as "DLMF 5.2.1" are stable locators.
- Each formula is a fact, so math2 can restate it in its own words or LaTeX, with a citation. It must not copy the text or the commentary.

---

## A.10 Fungrim (Mathematical Functions Grimoire)

**Size:** "457 symbols, 3130 entries (definitions, formulas, tables, plots)", 82 topics. https://fungrim.org/
- The page build timestamp is 2021-03-15. The repository was last pushed 2022-02-24. https://api.github.com/repos/fredrik-johansson/fungrim
- **It is dormant.**

**Licence:** **MIT** ("Fungrim is provided under the MIT license"). The repository `LICENSE` reads "Copyright (c) 2019 Fredrik Johansson".

**Bulk channel:** git. The entries are machine-readable Python/Grim expressions, with symbolic formulas and conditions.

**What math2 can take:** `open`. MIT is permissive: keep the notice in provenance.
- Fungrim is unusually valuable, because each formula has explicit assumptions. That makes it a good source for `statement` nodes, and later for Lean.

---

## A.11 Wolfram Functions Site

**Size:** **307,409 formulas** "as of October 26, 2026". https://functions.wolfram.com/

**Licence:** "© 1998–2026 Wolfram Research, Inc.".
- Neither the home page nor the About page states any terms of use.
- The About page says formulas are downloadable as notebooks and PDFs, and in MathML and ASCII. https://functions.wolfram.com/About/
- `robots.txt` allows crawling except `/webMathematica`. It lists 15 sitemaps.
- The general Wolfram terms that apply are UNVERIFIED (not found).

**What math2 can take:** **record only**: links, and formula IDs as locators. The formulas are facts, so math2 restates them in its own words.

---

## A.12 LMFDB

**Licence** (from the source template, because the site gates non-browser clients). https://raw.githubusercontent.com/LMFDB/lmfdb/main/lmfdb/templates/license.html
- "The database underpinning the LMFDB is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License (CC-BY-SA)".
- The code is GPL v2 or later.
- The page says copyright is in the collection: "completeness claims are protected … as are search results for queries. Individual facts … are not".
- On AI: "How does this apply to training an AI model using this data? See this guidance from Creative Commons. We also encourage you to use more efficient access options rather than scraping the website directly, and to rate limit your website requests".

**2025–2026 changes in access**
- **`www.lmfdb.org` now redirects requests to `beta.lmfdb.org/gate.html`.**
  - `gate.html` is a JavaScript page that sets a `human=1` cookie. It has `Last-Modified: Thu, 14 Aug 2025`.
  - A client without JavaScript gets "Enable JavaScript, then reload."
  - I observed this on 2026-10-26 with curl. I did not try to bypass it.
- **`robots.txt` disallows `/api/`** for all agents. It also has `Crawl-delay: 30` and fully blocks two named bots. https://www.lmfdb.org/robots.txt
  - So the "API over 26 collections" in spec 1.7 is off-limits to robots.
  - The homepage footer reads "LMFDB Release 1.2.1 · SageMath 10.5". https://www.lmfdb.org/

**Size:** not re-measured, because of the gate. The spec 1.7 figures are from 2022.

**What math2 can take:** `open` (CC BY-SA 4.0).
- Get the data from LMFDB's bulk downloads, or by agreement. Do not use the API or scrape the site.
- Knowl text, the LMFDB's short explanations, is part of "the database", so presumably CC BY-SA too. That knowls fall under the data licence is UNVERIFIED: the licence page does not mention them.

---

## A.13 The subwikis: groupprops, Topospaces, Commalg, Diffgeom, Market

These are MediaWiki `siteinfo` figures [measured].

| Wiki | Articles | Pages | Edits | Active users | Licence |
|---|---|---|---|---|---|
| groupprops.subwiki.org | 8,076 | 15,469 | 51,383 | 1 | CC BY-SA 3.0 |
| topospaces.subwiki.org | 825 | 1,450 | 3,862 | 1 | CC BY-SA 3.0 |
| commalg.subwiki.org | 429 | 708 | 1,630 | 1 | CC BY-SA 3.0 |
| diffgeom.subwiki.org | 435 | 716 | 1,521 | 1 | CC BY-SA 3.0 |
| market.subwiki.org | 223 | 464 | 1,638 | 1 | CC BY-SA 3.0 |

- **Source:** `https://<wiki>.subwiki.org/w/api.php?action=query&meta=siteinfo&siprop=statistics|rightsinfo`. The `rightsinfo` field is "https://creativecommons.org/licenses/by-sa/3.0/".
- **Bulk channel:** the MediaWiki API (Special:Export). No published dump was found.
- **AI terms:** none.
- **What math2 can take:** `open`.
  - Groupprops is the most useful: it has structured "property implications" and "facts" pages about groups, and it suits `statement` candidates.
  - Fetch it page by page and slowly. The site is small and maintained by one person.

---

## A.14 Manifold Atlas (MPIM Bonn)

**Size:** **215 articles**, 2,721 pages, 17,516 edits, 0 active users [measured, `siteinfo`].
- It runs MediaWiki 1.18.4 and is reachable over HTTP; HTTPS refused the connection on the read date. http://www.map.mpim-bonn.mpg.de/api.php
- It has an editorial board and refereed pages.

**Licence: self-contradictory.** The Manifold Atlas:Copyrights page says "the copyright rules are the same as for Wikipedia, the GNU Free Documentation License 1.3", and then that "The content of the Manifold Atlas is released into the public domain". The `rightsinfo` field is empty.

**What math2 can take:** treat it as `gfdl`, **link only**, until the editors clarify. The public-domain sentence is not a clean dedication: there is no CC0 and no waiver text. Records and links are fine.

---

## A.15 π-Base

**Size** [measured, `pi-base/data` tree at commit 2026-10-24T21:57:19+02:00, "Torus initial PR (#1777)"]: **225 spaces, 247 properties, 935 theorems, and 2,147 space–property assertions**. This matches spec 1.7.

**Licence:** **CC BY 4.0**. The GitHub licence of the `pi-base/data` repository is `CC-BY-4.0`. https://api.github.com/repos/pi-base/data

**Bulk channel:** git (Markdown plus YAML). It was pushed 2026-10-26.

**What math2 can take:** `open`. Theorems are implications between properties, and they are good `statement` candidates.

---

## A.16 House of Graphs

**Size:** "nearly 22 000 graphs" in the 2.0 paper (2022). https://arxiv.org/abs/2210.17253
- The site is a JavaScript application.
- The API endpoints I tried (`/api/statistics`, `/api/enums/invariants`) return `{"error":true,"message":"Unauthorized"}`.
- The news entry of 23 Apr 2026 added the invariant "independent domination number" (from a search summary of https://houseofgraphs.org/).
- **The current count of graphs and of invariants is UNVERIFIED.** The site shows nothing without JavaScript, and the API needs authentication. The 2022 paper says 44 invariants; spec 1.7 says 52.

**Licence:** none found. Users are asked to cite Coolsaet, D'hondt and Goedgebeur, *Discrete Appl. Math.* 325 (2023) 97–107.

**What math2 can take:** **record only**.
- graph6 strings and invariant values are facts, so they are fine as CC0 metadata.
- Wikidata has 213 House of Graphs IDs (P12934) [measured].

---

## A.17 ISGCI (Information System on Graph Classes and their Inclusions)

**Size:** **1,695 graph classes, 242,975 inclusions, 30,147 complexities, 52,029 bounds**, "updated 2026-10-22". The latest news item is dated 2025-11-16. https://www.graphclasses.org/

**Licence:** none stated. The site shows only "by H.N. de Ridder et al. 2001-2014" and an Impressum.

**Bulk channel:** none listed on the page. The accompanying Java application loads the data; the format and URL are UNVERIFIED.

**What math2 can take:** **record only**.
- Class IDs (`gc_N`) are stable locators.
- Inclusion facts are facts and can be CC0 metadata. The collection as a whole may still carry database rights in the EU (ISGCI is maintained in Germany).

---

## A.18 Database of Ring Theory (DaRT)

**Size** [measured, `rschwiebert/dart_data` tree at commit 2026-10-24]:
- **218 rings, 178 ring properties, 72 theorems, 46 modules, 94 module properties.**
- It also holds logic (implication) files, 574 ring citation files and 269 publication files.
- The repository was pushed 2026-10-25.

**Licence:** **CC BY 4.0** (`dart_data` repository licence). https://api.github.com/repos/rschwiebert/dart_data
- MathBases lists DaRT's licence as MIT, which probably refers to the app code. https://mathbases.org/d/ring-theory (from a search summary)

**Bulk channel:** git (YAML). The site is https://ringtheory.herokuapp.com/

**What math2 can take:** `open`.

---

## A.19 KnotInfo / LinkInfo and the Knot Atlas

**KnotInfo** (Livingston and Moore)
- It covers knots of crossing number 3 to 13. The site's option list goes up to "3-13".
- I counted about 116 invariant fields on the search page, which is my own tally.
- Citation form: "KnotInfo: Table of Knot Invariants, https://knotinfo.org/ (May 16, 2026)" (from search).
- The old host `knotinfo.math.indiana.edu` no longer resolves (DNS failure on 2026-10-26).
- **Licence:** none on the site. https://knotinfo.org/
- **Download:** the full invariant table is an Excel spreadsheet, about 16 MB zipped. There is also a polynomial file (8 MB), 3D coordinates (46 MB) and diagrams (about 9,000 files). https://knotinfo.org/homelinks/database_download.php
- **A licence surprise.** The `database_knotinfo` Python package (version 2026.10.5) redistributes "Complete contents of the KnotInfo and LinkInfo databases in CSV format" under **GPL-3.0**. https://api.github.com/repos/soehms/database_knotinfo ; https://pypi.org/pypi/database-knotinfo/json

**Knot Atlas** (Bar-Natan and Morrison)
- **2,710 articles** but 1,651,548 pages in all, because data pages are generated. 1,729,610 edits, 1 active user [measured, `siteinfo`]. https://katlas.org/api.php
- **Licence:** `rightsinfo` is empty. No licence page was found: `Knot_Atlas:General_disclaimer` returns 404.
- The data is produced by the `KnotTheory`` Mathematica package. https://katlas.org/wiki/Main_Page (last edited 6 May 2026)

**What math2 can take:** **record only** for both.
- Knot names (Rolfson notation, `11n_34` and similar) and invariant values are facts. Wikidata has 2,978 KnotInfo IDs and 2,228 Knot Atlas IDs [measured].
- Do not import the GPL-3.0 CSV package as text. GPL does not fit into the CC BY-SA or CC0 zones, and the facts can be taken without it.

---

## A.20 ATLAS of Finite Group Representations (v3)

**Size:** "information (including 5215 representations) on about 716 groups", version 3.004. https://brauer.maths.qmul.ac.uk/Atlas/v3/

**Licence:** none stated on the site.
- The GAP interface `AtlasRep` is GPL-3.0-or-later. Version 2.1.12 came out in August 2026 (from search). https://docs.gap-system.org/pkg/atlasrep/doc/chap0.html
- The package archives do **not** contain the data; it is fetched on demand (from search, AtlasRep documentation).
- The mirror is http://atlas.math.rwth-aachen.de/Atlas/v3 (from search).

**What math2 can take:** **record only**: group names, representation identifiers and facts.

---

## A.21 Open Problem Garden

- **Licence:** GFDL. The footer says "Content distributed under" with a GNU FDL badge whose image file is named `gnu-fdl3`, so probably version 1.3. The exact version is UNVERIFIED. http://www.openproblemgarden.org/
- **Size:** the subject counts sum to 713. Problems can appear in several subjects, so this is an upper bound on distinct problems. The Recent Activity list shows no dates. This matches spec 1.3.
- **What math2 can take:** `gfdl`, **link only**. This confirms the spec.

---

## A.22 Erdős problems

- **Size:** "There are 1221 problems in the database." https://www.erdosproblems.com/
- **Licence of the problem text: none stated on erdosproblems.com.** The site has no licence or terms of reuse.
- **The GitHub repository `teorth/erdosproblems` is Apache-2.0, but `data/problems.yaml` contains no problem statements.** It holds only these fields: number, prize, `informal_status`, `formal_status`, status, oeis, formalized and tags. Its 1,221 entries have zero `statement` fields [measured]. https://raw.githubusercontent.com/teorth/erdosproblems/main/data/problems.yaml (repository pushed 2026-10-25)
  - **Spec 9.2 says math2 takes "problem statements" from `teorth/erdosproblems`. That is wrong.** The statements are on Bloom's site with no licence. The only Apache-2.0 statements are the *formal* ones in DeepMind's formal-conjectures.
- **What math2 can take:**
  - from the repository: status and links, as `open` metadata;
  - from erdosproblems.com: the informal statements, as **record only**, by link and problem number;
  - from formal-conjectures: the formal statements, `open` (Apache-2.0).

  math2 writes its own informal statement for each problem.

---

## A.23 Kourovka Notebook (Unsolved Problems in Group Theory)

- It is on arXiv as **arXiv:1401.0300**. The latest version, **v49, is dated 20 Oct 2026**. It is the **21st edition**, "which contains 150 new problems". https://arxiv.org/abs/1401.0300 ; https://export.arxiv.org/api/query?id_list=1401.0300
- **Licence:** the arXiv non-exclusive distribution licence 1.0. https://arxiv.org/licenses/nonexclusive-distrib/1.0/
- **What math2 can take:** **record only**.
  - Problem numbers (such as "19.37") with the edition and arXiv version are locators.
  - math2 writes its own statements, and the `tamnd/arxiv` gate applies.

---

## A.24 MathOverflow and Mathematics Stack Exchange

**Size** [measured, Stack Exchange API `/2.3/info`, 2026-10-26]:
- **MathOverflow:** 168,603 questions, 201,920 answers, 1,107,746 comments, 516,907 users.
- **Math.SE:** **1,702,027 questions**, 2,236,177 answers, 9,483,312 comments, 1,617,966 users.

**Licence of the text:** CC BY-SA 2.5, 3.0 or 4.0 per post, by date. This is in the spec already.

**Terms**
- **API Terms of Use** were last updated 30 Nov 2023 and are short. Applications "must ensure they visually indicate that the Stack Exchange Network is the source". They incorporate the network Terms of Service. https://policies.stackoverflow.co/stack-overflow/api-terms-of-use/
- **Public Network Terms of Service** were "Last updated November 13, 2025". https://policies.stackoverflow.co/stack-overflow/public-network-terms-of-service/
  - The body is loaded by JavaScript and I could not read it. **Whether the November 2025 revision added AI or scraping clauses is UNVERIFIED.**
- **API throttles.** https://api.stackexchange.com/docs/throttle
  - More than 30 requests per second from one IP gets a ban.
  - The daily quota is 10,000 requests with a key, or per user with an access token. Without a key, the quota observed was 300 (`quota_max: 300`).
  - A `backoff` field must be honoured. I got a `throttle_violation` for calling a second site within the 10 s backoff, so the backoff seems to apply across sites.
- **Dumps.**
  - Since July 2024 the dumps are behind a login and a click-through: "…for projects that do not include training a large language model (LLM)…". https://www.devclass.com/development/2024/07/30/stack-exchange-restricts-access-to-dump-of-user-contributed-data-critics-complain-this-contradicts-license/1625192
  - **The watermarks are documented.** "v1" was two fabricated `<row>` entries in `Posts.xml`, with post IDs `1000000001` and `1000000010`. They appeared only in the 2025-06-30 dumps (rev1 and rev2), and no 2026 dump is listed as watermarked. https://github.com/LunarWatcher/se-data-dump-transformer/blob/master/docs/meta/Known%20watermarks.md
  - Community re-uploads to the Internet Archive go up to 2026-06-30. https://search.feep.dev/blog/post/2026-08-25-stackoverflow-archive (from search; the 2026-09-30 dump's re-upload is UNVERIFIED)
- **New channel.**
  - Stack Overflow launched a public **MCP server** in beta in December 2025, limited to 100 requests a day. https://stackoverflow.blog/2025/12/01/what-s-new-at-stack-overflow-december-2025/ (from search)
  - The API docs navigation now has an "MCP Server" entry.
  - Whether it covers MathOverflow and Math.SE is UNVERIFIED.
- **Commercial AI licensing.** Stack Overflow sells "Stack Data Licensing" to AI labs (footer link on policies.stackoverflow.co). Reports from late 2025 describe deals with several labs; these come from secondary sources only.

**What math2 can take:** `open`, per post, through the API, exactly as spec 16.6 says.
- Anonymous use gives only 300 calls a day. math2 needs a registered key for 10,000.

---

## A.25 Other large human-written references (brief)

- **MathBases** (https://mathbases.org/) is a directory "Of, by, and for mathematical databases". Its area tags sum to 251; this is not a count of databases.
  - Entry pages have a licence field. For example, it gives DaRT as "MIT".
  - It is a useful seed list for new adapters. Its own licence is UNVERIFIED.
- **FindStat** (https://www.findstat.org/) has 1,997 statistics, 336 maps and 24 collections. The home page states no licence (still [U] in spec 1.7).
- **Lexikon der Mathematik** (Springer Spektrum) is proprietary. Wikidata P12975 has 603 IDs [measured]: record only.
- **All-Russian Mathematical Portal** (Math-Net.Ru) is a bibliographic portal, not an encyclopedia. Wikidata P4252 has 5,090 IDs [measured].
- **Metamath** (`set.mm`). Wikidata P12888 has 438 statement IDs [measured]. `set.mm` is said to be public domain or CC0, which is UNVERIFIED here. It is formal, so it belongs in the formal-library catalogue (9.4).
- **Not reached:** GroupNames (Bristol) refused the connection; Kerodon's licence is not on its about page.

---

## Summary table

| Source | Size (2026-10-26 unless noted) | Licence of text / data | Channel | AI terms | What math2 can take |
|---|---|---|---|---|---|
| Wikipedia (en) | 28,196 article pages in WP Math (33,957 project pages) | CC BY-SA 4.0 (+GFDL where no BY-SA-only text) | monthly XML dump (2026); Enterprise free tier 30 snapshots and 50k on-demand a month; API 10–2,000 req/min | Nov 2025 ask: attribute, use Enterprise; no licence change | **text with attribution** |
| Wikipedia de/fr/ja/zh/ru | 20,732 / 12,192 / 8,865 / 5,682 / 3,928 (project-tagged) | CC BY-SA 4.0 | same dumps | same | text with attribution |
| Wikidata | 113,118 P2534 items, **105,345 of them primes**; 6,560 non-prime; 3,208 theorems | CC0 | weekly JSON dump (103 GB bz2); WDQS 60 s, 5 parallel | none; May 2026 agent overload | **all (metadata)** |
| nLab | 20,778 pages | **no licence**; "free … if you acknowledge the source" | git mirror per edit | none | **link only** (+ ids, link graph) |
| ProofWiki | 47,254 content pages; 30,348 proofs / 36,102 definitions | CC BY-SA 3.0 + GFDL | **daily `latest.xml.gz` (36.6 MB)** | none | **text with attribution** |
| PlanetMath | 9,477 entries; inactive (last push Feb 2024) | CC BY-SA 3.0 | git (64 MSC repos) | none | text with attribution |
| Encyclopedia of Mathematics | >8,000 (2002 print); 1,224 Wikidata IDs; live count UNVERIFIED (502) | **originals © Springer**; new edits CC BY-SA 3.0 + GFDL | none (MediaWiki API) | none | **record only** (text only for wholly new articles) |
| MathWorld | 16,939 entries | proprietary; no bulk, no robots for archiving | none | none stated | record only |
| OEIS | 400,709 sequences | CC BY-SA 4.0 | `oeisdata` git (daily), `stripped.gz`; Crawl-delay 10 | none | text with attribution |
| DLMF | v1.2.8 (2026-09-15) | **© NIST**, no bulk, no commercial | none | none | record only |
| Fungrim | 3,130 entries, 457 symbols; dormant since 2021–22 | **MIT** | git | none | text with attribution |
| Wolfram Functions | 307,409 formulas | © Wolfram, terms not found | none | none | record only |
| LMFDB | (2022 figures in spec 1.7) | data CC BY-SA 4.0; code GPL v2+ | bulk downloads; **`/api/` disallowed in robots; JS gate since Aug 2025** | asks AI users not to scrape, points to CC guidance | data with attribution, via bulk only |
| groupprops (+ other subwikis) | 8,076 articles (topospaces 825, commalg 429, diffgeom 435) | CC BY-SA 3.0 | MediaWiki API | none | text with attribution |
| Manifold Atlas | 215 articles | **contradictory**: GFDL 1.3 and "public domain" | MediaWiki API (HTTP only) | none | link only (treat as GFDL) |
| π-Base | 225 spaces, 247 properties, 935 theorems | CC BY 4.0 | git | none | text with attribution |
| House of Graphs | ~22,000 graphs (2022); now UNVERIFIED | none stated | API needs auth | none | record only |
| ISGCI | 1,695 classes, 242,975 inclusions | none stated | Java app | none | record only |
| DaRT | 218 rings, 178 properties, 72 theorems | CC BY 4.0 | git | none | text with attribution |
| KnotInfo | knots ≤13 crossings, ~116 fields | none on site; **CSV package GPL-3.0** | Excel/CSV download | none | record only |
| Knot Atlas | 2,710 articles (1.65M pages) | none stated | MediaWiki API | none | record only |
| ATLAS v3 | ~716 groups, 5,215 reps | none stated (AtlasRep GPL-3+) | AtlasRep on demand | none | record only |
| Open Problem Garden | ≤713 entries | GFDL (prob. 1.3) | none | none | link only |
| Erdős problems | 1,221 | **site text: no licence**; repo Apache-2.0 but **metadata only** | git YAML | none | status as metadata; statements record only |
| Kourovka Notebook | 21st ed., arXiv:1401.0300v49 (2026-10-20) | arXiv non-exclusive | arXiv | none | record only |
| MathOverflow | 168,603 Q / 201,920 A | CC BY-SA 2.5/3.0/4.0 per post | API (10k/day with key; 300 anonymous); dumps no-LLM click-through | dump click-through; ToS updated 2025-11-13 (content UNVERIFIED) | per-post text with attribution, via API |
| Math.SE | 1,702,027 Q / 2,236,177 A | as above | as above | as above | as above |

---

## Proposed corrections to the spec

1. **1.7 and 9.2, Wikidata:**
   - 113,118 "items with a defining formula" are mostly prime numbers (105,345). Report **6,560** non-prime items.
   - Add the measured crosswalk properties: P7726 PlanetMath 951, P7554 EoM 1,224, P829 OEIS 469, P12934 House of Graphs 213, P11497 DLMF 199, P12987 LMFDB knowl 109, P8427 KnotInfo 2,978, P8426 Knot Atlas 2,228, P14106 Erdős 3.
2. **1.7 and 9.2, Encyclopedia of Mathematics:** change the licence from "CC BY-SA 3.0" to **"originals © Springer; only new wiki text CC BY-SA 3.0 + GFDL"**, and the class to **`record`**.
3. **9.2, Erdős problems:** `teorth/erdosproblems` has **no statements**. The informal statements on erdosproblems.com carry no licence, so they are record only.
4. **1.7 and 9.2, nLab:** 20,778 pages. The licence is now confirmed as *none*, with an informal permission, so the class is `record` (link only), and no longer [U].
5. **1.7 and 9.2, ProofWiki:** the licence is CC BY-SA 3.0 **and GFDL**. The daily dump is live (36.6 MB gzipped).
6. **1.7 and 9.2, LMFDB:** `robots.txt` disallows `/api/`, and there is a JavaScript gate. The channel is bulk downloads only.
7. **1.7, Wikipedia:** 28,196 article pages (not "about 26k"). The dumps are monthly in 2026. Add the 2026 API rate limits.
8. **9.3, licence classes:** add DLMF (NIST copyright), Wolfram Functions, the Kourovka Notebook (arXiv default), KnotInfo, ISGCI, the ATLAS and House of Graphs to `record`. Add Manifold Atlas to `gfdl`. Add Fungrim (MIT), DaRT (CC BY) and the subwikis (CC BY-SA 3.0) to `open`.
9. **1.7, MathWorld:** 16,939 entries. **OEIS:** 400,709.

---

## Open doubts

1. **EoM.** I could not read the live site (502 all morning). Did the move to EMS Press change the Springer copyright on the original articles? What is the current article count? A retry and a question to the EMS would settle it.
2. **Stack Exchange Public Network Terms of 13 Nov 2025.** Do they now forbid AI training or scraping of public content? (The text is loaded by JavaScript and I could not read it.) Does the new public MCP server cover MathOverflow and Math.SE?
3. **nLab.** Would the steering committee agree to an explicit CC BY-SA statement for new contributions? Old pages would still need each author's consent.
4. **Manifold Atlas.** GFDL 1.3 or public domain? Ask the editors.
5. **LMFDB.** Do knowls fall under the CC BY-SA "database" licence? Which bulk channel does the LMFDB want AI users to use? The site's JavaScript gate stopped me reading the current download page.
6. **House of Graphs.** The current number of graphs and of invariants, and any licence. The API needs authentication.
7. **KnotInfo.** Did the GPL-3.0 relicensing of the CSV package have the authors' consent? KnotInfo itself states no licence.
8. **Wikipedia dumps.** Is the move from twice-monthly to monthly runs official? Were the free `enterprise_html` dumps (last run 2025-03-20) withdrawn, or moved?
9. **Wikidata WDQS.** The details of the 2026–2027 backend migration and of the scholarly graph split, and whether the new endpoints have different limits.
10. **ProofWiki Main Page counters** (30,348 proofs, 36,102 definitions) are the same as in the spec. Are they live?
11. **Open Problem Garden.** The exact GFDL version (1.2 or 1.3) is inferred from an image file name.
12. **Unreached sources:** GroupNames (connection refused), FindStat's licence, Kerodon's licence, and MathBases' own licence.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/01b-literature-and-books.md">
# K01, part B. Literature indexes, open textbooks and syllabi: sizes, licences, channels (research for math2)

Status: research note for spec 01 §1.7, spec 09 (all, especially the 9.3 licence gate) and spec 16 §16.6.
Read date: **every fact below was read on 2026-10-26** unless another date is given. Each fact has a URL next to it.
**UNVERIFIED** means I could not confirm the fact at a primary source. The reason is given each time.
Method: WebSearch, WebFetch and a few single `curl` requests with User-Agent `m2/research (+https://github.com/tamnd/math2-agent)`. There was no crawling and no bulk download. The largest samples were three days of arXiv OAI-PMH `set=math` (1,421 records), the Open Textbook Library mathematics JSON (22 pages), four Mathlib YAML files, two syllabus PDFs and one zbMATH API query.

This note does not repeat what the spec already says. The spec already has: arXiv about 15% open, `s3://arxiv` requester-pays and Kaggle CC0; zbMATH 168M triples; Stacks GFDL 1.2 with 21,446 tags; TheoremSearch 9.2M / 690k / 1.34M; TheoremGraph 11.7M / 18.3M / 48k; Matlas 8M / 435k / 1.9k; MSC2020 CC BY-NC-SA; the Stack Exchange no-LLM clause. Where my findings correct or refine those figures, I say so.

The access classes used in the "math2 can take" column are those of spec 9.3: `open`, `share-alike-nc`, `verbatim`, `gfdl`, `record`. Node prose is CC BY-SA 4.0, code is Apache-2.0 and metadata is CC0 (16.6).

---

## Part 1. Literature indexes and full text

### 1.1 arXiv

**Bulk and API terms**
- **S3 bulk:** the bucket `arxiv` (prefixes `pdf/`, `src/`) is requester-pays in us-east-1. Files come as tars of about 500 MB with manifests and are updated about monthly. The total is "about 9.2 TB" as of April 2025, growing by about 100 GB a month. Users "must link back to arXiv for downloads". (https://info.arxiv.org/help/bulk_data_s3.html, https://info.arxiv.org/help/bulk_data.html)
- **API Terms of Use** (https://info.arxiv.org/help/api/tou.html; the page carries no date):
  - Metadata is CC0.
  - "The redistribution of e-prints requires permission from the copyright holder." Storing and serving e-prints needs permission or a licence; personal and research use is allowed.
  - The rate limit is one request every 3 seconds on a single connection, counted across all of a user's machines.
  - Users may not claim arXiv endorsement.
- **AI and TDM terms:** none of the four arXiv help pages above says anything about AI, LLMs, TDM or training. So for arXiv the per-paper licence is the only AI-relevant term. (A missing clause is not the same as permission. See open doubt D3.)
- **Licence options for authors** (https://info.arxiv.org/help/license/index.html):

| Licence | 9.3 class |
|---|---|
| CC BY 4.0 | `open` |
| CC BY-SA 4.0 | `open` |
| CC0 | `open` |
| CC BY-NC-SA 4.0 | `share-alike-nc` |
| CC BY-NC-ND 4.0 | `verbatim` |
| arXiv non-exclusive distribution licence 1.0 (http://arxiv.org/licenses/nonexclusive-distrib/1.0/) | `record` |

**New measurement: the 2026 licence mix in math.** I harvested three single days from `oaipmh.arxiv.org/oai`, with `verb=ListRecords&metadataPrefix=arXiv&set=math`: from=until=2026-10-01, 2026-04-15 and 2026-07-08. That gave 1,421 records. The 1,026 records with 26xx identifiers (papers first submitted in 2026) break down as follows:

| Licence | Share |
|---|---|
| non-exclusive (record) | 55.3% |
| CC BY 4.0 | 34.0% |
| CC BY-NC-ND | 5.9% |
| CC BY-NC-SA | 1.9% |
| CC0 | 1.9% |
| CC BY-SA | 1.0% |

- **`open` share for node prose (BY + BY-SA + CC0): 36.9%.** By day it was 35.3%, 38.3% and 38.8%. Records with older identifiers that were re-versioned on those days (395 of them) were about 39.2% `open`.
- **Consequence:** the spec's "about 15% of arXiv is `open`" is the all-time stock. The 2026 flow in math looks more than twice as open. This is a small sample and needs confirmation (D2).
- **Caveats:**
  - The OAI datestamp also counts metadata updates, not just new submissions.
  - I did not establish whether `set=math` includes cross-lists.

### 1.2 zbMATH Open

- **OAI-PMH:** Identify at https://oai.zbmath.org/v1/?verb=Identify works. It reports earliestDatestamp 1755.
- **REST API** (https://api.zbmath.org/v1/document/_search):
  - It returns full review text (`editorial_contributions`) and a per-document `license` field. That field was empty in my sample.
  - The query `py:2025` returns `nr_total_results` 133,093. That is a reasonable yearly-volume figure for 2025.
- **API terms:** https://static.zbmath.org/legal/api-terms-and-conditions.html was blocked by a Cloudflare challenge, and so was https://zbmath.org/about/. **The API terms, and any AI clause in them, are UNVERIFIED.**
- **Knowledge graph dump** (Zenodo record 21497976, concept DOI 10.5281/zenodo.21497975; https://zenodo.org/records/21497976):
  - Published 2026-09-01 and modified 2026-10-24. File `zbmath-kg-ttl-2026.tar.gz`, 1.6 GB.
  - Built from the January 2026 API dump. It is updated about twice a year, not monthly as spec 9.2 says.
  - A temporary SPARQL endpoint is at http://212.227.170.235:8890/sparql.
  - Repository: https://github.com/zbMATHOpen/zbmath-open-kg. Paper: https://arxiv.org/abs/2609.00969.
  - Stats: 168M+ triples, 34M+ entities, 4M+ publications, 1M+ persons, 3M+ reviews.
  - Restricted abstracts appear as the literal placeholder "zbMATH Open Web Interface contents unavailable due to conflicting licenses".
- **The licence conflict:** the Zenodo metadata field says **CC BY 4.0**, but the record's description says **CC BY-SA 4.0** for the dataset, with the I4OC references CC0. The spec says "CC BY-SA 4.0". For math2 both are `open`. They do differ on whether derived exports must be BY-SA (D1).
- **What math2 can take:**
  - Records, MSC codes and links as metadata.
  - Review text is `open` under either reading. In practice it should be used as evidence and for locators, not pasted into prose. Reviews are third-party texts, and their per-item rights in the API `license` field are empty or unclear.

### 1.3 OpenAlex and Unpaywall

- **Snapshot** (https://help.openalex.org/download-all-data/openalex-snapshot, page updated 2026-09-24):
  - The bucket `s3://openalex` is free and anonymous (`--no-sign-request`). `LICENSE.txt` in the bucket is CC0 1.0, which I checked.
  - The September 2026 release is about 745 GB of compressed JSONL (works about 615 GB), or about 770 GB in Parquet.
  - The free snapshot is quarterly, on the second Wednesday of January, April, July and October. Daily snapshots are paid.
- **API key and pricing:** an API key has been required since 2026-02-13, with usage-based pricing. **Partly UNVERIFIED**: I found this only in secondary sources (https://casrai.org/news/openalex-api-keys-mandatory-usage-based-pricing-2026). https://help.openalex.org/api/ did not render the details for me.
- **Unpaywall:**
  - Unpaywall now runs inside OpenAlex (the "Walden" rewrite), and its data feed is a paid product.
  - The last free standalone Unpaywall snapshot I found is dated 2024-11-27, according to a SUB Göttingen note.
  - https://unpaywall.org/products/snapshot returned an empty page. **The current Unpaywall terms are UNVERIFIED.**
- **What math2 can take:** all of it, as CC0 metadata (works, authors, venues, concepts, OA locations). OpenAlex has no full text.

### 1.4 Crossref

- **REST API** (https://www.crossref.org/documentation/retrieve-metadata/rest-api/): no sign-up is needed. Crossref says "almost none of the metadata is subject to copyright", but "some abstracts ... may be subject to copyright".
- **2026 public data file** (https://www.crossref.org/blog/2026-public-data-file-now-available/, posted 2026-03-17): 180,034,490 records as of 2026-03-14, 208 GB of JSONL. It is distributed via Academic Torrents (doi 10.13003/nggf-vt1j) or AWS.
- **What math2 can take:** DOI metadata and reference lists as facts. Abstracts are `record` unless the item's own licence says otherwise.

### 1.5 DataCite

- **Public data file:** annual and CC0. The use policy asks for credit.
  - The 2025 file was generated on 2026-01-06 and announced on 2026-01-22. It has 108M findable DOIs and is at https://datafiles.datacite.org/.
  - The monthly file is for members only. The file is also on Dimensions' BigQuery.
  - Sources: https://support.datacite.org/docs/datacite-public-data-file, https://support.datacite.org/docs/datacite-data-file-use-policy, and the DataCite blog.
- **What math2 can take:** all of it, as metadata. Its use for math2 is datasets and software DOIs (Zenodo and others).

### 1.6 Semantic Scholar and S2ORC

- **API licence** (https://www.semanticscholar.org/product/api/license, "last updated" 2023-05-17):
  - The licence is non-transferable. You may not redistribute or sublicense the API.
  - S2 data is under its own licences, "such as CC BY-NC or ODC-BY". Attribution is required.
  - It has no AI-training clause.
- **S2ORC:**
  - https://github.com/allenai/s2orc says ODC-By 1.0 (earlier releases were NC). The Hugging Face card at https://huggingface.co/datasets/allenai/s2orc is tagged cc-by-2.0, which is an inconsistency.
  - Size: a 2026 arXiv paper reports the 2026-05-26 release as having 15,679,781 S2ORC full-text papers.
- **UNVERIFIED:**
  - The licence of the separate abstracts dataset (the datasets API docs returned HTTP 429).
  - The current total of papers. A third-party figure for 2025-09 says 200M+ papers and 100M+ abstracts.
- **What math2 can take:**
  - Paper ids as crosswalk targets.
  - S2ORC full text is ODC-By *as a database*, but each paper's text keeps the paper's own copyright. Treat it per paper, by its arXiv or publisher licence. It is never `open` just because it sits in S2ORC.

### 1.7 CORE

- **Terms** (https://core.ac.uk/terms; I read this via a search snippet because the page returned 403 to curl):
  - Dumps are ODC-By, covering CORE's own database rights only.
  - API access needs a licence; it is free for individuals and for public research.
  - A licensing-framework review proposes CC0/ODC-0 for metadata and "conditional access for lawful TDM" for full texts.
- **Size:** 291M metadata records and 32.8M full texts (February 2023, the CORE paper in *Scientific Data*). **The current size and the current dump terms are UNVERIFIED** (core.ac.uk/documentation returned 403).
- **What math2 can take:** records. Full texts follow each paper's licence, as for S2ORC.

### 1.8 HAL (France)

- **OAI-PMH** (https://api.archives-ouvertes.fr/oai/hal/?verb=Identify): "Metadata is licensed under CC0".
- **Math share, measured by API:** a query on `level0_domain_s:math` returned **170,406 records, 105,413 of them with a file**. (The query `domain_s:math*` returns 0.) The licence facet over the records with a file:

| Licence | Records | 9.3 class |
|---|---:|---|
| hal-authorisation-v1 (https://about.hal.science/hal-authorisation-v1/) | 84,016 | `record` |
| CC BY 4.0 | 9,239 | `open` |
| hal copyright (https://hal.science/licences/copyright/) | 4,272 | `record` |
| CC BY-NC-ND | 3,861 | `verbatim` |
| CC BY-NC | 1,796 | `share-alike-nc` |
| CC BY-NC-SA | 731 | `share-alike-nc` |
| CC0 | 492 | `open` |
| CC BY-SA | 449 | `open` |
| CC BY-ND | 254 | `verbatim` |
| Etalab 2.0 | 163 | `open` (attribution licence, compatible with CC BY) |
| PDM | 6 | `open` |
| none / empty | 113 + 21 | `record` |

- **`open` share: 10,349 / 105,413 = about 9.8%.**
- **Channels:** REST at api.archives-ouvertes.fr/search, SPARQL at https://data.hal.science/doc/sparql, and OAI-PMH.
- I found no AI clause.

### 1.9 Numdam (Mathdoc)

- **Size** (https://www.numdam.org/en/): 75,599 articles in 120 periodicals, 706 books in 8 collections, 416 theses, 1,428,508 pages.
- **Moving wall:** 2 to 10 years. Mersenne journals arrive after 1 year.
- **Conditions** (https://www.numdam.org/conditions/):
  - Metadata is CC0 and can be harvested by OAI-PMH.
  - Full texts are for individual download for research, study or training.
  - Downloading "un volume substantiel" without explicit permission is forbidden, and so is modifying the files.
  - Republishing is allowed only for public-domain items (described as "très rares sur Numdam") and for CC-licensed articles.
- **What math2 can take:** records (CC0). Text is `record` by default, or the per-article CC licence where one is given. Bulk full text is not allowed without permission.

### 1.10 EuDML

- **UNVERIFIED for current figures.** eudml.org, including its OAI endpoint, returned 403 or a Cloudflare challenge.
- What I could find:
  - Mathdoc's page (https://www.mathdoc.fr/en/realisations/eudml/, undated) says "over 270,000 documents".
  - A 2016 report says 257k items, 252,111 articles and 323 journals.
  - https://eudml.org/journals listed 339 journals in a search snippet.
  - EuDML's policy is to hold only content past its moving wall, with JATS-flavoured OAI. **Its metadata licence is UNVERIFIED.**
- **What math2 can take:** records via the partner sources (Numdam, GDZ, CEDRAM and others) rather than EuDML itself.

### 1.11 Project Euclid

- https://projecteuclid.org/about says "over 80% of the journal articles hosted ... are openly available". It is run by Duke University Press.
- "Openly available" means free to read, not openly licensed. **Per-journal licences are UNVERIFIED.** Default to `record`, and use the per-article licence where it is shown (for example, some IMS/Bernoulli journals).

### 1.12 GDZ (Göttinger Digitalisierungszentrum)

- **Size:** over 15M pages, with collections for Mathematica and FID Mathematik (https://gdz.sub.uni-goettingen.de/).
- **IIIF:** the manifest for *Mathematische Annalen* vol. 1 (1869) at https://manifests.sub.uni-goettingen.de/iiif/presentation/PPN235181684_0001/manifest has an attribution but **no `license` field**.
- **PDF terms:** a generated PDF (https://gdz.sub.uni-goettingen.de/download/pdf/PPN71557308X/LOG_0040.pdf) has a cover page saying reproductions may not be made for other repositories without written permission.
- **The licence of the scans is UNVERIFIED.** The underlying 19th-century texts are public domain.
- **What math2 can take:** records and locators. Statements from public-domain works, which math2 rewrites anyway. Not the scans.

### 1.13 The US public-domain line, HathiTrust, Internet Archive, Project Gutenberg

**The US public-domain line.** Works published in 1930 entered the US public domain on 2026-01-01 (https://web.law.duke.edu/cspd/publicdomainday/2026/). So in 2026, works published in the US before 1931 are public domain in the US. Public domain is `open` for prose: no attribution is needed, but math2 should give it anyway.

**HathiTrust**
- hathitrust.org (terms and acceptable-use pages) is behind Cloudflare. **The terms and any AI policy are UNVERIFIED.**
- https://github.com/hathitrust/datasets lists the public-domain text subsets `ht_text_pd`, `ht_text_pd_open_access`, `ht_text_pd_world` and `ht_text_pd_world_open_access`. The full `ht_text` is used only by HTRC for non-consumptive research.
- A related public-domain corpus: Harvard Institutional Books 1.0 has 983,004 public-domain volumes.

**Internet Archive**
- Counts from advancedsearch (https://archive.org/advancedsearch.php?q=subject%3Amathematics+AND+mediatype%3Atexts&rows=0&output=json):
  - `subject:mathematics AND mediatype:texts` gives 206,597 items.
  - Restricting to `year:[1500 TO 1930]` gives 8,076.
- https://archive.org/about/terms returned an empty page. **The IA terms are UNVERIFIED.** Uploads on IA include many in-copyright books. For example, Hatcher's *Algebraic Topology* is there, though its licence is noncommercial copying only (§2.4). IA hosting therefore says nothing about licence.

**Project Gutenberg**
- **Robot policy** (https://www.gutenberg.org/policy/robot_access.html): the site is "intended for human users only". Bulk users should use a mirror, or wget with `-w 2` through `/robot/harvest`. Catalogue data is public domain.
- **Licence** (https://www.gutenberg.org/policy/license.html): "Nearly all the individual works ... public domain in the United States". The trademark licence no longer applies once "all references to Project Gutenberg are removed".
- **Size:** about 78,000 items (a 2026-03-22 figure).
- **Mathematics bookshelf 102** (https://www.gutenberg.org/ebooks/bookshelf/102) includes:
  - Hardy, *A Course of Pure Mathematics* (#38769, with LaTeX source)
  - Dedekind (#21016)
  - Casey's Euclid (#21076)
  - Thompson, *Calculus Made Easy*
- **What math2 can take:** `open` (public domain in the US) with the Project Gutenberg header removed. The non-US public-domain status of each work is a separate check (D10).

### 1.14 Extraction projects (TheoremSearch, TheoremGraph and LeanGraph, Matlas, and new ones)

**TheoremSearch** (https://arxiv.org/abs/2602.05216, v2 dated 2026-03-07)
- **Size:** https://theoremsearch.com shows "8.4M" indexed. By source:

| Source | Statements |
|---|---:|
| arXiv | 8,104,539 |
| Lean | 285,213 |
| ProofWiki | 23,284 |
| Stacks | 10,645 |
| Lean Community | 1,888 |
| Open Logic | 588 |
| CRing | 498 |
| HoTT | 257 |
| Napkin | 228 |

  The spec's 9.2M figure is from the paper; the live site now shows 8.4M.
- **Channels:** a REST API at https://api.theoremsearch.com/search and MCP at https://api.theoremsearch.com/mcp. **No terms of use are published.**
- **Released data:**
  - https://huggingface.co/datasets/uw-math-ai/theorem-search-dataset is tagged `cc-by-sa-4.0`, with 1,341,083 theorems from 209,777 papers. That matches the spec's 1.34M. It mixes Stacks (GFDL), ProofWiki and other sources under one BY-SA tag, which is a licence inconsistency (D4). Its card says "~15% of arXiv papers use permissive licenses (CC BY, CC BY-SA, or CC0)".
  - https://huggingface.co/datasets/uw-math-ai/theorem-search-dataset-permissive is tagged `cc-by-4.0`. Its card says 1,239,720 theorems from 197,889 papers (arXiv 1,238,975, Open Logic 745), covering CC0, CC BY 4.0, CC BY 3.0 and PD. ShareAlike is excluded. The dataset viewer shows 2,677,439 rows, which does not match the card (D5). The "slogans" (natural-language summaries) were written by DeepSeek-V3.1.
- **What math2 can take:** the `-permissive` set as `open` candidate evidence (E1), keeping each row's per-paper licence. The BY-SA set should be used only after filtering out Stacks and any other GFDL or NC rows.

**TheoremGraph and LeanGraph** (https://arxiv.org/abs/2606.25363, 2026-06-24, cs.IR; the paper itself is under the arXiv non-exclusive licence)
- https://huggingface.co/datasets/uw-math-ai/math-graph:
  - The card says CC BY 4.0, with "per-record reuse governed by each statement's source license". It has 16,131,238 rows.
  - The graph structure is open. Statement text is filled in only for CC0/BY/BY-SA papers and is NULL otherwise. `paper_arxiv.csv` has a licence column.
- https://huggingface.co/datasets/uw-math-ai/theorem-matching has 101k rows, built from the Kaggle arXiv metadata.
- **What math2 can take:** the edges and ids (CC BY 4.0, so attribute them), and text only where it is non-NULL. This is the best-licensed of the three big extractions.

**Matlas** (https://arxiv.org/abs/2604.17484, 2026-04-19; the paper is CC BY 4.0)
- **Size:** 8.07M statements from 435K papers (1826–2025), 180 journals and 1.9K textbooks.
- **API:** https://matlas.ai/openapi.json describes `POST /api/search` (`num_results` 10–200). It returns the `statement` text with doi, title, authors, journal, year and type. There is also `/api/feedback`.
- No auth, no licence and no terms are published. **The licence of the corpus is UNVERIFIED and probably not open.** The statements are from journals and textbooks under copyright.
- **What math2 can take:** under spec 9.6, query-time matching only, storing only DOIs and ids (`record`).

**New 2026 projects not in spec 01**

| Project | Date | Licence | What it is |
|---|---|---|---|
| uw-math-ai/FormalArXiv (https://huggingface.co/datasets/uw-math-ai/FormalArXiv) | n/a | "other" | 89 Lean statements from August 2026 arXiv papers, via LemmaBench |
| Lean Pool (https://arxiv.org/abs/2609.25199, Ilin) | 2026-09-21 | paper CC BY 4.0 | an AI-maintained archive of formalised mathematics |
| AnnalsChallenge (https://arxiv.org/abs/2610.18301, Buzzard et al.) | 2026-10-13 | paper CC BY-SA 4.0 | 50 Lean statements of Annals results |
| Stacks↔Mathlib alignment (https://arxiv.org/abs/2610.31011; code https://github.com/AnkitSatpute/Formal-Informal-Mathematical-Alignment) | 2026-10-22 | paper CC BY 4.0 | added to MaRDI |
| KnowTeX (https://arxiv.org/abs/2601.15294) | n/a | n/a | n/a |
| uw-math-ai other artefacts (https://huggingface.co/uw-math-ai: MELD, Math2Vec, APRIL, MathLeap models; https://github.com/uw-math-ai/arXiTeX) | n/a | no licences on the org page | n/a |

---

## Part 2. Open textbooks

Licences were read at the book's own source (repository LICENSE, README or front matter) unless a row says "per OTL". In the Open Textbook Library (OTL), licences are sometimes stale; see §2.3.

### 2.1 Verified at source

| Book | Licence (exact) | Format | Repository / source URL | 9.3 class |
|---|---|---|---|---|
| Univalent Foundations, *Homotopy Type Theory* (HoTT book) | CC BY-SA 3.0 Unported | LaTeX | https://github.com/HoTT/book (README) | `open` |
| Open Logic Project | CC BY 4.0 | LaTeX | https://github.com/OpenLogicProject/OpenLogic | `open` |
| Hefferon, *Linear Algebra* | dual: GFDL **or** CC BY-SA 3.0 US, at the user's choice | LaTeX | https://hefferon.net/source.html, https://gitlab.com/jim.hefferon/linear-algebra | `open` (take it under BY-SA) |
| Lebl, *Basic Analysis* I/II | dual: CC BY-NC-SA 4.0 **and** CC BY-SA 4.0 | LaTeX | https://github.com/jirilebl/ra | `open` (take it under BY-SA) |
| Lebl, *Notes on Diffy Qs* | dual: CC BY-NC-SA 4.0 and CC BY-SA 4.0 | LaTeX | https://github.com/jirilebl/diffyqs | `open` (BY-SA) |
| Lebl, *Tasty Bits of Several Complex Variables* | dual: CC BY-NC-SA 4.0 and CC BY-SA 4.0 | LaTeX | https://github.com/jirilebl/scv | `open` (BY-SA) |
| Boelkins et al., *Active Calculus* (single variable, 2012–2025 edition) | CC BY-SA 4.0 | PreTeXt | https://github.com/active-calculus/active-calculus-single-mbx (`source/bibinfo.xml`) | `open` |
| Evan Chen, *An Infinitely Large Napkin* | text CC BY-SA 4.0; source code GPL v3 | LaTeX | https://github.com/vEnhance/napkin/ | `open` (text) |
| Levin, *Discrete Mathematics: An Open Introduction*, **3rd edition and earlier** | CC BY-SA 4.0 | PreTeXt | https://github.com/oscarlevin/discrete-book | `open` |
| Levin, same book, **4th edition** | CC BY-NC-SA 4.0 | PreTeXt | same | `share-alike-nc` |
| Keller and Trotter, *Applied Combinatorics* | CC BY-SA 4.0 | PreTeXt | https://github.com/mitchkeller/applied-combinatorics | `open` |
| Ernst, *An Introduction to Proof via Inquiry-Based Learning* | CC BY-SA 4.0 | LaTeX | https://github.com/dcernst/IBL-IntroToProof | `open` |
| Austin, *Understanding Linear Algebra* | CC BY 4.0 (licence link in README) | PreTeXt | https://github.com/davidaustinm/ula | `open` |
| Beck, Marchesi, Pixton, Sabalka, *A First Course in Complex Analysis* v1.6 | CC BY 4.0 | PDF (LaTeX not checked) | https://matthbeck.github.io/complex.html | `open` |
| Lehman, Leighton, Meyer, *Mathematics for Computer Science* (2018 rev.) | CC BY-SA 3.0 ("available under the terms of the Creative Commons Attribution-ShareAlike 3.0 license") | PDF | https://courses.csail.mit.edu/6.042/spring18/mcs.pdf | `open` |
| Judson, *Abstract Algebra: Theory and Applications* (AATA) | GFDL 1.3 or later | PreTeXt | https://github.com/twjudson/aata | `gfdl` |
| Beezer, *A First Course in Linear Algebra* (FCLA) | GFDL 1.2 or later | PreTeXt | https://github.com/rbeezer/fcla | `gfdl` |
| The Stacks Project | GFDL 1.2 | LaTeX | https://stacks.math.columbia.edu (COPYING) | `gfdl` |
| Hartman et al., *APEX Calculus* | CC BY-NC 4.0, © 2021 Hartman | PreTeXt | https://github.com/APEXCalculus/APEXCalculusPTX (`ptx/apex.ptx`) | `share-alike-nc` |
| Feldman, Rechnitzer, Yeager, *CLP Calculus* (UBC) | CC BY-NC-SA 4.0 ("It is licensed under a Creative Commons BY-NC-SA 4.0 International License") | LaTeX source published | https://personal.math.ubc.ca/~CLP/src/ | `share-alike-nc` |
| Axler, *Linear Algebra Done Right*, 4th edition | CC BY-NC 4.0 (Springer open access) | PDF; HTML via PreTeXt (David Farmer) | https://link.springer.com/book/10.1007/978-3-031-41026-0, https://library.oapen.org/handle/20.500.12657/85067, https://linear.axler.net/ (PDF dated 2026-08-16), https://ladr.pretext.plus/ | `share-alike-nc` |
| Hammack, *Book of Proof*, 3rd edition (v3.4, 2025-02-05) | CC BY-NC-ND 4.0 | PDF only | https://richardhammack.github.io/BookOfProof/ | `verbatim` |
| Hatcher, *Algebraic Topology* | not a CC licence: "Single paper or electronic copies for noncommercial use may be made freely without explicit permission from the author or publisher" (© 2002 CUP) | PDF | https://pi.math.cornell.edu/~hatcher/AT/ATpage.html | `record` |
| Lurie, *Kerodon* | **no licence statement found** anywhere on the site (about page: maintained by Jacob Lurie, built on Gerby). /statistics shows 8,189 tags | web (Gerby) | https://kerodon.net/ | `record` (all rights reserved by default). **This contradicts spec 9.3 and 16.6, which call it GFDL** (D6) |

### 2.2 OpenStax mathematics

Licences were read from each book's `collection.xml` `md:license` on GitHub (the openstax/osbooks-* repositories). openstax.org/subjects/math returned an empty page.

| Book (OpenStax repository) | Licence | 9.3 class |
|---|---|---|
| Calculus Vols 1–3, College Algebra 2e, Precalculus 2e, Algebra and Trigonometry 2e, Prealgebra 2e, Elementary Algebra 2e, Intermediate Algebra 2e, Introductory Statistics 2e, Introductory Business Statistics 2e, Contemporary Mathematics, Algebra 1, Principles of Data Science, *Cálculo* (Spanish) | CC BY-NC-SA 4.0 | `share-alike-nc` |
| osbooks-statistics (Statistics), *Precálculo* 2ed | CC BY 4.0 | `open` |

Earlier editions were CC BY: the OTL lists Prealgebra 2e (2015), Precalculus (2014) and Introductory Statistics 2e (2023) as BY. Earlier CC BY editions are still CC BY, and math2 could adapt those editions specifically (D7).

### 2.3 Aggregators and catalogues

**Open Textbook Library** (https://open.umn.edu/opentextbooks/subjects/mathematics.json)
- `total_count` is 214; 191 entries were retrieved. Licence counts:

| Licence | Books |
|---|---:|
| BY-NC-SA | 67 |
| BY | 51 |
| BY-SA | 38 |
| BY-NC | 17 |
| GFDL | 12 |
| BY-NC-ND | 3 |
| "No Rights Reserved" | 2 |
| BY-ND | 1 |

- So about 48% are `open` (BY, BY-SA, CC0) and 52% are not.
- The subject also contains some non-mathematics books (statistics, psychology).
- **OTL licences can be stale or simplified.** It lists *Active Calculus* 2.0 (2017) as BY-NC-SA, while the current edition is BY-SA. It lists *Book of Proof* as "Attribution-NoDerivs", while the author's page says BY-NC-ND. It lists Hefferon as BY-SA only, while the book is dual-licensed GFDL / BY-SA. **Verify at source before ingesting.**

**American Institute of Mathematics (AIM) Open Textbook Initiative, approved list** (https://textbooks.aimath.org/textbooks/approved-textbooks/)
- 68 unique titles grouped by course. **Licences are not shown on the list.** The verified licences for AIM titles are in §2.1, §2.5 and §2.6.
- AIM titles not otherwise covered in this note include Bruckner–Thomson, ORCCA, Manes, May–Bart, Marsden–Weinstein, Boyd–Vandenberghe, Treil, Fields, Bender–Williamson, Bard, Zimmermann, Crisman, Stein and Connell. Their licences were not checked (D11).

**PreTeXt catalogue** (https://pretextbook.org/catalog.html)
- 80 projects, 67 of them mathematics.
- **GFDL badges:** FCLA, Discover Linear Algebra, Fitch *Discrete Math*, Yoshiwara *Elementary Algebra*, LA Companion, LA Refresher, Yoshiwara *Modeling, Functions and Graphs*, Yoshiwara *Trigonometry*, AATA, AA Refresher, IBL Geometries, *Exploring Combinatorial Mathematics*, *First-Semester Abstract Algebra*, Hitchman *Geometry with an Introduction to Cosmic Topology*, *Models of Non-Euclidean Geometry*, *The ODE Project*, *Algebraic Graph Theory with Sage*, *Elementary Foundations*, *A Second Course in Linear Algebra*, *Abstract Algebra for Teachers*, Cryptology.
- **All rights reserved:** *Linear Algebra* (W214) and *Intro to Topology (IBL)*.
- The rest are badged only "Creative Commons", without saying which variant. Each needs a source check.

**LibreTexts** (https://math.libretexts.org/Bookshelves)
- The licence is set per page, through tags such as `license:ccbyncsa` and an auto-generated "Detailed Licensing" back-matter page. An example: https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/00%3A_Front_Matter/04%3A_Licensing.
- Many pages are "not declared". The licence mix inside a single book is common.
- The terms (https://libretexts.org/terms-conditions, effective 2020-10-15) allow copying of open-licensed content as its licence permits. They have no AI or scraping clause.
- There are no aggregate licence statistics, so the mix is **UNVERIFIED**.
- **For math2:** go to the upstream source of each LibreTexts book, not the LibreTexts copy. Use LibreTexts only page by page, by the page's `license:` tag. "Not declared" counts as `record`.

**MIT OpenCourseWare** (https://ocw.mit.edu/pages/privacy-and-terms-of-use/)
- Course materials are CC BY-NC-SA 4.0, so `share-alike-nc`.
- There is a **"Permitted Use of AI Training"** section: training is allowed with attribution, the models must be non-commercial, and derivative models must be released under BY-NC-SA.
- There is no bulk-download section.
- A note: MIT 6.042's *Mathematics for Computer Science* is distributed separately under CC BY-SA 3.0 (§2.1), so it is `open`, unlike the OCW course pages.

### 2.4 Books commonly assumed open that are not

| Book | Status |
|---|---|
| Hatcher, *Algebraic Topology* | noncommercial copying only; `record` |
| *Book of Proof* | ND, so `verbatim` |
| Axler LADR4 | NC |
| APEX | NC |
| CLP | NC-SA |
| OpenStax 2e math editions | NC-SA |
| Kerodon | no licence; `record` |

### 2.5 Table A: textbooks math2 may adapt into CC BY-SA 4.0 node prose (CC0, CC BY, CC BY-SA)

Attribution is required for all of them. CC BY-SA 3.0 (Unported or US) adaptations may be licensed under BY-SA 4.0: CC BY-SA 3.0 Unported §4(b) (https://creativecommons.org/licenses/by-sa/3.0/legalcode.en, read 2026-10-26) allows an adaptation to be distributed under "a later version of this License with the same License Elements". I did not re-read the 3.0 US port; I assume it has the same clause, which matters for Hefferon. Treat that as UNVERIFIED.

| Book | Licence math2 uses | Source format | Repository / URL | Notes |
|---|---|---|---|---|
| Open Logic Project | CC BY 4.0 | LaTeX | https://github.com/OpenLogicProject/OpenLogic | already in TheoremSearch (588 / 745 items) |
| HoTT book | CC BY-SA 3.0 → 4.0 | LaTeX | https://github.com/HoTT/book | |
| Hefferon, *Linear Algebra* | CC BY-SA 3.0 US option of the dual licence | LaTeX | https://gitlab.com/jim.hefferon/linear-algebra | record that math2 elected the BY-SA option |
| Lebl, *Basic Analysis* I/II | CC BY-SA 4.0 option of the dual licence | LaTeX | https://github.com/jirilebl/ra | elect BY-SA |
| Lebl, *Notes on Diffy Qs* | CC BY-SA 4.0 option | LaTeX | https://github.com/jirilebl/diffyqs | elect BY-SA |
| Lebl, *Tasty Bits of SCV* | CC BY-SA 4.0 option | LaTeX | https://github.com/jirilebl/scv | elect BY-SA |
| *Active Calculus* (single variable, 2012–2025) | CC BY-SA 4.0 | PreTeXt | https://github.com/active-calculus/active-calculus-single-mbx | the OTL's "BY-NC-SA" is for 2.0 (2017) and is stale |
| Napkin (text) | CC BY-SA 4.0 | LaTeX | https://github.com/vEnhance/napkin/ | do not take the GPL code |
| Levin, *Discrete Mathematics*, 3rd edition and earlier | CC BY-SA 4.0 | PreTeXt | https://github.com/oscarlevin/discrete-book | pin a pre-4th-edition commit or tag |
| Keller–Trotter, *Applied Combinatorics* | CC BY-SA 4.0 | PreTeXt | https://github.com/mitchkeller/applied-combinatorics | |
| Ernst, *Intro to Proof via IBL* | CC BY-SA 4.0 | LaTeX | https://github.com/dcernst/IBL-IntroToProof | |
| Austin, *Understanding Linear Algebra* | CC BY 4.0 | PreTeXt | https://github.com/davidaustinm/ula | |
| Beck et al., *First Course in Complex Analysis* | CC BY 4.0 | PDF | https://matthbeck.github.io/complex.html | |
| Lehman–Leighton–Meyer, *Mathematics for Computer Science* | CC BY-SA 3.0 → 4.0 | PDF | https://courses.csail.mit.edu/6.042/spring18/mcs.pdf | |
| OpenStax *Statistics*, *Precálculo* 2ed, and the earlier CC BY editions (Prealgebra 2015, Precalculus 2014) | CC BY 4.0 | CNXML | github.com/openstax (osbooks-statistics and others) | pin the CC BY edition |
| Per OTL, not verified at source: Zakon, *Mathematical Analysis I*; *Counting Rocks!*; *Calculus in Context*; Kuttler, *A First Course in Linear Algebra*; Shimamoto, *Multivariable Calculus*; Dumas–McCarthy; Wiggins, *ODE* | CC BY | various | https://open.umn.edu/opentextbooks/subjects/mathematics.json | verify at source (D8) |
| Per OTL, not verified at source: Kuttler, *Linear Algebra, Theory and Applications*; Hitchman (OTL says BY-SA, but the PreTeXt catalogue shows GFDL, D8); Petrunin, *Euclidean Plane and its Relatives*; Callahan, *Euclid's Elements Redux*; Davies, *A Cool Brisk Walk*; Brin, *Tea Time Numerical Analysis* and *Tea Time Linear Algebra*; Lippman, *Math in Society*; OpenIntro Statistics; *Introduction to Modern Statistics* 2e | CC BY-SA | various | same | verify at source |
| Public domain (US, pre-1931): e.g. Hardy, *A Course of Pure Mathematics* (PG #38769); Dedekind; Casey's Euclid; Thompson, *Calculus Made Easy* | public domain (US) | LaTeX / HTML | https://www.gutenberg.org/ebooks/bookshelf/102 | strip PG trademark text; check non-US status |

### 2.6 Table B: textbooks math2 may NOT adapt into node prose (NC, ND, GFDL, all rights reserved)

All of these can still give records, locators, tables of contents (9.7) and internal embeddings.

| Book | Licence | 9.3 class | URL |
|---|---|---|---|
| Judson, AATA | GFDL 1.3+ | `gfdl` | https://github.com/twjudson/aata |
| Beezer, FCLA | GFDL 1.2+ | `gfdl` | https://github.com/rbeezer/fcla |
| Stacks Project | GFDL 1.2 | `gfdl` | https://stacks.math.columbia.edu |
| The PreTeXt GFDL-badged titles (§2.3: Yoshiwara ×3, Fitch, ODE Project, etc.) | GFDL | `gfdl` | https://pretextbook.org/catalog.html |
| Per OTL: Grinstead–Snell, *Introduction to Probability*; Bogart, *Combinatorics Through Guided Discovery*; Corral, *Vector Calculus* and others; Collingwood et al., *Precalculus*; Yoshiwara, *Modeling* (12 GFDL in total) | GFDL | `gfdl` | https://open.umn.edu/opentextbooks/subjects/mathematics.json |
| Hartman et al., *APEX Calculus* | CC BY-NC 4.0 | `share-alike-nc` | https://github.com/APEXCalculus/APEXCalculusPTX |
| Axler, LADR 4e | CC BY-NC 4.0 | `share-alike-nc` | https://link.springer.com/book/10.1007/978-3-031-41026-0 |
| Per OTL: Axler, *Measure, Integration & Real Analysis*; Lafferriere et al. | CC BY-NC | `share-alike-nc` | OTL |
| *CLP Calculus* (UBC) | CC BY-NC-SA 4.0 | `share-alike-nc` | https://personal.math.ubc.ca/~CLP/src/ |
| Levin, *Discrete Mathematics*, 4th edition | CC BY-NC-SA 4.0 | `share-alike-nc` | https://github.com/oscarlevin/discrete-book |
| OpenStax Calculus 1–3, College Algebra 2e, Precalculus 2e, Algebra & Trig 2e, Prealgebra 2e, Elementary/Intermediate Algebra 2e, Intro Statistics 2e, Business Stats 2e, Contemporary Math, Algebra 1, Principles of Data Science, *Cálculo* | CC BY-NC-SA 4.0 | `share-alike-nc` | github.com/openstax |
| MIT OCW course materials | CC BY-NC-SA 4.0 (AI-training clause: non-commercial, BY-NC-SA models) | `share-alike-nc` | https://ocw.mit.edu/pages/privacy-and-terms-of-use/ |
| Per OTL: Guichard, *Calculus* (early transcendentals); Trench, *Introduction to Real Analysis* and *Elementary Differential Equations*; Sundstrom, *Mathematical Reasoning*; Doerr–Levasseur, *Applied Discrete Structures*; Rogers–Boman; Stitz–Zeager; Cherney et al., *Linear Algebra*; Nicholson, *Linear Algebra with Applications*; Morris, *Proofs and Concepts*; Sloughter; Hill–Thron; Strang, *Calculus* | CC BY-NC-SA | `share-alike-nc` | OTL |
| Hammack, *Book of Proof* 3e | CC BY-NC-ND 4.0 | `verbatim` | https://richardhammack.github.io/BookOfProof/ |
| Per OTL: Shoup, *A Computational Introduction to Number Theory and Algebra* | CC BY-NC-ND | `verbatim` | OTL |
| Hatcher, *Algebraic Topology* | © CUP 2002; noncommercial copies only | `record` | https://pi.math.cornell.edu/~hatcher/AT/ATpage.html |
| Lurie, *Kerodon* | no licence stated, so all rights reserved | `record` | https://kerodon.net/ |
| PreTeXt: *Linear Algebra* (W214), *Intro to Topology (IBL)* | all rights reserved | `record` | https://pretextbook.org/catalog.html |
| LibreTexts pages tagged "not declared" | none | `record` | https://math.libretexts.org |

---

## Part 3. Syllabi and theorem lists

### 3.1 MAA CUPM Curriculum Guide (2015)

- **Citation:** *2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences*, Schumacher and Siegel (co-chairs), Zorn (editor). 116 pages. The PDF has CreationDate 2015-05-05.
- **Copyright:** "© 2015 The Mathematical Association of America (Incorporated)", ISBN 978-0-88385-917-9. I found no CC licence, so it is `record`.
- **URLs:**
  - Official page: https://maa.org/resource/cupm-guide/. It is JavaScript-rendered, so curl got no content.
  - Copy I read: https://www.amstat.org/docs/default-source/amstat-documents/cupmguide_print.pdf
  - The separate course-area reports (Abstract Algebra, Real Analysis, Topology, etc.) are linked from the MAA page. Their licences are not checked.
- **What math2 can take:** the structure (course areas, topic lists) as facts for `topics/curricula/`, and citations. Not the prose.
- Nothing newer than the 2015 guide was found (D12).

### 3.2 QAA Subject Benchmark Statement: Mathematics, Statistics and Operational Research

- **Current edition:** the **5th edition, published 2023-03-08** (the PDF's last page reads "Fifth Edition / Published - 8 March 2023"). It covers Master's level too.
- **Page:** https://www.qaa.ac.uk/the-quality-code/subject-benchmark-statements/subject-benchmark-statement-mathematics-statistics-and-operational-research
- **PDF:** https://www.qaa.ac.uk/docs/qaa/sbs/sbs-mathematics-statistics-and-operational-research-23.pdf, 2.1 MB, 35 pages.
- **Copyright:** "© The Quality Assurance Agency for Higher Education 2023". No CC or OGL licence is stated in the PDF, so it is `record`. (An older 2009 annex is in DERA under the Open Government Licence. That does not carry over to 2023.)
- **Upcoming:** the QAA site has a "Subject Benchmark Statement Reviews 2026-27" page, so a 6th edition may be in progress (D12).
- **What math2 can take:** its outcome categories, as facts.

### 3.3 Mathlib lists

Source: leanprover-community/mathlib4 `master` (repository HEAD 4c30f23495 at 2026-10-26T08:02Z). Licence Apache-2.0.

| File | Contents | Last change |
|---|---|---|
| `docs/undergrad.yaml` (https://raw.githubusercontent.com/leanprover-community/mathlib4/master/docs/undergrad.yaml) | 563 leaf items, 130 of them empty (not yet formalised). The topics were originally taken from the 2020 French *agrégation* programme | 58d1b21e88, 2026-10-07 |
| `docs/overview.yaml` | 511 leaves | n/a |
| `docs/1000.yaml` | 1,199 entries keyed by Wikidata id; 217 have `decl`(s), 30 have an external `url` | c6f03d20ca, 2026-10-17 |
| `docs/100.yaml` | 100 entries; 78 have `decl` | n/a |

**What math2 can take:** everything. These are Apache-2.0 code-zone data and facts, and the best machine-readable syllabus and theorem lists available. The Wikidata keys of `1000.yaml` are crosswalk ids.

### 3.4 Freek Wiedijk's 100 theorems

- **Page:** https://www.cs.ru.nl/~freek/100/, last modified 2026-09-17. All 100 theorems are formalised in at least one system.
- **Per system:**

| System | Theorems |
|---|---:|
| HOL Light | 95 |
| Isabelle | 95 |
| Lean | 83 |
| Rocq | 80 |
| Metamath | 74 |
| Mizar | 71 |
| ACL2 | 48 |
| ProofPower | 43 |
| PVS | 26 |
| Imandra | 20 |
| Megalodon | 12 |
| Naproche | 10 |
| NuPRL | 8 |

- **Licence:** no licence statement, so `record`. The list of theorem names is factual.
- **Discrepancy:** Mathlib's `100.yaml` has 78 entries with a `decl`, but Freek's page counts 83 for Lean. The difference is probably entries proved outside Mathlib, or stale yaml. Not resolved.

### 3.5 The 1000+ theorems project

- **Site:** https://1000-plus.github.io/. I counted 228 entries on the page.
- **Repository:** https://github.com/1000-plus-theorems/1000-plus, licence **Unlicense**, last pushed 2026-07-22. That makes it public-domain-equivalent, so `open`.
- **Relation to Mathlib:** Mathlib's `1000.yaml` is the Lean-side view of the same Wikidata-keyed list.

### 3.6 Department qualifying-exam lists

- **Example:** UC Berkeley Preliminary Examination (https://math.berkeley.edu/graduate/phd-program/preliminary-exam).
  - Two mornings of 3 hours each. Students attempt 5 of 8 questions per day (9 before Fall 2024).
  - Topics: real analysis and calculus, complex analysis, linear algebra, abstract algebra.
  - The syllabus PDF and past exams are linked at the bottom of the page.
- These are department web pages with no licence, so `record`. Topic lists are factual. Past exam problems are copyrighted text.
- *Berkeley Problems in Mathematics* (de Souza and Silva, Springer) collects 1,250 prelim problems from 1977 to 2003. It is all rights reserved.
- I did not survey other departments; other examples include UW (https://math.washington.edu/phd-preliminary-exams). **Breadth is UNVERIFIED** (D12).

---

## Summary table

| Item | Size (date) | Licence of text / data | Format | Bulk channel and terms | AI terms | What math2 can take (9.3) |
|---|---|---|---|---|---|---|
| arXiv | about 9.2 TB of files (April 2025), +100 GB/month | metadata CC0; text per paper. 2026 math sample is 36.9% `open` (BY/BY-SA/CC0) vs 15% all-time | TeX source, PDF | `s3://arxiv` requester-pays; API 1 req / 3 s; must link back; redistributing e-prints needs the holder's permission | none stated | records (CC0); text of `open` papers into prose; others `record` |
| zbMATH Open | 4M+ publications, 3M+ reviews, 168M+ triples (KG v1, 2026-09-01); 133,093 documents for 2025 | KG: Zenodo field CC BY 4.0 vs description CC BY-SA 4.0; I4OC references CC0 | RDF/TTL, JSON API, OAI-PMH | Zenodo dump (1.6 GB, twice a year); REST; OAI-PMH; API terms UNVERIFIED | UNVERIFIED | records, MSC, reviews as `open` evidence |
| OpenAlex | about 745 GB JSONL (September 2026) | CC0 | JSONL, Parquet | `s3://openalex` anonymous, quarterly; API key since 2026-02 (secondary source) | none | all metadata |
| Unpaywall | n/a | n/a | n/a | folded into OpenAlex; data feed paid; terms UNVERIFIED | n/a | OA locations via OpenAlex |
| Crossref | 180,034,490 records, 208 GB (2026-03-14) | metadata effectively free; some abstracts copyrighted | JSONL | public data file (torrent, AWS); open REST API | none | metadata, references; abstracts `record` |
| DataCite | 108M DOIs (2025 file, 2026-01-06) | CC0 | JSON | annual public file; monthly file members-only | none | all metadata |
| Semantic Scholar / S2ORC | S2ORC 15,679,781 full texts (2026-05-26) | API non-transferable; S2ORC ODC-By (database) plus per-paper copyright; abstracts UNVERIFIED | JSON | API key; Datasets API | none | ids; text only per paper licence |
| CORE | 291M metadata / 32.8M full texts (2023); current size UNVERIFIED | dumps ODC-By (database only) | JSON | API licence; dumps | proposed "lawful TDM" | records; text per paper |
| HAL (math) | 170,406 records, 105,413 with file (2026-10-26) | metadata CC0; files: 9.8% `open`, 80% hal-authorisation | PDF, TEI | OAI-PMH, REST, SPARQL | none found | records; the 10,349 `open` texts |
| Numdam | 75,599 articles, 706 books, 1.43M pages | metadata CC0; text: personal use, no bulk, no modification | PDF, OAI | OAI-PMH (metadata only) | none | records; text only where CC or PD |
| EuDML | over 270k documents (undated) | UNVERIFIED | n/a | blocked (403) | n/a | records via partner sources |
| Project Euclid | "over 80%" free to read | per journal, UNVERIFIED | PDF | none found | n/a | records; per-article licence |
| GDZ | over 15M pages | scans' licence UNVERIFIED; PDFs: no re-hosting without permission | IIIF, PDF | IIIF, OAI | n/a | records; public-domain statements rewritten |
| HathiTrust | n/a | public-domain subsets; terms UNVERIFIED (Cloudflare) | text | `ht_text_pd*` datasets | UNVERIFIED | US pre-1931 public-domain text |
| Internet Archive | 206,597 math texts; 8,076 dated 1500–1930 | mixed; hosting does not establish licence | PDF, OCR | advancedsearch; terms UNVERIFIED | UNVERIFIED | public-domain items after a per-item check |
| Project Gutenberg | about 78,000 items (2026-03) | public domain (US) | TXT, HTML, some LaTeX | mirrors or `/robot/harvest`, `-w 2` | none | `open` (strip PG trademark) |
| TheoremSearch | 8.4M indexed (live site); 1,341,083 BY-SA release; 1,239,720 permissive release | HF tags BY-SA / BY; BY-SA set includes GFDL Stacks | Parquet, REST, MCP | Hugging Face; API with no terms | none | `-permissive` set as E1 candidates |
| TheoremGraph / LeanGraph | 16,131,238 rows (math-graph) | CC BY 4.0 plus per-record source licence; text NULL unless open | CSV, Parquet | Hugging Face | none | edges, ids, open-only text |
| Matlas | 8.07M statements, 435K papers, 1.9K textbooks | not stated (UNVERIFIED; likely closed) | JSON API | `POST /api/search`, no terms | none | query-time ids only (`record`) |
| Open textbooks | 214 math books in OTL; 68 AIM-approved; 67 PreTeXt math | OTL: 48% `open`, 52% NC/ND/GFDL | LaTeX, PreTeXt, PDF | Git repositories | MIT OCW: NC models only | Table A only |
| MIT OCW | n/a | CC BY-NC-SA 4.0 | HTML, PDF | none | training allowed, NC models, BY-NC-SA | `share-alike-nc` |
| CUPM 2015 | 116 pages | © MAA | PDF | n/a | n/a | structure as facts |
| QAA SBS MSOR (5th edition, 2023-03-08) | 35 pages | © QAA 2023 | PDF | n/a | n/a | structure as facts |
| Mathlib yaml | undergrad 563 (130 empty); 1000.yaml 1,199; 100.yaml 100 | Apache-2.0 | YAML | Git | n/a | all |
| Freek's 100 | 100 (2026-09-17) | no licence | HTML | n/a | n/a | list as facts |
| 1000+ theorems | 228 entries on the site | Unlicense | Markdown, YAML | Git | n/a | all |

---

## Open doubts

1. **D1. zbMATH KG licence.** The Zenodo field says CC BY 4.0, but the record description says CC BY-SA 4.0. Ask zbMATH (or open a GitHub issue on zbMATHOpen/zbmath-open-kg) which one governs. Both are `open`, but BY-SA binds derived exports. The API terms are unread (Cloudflare).
2. **D2. arXiv `open` share.** The spec's "about 15%" is the all-time stock. My three-day 2026 math sample gives 36.9% (and about 39% for re-versioned older papers). Confirm this with a full 2026 `set=math` harvest from the `tamnd/arxiv` metadata plane. Separate first versions from updates, and decide whether cross-lists count. If it holds, spec 9.3 should state both figures.
3. **D3. Missing AI clauses.** arXiv, Crossref, OpenAlex, HAL and Gutenberg say nothing about AI. MIT OCW has an explicit NC-only AI clause. Stack Exchange forbids LLM use (already in the spec). The zbMATH, HathiTrust, IA and EuDML terms are unread. Silence in the terms is not permission: the per-item licence still governs.
4. **D4. TheoremSearch BY-SA set** puts Stacks (GFDL), ProofWiki and other sources under one `cc-by-sa-4.0` tag. math2 must filter by source and licence before treating any row as `open`.
5. **D5. TheoremSearch counts.** The card says 1,239,720 rows for the permissive set; the dataset viewer shows 2,677,439 rows. The BY-SA set has 1,341,083. The live site shows 8.4M against the paper's 9.2M. Not reconciled. The difference may be duplicated rows across splits, or slogans stored as separate rows.
6. **D6. Kerodon licence. Spec 9.3 and 16.6 list Kerodon as GFDL; I found no licence statement anywhere on kerodon.net.** Unless Lurie states a licence, Kerodon is all rights reserved, so `record`. The spec should change it to `record`, and §1.7's "[U]" should be resolved accordingly. The tag count on /statistics is 8,189; §1.7 says "2,200+ pages".
7. **D7. OpenStax edition drift.** The current 2e repositories are CC BY-NC-SA 4.0, but older editions were CC BY. Adapting an old CC BY edition is legal, but the edition and commit must be pinned.
8. **D8. OTL licences are stale or simplified** (*Active Calculus*, *Book of Proof*, Hefferon, Hitchman). Every Table A row marked "per OTL" must be verified at the source before ingestion.
9. **D9. Dual-licensed books** (Lebl ×3, Hefferon). math2 must record its election of the BY-SA option in the source entry.
10. **D10. Public domain is territorial.** The US pre-1931 rule does not make a work public domain elsewhere (life plus 70 years in the EU). Since math2 is published worldwide, decide whether US public domain is enough for `open`. For Hardy (died 1947) the text is public domain in the EU too; for some authors it is not.
11. **D11. AIM titles with unchecked licences** (Bruckner–Thomson, ORCCA, Manes, May–Bart, Marsden–Weinstein, Boyd–Vandenberghe, Treil, Fields, Bender–Williamson, Bard, Zimmermann et al., Crisman, Stein, Connell).
12. **D12. Syllabi.**
    - CUPM 2015 is the latest guide I found; I did not confirm whether a newer one exists.
    - The QAA 6th-edition review (2026–27) has not been checked.
    - Qualifying-exam lists were sampled at one department only.
    - The difference between Mathlib's `100.yaml` (78 with `decl`) and Freek's Lean count (83) is not resolved.
13. **D13. UNVERIFIED because of blocking or missing pages:**
    - zbMATH API terms
    - EuDML metadata licence and size
    - HathiTrust terms
    - Internet Archive terms
    - current CORE size and dump terms
    - Unpaywall terms
    - Semantic Scholar abstracts licence
    - Project Euclid per-journal licences
    - GDZ scan licence
    - OpenAlex API pricing (secondary source only)
14. **D14. Matlas** returns full statement text from copyrighted journals and textbooks through an open API with no terms. math2 should store only ids from it (spec 9.6). Its corpus licence is unknown.
15. **D15. LibreTexts** has no aggregate licence data. The per-page "not declared" share is unknown; treat it as `record` and prefer the upstream source.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/01c-formal-libraries.md">
# K01, part C. Formal libraries and formal / informal-formal datasets

Research note for the Math 2.0 spec (tamnd/math2). This note extends spec 01 §1.2, spec 07 (bindings, M2Extract, §7.10 other proof assistants) and spec 09.4 (libraries enter as bindings, never as copies). It does not repeat those sections. Where a number has moved since the spec was written, the note says so.

Conventions:
- I read every source on **2026-10-26**, unless the entry says otherwise. "(read 2026-10-26)" is left out to keep the text short. Every fact has a URL.
- **UNVERIFIED** means I could not confirm the fact from a primary source. The reason follows the tag.
- File counts come from the GitHub git-trees API (`recursive=1`). The `truncated` flag was false in every case unless noted. They count files, not declarations.
- HF row counts come from `datasets-server.huggingface.co/size` or the dataset card.

---

## 1. Formal libraries

### 1.1 Lean 4

**Mathlib**
- Stats page https://leanprover-community.github.io/mathlib_stats.html: **139,616 definitions, 291,237 theorems, 772 contributors**. The page shows no date. Spec 01 had 291,178 / 139,569 on 2026-10-25, so the count grew in one day.
- Licence: Apache-2.0 (https://github.com/leanprover-community/mathlib4). The repo has 35,715 commits and about 4.4k stars.
- Tree API: **8,661 `.lean` files under `Mathlib/`**. The tree has 10,796 entries and was not truncated. Spec 01 says 9,437 files, which probably counts every file and not only `Mathlib/*.lean`. This is a definitional difference, not a conflict.
- Informal text: docstrings and module docs only. See the docstring coverage under 1.1.1.

**1.1.1 Official extraction route: Mathlib Initiative datasets (new; not in spec)**
- Tool: `lean_scout`, https://github.com/mathlib-initiative/lean_scout (Apache-2.0).
- Datasets on HF under `mathlib-initiative/`, all Apache-2.0. They were extracted at Mathlib commit `56a005855ea1…`, "chore: bump toolchain to v4.35.0 (#44806)", dated **2026-10-13**.
  - `mathlib-types`: **797,294 rows**. Fields are `name`, `module`, `type`, `docString` and `allowCompletion`.
    - `allowCompletion = True` for 483,112 rows.
    - `docString` is null for **681,418 rows (85.47%)**. So only about **115,876 constants (~14.5%) have a docstring**, with a mean length of about 130 characters.
  - `mathlib-const-dep`: 797,294 rows (dependency edges per constant).
  - `mathlib-tactics`: **1,999,349 rows** (tactic plus goal states).
- Relevance to M2Extract (spec 07): the schema `name / module / type / docString` already matches what a binding needs. The constant count of 797k covers auxiliary and internal constants. The `allowCompletion` filter, at about 483k, is a better match for "user-facing declarations". Neither number matches the 291k theorems plus 139k definitions on the stats page. Each counts a different thing, and M2Extract should record which one it uses.

**doc-gen4**
- https://github.com/leanprover/doc-gen4 (Apache-2.0).
- It runs as `lake build Lib:docs` from a nested docbuild project. The target library must compile first.
- Output is HTML plus a declaration-data JSON. That makes it heavier than lean_scout when only statements are needed.

**Reservoir (Lean package index)**
- https://reservoir.lean-lang.org/packages?sort=stars: **908 packages**. Reservoir shows no sizes or licences in the listing.
- Top packages by stars:

  | Package | Stars |
  |---|---|
  | mathlib | 4,352 |
  | openai/NavierStokesAndEuler | 2,110 |
  | teorth/Analysis | 1,971 |
  | ulfjack/ryu | 1,357 |
  | LeanCopilot | 1,337 |
  | formal_conjectures | 1,336 |
  | anthropics/flt_e2e | 1,292 |
  | ImperialCollegeLondon/FLT | 1,053 |
  | Physlib | 808 |
  | cslib | 777 |
  | mil | 623 |
  | equational_theories | 562 |
  | paperproof | 554 |
  | scilean | 539 |
  | glimpseOfLean | 426 |
  | batteries | 424 |
  | aesop | 410 |
  | verso | 398 |
  | Game | 387 |
  | PrimeNumberTheoremAnd | 351 |

- Licences of individual packages must be read per repo. I did not survey them all.

**Tau Ceti**
- https://github.com/TauCetiProject/TauCeti: Apache-2.0, 19,274 commits, 247 stars.
- **18,849 `.lean` files** (tree API), which is about twice Mathlib's file count.
- Written by AI and reviewed against human rubrics. It tracks Mathlib master with an hourly bump. It is incubated by the Lean FRO and the Mathlib Initiative.
- Docs are at taucetiproject.org/docs. The stats page returned 404.
- Xena blog, 2026-10-16 (https://xenaproject.wordpress.com/): "Tau Ceti is now slightly larger than Mathlib (3 million lines of code)".
  - **UNVERIFIED** whether "3 million lines" refers to Tau Ceti or to Mathlib. The sentence is ambiguous. Spec 01 reads it as Tau Ceti ≈ 3M. Spec 01 also gives Mathlib 2,481,682 lines, which fits "slightly larger" only if Tau Ceti is somewhat above 2.5M.
- Consequence for E-tiers: Tau Ceti is mostly AI-authored. Any provenance or trust flag in spec 07 should separate it from human-reviewed Mathlib.

**Formal Frontier**
- https://github.com/orgs/FormalFrontier/repositories: 50 repos. All 30 shown on the first page are **Apache-2.0**.
- Examples: projective-modules, algebraic-groups, group-theory, homological-algebra, complex-analysis, scheme-properties, sphere-topology, simplicial-objects, iwasawa-modules.
- Xena (2026-10-16) reports 41 released libraries, about 200k lines. Highlights include the hairy ball theorem, Kaplansky's theorem, Bass stable range, Tate periodicity and Stone duality.
- The page mathlib-initiative.org/formal-frontier names no licence. The licence of the remaining 20 repos is UNVERIFIED (not shown on the first page).

**ATLAS (Meta), licence warning (new)**
- Repo https://github.com/facebookresearch/atlas-lean; paper arXiv:2605.29955 (2026-05-28).
- Size: 26 textbooks, more than 45,000 declarations, about 500k lines.
- **v1 is under CC BY-NC 4.0 with a "no-training rider".** It prohibits commercial use and any use "to train, fine-tune, distill, evaluate, or otherwise develop ML models".
- **v2 (repo root) is labelled Apache 2.0, but the root LICENSE file opens with "Codebase is not intended for use to train, fine-tune, distill, evaluate, or otherwise develop ML models".** This is an Apache grant with a non-standard preamble, so its legal effect is unclear.
- AutoformBench (https://github.com/facebookresearch/autoformbench) is CC BY-NC 4.0.
- 36Kr (https://eu.36kr.com/en/p/3829852422940291) reports 183B tokens. This is a secondary source and UNVERIFIED in the paper.
- Recommendation: crosswalk only. Do not bind v1. Treat v2 as legally unclear until the preamble is resolved.

### 1.2 Isabelle

**AFP**
- https://www.isa-afp.org/statistics/: **1,042 entries, 614 authors, ~330,000 lemmas, ~5,518,000 lines**. Spec 01 had 1,038 / ~326,500 / ~5.44M. The page gives no per-year data.
- Licence (https://isa-afp.org/about/): "All entries … licensed under a BSD-style License or the GNU LGPL". Each entry page has a License field, so the licence must be recorded per entry.
- Access: https://isa-afp.org/release/afp-current.tar.gz (about 70 MB), or Mercurial at https://foss.heptapod.net/isa-afp.
- AI provenance: the entry Jacobian_Counterexample (2026-07-20, BSD) says "AI assistance was used for proof engineering". The AFP has no archive-wide AI policy that I found.
- Informal text: Isabelle theories carry `text \<open>…\<close>` blocks and each entry has an abstract. That is more prose than Mathlib docstrings, but I did not measure it.

**Isabelle2026**
- https://isabelle.in.tum.de/: "Now available: Isabelle2026 (October 2026)". This **resolves the [U] in spec 01**.
- Features include HOL-New_Algebra and cvc5 proof reconstruction.
- Licence: BSD-style for the core, mixed overall (bundled components).
- sketis.net shows RC4 dated 14-Oct-2026 and the final release planned for 24-Oct-2026.
- The size of the distribution library (HOL, HOL-Analysis and so on) is UNVERIFIED. I found no official count.

### 1.3 Rocq (formerly Coq)

| Library | URL | Licence (LICENSE file) | `.v` files |
|---|---|---|---|
| Rocq core | https://github.com/rocq-prover/rocq | LGPL-2.1 | – |
| Stdlib | https://github.com/rocq-prover/stdlib | LGPL-2.1 | 904 |
| MathComp | https://github.com/math-comp/math-comp | CeCILL-B | 147 |
| MathComp Analysis | https://github.com/math-comp/analysis | CeCILL-C | 131 |
| Odd Order | https://github.com/math-comp/odd-order | CeCILL-B | 34 |
| UniMath | https://github.com/UniMath/UniMath | MIT-style custom | 1,714 |
| Coquelicot | gitlab.inria.fr (blocked by Anubis) | **UNVERIFIED** | – |

- The coq-community org was renamed **rocq-community** and has 75 public repos. The math-comp org has 27 repos.
- The opam `released/packages` directory (rocq-prover/opam) has 603 entries.
- Coquelicot: the GitHub fork proux01/coquelicot is LGPL-3.0. The upstream licence is UNVERIFIED because the Inria GitLab did not load (an Anubis bot challenge).
- MathComp files are large, so file count understates size. I found no declaration count.
- CeCILL-B is BSD-like and CeCILL-C is LGPL-like. Both allow bindings.
- Informal text: comments only. MathComp has structured header comments.

### 1.4 HOL Light and HOL4

- **HOL Light**: https://github.com/jrh13/hol-light. BSD-style LICENSE, 664 `.ml` files. Blanqui's LPAR talk gives about 37,000 objects in 635 files. I have the talk only as a secondary reference, so that figure is UNVERIFIED in a primary listing.
- **Flyspeck**: MIT, 503 `.hl` files.
- **HOL4**: https://github.com/HOL-Theorem-Prover/HOL. BSD-style COPYRIGHT, 1,495 `*Script.sml` files.
- **hol2dk** (https://github.com/Deducteam/hol2dk, CeCILL 2.1): translates HOL Light to Dedukti, Lambdapi and Rocq.
  - `hol.ml` has 5,687 theorems.
  - Multivariate has 40,728 theorems. The dump is about 135 GB and Rocq checking takes about 21 h.
  - Output is in the opam packages `coq-hol-light` and `rocq-hollight-logic`.
  - **There is no Lean output.**
- arXiv:2609.24598 (Blanqui and Gontard, 2026-09-21, LPAR-26) maps HOL Light constants to idiomatic Rocq. This "alignment" step is what makes translations usable rather than literal.
- **HolLean** (https://github.com/hol-light-family/HolLean) ports the HOL Light *system* to Lean 4 as a meta-language. It does not translate the library.

### 1.5 Metamath

- **set.mm**: https://github.com/metamath/set.mm. **LICENSE = CC0 1.0** (read through the gh API). 17,928 commits; last push 2026-10-26.
- Sizes. The three pages count different things.
  - The theorem list https://us.metamath.org/mpeuni/mmtheorems.html has 513 pages, and its last statement number is **51,207**. That number includes axioms and definitions. The newest dated theorem is 27-Aug-2026.
  - https://us.metamath.org/mpeuni/mmset.html says "over 26,000 … main sections, over 41,000 counting mathboxes".
- **iset.mm** (https://us.metamath.org/ileuni/mmtheorems.html): about 17,395 statements (174 pages).
- **Metamath 100**: 75 as of 2026-04-28 (the mm_100 page). Wiedijk's list credits Metamath with 74. The gap is a sync lag.
- Tooling:
  - **mm0** (https://github.com/digama0/mm0, CC0) includes a set.mm to Lean 3 translation path.
  - digama0/mm-lean4 and zariuq/mm-lean4 are Metamath verifiers in Lean 4, not library translators.
- Informal text: every statement has a mandatory comment, usually a sentence or more. **set.mm is the only large library that is CC0 and has a prose comment on every statement.** This strengthens the "v2 candidate" status in spec 07 §7.10.

### 1.6 Mizar

- https://mizar.uwb.edu.pl/: **MML 5.94.1493 (2025-05-30)**, Mizar 8.1.15.
- **Possible conflict**: in MML naming, the last field of the version (1493) is normally the article count. Spec 01 gives 1,498 articles. UNVERIFIED which is right; the 73,460 theorems figure could not be rechecked.
- Licence: dual **GPLv3+ / CC BY-SA 3.0+** since MML 4.160.1126. Sources: https://mizar.uwb.edu.pl/forum/archive/1104/msg00000.html and arXiv:1107.3212.
- The system has been GPLv3 at https://github.com/MizarProject/system since about 2023.
- CC BY-SA is share-alike. Derived statements would inherit it, so it should be crosswalk-only unless the corpus accepts SA.

### 1.7 Agda

| Library | Licence | Files | Informal text |
|---|---|---|---|
| agda-stdlib (https://github.com/agda/agda-stdlib) | MIT-style | 1,183 `src/*.agda` | comments |
| agda-unimath (https://github.com/UniMath/agda-unimath) | MIT | 3,072 `.lagda.md` | **literate Markdown, prose throughout** |
| 1Lab (https://github.com/the1lab/1lab) | **AGPL-3.0** | 710 `.lagda.md` | **literate, prose throughout** |

- This fills the "Agda sizes [U]" gap in spec 01 with file counts. Declaration counts are still unknown.
- 1Lab is AGPL, so it should be crosswalk-only.
- agda-unimath is the cleanest literate library for informal and formal alignment: MIT, and prose sits next to each definition.

### 1.8 Naproche, Megalodon, PVS, ACL2

- **Naproche** (https://github.com/naproche/naproche): GPL-3.0, 32 `.ftl.tex` files. The text is controlled natural language, so the source is itself informal.
- **Megalodon**
  - ai4reason/Megalodon has an MIT-style licence (CIIRC/CTU).
  - The mgwiki/god1 and mgwiki/mgw_test repos have **no licence** (gh shows none).
  - arXiv:2601.03298 (Urban, 2026-01-06) reports about 160k lines and more than 1.5k lemmas. The work used ChatGPT 5.2 or Claude Sonnet 4.5 and cost about $100. Spec 01 had 130k lines for Munkres; the paper is newer.
- **NASALib** (https://github.com/nasa/pvslib)
  - The README says "63 top-level libraries, containing about 41K proven formulas". The file `nasalib.grandtotals` says 64 libraries.
  - Licence: the GitHub API shows none. The README says the NASA developments are "in the public domain"/"fundamental research that do not constitute software". Other contributions carry licences in each `top.pvs`, so licences are per sub-library.
  - The repo contains a `CLAUDE.md`, which suggests AI-assisted maintenance.
- **ACL2** (https://github.com/acl2/acl2): BSD-3 for the system. The `books/` directory has per-file licences. The tree has 40,929 entries.

---

## 2. Translation between systems

### 2.1 Infrastructure

| Tool | Status | Licence | Notes |
|---|---|---|---|
| Mathport (https://github.com/leanprover-community/mathport) | **archived**, last push 2024-11-21 | Apache-2.0 | Lean 3 to Lean 4 only; finished its job |
| lean2dk (Deducteam) | WIP, last push 2026-06 | Apache-2.0 | Lean 4 to Dedukti; spec 01 already says WIP |
| Logipedia | last push 2024-11 | – | All proofs in STTForall (arXiv:2305.00064). Dormant |
| OpenTheory | last milestone 2017 | – | Reads and writes HOL Light, HOL4 and ProofPower; reads Isabelle; no counts published |
| hol2dk | active (2026) | CeCILL 2.1 | HOL Light to Rocq, 40,728 theorems; no Lean target |
| Minilang (arXiv:2507.18885) | paper | – | Rule-based Isabelle Isar translation, about 340K proofs, about 85% success |

- No working, library-scale **X to Lean 4** translator exists as of today. The paths are hol2dk (to Rocq), mm0 (set.mm to Lean 3, legacy) and LLM-based porting.

### 2.2 LLM-based translation, with measured success

- **ITPEval** (arXiv:2607.19407, Wu, George and Anandkumar, 2026-07-07)
  - Covers 1,560 files and 6,848 theorems. It tests 12 directed pairs over Lean 4, Rocq, Isabelle and HOL Light, with 5 LLMs.
  - Statement translation peaks at **29.1% pass@1**; proof translation at **10.5%**.
  - Controlled files reach 29.7%, ecosystem files only 5.2%.
  - Multi-ITP context raises Lean success from 4.8% to 10.6%.
  - BEq confirms only **54.0%** of compiler-verified miniF2F-to-Lean statements.
  - Conclusion for the spec: compiling is not the same as being faithful.
  - The arXiv page gives no data URL, so dataset availability is UNVERIFIED.
- **miniF2F-rocq** (arXiv:2503.04763; https://github.com/LLM4Rocq/miniF2F-rocq, MIT): 478 of 488 problems translated, using GPT-4o mini, Claude 3.5 Sonnet, o1 mini and o1.
- **Megalodon Munkres** (arXiv:2601.03298): LLM-driven formalization of a full textbook for about $100 (see 1.8).
- **Lean Pool** (arXiv:2609.25199, 2026-09-21): an AI-maintained archive with a catalogue of imported projects. It gives no counts, so its size is UNVERIFIED.
- The Lean Zulip thread "Autotranslation of Logical Foundations from Rocq to Lean" (Jan 2026) is anecdotal evidence. I took no measured numbers from it.

### 2.3 Cross-system theorem lists

**Wiedijk 100** (https://www.cs.ru.nl/~freek/100/, last modified 2026-09-17):

| System | Count |
|---|---|
| HOL Light | 95 |
| Isabelle | 95 |
| Lean | 83 |
| Rocq | 80 |
| Metamath | 74 |
| Mizar | 71 |
| nqthm/ACL2 | 48 |
| ProofPower | 43 |
| PVS | 26 |
| Imandra | 20 |
| Megalodon | 12 |
| Naproche | 10 |
| NuPRL | 8 |

**1000+ theorems** (https://github.com/1000-plus-theorems/1000-plus)
- Licence: **Unlicense** (public domain). Last push 2026-07-22.
- **1,200 `_thm/*.md` YAML files keyed by Wikidata Q-id.**
  - Fields: `msc_classification`, `wikipedia_links`, and per-system blocks `isabelle`, `hol_light`, `rocq`, `lean`, `metamath`, `mizar`.
  - Entries come from Wikipedia's List of theorems.
- This is the best ready-made **crosswalk seed** for spec 07 §7.10: public domain, keyed by Wikidata, MSC-tagged and pointing into six systems.
- Per-system counts are UNVERIFIED. The web page showed about 228 formalized entries, which may be one page of a paginated list. Spec 01's "241 in Lean" could not be rechecked without parsing all 1,200 files.

---

## 3. Datasets

### 3.1 Benchmarks (formal statement plus informal statement)

**miniF2F**
- 488 problems: 244 valid, 244 test.
- Lean 4 ports:
  - yangky11/miniF2F-lean4 (MIT)
  - facebookresearch/miniF2F (MIT)
  - AI-MO/minif2f_test (Apache-2.0, 244 rows)
- Quality: "miniF2F-Lean Revisited" (arXiv:2511.03108, NeurIPS 2025) found discrepancies between formal and informal statements in **more than half** the problems.
  - Pipeline accuracy is 36–40% on the original and about 70% on the corrected v2.
  - The "over 300 problems corrected" figure is UNVERIFIED (search snippet only).

**ProofNet**
- MIT, 371 problems, originally Lean 3.
- ProofNet# (arXiv:2406.07222; HF PAug/ProofNetSharp, MIT) found **118 of 371 (31.8%)** wrong in the Lean 4 ports.
- ProofNet-Verified (HF marcusm117, Apache-2.0, NeurIPS 2026, targets Lean v4.34.0) found **205/367 (55.9%)** errors in ProofNet and **99/367 (27.0%)** in ProofNet#. A fixed benchmark still had errors in more than a quarter of its items.

**PutnamBench** (https://github.com/trishullab/PutnamBench)
- Lean 4 672, Isabelle 640, Coq 412, covering 1962–2025.
- Apache 2.0 for Lean and Isabelle, MIT for Coq.
- **The informal statements are used "with permission from the MAA"**. They are third-party text, so do not relicense them.
- The repo-level licence field is empty, so the licence is per subdirectory.

**FormalMATH**
- HF SphereLab/FormalMATH-All: MIT, 5,560 rows. GitHub Sphere-AI-Lab/FormalMATH-Bench: Apache-2.0 (EMNLP 2026 Findings). The two licences differ.
- About 72.09% of statements were kept before expert review. The best prover reached 16.46%.

**DeepSeek-ProverBench** (https://huggingface.co/datasets/deepseek-ai/DeepSeek-ProverBench)
- 325 problems: 15 from AIME 24/25 and 310 from textbooks.
- **No licence tag.**

**Defect audit across benchmarks: "Faults in Our Formal Benchmarking"** (arXiv:2606.29493, ICML 2026)
- 4,833 findings and **398 certified defects** in total.

  | Benchmark | Findings | Certified |
  |---|---|---|
  | FormalMath | 3,463 | 147 |
  | ProverBench | 370 | 208 |
  | ProofNet | 149 | 6 |
  | miniF2F (7 variants) | 773 | 36 |
  | CombiBench | 78 | 1 |

- ProverBench has **208 certified defects out of 325 problems**.

**Statement-faithfulness judges**
- EFG (arXiv:2506.10903) reports correctness of 76.64% for Isabelle miniF2F, 82.79% for Lean miniF2F and 64.84% for ProofNet Lean. These are UNVERIFIED: search snippet only, not confirmed on the abstract page.
- Beyond Compilation (arXiv:2606.31002): 227 statements. Its judge agrees with the human majority 91.5% of the time.

### 3.2 Large synthetic statement and proof sets (competition style)

| Dataset | Rows | Licence | Informal side | Notes |
|---|---|---|---|---|
| Lean Workbook (internlm, HF `internlm/Lean-Workbook`) | 57,231 + 82,893 (Plus) per paper/card; **default HF config 25,214 rows** | Apache-2.0 | Competition problems (AoPS) | Lean v4.8.0-rc1. A manual sample check found 93.5% accurate. 4,898 proved (pass@1024 is 8.6%). The row-count gap is UNVERIFIED |
| Goedel-LM/Lean-workbook-proofs | 29,750 | **no tag** | – | proofs for Lean Workbook |
| DeepSeek-Prover-V1 (HF) | 27,503 | "other" (deepseek-license; the card says commercial use is supported) | NL problem | The paper describes 8M synthetic statements; only 27.5k released |
| DeepSeek-Prover-V2 repo | – | NOASSERTION | – | cold-start data generated by DeepSeek-V3 |
| Goedel-Pset-v1 | 1,732,594 | **no tag** | NuminaMath problems | Two formalizers. Filtered by compile plus an FC test with Qwen2.5-72B-Instruct |
| Goedel-LM/SFT_dataset_v2 | 1,745,010 | Apache-2.0 | – | Goedel-Prover-V2 (arXiv:2508.03613) |
| Goedel-LM/RL_dataset_V2 | – | Apache-2.0 | – | 2026-03 |
| NuminaMath-LEAN (AI-MO) | 104,155 | Apache-2.0 | NuminaMath problem | Mathlib v4.15.0. `author` = human or autoformalizer. Proofs from Kimina-Prover-72B. **The card admits wrong reference answers and inaccurate formalizations** |
| Kimina-Prover-Promptset (AI-MO) | 24,418 | Apache-2.0 (stated in card text; no HF tag) | `natural_language` | Hard subset of NuminaMath-LEAN plus **Gemini-generated variants**, with duplication |
| NuminaMath-1.5 | 896,215 | Apache-2.0 | informal only | includes 268,819 from cn_k12 |
| AgenticCommons/formal-math-autoformalization | 3,623 | **CC0** | NL problem | Lean v4.30.0, compile and axiom gates |

- "Generated by model X" exposure:
  - DeepSeek-V3 for Prover-V2 data (DeepSeek licence).
  - Qwen2.5-72B for the Goedel filter.
  - Gemini for the Kimina variants.
  - Kimina-Prover-72B for the NuminaMath-LEAN proofs.
  - None of these cards adds a no-training clause. Whether the generating model's terms flow through to the data is a legal question I could not settle from the cards (UNVERIFIED).

### 3.3 Library-derived informal and formal pairs

| Dataset | Rows | Licence | Informal side | Measured quality |
|---|---|---|---|---|
| Herald (FrenzyMath; arXiv:2410.10878, ICLR 2025) | 579,883 statement pairs + 44,553 proofs (from about 110k Mathlib theorems) | Apache-2.0 | LLM informalization of Mathlib (Lean v4.11.0) | Translator gets 93.2% on miniF2F-test at pass@128. In a human check on ProofNet, 101/151 validated outputs were correct. **No score for the informalizations themselves.** The informalizing LLM is not named on the card |
| MMA (arXiv:2311.03755) | 244,238 AFP + 88,536 Mathlib4 (~332K) | **UNVERIFIED** (not on the arXiv page) | GPT-4 informalization | The paper calls them "noisy approximations"; no number |
| FrenzyMath/mathlib_informal_v4.19.0 | 218,071 | Apache-2.0 | `informal_name`, `informal_description`, plus `docstring` | none published (v4.16.0 also exists) |
| FrenzyMath/lsv2-mathlib-v4.28.0-rc1-jsonl | 310,579 | Apache-2.0 | LLM descriptions (LeanSearch v2, arXiv:2605.13137; https://github.com/frenzymath/LeanSearch-v2) | retrieval metrics only; no informalization accuracy |
| adeo1/mathlib_informal_v4.28.0 | 183,489 | MIT | LLM descriptions | none |
| phanerozoic/Lean4-Mathlib | 232,431 | Apache-2.0 | docstrings plus structure (imports, deps) | n/a; pinned to Mathlib commit `b9f1435…` |
| mathlib-initiative/mathlib-types | 797,294 | Apache-2.0 | human docstrings (~14.5% coverage) | human-written |
| TheoremGraph (arXiv:2606.25363; HF uw-math-ai/theorem-matching) | 100,831 candidate pairs (cosine ≥ 0.80); 388,105 Lean declarations from 25 projects vs 11.7M arXiv statements | **CC BY-SA 4.0** | real arXiv statements (not generated) | GPT-5.4 judge: 47,952 matches. DeepSeek-V4-Pro (lenient): 61,210. Acceptance about 48% overall, about 87% at similarity ≥ 0.9 |
| Stacks-to-Mathlib alignment (arXiv:2610.31011, CIKM '26; https://github.com/AnkitSatpute/Formal-Informal-Mathematical-Alignment) | – | MIT | Stacks Project tags | size UNVERIFIED |
| MathAtlas (arXiv:2605.14061; HF MathAtlas/MathAtlas) | 35,566 test + 698 hard (36,264), from 103 textbooks | **no tag** | informal only, with `mathlib_links` | best model 2.6% on the hard set |
| formalica/lean4oeis | – | CC BY-SA 4.0 | OEIS entries | Lean v4.34.0-rc1; size not recorded |

- TheoremGraph is the only large set that pairs Mathlib with **human-written** arXiv statements and not LLM paraphrases. Its share-alike licence must be propagated.

### 3.4 Proof-state and tracing datasets

- **LeanDojo** (https://github.com/lean-dojo/LeanDojo): MIT.
- **LeanDojo-v2**: licence inconsistent. The GitHub API says Apache-2.0 while the README says MIT. It traces repositories to JSONL and has a DynamicDatabase.
- The LeanAgent repo has **no licence**.
- HF tasksource/leandojo: cc-by-2.0, a re-upload.
- The size of LeanDojo Benchmark 4 is UNVERIFIED (not rechecked this session).
- l3lab/ntp-mathlib: 307,049 tactic examples (Lean v4.4.0).
- miniCTX and miniCTX-v2: Apache-2.0.
- MathlibPR: Apache-2.0, 15,895 PR snapshots.
- mathlib-initiative/mathlib-tactics: 1,999,349 rows, the newest (v4.35.0, 2026-10-13).

### 3.5 Pretraining corpora (informal math, some formal)

| Corpus | Size | Licence | Notes |
|---|---|---|---|
| FineMath (HuggingFaceTB) | 48,283,984 rows total. FineMath-3+: 34B tokens / 21.4M docs. 4+: 9.6B tokens | ODC-By + CommonCrawl ToU | classifier trained on Llama-3.1-70B annotations |
| OpenWebMath | 6.3M docs, 14.7B tokens | ODC-By 1.0 + CC ToU | |
| Proof-Pile-2 (EleutherAI) | 55B tokens: arXiv 29B, OWM 15B, AlgebraicStack 11B | "We do not alter the license of any of the underlying data" | AlgebraicStack formal part: Lean 285.6M, Isabelle 1,089.7M, Coq 281.9M, Agda 35.2M tokens |
| MathPile (GAIR) | 9.5B tokens (UNVERIFIED; the card is gated) | **CC BY-NC-SA 4.0**, gated | NC: exclude |
| AutoMathText | about 200 GB, 4,371,703 rows | CC BY-SA 4.0 | scored by Qwen-72B |

- Proof-Pile-2 has no licence of its own; every source keeps its upstream licence. Its AlgebraicStack Isabelle share (about 1.09B tokens) is larger than Lean, Coq and Agda combined.

---

## 4. Informalization tools and papers

| Tool / paper | What it does | Measured quality |
|---|---|---|
| doc-gen4 | HTML docs plus declaration JSON from compiled Lean | none needed (deterministic; human docstrings only) |
| lean_scout (Mathlib Initiative) | official extractor for types, docstrings, dependencies and tactics | deterministic |
| Herald pipeline (arXiv:2410.10878) | LLM informalization of Mathlib with dependency context | no direct informalization score; the downstream translator scores 93.2% on miniF2F |
| LeanSearch v2 (arXiv:2605.13137) | LLM descriptions of 310k Mathlib constants for retrieval | retrieval metrics only |
| MMA (arXiv:2311.03755) | GPT-4 informalization of AFP and Mathlib | "noisy"; no number |
| FormL4 (arXiv:2406.01940) | informalization judged by humans | Gemini-Pro-1.5 80% vs GPT-4 70% success |
| TheoremGraph (arXiv:2606.25363) | matches Lean declarations to real arXiv statements (not generated) | about 48% judge acceptance overall, about 87% at sim ≥ 0.9 |
| LeanAgent / LeanDojo-v2 | lifelong-learning prover and tracer | prover metrics only; not informalization |
| Beyond Compilation (arXiv:2606.31002) | faithfulness judge for formal statements | 91.5% agreement with human majority (n=227) |
| ITPEval BEq check (arXiv:2607.19407) | equivalence checking of translated statements | only 54.0% of compiled translations confirmed |

- **No published, measured error rate exists for LLM informalizations of Mathlib at library scale.** Herald, LeanSearch v2, the FrenzyMath datasets and adeo1 report downstream or retrieval metrics. FormL4 (80%) is the closest number, but it covers a different setting.
- If Math 2.0 generates informal text for bindings, it needs its own sampled human audit.

---

## 5. Summary tables

### 5.1 Libraries

| Library | Size (date) | Licence | Extraction route | Informal text |
|---|---|---|---|---|
| Mathlib | 291,237 thm / 139,616 def (stats page, read 2026-10-26); 797,294 constants at v4.35.0 (2026-10-13) | Apache-2.0 | lean_scout HF datasets; doc-gen4; LeanDojo | docstrings, ~14.5% of constants |
| Tau Ceti | 18,849 `.lean` files (2026-10-26); "3M lines" ambiguous | Apache-2.0 | same as Mathlib (Lean 4) | docs site; AI-written |
| Formal Frontier | 41 libraries, about 200k lines (Xena 2026-10-16) | Apache-2.0 (30/50 repos checked) | Lean 4 tools | per-repo |
| ATLAS | 26 books, more than 45k decls, about 500k lines (2026-05-28) | **v1 CC BY-NC + no-ML rider; v2 Apache with no-ML preamble** | Lean 4 | textbook-aligned |
| Reservoir | 908 packages (2026-10-26) | per package | Lake | per package |
| Isabelle AFP | 1,042 entries, ~330k lemmas, ~5.518M lines (2026-10-26) | BSD-style or LGPL, per entry | tarball / hg; Isabelle export | abstracts plus `text` blocks |
| Isabelle2026 | released Oct 2026; size UNVERIFIED | BSD-style core, mixed | Isabelle export | `text` blocks |
| Rocq stdlib | 904 `.v` files | LGPL-2.1 | coqdoc / SerAPI-style tools | comments |
| MathComp / Analysis / Odd Order | 147 / 131 / 34 `.v` files | CeCILL-B / CeCILL-C / CeCILL-B | opam | header comments |
| UniMath | 1,714 `.v` files | MIT-style | opam | comments |
| Coquelicot | – | UNVERIFIED (fork is LGPL-3.0) | opam | comments |
| HOL Light | 664 `.ml` files; about 37k objects (secondary) | BSD-style | hol2dk dump | comments |
| Flyspeck | 503 `.hl` files | MIT | hol2dk | comments |
| HOL4 | 1,495 `*Script.sml` files | BSD-style | HOL4 export | comments |
| set.mm | 51,207 statement numbers; over 41k theorems with mathboxes (2026-10-26) | **CC0 1.0** | plain-text `.mm` parse | **a comment on every statement** |
| iset.mm | about 17,395 statements | CC0 (same repo) | `.mm` parse | per statement |
| Mizar MML | 5.94.1493 (2025-05-30) | GPLv3+ / CC BY-SA 3.0+ | XML export | article prose limited |
| agda-stdlib | 1,183 files | MIT-style | Agda HTML/JSON | comments |
| agda-unimath | 3,072 `.lagda.md` | MIT | literate parse | **literate prose** |
| 1Lab | 710 `.lagda.md` | **AGPL-3.0** | literate parse | **literate prose** |
| Naproche | 32 `.ftl.tex` | GPL-3.0 | ForTheL | **the source is CNL** |
| Megalodon | about 160k lines, more than 1.5k lemmas (2026-01-06) | system MIT-style; mgwiki repos **no licence** | Megalodon | none |
| NASALib | 63 or 64 libraries, about 41K formulas | public domain (NASA) plus per-library | PVS | comments |
| ACL2 books | 40,929 tree entries | BSD-3 plus per-file | ACL2 | doc strings (XDOC) |

### 5.2 Datasets

| Dataset | Size | Licence | Informal side | Quality (measured) |
|---|---|---|---|---|
| miniF2F (Lean ports) | 488 (244 test) | MIT / Apache | competition NL | more than 50% discrepancies (arXiv:2511.03108); 36 certified defects (arXiv:2606.29493) |
| ProofNet | 371 | MIT | textbook NL | 55.9% erroneous (ProofNet-Verified) |
| ProofNet# | 371 | MIT | same | 27.0% still erroneous |
| ProofNet-Verified | 367 | Apache-2.0 | same | target Lean v4.34.0 |
| PutnamBench | 672 L / 640 I / 412 C | Apache (L, I), MIT (C); NL **by MAA permission** | Putnam NL | no audit number (not in the arXiv:2606.29493 table) |
| FormalMATH | 5,560 | MIT (HF) / Apache (GH) | Olympiad/college NL | 147 certified defects / 3,463 findings |
| DeepSeek-ProverBench | 325 | **none** | AIME/textbook NL | 208 certified defects |
| Lean Workbook (+Plus) | 57,231 + 82,893 (HF default 25,214) | Apache-2.0 | AoPS NL | 93.5% (sample) |
| Goedel-Pset-v1 | 1,732,594 | **none** | Numina NL | compile plus Qwen FC filter; no error rate |
| Goedel SFT v2 | 1,745,010 | Apache-2.0 | – | – |
| NuminaMath-LEAN | 104,155 | Apache-2.0 | Numina NL | card admits errors; no rate |
| Kimina-Prover-Promptset | 24,418 | Apache-2.0 (text only) | Numina NL plus Gemini variants | – |
| DeepSeek-Prover-V1 | 27,503 | DeepSeek "other" | NL | – |
| Herald | 579,883 + 44,553 proofs | Apache-2.0 | LLM informalization | no informalization score |
| MMA | ~332K | UNVERIFIED | GPT-4 informalization | "noisy" |
| FrenzyMath mathlib_informal v4.19.0 | 218,071 | Apache-2.0 | LLM | none |
| LeanSearch v2 JSONL | 310,579 | Apache-2.0 | LLM | retrieval only |
| mathlib-initiative/mathlib-types | 797,294 | Apache-2.0 | human docstrings | 14.5% coverage |
| TheoremGraph matching | 100,831 pairs (47,952 accepted) | **CC BY-SA 4.0** | real arXiv text | judge 48% / 87% at sim ≥ 0.9 |
| MathAtlas | 36,264 | **none** | textbook NL | best 2.6% hard |
| AgenticCommons autoformalization | 3,623 | CC0 | NL | compile and axiom gates |
| LeanDojo / LeanDojo-v2 | Benchmark 4 size UNVERIFIED | MIT / (Apache vs MIT conflict) | none | – |
| ntp-mathlib | 307,049 | – (not recorded) | none | – |
| FineMath | 48.3M rows; 34B tok (3+) | ODC-By + CC ToU | web math | classifier-filtered |
| OpenWebMath | 6.3M docs, 14.7B tok | ODC-By + CC ToU | web math | – |
| Proof-Pile-2 | 55B tok | upstream licences | arXiv/web/code | – |
| MathPile | 9.5B tok (UNVERIFIED) | **CC BY-NC-SA 4.0** | mixed | exclude |
| AutoMathText | 4,371,703 rows | CC BY-SA 4.0 | web/arXiv/code | Qwen-72B scored |

(The PutnamBench row: the defect study's table did not list PutnamBench, so I have no audit number for it.)

### 5.3 Changes against the spec

| Spec 01 value | Value now (read 2026-10-26) |
|---|---|
| Mathlib 291,178 thm / 139,569 def | 291,237 / 139,616 |
| Mathlib 9,437 files | 8,661 `.lean` under `Mathlib/` (a definitional difference) |
| AFP 1,038 / ~326,500 / ~5.44M | 1,042 / ~330,000 / ~5,518,000 |
| Isabelle2026 [U] | released (October 2026) |
| Agda sizes [U] | stdlib 1,183, agda-unimath 3,072, 1Lab 710 files |
| Megalodon Munkres 130k lines | about 160k lines, more than 1.5k lemmas (arXiv:2601.03298) |
| Mizar 1,498 articles | version string says 1493, a conflict |
| Metamath 100: 75 | 75 (mm_100); Wiedijk lists 74 |

### 5.4 Licence red flags for ingestion (spec 09.4)

- **No ML use or NC:** ATLAS v1 (CC BY-NC plus no-ML rider), ATLAS v2 preamble, AutoformBench (CC BY-NC), MathPile (CC BY-NC-SA).
- **Share-alike / copyleft:** Mizar (GPLv3 / CC BY-SA), 1Lab (AGPL-3.0), TheoremGraph (CC BY-SA 4.0), AutoMathText (CC BY-SA 4.0), lean4oeis (CC BY-SA 4.0), Naproche (GPL-3.0).
- **No licence at all:** Goedel-Pset-v1, Lean-workbook-proofs, DeepSeek-ProverBench, MathAtlas, LeanAgent, mgwiki Megalodon repos, NASALib (no SPDX; per-library).
- **Third-party text:** PutnamBench informal statements (MAA permission).
- **Clean and permissive:** Mathlib, the mathlib-initiative datasets, Tau Ceti, Formal Frontier, Herald, the FrenzyMath datasets, NuminaMath-LEAN, Goedel SFT v2.
- **Public domain:** set.mm and iset.mm (CC0), 1000+ list (Unlicense), AgenticCommons (CC0).

---

## 6. Open doubts

1. **Mizar article count.** The version string 5.94.1493 implies 1,493 articles; spec 01 says 1,498. The 73,460 theorems figure was not rechecked.
2. **Tau Ceti size.** It is unclear whether Xena's "3 million lines" refers to Tau Ceti or to Mathlib. The stats page returned 404. Only the file count (18,849) is confirmed.
3. **Formal Frontier licences.** Only 30 of 50 repos were checked, and all were Apache-2.0. The remaining 20 are unchecked.
4. **ATLAS v2 legal status.** The Apache-2.0 grant opens with a "not intended for … ML" sentence. Whether that is binding is a legal question.
5. **1000+ per-system counts.** The web page showed about 228 formalized entries, possibly one page of a paginated list. Spec 01's "241 in Lean" is unconfirmed; it needs a parse of the 1,200 YAML files (one tarball fetch).
6. **Metamath counts.** 51,207 statement numbers (including axioms and definitions) against "over 41,000 counting mathboxes" against Wiedijk's 74 / mm_100's 75. The pages count different things.
7. **Lean Workbook rows.** The paper and card give 57,231 + 82,893, but the default HF config has 25,214 rows. Other configs or splits may hold the rest.
8. **Coquelicot licence.** The Inria GitLab is blocked by Anubis. The fork is LGPL-3.0; upstream is unconfirmed.
9. **LeanDojo-v2 licence.** The GitHub API says Apache-2.0 and the README says MIT. The size of LeanDojo Benchmark 4 was not rechecked.
10. **MathPile token count.** 9.5B is from memory or secondary sources; the card is gated.
11. **EFG judge rates** (arXiv:2506.10903): search snippet only.
12. **miniF2F "over 300 corrected"**: search snippet only. The "more than half" figure is from the abstract.
13. **MMA licence**: not stated on the arXiv page.
14. **ITPEval data availability**: no data URL found.
15. **Model-terms pass-through.** Datasets generated with DeepSeek-V3, Qwen2.5-72B, Gemini or Kimina-72B carry permissive tags. Whether upstream model terms constrain the data is unresolved.
16. **Isabelle2026 distribution size and HOL-Analysis counts**: not found.
17. **HOL Light "~37,000 objects in 635 files"**: from Blanqui's talk, a secondary source. The repo shows 664 `.ml` files.
18. **Lean Pool** (arXiv:2609.25199): size and licence of imported projects are unknown.
19. **Mathlib declaration count definition.** 291k + 139k (stats page) against 483k `allowCompletion` against 797k constants (lean_scout). M2Extract must pick one definition and record it.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/01d-model-generation.md">
# K01, part D. LLM generation at scale: quality, prior projects, memorization, law, terms and cost

Research for the math2 spec (tamnd/math2, tamnd/math2-agent). Written 2026-10-26.

This document extends spec 01 (§1.4 to 1.6, §1.12), spec 10 (autoformalization and fidelity) and spec 12 (research agents). It does not repeat them. The items below that spec 01 already covers are cited there and only extended here:
- PutnamBench costs;
- the FLT project;
- Automatic Textbook Formalization;
- Lean Pool;
- Tau Ceti;
- the Aletheia 6.5% figure;
- the DeepSeekMath-V2 verifier.

**Conventions**

- Every fact has the URL where it was found. All pages were read on **2026-10-26** unless another date is given.
- **[P]** means a primary source: the paper, the official page, the licence text or the court docket.
- **[S]** means a secondary source: a law-firm note, a tracker or news.
- **[snippet]** means the fact was seen only in a search-result excerpt, not on the page itself.
- **UNVERIFIED** marks a fact that could not be confirmed. The reason is given next to it.
- "Derived" marks a figure we computed from a source's numbers. The source does not state it.
- Most numbers were taken from a fetch tool's extraction of the page, not from the raw PDF. Re-check any number against the PDF before the spec quotes it as normative.
- Model names are written as each source wrote them.
- No model API was called for this research.

**Contents**

1. Quality: measured error rates of LLM-written mathematics
2. Scale projects: throughput, cost, fidelity
3. Memorization and copyright
   - 3a. Reproduction and its detection
   - 3b. The legal position
4. Provider terms and open-weight licences
5. Cost of one million nodes
6. Community norms on LLM text (short)
7. What the spec should require because of this
8. Open doubts

## 1. Quality: measured error rates of LLM-written mathematics, 2024 to 2026

All sources were read on 2026-10-26. The numbers come from the fetch tool's extraction of each page. Anything that will be quoted in the spec should be checked against the PDF first. Model names are as the papers give them. "Derived" means our own calculation.

### 1.1 Informal proofs graded by humans

| Study | Setting | Key measured result | Source |
|---|---|---|---|
| Proof or Bluff (Petrov et al., arXiv:2503.21934, v5 4 Sep 2025) | USAMO 2025, expert graders | Best model Gemini-2.5-Pro 10.1/42 (about 25%). All others under 5%: R1 2.0, o3-mini 0.9, o1-pro 1.2 | https://arxiv.org/html/2503.21934v5 [P] |
| MathArena "Not Even Bronze" (17 Jul 2025) | IMO 2025, 2 judges per response, best-of-32 | Best model Gemini 2.5 Pro 31% (13/42), below bronze (19). Gemini "cite[s] non-existent theorems when it fails to find a valid proof" | https://matharena.ai/imo/ [P] |
| Open Proof Corpus (Dekoninck et al., arXiv:2506.21621, v2 14 Jan 2026) | over 5,000 human-graded proofs | **43% of proofs correct overall.** Only 59.5% of o3's correct final answers carried a correct proof. Models admitted failure in 114 of more than 1,700 wrong solutions | https://arxiv.org/html/2506.21621v2 [P] |
| IMO-Bench (Luong et al., arXiv:2511.01846, EMNLP 2025) | IMO-ProofBench, advanced set | Gemini Deep Think 65.7%; next best is 42.4 points lower | https://arxiv.org/abs/2511.01846 [P] |
| ProofBench / ProofGrader (Ma et al., arXiv:2510.13888, ICLR 2026) | 145 problems, 435 solutions, 0-7 expert scale | Best-of-16 picked by ProofGrader 4.14/7; human oracle 4.62; binary evaluator 2.48 | https://arxiv.org/abs/2510.13888 [P] |
| Hard2Verify (Pandit et al., arXiv:2510.13744) | 1,860 PhD-annotated steps (GPT-5, Gemini 2.5 Pro, Claude Sonnet 4) | 58% of steps correct. Claude Sonnet 4 had 4 of 72 solutions fully correct | https://arxiv.org/html/2510.13744v1 [P] |
| IneqMath (Lu et al., arXiv:2506.07927, NeurIPS 2025) | 29 LLMs, step-wise grading | o1 under 10% overall; step-wise grading lowers accuracy by up to 65.5 points from the final-answer score | https://arxiv.org/abs/2506.07927 [P] |
| QEDBench (Gonzalez et al., arXiv:2602.20629, ICML 2026) | university level, human experts | Gemini 3.0 Pro 0.91. GPT-5 Pro and Claude Sonnet 4.5: 0.72 and 0.63 in discrete math, 0.74 and 0.50 in graph theory | https://arxiv.org/abs/2602.20629 [P] |
| LemmaBench (Peyronnet et al., arXiv:2602.24173) | lemmas from recent arXiv papers | SOTA "around 10-15% accuracy" pass@1 | https://arxiv.org/abs/2602.24173 [P] |
| MathArena platform (arXiv:2605.00674, v2 15 May 2026) | USAMO 2026; research-level questions | GPT-5.5 "reaches 98% on the 2026 USA Math Olympiad and 74% on research-level questions" | https://arxiv.org/abs/2605.00674 [P] |
| First Proof, batch 1 (arXiv:2602.05192) | 10 unpublished research questions | No score is given; best answers were essentially correct on 2 of 10. Cases of "citing lemmas that did not exist" | https://arxiv.org/html/2602.05192v2 [P] |
| First Proof, batch 2 (arXiv:2606.18119, 16 Jun 2026) | referee verdicts on 4 systems | Derived and to be rechecked: about 6 of 9, 5 of 10, 5 of 10 and 1 of 10 rated flawless or minor-revision. One system "hallucinates a theorem from the literature which exactly matches the theorem it was asked to prove". Citation problems on 4 to 8 of 10 problems for most systems | https://arxiv.org/html/2606.18119v1 [P] |

**Proving false statements** is the measured proxy for the "misstated theorem" risk:
- **BrokenMath** (arXiv:2510.04721) [P]. https://arxiv.org/html/2510.04721v1
  - Sycophancy rates: GPT-5 29.0%, Gemini-2.5-Pro 37.5%, DeepSeek-V3.1 70.2%.
  - Rates are higher on proof problems, by up to 22.5 points. They are higher again when the false statement replaces the model's own theorem, by up to 15.6 points.
  - Self-verification lowers sycophancy by 7.6 to 12.5 points but does not remove it.
- **BrokenArXiv** (MathArena, 13 Mar 2026; 31 perturbed arXiv statements) [P]. https://matharena.ai/brokenarxiv/
  - Share of false statements rejected: GPT-5.4 "just under 40%"; Gemini-3.1-Pro 18.5%, rising to 71% with a "Prove or disprove" prompt; Claude-Opus-4.6 3.2%.
  - Later figures (GPT-5.5 72%) are [snippet].

### 1.2 Reliability of LLM judges

- **Proof or Bluff, Table 2** [P]. LLM judges inflated scores "by a factor of up to 20".
  - Examples: R1 scored 2.0 by humans and 19.3 by an o3-mini judge; QwQ 1.2 and 23.8.
  - Derived means: humans about 2.6/42, the o3-mini judge about 18.3, the Claude 3.7 judge about 16.6.
- **OPC judge benchmark** (293 human-labelled proofs) [P].
  - Human inter-annotator agreement: 90.4%.
  - Judge accuracy, pass@1 / maj@5:

    | Judge | pass@1 | maj@5 |
    |---|---|---|
    | GPT-5 | 89.3 | 90.8 |
    | Grok-4 | 88.3 | 89.8 |
    | Gemini-2.5-Pro | 85.4 | 88.1 |
    | o3 | 83.1 | 84.3 |
    | Claude-4-Sonnet | 70.6 | 75.0 |
    | GPT-4.1 | 61.4 | 60.8 |

  - All but one model judge their own proofs more harshly than other models' proofs.
- **Hard2Verify, step-level** [P].
  - Balanced accuracy: GPT-5 86.5, Gemini 2.5 Pro 83.4, Claude Sonnet 4 70.6, Qwen2.5-Math-PRM-72B 55.8.
  - True-negative rate on incorrect steps: GPT-5 78.7, Gemini 78.6, Claude Sonnet 4 43.7, Llama-3.3-70B 10.1.
  - Derived: **the best judges accept about 21% of wrong steps; weak judges accept 56 to 90%.** The paper concludes: "weaker verifiers cannot catch errors."
- **IMO-Bench** [P]. https://arxiv.org/html/2511.01846v1
  - ProofAutoGrader against humans: Pearson r of 0.96 (basic set) and 0.93 (advanced set).
  - 4-class IMO-GradingBench accuracy is only 44 to 54% for every model tested.
- **ProofGrader** [P]: mean absolute error of 0.926 on the 0-7 scale.
- **QEDBench** [P]: judges show positive bias, from +0.18 for Claude Opus 4.5 up to +0.36 for Llama 4 Maverick on a 0-1 scale.
- **Grayzel, "Cost-Effective Automated Judging..."** (arXiv:2608.00004, v2 5 Oct 2026) [P]. https://arxiv.org/abs/2608.00004
  - GPT-OSS-120B, DeepSeek-V4-Flash and their consensus are "statistically no worse than the frontier (Claude Opus 4.7, Gemini 3.1 Pro)" at "4-100x lower cost".
  - On the 1,000 IMO-GradingBench items, a unanimous vote has the highest precision (0.855) and a majority vote the highest recall (0.912).
- **"One Token to Fool LLM-as-a-Judge"** (arXiv:2507.08794) [P]: reference-based reward models "can consistently elicit false positive rewards without any substantive reasoning". No rate is given.

### 1.3 Wrong definitions, missing hypotheses, misnamed theorems

- **No study measures the rate at which LLMs write a wrong definition, drop a hypothesis or misname a standard theorem in informal text.** This is a gap. math2 will have to measure it on its own audit sample.
- The nearest measured evidence:
  - **Re²Math** (Lyu et al., arXiv:2605.09012, 9 May 2026) [P]. https://arxiv.org/html/2605.09012
    - Task: cite a source theorem that exists and also justifies the step. The best success rate is 7.0% (Claude Opus 4.5).
    - Breakdown for n=200 (no grounded support / grounded but insufficient / success):
      - GPT-5.2: 75.0 / 22.0 / 3.0%
      - Gemini 3.1 Pro: 88.0 / 10.0 / 2.0%
      - Claude Opus 4.5: 76.0 / 17.0 / 7.0%
      - Grok 4: 58.5 / 38.0 / 3.5%
    - "Models often retrieve or restate true mathematics without making it usable."
  - **Banerjee and Bhattacharjee** (arXiv:2606.24902) [P]. Every one of 8 Gemini 2.5 Flash research proofs contained an unsupported load-bearing "fundamental result" ("premise smuggling"). None had a confirmed fabricated citation. The sample is tiny.

### 1.4 Hallucinated citations

| Study | Measured | Source |
|---|---|---|
| Naser, arXiv:2603.03299 | 10 commercial LLMs, 69,557 citations: **11.4% to 56.8% hallucinated** | https://arxiv.org/abs/2603.03299 [P] |
| Rao et al., arXiv:2604.03173 | 3 to 13% of citation URLs hallucinated, 5 to 18% do not resolve | https://arxiv.org/abs/2604.03173 [P] |
| Linardon et al., JMIR Mental Health, 12 Nov 2025 | GPT-4o: 19.9% fabricated, 45.4% of the real ones have errors; 6% on a familiar topic against 28 to 29% on unfamiliar ones | https://pmc.ncbi.nlm.nih.gov/articles/PMC12658395/ [P] |
| Ansari, arXiv:2602.05930 | 100 hallucinated citations in 53 accepted NeurIPS 2025 papers (about 1%) | https://arxiv.org/abs/2602.05930 [P] |
| Russinovich et al., arXiv:2607.00738 | reference-level rate usually below 1%; about 1 in 20 NeurIPS and USENIX Security 2025 papers has 2 or more likely hallucinated references | https://arxiv.org/abs/2607.00738 [P] |
| Zhao et al. (with Ginsparg), arXiv:2605.07723 | 111M references: "conservative estimate of 146,932 hallucinated citations in 2025 alone"; no figure for mathematics alone | https://arxiv.org/abs/2605.07723 [P] |
| QED (An et al., arXiv:2604.24021) | in research-level math proofs, "citation hallucination was not a rare edge case"; no number given | https://arxiv.org/html/2604.24021 [P] |

The rate of unfamiliar-topic fabrication matters for math2. Niche research mathematics is the "unfamiliar topic" case.

### 1.5 Which checks catch the errors (measured)

| Check | Measured effect | Source |
|---|---|---|
| Strong judge plus majority vote | maj@5 adds 1.5 to 4.3 points over pass@1 (OPC). Unanimous vote gives precision 0.855; majority vote gives recall 0.912 | 2506.21621, 2608.00004 [P] |
| Pseudo-formalization (structured semi-formal rewriting) | FirstProof errors: recall 0.66 at precision 0.77, against 0.54 at 0.76 for an LLM judge (GPT-5.5). On arXiv errors, both "leave roughly 40% of annotated ground-truth errors uncaught" | https://arxiv.org/html/2605.20531 [P] |
| Lean formalization of an informal proof | LeanTutor: only 13.0 to 30.1% of 146 wrong student proofs were faithfully autoformalized so that Lean flagged the error. "Don't Trust: Verify": more than 12% gain on GSM8K (answers, not proofs) | https://arxiv.org/html/2506.08321v2, https://arxiv.org/abs/2403.18120 [P] |
| Verifier-gated agent | QED: 0 false positives among 17 accepted out of 214; false negatives not measured | 2604.24021 [P] |
| Cross-model agreement on citations | a work cited by more than 3 LLMs is real 95.6% of the time; within-prompt repetition 88.9%; classifier AUC 0.876 | 2603.03299 [P] |
| Retrieval or liveness check on citations | cuts non-resolving URLs "by 6-79x to under 1%" | 2604.03173 [P] |
| "Prove or disprove" prompting | Gemini-3.1-Pro rejects false statements 71% of the time, against 18.5% | matharena.ai/brokenarxiv [P] |
| Self-verification | the evidence is mixed. Hard2Verify finds models more lenient on their own work; OPC finds them harsher | 2510.13744, 2506.21621 [P] |

### 1.6 What section 1 says, in eight numbers

1. Unchecked LLM judges inflate proof scores by up to 20x (Proof or Bluff).
2. The best step judges accept about 21% of wrong steps; weak judges accept 56 to 90% (Hard2Verify).
3. Strong models "prove" false competition statements 29% to 70% of the time (BrokenMath). At research level they rejected under 40% of false statements (BrokenArXiv, Feb 2026).
4. A correct final answer is not a correct proof: 59.5% for o3 (OPC).
5. The ceiling for single-judge agreement with humans is about 90%, which is also the human inter-annotator rate (OPC).
6. Theorem references in research proofs are usable at most 7% of the time (Re²Math).
7. Even structured verification misses about 34 to 40% of known research-proof errors (Pseudo-Formalization).
8. 11.4 to 56.8% of generated citations are hallucinated. Agreement among more than 3 models gives 95.6% accuracy.

### 1.7 Could not verify (quality)

- Per-model proof-correctness rates in OPC and Hard2Verify; they appear only in figures.
- ProofBench OpenReview numbers [snippet].
- BrokenArXiv later figures [snippet].
- Phantom References per-venue shares [snippet].
- ProofGrid self-grading figures [snippet].
- IMProofBench per-model scores: v2 has none in the abstract, and v1 is [snippet].
- The First Proof batch 2 tally is derived and needs checking against its Tables 1 to 5.
- The Linardon abstract and results disagree on one subgroup.
- The FormalAlign arXiv id from search pointed to an unrelated paper, so it is excluded.

## 2. Scale projects: throughput, cost, fidelity

All sources were read on 2026-10-26. A figure marked "derived" is our own division, not the source's. **Main finding:**
- Measured fidelity numbers exist only for the autoformalized-statement datasets: DeepSeek-Prover V1, Goedel, AlphaProof, Lean Workbook, Herald and CriticLean.
- The large agent runs (Gauss, Textbook Formalization, Lean Pool, Tau Ceti) report size, and sometimes cost.
- **None of the agent runs measures a misformalization rate.**

### 2.1 Agent runs on research mathematics

- **Math Inc Gauss, Strong PNT.** https://github.com/math-inc/strongpnt, https://www.math.inc/gauss [P]
  - Output: about 25k lines of Lean and about 1.1k theorems and definitions, done in three weeks.
  - Compute: "thousands of concurrent agents", "multiple terabytes of cluster RAM".
  - Human input: "Humans provided the high-level blueprint, reviewed critical lemmas, and adapted prior work where needed."
  - Cost: not published.
  - Fidelity: no rate is published; humans reviewed the key lemmas.
  - Announcement date 11 Sep 2025 [snippet].
- **Math Inc sphere packing.** https://www.math.inc/sphere-packing [P]; the page is undated, and its latest milestone is February 2026.
  - Size: about 200k lines, after a peak of 500k before refactoring.
  - Dimension 8: "In just 5 days, Gauss automatically proved all remaining results".
  - Dimension 24: two weeks, "using only the original paper as input".
  - Cost and compute: not stated.
- **FormalQualBench.** https://www.math.inc/formalqualbench [P]
  - Setup: 23 graduate-level theorems, 4-hour timeout.
  - Results:

    | System | Solved (of 23) | Mean cost per solve |
    |---|---|---|
    | OpenGauss | 8 | $24.93 |
    | Claude Code | 4 | $23.16 |
    | Claude Code (Skills) | 5 | $13.47 |
    | Codex | 5 | not given |
    | opencode | 5 | $22.22 |
    | Aristotle | 6 | not given |

  - Aristotle ran with no timeout and was "not Comparator-validated", so it "may contain false positives".
  - The statements were "verified by a Lean expert".
  - Codex and opencode tried elaborator-level workarounds. **This is direct evidence that an agent will game a checker if it can.**
- **Automatic Textbook Formalization** (arXiv:2604.03071) [P]. The spec 01 entry is extended here with cost.
  - Cost: "around $100K, or around $200 per page and $300 per target theorem or definition". Without caching it would have been $430K.
  - Tokens: 83B input and 561M output. Output tokens cost $14K of the total.
  - Agents: about 3,300 of 30K succeeded.
  - Derived: about $17 per Lean declaration ($100K / 5,900).
  - Fidelity: there is no rate. Key statements were "manually spot-checked". The authors say it "remains a possibility that individual theorems and definitions have not been translated semantically faithfully".
- **Lean Pool** (arXiv:2609.25199, 21 Sep 2026) [P]
  - Size: 211 projects, 3,228,485 lines, 193,862 declarations.
  - Provenance: 70 human, 102 AI, 39 mixed.
  - Review cost estimates: $308.52 over 285 reports, median $0.20. These are "not cash payments", and generation cost was not logged.
  - Review agreement: repeated reviews of the same PR agreed on 37 of 69 pairs (about 54%).
- **Tau Ceti.** https://github.com/TauCetiProject/TauCeti [P]
  - Status: 19,275 commits.
  - Not published: line counts, cost and rejection rates.
  - Launch around July 2026 [snippet].
- **Erdős problems.** Tao's wiki "AI contributions to Erdős problems", frozen as of 30 Jun 2026. https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems [P]
  - Tally (approximate, from a summary of the tables rather than stated totals): about 90 full-solution rows and about 64 marked "(Lean)".
  - Problems found: some AI entries are marked incorrect, and some problems were misstated at the source.

### 2.2 Competition provers

- **Harmonic Aristotle** (arXiv:2510.01346) [P]
  - IMO 2025: verified solutions to 5 of 6 problems.
  - Human input: the statements were "translated … by hand", which the paper calls "the only human formalization step".
  - Model: the search model has more than 200B parameters.
  - Not reported: compute, wall time and cost. API price and throughput were not found (UNVERIFIED).
- **Seed-Prover 1.0** (arXiv:2507.23726) [P]
  - IMO 2025: 5 of 6 problems.
  - Time per problem, from the README: P3 took 3 days (a 2,000-line proof), P4 3 days (4,000 lines), P5 1 day.
- **Seed-Prover 1.5** (arXiv:2512.17260) [P]
  - Results: 88% on PutnamBench (580/660 [snippet]) and Putnam 2025 11/12 within 9 hours.
  - Compute: "10 H20 days / problem", against 18 H20-days for 1.0. The text elsewhere says 20 for 1.0, which is inconsistent. The Putnam cap was 40 H20-days per problem.
- **AlphaProof** (Nature, 12 Nov 2025). https://www.nature.com/articles/s41586-025-09833-y [P]
  - Dataset: about 1M natural-language problems became about 80M formal problems.
  - Compute:
    - Autoformalization: about 100,000 TPU-days, derived about 1.25 TPU-days per 1,000 formal problems.
    - Reinforcement learning: about 80,000 TPU-days.
  - Fidelity, judged by a panel of about three Lean experts, pass@1: **60% on 50 IMO problems** (combinatorics 33.3%) and **64% on 50 Putnam problems**.
  - The paper holds that an unfaithful statement still "provides a valid formal problem". That is fine for training, but not for a library.
  - IMO 2024: the statements were formalized by experts. Each solved problem needed 2 to 3 days of test-time RL.
  - The "500 TPU-days per problem" figure appears only in Seed-Prover 1.5's citation.

### 2.3 Autoformalized statement datasets (the only measured fidelity)

| Dataset | Size | Measured fidelity | Source |
|---|---|---|---|
| DeepSeek-Prover V1 | 712,073 statements from 869,659 problems; about 8M with proofs | "at least 20% … even after quality filtering, were incorrect" | https://arxiv.org/html/2405.14333 [P] |
| DeepSeek-Prover-V2 | corpus size not reported | 2 PutnamBench and 2 CombiBench statements found misformulated; no audit | https://arxiv.org/html/2504.21801 [P] |
| Goedel-Pset-v1 | 1.64M from 928K problems | compiles 88.5%; LLM faithfulness pass@1 80.4%. Goedel-V2: "over 80% of unsolved problems are incorrectly formalized" (sample size not given) | https://arxiv.org/html/2502.07640, https://arxiv.org/html/2508.03613 [P] |
| Goedel-Formalizer-V2 vs Kimina | 300 Omni-MATH problems | 228/300 against 161/300 (judging method unconfirmed) | 2508.03613 [P] |
| Kimina-Autoformalizer | about 100k (+10k human) | 90% compile, **66% accurate** on a human-curated set of about 1,000 | https://arxiv.org/html/2504.11354 [P] |
| NuminaMath-LEAN | 104,155 | admits "wrongly formalized" items; no rate | https://huggingface.co/datasets/AI-MO/NuminaMath-LEAN [P] |
| Lean Workbook | 57,231 + 82,893 | authors 93.5% (5 to 10 samples per tag); CriticLean re-audit **84%** (n=50) | arXiv:2406.03847, 2507.06181 [P] |
| Herald | 580k | 101/151 correct on ProofNet-valid (derived 67%); 24 minor and 26 major errors | arXiv:2410.10878 [P] |
| FineLeanCorpus (CriticLean) | 285,957 | human accuracy 69% to 100% by source; best critic (Gemini 2.5 Pro) 89.2% on the 500-item CriticLeanBench | arXiv:2507.06181 [P] |
| FormalMATH | 5,560 | 72.09% survive LLM semantic checks before expert review | arXiv:2505.02735 [P] |
| miniF2F revisited | 488 | discrepancies in "more than half of the problems" | arXiv:2511.03108 [P] |
| "Faults in Our Formal Benchmarking" | five benchmarks | 4,833 findings, 398 "mechanically certified issues" | arXiv:2606.29493 [P] |

Compute for Goedel V1:
- Pass@16 over 1.78M statements took about 6 hours on 64 H100s. Verification took about 10 hours on 8,000 CPUs.
- Derived: about 0.23 H100-hours per 1,000 statements per pass@16 sweep.

Compute for Kimina: the Lean server verifies up to 100 proofs per second on 64 cores.

**Reading.** Without a human in the loop, autoformalized statements are faithful about 60 to 90% of the time, depending on source difficulty. Competition-style problems sit at the upper end; graduate and research material is lower (Herald's textbook set: 22.5% proved). Compiling is not evidence of fidelity: Kimina compiles 90% of statements but only 66% are accurate.

### 2.4 LLM-written encyclopedias

- **Grokipedia**, from Triedman and Mantzarlis (arXiv:2511.09685, 12 Nov 2025) [P]:
  - 883,858 articles were scraped.
  - **56% carry the Wikipedia CC footer.** Similarity to Wikipedia is 0.90 for those and 0.77 for the rest.
  - 5.5% cite a blacklisted source, about 9x the rate on Wikipedia.
  - No factual-error rate was measured. Grokipedia's self-reported "92.5% factually correct" is UNVERIFIED (no evaluator is named).
- **AI-written Wikipedia articles.** Brooks et al. (arXiv:2410.08044) [P]
  - "Over 5%" of new English articles were flagged, at a 1% false-positive rate.
  - On Wikipedia's side, speedy-deletion criterion G15 for LLM-generated pages was adopted in August 2025 [S].
- **STORM** (arXiv:2402.14207) and **Co-STORM** (arXiv:2408.15232) [P]
  - Gains: +25% absolute on organization and +10% on coverage.
  - Failure modes: "source bias transfer" and "over-association of unrelated facts".
- No 2025 to 2026 LLM-written mathematics encyclopedia or textbook with a measured error rate was found.

### 2.5 Could not verify (scale)

- Not published:
  - Aristotle API price, throughput and training-set size;
  - Gauss agent counts, compute and dollar cost;
  - the DeepSeek-Prover-V2 synthetic data size;
  - the sample sizes behind Goedel-V2's ">80%" and CriticLean's per-source rates.
- [snippet] only:
  - the dates of the Gauss announcement and the Tau Ceti launch;
  - Seed-Prover 1.5's 580/660;
  - AlphaProof's "500 TPU-days per problem", which is only in Seed-Prover 1.5's citation.
- The Erdős wiki tallies are approximate.

## 3. Memorization and copyright

Part 3a covers what models reproduce and how to detect it. Part 3b covers what the law says about the output.

## 3a. Memorization and copy detection

All sources read 2026-10-26, mostly arXiv abstract or HTML pages. Numbers are as extracted from the page. "[snippet]" means the item was seen only in a search result.

### 3a.1 Ordinary, non-adversarial use

- **Aerni, Rando, Debenedetti, Carlini, Ippolito, Tramèr, "Measuring Non-Adversarial Reproduction of Training Data in LLMs"**, arXiv:2411.10242 (Nov 2024). https://arxiv.org/abs/2411.10242 [P]
  - **Definition.** A generated substring counts as reproduced if it is at least **50 characters** long and appears verbatim in AuxDataset, about 10 TB of public web text up to March 2022. The authors call the result a lower bound.
  - **Setup.** 3,696 prompts over 15 tasks, at temperature 0.7. Temperature had a negligible effect.
  - **Models.** GPT-4o, GPT-4o-mini, GPT-4 Turbo, Claude 3 Opus, Claude 3.5 Sonnet, Claude 3 Haiku, Llama 3.1 (8B, 70B, 405B), Gemini 1.5 Flash and Gemini 1.5 Pro. o1 is in an appendix.
  - **Overlap rate**, meaning the share of output characters inside such substrings:
    - It ranges from about 7% (Gemini) to over 15% (Claude 3 Opus).
    - Expository writing overlaps 3x to 10x more than creative writing.
    - **Encyclopedia-style prompts average over 27%.**
  - **Human baseline.** The human median is 0%. LLM medians go up to 7.5%.
  - **Tail.** Nearly all generations contain a 30-character match. About 2.5% contain a 100-character match, and about 0.01% a 1,000-character match.
  - **Example.** Claude 3 Opus reproduced 1,170 characters of the Wikipedia article on black holes.
  - **Mitigation.** An anti-reproduction system prompt cut Llama's mean overlap from about 16% to about 6%. GPT and Claude dropped by 3 points or more. Matches of 600 to 700 characters still occurred.
  - **Real chat logs.** WildChat and LMSYS-Chat-1M show similar rates.
- **Lu et al., "AI as Humanity's Salieri" (Creativity Index)**, arXiv:2410.04265. https://arxiv.org/abs/2410.04265 [P]
  - On average, professional human authors score 66.2% higher than LLMs.
  - The method, DJ Search, combines verbatim matches with near-verbatim matches under a Word Mover's Distance threshold δ. The value of δ was not seen.
- **Lee et al., "Deduplicating Training Data Makes LMs Better"**, arXiv:2107.06499. https://arxiv.org/abs/2107.06499 [P]
  - Over 1% of unprompted output is copied from training data.
  - Models trained on deduplicated data emit memorized text 10x less often.
- **Lee et al., "Do Language Models Plagiarize?"** (WWW'23, arXiv:2203.07618) [P]
  - Rates for GPT-2 outputs: 0.47 to 0.70% verbatim, 0.16 to 0.32% paraphrase, 0.05 to 0.38% idea plagiarism.
  - The detector favours precision, so these rates are lower bounds.

### 3a.2 Books and extraction

- **Cooper et al., arXiv:2505.12546** (v6 dated 20 Jul 2026; COLM 2026). https://arxiv.org/abs/2505.12546 [P]
  - Study: 200 books and 14 open-weight models.
  - Finding: most models do not memorize most books, but Llama 3.1 70B memorizes *Harry Potter 1* nearly entirely.
- **Ahmed, Cooper, Koyejo, Liang, "Extracting books from production language models"**, arXiv:2601.02671 (6 Jan 2026). https://arxiv.org/abs/2601.02671 [P]
  - **Metric (nv-recall).** Ordered difflib verbatim blocks, merged across small gaps. Blocks of 20 words or more are kept, then 100 words or more after a second merge. The rules are deliberately conservative.
  - **Harry Potter 1 results:**

    | Model | nv-recall | Jailbreak |
    |---|---|---|
    | Claude 3.7 Sonnet | 95.8% | Best-of-N jailbreak, N=258 |
    | Gemini 2.5 Pro | 76.8% | None |
    | Grok 3 | 70.3% | None |
    | GPT-4.1 | 4.0% | Refused after chapter 1 |

  - Most other runs scored 10% or less. The tests ran from August to September 2025.
- **Liu, Mireshghallah, Ginsburg, Chakrabarty, "Alignment Whack-a-Mole"**, arXiv:2603.20957 (v5 Sep 2026). https://arxiv.org/abs/2603.20957 [P]
  - Method: GPT-4o, Gemini 2.5 Pro and DeepSeek-V3.1 were fine-tuned to expand summaries.
  - Result: they reproduced up to 85 to 90% of held-out books, with single spans of more than 460 words.
- **Nasr et al.**, arXiv:2311.17035. https://arxiv.org/abs/2311.17035 [P]
  - Memorized is defined as a verbatim span of 50 tokens or more in AuxDataset.
  - About $200 of queries extracted more than 10,000 unique examples from gpt-3.5-turbo. The divergence attack raises the emission rate 150x.
- **Ippolito et al.**, arXiv:2210.17546. https://arxiv.org/abs/2210.17546 [P]
  - A 10-gram Bloom filter blocks all verbatim memorization.
  - Style-transfer prompts still leak near-verbatim text.
  - **So an exact-match filter alone gives false safety.**
- **Shilov, Meeus, de Montjoye, "The Mosaic Memory of LLMs"**, arXiv:2405.15523. https://arxiv.org/abs/2405.15523 [P]
  - A fuzzy duplicate contributes "as much as 0.8 of an exact duplicate" to memorization.
- **Benchmarks:**
  - **CopyBench** (arXiv:2407.07087), literal copying, defined as ROUGE-L > 0.8 on a 50-word continuation:

    | Model | Literal copying | Other |
    |---|---|---|
    | Llama3-8B to 70B | 0.2% to 10.5% | |
    | GPT-4-Turbo | 0.4% | 3.4% plot events |

  - **CoTaEval** (arXiv:2406.18664). [P]
    - It uses eight metrics: LCS, ROUGE, word ACS, Levenshtein, MinHash Jaccard over 3-grams, and MiniLM cosine.
    - The MemFree decoding filter (n=6) cost MMLU 53.9 to 47.3, and misspellings bypass it.
    - No mitigation won on all metrics.

### 3a.3 Mathematics specifically

- **Llemma** (arXiv:2310.10631). https://arxiv.org/html/2310.10631 [P]
  - 30-gram hits in OpenWebMath and AlgebraicStack cover about 7% of MATH test statements and 0.6% of solutions.
- **Wu et al.** (arXiv:2507.10532). https://arxiv.org/html/2507.10532v3 [P]
  - Prompted with 60% of a MATH-500 problem, Qwen2.5-Math-7B completes the rest exactly 54.6% of the time. Llama3.1-8B does so 3.8% of the time.
  - On newer sets (AIME2025, LiveMathBench) exact completion is about 0 to 5%.
- **GSM1k** (arXiv:2405.00332). https://arxiv.org/abs/2405.00332 [P]
  - Accuracy drops by up to 8% on fresh problems of the same kind. Frontier models show minimal overfitting.
- **Infini-gram mini** (arXiv:2506.12229). https://arxiv.org/abs/2506.12229 [P]
  - It indexes 83 TB, and the index is about 44% of the corpus size.
  - GSM8K contamination reaches up to 74.2%.
- **MathPile** (arXiv:2312.17120) reports near-duplicates within ProofWiki. [snippet; not read from the primary source]
- **No study was found that measures how much LLM-written mathematical prose copies Wikipedia, ProofWiki or textbook phrasing.**
  - The closest proxy is Aerni et al.'s encyclopedia category: more than 27% overlap at the 50-character threshold.
  - **Inference, not a measurement:** math prose plausibly sits at the high end, because definitions and statements are stereotyped and expository.

### 3a.4 Detection methods and parameters used in the literature

| Method | Parameters | Source |
|---|---|---|
| Exact substring (suffix array) | 50 BPE tokens (Lee ExactSubstr, Nasr); 50 characters (Aerni) | 2107.06499, 2311.17035, 2411.10242 [P] |
| Infini-gram | suffix-array n-gram counts over 5T tokens, millisecond queries | https://arxiv.org/abs/2401.17377 [P] |
| OLMoTrace | maximal verbatim spans in 4.6T tokens, keeping the K=ceil(0.05 L) lowest-unigram-probability spans (median 10 tokens); a tracing tool with no accuracy figure | https://arxiv.org/html/2504.07096 [P] |
| MinHash/LSH | Lee NearDup: 5-grams, 9,000 hashes in 450x20 bands, Jaccard 0.8, then edit similarity > 0.8. FineWeb: 5-grams, 112 hashes in 14x8 bands, about 0.75 target (match probability 77% at 0.75, 92% at 0.8) | 2107.06499, https://arxiv.org/abs/2406.17557 [P] |
| ROUGE-L / BLEU | ROUGE-L > 0.8 (CopyBench); BLEU > 0.75 (Ippolito) | as above [P] |
| Embeddings | MiniLM cosine with no threshold (CoTaEval); WMD below δ (DJ Search); no reported accuracy | as above [P] |
| Paraphrase plagiarism | BM25 top-10, then PAN-2014 alignment, then a RoBERTa classifier. Precision 0.96 to 1.0 on verbatim copies, recall 0.72 to 0.79 on paraphrases and 0.3 to 0.41 on ideas | 2203.07618 [P] |
| Human detection | People spot GPT-3 paraphrases with 53% accuracy; the best detector reaches 66% F1 | https://arxiv.org/abs/2210.03568 [P] |
| GitHub Copilot filter | blocks matches of 65 lexemes or more (about 150 characters) against public code | https://github.com/features/copilot [P] |
| Gemini API | `finishReason: RECITATION`; `citationMetadata.citationSources[]` with start and end index, uri and license; no published threshold | https://ai.google.dev/api/generate-content [P] |

No Anthropic or OpenAI documentation of an output recitation filter was found (UNVERIFIED, absence of evidence).

### 3a.5 Recommended detection recipe (proposal, not validated on math text)

1. **Reference corpus.** Use a reference corpus, not only the cited sources:
   - Wikipedia math pages, ProofWiki, nLab, PlanetMath;
   - Math.StackExchange and MathOverflow dumps;
   - Stacks and Kerodon (GFDL);
   - OpenWebMath or arXiv;
   - licensed textbooks where indexing is permitted;
   - optionally, the infini-gram or infini-gram mini web-scale API.
2. **Normalize.** Lowercase, collapse whitespace and canonicalize LaTeX. Mask display and inline formulas as a placeholder, so that standard formulas do not trigger matches.
3. **Exact check (suffix array).**
   - Flag any normalized prose span of **50 characters or more**.
   - Block for rewrite at **100 characters or more**, or when the 50-character overlap rate exceeds about 10% of the node.
   - Use an allowlist of stock phrases ("let G be a group"), calibrated on human-written math.
4. **Near-duplicate check.** Run 5-gram MinHash per paragraph, with FineWeb-style 112 hashes in 14x8 bands, taking candidates at Jaccard 0.5 or more. Confirm with either of these:
   - ROUGE-L of 0.8 or more;
   - difflib blocks of 20 words or more (nv-recall-style rules).
5. **Paraphrase check.** Retrieve the top-k candidates by embedding or BM25. Send pairs above a high cosine threshold, initially 0.9 and uncalibrated, to human review. Expect low recall.
6. **Calibrate** every threshold by running the same pipeline over human-written Wikipedia and ProofWiki pages checked against each other.
7. **Record match spans** (source, uri, licence, offsets), as Gemini's citation metadata does. When a match is to a CC BY-SA source, attribute it and keep the share-alike terms. Never keep a GFDL match.

### 3a.6 Could not verify (memorization)

- How ParaPO's regurgitation metric is defined (arXiv:2504.14452 reports 17.3 to 12.9 for Llama3.1-8B creative writing).
- The value of δ in DJ Search.
- Cooper et al.'s coverage figure of 90.89% [snippet, older version].
- The GPT-4 50-character and Llama 3 8-gram contamination rules [secondary snippet only].
- MathPile's ProofWiki duplication [snippet].
- BloomScrub numbers (arXiv:2504.16046) [snippet].
- Gemini's recitation threshold.
- The Nature Communications version of Mosaic Memory, which needs a login.

## 3b. The legal position of LLM output

### United States

- **Copyright Office, Part 2: Copyrightability (29 Jan 2025)** [P]. https://www.copyright.gov/ai/Copyright-and-Artificial-Intelligence-Part-2-Copyrightability-Report.pdf (read 2026-10-26). Its summarised conclusions:
  - "Copyright does not extend to purely AI-generated material, or material where there is insufficient human control over the expressive elements."
  - "Based on the functioning of current generally available technology, prompts do not alone provide sufficient control."
  - "Human authors are entitled to copyright in their works of authorship that are perceptible in AI-generated outputs, as well as the creative selection, coordination, or arrangement of material in the outputs, or creative modifications of the outputs."
  - "The case has not been made for additional copyright or sui generis protection for AI-generated content."
  - Whether a human contribution is enough is decided "on a case-by-case basis".
- **Copyright Office, Part 3: Generative AI Training** [P]. The pre-publication version came out on 9 May 2025. On 2026-10-26, https://www.copyright.gov/ai/ still lists it as pre-publication: "A final version of Part 3 will be published in the future, without any substantive changes expected." Two of its points matter here (PDF read 2026-10-26):
  - When a model "can generate verbatim or substantially similar copies of a training example, without that expression being provided externally in the form of a prompt or other input, it must exist in some form in the model's weights". There is then "a strong argument that copying the model's weights implicates the right of reproduction for the memorized examples."
  - "copying the resulting weights will only infringe where there is substantial similarity."
- No Part 4 and no 2026 report is listed on https://www.copyright.gov/ai/ (read 2026-10-26).
- **Thaler v. Perlmutter.** The D.C. Circuit held on 18 Mar 2025 (No. 23-5233) that the Copyright Act requires a human author. The Supreme Court denied certiorari on **2 Mar 2026** (No. 25-449; docket entry "Petition DENIED") [P]. https://www.supremecourt.gov/docket/docketfiles/html/public/25-449.html (read 2026-10-26). A denial is not a ruling on the merits. The human-authorship rule of the D.C. Circuit stands.
- **Training cases.** All of these concern inputs, not the copyrightability of outputs.
  - *Bartz v. Anthropic* (N.D. Cal.): Judge Alsup's order of 23 Jun 2025 held that training on lawfully acquired books was fair use ("exceedingly transformative"). Building a library of pirated copies was not fair use. The case settled for $1.5B. Final approval came on **20 Jul 2026** (3:24-cv-05417-AMO; about 482,460 works) [S: Authors Alliance https://www.authorsalliance.org/2026/07/21/bartz-v-anthropic-settlement-receives-final-approval/ ; vLex https://case-law.vlex.com/vid/bartz-v-anthropic-pbc-1129578441 ; read 2026-10-26].
  - *Kadrey v. Meta* (N.D. Cal., 25 Jun 2025, Chhabria J.): fair use on that record, because the plaintiffs showed no market dilution. The judge warned that "in many circumstances it will be illegal to copy copyright-protected works to train generative AI models without permission" [S: https://chatgptiseatingtheworld.com/2025/06/25/meta-prevails-on-fair-use-in-ai-training-in-kadrey-v-meta-but-judge-chhabria-cautions-a-better-record-of-dilution-or-market-harm-could-prevail-in-other-cases/ ; read 2026-10-26].
  - *Thomson Reuters v. Ross* (3d Cir., No. 25-2153, decided 29 Sep 2026; the opinion was first issued under seal): this is the first appellate ruling on fair use in AI training. Copying Westlaw headnotes to train a non-generative legal search tool was **not** fair use, and the headnotes were held original. The court said that generative AI is a different case [S: https://www.ballardspahr.com/insights/alerts-and-articles/2026/10/third-circuit-addresses-fair-use-in-ai-training-but-leaves-generative-ai-questions-unresolved ; https://patentlyo.com/patent/2026/10/ease-is-not-necessity-third-circuit-affirms-no-fair-use-in-thomson-reuters-v-ross.html ; read 2026-10-26].
  - *NYT v. OpenAI* (S.D.N.Y.): this is the main US case about regurgitated **outputs**. It had no decision on the merits by late October 2026 [S: https://www.nortonrosefulbright.com/en/knowledge/publications/ce8eaa5f/ai-in-litigation-series-an-update-on-ai-copyright-cases-in-2026 ; read 2026-10-26]. Search summaries said that the DOJ filed in support of OpenAI on 28 Sep 2026. UNVERIFIED: the filing was not seen.

### European Union, United Kingdom, Germany

- **GEMA v. OpenAI**, LG München I, 42 O 14139/24, 11 Nov 2025 [S]. The court held that memorisation of song lyrics in the model is a reproduction, and that so is the output of the lyrics in response to prompts. It held that the TDM exception covers making the training corpus but not "permanent embodiment of works in trained models". OpenAI was held liable for the outputs. OpenAI appealed, and no appellate decision was found [S: https://www.williamfry.com/knowledge/gema-v-openai-europes-direction-on-ai-infringement/ ; https://ailawsuittracker.com/cases/gema-v-openai/ ; read 2026-10-26]. The primary judgment text was not read (merlin.obs.coe.int and cms.law returned 403).
- **CJEU C-250/25 *Like Company v. Google*** is the first reference to the CJEU on generative AI. It asks whether chatbot output that partly matches press articles is a reproduction or a communication to the public. The Grand Chamber heard it on 10 Mar 2026. No Advocate General's opinion had been found by late September 2026, and a judgment is expected in 2027 [S: https://www.create.ac.uk/blog/2026/03/03/copyright-and-ai-case-c-250-25-like-company-before-the-grand-chamber-of-the-court-of-justice-of-the-european-union-opinion-by-the-european-copyright-society/ ; https://www.stibbe.com/publications-and-insights/ai-copyright-claims-in-europe-key-considerations ; read 2026-10-26]. UNVERIFIED: whether the opinion came out in October 2026. The CURIA page was not checked.
- **Getty Images v. Stability AI**, [2025] EWHC 2863 (Ch), 4 Nov 2025: the model weights "do not store an infringing copy". Permission to appeal was granted on 16 Dec 2025. No Court of Appeal judgment was found [S: https://www.taylorwessing.com/en/insights-and-events/insights/2026/01/next-steps-for-getty-v-stability-why-has-permission-to-appeal-been-granted ; read 2026-10-26].
- **EU AI Act, duties for GPAI providers (Art. 53 to 55).** They apply from **2 Aug 2025**. The Commission's powers to enforce and fine apply from **2 Aug 2026**. Models placed on the market before 2 Aug 2025 have until 2 Aug 2027 [P: https://digital-strategy.ec.europa.eu/en/policies/guidelines-gpai-providers ; read 2026-10-26]. The duties fall on **providers** of models: technical documentation, a copyright policy and a public summary of the training content. A user of a model's outputs has none of these duties. A "downstream modifier" becomes a provider only if its modification uses more than one third of the original model's training compute (Commission guidelines, July 2025, para 63) [S: https://digital-strategy.ec.europa.eu/en/faqs/guidelines-obligations-general-purpose-ai-providers via search summary; read 2026-10-26]. A math2 fine-tune of an open prover would be far below that threshold.
- **EU AI Act, Art. 50, transparency.** It applies from **2 Aug 2026** [S: https://artificialintelligenceact.eu/article/50/ ; read 2026-10-26].
  - Art. 50(2): providers of systems that generate text must mark outputs "in a machine-readable format ... detectable as artificially generated". This duty falls on the model provider.
  - Art. 50(4): **deployers** that publish AI-generated text "with the purpose of informing the public on matters of public interest" must disclose that the text is artificially generated. This duty does not apply "where the AI-generated content has undergone a process of human review or editorial control and where a natural or legal person holds editorial responsibility for the publication".
  - The Digital Omnibus on AI (Regulation (EU) 2026/1744, in force 27 Jul 2026) reportedly gives a grace period to 2 Dec 2026 for Art. 50(2) marking. Sources disagree on its scope [S: https://fpf.org/blog/the-ai-act-implementation-timeline-what-changes-under-the-ai-omnibus/ ; https://cdp.cooley.com/digital-ai-omnibus-delays-key-deadlines-introduces-new-rules/ ; read 2026-10-26].
  - The Commission's Art. 50 guidelines (20 Jul 2026) and the code of practice on transparency of AI-generated content (10 Jun 2026) are reported but were not read. UNVERIFIED.
  - Whether a mathematics library publishes text "on matters of public interest" is not settled. math2 already discloses AI use per field, so it meets Art. 50(4) whichever way the question is decided.

### What this means for CC BY-SA 4.0 on LLM-written text

1. **Text generated purely by an LLM has, in the US, no copyright for math2 to license.** Prompts alone are not enough (Copyright Office, Part 2). Mathematical facts, definitions and short functional statements have thin protection in any case. Applying CC BY-SA 4.0 to such text is not wrong, but it may be **inoperative**. CC BY-SA 4.0 §8(a) says that the licence does not "reduce, limit, restrict, or impose conditions on any use ... that could lawfully be made without permission", and §2(a)(2) says that "where Exceptions and Limitations apply to Your use, this Public License does not apply" [P: https://creativecommons.org/licenses/by-sa/4.0/legalcode.en ; read 2026-10-26]. Thus share-alike cannot be enforced on unedited machine text.
2. **Creative Commons' own position** [P: https://creativecommons.org/2023/08/18/understanding-cc-licenses-and-generative-ai/ ; read 2026-10-26]: "The CC license you choose will apply to the creative work that you contribute to the final product, even if the portion produced by the generative AI system itself may be uncopyrightable." It also says: "We encourage the use of CC0 for those works that do not involve a significant degree of human creativity." The 2026 CC FAQ does not answer the question for outputs (https://creativecommons.org/faq/ ; read 2026-10-26).
3. **What human work adds.** Human selection, arrangement and creative modification of the corpus are protectable. So are the human-written parts. The graph structure, the curation and the edited nodes can thus carry CC BY-SA. Unedited agent text cannot, at least in the US. In other jurisdictions the answer differs and is unsettled (e.g. the UK CDPA s.9(3) "computer-generated works" rule; not researched here).
4. **The real legal risk is regurgitation, not ownership.**
   - If an LLM reproduces a protected source (a textbook, or GFDL text from Stacks or Kerodon), the third party's copyright attaches to the output.
   - The GEMA judgment held the model provider liable for such output. Whoever republishes the output is also exposed.
   - CC BY-SA text that is reproduced closely needs attribution and share-alike. This is compatible with math2 only when the attribution is recorded.
   - GFDL and NC text are incompatible with math2.
   - Thus every generated node needs a copy check against a reference corpus. The check must cover more than the cited sources, because a node that a model writes from memory may have no cited source at all (see section 3a).
5. **Provider terms do not change this picture.** Each major provider assigns to the customer whatever rights it has in the outputs (section 4). An assignment cannot create a copyright that the law does not recognise.

## 4. Provider terms

### Hosted APIs

| Provider | Ownership of outputs | Publishing outputs | Using outputs to train other models | Source (read 2026-10-26) |
|---|---|---|---|---|
| Anthropic (Commercial Terms, effective 17 Jun 2025) | Customer "owns its Outputs". "Anthropic hereby assigns to Customer its right, title and interest (if any) in and to Outputs." "Anthropic may not train models on Customer Content from Services." | no restriction in the Commercial Terms. The AUP bans "Plagiarize or submit AI-assisted work without proper permission or attribution" and bans infringing third-party IP. | Commercial Terms D.4: may not "access the Services to build a competing product or service, including to train competing AI models". **AUP (effective 12 Nov 2026): may not "Use outputs to train an AI model (e.g., "model scraping" or "model distillation") without prior authorization from Anthropic".** | https://www.anthropic.com/legal/commercial-terms ; https://www.anthropic.com/legal/aup |
| OpenAI (Services Agreement ONLINE v.010126) | §4.1: Customer "owns all Output. OpenAI hereby assigns to Customer all OpenAI's right, title, and interest, if any, in and to Output." §4.4: "Output may not be unique, and other users may receive similar content". | no restriction found in the agreement | §3.3(e): may not, "except for a Permitted Exception, use Output to develop artificial intelligence models that compete with OpenAI's products and services". The Permitted Exceptions are (a) classifiers and embeddings that are "not distributed or made commercially available to third parties", and (b) fine-tuning on OpenAI's own services. | https://cdn.openai.com/osa/openai-services-agreement.pdf (openai.com returned 403; the CDN PDF was read with pdftotext) |
| Google (Gemini API Additional Terms, last updated 2026-04-28) | "Google won't claim ownership over that content." "Google may generate the same or similar content for others." | "You're responsible for your use of generated content, and for the use of that content by anyone you share it with." Applicable law "may require the provision of attribution to your users when returned as part of an API call". | "You may not use the Services to develop models that compete with the Services (e.g., Gemini API or Google AI Studio)." | https://ai.google.dev/gemini-api/terms |

The consequences for math2:

- **Publishing is allowed by all three providers.** None of them claims the outputs, and none forbids publication under an open licence.
- **The training clauses bind math2, and math2's downstream users, differently.**
  - math2 itself may not use Claude outputs to train or fine-tune a model without Anthropic's authorisation, under the AUP that takes effect on 12 Nov 2026. This rules out a math2 "fine-tune an open prover on our Claude-written proofs" plan unless it is authorised. The same holds for OpenAI and Google outputs in any model that "competes".
  - Third parties who later download math2 under CC BY-SA have no contract with Anthropic. Whether the AUP reaches them is a contract question, not a copyright question. The licence of math2 cannot pass a training ban on to them, and it should not try to.
- **Dataset releases need provenance.** The model id recorded for each field (06.5) lets a downstream user filter the corpus by model family when that user's own terms require it.
- **Indemnities** apply only to the customer's paid use. They are not a reason to skip copy checks.
  - Anthropic Commercial Terms K.1: Anthropic defends claims "alleging that Customer's paid use of the Services" violates third-party IP.
  - OpenAI reportedly extends its indemnity to output in its Service Terms of 12 Jun 2026 [S, not read].

### Open-weight model licences, checked on Hugging Face and GitHub on 2026-10-26

| Family (newest seen) | Licence | Extra duties on outputs or on models trained with outputs |
|---|---|---|
| Llama 4 (Scout/Maverick; no newer Meta LLM on HF) | Llama 4 Community License | **Yes.** "If you use the Llama Materials or any outputs or results of the Llama Materials to create, train, fine tune, or otherwise improve an AI model, which is distributed or made available, you shall also include "Llama" at the beginning of any such AI model name." "Built with Llama" must be displayed when such a model or derivative is distributed. Over 700M MAU needs a separate licence. The licence ends if you sue Meta claiming that Llama outputs infringe. https://raw.githubusercontent.com/meta-llama/llama-models/main/models/llama4/LICENSE |
| DeepSeek V4 / V4.1 (Flash, Pro; Sep to Oct 2026) | MIT (HF tag `license:mit`) | none. |
| DeepSeek-Prover-V2 (Apr 2025) | **conflicting.** The HF LICENSE file is MIT, but the README says "subject to the Model License" (DeepSeek Model Agreement). | The DeepSeek Model Agreement defines "Derivatives of the Model" to include any model trained on "synthetic data [generated] by the Model". Use restrictions (Attachment A) flow down to derivatives. §6: "DeepSeek claims no rights in the Output". https://raw.githubusercontent.com/deepseek-ai/DeepSeek-V3/main/LICENSE-MODEL |
| DeepSeek-Math-V2 (Nov 2025) | Apache-2.0 | none |
| Qwen3.5 (Feb 2026) | Apache-2.0 | none |
| **Qwen3.8 (Aug 2026)** | **Qwen Community License 1.0** (new in 2026) | Commercial products with more than 100M MAU or more than US$20M monthly revenue must display the model name. A "Model as a Service" or "AI Work Assistant" business needs a separate licence. "Internal Use" is exempt "provided that such Use does not make the Software, its outputs, or its underlying model capabilities available to any third party". It is unclear whether publishing a corpus of outputs is a non-internal use. https://huggingface.co/Qwen/Qwen3.8-Flash-Next-FP8/raw/main/LICENSE |
| Kimi K2 (2025) | Modified MIT | display "Kimi K2" above 100M MAU or $20M monthly revenue |
| **Kimi K3 (Jun 2026)** | Kimi K3 License | "Model as a Service" businesses above $20M revenue in 12 months need an agreement. Display "Kimi K3" above the MAU or revenue thresholds. The exemption for internal use is again tied to not making "the Software, its outputs, or its underlying capabilities available to third parties". https://huggingface.co/moonshotai/Kimi-K3/raw/main/LICENSE |
| GLM-4.6 / GLM-5 (Feb 2026) | MIT | none |
| Mistral Large 3 (Dec 2025), Leanstral 1.5 (Jul 2026) | Apache-2.0 | none |
| gpt-oss-120b/20b (Aug 2025) | Apache-2.0 plus a one-line usage policy ("comply with all applicable law") | none |
| Goedel-Prover-V2 | Apache-2.0 | none |
| Kimina-Prover-72B | MIT | none |

- In 2026, the Chinese labs moved from plain MIT or Apache to custom licences: Qwen 3.8 and Kimi K3.
- For a corpus that math2 publishes, the duties that matter are these. Llama's naming duty applies only if someone trains a distributed model on Llama outputs. The DeepSeek Model Agreement flows its restrictions down to any model trained on the outputs. The internal-use exemptions of Qwen 3.8 and Kimi K3 do not cover outputs made available to third parties, so the MaaS and display thresholds apply. math2 is far below the thresholds, but the terms should be recorded per model.
- MIT, Apache-2.0 and the gpt-oss terms add nothing.

## 5. Cost

### List prices on 2026-10-26 (USD per million tokens; official pages only)

Anthropic, from https://platform.claude.com/docs/en/about-claude/pricing (read 2026-10-26):

| Model | Input | Cache hit | Output | Batch input | Batch output |
|---|---|---|---|---|---|
| Claude Fable 5.1 | $10 | $0.25 (0.025×) | $50 | $5 | $25 |
| Claude Opus 5.5 | $4 | $0.20 (0.05×) | $20 | $2 | $10 |
| Claude Sonnet 5.5 | $2 | $0.10 (0.05×) | $10 | $1 | $5 |
| Claude Haiku 5.5, prompt ≤100k tokens | $0.10 | $0.01 | $0.50 | $0.05 | $0.25 |
| Claude Haiku 5.5, prompt >100k tokens | $0.50 | $0.05 | $2.50 | $0.25 | $1.25 |

- **Cache writes:** 1.25× input for a 5-minute TTL and 2× for 1 hour. "These multipliers stack with other pricing modifiers, including the Batch API discount".
- **Long context:** "Claude 4.6 and later models (except Claude Haiku 5.5) ... include the full 1M token context window at standard pricing."
- **Data residency:** `inference_geo: "us"` costs 1.1×.
- **Fast mode** for Opus 5.5 costs $8/$40, and it is not available with Batch.
- **Batch API** (https://platform.claude.com/docs/en/build-with-claude/batch-processing ; read 2026-10-26):
  - a 50% discount;
  - up to 100,000 requests or 256 MB per batch, "most batches completing within 1 hour", expiry at 24 h, results kept 29 days;
  - "cache hits are provided on a best-effort basis. Users typically experience cache hit rates ranging from 30 percent to 98 percent".
- The prices in spec 12.3 match this page.

OpenAI, from https://developers.openai.com/api/docs/pricing (platform.openai.com redirects there; read 2026-10-26), short context ≤272K input:

| Model | Input | Cached | Output | Batch in / cached / out |
|---|---|---|---|---|
| gpt-6-astra | $10 | $1.00 | $50 | $5 / $0.50 / $25 |
| gpt-6.1-sol | $2 | $0.10 | $10 | $1 / $0.05 / $5 |
| gpt-6-luna | $0.10 | $0.01 | $0.50 | $0.05 / $0.005 / $0.25 |

- Long context costs more: gpt-6-astra $20/$75, gpt-6.1-sol $4/$15.
- Flex rates equal the Batch rates.

Google, from https://ai.google.dev/gemini-api/docs/pricing ("Last updated 2026-10-22 UTC"; read 2026-10-26):

| Model | Input | Output | Batch |
|---|---|---|---|
| Gemini 4 Argon | $2.00 until 30 Nov 2026, then **$4.00** from 1 Dec 2026 | $10.00, then **$20.00** | 50% |
| Gemini 3.8 Flash | $0.75 until 31 Dec 2026, then $1.50 | $3.75, then $7.50 | 50% |
| Gemini 3.5 Flash-Lite | $0.30 | $2.50 | 50% |

Google's prices for cached input were not read.

### An estimate for one million nodes

The token counts below are **assumptions, not measurements**. Calibrate them on the first 1,000 real runs (the cost per merged output in 12.9). Tokenizers differ between providers, so the same text costs a different number of tokens.

- **Generation call:** 10k input, of which 8k is a cached shared prefix (system prompt, style guide, neighbourhood) and 2k is unique. 6k output, including thinking.
- **Three check calls** (grounding or judge, critic, back-translation or consistency), each with 8k input (2k cached, 6k unique) and 2k output.
- All calls use Batch. Cache reads are assumed to hit.

| Configuration | $ per node | $ per 1M nodes |
|---|---|---|
| all Claude Haiku 5.5 | 0.004 | ~$4k |
| all Claude Sonnet 5.5 (≈ all gpt-6.1-sol) | 0.081 | ~$81k |
| all Claude Opus 5.5 | 0.161 | ~$161k |
| all Claude Fable 5.1 (≈ all gpt-6-astra) | 0.40 | ~$400k |
| generate on Opus 5.5; check with 2 Sonnet 5.5 calls and 1 gpt-6.1-sol call (cross-family) | 0.113 | ~$113k |
| generate on Fable 5.1; checks as above | 0.209 | ~$209k |
| all Sonnet 5.5, zero cache hits | 0.094 | ~$94k |
| all Gemini 4 Argon, before / after 1 Dec 2026 (no cache price assumed) | 0.094 / 0.188 | ~$94k / ~$188k |

- **Divide by the acceptance rate.** At a 50% first-pass acceptance rate, the cost doubles.
- **Output tokens dominate:** 70 to 85% of the cost in every configuration. The thinking budget is the lever that matters most, not the size of the context.
- **Formal work is not in this table.** A Lean statement with its F1 battery, and above all a Lean proof, costs orders of magnitude more per item (section 2). Price it separately.
- At one million nodes, informal generation costs tens to hundreds of thousands of dollars. The cost is much smaller than the review capacity that math2 would need for the output (10.6), and much smaller than formal proof.

### 5.1 Model facts that bear on scale

Source: https://platform.claude.com/docs/en/about-claude/models/overview [P].

- **Model ids:**
  - `claude-fable-5-1`
  - `claude-opus-5-5`
  - `claude-sonnet-5-5`
  - `claude-haiku-5-5`
- **Context and output:** a 1M-token context window, and up to 128K output tokens per request. On the Batch API, output can go to 300K with the beta header `output-300k-2026-03-24`.
- **Training cutoff:** reliable knowledge and training data go to **June 2026**. Mathematics published after that date is unknown to the models unless it is retrieved into the prompt. This bears on explore and novelty checks (16.5 item 5): a model's "this is new" is worth nothing for results after its cutoff.
- **Retirement:** each model is retired no sooner than 1 Sep, 22 Sep, 28 Sep and 7 Oct 2027 respectively. A million-node campaign that runs past those dates will cross a model change. This is a further reason to record exact model ids (12.3) and to keep `m2 prov quarantine` working.
- **Size rule of thumb:** 1M tokens is about 555k words, or about 2.5M characters.
  - So a node of 1,500 words of prose is about 2.7k tokens, before LaTeX. LaTeX tokenizes worse than prose.
  - The token assumptions in 5 are plausible, but they still need calibration.

## 6. Community norms on LLM text (short)

- **Wikipedia.** Speedy-deletion criterion G15 for unreviewed LLM-generated pages was adopted in August 2025, and later renamed "Unambiguously LLM-generated pages". https://en.wikipedia.org/wiki/Wikipedia:Speedy_deletion [S, the policy page as summarised; the adoption discussion is in the talk archives]. WikiProject AI Cleanup exists to find and fix such text (https://en.wikipedia.org/wiki/Wikipedia:WikiProject_AI_Cleanup) [S].
  - Wikipedia is a source that math2 imports under CC BY-SA (16.6). math2 must not push unreviewed agent text back into it (16.5 "Upstream etiquette" already forbids bulk writes).
- **arXiv.** On 31 Oct 2025, arXiv said that review articles and position papers in the CS category must have been peer-reviewed before submission, citing the flood of LLM-written submissions. https://blog.arxiv.org/?p=1740 [S, read via summary].
- **Grokipedia** (section 2.4) is the large-scale counter-example: 56% of its articles carry Wikipedia's CC footer, and no error rate is published.

These norms agree with spec 16.5. They add one point: **communities sanction unreviewed machine text in bulk, not the use of AI as such.** math2's protection is the marker plus human confirmation (P04). It should not ship unconfirmed prose as if it were reviewed.

## 7. What the spec should require because of this

The numbers in brackets point to the evidence above. Each item names the spec section it changes.

### 7.1 Quality and checking (spec 10, 12, 15)

1. **No LLM verdict is evidence of correctness for an informal proof (12.6, 16.5 item 4).**
   - Evidence: LLM judges inflate scores up to 20x [1.2]. The best step judges accept about 21% of wrong steps, and weaker judges 56 to 90% [1.2].
   - Requirement: agent-written informal proofs stay E1, and a judge pass is recorded as triage only. The spec should say this explicitly for judges, as it already does for proofs.
2. **No Haiku-tier or weak open-model judge for proofs or statements (12.3).**
   - Evidence: Hard2Verify finds that "weaker verifiers cannot catch errors". Claude Sonnet 4-class judges flag only 43.7% of wrong steps [1.2].
   - Requirement: limit tier 0 to triage, classification and dedup pre-filters. Proof and statement judging starts at the tier that the run records show reaches a measured true-negative rate, for example 75% or more on a labelled set.
   - Grayzel's result supports a cheap cross-family consensus [1.2]. Treat that as something to measure, not to assume.
3. **Judges vote across families; accept by unanimity, reject by majority (10.4 check 9, 12.6).**
   - Evidence: a unanimous vote gives the best precision, and a majority vote the best recall [1.2, 1.5]. Cross-model agreement is the best citation check that has been measured [1.5].
   - Requirement: record each judge's verdict and model id in the run record. Disagreement goes to a human, and is never settled by a tiebreak call.
4. **An anti-sycophancy check in prove and formalize (10.4, 12.4).**
   - Evidence: strong models "prove" false statements 29 to 97% of the time [1.1]. A "prove or disprove" framing raised the rejection rate of false statements from 18.5% to 71% [1.5].
   - Requirements:
     - (a) The prompt of every prove and formalize job is framed as "prove or disprove".
     - (b) A periodic canary: inject perturbed false statements into the queue, and measure how many the agents "prove". Publish the rate in the weekly report (12.9).
     - Check 6 of the F1 battery (cheap negation) only catches cheaply false statements. It does not catch the subtle ones that BrokenMath and BrokenArXiv use.
5. **Compiling and back-translation are not fidelity (10.4).**
   - Evidence: Kimina compiles 90% of statements but only 66% are accurate. Unsupervised autoformalization is faithful 60 to 90% of the time [2.3], and less often on graduate material.
   - Requirements:
     - Set the prior error rate for F1 at 10 to 40% until math2's own sampled re-review (10.8) measures it.
     - Do not treat F1 as "probably right" in the workspace.
6. **Measure the informal error classes that no one has measured (10.8, 17.3).**
   - Evidence: there is no published rate for wrong definitions, missing hypotheses or misnamed theorems in informal LLM text [1.3].
   - Requirement: add an informal audit beside the F2 re-review. Each month, a blind expert audit of a seeded sample of agent-written definitions and statements, classified with M-* style classes for informal text:
     - wrong definition;
     - dropped hypothesis;
     - misattributed name;
     - non-standard convention unstated.
   - Without this, the published error rates in release notes cover only formal bindings.
7. **Source records written by agents must be resolved and checked (04 `sources:`, a new rule in G or P).**
   - Evidence: 11.4 to 56.8% of generated citations are hallucinated [1.4]. Usable theorem references in research proofs run at 7% or less [1.3]. Fabrication is worst on unfamiliar topics [1.4].
   - Requirement: a hard rule. A `sources:` entry added by an agent must resolve to a real record (DOI, zbMATH, arXiv id, ISBN, or a record-plane entry), and the resolution is logged. A locator ("Thm 3.1") written by an agent counts toward E2 or E3 only after it is checked against retrievable text, or after a person confirms it.
   - Also: theorem names in prose ("by the X–Y theorem") must link to a node or carry a source. A name that resolves to nothing fails.
8. **Prefer pseudo-formal structure for informal proofs (12.4, 04).**
   - Evidence: pseudo-formalization raised error recall from 0.54 to 0.66, but still missed about 40% of errors on arXiv proofs [1.5].
   - Requirement: agent-written informal proofs use a step-structured format with each step's justification as a link or a citation. This helps the judge, but it does not change E1.
9. **Watch for checker gaming (11.3, X rules).**
   - Evidence: in FormalQualBench, agents tried elaborator-level workarounds [2.1]. Aristotle's results were flagged as "not Comparator-validated" [2.1].
   - Requirement: the comparator-pinned compile (check 10) and the axiom audit are mandatory for every accepted proof, with no exception for "trusted" provers.

### 7.2 Scale planning (spec 12.3, 12.9, 10.6)

10. **Budget review, not generation.**
    - Evidence: a million informal nodes costs about $4k to $400k in model calls, depending on the tier mix [5]. The published formal campaigns cost about $17 per declaration, or $300 per target theorem [2.1]. No large agent run has published a misformalization rate [2].
    - Requirements:
      - The spec's throughput targets (10.6) must be set against review capacity.
      - The weekly report should publish **cost per accepted node and per F2 binding**, and not only spend.
11. **Publish a fidelity rate before claiming scale.** No prior project publishes a misformalization rate for its agent-written library [2]. math2's sampled re-review (10.8) is a differentiator. The spec should require that a release with more than N agent-written bindings is not tagged until the sample has been drawn and measured.
12. **Plan for model churn.**
    - Facts: the current models retire from Sep 2027, and their knowledge stops at June 2026 [5.1].
    - Requirements:
      - Novelty checks (16.5 item 5) must retrieve, and not rely on model memory.
      - Every campaign record states the model ids, so that `m2 prov quarantine` can act on them.

### 7.3 Copying and licence (spec 15 L-rules, 16.6)

13. **Extend L04: compare with a reference corpus, not only with cited sources (15 L04).**
    - Evidence: LLMs reproduce text without being asked. Over 27% of encyclopedia-style output sits in verbatim spans of 50 characters or more. One case reproduced 1,170 characters of Wikipedia [3a.1]. Text a model writes from memory has no cited source to compare against.
    - Requirement: a new rule `L05` (S, rising to H above a hard threshold), run on every agent-written prose field. Check it against an indexed reference corpus:
      - Wikipedia math, ProofWiki, nLab and PlanetMath;
      - Math.SE and MathOverflow, under the per-post rules of 16.6;
      - **Stacks and Kerodon (GFDL)**;
      - OpenWebMath or arXiv text, for detection only;
      - any licensed textbook index.
    - Parameters to start with, and then to calibrate [3a.5]:
      - Text normalization: LaTeX normalized and formulas masked.
      - Exact matching: flag a span of 50 characters or more; hard-fail at 100 characters or more against a GFDL, NC or all-rights-reserved source; flag at 10% or more covered.
      - Near-duplicate matching: 5-gram MinHash with Jaccard 0.5 as the candidate threshold, confirmed with ROUGE-L 0.8 or more, or with difflib blocks of 20 words or more.
      - A stock-phrase allowlist.
    - Every threshold is calibrated on human-written math pages, whose median overlap is 0% [3a.1].
14. **An exact-match filter is not enough.**
    - Evidence: a 10-gram filter blocks verbatim copying while near-verbatim text gets through, and fuzzy duplicates count about 0.8 of exact ones [3a.2].
    - Requirement: L05 needs a near-duplicate stage, and a paraphrase stage that is sampled and sent to humans, not blocking.
15. **Record match spans.** For each L05 hit, record the source, URI, licence and offsets in provenance. This follows Gemini's citation metadata [3a.4]. A hit against a CC BY-SA source becomes an attributed import (L03). A hit against GFDL, NC or arXiv-default text is rewritten (L02).
16. **Anti-reproduction instructions in every prose prompt (12.8 prompts).**
    - Evidence: such a system prompt cut mean overlap by more than half in one model, but long matches remained [3a.1].
    - This is a cheap first layer, not a control.
17. **Licence the unedited agent text honestly (16.6).**
    - Evidence: in the US, purely AI-generated text has no copyright, and CC BY-SA cannot bind uses that need no permission (§8(a), §2(a)(2)) [3b]. CC itself suggests CC0 for works without significant human creativity [3b].
    - The spec should choose one of these:
      - **(a)** Keep CC BY-SA for all prose. Add a notice that the licence applies to the extent that copyright subsists, and that agent-written, unconfirmed fields may be in the public domain in some jurisdictions. Provenance (P04) shows which fields these are.
      - **(b)** Release unconfirmed agent prose under CC0 until a person edits or confirms it. It then becomes CC BY-SA.
    - Option (a) is simpler and keeps one prose zone. Option (b) is more honest, but makes licence zones depend on provenance. **Recommendation: (a) with the notice.** The collection, its selection and arrangement, and human-edited text stay under CC BY-SA in any case.
    - Text an LLM regurgitates from a CC BY-SA source keeps that source's terms whatever math2 says. This is why item 15 matters.
18. **Art. 50(4) disclosure is satisfied by provenance (16.5).** EU deployer disclosure for published AI text applies from 2 Aug 2026, unless a human reviewed the text and someone holds editorial responsibility [3b]. math2's per-field markers (P04) and named responsible persons (P03) cover both routes. The spec should say this in 16.5 item 7, and the site should carry a machine-readable AI-generated flag on unconfirmed fields.

### 7.4 Provider and model terms (spec 12.3, 20.2, 06.5)

19. **No training on hosted-model outputs without authorisation (12.3, 20.2).**
    - Facts:
      - Anthropic's AUP, in force from 12 Nov 2026, forbids using outputs to train an AI model without prior authorisation.
      - OpenAI and Google forbid using outputs to build "competing" models [4].
    - Requirement: any plan to fine-tune the open provers of tier 0′ on math2 data must exclude, or get authorisation for, fields generated by hosted models. Provenance must make that filter mechanical. Add it to the provenance tests of 06.5: "list each field produced by model family X".
20. **Record each model's licence and its output terms (12.3 price table, a new `models.reg`).**
    - Several open-weight licences have output clauses:
      - Llama 4 naming;
      - the DeepSeek Model Agreement derivatives clause, and the LICENSE file that conflicts with the README for DeepSeek-Prover-V2;
      - the Qwen 3.8 and Kimi K3 internal-use exemptions, which exclude outputs made available to third parties [4].
    - Requirement: a register of models with these fields: licence, URL, date read, output clauses, and thresholds. The orchestrator refuses a model with no entry.
    - Prefer MIT or Apache models for tier 0′: DeepSeek V4, DeepSeek-Math-V2, Goedel-Prover-V2, Kimina, Leanstral, GLM, gpt-oss.
21. **math2's licence does not pass provider training bans on to users (16.6).** A CC BY-SA licence cannot add a no-training term, and math2 should not try. The release notes can state which fields came from which provider. Downstream users with their own contracts can then filter.

### 7.5 Cost accounting (12.3, 12.9)

22. **Price table and prices in `llm.Prices` must be dated, and they must model batch, cache read, cache write (1.25x / 2x), the Haiku >100k tier, and residency (1.1x)** [5].
    - Google's Gemini 4 Argon price doubles on 1 Dec 2026, and Gemini 3.8 Flash on 1 Jan 2027 [5]. A static table goes stale within weeks.
23. **Thinking tokens are the cost lever.** Output, including thinking, is 70 to 85% of the modelled cost [5]. The weekly report should give output tokens per accepted node for each role and tier. Effort settings (12.3) should be tuned on that number.

## 8. Open doubts

1. **No measured error rate for informal LLM definitions and statements.** Section 1.3 found none. The prior in 7.1 item 5 is borrowed from formal autoformalization, and may be wrong in either direction for prose.
2. **The mathematical reproduction rate is unmeasured.**
   - The figure of over 27% at 50 characters is for encyclopedia prose in general, and was measured on 2024 models against a 2022 web snapshot.
   - Math prose may be higher, because of stereotyped wording, or lower, because formulas are masked.
   - The L05 thresholds in 7.3 are untested on mathematics.
3. **What "copying" means for mathematical statements.** Standard definitions have few reasonable wordings, and the merger doctrine and thin protection may apply. A strict L05 could flood review with false positives. Calibration on human-written pages is required before L05 can block.
4. **Copyright in AI output outside the US.**
   - The UK CDPA s.9(3) "computer-generated works" rule and EU originality law were not researched.
   - The CJEU judgment in C-250/25 (expected 2027) and the GEMA appeal may change the EU position on memorized output.
5. **Whether the Anthropic AUP's training ban reaches third parties** who download math2. Their only relation is through CC BY-SA, and not through any contract. Legal advice is needed before math2 makes any claim either way.
6. **DeepSeek-Prover-V2's licence conflict** (an MIT LICENSE file, against a README pointing to the Model Agreement). Which one governs is unclear. Ask DeepSeek, or avoid using its outputs for training.
7. **Qwen 3.8 and Kimi K3 "internal use".** It is not clear whether publishing a corpus of outputs is a use that "makes outputs available to third parties" in a way that triggers the MaaS clauses. math2 is far below the revenue thresholds, but the reading is uncertain.
8. **Art. 50(4) scope.** Whether a mathematics library publishes "matters of public interest" is unsettled. The Commission's Art. 50 guidelines (20 Jul 2026) and the transparency code of practice were reported but not read (UNVERIFIED).
9. **Cost model.**
   - The token counts are assumptions, and cache-hit rates on Batch are "best-effort", between 30 and 98%.
   - Acceptance rates are unknown until math2 runs. The true cost per accepted node could be 2 to 5 times the table.
   - Formal work is priced only by analogy, at $17 per declaration in Automatic Textbook Formalization.
10. **Judge evidence comes from older models.** Most measurements are of 2025 models (GPT-5, Gemini 2.5, Claude Sonnet 4 and 4.5, Opus 4.5 and 4.6). Opus 5.5 and Fable 5.1 have no published proof-judging measurements that we found. math2 must measure its own judges on a labelled set before item 2 of 7.1 can be settled.
11. **Self-verification evidence conflicts.** Hard2Verify finds models more lenient on their own work, and OPC finds them harsher. Whether the generator may also act as a judge must be measured, not assumed. The safe default is a different family, as 10.4 check 9 already says.
12. **Snippet-only and derived figures** throughout sections 1 and 2 must be re-checked against the primary PDFs before any is quoted in the spec. This applies above all to the First Proof batch 2 tally, BrokenArXiv's later figures, Seed-Prover 1.5's 580/660, and the "500 TPU-days per problem" attributed to AlphaProof.
13. **No Anthropic or OpenAI recitation filter is documented.** We cannot rely on the provider to block regurgitation. Gemini has a RECITATION finish reason, but its threshold is not published.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/01e-representation-and-packaging.md">
# K01, part E. Representation, pedagogy, exercises, packaging and notation

Research for the Math 2.0 spec (tamnd/math2). This file builds on spec 01 §1.7 (semantic formats), 01 §1.2 (blueprints), 04 (object model) and 13 (research workspace), and tries not to repeat them. Facts those documents already state are only referred to.

Rules for this file:
- I read every source on **2026-10-26**, unless the entry says otherwise.
- A fact I could not confirm is marked **UNVERIFIED**.
- "From search" means I saw the claim only in a search-result excerpt, not on the page itself.

---

## 0. What the spec already says (so it is not repeated)

**Spec 04:**
- **Node kinds:** concept, definition, statement, conjecture, problem, example, counterexample, object, proof, computation, topic.
- **Tags:** 6-character Crockford base32, permanent.
- **Body sections:** fixed per kind.
- **Explanations:** levels L1 to L4, each marked human-written or agent-written.
- **Math and links:** math is LaTeX limited to the macros in `macros/m2.sty` and `m2.katex.json`; links are `[[text|TAG]]`.
- **Status:** computed, never written by hand.

**Spec 13:**
- a fixed page model;
- prerequisite paths that start from the reader's background;
- a "Translate notation" feature;
- project blueprints generated from the graph.

**Spec 01 §1.7 already covers:**
- MathML 4 is still a Working Draft and has `intent`;
- MathML Core is a CR (June 2025);
- OpenMath 2.0 is inactive;
- sTeX4 = RusTeX + FLAMS + MathHub;
- arXiv HTML is made with LaTeXML;
- Wikidata math counts and P14534;
- MathGloss.

**Spec 01 §1.2 already covers:** leanblueprint, Verso Blueprint, LeanArchitect, Naproche and Megalodon.

What follows is new detail, plus a decision for each item.

---

## 1. Representation

### 1.1 OMDoc and MMT

**What it is**
- OMDoc is an XML format for mathematical documents. It has three levels:
  - object: formulas, in OpenMath or Content MathML;
  - statement: definitions, theorems, proofs, examples;
  - theory: theories, imports and views.
- The project started in 1998.
- KWARC says "the formal part of OMDoc has been refined into the MMT format", with the "MMT System as the reference implementation". KWARC is "extending MMT to cover the full flexiformal coverage of OMDoc".
- Source: https://kwarc.info/systems/omdoc/

**MMT in 2026**
- MMT is written in Scala and stores content in "archives".
- Repo: https://github.com/UniFormal/MMT. It has 7,458 commits and no releases listed on GitHub.
- The licence is in `COPYING.txt`. I did not read it, so the licence is **UNVERIFIED**.
- KWARC is now at FAU Erlangen-Nürnberg.

**Take for math2**
- The **theory/view** idea: content lives in theories, and a view maps one theory into another (§4.6, Realms).
- The **object/statement/theory** split matches math2's formula / node / topic split. math2 needs no XML for this.
- Do not adopt OMDoc as a storage format. Its tooling is research-grade.

### 1.2 sTeX and SMGloM

**sTeX 4**
- Version 4.1.0, dated 2026-06-24, under LPPL. Maintainers: M. Kohlhase and D. Müller. https://www.ctan.org/pkg/stex
- The toolchain (https://kwarc.info/systems/sTeX/):
  - RusTeX converts LaTeX to FTML (HTML with semantic annotations);
  - FLAMS, the "Flexiformal Annotation Management System", is GPL-3.0 (https://github.com/FlexiFormal/flams);
  - MathHub hosts the archives;
  - there is a VS Code IDE.
- The older LaTeXML plugin is deprecated.
- mathhub.info now shows a FLAMS-branded interface.

**Syntax**

From the manual (https://ctan.math.illinois.edu/macros/latex/contrib/stex/doc/stex-manual.pdf). I saw these in excerpts, not by reading the whole PDF.
- **Modules:** `\begin{smodule}{Name}` opens a module. A symbol must belong to a module.
- **Symbols and notations:**
  - `\symdecl{name}` declares a symbol, which is an identity with no notation. `\symdecl*` declares one without a semantic macro.
  - `\notation{name}[variant]{…}` attaches a notation to a symbol. A symbol can have several notations.
  - `\symdef` is `\symdecl` plus `\notation` in one command. Example:
    `\symdef{plus}[name=addition,args=2,type=\funspace{\Nat,\Nat}{\Nat}]{#1 \mathbin{\maincomp{+}} #2}`
  - `prec=` sets precedence for bracketing. `args=a` marks a flexary (associative) argument.
- **Variables:** `\vardef` declares a variable.
- **Imports:**
  - `\importmodule` is transitive. Example: `\importmodule[sTeX/MathBase/Functions]{mod?Function}`.
  - `\requiremodule` is not transitive.
  - `\usemodule` is for use outside a module.
- **Definitions:** `sdefinition` with `for=`, plus `\definiendum` / `\definame`. `\defnotation` marks where a notation is introduced.
- The syntax changed incompatibly on 2023-04-25. This shows the cost of tying content to a LaTeX package's macro API.

**SMGloM**
- The CICM 2022 sTeX3 paper (https://kwarc.info/people/dmueller/pubs/cicm22stexsd.pdf) reports:
  - at least 2,250 concepts;
  - verbalisations in English for 93% of them, German 71%, Chinese 11%.
- **Signature modules** (language-independent symbols, "modsig") are separate from **natural-language modules** ("mhmodnl"), one per language.

**SMGloM licences**
- The early SMGloM Public License (SPL 0.1) "protects symbols against non-conservative changes while allowing derived works". https://kwarc.info/people/mkohlhase/submit/cicm14-smglom-system.pdf
- The current `smglom/mathsoft` archive is CC BY-SA 4.0: https://gl.mathhub.info/smglom/mathsoft/-/blob/main/LICENSE
- I did not check the other SMGloM archives. That they share this licence is **UNVERIFIED**.

**ALeA (Adaptive Learning Assistant, KWARC)**
- ALeA is built on sTeX content. It was funded through the VoLL-KI project (BMBF, 2021 to 2025).
- It keeps a learner model with values M(C,D) in [0,1] for each concept C and each cognitive dimension D (Bloom-like).
- It has been used in 6 courses with more than 1,000 FAU students. The platform is at https://courses.voll-ki.fau.de/.
- **Guided tours:** the system lists all prerequisites of a target, removes the ones the learner already knows, and shows the rest as a sequence of definitions. It also generates flashcards and quizzes.
- Sources: https://sme.uni-bamberg.de/ai4ai/p5.pdf ; CICM 2023 paper, https://www.researchgate.net/publication/373539971

**Take for math2**
1. **Separate a symbol's identity from its notations.** math2 macros are currently only render macros. Each macro should declare the concept tag it denotes, its arity and its precedence. One concept may have several notations, and one of them is the default.
2. **Keep the language-independent part apart from the prose part.** Spec 13 §13.4 already keeps formal statements untranslated. The SMGloM split suggests also keeping a per-concept "verbalisation" record per language: the noun phrase, plural and gender.
3. **ALeA's guided tour** is close to the spec 13 prerequisite path. ALeA also tracks a cognitive dimension per concept, and math2 can do the same with the MATH-taxonomy groups (§2.2).
4. Do not depend on the sTeX macros themselves (LPPL, API churn). Export to sTeX/FTML only if someone asks for it.

### 1.3 OpenMath Content Dictionaries

**What it is**
- 218 Content Dictionaries (CDs) are listed at https://openmath.org/cdnames/. Each has a status: official, experimental, private or obsolete.
- `arith1` (version 3, 2004-03-30) defines abs, divide, gcd, lcm, minus, plus, power, product, root, sum, times and unary_minus. https://openmath.org/cd/arith1
- `transc1` follows Abramowitz and Stegun for branch cuts. https://openmath.org/cd/transc1
- The repo https://github.com/OpenMath/CDs was last updated 2022-06-28.
- Content MathML's `cd` / `cdgroup` mechanism refers to OpenMath CDs.

**Licence**

The CD licence is custom:
- verbatim redistribution is allowed;
- a derived work must say that it is derived and is not the original;
- a derived CD must use a different CDName, for example `private_mathN` instead of `mathN`.

There is also the OpenMath royalty-free IPR policy of 2013-12-01: https://openmath.org/documents/openmath-ipr/

**Take for math2**
- Use OpenMath symbol URIs (`cd/name`, for example `arith1/plus`) as **alignment targets** for concept nodes, next to Wikidata QIDs. They are stable and well defined, they include branch-cut conventions, and they are what Content MathML refers to.
- Do not copy CD text into nodes, because of the renaming requirement. Link only.

### 1.4 MathML 4 (presentation, content, intent)

**Status**
- The Editor's Draft is dated 24 October 2026: https://w3c.github.io/mathml/
- The latest published version is still a Working Draft (agrees with spec 01 §1.7). The w3.org/TR page returned HTTP 403 to my fetch, so I read the editor's draft and the charter instead.

**Charter**

The proposed 2026 charter is at https://w3c.github.io/mathml-docs/charter-2026.html.
- **Targets in the timeline:**
  - MathML 4 CR, May 2026;
  - a community intent repository, August 2026;
  - MathML Core Recommendation, December 2026;
  - MathML 4 Recommendation, March 2027.
- **Deliverables table:** Core Recommendation about Q4 2026; MathML 4 complete about Q1 2027.
- **The May 2026 CR target was missed.** As of 2026-10-26 the spec is still a WD.
- **Content MathML:** MathML 4 "largely excludes" new Content MathML work, which is deferred to MathML 5.
- **i18n review:** ran from 2026-03-18 to 2026-05-14. https://github.com/w3c/i18n-request/issues/299

**Intent**
- Syntax example: `intent="power($base,$exponent)"`, with `arg="base"` on the child elements.
- Intent concepts come from two lists in the w3c/mathml-docs repository:
  - a **Core** list: https://w3c.github.io/mathml-docs/intent-core-concepts/ (from search);
  - an **Open** list, to which contributions are invited and which "will never be complete" (Soiffer et al., https://arxiv.org/abs/2407.06720, from search).
- There is also a Core *property* list generated from `core-properties.yml`: https://w3c.github.io/mathml-docs/intent-core-properties/ (from search).
- Intent is meant for speech and assistive technology. It is not a semantic definition.

**LaTeX support**
- The LaTeX Project ships `latex-lab-mathintent` with `\MathMLintent` and `\MathMLarg` (v0.1c, 2026-01-16). https://ftp.icm.edu.pl/packages/tex/macros/latex-dev/required/latex-lab/latex-lab-mathintent.pdf

**Take for math2**
1. Each notation declaration (§1.2, §5) gets an optional `intent:` field holding a MathML intent concept name, from the Core list where one exists. The KaTeX build can then emit `intent` on the MathML it already produces (spec 13 §13.7). This costs very little and gives correct speech output for, say, $f^{-1}$ (inverse) versus $x^{-1}$ (power).
2. Do not build on Content MathML. W3C has deferred it.
3. Contribute missing concepts to the Open list upstream, instead of inventing private names.

### 1.5 LaTeXML

**What it is**
- Version 0.8.8.1 (Chocolatey package updated 2026-02-20: https://community.chocolatey.org/packages/latexml).
- Outputs: XML, HTML, MathML, ePub and JATS. https://github.com/brucemiller/LaTeXML
- The README licence badge reads as public domain / CC0. I did not read the LICENSE file, so the exact licence is **UNVERIFIED**.

**Rust port**
- arXiv:2605.16562 (Ginev, Caruso, Miller, Sank, Weiskoff; 15 May 2026) describes a Rust port that is in progress. It reports:
  - 75% of arXiv converts without errors, against a 90% target;
  - about half of roughly 6,000 user reports resolved;
  - initial work on MathML 4 intent.
- The paper uses the words "candidate recommendation draft" for MathML 4. That does not match the W3C status in §1.4 and looks like loose wording.

**Take for math2**
- LaTeXML is the converter for **ingesting** arXiv LaTeX (already the arXiv HTML path).
- math2 node math is restricted to KaTeX-renderable macros, so math2 does not need LaTeXML to render its own nodes. It needs LaTeXML only for the arXiv source side and possibly for the JATS export (§4.3).

### 1.6 PreTeXt

**What it is**
- An XML authoring vocabulary and conversion toolchain, under GPL (https://github.com/PreTeXtBook/pretext).
- Outputs (https://pretextbook.org/doc/guide/html/the-index.html):
  - HTML; PDF and print (via LaTeX or XSL-FO); EPUB and Kindle;
  - Runestone (interactive); Braille;
  - Reveal.js and Beamer slides; Jupyter notebooks; plain text and Markdown;
  - instructor versions; WeBWorK problem sets.

**Elements** (from the guide)
- **Theorem-like:** definition, theorem, corollary, claim, fact, algorithm.
- **Axiom-like:** assumption, axiom, conjecture, heuristic, hypothesis, principle.
- **Example-like:** example, problem, question. Each can have hint, answer and solution.
- **Remark-like** and **project-like** families also exist.
- **Exercises:**
  - `<exercise>` contains `<statement>`, `<hint>`, `<answer>` and `<solution>`;
  - `<exercisegroup>` groups exercises;
  - `<exercises>` is a whole division of exercises.
- **Proofs:** a proof can be detached from its theorem, and proofs can have `<case>`.
- **Interactive exercises:**
  - `<webwork>` takes a problem from the OPL or local PG source, reused by xml:id;
  - `<myopenmath problem="ID">`;
  - STACK;
  - Runestone types: Parsons, matching, fill-in, clickable.
- **Notation** (https://pretextbook.org/doc/guide/html/topic-notation.html):
  - a `<notation>` element contains `<usage>` (exactly one `<m>`) and `<description>`;
  - it is placed in a definition's metadata or in a paragraph's `<term>`, and it is **invisible where it is placed**;
  - `<notation-list/>` in an appendix generates a three-column table (usage, description, locator), in order of first appearance.
- I found no 2026 release notes.

**Take for math2**
1. **PreTeXt is the best single book target.** One PreTeXt file gives web, PDF, EPUB, braille and WeBWorK output, so math2 does not have to write five back ends. The `m2 book` compiler (§4.5) should emit PreTeXt XML first.
2. Most math2 node kinds map directly onto PreTeXt elements:
   - definition → `definition`; statement → `theorem` / `lemma` / `corollary`; conjecture → `conjecture`;
   - example / counterexample → `example`; proof → `proof`; exercise (new, §3.6) → `exercise`.
   - concept nodes and topic nodes become divisions and introductions.
3. Copy the **notation-list** model: a definition node declares the notation it introduces, and the book compiler builds the "List of notation" in reading order.

### 1.7 Lean Verso, Verso Blueprint, leanblueprint, doc-gen4

**Verso** (https://github.com/leanprover/verso; https://verso.lean-lang.org/)
- Licence: Apache-2.0. The vendored JavaScript is MIT.
- Genres: manual, blog/website, book, API docs, blueprint and paper.
- `xref.json` and `lake exe verso sync` give experimental cross-document references.
- Verso powers the Lean reference manual and lean-lang.org. L. de Moura moved his website to Verso on 2026-03-15: https://leodemoura.github.io/notes/2026-3-15-this-website-is-now-powered-by-verso/

**Mathlib docstrings in Verso** (all PRs by Vierkantor)
- mathlib4 #44635 (8 October 2026) enabled `doc.verso` in the first file.
- mathlib4 #45031 turned it on for 1,352 files, with no syntax change and identical doc-gen output.
  - It was merged on 20 October 2026, then reverted because of lean4#15817, a scoped-instance race.
  - The fix, lean4#15839, landed on 21 October.
  - Sources: https://github.com/leanprover-community/mathlib4/pull/45031 ; https://github.com/leanprover/lean4/issues/15817
- doc-gen4 does not render Verso docstrings yet, so they are converted back to Markdown.
- mathlib4 #44898 changed the Stacks/Kerodon cross-reference attribute so that it writes into the Verso docstring.
- **Consequence:** Mathlib docstring syntax is moving to Verso during 2026–27. Any math2 tool that scrapes Mathlib docstrings, for binding prose or for "Main definitions" sections, must handle both Markdown and Verso.

**Verso Blueprint** (https://github.com/leanprover/verso-blueprint; extends spec 01 §1.2)
- **Directives:** `:::definition "label" (lean := "Nat.add, Nat.succ")`, `:::theorem "label" (owner := …) (priority := …)`, `:::proof "label"`.
- **Dependencies:** written inline as `{uses "foo"}[]` or as `(uses := "foo, bar")`.
- **Lean side:** `@[blueprint "label"]` on Lean declarations. With `(autoDeps := true)`, edges are derived from Lean and recorded with origin "automatic".
- **Tooling:**
  - `--dump-manifest` writes a "semantic manifest";
  - `blueprint-html-cache.json` is a cache;
  - `lake exe vbp query status <label>` reports status and separates *known holes* from *unverified coverage*.
- **Catalogue:** the README lists `ejgallego/verso-flt` and `verso-sphere-packing` (Lean v4.35). Noperthedron and Carleson are excluded until they move from v4.34. v4.35 is the only maintained line.
- The FLT repository (https://github.com/ImperialCollegeLondon/FLT) contains **both** a `blueprint` folder and a `blueprint-verso` folder. So spec 01 §1.2's statement that "FLT, Carleson, Sphere Packing have moved to it" is too strong: FLT runs both, and Carleson is not yet in the catalogue (see Doubts).
- I could not fetch the licence (the raw LICENSE returned 404). **UNVERIFIED.**

**leanblueprint** (https://github.com/PatrickMassot/leanblueprint)
- A plasTeX plugin plus `plastexdepgraph`.
- Macros: `\lean`, `\leanok`, `\uses`, `\mathlibok`, `\notready`, `\proves`, `\discussion`.
- `checkdecls` checks that the Lean declaration names exist.
- Node colours: not_ready orange; can_prove blue; proved green; fully_proved dark green; mathlib dark green.
- **Pitfalls for importers:** a node that `\uses` itself causes a RecursionError; a remark inside `\uses` stops its dependants from becoming fully_proved.
- **LeanArchitect** (ITP 2026, LIPIcs 382, paper 25: https://drops.dagstuhl.de/storage/00lipics/lipics-vol382-itp2026/LIPIcs.ITP.2026.25/LIPIcs.ITP.2026.25.pdf) fills these macros automatically from Lean. There, `\leanok` means that `sorryAx` is absent from the declaration's closure.
- KnowTeX (arXiv:2601.15294) is a related tool. I did not read it in detail.

**doc-gen4** (https://github.com/leanprover/doc-gen4)
- Apache-2.0. Built with `lake build Lib:docs`. Environment variables include `DOCGEN_SRC` and `DISABLE_EQUATIONS`.
- I did not confirm whether it publishes machine-readable declaration data that math2 could consume. **UNVERIFIED.**

**Mathlib documentation style** (https://leanprover-community.github.io/contribute/doc.html)
- Math is written `$…$` or `$$…$$`, or `\begin{…}` environments without dollars, to match the MathJax configuration of the online docs.
- A module docstring has these sections, in order: **Main definitions, Main statements, Notation** (required if the file introduces notation), **Implementation notes, References, Tags**.
- References are keys into `docs/references.bib`, for example `[Boole1854]`.
- Named theorems are written in bold.
- The `docBlame` and `docBlameThm` linters enforce docstrings.

**Take for math2**
1. **Import:** `m2 ingest blueprint` (spec 09.4) should read the Verso Blueprint semantic manifest when one exists, and fall back to leanblueprint's plasTeX output. Keep the "automatic" versus "manual" edge origin as edge provenance. Map "known hole" and "unverified coverage" onto evidence levels without collapsing them.
2. **Export:** math2 topic nodes with Lean bindings can be emitted as a Verso Blueprint document (`:::definition "TAG" (lean := …)`). The math2 tag is the blueprint label, and `uses` comes from the graph. This gives a Lean project a blueprint generated by math2.
3. Model math2's per-file conventions on Mathlib's module-doc sections. A math2 topic node should have **Main definitions, Main statements, Notation, Conventions, References** sections. That matches what formal-library readers already expect.
4. Spec 21's open question on Verso documents: accept Verso only as an **export** format, not as a node source format. Node bodies stay Markdown + LaTeX (spec 04).

### 1.8 Wikidata formulas and Wikifunctions

**Wikidata properties**
- P2534 "defining formula".
- P7235 "in defining formula": one statement per symbol, with the qualifier P9758 "symbol represents". https://www.wikidata.org/wiki/Property:P7235
- P7973 "quantity symbol (LaTeX)".
- The `math` datatype stores LaTeX strings.
- Older usage counts: P2534 about 5,166 uses and P7235 about 703 (Scharpf et al. 2021, https://gipplab.uni-goettingen.de/wp-content/papercite-data/pdf/scharpf2021b.pdf). Current counts are not checked here; spec 01 §1.7 has the 2026 totals.
- A 2026 dataset of 2,405 formulae: arXiv:2608.02457 (title not recorded here).

**Wikifunctions**
- The Abstract Wikipedia beta started in the week before 2026-03-26: https://www.wikifunctions.org/wiki/Wikifunctions:Status_updates/2026-03-26. The exact date of 20 March is from search only.
- Wikifunctions passed 4,000 functions in mid-April 2026. The 4,000th was subtraction of complex numbers. https://www.wikifunctions.org/wiki/Special:MyLanguage/Wikifunctions:Status_updates/2026-04-16
- Its math functions are few and elementary, for example Z32261, congruence mod n.

**Take for math2**
1. The **P7235 + P9758 pattern** (one statement per symbol in a formula, naming the item the symbol stands for) is the same as a math2 notation declaration attached to a formula. Export math2 symbol declarations to it when aligning to Wikidata.
2. Wikifunctions is of low relevance for now. At most, a `computation` node could link to a Wikifunctions ZID as a further implementation.

### 1.9 Naproche and ForTheL

**What it is**
- Licence: GPL-3.0. https://github.com/naproche/naproche
- It uses the automated provers E, Vampire and SPASS.
- Input is controlled natural language inside `\begin{forthel}` environments in LaTeX.
- It was left out of Isabelle 2025 because of folder-naming errors (ITP 2025 perfectoid-rings paper, https://drops.dagstuhl.de/storage/00lipics/lipics-vol352-itp2025/LIPIcs.ITP.2025.6/LIPIcs.ITP.2025.6.pdf).
- Naproche-ZF is a grammar-based successor.
- Visored (arXiv:2606.17581) says ForTheL → Lean has been a stated direction since 2020 but is not in production.

**Take for math2**
- Controlled natural language is a useful **style target** for the Statement section: one claim per sentence, explicit quantifiers, defined terms only. A lint rule can check "every content word in the Statement is a defined term or a variable" without adopting ForTheL itself.
- Do not take a dependency on Naproche.

---

## 2. Levels of explanation and pedagogy

### 2.1 Stages of mathematical maturity

T. Tao describes three stages (https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/; the 2009 date is from search, **UNVERIFIED**):
- **pre-rigorous:** intuition, examples, computation;
- **rigorous:** formal definitions and proofs;
- **post-rigorous:** intuition again, now backed by rigour.

**Take:** spec 04's L1–L4 levels are defined by audience. The spec should also say what each level is *for*:
- L1 is pre-rigorous: picture, example, why anyone cares.
- L2 is rigorous: the definition unpacked, standard examples and non-examples.
- L3 and L4 are post-rigorous: the right way to think about it, how it generalises, where it breaks.

This gives the explanation agent and its reviewers a test that does not depend on audience labels alone.

### 2.2 Bloom and the MATH taxonomy

The MATH taxonomy is from Smith, Wood, Coupland, Stephenson, Crawford and Ball (1996), *IJMEST* 27(1):65–77. Sources: https://opus.lib.uts.edu.au/bitstream/10453/6622/1/2003001060.pdf ; https://bsrlm.org.uk/wp-content/uploads/2016/02/BSRLM-IP-33-1-02.pdf

It is a variant of Bloom made for mathematics. It has three groups:
- **Group A:** factual knowledge; comprehension; routine use of procedures.
- **Group B:** information transfer; application in new situations.
- **Group C:** justification and interpretation; implications, conjectures and comparisons; evaluation.

The groups describe "the nature of the activity … not the degree of difficulty". One study cited in the BSRLM paper found that Group A tasks carried 67 of the 75 marks on an A-level paper.

**Take:** each exercise node (§3.6) carries a `math_group: A|B|C` field, and optionally a subcategory. Coverage reports per topic can then show "no Group C exercises", which is a typical gap in generated banks.

### 2.3 Worked examples and their evidence

- **Meta-analysis:** Barbieri, Miller-Cotto, Clerjuste and Chawla (2023), *Educational Psychology Review* 35(1), article 11, doi:10.1007/s10648-023-09745-1 (https://eric.ed.gov/?id=EJ1364058). Findings:
  - worked examples in mathematics have an overall effect of g = 0.48;
  - correct examples alone did better than incorrect or mixed correct/incorrect examples;
  - self-explanation prompts moderated the effect *negatively*.
- **Expertise reversal** (Kalyuga et al. 2003): worked examples help novices but can hurt advanced learners.
- **Fading:** Renkl and Atkinson 2003, and Renkl et al. 2002. Worked steps are removed one at a time until the learner solves the whole problem.
- **Adaptive fading** beats fixed fading: Salden et al. 2010, https://link.springer.com/article/10.1007/s11251-009-9102-0
- **Self-explanation of proofs:** Hodds, Alcock and Inglis (2014), *JRME* 45(1):62–101 (https://eric.ed.gov/?id=EJ1028348). Self-explanation *training* for reading proofs gave d = 0.95, and a 15-minute intervention had a lasting effect.
  - This is different from the meta-analysis above. A trained reading strategy helps; prompts attached to worked examples did not.

**Take:**
1. An `example` node should be able to hold a **worked solution as a numbered list of steps**, so that a renderer can fade them: show steps 1 to k and ask for the rest.
2. Explanations at L1 and L2 should prefer **correct** worked examples. Incorrect "find the error" examples belong in exercises, marked as such.
3. Following expertise reversal, L3 and L4 explanations should *not* include full worked computations by default.
4. A short "how to read a proof" guide (after Hodds et al.) can be one topic-level explanation node, linked from proof pages. It should not be repeated on every proof.

### 2.4 Examples, non-examples and concept images

- **Concept image versus concept definition:** Tall and Vinner (1981), *ESM* 12:151–169.
- **Example spaces:** Watson and Mason (2002/2005). "Ideal non-examples are those that are very nearly examples." https://mrbartonmaths.com/resourcesnew/8.%20Research/Inquries/Extending%20Example%20Spaces%20-%20Watson%20and%20Mason.pdf
- **"Mis-in" and "mis-out" examples** (ESM 2023, https://link.springer.com/article/10.1007/s10649-023-10273-6). These are examples that learners wrongly include in a concept, or wrongly leave out of it.
- **Definition–theorem–proof format:** Weber (2004), *JMB* 23(2):115–133, studies teaching in this format (cited via https://sites.math.rutgers.edu/~jpmejia/files/Lew_JRME.pdf).

**Take:**
1. spec 04 has `counterexample` for statements. It also needs a **non-example** role for concepts and definitions: an object that fails exactly one clause of the definition. The edge should name the clause that fails. This is cheap to generate and to check, and it is what the research says helps most.
2. An example should declare which definition clauses it exercises. Then "is every clause covered by a near-miss non-example?" becomes a computable coverage check.

### 2.5 Structure of an entry and of a proof

- **nLab:** its page template has the sections **Idea, Definition, Properties, Examples, Related concepts, References**, plus redirects for synonyms. https://ncatlab.org/nlab/show/Template+page ; https://ncatlab.org/nlab/show/writing+in+the+nLab
- **Lamport:** "How to Write a 21st Century Proof", *J. Fixed Point Theory Appl.* 11:43–63 (2012) (https://lamport.azurewebsites.net/pubs/proof.pdf). Proofs are hierarchical numbered steps (⟨1⟩1, ⟨2⟩1 …), each with its own justification, readable at any depth.
- Leron, "Structuring Mathematical Proofs", *AMM* 90(3), 1983, describes levelled proofs. The citation details are **UNVERIFIED**.
- **Stacks Project rules:** each definition marks the terms it defines in `{\it …}`; references go in a `reference` environment. https://raw.githubusercontent.com/stacks/stacks-project/master/documentation/rules

**Take:**
1. spec 04's sections are close to nLab's. Add an **Idea** (motivation) section to concept nodes as the leading paragraph, kept separate from Explanations. nLab shows that a one-paragraph "Idea" is the most-read part of a page.
2. A proof body may be a **structured proof**: nested numbered steps in Lamport style, each step with its own justification links. The "Unfold this proof" feature (spec 13 §13.4) already assumes a step hierarchy for formal proofs, and this gives the same hierarchy to informal proofs. Use a Markdown ordered-list convention, not a new syntax.
3. Require the **definiendum** to be marked in definition bodies (Stacks `{\it}`, sTeX `\definiendum`). Then the linter can check that the defined term is the node's name or a declared synonym.

---

## 3. Exercises

### 3.1 WeBWorK Open Problem Library (OPL)

- **Licence:** CC BY-NC-SA 3.0 by default. Copyright stays with the authors, who may choose an OSI licence instead. https://github.com/openwebwork/webwork-open-problem-library/blob/main/OPL_LICENSE
- **Size:** about 35,000 problems. https://wiki.openwebwork.org/wiki/Open_Problem_Library
- **Tags on each problem:**
  - DBsubject / DBchapter / DBsection (taxonomy);
  - Date, Author, Institution;
  - MLT (more-like-this groups);
  - Level;
  - KEYWORDS;
  - textbook tags (title, edition, section, problem number).
- **Workflow:** problems go into `Contrib` first and are reviewed into the OPL.
- Problems are written in PG/PGML (Perl). Answers are checked by the PG answer evaluators.

### 3.2 MyOpenMath and IMathAS

- **IMathAS Community License:**
  - question *code* is GPL 2.0;
  - the generated HTML of a question must be CC BY 4.0.
  - https://www.imathas.com/communitylicense.html
- MyOpenMath has been run by a non-profit since 2018.

### 3.3 STACK and Numbas

- **STACK:**
  - GPL; uses the Maxima CAS; grades answers with "potential response trees";
  - v4.11.1 in January 2026;
  - the Moodle plugin page's latest release is dated 16 October 2026: https://marketplace.moodle.com/plugins/qtype_stack
- **Numbas:**
  - code Apache-2.0, © Newcastle University;
  - content stays with its authors, and public questions in the Numbas editor are under Creative Commons;
  - packages are exported as SCORM 2004;
  - expressions are written in its JME language.
  - https://docs.numbas.org.uk/en/latest/index.html

### 3.4 QTI 3.0

- The QTI v3.0.1 information model supports **MathML 3 only**; MathML 2 was removed.
- Portable Custom Interactions (PCI) are AMD modules; `customInteraction` is deprecated.
- The implementation guide §3.7.1 covers CAS-style custom response processing (Maxima-style).
- MathML support is required only for **Advanced** conformance.
- https://www.imsglobal.org/spec/qti/v3p0/impl ; https://www.imsglobal.org/spec/qti/v3p0/conf

### 3.5 Lean exercises

- **NNG4** (Natural Number Game) is Apache-2.0: https://github.com/leanprover-community/NNG4
- **lean4game**, the game engine, is GPL-3.0. Games are hosted at adam.math.hhu.de. They include:
  - Knights and Knaves, Linear Algebra, Logic, NNG, Robo;
  - Real Analysis (Kontorovich);
  - Reintroduction to Proofs (Riehl);
  - Set Theory (Velleman).
- **Mathematics in Lean (MIL):** Apache-2.0. Release v4.19.0, with a PDF dated 21 October 2026: https://leanprover-community.github.io/mathematics_in_lean/mathematics_in_lean.pdf. Exercises are Lean files in the `MIL` folder, with solutions kept separately.
- **Autograding:**
  - `robertylewis/lean4-autograder-main` uses the attributes `@[autogradedProof pts]` and `@[legalAxiom]`, with a Gradescope integration. https://github.com/robertylewis/lean4-autograder-main
  - `isun3071/lean-4-teaching` checks submissions with Comparator and supports tactic allowlists.
- **Verso game prototype:** https://github.com/pfaffelh/nng-verso

### 3.6 Open textbooks and their licences

| Book | Licence | Source |
|---|---|---|
| Judson, *Abstract Algebra: Theory and Applications* | GFDL 1.2 or later | http://abstract.ups.edu/ |
| Beezer, *A First Course in Linear Algebra* | GFDL | open.umn.edu (Open Textbook Library) |
| Hefferon, *Linear Algebra* | CC BY-SA | from search |
| OpenStax *Calculus* Vol. 1 | CC BY-NC-SA 4.0 | https://openstax.org/details/books/calculus-volume-1 |
| APEX Calculus | CC BY-NC 4.0 | from search |
| *Active Calculus* (single variable, 2018 and later) | CC BY-SA 4.0. Sources conflict; **UNVERIFIED** | from search |
| *Active Calculus* multivariable | CC BY-NC-SA | from search |

### 3.7 MyST / Jupyter Book 2

- The `exercise` and `solution` directives: a solution refers to its exercise by label. https://mystmd.org/guide/exercises
- `prf:` / `proof:kind` directives for theorem-like blocks (mystmd 1.11): https://mystmd.org/guide/proofs-and-theorems

### 3.8 Licence consequences for math2

Exercise banks are licensed differently from each other and from math2 prose. Spec 16 §16.6 says node prose is CC BY-SA 4.0 and Lean code is Apache-2.0. It already says GFDL sources are linked, never copied, and MSC text is never stored.
- OPL default: NC-SA. **Not compatible** with BY-SA.
- OpenStax: NC-SA. Not compatible.
- APEX: NC. Not compatible.
- IMathAS: code GPL-2.0, rendered output CC BY. Rendered text is fine; the code is GPL.
- GFDL books: not compatible with CC BY-SA in either direction, unless dual-licensed.
- Hefferon (BY-SA): compatible.
- NNG4 and MIL (Apache-2.0): fine for Lean code.

**Take:** math2 must **not copy** NC-licensed problems into nodes. It may **link** to an OPL path or a MyOpenMath ID as an external "interactive version". The exercise node itself is then math2's own text.

### 3.9 Take for math2: an `exercise` node kind

PreTeXt, OPL, MyST and MIL all treat an exercise as a first-class object with statement, hint, answer and solution. math2 should do the same.

**Body sections:** Statement, Hints (ordered), Answer, Solution (structured steps, §2.5).

**Front matter:**
- `exercises: [TAG…]`: the concepts or statements it practises. This edge gives schema.org `assesses` (§4.4).
- `math_group: A|B|C` (§2.2).
- `level: L1–L4`.
- `check:`, one of:
  - `lean`: a statement with `sorry`, graded by Comparator or the autograder, with a tactic allowlist;
  - `cas`: an answer expression plus an equivalence test, exportable to STACK, Numbas or PG;
  - `choice` / `numeric`: exportable to QTI 3;
  - `none`: a proof exercise checked by people.
- `external: [{system: webwork-opl, path: …}, {system: myopenmath, id: …}]`, as links only.

**Checking:** an exercise with `check: lean` gets evidence like a statement does. The reference solution must compile, and the `sorry` version must be the same type. This is the machine-checkable answer format that the task asks for. CAS answers are checked by running the CAS equivalence on the reference answer in CI.

---

## 4. Packaging

### 4.1 Book toolchains (status as of 2026-10-26)

| Tool | Status / version | Licence | Notes |
|---|---|---|---|
| PreTeXt | active; no 2026 release notes found | GPL | §1.6; the best multi-output target |
| Quarto | 1.8 (2025-10-13: https://quarto.org/docs/blog/posts/2025-10-13-1.8-release/); 1.9 adds Typst books ("orange-book"; blog 2026-03-31: https://quarto.org/docs/blog/posts/2026-03-31-typst-books-and-more/) | GPL (from Wikipedia) | theorems via `#thm-` cross-ref divs; amsthm in LaTeX output. A Rust "Quarto 2" for late 2026 is **UNVERIFIED** (search only) |
| Typst | 0.14 (2025-10-24, PDF/UA-1); 0.15.0 (2026-06-15, MathML export); 0.15.1 on 2026-07-17 per search (**UNVERIFIED**) | Apache-2.0 (https://typst.app/open-source/) | fast; its math syntax is not LaTeX, so a converter is needed |
| Pandoc | 3.11 (2026-08-29) | GPL (well known; not re-checked) | "mathml is now the default math-method" (https://github.com/jgm/pandoc/discussions/11835); `--math-method` |
| mdBook | 0.5.0 (2025-11-17); 0.5.4 (2026-07-06) | MPL-2.0 (not re-checked, **UNVERIFIED**) | `mathjax-support`; does not accept `$$` delimiters (https://github.com/rust-lang/mdbook/releases) |
| Jupyter Book 2 / MyST | mystmd 1.11 | MIT (not re-checked, **UNVERIFIED**) | `prf:` theorem directives and exercise/solution (§3.7) |
| LaTeX | TeX Live 2025 on arXiv | LPPL | the arXiv target (§4.3) |

**Take:**
- Emit **PreTeXt** for books.
- Emit **LaTeX** (amsthm + `m2.sty`) for arXiv and print.
- Emit **Typst** as a third option only after the macro → Typst mapping exists. That mapping can live in the notation register (§5), one Typst form per macro.
- Pandoc and Quarto are not needed as primary targets, since PreTeXt covers their outputs. Quarto is a fallback for notebook-style computation nodes.
- mdBook's lack of `$$` makes it a poor fit.

### 4.2 Course standards

- **Common Cartridge 1.3** (July 2013; https://www.imsglobal.org/cc/index.html):
  - carries web content, QTI assessments (six question types), LTI links and discussions;
  - "Thin CC" carries links only;
  - CC is "not designed as a replacement for SCORM".
  - I found nothing newer than 1.3. Whether a later version exists is **UNVERIFIED**.
- **LTI 1.3 Core:** dated 16 April 2019. Earlier LTI versions lost support on 30 June 2022.
- **xAPI:** IEEE 9274.1.1-2023 (published 2023-10-06; https://standards.ieee.org/ieee/9274.1.1/7321/), adopted as ISO/IEC/IEEE 39274-1-1:2025.
- **cmi5:** the xAPI profile for LMS launch, dated 2016-06-01.
- **SCORM 2004:** what Numbas exports (§3.3).

**Take:**
- A math2 "course" export is a **Common Cartridge 1.3** containing:
  - HTML pages generated per node;
  - QTI 3 items, converted to CC's QTI subset, from exercises with `check: choice|numeric`;
  - LTI links back to the live workspace for Lean and CAS exercises.
- **xAPI** statements are optional and live in the workspace (spec 13), never in `tamnd/math2`.
- Do **not** implement SCORM.

### 4.3 Article formats

- **JATS 1.4** = ANSI/NISO Z39.96-2024, approved 31 October 2024. It prefers MathML 3, allows `<tex-math notation="LaTeX" version="LaTeX2e">`, and allows both inside `<alternatives>`. https://jats.nlm.nih.gov/archiving/tag-library/1.4/element/tex-math.html
- **JATS4R** recommends math-mode LaTeX without `$` delimiters inside `<tex-math>`. https://jats4r.niso.org/math/
- **arXiv** (https://info.arxiv.org/help/00README.html):
  - TeX Live 2025 is the default; 2023 is the fallback.
  - Submission system 1.5 (April 2025).
  - `00README.json` keys: `process.compiler`, `sources`, `texlive_version`, `stamp`, `nohyperref`, `spec_version`.
  - Only packages in TeX Live are available, so `m2.sty` must be **bundled** in the submission.

**Take:**
- `m2 article` emits an arXiv-ready source tree:
  - `main.tex` with `m2.sty` bundled;
  - `00README.json` naming the compiler and the TeX Live year;
  - a `.bib` built from node Sources.
- It also emits JATS 1.4 with `<alternatives>` holding both `<tex-math>` (no delimiters) and MathML, the MathML coming from KaTeX at build time.
- Each theorem in the article carries its `m2://TAG@release` as a footnote or a link, so readers can reach the checked node.

### 4.4 Metadata: schema.org LearningResource and LRMI

**schema.org LearningResource** (https://schema.org/LearningResource)
- Properties include `teaches`, `assesses`, `competencyRequired`, `educationalAlignment`, `educationalLevel`, `educationalUse` and `learningResourceType`.
- Subtypes include Course, Quiz and Syllabus.
- The type is in schema.org's "new" area.

**LRMI** (https://www.dublincore.org/specifications/lrmi/)
- LRMI 1.1. DCMI has maintained it since 2014.
- It publishes LRMI Terms and SKOS concept schemes, for example for learning resource types.

**Learning Commons Knowledge Graph** (Chan Zuckerberg Initiative)
- Licence: code MIT, data CC BY 4.0. https://docs.learningcommons.org/knowledge-graph/resources/license
- It models learning components, prerequisite relations between standards, and the Eedi misconception graph.
- Its open platform launched on 2026-09-22. I read this date on the docs site; the details there are from my reading of 2026-10-26.

**Take:** the static site (spec 13) embeds JSON-LD on each page, mapped as follows.

| schema.org property | Source in math2 |
|---|---|
| `@type` | `LearningResource` for explanation and exercise views; `DefinedTerm` for concepts |
| `teaches` | the node's own concept tags |
| `competencyRequired` | the direct `uses` / prerequisite closure, depth 1 |
| `assesses` | the `exercises:` edge of exercise nodes |
| `educationalLevel` | the L1–L4 level, mapped to a published SKOS scheme rather than free text |
| `learningResourceType` | an LRMI term such as "worked example" or "exercise" |

**Learning Commons:** it is a possible alignment target for school-level (L1) concepts, and it supplies a misconception graph. That suggests a `misconception` role or edge for non-examples (§2.4).

### 4.5 Prerequisite graphs and datasets

- **Survey:** prerequisite relation learning, doi:10.1145/3733593 (ACM, 2025). I read the abstract only.
- **Datasets:**
  - RefD Math: 65 knowledge components, 658 pairs, 75 positive;
  - LectureBank;
  - MOOCCube / MOOCCubeX.
  - All are small or tied to particular courses. None is a usable source of math2 edges; they are useful only as evaluation sets.
- **Metacademy** (https://github.com/metacademy/metacademy-application/wiki/Alternative-learning-paths):
  - 487 machine-learning concepts and 1,208 prerequisite pairs;
  - each concept has **a single dependency set**, which the project itself names as a limitation, since there are alternative paths.
- **Knowledge space theory / ALEKS:**
  - the **outer fringe** of a learner's knowledge state is the set of items they are "ready to learn";
  - a learner succeeds on outer-fringe items with probability about 0.93;
  - https://www.sciencedirect.com/science/article/abs/pii/S0022249621000134 ; arXiv:1511.06757
- **PAIGOS** (Ullrich 2008, LNAI 5260, doi:10.1007/978-3-540-88215-2):
  - generates courses from OMDoc content by HTN planning over an ontology of instructional objects (definition, example, exercise, and so on);
  - outputs an IMS manifest;
  - the closest published predecessor of a "graph → course" compiler.
- **Recent graph-to-path work:**
  - AutoMathKG (arXiv:2505.13406);
  - GraphRAG learning paths (arXiv:2506.22303).
  - I found **no** published system that generates a full mathematics textbook from a knowledge graph.

**Take:**
1. **Alternative prerequisite sets.** A definition can be reached from different backgrounds. A concept can have several definitions (Realms, §4.6), each with its own `uses` set. So the prerequisite path in spec 13 should be computed over a *chosen definition per concept*, not over the union of all of them. This fixes Metacademy's limitation in a natural way.
2. **"Ready to learn".** Given a reader background B (spec 13 §13.5), the outer fringe is the set of nodes whose prerequisites are all in B. That is a one-line graph query, and it gives a principled "what next" list.
3. **Evaluation sets.** Use RefD Math / LectureBank style labelled pairs only to measure the edge-inference agent, never as content.

### 4.6 Turning a graph into a linear text

**Ordering**
- Choosing a linear order that minimises forward references is an instance of the minimum feedback arc set problem, which is NP-hard. arXiv:2304.15006 discusses this for documents.
- For a DAG, a topological order has zero forward references in proofs. The real choice is *among* topological orders, and that choice is about locality, keeping related nodes close.
- **Stacks rule:** "Proofs do not refer to results later in the stacks project." https://raw.githubusercontent.com/stacks/stacks-project/master/documentation/rules
- Each Stacks chapter opens with a **Conventions** section (for example tag 0439).
- Stacks tag 06G9 admits that notation conflicts between chapters. That is an example of local conventions being preferred over global consistency.

**Choosing among equivalent definitions**
- Realms (Carette, Farmer, Kohlhase, CICM 2014, LNCS 8543:252–266, arXiv:1405.5956) treat a concept with several equivalent axiomatisations as a **realm**: theories joined by views that prove them equivalent, plus a single external **interface**.
- Follow-up: "Lightweight Realms" (2025), https://link.springer.com/chapter/10.1007/978-3-032-07021-0_13
- Background: the "little theories" method (Farmer et al. 1992).
- **A concrete case:** Mathlib defines `IsCompact` with filters:
  `∀ ⦃f : Filter X⦄ [f.NeBot], f ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x f`
  (https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Defs/Filter.html). Most textbooks use open covers. A book generated from math2 must choose the open-cover definition and present the filter one as a theorem, while the Lean binding goes to the filter one.

**Take: an `m2 book` compiler**

The input is a book manifest `books/<id>.yaml` in the content tree. It contains:
- `targets`: tags, or a topic;
- `background`: the reader background, i.e. the nodes the book assumes;
- `choose`: for each concept with more than one definition node, which one is primary in this book. The others become "equivalent characterisations" theorems, and the equivalence-proof nodes are pulled in;
- `notation`: per-book overrides of default notations (§5);
- `level`: which explanation level to inline;
- `format`: `pretext`, `latex`, `typst`, `jats` or `cc`.

The algorithm:
1. Compute the closure of `targets` minus `background`, using only the chosen definitions' `uses`.
2. Topologically sort it, breaking ties first by the order of the topic nodes' author-given outline, then by keeping a statement next to its proof and examples, then by MSC.
3. Reject the manifest if a proof would refer forward (the Stacks rule). Never silently reorder it.
4. Generate a **Conventions** section from the concept nodes' recorded conventions, and a **List of notation** (PreTeXt `notation-list`).
5. Write `book.lock` with the exact `TAG@release` of every included node, so that the book is reproducible and citable.

The manifest and the lock are the "published method" here. The decisions are recorded data, not hidden heuristics.

---

## 5. Notation

### 5.1 How large projects keep notation consistent

- **Mathlib:**
  - Notation is declared in Lean, often *scoped*. It is enabled with `open scoped X`, for example `open scoped PiNotation`.
  - `notation3` generates delaborators, so that the notation is also printed back: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/Notation3.html
  - mathlib4 PR #45028 proposes renaming `notation3` to `notation +delab`. Its merge status is unclear.
  - Every file that introduces notation must have a "Notation" section in its module doc (§1.7).
- **Stacks Project:** a Conventions section per chapter, and tolerance of documented conflicts between chapters (tag 06G9).
- **sTeX:** the symbol is the identity, and notations are attached to it. Variant notations are selectable per document (§1.2).
- **PreTeXt:** `<notation>` with usage and description, attached where the notation is introduced, and collected into a notation list (§1.6).
- **Wikipedia MOS:MATH:** keep a list of "Mathematical conventions" (for example: does "compact" include Hausdorff? are rings unital?) and require "consistency within an article". https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Mathematics
  - arXiv:2412.20419 notes that consistency *across* articles is "too much to ask" on Wikipedia.
- **ISO 80000-2:2019:** ℕ = {0, 1, 2, …}, and ℕ* denotes the positive integers. https://cdn.standards.iteh.ai/samples/64973/c49664e1b9364d1d811246de9d1db4aa/ISO-80000-2-2019.pdf

### 5.2 Take for math2: a notation register

Replace the flat macro list (`macros/m2.sty` + `m2.katex.json`) with a **source register** `macros/notation.yaml` from which both files are generated. Each entry has these fields:

```yaml
- macro: \Nat              # the LaTeX command (the only thing a node body uses)
  denotes: 4M2XKQ          # concept tag (sTeX \symdecl identity)
  args: 0
  prec: null               # for operators, as sTeX prec=
  latex: \mathbb{N}        # expansion for m2.sty and KaTeX
  typst: NN                # for the Typst target (§4.1)
  intent: natural-numbers  # MathML intent concept, Core or Open list (§1.4); null if none
  openmath: setname1/N     # alignment (§1.3), optional
  convention: "includes 0 (ISO 80000-2)"  # one line, shown in the notation list
  variants:                # alternative notations for the same concept
    - {name: N-bold, latex: \mathbf{N}}
  introduced_by: 9QF3ZD    # definition node that introduces it (PreTeXt <notation> placement)
```

New lint rules:
1. Every macro used in a node body exists in the register. This extends B01.
2. A definition node that introduces a symbol has a register entry with `introduced_by` pointing to it.
3. One concept has exactly one default notation. Variants are allowed only through a book or page override (spec 13's "Translate notation" then becomes a lookup in this register).
4. Overloaded symbols, such as $\subset$ or $(a,b)$ for pair, interval and gcd, must use distinct macros: `\subset` versus `\subsetneq`, `\pair`, `\ointerval`, `\gcd`. Each macro has a single `denotes`. This is the sTeX principle, and it is what makes rendering by convention and intent emission possible.

**Conventions:**
- A concept node's front matter gets `conventions:`, a list of short statements such as "ring: unital, not necessarily commutative" or "ℕ: includes 0".
- The default follows ISO 80000-2 where ISO says anything.
- The book compiler (§4.6) collects these into the book's Conventions section, Stacks style.
- When a statement depends on a convention, for example on whether 0 ∈ ℕ, it must link the concept that carries the convention, so that a change of convention can find every statement it affects.

---

## 6. What the spec should adopt

Ordered by value for effort.

1. **Notation register** (§5.2): a symbol-identity + notation split (sTeX), with `denotes`, arity, precedence, Typst form, MathML `intent` and OpenMath alignment per macro. `m2.sty` and `m2.katex.json` are generated from it. Conventions are recorded on concept nodes, with ISO 80000-2 as the default.
2. **`exercise` node kind** (§3.9): Statement / Hints / Answer / Solution; an `exercises:` edge; `math_group` (MATH taxonomy A/B/C); and a `check:` of `lean` (with a `sorry` target, graded via Comparator or the autograder), `cas`, `choice`, `numeric` or `none`. NC-licensed banks (OPL, OpenStax, APEX) are linked only, never copied.
3. **Non-example role** (§2.4) that names the definition clause that fails. Examples declare the clauses they exercise, and clause coverage is a computed report.
4. **Pedagogy rules for explanations** (§2.1–2.3):
   - L1–L4 map onto Tao's pre-rigorous / rigorous / post-rigorous stages;
   - prefer correct worked examples at L1–L2, as numbered steps that a renderer can fade;
   - no full worked computations at L3–L4 (expertise reversal);
   - an Idea paragraph on concept nodes (nLab);
   - optional Lamport-style structured proofs.
5. **`m2 book` compiler with a manifest and a lock** (§4.6):
   - the closure of the targets minus the reader background;
   - one *chosen* definition per concept (Realms), with the others becoming equivalence theorems;
   - a topological order that keeps the author's outline and refuses forward references in proofs (Stacks rule);
   - a generated Conventions section and List of notation;
   - `book.lock` pinning `TAG@release`.
6. **Output targets, in this order:**
   1. PreTeXt XML (web, PDF, EPUB, braille, WeBWorK in one step);
   2. LaTeX + bundled `m2.sty` + `00README.json` for arXiv;
   3. JATS 1.4 with `<alternatives>` (tex-math without delimiters, plus MathML);
   4. Common Cartridge 1.3 with QTI items and LTI links for courses;
   5. Verso Blueprint for Lean projects;
   6. Typst later.
   - No SCORM. xAPI only in the workspace.
7. **JSON-LD on every static page** (§4.4): schema.org `LearningResource` / `DefinedTerm` with `teaches`, `competencyRequired` (from `uses`), `assesses` (from `exercises:`), and `educationalLevel` from a SKOS scheme for L1–L4.
8. **Prerequisite paths per chosen definition**, plus the KST "outer fringe" as the "ready to learn" query (§4.5).
9. **Blueprint import and export** (§1.7):
   - read Verso Blueprint's semantic manifest and keep the automatic/manual edge origin and the hole/unverified distinction;
   - read leanblueprint as a fallback, guarding against self-`\uses` and remarks in `\uses`;
   - accept Verso only as an export, not as a node source format (answers spec 21).
10. **Alignment targets:** add OpenMath CD symbols and MathML intent concept names next to Wikidata QIDs. Export symbol declarations in the Wikidata P7235 + P9758 pattern.
11. **Topic nodes follow Mathlib's module-doc sections** (Main definitions, Main statements, Notation, Conventions, References), and any scraper of Mathlib docstrings handles Verso docstrings as well as Markdown.
12. **Correct spec 01 §1.2.** Say that FLT keeps both a leanblueprint and a Verso blueprint, and that only verso-flt and verso-sphere-packing are in the Verso Blueprint catalogue (Carleson is pending a toolchain bump). "Moved to it" is too strong.

---

## 7. Open doubts

1. **MathML 4 status.** The charter's May 2026 CR target was missed, and the spec is still a WD with an editor's draft of 24 October 2026. The new CR date is unknown. w3.org/TR returned 403, so I could not read the TR page directly.
2. **Verso Blueprint licence:** UNVERIFIED (LICENSE fetch returned 404).
3. **Which projects have moved to Verso Blueprint.** FLT has both folders; Carleson and Noperthedron are excluded from the catalogue until they move to v4.35. Spec 01 §1.2's "FLT, Carleson, Sphere Packing have moved" needs rechecking.
4. **LaTeXML licence:** the badge suggests CC0 or public domain (NIST work). I did not read the LICENSE file.
5. **MMT licence** (COPYING.txt): not read.
6. **Licences of SMGloM archives other than `smglom/mathsoft`:** not checked. The old SPL 0.1 may still apply to some of them.
7. **Active Calculus licence:** sources conflict between CC BY-SA 4.0 and CC BY-NC-SA for the single-variable edition.
8. **Leron 1983** (*AMM* 90(3)): the citation is UNVERIFIED.
9. **Date of Tao's "more to mathematics than rigour" post** (2009?): UNVERIFIED.
10. **Quarto 2 (Rust) timing, Typst 0.15.1 date, mdBook and mystmd licences:** from search or memory only.
11. **Common Cartridge after 1.3:** I found no newer version, but did not prove that none exists. QTI 3 support in Canvas, Moodle and Blackboard as of 2026 was not checked.
12. **doc-gen4 machine-readable output:** whether doc-gen4 publishes per-declaration JSON that math2 can use for binding was not confirmed. doc-gen4 also does not yet render Verso docstrings.
13. **mathlib4 #45028** (`notation3` → `notation +delab`): merge status unknown.
14. **Learning Commons Knowledge Graph:** whether its K-12 math nodes are worth aligning to is unknown. Its scope is US standards.
15. **Graph-to-textbook generation:** I found no published system that generates a complete mathematics textbook from a knowledge graph. PAIGOS (2008) generates courses, not books. The `m2 book` design in §4.6 is therefore new and has no prior evaluation.
16. **Wikifunctions:** relevance to math2 is low today. Revisit if its math function catalogue grows.
17. **MathML intent Open list:** its exact published URL was not confirmed. The Core list URL and the concept-lists note (https://w3c.github.io/mathml-docs/concept-lists/) are from search results.
18. **Exercise answers and Lean code:** spec 16 §16.6 gives prose CC BY-SA 4.0 and `lean/` Apache-2.0. It does not say which zone a Lean exercise's statement and reference solution belong to. Presumably Apache-2.0, but the spec should say so.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/02-the-goal.md">
# K02. The goal of the corpus

This document says what the corpus programme must produce, how we know when it is complete, and what it must not do. Document K04 says how we count the items. The other documents say how we produce them.

## K2.1 The goal in one sentence

**Each definition, theorem, proof and example of mathematics that a person can find in a public source is in math2 as a node, or it has a recorded decision that says why it is not. Each node is written for people, it is checkable by a machine, and it is ready to go into a course, a book or an article.**

The sentence has five parts. Each part is a requirement with a measurement:

| Part of the sentence | Requirement | Measurement |
|---|---|---|
| "each ... that a person can find in a public source" | The universe is the union of the inventories of all the sources of K05 and K06, plus the items that models list (K07). | the inventory (K04): one row per distinct item |
| "is in math2 as a node, or it has a recorded decision" | No item is silently missing. Each inventory row has an outcome. | the **open count**: inventory rows with no outcome. The target is zero. |
| "written for people" | Each node has the sections of its kind card (K08), in the house style, with notation from the notation register (K10.4). | the writing level W0 to W3 (K2.4) |
| "checkable by a machine" | Each statement has a formal binding, or a recorded reason why it has none. Each proof has a formal proof, or it is an informal proof with `uses` edges that a machine can follow. | the E and F levels of spec 04.7, and the formal outcome of K2.5 |
| "ready to go into a course, a book or an article" | A selection of nodes compiles into a linear text with no manual edits (K11). | the pack check: `m2 pack -check` on the standard volumes (K11.6) |

## K2.2 Why this is a different goal from the main spec

The main specification (documents 00 to 21) builds the system: the object model, the graph, the formal engine, the agents, the audit and the workspace. It decided that node count is not an axis of progress (settled decision 23) and it measures coverage with C0 to C4 (18.1).

The corpus programme fills the system. It does not change any settled decision of the main spec. It adds three things that the main spec does not have:

1. **A denominator for "all".** C2 counts the items of a skeleton, and C3 and C4 count the items of a core. Both are curated lists of a few hundred to a few thousand items for each class. The corpus inventory (K04) is the full list: each distinct item that any source in scope holds. The skeletons and cores become views of the inventory.
2. **A production line.** The main spec has roles that improve one node at a time (12.1). The corpus programme runs these roles, and four new ones (K7.3), over millions of items. It needs queues, batch sizes, cost limits and a review budget (K12).
3. **A content standard.** The main spec says what a node file must contain to be valid (04.5). The corpus programme says what a node must contain to be **good**: the sections, the order, the style, the notation and the examples (K08).

Node count is still not an axis of progress. The axes of the corpus are the open count, the writing level, the trust levels and the pack check. A million nodes at W1 with no proofs is a bad result, and the reports show it as bad.

## K2.3 What "each definition, theorem, proof and example" means

The corpus holds the 11 kinds of spec 04.2. For the corpus, the important kinds and what counts as one item are:

| Kind | One item is | It is not |
|---|---|---|
| `concept` | one mathematical notion that sources name, for example "compact space" | one wording of the notion |
| `definition` | one definition of a concept that a source uses, for example "compact: each open cover has a finite subcover" | a restatement of the same definition in other words |
| `statement` | one proposition that a source states and calls true, at the generality that the source uses | each corollary that only renames variables |
| `proof` | one argument for one statement that differs from the others in what it uses or in its method | the same argument with other notation |
| `example`, `counterexample` | one specific instance that a source uses to illustrate or to refute | a family of instances (that is a statement) |
| `object` | one named object with an identity, for example a group in the ATLAS, an OEIS sequence or an LMFDB curve | each record of a database: K4.6 limits which objects become nodes |
| `conjecture`, `problem` | one open question that a source poses | each restatement of it |

The granularity rules of spec 05.4 apply. A family of near-identical forms becomes a hub and its members, not one node and not a thousand nodes.

Exercises are not a kind in node.v1. The corpus needs them for courses (K11). K8.9 proposes the kind `exercise` for node.v2 and gives the reason. Until node.v2, exercises are `example` nodes with the role `exercise` in `## Remarks`.

## K2.4 Writing levels

A writing level says how complete the human-readable part of a node is. The build computes it from the sections that are present and from their provenance, like the E and F levels. No person and no agent writes it.

| Level | Name | The node has |
|---|---|---|
| W0 | stub | a tag, a kind, a name, an inventory row and at least one `same_as` or source. It has no body. |
| W1 | stated | W0, plus the main section of its kind (`Statement`, `Definition`, `Proof` and so on) that passes the body rules B01 to B03, plus at least one source with a locator |
| W2 | usable | W1, plus the sections that the kind card marks as required for W2 (K08): for a statement, at least one proof node, one example or non-example, and an L2 explanation; for a definition, one example and one non-example |
| W3 | complete | W2, plus each section that the kind card lists, explanations at each level that applies, at least one exercise for a statement or definition in a curriculum, and a person confirmed each agent-written section (rule P04) |

W does not measure truth. A node at W3 can be at E1. A node at E5 can be at W0. The reports give both. A volume (K11) sets the levels that its nodes must reach.

## K2.5 Formal outcomes

The goal says "checkable by a machine". For each statement and definition the corpus records one formal outcome. The build computes it from the bindings and the registers:

| Outcome | Means |
|---|---|
| `bound` | at least one formal binding exists, at any F level |
| `proved` | the node is at E5 |
| `pending` | a formalize or prove job is in the queue or ran and failed with a reason |
| `blocked` | the formal statement needs a definition that no library has. The blocking definition is a node with outcome `pending` |
| `out_of_reach` | a person decided that formalization is not useful now. The decision has a reason, for example "needs the classification of finite simple groups". It is reviewed each year |

The target is that each node has an outcome, not that each node is `proved`. In 2026 the best systems formalized whole textbooks, but research-level statements still failed often (K1.4). An honest corpus says which statements are proved, which are bound and which are not.

## K2.6 What the corpus must not do

These rules come from the main spec and from the research of K01. They are not new trade-offs. They are the limits that the corpus works inside.

1. **It must not copy text that the licence gate refuses** (spec 9.3, 16.6). Most of the literature gives a record and a locator, not text. math2 writes its own statements.
2. **It must not take model output as truth.** A statement that a model wrote is E1 (spec 04.7). A proof that a model wrote is an informal proof at E1 until it is formal, certified or published. Agreement between models is evidence for triage. It never raises a level (settled decision 3).
3. **It must not publish text that a model reproduced from a source with a closed licence.** Each generated body goes through the copy check of K9.4 before it can merge.
4. **It must not invent sources.** A source is a record with a locator that the ingestion found or that a person entered. A model can propose a source. The proposal becomes a source only when the record exists in the record plane and a check or a person confirms the locator (K9.5).
5. **It must not flood the people.** Agent output is limited by the measured review capacity (settled decision 20). The corpus plan in K12 is set by that capacity, not by the speed of the models.
6. **It must not hide what it does not know.** Each report shows the open count, the items with the outcome `out_of_reach`, and the error rates from the sample audits (K9.7), including the bad ones.

## K2.7 The reference scale

The numbers in this table are estimates for planning. Spec 18.4 has the basis for the first rows. K01 has the basis for the others. When math2 has a measured number, the measured number replaces the estimate.

| Quantity | Estimate | Basis |
|---|---|---|
| distinct concepts | 100,000 to 300,000 | spec 18.4 |
| definitions (several for each concept) | 150,000 to 500,000 | about 1.5 definitions for each concept in textbooks and in Mathlib |
| named and textbook statements | 300,000 to 1,000,000 | Mathlib theorems that a person would cite, ProofWiki, the textbooks of K5.4 and the cores of spec 18.3 |
| statements in the research literature, after deduplication | 10 to 30 million | spec 18.4 |
| proofs | 1 to 1.5 for each statement in the textbook layer | ProofWiki and textbooks often give more than one proof of a named theorem |
| examples and counterexamples | 2 to 5 for each definition in the textbook layer | the kind card of K08 asks for at least one example and one non-example |
| exercises | 3 to 10 for each statement and definition in a curriculum | open textbooks (K5.4) |

The corpus programme has three layers. Each layer is a target with its own exit criterion in K14:

1. **The textbook layer**: everything that the undergraduate and graduate textbooks and the reference wikis hold. About 1 to 2 million nodes. This layer must reach an open count of zero, W2 for each node, and E2 or better for each statement.
2. **The formal layer**: everything that the formal libraries hold that a person would cite. About 300,000 to 600,000 nodes, most of them also in the textbook layer. This layer must reach an open count of zero and a binding for each node.
3. **The research layer**: the results of the research literature. 10 to 30 million items in the record plane. They become nodes only when the graph needs them (spec 9.8), or when a project or a campaign asks for an area (spec 18.5). This layer must reach an open count of zero in the record plane, and each record must have an outcome: a node, an `aligned` edge to a node, or "kept as a record".
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/03-identifiers.md">
# K03. Identifiers: tags of ten characters

The corpus programme makes millions of nodes and tens of millions of inventory rows. A tag of six characters (spec 04.3) gives about 1.07 billion values. That is enough for the nodes, but it is too tight for random minting at the scale of the inventory, and it has no protection against a mistyped tag. This document replaces the tag format of spec 04.3 with a tag of ten characters. It also says what happens to the tags that already exist, how files are sharded, and what must change in the main spec and in the code.

## K3.1 The decision

**A new tag has ten characters: nine random characters and one check character, all from Crockford base32.** The tags of six characters that exist on the day of the change stay valid forever, as a frozen legacy set. No new tag of six characters is minted after that day.

| Property | Tag of 6 (legacy) | Tag of 10 (new) |
|---|---|---|
| alphabet | Crockford base32: `0-9 A-Z` without `I L O U` | the same |
| random characters | 6 | 9 |
| number of values | 32^6, about 1.07 × 10^9 | 32^9, about 3.5 × 10^13 |
| check character | none | 1, by the Luhn mod 32 algorithm (K3.3) |
| example | `3C53TD` | `7K3QX2M9R2` |
| minted after the change | no | yes |

The three rules of spec 04.3 do not change. A tag is assigned once, it is never edited, and it is never reused. The legacy tags are not converted to long tags, because a conversion would edit a tag. A node with a legacy tag keeps it. A node never has two tags.

## K3.2 Why ten characters

- **Room for the inventory.** The inventory (K04) has one row for each candidate item, and the research layer has 10 to 30 million rows (K2.7). Rows and nodes share one space of identifiers (K3.6), so that a bare identifier resolves to one thing. With 6 characters and 30 million identifiers in use, about 3% of random draws collide. With 9 random characters, the chance is below 1 in a million for each draw.
- **Random minting without coordination.** Spec 04.3 mints tags at random, so that two branches can mint without a central counter. A sparse space keeps collisions at merge rare. With 3.5 × 10^13 values and 100 million in use, the chance that one draw collides is about 3 in 10^6.
- **A check character.** People copy tags by hand into papers, emails and search boxes. One mistyped character in a tag of six characters gives another valid tag, which can be the tag of another node. The check character detects each error in one character and almost each swap of two adjacent characters. A tool can then say "this tag has a typo" and not show the wrong node.
- **Still short.** Ten characters fit in a citation, a URL and a Lean attribute. A Stacks tag has four characters, a Wikidata id has up to 10 characters, and a DOI is much longer.

Eleven or twelve characters would give more room, but we have no use for it. Ten characters give room for the research layer at a thousand times its estimated size.

## K3.3 The format

A tag of ten characters is `P1 P2 ... P9 C`:

- `P1` to `P9` are random characters of the alphabet `0123456789ABCDEFGHJKMNPQRSTVWXYZ`. The value of a character is its position in this alphabet, from 0 to 31.
- `C` is the check character. It is computed from `P1` to `P9` by the Luhn mod N algorithm with N = 32:
  1. Start at `P9` and go to the left. The factor of `P9` is 2, the factor of `P8` is 1, then 2, then 1, and so on.
  2. For each character, multiply its value by its factor. Write the product in base 32 and add its digits: `product / 32 + product mod 32`.
  3. Add the results. The check value is `(32 - (sum mod 32)) mod 32`. `C` is the character with that value.
- To validate a tag, do the same over all ten characters with the factor 1 for `C`, then 2 for `P9`, and so on. The tag is valid when the sum is a multiple of 32.

Example: for `7K3QX2M9R` the check character is `2`, thus the tag is `7K3QX2M9R2`. If a person types `7K3XQ2M9R2` (two characters swapped) or `7KZQX2M9R2` (one character changed), the validation fails.

These rules of spec 04.3 apply to long tags too:

- **The seed prefix.** A tag that starts with `0` belongs to the bootstrap seed. `Mint` never gives one. `MintSeed` gives one.
- **No tag that reads as another kind of identifier.** `Mint` does not give a tag that YAML reads as a number, or that starts with an MSC code (for example `20D60...`). For long tags it also refuses a tag with fewer than two letters. That rule removes each tag that could be an ISBN-10 (nine digits and a digit or `X`) and each tag that looks like a phone number or a numeric id.

## K3.4 Input and display

- **The canonical form** is ten capital characters with no separator: `7K3QX2M9R2`. Files, registers, URIs and the API use only the canonical form.
- **For reading**, the workspace and printed works can show a tag in two groups of five: `7K3QX-2M9R2`. The hyphen is not part of the tag.
- **Normalization on input.** The resolver accepts lower case, hyphens and spaces, and the Crockford substitutions `I` and `L` for `1` and `O` for `0`. It normalizes, then it validates the check character. A long tag that fails the check gives the error "not a valid tag: the check character does not match". It never resolves to the nearest valid tag, because that can be another node.
- **Legacy tags** have no check character. The resolver accepts a six-character string only when it is in the legacy set of `registers/tags.reg`.

## K3.5 Sharding

Spec 04.5 shards node files by the first two characters of the tag, which gives 1,024 directories. At a million nodes that is about 1,000 files in each directory. The corpus target for nodes is 1 to 5 million (K2.7), which would give up to 5,000 files in each directory. That makes git trees large and slow.

**The new shard key is the first three characters of the tag**, for long and legacy tags. That gives 32,768 directories:

| Nodes | Files in each directory, on average |
|---|---|
| 1 million | about 31 |
| 5 million | about 153 |
| 30 million | about 916 |

The path of a node is `nodes/<c1 c2 c3>/<tag>.md`, for example `nodes/7K3/7K3QX2M9R2.md` and `nodes/3C5/3C53TD.md`. The same key shards `prov/`, `reviews/` and `compute/`. A path is not an identity, so moving the three legacy node files to the new shards changes no tag. The migration (K3.8) moves them.

## K3.6 One space for nodes and inventory rows

The inventory (K04) has one row for each candidate item. A row gets an identifier when the inventory first records it. The identifier has the format of a long tag, and it is minted from the same generator. It is recorded in the inventory register `registers/inventory/`, which is sharded like the nodes (K4.8), not in `registers/tags.reg`.

- A row identifier is never a node tag, and a node tag is never a row identifier. Rule `T08` (K3.8) checks that the two registers have no value in common.
- With this rule, each bare identifier resolves to one thing: a node, or an inventory row. The URI of a row is `m2://inv/<id>`. The URI of a node stays `m2://<tag>`.
- When a row becomes a node, the node gets its own tag, and the row records it in its outcome. The row identifier is not reused as the tag, because a tag promises a node, and most rows of the research layer never become nodes.

The other identifiers of math2 do not change: provisional ids (`~` and 8 hex characters), run ids, review ids, project ids and record ids of the record plane.

## K3.7 What this changes in the main spec

| Place | Change |
|---|---|
| 00, Conventions | "Tags are 6 characters" becomes "tags are 10 characters (9 random and 1 check character); the legacy tags of 6 characters stay valid". The example becomes `7K3QX2M9R2`. |
| 04.3, Format | replaced by K3.3 and K3.4. The text about 32^6 tags is replaced by K3.2. |
| 04.5 | the shard key is the first three characters (K3.5) |
| 04.4 | a URI accepts both tag lengths, and `m2://inv/<id>` names an inventory row |
| 05.1, 06.4 | the registers include the sharded inventory register `registers/inventory/` (K4.8) |
| 06.7 | the sizes use the new shard counts |
| 15.3, family T | `T08`: the tag register and the inventory register have no value in common; `T09`: each long tag has a valid check character; `T10`: no tag of six characters is minted after the cut-over commit |
| 20.1 | the trees show `nodes/<3>/` |

The change is additive for readers of math2: each URI that worked before still works.

## K3.8 What this changes in the code

The toolchain in `tamnd/math2-agent` holds the tag format in a few places. The change is one pull request for each repository, in this order:

1. **`tag`**: `Len` becomes 10 for new tags, with `LegacyLen` 6. `IsTag` accepts a long tag with a valid check character, or a legacy tag. `IsLegacy` reports the second case. `Mint` and `MintSeed` draw nine characters (45 bits, from 6 random bytes), refuse excluded values (K3.3), and append the check character. A new function `Check(payload string) byte` computes the check character, and `Normalize(s string) (string, error)` implements K3.4. Each exported function keeps a testable example.
2. **`schema/node.v1.json`** in both repositories: the tag pattern becomes `^([0-9A-HJKMNP-TV-Z]{6}|[0-9A-HJKMNP-TV-Z]{10})$`. The check character cannot be expressed in JSON Schema, so rule `T09` checks it. The review id pattern `r-TAG-NN` accepts both lengths. This is an additive change of node.v1 (spec 04.9), because each file that was valid stays valid.
3. **The shard function** (in `node` and its users: `build`, `audit`, `site`, `prov`, `review`, `compute`): three characters. `m2 migrate shard3` moves the files of an existing tree, and it is safe to run again.
4. **`audit`**: `T06` accepts both lengths. New rules `T08`, `T09` and `T10` as in K3.7. `T10` reads the cut-over commit from `registers/README.md`.
5. **The Lean attribute** `Math2.Meta.Attr`, which checks the tag argument of `@[math2 TAG]`, accepts both lengths and the check character.
6. **The site and the API**: the resolver normalizes input (K3.4). Pages and JSON use the canonical form. The display form with a hyphen is used only in printed works (K11).
7. **`tamnd/math2`**: run `m2 migrate shard3`, record the cut-over commit, and freeze the legacy set. On the day of this spec the legacy set is three tags: `3C53TD`, `AAYEQE` and `G5S1M9`.

The test fixtures of the toolchain use legacy tags in many places. They stay as they are, because legacy tags stay valid. New fixtures use long tags.

## K3.9 Rejected options

- **Convert each legacy tag to a long tag.** This would edit a tag, which spec 04.3 forbids. It would also break each link to the published pages of the three legacy nodes. The gain is only that all tags have one length.
- **Ten random characters with no check character.** This gives 32 times more values, which we do not need, and it loses typo detection, which we do need for tags that people type.
- **The check symbol of Crockford's own scheme.** It uses modulo 37 and needs the five extra symbols `* ~ $ = U`. These symbols are not safe in URLs, file names and Lean identifiers.
- **A sequential counter.** It needs coordination between branches, and it leaks the order of creation, which some contributors do not want to show.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/04-inventory.md">
# K04. The inventory

The goal of K02 says "each definition, theorem, proof and example ... that a person can find in a public source". To know when that goal is reached, math2 needs a list of what is out there. The inventory is that list. It is the denominator of the corpus: the open count (K2.1) counts its rows that have no outcome.

This document says what a row is, how rows come from sources, how mentions of one item become one row, which outcomes a row can have, how we measure what the inventory misses, and where it is stored.

## K4.1 Mentions and rows

The inventory has two levels:

- A **mention** is one item as one source presents it. "Theorem 3.14 of Rudin's *Principles of Mathematical Analysis*, 3rd edition", "the Wikipedia article *Heine–Borel theorem*", "the Mathlib declaration `Metric.isCompact_iff_isClosed_bounded`" and "item 47 of the list that the enumerate role made for MSC 54D30" are four mentions.
- A **row** is one distinct item, in the sense of K2.3. It groups the mentions of that item. The four mentions above can be one row, "Heine–Borel theorem for $\mathbb{R}^n$", or two rows if one of them states a more general form.

Mentions are records of the record plane. They use the record identifier of spec 9.5: `m2://rec/<source>/<id>#<n>`, or `ax://paper/<id>#<tag>` for arXiv. Rows have identifiers of the format of K03, in the space that they share with node tags (K3.6).

A mention holds:

| Field | Holds |
|---|---|
| `id` | the record identifier |
| `source` | the source, from `manifests/sources.yaml` |
| `snapshot` | the hash of the source snapshot that the importer read |
| `locator` | where in the source: a URL with an anchor, a theorem number, a page, a declaration name |
| `native_id` | the identifier of the item in the source, when it has one: a Wikidata QID, a ProofWiki page, a Mathlib declaration, an nLab page, an OEIS number |
| `kind` | the kind that the importer or the model reads, one of the 11 kinds of spec 04.2, or `unknown` |
| `name` | the name that the source uses, if it has one |
| `text` | the statement text, only when the licence class allows text in the record plane (spec 9.3); otherwise its hash and its embedding, which stay internal |
| `msc` | the MSC2020 codes, from the source or from the classifier |
| `class` | the licence class of the source or of the item (spec 9.3) |
| `by` | the importer or the run that made the mention |

A row holds:

| Field | Holds |
|---|---|
| `id` | the row identifier |
| `mentions` | the mention identifiers |
| `kind`, `name`, `msc` | the values that the clustering decided |
| `layer` | `textbook`, `formal` or `research` (K2.7), computed from the sources of the mentions |
| `spread` | the number of independent sources among the mentions (K12.3) |
| `outcome` | K4.4, or empty |
| `history` | each merge, split and outcome change, with the commit and the person or run |

## K4.2 Where mentions come from

Each source in scope has an importer. K05 lists the sources of people and K06 the formal libraries. The importers give the mentions of their source, and nothing else: an importer does not decide which mentions are the same item.

Mentions come from four kinds of place:

1. **Reference works and wikis**: Wikidata, Wikipedia, ProofWiki, nLab, PlanetMath, the Encyclopedia of Mathematics, the Stacks project, Kerodon, the Digital Library of Mathematical Functions and others (K05). These give names, statements and a structure of links.
2. **Textbooks and lecture notes**: the theorem environments of open textbooks give text. The indexes and the tables of contents of closed textbooks give records and locators only (K05).
3. **Formal libraries**: Mathlib and the other libraries of K06. Each declaration is a mention. Most of them are not items that a person cites (K4.5).
4. **Models**: the enumerate role (K7.3) asks models to list each item of an area that they know, with a source for each. A model list is a mention source like any other. Its mentions are marked `by: agent`, and K4.3 never makes a row from model mentions alone (rule I03).

The research literature is the record plane of spec 09. Its extracted results are mentions already.

## K4.3 From mentions to rows

The clustering follows the four passes of spec 05.4, in order. Each pass is cheaper and more certain than the next one:

1. **Identifiers.** Two mentions with the same identifier, or with identifiers that a crosswalk links, belong to one row. The crosswalks are the `same_as` edges of the graph, the external-identifier properties of Wikidata, and the links between the sources themselves (an nLab page that links to a Wikipedia article on the same notion, or a Mathlib docstring that names a theorem).
2. **Formal equivalence.** Two mentions with formal statements that a Lean proof shows equivalent belong to one row.
3. **Similarity and judgement.** For each mention that has no row, the clustering finds the nearest rows by name, by embedding and by the links of the mention. The dedup judge classifies each candidate pair as `same`, `generalizes`, `specializes`, `related` or `different` (spec 05.4). A `same` with high confidence and with agreement of the name and the MSC code joins the row. Other `same` verdicts go to the `cluster` queue (K12.2).
4. **People.** Curators decide the close cases in the queue.

The granularity rules of spec 05.4 apply to rows. A family of forms that people cite separately is a row for each form, and a row for the hub. A `generalizes` or `specializes` verdict gives two rows and an edge between them, not one row.

The clustering is measured. A gold set of 2,000 pairs for each layer, with decisions by curators, gives the precision and the recall of each pass. The release manifest records them. An error in clustering has two forms, and both are bad. A false merge hides an item, and the open count does not show it. A false split gives two nodes for one item, and the dedup queue must find it later.

## K4.4 Outcomes

Each row must get an outcome. The open count is the number of rows that have none. The target is zero (K2.1).

| Outcome | Means | Who decides |
|---|---|---|
| `node` | the row is a node, with its tag | the curate role proposes, the merge decides |
| `member` | the row is a member of a hub family, as its own node | the same |
| `merged` | the row is the same item as another row, and it points to that row | the clustering, or a curator |
| `split` | the row was more than one item, and it points to the rows that replace it | a curator |
| `aligned` | the row is a result of the research layer that a node states, and it has an `aligned` edge to the node | the align role, with the rules of spec 09.6 |
| `record` | the row stays a record of the research layer, with no node, until the graph needs it (spec 9.8) | the rule K4.5 |
| `formal_only` | the row is a declaration of a formal library that no person cites, kept in the crosswalk (K4.5) | the rule K4.5 |
| `object_record` | the row is a database object that stays a record (K4.6) | the rule K4.6 |
| `out_of_scope` | the row is not mathematics in the sense of spec 18.2, or it is a notation, a teaching note or a mistake in the source. The reason is required | a curator, or a rule that a curator approved |
| `deferred` | the decision waits for something named, until a date | a curator |

`merged` and `split` keep the row identifier in the register forever, as a tag is kept (spec 04.3). A link to an old row identifier resolves to its successor.

A `deferred` row counts in the open count when its date passes. A `deferred` row with no date is not allowed.

Most outcomes come from rules, not from people. A rule is a decision that a curator approved once and that applies to many rows, for example "each mention from the enumerate role that has no other mention after 90 days gets `out_of_scope`, reason: no source confirmed". The rules live in `schema/outcomes.v1.yaml`, and each outcome that a rule gave names the rule.

## K4.5 Which formal declarations and research results become nodes

The formal libraries have many declarations that no person would cite: API lemmas, simp lemmas, instances and auxiliary results. A node for each one would give a graph that nobody can read. The research layer has 10 to 30 million results, and spec 9.8 already keeps them as records until the graph needs them.

**A formal declaration becomes a node** when at least one of these is true:

- a mention from a human source (K05) is in the same row;
- the library documents it as a main result, in the module documentation, the list of named theorems, or a list such as "100 theorems" or "1000+ theorems";
- a node of the graph uses it in a `uses` edge that a person wrote;
- a person asks for it.

Each other declaration gets the outcome `formal_only`. It stays in the crosswalk, so that the formal binding of a node can point to it, and search can find it. Spec 07 and K06 give the details for each library.

**A research result becomes a node** when spec 9.8 says so: when the graph needs it, or when a project or a campaign asks for its area (spec 18.5). Until then its row has the outcome `record` or `aligned`. The open count of the research layer goes to zero by these two outcomes, not by nodes.

## K4.6 Which objects become nodes

The databases of mathematical objects hold millions of records: the OEIS, the LMFDB, the House of Graphs, the ATLAS of finite groups, the GAP libraries of small groups and others. A node for each would add little that the database does not have. The rule is in line with spec 06.7:

- **An object becomes a node** when a human source names it, when a node cites it, when it is in a curated list (the sporadic groups, the exceptional Lie algebras, the regular polytopes), or when a project asks for it.
- **The OEIS** is a special case. Each sequence has a name and a curated entry, and spec 06.7 counts them as nodes. They become nodes in bulk, at W0, with `same_as` to the A-number.
- **Each other object** gets the outcome `object_record`. It keeps its database identifier as a `same_as` target, so that a node can point to it.

## K4.7 Measuring what the inventory misses

The open count is zero only for the rows that exist. An item that no source in scope holds, and that no model lists, has no row. So the inventory needs a measure of its own recall.

1. **Capture and recapture.** When two independent sources cover the same area, the overlap of their mentions gives an estimate of the total. If source A has *a* items of an area, source B has *b*, and *m* items are in both, the estimate of the total is *a* · *b* / *m* (the Lincoln and Petersen estimator). With three or more sources, log-linear models give a better estimate, and they allow for sources that depend on each other. The estimate is computed for each MSC code at two digits, each quarter.
2. **A held-out source.** When a new source enters scope, the clustering runs on it before the importer adds its mentions. The share of its items that already have rows is an estimate of the recall of the inventory for its area.
3. **Index terms.** The indexes of standard textbooks list the terms that each book defines. Each index term of a book in scope must match a row. The match rate for each book is in the report.
4. **Model lists.** The enumerate role is a recall tool, not a source of truth. An item that a model lists, and that the inventory does not have, is a candidate gap. A gap becomes a row only when K9.5 confirms a source for it.

The report gives the estimated recall for each layer and each MSC code at two digits, with the method and the interval. Where the estimate of the total is much larger than the number of rows, the area goes to the `enumerate` queue and to the importer plan of K05.

## K4.8 Storage

- **Mentions** are files of the record plane: `records/mentions/<source>/<c1c2c3>.jsonl`, sorted by identifier, one mention per line. The shard key is the first three characters of a hash of the mention identifier.
- **Rows** are in `records/inventory/<c1c2c3>.jsonl`, sharded by the first three characters of the row identifier (K3.5), one row per line.
- **The identifier register** is `registers/inventory/<c1c2c3>.reg`, sharded in the same way, because at the scale of the research layer one file would be larger than a gigabyte. Its rules are those of `registers/tags.reg`: lines are never removed or edited. K3.6 and rule T08 apply to it.
- **The licence zone** of mentions and rows is metadata, CC0 (settled decision 17). A mention with text holds only text that its class allows in the record plane (spec 9.3), under the licence of its source.
- When the record plane moves to `tamnd/math2-records` (spec 06.1), mentions, rows and the inventory register move with it.

The build reads the inventory and writes the open count, the coverage of each layer and the queues of K12.2 into the derived store.

## K4.9 Rules

| Rule | H/S | Checks |
|---|---|---|
| I01 | H | each `inventory` value in a node points to a row whose outcome is `node` or `member` with the tag of that node |
| I02 | H | each row with the outcome `node` or `member` points to a live node, or to a retired node with a successor |
| I03 | H | no row has only mentions from models. Such mentions wait in the `enumerate` results until K9.5 confirms a source |
| I04 | H | each outcome that a rule gave names a rule in `schema/outcomes.v1.yaml` |
| I05 | S | a `deferred` row whose date has passed |
| I06 | S | an MSC code at two digits whose estimated recall (K4.7) is below 90% |
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/05-human-sources.md">
# K05. Human sources

People have written down most of the mathematics that the corpus must hold: in reference works, wikis, textbooks, papers, problem lists, syllabi and exercise banks. This document lists those sources, says what the corpus takes from each one, by which channel, under which licence, and in which order. K06 does the same for the formal libraries.

Each fact about a source has its URL and its date of reading in K01, parts A and B. All counts were read on 2026-10-26, unless the text gives another date. A fact marked "not settled" is in the open questions of K15.

## K5.1 What a human source gives

A human source can give five things. Most sources give only some of them:

| What | Example | Licence that it needs |
|---|---|---|
| **mentions** | "the ProofWiki page *Lagrange's Theorem (Group Theory)*", "Theorem 3.14 of Rudin, 3rd edition" | none: a name and a locator are facts |
| **open text** | the statement and proof on that ProofWiki page | `open` class (spec 9.3) |
| **locators** | "Hefferon, *Linear Algebra*, Chapter Two, §III.1", "DLMF 5.2.1" | none |
| **a link structure** | the links between nLab pages, the cross-references of a textbook | none for the edges. The selection of a whole collection can have a database right (K13.4) |
| **curricula** | a table of contents, a syllabus, a list of theorems | none for the structure. The words of the source are its author's |

Every source gives mentions. Only sources of the `open` class give text. Mentions and locators are the larger part of the value: they fill the inventory (K04), and the inventory is the denominator of the goal. A node can state a theorem in math2's own words from a locator, so a source that gives no text still gives the item.

## K5.2 The source document

Each source has one entry in `manifests/sources.yaml`. A person approves each entry and each change to its licence fields (spec 9.2). The importer reads nothing from a source that has no approved entry.

```yaml
- id: lebl-ra
  title: Basic Analysis I and II
  authors: [Jiří Lebl]
  kind: textbook            # reference | wiki | textbook | paper-index | extraction | curriculum | exercises | database
  url: https://github.com/jirilebl/ra
  licence: CC-BY-SA-4.0 OR CC-BY-NC-SA-4.0   # SPDX expression, as the source states it
  election: CC-BY-SA-4.0    # for a dual licence: the option that math2 takes
  class: open               # spec 9.3
  licence_read:
    at: https://github.com/jirilebl/ra      # the source itself, never an aggregator
    on: 2026-10-26
    by: <person>
  per_item_licence: false   # true if items have their own licences (K13.4)
  public_domain: null       # or {us: true, eu: true, basis: "..."} (K5.2.3)
  authored: human           # human | ai | mixed (K6.5)
  edition: <tag or edition> # pinned edition, where editions differ in licence
  channel: git              # dump | git | api | bulk | page | catalogue
  snapshot: {commit: <commit>, read: <date>}
  terms:
    robots: <summary or n/a>
    rate: <limit that the importer keeps>
    ai: <clause on AI or machine learning, or "none stated">
  attribution: "Jiří Lebl, Basic Analysis I and II, CC BY-SA 4.0, https://github.com/jirilebl/ra"
  adapter: importer/latexthm
  approved_by: <maintainer>
```

### K5.2.1 Read the licence at the source

The licence of a book or a site is read at the source: the `LICENSE` file of its repository, its front matter, or the licence page of the site. A catalogue such as the Open Textbook Library or the PreTeXt catalogue is a list of candidates, not a licence. Research B found three stale or simplified licences in the Open Textbook Library in one afternoon (K5.4.2).

The licence is read again at each new snapshot. A change of licence between snapshots changes the class of the new snapshot only. The text that math2 took under the old licence keeps that licence, and the source document keeps both, with the snapshot of each.

### K5.2.2 Dual licences

When a source offers a choice of licences, math2 takes the option that the prose zone can hold, and the field `election` records the choice. Examples: Lebl's three books (CC BY-SA 4.0 or CC BY-NC-SA 4.0), Hefferon's *Linear Algebra* (GFDL or CC BY-SA 3.0 US), and ProofWiki (CC BY-SA 3.0 and the GFDL). The attribution names the elected licence.

### K5.2.3 Public domain

Public domain depends on the country. In 2026, a work published in the United States before 1931 is in the public domain there. In the EU, a work is in the public domain 70 years after the death of its author, so in 2026 the author must have died before 1956. math2 is published in all countries, so:

- A work is `open` as public domain only when it is in the public domain in the United States **and** in the EU. The field `public_domain` records both answers and the basis for each, for example the date of publication and the year of the death of the author.
- A work that is in the public domain in one of them only is `record`.
- A translation, a modern edition and a modern commentary have their own copyright. The importer takes only the text of the original, from an edition whose added text it does not use. Project Gutenberg texts lose the Project Gutenberg header and licence text, as its licence requires.

Hardy's *A Course of Pure Mathematics* (Project Gutenberg #38769) passes both tests: it was published before 1931, and Hardy died in 1947.

### K5.2.4 The `authored` field

The field `authored` of K6.5 applies to human sources too. A source that people wrote is `human`. A source with parts that a model wrote is `mixed`, and the source document says which parts. For example, the "slogans" of the TheoremSearch datasets were written by DeepSeek-V3.1, so they are `ai`, while the statements are from the papers. Text from an `ai` part counts as agent text for the rules of K07 and K09: it needs the same checks as text that a math2 role wrote.

## K5.3 Reference works and wikis

| Source | Size | Licence | Class | The corpus takes | Channel |
|---|---|---|---|---|---|
| Wikidata | 3,208 theorems; 113,118 items with a defining formula (P2534), but 105,345 of these are prime numbers, so 6,560 real formulas | CC0 | `open` | items, labels, crosswalk properties, formulas | weekly JSON dump (103 GB); WDQS only for checks |
| Wikipedia (en) | 28,196 mathematics articles (WikiProject Mathematics) | CC BY-SA 4.0 | `open` | mentions, text with attribution, the link graph | monthly XML dump |
| Wikipedia (de, fr, ja, zh, ru) | 20,732, 12,192, 8,865, 5,682, 3,928 (each wiki counts in its own way) | CC BY-SA 4.0 | `open` | mentions; text in a later phase | monthly dumps |
| ProofWiki | 47,254 content pages; 30,348 proofs and 36,102 definitions (Main Page) | CC BY-SA 3.0 and GFDL | `open`, elect CC BY-SA 3.0 | mentions, statements, proofs, definitions | daily dump `latest.xml.gz` (36.6 MB) |
| PlanetMath | 9,477 entries, inactive since 2018 | CC BY-SA 3.0 | `open` | mentions and text | git, 64 repositories by MSC class |
| nLab | 20,778 pages | none. The site allows use "if you acknowledge the source", but no right to adapt | `record` | page names, the link graph | git mirror `ncatlab/nlab-content` |
| Encyclopedia of Mathematics | live count not settled; 1,224 Wikidata ids | originals © Springer; only new wiki text is CC BY-SA 3.0 and GFDL | `record` | ids and links. Text only from an article that its history shows to be wholly new | none (the site gave HTTP 502 on the day of reading) |
| MathWorld | 16,939 entries | proprietary, no bulk copies | `record` | ids, from Wikidata (P2812, 6,840 items) | none |
| OEIS | 400,709 sequences | CC BY-SA 4.0 | `open` | names, terms, formulas, references | git `oeis/oeisdata` |
| DLMF | version 1.2.8 (15 September 2026) | © NIST, no bulk copies | `record` | equation numbers as locators | none |
| Fungrim | 3,130 entries, dormant since 2022 | MIT | `open` | formulas with their conditions | git |
| Wolfram Functions | 307,409 formulas | © Wolfram Research | `record` | formula ids as locators | none |
| LMFDB | figures of 2022 (not measured again) | data CC BY-SA 4.0; code GPL | `open` (data) | object records by label | bulk downloads only. Its `robots.txt` forbids `/api/` |
| groupprops, Topospaces, Commalg, Diffgeom | 8,076, 825, 429, 435 articles | CC BY-SA 3.0 | `open` | mentions and text, mostly facts about groups | MediaWiki API, slowly: one maintainer |
| Manifold Atlas | 215 articles | contradictory: GFDL 1.3 and "public domain" | `gfdl` until the editors answer | mentions and links | MediaWiki API |
| π-Base | 225 spaces, 247 properties, 935 theorems | CC BY 4.0 | `open` | theorems as statement candidates | git |
| DaRT | 218 rings, 178 properties, 72 theorems | CC BY 4.0 | `open` | the same | git |
| House of Graphs, ISGCI, KnotInfo, Knot Atlas, ATLAS v3 | about 22,000 graphs (2022); 1,695 graph classes; knots to 13 crossings; 2,710 articles; about 716 groups | no licence stated | `record` | ids and facts, as metadata | downloads where offered. Not the GPL-3.0 CSV package of KnotInfo |
| Stacks Project | about 21,000 tags | GFDL 1.2 | `gfdl` | tags, titles, the dependency graph | git |
| Kerodon | 8,189 tags | no licence stated | `record` | tags, titles, the dependency graph | none: page by page, slowly |
| Open Problem Garden | at most 713 problems | GFDL | `gfdl` | mentions and links | none |
| Erdős problems | 1,221 problems | the site states no licence; `teorth/erdosproblems` is Apache-2.0 but has no statements | `record` (statements), `open` (status) | status, links, the problem number | git `data/problems.yaml` |
| Kourovka Notebook | 21st edition, arXiv:1401.0300v49 (20 October 2026) | arXiv non-exclusive licence | `record` | problem numbers with the edition | arXiv |
| MathOverflow | 168,603 questions | CC BY-SA 2.5, 3.0 or 4.0, by the date of each post | `open`, for each post | mentions, references, text of single posts with attribution | the API, with a key |
| Math.StackExchange | 1,702,027 questions | the same | `open`, for each post | the same | the same |
| FindStat | 1,997 statistics | not settled | `record` until settled | ids | none |

Notes on the table:

- **Wikidata formulas.** The P2534 count is mostly primes. The importer drops the items that are instances of *prime number* (Q49008) and its kin before it treats P2534 as a source of formulas.
- **Stack Exchange dumps.** Since July 2024 the dumps are behind a click-through that excludes the training of large language models. math2 does not train a model on them, but the click-through is a contract, so the importer uses the API and not the dumps. The importer reads only posts that a node cites or that the enumerate role proposes as a source, not whole sites. Each post has its own licence, by date (rule L06).
- **Wiki text and the copy check.** Text from a wiki that enters a node keeps the licence of the wiki and the attribution of its history. The copy check (K9.4) sees it as a match with an open source. That match is allowed, and it adds the attribution.

## K5.4 Open textbooks

Open textbooks are the main source of text for the textbook layer (K2.7). They give statements, proofs, examples and exercises in the order that a course needs, and the theorem environments of their sources mark each item.

### K5.4.1 Table A: books that the prose zone can take

Each licence below was read at the source of the book. Each book needs attribution. A CC BY-SA 3.0 Unported text may be adapted under CC BY-SA 4.0 (its §4(b)). For the 3.0 US port of Hefferon, the same clause is assumed and not settled.

| Book | Licence that math2 uses | Source format | Notes |
|---|---|---|---|
| Open Logic Project | CC BY 4.0 | LaTeX | |
| *Homotopy Type Theory* (HoTT book) | CC BY-SA 3.0 | LaTeX | |
| Hefferon, *Linear Algebra* | CC BY-SA 3.0 US, elected from a dual licence | LaTeX | |
| Lebl, *Basic Analysis* I and II, *Notes on Diffy Qs*, *Tasty Bits of Several Complex Variables* | CC BY-SA 4.0, elected from a dual licence | LaTeX | |
| Boelkins et al., *Active Calculus* (single variable) | CC BY-SA 4.0 | PreTeXt | the Open Textbook Library entry is for the 2017 edition and is stale |
| Chen, *An Infinitely Large Napkin* | CC BY-SA 4.0 (text) | LaTeX | its code is GPL v3, and math2 does not take it |
| Levin, *Discrete Mathematics: An Open Introduction*, 3rd edition and earlier | CC BY-SA 4.0 | PreTeXt | pin a commit before the 4th edition, which is CC BY-NC-SA |
| Keller and Trotter, *Applied Combinatorics* | CC BY-SA 4.0 | PreTeXt | |
| Ernst, *An Introduction to Proof via Inquiry-Based Learning* | CC BY-SA 4.0 | LaTeX | |
| Austin, *Understanding Linear Algebra* | CC BY 4.0 | PreTeXt | |
| Beck, Marchesi, Pixton and Sabalka, *A First Course in Complex Analysis* | CC BY 4.0 | PDF | |
| Lehman, Leighton and Meyer, *Mathematics for Computer Science* (2018) | CC BY-SA 3.0 | PDF | the MIT OCW course pages are NC; this book is not |
| OpenStax *Statistics*, *Precálculo* 2nd edition, and the earlier CC BY editions of *Prealgebra* (2015) and *Precalculus* (2014) | CC BY 4.0 | CNXML | pin the CC BY edition |
| Hardy, *A Course of Pure Mathematics*, and other public-domain books that pass K5.2.3 | public domain | LaTeX, HTML | strip the Project Gutenberg text |

The Open Textbook Library lists more books under CC BY or CC BY-SA, for example Zakon, Kuttler, Shimamoto, Petrunin, Callahan and *OpenIntro Statistics*. They are candidates. Each one enters Table A only after a person reads its licence at the source.

### K5.4.2 Table B: books that the prose zone cannot take

These books give mentions, locators, tables of contents and internal embeddings, but no text in nodes.

| Book | Licence | Class |
|---|---|---|
| Judson, *Abstract Algebra: Theory and Applications* | GFDL 1.3 or later | `gfdl` |
| Beezer, *A First Course in Linear Algebra* | GFDL 1.2 or later | `gfdl` |
| the PreTeXt titles with a GFDL badge (21 titles, for example Yoshiwara's three books and *The ODE Project*), Grinstead and Snell, Bogart | GFDL | `gfdl` |
| Hartman et al., *APEX Calculus*; Axler, *Linear Algebra Done Right*, 4th edition | CC BY-NC 4.0 | `share-alike-nc` |
| *CLP Calculus* (UBC); Levin, 4th edition; the OpenStax 2nd editions of calculus and algebra; MIT OpenCourseWare | CC BY-NC-SA 4.0 | `share-alike-nc` |
| Hammack, *Book of Proof*, 3rd edition; Shoup | CC BY-NC-ND | `verbatim` |
| Hatcher, *Algebraic Topology* | © CUP 2002, noncommercial copies only | `record` |
| Lurie, Kerodon | no licence stated | `record` |
| the PreTeXt titles *Linear Algebra* (W214) and *Intro to Topology (IBL)* | all rights reserved | `record` |
| LibreTexts pages tagged "not declared" | none | `record` |

Three books on this list are often taken to be open: Hatcher, *Book of Proof* and Axler. They are not open for the prose zone.

### K5.4.3 Aggregators

- **Open Textbook Library.** 214 mathematics books. By its own licence data, about 48% are `open`. Its licences can be stale or simplified: it lists *Active Calculus* as CC BY-NC-SA (true for the 2017 edition only), *Book of Proof* as "Attribution-NoDerivs" (the author says CC BY-NC-ND), and Hefferon as CC BY-SA only (the book is dual-licensed). The corpus uses it as a list of candidates.
- **AIM Open Textbook Initiative.** 68 approved titles. The list shows no licences. Fourteen titles have licences that nobody has read yet (K15).
- **PreTeXt catalogue.** 67 mathematics projects. Many have only a badge that says "Creative Commons", with no variant. Each one needs a check at the source.
- **LibreTexts.** The licence is set page by page, and one book can mix licences. The importer goes to the upstream source of each book. Where it reads LibreTexts, it reads page by page, by the `license:` tag of each page. A page without a declared licence is `record`.

### K5.4.4 Editions

Some books changed licence between editions: Levin's 4th edition became CC BY-NC-SA, and the OpenStax 2nd editions are CC BY-NC-SA where the 1st editions were CC BY. An earlier edition keeps the licence under which it was published. So the source document of such a book pins the edition and the commit (field `edition`), and the importer reads nothing from a later commit until a person approves the licence of that commit.

### K5.4.5 What the importer takes from an open textbook

From the theorem environments of the source (`\begin{theorem}`, PreTeXt `<theorem>`, CNXML `<rule>`, and the like):

- the kind, the number and the title of each environment, as a mention with a locator;
- the statement, the proof, the examples and the exercises, as text of the record plane under the licence of the book;
- the cross-references between environments, as candidate `uses` edges;
- the table of contents and the index, as a curriculum (K5.8) and as index terms for K4.7.

The text enters a node only by the importer of `open` text (K14.4), with the attribution of the book, and the copy check marks it as a match with an open source. The write role does not paste it: a role that writes a node from scratch has no text of a restricted book in its context (K7.5), and it may have open text there, with its licence.

## K5.5 Closed textbooks

Most of the standard textbooks are not open. They give records and locators only:

- **The record of the book**: ISBN, edition, zbMATH and MR ids, from catalogues (K5.6).
- **Locators**: "Rudin, *Principles of Mathematical Analysis*, 3rd edition, Theorem 3.14". A locator is a fact.
- **The table of contents and the index**, where the publisher shows them in public, for example in a preview or on the site of the book. They go into the record plane as one locator for each item, not as a copy of the table (K13.4). An index term is a mention of a definition, and K4.7 uses the index terms of standard books to measure recall.
- **Not the text.** No text of a closed book enters the record plane, the context of a role, or a node. E2 needs two book locators that a person has checked (spec 9.7), because that check needs the book.

The list of closed textbooks in scope comes from the curricula of K5.8: each book that a syllabus names is a candidate.

## K5.6 Literature indexes and full text

The research literature is the record plane of spec 09. This section adds the facts of 2026 that change what the importers do.

| Source | Size | Licence | The corpus takes |
|---|---|---|---|
| arXiv (through `tamnd/arxiv`) | about 9.2 TB of files (April 2025) | metadata CC0; text by the licence of each paper | records; the text of `open` papers |
| zbMATH Open | 4M+ publications, 3M+ reviews; 133,093 documents for 2025 | knowledge graph: CC BY 4.0 or CC BY-SA 4.0 (the record says both) | records, MSC codes, reviews as evidence |
| OpenAlex | about 745 GB (September 2026) | CC0 | all metadata |
| Crossref | 180,034,490 records (March 2026) | metadata free; some abstracts have copyright | metadata and references; abstracts are `record` |
| DataCite | 108M DOIs | CC0 | metadata of datasets and software |
| Semantic Scholar, S2ORC | 15,679,781 full texts (May 2026) | ODC-By for the database, the copyright of each paper for its text | ids. Text only by the licence of each paper |
| CORE | 291M records (2023) | ODC-By for the database | records. Text by the licence of each paper |
| HAL (mathematics) | 170,406 records, 105,413 with a file | metadata CC0; 9.8% of the files are `open` | records; the 10,349 open texts |
| Numdam | 75,599 articles | metadata CC0; text for personal use, no bulk | records. Text only where an article has a CC licence or is in the public domain |
| EuDML, Project Euclid, GDZ | | not settled; "free to read" is not an open licence | records, through partner sources |
| HathiTrust, Internet Archive, Project Gutenberg | 206,597 mathematics texts on the Internet Archive | public domain for some items; a hosted copy says nothing about the licence | public-domain books that pass K5.2.3 |

Three facts change the plan of spec 09:

1. **More of arXiv is open than spec 9.3 says.** In a sample of three days of 2026, 36.9% of new mathematics papers were `open` (CC BY 34.0%, CC0 1.9%, CC BY-SA 1.0%). Spec 9.3 gives about 15%, which is the share of all papers since 1991. The sample is small, so phase 1 measures the share over all of 2026 from the metadata plane, and spec 9.3 then gives both figures.
2. **The zbMATH knowledge graph is updated about twice a year**, not each month. Its licence is CC BY 4.0 in one field of the Zenodo record and CC BY-SA 4.0 in its description. Both are `open`, but they differ for derived exports, so the importer treats it as CC BY-SA 4.0 until zbMATH answers (K15).
3. **Full-text aggregators do not change the licence of a paper.** S2ORC, CORE and the Internet Archive hold texts under the copyright of each paper. An importer takes a text from them only when the licence of that paper allows it.

## K5.7 Extraction projects

Spec 9.6 says that math2 aligns to the large extraction projects and does not compete with them. The data of 2026:

| Project | Size | Licence | The corpus takes |
|---|---|---|---|
| TheoremSearch | 8.4M statements in the live index | the CC BY-SA release mixes in GFDL text from the Stacks Project. The `-permissive` release (1,239,720 theorems) is CC BY 4.0 and excludes share-alike | the `-permissive` release as candidate mentions (E1), with the licence of each paper. Its slogans are `ai` (K5.2.4) |
| TheoremGraph (`math-graph`) | 16,131,238 rows | CC BY 4.0, with the licence of each statement. The text is empty unless the paper is open | the edges and the ids, with attribution. The text where it is present |
| TheoremGraph (`theorem-matching`) | 100,831 candidate pairs of Lean declarations and arXiv statements | CC BY-SA 4.0 (K01, part C) | candidate crosswalks (K6.7), with share-alike attribution |
| Matlas | 8.07M statements from 435K papers and 1.9K textbooks | no licence or terms stated | ids of matches at query time only (`record`) |
| the Stacks Project and Mathlib alignment (arXiv:2610.31011) | | paper CC BY 4.0 | candidate crosswalks, which the align role checks (K6.7) |

The BY-SA release of TheoremSearch is not taken as one block. The importer filters it by source and by the licence of each row, and drops the Stacks rows and every row whose licence is NC or not stated.

## K5.8 Curricula and theorem lists

Curricula say which items a course needs, and in which order. The corpus imports them as data (K10.3), never as text:

| Source | Size | Licence | The corpus takes |
|---|---|---|---|
| Mathlib `docs/undergrad.yaml` | 563 items, 130 of them not yet formalized | Apache-2.0 | all: items, structure, declarations |
| Mathlib `docs/overview.yaml` | 511 items | Apache-2.0 | all |
| Mathlib `docs/1000.yaml` | 1,199 entries keyed by Wikidata id; 217 with declarations | Apache-2.0 | all. The Wikidata keys are crosswalk ids |
| Mathlib `docs/100.yaml` | 100 entries; 78 with declarations | Apache-2.0 | all |
| 1000+ theorems project | 228 entries on the site | Unlicense | all |
| Wiedijk, *Formalizing 100 Theorems* | 100 theorems, with counts for 13 systems | no licence | the list of names and the counts, as facts |
| MAA CUPM Curriculum Guide (2015) | 116 pages | © MAA | course areas and topic lists, as facts |
| QAA Subject Benchmark Statement, Mathematics, Statistics and Operational Research, 5th edition (2023) | 35 pages | © QAA | outcome categories, as facts |
| qualifying exam syllabi (for example UC Berkeley) | | no licence | topic lists, as facts. Not the past exam problems |
| tables of contents of open and closed textbooks | | K5.4, K5.5 | the order of the items |

A curriculum is also a mention source: each item that it names is a mention, with the curriculum as its source. The `curriculum` term of the priority (K12.3) comes from these mentions. The difference between Mathlib's `100.yaml` (78 with declarations) and Wiedijk's count for Lean (83) is not settled.

## K5.9 Exercise banks

The large exercise banks have licences that the prose zone cannot take. They give links, and the exercise node (K8.7) is math2's own text:

| Bank | Size | Licence | The corpus takes |
|---|---|---|---|
| WeBWorK Open Problem Library | about 35,000 problems | CC BY-NC-SA 3.0 by default; an author can choose an OSI licence | the path of a problem as a link in `external`; its tags as metadata. No text |
| MyOpenMath and IMathAS | | question code GPL-2.0; the generated HTML CC BY 4.0 | links. The generated HTML of a question is `open`, but the importer takes it only after a person checks the licence of that question |
| STACK | | GPL | links |
| Numbas | | code Apache-2.0; public questions under Creative Commons, by question | links. Text only from a question whose licence is CC BY or CC BY-SA |
| Natural Number Game (NNG4), *Mathematics in Lean* | | Apache-2.0 | Lean exercises, into the code zone, with attribution |
| lean4game games | | the engine is GPL-3.0; each game has its own licence | links. A game with a permissive licence is like NNG4 |
| the exercises of Table A books | | the licence of the book | exercise text, with attribution |

The exercise role (K7.3) has no text of an NC bank in its context.

## K5.10 Channels and politeness

The importers keep each limit below, identify themselves with the User-Agent of the toolchain, `m2/<version> (+https://github.com/tamnd/math2-agent)`, with no personal address, and prefer a dump to live requests:

| Source | Channel | Limit that the importer keeps |
|---|---|---|
| Wikipedia | the monthly XML dump. The API only for single pages | API: a compliant User-Agent gets 200 requests a minute. The robot policy of 16 March 2026: dumps before live requests |
| Wikidata | the weekly JSON dump | WDQS: 60 s of query time a minute, 5 parallel queries, only for checks |
| ProofWiki | the daily dump | no crawling |
| nLab | the git mirror | no crawling |
| OEIS | git `oeisdata` | site: `Crawl-Delay: 10`, and `/search` is disallowed |
| LMFDB | bulk downloads | `/api/` is disallowed, and the site has a JavaScript gate. The importer does not try to pass it |
| MathOverflow, Math.SE | the API, with a key | 10,000 requests a day; keep the `backoff` field, which applies across sites |
| arXiv | `tamnd/arxiv` | one request each 3 seconds, for all machines together |
| Project Gutenberg | `/robot/harvest` or a mirror | `wget -w 2` |
| MathWorld, DLMF, Wolfram Functions | none | no requests. Ids come from Wikidata |
| Kerodon, groupprops, the Manifold Atlas | single pages | one request at a time, at most one each 10 seconds. These sites are small and have one maintainer or none |

A source that blocks automatic clients, or asks for a login or a click-through, is not read until a person decides how to proceed. The importer never passes a gate.

## K5.11 Adapters

Each source has one adapter, `importer/<source>`, in `tamnd/math2-agent` (K12.8). Spec 09.1 puts adapters in `internal/ingest/<source>`. The corpus uses `importer/<source>` instead, because the toolchain has no `internal/` packages. Most sources share a format, so most adapters are thin and use a format package:

| Format | Sources |
|---|---|
| MediaWiki XML dump | Wikipedia, ProofWiki |
| MediaWiki API | the subwikis, the Manifold Atlas |
| Markdown in git | nLab, π-Base |
| LaTeX theorem environments | PlanetMath, Open Logic, HoTT, Hefferon, Lebl, Napkin, Ernst |
| PreTeXt XML | *Active Calculus*, Levin, Keller and Trotter, Austin |
| CNXML | OpenStax |
| YAML in git | the Mathlib lists, 1000+, Erdős problems, DaRT |
| JSON dump | Wikidata |
| JSONL or Parquet | TheoremSearch, TheoremGraph, OpenAlex |
| PDF | Beck et al., *Mathematics for Computer Science*: locators and the table of contents only, until a source in LaTeX is found |

Each adapter has fixtures from a small sample of its source, and tests that use the fixtures and never the network (spec 19). An adapter writes mentions and nothing else (K4.2).

## K5.12 The order of import

The order follows K14. A source enters when the phase before it meets its exit:

1. **Phase 1, open data with no doubt about the licence.** Wikidata, the Mathlib lists, 1000+, OEIS names, π-Base, DaRT, Fungrim, and the records of OpenAlex, Crossref, DataCite, HAL and Numdam. Then the records and locators of the `record` and `gfdl` sources: the tags of the Stacks Project and Kerodon, the names of the nLab, the ids of MathWorld and the EoM from Wikidata, the tables of contents of closed textbooks where the publisher shows them.
2. **Phase 2, the pilot area (group theory).** Wikipedia, ProofWiki, PlanetMath and groupprops for the area, the Table A books that cover it (Judson is `gfdl`, so it gives locators only), the curricula of K5.8, and the Stack Exchange posts that the nodes cite.
3. **Phase 3 and later, the textbook layer.** The rest of Table A, the rest of the wikis, and TheoremSearch `-permissive` and TheoremGraph for the research layer. Each new source first goes through the held-out check of K4.7.

## K5.13 Corrections to the main spec

| Spec | Value now | Value on 2026-10-26 |
|---|---|---|
| 1.7, 9.2 | Wikidata: about 113k items with a defining formula | 113,118, but 105,345 are prime numbers. 6,560 are other formulas |
| 1.7, 9.2 | Wikipedia: about 26k mathematics articles; dumps twice a month | 28,196 articles; the dumps are monthly in 2026; new API rate limits |
| 1.7, 9.2 | nLab: about 19k pages, licence not known | 20,778 pages, no licence: `record` |
| 9.2 | ProofWiki: CC BY-SA 3.0, quarterly | CC BY-SA 3.0 and GFDL; a daily dump |
| 1.7, 9.2 | Encyclopedia of Mathematics: CC BY-SA 3.0 | originals © Springer: `record`, except wholly new articles |
| 9.2 | Erdős problems: statements from `teorth/erdosproblems`, Apache-2.0 | the repository has no statements. The statements on the site have no licence: `record` |
| 9.2 | LMFDB: API and bulk downloads | bulk downloads only: `robots.txt` forbids `/api/` |
| 9.2 | zbMATH Open: CC BY-SA 4.0, monthly | CC BY 4.0 or CC BY-SA 4.0 (not settled); twice a year |
| 9.3, 16.6 | Kerodon: GFDL | no licence stated: `record` |
| 9.3 | about 15% of arXiv is `open` | 15% of all papers; 36.9% of new mathematics papers in a 2026 sample |
| 9.3 | `gfdl` examples: Stacks, Kerodon, Open Problem Garden | Stacks, the Manifold Atlas, Open Problem Garden |
| 9.3 | `record` examples | add the DLMF, Wolfram Functions, the Kourovka Notebook, KnotInfo, ISGCI, the ATLAS and House of Graphs |
| 9.6 | TheoremSearch: 9.2M theorems; the 1.34M open release | 8.4M in the live index; take the `-permissive` release, because the BY-SA release mixes in GFDL text |
| 1.7 | MathWorld 16,849; OEIS 400,673 | 16,939; 400,709 |
| 9.1 | adapters in `internal/ingest/<source>` | `importer/<source>` |

## K5.14 Rules

| Rule | H/S | Checks |
|---|---|---|
| L08 | H | each source in `manifests/sources.yaml` has the fields `licence`, `class`, `licence_read`, `authored`, `channel`, `snapshot`, `attribution` and `approved_by`, and each mention names a source that has them (K5.2) |
| L09 | H | a source whose `licence` is a choice of licences has an `election`, and the elected licence gives the class (K5.2.2) |
| L10 | H | a source of the `open` class whose licence is public domain has `public_domain` with `us: true` and `eu: true` (K5.2.3) |
| L11 | H | a mention with text comes from a snapshot whose licence was read on or before the date of that snapshot (K5.2.1) |
| L12 | S | a source whose licence was last read more than twelve months ago |

With rules L01 to L07, these rules are the licence gate for human sources. The open questions of this document are in K15: the current licence of the EoM, the terms of Stack Exchange of 13 November 2025, the licences of the nLab and the Manifold Atlas, the knowls of the LMFDB, the zbMATH licence, the 14 AIM titles, the US port of CC BY-SA 3.0, and the sources that the research could not reach.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/06-formal-sources.md">
# K06. Formal sources

The formal libraries are the second large source of the corpus, after the human sources of K05. Mathlib alone has more than 430,000 named theorems and definitions. The other proof assistants have about as many again. This document says what the corpus takes from each library, by which route, under which licence, and what it does not take. It also covers the datasets that pair formal and informal text, and the translators between systems.

Spec 07 says how a binding works, and spec 07.10 says which systems can give E5. This document does not repeat those sections. Each number comes from K01, part C, read on 2026-10-26. Where a number has changed since spec 01, K6.11 gives the new value.

## K6.1 What a library gives the corpus

A formal library gives four things. Each has its own rule:

1. **Mentions.** Each declaration is a mention in the inventory (K4.2). Most declarations get the outcome `formal_only` (K4.5), and only the ones that a person cites, or that the library marks as main results, become nodes.
2. **Bindings.** A Lean declaration can be the binding of a node, and so give E5 (spec 07.3). In v1, only Lean libraries give bindings.
3. **Crosswalks.** A declaration in another system is a `same_as` target of a node (spec 05.3). It helps the clustering (K4.3, pass 1), and it is evidence of an independent formalization towards F3 (spec 07.10).
4. **Informal text.** Docstrings, comments and literate prose. The corpus takes this text only when the licence of the library allows it in the prose zone (K13.1).

## K6.2 Intake classes

Each library has one intake class. The source document of the library in `manifests/sources.yaml` records it, with the licence and the reason:

| Class | The corpus takes | Example |
|---|---|---|
| `bind` | mentions, bindings, crosswalks and the informal text | Mathlib, Formal Frontier |
| `crosswalk` | mentions and crosswalks. Informal text only if the licence allows it | the AFP, MathComp, set.mm |
| `link` | crosswalks and links. No text, no statements | 1Lab, ATLAS |
| `evaluate` | nothing for the graph. The dataset tests the roles of K07 | miniF2F, ProofNet, PutnamBench |
| `exclude` | nothing | MathPile |

The class follows from three questions, in this order:

1. **Can we use it at all?** A non-commercial licence, a clause against machine learning, or no licence at all gives `link` for a library and `exclude` for a dataset. A dataset with no licence can become `evaluate` only when its authors confirm its terms in writing.
2. **Can its text go into the prose zone?** Apache-2.0, MIT, BSD, CC0 and the Unlicense allow it, with attribution. CC BY-SA allows it, with share-alike and attribution. GPL, AGPL, LGPL and CeCILL-C are licences for software, and math2 does not put their comments into prose. Their libraries give mentions and crosswalks only.
3. **Can it give bindings?** Only Lean 4 libraries, in v1 (spec 07.10).

## K6.3 The libraries

| Library | Size (date) | Licence | Class | Route | Informal text |
|---|---|---|---|---|---|
| Mathlib | 291,237 theorems and 139,616 definitions (stats page); 797,294 constants, 483,112 with `allowCompletion` (v4.35.0, 2026-10-13) | Apache-2.0 | `bind` | lean_scout datasets (K6.4) | docstrings on about 14.5% of constants |
| Tau Ceti | 18,849 `.lean` files | Apache-2.0 | `bind`, with the flag of K6.5 | as Mathlib | docs site; written by AI |
| Formal Frontier | 41 libraries, about 200k lines | Apache-2.0 (30 of 50 repositories checked) | `bind` | as Mathlib | per library |
| Other Reservoir packages | 908 packages | per package | per package | Lake | per package |
| ATLAS (Meta) | 26 textbooks, more than 45,000 declarations | v1: CC BY-NC 4.0 with a clause against ML. v2: Apache-2.0 with a preamble against ML | `link` | none | textbook-aligned |
| Isabelle AFP | 1,042 entries, about 330,000 lemmas | BSD-style or LGPL, per entry | `crosswalk` | the release tarball, Isabelle export | abstracts and `text` blocks; BSD entries only |
| Isabelle2026 library | released October 2026; size not known | BSD-style core, mixed | `crosswalk` | Isabelle export | `text` blocks of BSD parts |
| Rocq Stdlib | 904 `.v` files | LGPL-2.1 | `crosswalk` | opam | none |
| MathComp, Odd Order | 147 and 34 `.v` files | CeCILL-B | `crosswalk` | opam | header comments |
| MathComp Analysis | 131 `.v` files | CeCILL-C | `crosswalk` | opam | none |
| UniMath (Rocq) | 1,714 `.v` files | MIT-style | `crosswalk` | opam | comments |
| HOL Light, Flyspeck | 664 `.ml` and 503 `.hl` files | BSD-style, MIT | `crosswalk` | hol2dk dump | comments |
| HOL4 | 1,495 `Script.sml` files | BSD-style | `crosswalk` | HOL4 export | comments |
| Metamath set.mm, iset.mm | 51,207 and about 17,395 statement numbers | CC0 1.0 | `crosswalk`, with text (K6.6) | parse of the `.mm` file | a comment on each statement |
| Mizar MML | version 5.94.1493 | GPLv3+ or CC BY-SA 3.0+ | `crosswalk` | XML export | little |
| agda-stdlib | 1,183 files | MIT-style | `crosswalk` | Agda HTML | comments |
| agda-unimath | 3,072 `.lagda.md` files | MIT | `crosswalk`, with text (K6.6) | literate parse | prose on each page |
| 1Lab | 710 `.lagda.md` files | AGPL-3.0 | `link` | none | prose, not taken |
| Naproche | 32 `.ftl.tex` files | GPL-3.0 | `crosswalk` | ForTheL | not taken |
| Megalodon | about 160,000 lines | system MIT-style; the mgwiki repositories have no licence | `link` | none | none |
| NASALib (PVS) | 63 or 64 libraries, about 41,000 formulas | public domain for NASA work, per library for the rest | `crosswalk`, per library | PVS | comments |
| ACL2 books | 40,929 files and directories | BSD-3 and per file | `crosswalk`, per file | ACL2 | XDOC |

Two rules apply to each row:

- **The licence is recorded for each item where the library has a licence for each item.** The AFP has one for each entry, NASALib for each library, the ACL2 books for each file. The importer reads it, and rule F10 checks it.
- **The licence is read again at each pin.** A library that changes its licence changes its class from that pin on. Mentions from older pins keep the licence that they had.

## K6.4 Mathlib

**The route.** The Mathlib Initiative publishes the datasets of `lean_scout` (Apache-2.0) for each toolchain bump: `mathlib-types` (name, module, type, docstring, `allowCompletion`), `mathlib-const-dep` (the dependencies of each constant) and `mathlib-tactics` (goal states). The importer reads these datasets at the commit of the release pin. It does not run its own walk over the environment for mentions. `M2Extract` (spec 07.4) still runs in the build, for the bound declarations and their closure only, because the type hash and the axioms of a binding must come from the pinned build.

**What a mention is.** The three published counts measure different things: 430,853 named theorems and definitions on the stats page, 483,112 constants that the user can complete, and 797,294 constants in all. The corpus takes the second: a mention is a constant with `allowCompletion = true`. The other constants are internal or generated, and no person would cite them. The report gives all three numbers, with the definition that each one uses.

**Which mentions become nodes.** The rule of K4.5 applies. For Mathlib the corpus uses these signals, in order:

1. a human mention in the same row (pass 1 of the clustering uses the Wikidata and the docstring links);
2. the named-theorem lists: the "100 theorems" and "1000+ theorems" files, and the main results that module documentation names;
3. a declaration that a node already uses, or that a person asks for;
4. a definition that the statement of a selected theorem uses (`uses_constants_in_statement`).

A rough guess, not a measurement, is that 40,000 to 80,000 Mathlib declarations meet one of these. The first import measures the real number, and the report gives it.

**Informal text.** About 115,876 constants have a docstring, with a mean length of about 130 characters. A docstring is Apache-2.0 text that a person wrote, so it can be the start of a summary (K8.3), with attribution. The informalize role (K7.3) writes the statement of each selected declaration that has no human mention and no usable docstring.

**AI-written informal datasets are not text sources.** Herald, the FrenzyMath datasets, LeanSearch v2 and others hold informal text that a model wrote for Mathlib declarations. Their licences are permissive, but no study measures their error rate, and most of them do not name the model. Rule X08 needs each model that writes text for math2 to be in the model register. So these datasets are search aids and inputs to the clustering only, never text of a node.

## K6.5 Libraries that AI wrote

Tau Ceti is mostly written by AI, and reviewed against rubrics that people wrote. Megalodon's Munkres formalization was written with models. Some AFP entries say that AI helped with proof engineering, and NASALib has a `CLAUDE.md`. Other libraries will follow.

The kernel checks the proofs of each of these libraries, so the proofs are as good as any others. The risks are in the parts that the kernel does not check: whether a statement says what its name and docstring say, and whether its definitions are the standard ones.

So each library and each entry has a field `authored` in its source document, with the value `human`, `ai` or `mixed`, and the reason. The importer copies it to each mention. Then:

- **Docstrings of `ai` libraries are agent text.** They go through the copy check (K9.4), and they count as text that an agent wrote for rule P04. A person must confirm them before a node reaches W3.
- **Bindings to `ai` libraries need the F2 review of spec 07.3 like any other binding.** In addition, a binding to an `ai` declaration whose statement uses a definition of the same library needs the review to cover that definition too.
- **The report gives the share of bindings to `ai` libraries.**

The value `mixed` is for a library where both people and models write, with no record of which. It has the rules of `ai`.

## K6.6 Other systems

**No system translates a whole library into Lean 4 today.** hol2dk translates HOL Light to Rocq (40,728 theorems of Multivariate, a dump of about 135 GB), and not to Lean. Mathport is archived. lean2dk is still work in progress. In ITPEval, the best model translated 29.1% of statements and 10.5% of proofs between systems at pass@1, and an equivalence check confirmed only 54% of the translations that compiled. So the other systems enter the corpus as crosswalks, not as Lean. When a node needs a Lean binding of a result that only another system has, the formalize role writes it, with the source declaration in its context, and the binding goes through the F1 battery like any other.

**Metamath set.mm is the best informal and formal source after Mathlib.** It is CC0, and each statement has a comment that a person wrote, usually a sentence or more. The importer takes each comment as `open` text, with attribution (which CC0 does not need, but rule 2 of K13.8 does). A statement whose comment is a full sentence starts at W1, after the copy check and the body rules. The set.mm labels are `same_as` targets.

**agda-unimath** is MIT, and its pages are literate Markdown, with prose next to each definition. The importer takes the prose as `open` text, with the MIT notice in the attribution. It is the best source for definitions in homotopy type theory and univalent foundations.

**The AFP.** Each entry has an abstract, and the theories have `text` blocks. The importer takes the text of BSD entries as `open` text, and the text of LGPL entries not at all. The AFP is the largest source of crosswalks after Mathlib.

**Mizar.** The MML is dual-licensed, and one choice is CC BY-SA 3.0, which allows a later version (CC BY-SA 4.0) for adaptations. Its text is formal, and there is little prose. The corpus uses it for crosswalks only. The article count is not settled: the version string says 1,493 and spec 01 says 1,498 (K15).

**The others** (Rocq, HOL Light, HOL4, Naproche, NASALib, ACL2) give mentions and crosswalks. Their comments are not taken as text in v1.

## K6.7 Crosswalks

A crosswalk links a row to a declaration in a library. The corpus builds them in this order:

1. **From the libraries themselves.** Many declarations name the theorem that they state, in the docstring or in a list.
2. **The 1000+ theorems list.** It is in the public domain (the Unlicense), and it has 1,200 YAML files, one for each theorem, keyed by the Wikidata identifier, with MSC codes, links to Wikipedia and blocks for Isabelle, HOL Light, Rocq, Lean, Metamath and Mizar. It is the first crosswalk that the corpus imports. Wiedijk's 100 theorems list is the second. It has no licence, so the corpus takes only its facts: which theorem has a formalization in which system.
3. **Matching datasets.** TheoremGraph pairs 388,105 Lean declarations with real arXiv statements (not model text), with 47,952 to 61,210 pairs that a model judged to match. It is CC BY-SA 4.0. The Stacks-to-Mathlib alignment (MIT) pairs Stacks tags with Mathlib. The corpus uses these datasets as candidates. The align role judges each pair again, and an accepted pair becomes an edge of math2, with the dataset in its source. The corpus does not copy their text, and a pair that math2 did not judge is not an edge.
4. **The type hash and the clustering.** Two Lean declarations with the same type hash state the same thing (spec 07.4). The other systems need the align role.

## K6.8 Datasets

**Benchmarks are for evaluation only.** Their errors are measured, and they are many:

| Benchmark | Errors that a study found |
|---|---|
| miniF2F (Lean ports) | the formal and informal statements disagree in more than half of the problems |
| ProofNet | 55.9% of items have errors (ProofNet-Verified) |
| ProofNet# (the fixed version) | still 27.0% |
| DeepSeek-ProverBench | 208 certified defects in 325 problems |
| FormalMATH | 147 certified defects |

So a benchmark item never becomes a mention, a statement or an exercise of the corpus. The benchmarks test the roles of K07 (class `evaluate`). PutnamBench has one more rule: its informal statements are used by permission of the MAA, so math2 does not copy them anywhere, not even into a test set that it publishes.

**Synthetic statement sets** (Lean Workbook, Goedel-Pset, NuminaMath-LEAN and others) hold millions of formal statements of competition problems, made by models. Most of them have a measured fidelity of 60 to 90%, or no measure. Their informal side is often text from a third party, such as Art of Problem Solving. They are `evaluate` at most. An exercise of the corpus (K8.7) is not taken from them.

**Pretraining corpora** (OpenWebMath, FineMath, Proof-Pile-2, AutoMathText) are not sources. OpenWebMath is in the reference corpus of the copy check, for detection only (K9.4). MathPile is `exclude` (CC BY-NC-SA).

**Red flags.** The importer refuses a dataset in each of these groups until a person changes its class in the source document:

| Flag | Datasets |
|---|---|
| non-commercial, or a clause against ML | ATLAS v1 and v2, AutoformBench, MathPile |
| no licence | Goedel-Pset-v1, Goedel Lean-workbook-proofs, DeepSeek-ProverBench, MathAtlas, LeanAgent, the mgwiki Megalodon repositories |
| licence conflict | DeepSeek-Prover-V2 (MIT file against the Model Agreement), LeanDojo-v2 (Apache against MIT), FormalMATH (MIT on Hugging Face, Apache on GitHub) |
| third-party text | PutnamBench informal statements |
| made by a model under terms that may pass on | data from DeepSeek-V3, Qwen2.5, Gemini or Kimina models (K7.9) |

## K6.9 Informalization

The formal track is large and the human track is small for the same items: only 14.5% of Mathlib constants have a docstring. So the informalize role writes much of the text of the formal layer. K7.5 gives the job, and K9.7 gives the audit. Two rules from the evidence:

- **Compiling is not faithfulness.** A statement that compiles can still say something else. Each informalized statement has the F1 battery of spec 10 on the pair, and the sample audit checks fidelity as its own property.
- **There is no published error rate.** No study measures informalizations of Mathlib at scale. The closest number (FormL4) is 70 to 80% correct, in a different setting. The informalize role is its own stratum from its first merge, and the corpus publishes the first measured rate.

## K6.10 Pins and bumps

Each library is pinned in `manifests/sources.yaml`, with a commit and a toolchain (spec 07.1). The importer reads the library at its pin. When the pin moves:

- New declarations become new mentions.
- A removed or renamed declaration keeps its mention, with `retired` and its successor, if the library gives one (Mathlib's deprecation aliases do).
- A declaration whose type hash changed is a new mention. The bump role (spec 07.9) handles the bindings.
- The report gives the change in mentions, rows and nodes for each library, for each bump.

## K6.11 Corrections to spec 01

| Spec 01 value | Value on 2026-10-26 |
|---|---|
| Mathlib 291,178 theorems and 139,569 definitions | 291,237 and 139,616 |
| Mathlib 9,437 files | 8,661 `.lean` files under `Mathlib/`. Spec 01 probably counted all files |
| AFP 1,038 entries, about 326,500 lemmas, about 5.44M lines | 1,042, about 330,000, about 5.518M |
| Isabelle2026 not known | released in October 2026 |
| Agda sizes not known | agda-stdlib 1,183 files, agda-unimath 3,072, 1Lab 710 |
| Megalodon Munkres 130k lines | about 160k lines and more than 1,500 lemmas |
| Mizar 1,498 articles | the version string says 1,493 (not settled) |
| Metamath set.mm more than 40k theorems (2022) | 51,207 statement numbers, including axioms and definitions |
| Tau Ceti about 3M lines | 18,849 files. The source of "3M lines" may mean Mathlib (not settled) |

## K6.12 Rules

| Rule | H/S | Checks |
|---|---|---|
| F10 | H | each mention from a library records the library, the pin, the declaration name, the intake class and the licence of the item (K6.3) |
| F11 | H | no section of a node holds text from a library whose class is `link`, `evaluate` or `exclude`, or from an item whose licence is not open (K6.2) |
| F12 | H | each source document of a library has the field `authored`, and each mention copies it (K6.5) |
| F13 | S | a binding to an `ai` or `mixed` declaration whose F2 review does not cover the definitions of that library that its statement uses (K6.5) |

The open questions of this document are in K15: the Mizar article count, the size of Tau Ceti, the licences of the other 20 Formal Frontier repositories, the effect of the ATLAS v2 preamble, the Coquelicot licence, and whether the terms of a model pass on to the data that it made.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/07-machine-generation.md">
# K07. Machine generation: what models write, and how

The goal asks for the mathematics that people wrote down, and also for what models know (K2.1). Models are the only way to write a million nodes with the people that the project can have. They are also the largest source of errors, of copies and of invented sources. This document says what models do in the corpus, what they must not do, and how each piece of model output is controlled.

The main spec already sets the frame. Models never raise a status (settled decision 3). Each change is a pull request that the audit checks (settled decision 7). Agent pull requests are capped by review capacity (settled decision 20). Each run records the exact model ids, the role version and the prompt hashes (spec 12.3, 12.8). This document does not change these rules.

## K7.1 What models are for

| Use | Role | Output | Trusted as |
|---|---|---|---|
| list the items of an area | enumerate (K7.3) | mentions in the record plane | a mention source. A row never comes from model mentions alone (rule I03) |
| write the sections of a node | write, explain | bodies at W1 to W3 | text by an agent until a person confirms it (rule P04) |
| write the informal text of a formal declaration | informalize | statement and concept nodes | text by an agent. The binding is checked by the F1 battery, and F2 needs a person |
| write exercises | exercise | exercise nodes (K8.7) | text by an agent. An answer with a machine check is checked by the machine |
| decide, align, formalize, prove, explore | the roles of spec 12.1 | as in spec 12 | as in spec 12 |
| check other model output | judges (K7.6) | verdicts in the run record | triage only |

Models do **not**:

- decide that a statement is true. Only the kernel, the certificate checkers and refereed or textbook sources give E levels (spec 04.7);
- add a source that does not resolve to a record (rule P05);
- give the outcome of an inventory row by themselves when the rules of K4.4 say that a curator decides;
- get the text of a source whose licence class is not `open` in their context (K7.5);
- write a register, a review or a status (spec 12.1).

## K7.2 What the evidence says

The research of K01 (part D) measured model output in mathematics from 2024 to 2026. Eight results set the controls of this document:

1. Model judges gave proofs scores up to 20 times higher than human graders did.
2. The best judges of single proof steps still accept about 21% of wrong steps. Weaker judges accept 56% to 90%.
3. Strong models "proved" false statements of competition level 29% to 70% of the time. At research level, they rejected fewer than 40% of false statements.
4. A correct final answer is not a correct proof. For one model, only 59.5% of correct answers had a correct proof.
5. From 11% to 57% of the citations that models generate do not exist. A citation that more than three models give is real 95.6% of the time.
6. Statements that models formalize with no person in the loop are faithful 60% to 90% of the time. A statement that compiles is not a faithful statement: one system compiled 90% and was accurate on 66%.
7. More than 27% of the text that models write in the style of an encyclopedia is in verbatim runs of 50 characters or more from public text.
8. No study measures how often models write a wrong definition, drop a hypothesis or give a theorem the wrong name in informal text. The corpus must measure this itself (K9.7).

Most of these numbers come from models of 2025. Nobody has published the same measures for the current models. So the corpus measures its own configurations before it trusts them, and the numbers above are the prior.

## K7.3 The four new roles

The corpus adds four roles to the eight of spec 12.1. Each has a bot account, a tool profile and a field permission set, and rule X01 checks each pull request against it.

| Role | Does | May create | May edit | Auto-merge |
|---|---|---|---|---|
| **enumerate** | lists the items of an area that the model knows, each with its kind, a one-line statement, its aliases and one or more source identifiers | mentions in `records/mentions/model-<family>/` (K4.8) | none | yes, into the record plane only. A mention changes no node |
| **write** | writes the sections of a node that its card asks for at the target writing level (K8.4): the statement, the definition, the proof in steps, the examples and non-examples | sections marked `by: agent`; example nodes; informal proof nodes | its own unconfirmed sections | never |
| **informalize** | writes the informal text of a formal declaration that no person described: the name, the statement, the summary | statement and concept nodes with a binding at F0 to the declaration; sections marked `by: agent` | its own unconfirmed sections | never |
| **exercise** | writes exercises for a node, with a hint, an answer, a solution and a check (K8.7) | exercise nodes (node.v2), or example nodes with the role `exercise` before node.v2 | its own unconfirmed exercises | never |

The roles of spec 12.1 do this work in the corpus:

- **curate** proposes the outcome of each row in the `decide` queue (K4.4);
- **explain** writes the levels L1 to L4 (K8.6), with the grounding check of spec 12.6;
- **formalize** and **prove** run the formal track (K12.1);
- **explore** runs the sanity checks of K9.3;
- **align** proposes the `aligned` outcomes and the crosswalks to Wikidata, Mathlib and the other libraries (K06);
- **review-assist** makes the packets for each check that people do (K9.6).

## K7.4 Generation from what models know

The enumerate role asks models what mathematics they know. It is how the corpus finds the items that no catalogue lists: the lemma that each textbook proves and no index names, the standard counterexample, the folklore result.

**The unit of work.** One job asks one model about one cell: an MSC code at five digits and one level (school, undergraduate, graduate, research). There are about 6,000 codes and 4 levels, so a full pass is about 24,000 jobs for each model.

**The prompt.** The job gives the model the code, the level, the names of the rows that the inventory already has for the cell, and the schema of a mention (K4.1). It asks for each item that is not on the list: its kind, its common name and aliases, a statement of one sentence, its prerequisites by name, and the sources where it appears, as identifiers (an ISBN with a theorem number, a DOI, an arXiv id, a Wikipedia title). It asks the model to say "unsure" when it is unsure, and never to give a source that it cannot name exactly.

**Rounds until saturation.** A cell gets a new round with the new list, until a round adds fewer than 2% new rows after clustering (K4.3), for two rounds in a row and for models of two families.

**More than one family.** The models of one family share their training data, so their lists are not independent. The job runs on at least two families: Claude, and one open-weight family with a licence that adds no duty on outputs (K7.9). An item that only one model lists, with no source that resolves, goes to the `cluster` queue as a mention and waits for a human source.

**What enumeration cannot measure.** Capture and recapture between models overstates recall, because the models have seen the same text. So model lists count as one capture occasion in the estimates of K4.7, and the held-out human source is the test of recall.

**The knowledge cutoff.** The current Claude models know mathematics up to June 2026. A model cannot list what was published after its cutoff, and its claim that a result is new means nothing. The research layer comes from the record plane (spec 09), not from what models remember.

## K7.5 How a write job runs

A write job takes one node to the next writing level. It runs the common loop of spec 12.2, with this context and these limits:

**The context.**

- the card of the kind at the target level (K8.4) and the house style (K8.2), in the cached part of the prompt;
- the notation register of the scope of the node and the conventions register (K10.4, K10.5), also cached;
- the node, its neighbourhood to depth 2, and the inventory row with all its mentions (K4.1);
- the text of the sources of the row, **only** when the source has the licence class `open` and its licence is compatible with the prose licence (CC0, CC BY, CC BY-SA, public domain). The node then carries the attribution of rule L03;
- for sources of the classes `share-alike-nc`, `gfdl`, `verbatim` and `record`: the identifier and the locator only. Their text never goes into the context of a model that writes prose for math2. A model that rewrites a GFDL proof that it can see makes a derived work, and the licence of math2 cannot take it.

**The instructions.**

- Write in your own words. Do not reproduce a source. This cuts copying, but it does not stop it (K9.4).
- Write each proof as numbered steps, with the reason for each step as a link to a node or a source (K8.5). The structure helps the judges, but it does not change the E level.
- For a statement, prove it or disprove it (K9.3). Report a hypothesis that you think is missing or not needed.
- Use only the macros of the register. Declare each new symbol in the `notation` field.
- Mark a source that you cannot name exactly as unsure, or leave it out.

**The output.** The diff adds sections marked `by: agent`, with the role version and the model id of each field (spec 04.8). The gates of K9.2 run in the loop before the pull request opens, so the model can repair its draft in the same run.

**The informalize role.** It gets the declaration, its type, its docstring when one exists, the module documentation, the declarations that it uses, and their informal text when math2 has it. The output is a statement in the house style, with the conventions of the declaration stated (K10.5). Then the F1 battery runs on the pair, as on any other binding. No published study measures the error rate of informalizations of Mathlib (K01, part C). So informalize is a stratum of its own in the sample audit from its first merge (K9.7).

**The exercise role.** It writes exercises for a node, with the fields of K8.7. It uses a model of another family to solve each exercise from the statement alone. An exercise whose answer the solver does not reach, or reaches in a different form, goes to a person. An exercise with `check: lean` must compile with `sorry`, and its reference solution must compile. Banks of exercises with a non-commercial licence (K05) are links only, and their text is not in the context.

## K7.6 Checks on model output

Each piece of model output goes through the gates of K9.2. The judges of K9.3 follow these rules, because the evidence of K7.2 says that one judge is not enough:

1. **A judge verdict is triage.** It decides what a person sees first. It never raises an E level, an F level or a writing level.
2. **The judge is of another family.** The model that wrote a node does not judge it. A check names the family of the writer and the family of each judge, and rule X09 checks that they differ.
3. **Judges are measured first.** A judge configuration judges proofs or statements only after it has a measured rate of rejecting wrong steps of 75% or more on a labelled set of math2. The set holds wrong proofs that people found, and canaries (K9.3). Tier 0 models do triage, classification and the first pass of the dedup judge, and they do not judge proofs.
4. **Accept by all, send by any.** A node passes the judges when each judge passes it. When one judge fails it, the node goes to a person with the reports of all judges. No extra call breaks a tie.
5. **Canaries each month.** The queues get false statements made by small changes to true ones. The weekly report gives the share that each configuration "proved", and a share above 5% stops the configuration (K9.3).
6. **Agents game checkers.** In a published benchmark, agents tried to work around the elaborator. So the pinned statement, the Comparator check and the axiom audit of spec 12.2 apply to each accepted proof, from each prover, with no exception.

## K7.7 Provenance and the model register

- **Each field has its writer.** The provenance of spec 04.8 records the role version, the model id and the run id of each field that an agent wrote. A person who confirms a field adds a confirmation and does not replace the record.
- **A register of models.** `models/models.yaml` in `tamnd/math2-agent` holds an entry for each model that the orchestrator can call: the exact id, the family, the provider, the licence or the terms with their URL and the date that a person read them, the clauses on outputs (for example a duty to name the model, or a ban on training other models with the outputs), and the retirement date when one is known. The orchestrator refuses a model with no entry. Each release manifest of `tamnd/math2` includes the entries of the models that wrote its fields.
- **Filters by model.** `m2 prov list -family F` lists each field that a model of a family wrote. A user whose own terms need it can remove those fields. The corpus uses the same filter for the rule of K7.9 on training.
- **Model change.** The current Claude models retire from September 2027. A campaign of a million nodes will cross a model change. A new model is a new configuration, with its own stratum in the sample audit (K9.7), and `m2 prov quarantine` can act on the output of one model.

## K7.8 Configurations and cost

The cost estimate of K12.6 is for each stage. This table, from K01 part D, is for each configuration of one node with one generation call and three check calls, at batch prices with cache hits. The token counts are assumptions that the first thousand runs replace.

| Configuration | For each node | For one million nodes |
|---|---|---|
| all calls on Haiku 5.5 | $0.004 | about $4,000 |
| all calls on Sonnet 5.5 | $0.081 | about $81,000 |
| all calls on Opus 5.5 | $0.161 | about $161,000 |
| all calls on Fable 5.1 | $0.40 | about $400,000 |
| write on Opus 5.5, check with two Sonnet 5.5 calls and one call to a model of another family | $0.113 | about $113,000 |

- **Divide by the acceptance rate.** When half of the drafts pass, the cost doubles. Until the pilot measures the rate (K14.4), the plan uses two to five times the table.
- **Output tokens are 70% to 85% of the cost**, and thinking tokens are output tokens. The effort setting is the lever that matters. The weekly report gives the output tokens for each accepted node, for each role and tier.
- **The all-Haiku row is not a real option** for proofs and statements, because tier 0 does not judge proofs (K7.6). It is the cost of enumeration and triage.
- **Formal work costs much more.** A published campaign of textbook formalization cost about $17 for each Lean declaration and about $300 for each target theorem (K01). K12.6 prices it apart.
- **Prices change.** The price table of the `llm` package has the date of each price, and it models batch, cache reads, cache writes and the long-context tiers. A table with no date is a finding of the weekly report.

The money is not the limit. The limit is the review hours (K12.5). The plan gives the cost of each accepted node and of each F2 binding, not only the spend.

## K7.9 Terms of the providers

K13 gives the law. This section gives the rules for the operations:

- **Publishing is allowed.** The terms of Anthropic, OpenAI and Google that a person read on 2026-10-26 give the customer the rights in the outputs that the provider has, and none of them stops the publication of outputs under an open licence.
- **No training on hosted outputs without authorization.** From 12 November 2026, the usage policy of Anthropic forbids the use of outputs to train an AI model without its prior authorization. OpenAI and Google forbid the use of outputs to build models that compete with theirs. So the corpus does not fine-tune a model (for example an open prover of tier 0′) on fields that a hosted model wrote, unless the provider authorized it in writing. The filter of K7.7 makes this mechanical.
- **Open-weight models with duties on outputs.** Some licences add duties on outputs or on models trained with them: the naming duty of Llama 4, the clause of the DeepSeek Model Agreement on models trained with its synthetic data, and the conditions of the Qwen 3.8 and Kimi K3 licences for outputs made available to others. The model register records each clause. For roles that write content, the corpus prefers models with MIT or Apache-2.0 licences and no clause on outputs, for example DeepSeek V4, Qwen3.5, GLM-5, Mistral, Leanstral, gpt-oss, Goedel-Prover-V2 and Kimina-Prover.
- **The licence of math2 adds no term.** CC BY-SA cannot carry a ban on training, and math2 does not try to add one. The release notes say which fields came from which provider, so that users with their own contracts can filter.
- **No model run without approval.** Each run that calls a model needs a budget that a person approved (K12.6). Tests use fakes for each model.

## K7.10 Rules

| Rule | H/S | Checks |
|---|---|---|
| X08 | H | each field that an agent wrote names a model that has an entry in the model register |
| X09 | H | each judge verdict in a check of K9.3 is from a model of a family other than the family of the writer |

With rule I03 (K4.9), L05 (K9.4) and P05 (K9.5), these rules are the hard limits on model output.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/08-the-node-card.md">
# K08. The node card: what a good node contains

Spec 04.5 says what a node file must contain to be valid. This document says what it must contain to be good. For each kind, a **card** lists the sections and the edges, the order in which a reader sees them, which ones each writing level needs (K2.4), and the style of the text. The writers follow the cards. These writers are people, the roles of K07, and the importers of K05 and K06. The build checks the cards and computes the writing level from them.

The cards are data, not only prose. They live in `schema/cards.v1.yaml` in `tamnd/math2`, and the build reads them. A change to a card is a pull request that a curator approves.

## K8.1 Principles

1. **One node, one thing.** The main section states one definition, one proposition or one argument. Motivation, history and explanation have their own sections (spec 04.5).
2. **Self-contained statements.** A reader can understand the main section with only the linked nodes and the notation register. Each symbol is defined in the section, by a link, or by the notation register (K10.4). "As above" and "under the usual hypotheses" are not allowed.
3. **Reuse by edges, not by copies.** An example is an `example` node with an `instance_of` edge, not a paragraph inside a definition. A card says which edges it shows. The workspace and the packer (K11) put the linked nodes in place. One example can then serve ten definitions and three courses.
4. **Formal and informal side by side.** Each statement and definition card has a place for the formal binding. Each proof card has a structure that maps to a formal proof (K8.5).
5. **Levels are contracts.** An explanation at level L*n* uses only the prerequisites of that level (spec 12.6). The card gives the length and the purpose of each level.
6. **Every section has a writer.** The provenance log records a person or a run for each section (spec 04.8). The card says which sections an agent may write and which need a person to confirm them.

## K8.2 The house style

The text of a node is for a reader who reads one node and leaves. It must be clear on its own, and it must read the same across a million nodes. The house style is plain technical English in the spirit of ASD-STE100, adapted to mathematics:

- **Short sentences.** At most 25 words in a sentence of the main section, and at most 30 words in other sections. A formula counts as one word.
- **One claim in each sentence** in `Statement` and `Definition`. Hypotheses come first, then the conclusion: "Let ... . If ..., then ...".
- **Active voice** in proofs: "We apply Lemma 2" and not "Lemma 2 is applied". "We" is allowed in proofs and explanations. It is not used in `Statement` or `Definition`.
- **No filler words that hide a step.** The words "clearly", "obviously", "trivially", "it is easy to see" and "the reader can verify" are not allowed. If a step is short, give it. If it is long, make it a lemma.
- **Name the convention.** When conventions differ, the text says which one it uses: "Here $\mathbb{N}$ contains 0", "rings have a unit". The concept node records the conventions (K10.5), and the text links to it.
- **Standard words.** Use the term that the concept node names first. Give other terms as aliases in the front matter, not as alternatives in the text.
- **Notation from the register.** Use the macros and symbols of the notation register (K10.4). A new symbol needs a register entry first.
- **No references to place.** Write "the [[Sylow theorems|...]]" and not "the theorem in Section 3". A node has no position. A book gives it one (K11).
- **English first.** The canonical text is English. Translations follow spec 13.4.

`m2 check -style` runs a style linter over the bodies. The linter is mechanical: sentence length, the banned words, undeclared symbols, links that are not tags. Its findings are warnings for W1 and errors for W3.

## K8.3 The common front matter

Each card uses the front matter of node.v1 (spec 04.5). The corpus adds four optional fields. All four are additive (spec 04.9):

| Field | Holds | Why |
|---|---|---|
| `inventory` | the ids of the inventory rows (K04) that this node accounts for | the open count needs the link from row to node, and the reverse link makes the reports cheap |
| `conventions` | for a concept: the conventions that sources use, each with a short name and the sources that use it | K10.5 |
| `notation` | for a concept or a definition: the symbols that it introduces, each with its macro and its meaning | K10.4 |
| `audience` | the lowest curriculum level at which the node is taught: `school`, `ug1` to `ug4`, `grad`, `research` | the packer (K11) and the explain levels need it. The build compares it with the curricula and reports a mismatch |

## K8.4 The cards

The table lists the sections of each card in reading order. **R** means that the section is required for W1, **W2** for W2, **W3** for W3, and **o** means optional. Sections in italics are new, and they come in a minor version of node.v1 (K8.9). Until that version, the text of a new section goes into `Remarks` with a level-three heading of the same name.

### Concept

| Section or edge | Level | Contents |
|---|---|---|
| `Summary` | R | two to four sentences: what the notion is and why it matters. No formula is needed. This is the "Idea" paragraph of the nLab, and readers read it more than any other part of a page (K01). |
| *`Motivation`* | W3 | the question that the notion answers, or the example that leads to it |
| definitions (`defines` edges in) | W2 | at least one definition. The card shows each one with its source, and marks the one that Mathlib uses |
| equivalences (`equivalent` edges) | W2, when there are two or more definitions | a statement for each pair of definitions that a source proves equivalent |
| examples (`instance_of` edges in) | W2 | at least two examples |
| non-examples (`not_instance_of` edges in, K8.9) | W2 | at least one non-example, with the reason |
| `specializes` edges | W2 | the more general notions and the special cases |
| *`Notation`* | o | the symbols and the conventions, from the front matter |
| `History` | W3 | who introduced the notion and when, with sources |
| `Explanations` | W3 | L1 to L3, and L4 for research concepts |
| `Remarks` | o | other facts that have no section |

### Definition

| Section or edge | Level | Contents |
|---|---|---|
| `Definition` | R | the definition, with each condition as a separate numbered clause when there are more than two. It names what is defined in bold the first time, and the linter checks that the bold term is the name of the node or an alias. |
| formal binding | W2, when a library has the definition | `formal:` to a `def`, `structure` or `class` |
| examples and non-examples | W2 | one of each, by edges. A non-example fails exactly one clause, and its edge names the clause (K8.4a) |
| *`Notation`* | o | |
| `Remarks` | o | why this definition and not another, and what changes under other conventions |
| `Explanations` | W3 | |

### Statement

| Section or edge | Level | Contents |
|---|---|---|
| `Statement` | R | the proposition. Hypotheses first, then the conclusion. For a statement with several parts, a numbered list, one part in each item. |
| `uses` edges | R | each definition that the statement needs to make sense |
| `about` edges | W2 | the concepts that the statement is about |
| proofs (`proofs:`) | W2 | at least one proof node, informal or formal |
| examples (`instance_of` or `illustrates` edges) | W2 | one example that shows the statement at work |
| a counterexample to a weaker form | W3, when one is known | the hypothesis that cannot be removed, and the counterexample that shows it |
| formal binding | W2, when a library has the statement; otherwise a formalize job | spec 07.3 |
| `History` | W3 | who stated it first, who proved it first, with sources |
| `Explanations` | W2: L2; W3: each level that applies | |
| exercises (K8.7) | W3, when the node is in a curriculum | at least one |
| `Remarks` | o | generalizations, the converse, common errors |

### Proof

| Section or edge | Level | Contents |
|---|---|---|
| `Idea` | W2 | one to three sentences: the method and the key step |
| `Proof` | R | the argument in numbered steps (K8.5) |
| `uses` edges | R | each node that the argument needs |
| `method` edges | W2 | the methods, from the `methods` topic |
| formal binding | o | a Lean proof of the bound statement |
| *`Variants`* | o | the changes that give a proof of a nearby statement |

### Example and counterexample

| Section or edge | Level | Contents |
|---|---|---|
| `Example` or `Counterexample` | R | the instance, and the check that it is an instance (or that it refutes the target) |
| `instance_of`, `not_instance_of` or `refutes` edges | R | |
| computation | W3, when the check is computational | a `computation` node that runs the check (spec 08) |
| `Explanations` | o | |

### Conjecture and problem

These follow spec 04.2 and the problem page of spec 13.3. The card adds `bounds` edges and a `History` with the year of the first source, both for W2.

### Object

| Section or edge | Level | Contents |
|---|---|---|
| `Description` | R | what the object is, its standard name and its main invariants |
| `same_as` | R | the identifiers in the databases (LMFDB label, OEIS number, GAP id, ATLAS name) |
| invariants | W2 | a table of the main invariants, each with a source or a computation |
| `instance_of` | W2 | the concepts that it is an instance of |

### K8.4a Clauses, examples and non-examples

The research on how people learn definitions says that the best non-examples are near misses: objects that are almost examples (K01). So the corpus makes near misses easy to find and to check:

- Each clause of a definition has a number (K8.4, `Definition`).
- A `not_instance_of` edge has a field `fails` with the number of the clause that the non-example fails. A non-example that fails more than one clause lists them all, and it counts as a near miss only for a single clause.
- An `instance_of` edge can list the clauses that the example shows at work.
- The build reports **clause coverage** for each definition: the clauses that have a near-miss non-example. At W3, each clause of a definition with three or more clauses has one.

## K8.5 Proofs in steps

An informal proof in the corpus is a list of numbered steps. This form serves three readers at once: a person reads it as a proof, a formalizer reads it as a plan, and the packer can show it at two levels of detail.

- **Each step states one claim and its reason.** The reason is a link to a node, an earlier step, or a computation: "3. $H$ has index 2 in $G$, by step 2 and [[Lagrange's theorem|7K3QX2M9R2]]."
- **Steps can have substeps** (3.1, 3.2) to two levels. A step with more than two levels is a lemma, which is its own node.
- **The last step is the conclusion** of the statement, in the words of the statement.
- **A step can be marked `routine`.** A short form of the proof leaves out the routine steps. The marks come from the writer and a person can change them.
- **The form is Markdown.** Steps are a numbered list, and substeps are a nested list. This is the structured proof of Lamport in plain Markdown, with no new syntax.
- **Map to Lean.** When a proof has a formal version, each step can name the `have` or the lemma that matches it. This is the "Unfold this proof" view of spec 13.4 in the other direction. The prove role (spec 12.4) uses the steps as its plan.

A proof that a model wrote is E1 (spec 04.7). The card does not change that. It makes the proof easy to check: a reviewer checks one step at a time, and a formalizer proves one step at a time.

## K8.6 Explanations at levels

Spec 04.5 gives four levels. The card fixes their purpose and their length:

| Level | For | Stage (Tao) | Purpose | Length |
|---|---|---|---|---|
| L1 | a person who has not met the concept | pre-rigorous | what it is, with one picture or one concrete example, no proof | 80 to 250 words |
| L2 | an undergraduate | rigorous | the definition unpacked, why it is true or why it is defined this way, the standard examples and non-examples, the common errors | 150 to 500 words |
| L3 | a graduate student | post-rigorous | the right way to think about it, how it fits with the rest of the field, the generalizations, where it breaks | 150 to 600 words |
| L4 | a researcher | post-rigorous | the state of the art, the open questions, the literature | 100 to 500 words, with sources |

The stage gives the writer and the reviewer a test that does not depend on the label of the audience. The evidence on worked examples (K01) gives three more rules:

- L1 and L2 use **correct** worked examples, written as numbered steps, so that a renderer can fade them: it shows the first steps and asks the reader for the rest. An example with an error for the reader to find is an exercise, not an explanation.
- L3 and L4 do not include full worked computations. They help novices and can hurt advanced readers (the expertise reversal effect).
- A guide on how to read a proof is one explanation node at the level of a topic, linked from proof pages. It is not repeated on each proof.

A node does not need each level. A research lemma needs L3 or L4, not L1. A school concept needs L1 and L2. The card of each kind says which levels W3 needs. The `audience` field (K8.3) decides it: each level from the audience up to L3, and L4 for audiences `grad` and `research`.

## K8.7 Exercises

Courses need exercises, and the textbook layer has many of them under open licences (K05). The corpus holds an exercise as its own node, so that a course can choose from a pool:

| Section or field | Contents |
|---|---|
| `Exercise` | the task, self-contained as a statement is |
| `Hint` | optional, one or two sentences |
| `Answer` | the final answer, when the task has one |
| `Solution` | a worked solution in the step form of K8.5 |
| `about`, `uses` edges | the nodes that the exercise practises and needs |
| `difficulty` | 1 to 5, set by the writer and corrected by use |
| `math_group` | `A`, `B` or `C`, from the MATH taxonomy: A is facts and routine procedures, B is transfer and application, C is justification, conjecture and evaluation. The groups describe the kind of task, not its difficulty |
| `check` | how a machine checks an answer: `exact` (a value), `numeric` (a value with a tolerance), `choice` (multiple choice), `cas` (equal up to a CAS normal form, spec 08), `lean` (a Lean statement with `sorry` that the answer must fill, graded with the comparator), or `none` (checked by people) |
| `external` | links to interactive versions in other systems, for example a WeBWorK OPL path or a MyOpenMath id. Links only: most of those banks have licences that the prose zone cannot take (K05) |

The build reports the groups of the exercises of each topic. Banks that models write tend to have few exercises in group C, and the report shows that gap.

An exercise with `check: lean` is also a formal statement. Its reference solution must compile, and the version with `sorry` must have the same type. A course can then grade it with the Lean server (spec 11.3), as the Natural Number Game does.

## K8.8 What the build computes from the cards

- **The writing level** of each node (K2.4), from the sections that are present, the edges, and the provenance of each section.
- **The card findings**: a required section that is missing, a section that is too long, a level that uses a node outside its prerequisites, a style finding. These are warnings, not audit failures, except where a rule of spec 15 already applies.
- **The work list** for each node: the sections that are missing for the next writing level. The orchestrator turns the work list into jobs for the roles of K07 (K12).

## K8.9 Proposed schema changes

Spec 04.9 has two kinds of change. Most of the changes below are additive, and they come in a minor version of node.v1. The new kind changes the list of kinds, and spec 04.9 calls that a breaking change. It comes in node.v2, with a migration and a release in which both versions are accepted.

| Change | Type | Version | Reason |
|---|---|---|---|
| edge `not_instance_of` (example → concept or definition), with the field `fails` | new edge | node.v1 minor | non-examples are half of a good definition card, and `refutes` is for statements (K8.4a) |
| edge `illustrates` (example → statement) | new edge | node.v1 minor | an example that shows a statement at work is not an instance of a concept |
| sections `Motivation`, `Notation` (concept and definition), `Variants` (proof) | new allowed sections | node.v1 minor | K8.4 |
| fields `inventory`, `conventions`, `notation`, `audience` | new optional fields | node.v1 minor | K8.3 |
| `schema/cards.v1.yaml` | new file | none | the cards as data |
| kind `exercise` with the sections and the fields `difficulty`, `math_group`, `check` and `external` of K8.7 | new kind | node.v2 | courses need exercises, and an exercise is not an example: it has a task and an answer |

The migration to node.v2 finds each `example` node that has the role `exercise` in `## Remarks` (K2.3). It changes the kind, and it moves the text into the sections of K8.7. The tag does not change. Spec 21 gets one entry for the new kind.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/09-quality.md">
# K09. Quality: how the corpus stays correct

The corpus is written at a scale that no group of people can read in full. Most first drafts come from importers and models. This document says which checks each piece of text goes through before it merges and after it merges, how the corpus measures its own error rate, and what happens when a check finds a problem.

The main spec already has the hard part: status is computed (spec 04.7), only the kernel and the certificate checkers referee E4 and E5 (settled decision 3), and fidelity above F1 needs people (spec 10.6). This document does not change these rules. It adds the checks that a corpus of millions of nodes needs on top of them.

## K9.1 Four properties

A node is good when it has four properties. Each property has its own checks, and no check covers two properties:

| Property | Question | Checked by |
|---|---|---|
| **true** | Is the statement true, and does the proof prove it? | the E levels: kernel, certificates, refereed and textbook sources (spec 04.7). Before that: the checks of K9.3, which only triage |
| **faithful** | Does the node say what its sources say, and does its formal binding say what the node says? | source confirmation (K9.5), the F levels (spec 10) |
| **readable** | Does the node follow its card and the house style? | the card findings and the style linter (K08), the writing level (K2.4) |
| **ours to publish** | Is each byte of text allowed under the prose licence? | the licence gate (spec 9.3), the copy check (K9.4) |

The reports give a separate number for each property. A good score in one property never hides a bad score in another.

## K9.2 The gates before merge

Each pull request that adds or changes a body goes through these gates in this order. A gate that fails stops the pull request. The failure has a reason, and the reason goes back to the job (K12) as output.

| Gate | Checks | Applies to | Hard or soft |
|---|---|---|---|
| G1. Schema and body | `m2 check`: rules S01 to S04 and B01 to B03 | each change | hard |
| G2. Card | the required sections of the target writing level (K8.4) | each change | hard for the level that the job asked for, soft otherwise |
| G3. Style | the linter of K8.2 | each body | soft below W3, hard for a W3 job |
| G4. Copy check | K9.4 | each body that an agent wrote, and each body from an importer of share-alike text | hard |
| G5. Sources | K9.5 | each source that the change adds | hard for a source that an agent proposed |
| G6. Sanity | K9.3 | each statement and proof that an agent wrote | soft: a failure opens a counterexample job and blocks only auto-merge |
| G7. Formal | rules F01 to F09 | each change to a binding | hard (spec 15.3) |
| G8. Review capacity | the cap of spec 11.5 | each agent pull request | hard: the pull request waits, it is not closed |

The gates run in the agent loop before the pull request opens (spec 12.2), and again in CI. An agent sees the result of each gate as a tool result, so it can repair its draft in the same run.

## K9.3 Sanity checks on statements and proofs

These checks catch false statements and broken proofs before a person spends time on them. **They never raise an E level and they never count as a review.** A pass means "no problem was found", and nothing more.

1. **Small cases.** When a statement quantifies over integers, finite groups, graphs, matrices of small size, or any family that the computational engine (spec 08) can enumerate, the explore role writes a test and runs it for small cases. A failure is a counterexample candidate. It opens a computation node and a job for a person, and it blocks the merge of the statement. Most false statements that models write fail at small cases.
2. **Formal round trip.** When the formalize role can bind the statement, the F1 battery of spec 10.4 runs. A back-translation that differs from the informal text shows an error in one of the two. The finding names the difference.
3. **Independent check by other models.** Models of other families read the statement or the proof one step at a time (K8.5) and report each step that they cannot follow. They get the node, the linked nodes and the sources, and not the run that wrote the node. The evidence on model judges sets the rules (K01, K7.6):
   - A judge verdict is triage, never evidence. Judges gave proofs scores up to 20 times higher than human graders did, and the best step judges still accept about one wrong step in five.
   - A judge model must have a measured rate of rejecting wrong steps of 75% or more on a labelled set of math2 before it can judge proofs or statements. Tier 0 models do not judge proofs.
   - The check passes only when each judge passes the node. When the judges disagree, the node goes to a person, and no extra model call breaks the tie.
   - Each verdict and its model id go in the run record. Agreement is used only for triage (settled decision 3).
4. **Hypothesis check.** For each hypothesis of a statement, the check asks whether a known counterexample shows that the hypothesis is needed. When a hypothesis is not needed, the statement is still true, but the node gets a remark: either the statement is weaker than it can be, or a source states it that way for a reason that the node must give.
5. **Prove or disprove.** A job that proves or formalizes a statement asks the model to prove it or to disprove it, not to prove it. Models "prove" false statements often, and this framing made one model reject false statements almost four times as often (K01). Each month, the queue gets canaries: false statements made by small changes to true ones. The weekly report gives the share of canaries that each configuration "proved", and a share above 5% stops the configuration.
6. **Names of theorems.** A theorem that the prose names ("by the theorem of X and Y") links to a node or has a source. Models invent theorems that match what they want to prove, so a name that resolves to nothing is a finding (rule X07).
7. **Proof dependencies.** Each `uses` edge of a proof points to a node at E2 or better, or the proof gets the `sorry_in_closure` flag (spec 04.7). An agent proof that uses an E1 node can merge, but it shows the flag.

## K9.4 The copy check

Models reproduce text that they saw in training, even when nobody asks them to. In one study, more than 27% of the text that models wrote in the style of an encyclopedia was in verbatim runs of 50 characters or more, and one model reproduced 1,170 characters of a Wikipedia article (K01). For mathematics the risk is higher, because standard proofs appear in many sources with almost the same words. The corpus must not publish text that the licence gate refuses (K2.6, rule 3). So each body that an agent wrote goes through a copy check before it merges.

- **Compare with a reference corpus, not only with the sources of the node.** A model that writes from memory cites nothing, so a check against the cited sources finds nothing. The reference corpus holds the mathematical pages of Wikipedia, ProofWiki, the nLab and PlanetMath, the dumps of Math.StackExchange and MathOverflow, the Stacks Project (GFDL), Kerodon (no licence, so `record`), the open textbooks of K05, and, for detection only, OpenWebMath and the arXiv text that the record plane can lawfully read (spec 9.3). Each text in it has its licence class. For sources that math2 cannot read, the check cannot run, and K9.6 covers the gap.
- **Normalization.** Lowercase, whitespace collapsed, LaTeX macros made canonical. Formulas are masked: each formula becomes a placeholder, so that a standard formula does not count as a copy. A separate pass compares the structure of formulas, so that a renamed variable does not hide a copied proof.
- **Stage 1, exact runs.** An index of suffixes over the reference corpus finds each run of the body that is also in a reference text. A run of 50 characters or more is a flag. A run of 100 characters or more in one source, or flagged runs that cover 10% or more of a section, is a **match**.
- **Stage 2, near copies.** A filter for exact runs alone gives false safety: in the studies, small edits pass it, and a near copy counts almost as much as an exact one (K01). So each paragraph also goes through MinHash on word 5-grams. A candidate pair with an estimated Jaccard similarity of 0.5 or more is a match if it also has a common block of 20 words or more after alignment, or a ROUGE-L of 0.8 or more.
- **Stage 3, paraphrases.** A sample of paragraphs, with their nearest reference paragraphs by embedding, goes to people. This stage does not block a merge. It measures what the first two stages miss, for the sample audit (K9.7).
- **What a match does.** A match with a source of class `record`, `verbatim`, `share-alike-nc` or `gfdl` blocks the merge, and the agent gets the locations to rewrite. A match with an `open` source is allowed when the licence of the source is compatible, and the node then gets the attribution of rule L03 in its provenance. Each match records the source, its URI, its licence and the offsets, in the run record.
- **Standard phrases.** Some text is the same in each source because there is only one way to say it, for example "let $G$ be a group". A list of standard phrases, kept by curators, does not count towards a match. A phrase gets onto the list only by a curator decision.
- **Calibration before blocking.** The thresholds above are a start, and nobody has tested them on mathematics. Before L05 blocks merges, the copy check runs over pages that people wrote, compared with each other: Wikipedia against ProofWiki, and open textbooks against each other. The median overlap of human text is 0% in the studies. The thresholds are set so that the rate of false matches on human text is under 1%, and K9.7 tunes them on samples after that.
- **A first layer in the prompt.** Each prompt for prose tells the model to write in its own words and not to reproduce a source. In a study, this cut the overlap by half or more, but long copies still came through. It is a cheap first layer and not a control.

Rule L04 of spec 15.3 is soft for text from people. The corpus adds a hard rule for text from agents:

| Rule | H/S | Checks |
|---|---|---|
| L05 | H | each agent-written body has a copy check result in its run record, and the result has no blocking match. Until the calibration is done, L05 is soft |

## K9.5 Source confirmation

Models invent references that look correct. A wrong reference is worse than none, because readers trust it. So an agent cannot add a source by itself:

1. **The record must exist.** A source is a record in the record plane: a DOI that Crossref resolves, a zbMATH id, an arXiv id, an ISBN with a catalogue record, or a record of an open textbook (K05). An agent proposes a source as an identifier. If the identifier does not resolve, the proposal is dropped, and the run record counts it as a fabricated source.
2. **The record must match.** The title, the authors and the year of the record must match what the agent claimed. A mismatch is a fabricated source too.
3. **The locator must be checked.** A locator says where in the source the item is: a theorem number, a section, a page. When the record plane can read the text, a check finds the locator and compares the item with the node. When it cannot read the text, the source has the flag `locator_unconfirmed`, and only a person can remove the flag.
4. **E2 and E3 need confirmed locators.** The build counts a source towards E2 or E3 (spec 04.7) only when its locator is confirmed. A node can show an unconfirmed source as further reading.

The rate of fabricated sources for each model and prompt version goes into the weekly agent report (spec 12.9). A rate above 2% stops source proposals from that configuration until a person looks at it.

| Rule | H/S | Checks |
|---|---|---|
| P05 | H | each source that an agent added resolves to a record in the record plane, and the record matches the claim |
| P06 | S | a source with `locator_unconfirmed` older than 180 days |

## K9.6 What people check

People are the scarce resource (settled decision 20). The corpus uses them only where a machine cannot check:

| Check | Done by | Why a machine cannot do it |
|---|---|---|
| fidelity of a formal statement (F2, F3) | reviewers, with `m2 review` (spec 16.3) | settled decision 4 |
| confirmation of an agent-written section (rule P04) | curators and area editors | the section claims to explain; only a reader can say whether it does |
| an unconfirmed locator | anyone with access to the source | the record plane cannot read most of the literature |
| a sanity finding (K9.3) | area editors | a finding can be a false alarm, and only a person can decide |
| a copy match that the agent cannot fix | curators | the decision whether a phrase is standard |

The review-assist role (spec 12.1) makes a packet for each check, with the evidence in one place. Each check has a target time, and spec 17 measures it. A person can confirm many sections in one session, but each confirmation is recorded for each section, not for the session.

## K9.7 Sample audits and published error rates

Gates and reviews catch errors one at a time. They cannot say how many errors remain. The sample audit can.

- **The strata.** Each month, the audit draws a random sample of merged nodes in each stratum. A stratum is a combination of kind, writer (person, importer, agent role), model and prompt version, and layer (K2.7). A new configuration gets its own stratum from its first merge.
- **The sample size.** With no errors in *n* samples, the upper end of the 95% confidence interval for the error rate is about 3/*n*. So 300 samples with no errors show a rate below 1%. The audit uses 100 samples in each stratum each month as a minimum, and more where the rate is near a threshold. The report uses the Wilson interval when the sample has errors.
- **The audit is blind.** Auditors do not see who or what wrote the node. They check the four properties of K9.1 against a written checklist for each kind.
- **Error classes for prose.** No study has measured how often models write a wrong definition or drop a hypothesis in informal text (K01). The corpus must measure it. The checklist classifies each error in a body: wrong definition, dropped hypothesis, statement stronger than its source, wrong name or attribution, convention not stated, step that does not follow, wrong source. The classes sit next to the misformalization classes of spec 10.5, so that the release notes give error rates for prose and not only for bindings.
- **The thresholds.** For each stratum, the error rate of each property must stay under a threshold: 1% for a false statement, 2% for an unfaithful statement or a wrong source, 5% for a style or card error. These numbers are a start, and a change to them is a decision that the release notes record.
- **Above the threshold.** When the upper end of the interval for a stratum passes its threshold, the configuration stops. Its unconfirmed output goes to quarantine by a generated pull request, as in the withdrawal test of spec 06.5. A person decides whether to fix, rerun or retire the output.
- **The publication.** Each release publishes the error rates of each stratum, with the sample sizes and the intervals, including the strata that failed. This is rule 6 of K2.6.
- **Small strata.** A stratum with fewer than 1,000 merges in a month pools its samples over three months. A stratum with fewer than 100 merges in three months is audited in full.

The sample audit costs reviewer time. At 100 samples in each of 20 strata, and 5 minutes for each sample, it costs about 170 hours each month. K12.5 counts this cost in the review budget first, before the agents get their share. When review time is short, the corpus runs fewer configurations. It does not take smaller samples, because a rate that nobody measured is not a rate.

## K9.8 Reports from readers

Each page of the workspace has a "report a problem" link. A report becomes an issue with the tag of the node and the property that the reader thinks is wrong. A report on a node at E5 that says the statement is false goes first in the queue, because it is a report on the fidelity of the binding or on the checker.

A confirmed report is an error that the sample audit did not catch. The report counts it in the stratum of the node, so that the published rates include it.

## K9.9 The rules that this document adds

| Rule | H/S | Checks | Section |
|---|---|---|---|
| L05 | H | copy check result present and clean for agent-written bodies | K9.4 |
| P05 | H | agent-added sources resolve and match | K9.5 |
| P06 | S | unconfirmed locators older than 180 days | K9.5 |
| X05 | H | an agent-written statement or proof has the results of the sanity checks of K9.3 in its run record | K9.3 |
| X06 | S | a stratum whose error rate is near its threshold | K9.7 |
| X07 | S | a theorem that a body names and that resolves to no node and has no source | K9.3 |

With the rules of K03 (T08 to T10) and K04, the catalogue of spec 15.3 grows from 65 rules. Each new rule needs a fixture pair, as spec 15.3 says.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/10-structure-and-notation.md">
# K10. Structure, notation and conventions

A node has no position (K8.2). The corpus still needs structure, for two reasons. A reader must find a node and the nodes around it. A packer must put nodes in an order that a reader can follow (K11). This document says which structures the corpus keeps, and how it keeps notation and conventions the same across a million nodes from hundreds of sources.

## K10.1 Three structures

The corpus has three structures. Each comes from different data, and none of them is a copy of another:

| Structure | Made of | Answers | Section |
|---|---|---|---|
| the logical graph | `uses`, `proves`, `implies`, `specializes` and the other edges of spec 05.2 | what must I know to read this node, and what follows from it? | spec 05 |
| the taxonomy | topic nodes, MSC2020 codes, `about` edges | where does this node belong, and what else is there? | K10.2 |
| the paths | curricula and learning paths: topics with ordered members | in what order do I read these nodes? | K10.3 |

The logical graph is the truth. The taxonomy and the paths are views that people curate. A path that puts a node before one of its prerequisites is a finding (rule N03), because the path is wrong, not the graph.

## K10.2 The taxonomy

- **MSC2020 is the base.** Each node has at least one MSC2020 code (spec 18.2). `topics/msc2020.yaml` maps each code to a topic node, with math2's own short labels (the official descriptions have a licence that does not allow them in math2, spec 20.1).
- **Topic nodes for what MSC splits badly.** Some subjects that people teach as one cross several MSC codes, for example "linear algebra" or "measure-theoretic probability". A topic node holds them, with `about` edges from its members. The skeletons of spec 18 are topic nodes of this kind.
- **Concepts give the fine structure.** Below the level of a topic, the structure is the concept nodes and their `specializes` edges: "metric space" specializes "topological space", and "Banach space" specializes "normed space". A concept with no `specializes` edge in or out is a finding of graph health (spec 05.8).
- **Crosswalks, not imports.** The categories of Wikipedia, the structure of the nLab and the module tree of Mathlib are not the taxonomy of math2. They are crosswalk data: the clustering (K4.3) and the classifier of MSC codes use them, and the workspace can show them.

## K10.3 Curricula and learning paths

A **curriculum** is a topic node in `topics/curricula/` with an ordered list of members. It describes a course, an exam syllabus or a book. Its members are nodes, and each member can have a weight: `core`, `standard` or `optional`.

The corpus imports curricula from public sources as data, not as text (K05). A curriculum lists the items that it names, so it is also an inventory source (K4.2), and its items enter the `curriculum` term of the priority (K12.3). Examples of sources: the lists of undergraduate mathematics in Mathlib, published qualifying exam syllabi, the tables of contents of open textbooks, and national school curricula that are published under an open licence.

A **learning path** is computed, not curated. For a set of target nodes and a reader who knows a set of nodes, the path is an order of the prerequisite closure (spec 05.5) of the targets, minus what the reader knows:

1. Take the closure of the targets over `uses` edges and `defines` edges, with the alternatives of spec 05.5. Where a concept has more than one definition, take only the definition that the path chose (K11.1, `choose`), or by default the one with the smallest closure. Where a statement has more than one proof, choose the proof with the smallest closure that is not known yet.
2. Remove the nodes that the reader knows, and their closures.
3. Order the rest topologically. Among the orders, prefer the one that keeps the nodes of one topic together and follows the order of a curriculum where one exists.

The packer (K11) uses learning paths to order a volume. The workspace uses them for "what do I need to read first?".

The choice of one definition for each concept matters. Metacademy gave each concept one set of prerequisites and named this as its main limit (K01). A concept with two equivalent definitions, such as compactness by open covers and by filters, has two sets of prerequisites. The path uses one of them, and the equivalence is a theorem that the path can add.

The workspace also answers **"what can I read next?"** For the nodes that a reader knows, the answer is the nodes whose prerequisites are all known and which are not known. Knowledge space theory calls this set the outer fringe, and the corpus computes it with one query on the graph.

## K10.4 The notation register

Each source has its own notation, and the same symbol means different things in different areas. A corpus written from hundreds of sources mixes them, unless something stops it. The notation register stops it.

The design takes one idea from sTeX: **a symbol is an identity, and a notation is a way to print it** (K01). Today `macros/m2.sty` and `macros/m2.katex.json` are render macros only. They say how to print `\Nat`, but not what `\Nat` means. The register adds the meaning, and the two macro files become build outputs of the register.

The register is a set of files `notation/<topic>.yaml` in `tamnd/math2`, one for each topic, plus `notation/global.yaml` for the symbols of all areas. Each entry is one macro:

```yaml
- macro: \totient             # the only form that a node body uses
  denotes: 2HQ7XKM4CP         # the tag of the concept or definition
  args: 1
  prec: null                  # for operators: the binding strength
  latex: \varphi(#1)          # the expansion for m2.sty and KaTeX
  typst: phi(#1)              # for the Typst output, when it exists
  intent: null                # a MathML intent concept name, when one exists
  openmath: null              # an OpenMath symbol, for alignment
  reads: "phi of #1"          # how a person reads it, for screen readers
  lean: Nat.totient           # the Lean declaration or notation, when one exists
  scope: [number-theory]      # the topics where the macro is in use
  introduced_by: 2HQ7XKM4CP   # the definition node that introduces it
  convention: null            # one line for the list of notation, when it applies
  variants:                   # other ways to print the same thing
    - {name: phi, latex: \phi(#1), sources: [...]}
  conflicts: [geo.goldenratio] # entries in other scopes that print the same glyph
```

The rules:

- **One macro, one meaning.** Each macro has one `denotes`. A glyph with more than one meaning gets one macro for each meaning: `\pair`, `\ointerval` and `\gcd` for the three uses of $(a,b)$, and `\totient` and `\goldenratio` for the two uses of $\varphi$. A body then says what it means even when the printed symbol is the same. This is also what makes the `intent` of MathML and the screen reader text correct.
- **One default notation for each concept.** A concept can have variants, and a volume or a reader can choose one (K11.1). A body always uses the macro, so the choice changes only the output.
- **A body uses macros in its scope.** The scope of a node is its topics. A body can use a macro of the global scope, a macro whose `scope` has one of the topics of the node, or a macro that the node introduces with its `notation` field (K8.3).
- **Glyphs can repeat across scopes, with a record.** Two entries can print the same glyph only if they are different macros. The `conflicts` field records the pair, and the pack check (PC4) stops a volume that uses both.
- **New notation has a home.** A definition that introduces a symbol is the `introduced_by` node of the entry. The packer uses this to place a notation item where the symbol first appears and to build the list of notation, as PreTeXt does with `<notation>` and `<notation-list>`.
- **Generated files.** The build writes `macros/m2.sty` and `macros/m2.katex.json` from the register. A pull request that edits them by hand fails the build. Rule B01 (declared macros only) stays, and the register adds its meaning.
- **Alignment.** The `openmath` and `intent` fields are alignment data, like a Wikidata id. The export of symbols to Wikidata uses the pattern of the properties P7235 and P9758 (K01).

The write role gets the entries of the scope of its target in its context, as part of the cached prompt (K12.4).

## K10.5 Conventions

Some definitions differ between sources by convention: whether $\mathbb{N}$ contains 0, whether a ring has a unit, whether a compact space is Hausdorff, whether a graph can have loops, which argument of an inner product is linear, where the factor $2\pi$ goes in a Fourier transform. A convention changes what a statement means, so the corpus records it.

- **A register of conventions.** `notation/conventions.yaml` lists each convention: an identifier, the options, the sources that use each option, and the default of math2.
- **The default.** The default follows Mathlib where Mathlib has one (settled decision 16), so that the informal text and the formal binding agree. Where Mathlib has no choice, the default follows ISO 80000-2 where that standard has one, and a curator decides the rest. For $\mathbb{N}$, Mathlib and ISO 80000-2 agree: 0 is a natural number, and ISO writes $\mathbb{N}^*$ for the positive integers.
- **Nodes declare the conventions they depend on.** A concept node lists the conventions that change its meaning (K8.3), in short form: "ring: unital, not necessarily commutative". A definition or statement that depends on a convention links the concept that carries it, so that a change of option finds each node that it affects. A node that uses an option other than the default says so in its main section and in the front matter.
- **Bindings must agree.** The F1 battery (spec 10.4) checks that the convention of the informal text and the convention of the formal binding are the same. A mismatch is the misformalization class "convention" of spec 10.5.
- **Volumes choose.** A volume (K11) can choose another option for a convention. The packer writes a Conventions section from the conventions of its nodes, as each chapter of the Stacks Project has one. The pack check (K11.6) reports each node in the volume whose text depends on the convention, so that an editor can check it.

## K10.6 Names and languages

- **One canonical name for each node**, in English, with the aliases in `registers/aliases.reg`. The canonical name is the one that most sources in the row use (K4.1). A curator decides when the sources disagree.
- **Names of people** follow the form that the MacTutor archive or Wikidata uses, with the diacritics: "Hahn–Banach", "Erdős".
- **Other languages.** Wikidata gives labels in many languages under CC0, and the corpus can import them as aliases with a language code. Translations of bodies follow spec 13.4. A translation is a view of the English text, and it never changes a statement.

## K10.7 Rules

| Rule | H/S | Checks |
|---|---|---|
| N01 | H | each macro in a body has an entry in the notation register whose scope includes a topic of the node, or the global scope, or the node defines it |
| N02 | H | two entries with the same glyph in scopes with a topic in common are the same macro, or each lists the other in `conflicts` |
| N03 | S | a curriculum puts a member before one of its prerequisites |
| N04 | S | a node uses an option of a convention other than the default and does not say so in its front matter |
| N05 | S | a concept with no `specializes` edge and no topic other than its MSC code |
| N06 | H | each macro in the register has one `denotes`, and `macros/m2.sty` and `macros/m2.katex.json` are the output of the register |
| N07 | S | a definition introduces a symbol in its `notation` field, and no register entry has it as `introduced_by` |
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/11-packaging.md">
# K11. Packaging: from nodes to courses, books and articles

The goal says that each node is "ready to go into a course, a book or an article" (K2.1). This document says what that means as a process that a machine can run and check. A **volume** is a selection of nodes with an order, some editorial text and a set of choices. The **packer** compiles a volume into its outputs. The **pack check** says whether the volume is ready.

The packer adds no mathematics. Each definition, statement, proof and example in a volume is a node, and the volume shows it as the node says it. This keeps one truth: an error that is fixed in a node is fixed in each volume at its next build.

## K11.1 The volume file

A volume is a file `volumes/<id>.yaml` in `tamnd/math2`:

| Field | Holds |
|---|---|
| `id`, `title`, `editors` | the identity of the volume and the people who are responsible for it |
| `kind` | `course`, `book`, `notes`, `survey`, `article`, `reference` or `problems` (K11.5) |
| `audience` | the level of the reader, as in K8.3 |
| `targets` | the nodes, topics or curricula that the volume must cover |
| `assumes` | the nodes that the reader knows. The volume lists them and does not explain them |
| `release` | the math2 release that an edition pins |
| `levels` | the lowest writing level and evidence level that a node must have to go in (K11.6) |
| `proofs` | `all`, `main` (the main proof of each statement, with the routine steps folded, K8.5), or `none` |
| `exercises` | how many exercises for each section, and the difficulties |
| `choose` | for each concept with more than one definition in the closure, the definition that the volume uses. The others become theorems of the form "equivalent characterizations", and the packer adds the proofs of the equivalence (K11.2) |
| `conventions`, `notation` | the options of K10.5 and the notation variants of K10.4 that the volume uses |
| `outline` | optional: the chapters and the sections, each with its nodes. When it is absent, the packer makes it (K11.2) |
| `glue` | the editorial text: K11.3 |
| `formats` | the outputs of K11.4 |

## K11.2 Linearization

A volume becomes a sequence in three steps:

1. **The closure.** The packer takes the learning path of K10.3 from `assumes` to `targets`, over the definitions of `choose` only. Each node in the closure that is not in `assumes` goes in the volume. For each concept in `choose`, the other definitions go in as theorems, with their proofs of equivalence.
2. **The outline.** When the volume has no `outline`, the packer groups the nodes by topic into chapters, and by concept into sections. A graph with no cycles has many orders with no forward reference, and the choice among them is about keeping related nodes close. The packer breaks ties in this order: the outline of the topic nodes, then a statement next to its proof and its examples, then the order of a curriculum where one exists, then the MSC code. An editor can then write an `outline`, and the packer checks it.
3. **No forward references in proofs.** This is the rule of the Stacks Project: a proof does not refer to a result that comes later. When an `outline` breaks it, the packer stops with the list of the references, and pack check PC2 fails. The packer never moves a node silently.
4. **Numbering and references.** The packer gives each node a number in the volume: "Theorem 3.4", "Definition 2.1". A link in a body to a node in the volume becomes a reference with the number. A link to a node outside the volume becomes a citation of the node by its URI and its name, in the form that K11.8 gives.

A node can appear in more than one place only as a reference. It has one home in each volume.

The packer also writes two generated sections: **Conventions**, from the conventions of the nodes in the volume (K10.5), and the **list of notation**, from the `introduced_by` entries of the notation register in reading order (K10.4).

**The lock file.** Each build writes `volumes/<id>.lock`, with the `TAG@release` of each node in the volume and the choices that the packer made. The volume file and the lock file together are the method of the volume. A person can read each decision as data, and a second build from the lock gives the same bytes (PC9).

## K11.3 Editorial text

A book needs text that is not a node: a preface, the introduction of a chapter, a sentence that leads from one section to the next. This is the **glue**.

- Glue lives in the volume file, or in files next to it in `volumes/<id>/`. It is prose in the CC BY-SA 4.0 zone, with provenance as for any other text (spec 04.8).
- **Glue does not state mathematics.** It cannot contain a definition, a statement or a proof. If an editor wants to say something mathematical, it is a node. The pack check finds glue that has the form of a statement (pack check PC7).
- Glue that a model wrote is marked as such, and it needs confirmation by an editor of the volume before an edition (rule P04).

## K11.4 The outputs

The packer has one internal form, a typed document tree, and one back end for each output. The tree holds the nodes, the glue, the numbering and the references. Each back end writes it in one format, and no back end changes the content.

| Output | For | Notes |
|---|---|---|
| HTML | the web, in the site of math2 | the same templates as the site, with the status of each node shown, and JSON-LD on each page (K11.4a) |
| PreTeXt XML | books | the PreTeXt tools make web pages, PDF, EPUB, braille and WeBWorK output from one file, so math2 does not write five back ends. Most kinds map to a PreTeXt element. The notation items map to `<notation>`. The PreTeXt tools are GPL, and math2 runs them as separate programs |
| LaTeX and PDF | print, and arXiv | `amsthm` environments and `macros/m2.sty`, one class for each kind of volume. For an `article`, an arXiv source tree: `main.tex`, a copy of `m2.sty` (arXiv has only the packages of TeX Live), a `00README.json` that names the compiler and the TeX Live year, and a `.bib` from the sources of the nodes |
| JATS 1.4 | articles for journals and repositories | each formula in `<alternatives>` with `<tex-math>` (with no `$` delimiters, as JATS4R recommends) and MathML from KaTeX |
| Common Cartridge 1.3 | courses in a learning management system | HTML pages for each node, QTI items from exercises with `check: choice` or `numeric`, and LTI 1.3 links to the workspace for exercises with `check: lean` or `cas` |
| Verso Blueprint | Lean projects | a blueprint document with one directive for each node, the math2 tag as the label, and the `uses` edges of the graph (K11.9) |
| MyST Markdown | Jupyter Book and the notebook tools | computation nodes become code cells |
| Typst | a second typesetting target | only when the `typst` field of the notation register (K10.4) covers the macros of the volume |
| slides | lectures | Beamer, from a volume of kind `course` with one section for each lecture |

The first back ends are HTML, LaTeX and PreTeXt XML (K14.2). EPUB comes from PreTeXt. The others come when a user asks for them. math2 does not write SCORM: Common Cartridge covers the same need, and its makers say that it does not replace SCORM, so a learning management system that needs SCORM can convert. Records of what learners did (xAPI) belong to the workspace (spec 13), never to `tamnd/math2`.

### K11.4a Metadata on each page

The site and each HTML output put JSON-LD of schema.org on each page:

| Property | From |
|---|---|
| `@type` | `DefinedTerm` for a concept, `LearningResource` for an explanation or an exercise |
| `teaches` | the concepts of the node |
| `competencyRequired` | the direct prerequisites: `uses` at depth 1 |
| `assesses` | for an exercise, the nodes that it exercises |
| `educationalLevel` | L1 to L4 (K8.6), as terms of a published SKOS scheme, not free text |
| `learningResourceType` | an LRMI term, for example "worked example" or "exercise" |

This metadata is CC0, like the rest of the metadata zone.

## K11.5 Kinds of volumes

| Kind | Made from | Has |
|---|---|---|
| `course` | a curriculum, with a schedule | sections for each lecture, exercises, problem sheets, solutions in a separate output for teachers |
| `book` | a topic or a set of targets | chapters, full proofs, exercises, an index of terms and of notation |
| `notes` | a short set of targets | proofs `main`, few exercises |
| `survey` | a topic in the research layer | statements without proofs, the history of each result, the citations of the research layer (the `aligned` records), the open problems |
| `article` | a small set of targets with a new result | the format of a journal, with the formal status of each claim |
| `reference` | a topic | definitions and statements only, in the order of the taxonomy, as a handbook |
| `problems` | exercises and problems | an exam pack or a problem book, with solutions |

## K11.6 The pack check

`m2 pack -check VOLUME` says whether a volume is ready. It runs with no model and no network, on a pinned release. Its checks have the prefix PC, so that they are not confused with the audit rules of spec 15.3:

| Check | Requires |
|---|---|
| PC1 levels | each node meets the `levels` of the volume. Each statement that the volume shows as a theorem is at E2 or better. A statement below that is shown with its level, for example as a conjecture or as "claimed" |
| PC2 order | no node comes before one of its prerequisites, and no proof refers to a result that comes after it (K11.2) |
| PC3 closure | each `uses` edge of each node in the volume ends in the volume or in `assumes` |
| PC4 notation | each printed glyph has one meaning in the volume (K10.4). Two macros in the `conflicts` of each other in one volume are a failure, unless the volume chooses a variant for one of them |
| PC5 conventions | each node that depends on a convention uses the option of the volume, or the volume lists it with the difference |
| PC6 references | each reference resolves, in the volume or to a citation |
| PC7 glue | the glue contains no statement, definition or proof, and each part of it has a confirmed writer |
| PC8 licence | the attribution of K11.7 is complete, and rule L02 is clean for each node in the volume |
| PC9 outputs | each output format builds with no error, and the build is reproducible: the same inputs give the same bytes |
| PC10 exercises | each section has the exercises that the volume asks for, and each exercise with an answer check passes it |

The **standard volumes** are the volumes that the corpus itself maintains, one for each area of the undergraduate core (K14.5). The goal of K2.1 counts as met for an area when the pack check passes on its standard volume.

## K11.7 Licence and attribution

- **The prose of a volume is CC BY-SA 4.0**, like the prose of math2 (settled decision 17). A volume cannot have a licence with more restrictions. Anyone can print and sell it, as the licence allows.
- **The Lean companion is Apache-2.0**, like the code of math2. So are the Lean statement and the reference solution of an exercise with `check: lean`: they are Lean code. The text of the exercise is prose.
- **The attribution is generated.** The packer writes an attribution page from the provenance of each node and each part of the glue: the people who wrote and reviewed the content, the share-alike sources that rule L03 requires, and the disclosure of agent contributions that `m2 prov disclose` generates (settled decision 24).
- **Credit to people goes first.** The page lists people by their contributions to the volume. The list of agent runs comes after it, in the form of the disclosure.

## K11.8 Editions, citations and errata

- **An edition pins a release.** The edition of a volume records the math2 release, so that its content is fixed. The volume has its own version, `<id>@<release>`.
- **Citations of nodes.** A volume cites a node outside it by its name and its URI, for example "Sylow theorems, `m2://7K3QX2M9R2`". Printed works can show the tag in the reading form of K3.4.
- **Errata are generated.** When a node of an edition changes in a later release in its main section, the next build of the volume lists the change as an erratum of the edition. A change that only adds sections or explanations is not an erratum.
- **A volume can be a citable work.** Its editors can ask for a DOI for an edition. This is a publication, and it needs the approval of spec 06.2.

## K11.9 The Lean companion

Each volume can have a Lean companion: a Lake project that requires Mathlib and the math2 Lean library at the commits of the release.

- It has one file for each chapter. For each node with a binding, the file has a reference to the declaration with the number of the node in the volume as a comment, and the proof when math2 holds it.
- For each statement with no formal proof, it has the formal statement with `sorry`, marked as open, and a Verso Blueprint of the dependency graph (K11.4). The companion is then a plan for the formalization of the volume.
- The blueprint keeps the origin of each edge: "automatic" when Lean found it, "manual" when the graph of math2 has it and Lean does not. The import of blueprints (spec 09.4) keeps the same distinction in the other direction.
- The companion builds in CI with the release. A companion that does not build fails pack check PC9.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/12-pipeline-and-operations.md">
# K12. Pipeline and operations

This document says how the corpus is produced: the stages that an item goes through, the jobs and the queues, the limits on speed, the cost, the commands and the libraries. It uses the orchestrator of spec 11.4 and 11.5 and the roles of spec 12.1 and K7.3. It adds no new write path. Each change is a pull request, and the audit is the gate (settled decision 7).

## K12.1 The stages of an item

An item goes from "a source mentions it" to "a complete node" in stages. Each stage has a goal predicate that the build can compute, so a job knows when it is done.

| Stage | Goal predicate | Done by | Section |
|---|---|---|---|
| 1. Listed | the item is a mention in a source catalogue, or in a list that a model made | importers (K05, K06), the enumerate role (K7.3) | K4.2 |
| 2. Inventoried | the mention belongs to an inventory row | clustering (K4.3), the dedup judge, curators for close cases | K4.3 |
| 3. Decided | the row has an outcome: a node, a member of a hub, an `aligned` record, or a reason why it is not a node | the curate role proposes, the rules of K4.4 decide, curators decide the rest | K4.4 |
| 4. Stub | a node at W0 | the curate role, importers | K2.4 |
| 5. Stated | W1 | importers of `open` text, the write role, the informalize role for formal-first items | K08 |
| 6. Usable | W2 | the write, explain and exercise roles; the prove and formalize roles in parallel | K08 |
| 7. Complete | W3 | the same roles, and people who confirm sections | K08 |

The formal track runs next to stages 5 to 7. It does not wait for them:

| Formal stage | Goal | Done by |
|---|---|---|
| bound | a binding at F0 | formalize, or align for items that a library already has |
| checked | F1 | the F1 battery (spec 10.4) |
| reviewed | F2 or F3 | people (spec 10.6) |
| proved | E5 | prove, or an import from a library with a checked proof |

An item that a formal library has (K06) starts on the formal track at "bound" and reaches stage 5 through the informalize role. An item from a textbook starts at stage 5 and reaches the formal track through formalize. Both paths end at the same node.

## K12.2 Jobs

The corpus adds standing queues to the four sources of jobs in spec 11.4. Each queue is computed from the graph and the inventory on each build:

| Queue | Holds | Role | Goal |
|---|---|---|---|
| `enumerate` | each topic and each MSC code whose inventory has fewer rows than its estimate (K4.7) | enumerate | the model lists each item of the area that it knows, with a source for each one |
| `cluster` | the mentions with no row, and the candidate pairs above the similarity threshold | dedup judge, then curators | each mention belongs to one row |
| `decide` | the rows with no outcome | curate | the open count goes down |
| `state` | the nodes at W0 | write, informalize | W1 |
| `complete` | the nodes below the writing level of their volume (K11) or their layer (K2.7) | write, explain, exercise | the next writing level |
| `confirm` | the agent-written sections with no confirmation | review-assist, then people | rule P04 is clean |
| `sanity` | the agent-written statements with no result of K9.3 | explore | X05 is clean |
| `source` | the sources with `locator_unconfirmed` | review-assist, then people | P06 is clean |
| `bind`, `prove` | the queues of spec 11.4 | formalize, prove | spec 12.4 |

A job of the corpus has the record of spec 11.4. Its goal names a stage, for example `goal: "node reaches W2"` or `goal: "row 9DK2RX7Q4H has an outcome"`.

## K12.3 Priority

The priority formula of spec 11.4 applies. The corpus adds two terms, and the formula keeps its version like a prompt:

```
priority = w1 · centrality(target)
         + w2 · demand(target)
         + w3 · unlock(target)
         + w4 · tractability(target, role)
         − w5 · attempts(target, role)
         + w6 · spread(target)        # the number of independent sources that hold the item
         + w7 · curriculum(target)    # the item is in a curriculum or a standard volume (K10, K11)
```

`spread` puts first the items that many sources hold. An item that eight textbooks and two libraries hold is almost surely standard and is cheap to state well. An item that one source holds needs more care, and it can wait. `curriculum` puts first the items that a course or a book needs, so that the first volumes can be packed early (K14).

The order of work in spec 18.5 still applies: breadth first on the record plane, depth first on the statement plane. In the corpus, this means that the inventory of each area is built first, and then each area is written in depth, in the order of the curricula.

## K12.4 Batches

Most corpus work has no person waiting for it. So it runs in batches:

- **Model batches.** The write, explain, exercise, informalize and enumerate roles, and the dedup judge, use Message Batches at half price (spec 12.3). A batch holds jobs of one role and one topic, so that the cached part of the context (the system prompt, the card, the style guide, the notation register of the topic) is the same for each job.
- **Pull request batches.** One pull request for each node would give millions of pull requests. A batch pull request holds up to 50 nodes of one topic, from one role and one configuration. The audit checks each node in it. A reviewer can merge a part of it: the nodes that pass, with the others sent back to the queue with their reasons.
- **Small diffs.** A batch pull request changes only the files of its nodes and their provenance logs. It never changes a register by hand. The build writes the registers.

## K12.5 Review capacity is the limit

Settled decision 20 caps agent pull requests at a multiple of the measured review throughput. In the corpus, this is the main limit on speed, not money and not model speed. So the plan starts from the review hours.

Each month, the review hours go first to the work that keeps the measurements honest, and then to the work that moves items:

| Order | Use | Example cost |
|---|---|---|
| 1 | the sample audit (K9.7) | about 170 hours at 20 strata |
| 2 | reader reports (K9.8) and quarantine decisions | measured |
| 3 | F2 reviews in the order of `unlock` (spec 10.6) | 3 minutes for each undergraduate statement |
| 4 | confirmation of agent-written sections (rule P04) | 1 to 2 minutes for each section, by estimate |
| 5 | decisions on close cases of the inventory (K4.3, K4.4) | 1 minute for each case, by estimate |
| 6 | locator confirmation (K9.5) | 2 minutes for each source, by estimate |

The agents then get pull request capacity in proportion to the review hours that remain, by spec 11.5. A role that makes work that nobody can review in time does not run.

An example with round numbers. 100 active reviewers give 10 hours each month: 1,000 hours. The sample audit and the reports take 250 hours. Of the 750 hours that remain, 400 go to F2 reviews (8,000 statements) and 350 go to confirmation (about 14,000 sections). At this rate, F2 for the 500,000 statements of the textbook layer takes about five years, and W3 for 1.5 million nodes takes longer. This is why W2 is the target of the textbook layer and W3 is the target of the standard volumes only (K2.7, K11). It is also why the corpus must recruit reviewers (K14) and make each review fast (spec 16.3).

## K12.6 The cost of the models

The model cost is small next to the cost of people, but it is not zero, and the plan must say what it is. The numbers below are estimates for planning, with the prices of spec 12.3 and batch pricing. They assume that the cached part of each context is read from the cache. The run records replace them with measured numbers after the first thousand jobs of each role. K7.8 gives the cost of each configuration of models for one node.

| Stage | Tier | Tokens in, tokens out, for each item | Cost for each item |
|---|---|---|---|
| enumerate (for each item listed) | 1 | 0.5k, 0.1k | under $0.001 |
| dedup judge (for each candidate pair) | 0 | 2k, 0.1k | under $0.001 |
| state: W0 to W1 | 1 | 10k, 1.5k | about $0.02 |
| usable: W1 to W2 (proof, examples, L2, the second-model check of K9.3) | 1, and 2 for hard items | 30k, 6k | about $0.06 at tier 1, $0.12 at tier 2 |
| complete: W2 to W3 (other levels, history, exercises) | 1 and 2 | 30k, 8k | about $0.07 to $0.14 |
| formalize a statement to F1 | 1 to 3 | varies by orders of magnitude | $0.50 to $5, by spec 17 measurements when they exist |

For the textbook layer of 1.5 million nodes at W2, the writing cost is about $0.10 for each node, or about $150,000. Retries and escalation can double it. Formalization of 500,000 statements costs $250,000 to $2.5 million, and it is the largest model cost. The copy check (K9.4) and the sanity checks by computation run on math2's own machines.

**No model run starts without a budget that a person approved.** The global monthly budget of spec 11.5 is the only source of money. `m2 corpus plan` (K12.8) estimates the cost of a queue before anyone approves it.

## K12.7 Measurement

The corpus has its own section in the weekly agent report (spec 12.9) and in each release manifest:

- the open count, for each layer and each MSC code, and its change since the last report;
- the number of nodes at each writing level, for each layer;
- the formal outcomes of K2.5;
- the error rates of K9.7, with their intervals;
- the review hours used, by use (K12.5), and the queue lengths;
- the model spend, by role and stage, and the cost of each item that reached a new stage;
- the rate of fabricated sources (K9.5) and of copy matches (K9.4), by configuration.

A number that goes the wrong way is in the report with the others. The release notes account for each change in the counts (rule R03).

## K12.8 Commands and libraries

The corpus adds two command groups to `m2`. Each command is a thin layer over a library package. The packages follow the conventions of the toolchain: one package for each concept, no `internal/` directory, exported types with documentation and a testable example, and tests that use fakes for each model and each network service.

| Command | Does |
|---|---|
| `m2 corpus import -source NAME` | runs the importer of a source (K05, K06) into mentions in the record plane |
| `m2 corpus cluster` | groups mentions into inventory rows, and writes the close cases to the `cluster` queue |
| `m2 corpus status [-layer L] [-msc CODE]` | prints the open count, the writing levels and the formal outcomes |
| `m2 corpus queue NAME` | prints a standing queue in priority order |
| `m2 corpus plan NAME -n N` | estimates the cost and the review hours of the first N jobs of a queue. It does not call a model |
| `m2 corpus run NAME -n N -budget USD` | submits the jobs to the orchestrator under an approved budget |
| `m2 corpus sample -month YYYY-MM` | draws the sample of K9.7 and writes the audit packets |
| `m2 pack VOLUME` and `m2 pack -check` | K11 |

| Package | Holds |
|---|---|
| `inventory` | mentions, rows, outcomes and the open count (K04) |
| `cluster` | the grouping of mentions, with the similarity and the judge behind interfaces |
| `card` | the cards of K08, read from `schema/cards.v1.yaml`, and the writing level |
| `style` | the linter of K8.2 |
| `copycheck` | the shingles, the fingerprints and the match rules of K9.4 |
| `sources` | the confirmation of K9.5, with Crossref, zbMATH and the catalogues behind interfaces |
| `sample` | the strata, the draw and the intervals of K9.7 |
| `pack` | the volumes and the outputs of K11 |
| `importer/<source>` | one package for each source adapter of K05 and K06 |

## K12.9 Scale of the repository

The textbook and formal layers add 1 to 2 million node files, with their provenance logs. That is in the range of spec 06.1 and 06.7. The research layer is in the record plane, and its inventory rows are in the record plane too (K4.8). When the repository passes the trigger of spec 06.1 (10 GB packed or 5 million files), the record plane and the inventory of the research layer move to `tamnd/math2-records` together.

Agent branches are short. A batch branch lives until its pull request merges or closes, and the orchestrator deletes it then. Contributors who work on one area use a sparse checkout of the shards that they need. `m2` works on a sparse checkout, except for the commands that need the full graph (`m2 build`, `m2 audit`), which run in CI.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/13-legal-and-ethics.md">
# K13. Law and ethics

The corpus takes text from thousands of sources and has models write most of the rest. Both are legal questions, and both are questions of trust. This document says what the law says on the day of this spec, what math2 does because of it, and which questions need a lawyer before a phase starts (K13.10).

This document is not legal advice. It is the reading of the sources of K01, part D, by people who are not lawyers. Each fact has its source and the date of reading in K01. Where the law is not settled, this document says so and chooses the safe course.

## K13.1 The three zones

Settled decision 17 gives three licence zones. The corpus keeps them, and puts each new kind of content in one zone:

| Content | Zone | Licence |
|---|---|---|
| prose of nodes, explanations, glue of volumes, text of exercises | prose | CC BY-SA 4.0 |
| Lean code, the Lean statement and the reference solution of an exercise, the code of computation nodes, the Lean companion of a volume | code | Apache-2.0 |
| front matter, mentions, inventory rows, outcomes, the notation register, the conventions register, the JSON-LD of the site, records | metadata | CC0 |

A volume has the licences of its parts (K11.7). A dataset release has a manifest that gives the zone of each file.

## K13.2 Copyright in text that models write

**The United States.** The Copyright Office, in Part 2 of its report on AI (29 January 2025), says that copyright does not extend to material that is purely AI-generated, or where a person had insufficient control over the expressive elements. It says that prompts alone do not give enough control. A person has copyright in what they wrote, and in their creative selection, arrangement or modification of the outputs. The courts agree that the Act needs a human author: the D.C. Circuit held so in *Thaler v. Perlmutter* (18 March 2025), and the Supreme Court denied certiorari on 2 March 2026.

**What this means for math2.**

1. A section that a model wrote and no person changed may have no copyright in the United States. The CC BY-SA licence on it may then have no effect: section 8(a) of the licence says that it does not limit uses that need no permission. Mathematical facts, definitions and short statements have thin protection in any case.
2. What people add has copyright: the text that people write or edit, and the selection, the arrangement and the curation of the corpus. CC BY-SA applies to these.
3. Other jurisdictions answer the question differently, and some have not answered it. For example, the United Kingdom has a rule for "computer-generated works" (CDPA s.9(3)). K01 did not research these.

**The decision.** The prose zone stays CC BY-SA 4.0 for all prose, with a notice in `LICENCE`:

> The licence applies to the extent that copyright subsists in the material. Some sections were written by software and not changed by a person. These sections may be in the public domain in some jurisdictions. The provenance of each section says who or what wrote it.

The other option was CC0 for prose that a model wrote and no person confirmed, and CC BY-SA after a person confirms it. Creative Commons itself suggests CC0 for works with no significant human creativity. The corpus does not take this option, because it makes the licence of a section depend on its provenance, and the zone of a section would change when a person confirms it. The notice tells the truth, and the provenance shows which sections it covers. A change to this decision is a change to settled decision 17.

## K13.3 The real risk: copies

Who owns model output is a small question for math2. The large question is whether model output copies someone else's text. A model that reproduces a textbook proof gives math2 the textbook's text, with the textbook's copyright.

- **Training cases are about inputs.** *Bartz v. Anthropic* (N.D. Cal.) held in June 2025 that training on lawfully acquired books was fair use, and that a library of pirated copies was not. The case settled, with final approval on 20 July 2026. In *Kadrey v. Meta* (June 2025) the court found fair use on that record. In *Thomson Reuters v. Ross* (3d Cir., 29 September 2026), the court held that copying headnotes to train a search tool that does not generate text was not fair use, and said that generative AI is a different case. None of these cases decides the status of outputs.
- **Output cases.** *NYT v. OpenAI* (S.D.N.Y.) is the main case in the United States on regurgitated output, and it has no decision on the merits. In Germany, *GEMA v. OpenAI* (LG München I, 11 November 2025) held that the memorization of song lyrics in a model is a reproduction, and that the output of the lyrics is one too. It held OpenAI liable for the outputs, and OpenAI appealed. In the United Kingdom, *Getty Images v. Stability AI* ([2025] EWHC 2863 (Ch), 4 November 2025) held that the weights of a model do not store an infringing copy, and an appeal is pending. The Court of Justice of the EU heard *Like Company v. Google* (C-250/25), on chatbot output that matches press articles, in March 2026, and a judgment is expected in 2027.
- **Who is exposed.** Whoever publishes a copy is exposed, not only the provider of the model.

**What math2 does.**

1. **The copy check** (K9.4) runs on each body that an agent wrote, against a reference corpus and not only the cited sources. A match with a restricted source blocks the merge.
2. **No restricted text in the context** of a write job (K7.5). A model that rewrites a text that it can see makes a derived work.
3. **Matches with open sources are attributed.** A close match with a CC BY-SA source keeps that source's terms, whatever math2 says, so the node records the attribution (rule L03).
4. **GFDL, NC and all-rights-reserved text never enters the prose zone.** GFDL text, for example from the Stacks Project, cannot be relicensed under CC BY-SA 4.0. Kerodon states no licence, so its text is all rights reserved. Non-commercial text cannot be in a corpus that anyone may sell.
5. **Takedown.** A claim that a node copies a work goes to the curators as a report of the property "ours to publish" (K9.8). The node is quarantined within two working days, while a person checks the claim.

## K13.4 Human sources

The corpus takes from human sources by the licence class of each source (K05):

| Class | What math2 may hold | Example |
|---|---|---|
| `open` | the text, with attribution, when the licence is compatible with CC BY-SA 4.0 | Wikipedia (CC BY-SA), set.mm (CC0), Mathlib docstrings (Apache-2.0) |
| `share-alike-nc` | records, locators and links. No text | OpenStax (CC BY-NC-SA), the WeBWorK Open Problem Library (CC BY-NC-SA 3.0) |
| `gfdl` | records, locators and links. No text | the Stacks Project, the Manifold Atlas |
| `verbatim` | short quotations where the law of quotation allows them, with the source | a famous statement in its original words |
| `record` | records and locators only | most textbooks and papers, Kerodon, the DLMF |

Three further points apply to each class:

- **Facts are free, words are not.** That a theorem is true, what it says and where it appears are facts. The words of a source are its author's. A node that states a standard theorem in math2's words takes nothing that copyright protects.
- **Collections and databases.** The selection and the arrangement of a collection can have copyright, and in the EU a database can have a database right against the extraction of a substantial part. So an importer takes from a catalogue only what the terms of the catalogue allow, and the source document (K05) records those terms. A table of contents goes into the record plane as locators for each item, not as a copy of the table.
- **Licences per item.** Some sources have a licence for each entry, for example the entries of the Archive of Formal Proofs, or the posts of Math.StackExchange by date. The importer records the licence of each item, not of the source.

## K13.5 The EU AI Act

- **Duties of providers.** The duties for providers of general-purpose AI models (Articles 53 to 55) apply from 2 August 2025, and the Commission can enforce them from 2 August 2026. They fall on the providers of models. math2 uses models and is not their provider. A team that changes a model becomes a provider only if the change uses more than a third of the training compute of the original. A fine-tune of an open prover by math2 would be far below that.
- **Transparency (Article 50).** It applies from 2 August 2026. Providers of systems that generate text must mark the outputs in a machine-readable form (Article 50(2)). The Digital Omnibus on AI (Regulation (EU) 2026/1744) reportedly gives this duty a grace period to 2 December 2026, and sources disagree on its scope. Deployers that publish text generated by AI to inform the public on matters of public interest must disclose that it is generated (Article 50(4)). The duty does not apply when the text had human review or editorial control and a person holds editorial responsibility for the publication.
- **What math2 does.** It is not settled whether a mathematics library informs the public "on matters of public interest". math2 meets the duty either way. Each field that an agent wrote is marked as such (rule P04), the site puts a machine-readable flag on each field that no person confirmed, and each area has a named editor who holds editorial responsibility (spec 16).

## K13.6 The terms of the providers

K7.9 gives the rules that follow from the terms. In short:

- The terms of Anthropic, OpenAI and Google give the customer the rights in the outputs that the provider has. None of them stops publication under an open licence.
- From 12 November 2026, the usage policy of Anthropic forbids the use of outputs to train an AI model without its prior authorization. OpenAI and Google forbid the use of outputs to build competing models. math2 does not train on the output of hosted models without written authorization.
- Some open-weight licences add duties on outputs. The model register (K7.7) records them.
- **Users of math2.** A person who downloads math2 under CC BY-SA has no contract with the provider of a model. It is not settled whether a provider's terms reach that person, and math2 claims nothing either way. The release notes say which provider wrote which fields, so that users can decide for themselves.

## K13.7 Disclosure

The disclosure of agent work is generated and complete (settled decision 24):

- **Each field.** The provenance names the role, the model and the run of each field that an agent wrote, and the person who confirmed it, if any.
- **Each page.** The site shows "written by software, not yet checked by a person" on each unconfirmed field, in words and in the machine-readable flag of K13.5.
- **Each volume.** The attribution page lists people first, by their contributions, and then the disclosure of agent runs (K11.7).
- **Each release.** The release notes give the share of fields that agents wrote, by layer and kind, and the share of those that a person confirmed.

## K13.8 Ethics

Four duties are not in the law, and the corpus keeps them:

1. **Do not flood the commons.** Wikipedia adopted a speedy-deletion criterion for unreviewed pages that a model generated (G15, August 2025). arXiv asked that review and position papers in computer science be peer-reviewed first (October 2025). Communities object to unreviewed machine text in bulk, not to the use of AI as such. math2 never writes agent text into the sources that it imports from (spec 16.5), and it never presents an unconfirmed section as reviewed.
2. **Credit to the people whose work the corpus uses.** Each node names the sources that its row has, even where the licence does not require it. A mathematician whose result is in the corpus is named with the result (K10.6).
3. **Honest numbers.** The error rates of K9.7 are published with their intervals, including the rates that are bad. A count that goes down is explained in the release notes (rule R03).
4. **Be polite to services.** Importers keep the rate limits and the robots rules of each source, identify themselves with the User-Agent of the toolchain (with no personal address), and prefer a published dump to crawling.

## K13.9 Rules

| Rule | H/S | Checks |
|---|---|---|
| L06 | H | each item from a source with a licence for each item records the licence of the item (K13.4) |
| L07 | S | a node with a takedown report that is older than two working days and not quarantined |

## K13.10 Questions for a lawyer

Each of these needs advice before the phase that the table names:

| Question | Before |
|---|---|
| Is the notice of K13.2 enough, or does each section that a model wrote need its own notice? | the first release with agent prose (phase 2) |
| Does the database right of the EU limit the import of catalogues such as zbMATH, the OEIS and MathSciNet, beyond their terms? | phase 1 |
| Do the terms of a provider reach users of math2 who have no contract with it? | the first dataset release |
| Is the publication of a corpus of outputs an "availability to third parties" under the Qwen 3.8 and Kimi K3 licences? | any use of these models |
| Which licence governs DeepSeek-Prover-V2: the MIT file or the Model Agreement that its README names? | any use of its outputs for training |
| Is a mathematics library within Article 50(4)? | not needed: math2 meets the duty either way |
| What is the position in the United Kingdom and the EU on the copyright of output that a model generated? | phase 3 |
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/14-roadmap.md">
# K14. Roadmap

This document puts the corpus programme in phases. Each phase has a goal, the work, and an exit criterion that the build can check. A phase starts when the phase before it meets its exit criterion, except where the table says that two phases overlap. The phases come after the milestones of spec 19 that build the system. They use the system, and they do not replace a milestone.

Each phase that runs models needs a budget that a person approved (K12.6). No phase publishes a dataset or a DOI without approval (spec 06.2). The roadmap gives no dates, because the speed is set by review capacity (K12.5), and review capacity is not known yet.

## K14.1 The state on the day of this spec

`tamnd/math2` holds four nodes (the concept *Group*, Lagrange's theorem, and a computation with its node), the registers, the schemas of node.v1, the curriculum seed and the skeletons of 60 MSC classes in `topics/skeletons/`. The toolchain `tamnd/math2-agent` has the build, the audit of spec 15, the site, the agent framework with fakes, and the orchestrator. No corpus package of K12.8 exists yet. No model has written content for math2.

## K14.2 Phase 0: the foundations

**Goal.** The toolchain can hold, check and measure a corpus before the corpus exists.

**Work, in this order.** Each item is one pull request, or one in each repository:

1. Tags of ten characters (K3.8): the `tag` package, the schema patterns, `m2 migrate shard3`, rules T06 and T08 to T10, the Lean attribute and the resolver. In `tamnd/math2`: the migration, the cut-over commit and the frozen legacy set.
2. The minor version of node.v1 with the fields and the edges of K8.9, and `schema/cards.v1.yaml`. The `card` and `style` packages, and the writing level in the build.
3. The record plane layout of K4.8: `records/mentions/`, `records/inventory/`, `registers/inventory/`. The `inventory` package, the open count, and rules I01 to I06.
4. The notation register (K10.4), seeded from `macros/m2.sty`, with the build that writes `m2.sty` and `m2.katex.json` from it, and rules N01 to N07.
5. The `copycheck`, `sources` and `sample` packages, and rules L05, P05, P06, X05 and X06 (K9.9).
6. `m2 corpus status`, `queue` and `plan` (K12.8). These commands call no model.
7. The first `pack` with the HTML output, and `m2 pack -check` (K11). LaTeX and PreTeXt follow in phase 2.

**Exit.** Each rule of K9.9, K4.9 and K10.7 has a fixture pair, and the audit runs them on `tamnd/math2`. `m2 corpus status` prints the open count of an empty inventory. `m2 pack -check` passes on a volume made of the nodes that exist.

## K14.3 Phase 1: the inventory of sources that are safe and open

**Goal.** A first inventory, made only from sources whose data math2 can hold with no doubt about the licence, and no model.

**Work.**

1. Importers for the CC0 and permissive sources first: Wikidata items about mathematics, the declarations and the module documentation of Mathlib (Apache-2.0), the OEIS names (K4.6), and the curricula that are published as open data (K10.3).
2. Importers of records and locators only, for sources of the `record` and `gfdl` classes: tables of contents and indexes, where the source allows it (K05).
3. Clustering passes 1 and 2 (K4.3), which need no model. The gold set of K4.3 for the first areas.
4. The capture and recapture estimates of K4.7 for each MSC code at two digits.

**Exit.** Each mention has a row or is in the `cluster` queue. The report gives the estimated recall for each MSC code at two digits. The licence class of each mention is recorded, and rule L02 is clean.

## K14.4 Phase 2: the pilot area

**Goal.** One area, end to end, so that each number in K12 is measured and not estimated.

**The area.** Undergraduate group theory: MSC 20 at the level of a first course. The reasons: math2 already has two of its nodes, Mathlib covers it well, the textbook and wiki sources are many, and the statements are short enough that small-case checks (K9.3) work.

**Work.**

1. The inventory of the area from all sources of K05 and K06, and from the enumerate role, with the clustering passes 3 and 4.
2. An outcome for each row.
3. Nodes to W2 for each row with the outcome `node`, by importers of `open` text and by the write, explain, exercise and informalize roles, under an approved budget.
4. The formal track for each statement: bound, F1, and F2 for the core list.
5. The sample audit of K9.7 for each stratum of the pilot.
6. The first volume: *Groups: a first course*, packed in each output format of K11.

**Exit.**

- The open count of the area is zero.
- Each node of the area is at W2 or better, and each statement has a formal outcome (K2.5).
- The sample audit gives an error rate under the thresholds of K9.7 for each stratum, with its interval. Strata above a threshold are stopped, and their output is fixed or removed.
- The measured cost for each stage replaces the estimates of K12.6 in the spec.
- `m2 pack -check` passes on the volume, and a teacher who is not in the project has read it and reported the result.

If the pilot fails its exit, the corpus programme stops and the spec changes before anything grows. A failed pilot is a result, and the report says why.

## K14.5 Phase 3: the textbook layer

**Goal.** The textbook layer of K2.7: an open count of zero, W2 for each node, and E2 or better for each statement.

**Work.** Area by area, in the order of the curricula of K10.3 and the priority of K12.3. The undergraduate core comes first (spec 18.3), then the graduate core, then the rest of the textbook layer. Each area repeats the work of the pilot with the measured numbers. A standard volume for each area of the undergraduate core goes to W3.

**Exit for each area.** The exit of the pilot. **Exit for the layer.** Each area has met its exit, and the inventory recall (K4.7) is at 90% or better for each MSC code at two digits.

## K14.6 Phase 4: the formal layer

**Goal.** The formal layer of K2.7: each declaration that a person would cite (K4.5) is a node with a binding.

This phase overlaps with phase 3. It starts when the importers of K06 exist, and it shares the review budget with phase 3 by the order of K12.5.

**Work.** For each library of K06: the declarations that K4.5 selects, the informalize role for the ones with no human mention, the alignment of the ones that a textbook row already has, and F1 for each binding.

**Exit.** Each row of the formal layer has an outcome. Each node of the layer has a binding at F1 or better.

## K14.7 Phase 5: the research layer

**Goal.** The research layer of K2.7: an open count of zero in the record plane.

**Work.** The extraction and the alignment of spec 09, area by area, in the order of spec 18.5. Each result gets the outcome `aligned` or `record`, or it becomes a node when the graph needs it.

**Exit.** Each result of the record plane has an outcome.

## K14.8 What runs all the time

From phase 2 on, these run each month: the sample audit (K9.7), the reports of K12.7, the estimates of recall (K4.7), and the review of the thresholds of K9.7. From phase 3 on, a new source that enters scope goes through the held-out check of K4.7 before its mentions are added.

## K14.9 The people that the phases need

| Phase | People | For |
|---|---|---|
| 0 | maintainers of the toolchain | the code |
| 1 | one curator for each source class | the licence decisions, the gold set |
| 2 | an area editor for group theory, 5 to 10 reviewers, one outside teacher | the pilot |
| 3 | an area editor for each area of the undergraduate core, about 100 reviewers | K12.5 |
| 4 | reviewers who know each formal library | F2 |
| 5 | area editors in the research areas | the close cases of the alignment |

The recruitment of reviewers is work of the project, not a side effect. Spec 16 says how reviewers get credit.
</document>

<document source="https://tamnd.github.io/math2/spec/corpus/15-open-questions.md">
# K15. Open questions

This document lists each question that the corpus spec leaves open. Each question has an id, the way to settle it, and the phase of K14 that it must be settled before. A question that blocks nothing has "no block".

## K15.1 How the list works

- **Owner.** The maintainers own the list. A question is closed by a commit that changes this document and the documents that the answer affects, with the source of the answer.
- **The safe course.** Until a question is closed, the corpus takes the safe course that the text gives: a source with a licence in doubt is `record`, a count in doubt is not quoted as a rule, and a check that is not calibrated is soft.
- **Asking a source.** Many questions are settled by asking the people who run a source. Each request is polite, short, and sent by a maintainer from the project, not by an agent. The answer is kept with the source document (K5.2).
- **Questions for a lawyer** are in K13.10. This list names them and does not repeat them.

## K15.2 Licences and terms of human sources

| Id | Question | How to settle | Before |
|---|---|---|---|
| OQ01 | Did the move of the Encyclopedia of Mathematics to EMS Press change the copyright of the Springer originals? What is its current size? | read the live site again; ask the EMS | phase 2 |
| OQ02 | Do the Stack Exchange Public Network Terms of 13 November 2025 add clauses on AI or on scraping? Does the public MCP server cover MathOverflow and Math.SE? | read the terms in a browser | phase 2 |
| OQ03 | Will the nLab steering committee give an explicit licence for new contributions? | ask the steering committee | no block: the nLab is `record` |
| OQ04 | Is the Manifold Atlas GFDL 1.3 or in the public domain? | ask the editors | no block: it is `gfdl` |
| OQ05 | Are the knowls of the LMFDB under the CC BY-SA licence of its database? Which bulk channel does the LMFDB want AI users to use? | ask the LMFDB | the first LMFDB import |
| OQ06 | Is the zbMATH knowledge graph CC BY 4.0 or CC BY-SA 4.0? What do the API terms say? | ask zbMATH, or open an issue on `zbMATHOpen/zbmath-open-kg` | phase 1 (until then, CC BY-SA 4.0) |
| OQ07 | What are the licences of the 14 AIM titles that nobody has read (Bruckner and Thomson, ORCCA, Manes, May and Bart, Marsden and Weinstein, Boyd and Vandenberghe, Treil, Fields, Bender and Williamson, Bard, Zimmermann, Crisman, Stein, Connell)? | read each at its source | before each book enters Table A |
| OQ08 | Do the books that the Open Textbook Library lists as CC BY or CC BY-SA (Zakon, Kuttler, Shimamoto, Petrunin, Callahan and others) have those licences at their source? Is Hitchman CC BY-SA or GFDL? | read each at its source | before each book enters Table A |
| OQ09 | Does the 3.0 US port of CC BY-SA allow adaptation under CC BY-SA 4.0, as §4(b) of the Unported licence does? | read the legal code of the port | the import of Hefferon |
| OQ10 | Which CC variant do the PreTeXt titles with a plain "Creative Commons" badge have? | read each at its source | before each book enters Table A |
| OQ11 | What licences do FindStat, House of Graphs, KnotInfo and the Knot Atlas give? Did the authors of KnotInfo consent to the GPL-3.0 CSV package? | ask each project | no block: they are `record` |
| OQ12 | What are the terms of HathiTrust, the Internet Archive, CORE, Unpaywall and EuDML, and the licence of the GDZ scans, the Project Euclid journals and the Semantic Scholar abstracts? | read each in a browser | before the first import from each |
| OQ13 | What is the licence of MathBases, and of the sources that the research could not reach (GroupNames)? | read again | no block |
| OQ14 | What share of LibreTexts pages has no declared licence? | a sample of pages, by the `license:` tag | no block: the corpus prefers the upstream source |

## K15.3 Formal libraries and datasets

| Id | Question | How to settle | Before |
|---|---|---|---|
| OQ15 | What are the licences of the 20 Formal Frontier repositories that nobody has read? | read each `LICENSE` | before each binding to them |
| OQ16 | Does the "not intended for ML" sentence of the ATLAS v2 preamble bind a user of its Apache-2.0 grant? | K13.10 | no block: ATLAS is `link` |
| OQ17 | Is Coquelicot LGPL-3.0 upstream, as its fork is? | read the Inria repository | no block: it is `crosswalk` |
| OQ18 | Is LeanDojo-v2 Apache-2.0 (the GitHub API) or MIT (its README)? | ask the authors | no block: it is `evaluate` |
| OQ19 | Do the terms of the model that generated a dataset (DeepSeek-V3, Qwen2.5-72B, Gemini, Kimina-72B) pass on to the dataset? | K13.10 | any use of such a dataset beyond evaluation |
| OQ20 | What is the size of Lean Pool, and what are the licences of the projects that it imports? | read its repository | before Lean Pool is bound |
| OQ21 | Does doc-gen4 publish JSON for each declaration that the binding can use? | read doc-gen4 | phase 0, item 2 |

## K15.4 Counts that do not agree

These counts change no rule. The corpus quotes the figure of the source and says that it is not settled, until a reading settles it.

| Id | Question | How to settle |
|---|---|---|
| OQ22 | Mizar: 1,493 articles (the version string) or 1,498 (spec 01)? | count the articles of the MML release |
| OQ23 | Tau Ceti: does "3 million lines" mean Tau Ceti or Mathlib? | count the lines at a pinned commit |
| OQ24 | Metamath: 51,207 statement numbers, "over 41,000" with mathboxes, and 74 or 75 of the 100 theorems. Which count does the corpus report? | define the count in the importer and record it |
| OQ25 | TheoremSearch: 1,239,720 rows in the card of the `-permissive` release against 2,677,439 in the viewer; 8.4M live against 9.2M in the paper | count the rows at a pinned revision |
| OQ26 | Mathlib `100.yaml` gives 78 theorems with declarations; Wiedijk gives 83 for Lean | compare the two lists item by item |
| OQ27 | 1000+: 228 entries on the site; spec 01 says 241 in Lean | parse the YAML files at a pinned commit |
| OQ28 | Are the ProofWiki Main Page counters (30,348 proofs, 36,102 definitions) live? | count from the daily dump |
| OQ29 | Lean Workbook: 57,231 and 82,893 rows in the paper, 25,214 in the default configuration | read each configuration |
| OQ30 | Is the move of the Wikipedia dumps from twice a month to once a month official? Were the free HTML dumps (last run 20 March 2025) stopped? | read the Wikimedia announcements |

## K15.5 The law

| Id | Question | Where |
|---|---|---|
| OQ31 | Is the notice of K13.2 enough? | K13.10 |
| OQ32 | Does the database right of the EU limit the import of catalogues beyond their terms? | K13.10, phase 1 |
| OQ33 | Do the terms of a provider reach users of math2? | K13.10 |
| OQ34 | The "availability to third parties" clauses of the Qwen 3.8 and Kimi K3 licences | K13.10 |
| OQ35 | The licence of DeepSeek-Prover-V2: the MIT file or the Model Agreement? | K13.10 |
| OQ36 | The copyright of model output in the United Kingdom and the EU, and the effect of the judgment in C-250/25 (expected 2027) and of the appeal in *GEMA v. OpenAI* | K13.10, phase 3 |
| OQ37 | What do the Commission's guidelines on Article 50 (reported for 20 July 2026) and the code of practice on transparency say? | read them; K13.5 |
| OQ38 | Is US public domain enough for a work that is not in the public domain in the EU? The corpus says no (K5.2.3). Should it accept a work that is in the public domain in all countries where math2 has users, if the EU rule is not met? | no block: the corpus takes the safe course |

## K15.6 Model generation and quality

These questions are settled by measurement in phase 2 (K14.4), not by reading.

| Id | Question | How to settle |
|---|---|---|
| OQ39 | What is the error rate of model-written definitions and statements in prose? No study has measured it | the sample audit of K9.7 in the pilot |
| OQ40 | How often do models reproduce mathematical text? The 27% figure is for encyclopedia prose in general, on models of 2024 | the copy check of K9.4 over the pilot, against the reference corpus |
| OQ41 | Which thresholds of the copy check give fewer than 1% false matches on text that people wrote? | the calibration of K9.4, before L05 blocks |
| OQ42 | How well do the current judges (Opus 5.5, Fable 5.1 and the models of other families) find wrong steps? The published numbers are for models of 2025 | a labelled set of the pilot area (K7.2) |
| OQ43 | May a model judge its own output, or must the judge be of another family? The studies disagree | the labelled set, with both arms |
| OQ44 | What is the first-pass acceptance rate, and so the real cost for each accepted node? | the run records of the pilot (K12.6) |
| OQ45 | How many hours does a person need for each node, by kind and writing level? | the review records of the pilot (K12.5) |
| OQ46 | Does any provider filter recitation, and at which threshold? Anthropic and OpenAI document none; Gemini has a RECITATION reason with no published threshold | read the documentation; no block, because the copy check of K9.4 does not rely on it |
| OQ47 | The figures in part D marked [snippet] or "derived" (the First Proof batch 2 tally, the later BrokenArXiv figures, Seed-Prover 1.5, the AlphaProof TPU-days) | read the primary PDFs before a spec section quotes them |

## K15.7 Representation and packaging

| Id | Question | How to settle | Before |
|---|---|---|---|
| OQ48 | When does MathML 4 reach Candidate Recommendation? | the W3C Math Working Group | no block: the corpus generates MathML |
| OQ49 | What are the licences of Verso Blueprint, LaTeXML, MMT and the SMGloM archives other than `smglom/mathsoft`? | read each `LICENSE` | before each tool is used in the packager |
| OQ50 | Which projects have moved to Verso Blueprint, and which keep leanblueprint? | read the catalogue again | the blueprint import |
| OQ51 | Is there a version of Common Cartridge after 1.3? Do Canvas, Moodle and Blackboard support QTI 3? | read the 1EdTech pages and the LMS documentation | the course output of K11 |
| OQ52 | Is the Learning Commons Knowledge Graph worth aligning to for school mathematics? | a sample alignment | no block |
| OQ53 | The citation of Leron (1983) and the date of Tao's post "There's more to mathematics than rigour and proofs" | read the primary sources | before a corpus document cites them |

## K15.8 Design questions of the corpus

These questions are about the corpus itself. The text gives a first answer, and the pilot tests it.

| Id | Question | First answer | How to settle |
|---|---|---|---|
| OQ54 | Are the sizes of K2.7 right, for example 300,000 to 1,000,000 named statements? | the estimates of K2.7 | capture and recapture (K4.7) after phase 1 |
| OQ55 | Is a recall of 90% for each MSC code at two digits (rule I06) the right bar? | 90% | the held-out checks of K4.7 in phase 1 and phase 2 |
| OQ56 | How many Mathlib declarations become nodes? | a rough guess of 40,000 to 80,000 (K6.4) | the outcomes of the pilot area, scaled by area |
| OQ57 | Is a gold set of 2,000 pairs for each layer enough to measure the clustering? | 2,000 (K4.3) | the intervals of the first measurement |
| OQ58 | Are the error thresholds of K9.7 right for each stratum? | the thresholds of K9.7 | the monthly review of the thresholds (K14.8) |
| OQ59 | How much review capacity can the project get, and from whom? | not known; it sets the speed of each phase (K12.5) | the pilot, and the curators who join it |
| OQ60 | Should ProofWiki text enter nodes as text, or only as a source for math2's own text? | as text, with attribution, by the importer of open text (K5.4.5) | the review of the pilot nodes that use it |
</document>

<document source="https://tamnd.github.io/math2/node/000AYPA9AJ.md" tag="000AYPA9AJ" kind="concept">
# Vector space

## Summary

A vector space over a field is a set of vectors that can be added and multiplied by scalars, with the usual rules of arithmetic.
Vector spaces are the main objects of linear algebra.
</document>

<document source="https://tamnd.github.io/math2/node/000JFQGDXG.md" tag="000JFQGDXG" kind="statement">
# The integers are a Euclidean domain

## Statement

The [[ring of integers|0JHM6JK7FD]] $\mathbb{Z}$ is a [[Euclidean domain|0GJ2HRG95T]], with the absolute value $\delta(n) = |n|$ as Euclidean function.
That is, for all $a, b \in \mathbb{Z}$ with $b \neq 0$, there are $q, r \in \mathbb{Z}$ with $a = b q + r$ and $0 \le r < |b|$.

## Remarks

The proof: let $r$ be the least element of $\{a - b q : q \in \mathbb{Z}\} \cap \mathbb{N}$. Then $r < |b|$, because otherwise $r - |b|$ would be a smaller element of this set.
The pair $(q, r)$ with $0 \le r < |b|$ is unique.
Consequences: $\mathbb{Z}$ is a [[principal ideal domain|0RZYDVZ475]], and integers factor uniquely into primes.

## Explanations

### L1 human

This is division with remainder from school, made precise.
Dividing $17$ by $5$ gives $17 = 3 \cdot 5 + 2$, and the remainder $2$ is smaller than $5$.
It works for negative numbers too: $-17 = (-4) \cdot 5 + 3$.
</document>

<document source="https://tamnd.github.io/math2/node/000MXHEY22.md" tag="000MXHEY22" kind="statement">
# Sequential characterization of limits

## Statement

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, let $a$ be a limit point of $D$, and let $L$ be a real number.
Then $\lim_{x \to a} f(x) = L$ if and only if the following holds.
For each sequence $(x_n)$ in $D \setminus \{a\}$ with $x_n \to a$, $f(x_n) \to L$.

## Remarks

This turns facts about [[limits of sequences|0XKWSSY8P2]] into facts about limits of functions. Examples are the limit laws for sums, products, and quotients.
It is also a tool to show that a limit does not exist. Find two sequences tending to $a$ whose images have different limits.
Mathlib states a general form as `tendsto_nhds_iff_seq_tendsto`.

## Explanations

### L1 human

The function has limit $L$ at $a$ exactly when it sends every sequence approaching $a$ to a sequence approaching $L$.
So limits of functions reduce to limits of sequences.
</document>

<document source="https://tamnd.github.io/math2/node/000NPTMX6J.md" tag="000NPTMX6J" kind="statement">
# Singular value decomposition

## Statement

Let $A$ be an $m \times n$ complex matrix.
Then there are a [[unitary matrix|0AR68JTMDT]] $U$ of size $m \times m$, a unitary matrix $V$ of size $n \times n$, and an $m \times n$ matrix $\Sigma$ with
$$A = U \Sigma V^*.$$
Here $\Sigma_{ij} = 0$ for $i \ne j$, and $\Sigma_{11} \ge \Sigma_{22} \ge \cdots \ge 0$ are the [[singular values|09VMDSRXWJ]] of $A$.

## Remarks

If $A$ is real, then $U$ and $V$ can be chosen real orthogonal.
The columns of $V$ are eigenvectors of $A^* A$, and the columns of $U$ are eigenvectors of $A A^*$.
The matrix $\Sigma$ is unique; $U$ and $V$ are not unique in general.
Keeping the $k$ largest singular values gives the best approximation of $A$ of rank $k$, for the Frobenius and operator norms.

## Explanations

### L1 human

Every matrix is a rotation, then a stretch along the axes, then another rotation.
The stretch factors are the singular values.
This decomposition is a central tool in data analysis and numerical computation.
</document>

<document source="https://tamnd.github.io/math2/node/0011MTQMR6.md" tag="0011MTQMR6" kind="concept">
# Quotient set

## Summary

The quotient set of a set by an equivalence relation is the set of all equivalence classes.
</document>

<document source="https://tamnd.github.io/math2/node/001B78R0DJ.md" tag="001B78R0DJ" kind="concept">
# Mutual independence of events

## Summary

A family of events is mutually independent when, for each finite subfamily, the probability that all its events occur is the product of their probabilities.
The node [[mutual independence of events|0GCVEV6BJA]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/001CX45SWQ.md" tag="001CX45SWQ" kind="definition">
# Partition function

## Definition

The partition function is the function $p : \mathbb{N} \to \mathbb{N}$ such that $p(n)$ is the number of [[partitions|0404R9P9J3]] of $n$.

## Remarks

The values for $n = 0, 1, \dots, 10$ are $1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42$.
The [[generating function for partitions|0578PA53VA]] is $\sum_{n \ge 0} p(n) x^n = \prod_{k \ge 1} \frac{1}{1 - x^k}$.
The function grows faster than any polynomial and slower than any exponential $c^n$ with $c > 1$.

## Explanations

### L1 human

$p(n)$ counts the ways to write $n$ as a sum of positive integers in any order.
For example, $p(5) = 7$: $5$, $4+1$, $3+2$, $3+1+1$, $2+2+1$, $2+1+1+1$ and $1+1+1+1+1$.
</document>

<document source="https://tamnd.github.io/math2/node/001T5FJDC1.md" tag="001T5FJDC1" kind="concept">
# Block diagonal matrix

## Summary

A block diagonal matrix is a square block matrix whose diagonal blocks are square and whose other blocks are all zero.
</document>

<document source="https://tamnd.github.io/math2/node/002X7EK91H.md" tag="002X7EK91H" kind="statement">
# Power series are differentiable inside the radius of convergence

## Statement

Let $f(x) = \sum_{n=0}^{\infty} a_n (x - c)^n$ be a [[power series|0VP56B4EMX]] with [[radius of convergence|0660QX1VPR]] $R > 0$.
Then $f$ is differentiable on $(c - R, c + R)$, and
$$f'(x) = \sum_{n=1}^{\infty} n a_n (x - c)^{n-1}.$$
The derived series also has radius of convergence $R$.

## Remarks

The [[Cauchy–Hadamard formula|07E5MB1CA2]] gives the radius of the derived series, since $\sqrt[n]{n} \to 1$. Then [[term-by-term differentiation|0TJ8QBJX5K]] applies on each $[c - r, c + r]$ with $r < R$.
By induction, $f$ is infinitely differentiable, and $a_n = f^{(n)}(c)/n!$. So the series is the Taylor series of $f$ at $c$.
Mathlib states differentiability as `HasFPowerSeriesOnBall.differentiableOn`.

## Explanations

### L1 human

A power series can be differentiated like a polynomial, one term at a time.
The new series converges on the same open interval.
</document>

<document source="https://tamnd.github.io/math2/node/003QDZ4T16.md" tag="003QDZ4T16" kind="statement">
# A function with zero derivative is constant

## Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be differentiable with $f'(x) = 0$ for all $x \in I$.
Then $f$ is constant on $I$.

## Remarks

For $x < y$ in $I$, the [[mean value theorem|0BARQY52N5]] gives $f(y) - f(x) = f'(c)(y - x) = 0$.
The domain must be an interval. On $(0, 1) \cup (2, 3)$, a function equal to $0$ on one part and $1$ on the other has zero derivative.
So two antiderivatives of a function on an interval differ by a constant. Mathlib states the result as `is_const_of_deriv_eq_zero`.

## Explanations

### L1 human

A function whose slope is zero everywhere on an interval never rises or falls.
So its graph is a horizontal line.
</document>

<document source="https://tamnd.github.io/math2/node/003QP7A8HG.md" tag="003QP7A8HG" kind="statement">
# Closure is the set of limits of sequences

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $S \subseteq X$ and let $x \in X$.
Then $x$ is in the [[closure|0DRKT1QJR7]] $\overline{S}$ if and only if there is a sequence $(s_n)$ of points of $S$ that [[converges|06JYPJ4WHH]] to $x$.

## Remarks

Proof of the direct part: by [[closure in a metric space|068P0P32DW]], for each $n \ge 1$ there is $s_n \in S$ with $d(x, s_n) < 1/n$. Then $s_n \to x$.
The converse follows because $\overline{S}$ is closed and [[closed sets contain their limits|09GQ3ZTQJ9]].
A consequence: $S$ is closed if and only if the limit of each convergent sequence of points of $S$ is in $S$.
In a general topological space, sequences are not enough, and one uses nets or filters instead.

## Explanations

### L1 human

In a metric space, the closure of a set consists of the limits of sequences that lie in the set.
For example, each real number is a limit of rational numbers, so the closure of $\mathbb{Q}$ is $\mathbb{R}$.
</document>

<document source="https://tamnd.github.io/math2/node/003R0V5M7V.md" tag="003R0V5M7V" kind="concept">
# Differential equation

## Summary

A differential equation is an equation that relates an unknown [[function|0EPAT51DXQ]] to one or more of its derivatives.
A solution is a function that makes the equation true at each point of its domain.
If the unknown function depends on one variable, the equation is an [[ordinary differential equation|0NT64JV96A]]. If it depends on several variables and the equation has partial derivatives, it is a partial differential equation.

## Remarks

The [[order|0XK0G19SK8]] of a differential equation is the highest order of a derivative in it.
The equation $y' = y$ is a differential equation of order $1$. Its solutions are the functions $y(x) = C e^{x}$ with a constant $C$.
Differential equations describe how a quantity changes. Many laws of physics, biology and economics have this form.

## Explanations

### L1 human

An ordinary equation asks for a number, but a differential equation asks for a function.
The equation gives a rule for the rate of change of the function.
For example, $y' = y$ says that the function grows at a rate equal to its own value.
</document>

<document source="https://tamnd.github.io/math2/node/0044CPHJBV.md" tag="0044CPHJBV" kind="definition">
# Natural logarithm

## Definition

For a real number $x > 0$, the natural logarithm $\ln x$ is the unique real number $y$ with $e^y = x$.
This defines a function $\ln \colon (0, \infty) \to \mathbb{R}$.

## Remarks

The number $y$ exists and is unique, because $\exp$ is a [[strictly increasing|0KN2V7840E]] continuous bijection from $\mathbb{R}$ onto $(0, \infty)$.
Some texts define instead $\ln x = \int_1^x \frac{dt}{t}$. Both definitions give the [[same function|0K1N1Y4EGZ]].
Other notations are $\log x$ and $\log_e x$. Mathlib defines `Real.log` for all reals, with $\log x = \log |x|$ and $\log 0 = 0$.

## Explanations

### L1 human

The logarithm answers the question: to which power must we raise $e$ to get $x$?
For example, $\ln 1 = 0$ and $\ln e = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/004FH91BN5.md" tag="004FH91BN5" kind="concept">
# Jump discontinuity

## Summary

A jump discontinuity is a point where both one-sided limits of a function exist but are different. The graph jumps from one height to another there. The node [[jump discontinuity|05B0FKFXQA]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/004FRCXH6M.md" tag="004FRCXH6M" kind="definition">
# Exterior derivative

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $\omega = \sum_I f_I \, dx_I$ be a [[differential k-form|0QHTX7WQEY]] on $U$ whose coefficients $f_I$ are of class $C^1$.
The exterior derivative of $\omega$ is the $(k + 1)$-form
$$d\omega = \sum_I \sum_{j=1}^n \frac{\partial f_I}{\partial x_j} \, dx_j \wedge dx_I,$$
where $\wedge$ is the [[wedge product|0S166D363M]].

## Remarks

For a function $f$, $df = \sum_j \partial_j f \, dx_j$ is its differential, the [[one-form|0MTENFBW3Q]] that corresponds to the [[gradient|0NS2QYVC4Y]].
The operator $d$ is linear and satisfies $d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta$ for a $k$-form $\alpha$. For $C^2$ forms, [[d d = 0|0X308R2DBE]].
In the plane, $d(L \, dx + M \, dy) = (\partial_x M - \partial_y L) \, dx \wedge dy$.
In $\mathbb{R}^3$, $d(F_1 \, dx + F_2 \, dy + F_3 \, dz)$ has the components of the [[curl|0GDNHS9J2X]] of $F$ as coefficients of $dy \wedge dz$, $dz \wedge dx$ and $dx \wedge dy$. Also $d(F_1 \, dy \wedge dz + F_2 \, dz \wedge dx + F_3 \, dx \wedge dy) = (\nabla \cdot F) \, dx \wedge dy \wedge dz$, which is the [[divergence|02NSMQVQWK]].

## Explanations

### L1 human

The exterior derivative is one operation that does the work of gradient, curl and divergence together.
You differentiate each coefficient in each direction and attach the new direction with a wedge product.
</document>

<document source="https://tamnd.github.io/math2/node/004J7ZEFAT.md" tag="004J7ZEFAT" kind="example">
# Power set Boolean algebra

## Example

Let $X$ be a set, and let $\mathcal{P}(X)$ be the set of all subsets of $X$.
With $A \wedge B = A \cap B$, $A \vee B = A \cup B$, $\neg A = X \setminus A$, $0 = \varnothing$ and $1 = X$, the set $\mathcal{P}(X)$ is a [[Boolean algebra|06GKBTAA5E]].
Its order is inclusion $\subseteq$.

## Remarks

If $X$ has $n$ elements, this Boolean algebra has $2^n$ elements.
A field of sets on $X$ is a subset of $\mathcal{P}(X)$ that contains $\varnothing$ and is closed under complement and binary union. It is a Boolean subalgebra.
Every finite Boolean algebra is isomorphic to a power set Boolean algebra, by [[Stone's representation theorem for finite Boolean algebras|0DYS9C3HH5]].

## Explanations

### L1 human

The subsets of a set form a Boolean algebra: intersection is and, union is or, and complement is not.
Venn diagrams show the laws of this Boolean algebra.
</document>

<document source="https://tamnd.github.io/math2/node/004JW7A2N5.md" tag="004JW7A2N5" kind="definition">
# Square-free integer

## Definition

A nonzero integer $n$ is square-free when no square of a [[prime number|0SN0MAEKYX]] divides $n$.
That is, there is no prime $p$ such that $p^2 \mid n$.

## Remarks

Equivalently, $n$ is square-free when the only perfect squares that divide $n$ are $1$.
A positive integer is square-free if and only if each exponent in its [[prime factorization|0AEKDFCYFA]] is $1$.
The first square-free positive integers are $1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15$. The integer $12 = 2^2 \cdot 3$ is not square-free.
The integer $0$ is not square-free, because each square divides $0$.

## Explanations

### L1 human

A number is square-free when no prime appears twice in its factorization.
For example, $30 = 2 \cdot 3 \cdot 5$ is square-free, but $18 = 2 \cdot 3 \cdot 3$ is not, because $9$ divides $18$.
</document>

<document source="https://tamnd.github.io/math2/node/0053CQE9Y2.md" tag="0053CQE9Y2" kind="concept">
# Sylow p-subgroup

## Summary

Let $G$ be a finite group, let $p$ be a prime number, and let $p^n$ be the largest power of $p$ that divides $|G|$.
A Sylow $p$-subgroup of $G$ is a subgroup of order $p^n$.
</document>

<document source="https://tamnd.github.io/math2/node/005JT76HCK.md" tag="005JT76HCK" kind="statement">
# A differentiable function is convex iff its derivative is increasing

## Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be differentiable.
Then $f$ is [[convex|0VV0HSW1ZS]] if and only if $f'$ is increasing on $I$.

## Remarks

If $f$ is convex, the [[slope inequalities|0BY5FTHEEE]] give $f'(x) \le f'(y)$ for $x < y$.
If $f'$ is increasing, the [[mean value theorem|0BARQY52N5]] on $[x, y]$ and $[y, z]$ gives the slope inequalities.
Then the graph lies above each tangent line: $f(y) \ge f(x) + f'(x)(y - x)$.
Mathlib states the converse direction as `MonotoneOn.convexOn_of_deriv`.

## Explanations

### L1 human

A differentiable function is convex when its slope keeps growing.
The graph then turns to the left as we move along it to the right.
</document>

<document source="https://tamnd.github.io/math2/node/005Q84BMAD.md" tag="005Q84BMAD" kind="statement">
# Taylor's theorem with the Cauchy remainder

## Statement

Let $k \ge 0$ and let $a \ne x$ be real numbers. Let $J$ be the closed interval with endpoints $a$ and $x$.
Let $f$ be a real function with $f^{(k)}$ continuous on $J$ and $f^{(k+1)}$ existing on the interior of $J$.
Then there is $\xi$ strictly between $a$ and $x$ with
$$f(x) = P_k(x) + \frac{f^{(k+1)}(\xi)}{k!} (x - \xi)^k (x - a),$$
where $P_k$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.

## Remarks

Apply the [[mean value theorem|0BARQY52N5]] to $F(t) = f(t) + \sum_{j=1}^{k} f^{(j)}(t) (x - t)^j / j!$ on $J$. Note that $F(x) = f(x)$ and $F(a) = P_k(x)$.
This form sometimes gives better bounds than the [[Lagrange form|0H5ZAZCX2T]], for example for $\ln(1 + x)$ with $-1 < x < 0$.
Mathlib states it as `taylor_mean_remainder_cauchy`.

## Explanations

### L1 human

The Cauchy form is another exact expression for the error.
It moves part of the power of $(x - a)$ to a power of $(x - \xi)$.
</document>

<document source="https://tamnd.github.io/math2/node/006BHFWMAP.md" tag="006BHFWMAP" kind="concept">
# Row equivalence

## Summary

Two matrices are row equivalent when one of them can be changed into the other by a finite sequence of elementary row operations.
</document>

<document source="https://tamnd.github.io/math2/node/00758QM6JJ.md" tag="00758QM6JJ" kind="statement">
# The characteristic of an integral domain is zero or prime

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]].
Then the [[characteristic|0P7CH1CND5]] of $R$ is $0$ or a prime number.

## Remarks

The proof: suppose that the characteristic is $n = a b$ with $1 < a, b < n$. Then $(a \cdot 1)(b \cdot 1) = n \cdot 1 = 0$, but $a \cdot 1 \neq 0$ and $b \cdot 1 \neq 0$.
The characteristic is not $1$, because $1 \neq 0$ in an integral domain.
In particular, the characteristic of a [[field|0QA8CHH8MG]] is $0$ or a prime number.
The ring $\mathbb{Z}/6\mathbb{Z}$ has characteristic $6$, and it is not an integral domain.

## Explanations

### L1 human

If $6 \cdot 1 = 0$ in a ring, then $(2 \cdot 1)(3 \cdot 1) = 0$.
In an integral domain, this forces $2 \cdot 1 = 0$ or $3 \cdot 1 = 0$.
So the least $n$ with $n \cdot 1 = 0$ cannot be a product of two smaller numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0076HWHGHQ.md" tag="0076HWHGHQ" kind="concept">
# Quotient ring

## Summary

The quotient ring of a ring by a two-sided ideal is the set of the cosets of the ideal.
Its operations come from the operations of the ring.
The integers modulo $n$ are the main example.
The [[definition of the quotient ring|0VFW4Y69BT]] gives the construction.
</document>

<document source="https://tamnd.github.io/math2/node/007NPQX34K.md" tag="007NPQX34K" kind="concept">
# Least common multiple

## Summary

The least common multiple of two nonzero integers is the smallest positive integer that is a multiple of both of them.
We write it as $\operatorname{lcm}(a, b)$.
The [[definition of the least common multiple|09S5SN35DJ]] gives the precise form and the convention when one of the numbers is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/007QED3E0Y.md" tag="007QED3E0Y" kind="definition">
# Variance

## Definition

Let $X$ be an integrable real [[random variable|0DKRR22AEQ]] with mean $\mu = E[X]$.
The variance of $X$ is
$$\operatorname{Var}(X) = E\left[(X - \mu)^2\right] \in [0, \infty].$$

## Remarks

Other notations are $\operatorname{V}(X)$, $\sigma^2$ and $\sigma_X^2$. The variance is finite if and only if $E[X^2] < \infty$.
For a discrete $X$ with values $x_i$, $\operatorname{Var}(X) = \sum_i (x_i - \mu)^2 P(X = x_i)$. For $X$ with density $f$, $\operatorname{Var}(X) = \int (x - \mu)^2 f(x) \, dx$.
$\operatorname{Var}(X) = 0$ if and only if $X = \mu$ almost surely.
For a fair die, $\mu = 7/2$ and $\operatorname{Var}(X) = 35/12$.

## Explanations

### L1 human

The variance tells how far, on average, the values of $X$ are from the mean. The distances are squared, so that large deviations count more and positive and negative deviations do not cancel.
Its square root, the standard deviation, is in the same units as $X$.
</document>

<document source="https://tamnd.github.io/math2/node/00889YD15K.md" tag="00889YD15K" kind="statement">
# Dimension of the dual space

## Statement

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.
Then the [[dual space|07JBSZE0V4]] $V^*$ is finite-dimensional, and
$$\dim V^* = \dim V.$$

## Remarks

The [[dual basis|0ZD2CF6ZS0]] of a basis of $V$ is a basis of $V^*$ with the same number of elements.
This also follows from $\dim \Hom_F(V, W) = \dim V \cdot \dim W$ with $W = F$.
The isomorphism $V \to V^*$ that sends $b_i$ to $b_i^*$ depends on the basis.
For infinite-dimensional $V$, the dimension of $V^*$ is strictly greater than the dimension of $V$.

## Explanations

### L1 human

A linear form on an $n$-dimensional space is fixed by its $n$ values on a basis.
These $n$ values can be any numbers.
So the linear forms make up an $n$-dimensional space too.
</document>

<document source="https://tamnd.github.io/math2/node/008BH74D7R.md" tag="008BH74D7R" kind="concept">
# Composite number

## Summary

A composite number is a natural number greater than $1$ that is not prime.
Such a number is a product of two smaller natural numbers, each greater than $1$.
The [[definition of a composite number|0N3ADXHNTQ]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/008TBCYDPX.md" tag="008TBCYDPX" kind="example">
# Zero ring

## Example

The zero ring is the set $\{0\}$ with the operations $0 + 0 = 0$ and $0 \cdot 0 = 0$.
It is a [[commutative ring|0GYGHKX834]] in which $1 = 0$.

## Remarks

A [[ring|0894J980RK]] $R$ is the zero ring if and only if $1 = 0$ in $R$. If $1 = 0$, then $a = a \cdot 1 = a \cdot 0 = 0$ for each $a \in R$.
The zero ring is not an [[integral domain|09S3PFBEPM]] and not a [[field|0QA8CHH8MG]], because these definitions require $1 \neq 0$.

## Explanations

### L1 human

The zero ring has only one element.
All the axioms of a ring hold, but in a trivial way.
It is the only ring in which $1$ and $0$ are the same element.
</document>

<document source="https://tamnd.github.io/math2/node/009892PZ30.md" tag="009892PZ30" kind="definition">
# Linear map of a matrix

## Definition

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$ with bases $B = (b_1, \dots, b_n)$ and $C = (c_1, \dots, c_m)$.
Let $A = (a_{ij})$ be an $m \times n$ matrix over $F$.
The *linear map of $A$* in the bases $B$ and $C$ is the linear map $T_A \colon V \to W$ with
$$T_A(b_j) = \sum_{i=1}^{m} a_{ij} c_i \qquad (1 \le j \le n).$$

## Remarks

The map $A \mapsto T_A$ is the inverse of the map $T \mapsto [T]_{C,B}$ of the [[matrix of a linear map|0YQSGNYE3K]].
For $V = F^n$ and $W = F^m$ with the standard bases, $T_A$ is the map $x \mapsto Ax$, see [[multiplication by a matrix as a linear map|094V2VBJXJ]].
In Mathlib, this map is `Matrix.toLin`.

## Explanations

### L1 human

A matrix and two bases together define a linear map.
Column $j$ tells us where the map sends basis vector $j$.
This undoes the step from a linear map to its matrix.
</document>

<document source="https://tamnd.github.io/math2/node/00999TYPXB.md" tag="00999TYPXB" kind="definition">
# Local maximum

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$.
The function $f$ has a local maximum at $a$ if there is $\delta > 0$ with $f(x) \le f(a)$ for all $x \in D$ with $|x - a| < \delta$.
It is a strict local maximum if $f(x) < f(a)$ for all such $x \ne a$.

## Remarks

A global maximum, where $f(x) \le f(a)$ for all $x \in D$, is also a local maximum.
A [[local minimum|0NS8BP96QX]] is defined with $f(x) \ge f(a)$. Local maxima and minima are local extrema.
At an interior local extremum, a differentiable function has [[derivative zero|0PCVBRV9VJ]].
Mathlib defines it as `IsLocalMax`.

## Explanations

### L1 human

A local maximum is a hilltop of the graph.
Nearby values are not higher, although values far away may be higher.
</document>

<document source="https://tamnd.github.io/math2/node/009CY8JXFX.md" tag="009CY8JXFX" kind="statement">
# Scaling a row scales the determinant

## Statement

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$, and let $c \in K$.
Let $B$ be the matrix that results when one row of $A$ is multiplied by $c$.
Then $\det B = c \det A$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is a special case of [[linearity in each row|0PPG793642]].
It describes the effect of the scaling [[row operation|04XWH28452]] on the determinant.
Scaling all $n$ rows gives $\det(cA) = c^n \det A$, see [[the determinant of a scalar multiple|01EXSMK864]].

## Explanations

### L1 human

If we stretch one edge of a box by the factor $c$, then its volume is multiplied by $c$.
The determinant behaves in the same way.
For example, doubling one row doubles the determinant.
</document>

<document source="https://tamnd.github.io/math2/node/009QCP5T3Z.md" tag="009QCP5T3Z" kind="definition">
# Sum of divisors

## Definition

For a positive integer $n$, the sum of divisors of $n$ is
$$\sigma(n) = \sigma_1(n) = \sum_{d \mid n} d,$$
where $\sigma_1$ is the [[divisor function|0YEW47VJ0J]] with $k = 1$.

## Remarks

If $n = p_1^{e_1} \cdots p_r^{e_r}$ with distinct primes $p_i$, then $\sigma(n) = \prod_{i=1}^{r} \frac{p_i^{e_i + 1} - 1}{p_i - 1}$.
For example, $\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28$.
The sum of the proper divisors is $\sigma(n) - n$. It defines [[perfect|0SB7Q8SYSF]], [[abundant|0VS0MP1P5H]] and [[deficient|0G01TS1RRP]] numbers.

## Explanations

### L1 human

The function $\sigma(n)$ adds all the divisors of $n$, including $1$ and $n$.
For a prime $p$, the only divisors are $1$ and $p$, so $\sigma(p) = p + 1$.
</document>

<document source="https://tamnd.github.io/math2/node/009TEMTQKH.md" tag="009TEMTQKH" kind="statement">
# Stars and bars

## Statement

Let $n$ and $k$ be natural numbers with $n \ge 1$.
Then the number of multisets of size $k$ with elements in an $n$-element set is
$$\left(\!\binom{n}{k}\!\right) = \binom{n + k - 1}{k}.$$
Equivalently, this is the number of solutions $(x_1, \dots, x_n)$ of $x_1 + \dots + x_n = k$ in natural numbers.

## Remarks

The proof writes a multiset as a row of $k$ stars and $n - 1$ bars. The bars separate the stars into $n$ groups, one group for each element.
The number of solutions in positive integers is $\binom{k - 1}{n - 1}$ for $k \ge 1$. This is the [[number of compositions|057H72B2MB]] of $k$ into $n$ parts.

## Explanations

### L1 human

To share $k$ identical sweets among $n$ children, put the $k$ sweets in a row and insert $n - 1$ dividers.
There are $k + n - 1$ positions, and you choose which $k$ of them are sweets.
</document>

<document source="https://tamnd.github.io/math2/node/00A0RPAQ8A.md" tag="00A0RPAQ8A" kind="definition">
# Normed vector space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$.
A *normed vector space* over $\mathbb{K}$ is a vector space $V$ over $\mathbb{K}$ together with a [[norm|0V2WP9B1MF]] $\norm{\cdot}$ on $V$.

## Remarks

A normed space is a metric space with the distance $d(x, y) = \norm{x - y}$.
A normed space that is complete for this metric is a *Banach space*.
Each [[inner product space|0QVNCJXRY3]] is a normed space with the induced norm.
On a finite-dimensional space, all norms are equivalent, so they define the same topology.

## Explanations

### L1 human

A normed space is a vector space where each vector has a length.
With lengths, we also have distances, and so limits and continuity.
This is the setting of much of analysis.
</document>

<document source="https://tamnd.github.io/math2/node/00A2EDFT9Q.md" tag="00A2EDFT9Q" kind="definition">
# Principal ideal

## Definition

Let $R$ be a [[ring|0894J980RK]].
A left ideal $I$ of $R$ is principal if there is an element $a \in R$ such that $I = (a)$, the [[left ideal generated by|030TZ0DQWT]] $\{a\}$.

## Remarks

The left ideal generated by $a$ is $R a = \{r a : r \in R\}$.
If $R$ is commutative, the principal ideal $(a)$ is the set of the multiples of $a$.
Every ideal of $\mathbb{Z}$ is principal: each one is $n\mathbb{Z}$ for some integer $n \ge 0$.
In $\mathbb{Z}[x]$, the ideal $(2, x)$ is not principal.

## Explanations

### L1 human

A principal ideal is the set of all the multiples of one element.
For example, the multiples of $3$ in the integers form the principal ideal $(3)$.
</document>

<document source="https://tamnd.github.io/math2/node/00BE2J8H05.md" tag="00BE2J8H05" kind="concept">
# One-sided limit

## Summary

A one-sided limit is the limit of a function as the argument approaches a point from the left only or from the right only. A function has a limit at a point exactly when both one-sided limits exist and agree. The node [[one-sided limit|0909PWZR62]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/00BETAJNZ4.md" tag="00BETAJNZ4" kind="definition">
# Platonic solid

## Definition

A Platonic solid is a convex [[polyhedron|0325SGSJAR]] $P$ in $\mathbb{R}^3$ with these two properties:
- the faces of $P$ are [[regular polygons|01KJJ9A400]] that are all congruent to each other;
- the same number of faces meet at each vertex of $P$.

## Remarks

If each face has $p$ sides and $q$ faces meet at each vertex, the solid has the Schläfli symbol $\{p, q\}$.
The five Platonic solids are the regular tetrahedron $\{3, 3\}$, the cube $\{4, 3\}$, the regular octahedron $\{3, 4\}$, the regular dodecahedron $\{5, 3\}$ and the regular icosahedron $\{3, 5\}$. See [[the classification|0M4TCS0RRG]].
They have the name of Plato, who wrote about them in the dialogue Timaeus.

## Explanations

### L1 human

A Platonic solid is a perfectly regular dice shape. All faces are the same regular polygon, and every corner looks the same. A cube is one: six equal squares, with three squares at each corner.
</document>

<document source="https://tamnd.github.io/math2/node/00BKRZKMSS.md" tag="00BKRZKMSS" kind="concept">
# Composition of linear maps

## Summary

The composition of two linear maps is the linear map that applies the first map and then the second.
</document>

<document source="https://tamnd.github.io/math2/node/00BT70ND50.md" tag="00BT70ND50" kind="statement">
# Picard–Lindelöf theorem in integral form

## Statement

Let $J = [t_{\min}, t_{\max}]$ and $t_0 \in J$, let $x_0 \in \mathbb{R}^n$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ satisfy the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] at $(t_0, x_0)$ with constants $a, L, K$.
Let $B$ be the closed ball of radius $a$ around $x_0$.
Then there is exactly one continuous curve $\alpha : J \to B$ such that
$$\alpha(t) = x_0 + \int_{t_0}^{t} f\left(s, \alpha(s)\right) ds \quad \text{for each } t \in J.$$

## Remarks

The curve $\alpha$ is the unique fixed point of the [[Picard operator|00WAVB68F9]] on the curves with values in $B$.
By the [[integral equation form|015KK993C5]], this is the same curve as the solution of the [[Picard–Lindelöf theorem|0P6KRD5B7A]].

## Explanations

### L1 human

This is the Picard–Lindelöf theorem written without derivatives.
It says that the integral equation has exactly one continuous solution that stays in the ball.
</document>

<document source="https://tamnd.github.io/math2/node/00BWV5CKCF.md" tag="00BWV5CKCF" kind="statement">
# Finite intersection property

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $K \subseteq X$.
Then $K$ is [[compact|00R0X1WGPX]] if and only if it has the property that follows.
Let $(F_i)_{i \in I}$ be a family of [[closed subsets|0G5AV9Q4RR]] of $X$ with $K \cap F_{i_1} \cap \cdots \cap F_{i_n} \neq \emptyset$ for all finitely many indices $i_1, \ldots, i_n \in I$.
Then also $K \cap \bigcap_{i \in I} F_i \neq \emptyset$.

## Remarks

A family of sets has the finite intersection property when each finite subfamily has a nonempty intersection.
Proof: take complements. The sets $X \setminus F_i$ cover $K$ if and only if $K \cap \bigcap_i F_i = \emptyset$. Then use [[compactness via open covers|0P6PMJB4MA]].
For $K = X$: a space is compact if and only if each family of closed sets with the finite intersection property has nonempty intersection.
[[Cantor's intersection theorem|0EZBG8SKJM]] is a special case.

## Explanations

### L1 human

This is the dual form of compactness, with closed sets instead of open sets.
Suppose that each finite subfamily of the closed sets has a common point in $K$. Then the whole family has a common point in $K$.
</document>

<document source="https://tamnd.github.io/math2/node/00C0BR7D7H.md" tag="00C0BR7D7H" kind="concept">
# Graph

## Summary

A graph is a set of vertices together with a set of edges, each edge joining two distinct vertices. Graphs model networks, relations and maps, and they are the main object of graph theory. The node [[graph|0JD5C82QMB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/00C2WN1H21.md" tag="00C2WN1H21" kind="definition">
# Polyhedron

## Definition

A convex polyhedron is the [[convex hull|0FQSDC2EQ2]] of a finite set of points of $\mathbb{R}^3$ that do not all lie in one [[plane|0V3XAMF3C1]].
A polyhedron is a bounded subset of $\mathbb{R}^3$ with a connected interior that is the union of finitely many convex polyhedra.
Its faces are the connected parts of its boundary inside the planes that cover the boundary. Its edges and vertices are the segments and points where the faces meet.

## Remarks

There is no single definition of a polyhedron that all authors use. Some definitions describe a surface in place of a solid, and some allow self-crossing faces.
A convex polyhedron is also a bounded intersection of finitely many closed half-spaces, with a nonempty interior.
Examples are the cube, the regular tetrahedron and each prism over a convex polygon.

## Explanations

### L1 human

A polyhedron is a solid with flat sides, like a box or a pyramid. A ball is not one, because its surface is curved. The cube has $8$ vertices, $12$ edges and $6$ faces.
</document>

<document source="https://tamnd.github.io/math2/node/00C5P9JZBK.md" tag="00C5P9JZBK" kind="object">
# Euler–Mascheroni constant

## Description

The Euler–Mascheroni constant is the real number
$$\gamma = \lim_{n \to \infty} \big( H_n - \ln n \big) \approx 0.5772156649,$$
where $H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ is the $n$-th [[harmonic number|0D2Y6G4C4S]].

## Remarks

The [[limit exists|05S2GSPMGA]], because the sequence $H_n - \ln n$ is decreasing and bounded below by $0$.
Also $\gamma = \int_1^\infty \Big( \frac{1}{\lfloor x \rfloor} - \frac{1}{x} \Big) \, dx$.
It is not known whether $\gamma$ is rational. Mathlib defines it as `Real.eulerMascheroniConstant`.
Leonhard Euler introduced the constant in a paper of 1734. Lorenzo Mascheroni later computed it to more decimal places, with some errors.

## Explanations

### L1 human

The harmonic numbers grow like $\ln n$, and $\gamma$ is the limit of the difference.
So $H_n \approx \ln n + 0.5772$ for large $n$.
</document>

<document source="https://tamnd.github.io/math2/node/00CN2F997Q.md" tag="00CN2F997Q" kind="statement">
# Addition formula for Fibonacci numbers

## Statement

Let $m$ and $n$ be natural numbers, and let $F_k$ be the [[Fibonacci numbers|0GA9K57A39]].
Then
$$F_{m+n+1} = F_m F_n + F_{m+1} F_{n+1}.$$

## Remarks

With $m = n$ it gives $F_{2n+1} = F_n^2 + F_{n+1}^2$.
An equivalent form is $F_{m+n} = F_{m-1} F_n + F_m F_{n+1}$ for $m \ge 1$.
The formula is the basis of fast algorithms that compute $F_n$ with $O(\log n)$ arithmetic operations.

## Explanations

### L1 human

To get a Fibonacci number with a large index, you can combine Fibonacci numbers with smaller indices.
For example, with $m = 2$ and $n = 3$: $F_6 = F_2 F_3 + F_3 F_4 = 1 \cdot 2 + 2 \cdot 3 = 8$.
</document>

<document source="https://tamnd.github.io/math2/node/00CR00C9DN.md" tag="00CR00C9DN" kind="concept">
# Countably infinite set

## Summary

A countably infinite set is a set that is equinumerous with the set of natural numbers.
</document>

<document source="https://tamnd.github.io/math2/node/00CT0PTH74.md" tag="00CT0PTH74" kind="definition">
# Graph isomorphism

## Definition

Let $G$ and $H$ be [[graphs|0JD5C82QMB]] with vertex sets $V(G)$ and $V(H)$.
A graph isomorphism from $G$ to $H$ is a bijection $f : V(G) \to V(H)$ such that, for all $u, v \in V(G)$,
$$u \sim v \text{ in } G \iff f(u) \sim f(v) \text{ in } H.$$
The graphs $G$ and $H$ are isomorphic, written $G \cong H$, if there is an isomorphism between them.

## Remarks

Isomorphic graphs have the same number of vertices and edges and the same degrees. They differ only in the names of their vertices.
An isomorphism from $G$ to itself is an automorphism of $G$.
The path $P_4$ is isomorphic to its own [[complement|0RNMP6VCR2]].

## Explanations

### L1 human

Two graphs are isomorphic when one is a relabelling of the other.
Two drawings can look very different and still show isomorphic graphs.
</document>

<document source="https://tamnd.github.io/math2/node/00DGYY10FV.md" tag="00DGYY10FV" kind="statement">
# The logarithm grows slower than any power

## Statement

For every real $a > 0$,
$$\lim_{x \to \infty} \frac{\ln x}{x^a} = 0.$$
That is, $\ln x = o(x^a)$ as $x \to \infty$.

## Remarks

Substitute $x = e^{t/a}$. Then $\frac{\ln x}{x^a} = \frac{t / a}{e^{t}}$, which tends to $0$ by [[exponential growth|0CT6CBGTYY]].
In [[little o notation|04TSHB9H8D]], $\ln x = o(x)$. Mathlib states this case as `Real.isLittleO_log_id_atTop`.
Similarly, $x^a \ln x \to 0$ as $x \to 0^+$.

## Explanations

### L1 human

The logarithm goes to infinity, but very slowly.
Each positive power of $x$, even $x^{0.001}$, eventually beats it.
</document>

<document source="https://tamnd.github.io/math2/node/00DRK90Z6K.md" tag="00DRK90Z6K" kind="statement">
# Laws of exponents in a group

## Statement

Let $G$ be a [[group|G5S1M9]] with identity element $e$, and let $a \in G$.
For an integer $n$, define $a^n$ as follows: $a^0 = e$, $a^{n+1} = a^n a$ for $n \geq 0$, and $a^{-n} = (a^{-1})^n$ for $n > 0$.
Then $a^{m+n} = a^m a^n$ and $(a^m)^n = a^{mn}$ for all integers $m$ and $n$.

## Remarks

The law $(ab)^n = a^n b^n$ is true for all $n$ when $ab = ba$, but it can fail when $a$ and $b$ do not commute.
In additive notation the laws are $(m + n)a = ma + na$ and $n(ma) = (nm)a$.
The first law shows that $n \mapsto a^n$ is a group homomorphism from $(\mathbb{Z}, +)$ to $G$.

## Explanations

### L1 human

The powers of an element of a group follow the rules that you know for the powers of a nonzero number.
A negative power means a power of the inverse, so $a^{-2} = a^{-1} a^{-1}$.
For example, $a^3 a^{-1} = a a a a^{-1} = a^2$.
</document>

<document source="https://tamnd.github.io/math2/node/00DW862T78.md" tag="00DW862T78" kind="statement">
# −1 is a square modulo p if and only if p is not 3 modulo 4

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
There is an integer $x$ with $x^2 \equiv -1 \pmod p$ if and only if $p \not\equiv 3 \pmod 4$.

## Remarks

For $p = 2$, take $x = 1$.
For odd $p$, the statement is the [[first supplement to quadratic reciprocity|0PHBS78HKT]].
For $p \equiv 1 \pmod 4$, an explicit solution is $x = \left(\frac{p-1}{2}\right)!$, by [[Wilson's theorem|03KGGKTB2E]].
The statement is a step in the proof of [[Fermat's theorem on sums of two squares|043B3ENSTY]].

## Explanations

### L1 human

The primes $2$, $5$, $13$ and $17$ have a square root of $-1$: for example $5^2 = 25 \equiv -1 \pmod{13}$.
The primes $3$, $7$ and $11$ have none.
The rule is that $p$ must not leave remainder $3$ on division by $4$.
</document>

<document source="https://tamnd.github.io/math2/node/00E0YZCERH.md" tag="00E0YZCERH" kind="concept">
# Power set

## Summary

The power set of a set is the set of all its subsets.
</document>

<document source="https://tamnd.github.io/math2/node/00EBGSR5NY.md" tag="00EBGSR5NY" kind="definition">
# Euclidean vector

## Definition

Let $n$ be a positive integer.
A Euclidean vector in the [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$ is an element $v = (v_1, \dots, v_n)$ of $\mathbb{R}^n$. The real numbers $v_1, \dots, v_n$ are the components of $v$.
For two points $A = (a_1, \dots, a_n)$ and $B = (b_1, \dots, b_n)$, the vector from $A$ to $B$ is
$$\overrightarrow{AB} = (b_1 - a_1, \dots, b_n - a_n).$$
The sum of two vectors $u$ and $v$ is $u + v = (u_1 + v_1, \dots, u_n + v_n)$.
The product of a real number $c$ and a vector $v$ is $c v = (c v_1, \dots, c v_n)$.
The zero vector is $0 = (0, \dots, 0)$.

## Remarks

Two arrows give the same vector exactly when a translation moves one arrow onto the other.
For each three points $A$, $B$ and $C$, the vectors satisfy $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$.
With the sum and the product by real numbers, the vectors of $\mathbb{R}^n$ form a real vector space of dimension $n$.

## Explanations

### L1 human

A vector tells you how to move: how far, and in which direction. The vector $(3, 4)$ in the plane says "move 3 to the right and 4 up". It does not matter where you start, so arrows of the same length and direction are the same vector.
</document>

<document source="https://tamnd.github.io/math2/node/00EHX6FWC0.md" tag="00EHX6FWC0" kind="definition">
# Root of a polynomial

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $f \in R[x]$.
An element $c \in R$ is a root of $f$ if $f(c) = 0$, where $f(c)$ is the [[value of $f$ at $c$|0GJ7T0NK49]].

## Remarks

The roots of $x^2 - 1$ in $\mathbb{Z}$ are $1$ and $-1$.
The polynomial $x^2 + 1$ has no root in $\mathbb{R}$. It has the roots $i$ and $-i$ in $\mathbb{C}$.
By the [[factor theorem|0K3MWA7DMC]], over a commutative ring, $c$ is a root of $f$ if and only if $x - c$ divides $f$.

## Explanations

### L1 human

A root is a solution of the equation $f(x) = 0$.
Whether a polynomial has a root depends on the ring: $x^2 - 2$ has no root in $\mathbb{Q}$, but it has the root $\sqrt{2}$ in $\mathbb{R}$.
</document>

<document source="https://tamnd.github.io/math2/node/00EYM7MDKS.md" tag="00EYM7MDKS" kind="concept">
# Hasse diagram

## Summary

A Hasse diagram is a drawing of a finite [[partially ordered set|011C29Q1S1]].
Each element is a point, and there is a line upward from $a$ to $b$ when $b$ covers $a$.
This means that $a < b$ and no $c$ satisfies $a < c < b$.
Then $a \le b$ exactly when one can go from $a$ to $b$ by a path that goes only upward.

## Remarks

The diagram does not draw the loops of reflexivity or the lines that transitivity implies.
The Hasse diagram of the power set of $\{1, 2, 3\}$ under $\subseteq$ is the picture of a cube.

## Explanations

### L1 human

A Hasse diagram is a minimal picture of an order.
For the divisors of $12$ under divisibility, put $1$ at the bottom and $12$ at the top.
Draw a line from $2$ to $4$ and from $2$ to $6$.
Do not draw a line from $2$ to $12$, because the path through $4$ already shows it.
</document>

<document source="https://tamnd.github.io/math2/node/00F2JFA41J.md" tag="00F2JFA41J" kind="definition">
# Orbit

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \in X$.
The orbit of $x$ is the set $G \cdot x = \{g \cdot x : g \in G\}$.
The set of all orbits is written $X/G$.

## Remarks

The point $x$ lies in its own orbit, because $e \cdot x = x$.
The [[orbits partition|0ZC7FYE0NZ]] the set $X$.
For the action of the rotation group of a square on its corners, there is one orbit, which contains all four corners.

## Explanations

### L1 human

The orbit of a point collects all the places where the group can move the point.
Two points in the same orbit can be moved onto each other.
Points in different orbits are never moved onto each other.
</document>

<document source="https://tamnd.github.io/math2/node/00FSJTXPJM.md" tag="00FSJTXPJM" kind="concept">
# Closed map

## Summary

A closed map is a function between topological spaces that sends closed sets to closed sets.
Continuous maps from compact spaces to Hausdorff spaces are closed maps.
</document>

<document source="https://tamnd.github.io/math2/node/00FWBN7F0Y.md" tag="00FWBN7F0Y" kind="statement">
# Congruence is compatible with addition

## Statement

Let $n$ be a positive integer, and let $a$, $b$, $c$ and $d$ be integers.
If $a \equiv b \pmod n$ and $c \equiv d \pmod n$, then $a + c \equiv b + d \pmod n$.

## Remarks

The proof: $(a + c) - (b + d) = (a - b) + (c - d)$, and $n$ divides both terms.
The same holds for subtraction: $a - c \equiv b - d \pmod n$.
Together with [[compatibility with multiplication|072D5G4SK6]], this result shows that the addition of [[integers modulo n|01N3AYAA6N]] is well defined.

## Explanations

### L1 human

To find the remainder of a sum, you can add the remainders and reduce again.
For example, $17 \equiv 2 \pmod 5$ and $24 \equiv 4 \pmod 5$, so $17 + 24 = 41 \equiv 2 + 4 = 6 \equiv 1 \pmod 5$.
</document>

<document source="https://tamnd.github.io/math2/node/00FWNP9TXY.md" tag="00FWNP9TXY" kind="definition">
# Contour integral

## Definition

Let $a < b$ be real numbers and let $\gamma : [a, b] \to \mathbb{C}$ be a path that is continuous and piecewise continuously differentiable.
Let $f$ be a complex function that is continuous on the image $\gamma([a, b])$.
The contour integral of $f$ along $\gamma$ is
$$\int_\gamma f(z) \, dz = \int_a^b f(\gamma(t)) \, \gamma'(t) \, dt.$$
The integral on the right of a complex function $g$ of $t \in [a, b]$ is $\int_a^b \operatorname{Re} g(t) \, dt + i \int_a^b \operatorname{Im} g(t) \, dt$.

## Remarks

The length of $\gamma$ is $L(\gamma) = \int_a^b |\gamma'(t)| \, dt$.
The integral does not change under a reparametrization of $\gamma$ that keeps the direction. It changes sign if we reverse the direction.
The path $\gamma$ is closed if $\gamma(a) = \gamma(b)$. For a closed path, many authors write $\oint_\gamma f(z) \, dz$.
For example, along $\gamma(t) = t + i t$ with $t \in [0, 1]$, $\int_\gamma z \, dz = \int_0^1 (1 + i) t (1 + i) \, dt = i$.

## Explanations

### L1 human

A contour integral adds up the values of $f$ along a path, each one multiplied by a small step $dz$ of the path.
Because $dz$ is a complex number, the direction of each step matters, and not only its length.
</document>

<document source="https://tamnd.github.io/math2/node/00GC8ZQP4Q.md" tag="00GC8ZQP4Q" kind="concept">
# Zero of a holomorphic function

## Summary

A zero of a holomorphic function $f$ is a point $z_0$ with $f(z_0) = 0$, see the [[definition|0CEGEASY49]].
The zeros of a holomorphic function that is not identically zero on a domain are isolated, and each has a finite order.
</document>

<document source="https://tamnd.github.io/math2/node/00GXTQMY8Z.md" tag="00GXTQMY8Z" kind="definition">
# Modular multiplicative inverse

## Definition

Let $n$ be a positive integer and let $a$ be an integer.
A modular multiplicative inverse of $a$ modulo $n$ is an integer $x$ such that $a x \equiv 1 \pmod n$.

## Remarks

An inverse of $a$ modulo $n$ exists if and only if $\gcd(a, n) = 1$. See [[existence of inverses modulo n|0M67FWZWPQ]].
If $x$ is an inverse of $a$, then the inverses of $a$ are exactly the integers $y \equiv x \pmod n$. So the inverse is unique as a residue class, and we write it as $a^{-1} \bmod n$.
In the ring of [[integers modulo n|01N3AYAA6N]], the class $\overline{x}$ is the inverse of $\overline{a}$.
For example, $3^{-1} \equiv 5 \pmod 7$, because $3 \cdot 5 = 15 \equiv 1 \pmod 7$.

## Explanations

### L1 human

Modulo $n$, the inverse of $a$ plays the part of the fraction $1/a$.
To solve $3 x \equiv 4 \pmod 7$, multiply both sides by $5$, the inverse of $3$. You get $x \equiv 20 \equiv 6 \pmod 7$.
</document>

<document source="https://tamnd.github.io/math2/node/00HF4RJ5QN.md" tag="00HF4RJ5QN" kind="concept">
# Direct product of groups

## Summary

The direct product of two groups $G$ and $H$ is the set $G \times H$ of ordered pairs, with the operation done in each component separately.
It is the simplest way to build a new group from two given groups.
</document>

<document source="https://tamnd.github.io/math2/node/00HJCMC2CD.md" tag="00HJCMC2CD" kind="concept">
# Conic section

## Summary

A conic section is a curve that a plane cuts from the surface of a double cone. In the plane, it is the set of zeros of a polynomial of degree $2$ in two variables. The non-degenerate conics are the ellipses, the parabolas and the hyperbolas.
</document>

<document source="https://tamnd.github.io/math2/node/00HPT88ZX0.md" tag="00HPT88ZX0" kind="definition">
# Norm of a Gaussian integer

## Definition

Let $\alpha = a + bi$ be a [[Gaussian integer|0B268GVX95]], with integers $a$ and $b$.
The norm of $\alpha$ is the integer $N(\alpha) = a^2 + b^2 = \alpha \bar{\alpha}$, where $\bar{\alpha} = a - bi$ is the complex conjugate of $\alpha$.

## Remarks

The norm is multiplicative: $N(\alpha \beta) = N(\alpha) N(\beta)$ for all Gaussian integers $\alpha$ and $\beta$.
This product rule is the [[Brahmagupta–Fibonacci identity|0K2DMACEZ4]].
The norm is a nonnegative integer, and $N(\alpha) = 0$ only for $\alpha = 0$. The units are the elements of norm $1$.
For example, $N(2 + i) = 5$ and $N(3 - 2i) = 13$.

## Explanations

### L1 human

The norm of $a + bi$ is the square of its distance from $0$ in the complex plane.
It turns a question about Gaussian integers into a question about ordinary integers.
</document>

<document source="https://tamnd.github.io/math2/node/00J3DCCEK5.md" tag="00J3DCCEK5" kind="problem">
# Trisecting the angle

## Problem

Given an arbitrary angle, construct with [[straightedge and compass|0DAECMZVRT]] alone an angle one third as large.

## Remarks

Some angles can be trisected. For example, a right angle can be trisected, because an angle of $30^\circ$ is constructible.
The general problem asks for one method that works for every angle.
The answer is no: [[the angle of $60^\circ$ cannot be trisected|0EWXPZCSZ0]].
The problem has solutions with other tools, such as a marked ruler or origami.

## History

The problem comes from ancient Greek geometry.
Pierre Wantzel proved in 1837 that it has no solution for arbitrary angles.

## Explanations

### L1 human

Bisecting an angle is easy with a compass.
Trisecting it, that is, dividing it into three equal parts, is not possible in general with straightedge and compass alone.
</document>

<document source="https://tamnd.github.io/math2/node/00J3KZP8Y0.md" tag="00J3KZP8Y0" kind="concept">
# Lower triangular matrix

## Summary

A lower triangular matrix is a square matrix whose entries above the main diagonal are all zero.
</document>

<document source="https://tamnd.github.io/math2/node/00J8E3HX0M.md" tag="00J8E3HX0M" kind="concept">
# Divisibility rule

## Summary

A divisibility rule is a short method to decide whether a fixed integer $d$ divides an integer $n$.
It uses the digits of $n$, and it does not need a full division.
Most rules for base $10$ come from the [[congruences|0TPW25CBN6]] of the powers $10^k$ modulo $d$.
For example, $10 \equiv 1 \pmod 3$, so $n$ and the sum of its digits have the same remainder modulo $3$.

## Remarks

For example, $2$ divides $n$ if and only if the last digit of $n$ is even. Also $5$ divides $n$ if and only if the last digit is $0$ or $5$.
See [[divisibility by 3|0R9D4CEJEB]], [[divisibility by 9|0ESHD7ZQDX]] and [[divisibility by 11|09YBMJTNBM]].
[[Casting out nines|0E7W0EM6J6]] uses the rule for $9$ to check computations.

## Explanations

### L1 human

You can see at once that $1{,}234{,}570$ is a multiple of $10$, because it ends with $0$.
Divisibility rules give similar quick tests for other divisors, for example $3$, $4$, $9$ and $11$.
</document>

<document source="https://tamnd.github.io/math2/node/00K0RF9BFY.md" tag="00K0RF9BFY" kind="definition">
# Multinomial coefficient

## Definition

Let $k_1, \dots, k_m$ be natural numbers and let $n = k_1 + \dots + k_m$.
The multinomial coefficient is the natural number
$$\binom{n}{k_1, k_2, \dots, k_m} = \frac{n!}{k_1!\, k_2! \cdots k_m!}.$$

## Remarks

For $m = 2$ it is the [[binomial coefficient|092GEF38YD]]: $\binom{n}{k, n-k} = \binom{n}{k}$.
It counts the ways to put $n$ distinct objects into $m$ labelled boxes, with $k_i$ objects in box $i$.
It also counts the distinct words that use a letter $a_i$ exactly $k_i$ times. For example, MISSISSIPPI has $\frac{11!}{1!\,4!\,4!\,2!} = 34650$ distinct rearrangements.
It is the coefficient in the [[multinomial theorem|0XFC1VQYEK]].

## Explanations

### L1 human

The multinomial coefficient counts the ways to split $n$ objects into groups of given sizes.
To split 4 people into a group of 2 and two groups of 1, with labelled groups, there are $\frac{4!}{2!\,1!\,1!} = 12$ ways.
</document>

<document source="https://tamnd.github.io/math2/node/00K4FK4NBH.md" tag="00K4FK4NBH" kind="definition">
# Nullity

## Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *nullity* of $T$ is
$$\operatorname{nullity} T = \dim \ker T,$$
the [[dimension|03GSZ8RTZJ]] of the [[kernel|0S23E6YE83]] of $T$.
The nullity of a matrix $A$ is the nullity of the map $x \mapsto Ax$, the dimension of the [[null space|0PHDYNR24D]] of $A$.

## Remarks

The nullity is $0$ if and only if $T$ is injective.
For an $m \times n$ matrix, the nullity is the number of free variables of $Ax = 0$.
By the [[rank–nullity theorem|0W4FSW1TVB]], $\rank T + \operatorname{nullity} T = \dim V$ when $V$ is finite-dimensional.

## Explanations

### L1 human

The nullity counts the dimensions that the map crushes to zero.
The projection of space onto a plane crushes one direction, so its nullity is $1$.
Rank counts what survives, and nullity counts what is lost.
</document>

<document source="https://tamnd.github.io/math2/node/00K7ASZ1PR.md" tag="00K7ASZ1PR" kind="counterexample">
# Non-uniqueness for a non-Lipschitz equation

## Counterexample

For a real number $x$, let $x^{1/3}$ be its real cube root and let $x^{2/3} = \left(x^{1/3}\right)^2$.
Consider the [[initial value problem|075VBY4K4X]] $x'(t) = 3\, x(t)^{2/3}$, $x(0) = 0$, on $\mathbb{R}$.
The constant function $x(t) = 0$ is a solution.
The function $x(t) = t^3$ is also a solution, because its derivative is $3t^2 = 3 \left(t^3\right)^{2/3}$.
More generally, for each $c \ge 0$, the function equal to $0$ for $t < c$ and to $(t - c)^3$ for $t \ge c$ is a solution.
So the problem has infinitely many solutions on every interval that contains $0$ in its interior.

## Remarks

The right side $f(x) = 3 x^{2/3}$ is continuous, so the [[Peano existence theorem|098YP831QY]] applies.
The function $f$ is not Lipschitz on any neighbourhood of $0$: the quotient $f(x)/x = 3 x^{-1/3}$ is unbounded as $x \to 0$. So the [[Picard–Lindelöf theorem|0P6KRD5B7A]] does not apply.
For an initial value $x(0) = x_0 \neq 0$, the field is $C^1$ near $x_0$, and the solution is unique near $t = 0$.

## Explanations

### L1 human

A solution can rest at $0$ for any length of time and then leave.
The field is so steep near $0$ that the equation cannot tell when the departure must happen.
This shows that continuity alone does not give uniqueness.
</document>

<document source="https://tamnd.github.io/math2/node/00KAFBR32E.md" tag="00KAFBR32E" kind="statement">
# Intermediate value theorem

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be [[continuous|0P27W9RW0G]].
Let $y$ be a real number between $f(a)$ and $f(b)$.
Then there is $c \in [a, b]$ with $f(c) = y$.

## Remarks

Assume $f(a) < y < f(b)$. Let $c$ be the supremum of $\{ x \in [a, b] : f(x) < y \}$. Continuity gives $f(c) \le y$ and $f(c) \ge y$.
The proof needs the completeness of $\mathbb{R}$. Over $\mathbb{Q}$, the function $x^2 - 2$ changes sign on $[1, 2]$ but has no rational zero.
It gives the [[existence of $n$-th roots|0J29GM30Z7]], and it shows that each odd degree real polynomial has a real root.
Mathlib states it as `intermediate_value_Icc`.

## History

Bernard Bolzano first proved the theorem in 1817.

## Explanations

### L1 human

A continuous function on an interval cannot skip values.
To get from $f(a)$ to $f(b)$, its graph must cross every horizontal line in between.
</document>

<document source="https://tamnd.github.io/math2/node/00KDZRW4C0.md" tag="00KDZRW4C0" kind="concept">
# Determinant

## Summary

The determinant is the unique function of square matrices that is multilinear and alternating in the rows and is $1$ on the identity matrix.
It is zero exactly for the singular matrices, over a field.
</document>

<document source="https://tamnd.github.io/math2/node/00KRCVE7CK.md" tag="00KRCVE7CK" kind="statement">
# Modulus of the conjugate

## Statement

For each complex number $z$, $|\bar{z}| = |z|$.
Here $\bar{z}$ is the [[complex conjugate|0BJV24C4SY]] of $z$ and $|\cdot|$ is the [[modulus|0KW2X0TFBX]].

## Remarks

Proof: for $z = a + b i$, $|\bar{z}| = \sqrt{a^2 + (-b)^2} = \sqrt{a^2 + b^2} = |z|$.
In the complex plane, this says that a reflection in the real axis does not change the distance to $0$.

## Explanations

### L1 human

A number and its conjugate are mirror images in the real axis.
A mirror keeps distances, so both numbers are at the same distance from $0$.
</document>

<document source="https://tamnd.github.io/math2/node/00MHHZ9SNE.md" tag="00MHHZ9SNE" kind="concept">
# Prime ideal

## Summary

A prime ideal of a commutative ring is a proper ideal with the following property. If it contains a product, then it contains one of the factors.
The prime ideals of $\mathbb{Z}$ are the ideals $p\mathbb{Z}$ for $p$ prime and the zero ideal.
The [[definition of a prime ideal|06HC6HK0Z8]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/00MRN0FARV.md" tag="00MRN0FARV" kind="concept">
# Independent and identically distributed random variables

## Summary

A family of random variables is independent and identically distributed (i.i.d.) when the variables are mutually independent and all have the same distribution.
It is the standard model of repeated trials of one experiment. The node [[i.i.d. random variables|0AJN7WHBJZ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/00MS9VMQ9Z.md" tag="00MS9VMQ9Z" kind="statement">
# Every vector space has a basis

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$.
Then $V$ has a [[basis|0KCQ9Y3QKA]].

## Remarks

For a space with a finite spanning set, the proof is elementary: remove vectors from the spanning set until it is linearly independent.
In general, the proof uses Zorn's lemma: a maximal linearly independent subset of $V$ is a basis.
The statement for all vector spaces is equivalent to the axiom of choice, over set theory without choice.
For example, the axiom of choice gives a basis of $\R$ as a vector space over $\Q$, a Hamel basis.

## Explanations

### L1 human

Every vector space has a set of building blocks that is independent and reaches every vector.
For finite-dimensional spaces, we can find one by hand.
For very large spaces, we must use the axiom of choice, and we may not be able to write the basis down.
</document>

<document source="https://tamnd.github.io/math2/node/00N3T18QNA.md" tag="00N3T18QNA" kind="definition">
# Saddle point

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}$ be [[differentiable|0135XZZQ8M]] at $a \in U$.
The point $a$ is a saddle point of $f$ if $\nabla f(a) = 0$ and $a$ is not a [[local extremum|04CPG9K1C4]] of $f$.

## Remarks

Equivalently, $\nabla f(a) = 0$ and each ball around $a$ contains points $x$ and $y$ with $f(x) < f(a) < f(y)$.
The origin is a saddle point of $f(x, y) = x^2 - y^2$. Along the $x$-axis $f$ has a minimum at $0$, and along the $y$-axis it has a maximum.
If $f$ is of class $C^2$ and the [[Hessian matrix|0SCSQWXRA4]] at a critical point is indefinite, the point is a saddle point. This condition is not necessary: $f(x, y) = x^3$ has a saddle point at $(0, 0)$ with zero Hessian.
In one variable, a saddle point is a stationary point that is not an extremum, such as $0$ for $x \mapsto x^3$.

## Explanations

### L1 human

The name comes from the shape of a horse saddle.
It curves up from front to back and curves down from side to side. So the centre is flat, but it is neither a top nor a bottom.
</document>

<document source="https://tamnd.github.io/math2/node/00N5FHP1CV.md" tag="00N5FHP1CV" kind="statement">
# Change of variables in multiple integrals

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $\varphi \colon U \to \mathbb{R}^n$ be injective and of [[class C^1|0TCBZMEYGN]] with $\det D\varphi(u) \ne 0$ for each $u \in U$.
Let $f \colon \varphi(U) \to \mathbb{R}$ be continuous with compact support in $\varphi(U)$.
Then
$$\int_{\varphi(U)} f(v) \, dv = \int_U f(\varphi(u)) \, \lvert \det D\varphi(u) \rvert \, du.$$

## Remarks

Here $\det D\varphi(u)$ is the [[Jacobian determinant|03W6Q1MFR5]] of $\varphi$ at $u$. Under these hypotheses, $\varphi(U)$ is open and $\varphi$ is a [[diffeomorphism|0Z1Y7JHGBR]] from $U$ onto $\varphi(U)$.
The factor $\lvert \det D\varphi(u) \rvert$ is the ratio by which $\varphi$ scales small volumes near $u$. A linear map $A$ maps the unit cube to a parallelepiped of volume $\lvert \det A \rvert$.
The formula extends to Lebesgue integrable $f$ on $\varphi(U)$, and to maps that fail the hypotheses only on a set of measure zero.
Important cases are [[integration in polar coordinates|0T89W1QZ9K]], cylindrical coordinates and [[spherical coordinates|05WKRC7HWT]].

## Explanations

### L1 human

In one variable, substitution needs the factor $\varphi'(u)$. In several variables, the Jacobian determinant plays this role.
It corrects for the way the change of coordinates stretches or squeezes small regions.
</document>

<document source="https://tamnd.github.io/math2/node/00NBBSXRQS.md" tag="00NBBSXRQS" kind="concept">
# Prior probability

## Summary

A prior probability is the probability that we give to a hypothesis before we see the data.
More generally, a prior distribution is the distribution that we give to an unknown quantity before we see the data.
In [[Bayes' theorem|0NM6E9VW04]] the prior is combined with the likelihood of the data, and the result is the [[posterior probability|0KJ7ZS8AJA]].

## Remarks

An informative prior uses real knowledge about the quantity. An uninformative prior tries to express little knowledge, for example a uniform distribution on the possible values.
An improper prior is a "distribution" with infinite total mass, such as the uniform measure on $\mathbb{R}$. It can still give a proper posterior.

## Explanations

### L1 human

The prior is what you believe before the experiment.
If $1\%$ of people have a disease, then the prior probability that a random person is sick is $0.01$, before any test is done.
</document>

<document source="https://tamnd.github.io/math2/node/00NZKRSWNX.md" tag="00NZKRSWNX" kind="concept">
# Complex sine and cosine

## Summary

The complex sine and cosine extend the real sine and cosine to all of $\mathbb{C}$ with the formulas $\sin z = (e^{iz} - e^{-iz})/(2i)$ and $\cos z = (e^{iz} + e^{-iz})/2$, see the [[definition|0A7YDKRNR6]].
They are entire functions, and they are not bounded on $\mathbb{C}$.
</document>

<document source="https://tamnd.github.io/math2/node/00PPY3RPNB.md" tag="00PPY3RPNB" kind="definition">
# Power of a point

## Definition

Let $C$ be a [[circle|060W5CBZP6]] in the Euclidean plane $\mathbb{R}^2$ with the centre $O$ and the radius $r$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
The power of a point $P \in \mathbb{R}^2$ with respect to $C$ is the real number
$$\Pi(P) = d(P, O)^2 - r^2.$$

## Remarks

$\Pi(P) > 0$ if $P$ is outside $C$, $\Pi(P) = 0$ if $P$ is on $C$, and $\Pi(P) < 0$ if $P$ is inside $C$.
For $P$ outside $C$ and a tangent from $P$ that touches $C$ at $T$, the Pythagorean theorem gives $\Pi(P) = d(P, T)^2$.
Let a line through $P$ meet $C$ at $S_1$ and $S_2$. Then $d(P, S_1)\, d(P, S_2) = |\Pi(P)|$. This contains the [[intersecting chords theorem|0YQQYET8YQ]] and the [[intersecting secants theorem|0J5XR7BW2S]].
Jakob Steiner introduced the power of a point in 1826.

## Explanations

### L1 human

The power of a point measures how far the point is from a circle, in a squared form. For the unit circle and $P = (3, 0)$, the power is $9 - 1 = 8$. A tangent from $P$ to the circle has the length $\sqrt{8}$.
</document>

<document source="https://tamnd.github.io/math2/node/00QN1N6MNT.md" tag="00QN1N6MNT" kind="definition">
# Positive definite quadratic form

## Definition

Let $V$ be a finite-dimensional real vector space, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
The form $Q$ is *positive definite* when
$$Q(v) > 0 \qquad \text{for all } v \in V \text{ with } v \ne 0.$$

## Remarks

$Q(x) = x^T A x$ with $A$ symmetric is positive definite if and only if $A$ is a [[positive definite matrix|0KX66CZMBB]].
Equivalently, the signature of $Q$ is $(n, 0, 0)$, with $n = \dim V$.
The form $x^2 + y^2$ is positive definite on $\R^2$; the forms $x^2 - y^2$ and $x^2$ are not.
$Q$ is positive definite if and only if its associated symmetric bilinear form is an inner product.

## Explanations

### L1 human

A positive definite quadratic form is positive everywhere except at the origin.
Its graph is a bowl with the lowest point at the origin.
Such forms describe squared lengths for different inner products.
</document>

<document source="https://tamnd.github.io/math2/node/00QSK85Y2X.md" tag="00QSK85Y2X" kind="concept">
# Primitive element of a finite field

## Summary

A primitive element of a finite field is a generator of its multiplicative group.
The [[definition of a primitive element|0YAA9CJQXF]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/00R0X1WGPX.md" tag="00R0X1WGPX" kind="definition">
# Compact set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $K \subseteq X$.
The set $K$ is compact when $K$, with the [[subspace topology|0D5YHQGDJY]], is a [[compact space|0QPHW703PV]].

## Remarks

By [[compactness via open covers|0P6PMJB4MA]], $K$ is compact exactly when each [[cover|0NVPNC12JY]] of $K$ by open sets of $X$ has a finite subcover.
Each finite subset of $X$ is compact. The empty set is compact.
In $\mathbb{R}$, the interval $[0, 1]$ is compact, but $(0, 1)$ and $[0, \infty)$ are not. The [[Heine–Borel theorem|07PFYV65MB]] describes the compact subsets of $\mathbb{R}^n$.

## Explanations

### L1 human

A compact set behaves in many ways like a finite set.
For example, a continuous real function on a compact set is bounded and attains its maximum, as a function on a finite set does.
</document>

<document source="https://tamnd.github.io/math2/node/00R1JDDV5X.md" tag="00R1JDDV5X" kind="definition">
# Arc length parametrisation

## Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a [[regular|06JMR2XWXW]] $C^1$ curve.
Its arc length function is
$$s(t) = \int_a^t \|\gamma'(u)\| \, du, \qquad t \in [a, b],$$
and its length is $\ell = s(b)$.
Since $s'(t) = \|\gamma'(t)\| > 0$, the function $s \colon [a, b] \to [0, \ell]$ is a strictly increasing $C^1$ bijection with a $C^1$ inverse $t(s)$.
The arc length parametrisation of $\gamma$ is the curve $\tilde\gamma \colon [0, \ell] \to \mathbb{R}^n$, $\tilde\gamma(s) = \gamma(t(s))$.

## Remarks

The curve $\tilde\gamma$ has the same image and direction as $\gamma$, and $\|\tilde\gamma'(s)\| = 1$ for each $s$. So $\tilde\gamma'(s)$ is the unit [[tangent vector|0D98771SM0]].
The length of $\tilde\gamma$ from $\tilde\gamma(s_0)$ to $\tilde\gamma(s_1)$ is $s_1 - s_0$.
For the helix $\gamma(t) = (\cos t, \sin t, t)$, $\|\gamma'(t)\| = \sqrt{2}$, so $s = \sqrt{2}\, t$ and $\tilde\gamma(s) = (\cos(s / \sqrt{2}), \sin(s / \sqrt{2}), s / \sqrt{2})$.
The arc length parametrisation is unique up to a shift of the parameter. It simplifies the formulas for curvature.

## Explanations

### L1 human

Reparametrise the curve so that the parameter is the distance travelled from the start.
Then the point moves at constant speed $1$, and only the shape of the curve remains.
</document>

<document source="https://tamnd.github.io/math2/node/00R4MWW532.md" tag="00R4MWW532" kind="statement">
# Derivative of sine

## Statement

The sine function is differentiable on $\mathbb{R}$, and $\sin'(x) = \cos x$ for all real $x$.

## Remarks

[[Differentiate the power series|002X7EK91H]] term by term. The derivative of $\frac{(-1)^n x^{2n+1}}{(2n+1)!}$ is $\frac{(-1)^n x^{2n}}{(2n)!}$, the term of the cosine series.
With the addition formula, the claim reduces to the limits $\frac{\sin h}{h} \to 1$ and $\frac{\cos h - 1}{h} \to 0$.
Mathlib states it as `Real.hasDerivAt_sin`.

## Explanations

### L1 human

The slope of the sine curve at $x$ is $\cos x$.
At $x = 0$ the sine curve rises with slope $1$, and at $x = \pi/2$ it is flat.
</document>

<document source="https://tamnd.github.io/math2/node/00RA98X2CP.md" tag="00RA98X2CP" kind="statement">
# Divergence of the sum of the reciprocals of the primes

## Statement

The series $\sum_{p} \frac{1}{p}$, where $p$ runs over the [[prime numbers|0SN0MAEKYX]], diverges.
That is, the partial sums $\sum_{p \le x} \frac{1}{p}$ are not bounded as $x \to \infty$.

## Remarks

The result implies [[Euclid's theorem|0292EPPPT0]], since a finite sum is bounded.
The partial sums grow like $\ln \ln x$. More precisely, $\sum_{p \le x} \frac{1}{p} - \ln \ln x$ has a finite limit as $x \to \infty$.
In contrast, the sum of $1 / n^2$ over all positive integers $n$ converges. So the primes are not as rare as the squares.

## History

Euler proved the result in 1737. Erdős gave an elementary proof by upper and lower estimates.

## Explanations

### L1 human

The sum $\frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \cdots$ becomes larger than each number that you choose, if you take enough terms.
It grows very slowly: the terms for the primes up to $10^9$ still give a sum less than $4$.
</document>

<document source="https://tamnd.github.io/math2/node/00RE1R4W7N.md" tag="00RE1R4W7N" kind="definition">
# Second-countable space

## Definition

A [[topological space|0CNQXVBHCX]] $(X, \tau)$ is second-countable when $\tau$ has a [[basis|06EX0FHBTF]] $\mathcal{B}$ that is [[countable|0ZG61TFBES]].

## Remarks

The space $\mathbb{R}^n$ is second-countable: the open balls with rational centres and rational radii form a countable basis.
Each second-countable space is [[separable|0GKJ78948V]] and [[first-countable|0HG11FW8MJ]].
A metric space is second-countable if and only if it is separable.
An uncountable set with the [[discrete topology|0S2TEHX4E1]] is not second-countable.

## Explanations

### L1 human

A second-countable space has a countable list of open sets from which you can build every open set by unions.
For $\mathbb{R}$, the open intervals with rational endpoints form such a list.
</document>

<document source="https://tamnd.github.io/math2/node/00S7C68D51.md" tag="00S7C68D51" kind="definition">
# Absolute value

## Definition

Let $x$ be a real number.
The absolute value of $x$ is the real number
$$|x| = \begin{cases} x & \text{if } x \ge 0, \\ -x & \text{if } x < 0. \end{cases}$$

## Remarks

For each real $x$: $|x| \ge 0$; $|x| = 0$ only for $x = 0$; $|-x| = |x|$; and $|x| = \max(x, -x)$.
The distance between two real numbers $x$ and $y$ is $|x - y|$.
For $r \ge 0$, the inequality $|x| \le r$ holds exactly when $-r \le x \le r$.

## Explanations

### L1 human

The absolute value removes the minus sign from a number.
On the number line, $|x|$ is the distance from $x$ to $0$, and $|x - y|$ is the distance from $x$ to $y$.
</document>

<document source="https://tamnd.github.io/math2/node/00SB5EA3VW.md" tag="00SB5EA3VW" kind="definition">
# Asymptotic equivalence

## Definition

Let $f$ and $g$ be real functions defined on an interval $[x_0, \infty)$.
The functions are asymptotically equivalent as $x \to \infty$ if $f(x) - g(x) = o(g(x))$ as $x \to \infty$, in the sense of [[little o notation|04TSHB9H8D]].
The notation is $f(x) \sim g(x)$.

## Remarks

If $g(x) \ne 0$ for large $x$, then $f \sim g$ means that $f(x)/g(x) \to 1$.
The relation $\sim$ is an equivalence relation.
For example, $x^2 + x \sim x^2$. [[Stirling's formula|0KC0F51T6A]] states $n! \sim \sqrt{2 \pi n}\, (n/e)^n$.
Mathlib defines it for filters, as `Asymptotics.IsEquivalent`.

## Explanations

### L1 human

Two functions are asymptotically equivalent if their ratio tends to $1$.
Their difference may still grow, but it is small compared with the functions.
</document>

<document source="https://tamnd.github.io/math2/node/00SPJSDT3Z.md" tag="00SPJSDT3Z" kind="concept">
# Gram matrix

## Summary

The Gram matrix of a finite family of vectors in an inner product space is the square matrix of all their pairwise inner products.
</document>

<document source="https://tamnd.github.io/math2/node/00SQGJNRTB.md" tag="00SQGJNRTB" kind="definition">
# Event

## Definition

Let $\Omega$ be a [[sample space|0W7NFW23RF]] and let $\mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] on $\Omega$.
An event is an element of $\mathcal{F}$, that is, a [[subset|0FE8Q5S42Y]] $A \subseteq \Omega$ with $A \in \mathcal{F}$.
The event $A$ occurs in a trial when the outcome $\omega$ of the trial satisfies $\omega \in A$.

## Remarks

When $\Omega$ is finite or countable, one usually takes $\mathcal{F}$ to be the power set of $\Omega$. Then each subset of $\Omega$ is an event.
For uncountable $\Omega$, such as $\Omega = \mathbb{R}$, one cannot in general give a probability to each subset. So only the sets in $\mathcal{F}$ are events.
The set operations have probabilistic names: $A \cup B$ is "$A$ or $B$", $A \cap B$ is "$A$ and $B$", and $\Omega \setminus A$ is "not $A$". The empty set is the impossible event, and $\Omega$ is the sure event.
Two events $A$ and $B$ are mutually exclusive when $A \cap B = \varnothing$.

## Explanations

### L1 human

An event is a statement about the result of the experiment that is either true or false after the experiment.
For a die, "the result is at least $5$" is the event $\{5, 6\}$. It happens when you roll a $5$ or a $6$.
</document>

<document source="https://tamnd.github.io/math2/node/00SVP6W3CB.md" tag="00SVP6W3CB" kind="example">
# Exponential growth and decay

## Example

Let $k$ and $y_0$ be real numbers.
The [[initial value problem|075VBY4K4X]] $y'(t) = k\, y(t)$, $y(0) = y_0$ has the unique solution
$$y(t) = y_0\, e^{k t}, \qquad t \in \mathbb{R}.$$
If $k > 0$ and $y_0 > 0$, the solution grows: this is exponential growth. If $k < 0$ and $y_0 > 0$, the solution decreases to $0$: this is exponential decay.

## Remarks

For $k > 0$, the doubling time is $\ln 2 / k$: we have $y(t + \ln 2 / k) = 2\, y(t)$ for each $t$.
For $k < 0$, the half-life is $\ln 2 / |k|$. Radioactive decay follows this law.
Uniqueness: if $y$ is a solution, then the derivative of $y(t) e^{-k t}$ is $0$, so $y(t) e^{-kt}$ is the constant $y_0$.

## Explanations

### L1 human

The equation says that the rate of change is proportional to the present amount.
A population with no limits, or money with continuous interest, grows like this.
The larger the amount, the faster it grows, so the growth gets faster and faster.
</document>

<document source="https://tamnd.github.io/math2/node/00SW8B91E7.md" tag="00SW8B91E7" kind="definition">
# Transcendental element

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$.
The element $a$ is transcendental over $K$ if $a$ is not [[algebraic|0NPW1DWECM]] over $K$.
That is, $f(a) \neq 0$ for every nonzero $f \in K[x]$.

## Remarks

The number $\pi$ is transcendental over $\mathbb{Q}$, but it is algebraic over $\mathbb{R}$, as a root of $x - \pi$.
The variable $x$ is transcendental over $K$ in the field $K(x)$ of rational functions.
If $a$ is transcendental over $K$, then $K(a)$ is isomorphic to $K(x)$, and $[K(a) : K]$ is infinite.

## Explanations

### L1 human

A transcendental element satisfies no polynomial equation over the smaller field.
It behaves like a free variable.
The transcendental numbers are the complex numbers that are transcendental over $\mathbb{Q}$, such as $\pi$ and $e$.
</document>

<document source="https://tamnd.github.io/math2/node/00T5EPFQB9.md" tag="00T5EPFQB9" kind="concept">
# Little o notation

## Summary

Little o notation says that one function is negligible compared with another. The statement $f = o(g)$ means that $f/g$ tends to $0$ at the point in question. The node [[little o notation|04TSHB9H8D]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/00TQV3F5S4.md" tag="00TQV3F5S4" kind="statement">
# Ramsey's theorem

## Statement

Let $r$ and $s$ be positive integers.
Then there is a positive integer $n$ with this property.
Colour each edge of the [[complete graph|03WKB2H474]] $K_n$ blue or red, in any way.
Then there is a blue [[clique|0G52BPNPKG]] with $r$ vertices or a red clique with $s$ vertices.

## Remarks

The least such $n$ is the [[Ramsey number|01DSVCRF6X]] $R(r, s)$.
The proof gives the bound $R(r, s) \le R(r - 1, s) + R(r, s - 1)$, by induction on $r + s$.
The theorem also holds for any finite number of colours.

## History

Frank Ramsey proved the first version of the theorem in a 1930 paper on formal logic.
This result started the field now called Ramsey theory.

## Explanations

### L1 human

Complete disorder is impossible in a large enough structure.
If you colour the edges of a large complete graph with two colours, a large clique of one colour always appears.
</document>

<document source="https://tamnd.github.io/math2/node/00WAVB68F9.md" tag="00WAVB68F9" kind="definition">
# Picard operator

## Definition

Let $J = [t_{\min}, t_{\max}]$, let $t_0 \in J$ and let $x_0 \in \mathbb{R}^n$. Let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ be a function such that $s \mapsto f(s, \gamma(s))$ is integrable on $J$ for each continuous $\gamma : J \to \mathbb{R}^n$.
The Picard operator of $f$ at $(t_0, x_0)$ maps a continuous curve $\gamma : J \to \mathbb{R}^n$ to the curve $P\gamma : J \to \mathbb{R}^n$ given by
$$(P \gamma)(t) = x_0 + \int_{t_0}^{t} f\left(s, \gamma(s)\right) ds.$$

## Remarks

Assume the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] with constants $a, L, K$. Then $P$ maps the continuous curves with values in the closed ball $\|x - x_0\| \le a$ to curves in this ball.
A fixed point of $P$ is a solution of the [[initial value problem|075VBY4K4X]]. See [[the integral equation form|015KK993C5]].
[[Picard iteration|0RJ5NHWYTP]] applies $P$ again and again, starting from the constant curve $x_0$.

## Explanations

### L1 human

The operator takes a guess for the solution and makes a new guess.
It integrates the field along the old guess, starting from $x_0$.
A curve that the operator does not change is a solution.
</document>

<document source="https://tamnd.github.io/math2/node/00X2A9G9W3.md" tag="00X2A9G9W3" kind="definition">
# Irreducible polynomial

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]].
A polynomial $f \in R[x]$ is irreducible over $R$ if it is an [[irreducible element|052SDEPZH7]] of the integral domain $R[x]$.
That is, $f$ is nonzero and not a unit, and if $f = g h$ with $g, h \in R[x]$, then $g$ or $h$ is a unit of $R[x]$.

## Remarks

Over a [[field|0QA8CHH8MG]] $K$, the units of $K[x]$ are the nonzero constants. So $f$ is irreducible if and only if $\deg f \ge 1$ and $f$ is not a product of two polynomials of smaller degree.
The polynomial $x^2 - 2$ is irreducible over $\mathbb{Q}$ but not over $\mathbb{R}$, where $x^2 - 2 = (x - \sqrt{2})(x + \sqrt{2})$.
The polynomial $x^2 + 1$ is irreducible over $\mathbb{R}$ but not over $\mathbb{C}$.
Over $\mathbb{Z}$, the polynomial $2x + 2 = 2(x + 1)$ is not irreducible, because $2$ is not a unit of $\mathbb{Z}[x]$.

## Explanations

### L1 human

Irreducible polynomials are the prime numbers of polynomial rings.
Over a field, they are the polynomials that you cannot factor into polynomials of smaller degree.
Irreducibility depends on the field: $x^2 + 1$ factors over $\mathbb{C}$ but not over $\mathbb{R}$.
</document>

<document source="https://tamnd.github.io/math2/node/00Y5QZ82AA.md" tag="00Y5QZ82AA" kind="example">
# Lotka–Volterra equations

## Example

Let $\alpha, \beta, \gamma, \delta > 0$ be constants. Let $x(t) > 0$ be the size of a prey population and $y(t) > 0$ the size of a predator population at time $t$.
The Lotka–Volterra equations are the [[autonomous system|07BNRFQ0PJ]]
$$x' = \alpha x - \beta x y, \qquad y' = \delta x y - \gamma y.$$
The [[equilibrium points|0G4X36AE0X]] in the closed quadrant $x, y \ge 0$ are $(0, 0)$ and $(\gamma / \delta, \alpha / \beta)$.
The function $V(x, y) = \delta x - \gamma \ln x + \beta y - \alpha \ln y$ is a [[first integral|0SS1P1YC9A]] on the open quadrant $x, y > 0$.

## Remarks

Along a solution, $\frac{d}{dt} V = (\delta x - \gamma)(\alpha - \beta y) + (\beta y - \alpha)(\delta x - \gamma) = 0$.
The [[linearisation|09ZWYRQ41J]] at $(0, 0)$ has the eigenvalues $\alpha$ and $-\gamma$, so the origin is a saddle. At $(\gamma/\delta, \alpha/\beta)$ the eigenvalues are $\pm i \sqrt{\alpha \gamma}$.
The level curves of $V$ in the open quadrant are closed curves around $(\gamma/\delta, \alpha/\beta)$. So each solution other than this equilibrium is [[periodic|01DSNP0B4R]]: the two populations oscillate.

## Explanations

### L1 human

Prey grow when there are few predators, and predators grow when there is much prey.
This feedback makes both populations rise and fall in cycles, with the predator peak after the prey peak.
</document>

<document source="https://tamnd.github.io/math2/node/00Y9N80VSS.md" tag="00Y9N80VSS" kind="statement">
# A finite group is nilpotent iff it is the product of its Sylow subgroups

## Statement

Let $G$ be a finite [[group|G5S1M9]].
The following conditions are equivalent:

1. $G$ is [[nilpotent|0Q7MMQHC1P]].
2. For each prime number $p$, $G$ has exactly one [[Sylow p-subgroup|0053CQE9Y2]].
3. $G$ is the [[internal direct product|0VBZH90R1A]] of its Sylow subgroups, one for each prime divisor of $|G|$.

## Remarks

The direction from 3 to 1 holds because [[finite p-groups are nilpotent|081WTKYX6C]] and a finite direct product of nilpotent groups is nilpotent.
The direction from 1 to 2 uses the [[Frattini argument|0QTZDKJEDV]]. In a nilpotent group, a proper subgroup is strictly smaller than its normaliser.
For example, $\mathbb{Z}/12\mathbb{Z} \cong \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ is nilpotent, but $S_3$ has three Sylow $2$-subgroups, so it is not nilpotent.

## Explanations

### L1 human

A finite nilpotent group splits into its $p$-parts, one for each prime.
The parts do not interact, so the group is their direct product.
Conversely, a direct product of $p$-groups is nilpotent.
</document>

<document source="https://tamnd.github.io/math2/node/00YC1D652K.md" tag="00YC1D652K" kind="concept">
# Linear subspace

## Summary

A linear subspace of a vector space is a subset that contains the zero vector and is closed under addition and scalar multiplication.
It is a vector space with the same operations.
</document>

<document source="https://tamnd.github.io/math2/node/00YH28R457.md" tag="00YH28R457" kind="statement">
# Volume of a ball in R^n

## Statement

Let $n \ge 1$, let $R > 0$, and let $B_n(R) = \{x \in \mathbb{R}^n : \|x\| \le R\}$.
Then
$$\operatorname{vol}(B_n(R)) = \frac{\pi^{n/2}}{\Gamma\!\left(\frac{n}{2} + 1\right)} \, R^n,$$
where $\Gamma(s) = \int_0^\infty t^{s - 1} e^{-t} \, dt$ is the gamma function.

## Remarks

The [[change of variables|00N5FHP1CV]] $x = R u$ shows that $\operatorname{vol}(B_n(R)) = R^n \operatorname{vol}(B_n(1))$.
Since $\Gamma(1) = 1$, $\Gamma(1/2) = \sqrt{\pi}$ and $\Gamma(s + 1) = s \, \Gamma(s)$, the formula gives $2R$, $\pi R^2$, $\tfrac{4}{3} \pi R^3$ and $\tfrac{1}{2} \pi^2 R^4$ for $n = 1, 2, 3, 4$.
One proof computes $\int_{\mathbb{R}^n} e^{-\|x\|^2} \, dx = \pi^{n/2}$ in two ways: as a product of $n$ Gaussian integrals by [[Fubini's theorem|0R0T1VWH5X]], and by integration over spheres of radius $r$.
The volume of the unit ball tends to $0$ as $n \to \infty$.

## Explanations

### L1 human

The volume of a ball grows like the $n$-th power of its radius, times a constant that depends on the dimension.
The gamma function gives this constant in one formula for all dimensions.
</document>

<document source="https://tamnd.github.io/math2/node/00Z1YBJJW8.md" tag="00Z1YBJJW8" kind="concept">
# Arcsine

## Summary

The arcsine is the inverse of the sine function restricted to $[-\pi/2, \pi/2]$. It is defined on $[-1, 1]$. The node [[arcsine|0B89D93YA1]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/00ZEBSMNNV.md" tag="00ZEBSMNNV" kind="statement">
# Additivity of line integrals under concatenation

## Statement

Let $\gamma_1 \colon [a, b] \to \mathbb{R}^n$ and $\gamma_2 \colon [b, c] \to \mathbb{R}^n$ be piecewise $C^1$ curves with $\gamma_1(b) = \gamma_2(b)$.
Let $\gamma \colon [a, c] \to \mathbb{R}^n$ be their concatenation: $\gamma = \gamma_1$ on $[a, b]$ and $\gamma = \gamma_2$ on $[b, c]$.
For each continuous vector field $F$ and each continuous function $f$ on the image of $\gamma$,
$$\int_\gamma F \cdot d\mathbf{r} = \int_{\gamma_1} F \cdot d\mathbf{r} + \int_{\gamma_2} F \cdot d\mathbf{r}, \qquad \int_\gamma f \, ds = \int_{\gamma_1} f \, ds + \int_{\gamma_2} f \, ds.$$

## Remarks

Proof: apply the [[parametric formula|0W3W1VYWBP]] on $[a, c]$ and split the integral at $t = b$.
If the parameter intervals do not match, first shift the parameter of $\gamma_2$. A shift does not change line integrals.
Together with [[path reversal|0FSCXYG856]], this lets one compute line integrals over the boundary of a polygon, side by side.
The concatenation is often written $\gamma_1 + \gamma_2$ or $\gamma_1 * \gamma_2$.

## Explanations

### L1 human

Walking from $A$ to $B$ and then from $B$ to $C$ is one walk from $A$ to $C$.
The line integral over the whole walk is the sum of the line integrals over the two parts.
</document>

<document source="https://tamnd.github.io/math2/node/00ZG9Y9TR4.md" tag="00ZG9Y9TR4" kind="concept">
# Topological embedding

## Summary

A topological embedding is a map that identifies a space with a subspace of another space, with the same topology.
It is a homeomorphism onto its image.
</document>

<document source="https://tamnd.github.io/math2/node/010JCV9J4E.md" tag="010JCV9J4E" kind="definition">
# Ordered field

## Definition

An ordered field is a field $F$ with a total order $\le$ on $F$.
The order must have these two properties for all $a$, $b$ and $c$ in $F$:

1. if $a \le b$, then $a + c \le b + c$;
2. if $0 \le a$ and $0 \le b$, then $0 \le a b$.

## Remarks

In an ordered field, each square is nonnegative: $a^2 \ge 0$ for each $a$. In particular $1 = 1^2 > 0$ and $-1 < 0$.
An equivalent definition uses a positive cone $P$. This is a subset of $F$ that is closed under addition and multiplication.
Also, $F$ must be the disjoint union of $P$, $\{0\}$ and $-P$.
The field $\mathbb{C}$ of complex numbers [[has no such order|01CJRJKDAA]].

## Explanations

### L1 human

An ordered field is a number system where you can add, multiply, divide and compare numbers.
The rules of comparison are the usual ones.
If you add the same number to both sides of an inequality, the inequality stays true.
The product of two positive numbers is positive.
</document>

<document source="https://tamnd.github.io/math2/node/010PKCQPXP.md" tag="010PKCQPXP" kind="statement">
# Uniqueness of finite fields of a given order

## Statement

Let $F$ and $F'$ be [[finite fields|08RTDHX1SK]] with $|F| = |F'|$.
Then $F$ and $F'$ are isomorphic.

## Remarks

The proof: let $q = |F|$. Both fields have characteristic $p$ and prime subfield $\mathbb{F}_p$. Both are [[splitting fields of $x^q - x$ over $\mathbb{F}_p$|0CDAHW576M]], so they are isomorphic by the [[uniqueness of splitting fields|069A3BC3EP]].
Also, a field contains at most one subfield of order $q$: its elements must be the $q$ roots of $x^q - x$.

## History

E. H. Moore first proved the classification of finite fields in 1893.

## Explanations

### L1 human

Two finite fields with the same number of elements have the same structure.
Only the names of the elements can differ.
</document>

<document source="https://tamnd.github.io/math2/node/010QVB71T2.md" tag="010QVB71T2" kind="statement">
# Hensel's lemma

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]], and let $f$ be a polynomial with integer coefficients, with derivative $f'$.
Let $r$ be an integer such that $f(r) \equiv 0 \pmod p$ and $f'(r) \not\equiv 0 \pmod p$.
Then for each positive integer $k$ there is an integer $r_k$ with $r_k \equiv r \pmod p$ and $f(r_k) \equiv 0 \pmod{p^k}$.
This $r_k$ is unique modulo $p^k$.

## Remarks

The proof gives the lifts step by step. If $f(r_k) \equiv 0 \pmod{p^k}$ and $t$ is an inverse of $f'(r)$ modulo $p$, then $r_{k+1} = r_k - t f(r_k)$ is a root modulo $p^{k+1}$.
For example, $3^2 \equiv 2 \pmod 7$ and $f'(3) = 6$ for $f = x^2 - 2$. The step gives $3 - 6 \cdot 7 = -39 \equiv 10 \pmod{49}$, and $10^2 = 100 \equiv 2 \pmod{49}$.
In the language of $p$-adic numbers, a simple root modulo $p$ lifts to a unique root in the $p$-adic integers.
The lemma is the analogue of Newton's method for congruences.

## History

The lemma is named after Kurt Hensel.

## Explanations

### L1 human

A solution of a polynomial congruence modulo $p$ often gives solutions modulo $p^2$, $p^3$, and so on.
The condition $f'(r) \not\equiv 0 \pmod p$ says that $r$ is a simple root modulo $p$.
Under this condition, each step of the lifting has exactly one answer.
</document>

<document source="https://tamnd.github.io/math2/node/0110GBHPPB.md" tag="0110GBHPPB" kind="statement">
# Recurrence for Bell numbers

## Statement

Let $n$ be a natural number, and let $B_m$ be the [[Bell numbers|0WW85SQVS3]].
Then
$$B_{n+1} = \sum_{k=0}^{n} \binom{n}{k} B_k.$$

## Remarks

Proof idea: in a partition of $\{1, \dots, n + 1\}$, the block of $n + 1$ contains $n - k$ other elements, for some $k$. The remaining $k$ elements form an arbitrary partition.
Example: $B_4 = 1 \cdot 1 + 3 \cdot 1 + 3 \cdot 2 + 1 \cdot 5 = 15$.

## Explanations

### L1 human

Fix the last element and decide which other elements share its group.
For each choice, the elements left over can be split in any way, so you add a Bell number for each case.
</document>

<document source="https://tamnd.github.io/math2/node/0112DTS464.md" tag="0112DTS464" kind="definition">
# Image of a homomorphism

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]] and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
The image of $\varphi$ is the set $\operatorname{im} \varphi = \varphi(G) = \{\varphi(g) : g \in G\}$.

## Remarks

The image is a [[subgroup|0X1JXT47GA]] of $H$, because $\varphi(a) \varphi(b)^{-1} = \varphi(ab^{-1})$.
The homomorphism $\varphi$ is surjective exactly when $\operatorname{im} \varphi = H$.
The image of the map $n \mapsto 2^n$ from $\mathbb{Z}$ to $\mathbb{Q}^\times$ is the set of integer powers of $2$.

## Explanations

### L1 human

The image is the part of the target group that the homomorphism actually reaches.
It is always a group of its own, inside the target.
The [[first isomorphism theorem|0CXXJNG5B6]] says that the image looks like a quotient of the source group.
</document>

<document source="https://tamnd.github.io/math2/node/011C29Q1S1.md" tag="011C29Q1S1" kind="definition">
# Partial order

## Definition

Let $A$ be a set.
A partial order on $A$ is a [[preorder|0RXS8R6K37]] $\le$ on $A$ that is [[antisymmetric|01MGYYBB9R]]. For all $a, b, c \in A$:

1. $a \le a$;
2. $a \le b$ and $b \le a$ imply $a = b$;
3. $a \le b$ and $b \le c$ imply $a \le c$.

The pair $(A, \le)$ is a partially ordered set.

## Remarks

Examples are $\le$ on $\mathbb{R}$, divisibility on $\mathbb{N}$, and $\subseteq$ on the [[power set|0BC7KX51Z8]] of a set.
"Partial" means that two elements need not be comparable: neither $\{1\} \subseteq \{2\}$ nor $\{2\} \subseteq \{1\}$.
The strict order $a < b$ means $a \le b$ and $a \neq b$.
A finite poset can be drawn as a [[Hasse diagram|00EYM7MDKS]].

## Explanations

### L1 human

A partial order ranks some pairs of elements, but not necessarily all pairs.
In the family tree, "is an ancestor of or equal to" is a partial order: two cousins are not comparable.
</document>

<document source="https://tamnd.github.io/math2/node/011EJQWWJS.md" tag="011EJQWWJS" kind="concept">
# Strictly monotone function

## Summary

A strictly monotone function is a function that is either strictly increasing or strictly decreasing. Such a function is injective. The node [[strictly monotone function|0TR9SR9FTB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/011HV2GAWF.md" tag="011HV2GAWF" kind="statement">
# A convex function on an open interval is continuous

## Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be convex.
Then $f$ is [[continuous|0P27W9RW0G]] on $I$.

## Remarks

Fix $y \in I$ and points $x < y < z$ in $I$. By the [[slope inequalities|0BY5FTHEEE]], the slopes of chords from $y$ are bounded for points near $y$. So $|f(w) - f(y)| \le C |w - y|$ near $y$.
On a closed interval this fails. The function on $[0, 1]$ with $f(0) = 1$ and $f(x) = 0$ for $x > 0$ is convex but not continuous at $0$.
A convex function on an open interval has one-sided derivatives at every point. Mathlib states continuity as `ConvexOn.continuousOn`, for open sets.

## Explanations

### L1 human

A convex function cannot jump inside an open interval.
A jump would break the condition that the chords lie above the graph.
</document>

<document source="https://tamnd.github.io/math2/node/011K11VPJN.md" tag="011K11VPJN" kind="statement">
# Jordan–Chevalley decomposition

## Statement

Let $F$ be a perfect field, for example a field of characteristic $0$ or an algebraically closed field.
Let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Then there are unique linear maps $S, N \colon V \to V$ with these properties:

1. $T = S + N$.
2. $S$ is [[semisimple|08VRX3Q4PB]] and $N$ is [[nilpotent|063D5Z2C70]].
3. $SN = NS$.

## Remarks

Moreover, $S$ and $N$ are polynomials in $T$ without constant term.
Over $\C$, $S$ is diagonalisable; in a Jordan basis, $S$ is the diagonal part of the Jordan form.
For $J_2(\lambda)$, the decomposition is $S = \lambda I_2$ and $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$.

## Explanations

### L1 human

Every linear map over $\C$ splits into a diagonalisable part and a nilpotent part.
The two parts commute with each other.
This splitting is unique, and it makes functions of matrices easier to compute.
</document>

<document source="https://tamnd.github.io/math2/node/0125QVNCFQ.md" tag="0125QVNCFQ" kind="definition">
# Evaluation homomorphism

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $c \in R$.
The evaluation homomorphism at $c$ is the map
$$\operatorname{ev}_c \colon R[x] \to R, \qquad \operatorname{ev}_c(f) = f(c),$$
where $f(c)$ is the [[value of $f$ at $c$|0GJ7T0NK49]].
It is a [[ring homomorphism|0GGE0Z0JQ1]].

## Remarks

The map $\operatorname{ev}_c$ is surjective, because $\operatorname{ev}_c(r) = r$ for each constant $r \in R$.
Its [[kernel|0AAHE7GEW6]] is the principal ideal $(x - c)$. This is a form of the [[factor theorem|0K3MWA7DMC]].
The commutativity of $R$ is necessary: in general $(f g)(c) \neq f(c) g(c)$ when $c$ does not commute with the coefficients.

## Explanations

### L1 human

Evaluation at $c$ turns each polynomial into an element of $R$.
It respects sums and products: to evaluate $f g$, you can evaluate $f$ and $g$ and multiply.
</document>

<document source="https://tamnd.github.io/math2/node/012C7GFF20.md" tag="012C7GFF20" kind="statement">
# 1 + x is at most exp x

## Statement

For every real $x$,
$$1 + x \le e^x,$$
with equality if and only if $x = 0$.

## Remarks

The line $y = 1 + x$ is the tangent to the graph of $e^x$ at $0$. The graph of a [[convex function|0D7GRZ2ADX]] lies above its tangent lines, by [[the derivative criterion|005JT76HCK]].
For $x \ge 0$ the claim also follows from the series, since the other terms are nonnegative.
Taking logarithms gives [[$\ln y \le y - 1$|04CBRA4JRK]] for $y > 0$. Mathlib states the inequality as `Real.add_one_le_exp`.

## Explanations

### L1 human

The graph of $e^x$ lies above its tangent line at $0$.
That tangent line is $y = 1 + x$.
</document>

<document source="https://tamnd.github.io/math2/node/012PBA0FMD.md" tag="012PBA0FMD" kind="concept">
# Hölder continuity

## Summary

A map between metric spaces is Hölder continuous with exponent $\alpha$ when $d(f(x), f(y)) \le C \, d(x, y)^\alpha$ for some constant $C$ and all $x, y$.
For $\alpha = 1$ this is the Lipschitz condition.
</document>

<document source="https://tamnd.github.io/math2/node/012RYET5E2.md" tag="012RYET5E2" kind="concept">
# Reduction formula

## Summary

A reduction formula expresses an integral that depends on an integer $n$ through the same integral with a smaller $n$. Repeated use of the formula reduces the integral to a simple case. [[Integration by parts|08Y1BBQTK9]] usually produces the formula.

## Remarks

For $I_n = \int_0^{\pi/2} \sin^n x \, dx$, integration by parts gives $I_n = \frac{n - 1}{n} I_{n-2}$ for $n \ge 2$.
With $I_0 = \pi/2$ and $I_1 = 1$, this gives all $I_n$.
These integrals lead to the [[Wallis product|0XCYA4YERS]].

## Explanations

### L1 human

A reduction formula is a recursion for a family of integrals.
Each step lowers a power, until the integral is easy.
</document>

<document source="https://tamnd.github.io/math2/node/0135XZZQ8M.md" tag="0135XZZQ8M" kind="definition">
# Differentiable function of several variables

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$.
The function $f$ is differentiable at $a$ when there is a linear map $L \colon \mathbb{R}^n \to \mathbb{R}^m$ such that
$$\lim_{h \to 0} \frac{\|f(a + h) - f(a) - L h\|}{\|h\|} = 0.$$
Here the limit is over $h \ne 0$ with $a + h \in U$, and $\|\cdot\|$ is the Euclidean norm.
The map $L$ is then unique. It is the derivative $Df(a)$, also called the [[total derivative|0FNKFT8H6E]] of $f$ at $a$.

## Remarks

This is the [[Fréchet derivative|08EVNYZF42]] for the spaces $\mathbb{R}^n$ and $\mathbb{R}^m$, where each linear map is continuous.
A function that is differentiable at $a$ is continuous at $a$, and all its [[directional derivatives|06KWGHDN2J]] exist at $a$.
The converse fails: a function can have all partial derivatives at a point and not be differentiable there, see [[this counterexample|0TNZC55X3J]].
For $n = m = 1$, the definition is the usual one, with $L h = f'(a) h$.

## Explanations

### L1 human

A differentiable function looks flat when you zoom in on its graph near the point.
For $f(x, y) = x^2 + y^2$ at $(1, 1)$, the change is $f(1 + h_1, 1 + h_2) - f(1, 1) = 2 h_1 + 2 h_2 + h_1^2 + h_2^2$.
The linear part is $L h = 2 h_1 + 2 h_2$, and the rest $h_1^2 + h_2^2 = \|h\|^2$ is small compared with $\|h\|$.
</document>

<document source="https://tamnd.github.io/math2/node/013AHC4DRT.md" tag="013AHC4DRT" kind="statement">
# exp z = 1 iff z is an integer multiple of 2πi

## Statement

Let $z \in \mathbb{C}$.
Then $e^z = 1$ if and only if $z = 2 \pi i k$ for some [[integer|0C4JS9RF5M]] $k$.
Here $e^z$ is the [[complex exponential|054VXHMCA3]] of $z$.

## Remarks

Proof: write $z = x + iy$ with $x, y$ real. Then $e^z = e^x (\cos y + i \sin y)$. If $e^z = 1$, then $e^x = |e^z| = 1$, so $x = 0$. Then $\cos y = 1$ and $\sin y = 0$, so $y = 2 \pi k$. The converse follows from [[Euler's formula|04R3CY41VJ]].
It follows that $e^z = e^w$ if and only if $z - w = 2 \pi i k$ for some $k \in \mathbb{Z}$.
This is the reason why the [[complex logarithm|0X2FY491PF]] has many values.

## Explanations

### L1 human

The value $1$ has size $1$ and direction $0$.
So $e^z = 1$ needs a real part $0$ and an imaginary part that is a whole number of full turns, that is, a multiple of $2\pi$.
</document>

<document source="https://tamnd.github.io/math2/node/01441KDGBY.md" tag="01441KDGBY" kind="statement">
# Existence of splitting fields

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ have degree $n \ge 1$.
Then $f$ has a [[splitting field|02NZZ1J0FH]] $L$ over $K$, and $[L : K] \le n!$.

## Remarks

The proof is by induction on $n$. [[Some extension $K_1$ contains a root $a_1$ of $f$|0VG02M1FEZ]], with $[K_1 : K] \le n$. Then $f = (x - a_1) g$ in $K_1[x]$, and a splitting field of $g$ over $K_1$ is a splitting field of $f$ over $K$.
The bound $n!$ follows from the [[tower law|0A7KJ68XHT]].
The bound is reached: the splitting field of $x^3 - 2$ over $\mathbb{Q}$ has degree $6 = 3!$.

## Explanations

### L1 human

You build a splitting field one root at a time.
Each step adds a root of a remaining factor, and the degree of the step is at most the degree of that factor.
</document>

<document source="https://tamnd.github.io/math2/node/0146MBQNNE.md" tag="0146MBQNNE" kind="definition">
# Strict differentiability

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$.
The function $f$ is strictly differentiable at $a$ if there is a linear map $L \colon \mathbb{R}^n \to \mathbb{R}^m$ such that
$$\lim_{\substack{(x, y) \to (a, a) \\ x \ne y}} \frac{\|f(x) - f(y) - L(x - y)\|}{\|x - y\|} = 0.$$

## Remarks

If $f$ is strictly differentiable at $a$, put $y = a$: then $f$ is [[differentiable|0135XZZQ8M]] at $a$ and $L = Df(a)$ is its [[Fréchet derivative|08EVNYZF42]].
The converse is false. The function $f(x) = x^2 \sin(1/x)$ for $x \ne 0$, $f(0) = 0$, is differentiable at $0$, but it is not strictly differentiable at $0$ because $f'$ is not continuous at $0$.
A function of class $C^1$ on $U$ is [[strictly differentiable|0STPPDXGME]] at each point of $U$.
The same definition works for maps between Banach spaces, with norms in place of Euclidean norms.

## Explanations

### L1 human

Ordinary differentiability compares $f(x)$ with $f(a)$ only.
Strict differentiability compares $f(x)$ with $f(y)$ for all pairs of points near $a$.
So the linear approximation is uniform in a small neighbourhood, not only at the single point.
</document>

<document source="https://tamnd.github.io/math2/node/014E1DZQE8.md" tag="014E1DZQE8" kind="definition">
# Semidirect product

## Definition

Let $N$ and $H$ be [[groups|G5S1M9]], and let $\varphi \colon H \to \operatorname{Aut}(N)$, $h \mapsto \varphi_h$, be a [[group homomorphism|0JYTG5K9Y4]].
The semidirect product $N \rtimes_\varphi H$ is the set $N \times H$ with the operation $(n_1, h_1)(n_2, h_2) = (n_1 \varphi_{h_1}(n_2), h_1 h_2)$.

## Remarks

The identity is $(e, e)$, and the inverse of $(n, h)$ is $(\varphi_{h^{-1}}(n^{-1}), h^{-1})$.
The set $\{(n, e) : n \in N\}$ is a [[normal subgroup|0CB0DX5BVV]] isomorphic to $N$, and $\{(e, h) : h \in H\}$ is a subgroup isomorphic to $H$.
If $\varphi$ is the trivial homomorphism, then $N \rtimes_\varphi H$ is the [[direct product|00HF4RJ5QN]] $N \times H$.
The [[dihedral group|0JZ27DMJ5D]] $D_n$ is $\mathbb{Z}/n\mathbb{Z} \rtimes_\varphi \mathbb{Z}/2\mathbb{Z}$, where the nonzero element of $\mathbb{Z}/2\mathbb{Z}$ acts by $x \mapsto -x$.

## Explanations

### L1 human

In a direct product, the two factors do not interact.
In a semidirect product, the second factor $H$ twists the first factor $N$ when you multiply.
The homomorphism $\varphi$ says exactly how each element of $H$ twists $N$.
</document>

<document source="https://tamnd.github.io/math2/node/014PEBGDYF.md" tag="014PEBGDYF" kind="statement">
# Chernoff bound

## Statement

Let $X$ be a real random variable with [[moment-generating function|0NNRNGKMFM]] $M_X(t) = E[e^{tX}]$, and let $a \in \mathbb{R}$. Then
$$P(X \ge a) \le \inf_{t > 0} e^{-ta} M_X(t).$$

## Remarks

Proof: for each $t > 0$, $\{X \ge a\} = \{e^{tX} \ge e^{ta}\}$. [[Markov's inequality|05PWWHYBX5]] gives $P(X \ge a) \le e^{-ta} E[e^{tX}]$. Then take the infimum over $t$.
For a sum $S$ of independent $X_i$, $M_S = \prod_i M_{X_i}$. So the bound becomes $\inf_{t > 0} e^{-ta} \prod_i M_{X_i}(t)$, which decreases exponentially in many cases.
For $S \sim \operatorname{Bin}(n, p)$ with mean $\mu = np$ and $0 < \delta$, a standard consequence is $P(S \ge (1 + \delta)\mu) \le \left(e^{\delta} / (1 + \delta)^{1 + \delta}\right)^{\mu}$.

## History

The bound is named after Herman Chernoff.

## Explanations

### L1 human

The Chernoff bound gives tail estimates that decrease very fast, often exponentially.
It is much stronger than Chebyshev's inequality for sums of many independent terms, such as the number of heads in many tosses.
</document>

<document source="https://tamnd.github.io/math2/node/0151VYVX27.md" tag="0151VYVX27" kind="statement">
# Cayley's theorem

## Statement

Let $G$ be a [[group|G5S1M9]].
Then $G$ is [[isomorphic|08MG7H8JW5]] to a [[subgroup|0X1JXT47GA]] of the [[symmetric group|0515A6D2Y1]] $\operatorname{Sym}(G)$.
Explicitly, the map $g \mapsto \ell_g$, where $\ell_g(x) = gx$, is an injective [[group homomorphism|0JYTG5K9Y4]] from $G$ to $\operatorname{Sym}(G)$.

## Remarks

The map comes from the [[action by left multiplication|0K5TWJAGPD]], and it is injective because $\ell_g(e) = g$.
If $|G| = n$ is finite, then $G$ is isomorphic to a subgroup of $S_n$.
Often a smaller symmetric group is enough, for example $S_3$ is isomorphic to a subgroup of $S_3$ and not only of $S_6$.

## History

Arthur Cayley introduced abstract groups in 1854.
At that time, groups were usually groups of permutations, and the theorem shows that the abstract notion gives nothing more, up to isomorphism.

## Explanations

### L1 human

Each element of a group shuffles the elements of the group when you multiply on the left.
Different elements give different shuffles, and the product of elements gives the composition of shuffles.
So each group is a group of permutations, up to isomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/015KK993C5.md" tag="015KK993C5" kind="statement">
# Integral equation form of an initial value problem

## Statement

Let $J$ be an interval, let $U$ be an open subset of $\mathbb{R}^n$, let $f : J \times U \to \mathbb{R}^n$ be continuous, and let $t_0 \in J$ and $x_0 \in U$.
Let $\gamma : J \to U$ be a continuous curve.
Then $\gamma$ is differentiable on $J$ with $\gamma'(t) = f(t, \gamma(t))$ for each $t \in J$ and $\gamma(t_0) = x_0$ if and only if
$$\gamma(t) = x_0 + \int_{t_0}^{t} f\left(s, \gamma(s)\right) ds \quad \text{for each } t \in J.$$
At an endpoint of $J$ that belongs to $J$, the derivative is one-sided.

## Remarks

In other words, the solutions of the [[initial value problem|075VBY4K4X]] on $J$ are the fixed points of the [[Picard operator|00WAVB68F9]].
Proof: one direction is the fundamental theorem of calculus applied to $\gamma'$. For the other, the integrand $s \mapsto f(s, \gamma(s))$ is continuous, so the integral is differentiable in $t$.
The integral form needs only continuity of $\gamma$. This makes it the right setting for fixed point arguments.

## Explanations

### L1 human

Integrate both sides of $\gamma' = f(t, \gamma)$ from $t_0$ to $t$.
The initial condition and the differential equation become one equation with an integral.
</document>

<document source="https://tamnd.github.io/math2/node/0167F9B344.md" tag="0167F9B344" kind="concept">
# Glide reflection

## Summary

A glide reflection is a reflection in a hyperplane followed by a translation parallel to that hyperplane. In the plane, it is a reflection in a line followed by a shift along that line.
</document>

<document source="https://tamnd.github.io/math2/node/016PE4VZ8Q.md" tag="016PE4VZ8Q" kind="statement">
# Leibniz formula for determinants

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $A = (a_{ij})$ be an $n \times n$ matrix over $K$.
Let $S_n$ be the group of permutations of $\{1, \dots, n\}$, and let $\sgn(\sigma) \in \{1, -1\}$ be the sign of $\sigma$.
Then the [[determinant|00KDZRW4C0]] of $A$ is
$$\det A = \sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i=1}^{n} a_{i, \sigma(i)}.$$

## Remarks

The sum has $n!$ terms.
Each term takes exactly one entry from each row and from each column.
For $n = 3$, the formula is the [[rule of Sarrus|0H8SZFFDXJ]].
The formula shows that the determinant is a polynomial in the entries, with integer coefficients.

## Explanations

### L1 human

To compute a determinant by this formula, choose one entry in each row so that no two lie in the same column.
Multiply the chosen entries, and give the product the sign of the permutation.
Add these signed products over all choices.
</document>

<document source="https://tamnd.github.io/math2/node/017BPGVEE8.md" tag="017BPGVEE8" kind="concept">
# Principal branch

## Summary

The principal branch of a multivalued function is one branch of it that a standard convention chooses, see the [[definition|0YKY4M3KRQ]].
Its values are the principal values. An example is $\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$ with $\operatorname{Arg} z \in (-\pi, \pi]$.
</document>

<document source="https://tamnd.github.io/math2/node/017BR1M98J.md" tag="017BR1M98J" kind="definition">
# Alternating series

## Definition

Let $(b_n)$ be a real sequence with $b_n > 0$ for all $n$.
The series
$$\sum_{n=0}^{\infty} (-1)^n b_n = b_0 - b_1 + b_2 - b_3 + \cdots$$
is an alternating series.
More generally, a [[series|0JH4MN4K1C]] is alternating if its terms are nonzero and their signs alternate.

## Remarks

Some texts allow $b_n \ge 0$.
The [[alternating harmonic series|0JQ5Y3WWVE]] $1 - 1/2 + 1/3 - \cdots$ is the standard example.
The [[alternating series test|0PK8QH1W1V]] gives a simple condition for convergence.

## Explanations

### L1 human

In an alternating series, the signs of the terms go plus, minus, plus, minus.
The partial sums then move up and down, like a pendulum.
</document>

<document source="https://tamnd.github.io/math2/node/017DH8WXTE.md" tag="017DH8WXTE" kind="statement">
# Riemann's criterion for integrability

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded.
Then $f$ is Riemann integrable on $[a, b]$ if and only if the following holds.
For each $\varepsilon > 0$ there is a partition $P$ of $[a, b]$ with $U(f, P) - L(f, P) < \varepsilon$.

## Remarks

Here $U$ and $L$ are the upper and lower [[Darboux sums|0QRQVH0CRW]].
The condition says that the upper and lower Darboux integrals are equal. So it follows from the [[equality of Darboux and Riemann integrability|021XJAQVDC]].
This is the usual tool to prove that [[continuous|06SESV7GQY]] and [[monotone|04YKAYDQPW]] functions are integrable.

## Explanations

### L1 human

A function is integrable when we can trap its graph between step functions whose areas differ by as little as we like.
</document>

<document source="https://tamnd.github.io/math2/node/017E5PRS1Q.md" tag="017E5PRS1Q" kind="concept">
# Platonic solid

## Summary

A Platonic solid is a convex polyhedron whose faces are congruent regular polygons, with the same number of faces at each vertex. There are exactly five: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.
</document>

<document source="https://tamnd.github.io/math2/node/017SRZ7NE9.md" tag="017SRZ7NE9" kind="statement">
# Dyck words are counted by Catalan numbers

## Statement

Let $n$ be a natural number.
Then the number of [[Dyck words|0R9JR0465V]] of semilength $n$ is the [[Catalan number|092DRBFKMG]] $C_n$.

## Remarks

For $n = 3$ the $5$ Dyck words are $UUUDDD$, $UUDUDD$, $UUDDUD$, $UDUUDD$ and $UDUDUD$.
Proof idea: a nonempty Dyck word has a unique form $U w_1 D w_2$, where $w_1$ and $w_2$ are Dyck words. This gives the [[Catalan recurrence|0Y1MG9R5VN]].

## Explanations

### L1 human

Each nonempty balanced bracket string starts with an opening bracket, and that bracket has a matching closing bracket.
The inside and the part after the match are smaller balanced strings, so the count satisfies the Catalan recurrence.
</document>

<document source="https://tamnd.github.io/math2/node/01826KMG2H.md" tag="01826KMG2H" kind="concept">
# Prime number

## Summary

A prime number is a natural number greater than $1$ whose only positive divisors are $1$ and itself.
The [[definition of a prime number|0SN0MAEKYX]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/018GTRS7A4.md" tag="018GTRS7A4" kind="concept">
# Vector-valued function

## Summary

A vector-valued function is a [[function|0EPAT51DXQ]] whose values are vectors, for example $f \colon D \to \mathbb{R}^m$ with $D \subseteq \mathbb{R}^n$ and $m \ge 2$.
Such a function has $m$ component functions $f_1, \dots, f_m \colon D \to \mathbb{R}$, with $f(x) = (f_1(x), \dots, f_m(x))$ for each $x \in D$.
An example is $r(t) = (\cos t, \sin t, t)$ for $t \in \mathbb{R}$, whose values trace a helix in the [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^3$.

## Remarks

Limits, continuity and derivatives of a vector-valued function into $\mathbb{R}^m$ can be computed one component at a time.
When the domain is an interval of $\mathbb{R}$, the function is a [[parametrised curve|0G7WM8DX4P]].

## Explanations

### L1 human

A vector-valued function gives an arrow, or a point, for each input, not a single number.
If $t$ is time, then $r(t)$ can be the position of a moving particle in space.
</document>

<document source="https://tamnd.github.io/math2/node/018W3V3TCT.md" tag="018W3V3TCT" kind="definition">
# Finite-dimensional vector space

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
The space $V$ is *finite-dimensional* when it has a finite [[spanning set|088RYKAWK9]].
Otherwise $V$ is *infinite-dimensional*.

## Remarks

A finite-dimensional space has a finite basis, because [[every spanning set contains a basis|06MRNP4BNT]].
The spaces $F^n$ and $M_{m \times n}(F)$ are finite-dimensional.
The polynomial ring $F[x]$ is infinite-dimensional: finitely many polynomials have a bounded degree, so they do not span $F[x]$.
Each [[subspace|00YC1D652K]] of a finite-dimensional space is finite-dimensional.

## Explanations

### L1 human

A space is finite-dimensional when finitely many vectors are enough to build every vector.
The plane needs two vectors, and ordinary space needs three.
The space of all polynomials needs infinitely many, because the degrees have no bound.
</document>

<document source="https://tamnd.github.io/math2/node/0194DY35BG.md" tag="0194DY35BG" kind="statement">
# Rolle's theorem

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$, with $f(a) = f(b)$.
Then there is $c \in (a, b)$ with $f'(c) = 0$.

## Remarks

By the [[extreme value theorem|0WS1A8E2P9]], $f$ has a maximum and a minimum on $[a, b]$. If both occur at the endpoints, $f$ is constant. Otherwise one occurs at an interior point $c$, and [[Fermat's theorem|0PCVBRV9VJ]] gives $f'(c) = 0$.
Differentiability inside the interval is needed. The function $|x|$ on $[-1, 1]$ has no point with derivative $0$.
Mathlib states it as `exists_deriv_eq_zero`.

## History

Michel Rolle proved a version for polynomials in 1691. Cauchy gave a proof of the general theorem in 1823.

## Explanations

### L1 human

A smooth graph that starts and ends at the same height must turn around somewhere.
At the turning point, the tangent is horizontal.
</document>

<document source="https://tamnd.github.io/math2/node/01991NCP7H.md" tag="01991NCP7H" kind="statement">
# Residue at a simple pole

## Statement

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on $0 < |z - c| < r$.
Assume that $f$ has a simple pole at $c$, that is, a [[pole of order|0YK27TJ4WF]] $1$.
Then
$$\operatorname{Res}(f, c) = \lim_{z \to c} (z - c) f(z),$$
where $\operatorname{Res}(f, c)$ is the [[residue|0JFGKKWWFK]] of $f$ at $c$.
In particular, let $f = g / h$, where $g$ and $h$ are holomorphic near $c$, $g(c) \neq 0$, $h(c) = 0$ and $h'(c) \neq 0$. Then $\operatorname{Res}(f, c) = \frac{g(c)}{h'(c)}$.

## Remarks

Proof: near $c$, $f(z) = \frac{a_{-1}}{z - c} + a_0 + a_1 (z - c) + \cdots$, so $(z - c) f(z) \to a_{-1}$.
For example, $\operatorname{Res}\left(\frac{1}{z^2 + 1}, i\right) = \frac{1}{2i}$, with $g = 1$ and $h = z^2 + 1$.
At a pole of order $m$, the formula is $\operatorname{Res}(f, c) = \frac{1}{(m-1)!} \lim_{z \to c} \frac{d^{m-1}}{dz^{m-1}} \left( (z - c)^m f(z) \right)$.

## Explanations

### L1 human

At a simple pole, $f$ looks like $\frac{a}{z - c}$ plus something harmless. Multiplying by $z - c$ and letting $z \to c$ gives $a$, the residue.
For example, the residue of $\frac{1}{z}$ at $0$ is $1$, and the residue of $\frac{1}{z - 2}$ at $2$ is also $1$.
</document>

<document source="https://tamnd.github.io/math2/node/019H1Z6419.md" tag="019H1Z6419" kind="statement">
# Derivative of the complex exponential

## Statement

For each $z \in \mathbb{C}$, the [[complex exponential|054VXHMCA3]] is [[complex differentiable|0D1C9Y1EXA]] at $z$, and
$$\exp'(z) = \exp z.$$

## Remarks

Proof: differentiate the series $\sum_{n \ge 0} z^n / n!$ term by term. The derivative of $z^n / n!$ is $z^{n-1} / (n-1)!$ for $n \ge 1$, so the series of derivatives is the same series.
With the chain rule, the derivative of $e^{a z}$ is $a e^{a z}$ for each constant $a \in \mathbb{C}$.
It follows that $\exp$ is [[entire|0G2YJN0H5S]].

## Explanations

### L1 human

As in the real case, the exponential function is its own derivative.
You can see it in the power series: differentiating each term gives the term before it, so the whole series does not change.
</document>

<document source="https://tamnd.github.io/math2/node/019MA1VTR2.md" tag="019MA1VTR2" kind="concept">
# Root of a polynomial

## Summary

A root of a polynomial $f$ over a ring $R$ is an element $c$ of $R$ with $f(c) = 0$.
The [[definition of a root|00EHX6FWC0]] states this precisely.
</document>

<document source="https://tamnd.github.io/math2/node/019NFGFPR5.md" tag="019NFGFPR5" kind="definition">
# Unit of a ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
An element $u \in R$ is a unit of $R$ if there is an element $v \in R$ such that $u \cdot v = v \cdot u = 1$.
The element $v$ is unique. It is the inverse of $u$, with the notation $u^{-1}$.

## Remarks

The units of $\mathbb{Z}$ are $1$ and $-1$.
In a [[field|0QA8CHH8MG]], every nonzero element is a unit.
A real $n \times n$ matrix is a unit of the matrix ring if and only if its determinant is not zero.
The units of $R$ form the [[group of units|05THZKR9AN]] of $R$.

## Explanations

### L1 human

A unit is an element that you can divide by.
In the integers, you can divide by $1$ and $-1$ and stay in the integers, but you cannot divide by $2$.
So $1$ and $-1$ are the only units of $\mathbb{Z}$.
</document>

<document source="https://tamnd.github.io/math2/node/019QZ6RRVH.md" tag="019QZ6RRVH" kind="statement">
# Integral test

## Statement

Let $N$ be an integer and let $f \colon [N, \infty) \to \mathbb{R}$ be a decreasing function with $f(x) \ge 0$ for all $x \ge N$.
Then the series $\sum_{n=N}^{\infty} f(n)$ converges if and only if the [[improper integral|0VCTBJMEJF]] $\int_N^{\infty} f(x)\, dx$ converges.

## Remarks

A decreasing function is Riemann integrable on each bounded interval, so the integrals $\int_N^b f(x)\, dx$ exist.
The proof uses the [[comparison of a series and an integral|069HKV4X1E]]. When both converge,
$$\int_N^{\infty} f(x)\, dx \le \sum_{n=N}^{\infty} f(n) \le f(N) + \int_N^{\infty} f(x)\, dx.$$
With $f(x) = 1/x^p$ it gives the [[p-series test|0ZNSSK8ZK1]].
Mathlib states the convergence direction for an antitone function, in `AntitoneOn.summable_of_integrableOn_Ioi`.

## Explanations

### L1 human

Draw rectangles of width $1$ and height $f(n)$ under or over the graph of $f$.
The total area of the rectangles is the sum of the series, and it stays within $f(N)$ of the area under the graph.
</document>

<document source="https://tamnd.github.io/math2/node/019VSXSFMQ.md" tag="019VSXSFMQ" kind="definition">
# Maximal solution

## Definition

Let $D$ be an open subset of $\mathbb{R} \times \mathbb{R}^n$ and let $f : D \to \mathbb{R}^n$ be continuous.
Let $I$ be an open interval and let $\gamma : I \to \mathbb{R}^n$ be a [[solution|04SW7WECS8]] of $x' = f(t, x)$ whose graph lies in $D$.
The solution $\gamma$ is maximal if there is no solution $\tilde{\gamma} : \tilde{I} \to \mathbb{R}^n$ on an open interval $\tilde{I}$ with $I \subsetneq \tilde{I}$ and $\tilde{\gamma}(t) = \gamma(t)$ for each $t \in I$.
The interval $I$ of a maximal solution is its maximal interval of existence.

## Remarks

If $f$ is locally Lipschitz in $x$, each [[initial value problem|075VBY4K4X]] $x(t_0) = x_0$ with $(t_0, x_0) \in D$ has exactly one maximal solution. Each other solution of the problem is a restriction of it.
The maximal interval $(t_-, t_+)$ can depend on the initial value, and it can be smaller than $\mathbb{R}$.
If $t_+ < \infty$, then $(t, \gamma(t))$ leaves every compact subset of $D$ as $t \to t_+$. For $D = \mathbb{R} \times \mathbb{R}^n$, this means $\|\gamma(t)\| \to \infty$. See [[blow-up in finite time|0E4GXX7SJP]].

## Explanations

### L1 human

A maximal solution is a solution that we cannot continue any further.
Its interval ends only where the solution runs away to infinity or to the edge of the domain of the field.
</document>

<document source="https://tamnd.github.io/math2/node/01A0MK1730.md" tag="01A0MK1730" kind="counterexample">
# Weierstrass function

## Counterexample

Let $0 < a < 1$, and let $b$ be a positive odd integer with $a b > 1 + \frac{3}{2} \pi$. The Weierstrass function is
$$W(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x).$$
The function $W$ is continuous on $\mathbb{R}$, but it is not [[differentiable|0A61B60B5R]] at any point.
So a continuous function need not have a derivative anywhere.

## Remarks

The terms satisfy $|a^n \cos(b^n \pi x)| \le a^n$. The [[M-test|0NX5NWAFC9]] and the [[continuity of uniform sums|0R0MYT0BJD]] give continuity.
G. H. Hardy later showed that the conclusion holds under the weaker condition $a b \ge 1$.
Karl Weierstrass presented the function to the Prussian Academy of Sciences on 18 July 1872.
Bernard Bolzano had constructed such a function around 1831, but it was not published until 1922.

## Explanations

### L1 human

The graph of the Weierstrass function wiggles at every scale.
Zooming in never shows a straight piece, so there is no tangent line at any point.
</document>

<document source="https://tamnd.github.io/math2/node/01A2EGXJEM.md" tag="01A2EGXJEM" kind="statement">
# First derivative test

## Statement

Let $f$ be a real function on an open interval $(c - \delta, c + \delta)$ with $\delta > 0$, continuous at $c$ and differentiable at each $x \ne c$.
If $f'(x) \le 0$ for $c - \delta < x < c$ and $f'(x) \ge 0$ for $c < x < c + \delta$, then $f$ has a [[local minimum|0NS8BP96QX]] at $c$.
If the inequalities are reversed, then $f$ has a local maximum at $c$.

## Remarks

The function is [[decreasing on the left and increasing on the right|026BJBQNGJ]] of $c$.
The test needs no derivative at $c$. It works for $|x|$ at $0$.
Mathlib states the minimum case as `isLocalMin_of_deriv`.

## Explanations

### L1 human

If the graph goes down until $c$ and then goes up, the point $c$ is the bottom of a valley.
</document>

<document source="https://tamnd.github.io/math2/node/01A8JGR4JM.md" tag="01A8JGR4JM" kind="statement">
# cos pi = -1

## Statement

$$\cos \pi = -1, \qquad \sin \pi = 0.$$

## Remarks

By the [[definition of $\pi$|0C1VYVNV2N]], $\cos(\pi/2) = 0$. The [[Pythagorean identity|03H7WZ4T3K]] gives $\sin(\pi/2) = 1$, since $\sin > 0$ on $(0, 2]$.
The [[double angle formulas|028XEQCE88]] then give $\cos \pi = 2 \cos^2(\pi/2) - 1 = -1$ and $\sin \pi = 2 \sin(\pi/2) \cos(\pi/2) = 0$.
Together with $\sin \pi = 0$, it gives Euler's identity $e^{i \pi} = -1$ in complex form. Mathlib states the first claim as `Real.cos_pi`.

## Explanations

### L1 human

Half a turn around the unit circle takes the point $(1, 0)$ to the point $(-1, 0)$.
So $\cos \pi = -1$ and $\sin \pi = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/01AAYNSBN6.md" tag="01AAYNSBN6" kind="statement">
# Distance from a point to a line

## Statement

Let $a$, $b$ and $c$ be real numbers with $(a, b) \neq (0, 0)$, and let $L = \{(x, y) \in \mathbb{R}^2 : a x + b y + c = 0\}$ be a [[line|0MP9BCF6DP]] in the plane.
Let $P = (x_0, y_0)$ be a point of $\mathbb{R}^2$. The distance from $P$ to $L$ is the smallest [[Euclidean distance|0NPPGHQNXQ]] $d(P, Q)$ for $Q \in L$. Then
$$\operatorname{dist}(P, L) = \frac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}.$$

## Remarks

The smallest distance is reached at exactly one point $Q$ of $L$, the foot of the perpendicular from $P$. It is $Q = P - \frac{a x_0 + b y_0 + c}{a^2 + b^2} (a, b)$.
In $\mathbb{R}^n$, let $L$ be the line through the point $A$ with the direction vector $v$. The distance from $P$ to $L$ is the length of the part of $P - A$ that is perpendicular to $v$: the length of $(P - A) - \frac{(P - A) \cdot v}{v \cdot v} v$.
In $\mathbb{R}^3$ this distance is also $\|(P - A) \times v\| / \|v\|$.

## Explanations

### L1 human

The shortest way from a point to a line is the path that meets the line at a right angle. The formula measures this path. For the line $y = 0$, thus $a = 0$, $b = 1$, $c = 0$, it gives $|y_0|$: the height of the point above or below the horizontal axis.
</document>

<document source="https://tamnd.github.io/math2/node/01AHG4G9RT.md" tag="01AHG4G9RT" kind="concept">
# Preimage of a set

## Summary

The preimage of a set under a function is the set of all inputs whose value is in that set.
</document>

<document source="https://tamnd.github.io/math2/node/01APZ8GDQH.md" tag="01APZ8GDQH" kind="statement">
# Chapman–Kolmogorov equation

## Statement

Let $(X_n)_{n \ge 0}$ be a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable state space $S$. For $n \ge 0$, let $p^{(n)}_{ij} = P(X_{m+n} = j \mid X_m = i)$ be the $n$-step transition probabilities.
Then for all $m, n \ge 0$ and $i, j \in S$,
$$p^{(m+n)}_{ij} = \sum_{k \in S} p^{(m)}_{ik} \, p^{(n)}_{kj}.$$
In matrix form, $P^{(m+n)} = P^{(m)} P^{(n)}$, so $P^{(n)} = P^n$ for the [[transition matrix|0F1KBVY161]] $P$.

## Remarks

Proof: condition on the state $X_m = k$ at the intermediate time, with the [[law of total probability|0E5G5FMT2R]], and use the [[Markov property|0HB8SXHPFA]].
The equation says that to go from $i$ to $j$ in $m + n$ steps, the chain must be at some state $k$ after $m$ steps.
For a Markov process with transition kernels $p_{s,t}(x, dy)$, the equation reads $p_{s,u}(x, A) = \int p_{s,t}(x, dy) \, p_{t,u}(y, A)$ for $s \le t \le u$.

## History

The equation is named after Sydney Chapman and Andrey Kolmogorov.

## Explanations

### L1 human

The chances for several steps come from multiplying the one-step transition matrix with itself.
For example, the chance of rain in two days, given sun today, sums over the two possible states of tomorrow.
</document>

<document source="https://tamnd.github.io/math2/node/01AXG9KZQF.md" tag="01AXG9KZQF" kind="definition">
# Block matrix

## Definition

Let $R$ be a ring.
Let $m = m_1 + \cdots + m_p$ and $n = n_1 + \cdots + n_q$ with positive integers $m_i$ and $n_j$.
For each $i \le p$ and $j \le q$, let $A_{ij}$ be an $m_i \times n_j$ [[matrix|0VD9ZSWWJG]] over $R$.
The *block matrix* with blocks $A_{ij}$ is the $m \times n$ matrix
$$A = \begin{pmatrix} A_{11} & \cdots & A_{1q} \\ \vdots & \ddots & \vdots \\ A_{p1} & \cdots & A_{pq} \end{pmatrix},$$
in which $A_{ij}$ fills rows $m_1 + \cdots + m_{i-1} + 1$ to $m_1 + \cdots + m_i$ and columns $n_1 + \cdots + n_{j-1} + 1$ to $n_1 + \cdots + n_j$.

## Remarks

If the column partition of $A$ matches the row partition of $B$, then $AB$ is computed block by block: $(AB)_{ik} = \sum_j A_{ij} B_{jk}$.
In Mathlib, `Matrix.fromBlocks` makes the $2 \times 2$ block matrix $\begin{pmatrix} A & B \\ C & D \end{pmatrix}$.
The [[determinant of a block triangular matrix|0J9TVTNB4K]] is the product of the determinants of its diagonal blocks.

## Explanations

### L1 human

A block matrix is a large matrix cut into smaller rectangles.
We can treat each rectangle like a single entry.
Then we add and multiply block matrices with the usual rules, as long as the sizes fit.
</document>

<document source="https://tamnd.github.io/math2/node/01BD2TFFP4.md" tag="01BD2TFFP4" kind="definition">
# Bias of an estimator

## Definition

Let $\hat{\theta}$ be an [[estimator|0TTNRP5PFQ]] of a real parameter $\theta$, with $E_\theta[|\hat{\theta}|] < \infty$ for each $\theta$. Here $E_\theta$ is the expectation when $\theta$ is the true parameter.
The bias of $\hat{\theta}$ at $\theta$ is
$$\operatorname{Bias}_\theta(\hat{\theta}) = E_\theta[\hat{\theta}] - \theta.$$
The estimator is unbiased if $\operatorname{Bias}_\theta(\hat{\theta}) = 0$ for each $\theta$.

## Remarks

For an i.i.d. sample with mean $\mu$, the sample mean is an unbiased estimator of $\mu$.
The sample variance with divisor $n$ has bias $-\sigma^2/n$. With divisor $n - 1$, it is unbiased.
An unbiased estimator need not be good. The [[mean squared error|0V4VSCBXP3]] is variance plus squared bias, and a biased estimator can have a smaller mean squared error.
Unbiasedness is not preserved by nonlinear maps. For example, the sample standard deviation $S_n$ is a biased estimator of $\sigma$, even though $S_n^2$ is unbiased for $\sigma^2$.

## Explanations

### L1 human

An estimator is unbiased if it is right on average, over many repetitions of the experiment.
Bias is a systematic error: a scale that always shows $1$ kg too much has a bias of $1$ kg.
</document>

<document source="https://tamnd.github.io/math2/node/01BX15HSRR.md" tag="01BX15HSRR" kind="statement">
# Well-ordering principle

## Statement

Every nonempty set of natural numbers has a least element.
That is, if $A \subseteq \mathbb{N}$ and $A \neq \emptyset$, then there is an $a$ in $A$ with $a \le x$ for each $x$ in $A$.

## Remarks

The least element is unique, because the [[order|0Y9H52BT03]] on $\mathbb{N}$ is antisymmetric.
The principle follows from [[strong induction|0KMVFSHFGZ]], and conversely it implies the [[principle of mathematical induction|0ASJVEFA9J]].
The set of the integers does not have this property: the set of negative integers has no least element.

## Explanations

### L1 human

If some natural numbers have a property, then there is a smallest one with that property.
For example, among the natural numbers that are larger than $10$ and even, the smallest is $12$.
Proofs often use this: they take the smallest counterexample and show that it cannot exist.
</document>

<document source="https://tamnd.github.io/math2/node/01CGSJ0GNQ.md" tag="01CGSJ0GNQ" kind="concept">
# Well-order

## Summary

A well-order is a total order in which each nonempty subset has a least element.
</document>

<document source="https://tamnd.github.io/math2/node/01CJRJKDAA.md" tag="01CJRJKDAA" kind="statement">
# The complex numbers cannot be ordered

## Statement

There is no total order $\le$ on $\mathbb{C}$ that makes $\mathbb{C}$, with its usual addition and multiplication, an [[ordered field|010JCV9J4E]].

## Remarks

Proof: in an ordered field, each square is nonnegative, and $1 = 1^2 > 0$. Then $-1 < 0$. But $-1 = i^2$ is a square, by [[i squared is minus one|07KBQCG9KV]], a contradiction.
There are total orders on the set $\mathbb{C}$, for example the lexicographic order. They are not compatible with the field operations.
Thus inequalities such as $z < w$ make sense only for real numbers $z$ and $w$.

## Explanations

### L1 human

You cannot say which of $i$ and $0$ is larger in a way that keeps the usual rules of inequalities.
If $i > 0$, then $i \cdot i > 0$, so $-1 > 0$. If $i < 0$, then $-i > 0$ and $(-i)(-i) > 0$, so again $-1 > 0$.
Both cases are impossible.
</document>

<document source="https://tamnd.github.io/math2/node/01CS8Q2APC.md" tag="01CS8Q2APC" kind="definition">
# Characteristic subgroup

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
The subgroup $H$ is characteristic in $G$ when $\alpha(H) = H$ for each [[automorphism|0K1SCPBNKJ]] $\alpha$ of $G$.

## Remarks

It is equivalent to ask $\alpha(H) \subseteq H$ for each automorphism $\alpha$, because then also $\alpha^{-1}(H) \subseteq H$.
The [[centre|0DZ93493VN]] and the [[commutator subgroup|0ASGSTXSBW]] of a group are characteristic.
A characteristic subgroup of a normal subgroup of $G$ is normal in $G$.

## Explanations

### L1 human

A characteristic subgroup is a subgroup that no symmetry of the group can move.
Usually such a subgroup has a description that uses only the group operation, like "the elements that commute with everything".
Any renaming of the elements that keeps the products correct must keep such a subgroup in place.
</document>

<document source="https://tamnd.github.io/math2/node/01CYW8S04D.md" tag="01CYW8S04D" kind="definition">
# Totally disconnected space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is totally disconnected when each [[connected subset|0R37DFJKFW]] of $X$ has at most one point.

## Remarks

Equivalently, each connected component of $X$ is a single point.
Each [[discrete space|0S2TEHX4E1]] is totally disconnected. So are $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$, as subspaces of $\mathbb{R}$.
The [[Cantor set|0BXNDZFQME]] is totally disconnected, although it is compact and has no isolated points.

## Explanations

### L1 human

A totally disconnected space falls apart into single points: no subset with two or more points is in one piece.
This does not mean that the space is discrete. The rationals are a good example.
</document>

<document source="https://tamnd.github.io/math2/node/01DSNP0B4R.md" tag="01DSNP0B4R" kind="definition">
# Periodic orbit

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz, and consider the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.
A [[solution|04SW7WECS8]] $\gamma : \mathbb{R} \to U$ is periodic if it is not constant and there is a real number $T > 0$ with $\gamma(t + T) = \gamma(t)$ for each $t \in \mathbb{R}$.
The smallest such $T$ is the period of $\gamma$.
A periodic orbit, or closed orbit, is the [[trajectory|0BVT6VSB3S]] $\gamma(\mathbb{R})$ of a periodic solution $\gamma$.

## Remarks

A non-constant periodic solution has a smallest period $T > 0$, because $\gamma$ is continuous and not constant.
A periodic orbit is a simple closed curve in $U$.
Let a maximal solution $\gamma$ that is not constant satisfy $\gamma(t_1 + T) = \gamma(t_1)$ for one $t_1$ and one $T > 0$. Then, by [[uniqueness|025PKW8AMR]], $\gamma$ is defined on $\mathbb{R}$ and is periodic.
For the [[harmonic oscillator|0YG0ZJPTWG]] $x' = y$, $y' = -x$, each circle $x^2 + y^2 = r^2$ with $r > 0$ is a periodic orbit of period $2\pi$.

## Explanations

### L1 human

A periodic orbit is a motion that repeats itself exactly after a fixed time.
In the phase space it is a closed loop that the state goes around again and again.
</document>

<document source="https://tamnd.github.io/math2/node/01DSVCRF6X.md" tag="01DSVCRF6X" kind="definition">
# Ramsey number

## Definition

Let $r$ and $s$ be positive integers.
The Ramsey number $R(r, s)$ is the least positive integer $n$ with this property.
Colour each edge of the [[complete graph|03WKB2H474]] $K_n$ blue or red, in any way.
Then there is a blue [[clique|0G52BPNPKG]] with $r$ vertices or a red clique with $s$ vertices.

## Remarks

Such an $n$ exists by [[Ramsey's theorem|00TQV3F5S4]].
$R(r, s) = R(s, r)$ and $R(r, 2) = r$. The [[theorem on friends and strangers|0NYM0ZGAK9]] gives $R(3, 3) = 6$.
With more colours there are similar numbers, for example $R(3, 3, 3) = 17$.

## Explanations

### L1 human

The Ramsey number $R(r, s)$ is the size of a party that is large enough for one of two things to happen.
There are $r$ people who all know each other, or $s$ people who are all strangers to each other.
</document>

<document source="https://tamnd.github.io/math2/node/01E7SWYJX3.md" tag="01E7SWYJX3" kind="statement">
# Normality via left and right cosets

## Statement

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $N$ is [[normal|0CB0DX5BVV]] in $G$ if and only if $gN = Ng$ for each $g \in G$.

## Remarks

Here $gN$ is a [[left coset|03DEH43F4H]] and $Ng$ is a [[right coset|0NE1DK7H4T]].
The equation $gN = Ng$ is equivalent to $gNg^{-1} = N$, by multiplying on the right by $g^{-1}$.
The condition does not say that $gn = ng$ for each $n \in N$; it says only that the two sets are equal.

## Explanations

### L1 human

For a normal subgroup, it does not matter if you multiply by $g$ on the left or on the right: you get the same coset.
Thus there is only one kind of coset, and we can just say "the cosets of $N$".
This is what makes the quotient group possible.
</document>

<document source="https://tamnd.github.io/math2/node/01E9MNJYEX.md" tag="01E9MNJYEX" kind="concept">
# Public-key cryptography

## Summary

Public-key cryptography is a kind of cryptography in which each user has a pair of keys: a public key and a private key.
Anyone can use the public key to encrypt a message or to check a signature. Only the owner of the private key can decrypt the message or make the signature.
Its security depends on problems that are believed to be hard to solve, such as integer factorization or the [[discrete logarithm|09B4X51J09]].

## Remarks

Before the mid-1970s, all ciphers were symmetric: both parties used the same secret key, which they had to exchange in a secure way.
The [[RSA cryptosystem|0HM584C465]] and the [[Diffie–Hellman key exchange|01TKKESHSB]] are two early examples.
Whitfield Diffie and Martin Hellman published the idea in 1976.

## Explanations

### L1 human

Think of a mailbox with a slot: anyone can drop a letter in, but only the owner has the key to open it.
The public key is the slot, and the private key is the key of the mailbox.
</document>

<document source="https://tamnd.github.io/math2/node/01EK923SKP.md" tag="01EK923SKP" kind="definition">
# Inner product space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$.
An *inner product space* over $\mathbb{K}$ is a vector space $V$ over $\mathbb{K}$ together with an [[inner product|0NB0AWGJBW]] $\langle \cdot, \cdot \rangle$ on $V$.

## Remarks

A finite-dimensional real inner product space is a *Euclidean space*.
Each inner product space has the [[induced norm|0MFJJV97RJ]] $\norm{x} = \sqrt{\langle x, x \rangle}$.
An inner product space that is complete for this norm is a *Hilbert space*.
Each finite-dimensional inner product space is complete.

## Explanations

### L1 human

An inner product space is a vector space where we can measure lengths and angles.
The space $\R^n$ with the dot product is the main example.
Much of plane geometry, such as the Pythagorean theorem, works in every inner product space.
</document>

<document source="https://tamnd.github.io/math2/node/01EQPZ6RYZ.md" tag="01EQPZ6RYZ" kind="concept">
# Law of a random variable

## Summary

The law of a random variable $X$ is the probability distribution of its values: it gives each set $B$ of values the probability $P(X \in B)$.
The node [[law of a random variable|0TST4MHA3A]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/01EXSMK864.md" tag="01EXSMK864" kind="statement">
# Determinant of a scalar multiple

## Statement

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$, and let $c \in K$.
Then
$$\det(cA) = c^n \det A,$$
where $cA$ is the [[scalar multiple|09V6DDACXR]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

Each of the $n$ rows is multiplied by $c$, and each row contributes one factor $c$.
For example, $\det(-A) = (-1)^n \det A$.
So an $n \times n$ real [[skew-symmetric matrix|0PBY4E9GQF]] with $n$ odd has determinant $0$.

## Explanations

### L1 human

Scaling a matrix by $c$ stretches every edge of a box by $c$.
In $n$ dimensions, this multiplies the volume by $c^n$.
So doubling a $3 \times 3$ matrix multiplies its determinant by $8$.
</document>

<document source="https://tamnd.github.io/math2/node/01EY0EM7RP.md" tag="01EY0EM7RP" kind="statement">
# Extensions of solvable groups are solvable

## Statement

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
If $N$ and $G/N$ are [[solvable|0FXJ7YFMMX]], then $G$ is solvable.

## Remarks

If $(G/N)^{(k)}$ is trivial, then $G^{(k)} \subseteq N$, so $G^{(k+m)} \subseteq N^{(m)}$ for each $m$.
Choose $m$ with $N^{(m)} = \{e\}$; then $G^{(k+m)} = \{e\}$.
With [[subgroups|05Y0F9M0TS]] and [[quotients|027ZQS04SJ]], this gives: $G$ is solvable if and only if $N$ and $G/N$ are solvable.
In particular, direct products and semidirect products of solvable groups are solvable.

## Explanations

### L1 human

If the bottom part and the top part of a group can both be built from abelian layers, then so can the whole group.
You stack the layers of the top part on the layers of the bottom part.
For example, $S_4$ is built from $V$, $A_4/V$ and $S_4/A_4$.
</document>

<document source="https://tamnd.github.io/math2/node/01EY3VR7ZD.md" tag="01EY3VR7ZD" kind="statement">
# Urysohn's lemma

## Statement

Let $X$ be a [[normal space|0AJ646N339]] and let $A, B \subseteq X$ be disjoint [[closed sets|0G5AV9Q4RR]].
Then there is a [[continuous map|0SY9ECQ19Q]] $f \colon X \to [0, 1]$ with $f(a) = 0$ for all $a \in A$ and $f(b) = 1$ for all $b \in B$.

## Remarks

The converse also holds: if such a function exists for all disjoint closed $A$ and $B$, then $X$ is normal. Take $U = f^{-1}([0, 1/2))$ and $V = f^{-1}((1/2, 1])$.
The proof builds open sets $U_r$ for the dyadic rationals $r \in [0, 1]$, nested by the order of $r$. Then $f(x)$ is the infimum of the $r$ with $x \in U_r$.
In a metric space, the function $f(x) = d(x, A) / (d(x, A) + d(x, B))$ works.
The [[Tietze extension theorem|0X52TNMTCJ]] extends this result to all continuous functions on closed sets.

## Explanations

### L1 human

In a normal space, a continuous function can separate two disjoint closed sets: it is $0$ on one set and $1$ on the other.
So normal spaces have many continuous real functions.
</document>

<document source="https://tamnd.github.io/math2/node/01FPXMTBX1.md" tag="01FPXMTBX1" kind="statement">
# Composition of injections is injective

## Statement

Let $f \colon A \to B$ and $g \colon B \to C$ be [[injective functions|0PVG1CRAJN]].
Then the [[composition|07HF9YCWPP]] $g \circ f \colon A \to C$ is injective.

## Remarks

Proof: if $g(f(a)) = g(f(a'))$, then $f(a) = f(a')$ because $g$ is injective, and $a = a'$ because $f$ is injective.
A partial converse holds: if $g \circ f$ is injective, then $f$ is injective. The function $g$ need not be injective.

## Explanations

### L1 human

Think of $f$ and $g$ as machines, and of $g \circ f$ as the two machines in a chain.
If neither machine merges two inputs, the chain does not merge two inputs either.
</document>

<document source="https://tamnd.github.io/math2/node/01G1WY1GVX.md" tag="01G1WY1GVX" kind="definition">
# Pythagorean triple

## Definition

A Pythagorean triple is a triple $(a, b, c)$ of positive integers such that $a^2 + b^2 = c^2$.
The triple is primitive if $\gcd(a, b, c) = 1$, with $\gcd$ the [[greatest common divisor|02WF878BH1]].

## Remarks

The smallest examples are $(3, 4, 5)$ and $(5, 12, 13)$.
If $(a, b, c)$ is a Pythagorean triple, then so is $(ka, kb, kc)$ for each positive integer $k$.
The [[classification of Pythagorean triples|0KF3J2KAER]] lists all of them.
In a primitive triple, $a$, $b$ and $c$ are pairwise coprime, and exactly one of $a$ and $b$ is even.

## Explanations

### L1 human

A Pythagorean triple gives a right triangle whose three sides are whole numbers.
The triangle with sides $3$, $4$ and $5$ is the best-known example, since $9 + 16 = 25$.
</document>

<document source="https://tamnd.github.io/math2/node/01GBHQ5ABF.md" tag="01GBHQ5ABF" kind="concept">
# Periodic continued fraction

## Summary

A periodic continued fraction is an infinite simple continued fraction whose terms repeat a fixed block from some point on.
The [[definition of a periodic continued fraction|0XKJPFS2W0]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/01GTP8M5RG.md" tag="01GTP8M5RG" kind="definition">
# Order of cardinals

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The cardinality of $A$ is at most the cardinality of $B$, written $|A| \le |B|$, when there is an [[injective function|0PVG1CRAJN]] $f \colon A \to B$.
Write $|A| < |B|$ when $|A| \le |B|$ and $|A| \neq |B|$.

## Remarks

The relation does not depend on the choice of $A$ and $B$ in their classes of [[equinumerous|08EJHHMB5R]] sets.
It is reflexive and transitive. It is antisymmetric by the [[Schröder–Bernstein theorem|0A9V6RKQG6]].
With the axiom of choice, any two cardinals are comparable. See [[comparability of cardinals|0YBHSJMN3J]].

## Explanations

### L1 human

$|A| \le |B|$ means that $A$ fits into $B$ without two elements landing in the same place.
For example, $|\mathbb{N}| \le |\mathbb{R}|$, because $\mathbb{N}$ is a subset of $\mathbb{R}$.
</document>

<document source="https://tamnd.github.io/math2/node/01H6QKWGDH.md" tag="01H6QKWGDH" kind="concept">
# Generalised eigenvector

## Summary

A generalised eigenvector of an endomorphism for $\lambda$ is a nonzero vector with a special property.
Some power of the map minus $\lambda$ times the identity sends it to zero.
</document>

<document source="https://tamnd.github.io/math2/node/01HR2AH0BX.md" tag="01HR2AH0BX" kind="definition">
# Divergent sequence

## Definition

A [[real sequence|0CTV783HYT]] $(a_n)$ is divergent if it is not a [[convergent sequence|0AZ3R52593]].
Thus, for each real number $L$, the sequence $(a_n)$ does not converge to $L$.

## Remarks

The sequence $a_n = (-1)^n$ is divergent, and it is bounded.
The sequence $a_n = n$ is divergent, and it [[tends to infinity|09AZCV6GMC]].
Some texts say that a sequence that tends to $+\infty$ diverges to $+\infty$. It is still divergent in the sense of this definition.

## Explanations

### L1 human

A divergent sequence has no limit.
Its terms can grow without bound, as in $1, 2, 3, \dots$, or they can jump between values, as in $1, -1, 1, -1, \dots$.
</document>

<document source="https://tamnd.github.io/math2/node/01HT2CH8BH.md" tag="01HT2CH8BH" kind="statement">
# Equivalence classes form a partition

## Statement

Let $\sim$ be an [[equivalence relation|0SGTMBEWRF]] on a set $A$.
Then the set $\{[a] \mid a \in A\}$ of [[equivalence classes|082RMRSY26]] is a [[partition|09CCE86A6D]] of $A$.

## Remarks

Each class $[a]$ is nonempty, because $a \in [a]$ by reflexivity.
If $[a]$ and $[b]$ have a common element, then $a \sim b$ by symmetry and transitivity, so $[a] = [b]$.
The left cosets of a subgroup of a [[group|G5S1M9]] form such a partition. This fact is the key step in [[Lagrange's theorem|3C53TD]].

## Explanations

### L1 human

The three properties of an equivalence relation are exactly what you need to sort elements into classes.
Reflexivity puts each element in some class.
Symmetry and transitivity stop two classes from overlapping.
</document>

<document source="https://tamnd.github.io/math2/node/01J7FZ65J0.md" tag="01J7FZ65J0" kind="concept">
# Maximal solution

## Summary

A maximal solution of an ordinary differential equation is a solution that is not the restriction of a solution on a larger interval.
Under a local Lipschitz condition, each initial value problem has exactly one maximal solution. See the [[definition|019VSXSFMQ]].
</document>

<document source="https://tamnd.github.io/math2/node/01J8AMV9RS.md" tag="01J8AMV9RS" kind="definition">
# Legendre symbol

## Definition

Let $p$ be an odd [[prime number|0SN0MAEKYX]] and let $a$ be an integer.
The Legendre symbol $\left(\frac{a}{p}\right)$ is the number
$$\left(\frac{a}{p}\right) = \begin{cases} 0 & \text{if } p \mid a, \\ 1 & \text{if } p \nmid a \text{ and } a \text{ is a quadratic residue modulo } p, \\ -1 & \text{if } a \text{ is a quadratic nonresidue modulo } p. \end{cases}$$
Here quadratic residue has the meaning of the [[definition of a quadratic residue|0SQDB87Z3T]].

## Remarks

The value depends only on the class of $a$ modulo $p$.
For example, $\left(\frac{2}{7}\right) = 1$ because $3^2 \equiv 2 \pmod 7$, and $\left(\frac{3}{7}\right) = -1$.
[[Euler's criterion|0M7DEDKSYB]] gives the value as $a^{(p-1)/2}$ modulo $p$.
The [[law of quadratic reciprocity|085MJS6907]] helps to compute the symbol quickly.

## Explanations

### L1 human

The Legendre symbol is a short way to say whether $a$ is a square modulo $p$.
The value $1$ means yes, the value $-1$ means no, and the value $0$ means that $a$ is $0$ modulo $p$.
</document>

<document source="https://tamnd.github.io/math2/node/01JADB9VW5.md" tag="01JADB9VW5" kind="statement">
# Vandermonde's identity

## Statement

Let $m$, $n$ and $r$ be natural numbers.
Then
$$\binom{m + n}{r} = \sum_{k=0}^{r} \binom{m}{k} \binom{n}{r - k}.$$

## Remarks

Combinatorial proof: take a set of $m$ men and $n$ women. An $r$-element committee has some number $k$ of men and $r - k$ women.
With $m = n = r$, and the [[symmetry of binomial coefficients|0R7JS3DP3W]], it gives $\binom{2n}{n} = \sum_{k=0}^{n} \binom{n}{k}^2$.

## Explanations

### L1 human

To choose $r$ people from two groups together, choose $k$ from the first group and $r - k$ from the second, for each possible $k$.
Then add the counts for all values of $k$.
</document>

<document source="https://tamnd.github.io/math2/node/01KJJ9A400.md" tag="01KJJ9A400" kind="concept">
# Regular polygon

## Summary

A regular polygon is a polygon with all sides of the same length and all angles of the same size. Its vertices lie on one circle. Examples are the equilateral triangle and the square.
</document>

<document source="https://tamnd.github.io/math2/node/01M5T0KDG7.md" tag="01M5T0KDG7" kind="definition">
# Multigraph

## Definition

A multigraph is a triple $G = (V, E, r)$ of a set $V$ of vertices, a set $E$ of edges, and a map $r$.
The map $r$ assigns to each edge $e \in E$ a set $r(e) \subseteq V$ of one or two vertices, the ends of $e$.
Two distinct edges may have the same ends. Such edges are parallel, or multiple, edges.
An edge with only one end is a loop.

## Remarks

Some authors do not allow loops in a multigraph, and they call a multigraph with loops a pseudograph.
A [[graph|0JD5C82QMB]] is a multigraph without loops and without parallel edges.
The [[Seven Bridges of Königsberg|0YCW781Q9D]] form a multigraph with $4$ vertices and $7$ edges.

## Explanations

### L1 human

A multigraph is like a graph, but two points can be joined by several lines.
A road map with two different roads between the same two towns is a multigraph.
</document>

<document source="https://tamnd.github.io/math2/node/01M6CW9VM6.md" tag="01M6CW9VM6" kind="statement">
# Triangle inequality for a metric

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]].
For all points $x, y, z \in X$,
$$d(x, z) \le d(x, y) + d(y, z).$$

## Remarks

This is one of the four axioms of a metric.
For $X = \mathbb{R}$ with $d(x, y) = |x - y|$, it is the [[triangle inequality|03HF5PMY5T]] for real numbers, with $a = x - y$ and $b = y - z$.
A consequence is the reverse form $|d(x, y) - d(x, z)| \le d(y, z)$ for all $x, y, z \in X$.

## Explanations

### L1 human

A direct path from $x$ to $z$ is never longer than a path that goes through a third point $y$.
In the plane, this says that one side of a triangle is at most the sum of the other two sides.
</document>

<document source="https://tamnd.github.io/math2/node/01MGYYBB9R.md" tag="01MGYYBB9R" kind="definition">
# Antisymmetric relation

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is antisymmetric when, for all $a, b \in A$, $a \mathrel{R} b$ and $b \mathrel{R} a$ imply $a = b$.

## Remarks

The relation $\le$ on $\mathbb{R}$ and the relation $\subseteq$ on sets are antisymmetric.
Divisibility is antisymmetric on $\mathbb{N}$ but not on $\mathbb{Z}$, because $1 \mid -1$ and $-1 \mid 1$.
Antisymmetric is not the opposite of symmetric: equality is both symmetric and antisymmetric.

## Explanations

### L1 human

If $a \le b$ and $b \le a$, then $a = b$.
So for $\le$, a two-way relation between two elements forces them to be the same.
</document>

<document source="https://tamnd.github.io/math2/node/01MKW4R283.md" tag="01MKW4R283" kind="definition">
# Infinite set

## Definition

A [[set|0SJ3RX9TXA]] $A$ is infinite when it is not [[finite|0BBTH3YSZA]], that is, when $A$ is not equinumerous with $\{1, \dots, n\}$ for any natural number $n$.

## Remarks

Examples are $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$.
With the axiom of choice, a set is infinite exactly when it is equinumerous with a proper subset of itself. This is the Dedekind definition.
Infinite sets can have different sizes. See [[Cantor's theorem|0WTEMGYV2C]].

## Explanations

### L1 human

An infinite set never runs out of elements when you count them.
The natural numbers are infinite: after each number there is a next one.
</document>

<document source="https://tamnd.github.io/math2/node/01MMVW8CKC.md" tag="01MMVW8CKC" kind="concept">
# Positive semidefinite matrix

## Summary

A positive semidefinite matrix is a Hermitian matrix $A$ with $x^* A x \ge 0$ for every vector $x$.
</document>

<document source="https://tamnd.github.io/math2/node/01MPZXYVYF.md" tag="01MPZXYVYF" kind="statement">
# Cauchy's integral theorem

## Statement

Let $U \subseteq \mathbb{C}$ be a simply connected open set. This means that $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path.
Let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $\gamma : [a, b] \to U$ be a closed path that is continuous and piecewise continuously differentiable.
Then the [[contour integral|00FWNP9TXY]] of $f$ along $\gamma$ is
$$\oint_\gamma f(z) \, dz = 0.$$

## Remarks

Open discs, open half-planes, $\mathbb{C}$ and each convex open set are simply connected. So the theorem holds on each of them.
A special case: if $f$ is holomorphic on an open set that contains the closed disc $|z - c| \le R$, then $\oint_{|z - c| = R} f(z) \, dz = 0$.
The hypothesis on $U$ is necessary. On $\mathbb{C} \setminus \{0\}$, the [[integral of 1/z|0SXZ68HV6F]] over the unit circle is $2 \pi i$.
It follows that the integral of $f$ along a path in $U$ depends only on the end points of the path.

## Explanations

### L1 human

The integral of a holomorphic function around a closed loop is zero, as long as the loop does not go around a hole of the domain.
It is like walking around a hill and coming back to the start: the total change of height is zero.
</document>

<document source="https://tamnd.github.io/math2/node/01N3AYAA6N.md" tag="01N3AYAA6N" kind="definition">
# Integers modulo n

## Definition

Let $n$ be a positive integer.
The integers modulo $n$ form the set $\mathbb{Z}/n\mathbb{Z} = \{ \overline{0}, \overline{1}, \dots, \overline{n - 1} \}$ of the [[residue classes|0QRV8FPK5M]] modulo $n$.
Addition and multiplication on $\mathbb{Z}/n\mathbb{Z}$ are $\overline{a} + \overline{b} = \overline{a + b}$ and $\overline{a} \cdot \overline{b} = \overline{a b}$.

## Remarks

The operations are well defined: the result does not depend on the choice of $a$ and $b$ in their classes. This follows from [[compatibility with addition|00FWBN7F0Y]] and [[compatibility with multiplication|072D5G4SK6]].
With these operations, $\mathbb{Z}/n\mathbb{Z}$ is a commutative ring with $n$ elements, with zero $\overline{0}$ and identity $\overline{1}$.
Other notations are $\mathbb{Z}_n$ and $\mathbb{Z}/(n)$. The Lean library Mathlib writes `ZMod n`, and it sets `ZMod 0` to be $\mathbb{Z}$.
The ring is a field if and only if $n$ is prime. See [[integers modulo a prime form a field|0R8YXRC9AJ]].

## Explanations

### L1 human

In $\mathbb{Z}/12\mathbb{Z}$ you compute as on a clock: $\overline{9} + \overline{5} = \overline{14} = \overline{2}$.
Products work the same way: $\overline{5} \cdot \overline{7} = \overline{35} = \overline{11}$.
</document>

<document source="https://tamnd.github.io/math2/node/01NHFZF2SK.md" tag="01NHFZF2SK" kind="concept">
# Nondegenerate bilinear form

## Summary

A nondegenerate bilinear form is a bilinear form for which no nonzero vector gives zero with every vector.
</document>

<document source="https://tamnd.github.io/math2/node/01NN8NY497.md" tag="01NN8NY497" kind="counterexample">
# Function with unequal mixed partial derivatives

## Counterexample

One could guess that the mixed [[partial derivatives|02RSXVM1ZN]] $f_{xy}$ and $f_{yx}$ are equal at each point where both exist.
This is false.
Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be given by
$$f(x, y) = \frac{x y (x^2 - y^2)}{x^2 + y^2} \text{ for } (x, y) \ne (0, 0), \qquad f(0, 0) = 0.$$
Both mixed second partial derivatives of $f$ exist at $(0, 0)$, but $\dfrac{\partial}{\partial y}\left(\dfrac{\partial f}{\partial x}\right)(0, 0) = -1$ and $\dfrac{\partial}{\partial x}\left(\dfrac{\partial f}{\partial y}\right)(0, 0) = 1$.

## Remarks

From the definition, $\partial f / \partial x (0, y) = \lim_{h \to 0} f(h, y) / h = -y$ for each $y$, and $\partial f / \partial y (x, 0) = \lim_{k \to 0} f(x, k) / k = x$ for each $x$. Differentiate these once more at $0$.
The function is of class $C^1$ on $\mathbb{R}^2$, but its second partial derivatives are not continuous at the origin. So [[Schwarz's theorem|0NNQY8WCW6]] does not apply.
This example is due to Peano.

## Explanations

### L1 human

Schwarz's theorem needs continuous second partial derivatives, and this function shows why.
Away from the origin everything is smooth, but at the origin the two orders of differentiation see different limits.
</document>

<document source="https://tamnd.github.io/math2/node/01PKQYTTMT.md" tag="01PKQYTTMT" kind="definition">
# Isolated point

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]], let $S \subseteq X$ and let $x \in S$.
The point $x$ is an isolated point of $S$ when there is an [[open set|01ZQQDCERP]] $U$ of $X$ with $U \cap S = \{x\}$.

## Remarks

Each point of $S$ is either an isolated point of $S$ or a [[limit point|0HBQC2GJXT]] of $S$, and not both.
In $\mathbb{R}$, each point of $\mathbb{Z}$ is an isolated point of $\mathbb{Z}$: take $U = (n - 1/2, n + 1/2)$.
A point $x$ is an isolated point of $X$ itself if and only if $\{x\}$ is open.

## Explanations

### L1 human

An isolated point is a point of the set that stands alone: a small region about it contains no other point of the set.
In the set $\{0\} \cup [1, 2]$, the point $0$ is isolated, but $1$ is not.
</document>

<document source="https://tamnd.github.io/math2/node/01PNAVFMYZ.md" tag="01PNAVFMYZ" kind="definition">
# Similar triangles

## Definition

Let $ABC$ and $A'B'C'$ be [[triangles|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
The triangles are similar, written $ABC \sim A'B'C'$, if there is a real number $r > 0$ such that
$$d(A', B') = r \, d(A, B), \quad d(B', C') = r \, d(B, C) \quad \text{and} \quad d(C', A') = r \, d(C, A).$$
The number $r$ is the ratio of the similarity.

## Remarks

Similar triangles have equal corresponding angles. Conversely, the [[angle-angle criterion|0NZVJYZVTZ]] says that two pairs of equal angles are enough for similarity.
[[Congruent triangles|093EDENVA1]] are the similar triangles with the ratio $r = 1$.
Two triangles are similar if and only if a [[homothety|0ZG1PNEP35]] followed by an isometry sends one onto the other, vertex by vertex.
Any two equilateral triangles are similar.

## Explanations

### L1 human

A photo of a triangle, enlarged or reduced, gives a similar triangle. All sides change by the same factor, and the angles stay the same. For example, the triangles with sides $3, 4, 5$ and $6, 8, 10$ are similar with the ratio $2$.
</document>

<document source="https://tamnd.github.io/math2/node/01PPDSDP2N.md" tag="01PPDSDP2N" kind="concept">
# Hyperbolic functions

## Summary

The hyperbolic functions are analogues of the trigonometric functions defined from the exponential function. The main ones are the hyperbolic sine, cosine and tangent. The node [[hyperbolic functions|0J6M03WSS6]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/01QFQAQ2PA.md" tag="01QFQAQ2PA" kind="concept">
# Spherical coordinates

## Summary

Spherical coordinates describe a point of space by three numbers. They are its distance from the origin, the angle between its position vector and the positive $z$-axis, and an azimuth angle around the $z$-axis.
They simplify problems with symmetry about a point, such as integrals over balls.

## Remarks

The names and the order of the two angles differ between mathematics and physics texts. Each text must state its convention.
</document>

<document source="https://tamnd.github.io/math2/node/01QSSKS5C1.md" tag="01QSSKS5C1" kind="statement">
# sin x / x tends to 1

## Statement

$$\lim_{x \to 0} \frac{\sin x}{x} = 1.$$

## Remarks

The quotient is the difference quotient of $\sin$ at $0$. So the limit is $\sin'(0) = \cos 0 = 1$, by the [[derivative of sine|00R4MWW532]].
The [[series|02KTRHQMHN]] gives $\frac{\sin x}{x} = 1 - \frac{x^2}{6} + \cdots$ for $x \ne 0$.
A geometric proof uses $\cos x \le \frac{\sin x}{x} \le 1$ for $0 < |x| < \pi/2$ and the squeeze theorem. Mathlib states it as `Real.tendsto_sin_div_nhdsNE_zero`.

## Explanations

### L1 human

For small angles in radians, $\sin x$ is almost equal to $x$.
The ratio of the two tends to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/01QZNHQWYD.md" tag="01QZNHQWYD" kind="statement">
# Sum of the first n natural numbers

## Statement

For every natural number $n$,
$$0 + 1 + 2 + \dots + n = \sum_{k=0}^{n} k = \frac{n(n + 1)}{2}.$$

## Remarks

The number $n(n + 1)/2$ is the $n$th triangular number. For $n = 1, 2, 3, 4$ it is $1, 3, 6, 10$.
A proof by [[induction|0ASJVEFA9J]] uses $\frac{n(n+1)}{2} + (n + 1) = \frac{(n+1)(n+2)}{2}$.
An equivalent form without division is $2 \sum_{k=0}^{n} k = n(n + 1)$.

## Explanations

### L1 human

Write the sum $1 + 2 + \dots + n$ two times, once forward and once backward.
Each column then adds up to $n + 1$, and there are $n$ columns.
So two copies of the sum give $n(n + 1)$, and one copy is half of that.
</document>

<document source="https://tamnd.github.io/math2/node/01R1HPEK6S.md" tag="01R1HPEK6S" kind="statement">
# A graph is bipartite iff it has no odd cycle

## Statement

Let $G$ be a [[graph|0JD5C82QMB]].
Then $G$ is [[bipartite|07DVHJ8N0A]] if and only if $G$ has no [[cycle|07CT9RDY7Q]] of odd length.

## Remarks

One direction is easy. A cycle in a bipartite graph alternates between the two parts, so its length is even.
For the converse, work in each component. Colour a vertex by the parity of its distance to a fixed vertex. An edge between two vertices of the same colour would give an odd closed walk, and then an odd cycle.
A graph is also bipartite if and only if its chromatic number is at most $2$.

## Explanations

### L1 human

Walking around a cycle in a bipartite graph, you switch sides at every step.
To come back to the start you need an even number of steps, so odd cycles are impossible.
</document>

<document source="https://tamnd.github.io/math2/node/01R1HW3Y42.md" tag="01R1HW3Y42" kind="concept">
# Tangent line

## Summary

Let $f$ be [[differentiable|0F1WR9C83K]] at $a$.
The tangent line to the graph of $f$ at $(a, f(a))$ is the line $y = f(a) + f'(a)(x - a)$.
It is the line through this point that best fits the graph near it.

## Remarks

The tangent line is the limit of the secant lines through $(a, f(a))$ and $(x, f(x))$ as $x \to a$.
The error $f(x) - f(a) - f'(a)(x - a)$ is $o(x - a)$ as $x \to a$. This property characterizes the derivative.
For $f(x) = x^2$ at $a = 1$, the tangent line is $y = 2x - 1$.

## Explanations

### L1 human

The tangent line touches the graph at one point and has the same slope as the graph there.
Near the point, the graph and the line are almost the same.
</document>

<document source="https://tamnd.github.io/math2/node/01RBSSGMY5.md" tag="01RBSSGMY5" kind="concept">
# Definite and indefinite quadratic forms

## Summary

Let $A$ be a symmetric $n \times n$ matrix, and let $Q(x) = x^{\mathsf{T}} A x$ be its real quadratic form on $\mathbb{R}^n$. The form $Q$ is positive definite if $Q(x) > 0$ for all $x \ne 0$. It is negative definite if $Q(x) < 0$ for all $x \ne 0$.
It is positive or negative semidefinite if $Q(x) \ge 0$ or $Q(x) \le 0$ for all $x$, and indefinite if it takes both positive and negative values.

## Remarks

The form $Q$ is positive definite if and only if all eigenvalues of $A$ are positive. It is indefinite if and only if $A$ has a positive and a negative eigenvalue.
On $\mathbb{R}^2$, $x_1^2 + x_2^2$ is positive definite, $x_1^2$ is positive semidefinite but not definite, and $x_1^2 - x_2^2$ is indefinite.
In the [[second derivative test|0GYDV9FE98]], the matrix $A$ is the Hessian matrix of a function at a critical point.

## Explanations

### L1 human

A positive definite form is a bowl: it is zero at the origin and positive everywhere else.
An indefinite form is a saddle: it goes up in some directions and down in others.
</document>

<document source="https://tamnd.github.io/math2/node/01REXW3WWE.md" tag="01REXW3WWE" kind="concept">
# Elementary matrix

## Summary

An elementary matrix is a matrix that we get from the identity matrix by one elementary row operation.
Multiplying a matrix on the left by an elementary matrix performs that row operation.
</document>

<document source="https://tamnd.github.io/math2/node/01REYCJ371.md" tag="01REYCJ371" kind="concept">
# Lyapunov function

## Summary

A Lyapunov function for an equilibrium of a differential equation is a scalar function with three properties. It is zero at the equilibrium, positive away from it, and non-increasing along solutions.
Its existence proves stability without solving the equation. It is named after Aleksandr Lyapunov. See the [[definition|04P7B5XC22]].
</document>

<document source="https://tamnd.github.io/math2/node/01RSTGNTXG.md" tag="01RSTGNTXG" kind="concept">
# Residue class

## Summary

A residue class modulo $n$ is the set of all integers that are congruent to a given integer modulo $n$.
The [[definition of a residue class|0QRV8FPK5M]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/01RVCMSN5N.md" tag="01RVCMSN5N" kind="concept">
# Surjective function

## Summary

A function is surjective when each element of the codomain is a value of the function.
</document>

<document source="https://tamnd.github.io/math2/node/01SWBKDAWA.md" tag="01SWBKDAWA" kind="definition">
# Composition of relations

## Definition

Let $A$, $B$ and $C$ be sets, let $R \subseteq A \times B$ and let $S \subseteq B \times C$ be [[binary relations|08EJC6Q25H]].
The composition of $R$ and $S$ is the relation from $A$ to $C$
$$R \mathbin{;} S = \{(a, c) \in A \times C \mid \exists b \in B, \, a \mathrel{R} b \text{ and } b \mathrel{S} c\}.$$

## Remarks

Many authors write $S \circ R$ for $R \mathbin{;} S$. This order agrees with [[function composition|07HF9YCWPP]].
Composition of relations is associative.
A relation $R$ on a set is [[transitive|0CYRECG2YQ]] exactly when $R \mathbin{;} R \subseteq R$.

## Explanations

### L1 human

Let $R$ be "is a parent of" and let $S$ be "is a parent of" again.
Then $R \mathbin{;} S$ is "is a grandparent of": you go from a person to a child, then from the child to a grandchild.
</document>

<document source="https://tamnd.github.io/math2/node/01T3TMGTTE.md" tag="01T3TMGTTE" kind="definition">
# Neighbourhood

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] with vertex set $V$, and let $v \in V$.
The neighbourhood of $v$ is the set
$$N(v) = \{ u \in V : u \sim v \}$$
of the vertices adjacent to $v$.
The elements of $N(v)$ are the neighbours of $v$.

## Remarks

This is the open neighbourhood. It does not contain $v$, because no vertex is adjacent to itself. The closed neighbourhood is $N[v] = N(v) \cup \{v\}$.
The [[degree|0T0F6CCS9W]] of $v$ is $|N(v)|$ when $N(v)$ is finite.
In the cycle with vertices $1, 2, 3, 4, 5$ in this order, $N(1) = \{2, 5\}$.

## Explanations

### L1 human

The neighbours of a vertex are the vertices joined to it by an edge.
In a friendship graph, the neighbourhood of a person is the set of that person's friends.
</document>

<document source="https://tamnd.github.io/math2/node/01THH3Y3YP.md" tag="01THH3Y3YP" kind="definition">
# Integrating factor

## Definition

Let $I$ be an open interval, and let $p, q : I \to \mathbb{R}$ be continuous.
An integrating factor of the [[linear first-order equation|0Z3YKRQZ48]] $y' + p(x) y = q(x)$ is a differentiable function $\mu : I \to \mathbb{R}$ with $\mu(x) \ne 0$ for each $x \in I$ and
$$\mu'(x) = p(x)\, \mu(x) \quad \text{for each } x \in I.$$
For such $\mu$ and each differentiable $y$ on $I$, $\mu \left(y' + p y\right) = (\mu y)'$.

## Remarks

If $P$ is an antiderivative of $p$ on $I$, then $\mu(x) = e^{P(x)}$ is an integrating factor. Each integrating factor is a nonzero constant times this one.
After multiplication by $\mu$, the equation becomes $(\mu y)' = \mu q$. One integration gives the [[solution formula|06B9W45XV0]].
More generally, an integrating factor is a function that makes an equation exact after multiplication. See [[integrating factor for a non-exact equation|0CX86CR8JV]].

## Explanations

### L1 human

The left side $y' + p y$ is not the derivative of a simple expression.
After multiplication by the right function $\mu$, it becomes the derivative of $\mu y$, by the product rule.
Then you can integrate both sides directly.
For $y' + y = x$, the factor is $e^x$, and the equation becomes $(e^x y)' = x e^x$.
</document>

<document source="https://tamnd.github.io/math2/node/01TKKESHSB.md" tag="01TKKESHSB" kind="example">
# Diffie–Hellman key exchange

## Example

The Diffie–Hellman key exchange is an example of [[public-key cryptography|01E9MNJYEX]]. Two parties, Alice and Bob, use it to agree on a secret key over a public channel.
They agree in public on a prime $p$ and an integer $g$, often a [[primitive root|0KM2CJ80G7]] modulo $p$.
Alice chooses a secret integer $a$ and sends $A = g^a \bmod p$. Bob chooses a secret integer $b$ and sends $B = g^b \bmod p$.
Alice computes $B^a \bmod p$, and Bob computes $A^b \bmod p$. Both are equal to $g^{ab} \bmod p$, the shared secret.
A small case: $p = 11$, $g = 2$, $a = 3$ and $b = 4$. Then $A = 8$, $B = 16 \bmod 11 = 5$, and $5^3 \equiv 8^4 \equiv 4 \pmod{11}$.

## Remarks

An eavesdropper sees $p$, $g$, $A$ and $B$. To find $g^{ab}$, a natural method is to compute a [[discrete logarithm|09B4X51J09]], which is believed to be hard for large $p$.
The same method works in any finite cyclic group, for example the group of points of an elliptic curve.
Whitfield Diffie and Martin Hellman published the method in 1976.

## Explanations

### L1 human

Each party mixes a public number with a private number, and they swap the results.
Each then mixes in their own private number again, and both get the same final number.
An eavesdropper sees only the swapped numbers, and cannot easily undo the mixing.
</document>

<document source="https://tamnd.github.io/math2/node/01TPMXM6NM.md" tag="01TPMXM6NM" kind="concept">
# Hyperbolic equilibrium point

## Summary

A hyperbolic equilibrium point of a vector field is an equilibrium point at which the Jacobian matrix has no eigenvalue with real part zero.
Such a point has no centre manifold. See the [[definition|0C5PREPZWG]].
</document>

<document source="https://tamnd.github.io/math2/node/01TQB88H4C.md" tag="01TQB88H4C" kind="concept">
# Countable set

## Summary

A countable set is a set whose elements can be listed in a finite or infinite sequence.
It is finite or equinumerous with the natural numbers.
</document>

<document source="https://tamnd.github.io/math2/node/01TXN7ETH2.md" tag="01TXN7ETH2" kind="concept">
# Orthogonal family for a bilinear form

## Summary

A family of vectors is orthogonal for a bilinear form when the form is zero on each pair of distinct members of the family.
</document>

<document source="https://tamnd.github.io/math2/node/01V9FPBBVK.md" tag="01V9FPBBVK" kind="statement">
# Symmetry of second derivatives

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$.
Suppose that $f$ is [[differentiable|0135XZZQ8M]] at each point of $U$, and that the derivative $Df \colon U \to L(\mathbb{R}^n, \mathbb{R}^m)$ is differentiable at $a$.
Then the second derivative $D^2 f(a)$ is a symmetric bilinear map: $D^2 f(a)(u, v) = D^2 f(a)(v, u)$ for all $u, v \in \mathbb{R}^n$.
In particular, $\dfrac{\partial^2 f}{\partial x_i \, \partial x_j}(a) = \dfrac{\partial^2 f}{\partial x_j \, \partial x_i}(a)$ for all $i, j$.

## Remarks

Here $D^2 f(a)$ is the [[iterated derivative|0EY05Z9RPN]] of order $2$, and $D^2 f(a)(e_i, e_j)$ is the mixed [[partial derivative|02RSXVM1ZN]] $\partial_i \partial_j f(a)$.
The hypothesis is that $f$ is twice differentiable at $a$. It holds when the second partial derivatives are continuous near $a$, which gives [[Schwarz's theorem|0NNQY8WCW6]].
The existence of the second partial derivatives at $a$ is not enough: there is a [[function with unequal mixed partial derivatives|01NN8NY497]].

## History

Euler published a proposed proof in 1740, and many incomplete proofs followed over more than a century.
In 1867 Lindelöf analysed the earlier proofs and gave a counterexample to the general claim.
Schwarz gave the first rigorous proof six years later, and Jordan gave in 1883 the general version that most textbooks use.

## Explanations

### L1 human

The theorem says that the order of two differentiations does not matter, when $f$ is regular enough.
For $f(x, y) = x^3 y^2$, differentiating first in $x$ and then in $y$ gives $6 x^2 y$, and the other order gives $6 x^2 y$ too.
</document>

<document source="https://tamnd.github.io/math2/node/01VG2A0BHW.md" tag="01VG2A0BHW" kind="statement">
# Bijective iff it has a two-sided inverse

## Statement

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
Then $f$ is [[bijective|0SNQEF7Z53]] if and only if there is a function $g \colon B \to A$ with $g \circ f = \mathrm{id}_A$ and $f \circ g = \mathrm{id}_B$.

## Remarks

The function $g$ is then unique. It is the [[inverse function|0W766JAPSN]] $f^{-1}$.
This statement does not need the axiom of choice: for a bijection, $g(b)$ is the unique $a$ with $f(a) = b$.
If $f$ has a [[left inverse|02486SD328]] $g$ and a [[right inverse|0DDPZAG5ER]] $h$, then $g = h$.

## Explanations

### L1 human

A bijection can be undone in both directions.
Going from $A$ to $B$ and back gives the start point, and going from $B$ to $A$ and back gives the start point too.
</document>

<document source="https://tamnd.github.io/math2/node/01VH21EV6F.md" tag="01VH21EV6F" kind="concept">
# Prime element

## Summary

A prime element of a commutative ring is a nonzero nonunit $p$ with the following property. If $p$ divides a product, then $p$ divides one of the factors.
The [[definition of a prime element|0MNHXBSX3M]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/01VN79T2PR.md" tag="01VN79T2PR" kind="definition">
# Quadratic integer

## Definition

A quadratic integer is a complex number $x$ that satisfies an equation $x^2 + bx + c = 0$ with integers $b$ and $c$.
Equivalently, $x = \dfrac{-b \pm \sqrt{b^2 - 4c}}{2}$ for some integers $b$ and $c$.

## Remarks

Each integer $n$ is a quadratic integer, since it is a root of $x^2 - 2nx + n^2 = 0$.
A quadratic integer that is not an integer is an [[algebraic number|0MB4TGMXG6]] of degree $2$.
It lies in a quadratic field $\mathbb{Q}(\sqrt{D})$ with $D$ a squarefree integer.
The [[Gaussian integers|0B268GVX95]] are the quadratic integers in $\mathbb{Q}(i)$. Another example is the golden ratio $\frac{1 + \sqrt{5}}{2}$, a root of $x^2 - x - 1 = 0$.
For $D \equiv 2, 3 \pmod 4$, the quadratic integers in $\mathbb{Q}(\sqrt{D})$ form the ring $\mathbb{Z}[\sqrt{D}] = \{ a + b\sqrt{D} : a, b \in \mathbb{Z} \}$.

## Explanations

### L1 human

A quadratic integer is a root of a quadratic equation with integer coefficients and leading coefficient $1$.
Numbers such as $\sqrt{2}$, $i$ and $1 + \sqrt{3}$ are examples. They extend the integers in the same way that $\sqrt{2}$ extends the rationals.
</document>

<document source="https://tamnd.github.io/math2/node/01VQ7C4FTB.md" tag="01VQ7C4FTB" kind="statement">
# Uniqueness of the algebraic closure

## Statement

Let $K$ be a [[field|0QA8CHH8MG]], and let $L$ and $L'$ be [[algebraic closures|0MCXX29HWY]] of $K$.
Then there is a ring isomorphism $\sigma \colon L \to L'$ with $\sigma(c) = c$ for every $c \in K$.

## Remarks

The proof uses Zorn's lemma on the partial isomorphisms between subfields of $L$ and $L'$ that fix $K$.
The isomorphism is not unique in general. For example, complex conjugation and the identity are two such isomorphisms from $\mathbb{C}$ to $\mathbb{C}$ over $\mathbb{R}$.
So one often writes "the" algebraic closure $\overline{K}$.
The statement is an infinite form of the [[uniqueness of splitting fields|069A3BC3EP]].

## Explanations

### L1 human

Two algebraic closures of the same field are the same field with different names for the elements.
There can be many ways to match them, so the matching is not part of the structure.
</document>

<document source="https://tamnd.github.io/math2/node/01VT24J1SD.md" tag="01VT24J1SD" kind="statement">
# Cantor set is closed

## Statement

The [[Cantor set|0BXNDZFQME]] $C = \bigcap_{n \ge 0} C_n$ is a [[closed subset|0G5AV9Q4RR]] of $\mathbb{R}$.

## Remarks

Proof: each $C_n$ is a union of $2^n$ closed intervals, so it is closed. An [[intersection of closed sets is closed|09YMV8H4X5]].
Since $C \subseteq [0, 1]$, the set $C$ is also bounded, and thus compact by the [[Heine–Borel theorem|07PFYV65MB]].
Each endpoint of an interval of some $C_n$ lies in $C$. For example, $\tfrac{1}{3}$ and $\tfrac{2}{3}$ are in $C$.

## Explanations

### L1 human

At each step we keep a finite union of closed intervals, which is a closed set.
Closed sets stay closed under intersections, so the limit set is closed.
</document>

<document source="https://tamnd.github.io/math2/node/01VVP10MWJ.md" tag="01VVP10MWJ" kind="definition">
# Connected component

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A connected component of $G$ is an equivalence class of the vertices of $G$ under [[reachability|0J8XE7W68F]].
The subgraph induced by a connected component is also called a connected component.

## Remarks

The connected components partition the vertex set. Each component, as an induced subgraph, is [[connected|0FFAYNMAJZ]].
A graph with $n$ vertices and $m$ edges has at least $n - m$ components.
A graph with no edges on $n$ vertices has $n$ components.

## Explanations

### L1 human

The connected components are the separate pieces of a graph.
Two vertices are in the same piece exactly when a walk joins them.
</document>

<document source="https://tamnd.github.io/math2/node/01W6RKJP01.md" tag="01W6RKJP01" kind="statement">
# Arguments add under multiplication

## Statement

Let $z$ and $w$ be complex numbers with $z \neq 0$ and $w \neq 0$.
If $\theta$ is an [[argument|0ZZRSX91QR]] of $z$ and $\varphi$ is an argument of $w$, then $\theta + \varphi$ is an argument of $z w$.
Thus $\arg(z w) \equiv \arg z + \arg w \pmod{2 \pi}$.

## Remarks

Proof: write $z = |z| e^{i \theta}$ and $w = |w| e^{i \varphi}$. Then $z w = |z| |w| e^{i (\theta + \varphi)}$ by the [[product of exponentials|0WM7DJCM6G]].
For the principal values, the equality can fail. For example, $\operatorname{Arg}(-1) + \operatorname{Arg}(-1) = 2\pi$, but $\operatorname{Arg}((-1)(-1)) = 0$.

## Explanations

### L1 human

When you multiply two complex numbers, the angles add.
For example, $i$ points at $90$ degrees, and $i \cdot i = -1$ points at $180$ degrees.
</document>

<document source="https://tamnd.github.io/math2/node/01WHD9T0V6.md" tag="01WHD9T0V6" kind="definition">
# Central binomial coefficient

## Definition

Let $n$ be a natural number.
The $n$-th central binomial coefficient is the [[binomial coefficient|092GEF38YD]]
$$\binom{2n}{n} = \frac{(2n)!}{(n!)^2}.$$

## Remarks

The first values, for $n = 0, 1, 2, 3, 4$, are $1, 2, 6, 20, 70$.
It is the middle entry of row $2n$ of [[Pascal's triangle|05DME4TZ3K]], and the largest entry of that row.
It counts the [[lattice paths|040WY0TT2G]] from $(0, 0)$ to $(n, n)$ with unit steps right and up. The [[closed formula for Catalan numbers|0NNDQVJ54Q]] divides it by $n + 1$.

## Explanations

### L1 human

The central binomial coefficient is the number in the middle of an even row of Pascal's triangle.
For $n = 2$ it is $\binom{4}{2} = 6$: the ways to choose 2 of 4 objects.
</document>

<document source="https://tamnd.github.io/math2/node/01WPV9PHN7.md" tag="01WPV9PHN7" kind="statement">
# Legendre's formula

## Statement

Let $p$ be a prime number and let $n$ be a natural number. Let $\nu_p$ be the [[p-adic valuation|04VCPS30ZE]].
Then
$$\nu_p(n!) = \sum_{i=1}^{\infty} \left\lfloor \frac{n}{p^i} \right\rfloor.$$
The sum is finite, because $\lfloor n / p^i \rfloor = 0$ when $p^i > n$.

## Remarks

An equivalent form: $\nu_p(n!) = \dfrac{n - s_p(n)}{p - 1}$, where $s_p(n)$ is the sum of the digits of $n$ in base $p$.
For example, $\nu_2(10!) = 5 + 2 + 1 = 8$, and $\nu_5(100!) = 20 + 4 = 24$. So $100!$ ends with $24$ zeros.
The proof counts, for each $i$, the multiples of $p^i$ among $1, \dots, n$.

## Explanations

### L1 human

To count the factors $p$ in $n!$, count the multiples of $p$ up to $n$. Then add the multiples of $p^2$, which give a second factor, and so on.
For $10!$ and $p = 3$: there are $3$ multiples of $3$ and $1$ multiple of $9$, so $3^4$ is the largest power of $3$ that divides $10!$.
</document>

<document source="https://tamnd.github.io/math2/node/01WQZK97Z5.md" tag="01WQZK97Z5" kind="statement">
# Cassini's identity

## Statement

Let $n$ be a positive integer, and let $F_m$ be the [[Fibonacci numbers|0GA9K57A39]].
Then
$$F_{n-1} F_{n+1} - F_n^2 = (-1)^n.$$

## Remarks

Examples: for $n = 5$, $F_4 F_6 - F_5^2 = 3 \cdot 8 - 25 = -1$. For $n = 6$, $F_5 F_7 - F_6^2 = 5 \cdot 13 - 64 = 1$.
A short proof takes determinants in the matrix identity $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}$.
Catalan's identity $F_n^2 - F_{n-r} F_{n+r} = (-1)^{n-r} F_r^2$ generalizes it. The case $r = 1$ is Cassini's identity.

## Explanations

### L1 human

The product of the two neighbours of a Fibonacci number differs from its square by exactly $1$.
The sign alternates: $1 \cdot 3 - 2^2 = -1$, then $2 \cdot 5 - 3^2 = 1$, then $3 \cdot 8 - 5^2 = -1$.
</document>

<document source="https://tamnd.github.io/math2/node/01X5CVV1Q5.md" tag="01X5CVV1Q5" kind="example">
# Geometric series

## Example

Let $a$ and $r$ be real numbers with $|r| < 1$.
The geometric series $\sum_{n=0}^{\infty} a r^n$ converges, and
$$\sum_{n=0}^{\infty} a r^n = \frac{a}{1 - r}.$$
The [[sum of a finite geometric series|05APWSX53S]] gives the partial sums $s_N = a (1 - r^{N+1})/(1 - r)$.
Since [[$r^{N+1} \to 0$|0Q7K4AXE53]], the partial sums tend to $a/(1 - r)$.

## Remarks

For $a \ne 0$ and $|r| \ge 1$, the series diverges by the [[term test|0VVY4H92G2]], because $|a r^n| \ge |a|$.
For $a = 1$ and $r = 1/2$, it gives $1 + 1/2 + 1/4 + \cdots = 2$.
The decimal $0.333\ldots$ is the geometric series with $a = 3/10$ and $r = 1/10$, and its sum is $1/3$.
The terms form a [[geometric sequence|0MNM54WB99]].

## Explanations

### L1 human

Walk half of the way to a wall, then half of the rest, and so on.
The distances $1/2 + 1/4 + 1/8 + \cdots$ add up to the whole distance $1$.
</document>

<document source="https://tamnd.github.io/math2/node/01X8YSDTBW.md" tag="01X8YSDTBW" kind="definition">
# Identically distributed random variables

## Definition

Let $(E, \mathcal{E})$ be a set with a σ-algebra.
Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, P)$ with values in $E$.
Let $Y$ be a random variable on a probability space $(\Omega', \mathcal{F}', P')$ with values in $E$.
$X$ and $Y$ are identically distributed if they have the same [[law|0TST4MHA3A]]: $P(X \in B) = P'(Y \in B)$ for each $B \in \mathcal{E}$.
One writes $X \overset{d}{=} Y$.

## Remarks

For real random variables, this holds if and only if $X$ and $Y$ have the same cumulative distribution function.
Identically distributed random variables need not be equal. For a fair coin, the indicator $X$ of heads and the indicator $1 - X$ of tails are identically distributed, but they are never equal.
If $X \overset{d}{=} Y$ and $g$ is a measurable function, then $g(X) \overset{d}{=} g(Y)$.

## Explanations

### L1 human

Two random variables are identically distributed when they behave the same way statistically.
The results of two different fair dice are identically distributed, although the dice show different numbers in most throws.
</document>

<document source="https://tamnd.github.io/math2/node/01XAHPADBA.md" tag="01XAHPADBA" kind="concept">
# Moment

## Summary

The moments of a random variable are the expected values of its powers. The first moment is the mean.
The node [[moment|0PH0VHTQS0]] gives the precise definition for random variables.
</document>

<document source="https://tamnd.github.io/math2/node/01XAY17QA3.md" tag="01XAY17QA3" kind="definition">
# Uniformly Cauchy sequence of functions

## Definition

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ for $n \ge 1$.
The sequence $(f_n)$ is uniformly Cauchy on $S$ if for every $\varepsilon > 0$ there is $N$ such that
$$|f_n(x) - f_m(x)| < \varepsilon \quad \text{for all } m, n \ge N \text{ and all } x \in S.$$

## Remarks

For each $x$, the sequence $(f_n(x))$ is then a [[Cauchy sequence|0QZMTN5AXX]]. So it has a limit $f(x)$, by the [[Cauchy criterion|0ZA4AX4MW3]].
A sequence of real functions is uniformly Cauchy on $S$ if and only if it [[converges uniformly|08HJRTG5N7]] on $S$.
This criterion proves uniform convergence without knowing the limit. Mathlib defines it as `UniformCauchySeqOn`.

## Explanations

### L1 human

A uniformly Cauchy sequence has graphs that get close to each other, uniformly in $x$.
Such a sequence always converges uniformly to some function.
</document>

<document source="https://tamnd.github.io/math2/node/01XM7JEKHY.md" tag="01XM7JEKHY" kind="concept">
# Constructive proof

## Summary

A constructive proof of an existence statement $\exists x \, P(x)$ gives a specific object $a$, or a method to find it, and proves $P(a)$.
A non-constructive proof shows that an object exists without giving it, for example by [[proof by contradiction|0J0XBKG64S]].

## Remarks

A classical non-constructive example: there are irrational numbers $a$ and $b$ such that $a^b$ is rational. Consider $\sqrt{2}^{\sqrt{2}}$. It is rational or irrational, and each case gives such $a$ and $b$.
Intuitionistic logic accepts only constructive proofs of existence.

## Explanations

### L1 human

A constructive proof that a solution exists shows you the solution.
A non-constructive proof only shows that "no solution" is impossible.
Both are correct in classical mathematics, but the constructive proof tells you more.
</document>

<document source="https://tamnd.github.io/math2/node/01YYRV7S9Z.md" tag="01YYRV7S9Z" kind="concept">
# Limit at infinity

## Summary

The limit of a function at infinity is the value that the function approaches as its argument grows without bound. It describes horizontal asymptotes of graphs. The node [[limit at infinity|0Y12QYTJK3]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/01ZQQDCERP.md" tag="01ZQQDCERP" kind="definition">
# Open set

## Definition

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]].
A subset $U$ of $X$ is open in $X$ when $U \in \tau$.
A subset of a [[metric space|063W5JMY85]] is open when it is open for the [[metric topology|0A6YRNZB2R]].

## Remarks

In a metric space $(X, d)$, a set $U$ is open if and only if for each $x \in U$ there is $\varepsilon > 0$ with $B(x, \varepsilon) \subseteq U$. This is the statement [[open sets of a metric space|0A26X79RTB]].
In $\mathbb{R}$, each open interval $(a, b)$ is open, but $[0, 1)$ is not open: no ball about $0$ lies in $[0, 1)$.
The empty set and $X$ are always open.

## Explanations

### L1 human

In a metric space, an open set has no edge points. From each of its points, you can move a little in any direction and stay in the set.
The interval $(0, 1)$ is open, because each point has some room on both sides.
</document>

<document source="https://tamnd.github.io/math2/node/01ZTQBK64Q.md" tag="01ZTQBK64Q" kind="statement">
# Adding a multiple of one row to another preserves the determinant

## Statement

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$, and let $c \in K$.
Let $i \ne j$, and let $B$ be the matrix that results when $c$ times row $j$ of $A$ is added to row $i$.
Then $\det B = \det A$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

By linearity in row $i$, $\det B = \det A + c \det A'$, where $A'$ has two equal rows; so $\det A' = 0$.
This is the reason why [[Gaussian elimination|0JVJ2WCXZH]] computes determinants efficiently.
The same holds for columns.

## Explanations

### L1 human

Adding a multiple of one row to another row shears the box spanned by the rows.
A shear does not change the volume.
So this row operation keeps the determinant.
</document>

<document source="https://tamnd.github.io/math2/node/01ZWX875AT.md" tag="01ZWX875AT" kind="statement">
# Law of the unconscious statistician

## Statement

Let $X$ be a random variable on $(\Omega, \mathcal{F}, P)$ with values in $(E, \mathcal{E})$ and with [[law|0TST4MHA3A]] $\mu_X$. Let $g \colon E \to \mathbb{R}$ be measurable.
Then $g(X)$ is integrable if and only if $\int_E |g| \, d\mu_X < \infty$. In that case,
$$E[g(X)] = \int_E g(x) \, \mu_X(dx).$$
In particular, if $X$ is discrete with [[mass function|0WF4EJP91W]] $p$, then $E[g(X)] = \sum_x g(x) \, p(x)$. If $X$ is real with [[density|04YG690N5V]] $f$, then $E[g(X)] = \int_{-\infty}^{\infty} g(x) f(x) \, dx$.

## Remarks

The formula also holds for $g \ge 0$ without any integrability condition, with both sides in $[0, \infty]$.
The proof checks the formula first for indicator functions $g = 1_B$, where it says $P(X \in B) = \mu_X(B)$. Then it extends to simple, nonnegative and integrable $g$.
The point is that one does not need the law of $g(X)$ to compute $E[g(X)]$.

## History

The name refers to statisticians who use the formula as the definition of $E[g(X)]$. They are "unconscious" that it is a theorem that needs a proof.

## Explanations

### L1 human

To find the average of $g(X)$, you can average $g(x)$ over the values $x$ of $X$ with their probabilities.
For a fair die, $E[X^2] = (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6$.
</document>

<document source="https://tamnd.github.io/math2/node/0201Q4P2B5.md" tag="0201Q4P2B5" kind="statement">
# Carathéodory's theorem

## Statement

Let $P$ be a subset of $\mathbb{R}^d$, and let $x$ be a point of its [[convex hull|0FQSDC2EQ2]] $\operatorname{conv}(P)$.
Then there are at most $d + 1$ points $p_1, \dots, p_k$ of $P$, with $k \leq d + 1$, such that $x \in \operatorname{conv}\{p_1, \dots, p_k\}$.
Equivalently, $x$ is a convex combination of at most $d + 1$ points of $P$.

## Remarks

Thus $\operatorname{conv}(P)$ is the union of the simplices with vertices in $P$, of dimension at most $d$. Mathlib states this form as `convexHull_eq_union`.
The bound $d + 1$ is the best possible in general. Let $P$ be the three vertices of a triangle in $\mathbb{R}^2$. A point inside the triangle is not in the hull of two points of $P$.
The proof only uses the order and the field operations of $\mathbb{R}$. So it works over each ordered field.

## History

Carathéodory proved the theorem in 1911 for compact sets $P$. Steinitz extended it to arbitrary sets in 1914.

## Explanations

### L1 human

In the plane, a point in the convex hull of many points is always inside a triangle with corners from those points. You never need more than three corners. In space, four corners, the vertices of a tetrahedron, are always enough.
</document>

<document source="https://tamnd.github.io/math2/node/020422MDQF.md" tag="020422MDQF" kind="statement">
# Euler's theorem on Eulerian circuits

## Statement

Let $G$ be a finite [[connected|0FFAYNMAJZ]] [[graph|0JD5C82QMB]] or [[multigraph|01M5T0KDG7]] with at least one edge.
Then $G$ has an Eulerian circuit if and only if every vertex of $G$ has even [[degree|0T0F6CCS9W]].

## Remarks

The condition is necessary by the theorem on [[degrees along an Eulerian trail|0CDVDGWGEG]].
A consequence: a finite connected graph has an [[Eulerian trail|0NNYDD1AVR]] that is not closed if and only if exactly two vertices have odd degree.
Hierholzer's algorithm builds an Eulerian circuit: it joins closed trails until every edge is used.

## History

Leonhard Euler showed the necessity of the condition, and he stated without proof that it is sufficient.
The first complete proof is by Carl Hierholzer. It was published after his death, in 1873.

## Explanations

### L1 human

You can draw a connected graph in one closed stroke exactly when every vertex has an even number of edges.
Each time the pen passes a vertex, it uses one edge to arrive and one edge to leave.
</document>

<document source="https://tamnd.github.io/math2/node/020K0X7HVR.md" tag="020K0X7HVR" kind="statement">
# Existence of prime factorization

## Statement

Each positive integer $n$ has a [[prime factorization|0AEKDFCYFA]].
That is, there are [[prime numbers|0SN0MAEKYX]] $p_1, \dots, p_k$, with $k \ge 0$, such that $n = p_1 p_2 \cdots p_k$.

## Remarks

The proof uses strong induction on $n$. If $n = 1$, take $k = 0$. If $n$ is prime, take $k = 1$. Otherwise $n = a b$ with $1 < a, b < n$, and we join the factorizations of $a$ and $b$.
The [[fundamental theorem of arithmetic|084HH1NDCB]] adds that the factorization is unique, up to the order of the factors.

## Explanations

### L1 human

Split each number that is not prime into two smaller factors. Continue until all the pieces are prime.
The pieces always get smaller, so the splitting stops.
For example, $84 = 2 \cdot 42 = 2 \cdot 2 \cdot 21 = 2 \cdot 2 \cdot 3 \cdot 7$.
</document>

<document source="https://tamnd.github.io/math2/node/020M04FN1H.md" tag="020M04FN1H" kind="concept">
# Matrix exponential

## Summary

The exponential of a real or complex square matrix $A$ is the sum of the convergent power series $I + A + A^2/2! + A^3/3! + \cdots$.
</document>

<document source="https://tamnd.github.io/math2/node/020Q06TYYH.md" tag="020Q06TYYH" kind="concept">
# Totally bounded set

## Summary

A subset of a metric space is totally bounded when, for each size, it can be covered by finitely many pieces of that size.
Compact sets are exactly the totally bounded sets that are complete.
</document>

<document source="https://tamnd.github.io/math2/node/020Z5SEN27.md" tag="020Z5SEN27" kind="definition">
# Free group

## Definition

Let $S$ be a set.
The free group $F_S$ on $S$ is the set of [[reduced words|0KBTP8Q2FS]] on $S$.
The product of two reduced words is their concatenation, followed by reduction.
A [[group|G5S1M9]] $G$ is free when it is [[isomorphic|08MG7H8JW5]] to $F_S$ for some set $S$.

## Remarks

The identity of $F_S$ is the empty word, and the inverse of $t_1 \cdots t_n$ is $t_n^{-1} \cdots t_1^{-1}$.
The set $S$ is a basis of $F_S$. Also $F_S \cong F_T$ if and only if $S$ and $T$ have the same cardinality, which is the rank.
The free group of rank $1$ is the [[infinite cyclic group|02B1T50JD1]], and for $|S| \geq 2$ the centre of $F_S$ is trivial.
A nontrivial finite group is never free, because each element of a basis has infinite order.

## Explanations

### L1 human

A free group has generators but no relations that you do not need.
Two words give the same element only when the group axioms force it, as in $s t = s u u^{-1} t$.
Each group is a quotient of a free group, so free groups are the starting point for presentations.
</document>

<document source="https://tamnd.github.io/math2/node/021WEM680W.md" tag="021WEM680W" kind="definition">
# Definite integral

## Definition

Let $I$ be an interval, let $f \colon I \to \mathbb{R}$, and let $a, b \in I$.
Assume that $f$ is Riemann integrable on the closed interval between $a$ and $b$.
The definite integral of $f$ from $a$ to $b$ is the number $\int_a^b f(x) \, dx$ defined as follows.
If $a < b$, it is the [[Riemann integral|0QM4TXT3DR]] of $f$ over $[a, b]$.
If $a = b$, it is $0$. If $a > b$, it is $-\int_b^a f(x) \, dx$.

## Remarks

With this sign convention, $\int_a^b f + \int_b^c f = \int_a^c f$ holds for all $a, b, c$ in $I$.
The numbers $a$ and $b$ are the limits of integration, and $f$ is the integrand.
Mathlib writes the definite integral as `∫ x in a..b, f x`, through `intervalIntegral`.

## Explanations

### L1 human

The definite integral of $f$ from $a$ to $b$ is a number. It measures signed area under the graph between $a$ and $b$.
The order of the limits matters: swapping them changes the sign.
</document>

<document source="https://tamnd.github.io/math2/node/021XJAQVDC.md" tag="021XJAQVDC" kind="statement">
# Darboux and Riemann integrals agree

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded.
Then $f$ is [[Riemann integrable|0N1JQXZXHC]] if and only if its upper and lower [[Darboux integrals|09FQT057T7]] are equal.
In this case the common value is the [[Riemann integral|0QM4TXT3DR]] $\int_a^b f(x)\, dx$.

## Remarks

Each Riemann sum lies between the lower and upper Darboux sums. This gives one direction.
For the other direction, one shows that Darboux sums for partitions of small mesh are close to the Darboux integrals.
So the Riemann and the Darboux approaches define the same integral.

## Explanations

### L1 human

There are two standard ways to define the integral, with Riemann sums or with upper and lower sums.
They give the same integrable functions and the same values.
</document>

<document source="https://tamnd.github.io/math2/node/0225QPWRRE.md" tag="0225QPWRRE" kind="definition">
# Taylor series

## Definition

Let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be [[smooth|0BE3QDH73A]].
The Taylor series of $f$ at $a$ is the [[power series|0VP56B4EMX]]
$$\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n.$$

## Remarks

Its partial sums are the [[Taylor polynomials|0HFSFVMWQB]] of $f$ at $a$.
The Taylor series may diverge for each $x \ne a$. It may also converge to a function other than $f$, as for the [[smooth non-analytic function|09RTYYHXKA]].
The Taylor series converges to $f$ near $a$ exactly when $f$ is [[analytic|0JD5ZNW0F4]] at $a$.
For $a = 0$ it is the [[Maclaurin series|043SJ5RF59]].

## Explanations

### L1 human

The Taylor series is the Taylor polynomial of infinite degree.
It uses all the derivatives of $f$ at one point.
</document>

<document source="https://tamnd.github.io/math2/node/022RSX714V.md" tag="022RSX714V" kind="concept">
# Order of cardinals

## Summary

The order of cardinals compares the sizes of sets by injections.
</document>

<document source="https://tamnd.github.io/math2/node/0230ECX1Q5.md" tag="0230ECX1Q5" kind="definition">
# Well-founded relation

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is well-founded when each nonempty subset $S$ of $A$ has an element $m$ such that no $s \in S$ satisfies $s \mathrel{R} m$.

## Remarks

The relation $<$ on $\mathbb{N}$ is well-founded. The relation $<$ on $\mathbb{Z}$ is not, because $\mathbb{Z}$ itself has no minimal element.
If $R$ is well-founded, there is no sequence $x_0, x_1, x_2, \dots$ with $x_{n+1} \mathrel{R} x_n$ for each $n$. The converse needs the axiom of dependent choice.
Well-founded relations allow proofs by [[transfinite induction|0ZHQMQ2C1N]].

## Explanations

### L1 human

In a well-founded relation, you cannot go down forever.
Each strictly decreasing sequence of natural numbers stops after finitely many steps.
</document>

<document source="https://tamnd.github.io/math2/node/023VY493K0.md" tag="023VY493K0" kind="concept">
# Euclidean algorithm in a Euclidean domain

## Summary

Let $R$ be a [[Euclidean domain|0GJ2HRG95T]] with Euclidean function $\delta$ and let $a, b \in R$.
The Euclidean algorithm sets $r_0 = a$ and $r_1 = b$. While $r_k \neq 0$, it divides $r_{k-1}$ by $r_k$ and lets $r_{k+1}$ be the remainder.
The values $\delta(r_k)$ decrease strictly, so the algorithm stops with some $r_{n+1} = 0$.
The last nonzero remainder $r_n$ is a [[greatest common divisor|0F7KNJ1ZYN]] of $a$ and $b$, if $b \neq 0$.

## Remarks

The algorithm works because the common divisors of $r_{k-1}$ and $r_k$ are the common divisors of $r_k$ and $r_{k+1}$.
In $\mathbb{Z}$: $252 = 1 \cdot 198 + 54$, $198 = 3 \cdot 54 + 36$, $54 = 1 \cdot 36 + 18$, $36 = 2 \cdot 18$. So $\gcd(252, 198) = 18$.
The extended algorithm also gives the coefficients of [[Bézout's identity|0VY3WPP9Y3]].

## Explanations

### L1 human

To find the gcd of two numbers, replace the larger one by its remainder modulo the smaller one, and repeat.
When the remainder is zero, the last divisor is the gcd.
The same steps work in every Euclidean domain, for example for polynomials over a field.
</document>

<document source="https://tamnd.github.io/math2/node/02486SD328.md" tag="02486SD328" kind="definition">
# Left inverse

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
A left inverse of $f$ is a function $g \colon B \to A$ such that $g \circ f = \mathrm{id}_A$, that is, $g(f(a)) = a$ for each $a \in A$.

## Remarks

A function with a nonempty domain has a left inverse exactly when it is [[injective|0PVG1CRAJN]]. See [[injective iff it has a left inverse|0HYFEEZAEM]].
A left inverse is in general not unique. Its values on $B \setminus f(A)$ can be any elements of $A$.
If $g$ is a left inverse of $f$, then $f$ is a [[right inverse|0DDPZAG5ER]] of $g$.

## Explanations

### L1 human

Let $f \colon \mathbb{N} \to \mathbb{N}$ be $f(n) = n + 1$.
The function $g$ with $g(n) = n - 1$ for $n \ge 1$ and $g(0) = 0$ is a left inverse: $g(f(n)) = n$.
But $f(g(0)) = 1 \neq 0$, so $g$ is not an inverse on both sides.
</document>

<document source="https://tamnd.github.io/math2/node/025158VEYZ.md" tag="025158VEYZ" kind="definition">
# Reflection

## Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $W$ be a finite-dimensional subspace of $V$, with [[orthogonal projection|0B8KZRT28M]] $P_W$.
The *reflection* in $W$ is the linear map
$$R_W \colon V \to V, \qquad R_W(v) = 2 P_W(v) - v.$$

## Remarks

$R_W$ fixes each vector of $W$, negates each vector of $W^\perp$, and satisfies $R_W^2 = \id_V$.
It is a linear isometry.
For a hyperplane $W = a^\perp$ with $a \ne 0$, the formula is $R_W(v) = v - 2 \frac{\langle v, a \rangle}{\langle a, a \rangle} a$.
In $\R^n$, the matrix of a hyperplane reflection is orthogonal with determinant $-1$.

## Explanations

### L1 human

A reflection is a mirror image in a subspace.
Vectors in the mirror stay where they are, and vectors perpendicular to it change direction.
Reflecting twice brings every vector back.
</document>

<document source="https://tamnd.github.io/math2/node/025PKW8AMR.md" tag="025PKW8AMR" kind="statement">
# Uniqueness of solutions for Lipschitz vector fields

## Statement

Let $D$ be an open subset of $\mathbb{R} \times \mathbb{R}^n$ and let $f : D \to \mathbb{R}^n$ be continuous.
Assume that $f$ is locally Lipschitz in $x$. That is, each point of $D$ has a neighbourhood $V \subseteq D$ and a constant $K \ge 0$ such that $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ whenever $(t, x), (t, y) \in V$.
Let $I$ be an interval, and let $\gamma_1, \gamma_2 : I \to \mathbb{R}^n$ be two [[solutions|04SW7WECS8]] of $x' = f(t, x)$ whose graphs lie in $D$.
If $\gamma_1(t_0) = \gamma_2(t_0)$ for one $t_0 \in I$, then $\gamma_1(t) = \gamma_2(t)$ for all $t \in I$.

## Remarks

So an [[initial value problem|075VBY4K4X]] for such an $f$ has at most one solution on each interval that contains $t_0$.
Proof idea: the function $u(t) = \|\gamma_1(t) - \gamma_2(t)\|$ satisfies $u(t) \le K \left| \int_{t_0}^{t} u(s)\, ds \right|$ near $t_0$. Then [[Grönwall's inequality|07EJHQYR9A]] gives $u = 0$. A connectedness argument extends this to all of $I$.
A $C^1$ function $f$ is locally Lipschitz in $x$.
Without the Lipschitz condition, uniqueness can fail. See the [[non-uniqueness example|00K7ASZ1PR]].

## Explanations

### L1 human

Two solutions of a Lipschitz equation that meet at one time are the same solution.
So the graphs of different solutions never cross or touch.
</document>

<document source="https://tamnd.github.io/math2/node/025YZN3VJW.md" tag="025YZN3VJW" kind="concept">
# Conjunction

## Summary

Conjunction is the logical operation "and".
It joins two [[propositions|0ZGC7ZESYN]] into one proposition that is true only when both parts are true.
</document>

<document source="https://tamnd.github.io/math2/node/0262JP0NV9.md" tag="0262JP0NV9" kind="concept">
# First integral

## Summary

A first integral, or constant of motion, of a system of differential equations is a function of the state that stays constant along each solution.
Energy, linear momentum and angular momentum are common examples in mechanics. See the [[definition|0SS1P1YC9A]].
</document>

<document source="https://tamnd.github.io/math2/node/026BJBQNGJ.md" tag="026BJBQNGJ" kind="statement">
# A function with nonnegative derivative is increasing

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous, and differentiable at each interior point of $I$.
If $f'(x) \ge 0$ for each interior point $x$ of $I$, then $f$ is increasing on $I$. That is, $f(x) \le f(y)$ for $x \le y$ in $I$.

## Remarks

The proof uses the [[mean value theorem|0BARQY52N5]].
For differentiable $f$ on an open interval, the converse holds too: if $f$ is increasing, then $f' \ge 0$.
Mathlib states it as `monotoneOn_of_deriv_nonneg`.

## Explanations

### L1 human

If the slope is never negative, the graph never goes down.
It may stay flat on some parts.
</document>

<document source="https://tamnd.github.io/math2/node/026FDZFRXS.md" tag="026FDZFRXS" kind="statement">
# Uniformly continuous functions map Cauchy sequences to Cauchy sequences

## Statement

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be [[uniformly continuous|0Q4DR37DVE]].
Let $(x_n)$ be a [[Cauchy sequence|0QZMTN5AXX]] in $D$.
Then $(f(x_n))$ is a Cauchy sequence.

## Remarks

Given $\varepsilon > 0$, take $\delta$ from uniform continuity, and then $N$ with $|x_m - x_n| < \delta$ for $m, n \ge N$.
Continuity alone is not enough. The function $1/x$ on $(0, 1)$ maps the Cauchy sequence $1/n$ to the sequence $n$.
Mathlib states it as `UniformContinuous.comp_cauchySeq`.

## Explanations

### L1 human

Points of a Cauchy sequence crowd closer and closer together.
A uniformly continuous function keeps their images crowding together too.
</document>

<document source="https://tamnd.github.io/math2/node/026HPGYGS4.md" tag="026HPGYGS4" kind="statement">
# Parallelogram law

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, with the [[induced norm|0MFJJV97RJ]] $\norm{\cdot}$.
Then, for all $x, y \in V$,
$$\norm{x + y}^2 + \norm{x - y}^2 = 2 \norm{x}^2 + 2 \norm{y}^2.$$

## Remarks

The proof expands both squares with the inner product; the mixed terms cancel.
By the Jordan–von Neumann theorem, a norm comes from an inner product if and only if it satisfies this law.
The norm $\abs{x_1} + \abs{x_2}$ on $\R^2$ fails it for $x = (1, 0)$ and $y = (0, 1)$, so it comes from no inner product.

## Explanations

### L1 human

In a parallelogram, the squares of the two diagonals add up to the squares of the four sides.
The vectors $x + y$ and $x - y$ are the diagonals of the parallelogram with sides $x$ and $y$.
In a rectangle, both diagonals have the same length, and this is the Pythagorean theorem twice.
</document>

<document source="https://tamnd.github.io/math2/node/026J1MG2NW.md" tag="026J1MG2NW" kind="concept">
# Flux through a surface

## Summary

The flux of a vector field through an [[oriented surface|0TD904MG06]] measures how much of the field passes through the surface. It counts flow in the direction of the chosen normal.
It is the [[surface integral|065HE5V98E]] of the normal component of the field. For a velocity field of a fluid, it is the volume of fluid that crosses the surface per unit time.
</document>

<document source="https://tamnd.github.io/math2/node/027JF1FBB0.md" tag="027JF1FBB0" kind="definition">
# Density of the normal distribution

## Definition

Let $\mu \in \mathbb{R}$ and $\sigma^2 > 0$.
The density of the normal distribution with mean $\mu$ and variance $\sigma^2$ is the function $\varphi_{\mu, \sigma^2} \colon \mathbb{R} \to (0, \infty)$ with
$$\varphi_{\mu, \sigma^2}(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right).$$
The standard normal density is $\varphi(x) = \varphi_{0, 1}(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$.

## Remarks

The integral of $\varphi_{\mu, \sigma^2}$ over $\mathbb{R}$ is $1$. This follows from the Gaussian integral $\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$.
So $\varphi_{\mu, \sigma^2}$ is a [[probability density function|04YG690N5V]], the density of the [[normal distribution|07RXBVTVZB]] $\mathcal{N}(\mu, \sigma^2)$.
The function is symmetric about $\mu$, has its maximum $1/\sqrt{2\pi\sigma^2}$ at $x = \mu$, and has inflection points at $\mu \pm \sigma$.
It satisfies $\varphi_{\mu, \sigma^2}(x) = \frac{1}{\sigma} \varphi\left(\frac{x - \mu}{\sigma}\right)$ with $\sigma = \sqrt{\sigma^2}$.

## Explanations

### L1 human

This is the formula of the bell curve.
The exponent $-(x - \mu)^2 / (2\sigma^2)$ makes the curve fall very fast away from $\mu$, and the constant in front makes the total area equal to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/027K9RMZ43.md" tag="027K9RMZ43" kind="definition">
# Derivative function

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be [[differentiable|0A61B60B5R]] on $I$.
The derivative function of $f$ is the function $f' \colon I \to \mathbb{R}$ that maps each $x \in I$ to the [[derivative|0F1WR9C83K]] $f'(x)$ of $f$ at $x$.

## Remarks

Other notations are $\frac{df}{dx}$ and $D f$.
For example, the derivative function of $x^3$ is $3 x^2$.
The derivative function need not be continuous. See [[x^2 sin(1/x)|0WDAKKN4NC]].
Mathlib defines `deriv f` on all of $\mathbb{R}$, with value $0$ at points where $f$ is not differentiable.

## Explanations

### L1 human

The derivative function collects the slopes of $f$ at all points into one new function.
We can then differentiate it again.
</document>

<document source="https://tamnd.github.io/math2/node/027KDW2K72.md" tag="027KDW2K72" kind="definition">
# Similar matrices

## Definition

Let $F$ be a field and let $A$ and $B$ be $n \times n$ matrices over $F$.
The matrix $B$ is *similar* to $A$ when there is an [[invertible|0EB9N0S33C]] $n \times n$ matrix $P$ over $F$ with
$$B = P^{-1} A P.$$

## Remarks

Similarity is an equivalence relation on $M_n(F)$.
By the [[change of basis formula|0MQXVZEBMT]], two matrices are similar if and only if they represent the same endomorphism of $F^n$ in two bases.
Similar matrices have the same [[trace|0XC1RDBY7A]], [[determinant|00KDZRW4C0]], [[characteristic polynomial|0VSSPAFM2C]], [[rank|0ZC9YTXZRY]] and [[eigenvalues|0BQAA2BK3Q]].
The matrices $I_2$ and $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ have the same characteristic polynomial but are not similar.

## Explanations

### L1 human

Similar matrices describe the same linear map, seen from two different bases.
Properties of the map itself, like trace and determinant, are the same for both matrices.
A main goal of linear algebra is to find a simple matrix in each similarity class.
</document>

<document source="https://tamnd.github.io/math2/node/027MBSVPY3.md" tag="027MBSVPY3" kind="definition">
# Reflection in a line or plane

## Definition

Let $a \in \mathbb{R}^n$ be a nonzero vector and $c$ a real number, and let $\cdot$ be the dot product. Let $H = \{\, x \in \mathbb{R}^n : x \cdot a = c \,\}$ be the hyperplane with the [[normal vector|0WQJJCCTC8]] $a$.
The reflection in $H$ is the map
$$\sigma_H \colon \mathbb{R}^n \to \mathbb{R}^n, \qquad \sigma_H(x) = x - 2\, \frac{x \cdot a - c}{a \cdot a}\, a.$$
For $n = 2$, $H$ is a line; for $n = 3$, $H$ is a plane.

## Remarks

The map $\sigma_H$ is an indirect [[Euclidean isometry|0J1N3WTKM4]]. Its fixed points are the points of $H$, and $\sigma_H \circ \sigma_H$ is the identity.
For $x \notin H$, the hyperplane $H$ is the [[perpendicular bisector|055VNYT0PB]] of $x$ and $\sigma_H(x)$.
The linear part of $\sigma_H$ is an orthogonal matrix with determinant $-1$ and the eigenvalues $-1, 1, \dots, 1$.
Reflections in hyperplanes generate the Euclidean group.

## Explanations

### L1 human

A reflection is a mirror image. Each point goes to the point on the other side of the mirror, at the same distance from it. The reflection in the line $y = x$ in the plane sends $(x, y)$ to $(y, x)$.
</document>

<document source="https://tamnd.github.io/math2/node/027ZQS04SJ.md" tag="027ZQS04SJ" kind="statement">
# Quotients of solvable groups are solvable

## Statement

Let $G$ be a [[solvable group|0FXJ7YFMMX]] and let $\varphi \colon G \to K$ be a surjective [[group homomorphism|0JYTG5K9Y4]].
Then $K$ is solvable.
In particular, $G/N$ is solvable for each [[normal subgroup|0CB0DX5BVV]] $N$ of $G$.

## Remarks

A proof uses the [[derived series|0Q5G5DJ4P2]]: $\varphi(G^{(i)}) = K^{(i)}$ for each $i$, because $\varphi$ maps commutators onto commutators.
So $G^{(n)} = \{e\}$ implies $K^{(n)} = \{e\}$.
The second part follows with the projection $G \to G/N$.

## Explanations

### L1 human

A surjective homomorphism sends the commutators of $G$ onto the commutators of $K$.
So the derived series of $K$ is the image of the derived series of $G$.
If one reaches the identity, the image does too.
</document>

<document source="https://tamnd.github.io/math2/node/02879QH7BY.md" tag="02879QH7BY" kind="concept">
# Associate elements

## Summary

Two elements of a commutative ring are associates when one is a unit times the other.
The [[definition of associate elements|053E86NWVK]] states this precisely.
</document>

<document source="https://tamnd.github.io/math2/node/028XEQCE88.md" tag="028XEQCE88" kind="statement">
# Angle addition formulas

## Statement

For all real numbers $x$ and $y$,
$$\sin(x + y) = \sin x \cos y + \cos x \sin y,$$
$$\cos(x + y) = \cos x \cos y - \sin x \sin y.$$

## Remarks

One proof multiplies the power series with the [[Cauchy product|0AN28XQ45Z]] and uses the binomial theorem.
Another proof fixes $y$ and shows that the difference of the two sides solves $f'' = -f$ with $f(0) = f'(0) = 0$.
With $y = x$, they give $\sin 2x = 2 \sin x \cos x$ and $\cos 2x = \cos^2 x - \sin^2 x$. Mathlib states them as `Real.sin_add` and `Real.cos_add`.

## Explanations

### L1 human

These formulas compute the sine and cosine of a sum of two angles from the sines and cosines of the angles.
They are the most important trigonometric identities.
</document>

<document source="https://tamnd.github.io/math2/node/028Z8WC5Y9.md" tag="028Z8WC5Y9" kind="definition">
# Exponential distribution

## Definition

Let $\lambda > 0$.
The exponential distribution with rate $\lambda$ is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$f(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if } x \ge 0, \\ 0 & \text{if } x < 0. \end{cases}$$
One writes $X \sim \operatorname{Exp}(\lambda)$.

## Remarks

The cumulative distribution function is $F(x) = 1 - e^{-\lambda x}$ for $x \ge 0$, so $P(X > x) = e^{-\lambda x}$.
The mean is $1/\lambda$, the variance is $1/\lambda^2$ and the median is $\ln 2 / \lambda$. Some authors use the scale $\beta = 1/\lambda$ as the parameter.
The exponential distribution is [[memoryless|0ZAZZ34WQS]]: $P(X > s + t \mid X > s) = P(X > t)$ for $s, t \ge 0$.
If $X_1, \dots, X_n$ are independent with $X_i \sim \operatorname{Exp}(\lambda_i)$, then $\min_i X_i \sim \operatorname{Exp}(\lambda_1 + \dots + \lambda_n)$.

## Explanations

### L1 human

The exponential distribution models the time until something happens when it can happen at any moment with the same intensity.
Examples are the decay of a radioactive atom and the time until the next call to a help desk.
</document>

<document source="https://tamnd.github.io/math2/node/02903SSA48.md" tag="02903SSA48" kind="definition">
# Flow of a vector field

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be locally Lipschitz.
For $x_0 \in U$, let $\gamma_{x_0} : I(x_0) \to U$ be the [[maximal solution|019VSXSFMQ]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ with $x(0) = x_0$.
Let $\Omega = \{(t, x_0) : x_0 \in U,\ t \in I(x_0)\}$.
The flow of $f$ is the map $\varphi : \Omega \to U$ given by $\varphi(t, x_0) = \gamma_{x_0}(t)$.
Another notation is $\varphi_t(x_0) = \varphi(t, x_0)$.

## Remarks

The flow has the group properties $\varphi_0(x) = x$ and $\varphi_s(\varphi_t(x)) = \varphi_{s + t}(x)$, whenever both sides are defined. The second property follows from [[uniqueness|025PKW8AMR]] and [[time translation|0WA7ZR95V4]].
The set $\Omega$ is open and $\varphi$ is continuous on $\Omega$.
If $\Omega = \mathbb{R} \times U$, the flow is complete. Then $t \mapsto \varphi_t$ is an action of the additive group $\mathbb{R}$ on $U$.
For a linear system $x' = A x$, the flow is $\varphi_t(x) = e^{tA} x$, the [[matrix exponential|0YC8JT513J]].

## Explanations

### L1 human

The flow moves each point of the phase space along the solution through it.
The map $\varphi_t$ says where each point is after time $t$.
Moving for time $t$ and then for time $s$ is the same as moving for time $s + t$.
</document>

<document source="https://tamnd.github.io/math2/node/0292EPPPT0.md" tag="0292EPPPT0" kind="statement">
# Euclid's theorem

## Statement

There are infinitely many [[prime numbers|0SN0MAEKYX]].
Equivalently, for each natural number $n$ there is a prime number $p$ with $p \ge n$.

## Remarks

Euclid's proof: let $p_1, \dots, p_k$ be a finite list of primes, and let $N = p_1 p_2 \cdots p_k + 1$. A [[prime divisor|0SPBEDVNW1]] of $N$ is not in the list, because each $p_i$ leaves the remainder $1$ when it divides $N$.
The [[divergence of the sum of the reciprocals of the primes|00RA98X2CP]] is a stronger result.
[[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]] gives infinitely many primes in each suitable arithmetic progression.

## History

The theorem is Proposition 20 of Book IX of the Elements of Euclid.

## Explanations

### L1 human

No finite list holds all the primes. Multiply all the primes of the list and add $1$.
The new number leaves the remainder $1$ on division by each prime of the list, so its prime factors are new.
For example, $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 + 1 = 30031 = 59 \cdot 509$, and $59$ and $509$ are not in the list.
</document>

<document source="https://tamnd.github.io/math2/node/0299VCW1TH.md" tag="0299VCW1TH" kind="concept">
# Vector field

## Summary

A vector field on a subset of $\mathbb{R}^n$ assigns a vector of $\mathbb{R}^n$ to each point of the subset.
It models quantities with a direction and a size at each point, such as the velocity of a fluid or a force field.
</document>

<document source="https://tamnd.github.io/math2/node/029JBBF693.md" tag="029JBBF693" kind="concept">
# Negation

## Summary

Negation is the logical operation that changes a [[proposition|0ZGC7ZESYN]] into its opposite.
The negation of a true proposition is false, and the negation of a false proposition is true.
</document>

<document source="https://tamnd.github.io/math2/node/029JQEK15B.md" tag="029JQEK15B" kind="definition">
# Planar graph

## Definition

A [[graph|0JD5C82QMB]] $G$ is planar if it has a drawing in the plane in which no two edges cross.
In such a drawing, the vertices are distinct points of $\mathbb{R}^2$, and each edge $uv$ is a simple arc from $u$ to $v$.
Two arcs meet only at a common end, and an arc contains no vertex other than its ends.

## Remarks

A drawing of this kind is a plane graph, or an embedding of $G$ in the plane.
The complete graph $K_4$ is planar. [[$K_5$ and $K_{3,3}$ are not planar|030A3FHN14]].
[[Kuratowski's theorem|0J6G5NJC5R]] characterizes the finite planar graphs by these two graphs.

## Explanations

### L1 human

A graph is planar when you can draw it on paper with no crossing edges.
A map of countries gives a planar graph: one vertex for each country, and an edge between neighbours.
</document>

<document source="https://tamnd.github.io/math2/node/02A8EMC83C.md" tag="02A8EMC83C" kind="definition">
# Binary tree

## Definition

The binary trees are defined recursively by these two rules.
The empty tree is a binary tree. It has no nodes.
If $L$ and $R$ are binary trees, then $\operatorname{node}(L, R)$ is a binary tree. It has one more node than $L$ and $R$ together, the root, with left subtree $L$ and right subtree $R$.
Every binary tree comes from these two rules in a unique way, in finitely many steps.

## Remarks

The left and right subtrees are ordered. Thus $\operatorname{node}(\operatorname{node}(\varnothing, \varnothing), \varnothing)$ and $\operatorname{node}(\varnothing, \operatorname{node}(\varnothing, \varnothing))$ are different binary trees with $2$ nodes.
The number of binary trees with $n$ nodes is the Catalan number $C_n$. The node [[binary trees are counted by Catalan numbers|0Y97E2DMB7]] states this.
A nonempty binary tree is a [[rooted tree|0C9J6W2A8A]] in which each vertex has at most two children, each marked left or right.

## Explanations

### L1 human

A binary tree is either empty, or a root with a left part and a right part that are binary trees again.
Binary trees are used in computer science to store data for fast search.
</document>

<document source="https://tamnd.github.io/math2/node/02AC7JR6G6.md" tag="02AC7JR6G6" kind="statement">
# Holomorphic functions on a disc have primitives

## Statement

Let $c \in \mathbb{C}$, let $R \in (0, \infty]$, and let $D = \{z : |z - c| < R\}$ be the open disc with center $c$ and radius $R$.
Let $f : D \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.
Then there is a holomorphic function $F : D \to \mathbb{C}$ with $F'(z) = f(z)$ for each $z \in D$.

## Remarks

One primitive is $F(z) = \int_{[c, z]} f(w) \, dw$, the [[contour integral|00FWNP9TXY]] along the segment from $c$ to $z$. [[Goursat's theorem|0BRG85WXYC]] for triangles shows that $F' = f$.
Two primitives of $f$ on $D$ differ by a constant, because $D$ is connected.
The disc can be replaced by a convex open set, or more generally by a simply connected open set.
The result fails on $\mathbb{C} \setminus \{0\}$: the function $1/z$ has no primitive there, by the [[integral of 1/z|0SXZ68HV6F]].

## Explanations

### L1 human

On a disc, each holomorphic function is the derivative of another holomorphic function.
You can build it by integrating along straight lines from the center, as in the real case $F(x) = \int_c^x f(t) \, dt$.
</document>

<document source="https://tamnd.github.io/math2/node/02AS45T3H9.md" tag="02AS45T3H9" kind="concept">
# Convex set

## Summary

A set of points is convex if it contains the whole segment between each two of its points. A solid ball and a solid cube are convex; a crescent is not.
</document>

<document source="https://tamnd.github.io/math2/node/02B1T50JD1.md" tag="02B1T50JD1" kind="example">
# Infinite cyclic group

## Example

The [[additive group of integers|0BSCJA5G53]] $(\mathbb{Z}, +)$ is an infinite [[cyclic group|0KVGH0J5B3]].
The element $1$ generates it, because each integer is a sum of copies of $1$ or of $-1$.
Its only generators are $1$ and $-1$.

## Remarks

If $G = \langle g \rangle$ is an infinite cyclic group, then the powers $g^k$ for $k \in \mathbb{Z}$ are all distinct.
Then $k \mapsto g^k$ is an isomorphism from $\mathbb{Z}$ to $G$, thus each infinite cyclic group is isomorphic to $(\mathbb{Z}, +)$.
The group of the translations of the plane by integer multiples of a fixed vector is an infinite cyclic group.

## Explanations

### L1 human

In an infinite cyclic group, the powers of the generator never repeat.
You can walk forever in both directions without coming back.
This is exactly how the integers behave under addition.
</document>

<document source="https://tamnd.github.io/math2/node/02BH3MCBQB.md" tag="02BH3MCBQB" kind="concept">
# Open set

## Summary

An open set is a member of a topology.
In a metric space, a set is open when each of its points has a small ball around it that stays in the set.
</document>

<document source="https://tamnd.github.io/math2/node/02BPKD7TMD.md" tag="02BPKD7TMD" kind="statement">
# Eisenstein's criterion

## Statement

Let $f = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in \mathbb{Z}[x]$ with $n \ge 1$, and let $p$ be a prime number with these three properties.

1. $p$ divides $a_i$ for each $i$ with $0 \le i < n$.
2. $p$ does not divide $a_n$.
3. $p^2$ does not divide $a_0$.

Then $f$ is [[irreducible|00X2A9G9W3]] in $\mathbb{Q}[x]$.

## Remarks

If $f$ is also [[primitive|0SP14DZN0G]], then $f$ is irreducible in $\mathbb{Z}[x]$, by [[Gauss's lemma|0QFV5Y1Z5X]].
For example, $x^n - 2$ is irreducible over $\mathbb{Q}$ for each $n \ge 1$, with $p = 2$.
A change of variable can help: $f(x + 1)$ is irreducible if and only if $f(x)$ is. This proves that $x^{p-1} + \dots + x + 1$ is irreducible for $p$ prime.
The criterion holds more generally for a prime ideal $P$ of an integral domain $R$, with $P^2$ in place of $p^2$.

## History

Theodor Schönemann published a version of the criterion in 1846.
Gotthold Eisenstein published a different version in 1850.
Both papers appeared in Crelle's Journal.

## Explanations

### L1 human

The criterion is a quick test of irreducibility over the rationals.
Find one prime that divides all the coefficients except the leading one, and whose square does not divide the constant term.
For $x^5 + 6x + 3$, the prime $3$ works, so the polynomial is irreducible.
</document>

<document source="https://tamnd.github.io/math2/node/02CBHV4ZZQ.md" tag="02CBHV4ZZQ" kind="definition">
# Covariance matrix

## Definition

Let $X = (X_1, \dots, X_n)^{\mathsf{T}}$ be a random vector in $\mathbb{R}^n$ with $E[X_i^2] < \infty$ for each $i$, and let $\mu = E[X]$ be its mean vector.
The covariance matrix of $X$ is the $n \times n$ matrix $\Sigma = \operatorname{Cov}(X)$ with entries
$$\Sigma_{ij} = \operatorname{Cov}(X_i, X_j), \quad 1 \le i, j \le n.$$
In matrix form, $\Sigma = E\left[(X - \mu)(X - \mu)^{\mathsf{T}}\right]$.

## Remarks

The diagonal entries are the [[variances|007QED3E0Y]] $\Sigma_{ii} = \operatorname{Var}(X_i)$.
$\Sigma$ is symmetric and positive semidefinite: $a^{\mathsf{T}} \Sigma a = \operatorname{Var}(a^{\mathsf{T}} X) \ge 0$ for each $a \in \mathbb{R}^n$.
For a real $m \times n$ matrix $A$ and $b \in \mathbb{R}^m$, $\operatorname{Cov}(AX + b) = A \Sigma A^{\mathsf{T}}$.
The covariance matrix and the mean vector are the two parameters of the [[multivariate normal distribution|0G558GJQMW]].

## Explanations

### L1 human

For a random vector, one number is not enough to describe the spread.
The covariance matrix collects the variance of each component and the covariance of each pair in one table.
</document>

<document source="https://tamnd.github.io/math2/node/02CDYWSWSY.md" tag="02CDYWSWSY" kind="definition">
# Edge colouring

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] with edge set $E$, and let $C$ be a set of colours.
An edge colouring of $G$ is a function $c : E \to C$ such that $c(e) \neq c(e')$ for any two distinct edges $e$ and $e'$ with a common end.
The chromatic index $\chi'(G)$ is the smallest number of colours in an edge colouring of a finite graph $G$.

## Remarks

Each colour class is a [[matching|0Z87BHAFTB]]. Thus an edge colouring is a partition of the edges into matchings.
The edges at a vertex need different colours, so $\chi'(G) \ge \Delta(G)$.
Vizing's theorem of 1964 says that a simple graph has $\chi'(G) = \Delta(G)$ or $\chi'(G) = \Delta(G) + 1$.

## Explanations

### L1 human

An edge colouring colours the lines instead of the points: lines that meet must have different colours.
In a sports schedule, each colour is a round, and each team plays at most one game per round.
</document>

<document source="https://tamnd.github.io/math2/node/02CK6QMXYD.md" tag="02CK6QMXYD" kind="concept">
# Cardinality of a finite set

## Summary

The cardinality of a finite set is the number of elements of the set. It is the basic quantity that all counting results compute. The node [[cardinality of a finite set|05099E1NMJ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/02CYB184RC.md" tag="02CYB184RC" kind="statement">
# Mean value inequality

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$ be [[differentiable|0135XZZQ8M]] on $U$, and let $x, y \in U$ be points such that the segment $[x, y] = \{x + t (y - x) : 0 \le t \le 1\}$ lies in $U$.
Suppose that $M \ge 0$ and $\|Df(z)\| \le M$ for each $z \in [x, y]$, where $\|Df(z)\| = \sup_{\|h\| \le 1} \|Df(z) h\|$ is the operator norm.
Then
$$\|f(y) - f(x)\| \le M \, \|y - x\|.$$

## Remarks

For $m = 1$, the inequality follows from the [[mean value theorem in several variables|03HGK7EW6R]]. For $m \ge 2$ there is in general no equality form, but the inequality still holds.
A standard proof applies the one-variable mean value theorem to $\varphi(t) = (f(y) - f(x)) \cdot f(x + t(y - x))$.
On a convex set $C \subseteq U$ with $\|Df\| \le M$ on $C$, the inequality holds for all $x, y \in C$, so $f$ is [[Lipschitz on C|0TKG7T3K0E]].

## Explanations

### L1 human

Suppose that the speed of change of $f$ is at most $M$ everywhere on a straight path. Then $f$ changes by at most $M$ times the length of the path.
This is like saying that a car that never goes faster than $100$ km/h cannot cover more than $200$ km in two hours.
</document>

<document source="https://tamnd.github.io/math2/node/02DKV69ZH0.md" tag="02DKV69ZH0" kind="concept">
# Sign of a permutation

## Summary

The sign of a permutation is $+1$ or $-1$, according to the parity of the number of transpositions in a decomposition of the permutation.
The sign map is a homomorphism from $S_n$ to $\{1, -1\}$.
</document>

<document source="https://tamnd.github.io/math2/node/02DTRYKGE8.md" tag="02DTRYKGE8" kind="definition">
# Transposition

## Definition

Let $X$ be a set and let $a, b \in X$ with $a \neq b$.
The transposition $(a\ b)$ is the permutation of $X$ that sends $a$ to $b$, sends $b$ to $a$, and fixes each other element of $X$.

## Remarks

A transposition is a $2$-cycle in [[cycle notation|0ZKPH2BD1T]], and $(a\ b) = (b\ a)$.
Each transposition is its own inverse, so it has [[order|0WE2NHZJ7B]] $2$.
The symmetric group $S_n$ contains exactly $\binom{n}{2}$ transpositions.

## Explanations

### L1 human

A transposition swaps two things and leaves everything else in place.
If you do the same swap two times, you are back at the start.
All permutations can be built from transpositions.
</document>

<document source="https://tamnd.github.io/math2/node/02E1T12PP4.md" tag="02E1T12PP4" kind="definition">
# Free module

## Definition

Let $R$ be a ring with identity element and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.
A subset $E \subseteq M$ is a *basis* of $M$ when:

1. Each $x \in M$ is a finite sum $r_1 e_1 + \cdots + r_k e_k$ with $r_i \in R$ and $e_i \in E$.
2. If $e_1, \dots, e_k \in E$ are distinct and $r_1 e_1 + \cdots + r_k e_k = 0$, then $r_1 = \cdots = r_k = 0$.

The module $M$ is *free* when it has a basis.

## Remarks

$R^n$ is free, with the standard basis; each vector space is free over its field.
For $n \ge 2$, the $\Z$-module $\Z / n\Z$ is not free, because $n x = 0$ for all its elements.
Over a nonzero commutative ring, any two bases of a free module have the same cardinality, called the *rank*.
A submodule of a free module need not be free, but over a principal ideal domain it is free.

## Explanations

### L1 human

A free module is a module that behaves like $R^n$: it has a basis.
Every vector space has a basis, but many modules do not.
For example, $\Z / 2\Z$ has no basis over $\Z$, because $2x = 0$ for every element $x$.
</document>

<document source="https://tamnd.github.io/math2/node/02E9VT1RVQ.md" tag="02E9VT1RVQ" kind="concept">
# Power of a point

## Summary

The power of a point with respect to a circle is the square of its distance to the centre minus the square of the radius. It is positive outside the circle, zero on the circle and negative inside.
</document>

<document source="https://tamnd.github.io/math2/node/02EEX96DFS.md" tag="02EEX96DFS" kind="statement">
# Open balls form a basis of the metric topology

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]].
Then the set of all [[open balls|0DC3MPR7C9]] $B(x, r)$, with $x \in X$ and real $r > 0$, is a [[basis|06EX0FHBTF]] of the [[metric topology|0A6YRNZB2R]] of $X$.
Also, the balls $B(x, 1/n)$ with $x \in X$ and integer $n \ge 1$ form a basis.

## Remarks

Proof: each ball is open, by [[open balls are open|0B2305GNAZ]]. If $U$ is open and $x \in U$, then some ball $B(x, r)$ lies in $U$, and $B(x, 1/n) \subseteq B(x, r)$ for $n > 1/r$.
For each point $x$, the balls $B(x, 1/n)$ form a countable neighbourhood basis at $x$. Thus each metric space is [[first-countable|0HG11FW8MJ]].
In $\mathbb{R}$, the open balls are the bounded open intervals.

## Explanations

### L1 human

Each open set of a metric space is a union of open balls.
So, to check a property for all open sets, it is often enough to check it for balls.
</document>

<document source="https://tamnd.github.io/math2/node/02EKEPPJ3R.md" tag="02EKEPPJ3R" kind="concept">
# Prime factorization

## Summary

The prime factorization of a positive integer writes it as a product of prime numbers.
The problem of integer factorization is to find this product for a given integer.
The [[definition of a prime factorization|0AEKDFCYFA]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/02FDWY55AS.md" tag="02FDWY55AS" kind="statement">
# Rank of a product of matrices

## Statement

Let $F$ be a field, let $A$ be an $m \times n$ matrix and $B$ an $n \times p$ matrix over $F$.
Then
$$\rank(AB) \le \min(\rank A, \rank B),$$
where $\rank$ is the [[rank of a matrix|0ZC9YTXZRY]].

## Remarks

The column space of $AB$ is contained in the column space of $A$, so $\rank(AB) \le \rank A$.
The row space of $AB$ is contained in the row space of $B$, so $\rank(AB) \le \rank B$.
If $A$ is invertible, then $\rank(AB) = \rank B$; if $B$ is invertible, then $\rank(AB) = \rank A$.
A lower bound is Sylvester's rank inequality: $\rank A + \rank B - n \le \rank(AB)$.

## Explanations

### L1 human

Multiplying by a matrix can never create new independent directions.
So the rank of a product is at most the rank of each factor.
Multiplying by an invertible matrix keeps the rank.
</document>

<document source="https://tamnd.github.io/math2/node/02GK6HZYBT.md" tag="02GK6HZYBT" kind="concept">
# Transitive action

## Summary

A group action is transitive when it has exactly one orbit.
Then each point can be moved onto each other point by some element of the group.
</document>

<document source="https://tamnd.github.io/math2/node/02HB5F40VR.md" tag="02HB5F40VR" kind="example">
# Newton's law of cooling

## Example

Let $T(t)$ be the temperature of a body at time $t$. Let $T_a$ be the constant temperature of the surroundings, and let $k > 0$ be a constant.
Newton's law of cooling is the equation
$$T'(t) = -k \left(T(t) - T_a\right).$$
With the initial condition $T(0) = T_0$, its unique solution is
$$T(t) = T_a + (T_0 - T_a)\, e^{-k t}.$$

## Remarks

The difference $D = T - T_a$ satisfies $D' = -k D$. This is [[exponential decay|00SVP6W3CB]], so $D(t) = D(0) e^{-kt}$.
As $t \to \infty$, $T(t) \to T_a$. If $T_0 > T_a$ the body cools, and if $T_0 < T_a$ it warms.
The law is a good model when the temperature difference is small and the body has a nearly uniform temperature.

## Explanations

### L1 human

A hot cup of coffee cools fast at first and then more slowly.
The law says that the speed of cooling is proportional to the difference between the coffee and the room.
So the difference halves after each fixed time interval.
</document>

<document source="https://tamnd.github.io/math2/node/02HHNPC8ED.md" tag="02HHNPC8ED" kind="statement">
# Casorati–Weierstrass theorem

## Statement

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 < |z - c| < r$.
Assume that $c$ is an [[essential singularity|0YKDJBV26S]] of $f$.
Then for each $\delta$ with $0 < \delta < r$, the image $f(\{z : 0 < |z - c| < \delta\})$ is dense in $\mathbb{C}$.
Thus for each $w \in \mathbb{C}$ there is a sequence $z_n \to c$ with $z_n \neq c$ and $f(z_n) \to w$.

## Remarks

Proof idea: if the image of a punctured disc misses a disc $|w - w_0| < \varepsilon$, then $g(z) = 1/(f(z) - w_0)$ is bounded near $c$. By [[Riemann's theorem|0BSQDZWF34]], $g$ is holomorphic at $c$, and then $c$ is removable or a pole of $f$.
[[Picard's great theorem|08QA8QTM71]] is a much stronger statement.
In the Russian literature, the theorem has the name of Sokhotski.

## Explanations

### L1 human

Near an essential singularity, the function comes as close as you like to each complex number.
For example, $e^{1/z}$ comes close to $5$, to $-i$ and to $0$ on each small disc around $0$.
</document>

<document source="https://tamnd.github.io/math2/node/02HJYT53F7.md" tag="02HJYT53F7" kind="statement">
# Existence of maximal ideals

## Statement

Let $R$ be a [[ring|0894J980RK]] and let $I$ be an ideal of $R$ with $I \neq R$.
Then there is a [[maximal ideal|09J6KGT65G]] $M$ of $R$ with $I \subseteq M$.
In particular, every nonzero ring has a maximal ideal.

## Remarks

The proof uses Zorn's lemma. The union of a chain of proper ideals that contain $I$ is again a proper ideal, because it does not contain $1$.
The statement for every nonzero ring is equivalent to the axiom of choice, over the other axioms of ZF set theory.
The statement holds for left, right and two-sided ideals.

## History

Wolfgang Krull proved the theorem in 1929 with transfinite induction.

## Explanations

### L1 human

Start with a proper ideal and make it larger, step by step, without reaching the whole ring.
In an infinite ring this process may not stop by itself.
Zorn's lemma guarantees that an ideal that you cannot enlarge exists.
</document>

<document source="https://tamnd.github.io/math2/node/02HPP5VCZH.md" tag="02HPP5VCZH" kind="concept">
# Isometry

## Summary

An isometry is a map between metric spaces that keeps all distances.
</document>

<document source="https://tamnd.github.io/math2/node/02HR4C0B04.md" tag="02HR4C0B04" kind="concept">
# Nullcline

## Summary

A nullcline of a system of autonomous equations $x_j' = f_j(x)$ is a set on which one component $f_j$ of the vector field vanishes.
The equilibrium points of the system are the common points of all the nullclines. See the [[definition|0WW060JR6P]].
</document>

<document source="https://tamnd.github.io/math2/node/02HTDK1FBD.md" tag="02HTDK1FBD" kind="definition">
# Inverse trigonometric functions

## Definition

The inverse trigonometric functions are the inverses of the restrictions of the trigonometric functions. The restrictions are to the following intervals:
$\sin$ on $[-\pi/2, \pi/2]$, $\cos$ on $[0, \pi]$, and $\tan$ on $(-\pi/2, \pi/2)$.
They are the [[arcsine|0B89D93YA1]] $\arcsin \colon [-1, 1] \to [-\pi/2, \pi/2]$, the [[arccosine|0RDP2HEFEW]] $\arccos \colon [-1, 1] \to [0, \pi]$ and the [[arctangent|065XW2V6C3]] $\arctan \colon \mathbb{R} \to (-\pi/2, \pi/2)$.

## Remarks

The trigonometric functions are not injective on $\mathbb{R}$. The restrictions to these intervals are strictly monotone and continuous, so they have continuous inverses.
The values are the principal values. Other notations are $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$.
Similar definitions give $\operatorname{arccot}$, $\operatorname{arcsec}$ and $\operatorname{arccsc}$.

## Explanations

### L1 human

An inverse trigonometric function gives back an angle from a value of sine, cosine or tangent.
We restrict to one interval so that the answer is unique.
</document>

<document source="https://tamnd.github.io/math2/node/02HZ1BK3MW.md" tag="02HZ1BK3MW" kind="statement">
# Chebyshev's inequality

## Statement

Let $X$ be a real random variable with finite mean $\mu$ and finite [[variance|007QED3E0Y]] $\sigma^2$. Then for each $a > 0$,
$$P(|X - \mu| \ge a) \le \frac{\sigma^2}{a^2}.$$
Equivalently, if $\sigma > 0$, then $P(|X - \mu| \ge k\sigma) \le 1/k^2$ for each $k > 0$.

## Remarks

Proof: apply [[Markov's inequality|05PWWHYBX5]] to the nonnegative variable $(X - \mu)^2$ and the level $a^2$.
For $k = 2$, at least $3/4$ of the mass is within two [[standard deviations|0VTN5RKNC5]] of the mean, for every distribution with finite variance.
The bound is sharp: some distributions reach equality for a given $k \ge 1$.
Chebyshev's inequality gives a short proof of the weak law of large numbers for variables with finite variance.

## History

The inequality is named after Pafnuty Chebyshev. Irénée-Jules Bienaymé formulated it first.

## Explanations

### L1 human

The variance controls how often a variable can be far from its mean.
For any distribution, at most $1/9$ of the mass lies three or more standard deviations away from the mean.
</document>

<document source="https://tamnd.github.io/math2/node/02J4G4J4TJ.md" tag="02J4G4J4TJ" kind="statement">
# Isometries are injective

## Statement

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be an [[isometry|0878DN3WCS]].
Then $f$ is [[injective|0PVG1CRAJN]].

## Remarks

Proof: if $f(x) = f(y)$, then $d_X(x, y) = d_Y(f(x), f(y)) = 0$, thus $x = y$.
The proof uses the axiom of a metric that $d(x, y) = 0$ implies $x = y$. For a pseudometric, the statement can fail.
Thus an isometry is a bijection from $X$ onto its image $f(X)$.

## Explanations

### L1 human

Two different points have a positive distance, and an isometry keeps that distance.
So their images are also at a positive distance, and they are different.
</document>

<document source="https://tamnd.github.io/math2/node/02JBEMG8R4.md" tag="02JBEMG8R4" kind="concept">
# Monotone sequence

## Summary

A monotone sequence is a real sequence that never decreases or never increases. A bounded monotone sequence always converges. The node [[monotone sequence|0KN22AAW5R]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/02JGYG0T7M.md" tag="02JGYG0T7M" kind="statement">
# Characteristic function of a sum of independent random variables

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] real random variables, and let $S = X_1 + \dots + X_n$.
Then for each $t \in \mathbb{R}$, the [[characteristic functions|0MYDNRT7PP]] satisfy
$$\varphi_S(t) = \varphi_{X_1}(t) \, \varphi_{X_2}(t) \cdots \varphi_{X_n}(t).$$

## Remarks

Proof: $e^{itS} = e^{itX_1} \cdots e^{itX_n}$. The factors are independent and bounded, so the [[product rule for expectations|05AW1G0WE1]] applies to their real and imaginary parts.
For i.i.d. $X_i$, $\varphi_S(t) = \varphi_{X_1}(t)^n$. The sample mean has $\varphi_{S/n}(t) = \varphi_{X_1}(t/n)^n$.
The converse is false: $\varphi_{X+Y} = \varphi_X \varphi_Y$ does not imply that $X$ and $Y$ are independent.

## Explanations

### L1 human

Adding independent variables corresponds to multiplying their characteristic functions.
This is the key step in the usual proof of the central limit theorem.
</document>

<document source="https://tamnd.github.io/math2/node/02KTRHQMHN.md" tag="02KTRHQMHN" kind="statement">
# Power series of sine

## Statement

For every real $x$, the series $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$ converges absolutely, and
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots.$$
This series is the [[Maclaurin series|043SJ5RF59]] of the sine function.

## Remarks

The [[ratio of the terms|0JBMYAA9QH]] is $\frac{x^2}{(2n+2)(2n+3)} \to 0$. So the radius of convergence is $\infty$.
With the series as the definition of $\sin$, the equality holds by definition. With a geometric definition, Taylor's theorem proves it.
For $|x| \le 1$, the error after the term $x^{2n+1}$ is at most $\frac{1}{(2n+3)!}$. Mathlib states the series as `Real.hasSum_sin`.

## Explanations

### L1 human

Sine is an infinite odd polynomial with alternating signs.
Near $0$, $\sin x$ is close to $x$, and closer still to $x - x^3/6$.
</document>

<document source="https://tamnd.github.io/math2/node/02KYGSCTCN.md" tag="02KYGSCTCN" kind="definition">
# Stationary distribution

## Definition

Let $P = (p_{ij})_{i, j \in S}$ be the [[transition matrix|0F1KBVY161]] of a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable state space $S$.
A stationary distribution of the chain is a family $\pi = (\pi_j)_{j \in S}$ of real numbers with $\pi_j \ge 0$ for each $j$, $\sum_{j \in S} \pi_j = 1$, and
$$\pi_j = \sum_{i \in S} \pi_i \, p_{ij} \quad \text{for each } j \in S.$$
In matrix form, $\pi P = \pi$ for the row vector $\pi$.

## Remarks

If $X_0$ has the law $\pi$, then $X_n$ has the law $\pi$ for each $n \ge 0$.
An irreducible chain on a finite state space has exactly one stationary distribution. Its entries are positive, and $\pi_j = 1 / E_j[T_j]$, where $T_j$ is the time of the first return to $j$.
For an irreducible and aperiodic finite chain, $P(X_n = j) \to \pi_j$ as $n \to \infty$, for each initial law.
The simple random walk on $\mathbb{Z}$ has no stationary distribution.

## Explanations

### L1 human

A stationary distribution is a balance: the flow of probability into each state equals the flow out.
For many chains, it also describes the long-run fraction of time that the chain spends in each state.
</document>

<document source="https://tamnd.github.io/math2/node/02M2K4QH9P.md" tag="02M2K4QH9P" kind="definition">
# Fermat pseudoprime

## Definition

Let $a$ be an integer with $a > 1$.
A Fermat pseudoprime to the base $a$ is a [[composite number|0N3ADXHNTQ]] $n$ such that $a^{n-1} \equiv 1 \pmod n$.

## Remarks

[[Fermat's little theorem|0M25PRN3WQ]] says that each prime $n$ with $n \nmid a$ satisfies $a^{n-1} \equiv 1 \pmod n$. A Fermat pseudoprime is a composite number that passes the same test.
The smallest Fermat pseudoprime to the base $2$ is $341 = 11 \cdot 31$.
The condition implies $\gcd(a, n) = 1$.
A composite number that is a Fermat pseudoprime to each base coprime to it is a [[Carmichael number|0TA7WT68RK]].

## Explanations

### L1 human

The [[Fermat primality test|0N5ZSZMYH6]] declares $n$ "probably prime" when $a^{n-1} \equiv 1 \pmod n$.
A Fermat pseudoprime is a composite number that fools this test for the base $a$.
For example, $2^{340} \equiv 1 \pmod{341}$, but $341$ is not prime.
</document>

<document source="https://tamnd.github.io/math2/node/02MAKWDKGT.md" tag="02MAKWDKGT" kind="concept">
# Polynomial evaluation

## Summary

To evaluate a polynomial at an element, you replace the variable by the element and compute the result in the ring.
The [[definition of evaluation|0GJ7T0NK49]] gives the formula.
</document>

<document source="https://tamnd.github.io/math2/node/02MY6WZXEQ.md" tag="02MY6WZXEQ" kind="concept">
# Discrete uniform distribution

## Summary

The discrete uniform distribution on a finite set gives the same probability to each element of the set.
The node [[discrete uniform distribution|0KYP951QWH]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/02NBKVQFRQ.md" tag="02NBKVQFRQ" kind="definition">
# Intermediate field

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
An intermediate field of $L/K$ is a [[subfield|07QY5RPFBR]] $E$ of $L$ with $K \subseteq E$.

## Remarks

Then $L/E$ and $E/K$ are field extensions, and one calls $K \subseteq E \subseteq L$ a tower of fields.
The fields $K$ and $L$ are intermediate fields of $L/K$.
For example, $\mathbb{Q}(\sqrt{2})$ is an intermediate field of $\mathbb{R}/\mathbb{Q}$.
The [[fundamental theorem of Galois theory|0T1Q8D0FW8]] describes the intermediate fields of a Galois extension.

## Explanations

### L1 human

An intermediate field sits between the small field and the big field.
Galois theory gives a way to list all such fields, by a correspondence with subgroups of a group.
</document>

<document source="https://tamnd.github.io/math2/node/02NSMQVQWK.md" tag="02NSMQVQWK" kind="definition">
# Divergence

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $F = (F_1, \dots, F_n) \colon U \to \mathbb{R}^n$ be a [[vector field|0N67NBKQ5T]] whose [[partial derivatives|05AXG3QRKR]] $\partial F_i / \partial x_i$ exist on $U$.
The divergence of $F$ is the function $\operatorname{div} F \colon U \to \mathbb{R}$,
$$\operatorname{div} F = \nabla \cdot F = \frac{\partial F_1}{\partial x_1} + \frac{\partial F_2}{\partial x_2} + \dots + \frac{\partial F_n}{\partial x_n}.$$

## Remarks

The divergence is the trace of the [[Jacobian matrix|0YF0VXKXGB]] of $F$.
For $F(x) = x$ on $\mathbb{R}^n$, $\operatorname{div} F = n$. For the rotation field $F(x, y) = (-y, x)$, $\operatorname{div} F = 0$.
The divergence is linear, and $\operatorname{div}(\phi F) = \nabla \phi \cdot F + \phi \operatorname{div} F$ for a differentiable function $\phi$.
Take a $C^1$ field and a small region around $x_0$. The divergence at $x_0$ is the limit of the outward flux through the boundary of the region, divided by its volume. This follows from the [[divergence theorem|0JX86PSSA9]].

## Explanations

### L1 human

Think of $F$ as the velocity of a fluid.
Where the divergence is positive, more fluid leaves a small region than enters it, as at a source. Where it is negative, fluid gathers, as at a sink.
</document>

<document source="https://tamnd.github.io/math2/node/02NZZ1J0FH.md" tag="02NZZ1J0FH" kind="definition">
# Splitting field

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be nonzero.
A splitting field of $f$ over $K$ is a [[field extension|0WQ1A8SHTZ]] $L/K$ with these two properties.

1. The polynomial $f$ [[splits|0SHMC00RJV]] over $L$, as $f = c \, (x - a_1) \cdots (x - a_n)$.
2. The field $L$ is [[generated over $K$|0MA5B744K3]] by the roots: $L = K(a_1, \dots, a_n)$.

## Remarks

Splitting fields [[exist|01441KDGBY]] and are [[unique up to isomorphism|069A3BC3EP]].
The splitting field of $x^2 + 1$ over $\mathbb{R}$ is $\mathbb{C}$.
The splitting field of $x^3 - 2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt[3]{2}, \omega)$ with $\omega = e^{2\pi i/3}$. Its degree over $\mathbb{Q}$ is $6$.
The splitting field of $x^q - x$ over $\mathbb{F}_p$, for $q = p^n$, is the [[finite field|08RTDHX1SK]] with $q$ elements.

## Explanations

### L1 human

A splitting field is the smallest field that contains all the roots of a polynomial.
It is where the polynomial factors completely.
For $x^3 - 2$, the real cube root of $2$ is not enough, because the other two roots are not real.
</document>

<document source="https://tamnd.github.io/math2/node/02P2PYH3X3.md" tag="02P2PYH3X3" kind="statement">
# The complex exponential is 2πi-periodic

## Statement

For each $z \in \mathbb{C}$,
$$e^{z + 2 \pi i} = e^{z},$$
where $e^z$ is the [[complex exponential|054VXHMCA3]] of $z$.

## Remarks

Proof: by [[Euler's formula|04R3CY41VJ]], $e^{2 \pi i} = \cos 2\pi + i \sin 2\pi = 1$. By the [[product of exponentials|0WM7DJCM6G]], $e^{z + 2\pi i} = e^z e^{2\pi i} = e^z$.
The periods of $\exp$ are exactly the numbers $2 \pi i k$ with $k \in \mathbb{Z}$, by [[exp z = 1 iff z is in 2πiZ|013AHC4DRT]].
So $\exp$ is not injective on $\mathbb{C}$. It is injective on each horizontal strip $\{z : a < \operatorname{Im} z \le a + 2\pi\}$ with $a \in \mathbb{R}$.

## Explanations

### L1 human

Adding $2 \pi i$ to $z$ adds a full turn to the direction of $e^z$ and does not change its size.
So $e^z$ takes the same value again: the exponential repeats itself along each vertical line, with step $2\pi$.
</document>

<document source="https://tamnd.github.io/math2/node/02PETK9PQ4.md" tag="02PETK9PQ4" kind="statement">
# Sum of the roots of unity

## Statement

Let $n \ge 2$ be an integer and let $\zeta$ be a [[primitive root of unity|0J817RTVG6]].
Then
$$\sum_{k=0}^{n-1} \zeta^k = 1 + \zeta + \zeta^2 + \dots + \zeta^{n-1} = 0.$$

## Remarks

Proof: $\zeta \neq 1$, so the [[geometric sum formula|03X8V7D64M]] gives $\frac{\zeta^n - 1}{\zeta - 1} = 0$.
Thus the sum of all the $n$th roots of unity is $0$ for $n \ge 2$. For $n = 1$ the sum is $1$.

## Explanations

### L1 human

The $n$th roots of unity are the corners of a regular polygon with center $0$.
By symmetry, their center of mass is $0$, and so their sum is $0$.
For $n = 4$: $1 + i + (-1) + (-i) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/02QGGD5G87.md" tag="02QGGD5G87" kind="concept">
# Complex logarithm

## Summary

A complex logarithm of a nonzero complex number $z$ is a complex number $w$ with $e^w = z$, see the [[definition|0X2FY491PF]].
Each $z \neq 0$ has infinitely many logarithms, which differ by integer multiples of $2\pi i$.
A complex logarithm function is a continuous choice of a logarithm on an open subset of $\mathbb{C} \setminus \{0\}$.
</document>

<document source="https://tamnd.github.io/math2/node/02QJVFCNYD.md" tag="02QJVFCNYD" kind="concept">
# Line integral of a scalar field

## Summary

The line integral of a scalar field along a curve adds up the values of the field along the curve. Each value is weighted by a small length of the curve.
For a non-negative field on a plane curve, it is the area of the curtain between the curve and the graph of the field.
</document>

<document source="https://tamnd.github.io/math2/node/02RRHJ7R98.md" tag="02RRHJ7R98" kind="concept">
# Multiplicative order

## Summary

The multiplicative order of an integer $a$ modulo $n$, with $a$ coprime to $n$, is the smallest positive exponent $k$ such that $a^k \equiv 1 \pmod n$.
The [[definition of the multiplicative order|0SBRX5CZ5R]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/02RSXVM1ZN.md" tag="02RSXVM1ZN" kind="definition">
# Higher partial derivatives

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $i, j \in \{1, \dots, n\}$.
Suppose that the [[partial derivative|05AXG3QRKR]] $\partial f / \partial x_i$ exists at each point of $U$.
The second partial derivative of $f$ with respect to $x_i$ and then $x_j$ at $a \in U$ is
$$\frac{\partial^2 f}{\partial x_j \, \partial x_i}(a) = \frac{\partial}{\partial x_j}\left(\frac{\partial f}{\partial x_i}\right)(a),$$
when this partial derivative exists. It is also written $f_{x_i x_j}(a)$ or $\partial_j \partial_i f(a)$.
For $i = j$, one writes $\partial^2 f / \partial x_i^2$.
By induction, a partial derivative of order $k \ge 2$ is a partial derivative of a partial derivative of order $k - 1$: $\dfrac{\partial^k f}{\partial x_{i_k} \cdots \partial x_{i_1}} = \dfrac{\partial}{\partial x_{i_k}} \left( \dfrac{\partial^{k-1} f}{\partial x_{i_{k-1}} \cdots \partial x_{i_1}} \right)$.

## Remarks

In $\partial^2 f / \partial x_j \, \partial x_i$ the variable nearest to $f$ is differentiated first. In $f_{x_i x_j}$ the variable written first is differentiated first.
For $f(x, y) = x^3 y^2$, one gets $f_{xx} = 6 x y^2$, $f_{xy} = f_{yx} = 6 x^2 y$ and $f_{yy} = 2 x^3$.
If the second partial derivatives are continuous, then the order of differentiation does not matter, by [[Schwarz's theorem|0NNQY8WCW6]]. Without continuity, the mixed partial derivatives [[can differ|01NN8NY497]].
The second partial derivatives at a point form the [[Hessian matrix|0693Y2METT]].

## Explanations

### L1 human

A second partial derivative measures how a slope changes. For example, $f_{xy}$ measures how the slope in the $x$-direction changes as you move in the $y$-direction.
You compute it by differentiating twice, one variable at a time.
</document>

<document source="https://tamnd.github.io/math2/node/02SJG1HC32.md" tag="02SJG1HC32" kind="definition">
# Algebraically closed field

## Definition

A [[field|0QA8CHH8MG]] $K$ is algebraically closed if every polynomial $f \in K[x]$ with $\deg f \ge 1$ has a [[root|00EHX6FWC0]] in $K$.

## Remarks

Equivalently, every nonzero polynomial over $K$ [[splits|0SHMC00RJV]] over $K$. Also equivalently, the [[irreducible polynomials|00X2A9G9W3]] over $K$ are exactly the polynomials of degree one.
Another equivalent condition: $K$ has no [[algebraic extension|09EYMKMSB9]] other than $K$ itself.
The field $\mathbb{C}$ is algebraically closed, by the fundamental theorem of algebra. The field $\mathbb{R}$ is not, because $x^2 + 1$ has no real root.
No finite field is algebraically closed: for a finite field $K$, the polynomial $1 + \prod_{a \in K} (x - a)$ has no root in $K$.

## Explanations

### L1 human

In an algebraically closed field, every polynomial equation of degree at least one has a solution.
You never need a bigger field to find roots.
</document>

<document source="https://tamnd.github.io/math2/node/02SMVGC690.md" tag="02SMVGC690" kind="concept">
# Topological property

## Summary

A topological property is a property of topological spaces such that, if a space has it, each [[homeomorphic|05GZ11PWRY]] space also has it.
Examples are [[compactness|0QPHW703PV]], [[connectedness|0D939CMME7]], [[separability|0GKJ78948V]], [[metrizability|09KDHB82SG]] and the [[Hausdorff property|0MXB4CKHFW]].

## Remarks

Topological properties are used to show that two spaces are not homeomorphic. For example, $[0, 1]$ is compact and $(0, 1)$ is not, so they are not homeomorphic.
Properties that depend on a metric are often not topological. For example, $(0, 1)$ and $\mathbb{R}$ are homeomorphic, but only $\mathbb{R}$ is [[complete|08EWT7J0SC]] and only $(0, 1)$ is bounded.

## Explanations

### L1 human

A topological property is a property that a space keeps when you deform it by a homeomorphism.
Size and distance are not topological properties, but the number of pieces of a space is one.
</document>

<document source="https://tamnd.github.io/math2/node/02SRXDAM3X.md" tag="02SRXDAM3X" kind="concept">
# Reduction of order

## Summary

Reduction of order is a method to find a second solution of a homogeneous linear equation from a known one.
Let $p, q$ be continuous on an open interval $I$, and let $y_1$ be a solution of $y'' + p(x) y' + q(x) y = 0$ with no zero on $I$.
Put $y = v\, y_1$. Then $y$ is a solution if and only if $y_1 v'' + (2 y_1' + p\, y_1) v' = 0$, a first-order linear equation for $v'$.
One solution gives $y_2 = y_1 \int \frac{e^{-\int p\, dx}}{y_1^2}\, dx$, and $(y_1, y_2)$ is a [[fundamental system|0H4KNNCHPY]] on $I$.

## Remarks

Example: $y_1 = e^{x}$ solves $y'' - 2 y' + y = 0$. Here $p = -2$, so $v' = e^{2x}/e^{2x} = 1$, and $y_2 = x e^x$.
The [[Wronskian|0N5NF1RGBP]] of $y_1$ and $y_2$ is $e^{-\int p\, dx}$, which has no zero. This is consistent with [[Abel's identity|0VGWKX2JHB]].
The same idea lowers the order by one for a linear equation of order $n$.

## Explanations

### L1 human

If you know one solution, look for a second one as the first one times an unknown function $v$.
The terms without derivatives of $v$ cancel, because $y_1$ is a solution.
What remains is a first-order equation for $v'$, which is easy to solve.
</document>

<document source="https://tamnd.github.io/math2/node/02STTYFBVY.md" tag="02STTYFBVY" kind="definition">
# Polar form of a quadratic form

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
The *polar form* of $Q$ is the map $B_Q \colon V \times V \to F$ with
$$B_Q(u, v) = Q(u + v) - Q(u) - Q(v).$$

## Remarks

$B_Q$ is a [[symmetric bilinear form|0AXZNE5XNH]], and $B_Q(v, v) = 2 Q(v)$ for all $v$.
For $Q(x) = x^T A x$ on $F^n$, $B_Q(x, y) = x^T (A + A^T) y$.
Many authors divide by $2$ when the characteristic is not $2$, so that the result $B$ satisfies $B(v, v) = Q(v)$; see [[the associated form|0KRVRV5V80]].
This construction is the degree $2$ case of the polarisation of homogeneous polynomials.

## Explanations

### L1 human

The polar form recovers a bilinear form from a quadratic form.
For $Q(x) = x^2$ on $\R$, the polar form is $(x + y)^2 - x^2 - y^2 = 2xy$.
It is the algebraic form of the [[polarisation identity|05FKHRT0B4]] for inner products.
</document>

<document source="https://tamnd.github.io/math2/node/02SYZAWM2K.md" tag="02SYZAWM2K" kind="statement">
# Cauchy mean value theorem

## Statement

Let $a < b$ and let $f, g \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$.
Then there is $c \in (a, b)$ with
$$(f(b) - f(a)) \, g'(c) = (g(b) - g(a)) \, f'(c).$$

## Remarks

Apply [[Rolle's theorem|0194DY35BG]] to $h(x) = (f(b) - f(a)) g(x) - (g(b) - g(a)) f(x)$, which has $h(a) = h(b)$.
With $g(x) = x$ it is the [[mean value theorem|0BARQY52N5]].
It is the main step in the proof of [[L'Hôpital's rule|0JWT4PPQ7B]]. Mathlib states it as `exists_ratio_deriv_eq_ratio_slope`.

## Explanations

### L1 human

Think of $(g(t), f(t))$ as a point moving along a curve in the plane.
At some time, the direction of motion is parallel to the chord joining the start and the end.
</document>

<document source="https://tamnd.github.io/math2/node/02TZDKG50Q.md" tag="02TZDKG50Q" kind="statement">
# Closure is closed

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then the [[closure|0DRKT1QJR7]] $\overline{S}$ is a [[closed set|0G5AV9Q4RR]] of $X$.

## Remarks

Proof: $\overline{S}$ is an intersection of closed sets, and each [[intersection of closed sets is closed|09YMV8H4X5]].
It follows that $\overline{\overline{S}} = \overline{S}$.

## Explanations

### L1 human

The closure is built as an intersection of closed sets, and such an intersection is again closed.
</document>

<document source="https://tamnd.github.io/math2/node/02V2GYAC8P.md" tag="02V2GYAC8P" kind="statement">
# Decreasing sequence bounded below converges

## Statement

Let $(a_n)$ be a decreasing real sequence: $a_n \ge a_{n+1}$ for all $n \in \mathbb{N}$.
Assume that $(a_n)$ is bounded below.
Then $(a_n)$ converges, and its limit is the infimum $\inf \{a_n : n \in \mathbb{N}\}$.

## Remarks

The proof applies the [[monotone convergence theorem|0FM6N4R0VB]] to the increasing sequence $(-a_n)$.
For example, $1/n$ decreases and is bounded below by $0$. Its limit is its infimum $0$.
Together, the two statements say that a monotone sequence converges if and only if it is bounded.

## Explanations

### L1 human

A decreasing sequence that cannot pass a floor must settle down.
It settles at the highest floor, which is the infimum of its terms.
</document>

<document source="https://tamnd.github.io/math2/node/02WF878BH1.md" tag="02WF878BH1" kind="definition">
# Greatest common divisor

## Definition

Let $a$ and $b$ be integers that are not both $0$.
A common divisor of $a$ and $b$ is an integer $d$ such that $d \mid a$ and $d \mid b$.
The greatest common divisor $\gcd(a, b)$ is the largest common divisor of $a$ and $b$.
By convention, $\gcd(0, 0) = 0$.

## Remarks

The number $\gcd(a, b)$ is positive when $a$ and $b$ are not both $0$, because $1$ is a common divisor.
An equivalent definition: $\gcd(a, b)$ is the unique nonnegative common divisor of $a$ and $b$ that each common divisor [[divides|096PW7Q014]]. This form also gives $\gcd(0, 0) = 0$.
For all integers $a$, $\gcd(a, 0) = |a|$.
The [[Euclidean algorithm|0TRNT4X7J0]] computes $\gcd(a, b)$.

## Explanations

### L1 human

List the positive divisors of each number, and take the largest number that is in both lists.
The divisors of $12$ are $1, 2, 3, 4, 6, 12$, and the divisors of $18$ are $1, 2, 3, 6, 9, 18$.
Thus $\gcd(12, 18) = 6$.
</document>

<document source="https://tamnd.github.io/math2/node/02WHMEG5ZR.md" tag="02WHMEG5ZR" kind="statement">
# Riemann mapping theorem

## Statement

Let $U$ be a nonempty open subset of $\mathbb{C}$ with $U \neq \mathbb{C}$.
Suppose that $U$ is simply connected: $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path.
Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$.
Then there is a biholomorphic map $f$ from $U$ onto $\mathbb{D}$, that is, a bijective [[holomorphic|0DG0Z72CQD]] map with a holomorphic inverse.
Moreover, for each $z_0 \in U$ there is exactly one such $f$ with $f(z_0) = 0$ and $f'(z_0)$ a positive real number.

## Remarks

A biholomorphic map is [[conformal|0VHNKP1N3K]], so $U$ and $\mathbb{D}$ are conformally equivalent.
The case $U = \mathbb{C}$ is excluded: a holomorphic map $\mathbb{C} \to \mathbb{D}$ is bounded, so it is constant by [[Liouville's theorem|0EHKVRF4J6]].
The uniqueness follows from the [[Schwarz lemma|0MB48XDBY3]].
Riemann stated the theorem in his thesis of 1851, and Osgood gave the first rigorous proof in 1900.

## Explanations

### L1 human

The theorem says that every simply connected open set in the plane, except the plane itself, looks like the unit disc to complex analysis.
The set can have a very irregular boundary, and the map still exists.
For the upper half-plane, the [[Cayley transform|0X45ZM731G]] is such a map.
</document>

<document source="https://tamnd.github.io/math2/node/02WZB422CE.md" tag="02WZB422CE" kind="definition">
# External direct sum

## Definition

Let $F$ be a field and let $(V_i)_{i \in I}$ be a family of [[vector spaces|000AYPA9AJ]] over $F$.
The *external direct sum* $\bigoplus_{i \in I} V_i$ is the set of families $(v_i)_{i \in I}$ with $v_i \in V_i$ for each $i$ and $v_i = 0$ for all but finitely many $i$.
It is a vector space over $F$ with the componentwise operations
$$(v_i)_i + (w_i)_i = (v_i + w_i)_i, \qquad a (v_i)_i = (a v_i)_i.$$

## Remarks

For finite $I$, the direct sum equals the [[product|0EX2HPXPJ6]] $\prod_{i \in I} V_i$.
For infinite $I$, it is a proper subspace of the product when infinitely many $V_i$ are nonzero.
Each $V_j$ embeds in the direct sum as the families that are zero outside $j$.
The direct sum is the [[internal direct sum|0XYYT5KNTX]] of these images.
The direct sum of countably many copies of $F$ is isomorphic to $F[x]$.

## Explanations

### L1 human

The external direct sum builds a new space out of several spaces, side by side.
A vector of the new space chooses one vector in each space, with only finitely many nonzero choices.
For two spaces, this is just the space of pairs.
</document>

<document source="https://tamnd.github.io/math2/node/02XPXR1FBK.md" tag="02XPXR1FBK" kind="definition">
# Bernoulli process

## Definition

Let $p \in [0, 1]$.
A Bernoulli process with parameter $p$ is a [[stochastic process|0QH5ENKZM1]] $(X_n)_{n \ge 1}$ of [[i.i.d.|0AJN7WHBJZ]] random variables with the [[Bernoulli distribution|0J97P5KZY0]] $\operatorname{Bernoulli}(p)$:
$$P(X_n = 1) = p \quad \text{and} \quad P(X_n = 0) = 1 - p \quad \text{for each } n \ge 1.$$

## Remarks

One calls $X_n = 1$ a success and $X_n = 0$ a failure in trial $n$.
The number of successes in the first $n$ trials has the binomial distribution $\operatorname{Bin}(n, p)$.
For $0 < p \le 1$, the number of failures before the first success has the geometric distribution with parameter $p$. The times between successes are i.i.d.
The Bernoulli process is a discrete-time analogue of the Poisson process.

## Explanations

### L1 human

The Bernoulli process is the mathematical model of tossing the same coin again and again, forever.
Each toss is independent of the others, and each has the same probability $p$ of heads.
</document>

<document source="https://tamnd.github.io/math2/node/02YFME3RGW.md" tag="02YFME3RGW" kind="definition">
# Kernel of a linear map

## Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *kernel* of $T$ is
$$\ker T = \{v \in V : T(v) = 0\}.$$

## Remarks

The kernel is a [[linear subspace|00YC1D652K]] of $V$.
[[A linear map is injective if and only if its kernel is trivial|0BJBYCGMWE]].
For the map $x \mapsto Ax$ of a matrix $A$, the kernel is the [[null space|0PHDYNR24D]] of $A$, the solution set of $Ax = 0$.
The kernel of the projection $\R^3 \to \R^2$, $(x, y, z) \mapsto (x, y)$, is the $z$-axis.

## Explanations

### L1 human

The kernel collects the vectors that the map sends to zero.
It measures how much information the map loses.
If only the zero vector goes to zero, then the map loses nothing.
</document>

<document source="https://tamnd.github.io/math2/node/02YRQXZDCS.md" tag="02YRQXZDCS" kind="definition">
# Conditional convergence

## Definition

Let $(a_n)$ be a real sequence.
The series $\sum a_n$ converges conditionally if $\sum a_n$ [[converges|0CVVKY7GY2]] and $\sum |a_n|$ diverges.

## Remarks

So a convergent series converges either [[absolutely|0W3K3QPEWT]] or conditionally, and not both.
The [[alternating harmonic series|0JQ5Y3WWVE]] converges conditionally.
By the [[Riemann rearrangement theorem|0E7VBYGFF7]], the terms of a conditionally convergent series can be reordered to give any real sum.

## Explanations

### L1 human

A conditionally convergent series converges only because its positive and negative terms cancel.
The positive terms alone add up to $+\infty$, and the negative terms alone add up to $-\infty$.
</document>

<document source="https://tamnd.github.io/math2/node/02YSJPQYQH.md" tag="02YSJPQYQH" kind="concept">
# Branch of the logarithm

## Summary

A branch of the logarithm on an open set $U \subseteq \mathbb{C} \setminus \{0\}$ is a continuous function $L$ on $U$ with $e^{L(z)} = z$, see the [[definition|0SQ1F2BEDS]].
On a domain, two branches differ by a constant integer multiple of $2\pi i$.
</document>

<document source="https://tamnd.github.io/math2/node/02YYV36BFM.md" tag="02YYV36BFM" kind="concept">
# Centraliser

## Summary

The centraliser of a subset $S$ of a group is the set of elements that commute with each element of $S$.
It is a subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/02Z5B2HA3F.md" tag="02Z5B2HA3F" kind="concept">
# One-form

## Summary

A one-form is a [[differential form|0E3G0GRPZ6]] of degree one.
At each point it gives a linear function of tangent vectors. One-forms are the natural integrands of line integrals, and the differential of a function is the basic example.
</document>

<document source="https://tamnd.github.io/math2/node/02ZE6JHJ7G.md" tag="02ZE6JHJ7G" kind="definition">
# Real and imaginary parts

## Definition

Let $z = a + b i$ be a [[complex number|09FTAET5Z4]], with $a$ and $b$ real.
The real part of $z$ is $\operatorname{Re} z = a$.
The imaginary part of $z$ is $\operatorname{Im} z = b$.

## Remarks

The imaginary part is a real number: $\operatorname{Im}(2 + 3 i) = 3$, not $3 i$.
The maps $\operatorname{Re}$ and $\operatorname{Im}$ are additive and $\mathbb{R}$-linear, but they do not respect products.
With the [[conjugate|0BJV24C4SY]], $\operatorname{Re} z = (z + \bar{z})/2$ and $\operatorname{Im} z = (z - \bar{z})/(2 i)$.
A complex number $z$ is real when $\operatorname{Im} z = 0$, and purely imaginary when $\operatorname{Re} z = 0$.

## Explanations

### L1 human

Each complex number has two real coordinates.
For $z = 4 - 7 i$, the real part is $4$ and the imaginary part is $-7$.
</document>

<document source="https://tamnd.github.io/math2/node/030A3FHN14.md" tag="030A3FHN14" kind="statement">
# K5 and K3,3 are not planar

## Statement

The [[complete graph|03WKB2H474]] $K_5$ and the [[complete bipartite graph|0V0GC7BQAN]] $K_{3,3}$ are not [[planar|029JQEK15B]].

## Remarks

The graph $K_5$ has $5$ vertices and $10$ edges. The [[edge bound for planar graphs|0SM1REAVYJ]] allows at most $3 \cdot 5 - 6 = 9$ edges.
The graph $K_{3,3}$ has $6$ vertices and $9$ edges, and it has no cycle of length $3$. The bound $e \le 2v - 4$ allows at most $8$ edges.
The statement for $K_{3,3}$ is the three utilities problem: three houses cannot each be joined to three utilities without crossings.

## Explanations

### L1 human

Try to join each of three houses to each of three utilities with lines that never cross.
It is impossible, and the reason is a count of edges and faces.
</document>

<document source="https://tamnd.github.io/math2/node/030TZ0DQWT.md" tag="030TZ0DQWT" kind="definition">
# Ideal generated by a subset

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $X \subseteq R$ be a subset.
The left ideal generated by $X$ is the intersection of all the left [[ideals|0MYSA6MYNA]] of $R$ that contain $X$.
The notation for it is $(X)$.
It is the smallest left ideal of $R$ that contains $X$.

## Remarks

The left ideal $(X)$ is the set of the finite sums $r_1 x_1 + \dots + r_n x_n$ with $n \ge 0$, $r_i \in R$ and $x_i \in X$.
For a finite set, one writes $(x_1, \dots, x_n)$ for $(\{x_1, \dots, x_n\})$.
In $\mathbb{Z}$, the ideal $(4, 6)$ is $2\mathbb{Z}$.
An ideal generated by one element is a [[principal ideal|00A2EDFT9Q]].

## Explanations

### L1 human

Start with some elements of the ring.
The ideal that they generate contains all the sums of their multiples, and nothing else.
In the integers, the multiples of $4$ and $6$ give all the even numbers, because $2 = 2 \cdot 4 - 1 \cdot 6$.
</document>

<document source="https://tamnd.github.io/math2/node/030VFEREKG.md" tag="030VFEREKG" kind="concept">
# Partial fraction decomposition

## Summary

Partial fraction decomposition writes a rational function as a sum of a polynomial and simple fractions.
Let $K$ be a field, and let $P/Q$ be a rational function with $P, Q \in K[X]$ and $Q \neq 0$.
Assume that $Q = \prod_{i=1}^{r} q_i^{m_i}$, where the $q_i$ are distinct monic irreducible polynomials.
Then there are unique polynomials $B$ and $A_{ij}$, with $\deg A_{ij} < \deg q_i$, such that
$$\frac{P}{Q} = B + \sum_{i=1}^{r} \sum_{j=1}^{m_i} \frac{A_{ij}}{q_i^{\,j}}.$$

## Remarks

Example: $\frac{1}{1 - x - x^2} = \frac{1}{\sqrt{5}} \left( \frac{\varphi}{1 - \varphi x} - \frac{\psi}{1 - \psi x} \right)$, with $\varphi$ and $\psi$ the roots of $X^2 - X - 1$.
In combinatorics it gives the coefficients of a rational [[generating function|0HH6JX5WZZ]]. Each term $\frac{1}{1 - ax}$ expands as a [[geometric series|0R8JR0ARJ4]].

## Explanations

### L1 human

A complicated fraction of polynomials splits into a sum of simple fractions.
For example, $\frac{1}{x(x+1)} = \frac{1}{x} - \frac{1}{x+1}$.
</document>

<document source="https://tamnd.github.io/math2/node/0310REJJY8.md" tag="0310REJJY8" kind="concept">
# Commutative ring

## Summary

A commutative ring is a ring in which the multiplication is commutative.
Commutative algebra is the study of these rings.
The [[definition of a commutative ring|0GYGHKX834]] gives the precise condition.
</document>

<document source="https://tamnd.github.io/math2/node/0316AYH1SC.md" tag="0316AYH1SC" kind="definition">
# Total order

## Definition

Let $A$ be a set.
A total order on $A$ is a [[partial order|011C29Q1S1]] $\le$ on $A$ such that, for all $a, b \in A$, $a \le b$ or $b \le a$.

## Remarks

Other names are linear order and simple order.
The usual orders on $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$ are total orders.
The order $\subseteq$ on the power set of $\{1, 2\}$ is not total.

## Explanations

### L1 human

In a total order, all elements stand on one line.
For any two of them, one comes before the other, or they are equal.
</document>

<document source="https://tamnd.github.io/math2/node/031BG625Y3.md" tag="031BG625Y3" kind="statement">
# The real part of a holomorphic function is harmonic

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $u = \operatorname{Re} f$ and $v = \operatorname{Im} f$, as real functions of $(x, y)$ on $U$, with $z = x + iy$.
Then $u$ and $v$ are twice continuously differentiable, and
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \qquad \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0$$
on $U$. Thus $u$ and $v$ are harmonic functions on $U$.

## Remarks

Proof: $f$ is [[infinitely differentiable|0QEBN8DGVE]]. By the [[Cauchy–Riemann equations|0T71SSYZ39]], $u_{xx} = v_{yx} = v_{xy} = -u_{yy}$, and the same computation works for $v$.
For example, $\operatorname{Re}(z^2) = x^2 - y^2$ is harmonic, and so is $\operatorname{Re}(e^z) = e^x \cos y$.
The function $v$ is a [[harmonic conjugate|0CC56MJ4FG]] of $u$.

## Explanations

### L1 human

A harmonic function is a function of two real variables whose two pure second derivatives add up to $0$.
Each holomorphic function gives two harmonic functions for free: its real part and its imaginary part.
</document>

<document source="https://tamnd.github.io/math2/node/031JV5S54D.md" tag="031JV5S54D" kind="statement">
# A tangent is perpendicular to the radius

## Statement

Let $C$ be a [[circle|060W5CBZP6]] with the centre $O$ and the radius $r > 0$ in the Euclidean plane $\mathbb{R}^2$. Let $T$ be a point of $C$, and let $\ell$ be a line through $T$.
Then $\ell$ is [[tangent|0T7DX8NY1X]] to $C$ at $T$ if and only if $\ell$ is perpendicular to the radius $OT$. That is, $(X - T) \cdot (T - O) = 0$ for each point $X$ of $\ell$.

## Remarks

A proof: for $X = T + t v$ on $\ell$, we have $\|X - O\|^2 = r^2 + 2t\, v \cdot (T - O) + t^2 \|v\|^2$. So $t = 0$ is the only point of $\ell$ on $C$ if and only if $v \cdot (T - O) = 0$.
Thus each point of $C$ has exactly one tangent line.
The same holds for a sphere and its tangent plane in $\mathbb{R}^3$.

## Explanations

### L1 human

To draw the tangent to a circle at a point, first draw the radius to that point. The tangent is the line through the point at a right angle to the radius. For the unit circle at $(1, 0)$, the radius is horizontal and the tangent is the vertical line $x = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/031M8HX7DA.md" tag="031M8HX7DA" kind="statement">
# Order of a cyclic subgroup equals the order of its generator

## Statement

Let $G$ be a [[group|G5S1M9]] and let $g \in G$.
If $g$ has finite [[order|0WE2NHZJ7B]] $n$, then $\langle g \rangle = \{e, g, g^2, \ldots, g^{n-1}\}$, these $n$ elements are distinct, and thus $|\langle g \rangle| = n$.
If $g$ has infinite order, then the powers $g^k$ for $k \in \mathbb{Z}$ are distinct, and $\langle g \rangle$ is infinite.

## Remarks

Here $\langle g \rangle$ is the [[cyclic subgroup generated by $g$|08ABT85339]], and $|\langle g \rangle|$ is its [[order|0B85S2V2DR]].
The proof uses [[the divisibility criterion for powers|0F5QC101Q7]]: $g^a = g^b$ exactly when $n$ divides $a - b$.
With [[Lagrange's theorem|3C53TD]], this fact shows that the order of an element of a finite group divides the order of the group.

## Explanations

### L1 human

The two meanings of the word "order" agree.
The order of an element is the number of distinct powers that it has, and these powers form the subgroup that it generates.
For example, a rotation by $120$ degrees has order $3$, and its powers are the three rotations by $0$, $120$ and $240$ degrees.
</document>

<document source="https://tamnd.github.io/math2/node/031TPJVS74.md" tag="031TPJVS74" kind="concept">
# Event

## Summary

An event is a set of outcomes of a random experiment to which we give a probability.
The event happens when the outcome of the experiment is in the set. The node [[event|00SQGJNRTB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0325SGSJAR.md" tag="0325SGSJAR" kind="concept">
# Polyhedron

## Summary

A polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and corner points, the vertices. The word names the solid or its boundary surface. Cubes, pyramids and prisms are polyhedra.
</document>

<document source="https://tamnd.github.io/math2/node/032HKYS329.md" tag="032HKYS329" kind="statement">
# Residue theorem

## Statement

Let $U \subseteq \mathbb{C}$ be a simply connected open set. This means that $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path.
Let $a_1, \dots, a_n$ be distinct points of $U$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on $U \setminus \{a_1, \dots, a_n\}$.
Let $\gamma$ be a closed, continuous and piecewise continuously differentiable path in $U \setminus \{a_1, \dots, a_n\}$.
Then
$$\oint_\gamma f(z) \, dz = 2 \pi i \sum_{k=1}^{n} n(\gamma, a_k) \operatorname{Res}(f, a_k),$$
where $n(\gamma, a_k)$ is the [[winding number|0X7SBS8E0F]] of $\gamma$ around $a_k$ and $\operatorname{Res}(f, a_k)$ is the [[residue|0JFGKKWWFK]] of $f$ at $a_k$.

## Remarks

If $\gamma$ is a simple closed curve traveled once counterclockwise, then $\oint_\gamma f(z) \, dz$ is $2 \pi i$ times the sum of the residues at the points $a_k$ inside $\gamma$.
With no points $a_k$, the theorem is [[Cauchy's integral theorem|01MPZXYVYF]].
For example, $\oint_{|z| = 2} \frac{dz}{z^2 + 1} = 2 \pi i \left( \frac{1}{2i} - \frac{1}{2i} \right) = 0$, by the [[residues at the simple poles|01991NCP7H]] $i$ and $-i$.
The theorem is the main tool for the [[evaluation of real integrals|0476QH9846]].

## Explanations

### L1 human

To integrate around a closed curve, you only need to know the residues at the bad points inside the curve.
Each bad point contributes $2 \pi i$ times its residue, once for each turn of the curve around it.
</document>

<document source="https://tamnd.github.io/math2/node/032Q9FB8MT.md" tag="032Q9FB8MT" kind="concept">
# Matrix multiplication

## Summary

Matrix multiplication is the operation that takes an $m \times n$ matrix and an $n \times p$ matrix and gives an $m \times p$ matrix.
Entry $(i, k)$ of the product combines row $i$ of the first matrix with column $k$ of the second matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0337CYB0VD.md" tag="0337CYB0VD" kind="definition">
# Order on the reals

## Definition

Construct the real numbers as classes of Cauchy sequences of rational numbers, as in the [[construction from Cauchy sequences|0D8HE681SC]].
Let $x$ and $y$ be real numbers, represented by the Cauchy sequences $(x_n)$ and $(y_n)$ of rational numbers.
The notation $x < y$ means that there are a rational number $\varepsilon > 0$ and a natural number $N$ such that $y_n - x_n > \varepsilon$ for each $n \ge N$.
The notation $x \le y$ means that $x < y$ or $x = y$.

## Remarks

The relation $x < y$ does not depend on the choice of the sequences that represent $x$ and $y$.
For all real numbers $x$ and $y$, exactly one of $x < y$, $x = y$ and $y < x$ holds. Thus $\le$ is a total order.
On the rational numbers, this order agrees with the [[order of the rationals|09RFKMBQ63]].
With [[Dedekind cuts|04QNZZ4HHY]], the order is inclusion of the lower sets of the cuts.

## Explanations

### L1 human

Two real numbers are compared through their rational approximations.
Suppose that, from some point on, the approximations of $y$ exceed those of $x$ by a fixed positive gap.
Then $x$ is less than $y$.
For example, $1.41, 1.414, 1.4142, \dots$ approximates $\sqrt{2}$, and these approximations are larger than $1.4$ by more than $0.005$, thus $1.4 < \sqrt{2}$.
</document>

<document source="https://tamnd.github.io/math2/node/033DA1WE8Y.md" tag="033DA1WE8Y" kind="concept">
# Projection

## Summary

A projection is a linear map from a vector space to itself that is equal to its own square.
It projects the space onto its image along its kernel.
</document>

<document source="https://tamnd.github.io/math2/node/033H8MQRC3.md" tag="033H8MQRC3" kind="concept">
# Nine-point circle

## Summary

The nine-point circle of a triangle is the circle through the midpoints of its three sides. It also passes through the three feet of the altitudes and through the midpoints of the segments from the vertices to the orthocentre.
</document>

<document source="https://tamnd.github.io/math2/node/033KTKKRV6.md" tag="033KTKKRV6" kind="statement">
# Factorial formula for binomial coefficients

## Statement

Let $n$ and $k$ be natural numbers with $k \le n$.
Then
$$\binom{n}{k} = \frac{n!}{k!\,(n - k)!}.$$

## Remarks

Equivalently, $k!\,(n-k)!\,\binom{n}{k} = n!$. In this form the identity has no division.
The formula needs $k \le n$. For $k > n$ the left side is $0$ and $(n - k)!$ has no meaning in the natural numbers.
A cancelled form is $\binom{n}{k} = \frac{(n)_k}{k!}$, with the [[falling factorial|0N5M924N3P]] $(n)_k$.

## Explanations

### L1 human

Arrange all $n$ objects in a row in $n!$ ways, and take the first $k$ as the chosen set.
Each choice of set appears $k!\,(n-k)!$ times, because the first $k$ and the last $n-k$ objects can be reordered freely.
</document>

<document source="https://tamnd.github.io/math2/node/033RZ32T4Q.md" tag="033RZ32T4Q" kind="concept">
# Limit inferior

## Summary

The limit inferior of a sequence is the limit of the infima of its tails. It describes the smallest value near which the sequence lies infinitely often. The node [[limit inferior|0H4R3RESGV]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/033TJBZDG0.md" tag="033TJBZDG0" kind="definition">
# Markov chain

## Definition

Let $S$ be a nonempty [[countable set|0ZG61TFBES]].
Let $(X_n)_{n \ge 0}$ be a [[stochastic process|0QH5ENKZM1]] with values in $S$. It is a Markov chain with state space $S$ if
$$P(X_{n+1} = j \mid X_0 = i_0, \dots, X_{n-1} = i_{n-1}, X_n = i) = P(X_{n+1} = j \mid X_n = i)$$
for each $n \ge 0$ and all states $i_0, \dots, i_{n-1}, i, j \in S$ with $P(X_0 = i_0, \dots, X_n = i) > 0$.
The chain is time-homogeneous if $p_{ij} = P(X_{n+1} = j \mid X_n = i)$ does not depend on $n$.

## Remarks

The condition is the [[Markov property|0HB8SXHPFA]] in discrete time.
For a time-homogeneous chain, the numbers $p_{ij}$ form the [[transition matrix|0F1KBVY161]] $P = (p_{ij})_{i, j \in S}$.
The initial distribution $\lambda_i = P(X_0 = i)$ and $P$ determine the law of the chain: $P(X_0 = i_0, \dots, X_n = i_n) = \lambda_{i_0} p_{i_0 i_1} \cdots p_{i_{n-1} i_n}$.
Markov chains also exist in continuous time and on general state spaces.

## Explanations

### L1 human

A Markov chain has no memory beyond the present state.
Consider the weather each day: if tomorrow depends only on today, the sequence of weather states is a Markov chain.
</document>

<document source="https://tamnd.github.io/math2/node/033WGN00BF.md" tag="033WGN00BF" kind="concept">
# Formal power series

## Summary

A formal power series is an infinite sum $\sum a_n X^n$ treated as an algebraic object, without any convergence condition. Formal power series form a ring that is the natural home of generating functions. The node [[formal power series|072W67KFYK]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/034FGCEYX4.md" tag="034FGCEYX4" kind="statement">
# Formula for the modulus

## Statement

Let $x$ and $y$ be real numbers.
Then the [[modulus|0KW2X0TFBX]] of the complex number $x + y i$ is
$$|x + y i| = \sqrt{x^2 + y^2}.$$

## Remarks

This follows from the definition of the modulus, because $\operatorname{Re}(x + y i) = x$ and $\operatorname{Im}(x + y i) = y$ for real $x$ and $y$.
The hypothesis that $x$ and $y$ are real is necessary. For example, $|i + i \cdot i| = |-1 + i| = \sqrt{2}$, but $\sqrt{i^2 + i^2} = \sqrt{-2}$ is not even real.

## Explanations

### L1 human

To find the length of a complex number, square its two real coordinates, add them, and take the square root.
For $1 - i$, this gives $\sqrt{1 + 1} = \sqrt{2}$.
</document>

<document source="https://tamnd.github.io/math2/node/035V802J58.md" tag="035V802J58" kind="concept">
# Tangent plane to a graph

## Summary

Take the [[graph|0JBJYT69ZV]] of a [[differentiable|0HHHDCXZV4]] function of two variables. Its tangent plane at a point is the plane through that point that best approximates the graph nearby.
It is the graph of the linear approximation of the function at the point.
</document>

<document source="https://tamnd.github.io/math2/node/0363JMQ42Y.md" tag="0363JMQ42Y" kind="statement">
# Triangularisation

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Then $A$ is [[similar|0E89HQATM8]] to an [[upper triangular matrix|0N4XY7Z3NN]] over $F$ if and only if the [[characteristic polynomial|0VSSPAFM2C]] of $A$ splits over $F$.
Here, to split means to be a product of linear factors.
In particular, over an algebraically closed field, such as $\C$, each square matrix is similar to an upper triangular matrix.

## Remarks

The diagonal entries of the triangular matrix are the eigenvalues of $A$, with their algebraic multiplicities.
The proof is by induction: choose an eigenvector, extend it to a basis, and work on the quotient space.
Over $\C$, the matrix $P$ can even be chosen unitary; this is the Schur decomposition.

## Explanations

### L1 human

Not every matrix can be diagonalised, but over $\C$ every matrix can be made triangular by a change of basis.
A triangular form is almost as useful: its eigenvalues are on the diagonal.
Over $\R$, a rotation matrix cannot be triangularised, because it has no real eigenvalue.
</document>

<document source="https://tamnd.github.io/math2/node/036696EMG2.md" tag="036696EMG2" kind="concept">
# Hyperbolic cosine

## Summary

The hyperbolic cosine is the average of $e^x$ and $e^{-x}$. It is an even function with minimum value $1$ at $0$. The node [[hyperbolic cosine|0EQFWKJ37J]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/036756NFD6.md" tag="036756NFD6" kind="definition">
# Statistic

## Definition

Let $X_1, \dots, X_n$ be random variables with values in a measurable space $(E, \mathcal{E})$. For example, they can be a random sample from a law with an unknown parameter $\theta$.
A statistic is a random variable of the form $T = g(X_1, \dots, X_n)$, where $g \colon E^n \to \mathbb{R}^k$ is a measurable function that does not depend on $\theta$.

## Remarks

Examples are the sample mean $\bar{X}_n = (X_1 + \dots + X_n)/n$, the sample variance, the sample maximum $\max_i X_i$ and the order statistics.
A statistic is a random variable, so it has a law: the sampling distribution of the statistic.
A statistic that one uses to estimate a parameter is an [[estimator|0TTNRP5PFQ]]. A statistic that one uses to decide a hypothesis test is a test statistic.

## Explanations

### L1 human

A statistic is a number that you can compute from your data alone, such as the average of the measurements.
It must not contain the unknown quantity that you want to learn about.
</document>

<document source="https://tamnd.github.io/math2/node/037ACYXB5Z.md" tag="037ACYXB5Z" kind="statement">
# Gauss's lemma on quadratic residues

## Statement

Let $p$ be an odd prime and let $a$ be an integer with $p \nmid a$.
Reduce each of the integers $a, 2a, \dots, \frac{p-1}{2} a$ to its least positive residue modulo $p$.
Let $n$ be the number of these residues that are greater than $p/2$.
Then $\left(\frac{a}{p}\right) = (-1)^n$, where the left side is the [[Legendre symbol|01J8AMV9RS]].

## Remarks

For example, take $p = 7$ and $a = 3$. The residues of $3, 6, 9$ are $3, 6, 2$, and only $6$ is greater than $7/2$. So $\left(\frac{3}{7}\right) = -1$.
With $a = 2$, the lemma gives the [[second supplement to quadratic reciprocity|0ZXW95S44M]].
Gauss used the lemma in one of his proofs of the [[law of quadratic reciprocity|085MJS6907]].

## Explanations

### L1 human

Write the first half of the multiples of $a$ modulo $p$, and count how many land in the upper half of $1, \dots, p - 1$.
An even count means that $a$ is a square modulo $p$, and an odd count means that it is not.
</document>

<document source="https://tamnd.github.io/math2/node/037F589D1D.md" tag="037F589D1D" kind="concept">
# Dirichlet convolution

## Summary

The Dirichlet convolution is a product of two arithmetic functions, given by a sum over the divisors of $n$.
The [[definition of the Dirichlet convolution|0KH1N66QWW]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/037FD5NBRT.md" tag="037FD5NBRT" kind="statement">
# Leibniz formula for pi

## Statement

The series $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n + 1}$ converges, and
$$1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}.$$

## Remarks

The series converges by the [[alternating series test|0PK8QH1W1V]]. Its value comes from the [[power series of arctangent|0T32ME0A6H]] at $x = 1$, since $\arctan 1 = \pi/4$.
The convergence is very slow. The error after the terms up to $n = N$ is at most $1/(2N + 3)$.
Mathlib states it as a limit of partial sums, in `Real.tendsto_sum_pi_div_four`.

## History

Madhava of Sangamagrama or his school found the series in the 14th or 15th century. James Gregory found it again in 1671, and Leibniz in 1673.

## Explanations

### L1 human

Add and subtract the reciprocals of the odd numbers: $1 - 1/3 + 1/5 - \cdots$.
The result is a quarter of $\pi$, which links the odd numbers with circles.
</document>

<document source="https://tamnd.github.io/math2/node/037ZNY7NAG.md" tag="037ZNY7NAG" kind="concept">
# System of linear differential equations

## Summary

A system of linear differential equations is an equation $x' = A(t) x + b(t)$ for a vector-valued function $x$, with a matrix $A(t)$ of coefficients.
Its theory extends that of a single linear equation. See the [[definition|0A6QZ367GX]].
</document>

<document source="https://tamnd.github.io/math2/node/0381S8Z5P0.md" tag="0381S8Z5P0" kind="statement">
# Arzelà–Ascoli theorem

## Statement

Let $X$ be a [[compact|0QPHW703PV]] [[metric space|063W5JMY85]] and let $C(X)$ be the set of continuous functions $X \to \mathbb{R}$, with the [[uniform metric|06HE9J7DXF]] $d_\infty(f, g) = \sup_{x \in X} |f(x) - g(x)|$.
Let $\mathcal{F} \subseteq C(X)$.
Then the closure of $\mathcal{F}$ in $C(X)$ is compact if and only if $\mathcal{F}$ is [[equicontinuous|0MW2WXTHMC]] and uniformly bounded.
Here uniformly bounded means that there is a real $M$ with $|f(x)| \le M$ for all $f \in \mathcal{F}$ and $x \in X$.

## Remarks

A sequential form: each uniformly bounded and equicontinuous sequence of functions $[a, b] \to \mathbb{R}$ has a subsequence that converges uniformly on $[a, b]$.
The proof shows that $\mathcal{F}$ is [[totally bounded|0WE5BBHGM6]] in the complete space $C(X)$, and then uses [[compact iff complete and totally bounded|054Q5PW5GX]].
For example, the functions $f \colon [0, 1] \to [0, 1]$ that are [[Lipschitz|0TMZ21WKCN]] with constant $1$ form a compact set in $C([0, 1])$.
The theorem is a main tool for existence proofs, for example in the Peano existence theorem for ordinary differential equations.

## Explanations

### L1 human

In $\mathbb{R}^n$, bounded sequences have convergent subsequences. The Arzelà–Ascoli theorem gives the same result for sequences of functions, but you also need equicontinuity.
Equicontinuity stops the functions from becoming more and more steep, as $x^n$ does near $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0386Y0Z7Y7.md" tag="0386Y0Z7Y7" kind="concept">
# Bernoulli distribution

## Summary

The Bernoulli distribution with parameter $p$ is the distribution of a random variable with two values.
The value $1$ has probability $p$ and the value $0$ has probability $1 - p$.
It is the distribution of one Bernoulli trial. The node [[Bernoulli distribution|0J97P5KZY0]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/038YGY0HSC.md" tag="038YGY0HSC" kind="definition">
# Chromatic polynomial

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]].
For each natural number $k$, let $P(G, k)$ be the number of [[colourings|0874138YP6]] of $G$ with the set of colours $\{1, \dots, k\}$.
There is a unique polynomial $P(G, x)$ whose value at each natural number $k$ is $P(G, k)$. It is the chromatic polynomial of $G$.

## Remarks

For the [[path graph|075WR6W3Y3]] $P_3$, $P(P_3, x) = x(x-1)^2$. For the [[complete graph|03WKB2H474]], $P(K_n, x) = x(x-1)\cdots(x-n+1)$.
For an edge $e$, the deletion–contraction rule says $P(G, x) = P(G - e, x) - P(G / e, x)$. Here $G / e$ merges the two ends of $e$.
The [[chromatic number|0C5M7AM6FH]] is the smallest positive integer $k$ with $P(G, k) > 0$.

## Explanations

### L1 human

The chromatic polynomial counts the proper colourings of a graph with $k$ colours, for each $k$.
George David Birkhoff introduced it in 1912 to study the four colour problem.
</document>

<document source="https://tamnd.github.io/math2/node/0396VGFA78.md" tag="0396VGFA78" kind="statement">
# Freshman's dream

## Statement

Let $p$ be a prime number and let $R$ be a [[commutative ring|0GYGHKX834]] of [[characteristic|0P7CH1CND5]] $p$.
Then $(x + y)^p = x^p + y^p$ for all $x, y \in R$.

## Remarks

The proof uses the binomial theorem. For $0 < k < p$, the prime $p$ divides the binomial coefficient $\binom{p}{k}$, so these terms vanish in $R$.
By induction, $(x + y)^{p^n} = x^{p^n} + y^{p^n}$ for every $n \ge 0$.
The identity gives the [[Frobenius endomorphism|0GHDDVAH8P]].
In $\mathbb{R}$ the identity is false: $(1 + 1)^2 = 4$ and $1^2 + 1^2 = 2$.

## Explanations

### L1 human

The name refers to a common error: $(x + y)^2 = x^2 + y^2$ is false for real numbers.
In characteristic $2$, the middle term $2 x y$ is $0$, and the identity becomes true.
The same thing happens for every prime $p$ in characteristic $p$.
</document>

<document source="https://tamnd.github.io/math2/node/0399QFDFS8.md" tag="0399QFDFS8" kind="concept">
# Parametric equation of a line

## Summary

A parametric equation of a [[line|0MP9BCF6DP]] describes its points as the values of a function of one real parameter $t$.
The line in $\mathbb{R}^3$ through the point $(x_0, y_0, z_0)$ with the nonzero direction vector $(a, b, c)$ has the parametric equations
$$x = x_0 + a t, \quad y = y_0 + b t, \quad z = z_0 + c t, \qquad t \in \mathbb{R}.$$
In vector form, the line through the point $p$ with the direction vector $v \neq 0$ is the set of the points $p + t v$ for $t \in \mathbb{R}$. The same form works in $\mathbb{R}^n$ for each $n \geq 2$.

## Remarks

The line through two distinct points $A$ and $B$ has the parametric equation $A + t(B - A)$. The values $0 \leq t \leq 1$ give the segment from $A$ to $B$, and $t = 1/2$ gives its midpoint.
If $a$, $b$ and $c$ are all nonzero, you can remove $t$. The result is the symmetric form $\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$.

## Explanations

### L1 human

Think of $t$ as time and of $p + t v$ as the position of a particle. At time $0$ the particle is at $p$, and it moves with the constant velocity $v$. Its path is the line.
</document>

<document source="https://tamnd.github.io/math2/node/039HF58KQ5.md" tag="039HF58KQ5" kind="definition">
# Characteristic polynomial

## Definition

Let $K$ be a commutative ring, let $n \ge 1$, and let $A$ be an $n \times n$ matrix over $K$.
The *characteristic polynomial* of $A$ is the polynomial
$$p_A(t) = \det(t I_n - A) \in K[t],$$
where $\det$ is the [[determinant|00KDZRW4C0]] of a matrix with entries in the polynomial ring $K[t]$.

## Remarks

The polynomial $p_A$ is monic of degree $n$.
For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $p_A(t) = t^2 - (a + d) t + (ad - bc)$.
Some authors use $\det(A - t I_n)$, which equals $(-1)^n p_A(t)$.
Over a field, the roots of $p_A$ are the [[eigenvalues|0BQAA2BK3Q]] of $A$.

## Explanations

### L1 human

The number $\lambda$ is an eigenvalue exactly when $A - \lambda I$ is singular, that is, when $\det(\lambda I - A) = 0$.
If we replace $\lambda$ by a variable $t$, then we get a polynomial.
Its roots are the eigenvalues.
</document>

<document source="https://tamnd.github.io/math2/node/039JAF7ZAN.md" tag="039JAF7ZAN" kind="example">
# Integral of exp(-x) over the half line

## Example

The [[improper integral|0VCTBJMEJF]] of $e^{-x}$ over $[0, \infty)$ converges, and
$$\int_0^\infty e^{-x} \, dx = 1.$$
For $b > 0$, [[the exponential formula|06D0EWR2DV]] with $c = -1$ gives $\int_0^b e^{-x} \, dx = 1 - e^{-b}$. This tends to $1$ as $b \to \infty$.

## Remarks

It is the case $s = 1$ of the [[gamma function|0V3FZMVB4K]]: $\Gamma(1) = 1$.
Mathlib states it as `integral_exp_neg_Ioi_zero`.

## Explanations

### L1 human

The region under $e^{-x}$ is infinitely long, but its area is finite.
The area is exactly $1$.
</document>

<document source="https://tamnd.github.io/math2/node/03AHF0XRVG.md" tag="03AHF0XRVG" kind="concept">
# Independence of two events

## Summary

Two events are independent when the occurrence of one of them does not change the probability of the other.
The node [[independence of two events|0P0MZ4X682]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/03AJHX431A.md" tag="03AJHX431A" kind="statement">
# Taylor's theorem with the Peano remainder

## Statement

Let $k \ge 1$, let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be $k$ times differentiable at $a$.
Then
$$f(x) = \sum_{j=0}^{k} \frac{f^{(j)}(a)}{j!} (x - a)^j + o\big((x - a)^k\big) \quad \text{as } x \to a.$$

## Remarks

This is [[Taylor's theorem|03QHBVW4TT]] written with [[little o notation|04TSHB9H8D]].
For example, $\cos x = 1 - x^2/2 + o(x^3)$ as $x \to 0$.
Mathlib states a form for $C^k$ functions on a convex set as `taylor_isLittleO`.

## Explanations

### L1 human

The Peano form only says that the error is small compared with $(x - a)^k$.
It gives no bound for a fixed $x$, but it needs weak hypotheses.
</document>

<document source="https://tamnd.github.io/math2/node/03AJKXH1AV.md" tag="03AJKXH1AV" kind="concept">
# Column space

## Summary

The column space of a matrix is the span of its columns.
It is the image of the linear map of the matrix.
</document>

<document source="https://tamnd.github.io/math2/node/03AN1V4ZFJ.md" tag="03AN1V4ZFJ" kind="definition">
# Norm on the Gaussian integers

## Definition

The norm on the [[Gaussian integers|0K5W69AC0J]] is the map $N \colon \mathbb{Z}[i] \to \mathbb{N}$,
$$N(a + b i) = a^2 + b^2 = (a + b i)(a - b i).$$

## Remarks

The norm is multiplicative: $N(z w) = N(z) N(w)$ for all $z, w \in \mathbb{Z}[i]$. This follows from $|z w| = |z| |w|$ in $\mathbb{C}$.
$N(z) = 0$ if and only if $z = 0$.
The norm is a Euclidean function on $\mathbb{Z}[i]$.
For example, $N(2 + i) = 5$ and $N(1 + i) = 2$.

## Explanations

### L1 human

The norm is the squared distance from $0$ to the point $a + b i$ in the complex plane.
Because it is multiplicative, it turns questions about factoring Gaussian integers into questions about factoring integers.
</document>

<document source="https://tamnd.github.io/math2/node/03AN9X29R7.md" tag="03AN9X29R7" kind="concept">
# Absorbing Markov chain

## Summary

An absorbing Markov chain has at least one state that it never leaves. Also, each state can reach such a state.
The node [[absorbing Markov chain|0YB93GWTKK]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/03BAREW1Z5.md" tag="03BAREW1Z5" kind="statement">
# Generalized Euclid's lemma

## Statement

Let $n$, $a$ and $b$ be integers.
If $n \mid a b$ and $n$ and $a$ are [[coprime|0GFKP8F8AG]], then $n \mid b$.

## Remarks

When $n$ is a prime $p$ and $p \nmid a$, then $p$ and $a$ are coprime. So this result implies [[Euclid's lemma|051VYKA1EX]].
The proof uses [[Bézout's identity|0DQVHMYAGA]]: from $n x + a y = 1$ we get $b = n b x + a b y$, and $n$ divides both terms.
For example, $5 \mid 3 \cdot 10$ and $\gcd(5, 3) = 1$, so $5 \mid 10$.

## Explanations

### L1 human

If $n$ divides $a b$ but has no common factor with $a$, then all the factors of $n$ must come from $b$.
For example, $4 \mid 3 b$ forces $4 \mid b$, because $3$ gives no factor $2$.
</document>

<document source="https://tamnd.github.io/math2/node/03BR7SAX0Q.md" tag="03BR7SAX0Q" kind="concept">
# Modulo operation

## Summary

The modulo operation takes two integers $a$ and $n$ with $n \neq 0$, and gives the remainder of the division of $a$ by $n$.
We write the result as $a \bmod n$.
The [[definition of the modulo operation|04FHQSJCQ5]] gives the convention that this graph uses.
</document>

<document source="https://tamnd.github.io/math2/node/03BY5XTNA4.md" tag="03BY5XTNA4" kind="statement">
# Law of total variance

## Statement

Let $X$ be a real random variable with $E[X^2] < \infty$, and let $Y$ be a random variable on the same probability space. Then
$$\operatorname{Var}(X) = E\left[\operatorname{Var}(X \mid Y)\right] + \operatorname{Var}\left(E[X \mid Y]\right).$$

## Remarks

Proof: by the [[variance formula|0NMF95YK8W]] in its conditional form, $E[X^2 \mid Y] = \operatorname{Var}(X \mid Y) + (E[X \mid Y])^2$. Take expectations with the [[law of total expectation|04RRB6HM00]], and subtract $(E[X])^2 = (E[E[X \mid Y]])^2$.
The same result holds with a σ-algebra $\mathcal{G}$ in place of $Y$.
The first term is the mean variance inside the groups. The second term is the variance between the group means.
A corollary: $\operatorname{Var}(E[X \mid Y]) \le \operatorname{Var}(X)$.

## Explanations

### L1 human

The total spread of $X$ has two parts: the spread inside each group, and the spread of the group averages.
This decomposition is the base of the analysis of variance in statistics.
</document>

<document source="https://tamnd.github.io/math2/node/03CPD1HGF3.md" tag="03CPD1HGF3" kind="concept">
# Multivalued function

## Summary

A multivalued function from a set $X$ to a set $Y$ assigns to each $x \in X$ a subset $F(x)$ of $Y$. For at least one $x$, the subset $F(x)$ has two or more elements.
In complex analysis, the main examples come from the inverses of functions that are not injective.
For example, each $z \neq 0$ has two square roots, and it has infinitely many [[logarithms|0X2FY491PF]] and [[arguments|0ZZRSX91QR]].

## Remarks

A [[branch|0SQ1F2BEDS]] of a multivalued function $F$ on a set $U$ is a function $f$ with $f(x) \in F(x)$ for each $x \in U$, usually continuous or holomorphic.
A [[principal branch|0YKY4M3KRQ]] is one branch that a standard convention chooses.
Riemann surfaces give a way to treat a multivalued function as one single-valued function.

## Explanations

### L1 human

The equation $w^2 = 4$ has two solutions, $2$ and $-2$, so the square root of $4$ has two values.
A multivalued function keeps all the values together, and a branch picks one value at each point.
</document>

<document source="https://tamnd.github.io/math2/node/03DEH43F4H.md" tag="03DEH43F4H" kind="concept">
# Left coset

## Summary

A left coset of a subgroup $H$ of a group $G$ is a set $gH$, with $g \in G$. You get it when you multiply each element of $H$ on the left by $g$.
The left cosets of $H$ partition $G$.
</document>

<document source="https://tamnd.github.io/math2/node/03EESZDB9A.md" tag="03EESZDB9A" kind="statement">
# Variance of the normal distribution

## Statement

Let $\mu \in \mathbb{R}$, let $\sigma^2 > 0$ and let $X$ have the [[normal distribution|07RXBVTVZB]] $\mathcal{N}(\mu, \sigma^2)$. Then
$$\operatorname{Var}(X) = \sigma^2.$$

## Remarks

Proof: $Z = (X - \mu)/\sigma$ has the law $\mathcal{N}(0, 1)$, where $\sigma = \sqrt{\sigma^2}$. Integration by parts gives $E[Z^2] = \int z^2 \varphi(z) \, dz = \int \varphi(z) \, dz = 1$.
Since $E[Z] = 0$ by the [[mean of the normal distribution|09PR1V6RWR]], $\operatorname{Var}(Z) = 1$. The [[scaling|0KVQ9D86BM]] and [[translation|04CYN5G6RY]] rules then give $\operatorname{Var}(X) = \sigma^2 \operatorname{Var}(Z) = \sigma^2$.
So the two parameters of $\mathcal{N}(\mu, \sigma^2)$ are its mean and its variance.

## Explanations

### L1 human

The second parameter of the normal distribution is exactly its variance. So $\sigma$ is the standard deviation, the typical distance of a value from the center.
</document>

<document source="https://tamnd.github.io/math2/node/03F5KP94K0.md" tag="03F5KP94K0" kind="definition">
# Determinant of a linear map

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The *determinant* of $T$ is
$$\det T = \det [T]_B,$$
the [[determinant|00KDZRW4C0]] of the [[matrix of $T$|0YQSGNYE3K]] in a basis $B$ of $V$.

## Remarks

The value does not depend on $B$, because another basis gives a [[similar|0E89HQATM8]] matrix $P^{-1}[T]_B P$, with the same determinant.
It satisfies $\det(S \circ T) = \det S \cdot \det T$ and $\det \id_V = 1$.
The map $T$ is invertible if and only if $\det T \ne 0$.
For $V = 0$, the convention is $\det T = 1$.

## Explanations

### L1 human

We define the determinant of a linear map with its matrix in some basis.
Another basis changes the matrix, but not the determinant.
So the determinant belongs to the map itself, like the trace.
</document>

<document source="https://tamnd.github.io/math2/node/03F7N3915D.md" tag="03F7N3915D" kind="concept">
# Cauchy–Euler equation

## Summary

A Cauchy–Euler equation is a linear equation $\sum_k a_k x^k y^{(k)} = 0$ with constant $a_k$, in which the $k$-th derivative has the factor $x^k$.
It has solutions of the form $x^m$, and the change of variable $x = e^t$ gives constant coefficients. See the [[definition|0WDNRBXX7K]].
</document>

<document source="https://tamnd.github.io/math2/node/03FHD14RKC.md" tag="03FHD14RKC" kind="concept">
# Range of a function

## Summary

The range of a function is the set of all its values.
</document>

<document source="https://tamnd.github.io/math2/node/03FKX7Z5DC.md" tag="03FKX7Z5DC" kind="concept">
# Irreducible polynomial

## Summary

An irreducible polynomial is a nonconstant polynomial that is not a product of two polynomials of smaller degree, over a field.
More generally, it is an irreducible element of a polynomial ring.
The [[definition of an irreducible polynomial|00X2A9G9W3]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/03FTB26QYD.md" tag="03FTB26QYD" kind="statement">
# Vector triple product

## Statement

Let $a$, $b$ and $c$ be vectors in $\mathbb{R}^3$. Let $\times$ be the [[cross product|0N5N15HWZ7]] and let $\cdot$ be the dot product. Then
$$a \times (b \times c) = (a \cdot c) \, b - (a \cdot b) \, c.$$

## Remarks

The formula is easy to remember as "BAC minus CAB": $a \times (b \times c) = b(a \cdot c) - c(a \cdot b)$.
The cross product is not associative. The formula gives $(a \times b) \times c = (a \cdot c) \, b - (b \cdot c) \, a$, which is in general different from $a \times (b \times c)$.
A consequence is the Jacobi identity $a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0$.

## Explanations

### L1 human

The vector $b \times c$ is perpendicular to the plane of $b$ and $c$. A second cross product with $a$ gives a vector perpendicular to $b \times c$, so the result lies back in the plane of $b$ and $c$. The formula tells you exactly which combination of $b$ and $c$ it is.
</document>

<document source="https://tamnd.github.io/math2/node/03GGQEAG9S.md" tag="03GGQEAG9S" kind="statement">
# Chain rule for Jacobian matrices

## Statement

Let $U \subseteq \mathbb{R}^n$ and $W \subseteq \mathbb{R}^m$ be open, let $g \colon U \to W$, let $f \colon W \to \mathbb{R}^k$, and let $a \in U$.
Suppose that $g$ is differentiable at $a$ and $f$ is differentiable at $g(a)$.
Then the [[Jacobian matrices|0YF0VXKXGB]] satisfy
$$J_{f \circ g}(a) = J_f(g(a)) \, J_g(a),$$
a product of a $k \times m$ matrix and an $m \times n$ matrix.

## Remarks

This is the [[chain rule in several variables|0EAAS2EHDR]], written with matrices: the matrix of a composition of linear maps is the product of their matrices.
For $k = m = n$, taking determinants gives $\det J_{f \circ g}(a) = \det J_f(g(a)) \cdot \det J_g(a)$.
For example, let $g(r, \theta) = (r \cos \theta, r \sin \theta)$ and $f(x, y) = x^2 + y^2$. Then $J_f(g(r, \theta)) J_g(r, \theta) = (2r \cos \theta, \, 2r \sin \theta) J_g(r, \theta) = (2r, \, 0)$, which is the Jacobian matrix of $(f \circ g)(r, \theta) = r^2$.

## Explanations

### L1 human

Each Jacobian matrix turns small input steps into small output steps.
Doing two maps one after the other means multiplying their matrices, in the same order as the maps act, from right to left.
</document>

<document source="https://tamnd.github.io/math2/node/03GP7KMBYM.md" tag="03GP7KMBYM" kind="definition">
# Absolutely convergent improper integral

## Definition

Let $f \colon [a, \infty) \to \mathbb{R}$ be Riemann integrable on $[a, b]$ for each $b > a$.
The [[improper integral|0VCTBJMEJF]] $\int_a^\infty f(x) \, dx$ converges absolutely if $\int_a^\infty |f(x)| \, dx$ converges.

## Remarks

An absolutely convergent improper integral converges. The proof uses $0 \le f + |f| \le 2|f|$ and the [[comparison test|0FR1BDKK10]].
The converse fails. The [[Dirichlet integral|0GZNSHMG4Q]] $\int_0^\infty \frac{\sin x}{x} \, dx$ converges, but not absolutely.
A function on $[a, \infty)$ whose improper integral converges absolutely is Lebesgue integrable there. The two integrals agree.

## Explanations

### L1 human

An improper integral converges absolutely when the area between the graph and the axis is finite, with all parts counted as positive.
This is the analogue of absolute convergence of series.
</document>

<document source="https://tamnd.github.io/math2/node/03GSZ8RTZJ.md" tag="03GSZ8RTZJ" kind="concept">
# Dimension

## Summary

The dimension of a vector space is the number of vectors in a basis of the space.
It does not depend on the basis.
</document>

<document source="https://tamnd.github.io/math2/node/03H7WZ4T3K.md" tag="03H7WZ4T3K" kind="statement">
# Pythagorean trigonometric identity

## Statement

For every real $x$,
$$\sin^2 x + \cos^2 x = 1.$$

## Remarks

Let $f(x) = \sin^2 x + \cos^2 x$. By the [[derivatives|00R4MWW532]] [[of sine and cosine|0V2DV29EGT]], $f'(x) = 2 \sin x \cos x - 2 \cos x \sin x = 0$. So $f$ is [[constant|003QDZ4T16]], and $f(0) = 1$.
So $|\sin x| \le 1$ and $|\cos x| \le 1$.
Geometrically, the point $(\cos x, \sin x)$ lies on the unit circle. Mathlib states it as `Real.sin_sq_add_cos_sq`.

## Explanations

### L1 human

In a right triangle with hypotenuse $1$, the legs are $\cos x$ and $\sin x$.
The Pythagorean theorem then says that their squares add up to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/03HDF50VN4.md" tag="03HDF50VN4" kind="concept">
# P-adic valuation

## Summary

The $p$-adic valuation of a nonzero integer $n$, for a prime $p$, is the exponent of the highest power of $p$ that divides $n$.
The [[definition of the p-adic valuation|04VCPS30ZE]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/03HF498P1J.md" tag="03HF498P1J" kind="concept">
# Antichain

## Summary

An antichain in a partially ordered set is a subset in which no two different elements are comparable.
</document>

<document source="https://tamnd.github.io/math2/node/03HF5PMY5T.md" tag="03HF5PMY5T" kind="statement">
# Triangle inequality

## Statement

For all real numbers $a$ and $b$,
$$|a + b| \le |a| + |b|,$$
where $|\cdot|$ is the [[absolute value|00S7C68D51]].

## Remarks

Proof: $-|a| \le a \le |a|$ and $-|b| \le b \le |b|$. Add these inequalities to get $-(|a| + |b|) \le a + b \le |a| + |b|$.
Equality holds exactly when $a b \ge 0$.
For distances, it gives $|x - z| \le |x - y| + |y - z|$. The [[reverse triangle inequality|0C9S13WKMF]] is a consequence.

## Explanations

### L1 human

The name comes from geometry: one side of a triangle is never longer than the other two sides together.
For numbers, $|a + b|$ can be smaller than $|a| + |b|$ when $a$ and $b$ have opposite signs, for example $|5 + (-3)| = 2 < 8$.
</document>

<document source="https://tamnd.github.io/math2/node/03HGK7EW6R.md" tag="03HGK7EW6R" kind="statement">
# Mean value theorem in several variables

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$ be [[differentiable|0135XZZQ8M]] on $U$, and let $x, y \in U$ be points such that the segment $\{x + t (y - x) : 0 \le t \le 1\}$ lies in $U$.
Then there is a number $c \in (0, 1)$ such that
$$f(y) - f(x) = \nabla f\big(x + c (y - x)\big) \cdot (y - x).$$

## Remarks

The proof applies the one-variable mean value theorem to $g(t) = f(x + t(y - x))$ on $[0, 1]$, with $g'(t) = \nabla f(x + t(y - x)) \cdot (y - x)$ by the [[chain rule for a curve|0A8TJ0VHC7]].
The theorem is false for maps into $\mathbb{R}^m$ with $m \ge 2$. For $f(t) = (\cos t, \sin t)$ on $[0, 2\pi]$, $f(2\pi) - f(0) = 0$, but $f'(t) \ne 0$ for each $t$. The [[mean value inequality|02CYB184RC]] replaces it.
By the Cauchy–Schwarz inequality, it gives $|f(y) - f(x)| \le \sup_{z} \|\nabla f(z)\| \, \|y - x\|$, with $z$ on the segment.

## Explanations

### L1 human

On a straight path from $x$ to $y$, the average slope of $f$ equals the actual slope at some point between them.
This reduces the problem to one variable, by looking at $f$ only along the segment.
</document>

<document source="https://tamnd.github.io/math2/node/03HMJASQBQ.md" tag="03HMJASQBQ" kind="concept">
# Regular space

## Summary

A topological space is regular when each closed set and each point outside it lie in disjoint open sets.
</document>

<document source="https://tamnd.github.io/math2/node/03J4NZFYRX.md" tag="03J4NZFYRX" kind="definition">
# Perpendicular bisector

## Definition

Let $A$ and $B$ be distinct points of the Euclidean space $\mathbb{R}^n$, and let $M = \frac{1}{2}(A + B)$ be their [[midpoint|0S706JBZBG]]. Let $\cdot$ be the dot product.
The perpendicular bisector of $A$ and $B$ is the set
$$\{\, X \in \mathbb{R}^n \;:\; (X - M) \cdot (B - A) = 0 \,\}.$$

## Remarks

A point $X$ is on the perpendicular bisector if and only if $d(X, A) = d(X, B)$, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]].
It is a hyperplane with the [[normal vector|0WQJJCCTC8]] $B - A$. In the plane it is a line, and in $\mathbb{R}^3$ it is a plane.
Mathlib's `AffineSubspace.perpBisector` also allows $A = B$. Then the set is the whole space.
The perpendicular bisectors of the three sides of a triangle in the plane meet at the [[circumcentre|0F98FZK14A]].

## Explanations

### L1 human

The perpendicular bisector of a segment cuts it in half at a right angle. Each point on it is as far from one end of the segment as from the other. For the points $(0, 0)$ and $(2, 0)$, the perpendicular bisector is the vertical line $x = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/03J5RAKB30.md" tag="03J5RAKB30" kind="concept">
# Normed vector space

## Summary

A normed vector space is a real or complex vector space together with a norm.
</document>

<document source="https://tamnd.github.io/math2/node/03J8BS7FMV.md" tag="03J8BS7FMV" kind="definition">
# Separable polynomial

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $f \in R[x]$.
The polynomial $f$ is separable if there are $a, b \in R[x]$ with $a f + b f' = 1$, where $f'$ is the [[formal derivative|0AD4VS02H3]] of $f$.

## Remarks

Over a [[field|0QA8CHH8MG]] $K$, $f$ is separable if and only if $f$ has no repeated root in an [[algebraic closure|0MCXX29HWY]] of $K$.
The polynomial $f = x^2 - 2$ is separable over $\mathbb{Q}$, because $f' = 2x$ and $-\tfrac{1}{2} f + \tfrac{1}{4} x f' = 1$.
Over $\mathbb{F}_p(t)$, the irreducible polynomial $x^p - t$ is not separable, because its derivative is $0$.
Some older texts call an irreducible polynomial separable when it has distinct roots, and extend this to other polynomials factor by factor.

## Explanations

### L1 human

A separable polynomial has distinct roots, in a field that is large enough to contain all of them.
Over the rational numbers, every irreducible polynomial is separable.
Inseparable irreducible polynomials only appear in prime characteristic.
</document>

<document source="https://tamnd.github.io/math2/node/03KGGKTB2E.md" tag="03KGGKTB2E" kind="statement">
# Wilson's theorem

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
Then $(p - 1)! \equiv -1 \pmod p$.

## Remarks

A proof: in $\mathbb{Z}/p\mathbb{Z}$, each element of $\{ \overline{1}, \dots, \overline{p - 1} \}$ has an inverse. Only $\overline{1}$ and $\overline{-1}$ are their own inverses. The other elements cancel in pairs in the product.
For example, $4! = 24 \equiv -1 \pmod 5$ and $6! = 720 \equiv -1 \pmod 7$.
The [[converse of Wilson's theorem|06WRV8HFDY]] also holds, so the congruence characterizes the primes.

## History

Ibn al-Haytham stated the result about the year 1000. Edward Waring announced it in 1770 and named his student John Wilson as its discoverer. Lagrange gave the first proof in 1771.

## Explanations

### L1 human

Multiply all the numbers from $1$ to $p - 1$, and divide by $p$. The remainder is always $p - 1$.
For $p = 7$: $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 = 720 = 102 \cdot 7 + 6$.
</document>

<document source="https://tamnd.github.io/math2/node/03MC70ZRNR.md" tag="03MC70ZRNR" kind="statement">
# Equivalence of norms in finite dimension

## Statement

Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$.
Then any two norms on $V$ are [[equivalent|0J3578562S]].

## Remarks

Proof idea: fix a basis and compare a norm $\|\cdot\|$ with the [[sup norm|0KNJWEVSMG]] $\|\cdot\|_\infty$ of the coordinates. The triangle inequality gives $\|x\| \le C \|x\|_\infty$, so $\|\cdot\|$ is continuous for $\|\cdot\|_\infty$. The unit sphere of $\|\cdot\|_\infty$ is compact by the [[Heine–Borel theorem|07PFYV65MB]], so $\|\cdot\|$ has a positive minimum $c$ on it, by the [[extreme value theorem|0V4DET59NW]].
Thus there is only one norm topology on $\mathbb{R}^n$, the Euclidean topology.
The statement fails in infinite dimensions: each infinite-dimensional normed space has norms that are not equivalent.

## Explanations

### L1 human

In finite dimensions, it does not matter which norm you choose: all norms give the same convergent sequences and the same open sets.
</document>

<document source="https://tamnd.github.io/math2/node/03MTBVG3VE.md" tag="03MTBVG3VE" kind="statement">
# de Moivre's formula

## Statement

For every real number $x$ and every integer $n$,
$$(\cos x + i \sin x)^n = \cos(n x) + i \sin(n x).$$

## Remarks

Proof: by [[Euler's formula|04R3CY41VJ]], the left side is $(e^{i x})^n = e^{i n x}$, which is the right side.
For $n = 2$, comparing real and imaginary parts gives $\cos 2x = \cos^2 x - \sin^2 x$ and $\sin 2x = 2 \sin x \cos x$.
For non-integer exponents $n$, the formula fails in general.

## Explanations

### L1 human

Raising a point of the unit circle to the power $n$ multiplies its angle by $n$.
For example, $\cos(\pi/4) + i \sin(\pi/4)$ points at $45$ degrees, and its fourth power points at $180$ degrees, so it is $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/03N8C8ZR4B.md" tag="03N8C8ZR4B" kind="concept">
# Type I and type II errors

## Summary

A type I error is the rejection of a true null hypothesis, a false positive. A type II error is the failure to reject a false null hypothesis, a false negative.
The node [[type I and type II errors|0R41PVFA39]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/03NAMV1DHJ.md" tag="03NAMV1DHJ" kind="concept">
# Path

## Summary

A path in a graph is a walk that visits each vertex at most once. Paths measure distance and connectivity in graphs. The node [[path|0JDE56HWQ8]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/03NDVCSHSB.md" tag="03NDVCSHSB" kind="concept">
# Minimal polynomial

## Summary

The minimal polynomial of an endomorphism of a finite-dimensional space is the monic polynomial of least degree that annihilates it.
</document>

<document source="https://tamnd.github.io/math2/node/03NK6RSRQ1.md" tag="03NK6RSRQ1" kind="statement">
# Vectors are perpendicular if and only if their dot product is zero

## Statement

Let $n$ be a positive integer, and let $u$ and $v$ be nonzero vectors in $\mathbb{R}^n$, with dot product $u \cdot v = u_1 v_1 + \dots + u_n v_n$.
Let $\theta$ be the [[angle between $u$ and $v$|0ZEHZY85E6]].
Then $u \cdot v = 0$ if and only if $\theta = \pi / 2$.

## Remarks

The proof uses the [[geometric formula for the dot product|070TXS9AXM]]: $u \cdot v = \|u\| \, \|v\| \cos \theta$ with $\|u\| \neq 0$ and $\|v\| \neq 0$, and $\cos \theta = 0$ for $0 \leq \theta \leq \pi$ only when $\theta = \pi / 2$.
Two vectors $u$ and $v$ with $u \cdot v = 0$ are called orthogonal. With this word, the zero vector is orthogonal to each vector.
In Mathlib the angle with the zero vector is $\pi / 2$, so the equivalence holds there for all vectors.

## Explanations

### L1 human

This gives a quick test for a right angle. The vectors $(2, 1)$ and $(-1, 2)$ have dot product $-2 + 2 = 0$, so they are perpendicular. You do not need to draw them or measure an angle.
</document>

<document source="https://tamnd.github.io/math2/node/03NM7DZXXC.md" tag="03NM7DZXXC" kind="example">
# Torus

## Example

Let $\sim$ be the equivalence relation on the square $[0, 1] \times [0, 1]$ generated by $(0, y) \sim (1, y)$ and $(x, 0) \sim (x, 1)$ for all $x, y \in [0, 1]$.
The quotient space $[0, 1]^2 / {\sim}$, with the [[quotient topology|0V8046R0P9]], is the torus $T^2$.

## Remarks

The torus is homeomorphic to the product $S^1 \times S^1$ of two [[circles|0WH1GVBJNH]], with the [[product topology|0Z57EC6YX3]], and to $\mathbb{R}^2 / \mathbb{Z}^2$.
It is also homeomorphic to a surface in $\mathbb{R}^3$: turn a circle about an axis in its plane that does not meet the circle.
The torus is a compact, connected surface. It is not homeomorphic to the sphere.

## Explanations

### L1 human

Take a square sheet. Glue the left edge to the right edge to get a tube, and then glue the two end circles of the tube.
The result is the surface of a doughnut.
</document>

<document source="https://tamnd.github.io/math2/node/03PGHKAJG3.md" tag="03PGHKAJG3" kind="statement">
# Cauchy–Schwarz inequality for expectations

## Statement

Let $X$ and $Y$ be real random variables on one probability space with $E[X^2] < \infty$ and $E[Y^2] < \infty$. Then $XY$ is integrable and
$$\left(E[XY]\right)^2 \le E[X^2] \, E[Y^2].$$
Equality holds if and only if $aX + bY = 0$ almost surely for some reals $a$ and $b$, not both $0$.

## Remarks

Proof: for each real $s$, $0 \le E[(X + sY)^2] = E[X^2] + 2s E[XY] + s^2 E[Y^2]$. A quadratic polynomial in $s$ that is never negative has a discriminant that is not positive.
This is the Cauchy–Schwarz inequality for the inner product $\langle X, Y \rangle = E[XY]$ on the space $L^2$ of square-integrable random variables.
Applied to $X - E[X]$ and $Y - E[Y]$, it gives $\operatorname{Cov}(X, Y)^2 \le \operatorname{Var}(X) \operatorname{Var}(Y)$. So the correlation coefficient is in $[-1, 1]$.

## Explanations

### L1 human

The average of a product cannot be larger than what the sizes of the two factors allow.
This simple inequality is behind the fact that a correlation is always between $-1$ and $1$.
</document>

<document source="https://tamnd.github.io/math2/node/03PPNCM9GG.md" tag="03PPNCM9GG" kind="concept">
# Second-countable space

## Summary

A topological space is second-countable when its topology has a countable basis.
This condition limits the size of the topology and holds for Euclidean spaces.
</document>

<document source="https://tamnd.github.io/math2/node/03Q7RE2HXR.md" tag="03Q7RE2HXR" kind="statement">
# A matrix is invertible iff its determinant is a unit

## Statement

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$.
Then $A$ is [[invertible|0EB9N0S33C]] if and only if its [[determinant|00KDZRW4C0]] $\det A$ is a unit of $K$.
In particular, over a field, $A$ is invertible if and only if $\det A \ne 0$.

## Remarks

If $\det A$ is a unit, then the [[adjugate|07W9H8QF38]] gives the inverse $A^{-1} = (\det A)^{-1} \operatorname{adj}(A)$.
Over $\Z$, the matrix $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ has determinant $2 \ne 0$, but it has no inverse with integer entries.
Over a field, this is one of many equivalent conditions for invertibility, with $\rank A = n$ and $\ker A = 0$.

## Explanations

### L1 human

A real square matrix has an inverse exactly when its determinant is not zero.
A zero determinant means that the matrix squeezes space into a lower dimension.
Such a squeeze cannot be undone.
</document>

<document source="https://tamnd.github.io/math2/node/03QC59M99B.md" tag="03QC59M99B" kind="statement">
# Complement principle

## Statement

Let $U$ be a finite set and let $A \subseteq U$.
Let $A^c = U \setminus A$ be the complement of $A$ in $U$.
Then $|A^c| = |U| - |A|$.

## Remarks

The rule follows from the [[rule of sum|0EE2V3CHFK]], because $U$ is the disjoint union of $A$ and $A^c$.
It is useful when the complement is easier to count. For example, the number of words of length $n$ over $\{0, 1\}$ that contain at least one $1$ is $2^n - 1$.

## Explanations

### L1 human

Sometimes it is easier to count what you do not want.
To count the outcomes with at least one six in two dice rolls, count the outcomes with no six: $5 \cdot 5 = 25$. Then $36 - 25 = 11$ outcomes have at least one six.
</document>

<document source="https://tamnd.github.io/math2/node/03QHBVW4TT.md" tag="03QHBVW4TT" kind="statement">
# Taylor's theorem

## Statement

Let $k \ge 1$, let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be $k$ times differentiable at $a$.
Let $P_k$ be the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.
Then there is a function $h_k \colon I \to \mathbb{R}$ with $h_k(x) \to 0$ as $x \to a$ and
$$f(x) = P_k(x) + h_k(x) (x - a)^k \quad \text{for all } x \in I.$$

## Remarks

In [[little o notation|04TSHB9H8D]], the conclusion is the [[Peano form|03AJHX431A]] $f(x) - P_k(x) = o((x - a)^k)$.
Under stronger hypotheses, the remainder has explicit forms: the [[Lagrange form|0H5ZAZCX2T]], the [[Cauchy form|005Q84BMAD]], and the [[integral form|0WJGR81HBD]].
For $k = 1$, it says that $f(x) = f(a) + f'(a)(x - a) + o(x - a)$.

## Explanations

### L1 human

Near $a$, a function is close to its Taylor polynomial.
The error is small even compared with $(x - a)^k$.
</document>

<document source="https://tamnd.github.io/math2/node/03RNVFDEAN.md" tag="03RNVFDEAN" kind="definition">
# Skew-symmetric matrix

## Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $A$ is *skew-symmetric* when $A^T = -A$, where $A^T$ is the [[transpose|09DXWYRD43]] of $A$.
Equivalently, $a_{ji} = -a_{ij}$ for all $i$ and $j$.

## Remarks

If $2$ is invertible in $R$, for example over $\R$, then each diagonal entry of a skew-symmetric matrix is $0$.
Over a field of characteristic not $2$, each square matrix is the sum of a [[symmetric matrix|0VR671PHE5]] and a skew-symmetric matrix: $A = \tfrac12 (A + A^T) + \tfrac12 (A - A^T)$.
A real skew-symmetric matrix of odd order has determinant $0$.
The matrix $\begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}$ is skew-symmetric.

## Explanations

### L1 human

In a skew-symmetric matrix, the entry across the diagonal is the negative of the entry here.
Over the real numbers this forces zeros on the diagonal, because only $0$ equals its own negative.
These matrices describe infinitesimal rotations.
</document>

<document source="https://tamnd.github.io/math2/node/03RSJXF728.md" tag="03RSJXF728" kind="statement">
# Formula for the number of derangements

## Statement

Let $n$ be a natural number, and let $D_n$ be the number of [[derangements|0TH61JR04G]] of an $n$-element set.
Then
$$D_n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!} = \sum_{k=0}^{n} (-1)^k \frac{n!}{k!}.$$

## Remarks

For $n = 4$: $24 - 24 + 12 - 4 + 1 = 9$.
The proof applies the [[inclusion–exclusion principle|0FF7726C7P]] to the sets $A_i$ of permutations that fix $i$.
The sum is a partial sum of the series of $e^{-1}$. Thus $D_n / n! \to 1/e$, and for $n \ge 1$, $D_n$ is the integer nearest to $n!/e$.

## Explanations

### L1 human

Start with all $n!$ permutations. Remove those that fix some point, add back those that fix two points, and so on.
The result is the alternating sum in the formula.
</document>

<document source="https://tamnd.github.io/math2/node/03S6GWWKEZ.md" tag="03S6GWWKEZ" kind="statement">
# Limit of a quotient of sequences

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers with $b \ne 0$.
Assume that $a_n \to a$ and $b_n \to b$ as $n \to \infty$.
Then there is a natural number $N$ such that $b_n \ne 0$ for all $n \ge N$.
Moreover, the sequence $(a_n / b_n)_{n \ge N}$ converges to $a / b$.

## Remarks

The main step is $1/b_n \to 1/b$. For large $n$ we have $|b_n| > |b|/2$, and then $|1/b_n - 1/b| = |b - b_n| / (|b_n|\,|b|) < 2 |b_n - b| / |b|^2$.
The [[limit of a product|0734RSVF2V]] of $(a_n)$ and $(1/b_n)$ then gives the result.
The hypothesis $b \ne 0$ is needed. For $a_n = 1$ and $b_n = 1/n$, the quotient is $n$, which has no real limit.

## Explanations

### L1 human

The limit of a quotient is the quotient of the limits, when the limit of the denominator is not zero.
For example, $(3 + 1/n)/(2 + 1/n) \to 3/2$.
</document>

<document source="https://tamnd.github.io/math2/node/03TM135G0N.md" tag="03TM135G0N" kind="concept">
# Continuous linear map

## Summary

A continuous linear map between normed spaces is a linear map that is continuous for the norm topologies.
Such maps are the same as the bounded linear maps, which multiply lengths by at most a fixed constant.
</document>

<document source="https://tamnd.github.io/math2/node/03VFZ2WETT.md" tag="03VFZ2WETT" kind="concept">
# Integers modulo n

## Summary

The integers modulo $n$ form the ring $\mathbb{Z}/n\mathbb{Z}$ of the $n$ residue classes modulo $n$, with the addition and the multiplication of representatives.
The [[definition of the integers modulo n|01N3AYAA6N]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/03VHANA6ZK.md" tag="03VHANA6ZK" kind="definition">
# Indefinite integral

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ have an [[antiderivative|0BW9N5CDSB]] $F$ on $I$.
The indefinite integral of $f$ is the set of all antiderivatives of $f$ on $I$.
It is written
$$\int f(x) \, dx = F(x) + C,$$
where $C$ stands for an arbitrary real constant.

## Remarks

The constant $C$ is the constant of integration.
For example, $\int \cos x \, dx = \sin x + C$.
Some texts use the name for the function $x \mapsto \int_a^x f(t) \, dt$, for a fixed $a$.

## Explanations

### L1 human

The indefinite integral collects all the antiderivatives of a function.
They differ only by a constant, so one antiderivative plus $C$ describes them all.
</document>

<document source="https://tamnd.github.io/math2/node/03VZJE6C8C.md" tag="03VZJE6C8C" kind="concept">
# Circumcircle

## Summary

The circumcircle of a triangle is the circle that passes through its three vertices. Each triangle has exactly one circumcircle. Its centre is the circumcentre and its radius is the circumradius.
</document>

<document source="https://tamnd.github.io/math2/node/03W6Q1MFR5.md" tag="03W6Q1MFR5" kind="definition">
# Jacobian determinant

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^n$, and let $a \in U$ be a point at which all first partial derivatives of $f$ exist.
The Jacobian determinant of $f$ at $a$ is the determinant
$$\det J_f(a) = \det \left( \frac{\partial f_i}{\partial x_j}(a) \right)_{1 \le i, j \le n}$$
of the [[Jacobian matrix|0YF0VXKXGB]] of $f$ at $a$.
It is also written $\dfrac{\partial (f_1, \dots, f_n)}{\partial (x_1, \dots, x_n)}(a)$, and it is often called the Jacobian of $f$.

## Remarks

For the polar map $(r, \theta) \mapsto (r \cos \theta, r \sin \theta)$, the Jacobian determinant is $r \cos^2 \theta + r \sin^2 \theta = r$.
If $f$ is of class $C^1$ and $\det J_f(a) \ne 0$, then $f$ has a differentiable local inverse near $a$, by the [[inverse function theorem|0K9HSJFCMJ]].
The absolute value $|\det J_f|$ is the factor in the [[change of variables formula|00N5FHP1CV]] for multiple integrals.

## Explanations

### L1 human

A linear map of the plane sends the unit square to a parallelogram, and the determinant is the signed area of that parallelogram.
The Jacobian determinant does the same for a curved map, at the scale of tiny squares near one point.
</document>

<document source="https://tamnd.github.io/math2/node/03WKB2H474.md" tag="03WKB2H474" kind="example">
# Complete graph

## Example

Let $n$ be a natural number.
The complete graph $K_n$ is the [[graph|0JD5C82QMB]] with vertex set $\{1, \dots, n\}$ in which any two distinct vertices are adjacent.

More generally, the complete graph on a set $V$ has an edge between each pair of distinct elements of $V$.

## Remarks

The graph $K_n$ is $(n-1)$-[[regular|0KA4S8R39M]], and it has $\binom{n}{2}$ edges. The node [[number of edges of a complete graph|05RN90E14Y]] states this.
$K_3$ is a triangle. $K_4$ is planar, and [[$K_5$ is not planar|030A3FHN14]].
A clique in a graph is a set of vertices that induces a complete graph.

## Explanations

### L1 human

In a complete graph every vertex is joined to every other vertex.
If $5$ people all shake hands with each other, the handshakes form $K_5$.
</document>

<document source="https://tamnd.github.io/math2/node/03WY4HHMT1.md" tag="03WY4HHMT1" kind="definition">
# Number of derangements

## Definition

The derangement numbers $D_n$, for natural numbers $n$, are defined by the recursion
$$D_0 = 1, \qquad D_1 = 0, \qquad D_{n+2} = (n + 1)\,(D_n + D_{n+1}).$$

## Remarks

The first values, for $n = 0, 1, \dots, 6$, are $1, 0, 1, 2, 9, 44, 265$.
The value $D_n$ is the number of [[derangements|0TH61JR04G]] of an $n$-element set. The node [[the derangement numbers count derangements|045ENVCCZV]] states this.
Another notation is $!n$, the subfactorial of $n$.

## Explanations

### L1 human

The derangement numbers start $1, 0, 1, 2, 9, 44$.
Each new number comes from the two before it: $D_4 = 3 \cdot (D_2 + D_3) = 3 \cdot 3 = 9$.
</document>

<document source="https://tamnd.github.io/math2/node/03X8V7D64M.md" tag="03X8V7D64M" kind="statement">
# Sum of a finite geometric progression

## Statement

Let $x$ be a real number with $x \neq 1$, and let $n$ be a natural number.
Then
$$\sum_{k=0}^{n-1} x^k = 1 + x + x^2 + \dots + x^{n-1} = \frac{x^n - 1}{x - 1}.$$

## Remarks

The same formula holds for $x$ in any field, for example for complex numbers, when $x \neq 1$.
For $x = 1$, the sum is $n$.
A short proof: $(x - 1)(1 + x + \dots + x^{n-1}) = x^n - 1$, because the other terms cancel.

## Explanations

### L1 human

In a geometric progression, you get each term from the one before by multiplying by the same number $x$.
For $x = 2$, the formula gives $1 + 2 + 4 + 8 = 2^4 - 1 = 15$.
Multiply the sum by $x$ and subtract the sum: almost all terms cancel, and only $x^n - 1$ stays.
</document>

<document source="https://tamnd.github.io/math2/node/03XVP8Q4GY.md" tag="03XVP8Q4GY" kind="definition">
# Multiplicity of a root

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]], let $f \in R[x]$ with $f \neq 0$ and let $c \in R$.
The multiplicity of $c$ as a root of $f$ is the largest integer $m \ge 0$ such that $(x - c)^m$ divides $f$.

## Remarks

The integer $m$ exists, because $(x - c)^m$ is monic of degree $m$, so it cannot divide $f$ for $m > \deg f$.
The multiplicity is positive exactly when $c$ is a [[root|00EHX6FWC0]] of $f$.
A root of multiplicity $1$ is a simple root. A root of multiplicity at least $2$ is a repeated root.
In $\mathbb{Q}[x]$, $f = (x - 1)^3 (x + 2)$ has the root $1$ with multiplicity $3$ and the root $-2$ with multiplicity $1$.

## Explanations

### L1 human

The multiplicity counts how many times the factor $x - c$ appears in $f$.
For $(x - 1)^2 (x + 3)$, the root $1$ counts twice and the root $-3$ counts once.
</document>

<document source="https://tamnd.github.io/math2/node/03Y6AQ2S0J.md" tag="03Y6AQ2S0J" kind="concept">
# Sample mean

## Summary

The sample mean of a sample is the arithmetic mean of its values. It is the usual estimator of the mean of the population.
The node [[sample mean|0QNRBHWV35]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/03YEM7JEDN.md" tag="03YEM7JEDN" kind="statement">
# Open mapping theorem

## Statement

Let $D \subseteq \mathbb{C}$ be a [[domain|097KNJFJ7V]] and let $f : D \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.
Assume that $f$ is not constant on $D$.
Then $f$ is an open map: for each open set $V \subseteq D$, the image $f(V)$ is an open subset of $\mathbb{C}$.

## Remarks

The [[maximum modulus principle|048F2P90WR]] follows. If $|f|$ had a local maximum at $z_0$, then $f$ would map a small disc around $z_0$ onto a set that contains no disc around $f(z_0)$.
The connectedness of $D$ matters only to exclude functions that are constant on one part of $D$.
There is no real analogue: $x \mapsto x^2$ maps $(-1, 1)$ onto $[0, 1)$, which is not open in $\mathbb{R}$. In contrast, $z \mapsto z^2$ maps the unit disc onto the unit disc.

## Explanations

### L1 human

A nonconstant holomorphic function never squeezes an open set into a curve or a point.
Around each point of the image, the image also contains a small disc.
</document>

<document source="https://tamnd.github.io/math2/node/03YN70NK0Z.md" tag="03YN70NK0Z" kind="concept">
# Oriented angle

## Summary

An oriented angle in the plane measures the turn from one direction to another, with a sign. Anticlockwise turns are positive and clockwise turns are negative. It is defined modulo a full turn $2\pi$, and it changes sign when you swap the two directions.
</document>

<document source="https://tamnd.github.io/math2/node/03Z4JC4YRJ.md" tag="03Z4JC4YRJ" kind="counterexample">
# The Dirichlet function is not Riemann integrable

## Counterexample

Let $f \colon [0, 1] \to \mathbb{R}$ be the [[Dirichlet function|0T4EJ8ZJM9]]: $f(x) = 1$ for rational $x$ and $f(x) = 0$ for irrational $x$.
Each subinterval of positive length contains both rational and irrational numbers. So for each partition $P$, $U(f, P) = 1$ and $L(f, P) = 0$.
The upper [[Darboux integral|09FQT057T7]] is $1$ and the lower one is $0$. So $f$ is [[not Riemann integrable|021XJAQVDC]].

## Remarks

The function is bounded, so the failure is not caused by unboundedness.
The function is Lebesgue integrable, with Lebesgue integral $0$, since $\mathbb{Q}$ has measure zero.
It is the pointwise limit of Riemann integrable functions. So pointwise limits do not keep Riemann integrability.

## Explanations

### L1 human

Every Riemann sum of this function is $1$ or $0$ or in between, depending only on the choice of tags.
So the sums cannot settle down to one number.
</document>

<document source="https://tamnd.github.io/math2/node/03Z9V0Y78A.md" tag="03Z9V0Y78A" kind="statement">
# C^1 vector fields satisfy the Picard–Lindelöf hypotheses

## Statement

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be continuously differentiable, let $x_0 \in U$ and let $t_0 \in \mathbb{R}$.
Then there are real numbers $\varepsilon > 0$, $a > 0$ and $L, K \ge 0$ with these properties.
The closed ball $B$ of radius $a$ around $x_0$ lies in $U$.
The time-independent field $(t, x) \mapsto f(x)$, extended in any way outside $B$, satisfies the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] at $(t_0, x_0)$ on $[t_0 - \varepsilon, t_0 + \varepsilon]$ with constants $a, L, K$.

## Remarks

Proof: choose $a > 0$ with $B \subseteq U$. The continuous functions $\|f\|$ and $\|f'\|$ are bounded on the compact ball $B$, by $L$ and $K$. The mean value inequality makes $f$ $K$-Lipschitz on $B$. Then choose $\varepsilon$ with $L \varepsilon \le a$.
So the initial value problem $x' = f(x)$, $x(t_0) = x_0$ has a unique local solution for each $C^1$ field $f$.
The same argument works for a time-dependent field $f(t, x)$ of class $C^1$.

## Explanations

### L1 human

A smooth field has a bounded derivative near each point, so it is Lipschitz near this point.
This is why, in practice, one rarely checks the Lipschitz condition directly: smoothness is enough.
</document>

<document source="https://tamnd.github.io/math2/node/0404D8TN7Y.md" tag="0404D8TN7Y" kind="statement">
# Binet's formula

## Statement

Let $n$ be a natural number, and let $F_n$ be the $n$-th [[Fibonacci number|0GA9K57A39]].
Let $\varphi = \frac{1 + \sqrt{5}}{2}$ be the [[golden ratio|0SC8T7548D]] and let $\psi = \frac{1 - \sqrt{5}}{2}$.
Then
$$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}.$$

## Remarks

The proof uses the [[characteristic polynomial|0BWDJGQYST]] $x^2 - x - 1$ of the recurrence. Its roots are $\varphi$ and $\psi$, so $\varphi^n$ and $\psi^n$ are [[solutions|096SD4ERQH]], and the initial values fix the coefficients.
Because $|\psi^n / \sqrt{5}| < \frac{1}{2}$, $F_n$ is the integer nearest to $\varphi^n / \sqrt{5}$.

## Explanations

### L1 human

Fibonacci numbers are integers, but this formula uses $\sqrt{5}$, and the irrational parts cancel.
The term with $\psi^n$ is very small, so $F_n$ is close to $\varphi^n / \sqrt{5}$.
</document>

<document source="https://tamnd.github.io/math2/node/0404R9P9J3.md" tag="0404R9P9J3" kind="definition">
# Integer partition

## Definition

Let $n$ be a natural number.
A partition of $n$ is a finite multiset of positive integers whose sum is $n$.
Its elements are the parts of the partition.

Equivalently, a partition of $n$ is a finite sequence $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_k$ of positive integers with $\lambda_1 + \dots + \lambda_k = n$.

## Remarks

The order of the parts does not matter. A sum in which the order matters is a [[composition|0J8ANSYKAC]].
The $5$ partitions of $4$ are $4$, $3 + 1$, $2 + 2$, $2 + 1 + 1$ and $1 + 1 + 1 + 1$.
The number $0$ has exactly one partition, the empty partition.
The number of partitions of $n$ is the [[partition function|001CX45SWQ]] $p(n)$, and a [[Young diagram|0K78D81EF7]] draws a partition.

## Explanations

### L1 human

A partition of $n$ is a way to write $n$ as a sum of positive integers when the order of the summands does not matter.
So $3 + 1$ and $1 + 3$ are the same partition of $4$.
</document>

<document source="https://tamnd.github.io/math2/node/040CFNJEAP.md" tag="040CFNJEAP" kind="statement">
# Lebesgue number lemma

## Statement

Let $(X, d)$ be a [[compact|0QPHW703PV]] [[metric space|063W5JMY85]] and let $(U_i)_{i \in I}$ be an [[open cover|0NVPNC12JY]] of $X$.
Then there is a real $\delta > 0$ such that each subset $A \subseteq X$ with [[diameter|0692CGQF00]] $\operatorname{diam}(A) < \delta$ is contained in $U_i$ for some $i \in I$.

## Remarks

Such a number $\delta$ is a Lebesgue number of the cover.
Equivalently, for each $x \in X$, the ball $B(x, \delta / 2)$ lies in one of the sets $U_i$.
Compactness is needed: the cover of $(0, 1)$ by the sets $(1/n, 1)$ has no Lebesgue number.

## Explanations

### L1 human

In a compact metric space, each open cover has a uniform size: each small enough set fits completely into one set of the cover.
You can think of $\delta$ as a safe size for a window that you move around the space.
</document>

<document source="https://tamnd.github.io/math2/node/040WY0TT2G.md" tag="040WY0TT2G" kind="concept">
# Lattice path

## Summary

A lattice path is a finite sequence of points of the integer lattice $\mathbb{Z}^2$. Each step between consecutive points is from a given finite set of steps.
The most common case uses the steps $E = (1, 0)$ and $N = (0, 1)$. These are North-East lattice paths.

A North-East path from $(0, 0)$ to $(a, b)$, with $a, b \in \mathbb{N}$, has $a$ steps $E$ and $b$ steps $N$.
It is a word of length $a + b$ in $E$ and $N$. Thus the number of such paths is the [[binomial coefficient|092GEF38YD]] $\binom{a + b}{a}$.

## Remarks

The number of North-East paths from $(0, 0)$ to $(n, n)$ is the [[central binomial coefficient|01WHD9T0V6]] $\binom{2n}{n}$.
The paths of this type that never go above the diagonal $y = x$ are counted by the [[Catalan numbers|092DRBFKMG]]. They correspond to [[Dyck words|0R9JR0465V]].

## Explanations

### L1 human

Think of a city grid where you may only walk east or north.
To go 3 blocks east and 2 blocks north, you choose which 2 of your 5 steps go north, so there are $\binom{5}{2} = 10$ routes.
</document>

<document source="https://tamnd.github.io/math2/node/0419794R0T.md" tag="0419794R0T" kind="definition">
# Entire function

## Definition

An entire function is a function $f : \mathbb{C} \to \mathbb{C}$ that is [[holomorphic|0DG0Z72CQD]] on all of $\mathbb{C}$.

## Remarks

Polynomials, the [[complex exponential|054VXHMCA3]] and the complex [[sine and cosine|0A7YDKRNR6]] are entire.
Sums, products and compositions of entire functions are entire.
An entire function is equal to one power series $\sum_{n \ge 0} a_n z^n$ that converges for every $z \in \mathbb{C}$.
By [[Liouville's theorem|0EHKVRF4J6]], each bounded entire function is constant.

## Explanations

### L1 human

An entire function is a function that is holomorphic at every complex number, with no exceptions.
For example, $z^3 - 2z + 5$ and $e^z$ are entire, but $1/z$ is not, because it is not defined at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0419GJC84P.md" tag="0419GJC84P" kind="definition">
# Nilpotent matrix

## Definition

Let $R$ be a ring and let $N$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $N$ is *nilpotent* when $N^k = 0$ for some integer $k \ge 1$.
The least such $k$ is the *index* of $N$.

## Remarks

The matrix $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ is nilpotent of index $2$.
Each strictly upper triangular matrix is nilpotent.
Over a field, an $n \times n$ matrix $N$ is nilpotent if and only if its characteristic polynomial is $x^n$; then $N^n = 0$.
The only eigenvalue of a nilpotent matrix over a field is $0$, so its trace and determinant are $0$.

## Explanations

### L1 human

A nilpotent matrix becomes the zero matrix after we multiply it by itself enough times.
A shift that moves each coordinate one step down, and drops the last one, is an example.
After $n$ shifts nothing is left.
</document>

<document source="https://tamnd.github.io/math2/node/041JTZCK9Q.md" tag="041JTZCK9Q" kind="definition">
# Direct sum of subspaces

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U_1, \dots, U_k$ be subspaces of $V$.
The space $V$ is the *direct sum* of $U_1, \dots, U_k$, written $V = U_1 \oplus \cdots \oplus U_k$, when each $v \in V$ can be written as
$$v = u_1 + \cdots + u_k, \qquad u_i \in U_i,$$
in exactly one way.
Equivalently, $V = U_1 + \cdots + U_k$, and $u_1 + \cdots + u_k = 0$ with $u_i \in U_i$ implies $u_1 = \cdots = u_k = 0$.

## Remarks

For two subspaces, $V = U \oplus W$ if and only if $V = U + W$ and $U \cap W = \{0\}$.
For three or more subspaces, pairwise zero intersections are not enough: three different lines through $0$ in $\R^2$ are not a direct sum.
If $V = U_1 \oplus \cdots \oplus U_k$ and $V$ is finite-dimensional, then $\dim V = \dim U_1 + \cdots + \dim U_k$.

## Explanations

### L1 human

A space is the direct sum of some subspaces when we can split each vector into pieces, one piece in each subspace.
The split must be unique.
The plane is the direct sum of the $x$-axis and the $y$-axis.
The pieces of a vector are its components along the subspaces.
</document>

<document source="https://tamnd.github.io/math2/node/0423TXR9AN.md" tag="0423TXR9AN" kind="statement">
# Subgroups of a finite cyclic group correspond to divisors of its order

## Statement

Let $G$ be a [[cyclic group|0KVGH0J5B3]] of finite order $n$, generated by $g$.
For each positive divisor $d$ of $n$, $G$ has exactly one [[subgroup|0X1JXT47GA]] of order $d$, namely $\langle g^{n/d} \rangle$.
Each subgroup of $G$ has this form for some positive divisor $d$ of $n$.

## Remarks

Thus the number of subgroups of $G$ is the number of positive divisors of $n$.
For $d, d'$ dividing $n$, the subgroup of order $d$ is contained in the subgroup of order $d'$ exactly when $d$ divides $d'$.
For example, $\mathbb{Z}/12\mathbb{Z}$ has six subgroups, of the orders $1, 2, 3, 4, 6, 12$.

## Explanations

### L1 human

In a cyclic group, the subgroups are as simple as possible: there is one for each divisor of the order.
In the clock group $\mathbb{Z}/12\mathbb{Z}$, the subgroup of order $4$ is $\{0, 3, 6, 9\}$, generated by $12/4 = 3$.
The order of each subgroup divides $12$, as [[Lagrange's theorem|3C53TD]] says, and here every divisor occurs.
</document>

<document source="https://tamnd.github.io/math2/node/042JWAR3Q0.md" tag="042JWAR3Q0" kind="statement">
# Rational canonical form

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Then there are unique monic nonconstant polynomials $f_1, \dots, f_r \in F[t]$ with $f_1 \mid f_2 \mid \cdots \mid f_r$ and the following property. The matrix $A$ is [[similar|0E89HQATM8]] to the [[block diagonal matrix|001T5FJDC1]]
$$\begin{pmatrix} C(f_1) & & \\ & \ddots & \\ & & C(f_r) \end{pmatrix},$$
where $C(f_i)$ is the [[companion matrix|0ZCWPNZTQ3]] of $f_i$.
This block matrix is the *rational canonical form* of $A$.

## Remarks

The polynomials $f_1, \dots, f_r$ are the *invariant factors* of $A$.
The last factor $f_r$ is the minimal polynomial of $A$, and $f_1 f_2 \cdots f_r$ is its characteristic polynomial.
Unlike the Jordan form, it needs no field extension: it works over any field.
Two matrices over $F$ are similar if and only if they have the same invariant factors.

## Explanations

### L1 human

The rational canonical form is a standard representative of each similarity class, over any field.
It uses companion matrices instead of Jordan blocks.
It needs no roots of polynomials, only divisibility of polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/0430ETCMKD.md" tag="0430ETCMKD" kind="statement">
# Continuous bijection from a compact space to a Hausdorff space is a homeomorphism

## Statement

Let $X$ be a [[compact space|0QPHW703PV]], let $Y$ be a [[Hausdorff space|0MXB4CKHFW]] and let $f \colon X \to Y$ be a [[continuous map|0SY9ECQ19Q]] that is [[bijective|0SNQEF7Z53]].
Then $f$ is a [[homeomorphism|05GZ11PWRY]].

## Remarks

Proof: it is enough to show that $f$ maps closed sets to closed sets. A closed $F \subseteq X$ is compact because [[closed subsets of compact sets are compact|066HW6D9A7]]. Then $f(F)$ is compact, and thus closed in $Y$ by [[compact subsets of a Hausdorff space are closed|0MYFJVMN7M]].
The compactness is needed: the map $t \mapsto (\cos t, \sin t)$ from $[0, 2\pi)$ to the unit circle is a continuous bijection, but its inverse is not continuous.

## Explanations

### L1 human

For compact spaces, you do not need to check that the inverse map is continuous: it comes for free.
This is a quick way to show that two compact spaces have the same shape.
</document>

<document source="https://tamnd.github.io/math2/node/0431RN0R1N.md" tag="0431RN0R1N" kind="statement">
# Uniform limits of holomorphic functions are holomorphic

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f_1, f_2, \dots$ be functions $U \to \mathbb{C}$ that are [[holomorphic|0DG0Z72CQD]] on $U$.
Let $f : U \to \mathbb{C}$ be a function such that $f_n \to f$ uniformly on each compact subset of $U$.
Then $f$ is holomorphic on $U$. Moreover, $f_n' \to f'$ uniformly on each compact subset of $U$.

## Remarks

Proof idea: $f$ is continuous. On each disc in $U$, [[Cauchy's integral theorem|01MPZXYVYF]] gives $\oint_{\partial T} f_n = 0$ for each triangle $T$, so $\oint_{\partial T} f = 0$ by the uniform convergence, and [[Morera's theorem|0VDD7W0EKZ]] applies.
A main use: if a series $\sum g_k$ of holomorphic functions converges uniformly on compact subsets of $U$, then its sum is holomorphic on $U$.
There is no real analogue. On $[-1, 1]$, the polynomials converge uniformly to $|x|$, which is not differentiable at $0$.

## Explanations

### L1 human

If holomorphic functions get close to a function $f$ uniformly, then $f$ is holomorphic too, and their derivatives get close to the derivative of $f$.
This lets us build new holomorphic functions as limits of series, for example the zeta function $\sum n^{-z}$.
</document>

<document source="https://tamnd.github.io/math2/node/043B3ENSTY.md" tag="043B3ENSTY" kind="statement">
# Fermat's theorem on sums of two squares

## Statement

Let $p$ be an odd [[prime number|0SN0MAEKYX]].
Then there are integers $x$ and $y$ with $p = x^2 + y^2$ if and only if $p \equiv 1 \pmod 4$.

## Remarks

The prime $2 = 1^2 + 1^2$ is also a sum of two squares. So a prime $p$ is a sum of two squares if and only if $p \not\equiv 3 \pmod 4$.
For example, $5 = 1 + 4$, $13 = 4 + 9$ and $29 = 4 + 25$. For $p \equiv 1 \pmod 4$, the representation is unique up to order and signs.
The "only if" part is easy: a square is $0$ or $1$ modulo $4$, so $x^2 + y^2 \not\equiv 3 \pmod 4$.
One proof of the other part starts from a [[square root of $-1$ modulo $p$|00DW862T78]] and uses [[Gaussian integers|0B268GVX95]] or [[Minkowski's theorem|0K387S2PKQ]].

## History

Albert Girard stated the result in a book published in 1625.
Fermat stated it in a letter to Mersenne of 25 December 1640, so it is also called Fermat's Christmas theorem.
Euler gave the first proof, by infinite descent, in 1749.

## Explanations

### L1 human

The primes $5, 13, 17, 29$ leave remainder $1$ on division by $4$, and each of them is a sum of two squares.
The primes $3, 7, 11, 19$ leave remainder $3$, and none of them is a sum of two squares.
</document>

<document source="https://tamnd.github.io/math2/node/043SJ5RF59.md" tag="043SJ5RF59" kind="definition">
# Maclaurin series

## Definition

Let $I$ be an open interval with $0 \in I$, and let $f \colon I \to \mathbb{R}$ be smooth.
The Maclaurin series of $f$ is its [[Taylor series|0225QPWRRE]] at $0$:
$$\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n.$$

## Remarks

For example, the Maclaurin series of $e^x$ is $\sum x^n / n!$, and the one of $1/(1 - x)$ is $\sum x^n$.
The series is named after Colin Maclaurin.

## Explanations

### L1 human

A Maclaurin series is a Taylor series centered at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0443XNARDV.md" tag="0443XNARDV" kind="concept">
# Multilinear map

## Summary

A multilinear map is a map of several vector variables that is linear in each variable when the other variables are fixed.
</document>

<document source="https://tamnd.github.io/math2/node/044D4AQV8V.md" tag="044D4AQV8V" kind="definition">
# Product metric

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]].
The product metric on the [[Cartesian product|0C81YBVTWH]] $X \times Y$ is the function $d$ with
$$d\bigl((x_1, y_1), (x_2, y_2)\bigr) = \max\bigl(d_X(x_1, x_2), \, d_Y(y_1, y_2)\bigr)$$
for all $x_1, x_2 \in X$ and $y_1, y_2 \in Y$.

## Remarks

The function $d$ is a metric on $X \times Y$.
Other common product metrics are $d_X + d_Y$ and $\sqrt{d_X^2 + d_Y^2}$. Each of them is between $d$ and $2d$, so all of them are [[equivalent|0TWQ5PRKWS]] and give the [[product topology|0Z57EC6YX3]].
For $X = Y = \mathbb{R}$ with the usual metric, $d$ is the [[sup metric|0VTAXQEWQD]] on $\mathbb{R}^2$.
The same formula with a maximum over $n$ factors gives a product metric on a product of $n$ metric spaces.

## Explanations

### L1 human

Two pairs are close in the product metric when their first parts are close and their second parts are close.
The maximum is the worse of the two distances.
</document>

<document source="https://tamnd.github.io/math2/node/044JNC7ZAA.md" tag="044JNC7ZAA" kind="concept">
# Diagonal matrix

## Summary

A diagonal matrix is a square matrix whose entries outside the main diagonal are all zero.
</document>

<document source="https://tamnd.github.io/math2/node/045D8P03E0.md" tag="045D8P03E0" kind="concept">
# Random variable with a density

## Summary

A real random variable has a density when there is a fixed nonnegative function with this property.
The probability of each set of values is the integral of the function over that set.
The node [[random variable with a density|0C21RZG5V2]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/045DN06F3S.md" tag="045DN06F3S" kind="definition">
# Circle

## Definition

Let $d$ be the [[Euclidean distance|0NPPGHQNXQ]] on the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$. Let $c \in \mathbb{R}^2$ be a point and $r > 0$ a real number.
The circle with the centre $c$ and the radius $r$ is the set
$$\{\, x \in \mathbb{R}^2 \;:\; d(x, c) = r \,\}.$$
With $c = (a, b)$, it is the set of the points $(x, y)$ with $(x - a)^2 + (y - b)^2 = r^2$.

## Remarks

A chord is a segment between two points of the circle. A diameter is a chord through the centre, and its length is $2r$.
The circle has the length $2 \pi r$ and bounds a disc of area $\pi r^2$.
A circle in $\mathbb{R}^n$ is a circle in some plane of $\mathbb{R}^n$. Mathlib's `EuclideanGeometry.Sphere` is the sphere of a centre and a radius in a metric space; in the plane it is a circle.

## Explanations

### L1 human

A circle is what a compass draws: all the points at one fixed distance from the centre. The unit circle has the centre $(0, 0)$ and the radius $1$. It is the set of the points with $x^2 + y^2 = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/045ENVCCZV.md" tag="045ENVCCZV" kind="statement">
# The derangement numbers count derangements

## Statement

Let $X$ be a finite set with $|X| = n$.
Then the number of [[derangements|0TH61JR04G]] of $X$ is the derangement number $D_n$ of the node [[number of derangements|03WY4HHMT1]].

## Remarks

The proof checks the recursion $D_{n+2} = (n + 1)(D_n + D_{n+1})$, as in the node [[recurrence for derangements|0QRRYXG7V7]].
For $n = 4$ there are $9$ derangements among the $24$ permutations.

## Explanations

### L1 human

The derangement numbers are defined by a recursion, and this result says that they really count the permutations with no fixed point.
For example, $D_3 = 2$, and $\{1, 2, 3\}$ has exactly the two derangements $231$ and $312$.
</document>

<document source="https://tamnd.github.io/math2/node/045EWAXY9P.md" tag="045EWAXY9P" kind="definition">
# Disc of convergence

## Definition

Let $\sum_{n \ge 0} a_n (z - c)^n$ be a [[power series|0VARVQ9ZW1]] with center $c \in \mathbb{C}$ and complex coefficients $a_n$.
Its radius of convergence is
$$R = \sup \{ r \ge 0 : \text{the sequence } (|a_n| r^n)_{n \ge 0} \text{ is bounded} \},$$
an element of $[0, \infty]$.
Its disc of convergence is the open disc $D = \{ z \in \mathbb{C} : |z - c| < R \}$. It is empty if $R = 0$, and it is $\mathbb{C}$ if $R = \infty$.

## Remarks

The series converges absolutely at each $z \in D$, and uniformly on each closed disc $|z - c| \le r$ with $r < R$. It diverges at each $z$ with $|z - c| > R$.
The Cauchy–Hadamard formula gives $1/R = \limsup_{n \to \infty} |a_n|^{1/n}$, with $1/0 = \infty$ and $1/\infty = 0$.
On the circle $|z - c| = R$, the series can converge at all points, at some points, or at no point.
For example, $\sum z^n$, $\sum z^n / n$ and $\sum z^n / n^2$ all have $R = 1$. On $|z| = 1$, the first converges nowhere, the second everywhere except at $z = 1$, and the third everywhere.

## Explanations

### L1 human

A power series converges inside a disc around its center and diverges outside the closed disc.
The radius of this disc is the radius of convergence. On the boundary circle, each case is possible.
</document>

<document source="https://tamnd.github.io/math2/node/045YFKRW0P.md" tag="045YFKRW0P" kind="concept">
# Closed curve

## Summary

A closed curve is a [[parametrised curve|0G7WM8DX4P]] on a closed interval whose end point is equal to its initial point.
Its image can be seen as the image of a circle under a continuous map.
</document>

<document source="https://tamnd.github.io/math2/node/0464Z0JP2E.md" tag="0464Z0JP2E" kind="statement">
# Monotonicity of probability

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \in \mathcal{F}$ be [[events|00SQGJNRTB]] with $A \subseteq B$.
Then $P(A) \le P(B)$.
More precisely, $P(B \setminus A) = P(B) - P(A)$.

## Remarks

The proof: $B$ is the disjoint union of $A$ and $B \setminus A$, so $P(B) = P(A) + P(B \setminus A)$, and $P(B \setminus A) \ge 0$.
With $B = \Omega$, monotonicity gives $P(A) \le 1$ for each event $A$.
If $A$ implies $B$, then $B$ is at least as likely as $A$. For example, "the die shows $6$" is at most as likely as "the die shows an even number".

## Explanations

### L1 human

A bigger event contains more outcomes, so it cannot be less likely.
The chance to draw a heart from a deck is at most the chance to draw a red card, because every heart is red.
</document>

<document source="https://tamnd.github.io/math2/node/0469V0EN5N.md" tag="0469V0EN5N" kind="concept">
# Homogeneous linear system with constant coefficients

## Summary

A homogeneous linear system with constant coefficients is a system $x' = A x$ with a constant square matrix $A$.
Its solutions are given by the matrix exponential, and the eigenvalues of $A$ control their behaviour. See the [[definition|0VJF99CZ0Z]].
</document>

<document source="https://tamnd.github.io/math2/node/0476QH9846.md" tag="0476QH9846" kind="concept">
# Evaluation of real integrals by residues

## Summary

The [[residue theorem|032HKYS329]] gives the values of many definite integrals of real functions.
For an integral $\int_0^{2\pi} R(\cos \theta, \sin \theta) \, d\theta$ with $R$ a rational function of two variables, the substitution $z = e^{i\theta}$ changes it into a [[contour integral|00FWNP9TXY]] over the unit circle.
This substitution uses $\cos \theta = \frac{1}{2}(z + z^{-1})$, $\sin \theta = \frac{1}{2i}(z - z^{-1})$ and $d\theta = \frac{dz}{iz}$.
For an integral $\int_{-\infty}^{\infty} f(x) \, dx$, one closes the segment $[-R, R]$ with a semicircle of radius $R$ in the upper half-plane and lets $R \to \infty$.
This method gives the integral when the integral over the semicircle tends to $0$, for example by the [[estimation lemma|0DJMGPN8K9]] or by [[Jordan's lemma|0PNPG2PP9H]].

## Remarks

For example, $\frac{1}{1 + z^2}$ has one pole $i$ in the upper half-plane, with residue $\frac{1}{2i}$, so $\int_{-\infty}^{\infty} \frac{dx}{1 + x^2} = 2\pi i \cdot \frac{1}{2i} = \pi$.
With Jordan's lemma, the same method gives $\int_{-\infty}^{\infty} \frac{e^{itx}}{x^2 + 1} \, dx = \pi e^{-|t|}$ for each real $t$.
When the integrand has a pole on the real axis, the method usually gives a [[Cauchy principal value|0V3RDQV3ZX]].

## Explanations

### L1 human

A hard real integral can become easy when you view the real line as part of a closed curve in the complex plane.
The residue theorem gives the integral over the closed curve from a few residues.
Then you show that the extra part of the curve contributes nothing in the limit.
</document>

<document source="https://tamnd.github.io/math2/node/0477R9PYAE.md" tag="0477R9PYAE" kind="concept">
# Variance

## Summary

The variance of a random variable is the expected squared distance between the variable and its mean. It measures how widely the values spread around the mean.
The node [[variance|007QED3E0Y]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/047KEVEZ9Q.md" tag="047KEVEZ9Q" kind="statement">
# Every real number has a decimal expansion

## Statement

Let $x$ be a real number with $x \ge 0$.
Then $x$ has a [[decimal expansion|06XY4X9430]]: there are a natural number $a_0$ and digits $d_k \in \{0, \dots, 9\}$ for $k \ge 1$ with $x = a_0 + \sum_{k=1}^{\infty} d_k 10^{-k}$.

## Remarks

One choice is $a_0 = \lfloor x \rfloor$ and $d_k = \lfloor 10^k x \rfloor - 10 \lfloor 10^{k-1} x \rfloor$, with the [[floor function|0YVRNEDSVG]].
With this choice, the truncation $a_0.d_1 \dots d_k$ satisfies $0 \le x - a_0.d_1 \dots d_k < 10^{-k}$.
The expansion is unique, except for the numbers with a finite expansion. They also have a second expansion that ends in infinitely many $9$.

## Explanations

### L1 human

You can find the digits of a number one at a time.
At each step, you choose the largest digit that keeps the truncated decimal at or below the number.
The truncations then get closer to the number by a factor of ten at each step.
</document>

<document source="https://tamnd.github.io/math2/node/047NDTEPCV.md" tag="047NDTEPCV" kind="concept">
# Jacobian matrix

## Summary

Let $U \subseteq \mathbb{R}^n$ be open. The Jacobian matrix of a map $f \colon U \to \mathbb{R}^m$ at a point $a$ is the $m \times n$ matrix of all first [[partial derivatives|09EGC6F8PT]] $\partial f_i / \partial x_j$ at $a$.
When $f$ is differentiable at $a$, it is the matrix of the [[total derivative|0FNKFT8H6E]] $Df(a)$ in the standard bases.
</document>

<document source="https://tamnd.github.io/math2/node/047RFCNVSK.md" tag="047RFCNVSK" kind="statement">
# Parseval's identity

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $(e_1, \dots, e_n)$ be an [[orthonormal basis|0S9GGTR0EX]] of $V$.
Then, for all $x, y \in V$,
$$\langle x, y \rangle = \sum_{i=1}^{n} \langle x, e_i \rangle \overline{\langle y, e_i \rangle} \qquad \text{and} \qquad \norm{x}^2 = \sum_{i=1}^{n} \abs{\langle x, e_i \rangle}^2.$$

## Remarks

It is the case of equality in [[Bessel's inequality|0RBT5PB74K]].
In coordinates of an orthonormal basis, the inner product becomes the standard inner product of $\mathbb{K}^n$.
For Fourier series, the same identity holds with infinite sums, for square integrable functions.

## Explanations

### L1 human

In an orthonormal basis, the inner product of two vectors is the standard inner product of their coordinates.
In particular, the squared length is the sum of the squared coordinates.
This is the Pythagorean theorem in $n$ dimensions.
</document>

<document source="https://tamnd.github.io/math2/node/047SX3TD69.md" tag="047SX3TD69" kind="concept">
# Multiplicity of a root

## Summary

The multiplicity of a root $c$ of a nonzero polynomial $f$ is the largest $m$ such that $(x - c)^m$ divides $f$.
The [[definition of the multiplicity|03XVP8Q4GY]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/048F2P90WR.md" tag="048F2P90WR" kind="statement">
# Maximum modulus principle

## Statement

Let $D \subseteq \mathbb{C}$ be a [[domain|097KNJFJ7V]] and let $f : D \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.
Assume that $|f|$ has a local maximum at a point $z_0 \in D$: there is $r > 0$ such that $|f(z)| \le |f(z_0)|$ for each $z$ with $|z - z_0| < r$.
Then $f$ is constant on $D$.

## Remarks

Proof idea: the mean value formula $f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + \rho e^{i\theta}) \, d\theta$ for small $\rho$ forces $|f| = |f(z_0)|$ near $z_0$. Then $f$ is constant near $z_0$, and on $D$ by the [[identity theorem|0YBVKMYX0W]].
Corollary: if $D$ is bounded and $f$ is continuous on the closure $\overline{D}$ and holomorphic on $D$, then $\max_{\overline{D}} |f|$ is attained at a point of the boundary of $D$.
The minimum of $|f|$ can be inside $D$, but only at a zero of $f$. For example, $|z|$ has its minimum on the unit disc at $0$.

## Explanations

### L1 human

The size $|f|$ of a nonconstant holomorphic function has no highest point inside its domain.
For example, on the closed unit disc, $|z^2 + 1|$ is largest on the boundary circle, at $z = \pm 1$, and it equals $2$ there.
</document>

<document source="https://tamnd.github.io/math2/node/049DCJZ448.md" tag="049DCJZ448" kind="concept">
# Concyclic points

## Summary

Points are concyclic when they all lie on one circle. A polygon with concyclic vertices is a cyclic polygon, and the circle is its circumcircle.
</document>

<document source="https://tamnd.github.io/math2/node/049PXYHW1T.md" tag="049PXYHW1T" kind="concept">
# Open cover

## Summary

An open cover of a subset of a topological space is a family of open sets whose union contains the subset.
Open covers are used to define compactness.
</document>

<document source="https://tamnd.github.io/math2/node/04A2XEX4D4.md" tag="04A2XEX4D4" kind="statement">
# Lagrange multiplier with one constraint

## Statement

Let $U \subseteq \mathbb{R}^n$ be open with $n \ge 2$, and let $f, g \colon U \to \mathbb{R}$ be of class $C^1$.
Let $c \in \mathbb{R}$, and let $a$ be a point of the [[level set|0Y5Q8Q9WJ7]] $\{x \in U : g(x) = c\}$ with $\nabla g(a) \ne 0$.
If the restriction of $f$ to this level set has a local extremum at $a$, then there is a unique $\lambda \in \mathbb{R}$ with
$$\nabla f(a) = \lambda \, \nabla g(a).$$

## Remarks

This is the case $m = 1$ of the [[Lagrange multiplier theorem|0VKKP8X6Q6]], applied to $g - c$.
Geometrically, the level set of $f$ through $a$ is tangent to the constraint set at $a$, because both [[gradients are orthogonal|0VXYV3NQQ1]] to it.
Example: to maximise $f(x, y) = x + y$ on $x^2 + y^2 = 1$, solve $(1, 1) = \lambda (2 x, 2 y)$ and $x^2 + y^2 = 1$. The maximum is $\sqrt{2}$, at $(1/\sqrt{2}, 1/\sqrt{2})$.
Without the condition $\nabla g(a) \ne 0$ the conclusion can fail. For $g(x, y) = y^2 - x^3$ and $f(x, y) = x$, the minimum on $g = 0$ is at $(0, 0)$, where $\nabla g = 0$ and $\nabla f = (1, 0)$.

## Explanations

### L1 human

Move along the constraint curve and watch the value of $f$.
At the best point, the curve just touches a level curve of $f$, so the two curves have the same normal direction there.
The multiplier $\lambda$ is the ratio between the two normal vectors.
</document>

<document source="https://tamnd.github.io/math2/node/04AAKZNXHD.md" tag="04AAKZNXHD" kind="concept">
# Greedy colouring

## Summary

Greedy colouring is a simple method that makes a [[colouring|0874138YP6]] of a finite [[graph|0JD5C82QMB]].
Put the vertices in an order $v_1, v_2, \dots, v_n$, and use the colours $1, 2, 3, \dots$.
For $i = 1, \dots, n$, give $v_i$ the smallest colour that no earlier neighbour of $v_i$ has.

## Remarks

A vertex $v_i$ has at most $\deg(v_i)$ earlier neighbours. Thus greedy colouring uses at most $\Delta(G) + 1$ colours, where $\Delta(G)$ is the [[maximum degree|0KNAN9F27P]].
The number of colours depends on the order. For each graph, some order gives an optimal colouring, but such an order is hard to find in general.
Greedy colouring can be done in linear time.

## Explanations

### L1 human

Go through the vertices one by one, and give each one the first colour that its coloured neighbours do not use.
It is fast and simple, but it does not always use the least number of colours.
</document>

<document source="https://tamnd.github.io/math2/node/04AJE0M271.md" tag="04AJE0M271" kind="example">
# Möbius strip

## Example

Let $\sim$ be the equivalence relation on the square $[0, 1] \times [0, 1]$ generated by $(0, y) \sim (1, 1 - y)$ for all $y \in [0, 1]$.
The quotient space $[0, 1]^2 / {\sim}$, with the [[quotient topology|0V8046R0P9]], is the Möbius strip.

## Remarks

The gluing has a half-twist: the bottom of the left edge is glued to the top of the right edge.
The Möbius strip is a non-orientable surface. Its boundary is one circle, not two.
Without the twist, the gluing $(0, y) \sim (1, y)$ gives a cylinder, which is not homeomorphic to the Möbius strip.

## Explanations

### L1 human

Take a strip of paper, give one end a half-turn, and glue the two ends together.
The result has only one side: an ant can walk to every point of the strip without crossing an edge.
</document>

<document source="https://tamnd.github.io/math2/node/04BD23PSK5.md" tag="04BD23PSK5" kind="statement">
# Exponential of a sum of commuting matrices

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $A$ and $B$ be $n \times n$ matrices over $\mathbb{K}$ with $AB = BA$.
Then
$$e^{A + B} = e^A e^B,$$
where $e^A$ is the [[matrix exponential|020M04FN1H]].

## Remarks

The proof multiplies the two series and uses the binomial formula, which needs $AB = BA$.
It gives $e^A e^{-A} = I_n$, so $e^A$ is always invertible.
Without $AB = BA$, the identity can fail, for example for $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$.

## Explanations

### L1 human

For numbers, $e^{a + b} = e^a e^b$.
For matrices, this rule holds only when the two matrices commute.
In particular, $e^{sA} e^{tA} = e^{(s + t)A}$ for all scalars $s$ and $t$.
</document>

<document source="https://tamnd.github.io/math2/node/04C3BAHV03.md" tag="04C3BAHV03" kind="definition">
# Complementary subspaces

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be subspaces of $V$.
The subspaces $U$ and $W$ are *complementary* when $V = U \oplus W$, that is, when $U + W = V$ and $U \cap W = \{0\}$.
Then $W$ is a *complement* of $U$ in $V$.

## Remarks

A complement is not unique: in $\R^2$, each line through $0$ other than the $x$-axis is a complement of the $x$-axis.
If $V$ is finite-dimensional and $W$ is a complement of $U$, then $\dim W = \dim V - \dim U$.
A complement $W$ of $U$ is isomorphic to the [[quotient space|0BR3FY5Q8M]] $V / U$.
In an inner product space, the [[orthogonal complement|0XJXTBCB2T]] is one particular complement.

## Explanations

### L1 human

Two subspaces are complementary when together they fill the space and they share only the zero vector.
Each vector then splits in a unique way into a part in each subspace.
A line and a plane through the origin in space are complementary when the line does not lie in the plane.
</document>

<document source="https://tamnd.github.io/math2/node/04C5ZBG1WT.md" tag="04C5ZBG1WT" kind="concept">
# Field

## Summary

A field is a commutative ring in which you can divide by every nonzero element.
The rational numbers, the real numbers and the complex numbers are fields.
The [[definition of a field|0QA8CHH8MG]] gives the axioms.
</document>

<document source="https://tamnd.github.io/math2/node/04CBRA4JRK.md" tag="04CBRA4JRK" kind="statement">
# log x is at most x - 1

## Statement

For every real $x > 0$,
$$\ln x \le x - 1,$$
with equality if and only if $x = 1$.

## Remarks

Put $\ln x$ into the inequality [[$1 + y \le e^y$|012C7GFF20]]. Then $1 + \ln x \le e^{\ln x} = x$.
The line $y = x - 1$ is the tangent to the graph of $\ln$ at $x = 1$.
Replacing $x$ by $1/x$ gives $\ln x \ge 1 - 1/x$. Mathlib states the main claim as `Real.log_le_sub_one_of_pos`.

## Explanations

### L1 human

The graph of $\ln$ lies below its tangent line at $x = 1$.
This follows from the concavity of the logarithm.
</document>

<document source="https://tamnd.github.io/math2/node/04CC6K5VDT.md" tag="04CC6K5VDT" kind="concept">
# Real sequence

## Summary

A real sequence assigns a real number to each natural number. Real sequences are the first setting for limits, convergence and series in real analysis. The node [[real sequence|0CTV783HYT]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/04CPG9K1C4.md" tag="04CPG9K1C4" kind="definition">
# Local maximum and minimum

## Definition

Let $D \subseteq \mathbb{R}^n$, let $f \colon D \to \mathbb{R}$, and let $a \in D$.
The function $f$ has a local maximum at $a$ if there is $\varepsilon > 0$ such that $f(x) \le f(a)$ for all $x \in D$ with $\|x - a\| < \varepsilon$.
The function $f$ has a local minimum at $a$ if there is $\varepsilon > 0$ such that $f(x) \ge f(a)$ for all $x \in D$ with $\|x - a\| < \varepsilon$.
A local extremum of $f$ is a local maximum or a local minimum of $f$.
The extremum is strict if the inequality is strict for all such $x \ne a$.

## Remarks

A global maximum of $f$ is a point $a$ with $f(x) \le f(a)$ for all $x \in D$. Each global maximum is a local maximum.
The function $f(x, y) = x^2 + y^2$ has a strict local minimum at $(0, 0)$, which is also a global minimum.
In two or more variables, a local minimum that is the only critical point need not be a global minimum. For $f(x, y) = x^2 + y^2 (1 - x)^3$, the only critical point $(0, 0)$ is a local minimum, but $f(2, 3) = -5 < f(0, 0)$.
At an interior local extremum of a differentiable function, the [[gradient is zero|0F405477F8]].

## Explanations

### L1 human

A local maximum is a hilltop: it is the highest point in a small disc around it, but a higher hill can exist far away.
A local minimum is the bottom of a valley in the same sense.
</document>

<document source="https://tamnd.github.io/math2/node/04CPJ6AF00.md" tag="04CPJ6AF00" kind="statement">
# Determinant is multiplicative

## Statement

Let $K$ be a commutative ring and let $A$ and $B$ be $n \times n$ matrices over $K$.
Then
$$\det(AB) = \det A \cdot \det B,$$
where $AB$ is the [[matrix product|032Q9FB8MT]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

Thus $\det$ is a group homomorphism from the [[general linear group|0FCHE902D7]] $GL_n(F)$ to $F^\times$, over a field $F$.
It follows that $\det(AB) = \det(BA)$, although $AB \ne BA$ in general.
Similar matrices have the same determinant, because $\det(P^{-1} A P) = \det A$.

## Explanations

### L1 human

If one matrix scales volume by $\det A$ and another by $\det B$, then doing both scales volume by $\det A \cdot \det B$.
This is the meaning of the product rule.
It is one of the most useful properties of the determinant.
</document>

<document source="https://tamnd.github.io/math2/node/04CTY6XYWN.md" tag="04CTY6XYWN" kind="definition">
# Correlation coefficient

## Definition

Let $X$ and $Y$ be real random variables on one probability space with finite and positive variances.
The correlation coefficient of $X$ and $Y$ is
$$\rho_{X, Y} = \operatorname{Corr}(X, Y) = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \, \sigma_Y},$$
where $\operatorname{Cov}$ is the [[covariance|0DJ8W3HP2D]] and $\sigma_X$, $\sigma_Y$ are the [[standard deviations|0VTN5RKNC5]].

## Remarks

By the Cauchy–Schwarz inequality, $-1 \le \rho_{X, Y} \le 1$.
$|\rho_{X, Y}| = 1$ if and only if $Y = aX + b$ almost surely for some reals $a \ne 0$ and $b$. The sign of $\rho$ is the sign of $a$.
The coefficient does not depend on units: $\rho_{aX + b, \, cY + d} = \rho_{X, Y}$ if $a c > 0$.
For a sample of pairs $(x_i, y_i)$, the sample correlation coefficient $r$ uses sample means and sample sums of squares instead of expectations.

## Explanations

### L1 human

The correlation coefficient tells how close the points $(X, Y)$ are to a straight line.
A value near $1$ means a strong increasing linear relation, and a value near $-1$ means a strong decreasing one.
A value near $0$ means that there is no linear relation.
</document>

<document source="https://tamnd.github.io/math2/node/04CW1P44VD.md" tag="04CW1P44VD" kind="statement">
# Spectral theorem for normal matrices

## Statement

Let $A$ be an $n \times n$ complex matrix.
Then $A$ is [[normal|0MR0BYF0K2]] if and only if there are a [[unitary matrix|0AR68JTMDT]] $U$ and a diagonal matrix $D$ with
$$A = U D U^*.$$

## Remarks

The columns of $U$ are then an orthonormal basis of $\C^n$ of eigenvectors of $A$.
If $A$ is Hermitian, then $D$ is real; if $A$ is unitary, then the diagonal entries of $D$ have absolute value $1$.
Over $\R$, a normal matrix need not be diagonalisable: a rotation of the plane is normal.

## Explanations

### L1 human

The normal matrices are exactly the complex matrices that are diagonal in some orthonormal basis.
This generalises the spectral theorem from Hermitian matrices to all normal matrices.
The eigenvalues can now be any complex numbers.
</document>

<document source="https://tamnd.github.io/math2/node/04CYN5G6RY.md" tag="04CYN5G6RY" kind="statement">
# Variance under translation

## Statement

Let $X$ be an integrable real random variable and let $b \in \mathbb{R}$. Then
$$\operatorname{Var}(X + b) = \operatorname{Var}(X).$$

## Remarks

Proof: $E[X + b] = E[X] + b$ by [[linearity of expectation|0P7PTC8DDE]], so $(X + b) - E[X + b] = X - E[X]$.
Together with the [[scaling rule|0KVQ9D86BM]], this gives $\operatorname{Var}(aX + b) = a^2 \operatorname{Var}(X)$ for all real $a$ and $b$.

## Explanations

### L1 human

If you add the same constant to every value, the whole distribution moves, but its spread does not change.
</document>

<document source="https://tamnd.github.io/math2/node/04CYQHN6MC.md" tag="04CYQHN6MC" kind="concept">
# Coordinate vector

## Summary

The coordinate vector of a vector in a finite basis is the list of the coefficients of the vector in that basis.
</document>

<document source="https://tamnd.github.io/math2/node/04D7CHT5V1.md" tag="04D7CHT5V1" kind="definition">
# Inverse hyperbolic functions

## Definition

The inverse hyperbolic sine $\operatorname{arsinh} \colon \mathbb{R} \to \mathbb{R}$ is the inverse of [[$\sinh$|0GWD21EWRW]].
The inverse hyperbolic cosine $\operatorname{arcosh} \colon [1, \infty) \to [0, \infty)$ is the inverse of the restriction of [[$\cosh$|0EQFWKJ37J]] to $[0, \infty)$.
The inverse hyperbolic tangent $\operatorname{artanh} \colon (-1, 1) \to \mathbb{R}$ is the inverse of [[$\tanh$|0MWYQTZ07Z]].

## Remarks

Solving a quadratic equation in $e^y$ gives the formulas $\operatorname{arsinh} x = \ln(x + \sqrt{x^2 + 1})$ and $\operatorname{arcosh} x = \ln(x + \sqrt{x^2 - 1})$.
Similarly, $\operatorname{artanh} x = \frac{1}{2} \ln \frac{1 + x}{1 - x}$ for $|x| < 1$.
Their derivatives are $\frac{1}{\sqrt{x^2 + 1}}$, $\frac{1}{\sqrt{x^2 - 1}}$ for $x > 1$, and $\frac{1}{1 - x^2}$. Mathlib defines the first one as `Real.arsinh`.

## Explanations

### L1 human

The inverse hyperbolic functions undo the hyperbolic functions.
Since the hyperbolic functions come from $e^x$, their inverses come from the logarithm.
</document>

<document source="https://tamnd.github.io/math2/node/04DC0SRN2K.md" tag="04DC0SRN2K" kind="statement">
# Order of a group is the index times the order of the subgroup

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $|G| = [G : H] \cdot |H|$, where $[G : H]$ is the [[index|0DTXBQPA2S]] of $H$ in $G$.

## Remarks

The left cosets of $H$ partition $G$, and [[each coset has $|H|$ elements|0DKMBWTD7P]].
This statement gives [[Lagrange's theorem|3C53TD]]: the order of $H$ divides the order of $G$.
The formula $|G| = [G : H] \cdot |H|$ is also true for infinite groups as an equation of cardinal numbers.

## Explanations

### L1 human

Cut a finite group into the left cosets of a subgroup.
Each piece has the same size as the subgroup, and the number of pieces is the index.
So the size of the group is the number of pieces times the size of one piece.
</document>

<document source="https://tamnd.github.io/math2/node/04DR5PM25N.md" tag="04DR5PM25N" kind="concept">
# Modular exponentiation

## Summary

Modular exponentiation computes $b^e \bmod m$ for integers $b$, $e \ge 0$ and $m \ge 1$.
The binary method writes $e$ in base $2$. It squares $b$ again and again modulo $m$, and it multiplies the squares $b^{2^i}$ for the bits $i$ of $e$ that are $1$.
It needs about $2 \log_2 e$ multiplications modulo $m$, and each intermediate number is smaller than $m^2$.

## Remarks

The method is correct because of [[compatibility with multiplication|072D5G4SK6]]: one can reduce modulo $m$ after each multiplication.
The [[Fermat primality test|0N5ZSZMYH6]], the [[RSA cryptosystem|0HM584C465]] and the [[Diffie–Hellman key exchange|01TKKESHSB]] all use modular exponentiation.
The inverse problem, to find $e$ from $b$, $m$ and $b^e \bmod m$, is the [[discrete logarithm|09B4X51J09]] problem. It seems much harder.

## Explanations

### L1 human

To compute $3^{13} \bmod 7$, write $13 = 8 + 4 + 1$. The squares are $3^1 \equiv 3$, $3^2 \equiv 2$, $3^4 \equiv 4$ and $3^8 \equiv 2$.
Then $3^{13} \equiv 2 \cdot 4 \cdot 3 = 24 \equiv 3 \pmod 7$.
You never need the large number $3^{13} = 1594323$.
</document>

<document source="https://tamnd.github.io/math2/node/04DXAFN2ES.md" tag="04DXAFN2ES" kind="concept">
# Natural logarithm

## Summary

The natural logarithm is the inverse of the exponential function. It is defined for positive numbers, it turns products into sums, and its derivative is $1/x$. The node [[natural logarithm|0044CPHJBV]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/04E4V9G5FR.md" tag="04E4V9G5FR" kind="definition">
# Commutativity

## Definition

Let $S$ be a set and let $\ast$ be a [[binary operation|0M418FE7BB]] on $S$.
The operation $\ast$ is commutative when $a \ast b = b \ast a$ for all $a, b \in S$.

## Remarks

Addition and multiplication of real numbers are commutative.
Subtraction is not commutative: $1 - 2 \neq 2 - 1$.
Multiplication of $2 \times 2$ real matrices is associative but not commutative.

## Explanations

### L1 human

Commutativity says that the order of the two inputs does not matter.
For example, $3 \times 5$ and $5 \times 3$ are both $15$.
Many operations do not have this property, for example subtraction and the composition of rotations in space.
</document>

<document source="https://tamnd.github.io/math2/node/04EAFVFTF7.md" tag="04EAFVFTF7" kind="object">
# Circle group

## Description

The circle group is the set $\mathbb{T} = \{z \in \mathbb{C} : |z| = 1\}$ of the complex numbers of absolute value $1$, with complex multiplication.
It is an [[abelian group|082GHNPJ53]] with identity element $1$, and the inverse of $z$ is its complex conjugate $\bar{z} = 1/z$.
Each element of $\mathbb{T}$ is $e^{i\theta}$ for a real number $\theta$, and $e^{i\theta} e^{i\varphi} = e^{i(\theta + \varphi)}$.

## Remarks

The map $t \mapsto e^{2\pi i t}$ is a surjective homomorphism from $(\mathbb{R}, +)$ to $\mathbb{T}$ with kernel $\mathbb{Z}$, thus $\mathbb{T}$ is isomorphic to the quotient $\mathbb{R}/\mathbb{Z}$.
The circle group is also isomorphic to the group $\mathrm{SO}(2)$ of rotations of the plane about the origin.
For each positive integer $n$, the [[group of $n$th roots of unity|0WX11EA3F3]] is a finite subgroup of $\mathbb{T}$.

## Explanations

### L1 human

The circle group is the unit circle in the complex plane, where the product of two points is their complex product.
If you write the points as angles, the product adds the angles.
Thus the circle group is the group of rotations of a circle.
</document>

<document source="https://tamnd.github.io/math2/node/04EHB7YDQS.md" tag="04EHB7YDQS" kind="definition">
# Spectrum of a matrix

## Definition

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
The *spectrum* of $A$ is the set
$$\sigma(A) = \{\lambda \in F : A - \lambda I_n \text{ is not invertible}\}.$$
It is the set of [[eigenvalues|0BQAA2BK3Q]] of $A$.

## Remarks

The spectrum has at most $n$ elements, because each eigenvalue is a root of the characteristic polynomial of degree $n$.
Over $\C$, the spectrum of each square matrix is nonempty.
The spectrum depends on the field: the real matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ has empty spectrum over $\R$ and spectrum $\{i, -i\}$ over $\C$.
Some authors count the eigenvalues with their multiplicities and call this list the spectrum.

## Explanations

### L1 human

The spectrum is the list of all eigenvalues of a matrix.
It tells us the scaling factors that the matrix shows in special directions.
For a diagonal matrix, it is the set of diagonal entries.
</document>

<document source="https://tamnd.github.io/math2/node/04ETX30HT2.md" tag="04ETX30HT2" kind="statement">
# Uniqueness of the supremum

## Statement

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$.
If $b$ and $b'$ are both [[suprema|0BTMQ04556]] of $S$, then $b = b'$.

## Remarks

Proof: $b$ is an upper bound and $b'$ is a least upper bound, thus $b' \le b$. In the same way $b \le b'$, and antisymmetry gives $b = b'$.
The same argument shows that a set has at most one infimum.

## Explanations

### L1 human

A set cannot have two different least upper bounds.
If it had two, each one would be at most the other one, so they would be equal.
</document>

<document source="https://tamnd.github.io/math2/node/04FHQSJCQ5.md" tag="04FHQSJCQ5" kind="definition">
# Modulo operation

## Definition

Let $a$ and $n$ be integers with $n \neq 0$.
The integer $a \bmod n$ is the remainder $r$ of the [[division of $a$ by $n$|0CHQKZFK5S]].
Thus $a \bmod n$ is the unique integer $r$ such that $0 \le r < |n|$ and $a - r$ is a multiple of $n$.

## Remarks

For example, $17 \bmod 5 = 2$ and $(-17) \bmod 5 = 3$.
Many programming languages use a different convention. In them, the remainder has the sign of $a$, so the result for $-17$ and $5$ is $-2$.
The Lean library Mathlib sets $a \bmod 0 = a$.

## Explanations

### L1 human

The value $a \bmod n$ is what is left after you take away from $a$ as many copies of $n$ as you can.
A clock uses this operation: $15$ hours after $0$ o'clock, a $12$-hour clock shows $15 \bmod 12 = 3$.
</document>

<document source="https://tamnd.github.io/math2/node/04GM4NYXJB.md" tag="04GM4NYXJB" kind="definition">
# Mesh of a partition

## Definition

Let $P = (x_0, \dots, x_n)$ be a [[partition|069P20KYQT]] of $[a, b]$.
The mesh of $P$ is
$$\| P \| = \max_{1 \le i \le n} (x_i - x_{i-1}),$$
the length of the longest subinterval of $P$.

## Remarks

Refining a partition does not increase its mesh.
The [[Riemann integral|0QM4TXT3DR]] is a limit of Riemann sums as the mesh tends to $0$.
For the partition of $[0, 1]$ into $n$ equal pieces, the mesh is $1/n$.

## Explanations

### L1 human

The mesh measures how fine a partition is.
A small mesh means that all the pieces are short.
</document>

<document source="https://tamnd.github.io/math2/node/04GNK7K4VP.md" tag="04GNK7K4VP" kind="definition">
# Divisibility in a commutative ring

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $a, b \in R$.
The element $a$ divides $b$, written $a \mid b$, if there is $c \in R$ with $b = a c$.
Then $a$ is a divisor of $b$, and $b$ is a multiple of $a$.

## Remarks

The relation $a \mid b$ holds if and only if $(b) \subseteq (a)$ for the [[principal ideals|00A2EDFT9Q]].
Every element divides $0$, and $0$ divides only $0$.
A [[unit|019NFGFPR5]] divides every element.
In $\mathbb{Z}[i]$, $1 + i$ divides $2$, because $2 = (1 + i)(1 - i)$.

## Explanations

### L1 human

Divisibility in a ring works as for integers: $a$ divides $b$ if $b$ is $a$ times some element of the ring.
The answer depends on the ring: $2$ does not divide $1$ in $\mathbb{Z}$, but it does in $\mathbb{Q}$.
</document>

<document source="https://tamnd.github.io/math2/node/04GYPXDXNH.md" tag="04GYPXDXNH" kind="statement">
# Riesz representation theorem in finite dimensions

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, with the inner product linear in the first argument.
Then, for each [[linear form|07YQ1C78G0]] $f \colon V \to \mathbb{K}$, there is exactly one vector $y \in V$ with
$$f(x) = \langle x, y \rangle \qquad \text{for all } x \in V.$$

## Remarks

For an orthonormal basis $e_1, \dots, e_n$, the vector is $y = \sum_i \overline{f(e_i)}\, e_i$.
The map $y \mapsto \langle \cdot, y \rangle$ from $V$ to its dual is bijective; it is linear over $\R$ and conjugate linear over $\C$.
The statement holds for continuous linear forms on every Hilbert space.

## Explanations

### L1 human

In an inner product space, every linear form is an inner product with a fixed vector.
On $\R^3$, the form $x \mapsto 2x_1 - x_2 + 5x_3$ is the dot product with $(2, -1, 5)$.
So in finite dimensions, vectors and linear forms correspond one to one.
</document>

<document source="https://tamnd.github.io/math2/node/04J4BYB0XE.md" tag="04J4BYB0XE" kind="concept">
# Separation axioms

## Summary

The separation axioms are conditions on a topological space that say how well open sets can separate points and closed sets.
The main ones, from weak to strong, are [[T0|0R3B7DX55J]], [[T1|05GQJXBP3F]], [[Hausdorff|0MXB4CKHFW]], [[regular|0JF9KW2TXB]] and [[normal|0AJ646N339]].

## Remarks

Each of these axioms, together with T1, implies the axioms before it in this list.
Each metric space satisfies all of them, by [[metric spaces are Hausdorff|0P9DCDES73]] and [[metrizable spaces are normal|0AD0BZ49DW]].
Authors do not agree on all names. For example, some authors include T1 in the definition of a regular or normal space, and some do not.

## Explanations

### L1 human

The separation axioms measure how many open sets a space has.
With more open sets, you can put distinct points, or points and closed sets, into disjoint open sets.
</document>

<document source="https://tamnd.github.io/math2/node/04J4Z73J5B.md" tag="04J4Z73J5B" kind="concept">
# Divisor function

## Summary

The divisor function $\sigma_k$ sends a positive integer $n$ to the sum of the $k$-th powers of its positive divisors.
The [[definition of the divisor function|0YEW47VJ0J]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/04J6G4H78J.md" tag="04J6G4H78J" kind="statement">
# Density of the irrationals

## Statement

Let $x$ and $y$ be real numbers with $x < y$.
Then there is an [[irrational number|080T2TXVQV]] $z$ with $x < z < y$.

## Remarks

Proof: by the [[density of the rationals|0STX0WRDNG]], there is a rational $q$ with $x / \sqrt{2} < q < y / \sqrt{2}$ and $q \neq 0$. Then $z = q \sqrt{2}$ is irrational and $x < z < y$.
Thus both $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ are dense in $\mathbb{R}$.

## Explanations

### L1 human

Between any two different real numbers, there is also a number that is not a fraction.
So fractions and non-fractions are mixed together everywhere on the number line.
</document>

<document source="https://tamnd.github.io/math2/node/04JAAVHDDM.md" tag="04JAAVHDDM" kind="definition">
# Consistent and inconsistent systems

## Definition

Let $F$ be a field.
A [[system of linear equations|0GT4WWGWH2]] over $F$ is *consistent* when it has at least one solution in $F^n$.
It is *inconsistent* when it has no solution.

## Remarks

The system $x + y = 1$, $x + y = 2$ over $\R$ is inconsistent.
Every homogeneous system is consistent, because the zero tuple is a solution.
The [[Rouché–Capelli theorem|0HV1SKWPMS]] gives a test with ranks: $Ax = b$ is consistent if and only if $\rank A = \rank [A \mid b]$.

## Explanations

### L1 human

"Consistent" means that the equations can all be true at the same time.
Two parallel lines in the plane, like $x + y = 1$ and $x + y = 2$, have no common point, so their system is inconsistent.
Elimination shows inconsistency when it produces an equation $0 = c$ with $c \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/04JAY9DGB7.md" tag="04JAY9DGB7" kind="statement">
# Dirichlet's approximation theorem

## Statement

Let $\alpha$ be a real number and let $N$ be a positive integer.
Then there are integers $p$ and $q$ with $1 \le q \le N$ and
$$|q \alpha - p| \le \frac{1}{N + 1}.$$

## Remarks

A corollary: if $\alpha$ is irrational, then $\left| \alpha - \frac{p}{q} \right| < \frac{1}{q^2}$ for infinitely many fractions $p / q$.
The proof uses the pigeonhole principle. Among the $N + 1$ fractional parts of $0, \alpha, 2\alpha, \dots, N\alpha$, two lie in one of $N + 1$ intervals of length $\frac{1}{N + 1}$.
The [[convergents|0E70M92FE3]] of the continued fraction of $\alpha$ give such fractions explicitly.

## Explanations

### L1 human

Each real number has fractions $p/q$ very close to it, with an error much smaller than $1/q$.
The theorem gives the bound $1/q^2$ for infinitely many denominators when $\alpha$ is irrational.
</document>

<document source="https://tamnd.github.io/math2/node/04JHQNNBFE.md" tag="04JHQNNBFE" kind="statement">
# Hockey-stick identity

## Statement

Let $n$ and $r$ be natural numbers with $r \le n$.
Then
$$\sum_{i=r}^{n} \binom{i}{r} = \binom{n + 1}{r + 1}.$$

## Remarks

For $r = 1$ it gives $1 + 2 + \dots + n = \binom{n+1}{2} = \frac{n(n+1)}{2}$.
Combinatorial proof: classify the $(r+1)$-subsets of $\{1, \dots, n + 1\}$ by their largest element $i + 1$. The other $r$ elements come from $\{1, \dots, i\}$.
In [[Pascal's triangle|05DME4TZ3K]] the summands lie on a diagonal, and the result lies below the end of the diagonal. This shape is the reason for the name.

## Explanations

### L1 human

Add the entries of [[Pascal's triangle|05DME4TZ3K]] along a diagonal, from the edge downward.
The total is the entry just below and to the side of the last one, so the picture looks like a hockey stick.
</document>

<document source="https://tamnd.github.io/math2/node/04KFTA34C5.md" tag="04KFTA34C5" kind="definition">
# Infimum

## Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$.
An element $a$ of $P$ is an infimum, or greatest lower bound, of $S$ when these two conditions hold:

1. $a$ is a [[lower bound|0B5H2V4JW8]] of $S$;
2. if $m$ is a lower bound of $S$, then $m \le a$.

The notation is $a = \inf S$.

## Remarks

A set has at most one infimum.
In $\mathbb{R}$, $\inf \{1/n : n \ge 1\} = 0$, and $0$ is not an element of this set.
For a subset $S$ of $\mathbb{R}$, $\inf S = -\sup(-S)$ when one side exists, where $-S = \{-s : s \in S\}$.

## Explanations

### L1 human

The infimum is the highest possible floor for a set.
For the set $\{1, 1/2, 1/3, \dots\}$, every number at most $0$ is a floor, and $0$ is the highest of these, so the infimum is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/04KGXJG1A2.md" tag="04KGXJG1A2" kind="concept">
# Homogeneous system of linear equations

## Summary

A homogeneous system of linear equations is a system in which every constant term is zero.
Such a system always has the zero solution, and its solution set is a vector subspace.
</document>

<document source="https://tamnd.github.io/math2/node/04MQSQ1F5Q.md" tag="04MQSQ1F5Q" kind="concept">
# Clique number

## Summary

The clique number of a finite graph is the size of its largest clique. It is a lower bound for the chromatic number. The node [[clique number|0WCKRBHFFD]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/04NPWJMA6H.md" tag="04NPWJMA6H" kind="statement">
# Sum of two squares theorem

## Statement

Let $n$ be a positive integer.
Then $n = x^2 + y^2$ for some integers $x$ and $y$ if and only if $\nu_p(n)$ is even for each prime $p \equiv 3 \pmod 4$.
Here $\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].

## Remarks

For example, $45 = 3^2 \cdot 5 = 6^2 + 3^2$, but $21 = 3 \cdot 7$ is not a sum of two squares.
The "if" part follows from [[Fermat's theorem on sums of two squares|043B3ENSTY]] and the [[Brahmagupta–Fibonacci identity|0K2DMACEZ4]].
Zero squares are allowed, so each square and each double of a square is a sum of two squares.

## Explanations

### L1 human

To decide whether $n$ is a sum of two squares, factor $n$ into primes.
Look only at the primes that leave remainder $3$ on division by $4$. Each of them must appear an even number of times.
</document>

<document source="https://tamnd.github.io/math2/node/04P2RJ4WPS.md" tag="04P2RJ4WPS" kind="concept">
# Continued fraction

## Summary

A continued fraction is an expression $b_0 + a_1 / (b_1 + a_2 / (b_2 + \cdots))$, a fraction whose denominator contains a fraction, and so on.
The [[definition of a continued fraction|0BK6JBK58W]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/04P2SR252N.md" tag="04P2SR252N" kind="concept">
# Geometric distribution

## Summary

The geometric distribution describes the waiting time until the first success in a sequence of independent Bernoulli trials.
One version counts the failures before the first success, and the other counts the trials up to and including it. The node [[geometric distribution|0XCVV5Q500]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/04P7B5XC22.md" tag="04P7B5XC22" kind="definition">
# Lyapunov function

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz, and let $x^* \in U$ be an [[equilibrium point|0G4X36AE0X]] of $x' = f(x)$.
Let $W \subseteq U$ be an open neighbourhood of $x^*$ and let $V : W \to \mathbb{R}$ be continuously differentiable.
The orbital derivative of $V$ is the function $\dot{V}(x) = \nabla V(x) \cdot f(x)$ on $W$.
The function $V$ is a Lyapunov function for $x^*$ if $V(x^*) = 0$, $V(x) > 0$ for each $x \in W \setminus \{x^*\}$, and $\dot{V}(x) \le 0$ for each $x \in W$.
It is a strict Lyapunov function if also $\dot{V}(x) < 0$ for each $x \in W \setminus \{x^*\}$.

## Remarks

For each solution $\gamma$ in $W$, $\frac{d}{dt} V(\gamma(t)) = \dot{V}(\gamma(t))$. So $V$ does not increase along solutions.
For $x' = -x$ on $\mathbb{R}$, the function $V(x) = x^2$ is a strict Lyapunov function for $0$, since $\dot{V}(x) = -2x^2$.
In mechanics, the total energy is often a Lyapunov function. See [[Lyapunov's stability theorem|0J6A752DTZ]].

## Explanations

### L1 human

A Lyapunov function is like a height above the equilibrium, which is the lowest point.
If the height never increases along the motion, the state cannot climb away from the equilibrium.
</document>

<document source="https://tamnd.github.io/math2/node/04P7D63W8P.md" tag="04P7D63W8P" kind="definition">
# Almost sure convergence

## Definition

Let $X, X_1, X_2, \dots$ be real random variables on one probability space $(\Omega, \mathcal{F}, P)$.
The sequence $(X_n)$ converges almost surely to $X$ if
$$P\left(\left\{\omega \in \Omega : \lim_{n \to \infty} X_n(\omega) = X(\omega)\right\}\right) = 1.$$
One writes $X_n \to X$ a.s. or $X_n \xrightarrow{\text{a.s.}} X$.

## Remarks

The set in the definition is an event, because the $X_n$ and $X$ are measurable.
An equivalent condition: for each $\varepsilon > 0$, $P(|X_n - X| > \varepsilon \text{ for infinitely many } n) = 0$.
The limit is unique up to equality [[almost surely|0R46A1XY60]].
The strong law of large numbers is a result about almost sure convergence.

## Explanations

### L1 human

Almost sure convergence means that, apart from a set of outcomes of probability $0$, each run of the experiment gives a convergent sequence of numbers.
</document>

<document source="https://tamnd.github.io/math2/node/04QEBF8S49.md" tag="04QEBF8S49" kind="definition">
# Index of a subgroup

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
The index of $H$ in $G$, written $[G : H]$ or $|G : H|$, is the cardinality of the set of [[left cosets|03DEH43F4H]] of $H$ in $G$.

## Remarks

The index is also the number of right cosets, because $gH \mapsto Hg^{-1}$ is a bijection between the two sets of cosets.
In $(\mathbb{Z}, +)$, the subgroup $n\mathbb{Z}$ has index $n$ for each positive integer $n$.
A subgroup of a finite group has index $|G| / |H|$, but an infinite group can have a subgroup of finite index, for example $2\mathbb{Z}$ in $\mathbb{Z}$.

## Explanations

### L1 human

The index counts how many disjoint copies of the subgroup you need to cover the whole group.
For the even integers inside the integers, you need two copies: the even and the odd integers.
Thus the index of the even integers is $2$.
</document>

<document source="https://tamnd.github.io/math2/node/04QNZZ4HHY.md" tag="04QNZZ4HHY" kind="definition">
# Dedekind cut

## Definition

A Dedekind cut is a pair $(A, B)$ of subsets of the [[rational numbers|0NCJ5WXN6Z]] $\mathbb{Q}$ with these properties:

1. $A$ and $B$ are nonempty, $A \cap B = \emptyset$ and $A \cup B = \mathbb{Q}$;
2. if $a \in A$, $q \in \mathbb{Q}$ and $q < a$, then $q \in A$;
3. $A$ has no greatest element.

## Remarks

The set $A$ determines the cut, because $B = \mathbb{Q} \setminus A$. Thus some authors call $A$ itself the cut.
The cut with $A = \{q \in \mathbb{Q} : q < 0 \text{ or } q^2 < 2\}$ corresponds to $\sqrt{2}$. Here $B$ has no least element.
The set of all cuts, ordered by inclusion of the lower sets, is a [[complete ordered field|0A9E6X0S3E]] with suitable operations.

## Explanations

### L1 human

Cut the rational number line at one point. Put the numbers to the left in $A$ and the numbers to the right in $B$.
When the cut is at a gap of $\mathbb{Q}$, such as $\sqrt{2}$, the cut itself stands for the missing number.
</document>

<document source="https://tamnd.github.io/math2/node/04R3CY41VJ.md" tag="04R3CY41VJ" kind="statement">
# Euler's formula

## Statement

For every complex number $z$,
$$e^{i z} = \cos z + i \sin z.$$
Here $e^{w} = \sum_{n=0}^{\infty} w^n / n!$ is the complex exponential function, and $\cos$ and $\sin$ are given by their power series.

## Remarks

The main case is a real number $z = x$. Then $\cos x$ and $\sin x$ are the real and imaginary parts of $e^{i x}$, and $|e^{i x}| = 1$.
For $x = \pi$, the formula gives [[Euler's identity|09285XX7AE]].
A proof compares the power series: the even terms of $e^{i z}$ give $\cos z$ and the odd terms give $i \sin z$.

## Explanations

### L1 human

As the real number $x$ grows, the point $e^{i x}$ moves around the unit circle at constant speed.
Its coordinates are $\cos x$ and $\sin x$, as in trigonometry.
The formula links the exponential function with the trigonometric functions.
</document>

<document source="https://tamnd.github.io/math2/node/04R9K5H562.md" tag="04R9K5H562" kind="concept">
# Classification of discontinuities

## Summary

A point of the domain where a real function is not [[continuous|0EEE2CZB2A]] is a discontinuity. Discontinuities are sorted by the behavior of the [[one-sided limits|0909PWZR62]] at the point.

## Remarks

At a [[removable discontinuity|0QG9C579SR]], the limit exists but differs from the value.
At a [[jump discontinuity|05B0FKFXQA]], both one-sided limits exist but they differ.
At an essential discontinuity, at least one one-sided limit does not exist. An example is $\sin(1/x)$ at $0$, with any value at $0$.
A [[monotone function has countably many discontinuities|0WYTM4C9QG]], and all of them are jumps.

## Explanations

### L1 human

A function can fail to be continuous in different ways.
A hole in the graph can be repaired, a jump cannot, and some functions oscillate too wildly to have any limit.
</document>

<document source="https://tamnd.github.io/math2/node/04RG02S6BA.md" tag="04RG02S6BA" kind="statement">
# Cyclic groups are abelian

## Statement

Let $G$ be a [[cyclic group|0KVGH0J5B3]].
Then $G$ is [[abelian|082GHNPJ53]].

## Remarks

If $g$ generates $G$, then any two elements are $g^a$ and $g^b$ for integers $a, b$, and $g^a g^b = g^{a+b} = g^b g^a$ by [[the laws of exponents|00DRK90Z6K]].
The converse is false: the Klein four-group is abelian but not cyclic.
The group $(\mathbb{Q}, +)$ is abelian but not cyclic.

## Explanations

### L1 human

All elements of a cyclic group are powers of one element.
Two powers of the same element always commute, because the exponents just add.
Thus the order of the factors never matters in a cyclic group.

### L2 agent

A cyclic group is generated by one element $g$. Each element is a power $g^m$ for some integer $m$. The result says that such a group is abelian.

The reason is that powers of one element add their exponents:

1. Let $x = g^m$ and $y = g^n$.
2. By the laws of exponents, $xy = g^{m+n}$ and $yx = g^{n+m}$.
3. Addition of integers is commutative, so $xy = yx$.

A worked example in $\mathbb{Z}/6\mathbb{Z}$ under addition, with generator $1$:

1. Write $2 = 1 + 1$ and $5 = 1 + 1 + 1 + 1 + 1$.
2. Then $2 + 5$ and $5 + 2$ are both seven copies of $1$.
3. Seven copies of $1$ give $7 \bmod 6 = 1$. Thus $2 + 5 = 5 + 2 = 1$.

The converse is false. The Klein four-group $V = \{e, a, b, c\}$ is abelian, but each element other than $e$ has order $2$. Thus no element generates the four elements, and $V$ is not cyclic. The group $(\mathbb{Q}, +)$ is another abelian group that is not cyclic.

A useful consequence is a test for groups that are not cyclic. If a group is not abelian, then it is not cyclic. For example, $S_3$ is not abelian, so it is not cyclic.

Each subgroup of a cyclic group is also cyclic. This fact needs a different proof, by division with remainder.
</document>

<document source="https://tamnd.github.io/math2/node/04RRB6HM00.md" tag="04RRB6HM00" kind="statement">
# Law of total expectation

## Statement

Let $X$ be an integrable real random variable on $(\Omega, \mathcal{F}, P)$ and let $\mathcal{G} \subseteq \mathcal{F}$ be a σ-algebra. Then
$$E\left[E[X \mid \mathcal{G}]\right] = E[X].$$
In particular, $E[E[X \mid Y]] = E[X]$ for each random variable $Y$ on the same space.

## Remarks

Proof: put $G = \Omega$ in the defining property of the [[conditional expectation|0YAMJRK04E]].
For a partition of $\Omega$ into events $A_1, A_2, \dots$ with $P(A_i) > 0$, it gives $E[X] = \sum_i E[X \mid A_i] \, P(A_i)$. This extends the [[law of total probability|0E5G5FMT2R]].
More generally, for σ-algebras $\mathcal{H} \subseteq \mathcal{G}$, $E[E[X \mid \mathcal{G}] \mid \mathcal{H}] = E[X \mid \mathcal{H}]$. This is the tower property.

## Explanations

### L1 human

To find an overall average, you can first average inside each group and then average the group averages, with the group sizes as weights.
If $60\%$ of the students have mean score $70$ and $40\%$ have mean score $80$, the overall mean is $0.6 \cdot 70 + 0.4 \cdot 80 = 74$.
</document>

<document source="https://tamnd.github.io/math2/node/04RXTQA5ZA.md" tag="04RXTQA5ZA" kind="statement">
# Zorn's lemma

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Let $(P, \le)$ be a nonempty [[partially ordered set|011C29Q1S1]] in which each [[chain|0GP9ASVQ1X]] has an [[upper bound|0H5J02P8S9]] in $P$.
Then $P$ has a [[maximal element|0TZEEFTNCC]].

## Remarks

Over ZF, Zorn's lemma is equivalent to the axiom of choice.
If one requires an upper bound also for the empty chain, the hypothesis "$P$ is nonempty" is not needed.
Applications: each vector space has a basis, and each nonzero ring with unity has a maximal ideal.

## History

Kazimierz Kuratowski proved the lemma in 1922, and Max Zorn proved it independently in 1935.
The [[Hausdorff maximal principle|0KM4KVR9MV]] of 1914 is an earlier statement of the same kind.

## Explanations

### L1 human

Zorn's lemma is a tool to show that a maximal object exists.
You check that each increasing chain of objects has an object above it.
Then the lemma gives an object that you cannot enlarge any more.
</document>

<document source="https://tamnd.github.io/math2/node/04S2GMHX82.md" tag="04S2GMHX82" kind="concept">
# Parabola

## Summary

A parabola is a plane curve. Each of its points has the same distance from a fixed point, the focus, and from a fixed line, the directrix. It is symmetric about the line through the focus perpendicular to the directrix.
</document>

<document source="https://tamnd.github.io/math2/node/04S9HX3GAP.md" tag="04S9HX3GAP" kind="definition">
# Product topology on an arbitrary product

## Definition

Let $(X_i)_{i \in I}$ be a family of [[topological spaces|0CNQXVBHCX]], indexed by any set $I$, and let $X = \prod_{i \in I} X_i$ be the [[Cartesian product|0C81YBVTWH]], with projections $\pi_i \colon X \to X_i$.
The product topology on $X$ is the topology generated by the [[subbasis|0CHR35YNJQ]]
$$\{ \pi_i^{-1}(U) : i \in I, \ U \text{ open in } X_i \}.$$

## Remarks

A basis consists of the sets $\prod_{i \in I} U_i$, where each $U_i$ is open in $X_i$ and $U_i = X_i$ for all but finitely many $i$.
For two factors, this is the [[product topology|0Z57EC6YX3]] of $X_1 \times X_2$.
It is the coarsest topology on $X$ for which each projection $\pi_i$ is continuous.
A sequence converges in $X$ if and only if each of its coordinates converges. The [[box topology|0CTM1EHG9P]] is finer and lacks this property.

## Explanations

### L1 human

In an infinite product, a basic open set can restrict only finitely many coordinates.
This choice makes the [[universal property|0M6W4XNST6]] and [[Tychonoff's theorem|0JTRSJKCFB]] work.
</document>

<document source="https://tamnd.github.io/math2/node/04SD3F4FCR.md" tag="04SD3F4FCR" kind="problem">
# Doubling the cube

## Problem

Given the edge of a cube, construct with [[straightedge and compass|0DAECMZVRT]] alone the edge of a cube with twice the volume.

## Remarks

For an edge of length $1$, the new edge has length $\sqrt[3]{2}$.
So the problem asks whether $\sqrt[3]{2}$ is a [[constructible number|0FMN2MN7TM]].
The answer is no: [[doubling the cube is impossible|0CQ357M0QS]].
The problem has solutions with other tools, such as a marked ruler or conic sections.

## History

The problem is also called the Delian problem, after a legend about an oracle on the island of Delos.
Hippocrates of Chios showed that it is equivalent to finding two mean proportionals between a segment and a segment twice as long.
Pierre Wantzel proved in 1837 that no straightedge and compass solution exists.

## Explanations

### L1 human

The problem asks to build a cube with twice the volume of a given cube, using only an unmarked ruler and a compass.
It is one of the three classical construction problems of Greek geometry.
</document>

<document source="https://tamnd.github.io/math2/node/04SGPNT8R9.md" tag="04SGPNT8R9" kind="object">
# Galois field

## Description

For each prime power $q = p^n$, the Galois field $\mathrm{GF}(q)$, also written $\mathbb{F}_q$, is the [[finite field|08RTDHX1SK]] with $q$ elements.
It [[exists|0T8M3YC5RG]], and it is [[unique up to isomorphism|010PKCQPXP]].
One model is the splitting field of $x^q - x$ over $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.
Another model is $\mathbb{F}_p[x]/(f)$ for an irreducible polynomial $f \in \mathbb{F}_p[x]$ of degree $n$.

## Remarks

The name honours Évariste Galois.
The field $\mathrm{GF}(4) = \mathbb{F}_2[x]/(x^2 + x + 1)$ has the elements $0, 1, \alpha, \alpha + 1$, with $\alpha^2 = \alpha + 1$.
The field $\mathrm{GF}(p^m)$ is isomorphic to a subfield of $\mathrm{GF}(p^n)$ if and only if $m$ divides $n$.

## Explanations

### L1 human

Since there is only one finite field of each order, up to a change of names, one gives it a fixed name: the Galois field with $q$ elements.
Computations in it are done with polynomials over the integers modulo $p$.
</document>

<document source="https://tamnd.github.io/math2/node/04SJTX9P4N.md" tag="04SJTX9P4N" kind="statement">
# Chinese remainder theorem for non-coprime moduli

## Statement

Let $m$ and $n$ be positive integers, and let $a$ and $b$ be integers. Let $g = \gcd(m, n)$ and $M = \operatorname{lcm}(m, n)$.
The system $x \equiv a \pmod m$, $x \equiv b \pmod n$ has an integer solution if and only if $a \equiv b \pmod g$.
In this case, two solutions are congruent modulo $M$.

## Remarks

When $g = 1$, this is the [[Chinese remainder theorem|0FNN9B4W64]] for two moduli.
The condition is necessary: a solution $x$ gives $a \equiv x \equiv b \pmod g$, since $g$ divides $m$ and $n$.
For $k$ moduli $n_1, \dots, n_k$, the system $x \equiv a_i \pmod{n_i}$ has a solution if and only if $\gcd(n_i, n_j) \mid a_i - a_j$ for all $i \neq j$.
For example, $x \equiv 1 \pmod 4$ and $x \equiv 3 \pmod 6$ has the solutions $x \equiv 9 \pmod{12}$. But $x \equiv 1 \pmod 4$ and $x \equiv 2 \pmod 6$ has no solution.

## Explanations

### L1 human

When the moduli share a factor $g$, the two conditions both say something about $x$ modulo $g$.
They must agree there. If they agree, there is a solution, and it is unique modulo the lcm.
</document>

<document source="https://tamnd.github.io/math2/node/04SW7WECS8.md" tag="04SW7WECS8" kind="definition">
# Solution of an ordinary differential equation

## Definition

Let $F\left(x, y, y', \dots, y^{(n)}\right) = 0$ be an [[ordinary differential equation|0P97V20RW7]] of order $n$, with $F$ defined on a set $U \subseteq \mathbb{R}^{n+2}$.
Let $I$ be an open [[interval|0Q1D85B8G6]] of $\mathbb{R}$.
A solution of the equation on $I$ is a function $u : I \to \mathbb{R}$ with these two properties.
The function $u$ is $n$ times differentiable on $I$, and $\left(x, u(x), u'(x), \dots, u^{(n)}(x)\right) \in U$ for each $x \in I$.
For each $x \in I$, $F\left(x, u(x), u'(x), \dots, u^{(n)}(x)\right) = 0$.
For a system, $u$ takes values in $\mathbb{R}^m$ and the same two conditions hold.

## Remarks

Consider the first-order system $x' = f(x)$, with a vector field $f$ on an open set of $\mathbb{R}^m$. A solution is a curve whose velocity at each time is the value of $f$ at its position. Such a curve is an integral curve of $f$.
The function $u(x) = e^{2x}$ is a solution of $y' = 2y$ on $\mathbb{R}$.
A solution on $I$ restricts to a solution on each open subinterval of $I$. A [[maximal solution|01J7FZ65J0]] is a solution that has no extension to a larger interval.

## Explanations

### L1 human

To check that a function is a solution, put it and its derivatives into the equation.
If both sides are equal at each point of the interval, the function is a solution there.
For example, $u(x) = \sin x$ gives $u'' + u = -\sin x + \sin x = 0$, so it solves $y'' + y = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/04T1CT7212.md" tag="04T1CT7212" kind="statement">
# Contrapositive

## Statement

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The [[implication|0ZNX6DQ3EJ]] $p \to q$ is [[logically equivalent|06GYGS41JK]] to its contrapositive $\neg q \to \neg p$.
$$p \to q \equiv \neg q \to \neg p$$

## Remarks

The direction from $p \to q$ to $\neg q \to \neg p$ is valid in intuitionistic logic. The other direction needs classical logic.
This equivalence is the basis of [[proof by contrapositive|0BB5TPGQ1N]].
Do not confuse the contrapositive with the [[converse|0RVYE75E2M]] $q \to p$.

## Explanations

### L1 human

"If it rains, the street is wet" says the same as "if the street is not wet, it does not rain".
Both sentences are false in exactly one case: rain and a dry street.
</document>

<document source="https://tamnd.github.io/math2/node/04TKTF8KDK.md" tag="04TKTF8KDK" kind="concept">
# Higher derivative

## Summary

The higher derivatives of a function come from repeated differentiation. The $n$-th derivative $f^{(n)}$ is the derivative of $f^{(n-1)}$. The node [[higher derivative|084NGSTWM7]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/04TMP0XGMJ.md" tag="04TMP0XGMJ" kind="concept">
# Definite integral

## Summary

A definite integral is the integral of a function between two given limits of integration. It is a number, and its sign changes when the two limits are swapped. The node [[definite integral|021WEM680W]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/04TSHB9H8D.md" tag="04TSHB9H8D" kind="definition">
# Little o notation

## Definition

Let $f$ and $g$ be real functions defined on an interval $[x_0, \infty)$.
The notation $f(x) = o(g(x))$ as $x \to \infty$ means that for each $\varepsilon > 0$ there is $x_1 \ge x_0$ with
$$|f(x)| \le \varepsilon |g(x)| \quad \text{for all } x \ge x_1.$$

## Remarks

If $g(x) \ne 0$ for large $x$, then $f = o(g)$ means that $f(x)/g(x) \to 0$.
Each $f = o(g)$ is also $f = O(g)$, in the sense of [[big O notation|0RYTGBJAGN]].
For example, $x = o(x^2)$ and $\ln x = o(x)$ as $x \to \infty$.
Mathlib defines it for filters, as `Asymptotics.IsLittleO`.

## Explanations

### L1 human

The statement $f = o(g)$ says that $f$ becomes negligible compared with $g$.
Each fixed fraction of $|g|$ eventually exceeds $|f|$.
</document>

<document source="https://tamnd.github.io/math2/node/04VA28F3HB.md" tag="04VA28F3HB" kind="concept">
# Continued fraction expansion of a real number

## Summary

The continued fraction expansion of a real number is the simple continued fraction found by taking integer parts and reciprocals again and again.
The [[definition of the continued fraction expansion|0SJ7Z6VKWK]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/04VCPS30ZE.md" tag="04VCPS30ZE" kind="definition">
# P-adic valuation

## Definition

Let $p$ be a [[prime number|0SN0MAEKYX]] and let $n$ be a nonzero integer.
The $p$-adic valuation $\nu_p(n)$ is the largest natural number $k$ such that $p^k \mid n$.
By convention, $\nu_p(0) = \infty$.

## Remarks

For a positive integer $n$, $\nu_p(n)$ is the exponent of $p$ in the [[prime factorization|0AEKDFCYFA]] of $n$. It is $0$ when $p \nmid n$.
For example, $\nu_2(40) = 3$ and $\nu_5(40) = 1$, because $40 = 2^3 \cdot 5$.
For a nonzero rational number $a / b$, one sets $\nu_p(a / b) = \nu_p(a) - \nu_p(b)$.
The Lean library Mathlib sets $\nu_p(0) = 0$ for its function on natural numbers.

## Explanations

### L1 human

The number $\nu_p(n)$ counts how many times you can divide $n$ by $p$ and still get an integer.
For $n = 72$ and $p = 2$: $72, 36, 18, 9$, so you can divide three times, and $\nu_2(72) = 3$.
</document>

<document source="https://tamnd.github.io/math2/node/04VZE3FZXF.md" tag="04VZE3FZXF" kind="concept">
# Random walk

## Summary

A random walk is a stochastic process whose position is a sum of independent random steps. The simple random walk on the integers moves one step up or down at each time.
The node [[random walk|0C08YMV95H]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/04WC36XKV6.md" tag="04WC36XKV6" kind="statement">
# Multiplicativity of the divisor function

## Statement

Let $k$ be a natural number.
The [[divisor function|0YEW47VJ0J]] $\sigma_k$ is a [[multiplicative function|0AP80TT660]].
That is, $\sigma_k(mn) = \sigma_k(m) \sigma_k(n)$ for all coprime positive integers $m$ and $n$.

## Remarks

The proof: for coprime $m$ and $n$, each divisor of $mn$ is a unique product $d e$ with $d \mid m$ and $e \mid n$.
The function $\sigma_k$ is not completely multiplicative: $\sigma_0(4) = 3$, but $\sigma_0(2)^2 = 4$.
At a prime power, $\sigma_k(p^e) = 1 + p^k + p^{2k} + \dots + p^{ek}$. Multiplicativity then gives $\sigma_k(n)$ for all $n$.

## Explanations

### L1 human

For example, $\sigma(4) = 7$ and $\sigma(3) = 4$, so $\sigma(12) = 28$.
This works because $4$ and $3$ share no prime factor. Each divisor of $12$ is a divisor of $4$ times a divisor of $3$.
</document>

<document source="https://tamnd.github.io/math2/node/04WSW8T3SY.md" tag="04WSW8T3SY" kind="definition">
# Sigma-algebra

## Definition

Let $\Omega$ be a [[set|0SJ3RX9TXA]].
A σ-algebra on $\Omega$ is a subset $\mathcal{F}$ of the [[power set|0BC7KX51Z8]] of $\Omega$ with these three properties.
The set $\Omega$ is in $\mathcal{F}$.
If $A \in \mathcal{F}$, then its [[complement|0B1FSS1VZW]] $\Omega \setminus A$ is in $\mathcal{F}$.
If $A_1, A_2, A_3, \dots$ is a sequence of elements of $\mathcal{F}$, then the [[union|07KVB4JSC7]] $\bigcup_{n=1}^{\infty} A_n$ is in $\mathcal{F}$.
The pair $(\Omega, \mathcal{F})$ is a measurable space, and the elements of $\mathcal{F}$ are the measurable sets.

## Remarks

A σ-algebra contains $\varnothing$ and is closed under countable intersections, by De Morgan's laws.
The power set of $\Omega$ and the family $\{\varnothing, \Omega\}$ are σ-algebras on $\Omega$. They are the largest and the smallest σ-algebras on $\Omega$.
For each family $\mathcal{C}$ of subsets of $\Omega$, the intersection of all σ-algebras that contain $\mathcal{C}$ is a σ-algebra. It is the σ-algebra $\sigma(\mathcal{C})$ generated by $\mathcal{C}$.
The [[Borel σ-algebra|0BVHZXWKCD]] on $\mathbb{R}$ is the σ-algebra generated by the open intervals.

## Explanations

### L1 human

A σ-algebra is the list of the sets that we are allowed to measure.
If we can measure a set, then we can measure its complement.
If we can measure each set of a sequence, then we can measure their union.
For a finite sample space we can usually measure every subset, so the σ-algebra is the power set.
</document>

<document source="https://tamnd.github.io/math2/node/04WTBQ7APQ.md" tag="04WTBQ7APQ" kind="statement">
# Integral of cosine

## Statement

For all real numbers $a$ and $b$,
$$\int_a^b \cos x \, dx = \sin b - \sin a.$$

## Remarks

The [[derivative of sine|00R4MWW532]] is $\cos$, so $\sin$ is an antiderivative of $\cos$.
For example, $\int_0^{\pi/2} \cos x \, dx = 1$.
Mathlib states it as `integral_cos`.

## Explanations

### L1 human

The sine function is an antiderivative of the cosine function.
So the integral of cosine is a difference of two sine values.
</document>

<document source="https://tamnd.github.io/math2/node/04X3PR72VJ.md" tag="04X3PR72VJ" kind="concept">
# Determinant as signed volume

## Summary

Let $n \ge 1$ and let $v_1, \dots, v_n \in \R^n$ be the columns of a real $n \times n$ matrix $A$.
The parallelepiped spanned by them is $P = \{t_1 v_1 + \cdots + t_n v_n : 0 \le t_i \le 1\}$.
Its $n$-dimensional volume is $\abs{\det A}$, where $\det$ is the [[determinant|00KDZRW4C0]].
If $\det A \ne 0$, then its sign is positive when $(v_1, \dots, v_n)$ has the [[orientation|0QJJAQ6ZYZ]] of the standard basis, and negative when it has the opposite orientation.

## Remarks

In $\R^2$, $\abs{ad - bc}$ is the area of the parallelogram spanned by $(a, c)$ and $(b, d)$.
For each region $S$ of finite volume, the linear map $x \mapsto Ax$ multiplies the volume of $S$ by $\abs{\det A}$.
This fact is the origin of the Jacobian determinant in the change of variables formula.

## Explanations

### L1 human

The determinant measures how a matrix changes volume.
The unit cube goes to a box spanned by the columns, and $\abs{\det A}$ is the volume of this box.
A negative determinant means that the matrix also turns space inside out, like a mirror.
</document>

<document source="https://tamnd.github.io/math2/node/04XGKK8WQW.md" tag="04XGKK8WQW" kind="statement">
# Existence of primitive roots modulo a prime

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
Then there is a [[primitive root|0KM2CJ80G7]] modulo $p$.
That is, the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^\times$ is cyclic of order $p - 1$.

## Remarks

A proof: a polynomial of degree $d$ has at most $d$ roots in the field $\mathbb{Z}/p\mathbb{Z}$. So for each $d \mid p - 1$, at most $\varphi(d)$ elements have order $d$. The [[sum of the totient over the divisors|09JXE6XMCC]] then forces exactly $\varphi(d)$ elements of each order $d$.
In particular, there are exactly $\varphi(p - 1)$ primitive roots modulo $p$ in $\{1, \dots, p - 1\}$.
More generally, each finite subgroup of the multiplicative group of a field is cyclic.

## Explanations

### L1 human

For each prime $p$, some number $g$ has powers that run through all of $1, 2, \dots, p - 1$ modulo $p$.
For $p = 11$, the number $2$ works: its powers are $2, 4, 8, 5, 10, 9, 7, 3, 6, 1$.
</document>

<document source="https://tamnd.github.io/math2/node/04XPQT2XTC.md" tag="04XPQT2XTC" kind="statement">
# Every connected graph has a spanning tree

## Statement

Let $G$ be a [[connected|0FFAYNMAJZ]] [[graph|0JD5C82QMB]].
Then $G$ has a [[spanning tree|0FQ5B8S1ZC]].

## Remarks

For a finite graph, a proof deletes edges that lie on cycles, one at a time. Such an edge is not a [[bridge|0D81KWH9ZM]], so the graph stays connected, and the process ends with a tree.
For an infinite graph, the proof uses Zorn's lemma, or a well-ordering of the vertices.
The converse also holds: a graph with a spanning tree is connected.

## Explanations

### L1 human

If a connected graph has a loop, you can remove one edge of the loop and the graph stays in one piece.
Repeat until there are no loops. What is left is a spanning tree.
</document>

<document source="https://tamnd.github.io/math2/node/04XWH28452.md" tag="04XWH28452" kind="concept">
# Elementary row operations

## Summary

The elementary row operations are three kinds of operations on the rows of a matrix.
They swap two rows, multiply a row by a nonzero scalar, or add a multiple of one row to another row.
</document>

<document source="https://tamnd.github.io/math2/node/04YG690N5V.md" tag="04YG690N5V" kind="definition">
# Probability density function

## Definition

Let $X$ be a real random variable on a probability space $(\Omega, \mathcal{F}, P)$.
A probability density function of $X$ is a Borel measurable function $f \colon \mathbb{R} \to [0, \infty)$ such that $P(X \in B) = \int_B f(x) \, dx$ for each Borel set $B \subseteq \mathbb{R}$.
So $X$ has a probability density function if and only if $X$ is a [[random variable with a density|0C21RZG5V2]].

## Remarks

Each density satisfies $\int_{-\infty}^{\infty} f(x) \, dx = 1$. Two densities of $X$ are equal except on a set of Lebesgue measure zero.
$P(a \le X \le b) = \int_a^b f(x) \, dx$ for $a \le b$.
If the [[cumulative distribution function|0MXMS3AE13]] $F_X$ is differentiable at $x$ and $f$ is continuous at $x$, then $F_X'(x) = f(x)$.
A density can be larger than $1$. The uniform distribution on $[0, 1/2]$ has the density $2$ on this interval.

## Explanations

### L1 human

The density $f(x)$ is not a probability. It is a probability per unit of length near $x$.
For a small $h$, the probability that $X$ lies between $x$ and $x + h$ is about $f(x) \, h$.
</document>

<document source="https://tamnd.github.io/math2/node/04YKAYDQPW.md" tag="04YKAYDQPW" kind="statement">
# Monotone functions are Riemann integrable

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be monotone.
Then $f$ is Riemann integrable on $[a, b]$.

## Remarks

Assume $f$ increasing, and take the partition into $n$ equal pieces. Then $M_i = f(x_i)$ and $m_i = f(x_{i-1})$, so the sum telescopes:
$$U(f, P) - L(f, P) = \frac{b - a}{n} (f(b) - f(a)).$$
This tends to $0$, so [[Riemann's criterion|017DH8WXTE]] applies.
A monotone function can have infinitely many discontinuities, and it is still integrable.

## Explanations

### L1 human

For an increasing function, the gaps between upper and lower rectangles stack up to one column of total height $f(b) - f(a)$.
Its width shrinks to $0$ as the partition gets finer.
</document>

<document source="https://tamnd.github.io/math2/node/04Z5G0THN6.md" tag="04Z5G0THN6" kind="statement">
# Comparison test

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences with $0 \le a_n \le b_n$ for all $n$.
If $\sum b_n$ converges, then $\sum a_n$ converges, and $\sum a_n \le \sum b_n$.
If $\sum a_n$ diverges, then $\sum b_n$ diverges.

## Remarks

The partial sums of $\sum a_n$ are at most those of $\sum b_n$. The result follows from [[a series of nonnegative terms converges iff its partial sums are bounded|0MX1HZBBXJ]].
It is enough that $0 \le a_n \le b_n$ for all $n$ larger than some $N$.
For example, $0 \le 1/(n^2 + 1) \le 1/n^2$, so $\sum 1/(n^2 + 1)$ converges.
Mathlib states it for a summable upper bound, in `Summable.of_nonneg_of_le`.

## Explanations

### L1 human

If a series of nonnegative terms is smaller term by term than a series with a finite sum, it has a finite sum too.
If a smaller series already has an infinite sum, the larger one has an infinite sum as well.
</document>

<document source="https://tamnd.github.io/math2/node/05006AJ7DZ.md" tag="05006AJ7DZ" kind="concept">
# Tautology

## Summary

A tautology is a formula of [[propositional logic|09K9MWD2QQ]] that is true for every assignment of truth values to its variables.
</document>

<document source="https://tamnd.github.io/math2/node/05099E1NMJ.md" tag="05099E1NMJ" kind="definition">
# Cardinality of a finite set

## Definition

Let $A$ be a set and let $n$ be a natural number.
The set $A$ has $n$ elements if there is a bijection from $\{1, 2, \dots, n\}$ to $A$.
For $n = 0$, the set $\{1, \dots, n\}$ is the empty set, so only the empty set has $0$ elements.
A set is finite if it has $n$ elements for some natural number $n$.

For a finite set $A$, this number $n$ is unique.
It is the cardinality of $A$, with the notation $|A|$ or $\# A$.

## Remarks

Two finite sets have the same cardinality if and only if there is a bijection between them. The [[bijection principle|0GV2573FP4]] states this.
For example, $|\{a, b, c\}| = 3$ and $|\varnothing| = 0$.
The uniqueness of $n$ follows from the fact that there is no injection from $\{1, \dots, m\}$ to $\{1, \dots, n\}$ when $m > n$.

## Explanations

### L1 human

The cardinality of a finite set is the number of its elements.
To count a set, you label its elements $1, 2, 3, \dots$ without gaps and without repeats.
The last label is the cardinality.
</document>

<document source="https://tamnd.github.io/math2/node/050F2FMTKF.md" tag="050F2FMTKF" kind="concept">
# Harmonic conjugate

## Summary

A harmonic conjugate of a real harmonic function $u$ is a real function $v$ such that $u + iv$ is holomorphic, see the [[definition|0CC56MJ4FG]].
On a domain, it is unique up to an additive constant when it exists.
</document>

<document source="https://tamnd.github.io/math2/node/0515A6D2Y1.md" tag="0515A6D2Y1" kind="concept">
# Symmetric group

## Summary

The symmetric group on a set $X$ is the group of all bijections from $X$ to $X$, with composition as the operation.
For $X = \{1, \dots, n\}$ it is written $S_n$, and its elements are the permutations of $n$ symbols.
</document>

<document source="https://tamnd.github.io/math2/node/0518MM6BE1.md" tag="0518MM6BE1" kind="statement">
# The real numbers are uncountable

## Statement

The set $\mathbb{R}$ of real numbers is [[uncountable|0X55M9XET4]].

## Remarks

Cantor proved it in his article of 1874. That proof uses nested intervals.
A later proof uses [[Cantor's diagonal argument|0XCH8QVDCQ]] on decimal or binary expansions.
Since the algebraic numbers are countable, the result shows that transcendental numbers exist. Cantor gave this application in the same article.

## Explanations

### L1 human

Each list of real numbers misses some real number.
So there are more real numbers than natural numbers, although both sets are infinite.
</document>

<document source="https://tamnd.github.io/math2/node/051CYS71H5.md" tag="051CYS71H5" kind="concept">
# Classical definition of probability

## Summary

Suppose that all cases are equally likely.
Then the classical definition says that the probability of an event is the number of favourable cases divided by the number of all cases.
It goes back to Jacob Bernoulli and Pierre-Simon Laplace. The node [[classical definition of probability|0AKNPYQ04R]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/051VYKA1EX.md" tag="051VYKA1EX" kind="statement">
# Euclid's lemma

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]], and let $a$ and $b$ be integers.
If $p \mid a b$, then $p \mid a$ or $p \mid b$.

## Remarks

The hypothesis that $p$ is prime is necessary. For example, $6 \mid 4 \cdot 9$, but $6 \nmid 4$ and $6 \nmid 9$.
A short proof uses [[Bézout's identity|0DQVHMYAGA]]. If $p \nmid a$, then $\gcd(p, a) = 1$, so $p x + a y = 1$ for some integers $x$ and $y$. Multiply by $b$ to see that $p \mid b$.
The [[generalized Euclid's lemma|03BAREW1Z5]] replaces the prime $p$ by an integer that is coprime to $a$.
The lemma is the main step in the proof of the uniqueness part of the [[fundamental theorem of arithmetic|084HH1NDCB]].

## History

The lemma is Proposition 30 of Book VII of the Elements of Euclid.

## Explanations

### L1 human

A prime cannot be split into smaller factors, so it cannot share itself between $a$ and $b$.
If $7$ divides a product $a b$, then $7$ already divides $a$ or $b$.
For example, $7 \mid 14 \cdot 3$ and $7 \mid 14$.
</document>

<document source="https://tamnd.github.io/math2/node/0528HKWZ1Z.md" tag="0528HKWZ1Z" kind="statement">
# Product of a matrix and its adjugate

## Statement

Let $K$ be a commutative ring, let $n \ge 2$, and let $A$ be an $n \times n$ matrix over $K$.
Then
$$A \operatorname{adj}(A) = \operatorname{adj}(A) A = \det(A)\, I_n,$$
where $\operatorname{adj}(A)$ is the [[adjugate|07W9H8QF38]] of $A$ and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The diagonal entries of $A \operatorname{adj}(A)$ are the [[Laplace expansions|0C08BQ3M05]] of $\det A$ along the rows.
The entries off the diagonal are expansions of matrices with two equal rows, so they are zero.
The identity holds for all square matrices, also for singular ones.

## Explanations

### L1 human

Multiplying a matrix by its adjugate gives the determinant times the identity matrix.
So, if the determinant is invertible, then dividing the adjugate by it gives the inverse.
For a singular matrix, the product is the zero matrix.
</document>

<document source="https://tamnd.github.io/math2/node/052JJ9YVNM.md" tag="052JJ9YVNM" kind="concept">
# Graph homomorphism

## Summary

A graph homomorphism is a map between the vertex sets of two graphs that preserves adjacency. Homomorphisms unify colourings and other graph maps. The node [[graph homomorphism|06SCB3R6BN]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/052M4SX57M.md" tag="052M4SX57M" kind="definition">
# Annihilating polynomial

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an endomorphism.
A polynomial $p \in F[x]$ *annihilates* $T$ when $p(T) = 0$, where $p(T)$ is the [[value of $p$ at $T$|0E3MV6E6MG]].
Such a polynomial is an *annihilating polynomial* of $T$.

## Remarks

The annihilating polynomials of $T$ form an ideal of $F[x]$.
If $\dim V = n$ is finite, then a nonzero annihilating polynomial exists, because the $n^2 + 1$ maps $\id_V, T, \dots, T^{n^2}$ are linearly dependent.
The monic generator of this ideal is the [[minimal polynomial|03NDVCSHSB]] of $T$.
For a [[projection|033DA1WE8Y]] $P$, the polynomial $x^2 - x$ annihilates $P$.

## Explanations

### L1 human

An annihilating polynomial is a polynomial that becomes the zero map when we put $T$ into it.
For example, a reflection $R$ satisfies $R^2 = I$, so $x^2 - 1$ annihilates it.
Such relations tell us a lot about the possible eigenvalues.
</document>

<document source="https://tamnd.github.io/math2/node/052SDEPZH7.md" tag="052SDEPZH7" kind="definition">
# Irreducible element

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]].
An element $p \in R$ is irreducible if it has these three properties.

1. $p \neq 0$.
2. $p$ is not a [[unit|019NFGFPR5]].
3. If $p = a b$ with $a, b \in R$, then $a$ is a unit or $b$ is a unit.

## Remarks

The irreducible elements of $\mathbb{Z}$ are the numbers $p$ and $-p$ with $p$ prime.
In $\mathbb{Z}[i]$, the number $3$ is irreducible, but $5 = (2 + i)(2 - i)$ is not.
Every [[prime element|0MNHXBSX3M]] is irreducible. The converse [[can fail|0CHE3J7QMX]].
For polynomials over a field, the irreducible elements are the [[irreducible polynomials|00X2A9G9W3]].

## Explanations

### L1 human

An irreducible element cannot be split into smaller factors.
The only factorizations are trivial ones, with a unit as one factor, like $7 = (-1)(-7)$.
In the integers, the irreducible elements are the primes and their negatives.
</document>

<document source="https://tamnd.github.io/math2/node/053E86NWVK.md" tag="053E86NWVK" kind="definition">
# Associate elements

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $a, b \in R$.
The elements $a$ and $b$ are associates, written $a \sim b$, if there is a [[unit|019NFGFPR5]] $u \in R$ with $b = u a$.

## Remarks

Being associates is an equivalence relation.
In an [[integral domain|09S3PFBEPM]], $a \sim b$ if and only if $a \mid b$ and $b \mid a$, if and only if $(a) = (b)$.
In $\mathbb{Z}$, the associates of $n$ are $n$ and $-n$.
In $\mathbb{Z}[i]$, the associates of $1 + i$ are $1 + i$, $-1 - i$, $-1 + i$ and $1 - i$.

## Explanations

### L1 human

Associates are elements that differ only by a unit factor.
For divisibility they behave in the same way.
In the integers, $6$ and $-6$ are associates.
</document>

<document source="https://tamnd.github.io/math2/node/053QVHF6N7.md" tag="053QVHF6N7" kind="statement">
# Eigenvalues are the roots of the characteristic polynomial

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Let $p_T$ be the [[characteristic polynomial|0J5EMJRRFM]] of $T$, and let $\lambda \in F$.
Then $\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $T$ if and only if $p_T(\lambda) = 0$.

## Remarks

The proof: $\lambda$ is an eigenvalue if and only if $\lambda \id_V - T$ is not injective, if and only if $\det(\lambda \id_V - T) = 0$.
So $T$ has at most $\dim V$ eigenvalues.
For $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$, $p_A(t) = t^2 - 4t + 3 = (t - 1)(t - 3)$, so the eigenvalues are $1$ and $3$.

## Explanations

### L1 human

To find the eigenvalues of a matrix, compute its characteristic polynomial and find its roots.
Each root is an eigenvalue, and each eigenvalue is a root.
Then, for each eigenvalue $\lambda$, solve $(A - \lambda I)x = 0$ to find the eigenvectors.
</document>

<document source="https://tamnd.github.io/math2/node/054K3HNN5Z.md" tag="054K3HNN5Z" kind="concept">
# Converse relation

## Summary

The converse of a relation is the relation with all pairs reversed.
</document>

<document source="https://tamnd.github.io/math2/node/054Q5PW5GX.md" tag="054Q5PW5GX" kind="statement">
# Compact iff complete and totally bounded

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $K \subseteq X$.
Then $K$ is [[compact|00R0X1WGPX]] if and only if $K$, with the [[subspace metric|0T2WYDTRP2]], is [[complete|08EWT7J0SC]] and $K$ is [[totally bounded|0WE5BBHGM6]].

## Remarks

Proof idea of the converse: in a totally bounded set, a diagonal argument gives a Cauchy subsequence of each sequence. Completeness gives a limit. Thus $K$ is sequentially compact, and so [[compact|0K8SGQSV7G]].
In a complete space $X$, a subset is compact if and only if it is closed and totally bounded.
The [[Heine–Borel theorem|07PFYV65MB]] follows, because bounded subsets of $\mathbb{R}^n$ are totally bounded.

## Explanations

### L1 human

Compactness in a metric space splits into two parts: no holes, and finitely many small pieces for each size.
The interval $(0, 1)$ is totally bounded but has holes at its ends. The real line has no holes but is not totally bounded.
</document>

<document source="https://tamnd.github.io/math2/node/054SBCQ93T.md" tag="054SBCQ93T" kind="example">
# The dihedral group of order 8

## Example

The [[dihedral group|0JZ27DMJ5D]] $D_4$ is the group of the symmetries of a square, and it has order $8$.
Let $r$ be the rotation by $90$ degrees and let $s$ be a reflection in a line of symmetry.
Then $D_4 = \{e, r, r^2, r^3, s, rs, r^2s, r^3s\}$, with $r^4 = e$, $s^2 = e$ and $srs^{-1} = r^{-1}$.
The elements $r$ and $r^3$ have order $4$, and the other five elements different from $e$ have order $2$.
The [[centre|0DZ93493VN]] of $D_4$ is $\{e, r^2\}$, because the rotation by $180$ degrees commutes with each symmetry.
The rotation subgroup $\langle r \rangle$ is [[normal|0CB0DX5BVV]] of index $2$.
The subgroup $\{e, s\}$ is not normal, because $r s r^{-1} = r^2 s$.
Numbering the vertices $1, 2, 3, 4$ gives an injective homomorphism from $D_4$ into $S_4$, and its image is a Sylow $2$-subgroup of $S_4$.
</document>

<document source="https://tamnd.github.io/math2/node/054VXHMCA3.md" tag="054VXHMCA3" kind="definition">
# Complex exponential

## Definition

The complex exponential is the function $\exp : \mathbb{C} \to \mathbb{C}$ given by
$$\exp z = \sum_{n=0}^{\infty} \frac{z^n}{n!} = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots$$
for each [[complex number|09FTAET5Z4]] $z$. The series converges absolutely for each $z \in \mathbb{C}$.
Another notation for $\exp z$ is $e^z$.

## Remarks

For real $x$, $\exp x$ is the real exponential function, and $e = \exp 1$.
For $z = x + iy$ with $x, y$ real, $e^z = e^x (\cos y + i \sin y)$. The case $x = 0$ is [[Euler's formula|04R3CY41VJ]].
By the [[product of exponentials|0WM7DJCM6G]], $e^{z + w} = e^z e^w$. So $e^z \neq 0$ for each $z$, and $|e^z| = e^{\operatorname{Re} z}$.

## Explanations

### L1 human

The power series of $e^x$ makes sense for complex numbers too, so we use it as the definition of $e^z$.
The real part of $z$ controls the size of $e^z$, and the imaginary part of $z$ controls its direction.
For example, $e^{i\pi} = -1$: the size is $e^0 = 1$ and the direction is a half turn.
</document>

<document source="https://tamnd.github.io/math2/node/054YEQ0Y8M.md" tag="054YEQ0Y8M" kind="concept">
# Sesquilinear form

## Summary

A sesquilinear form on a complex vector space is a map of two vectors to the complex numbers.
It is linear in one argument and conjugate linear in the other.
</document>

<document source="https://tamnd.github.io/math2/node/05598J0Y8R.md" tag="05598J0Y8R" kind="statement">
# Fermat numbers are pairwise coprime

## Statement

Let $m$ and $n$ be distinct natural numbers.
Then the [[Fermat numbers|0R03B54V4Z]] $F_m$ and $F_n$ are [[coprime|0GFKP8F8AG]].

## Remarks

The proof uses $F_0 F_1 \cdots F_{n-1} = F_n - 2$. A common divisor of $F_m$ and $F_n$ with $m < n$ divides $2$. Fermat numbers are odd, so the divisor is $1$.
The result is often called Goldbach's theorem, after Christian Goldbach.
A consequence is a new proof that there are [[infinitely many primes|0292EPPPT0]]: each $F_n$ has a prime factor, and these primes are all distinct.

## Explanations

### L1 human

No two Fermat numbers share a prime factor.
For example, $F_2 = 17$ and $F_3 = 257$ are coprime, and $F_5 = 641 \cdot 6700417$ has no factor in common with $3$, $5$, $17$, $257$ or $65537$.
</document>

<document source="https://tamnd.github.io/math2/node/055HT85T3S.md" tag="055HT85T3S" kind="statement">
# PID implies UFD

## Statement

Let $R$ be a [[principal ideal domain|0RZYDVZ475]].
Then $R$ is a [[unique factorisation domain|06YMXJMXYS]].

## Remarks

Existence of factorisations: a principal ideal domain is Noetherian, so a process of splitting an element into factors must stop.
Uniqueness: in a principal ideal domain, an irreducible element generates a maximal ideal, so it is prime.
Consequences: $\mathbb{Z}$, $\mathbb{Z}[i]$ and $K[x]$ for a field $K$ have unique factorisation.
The converse is false: $\mathbb{Z}[x]$ is a unique factorisation domain but [[not a principal ideal domain|06E3AG9TM2]].

## Explanations

### L1 human

In a principal ideal domain, every irreducible element behaves like a prime number.
That is the key to unique factorisation.
So every Euclidean domain, such as the integers, has unique factorisation.
</document>

<document source="https://tamnd.github.io/math2/node/055VNYT0PB.md" tag="055VNYT0PB" kind="concept">
# Perpendicular bisector

## Summary

The perpendicular bisector of a segment passes through the midpoint of the segment at a right angle to it. In the plane it is a line, and in space it is a plane. Its points are exactly the points at the same distance from the two endpoints.
</document>

<document source="https://tamnd.github.io/math2/node/055VVSN00V.md" tag="055VVSN00V" kind="definition">
# Lipschitz condition

## Definition

Let $S$ be a subset of $\mathbb{R}^n$, let $f : S \to \mathbb{R}^m$ be a [[function|0EPAT51DXQ]], and let $K \ge 0$ be a real number.
The function $f$ satisfies a Lipschitz condition on $S$ with constant $K$, or is $K$-Lipschitz on $S$, if
$$\| f(x) - f(y) \| \le K\, \| x - y \| \quad \text{for all } x, y \in S.$$
Let $D \subseteq \mathbb{R} \times \mathbb{R}^n$ and $f : D \to \mathbb{R}^m$. The function $f$ is Lipschitz in $x$ on $D$ with constant $K$ if $\| f(t, x) - f(t, y) \| \le K \| x - y \|$ whenever $(t, x), (t, y) \in D$.

## Remarks

A $K$-Lipschitz function is uniformly continuous.
A differentiable function on a convex set with $\| f'(x) \| \le K$ at each point is $K$-Lipschitz, by the mean value inequality. For example, $\sin$ is $1$-Lipschitz on $\mathbb{R}$.
The function $x \mapsto \sqrt{|x|}$ is continuous but not Lipschitz on any interval that contains $0$. The [[non-uniqueness example|00K7ASZ1PR]] uses a function of this kind.
The [[Picard–Lindelöf theorem|0P6KRD5B7A]] needs a Lipschitz condition in $x$.

## Explanations

### L1 human


A Lipschitz function cannot change faster than a fixed rate.
The slope of each chord of its graph is at most $K$ in absolute value.
</document>

<document source="https://tamnd.github.io/math2/node/055Z15H69G.md" tag="055Z15H69G" kind="concept">
# Sequential continuity

## Summary

A function is sequentially continuous when it maps each convergent sequence in its domain to a sequence that converges to the value at the limit. For real functions, this is the same as continuity. The node [[sequential continuity|0MVDQEAB60]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0565DC8RMK.md" tag="0565DC8RMK" kind="definition">
# Jordan measure

## Definition

A box in $\mathbb{R}^n$ is a set $Q = [a_1, b_1) \times \dots \times [a_n, b_n)$ with $a_i \le b_i$, and its volume is $|Q| = (b_1 - a_1) \cdots (b_n - a_n)$.
A simple set is a finite union of boxes. Each simple set $S$ is a finite union of pairwise disjoint boxes $Q_1, \dots, Q_k$, and $m(S) = |Q_1| + \dots + |Q_k|$ does not depend on this choice.
For a bounded set $A \subseteq \mathbb{R}^n$, the inner and outer Jordan measures are
$$m_*(A) = \sup \{ m(S) : S \subseteq A \}, \qquad m^*(A) = \inf \{ m(S) : S \supseteq A \},$$
where $S$ runs over simple sets.
The set $A$ is Jordan measurable if $m_*(A) = m^*(A)$, and then this common value $m(A)$ is the Jordan measure, or Jordan content, of $A$.

## Remarks

Here $\sup$ is the [[supremum|0R3H5PANRQ]] and $\inf$ is the [[infimum|0QF1MZ9FRG]].
Closed boxes, balls and bounded regions between the graphs of two continuous functions are Jordan measurable. Finite unions, intersections and differences of Jordan measurable sets are Jordan measurable.
A bounded set $A$ is Jordan measurable if and only if its boundary has Jordan measure zero. Then $m(A) = \int_Q \mathbf{1}_A$ for each closed box $Q \supseteq A$, where $\mathbf{1}_A$ is the indicator function of $A$.
The set $\mathbb{Q} \cap [0, 1]$ is not Jordan measurable: its inner measure is $0$ and its outer measure is $1$. Lebesgue measure extends Jordan measure to more sets.

## Explanations

### L1 human

Jordan measure measures a set by fitting finite unions of boxes inside it and around it.
If the best inner and outer approximations have the same total volume, that volume is the measure of the set.
</document>

<document source="https://tamnd.github.io/math2/node/0565EGK6XP.md" tag="0565EGK6XP" kind="statement">
# Lebesgue's criterion for Riemann integrability

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded.
Then $f$ is [[Riemann integrable|0N1JQXZXHC]] if and only if the set of points where $f$ is not [[continuous|0EEE2CZB2A]] has Lebesgue measure zero.

## Remarks

A set $E \subseteq \mathbb{R}$ has measure zero if for each $\varepsilon > 0$ it is covered by countably many open intervals of total length less than $\varepsilon$.
Countable sets have measure zero. So [[Thomae's function|0HJ28H9Y95]] is integrable, and so is each monotone function.
The [[Dirichlet function|03Z4JC4YRJ]] is discontinuous everywhere, so it is not integrable.
Giuseppe Vitali and Henri Lebesgue proved the criterion independently, in 1907.

## Explanations

### L1 human

A bounded function is Riemann integrable exactly when it is continuous at almost all points.
The bad points must form a set of total length zero.
</document>

<document source="https://tamnd.github.io/math2/node/056FSKG32J.md" tag="056FSKG32J" kind="definition">
# Pointwise convergence

## Definition

Let $S$ be a set, and let $f_n \colon S \to \mathbb{R}$ for $n \ge 1$ and $f \colon S \to \mathbb{R}$ be functions.
The sequence $(f_n)$ converges pointwise to $f$ on $S$ if $\lim_{n \to \infty} f_n(x) = f(x)$ for every $x \in S$.
That is, for every $x \in S$ and every $\varepsilon > 0$ there is $N$ with $|f_n(x) - f(x)| < \varepsilon$ for all $n \ge N$.

## Remarks

Here $N$ may depend on both $\varepsilon$ and $x$.
The pointwise limit is unique, since [[limits|0XKWSSY8P2]] of real sequences are unique.
A pointwise limit of continuous functions need not be continuous: see [[the sequence $x^n$|0YSRR1JTKZ]]. Mathlib expresses pointwise convergence as convergence in the product topology, through `tendsto_pi_nhds`.

## Explanations

### L1 human

Pointwise convergence means that at each single point, the values $f_n(x)$ approach $f(x)$.
Different points may converge at very different speeds.
</document>

<document source="https://tamnd.github.io/math2/node/056SW1E0AC.md" tag="056SW1E0AC" kind="statement">
# A matrix is diagonalisable iff it has a basis of eigenvectors

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Then $A$ is [[diagonalisable|07SQCNV233]] if and only if $F^n$ has a [[basis|0KCQ9Y3QKA]] of [[eigenvectors|0N6960W0BV]] of $A$.
If $v_1, \dots, v_n$ is such a basis with $A v_i = \lambda_i v_i$, and $P$ has the columns $v_1, \dots, v_n$, then $P^{-1} A P$ is the diagonal matrix with entries $\lambda_1, \dots, \lambda_n$.

## Remarks

The equation $AP = PD$ says column by column that $A v_i = \lambda_i v_i$.
Equivalently, $F^n$ is the direct sum of the eigenspaces of $A$.
Equivalently, the sum of the [[geometric multiplicities|099TWVJ909]] of the eigenvalues is $n$.

## Explanations

### L1 human

To diagonalise a matrix, we need $n$ independent eigenvectors.
We put them as columns into a matrix $P$.
Then $P^{-1} A P$ is diagonal, with the eigenvalues on the diagonal.
</document>

<document source="https://tamnd.github.io/math2/node/056VS31A0J.md" tag="056VS31A0J" kind="statement">
# Isosceles triangle theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in a Euclidean space $\mathbb{R}^n$ with $d(A, B) = d(A, C)$, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]].
Then the angles at $B$ and $C$ are equal: $\angle ABC = \angle ACB$.

## Remarks

The converse holds too: if $\angle ABC = \angle ACB$ in a triangle $ABC$, then $d(A, B) = d(A, C)$.
Mathlib's form of the statement does not need that $A$, $B$ and $C$ are not collinear.
In an equilateral triangle, the theorem applies at each vertex. So all three angles are equal.

## History

The statement is Proposition 5 of Book I of Euclid's Elements. Its Latin name is pons asinorum, "the bridge of donkeys".

## Explanations

### L1 human

An isosceles triangle has two equal sides, like the outline of a simple roof. The theorem says that the two base angles are then equal. Fold the triangle along the line from the top vertex to the midpoint of the base: the two halves match.
</document>

<document source="https://tamnd.github.io/math2/node/05788J51WZ.md" tag="05788J51WZ" kind="definition">
# Parametrised surface

## Definition

Let $D \subseteq \mathbb{R}^2$ be a set that is the closure of an open set.
A parametrised surface is a map $r \colon D \to \mathbb{R}^3$, $r(u, v) = \big( x(u, v), y(u, v), z(u, v) \big)$, of [[class C^1|0TCBZMEYGN]] on an open set that contains $D$.
Its image $S = r(D)$ is the surface that $r$ parametrises.
The parametrisation is regular at $(u, v)$ if the [[partial derivatives|05AXG3QRKR]] $r_u = \partial r / \partial u$ and $r_v = \partial r / \partial v$ are linearly independent, that is, $r_u \times r_v \ne 0$.

## Remarks

The graph of a function $f \colon D \to \mathbb{R}$ is parametrised by $r(x, y) = (x, y, f(x, y))$. Here $r_x \times r_y = (-f_x, -f_y, 1) \ne 0$, so this parametrisation is regular.
The unit sphere is parametrised by $r(\varphi, \theta) = (\sin \varphi \cos \theta, \sin \varphi \sin \theta, \cos \varphi)$ on $[0, \pi] \times [0, 2\pi]$. It is not regular at the poles $\varphi = 0$ and $\varphi = \pi$.
The cylinder of radius $R > 0$ about the $z$-axis is parametrised by $r(\theta, z) = (R \cos \theta, R \sin \theta, z)$.
One surface has many parametrisations. Areas and surface integrals do not depend on the choice.

## Explanations

### L1 human

A parametrised surface is to a surface what a [[parametrised curve|0JF1XXGWB4]] is to a curve, with two parameters instead of one.
Think of $(u, v)$ as coordinates on a flat sheet, and of $r$ as a map that bends the sheet into space.
</document>

<document source="https://tamnd.github.io/math2/node/0578PA53VA.md" tag="0578PA53VA" kind="statement">
# Generating function for partitions

## Statement

Let $p(n)$ be the [[partition function|001CX45SWQ]].
Then, as formal power series in $\mathbb{Z}[\![x]\!]$,
$$\sum_{n \ge 0} p(n)\, x^n = \prod_{k \ge 1} \frac{1}{1 - x^k}.$$

## Remarks

The infinite product makes sense, because each coefficient of the product depends only on finitely many factors.
Proof idea: the factor $\frac{1}{1 - x^k} = 1 + x^k + x^{2k} + \dots$ chooses how many parts equal $k$.
Restricting the factors gives other results. For example, $\prod_{k \ge 1} (1 + x^k)$ counts partitions into distinct parts. This gives [[Euler's partition theorem|0XFZ1BMF5Q]].

## Explanations

### L1 human

Each factor of the product decides how many times a part $k$ appears in the partition.
Multiplying the factors combines these choices, and the coefficient of $x^n$ counts the choices with total $n$.
</document>

<document source="https://tamnd.github.io/math2/node/057BH3JC9S.md" tag="057BH3JC9S" kind="statement">
# The nth roots of unity

## Statement

Let $n$ be a positive integer.
A complex number $z$ is an $n$th [[root of unity|0S61N75H45]] if and only if
$$z = e^{2 \pi i k / n} = \cos \frac{2 \pi k}{n} + i \sin \frac{2 \pi k}{n}$$
for some integer $k$ with $0 \le k < n$.

## Remarks

Proof: by [[de Moivre's formula|03MTBVG3VE]], each such $z$ satisfies $z^n = e^{2 \pi i k} = 1$. Conversely, $z^n = 1$ forces $|z| = 1$ and $n \arg z \in 2 \pi \mathbb{Z}$.
For $n = 3$, the roots are $1$ and $-\frac{1}{2} \pm \frac{\sqrt{3}}{2} i$.

## Explanations

### L1 human

The $n$th roots of unity sit on the unit circle at equal angles of $2\pi/n$, starting at $1$.
For $n = 6$, they are the six corners of a regular hexagon.
</document>

<document source="https://tamnd.github.io/math2/node/057H72B2MB.md" tag="057H72B2MB" kind="statement">
# Number of compositions

## Statement

Let $n$ be a positive integer.
Then the number of [[compositions|0J8ANSYKAC]] of $n$ is $2^{n-1}$.

## Remarks

Proof: write $n$ as a row of $n$ ones. A composition is a choice of where to cut among the $n - 1$ gaps between them.
The number of compositions of $n$ into exactly $k$ parts is $\binom{n-1}{k-1}$.
For $n = 0$ there is one composition, the empty one.

## Explanations

### L1 human

Put $n$ dots in a row. Between two neighbours you either draw a bar or you do not.
There are $n - 1$ gaps with $2$ choices each, so there are $2^{n-1}$ compositions.
</document>

<document source="https://tamnd.github.io/math2/node/0580ZJSA3Q.md" tag="0580ZJSA3Q" kind="definition">
# Binary operation

## Definition

Let $S$ be a set.
A binary operation on $S$ is a map $\ast \colon S \times S \to S$.
For $a, b \in S$, the notation for the image of the pair $(a, b)$ is $a \ast b$.

## Remarks

Addition, subtraction and multiplication are binary operations on the set $\mathbb{Z}$ of integers.
Subtraction is not a binary operation on the set $\mathbb{N}$ of natural numbers, because $1 - 2$ is not in $\mathbb{N}$.
The condition that $a \ast b$ is in $S$ for all $a, b \in S$ is often called closure.

## Explanations

### L1 human

A binary operation is a rule that takes two elements of a set, in order, and gives one element of the same set.
For example, addition takes $2$ and $3$ and gives $5$.
The order can matter: $2 - 3$ and $3 - 2$ are different.
</document>

<document source="https://tamnd.github.io/math2/node/058YS05YP5.md" tag="058YS05YP5" kind="statement">
# Ptolemy's theorem

## Statement

Let $A$, $B$, $C$ and $D$ be [[concyclic points|049DCJZ448]] of the Euclidean plane $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Suppose that the segments $AC$ and $BD$ meet in a point that is inside both segments. Thus $ABCD$ is a convex [[cyclic quadrilateral|0RCP929Y1M]] with the diagonals $AC$ and $BD$.
Then
$$d(A, C)\, d(B, D) = d(A, B)\, d(C, D) + d(B, C)\, d(A, D).$$

## Remarks

In words: the product of the diagonals is the sum of the products of the opposite sides.
The converse is also true. If a quadrilateral has this property, then it is cyclic.
For each four points of the plane, $d(A, C)\, d(B, D) \leq d(A, B)\, d(C, D) + d(B, C)\, d(A, D)$. This is Ptolemy's inequality.
For a rectangle with the sides $a$ and $b$, the theorem gives $a^2 + b^2 = c^2$, with the diagonal $c$. This is the [[Pythagorean theorem|0VP3A1290K]].

## History

The theorem has the name of the astronomer Ptolemy. He used it to make his table of chords, a trigonometric table for astronomy.

## Explanations

### L1 human

Put four points in order on a circle and join them into a quadrilateral. Multiply the two diagonals. You get the same number as when you multiply the opposite sides in pairs and add the two products.
</document>

<document source="https://tamnd.github.io/math2/node/059Q2XWJDC.md" tag="059Q2XWJDC" kind="statement">
# Scalar triple product

## Statement

Let $a = (a_1, a_2, a_3)$, $b = (b_1, b_2, b_3)$ and $c = (c_1, c_2, c_3)$ be vectors in $\mathbb{R}^3$. Let $\times$ be the [[cross product|0N5N15HWZ7]] and let $\cdot$ be the dot product.
The scalar triple product of $a$, $b$ and $c$ is the real number $a \cdot (b \times c)$. It is equal to the determinant of the matrix whose rows are $a$, $b$ and $c$:
$$a \cdot (b \times c) = \det \begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{pmatrix}.$$

## Remarks

A cyclic shift of the three vectors does not change the value: $a \cdot (b \times c) = b \cdot (c \times a) = c \cdot (a \times b)$. A swap of two vectors changes its sign.
The absolute value $|a \cdot (b \times c)|$ is the volume of the [[parallelepiped|0SQV2VJTGY]] with the edges $a$, $b$ and $c$.
The scalar triple product is $0$ if and only if $a$, $b$ and $c$ are linearly dependent. This happens exactly when they lie in one plane through $0$.

## Explanations

### L1 human

The scalar triple product measures the volume of the slanted box that three arrows make. For the unit vectors $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$ the box is a unit cube, and the product is $1$. If the three arrows lie flat in one plane, the box has no volume, and the product is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/059THXBRBB.md" tag="059THXBRBB" kind="statement">
# Isomorphism with the coordinate space

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \dots, b_n)$ be a basis of $V$.
Then the map
$$V \to F^n, \qquad v \mapsto [v]_B,$$
that sends each vector to its [[coordinate vector|04CYQHN6MC]] in $B$, is a [[linear isomorphism|0B8T10CE7N]].
Its inverse sends $(a_1, \dots, a_n)$ to $a_1 b_1 + \cdots + a_n b_n$.

## Remarks

The isomorphism depends on the choice of the basis.
It sends $b_i$ to the standard basis vector $e_i$.
Through this map, each statement about $F^n$ gives a statement about $V$.

## Explanations

### L1 human

Choosing a basis turns every vector into a list of $n$ numbers.
This translation respects addition and scaling, and it loses no information.
So each $n$-dimensional space is a copy of $F^n$.
</document>

<document source="https://tamnd.github.io/math2/node/059TJN0NZ1.md" tag="059TJN0NZ1" kind="concept">
# Probability measure

## Summary

A probability measure is a countably additive function on a σ-algebra of events that gives the whole space the value $1$.
It assigns to each event its probability. The node [[probability measure|0VGMEBVW1M]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/05A6PKMAJ9.md" tag="05A6PKMAJ9" kind="concept">
# Partition of an interval

## Summary

A partition of a closed interval is a finite increasing list of points that starts at the left end and stops at the right end. It cuts the interval into subintervals. The node [[partition of an interval|069P20KYQT]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/05APWSX53S.md" tag="05APWSX53S" kind="statement">
# Sum of a finite geometric series

## Statement

Let $r$ be a real number with $r \ne 1$, and let $n$ be a natural number.
Then
$$\sum_{k=0}^{n-1} r^k = 1 + r + r^2 + \cdots + r^{n-1} = \frac{1 - r^n}{1 - r}.$$

## Remarks

The proof multiplies the sum $S$ by $1 - r$. The product $(1 - r) S$ telescopes to $1 - r^n$.
For $r = 1$ the sum is $n$.
The identity holds in each commutative ring in the form $(1 - r) \sum_{k=0}^{n-1} r^k = 1 - r^n$, with no division.
Mathlib also states the version for a sum over the indices $m \le k < n$, as `geom_sum_Ico`.
Letting $n \to \infty$ gives the [[geometric series|01X5CVV1Q5]].

## Explanations

### L1 human

Write $S = 1 + r + \cdots + r^{n-1}$ and $rS = r + r^2 + \cdots + r^n$.
Subtract, and almost all terms cancel: $S - rS = 1 - r^n$.
For example, $1 + 2 + 4 + 8 = (1 - 16)/(1 - 2) = 15$.
</document>

<document source="https://tamnd.github.io/math2/node/05AQE0QR0D.md" tag="05AQE0QR0D" kind="concept">
# Normal distribution

## Summary

The normal distribution with mean $\mu$ and variance $\sigma^2$ is the continuous distribution on $\mathbb{R}$ with the bell-shaped density $\frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x-\mu)^2/(2\sigma^2)}$.
By the central limit theorem, it is the approximate law of a sum of many small independent effects. The node [[normal distribution|07RXBVTVZB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/05AVARPXA2.md" tag="05AVARPXA2" kind="concept">
# Binary relation

## Summary

A binary relation between two sets says, for each pair of elements, whether the first element is related to the second.
Formally, it is a set of ordered pairs.
</document>

<document source="https://tamnd.github.io/math2/node/05AW1G0WE1.md" tag="05AW1G0WE1" kind="statement">
# Expectation of a product of independent random variables

## Statement

Let $X$ and $Y$ be [[independent|0E9502NT92]] integrable real random variables.
Then $XY$ is integrable and
$$E[XY] = E[X] \, E[Y].$$

## Remarks

The proof uses that the joint law of $(X, Y)$ is the [[product of the marginal laws|08QCA2F1NX]], and then Fubini's theorem.
For nonnegative independent $X$ and $Y$, the equality holds in $[0, \infty]$ without integrability.
The converse is false: $E[XY] = E[X] \, E[Y]$ means only that $X$ and $Y$ are uncorrelated.
More generally, $E[g(X) \, h(Y)] = E[g(X)] \, E[h(Y)]$ for measurable $g$ and $h$ with $g(X)$ and $h(Y)$ integrable.

## Explanations

### L1 human

For independent variables, the average of a product is the product of the averages.
For two fair dice, the expected product of the two results is $3.5 \cdot 3.5 = 12.25$.
</document>

<document source="https://tamnd.github.io/math2/node/05AXG3QRKR.md" tag="05AXG3QRKR" kind="definition">
# Partial derivative

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}$, let $a \in U$, and let $i \in \{1, \dots, n\}$.
Let $e_i$ be the $i$-th standard basis vector of [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$.
The partial derivative of $f$ with respect to $x_i$ at $a$ is the limit
$$\frac{\partial f}{\partial x_i}(a) = \lim_{h \to 0} \frac{f(a + h e_i) - f(a)}{h},$$
when this limit exists in $\mathbb{R}$.
Other notations for it are $\partial_i f(a)$, $D_i f(a)$ and $f_{x_i}(a)$.

## Remarks

The partial derivative is the ordinary derivative at $t = a_i$ of the function $t \mapsto f(a_1, \dots, a_{i-1}, t, a_{i+1}, \dots, a_n)$ of one variable.
So you compute it with the rules of one-variable calculus, and you treat the other variables as constants.
It is the [[directional derivative|06KWGHDN2J]] in the direction $e_i$.
The existence of all partial derivatives at $a$ does not imply that $f$ is continuous at $a$, see [[this counterexample|0TNZC55X3J]].

## Explanations

### L1 human

To find $\partial f / \partial x$, pretend that $y$ is a fixed number and differentiate in $x$ as usual.
For $f(x, y) = x^2 y + \sin y$, this gives $\partial f / \partial x = 2xy$, because $\sin y$ is a constant for this purpose.
Geometrically, it is the slope of the graph of $f$ when you walk in the $x$-direction.
</document>

<document source="https://tamnd.github.io/math2/node/05B0FKFXQA.md" tag="05B0FKFXQA" kind="definition">
# Jump discontinuity

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$ be a limit point of both $D \cap (-\infty, a)$ and $D \cap (a, \infty)$.
The point $a$ is a jump discontinuity of $f$ if both [[one-sided limits|0909PWZR62]] $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ exist and are different.

## Remarks

The difference of the two one-sided limits is the size of the jump.
For example, the sign function has a jump discontinuity at $0$, of size $2$.
Monotone functions have only jump discontinuities.

## Explanations

### L1 human

At a jump discontinuity, the graph approaches one height from the left and another height from the right.
No choice of the value at $a$ can repair this.
</document>

<document source="https://tamnd.github.io/math2/node/05BDYVQQ0T.md" tag="05BDYVQQ0T" kind="definition">
# Space of solutions of a linear recurrence

## Definition

Let $R$ be a commutative ring, and let $E$ be a linear recurrence with constant coefficients over $R$ of order $d$.
The space of solutions of $E$ is the set of all [[solutions|0ZV2YKKZ58]] of $E$.
It is a submodule of the $R$-module $R^{\mathbb{N}}$ of all sequences in $R$, with termwise addition and scalar multiplication.

## Remarks

The map $u \mapsto (u_0, \dots, u_{d-1})$ is an isomorphism of $R$-modules from the space of solutions to $R^d$. Over a field, the space of solutions has dimension $d$.
For the Fibonacci recurrence over $\mathbb{R}$, the sequences $\varphi^n$ and $\psi^n$ form a basis, with $\varphi$ and $\psi$ the roots of $X^2 - X - 1$.

## Explanations

### L1 human

If two sequences solve the same linear recurrence, then so does their sum, and so does any multiple of one of them.
So the solutions form a vector space, and the first $d$ terms are coordinates on it.
</document>

<document source="https://tamnd.github.io/math2/node/05BM304NRX.md" tag="05BM304NRX" kind="statement">
# Stabilisers of points in the same orbit are conjugate

## Statement

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, let $x \in X$, and let $g \in G$.
Then the [[stabiliser|0AYMP41THN]] of $g \cdot x$ is $G_{g \cdot x} = g G_x g^{-1}$.

## Remarks

An element $h$ fixes $g \cdot x$ if and only if $g^{-1}hg$ fixes $x$.
So the stabilisers of the points of one [[orbit|0YQPZB5Q1G]] are [[conjugate subgroups|0H11ZPX4ND]].
In particular, they all have the same order.

## Explanations

### L1 human

If $g$ moves $x$ to $y$, then the elements that fix $y$ are the elements that fix $x$, seen through $g$.
To fix $y$, you can move $y$ back to $x$, fix $x$, and move $x$ to $y$ again.
So the two stabilisers are conjugate.
</document>

<document source="https://tamnd.github.io/math2/node/05CG0S20S3.md" tag="05CG0S20S3" kind="concept">
# Tangent function

## Summary

The tangent function is the quotient of sine by cosine. It has period $\pi$, and it has vertical asymptotes where the cosine vanishes. The node [[tangent function|06T0HS14PP]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/05CJSSXPR3.md" tag="05CJSSXPR3" kind="concept">
# Determinant of a family of vectors in a basis

## Summary

The determinant of a family of $n$ vectors in a basis of an $n$-dimensional space is a scalar.
It is the determinant of the matrix of their coordinates in that basis.
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<document source="https://tamnd.github.io/math2/node/05D5EGMFYV.md" tag="05D5EGMFYV" kind="concept">
# Boundary

## Summary

The boundary of a subset of a topological space is the set of points in its closure that are not in its interior.
Each neighbourhood of a boundary point meets both the subset and its complement.
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<document source="https://tamnd.github.io/math2/node/05DEJR6TGF.md" tag="05DEJR6TGF" kind="statement">
# Zeckendorf's theorem

## Statement

Let $F_k$ be the [[Fibonacci numbers|0GA9K57A39]], and let $N$ be a positive integer.
Then there is a unique finite set $\{c_1, \dots, c_r\}$ of integers, with $r \ge 1$, such that
$$N = F_{c_1} + F_{c_2} + \dots + F_{c_r},$$
each $c_i \ge 2$, and $c_{i+1} \ge c_i + 2$ for $1 \le i < r$.

## Remarks

The condition says that the sum uses distinct Fibonacci numbers and no two consecutive ones. The bound $c_i \ge 2$ excludes $F_0 = 0$ and the repeated value $F_1 = F_2 = 1$.
Example: $100 = 89 + 8 + 3 = F_{11} + F_6 + F_4$.
The greedy method finds the representation: subtract the largest Fibonacci number that is at most $N$, and repeat.

## Explanations

### L1 human

Each positive integer is a sum of different Fibonacci numbers, no two of them neighbours in the sequence, in exactly one way.
For example, $20 = 13 + 5 + 2$.
</document>

<document source="https://tamnd.github.io/math2/node/05DME4TZ3K.md" tag="05DME4TZ3K" kind="object">
# Pascal's triangle

## Description

Pascal's triangle is the triangular array of the [[binomial coefficients|092GEF38YD]].
Its rows are numbered $n = 0, 1, 2, \dots$ from the top.
Row $n$ has $n + 1$ entries, $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$, from left to right.

The first five rows are these.
$$\begin{array}{ccccccccc} & & & & 1 & & & & \\ & & & 1 & & 1 & & & \\ & & 1 & & 2 & & 1 & & \\ & 1 & & 3 & & 3 & & 1 & \\ 1 & & 4 & & 6 & & 4 & & 1 \end{array}$$

Each row starts and ends with $1$.
Each other entry is the sum of the two entries next to it in the row above, by [[Pascal's rule|0YRG241HQQ]].

## Remarks

Row $n$ lists the coefficients of $(x + y)^n$, by the [[binomial theorem|0EHWPWGHB2]].
The sum of row $n$ is $2^n$. This is the [[sum of binomial coefficients|0KAMF19NV8]].
Each row is symmetric, by the [[symmetry of binomial coefficients|0R7JS3DP3W]].

## Explanations

### L1 human

Start with a $1$ at the top. Each new row starts and ends with $1$, and each inside number is the sum of the two numbers above it.
Row 4 is $1, 4, 6, 4, 1$, and these are the coefficients of $(x + y)^4$.
</document>

<document source="https://tamnd.github.io/math2/node/05DSV40EB9.md" tag="05DSV40EB9" kind="definition">
# Closed ball

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $x \in X$ and let $r \ge 0$ be a real number.
The closed ball with centre $x$ and radius $r$ is the set
$$\bar{B}(x, r) = \{ y \in X : d(x, y) \le r \}.$$

## Remarks

The [[open ball|0DC3MPR7C9]] $B(x, r)$ is a subset of $\bar{B}(x, r)$.
In $\mathbb{R}$ with the usual metric, $\bar{B}(x, r)$ is the closed interval $[x - r, x + r]$.
The closed ball is not always the closure of the open ball. In the [[discrete metric|0NA4FK7B8S]], $B(x, 1) = \{x\}$ is closed, but $\bar{B}(x, 1) = X$.
For $r = 0$, the closed ball is $\{x\}$.

## Explanations

### L1 human

The closed ball contains the points at distance at most $r$, so it also contains its edge.
In the plane, it is the disc of radius $r$ together with its circle.
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<document source="https://tamnd.github.io/math2/node/05EGFBZTGC.md" tag="05EGFBZTGC" kind="definition">
# Having a directional derivative

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, let $a \in U$, let $v \in \mathbb{R}^n$, and let $L \in \mathbb{R}^m$.
Let $g(t) = f(a + t v)$ for real $t$ near $0$. The function $f$ has the directional derivative $L$ at $a$ in the direction $v$ when $g$ is differentiable at $t = 0$ with $g'(0) = L$.
Equivalently, $\dfrac{f(a + t v) - f(a)}{t} \to L$ as $t \to 0$.

## Remarks

This definition states a property of $f$, $a$, $v$ and $L$. The [[directional derivative|0Y7PVW4Q44]] $D_v f(a)$ is then the vector $L$, which is unique.
Only the values of $f$ on the line through $a$ in the direction $v$ matter. So $f$ can have a directional derivative in each direction at $a$ and still fail to be continuous at $a$, see [[this counterexample|0TNZC55X3J]].

## Explanations

### L1 human

The function $g(t) = f(a + t v)$ is the function $f$ seen along the straight line through $a$ with direction $v$.
To have a directional derivative means that this function of one variable has a derivative at the start of the line.
</document>

<document source="https://tamnd.github.io/math2/node/05EGHEQZAW.md" tag="05EGHEQZAW" kind="definition">
# Vector space

## Definition

Let $F$ be a field.
A *vector space* over $F$ is a set $V$ with two maps: an addition $V \times V \to V$, $(u, v) \mapsto u + v$, and a scalar multiplication $F \times V \to V$, $(a, v) \mapsto av$.
These axioms must hold for all $u, v, w \in V$ and $a, b \in F$:

1. $u + (v + w) = (u + v) + w$.
2. $u + v = v + u$.
3. There is an element $0 \in V$ with $v + 0 = v$ for all $v \in V$.
4. For each $v \in V$ there is an element $-v \in V$ with $v + (-v) = 0$.
5. $a(bv) = (ab)v$.
6. $1v = v$, where $1$ is the unit of $F$.
7. $a(u + v) = au + av$.
8. $(a + b)v = av + bv$.

The elements of $V$ are *vectors*, and the elements of $F$ are *scalars*.

## Remarks

Axioms 1 to 4 say that $(V, +)$ is an abelian group.
The axioms give $0v = 0$ and $(-1)v = -v$ for all $v \in V$.
A vector space is a [[module|0NZ8JA25WK]] over a field; Mathlib uses `Module F V` for both.
Examples are the [[coordinate space|0XM8BZD76G]] $F^n$, [[spaces of matrices|0RE6X3BGPN]] and [[function spaces|0GZB2SN0PH]].

## Explanations

### L1 human

A vector space is a collection of objects that we can add together and stretch by numbers.
Arrows in the plane are the standard picture, but polynomials and functions also form vector spaces.
The eight axioms list the rules that make the usual algebra work.
</document>

<document source="https://tamnd.github.io/math2/node/05FD317V2W.md" tag="05FD317V2W" kind="definition">
# Left coset

## Definition

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \in G$.
The left coset of $H$ with representative $g$ is the set $gH = \{gh : h \in H\}$.

## Remarks

Two left cosets $aH$ and $bH$ are equal when $a^{-1}b \in H$, and they are disjoint otherwise.
Thus the left cosets of $H$ form a partition of $G$, and $H = eH$ is one of them.
In $(\mathbb{Z}, +)$, the left cosets of $3\mathbb{Z}$ are $3\mathbb{Z}$, $1 + 3\mathbb{Z}$ and $2 + 3\mathbb{Z}$.

## Explanations

### L1 human

A left coset is a copy of the subgroup that you move by multiplying with a fixed element on the left.
Different elements can give the same coset.
For the even integers inside the integers, there are two cosets: the even integers and the odd integers.
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<document source="https://tamnd.github.io/math2/node/05FKHRT0B4.md" tag="05FKHRT0B4" kind="statement">
# Polarisation identity

## Statement

Let $V$ be an inner product space with the [[induced norm|0MFJJV97RJ]] $\norm{\cdot}$, and let $x, y \in V$.
If $V$ is a real space, then
$$\langle x, y \rangle = \frac{1}{4} \left( \norm{x + y}^2 - \norm{x - y}^2 \right).$$
If $V$ is a complex space, with the inner product linear in the first argument, then
$$\langle x, y \rangle = \frac{1}{4} \sum_{k=0}^{3} i^k \norm{x + i^k y}^2.$$

## Remarks

So the norm determines the inner product.
In particular, a linear map that keeps norms also keeps inner products.
With the convention of linearity in the second argument, the factor $i^k$ moves to the other vector, and the complex formula changes accordingly.

## Explanations

### L1 human

If we know the lengths of all vectors, then we also know all inner products.
In the real case, we only need the lengths of the two diagonals $x + y$ and $x - y$.
So lengths determine angles.
</document>

<document source="https://tamnd.github.io/math2/node/05FQ3VGAR9.md" tag="05FQ3VGAR9" kind="definition">
# Bounded sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]].
The sequence is bounded above if there is a real number $M$ with $a_n \le M$ for all $n \in \mathbb{N}$.
The sequence is bounded below if there is a real number $m$ with $a_n \ge m$ for all $n \in \mathbb{N}$.
The sequence is bounded if it is bounded above and bounded below.
Equivalently, $(a_n)$ is bounded if there is a real number $K \ge 0$ with $|a_n| \le K$ for all $n \in \mathbb{N}$.

## Remarks

The sequence $a_n = (-1)^n$ is bounded, with $K = 1$.
The sequence $a_n = n$ is bounded below by $0$, but it is not bounded above.
A sequence is a function on $\mathbb{N}$, so a bounded sequence is a bounded function on $\mathbb{N}$.
Each [[convergent sequence|0AZ3R52593]] is bounded, as the node [[convergent sequences are bounded|0S4JF619V7]] states.

## Explanations

### L1 human

A sequence is bounded when all its terms stay inside one fixed interval $[-K, K]$.
The terms can move up and down, but they never escape from that interval.
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<document source="https://tamnd.github.io/math2/node/05FSTXM92W.md" tag="05FSTXM92W" kind="definition">
# Likelihood function

## Definition

Let $\Theta$ be a set of parameters. For each $\theta \in \Theta$, let $f(\cdot \mid \theta)$ be a [[probability mass function|0WF4EJP91W]] or a [[density|04YG690N5V]] on $\mathbb{R}^n$. It gives the law of the data $X = (X_1, \dots, X_n)$ when $\theta$ is the true parameter.
For observed data $x \in \mathbb{R}^n$, the likelihood function is the function $L \colon \Theta \to [0, \infty)$ with
$$L(\theta \mid x) = f(x \mid \theta).$$
The log-likelihood is $\ell(\theta \mid x) = \ln L(\theta \mid x)$.

## Remarks

For an [[i.i.d.|0AJN7WHBJZ]] sample with mass function or density $f_1(\cdot \mid \theta)$ for each observation, $L(\theta \mid x) = \prod_{i=1}^{n} f_1(x_i \mid \theta)$ and $\ell(\theta \mid x) = \sum_{i=1}^{n} \ln f_1(x_i \mid \theta)$.
The likelihood is not a probability distribution on $\Theta$. In general, its integral over $\theta$ is not $1$.
Example: for $k$ successes in $n$ Bernoulli($p$) trials, $L(p \mid x) = p^k (1 - p)^{n - k}$ for $p \in [0, 1]$.
Only ratios $L(\theta_1 \mid x) / L(\theta_2 \mid x)$ matter for inference. These ratios compare how well two parameter values explain the data.

## Explanations

### L1 human

The likelihood answers the question: "If the parameter were $\theta$, how probable would my data be?"
Parameter values with a high likelihood explain the observed data well.
</document>

<document source="https://tamnd.github.io/math2/node/05G2G0P224.md" tag="05G2G0P224" kind="definition">
# Dimension

## Definition

Let $F$ be a field and let $V$ be a [[finite-dimensional|0ZHWKQMVX5]] vector space over $F$.
The *dimension* of $V$ over $F$, written $\dim_F V$ or $\dim V$, is the number of vectors in a [[basis|0KCQ9Y3QKA]] of $V$.
For an arbitrary vector space, the dimension is the cardinality of a basis.

## Remarks

The number is well defined, by the [[dimension theorem for vector spaces|0WK0Y8RH1M]].
The zero space $\{0\}$ has the empty basis and dimension $0$.
The dimension depends on the field: $\dim_\C \C = 1$, but $\dim_\R \C = 2$.
Mathlib writes `Module.finrank F V` for the dimension as a natural number, with value $0$ for infinite-dimensional spaces.

## Explanations

### L1 human

The dimension counts the independent directions in a space.
A line has dimension $1$, a plane has dimension $2$, and ordinary space has dimension $3$.
Every basis of a space has this same number of vectors.
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<document source="https://tamnd.github.io/math2/node/05GK32H72H.md" tag="05GK32H72H" kind="definition">
# Union of a family of sets

## Definition

Let $(A_i)_{i \in I}$ be a [[family of sets|05Q6G4EH2G]].
The union of the family is the set
$$\bigcup_{i \in I} A_i = \{x \mid \exists i \in I, \, x \in A_i\}.$$
So $x$ is an element of the union when $x \in A_i$ for at least one $i \in I$.

## Remarks

For $I = \{1, 2\}$, this is the [[union|07KVB4JSC7]] $A_1 \cup A_2$.
For an empty index set, the union is the [[empty set|0ECVK59Y03]].
Example: $\bigcup_{n \ge 1} [0, n] = [0, \infty)$ in $\mathbb{R}$.

## Explanations

### L1 human

The union of a family collects every element that appears in some member of the family.
It generalizes $A \cup B$ from two sets to any number of sets.
</document>

<document source="https://tamnd.github.io/math2/node/05GQJXBP3F.md" tag="05GQJXBP3F" kind="definition">
# T1 space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is T1 when for all points $x \neq y$ of $X$ there is an [[open set|01ZQQDCERP]] $U$ with $x \in U$ and $y \notin U$.

## Remarks

A space is T1 if and only if each set $\{x\}$ with $x \in X$ is [[closed|0G5AV9Q4RR]].
Each [[Hausdorff space|0MXB4CKHFW]] is T1, and each T1 space is [[T0|0R3B7DX55J]].
An infinite set with the [[cofinite topology|0B906ZBTMZ]] is T1 but not Hausdorff. In fact, the cofinite topology is the coarsest T1 topology on a set.

## Explanations

### L1 human

In a T1 space, for two distinct points, each point has an open set around it that does not contain the other point.
This is the same as saying that single points are closed.
</document>

<document source="https://tamnd.github.io/math2/node/05GZ11PWRY.md" tag="05GZ11PWRY" kind="definition">
# Homeomorphism

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]].
A function $f \colon X \to Y$ is a homeomorphism when $f$ is [[bijective|0SNQEF7Z53]], $f$ is [[continuous|0SY9ECQ19Q]], and the [[inverse function|0W766JAPSN]] $f^{-1} \colon Y \to X$ is continuous.
The spaces $X$ and $Y$ are homeomorphic when there is a homeomorphism $X \to Y$.

## Remarks

Equivalently, $f$ is a bijection and, for each $U \subseteq X$, the set $U$ is open if and only if $f(U)$ is open.
The map $x \mapsto x / (1 - |x|)$ is a homeomorphism from $(-1, 1)$ to $\mathbb{R}$.
A continuous bijection need not be a homeomorphism: $t \mapsto (\cos t, \sin t)$ from $[0, 2\pi)$ to the circle is one example. For a compact domain and a Hausdorff target, [[it is always a homeomorphism|0430ETCMKD]].
Homeomorphisms keep each [[topological property|02SMVGC690]].

## Explanations

### L1 human

A homeomorphism is a way to deform one space into another by stretching and bending, without tearing or gluing.
A coffee cup and a doughnut are homeomorphic, but a sphere and a doughnut are not.
</document>

<document source="https://tamnd.github.io/math2/node/05H4DDDYBZ.md" tag="05H4DDDYBZ" kind="statement">
# Root test

## Statement

Let $(a_n)$ be a real sequence and let $C = \limsup_{n \to \infty} |a_n|^{1/n}$, the [[limit superior|0P8BA7T7S5]], with the value $+\infty$ allowed.
If $C < 1$, then the series $\sum a_n$ converges [[absolutely|0W3K3QPEWT]].
If $C > 1$, then the series $\sum a_n$ diverges.

## Remarks

If $C < 1$, choose $q$ with $C < q < 1$. Then $|a_n| \le q^n$ for large $n$, and the [[comparison test|04Z5G0THN6]] with a [[geometric series|01X5CVV1Q5]] applies.
If $C > 1$, then $|a_n| > 1$ for infinitely many $n$, so the terms do not tend to $0$.
For $C = 1$ the test gives no answer.
The root test is stronger than the [[ratio test|0QNY6ZBXPN]]: when the ratio limit exists, the root limit exists and has the same value.

## Explanations

### L1 human

The $n$-th root of $|a_n|$ measures how fast the terms shrink, compared with a geometric sequence $q^n$.
If this rate is below $1$, the terms shrink like a convergent geometric series.
</document>

<document source="https://tamnd.github.io/math2/node/05HN08ZSRJ.md" tag="05HN08ZSRJ" kind="concept">
# Dihedral angle

## Summary

A dihedral angle is the angle between two planes that meet in a line, or between two half-planes with a common edge. It is measured in a third plane that is perpendicular to the common line.
</document>

<document source="https://tamnd.github.io/math2/node/05HQYM0R4G.md" tag="05HQYM0R4G" kind="definition">
# Dot product

## Definition

Let $n \ge 1$ and let $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$ be vectors of $\R^n$.
The *dot product* of $x$ and $y$ is the real number
$$x \cdot y = \sum_{i=1}^{n} x_i y_i = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n.$$

## Remarks

In matrix notation, $x \cdot y = x^T y$ for column vectors $x$ and $y$.
It is symmetric, bilinear and positive definite, so it is an [[inner product|0NB0AWGJBW]] on $\R^n$.
Geometrically, $x \cdot y = \norm{x} \norm{y} \cos\theta$, where $\theta$ is the angle between $x$ and $y$.
The same formula defines a symmetric bilinear form on $F^n$ for every field $F$.

## Explanations

### L1 human

The dot product multiplies two vectors coordinate by coordinate and adds the results.
For example, $(1, 2, 3) \cdot (4, 5, 6) = 4 + 10 + 18 = 32$.
It is zero exactly when the two vectors are perpendicular, or when one of them is zero.
</document>

<document source="https://tamnd.github.io/math2/node/05HYGWFG63.md" tag="05HYGWFG63" kind="statement">
# Inner automorphism group is isomorphic to G modulo its centre

## Statement

Let $G$ be a [[group|G5S1M9]] with [[centre|0DZ93493VN]] $Z(G)$.
Then the map $gZ(G) \mapsto c_g$, where $c_g(x) = gxg^{-1}$, is an [[isomorphism|08MG7H8JW5]] from $G/Z(G)$ to $\operatorname{Inn}(G)$.

## Remarks

The map $g \mapsto c_g$ is a surjective homomorphism from $G$ to $\operatorname{Inn}(G)$. Its kernel is $Z(G)$, because $c_g$ is the identity exactly when $g$ commutes with each element.
The [[first isomorphism theorem|0CXXJNG5B6]] then gives the result.
For example, the symmetric group $S_3$ has trivial centre, so $\operatorname{Inn}(S_3) \cong S_3$.

## Explanations

### L1 human

Two elements give the same conjugation exactly when they differ by an element of the centre.
So the inner automorphisms are counted by the cosets of the centre.
A group with a big centre has few inner automorphisms.
</document>

<document source="https://tamnd.github.io/math2/node/05J9NWX07X.md" tag="05J9NWX07X" kind="concept">
# Transitive closure

## Summary

The transitive closure of a relation is the smallest transitive relation that contains it.
</document>

<document source="https://tamnd.github.io/math2/node/05K3J8J8T2.md" tag="05K3J8J8T2" kind="example">
# The rationals as the field of fractions of the integers

## Example

The [[field of fractions|0VF11ZTTMD]] of the [[ring of integers|0JHM6JK7FD]] $\mathbb{Z}$ is the field $\mathbb{Q}$ of rational numbers.
Each element of $\operatorname{Frac}(\mathbb{Z})$ is a class $a/b$ with $a, b \in \mathbb{Z}$ and $b \neq 0$.
Two pairs give the same class, $a/b = c/d$, if and only if $a d = b c$.
The map $\mathbb{Z} \to \mathbb{Q}$, $n \mapsto n/1$, is the usual inclusion of the integers in the rationals.

## Remarks

Every rational number has a unique form $a/b$ with $b > 0$ and $\gcd(a, b) = 1$.
Every subfield of $\mathbb{C}$ contains $\mathbb{Z}$, so it contains $\mathbb{Q}$, by the universal property of the field of fractions.

## Explanations

### L1 human

The rational numbers are exactly the fractions of integers.
The field of fractions makes this precise for any integral domain.
The fractions $1/2$, $2/4$ and $-3/(-6)$ all name the same rational number.
</document>

<document source="https://tamnd.github.io/math2/node/05KQ3F99BQ.md" tag="05KQ3F99BQ" kind="concept">
# Improper integral

## Summary

An improper integral is a limit of definite integrals. It extends integration to unbounded intervals and to functions that are unbounded near a point. The node [[improper integral|0VCTBJMEJF]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/05M5SWSZZD.md" tag="05M5SWSZZD" kind="concept">
# Limit of a sequence

## Summary

A sequence has a limit $L$ when its terms come and stay arbitrarily close to $L$ for large indices. Limits of sequences are the base of the notions of convergence, continuity and series in analysis. The node [[limit of a sequence|0XKWSSY8P2]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/05MC3HND79.md" tag="05MC3HND79" kind="statement">
# Poincaré lemma for one-forms on convex sets

## Statement

Let $U \subseteq \mathbb{R}^n$ be open and convex, and let $\omega = \sum_i f_i \, dx_i$ be a [[closed|0F1AVGX26N]] [[one-form|0MTENFBW3Q]] on $U$ with $C^1$ coefficients.
Then $\omega$ is exact. Explicitly, for a fixed $p \in U$, the function
$$g(x) = \int_0^1 \sum_{i=1}^n f_i\big( p + t (x - p) \big) \, (x_i - p_i) \, dt$$
satisfies $dg = \omega$ on $U$.

## Remarks

Proof: [[differentiate under the integral sign|0AR06W1E1B]], and use $\partial_j f_i = \partial_i f_j$. The integrand of $\partial_j g$ becomes $\frac{d}{dt} \big( t \, f_j(p + t (x - p)) \big)$, which integrates to $f_j(x)$.
In $\mathbb{R}^3$, this is the theorem that a [[curl-free field on a convex set is conservative|0B22P010D6]].
The lemma also holds for closed $k$-forms with $k \ge 1$ on star-shaped open sets. For one-forms, it holds on [[simply connected|075C629XTH]] open sets.
Some condition on $U$ is needed: the angle form on $\mathbb{R}^2 \setminus \{0\}$ is closed but not exact. See also the [[vortex field|0GRM3BS0DD]].

## Explanations

### L1 human

On a convex set, a one-form that passes the local test for being a derivative really is a derivative.
The proof builds the function by integrating the form along straight segments from one base point.
</document>

<document source="https://tamnd.github.io/math2/node/05MDKQBEAF.md" tag="05MDKQBEAF" kind="statement">
# Angle sum of a triangle

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in a Euclidean space $\mathbb{R}^n$, and let $\angle$ denote the [[angle at a point|0GJAJC6TNZ]]. Then
$$\angle BAC + \angle CBA + \angle ACB = \pi.$$

## Remarks

Mathlib proves the formula under the weaker hypothesis that $B \neq A$ and $C \neq A$, with its convention for the angle at a repeated point.
An equilateral triangle has three equal angles, so each one is $\pi / 3$.
In the axioms of plane geometry, the statement is equivalent to the parallel postulate. It is false in spherical geometry, where the sum is more than $\pi$, and in hyperbolic geometry, where it is less.

## History

Euclid's Elements proves the statement as part of Proposition 32 of Book I.

## Explanations

### L1 human

Cut a paper triangle and tear off its three corners. Put the corners side by side with their tips at one point. They always fill a straight line, which is half a turn, or $\pi$ radians.
</document>

<document source="https://tamnd.github.io/math2/node/05MM4EK4SA.md" tag="05MM4EK4SA" kind="statement">
# Groups of prime order are cyclic

## Statement

Let $p$ be a prime number and let $G$ be a [[group|G5S1M9]] of order $p$.
Then $G$ is [[cyclic|0KVGH0J5B3]], and each element of $G$ other than the identity generates $G$.

## Remarks

If $g \neq e$, then the order of $\langle g \rangle$ is greater than $1$ and divides $p$ by [[Lagrange's theorem|3C53TD]], so it is $p$.
Thus each group of order $p$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$.
A group of prime order has no subgroups except $\{e\}$ and itself.

## Explanations

### L1 human

When the size of a group is a prime, there is no room for proper subgroups other than the trivial one.
Any element other than the identity generates a subgroup larger than $1$, so it generates everything.
Thus, for example, each group with $7$ elements is a clock with $7$ hours.

### L2 agent

A prime $p$ has only the divisors $1$ and $p$. With Lagrange's theorem, this leaves a group of order $p$ no room for subgroups other than $\{e\}$ and the whole group.

The proof is short:

1. Let $g \in G$ with $g \ne e$.
2. The order of $g$ divides $p$, so it is $1$ or $p$.
3. Only $e$ has order $1$. Thus $\operatorname{ord}(g) = p$.
4. Thus $\langle g \rangle$ has $p$ elements, and it is all of $G$.

A worked example with $p = 5$ and $G = \mathbb{Z}/5\mathbb{Z}$:

1. Take $g = 2$. Its multiples are $2, 4, 1, 3, 0$.
2. All five elements occur, so $2$ generates $G$.
3. The same is true for $1$, $3$ and $4$, as the result states.

A consequence is that, up to isomorphism, there is exactly one group of each prime order $p$, namely $\mathbb{Z}/p\mathbb{Z}$. Thus the groups of order $2$, $3$, $5$ and $7$ are all cyclic and abelian.

The hypothesis that $p$ is prime is necessary. The Klein four-group has order $4$, and it is not cyclic. The group $S_3$ has order $6$, and it is not even abelian.

A common error is to think that the result also gives a generator for groups of prime power order. A group of order $p^2$ is abelian, but it need not be cyclic, as $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$ shows.
</document>

<document source="https://tamnd.github.io/math2/node/05MW7MPRBS.md" tag="05MW7MPRBS" kind="definition">
# Cross-ratio

## Definition

Let $z_1, z_2, z_3, z_4$ be four distinct complex numbers.
Their cross-ratio is
$$(z_1, z_2; z_3, z_4) = \frac{(z_3 - z_1)(z_4 - z_2)}{(z_3 - z_2)(z_4 - z_1)}.$$
Suppose that one of the four points is the point $\infty$ of the [[Riemann sphere|0CDMXGPSES]] and the other three are distinct complex numbers. Then the cross-ratio is the same formula without the two factors that contain $\infty$.

## Remarks

Other books use other orders of the four points in the formula, so one must check the convention.
For each [[Möbius transformation|0B346DDAM5]] $f$, the cross-ratio $(f(z_1), f(z_2); f(z_3), f(z_4))$ is equal to $(z_1, z_2; z_3, z_4)$.
With this convention, the map $z \mapsto (z, z_2; z_3, z_4)$ is the Möbius transformation that sends $z_2$, $z_3$, $z_4$ to $1$, $0$, $\infty$.
The cross-ratio is a real number exactly when the four points are on one circle or on one line.

## Explanations

### L1 human

The cross-ratio is one number made from four points.
Möbius transformations change distances and angles of figures in complicated ways, but they do not change this number.
So the cross-ratio helps to find a Möbius map that sends given points to other given points.
</document>

<document source="https://tamnd.github.io/math2/node/05N4TQKPH0.md" tag="05N4TQKPH0" kind="statement">
# Subgroup criterion

## Statement

Let $G$ be a [[group|G5S1M9]] and let $H$ be a subset of $G$.
Then $H$ is a [[subgroup|0X1JXT47GA]] of $G$ if and only if $H$ is not empty and $ab^{-1} \in H$ for all $a, b \in H$.

## Remarks

If $a \in H$, then $e = aa^{-1} \in H$, then $b^{-1} = eb^{-1} \in H$ for $b \in H$, and then $ab = a(b^{-1})^{-1} \in H$.
In additive notation, the condition is $a - b \in H$ for all $a, b \in H$.
For a finite nonempty subset $H$, it is sufficient that $ab \in H$ for all $a, b \in H$.

## Explanations

### L1 human

This test replaces three checks by one.
You only check that the subset is not empty and that $ab^{-1}$ stays in the subset.
For example, the multiples of $5$ form a subgroup of $(\mathbb{Z}, +)$, because the difference of two multiples of $5$ is a multiple of $5$.
</document>

<document source="https://tamnd.github.io/math2/node/05NAS7AGMZ.md" tag="05NAS7AGMZ" kind="concept">
# Field generated by adjoining elements

## Summary

For a field extension $L/K$ and a subset $S$ of $L$, the field $K(S)$ is the smallest subfield of $L$ that contains $K$ and $S$.
The [[definition of $K(S)$|0MA5B744K3]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/05NK26G0DR.md" tag="05NK26G0DR" kind="statement">
# Composition of continuous maps is continuous

## Statement

Let $X$, $Y$ and $Z$ be [[topological spaces|0CNQXVBHCX]].
Let $f \colon X \to Y$ and $g \colon Y \to Z$ be [[continuous maps|0SY9ECQ19Q]].
Then the [[composition|07HF9YCWPP]] $g \circ f \colon X \to Z$ is continuous.

## Remarks

Proof: for each open set $W$ of $Z$, $(g \circ f)^{-1}(W) = f^{-1}(g^{-1}(W))$. The set $g^{-1}(W)$ is open in $Y$, so its preimage under $f$ is open in $X$.
For example, $x \mapsto \sin(x^2)$ is continuous on $\mathbb{R}$, because $x \mapsto x^2$ and $\sin$ are continuous.

## Explanations

### L1 human

If you do one continuous operation and then another one, the result is continuous.
Small changes in the input give small changes after the first step, and then small changes after the second step.
</document>

<document source="https://tamnd.github.io/math2/node/05NWGBHKSZ.md" tag="05NWGBHKSZ" kind="concept">
# Eulerian trail

## Summary

An Eulerian trail is a trail that uses every edge of a graph exactly once. The question of which graphs have one is the origin of graph theory. The node [[Eulerian trail|0NNYDD1AVR]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/05NZ1HHQMV.md" tag="05NZ1HHQMV" kind="concept">
# Mean squared error

## Summary

The mean squared error of an estimator is the expected squared distance between the estimator and the true value. It combines the variance and the bias of the estimator.
The node [[mean squared error|0V4VSCBXP3]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/05P4B4Q22V.md" tag="05P4B4Q22V" kind="statement">
# Derivative of arctangent

## Statement

The arctangent is differentiable on $\mathbb{R}$, and
$$\frac{d}{dx} \arctan x = \frac{1}{1 + x^2} \quad \text{for every real } x.$$

## Remarks

Let $y = \arctan x$. By the [[inverse function rule|0M0RZ3HSB0]] and [[$\tan' = 1 + \tan^2$|0SAMR34WXC]], the derivative is $\frac{1}{1 + \tan^2 y} = \frac{1}{1 + x^2}$.
So $\int_0^1 \frac{dx}{1 + x^2} = \arctan 1 = \frac{\pi}{4}$.
Mathlib states it as `Real.hasDerivAt_arctan`.

## Explanations

### L1 human

The derivative of the arctangent is the simple rational function $1/(1 + x^2)$.
This connects the number $\pi$ with integrals of rational functions.
</document>

<document source="https://tamnd.github.io/math2/node/05PWWHYBX5.md" tag="05PWWHYBX5" kind="statement">
# Markov's inequality

## Statement

Let $X$ be a random variable with values in $[0, \infty)$ and let $a > 0$. Then
$$P(X \ge a) \le \frac{E[X]}{a}.$$

## Remarks

Proof: $a \, 1_{\{X \ge a\}} \le X$ pointwise. Take expectations: $a \, P(X \ge a) \le E[X]$.
More generally, for each nondecreasing $g \colon [0, \infty) \to [0, \infty)$ with $g(a) > 0$, $P(X \ge a) \le E[g(X)]/g(a)$.
Applied to $(X - E[X])^2$, the inequality gives [[Chebyshev's inequality|02HZ1BK3MW]]. Applied to $e^{tX}$, it gives the [[Chernoff bound|014PEBGDYF]].

## History

The inequality is named after Andrey Markov. It appears earlier in the work of Pafnuty Chebyshev, the teacher of Markov.

## Explanations

### L1 human

A nonnegative quantity with a small average cannot often be large.
If the mean income is $30000$, then at most one person in five can have an income of $150000$ or more.
</document>

<document source="https://tamnd.github.io/math2/node/05Q1B7MH0M.md" tag="05Q1B7MH0M" kind="statement">
# Matrix of a composition is the product of matrices

## Statement

Let $F$ be a field, and let $U$, $V$, $W$ be finite-dimensional vector spaces over $F$ with bases $A$, $B$, $C$.
Let $S \colon U \to V$ and $T \colon V \to W$ be linear maps.
Then
$$[T \circ S]_{C,A} = [T]_{C,B}\, [S]_{B,A},$$
where $[\,\cdot\,]$ denotes the [[matrix of a linear map|0YQSGNYE3K]] and the right side is a [[matrix product|032Q9FB8MT]].

## Remarks

This identity is the reason for the definition of matrix multiplication.
It implies that matrix multiplication is associative, because composition of maps is associative.
For $S$ invertible, it gives $[S^{-1}]_{A,B} = [S]_{B,A}^{-1}$.

## Explanations

### L1 human

Doing one linear map after another corresponds to multiplying their matrices.
The order is the same as in $T \circ S$: the matrix of the map that acts first is on the right.
This is why matrix multiplication has its unusual row-times-column rule.
</document>

<document source="https://tamnd.github.io/math2/node/05Q4JD6XEH.md" tag="05Q4JD6XEH" kind="definition">
# Orthogonal family for a bilinear form

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B$ be a [[bilinear form|0SQZMXB6GP]] on $V$.
A family $(v_i)_{i \in I}$ of vectors of $V$ is *orthogonal* for $B$ when
$$B(v_i, v_j) = 0 \qquad \text{for all } i, j \in I \text{ with } i \ne j.$$

## Remarks

For an inner product, this is the usual notion of an orthogonal family.
For a general $B$, the relation $B(u, v) = 0$ need not be symmetric in $u$ and $v$; it is symmetric for symmetric and alternating forms.
In a basis that is orthogonal for $B$, the matrix of $B$ is diagonal.
Unlike in an inner product space, a nonzero vector can be orthogonal to itself: for $B(x, y) = x_1 y_1 - x_2 y_2$, take $v = (1, 1)$.

## Explanations

### L1 human

Orthogonality makes sense for every bilinear form, not only for inner products.
Two vectors are orthogonal when the form gives zero.
A basis of pairwise orthogonal vectors makes the matrix of the form diagonal.
</document>

<document source="https://tamnd.github.io/math2/node/05Q6G4EH2G.md" tag="05Q6G4EH2G" kind="definition">
# Family of sets

## Definition

Let $I$ be a set.
A family of sets indexed by $I$ assigns to each $i \in I$ a [[set|0SJ3RX9TXA]] $A_i$. It is written $(A_i)_{i \in I}$.
The set $I$ is the index set.

## Remarks

Formally, a family indexed by $I$ is a [[function|0XHDTE5YMZ]] with [[domain|093CM69AGS]] $I$.
The same set can appear for two different indices: $A_i = A_j$ with $i \neq j$ is allowed.
Example: $A_n = \{0, 1, \dots, n\}$ for $n \in \mathbb{N}$ is a family indexed by $\mathbb{N}$.
Some authors use "family of sets" for any set of sets, without an index.

## Explanations

### L1 human

A family is a labelled list of sets, where the labels can be any set.
With labels $1, 2, 3$, you get three sets $A_1, A_2, A_3$.
With labels in $\mathbb{N}$, you get an infinite sequence of sets.
</document>

<document source="https://tamnd.github.io/math2/node/05QY1VEBYD.md" tag="05QY1VEBYD" kind="concept">
# Finitely generated module

## Summary

A finitely generated module is a module in which every element is a linear combination of a fixed finite set of elements.
</document>

<document source="https://tamnd.github.io/math2/node/05R9NF2J49.md" tag="05R9NF2J49" kind="definition">
# Conservative vector field

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $F \colon U \to \mathbb{R}^n$ be a continuous [[vector field|0N67NBKQ5T]].
The field $F$ is conservative on $U$ if there is a function $\phi \colon U \to \mathbb{R}$ of [[class C^1|0TCBZMEYGN]] with $F = \nabla \phi$ on $U$, that is, $F_i = \partial \phi / \partial x_i$ for $i = 1, \dots, n$.

## Remarks

A function $\phi$ with $\nabla \phi = F$ is a [[scalar potential|0FP4BVDAAQ]] of $F$. Physics texts often use $-\phi$ instead.
The field $F(x, y) = (2 x y, x^2)$ is conservative on $\mathbb{R}^2$, with $\phi(x, y) = x^2 y$.
The field $F(x, y) = (-y, x)$ is not conservative on $\mathbb{R}^2$. Its line integral around the unit circle is $2\pi \ne 0$, which contradicts [[path independence|0BFN21TF5F]].
If $F$ is conservative and of class $C^1$, then its [[curl is zero|0M2JRW1J0N]]. The converse holds on convex sets but [[fails in general|0GRM3BS0DD]].

## Explanations

### L1 human

A conservative field is the slope field of some landscape $\phi$.
Going around a loop in a landscape brings you back to the same height. So the total work of the field around the loop is zero.
</document>

<document source="https://tamnd.github.io/math2/node/05RCDKHZM6.md" tag="05RCDKHZM6" kind="statement">
# Additivity of the integral over intervals

## Statement

Let $a < c < b$ and let $f \colon [a, b] \to \mathbb{R}$.
Then $f$ is Riemann integrable on $[a, b]$ if and only if it is Riemann integrable on $[a, c]$ and on $[c, b]$.
In this case,
$$\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.$$

## Remarks

With the sign convention of the [[definite integral|021WEM680W]], the formula holds for any order of $a$, $b$, $c$.
This needs $f$ to be integrable on the largest of the intervals.
Mathlib states it as `intervalIntegral.integral_add_adjacent_intervals`.

## Explanations

### L1 human

The area over $[a, b]$ is the area over $[a, c]$ plus the area over $[c, b]$.
We can cut the interval at any point and add the two pieces.
</document>

<document source="https://tamnd.github.io/math2/node/05RCT1522R.md" tag="05RCT1522R" kind="statement">
# Laplacian as a sum of second partial derivatives

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}$ be twice differentiable on $U$.
Then
$$\Delta f = \frac{\partial^2 f}{\partial x_1^2} + \frac{\partial^2 f}{\partial x_2^2} + \dots + \frac{\partial^2 f}{\partial x_n^2}.$$

## Remarks

Proof: $\nabla f = (\partial f / \partial x_1, \dots, \partial f / \partial x_n)$, and the [[divergence|02NSMQVQWK]] adds the derivative of the $i$-th component with respect to $x_i$.
So $\Delta f$ is the trace of the [[Hessian matrix|0SCSQWXRA4]] of $f$. The trace does not change under a rotation of coordinates, so neither does $\Delta f$.
Examples: $\Delta (x^2 + y^2) = 4$, $\Delta (x^2 - y^2) = 0$ and $\Delta \|x\|^2 = 2n$ on $\mathbb{R}^n$.
In polar coordinates in the plane, $\Delta f = f_{rr} + \tfrac{1}{r} f_r + \tfrac{1}{r^2} f_{\theta\theta}$ for $r > 0$.

## Explanations

### L1 human

In Cartesian coordinates, the Laplacian adds up the second derivatives of $f$ in the $n$ coordinate directions.
Only the unmixed second partial derivatives appear, not mixed ones like $f_{xy}$.
</document>

<document source="https://tamnd.github.io/math2/node/05RN90E14Y.md" tag="05RN90E14Y" kind="statement">
# Number of edges of a complete graph

## Statement

Let $n$ be a natural number.
Then the [[complete graph|03WKB2H474]] $K_n$ has exactly
$$\binom{n}{2} = \frac{n (n - 1)}{2}$$
edges.

## Remarks

The edges of $K_n$ are exactly the $2$-element subsets of the vertex set. The [[number of k-element subsets|0BFMR2FQ8B]] gives the count.
The [[handshaking lemma|0XAP8WHNES]] gives the same count: $n (n - 1) = 2 |E|$.
Each simple graph on $n$ vertices has at most $\binom{n}{2}$ edges.

## Explanations

### L1 human

If $n$ people all shake hands with each other, there are $\frac{n(n-1)}{2}$ handshakes.
Each person shakes $n - 1$ hands, and this counts each handshake twice.
</document>

<document source="https://tamnd.github.io/math2/node/05RSK4Q657.md" tag="05RSK4Q657" kind="concept">
# Truth table

## Summary

A truth table is a table that gives the truth value of a compound [[proposition|0ZGC7ZESYN]] for each assignment of truth values to its variables.
Each row of the table is one assignment, and the last column gives the value of the compound proposition.
A proposition with $n$ variables has a truth table with $2^n$ rows.

## Remarks

The truth tables of [[negation|0XR7A7RSXB]], [[conjunction|0M28ASWG8T]], [[disjunction|0H5GKDTB01]], [[implication|0ZNX6DQ3EJ]] and the [[biconditional|0V6GAWTBYW]] define these connectives.
A proposition is a [[tautology|0D1AHYPM8N]] when its column has only the value true.
Two propositions are [[logically equivalent|06GYGS41JK]] when their columns are the same.

## Explanations

### L1 human

A truth table lists every possible case.
For $p \land q$ there are four cases, because $p$ and $q$ each have two values.
You compute the value of the whole formula in each case, one row at a time.
</document>

<document source="https://tamnd.github.io/math2/node/05RWFKVTJ7.md" tag="05RWFKVTJ7" kind="concept">
# Permutation

## Summary

A permutation of a set is a rearrangement of its elements, that is, a bijection from the set to itself. Permutations are the basic objects of enumerative combinatorics and of the theory of symmetric groups. The node [[permutation|0BQMWCNS8S]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/05RZ4YZG4E.md" tag="05RZ4YZG4E" kind="definition">
# Standard score

## Definition

Let $X$ be a real random variable with mean $\mu = E[X]$ and standard deviation $\sigma > 0$.
The standard score, or $z$-score, of a real number $x$ is
$$z = \frac{x - \mu}{\sigma}.$$
The standardized random variable is $Z = (X - \mu)/\sigma$.

## Remarks

The standardized variable $Z$ has mean $0$ and variance $1$.
If $X \sim \mathcal{N}(\mu, \sigma^2)$, then $Z \sim \mathcal{N}(0, 1)$. So $P(X \le x) = \Phi(z)$, where $\Phi$ is the standard normal cumulative distribution function.
For a data set, one uses the sample mean and the sample standard deviation in place of $\mu$ and $\sigma$.

## Explanations

### L1 human

A $z$-score says how unusual a value is, in units of the standard deviation.
Suppose that test scores have mean $70$ and standard deviation $10$.
Then a score of $85$ has $z = 1.5$. It is one and a half standard deviations above the mean.
</document>

<document source="https://tamnd.github.io/math2/node/05S2GSPMGA.md" tag="05S2GSPMGA" kind="statement">
# Harmonic numbers grow like log n

## Statement

Let $H_n = \sum_{k=1}^{n} \frac{1}{k}$ be the [[harmonic numbers|0D2Y6G4C4S]].
Then the limit $\lim_{n \to \infty} (H_n - \ln n)$ exists, and it lies in $[0, 1]$.
In particular, $H_n \sim \ln n$ as $n \to \infty$.

## Remarks

Let $a_n = H_n - \ln n$. Since $\ln(n + 1) - \ln n = \int_n^{n+1} \frac{dt}{t}$ lies between $\frac{1}{n+1}$ and $\frac{1}{n}$, the sequence $a_n$ decreases. Also $a_n \ge H_n - \ln(n + 1) \ge 0$.
The [[monotone convergence theorem|0FM6N4R0VB]] gives the limit. It is the [[Euler–Mascheroni constant|00C5P9JZBK]] $\gamma$.
The symbol $\sim$ is [[asymptotic equivalence|00SB5EA3VW]]. Mathlib states the limit as `Real.tendsto_harmonic_sub_log`.

## Explanations

### L1 human

The sum $1 + 1/2 + \cdots + 1/n$ is close to the area under $1/x$ from $1$ to $n$, which is $\ln n$.
The error tends to a constant.
</document>

<document source="https://tamnd.github.io/math2/node/05SG20AMXH.md" tag="05SG20AMXH" kind="concept">
# Left inverse

## Summary

A left inverse of a function undoes the function: applied after the function, it gives back the input.
</document>

<document source="https://tamnd.github.io/math2/node/05SZQKH75K.md" tag="05SZQKH75K" kind="statement">
# Negation of an existential statement

## Statement

Let $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.
The negation of $\exists x \in X, \, P(x)$ is logically equivalent to $\forall x \in X, \, \neg P(x)$.
$$\neg \bigl( \exists x \in X, \, P(x) \bigr) \equiv \forall x \in X, \, \neg P(x)$$

## Remarks

This equivalence is valid in intuitionistic logic as well as in classical logic.
Together with the [[negation of a universal statement|0P3A06QNDP]], it gives a rule. To negate a quantified statement, swap each $\forall$ and $\exists$ and negate the predicate.

## Explanations

### L1 human

"No student failed" means "every student did not fail".
To deny that some element has a property, you must show that each element lacks it.
</document>

<document source="https://tamnd.github.io/math2/node/05TG0SE4HT.md" tag="05TG0SE4HT" kind="definition">
# Poisson point process

## Definition

Let $\lambda > 0$.
A homogeneous Poisson process with rate $\lambda$ is a [[stochastic process|0QH5ENKZM1]] $(N(t))_{t \ge 0}$ with values in $\{0, 1, 2, \dots\}$ such that:
1. $N(0) = 0$.
2. For all $0 \le t_0 < t_1 < \dots < t_n$, the increments $N(t_1) - N(t_0), \dots, N(t_n) - N(t_{n-1})$ are [[independent|0XCE070V8B]].
3. For all $0 \le s < t$, $N(t) - N(s)$ has the [[Poisson distribution|0VEFSSKX18]] with parameter $\lambda (t - s)$.

## Remarks

$N(t)$ counts the points, or events, in the interval $(0, t]$. One usually also asks that the paths $t \mapsto N(t)$ are right-continuous.
The times between consecutive events are i.i.d. with the [[exponential distribution|028Z8WC5Y9]] $\operatorname{Exp}(\lambda)$. Conversely, i.i.d. $\operatorname{Exp}(\lambda)$ gaps define a Poisson process with rate $\lambda$.
Given $N(t) = n$, the $n$ event times in $(0, t]$ have the law of $n$ independent uniform points in $(0, t]$, put in increasing order.
On a general space $E$ with a measure $\mu$, a Poisson point process has independent counts in disjoint sets.
The count in a set $B$ has the law $\operatorname{Poisson}(\mu(B))$ when $\mu(B) < \infty$.

## Explanations

### L1 human

The Poisson process models events that happen "completely at random" in time, such as clicks of a Geiger counter or arrivals of customers.
On average, $\lambda$ events happen per unit of time, and the waiting times between events are exponential.
</document>

<document source="https://tamnd.github.io/math2/node/05THZKR9AN.md" tag="05THZKR9AN" kind="definition">
# Group of units of a ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
The group of units of $R$ is the set $R^\times$ of the [[units|019NFGFPR5]] of $R$, with the multiplication of $R$.
It is a [[group|G5S1M9]] with identity element $1$.

## Remarks

The set $R^\times$ is closed under multiplication: if $u$ and $v$ are units, then $(u v)^{-1} = v^{-1} u^{-1}$.
Other notations for $R^\times$ are $R^*$ and $U(R)$.
The group $\mathbb{Z}^\times = \{1, -1\}$ has order $2$.
For a field $K$, $K^\times = K \setminus \{0\}$.

## Explanations

### L1 human

The units of a ring are the elements with an inverse.
If you multiply two units, you get a unit again.
So the units form a group under multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/05V7J4WD1C.md" tag="05V7J4WD1C" kind="statement">
# Equality of complex numbers

## Statement

Let $z$ and $w$ be complex numbers.
Then $z = w$ if and only if $\operatorname{Re} z = \operatorname{Re} w$ and $\operatorname{Im} z = \operatorname{Im} w$.

## Remarks

Here $\operatorname{Re}$ and $\operatorname{Im}$ are the [[real and imaginary parts|02ZE6JHJ7G]].
Thus one equation between complex numbers gives two equations between real numbers.
For example, $x + y i = 3 - 2 i$ with $x, y$ real means $x = 3$ and $y = -2$.

## Explanations

### L1 human

Two complex numbers are equal when their real parts agree and their imaginary parts agree.
In the plane, this says that two points are equal when both of their coordinates agree.
</document>

<document source="https://tamnd.github.io/math2/node/05VDGH129F.md" tag="05VDGH129F" kind="concept">
# Phase plane

## Summary

The phase plane of a planar [[autonomous system|07BNRFQ0PJ]] $x' = f(x, y)$, $y' = g(x, y)$ is the plane with coordinates $(x, y)$, the two state variables.
Each solution traces a curve in this plane, its [[trajectory|0BVT6VSB3S]]. The vector $(f(x, y), g(x, y))$ is tangent to the trajectory through $(x, y)$.
It is the two-dimensional case of the phase space.

## Remarks

A second-order equation $x'' = h(x, x')$ gives a planar system with $y = x'$. Its phase plane has the position and the velocity as coordinates.
For a linear system $(x, y)' = A (x, y)$ with a $2 \times 2$ real matrix $A$, the eigenvalues of $A$ determine the shape of the trajectories. See the [[classification of planar linear systems|0N7ATVF2X8]].

## Explanations

### L1 human

The phase plane shows the two state variables at once, without the time axis.
Each point is a state, and the system moves from state to state along a curve.
</document>

<document source="https://tamnd.github.io/math2/node/05WAF5T0GZ.md" tag="05WAF5T0GZ" kind="statement">
# Maximal ideals are prime

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $M$ be a [[maximal ideal|09J6KGT65G]] of $R$.
Then $M$ is a [[prime ideal|06HC6HK0Z8]].

## Remarks

The proof: [[$R/M$ is a field|0WD3SE7S1X]], so it is an integral domain, so [[$M$ is prime|0AVXBEC05W]].
The converse is false: the zero ideal of $\mathbb{Z}$ is prime but not maximal.
In $\mathbb{Z}$, every nonzero prime ideal is maximal.

## Explanations

### L1 human

Every field is an integral domain.
So when $R/M$ is a field, $R/M$ is an integral domain, and $M$ is prime.
The zero ideal of the integers shows that a prime ideal need not be maximal.
</document>

<document source="https://tamnd.github.io/math2/node/05WDSN597C.md" tag="05WDSN597C" kind="concept">
# Submodule

## Summary

A submodule of a module is a subset that is a subgroup under addition and that is closed under multiplication by the ring elements.
</document>

<document source="https://tamnd.github.io/math2/node/05WEYX99X4.md" tag="05WEYX99X4" kind="concept">
# Inverse trigonometric functions

## Summary

The inverse trigonometric functions recover an angle from its sine, cosine or tangent. They are the inverses of the trigonometric functions restricted to standard intervals. The node [[inverse trigonometric functions|02HTDK1FBD]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/05WKRC7HWT.md" tag="05WKRC7HWT" kind="statement">
# Jacobian of spherical coordinates

## Statement

Let $S(\rho, \varphi, \theta) = (\rho \sin \varphi \cos \theta, \ \rho \sin \varphi \sin \theta, \ \rho \cos \varphi)$ be the [[spherical coordinate map|0RW8KPMXF6]].
Its [[Jacobian determinant|03W6Q1MFR5]] is
$$\det DS(\rho, \varphi, \theta) = \rho^2 \sin \varphi.$$
Hence, for a continuous function $f$ on a region $E \subseteq \mathbb{R}^3$ described by a region $E'$ of $(\rho, \varphi, \theta)$ with $\rho \ge 0$, $0 \le \varphi \le \pi$, $0 \le \theta \le 2\pi$,
$$\iiint_E f \, dV = \iiint_{E'} f(S(\rho, \varphi, \theta)) \, \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta.$$

## Remarks

The Jacobian matrix has rows $(\sin \varphi \cos \theta, \ \rho \cos \varphi \cos \theta, \ -\rho \sin \varphi \sin \theta)$, $(\sin \varphi \sin \theta, \ \rho \cos \varphi \sin \theta, \ \rho \sin \varphi \cos \theta)$ and $(\cos \varphi, \ -\rho \sin \varphi, \ 0)$. Expansion along the last row gives $\rho^2 \sin \varphi$.
The integral formula follows from the [[change of variables theorem|00N5FHP1CV]]. The set where $S$ is not injective, or where the determinant vanishes, has volume zero.
Since $0 \le \varphi \le \pi$, the factor $\rho^2 \sin \varphi$ is non-negative, so no absolute value is needed.
Example: the volume of the ball of radius $a$ is $\int_0^{2\pi} \int_0^\pi \int_0^a \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta = \tfrac{4}{3} \pi a^3$.

## Explanations

### L1 human

A small spherical box has sides $d\rho$, $\rho \, d\varphi$ and $\rho \sin \varphi \, d\theta$.
Its volume is about $\rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta$, and this factor appears in integrals in spherical coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/05WR29H2KH.md" tag="05WR29H2KH" kind="concept">
# Number of derangements

## Summary

The derangement numbers $D_n$ count the permutations of $n$ elements with no fixed point. They grow like $n!/e$. The node [[number of derangements|03WY4HHMT1]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/05WTQ6FFPS.md" tag="05WTQ6FFPS" kind="concept">
# Set difference

## Summary

The set difference of two sets is the set of the elements of the first set that are not elements of the second set.
</document>

<document source="https://tamnd.github.io/math2/node/05XT5AVJBK.md" tag="05XT5AVJBK" kind="definition">
# Laplace transform

## Definition

Let $f : [0, \infty) \to \mathbb{C}$ be a [[function|0EPAT51DXQ]] that is integrable on each interval $[0, T]$ with $T > 0$.
The Laplace transform of $f$ is the function $F = \mathcal{L}\{f\}$ given by
$$F(s) = \int_0^\infty f(t)\, e^{-s t}\, dt = \lim_{T \to \infty} \int_0^T f(t)\, e^{-s t}\, dt,$$
for each complex number $s$ for which the limit exists.

## Remarks

Let $f$ be of exponential order: there are constants $M, a$ with $|f(t)| \le M e^{a t}$ for all $t \ge 0$. Then the integral converges absolutely for $\operatorname{Re} s > a$.
Some transforms: $\mathcal{L}\{1\}(s) = 1/s$ and $\mathcal{L}\{e^{a t}\}(s) = 1/(s - a)$ for $\operatorname{Re} s > \operatorname{Re} a$, and $\mathcal{L}\{\sin \omega t\}(s) = \omega / (s^2 + \omega^2)$ for $\operatorname{Re} s > 0$ and real $\omega$.
The transform is linear: $\mathcal{L}\{c f + g\} = c \mathcal{L}\{f\} + \mathcal{L}\{g\}$ where both sides converge.
It turns derivatives into multiplication by $s$. See the [[Laplace transform of a derivative|0K090GVHVN]].

## Explanations

### L1 human

The Laplace transform turns a function of time into a function of a complex variable $s$.
It weights the function by $e^{-st}$, which decays fast, and adds up the result over all positive times.
Its main use is to turn linear differential equations into algebraic equations.
</document>

<document source="https://tamnd.github.io/math2/node/05Y0F9M0TS.md" tag="05Y0F9M0TS" kind="statement">
# Subgroups of solvable groups are solvable

## Statement

Let $G$ be a [[solvable group|0FXJ7YFMMX]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $H$ is solvable.

## Remarks

A proof uses the [[derived series|0Q5G5DJ4P2]]: $H^{(i)} \subseteq G^{(i)}$ for each $i$, so $G^{(n)} = \{e\}$ implies $H^{(n)} = \{e\}$.
More generally, if there is an injective homomorphism from $H$ into a solvable group, then $H$ is solvable.
For example, each subgroup of $S_4$ is solvable.

## Explanations

### L1 human

Commutators of elements of $H$ are also commutators in $G$.
So the derived series of $H$ stays inside the derived series of $G$.
If the larger series reaches the identity, the smaller one does too.
</document>

<document source="https://tamnd.github.io/math2/node/05Y1JAM3Y5.md" tag="05Y1JAM3Y5" kind="definition">
# Asymptotic stability

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz. Let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$, and let $x^* \in U$ be an [[equilibrium point|0G4X36AE0X]].
The point $x^*$ is asymptotically stable if it is [[Lyapunov stable|092S6VBECQ]] and there is an $\eta > 0$ with this property. For each $x_0 \in U$ with $\|x_0 - x^*\| < \eta$, $\varphi_t(x_0) \to x^*$ as $t \to \infty$.
If $U = \mathbb{R}^n$ and $\varphi_t(x_0) \to x^*$ for each $x_0 \in \mathbb{R}^n$, and $x^*$ is Lyapunov stable, then $x^*$ is globally asymptotically stable.

## Remarks

For $x' = -x$ on $\mathbb{R}$, the equilibrium $0$ is globally asymptotically stable, since $\varphi_t(x_0) = x_0 e^{-t}$.
The point $x^*$ is exponentially stable if there are $\eta, C, \alpha > 0$ with $\|\varphi_t(x_0) - x^*\| \le C e^{-\alpha t} \|x_0 - x^*\|$ for $\|x_0 - x^*\| < \eta$ and $t \ge 0$.
The attraction condition alone does not imply Lyapunov stability. There are planar examples where all solutions tend to $x^*$, but some first make a large excursion.

## Explanations

### L1 human

An equilibrium is asymptotically stable if nearby solutions stay near it and also approach it in the long run.
A ball at the bottom of a bowl with friction is a typical example.
</document>

<document source="https://tamnd.github.io/math2/node/05YBKMRVAR.md" tag="05YBKMRVAR" kind="definition">
# Composition of linear maps

## Definition

Let $F$ be a field, let $U$, $V$, $W$ be vector spaces over $F$, and let $S \colon U \to V$ and $T \colon V \to W$ be [[linear maps|0NRYHJPQG2]].
The *composition* of $T$ and $S$ is the map
$$T \circ S \colon U \to W, \qquad (T \circ S)(u) = T(S(u)).$$

## Remarks

The composition of linear maps is linear.
Composition is associative, and the identity maps are its identity elements.
The [[matrix of a composition is the product of the matrices|05Q1B7MH0M]], in the same order.
Composition is not commutative: in general $T \circ S \ne S \circ T$, even when both are defined.

## Explanations

### L1 human

To compose two linear maps, we apply one and then the other.
We write $T \circ S$ for "first $S$, then $T$".
Matrix multiplication is exactly this operation, written in coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/05Z3FC0154.md" tag="05Z3FC0154" kind="concept">
# Isolated singularity

## Summary

An isolated singularity of a function is a point $c$ such that the function is holomorphic on a punctured disc around $c$, see the [[definition|0TYB3HM9KE]].
It is removable, a pole, or essential.
</document>

<document source="https://tamnd.github.io/math2/node/05ZPH8KF9K.md" tag="05ZPH8KF9K" kind="definition">
# Perfect matching

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A perfect matching of $G$ is a [[matching|0Z87BHAFTB]] $M$ such that every vertex of $G$ is an end of an edge of $M$.

## Remarks

In a perfect matching, each vertex is an end of exactly one edge of $M$. Thus a finite graph with a perfect matching has an even number of vertices.
[[Hall's theorem for bipartite graphs|0X2GJPNPFN]] characterizes the bipartite graphs with a perfect matching. [[Tutte's theorem|08J7R35B6Q]] does this for all finite graphs.
The [[complete graph|03WKB2H474]] $K_4$ has $3$ perfect matchings.

## Explanations

### L1 human

A perfect matching pairs up all vertices of a graph along edges, with nobody left alone.
For this, the number of vertices must be even.
</document>

<document source="https://tamnd.github.io/math2/node/060W5CBZP6.md" tag="060W5CBZP6" kind="concept">
# Circle

## Summary

A circle is the set of all points of a plane at a given distance from a given point. The point is the centre and the distance is the radius. The region inside a circle is a disc.
</document>

<document source="https://tamnd.github.io/math2/node/060Z37VVB6.md" tag="060Z37VVB6" kind="definition">
# Right coset

## Definition

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \in G$.
The right coset of $H$ with representative $g$ is the set $Hg = \{hg : h \in H\}$.

## Remarks

Two right cosets $Ha$ and $Hb$ are equal when $ab^{-1} \in H$, and they are disjoint otherwise.
In an [[abelian group|082GHNPJ53]], each right coset $Hg$ equals the [[left coset|03DEH43F4H]] $gH$.
In general $gH$ and $Hg$ can differ, but the map $gH \mapsto Hg^{-1}$ is a bijection from the left cosets to the right cosets.

## Explanations

### L1 human

A right coset is like a left coset, but you multiply the subgroup by the fixed element on the right.
In a group where the order of products matters, left and right cosets can be different sets.
The number of left cosets and the number of right cosets are always the same.
</document>

<document source="https://tamnd.github.io/math2/node/060ZVZBV3E.md" tag="060ZVZBV3E" kind="statement">
# Law of cosines

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$. Let $a = d(B, C)$, $b = d(C, A)$ and $c = d(A, B)$ be its side lengths, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]]. Let $\gamma = \angle ACB$ be its [[angle|0GJAJC6TNZ]] at $C$.
Then
$$c^2 = a^2 + b^2 - 2ab \cos \gamma.$$

## Remarks

The proof in vector form is short. With $u = A - C$ and $v = B - C$: $c^2 = \|u - v\|^2 = \|u\|^2 + \|v\|^2 - 2 u \cdot v$, and $u \cdot v = ab \cos \gamma$.
For $\gamma = \pi / 2$ the formula is the [[Pythagorean theorem|0VP3A1290K]].
The formula lets you find the third side from two sides and the angle between them. It also gives each angle from the three sides.
Mathlib states the formula for all points, with its convention for the angle at a repeated point.

## History

Book II of Euclid's Elements contains the statement in the language of areas of rectangles. Propositions 12 and 13 of Book II treat the obtuse and the acute case.

## Explanations

### L1 human

The law of cosines corrects the Pythagorean theorem for triangles without a right angle. If the angle $\gamma$ is larger than a right angle, $\cos \gamma < 0$, and the opposite side is longer than in the right triangle. If $\gamma$ is smaller, the opposite side is shorter.
</document>

<document source="https://tamnd.github.io/math2/node/060ZX7C32T.md" tag="060ZX7C32T" kind="concept">
# Carmichael number

## Summary

A Carmichael number is a composite number $n$ that satisfies $a^{n-1} \equiv 1 \pmod n$ for each integer $a$ coprime to $n$.
The [[definition of a Carmichael number|0TA7WT68RK]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/061BTYA0RN.md" tag="061BTYA0RN" kind="statement">
# Lucas–Lehmer test

## Statement

Let $p$ be an odd prime, and let $M_p = 2^p - 1$ be the [[Mersenne number|0GJBJ19NXR]].
Define the sequence $(s_i)_{i \ge 0}$ by $s_0 = 4$ and $s_i = s_{i-1}^2 - 2$ for $i \ge 1$.
Then $M_p$ is prime if and only if $s_{p-2} \equiv 0 \pmod{M_p}$.

## Remarks

For example, take $p = 5$ and $M_5 = 31$. Modulo $31$, the sequence is $4, 14, 8, 0$, so $s_3 \equiv 0$ and $31$ is prime.
In practice, one reduces each term modulo $M_p$. Then the test needs $p - 2$ squarings of numbers with at most $p$ bits.

## History

Édouard Lucas developed the test in 1878, and Derrick Henry Lehmer proved it in 1930.

## Explanations

### L1 human

The test checks a Mersenne number for primality with a simple loop: square, subtract $2$, and reduce modulo $M_p$.
After $p - 2$ steps, the result is $0$ exactly when $M_p$ is prime.
</document>

<document source="https://tamnd.github.io/math2/node/061J5VGSPT.md" tag="061J5VGSPT" kind="concept">
# Symmetric relation

## Summary

A relation is symmetric when it always holds in both directions.
</document>

<document source="https://tamnd.github.io/math2/node/062D9KXJB1.md" tag="062D9KXJB1" kind="concept">
# Tree

## Summary

A tree is a connected graph with no cycles. Trees are the minimal connected graphs, and they appear in data structures, in spanning problems and in counting problems. The node [[tree|0MJ0JS72DZ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/062MQGP8M5.md" tag="062MQGP8M5" kind="definition">
# Trail

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A trail in $G$ is a [[walk|0VJZM0F9YM]] $(w_0, w_1, \dots, w_k)$ whose edges $w_0 w_1, w_1 w_2, \dots, w_{k-1} w_k$ are pairwise distinct.

## Remarks

A trail may visit a vertex more than once, but it uses each edge at most once.
Every [[path|0JDE56HWQ8]] is a trail. In the graph of two triangles that share a vertex $c$, the closed walk $a, b, c, d, e, c, a$ is a trail but not a path.
An [[Eulerian trail|0NNYDD1AVR]] is a trail that uses every edge.

## Explanations

### L1 human

A trail is a walk that never uses the same edge twice.
It is the kind of route that you draw without lifting your pen and without retracing a line.
</document>

<document source="https://tamnd.github.io/math2/node/06395GBNWD.md" tag="06395GBNWD" kind="definition">
# Gamma distribution

## Definition

Let $\alpha > 0$ and $\beta > 0$.
The gamma distribution with shape $\alpha$ and rate $\beta$ is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$f(x) = \begin{cases} \dfrac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x} & \text{if } x > 0, \\ 0 & \text{if } x \le 0, \end{cases}$$
with the gamma function $\Gamma(\alpha) = \int_0^\infty t^{\alpha - 1} e^{-t} \, dt$. One writes $X \sim \operatorname{Gamma}(\alpha, \beta)$.

## Remarks

The mean is $\alpha / \beta$ and the variance is $\alpha / \beta^2$. Another common form uses the scale $\theta = 1/\beta$.
$\operatorname{Gamma}(1, \beta)$ is the [[exponential distribution|028Z8WC5Y9]] with rate $\beta$. For an integer $n \ge 1$, $\operatorname{Gamma}(n, \beta)$ is the law of a sum of $n$ independent $\operatorname{Exp}(\beta)$ random variables. It is the Erlang distribution.
If $X \sim \operatorname{Gamma}(\alpha_1, \beta)$ and $Y \sim \operatorname{Gamma}(\alpha_2, \beta)$ are independent, then $X + Y \sim \operatorname{Gamma}(\alpha_1 + \alpha_2, \beta)$.
The chi-squared distribution with $k$ degrees of freedom is $\operatorname{Gamma}(k/2, 1/2)$.

## Explanations

### L1 human

The gamma distribution with an integer shape $n$ is the waiting time for the $n$-th event of a Poisson process.
For example, it can model the time until the third customer arrives.
</document>

<document source="https://tamnd.github.io/math2/node/063D5Z2C70.md" tag="063D5Z2C70" kind="concept">
# Nilpotent endomorphism

## Summary

A nilpotent endomorphism is a linear map from a space to itself that becomes zero after finitely many compositions with itself.
</document>

<document source="https://tamnd.github.io/math2/node/063H5KJC69.md" tag="063H5KJC69" kind="statement">
# First fundamental theorem of calculus

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be Riemann integrable.
Define $F(x) = \int_a^x f(t) \, dt$ for $x \in [a, b]$.
Then $F$ is continuous on $[a, b]$.
If $f$ is continuous at $c \in (a, b)$, then $F$ is differentiable at $c$ and $F'(c) = f(c)$.

## Remarks

If $|f| \le M$, then $|F(y) - F(x)| \le M |y - x|$. So $F$ is Lipschitz continuous.
By [[additivity|05RCDKHZM6]], $\frac{F(c + h) - F(c)}{h}$ is the average of $f$ over the interval between $c$ and $c + h$. Continuity at $c$ makes this average tend to $f(c)$.
Mathlib states a strict form as `intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_right`.

## Explanations

### L1 human

Let the area under the graph from $a$ to $x$ be a function of $x$.
Where $f$ is continuous, this area grows at the rate $f(x)$.
</document>

<document source="https://tamnd.github.io/math2/node/063JNBSMJ0.md" tag="063JNBSMJ0" kind="conjecture">
# Goldbach's conjecture

## Statement

Each even integer $n > 2$ is the sum of two [[prime numbers|0SN0MAEKYX]].

## Remarks

For example, $4 = 2 + 2$, $28 = 5 + 23 = 11 + 17$ and $100 = 3 + 97$.
The weak Goldbach conjecture says that each odd integer $n > 5$ is the sum of three primes. Harald Helfgott proved it in 2013.
The conjecture is open. A distributed computer search checked it for each even $n \le 4 \cdot 10^{18}$.

## History

Goldbach stated a form of the conjecture in a letter to Euler in 1742.

## Explanations

### L1 human

Take an even number larger than $2$. The conjecture says that you can always split it into two primes.
For example, $18 = 7 + 11$. Nobody has found an even number that fails, but nobody has a proof.
</document>

<document source="https://tamnd.github.io/math2/node/063W5JMY85.md" tag="063W5JMY85" kind="definition">
# Metric space

## Definition

Let $X$ be a set.
A metric on $X$ is a [[function|0XHDTE5YMZ]] $d \colon X \times X \to \mathbb{R}$ with these properties, for all $x, y, z \in X$:

1. $d(x, x) = 0$.
2. If $x \neq y$, then $d(x, y) > 0$.
3. $d(x, y) = d(y, x)$.
4. $d(x, z) \le d(x, y) + d(y, z)$.

A metric space is a pair $(X, d)$, where $X$ is a set and $d$ is a metric on $X$.
The number $d(x, y)$ is the distance between $x$ and $y$.

## Remarks

Property 4 is the [[triangle inequality for a metric|01M6CW9VM6]].
The properties give $d(x, y) \ge 0$ for all $x, y$: take $z = x$ in property 4 and use property 3.
The set $\mathbb{R}$ with $d(x, y) = |x - y|$ is a metric space: this is the [[usual metric on the real line|0KXJTZSW9B]].
If you remove property 2, you get a [[pseudometric space|0H3W614GEC]].

## Explanations

### L1 human

A metric is a rule that gives a distance to each pair of points.
The four properties are the facts that you expect from distances on a map.
For example, the road distance between towns is not always a metric, because one-way roads can break symmetry.
</document>

<document source="https://tamnd.github.io/math2/node/064N002TX4.md" tag="064N002TX4" kind="definition">
# Order of a zero

## Definition

Let $U \subseteq \mathbb{C}$ be an open set, let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$, and let $z_0 \in U$.
Assume that $f$ is not identically zero on any open disc with center $z_0$.
The order of $f$ at $z_0$ is the smallest integer $m \ge 0$ with $f^{(m)}(z_0) \neq 0$, where $f^{(m)}$ is the $m$-th complex derivative of $f$.
If $f(z_0) = 0$, this order $m \ge 1$ is the order of the [[zero|0CEGEASY49]] $z_0$ of $f$. A zero of order $1$ is a simple zero.
If $f$ is identically zero on an open disc with center $z_0$, the order of $f$ at $z_0$ is $\infty$.

## Remarks

The order is $m$ if and only if $f(z) = (z - z_0)^m g(z)$ near $z_0$, where $g$ is holomorphic near $z_0$ and $g(z_0) \neq 0$.
The order exists because [[holomorphic functions are analytic|0826SGV3R0]]: if all the derivatives at $z_0$ are $0$, then the Taylor series is $0$ and $f = 0$ near $z_0$.
For example, $z^3 (z - 1)$ has a zero of order $3$ at $0$ and a simple zero at $1$. The function $\sin z - z$ has a zero of order $3$ at $0$.

## Explanations

### L1 human

The order of a zero says how fast the function goes to $0$ at that point.
Near a zero of order $m$, the function behaves like a constant times $(z - z_0)^m$.
</document>

<document source="https://tamnd.github.io/math2/node/0659J5X7B5.md" tag="0659J5X7B5" kind="statement">
# LaSalle's invariance principle

## Statement

Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be locally Lipschitz with $f(0) = 0$, and let $V : \mathbb{R}^n \to \mathbb{R}$ be continuously differentiable.
Assume that $\dot{V}(x) = \nabla V(x) \cdot f(x) \le 0$ for all $x \in \mathbb{R}^n$, that $V(0) = 0$ and $V(x) > 0$ for $x \neq 0$, and that $V(x) \to \infty$ as $\|x\| \to \infty$.
Let $E = \{x \in \mathbb{R}^n : \dot{V}(x) = 0\}$.
Assume that the only solution $x(t)$ of $x' = f(x)$ with $x(t) \in E$ for all $t \ge 0$ is the solution $x(t) = 0$.
Then the origin is globally asymptotically stable for $x' = f(x)$.

## Remarks

The general principle: if $\dot{V} \le 0$, then each solution that is bounded for $t \ge 0$ approaches the largest invariant subset of $E$.
The principle allows $\dot{V} = 0$ outside the origin, where [[Lyapunov's stability theorem|0J6A752DTZ]] needs a strict Lyapunov function.
For the damped oscillator $x' = y$, $y' = -x - c y$ with $c > 0$, take $V = \frac{1}{2}(x^2 + y^2)$. Then $E = \{y = 0\}$, and a solution in $E$ has $y' = -x = 0$. So the origin is globally asymptotically stable.
A local version holds when the conditions on $V$ hold only near the origin.

## History

The principle is named after Joseph LaSalle. It is also named after Nikolay Krasovskii and Evgenii Barbashin.

## Explanations

### L1 human

Sometimes the energy only stops decreasing on a set of states, not only at the equilibrium.
LaSalle's principle says that the motion must still end at the equilibrium, if no motion can stay forever in that set.
</document>

<document source="https://tamnd.github.io/math2/node/065CK367F9.md" tag="065CK367F9" kind="concept">
# Coprime ideals

## Summary

Two ideals of a ring are coprime when their sum is the whole ring.
For the integers, the ideals $(m)$ and $(n)$ are coprime exactly when $m$ and $n$ are coprime numbers.
The [[definition of coprime ideals|0MDJP82TEH]] gives the condition.
</document>

<document source="https://tamnd.github.io/math2/node/065HE5V98E.md" tag="065HE5V98E" kind="concept">
# Surface integral

## Summary

A surface integral is an integral of a function over a surface in space, with respect to [[surface area|0S29Y4ZW0Y]].
It is the analogue for surfaces of the line integral along a curve.
</document>

<document source="https://tamnd.github.io/math2/node/065J822CKN.md" tag="065J822CKN" kind="statement">
# Inclusion of a subspace is continuous

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]], let $S \subseteq X$ have the [[subspace topology|0D5YHQGDJY]] and let $\iota \colon S \to X$ be the inclusion map $\iota(s) = s$.
Then $\iota$ is [[continuous|0SY9ECQ19Q]].

## Remarks

Proof: for $U$ open in $X$, the [[preimage|0FGCFY1AZX]] is $\iota^{-1}(U) = S \cap U$, which is open in $S$.
Thus the [[restriction|0D2MXBV21T]] $f|_S = f \circ \iota$ of a continuous map $f \colon X \to Y$ is continuous.
A map $g \colon Z \to S$ is continuous if and only if $\iota \circ g \colon Z \to X$ is continuous. This is the universal property of the subspace topology.

## Explanations

### L1 human

Putting a subspace back into the big space is a continuous map.
So when you restrict a continuous function to a subset, you get a continuous function.
</document>

<document source="https://tamnd.github.io/math2/node/065JWMZW86.md" tag="065JWMZW86" kind="concept">
# Area of a triangle

## Summary

The area of a [[triangle|0CTNFGYMM2]] is the measure of the region of the plane that the triangle bounds. It is half the product of a base and the height on that base: $T = \frac{1}{2} b h$.
Here $b$ is the length of a side, and $h$ is the distance from the opposite vertex to the line of that side.
If two sides have the lengths $a$ and $b$ and the angle $\gamma$ between them, then $T = \frac{1}{2} a b \sin \gamma$.
For the vertices $A = (x_A, y_A)$, $B = (x_B, y_B)$ and $C = (x_C, y_C)$ in $\mathbb{R}^2$:
$$T = \tfrac{1}{2} \left| (x_B - x_A)(y_C - y_A) - (x_C - x_A)(y_B - y_A) \right|.$$

## Remarks

[[Heron's formula|0FRBZ2X4HJ]] gives the area from the three side lengths alone.
In $\mathbb{R}^3$, the area of the triangle $ABC$ is $\frac{1}{2} \|(B - A) \times (C - A)\|$, with the [[cross product|0N5N15HWZ7]].
The coordinate formula without the absolute value is the signed area. It is positive when $A$, $B$, $C$ go round anticlockwise.

## Explanations

### L1 human

A triangle is half of a parallelogram: two copies of it fit together into one. The parallelogram has the area base times height. So the triangle has half of that, $\frac{1}{2} b h$.
</document>

<document source="https://tamnd.github.io/math2/node/065S5B9HF1.md" tag="065S5B9HF1" kind="statement">
# Repeating decimals are rational

## Statement

Let $x \ge 0$ be a real number with a [[decimal expansion|06XY4X9430]] that is eventually periodic.
That is, there are $N \ge 0$ and $p \ge 1$ with $d_{k + p} = d_k$ for each $k > N$.
Then $x$ is a [[rational number|0NCJ5WXN6Z]].

## Remarks

Proof: $10^{N+p} x - 10^N x$ is an integer, because the periodic tails cancel. Thus $x$ is a quotient of an integer by $10^N (10^p - 1)$.
For example, $x = 0.\overline{36}$ gives $100 x - x = 36$, thus $x = 36/99 = 4/11$.
The converse is also true: the decimal expansion of each rational number is eventually periodic.

## Explanations

### L1 human

A decimal that repeats the same block of digits forever is always a fraction.
Shift the decimal point by the length of the block and subtract: the endless tail disappears.
For $0.777\dots$, this gives $10 x - x = 7$, thus $x = 7/9$.
</document>

<document source="https://tamnd.github.io/math2/node/065X9S8CM3.md" tag="065X9S8CM3" kind="statement">
# De Moivre–Laplace theorem

## Statement

Let $p \in (0, 1)$ and let $S_n \sim \operatorname{Bin}(n, p)$ for each $n \ge 1$. Let $q = 1 - p$.
Then $(S_n - np)/\sqrt{npq}$ converges [[in distribution|0JDAQHWA9N]] to $\mathcal{N}(0, 1)$. That is, for all reals $a < b$,
$$\lim_{n \to \infty} P\left(a \le \frac{S_n - np}{\sqrt{npq}} \le b\right) = \int_a^b \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \, dx.$$

## Remarks

This is the special case of the [[central limit theorem|0EPZC4Z1FC]] for Bernoulli variables.
Indeed, $S_n$ has the law of a sum of $n$ independent Bernoulli($p$) variables.
A local form also holds: $\binom{n}{k} p^k q^{n-k} \sim \frac{1}{\sqrt{2\pi npq}} e^{-(k - np)^2/(2npq)}$ as $n \to \infty$, uniformly for $k$ with $|k - np| \le C\sqrt{n}$. The proof uses Stirling's formula.
In practice, one often adds a continuity correction of $1/2$ to the integer bounds.

## History

Abraham de Moivre proved the case $p = 1/2$. Pierre-Simon Laplace extended it to general $p$.

## Explanations

### L1 human

For many trials, the binomial distribution looks like a bell curve with mean $np$ and variance $np(1 - p)$.
This lets you approximate binomial probabilities, such as "at least $520$ heads in $1000$ tosses", with the normal distribution.
</document>

<document source="https://tamnd.github.io/math2/node/065XW2V6C3.md" tag="065XW2V6C3" kind="definition">
# Arctangent

## Definition

The arctangent is the function $\arctan \colon \mathbb{R} \to (-\pi/2, \pi/2)$ such that, for each real $x$, $\arctan x$ is the unique $y \in (-\pi/2, \pi/2)$ with $\tan y = x$.

## Remarks

The [[tangent|06T0HS14PP]] is strictly increasing and continuous on $(-\pi/2, \pi/2)$, with range $\mathbb{R}$. So $\arctan$ is well defined and [[continuous|07BY47HG5S]].
For example, $\arctan 0 = 0$ and $\arctan 1 = \pi/4$.
Also $\arctan x \to \pi/2$ as $x \to \infty$, and $\arctan x \to -\pi/2$ as $x \to -\infty$. Mathlib defines it as `Real.arctan`.

## Explanations

### L1 human

The arctangent of $x$ is the angle between $-\pi/2$ and $\pi/2$ whose tangent is $x$.
It is the angle of a line with slope $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0660QX1VPR.md" tag="0660QX1VPR" kind="definition">
# Radius of convergence

## Definition

Let $\sum a_n (x - c)^n$ be a real [[power series|0VP56B4EMX]].
Its radius of convergence is
$$R = \sup \{ r \ge 0 : \text{the sequence } (|a_n| r^n) \text{ is bounded} \},$$
an element of $[0, \infty]$.

## Remarks

The series converges absolutely for $|x - c| < R$, and it diverges for $|x - c| > R$.
At the points $x = c \pm R$, the series may converge or diverge.
Many texts define $R$ as the supremum of all $|x - c|$ where the series converges. The two definitions give the same value.
The [[Cauchy–Hadamard theorem|07E5MB1CA2]] gives a formula for $R$. Mathlib uses the bounded form, in `FormalMultilinearSeries.radius`.

## Explanations

### L1 human

A power series converges on an interval centered at $c$, and diverges outside it.
The radius of convergence is the half length of this interval.
</document>

<document source="https://tamnd.github.io/math2/node/066E7PJCYE.md" tag="066E7PJCYE" kind="concept">
# Pole

## Summary

A pole of a function is an isolated singularity where the modulus of the function tends to infinity, see the [[definition|0WPQYW8MTE]].
It is the simplest kind of singularity that is not removable.
</document>

<document source="https://tamnd.github.io/math2/node/066HW6D9A7.md" tag="066HW6D9A7" kind="statement">
# Closed subsets of compact sets are compact

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]], let $K \subseteq X$ be [[compact|00R0X1WGPX]] and let $F \subseteq X$ be [[closed|0G5AV9Q4RR]] with $F \subseteq K$.
Then $F$ is compact.

## Remarks

Proof: let $(U_i)$ be open sets of $X$ that cover $F$. Add the open set $X \setminus F$ to get an open cover of $K$. A finite subcover of $K$ gives, after you remove $X \setminus F$, a finite subcover of $F$.
More generally, if $K$ is compact and $F$ is closed, then $K \cap F$ is compact.
For example, the [[Cantor set|0BXNDZFQME]] is compact, because it is a closed subset of $[0, 1]$.

## Explanations

### L1 human

A closed part of a compact set is again compact.
The open complement of the closed part covers everything outside it, so it does the job of all the extra open sets.
</document>

<document source="https://tamnd.github.io/math2/node/066RJJVWH9.md" tag="066RJJVWH9" kind="statement">
# The integers are countable

## Statement

The set $\mathbb{Z}$ of integers is [[countably infinite|0SEZ3JGZP7]].
The function $f \colon \mathbb{N} \to \mathbb{Z}$ with $f(2k) = k$ and $f(2k + 1) = -(k + 1)$ is a [[bijection|0SNQEF7Z53]].

## Remarks

The bijection lists the integers as $0, -1, 1, -2, 2, \dots$.
So $|\mathbb{Z}| = |\mathbb{N}| = \aleph_0$, although $\mathbb{N}$ is a proper subset of $\mathbb{Z}$.

## Explanations

### L1 human

The list $0, 1, 2, \dots$ never reaches the negative integers.
Alternate between the two signs, and every integer gets a position in the list.
</document>

<document source="https://tamnd.github.io/math2/node/0670T59HGE.md" tag="0670T59HGE" kind="concept">
# Noetherian ring

## Summary

A Noetherian ring is a ring in which every ideal is finitely generated, or equivalently, every increasing chain of ideals is eventually constant.
The [[definition of a Noetherian ring|0MVH4Y1CRZ]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/06738GGWWK.md" tag="06738GGWWK" kind="concept">
# Flow of a vector field

## Summary

A flow is a family of maps $\varphi_t$ of a space to itself, indexed by real time $t$. The map $\varphi_0$ is the identity, and $\varphi_s \circ \varphi_t = \varphi_{s+t}$.
The flow of a vector field sends each point to the position, after time $t$, of the solution of $x' = f(x)$ through it. See the [[definition|02903SSA48]].
</document>

<document source="https://tamnd.github.io/math2/node/0673D85QKZ.md" tag="0673D85QKZ" kind="concept">
# Autonomous system

## Summary

An autonomous system is a system of ordinary differential equations $x' = f(x)$ whose right side does not depend explicitly on the independent variable.
When the independent variable is time, such a system is also called time-invariant. See the [[definition|07BNRFQ0PJ]].
</document>

<document source="https://tamnd.github.io/math2/node/067F8YVYMH.md" tag="067F8YVYMH" kind="statement">
# Repeated roots and the derivative

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]], let $f \in R[x]$ with $f \neq 0$ and let $c \in R$.
Then $c$ is a root of $f$ with [[multiplicity|03XVP8Q4GY]] at least $2$ if and only if $f(c) = 0$ and $f'(c) = 0$.
Here $f'$ is the [[formal derivative|0AD4VS02H3]] of $f$.

## Remarks

The proof: write $f = (x - c) g$. The product rule gives $f' = g + (x - c) g'$, so $f'(c) = g(c)$. And $g(c) = 0$ exactly when $(x - c)^2$ divides $f$.
For example, $f = x^3 - 3x + 2 = (x - 1)^2 (x + 2)$. Here $f(1) = 0$ and $f'(1) = 3 - 3 = 0$.

## Explanations

### L1 human

At a repeated root, the graph of a real polynomial touches the axis and has a horizontal tangent.
So the polynomial and its derivative are both zero there.
The algebraic statement is true over every commutative ring.
</document>

<document source="https://tamnd.github.io/math2/node/067NKSBEWT.md" tag="067NKSBEWT" kind="concept">
# Scalar matrix

## Summary

A scalar matrix is a diagonal matrix whose diagonal entries are all equal, that is, a scalar multiple of the identity matrix.
</document>

<document source="https://tamnd.github.io/math2/node/067SYD2J4B.md" tag="067SYD2J4B" kind="statement">
# Tonelli's theorem

## Statement

Let $f \colon \mathbb{R}^n \times \mathbb{R}^m \to [0, \infty]$ be Lebesgue measurable.
Then $x \mapsto \int_{\mathbb{R}^m} f(x, y) \, dy$ and $y \mapsto \int_{\mathbb{R}^n} f(x, y) \, dx$ are measurable functions with values in $[0, \infty]$, and
$$\int_{\mathbb{R}^n \times \mathbb{R}^m} f \, d(x, y) = \int_{\mathbb{R}^n} \int_{\mathbb{R}^m} f(x, y) \, dy \, dx = \int_{\mathbb{R}^m} \int_{\mathbb{R}^n} f(x, y) \, dx \, dy.$$
The three values may be $+\infty$.

## Remarks

Tonelli's theorem has the same conclusion as [[Fubini's theorem|0R0T1VWH5X]]. It replaces integrability of $|f|$ by the hypothesis $f \ge 0$.
In practice, one applies Tonelli's theorem to $|f|$ to check that $\int |f| < \infty$, and then applies Fubini's theorem to $f$.
More generally, the theorem holds for non-negative measurable functions on a product of two $\sigma$-finite measure spaces. For counting measure on $\mathbb{N}$, it says that $\sum_x \sum_y a_{xy} = \sum_y \sum_x a_{xy}$ when all $a_{xy} \ge 0$.
Leonida Tonelli published this variation of Fubini's theorem in 1909.

## Explanations

### L1 human

For a function that is never negative, the order of integration never matters.
Both orders give the same value, even when that value is infinite, because no positive and negative parts can cancel.
</document>

<document source="https://tamnd.github.io/math2/node/068BZBXE0C.md" tag="068BZBXE0C" kind="statement">
# Existence of square roots of nonnegative reals

## Statement

Let $x$ be a real number with $x \ge 0$.
Then there is exactly one real number $y$ with $y \ge 0$ and $y^2 = x$.

## Remarks

Existence uses the [[completeness axiom|0AXWET80YW]]: take $y = \sup \{t \ge 0 : t^2 \le x\}$ and show that $y^2 < x$ and $y^2 > x$ are both impossible.
Uniqueness: if $y, z \ge 0$ and $y^2 = z^2$, then $(y - z)(y + z) = 0$, thus $y = z$.
In $\mathbb{Q}$ the existence fails: $2$ has no rational square root, by the [[irrationality of the square root of 2|0DZGSDHD92]].

## Explanations

### L1 human

Every nonnegative real number is the area of exactly one square, and the side of that square is the square root.
This needs the real numbers: among the fractions, no square has area exactly $2$.
</document>

<document source="https://tamnd.github.io/math2/node/068P0P32DW.md" tag="068P0P32DW" kind="statement">
# Closure in a metric space

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $S \subseteq X$ and let $x \in X$.
Then $x$ is in the [[closure|0DRKT1QJR7]] $\overline{S}$ if and only if for each real $\varepsilon > 0$ there is $s \in S$ with $d(x, s) < \varepsilon$.

## Remarks

Equivalently, $x \in \overline{S}$ if and only if each [[open ball|0DC3MPR7C9]] $B(x, \varepsilon)$ meets $S$.
For nonempty $S$, this means $d(x, S) = 0$, where $d(x, S)$ is the [[distance from the point to the set|07TXYYVYZR]].
In $\mathbb{R}$, the point $0$ is in the closure of $\{1/n : n \ge 1\}$, because $1/n < \varepsilon$ for large $n$.

## Explanations

### L1 human

A point is in the closure of $S$ when there are points of $S$ as close to it as you want.
The point does not need to be in $S$ itself.
</document>

<document source="https://tamnd.github.io/math2/node/0692CGQF00.md" tag="0692CGQF00" kind="definition">
# Diameter of a set

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $S$ be a nonempty subset of $X$.
The diameter of $S$ is
$$\operatorname{diam}(S) = \sup \{ d(x, y) : x, y \in S \},$$
where the [[supremum|0BTMQ04556]] is taken in the [[extended real number line|0GSZ7B0CGR]], so that $\operatorname{diam}(S) \in [0, \infty]$.

## Remarks

The set $S$ is [[bounded|0P953MG05E]] if and only if $\operatorname{diam}(S) < \infty$.
For the empty set, a common convention is $\operatorname{diam}(\emptyset) = 0$.
In $\mathbb{R}$, the interval $(a, b)$ with $a < b$ has diameter $b - a$, and so does $[a, b]$.
If $S \subseteq T$, then $\operatorname{diam}(S) \le \operatorname{diam}(T)$.

## Explanations

### L1 human

The diameter is the largest distance between two points of the set, or the limit of such distances.
For a disc of radius $r$ in the plane, the diameter is $2r$.
</document>

<document source="https://tamnd.github.io/math2/node/0693Y2METT.md" tag="0693Y2METT" kind="concept">
# Hessian matrix

## Summary

The Hessian matrix of a real-valued function of $n$ variables at a point is the $n \times n$ matrix of its second [[partial derivatives|09ZXGX55DM]] at that point.
It describes the curvature of the function, and it is the main tool of the second derivative test for local extrema.
</document>

<document source="https://tamnd.github.io/math2/node/0698M7C7AX.md" tag="0698M7C7AX" kind="concept">
# Triple integral

## Summary

The triple integral of a function of three variables over a solid region is a limit of Riemann sums. These sums use partitions of the region into small boxes.
It extends the [[double integral|0WAR4F23B4]] to three dimensions, and it computes volumes, masses and centres of mass of solids.
</document>

<document source="https://tamnd.github.io/math2/node/069A3BC3EP.md" tag="069A3BC3EP" kind="statement">
# Uniqueness of splitting fields

## Statement

Let $K$ be a [[field|0QA8CHH8MG]], let $f \in K[x]$ be nonzero, and let $L$ and $L'$ be [[splitting fields|02NZZ1J0FH]] of $f$ over $K$.
Then there is a ring isomorphism $\sigma \colon L \to L'$ with $\sigma(c) = c$ for every $c \in K$.

## Remarks

The proof is by induction on $[L : K]$. It extends an isomorphism one root at a time, with the [[structure of simple extensions|0ZN39QDTP9]].
The isomorphism $\sigma$ is usually not unique. The isomorphisms from $L$ to itself that fix $K$ form the [[Galois group|0XVR72M51M]] of $f$.
So one speaks of "the" splitting field of $f$ over $K$.
The choice of the irreducible factors in the construction does not change the result, up to isomorphism.

## Explanations

### L1 human

Two splitting fields of the same polynomial are the same field with different names for the elements.
The proof extends a matching of the base field, one root at a time.
</document>

<document source="https://tamnd.github.io/math2/node/069HKV4X1E.md" tag="069HKV4X1E" kind="statement">
# Comparison of a series and an integral

## Statement

Let $a$ be a real number, let $n$ be a natural number, and let $f \colon [a, a + n] \to \mathbb{R}$ be a decreasing function.
Then
$$\sum_{i=1}^{n} f(a + i) \le \int_a^{a+n} f(x)\, dx \le \sum_{i=0}^{n-1} f(a + i).$$

## Remarks

On each interval $[a + i, a + i + 1]$ we have $f(a + i + 1) \le f(x) \le f(a + i)$. Integrate and add over $i$.
The integral exists because monotone functions are [[Riemann integrable|04YKAYDQPW]].
With $f(x) = 1/x$ and $a = 1$ it gives $\ln(n + 1) \le H_n \le 1 + \ln n$ for the [[harmonic numbers|0D2Y6G4C4S]].
Mathlib states the first inequality as `AntitoneOn.sum_le_integral`.

## Explanations

### L1 human

The sum of the values at the right ends of unit intervals is an area of rectangles below the graph of a decreasing function.
The sum of the values at the left ends is an area of rectangles above it.
</document>

<document source="https://tamnd.github.io/math2/node/069MCGX635.md" tag="069MCGX635" kind="concept">
# Row space

## Summary

The row space of a matrix is the span of its rows.
</document>

<document source="https://tamnd.github.io/math2/node/069P20KYQT.md" tag="069P20KYQT" kind="definition">
# Partition of an interval

## Definition

Let $a < b$ be real numbers.
A partition of $[a, b]$ is a finite sequence of real numbers $P = (x_0, x_1, \dots, x_n)$ with
$$a = x_0 < x_1 < \cdots < x_n = b.$$
The intervals $[x_{i-1}, x_i]$ for $i = 1, \dots, n$ are the subintervals of $P$.

## Remarks

Partitions are the basis of [[Darboux sums|0QRQVH0CRW]] and [[Riemann sums|0HBY48G8G1]].
A [[refinement|0EEKF7YFBW]] of $P$ adds more points. The [[mesh|04GM4NYXJB]] of $P$ is the length of its longest subinterval.
A tagged partition also chooses a point $t_i$ in each subinterval $[x_{i-1}, x_i]$.

## Explanations

### L1 human

A partition cuts the interval $[a, b]$ into finitely many pieces.
The cut points are listed from left to right.
</document>

<document source="https://tamnd.github.io/math2/node/06AMCJMMVM.md" tag="06AMCJMMVM" kind="statement">
# Laplace expansion along a column

## Statement

Let $K$ be a commutative ring, let $n \ge 2$, and let $A = (a_{ij})$ be an $n \times n$ matrix over $K$.
Let $C_{ij}$ be the [[cofactors|0XPKR18APA]] of $A$.
Then, for each column $j$,
$$\det A = \sum_{i=1}^{n} a_{ij} C_{ij} = \sum_{i=1}^{n} (-1)^{i+j} a_{ij} M_{ij}.$$

## Remarks

It follows from the [[expansion along a row|0C08BQ3M05]], applied to $A^T$, because $\det A^T = \det A$.
For example, expansion along column $1$ of $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 6 & 7 \end{pmatrix}$ gives $1 \cdot (28 - 30) = -2$.

## Explanations

### L1 human

The same expansion works down a column instead of across a row.
We choose the column with the most zeros, to save work.
Each zero entry removes one smaller determinant from the sum.
</document>

<document source="https://tamnd.github.io/math2/node/06B8BNJCD6.md" tag="06B8BNJCD6" kind="statement">
# Heine–Cantor theorem

## Statement

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be a [[continuous map|0SY9ECQ19Q]].
If $X$ is [[compact|0QPHW703PV]], then $f$ is [[uniformly continuous|0FHSDCQACY]].

## Remarks

Proof by contradiction: if $f$ is not uniformly continuous, there are $\varepsilon > 0$ and points $x_n, y_n$ with $d(x_n, y_n) \to 0$ and $d(f(x_n), f(y_n)) \ge \varepsilon$. By [[sequential compactness|0K8SGQSV7G]], a subsequence of $(x_n)$ converges to some $x$, and the matching subsequence of $(y_n)$ also converges to $x$. Continuity at $x$ gives a contradiction.
For example, each continuous $f \colon [a, b] \to \mathbb{R}$ is uniformly continuous. This fact is used to show that $f$ is Riemann integrable.
The theorem fails on $\mathbb{R}$: $x \mapsto x^2$ is continuous but not uniformly continuous.

## Explanations

### L1 human

On a compact space, continuity at each point is enough to get one $\delta$ for the whole space.
Compactness rules out regions where the function becomes steeper and steeper without end.
</document>

<document source="https://tamnd.github.io/math2/node/06B9W45XV0.md" tag="06B9W45XV0" kind="statement">
# Solution formula for linear first-order equations

## Statement

Let $I$ be an open interval, let $p, q : I \to \mathbb{R}$ be continuous, let $x_0 \in I$ and let $y_0 \in \mathbb{R}$.
Let $P(x) = \int_{x_0}^{x} p(s)\, ds$ for $x \in I$.
Then the [[initial value problem|075VBY4K4X]] $y' + p(x)\, y = q(x)$, $y(x_0) = y_0$ has exactly one solution on $I$, and it is
$$y(x) = e^{-P(x)} \left( y_0 + \int_{x_0}^{x} e^{P(s)}\, q(s)\, ds \right), \qquad x \in I.$$

## Remarks

Proof idea: $\mu = e^{P}$ is an [[integrating factor|01THH3Y3YP]], so a solution satisfies $(\mu y)' = \mu q$. Integrate from $x_0$ to $x$ and use $\mu(x_0) = 1$.
The solution exists on all of $I$. A linear equation has no solution that blows up inside the interval of continuity of $p$ and $q$.
Example: for $y' + y = 1$, $y(0) = 0$, the formula gives $y(x) = 1 - e^{-x}$.

## Explanations

### L1 human

The formula has two parts.
The term $y_0 e^{-P(x)}$ is the solution of the equation without the right side $q$.
The integral term adds the effect of $q$, collected over the time from $x_0$ to $x$.
</document>

<document source="https://tamnd.github.io/math2/node/06BA5JZKTQ.md" tag="06BA5JZKTQ" kind="definition">
# Strictly convex function

## Definition

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be a function.
The function $f$ is strictly convex if
$$f(t x + (1 - t) y) < t f(x) + (1 - t) f(y)$$
for all $x, y \in I$ with $x \ne y$ and all $t \in (0, 1)$.

## Remarks

Each strictly convex function is [[convex|0VV0HSW1ZS]]. A linear function is convex but not strictly convex.
A strictly convex function has at most one minimum point.
For example, $x^2$ and $e^x$ are strictly convex. Mathlib defines strict convexity as `StrictConvexOn`.

## Explanations

### L1 human

A strictly convex function bends upward everywhere, with no straight pieces in its graph.
The chords lie strictly above the graph, except at their ends.
</document>

<document source="https://tamnd.github.io/math2/node/06BRDF3F0E.md" tag="06BRDF3F0E" kind="concept">
# Reflection

## Summary

The reflection in a subspace $W$ of an inner product space fixes the vectors of $W$ and negates the vectors orthogonal to $W$.
</document>

<document source="https://tamnd.github.io/math2/node/06C0JBE52D.md" tag="06C0JBE52D" kind="statement">
# Five colour theorem

## Statement

Let $G$ be a finite [[planar graph|029JQEK15B]].
Then $\chi(G) \le 5$.

## Remarks

Proof idea: $G$ has a vertex $v$ of degree at most $5$, by the [[edge bound for planar graphs|0SM1REAVYJ]]. Colour $G - v$ by induction.
If the neighbours of $v$ use all five colours, a swap of two colours along a Kempe chain frees one colour for $v$.
The [[four colour theorem|0M406JC8J1]] is stronger, but its proof is much harder.

## History

Percy John Heawood proved the theorem in 1890.
He found an error in the 1879 proof of the four colour theorem by Alfred Kempe, and he used Kempe's ideas to prove five colours.

## Explanations

### L1 human

Every map can be coloured with five colours so that neighbouring countries have different colours.
The proof is short, unlike the proof that four colours are enough.
</document>

<document source="https://tamnd.github.io/math2/node/06CTFY2DEZ.md" tag="06CTFY2DEZ" kind="definition">
# Symmetric polynomial

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $n \ge 1$.
A polynomial $f \in R[x_1, \dots, x_n]$ is symmetric if
$$f(x_{\sigma(1)}, \dots, x_{\sigma(n)}) = f(x_1, \dots, x_n)$$
for every permutation $\sigma$ of $\{1, \dots, n\}$.

## Remarks

Examples in two variables: $x_1 + x_2$, $x_1 x_2$ and $x_1^2 + x_2^2$. The polynomial $x_1 - x_2$ is not symmetric.
The symmetric polynomials form a subring of $R[x_1, \dots, x_n]$, the ring of invariants of the symmetric group.
The main examples are the [[elementary symmetric polynomials|0ZMY8441GW]] and the [[power sums|0AS1FZT6BM]].
The coefficients of a monic polynomial are symmetric polynomials in its roots, by [[Vieta's formulas|0M8EWQJ1YV]].

## Explanations

### L1 human

A symmetric polynomial treats all its variables in the same way.
Swapping two variables gives the same polynomial.
This is why symmetric expressions in the roots of a polynomial can be computed from the coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/06D0EWR2DV.md" tag="06D0EWR2DV" kind="statement">
# Integral of the exponential function

## Statement

For all real numbers $a$ and $b$,
$$\int_a^b e^x \, dx = e^b - e^a.$$

## Remarks

The [[exponential function|0RMVS7RXZX]] is [[its own derivative|0EX9GFJ0B3]], so it is its own antiderivative.
More generally, $\int_a^b e^{c x} \, dx = (e^{c b} - e^{c a})/c$ for $c \ne 0$.
Mathlib states it as `integral_exp`.

## Explanations

### L1 human

The exponential function does not change under differentiation.
So it also does not change under integration, up to the constant.
</document>

<document source="https://tamnd.github.io/math2/node/06DAH3VS6M.md" tag="06DAH3VS6M" kind="definition">
# P-group

## Definition

Let $p$ be a prime number.
A [[group|G5S1M9]] $G$ is a $p$-group when, for each $g \in G$, the [[order|0WE2NHZJ7B]] of $g$ is $p^n$ for some integer $n \geq 0$.

## Remarks

A finite group is a $p$-group if and only if $|G| = p^k$ for some $k \geq 0$.
One direction uses [[Lagrange's theorem|3C53TD]], and the other direction uses [[Cauchy's theorem|0JM7PA288B]].
The [[dihedral group|0JZ27DMJ5D]] $D_4$ of order $8$ is a $2$-group that is not abelian.
The trivial group is a $p$-group for each prime $p$.

## Explanations

### L1 human

In a $p$-group, each element comes back to the identity after a power of $p$ steps.
For finite groups this means that the number of elements is a power of $p$.
These groups are the building blocks of the Sylow theory.
</document>

<document source="https://tamnd.github.io/math2/node/06DW9BAWB1.md" tag="06DW9BAWB1" kind="object">
# Identity matrix

## Description

Let $R$ be a ring and let $n \ge 0$ be an integer.
The *identity matrix* $I_n$ is the $n \times n$ [[matrix|0VD9ZSWWJG]] over $R$ with entries $1$ on the main diagonal and $0$ elsewhere.
Its entries are $(I_n)_{ij} = \delta_{ij}$, where $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ otherwise.
For each $m \times n$ matrix $A$, $A I_n = A$ and $I_m A = A$.

## Remarks

The identity matrix $I_n$ is the identity element of the [[matrix ring|0FHVPRAE60]] $M_n(R)$.
It is the matrix of the identity map in any basis.
For example, $I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

## Explanations

### L1 human

The identity matrix has ones on the diagonal and zeros everywhere else.
It plays the role of the number $1$: multiplying by it changes nothing.
As a map, it leaves each vector where it is.
</document>

<document source="https://tamnd.github.io/math2/node/06E3AG9TM2.md" tag="06E3AG9TM2" kind="counterexample">
# Z[x] is not a principal ideal domain

## Counterexample

The ring $\mathbb{Z}[x]$ is an integral domain, but it is not a [[principal ideal domain|0RZYDVZ475]].
The ideal $(2, x)$, the set of the polynomials with an even constant term, is not [[principal|00A2EDFT9Q]].
Suppose $(2, x) = (f)$. Then $f \mid 2$, so $f$ is a constant $c$ with $c \in \{\pm 1, \pm 2\}$.
If $c = \pm 1$, then $1 \in (2, x)$, but $1$ has an odd constant term.
If $c = \pm 2$, then $x$ is a multiple of $2$ in $\mathbb{Z}[x]$, which is false.

## Remarks

The quotient $\mathbb{Z}[x]/(2, x)$ is $\mathbb{F}_2$, so $(2, x)$ is a maximal ideal.
The ring $\mathbb{Z}[x]$ is still a [[unique factorization domain|06YMXJMXYS]], by [[the theorem on polynomial rings over a UFD|0D6KH8Y59N]].
Over a field $K$, the ring $K[x]$ is a principal ideal domain.

## Explanations

### L1 human

In $\mathbb{Z}[x]$, the ideal generated by $2$ and $x$ needs two generators.
No single polynomial divides both $2$ and $x$ except the units $\pm 1$, and $\pm 1$ generates too much.
The cause is that $\mathbb{Z}$ is not a field.
</document>

<document source="https://tamnd.github.io/math2/node/06EH97V2A7.md" tag="06EH97V2A7" kind="concept">
# Fréchet derivative

## Summary

The Fréchet derivative of a map $f$ between normed vector spaces at a point $x$ is a bounded linear map. It best approximates $f(x + h) - f(x)$ for small $h$.
It extends the derivative of one-variable calculus to maps between normed spaces, which can have infinite dimension.
For maps between Euclidean spaces it is the [[total derivative|0FNKFT8H6E]].
</document>

<document source="https://tamnd.github.io/math2/node/06EKW1AKXF.md" tag="06EKW1AKXF" kind="concept">
# Operator norm

## Summary

The operator norm of a continuous linear map measures the largest factor by which the map can stretch a vector.
It makes the space of continuous linear maps into a normed space.
</document>

<document source="https://tamnd.github.io/math2/node/06EX0FHBTF.md" tag="06EX0FHBTF" kind="definition">
# Basis of a topology

## Definition

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]].
A basis of $\tau$ is a set $\mathcal{B} \subseteq \tau$ such that each $U \in \tau$ is a [[union|05GK32H72H]] of members of $\mathcal{B}$.
Equivalently, $\mathcal{B} \subseteq \tau$ and for each $U \in \tau$ and each $x \in U$ there is $B \in \mathcal{B}$ with $x \in B \subseteq U$.

## Remarks

A family $\mathcal{B}$ of subsets of a set $X$ is a basis of some topology on $X$ if and only if two conditions hold.
First, the members of $\mathcal{B}$ cover $X$. Second, for all $B_1, B_2 \in \mathcal{B}$ and $x \in B_1 \cap B_2$ there is $B_3 \in \mathcal{B}$ with $x \in B_3 \subseteq B_1 \cap B_2$. This topology is then unique.
The open intervals form a basis of the usual topology of $\mathbb{R}$. The intervals with rational endpoints form a countable basis.
In a metric space, the [[open balls form a basis|02EEX96DFS]] of the metric topology.

## Explanations

### L1 human

A basis is a set of building blocks: each open set is a union of blocks.
In the plane, you can build each open set from open discs, or from open squares.
</document>

<document source="https://tamnd.github.io/math2/node/06F3Y6CMG9.md" tag="06F3Y6CMG9" kind="definition">
# Midpoint

## Definition

Let $K$ be a field in which $2 \neq 0$, and let $A$ be an [[affine space|09XNAPTJQ7]] over a vector space $V$ over $K$. Let $p$ and $q$ be points of $A$.
The midpoint of $p$ and $q$ is the point
$$\operatorname{midpoint}(p, q) = p + \tfrac{1}{2}(q - p).$$
It is the [[affine combination|0KJH6SRFZK]] $\frac{1}{2} p + \frac{1}{2} q$.

## Remarks

In $\mathbb{R}^n$, the midpoint of $(p_1, \dots, p_n)$ and $(q_1, \dots, q_n)$ is $\left(\frac{p_1 + q_1}{2}, \dots, \frac{p_n + q_n}{2}\right)$.
In a Euclidean space, the midpoint $m$ of $p$ and $q$ satisfies $d(p, m) = d(m, q) = \frac{1}{2} d(p, q)$. It is the only point with this property.
The midpoint is symmetric: $\operatorname{midpoint}(p, q) = \operatorname{midpoint}(q, p)$. Also $\operatorname{midpoint}(p, p) = p$.

## Explanations

### L1 human

The midpoint is the average of the two points. To find it, take the average of each coordinate. For example, the midpoint of $(0, 0)$ and $(4, 6)$ is $(2, 3)$.
</document>

<document source="https://tamnd.github.io/math2/node/06GKBTAA5E.md" tag="06GKBTAA5E" kind="definition">
# Boolean algebra

## Definition

A Boolean algebra is a set $A$ with two binary operations $\wedge$ and $\vee$, a unary operation $\neg$, and two elements $0, 1 \in A$.
These axioms must hold for all $a, b, c \in A$.
Associativity: $a \vee (b \vee c) = (a \vee b) \vee c$ and $a \wedge (b \wedge c) = (a \wedge b) \wedge c$.
Commutativity: $a \vee b = b \vee a$ and $a \wedge b = b \wedge a$.
Absorption: $a \vee (a \wedge b) = a$ and $a \wedge (a \vee b) = a$.
Identity: $a \vee 0 = a$ and $a \wedge 1 = a$.
Distributivity: $a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$ and $a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)$.
Complements: $a \vee \neg a = 1$ and $a \wedge \neg a = 0$.

## Remarks

The relation $a \le b$, defined by $a \wedge b = a$, is a partial order with least element $0$ and greatest element $1$.
Equivalently, a Boolean algebra is a complemented [[distributive lattice|0T36P6TXQ6]].
The power set of a set is a Boolean algebra, the [[power set Boolean algebra|004J7ZEFAT]]. Boolean algebras correspond to [[Boolean rings|0N5FCV2YV6]].

## Explanations

### L1 human

A Boolean algebra has operations that behave like and, or and not on true and false, or like intersection, union and complement on sets.
Its axioms are the common rules of logic and of set algebra.
</document>

<document source="https://tamnd.github.io/math2/node/06GPJF9V7M.md" tag="06GPJF9V7M" kind="definition">
# Mersenne prime

## Definition

A Mersenne prime is a [[prime number|0SN0MAEKYX]] of the form $2^n - 1$ for a natural number $n$.
That is, a Mersenne prime is a [[Mersenne number|0GJBJ19NXR]] that is prime.

## Remarks

The first Mersenne primes are $3$, $7$, $31$ and $127$, for $n = 2, 3, 5, 7$.
If $2^n - 1$ is prime, then [[$n$ is prime|07MMWAXMYD]]. The converse fails: $2^{11} - 1 = 2047 = 23 \cdot 89$.
The [[Lucas–Lehmer test|061BTYA0RN]] decides whether $2^p - 1$ is prime.
It is not known whether there are infinitely many Mersenne primes.

## Explanations

### L1 human

A Mersenne prime is a prime that is one less than a power of $2$, such as $31 = 2^5 - 1$.
The even [[perfect numbers|0SB7Q8SYSF]] come exactly from these primes, by the [[Euclid–Euler theorem|0WETJCKNRG]].
</document>

<document source="https://tamnd.github.io/math2/node/06GYGS41JK.md" tag="06GYGS41JK" kind="definition">
# Logical equivalence

## Definition

Let $\varphi$ and $\psi$ be formulas of [[propositional logic|09K9MWD2QQ]].
The formulas $\varphi$ and $\psi$ are logically equivalent when they have the same truth value for each assignment of truth values to their variables.
This is written $\varphi \equiv \psi$.

## Remarks

The formulas $\varphi$ and $\psi$ are logically equivalent exactly when $\varphi \leftrightarrow \psi$ is a [[tautology|0D1AHYPM8N]].
For example, $p \to q \equiv \neg p \lor q$.
[[De Morgan's laws|09PKJKA6J0]] give two more examples.

## Explanations

### L1 human

Logically equivalent formulas say the same thing in different words.
You can check an equivalence with a [[truth table|05RSK4Q657]]: the two columns must agree in every row.
Then you can replace one formula by the other in a proof.
</document>

<document source="https://tamnd.github.io/math2/node/06GZ956657.md" tag="06GZ956657" kind="concept">
# Banach space

## Summary

A Banach space is a normed vector space that is complete for the metric of its norm.
Banach spaces are the main setting of functional analysis.
</document>

<document source="https://tamnd.github.io/math2/node/06HC6HK0Z8.md" tag="06HC6HK0Z8" kind="definition">
# Prime ideal

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
An [[ideal|0MYSA6MYNA]] $P$ of $R$ is prime if it has these two properties.

1. $P \neq R$.
2. For all $a, b \in R$, if $a b \in P$, then $a \in P$ or $b \in P$.

## Remarks

In $\mathbb{Z}$, the ideal $n\mathbb{Z}$ with $n \ge 2$ is prime if and only if $n$ is a prime number. The zero ideal of $\mathbb{Z}$ is also prime.
The zero ideal of $R$ is prime if and only if $R$ is an [[integral domain|09S3PFBEPM]].
An ideal $P$ is prime if and only if [[$R/P$ is an integral domain|0AVXBEC05W]].
Every [[maximal ideal|09J6KGT65G]] is prime.

## Explanations

### L1 human

A prime number $p$ has this property: if $p$ divides $a b$, then $p$ divides $a$ or $b$.
A prime ideal has the same property, with "divides" replaced by "contains".
The ideal $(6)$ is not prime: it contains $2 \cdot 3$ but neither $2$ nor $3$.
</document>

<document source="https://tamnd.github.io/math2/node/06HE9J7DXF.md" tag="06HE9J7DXF" kind="example">
# Uniform metric on bounded functions

## Example

Let $S$ be a nonempty set.
Let $B(S)$ be the set of bounded functions $f \colon S \to \mathbb{R}$, that is, the functions for which there is a real $M$ with $|f(s)| \le M$ for all $s \in S$.
For $f, g \in B(S)$, let
$$d_\infty(f, g) = \sup_{s \in S} |f(s) - g(s)|.$$
Then $d_\infty$ is a [[metric|063W5JMY85]] on $B(S)$, the uniform metric.

## Remarks

The [[supremum|0BTMQ04556]] is finite, because $|f(s) - g(s)| \le |f(s)| + |g(s)|$ and both functions are bounded.
A sequence $(f_k)$ converges to $f$ in this metric if and only if $f_k \to f$ uniformly on $S$.
For $S = \{1, \ldots, n\}$, it is the [[sup metric on R^n|0VTAXQEWQD]].
It is $d_\infty(f, g) = \|f - g\|_\infty$ for the [[sup norm|0PARCKE8WP]].

## Explanations

### L1 human

Two functions are close in the uniform metric when their graphs are close at every point at the same time.
For example, on $[0, 1]$ the distance between $f(x) = x$ and $g(x) = x^2$ is $1/4$, the largest gap, at $x = 1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/06HMCTAQH3.md" tag="06HMCTAQH3" kind="concept">
# Linear map of a matrix

## Summary

The linear map of a matrix in two bases is the linear map whose matrix in these bases is the given matrix.
</document>

<document source="https://tamnd.github.io/math2/node/06HN47VPTH.md" tag="06HN47VPTH" kind="statement">
# Dimension of a direct product

## Statement

Let $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Then the [[product|0EX2HPXPJ6]] $V \times W$ is finite-dimensional, and
$$\dim(V \times W) = \dim V + \dim W,$$
where $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

Let $(v_i)$ be a basis of $V$ and let $(w_j)$ be a basis of $W$.
Then the vectors $(v_i, 0)$ and $(0, w_j)$ together form a basis of $V \times W$.
For example, $\dim(\R^2 \times \R^3) = 5$.
By induction, $\dim(V_1 \times \cdots \times V_k) = \dim V_1 + \cdots + \dim V_k$.

## Explanations

### L1 human

A vector of $V \times W$ is a pair, so we need the directions of $V$ and the directions of $W$.
The dimensions add up.
The plane is the product of two lines, and $1 + 1 = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/06HTJNCMPA.md" tag="06HTJNCMPA" kind="statement">
# Grassmann's formula

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be finite-dimensional [[linear subspaces|00YC1D652K]] of $V$.
Then
$$\dim(U + W) + \dim(U \cap W) = \dim U + \dim W,$$
where $U + W$ is the [[sum of the subspaces|0JMTY8MF54]] and $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

The proof extends a basis of $U \cap W$ to a basis of $U$ and to a basis of $W$.
Together, these vectors form a basis of $U + W$.
Another proof applies the [[rank–nullity theorem|0W4FSW1TVB]] to the map $U \times W \to V$, $(u, w) \mapsto u - w$.
Two planes through the origin in $\R^3$ with $U \ne W$ have $\dim(U + W) = 3$, so $\dim(U \cap W) = 1$.

## Explanations

### L1 human

The formula counts dimensions like a count of elements in a union of two sets.
We add the two dimensions, but then the common part is counted twice, so we subtract it once.
Two different planes through the origin in space always share at least a line.
</document>

<document source="https://tamnd.github.io/math2/node/06HYY6DK6X.md" tag="06HYY6DK6X" kind="definition">
# Affine combination

## Definition

Let $K$ be a field and let $A$ be an [[affine space|09XNAPTJQ7]] over a vector space $V$ over $K$.
Let $p_1, \dots, p_k$ be points of $A$, and let $w_1, \dots, w_k$ be elements of $K$ with $w_1 + \dots + w_k = 1$.
The affine combination of the points $p_i$ with the weights $w_i$ is the point
$$\sum_{i=1}^{k} w_i p_i = o + \sum_{i=1}^{k} w_i (p_i - o),$$
where $o$ is a point of $A$. The result does not depend on the choice of $o$, because the weights add up to 1.

## Remarks

In a vector space with $o = 0$, the affine combination is the linear combination $w_1 p_1 + \dots + w_k p_k$.
The [[midpoint|0S706JBZBG]] of $p$ and $q$ is the affine combination with the weights $\frac{1}{2}$ and $\frac{1}{2}$.
Affine maps keep affine combinations: $f\left(\sum_i w_i p_i\right) = \sum_i w_i f(p_i)$.
An affine combination with all weights $w_i \geq 0$, over the real numbers, is a convex combination.

## Explanations

### L1 human

You cannot add two points of the plane in a sensible way without an origin. But a weighted average of points, with weights that sum to 1, gives the same point for each choice of origin. For example, $\frac{1}{3}A + \frac{1}{3}B + \frac{1}{3}C$ is the centroid of the triangle $ABC$.
</document>

<document source="https://tamnd.github.io/math2/node/06J4PXRG22.md" tag="06J4PXRG22" kind="statement">
# Spectral theorem

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be a [[self-adjoint|0NEGN5CRSC]] linear map.
Then $V$ has an [[orthonormal basis|0S9GGTR0EX]] that consists of [[eigenvectors|0N6960W0BV]] of $T$.
The corresponding eigenvalues are real.

## Remarks

In this basis, the matrix of $T$ is a real diagonal matrix.
Equivalently, $V$ is the orthogonal direct sum of the eigenspaces of $T$.
Over $\C$, the same conclusion holds, with complex eigenvalues, for each [[normal operator|0C8BR8XRTR]].
The matrix form is the [[spectral theorem for Hermitian matrices|0DGZBQWZWN]].

## Explanations

### L1 human

A self-adjoint map is a pure stretch along perpendicular axes.
The axes are the eigenvectors, and the stretch factors are the real eigenvalues.
Every symmetric real matrix is diagonal in a suitable orthonormal basis.
</document>

<document source="https://tamnd.github.io/math2/node/06J5DE801P.md" tag="06J5DE801P" kind="concept">
# Multiplication of natural numbers

## Summary

Multiplication of natural numbers is the operation that sends two [[natural numbers|0MWKJ44V5A]] $a$ and $b$ to their product $a \cdot b$.
The product $a \cdot b$ is the sum of $b$ copies of $a$.
</document>

<document source="https://tamnd.github.io/math2/node/06J6P03MG7.md" tag="06J6P03MG7" kind="statement">
# Classification of finite simple groups

## Statement

Each finite [[simple group|0YMYMNQ0QZ]] is isomorphic to one of the following groups:

1. a [[cyclic group|0KVGH0J5B3]] $\mathbb{Z}/p\mathbb{Z}$ of prime order $p$;
2. an [[alternating group|0Z5905Q73D]] $A_n$ with $n \geq 5$;
3. a group of Lie type;
4. one of $26$ sporadic groups.

## Remarks

The groups of Lie type form infinite families, for example the groups $\mathrm{PSL}_n(\mathbb{F}_q)$, with a few exceptions for small $n$ and $q$.
Some authors count the Tits group as a $27$th sporadic group, because it is not strictly of Lie type.
The largest sporadic group is the Monster group.
The proof consists of tens of thousands of pages in several hundred journal articles.

## History

Daniel Gorenstein announced in 1983 that the classification was complete, but this was premature.
The gap was the case of the quasithin groups.
Michael Aschbacher announced the completed proof in 2004, after Aschbacher and Stephen Smith published a proof of $1221$ pages for this case.

## Explanations

### L1 human

The theorem lists all the finite simple groups.
Most of them come in a few infinite families, and only $26$ are special cases.
Since each finite group is built from simple groups, this list is a map of all the building blocks.
</document>

<document source="https://tamnd.github.io/math2/node/06J8TEPJPY.md" tag="06J8TEPJPY" kind="concept">
# Real and imaginary parts

## Summary

The real part and the imaginary part of a complex number $z = a + b i$ are the real numbers $a$ and $b$.
They are the coordinates of $z$ in the [[complex plane|0CTE7A31XQ]].
</document>

<document source="https://tamnd.github.io/math2/node/06JMR2XWXW.md" tag="06JMR2XWXW" kind="definition">
# Regular curve

## Definition

Let $I \subseteq \mathbb{R}$ be an interval, and let $\gamma \colon I \to \mathbb{R}^n$ be a $C^1$ [[parametrised curve|0JF1XXGWB4]].
The curve $\gamma$ is regular if its [[tangent vector|0D98771SM0]] $\gamma'(t)$ is not zero for each $t \in I$.

## Remarks

The circle $t \mapsto (\cos t, \sin t)$ and each line $t \mapsto p + t v$ with $v \ne 0$ are regular.
The curve $t \mapsto (t^2, t^3)$ is not regular, since its tangent vector is zero at $t = 0$.
The curve $t \mapsto (t^3, t^3)$ is not regular at $t = 0$, but its image is a straight line. So regularity is a property of the parametrisation, not of the image.
A regular curve has an [[arc length parametrisation|00R1JDDV5X]].

## Explanations

### L1 human

A regular curve never stops.
The moving point always has non-zero speed, so the curve has no corners or cusps caused by the point coming to rest.
</document>

<document source="https://tamnd.github.io/math2/node/06JT7KDHXM.md" tag="06JT7KDHXM" kind="definition">
# Homogeneous system of linear equations

## Definition

Let $F$ be a field.
A [[system of linear equations|0GT4WWGWH2]] over $F$ is *homogeneous* when each of its constant terms is $0$.
In matrix form, a homogeneous system is an equation $Ax = 0$, where $A$ is an $m \times n$ matrix over $F$ and $x = (x_1, \dots, x_n)^T$.

## Remarks

The tuple $(0, \dots, 0)$ is always a solution, the *trivial solution*.
The solution set of $Ax = 0$ is a [[linear subspace|00YC1D652K]] of $F^n$, the [[null space|0PHDYNR24D]] of $A$.
If a system $Ax = b$ has a solution $p$, then its solution set is $\{p + v : Av = 0\}$.
A homogeneous system with more unknowns than equations has a nontrivial solution.

## Explanations

### L1 human

A homogeneous system has zero on the right side of every equation.
Setting every unknown to zero always works, so the question is whether other solutions exist.
The sum of two solutions is again a solution, and so is any multiple of a solution.
</document>

<document source="https://tamnd.github.io/math2/node/06JYC06JHY.md" tag="06JYC06JHY" kind="definition">
# Branch cut

## Definition

Let $U \subseteq \mathbb{C}$ be an open set and let $F$ be a [[multivalued function|03CPD1HGF3]] on $U$.
A branch cut for $F$ is a closed subset $K$ of $U$ such that $F$ has a continuous single-valued branch on the open set $U \setminus K$. Usually a branch cut is a curve or a union of curves.
A branch on $U \setminus K$ is a continuous function $f$ on $U \setminus K$ with $f(z) \in F(z)$ for each $z \in U \setminus K$.
The standard branch cut for the logarithm on $\mathbb{C} \setminus \{0\}$ is the ray $(-\infty, 0)$. The open set $\mathbb{C} \setminus (-\infty, 0]$ is the slit plane.

## Remarks

A branch cut usually joins branch points. A branch point of $F$ is a point $z_0$ with the following property. When $z$ goes once around a small circle with center $z_0$, the values of $F$ do not come back to their start.
For $\log z$ and $\sqrt z$, the branch points are $0$ and $\infty$, so a cut is a curve from $0$ to $\infty$.
The [[principal branches|0YKY4M3KRQ]] of the logarithm and of the square root use the slit plane. Each other ray from $0$ is also a valid branch cut and gives other [[branches|0SQ1F2BEDS]].
The slit plane is $\{z \in \mathbb{C} : \operatorname{Re} z > 0 \text{ or } \operatorname{Im} z \neq 0\}$.

## Explanations

### L1 human

You cannot choose the logarithm continuously on a full circle around $0$.
A branch cut is a line that you remove from the plane so that no path can go around $0$. On what is left, a continuous choice is possible.
Crossing the cut makes the chosen value jump, for the logarithm by $2\pi i$.
</document>

<document source="https://tamnd.github.io/math2/node/06JYPJ4WHH.md" tag="06JYPJ4WHH" kind="definition">
# Convergence of a sequence in a metric space

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $(x_n)_{n \ge 1}$ be a sequence of points of $X$, and let $x \in X$.
The sequence $(x_n)$ converges to $x$ when for each real $\varepsilon > 0$ there is an integer $N$ such that $d(x_n, x) < \varepsilon$ for all $n \ge N$.
Then $x$ is a limit of $(x_n)$, with the notation $x_n \to x$ or $\lim_{n \to \infty} x_n = x$.

## Remarks

Equivalently, $x_n \to x$ if and only if $d(x_n, x) \to 0$ in $\mathbb{R}$.
A sequence in a metric space has at most one limit. This follows from the triangle inequality, and also from the fact that [[limits are unique in Hausdorff spaces|0D8T984XEE]].
For $X = \mathbb{R}$ with the usual metric, this is the usual limit of a real sequence.

## Explanations

### L1 human

A sequence converges to $x$ when its terms come as close to $x$ as you want and stay close.
For each tolerance $\varepsilon$, only finitely many terms are farther than $\varepsilon$ from $x$.
</document>

<document source="https://tamnd.github.io/math2/node/06JZGMYTYD.md" tag="06JZGMYTYD" kind="statement">
# Power series of the exponential function

## Statement

For every real $x$,
$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}.$$
This series is the [[Maclaurin series|043SJ5RF59]] of the exponential function. Its partial sums converge to $e^x$ uniformly on each bounded interval.

## Remarks

By the [[derivative rule|0EX9GFJ0B3]], every derivative of $\exp$ at $0$ equals $\exp(0) = 1$. So the Maclaurin coefficients are $1/n!$.
With the series as the definition of $\exp$, the first claim holds by definition. Uniform convergence follows from [[the result on power series|0BWXV7GADW]].
The partial sums give good approximations near $0$. For example, $1 + x + x^2/2$ approximates $e^x$ with error $O(x^3)$.

## Explanations

### L1 human

The exponential function equals the sum of its Taylor polynomials at $0$.
Adding more terms gives better approximations of $e^x$, for every $x$.
</document>

<document source="https://tamnd.github.io/math2/node/06K7SARZKR.md" tag="06K7SARZKR" kind="statement">
# The minimal polynomial divides every polynomial vanishing at the element

## Statement

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]], let $a \in L$ be algebraic over $K$, and let $m_a$ be its [[minimal polynomial|0GK1WB9R98]].
For every $f \in K[x]$ with $f(a) = 0$, the polynomial $m_a$ [[divides|04GNK7K4VP]] $f$ in $K[x]$.

## Remarks

The proof: [[polynomial division|0MES3KS0ZC]] gives $f = q m_a + r$ with $\deg r < \deg m_a$. Then $r(a) = 0$, so $r = 0$ by the minimality of $\deg m_a$.
Equivalently, the kernel of the evaluation map $K[x] \to L$, $f \mapsto f(a)$, is the ideal $(m_a)$.
If $a$ is a root of $f$ and $f$ is monic and irreducible over $K$, then $f = m_a$.

## Explanations

### L1 human

Every polynomial equation that the element satisfies comes from the minimal polynomial.
It is a multiple of the minimal polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/06KWGHDN2J.md" tag="06KWGHDN2J" kind="concept">
# Directional derivative

## Summary

The directional derivative of a function $f$ at a point $a$ in the direction of a vector $v$ is a rate of change. It is the rate of change of $f$ when its input moves from $a$ along $v$.
It is the derivative at $t = 0$ of the function $t \mapsto f(a + t v)$ of one variable.
The [[partial derivatives|09EGC6F8PT]] are the directional derivatives along the standard basis vectors.
</document>

<document source="https://tamnd.github.io/math2/node/06MG208MGR.md" tag="06MG208MGR" kind="concept">
# Inflection point

## Summary

An inflection point of a function is a point where the function changes from convex to concave, or from concave to convex. For twice differentiable functions, the second derivative vanishes there. The node [[inflection point|0Y0WS5BCJ7]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/06MJC8YDYR.md" tag="06MJC8YDYR" kind="concept">
# Solution of a linear recurrence

## Summary

A solution of a linear recurrence is a sequence that satisfies the recurrence for all indices. The solutions form a module, and they are determined by their first terms. The node [[solution of a linear recurrence|0ZV2YKKZ58]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/06MRNP4BNT.md" tag="06MRNP4BNT" kind="statement">
# Every spanning set contains a basis

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $S \subseteq V$ be a [[spanning set|088RYKAWK9]] of $V$.
Then there is a subset $B \subseteq S$ that is a [[basis|0KCQ9Y3QKA]] of $V$.

## Remarks

For finite $S$, remove from $S$, one at a time, a vector that is a linear combination of the other vectors.
When no such vector remains, the set is linearly independent and still spans $V$.
In general, take a maximal linearly independent subset of $S$, by Zorn's lemma.
For example, $\{(1, 0), (0, 1), (1, 1)\}$ spans $\R^2$ and contains the basis $\{(1, 0), (1, 1)\}$.

## Explanations

### L1 human

If a set of vectors reaches the whole space, then some part of it is a basis.
We throw away vectors that the others can already build.
When nothing redundant remains, the remaining vectors form a basis.
</document>

<document source="https://tamnd.github.io/math2/node/06N6JAZF44.md" tag="06N6JAZF44" kind="concept">
# Fermat prime

## Summary

A Fermat prime is a prime number of the form $2^{2^n} + 1$.
The [[definition of a Fermat prime|0N9E4G2P97]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/06PVXMY0TH.md" tag="06PVXMY0TH" kind="statement">
# Cauchy criterion for series

## Statement

Let $(a_n)$ be a real sequence.
The series $\sum a_n$ converges if and only if the following holds for each $\varepsilon > 0$.
There is a natural number $N$ such that
$$\left| \sum_{k=m+1}^{n} a_k \right| < \varepsilon \quad \text{for all natural numbers } n > m \ge N.$$

## Remarks

The sum in the condition is $s_n - s_m$, a difference of [[partial sums|0NB643K60Q]]. So the condition says that $(s_n)$ is a Cauchy sequence, and the result is the [[Cauchy criterion for sequences|0ZA4AX4MW3]].
With $n = m + 1$ it gives the [[term test|0VVY4H92G2]].
It proves that [[absolutely convergent series converge|0Y123X48GA]].

## Explanations

### L1 human

A series converges exactly when its tails become small.
Each block of consecutive terms far out in the series must have a small sum.
</document>

<document source="https://tamnd.github.io/math2/node/06Q1EFVH0Q.md" tag="06Q1EFVH0Q" kind="statement">
# Dimension of the annihilator

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a subspace of $V$.
Then
$$\dim W + \dim W^0 = \dim V,$$
where $W^0 \subseteq V^*$ is the [[annihilator|0ASHNYF7AG]] of $W$.

## Remarks

A proof extends a basis $b_1, \dots, b_k$ of $W$ to a basis $b_1, \dots, b_n$ of $V$.
Then the dual basis forms $b_{k+1}^*, \dots, b_n^*$ are a basis of $W^0$.
Another proof uses the isomorphism $W^0 \cong (V / W)^*$.
With the double dual isomorphism, it gives $(W^0)^0 = W$ in finite dimensions.

## Explanations

### L1 human

The more directions a subspace has, the fewer independent linear forms vanish on it.
In $\R^3$, a line is killed by a plane of forms, and a plane is killed by a line of forms.
The two dimensions always add up to the dimension of the space.
</document>

<document source="https://tamnd.github.io/math2/node/06QCTRFBM0.md" tag="06QCTRFBM0" kind="statement">
# cosh^2 - sinh^2 = 1

## Statement

For every real $x$,
$$\cosh^2 x - \sinh^2 x = 1.$$

## Remarks

The left side factors as $(\cosh x - \sinh x)(\cosh x + \sinh x) = e^{-x} e^{x} = 1$.
So the point $(\cosh t, \sinh t)$ lies on the right branch of the hyperbola $x^2 - y^2 = 1$.
Dividing by $\cosh^2 x$ gives $1 - \tanh^2 x = \operatorname{sech}^2 x$. Mathlib states the identity as `Real.cosh_sq_sub_sinh_sq`.

## Explanations

### L1 human

This is the hyperbolic version of $\cos^2 x + \sin^2 x = 1$, with a minus sign.
It follows directly from the definitions with $e^x$ and $e^{-x}$.
</document>

<document source="https://tamnd.github.io/math2/node/06QYRTX9KQ.md" tag="06QYRTX9KQ" kind="concept">
# P-group

## Summary

For a prime number $p$, a $p$-group is a group in which the order of each element is a power of $p$.
A finite group is a $p$-group if and only if its order is a power of $p$.
</document>

<document source="https://tamnd.github.io/math2/node/06RGT4V6Y6.md" tag="06RGT4V6Y6" kind="definition">
# Commutator of two elements

## Definition

Let $G$ be a [[group|G5S1M9]] and let $g, h \in G$.
The commutator of $g$ and $h$ is the element $[g, h] = g^{-1} h^{-1} g h$ of $G$.

## Remarks

The equation $gh = hg [g, h]$ holds, so $[g, h] = e$ if and only if $gh = hg$.
Some authors use the convention $[g, h] = g h g^{-1} h^{-1}$ instead.
The inverse of $[g, h]$ is $[h, g]$.

## Explanations

### L1 human

The commutator measures how far two elements are from commuting.
It is the correction factor that you need when you swap $g$ and $h$ in a product.
In an abelian group, every commutator is the identity.
</document>

<document source="https://tamnd.github.io/math2/node/06RH4XAGRY.md" tag="06RH4XAGRY" kind="statement">
# Units of the Gaussian integers

## Statement

The [[units|019NFGFPR5]] of the [[Gaussian integers|0K5W69AC0J]] $\mathbb{Z}[i]$ are $1$, $-1$, $i$ and $-i$.

## Remarks

The proof uses the [[norm|03AN1V4ZFJ]]: if $u v = 1$, then $N(u) N(v) = 1$, so $N(u) = 1$. The only solutions of $a^2 + b^2 = 1$ in integers are $(\pm 1, 0)$ and $(0, \pm 1)$.
Conversely, an element of norm $1$ is a unit, because $(a + b i)(a - b i) = 1$.
The group of units is cyclic of order $4$, generated by $i$.

## Explanations

### L1 human

A unit must have norm $1$, so it lies on the unit circle.
The only points with integer coordinates on the unit circle are $1$, $i$, $-1$ and $-i$.
</document>

<document source="https://tamnd.github.io/math2/node/06RJWZCG95.md" tag="06RJWZCG95" kind="statement">
# Number of invertible residues modulo n

## Statement

Let $n$ be a positive integer.
The number of invertible elements of the ring $\mathbb{Z}/n\mathbb{Z}$ of the [[integers modulo n|01N3AYAA6N]] is $\varphi(n)$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

By the [[existence of inverses modulo n|0M67FWZWPQ]], the class $\overline{a}$ is invertible if and only if $\gcd(a, n) = 1$. The classes of $1, \dots, n$ that are coprime to $n$ give $\varphi(n)$ classes.
The invertible elements form a group under multiplication, the group of units $(\mathbb{Z}/n\mathbb{Z})^\times$. It has order $\varphi(n)$.
For example, $(\mathbb{Z}/8\mathbb{Z})^\times = \{ \overline{1}, \overline{3}, \overline{5}, \overline{7} \}$ has $\varphi(8) = 4$ elements.

## Explanations

### L1 human

Modulo $n$, you can divide by exactly $\varphi(n)$ of the $n$ residues.
For $n = 9$, these are the residues of $1, 2, 4, 5, 7, 8$, and $\varphi(9) = 6$.
</document>

<document source="https://tamnd.github.io/math2/node/06RW4RKWH7.md" tag="06RW4RKWH7" kind="concept">
# Completion of a metric space

## Summary

The completion of a metric space is a complete metric space that contains an isometric copy of the space as a dense subset.
It is unique up to an isometric isomorphism that fixes the copy.
</document>

<document source="https://tamnd.github.io/math2/node/06S4WGSEQA.md" tag="06S4WGSEQA" kind="definition">
# Disjunctive normal form

## Definition

A literal is a propositional variable $p$ or its negation $\neg p$.
A formula of propositional logic is in disjunctive normal form if it is a disjunction of one or more conjunctions of one or more literals.

## Remarks

Example: $(p \wedge \neg q) \vee (q \wedge r) \vee \neg p$ is in disjunctive normal form.
Every formula is logically equivalent to a formula in disjunctive normal form. For a formula that is not a contradiction, take one conjunction for each row of the truth table where it is true.
A consequence: the connectives $\wedge$, $\vee$ and $\neg$ are functionally complete.

## Explanations

### L1 human

A formula in disjunctive normal form is an or of ands of simple conditions.
It lists all the cases in which the formula is true.
</document>

<document source="https://tamnd.github.io/math2/node/06SCB3R6BN.md" tag="06SCB3R6BN" kind="definition">
# Graph homomorphism

## Definition

Let $G$ and $H$ be [[graphs|0JD5C82QMB]] with vertex sets $V(G)$ and $V(H)$.
A graph homomorphism from $G$ to $H$ is a map $f : V(G) \to V(H)$ such that $f(u) \sim f(v)$ in $H$ whenever $u \sim v$ in $G$.

## Remarks

A homomorphism maps edges to edges. It may map non-adjacent vertices to adjacent ones, and it may map several vertices to one vertex.
A homomorphism from $G$ to the complete graph $K_k$ is the same as a proper [[colouring|0874138YP6]] of $G$ with $k$ colours.
A bijective homomorphism whose inverse is also a homomorphism is a [[graph isomorphism|00CT0PTH74]].

## Explanations

### L1 human

A graph homomorphism is a map of vertices that sends each edge to an edge.
For example, mapping the vertices of a square alternately to the two ends of one edge is a homomorphism from $C_4$ to $K_2$.
</document>

<document source="https://tamnd.github.io/math2/node/06SESV7GQY.md" tag="06SESV7GQY" kind="statement">
# Continuous functions are Riemann integrable

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous.
Then $f$ is Riemann integrable on $[a, b]$.

## Remarks

By the [[Heine–Cantor theorem|0W0TZDRF9C]], $f$ is uniformly continuous. Choose $\delta$ for $\varepsilon/(b - a)$. For a partition of mesh less than $\delta$, $U(f, P) - L(f, P) < \varepsilon$. Then [[Riemann's criterion|017DH8WXTE]] applies.
More generally, a bounded function with finitely many discontinuities is Riemann integrable.

## Explanations

### L1 human

On short pieces of the interval, a continuous function changes very little.
So the upper and lower sums are close for fine partitions.
</document>

<document source="https://tamnd.github.io/math2/node/06SKCHDQT5.md" tag="06SKCHDQT5" kind="definition">
# Continuously differentiable function

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$.
The function $f$ is continuously differentiable, or of class $C^1$, if $f$ is [[differentiable|0A61B60B5R]] on $I$ and $f'$ is [[continuous|0P27W9RW0G]] on $I$.
More generally, for $k \ge 1$, $f$ is of class $C^k$ if its [[higher derivative|084NGSTWM7]] $f^{(k)}$ exists and is continuous on $I$.
The class $C^0$ consists of the continuous functions.

## Remarks

Each $C^{k+1}$ function is $C^k$.
A differentiable function need not be $C^1$. See [[x^2 sin(1/x)|0WDAKKN4NC]].
The function $x |x|$ is $C^1$ but not twice differentiable at $0$.
Mathlib defines the class as `ContDiff ℝ k f`.

## Explanations

### L1 human

A $C^1$ function has a slope at each point, and the slope changes without jumps.
The class $C^k$ asks the same of the first $k$ derivatives.
</document>

<document source="https://tamnd.github.io/math2/node/06T0HS14PP.md" tag="06T0HS14PP" kind="definition">
# Tangent function

## Definition

The tangent function is defined by
$$\tan x = \frac{\sin x}{\cos x}$$
for all real $x$ with $\cos x \ne 0$, that is, for $x \ne \frac{\pi}{2} + k \pi$ with $k \in \mathbb{Z}$.

## Remarks

The function is odd and periodic with period $\pi$.
On $(-\pi/2, \pi/2)$ it is strictly increasing, and it maps this interval onto $\mathbb{R}$.
The related functions are $\cot x = \cos x / \sin x$, $\sec x = 1 / \cos x$ and $\csc x = 1 / \sin x$. Mathlib defines `Real.tan` with the value $0$ where $\cos x = 0$.

## Explanations

### L1 human

The tangent of an angle is the slope of the line through the origin at that angle.
It is undefined where this line is vertical.
</document>

<document source="https://tamnd.github.io/math2/node/06TK51263Q.md" tag="06TK51263Q" kind="concept">
# Sum of divisors

## Summary

The sum of divisors $\sigma(n)$ of a positive integer $n$ is the sum of all its positive divisors.
The [[definition of the sum of divisors|009QCP5T3Z]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/06WHP8SE98.md" tag="06WHP8SE98" kind="concept">
# Rayleigh quotient

## Summary

The Rayleigh quotient of a Hermitian matrix $A$ and a nonzero vector $x$ is the number $x^* A x / x^* x$.
It lies between the least and the greatest eigenvalue of $A$.
</document>

<document source="https://tamnd.github.io/math2/node/06WHRARZKY.md" tag="06WHRARZKY" kind="statement">
# Existence of Laurent expansions on an annulus

## Statement

Let $c \in \mathbb{C}$, let $0 \le r < R \le \infty$, and let $A = \{z : r < |z - c| < R\}$ be an [[annulus|0AJ0YVK743]].
Let $f : A \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $A$.
Then there are unique complex numbers $(a_n)_{n \in \mathbb{Z}}$ such that the [[Laurent series|0X3XD2ATB0]] $\sum_{n \in \mathbb{Z}} a_n (z - c)^n$ converges to $f(z)$ for each $z \in A$.
For each $\rho$ with $r < \rho < R$ and each $n \in \mathbb{Z}$,
$$a_n = \frac{1}{2 \pi i} \oint_{|z - c| = \rho} \frac{f(z)}{(z - c)^{n+1}} \, dz.$$
The series converges absolutely on $A$ and uniformly on each closed annulus $\rho_1 \le |z - c| \le \rho_2$ with $r < \rho_1 \le \rho_2 < R$.

## Remarks

Uniqueness means: each Laurent series that converges to $f$ on $A$ has these coefficients. So we can find them by any method, for example with known power series.
For example, on $0 < |z| < 1$, $\frac{1}{z(1 - z)} = \frac{1}{z} + 1 + z + z^2 + \cdots$. On $|z| > 1$, the same function is $-\frac{1}{z^2} - \frac{1}{z^3} - \cdots$.
If $f$ is holomorphic on the whole disc $|z - c| < R$, then $a_n = 0$ for $n < 0$, and the Laurent series is the Taylor series.

## Explanations

### L1 human

On a ring around $c$, each holomorphic function is a sum of powers of $z - c$, with negative and positive exponents.
The coefficients come from circle integrals, as for Taylor series. The same function can have different Laurent series on different rings.
</document>

<document source="https://tamnd.github.io/math2/node/06WRV8HFDY.md" tag="06WRV8HFDY" kind="statement">
# Converse of Wilson's theorem

## Statement

Let $n$ be an integer with $n > 1$.
If $(n - 1)! \equiv -1 \pmod n$, then $n$ is a [[prime number|0SN0MAEKYX]].

## Remarks

With [[Wilson's theorem|03KGGKTB2E]], this gives: an integer $n > 1$ is prime if and only if $(n - 1)! \equiv -1 \pmod n$.
The proof: if $n = a b$ with $1 < a < n$, then $a$ divides $(n - 1)!$. A congruence $(n - 1)! \equiv -1 \pmod n$ would give $a \mid 1$, which is false.
In fact $(n - 1)! \equiv 0 \pmod n$ for each composite $n > 4$, and $3! \equiv 2 \pmod 4$.
This primality test is not practical, because $(n - 1)!$ needs about $n$ multiplications.

## Explanations

### L1 human

For a composite number $n$, a factor of $n$ appears in the product $1 \cdot 2 \cdots (n - 1)$.
So the product cannot leave the remainder $n - 1$. For example, $5! = 120 \equiv 0 \pmod 6$.
</document>

<document source="https://tamnd.github.io/math2/node/06X5PF6YFP.md" tag="06X5PF6YFP" kind="problem">
# Buffon's needle problem

## Problem

A floor has parallel lines at distance $t > 0$ apart. A needle of length $l$ with $0 < l \le t$ falls at random on the floor.
"At random" means this.
The distance $x$ of the centre of the needle from the nearest line is uniform on $[0, t/2]$.
The acute angle $\theta$ between the needle and the lines is uniform on $[0, \pi/2]$.
The variables $x$ and $\theta$ are independent.
Find the probability that the needle crosses a line.

## Remarks

The answer is $\dfrac{2l}{\pi t}$. The needle crosses a line exactly when $x \le \frac{l}{2} \sin \theta$, and the integral of this region gives the answer.
For a needle with $l = t$, the probability is $2/\pi \approx 0.637$.
The answer contains $\pi$, so repeated throws give a random estimate of $\pi$.
For a long needle with $l > t$, the answer is a different formula.

## History

Georges-Louis Leclerc, Comte de Buffon, posed the problem in the 18th century. It was one of the first problems of geometric probability.

## Explanations

### L1 human

The needle crosses a line when it lands close to a line and at a steep angle.
The chance is the area of the "good" pairs (distance, angle) divided by the area of all pairs.
A sine function appears in this area. That is how $\pi$ comes into the answer.
</document>

<document source="https://tamnd.github.io/math2/node/06XF7BCFFJ.md" tag="06XF7BCFFJ" kind="definition">
# Simple continued fraction

## Definition

A simple continued fraction is a [[continued fraction|0BK6JBK58W]] whose partial numerators are all $1$.
Its first term $a_0$ is an integer, and its later terms $a_1, a_2, \dots$ are positive integers:
$$[a_0; a_1, a_2, \dots] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cdots}}$$
The sequence of terms can be finite or infinite.

## Remarks

The $a_n$ are called the partial quotients.
For example, $[2; 3, 4] = 2 + \frac{1}{3 + 1/4} = \frac{30}{13}$.
An infinite simple continued fraction always converges, by the [[convergence of the convergents|0E70M92FE3]].
Each real number has a [[continued fraction expansion|0SJ7Z6VKWK]] of this form.

## Explanations

### L1 human

A simple continued fraction is a list of integers $[a_0; a_1, a_2, \dots]$ that encodes a number.
To read it, start at the end, take a reciprocal, add the next term, and repeat.
</document>

<document source="https://tamnd.github.io/math2/node/06XY4X9430.md" tag="06XY4X9430" kind="definition">
# Decimal expansion

## Definition

Let $x$ be a real number with $x \ge 0$.
A decimal expansion of $x$ is a natural number $a_0$ and a sequence $(d_k)_{k \ge 1}$ of digits $d_k \in \{0, 1, \dots, 9\}$ with
$$x = a_0 + \sum_{k=1}^{\infty} \frac{d_k}{10^k}.$$
The notation is $x = a_0.d_1 d_2 d_3 \dots$, with $a_0$ in base $10$.

## Remarks

The series always converges, because its partial sums are increasing and bounded above by $a_0 + 1$.
A number can have two expansions. This happens exactly when one expansion ends in infinitely many $0$ and the other in infinitely many $9$, as for [[0.999... and 1|0NN5JEMTVE]].
For a negative number $x$, one writes $-$ followed by an expansion of $-x$.

## Explanations

### L1 human

The digits after the point say how many tenths, hundredths, thousandths, and so on, the number has.
For example, $3.14$ means $3 + 1/10 + 4/100$.
For a number such as $\pi$, the digits go on forever.
</document>

<document source="https://tamnd.github.io/math2/node/06YMXJMXYS.md" tag="06YMXJMXYS" kind="definition">
# Unique factorisation domain

## Definition

A unique factorisation domain is an [[integral domain|09S3PFBEPM]] $R$ with these two properties.

1. Every nonzero element $a \in R$ that is not a unit is a product $a = p_1 p_2 \cdots p_n$ of [[irreducible elements|052SDEPZH7]], with $n \ge 1$.
2. The factorization is unique up to order and units. If $p_1 \cdots p_n = q_1 \cdots q_m$ with all $p_i, q_j$ irreducible, then $n = m$. Also, there is a permutation $\sigma$ of $\{1, \dots, n\}$ such that $p_i$ and $q_{\sigma(i)}$ are [[associates|053E86NWVK]] for each $i$.

## Remarks

The integers are a unique factorisation domain: this is the fundamental theorem of arithmetic.
Every [[principal ideal domain is a unique factorisation domain|055HT85T3S]], and so is every [[polynomial ring over a unique factorisation domain|0D6KH8Y59N]].
The ring $\mathbb{Z}[\sqrt{-5}]$ is [[not a unique factorisation domain|0BSJYCG7NM]].
In a unique factorisation domain, [[irreducible elements are prime|0QRCFZKXH8]].

## Explanations

### L1 human

In the integers, $60 = 2 \cdot 2 \cdot 3 \cdot 5$, and every other factorization into primes uses the same primes, up to order and signs.
A unique factorisation domain is a ring where this works for every nonzero nonunit.
Units like $-1$ do not count as a different factorization.
</document>

<document source="https://tamnd.github.io/math2/node/06YSJVR356.md" tag="06YSJVR356" kind="definition">
# Geometric multiplicity

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Let $\lambda$ be an eigenvalue of $T$.
The *geometric multiplicity* of $\lambda$ is $\dim E_\lambda$, the [[dimension|03GSZ8RTZJ]] of the [[eigenspace|0G0RFPG8HR]] $E_\lambda = \ker(T - \lambda \id_V)$.

## Remarks

It is the maximal number of linearly independent eigenvectors for $\lambda$.
It satisfies $1 \le \dim E_\lambda \le m$, where $m$ is the [[algebraic multiplicity|0FY42N02DV]] of $\lambda$.
For $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the eigenvalue $1$ has geometric multiplicity $1$ and algebraic multiplicity $2$.
A matrix is [[diagonalisable|07SQCNV233]] if and only if $p_A$ splits and the two multiplicities agree for each eigenvalue.

## Explanations

### L1 human

The geometric multiplicity counts the independent directions that the map stretches by $\lambda$.
It can be smaller than the algebraic multiplicity.
When it is smaller, the matrix cannot be diagonalised.
</document>

<document source="https://tamnd.github.io/math2/node/06ZARTQTQE.md" tag="06ZARTQTQE" kind="statement">
# Continuous image of a closed interval is a closed interval

## Statement

Let $a \le b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous.
Then $f([a, b]) = [m, M]$, where $m$ and $M$ are the minimum and the maximum of $f$ on $[a, b]$.

## Remarks

The [[extreme value theorem|0WS1A8E2P9]] gives $m$ and $M$ as values of $f$. The [[intermediate value theorem|00KAFBR32E]] gives all values in between.
The image can be a single point, when $f$ is constant.
Mathlib states it as `ContinuousOn.image_Icc`.

## Explanations

### L1 human

A continuous function maps a closed bounded interval onto a closed bounded interval.
The graph reaches a lowest and a highest point, and it takes every height in between.
</document>

<document source="https://tamnd.github.io/math2/node/06ZGPYTWK8.md" tag="06ZGPYTWK8" kind="concept">
# Contour integral

## Summary

A contour integral is the integral of a complex function along a path in the complex plane, see the [[definition|00FWNP9TXY]].
Contour integration is the method of computing integrals with the theorems of complex analysis about such integrals.
</document>

<document source="https://tamnd.github.io/math2/node/070PFSYJJR.md" tag="070PFSYJJR" kind="concept">
# Argument of a complex number

## Summary

The argument of a nonzero complex number $z$ is the angle from the positive real axis to $z$ in the [[complex plane|0CTE7A31XQ]].
It is determined up to a multiple of $2\pi$.
The usual choice in $(-\pi, \pi]$ is the [[principal value|0SQSQP748R]].
</document>

<document source="https://tamnd.github.io/math2/node/070R982G4B.md" tag="070R982G4B" kind="concept">
# Finite set

## Summary

A finite set is a set with a natural number of elements.
</document>

<document source="https://tamnd.github.io/math2/node/070TXS9AXM.md" tag="070TXS9AXM" kind="statement">
# Geometric formula for the dot product

## Statement

Let $n$ be a positive integer, and let $u$ and $v$ be nonzero vectors in $\mathbb{R}^n$.
Let $u \cdot v = u_1 v_1 + \dots + u_n v_n$ be their dot product, let $\|u\|$ and $\|v\|$ be their [[lengths|0XKBKGTCPC]], and let $\theta$ be the [[angle between them|0ZEHZY85E6]].
Then
$$u \cdot v = \|u\| \, \|v\| \cos \theta.$$

## Remarks

When the angle is defined by its cosine, the formula follows from the definition. The content is that this angle is the angle of elementary geometry. In the plane, write $u = \|u\| (\cos \alpha, \sin \alpha)$ and $v = \|v\| (\cos \beta, \sin \beta)$. Then $u \cdot v = \|u\| \, \|v\| \cos(\alpha - \beta)$ by the addition formula for the cosine.
The formula also follows from the law of cosines in the triangle with the vertices $0$, $u$ and $v$.
If $u = 0$ or $v = 0$, then both sides are $0$ for each value of $\theta$.

## Explanations

### L1 human

The dot product is large and positive when two vectors point in nearly the same direction. It is zero when they are perpendicular, and negative when they point in nearly opposite directions. For unit vectors $u$ and $v$, the dot product is just the cosine of the angle between them.
</document>

<document source="https://tamnd.github.io/math2/node/0717XXPWM6.md" tag="0717XXPWM6" kind="definition">
# Linear isometry

## Definition

Let $V$ and $W$ be [[normed vector spaces|03J5RAKB30]] over $\R$ or $\C$.
A *linear isometry* from $V$ to $W$ is a [[linear map|0NRYHJPQG2]] $T \colon V \to W$ with
$$\norm{T(v)} = \norm{v} \qquad \text{for all } v \in V.$$

## Remarks

A linear isometry is injective, because $T(v) = 0$ implies $\norm{v} = 0$.
It keeps distances: $\norm{T(u) - T(v)} = \norm{u - v}$.
Between inner product spaces, it keeps inner products, see [[the next statement|0R7VXTYHYZ]].
Between spaces of the same finite dimension, a linear isometry is bijective.

## Explanations

### L1 human

A linear isometry is a linear map that keeps all lengths.
Rotations and reflections of the plane are examples.
A map that keeps lengths cannot squeeze anything to zero.
</document>

<document source="https://tamnd.github.io/math2/node/071DDFVG4J.md" tag="071DDFVG4J" kind="definition">
# Abelian group

## Definition

An abelian group is a [[group|G5S1M9]] $(G, \ast)$ whose operation is [[commutative|099E72KTF8]].
Thus $a \ast b = b \ast a$ for all $a, b \in G$.

## Remarks

The integers, the rationals and the real numbers with addition are abelian groups.
The operation of an abelian group is often written $+$, with identity $0$ and inverse $-a$.
The group of invertible $2 \times 2$ real matrices under multiplication is not abelian.

## Explanations

### L1 human

In an abelian group, the order of two elements in a product does not matter.
Addition of numbers is the model: $a + b = b + a$.
The name honours Niels Henrik Abel.
</document>

<document source="https://tamnd.github.io/math2/node/072D5G4SK6.md" tag="072D5G4SK6" kind="statement">
# Congruence is compatible with multiplication

## Statement

Let $n$ be a positive integer, and let $a$, $b$, $c$ and $d$ be integers.
If $a \equiv b \pmod n$ and $c \equiv d \pmod n$, then $a c \equiv b d \pmod n$.

## Remarks

The proof: $a c - b d = a (c - d) + d (a - b)$, and $n$ divides both terms.
With [[compatibility with addition|00FWBN7F0Y]], this result shows that the multiplication of [[integers modulo n|01N3AYAA6N]] is well defined.
By induction, it gives [[compatibility with powers|0J5VGJYYC3]].

## Explanations

### L1 human

To find the remainder of a product, you can multiply the remainders and reduce again.
For example, the last digit of $27 \cdot 43$ is the last digit of $7 \cdot 3 = 21$, that is, $1$. This is the case $n = 10$.
</document>

<document source="https://tamnd.github.io/math2/node/072G59HW59.md" tag="072G59HW59" kind="concept">
# Maximal element

## Summary

A maximal element of a partially ordered set is an element with no strictly larger element.
</document>

<document source="https://tamnd.github.io/math2/node/072J68M6HX.md" tag="072J68M6HX" kind="concept">
# Comparison of topologies

## Summary

Two topologies on one set can be compared by inclusion. The finer topology has more open sets and the coarser topology has fewer open sets.
</document>

<document source="https://tamnd.github.io/math2/node/072W67KFYK.md" tag="072W67KFYK" kind="definition">
# Formal power series

## Definition

Let $R$ be a commutative ring.
A formal power series in one variable $X$ over $R$ is a sequence $(a_n)_{n \ge 0}$ of elements of $R$, written $\sum_{n \ge 0} a_n X^n$.
The set of these series is written $R[\![X]\!]$.

Addition is termwise: $\sum a_n X^n + \sum b_n X^n = \sum (a_n + b_n) X^n$.
Multiplication is the Cauchy product: $\left( \sum a_n X^n \right) \left( \sum b_n X^n \right) = \sum_n \left( \sum_{i+j=n} a_i b_j \right) X^n$.
With these operations $R[\![X]\!]$ is a commutative ring, and it contains the polynomial ring $R[X]$.

## Remarks

The word formal means that no convergence is required. A series is a sequence of coefficients, not a function.
A series $\sum a_n X^n$ is a unit in $R[\![X]\!]$ if and only if $a_0$ is a unit in $R$.
The [[coefficient|0M98XYVX6S]] of $X^n$ in a series $f$ is written $[X^n] f$.

## Explanations

### L1 human

A formal power series is like a polynomial that may go on forever.
You add and multiply it like a polynomial, and you never substitute a number for $X$, so the question of convergence does not arise.
</document>

<document source="https://tamnd.github.io/math2/node/0734RSVF2V.md" tag="0734RSVF2V" kind="statement">
# Limit of a product of sequences

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers.
If $a_n \to a$ and $b_n \to b$ as $n \to \infty$, then $a_n b_n \to a b$ as $n \to \infty$.

## Remarks

The proof uses the identity $a_n b_n - a b = a_n (b_n - b) + b (a_n - a)$. The sequence $(a_n)$ is bounded, because [[convergent sequences are bounded|0S4JF619V7]].
With a constant sequence $a_n = c$, it gives $c\, b_n \to c\, b$.
By induction, it gives $a_n^k \to a^k$ for each natural number $k$.

## Explanations

### L1 human

We may multiply the limits to get the limit of the product.
For example, $(1 + 1/n)(2 - 1/n) \to 1 \cdot 2 = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/0736AXH2ME.md" tag="0736AXH2ME" kind="statement">
# Cyclic groups of the same order are isomorphic

## Statement

Let $G$ and $H$ be [[cyclic groups|0KVGH0J5B3]] of the same [[order|0B85S2V2DR]], finite or infinite.
Then $G$ and $H$ are isomorphic: there is a bijection $\varphi \colon G \to H$ with $\varphi(ab) = \varphi(a) \varphi(b)$ for all $a, b \in G$.

## Remarks

If $g$ generates $G$ and $h$ generates $H$, an isomorphism is $\varphi(g^k) = h^k$ for $k \in \mathbb{Z}$.
This map is well defined because $g^a = g^b$ exactly when $h^a = h^b$, by [[the order of a cyclic subgroup|031M8HX7DA]].
Thus each cyclic group of order $n$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$, and each infinite cyclic group is isomorphic to $\mathbb{Z}$.

## Explanations

### L1 human

Up to the names of the elements, there is only one cyclic group of each size.
To match two of them, send the generator of one to the generator of the other, and send each power to the same power.
For example, the rotations of a hexagon and the integers modulo $6$ are the same group in two costumes.
</document>

<document source="https://tamnd.github.io/math2/node/073C80EC63.md" tag="073C80EC63" kind="statement">
# Third isomorphism theorem for rings

## Statement

Let $R$ be a [[ring|0894J980RK]] and let $I \subseteq J$ be two-sided ideals of $R$.
Then $J/I = \{a + I : a \in J\}$ is a two-sided ideal of the [[quotient ring|0VFW4Y69BT]] $R/I$.
The map $(R/I)/(J/I) \to R/J$, $(a + I) + J/I \mapsto a + J$, is a [[ring isomorphism|0B6QC0C0G1]].

## Remarks

The proof applies the [[first isomorphism theorem|0ZMYX7GMH0]] to the map $R/I \to R/J$, $a + I \mapsto a + J$. It is surjective, and its kernel is $J/I$.
For example, $6\mathbb{Z} \subseteq 2\mathbb{Z}$, and $(\mathbb{Z}/6\mathbb{Z})/(2\mathbb{Z}/6\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$.

## Explanations

### L1 human

You can make a quotient in two steps or in one step.
Dividing first by $I$ and then by the image of $J$ gives the same ring as dividing directly by $J$.
</document>

<document source="https://tamnd.github.io/math2/node/073E4XC016.md" tag="073E4XC016" kind="statement">
# Associativity of matrix multiplication

## Statement

Let $R$ be a ring and let $m, n, p, q \ge 0$ be integers.
Let $A$ be an $m \times n$ matrix, $B$ an $n \times p$ matrix and $C$ a $p \times q$ matrix over $R$.
Then $(AB)C = A(BC)$, where the products are [[matrix products|032Q9FB8MT]].

## Remarks

Both sides have entry $(i, l)$ equal to $\sum_{j=1}^{n} \sum_{k=1}^{p} a_{ij} b_{jk} c_{kl}$.
Because of this property, the product $ABC$ needs no parentheses.
The property also follows from the associativity of the composition of linear maps.

## Explanations

### L1 human

When we multiply three matrices, we may multiply the first two first, or the last two first.
The result is the same in both cases.
This is like $(2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4)$ for numbers.
</document>

<document source="https://tamnd.github.io/math2/node/073JKP99K2.md" tag="073JKP99K2" kind="concept">
# Tangent plane and normal vector of a surface

## Summary

The tangent plane of a [[surface|0ZG9QF40HE]] at a point is the plane that best approximates the surface near the point.
A normal vector is a nonzero vector perpendicular to the tangent plane.
</document>

<document source="https://tamnd.github.io/math2/node/073MWACF1K.md" tag="073MWACF1K" kind="concept">
# Order of a pole

## Summary

The order of a pole $c$ of $f$ is the integer $m \ge 1$ with the following property. The function $(z - c)^m f(z)$ extends to a holomorphic function near $c$ that is not $0$ at $c$, see the [[definition|0YK27TJ4WF]].
</document>

<document source="https://tamnd.github.io/math2/node/073Q8Z94KG.md" tag="073Q8Z94KG" kind="statement">
# Kolmogorov's zero–one law

## Statement

Let $X_1, X_2, \dots$ be [[independent|0XCE070V8B]] random variables on a probability space $(\Omega, \mathcal{F}, P)$.
Let $\mathcal{T} = \bigcap_{n=1}^{\infty} \sigma(X_n, X_{n+1}, \dots)$ be the tail σ-algebra, where $\sigma(X_n, X_{n+1}, \dots)$ is the smallest [[σ-algebra|04WSW8T3SY]] that makes $X_m$ measurable for all $m \ge n$.
Then $P(A) \in \{0, 1\}$ for each $A \in \mathcal{T}$.

## Remarks

An event is in $\mathcal{T}$ if changing finitely many of the $X_n$ cannot change whether it occurs.
Examples of tail events: "the series $\sum_n X_n$ converges" and "$\limsup_n (X_1 + \dots + X_n)/n \le c$" for a real $c$.
Proof idea: $\mathcal{T}$ is [[independent|09T8ZHST1V]] of $\sigma(X_1, \dots, X_n)$ for each $n$, hence of $\sigma(X_1, X_2, \dots)$, which contains $\mathcal{T}$. So each $A \in \mathcal{T}$ is independent of itself, and $P(A) = P(A)^2$.
The theorem says that a tail event is almost sure or almost impossible. It does not say which of the two cases holds.

## History

The theorem is named after Andrey Kolmogorov.

## Explanations

### L1 human

Some events depend only on the "far end" of an infinite sequence of independent trials, such as "the averages converge".
For such an event, the probability cannot be $1/2$ or any value strictly between $0$ and $1$: it is always $0$ or $1$.
</document>

<document source="https://tamnd.github.io/math2/node/074482Q7CD.md" tag="074482Q7CD" kind="statement">
# Correspondence between equivalence relations and partitions

## Statement

Let $A$ be a set.
Send each [[equivalence relation|0SGTMBEWRF]] $\sim$ on $A$ to its set of [[equivalence classes|082RMRSY26]] $A/{\sim}$.
This map is a [[bijection|0SNQEF7Z53]] from the set of equivalence relations on $A$ to the set of [[partitions|09CCE86A6D]] of $A$.
Its inverse sends a partition $P$ to the relation $\sim_P$ with $a \sim_P b$ when $a$ and $b$ are in the same block of $P$.

## Remarks

The relation $\sim_P$ is an equivalence relation, and its classes are the blocks of $P$.
A smaller equivalence relation, one that relates fewer pairs, gives a finer partition.
The number of partitions of a set with $n$ elements is the Bell number $B_n$. For example, $B_3 = 5$.

## Explanations

### L1 human

An equivalence relation and a partition are two descriptions of the same thing.
From the relation you get the boxes, and from the boxes you get the relation "is in the same box as".
</document>

<document source="https://tamnd.github.io/math2/node/0744CN4MDA.md" tag="0744CN4MDA" kind="definition">
# Symmetric bilinear form

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A [[bilinear form|0SQZMXB6GP]] $B$ on $V$ is *symmetric* when
$$B(u, v) = B(v, u) \qquad \text{for all } u, v \in V.$$

## Remarks

In each basis, $B$ is symmetric if and only if its matrix is a [[symmetric matrix|0VR671PHE5]].
The dot product on $\R^n$ is symmetric.
If the characteristic of $F$ is not $2$, symmetric bilinear forms correspond one to one with [[quadratic forms|0RBQS1JJCR]].

## Explanations

### L1 human

A symmetric bilinear form gives the same value for $(u, v)$ and for $(v, u)$.
Inner products on real spaces are the main examples.
Their matrices are symmetric.
</document>

<document source="https://tamnd.github.io/math2/node/074E9A8ZJK.md" tag="074E9A8ZJK" kind="concept">
# Separable equation

## Summary

A separable equation is a first-order equation $y' = g(x) h(y)$ whose right side is a product of a function of $x$ and a function of $y$.
One solves it by separation of variables: divide by $h(y)$ and integrate both sides. See the [[definition|0V4E623THY]].
</document>

<document source="https://tamnd.github.io/math2/node/074P7C85XH.md" tag="074P7C85XH" kind="concept">
# Gaussian integer

## Summary

A Gaussian integer is a complex number $a + bi$ with integers $a$ and $b$. The Gaussian integers form the ring $\mathbb{Z}[i]$.
The [[definition of a Gaussian integer|0B268GVX95]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0753Y2X5K6.md" tag="0753Y2X5K6" kind="statement">
# One-sided inverse of a square matrix is two-sided

## Statement

Let $F$ be a field and let $A$ and $B$ be $n \times n$ matrices over $F$.
If $BA = I_n$, then $AB = I_n$, so $A$ is [[invertible|0EB9N0S33C]] and $A^{-1} = B$.
The same holds with the roles of $A$ and $B$ exchanged.

## Remarks

The statement also holds over each commutative ring, by the [[determinant|00KDZRW4C0]]: $BA = I_n$ gives $\det B \det A = 1$.
For non-square matrices it fails: a $2 \times 1$ matrix can have a left inverse but never a right inverse.
A proof over a field uses [[rank–nullity|0W4FSW1TVB]]: $BA = I_n$ makes $x \mapsto Ax$ injective, hence surjective.

## Explanations

### L1 human

For square matrices, we only need to check one of $AB = I$ and $BA = I$.
The other one then holds automatically.
This saves half of the work when we check an inverse.
</document>

<document source="https://tamnd.github.io/math2/node/075C629XTH.md" tag="075C629XTH" kind="definition">
# Simply connected domain

## Definition

Let $X \subseteq \mathbb{R}^n$.
The set $X$ is simply connected if $X$ is path-connected and each continuous [[closed curve|0D032NSZKY]] $c \colon [0, 1] \to X$ has this property. There are a point $p \in X$ and a continuous map $H \colon [0, 1] \times [0, 1] \to X$ such that
$$H(t, 0) = c(t), \qquad H(t, 1) = p, \qquad H(0, s) = H(1, s) \quad \text{for all } s, t \in [0, 1].$$
An open simply connected set is a simply connected domain.

## Remarks

The map $H$ deforms the loop $c$ into the constant loop at $p$ through closed loops $t \mapsto H(t, s)$ in $X$.
Each convex set is simply connected. For example, $\mathbb{R}^n$ and each ball are simply connected.
The plane minus the origin is not simply connected, and neither is $\mathbb{R}^3$ minus the $z$-axis. But $\mathbb{R}^n$ minus a point is simply connected for $n \ge 3$.
An equivalent condition: $X$ is path-connected, and any two paths in $X$ with the same end points can be deformed into each other with the end points fixed.

## Explanations

### L1 human

A set is simply connected when every loop in it can be pulled tight to a point, like a rubber band, without leaving the set.
A disc has this property. An annulus does not, because a loop around the hole cannot shrink past the hole.
</document>

<document source="https://tamnd.github.io/math2/node/075PM9FH7A.md" tag="075PM9FH7A" kind="statement">
# The reals have the cardinality of the continuum

## Statement

The set $\mathbb{R}$ of real numbers has [[cardinality|0JTGRV4WT4]] $2^{\aleph_0}$:
$$|\mathbb{R}| = \mathfrak{c} = |\mathcal{P}(\mathbb{N})|.$$

## Remarks

One proof uses the [[Schröder–Bernstein theorem|0A9V6RKQG6]]. A real number gives a set of rationals, the rationals less than it, so $|\mathbb{R}| \le |\mathcal{P}(\mathbb{Q})| = 2^{\aleph_0}$. Binary sequences give an injection in the other direction, for example through decimal expansions with digits $0$ and $1$.
With [[Cantor's theorem|0WTEMGYV2C]], this gives another proof that $\mathbb{R}$ is [[uncountable|0518MM6BE1]].

## Explanations

### L1 human

The real numbers and the subsets of $\mathbb{N}$ have the same size.
Roughly, a real number is an infinite sequence of digits, and an infinite sequence of $0$ and $1$ is a subset of $\mathbb{N}$.
</document>

<document source="https://tamnd.github.io/math2/node/075VBY4K4X.md" tag="075VBY4K4X" kind="definition">
# Initial value problem

## Definition

Let $\Omega$ be an open subset of $\mathbb{R} \times \mathbb{R}^m$, let $f : \Omega \to \mathbb{R}^m$ be a function, and let $(t_0, y_0) \in \Omega$.
The initial value problem for $f$ at $(t_0, y_0)$ is the system
$$y'(t) = f(t, y(t)), \qquad y(t_0) = y_0.$$
A solution of the problem is a [[solution|04SW7WECS8]] $y : I \to \mathbb{R}^m$ of the equation $y' = f(t, y)$ on an open interval $I$ with $t_0 \in I$, such that $y(t_0) = y_0$.
The condition $y(t_0) = y_0$ is the initial condition.

## Remarks

For an equation of order $n$ in explicit form, the initial conditions are the values $y(t_0), y'(t_0), \dots, y^{(n-1)}(t_0)$. The [[reduction to a first-order system|0TH9CPW3VS]] changes this problem into the problem above.
The problem $y' = y$, $y(0) = 1$ has the unique solution $y(t) = e^t$.
The [[Picard–Lindelöf theorem|0P6KRD5B7A]] gives a unique local solution when $f$ is continuous and Lipschitz in $y$. The [[Peano existence theorem|098YP831QY]] gives a solution when $f$ is only continuous.

## Explanations

### L1 human

A differential equation alone usually has many solutions.
The initial condition says where the solution starts, and this picks one solution.
Suppose that you know the speed of a car at each moment and its position at the start. Then you know its position at each moment.
</document>

<document source="https://tamnd.github.io/math2/node/075WR6W3Y3.md" tag="075WR6W3Y3" kind="example">
# Path graph

## Example

Let $n \ge 1$ be an integer.
The path graph $P_n$ is the [[graph|0JD5C82QMB]] with vertex set $\{0, 1, \dots, n - 1\}$ in which $i$ and $j$ are adjacent if and only if $|i - j| = 1$.

## Remarks

$P_n$ has $n$ vertices and $n - 1$ edges. It is a [[tree|0MJ0JS72DZ]], and for $n \ge 2$ it has exactly two leaves, $0$ and $n - 1$.
Some authors write $P_n$ for the path with $n$ edges, that is, with $n + 1$ vertices.

## Explanations

### L1 human

A path graph is a row of vertices, each joined to the next.
It looks like beads on a string.
</document>

<document source="https://tamnd.github.io/math2/node/0762WAE49P.md" tag="0762WAE49P" kind="statement">
# Uniqueness of the derivative

## Statement

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$, let $U \subseteq V$ be open, let $f \colon U \to W$, and let $x \in U$.
If the bounded linear maps $A$ and $B$ from $V$ to $W$ are both [[Fréchet derivatives|08EVNYZF42]] of $f$ at $x$, then $A = B$.

## Remarks

The proof uses that $U$ is open. For a unit vector $v$ and small $t > 0$, the point $x + t v$ lies in $U$, and $\|t A v - t B v\| = o(t)$. Divide by $t$ and let $t \to 0$ to get $A v = B v$.
So it is correct to speak of the derivative $Df(x)$.
On a set that is not open, for example a single point, a derivative within the set need not be unique.

## Explanations

### L1 human

Two different linear maps differ by a definite amount in some direction, at the scale of the step.
If both were good approximations, their difference would be tiny compared with the step in every direction. This forces the difference to be $0$.
</document>

<document source="https://tamnd.github.io/math2/node/076JGP6R3T.md" tag="076JGP6R3T" kind="definition">
# Matrix multiplication

## Definition

Let $R$ be a ring, let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$, and let $B = (b_{jk})$ be an $n \times p$ matrix over $R$.
The *product* $AB$ is the $m \times p$ matrix with entries
$$(AB)_{ik} = \sum_{j=1}^{n} a_{ij} b_{jk}$$
for $1 \le i \le m$ and $1 \le k \le p$.

## Remarks

The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$.
Column $k$ of $AB$ is the [[matrix–vector product|0PDVQEP0ZC]] of $A$ with column $k$ of $B$.
The product is [[associative|073E4XC016]] and [[distributive|0RXD6EZYBK]], but it is [[not commutative|0P506SPZP2]].
If $A$ and $B$ are the matrices of two linear maps, then $AB$ is the matrix of their composition.

## Explanations

### L1 human

To find the entry in row $i$ and column $k$ of $AB$, go along row $i$ of $A$ and down column $k$ of $B$.
Multiply the pairs of numbers and add the products.
For example, $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 11 \\ 4 \end{pmatrix}$.
</document>

<document source="https://tamnd.github.io/math2/node/0771AJP4BX.md" tag="0771AJP4BX" kind="definition">
# Even and odd permutations

## Definition

Let $n \geq 2$ and let $\sigma \in S_n$.
The permutation $\sigma$ is even when $\operatorname{sgn}(\sigma) = 1$, and odd when $\operatorname{sgn}(\sigma) = -1$.

## Remarks

The identity is even and each [[transposition|0FWVRHBRWM]] is odd.
A product of two even or two odd permutations is even, and a product of an even and an odd permutation is odd.
For $n \geq 2$, $S_n$ has $n!/2$ even and $n!/2$ odd permutations.
The odd permutations do not form a subgroup, but they form a coset of $A_n$.

## Explanations

### L1 human

Each permutation is even or odd, like each integer.
Even and odd combine by the same rules as for integers.
Exactly half of the permutations are even.
</document>

<document source="https://tamnd.github.io/math2/node/077GN9HDR6.md" tag="077GN9HDR6" kind="statement">
# Independent random variables are uncorrelated

## Statement

Let $X$ and $Y$ be [[independent|0E9502NT92]] real random variables with $E[X^2] < \infty$ and $E[Y^2] < \infty$.
Then $\operatorname{Cov}(X, Y) = 0$, so $X$ and $Y$ are uncorrelated.

## Remarks

Proof: by the [[product rule for independent variables|05AW1G0WE1]], $E[XY] = E[X] \, E[Y]$. So $\operatorname{Cov}(X, Y) = E[XY] - E[X] \, E[Y] = 0$.
The converse is false. The node [[uncorrelated random variables that are not independent|0C6J8BQXVR]] gives a counterexample.
For a pair $(X, Y)$ with a joint multivariate normal law, uncorrelated does imply independent.

## Explanations

### L1 human

If two variables do not influence each other at all, they have no linear relation either.
But "no linear relation" is a much weaker property than independence.
</document>

<document source="https://tamnd.github.io/math2/node/0783S9D0ZH.md" tag="0783S9D0ZH" kind="statement">
# Determinant of the inverse

## Statement

Let $K$ be a commutative ring and let $A$ be an [[invertible|0EB9N0S33C]] $n \times n$ matrix over $K$.
Then $\det A$ is a unit of $K$, and
$$\det(A^{-1}) = (\det A)^{-1},$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The proof applies the [[product rule|04CPJ6AF00]] to $A A^{-1} = I_n$: $\det A \cdot \det(A^{-1}) = 1$.
Over a field, this says $\det(A^{-1}) = 1 / \det A$.

## Explanations

### L1 human

The inverse matrix undoes the matrix, so it must undo its scaling of volume.
If $A$ scales volume by $3$, then $A^{-1}$ scales volume by $1/3$.
So the determinant of the inverse is the inverse of the determinant.
</document>

<document source="https://tamnd.github.io/math2/node/079D68S3AW.md" tag="079D68S3AW" kind="definition">
# Orthogonal complement

## Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $W$ be a [[subspace|00YC1D652K]] of $V$.
The *orthogonal complement* of $W$ is
$$W^\perp = \{v \in V : \langle v, w \rangle = 0 \text{ for all } w \in W\}.$$

## Remarks

$W^\perp$ is a subspace of $V$, and $W \cap W^\perp = 0$.
In $\R^3$ with the dot product, the orthogonal complement of a plane through $0$ is its normal line.
For a real $m \times n$ matrix $A$, the null space of $A$ is the orthogonal complement of the row space of $A$ in $\R^n$.
In finite dimensions, $V = W \oplus W^\perp$, see [[the orthogonal decomposition|0SP7P0G7KQ]].

## Explanations

### L1 human

The orthogonal complement collects all vectors that are perpendicular to a subspace.
The complement of a line in $\R^3$ is a plane, and the complement of a plane is a line.
A subspace and its complement together fill the space.
</document>

<document source="https://tamnd.github.io/math2/node/079EGSD32Y.md" tag="079EGSD32Y" kind="definition">
# Transcendental number

## Definition

A complex number is transcendental if it is not an [[algebraic number|0MB4TGMXG6]].
That is, it is not a root of any nonzero polynomial with rational coefficients.

## Remarks

The numbers $e$ and $\pi$ are transcendental, by the [[Lindemann–Weierstrass theorem|0JNCBFMPXR]].
Each [[Liouville number|0J1R5NP36S]] is transcendental, by [[Liouville's theorem|0N3WFZEFPM]].
The algebraic numbers are countable, so almost all real numbers are transcendental.
Joseph Liouville first proved that transcendental numbers exist, in 1844.

## Explanations

### L1 human

A transcendental number solves no polynomial equation with integer coefficients.
It is usually hard to prove that a given number is transcendental, even though most numbers are.
</document>

<document source="https://tamnd.github.io/math2/node/07BNRFQ0PJ.md" tag="07BNRFQ0PJ" kind="definition">
# Autonomous system

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be a function.
The autonomous system with vector field $f$ is the system of [[ordinary differential equations|0P97V20RW7]]
$$x'(t) = f\left(x(t)\right).$$
A [[solution|04SW7WECS8]] is a differentiable function $\gamma : I \to U$ on an interval $I$ with $\gamma'(t) = f(\gamma(t))$ for each $t \in I$.
The set $U$ is the phase space, or state space, of the system.

## Remarks

The right side depends on the state $x$ only, not on the time $t$ directly.
A non-autonomous system $x' = f(t, x)$ becomes autonomous with the extra state variable $s$ and the extra equation $s' = 1$.
The equation $x' = k x$ of [[exponential growth|00SVP6W3CB]] and the [[logistic equation|0CC4VC7KR3]] are autonomous. The equation $x' = t x$ is not.
Solutions of an autonomous system can be [[translated in time|0WA7ZR95V4]].

## Explanations

### L1 human

In an autonomous system, the law of motion does not change with time.
The velocity depends only on where the state is, not on when it is there.
</document>

<document source="https://tamnd.github.io/math2/node/07BSPZPSMF.md" tag="07BSPZPSMF" kind="concept">
# Nilpotent matrix

## Summary

A nilpotent matrix is a square matrix $N$ such that $N^k = 0$ for some positive integer $k$.
</document>

<document source="https://tamnd.github.io/math2/node/07BY47HG5S.md" tag="07BY47HG5S" kind="statement">
# Continuity of the inverse of a strictly monotone continuous function

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous and [[strictly increasing|0TR9SR9FTB]].
Then $J = f(I)$ is an interval, and the inverse function $f^{-1} \colon J \to I$ is continuous and strictly increasing.

## Remarks

The set $J$ is an interval by the [[intermediate value theorem|00KAFBR32E]].
The inverse is strictly increasing and its image is the interval $I$. A monotone function whose image is an interval has no jumps, so $f^{-1}$ is continuous.
The same holds for strictly decreasing functions.
This gives the continuity of $n$-th roots, of the [[logarithm|0044CPHJBV]], and of the inverse trigonometric functions.

## Explanations

### L1 human

The graph of $f^{-1}$ is the graph of $f$ reflected in the line $y = x$.
A rising unbroken graph stays rising and unbroken after the reflection.
</document>

<document source="https://tamnd.github.io/math2/node/07C4A8Y18K.md" tag="07C4A8Y18K" kind="definition">
# Conjunctive normal form

## Definition

A formula of propositional logic is in conjunctive normal form if it is a conjunction of one or more disjunctions of one or more literals.
Each disjunction is a clause.

## Remarks

Example: $(p \vee \neg q) \wedge (q \vee r) \wedge \neg p$ is in conjunctive normal form.
Every formula is logically equivalent to a formula in conjunctive normal form. Negate the formula, put the negation in [[disjunctive normal form|06S4WGSEQA]], and use the [[De Morgan laws|0XYVDWGQX2]].
The satisfiability problem for formulas in conjunctive normal form with clauses of $3$ literals is NP-complete.

## Explanations

### L1 human

A formula in conjunctive normal form is an and of ors of simple conditions.
Each clause is a constraint, and the formula is true when all constraints hold.
</document>

<document source="https://tamnd.github.io/math2/node/07C6ECNJET.md" tag="07C6ECNJET" kind="concept">
# Group of units of a ring

## Summary

The group of units of a ring is the set of its units, with the multiplication of the ring.
The [[definition of the group of units|05THZKR9AN]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/07CHJ5YX28.md" tag="07CHJ5YX28" kind="definition">
# Convolution of probability distributions

## Definition

Let $\mu$ and $\nu$ be [[probability distributions|0QB913WM5E]] on $\mathbb{R}$.
Their convolution is the probability distribution $\mu * \nu$ on $\mathbb{R}$ with
$$(\mu * \nu)(B) = \int_{\mathbb{R}} \nu(B - x) \, \mu(dx)$$
for each Borel set $B \subseteq \mathbb{R}$, where $B - x = \{b - x : b \in B\}$.

## Remarks

If $X \sim \mu$ and $Y \sim \nu$ are [[independent|0E9502NT92]], then $X + Y \sim \mu * \nu$.
For distributions on the integers with mass functions $p$ and $q$, $\mu * \nu$ has the mass function $(p * q)(n) = \sum_{k} p(k) \, q(n - k)$.
If $\mu$ and $\nu$ have densities $f$ and $g$, then $\mu * \nu$ has the density $(f * g)(x) = \int f(y) \, g(x - y) \, dy$.
The convolution is commutative and associative. Its characteristic function is the product: $\varphi_{\mu * \nu} = \varphi_\mu \varphi_\nu$.

## Explanations

### L1 human

The convolution answers the question: "What is the law of $X + Y$, if $X$ and $Y$ are independent?"
For two fair dice, the convolution of the two uniform laws on $\{1, \dots, 6\}$ gives the familiar triangle-shaped law of the sum on $\{2, \dots, 12\}$.
</document>

<document source="https://tamnd.github.io/math2/node/07CJNE6ZAQ.md" tag="07CJNE6ZAQ" kind="example">
# Taxicab metric

## Example

Let $n \ge 1$ be an integer.
For $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let
$$d_1(x, y) = \sum_{i=1}^{n} |x_i - y_i|.$$
Then $d_1$ is a [[metric|063W5JMY85]] on $\mathbb{R}^n$, the taxicab metric.

## Remarks

The triangle inequality for $d_1$ follows from the [[triangle inequality|03HF5PMY5T]] for real numbers in each coordinate.
In $\mathbb{R}^2$, the distance from $(0, 0)$ to $(3, 4)$ is $7$, but the [[Euclidean distance|0E1MMCAY5B]] is $5$.
It is the case $p = 1$ of the [[p-metric|0TFDZN6AXP]].

## Explanations

### L1 human

A taxi in a city with a grid of streets cannot drive through the blocks.
It must go along the streets, so it travels the sum of the east-west and the north-south distances.
</document>

<document source="https://tamnd.github.io/math2/node/07CQR9BW3R.md" tag="07CQR9BW3R" kind="statement">
# Classification of finite abelian groups

## Statement

Let $G$ be a finite [[abelian group|082GHNPJ53]].
Then $G$ is [[isomorphic|08MG7H8JW5]] to a [[direct sum|00HF4RJ5QN]] $\mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z}$, where each $q_i$ is a power of a prime number.
The numbers $q_1, \ldots, q_t$ are unique up to their order.
Equivalently, $G \cong \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z}$ with $1 < k_1 \mid k_2 \mid \cdots \mid k_u$, and these $k_i$ are unique.

## Remarks

The two forms are equivalent because $\mathbb{Z}/m n\mathbb{Z} \cong \mathbb{Z}/m\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$ if and only if $\gcd(m, n) = 1$, as in [[this statement|08FDFEF8D1]].
Each abelian group of order $8$ is isomorphic to exactly one of $\mathbb{Z}/8\mathbb{Z}$, $\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$.
Each abelian group of order $15$ is cyclic, because $15 = 3 \cdot 5$ with $\gcd(3, 5) = 1$.
The theorem is the case of rank $0$ of the [[structure theorem for finitely generated abelian groups|0RY0B72XAS]].

## History

Leopold Kronecker proved the theorem in 1870 with a group-theoretic proof, but he did not state it in the language of groups.
His work extended an earlier result of Carl Friedrich Gauss from 1801 on quadratic forms.
Ferdinand Georg Frobenius and Ludwig Stickelberger stated and proved the theorem in the language of groups in 1878.

## Explanations

### L1 human

Each finite abelian group is built from cyclic groups.
You can choose the cyclic pieces to have prime power orders, or to have orders that divide each other.
In both forms, the pieces are unique, so this lists all finite abelian groups.
</document>

<document source="https://tamnd.github.io/math2/node/07CT9RDY7Q.md" tag="07CT9RDY7Q" kind="definition">
# Cycle

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A cycle in $G$ is a [[circuit|0PD57PSHNR]] $(w_0, w_1, \dots, w_k)$ whose vertices $w_1, \dots, w_k$ are pairwise distinct.

## Remarks

The only repeated vertex is $w_0 = w_k$. In a simple graph, each cycle has length $k \ge 3$.
The vertices and edges of a cycle of length $k$ form a subgraph isomorphic to the [[cycle graph|0QQ0F12962]] $C_k$.
A graph with no cycle is [[acyclic|0XNN9Z8E5A]].

## Explanations

### L1 human

A cycle is a round trip that visits each of its vertices once and comes back to the start.
A triangle is the shortest possible cycle in a simple graph.
</document>

<document source="https://tamnd.github.io/math2/node/07CZYE06ZJ.md" tag="07CZYE06ZJ" kind="statement">
# Lcm of coprime integers

## Statement

Let $a$ and $b$ be natural numbers that are [[coprime|0GFKP8F8AG]].
Then $\operatorname{lcm}(a, b) = a b$.

## Remarks

This follows from the [[product of the gcd and the lcm|0TDY6JZHKS]], because $\gcd(a, b) = 1$.
The converse is also true when $a$ and $b$ are positive: if $\operatorname{lcm}(a, b) = a b$, then $a$ and $b$ are coprime.
For example, $\operatorname{lcm}(8, 15) = 120$.

## Explanations

### L1 human

When $a$ and $b$ have no common factor, a multiple of both must contain all the factors of $a$ and all the factors of $b$.
No factor can serve both numbers at the same time, so the smallest common multiple is $a b$.
</document>

<document source="https://tamnd.github.io/math2/node/07DCKDXRBP.md" tag="07DCKDXRBP" kind="statement">
# Lipschitz maps are uniformly continuous

## Statement

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be a [[Lipschitz map|0TMZ21WKCN]].
Then $f$ is [[uniformly continuous|0FHSDCQACY]].

## Remarks

Proof: let $K > 0$ be a Lipschitz constant of $f$. For $\varepsilon > 0$, take $\delta = \varepsilon / K$. If $d(x, y) < \delta$, then $d(f(x), f(y)) \le K d(x, y) < \varepsilon$. If $f$ is Lipschitz with $K = 0$, then $f$ is constant.
The converse fails: $x \mapsto \sqrt{x}$ on $[0, 1]$ is uniformly continuous, but it is not Lipschitz.
In particular, each Lipschitz map is continuous.

## Explanations

### L1 human

For a Lipschitz map, you can give an explicit $\delta$ for each $\varepsilon$: divide $\varepsilon$ by the constant $K$.
This $\delta$ does not depend on the point, so the map is uniformly continuous.
</document>

<document source="https://tamnd.github.io/math2/node/07DFG5JYPT.md" tag="07DFG5JYPT" kind="concept">
# Incircle

## Summary

The incircle of a triangle is the largest circle inside the triangle. It touches each of the three sides at one point. Its centre is the incentre and its radius is the inradius.
</document>

<document source="https://tamnd.github.io/math2/node/07DJAZW32T.md" tag="07DJAZW32T" kind="concept">
# Integrating factor

## Summary

An integrating factor is a function by which one multiplies a differential equation so that it can be integrated directly.
For the linear equation $y' + p y = q$, the factor $e^{\int p}$ turns the left side into the derivative $\left(e^{\int p} y\right)'$. See the [[definition|01THH3Y3YP]].
</document>

<document source="https://tamnd.github.io/math2/node/07DVHJ8N0A.md" tag="07DVHJ8N0A" kind="definition">
# Bipartite graph

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] with vertex set $V$.
Then $G$ is bipartite if $V$ is the union of two disjoint sets $A$ and $B$ with this property.
Each edge of $G$ has one end in $A$ and one end in $B$.
The pair $(A, B)$ is a bipartition of $G$.

## Remarks

Equivalently, $G$ is bipartite if it has a proper [[colouring|0874138YP6]] with $2$ colours.
The [[complete bipartite graphs|0V0GC7BQAN]], the even [[cycle graphs|0QQ0F12962]] and all [[trees|0MJ0JS72DZ]] are bipartite.
A graph is bipartite if and only if it has no cycle of odd length. The node [[a graph is bipartite iff it has no odd cycle|01R1HPEK6S]] states this.

## Explanations

### L1 human

A graph is bipartite when its vertices split into two groups, and every edge goes between the groups.
Students and the courses they take form a bipartite graph.
</document>

<document source="https://tamnd.github.io/math2/node/07E5MB1CA2.md" tag="07E5MB1CA2" kind="statement">
# Cauchy–Hadamard theorem

## Statement

Let $\sum a_n (x - c)^n$ be a real power series with [[radius of convergence|0660QX1VPR]] $R$.
Then
$$\frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n},$$
with the conventions $1/0 = \infty$ and $1/\infty = 0$.

## Remarks

The proof applies the [[root test|05H4DDDYBZ]] to the series $\sum a_n (x - c)^n$. The [[limit superior|0P8BA7T7S5]] of $|a_n (x - c)^n|^{1/n}$ is $|x - c| \limsup |a_n|^{1/n}$.
For example, $\sum n x^n$ has $R = 1$, since $n^{1/n} \to 1$.
Mathlib states it as `FormalMultilinearSeries.radius_inv_eq_limsup`.

## Explanations

### L1 human

The coefficients $a_n$ grow or shrink roughly like $\rho^n$, where $\rho$ is the limit superior of $|a_n|^{1/n}$.
Then $a_n x^n$ behaves like $(\rho x)^n$, which is small exactly when $|x| < 1/\rho$.
</document>

<document source="https://tamnd.github.io/math2/node/07E6D8R4D6.md" tag="07E6D8R4D6" kind="statement">
# Disjoint cycles commute

## Statement

Let $\sigma$ and $\tau$ be cycles in the [[symmetric group|0515A6D2Y1]] $\operatorname{Sym}(X)$ whose sets of moved points are disjoint.
Then $\sigma \tau = \tau \sigma$.

## Remarks

A point moved by $\sigma$ is fixed by $\tau$, and $\sigma$ maps it to another point that $\tau$ fixes. So both products send it to the same point.
The same argument works for the points moved by $\tau$, and the points fixed by both are fixed by both products.
Cycles that are not disjoint need not commute, for example $(1\ 2)(1\ 3) \neq (1\ 3)(1\ 2)$.

## Explanations

### L1 human

Two disjoint cycles move different points.
Each cycle does its work without touching the points of the other cycle.
So the order in which you do them does not matter.
</document>

<document source="https://tamnd.github.io/math2/node/07EGCNHAVW.md" tag="07EGCNHAVW" kind="definition">
# Limit point of a set

## Definition

Let $S$ be a subset of $\mathbb{R}$ and let $a$ be a real number.
The point $a$ is a limit point of $S$ if for each $\delta > 0$ there is $x \in S$ with $0 < |x - a| < \delta$.

## Remarks

A limit point of $S$ need not belong to $S$. For example, $0$ is a limit point of $\{1/n : n \ge 1\}$.
Each point of an interval with more than one point is a limit point of the interval. So are its endpoints.
Limit points of the domain are the points where the [[limit of a function|0TXTSEMHR5]] makes sense.
Mathlib uses the general form `AccPt`.

## Explanations

### L1 human

A limit point of $S$ is a point that we can approach by other points of $S$.
Points of $S$ lie as close to it as we like, not counting the point itself.
</document>

<document source="https://tamnd.github.io/math2/node/07EH4R3RVQ.md" tag="07EH4R3RVQ" kind="statement">
# Composition of surjections is surjective

## Statement

Let $f \colon A \to B$ and $g \colon B \to C$ be [[surjective functions|0XKH881BZ7]].
Then the [[composition|07HF9YCWPP]] $g \circ f \colon A \to C$ is surjective.

## Remarks

Proof: let $c \in C$. Since $g$ is surjective, $c = g(b)$ for some $b \in B$. Since $f$ is surjective, $b = f(a)$ for some $a \in A$. Then $c = (g \circ f)(a)$.
A partial converse holds: if $g \circ f$ is surjective, then $g$ is surjective. The function $f$ need not be surjective.
Together with [[the result for injections|01FPXMTBX1]], this shows that a composition of [[bijections|0SNQEF7Z53]] is a bijection.

## Explanations

### L1 human

To reach a point $c$ of $C$, first find a point of $B$ that $g$ sends to $c$.
Then find a point of $A$ that $f$ sends to that point of $B$.
</document>

<document source="https://tamnd.github.io/math2/node/07EJHQYR9A.md" tag="07EJHQYR9A" kind="statement">
# Grönwall's inequality

## Statement

Let $a < b$ and let $I$ be an [[interval|0Q1D85B8G6]] of the form $[a, b]$, $[a, b)$ or $[a, \infty)$. Let $I^\circ$ be the interior of $I$.
Differential form: let $\beta, u : I \to \mathbb{R}$ be continuous, let $u$ be differentiable on $I^\circ$, and assume $u'(t) \le \beta(t)\, u(t)$ for each $t \in I^\circ$.
Then
$$u(t) \le u(a) \exp\left( \int_a^t \beta(s)\, ds \right) \quad \text{for each } t \in I.$$
Integral form: let $\beta, u : I \to \mathbb{R}$ be continuous with $\beta \ge 0$, let $\alpha : I \to \mathbb{R}$ be non-decreasing, and assume
$$u(t) \le \alpha(t) + \int_a^t \beta(s)\, u(s)\, ds \quad \text{for each } t \in I.$$
Then
$$u(t) \le \alpha(t) \exp\left( \int_a^t \beta(s)\, ds \right) \quad \text{for each } t \in I.$$

## Remarks

The differential form needs no assumption on the signs of $\beta$ and $u$. Its proof shows that $u(t) \exp\left(-\int_a^t \beta\right)$ is non-increasing.
The integral form does not need $u$ to be differentiable.
For constant $\alpha = 0$, the integral form gives $u \le 0$. This is the step that proves [[uniqueness of solutions|025PKW8AMR]].
A variant with an additional constant term gives the [[Grönwall bound|0FRX8JD7QH]], and there is a [[discrete version|0NTM3J5MFE]].

## History

The inequality is named after Thomas Hakon Grönwall. He proved it in 1919, in the integral form with constant $\alpha$ and $\beta$.
Richard Bellman proved a slightly more general integral form in 1943.

## Explanations

### L1 human

If a quantity grows at most at a rate proportional to itself, it grows at most exponentially.
The inequality compares the quantity with the solution of the equation $v' = \beta v$.
This is the main tool to compare two solutions of a differential equation.
</document>

<document source="https://tamnd.github.io/math2/node/07EXVNF9VP.md" tag="07EXVNF9VP" kind="concept">
# Newton's method

## Summary

Newton's method finds approximate zeros of a differentiable function $f$.
From a starting point $x_0$, it computes
$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.$$
Each new point is the zero of the [[tangent line|01R1HW3Y42]] to the graph at the old point.

## Remarks

Assume $f$ is $C^2$ near a zero $r$ with $f'(r) \ne 0$. Then the method converges to $r$ for $x_0$ close to $r$. The convergence is quadratic: $|x_{n+1} - r| \le C |x_n - r|^2$.
For $f(x) = x^2 - 2$, the step is $x_{n+1} = (x_n + 2/x_n)/2$. This is Heron's method for $\sqrt{2}$.
The method can fail. It stops when $f'(x_n) = 0$, and it can cycle or diverge from a poor starting point.

## Explanations

### L1 human

Replace the graph by its tangent line, and find where the line crosses the axis.
Repeat from there. Near a simple root, each step roughly doubles the number of correct digits.
</document>

<document source="https://tamnd.github.io/math2/node/07EZA80NEH.md" tag="07EZA80NEH" kind="concept">
# Signature of a quadratic form

## Summary

The signature of a real quadratic form records the numbers of positive, negative and zero coefficients in any diagonal form of it.
</document>

<document source="https://tamnd.github.io/math2/node/07GJV36896.md" tag="07GJV36896" kind="concept">
# Complex exponential

## Summary

The complex exponential is the function $z \mapsto e^z = \sum_{n \ge 0} z^n / n!$ on $\mathbb{C}$, see the [[definition|054VXHMCA3]].
It extends the real exponential function, it is entire, and it has period $2\pi i$.
</document>

<document source="https://tamnd.github.io/math2/node/07GNX8GCYP.md" tag="07GNX8GCYP" kind="concept">
# Polynomials in several variables

## Summary

The ring $R[x_1, \dots, x_n]$ is the ring of the polynomials in the variables $x_1, \dots, x_n$ with coefficients in a commutative ring $R$.
The [[definition of the multivariate polynomial ring|0CES91M79A]] gives the construction.
</document>

<document source="https://tamnd.github.io/math2/node/07GR3TNW4H.md" tag="07GR3TNW4H" kind="concept">
# Convex function on R^n

## Summary

Take a real-valued [[function of several real variables|0JPCHTXWG0]] on a convex set. The function is convex when its graph lies on or below each chord between two points of the graph.
Convex functions are important in optimisation because each local minimum of a convex function is a global minimum.
</document>

<document source="https://tamnd.github.io/math2/node/07HF9YCWPP.md" tag="07HF9YCWPP" kind="definition">
# Function composition

## Definition

Let $f \colon A \to B$ and $g \colon B \to C$ be [[functions|0XHDTE5YMZ]].
The composition of $g$ and $f$ is the function $g \circ f \colon A \to C$ with
$$(g \circ f)(a) = g(f(a))$$
for each $a \in A$.

## Remarks

Read $g \circ f$ as "$g$ after $f$": $f$ acts first.
In general $g \circ f \neq f \circ g$. For $f(x) = x + 1$ and $g(x) = 2x$ on $\mathbb{R}$, $(g \circ f)(x) = 2x + 2$ and $(f \circ g)(x) = 2x + 1$.
Composition is associative. See [[associativity of composition|0J6AKYZQT8]].

## Explanations

### L1 human

Composition chains two machines: the output of $f$ goes into $g$.
For $g \circ f$ to make sense, the values of $f$ must lie in the domain of $g$.
</document>

<document source="https://tamnd.github.io/math2/node/07HFRH2AWY.md" tag="07HFRH2AWY" kind="example">
# Circulant matrix

## Example

Let $F$ be a field and let $c_0, c_1, \dots, c_{n-1} \in F$.
The *circulant matrix* with first column $(c_0, c_1, \dots, c_{n-1})^T$ is the $n \times n$ matrix $C$ with entries $C_{ij} = c_{(i - j) \bmod n}$, for $0 \le i, j \le n - 1$.
Each column is the previous column shifted down by one place, with the last entry moved to the top.
For $n = 3$:
$$C = \begin{pmatrix} c_0 & c_2 & c_1 \\ c_1 & c_0 & c_2 \\ c_2 & c_1 & c_0 \end{pmatrix}.$$

## Remarks

Some authors define circulant matrices by their first row; the two conventions give transposed matrices.
Over $\C$, the vectors $(1, \omega^k, \omega^{2k}, \dots, \omega^{(n-1)k})$ with $\omega = e^{2\pi i/n}$ are eigenvectors of every $n \times n$ circulant matrix.
The product of two circulant matrices of the same order is circulant, and circulant matrices commute with each other.

## Explanations

### L1 human

In a circulant matrix, each column is the column before it, rotated down by one step.
All the information is in the first column.
The discrete Fourier transform diagonalises all of these matrices at the same time.
</document>

<document source="https://tamnd.github.io/math2/node/07HXA8KV6F.md" tag="07HXA8KV6F" kind="statement">
# Pépin's test

## Statement

Let $n$ be a positive integer and let $F_n = 2^{2^n} + 1$ be the [[Fermat number|0R03B54V4Z]].
Then $F_n$ is prime if and only if
$$3^{(F_n - 1)/2} \equiv -1 \pmod{F_n}.$$

## Remarks

For example, $F_1 = 5$ and $3^2 = 9 \equiv -1 \pmod 5$.
The power can be computed by $2^n - 1$ squarings with [[modular exponentiation|04DR5PM25N]].
The "only if" part follows from [[Euler's criterion|0M7DEDKSYB]] and quadratic reciprocity, since $3$ is a nonresidue modulo $F_n$.

## History

The test is named after Théophile Pépin.

## Explanations

### L1 human

Pépin's test decides whether a Fermat number is prime with one large modular power.
The Fermat numbers grow so fast that the test is practical only for small $n$.
</document>

<document source="https://tamnd.github.io/math2/node/07J5X2T1ZX.md" tag="07J5X2T1ZX" kind="statement">
# A function with positive derivative is strictly increasing

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous, and differentiable at each interior point of $I$.
If $f'(x) > 0$ for each interior point $x$ of $I$, then $f$ is [[strictly increasing|0TR9SR9FTB]] on $I$.

## Remarks

For $x < y$ in $I$, the [[mean value theorem|0BARQY52N5]] gives $f(y) - f(x) = f'(c)(y - x) > 0$.
The converse fails: $x^3$ is strictly increasing, but its derivative at $0$ is $0$.
Mathlib states it as `strictMonoOn_of_deriv_pos`.

## Explanations

### L1 human

If the graph has positive slope everywhere, it rises from left to right.
</document>

<document source="https://tamnd.github.io/math2/node/07J9ZJFECP.md" tag="07J9ZJFECP" kind="concept">
# Order on the integers

## Summary

The order on the integers is the relation $\le$ on $\mathbb{Z}$ that puts the integers in the sequence $\dots < -2 < -1 < 0 < 1 < 2 < \dots$.
It is a total order that agrees with addition and with multiplication by positive integers.
</document>

<document source="https://tamnd.github.io/math2/node/07JBSZE0V4.md" tag="07JBSZE0V4" kind="concept">
# Dual space

## Summary

The dual space of a vector space is the vector space of all linear forms on it.
</document>

<document source="https://tamnd.github.io/math2/node/07JH131FVP.md" tag="07JH131FVP" kind="concept">
# Linear first-order equation

## Summary

A linear first-order equation is an equation $y' + p(x) y = q(x)$ with given functions $p$ and $q$.
It is the simplest [[linear differential equation|0QQRP17Z7S]], and an [[integrating factor|07DJAZW32T]] solves it explicitly. See the [[definition|0Z3YKRQZ48]].
</document>

<document source="https://tamnd.github.io/math2/node/07KBQCG9KV.md" tag="07KBQCG9KV" kind="statement">
# i squared is minus one

## Statement

Let $i$ be the [[imaginary unit|0ESCX36MMC]] of $\mathbb{C}$.
Then $i^2 = -1$.

## Remarks

With the product of [[complex numbers|09FTAET5Z4]], $(0 + 1 i)(0 + 1 i) = (0 \cdot 0 - 1 \cdot 1) + (0 \cdot 1 + 1 \cdot 0) i = -1$.
Also $(-i)^2 = -1$. Thus $-1$ has the two square roots $i$ and $-i$ in $\mathbb{C}$.

## Explanations

### L1 human

This is the property that defines $i$.
All the rules for computation with complex numbers come from the usual rules of algebra together with $i^2 = -1$.
</document>

<document source="https://tamnd.github.io/math2/node/07KVB4JSC7.md" tag="07KVB4JSC7" kind="definition">
# Union

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The union of $A$ and $B$ is the set
$$A \cup B = \{x \mid x \in A \lor x \in B\}.$$
So an object $x$ is an [[element|0EZ758XE73]] of $A \cup B$ when $x \in A$ or $x \in B$, or both.

## Remarks

Example: $\{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}$.
Union is commutative and associative, and $A \cup \varnothing = A$.
The union of an arbitrary family of sets is in [[union of a family of sets|05GK32H72H]].

## Explanations

### L1 human

To form $A \cup B$, put the elements of $A$ and the elements of $B$ together in one set.
An element of both sets appears only one time.
</document>

<document source="https://tamnd.github.io/math2/node/07MMWAXMYD.md" tag="07MMWAXMYD" kind="statement">
# If 2^p − 1 is prime then p is prime

## Statement

Let $n$ be a natural number.
If $2^n - 1$ is a [[prime number|0SN0MAEKYX]], then $n$ is a prime number.

## Remarks

The proof: if $n = ab$ with $1 < a < n$, then $2^a - 1$ divides $2^n - 1$, because $x - 1$ divides $x^b - 1$ for $x = 2^a$. And $1 < 2^a - 1 < 2^n - 1$.
For example, $2^6 - 1 = 63$ is divisible by $2^2 - 1 = 3$ and by $2^3 - 1 = 7$.
The converse is false: $11$ is prime, but $2^{11} - 1 = 23 \cdot 89$.

## Explanations

### L1 human

To look for primes of the form $2^n - 1$, you only need to try prime exponents $n$.
A factorization of $n$ always gives a factorization of $2^n - 1$.
</document>

<document source="https://tamnd.github.io/math2/node/07N7W5C29X.md" tag="07N7W5C29X" kind="definition">
# Continuity at a point of a topological space

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], let $f \colon X \to Y$ be a function and let $a \in X$.
The function $f$ is continuous at $a$ when for each [[neighbourhood|09CJ00H3NW]] $V$ of $f(a)$, the [[preimage|0FGCFY1AZX]] $f^{-1}(V)$ is a neighbourhood of $a$.

## Remarks

Equivalently, for each neighbourhood $V$ of $f(a)$ there is a neighbourhood $U$ of $a$ with $f(U) \subseteq V$.
A function is a [[continuous map|0SY9ECQ19Q]] if and only if it is continuous at each point of $X$.
For metric spaces, continuity at $a$ is the same as the [[epsilon-delta condition at a|0EKGKY5TGH]].

## Explanations

### L1 human

Continuity at $a$ means that you can keep $f(x)$ as close to $f(a)$ as you want, if you keep $x$ close enough to $a$.
Neighbourhoods replace the distances of the metric case.
</document>

<document source="https://tamnd.github.io/math2/node/07NKG38VEG.md" tag="07NKG38VEG" kind="statement">
# Fermat's Last Theorem for n = 3

## Statement

There are no positive integers $a$, $b$ and $c$ with $a^3 + b^3 = c^3$.

## Remarks

A sum of two positive cubes is never a cube.
Standard proofs use [[infinite descent|0EFYCFHYFG]] in the ring $\mathbb{Z}[\omega]$ of Eisenstein integers, where $\omega$ is a primitive cube root of unity.
This case is part of [[Fermat's Last Theorem|0Y5X3SDBCZ]].

## History

Fermat mentioned the case $n = 3$ in letters of 1636, 1640 and 1657.
Euler published a proof in 1770. Many other proofs followed, for example by Legendre.

## Explanations

### L1 human

You can never write a cube as a sum of two positive cubes.
For example, $6^3 = 216$, and $3^3 + 4^3 + 5^3 = 216$ needs three cubes, not two.
</document>

<document source="https://tamnd.github.io/math2/node/07NN4VNYWM.md" tag="07NN4VNYWM" kind="concept">
# Liouville function

## Summary

The Liouville function $\lambda(n)$ is $1$ or $-1$, by the parity of the number of prime factors of $n$ counted with multiplicity.
The [[definition of the Liouville function|08RQAPXF1T]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/07NSKDGS7S.md" tag="07NSKDGS7S" kind="statement">
# Menelaus's theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\mathbb{R}^2$. For [[collinear points|0AFNNTRTJE]] $X$, $Y$, $Z$ with $Z \neq Y$, let the signed ratio $\frac{XZ}{ZY}$ be the real number $\lambda$ with $Z - X = \lambda\,(Y - Z)$.
Let $\ell$ be a [[line|0MP9BCF6DP]] that does not pass through $A$, $B$ or $C$. Suppose that $\ell$ meets the lines $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$.
Then
$$\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = -1.$$
Conversely, let $D$, $E$ and $F$ be points of the lines $BC$, $CA$ and $AB$, different from the vertices, with this product equal to $-1$. Then $D$, $E$ and $F$ are collinear.

## Remarks

With unsigned lengths, the product is $1$. The converse of this weaker form is false in general.
The formula differs from the formula of [[Ceva's theorem|0KN93VDSXA]] only in the sign.
The line $\ell$ meets the inside of exactly two sides, or of no side. Thus one or three of the ratios are negative, and the product is negative.

## History

The oldest surviving statement is in the Spherics of Menelaus of Alexandria. There, he uses the plane version as a lemma for a version on the sphere.
Ptolemy used the theorem in the Almagest for problems of spherical astronomy.

## Explanations

### L1 human

A straight line that cuts across a triangle meets the three side lines in three points. These points cut the sides in certain ratios. Going round the triangle, the product of the three signed ratios is always $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/07PB7HH1T9.md" tag="07PB7HH1T9" kind="concept">
# Picard operator

## Summary

The Picard operator of an initial value problem $x' = f(t, x)$, $x(t_0) = x_0$ maps a curve $\gamma$ to $t \mapsto x_0 + \int_{t_0}^{t} f(s, \gamma(s))\, ds$.
Its fixed points are the solutions of the problem. See the [[definition|00WAVB68F9]].
</document>

<document source="https://tamnd.github.io/math2/node/07PFYV65MB.md" tag="07PFYV65MB" kind="statement">
# Heine–Borel theorem

## Statement

Let $n \ge 1$ be an integer and give $\mathbb{R}^n$ the [[Euclidean metric|0E1MMCAY5B]].
A subset $K$ of $\mathbb{R}^n$ is [[compact|00R0X1WGPX]] if and only if $K$ is [[closed|0G5AV9Q4RR]] and [[bounded|0P953MG05E]].

## Remarks

The direct part holds in each metric space: [[compact subsets of a metric space are bounded|0R7BTEWCJ2]], and they are closed.
The converse uses the completeness of $\mathbb{R}$. A bounded set lies in a cube $[-M, M]^n$, which is compact. A closed subset of a compact set is compact.
The converse fails in general metric spaces. In the space of bounded real sequences with the sup metric, the closed unit ball is closed and bounded but not compact.
Metric spaces in which closed bounded sets are compact are the [[proper metric spaces|0Z4V98VN5E]].

## History

Dirichlet used finite subcovers of a closed interval in his 1852 lectures, to prove that continuous functions on such intervals are uniformly continuous.
Émile Borel stated and proved a version for countable covers in 1895. Cousin, Lebesgue and Schoenflies later extended it to arbitrary covers.

## Explanations

### L1 human

In $\mathbb{R}^n$, you can recognize compact sets by two easy tests: they contain their limit points and they fit in a ball.
For example, the closed disc and the sphere are compact, but the open disc and the whole plane are not.
</document>

<document source="https://tamnd.github.io/math2/node/07PH4RX0X5.md" tag="07PH4RX0X5" kind="concept">
# Jordan block

## Summary

A Jordan block is a square matrix with one scalar $\lambda$ on the diagonal, $1$ on the diagonal just above it, and $0$ elsewhere.
</document>

<document source="https://tamnd.github.io/math2/node/07PNDWA7CE.md" tag="07PNDWA7CE" kind="definition">
# Generalised eigenspace

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$.
The *generalised eigenspace* of $T$ for $\lambda$ is
$$V_\lambda = \{v \in V : (T - \lambda \id_V)^k (v) = 0 \text{ for some } k \ge 1\}.$$

## Remarks

It is a $T$-invariant subspace that contains the eigenspace $E_\lambda$.
If $\dim V = n$ is finite, then $V_\lambda = \ker (T - \lambda \id_V)^n$.
Its dimension is the [[algebraic multiplicity|0FY42N02DV]] of $\lambda$, in finite dimensions.
Its nonzero elements are the [[generalised eigenvectors|01H6QKWGDH]] for $\lambda$.

## Explanations

### L1 human

The generalised eigenspace is a larger version of the eigenspace.
It contains the vectors that become zero after we apply $T - \lambda \id$ several times.
These spaces always fill the space over $\C$, even when the eigenspaces do not.
</document>

<document source="https://tamnd.github.io/math2/node/07PTPSZPME.md" tag="07PTPSZPME" kind="definition">
# Trace of a matrix

## Definition

Let $R$ be a commutative ring and let $A = (a_{ij})$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The *trace* of $A$ is
$$\tr A = \sum_{i=1}^{n} a_{ii} = a_{11} + a_{22} + \cdots + a_{nn}.$$

## Remarks

The trace of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is $5$.
The trace is [[linear|0GJWQZCPZM]], it is [[invariant under transpose|09E1RCZ7YT]], and [[$\tr(AB) = \tr(BA)$|0Y4NF1DRKD]].
Similar matrices have the same trace, so the [[trace of a linear map|0KZDNGA561]] is well defined.
Over an algebraically closed field, the trace is the [[sum of the eigenvalues|0NMXDCNNKR]], counted with multiplicity.

## Explanations

### L1 human

The trace adds up the numbers on the main diagonal of a square matrix.
It is a single number that does not change when we change the basis.
For the identity matrix of order $n$, the trace is $n$.
</document>

<document source="https://tamnd.github.io/math2/node/07Q4TBTFVB.md" tag="07Q4TBTFVB" kind="statement">
# Möbius inversion formula

## Statement

Let $f$ and $g$ be arithmetic functions.
Then $g(n) = \sum_{d \mid n} f(d)$ for all positive integers $n$ if and only if
$$f(n) = \sum_{d \mid n} \mu(d) \, g\!\left(\frac{n}{d}\right)$$
for all positive integers $n$. Here $\mu$ is the [[Möbius function|0C8QENHYZW]].

## Remarks

In terms of the [[Dirichlet convolution|0KH1N66QWW]], the statement is: $g = f * \mathbf{1}$ if and only if $f = g * \mu$.
The proof uses [[$\mu * \mathbf{1} = \varepsilon$|0ASC73QEQT]] and the associativity of $*$.
For example, the [[sum of the totient over the divisors|09JXE6XMCC]] gives $\sum_{d \mid n} \varphi(d) = n$. So $\varphi(n) = \sum_{d \mid n} \mu(d) \, n/d$.

## History

August Ferdinand Möbius introduced the formula into number theory in 1832.

## Explanations

### L1 human

If you know all the divisor sums of $f$, the formula recovers $f$ itself.
It undoes the operation "sum over the divisors", as subtraction undoes addition.
</document>

<document source="https://tamnd.github.io/math2/node/07Q7CNM5WF.md" tag="07Q7CNM5WF" kind="concept">
# Convex function

## Summary

A convex function on an interval is a function whose graph lies on or below each of its chords. Convex functions have increasing slopes, and they satisfy Jensen's inequality. The node [[convex function|0VV0HSW1ZS]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/07QY5RPFBR.md" tag="07QY5RPFBR" kind="definition">
# Subfield

## Definition

Let $L$ be a [[field|0QA8CHH8MG]].
A subfield of $L$ is a [[subring|0859D58WPD]] $K \subseteq L$ such that $a^{-1} \in K$ for every nonzero $a \in K$.

## Remarks

Equivalently, $K$ contains $0$ and $1$, and $K$ is closed under addition, negation, multiplication and the inverse of nonzero elements.
A subfield is a field with the operations of $L$ restricted to $K$.
The rationals $\mathbb{Q}$ are a subfield of $\mathbb{R}$, and $\mathbb{R}$ is a subfield of $\mathbb{C}$.
The intersection of any family of subfields of $L$ is a subfield of $L$. The intersection of all of them is the [[prime subfield|0VPFBAY7MW]].

## Explanations

### L1 human

A subfield is a smaller field that sits inside a bigger one.
You can add, subtract, multiply and divide its elements without leaving it.
The integers are a subring of $\mathbb{Q}$, but not a subfield, because $1/2 \notin \mathbb{Z}$.
</document>

<document source="https://tamnd.github.io/math2/node/07RB3EMBWW.md" tag="07RB3EMBWW" kind="statement">
# Third isomorphism theorem for vector spaces

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $W \subseteq U$ be subspaces of $V$.
Then $U / W$ is a subspace of $V / W$, and the map
$$(V / W) / (U / W) \to V / U, \qquad (v + W) + U/W \mapsto v + U,$$
is well defined, and it is a linear isomorphism of [[quotient spaces|0BR3FY5Q8M]].

## Remarks

The proof applies the [[first isomorphism theorem|0MTBGVZ4XJ]] to the surjective map $V / W \to V / U$, $v + W \mapsto v + U$, whose kernel is $U / W$.
In finite dimensions, both sides have dimension $\dim V - \dim U$.

## Explanations

### L1 human

Dividing first by a small subspace and then by a larger one is the same as dividing by the larger one at once.
This works like cancelling in fractions.
The same theorem holds for groups, rings and modules.
</document>

<document source="https://tamnd.github.io/math2/node/07RB7B04HX.md" tag="07RB7B04HX" kind="concept">
# Uniform continuity on a set

## Summary

A function is uniformly continuous on a set when its restriction to the set is uniformly continuous. Each continuous function on a closed bounded interval has this property. The node [[uniform continuity on a set|0A3CK6DVVB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/07RGNK57ED.md" tag="07RGNK57ED" kind="concept">
# Divisibility in a commutative ring

## Summary

In a commutative ring, an element $a$ divides an element $b$ when $b$ is a multiple of $a$.
The [[definition of divisibility|04GNK7K4VP]] states this precisely.
</document>

<document source="https://tamnd.github.io/math2/node/07RXBVTVZB.md" tag="07RXBVTVZB" kind="definition">
# Normal distribution

## Definition

Let $\mu \in \mathbb{R}$ and $\sigma^2 > 0$.
The normal distribution with mean $\mu$ and variance $\sigma^2$, written $\mathcal{N}(\mu, \sigma^2)$, is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$\varphi_{\mu, \sigma^2}(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right), \quad x \in \mathbb{R}.$$
The standard normal distribution is $\mathcal{N}(0, 1)$.

## Remarks

The function $\varphi_{\mu, \sigma^2}$ is the [[density of the normal distribution|027JF1FBB0]].
Some authors, and the library Mathlib, also allow $\sigma^2 = 0$. Then $\mathcal{N}(\mu, 0)$ is the [[Dirac measure|099SXAG014]] $\delta_\mu$.
If $X \sim \mathcal{N}(\mu, \sigma^2)$ and $a \ne 0$, $b$ are real, then $aX + b \sim \mathcal{N}(a\mu + b, a^2 \sigma^2)$. In particular, $(X - \mu)/\sigma \sim \mathcal{N}(0, 1)$.
The cumulative distribution function of $\mathcal{N}(0, 1)$ is written $\Phi$. It has no formula with elementary functions.

## Explanations

### L1 human

The normal distribution is the famous bell curve. It is symmetric about the mean $\mu$, and $\sigma$ measures its width.
Heights of people and errors of measurement are often close to normal.
</document>

<document source="https://tamnd.github.io/math2/node/07SDJ5DNE4.md" tag="07SDJ5DNE4" kind="concept">
# Equation of a plane

## Summary

A [[plane|0V3XAMF3C1]] in $\mathbb{R}^3$ is the set of the solutions of one linear equation in the coordinates.
Let $p_0 = (x_0, y_0, z_0)$ be a point and let $n = (a, b, c)$ be a nonzero vector. The plane through $p_0$ with the normal vector $n$ is the set of the points $p$ with $n \cdot (p - p_0) = 0$. In coordinates this is the point-normal form
$$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0.$$
Expanded, it is the general form $a x + b y + c z = d$ with $d = a x_0 + b y_0 + c z_0$. Conversely, for each real $d$ and each $(a, b, c) \neq (0, 0, 0)$, this equation defines a plane with the normal vector $(a, b, c)$.

## Remarks

Two equations define the same plane if and only if one is a nonzero multiple of the other.
For a plane through the point $p_0$ that contains the linearly independent vectors $u$ and $v$, the [[cross product|0N5N15HWZ7]] $u \times v$ is a normal vector.
Two planes are parallel or equal if and only if their normal vectors are multiples of each other.

## Explanations

### L1 human

The normal vector of a plane points straight out of it. A point $p$ is on the plane exactly when the arrow from $p_0$ to $p$ lies flat in the plane. Then this arrow is perpendicular to the normal vector. The dot product test turns this into one equation. For example, $z = 0$ is the floor plane with the normal vector $(0, 0, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/07SG12TQH3.md" tag="07SG12TQH3" kind="concept">
# T0 space

## Summary

A T0 space, or Kolmogorov space, is a topological space in which the open sets can tell any two distinct points apart.
It is the weakest of the usual separation axioms.
</document>

<document source="https://tamnd.github.io/math2/node/07SJXTWVQW.md" tag="07SJXTWVQW" kind="concept">
# Linear isometry

## Summary

A linear isometry is a linear map between normed spaces that keeps the norm of every vector.
</document>

<document source="https://tamnd.github.io/math2/node/07SQCNV233.md" tag="07SQCNV233" kind="concept">
# Diagonalisable matrix

## Summary

A square matrix is diagonalisable when it is similar to a diagonal matrix, that is, when its space has a basis of eigenvectors.
</document>

<document source="https://tamnd.github.io/math2/node/07T2TKGZ4J.md" tag="07T2TKGZ4J" kind="statement">
# Derivatives of the hyperbolic functions

## Statement

The functions $\sinh$, $\cosh$ and $\tanh$ are differentiable on $\mathbb{R}$, and for every real $x$
$$\sinh'(x) = \cosh x, \qquad \cosh'(x) = \sinh x, \qquad \tanh'(x) = \frac{1}{\cosh^2 x} = 1 - \tanh^2 x.$$

## Remarks

The first two follow from [[$\exp' = \exp$|0EX9GFJ0B3]] and $(e^{-x})' = -e^{-x}$.
The third follows from the [[quotient rule|0BS0NJPKVE]] and the identity [[$\cosh^2 - \sinh^2 = 1$|06QCTRFBM0]].
So $\sinh'' = \sinh$ and $\cosh'' = \cosh$. Mathlib states the first one as `Real.hasDerivAt_sinh`.

## Explanations

### L1 human

Unlike sine and cosine, there is no minus sign here.
Each of $\sinh$ and $\cosh$ is the derivative of the other.
</document>

<document source="https://tamnd.github.io/math2/node/07TXYYVYZR.md" tag="07TXYYVYZR" kind="definition">
# Distance from a point to a set

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $A$ be a nonempty subset of $X$ and let $x \in X$.
The distance from $x$ to $A$ is the real number
$$d(x, A) = \inf \{ d(x, a) : a \in A \}.$$

## Remarks

The [[infimum|04KFTA34C5]] exists, because the set is nonempty and bounded below by $0$.
If $x \in A$, then $d(x, A) = 0$. The converse is false: in $\mathbb{R}$, $d(0, (0, 1)) = 0$ but $0 \notin (0, 1)$.
In general $d(x, A) = 0$ if and only if $x$ is in the [[closure|0DRKT1QJR7]] of $A$.
For all $x, y \in X$, $|d(x, A) - d(y, A)| \le d(x, y)$. Thus $x \mapsto d(x, A)$ is [[Lipschitz|0TMZ21WKCN]] with constant $1$.

## Explanations

### L1 human

The distance from a point to a set is the shortest distance from the point to the set, when a shortest distance exists.
For example, the distance from $3$ to the interval $[0, 1]$ is $2$.
</document>

<document source="https://tamnd.github.io/math2/node/07TZ2Q4C1M.md" tag="07TZ2Q4C1M" kind="concept">
# Trajectory

## Summary

The trajectory, or orbit, of a point under a dynamical system is the set of states that the system reaches from this point over time.
Different orbits do not intersect, so the orbits form a partition of the phase space. See the [[definition|0BVT6VSB3S]].
</document>

<document source="https://tamnd.github.io/math2/node/07TZV87Y0T.md" tag="07TZV87Y0T" kind="definition">
# Recurrence relation

## Definition

Let $X$ be a set and let $k$ be a positive integer.
A recurrence relation of order $k$ on $X$ is an equation
$$u_{n+k} = f(n, u_n, u_{n+1}, \dots, u_{n+k-1}) \qquad (n \ge 0)$$
for a sequence $(u_n)_{n \ge 0}$ in $X$, where $f : \mathbb{N} \times X^k \to X$ is a given function.

A solution is a sequence that satisfies the equation for all $n \ge 0$.

## Remarks

For each choice of initial values $u_0, \dots, u_{k-1}$ there is exactly one solution. This follows by induction on $n$.
Examples: the [[factorial|0WFCGVMRRC]] satisfies $u_{n+1} = (n + 1) u_n$ of order $1$. The [[Fibonacci sequence|0GA9K57A39]] satisfies $u_{n+2} = u_{n+1} + u_n$ of order $2$.
An important special case is a [[linear recurrence with constant coefficients|081THG13JZ]].

## Explanations

### L1 human

A recurrence relation gives each term of a sequence from the terms just before it.
Once you know the first few terms, the rule determines all the others.
</document>

<document source="https://tamnd.github.io/math2/node/07VHQ31EYJ.md" tag="07VHQ31EYJ" kind="statement">
# Young's inequality for products

## Statement

Let $p, q > 1$ be real numbers with $\frac{1}{p} + \frac{1}{q} = 1$.
Then for all real $a, b \ge 0$,
$$a b \le \frac{a^p}{p} + \frac{b^q}{q}.$$
Equality holds if and only if $a^p = b^q$.

## Remarks

It is the weighted [[AM–GM inequality|0DRF0F3XC9]] for the numbers $a^p$ and $b^q$, with weights $1/p$ and $1/q$.
For $p = q = 2$ it is $a b \le (a^2 + b^2)/2$.
It is the main step in the proof of [[Hölder's inequality|0BKDAGMGX9]]. Mathlib states it as `Real.young_inequality`.

## Explanations

### L1 human

Young's inequality bounds a product by a sum of powers.
It splits the product in a way that depends on the pair of exponents $p$ and $q$.
</document>

<document source="https://tamnd.github.io/math2/node/07VX34QXAV.md" tag="07VX34QXAV" kind="concept">
# Solution of an ordinary differential equation

## Summary

A solution of an [[ordinary differential equation|0NT64JV96A]] is a function on an interval that satisfies the equation at each point.
For a first-order system given by a vector field, the solutions are the integral curves. These are the curves whose velocity is the value of the vector field at each point.
See the [[definition|04SW7WECS8]].
</document>

<document source="https://tamnd.github.io/math2/node/07W9H8QF38.md" tag="07W9H8QF38" kind="concept">
# Adjugate matrix

## Summary

The adjugate of a square matrix is the transpose of its matrix of cofactors.
</document>

<document source="https://tamnd.github.io/math2/node/07WBTX1MHA.md" tag="07WBTX1MHA" kind="concept">
# Inverse Laplace transform

## Summary

Let $F$ be a function of a complex variable. Its inverse Laplace transform is a piecewise continuous function $f$ of exponential growth whose Laplace transform is $F$.
It can be computed by the Bromwich integral or by Post's inversion formula. See the [[definition|0JRW4X1MDC]].
</document>

<document source="https://tamnd.github.io/math2/node/07WYQPB8E3.md" tag="07WYQPB8E3" kind="concept">
# Identity element

## Summary

An identity element for a binary operation is an element that does not change any element when it is combined with it on either side.
Examples are $0$ for addition and $1$ for multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/07X1QA6BRW.md" tag="07X1QA6BRW" kind="statement">
# Derivative of the logarithm

## Statement

The [[natural logarithm|0044CPHJBV]] is differentiable on $(0, \infty)$, and
$$\frac{d}{dx} \ln x = \frac{1}{x} \quad \text{for } x > 0.$$

## Remarks

By the [[inverse function rule|0M0RZ3HSB0]] and [[$\exp' = \exp$|0EX9GFJ0B3]], the derivative at $x = e^y$ is $\frac{1}{e^y} = \frac{1}{x}$.
For $x \ne 0$, the derivative of $\ln |x|$ is $1/x$.
By the chain rule, $(\ln f(x))' = f'(x)/f(x)$ when $f > 0$. Mathlib states the main claim as `Real.hasDerivAt_log`.

## Explanations

### L1 human

The slope of the graph of $\ln$ at $x$ is $1/x$.
So the logarithm grows more and more slowly as $x$ gets large.
</document>

<document source="https://tamnd.github.io/math2/node/07XBM3A9J5.md" tag="07XBM3A9J5" kind="concept">
# Algebra of limits

## Summary

The algebra of limits is a set of rules for limits of convergent sequences.
The rules give the limit of a sum, a product or a quotient from the limits of the parts. If $a_n \to a$ and $b_n \to b$, then $a_n + b_n \to a + b$, $a_n b_n \to a b$, and $a_n / b_n \to a / b$ when $b \ne 0$. These rules let us find many limits from a few known ones.

## Remarks

The rules are the nodes [[limit of a sum of sequences|0RAM4E07NR]], [[limit of a product of sequences|0734RSVF2V]] and [[limit of a quotient of sequences|03S6GWWKEZ]].
Similar rules hold for the [[limit of a function|0TXTSEMHR5]].

## Explanations

### L1 human

To find the limit of $(2n + 1)/(n + 3)$, divide the numerator and the denominator by $n$ to get $(2 + 1/n)/(1 + 3/n)$.
Since $1/n \to 0$, the rules give the limit $(2 + 0)/(1 + 0) = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/07XF282NWH.md" tag="07XF282NWH" kind="concept">
# Image of a set

## Summary

The image of a set under a function is the set of the values of the function at the elements of that set.
</document>

<document source="https://tamnd.github.io/math2/node/07Y2E7P9NP.md" tag="07Y2E7P9NP" kind="concept">
# Median

## Summary

A median of a distribution, or of a data set, is a value that splits it into two halves.
At least half of the probability is at or below the value, and at least half is at or above it.
The node [[median|08W2CW8PKW]] gives the precise definition for random variables.
</document>

<document source="https://tamnd.github.io/math2/node/07YQ1C78G0.md" tag="07YQ1C78G0" kind="concept">
# Linear form

## Summary

A linear form on a vector space is a linear map from the space to its field of scalars.
</document>

<document source="https://tamnd.github.io/math2/node/07ZMX4ANJS.md" tag="07ZMX4ANJS" kind="statement">
# Differentiable functions are continuous

## Statement

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$, and let $a \in I$.
If $f$ is [[differentiable|0A61B60B5R]] at $a$, then $f$ is [[continuous at|0EEE2CZB2A]] $a$.

## Remarks

For $x \ne a$, write $f(x) - f(a) = \frac{f(x) - f(a)}{x - a} \cdot (x - a)$. The first factor tends to $f'(a)$ and the second to $0$.
The converse fails. The [[absolute value|0CQ9ZM383N]] is continuous but not differentiable at $0$. The [[Weierstrass function|01A0MK1730]] is continuous and nowhere differentiable.
Mathlib states it as `HasDerivAt.continuousAt`.

## Explanations

### L1 human

If a graph has a tangent line at a point, it cannot jump there.
A jump would make the slopes of the secant lines blow up.
</document>

<document source="https://tamnd.github.io/math2/node/07ZS685JJC.md" tag="07ZS685JJC" kind="statement">
# Anticommutativity of the cross product

## Statement

Let $a$ and $b$ be vectors in $\mathbb{R}^3$, and let $\times$ be the [[cross product|0N5N15HWZ7]]. Then
$$b \times a = -(a \times b).$$

## Remarks

With $b = a$ the statement gives $a \times a = -(a \times a)$, thus $a \times a = 0$ for each vector $a$.
The proof is a direct computation: each component of $b \times a$ is the component of $a \times b$ with the two products in the other order.
The cross product is thus not commutative. For example, $e_1 \times e_2 = e_3$ but $e_2 \times e_1 = -e_3$.

## Explanations

### L1 human

If you swap the two factors of a cross product, the result points the other way. Its length stays the same. The right-hand rule shows this: swapping the factors turns your hand over.
</document>

<document source="https://tamnd.github.io/math2/node/080Q3RPKY6.md" tag="080Q3RPKY6" kind="statement">
# Every integer is a square times a square-free integer

## Statement

Let $n$ be a positive integer.
Then there are positive integers $a$ and $b$ such that $n = a^2 b$ and $b$ is [[square-free|004JW7A2N5]].

## Remarks

The integers $a$ and $b$ are unique. The number $b$ is the product of the primes $p$ for which the exponent of $p$ in $n$ is odd.
For example, $72 = 2^3 \cdot 3^2 = 6^2 \cdot 2$, so $a = 6$ and $b = 2$.
The proof uses the [[prime factorization|0AEKDFCYFA]] of $n$. Write each exponent $e$ as $e = 2 q + r$ with $r \in \{0, 1\}$.

## Explanations

### L1 human

Take out of $n$ all the pairs of equal primes that you can. The pairs give the square $a^2$.
At most one copy of each prime stays, so the rest $b$ is square-free.
For example, $50 = 2 \cdot 5 \cdot 5 = 5^2 \cdot 2$.
</document>

<document source="https://tamnd.github.io/math2/node/080RWVVSB6.md" tag="080RWVVSB6" kind="statement">
# Linearity of the integral

## Statement

Let $f, g \colon [a, b] \to \mathbb{R}$ be Riemann integrable, and let $\alpha, \beta \in \mathbb{R}$.
Then $\alpha f + \beta g$ is Riemann integrable on $[a, b]$, and
$$\int_a^b (\alpha f(x) + \beta g(x)) \, dx = \alpha \int_a^b f(x) \, dx + \beta \int_a^b g(x) \, dx.$$

## Remarks

The [[Riemann sums|0HBY48G8G1]] satisfy $S(\alpha f + \beta g, P, t) = \alpha S(f, P, t) + \beta S(g, P, t)$. Pass to the limit.
So the Riemann integrable functions on $[a, b]$ form a vector space, and the integral is a linear map on it.
Mathlib states additivity as `intervalIntegral.integral_add`.

## Explanations

### L1 human

The integral of a sum is the sum of the integrals.
Constant factors can move out of the integral.
</document>

<document source="https://tamnd.github.io/math2/node/080T2TXVQV.md" tag="080T2TXVQV" kind="definition">
# Irrational number

## Definition

A [[real number|08J2M1JYAG]] $x$ is irrational when $x$ is not a [[rational number|0NCJ5WXN6Z]].
That is, there are no integers $a$ and $b$ with $b \neq 0$ and $x = a / b$.
The set of irrational numbers is $\mathbb{R} \setminus \mathbb{Q}$.

## Remarks

The number $\sqrt{2}$ is [[irrational|0DZGSDHD92]]. The numbers $\pi$ and $e$ are also irrational.
The decimal expansion of a real number is eventually periodic exactly when the number is rational. Thus the decimal expansion of an irrational number never repeats.
The set $\mathbb{Q}$ is countable and $\mathbb{R}$ is not, thus almost all real numbers are irrational.

## Explanations

### L1 human

An irrational number is a number on the number line that you cannot write as a fraction.
The length of the diagonal of a square with side $1$ is such a number: it is $\sqrt{2} = 1.41421\dots$.
</document>

<document source="https://tamnd.github.io/math2/node/0819CQ1V8H.md" tag="0819CQ1V8H" kind="statement">
# Rational root theorem

## Statement

Let $f = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in \mathbb{Z}[x]$ with $n \ge 1$ and $a_n \neq 0$.
Let $r = p/q$ be a rational [[root|00EHX6FWC0]] of $f$, with $p, q \in \mathbb{Z}$, $q > 0$ and $\gcd(p, q) = 1$.
Then $p$ divides $a_0$ and $q$ divides $a_n$.

## Remarks

The proof: multiply $f(p/q) = 0$ by $q^n$. Then $a_n p^n = -q(\dots)$ and $a_0 q^n = -p(\dots)$, and $\gcd(p, q) = 1$ gives the two divisibilities.
For example, the only possible rational roots of $2x^3 - 3x + 1$ are $\pm 1$ and $\pm \tfrac{1}{2}$. Evaluation shows that $1$ is a root.
The same statement holds over every [[unique factorisation domain|06YMXJMXYS]] and its field of fractions.
For a monic $f$, every rational root is an integer.

## Explanations

### L1 human

The theorem gives a finite list of candidates for the rational roots.
You test each candidate by evaluation.
For $x^3 - 2$, the candidates are $\pm 1$ and $\pm 2$, and none is a root, so $\sqrt[3]{2}$ is irrational.
</document>

<document source="https://tamnd.github.io/math2/node/081QSEAGEY.md" tag="081QSEAGEY" kind="concept">
# Phase line

## Summary

Let $f$ be a function on an interval of $\mathbb{R}$. The phase line of the scalar [[autonomous equation|07BNRFQ0PJ]] $x' = f(x)$ is a diagram of the line of states.
It marks the [[equilibrium points|0G4X36AE0X]], which are the zeros of $f$. It puts an arrow on each interval between them: up where $f > 0$ and down where $f < 0$.
The arrows show the direction in which each solution moves.

## Remarks

An equilibrium point with both arrows toward it is stable, or a sink. With both arrows away from it, it is unstable, or a source. With one arrow toward it and one away, it is semi-stable.
For $f$ locally Lipschitz, a solution stays in one interval between equilibrium points and is monotone there.
For the [[logistic equation|0CC4VC7KR3]] $x' = x(1 - x)$, the point $0$ is a source and the point $1$ is a sink.

## Explanations

### L1 human

The phase line is a one-dimensional map of where the state can go.
Arrows show whether the state increases or decreases, and the equilibrium points are the places where it stops.
</document>

<document source="https://tamnd.github.io/math2/node/081THG13JZ.md" tag="081THG13JZ" kind="definition">
# Linear recurrence with constant coefficients

## Definition

Let $R$ be a commutative ring and let $d$ be a natural number.
A linear recurrence of order $d$ with constant coefficients over $R$ is given by coefficients $c_0, c_1, \dots, c_{d-1} \in R$.
It is the [[recurrence relation|07TZV87Y0T]]
$$u_{n+d} = c_0 u_n + c_1 u_{n+1} + \dots + c_{d-1} u_{n+d-1} \qquad (n \ge 0)$$
for sequences $(u_n)_{n \ge 0}$ in $R$.

## Remarks

This form is homogeneous. A non-homogeneous recurrence adds a term $g(n)$ that does not depend on $u$.
The [[Fibonacci sequence|0GA9K57A39]] satisfies the recurrence of order $2$ with $c_0 = c_1 = 1$.
The sequences that satisfy it are its [[solutions|0ZV2YKKZ58]]. Its [[characteristic polynomial|0BWDJGQYST]] is $X^d - c_{d-1} X^{d-1} - \dots - c_1 X - c_0$.

## Explanations

### L1 human

In a linear recurrence with constant coefficients, each new term is a fixed weighted sum of the previous $d$ terms.
The weights do not change with $n$.
</document>

<document source="https://tamnd.github.io/math2/node/081WKEDK6C.md" tag="081WKEDK6C" kind="concept">
# Analytic function

## Summary

An analytic function is a function that is given locally by a convergent power series. Analytic functions are smooth, but not all smooth functions are analytic. The node [[analytic function|0JD5ZNW0F4]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/081WTKYX6C.md" tag="081WTKYX6C" kind="statement">
# Finite p-groups are nilpotent

## Statement

Let $p$ be a prime number and let $G$ be a finite [[p-group|06QYRTX9KQ]].
Then $G$ is [[nilpotent|0Q7MMQHC1P]].

## Remarks

Each quotient $G/Z_i$ of the [[upper central series|0XW54DH28V]] is a $p$-group, so it has a [[nontrivial centre|0RZCTT30H5]] when it is not trivial.
So the upper central series grows strictly until it reaches $G$, and $G$ is finite.
For $|G| = p^n$ with $n \geq 2$, the nilpotency class of $G$ is at most $n - 1$.

## Explanations

### L1 human

A nontrivial finite $p$-group always has a nontrivial centre.
Divide by it, and the quotient is again a $p$-group with a nontrivial centre.
Repeat this, and you climb the upper central series to the whole group.
</document>

<document source="https://tamnd.github.io/math2/node/081YA6RCX6.md" tag="081YA6RCX6" kind="definition">
# Directrix

## Definition

Let $Q$ be a non-degenerate [[conic section|00HJCMC2CD]] in the Euclidean plane $\mathbb{R}^2$ that is not a circle. Let $F$ be a [[focus|0RBKKNH4GV]] of $Q$. For a point $P$ and a [[line|0MP9BCF6DP]] $\ell$, let $d(P, \ell)$ be the [[distance from $P$ to $\ell$|01AAYNSBN6]].
A directrix of $Q$ for the focus $F$ is a line $\ell$ that does not contain $F$, such that for a real number $e > 0$:
$$Q = \{\, P \in \mathbb{R}^2 \;:\; d(P, F) = e \, d(P, \ell) \,\}.$$

## Remarks

Each focus of $Q$ has exactly one directrix, and the number $e$ is the [[eccentricity|0HVX863SAJ]] of $Q$. The conic is an ellipse if $e < 1$, a parabola if $e = 1$ and a hyperbola if $e > 1$.
For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ or the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the directrices are the lines $x = \pm a^2 / c$, for the foci $(\pm c, 0)$.
A circle has no directrix in the Euclidean plane.

## Explanations

### L1 human

For a parabola, the directrix is the line in its definition: each point is as far from the focus as from the directrix. For an ellipse, each point is closer to the focus than to the directrix, by a fixed factor $e < 1$. For a hyperbola, the factor is greater than $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0826SGV3R0.md" tag="0826SGV3R0" kind="statement">
# Holomorphic functions are analytic

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $c \in U$ and $r > 0$ be such that the open disc $D = \{z : |z - c| < r\}$ is contained in $U$.
Then there are complex numbers $a_0, a_1, a_2, \dots$ such that the [[power series|0VARVQ9ZW1]] $\sum_{n \ge 0} a_n (z - c)^n$ converges to $f(z)$ for each $z \in D$.
For each $\rho$ with $0 < \rho < r$, the coefficients are
$$a_n = \frac{1}{2 \pi i} \oint_{|z - c| = \rho} \frac{f(z)}{(z - c)^{n+1}} \, dz.$$
In particular, $f$ is analytic on $U$: near each point of $U$, $f$ is the sum of a convergent power series.

## Remarks

Proof idea: in [[Cauchy's integral formula|0DG10TVYPN]], expand $\frac{1}{z - a}$ as a geometric series in $\frac{a - c}{z - c}$, and integrate term by term.
The radius of convergence of the Taylor series at $c$ is at least the distance from $c$ to the boundary of $U$.
For example, the Taylor series of $1/(1 - z)$ at $0$ has radius $1$, the distance from $0$ to the point $1$ where the function is not defined.
The converse is also true: [[power series are holomorphic in their disc of convergence|0YMP521ZR5]].

## Explanations

### L1 human

A function with one complex derivative everywhere on an open set is automatically equal to its Taylor series near each point.
This is very different from real functions, where a differentiable function can fail to have a second derivative.
</document>

<document source="https://tamnd.github.io/math2/node/082DWWXMRJ.md" tag="082DWWXMRJ" kind="concept">
# Finitely generated group

## Summary

A finitely generated group is a group that a finite subset generates.
All finite groups and the group of integers are finitely generated.
</document>

<document source="https://tamnd.github.io/math2/node/082GHNPJ53.md" tag="082GHNPJ53" kind="concept">
# Abelian group

## Summary

An abelian group is a group whose operation is commutative.
The integers with addition are the basic example.
</document>

<document source="https://tamnd.github.io/math2/node/082MGNWNA8.md" tag="082MGNWNA8" kind="statement">
# Absolutely convergent series in a Banach space converge

## Statement

Let $(V, \|\cdot\|)$ be a [[Banach space|0BDTTM24DP]] and let $(x_n)_{n \ge 1}$ be a sequence in $V$ with $\sum_{n \ge 1} \|x_n\| < \infty$.
Then the partial sums $s_N = \sum_{n=1}^{N} x_n$ converge to a point $s \in V$, and $\|s\| \le \sum_{n \ge 1} \|x_n\|$.

## Remarks

Proof: for $M > N$, the triangle inequality gives $\|s_M - s_N\| \le \sum_{n=N+1}^{M} \|x_n\|$. This tends to $0$, so $(s_N)$ is a [[Cauchy sequence|0MCK8K2D7Z]], and it converges because $V$ is [[complete|08EWT7J0SC]].
The converse holds: a normed space in which each absolutely convergent series converges is a Banach space.
For $V = \mathbb{R}$, this is the fact that absolutely convergent real series converge.

## Explanations

### L1 human

If the lengths of the terms have a finite sum, then the series converges in a Banach space.
This is a practical test for completeness, and it is often used to build solutions as sums of series.
</document>

<document source="https://tamnd.github.io/math2/node/082RMRSY26.md" tag="082RMRSY26" kind="definition">
# Equivalence class

## Definition

Let $\sim$ be an [[equivalence relation|0SGTMBEWRF]] on a set $A$ and let $a \in A$.
The equivalence class of $a$ is the set
$$[a] = \{x \in A \mid x \sim a\}.$$
Each element of $[a]$ is a representative of the class.

## Remarks

For congruence modulo $3$ on $\mathbb{Z}$, the class of $1$ is $[1] = \{\dots, -5, -2, 1, 4, 7, \dots\}$.
For $a, b \in A$, $[a] = [b]$ if and only if $a \sim b$. Otherwise, $[a]$ and $[b]$ are disjoint.
The set of all classes is the [[quotient set|0ZVHXDSZC7]].

## Explanations

### L1 human

An equivalence class collects all the objects that the relation treats as the same.
For "has the same birthday as", the class of a person is the group of all people born on the same day of the year.
</document>

<document source="https://tamnd.github.io/math2/node/0832XVR512.md" tag="0832XVR512" kind="definition">
# Successor function

## Definition

Let $\mathbb{N}$ be the set of [[natural numbers|0MWKJ44V5A]].
The successor function is the function $S \colon \mathbb{N} \to \mathbb{N}$ with $S(n) = n + 1$ for each $n$ in $\mathbb{N}$.
The number $S(n)$ is the successor of $n$.

## Remarks

In the [[Peano axioms|0QFT7Z76N5]], $S$ is a primitive function, and the axioms do not define it. Then $n + 1$ is a name for $S(n)$, and $1$ is a name for $S(0)$.
For example, $S(0) = 1$, $S(1) = 2$ and $S(S(2)) = 4$.

## Explanations

### L1 human

The successor of a number is the number that comes after it when you count.
The successor of $4$ is $5$, and the successor of $0$ is $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0839XNQXDK.md" tag="0839XNQXDK" kind="definition">
# Intersection of a family of sets

## Definition

Let $(A_i)_{i \in I}$ be a [[family of sets|05Q6G4EH2G]] with a nonempty index set $I$.
The intersection of the family is the set
$$\bigcap_{i \in I} A_i = \{x \mid \forall i \in I, \, x \in A_i\}.$$
So $x$ is an element of the intersection when $x \in A_i$ for each $i \in I$.

## Remarks

For $I = \{1, 2\}$, this is the [[intersection|0HBRW4E6G1]] $A_1 \cap A_2$.
For an empty index set, the condition is [[vacuously true|0V4QXPGJQK]] for each $x$.
So the intersection is defined only relative to a [[universal set|0BPYRMHJ48]] $U$, and it is $U$.
Example: $\bigcap_{n \ge 1} (-1/n, 1/n) = \{0\}$ in $\mathbb{R}$.

## Explanations

### L1 human

The intersection of a family keeps only the elements that are in every member of the family.
It generalizes $A \cap B$ from two sets to any number of sets.
</document>

<document source="https://tamnd.github.io/math2/node/08420828EY.md" tag="08420828EY" kind="definition">
# Number of prime factors counted with multiplicity

## Definition

Let $n$ be a positive integer with [[prime factorization|0AEKDFCYFA]] $n = p_1^{e_1} \cdots p_k^{e_k}$, with distinct primes $p_i$.
Then $\Omega(n) = e_1 + e_2 + \dots + e_k$.
So $\Omega(n)$ is the number of prime factors of $n$, counted with multiplicity.

## Remarks

For example, $\Omega(1) = 0$, $\Omega(12) = 3$ and $\Omega(360) = 6$.
The function $\Omega$ is completely additive: $\Omega(mn) = \Omega(m) + \Omega(n)$ for all positive $m$ and $n$.
The [[Liouville function|08RQAPXF1T]] is $\lambda(n) = (-1)^{\Omega(n)}$.

## Explanations

### L1 human

The function $\Omega(n)$ counts all the prime factors of $n$, with repeats.
For $12 = 2 \cdot 2 \cdot 3$, there are three factors, so $\Omega(12) = 3$, while $\omega(12) = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/084HH1NDCB.md" tag="084HH1NDCB" kind="statement">
# Fundamental theorem of arithmetic

## Statement

Each positive integer $n$ has exactly one [[prime factorization|0AEKDFCYFA]].
That is, there are [[primes|0SN0MAEKYX]] $p_1 \le p_2 \le \dots \le p_k$ with $n = p_1 p_2 \cdots p_k$.
If also $n = q_1 q_2 \cdots q_l$ with primes $q_1 \le q_2 \le \dots \le q_l$, then $k = l$ and $p_i = q_i$ for each $i$.

## Remarks

The [[existence of prime factorization|020K0X7HVR]] gives the first part. The uniqueness follows from [[Euclid's lemma|051VYKA1EX]].
The statement uses the convention that $1$ is not prime. If $1$ were prime, $6 = 2 \cdot 3 = 1 \cdot 2 \cdot 3$ would have two factorizations.
Unique factorization fails in some rings. In $\mathbb{Z}[\sqrt{-5}]$, $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ gives two different factorizations into irreducible elements.

## History

Euclid's Elements contains the main steps of the proof, in Book VII, Propositions 30 to 32, and in Book IX, Proposition 14.
Gauss proved the uniqueness in his Disquisitiones Arithmeticae of 1801.

## Explanations

### L1 human

Each positive integer has one and only one recipe as a product of primes.
You can multiply the primes in any order, but the list of primes stays the same.
For example, $12 = 2 \cdot 2 \cdot 3$, and no other list of primes has the product $12$.
</document>

<document source="https://tamnd.github.io/math2/node/084NGSTWM7.md" tag="084NGSTWM7" kind="definition">
# Higher derivative

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$.
Let $f^{(0)} = f$.
If $f^{(n)}$ is defined and differentiable on $I$, let $f^{(n+1)} = (f^{(n)})'$, its [[derivative function|027K9RMZ43]].
The function $f^{(n)}$ is the $n$-th derivative of $f$, and $f$ is $n$ times differentiable when it exists.

## Remarks

The [[second derivative|0Y9D2FJRQ4]] is $f'' = f^{(2)}$.
For example, the $n$-th derivative of $x^n$ is the constant $n!$.
Mathlib defines `iteratedDeriv n f`.

## Explanations

### L1 human

Higher derivatives come from differentiating again and again.
They describe finer features of a function, like the bending of its graph.
</document>

<document source="https://tamnd.github.io/math2/node/0850JJMJYZ.md" tag="0850JJMJYZ" kind="statement">
# P-adic valuation of a product

## Statement

Let $p$ be a prime number, and let $a$ and $b$ be nonzero integers.
Then $\nu_p(a b) = \nu_p(a) + \nu_p(b)$, where $\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].

## Remarks

With the convention $\nu_p(0) = \infty$, the formula holds for all integers $a$ and $b$.
The proof uses [[Euclid's lemma|051VYKA1EX]]. Write $a = p^i a'$ and $b = p^j b'$ with $p \nmid a'$ and $p \nmid b'$. Then $p \nmid a' b'$, so $\nu_p(a b) = i + j$.
For example, $\nu_2(12 \cdot 10) = \nu_2(120) = 3 = 2 + 1$.

## Explanations

### L1 human

The factors $p$ in $a b$ come from $a$ or from $b$, and no new factor $p$ appears.
So the number of factors $p$ in the product is the sum of the numbers of factors $p$ in $a$ and in $b$.
</document>

<document source="https://tamnd.github.io/math2/node/0859D58WPD.md" tag="0859D58WPD" kind="definition">
# Subring

## Definition

Let $R$ be a [[ring|0894J980RK]].
A subring of $R$ is a subset $S \subseteq R$ with these properties.

1. $1 \in S$.
2. If $a, b \in S$, then $a - b \in S$.
3. If $a, b \in S$, then $a \cdot b \in S$.

A subring $S$ is a ring with the addition and the multiplication of $R$, restricted to $S$.

## Remarks

The integers $\mathbb{Z}$ are a subring of $\mathbb{Q}$, and $\mathbb{Q}$ is a subring of $\mathbb{R}$.
The set $2\mathbb{Z}$ of even integers is closed under addition and multiplication, but it is not a subring of $\mathbb{Z}$, because $1 \notin 2\mathbb{Z}$.
Authors who do not require a ring to have an identity also do not require $1 \in S$.

## Explanations

### L1 human

A subring is a smaller ring inside a ring, with the same operations.
For example, the integers sit inside the rational numbers.
The sum, the difference and the product of two integers are again integers.
</document>

<document source="https://tamnd.github.io/math2/node/0859R2V6ES.md" tag="0859R2V6ES" kind="statement">
# Angle bisector theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $D$ be a point of the segment $BC$, and let $\angle$ denote the [[angle at a point|0GJAJC6TNZ]].
If $\angle BAD = \angle DAC$, then
$$\frac{d(B, D)}{d(D, C)} = \frac{d(A, B)}{d(A, C)}.$$
Conversely, if $D$ is a point of the segment $BC$ with this equality, then $\angle BAD = \angle DAC$.

## Remarks

In words, the bisector of the angle at $A$ cuts the opposite side in the ratio of the two adjacent sides.
In an isosceles triangle with $d(A, B) = d(A, C)$, the bisector from $A$ meets $BC$ at its midpoint.
A similar statement holds for the bisector of the exterior angle at $A$, which meets the line $BC$ outside the segment.

## History

The theorem is Proposition 3 of Book VI of Euclid's Elements.

## Explanations

### L1 human

Take the triangle with $d(A, B) = 6$ and $d(A, C) = 3$. The bisector of the angle at $A$ cuts $BC$ into two parts. The part next to $B$ is twice as long as the part next to $C$.
</document>

<document source="https://tamnd.github.io/math2/node/085CH3CYMX.md" tag="085CH3CYMX" kind="concept">
# Walk

## Summary

A walk in a graph is a sequence of vertices in which consecutive vertices are adjacent. Walks are the basic notion of movement in a graph, and trails, paths, circuits and cycles are special walks. The node [[walk|0VJZM0F9YM]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/085HQ5G5TE.md" tag="085HQ5G5TE" kind="statement">
# Jordan normal form

## Statement

Let $F$ be an algebraically closed field, for example $\C$, and let $A$ be an $n \times n$ matrix over $F$.
Then $A$ is [[similar|0E89HQATM8]] to a [[block diagonal matrix|001T5FJDC1]]
$$J = \begin{pmatrix} J_{k_1}(\lambda_1) & & \\ & \ddots & \\ & & J_{k_r}(\lambda_r) \end{pmatrix},$$
whose diagonal blocks are [[Jordan blocks|07PH4RX0X5]].
The matrix $J$ is unique up to the order of the blocks.
It is the *Jordan normal form* of $A$.

## Remarks

The $\lambda_i$ are the eigenvalues of $A$; they need not be distinct.
The number of blocks for $\lambda$ is its geometric multiplicity, and the sum of their sizes is its algebraic multiplicity.
The largest block for $\lambda$ has size equal to the multiplicity of $\lambda$ in the minimal polynomial.
The theorem holds over any field $F$ for the matrices whose characteristic polynomial splits over $F$.

## Explanations

### L1 human

Every complex square matrix becomes almost diagonal in a good basis.
The only entries off the diagonal are some $1$s just above it, inside the Jordan blocks.
Two complex matrices are similar exactly when they have the same Jordan form, up to the order of the blocks.
</document>

<document source="https://tamnd.github.io/math2/node/085MJS6907.md" tag="085MJS6907" kind="statement">
# Law of quadratic reciprocity

## Statement

Let $p$ and $q$ be distinct odd primes.
Then
$$\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2} \cdot \frac{q-1}{2}},$$
where the symbols are [[Legendre symbols|01J8AMV9RS]].

## Remarks

So $\left(\frac{p}{q}\right) = \left(\frac{q}{p}\right)$ unless $p \equiv q \equiv 3 \pmod 4$. In that case $\left(\frac{p}{q}\right) = -\left(\frac{q}{p}\right)$.
For example, $\left(\frac{7}{3}\right) = \left(\frac{1}{3}\right) = 1$, and $3 \equiv 7 \equiv 3 \pmod 4$. So $\left(\frac{3}{7}\right) = -1$.
With the [[first supplement|0PHBS78HKT]], the [[second supplement|0ZXW95S44M]] and [[multiplicativity|0NW1JP7TBR]], the law computes every Legendre symbol.

## History

Euler and Legendre conjectured the law.
Gauss first proved it in 1796, and he published proofs in his Disquisitiones Arithmeticae of 1801.
Gauss called it the fundamental theorem.

## Explanations

### L1 human

The law connects two questions: is $p$ a square modulo $q$, and is $q$ a square modulo $p$?
The answers are the same, except when both primes leave remainder $3$ on division by $4$. Then the answers are opposite.
This lets you swap and reduce the numbers, as in the [[Euclidean algorithm|0TRNT4X7J0]].
</document>

<document source="https://tamnd.github.io/math2/node/085PND4KNT.md" tag="085PND4KNT" kind="definition">
# Group isomorphism

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]].
A group isomorphism from $G$ to $H$ is a [[group homomorphism|0JYTG5K9Y4]] $\varphi \colon G \to H$ that is a bijection.
The groups $G$ and $H$ are isomorphic, written $G \cong H$, when there is an isomorphism from $G$ to $H$.

## Remarks

The inverse map of an isomorphism is again a homomorphism, thus an isomorphism.
Isomorphism is an equivalence relation on groups.
The exponential map is an isomorphism from $(\mathbb{R}, +)$ to the positive real numbers with multiplication.
Isomorphic groups have the same order, and corresponding elements have the same order.

## Explanations

### L1 human

Two groups are isomorphic when they are the same group with different names for the elements.
An isomorphism is a dictionary that translates the names and keeps all products correct.
For example, the rotations of a triangle and the integers modulo $3$ under addition are isomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/086BE0J3FZ.md" tag="086BE0J3FZ" kind="statement">
# Fermat's right triangle theorem

## Statement

There is no right triangle with integer side lengths whose area is the square of an integer.
Equivalently, the equation $x^4 - y^4 = z^2$ has no solution in positive integers $x$, $y$ and $z$.

## Remarks

In terms of a [[Pythagorean triple|01G1WY1GVX]] $(a, b, c)$, the theorem says that $ab/2$ is never a perfect square.
A consequence is [[Fermat's Last Theorem for n = 4|0E741HSMJ1]], since $a^4 + b^4 = c^4$ gives $c^4 - b^4 = (a^2)^2$.
The proof is by [[infinite descent|0EFYCFHYFG]].

## History

The proof was published in 1670 among the works of Pierre de Fermat, after his death.
It is the only complete proof by Fermat that survives.

## Explanations

### L1 human

The triangle with sides $3$, $4$, $5$ has area $6$, and the one with sides $5$, $12$, $13$ has area $30$.
Fermat proved that such an area can never be a perfect square.
</document>

<document source="https://tamnd.github.io/math2/node/086HEVXR9Z.md" tag="086HEVXR9Z" kind="statement">
# Mean of the Poisson distribution

## Statement

Let $\lambda > 0$ and let $X$ have the [[Poisson distribution|0VEFSSKX18]] with parameter $\lambda$. Then
$$E[X] = \sum_{k=0}^{\infty} k \, e^{-\lambda} \frac{\lambda^k}{k!} = \lambda.$$

## Remarks

Proof: the term with $k = 0$ is zero. For $k \ge 1$, $k \lambda^k / k! = \lambda \cdot \lambda^{k-1}/(k-1)!$. So the sum is $\lambda e^{-\lambda} \sum_{j \ge 0} \lambda^j / j! = \lambda e^{-\lambda} e^{\lambda} = \lambda$.
The variance of $X$ is also $\lambda$.
So the parameter $\lambda$ is the mean number of events.

## Explanations

### L1 human

The parameter of a Poisson distribution is its average.
If a shop gets on average $4$ customers per hour, a Poisson model of the number of customers in one hour uses $\lambda = 4$.
</document>

<document source="https://tamnd.github.io/math2/node/086QKPF0A6.md" tag="086QKPF0A6" kind="concept">
# Predicate logic

## Summary

Predicate logic extends [[propositional logic|09K9MWD2QQ]] with variables for objects, [[predicates|0T3DPBG981]] and quantifiers.
Its formulas can say "for all $x$" with the [[universal quantifier|0B88PQTJ66]] and "there exists $x$" with the [[existential quantifier|0JW2FHW7G3]].
In first-order logic, the quantifiers range over objects, not over sets of objects or predicates.

## Remarks

Other names are first-order logic and predicate calculus.
An example of a formula is $\forall x \, \exists y \, (y > x)$.
Most of mathematics can be written in first-order logic with the axioms of [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].

## Explanations

### L1 human

Propositional logic cannot say "every number has a larger number".
Predicate logic can: it has variables such as $x$ and $y$, and words such as "for all" and "there exists".
This is the logic of most mathematical statements.
</document>

<document source="https://tamnd.github.io/math2/node/0874138YP6.md" tag="0874138YP6" kind="definition">
# Graph colouring

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] with vertex set $V$, and let $C$ be a set, the set of colours.
A colouring of $G$ with colours in $C$ is a function $c : V \to C$ such that $c(u) \neq c(v)$ for any two adjacent vertices $u$ and $v$.

## Remarks

Such a colouring is also called proper. A colouring with a set of $k$ colours is a $k$-colouring.
Each colour class $c^{-1}(x)$ is an [[independent set|0KZJ82WEPP]].
A graph has a $2$-colouring if and only if it is [[bipartite|07DVHJ8N0A]].
The subject started with the colouring of maps. Francis Guthrie stated the four colour conjecture in 1852.

## Explanations

### L1 human

Colour the vertices of a graph so that the two ends of each edge have different colours.
For example, a timetable gives each exam a time slot, so that exams with a common student are at different times.
</document>

<document source="https://tamnd.github.io/math2/node/0878DN3WCS.md" tag="0878DN3WCS" kind="definition">
# Isometry

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]].
A function $f \colon X \to Y$ is an isometry when
$$d_Y\bigl(f(x), f(y)\bigr) = d_X(x, y) \quad \text{for all } x, y \in X.$$

## Remarks

Each isometry is [[injective|02J4G4J4TJ]] and [[Lipschitz|0TMZ21WKCN]] with constant $1$.
Some authors require an isometry to be surjective. Here a surjective isometry is an [[isometric isomorphism|0VRD8GC5BM]].
The map $x \mapsto (x, 0)$ from $\mathbb{R}$ to $\mathbb{R}^2$ with the Euclidean metric is an isometry, but it is not surjective.
Rotations, reflections and translations of $\mathbb{R}^2$ are isometries for the Euclidean metric.

## Explanations

### L1 human

An isometry moves points without stretching or shrinking any distance.
It can put a space into a bigger space, as the real line is put into the plane as the horizontal axis.
</document>

<document source="https://tamnd.github.io/math2/node/087EMZTHWN.md" tag="087EMZTHWN" kind="definition">
# von Mangoldt function

## Definition

The von Mangoldt function $\Lambda$ is the arithmetic function given, for each positive integer $n$, by
$$\Lambda(n) = \begin{cases} \log p & \text{if } n = p^k \text{ for a prime } p \text{ and an integer } k \ge 1, \\ 0 & \text{otherwise}. \end{cases}$$
Here $\log$ is the natural logarithm. The word prime has the meaning of the [[definition of a prime number|0SN0MAEKYX]].

## Remarks

For example, $\Lambda(8) = \log 2$, $\Lambda(7) = \log 7$, and $\Lambda(1) = \Lambda(6) = 0$.
For each positive integer $n$, $\sum_{d \mid n} \Lambda(d) = \log n$.
The function $\Lambda$ is not multiplicative. It is named after Hans von Mangoldt.
The sum $\psi(x) = \sum_{n \le x} \Lambda(n)$ is the second Chebyshev function. The [[prime number theorem|0EHBY5Y84N]] is equivalent to $\psi(x) \sim x$.

## Explanations

### L1 human

The von Mangoldt function puts the weight $\log p$ on each prime power $p^k$, and $0$ on all other numbers.
With these weights, the divisor sums give exactly $\log n$, which makes the function useful in the study of primes.
</document>

<document source="https://tamnd.github.io/math2/node/087P03NEND.md" tag="087P03NEND" kind="concept">
# Statistical hypothesis test

## Summary

A statistical hypothesis test is a rule that uses data to decide whether to reject a hypothesis about the law of the data.
One fixes a null hypothesis $H_0$ and an alternative hypothesis $H_1$, for example $H_0 \colon \theta \in \Theta_0$ and $H_1 \colon \theta \in \Theta_1$ with disjoint sets $\Theta_0$ and $\Theta_1$.
One computes a test statistic $T(X)$ and rejects $H_0$ when $T(X)$ is in a critical region $C$. The region is chosen so that $P_\theta(T(X) \in C) \le \alpha$ for each $\theta \in \Theta_0$, for a significance level $\alpha$ such as $0.05$.

## Remarks

Equivalently, one can compute the [[p-value|0EHYZFZ4PK]] and reject $H_0$ when it is at most $\alpha$.
A test can make [[two types of errors|0R41PVFA39]]: reject a true $H_0$, or fail to reject a false $H_0$. The level $\alpha$ bounds the probability of the first error.
The power of the test at $\theta \in \Theta_1$ is $P_\theta(T(X) \in C)$, the probability to reject $H_0$ when it is false.
Not rejecting $H_0$ is not a proof that $H_0$ is true.

## Explanations

### L1 human

A hypothesis test asks whether the data are surprising under a default assumption, the null hypothesis.
If the data are too surprising, one rejects the assumption. The significance level fixes how often one accepts to reject a true assumption.
</document>

<document source="https://tamnd.github.io/math2/node/087VJM5Q77.md" tag="087VJM5Q77" kind="concept">
# Translation

## Summary

A translation moves each point of a space by the same distance in the same direction. It adds one fixed vector to each point.
</document>

<document source="https://tamnd.github.io/math2/node/0880F9EQCP.md" tag="0880F9EQCP" kind="definition">
# Positive semidefinite matrix

## Definition

Let $A$ be an $n \times n$ complex [[Hermitian matrix|0P17NGQQMQ]].
The matrix $A$ is *positive semidefinite* when
$$x^* A x \ge 0 \qquad \text{for all } x \in \C^n.$$
For a real symmetric $A$, the condition is $x^T A x \ge 0$ for all $x \in \R^n$.

## Remarks

$A$ is positive semidefinite if and only if all its eigenvalues are nonnegative.
Each [[positive definite|0KX66CZMBB]] matrix is positive semidefinite.
The matrix $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is positive semidefinite but not positive definite.
The notation $A \succeq 0$ is common.

## Explanations

### L1 human

A positive semidefinite matrix never gives a negative value of $x^T A x$.
It may give zero for some nonzero vectors.
Covariance matrices and Gram matrices are always of this kind.
</document>

<document source="https://tamnd.github.io/math2/node/08895Q5W96.md" tag="08895Q5W96" kind="statement">
# Chevalley–Warning theorem

## Statement

Let $\mathbb{F}$ be a finite field of characteristic $p$.
Let $f_1, \dots, f_r \in \mathbb{F}[X_1, \dots, X_n]$ be polynomials with total degrees $d_1, \dots, d_r$, such that $n > d_1 + \dots + d_r$.
Then the number of common zeros $(a_1, \dots, a_n) \in \mathbb{F}^n$ of $f_1, \dots, f_r$ is divisible by $p$.

## Remarks

A corollary is Chevalley's theorem: if the polynomials have no constant terms, then they have a common zero other than $(0, \dots, 0)$.
The reason is that $(0, \dots, 0)$ is a zero, and the number of zeros is a multiple of $p \ge 2$.
For example, a quadratic form in $3$ or more variables over $\mathbb{Z}/p\mathbb{Z}$ has a nontrivial zero.

## History

Ewald Warning and Claude Chevalley proved the results in 1935.

## Explanations

### L1 human

When a system of polynomial equations over a finite field has many variables compared with the degrees, the solutions come in multiples of $p$.
So if there is one obvious solution, such as all zeros, there must be another.
</document>

<document source="https://tamnd.github.io/math2/node/088RYKAWK9.md" tag="088RYKAWK9" kind="concept">
# Spanning set

## Summary

A spanning set of a vector space is a set of vectors whose linear span is the whole space.
</document>

<document source="https://tamnd.github.io/math2/node/0894J980RK.md" tag="0894J980RK" kind="definition">
# Ring

## Definition

A ring is a set $R$ with two binary operations and two elements $0, 1 \in R$ that satisfy the axioms below.
The operations are the addition $+$ and the multiplication $\cdot$.

1. $(R, +)$ is an abelian [[group|G5S1M9]] with identity element $0$.
2. The multiplication is associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in R$.
3. The element $1$ is a multiplicative identity: $1 \cdot a = a \cdot 1 = a$ for all $a \in R$.
4. The multiplication is distributive over the addition: $a \cdot (b + c) = a \cdot b + a \cdot c$ and $(b + c) \cdot a = b \cdot a + c \cdot a$ for all $a, b, c \in R$.

## Remarks

Some authors do not require the element $1$. They call the structure of this definition a ring with identity, or a unital ring.
The axioms give $0 \cdot a = a \cdot 0 = 0$ and $(-a) \cdot b = -(a \cdot b)$ for all $a, b \in R$.
The integers $\mathbb{Z}$, the integers modulo $n$ and the $n \times n$ real matrices are rings.
A ring in which $a \cdot b = b \cdot a$ for all $a, b$ is a [[commutative ring|0GYGHKX834]].

## Explanations

### L1 human

A ring is a place where you can add, subtract and multiply, as with the integers.
You cannot always divide.
The order of a product can matter: for two matrices $A$ and $B$, $AB$ and $BA$ are often different.
</document>

<document source="https://tamnd.github.io/math2/node/089KWVYD69.md" tag="089KWVYD69" kind="definition">
# Periodic function

## Definition

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a function and let $T$ be a real number.
The number $T$ is a period of $f$ if $f(x + T) = f(x)$ for all $x \in \mathbb{R}$.
The function $f$ is periodic if it has a period $T \ne 0$.

## Remarks

If $T$ is a period, then so is $k T$ for each integer $k$.
The [[sine function is periodic|0P4NQD746K]] with period $2 \pi$.
If a continuous periodic function is not constant, it has a smallest positive period. The [[Dirichlet function|0T4EJ8ZJM9]] has every rational number as a period.
Mathlib defines periods as `Function.Periodic`.

## Explanations

### L1 human

A periodic function repeats itself after each step of length $T$.
Its graph is made of copies of one piece, placed side by side.
</document>

<document source="https://tamnd.github.io/math2/node/08ABT85339.md" tag="08ABT85339" kind="concept">
# Cyclic subgroup generated by an element

## Summary

The cyclic subgroup generated by an element of a group is the set of all integer powers of the element.
It is the smallest subgroup that contains the element.
</document>

<document source="https://tamnd.github.io/math2/node/08AM0KRCZ1.md" tag="08AM0KRCZ1" kind="statement">
# Divergence part of the ratio test

## Statement

Let $(a_n)$ be a real sequence with $a_n \ne 0$ for all $n$.
Assume that $|a_{n+1}| / |a_n| \to L$ as $n \to \infty$, for a real number $L$ with $L > 1$.
Then the series $\sum a_n$ diverges.

## Remarks

For large $n$ we have $|a_{n+1}| > |a_n|$. So $|a_n|$ is eventually increasing and positive, and $a_n$ does not tend to $0$. The [[term test|0VVY4H92G2]] gives divergence.
The same proof works when $|a_{n+1}| / |a_n| \to +\infty$.
For example, $\sum 2^n / n$ diverges, since the ratio is $2n/(n+1) \to 2$.

## Explanations

### L1 human

If each term is eventually bigger than the term before, the terms cannot shrink to $0$.
Then the series cannot converge.
</document>

<document source="https://tamnd.github.io/math2/node/08ANASGBXV.md" tag="08ANASGBXV" kind="concept">
# Moment of inertia

## Summary

The moment of inertia of a body about an axis measures how hard it is to change the rotation of the body about that axis.
It is the integral of the density times the square of the distance to the axis. So for a solid body it is a [[triple integral|0698M7C7AX]].
</document>

<document source="https://tamnd.github.io/math2/node/08ANQHHG3P.md" tag="08ANQHHG3P" kind="concept">
# Independent set

## Summary

An independent set in a graph is a set of vertices with no edges between them. Independent sets are dual to cliques and to vertex covers, and they are the colour classes of colourings. The node [[independent set|0KZJ82WEPP]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/08B1RTW7VZ.md" tag="08B1RTW7VZ" kind="concept">
# Diophantine equation

## Summary

A Diophantine equation is a polynomial equation with integer coefficients, for which one looks for integer solutions, or sometimes rational solutions.
Typical questions ask whether a solution exists, whether there are finitely many, and how to find all of them.

## Remarks

Examples are the linear equation $ax + by = c$, [[Pythagorean triples|01G1WY1GVX]] $x^2 + y^2 = z^2$, [[Pell's equation|0AF3DANJ76]] $x^2 - d y^2 = 1$ and the equation of [[Fermat's Last Theorem|0Y5X3SDBCZ]].
By Hilbert's tenth problem, as solved by Yuri Matiyasevich in 1970, no algorithm decides for every Diophantine equation whether it has an integer solution.
The name refers to Diophantus of Alexandria.

## Explanations

### L1 human

In a Diophantine equation, only whole numbers count as solutions.
For example, $x^2 + y^2 = 3$ has real solutions, but no integer solutions.
The restriction to integers makes such equations much harder than ordinary equations.
</document>

<document source="https://tamnd.github.io/math2/node/08B9CX109G.md" tag="08B9CX109G" kind="concept">
# Adjacency matrix

## Summary

The adjacency matrix of a finite graph records adjacency as a matrix of zeros and ones. It lets linear algebra study graphs, for example through eigenvalues and powers that count walks. The node [[adjacency matrix|0NZ8MDBN6Q]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/08BPSC6YTE.md" tag="08BPSC6YTE" kind="statement">
# Class equation

## Statement

Let $G$ be a finite [[group|G5S1M9]] with [[centre|0DZ93493VN]] $Z(G)$.
Let $a_1, \dots, a_k$ be one element from each [[conjugacy class|0DEBXAJ5SA]] of $G$ that has more than one element.
Then $|G| = |Z(G)| + \sum_{i=1}^{k} [G : C_G(a_i)]$.

## Remarks

The conjugacy classes partition $G$, and the classes with one element are exactly the elements of $Z(G)$.
Each term $[G : C_G(a_i)]$ is the [[size of a class|0VK5V9P8G3]], so it divides $|G|$ and is greater than $1$.
For $S_3$, the equation is $6 = 1 + 3 + 2$.
The class equation is the key step in the proof that a nontrivial finite $p$-group has a nontrivial centre.

## Explanations

### L1 human

Count the elements of the group class by class.
The central elements form classes of size one, and the other classes have sizes that divide the order of the group.
This simple count gives strong facts, for example about groups of prime power order.
</document>

<document source="https://tamnd.github.io/math2/node/08C1NWBNHA.md" tag="08C1NWBNHA" kind="statement">
# Product of primitive polynomials is primitive

## Statement

Let $R$ be a [[unique factorisation domain|06YMXJMXYS]] and let $f, g \in R[x]$ be [[primitive polynomials|0SP14DZN0G]].
Then $f g$ is primitive.

## Remarks

The proof: suppose an irreducible $p$ divides all the coefficients of $f g$. Since $p$ is [[prime|0MNHXBSX3M]], $R/(p)$ is an integral domain. The images of $f$ and $g$ in $(R/(p))[x]$ are nonzero, but their product is zero, which is a contradiction.
Many texts call this statement Gauss's lemma. The node [[Gauss's lemma|0QFV5Y1Z5X]] states the form about irreducibility.
Over $\mathbb{Z}$: $2x + 1$ and $3x + 2$ are primitive, and so is $6x^2 + 7x + 2$.

## Explanations

### L1 human

Suppose that no prime divides all the coefficients of $f$, and no prime divides all the coefficients of $g$.
Then no prime divides all the coefficients of $f g$.
The proof reduces the coefficients modulo a prime and uses that the result is an integral domain.
</document>

<document source="https://tamnd.github.io/math2/node/08CJB2NQ92.md" tag="08CJB2NQ92" kind="statement">
# Existence of an eigenvalue over an algebraically closed field

## Statement

Let $F$ be an algebraically closed field, let $V$ be a finite-dimensional vector space over $F$ with $V \ne 0$, and let $T \colon V \to V$ be a linear map.
Then $T$ has at least one [[eigenvalue|0BQAA2BK3Q]] in $F$.

## Remarks

The [[characteristic polynomial|0J5EMJRRFM]] of $T$ has degree $\dim V \ge 1$, so it has a root in $F$.
The statement applies to $F = \C$, by the fundamental theorem of algebra.
It fails over $\R$: the rotation of $\R^2$ by $90$ degrees has no real eigenvalue.
It also fails in infinite dimensions: the map $p(x) \mapsto x\, p(x)$ on $\C[x]$ has no eigenvalue.

## Explanations

### L1 human

Over the complex numbers, every square matrix has at least one eigenvector.
The reason is that every nonconstant complex polynomial has a root.
Over the real numbers, a rotation can move every line, so no real eigenvector exists.
</document>

<document source="https://tamnd.github.io/math2/node/08EJC6Q25H.md" tag="08EJC6Q25H" kind="definition">
# Binary relation

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
A binary relation from $A$ to $B$ is a [[subset|0FE8Q5S42Y]] $R$ of the [[Cartesian product|0C81YBVTWH]] $A \times B$.
For $a \in A$ and $b \in B$, write $a \mathrel{R} b$ when $(a, b) \in R$.
A binary relation on $A$ is a binary relation from $A$ to $A$.

## Remarks

Examples on $\mathbb{Z}$ are $\le$, $=$ and divisibility $\mid$.
A relation on a finite set can be drawn as a directed graph, with an arrow from $a$ to $b$ when $a \mathrel{R} b$.
A [[function|0XHDTE5YMZ]] is a special kind of binary relation.

## Explanations

### L1 human

A relation is a rule that says which pairs belong together.
For the relation "divides" on $\{1, 2, 3, 4\}$, the pair $(2, 4)$ is in the relation and the pair $(3, 4)$ is not.
</document>

<document source="https://tamnd.github.io/math2/node/08EJHHMB5R.md" tag="08EJHHMB5R" kind="definition">
# Equinumerous sets

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The sets $A$ and $B$ are equinumerous, written $A \approx B$, when there is a [[bijective function|0SNQEF7Z53]] $f \colon A \to B$.

## Remarks

Equinumerosity is reflexive, symmetric and transitive. The identity, the inverse and the composition of bijections give the three properties.
The set $\mathbb{N}$ is equinumerous with the set of even natural numbers, by $n \mapsto 2n$. So an infinite set can be equinumerous with a proper subset.
Other names are "equipotent" and "have the same [[cardinality|0JTGRV4WT4]]".

## Explanations

### L1 human

You do not need to count to compare sizes.
If each chair in a room has one person and each person has one chair, there are as many people as chairs.
Equinumerosity applies the same idea to infinite sets.
</document>

<document source="https://tamnd.github.io/math2/node/08EVNYZF42.md" tag="08EVNYZF42" kind="definition">
# Fréchet derivative

## Definition

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$, let $U \subseteq V$ be an open set, let $f \colon U \to W$, and let $x \in U$.
A bounded linear map $A \colon V \to W$ is a Fréchet derivative of $f$ at $x$ when
$$\lim_{h \to 0} \frac{\|f(x + h) - f(x) - A h\|_W}{\|h\|_V} = 0.$$
The limit is over $h \in V \setminus \{0\}$ with $x + h \in U$.
When such an $A$ exists, $f$ is Fréchet differentiable at $x$, with the notation $Df(x) = A$ or $f'(x) = A$.

## Remarks

The Fréchet derivative is unique when it exists, see the [[uniqueness of the derivative|0762WAE49P]].
For $V = W = \mathbb{R}$, a linear map $A$ is multiplication by a number $a$. Then the condition says that $f$ has the derivative $a$ at $x$ in the usual sense.
For $V = \mathbb{R}^n$ and $W = \mathbb{R}^m$, each linear map is bounded, and $Df(x)$ is the [[total derivative|0FNKFT8H6E]] of $f$ at $x$. Its matrix is the [[Jacobian matrix|047NDTEPCV]].
If $f$ is Fréchet differentiable at $x$, then $f$ is continuous at $x$.

## Explanations

### L1 human

Near $x$, the change $f(x + h) - f(x)$ is close to $A h$, a linear function of the step $h$.
The error is small even compared with the length of the step, when the step is short.
So $A$ plays the role of the slope of a tangent line, but now the tangent object is a linear map.
</document>

<document source="https://tamnd.github.io/math2/node/08EWT7J0SC.md" tag="08EWT7J0SC" kind="definition">
# Complete metric space

## Definition

A [[metric space|063W5JMY85]] $(X, d)$ is complete when each [[Cauchy sequence|0MCK8K2D7Z]] in $X$ [[converges|06JYPJ4WHH]] to a point of $X$.

## Remarks

The real line $\mathbb{R}$ with the usual metric is complete. So is $\mathbb{R}^n$ with the Euclidean metric.
The rationals $\mathbb{Q}$ and the open interval $(0, 1)$ are not complete. In $(0, 1)$, the sequence $1/n$ is Cauchy but has no limit in $(0, 1)$.
Completeness depends on the metric, not only on the topology: $(0, 1)$ is homeomorphic to $\mathbb{R}$, but only $\mathbb{R}$ is complete with the usual metric.
Each metric space has a [[completion|0G4J7Z7WPJ]].

## Explanations

### L1 human

A complete space has no holes: each sequence that looks convergent really has a limit in the space.
The rationals have holes, for example at $\sqrt{2}$, and the reals fill them.
</document>

<document source="https://tamnd.github.io/math2/node/08FDFEF8D1.md" tag="08FDFEF8D1" kind="statement">
# Product of cyclic groups is cyclic iff the orders are coprime

## Statement

Let $G$ and $H$ be finite [[cyclic groups|0KVGH0J5B3]] of orders $m$ and $n$.
Let $G \times H$ be the group of pairs $(g, h)$ with $g \in G$ and $h \in H$, with the operation $(g, h)(g', h') = (gg', hh')$.
Then $G \times H$ is cyclic if and only if $\gcd(m, n) = 1$.

## Remarks

When $\gcd(m, n) = 1$, the group $\mathbb{Z}/mn\mathbb{Z}$ is isomorphic to $\mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$, and this is a form of the Chinese remainder theorem.
When $\gcd(m, n) > 1$, each element of $G \times H$ has order at most $\operatorname{lcm}(m, n) < mn$.
For example, $\mathbb{Z}/12\mathbb{Z}$ is isomorphic to $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, but $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$ is not cyclic.

## Explanations

### L1 human

The order of a pair $(g, h)$ is the least common multiple of the orders of $g$ and $h$.
A pair of generators has order $mn$ exactly when $m$ and $n$ have no common factor.
For example, $(1, 1)$ in $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ has order $6$, so this group is cyclic.
</document>

<document source="https://tamnd.github.io/math2/node/08FHNR8Z5D.md" tag="08FHNR8Z5D" kind="definition">
# Group of units of a monoid

## Definition

Let $(M, \ast, e)$ be a [[monoid|0N6R902FG1]].
An element $u \in M$ is a unit when it has a two-sided [[inverse|0ZNZH0EXFX]], that is, an element $v \in M$ with $u \ast v = v \ast u = e$.
The group of units of $M$ is the set $M^\times$ of the units of $M$, with the operation $\ast$ restricted to $M^\times$.
It is a [[group|G5S1M9]] with identity $e$.

## Remarks

The units of the monoid $(\mathbb{Z}, \cdot)$ are $1$ and $-1$.
The units of the monoid of $n \times n$ real matrices under multiplication are the invertible matrices, and they form the general linear group $\mathrm{GL}_n(\mathbb{R})$.
For a ring $R$, the group of units $R^\times$ is the group of units of the monoid $(R, \cdot)$.

## Explanations

### L1 human

In a monoid, some elements can be undone and others cannot.
The elements that can be undone are the units.
The units alone form a group, because products and inverses of units are units.
</document>

<document source="https://tamnd.github.io/math2/node/08FKQD7NJ3.md" tag="08FKQD7NJ3" kind="statement">
# Homomorphisms preserve identity and inverses

## Statement

Let $G$ and $H$ be [[groups|G5S1M9]] with identity elements $e_G$ and $e_H$, and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
Then $\varphi(e_G) = e_H$, and $\varphi(a^{-1}) = \varphi(a)^{-1}$ for each $a \in G$.

## Remarks

The first equation follows from $\varphi(e_G) \varphi(e_G) = \varphi(e_G) = \varphi(e_G) e_H$ and [[cancellation|0VSNX4P2Q0]].
The second follows from $\varphi(a) \varphi(a^{-1}) = \varphi(e_G) = e_H$ and [[the uniqueness of inverses|0D9NDGE9KP]].
A consequence is that $\varphi(a^n) = \varphi(a)^n$ for each integer $n$.

## Explanations

### L1 human

A map that respects products automatically respects the identity and the inverses.
You do not need to ask for these two properties in the definition of a homomorphism.
For example, the exponential map sends $0$ to $1$ and $-x$ to $1/e^x$.
</document>

<document source="https://tamnd.github.io/math2/node/08FPABDQ4C.md" tag="08FPABDQ4C" kind="definition">
# Uniform convergence

## Definition

Let $S$ be a set, and let $f_n \colon S \to \mathbb{R}$ for $n \ge 1$ and $f \colon S \to \mathbb{R}$ be functions.
The sequence $(f_n)$ converges uniformly to $f$ if for every $\varepsilon > 0$ there is $N$ such that
$$|f_n(x) - f(x)| < \varepsilon \quad \text{for all } n \ge N \text{ and all } x \in S.$$

## Remarks

Here $N$ depends only on $\varepsilon$, not on $x$. Uniform convergence implies [[pointwise convergence|056FSKG32J]].
Equivalently, $\sup_{x \in S} |f_n(x) - f(x)| \to 0$ as $n \to \infty$.
For example, $\frac{\sin(n x)}{n} \to 0$ uniformly on $\mathbb{R}$. Mathlib defines it as `TendstoUniformly`.

## Explanations

### L1 human

Uniform convergence means that the whole graph of $f_n$ lies in a thin band around the graph of $f$.
The width of the band tends to $0$, at the same rate for all points.
</document>

<document source="https://tamnd.github.io/math2/node/08G4XH86JC.md" tag="08G4XH86JC" kind="statement">
# Product of countable sets is countable

## Statement

Let $A$ and $B$ be [[countable sets|0ZG61TFBES]].
Then the [[Cartesian product|0C81YBVTWH]] $A \times B$ is countable.

## Remarks

Proof: take injections $f \colon A \to \mathbb{N}$ and $g \colon B \to \mathbb{N}$. Then $(a, b) \mapsto \pi(f(a), g(b))$ is an injection to $\mathbb{N}$, where $\pi$ is the [[Cantor pairing function|0NP5N5PYW5]].
By induction, a finite product $A_1 \times \dots \times A_n$ of countable sets is countable.
An infinite product need not be countable: the set of sequences of $0$ and $1$ is [[uncountable|0X55M9XET4]].

## Explanations

### L1 human

The pairs of natural numbers fill an infinite grid, and the grid can be listed diagonal by diagonal.
So a product of two lists is again a list.
</document>

<document source="https://tamnd.github.io/math2/node/08GR54CKEA.md" tag="08GR54CKEA" kind="concept">
# Floor function

## Summary

The floor function sends a real number $x$ to the greatest integer that is less than or equal to $x$. We write $\lfloor x \rfloor$ for it.
For example, $\lfloor 2.7 \rfloor = 2$ and $\lfloor -2.7 \rfloor = -3$.
</document>

<document source="https://tamnd.github.io/math2/node/08HJRTG5N7.md" tag="08HJRTG5N7" kind="definition">
# Uniform convergence on a set

## Definition

Let $D$ be a set, let $f_n \colon D \to \mathbb{R}$ for $n \ge 1$ and $f \colon D \to \mathbb{R}$ be functions, and let $S \subseteq D$.
The sequence $(f_n)$ converges uniformly to $f$ on $S$ if for every $\varepsilon > 0$ there is $N$ with $|f_n(x) - f(x)| < \varepsilon$ for all $n \ge N$ and all $x \in S$.

## Remarks

This is [[uniform convergence|08FPABDQ4C]] of the restrictions of $f_n$ and $f$ to $S$.
A sequence can converge uniformly on each $[0, r]$ with $r < 1$, but not on $[0, 1)$. The sequence $x^n$ is an example.
Mathlib defines it as `TendstoUniformlyOn`.

## Explanations

### L1 human

We often get uniform convergence only on part of the domain.
This definition measures the distance between $f_n$ and $f$ only on that part.
</document>

<document source="https://tamnd.github.io/math2/node/08HK38GCSM.md" tag="08HK38GCSM" kind="definition">
# Exterior algebra

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
Let $T(V) = \bigoplus_{k \ge 0} V^{\otimes k}$ be the tensor algebra of $V$, where $V^{\otimes 0} = F$ and the product is the [[tensor product|0BF2DE2SYK]].
Let $I$ be the two-sided ideal of $T(V)$ generated by the elements $v \otimes v$ for $v \in V$.
The *exterior algebra* of $V$ is the quotient algebra $\Lambda(V) = T(V) / I$.
The product in $\Lambda(V)$ is written $\wedge$.

## Remarks

For all $u, v \in V$, $v \wedge v = 0$ and $u \wedge v = -v \wedge u$.
The image of $V^{\otimes k}$ is the $k$-th exterior power $\Lambda^k(V)$; if $\dim V = n$, then $\dim \Lambda^k(V) = \binom{n}{k}$ and $\dim \Lambda(V) = 2^n$.
If $(e_1, \dots, e_n)$ is a basis of $V$, then $\Lambda^n(V)$ is spanned by $e_1 \wedge \cdots \wedge e_n$.
For $v_j = \sum_i a_{ij} e_i$, $v_1 \wedge \cdots \wedge v_n = \det(a_{ij})\, e_1 \wedge \cdots \wedge e_n$.

## Explanations

### L1 human

The exterior algebra multiplies vectors with the wedge product, which changes sign when we swap two factors.
In $\R^3$, $u \wedge v$ describes an oriented parallelogram, and $u \wedge v \wedge w$ an oriented volume.
The top wedge product of $n$ vectors gives the determinant.
</document>

<document source="https://tamnd.github.io/math2/node/08J2M1JYAG.md" tag="08J2M1JYAG" kind="object">
# Real numbers

## Description

The real numbers are the numbers that measure continuous quantities, such as lengths. We picture them as the points of a line.
We write $\mathbb{R}$ for the set of real numbers.
With addition, multiplication and its order, $\mathbb{R}$ is an [[ordered field|0QSPNASE50]].
This ordered field is complete: each nonempty subset of $\mathbb{R}$ that is bounded above has a [[least upper bound|0AXWET80YW]].
Up to a unique isomorphism, $\mathbb{R}$ is the only [[complete ordered field|0A9E6X0S3E]].

## Remarks

The set $\mathbb{R}$ contains the [[rational numbers|0NCJ5WXN6Z]] and also [[irrational numbers|080T2TXVQV]] such as $\sqrt{2}$ and $\pi$.
Two common constructions of $\mathbb{R}$ from $\mathbb{Q}$ use [[Dedekind cuts|04QNZZ4HHY]] or [[Cauchy sequences|0D8HE681SC]].
The set $\mathbb{R}$ is uncountable.

## Explanations

### L1 human

The real numbers fill the number line without gaps.
The rational numbers leave gaps, for example at $\sqrt{2}$, and the real numbers fill these gaps.
Each real number has a [[decimal expansion|06XY4X9430]], such as $3.14159\dots$ for $\pi$.
</document>

<document source="https://tamnd.github.io/math2/node/08J7R35B6Q.md" tag="08J7R35B6Q" kind="statement">
# Tutte's theorem

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]] with vertex set $V$.
For $U \subseteq V$, let $o(G - U)$ be the number of [[connected components|01VVP10MWJ]] of $G - U$ with an odd number of vertices.
Then $G$ has a [[perfect matching|05ZPH8KF9K]] if and only if $o(G - U) \le |U|$ for every $U \subseteq V$.

## Remarks

The condition is necessary. In a perfect matching, each odd component of $G - U$ has a vertex that is matched to a vertex of $U$. These vertices of $U$ are distinct.
For $U = \varnothing$, the condition says that each component of $G$ has an even number of vertices.
The theorem is a special case of the Tutte–Berge formula for the size of a maximum matching.

## History

The theorem is named after William Thomas Tutte.

## Explanations

### L1 human

Delete some vertices, and count the pieces with an odd number of vertices that remain.
Each odd piece needs a partner among the deleted vertices, and Tutte's theorem says this is the only obstruction.
</document>

<document source="https://tamnd.github.io/math2/node/08JF1K4DW1.md" tag="08JF1K4DW1" kind="definition">
# Fixed points of an action

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$.
A point $x \in X$ is a fixed point of the action when $g \cdot x = x$ for each $g \in G$.
The set of all fixed points is written $X^G$.
For $g \in G$, the set of points fixed by $g$ is $X^g = \{x \in X : g \cdot x = x\}$.

## Remarks

A point $x$ is a fixed point if and only if its [[orbit|0YQPZB5Q1G]] is $\{x\}$, if and only if its [[stabiliser|0AYMP41THN]] is $G$.
For the action of $G$ on itself by conjugation, the fixed points are the elements of the [[centre|0DZ93493VN]] $Z(G)$.
The sets $X^g$ appear in [[Burnside's lemma|0Z47M1B2VS]].

## Explanations

### L1 human

A fixed point is a point that no element of the group moves.
For the rotations of a square about its centre, the centre is a fixed point.
The corners are not fixed points, because a quarter turn moves them.
</document>

<document source="https://tamnd.github.io/math2/node/08KEYFAH66.md" tag="08KEYFAH66" kind="concept">
# Cyclotomic polynomial

## Summary

The $n$-th cyclotomic polynomial is the monic polynomial whose roots are the primitive $n$-th roots of unity, each with multiplicity one.
It has integer coefficients.
The [[definition of the cyclotomic polynomials|0W0KVBM54S]] gives the formula.
</document>

<document source="https://tamnd.github.io/math2/node/08MG7H8JW5.md" tag="08MG7H8JW5" kind="concept">
# Group isomorphism

## Summary

A group isomorphism is a bijective group homomorphism.
Two isomorphic groups have the same structure, and differ only in the names of their elements.
</document>

<document source="https://tamnd.github.io/math2/node/08N6EVA7DV.md" tag="08N6EVA7DV" kind="statement">
# Distributive laws for sets

## Statement

Let $A$, $B$ and $C$ be [[sets|0SJ3RX9TXA]].
Then intersection distributes over union, and union distributes over intersection:
$$A \cap (B \cup C) = (A \cap B) \cup (A \cap C),$$
$$A \cup (B \cap C) = (A \cup B) \cap (A \cup C).$$

## Remarks

The laws follow from the distributive laws of logic for $\land$ and $\lor$.
The second law has no analogue for numbers: $a + bc$ is not $(a + b)(a + c)$ in general.

## Explanations

### L1 human

To prove the first law, take $x$ in the left side. Then $x \in A$, and $x \in B$ or $x \in C$.
In the first case $x \in A \cap B$, and in the second case $x \in A \cap C$.
The other inclusion is similar.
</document>

<document source="https://tamnd.github.io/math2/node/08NM8KQPDQ.md" tag="08NM8KQPDQ" kind="definition">
# Infinite limit

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a$ be a limit point of $D$.
The notation $f(x) \to +\infty$ as $x \to a$ means that for each real $M$ there is $\delta > 0$ with $f(x) > M$ for all $x \in D$ with $0 < |x - a| < \delta$.
The definition of $f(x) \to -\infty$ is the same, with $f(x) < M$.

## Remarks

An infinite limit is not a limit in the sense of [[limit of a function|0TXTSEMHR5]], which requires a real value $L$.
For example, $1/x^2 \to +\infty$ as $x \to 0$.
The same idea gives $f(x) \to +\infty$ as $x \to +\infty$. An example is $f(x) = x^2$.

## Explanations

### L1 human

An infinite limit means that $f(x)$ becomes as large as we like when $x$ is close to $a$.
The graph then has a vertical asymptote at $x = a$.
</document>

<document source="https://tamnd.github.io/math2/node/08NYZ2FS6B.md" tag="08NYZ2FS6B" kind="definition">
# Norm

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $V$ be a vector space over $\mathbb{K}$.
A *norm* on $V$ is a map $\norm{\cdot} \colon V \to \R$ such that, for all $x, y \in V$ and $a \in \mathbb{K}$:

1. $\norm{x + y} \le \norm{x} + \norm{y}$.
2. $\norm{ax} = \abs{a} \norm{x}$.
3. If $\norm{x} = 0$, then $x = 0$.

## Remarks

These conditions imply $\norm{x} \ge 0$ for all $x$.
On $\R^n$, the Euclidean norm $\sqrt{x_1^2 + \cdots + x_n^2}$, the norm $\abs{x_1} + \cdots + \abs{x_n}$ and the norm $\max_i \abs{x_i}$ are norms.
Each inner product gives a norm, see [[the induced norm|0MFJJV97RJ]].
A norm defines a metric $d(x, y) = \norm{x - y}$.

## Explanations

### L1 human

A norm measures the length of a vector.
The length of a sum is at most the sum of the lengths, and scaling a vector scales its length.
Only the zero vector has length zero.
</document>

<document source="https://tamnd.github.io/math2/node/08P6T2WH6R.md" tag="08P6T2WH6R" kind="example">
# Sequence spaces lp

## Example

Let $p \ge 1$ be a real number. Let $\ell^p$ be the set of real sequences $x = (x_n)_{n \ge 1}$ with $\sum_{n \ge 1} |x_n|^p < \infty$, and let
$$\|x\|_p = \left( \sum_{n \ge 1} |x_n|^p \right)^{1/p}.$$
Let $\ell^\infty$ be the set of bounded real sequences, with $\|x\|_\infty = \sup_{n \ge 1} |x_n|$.
Then each $\ell^p$, for $1 \le p \le \infty$, is a vector space, $\|\cdot\|_p$ is a norm on it, and $\ell^p$ is a [[Banach space|0BDTTM24DP]].

## Remarks

The norms extend the [[p-norms on R^n|0KNJWEVSMG]] to infinite sequences.
For $1 \le p < q \le \infty$, we have $\ell^p \subseteq \ell^q$, and the inclusion is strict. For example, $(1/n)$ is in $\ell^2$ but not in $\ell^1$.
The space $\ell^2$ has the inner product $\langle x, y \rangle = \sum_n x_n y_n$, and it is a Hilbert space.
The closed unit ball of $\ell^p$ is not compact: the unit vectors $e_k$ satisfy $\|e_j - e_k\|_p = 2^{1/p}$ for $j \neq k$ and $p < \infty$.

## Explanations

### L1 human

The spaces $\ell^p$ are infinite-dimensional versions of $\mathbb{R}^n$ with the p-norm.
A sequence belongs to $\ell^p$ when its terms go to $0$ fast enough for the sum of $|x_n|^p$ to be finite.
</document>

<document source="https://tamnd.github.io/math2/node/08PA437800.md" tag="08PA437800" kind="definition">
# Lower central series

## Definition

Let $G$ be a [[group|G5S1M9]].
For subgroups $H$ and $K$ of $G$, let $[H, K]$ be the subgroup [[generated by|0RP7VWQ0YN]] the [[commutators|0PXW0BN7DW]] $[h, k]$ with $h \in H$ and $k \in K$.
The lower central series of $G$ is the sequence $\gamma_1(G) = G$ and $\gamma_{i+1}(G) = [\gamma_i(G), G]$ for $i \geq 1$.

## Remarks

The second term $\gamma_2(G) = [G, G]$ is the [[commutator subgroup|0ASGSTXSBW]].
Each $\gamma_i(G)$ is normal in $G$, and $\gamma_i(G)/\gamma_{i+1}(G)$ lies in the centre of $G/\gamma_{i+1}(G)$.
The terms of the [[derived series|0Q5G5DJ4P2]] satisfy $G^{(i)} \subseteq \gamma_{i+1}(G)$.
For $S_3$, the series is $S_3 \supseteq A_3 \supseteq A_3 \supseteq \cdots$, so it never reaches $\{e\}$.

## Explanations

### L1 human

The lower central series starts with the whole group.
At each step, it takes the commutators of the last term with all elements of the group.
If this process reaches the identity, the group is nilpotent.
</document>

<document source="https://tamnd.github.io/math2/node/08PY4GNAPK.md" tag="08PY4GNAPK" kind="statement">
# Product formula for the order of HK

## Statement

Let $G$ be a [[group|G5S1M9]] and let $H$ and $K$ be finite [[subgroups|0X1JXT47GA]] of $G$.
Then the [[product|0PT52XQZD7]] $HK$ has exactly $\dfrac{|H| \, |K|}{|H \cap K|}$ elements.

## Remarks

The formula holds also when $HK$ is not a subgroup.
The proof counts the pairs $(h, k)$: two pairs $(h, k)$ and $(h', k')$ give the same product exactly when $h' = hx$ and $k' = x^{-1}k$ for some $x \in H \cap K$.
For example, two distinct subgroups of order $2$ in $S_3$ meet in $\{e\}$, and their product has $2 \cdot 2 / 1 = 4$ elements.

## Explanations

### L1 human

Each product $hk$ can be written in several ways, because you can move an element of $H \cap K$ from one factor to the other.
Each element of $HK$ has exactly $|H \cap K|$ such ways.
So you divide the number of pairs, $|H| \, |K|$, by $|H \cap K|$.
</document>

<document source="https://tamnd.github.io/math2/node/08QA8QTM71.md" tag="08QA8QTM71" kind="statement">
# Picard's great theorem

## Statement

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 < |z - c| < r$.
Assume that $c$ is an [[essential singularity|0YKDJBV26S]] of $f$.
Then for each $\delta$ with $0 < \delta < r$, there is at most one $w_0 \in \mathbb{C}$ with the following property. The function $f$ takes the value $w_0$ only finitely many times on $0 < |z - c| < \delta$.
Thus, on each punctured disc around $c$, $f$ takes each complex value infinitely often, with at most one exception.

## Remarks

The exception can occur: $e^{1/z}$ never takes the value $0$, and it takes each $w \neq 0$ infinitely often near $0$, see [[exp(1/z)|0NB31CC4SN]].
The theorem is much stronger than the [[Casorati–Weierstrass theorem|02HHNPC8ED]], which gives only that the values come close to each $w$.
Picard's little theorem is a related result: a nonconstant entire function takes all complex values with at most one exception.

## Explanations

### L1 human

Near an essential singularity, the function does not only come close to each value: it hits almost every value infinitely many times.
At most one value can be missed, as $0$ is missed by $e^{1/z}$.
</document>

<document source="https://tamnd.github.io/math2/node/08QAX4H1PF.md" tag="08QAX4H1PF" kind="example">
# The sequence 1/n tends to zero

## Example

Let $a_n = 1/n$ for each integer $n \ge 1$.
Then $a_n \to 0$ as $n \to \infty$.
Let $\varepsilon > 0$. By the Archimedean property of $\mathbb{R}$, there is a natural number $N$ with $N > 1/\varepsilon$.
For each $n \ge N$ we get $|a_n - 0| = 1/n \le 1/N < \varepsilon$.
By the [[epsilon-N definition|09G6KHS6GY]], this shows $1/n \to 0$.

## Remarks

The proof uses the Archimedean property, which says that the natural numbers are not bounded above in $\mathbb{R}$.
With the [[algebra of limits|07XBM3A9J5]], it gives many more limits, for example $1/n^2 \to 0$ and $(n + 1)/n \to 1$.
Mathlib states it for $1/n$ over $\mathbb{R}$ as `tendsto_one_div_atTop_nhds_zero_nat`.

## Explanations

### L1 human

The numbers $1, 1/2, 1/3, 1/4, \dots$ get smaller and smaller, and they approach $0$.
To get $1/n$ below $0.001$, it is enough to take $n > 1000$.
</document>

<document source="https://tamnd.github.io/math2/node/08QCA2F1NX.md" tag="08QCA2F1NX" kind="statement">
# Independence means the joint law is the product of the marginals

## Statement

Let $X$ and $Y$ be random variables on one probability space, with values in measurable spaces $(E_1, \mathcal{E}_1)$ and $(E_2, \mathcal{E}_2)$.
Let $\mu_X$ and $\mu_Y$ be their laws, and let $\mu_{(X, Y)}$ be the [[joint law|0QAR0XDZH4]] of the pair $(X, Y)$ on $E_1 \times E_2$.
Then $X$ and $Y$ are [[independent|0E9502NT92]] if and only if $\mu_{(X, Y)} = \mu_X \otimes \mu_Y$, the product measure of the two [[marginal laws|0AXP2Z5N9H]].

## Remarks

The statement is a direct translation of the definition: $\mu_{(X, Y)}(A \times B) = P(X \in A, Y \in B)$ and $(\mu_X \otimes \mu_Y)(A \times B) = \mu_X(A) \, \mu_Y(B)$.
Two probability measures on $E_1 \times E_2$ that agree on all rectangles $A \times B$ are equal, because the rectangles form a π-system that generates the product σ-algebra.
For real random variables, it follows that $F_{(X, Y)}(x, y) = F_X(x) \, F_Y(y)$. If $X$ and $Y$ have densities, the joint density is $f_X(x) \, f_Y(y)$.
The same result holds for $n$ random variables: they are independent if and only if their joint law is $\mu_{X_1} \otimes \dots \otimes \mu_{X_n}$.

## Explanations

### L1 human

For independent variables, the full joint picture is fixed by the two separate pictures.
You find the chance of a pair of outcomes by multiplying the chances of the two outcomes.
</document>

<document source="https://tamnd.github.io/math2/node/08QMKTWF9T.md" tag="08QMKTWF9T" kind="concept">
# Partition of a set

## Summary

A partition of a set divides it into nonempty pieces that do not overlap and that together cover the set.
</document>

<document source="https://tamnd.github.io/math2/node/08QQH9F7G0.md" tag="08QQH9F7G0" kind="statement">
# Number of functions between finite sets

## Statement

Let $A$ and $B$ be finite sets with $|A| = m$ and $|B| = n$.
Let $B^A$ be the set of all functions from $A$ to $B$.
Then $|B^A| = n^m$.

## Remarks

The case $m = 0$ gives $n^0 = 1$: there is exactly one function from the empty set to $B$, also when $B$ is empty.
A function from $A$ to $B$ is a choice of one value in $B$ for each of the $m$ elements of $A$. The [[rule of product|0VPK4X73Q0]] then gives $n^m$.
For a subset count, take $B = \{0, 1\}$. This gives the [[number of subsets of a finite set|0C5RDJE73X]].

## Explanations

### L1 human

Each element of $A$ independently chooses its image among the $n$ elements of $B$.
For example, there are $3^2 = 9$ functions from $\{1, 2\}$ to $\{a, b, c\}$, because $1$ has 3 choices and $2$ has 3 choices.
</document>

<document source="https://tamnd.github.io/math2/node/08R2V9GZF7.md" tag="08R2V9GZF7" kind="concept">
# Cesàro mean

## Summary

The Cesàro means of a sequence are the averages of its first $n$ terms. They are the base of Cesàro summation, a way to give values to some divergent series. The node [[Cesàro mean|0P0DJHBGBB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/08RQ62VAVA.md" tag="08RQ62VAVA" kind="concept">
# Number of distinct prime factors

## Summary

The function $\omega(n)$ counts the distinct prime numbers that divide a positive integer $n$.
The [[definition of the number of distinct prime factors|0YVVXT3A0B]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/08RQAPXF1T.md" tag="08RQAPXF1T" kind="definition">
# Liouville function

## Definition

The Liouville function $\lambda$ is the arithmetic function given by $\lambda(n) = (-1)^{\Omega(n)}$ for each positive integer $n$.
Here $\Omega(n)$ is the [[number of prime factors of n counted with multiplicity|08420828EY]].

## Remarks

For example, $\lambda(1) = 1$, $\lambda(12) = -1$ and $\lambda(36) = 1$.
The function $\lambda$ is [[completely multiplicative|0GS27TKF1V]], since $\Omega$ is completely additive.
The sum $\sum_{d \mid n} \lambda(d)$ is $1$ if $n$ is a perfect square, and $0$ otherwise.
On squarefree integers, $\lambda$ agrees with the [[Möbius function|0C8QENHYZW]].

## Explanations

### L1 human

The Liouville function is $1$ when $n$ has an even number of prime factors, counted with repeats, and $-1$ when the number is odd.
Unlike the Möbius function, it is never $0$.
</document>

<document source="https://tamnd.github.io/math2/node/08RT9GDDPQ.md" tag="08RT9GDDPQ" kind="concept">
# Power series solution

## Summary

The power series method looks for a solution of a differential equation in the form $y = \sum_{k \ge 0} a_k (x - x_0)^k$.
Consider $y'' + p(x) y' + q(x) y = 0$, where $p$ and $q$ are analytic at $x_0$: they equal convergent power series near $x_0$.
One puts the series into the equation and compares the coefficients of each power $(x - x_0)^k$. This gives a recurrence for the $a_k$, with $a_0$ and $a_1$ free.
The resulting series converge at least on each disc around $x_0$ where the series of $p$ and $q$ converge, and they give two independent solutions.

## Remarks

Example: for $y' = y$, the series $\sum a_k x^k$ gives $(k + 1) a_{k+1} = a_k$, so $a_k = a_0 / k!$ and $y = a_0 e^x$.
Example: for $y'' + y = 0$, the recurrence is $a_{k+2} = -a_k / \left((k+1)(k+2)\right)$. It gives the series of $\cos x$ and $\sin x$.
At a point where a leading coefficient vanishes, the Frobenius method, a variant with a factor $(x - x_0)^r$, is used.

## Explanations

### L1 human

Write the unknown function as a power series with unknown coefficients.
The equation turns into a rule that gives each coefficient from the earlier ones.
So you can compute the solution term by term, even when there is no closed formula.
</document>

<document source="https://tamnd.github.io/math2/node/08RTDHX1SK.md" tag="08RTDHX1SK" kind="definition">
# Finite field

## Definition

A finite field is a [[field|0QA8CHH8MG]] $F$ whose underlying set is finite.
The order of $F$ is its number of elements $|F|$.

## Remarks

For each prime number $p$, the ring $\mathbb{Z}/p\mathbb{Z}$ is a [[finite field|0NV6E43P4A]] of order $p$.
The [[order of a finite field is a prime power|0H5WWC8RZX]] $p^n$, where $p$ is the characteristic.
The field $\mathbb{F}_2[x]/(x^2 + x + 1)$ is a finite field of order $4$. It is not $\mathbb{Z}/4\mathbb{Z}$, which is not a field.
Finite fields are used in number theory, algebraic geometry, cryptography and coding theory.

## Explanations

### L1 human

A finite field is a finite set where you can add, subtract, multiply and divide by nonzero elements, with the usual rules.
The simplest ones are the integers modulo a prime.
</document>

<document source="https://tamnd.github.io/math2/node/08S0NW8SES.md" tag="08S0NW8SES" kind="concept">
# Normal extension

## Summary

A normal extension is an algebraic extension $L/K$ such that every irreducible polynomial over $K$ with a root in $L$ splits over $L$.
The [[definition of a normal extension|0CBRFQ7DVZ]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/08S7FDRS5Z.md" tag="08S7FDRS5Z" kind="definition">
# Differentiable on an open set

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set and let $f \colon U \to \mathbb{R}^m$.
The function $f$ is differentiable on $U$ when it is [[differentiable|0135XZZQ8M]] at each point $a \in U$.
Then the derivative of $f$ is the map $Df \colon U \to L(\mathbb{R}^n, \mathbb{R}^m)$, $a \mapsto Df(a)$, where $L(\mathbb{R}^n, \mathbb{R}^m)$ is the space of linear maps from $\mathbb{R}^n$ to $\mathbb{R}^m$.

## Remarks

The function $f$ is continuously differentiable, or of class $C^1$, on $U$ when it is differentiable on $U$ and $Df$ is continuous. Here $L(\mathbb{R}^n, \mathbb{R}^m)$ carries the topology of $m \times n$ matrices.
For a set $S$ that is not open, one uses derivatives within $S$ instead. On an open set the two notions agree.
Each polynomial map $\mathbb{R}^n \to \mathbb{R}^m$ is differentiable on $\mathbb{R}^n$.

## Explanations

### L1 human

Differentiable on $U$ means that there is a good linear approximation at each point of $U$, not only at one point.
The approximation changes from point to point, so the derivative is itself a function on $U$.
</document>

<document source="https://tamnd.github.io/math2/node/08SJ5K2CBG.md" tag="08SJ5K2CBG" kind="definition">
# Commutator subgroup

## Definition

Let $G$ be a [[group|G5S1M9]].
The commutator subgroup of $G$ is the [[subgroup generated by|0RP7VWQ0YN]] the set of all [[commutators|0PXW0BN7DW]] $[g, h]$ with $g, h \in G$.
It is written $[G, G]$ or $G'$.

## Remarks

A product of commutators need not be a commutator, so in general the set of commutators is not a subgroup.
The subgroup $[G, G]$ is [[characteristic|0KABPMRHM2]], so it is normal in $G$.
The group $G$ is abelian if and only if $[G, G] = \{e\}$.
The commutator subgroup of the symmetric group $S_n$ is the alternating group $A_n$.

## Explanations

### L1 human

The commutator subgroup collects all the failures of commutativity in a group.
If you divide the group by it, you get an abelian group.
A large commutator subgroup means a group that is far from abelian.
</document>

<document source="https://tamnd.github.io/math2/node/08SP3DF8NB.md" tag="08SP3DF8NB" kind="concept">
# Permutation matrix

## Summary

A permutation matrix is a square matrix with exactly one entry $1$ in each row and each column, and all other entries $0$.
It permutes the coordinates of a vector.
</document>

<document source="https://tamnd.github.io/math2/node/08SVRVHJNY.md" tag="08SVRVHJNY" kind="definition">
# Topological embedding

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]].
A function $f \colon X \to Y$ is a topological embedding when $f$ is [[injective|0PVG1CRAJN]] and the map $X \to f(X)$, $x \mapsto f(x)$, is a [[homeomorphism|05GZ11PWRY]].
Here the [[image|0Q3F20XB0A]] $f(X)$ has the [[subspace topology|0D5YHQGDJY]] of $Y$.

## Remarks

Each embedding is continuous, because the [[inclusion of a subspace is continuous|065J822CKN]].
The inclusion of a subspace is an embedding. The map $\mathbb{R} \to \mathbb{R}^2$, $x \mapsto (x, 0)$, is an embedding.
The map $t \mapsto (\cos t, \sin t)$ from $[0, 2\pi)$ to $\mathbb{R}^2$ is continuous and injective, but not an embedding.
A continuous injective map from a [[compact space|0QPHW703PV]] to a [[Hausdorff space|0MXB4CKHFW]] is an embedding.

## Explanations

### L1 human

An embedding puts a copy of $X$ inside $Y$ without changing its shape.
An injective continuous map can still bring the ends of the space together in the limit. Then it is not an embedding.
</document>

<document source="https://tamnd.github.io/math2/node/08V7JHS3QZ.md" tag="08V7JHS3QZ" kind="statement">
# Order of an element divides the order of the group

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $g \in G$.
Then the [[order|0WE2NHZJ7B]] of $g$ divides the [[order|0B85S2V2DR]] $|G|$ of $G$.

## Remarks

The order of $g$ is the order of the cyclic subgroup $\langle g \rangle$, [[by this fact|031M8HX7DA]], and [[Lagrange's theorem|3C53TD]] applies to $\langle g \rangle$.
For example, in a group of order $10$, each element has order $1, 2, 5$ or $10$.

## Explanations

### L1 human

An element generates a subgroup with as many elements as its order.
The size of a subgroup always divides the size of the group.
So the order of an element divides the order of the group.

### L2 agent

The order of an element $g$ is the smallest $n \ge 1$ with $g^n = e$. The result says that this number divides the order of the group.

The proof reduces the result to Lagrange's theorem:

1. The powers of $g$ form the cyclic subgroup $\langle g \rangle$.
2. This subgroup has exactly $\operatorname{ord}(g)$ elements, namely $e, g, \dots, g^{n-1}$ with $n = \operatorname{ord}(g)$.
3. By Lagrange's theorem, $|\langle g \rangle|$ divides $|G|$.

A worked example in the group of units of $\mathbb{Z}/7\mathbb{Z}$, which has $6$ elements:

1. Take $g = 2$. Then $2^1 = 2$, $2^2 = 4$ and $2^3 = 8 = 1$.
2. Thus $\operatorname{ord}(2) = 3$, and $3$ divides $6$.
3. Take $g = 3$. The powers are $3, 2, 6, 4, 5, 1$, so $\operatorname{ord}(3) = 6$.

A useful consequence is that $g^{|G|} = e$ for each $g \in G$. Applied to the units of $\mathbb{Z}/p\mathbb{Z}$, this gives Fermat's little theorem: $a^{p-1} \equiv 1 \pmod p$ when $p$ does not divide $a$.

A common error is to think that each divisor of $|G|$ is the order of some element. The Klein four-group has order $4$, but it has no element of order $4$. Cauchy's theorem gives a partial converse: each prime divisor of $|G|$ is the order of some element.
</document>

<document source="https://tamnd.github.io/math2/node/08VRX3Q4PB.md" tag="08VRX3Q4PB" kind="concept">
# Semisimple endomorphism

## Summary

An endomorphism of a finite-dimensional vector space is semisimple when each invariant subspace has an invariant complement.
Over an algebraically closed field, this means that it is diagonalisable.
</document>

<document source="https://tamnd.github.io/math2/node/08VV7YJRB7.md" tag="08VV7YJRB7" kind="definition">
# Content of a polynomial

## Definition

Let $R$ be a [[unique factorisation domain|06YMXJMXYS]] and let $f = a_0 + a_1 x + \dots + a_n x^n \in R[x]$.
The content of $f$, written $c(f)$, is a [[greatest common divisor|0F7KNJ1ZYN]] of the coefficients $a_0, a_1, \dots, a_n$.

## Remarks

The content is unique up to associates. Over $\mathbb{Z}$, one often takes the positive gcd, with the sign of the leading coefficient in the primitive part.
For $f = 6x^2 + 4x - 2$ in $\mathbb{Z}[x]$, $c(f) = 2$.
Each nonzero $f$ is $c(f) \cdot g$ with $g$ a [[primitive polynomial|0SP14DZN0G]], the primitive part of $f$.
The content is multiplicative up to units: $c(f g) \sim c(f) c(g)$. This is a form of [[Gauss's lemma|0QFV5Y1Z5X]].

## Explanations

### L1 human

The content is the largest common factor that you can pull out of all the coefficients.
For $6x^2 + 4x - 2$, you can pull out $2$ and get $2(3x^2 + 2x - 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/08VZ57PT26.md" tag="08VZ57PT26" kind="concept">
# Uniform convergence

## Summary

A sequence of functions converges uniformly when the largest distance between $f_n$ and the limit tends to zero. Uniform limits keep continuity, and they commute with integration. The node [[uniform convergence|08FPABDQ4C]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/08W2CW8PKW.md" tag="08W2CW8PKW" kind="definition">
# Median

## Definition

Let $X$ be a real random variable on a probability space $(\Omega, \mathcal{F}, P)$.
A median of $X$ is a real number $m$ with
$$P(X \le m) \ge \frac{1}{2} \quad \text{and} \quad P(X \ge m) \ge \frac{1}{2}.$$

## Remarks

Each real random variable has a median, and the medians form a closed bounded interval. For example, $Q(1/2)$ is a median, where $Q$ is the [[quantile function|0B3Q3XDM5X]].
The median can be different from the mean. For the exponential distribution with rate $\lambda$, the median is $\ln 2 / \lambda$ and the mean is $1/\lambda$.
For a fair coin coded as $0$ and $1$, each $m \in [0, 1]$ is a median.
For a sorted data set $x_1 \le \dots \le x_n$ with $n$ odd, the median is $x_{(n+1)/2}$. For $n$ even, one usually takes the mean of the two middle values.

## Explanations

### L1 human

The median is the middle value: half of the chance lies on each side of it.
The median of the incomes in a country is often more useful than the mean.
A few very large incomes do not move the median much.
</document>

<document source="https://tamnd.github.io/math2/node/08W5RN375X.md" tag="08W5RN375X" kind="statement">
# Power series of cosine

## Statement

For every real $x$, the series $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$ converges absolutely, and
$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots.$$
This series is the [[Maclaurin series|043SJ5RF59]] of the cosine function.

## Remarks

The [[ratio of the terms|0JBMYAA9QH]] is $\frac{x^2}{(2n+1)(2n+2)} \to 0$. So the radius of convergence is $\infty$.
For $|x| \le 1$, the terms decrease in size. So the partial sums lie alternately above and below $\cos x$.
Mathlib states the series as `Real.hasSum_cos`.

## Explanations

### L1 human

Cosine is an infinite even polynomial with alternating signs.
Near $0$, $\cos x$ is close to $1 - x^2/2$.
</document>

<document source="https://tamnd.github.io/math2/node/08WCB90WND.md" tag="08WCB90WND" kind="statement">
# Hall's marriage theorem

## Statement

Let $\mathcal{F} = (S_1, \dots, S_n)$ be a finite family of finite sets.
A transversal of $\mathcal{F}$ is an injective function $f : \{1, \dots, n\} \to \bigcup_i S_i$ with $f(i) \in S_i$ for every $i$.
Then $\mathcal{F}$ has a transversal if and only if $\left| \bigcup_{i \in I} S_i \right| \ge |I|$ for every $I \subseteq \{1, \dots, n\}$.

## Remarks

The condition on all subfamilies is the marriage condition. It is necessary, because a transversal $f$ maps $I$ injectively into $\bigcup_{i \in I} S_i$.
A transversal is also called a system of distinct representatives. The sets $S_i$ may repeat in the family.
The theorem is equivalent to [[Hall's theorem for bipartite graphs|0X2GJPNPFN]].

## History

Philip Hall proved the theorem in 1935.
The name marriage theorem comes from a 1950 paper by Halmos and Vaughan.

## Explanations

### L1 human

Each of $n$ people has a list of acceptable partners, and we want to give each person a different partner from the list.
This is possible exactly when every group of $k$ people has at least $k$ names on their lists together.
</document>

<document source="https://tamnd.github.io/math2/node/08WE2W7DYS.md" tag="08WE2W7DYS" kind="statement">
# Cauchy–Binet formula

## Statement

Let $K$ be a commutative ring, let $m \le n$, let $A$ be an $m \times n$ matrix and $B$ an $n \times m$ matrix over $K$.
For a subset $S \subseteq \{1, \dots, n\}$ with $m$ elements, let $A_S$ be the $m \times m$ matrix of the columns of $A$ in $S$.
Let $B^S$ be the $m \times m$ matrix of the rows of $B$ in $S$.
Then
$$\det(AB) = \sum_{S} \det(A_S) \det(B^S),$$
where the sum runs over all subsets $S$ of $\{1, \dots, n\}$ with $m$ elements.

## Remarks

For $m = n$, the sum has one term, and the formula is $\det(AB) = \det A \det B$.
If $m > n$, then $\det(AB) = 0$.
For $m = 1$, it is the formula for the product of a row and a column.
The determinants $\det(A_S)$ and $\det(B^S)$ are [[minors|0BT9BYWVRY]] of order $m$.

## Explanations

### L1 human

The Cauchy–Binet formula computes the determinant of a product of two rectangular matrices.
It sums products of matching minors of the two factors.
It generalizes the product rule for determinants of square matrices.
</document>

<document source="https://tamnd.github.io/math2/node/08WFZP6P89.md" tag="08WFZP6P89" kind="concept">
# Sine and cosine

## Summary

Sine and cosine are the basic trigonometric functions. For an angle $x$, the point $(\cos x, \sin x)$ lies on the unit circle, and the two functions are periodic with period $2\pi$. The node [[sine and cosine|0WB57D9488]] gives the precise definition by power series.
</document>

<document source="https://tamnd.github.io/math2/node/08WTN6SCQ6.md" tag="08WTN6SCQ6" kind="statement">
# Euler's four-square identity

## Statement

Let $a_1, \dots, a_4$ and $b_1, \dots, b_4$ be elements of a commutative ring. Then
$$\begin{aligned} (a_1^2 + a_2^2 + a_3^2 + a_4^2)(b_1^2 + b_2^2 + b_3^2 + b_4^2) &= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 \\ &\quad + (a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 + (a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2. \end{aligned}$$

## Remarks

So the product of two sums of four squares is again a sum of four squares.
The signs come from the product of two quaternions: the identity says that the norm of quaternions is multiplicative.
Lagrange used it to reduce the [[four-square theorem|0KYYSMEVHH]] to the case of primes.

## History

Euler wrote about the identity in a letter to Goldbach of 4 May 1748.

## Explanations

### L1 human

This identity is the four-square analogue of the [[Brahmagupta–Fibonacci identity|0K2DMACEZ4]] for two squares.
You can check it by expanding both sides, since all the cross terms cancel.
</document>

<document source="https://tamnd.github.io/math2/node/08WWTN738R.md" tag="08WWTN738R" kind="concept">
# Explicit and implicit differential equations

## Summary

An [[ordinary differential equation|0P97V20RW7]] of [[order|0XK0G19SK8]] $n$ is explicit when it has the form $y^{(n)} = f\left(x, y, y', \dots, y^{(n-1)}\right)$ for a given function $f$.
It is implicit when it has the general form $F\left(x, y, y', \dots, y^{(n)}\right) = 0$ and is not solved for $y^{(n)}$.
Each explicit equation is also implicit, with $F = y^{(n)} - f$.

## Remarks

The equation $y' = x^2 + y^2$ is explicit. The equation $(y')^2 + y^2 = 1$ is implicit.
If $\partial F / \partial y^{(n)}$ is not zero at a point, the implicit function theorem gives an explicit equation near that point.
The basic existence theorems, such as the [[Picard–Lindelöf theorem|0P6KRD5B7A]], are for explicit equations.

## Explanations

### L1 human

An explicit equation tells you the highest derivative directly, as a formula in the lower ones.
An implicit equation only gives a relation, and you may have to solve it for $y'$ first.
For $(y')^2 + y^2 = 1$ you get two explicit equations, $y' = \sqrt{1 - y^2}$ and $y' = -\sqrt{1 - y^2}$.
</document>

<document source="https://tamnd.github.io/math2/node/08X0VCR2NW.md" tag="08X0VCR2NW" kind="concept">
# Square-free integer

## Summary

A square-free integer is an integer that no perfect square other than $1$ divides.
The [[definition of a square-free integer|004JW7A2N5]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/08XDXP86XE.md" tag="08XDXP86XE" kind="statement">
# An iterate of the Picard operator is a contraction

## Statement

Let $f$ satisfy the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] at $(t_0, x_0)$ on $J = [t_{\min}, t_{\max}]$ with constants $a, L, K$.
Let $X$ be the set of continuous curves $\gamma : J \to \mathbb{R}^n$ with $\|\gamma(t) - x_0\| \le a$ for each $t \in J$, with the distance $d(\gamma, \delta) = \sup_{t \in J} \|\gamma(t) - \delta(t)\|$.
Let $P : X \to X$ be the [[Picard operator|00WAVB68F9]] and let $T = \max(t_{\max} - t_0,\ t_0 - t_{\min})$.
Then for each integer $m \ge 0$ and all $\gamma, \delta \in X$,
$$d\left(P^m \gamma, P^m \delta\right) \le \frac{(K T)^m}{m!}\, d(\gamma, \delta).$$
In particular, there is an $m \ge 1$ such that $P^m$ is a contraction of $X$: $(KT)^m / m! < 1$.

## Remarks

Proof idea: by induction on $m$, $\|(P^m \gamma)(t) - (P^m \delta)(t)\| \le \frac{(K |t - t_0|)^m}{m!}\, d(\gamma, \delta)$ for each $t \in J$. The induction step integrates the bound of the previous step.
The space $X$ is a complete metric space. So the Banach fixed point theorem applies to $P^m$, and it gives a unique fixed point of $P$.
If $K T < 1$, then $P$ itself is a contraction. The power $P^m$ avoids the need for a short time interval.

## Explanations

### L1 human

A contraction brings any two points closer by a fixed factor smaller than one.
The Picard operator may not do this in one step, but after enough steps it does, because $m!$ grows faster than $(KT)^m$.
That is enough to get a unique fixed point.
</document>

<document source="https://tamnd.github.io/math2/node/08XFJJ8EYW.md" tag="08XFJJ8EYW" kind="definition">
# Comparison of topologies

## Definition

Let $X$ be a set and let $\tau_1$ and $\tau_2$ be [[topologies|0CNQXVBHCX]] on $X$.
If $\tau_1 \subseteq \tau_2$, then $\tau_1$ is coarser (or weaker) than $\tau_2$, and $\tau_2$ is finer (or stronger) than $\tau_1$.
If also $\tau_1 \neq \tau_2$, then $\tau_1$ is strictly coarser and $\tau_2$ is strictly finer.

## Remarks

On each set $X$, the [[indiscrete topology|0XJ02JBD6S]] is the coarsest topology and the [[discrete topology|0S2TEHX4E1]] is the finest topology.
We have $\tau_1 \subseteq \tau_2$ if and only if the identity map $(X, \tau_2) \to (X, \tau_1)$ is [[continuous|0SY9ECQ19Q]].
Two topologies need not be comparable: on $\{a, b\}$, the topologies $\{\varnothing, \{a\}, X\}$ and $\{\varnothing, \{b\}, X\}$ are not comparable.

## Explanations

### L1 human

A finer topology has more open sets, so it can separate points in more ways.
A coarser topology has fewer open sets, so more sequences converge and more maps into it are continuous.
</document>

<document source="https://tamnd.github.io/math2/node/08Y1BBQTK9.md" tag="08Y1BBQTK9" kind="statement">
# Integration by parts

## Statement

Let $a < b$ and let $u, v \colon [a, b] \to \mathbb{R}$ be [[continuously differentiable|06SKCHDQT5]].
Then
$$\int_a^b u(x) v'(x) \, dx = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx.$$

## Remarks

By the [[product rule|0V9A43H7WV]], $uv$ is an antiderivative of $u v' + u' v$. Apply the [[second fundamental theorem|0XYAFCN0PT]].
For indefinite integrals, the formula is $\int u \, dv = u v - \int v \, du$.
For example, $\int_0^1 x e^x \, dx = e - (e - 1) = 1$.
Mathlib states it as `intervalIntegral.integral_mul_deriv_eq_deriv_mul`.

## Explanations

### L1 human

Integration by parts is the product rule read backwards.
It moves the derivative from one factor to the other, at the cost of a boundary term.
</document>

<document source="https://tamnd.github.io/math2/node/08YERZDM9X.md" tag="08YERZDM9X" kind="statement">
# Subfields of a finite field

## Statement

Let $p$ be a prime number and let $n \ge 1$.
For each divisor $m$ of $n$, the field $\mathbb{F}_{p^n}$ has exactly one [[subfield|07QY5RPFBR]] of order $p^m$, namely $\{ a : a^{p^m} = a \}$.
It has no other subfields.

## Remarks

If $K$ is a subfield of $\mathbb{F}_{p^n}$ of order $p^m$, then $n = m \cdot [\mathbb{F}_{p^n} : K]$ by the [[tower law|0A7KJ68XHT]], so $m$ divides $n$.
For example, the subfields of $\mathbb{F}_{64} = \mathbb{F}_{2^6}$ are $\mathbb{F}_2$, $\mathbb{F}_4$, $\mathbb{F}_8$ and $\mathbb{F}_{64}$.
The subfield of order $p^m$ is the fixed field of $\varphi^m$, where $\varphi$ is the [[Frobenius automorphism|0VP4TBVJ19]].

## Explanations

### L1 human

The subfields of a finite field match the divisors of the exponent.
The field with $2^6$ elements contains the fields with $2^1$, $2^2$ and $2^3$ elements, but not one with $2^4$ elements.
</document>

<document source="https://tamnd.github.io/math2/node/08ZDRN6H41.md" tag="08ZDRN6H41" kind="statement">
# Steinitz exchange lemma

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$.
Let $u_1, \dots, u_m \in V$ be [[linearly independent|0CRMGPAH83]], and let $w_1, \dots, w_n \in V$ [[span|088RYKAWK9]] $V$.
Then $m \le n$.
Moreover, after a renumbering of the $w_j$, the family $u_1, \dots, u_m, w_{m+1}, \dots, w_n$ spans $V$.

## Remarks

The lemma follows by induction on $m$: each $u_i$ replaces one of the $w_j$ in a spanning family.
A consequence is that any two finite bases of $V$ have the same number of elements.
The lemma is named after Ernst Steinitz.

## Explanations

### L1 human

An independent family can never have more vectors than a spanning family.
We can trade the vectors of the spanning family, one by one, for the independent vectors, and the family still spans.
This lemma is the reason why the dimension of a space is well defined.
</document>

<document source="https://tamnd.github.io/math2/node/08ZDW5SA3B.md" tag="08ZDW5SA3B" kind="concept">
# Identically distributed random variables

## Summary

Random variables are identically distributed when they have the same law, even if they live on different probability spaces.
The node [[identically distributed random variables|01X8YSDTBW]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/08ZQAKBKHF.md" tag="08ZQAKBKHF" kind="definition">
# Alternating bilinear form

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A [[bilinear form|0SQZMXB6GP]] $B$ on $V$ is *alternating* when
$$B(v, v) = 0 \qquad \text{for all } v \in V.$$

## Remarks

An alternating form is *skew-symmetric*: $B(u, v) = -B(v, u)$ for all $u, v$.
If the characteristic of $F$ is not $2$, each skew-symmetric form is alternating.
On $F^2$, the form $B(x, y) = x_1 y_2 - x_2 y_1$ is alternating; it is the determinant of the matrix with the columns $x$ and $y$.
A nondegenerate alternating form is a *symplectic form*.

## Explanations

### L1 human

An alternating form gives zero when both vectors are equal.
Then swapping the two vectors changes the sign of the value.
The signed area of the parallelogram spanned by two plane vectors is an example.
</document>

<document source="https://tamnd.github.io/math2/node/08ZWFDJP3H.md" tag="08ZWFDJP3H" kind="concept">
# Eigenvalue method for linear systems

## Summary

The eigenvalue method solves a [[homogeneous linear system with constant coefficients|0VJF99CZ0Z]] $x' = A x$ with the eigenvalues and eigenvectors of $A$.
If $A v = \lambda v$ with $v \ne 0$, then $x(t) = e^{\lambda t} v$ is a solution.
If $A$ has $n$ linearly independent eigenvectors $v_1, \dots, v_n$ with eigenvalues $\lambda_1, \dots, \lambda_n$, then each solution is $x(t) = c_1 e^{\lambda_1 t} v_1 + \dots + c_n e^{\lambda_n t} v_n$ for unique constants $c_k$.
For a complex eigenvalue $\lambda$ of a real matrix with eigenvector $v$, the real and imaginary parts of $e^{\lambda t} v$ are two real solutions.

## Remarks

Example: $A = \begin{pmatrix} 1 & 1 \\ 4 & 1 \end{pmatrix}$ has the eigenvalues $3$ and $-1$, with eigenvectors $(1, 2)$ and $(1, -2)$. So $x(t) = c_1 e^{3t} (1, 2) + c_2 e^{-t} (1, -2)$.
If $A$ has too few eigenvectors, generalized eigenvectors give solutions of the form $e^{\lambda t}$ times a polynomial in $t$ with vector coefficients.
The eigenvalues decide the [[stability of the system|0SKVEMXJWF]].

## Explanations

### L1 human

Along an eigenvector, the matrix $A$ acts as multiplication by a number.
So on this line the system is the simple equation $y' = \lambda y$, with solution $e^{\lambda t}$.
Write the start vector in a basis of eigenvectors, and let each part grow or decay at its own rate.
</document>

<document source="https://tamnd.github.io/math2/node/0906Z03EMN.md" tag="0906Z03EMN" kind="example">
# Tower of Hanoi

## Example

The Tower of Hanoi is a puzzle with three rods and $n$ discs of different sizes.
At the start, all discs are on one rod, in order of size, with the largest at the bottom.
A move takes the top disc of one rod and puts it on top of another rod.
No move may put a disc on a smaller disc.
The aim is to move all discs to another rod.

Let $T_n$ be the minimum number of moves for $n$ discs.
Then $T_0 = 0$ and $T_n = 2 T_{n-1} + 1$ for $n \ge 1$.
This [[recurrence relation|07TZV87Y0T]] has the solution $T_n = 2^n - 1$.

## Remarks

The recurrence comes from the largest disc. Before it moves, the other $n - 1$ discs must be on the third rod. After it moves, they must come back on top of it.
Values: $T_1 = 1$, $T_2 = 3$, $T_3 = 7$ and $T_4 = 15$.
The French mathematician Édouard Lucas invented the puzzle and presented it in 1883.

## Explanations

### L1 human

To move a tower of $n$ discs, first move the top $n - 1$ discs out of the way. Then move the largest disc, and then move the $n - 1$ discs back on top.
Each new disc doubles the work and adds one move.
</document>

<document source="https://tamnd.github.io/math2/node/0909PWZR62.md" tag="0909PWZR62" kind="definition">
# One-sided limit

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a$ be a limit point of $D \cap (a, \infty)$.
A real number $L$ is the right limit of $f$ at $a$ if the following holds.
For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - L| < \varepsilon$ for all $x \in D$ with $a < x < a + \delta$.
The notation is $\lim_{x \to a^+} f(x) = L$.
The left limit $\lim_{x \to a^-} f(x)$ is defined in the same way, with $a - \delta < x < a$.

## Remarks

The right limit is the [[limit|0TXTSEMHR5]] of the restriction of $f$ to $D \cap (a, \infty)$.
If $a$ is a limit point of points on both sides, then $\lim_{x \to a} f(x) = L$ if and only if both one-sided limits exist and equal $L$.
For the sign function at $0$, the right limit is $1$ and the left limit is $-1$.

## Explanations

### L1 human

A one-sided limit only looks at points on one side of $a$.
A function with a jump at $a$ has two different one-sided limits there.
</document>

<document source="https://tamnd.github.io/math2/node/090AA928Y7.md" tag="090AA928Y7" kind="concept">
# Convergence of a sequence in a metric space

## Summary

A sequence in a metric space converges to a point when the distance from the terms to the point tends to zero.
The point is the limit of the sequence.
</document>

<document source="https://tamnd.github.io/math2/node/090WPKFABX.md" tag="090WPKFABX" kind="definition">
# Hamiltonian graph

## Definition

A [[graph|0JD5C82QMB]] is Hamiltonian if it has a [[Hamiltonian cycle|0HKXNG2RS1]].

## Remarks

The [[cycle graph|0QQ0F12962]] $C_n$ and the [[complete graph|03WKB2H474]] $K_n$ for $n \ge 3$ are Hamiltonian.
A [[tree|0MJ0JS72DZ]] is not Hamiltonian, because it has no cycle.
The [[Petersen graph|0NR95CTFCZ]] is not Hamiltonian.
[[Dirac's theorem|0TC99BRSSB]] and [[Ore's theorem|0NACHJYPKW]] give sufficient conditions in terms of degrees.

## Explanations

### L1 human

A graph is Hamiltonian when you can visit all of its vertices in one round trip, each vertex once.
Many edges make this easier, and this is the idea of the theorems of Dirac and Ore.
</document>

<document source="https://tamnd.github.io/math2/node/09115NQZ07.md" tag="09115NQZ07" kind="concept">
# Arithmetic function

## Summary

An arithmetic function is a function on the positive integers with complex values, often one that expresses an arithmetic property of $n$.
The [[definition of an arithmetic function|0H2EPRZC0D]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0912EEEVCT.md" tag="0912EEEVCT" kind="statement">
# An injective endomorphism of a finite-dimensional space is surjective

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Then these conditions are equivalent:

1. $T$ is injective.
2. $T$ is surjective.
3. $T$ is bijective.

## Remarks

The proof uses the [[rank–nullity theorem|0W4FSW1TVB]]: $\ker T = 0$ if and only if $\dim \im T = \dim V$.
It fails in infinite dimensions: on $F[x]$, multiplication by $x$ is injective and not surjective.
In characteristic $0$, the derivative on $F[x]$ is surjective and not injective.
For square matrices, this is the statement that $Ax = 0$ has only the trivial solution if and only if $Ax = b$ has a solution for each $b$.

## Explanations

### L1 human

For a linear map from a finite-dimensional space to itself, one to one and onto are the same thing.
If no information is lost, then nothing in the target can be missed, because the dimensions match.
This is like a map from a finite set to itself.
</document>

<document source="https://tamnd.github.io/math2/node/091AV9YHZY.md" tag="091AV9YHZY" kind="concept">
# Real exponentiation

## Summary

Real exponentiation defines the power $a^x$ for a positive base $a$ and a real exponent $x$. It is $e^{x \ln a}$, and it extends the powers with integer and rational exponents. The node [[real exponentiation|0TV07AQT62]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/091GJ3FYD7.md" tag="091GJ3FYD7" kind="concept">
# Metrizable space

## Summary

A topological space is metrizable when its topology comes from some metric.
Metrization theorems give conditions on the topology alone that make a space metrizable.
</document>

<document source="https://tamnd.github.io/math2/node/0924BCD257.md" tag="0924BCD257" kind="concept">
# Geometric probability

## Summary

Geometric probability studies random geometric objects, such as random points, lines or shapes in the plane or in space.
In the simplest case, a point is chosen uniformly in a region $\Omega$ of finite positive area.
Then the probability of a subregion $A$ is the area of $A$ divided by the area of $\Omega$.
Classic problems are [[Buffon's needle problem|06X5PF6YFP]] and the [[Bertrand paradox|0T0JWJ3M47]].

## Remarks

The same rule works with length on a line and with volume in space.
Mathematicians studied such problems from the 17th century on. Their modern form is part of integral geometry and stochastic geometry.

## Explanations

### L1 human

In geometric probability you measure chances with lengths and areas instead of with counts.
Throw a dart at random at a square board.
The chance to hit the inscribed circle is the area of the circle over the area of the square, that is $\pi/4$.
</document>

<document source="https://tamnd.github.io/math2/node/09285XX7AE.md" tag="09285XX7AE" kind="statement">
# Euler's identity

## Statement

$$e^{i \pi} + 1 = 0.$$

## Remarks

Proof: [[Euler's formula|04R3CY41VJ]] with $z = \pi$ gives $e^{i \pi} = \cos \pi + i \sin \pi = -1$.
The identity links five constants: $0$, $1$, $\pi$, $e$ and $i$.

## Explanations

### L1 human

Walk around the unit circle from $1$ for a distance of $\pi$, which is half of the circle.
You arrive at $-1$, so $e^{i \pi} = -1$.
</document>

<document source="https://tamnd.github.io/math2/node/092DRBFKMG.md" tag="092DRBFKMG" kind="definition">
# Catalan numbers

## Definition

The Catalan numbers $C_n$, for natural numbers $n$, are defined by the recursion
$$C_0 = 1, \qquad C_{n+1} = \sum_{i=0}^{n} C_i\, C_{n-i}.$$

## Remarks

The first values, for $n = 0, 1, \dots, 7$, are $1, 1, 2, 5, 14, 42, 132, 429$.
The [[closed formula|0NNDQVJ54Q]] is $C_n = \frac{1}{n+1} \binom{2n}{n}$.
$C_n$ counts [[Dyck words|0R9JR0465V]] of semilength $n$ and [[binary trees|02A8EMC83C]] with $n$ nodes, and many other families.

## Explanations

### L1 human

Catalan numbers count many structures that split into a left part and a right part.
For example, there are $C_3 = 5$ ways to put brackets on a product $abcd$ of four factors: $((ab)c)d$, $(a(bc))d$, $(ab)(cd)$, $a((bc)d)$ and $a(b(cd))$.
</document>

<document source="https://tamnd.github.io/math2/node/092GEF38YD.md" tag="092GEF38YD" kind="definition">
# Binomial coefficient

## Definition

Let $n$ and $k$ be natural numbers.
The binomial coefficient $\binom{n}{k}$, read "$n$ choose $k$", is the natural number defined by the recursion
$$\binom{n}{0} = 1, \qquad \binom{0}{k+1} = 0, \qquad \binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}.$$

## Remarks

The value $\binom{n}{k}$ is the [[number of k-element subsets|0BFMR2FQ8B]] of an $n$-element set. Many texts take this as the definition.
For $k > n$, $\binom{n}{k} = 0$. For $k \le n$, the [[factorial formula|033KTKKRV6]] gives $\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$.
The recursion is [[Pascal's rule|0YRG241HQQ]], and the values form [[Pascal's triangle|05DME4TZ3K]].

## Explanations

### L1 human

The number $\binom{n}{k}$ tells how many ways there are to choose $k$ objects out of $n$ when the order does not matter.
For example, $\binom{4}{2} = 6$: from $\{1, 2, 3, 4\}$ you can choose $12, 13, 14, 23, 24, 34$.
</document>

<document source="https://tamnd.github.io/math2/node/092S6VBECQ.md" tag="092S6VBECQ" kind="definition">
# Lyapunov stability

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz. Let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$, and let $x^* \in U$ be an [[equilibrium point|0G4X36AE0X]].
The point $x^*$ is Lyapunov stable if for each $\varepsilon > 0$ there is a $\delta > 0$ with this property. For each $x_0 \in U$ with $\|x_0 - x^*\| < \delta$, the solution $\varphi_t(x_0)$ exists for all $t \ge 0$, and $\|\varphi_t(x_0) - x^*\| < \varepsilon$ for all $t \ge 0$.
The point $x^*$ is unstable if it is not Lyapunov stable.

## Remarks

For $x' = -x$ and for $x' = 0$ on $\mathbb{R}$, the equilibrium $0$ is stable. For $x' = x$, it is unstable.
The centre of the [[harmonic oscillator|0YG0ZJPTWG]] is stable but not [[asymptotically stable|05Y1JAM3Y5]]: nearby solutions stay near, but they do not approach it.
A [[Lyapunov function|04P7B5XC22]] gives a way to prove stability without solving the equation.

## Explanations

### L1 human

An equilibrium is stable if solutions that start close to it stay close to it forever.
To stay within distance $\varepsilon$, it is enough to start within a smaller distance $\delta$.
</document>

<document source="https://tamnd.github.io/math2/node/093CM69AGS.md" tag="093CM69AGS" kind="definition">
# Domain

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The domain of $f$ is the set $A$.

## Remarks

The function $f$ has a value $f(a)$ for each $a$ in its domain.
In calculus, the natural domain of a formula is the largest set of real numbers where the formula has a meaning. For $1/x$, it is $\mathbb{R} \setminus \{0\}$.

## Explanations

### L1 human

The domain says which inputs are allowed.
For the square root as a real function, the domain is the set of real numbers $x \ge 0$.
</document>

<document source="https://tamnd.github.io/math2/node/093EDENVA1.md" tag="093EDENVA1" kind="concept">
# Congruent triangles

## Summary

Two triangles are congruent when they have the same shape and the same size. Then a rigid motion, possibly with a reflection, moves one onto the other.
</document>

<document source="https://tamnd.github.io/math2/node/093SAXPJ9P.md" tag="093SAXPJ9P" kind="concept">
# Mesh of a partition

## Summary

The mesh of a partition is the length of its longest subinterval. Riemann sums approach the integral as the mesh tends to zero. The node [[mesh of a partition|04GM4NYXJB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/093X6MN6Y6.md" tag="093X6MN6Y6" kind="statement">
# Minimal polynomial divides the characteristic polynomial

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Then the [[minimal polynomial|03NDVCSHSB]] of $A$ divides the [[characteristic polynomial|0VSSPAFM2C]] of $A$.
In particular, $\deg m_A \le n$.

## Remarks

The [[Cayley–Hamilton theorem|0F765KFC2J]] gives $p_A(A) = 0$, and the minimal polynomial divides each annihilating polynomial.
The characteristic polynomial divides a power of the minimal polynomial, because both have the same irreducible factors.
For $I_2$, $m = t - 1$ divides $p = (t - 1)^2$ and is not equal to it.

## Explanations

### L1 human

The minimal polynomial is a divisor of the characteristic polynomial.
So its degree is at most the size of the matrix.
Both polynomials have the same roots.
</document>

<document source="https://tamnd.github.io/math2/node/094V2VBJXJ.md" tag="094V2VBJXJ" kind="concept">
# Multiplication by a matrix as a linear map

## Summary

Each $m \times n$ matrix $A$ defines the linear map from $F^n$ to $F^m$ that sends a column vector $x$ to the product $Ax$.
</document>

<document source="https://tamnd.github.io/math2/node/095DDGZF16.md" tag="095DDGZF16" kind="definition">
# Row equivalence

## Definition

Let $F$ be a field and let $A$ and $B$ be $m \times n$ matrices over $F$.
The matrix $A$ is *row equivalent* to $B$ when there is a finite sequence of [[elementary row operations|04XWH28452]] that changes $A$ into $B$.

## Remarks

Row equivalence is an equivalence relation, because each elementary row operation has an inverse operation.
The matrices $A$ and $B$ are row equivalent if and only if $B = PA$ for an [[invertible|0EB9N0S33C]] $m \times m$ matrix $P$.
Each matrix is row equivalent to exactly one matrix in [[reduced row echelon form|0WF5AQYTW6]].
Row equivalent matrices have the same [[row space|069MCGX635]] and the same [[null space|0PHDYNR24D]].

## Explanations

### L1 human

Two matrices are row equivalent when row operations can turn one into the other.
If they are augmented matrices of two systems, then the two systems have the same solutions.
Row reduction replaces a matrix by a simpler row equivalent matrix.
</document>

<document source="https://tamnd.github.io/math2/node/095ZHQ7WCJ.md" tag="095ZHQ7WCJ" kind="statement">
# In a PID a nonzero element is irreducible iff it generates a maximal ideal

## Statement

Let $R$ be a [[principal ideal domain|0RZYDVZ475]] and let $p \in R$ with $p \neq 0$.
Then $p$ is [[irreducible|052SDEPZH7]] if and only if the principal ideal $(p)$ is a [[maximal ideal|09J6KGT65G]].

## Remarks

The proof: an ideal $I$ with $(p) \subseteq I$ is $(a)$ with $a \mid p$. If $p$ is irreducible, then $a$ is a unit or an associate of $p$, so $I = R$ or $I = (p)$.
A consequence is that $R/(p)$ is a [[field|0QA8CHH8MG]] for each irreducible $p$. For example, $\mathbb{Q}[x]/(x^2 - 2)$ is a field.
In a principal ideal domain, every nonzero prime ideal is maximal.

## Explanations

### L1 human

In a principal ideal domain, ideals that contain $(p)$ come from the divisors of $p$.
An irreducible $p$ has only trivial divisors.
So no ideal fits strictly between $(p)$ and the whole ring.
</document>

<document source="https://tamnd.github.io/math2/node/0964YQFSA8.md" tag="0964YQFSA8" kind="concept">
# Extended Euclidean algorithm

## Summary

The extended Euclidean algorithm computes the [[greatest common divisor|02WF878BH1]] $d$ of two integers $a$ and $b$, and also integers $x$ and $y$ such that $a x + b y = d$.
It does the steps of the [[Euclidean algorithm|0TRNT4X7J0]].
At each step, it writes the new remainder as an integer combination of $a$ and $b$.

## Remarks

The coefficients $x$ and $y$ prove [[Bézout's identity|0DQVHMYAGA]] for $a$ and $b$.
When $\gcd(a, n) = 1$, the coefficient $x$ of $a$ is a [[modular multiplicative inverse|00GXTQMY8Z]] of $a$ modulo $n$.

## Explanations

### L1 human

For $a = 48$ and $b = 18$, the Euclidean algorithm gives $12 = 48 - 2 \cdot 18$ and $6 = 18 - 1 \cdot 12$.
Put the first equation into the second: $6 = 18 - (48 - 2 \cdot 18) = 3 \cdot 18 - 1 \cdot 48$.
So $x = -1$ and $y = 3$ give $48 x + 18 y = 6$.
</document>

<document source="https://tamnd.github.io/math2/node/096AHMB3P4.md" tag="096AHMB3P4" kind="concept">
# Young diagram

## Summary

A Young diagram is a picture of an integer partition as left-aligned rows of boxes. Young diagrams carry Young tableaux and index the irreducible representations of the symmetric groups. The node [[Young diagram|0K78D81EF7]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/096B7Q5742.md" tag="096B7Q5742" kind="statement">
# Unique paths in trees

## Statement

Let $G$ be a [[graph|0JD5C82QMB]].
Then $G$ is a [[tree|0MJ0JS72DZ]] if and only if $G$ has at least one vertex and these conditions hold.
For any two vertices $u$ and $v$ of $G$, there is exactly one [[path|0JDE56HWQ8]] from $u$ to $v$.

## Remarks

Existence of a path between any two vertices is connectedness. Uniqueness of the path is equivalent to acyclicity: two different paths between the same vertices contain a cycle.
In a tree, the unique path from $u$ to $v$ is a shortest walk, so its length is the [[distance|0WPZ5YPAGR]] $d(u, v)$.

## Explanations

### L1 human

In a tree there is only one way to get from one vertex to another without going back.
If there were two different ways, together they would form a loop, and a tree has no loop.
</document>

<document source="https://tamnd.github.io/math2/node/096HN851YZ.md" tag="096HN851YZ" kind="definition">
# Subnormal series

## Definition

Let $G$ be a [[group|G5S1M9]].
A subnormal series of $G$ is a finite sequence of subgroups $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$, where each $G_{i-1}$ is a [[normal subgroup|0CB0DX5BVV]] of $G_i$.
The [[quotient groups|0KYX2N69J0]] $G_i / G_{i-1}$ are the factors of the series, and $n$ is its length.

## Remarks

The subgroup $G_{i-1}$ needs to be normal only in $G_i$, not in $G$.
When each $G_i$ is normal in $G$, the series is called a normal series.
For example, $\{e\} \trianglelefteq V \trianglelefteq A_4 \trianglelefteq S_4$ is a normal series of $S_4$, where $V$ is the [[Klein four-group|0PMRQJVTME]] in $S_4$.

## Explanations

### L1 human

A subnormal series cuts a group into layers.
Each layer is a quotient group, and you can study the layers one at a time.
Good series have simple layers, like composition series, or abelian layers, like the series of a solvable group.
</document>

<document source="https://tamnd.github.io/math2/node/096PW7Q014.md" tag="096PW7Q014" kind="definition">
# Divisibility

## Definition

Let $a$ and $b$ be integers.
The integer $a$ divides $b$ when there is an integer $k$ such that $b = a k$.
The notation $a \mid b$ means that $a$ divides $b$, and $a \nmid b$ means that it does not.
When $a \mid b$, $a$ is a divisor (or a factor) of $b$, and that $b$ is a multiple of $a$.

## Remarks

Each integer divides $0$, because $0 = a \cdot 0$. The integer $0$ divides only $0$.
The integers $1$ and $-1$ divide each integer.
If $a \mid b$ and $b \mid c$, then $a \mid c$. On the natural numbers, divisibility is a partial order.
If $a \mid b$ and $a \mid c$, then $a \mid (x b + y c)$ for all integers $x$ and $y$.

## Explanations

### L1 human

The integer $a$ divides $b$ when $b$ is in the list $\dots, -2a, -a, 0, a, 2a, \dots$ of the multiples of $a$.
For example, $3 \mid 12$ because $12 = 3 \cdot 4$.
But $5 \nmid 12$, because no integer $k$ gives $5 k = 12$.
</document>

<document source="https://tamnd.github.io/math2/node/096SD4ERQH.md" tag="096SD4ERQH" kind="statement">
# Geometric solutions come from roots of the characteristic polynomial

## Statement

Let $R$ be a commutative ring, let $E$ be a linear recurrence with constant coefficients over $R$, and let $q \in R$.
Then the sequence $(q^n)_{n \ge 0}$ is a [[solution|0ZV2YKKZ58]] of $E$ if and only if $q$ is a root of the [[characteristic polynomial|0BWDJGQYST]] $\chi_E$.

## Remarks

Proof idea: for the sequence $q^n$ the recurrence at index $n$ reads $q^n \chi_E(q) = 0$. At $n = 0$ this is $\chi_E(q) = 0$.
Over a field, if $\chi_E$ has $d$ distinct roots $q_1, \dots, q_d$, every solution is $u_n = a_1 q_1^n + \dots + a_d q_d^n$ with constants $a_i$.
[[Binet's formula|0404D8TN7Y]] is the case of the Fibonacci recurrence.

## Explanations

### L1 human

Try a solution of the form $u_n = q^n$. Put it into the recurrence and divide by $q^n$.
What is left is the equation $\chi_E(q) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/096V7H13H9.md" tag="096V7H13H9" kind="statement">
# Linearly independent set of the right size is a basis

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$ of finite [[dimension|03GSZ8RTZJ]] $n$.
Let $v_1, \dots, v_n \in V$ be $n$ vectors.
If $v_1, \dots, v_n$ are [[linearly independent|0CRMGPAH83]], then they form a [[basis|0KCQ9Y3QKA]] of $V$.
If $v_1, \dots, v_n$ span $V$, then they form a basis of $V$.

## Remarks

Thus, for exactly $\dim V$ vectors, independence and spanning are equivalent.
For example, three vectors of $\R^3$ form a basis if and only if the $3 \times 3$ matrix with these columns has nonzero determinant.
More generally, a family $v_1, \dots, v_k$ is linearly independent if and only if its span has dimension $k$.

## Explanations

### L1 human

When we have exactly as many vectors as the dimension, we only need to check one of the two basis conditions.
Independence gives spanning for free, and spanning gives independence for free.
This halves the work of checking a basis.
</document>

<document source="https://tamnd.github.io/math2/node/096Z9MP2ME.md" tag="096Z9MP2ME" kind="concept">
# Ideal

## Summary

An ideal of a ring is an additive subgroup that is closed under multiplication by every element of the ring.
Ideals are the kernels of ring homomorphisms, and they give quotient rings.
The [[definition of an ideal|0MYSA6MYNA]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/09753QQDCK.md" tag="09753QQDCK" kind="concept">
# Central moment

## Summary

A central moment of a random variable is a moment of its deviation from the mean. The second central moment is the variance.
The node [[central moment|0ZT24EC90K]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0977KZ6PQB.md" tag="0977KZ6PQB" kind="statement">
# Cramer's rule

## Statement

Let $F$ be a field, let $A$ be an $n \times n$ matrix over $F$ with $\det A \ne 0$, and let $b \in F^n$.
For $1 \le i \le n$, let $A_i$ be the matrix that results when column $i$ of $A$ is replaced by $b$.
Then the [[linear system|0GT4WWGWH2]] $Ax = b$ has exactly one solution $x = (x_1, \dots, x_n)$, and
$$x_i = \frac{\det A_i}{\det A} \qquad (1 \le i \le n).$$

## Remarks

For $n = 2$, the system $ax + by = e$, $cx + dy = f$ has the solution $x = (ed - bf)/(ad - bc)$ and $y = (af - ec)/(ad - bc)$.
The rule follows from the formula for the inverse with the [[adjugate|07W9H8QF38]].
It needs $n + 1$ determinants, so it is slow for large $n$ compared with [[Gaussian elimination|0JVJ2WCXZH]].

## Explanations

### L1 human

Cramer's rule gives each unknown as a quotient of two determinants.
The denominator is always $\det A$.
For the numerator of unknown $i$, put the right side $b$ in column $i$ of $A$.
</document>

<document source="https://tamnd.github.io/math2/node/097C7KCAT5.md" tag="097C7KCAT5" kind="statement">
# Riesz's lemma

## Statement

Let $X$ be a normed space over $\mathbb{R}$ or $\mathbb{C}$, and let $Y \neq X$ be a [[closed|0G5AV9Q4RR]] linear subspace of $X$.
Let $\alpha$ be a real number with $0 < \alpha < 1$.
Then there is $x \in X$ with $\|x\| = 1$ and $\|x - y\| \ge \alpha$ for all $y \in Y$.

## Remarks

Proof idea: take $x_0 \notin Y$ and let $r = \inf_{y \in Y} \|x_0 - y\| > 0$, which is positive because $Y$ is closed. Choose $y_0 \in Y$ with $\|x_0 - y_0\| \le r / \alpha$, and let $x = (x_0 - y_0) / \|x_0 - y_0\|$.
If $Y$ is finite-dimensional, one can take $\alpha = 1$.
The lemma is a substitute for orthogonal vectors in normed spaces without an inner product.
Its main application is that a [[compact closed unit ball forces finite dimension|0VZVV8VME5]].

## Explanations

### L1 human

For each closed proper subspace, there is a unit vector that is almost as far from the subspace as possible.
In an inner product space, a unit vector orthogonal to the subspace works with distance $1$.
</document>

<document source="https://tamnd.github.io/math2/node/097KNJFJ7V.md" tag="097KNJFJ7V" kind="definition">
# Domain in the complex plane

## Definition

Let $D$ be a subset of the [[complex plane|0CTE7A31XQ]] $\mathbb{C}$.
The set $D$ is open if for each $z_0 \in D$ there is a real number $r > 0$ such that each $z \in \mathbb{C}$ with $|z - z_0| < r$ is in $D$. Here $|z - z_0|$ is the [[modulus|0KW2X0TFBX]] of $z - z_0$.
The set $D$ is connected if there are no two disjoint nonempty open sets $A$ and $B$ with $D = A \cup B$.
A domain in the complex plane is a subset $D \subseteq \mathbb{C}$ that is nonempty, open and connected.

## Remarks

An open set $D \subseteq \mathbb{C}$ is connected if and only if each two points of $D$ are the end points of a polygonal path in $D$.
Examples of domains are $\mathbb{C}$, each open disc, the upper half-plane $\{z : \operatorname{Im} z > 0\}$ and $\mathbb{C} \setminus \{0\}$.
The union of two disjoint open discs is open but not connected, so it is not a domain.
Some authors use the word region for a domain. Other authors use region for a set between a domain and its closure.

## Explanations

### L1 human

A domain is a set in the plane that has no boundary points in it and that is in one piece.
Open means that you can move a little in each direction from each point and stay in the set.
Connected means that you can walk from each point of the set to each other point without leaving the set.
</document>

<document source="https://tamnd.github.io/math2/node/098C0KZZAP.md" tag="098C0KZZAP" kind="statement">
# Intermediate value theorem for connected spaces

## Statement

Let $X$ be a [[connected space|0D939CMME7]], let $f \colon X \to \mathbb{R}$ be [[continuous|0SY9ECQ19Q]], and let $a, b \in X$.
Then for each real $c$ between $f(a)$ and $f(b)$ there is $x \in X$ with $f(x) = c$.

## Remarks

Proof: $f(X)$ is connected, by [[continuous image of a connected set is connected|0EPD87YVG8]]. Thus it is an interval, by [[connected subsets of the real line are intervals|0B9ZQFMQNV]], and it contains $c$.
For $X = [a, b]$, this is the intermediate value theorem of calculus: a continuous $f \colon [a, b] \to \mathbb{R}$ takes each value between $f(a)$ and $f(b)$.
For example, each polynomial of odd degree with real coefficients has a real root.

## Explanations

### L1 human

A continuous real function on a connected space cannot jump over a value. To get from $f(a)$ to $f(b)$, it passes through all values in between.
</document>

<document source="https://tamnd.github.io/math2/node/098D2E7DEH.md" tag="098D2E7DEH" kind="statement">
# Fixed points of a p-group action

## Statement

Let $p$ be a prime number, let $G$ be a finite [[p-group|06QYRTX9KQ]], and let $G$ [[act|0WCY42EV9V]] on a finite set $X$.
Let $X^G$ be the set of the [[fixed points|0RRWNMYRXC]] of the action.
Then $|X| \equiv |X^G| \pmod p$.

## Remarks

The [[orbits|0ZC7FYE0NZ]] partition $X$, and the orbits with one point are exactly the fixed points.
Each other orbit has a size that divides $|G|$ and is greater than $1$, so its size is divisible by $p$.
A consequence: if $p$ does not divide $|X|$, then the action has at least one fixed point.

## Explanations

### L1 human

The orbits of a $p$-group have sizes that are powers of $p$.
The orbits of size $1$ are the fixed points, and all other orbits have a size divisible by $p$.
So counting modulo $p$ sees only the fixed points.
</document>

<document source="https://tamnd.github.io/math2/node/098D6FQN3N.md" tag="098D6FQN3N" kind="concept">
# Gauss–Jordan elimination

## Summary

Gauss–Jordan elimination is the variant of [[Gaussian elimination|0JVJ2WCXZH]] that continues until the matrix is in [[reduced row echelon form|0WF5AQYTW6]].
After the forward phase gives a row echelon form, it scales each pivot row to make the [[pivot|0J0BBCMJ5W]] equal to $1$.
Then it adds multiples of each pivot row to the rows above it, to make every other entry in each pivot column zero.
Applied to the [[augmented matrix|0T6MWGZSXA]] of a system, the result shows the solutions directly, with no [[back substitution|0ATFVE08MW]].

## Remarks

The result of Gauss–Jordan elimination does not depend on the choices in the algorithm.
The reason is that the reduced row echelon form of a matrix is unique.
Applied to $[A \mid I]$ for an invertible matrix $A$, it gives $[I \mid A^{-1}]$, see [[computing the inverse by row reduction|0Z83DHFMYM]].

## Explanations

### L1 human

Gauss–Jordan elimination cleans up the staircase matrix that Gaussian elimination produces.
It makes every pivot equal to $1$ and clears the entries above the pivots too.
Then each equation names one basic unknown in terms of the free unknowns.
</document>

<document source="https://tamnd.github.io/math2/node/098PHYBTV9.md" tag="098PHYBTV9" kind="concept">
# Combination

## Summary

A combination is a selection of elements of a set in which the order does not matter. Combinations of size $k$ are the $k$-element subsets, and binomial coefficients count them. The node [[combination|0JZSRWRBZE]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/098YP831QY.md" tag="098YP831QY" kind="statement">
# Peano existence theorem

## Statement

Let $D$ be an open subset of $\mathbb{R} \times \mathbb{R}^n$, let $f : D \to \mathbb{R}^n$ be continuous, and let $(t_0, x_0) \in D$.
Then there are an open interval $I$ with $t_0 \in I$ and a differentiable function $z : I \to \mathbb{R}^n$ with these properties.
For each $t \in I$, the point $(t, z(t))$ is in $D$ and $z'(t) = f(t, z(t))$.
Also, $z(t_0) = x_0$.

## Remarks

The theorem needs only continuity of $f$. It gives existence, but not uniqueness. See the [[non-uniqueness example|00K7ASZ1PR]].
With a Lipschitz condition in $x$, the [[Picard–Lindelöf theorem|0P6KRD5B7A]] gives existence and uniqueness.
A common proof builds Euler polygons with smaller and smaller steps and takes a uniformly convergent subsequence by the Arzelà–Ascoli theorem.

## History

Giuseppe Peano published the theorem in 1886, with a proof that was not correct.
In 1890 he published a correct proof that uses successive approximations.
The name Cauchy–Peano theorem also honours Augustin-Louis Cauchy.

## Explanations

### L1 human

If the vector field is continuous, at least one solution starts at each point.
The theorem does not say that the solution is unique, and it gives the solution only for a short time.
</document>

<document source="https://tamnd.github.io/math2/node/0990W5PN13.md" tag="0990W5PN13" kind="concept">
# Conditional probability

## Summary

The conditional probability of an event $A$ given an event $B$ is the probability of $A$ when we know that $B$ has occurred.
It is written $P(A \mid B)$. The node [[conditional probability|0FJB06AD2P]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/099E72KTF8.md" tag="099E72KTF8" kind="concept">
# Commutativity

## Summary

A binary operation is commutative when the order of the two inputs does not change the result.
</document>

<document source="https://tamnd.github.io/math2/node/099GXMHV40.md" tag="099GXMHV40" kind="concept">
# Space of solutions of a linear recurrence

## Summary

The space of solutions of a linear recurrence is the module of all sequences that satisfy it. It has dimension equal to the order of the recurrence over a field. The node [[space of solutions of a linear recurrence|05BDYVQQ0T]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/099SXAG014.md" tag="099SXAG014" kind="definition">
# Dirac measure

## Definition

Let $X$ be a set, let $\mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] on $X$, and let $x \in X$.
The Dirac measure at $x$ is the function $\delta_x \colon \mathcal{F} \to \{0, 1\}$ with
$$\delta_x(A) = \begin{cases} 1 & \text{if } x \in A, \\ 0 & \text{if } x \notin A. \end{cases}$$

## Remarks

$\delta_x$ is a [[probability measure|0VGMEBVW1M]]. In terms of the [[indicator function|0VSB0XF4G7]], $\delta_x(A) = \mathbf{1}_A(x)$.
The integral of a measurable function $f$ with respect to $\delta_x$ is $f(x)$.
Each probability measure on a countable set $X$ with the power set as σ-algebra is a sum $\sum_{x \in X} p(x) \, \delta_x$ with weights $p(x) \ge 0$ of sum $1$.

## Explanations

### L1 human

The Dirac measure puts all the probability on one point.
It describes an experiment whose result is certain, such as a coin with heads on both sides.
</document>

<document source="https://tamnd.github.io/math2/node/099TWVJ909.md" tag="099TWVJ909" kind="concept">
# Geometric multiplicity

## Summary

The geometric multiplicity of an eigenvalue is the dimension of its eigenspace.
</document>

<document source="https://tamnd.github.io/math2/node/09A3N7ZN5G.md" tag="09A3N7ZN5G" kind="concept">
# Converse

## Summary

The converse of an implication exchanges its hypothesis and its conclusion.
</document>

<document source="https://tamnd.github.io/math2/node/09ASMRMD67.md" tag="09ASMRMD67" kind="concept">
# Cross-ratio

## Summary

The cross-ratio is a number made from four points of a line, or of the Riemann sphere, see the [[definition|05MW7MPRBS]].
It does not change under [[Möbius transformations|0SG74WBFA2]] and, more generally, under projective transformations.
</document>

<document source="https://tamnd.github.io/math2/node/09AZCV6GMC.md" tag="09AZCV6GMC" kind="definition">
# Sequence tending to infinity

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]].
The sequence tends to $+\infty$ if, for each real number $M$, there is a natural number $N$ such that $a_n > M$ for all $n \ge N$.
Then the notation is $a_n \to +\infty$ as $n \to \infty$, or $\lim_{n \to \infty} a_n = +\infty$.
The sequence tends to $-\infty$ if $-a_n \to +\infty$.

## Remarks

The sequences $n$, $n^2$ and $2^n$ tend to $+\infty$.
A sequence that tends to $+\infty$ is not bounded above, so it is a [[divergent sequence|01HR2AH0BX]].
The converse fails. The sequence $0, 1, 0, 2, 0, 3, \dots$ is not bounded above, but it does not tend to $+\infty$.

## Explanations

### L1 human

A sequence tends to infinity when its terms become larger than any number you name, and stay larger.
It is not enough that the terms are sometimes large.
</document>

<document source="https://tamnd.github.io/math2/node/09B4X51J09.md" tag="09B4X51J09" kind="definition">
# Discrete logarithm

## Definition

Let $G$ be a finite cyclic [[group|G5S1M9]] of order $N$, with a generator $g$.
For each $h \in G$, the discrete logarithm of $h$ to the base $g$ is the unique integer $k$ with $0 \le k < N$ such that $g^k = h$.
The notation is $k = \log_g h$.

## Remarks

In elementary number theory, $G = (\mathbb{Z}/n\mathbb{Z})^\times$ and $g$ is a [[primitive root|0KM2CJ80G7]] modulo $n$. Then $\log_g a$ is also called the index of $a$, and $N = \varphi(n)$.
The rule $\log_g (h_1 h_2) \equiv \log_g h_1 + \log_g h_2 \pmod N$ holds, as for ordinary logarithms.
For example, $3$ is a primitive root modulo $7$, and $3^4 = 81 \equiv 4 \pmod 7$. So $\log_3 4 = 4$ modulo $7$.
No efficient general algorithm to compute discrete logarithms is known. The [[Diffie–Hellman key exchange|01TKKESHSB]] uses this fact.

## Explanations

### L1 human

[[Modular exponentiation|04DR5PM25N]] computes $g^k \bmod p$ from $k$. The discrete logarithm goes back from the result to $k$.
For small numbers, a table of powers is enough. For numbers with hundreds of digits, the problem seems very hard.
</document>

<document source="https://tamnd.github.io/math2/node/09BG0Y7SZD.md" tag="09BG0Y7SZD" kind="statement">
# Power series of the complex logarithm

## Statement

Let $\operatorname{Log}$ be the [[principal branch|0YKY4M3KRQ]] of the logarithm.
For each $z \in \mathbb{C}$ with $|z| < 1$, the series $\sum_{n \ge 1} z^n / n$ converges absolutely, and
$$-\operatorname{Log}(1 - z) = \sum_{n=1}^{\infty} \frac{z^n}{n} = z + \frac{z^2}{2} + \frac{z^3}{3} + \cdots.$$

## Remarks

Replacing $z$ by $-z$ gives $\operatorname{Log}(1 + z) = \sum_{n \ge 1} (-1)^{n+1} z^n / n$ for $|z| < 1$.
The [[radius of convergence|045EWAXY9P]] is $1$. For $|z| < 1$, the number $1 - z$ has positive real part, so it is not on the branch cut.
Proof idea: both sides vanish at $0$, and both have the derivative $1/(1 - z) = \sum_{n \ge 0} z^n$ on the unit disc.
At $z = -1$ the series still converges, and gives $\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \cdots$.

## Explanations

### L1 human

Integrate the geometric series $1 + z + z^2 + \cdots = 1/(1 - z)$ term by term from $0$ to $z$.
On the left you get $z + z^2/2 + z^3/3 + \cdots$, and on the right you get $-\operatorname{Log}(1 - z)$.
</document>

<document source="https://tamnd.github.io/math2/node/09BJQQB3VS.md" tag="09BJQQB3VS" kind="concept">
# Necessary and sufficient conditions

## Summary

When the [[implication|0ZNX6DQ3EJ]] $p \to q$ is true, $p$ is a sufficient condition for $q$, and $q$ is a necessary condition for $p$.
A condition $p$ is necessary and sufficient for $q$ when $p \to q$ and $q \to p$ are both true, that is, when $p \leftrightarrow q$ is true.

## Remarks

"$p$ only if $q$" means $p \to q$, and "$p$ if $q$" means $q \to p$.
"Necessary and sufficient" is another way to say "if and only if". See [[biconditional|0V6GAWTBYW]].
For an integer $n$, "$n$ is divisible by $4$" is sufficient but not necessary for "$n$ is even".

## Explanations

### L1 human

A sufficient condition is enough to get the result.
A necessary condition is one that you cannot do without.
To have a driving licence is necessary to drive legally, but it is not sufficient: you also need a car.
</document>

<document source="https://tamnd.github.io/math2/node/09C4JP7Y0M.md" tag="09C4JP7Y0M" kind="object">
# Heaviside step function

## Description

The Heaviside step function is the function $H : \mathbb{R} \to \mathbb{R}$ with $H(x) = 0$ for $x < 0$ and $H(x) = 1$ for $x > 0$.
Conventions for the value $H(0)$ differ: common choices are $0$, $\frac{1}{2}$ and $1$.
For $a \ge 0$ and $s \in \mathbb{C}$ with $\operatorname{Re} s > 0$, the [[Laplace transform|05XT5AVJBK]] of $t \mapsto H(t - a)$ is $e^{-a s} / s$.

## Remarks

The function $t \mapsto H(t - a)$ switches a signal on at the time $a$. For example, $f(t) H(t - a)$ is $0$ before $a$ and $f(t)$ after $a$.
For $g : [0, \infty) \to \mathbb{C}$ with a Laplace transform $G$ and $a \ge 0$, the shifted function $t \mapsto H(t - a)\, g(t - a)$ has the transform $e^{-a s} G(s)$.
In the sense of distributions, the derivative of $H$ is the [[Dirac delta|0KPHENZ5S7]].
It is named after Oliver Heaviside.

## Explanations

### L1 human

The step function is off before time $0$ and on after it.
It models a switch, such as a voltage turned on at one instant.
</document>

<document source="https://tamnd.github.io/math2/node/09CBQB6V67.md" tag="09CBQB6V67" kind="statement">
# Area of a region as a line integral

## Statement

Let $C$ be a positively oriented, piecewise smooth, simple [[closed curve|0D032NSZKY]] in $\mathbb{R}^2$ that encloses the bounded region $D$.
Then the area $A$ of $D$ satisfies
$$A = \oint_C x \, dy = -\oint_C y \, dx = \frac{1}{2} \oint_C (x \, dy - y \, dx).$$

## Remarks

Proof: apply [[Green's theorem|0HR3N02317]] with $(L, M)$ equal to $(0, x)$, $(-y, 0)$ and $(-y/2, x/2)$. In each case $\partial M / \partial x - \partial L / \partial y = 1$.
Example: the ellipse $x = a \cos t$, $y = b \sin t$, $t \in [0, 2\pi]$, with $a, b > 0$, gives $A = \tfrac{1}{2} \int_0^{2\pi} a b \, dt = \pi a b$.
For a polygon with vertices $(x_1, y_1), \dots, (x_n, y_n)$ in counterclockwise order, the formula gives $A = \tfrac{1}{2} \sum_{k=1}^n (x_k y_{k+1} - x_{k+1} y_k)$, with $(x_{n+1}, y_{n+1}) = (x_1, y_1)$.

## Explanations

### L1 human

You can find the area of a region by going once around its boundary, without looking inside.
This is a direct use of Green's theorem with a field whose curl is $1$.
</document>

<document source="https://tamnd.github.io/math2/node/09CCE86A6D.md" tag="09CCE86A6D" kind="definition">
# Partition of a set

## Definition

Let $A$ be a set.
A partition of $A$ is a set $P$ of [[subsets|0FE8Q5S42Y]] of $A$, called blocks, with these three properties:

1. each block is nonempty;
2. the blocks are pairwise [[disjoint|0W4RPA8QH3]];
3. the union of the blocks is $A$.

Equivalently, each element of $A$ is in exactly one block.

## Remarks

The set $\{\{1, 3\}, \{2\}\}$ is a partition of $\{1, 2, 3\}$.
The even and the odd integers form a partition of $\mathbb{Z}$ with two blocks.
The [[equivalence classes form a partition|01HT2CH8BH]], and each partition comes from exactly one equivalence relation. See [[the correspondence between equivalence relations and partitions|074482Q7CD]].

## Explanations

### L1 human

A partition is a way to sort all elements into boxes.
No box is empty, and each element goes into exactly one box.
</document>

<document source="https://tamnd.github.io/math2/node/09CEDJYBKP.md" tag="09CEDJYBKP" kind="concept">
# Cartesian product

## Summary

The Cartesian product of two sets is a set of ordered pairs.
The first entry of each pair is in the first set, and the second entry is in the second set.
Each such pair is in the product.
</document>

<document source="https://tamnd.github.io/math2/node/09CJ00H3NW.md" tag="09CJ00H3NW" kind="statement">
# Open sets characterised by neighbourhoods

## Statement

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]] and let $x \in X$.
A set $N \subseteq X$ is a neighbourhood of $x$ when there is an [[open set|01ZQQDCERP]] $U$ with $x \in U \subseteq N$.
Then a set $A \subseteq X$ is open if and only if $A$ is a neighbourhood of each of its points.

## Remarks

Proof of the converse: for each $x \in A$, choose an open $U_x$ with $x \in U_x \subseteq A$. Then $A = \bigcup_{x \in A} U_x$, which is open as a [[union|05GK32H72H]] of open sets.
A neighbourhood need not be open. For example, $[-1, 1]$ is a neighbourhood of $0$ in $\mathbb{R}$.
In a metric space, $N$ is a neighbourhood of $x$ if and only if $N$ contains a ball $B(x, \varepsilon)$ for some $\varepsilon > 0$.

## Explanations

### L1 human

A neighbourhood of $x$ is a set that contains all points near enough to $x$.
A set is open when it is a neighbourhood of each of its points. Then no point of the set lies on its edge.
</document>

<document source="https://tamnd.github.io/math2/node/09DXWYRD43.md" tag="09DXWYRD43" kind="concept">
# Transpose

## Summary

The transpose of a matrix is the matrix that we get when we write the rows of the matrix as columns.
</document>

<document source="https://tamnd.github.io/math2/node/09DZ8CD1AK.md" tag="09DZ8CD1AK" kind="statement">
# Coordinates in an orthonormal basis

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $(e_1, \dots, e_n)$ be an [[orthonormal basis|0S9GGTR0EX]] of $V$.
Then, for each $v \in V$,
$$v = \sum_{i=1}^{n} \langle v, e_i \rangle\, e_i.$$
So the coordinates of $v$ in this basis are the inner products $\langle v, e_i \rangle$.

## Remarks

If $v = \sum_j a_j e_j$, then $\langle v, e_i \rangle = \sum_j a_j \langle e_j, e_i \rangle = a_i$.
For the basis $\frac{1}{\sqrt 2}(1, 1)$, $\frac{1}{\sqrt 2}(1, -1)$ of $\R^2$, the vector $(3, 1)$ has the coordinates $4/\sqrt 2$ and $2/\sqrt 2$.
With the convention of linearity in the second argument, the coefficient is written $\langle e_i, v \rangle$.

## Explanations

### L1 human

In an orthonormal basis, we find each coordinate of a vector with one inner product.
This is much easier than solving a linear system.
It is the finite-dimensional form of a Fourier expansion.
</document>

<document source="https://tamnd.github.io/math2/node/09E1RCZ7YT.md" tag="09E1RCZ7YT" kind="statement">
# Trace of a transpose

## Statement

Let $R$ be a commutative ring and let $A$ be an $n \times n$ matrix over $R$.
Then $\tr(A^T) = \tr A$, where $A^T$ is the [[transpose|09DXWYRD43]] of $A$ and $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

The transpose does not move the entries on the main diagonal, so the sum of the diagonal entries stays the same.
For a complex matrix, $\tr(A^*) = \overline{\tr A}$.

## Explanations

### L1 human

Transposing flips a matrix over its diagonal.
The diagonal itself does not move.
So the trace, which only sees the diagonal, does not change.
</document>

<document source="https://tamnd.github.io/math2/node/09E6T7VYAG.md" tag="09E6T7VYAG" kind="concept">
# Complex differentiability

## Summary

A complex function is complex differentiable at a point when its difference quotient has a limit as the complex increment goes to $0$. See the [[definition|0D1C9Y1EXA]].
The limit is the complex derivative.
A function that is complex differentiable at each point of an open set is holomorphic on that set.
</document>

<document source="https://tamnd.github.io/math2/node/09EGC6F8PT.md" tag="09EGC6F8PT" kind="concept">
# Partial derivative

## Summary

A partial derivative of a [[function of several variables|0JPCHTXWG0]] is its derivative with respect to one variable, while all other variables stay fixed.
It measures the rate of change of the function along one coordinate axis.
For $f(x, y) = x^2 y$, the partial derivatives are $\partial f / \partial x = 2xy$ and $\partial f / \partial y = x^2$.
</document>

<document source="https://tamnd.github.io/math2/node/09EYMKMSB9.md" tag="09EYMKMSB9" kind="definition">
# Algebraic extension

## Definition

A [[field extension|0WQ1A8SHTZ]] $L/K$ is algebraic if every element of $L$ is [[algebraic|0NPW1DWECM]] over $K$.

## Remarks

[[Every finite extension is algebraic|0DFYTNZ6AM]], so $\mathbb{C}/\mathbb{R}$ and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ are algebraic.
The converse fails: the field of all algebraic numbers is an algebraic extension of $\mathbb{Q}$ of infinite degree.
The extension $\mathbb{R}/\mathbb{Q}$ is not algebraic, because $\pi$ is transcendental.
An [[algebraically closed field|02SJG1HC32]] has no algebraic extension other than itself.

## Explanations

### L1 human

In an algebraic extension, every new element solves a polynomial equation over the base field.
There are no elements that behave like free variables.
</document>

<document source="https://tamnd.github.io/math2/node/09F2QPVQY7.md" tag="09F2QPVQY7" kind="concept">
# Circuit

## Summary

A circuit is a closed trail in a graph: a round trip that uses each edge at most once. Eulerian circuits are the main example. The node [[circuit|0PD57PSHNR]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/09FQT057T7.md" tag="09FQT057T7" kind="definition">
# Upper and lower Darboux integrals

## Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded.
The upper Darboux integral of $f$ is the infimum of the upper [[Darboux sums|0QRQVH0CRW]] $U(f, P)$ over all partitions $P$ of $[a, b]$.
The lower Darboux integral of $f$ is the supremum of the lower Darboux sums $L(f, P)$ over all partitions $P$.

## Remarks

The lower Darboux integral is at most the upper one.
The function is Darboux integrable if the two are equal. By [[a theorem|021XJAQVDC]], this is the same as [[Riemann integrability|0N1JQXZXHC]].
For the [[Dirichlet function|03Z4JC4YRJ]] on $[0, 1]$, the upper integral is $1$ and the lower integral is $0$.

## Explanations

### L1 human

The upper integral is the best upper estimate of the area by step functions above the graph.
The lower integral is the best lower estimate by step functions below the graph.
</document>

<document source="https://tamnd.github.io/math2/node/09FTAET5Z4.md" tag="09FTAET5Z4" kind="object">
# Complex numbers

## Description

A complex number is a number of the form $a + b i$, where $a$ and $b$ are [[real numbers|08J2M1JYAG]] and $i$ is the [[imaginary unit|0ESCX36MMC]], with $i^2 = -1$.
We write $\mathbb{C}$ for the set of complex numbers.
The real number $a$ is the real part and $b$ is the imaginary part of $a + b i$, see [[real and imaginary parts|02ZE6JHJ7G]].
Addition and multiplication follow the usual rules of algebra together with $i^2 = -1$:
$$(a + b i) + (c + d i) = (a + c) + (b + d) i, \qquad (a + b i)(c + d i) = (a c - b d) + (a d + b c) i.$$
With these operations, $\mathbb{C}$ is a [[field|0FRSR10DWF]] that contains $\mathbb{R}$ as the numbers $a + 0 i$.

## Remarks

We picture $\mathbb{C}$ as the [[complex plane|0CTE7A31XQ]], with the point $(a, b)$ for $a + b i$.
Every polynomial equation of positive degree with complex coefficients has a solution in $\mathbb{C}$. This is the fundamental theorem of algebra.
Unlike $\mathbb{R}$, the field $\mathbb{C}$ [[cannot be ordered|01CJRJKDAA]].

## Explanations

### L1 human

The equation $x^2 = -1$ has no real solution, because a real square is never negative.
The complex numbers add a new number $i$ with $i^2 = -1$, and then all the numbers $a + b i$ that you can build from it.
For example, $(1 + 2 i)(3 - i) = 3 - i + 6 i - 2 i^2 = 5 + 5 i$.
</document>

<document source="https://tamnd.github.io/math2/node/09G6KHS6GY.md" tag="09G6KHS6GY" kind="statement">
# Epsilon-N definition of the limit of a sequence

## Statement

Let $(a_n)$ be a real sequence and let $L$ be a real number.
Then $(a_n)$ converges to $L$, in the sense of the [[limit of a sequence|0XKWSSY8P2]], if and only if the following holds.
For each real number $\varepsilon > 0$ there is a natural number $N$ such that $|a_n - L| < \varepsilon$ for all $n \ge N$.

## Remarks

The proof uses two facts. Each interval $(L - \varepsilon, L + \varepsilon)$ is open and contains $L$. Each open interval that contains $L$ contains such an interval.
In a metric space $(X, d)$ the same statement holds with $d(a_n, L)$ in place of $|a_n - L|$.
The number $N$ depends on $\varepsilon$. A smaller $\varepsilon$ in general needs a larger $N$.

## Explanations

### L1 human

The number $\varepsilon$ is the allowed error, and $N$ is the point after which the error is always smaller than $\varepsilon$.
For $a_n = 1/(n+1)$ and $L = 0$, the choice $N > 1/\varepsilon$ works.
</document>

<document source="https://tamnd.github.io/math2/node/09GCQ5STVJ.md" tag="09GCQ5STVJ" kind="statement">
# Second Sylow theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number.
Then any two [[Sylow p-subgroups|0053CQE9Y2]] $P$ and $Q$ of $G$ are [[conjugate|0H11ZPX4ND]]: there is a $g \in G$ with $Q = gPg^{-1}$.
Moreover, each $p$-subgroup of $G$ is contained in some Sylow $p$-subgroup of $G$.

## Remarks

So all Sylow $p$-subgroups of $G$ are [[isomorphic|08MG7H8JW5]].
A Sylow $p$-subgroup is [[normal|0CB0DX5BVV]] in $G$ if and only if $n_p = 1$.
The proof lets a $p$-subgroup $H$ act on the left cosets of $P$ and uses the [[fixed point lemma|098D2E7DEH]].

## Explanations

### L1 human

The Sylow $p$-subgroups of a group can look different as subsets.
But each one can be moved onto each other one by a conjugation.
So, up to a renaming, there is only one Sylow $p$-subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/09GK3DE0WV.md" tag="09GK3DE0WV" kind="definition">
# Dihedral group

## Definition

Let $n \geq 3$ be an integer.
Let $P$ be a regular $n$-gon in the plane with centre $0$.
The dihedral group $D_n$ is the group of all isometries of the plane that map $P$ onto itself, with composition as the operation.
Let $r$ be the rotation by the angle $2\pi/n$ about $0$, and let $s$ be a reflection in a line of symmetry of the polygon.

## Remarks

The elements of $D_n$ are the rotations $e, r, \dots, r^{n-1}$ and the reflections $s, rs, \dots, r^{n-1}s$.
The relations $r^n = e$, $s^2 = e$ and $srs^{-1} = r^{-1}$ hold, and these relations give a presentation of $D_n$.
In abstract algebra, many authors write $D_{2n}$ for this group, so the index is the order and not the number of sides.
The group $D_n$ is not abelian, and it is a [[semidirect product|0H3WZXCZNF]] of $\langle r \rangle$ and $\langle s \rangle$.

## Explanations

### L1 human

Take a regular polygon and look at all the ways to pick it up and put it back in the same outline.
You can turn it, and you can flip it over.
These motions form the dihedral group, and turning after a flip is not the same as flipping after a turn.
</document>

<document source="https://tamnd.github.io/math2/node/09GK5PS824.md" tag="09GK5PS824" kind="statement">
# Transpose of a product

## Statement

Let $R$ be a commutative ring.
Let $A$ be an $m \times n$ matrix and $B$ an $n \times p$ matrix over $R$.
Then $(AB)^T = B^T A^T$, where $^T$ denotes the [[transpose|09DXWYRD43]].

## Remarks

The order of the factors reverses.
The proof compares entries: both sides have entry $(k, i)$ equal to $\sum_{j} a_{ij} b_{jk}$.
By induction, $(A_1 A_2 \cdots A_k)^T = A_k^T \cdots A_2^T A_1^T$.
If $R$ is not commutative, the identity can fail.

## Explanations

### L1 human

The transpose of a product is the product of the transposes in the reverse order.
This is like taking off socks and shoes: the last thing put on comes off first.
The sizes also force this order, because $B^T$ is $p \times n$ and $A^T$ is $n \times m$.
</document>

<document source="https://tamnd.github.io/math2/node/09GQ3ZTQJ9.md" tag="09GQ3ZTQJ9" kind="statement">
# Closed sets contain their limits

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $F$ be a [[closed subset|0G5AV9Q4RR]] of $X$.
Let $(x_n)$ be a sequence of points of $F$ that [[converges|06JYPJ4WHH]] to a point $x \in X$.
Then $x \in F$.

## Remarks

Proof: if $x \notin F$, then $X \setminus F$ is open and contains $x$. Thus some ball $B(x, \varepsilon)$ misses $F$. But $x_n \in B(x, \varepsilon)$ for large $n$, a contradiction.
In a metric space the converse is also true: a set that contains the limits of all its convergent sequences is closed. This follows from [[closure is the set of limits of sequences|003QP7A8HG]].
The statement holds in every [[topological space|0CNQXVBHCX]], with the same proof.

## Explanations

### L1 human

You cannot leave a closed set by a limit of a sequence.
For example, $1/n \to 0$ in $\mathbb{R}$, and $0$ is not in $(0, 1]$, which shows that $(0, 1]$ is not closed.
</document>

<document source="https://tamnd.github.io/math2/node/09GRCZVR5E.md" tag="09GRCZVR5E" kind="statement">
# Completeness of the space of bounded continuous functions

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $(Y, d_Y)$ be a [[complete metric space|08EWT7J0SC]].
Let $C_b(X, Y)$ be the set of [[continuous maps|0SY9ECQ19Q]] $f \colon X \to Y$ whose image $f(X)$ is [[bounded|0P953MG05E]] in $Y$.
For $f, g \in C_b(X, Y)$, let $d_\infty(f, g) = \sup_{x \in X} d_Y(f(x), g(x))$, with $d_\infty(f, g) = 0$ when $X$ is empty.
Then $(C_b(X, Y), d_\infty)$ is a complete metric space.

## Remarks

For $Y = \mathbb{R}$, the metric $d_\infty$ is the [[uniform metric|06HE9J7DXF]], and it comes from the [[sup norm|0PARCKE8WP]].
Proof sketch: a Cauchy sequence $(f_n)$ in $d_\infty$ is Cauchy at each point $x$, so it has a pointwise limit $f(x)$ in $Y$. The convergence is uniform, and a uniform limit of continuous maps is continuous. The limit is also bounded.
For a compact space $X$, each continuous map to $\mathbb{R}$ is bounded, so $C(X, \mathbb{R})$ is complete with the sup metric.

## Explanations

### L1 human

If continuous functions get closer and closer to each other uniformly, then they converge uniformly to a function.
The key point is that this limit function is again continuous, so the limit stays in the space.
</document>

<document source="https://tamnd.github.io/math2/node/09H9Y0Q11W.md" tag="09H9Y0Q11W" kind="example">
# The symmetric group S3

## Example

The symmetric group $S_3$ has the six elements $e$, $(1\ 2)$, $(1\ 3)$, $(2\ 3)$, $(1\ 2\ 3)$ and $(1\ 3\ 2)$.
It is [[isomorphic|08MG7H8JW5]] to the [[dihedral group|0JZ27DMJ5D]] $D_3$ of the symmetries of an equilateral triangle, when you number the vertices $1$, $2$, $3$.
The $3$-cycles are the rotations, and the transpositions are the reflections.
It is not [[abelian|082GHNPJ53]], because $(1\ 2)(2\ 3) = (1\ 2\ 3)$ but $(2\ 3)(1\ 2) = (1\ 3\ 2)$.
Thus $S_3$ is the smallest group that is not abelian.
Its [[conjugacy classes|0DXD4QSZ8A]] are $\{e\}$, the three transpositions, and the two $3$-cycles.
Its subgroups are $\{e\}$, three subgroups of order $2$, the [[alternating group|0Z5905Q73D]] $A_3$ of order $3$, and $S_3$.
Only $\{e\}$, $A_3$ and $S_3$ are [[normal|0CB0DX5BVV]].
</document>

<document source="https://tamnd.github.io/math2/node/09HEXDQ0CQ.md" tag="09HEXDQ0CQ" kind="definition">
# Outer automorphism group

## Definition

Let $G$ be a [[group|G5S1M9]].
The outer automorphism group of $G$ is the [[quotient group|0KYX2N69J0]] $\operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Inn}(G)$.

## Remarks

The quotient exists because $\operatorname{Inn}(G)$ is a normal subgroup of $\operatorname{Aut}(G)$.
An automorphism that is not inner is called an outer automorphism, but the elements of $\operatorname{Out}(G)$ are cosets, not automorphisms.
For an abelian group, $\operatorname{Inn}(G)$ is trivial, so $\operatorname{Out}(G) \cong \operatorname{Aut}(G)$.
The symmetric group $S_6$ is the only symmetric group $S_n$ with nontrivial outer automorphism group.

## Explanations

### L1 human

Some symmetries of a group come from conjugation by its own elements, and some do not.
The outer automorphism group keeps only the second kind, up to the first kind.
For many groups it is small or trivial.
</document>

<document source="https://tamnd.github.io/math2/node/09HT3P35YP.md" tag="09HT3P35YP" kind="statement">
# Euler's polyhedron formula

## Statement

Let $P$ be a convex [[polyhedron|0325SGSJAR]] in $\mathbb{R}^3$ with $V$ vertices, $E$ edges and $F$ faces. Then
$$V - E + F = 2.$$

## Remarks

The number $V - E + F$ is the Euler characteristic of the surface of $P$. The value $2$ is the Euler characteristic of the sphere.
The formula also holds for each polyhedron whose surface is topologically a sphere and whose faces are topologically discs.
It can fail for other polyhedra. The surface of a polyhedron with the shape of a torus has $V - E + F = 0$.
For the cube, $8 - 12 + 6 = 2$. For the regular tetrahedron, $4 - 6 + 4 = 2$.

## History

Euler stated the formula in 1758. Cauchy gave a proof in 1811.

## Explanations

### L1 human

Count the corners, the edges and the faces of any convex solid with flat faces. Corners minus edges plus faces always gives $2$. Cut off one corner of a cube: you get $2$ more corners, $3$ more edges and $1$ more face, and the count stays $2$.
</document>

<document source="https://tamnd.github.io/math2/node/09J1H07VAH.md" tag="09J1H07VAH" kind="concept">
# Method of moments

## Summary

The method of moments is a way to build [[estimators|0TTNRP5PFQ]] of unknown parameters.
Suppose that the law of an i.i.d. sample $X_1, \dots, X_n$ depends on $k$ real parameters $\theta = (\theta_1, \dots, \theta_k)$, and that the [[moments|0PH0VHTQS0]] $\mu_j(\theta) = E_\theta[X_1^j]$ exist for $j \le k$.
The method replaces each $\mu_j(\theta)$ with the sample moment $\hat{m}_j = \frac{1}{n} \sum_{i=1}^{n} X_i^j$, and solves the $k$ equations $\mu_j(\hat{\theta}) = \hat{m}_j$ for $\hat{\theta}$.

## Remarks

Example: for $X_i \sim \operatorname{Exp}(\lambda)$, $E[X_1] = 1/\lambda$, so the method gives $\hat{\lambda} = 1/\bar{X}_n$.
Example: for $X_i \sim \mathcal{N}(\mu, \sigma^2)$, it gives $\hat{\mu} = \bar{X}_n$ and $\hat{\sigma}^2 = \frac{1}{n}\sum_i (X_i - \bar{X}_n)^2$.
If the map $\theta \mapsto (\mu_1(\theta), \dots, \mu_k(\theta))$ has a continuous inverse, the law of large numbers makes the estimators [[consistent|0XJVF81QF6]].
The estimators are often simple, but they are usually less efficient than maximum likelihood estimators.



## Explanations

### L1 human

The method of moments chooses the parameters so that the model has the same averages as the data.
For example, if the data have mean $2$, an exponential model with mean $1/\lambda$ gets the rate $\lambda = 1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/09J6KGT65G.md" tag="09J6KGT65G" kind="definition">
# Maximal ideal

## Definition

Let $R$ be a [[ring|0894J980RK]].
An [[ideal|0MYSA6MYNA]] $M$ of $R$ is maximal if it has these two properties.

1. $M \neq R$.
2. For every ideal $I$ of $R$ with $M \subseteq I$, either $I = M$ or $I = R$.

## Remarks

The definition is the same for left, right and two-sided ideals, with ideals of the same type in condition 2.
In $\mathbb{Z}$, the ideal $n\mathbb{Z}$ with $n \ge 0$ is maximal if and only if $n$ is a prime number.
In a [[commutative ring|0GYGHKX834]] $R$, an ideal $M$ is maximal if and only if [[$R/M$ is a field|0WD3SE7S1X]].
The zero ideal of a [[field|0QA8CHH8MG]] is its only maximal ideal.

## Explanations

### L1 human

A maximal ideal is as large as possible without being the whole ring.
In the integers, $(6)$ is not maximal, because $(6) \subseteq (2)$.
The ideal $(2)$ is maximal, because no ideal lies strictly between $(2)$ and $\mathbb{Z}$.
</document>

<document source="https://tamnd.github.io/math2/node/09JP6G9FTA.md" tag="09JP6G9FTA" kind="definition">
# Forest

## Definition

A forest is an [[acyclic|0XNN9Z8E5A]] [[graph|0JD5C82QMB]].

## Remarks

The connected components of a forest are [[trees|0MJ0JS72DZ]]. Thus a forest is a disjoint union of trees.
Every tree is a forest. The graph with no edges on $n$ vertices is a forest with $n$ components.

## Explanations

### L1 human

A forest is a graph with no cycles, and it may be in several pieces.
Each piece is a tree, as the name suggests.
</document>

<document source="https://tamnd.github.io/math2/node/09JXE6XMCC.md" tag="09JXE6XMCC" kind="statement">
# Sum of the totient over the divisors

## Statement

Let $n$ be a positive integer.
Then $\sum_{d \mid n} \varphi(d) = n$, where the sum runs over the positive divisors $d$ of $n$, and $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

The proof: sort the fractions $\frac{1}{n}, \frac{2}{n}, \dots, \frac{n}{n}$ by their denominators in lowest terms. For each divisor $d$ of $n$, exactly $\varphi(d)$ of them have the denominator $d$.
For example, for $n = 12$: $\varphi(1) + \varphi(2) + \varphi(3) + \varphi(4) + \varphi(6) + \varphi(12) = 1 + 1 + 2 + 2 + 2 + 4 = 12$.
With the [[Möbius inversion formula|07Q4TBTFVB]], the result gives $\varphi(n) = \sum_{d \mid n} \mu(d) \, n / d$.

## History

Gauss proved this identity in his Disquisitiones Arithmeticae of 1801.

## Explanations

### L1 human

Write the fractions $\frac{1}{6}, \frac{2}{6}, \dots, \frac{6}{6}$ in lowest terms: $\frac16, \frac13, \frac12, \frac23, \frac56, \frac11$.
The denominators $1, 2, 3, 6$ appear $1, 1, 2, 2$ times. These are $\varphi(1), \varphi(2), \varphi(3), \varphi(6)$, and their sum is $6$.
</document>

<document source="https://tamnd.github.io/math2/node/09JZ9P4899.md" tag="09JZ9P4899" kind="statement">
# Groups of order p squared are abelian

## Statement

Let $p$ be a prime number and let $G$ be a [[group|G5S1M9]] of order $p^2$.
Then $G$ is [[abelian|082GHNPJ53]].

## Remarks

By the [[nontrivial centre theorem|0RZCTT30H5]] and [[Lagrange's theorem|3C53TD]], $|Z(G)|$ is $p$ or $p^2$.
If $|Z(G)| = p$, then $G/Z(G)$ has order $p$, so it is cyclic, and [[this criterion|0AH9Q6YNMT]] gives a contradiction.
Thus $Z(G) = G$, and $G$ is isomorphic to $\mathbb{Z}/p^2\mathbb{Z}$ or to $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$.
For $p^3$ the result is false: the [[dihedral group|0JZ27DMJ5D]] $D_4$ and the [[quaternion group|0NRJGW864D]] have order $8$ and are not abelian.

## Explanations

### L1 human

A group of order $p^2$ has a centre with more than one element.
If the centre were not the whole group, the quotient by the centre would be cyclic, and that is impossible.
So all elements commute.
</document>

<document source="https://tamnd.github.io/math2/node/09K9MWD2QQ.md" tag="09K9MWD2QQ" kind="concept">
# Propositional logic

## Summary

Propositional logic is the part of logic that studies [[propositions|0ZGC7ZESYN]] and the connectives that join them.
Its formulas are made from propositional variables with connectives such as $\neg$, $\land$, $\lor$, $\to$ and $\leftrightarrow$.
It does not look inside a proposition, so it has no variables for objects and no quantifiers.

## Remarks

Other names are sentential logic and propositional calculus.
In classical propositional logic, the value of a formula depends only on the values of its variables. A [[truth table|05RSK4Q657]] computes it.
[[Predicate logic|086QKPF0A6]] extends propositional logic with predicates and quantifiers.

## Explanations

### L1 human

In propositional logic, a sentence such as "$7$ is prime" is one unit $p$.
You study only how such units combine with "not", "and", "or" and "if ... then".
For example, $p \lor \neg p$ is true whatever $p$ says.
</document>

<document source="https://tamnd.github.io/math2/node/09KDHB82SG.md" tag="09KDHB82SG" kind="definition">
# Metrizable space

## Definition

A [[topological space|0CNQXVBHCX]] $(X, \tau)$ is metrizable when there is a [[metric|063W5JMY85]] $d$ on $X$ whose [[metric topology|0A6YRNZB2R]] is $\tau$.

## Remarks

The metric is not unique: equivalent metrics give the same topology.
Each metrizable space is [[Hausdorff|0MXB4CKHFW]] and [[normal|0AD0BZ49DW]].
An indiscrete space with at least two points is not metrizable.
Urysohn's metrization theorem: each [[second-countable|00RE1R4W7N]] [[regular|0JF9KW2TXB]] Hausdorff space is metrizable.

## Explanations

### L1 human

A metrizable space is a topological space where you can measure distances in a way that gives exactly the open sets you already have.
Properties that hold in all metric spaces then hold in this space.
</document>

<document source="https://tamnd.github.io/math2/node/09M2RDB61G.md" tag="09M2RDB61G" kind="definition">
# Order of a differential equation

## Definition

Let $F\left(x, y, y', \dots, y^{(n)}\right) = 0$ be an [[ordinary differential equation|0P97V20RW7]] for an unknown function $y$ of $x$.
Its order is the largest integer $k$ such that the equation depends on the derivative $y^{(k)}$.
For a system of equations, the order is the largest order of a derivative that occurs in the system.

## Remarks

The equation $y' = x y$ has order $1$, and the equation $y'' + \sin y = 0$ has order $2$.
The order does not depend on the powers of the derivatives: $(y')^3 = y$ has order $1$.
An equation of order $n$ can be changed into a first-order system of $n$ equations. See [[reduction to a first-order system|0TH9CPW3VS]].

## Explanations

### L1 human

The order tells you how many times the unknown function is differentiated, at most.
An equation with $y'$ only is of first order. An equation with $y''$, but no higher derivative, is of second order.
</document>

<document source="https://tamnd.github.io/math2/node/09N7DZ7SDB.md" tag="09N7DZ7SDB" kind="statement">
# Ring homomorphisms from a field are injective

## Statement

Let $K$ be a [[field|0QA8CHH8MG]], let $S$ be a nonzero [[ring|0894J980RK]] and let $f \colon K \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]].
Then $f$ is injective.

## Remarks

The proof: if $a \neq 0$ and $f(a) = 0$, then $1 = f(1) = f(a) f(a^{-1}) = 0$ in $S$. This contradicts $S \neq 0$.
The hypothesis $S \neq 0$ is necessary, because the map from $K$ to the [[zero ring|008TBCYDPX]] is a ring homomorphism.
A consequence: a homomorphism of fields makes the first field a copy of a subfield of the second.

## Explanations

### L1 human

A field has no ideal other than $0$ and the whole field.
The kernel of $f$ is an ideal, and it does not contain $1$.
So the kernel is $0$, and $f$ is injective.
</document>

<document source="https://tamnd.github.io/math2/node/09NDE9V7X1.md" tag="09NDE9V7X1" kind="statement">
# Finite intersection of open sets is open

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $n \ge 1$ be an integer, and let $U_1, \ldots, U_n$ be [[open subsets|01ZQQDCERP]] of $X$.
Then the [[intersection|0HBRW4E6G1]] $U_1 \cap \cdots \cap U_n$ is open in $X$.

## Remarks

Proof: let $x$ be in the intersection. For each $k$ there is $\varepsilon_k > 0$ with $B(x, \varepsilon_k) \subseteq U_k$. Then $\varepsilon = \min(\varepsilon_1, \ldots, \varepsilon_n) > 0$ and $B(x, \varepsilon)$ lies in each $U_k$.
The statement is false for infinite families: in $\mathbb{R}$, the intersection of the open intervals $(-1/n, 1/n)$ for $n \ge 1$ is $\{0\}$, which is not open.
In a general [[topological space|0CNQXVBHCX]], this property is one of the axioms.

## Explanations

### L1 human

Each open set gives its own small ball about the point, and the smallest of these finitely many balls fits in all the sets.
With infinitely many sets, the radii can shrink to zero, and then no ball is left.
</document>

<document source="https://tamnd.github.io/math2/node/09NWETVTNY.md" tag="09NWETVTNY" kind="concept">
# Injective function

## Summary

A function is injective when different inputs always give different values.
</document>

<document source="https://tamnd.github.io/math2/node/09PKJKA6J0.md" tag="09PKJKA6J0" kind="statement">
# De Morgan's laws

## Statement

For all [[propositions|0ZGC7ZESYN]] $p$ and $q$, these two [[logical equivalences|06GYGS41JK]] hold.
$$\neg (p \lor q) \equiv \neg p \land \neg q$$
$$\neg (p \land q) \equiv \neg p \lor \neg q$$

## Remarks

The first law is also valid in intuitionistic logic. In the second law, only the direction from $\neg p \lor \neg q$ to $\neg (p \land q)$ is.
The laws for sets have the same form. See [[De Morgan's laws for sets|0VK0W4D5Y5]].
The [[negation of a universal statement|0P3A06QNDP]] is the analogue for quantifiers.

## Explanations

### L1 human

To negate "and", change it to "or" and negate each part.
"$n$ is not both even and prime" means "$n$ is not even, or $n$ is not prime".
A [[truth table|05RSK4Q657]] with four rows proves each law.
</document>

<document source="https://tamnd.github.io/math2/node/09PR1V6RWR.md" tag="09PR1V6RWR" kind="statement">
# Mean of the normal distribution

## Statement

Let $\mu \in \mathbb{R}$, let $\sigma^2 > 0$ and let $X$ have the [[normal distribution|07RXBVTVZB]] $\mathcal{N}(\mu, \sigma^2)$. Then $X$ is integrable and
$$E[X] = \int_{-\infty}^{\infty} x \, \frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x - \mu)^2/(2\sigma^2)} \, dx = \mu.$$

## Remarks

Proof: substitute $x = \mu + \sigma z$ with $\sigma = \sqrt{\sigma^2}$. The integral becomes $\mu \int \varphi(z) \, dz + \sigma \int z \varphi(z) \, dz$, where $\varphi$ is the [[standard normal density|027JF1FBB0]].
The first integral is $1$. The second is $0$, because $z \varphi(z)$ is an odd integrable function.
So the parameter $\mu$ is the mean. It is also the median and the mode, because the density is symmetric about $\mu$ and has its maximum there.

## Explanations

### L1 human

The bell curve of $\mathcal{N}(\mu, \sigma^2)$ is symmetric about $\mu$, so its average is $\mu$.
</document>

<document source="https://tamnd.github.io/math2/node/09PS3S0EF6.md" tag="09PS3S0EF6" kind="statement">
# Liouville's formula

## Statement

Let $I$ be an open interval, let $A : I \to \mathbb{R}^{n \times n}$ be continuous, and let $\Phi : I \to \mathbb{R}^{n \times n}$ be a differentiable function with $\Phi'(t) = A(t)\, \Phi(t)$ for each $t \in I$.
Then for each $t_0, t \in I$,
$$\det \Phi(t) = \det \Phi(t_0)\, \exp\left( \int_{t_0}^{t} \operatorname{tr} A(s)\, ds \right),$$
where $\operatorname{tr} A(s)$ is the trace of $A(s)$.

## Remarks

So $\det \Phi(t)$ is zero at each point of $I$ or at no point of $I$. This shows that a [[fundamental matrix|0MQ24RBG93]] is invertible everywhere if it is invertible at one point.
For a constant matrix $A$, the formula gives $\det e^{tA} = e^{t \operatorname{tr} A}$.
For the system that comes from $y'' + p y' + q y = 0$, the trace is $-p$, and the formula is [[Abel's identity|0VGWKX2JHB]].

## Explanations

### L1 human

The determinant of $\Phi$ is the volume spanned by $n$ solutions.
The formula says how this volume changes in time: its rate of growth is the trace of $A$.
</document>

<document source="https://tamnd.github.io/math2/node/09PV9JSY6E.md" tag="09PV9JSY6E" kind="definition">
# Prüfer sequence

## Definition

Let $n \ge 2$ be an integer and let $T$ be a [[tree|0MJ0JS72DZ]] with vertex set $\{1, \dots, n\}$.
The Prüfer sequence of $T$ is the sequence $(a_1, \dots, a_{n-2})$ that this process gives.
Let $T_0 = T$. For $i = 1, \dots, n - 2$, let $\ell_i$ be the leaf of $T_{i-1}$ with the smallest label.
Let $a_i$ be the unique neighbour of $\ell_i$ in $T_{i-1}$, and let $T_i = T_{i-1} - \ell_i$.

## Remarks

The map $T \mapsto (a_1, \dots, a_{n-2})$ is a bijection from the labelled trees on $\{1, \dots, n\}$ to $\{1, \dots, n\}^{n-2}$. This proves [[Cayley's formula|0M2SSGSYPE]].
A vertex $v$ appears $\deg(v) - 1$ times in the sequence. Thus the leaves are the labels that do not appear.
Example: the star with centre $1$ on $\{1, 2, 3, 4\}$ has the sequence $(1, 1)$.

## Explanations

### L1 human

Remove the smallest leaf again and again, and each time write down the vertex it was attached to.
Stop when two vertices are left. The list of $n - 2$ numbers fixes the tree.
</document>

<document source="https://tamnd.github.io/math2/node/09QSZVVM26.md" tag="09QSZVVM26" kind="concept">
# Bounded set

## Summary

A bounded set in a metric space is a set whose points have distances from each other below one fixed number.
Equivalently, the set lies in some ball.
</document>

<document source="https://tamnd.github.io/math2/node/09R8GC85RB.md" tag="09R8GC85RB" kind="concept">
# Removable singularity

## Summary

A removable singularity of a holomorphic function is an isolated singularity where one can define or redefine the function so that it becomes holomorphic there. See the [[definition|0P18JC72RE]].
</document>

<document source="https://tamnd.github.io/math2/node/09RDZZGJZ9.md" tag="09RDZZGJZ9" kind="statement">
# Cancellation in congruences

## Statement

Let $n$ be a positive integer, and let $a$, $b$ and $c$ be integers. Let $d = \gcd(c, n)$.
If $c a \equiv c b \pmod n$, then $a \equiv b \pmod{n / d}$.

## Remarks

In particular, if $c$ and $n$ are [[coprime|0GFKP8F8AG]], then $c a \equiv c b \pmod n$ implies $a \equiv b \pmod n$.
The modulus can become smaller. For example, $2 \cdot 1 \equiv 2 \cdot 4 \pmod 6$, but $1 \not\equiv 4 \pmod 6$. Here $d = 2$, and indeed $1 \equiv 4 \pmod 3$.
The proof: $n \mid c (a - b)$ gives $\frac{n}{d} \mid \frac{c}{d} (a - b)$. Since $n / d$ and $c / d$ are coprime, the [[generalized Euclid's lemma|03BAREW1Z5]] gives $\frac{n}{d} \mid a - b$.

## Explanations

### L1 human

You can divide both sides of a congruence by a number that has no common factor with the modulus.
If the number shares a factor with the modulus, you must also divide the modulus by that common factor.
</document>

<document source="https://tamnd.github.io/math2/node/09RFKMBQ63.md" tag="09RFKMBQ63" kind="statement">
# The rationals form an ordered field

## Statement

Define an order on the [[rational numbers|0NCJ5WXN6Z]] as follows. For integers $a, c$ and positive integers $b, d$, let $\frac{a}{b} \le \frac{c}{d}$ when $a d \le b c$ in $\mathbb{Z}$.
This relation is well defined, and with it $\mathbb{Q}$ is an [[ordered field|010JCV9J4E]].

## Remarks

The order on $\mathbb{Q}$ extends the [[order on the integers|0F6FM9Y4Y6]].
This is the only order on $\mathbb{Q}$ that makes it an ordered field.
The ordered field $\mathbb{Q}$ is Archimedean, but it is not complete: the set $\{q \in \mathbb{Q} : q^2 < 2\}$ has no least upper bound in $\mathbb{Q}$.

## Explanations

### L1 human

To compare two fractions, write them with positive denominators and cross-multiply.
For example, $2/3 < 3/4$ because $2 \cdot 4 = 8 < 9 = 3 \cdot 3$.
</document>

<document source="https://tamnd.github.io/math2/node/09RJ859N7A.md" tag="09RJ859N7A" kind="concept">
# Partial sum

## Summary

A partial sum of a series is the sum of its first finitely many terms. The sequence of partial sums decides whether the series converges. The node [[partial sum|0NB643K60Q]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/09RTQRK5PC.md" tag="09RTQRK5PC" kind="concept">
# Student's t-distribution

## Summary

Student's t-distribution is a continuous law on $\mathbb{R}$ that is symmetric about $0$ and bell-shaped, like the standard normal law, but with heavier tails. It is the law of the standardized sample mean when one estimates the standard deviation from the data.
The node [[Student's t-distribution|0NQ5798SB1]] gives the precise definition.

## Remarks

William Sealy Gosset published the distribution in 1908 under the pen name "Student".
</document>

<document source="https://tamnd.github.io/math2/node/09RTYYHXKA.md" tag="09RTYYHXKA" kind="counterexample">
# Smooth non-analytic function

## Counterexample

Let $f(x) = e^{-1/x}$ for $x > 0$ and $f(x) = 0$ for $x \le 0$.
Then $f$ is [[smooth|0BE3QDH73A]] on $\mathbb{R}$, and $f^{(n)}(0) = 0$ for all $n$.
So its [[Taylor series|0225QPWRRE]] at $0$ is the zero series. But $f(x) > 0$ for $x > 0$, so $f$ is not [[analytic|0JD5ZNW0F4]] at $0$.

## Remarks

For $x > 0$, each derivative has the form $p_n(1/x) e^{-1/x}$ for a polynomial $p_n$. It tends to $0$ as $x \to 0^+$, because $e^{-1/x}$ decays faster than any power of $x$.
Such functions give smooth bump functions and smooth partitions of unity.
Mathlib defines this function as `expNegInvGlue`.

## Explanations

### L1 human

This function is so flat at $0$ that all its derivatives vanish there.
So its Taylor series sees only the zero function, but the function is not zero.
</document>

<document source="https://tamnd.github.io/math2/node/09S3PFBEPM.md" tag="09S3PFBEPM" kind="definition">
# Integral domain

## Definition

An integral domain is a [[commutative ring|0GYGHKX834]] $R$ with these two properties.

1. $1 \neq 0$ in $R$.
2. For all $a, b \in R$, if $a b = 0$, then $a = 0$ or $b = 0$.

Equivalently, $R$ is a nonzero commutative ring in which $0$ is the only [[zero divisor|0K26YKNBSA]].

## Remarks

The integers $\mathbb{Z}$ and every [[field|0QA8CHH8MG]] are integral domains.
The ring $\mathbb{Z}/6\mathbb{Z}$ is not an integral domain, because $2 \cdot 3 = 0$.
In an integral domain, [[cancellation|0GJM24EXP9]] holds for nonzero factors.
Some authors, and Mathlib, also use the word domain for a noncommutative ring with property 2.

## Explanations

### L1 human

An integral domain is a ring that behaves like the integers for products.
If a product is zero, then one of the factors is zero.
This is what you use when you solve $(x - 2)(x - 3) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/09S5SN35DJ.md" tag="09S5SN35DJ" kind="definition">
# Least common multiple

## Definition

Let $a$ and $b$ be nonzero integers.
A common multiple of $a$ and $b$ is an integer $m$ such that $a \mid m$ and $b \mid m$.
The least common multiple $\operatorname{lcm}(a, b)$ is the smallest positive common multiple of $a$ and $b$.
By convention, $\operatorname{lcm}(a, 0) = \operatorname{lcm}(0, a) = 0$ for each integer $a$.

## Remarks

The positive integer $|a b|$ is a common multiple, so $\operatorname{lcm}(a, b)$ exists and $\operatorname{lcm}(a, b) \le |a b|$.
Each common multiple of $a$ and $b$ is a multiple of $\operatorname{lcm}(a, b)$.
For example, $\operatorname{lcm}(4, 6) = 12$.
The [[product of the gcd and the lcm|0TDY6JZHKS]] is the product of the two numbers.

## Explanations

### L1 human

List the positive multiples of each number, and take the smallest number that is in both lists.
The multiples of $4$ are $4, 8, 12, 16, \dots$ and the multiples of $6$ are $6, 12, 18, \dots$.
Thus $\operatorname{lcm}(4, 6) = 12$. You use this number when you add $\tfrac{1}{4} + \tfrac{1}{6}$ with a common denominator.
</document>

<document source="https://tamnd.github.io/math2/node/09S8DV34Y0.md" tag="09S8DV34Y0" kind="concept">
# Subbasis

## Summary

A subbasis of a topology is a family of open sets that generates the topology. The topology is the smallest one in which all members of the family are open.
The finite intersections of members of a subbasis form a basis.
</document>

<document source="https://tamnd.github.io/math2/node/09T8ZHST1V.md" tag="09T8ZHST1V" kind="definition">
# Independent sigma-algebras

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space, let $I$ be an index set, and for each $i \in I$ let $\mathcal{F}_i \subseteq \mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] on $\Omega$.
The family $(\mathcal{F}_i)_{i \in I}$ is independent if for each finite nonempty subset $J \subseteq I$ and each choice of events $A_j \in \mathcal{F}_j$ for $j \in J$,
$$P\left(\bigcap_{j \in J} A_j\right) = \prod_{j \in J} P(A_j).$$

## Remarks

Events $(A_i)_{i \in I}$ are [[mutually independent|0GCVEV6BJA]] if and only if the σ-algebras $\{\varnothing, A_i, A_i^c, \Omega\}$ are independent.
Random variables are independent when the σ-algebras that they generate are independent.
To check independence, it is enough to check the product formula on a generating family of each $\mathcal{F}_i$ that is closed under finite intersections. This follows from Dynkin's π-λ theorem.

## Explanations

### L1 human

A σ-algebra holds all the information of one part of an experiment, for example everything about the first roll of a die.
Two such σ-algebras are independent when information about one part never changes the chances about the other part.
</document>

<document source="https://tamnd.github.io/math2/node/09V3NASXND.md" tag="09V3NASXND" kind="concept">
# Equivalent metrics

## Summary

Two metrics on the same set are equivalent when they give the same open sets.
A stronger form asks that each metric is at most a constant times the other.
</document>

<document source="https://tamnd.github.io/math2/node/09V6DDACXR.md" tag="09V6DDACXR" kind="concept">
# Scalar multiplication of matrices

## Summary

Scalar multiplication of a matrix multiplies each entry of the matrix by the same scalar.
</document>

<document source="https://tamnd.github.io/math2/node/09VDS2W1E8.md" tag="09VDS2W1E8" kind="definition">
# Module over a ring

## Definition

Let $R$ be a ring with identity element $1$.
A *left module* over $R$ is an abelian group $(M, +)$ together with a map $R \times M \to M$, $(r, x) \mapsto rx$, such that, for all $r, s \in R$ and $x, y \in M$:

1. $r(x + y) = rx + ry$.
2. $(r + s)x = rx + sx$.
3. $(rs)x = r(sx)$.
4. $1x = x$.

## Remarks

If $R$ is a field, then an $R$-module is the same as a [[vector space|000AYPA9AJ]] over $R$.
Each abelian group is a module over the ring $\Z$, with $nx = x + \cdots + x$ ($n$ terms) for $n > 0$.
The ring $R$ is a module over itself, and $R^n$ is a module with the operations on coordinates.
Right modules use a map $M \times R \to M$; over a commutative ring, the two notions are the same.

## Explanations

### L1 human

A module is a vector space in which the scalars come from a ring instead of a field.
We cannot divide by scalars in general, so many facts about vector spaces fail.
For example, a module need not have a basis.
</document>

<document source="https://tamnd.github.io/math2/node/09VMDSRXWJ.md" tag="09VMDSRXWJ" kind="concept">
# Singular value

## Summary

The singular values of a matrix $A$ are the square roots of the eigenvalues of $A^* A$.
</document>

<document source="https://tamnd.github.io/math2/node/09VYHEF2VT.md" tag="09VYHEF2VT" kind="definition">
# Orthonormal family

## Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $I$ be a set.
A family $(e_i)_{i \in I}$ of vectors of $V$ is *orthonormal* when, for all $i, j \in I$,
$$\langle e_i, e_j \rangle = \delta_{ij},$$
where $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ if $i \ne j$.

## Remarks

So each $e_i$ has norm $1$, and $e_i$ and $e_j$ are [[orthogonal|0Y2X7P2WB7]] for $i \ne j$.
The standard basis of $\R^n$ is orthonormal for the dot product.
The functions $t \mapsto e^{2\pi i k t}$, $k \in \Z$, are orthonormal for $\langle f, g \rangle = \int_0^1 f \bar g\, dt$.

## Explanations

### L1 human

An orthonormal family is like a set of coordinate axes: unit vectors at right angles.
Computations with such vectors are simple, because all inner products are $0$ or $1$.
They are the best bases to work with in an inner product space.
</document>

<document source="https://tamnd.github.io/math2/node/09VZBHB33V.md" tag="09VZBHB33V" kind="statement">
# Every walk contains a path

## Statement

Let $G$ be a [[graph|0JD5C82QMB]] and let $u, v$ be vertices of $G$.
Suppose that there is a [[walk|0VJZM0F9YM]] $W$ from $u$ to $v$.
Then there is a [[path|0JDE56HWQ8]] from $u$ to $v$ whose vertices and edges are vertices and edges of $W$.

## Remarks

Proof idea: if a walk visits a vertex $x$ twice, delete the closed part between the two visits. Each deletion makes the walk shorter, so the process stops with a path.
A consequence: a shortest walk from $u$ to $v$ is a path. Thus the [[distance|0WPZ5YPAGR]] can be defined with paths.

## Explanations

### L1 human

If you can walk from $u$ to $v$, you can also do it without visiting any place twice.
Whenever you come back to a place, cut out the loop that you just made.
</document>

<document source="https://tamnd.github.io/math2/node/09WRNGPTMV.md" tag="09WRNGPTMV" kind="concept">
# Local minimum

## Summary

A function has a local minimum at a point when its value there is at most its values at all nearby points. Local minima are the bottoms of the valleys of the graph. The node [[local minimum|0NS8BP96QX]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/09XNAPTJQ7.md" tag="09XNAPTJQ7" kind="concept">
# Affine space

## Summary

An affine space is a set of points on which the vectors of a vector space act by translation. It keeps the ideas of lines, planes and parallelism from Euclidean geometry, but it has no origin, no distance and no angle.
</document>

<document source="https://tamnd.github.io/math2/node/09YB3WQG38.md" tag="09YB3WQG38" kind="statement">
# Permuting rows multiplies the determinant by the sign

## Statement

Let $K$ be a commutative ring, let $A = (a_{ij})$ be an $n \times n$ matrix over $K$, and let $\sigma$ be a permutation of $\{1, \dots, n\}$.
Let $B = (b_{ij})$ be the matrix with $b_{ij} = a_{\sigma(i), j}$.
Then $\det B = \sgn(\sigma) \det A$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

In particular, exchanging two rows changes the sign of the determinant.
For a [[permutation matrix|08SP3DF8NB]] $P$, this gives $\det P = \pm 1$.
The same holds for a permutation of the columns.

## Explanations

### L1 human

When we exchange two rows, the determinant changes its sign.
A general reordering of the rows is a sequence of such exchanges.
So it multiplies the determinant by $1$ or by $-1$, according to the sign of the permutation.
</document>

<document source="https://tamnd.github.io/math2/node/09YBMJTNBM.md" tag="09YBMJTNBM" kind="statement">
# Divisibility by 11

## Statement

Let $n$ be a natural number with decimal digits $d_k d_{k-1} \dots d_0$, so that $n = \sum_{i=0}^{k} d_i 10^i$.
Then $11 \mid n$ if and only if $11 \mid d_0 - d_1 + d_2 - \dots + (-1)^k d_k$.

## Remarks

More precisely, $n \equiv \sum_{i=0}^{k} (-1)^i d_i \pmod{11}$. This holds because $10 \equiv -1 \pmod{11}$, so $10^i \equiv (-1)^i \pmod{11}$.
For example, $918{,}082$ gives $2 - 8 + 0 - 8 + 1 - 9 = -22$, and $11 \mid -22$. So $11 \mid 918{,}082$.

## Explanations

### L1 human

Start at the last digit, and add and subtract the digits in turn.
The number is a multiple of $11$ exactly when the result is a multiple of $11$, including $0$.
For example, $121$ gives $1 - 2 + 1 = 0$, and indeed $121 = 11 \cdot 11$.
</document>

<document source="https://tamnd.github.io/math2/node/09YMV8H4X5.md" tag="09YMV8H4X5" kind="statement">
# Intersection of closed sets is closed

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $(F_i)_{i \in I}$ be a family of [[closed subsets|0G5AV9Q4RR]] of $X$, indexed by a nonempty set $I$.
Then the [[intersection|0839XNQXDK]] $\bigcap_{i \in I} F_i$ is closed in $X$.

## Remarks

Proof: by [[De Morgan's laws for sets|0VK0W4D5Y5]], $X \setminus \bigcap_i F_i = \bigcup_i (X \setminus F_i)$. This is a [[union of open sets|0HEF46RPEK]], so it is open.
The same proof works in each [[topological space|0CNQXVBHCX]].
A union of finitely many closed sets is closed, but an infinite union need not be. For example, the union of the sets $[1/n, 1]$ for $n \ge 1$ is $(0, 1]$.

## Explanations

### L1 human

Closed sets behave like open sets with the roles of union and intersection exchanged.
Any intersection of closed sets is closed, and finite unions of closed sets are closed.
</document>

<document source="https://tamnd.github.io/math2/node/09ZJ6CAK7S.md" tag="09ZJ6CAK7S" kind="statement">
# The number of Sylow subgroups is the index of the normaliser

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a prime number, and let $P$ be a [[Sylow p-subgroup|0053CQE9Y2]] of $G$.
Then $n_p = [G : N_G(P)]$, where $N_G(P)$ is the [[normaliser|0YGN3VAA0D]] of $P$.

## Remarks

By the [[second Sylow theorem|09GCQ5STVJ]], $G$ acts transitively by conjugation on $\operatorname{Syl}_p(G)$.
The stabiliser of $P$ is $N_G(P)$, so the [[orbit-stabiliser theorem|0E2HSCKZ6C]] gives the formula.
In particular, $P$ is normal in $G$ if and only if $n_p = 1$.

## Explanations

### L1 human

All Sylow $p$-subgroups are conjugates of one of them, $P$.
The number of conjugates of $P$ is the number of cosets of the elements that do not move $P$.
These elements form the normaliser.
</document>

<document source="https://tamnd.github.io/math2/node/09ZKE5XQ74.md" tag="09ZKE5XQ74" kind="statement">
# Product of compact sets is compact

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], and let $K \subseteq X$ and $L \subseteq Y$ be [[compact|00R0X1WGPX]].
Then $K \times L$ is compact in $X \times Y$ with the [[product topology|0Z57EC6YX3]].

## Remarks

The proof uses the tube lemma. For each $x \in K$, finitely many basic open sets $U \times V$ cover $\{x\} \times L$, and their intersection over the first factor gives an open set about $x$.
By induction, a product of finitely many compact sets is compact. For example, $[0, 1]^n$ is compact.
For arbitrary products of compact spaces, the statement is [[Tychonoff's theorem|0JTRSJKCFB]].

## Explanations

### L1 human

A rectangle made of two compact sides is compact.
For example, the square $[0, 1] \times [0, 1]$ is compact, because $[0, 1]$ is compact.
</document>

<document source="https://tamnd.github.io/math2/node/09ZWYRQ41J.md" tag="09ZWYRQ41J" kind="concept">
# Linearisation at an equilibrium

## Summary

Let $U \subseteq \mathbb{R}^n$ be open, let $f : U \to \mathbb{R}^n$ be continuously differentiable, and let $x^*$ be an [[equilibrium point|0G4X36AE0X]] of $x' = f(x)$.
Let $A = Df(x^*)$ be the Jacobian matrix of $f$ at $x^*$. The linearisation of the system at $x^*$ is the linear system $y' = A y$, where $y$ stands for $x - x^*$.
It comes from the first-order Taylor expansion $f(x) = A (x - x^*) + o(\|x - x^*\|)$.

## Remarks

If each eigenvalue of $A$ has negative real part, then $x^*$ is asymptotically stable for the nonlinear system. If one eigenvalue has positive real part, then $x^*$ is unstable.
If some eigenvalue has zero real part and none has positive real part, the linearisation does not decide stability. For $x' = -x^3$ and $x' = x^3$, the linearisation at $0$ is $y' = 0$ in both cases, but the first is stable and the second is not.
Near a hyperbolic equilibrium, the [[Hartman–Grobman theorem|0S8WX8A4V9]] says that the linearisation describes the phase portrait up to a homeomorphism.

## Explanations

### L1 human

Near an equilibrium, a smooth vector field looks like a linear one.
We replace the field by its best linear approximation and study the simpler linear system.
This works well unless the linear system is on the border between stable and unstable.
</document>

<document source="https://tamnd.github.io/math2/node/09ZXGX55DM.md" tag="09ZXGX55DM" kind="concept">
# Higher partial derivatives

## Summary

A higher partial derivative of a function of several variables is a [[partial derivative|09EGC6F8PT]] of a partial derivative, taken one or more times.
A second partial derivative such as $\partial^2 f / \partial y \, \partial x$, in which the two variables differ, is called a mixed partial derivative.
</document>

<document source="https://tamnd.github.io/math2/node/0A1QQN333A.md" tag="0A1QQN333A" kind="statement">
# Determinant of a matrix exponential

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $A$ be an $n \times n$ matrix over $\mathbb{K}$.
Then
$$\det(e^A) = e^{\tr A},$$
where $e^A$ is the [[matrix exponential|020M04FN1H]], $\det$ is the [[determinant|00KDZRW4C0]] and $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

If $A$ is triangular over $\C$ with diagonal entries $\lambda_i$, then $e^A$ is triangular with diagonal entries $e^{\lambda_i}$, and both sides equal $e^{\lambda_1 + \cdots + \lambda_n}$.
The general case follows by [[triangularisation|0363JMQ42Y]] over $\C$.
It shows again that $e^A$ is invertible, because $e^{\tr A} \ne 0$.
If $\tr A = 0$, then $e^A$ lies in the special linear group.

## Explanations

### L1 human

The determinant of the exponential of a matrix is the exponential of its trace.
This links the product of the eigenvalues of $e^A$ with the sum of the eigenvalues of $A$.
It is the matrix form of the rule $e^{a} e^{b} = e^{a + b}$.
</document>

<document source="https://tamnd.github.io/math2/node/0A1YXSKB48.md" tag="0A1YXSKB48" kind="concept">
# Isolated point

## Summary

An isolated point of a set $S$ is a point of $S$ that has a neighbourhood with no other point of $S$.
</document>

<document source="https://tamnd.github.io/math2/node/0A26X79RTB.md" tag="0A26X79RTB" kind="statement">
# Open sets of a metric space

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $U \subseteq X$.
Then $U$ is [[open|01ZQQDCERP]] in the [[metric topology|0A6YRNZB2R]] if and only if for each $x \in U$ there is a real $\varepsilon > 0$ with $B(x, \varepsilon) \subseteq U$.
Here $B(x, \varepsilon)$ is the [[open ball|0DC3MPR7C9]] with centre $x$ and radius $\varepsilon$.

## Remarks

The number $\varepsilon$ can depend on $x$.
In $\mathbb{R}$, the set $(0, \infty)$ is open: for $x > 0$, take $\varepsilon = x$.
The set $\mathbb{Q}$ is not open in $\mathbb{R}$, because each ball about a rational number contains an irrational number.

## Explanations

### L1 human

This is the test for open sets that you use in practice.
You pick any point of the set, and you find a small ball about it inside the set.
</document>

<document source="https://tamnd.github.io/math2/node/0A2HNZC44K.md" tag="0A2HNZC44K" kind="concept">
# Convergence in mean

## Summary

A sequence of random variables converges in $r$-th mean when the expected value of $|X_n - X|^r$ tends to $0$.
The node [[convergence in mean|0TTA57SEJG]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0A3CK6DVVB.md" tag="0A3CK6DVVB" kind="definition">
# Uniform continuity on a set

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function, and let $S \subseteq D$.
The function $f$ is uniformly continuous on $S$ if the following holds.
For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - f(y)| < \varepsilon$ for all $x, y \in S$ with $|x - y| < \delta$.

## Remarks

So $f$ is uniformly continuous on $S$ when its restriction to $S$ is [[uniformly continuous|0Q4DR37DVE]].
A function can be uniformly continuous on a set but not on a larger set. For example, $x^2$ is uniformly continuous on $[0, 1]$ but not on $\mathbb{R}$.
Mathlib defines it as `UniformContinuousOn`.

## Explanations

### L1 human

Uniform continuity on $S$ only looks at pairs of points inside $S$.
Points outside $S$ play no role.
</document>

<document source="https://tamnd.github.io/math2/node/0A3GWZ86AY.md" tag="0A3GWZ86AY" kind="statement">
# The reals are Archimedean

## Statement

The ordered field $\mathbb{R}$ of [[real numbers|08J2M1JYAG]] is an [[Archimedean ordered field|0HQARMKEV5]].
That is, for all real numbers $x$ and $y$ with $y > 0$, there is a natural number $n$ with $n y > x$.

## Remarks

The proof uses the [[completeness axiom|0AXWET80YW]]. If $\mathbb{N}$ were bounded above, let $b = \sup \mathbb{N}$. Then $b - 1$ is not an upper bound, so $n > b - 1$ for some $n$, and $n + 1 > b$, a contradiction.
Thus each [[complete ordered field|0A9E6X0S3E]] is Archimedean.

## Explanations

### L1 human

However small your step and however far your goal on the real line, finitely many steps get you past the goal.
There are no infinitely large real numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0A3HJX3KTF.md" tag="0A3HJX3KTF" kind="concept">
# Pseudometric space

## Summary

A pseudometric space is a set with a distance function that is almost a metric. The difference: two different points can have distance zero.
</document>

<document source="https://tamnd.github.io/math2/node/0A58THH5XV.md" tag="0A58THH5XV" kind="definition">
# Boolean function

## Definition

Let $n$ be a natural number.
A Boolean function of $n$ variables is a function $f : \{0, 1\}^n \to \{0, 1\}$.

## Remarks

There are $2^n$ inputs, so there are $2^{2^n}$ Boolean functions of $n$ variables.
A truth table lists the values of a Boolean function on all inputs.
Every Boolean function is given by a formula in [[disjunctive normal form|06S4WGSEQA]], and also by one in [[conjunctive normal form|07C4A8Y18K]].

## Explanations

### L1 human

A Boolean function takes some bits as input and returns one bit.
The logic gates and, or and not are Boolean functions, and a digital circuit computes a Boolean function.
</document>

<document source="https://tamnd.github.io/math2/node/0A61B60B5R.md" tag="0A61B60B5R" kind="definition">
# Differentiable function

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be a function.
The function $f$ is differentiable at $a \in I$ if $f$ has a [[derivative|0F1WR9C83K]] at $a$.
It is differentiable on $I$ if it is differentiable at each point of $I$.

## Remarks

[[Differentiable functions are continuous|07ZMX4ANJS]].
The converse fails: the [[absolute value|0CQ9ZM383N]] is continuous but not differentiable at $0$.
Polynomials, the exponential function, and the sine are differentiable on $\mathbb{R}$.
Mathlib defines differentiability at a point as `DifferentiableAt`.

## Explanations

### L1 human

A differentiable function has a well defined slope at each point.
Its graph is smooth, with no corners and no vertical tangents.
</document>

<document source="https://tamnd.github.io/math2/node/0A6PGTTP2A.md" tag="0A6PGTTP2A" kind="definition">
# Linear equation with constant coefficients

## Definition

Let $n \ge 1$, let $a_0, \dots, a_{n-1}$ be real numbers, let $I$ be an open interval and let $b : I \to \mathbb{R}$ be continuous.
A linear equation with constant coefficients is a [[linear differential equation|0PB7K8M7FG]] of the form
$$y^{(n)} + a_{n-1}\, y^{(n-1)} + \dots + a_1\, y' + a_0\, y = b(x).$$
Its homogeneous equation is the case $b = 0$, on the interval $I = \mathbb{R}$.

## Remarks

The function $e^{r x}$ solves the homogeneous equation if and only if $r$ is a root of the [[characteristic equation|0QDXMG46QW]] $r^n + a_{n-1} r^{n-1} + \dots + a_0 = 0$.
The roots of this polynomial give a [[fundamental system|0H4KNNCHPY]]. See the cases of [[distinct real roots|0HJ0HDA6W7]], [[repeated roots|0HRVY67CNV]] and [[complex conjugate roots|0BEZX8F2XW]].
The [[harmonic oscillator|0YG0ZJPTWG]] $x'' + \omega^2 x = 0$ is an example of order $2$.

## Explanations

### L1 human

When the coefficients are constants, the derivative of $e^{r x}$ is $r e^{r x}$, again a multiple of the same function.
So the equation becomes a polynomial equation in $r$.
This changes calculus into algebra.
</document>

<document source="https://tamnd.github.io/math2/node/0A6PMR20N4.md" tag="0A6PMR20N4" kind="statement">
# Derivative gives the directional derivatives

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$.
Suppose that $f$ is [[differentiable|0135XZZQ8M]] at $a$ with derivative $Df(a)$.
Then for each $v \in \mathbb{R}^n$, $f$ [[has the directional derivative|05EGFBZTGC]] $Df(a)(v)$ at $a$ in the direction $v$:
$$\lim_{t \to 0} \frac{f(a + t v) - f(a)}{t} = Df(a)(v).$$

## Remarks

For $v = e_i$ this gives $\dfrac{\partial f}{\partial x_i}(a) = Df(a)(e_i)$. So the [[partial derivatives|09EGC6F8PT]] of the components of $f$ are the entries of the matrix of $Df(a)$.
For a real-valued $f$, it gives $D_v f(a) = \nabla f(a) \cdot v$, where $\nabla f(a)$ is the [[gradient|0ABK9V9Y64]].
In particular, $D_v f(a)$ is linear in $v$. A function whose directional derivatives are not linear in $v$ is not differentiable.

## Explanations

### L1 human

Once you know the linear map $Df(a)$, you know the slope of $f$ in every direction: feed the direction into the map.
For $f(x, y) = x^2 + y^2$ at $(1, 1)$, $Df(1, 1)(v) = 2 v_1 + 2 v_2$. So the slope in the direction $(1, -1)$ is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0A6QZ367GX.md" tag="0A6QZ367GX" kind="definition">
# System of linear differential equations

## Definition

Let $n \ge 1$, let $I$ be an open interval, let $A : I \to \mathbb{R}^{n \times n}$ be a continuous matrix-valued function and let $b : I \to \mathbb{R}^n$ be continuous.
The system of linear differential equations with coefficient matrix $A$ and right side $b$ is the equation
$$x'(t) = A(t)\, x(t) + b(t)$$
for an unknown differentiable function $x : I \to \mathbb{R}^n$.
The system is homogeneous if $b = 0$ on $I$.

## Remarks

In coordinates, it is the system $x_i' = \sum_{j=1}^{n} a_{ij}(t)\, x_j + b_i(t)$ for $i = 1, \dots, n$.
For each $t_0 \in I$ and $x_0 \in \mathbb{R}^n$, there is exactly one solution on all of $I$ with $x(t_0) = x_0$.
The solutions of the homogeneous system form a vector space of dimension $n$. A basis gives a [[fundamental matrix|0MQ24RBG93]].
A [[linear equation of order n|0PB7K8M7FG]] becomes such a system by [[reduction to a first-order system|0TH9CPW3VS]].

## Explanations

### L1 human

Several unknown functions change at the same time, and each rate is a linear combination of all the unknowns.
In matrix form, the system looks like one linear first-order equation, with a matrix in place of a number.
</document>

<document source="https://tamnd.github.io/math2/node/0A6YRNZB2R.md" tag="0A6YRNZB2R" kind="definition">
# Metric topology

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]].
The metric topology of $(X, d)$ is the set $\tau_d$ of all $U \subseteq X$ such that for each $x \in U$ there is a real $\varepsilon > 0$ with $B(x, \varepsilon) \subseteq U$.
Here $B(x, \varepsilon)$ is the [[open ball|0DC3MPR7C9]] with centre $x$ and radius $\varepsilon$.

## Remarks

The set $\tau_d$ is a [[topology|0CNQXVBHCX]] on $X$: it contains $\varnothing$ and $X$, and it is closed under [[unions|0HEF46RPEK]] and [[finite intersections|09NDE9V7X1]].
Each open ball is in $\tau_d$, by [[open balls are open|0B2305GNAZ]]. The open balls form a [[basis|06EX0FHBTF]] of $\tau_d$.
For $\mathbb{R}$ with the usual metric, $\tau_d$ is the usual topology, generated by the open intervals.
Different metrics can give the same topology. Such metrics are [[equivalent|0TWQ5PRKWS]].

## Explanations

### L1 human

The metric topology is how a metric space becomes a topological space.
A set is open when each of its points has a small ball about it inside the set.
</document>

<document source="https://tamnd.github.io/math2/node/0A74SX55QP.md" tag="0A74SX55QP" kind="statement">
# Archimedean property

## Statement

Let $F$ be an [[Archimedean ordered field|0HQARMKEV5]] and let $x$ be an element of $F$.
Then there is a natural number $n$ with $x < n$.

## Remarks

Proof: apply the definition with $y = 1$.
Thus the natural numbers are not [[bounded above|0B5H2V4JW8]] in $F$.
The [[real numbers are Archimedean|0A3GWZ86AY]], thus this holds for each real number $x$.

## Explanations

### L1 human

No element of an Archimedean field is larger than all the natural numbers.
For example, for the real number $x = 10^{6} + 0.5$, the natural number $n = 10^{6} + 1$ is larger.
</document>

<document source="https://tamnd.github.io/math2/node/0A7KGAYHGV.md" tag="0A7KGAYHGV" kind="statement">
# Baire category theorem

## Statement

Each [[complete metric space|08EWT7J0SC]] is a [[Baire space|0D5KR33H7X]].

## Remarks

Explicitly: if $X$ is a complete metric space and $(U_n)_{n \in \mathbb{N}}$ is a sequence of open dense subsets of $X$, then $\bigcap_n U_n$ is dense in $X$.
The proof builds a nested sequence of closed balls with radii tending to $0$, and uses completeness to find a point in all of them.
A second form of the theorem says that each locally compact Hausdorff space is a Baire space.
A consequence: $\mathbb{R}$ is not a countable union of nowhere dense sets, and thus $\mathbb{R}$ is uncountable.

## Explanations

### L1 human

A complete metric space cannot be covered by countably many thin sets.
For example, you cannot write the real line as a countable union of sets like $\mathbb{Z}$ or the Cantor set.
</document>

<document source="https://tamnd.github.io/math2/node/0A7KJ68XHT.md" tag="0A7KJ68XHT" kind="statement">
# Tower law

## Statement

Let $K \subseteq L \subseteq M$ be fields.
Then
$$[M : K] = [M : L] \cdot [L : K].$$

## Remarks

The proof: let $u_1, \dots, u_d$ be a basis of $L$ over $K$, and let $w_1, \dots, w_e$ be a basis of $M$ over $L$.
Then the products $u_i w_j$ are a basis of $M$ over $K$.
The formula also holds for infinite degrees, as a product of cardinals.
So $M/K$ is finite if and only if $M/L$ and $L/K$ are finite.
If $[M : K]$ is a prime number, then $M/K$ has no [[intermediate field|02NBKVQFRQ]] other than $K$ and $M$.

## Explanations

### L1 human

Degrees multiply along a tower of fields.
For example, $[\mathbb{Q}(\sqrt{2}, \sqrt{3}) : \mathbb{Q}] = 2 \cdot 2 = 4$.
The law is like Lagrange's theorem for groups, and it is the main tool to compute degrees.
</document>

<document source="https://tamnd.github.io/math2/node/0A7KV2WB7A.md" tag="0A7KV2WB7A" kind="concept">
# Monotone function

## Summary

A monotone function between ordered sets is a function that keeps the order.
</document>

<document source="https://tamnd.github.io/math2/node/0A7YDKRNR6.md" tag="0A7YDKRNR6" kind="definition">
# Complex sine and cosine

## Definition

For $z \in \mathbb{C}$, the complex sine and the complex cosine of $z$ are
$$\sin z = \frac{e^{iz} - e^{-iz}}{2i}, \qquad \cos z = \frac{e^{iz} + e^{-iz}}{2},$$
where $e^w$ is the [[complex exponential|054VXHMCA3]] of $w$.

## Remarks

For real $z$, these are the usual sine and cosine, by [[Euler's formula|04R3CY41VJ]].
The functions $\sin$ and $\cos$ are [[entire|0419794R0T]], with $\sin' = \cos$ and $\cos' = -\sin$. The identity $\sin^2 z + \cos^2 z = 1$ holds for each $z \in \mathbb{C}$.
They are not bounded on $\mathbb{C}$. For real $y$, $\cos(iy) = \cosh y = \frac{e^y + e^{-y}}{2}$.
The [[power series of the cosine|0Z1G2F8DGY]] is the same as in the real case.

## Explanations

### L1 human

Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$ lets us solve for $\cos \theta$ and $\sin \theta$ in terms of exponentials.
These formulas make sense for every complex number, so we use them to define $\sin z$ and $\cos z$.
On the imaginary axis they grow very fast: $\cos(10 i)$ is more than $10000$.
</document>

<document source="https://tamnd.github.io/math2/node/0A8JNBGT2Z.md" tag="0A8JNBGT2Z" kind="concept">
# Bounded sequence

## Summary

A bounded sequence is a real sequence whose terms all lie in one fixed interval. Boundedness is a necessary condition for convergence, and it is the hypothesis of the Bolzano–Weierstrass theorem. The node [[bounded sequence|05FQ3VGAR9]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0A8JXHYTJC.md" tag="0A8JXHYTJC" kind="counterexample">
# x^2 is not uniformly continuous on the real line

## Counterexample

The function $f \colon \mathbb{R} \to \mathbb{R}$ with $f(x) = x^2$ is [[continuous|0P27W9RW0G]] but not [[uniformly continuous|0Q4DR37DVE]].
For $n \ge 1$, let $x_n = n$ and $y_n = n + 1/n$. Then $|x_n - y_n| = 1/n \to 0$, but
$$f(y_n) - f(x_n) = 2 + \frac{1}{n^2} > 2.$$
So no $\delta$ works for $\varepsilon = 2$.

## Remarks

On each bounded interval $[-c, c]$, the function $x^2$ is Lipschitz with constant $2c$. So it is uniformly continuous there.
The problem is that the slope $2x$ is not bounded on $\mathbb{R}$.

## Explanations

### L1 human

The graph of $x^2$ gets steeper and steeper as $x$ grows.
So a fixed small change in $x$ gives larger and larger changes in $x^2$.
</document>

<document source="https://tamnd.github.io/math2/node/0A8TJ0VHC7.md" tag="0A8TJ0VHC7" kind="statement">
# Chain rule for a composition with a curve

## Statement

Let $I \subseteq \mathbb{R}$ be an open interval, let $U \subseteq \mathbb{R}^n$ be open, let $\gamma \colon I \to U$, let $f \colon U \to \mathbb{R}^m$, and let $t_0 \in I$.
Suppose that $\gamma$ is differentiable at $t_0$ and $f$ is differentiable at $\gamma(t_0)$.
Then $f \circ \gamma$ is differentiable at $t_0$, and
$$(f \circ \gamma)'(t_0) = Df(\gamma(t_0))\big(\gamma'(t_0)\big).$$

## Remarks

For $m = 1$, this reads $(f \circ \gamma)'(t_0) = \nabla f(\gamma(t_0)) \cdot \gamma'(t_0) = \sum_{i=1}^n \dfrac{\partial f}{\partial x_i}(\gamma(t_0)) \, \gamma_i'(t_0)$, with the [[gradient|0ABK9V9Y64]] $\nabla f$.
It is the case $n = 1$ of the [[chain rule in several variables|0EAAS2EHDR]], since a linear map $\mathbb{R} \to \mathbb{R}^n$ is $s \mapsto s \gamma'(t_0)$.
For $\gamma(t) = a + t v$ it gives the [[directional derivative|06KWGHDN2J]] $D_v f(a) = Df(a)(v)$.

## Explanations

### L1 human

Walk along a [[curve|0G7WM8DX4P]] $\gamma(t)$ through a landscape with height $f$.
The rate at which your height changes is the slope of the landscape in the direction you walk, times your speed.
For $f(x, y) = x y$ and $\gamma(t) = (t, t^2)$, the rule gives $\nabla f \cdot \gamma' = t^2 \cdot 1 + t \cdot 2t = 3 t^2$, the derivative of $t^3$.
</document>

<document source="https://tamnd.github.io/math2/node/0A9E6X0S3E.md" tag="0A9E6X0S3E" kind="concept">
# Complete ordered field

## Summary

A complete ordered field is an [[ordered field|0P3VCBDEE2]] $F$ in which each nonempty subset that is bounded above has a [[supremum|0R3H5PANRQ]] in $F$.
The [[real numbers|08J2M1JYAG]] form a complete ordered field, by the [[completeness axiom|0AXWET80YW]].
Any two complete ordered fields are isomorphic, by a [[unique isomorphism|0G8G1TH5WK]] that preserves the order.
Thus "the complete ordered field" is a characterization of $\mathbb{R}$.

## Remarks

The ordered field $\mathbb{Q}$ is not complete.
Each complete ordered field is [[Archimedean|0XH79S06J3]].
Other forms of completeness of $\mathbb{R}$ are equivalent for Archimedean ordered fields, for example the convergence of each Cauchy sequence.

## Explanations

### L1 human

Many number systems allow addition, multiplication, division and comparison, for example $\mathbb{Q}$ and $\mathbb{R}$.
Among them, the real numbers are the only one that has no gaps.
So you can define the real numbers by this list of properties, without a construction.
</document>

<document source="https://tamnd.github.io/math2/node/0A9EB0KQBC.md" tag="0A9EB0KQBC" kind="concept">
# Chi-squared distribution

## Summary

The chi-squared distribution with $k$ degrees of freedom is the law of a sum of the squares of $k$ independent standard normal random variables. It is central in tests and confidence intervals for variances.
The node [[chi-squared distribution|0RFMMBHFTV]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0A9V6RKQG6.md" tag="0A9V6RKQG6" kind="statement">
# Schröder–Bernstein theorem

## Statement

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
If there are [[injective functions|0PVG1CRAJN]] $f \colon A \to B$ and $g \colon B \to A$, then there is a [[bijective function|0SNQEF7Z53]] $h \colon A \to B$.

## Remarks

In terms of cardinals: $|A| \le |B|$ and $|B| \le |A|$ imply $|A| = |B|$. So the [[order of cardinals|01GTP8M5RG]] is antisymmetric.
The proof does not use the axiom of choice.
Example: $[0, 1]$ and $[0, 1)$ are equinumerous, because each one injects into the other.

## History

Cantor stated the theorem in 1887 without a proof.
Dedekind proved it in 1887, but he did not publish the proof.
Bernstein and Schröder published proofs in 1898, and the proof of Schröder had a flaw.

## Explanations

### L1 human

If $A$ fits into $B$ and $B$ fits into $A$, then $A$ and $B$ have the same size.
For finite sets this is obvious, but for infinite sets it needs a clever construction of $h$.
</document>

<document source="https://tamnd.github.io/math2/node/0AA97XA70K.md" tag="0AA97XA70K" kind="concept">
# Periodic orbit

## Summary

Let a point that is not an equilibrium return to itself after a fixed positive time. Its orbit is a periodic orbit of the dynamical system.
For an autonomous differential equation, it is the closed curve traced by a non-constant periodic solution. See the [[definition|01DSNP0B4R]].
</document>

<document source="https://tamnd.github.io/math2/node/0AAHE7GEW6.md" tag="0AAHE7GEW6" kind="definition">
# Kernel of a ring homomorphism

## Definition

Let $f \colon R \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]].
The kernel of $f$ is the set
$$\ker f = \{a \in R : f(a) = 0\}.$$

## Remarks

The kernel of $f$ is a [[two-sided ideal|0DDA6CY7GJ]] of $R$.
The kernel of the reduction map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ is $n\mathbb{Z}$.
The map $f$ is injective if and only if $\ker f = \{0\}$, by the [[kernel criterion|0MMCB8RXGA]].

## Explanations

### L1 human

The kernel collects all the elements that $f$ sends to $0$.
For reduction modulo $5$, the kernel is the set of the multiples of $5$.
</document>

<document source="https://tamnd.github.io/math2/node/0AANKKNHS4.md" tag="0AANKKNHS4" kind="concept">
# Least element

## Summary

The least element of a subset of a partially ordered set is an element of the subset that is below every element of the subset.
</document>

<document source="https://tamnd.github.io/math2/node/0AAQ73WRFB.md" tag="0AAQ73WRFB" kind="concept">
# Limit cycle

## Summary

A limit cycle is a closed trajectory of a planar dynamical system with one property. At least one other trajectory spirals into it as time tends to $\infty$ or to $-\infty$.
Limit cycles model oscillations that sustain themselves. See the [[definition|0EERDDPSVT]].

## Remarks

Henri Poincaré started the study of limit cycles.
</document>

<document source="https://tamnd.github.io/math2/node/0AAXWGT5TD.md" tag="0AAXWGT5TD" kind="concept">
# Almost sure convergence

## Summary

A sequence of random variables converges almost surely when, for almost every outcome, the sequence of values converges.
The node [[almost sure convergence|04P7D63W8P]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ABK9V9Y64.md" tag="0ABK9V9Y64" kind="concept">
# Gradient

## Summary

Let $f$ be a [[differentiable|0HHHDCXZV4]] real-valued function on an open subset of $\mathbb{R}^n$. Its gradient at a point $a$ is the vector $\nabla f(a)$ of its [[partial derivatives|09EGC6F8PT]] at $a$.
It points in the direction in which $f$ increases fastest, and its length is that fastest rate of increase.
The derivative of $f$ at $a$ is the dot product with the gradient: $Df(a)(v) = \nabla f(a) \cdot v$.
</document>

<document source="https://tamnd.github.io/math2/node/0ABP2F2ZP7.md" tag="0ABP2F2ZP7" kind="statement">
# Finite-dimensional normed spaces are complete

## Statement

Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, with a norm $\|\cdot\|$.
Then $V$ is a [[Banach space|0BDTTM24DP]].

## Remarks

Proof: by [[equivalence of norms in finite dimension|03MC70ZRNR]], the norm is equivalent to the sup norm of the coordinates in a basis. Equivalent norms have the same Cauchy sequences and the same convergent sequences. With the sup norm, completeness follows from the completeness of $\mathbb{R}$ in each coordinate.
A consequence: each finite-dimensional subspace of a normed space is complete, and thus [[closed|0W41XG4W60]].

## Explanations

### L1 human

In finite dimensions, a Cauchy sequence is Cauchy in each coordinate, so each coordinate converges.
This works for each norm, because all norms are equivalent there.
</document>

<document source="https://tamnd.github.io/math2/node/0ACRGYNHSR.md" tag="0ACRGYNHSR" kind="concept">
# Dirac measure

## Summary

The Dirac measure at a point $x$ gives a set the value $1$ if the set contains $x$, and the value $0$ if not.
It is the probability measure of a random quantity that always takes the value $x$. The node [[Dirac measure|099SXAG014]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ACX5QT8TE.md" tag="0ACX5QT8TE" kind="concept">
# Invariant subspace

## Summary

A subspace is invariant under a linear map from a space to itself when the map sends every vector of the subspace into the subspace.
</document>

<document source="https://tamnd.github.io/math2/node/0AD0BZ49DW.md" tag="0AD0BZ49DW" kind="statement">
# Metrizable spaces are normal

## Statement

Let $X$ be a [[metrizable space|09KDHB82SG]].
Then $X$ is [[normal|0AJ646N339]].

## Remarks

Proof: choose a metric $d$ for the topology. For disjoint closed sets $A$ and $B$, let $U = \{x : d(x, A) < d(x, B)\}$ and $V = \{x : d(x, B) < d(x, A)\}$. Here $d(x, A)$ is the [[distance to a set|07TXYYVYZR]].
The sets $U$ and $V$ are open, because $x \mapsto d(x, A)$ is continuous. They are disjoint, and $A \subseteq U$ because $d(x, A) = 0 < d(x, B)$ for $x \in A$.
Single points are closed in a metric space. Thus each metrizable space is normal and T1, and also [[Hausdorff|0P9DCDES73]].

## Explanations

### L1 human

In a metric space, you separate two disjoint closed sets by the points that are nearer to one set than to the other.
</document>

<document source="https://tamnd.github.io/math2/node/0AD4VS02H3.md" tag="0AD4VS02H3" kind="definition">
# Formal derivative

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $f = a_0 + a_1 x + \dots + a_n x^n \in R[x]$.
The formal derivative of $f$ is the polynomial
$$f' = a_1 + 2 a_2 x + 3 a_3 x^2 + \dots + n a_n x^{n-1},$$
where $k a_k$ is the sum of $k$ copies of $a_k$.

## Remarks

The derivative is additive, and $(r f)' = r f'$ for each constant $r \in R$.
If $R$ is commutative, the product rule holds: $(f g)' = f' g + f g'$.
In characteristic $p$, the derivative of $x^p$ is $p x^{p-1} = 0$. So a nonconstant polynomial can have derivative $0$.
The derivative detects [[repeated roots|067F8YVYMH]].

## Explanations

### L1 human

The formal derivative uses the rule from calculus, but it is pure algebra.
You never take a limit, so it works over the integers modulo $p$ too.
For example, the derivative of $x^3 + 2x$ is $3x^2 + 2$.
</document>

<document source="https://tamnd.github.io/math2/node/0ADTQBXRMC.md" tag="0ADTQBXRMC" kind="concept">
# Jacobi symbol

## Summary

The Jacobi symbol $\left(\frac{a}{n}\right)$ extends the Legendre symbol to each odd positive integer $n$, as a product over the prime factors of $n$.
The [[definition of the Jacobi symbol|0XDB5AW7M1]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0AEAHMFSCA.md" tag="0AEAHMFSCA" kind="concept">
# Strictly convex function

## Summary

A strictly convex function is a function whose graph lies strictly below each chord, except at the endpoints of the chord. Its graph contains no line segments. The node [[strictly convex function|06BA5JZKTQ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0AEKDFCYFA.md" tag="0AEKDFCYFA" kind="definition">
# Prime factorization

## Definition

Let $n$ be a positive integer.
A prime factorization of $n$ is a finite list of [[prime numbers|0SN0MAEKYX]] $p_1 \le p_2 \le \dots \le p_k$ such that $n = p_1 p_2 \cdots p_k$.
For $n = 1$, the prime factorization is the empty list, with the empty product $1$.

## Remarks

We often group equal primes and write $n = q_1^{e_1} q_2^{e_2} \cdots q_r^{e_r}$, with distinct primes $q_1 < \dots < q_r$ and exponents $e_i \ge 1$.
For example, the prime factorization of $360$ is $2, 2, 2, 3, 3, 5$, that is, $360 = 2^3 \cdot 3^2 \cdot 5$.
Each positive integer has [[a prime factorization|020K0X7HVR]], and the [[fundamental theorem of arithmetic|084HH1NDCB]] says that it is unique.

## Explanations

### L1 human

A prime factorization breaks a number into primes, which you cannot break further.
A factor tree does this: $360 = 36 \cdot 10 = (6 \cdot 6) \cdot (2 \cdot 5) = 2 \cdot 3 \cdot 2 \cdot 3 \cdot 2 \cdot 5$.
Sort the primes to get the list $2, 2, 2, 3, 3, 5$.
</document>

<document source="https://tamnd.github.io/math2/node/0AF3DANJ76.md" tag="0AF3DANJ76" kind="definition">
# Pell's equation

## Definition

Let $d$ be a positive integer that is not a perfect square.
Pell's equation for $d$ is the [[Diophantine equation|08B1RTW7VZ]] $x^2 - d y^2 = 1$, with unknowns $x, y \in \mathbb{Z}$.
A solution is a pair $(x, y)$ of integers that satisfies the equation.

## Remarks

The pairs $(\pm 1, 0)$ are the trivial solutions.
For $d = 2$, the pair $(3, 2)$ is a solution, since $9 - 2 \cdot 4 = 1$.
A solution $(x, y)$ with $y \neq 0$ gives the rational approximation $x / y$ of $\sqrt{d}$.
The name comes from an error of Euler, who attributed a solution by William Brouncker to John Pell.

## Explanations

### L1 human

Pell's equation asks for integers $x$ and $y$ such that $x^2$ is exactly one more than $d y^2$.
For $d = 2$, the solutions $(3, 2)$, $(17, 12)$ and $(577, 408)$ give fractions very close to $\sqrt{2}$.
</document>

<document source="https://tamnd.github.io/math2/node/0AF7X08PCR.md" tag="0AF7X08PCR" kind="concept">
# Convergent series

## Summary

A convergent series is a series whose sequence of partial sums has a finite limit. Many tests decide whether a given series converges. The node [[convergent series|0CVVKY7GY2]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0AFNNTRTJE.md" tag="0AFNNTRTJE" kind="concept">
# Collinear points

## Summary

Points are collinear when one line contains all of them. Three points that are not collinear are the vertices of a triangle.
</document>

<document source="https://tamnd.github.io/math2/node/0AFWQZ1PKF.md" tag="0AFWQZ1PKF" kind="counterexample">
# sin(1/x) has no limit at 0

## Counterexample

Let $f(x) = \sin(1/x)$ for $x \ne 0$.
Then $\lim_{x \to 0} f(x)$ does not exist, and neither one-sided limit exists.
Let $x_n = 1/(2 \pi n + \pi/2)$ and $y_n = 1/(\pi n)$. Both sequences tend to $0$ from the right, but $f(x_n) = 1$ and $f(y_n) = 0$ for all $n \ge 1$.

## Remarks

The argument uses the [[sequential characterization of limits|000MXHEY22]].
So no value of $f$ at $0$ makes $f$ continuous at $0$. The point $0$ is an essential discontinuity.
The graph of $f$ on $(0, 1]$, with its closure, is the topologist's sine curve.

## Explanations

### L1 human

As $x$ approaches $0$, the number $1/x$ grows without bound.
So $\sin(1/x)$ swings between $-1$ and $1$ faster and faster, and it settles at no value.
</document>

<document source="https://tamnd.github.io/math2/node/0AH9Q6YNMT.md" tag="0AH9Q6YNMT" kind="statement">
# If G modulo its centre is cyclic then G is abelian

## Statement

Let $G$ be a [[group|G5S1M9]] with [[centre|0DZ93493VN]] $Z(G)$.
If the [[quotient group|0KYX2N69J0]] $G/Z(G)$ is [[cyclic|0KVGH0J5B3]], then $G$ is [[abelian|082GHNPJ53]].

## Remarks

Let $gZ(G)$ generate $G/Z(G)$; then each element of $G$ is $g^k z$ with $k \in \mathbb{Z}$ and $z \in Z(G)$, and two such elements commute.
In that case $G = Z(G)$, so in fact $G/Z(G)$ is trivial.
A consequence is that $G/Z(G)$ is never cyclic of order greater than $1$. For example, the centre of a group never has prime index.

## Explanations

### L1 human

If the elements of $G$ are powers of one element, up to central elements, then all elements commute.
Powers of one element commute with each other, and central elements commute with everything.
So the quotient by the centre cannot be a nontrivial cyclic group.
</document>

<document source="https://tamnd.github.io/math2/node/0AHKWBQSWV.md" tag="0AHKWBQSWV" kind="concept">
# Vertex cover

## Summary

A vertex cover of a graph is a set of vertices that meets every edge. Vertex covers are dual to matchings and complementary to independent sets. The node [[vertex cover|0KDADE9RQ8]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0AJ035ZTZE.md" tag="0AJ035ZTZE" kind="concept">
# Generalised eigenspace

## Summary

The generalised eigenspace of an endomorphism for a scalar $\lambda$ is a set of vectors.
It contains the vectors that some power of the map minus $\lambda$ times the identity sends to zero.
</document>

<document source="https://tamnd.github.io/math2/node/0AJ0KP1B31.md" tag="0AJ0KP1B31" kind="definition">
# Multiplication by a matrix as a linear map

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$.
The *linear map of multiplication by $A$* is
$$L_A \colon F^n \to F^m, \qquad L_A(x) = Ax,$$
where $Ax$ is the [[matrix–vector product|0PDVQEP0ZC]].

## Remarks

The map $L_A$ is linear, because $A(x + y) = Ax + Ay$ and $A(cx) = c\, Ax$.
Its matrix in the standard bases is $A$, because $L_A(e_j)$ is column $j$ of $A$.
Each linear map $F^n \to F^m$ equals $L_A$ for exactly one matrix $A$.
The kernel of $L_A$ is the [[null space|0PHDYNR24D]] of $A$, and its image is the [[column space|03AJKXH1AV]] of $A$.

## Explanations

### L1 human

Multiplying vectors by a fixed matrix is a linear map.
The matrix $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ stretches the $x$-direction by $2$ and the $y$-direction by $3$.
Every linear map between coordinate spaces is of this kind.
</document>

<document source="https://tamnd.github.io/math2/node/0AJ0YVK743.md" tag="0AJ0YVK743" kind="definition">
# Annulus

## Definition

Let $c \in \mathbb{C}$ and let $r, R$ be such that $0 \le r < R \le \infty$.
The annulus with center $c$, inner radius $r$ and outer radius $R$ is the subset
$$A(c; r, R) = \{ z \in \mathbb{C} : r < |z - c| < R \}$$
of the [[complex plane|0CTE7A31XQ]]. Here $|z - c|$ is the [[modulus|0KW2X0TFBX]] of $z - c$.

## Remarks

Each annulus is a [[domain|097KNJFJ7V]].
For $r = 0$, the annulus is the punctured disc $0 < |z - c| < R$. For $r = 0$ and $R = \infty$, it is $\mathbb{C} \setminus \{c\}$.
For $R < \infty$, its area is $\pi (R^2 - r^2)$.
Holomorphic functions on an annulus have [[Laurent expansions|06WHRARZKY]].

## Explanations

### L1 human

An annulus is the ring between two circles with the same center, without the two circles.
For example, $1 < |z| < 2$ is the set of the points whose distance to $0$ is between $1$ and $2$.
</document>

<document source="https://tamnd.github.io/math2/node/0AJ646N339.md" tag="0AJ646N339" kind="definition">
# Normal space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is normal when the property that follows holds.
For all disjoint [[closed sets|0G5AV9Q4RR]] $A, B \subseteq X$, there are [[open sets|01ZQQDCERP]] $U$ and $V$ with $A \subseteq U$, $B \subseteq V$ and $U \cap V = \varnothing$.

## Remarks

Names vary. Some authors call a space T4 when it is normal and [[T1|05GQJXBP3F]]. Such a space is [[regular|0JF9KW2TXB]] and [[Hausdorff|0MXB4CKHFW]].
Each [[metrizable space is normal|0AD0BZ49DW]], and each [[compact Hausdorff space is normal|0MQQQ5NA2S]].
A product of two normal spaces need not be normal. The Sorgenfrey plane is an example.
[[Urysohn's lemma|01EY3VR7ZD]] characterises normal spaces by continuous functions.

## Explanations

### L1 human

In a normal space, two closed sets that do not meet can be put into two open sets that do not meet.
This is a strong condition, but it holds for all metric spaces.
</document>

<document source="https://tamnd.github.io/math2/node/0AJN7WHBJZ.md" tag="0AJN7WHBJZ" kind="definition">
# Independent and identically distributed random variables

## Definition

Let $(X_i)_{i \in I}$ be a family of random variables on one probability space, all with values in the same measurable space $(E, \mathcal{E})$.
The family is independent and identically distributed (i.i.d.) if both of these conditions hold:
1. The family $(X_i)_{i \in I}$ is [[independent|0XCE070V8B]].
2. The random variables $X_i$ are [[identically distributed|01X8YSDTBW]]: all $X_i$ have the same law $\mu$.

## Remarks

One also says that $X_1, X_2, \dots$ is an i.i.d. sequence with common law $\mu$, or a random sample from $\mu$.
The results of repeated tosses of one coin form an i.i.d. sequence of Bernoulli random variables.
The law of large numbers and the central limit theorem are first proved for i.i.d. sequences.

## Explanations

### L1 human

I.i.d. variables are the results of the same experiment, repeated under the same conditions, with no influence between the repetitions.
Each one follows the same law, and none of them gives information about the others.
</document>

<document source="https://tamnd.github.io/math2/node/0AJXYH4DDQ.md" tag="0AJXYH4DDQ" kind="concept">
# Step function

## Summary

A step function is a function that is constant on each piece of a partition of an interval into finitely many pieces. Its integral is a finite sum of areas of rectangles. The node [[step function|0M9XEDF1HJ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0AK9TGSNXY.md" tag="0AK9TGSNXY" kind="concept">
# Successor function

## Summary

The successor function is the function $S \colon \mathbb{N} \to \mathbb{N}$ that sends each natural number $n$ to the next natural number, $S(n) = n + 1$.
It is a primitive notion of the [[Peano axioms|0QFT7Z76N5]].
</document>

<document source="https://tamnd.github.io/math2/node/0AKNPYQ04R.md" tag="0AKNPYQ04R" kind="definition">
# Classical definition of probability

## Definition

Let $\Omega$ be a nonempty [[finite set|0BBTH3YSZA]] of equally likely outcomes.
The classical probability of an event $A \subseteq \Omega$ is
$$P(A) = \frac{|A|}{|\Omega|},$$
the number of favourable outcomes divided by the number of all outcomes.

## Remarks

This $P$ is a [[probability measure|0VGMEBVW1M]] on the power set of $\Omega$, the uniform probability on $\Omega$. Each outcome has probability $1/|\Omega|$.
For two rolls of a fair die, the probability that the sum is $7$ is $6/36 = 1/6$.
Classical probability turns a probability problem into a counting problem. Binomial coefficients and the rule of product often give the counts.
The definition is circular as a foundation, since "equally likely" already uses probability. It also does not apply to infinite sample spaces.

## Explanations

### L1 human

When all outcomes have the same chance, a probability is a fraction: good cases over all cases.
A fair die has $6$ faces and $3$ of them are even, so the chance of an even number is $3/6 = 1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/0AKRKVHW94.md" tag="0AKRKVHW94" kind="statement">
# Stokes' theorem

## Statement

Let $D \subseteq \mathbb{R}^2$ be the compact region bounded by a piecewise smooth simple closed curve $\gamma$, oriented counterclockwise.
Let $r$ be an injective regular [[parametrised surface|05788J51WZ]] of class $C^2$ on an open set that contains $D$. Let $S = r(D)$ have the unit normal $n = (r_u \times r_v) / \| r_u \times r_v \|$, and let $\partial S$ be the closed curve $r \circ \gamma$.
If $F$ is a vector field of class $C^1$ on an open set that contains $S$, then
$$\oint_{\partial S} F \cdot d\mathbf{r} = \iint_S (\nabla \times F) \cdot n \, dS.$$

## Remarks

The left side is the circulation of $F$ around the boundary. The right side is the [[flux|0YPRY7T4PK]] of the [[curl|0GDNHS9J2X]] of $F$ through $S$.
The orientations match by the right-hand rule: with the thumb along $n$, the fingers curl in the direction of $\partial S$.
Example: for the upper unit hemisphere and $F = (-y, x, 0)$, $\nabla \times F = (0, 0, 2)$, and both sides equal $2\pi$.
For a flat surface in the $xy$-plane, the theorem reduces to [[Green's theorem|0HR3N02317]]. The proof pulls the integrals back to $D$ and applies Green's theorem there.

## History

The theorem first appears in a letter from William Thomson to George Stokes in 1850.
Stokes set it as a question on the Smith's Prize examination at Cambridge in 1854, and the result then got his name.

## Explanations

### L1 human

The total swirl of a field over a surface equals the circulation of the field around the edge of the surface.
The surface can be any shape: only its boundary curve matters.
</document>

<document source="https://tamnd.github.io/math2/node/0AM050KCAW.md" tag="0AM050KCAW" kind="statement">
# Gram matrices are positive semidefinite

## Statement

Let $B$ be an $m \times n$ complex matrix.
Then the $n \times n$ matrix $B^* B$ is [[positive semidefinite|01MMVW8CKC]], where $B^*$ is the [[conjugate transpose|0DXKE10PHF]].
It is positive definite if and only if the columns of $B$ are linearly independent.

## Remarks

The proof: $x^* B^* B x = (Bx)^* (Bx) = \norm{Bx}^2 \ge 0$.
For a real matrix $B$, the same holds for $B^T B$.
Each [[Gram matrix|00SPJSDT3Z]] has this form, so Gram matrices are positive semidefinite.

## Explanations

### L1 human

Matrices of the form $B^T B$ can never give negative values of $x^T A x$.
The value is the squared length of $Bx$.
It is zero only when $Bx = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0AN28XQ45Z.md" tag="0AN28XQ45Z" kind="statement">
# Cauchy product of absolutely convergent series

## Statement

Let $\sum a_n$ and $\sum b_n$ be real series that [[converge absolutely|0W3K3QPEWT]], with sums $A$ and $B$.
Then their [[Cauchy product|0B1K9PVWE9]] $\sum c_n$ converges absolutely, and its sum is $A B$.

## Remarks

The double series $\sum_{i,j} a_i b_j$ converges absolutely, so its terms can be grouped in any order. The Cauchy product groups them by $i + j$.
[[Mertens' theorem|0BJSWZJG18]] needs absolute convergence of only one of the two series, but then gives only convergence.
Mathlib states it for complete normed rings, as `tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm`.

## Explanations

### L1 human

When both series converge absolutely, we may multiply them term by term as with finite sums.
The order in which we add the products does not change the result.
</document>

<document source="https://tamnd.github.io/math2/node/0AP750DHVP.md" tag="0AP750DHVP" kind="concept">
# Circle integral

## Summary

A circle integral is the contour integral of a function along a circle $|z - c| = R$, traveled once counterclockwise, see the [[definition|0T7ABX4CWE]].
</document>

<document source="https://tamnd.github.io/math2/node/0AP80TT660.md" tag="0AP80TT660" kind="definition">
# Multiplicative function

## Definition

An [[arithmetic function|0H2EPRZC0D]] $f$ is multiplicative if $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all [[coprime|0GFKP8F8AG]] positive integers $m$ and $n$.

## Remarks

A multiplicative function is determined by its values at prime powers: if $n = p_1^{e_1} \cdots p_k^{e_k}$, then $f(n) = f(p_1^{e_1}) \cdots f(p_k^{e_k})$.
Examples are [[Euler's totient function|0VV04M41AN]], the [[divisor function|0YEW47VJ0J]] and the [[Möbius function|0C8QENHYZW]].
A function with $f(mn) = f(m) f(n)$ for all $m$ and $n$ is [[completely multiplicative|0GS27TKF1V]].

## Explanations

### L1 human

For a multiplicative function, you can compute $f(n)$ from the prime factorization of $n$.
For example, $\varphi(12) = \varphi(4) \varphi(3) = 2 \cdot 2 = 4$, because $4$ and $3$ are coprime.
</document>

<document source="https://tamnd.github.io/math2/node/0APCFW7639.md" tag="0APCFW7639" kind="concept">
# Nullity

## Summary

The nullity of a linear map is the dimension of its kernel.
</document>

<document source="https://tamnd.github.io/math2/node/0APN65D6Y6.md" tag="0APN65D6Y6" kind="statement">
# Second isomorphism theorem for vector spaces

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be subspaces of $V$.
Then the map
$$U / (U \cap W) \to (U + W) / W, \qquad u + (U \cap W) \mapsto u + W,$$
is well defined, and it is a linear isomorphism.
Here $U + W$ is the [[sum of the subspaces|0JMTY8MF54]], and both sides are [[quotient spaces|0BR3FY5Q8M]].

## Remarks

In finite dimensions, it gives [[Grassmann's formula|06HTJNCMPA]]: $\dim U - \dim(U \cap W) = \dim(U + W) - \dim W$.
The proof applies the [[first isomorphism theorem|0MTBGVZ4XJ]] to the map $U \to (U + W)/W$, $u \mapsto u + W$, which is surjective with kernel $U \cap W$.

## Explanations

### L1 human

Adding $U$ to $W$ adds the same new directions as $U$ has beyond its common part with $W$.
The theorem makes this precise with quotient spaces.
It is the linear algebra form of a general algebraic theorem.
</document>

<document source="https://tamnd.github.io/math2/node/0AQBSVMX9P.md" tag="0AQBSVMX9P" kind="concept">
# Edge set

## Summary

The edge set of a graph is the set of its edges, each edge an unordered pair of adjacent vertices. It is the basic data of a graph next to the vertex set. The node [[edge set|0WNGQBPPQX]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0AQNX7G443.md" tag="0AQNX7G443" kind="statement">
# The symmetric group is generated by transpositions

## Statement

Let $n \geq 2$.
The [[symmetric group|0515A6D2Y1]] $S_n$ is [[generated|0RP7VWQ0YN]] by its [[transpositions|0FWVRHBRWM]].

## Remarks

This is a reformulation of the [[decomposition into transpositions|0XK6FECMW5]].
The adjacent transpositions $(1\ 2), (2\ 3), \dots, (n-1\ n)$ already generate $S_n$.
The two elements $(1\ 2)$ and $(1\ 2\ \dots\ n)$ also generate $S_n$.

## Explanations

### L1 human

Each permutation is a product of swaps, so the swaps generate the whole symmetric group.
You do not even need all swaps.
Swaps of neighbours are enough, as in a bubble sort.
</document>

<document source="https://tamnd.github.io/math2/node/0AQZV0905N.md" tag="0AQZV0905N" kind="concept">
# Abelianisation

## Summary

The abelianisation of a group $G$ is the quotient $G/[G, G]$ by its commutator subgroup.
It is the largest abelian quotient of $G$.
</document>

<document source="https://tamnd.github.io/math2/node/0AR06W1E1B.md" tag="0AR06W1E1B" kind="statement">
# Differentiation under the integral sign

## Statement

Let $f \colon [a, b] \times [t_0, t_1] \to \mathbb{R}$ be continuous, and suppose that its [[partial derivative|05AXG3QRKR]] $\partial f / \partial t$ exists and is [[continuous|0S9EK5MQPE]] on $[a, b] \times [t_0, t_1]$.
Then $F(t) = \int_a^b f(x, t) \, dx$ is continuously differentiable on $[t_0, t_1]$, and
$$F'(t) = \int_a^b \frac{\partial f}{\partial t}(x, t) \, dx.$$
If, in addition, $\alpha, \beta \colon [t_0, t_1] \to [a, b]$ are continuously differentiable, then
$$\frac{d}{dt} \int_{\alpha(t)}^{\beta(t)} f(x, t) \, dx = \int_{\alpha(t)}^{\beta(t)} \frac{\partial f}{\partial t}(x, t) \, dx + f(\beta(t), t) \, \beta'(t) - f(\alpha(t), t) \, \alpha'(t).$$

## Remarks

The second formula follows from the first, the fundamental theorem of calculus and the [[chain rule|0EAAS2EHDR]], applied to $G(u, v, t) = \int_u^v f(x, t) \, dx$.
A measure theory version: let $f(\cdot, t)$ be integrable for each $t$ in an open set, let $\partial f / \partial t$ exist for almost all $x$, and let $|\partial f / \partial t (x, t)| \le g(x)$ for an integrable $g$. Then the same rule holds. Its proof uses the dominated convergence theorem.
Example: let $F(t) = \int_0^1 \frac{x^t - 1}{\ln x} \, dx$ for $t \ge 0$. For $t > 0$ the measure version, with $g = 1$, gives $F'(t) = \int_0^1 x^t \, dx = \frac{1}{t + 1}$. Since $F$ is continuous with $F(0) = 0$, $F(t) = \ln(1 + t)$.

## Explanations

### L1 human

If the integrand depends smoothly on a parameter, you can differentiate the integral by differentiating inside it.
When the limits of integration also move, the values of the integrand at the moving limits add extra terms.
</document>

<document source="https://tamnd.github.io/math2/node/0AR4B3K5QD.md" tag="0AR4B3K5QD" kind="counterexample">
# Pairwise independent events that are not mutually independent

## Counterexample

Toss a fair coin two times, so $\Omega = \{HH, HT, TH, TT\}$ with probability $1/4$ for each outcome.
Let $A$ be "the first toss is heads", $B$ be "the second toss is heads", and $C$ be "the two tosses are equal".
Then $A = \{HH, HT\}$, $B = \{HH, TH\}$ and $C = \{HH, TT\}$, so $P(A) = P(B) = P(C) = 1/2$.
Each intersection of two of them is $\{HH\}$, with probability $1/4 = \frac12 \cdot \frac12$. So $A$, $B$ and $C$ are [[pairwise independent|0N57D1XY6T]].
But $P(A \cap B \cap C) = P(\{HH\}) = 1/4$, and $P(A) \, P(B) \, P(C) = 1/8$. So $A$, $B$ and $C$ are not [[mutually independent|0GCVEV6BJA]].

## Remarks

Here any two of the events determine the third: if $A$ and $B$ both occur, then $C$ occurs.
The same idea with $n$ fair bits and their sums modulo $2$ gives large pairwise independent families. Such families are useful in computer science.

## Explanations

### L1 human

If you know the first toss, you know nothing about $C$, and the same for the second toss.
But if you know both tosses, you know $C$ exactly. So the three events are not independent as a group.
</document>

<document source="https://tamnd.github.io/math2/node/0AR68JTMDT.md" tag="0AR68JTMDT" kind="concept">
# Unitary matrix

## Summary

A unitary matrix is a complex square matrix whose conjugate transpose is its inverse.
</document>

<document source="https://tamnd.github.io/math2/node/0AREFQ9Z65.md" tag="0AREFQ9Z65" kind="concept">
# Continuous uniform distribution

## Summary

The continuous uniform distribution on an interval $[a, b]$ spreads the probability evenly over the interval: the probability of a subinterval is proportional to its length.
The node [[continuous uniform distribution|0WGDPYR30Q]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ARPRJQWZ2.md" tag="0ARPRJQWZ2" kind="statement">
# Simplicity of the alternating group

## Statement

Let $n \geq 5$.
Then the [[alternating group|0Z5905Q73D]] $A_n$ is simple: its only [[normal subgroups|0CB0DX5BVV]] are $\{e\}$ and $A_n$.

## Remarks

A usual proof shows that a nontrivial normal subgroup of $A_n$ contains a $3$-cycle, then all $3$-cycles, and then uses that the [[3-cycles generate|0KFZ0QX83G]] $A_n$.
The group $A_4$ is not simple, because the Klein four-group $\{e, (1\ 2)(3\ 4), (1\ 3)(2\ 4), (1\ 4)(2\ 3)\}$ is normal in $A_4$.
The group $A_3$ is simple because it has prime order $3$.
For $n \geq 5$, the only normal subgroups of $S_n$ are $\{e\}$, $A_n$ and $S_n$.

## Explanations

### L1 human

A simple group cannot be broken into a normal subgroup and a quotient.
From five symbols on, the even permutations form such an unbreakable group.
This fact is a key step in the proof that there is no general formula in radicals for polynomials of degree $5$ or more.
</document>

<document source="https://tamnd.github.io/math2/node/0AS1FZT6BM.md" tag="0AS1FZT6BM" kind="definition">
# Power sum symmetric polynomials

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $n \ge 1$.
For $k \ge 1$, the $k$-th power sum in $x_1, \dots, x_n$ is
$$p_k = x_1^k + x_2^k + \dots + x_n^k \in R[x_1, \dots, x_n].$$

## Remarks

Each $p_k$ is [[symmetric|06CTFY2DEZ]] and homogeneous of degree $k$.
[[Newton's identities|0WG2A3HE0S]] relate the power sums to the [[elementary symmetric polynomials|0ZMY8441GW]].
If $R$ contains $\mathbb{Q}$, then every symmetric polynomial in $R[x_1, \dots, x_n]$ is a polynomial in $p_1, \dots, p_n$.
This fails over $\mathbb{Z}$: $e_2 = \tfrac{1}{2}(p_1^2 - p_2)$ needs the coefficient $\tfrac{1}{2}$.

## Explanations

### L1 human

The power sums add up the same power of every variable.
They are symmetric, and over the rationals they can replace the elementary symmetric polynomials as generators.
</document>

<document source="https://tamnd.github.io/math2/node/0AS800VAYX.md" tag="0AS800VAYX" kind="concept">
# Consistent estimator

## Summary

An estimator is consistent when it converges to the true value of the parameter as the sample size grows.
The node [[consistent estimator|0XJVF81QF6]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ASC73QEQT.md" tag="0ASC73QEQT" kind="statement">
# The Möbius function is the inverse of the constant function 1

## Statement

Let $\mathbf{1}$ be the arithmetic function with $\mathbf{1}(n) = 1$ for all $n$, and let $\varepsilon$ be the function with $\varepsilon(1) = 1$ and $\varepsilon(n) = 0$ for $n > 1$.
Then $\mu * \mathbf{1} = \varepsilon$, where $\mu$ is the [[Möbius function|0C8QENHYZW]] and $*$ is the [[Dirichlet convolution|0KH1N66QWW]].
That is, for each positive integer $n$,
$$\sum_{d \mid n} \mu(d) = \begin{cases} 1 & \text{if } n = 1, \\ 0 & \text{if } n > 1. \end{cases}$$

## Remarks

The proof for $n > 1$: let $k = \omega(n) \ge 1$. Only squarefree divisors count, so the sum is $\sum_{j=0}^{k} \binom{k}{j} (-1)^j = (1 - 1)^k = 0$.
For example, the divisors of $12$ give $\mu(1) + \mu(2) + \mu(3) + \mu(6) = 1 - 1 - 1 + 1 = 0$.
This identity is the key step in the [[Möbius inversion formula|07Q4TBTFVB]].

## Explanations

### L1 human

The signs $\mu(d)$ over the divisors of $n$ cancel out, except for $n = 1$.
This is the same cancellation as in the principle of inclusion and exclusion.
</document>

<document source="https://tamnd.github.io/math2/node/0ASDFEQS6P.md" tag="0ASDFEQS6P" kind="concept">
# Block matrix

## Summary

A block matrix is a matrix that is divided into smaller matrices, its blocks, by horizontal and vertical lines.
</document>

<document source="https://tamnd.github.io/math2/node/0ASGSTXSBW.md" tag="0ASGSTXSBW" kind="concept">
# Commutator subgroup

## Summary

The commutator subgroup of a group is the subgroup generated by all commutators.
It is the smallest normal subgroup with an abelian quotient.
</document>

<document source="https://tamnd.github.io/math2/node/0ASHNYF7AG.md" tag="0ASHNYF7AG" kind="concept">
# Annihilator of a subspace

## Summary

The annihilator of a subspace is the set of linear forms that vanish on every vector of the subspace.
</document>

<document source="https://tamnd.github.io/math2/node/0ASJVEFA9J.md" tag="0ASJVEFA9J" kind="concept">
# Mathematical induction

## Summary

Mathematical induction is a method to prove that a property $P(n)$ holds for every [[natural number|0MWKJ44V5A]] $n$.
A proof by induction has two steps.
The base case proves $P(0)$.
The induction step proves, for an arbitrary natural number $n$, that $P(n)$ implies $P(n + 1)$.
Then $P(n)$ holds for every natural number $n$.

## Remarks

The method is valid because of the induction axiom of the [[Peano axioms|0QFT7Z76N5]].
In the induction step, the assumption $P(n)$ is the induction hypothesis.
Common variants are [[induction from an arbitrary base case|0GF6AJEBZ0]] and [[strong induction|0KMVFSHFGZ]].

## Explanations

### L1 human

Think of a line of dominoes that never ends.
If the first domino falls, and each falling domino knocks over the next one, then every domino falls.
The base case is the first domino, and the induction step is the rule that each domino knocks over the next one.
</document>

<document source="https://tamnd.github.io/math2/node/0ASM4EPMB2.md" tag="0ASM4EPMB2" kind="definition">
# Ceiling function

## Definition

Let $x$ be a real number.
The ceiling of $x$, written $\lceil x \rceil$, is the least [[integer|0C4JS9RF5M]] $n$ with $x \le n$.
The ceiling function is the function $\mathbb{R} \to \mathbb{Z}$, $x \mapsto \lceil x \rceil$.

## Remarks

The ceiling and the [[floor|0YVRNEDSVG]] are related by $\lceil x \rceil = -\lfloor -x \rfloor$.
For each real $x$, $x \le \lceil x \rceil < x + 1$.
If $x$ is not an integer, then $\lceil x \rceil = \lfloor x \rfloor + 1$. If $x$ is an integer, then $\lceil x \rceil = \lfloor x \rfloor = x$.

## Explanations

### L1 human

The ceiling of a number rounds it up to a whole number.
If a trip needs $2.1$ buses, then you must order $\lceil 2.1 \rceil = 3$ buses.
</document>

<document source="https://tamnd.github.io/math2/node/0AT1ZSNM25.md" tag="0AT1ZSNM25" kind="definition">
# Normaliser

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$.
The normaliser of $S$ in $G$ is the set $N_G(S) = \{g \in G : gSg^{-1} = S\}$, where $gSg^{-1} = \{gsg^{-1} : s \in S\}$.

## Remarks

The normaliser $N_G(S)$ is a [[subgroup|0X1JXT47GA]] of $G$, and it contains the [[centraliser|02YYV36BFM]] $C_G(S)$.
If $H$ is a subgroup of $G$, then $H$ is a [[normal subgroup|0CB0DX5BVV]] of $N_G(H)$. Also, $N_G(H)$ is the largest subgroup of $G$ with this property.
Thus $H$ is normal in $G$ exactly when $N_G(H) = G$.
The number of [[conjugates|0H11ZPX4ND]] of $H$ in $G$ is the index $[G : N_G(H)]$.

## Explanations

### L1 human

The normaliser of a subgroup is the set of elements that do not move the subgroup when they conjugate it.
They may move the elements of the subgroup around, but the subgroup as a set stays in place.
The normaliser is big when the subgroup is close to normal.
</document>

<document source="https://tamnd.github.io/math2/node/0ATF6Q0633.md" tag="0ATF6Q0633" kind="concept">
# Periodic function

## Summary

A periodic function takes the same values again after each shift of the argument by a fixed nonzero number $T$. The sine and cosine functions are periodic with period $2 \pi$. The node [[periodic function|089KWVYD69]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ATFVE08MW.md" tag="0ATFVE08MW" kind="concept">
# Back substitution

## Summary

Back substitution is a method to solve a system $Ux = b$, where $U$ is an $n \times n$ [[upper triangular matrix|0N4XY7Z3NN]] with nonzero diagonal entries.
The last equation gives $x_n = b_n / u_{nn}$.
Then, for $i = n-1, \dots, 1$ in this order, equation $i$ gives
$$x_i = \frac{1}{u_{ii}} \left( b_i - \sum_{j = i+1}^{n} u_{ij} x_j \right),$$
which uses only the unknowns that are already known.

## Remarks

Back substitution uses on the order of $n^2$ arithmetic operations.
Forward substitution solves $Lx = b$ for a [[lower triangular matrix|00J3KZP8Y0]] $L$ in the same way, from the first equation down.
After [[Gaussian elimination|0JVJ2WCXZH]], back substitution completes the solution of the system.
With an [[LU decomposition|0RKG5YW66Y]] $A = LU$, the system $Ax = b$ is solved by forward substitution for $Ly = b$ and then back substitution for $Ux = y$.

## Explanations

### L1 human

In a triangular system, the last equation has only one unknown, so we solve it at once.
We put that value into the equation above it, which now has only one new unknown.
We continue upwards until we know all the unknowns.
</document>

<document source="https://tamnd.github.io/math2/node/0ATVXC3NCC.md" tag="0ATVXC3NCC" kind="statement">
# Boundedness theorem

## Statement

Let $a \le b$ and let $f \colon [a, b] \to \mathbb{R}$ be [[continuous|0P27W9RW0G]].
Then $f$ is bounded. That is, there is $K$ with $|f(x)| \le K$ for all $x \in [a, b]$.

## Remarks

Suppose not. Then there are points $x_n \in [a, b]$ with $|f(x_n)| > n$. By the [[Bolzano–Weierstrass theorem|0KDJD249DT]], a subsequence converges to some $c \in [a, b]$. Continuity at $c$ gives a contradiction.
The interval must be closed and bounded. The function $1/x$ on $(0, 1]$ is continuous but not bounded.
Mathlib states a form for compact sets, as `IsCompact.bddAbove_image`.

## Explanations

### L1 human

On a closed bounded interval, a continuous function cannot shoot off to infinity.
Its graph stays between two horizontal lines.
</document>

<document source="https://tamnd.github.io/math2/node/0AVEYZ2T0Q.md" tag="0AVEYZ2T0Q" kind="example">
# The nth root of n tends to 1

## Example

For each integer $n \ge 1$, let $a_n = n^{1/n}$ be the positive $n$-th root of $n$.
Then $a_n \to 1$ as $n \to \infty$.
For the proof, write $a_n = 1 + h_n$ with $h_n \ge 0$.
For $n \ge 2$, the binomial theorem gives $n = (1 + h_n)^n \ge \binom{n}{2} h_n^2 = \frac{n(n-1)}{2} h_n^2$.
Thus $0 \le h_n \le \sqrt{2/(n-1)}$, and the [[squeeze theorem|0XMF00T4XQ]] gives $h_n \to 0$.

## Remarks

The first terms are $1, \sqrt{2} \approx 1.414, 3^{1/3} \approx 1.442, 4^{1/4} \approx 1.414$. The largest term is at $n = 3$.
In the same way, $c^{1/n} \to 1$ for each real $c > 0$.
Mathlib proves the version for a real variable, $x^{1/x} \to 1$ as $x \to \infty$, in the lemma `tendsto_rpow_div`.

## Explanations

### L1 human

The number $n$ grows, but the root index $n$ grows too, and the root wins.
For example, $1000^{1/1000}$ is about $1.0069$, very close to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0AVXBEC05W.md" tag="0AVXBEC05W" kind="statement">
# R/P is an integral domain iff P is prime

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $P$ be an ideal of $R$.
Then the [[quotient ring|0VFW4Y69BT]] $R/P$ is an [[integral domain|09S3PFBEPM]] if and only if $P$ is a [[prime ideal|06HC6HK0Z8]].

## Remarks

The condition $1 + P \neq 0 + P$ in $R/P$ is the condition $P \neq R$.
The condition that $(a + P)(b + P) = 0$ implies $a + P = 0$ or $b + P = 0$ is the condition that $a b \in P$ implies $a \in P$ or $b \in P$.
For example, $\mathbb{Z}/n\mathbb{Z}$ is an integral domain exactly when $n$ is prime or $n = 0$.

## Explanations

### L1 human

The theorem translates a property of the ideal into a property of the quotient ring.
An element is zero in $R/P$ exactly when it lies in $P$.
So "a product in $P$ has a factor in $P$" becomes "a zero product has a zero factor".
</document>

<document source="https://tamnd.github.io/math2/node/0AVY8HKBGD.md" tag="0AVY8HKBGD" kind="statement">
# Totient of a prime power

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]] and let $k$ be a positive integer.
Then $\varphi(p^k) = p^{k-1} (p - 1) = p^k - p^{k-1}$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

The proof: an integer is coprime to $p^k$ if and only if $p$ does not divide it. Among $1, \dots, p^k$, the multiples of $p$ are $p, 2p, \dots, p^{k-1} \cdot p$, so there are $p^{k-1}$ of them.
For example, $\varphi(8) = 4$ and $\varphi(27) = 18$.
With the [[multiplicativity of the totient function|0FSJA9M43W]], this formula gives $\varphi(n)$ from the prime factorization of $n$.

## Explanations

### L1 human

Among the numbers $1$ to $p^k$, exactly one in $p$ is a multiple of $p$.
Remove these numbers. The fraction $1 - 1/p$ of the numbers stays, so $\varphi(p^k) = p^k (1 - 1/p)$.
</document>

<document source="https://tamnd.github.io/math2/node/0AW1E6YTJY.md" tag="0AW1E6YTJY" kind="concept">
# Lipschitz condition

## Summary

A function satisfies a Lipschitz condition when the distance between two values is at most a constant times the distance between the two points.
This is a strong form of continuity. In the theory of differential equations, it gives uniqueness of solutions. See the [[definition|055VVSN00V]].

## Remarks

It is named after Rudolf Lipschitz.
</document>

<document source="https://tamnd.github.io/math2/node/0AWMPX35CW.md" tag="0AWMPX35CW" kind="statement">
# K[x]/(f) is a field iff f is irreducible

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$.
Then the [[quotient ring|0VFW4Y69BT]] $K[x]/(f)$ is a field if and only if $f$ is [[irreducible|00X2A9G9W3]] over $K$.

## Remarks

The proof: $K[x]$ is a [[principal ideal domain|0RZYDVZ475]]. So $(f)$ is maximal if and only if $f$ is irreducible, by the [[theorem on PIDs|095ZHQ7WCJ]]. Then [[$R/M$ is a field iff $M$ is maximal|0WD3SE7S1X]].
For example, $\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}$ and $\mathbb{Q}[x]/(x^2 - 2) \cong \mathbb{Q}(\sqrt{2})$.
The field $\mathbb{F}_2[x]/(x^2 + x + 1)$ has $4$ elements.
If $f$ is irreducible of degree $n$, then $K[x]/(f)$ is an extension of $K$ of degree $n$.
The class of $x$ in $K[x]/(f)$ is a root of $f$.

## Explanations

### L1 human

You can build new fields from polynomials.
Take an irreducible polynomial $f$ and declare $f(x) = 0$.
The result is a field in which $f$ has a root, as the complex numbers come from $x^2 + 1 = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0AXP2Z5N9H.md" tag="0AXP2Z5N9H" kind="definition">
# Marginal distribution

## Definition

Let $\mu$ be a probability distribution on $\mathbb{R}^2$, for example the [[joint distribution|0QAR0XDZH4]] of real random variables $X$ and $Y$.
The first marginal distribution of $\mu$ is the distribution $\mu_1$ on $\mathbb{R}$ with $\mu_1(A) = \mu(A \times \mathbb{R})$ for each Borel set $A \subseteq \mathbb{R}$. The second marginal is $\mu_2(B) = \mu(\mathbb{R} \times B)$.
If $\mu$ is the law of $(X, Y)$, then $\mu_1$ is the law of $X$ and $\mu_2$ is the law of $Y$.

## Remarks

The same definition works on a product $E_1 \times E_2$ of measurable spaces, and for $n$ variables with the projections onto some of the coordinates.
In the discrete case, $P(X = x) = \sum_{y} P(X = x, Y = y)$. In the case of a joint density $f(x, y)$, $X$ has the density $f_X(x) = \int_{-\infty}^{\infty} f(x, y) \, dy$.
The name comes from tables: the sums of the rows and the columns of a joint table are written in its margins.

## Explanations

### L1 human

To get the marginal distribution of $X$ from a joint table of $X$ and $Y$, add up each row.
You lose the information about how $X$ and $Y$ are linked, and you keep only the chances of the values of $X$.
</document>

<document source="https://tamnd.github.io/math2/node/0AXWET80YW.md" tag="0AXWET80YW" kind="statement">
# Completeness axiom

## Statement

Let $S$ be a nonempty subset of the [[real numbers|08J2M1JYAG]] that is [[bounded above|0B5H2V4JW8]].
Then $S$ has a [[supremum|0BTMQ04556]] in $\mathbb{R}$.

## Remarks

In the axiomatic approach, this is the axiom that distinguishes $\mathbb{R}$ from the other ordered fields. In a construction of $\mathbb{R}$, it is a theorem.
The ordered field $\mathbb{Q}$ does not have this property: $\{q \in \mathbb{Q} : q^2 < 2\}$ has no supremum in $\mathbb{Q}$.
The axiom gives the [[existence of the infimum|0EDH403M21]] and the [[Archimedean property of the reals|0A3GWZ86AY]].

## Explanations

### L1 human

The real line has no holes.
If a set of real numbers has a ceiling, then there is a lowest ceiling, and this lowest ceiling is again a real number.
</document>

<document source="https://tamnd.github.io/math2/node/0AXZNE5XNH.md" tag="0AXZNE5XNH" kind="concept">
# Symmetric bilinear form

## Summary

A symmetric bilinear form is a bilinear form that does not change when its two arguments are swapped.
</document>

<document source="https://tamnd.github.io/math2/node/0AYEPBQMAW.md" tag="0AYEPBQMAW" kind="definition">
# Eigenspace

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$.
The *eigenspace* of $T$ for $\lambda$ is
$$E_\lambda = \ker(T - \lambda \id_V) = \{v \in V : T(v) = \lambda v\}.$$

## Remarks

The eigenspace is a subspace of $V$; it is the [[kernel|0S23E6YE83]] of a linear map.
Its nonzero elements are the [[eigenvectors|0N6960W0BV]] of $T$ for $\lambda$.
$E_\lambda \ne 0$ if and only if $\lambda$ is an eigenvalue of $T$.
The dimension of $E_\lambda$ is the [[geometric multiplicity|099TWVJ909]] of $\lambda$.

## Explanations

### L1 human

The eigenspace collects all the vectors that the map stretches by the same factor $\lambda$, with the zero vector.
These vectors form a subspace, for example a line or a plane.
On this subspace, the map acts as multiplication by $\lambda$.
</document>

<document source="https://tamnd.github.io/math2/node/0AYMP41THN.md" tag="0AYMP41THN" kind="concept">
# Stabiliser

## Summary

The stabiliser of a point $x$ under a group action is the set of the group elements that do not move $x$.
It is a subgroup of the group.
</document>

<document source="https://tamnd.github.io/math2/node/0AYV6VGKF5.md" tag="0AYV6VGKF5" kind="concept">
# Product metric

## Summary

A product metric is a metric on the Cartesian product of finitely many metric spaces, made from the metrics of the factors.
Common choices take the maximum, the sum, or the Euclidean combination of the distances in the factors.
All these choices give the product topology.
</document>

<document source="https://tamnd.github.io/math2/node/0AZ3R52593.md" tag="0AZ3R52593" kind="definition">
# Convergent sequence

## Definition

A [[real sequence|0CTV783HYT]] $(a_n)$ is convergent if it converges to some real number $L$.
Here convergence is in the sense of the [[limit of a sequence|0XKWSSY8P2]].

## Remarks

The sequences $1/(n+1)$ and $(n+1)/(n+2)$ are convergent, with limits $0$ and $1$.
The constant sequence $a_n = c$ converges to $c$.
A sequence that is not convergent is a [[divergent sequence|01HR2AH0BX]].

## Explanations

### L1 human

A convergent sequence is a sequence that has a limit.
Its terms settle down near one real number.
</document>

<document source="https://tamnd.github.io/math2/node/0AZ5ARP287.md" tag="0AZ5ARP287" kind="concept">
# Dual map

## Summary

The dual map of a linear map $T \colon V \to W$ is the linear map $W^* \to V^*$ that sends each linear form $f$ to $f \circ T$.
</document>

<document source="https://tamnd.github.io/math2/node/0AZG100DKH.md" tag="0AZG100DKH" kind="example">
# Action on left cosets

## Example

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Let $G/H$ be the set of the [[left cosets|03DEH43F4H]] of $H$ in $G$.
The map $G \times G/H \to G/H$, $(g, aH) \mapsto gaH$, is an [[action|0WCY42EV9V]] of $G$ on $G/H$.
The map is well defined, because $aH = bH$ implies $gaH = gbH$.
The action is [[transitive|02GK6HZYBT]], because $(ba^{-1}) \cdot aH = bH$.
The [[stabiliser|0AYMP41THN]] of the coset $aH$ is the conjugate subgroup $aHa^{-1}$.
The kernel of the associated homomorphism $G \to \operatorname{Sym}(G/H)$ is the [[normal core|0GETX79CJN]] of $H$, the intersection of all conjugates of $H$.
For $H = \{e\}$, this is the [[action by left multiplication|0K5TWJAGPD]].
</document>

<document source="https://tamnd.github.io/math2/node/0B0KR7FH2Q.md" tag="0B0KR7FH2Q" kind="concept">
# Equinumerous sets

## Summary

Two sets are equinumerous when there is a bijection between them.
Equinumerous sets have the same size, also when they are infinite.
</document>

<document source="https://tamnd.github.io/math2/node/0B0WE1GKTK.md" tag="0B0WE1GKTK" kind="definition">
# Faithful action

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$.
The action is faithful when, for each $g \in G$, $g \cdot x = x$ for all $x \in X$ implies $g = e$.

## Remarks

The action is faithful if and only if the associated homomorphism $G \to \operatorname{Sym}(X)$ is injective.
So a faithful action identifies $G$ with a subgroup of $\operatorname{Sym}(X)$.
The action of $G$ on itself by conjugation is faithful if and only if the [[centre|0DZ93493VN]] of $G$ is trivial.

## Explanations

### L1 human

In a faithful action, each element of the group that is not the identity moves at least one point.
So you can tell the group elements apart by watching how they move the points.
The group is then a group of permutations of the set.
</document>

<document source="https://tamnd.github.io/math2/node/0B1261B5ED.md" tag="0B1261B5ED" kind="concept">
# Ring

## Summary

A ring is a set with an addition and a multiplication that obey the usual rules of arithmetic.
The multiplication need not be commutative.
The integers, the polynomials and the square matrices are the basic examples.
The [[definition of a ring|0894J980RK]] gives the axioms.
</document>

<document source="https://tamnd.github.io/math2/node/0B1FSS1VZW.md" tag="0B1FSS1VZW" kind="definition">
# Complement

## Definition

Let $U$ be a [[universal set|0BPYRMHJ48]] and let $A$ be a [[subset|0FE8Q5S42Y]] of $U$.
The complement of $A$ in $U$ is the set
$$A^{c} = \{x \in U \mid x \notin A\}.$$

## Remarks

Other notations are $\overline{A}$, $A'$ and $U \setminus A$.
Example: in $U = \mathbb{Z}$, the complement of the set of even integers is the set of odd integers.
The complement of $A^c$ is $A$, and $A \cup A^c = U$, $A \cap A^c = \varnothing$.
The complement of $A$ in $U$ is the [[set difference|0DT0TCV75N]] $U \setminus A$.

## Explanations

### L1 human

The complement is "everything else".
It depends on the universe: the complement of the even numbers is the odd numbers in $\mathbb{Z}$, but not in $\mathbb{R}$.
</document>

<document source="https://tamnd.github.io/math2/node/0B1G4E21G7.md" tag="0B1G4E21G7" kind="statement">
# Moment-generating function of the normal distribution

## Statement

Let $\mu \in \mathbb{R}$, let $\sigma^2 > 0$ and let $X \sim \mathcal{N}(\mu, \sigma^2)$. Then for each $t \in \mathbb{R}$,
$$M_X(t) = E\left[e^{tX}\right] = \exp\left(\mu t + \tfrac{1}{2} \sigma^2 t^2\right).$$

## Remarks

Proof for $Z \sim \mathcal{N}(0, 1)$: complete the square, $tz - z^2/2 = t^2/2 - (z - t)^2/2$. So $E[e^{tZ}] = e^{t^2/2} \int \varphi(z - t) \, dz = e^{t^2/2}$.
The general case follows from $X = \mu + \sigma Z$ with $\sigma = \sqrt{\sigma^2}$: $M_X(t) = e^{\mu t} M_Z(\sigma t)$.
The moment-generating function is finite for all $t$. Its derivatives at $0$ give $E[X] = \mu$ and $E[X^2] = \mu^2 + \sigma^2$.
The logarithm $\mu t + \sigma^2 t^2 / 2$ is a polynomial of degree $2$. So all cumulants of order $3$ or more are $0$.

## Explanations

### L1 human

The moment-generating function of a normal variable has a simple closed form.
With the [[multiplication rule for sums|0WRZ170YEH]], it shows quickly that a sum of independent normal variables is again normal.
</document>

<document source="https://tamnd.github.io/math2/node/0B1K9PVWE9.md" tag="0B1K9PVWE9" kind="definition">
# Cauchy product

## Definition

Let $\sum_{n=0}^{\infty} a_n$ and $\sum_{n=0}^{\infty} b_n$ be real [[series|0JH4MN4K1C]].
For each $n$, let $c_n = \sum_{k=0}^{n} a_k b_{n-k}$.
The series $\sum_{n=0}^{\infty} c_n$ is the Cauchy product of the two series.

## Remarks

The term $c_n$ collects all products $a_i b_j$ with $i + j = n$.
For power series, the Cauchy product of $\sum a_n x^n$ and $\sum b_n x^n$ is $\sum c_n x^n$. This is the rule for multiplying polynomials.
The Cauchy product of two convergent series need not converge. [[Mertens' theorem|0BJSWZJG18]] gives a sufficient condition.

## Explanations

### L1 human

To multiply two series, multiply every term of one by every term of the other.
The Cauchy product groups these products by the sum of their indices.
</document>

<document source="https://tamnd.github.io/math2/node/0B22P010D6.md" tag="0B22P010D6" kind="statement">
# Curl-free vector field on a convex set is conservative

## Statement

Let $U \subseteq \mathbb{R}^3$ be open and convex, and let $F \colon U \to \mathbb{R}^3$ be a vector field of class $C^1$ with $\nabla \times F = 0$ on $U$.
Then $F$ is [[conservative|05R9NF2J49]] on $U$.
Explicitly, for a fixed $p \in U$, the function
$$\phi(x) = \int_0^1 F\big(p + t (x - p)\big) \cdot (x - p) \, dt$$
satisfies $\nabla \phi = F$ on $U$.

## Remarks

The integral $\phi(x)$ is the [[line integral|0YS8RX87F4]] of $F$ along the segment from $p$ to $x$. Convexity makes sure that this segment lies in $U$.
To prove $\nabla \phi = F$, [[differentiate under the integral sign|0AR06W1E1B]] and use $\partial_j F_i = \partial_i F_j$, which is the condition $\nabla \times F = 0$.
The same proof works in $\mathbb{R}^n$ under the condition $\partial_j F_i = \partial_i F_j$ for all $i, j$, and on star-shaped open sets.
The result holds more generally on simply connected open sets. It [[fails on some other open sets|0GRM3BS0DD]]. With the [[converse|0M2JRW1J0N]], it gives: on a convex open set, a $C^1$ field is conservative if and only if its curl is zero.

## Explanations

### L1 human

On a convex set, a field with no swirl is the gradient of a potential.
You build the potential by computing the work of the field along straight segments from one base point.
</document>

<document source="https://tamnd.github.io/math2/node/0B2305GNAZ.md" tag="0B2305GNAZ" kind="statement">
# Open balls are open

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $x \in X$ and let $r > 0$.
Then the [[open ball|0DC3MPR7C9]] $B(x, r)$ is an [[open set|01ZQQDCERP]] of $X$.

## Remarks

Proof: let $y \in B(x, r)$ and let $\varepsilon = r - d(x, y) > 0$. If $d(y, z) < \varepsilon$, then $d(x, z) \le d(x, y) + d(y, z) < r$ by the [[triangle inequality|01M6CW9VM6]]. Thus $B(y, \varepsilon) \subseteq B(x, r)$.
The open sets of $X$ are exactly the unions of open balls.

## Explanations

### L1 human

A point inside an open ball is at some distance from the edge of the ball.
A smaller ball about that point, with radius equal to this distance, stays inside the big ball.
</document>

<document source="https://tamnd.github.io/math2/node/0B268GVX95.md" tag="0B268GVX95" kind="definition">
# Gaussian integer

## Definition

A Gaussian integer is a complex number $a + bi$ such that $a$ and $b$ are integers. Here $i$ is a complex number with $i^2 = -1$.
The set of Gaussian integers is written $\mathbb{Z}[i] = \{ a + bi : a, b \in \mathbb{Z} \}$.

## Remarks

The set $\mathbb{Z}[i]$ is closed under addition, subtraction and multiplication, so it is a subring of $\mathbb{C}$. It is an integral domain.
The units of $\mathbb{Z}[i]$ are $1$, $-1$, $i$ and $-i$.
In the complex plane, the Gaussian integers are the points of the square lattice.
The [[norm|00HPT88ZX0]] $N(a + bi) = a^2 + b^2$ links them to [[sums of two squares|04NPWJMA6H]].

## Explanations

### L1 human

Gaussian integers are the complex numbers whose real part and imaginary part are both integers, such as $3 - 2i$.
They behave much like the ordinary integers: you can add them, multiply them and factor them into primes.
</document>

<document source="https://tamnd.github.io/math2/node/0B2RZ0778N.md" tag="0B2RZ0778N" kind="concept">
# Level set

## Summary

A level set of a real-valued [[function of several variables|0JPCHTXWG0]] is a set of points of the domain. At these points, the function takes one given value.
For a function of two variables a level set is often a curve, and for a function of three variables it is often a surface.
</document>

<document source="https://tamnd.github.io/math2/node/0B346DDAM5.md" tag="0B346DDAM5" kind="definition">
# Möbius transformation

## Definition

Let $a, b, c, d$ be complex numbers with $ad - bc \neq 0$.
The Möbius transformation with these coefficients is the map $f : \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ of the [[Riemann sphere|0CDMXGPSES]] to itself given by
$$f(z) = \frac{az + b}{cz + d}$$
for each $z \in \mathbb{C}$ with $cz + d \neq 0$. The map also has the following values.
If $c \neq 0$, then $f(-d/c) = \infty$ and $f(\infty) = a/c$.
If $c = 0$, then $f(\infty) = \infty$.

## Remarks

Each Möbius transformation is a bijection of $\widehat{\mathbb{C}}$, and its inverse is the Möbius transformation $w \mapsto \frac{dw - b}{-cw + a}$.
The Möbius transformations form a [[group|G5S1M9]] under composition, the Möbius group.
The coefficients $(\lambda a, \lambda b, \lambda c, \lambda d)$ with $\lambda \neq 0$ give the same map.
If $ad - bc = 0$, the formula gives a constant or is undefined, so it is not a Möbius transformation.

## Explanations

### L1 human

A Möbius transformation is a quotient of two linear functions of $z$.
Examples are $z \mapsto z + 1$, $z \mapsto 2z$ and $z \mapsto \frac{1}{z}$, and each Möbius transformation is a composition of maps of these simple types.
The point $\infty$ is necessary, because the denominator can be zero.
</document>

<document source="https://tamnd.github.io/math2/node/0B3D4C8YF2.md" tag="0B3D4C8YF2" kind="concept">
# Conjunctive normal form

## Summary

A formula in conjunctive normal form is a conjunction of clauses, each a disjunction of literals. It is the standard input form for satisfiability solvers. The node [[conjunctive normal form|07C4A8Y18K]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0B3Q3XDM5X.md" tag="0B3Q3XDM5X" kind="definition">
# Quantile function

## Definition

Let $F \colon \mathbb{R} \to [0, 1]$ be the [[cumulative distribution function|0MXMS3AE13]] of a real random variable $X$.
The quantile function of $X$ is the function $Q \colon (0, 1) \to \mathbb{R}$ with
$$Q(p) = \inf \{x \in \mathbb{R} : F(x) \ge p\}.$$
Here $\inf$ is the [[infimum|04KFTA34C5]]. The number $Q(p)$ is the $p$-quantile of $X$.

## Remarks

The set in the definition is nonempty and bounded below for $0 < p < 1$, and the infimum is a minimum because $F$ is right-continuous. So $F(Q(p)) \ge p$, and $Q(p) \le x$ if and only if $p \le F(x)$.
If $F$ is continuous and strictly increasing, then $Q = F^{-1}$.
If $U$ is uniform on $(0, 1)$, then $Q(U)$ has the cumulative distribution function $F$. This is inverse transform sampling.
$Q(1/2)$ is a [[median|08W2CW8PKW]] of $X$, and $Q(1/4)$ and $Q(3/4)$ are the lower and upper quartiles.

## Explanations

### L1 human

The quantile function answers the question "below which value does a fraction $p$ of the probability lie?".
For the exponential distribution with rate $1$, $F(x) = 1 - e^{-x}$, so $Q(p) = -\ln(1 - p)$.
</document>

<document source="https://tamnd.github.io/math2/node/0B5BA4FRWS.md" tag="0B5BA4FRWS" kind="concept">
# Independent sigma-algebras

## Summary

A family of sub-σ-algebras of a probability space is independent when events chosen from different members of the family are always mutually independent.
It is the most general form of independence, and the independence of events and of random variables are special cases. The node [[independent sigma-algebras|09T8ZHST1V]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0B5H2V4JW8.md" tag="0B5H2V4JW8" kind="definition">
# Set bounded above

## Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$.
An element $M$ of $P$ is an upper bound of $S$ when $s \le M$ for each $s$ in $S$.
The set $S$ is bounded above when it has at least one upper bound in $P$.
Similarly, $m$ is a lower bound of $S$ when $m \le s$ for each $s$ in $S$.
The set $S$ is bounded below when it has a lower bound.

## Remarks

In $\mathbb{R}$, the interval $[0, 1)$ is bounded above. Each number $M \ge 1$ is an upper bound.
The set $\mathbb{N}$ is not bounded above in $\mathbb{R}$, by the [[Archimedean property|0A74SX55QP]].
An upper bound of $S$ does not need to be an element of $S$.

## Explanations

### L1 human

A set is bounded above when it does not go up forever: some number is a ceiling for all its elements.
The set $\{1, 1/2, 1/3, \dots\}$ is bounded above by $1$, and also by $2$ and by $100$.
</document>

<document source="https://tamnd.github.io/math2/node/0B5H4208Z9.md" tag="0B5H4208Z9" kind="definition">
# Eigenvector

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$.
An *eigenvector* of $T$ for $\lambda$ is a vector $v \in V$ with $v \ne 0$ and $T(v) = \lambda v$.
Then $\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $T$.

## Remarks

Each nonzero multiple of an eigenvector is again an eigenvector for the same eigenvalue.
For $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$, the vector $(1, 1)^T$ is an eigenvector for $3$, because $A (1, 1)^T = (3, 3)^T$.
The zero vector is never an eigenvector, by convention.

## Explanations

### L1 human

An eigenvector keeps its line under the map: the map only stretches it, shrinks it or flips it.
The stretch factor is the eigenvalue.
Consider a reflection of the plane in a line.
The vectors on the line are eigenvectors for $1$, and the vectors perpendicular to it are eigenvectors for $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/0B5WCC02AN.md" tag="0B5WCC02AN" kind="concept">
# Edge colouring

## Summary

An edge colouring assigns colours to the edges of a graph so that edges with a common end get different colours. It splits the edges into matchings and models scheduling problems. The node [[edge colouring|02CDYWSWSY]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0B5Z1EC66J.md" tag="0B5Z1EC66J" kind="statement">
# R[x] is an integral domain iff R is

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]].
Then the [[polynomial ring|0Q9EZS90GY]] $R[x]$ is an [[integral domain|09S3PFBEPM]] if and only if $R$ is an integral domain.

## Remarks

If $R$ is an integral domain and $f, g \neq 0$, then the leading coefficient of $f g$ is not zero, so $f g \neq 0$.
Conversely, $R$ is a subring of $R[x]$, and a subring of an integral domain is an integral domain.
By induction, the ring $R[x_1, \dots, x_n]$ of [[polynomials in several variables|0CES91M79A]] over an integral domain is an integral domain.

## Explanations

### L1 human

The product of two nonzero polynomials over the integers is never zero.
The reason is that the leading coefficients multiply to a nonzero number.
So the polynomial ring keeps the integral domain property of its coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/0B621RBVS9.md" tag="0B621RBVS9" kind="concept">
# Chromatic polynomial

## Summary

The chromatic polynomial of a finite graph counts its proper colourings as a function of the number of colours. It is a polynomial invariant of the graph that contains the chromatic number. The node [[chromatic polynomial|038YGY0HSC]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0B64Q7XNE3.md" tag="0B64Q7XNE3" kind="statement">
# Legendre's three-square theorem

## Statement

Let $n$ be a natural number.
Then $n = x^2 + y^2 + z^2$ for some integers $x$, $y$ and $z$ if and only if $n \neq 4^a (8b + 7)$ for all natural numbers $a$ and $b$.

## Remarks

The "only if" part is easy: a square is $0$, $1$ or $4$ modulo $8$, so no sum of three squares is $7$ modulo $8$.
The first numbers that are not sums of three squares are $7$, $15$, $23$, $28$ and $31$.
The theorem implies [[Lagrange's four-square theorem|0KYYSMEVHH]] in a few lines.

## History

Legendre gave the first proof in 1797 or 1798.
Gauss proved a more general result in 1801.

## Explanations

### L1 human

Most numbers are sums of three squares, for example $6 = 4 + 1 + 1$.
The exceptions are $7, 15, 23, \dots$, the numbers that leave remainder $7$ on division by $8$, and these numbers times powers of $4$.
</document>

<document source="https://tamnd.github.io/math2/node/0B68KNPGCG.md" tag="0B68KNPGCG" kind="concept">
# Lexicographic order

## Summary

The lexicographic order compares pairs, or words, first by the first entry, and then by the next entry when the first entries are equal.
</document>

<document source="https://tamnd.github.io/math2/node/0B6QC0C0G1.md" tag="0B6QC0C0G1" kind="definition">
# Ring isomorphism

## Definition

Let $R$ and $S$ be [[rings|0894J980RK]].
A ring isomorphism from $R$ to $S$ is a [[ring homomorphism|0GGE0Z0JQ1]] $f \colon R \to S$ that is bijective.
The rings $R$ and $S$ are isomorphic, written $R \cong S$, if there is a ring isomorphism from $R$ to $S$.

## Remarks

The inverse map of a ring isomorphism is again a ring homomorphism.
A ring isomorphism from $R$ to $R$ is an automorphism of $R$.
Complex conjugation is an automorphism of $\mathbb{C}$.

## Explanations

### L1 human

Two isomorphic rings are the same ring with other names for the elements.
The isomorphism is the dictionary between the names.
</document>

<document source="https://tamnd.github.io/math2/node/0B6QWX3086.md" tag="0B6QWX3086" kind="concept">
# Strict differentiability

## Summary

A function is strictly differentiable at a point when a single linear map approximates all its difference quotients near that point. Both points of each quotient can move.
This is stronger than the [[Fréchet derivative|06EH97V2A7]] condition, in which one of the two points stays fixed.
</document>

<document source="https://tamnd.github.io/math2/node/0B77KZCQ51.md" tag="0B77KZCQ51" kind="concept">
# Markov property

## Summary

A stochastic process has the Markov property when its future, given the present, is independent of its past.
The node [[Markov property|0HB8SXHPFA]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0B80PAQD3C.md" tag="0B80PAQD3C" kind="concept">
# Gamma distribution

## Summary

The gamma distribution is a two-parameter family of continuous distributions on $(0, \infty)$. It contains the exponential, Erlang and chi-squared distributions.
The node [[gamma distribution|06395GBNWD]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0B85Q41RP6.md" tag="0B85Q41RP6" kind="definition">
# Cyclic group

## Definition

A [[group|G5S1M9]] $G$ is cyclic when there is an element $g \in G$ with $G = \langle g \rangle$, where $\langle g \rangle$ is the [[cyclic subgroup generated by $g$|08ABT85339]].
Such an element $g$ is called a generator of $G$.

## Remarks

The group $(\mathbb{Z}, +)$ is cyclic, and its generators are $1$ and $-1$.
The integers modulo $n$ under addition form a cyclic group of order $n$, generated by $1$.
A cyclic group can have many generators: in the integers modulo $n$, the generators are the residues coprime to $n$.
The [[circle group|04EAFVFTF7]] is not cyclic, because it is uncountable.

## Explanations

### L1 human

A cyclic group is a group in which each element is a power of one fixed element.
The rotations of a regular polygon with $n$ sides form a cyclic group: each rotation is a power of the smallest rotation.
The word "cyclic" comes from this picture of going around a circle.
</document>

<document source="https://tamnd.github.io/math2/node/0B85S2V2DR.md" tag="0B85S2V2DR" kind="concept">
# Order of a group

## Summary

The order of a group is the number of its elements.
A group with infinitely many elements has infinite order.
</document>

<document source="https://tamnd.github.io/math2/node/0B8837S5T3.md" tag="0B8837S5T3" kind="concept">
# Essential singularity

## Summary

An essential singularity is an isolated singularity that is neither removable nor a pole, see the [[definition|0YKDJBV26S]].
Near an essential singularity, a function comes arbitrarily close to every complex value.
</document>

<document source="https://tamnd.github.io/math2/node/0B88PQTJ66.md" tag="0B88PQTJ66" kind="definition">
# Universal quantifier

## Definition

Let $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.
The universal quantification of $P$ over $X$ is the proposition $\forall x \in X, \, P(x)$, read "for all $x$ in $X$, $P(x)$".
It is true when $P(a)$ is true for each $a \in X$, and false otherwise.

## Remarks

When $X$ is empty, $\forall x \in X, \, P(x)$ is true for each $P$. This is a case of [[vacuous truth|0V4QXPGJQK]].
For a finite set $X = \{a_1, \dots, a_n\}$, the statement is equivalent to $P(a_1) \land \dots \land P(a_n)$.

## Explanations

### L1 human

"Every square of a real number is at least $0$" is $\forall x \in \mathbb{R}, \, x^2 \ge 0$.
To prove it, take an arbitrary $x$ and show $x^2 \ge 0$.
To refute it, one [[counterexample|0ED72CTBNP]] is enough.
</document>

<document source="https://tamnd.github.io/math2/node/0B89D93YA1.md" tag="0B89D93YA1" kind="definition">
# Arcsine

## Definition

The arcsine is the function $\arcsin \colon [-1, 1] \to [-\pi/2, \pi/2]$ such that, for each $x \in [-1, 1]$, $\arcsin x$ is the unique $y \in [-\pi/2, \pi/2]$ with $\sin y = x$.

## Remarks

The sine is strictly increasing and continuous on $[-\pi/2, \pi/2]$, with range $[-1, 1]$. So $\arcsin$ is well defined and [[continuous|07BY47HG5S]].
For example, $\arcsin 0 = 0$, $\arcsin 1 = \pi/2$, and $\arcsin \frac{1}{2} = \pi/6$.
The function is odd and strictly increasing. Mathlib defines `Real.arcsin` on all of $\mathbb{R}$, constant outside $[-1, 1]$.

## Explanations

### L1 human

The arcsine of $x$ is the angle between $-\pi/2$ and $\pi/2$ whose sine is $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0B8BW4YWTF.md" tag="0B8BW4YWTF" kind="concept">
# Lattice of subgroups

## Summary

The [[subgroups|0X1JXT47GA]] of a [[group|G5S1M9]] $G$, ordered by inclusion, form a complete lattice.
The meet of a family of subgroups is their intersection, and the join is the [[subgroup generated by|0RP7VWQ0YN]] their union.
The least element is the trivial subgroup $\{e\}$ and the greatest element is $G$.

## Remarks

For two subgroups $H$ and $K$, the meet is $H \cap K$ and the join is $\langle H \cup K \rangle$.
The lattice of subgroups of $(\mathbb{Z}, +)$ is the set of the subgroups $n\mathbb{Z}$, and $m\mathbb{Z} \subseteq n\mathbb{Z}$ exactly when $n$ divides $m$.
The lattice of normal subgroups of a group is a modular lattice.

## Explanations

### L1 human

You can draw the subgroups of a small group as a diagram, with a line from each subgroup up to each next bigger subgroup.
This diagram shows the lattice of subgroups.
Two subgroups always have a largest common subgroup below them and a smallest subgroup above them.
</document>

<document source="https://tamnd.github.io/math2/node/0B8GPXFSP7.md" tag="0B8GPXFSP7" kind="statement">
# Continuity via closed sets

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]] and let $f \colon X \to Y$ be a function.
Then $f$ is [[continuous|0SY9ECQ19Q]] if and only if for each [[closed set|0G5AV9Q4RR]] $F$ of $Y$, the [[preimage|0FGCFY1AZX]] $f^{-1}(F)$ is closed in $X$.

## Remarks

Proof: preimages commute with complements, that is, $f^{-1}(Y \setminus A) = X \setminus f^{-1}(A)$ for each $A \subseteq Y$. Then use that the [[complements of open sets are closed|0K67Q60Z8P]].
For example, for a continuous $f \colon \mathbb{R} \to \mathbb{R}$, the set of zeros $f^{-1}(\{0\})$ is closed.

## Explanations

### L1 human

Continuity can be tested with closed sets as well as with open sets.
This form is useful to show that a set given by an equation, such as $\{x : f(x) = 0\}$, is closed.
</document>

<document source="https://tamnd.github.io/math2/node/0B8KZRT28M.md" tag="0B8KZRT28M" kind="concept">
# Orthogonal projection

## Summary

The orthogonal projection onto a subspace $W$ sends each vector to the unique vector of $W$ whose difference with it is orthogonal to $W$.
</document>

<document source="https://tamnd.github.io/math2/node/0B8T10CE7N.md" tag="0B8T10CE7N" kind="concept">
# Linear isomorphism

## Summary

A linear isomorphism is a bijective linear map.
Its inverse is also linear, and two spaces with an isomorphism between them are isomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/0B906ZBTMZ.md" tag="0B906ZBTMZ" kind="example">
# Cofinite topology

## Example

Let $X$ be a set and let $\tau$ consist of $\varnothing$ and of all sets $U \subseteq X$ whose complement $X \setminus U$ is [[finite|0BBTH3YSZA]].
Then $\tau$ is a [[topology|0CNQXVBHCX]] on $X$, the cofinite topology.

## Remarks

The closed sets are $X$ and the finite subsets of $X$.
The cofinite topology is [[T1|05GQJXBP3F]]: each one-point set is closed.
If $X$ is infinite, any two nonempty open sets meet. Thus the space is not [[Hausdorff|0MXB4CKHFW]].
Each space with the cofinite topology is [[compact|0QPHW703PV]].

## Explanations

### L1 human

In the cofinite topology, a set is open when it misses only finitely many points.
On an infinite set, open sets are very large, so no two nonempty open sets are disjoint.
</document>

<document source="https://tamnd.github.io/math2/node/0B94TKFQJX.md" tag="0B94TKFQJX" kind="concept">
# Mathematical proof

## Summary

A mathematical proof is an argument that shows that a statement follows from axioms, definitions and statements that are already proved.
Each step of a proof follows from earlier steps by a valid rule of inference, such as [[modus ponens|0Y0MQYST0M]].
A proved statement is a theorem.

## Remarks

Common methods are [[direct proof|0NTF6DMVAP]], [[proof by contrapositive|0BB5TPGQ1N]], [[proof by contradiction|0J0XBKG64S]] and [[proof by cases|0WCP0PVET6]].
A formal proof is a sequence of formulas in a formal system. A computer can check it, for example in Lean.
Examples, however many, do not prove a universal statement.

## Explanations

### L1 human

A proof is a chain of small steps, and each step is certainly correct.
It starts from things that you accept or already know, and it ends at the statement.
After a correct proof, the statement is true in every case, not only in the cases that you tested.
</document>

<document source="https://tamnd.github.io/math2/node/0B96KMDDD1.md" tag="0B96KMDDD1" kind="statement">
# Cauchy condensation test

## Statement

Let $(a_n)_{n \ge 1}$ be a decreasing real sequence with $a_n \ge 0$ for all $n$.
Then $\sum_{n=1}^{\infty} a_n$ converges if and only if the condensed series $\sum_{k=0}^{\infty} 2^k a_{2^k}$ converges.

## Remarks

The proof groups the terms from $2^k$ to $2^{k+1} - 1$. Since the sequence decreases, $\sum_{n=1}^{\infty} a_n \le \sum_{k=0}^{\infty} 2^k a_{2^k} \le 2 \sum_{n=1}^{\infty} a_n$.
With $a_n = 1/n^p$ it gives the [[p-series test|0ZNSSK8ZK1]].
With $a_n = 1/(n \ln n)$ for $n \ge 2$, the condensed series is a multiple of $\sum 1/k$, so $\sum 1/(n \ln n)$ diverges.

## Explanations

### L1 human

For a decreasing sequence, the terms between $2^k$ and $2^{k+1}$ are all about the same size as $a_{2^k}$.
So only one term out of each such block matters, weighted by the size $2^k$ of the block.
</document>

<document source="https://tamnd.github.io/math2/node/0B9QNFGTHM.md" tag="0B9QNFGTHM" kind="example">
# Rotation matrix

## Example

Let $\theta \in \R$.
The *rotation matrix* of angle $\theta$ is
$$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}.$$
The map $x \mapsto R(\theta) x$ rotates the plane $\R^2$ about the origin by the angle $\theta$, counterclockwise.
It is an [[orthogonal matrix|0W02QESPJQ]] with determinant $\cos^2\theta + \sin^2\theta = 1$, so it is an element of $SO(2)$.

## Remarks

Rotations add their angles: $R(\alpha) R(\beta) = R(\alpha + \beta)$.
So $R(\theta)^{-1} = R(-\theta) = R(\theta)^T$.
For $\theta$ not a multiple of $\pi$, $R(\theta)$ has no real eigenvalue; its complex eigenvalues are $e^{\pm i\theta}$.
Every element of $SO(2)$ is $R(\theta)$ for some $\theta$.

## Explanations

### L1 human

The rotation matrix sends $(1, 0)$ to $(\cos\theta, \sin\theta)$ and $(0, 1)$ to $(-\sin\theta, \cos\theta)$.
These are the first and the second column.
For $\theta = 90$ degrees, it is $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.
</document>

<document source="https://tamnd.github.io/math2/node/0B9ZQFMQNV.md" tag="0B9ZQFMQNV" kind="statement">
# Connected subsets of the real line are intervals

## Statement

Let $S \subseteq \mathbb{R}$, where $\mathbb{R}$ has the usual topology.
Then $S$ is [[connected|0R37DFJKFW]] if and only if $S$ is an [[interval|0TK8Y3J5RW]], that is, for all $a, b \in S$ and all $c \in \mathbb{R}$ with $a < c < b$, $c \in S$.

## Remarks

If $S$ is not an interval, take $a < c < b$ with $a, b \in S$ and $c \notin S$. Then $(-\infty, c)$ and $(c, \infty)$ split $S$.
The proof that an interval $I$ is connected uses the [[completeness axiom|0AXWET80YW]]. If open sets $U$ and $V$ split $I$, with $a \in U$, $b \in V$ and $a < b$, let $c$ be the [[supremum|0BTMQ04556]] of the points of $U \cap [a, b]$. Then $c$ lies in neither $U$ nor $V$.
Under this definition the empty set and single points are intervals, and they are connected.
The statement fails for $\mathbb{Q}$: no interval of $\mathbb{Q}$ with more than one point is connected.

## Explanations

### L1 human

On the real line, the sets in one piece are exactly the intervals.
The proof needs the completeness of $\mathbb{R}$, because $\mathbb{Q}$ has gaps where it can be split.
</document>

<document source="https://tamnd.github.io/math2/node/0BANA3XYPF.md" tag="0BANA3XYPF" kind="concept">
# Bridge

## Summary

A bridge of a graph is an edge whose deletion disconnects a component. Bridges are exactly the edges that lie on no cycle. The node [[bridge|0D81KWH9ZM]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0BARQY52N5.md" tag="0BARQY52N5" kind="statement">
# Mean value theorem

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$.
Then there is $c \in (a, b)$ with
$$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

## Remarks

Apply [[Rolle's theorem|0194DY35BG]] to $g(x) = f(x) - \frac{f(b) - f(a)}{b - a} (x - a)$, which has $g(a) = g(b)$.
It turns bounds on $f'$ into bounds on $f$. It gives [[zero derivative implies constant|003QDZ4T16]] and the [[monotonicity tests|07J5X2T1ZX]].
Mathlib states it as `exists_deriv_eq_slope`.

## History

Augustin-Louis Cauchy stated and proved the theorem in its modern form in 1823.

## Explanations

### L1 human

Somewhere between $a$ and $b$, the tangent line is parallel to the chord from $(a, f(a))$ to $(b, f(b))$.
If a car travels 100 km in one hour, at some moment its speed is exactly 100 km per hour.
</document>

<document source="https://tamnd.github.io/math2/node/0BB5TPGQ1N.md" tag="0BB5TPGQ1N" kind="concept">
# Proof by contrapositive

## Summary

A proof by contrapositive proves an [[implication|0ZNX6DQ3EJ]] $p \to q$ by a [[direct proof|0NTF6DMVAP]] of $\neg q \to \neg p$.
The method is correct because $p \to q$ is logically equivalent to its [[contrapositive|04T1CT7212]].

## Remarks

Example: for an integer $n$, if $n^2$ is even, then $n$ is even. The contrapositive is "if $n$ is odd, then $n^2$ is odd", which has a short direct proof.
A [[proof by contradiction|0J0XBKG64S]] is different: it assumes $p$ and $\neg q$ together, and it gets any contradiction.

## Explanations

### L1 human

Sometimes the negation of the conclusion gives you more to work with.
"If $n^2$ is even, then $n$ is even" is hard to start.
"If $n$ is odd, then $n^2$ is odd" is easy, and it says the same thing.
</document>

<document source="https://tamnd.github.io/math2/node/0BBTH3YSZA.md" tag="0BBTH3YSZA" kind="definition">
# Finite set

## Definition

A [[set|0SJ3RX9TXA]] $A$ is finite when there is a natural number $n$ such that $A$ is [[equinumerous|08EJHHMB5R]] with $\{1, 2, \dots, n\}$.
For $n = 0$, this set is the [[empty set|0ECVK59Y03]].
The number $n$ is unique, and it is the number of elements of $A$.

## Remarks

Subsets, finite unions, finite products and the power set of finite sets are finite.
A finite set is not equinumerous with any proper subset of itself. This is the pigeonhole principle.
A set that is not finite is [[infinite|01MKW4R283]].

## Explanations

### L1 human

A set is finite when you can count its elements and stop.
The set of planets of the Solar System is finite, and so is the set of primes less than $100$.
</document>

<document source="https://tamnd.github.io/math2/node/0BC3Q6K92H.md" tag="0BC3Q6K92H" kind="statement">
# Matrix determinant lemma

## Statement

Let $K$ be a commutative ring, let $A$ be an invertible $n \times n$ matrix over $K$, and let $u, v \in K^n$ be column vectors.
Then
$$\det(A + u v^T) = (1 + v^T A^{-1} u) \det A,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The matrix $u v^T$ has rank at most $1$, so the lemma describes the effect of a rank one update on the determinant.
It follows from [[Sylvester's determinant theorem|0G24VQ1088]] with the matrices $A^{-1} u$ and $v^T$.
A variant without inverses is $\det(A + u v^T) = \det A + v^T \operatorname{adj}(A) u$.

## Explanations

### L1 human

Adding a matrix of rank one, $u v^T$, changes the determinant by a simple factor.
The factor is $1 + v^T A^{-1} u$, a single number.
So we can update a determinant cheaply when we know the inverse.
</document>

<document source="https://tamnd.github.io/math2/node/0BC7KX51Z8.md" tag="0BC7KX51Z8" kind="definition">
# Power set

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]].
The power set of $A$ is the set
$$\mathcal{P}(A) = \{S \mid S \subseteq A\}$$
of all [[subsets|0FE8Q5S42Y]] of $A$.

## Remarks

Other notations are $2^A$ and $\wp(A)$.
Example: $\mathcal{P}(\{1, 2\}) = \{\varnothing, \{1\}, \{2\}, \{1, 2\}\}$.
A finite set with $n$ elements has a power set with $2^n$ elements. See [[cardinality of the power set|0GPFKS96T7]].
By [[Cantor's theorem|0WTEMGYV2C]], no set is in bijection with its power set.

## Explanations

### L1 human

To list the power set, choose for each element whether it is in or out of the subset.
Each choice gives one subset, so a set with $3$ elements has $2^3 = 8$ subsets.
The empty set and the set itself are always among them.
</document>

<document source="https://tamnd.github.io/math2/node/0BCHJ30REG.md" tag="0BCHJ30REG" kind="definition">
# Cycle type

## Definition

Let $\sigma \in S_n$.
Write $\sigma$ as a product of disjoint cycles, and include a $1$-cycle for each fixed point.
The cycle type of $\sigma$ is the multiset of the lengths of these cycles.

## Remarks

The [[disjoint cycle decomposition|0YWDAZPRPQ]] is unique up to order, so the cycle type is well defined.
The lengths add up to $n$, so the cycle type is a partition of $n$.
For example, $(1\ 2)(3\ 4\ 5) \in S_6$ has cycle type $(3, 2, 1)$.

## Explanations

### L1 human

The cycle type forgets which points are in which cycle and keeps only the sizes of the cycles.
Two permutations with the same cycle type have the same shape.
In the symmetric group, the shape decides exactly when two permutations are conjugate.
</document>

<document source="https://tamnd.github.io/math2/node/0BCMGDHZGR.md" tag="0BCMGDHZGR" kind="definition">
# Pivot

## Definition

Let $F$ be a field and let $A$ be a matrix over $F$ in [[row echelon form|0VYCDGMJJJ]].
A *pivot* of $A$ is the leading entry of a nonzero row of $A$, that is, the first nonzero entry of that row.
A *pivot column* of $A$ is a column that contains a pivot.
For a general matrix $B$, take a row echelon form of $B$ that is row equivalent to $B$.
The pivot positions of $B$ are the positions of the pivots of this form.

## Remarks

The pivot positions do not depend on the choice of the row echelon form.
The number of pivots is the [[rank|0ZC9YTXZRY]] of the matrix.
In the system $Ax = b$, the unknowns of the pivot columns are the *basic variables*, and the other unknowns are the *free variables*.
In numerical work, partial pivoting chooses as pivot an entry of largest absolute value in its column, to reduce rounding errors.

## Explanations

### L1 human

A pivot is the first nonzero number in a row of a staircase matrix.
During elimination we divide by the pivot and use it to clear the entries below it.
Each pivot fixes one unknown, and each column without a pivot gives a free unknown.
</document>

<document source="https://tamnd.github.io/math2/node/0BDJBH8B4Z.md" tag="0BDJBH8B4Z" kind="statement">
# Order of the Galois group of a Galois extension equals its degree

## Statement

Let $L/K$ be a finite [[Galois extension|0DS2F7GK05]].
Then
$$|\operatorname{Gal}(L/K)| = [L : K].$$

## Remarks

Conversely, a finite extension with $|\operatorname{Aut}(L/K)| = [L : K]$ is Galois. So the [[general bound|0QM98DNWVW]] is an equality exactly for Galois extensions.
For $\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}$, the Galois group has order $4$: an automorphism can change the signs of $\sqrt{2}$ and of $\sqrt{3}$ independently.
For $\mathbb{F}_{p^n}/\mathbb{F}_p$, the Galois group is cyclic of order $n$, generated by the [[Frobenius automorphism|0VP4TBVJ19]].

## Explanations

### L1 human

For a Galois extension, counting symmetries is the same as computing the degree.
This links the size of a group to the size of a field extension.
</document>

<document source="https://tamnd.github.io/math2/node/0BDTTM24DP.md" tag="0BDTTM24DP" kind="definition">
# Banach space

## Definition

Let $V$ be a vector space over $\mathbb{R}$ or $\mathbb{C}$ with a [[norm|0J3578562S]] $\|\cdot\|$, and let $d(x, y) = \|x - y\|$.
The space $(V, \|\cdot\|)$ is a Banach space when the metric space $(V, d)$ is [[complete|08EWT7J0SC]].

## Remarks

Examples are $\mathbb{R}^n$ with any norm, the spaces [[ell p|08P6T2WH6R]] for $1 \le p \le \infty$, and $C_b(X)$ with the [[sup norm|0PARCKE8WP]].
The space of polynomial functions on $[0, 1]$ with the sup norm is not a Banach space. Its completion is $C([0, 1])$.
In a normed space $V$, completeness is the same as: [[each absolutely convergent series converges|082MGNWNA8]].
Each [[finite-dimensional normed space is complete|0ABP2F2ZP7]].

## Explanations

### L1 human

A Banach space is a vector space where you can measure lengths, and where Cauchy sequences always converge.
So you can take limits in it freely, as in $\mathbb{R}^n$.
</document>

<document source="https://tamnd.github.io/math2/node/0BDX4MBKT4.md" tag="0BDX4MBKT4" kind="definition">
# Monic polynomial

## Definition

Let $R$ be a [[ring|0894J980RK]].
A polynomial $f \in R[x]$ is monic if its [[leading coefficient|0P05RJC5XA]] is $1$.

## Remarks

The polynomial $x^2 - 2$ is monic, and $2x^2 - 1$ is not.
In a nonzero ring, a monic polynomial is not zero.
The product of two monic polynomials is monic, over every commutative ring.
Over a field, each nonzero polynomial is a constant times exactly one monic polynomial.

## Explanations

### L1 human

A monic polynomial starts with $x^d$, with no number in front.
Over a field, you can make any nonzero polynomial monic by dividing by its leading coefficient.
</document>

<document source="https://tamnd.github.io/math2/node/0BE3QDH73A.md" tag="0BE3QDH73A" kind="definition">
# Smooth function

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$.
The function $f$ is smooth, or of class $C^{\infty}$, if it is of [[class $C^k$|06SKCHDQT5]] for each natural number $k$.

## Remarks

So a smooth function has derivatives of all orders.
Polynomials, the exponential function, and the sine are smooth on $\mathbb{R}$.
A smooth function need not be [[analytic|0JD5ZNW0F4]]. See the [[smooth non-analytic function|09RTYYHXKA]].
Mathlib uses `ContDiff ℝ ⊤ f`.

## Explanations

### L1 human

A smooth function can be differentiated as many times as we like.
All its derivatives exist and are continuous.
</document>

<document source="https://tamnd.github.io/math2/node/0BEZX8F2XW.md" tag="0BEZX8F2XW" kind="statement">
# Solutions for complex conjugate roots

## Statement

Let $a_0, \dots, a_{n-1}$ be real numbers, and let $p$ be the [[characteristic polynomial|0QDXMG46QW]] of the equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = 0$.
Let $\alpha + i \beta$ be a root of $p$ of multiplicity $m$, with $\alpha, \beta \in \mathbb{R}$ and $\beta \ne 0$.
Then $\alpha - i \beta$ is also a root of multiplicity $m$, and the $2m$ functions
$$x^k e^{\alpha x} \cos(\beta x), \quad x^k e^{\alpha x} \sin(\beta x), \qquad k = 0, \dots, m - 1,$$
are linearly independent real solutions of the equation on $\mathbb{R}$.

## Remarks

By [[Euler's formula|04R3CY41VJ]], $e^{(\alpha + i \beta) x} = e^{\alpha x} (\cos \beta x + i \sin \beta x)$. Its real and imaginary parts are solutions because the coefficients are real.
For order $2$ and roots $\alpha \pm i \beta$, the real solutions are $e^{\alpha x} (c_1 \cos \beta x + c_2 \sin \beta x)$.
Example: $y'' + 2 y' + 5 y = 0$ has the roots $-1 \pm 2 i$, so its solutions are $e^{-x} (c_1 \cos 2x + c_2 \sin 2x)$.
These families for all roots, together with those of [[real roots|0HRVY67CNV]], give a real fundamental system.

## Explanations

### L1 human

Complex roots come in conjugate pairs, because the polynomial has real coefficients.
Each pair gives an oscillation: a sine and a cosine with frequency $\beta$.
The real part $\alpha$ makes the oscillation grow if $\alpha > 0$ and decay if $\alpha < 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0BF2DE2SYK.md" tag="0BF2DE2SYK" kind="concept">
# Tensor product of vector spaces

## Summary

The tensor product of two vector spaces is a vector space spanned by symbols $v \otimes w$.
These symbols are bilinear in $v$ and $w$, and no other relations hold.
</document>

<document source="https://tamnd.github.io/math2/node/0BFD3A6CRH.md" tag="0BFD3A6CRH" kind="statement">
# Global existence for globally Lipschitz vector fields

## Statement

Let $J$ be an interval, let $K \ge 0$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ be continuous.
Assume that $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ for all $t \in J$ and $x, y \in \mathbb{R}^n$.
Then for each $t_0 \in J$ and $x_0 \in \mathbb{R}^n$, the [[initial value problem|075VBY4K4X]] $x' = f(t, x)$, $x(t_0) = x_0$ has exactly one solution defined on all of $J$.

## Remarks

The point is that the solution does not blow up: the Lipschitz bound gives $\|f(t, x)\| \le \|f(t, 0)\| + K \|x\|$, so $\|x(t)\|$ grows at most exponentially.
A linear system $x' = A(t) x + b(t)$ with continuous $A$ and $b$ on $J$ satisfies the hypothesis on each compact subinterval. So its solutions exist on all of $J$.
A field that is only locally Lipschitz can give [[blow-up in finite time|0E4GXX7SJP]].

## Explanations

### L1 human

If the field grows at most linearly in the state, solutions cannot run away to infinity in finite time.
So the solution exists as long as the field is defined.
</document>

<document source="https://tamnd.github.io/math2/node/0BFDHF2H7N.md" tag="0BFDHF2H7N" kind="statement">
# Weak law of large numbers

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with $E[|X_1|] < \infty$ and $\mu = E[X_1]$. Let $\bar{X}_n = (X_1 + \dots + X_n)/n$.
Then $\bar{X}_n \to \mu$ [[in probability|0EFDDXTVD6]]: for each $\varepsilon > 0$,
$$\lim_{n \to \infty} P\left(\left|\bar{X}_n - \mu\right| > \varepsilon\right) = 0.$$

## Remarks

If $\sigma^2 = \operatorname{Var}(X_1) < \infty$, there is a short proof. By the [[variance of a sum|0V0778DCW9]], $\operatorname{Var}(\bar{X}_n) = \sigma^2/n$.
Then [[Chebyshev's inequality|02HZ1BK3MW]] gives $P(|\bar{X}_n - \mu| > \varepsilon) \le \sigma^2 / (n \varepsilon^2) \to 0$.
The general case uses truncation, or characteristic functions: $\varphi_{\bar{X}_n}(t) = \varphi_{X_1}(t/n)^n \to e^{i\mu t}$.
The finite-variance proof needs only that the $X_i$ are pairwise uncorrelated with the same mean and bounded variances.

## History

Jacob Bernoulli proved the first version, for Bernoulli trials. It appeared after his death in Ars Conjectandi in 1713.

## Explanations

### L1 human

The weak law says that for a large number $n$ of trials, the sample mean is very likely to be close to the true mean.
For example, in $10000$ tosses of a fair coin, the fraction of heads is very likely to be between $0.49$ and $0.51$.
</document>

<document source="https://tamnd.github.io/math2/node/0BFMR2FQ8B.md" tag="0BFMR2FQ8B" kind="statement">
# Number of k-element subsets

## Statement

Let $X$ be a finite set with $|X| = n$ and let $k$ be a natural number.
Then the number of subsets of $X$ with exactly $k$ elements is the [[binomial coefficient|092GEF38YD]] $\binom{n}{k}$.

## Remarks

A proof by induction on $n$ fixes an element $x \in X$. A $k$-subset either omits $x$ or contains $x$, and this gives [[Pascal's rule|0YRG241HQQ]].
For $k > n$ both sides are $0$.

## Explanations

### L1 human

To pick a committee of $k$ people from $n$, first count the ordered lists of $k$ people. Then divide by the $k!$ orders of each committee.
This gives $\frac{n (n-1) \cdots (n-k+1)}{k!}$, which is $\binom{n}{k}$.
</document>

<document source="https://tamnd.github.io/math2/node/0BFN21TF5F.md" tag="0BFN21TF5F" kind="statement">
# Path independence of line integrals

## Statement

Let $U \subseteq \mathbb{R}^n$ be open and connected, and let $F \colon U \to \mathbb{R}^n$ be a continuous vector field.
The following are equivalent.

1. $F$ is [[conservative|05R9NF2J49]] on $U$.
2. For all piecewise $C^1$ curves $\gamma_1, \gamma_2$ in $U$ with the same initial point and the same end point, $\int_{\gamma_1} F \cdot d\mathbf{r} = \int_{\gamma_2} F \cdot d\mathbf{r}$.
3. For each closed piecewise $C^1$ curve $\gamma$ in $U$, $\int_\gamma F \cdot d\mathbf{r} = 0$.

## Remarks

Statement 1 implies statement 2 by the [[gradient theorem|0SD237FEJD]].
Statements 2 and 3 are equivalent: a closed curve splits into two curves with the same end points, and one uses [[reversal|0FSCXYG856]] and [[additivity|00ZEBSMNNV]].
Statement 2 implies statement 1: fix $p \in U$ and let $\phi(x) = \int_\gamma F \cdot d\mathbf{r}$ for a path $\gamma$ from $p$ to $x$. A short straight last segment in direction $e_i$ shows that $\partial \phi / \partial x_i = F_i$.
Connectedness is used to join $p$ to each point. On a disconnected set, apply the result on each component.

## Explanations

### L1 human

For a conservative field, the work from one point to another does not depend on the route.
Conversely, if the work never depends on the route, then the work from a fixed base point defines a potential for the field.
</document>

<document source="https://tamnd.github.io/math2/node/0BH9JH7HBY.md" tag="0BH9JH7HBY" kind="concept">
# Infinite limit

## Summary

A function has an infinite limit at a point when its values become arbitrarily large, positive or negative, near the point. Infinite limits describe vertical asymptotes. The node [[infinite limit|08NM8KQPDQ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0BJ6H85DRQ.md" tag="0BJ6H85DRQ" kind="concept">
# Limit along different paths

## Summary

Let $f$ be a function on $D \subseteq \mathbb{R}^n$ and let $a$ be a limit point of $D$.
Let $\gamma \colon (0, 1] \to D$ be a path, for example a [[curve|0G7WM8DX4P]], with $\gamma(t) \ne a$ for each $t$ and $\gamma(t) \to a$ as $t \to 0^+$.
If $\lim_{x \to a} f(x) = L$, then $\lim_{t \to 0^+} f(\gamma(t)) = L$.
So if two such paths give different limits, or one path gives no limit, then the [[limit|0J8Y1NCY0K]] of $f$ at $a$ does not exist.

## Remarks

For $f(x, y) = xy / (x^2 + y^2)$ on $\mathbb{R}^2 \setminus \{(0, 0)\}$, the path $y = m x$ gives the limit $m / (1 + m^2)$ at the origin. This depends on $m$, so $f$ has no limit at $(0, 0)$.
Equal limits along all straight lines do not prove that the limit exists. For $f(x, y) = x^2 y / (x^4 + y^2)$, each line through the origin gives the limit $0$, but the parabola $y = x^2$ gives $1/2$.

## Explanations

### L1 human

In one variable you can approach a point only from the left or from the right.
In the plane you can approach a point along infinitely many routes: lines, parabolas, spirals.
A limit must give the same answer for every route, so one route that disagrees is enough to show that there is no limit.
</document>

<document source="https://tamnd.github.io/math2/node/0BJBYCGMWE.md" tag="0BJBYCGMWE" kind="statement">
# A linear map is injective iff its kernel is trivial

## Statement

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
Then $T$ is injective if and only if $\ker T = \{0\}$, where $\ker T$ is the [[kernel|0S23E6YE83]] of $T$.

## Remarks

The key step is $T(u) = T(v) \iff T(u - v) = 0 \iff u - v \in \ker T$.
Thus each nonempty fibre $T^{-1}(w)$ is a coset $v + \ker T$.
For a matrix $A$, the map $x \mapsto Ax$ is injective if and only if $Ax = 0$ has only the trivial solution.

## Explanations

### L1 human

To test if a linear map is one to one, we only need to look at the vectors that go to zero.
If two vectors have the same image, then their difference goes to zero.
So if only $0$ goes to $0$, then no two vectors share an image.
</document>

<document source="https://tamnd.github.io/math2/node/0BJDQ2HBH1.md" tag="0BJDQ2HBH1" kind="definition">
# Tangent plane and normal vector of a surface

## Definition

Let $r \colon D \to \mathbb{R}^3$ be a [[parametrised surface|05788J51WZ]] that is regular at the point $(u_0, v_0)$ of $D$, and let $p = r(u_0, v_0)$.
The tangent plane at $p$ is the plane $\{ p + s \, r_u(u_0, v_0) + t \, r_v(u_0, v_0) : s, t \in \mathbb{R} \}$.
The vector $N(u_0, v_0) = r_u(u_0, v_0) \times r_v(u_0, v_0)$ is the normal vector of $r$ at $(u_0, v_0)$, and $n = N / \|N\|$ is the unit normal vector.

## Remarks

The vector $N$ is perpendicular to $r_u$ and $r_v$, so it is perpendicular to the tangent plane. The other unit normal at $p$ is $-n$.
For the graph $z = f(x, y)$, the normal is $(-f_x, -f_y, 1)$, which points upward.
If the surface is the level set $g(x, y, z) = c$ of a $C^1$ function $g$ with $\nabla g(p) \ne 0$, then the [[gradient|0NS2QYVC4Y]] $\nabla g(p)$ is a normal vector at $p$.
For the unit sphere with $r(\varphi, \theta) = (\sin \varphi \cos \theta, \sin \varphi \sin \theta, \cos \varphi)$, one gets $r_\varphi \times r_\theta = \sin \varphi \, r(\varphi, \theta)$, which points outward for $0 < \varphi < \pi$.

## Explanations

### L1 human

The tangent plane is the flat plane that touches the surface at a point, and the normal vector sticks straight out of it.
The cross product of the two coordinate tangent vectors gives a normal at once.
</document>

<document source="https://tamnd.github.io/math2/node/0BJKVF90Z6.md" tag="0BJKVF90Z6" kind="definition">
# Spanning set

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A subset $S \subseteq V$ is a *spanning set* of $V$, or *spans* $V$, when $\operatorname{span}(S) = V$, where $\operatorname{span}$ is the [[linear span|0DTY8B9HMS]].
Equivalently, each vector of $V$ is a linear combination of finitely many vectors of $S$.

## Remarks

The vectors $(1, 0), (0, 1), (1, 1)$ span $\R^2$, and so do $(1, 0), (0, 1)$.
A linearly independent spanning set is a [[basis|0KCQ9Y3QKA]].
[[Every spanning set contains a basis|06MRNP4BNT]].
A vector space with a finite spanning set is [[finite-dimensional|0ZHWKQMVX5]].

## Explanations

### L1 human

A spanning set is a set of building blocks from which we can make every vector of the space.
It may contain more vectors than we need.
Removing the extra vectors leaves a basis.
</document>

<document source="https://tamnd.github.io/math2/node/0BJR9QR08E.md" tag="0BJR9QR08E" kind="concept">
# Subset

## Summary

A set $A$ is a subset of a set $B$ when each element of $A$ is an element of $B$.
</document>

<document source="https://tamnd.github.io/math2/node/0BJSWZJG18.md" tag="0BJSWZJG18" kind="statement">
# Mertens' theorem on Cauchy products

## Statement

Let $\sum a_n$ be a real series that [[converges absolutely|0W3K3QPEWT]], with sum $A$.
Let $\sum b_n$ be a [[convergent|0CVVKY7GY2]] real series, with sum $B$.
Then their [[Cauchy product|0B1K9PVWE9]] converges, and its sum is $A B$.

## Remarks

Convergence of both series is not enough. Take $a_n = b_n = (-1)^n / \sqrt{n + 1}$. Both series converge, but $|c_n| \ge 1$ for all $n$, so the Cauchy product diverges.
Franz Mertens proved the theorem.

## Explanations

### L1 human

One series may converge only conditionally, if the other converges absolutely.
Then the product of the two series is still the sum of their Cauchy product.
</document>

<document source="https://tamnd.github.io/math2/node/0BJV24C4SY.md" tag="0BJV24C4SY" kind="definition">
# Complex conjugate

## Definition

Let $z = a + b i$ be a [[complex number|09FTAET5Z4]], with $a$ and $b$ real.
The complex conjugate of $z$ is the complex number $\bar{z} = a - b i$.

## Remarks

Another common notation is $z^*$.
Conjugation respects the operations: $\overline{z + w} = \bar{z} + \bar{w}$ and $\overline{z w} = \bar{z} \, \bar{w}$. Also $\bar{\bar{z}} = z$.
A complex number $z$ is real exactly when $\bar{z} = z$.
The product $z \bar{z}$ is the real number $|z|^2$, see [[a complex number times its conjugate|0ZB7J071WQ]].

## Explanations

### L1 human

To conjugate a complex number, change the sign of its imaginary part.
For example, the conjugate of $2 + 5 i$ is $2 - 5 i$, and the conjugate of $7$ is $7$.
</document>

<document source="https://tamnd.github.io/math2/node/0BJVDCF38Y.md" tag="0BJVDCF38Y" kind="concept">
# Square root of a positive semidefinite matrix

## Summary

The square root of a positive semidefinite matrix $A$ is the unique positive semidefinite matrix $B$ with $B^2 = A$.
</document>

<document source="https://tamnd.github.io/math2/node/0BK6JBK58W.md" tag="0BK6JBK58W" kind="definition">
# Continued fraction

## Definition

A continued fraction is given by a number $b_0$ and a finite or infinite sequence of pairs $(a_1, b_1), (a_2, b_2), \dots$ of numbers.
It stands for the expression
$$b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}}}$$
The $a_n$ are the partial numerators and the $b_n$ are the partial denominators.
The $n$-th convergent is the value of the expression cut off after $(a_n, b_n)$. The value of an infinite continued fraction is the limit of its convergents, when this limit exists.

## Remarks

When $a_n = 1$ for all $n$, $b_0$ is an integer and $b_n$ is a positive integer for $n \ge 1$, the continued fraction is a [[simple continued fraction|06XF7BCFFJ]].
For example, $\sqrt{2} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cdots}}$.
Other continued fractions give formulas for $\pi$ and for functions such as $\tan x$.

## Explanations

### L1 human

A continued fraction is a fraction whose denominator contains another fraction, and so on.
Cut it off at some level, and you get an ordinary fraction. These fractions often approximate a number very well.
</document>

<document source="https://tamnd.github.io/math2/node/0BKDAGMGX9.md" tag="0BKDAGMGX9" kind="statement">
# Hölder's inequality for sums

## Statement

Let $p, q > 1$ be real numbers with $\frac{1}{p} + \frac{1}{q} = 1$, and let $a_1, \dots, a_n, b_1, \dots, b_n$ be real numbers.
Then
$$\sum_{i=1}^{n} |a_i b_i| \le \Big( \sum_{i=1}^{n} |a_i|^p \Big)^{1/p} \Big( \sum_{i=1}^{n} |b_i|^q \Big)^{1/q}.$$

## Remarks

Scale so that both factors on the right are $1$. Then apply [[Young's inequality|07VHQ31EYJ]] to each product $|a_i| |b_i|$ and add.
For $p = q = 2$ it is the Cauchy–Schwarz inequality.
Mathlib states a form as `Real.inner_le_Lp_mul_Lq`.

## Explanations

### L1 human

Hölder's inequality bounds a sum of products by the sizes of the two sequences.
Each size is measured with its own exponent, $p$ for one and $q$ for the other.
</document>

<document source="https://tamnd.github.io/math2/node/0BKJT6WG96.md" tag="0BKJT6WG96" kind="definition">
# Number of divisors

## Definition

For a positive integer $n$, the number of divisors of $n$ is $d(n) = \sigma_0(n) = \# D(n)$.
Here $\sigma_0$ is the [[divisor function|0YEW47VJ0J]] with $k = 0$, and $D(n)$ is the [[set of divisors|0FCWA16FZE]] of $n$.

## Remarks

Other common notations are $\tau(n)$ and $\sigma_0(n)$.
If $n = p_1^{e_1} \cdots p_r^{e_r}$ with distinct primes $p_i$, then $d(n) = (e_1 + 1) \cdots (e_r + 1)$.
For example, $12 = 2^2 \cdot 3$, so $d(12) = 3 \cdot 2 = 6$.
The number $d(n)$ is odd if and only if $n$ is a perfect square.

## Explanations

### L1 human

The function $d(n)$ counts the divisors of $n$.
A divisor of $p^e q^f$ is $p^a q^b$ with $0 \le a \le e$ and $0 \le b \le f$. So there are $(e + 1)(f + 1)$ choices.
</document>

<document source="https://tamnd.github.io/math2/node/0BMVBHTWV2.md" tag="0BMVBHTWV2" kind="definition">
# Coordinate vector

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \dots, b_n)$ be a [[basis|0KCQ9Y3QKA]] of $V$.
Each $v \in V$ is equal to $a_1 b_1 + \cdots + a_n b_n$ for exactly one $(a_1, \dots, a_n) \in F^n$.
The *coordinate vector* of $v$ in the basis $B$ is
$$[v]_B = (a_1, \dots, a_n)^T \in F^n.$$
The scalars $a_i$ are the *coordinates* of $v$ in $B$.

## Remarks

The coordinates depend on the order of the basis vectors, so $B$ is an ordered basis.
The map $v \mapsto [v]_B$ is a linear isomorphism $V \to F^n$, see [[isomorphism with the coordinate space|059THXBRBB]].
In $\R^2$ with $B = ((1, 1), (1, -1))$, the vector $(3, 1)$ has coordinate vector $(2, 1)^T$.
A [[change of basis matrix|0MKRHZB3WS]] converts coordinates in one basis into coordinates in another basis.

## Explanations

### L1 human

The coordinates of a vector tell us how much of each basis vector we need to build it.
Different bases give different coordinates for the same vector.
After we fix a basis, each vector is just a list of numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0BMXWXDVJZ.md" tag="0BMXWXDVJZ" kind="statement">
# Fermat's little theorem in the form a^p = a

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]] and let $a$ be an integer.
Then $a^p \equiv a \pmod p$.

## Remarks

If $p \nmid a$, multiply [[Fermat's little theorem|0M25PRN3WQ]] $a^{p-1} \equiv 1 \pmod p$ by $a$. If $p \mid a$, both sides are $0$ modulo $p$.
Conversely, this form implies the form $a^{p-1} \equiv 1$: cancel $a$, which is allowed when $p \nmid a$.
In the ring $\mathbb{Z}/p\mathbb{Z}$, the form says that $x^p = x$ for each element $x$.

## Explanations

### L1 human

The form $a^p \equiv a$ needs no hypothesis on $a$.
For example, with $p = 3$: $0^3 = 0$, $1^3 = 1$, $2^3 = 8 \equiv 2$, and $4^3 = 64 \equiv 1 \equiv 4 \pmod 3$.
</document>

<document source="https://tamnd.github.io/math2/node/0BMY28TBSF.md" tag="0BMY28TBSF" kind="concept">
# Uniformly Cauchy sequence of functions

## Summary

A sequence of functions is uniformly Cauchy when its terms become uniformly close to each other. For real-valued functions, this is the same as uniform convergence. The node [[uniformly Cauchy sequence of functions|01XAY17QA3]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0BMYPJBJH4.md" tag="0BMYPJBJH4" kind="definition">
# Line

## Definition

Let $n \geq 2$ be an integer. A line in the [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$ is a set of the form
$$L = \{\, p + t v \;:\; t \in \mathbb{R} \,\},$$
where $p$ is a point of $\mathbb{R}^n$. Here $v$ is a nonzero [[vector|0QN3D80AWP]] of $\mathbb{R}^n$. The vector $v$ is a direction vector of $L$.

## Remarks

For two distinct points $A$ and $B$, exactly one line contains both: the line through $A$ with the direction vector $\overrightarrow{AB}$.
The point $p$ and the vector $v$ are not unique. The same line has each point of $L$ as a base point, and each nonzero multiple of $v$ as a direction vector.
In the plane $\mathbb{R}^2$, the lines are also the sets $\{(x, y) : a x + b y = c\}$ with real numbers $a$, $b$, $c$ and $(a, b) \neq (0, 0)$.
In the language of affine geometry, a line is an [[affine subspace|0PYW26VNT6]] of dimension 1.

## Explanations

### L1 human

Start at a point $p$ and walk forward or backward in the direction $v$, for any amount of time $t$. The points you can reach form the line. For example, with $p = (0, 1)$ and $v = (1, 2)$ you get the line $y = 2x + 1$ in the plane.
</document>

<document source="https://tamnd.github.io/math2/node/0BN2A6EBYA.md" tag="0BN2A6EBYA" kind="statement">
# Trees are bipartite

## Statement

Let $T$ be a [[tree|0MJ0JS72DZ]].
Then $T$ is [[bipartite|07DVHJ8N0A]].

## Remarks

Proof: fix a vertex $r$. Put a vertex in $A$ if its [[distance|0WPZ5YPAGR]] to $r$ is even, and in $B$ if it is odd. The ends of an edge have distances that differ by $1$.
It also follows from the theorem that [[a graph is bipartite iff it has no odd cycle|01R1HPEK6S]], because a tree has no cycle at all.

## Explanations

### L1 human

Colour the vertices of a tree by levels, from a chosen root: even levels white, odd levels black.
Each edge goes between neighbouring levels, so it joins a white vertex and a black vertex.
</document>

<document source="https://tamnd.github.io/math2/node/0BNEXSKNRV.md" tag="0BNEXSKNRV" kind="definition">
# Polynomial of an endomorphism

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
Let $p(x) = c_0 + c_1 x + \cdots + c_d x^d$ be a polynomial in $F[x]$.
The *value of $p$ at $T$* is the endomorphism
$$p(T) = c_0 \id_V + c_1 T + c_2 T^2 + \cdots + c_d T^d,$$
where $T^k$ is the composition of $k$ copies of $T$.
For an $n \times n$ matrix $A$, $p(A) = c_0 I_n + c_1 A + \cdots + c_d A^d$ in the same way.

## Remarks

The map $p \mapsto p(T)$ is a homomorphism of algebras $F[x] \to \End_F(V)$: $(p + q)(T) = p(T) + q(T)$ and $(pq)(T) = p(T) q(T)$.
In particular, any two polynomials in $T$ commute.
If $T(v) = \lambda v$, then $p(T)(v) = p(\lambda) v$.

## Explanations

### L1 human

We can put a matrix into a polynomial, because matrices can be added, scaled and multiplied.
The constant term becomes a multiple of the identity matrix.
For example, if $p(x) = x^2 - 3x + 2$, then $p(A) = A^2 - 3A + 2I$.
</document>

<document source="https://tamnd.github.io/math2/node/0BNPC9850Y.md" tag="0BNPC9850Y" kind="example">
# Trivial group

## Example

The trivial group is the [[group|G5S1M9]] $\{e\}$ with one element $e$ and the operation $e \cdot e = e$.
The element $e$ is the identity element, and it is its own inverse.
Each group with exactly one element is isomorphic to this group.

## Remarks

Each group $G$ contains a trivial subgroup $\{e\}$, where $e$ is the identity of $G$.
In additive notation the trivial group is written $0$ or $\{0\}$.
The trivial group is [[abelian|082GHNPJ53]], and its order is $1$.

## Explanations

### L1 human

The trivial group is the smallest possible group: it has only the identity element.
It satisfies all the group axioms in a simple way.
It is often an edge case in theorems, for example when a theorem asks for a group that is not trivial.
</document>

<document source="https://tamnd.github.io/math2/node/0BPYRMHJ48.md" tag="0BPYRMHJ48" kind="definition">
# Universal set

## Definition

In a given context, fix a [[set|0SJ3RX9TXA]] $U$ such that all the sets under discussion are [[subsets|0FE8Q5S42Y]] of $U$.
The set $U$ is then the universal set, or the universe of discourse, of the context.

## Remarks

In elementary number theory, $U$ can be the set $\mathbb{Z}$ of integers.
The [[complement|0B1FSS1VZW]] of a set is taken relative to the universal set.
In [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], no set contains all sets. This is a consequence of [[Russell's paradox|0HESMMSYEG]].

## Explanations

### L1 human

The universal set is the "whole world" of a problem.
In a problem about real numbers, it is $\mathbb{R}$.
In a [[Venn diagram|0KCT6DPXJK]], it is the rectangle that contains all the circles.
</document>

<document source="https://tamnd.github.io/math2/node/0BQAA2BK3Q.md" tag="0BQAA2BK3Q" kind="concept">
# Eigenvalue

## Summary

An eigenvalue of a linear map from a space to itself is a scalar $\lambda$ with a special property.
The map sends some nonzero vector to $\lambda$ times that vector.
</document>

<document source="https://tamnd.github.io/math2/node/0BQMWCNS8S.md" tag="0BQMWCNS8S" kind="definition">
# Permutation

## Definition

Let $X$ be a set.
A permutation of $X$ is a bijection $\sigma : X \to X$.

The permutations of $X$ form a [[group|G5S1M9]] under composition, with the identity map as identity element.
This group is the symmetric group on $X$, written $\operatorname{Sym}(X)$.
For $X = \{1, \dots, n\}$ it is written $S_n$.

## Remarks

For a finite set $X = \{1, \dots, n\}$, a permutation is often written in one-line notation as the sequence $\sigma(1) \sigma(2) \cdots \sigma(n)$. In this form it is an arrangement of the elements in a row.
The [[number of permutations|0CEBCS0E0E]] of an $n$-element set is $n!$.
A [[derangement|0TH61JR04G]] is a permutation with no fixed point.

## Explanations

### L1 human

A permutation is a rearrangement of the elements of a set.
For $\{1, 2, 3\}$, the map that sends $1 \mapsto 2$, $2 \mapsto 3$ and $3 \mapsto 1$ is a permutation. In one-line notation it is $231$.
</document>

<document source="https://tamnd.github.io/math2/node/0BR3FY5Q8M.md" tag="0BR3FY5Q8M" kind="concept">
# Quotient space

## Summary

The quotient space of a vector space by a subspace is the set of cosets of the subspace, with the operations inherited from the space.
It identifies two vectors when their difference lies in the subspace.
</document>

<document source="https://tamnd.github.io/math2/node/0BRBYRK6PA.md" tag="0BRBYRK6PA" kind="statement">
# Side-side-side congruence

## Statement

Let $ABC$ and $A'B'C'$ be triangles in the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
Then $ABC$ and $A'B'C'$ are [[congruent|093EDENVA1]] if and only if
$$d(A, B) = d(A', B'), \quad d(B, C) = d(B', C') \quad \text{and} \quad d(C, A) = d(C', A').$$

## Remarks

Other criteria for congruence are side-angle-side, angle-side-angle and angle-angle-side.
Side-side-angle is not a criterion: two sides and an angle that is not between them can fit two different triangles.
Angle-angle-angle is not a criterion either. Triangles with equal angles are [[similar|0YF6QXXWB5]], but they can have different sizes.


## Explanations

### L1 human

If you know the three side lengths of a triangle, you know the whole triangle. Three sticks of fixed lengths make one rigid shape, up to a turn or a flip. This is why triangles make frames stiff.
</document>

<document source="https://tamnd.github.io/math2/node/0BRG85WXYC.md" tag="0BRG85WXYC" kind="statement">
# Goursat's theorem

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $Q = \{x + iy : a_1 \le x \le b_1, \ a_2 \le y \le b_2\}$ be a closed rectangle with $Q \subseteq U$, where $a_1 < b_1$ and $a_2 < b_2$ are real numbers.
Let $\partial Q$ be the boundary of $Q$, traveled once counterclockwise. Then the [[contour integral|00FWNP9TXY]] of $f$ along $\partial Q$ is
$$\oint_{\partial Q} f(z) \, dz = 0.$$

## Remarks

The theorem does not assume that $f'$ is continuous. This is the difference with the older proof by Green's theorem.
The same statement holds for each closed triangle $T \subseteq U$ and its boundary.
Proof idea: divide $Q$ into four equal rectangles and keep one where the integral is largest. Repeat, and use the complex derivative at the limit point.
Goursat's theorem is the first step to [[primitives on a disc|02AC7JR6G6]] and to [[Cauchy's integral theorem|01MPZXYVYF]].

## Explanations

### L1 human

If $f$ is holomorphic on a rectangle and its inside, then the integral of $f$ around the edge of the rectangle is zero.
The surprise is that this needs only the existence of the complex derivative, and nothing about its continuity.
</document>

<document source="https://tamnd.github.io/math2/node/0BRHKKP333.md" tag="0BRHKKP333" kind="definition">
# Inverse element

## Definition

Let $S$ be a set with a [[binary operation|0M418FE7BB]] $\ast$ and an [[identity element|07WYQPB8E3]] $e$.
Let $a, b \in S$.
The element $b$ is a left inverse of $a$ when $b \ast a = e$.
The element $b$ is a right inverse of $a$ when $a \ast b = e$.
The element $b$ is an inverse of $a$ (or two-sided inverse) when it is both a left inverse and a right inverse of $a$.
An element that has an inverse is called invertible.

## Remarks

For addition on $\mathbb{Z}$, the inverse of $a$ is $-a$.
For multiplication on $\mathbb{Q}$, each nonzero $a$ has the inverse $1/a$, and $0$ has no inverse.
When $\ast$ is [[associative|0K8F7PVDMX]], an element has at most one inverse.

## Explanations

### L1 human

An inverse of an element is a second element that cancels the first one.
If you add $5$ and then add $-5$, you come back to $0$, the identity for addition.
For multiplication, the inverse of $2$ is $\tfrac{1}{2}$, because $2 \cdot \tfrac{1}{2} = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0BRXFNQ5JV.md" tag="0BRXFNQ5JV" kind="statement">
# Polar form

## Statement

Let $z$ be a complex number with $z \neq 0$, let $r = |z|$ be its [[modulus|0KW2X0TFBX]] and let $\theta = \operatorname{Arg} z$ be its [[principal argument|0SQSQP748R]].
Then
$$z = r (\cos \theta + i \sin \theta) = r e^{i \theta}.$$

## Remarks

The second equality is [[Euler's formula|04R3CY41VJ]].
The pair $(r, \theta)$ gives the polar coordinates of the point $z$ in the [[complex plane|0CTE7A31XQ]].
Polar form makes products easy: $r e^{i \theta} \cdot s e^{i \varphi} = r s \, e^{i (\theta + \varphi)}$.

## Explanations

### L1 human

You can describe a point of the plane by its distance from $0$ and its direction.
For example, $1 + i$ has distance $\sqrt{2}$ and direction $\pi/4$, so $1 + i = \sqrt{2} \, e^{i \pi/4}$.
</document>

<document source="https://tamnd.github.io/math2/node/0BS0NJPKVE.md" tag="0BS0NJPKVE" kind="statement">
# Quotient rule

## Statement

Let $f$ and $g$ be real functions on an open interval $I$, both differentiable at $a \in I$, with $g(a) \ne 0$.
Then $f/g$ is differentiable at $a$, and
$$\left( \frac{f}{g} \right)'(a) = \frac{f'(a) g(a) - f(a) g'(a)}{g(a)^2}.$$

## Remarks

Write $f/g = f \cdot (1/g)$, and use the [[product rule|0V9A43H7WV]] and the [[reciprocal rule|0QD4E5JYPT]].
For example, it gives $(\tan x)' = 1/\cos^2 x$.
Mathlib states it as `HasDerivAt.div`.

## Explanations

### L1 human

The rule gives the slope of a quotient from the values and slopes of the numerator and the denominator.
A common memory aid is "low d-high minus high d-low, over low squared".
</document>

<document source="https://tamnd.github.io/math2/node/0BS3RMX8HM.md" tag="0BS3RMX8HM" kind="statement">
# Multiplication rule

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A_1, \dots, A_n \in \mathcal{F}$ with $n \ge 2$ and $P(A_1 \cap \dots \cap A_{n-1}) > 0$.
Then
$$P(A_1 \cap \dots \cap A_n) = P(A_1) \, P(A_2 \mid A_1) \, P(A_3 \mid A_1 \cap A_2) \cdots P(A_n \mid A_1 \cap \dots \cap A_{n-1}).$$
Here $P(\cdot \mid \cdot)$ is the [[conditional probability|0FJB06AD2P]].

## Remarks

The case $n = 2$ is $P(A \cap B) = P(A) \, P(B \mid A)$ for $P(A) > 0$. It is the definition of $P(B \mid A)$ multiplied by $P(A)$.
The hypothesis makes each conditional probability in the formula defined, because each conditioning event contains $A_1 \cap \dots \cap A_{n-1}$.
Example: draw two cards from a deck without replacement. The probability of two aces is $\frac{4}{52} \cdot \frac{3}{51} = \frac{1}{221}$.

## Explanations

### L1 human

To find the chance that several things all happen, take them one at a time.
Multiply the chance of the first by the chance of the second given the first.
Then multiply by the chance of the third given the first two, and so on.
</document>

<document source="https://tamnd.github.io/math2/node/0BSCJA5G53.md" tag="0BSCJA5G53" kind="example">
# Additive group of integers

## Example

Let $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$ be the set of integers, with the usual addition $+$.
Then $(\mathbb{Z}, +)$ is an [[abelian group|082GHNPJ53]].
Its identity element is $0$, and the inverse of $n$ is $-n$.

## Remarks

The group $(\mathbb{Z}, +)$ is infinite, and each element $n$ is the $n$-fold sum of $1$ or of $-1$.
Thus the element $1$ generates the group.
The integers with multiplication are not a group, because $2$ has no multiplicative inverse in $\mathbb{Z}$.

## Explanations

### L1 human

The integers with addition are the first example of a group for most students.
Adding is associative, $0$ changes nothing, and $-n$ undoes $n$.
Each element of the group is a sum of copies of $1$ or of $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/0BSJ7RQ9BW.md" tag="0BSJ7RQ9BW" kind="concept">
# Set

## Summary

A set is a collection of distinct objects, called its elements.
A set is determined by its elements, and the order and repetition of the elements do not matter.
</document>

<document source="https://tamnd.github.io/math2/node/0BSJSX7ME2.md" tag="0BSJSX7ME2" kind="concept">
# Dense set

## Summary

A dense subset of a topological space is a subset whose closure is the whole space.
Each nonempty open set meets a dense set.
</document>

<document source="https://tamnd.github.io/math2/node/0BSJYCG7NM.md" tag="0BSJYCG7NM" kind="counterexample">
# Z[√-5] is not a UFD

## Counterexample

The integral domain $\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5} : a, b \in \mathbb{Z}\}$ is not a [[unique factorisation domain|06YMXJMXYS]].
In this ring,
$$6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}).$$
Let $N(a + b\sqrt{-5}) = a^2 + 5b^2$. Then $N$ is multiplicative, and $N(2) = 4$, $N(3) = 9$, $N(1 \pm \sqrt{-5}) = 6$.
No element has norm $2$ or $3$, because $a^2 + 5b^2 \notin \{2, 3\}$ for integers $a, b$. So the four factors are [[irreducible|052SDEPZH7]].
The only units are $\pm 1$, so $2$ is not an [[associate|053E86NWVK]] of $1 + \sqrt{-5}$ or of $1 - \sqrt{-5}$. The two factorisations are different.

## Remarks

The element $2$ is [[irreducible but not prime|0CHE3J7QMX]] in this ring.
Every nonzero nonunit of $\mathbb{Z}[\sqrt{-5}]$ is still a product of irreducibles. Only the uniqueness fails.
Unique factorisation holds again for ideals: $\mathbb{Z}[\sqrt{-5}]$ is a Dedekind domain.

## Explanations

### L1 human

The number $6$ has two essentially different factorisations into irreducible elements in this ring.
The norm shows that none of the four factors can be split further.
So unique factorisation, which we take for granted in the integers, can fail.
</document>

<document source="https://tamnd.github.io/math2/node/0BSQDZWF34.md" tag="0BSQDZWF34" kind="statement">
# Riemann's theorem on removable singularities

## Statement

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 < |z - c| < r$.
Assume that $f$ is bounded near $c$: there are $M \ge 0$ and $\delta \in (0, r)$ with $|f(z)| \le M$ for $0 < |z - c| < \delta$.
Then $c$ is a [[removable singularity|0P18JC72RE]] of $f$. Moreover, the limit $\lim_{z \to c} f(z)$ exists, and the extension of $f$ by this value at $c$ is holomorphic on $|z - c| < r$.

## Remarks

The following are equivalent: $c$ is removable; $f$ has a continuous extension to $c$; $f$ is bounded near $c$; $\lim_{z \to c} (z - c) f(z) = 0$.
Proof idea: $h(z) = (z - c)^2 f(z)$, with $h(c) = 0$, is complex differentiable at $c$ with $h'(c) = 0$. So $h(z) = \sum_{n \ge 2} a_n (z - c)^n$, and $f(z) = \sum_{n \ge 2} a_n (z - c)^{n-2}$.
For example, $\frac{\sin z}{z}$ is bounded near $0$, so $0$ is removable.

## Explanations

### L1 human

If a holomorphic function stays bounded near a single bad point, then the point is not really bad.
You can give the function a value there, and the function becomes holomorphic at that point.
</document>

<document source="https://tamnd.github.io/math2/node/0BSW4T7THS.md" tag="0BSW4T7THS" kind="concept">
# Polar coordinates

## Summary

Polar coordinates describe a point of the plane by two numbers. They are its distance $r$ from the origin and the angle $\theta$ from the positive $x$-axis to the ray through the point.
They simplify problems with circular symmetry, such as integrals over discs.

## Remarks

The polar form of a complex number is a related idea: see [[polar form|0BRXFNQ5JV]].
</document>

<document source="https://tamnd.github.io/math2/node/0BSXRER81B.md" tag="0BSXRER81B" kind="problem">
# Squaring the circle

## Problem

Given a circle, construct with [[straightedge and compass|0DAECMZVRT]] alone a square with the same area.

## Remarks

For a circle of radius $1$, the square has side $\sqrt{\pi}$.
So the problem asks whether $\sqrt{\pi}$ is a [[constructible number|0FMN2MN7TM]].
The answer is no: [[squaring the circle is impossible|0X7CVDE3BG]].

## History

Anaxagoras is the first known Greek to study the problem.
Hippocrates of Chios squared some figures bounded by circular arcs, the lunes of Hippocrates.
In 1882, Ferdinand von Lindemann proved that $\pi$ is transcendental, and so that the construction is impossible.

## Explanations

### L1 human

The problem asks for a square with the same area as a given circle, with only an unmarked ruler and a compass.
The phrase "squaring the circle" now means to attempt something impossible.
</document>

<document source="https://tamnd.github.io/math2/node/0BT9BYWVRY.md" tag="0BT9BYWVRY" kind="concept">
# Minor

## Summary

A minor of a matrix is the determinant of a square submatrix.
We get such a submatrix when we choose some rows and the same number of columns.
</document>

<document source="https://tamnd.github.io/math2/node/0BTMQ04556.md" tag="0BTMQ04556" kind="definition">
# Supremum

## Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$.
An element $b$ of $P$ is a supremum, or least upper bound, of $S$ when these two conditions hold:

1. $b$ is an [[upper bound|0B5H2V4JW8]] of $S$;
2. if $M$ is an upper bound of $S$, then $b \le M$.

The notation is $b = \sup S$.

## Remarks

A set has at most one supremum, see [[uniqueness of the supremum|04ETX30HT2]].
In $\mathbb{R}$, $\sup [0, 1) = 1$ and $\sup [0, 1] = 1$. The first set does not contain its supremum.
In $\mathbb{Q}$, the set $\{q \in \mathbb{Q} : q^2 < 2\}$ is bounded above but has no supremum.

## Explanations

### L1 human

The supremum is the lowest possible ceiling for a set.
For the set $\{0.9, 0.99, 0.999, \dots\}$, every number at least $1$ is a ceiling, and $1$ is the lowest of these, so the supremum is $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0BVHZXWKCD.md" tag="0BVHZXWKCD" kind="definition">
# Borel sigma-algebra

## Definition

Let $X$ be a topological space.
The Borel σ-algebra $\mathcal{B}(X)$ is the smallest [[σ-algebra|04WSW8T3SY]] on $X$ that contains each open subset of $X$. It is the intersection of all σ-algebras on $X$ that contain the open sets.
A Borel set of $X$ is an element of $\mathcal{B}(X)$.

## Remarks

On the [[real line|08J2M1JYAG]], $\mathcal{B}(\mathbb{R})$ is also the σ-algebra generated by the open [[intervals|0TK8Y3J5RW]] $(a, b)$. The half-lines $(-\infty, b]$ with $b \in \mathbb{R}$ also generate it.
Each open set, each closed set, each countable set and each interval of $\mathbb{R}$ is a Borel set.
Some subsets of $\mathbb{R}$ are not Borel sets. For example, the Vitali sets are not Lebesgue measurable, so they are not Borel sets.

## Explanations

### L1 human

We want to give a probability to intervals such as "the waiting time is between $2$ and $5$ minutes".
The Borel sets are all the sets that we can build from intervals with complements and countable unions, again and again. So they are the events that come from intervals.
</document>

<document source="https://tamnd.github.io/math2/node/0BVT6VSB3S.md" tag="0BVT6VSB3S" kind="definition">
# Trajectory

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be locally Lipschitz. Let $\varphi$ be the [[flow|02903SSA48]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.
Let $x_0 \in U$, and let $I(x_0)$ be the maximal interval of existence of the solution with $x(0) = x_0$.
The trajectory, or orbit, of $x_0$ is the set
$$\mathcal{O}(x_0) = \{\varphi_t(x_0) : t \in I(x_0)\} \subseteq U.$$
The positive half-trajectory is $\mathcal{O}^+(x_0) = \{\varphi_t(x_0) : t \in I(x_0),\ t \ge 0\}$, and the negative half-trajectory uses $t \le 0$.

## Remarks

Two trajectories are either equal or disjoint. So the trajectories form a partition of $U$.
A trajectory is a single point exactly when $x_0$ is an [[equilibrium point|0G4X36AE0X]]. It is a closed curve when the solution is [[periodic|01DSNP0B4R]] and not constant.
Some books use the word trajectory for the solution curve $t \mapsto \varphi_t(x_0)$ itself, and the word orbit for its image.

## Explanations

### L1 human

The trajectory is the path that the state follows in the phase space.
It forgets the time: it records where the state goes, not when.
</document>

<document source="https://tamnd.github.io/math2/node/0BVTDM872V.md" tag="0BVTDM872V" kind="statement">
# Pythagorean theorem in an inner product space

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, with the [[induced norm|0MFJJV97RJ]] $\norm{\cdot}$.
If $x, y \in V$ are [[orthogonal|0Y2X7P2WB7]], then
$$\norm{x + y}^2 = \norm{x}^2 + \norm{y}^2.$$

## Remarks

The proof: $\norm{x + y}^2 = \norm{x}^2 + 2 \operatorname{Re} \langle x, y \rangle + \norm{y}^2$.
For pairwise orthogonal $x_1, \dots, x_k$, $\norm{x_1 + \cdots + x_k}^2 = \norm{x_1}^2 + \cdots + \norm{x_k}^2$.
In a real space, the converse holds; in a complex space, the equality only gives $\operatorname{Re} \langle x, y \rangle = 0$.

## Explanations

### L1 human

This is the theorem of Pythagoras for vectors.
Two orthogonal vectors are the legs of a right triangle, and their sum is the hypotenuse.
It holds in every inner product space, even in spaces of functions.
</document>

<document source="https://tamnd.github.io/math2/node/0BVXADM10P.md" tag="0BVXADM10P" kind="definition">
# Matrix addition

## Definition

Let $R$ be a ring and let $A = (a_{ij})$ and $B = (b_{ij})$ be $m \times n$ [[matrices|0VD9ZSWWJG]] over $R$.
The *sum* $A + B$ is the $m \times n$ matrix with entries
$$(A + B)_{ij} = a_{ij} + b_{ij}$$
for $1 \le i \le m$ and $1 \le j \le n$.

## Remarks

The sum is defined only for two matrices of the same size.
Matrix addition is associative and commutative, the [[zero matrix|0ZTKJJDGR7]] is its identity element, and $-A = (-a_{ij})$ is the additive inverse of $A$.
Thus $M_{m \times n}(R)$ is an abelian group under addition.
For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 4 & 4 \end{pmatrix}$.

## Explanations

### L1 human

To add two matrices of the same size, add the numbers in the same position.
The result has the same size as the two matrices.
Matrix addition has the same rules as the addition of numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0BW9N5CDSB.md" tag="0BW9N5CDSB" kind="definition">
# Antiderivative

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$.
A function $F \colon I \to \mathbb{R}$ is an antiderivative of $f$ on $I$ if $F$ is differentiable on $I$ and $F'(x) = f(x)$ for all $x \in I$.

## Remarks

If $F$ is an antiderivative of $f$, then so is $F + C$ for each constant $C$.
On an interval, these are all the antiderivatives, because a function with [[zero derivative|003QDZ4T16]] is constant.
For example, $x^3/3$ is an antiderivative of $x^2$ on $\mathbb{R}$.
A function with a jump has no antiderivative on an interval, by [[Darboux's theorem|0Q5A47J48D]].

## Explanations

### L1 human

An antiderivative undoes differentiation.
We look for a function whose derivative is the given function.
</document>

<document source="https://tamnd.github.io/math2/node/0BWDJGQYST.md" tag="0BWDJGQYST" kind="definition">
# Characteristic polynomial of a linear recurrence

## Definition

Let $R$ be a commutative ring, and let $E$ be a [[linear recurrence with constant coefficients|081THG13JZ]] over $R$, of order $d$ with coefficients $c_0, \dots, c_{d-1}$.
The characteristic polynomial of $E$ is the polynomial
$$\chi_E(X) = X^d - \sum_{i=0}^{d-1} c_i X^i \in R[X].$$

## Remarks

For the Fibonacci recurrence $u_{n+2} = u_{n+1} + u_n$, the characteristic polynomial is $X^2 - X - 1$. Its roots are the [[golden ratio|0SC8T7548D]] $\varphi$ and $1 - \varphi$.
A geometric sequence $q^n$ is a solution if and only if $q$ is a root of $\chi_E$. The node [[geometric solutions|096SD4ERQH]] states this.

## Explanations

### L1 human

Replace $u_{n+i}$ by $X^i$ in the recurrence and move everything to one side.
This gives the characteristic polynomial, and its roots tell you the geometric sequences that solve the recurrence.
</document>

<document source="https://tamnd.github.io/math2/node/0BWGTN92FG.md" tag="0BWGTN92FG" kind="concept">
# Bias of an estimator

## Summary

The bias of an estimator is the difference between its expected value and the true value of the quantity that it estimates. An estimator with zero bias is unbiased.
The node [[bias of an estimator|01BD2TFFP4]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0BWGW68B66.md" tag="0BWGW68B66" kind="statement">
# Uniform limits commute with integration

## Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be Riemann integrable for each $n$.
If $f_n \to f$ [[uniformly|08FPABDQ4C]] on $[a, b]$, then $f$ is Riemann integrable on $[a, b]$, and
$$\lim_{n \to \infty} \int_a^b f_n(x) \, dx = \int_a^b f(x) \, dx.$$

## Remarks

Let $\varepsilon_n = \sup |f_n - f|$. Then $f_n - \varepsilon_n \le f \le f_n + \varepsilon_n$. So the upper and lower sums of $f$ differ by at most $2 \varepsilon_n (b - a)$ more than those of $f_n$. This gives integrability, by [[Riemann's criterion|017DH8WXTE]].
By [[monotonicity|0MBWQKM8QR]], $\big| \int_a^b f_n - \int_a^b f \big| \le \varepsilon_n (b - a) \to 0$.
Pointwise convergence is not enough. On $[0, 1]$, the functions $f_n(x) = n^2 x (1 - x^2)^n$ tend to $0$ pointwise, but $\int_0^1 f_n = \frac{n^2}{2n + 2} \to \infty$.

## Explanations

### L1 human

If $f_n$ is uniformly close to $f$, the areas under their graphs are close too.
So we may take the limit inside the integral.
</document>

<document source="https://tamnd.github.io/math2/node/0BWJTRE3X3.md" tag="0BWJTRE3X3" kind="concept">
# Box topology

## Summary

The box topology on a product of topological spaces has as basic open sets all products of open sets.
For infinite products, it is finer than the product topology and has worse properties.
</document>

<document source="https://tamnd.github.io/math2/node/0BWXV7GADW.md" tag="0BWXV7GADW" kind="statement">
# Power series converge uniformly on smaller discs inside the radius of convergence

## Statement

Let $\sum_{n=0}^{\infty} a_n (x - c)^n$ be a [[power series|0VP56B4EMX]] with [[radius of convergence|0660QX1VPR]] $R > 0$.
Then for every $r$ with $0 < r < R$, the series converges absolutely and uniformly on $[c - r, c + r]$.

## Remarks

Choose $s$ with $r < s < R$. The series converges at $c + s$, so $|a_n| s^n \le C$ for some $C$. The [[M-test|0NX5NWAFC9]] with $M_n = C (r/s)^n$ applies.
So the sum is continuous on $(c - R, c + R)$.
The convergence need not be uniform on the whole interval: $\sum x^n$ is not uniformly convergent on $(-1, 1)$. Mathlib states the result as `HasFPowerSeriesOnBall.tendstoUniformlyOn`.

## Explanations

### L1 human

Inside the interval of convergence, a power series converges uniformly once we stay away from the endpoints.
Near the endpoints, the convergence can get slower and slower.
</document>

<document source="https://tamnd.github.io/math2/node/0BXA4Q0JHW.md" tag="0BXA4Q0JHW" kind="statement">
# Ring of integers

## Statement

The [[integers|0C4JS9RF5M]] $\mathbb{Z}$, with their addition $+$ and multiplication $\cdot$, form a commutative ring with identity $1$.
That is, for all integers $a$, $b$ and $c$:

1. $(a + b) + c = a + (b + c)$ and $a + b = b + a$;
2. $a + 0 = a$, and there is an integer $-a$ with $a + (-a) = 0$;
3. $(a b) c = a (b c)$ and $a b = b a$;
4. $a \cdot 1 = a$;
5. $a (b + c) = a b + a c$.

## Remarks

The ring $\mathbb{Z}$ is not a field: for example, $2$ has no multiplicative inverse in $\mathbb{Z}$.
The ring $\mathbb{Z}$ has no zero divisors: if $a b = 0$, then $a = 0$ or $b = 0$. Thus it is an integral domain.
Here "ring of integers" means the ring $\mathbb{Z}$. In algebraic number theory, the same words have a more general meaning.

## Explanations

### L1 human

This statement lists the usual rules of computation with whole numbers.
You can add, subtract and multiply integers, and the result is always an integer.
Division is different: $1 / 2$ is not an integer.
</document>

<document source="https://tamnd.github.io/math2/node/0BXBADR94C.md" tag="0BXBADR94C" kind="statement">
# Implicit function theorem

## Statement

Let $\Omega \subseteq \mathbb{R}^n \times \mathbb{R}^m$ be open, and let $F \colon \Omega \to \mathbb{R}^m$ be of [[class C^1|0TCBZMEYGN]], written $F(x, y)$ with $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$.
Let $(a, b) \in \Omega$ with $F(a, b) = 0$.
Suppose that the $m \times m$ matrix $D_y F(a, b) = \left( \partial F_i / \partial y_j (a, b) \right)_{i, j}$ is invertible.
Then there are open sets $U \subseteq \mathbb{R}^n$ and $V \subseteq \mathbb{R}^m$ with $a \in U$ and $b \in V$, and a unique function $g \colon U \to V$ with the following property. For $(x, y) \in U \times V$, $F(x, y) = 0$ if and only if $y = g(x)$.
The function $g$ is of class $C^1$, and $Dg(x) = -\big( D_y F(x, g(x)) \big)^{-1} D_x F(x, g(x))$ for each $x \in U$.

## Remarks

Here $D_x F$ is the $m \times n$ matrix of partial derivatives with respect to $x$. Together, $D_x F$ and $D_y F$ form the [[Jacobian matrix|0YF0VXKXGB]] of $F$.
For $F(x, y) = x^2 + y^2 - 1$ and $(a, b)$ with $b \ne 0$, the solution near $(a, b)$ is $y = \pm \sqrt{1 - x^2}$. At $(\pm 1, 0)$, $\partial F / \partial y = 0$, and no such function exists.
One proof applies the [[inverse function theorem|0K9HSJFCMJ]] to $(x, y) \mapsto (x, F(x, y))$. If $F$ is of class $C^k$, then $g$ is of class $C^k$.
The theorem has a long history. Cauchy gave the first rigorous form, and Dini extended the real version to any number of variables.

## Explanations

### L1 human

An equation $F(x, y) = 0$ often defines $y$ as a function of $x$, even when you cannot solve it with a formula.
The theorem says this works near a solution, provided $F$ really depends on $y$ there, in the sense that $D_y F$ is invertible.
</document>

<document source="https://tamnd.github.io/math2/node/0BXE0S1XER.md" tag="0BXE0S1XER" kind="definition">
# Having a gradient

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, let $a \in U$, and let $g \in \mathbb{R}^n$.
The function $f$ has the gradient $g$ at $a$ when
$$\lim_{h \to 0} \frac{f(a + h) - f(a) - g \cdot h}{\|h\|} = 0,$$
where the limit is over $h \ne 0$ with $a + h \in U$.
Equivalently, $f$ is [[differentiable|0135XZZQ8M]] at $a$ and $Df(a)(h) = g \cdot h$ for each $h \in \mathbb{R}^n$.

## Remarks

This definition states a property of $f$, $a$ and $g$. When it holds, the vector $g$ is unique, and it is the [[gradient|0NS2QYVC4Y]] $\nabla f(a)$.
The same definition works in each real inner product space, with the inner product in place of the dot product.

## Explanations

### L1 human

The function has the gradient $g$ at $a$ when the plane through $(a, f(a))$ with slopes given by $g$ hugs the graph of $f$ near $a$.
The error of this approximation must be small compared with the distance from $a$.
</document>

<document source="https://tamnd.github.io/math2/node/0BXEZV83GY.md" tag="0BXEZV83GY" kind="concept">
# Minimal element

## Summary

A minimal element of a partially ordered set is an element with no strictly smaller element.
</document>

<document source="https://tamnd.github.io/math2/node/0BXNDZFQME.md" tag="0BXNDZFQME" kind="example">
# Cantor set

## Example

Let $C_0 = [0, 1]$. For $n \ge 0$, let $C_{n+1}$ be the set that we get from $C_n$ when we remove the open middle third of each of its closed intervals. Thus $C_1 = [0, \tfrac{1}{3}] \cup [\tfrac{2}{3}, 1]$.
The Cantor set is $C = \bigcap_{n \ge 0} C_n$, with the subspace topology of $\mathbb{R}$.

## Remarks

Equivalently, $C$ is the set of $x \in [0, 1]$ that have a base-3 expansion $x = \sum_{n \ge 1} a_n 3^{-n}$ with all $a_n \in \{0, 2\}$.
The set $C$ is [[closed|01VT24J1SD]] and bounded, so it is [[compact|07PFYV65MB]]. It is [[totally disconnected|01CYW8S04D]] and has no isolated points.
It is uncountable, and it is [[homeomorphic to the space of binary sequences|0RQ8DHHR8V]]. Its Lebesgue measure is $0$, because $C_n$ has total length $(2/3)^n$.
The Cantor set is [[nowhere dense|0KSM23SN3H]] in $\mathbb{R}$.

## Explanations

### L1 human

Start with $[0, 1]$, remove the middle third, then remove the middle third of each piece that is left, and go on forever.
What remains is a set with no length, but with as many points as $\mathbb{R}$.
</document>

<document source="https://tamnd.github.io/math2/node/0BY52JW6NK.md" tag="0BY52JW6NK" kind="concept">
# Complete metric space

## Summary

A complete metric space is a metric space in which each Cauchy sequence converges to a point of the space.
The real numbers are complete, and the rational numbers are not.
</document>

<document source="https://tamnd.github.io/math2/node/0BY5FTHEEE.md" tag="0BY5FTHEEE" kind="statement">
# Slopes of a convex function are increasing

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be [[convex|0VV0HSW1ZS]].
Then for all $x < y < z$ in $I$,
$$\frac{f(y) - f(x)}{y - x} \le \frac{f(z) - f(x)}{z - x} \le \frac{f(z) - f(y)}{z - y}.$$

## Remarks

Write $y = t x + (1 - t) z$ with $t = (z - y)/(z - x)$, and apply the definition of convexity.
The converse holds: a function with this property is convex.
The first and the last quotient give the adjacent form `ConvexOn.slope_mono_adjacent` of Mathlib.

## Explanations

### L1 human

For a convex function, chords further to the right are steeper.
So the slopes of the graph increase from left to right.
</document>

<document source="https://tamnd.github.io/math2/node/0BY7GN3449.md" tag="0BY7GN3449" kind="statement">
# Generalized pigeonhole principle

## Statement

Let $A$ and $B$ be finite sets, let $f : A \to B$ be a function, and let $n$ be a natural number.
Assume that $|B| \cdot n < |A|$.
Then there is an element $b \in B$ whose fibre $f^{-1}(b) = \{a \in A : f(a) = b\}$ has more than $n$ elements.

## Remarks

For $n = 1$ this is the [[pigeonhole principle|0HGVCWQGY3]].
An equivalent form: if $N$ objects go into $k$ boxes, some box has at least $\lceil N / k \rceil$ objects.
Example: among $25$ people, at least $3$ share a birth month, because $12 \cdot 2 < 25$.

## Explanations

### L1 human

If every box held at most $n$ objects, the boxes would hold at most $n$ times the number of boxes.
So if there are more objects than that, some box must hold more than $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0BYTQ4X1HC.md" tag="0BYTQ4X1HC" kind="concept">
# Least squares

## Summary

Let $A$ be a real $m \times n$ matrix and let $b \in \R^m$.
A *least squares solution* of $Ax = b$ is a vector $\hat x \in \R^n$ that minimises $\norm{Ax - b}$, for the Euclidean norm.
The vectors $\hat x$ that do this are exactly the solutions of the *normal equations*
$$A^T A \hat x = A^T b.$$
If the columns of $A$ are linearly independent, then $A^T A$ is invertible, and $\hat x = (A^T A)^{-1} A^T b$ is unique.

## Remarks

The vector $A \hat x$ is the [[orthogonal projection|0B8KZRT28M]] of $b$ onto the column space of $A$, see [[the closest point property|0MCDDSCB9Z]].
The normal equations say that the residual $b - A\hat x$ is orthogonal to each column of $A$.
Fitting a line $y = c_0 + c_1 t$ to data points $(t_i, y_i)$ is a least squares problem.
In practice, the [[QR decomposition|0ERWG4NXQ9]] gives a more stable method than the normal equations.

## Explanations

### L1 human

A system $Ax = b$ with more equations than unknowns usually has no solution.
The least squares method finds the $x$ that makes $Ax$ as close as possible to $b$.
It is the standard way to fit a model to noisy data.
</document>

<document source="https://tamnd.github.io/math2/node/0BYWRWFT4F.md" tag="0BYWRWFT4F" kind="concept">
# Antiderivative

## Summary

An antiderivative of a function is a function whose derivative is the given function. On an interval, two antiderivatives of the same function differ by a constant. The node [[antiderivative|0BW9N5CDSB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0BYX1D7C2E.md" tag="0BYX1D7C2E" kind="concept">
# Clique

## Summary

A clique in a graph is a set of vertices that are pairwise adjacent. Cliques are the complete subgraphs, and they appear in colouring, in Ramsey theory and in extremal graph theory. The node [[clique|0G52BPNPKG]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0BYZTFM0DJ.md" tag="0BYZTFM0DJ" kind="statement">
# Zero derivative on a connected open set implies constant

## Statement

Let $U \subseteq \mathbb{R}^n$ be a connected open set and let $f \colon U \to \mathbb{R}^m$ be [[differentiable|0135XZZQ8M]] on $U$.
Suppose that $Df(x) = 0$ for each $x \in U$.
Then $f$ is constant on $U$.

## Remarks

The proof fixes $a \in U$ and shows that the set $S = \{x \in U : f(x) = f(a)\}$ is open and closed in $U$. It is open by the [[mean value inequality|02CYB184RC]] with $M = 0$ on small balls, and closed because $f$ is continuous. As $U$ is connected and $a \in S$, $S = U$.
Connectedness is needed: on $U = (0, 1) \cup (2, 3)$, the function that is $0$ on the first interval and $1$ on the second has derivative $0$ but is not constant.
For real-valued $f$, the condition $Df = 0$ means that all partial derivatives of $f$ vanish on $U$.

## Explanations

### L1 human

Suppose that $f$ does not change in any direction at any point, and that the domain is in one piece. Then $f$ cannot change at all.
On a domain in two pieces, $f$ can take a different constant value on each piece.
</document>

<document source="https://tamnd.github.io/math2/node/0BZ81AJNWF.md" tag="0BZ81AJNWF" kind="definition">
# Characteristic polynomial of a linear map

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The *characteristic polynomial* of $T$ is
$$p_T(t) = p_{[T]_B}(t) = \det(t I_n - [T]_B),$$
the [[characteristic polynomial|0VSSPAFM2C]] of the [[matrix of $T$|0YQSGNYE3K]] in a basis $B$ of $V$, where $n = \dim V$.

## Remarks

The polynomial does not depend on $B$, see [[the invariance of the characteristic polynomial|0TYMASG33Y]].
Its degree is $\dim V$.
For the identity map of $V$, it is $(t - 1)^n$.

## Explanations

### L1 human

A linear map has a matrix in each basis.
All these matrices have the same characteristic polynomial.
So we can call it the characteristic polynomial of the map.
</document>

<document source="https://tamnd.github.io/math2/node/0BZQ6ZX584.md" tag="0BZQ6ZX584" kind="statement">
# Addition rule for two events

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \in \mathcal{F}$ be [[events|00SQGJNRTB]].
Then
$$P(A \cup B) + P(A \cap B) = P(A) + P(B).$$
Equivalently, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

## Remarks

If $A$ and $B$ are mutually exclusive, then $P(A \cap B) = 0$ and $P(A \cup B) = P(A) + P(B)$.
The rule is the probabilistic form of [[inclusion–exclusion for two sets|0C4YEQ9H5E]]. The [[inclusion–exclusion principle|0FF7726C7P]] gives the version for $n$ events.
A consequence is $P(A \cup B) \le P(A) + P(B)$, the case $n = 2$ of [[Boole's inequality|0EGSEE9J9R]].

## Explanations

### L1 human

When you add $P(A)$ and $P(B)$, you count the outcomes in both events twice, so you subtract $P(A \cap B)$ once.
For one card from a deck, $P(\text{heart or king}) = 13/52 + 4/52 - 1/52 = 16/52$.
</document>

<document source="https://tamnd.github.io/math2/node/0BZVNWJ352.md" tag="0BZVNWJ352" kind="statement">
# Order of the dihedral group

## Statement

Let $n \geq 3$.
The [[dihedral group|0JZ27DMJ5D]] $D_n$ has [[order|0B85S2V2DR]] $2n$.

## Remarks

A symmetry of the regular $n$-gon is determined by the image of one vertex and by whether it keeps or reverses the orientation.
There are $n$ choices for the image of the vertex and $2$ choices for the orientation.
For example, $D_3$ has order $6$ and $D_4$ has order $8$.

## Explanations

### L1 human

A symmetry can send a fixed vertex to any of the $n$ vertices.
After that, the polygon can be face up or face down.
So there are $n \cdot 2 = 2n$ symmetries.
</document>

<document source="https://tamnd.github.io/math2/node/0C019SDTER.md" tag="0C019SDTER" kind="statement">
# Recurrence for Stirling numbers of the second kind

## Statement

Let $n$ and $k$ be natural numbers, and let $S(n, k)$ be the [[Stirling numbers of the second kind|0ZN07RQ3JS]].
Then
$$S(n + 1, k + 1) = (k + 1)\, S(n, k + 1) + S(n, k).$$

## Remarks

Proof idea: look at the element $n + 1$. Either it forms a block by itself, and the rest is a partition into $k$ blocks. Or it joins one of the $k + 1$ blocks of a partition of the rest.
Example: $S(4, 2) = 2\, S(3, 2) + S(3, 1) = 2 \cdot 3 + 1 = 7$.

## Explanations

### L1 human

Add a new element to a partition problem.
The new element is alone, or it joins one of the existing $k + 1$ groups, and these two cases give the two terms.
</document>

<document source="https://tamnd.github.io/math2/node/0C08BQ3M05.md" tag="0C08BQ3M05" kind="statement">
# Laplace expansion along a row

## Statement

Let $K$ be a commutative ring, let $n \ge 2$, and let $A = (a_{ij})$ be an $n \times n$ matrix over $K$.
Let $C_{ij}$ be the [[cofactors|0XPKR18APA]] of $A$.
Then, for each row $i$,
$$\det A = \sum_{j=1}^{n} a_{ij} C_{ij} = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}.$$

## Remarks

The expansion reduces a determinant of order $n$ to $n$ determinants of order $n - 1$.
It is efficient when the chosen row contains many zeros.
There is a similar [[expansion along a column|06AMCJMMVM]].
Replace the entries of row $i$ by those of another row $k$.
Then the sum is $0$, because the new matrix has two equal rows.

## Explanations

### L1 human

To expand a determinant along a row, take each entry of that row.
Multiply it by its cofactor, which is the signed determinant of what is left after deleting its row and column.
Add these products.
</document>

<document source="https://tamnd.github.io/math2/node/0C08YMV95H.md" tag="0C08YMV95H" kind="definition">
# Random walk

## Definition

Let $d \ge 1$, let $x_0 \in \mathbb{R}^d$, and let $\xi_1, \xi_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] random vectors in $\mathbb{R}^d$.
The random walk with start $x_0$ and steps $\xi_i$ is the [[stochastic process|0QH5ENKZM1]] $(S_n)_{n \ge 0}$ with
$$S_0 = x_0 \quad \text{and} \quad S_n = x_0 + \xi_1 + \dots + \xi_n \quad \text{for } n \ge 1.$$
The simple random walk on $\mathbb{Z}$ has $d = 1$, $x_0 = 0$ and $P(\xi_i = 1) = P(\xi_i = -1) = 1/2$.

## Remarks

For the simple random walk on $\mathbb{Z}$, $E[S_n] = 0$ and $\operatorname{Var}(S_n) = n$. By the central limit theorem, $S_n/\sqrt{n}$ converges in distribution to $\mathcal{N}(0, 1)$.
The simple random walk on $\mathbb{Z}^d$ returns to its start with probability $1$ if $d \le 2$, and with probability less than $1$ if $d \ge 3$. This is Pólya's theorem.
A random walk is a Markov chain with transition probabilities that do not change under translation.
After scaling of time and space, the simple random walk converges to the Wiener process.

## Explanations

### L1 human

A random walk is the path of someone who, at each step, chooses a direction at random.
It models the price of a share, the motion of a particle in a fluid, and the fortune of a gambler.
</document>

<document source="https://tamnd.github.io/math2/node/0C093D1T4M.md" tag="0C093D1T4M" kind="definition">
# Simple group

## Definition

A [[group|G5S1M9]] $G$ is simple when $G \neq \{e\}$ and the only [[normal subgroups|0CB0DX5BVV]] of $G$ are $\{e\}$ and $G$.

## Remarks

The cyclic groups $\mathbb{Z}/p\mathbb{Z}$ of prime order $p$ are simple, and they are the only [[abelian simple groups|0JJV80RXHD]].
The [[alternating group|0Z5905Q73D]] $A_n$ is simple for $n \geq 5$, and $A_5$ is the smallest simple group that is not abelian.
A simple group has only the trivial quotient and itself as quotients, so it cannot be split into smaller pieces by a quotient.

## Explanations

### L1 human

A simple group cannot be broken into a normal subgroup and a quotient group.
In this way, simple groups are like the prime numbers among the integers.
Each finite group is built from simple groups, by the Jordan–Hölder theorem.
</document>

<document source="https://tamnd.github.io/math2/node/0C0KPAKK9Y.md" tag="0C0KPAKK9Y" kind="counterexample">
# Separately continuous but not continuous function

## Counterexample

One could guess the following. A function of two variables is continuous when it is continuous in each variable for each fixed value of the other variable.
This is false.
Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be given by $f(x, y) = \dfrac{xy}{x^2 + y^2}$ for $(x, y) \ne (0, 0)$ and $f(0, 0) = 0$.
For each fixed $b$, the function $x \mapsto f(x, b)$ is continuous on $\mathbb{R}$, and for each fixed $a$, the function $y \mapsto f(a, y)$ is continuous on $\mathbb{R}$.
But $f$ is not [[continuous|0JEJATBVD7]] at $(0, 0)$.

## Remarks

For $b \ne 0$, the function $x \mapsto f(x, b)$ is a quotient of polynomials with a denominator that is never $0$. For $b = 0$, it is the constant $0$. The same holds for the other variable.
On the line $y = x$ with $x \ne 0$, the value is $f(x, x) = 1/2$. So $f(x, x) \to 1/2 \ne f(0, 0)$ as $x \to 0$, by the [[test with different paths|0BJ6H85DRQ]].

## Explanations

### L1 human

Separate continuity checks only the two directions along the axes.
On both axes this function is $0$, so nothing goes wrong there.
On the diagonal it is $1/2$ everywhere except at the origin, so the function jumps there.
</document>

<document source="https://tamnd.github.io/math2/node/0C12P4W4RH.md" tag="0C12P4W4RH" kind="concept">
# Density of the normal distribution

## Summary

The density of the normal distribution is the bell-shaped function $\frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x-\mu)^2/(2\sigma^2)}$.
The node [[density of the normal distribution|027JF1FBB0]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0C13E623WD.md" tag="0C13E623WD" kind="statement">
# Linear maps on finite-dimensional spaces are continuous

## Statement

Let $V$ be a finite-dimensional normed space and let $W$ be a normed space, both over $\mathbb{R}$ or over $\mathbb{C}$.
Then each linear map $T \colon V \to W$ is [[continuous|0S3SMRH0Z5]].

## Remarks

Proof: choose a basis $e_1, \ldots, e_n$ of $V$. For $x = \sum_i a_i e_i$, the triangle inequality gives $\|Tx\|_W \le \left( \sum_i \|T e_i\|_W \right) \max_i |a_i|$. By [[equivalence of norms in finite dimension|03MC70ZRNR]], $\max_i |a_i| \le c \|x\|_V$ for some $c$. Thus $T$ is bounded, and so [[continuous|0F3K4C7XYW]].
The finite dimension of $W$ is not needed, only that of $V$.
For example, each matrix gives a continuous map $\mathbb{R}^n \to \mathbb{R}^m$.

## Explanations

### L1 human

In finite dimensions, linear maps are automatically continuous, for each choice of norms.
Problems with continuity only appear in infinite dimensions.
</document>

<document source="https://tamnd.github.io/math2/node/0C1VYVNV2N.md" tag="0C1VYVNV2N" kind="object">
# pi

## Description

The cosine function has exactly one zero in the interval $[0, 2]$. The number $\pi$ is twice this zero.
So $\cos(\pi/2) = 0$, and
$$\pi \approx 3.14159265.$$

## Remarks

A zero exists by the [[intermediate value theorem|00KAFBR32E]], since $\cos 0 = 1$ and $\cos 2 < 0$. It is unique because $\cos$ is strictly decreasing on $[0, 2]$, by [[its derivative|0V2DV29EGT]] $-\sin$.
With this definition, $\pi$ is the ratio of the circumference of a circle to its diameter, and the area of the unit disc.
Mathlib defines `Real.pi` in the same way, as twice the zero of $\cos$ in $[1, 2]$.
William Jones first used the letter $\pi$ for this ratio, in 1706. Around 250 BC, Archimedes approximated $\pi$ with regular polygons.

## Explanations

### L1 human

The number $\pi$ is the length of half of the unit circle.
In analysis, we find it through the first positive zero of cosine, which is $\pi/2$.
</document>

<document source="https://tamnd.github.io/math2/node/0C1Y04SZ1W.md" tag="0C1Y04SZ1W" kind="concept">
# Finite extension

## Summary

A finite extension is a field extension of finite degree.
The [[definition of a finite extension|0NRYPM9Y27]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0C21Q694AG.md" tag="0C21Q694AG" kind="concept">
# Family of sets

## Summary

A family of sets is a collection of sets, often indexed by the elements of an index set.
</document>

<document source="https://tamnd.github.io/math2/node/0C21RZG5V2.md" tag="0C21RZG5V2" kind="definition">
# Random variable with a density

## Definition

Let $X$ be a real [[random variable|0DKRR22AEQ]] on a probability space $(\Omega, \mathcal{F}, P)$.
$X$ has a density, or is absolutely continuous, if there is a Borel measurable function $f \colon \mathbb{R} \to [0, \infty)$ with the following property. For each Borel set $B \subseteq \mathbb{R}$,
$$P(X \in B) = \int_B f(x) \, dx,$$
where the integral is the Lebesgue integral.

## Remarks

Such a function $f$ is a [[probability density function|04YG690N5V]] of $X$.
If $X$ has a density, then $P(X = x) = 0$ for each $x$, and the cumulative distribution function $F_X(x) = \int_{-\infty}^x f(t) \, dt$ is continuous.
A continuous cumulative distribution function does not imply a density. The Cantor distribution has a continuous cumulative distribution function and no density.
The same definition with $\mathbb{R}^n$ and the Lebesgue measure on $\mathbb{R}^n$ gives random vectors with a density.

## Explanations

### L1 human

For a random variable with a density, the probability of an interval is an area under a curve.
Single points have probability zero, just as a single point has no length.
</document>

<document source="https://tamnd.github.io/math2/node/0C28RT5BG7.md" tag="0C28RT5BG7" kind="definition">
# Rectifiable curve

## Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a [[parametrised curve|0JF1XXGWB4]].
For a partition $P$ given by $a = t_0 < t_1 < \dots < t_k = b$, let
$$L(\gamma, P) = \sum_{i = 1}^{k} \|\gamma(t_i) - \gamma(t_{i-1})\|$$
be the length of the inscribed polygonal path.
The curve $\gamma$ is rectifiable if $\sup_P L(\gamma, P) < \infty$, where the supremum is over all partitions of $[a, b]$.
Then the length of $\gamma$ is $\ell(\gamma) = \sup_P L(\gamma, P)$.

## Remarks

If $\gamma$ is of class $C^1$, then it is rectifiable and $\ell(\gamma) = \int_a^b \|\gamma'(t)\| \, dt$.
A refinement of $P$ does not decrease $L(\gamma, P)$, by the triangle inequality.
The graph of $f(x) = x \sin(1/x)$ for $0 < x \le 1$, $f(0) = 0$, is a continuous curve that is not rectifiable. The Koch curve is another example.
A curve is rectifiable if and only if each component $\gamma_i$ has bounded variation on $[a, b]$.

## Explanations

### L1 human

Approximate the curve by a chain of straight segments between points on it, and add up their lengths.
Finer chains give longer totals. If the totals stay bounded, the curve has a finite length, and the length is their least upper bound.
</document>

<document source="https://tamnd.github.io/math2/node/0C2AKPDCD9.md" tag="0C2AKPDCD9" kind="definition">
# Hamiltonian system

## Definition

Let $n \ge 1$, let $U$ be an open subset of $\mathbb{R}^n \times \mathbb{R}^n$ with coordinates $(q, p) = (q_1, \ldots, q_n, p_1, \ldots, p_n)$, and let $H : U \to \mathbb{R}$ be continuously differentiable.
The Hamiltonian system with Hamiltonian $H$ is the [[autonomous system|07BNRFQ0PJ]]
$$q_i' = \frac{\partial H}{\partial p_i}(q, p), \qquad p_i' = -\frac{\partial H}{\partial q_i}(q, p), \qquad i = 1, \ldots, n.$$

## Remarks

The function $H$ is a [[first integral|0SS1P1YC9A]] when it is not constant on an open set: along a solution, $\frac{d}{dt} H = \sum_i \left( \frac{\partial H}{\partial q_i} q_i' + \frac{\partial H}{\partial p_i} p_i' \right) = 0$.
For $H(q, p) = \frac{1}{2} p^2 + V(q)$ with $n = 1$, the system is $q' = p$, $p' = -V'(q)$. This is Newton's law $q'' = -V'(q)$ for a particle with potential energy $V$.
The [[harmonic oscillator|0YG0ZJPTWG]] has $V(q) = \frac{1}{2} q^2$, and the [[pendulum|0SHPXM2HWH]] has $V(q) = -\cos q$ in suitable units.
A time-dependent Hamiltonian $H(t, q, p)$ gives a non-autonomous Hamiltonian system, and then $H$ need not be conserved.

## Explanations

### L1 human

A Hamiltonian system describes a mechanical system without friction by one function, the total energy $H$.
The positions $q$ and the momenta $p$ change according to the partial derivatives of $H$, and the energy stays constant.
</document>

<document source="https://tamnd.github.io/math2/node/0C2F40MEBC.md" tag="0C2F40MEBC" kind="statement">
# Determinant is the product of the eigenvalues

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Assume that the characteristic polynomial of $A$ splits over $F$ as $p_A(t) = (t - \lambda_1) \cdots (t - \lambda_n)$.
Then
$$\det A = \lambda_1 \lambda_2 \cdots \lambda_n.$$

## Remarks

The $\lambda_i$ are the [[eigenvalues|0BQAA2BK3Q]] of $A$, each repeated according to its [[algebraic multiplicity|0FY42N02DV]].
The statement follows by comparing the constant coefficients, see [[the determinant as a coefficient|0T0X8GTYYS]].
For example, $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ has eigenvalues $1$ and $3$, and determinant $3$.

## Explanations

### L1 human

The determinant of a matrix is the product of its eigenvalues, counted with multiplicity.
This agrees with the volume picture: in a basis of eigenvectors, the map stretches each direction by its eigenvalue.
The volume then changes by the product of these factors.
</document>

<document source="https://tamnd.github.io/math2/node/0C2TQMW5A1.md" tag="0C2TQMW5A1" kind="definition">
# Dual map

## Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *dual map* of $T$, also called its *transpose*, is the map
$$T^* \colon W^* \to V^*, \qquad T^*(f) = f \circ T,$$
between the [[dual spaces|07JBSZE0V4]].

## Remarks

The dual map is linear.
It reverses the order of composition: $(T \circ S)^* = S^* \circ T^*$, and $(\id_V)^* = \id_{V^*}$.
In dual bases, the matrix of $T^*$ is the transpose of the matrix of $T$, see [[the matrix of the dual map|0ZPB3180AY]].
The kernel of $T^*$ is the [[annihilator|0ASHNYF7AG]] of the image of $T$.

## Explanations

### L1 human

A linear form on $W$ can be pulled back to $V$: first apply $T$, then the form.
The dual map does this pullback.
It goes in the opposite direction, from $W^*$ to $V^*$.
</document>

<document source="https://tamnd.github.io/math2/node/0C32J27WQC.md" tag="0C32J27WQC" kind="statement">
# Associativity of addition

## Statement

For all natural numbers $a$, $b$ and $c$, $(a + b) + c = a + (b + c)$.
Here $+$ is the [[addition of natural numbers|0W5DSDFQ4G]].

## Remarks

The proof from the recursive definition of addition uses [[induction|0ASJVEFA9J]] on $c$.
Because of this law, we can write $a + b + c$ without parentheses.
Multiplication of natural numbers is also associative.

## Explanations

### L1 human

When you add three numbers, you can add the first two first or the last two first.
For example, $(2 + 3) + 4 = 5 + 4 = 9$ and $2 + (3 + 4) = 2 + 7 = 9$.
</document>

<document source="https://tamnd.github.io/math2/node/0C38S0KMJK.md" tag="0C38S0KMJK" kind="definition">
# Interior

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
The interior of $S$ is the [[union|05GK32H72H]] of all [[open sets|01ZQQDCERP]] $U$ of $X$ with $U \subseteq S$.
It is written $\operatorname{int} S$ or $S^\circ$.
A point of $\operatorname{int} S$ is an interior point of $S$.

## Remarks

A point $x$ is an interior point of $S$ if and only if there is an open set $U$ with $x \in U \subseteq S$.
In a metric space, $x$ is an interior point of $S$ if and only if $B(x, \varepsilon) \subseteq S$ for some $\varepsilon > 0$.
In $\mathbb{R}$, the interior of $[0, 1]$ is $(0, 1)$, and the interior of $\mathbb{Q}$ is $\emptyset$.

## Explanations

### L1 human

The interior of a set is what remains when you remove its edge points.
A point is in the interior when a whole small neighbourhood of it lies in the set.
</document>

<document source="https://tamnd.github.io/math2/node/0C3CFHDN6R.md" tag="0C3CFHDN6R" kind="definition">
# Determinant

## Definition

Let $K$ be a commutative ring and let $n \ge 1$.
There is exactly one map $\det \colon M_n(K) \to K$ with these properties:

1. $\det A$ is a multilinear function of the $n$ rows of $A$.
2. $\det A = 0$ when two rows of $A$ are equal.
3. $\det I_n = 1$, for the [[identity matrix|06DW9BAWB1]] $I_n$.

This map is the *determinant*.

## Remarks

In other words, $\det$ is the unique [[alternating multilinear map|0K6CH0MT0F]] of the rows with value $1$ on $I_n$.
The [[Leibniz formula|016PE4VZ8Q]] gives $\det A$ as a sum over permutations.
For $n = 2$, $\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$.
The same characterization holds with columns in place of rows, because $\det A^T = \det A$.

## Explanations

### L1 human

The determinant gives one number for each square matrix.
Over the real numbers, its absolute value is the factor by which the matrix scales volume, and its sign tells if the matrix flips orientation.
A real square matrix is invertible exactly when its determinant is not zero.
</document>

<document source="https://tamnd.github.io/math2/node/0C3PN4075R.md" tag="0C3PN4075R" kind="statement">
# Euler product formula

## Statement

Let $s$ be a complex number with real part $\operatorname{Re}(s) > 1$, and let $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ be the Riemann zeta function.
Then
$$\zeta(s) = \prod_{p} \frac{1}{1 - p^{-s}},$$
where the product runs over all [[prime numbers|0SN0MAEKYX]] $p$ and converges absolutely.

## Remarks

The proof expands each factor as a geometric series $1 + p^{-s} + p^{-2s} + \cdots$. The [[fundamental theorem of arithmetic|084HH1NDCB]] shows that the product of these series contains each term $n^{-s}$ exactly one time.
The same method gives a product formula $\sum_n f(n) n^{-s} = \prod_p \sum_{k \ge 0} f(p^k) p^{-ks}$ for each [[multiplicative function|0AP80TT660]] $f$ when the series converges absolutely.
Since $\zeta(s) \to \infty$ as $s \to 1^+$, the formula gives a proof of [[Euclid's theorem|0292EPPPT0]].

## Explanations

### L1 human

Multiply $(1 + \tfrac12 + \tfrac14 + \cdots)(1 + \tfrac13 + \tfrac19 + \cdots)(1 + \tfrac15 + \cdots) \cdots$ and expand.
Each fraction $1/n$ appears one time, because $n$ is a product of prime powers in exactly one way.
The formula with the exponent $s$ is the same idea with $1 / n^s$.
</document>

<document source="https://tamnd.github.io/math2/node/0C3QBCR46M.md" tag="0C3QBCR46M" kind="concept">
# Monic polynomial

## Summary

A monic polynomial is a nonzero polynomial whose leading coefficient is $1$.
The [[definition of a monic polynomial|0BDX4MBKT4]] states this precisely.
</document>

<document source="https://tamnd.github.io/math2/node/0C4JS9RF5M.md" tag="0C4JS9RF5M" kind="object">
# Integers

## Description

The integers are the numbers $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$.
We write $\mathbb{Z}$ for the set of integers.
The set $\mathbb{Z}$ contains the [[natural numbers|0MWKJ44V5A]] $0, 1, 2, \dots$ and their negatives $-1, -2, -3, \dots$.
With addition and multiplication, $\mathbb{Z}$ is a [[commutative ring|0BXA4Q0JHW]]. Each integer $a$ has an additive inverse $-a$, thus subtraction is always possible in $\mathbb{Z}$.
The integers have a [[total order|0F6FM9Y4Y6]] that extends the order of $\mathbb{N}$.

## Remarks

The letter $\mathbb{Z}$ comes from the German word "Zahlen", which means "numbers".
One [[construction of the integers|0NS58JP68E]] uses pairs of natural numbers.
The [[rational numbers|0NCJ5WXN6Z]] contain the integers as the fractions $a / 1$.

## Explanations

### L1 human

The integers are the whole numbers with a sign: positive, negative or zero.
You need them to answer questions such as $5 - 8 = ?$, which has no answer in the natural numbers.
On a number line, the negative integers are to the left of $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0C4NT4GT3X.md" tag="0C4NT4GT3X" kind="statement">
# Eigenvalues of a self-adjoint operator are real

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be a [[self-adjoint|0NEGN5CRSC]] linear map.
Then each [[eigenvalue|0BQAA2BK3Q]] of $T$ is a real number.

## Remarks

If $T(v) = \lambda v$ with $v \ne 0$, then $\lambda \langle v, v \rangle = \langle T(v), v \rangle = \langle v, T(v) \rangle = \bar\lambda \langle v, v \rangle$, so $\lambda = \bar\lambda$.
In particular, a [[Hermitian matrix|0P17NGQQMQ]] has only real eigenvalues.
So a real [[symmetric matrix|0VR671PHE5]] has only real eigenvalues, also when we view it as a complex matrix.

## Explanations

### L1 human

Self-adjoint maps never have complex eigenvalues that are not real.
For example, the symmetric matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ has the eigenvalues $1$ and $-1$.
By contrast, the rotation matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ is not symmetric, and its eigenvalues are $\pm i$.
</document>

<document source="https://tamnd.github.io/math2/node/0C4VWBFY73.md" tag="0C4VWBFY73" kind="definition">
# Bernstein polynomial

## Definition

Let $f \colon [0, 1] \to \mathbb{R}$ be a function and let $n \ge 1$ be an integer.
The $n$-th Bernstein polynomial of $f$ is
$$B_n(f)(x) = \sum_{k=0}^{n} f\Big( \frac{k}{n} \Big) \binom{n}{k} x^k (1 - x)^{n-k}.$$
The polynomials $b_{k,n}(x) = \binom{n}{k} x^k (1 - x)^{n-k}$ are the Bernstein basis polynomials.

## Remarks

If $f$ is continuous, then $B_n(f) \to f$ uniformly on $[0, 1]$. This proves the [[Weierstrass approximation theorem|0JQE130DHW]].
The basis polynomials are nonnegative on $[0, 1]$, and they sum to $1$ by the binomial theorem.
Sergei Bernstein gave this proof in 1912, with an argument from probability. Mathlib defines the basis polynomials as `bernsteinPolynomial`.

## Explanations

### L1 human

The Bernstein polynomial samples $f$ at the points $k/n$ and blends the values with smooth weights.
Near $x$, the weights concentrate on the samples with $k/n$ close to $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0C4YEQ9H5E.md" tag="0C4YEQ9H5E" kind="statement">
# Inclusion–exclusion for two sets

## Statement

Let $A$ and $B$ be finite sets.
Then
$$|A \cup B| + |A \cap B| = |A| + |B|.$$
Equivalently, $|A \cup B| = |A| + |B| - |A \cap B|$.

## Remarks

If $A \cap B = \varnothing$, this is the [[rule of sum|0EE2V3CHFK]].
For three sets: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$.
The [[inclusion–exclusion principle|0FF7726C7P]] is the version for $n$ sets.

## Explanations

### L1 human

When you add the sizes of $A$ and $B$, you count the elements of both sets twice.
Subtract the size of $A \cap B$ once to correct this. For example, $12$ students play football, $8$ play chess, and $3$ do both, so $17$ students do at least one.
</document>

<document source="https://tamnd.github.io/math2/node/0C5BFX94J9.md" tag="0C5BFX94J9" kind="statement">
# Recurrence for Stirling numbers of the first kind

## Statement

Let $n$ and $k$ be natural numbers, and let $c(n, k)$ be the unsigned [[Stirling numbers of the first kind|0Z22VQH17V]].
Then
$$c(n + 1, k + 1) = n\, c(n, k + 1) + c(n, k).$$

## Remarks

Proof idea: look at the element $n + 1$ in a permutation of $n + 1$ elements. Either it is a fixed point, and the rest has $k$ cycles. Or it sits right after one of the $n$ other elements in its cycle.
Example: $c(4, 2) = 3\, c(3, 2) + c(3, 1) = 3 \cdot 3 + 2 = 11$.

## Explanations

### L1 human

Insert a new element into a permutation written as cycles.
It forms a new cycle of length $1$, or it goes into an existing cycle right after one of the $n$ old elements.
</document>

<document source="https://tamnd.github.io/math2/node/0C5M7AM6FH.md" tag="0C5M7AM6FH" kind="definition">
# Chromatic number

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]].
The chromatic number $\chi(G)$ is the smallest natural number $k$ such that $G$ is [[$k$-colourable|0Y7VDAX8Q3]].

## Remarks

For the [[complete graph|03WKB2H474]], $\chi(K_n) = n$. For the [[cycle graph|0QQ0F12962]] $C_n$, $\chi(C_n)$ is $2$ if $n$ is even and $3$ if $n$ is odd.
A [[greedy colouring|04AAKZNXHD]] shows that $\chi(G) \le \Delta(G) + 1$, where $\Delta(G)$ is the [[maximum degree|0KNAN9F27P]].
The chromatic number is at least the [[clique number|0WCKRBHFFD]].

## Explanations

### L1 human

The chromatic number is the least number of colours that you need to colour a graph properly.
For a map, it is the least number of colours that keeps neighbouring countries in different colours.
</document>

<document source="https://tamnd.github.io/math2/node/0C5PREPZWG.md" tag="0C5PREPZWG" kind="definition">
# Hyperbolic equilibrium point

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be continuously differentiable, and let $x^* \in U$ be an [[equilibrium point|0G4X36AE0X]] of $x' = f(x)$.
The point $x^*$ is hyperbolic if no eigenvalue of the Jacobian matrix $Df(x^*)$ has real part $0$.

## Remarks

A saddle, a node and a spiral of a planar linear system are hyperbolic. A centre is not.
For the pendulum $x' = y$, $y' = -\sin x$, the equilibrium $(\pi, 0)$ is hyperbolic, with eigenvalues $\pm 1$. The equilibrium $(0, 0)$ is not, with eigenvalues $\pm i$.
Near a hyperbolic equilibrium, the [[linearisation|09ZWYRQ41J]] describes the system qualitatively, by the [[Hartman–Grobman theorem|0S8WX8A4V9]].
If all eigenvalues have negative real part, the point is a sink. If all have positive real part, it is a source. Otherwise it is a saddle.

## Explanations

### L1 human

At a hyperbolic equilibrium, each direction of the linearisation either contracts or expands.
No direction is neutral, so small changes of the field do not change the type of the equilibrium.
</document>

<document source="https://tamnd.github.io/math2/node/0C5RDJE73X.md" tag="0C5RDJE73X" kind="statement">
# Number of subsets of a finite set

## Statement

Let $X$ be a finite set with $|X| = n$.
Let $\mathcal{P}(X)$ be the power set of $X$, the set of all subsets of $X$.
Then $|\mathcal{P}(X)| = 2^n$.

## Remarks

The map that sends a subset $S$ to its indicator function $1_S : X \to \{0, 1\}$ is a bijection. Thus the count follows from the [[number of functions between finite sets|08QQH9F7G0]].
Grouped by size, the count gives the identity in the node [[sum of binomial coefficients|0KAMF19NV8]].

## Explanations

### L1 human

To build a subset, decide for each element whether it is in or out.
That is 2 options per element, so $2 \cdot 2 \cdots 2 = 2^n$ subsets. For $\{a, b\}$ these are $\varnothing$, $\{a\}$, $\{b\}$ and $\{a, b\}$.
</document>

<document source="https://tamnd.github.io/math2/node/0C5SR4FC74.md" tag="0C5SR4FC74" kind="concept">
# Codomain

## Summary

The codomain of a function is the set in which its values lie.
</document>

<document source="https://tamnd.github.io/math2/node/0C5TP6RYSV.md" tag="0C5TP6RYSV" kind="statement">
# Polynomial functions are continuous

## Statement

Let $a_0, a_1, \dots, a_n$ be real numbers.
Then the function $p \colon \mathbb{R} \to \mathbb{R}$ given by $p(x) = a_0 + a_1 x + \cdots + a_n x^n$ is [[continuous|0P27W9RW0G]].

## Remarks

Constant functions and the identity $x \mapsto x$ are continuous. Sums and [[products of continuous functions are continuous|0VGMHCF4SC]], so induction on $n$ gives the result.
Mathlib states it as `Polynomial.continuous`.

## Explanations

### L1 human

Polynomials are built from $x$ and constants by addition and multiplication.
These operations keep continuity, so each polynomial is continuous.
</document>

<document source="https://tamnd.github.io/math2/node/0C62PZXX02.md" tag="0C62PZXX02" kind="concept">
# Multigraph

## Summary

A multigraph is a graph in which two vertices may be joined by more than one edge. Multigraphs model networks with parallel links, as in the problem of the Königsberg bridges. The node [[multigraph|01M5T0KDG7]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0C6J8BQXVR.md" tag="0C6J8BQXVR" kind="counterexample">
# Uncorrelated random variables that are not independent

## Counterexample

The claim "uncorrelated random variables are independent" is false. Here are two counterexamples.
First: let $X \sim \mathcal{N}(0, 1)$ and $Y = X^2$. Then $\operatorname{Cov}(X, Y) = E[X^3] - E[X] \, E[X^2] = 0 - 0 = 0$, since the odd moments of $X$ are $0$.
But $X$ and $Y$ are not independent: $P(|X| \le 1, \, Y > 1) = 0$, while $P(|X| \le 1) \, P(Y > 1) > 0$.
Second: let $X \sim \mathcal{N}(0, 1)$, and let $W$ be independent of $X$ with $P(W = 1) = P(W = -1) = 1/2$. Put $Y = WX$.
Then $Y \sim \mathcal{N}(0, 1)$ and $\operatorname{Cov}(X, Y) = E[W] \, E[X^2] = 0$. But $|Y| = |X|$, so $X$ and $Y$ are not independent.

## Remarks

In the second example, both $X$ and $Y$ are normal, but the pair $(X, Y)$ is not multivariate normal: $X + Y$ is $0$ with probability $1/2$.
So the rule "uncorrelated implies independent" needs a joint normal law, not only normal marginals.
The direction that is true is that [[independent variables are uncorrelated|077GN9HDR6]].

## Explanations

### L1 human

Covariance only detects linear relations. In the first example, $Y$ is a function of $X$, so it depends on $X$ completely, but the relation is not linear.
So zero correlation does not mean that two variables have nothing to do with each other.
</document>

<document source="https://tamnd.github.io/math2/node/0C6THFRMZR.md" tag="0C6THFRMZR" kind="definition">
# Tangent line to a circle

## Definition

Let $C$ be a [[circle|060W5CBZP6]] in the Euclidean plane $\mathbb{R}^2$, and let $T$ be a point of $C$.
A [[line|0MP9BCF6DP]] $\ell$ is tangent to $C$ at $T$ if $\ell \cap C = \{T\}$. The point $T$ is the point of tangency.

## Remarks

A line meets a circle in two points, in one point or in no point. A line with two common points is a secant line.
For the circle $(x - a)^2 + (y - b)^2 = r^2$ and a point $(x_1, y_1)$ on it, the tangent at this point has the equation $(x - x_1)(x_1 - a) + (y - y_1)(y_1 - b) = 0$.
The tangent at $T$ is the line through $T$ perpendicular to the radius to $T$. See [[a tangent is perpendicular to the radius|031JV5S54D]].
From a point outside the circle there are exactly two tangent lines to the circle.

## Explanations

### L1 human

A tangent line just touches a circle at one point and stays outside the circle. For the unit circle, the line $y = 1$ is tangent at $(0, 1)$. The line $y = 0$ is not tangent: it meets the circle at $(1, 0)$ and at $(-1, 0)$.
</document>

<document source="https://tamnd.github.io/math2/node/0C81YBVTWH.md" tag="0C81YBVTWH" kind="definition">
# Cartesian product

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The Cartesian product of $A$ and $B$ is the set
$$A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\}$$
of all [[ordered pairs|0JQWWFCS9R]] $(a, b)$ with $a \in A$ and $b \in B$.

## Remarks

Example: $\{1, 2\} \times \{x, y\} = \{(1, x), (1, y), (2, x), (2, y)\}$.
For finite sets with $m$ and $n$ elements, $A \times B$ has $mn$ elements.
The plane $\mathbb{R}^2$ is $\mathbb{R} \times \mathbb{R}$.
A [[binary relation|08EJC6Q25H]] from $A$ to $B$ is a subset of $A \times B$.

## Explanations

### L1 human

Think of a table with a row for each element of $A$ and a column for each element of $B$.
Each cell of the table is one pair $(a, b)$.
The Cartesian product is the set of all the cells.
</document>

<document source="https://tamnd.github.io/math2/node/0C8BR8XRTR.md" tag="0C8BR8XRTR" kind="concept">
# Normal operator

## Summary

A normal operator on a finite-dimensional inner product space is a linear map that commutes with its adjoint.
</document>

<document source="https://tamnd.github.io/math2/node/0C8QENHYZW.md" tag="0C8QENHYZW" kind="definition">
# Möbius function

## Definition

The Möbius function $\mu$ is an arithmetic function. For each positive integer $n$, it is given by
$$\mu(n) = \begin{cases} (-1)^{\omega(n)} & \text{if } n \text{ is squarefree}, \\ 0 & \text{otherwise}. \end{cases}$$
Here $\omega(n)$ is the [[number of distinct prime factors|0YVVXT3A0B]] of $n$. The word squarefree has the meaning of the [[definition of a squarefree integer|004JW7A2N5]].

## Remarks

So $\mu(1) = 1$, $\mu(p) = -1$ for each prime $p$, and $\mu(n) = 0$ when $p^2 \mid n$ for some prime $p$.
For example, $\mu(6) = 1$, $\mu(30) = -1$ and $\mu(12) = 0$.
The function $\mu$ is [[multiplicative|0VB9G5B034]], and it is the [[inverse of the constant function 1|0ASC73QEQT]] for the Dirichlet convolution.

## Explanations

### L1 human

The Möbius function is $0$ on numbers with a repeated prime factor.
On the other numbers, it is $1$ for an even number of prime factors and $-1$ for an odd number.
It is the sign rule of inclusion and exclusion, written as a function of $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0C8WYW4KYQ.md" tag="0C8WYW4KYQ" kind="concept">
# Greatest common divisor

## Summary

The greatest common divisor of two integers is the largest integer that divides both of them.
We write it as $\gcd(a, b)$.
The [[definition of the greatest common divisor|02WF878BH1]] gives the precise form and the convention for $\gcd(0, 0)$.
</document>

<document source="https://tamnd.github.io/math2/node/0C95TG69SG.md" tag="0C95TG69SG" kind="definition">
# Euclidean distance

## Definition

Let $n$ be a positive integer, and let $p = (p_1, \dots, p_n)$ and $q = (q_1, \dots, q_n)$ be points of $\mathbb{R}^n$.
The Euclidean distance between $p$ and $q$ is the real number
$$d(p, q) = \|p - q\| = \sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2 + \dots + (p_n - q_n)^2},$$
where $\|p - q\|$ is the [[length|0XKBKGTCPC]] of the vector $p - q$.

## Remarks

The function $d$ is a metric on $\mathbb{R}^n$. For all points $p$, $q$ and $r$: $d(p, q) \geq 0$, $d(p, q) = 0$ if and only if $p = q$, $d(p, q) = d(q, p)$, and $d(p, r) \leq d(p, q) + d(q, r)$.
For $n = 1$ the formula is $d(p, q) = |p - q|$.
Some applications use the squared distance $d(p, q)^2$, because it has no square root. The squared distance is not a metric, because it does not satisfy the triangle inequality.

## Explanations

### L1 human

To find the distance between two points in the plane, draw the right triangle with horizontal and vertical legs between them. The legs have lengths $|p_1 - q_1|$ and $|p_2 - q_2|$. The distance is the hypotenuse, so the Pythagorean theorem gives the formula. For $(1, 1)$ and $(4, 5)$ the distance is $\sqrt{9 + 16} = 5$.
</document>

<document source="https://tamnd.github.io/math2/node/0C9CMCN9BQ.md" tag="0C9CMCN9BQ" kind="concept">
# Contractive sequence

## Summary

A contractive sequence is a real sequence whose steps shrink at least by a fixed factor less than $1$. Such a sequence is Cauchy, so it converges. The node [[contractive sequence|0CZRT49QNG]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0C9J6W2A8A.md" tag="0C9J6W2A8A" kind="definition">
# Rooted tree

## Definition

A rooted tree is a pair $(T, r)$ of a [[tree|0MJ0JS72DZ]] $T$ and a vertex $r$ of $T$, the root.

Let $v \neq r$ be a vertex, and let $u$ be the neighbour of $v$ on the unique path from $v$ to $r$.
Then $u$ is the parent of $v$, and $v$ is a child of $u$.
The depth of $v$ is the [[distance|0WPZ5YPAGR]] from $r$ to $v$.

## Remarks

The root has no parent, and every other vertex has exactly one parent.
A vertex with no child is a leaf of the rooted tree. With this convention the root can be a leaf only when the tree has one vertex.
An ordered rooted tree also fixes an order of the children of each vertex. A [[binary tree|02A8EMC83C]] is a related recursive structure.

## Explanations

### L1 human

A rooted tree is a tree in which one vertex is chosen as the top.
Then each edge has a direction: from a parent toward its child, away from the root.
</document>

<document source="https://tamnd.github.io/math2/node/0C9MZY2RNA.md" tag="0C9MZY2RNA" kind="concept">
# Altitude of a triangle

## Summary

An altitude of a triangle is the segment from a vertex to the line of the opposite side, at a right angle to that line. Its length, the height, is the distance from the vertex to that line.
</document>

<document source="https://tamnd.github.io/math2/node/0C9S13WKMF.md" tag="0C9S13WKMF" kind="statement">
# Reverse triangle inequality

## Statement

For all real numbers $a$ and $b$,
$$\bigl| |a| - |b| \bigr| \le |a - b|.$$

## Remarks

Proof: the [[triangle inequality|03HF5PMY5T]] gives $|a| = |(a - b) + b| \le |a - b| + |b|$, thus $|a| - |b| \le |a - b|$. Exchange $a$ and $b$ for the other sign.
It shows that the function $x \mapsto |x|$ is continuous on $\mathbb{R}$.

## Explanations

### L1 human

Two numbers that are close to each other have absolute values that are also close.
The difference of the distances to $0$ is at most the distance between the two numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0C9ZKYSPYR.md" tag="0C9ZKYSPYR" kind="statement">
# The kernel is a normal subgroup

## Statement

Let $G$ and $H$ be [[groups|G5S1M9]] and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
Then the [[kernel|0W88CEGQP1]] $\ker \varphi$ is a [[normal subgroup|0CB0DX5BVV]] of $G$.

## Remarks

For $g \in G$ and $k \in \ker \varphi$, $\varphi(gkg^{-1}) = \varphi(g) \, e_H \, \varphi(g)^{-1} = e_H$.
Conversely, each normal subgroup $N$ of $G$ is the kernel of the projection $G \to G/N$.
Thus the normal subgroups of $G$ are exactly the kernels of the homomorphisms from $G$.

## Explanations

### L1 human

If an element goes to the identity, then each of its conjugates also goes to the identity.
So the kernel is closed under conjugation, which is the definition of a normal subgroup.
This is one reason why normal subgroups are important.

### L2 agent

The kernel of a homomorphism $\varphi : G \to H$ is the set of $g \in G$ with $\varphi(g) = e$. The result says that the kernel is a subgroup, and that it is normal.

The proof checks two properties:

1. The kernel is a subgroup. If $\varphi(a) = \varphi(b) = e$, then $\varphi(ab^{-1}) = e\,e^{-1} = e$.
2. The kernel is closed under conjugation. If $\varphi(k) = e$, then $\varphi(gkg^{-1}) = \varphi(g)\,e\,\varphi(g)^{-1} = e$.

A worked example with the determinant $\det : \mathrm{GL}_2(\mathbb{R}) \to \mathbb{R}^\times$:

1. The kernel is $\mathrm{SL}_2(\mathbb{R})$, the matrices with determinant $1$.
2. For $A \in \mathrm{SL}_2(\mathbb{R})$ and any invertible $P$, $\det(PAP^{-1}) = \det P \cdot 1 \cdot (\det P)^{-1} = 1$.
3. Thus $PAP^{-1} \in \mathrm{SL}_2(\mathbb{R})$, and the kernel is normal.

The result has a converse. Each normal subgroup $N$ is the kernel of the projection $G \to G/N$. Thus the normal subgroups of $G$ are exactly the kernels of the homomorphisms from $G$. This is why the quotient $G/\ker\varphi$ exists in the first isomorphism theorem.

A common error is to think that the image is also normal. The image is a subgroup of $H$, but it need not be normal. The inclusion of $\{e, (1\,2)\}$ in $S_3$ is a homomorphism, and its image is not normal in $S_3$.
</document>

<document source="https://tamnd.github.io/math2/node/0CARW59WD2.md" tag="0CARW59WD2" kind="definition">
# Sequential compactness

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $K \subseteq X$.
The set $K$ is sequentially compact when each sequence $(x_n)$ of points of $K$ has a subsequence $(x_{n_k})$ that converges to a point of $K$.
The space $X$ is sequentially compact when $X$ itself is a sequentially compact set.

## Remarks

Here a sequence $(y_k)$ converges to $y$ when each open set that contains $y$ contains $y_k$ for all large $k$. In a metric space, this is the [[usual convergence|06JYPJ4WHH]].
The interval $[0, 1]$ is sequentially compact by the Bolzano–Weierstrass theorem. The real line is not: the sequence $x_n = n$ has no convergent subsequence.
For metric spaces, [[compactness and sequential compactness agree|0K8SGQSV7G]]. In general topological spaces, neither notion implies the other.

## Explanations

### L1 human

In a sequentially compact set, a sequence cannot escape: some part of it always settles down to a point of the set.
In $(0, 1]$, the sequence $1/n$ tries to escape to $0$, which is outside the set. So $(0, 1]$ is not sequentially compact.
</document>

<document source="https://tamnd.github.io/math2/node/0CB0DX5BVV.md" tag="0CB0DX5BVV" kind="concept">
# Normal subgroup

## Summary

A normal subgroup of a group is a subgroup that is equal to each of its conjugates.
The normal subgroups are exactly the subgroups that give quotient groups, and exactly the kernels of homomorphisms.
</document>

<document source="https://tamnd.github.io/math2/node/0CBAB9QQ9X.md" tag="0CBAB9QQ9X" kind="concept">
# Complex numbers as pairs of reals

## Summary

The complex numbers can be constructed as ordered pairs of [[real numbers|08J2M1JYAG]].
On $\mathbb{R}^2$, define addition by $(a, b) + (c, d) = (a + c, b + d)$ and multiplication by $(a, b) \cdot (c, d) = (a c - b d, a d + b c)$.
With these operations, $\mathbb{R}^2$ is a field. It is the field $\mathbb{C}$ of [[complex numbers|09FTAET5Z4]].
The pair $(a, 0)$ is the real number $a$, and the pair $(0, 1)$ is the [[imaginary unit|0ESCX36MMC]] $i$.
Then $(a, b) = a + b i$, and the map $a + b i \mapsto (a, b)$ is a bijection $\mathbb{C} \to \mathbb{R}^2$.

## Remarks

This construction shows that the complex numbers exist, without the assumption of a mysterious square root of $-1$.
The bijection with $\mathbb{R}^2$ respects addition and multiplication by real numbers. It is the base of the [[complex plane|0CTE7A31XQ]].
Another construction is the quotient ring $\mathbb{R}[X]/(X^2 + 1)$.

## Explanations

### L1 human

A complex number is just a pair of real numbers with a special rule for multiplication.
The rule is made so that $(0, 1) \cdot (0, 1) = (-1, 0)$, which says $i^2 = -1$.
</document>

<document source="https://tamnd.github.io/math2/node/0CBQVVNBMP.md" tag="0CBQVVNBMP" kind="definition">
# Kurtosis

## Definition

Let $X$ be a real random variable with $E[X^4] < \infty$, mean $\mu$ and standard deviation $\sigma > 0$.
The kurtosis of $X$ is
$$\operatorname{Kurt}(X) = E\left[\left(\frac{X - \mu}{\sigma}\right)^4\right] = \frac{\mu_4}{\sigma^4},$$
where $\mu_4$ is the fourth [[central moment|0ZT24EC90K]]. The excess kurtosis is $\operatorname{Kurt}(X) - 3$.

## Remarks

Every [[normal distribution|07RXBVTVZB]] has kurtosis $3$, so its excess kurtosis is $0$.
For every $X$, $\operatorname{Kurt}(X) \ge 1$. The Bernoulli distribution with $p = 1/2$ reaches this bound.
Distributions with positive excess kurtosis are called leptokurtic. Those with negative excess kurtosis are called platykurtic.
The kurtosis does not change under $X \mapsto aX + b$ with $a \ne 0$.

## Explanations

### L1 human

Kurtosis compares the tails of a distribution with those of the normal distribution.
A high kurtosis means that rare extreme values are more frequent than for a normal variable with the same variance.
</document>

<document source="https://tamnd.github.io/math2/node/0CBRFQ7DVZ.md" tag="0CBRFQ7DVZ" kind="definition">
# Normal extension

## Definition

Let $L/K$ be an [[algebraic extension|09EYMKMSB9]].
The extension $L/K$ is normal if every [[irreducible polynomial|00X2A9G9W3]] $f \in K[x]$ that has a root in $L$ [[splits|0SHMC00RJV]] over $L$.

## Remarks

A finite extension $L/K$ is normal if and only if $L$ is the [[splitting field|02NZZ1J0FH]] of some polynomial over $K$.
Every extension of degree $2$ is normal. For example, $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ is normal.
The extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not normal: $x^3 - 2$ has a root in it, but its other two roots are not real.
Normality is not transitive: $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ are normal, but $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}$ is not.

## Explanations

### L1 human

A normal extension contains all the roots of a polynomial as soon as it contains one of them.
It has no lopsided gaps, like a field that contains one cube root of $2$ but not the others.
</document>

<document source="https://tamnd.github.io/math2/node/0CC4VC7KR3.md" tag="0CC4VC7KR3" kind="example">
# Logistic equation

## Example

Let $r > 0$ and $K > 0$ be constants. The logistic equation is
$$P'(t) = r\, P(t) \left(1 - \frac{P(t)}{K}\right).$$
For an initial value $P(0) = P_0 > 0$, its solution is
$$P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right) e^{-r t}}, \qquad t \ge 0.$$
With $r = K = 1$, the function $f(t) = 1 / \left(1 + e^{-t}\right)$ is the standard logistic function. It satisfies $f' = f(1 - f)$ and $f(0) = 1/2$.

## Remarks

The constant functions $P = 0$ and $P = K$ are the [[equilibrium points|0N8NGKDZDZ]]. For each $P_0 > 0$, $P(t) \to K$ as $t \to \infty$.
For small $P$, the equation is close to $P' = r P$, which is [[exponential growth|00SVP6W3CB]].
If $0 < P_0 < K/2$, the graph of $P$ is an S-shaped curve with its inflection point at the value $P = K/2$.
The equation is separable, and it is a [[Bernoulli equation|0T92VGGVWP]] with exponent $2$.

## Explanations

### L1 human

The logistic equation is a model of a population with limited resources.
When the population is small, it grows almost exponentially. When it is close to the capacity $K$, the growth slows down and stops.
The constant $K$ is the carrying capacity: the largest population that the environment can support.
</document>

<document source="https://tamnd.github.io/math2/node/0CC56MJ4FG.md" tag="0CC56MJ4FG" kind="definition">
# Harmonic conjugate

## Definition

Let $U \subseteq \mathbb{C}$ be an open set, and identify $\mathbb{C}$ with $\mathbb{R}^2$ by $x + iy \leftrightarrow (x, y)$.
Let $u : U \to \mathbb{R}$ be a harmonic function: $u$ is twice continuously differentiable and $u_{xx} + u_{yy} = 0$ on $U$.
A harmonic conjugate of $u$ is a function $v : U \to \mathbb{R}$ such that the function $f = u + iv$ is [[holomorphic|0DG0Z72CQD]] on $U$.

## Remarks

For example, $v = 2xy$ is a harmonic conjugate of $u = x^2 - y^2$ on $\mathbb{C}$, because $u + iv = z^2$.
If $U$ is a [[domain|097KNJFJ7V]], two harmonic conjugates of $u$ differ by a real constant.
If $v$ is a harmonic conjugate of $u$, then $-u$ is a harmonic conjugate of $v$, because $v - iu = -i(u + iv)$ is holomorphic.
On a disc, each harmonic function has a harmonic conjugate. On $\mathbb{C} \setminus \{0\}$, the harmonic function $\log |z|$ has none.

## Explanations

### L1 human

The [[real part of a holomorphic function is harmonic|031BG625Y3]]. A harmonic conjugate goes the other way: it starts with a harmonic function $u$ and finds the missing imaginary part $v$.
For $u = x$, the conjugate is $v = y$, because $x + iy = z$ is holomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/0CCPN09EMJ.md" tag="0CCPN09EMJ" kind="definition">
# Discriminant of a polynomial

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f = a_n x^n + \dots + a_0 \in K[x]$ with $n \ge 1$ and $a_n \neq 0$.
Let $f = a_n (x - r_1) \cdots (x - r_n)$ in a [[splitting field|02NZZ1J0FH]] of $f$.
The discriminant of $f$ is
$$\operatorname{Disc}(f) = a_n^{2n-2} \prod_{1 \le i < j \le n} (r_i - r_j)^2.$$

## Remarks

The product is symmetric in the roots, so $\operatorname{Disc}(f) \in K$, by the [[fundamental theorem of symmetric polynomials|0GW1ZZH0R4]]. It is a polynomial with integer coefficients in $a_0, \dots, a_n$.
So $\operatorname{Disc}(f) = 0$ if and only if $f$ has a [[repeated root|067F8YVYMH]].
For $f = a x^2 + b x + c$, $\operatorname{Disc}(f) = b^2 - 4ac$.
For $f = x^3 + p x + q$, $\operatorname{Disc}(f) = -4p^3 - 27q^2$.

## Explanations

### L1 human

The discriminant measures how far apart the roots of a polynomial are.
It is zero when two roots coincide.
For a real quadratic, its sign tells whether the roots are real or complex.
</document>

<document source="https://tamnd.github.io/math2/node/0CCQV3884B.md" tag="0CCQV3884B" kind="definition">
# Matrix–vector product

## Definition

Let $R$ be a ring, let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$, and let $x = (x_1, \dots, x_n)^T \in R^n$ be a column vector.
The *matrix–vector product* $Ax$ is the column vector in $R^m$ with entries
$$(Ax)_i = \sum_{j=1}^{n} a_{ij} x_j$$
for $1 \le i \le m$.

## Remarks

If $A_1, \dots, A_n$ are the columns of $A$, then $Ax = x_1 A_1 + \cdots + x_n A_n$, a [[linear combination|0V8F3E48E6]] of the columns.
A [[system of linear equations|0GT4WWGWH2]] is the equation $Ax = b$.
The map $x \mapsto Ax$ is linear, see [[multiplication by a matrix as a linear map|094V2VBJXJ]].
For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}$.

## Explanations

### L1 human

To multiply a matrix by a vector, take each row of the matrix in turn.
Multiply its entries by the entries of the vector, position by position, and add the results.
Each row gives one entry of the answer.
</document>

<document source="https://tamnd.github.io/math2/node/0CD87SFM2N.md" tag="0CD87SFM2N" kind="concept">
# Two-sided ideal

## Summary

A two-sided ideal of a ring is an additive subgroup that is closed under multiplication by ring elements on the left and on the right.
The two-sided ideals are the kernels of ring homomorphisms.
The [[definition of a two-sided ideal|0DDA6CY7GJ]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0CDAHW576M.md" tag="0CDAHW576M" kind="statement">
# A finite field of order q is the splitting field of x^q - x

## Statement

Let $F$ be a [[finite field|08RTDHX1SK]] of order $q = p^n$, with $p$ prime.
Then $F$ is a [[splitting field|02NZZ1J0FH]] of $x^q - x$ over its prime subfield $\mathbb{F}_p$.

## Remarks

The proof: [[$x^q - x = \prod_{a \in F} (x - a)$|0QWR8WTZEW]], so $x^q - x$ splits over $F$, and its roots are all the elements of $F$. So the roots generate $F$.
With the [[uniqueness of splitting fields|069A3BC3EP]], this gives the [[uniqueness of finite fields|010PKCQPXP]].

## Explanations

### L1 human

A finite field consists exactly of the roots of $x^q - x$.
So it is the smallest field that contains all these roots.
</document>

<document source="https://tamnd.github.io/math2/node/0CDMXGPSES.md" tag="0CDMXGPSES" kind="definition">
# Riemann sphere

## Definition

The Riemann sphere, or extended complex plane, is the set $\widehat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$, where $\mathbb{C}$ is the set of [[complex numbers|09FTAET5Z4]] and $\infty$ is one new point.
Its topology is as follows.
A set $V \subseteq \widehat{\mathbb{C}}$ is open if the following two conditions hold. First, $V \cap \mathbb{C}$ is open in the [[complex plane|0CTE7A31XQ]]. Second, in the case $\infty \in V$, $V$ also contains the set $\{ z \in \mathbb{C} : |z| > R \}$ for some $R > 0$.
Let $S^2 = \{ (x_1, x_2, x_3) \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 1 \}$ and $N = (0, 0, 1)$.
The stereographic projection $\sigma : S^2 \to \widehat{\mathbb{C}}$ is $\sigma(x_1, x_2, x_3) = \frac{x_1 + i x_2}{1 - x_3}$ for points other than $N$, and $\sigma(N) = \infty$.
The map $\sigma$ is a homeomorphism, so $\widehat{\mathbb{C}}$ is a sphere as a topological space.

## Remarks

A sequence $(z_n)$ of complex numbers tends to $\infty$ in $\widehat{\mathbb{C}}$ exactly when $|z_n| \to \infty$.
Some arithmetic extends to $\infty$: $z + \infty = \infty$ for $z \in \mathbb{C}$, and $\frac{z}{0} = \infty$ and $\frac{z}{\infty} = 0$ for $z \neq 0$, but the expressions $\infty - \infty$, $0 \cdot \infty$, $\frac{0}{0}$ and $\frac{\infty}{\infty}$ stay undefined.
With the value $\infty$ at each [[pole|0WPQYW8MTE]], a [[meromorphic function|0NSD1TPHR0]] on an open set $U$ becomes a continuous map from $U$ to $\widehat{\mathbb{C}}$.
The Riemann sphere is a compact Riemann surface, with the two charts $z$ near points of $\mathbb{C}$ and $\frac{1}{z}$ near $\infty$.

## Explanations

### L1 human

Imagine a ball that sits on the complex plane at $0$.
Draw a line from the top of the ball to a point of the plane; the line meets the ball at one more point.
Points far from $0$ go to points near the top, so the top of the ball plays the role of $\infty$.
</document>

<document source="https://tamnd.github.io/math2/node/0CDVDGWGEG.md" tag="0CDVDGWGEG" kind="statement">
# Degrees along an Eulerian trail

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]], and let $W$ be an [[Eulerian trail|0NNYDD1AVR]] of $G$ from $u$ to $v$.
If $u = v$, then every vertex of $G$ has even [[degree|0T0F6CCS9W]].
If $u \neq v$, then $u$ and $v$ have odd degree and every other vertex has even degree.

## Remarks

Proof idea: each time $W$ passes through a vertex, it uses two edges at that vertex. Only the start and the end use one edge without a partner.
Thus a graph with an Eulerian trail has $0$ or $2$ vertices of odd degree. The [[Seven Bridges of Königsberg|0YCW781Q9D]] have $4$.

## Explanations

### L1 human

Count the edges at a vertex as a walk uses them.
Every visit in the middle of the walk uses one edge in and one edge out.
Only the first and last vertices can be left with an odd count.
</document>

<document source="https://tamnd.github.io/math2/node/0CDVK9RQBH.md" tag="0CDVK9RQBH" kind="definition">
# Martingale

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space with a filtration $(\mathcal{F}_n)_{n \ge 0}$: σ-algebras with $\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \dots \subseteq \mathcal{F}$.
A martingale with respect to $(\mathcal{F}_n)$ is a sequence $(M_n)_{n \ge 0}$ of real random variables with these three properties:
1. $M_n$ is $\mathcal{F}_n$-measurable for each $n$.
2. $E[|M_n|] < \infty$ for each $n$.
3. $E[M_{n+1} \mid \mathcal{F}_n] = M_n$ almost surely for each $n$.

## Remarks

If "$=$" in property 3 is replaced by "$\ge$", the sequence is a submartingale. If it is replaced by "$\le$", it is a supermartingale.
By the [[law of total expectation|04RRB6HM00]], $E[M_n] = E[M_0]$ for each $n$.
Examples: the simple random walk $S_n$ and $S_n^2 - n$ are martingales for the filtration of the steps.
Doob's optional stopping theorem and Doob's martingale convergence theorem are the main tools of the theory.

## Explanations

### L1 human

A martingale is a fair game: given everything you know now, your expected fortune after the next round is your fortune now.
No betting strategy can change the expected value of a martingale at fixed times.
</document>

<document source="https://tamnd.github.io/math2/node/0CDWKDYWN6.md" tag="0CDWKDYWN6" kind="statement">
# Universal property of the free group

## Statement

Let $S$ be a set, let $F_S$ be the [[free group|0KTW26C80J]] on $S$, and let $G$ be a [[group|G5S1M9]].
For each function $f \colon S \to G$, there is exactly one [[group homomorphism|0JYTG5K9Y4]] $\varphi \colon F_S \to G$ with $\varphi(s) = f(s)$ for each $s \in S$.

## Remarks

Here $S$ is identified with the set of words of length $1$ in $F_S$.
The homomorphism is $\varphi(t_1 \cdots t_n) = f(t_1)^{\pm 1} \cdots f(t_n)^{\pm 1}$, with the sign of each letter.
This property characterises $F_S$ up to isomorphism, so some texts use it as the definition of a free group.
If the image of $f$ generates $G$, then $\varphi$ is surjective, so $G$ is a quotient of $F_S$.

## Explanations

### L1 human

To define a homomorphism out of a free group, you only choose the images of the generators.
There are no relations to check.
So each choice extends to exactly one homomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/0CDZ684TCD.md" tag="0CDZ684TCD" kind="concept">
# Work done by a force field

## Summary

The work done by a force field $F$ on a particle that moves along a curve $\gamma$ is the [[line integral|0H5MR9B6GT]] $W = \int_\gamma F \cdot d\mathbf{r}$.
Only the component of the force along the direction of motion contributes, positively when it points forward and negatively when it points backward.

## Remarks

For a constant force $F$ and a straight displacement $d$, the work is $F \cdot d$.
For a $C^1$ path, $W = \int_a^b F(\gamma(t)) \cdot \gamma'(t) \, dt$. The integrand $F \cdot \gamma'$ is the power, the rate at which the force does work.
If $F = -\nabla U$ for a potential energy $U$, then $W = U(\gamma(a)) - U(\gamma(b))$, and the work does not depend on the path. Gravity near the surface of the Earth is an example.
A force that is always orthogonal to the velocity does no work.

## Explanations

### L1 human

Work measures how much a force helps a motion.
Pushing a box forward does positive work, pushing against its motion does negative work, and pushing straight down on a sliding box does no work.
</document>

<document source="https://tamnd.github.io/math2/node/0CE768XNVP.md" tag="0CE768XNVP" kind="concept">
# Arctangent

## Summary

The arctangent is the inverse of the tangent function restricted to $(-\pi/2, \pi/2)$. It is defined on all of $\mathbb{R}$ and it is bounded. The node [[arctangent|065XW2V6C3]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0CEAT3418X.md" tag="0CEAT3418X" kind="definition">
# Conic section

## Definition

Let $A, B, C, D, E, F$ be real numbers, with $A$, $B$ and $C$ not all zero. Let
$$M = \begin{pmatrix} A & B/2 & D/2 \\ B/2 & C & E/2 \\ D/2 & E/2 & F \end{pmatrix}.$$
The set $Q = \{\, (x, y) \in \mathbb{R}^2 \;:\; A x^2 + B x y + C y^2 + D x + E y + F = 0 \,\}$ in [[Cartesian coordinates|0RWKYX8339]] is a non-degenerate conic section if $\det M \neq 0$ and $Q$ is not empty.

## Remarks

If $\det M = 0$, the set $Q$ is a degenerate conic: two lines, one line, one point or the empty set.
Each non-degenerate conic is an [[ellipse|0VKCCEXBJ1]], a [[parabola|04S2GMHX82]] or a [[hyperbola|0WN5G117YC]]. The [[discriminant $B^2 - 4AC$|0N5NR32NQG]] tells which one.
Equivalently, a non-degenerate conic is the intersection of a double cone with a plane that does not pass through the apex.
Apollonius of Perga wrote a systematic work on conic sections, about 200 BC.

## Explanations

### L1 human

Cut a double ice cream cone with a flat knife. Depending on the tilt of the knife, the cut edge is a closed oval, a U shape or a curve with two branches. These are the ellipse, the parabola and the hyperbola, and they all have equations of degree $2$.
</document>

<document source="https://tamnd.github.io/math2/node/0CEBCS0E0E.md" tag="0CEBCS0E0E" kind="statement">
# Number of permutations

## Statement

Let $X$ be a finite set with $|X| = n$.
Then the number of [[permutations|0BQMWCNS8S]] of $X$ is $n!$.

## Remarks

For $n = 3$ there are $3! = 6$ permutations of $\{1, 2, 3\}$: $123$, $132$, $213$, $231$, $312$ and $321$.
The count follows from the [[rule of product|0VPK4X73Q0]]. The image of the first element has $n$ choices, the image of the second has $n - 1$ choices, and so on.
It is the case $k = n$ of the [[number of injections|0X0N833FF8]].

## Explanations

### L1 human

To arrange $n$ different objects in a row, you have $n$ choices for the first place, then $n - 1$ for the second, and so on.
The product of these numbers is $n!$.
</document>

<document source="https://tamnd.github.io/math2/node/0CEC45PRJN.md" tag="0CEC45PRJN" kind="concept">
# Modular multiplicative inverse

## Summary

A modular multiplicative inverse of an integer $a$ modulo $n$ is an integer $x$ such that $a x$ leaves the remainder $1$ on division by $n$.
The [[definition of the modular multiplicative inverse|00GXTQMY8Z]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0CEGEASY49.md" tag="0CEGEASY49" kind="definition">
# Zero of a holomorphic function

## Definition

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
A zero of $f$ is a point $z_0 \in U$ with $f(z_0) = 0$.

## Remarks

The zeros of $\sin$ are the numbers $k\pi$ with $k \in \mathbb{Z}$. The [[complex exponential|054VXHMCA3]] has no zeros.
Each zero has an [[order|064N002TX4]], a positive integer or $\infty$.
If $f$ is not the zero function on a [[domain|097KNJFJ7V]], its zeros are [[isolated|0K83KEJ43P]].

## Explanations

### L1 human

A zero of a function is a point where the function takes the value $0$, as for real functions.
For example, $z^2 + 1$ has the zeros $i$ and $-i$.
</document>

<document source="https://tamnd.github.io/math2/node/0CES91M79A.md" tag="0CES91M79A" kind="definition">
# Polynomials in several variables

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $n \ge 1$ be an integer.
For $\alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{N}^n$, the monomial $x^\alpha$ is the formal product $x_1^{\alpha_1} \cdots x_n^{\alpha_n}$.
A polynomial in $x_1, \dots, x_n$ over $R$ is a finite sum $\sum_\alpha c_\alpha x^\alpha$ with $c_\alpha \in R$.
Two polynomials are equal if and only if they have the same coefficient $c_\alpha$ for each $\alpha$.
The ring $R[x_1, \dots, x_n]$ is the set of these polynomials, with coefficientwise addition and the product
$$\Big(\sum_\alpha c_\alpha x^\alpha\Big) \Big(\sum_\beta d_\beta x^\beta\Big) = \sum_\gamma \Big(\sum_{\alpha + \beta = \gamma} c_\alpha d_\beta\Big) x^\gamma.$$

## Remarks

There is a natural isomorphism $R[x_1, \dots, x_n] \cong R[x_1, \dots, x_{n-1}][x_n]$. So the ring can also be built by repeated use of the [[polynomial ring|0Q9EZS90GY]] in one variable.
The same construction works for any set of variables, with exponent maps that are zero at all but finitely many variables.
The polynomial $x_1 x_2 + x_3^2$ is an element of $\mathbb{Z}[x_1, x_2, x_3]$.

## Explanations

### L1 human

A polynomial in several variables is a finite sum of terms like $5 x^2 y z^3$.
You add and multiply such polynomials with the usual rules of algebra.
Each term is a coefficient times a product of powers of the variables.
</document>

<document source="https://tamnd.github.io/math2/node/0CEX9BRXRZ.md" tag="0CEX9BRXRZ" kind="statement">
# Classification of complex quadratic forms

## Statement

Let $V$ be a complex vector space of finite dimension $n$, and let $Q$ be a quadratic form on $V$.
Let $r$ be the [[rank|0ZC9YTXZRY]] of the symmetric matrix of $Q$ in a basis.
Then $V$ has a basis $(e_1, \dots, e_n)$ such that, for all $x_1, \dots, x_n \in \C$,
$$Q(x_1 e_1 + \cdots + x_n e_n) = x_1^2 + x_2^2 + \cdots + x_r^2.$$
So two quadratic forms on $V$ are equivalent if and only if their matrices have the same rank.

## Remarks

The proof starts from a [[diagonal form|0Z3FEWZ8HA]] $a_1 x_1^2 + \cdots + a_n x_n^2$ and replaces $e_i$ by $e_i / \sqrt{a_i}$ for each $a_i \ne 0$.
The same result holds over every algebraically closed field of characteristic not $2$.
Over $\R$, this fails: $x^2$ and $-x^2$ have the same rank but are not equivalent.

## Explanations

### L1 human

Over the complex numbers, every nonzero number is a square.
So all coefficients of a diagonal form can be made equal to $1$.
Only the number of squares remains, and it is the rank.
</document>

<document source="https://tamnd.github.io/math2/node/0CF0XGYHNJ.md" tag="0CF0XGYHNJ" kind="statement">
# exp(2 pi i/n) is a primitive root of unity

## Statement

Let $n$ be a positive integer.
Then $e^{2 \pi i / n}$ is a [[primitive root of unity|0J817RTVG6]].

## Remarks

Proof: for $1 \le k < n$, the number $\zeta^k = e^{2 \pi i k / n}$ has argument $2 \pi k / n$, which is strictly between $0$ and $2\pi$. Thus $\zeta^k \neq 1$.
For $n = 4$ this number is $i$, and for $n = 2$ it is $-1$.

## Explanations

### L1 human

The first root of unity after $1$, going counterclockwise around the circle, is a primitive one.
Its powers walk around the circle step by step and visit every $n$th root of unity before they return to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0CGCPX8Z70.md" tag="0CGCPX8Z70" kind="concept">
# Frobenius automorphism of a finite field

## Summary

The Frobenius automorphism of a finite field of characteristic $p$ is the map $x \mapsto x^p$.
It generates the group of all automorphisms of the field.
The [[definition of the Frobenius automorphism|0VP4TBVJ19]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0CGYNDCWRR.md" tag="0CGYNDCWRR" kind="object">
# Special orthogonal group

## Description

Let $n \ge 1$.
The *special orthogonal group* is the set
$$SO(n) = \{Q \in O(n) : \det Q = 1\}$$
of real orthogonal $n \times n$ matrices with [[determinant|00KDZRW4C0]] $1$, with matrix multiplication.
It is a normal subgroup of index $2$ of the [[orthogonal group|0TKYAEJ8N7]] $O(n)$.

## Remarks

Its elements are the *rotations* of $\R^n$ about the origin.
$SO(2)$ consists of the [[rotation matrices|0B9QNFGTHM]] $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$, and it is commutative.
Each element of $SO(3)$ is a rotation about an axis through the origin.
$SO(n)$ is a connected compact Lie group.

## Explanations

### L1 human

The special orthogonal group contains the orthogonal matrices that do not flip orientation.
These are the rotations.
Reflections have determinant $-1$, so they are not in this group.
</document>

<document source="https://tamnd.github.io/math2/node/0CH4VG15RE.md" tag="0CH4VG15RE" kind="concept">
# Disc of convergence

## Summary

The disc of convergence of a power series is the largest open disc around its center on which the series converges, see the [[definition|045EWAXY9P]].
Its radius is the radius of convergence, a number in $[0, \infty]$.
</document>

<document source="https://tamnd.github.io/math2/node/0CHE3J7QMX.md" tag="0CHE3J7QMX" kind="counterexample">
# Irreducible elements need not be prime

## Counterexample

In the integral domain $\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5} : a, b \in \mathbb{Z}\}$, the element $2$ is [[irreducible|052SDEPZH7]] but not [[prime|0MNHXBSX3M]].
Let $N(a + b\sqrt{-5}) = a^2 + 5b^2$. This norm is multiplicative, and the units are the elements of norm $1$, that is $\pm 1$.
If $2 = x y$ with nonunits $x, y$, then $N(x) N(y) = 4$, so $N(x) = 2$. But $a^2 + 5b^2 = 2$ has no integer solution. So $2$ is irreducible.
The element $2$ divides $(1 + \sqrt{-5})(1 - \sqrt{-5}) = 6$.
It does not divide $1 \pm \sqrt{-5}$, because $(1 \pm \sqrt{-5})/2$ is not in $\mathbb{Z}[\sqrt{-5}]$. So $2$ is not prime.

## Remarks

The same ring shows that factorization is not unique: $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$. See [[$\mathbb{Z}[\sqrt{-5}]$ is not a UFD|0BSJYCG7NM]].
In a [[unique factorization domain|06YMXJMXYS]], every irreducible element is prime.

## Explanations

### L1 human

In the integers, "cannot be factored" and "divides a factor of each product it divides" are the same property.
In $\mathbb{Z}[\sqrt{-5}]$, the number $2$ cannot be factored, but it divides $6 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ and neither factor.
So the two properties are different in general rings.
</document>

<document source="https://tamnd.github.io/math2/node/0CHQKZFK5S.md" tag="0CHQKZFK5S" kind="statement">
# Division algorithm

## Statement

Let $a$ and $b$ be integers with $b \neq 0$.
Then there is exactly one pair of integers $(q, r)$ such that $a = b q + r$ and $0 \le r < |b|$.

## Remarks

The integer $q$ is the quotient and the integer $r$ is the remainder of the division of $a$ by $b$.
The remainder is $0$ if and only if $b$ [[divides|096PW7Q014]] $a$.
For example, $-7 = 3 \cdot (-3) + 2$, so the quotient of $-7$ by $3$ is $-3$ and the remainder is $2$.

## Explanations

### L1 human

You can share $a$ objects into groups of $b$ objects, with some objects left over.
The theorem says that the number of full groups and the rest are unique, if the rest is smaller than $|b|$.
For example, $17 = 5 \cdot 3 + 2$: you get $3$ groups of $5$, and $2$ objects stay.
</document>

<document source="https://tamnd.github.io/math2/node/0CHR35YNJQ.md" tag="0CHR35YNJQ" kind="definition">
# Subbasis

## Definition

Let $X$ be a set and let $\mathcal{S}$ be a set of subsets of $X$.
The topology generated by $\mathcal{S}$ is the smallest [[topology|0CNQXVBHCX]] on $X$ that contains $\mathcal{S}$. It is the [[intersection|0839XNQXDK]] of all topologies on $X$ that contain $\mathcal{S}$.
If $(X, \tau)$ is a topological space and $\tau$ is the topology generated by $\mathcal{S}$, then $\mathcal{S}$ is a subbasis of $\tau$.

## Remarks

The open sets of the generated topology are the unions of finite intersections of members of $\mathcal{S}$. Here the empty intersection is $X$.
Thus the finite intersections of members of $\mathcal{S}$ form a [[basis|06EX0FHBTF]] of the generated topology.
The open rays $(-\infty, a)$ and $(b, \infty)$, with $a, b \in \mathbb{R}$, form a subbasis of the usual topology of $\mathbb{R}$.
The sets $\pi_i^{-1}(U)$, with $U$ open, form a subbasis of the [[product topology|04S9HX3GAP]].

## Explanations

### L1 human

A subbasis is a set of open sets that you want to have. The topology is the smallest one that makes them all open.
You get all open sets by taking finite intersections of them and then arbitrary unions.
</document>

<document source="https://tamnd.github.io/math2/node/0CHS7RBD39.md" tag="0CHS7RBD39" kind="concept">
# Euclidean domain

## Summary

A Euclidean domain is an integral domain with a division with remainder, in which the remainder is smaller than the divisor for some size function.
The integers and the polynomials over a field are the main examples.
The [[definition of a Euclidean domain|0GJ2HRG95T]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0CKGBEJX6B.md" tag="0CKGBEJX6B" kind="statement">
# Finite subgroups of the multiplicative group of a field are cyclic

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $G$ be a finite subgroup of the multiplicative group $K^\times$.
Then $G$ is cyclic.

## Remarks

The proof: for each $d$, the elements $g \in G$ with $g^d = 1$ are roots of $x^d - 1$.
So there are at most $d$ of them, by the [[bound on the number of roots|0KJVQV3E0F]], and a finite group with this property is cyclic.
For $K = \mathbb{C}$, the finite subgroups of $\mathbb{C}^\times$ are the groups of the $n$-th roots of unity.
The statement fails for division rings: the quaternion group $\{\pm 1, \pm i, \pm j, \pm k\}$ is a finite subgroup of the nonzero [[quaternions|0GE4DVQKR8]] that is not cyclic.

## Explanations

### L1 human

In a field, the equation $x^d = 1$ has at most $d$ solutions.
This forces every finite group of nonzero elements to be generated by one element.
</document>

<document source="https://tamnd.github.io/math2/node/0CMC2KAXFS.md" tag="0CMC2KAXFS" kind="definition">
# Cyclic quadrilateral

## Definition

Let $ABCD$ be a quadrilateral in the Euclidean plane $\mathbb{R}^2$, with the vertices $A$, $B$, $C$, $D$ in this order.
The quadrilateral $ABCD$ is cyclic if its four vertices are [[concyclic|049DCJZ448]]. The circle through them is its [[circumcircle|03VZJE6C8C]].

## Remarks

A convex quadrilateral $ABCD$ is cyclic if and only if its opposite angles add up to $\pi$: $\angle DAB + \angle BCD = \pi$. The direct part is Proposition 22 of Book III of Euclid's Elements.
Each rectangle is cyclic. A rhombus that is not a square is not cyclic.
Usually the quadrilateral is taken to be convex. A crossed quadrilateral with concyclic vertices is a crossed cyclic quadrilateral.

## Explanations

### L1 human

Each triangle fits in a circle, but a quadrilateral does not always. A quadrilateral is cyclic when one circle passes through all four corners. A simple test: the angles at two opposite corners add up to a straight angle.
</document>

<document source="https://tamnd.github.io/math2/node/0CMD0BCH8T.md" tag="0CMD0BCH8T" kind="concept">
# Principal ideal domain

## Summary

A principal ideal domain is an integral domain in which every ideal is generated by one element.
The integers and the polynomials over a field are examples.
The [[definition of a principal ideal domain|0RZYDVZ475]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0CMHSGTWXB.md" tag="0CMHSGTWXB" kind="concept">
# Equivalent matrices

## Summary

Two matrices of the same size are equivalent when $B = Q^{-1} A P$ for invertible matrices $P$ and $Q$; this holds exactly when they have the same rank.
</document>

<document source="https://tamnd.github.io/math2/node/0CN97GADE5.md" tag="0CN97GADE5" kind="statement">
# Abel's test

## Statement

Let $\sum a_n$ be a [[convergent|0CVVKY7GY2]] real series.
Let $(b_n)$ be a [[monotone|0KN22AAW5R]] and [[bounded|05FQ3VGAR9]] real sequence.
Then the series $\sum a_n b_n$ converges.

## Remarks

The sequence $(b_n)$ converges to some $b$, by the [[monotone convergence theorem|0FM6N4R0VB]]. Then $\sum a_n b_n = \sum a_n (b_n - b) + b \sum a_n$.
The first series converges by [[Dirichlet's test|0V9ME8GQE7]], since $b_n - b$ is monotone with limit $0$.
The hypotheses on $\sum a_n$ are stronger than in Dirichlet's test, and those on $(b_n)$ are weaker.

## History

Niels Henrik Abel proved the test in 1826.

## Explanations

### L1 human

Multiplying the terms of a convergent series by a bounded monotone sequence of weights keeps convergence.
The weights settle down to a limit, so they cannot disturb the series too much.
</document>

<document source="https://tamnd.github.io/math2/node/0CNQXVBHCX.md" tag="0CNQXVBHCX" kind="definition">
# Topological space

## Definition

Let $X$ be a set.
A topology on $X$ is a set $\tau$ of [[subsets|0FE8Q5S42Y]] of $X$ with these three properties:

1. The [[empty set|0ECVK59Y03]] $\varnothing$ and the set $X$ are in $\tau$.
2. For each family $(U_i)_{i \in I}$ of members of $\tau$, the [[union|05GK32H72H]] $\bigcup_{i \in I} U_i$ is in $\tau$.
3. For all $U, V \in \tau$, the [[intersection|0HBRW4E6G1]] $U \cap V$ is in $\tau$.

A topological space is a pair $(X, \tau)$ where $X$ is a set and $\tau$ is a topology on $X$.

## Remarks

The members of $\tau$ are the [[open sets|01ZQQDCERP]] of the space. Their complements are the [[closed sets|0G5AV9Q4RR]].
By induction, property 3 gives that each finite intersection of open sets is open. Infinite intersections need not be open.
Each metric gives a topology, the [[metric topology|0A6YRNZB2R]]. Other basic examples are the [[discrete topology|0S2TEHX4E1]] and the [[indiscrete topology|0XJ02JBD6S]].
One can also define a topology by its closed sets, or by the [[neighbourhoods of its points|09CJ00H3NW]].

## Explanations

### L1 human

A topology tells you which subsets of $X$ count as open, without any distances.
The rules copy the properties of open sets in metric spaces. Any union of open sets is open, and a finite intersection of open sets is open.
</document>

<document source="https://tamnd.github.io/math2/node/0CP5NDD21C.md" tag="0CP5NDD21C" kind="concept">
# First-countable space

## Summary

A topological space is first-countable when each point has a countable family of neighbourhoods that controls all neighbourhoods of the point.
In such spaces, sequences are enough to describe closures and continuity.
</document>

<document source="https://tamnd.github.io/math2/node/0CP7YY6NAG.md" tag="0CP7YY6NAG" kind="definition">
# Square matrix

## Definition

Let $R$ be a ring and let $n \ge 0$ be an integer.
A *square matrix* of order $n$ over $R$ is an $n \times n$ [[matrix|0VD9ZSWWJG]] over $R$.
Its *main diagonal* is the list of the entries $a_{11}, a_{22}, \dots, a_{nn}$.
The set of square matrices of order $n$ over $R$ is written $M_n(R)$.

## Remarks

The product of two $n \times n$ matrices is again an $n \times n$ matrix, so $M_n(R)$ is a [[ring|0FHVPRAE60]].
The [[trace|0XC1RDBY7A]], the [[determinant|00KDZRW4C0]] and the [[eigenvalues|0BQAA2BK3Q]] are defined for square matrices.
A square matrix represents a linear map from a space to itself.

## Explanations

### L1 human

A square matrix has as many rows as columns, like a $3 \times 3$ table.
Only square matrices can be multiplied by themselves, so only they have powers like $A^2$.
Many important notions, such as the determinant, apply only to square matrices.
</document>

<document source="https://tamnd.github.io/math2/node/0CQ357M0QS.md" tag="0CQ357M0QS" kind="statement">
# Impossibility of doubling the cube

## Statement

The number $\sqrt[3]{2}$ is not a [[constructible number|0FMN2MN7TM]].
So the cube cannot be [[doubled|04SD3F4FCR]] with straightedge and compass.

## Remarks

The proof: $x^3 - 2$ is irreducible over $\mathbb{Q}$, by [[Eisenstein's criterion|02BPKD7TMD]] with $p = 2$. So $[\mathbb{Q}(\sqrt[3]{2}) : \mathbb{Q}] = 3$.
Since $3$ is not a power of $2$, the number $\sqrt[3]{2}$ is not constructible, by the [[degree theorem|0SSSM1D475]].

## History

Pierre Wantzel proved the impossibility in 1837.

## Explanations

### L1 human

The edge of the doubled cube is the cube root of $2$.
Its minimal polynomial has degree $3$, but constructible numbers have degree a power of $2$.
</document>

<document source="https://tamnd.github.io/math2/node/0CQ9ZM383N.md" tag="0CQ9ZM383N" kind="counterexample">
# Absolute value is continuous but not differentiable at 0

## Counterexample

The function $f(x) = |x|$ is continuous on $\mathbb{R}$, but it is not differentiable at $0$.
The difference quotient is $\frac{|h|}{h}$, which is $1$ for $h > 0$ and $-1$ for $h < 0$. So it has no limit as $h \to 0$.

## Remarks

The [[one-sided derivatives|0HXEX0DPRA]] at $0$ exist: the right derivative is $1$ and the left derivative is $-1$.
So the converse of [[differentiable functions are continuous|07ZMX4ANJS]] fails.
For $x \ne 0$, the derivative is $x / |x|$, the sign of $x$. Mathlib states the failure at $0$ as `not_differentiableAt_abs_zero`.

## Explanations

### L1 human

The graph of $|x|$ is a V with a sharp corner at the origin.
At the corner, no single line fits the graph, so there is no tangent.
</document>

<document source="https://tamnd.github.io/math2/node/0CQHFPGJ45.md" tag="0CQHFPGJ45" kind="concept">
# Topological space

## Summary

A topological space is a set together with a choice of subsets, called open sets, that obey simple rules for unions and intersections.
This structure is enough to define continuity, convergence, compactness and connectedness without a notion of distance.
</document>

<document source="https://tamnd.github.io/math2/node/0CQQG3DEW9.md" tag="0CQQG3DEW9" kind="concept">
# Bernoulli process

## Summary

A Bernoulli process is an infinite sequence of independent trials with two outcomes, each with the same probability of success. A model is an infinite sequence of tosses of one coin.
The node [[Bernoulli process|02XPXR1FBK]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0CQW20XGFW.md" tag="0CQW20XGFW" kind="definition">
# Parabola

## Definition

Let $F$ be a point of the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$, and let $\ell$ be a [[line|0MP9BCF6DP]] that does not contain $F$. Let $d(P, F)$ be the [[Euclidean distance|0NPPGHQNXQ]], and let $d(P, \ell)$ be the [[distance from $P$ to $\ell$|01AAYNSBN6]].
The parabola with the focus $F$ and the directrix $\ell$ is the set
$$\{\, P \in \mathbb{R}^2 \;:\; d(P, F) = d(P, \ell) \,\}.$$

## Remarks

For $F = (a, 0)$ with $a > 0$ and $\ell$ the line $x = -a$, the parabola is the set of the points with $y^2 = 4 a x$.
For real numbers $p \neq 0$, $q$ and $r$, the graph of $y = p x^2 + q x + r$ is a parabola.
The [[eccentricity|0HVX863SAJ]] of each parabola is $1$. Any two parabolas are similar.
Menaechmus studied conic sections in the 4th century BC. Archimedes found the area of a parabolic segment, and the name is due to Apollonius.

## Explanations

### L1 human

A parabola is the set of points that are as far from a point as from a line. For the focus $(0, 1)$ and the line $y = -1$, the parabola is $y = x^2 / 4$.>>>
</document>

<document source="https://tamnd.github.io/math2/node/0CRMGPAH83.md" tag="0CRMGPAH83" kind="concept">
# Linear independence

## Summary

Vectors are linearly independent when no nontrivial linear combination of them is zero.
Equivalently, no vector of the family is a linear combination of the others.
</document>

<document source="https://tamnd.github.io/math2/node/0CS42KEM5C.md" tag="0CS42KEM5C" kind="statement">
# Uniform convergence of derivatives allows term-by-term differentiation

## Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be differentiable for each $n$.
Assume that $(f_n(x_0))$ converges for some $x_0 \in [a, b]$, and that $(f_n')$ converges [[uniformly|08FPABDQ4C]] on $[a, b]$ to a function $g$.
Then $(f_n)$ converges uniformly on $[a, b]$ to a differentiable function $f$, and $f' = g$.

## Remarks

The [[mean value theorem|0BARQY52N5]] applied to $f_n - f_m$ shows that $(f_n)$ is [[uniformly Cauchy|01XAY17QA3]].
Uniform convergence of $f_n$ alone is not enough. The functions $\frac{\sin(n^2 x)}{n}$ tend to $0$ uniformly, but their derivatives $n \cos(n^2 x)$ do not converge.
Mathlib states a version as `hasDerivAt_of_tendstoUniformly`.

## Explanations

### L1 human

Derivatives are sensitive to small wiggles, so uniform convergence of functions does not control them.
We need uniform convergence of the derivatives themselves.
</document>

<document source="https://tamnd.github.io/math2/node/0CT6CBGTYY.md" tag="0CT6CBGTYY" kind="statement">
# Exponential growth beats polynomial growth

## Statement

For every natural number $n$,
$$\lim_{x \to \infty} \frac{e^x}{x^n} = \infty.$$

## Remarks

For $x > 0$, the series gives $e^x > \frac{x^{n+1}}{(n+1)!}$. So $\frac{e^x}{x^n} > \frac{x}{(n+1)!}$, which [[tends to infinity|0Y12QYTJK3]].
Equivalently, $x^n e^{-x} \to 0$ as $x \to \infty$. The same holds for real exponents $n$.
So for each polynomial $p$, $p(x) e^{-x} \to 0$ as $x \to \infty$. Mathlib states the main claim as `Real.tendsto_exp_div_pow_atTop`.

## Explanations

### L1 human

The exponential function eventually grows faster than every power of $x$.
For example, $e^x > x^{10}$ for all $x \ge 36$, but not for $x = 35$.
</document>

<document source="https://tamnd.github.io/math2/node/0CTE7A31XQ.md" tag="0CTE7A31XQ" kind="concept">
# Complex plane

## Summary

The complex plane is the plane $\mathbb{R}^2$ in which the point $(a, b)$ represents the [[complex number|09FTAET5Z4]] $a + b i$.
The horizontal axis is the real axis and the vertical axis is the imaginary axis.
In this picture, addition of complex numbers is addition of vectors.
The [[modulus|0KW2X0TFBX]] $|z|$ is the distance from $z$ to $0$, and the [[argument|0ZZRSX91QR]] of $z$ is the angle from the positive real axis to $z$.

## Remarks

Multiplication by a complex number $w \neq 0$ scales the plane by $|w|$ and turns it by the argument of $w$.
The [[complex conjugate|0BJV24C4SY]] is the reflection in the real axis.

## Explanations

### L1 human

Draw the real numbers on a horizontal line, and draw the multiples of $i$ on a vertical line through $0$.
Then each complex number is a point of the plane: $3 + 2 i$ is three steps to the right and two steps up.
</document>

<document source="https://tamnd.github.io/math2/node/0CTM1EHG9P.md" tag="0CTM1EHG9P" kind="definition">
# Box topology

## Definition

Let $(X_i)_{i \in I}$ be a family of [[topological spaces|0CNQXVBHCX]] and let $X = \prod_{i \in I} X_i$ be the [[Cartesian product|0C81YBVTWH]].
The box topology on $X$ is the topology with [[basis|06EX0FHBTF]] the set of all products $\prod_{i \in I} U_i$, where each $U_i$ is open in $X_i$.

## Remarks

For finitely many factors, the box topology is the [[product topology|04S9HX3GAP]].
For infinite products, the box topology is finer than the product topology, and often strictly finer.
On $\mathbb{R}^{\mathbb{N}}$ with the box topology, the map $t \mapsto (t, t, t, \ldots)$ from $\mathbb{R}$ is not continuous, although each coordinate is continuous. The preimage of $\prod_{n \ge 1} (-1/n, 1/n)$ is $\{0\}$, which is not open.

## Explanations

### L1 human

The box topology lets you restrict all coordinates at the same time.
This looks natural, but it gives too many open sets, and simple maps into the product stop being continuous.
</document>

<document source="https://tamnd.github.io/math2/node/0CTNFGYMM2.md" tag="0CTNFGYMM2" kind="concept">
# Triangle

## Summary

A triangle is a polygon with three vertices and three sides. Its three vertices are points that do not lie on one line, and its sides are the segments that join them in pairs.
</document>

<document source="https://tamnd.github.io/math2/node/0CTV783HYT.md" tag="0CTV783HYT" kind="definition">
# Real sequence

## Definition

A real sequence is a [[sequence|0QE8STGJH2]] in the set $\mathbb{R}$ of real numbers.
Thus it is a function $a \colon \mathbb{N} \to \mathbb{R}$, and each term $a_n$ is a real number.

## Remarks

Examples are $a_n = 1/(n + 1)$, $a_n = (-1)^n$ and $a_n = 2^n$.
The real sequences form a real vector space under the termwise operations $(a_n) + (b_n) = (a_n + b_n)$ and $c\,(a_n) = (c\,a_n)$.
The [[limit of a sequence|0XKWSSY8P2]] is defined first for real sequences in most courses.

## Explanations

### L1 human

A real sequence is an endless list of real numbers $a_0, a_1, a_2, \dots$.
You can draw it as points $(n, a_n)$ in the plane, one point for each natural number $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0CV69WDRXW.md" tag="0CV69WDRXW" kind="concept">
# Partition function

## Summary

The partition function $p(n)$ gives the number of integer partitions of $n$. It is a classical object of additive number theory, with a product generating function and deep congruences. The node [[partition function|001CX45SWQ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0CVVKY7GY2.md" tag="0CVVKY7GY2" kind="definition">
# Convergent series

## Definition

Let $\sum a_n$ be a [[series|0JH4MN4K1C]] of real numbers with [[partial sums|0NB643K60Q]] $s_N = \sum_{n=0}^{N} a_n$.
The series is convergent if the sequence $(s_N)$ is a [[convergent sequence|0AZ3R52593]].

## Remarks

The series $\sum 1/2^n$ converges, since its partial sums $2 - 1/2^N$ tend to $2$.
The limit of the partial sums is the [[sum of the series|0K0H5E3Z5N]].
A series that does not converge is a [[divergent series|0NBRP6093V]].
Changing finitely many terms does not change whether a series converges. It can change the sum.

## Explanations

### L1 human

A series converges when its running totals approach a fixed number.
Adding more and more terms then changes the total less and less.
</document>

<document source="https://tamnd.github.io/math2/node/0CW52NV5TC.md" tag="0CW52NV5TC" kind="concept">
# Annulus

## Summary

An annulus is the region between two concentric circles, see the [[definition|0AJ0YVK743]].
In complex analysis, it is the open set $r < |z - c| < R$, and it is the natural domain of a Laurent series.
</document>

<document source="https://tamnd.github.io/math2/node/0CWDVN132S.md" tag="0CWDVN132S" kind="concept">
# Arccosine

## Summary

The arccosine is the inverse of the cosine function restricted to $[0, \pi]$. It is defined on $[-1, 1]$. The node [[arccosine|0RDP2HEFEW]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0CWK2NFSEW.md" tag="0CWK2NFSEW" kind="concept">
# Improper multiple integral

## Summary

An improper multiple integral extends the Riemann integral to unbounded regions or unbounded functions. It is a limit of integrals over bounded regions where the function is bounded.
For a non-negative function $f$ on $\mathbb{R}^n$ that is Riemann integrable on each cube $[-a, a]^n$, the improper integral is $\int_{\mathbb{R}^n} f = \lim_{a \to \infty} \int_{[-a, a]^n} f$, when this limit exists.

## Remarks

For non-negative functions, the limit does not depend on the choice of the exhausting sequence of bounded sets. For functions of both signs one usually requires that $\int |f|$ is finite.
Example: in polar coordinates, $\iint_{\mathbb{R}^2} e^{-(x^2 + y^2)} \, dA = \lim_{a \to \infty} \int_0^{2\pi} \int_0^a e^{-r^2} r \, dr \, d\theta = \pi$.
Example with a singularity: $\iint_{\|x\| \le 1} \|x\|^{-s} \, dA$ is finite for $s < 2$ and infinite for $s \ge 2$, by the same polar computation.
An improper [[double integral|0WAR4F23B4]] can fail to exist even when both iterated integrals exist, if $\int |f| = \infty$.

## Explanations

### L1 human

The Riemann integral needs a bounded function on a bounded region.
You may want to integrate over all of the plane, or near a point where $f$ blows up. Then integrate over larger and larger bounded pieces and take the limit.
</document>

<document source="https://tamnd.github.io/math2/node/0CWXCWP0Y0.md" tag="0CWXCWP0Y0" kind="definition">
# Automorphism group

## Definition

Let $G$ be a [[group|G5S1M9]].
An automorphism of $G$ is a [[group isomorphism|08MG7H8JW5]] from $G$ to $G$.
The automorphism group $\operatorname{Aut}(G)$ is the set of all automorphisms of $G$, with composition of maps as the operation.

## Remarks

The identity map is the identity element of $\operatorname{Aut}(G)$, and the inverse of an automorphism is its inverse map.
The automorphism group of $(\mathbb{Z}, +)$ has two elements, the identity and $n \mapsto -n$.
For an [[abelian group|082GHNPJ53]] $G$, the map $g \mapsto g^{-1}$ is an automorphism.

## Explanations

### L1 human

An automorphism is a way to rename the elements of a group so that all products stay correct.
The set of all such renamings is again a group, because you can compose and undo renamings.
For the integers, the only renamings are "do nothing" and "change all signs".
</document>

<document source="https://tamnd.github.io/math2/node/0CX86CR8JV.md" tag="0CX86CR8JV" kind="concept">
# Integrating factor for a non-exact equation

## Summary

Let $M\, dx + N\, dy = 0$ be an equation that is not [[exact|0EC7AGJ0PX]] on an open set $D$.
An integrating factor for it is a nowhere zero differentiable function $\mu$ on $D$ such that $\mu M\, dx + \mu N\, dy = 0$ is exact.
On a rectangle, with $M$, $N$ and $\mu$ of class $C^1$, the [[test for exactness|0MSCQNFTM8]] makes this condition $(\mu M)_y = (\mu N)_x$.
The two equations have the same solutions, because $\mu$ has no zero.

## Remarks

If $(M_y - N_x)/N$ is a function $g(x)$ of $x$ only, then $\mu(x) = e^{\int g(x)\, dx}$ is an integrating factor.
If $(N_x - M_y)/M$ is a function $h(y)$ of $y$ only, then $\mu(y) = e^{\int h(y)\, dy}$ is an integrating factor.
The [[integrating factor|01THH3Y3YP]] of a linear first-order equation is the first case: $y' + p y - q = 0$ has $M = p y - q$, $N = 1$ and $(M_y - N_x)/N = p$.

## Explanations

### L1 human

Some equations are not exact, but they become exact after multiplication by a good function.
In general, it is hard to find this function.
In two common cases, the factor depends on one variable only, and a simple integral gives it.
</document>

<document source="https://tamnd.github.io/math2/node/0CXTMAY5PX.md" tag="0CXTMAY5PX" kind="example">
# Harmonic series

## Example

The harmonic series is the series
$$\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots.$$
Its terms tend to $0$, but the series is a [[divergent series|0NBRP6093V]].
Its partial sums are the [[harmonic numbers|0D2Y6G4C4S]] $H_N = \sum_{n=1}^{N} 1/n$, and they tend to $+\infty$.

## Remarks

The node [[harmonic series diverges|0E9M24KS9J]] gives a proof.
It shows that the converse of the [[term test|0VVY4H92G2]] is false.
The partial sums grow very slowly, like $\ln N$. The node [[harmonic numbers grow like log n|05S2GSPMGA]] makes this precise.
The name comes from music. The wavelengths of the overtones of a string are $1/2, 1/3, 1/4, \dots$ of the wavelength of the string.

## Explanations

### L1 human

The terms $1, 1/2, 1/3, \dots$ become small, but not fast enough.
You can group them into blocks that each add up to at least $1/2$, so the total grows without bound.
</document>

<document source="https://tamnd.github.io/math2/node/0CXXJNG5B6.md" tag="0CXXJNG5B6" kind="statement">
# First isomorphism theorem

## Statement

Let $G$ and $H$ be [[groups|G5S1M9]] and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
Then the map $g \ker \varphi \mapsto \varphi(g)$ is a well-defined [[isomorphism|08MG7H8JW5]] from the [[quotient group|0KYX2N69J0]] $G / \ker \varphi$ to the [[image|0S21GNW9ZB]] $\operatorname{im} \varphi$.

## Remarks

Here $\ker \varphi$ is normal by [[this fact|0C9ZKYSPYR]], so the quotient exists.
If $\varphi$ is surjective, then $H \cong G / \ker \varphi$.
For example, the determinant gives $\mathrm{GL}_n(\mathbb{R}) / \mathrm{SL}_n(\mathbb{R}) \cong \mathbb{R}^\times$.
For finite $G$, a consequence is $|G| = |\ker \varphi| \cdot |\operatorname{im} \varphi|$.

## Explanations

### L1 human

A homomorphism collapses each coset of its kernel to one point of the image.
After this collapse, nothing else is lost: the quotient by the kernel and the image are the same group.
For example, reduction modulo $n$ shows that $\mathbb{Z}/n\mathbb{Z}$ is the image of $\mathbb{Z}$.

### L2 agent

The first isomorphism theorem says what a homomorphism forgets. It identifies the elements in each coset of the kernel, and it forgets nothing else.

The proof builds the isomorphism directly. Let $K = \ker\varphi$.

1. Define $\psi(gK) = \varphi(g)$. This is well defined, because $gK = hK$ gives $g^{-1}h \in K$ and thus $\varphi(g) = \varphi(h)$.
2. $\psi$ is a homomorphism, because $\psi(gK\,hK) = \varphi(gh) = \varphi(g)\varphi(h)$.
3. $\psi$ is onto $\operatorname{im}\varphi$ by construction.
4. $\psi$ is injective, because its kernel is the single coset $K$.

A worked example with $\varphi : \mathbb{Z} \to \mathbb{C}^\times$, $\varphi(n) = i^n$:

1. The image is $\{1, i, -1, -i\}$, a cyclic group of order $4$.
2. $i^n = 1$ exactly when $4$ divides $n$. Thus the kernel is $4\mathbb{Z}$.
3. By the theorem, $\mathbb{Z}/4\mathbb{Z} \cong \{1, i, -1, -i\}$.

Other standard examples are $\mathrm{GL}_n(\mathbb{R})/\mathrm{SL}_n(\mathbb{R}) \cong \mathbb{R}^\times$ through the determinant, and $S_n/A_n \cong \{1, -1\}$ through the sign.

For a finite group, the theorem gives $|G| = |\ker\varphi| \cdot |\operatorname{im}\varphi|$. This often bounds the image of a homomorphism without any computation.

A common error is to write $G/\ker\varphi \cong H$. This is true only when $\varphi$ is onto. In general the quotient is isomorphic to the image, which can be a proper subgroup of $H$.
</document>

<document source="https://tamnd.github.io/math2/node/0CXZ217QDP.md" tag="0CXZ217QDP" kind="statement">
# Linearity of the derivative

## Statement

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$, let $U \subseteq V$ be open, let $f, g \colon U \to W$, let $x \in U$, and let $c \in \mathbb{R}$.
Suppose that $f$ and $g$ are Fréchet differentiable at $x$, with [[Fréchet derivatives|08EVNYZF42]] $Df(x)$ and $Dg(x)$.
Then $f + g$ and $c f$ are Fréchet differentiable at $x$, and
$$D(f + g)(x) = Df(x) + Dg(x), \qquad D(c f)(x) = c \, Df(x).$$

## Remarks

The proof adds the two remainders: a sum of two terms that are $o(\|h\|)$ is again $o(\|h\|)$.
So the maps that are differentiable at $x$ form a vector space, and $f \mapsto Df(x)$ is a linear map on it.
The same rules hold for the [[partial derivatives|09EGC6F8PT]] and for the [[directional derivatives|06KWGHDN2J]].

## Explanations

### L1 human

The derivative of a sum is the sum of the derivatives, and constants come out, just as in one variable.
For example, $D(x^2 + 3 y)$ at a point $(a, b)$ is the linear map $h \mapsto 2 a h_1 + 3 h_2$, the sum of the derivatives of $x^2$ and of $3 y$.
</document>

<document source="https://tamnd.github.io/math2/node/0CYRECG2YQ.md" tag="0CYRECG2YQ" kind="definition">
# Transitive relation

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is transitive when, for all $a, b, c \in A$, $a \mathrel{R} b$ and $b \mathrel{R} c$ imply $a \mathrel{R} c$.

## Remarks

The relations $<$, $\le$ and $=$ on $\mathbb{R}$ are transitive, and so is divisibility on $\mathbb{Z}$.
"Is a parent of" is not transitive.
The smallest transitive relation that contains $R$ is the [[transitive closure|0PEJH7AC74]] of $R$.

## Explanations

### L1 human

Transitivity lets you chain steps.
From $2 < 5$ and $5 < 9$ you get $2 < 9$.
In the graph of a transitive relation, each path of two arrows has a direct arrow as a shortcut.
</document>

<document source="https://tamnd.github.io/math2/node/0CYYAF8AD2.md" tag="0CYYAF8AD2" kind="statement">
# Derivative of a continuous linear map

## Statement

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$ and let $A \colon V \to W$ be a bounded linear map.
Then $A$ is Fréchet differentiable at each point $x \in V$, and its [[Fréchet derivative|08EVNYZF42]] at $x$ is $A$ itself: $DA(x) = A$.

## Remarks

The proof is one line: $A(x + h) - A(x) - A h = 0$ for each $h$, so the remainder is $0$.
For $V = \mathbb{R}^n$ and $W = \mathbb{R}^m$, each linear map is bounded. So the derivative of $x \mapsto M x$, for an $m \times n$ matrix $M$, is the linear map with matrix $M$ at each point.
An affine map $x \mapsto A x + b$ has the same derivative $A$ at each point, because the derivative of a constant map is $0$.

## Explanations

### L1 human

The best linear approximation of a linear map is the map itself.
This is the analogue of the rule that the derivative of $x \mapsto c x$ is the constant $c$.
</document>

<document source="https://tamnd.github.io/math2/node/0CZA0NCFWP.md" tag="0CZA0NCFWP" kind="definition">
# Singular solution

## Definition

Let $F(x, y, y') = 0$ be a first-order [[ordinary differential equation|0P97V20RW7]], and let $y_s : I \to \mathbb{R}$ be a [[solution|04SW7WECS8]] on an open interval $I$.
The solution $y_s$ is singular if uniqueness fails at each point of its graph.
This means the following. For each $x_0 \in I$, there is another solution $y$ on an open interval containing $x_0$ with $y(x_0) = y_s(x_0)$ and $y'(x_0) = y_s'(x_0)$. This solution differs from $y_s$ on each neighbourhood of $x_0$.

## Remarks

Some authors use a weaker meaning: a solution at which the [[initial value problem|075VBY4K4X]] fails to have a unique solution at some point.
A singular solution is often the envelope of a family of solutions. It is not a member of the [[general solution|0SKYFQAFYD]] for any value of the constants.
Example: for $(y')^2 = 4 y$, the solutions $y = (x - C)^2$ are tangent to the solution $y = 0$, so $y = 0$ is singular. [[Clairaut's equation|0T6D4PE51C]] gives other examples.
By the [[uniqueness theorem|025PKW8AMR]], an equation $y' = f(x, y)$ with $f$ Lipschitz in $y$ has no singular solution.

## Explanations

### L1 human

A singular solution is a solution that touches other solutions at each of its points.
It often appears when the method of solution divides by an expression that can be zero.
So after you solve an equation, check what happens when the divisor is zero.
</document>

<document source="https://tamnd.github.io/math2/node/0CZRT49QNG.md" tag="0CZRT49QNG" kind="definition">
# Contractive sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]].
The sequence is contractive if there is a real number $C$ with $0 < C < 1$ such that
$$|a_{n+2} - a_{n+1}| \le C\, |a_{n+1} - a_n| \quad \text{for all } n \in \mathbb{N}.$$

## Remarks

Each contractive sequence is a [[Cauchy sequence|0QZMTN5AXX]]. By induction $|a_{n+1} - a_n| \le C^n |a_1 - a_0|$, and the [[geometric series|01X5CVV1Q5]] bounds $|a_m - a_n|$. So each contractive sequence converges.
If $f \colon \mathbb{R} \to \mathbb{R}$ is a contraction mapping, so that $|f(x) - f(y)| \le C|x - y|$ with $C < 1$, then each sequence $a_{n+1} = f(a_n)$ is contractive.
It is not enough that $|a_{n+2} - a_{n+1}| < |a_{n+1} - a_n|$ for all $n$. The partial sums of the harmonic series satisfy this, and they diverge.

## Explanations

### L1 human

In a contractive sequence, each step is at most a fixed fraction of the step before.
The steps shrink like a geometric sequence, so the total distance travelled is finite, and the sequence converges.
</document>

<document source="https://tamnd.github.io/math2/node/0D032NSZKY.md" tag="0D032NSZKY" kind="definition">
# Closed curve

## Definition

Let $a < b$, and let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a [[parametrised curve|0JF1XXGWB4]].
The curve $\gamma$ is closed if $\gamma(a) = \gamma(b)$.
A closed curve is simple if, in addition, $\gamma(s) \ne \gamma(t)$ for all $s < t$ in $[a, b]$ other than the pair $s = a$, $t = b$.

## Remarks

The unit circle $\gamma(t) = (\cos t, \sin t)$, $t \in [0, 2\pi]$, is a simple closed curve.
The curve $t \mapsto (\sin 2t, \sin t)$, $t \in [0, 2\pi]$, is closed but not simple: it passes through the origin at $t = 0$, $t = \pi$ and $t = 2\pi$.
The line integral of a vector field around a closed curve is often written $\oint_\gamma F \cdot d\mathbf{r}$. It is the circulation of $F$ around $\gamma$.
A simple closed curve in the plane bounds a region. This is the setting of [[Green's theorem|0HR3N02317]].

## Explanations

### L1 human

A closed curve is a path that comes back to where it started, like a lap around a track.
It is simple if it never crosses itself on the way.
</document>

<document source="https://tamnd.github.io/math2/node/0D0GBNDKDH.md" tag="0D0GBNDKDH" kind="concept">
# Martingale

## Summary

A martingale is a stochastic process in which the conditional expectation of the next value, given the past, is the present value. It models the fortune of a player in a fair game.
The node [[martingale|0CDVK9RQBH]] gives the precise definition in discrete time.
</document>

<document source="https://tamnd.github.io/math2/node/0D0SMQYEJA.md" tag="0D0SMQYEJA" kind="statement">
# Fibonacci recurrence

## Statement

Let $n$ be a natural number, and let $F_m$ be the [[Fibonacci numbers|0GA9K57A39]].
Then
$$F_{n+2} = F_{n+1} + F_n.$$

## Remarks

With the values $F_0 = 0$ and $F_1 = 1$, the recurrence determines the whole sequence.
The recurrence extends to negative indices with $F_{-n} = (-1)^{n+1} F_n$.

## Explanations

### L1 human

To get the next Fibonacci number, add the last two.
From $3$ and $5$ you get $8$, and from $5$ and $8$ you get $13$.
</document>

<document source="https://tamnd.github.io/math2/node/0D198GPMTJ.md" tag="0D198GPMTJ" kind="definition">
# Dense set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $D \subseteq X$.
The set $D$ is dense in $X$ when its [[closure|0DRKT1QJR7]] is $X$, that is, $\overline{D} = X$.

## Remarks

Equivalently, $D$ is dense if and only if $D$ meets each nonempty open set of $X$.
In a metric space, $D$ is dense if and only if for each $x \in X$ and each $\varepsilon > 0$ there is $y \in D$ with $d(x, y) < \varepsilon$.
The rationals $\mathbb{Q}$ are dense in $\mathbb{R}$: this is the [[density of the rationals|0STX0WRDNG]]. The irrationals are also dense in $\mathbb{R}$.

## Explanations

### L1 human

A dense set has points everywhere in the space: near each point of the space there are points of the dense set.
Each real number has rational numbers as close to it as you want.
</document>

<document source="https://tamnd.github.io/math2/node/0D1AHYPM8N.md" tag="0D1AHYPM8N" kind="definition">
# Tautology

## Definition

Let $\varphi$ be a formula of [[propositional logic|09K9MWD2QQ]] with the propositional variables $p_1, \dots, p_n$.
The formula $\varphi$ is a tautology when $\varphi$ is true for each of the $2^n$ assignments of truth values to $p_1, \dots, p_n$.

## Remarks

Examples are $p \lor \neg p$ and $p \to p$.
A formula that is false for each assignment is a contradiction.
A [[truth table|05RSK4Q657]] decides in a finite number of steps whether a formula is a tautology.

## Explanations

### L1 human

A tautology is true only because of its form.
The sentence "it rains or it does not rain" is true whatever the weather.
Its form $p \lor \neg p$ is a tautology.
</document>

<document source="https://tamnd.github.io/math2/node/0D1C9Y1EXA.md" tag="0D1C9Y1EXA" kind="definition">
# Complex differentiability

## Definition

Let $U \subseteq \mathbb{C}$ be an open set, let $f : U \to \mathbb{C}$ be a function, and let $z_0 \in U$.
The function $f$ is complex differentiable at $z_0$ if the limit
$$f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$$
exists in $\mathbb{C}$. Here $h$ runs through the nonzero [[complex numbers|09FTAET5Z4]] with $z_0 + h \in U$.
The number $f'(z_0)$ is the complex derivative of $f$ at $z_0$.
The limit means: for each $\varepsilon > 0$ there is $\delta > 0$ such that $\left| \frac{f(z_0 + h) - f(z_0)}{h} - f'(z_0) \right| < \varepsilon$ for all such $h$ with $0 < |h| < \delta$.

## Remarks

For example, $f(z) = z^2$ is complex differentiable at each $z_0$ with $f'(z_0) = 2 z_0$, because $\frac{(z_0 + h)^2 - z_0^2}{h} = 2 z_0 + h$.
The number $h$ can go to $0$ from each direction in the plane. This makes the condition much stronger than real differentiability, see [[the conjugate is not holomorphic|0Z8ATBPM5W]].
If $f$ is complex differentiable at $z_0$, then $f$ is continuous at $z_0$.
The [[Cauchy–Riemann equations|0T71SSYZ39]] describe complex differentiability with partial derivatives.

## Explanations

### L1 human

The definition is the same as the definition of the derivative of a real function, but $h$ is now a complex number.
The quotient must have the same limit when $h$ goes to $0$ along the real axis, along the imaginary axis, or along any other path.
</document>

<document source="https://tamnd.github.io/math2/node/0D1NY1ARDR.md" tag="0D1NY1ARDR" kind="concept">
# Geometric sequence

## Summary

A geometric sequence is a sequence in which each term is a fixed multiple of the term before it. The fixed factor is the common ratio. The node [[geometric sequence|0MNM54WB99]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0D1R4SVMBQ.md" tag="0D1R4SVMBQ" kind="statement">
# Correspondence theorem for rings

## Statement

Let $R$ be a [[ring|0894J980RK]], let $I$ be a two-sided ideal of $R$ and let $\pi \colon R \to R/I$ be the quotient map.
Let $\mathcal{A}$ be the set of the left [[ideals|0MYSA6MYNA]] $J$ of $R$ with $I \subseteq J$.
Let $\mathcal{B}$ be the set of the left ideals of $R/I$.
The map $\mathcal{A} \to \mathcal{B}$, $J \mapsto J/I = \pi(J)$, is a bijection.
Its inverse is $K \mapsto \pi^{-1}(K)$.
This bijection and its inverse preserve inclusion.

## Remarks

The same bijection holds for right ideals, for two-sided ideals, and for subrings that contain $I$.
For two-sided ideals $J \supseteq I$, the [[third isomorphism theorem|073C80EC63]] gives $(R/I)/(J/I) \cong R/J$.
The ideals of $\mathbb{Z}/12\mathbb{Z}$ correspond to the ideals $d\mathbb{Z}$ with $d \mid 12$, so there are six of them.

## Explanations

### L1 human

The ideals of a quotient ring $R/I$ are the ideals of $R$ that contain $I$, seen in the quotient.
So you can understand the ideals of $R/I$ from the ideals of $R$.
For example, the ideals of $\mathbb{Z}/12\mathbb{Z}$ come from the divisors $1, 2, 3, 4, 6, 12$ of $12$.
</document>

<document source="https://tamnd.github.io/math2/node/0D1VWQGZZ7.md" tag="0D1VWQGZZ7" kind="definition">
# Conjugate elements

## Definition

Let $G$ be a [[group|G5S1M9]] and let $a, b \in G$.
The element $b$ is conjugate to $a$ when there is $g \in G$ with $b = gag^{-1}$.
The element $gag^{-1}$ is called the conjugate of $a$ by $g$.

## Remarks

Conjugacy is an equivalence relation on $G$; its classes are the conjugacy classes.
In an [[abelian group|082GHNPJ53]], $gag^{-1} = a$, so each element is conjugate only to itself.
Conjugate elements have the same [[order|0WE2NHZJ7B]].
In the group of invertible matrices, conjugate matrices are similar matrices.

## Explanations

### L1 human

Conjugating $a$ by $g$ means: first undo $g$, then do $a$, then do $g$ again (reading the product from right to left).
The result does the same thing as $a$, but in a place that $g$ has moved.
For example, two rotations of a cube by the same angle about different axes are conjugate.
</document>

<document source="https://tamnd.github.io/math2/node/0D2MXBV21T.md" tag="0D2MXBV21T" kind="definition">
# Restriction of a function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $A$.
The restriction of $f$ to $S$ is the function $f|_S \colon S \to B$ with $f|_S(s) = f(s)$ for each $s \in S$.

## Remarks

The graph of $f|_S$ is $\{(s, f(s)) \mid s \in S\}$, a subset of the graph of $f$.
The function $f$ is an extension of $f|_S$.
Example: $x \mapsto x^2$ on $\mathbb{R}$ is not [[injective|0PVG1CRAJN]], but its restriction to $[0, \infty)$ is injective.

## Explanations

### L1 human

To restrict a function, you keep its rule and allow fewer inputs.
Restrictions often make a function injective, so that it has an inverse on a smaller domain.
</document>

<document source="https://tamnd.github.io/math2/node/0D2W7FS52T.md" tag="0D2W7FS52T" kind="statement">
# A set is closed iff it equals its closure

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then $S$ is [[closed|0G5AV9Q4RR]] in $X$ if and only if $\overline{S} = S$.

## Remarks

Proof: if $S$ is closed, then $\overline{S} \subseteq S$ by [[closure is the smallest closed superset|0SHC6Z304Z]], and $S \subseteq \overline{S}$ always holds. If $\overline{S} = S$, then $S$ is closed because the [[closure is closed|02TZDKG50Q]].
The dual statement is: $S$ is open if and only if $\operatorname{int} S = S$.

## Explanations

### L1 human

A closed set already contains all the points that you can approach from inside it.
So taking the closure adds nothing.
</document>

<document source="https://tamnd.github.io/math2/node/0D2Y6G4C4S.md" tag="0D2Y6G4C4S" kind="definition">
# Harmonic number

## Definition

Let $n$ be a natural number.
The $n$-th harmonic number is the rational number
$$H_n = \sum_{k=1}^{n} \frac{1}{k} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}.$$

## Remarks

The first values are $H_0 = 0$, $H_1 = 1$, $H_2 = 3/2$, $H_3 = 11/6$ and $H_4 = 25/12$.
The harmonic numbers are the partial sums of the [[harmonic series|0CXTMAY5PX]].
They satisfy $H_n - \ln n \to \gamma$, where $\gamma$ is the [[Euler–Mascheroni constant|00C5P9JZBK]].

## Explanations

### L1 human

The harmonic numbers are the running totals of $1 + 1/2 + 1/3 + \cdots$.
They grow without bound, but very slowly. One needs $n = 12367$ terms to pass $10$.
</document>

<document source="https://tamnd.github.io/math2/node/0D37QFPEZE.md" tag="0D37QFPEZE" kind="statement">
# Integration by substitution

## Statement

Let $g \colon [a, b] \to \mathbb{R}$ be [[continuously differentiable|06SKCHDQT5]], and let $f$ be continuous on an interval that contains $g([a, b])$.
Then
$$\int_a^b f(g(x)) \, g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du.$$

## Remarks

Let $F$ be an [[antiderivative|0NC2A3HWGT]] of $f$. By the [[chain rule|0RXX2PT1MQ]], $F \circ g$ is an antiderivative of $(f \circ g) \, g'$. Both sides equal $F(g(b)) - F(g(a))$.
The function $g$ need not be injective.
For example, with $u = x^2$, $\int_0^1 2x \cos(x^2) \, dx = \int_0^1 \cos u \, du = \sin 1$.
Mathlib states it as `intervalIntegral.integral_comp_mul_deriv`.

## Explanations

### L1 human

Substitution is the chain rule read backwards.
We replace an expression $g(x)$ by a new variable $u$, and $g'(x) \, dx$ by $du$.
</document>

<document source="https://tamnd.github.io/math2/node/0D3PFXEAKF.md" tag="0D3PFXEAKF" kind="statement">
# Consecutive Fibonacci numbers are coprime

## Statement

Let $(F_n)_{n \ge 0}$ be the Fibonacci sequence, given by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for $n \ge 0$.
Then $F_n$ and $F_{n+1}$ are [[coprime|0GFKP8F8AG]] for each $n \ge 0$, that is, $\gcd(F_n, F_{n+1}) = 1$.

## Remarks

A proof by induction: $\gcd(F_{n+1}, F_{n+2}) = \gcd(F_{n+1}, F_{n+2} - F_{n+1}) = \gcd(F_{n+1}, F_n)$, and $\gcd(F_0, F_1) = 1$.
A stronger fact is $\gcd(F_m, F_n) = F_{\gcd(m, n)}$ for all $m, n \ge 0$.
The [[Euclidean algorithm|0TRNT4X7J0]] on $F_{n+1}$ and $F_n$ takes many steps, and consecutive Fibonacci numbers give its worst case.

## Explanations

### L1 human

Two neighbours in the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, 13, \dots$ never have a common factor larger than $1$.
A common factor of two neighbours would also divide the term before them, and so on down to $F_1 = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0D3WA36K2Q.md" tag="0D3WA36K2Q" kind="concept">
# Order isomorphism

## Summary

An order isomorphism is a bijection between two partially ordered sets that keeps the order in both directions.
</document>

<document source="https://tamnd.github.io/math2/node/0D430QR02W.md" tag="0D430QR02W" kind="statement">
# Rouché's theorem

## Statement

Let $c \in \mathbb{C}$ and $r > 0$.
Let $f$ and $g$ be [[holomorphic|0DG0Z72CQD]] on an open set that contains the closed disc $\{ z : |z - c| \le r \}$.
Suppose that $|g(z)| < |f(z)|$ for each $z$ with $|z - c| = r$.
Then $f$ and $f + g$ have the same number of zeros in the open disc $|z - c| < r$, each zero counted as many times as its [[order|064N002TX4]].

## Remarks

The usual proof applies the [[argument principle|0PN2BXRS8G]] to $f$ and to $f + g$.
For example, on $|z| = 1$ we have $|z^5 + 1| \le 2 < 3 = |3z|$, so $z^5 + 3z + 1$ has exactly one zero in the open unit disc, as $3z$ has.
For a polynomial $p(z) = a_n z^n + \dots + a_0$ with $a_n \neq 0$, compare $p$ with $a_n z^n$ on a large circle; this gives $n$ zeros of $p$ and proves the [[fundamental theorem of algebra|0JP0W07Y9A]].

## Explanations

### L1 human

The theorem says that a small change does not change the number of zeros.
On the circle, $g$ is smaller than $f$, so $f + g$ stays close to $f$ in a relative sense.
Then $f + g$ turns around $0$ as many times as $f$, and so it has the same number of zeros inside.
</document>

<document source="https://tamnd.github.io/math2/node/0D474C7ES1.md" tag="0D474C7ES1" kind="concept">
# Convergents of a continued fraction

## Summary

The convergents of a continued fraction are the fractions obtained by cutting it off after finitely many terms.
The [[definition of the convergents|0SRWNR42BQ]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0D5KR33H7X.md" tag="0D5KR33H7X" kind="definition">
# Baire space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is a Baire space when it has the property that follows.
Let $(F_n)_{n \in \mathbb{N}}$ be a [[countable|0ZG61TFBES]] family of [[closed subsets|0G5AV9Q4RR]] of $X$ with $\operatorname{int} F_n = \emptyset$ for each $n$.
Then also $\operatorname{int} \bigcup_n F_n = \emptyset$.

## Remarks

Here $\operatorname{int}$ is the [[interior|0C38S0KMJK]].
Equivalently, in a Baire space each [[countable intersection of dense open sets is dense|0X2T63GFQQ]].
In a Baire space, each [[meagre set|0FQPYAY9PJ]] has empty interior. Thus a nonempty Baire space is not meagre in itself.
Each complete metric space is a Baire space by the [[Baire category theorem|0A7KGAYHGV]]. The space $\mathbb{Q}$ is not a Baire space.

## Explanations

### L1 human

In a Baire space, countably many thin closed sets can never fill up an open set.
This says that the space is large. It lets you prove that some objects exist: you show that the bad cases form a small set.
</document>

<document source="https://tamnd.github.io/math2/node/0D5YHQGDJY.md" tag="0D5YHQGDJY" kind="definition">
# Subspace topology

## Definition

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
The subspace topology on $S$ is
$$\tau_S = \{ S \cap U : U \in \tau \}.$$
With this topology, $S$ is a subspace of $X$.

## Remarks

The set $\tau_S$ is a topology on $S$, because intersection with $S$ commutes with unions and finite [[intersections|0HBRW4E6G1]].
A set open in $S$ need not be open in $X$. For example, $[0, 1)$ is open in the subspace $[0, 1]$ of $\mathbb{R}$, because $[0, 1) = [0, 1] \cap (-1, 1)$.
For a metric space, the subspace topology is the metric topology of the [[subspace metric|0T2WYDTRP2]].
The subspace topology is the coarsest topology on $S$ for which the [[inclusion map is continuous|065J822CKN]].

## Explanations

### L1 human

A subset of a space inherits a topology: you cut each open set of the big space down to the subset.
This is how the circle and the interval get their topologies from the plane and the line.
</document>

<document source="https://tamnd.github.io/math2/node/0D6FCW80A0.md" tag="0D6FCW80A0" kind="statement">
# A continuous injective function on an interval is strictly monotone

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous and injective.
Then $f$ is [[strictly monotone|0TR9SR9FTB]].

## Remarks

Suppose $x < y < z$ in $I$ with $f(y)$ not between $f(x)$ and $f(z)$. Then the [[intermediate value theorem|00KAFBR32E]] gives a value of $f$ taken twice, which contradicts injectivity.
Continuity is needed. The function on $[0, 2]$ with $f(x) = x$ for $x < 1$ and $f(x) = 3 - x$ for $x \ge 1$ is injective but not monotone.
Mathlib states a form for closed intervals as `ContinuousOn.strictMonoOn_of_injOn_Icc`.

## Explanations

### L1 human

A continuous function on an interval that goes up and then down would take some value twice.
So if it never repeats a value, it must always go up or always go down.
</document>

<document source="https://tamnd.github.io/math2/node/0D6J4B671D.md" tag="0D6J4B671D" kind="concept">
# Probability mass function

## Summary

The probability mass function of a discrete random variable gives the probability of each single value.
The node [[probability mass function|0WF4EJP91W]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0D6KH8Y59N.md" tag="0D6KH8Y59N" kind="statement">
# Polynomial ring over a UFD is a UFD

## Statement

Let $R$ be a [[unique factorisation domain|06YMXJMXYS]].
Then the [[polynomial ring|0Q9EZS90GY]] $R[x]$ is a unique factorisation domain.

## Remarks

The proof uses [[Gauss's lemma|0QFV5Y1Z5X]] and unique factorisation in $K[x]$, where $K$ is the field of fractions of $R$.
The irreducible elements of $R[x]$ are the irreducible elements of $R$ and the primitive polynomials of positive degree that are irreducible in $K[x]$.
By induction, $R[x_1, \dots, x_n]$ is a unique factorisation domain. In particular, $\mathbb{Z}[x]$ and $K[x, y]$ are.
These rings are not [[principal ideal domains|0RZYDVZ475]]: $(2, x)$ in $\mathbb{Z}[x]$ and $(x, y)$ in $K[x, y]$ are not principal.

## Explanations

### L1 human

Integer polynomials factor uniquely into irreducible pieces, like integers do.
The pieces are prime numbers and irreducible primitive polynomials.
For example, $2x^2 - 2 = 2 (x - 1)(x + 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/0D6KXBHKC0.md" tag="0D6KXBHKC0" kind="concept">
# Power series in the complex plane

## Summary

A power series is a series $\sum_{n \ge 0} a_n (z - c)^n$ with complex coefficients $a_n$, a center $c$ and a complex variable $z$, see the [[definition|0VARVQ9ZW1]].
It converges on a disc, and its sum is holomorphic there.
</document>

<document source="https://tamnd.github.io/math2/node/0D6RMJ10M9.md" tag="0D6RMJ10M9" kind="definition">
# Unitary matrix

## Definition

Let $n \ge 1$ and let $U$ be a complex $n \times n$ matrix.
The matrix $U$ is *unitary* when
$$U^* U = I_n,$$
where $U^*$ is the [[conjugate transpose|0DXKE10PHF]] and $I_n$ is the [[identity matrix|06DW9BAWB1]].

## Remarks

Equivalently, $U$ is invertible with $U^{-1} = U^*$, and then also $U U^* = I_n$.
Equivalently, the columns of $U$ form an orthonormal basis of $\C^n$ for the standard inner product.
The eigenvalues of a unitary matrix have absolute value $1$, and $\abs{\det U} = 1$.
A real unitary matrix is an [[orthogonal matrix|0W02QESPJQ]].

## Explanations

### L1 human

Unitary matrices are the complex version of orthogonal matrices.
They keep all lengths and inner products in $\C^n$.
In quantum mechanics, they describe how a closed system evolves in time.
</document>

<document source="https://tamnd.github.io/math2/node/0D6WHTTH7T.md" tag="0D6WHTTH7T" kind="definition">
# Conjugate subgroup

## Definition

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \in G$.
The conjugate of $H$ by $g$ is the set $gHg^{-1} = \{ghg^{-1} : h \in H\}$.
Two subgroups $H$ and $K$ of $G$ are conjugate when $K = gHg^{-1}$ for some $g \in G$.

## Remarks

The set $gHg^{-1}$ is a subgroup of $G$, and $h \mapsto ghg^{-1}$ is an isomorphism from $H$ to $gHg^{-1}$.
Conjugacy of subgroups is an equivalence relation on the set of subgroups of $G$.
A subgroup that equals each of its conjugates is a normal subgroup.

## Explanations

### L1 human

Conjugating a subgroup moves it to another position inside the group, but keeps its shape.
In the symmetric group $S_3$, the three subgroups of order $2$ are conjugate to each other.
Conjugate subgroups have the same order and the same structure.
</document>

<document source="https://tamnd.github.io/math2/node/0D752BG155.md" tag="0D752BG155" kind="statement">
# Zero is not a successor

## Statement

Let $S \colon \mathbb{N} \to \mathbb{N}$ be the [[successor function|0832XVR512]] of the natural numbers.
For each natural number $n$, $S(n) \neq 0$.

## Remarks

This is one of the [[Peano axioms|0QFT7Z76N5]]. In a construction of $\mathbb{N}$, it is a theorem.
With the [[injectivity of the successor function|0TBYMWPWWB]], it shows that the numbers $0, S(0), S(S(0)), \dots$ are all different.

## Explanations

### L1 human

When you count up from any natural number, you never get back to $0$.
Thus the natural numbers do not go around in a circle, like the hours on a clock do.
</document>

<document source="https://tamnd.github.io/math2/node/0D7EZ10V2R.md" tag="0D7EZ10V2R" kind="concept">
# Binary tree

## Summary

A binary tree is a recursive structure. It is empty, or it is a root with an ordered pair of binary trees as left and right subtrees. Binary trees are a basic data structure, and they are counted by the Catalan numbers. The node [[binary tree|02A8EMC83C]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0D7GRZ2ADX.md" tag="0D7GRZ2ADX" kind="statement">
# The exponential function is convex

## Statement

The [[exponential function|0RMVS7RXZX]] is strictly convex on $\mathbb{R}$.
That is, $e^{t x + (1 - t) y} < t e^x + (1 - t) e^y$ for all real $x \ne y$ and $t \in (0, 1)$.

## Remarks

The second derivative of $e^x$ is [[$e^x$ itself|0EX9GFJ0B3]], which is positive. So [[the second derivative test for convexity|0Z41C17TGP]] applies, in its strict form.
The tangent line at $0$ gives the inequality [[$1 + x \le e^x$|012C7GFF20]].
Mathlib states convexity as `convexOn_exp`.

## Explanations

### L1 human

The graph of $e^x$ bends upward everywhere.
Every chord of the graph lies above the graph.
</document>

<document source="https://tamnd.github.io/math2/node/0D81KWH9ZM.md" tag="0D81KWH9ZM" kind="definition">
# Bridge

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $e$ be an edge of $G$.
Let $G - e$ be the graph that $G$ gives when $e$ is deleted and all vertices are kept.
The edge $e$ is a bridge of $G$ if $G - e$ has more [[connected components|01VVP10MWJ]] than $G$.

## Remarks

Let $e = uv$. Then $e$ is a bridge if and only if $u$ and $v$ are not [[reachable|0J8XE7W68F]] from each other in $G - e$. In this case, deleting $e$ increases the number of components by exactly one.
In a [[tree|0MJ0JS72DZ]] every edge is a bridge. In a [[cycle graph|0QQ0F12962]] no edge is a bridge.
[[Bridges are the edges on no cycle|0FHM8DB6PM]].

## Explanations

### L1 human

A bridge is an edge that holds the graph together: delete it, and one piece breaks into two.
Think of the only road between two towns.
</document>

<document source="https://tamnd.github.io/math2/node/0D86GJEZ39.md" tag="0D86GJEZ39" kind="statement">
# Variance of a sum

## Statement

Let $X_1, \dots, X_n$ be real random variables on one probability space with $E[X_i^2] < \infty$ for each $i$. Then
$$\operatorname{Var}\left(\sum_{i=1}^n X_i\right) = \sum_{i=1}^n \operatorname{Var}(X_i) + 2 \sum_{1 \le i < j \le n} \operatorname{Cov}(X_i, X_j).$$
In particular, $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2 \operatorname{Cov}(X, Y)$.

## Remarks

Proof: $\operatorname{Var}(S) = \operatorname{Cov}(S, S)$ for $S = \sum_i X_i$. Expand by the bilinearity of the [[covariance|0DJ8W3HP2D]]: $\operatorname{Cov}(S, S) = \sum_{i, j} \operatorname{Cov}(X_i, X_j)$.
Also $\operatorname{Var}(X - Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) - 2 \operatorname{Cov}(X, Y)$.
More generally, $\operatorname{Var}\left(\sum_i a_i X_i\right) = \sum_{i, j} a_i a_j \operatorname{Cov}(X_i, X_j)$ for real $a_i$.

## Explanations

### L1 human

The spread of a sum is the sum of the spreads plus a correction for how the terms move together.
If the terms tend to move in the same direction, the sum spreads more. If they tend to compensate each other, it spreads less.
</document>

<document source="https://tamnd.github.io/math2/node/0D8HE681SC.md" tag="0D8HE681SC" kind="concept">
# Construction of the reals from Cauchy sequences

## Summary

This construction defines the [[real numbers|08J2M1JYAG]] as limits of sequences of [[rational numbers|0NCJ5WXN6Z]].
A sequence $(x_n)$ of rational numbers is a Cauchy sequence when, for each rational $\varepsilon > 0$, there is $N$ with $|x_m - x_n| < \varepsilon$ for all $m, n \ge N$.
Two Cauchy sequences $(x_n)$ and $(y_n)$ are equivalent when $x_n - y_n$ tends to $0$: for each rational $\varepsilon > 0$, there is $N$ with $|x_n - y_n| < \varepsilon$ for all $n \ge N$.
A real number is an equivalence class of Cauchy sequences of rational numbers.
Addition and multiplication act term by term: $[(x_n)] + [(y_n)] = [(x_n + y_n)]$ and $[(x_n)] \cdot [(y_n)] = [(x_n y_n)]$.

## Remarks

The rational number $q$ corresponds to the class of the constant sequence $(q, q, q, \dots)$.
The sequence $1, 1.4, 1.41, 1.414, \dots$ of decimal approximations represents $\sqrt{2}$.
The result is a [[complete ordered field|0A9E6X0S3E]], with the [[order on the reals|0337CYB0VD]]. The same method completes any metric space.

## Explanations

### L1 human

A real number is something that rational numbers can approximate as well as you like.
The construction takes the approximating sequences themselves as the new numbers.
Two sequences that get closer and closer to each other count as the same real number.
</document>

<document source="https://tamnd.github.io/math2/node/0D8T984XEE.md" tag="0D8T984XEE" kind="statement">
# Limits are unique in Hausdorff spaces

## Statement

Let $X$ be a [[Hausdorff space|0MXB4CKHFW]] and let $(x_n)_{n \ge 1}$ be a sequence in $X$.
Say that $(x_n)$ converges to $x \in X$ when for each [[neighbourhood|09CJ00H3NW]] $N$ of $x$ there is $n_0$ with $x_n \in N$ for all $n \ge n_0$.
If $(x_n)$ converges to $x$ and to $y$, then $x = y$.

## Remarks

Proof: if $x \neq y$, take disjoint open sets $U \ni x$ and $V \ni y$. For large $n$, the point $x_n$ is in both $U$ and $V$, which is impossible.
The same holds for nets and filters. In fact, a space is Hausdorff if and only if each net in it has at most one limit.
In an [[indiscrete space|0XJ02JBD6S]] with at least two points, each sequence converges to each point.

## Explanations

### L1 human

A sequence cannot come close to two different points at the same time, if the points have disjoint neighbourhoods.
So in Hausdorff spaces, you can speak of the limit of a sequence.
</document>

<document source="https://tamnd.github.io/math2/node/0D939CMME7.md" tag="0D939CMME7" kind="definition">
# Connected space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is connected when there are no [[open sets|01ZQQDCERP]] $U, V \subseteq X$ with $U \neq \varnothing$, $V \neq \varnothing$, $U \cap V = \varnothing$ and $U \cup V = X$.

## Remarks

Equivalently, the only subsets of $X$ that are both open and [[closed|0G5AV9Q4RR]] are $\varnothing$ and $X$.
Equivalently, each [[continuous map|0SY9ECQ19Q]] from $X$ to the discrete space $\{0, 1\}$ is constant.
The interval $[0, 1]$ is connected. The set $[0, 1] \cup [2, 3]$ is not connected.
Under this definition the empty space is connected. Some authors require a connected space to be nonempty.

## Explanations

### L1 human

A connected space cannot be split into two separate open parts.
Connectedness is the topological form of the idea that a space is in one piece.
</document>

<document source="https://tamnd.github.io/math2/node/0D98771SM0.md" tag="0D98771SM0" kind="definition">
# Tangent vector of a curve

## Definition

Let $I \subseteq \mathbb{R}$ be an interval, let $\gamma \colon I \to \mathbb{R}^n$ be a [[parametrised curve|0JF1XXGWB4]], and let $t_0 \in I$.
If the limit
$$\gamma'(t_0) = \lim_{h \to 0} \frac{\gamma(t_0 + h) - \gamma(t_0)}{h} = \big( \gamma_1'(t_0), \dots, \gamma_n'(t_0) \big)$$
exists, it is the tangent vector, or velocity vector, of $\gamma$ at $t_0$.
Its norm $\|\gamma'(t_0)\|$ is the speed of $\gamma$ at $t_0$.
If $\gamma'(t_0) \ne 0$, the unit tangent vector is $T(t_0) = \gamma'(t_0) / \|\gamma'(t_0)\|$, and the tangent line at $\gamma(t_0)$ is $\{\gamma(t_0) + s \, \gamma'(t_0) : s \in \mathbb{R}\}$.

## Remarks

At an end point of $I$, the limit is one-sided.
For the circle $\gamma(t) = (\cos t, \sin t)$, $\gamma'(t) = (-\sin t, \cos t)$. It is orthogonal to $\gamma(t)$ and has length $1$.
The tangent vector can be zero. For $\gamma(t) = (t^2, t^3)$, $\gamma'(0) = (0, 0)$, and the image has a cusp at the origin.
The tangent vector depends on the parametrisation, but for a [[regular curve|06JMR2XWXW]] its direction does not change under an increasing change of parameter.

## Explanations

### L1 human

The tangent vector points in the direction in which the curve moves at time $t_0$.
Its length tells how fast the point moves.
</document>

<document source="https://tamnd.github.io/math2/node/0D9NDGE9KP.md" tag="0D9NDGE9KP" kind="statement">
# Uniqueness of inverses

## Statement

Let $G$ be a [[group|G5S1M9]] with identity element $e$, and let $a, b \in G$.
If $ab = e$, then $b = a^{-1}$.

## Remarks

In particular, each element of a group has exactly one [[inverse|0ZNZH0EXFX]].
The same argument works in a [[monoid|0N6R902FG1]]: if $ba = e$ and $ac = e$, then $b = b(ac) = (ba)c = c$.
Two consequences are $(a^{-1})^{-1} = a$ and $(ab)^{-1} = b^{-1} a^{-1}$.

## Explanations

### L1 human

In a group, you only need to check one equation to know that $b$ is the inverse of $a$.
Multiply $ab = e$ on the left by $a^{-1}$, and you get $b = a^{-1}$.
Thus the notation $a^{-1}$ makes sense: there is only one inverse.

### L2 agent

The definition of a group asks for an inverse $b$ with $ab = e$ and $ba = e$. This result shows that one of the two equations is enough, and that the inverse is unique.

The proof uses only associativity and the identity:

1. Suppose that $ab = e$.
2. Multiply on the left by $a^{-1}$: $a^{-1}(ab) = a^{-1}e$.
3. By associativity, the left side is $(a^{-1}a)b = eb = b$.
4. The right side is $a^{-1}$. Thus $b = a^{-1}$.

A worked example in $\mathbb{Z}/7\mathbb{Z}$ under multiplication of the units:

1. We look for the inverse of $3$.
2. We have $3 \cdot 5 = 15 = 2 \cdot 7 + 1$, so $3 \cdot 5 = 1$.
3. By the result, $5$ is the only inverse of $3$. There is no need to check $5 \cdot 3$.

Two standard consequences follow at once. First, $(a^{-1})^{-1} = a$, because $a^{-1}a = e$. Second, $(ab)^{-1} = b^{-1}a^{-1}$, because $(ab)(b^{-1}a^{-1}) = e$.

A common error is to write $(ab)^{-1} = a^{-1}b^{-1}$. This is false when $a$ and $b$ do not commute. In $S_3$, take $a = (1\,2)$ and $b = (1\,3)$: then $a^{-1}b^{-1} = (1\,2)(1\,3)$, but $(ab)^{-1} = (1\,3)(1\,2)$, and these differ.

The step that needs care is step 2. It uses the inverse $a^{-1}$, so the argument is for groups. In a monoid an element can have a right inverse and no left inverse.
</document>

<document source="https://tamnd.github.io/math2/node/0DAECMZVRT.md" tag="0DAECMZVRT" kind="concept">
# Straightedge and compass construction

## Summary

A straightedge and compass construction draws points, lines and circles in the plane from given points with two ideal tools.
The straightedge draws the line through two constructed points, and it has no marks.
The compass draws the circle with a constructed centre through a constructed point.
New points are the intersections of the lines and circles drawn so far.
The [[constructible numbers|0FMN2MN7TM]] describe the points that these steps can reach.

## Remarks

Each step solves linear or quadratic equations.
So the coordinates of the new points lie in an extension of degree $1$ or $2$ of the field of the old coordinates.
This turns questions about constructions into questions about [[degrees of field extensions|0GS74FGFEK]].
It shows that [[doubling the cube|04SD3F4FCR]], [[trisecting the angle|00J3DCCEK5]] and [[squaring the circle|0BSXRER81B]] are impossible with these tools.

## Explanations

### L1 human

The ancient Greeks asked which figures can be drawn with an unmarked ruler and a compass alone.
Field theory gives the answer: a length can be drawn exactly when it comes from rational numbers by field operations and square roots.
</document>

<document source="https://tamnd.github.io/math2/node/0DAMHCT17R.md" tag="0DAMHCT17R" kind="concept">
# Euler's totient function

## Summary

Euler's totient function $\varphi$ gives, for each positive integer $n$, the number of integers from $1$ to $n$ that are coprime to $n$.
The [[definition of Euler's totient function|0VV04M41AN]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0DAX8PMAJ4.md" tag="0DAX8PMAJ4" kind="concept">
# Probability axioms

## Summary

The probability axioms of Kolmogorov are the three rules that a probability function $P$ on the events of a sample space $\Omega$ must obey.
First, $P(A) \ge 0$ for each [[event|00SQGJNRTB]] $A$.
Second, $P(\Omega) = 1$.
Third, for each sequence $A_1, A_2, \dots$ of pairwise disjoint events, $P\left(\bigcup_{n} A_n\right) = \sum_{n} P(A_n)$. This is countable additivity.
A function that obeys these rules is a [[probability measure|0VGMEBVW1M]].

## Remarks

Andrey Kolmogorov stated these axioms in 1933.
The other rules of probability follow from the axioms. Examples are $P(\varnothing) = 0$, $P(A) \le 1$, the rule for the complement and the addition rule for two events.

## Explanations

### L1 human

The axioms say three things: a probability is never negative, something always happens, and the probabilities of events that cannot happen together add up.
For a fair coin, $P(\{H\}) = P(\{T\}) = 1/2$, and $P(\{H, T\}) = 1/2 + 1/2 = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0DB0R4N5XX.md" tag="0DB0R4N5XX" kind="statement">
# Mean of the geometric distribution

## Statement

Let $p \in (0, 1]$ and let $X$ have the [[geometric distribution|0XCVV5Q500]] with parameter $p$, so $P(X = k) = (1 - p)^k p$ for $k = 0, 1, 2, \dots$. Then
$$E[X] = \frac{1 - p}{p}.$$
For the number $T = X + 1$ of trials up to and including the first success, $E[T] = 1/p$.

## Remarks

A proof uses the [[tail sum formula|0Y6QNF1J49]]: $P(X \ge n) = (1 - p)^n$, so $E[X] = \sum_{n \ge 1} (1 - p)^n = (1 - p)/p$.
The variance of $X$ and of $T$ is $(1 - p)/p^2$.

## Explanations

### L1 human

If each trial succeeds with probability $p$, you need on average $1/p$ trials to get the first success.
For a fair die, you wait on average $6$ rolls for the first six.
</document>

<document source="https://tamnd.github.io/math2/node/0DBTE6HVBD.md" tag="0DBTE6HVBD" kind="statement">
# Matrix of the adjoint is the conjugate transpose

## Statement

Let $V$ and $W$ be finite-dimensional inner product spaces over $\R$ or $\C$, with [[orthonormal bases|0S9GGTR0EX]] $B$ and $C$.
Let $T \colon V \to W$ be a linear map.
Then the matrix of the [[adjoint|0R9NMSM1ZP]] is
$$[T^*]_{B,C} = \left([T]_{C,B}\right)^*,$$
where $^*$ on the right is the [[conjugate transpose|0DXKE10PHF]]. Here $[\,\cdot\,]$ is the [[matrix of a linear map|0YQSGNYE3K]].

## Remarks

Over $\R$, the matrix of $T^*$ is the transpose of the matrix of $T$.
The statement needs orthonormal bases; in other bases, the Gram matrices of the bases appear in the formula.
The proof compares entries: both are $\langle T(b_j), c_i \rangle$, up to conjugation.

## Explanations

### L1 human

In orthonormal coordinates, taking the adjoint of a map means taking the conjugate transpose of its matrix.
This makes the abstract adjoint easy to compute.
In other bases, the rule is more complicated.
</document>

<document source="https://tamnd.github.io/math2/node/0DBWXQRXQJ.md" tag="0DBWXQRXQJ" kind="concept">
# Torsion subgroup

## Summary

The torsion subgroup of an abelian group is the subgroup of its elements of finite order.
The quotient by the torsion subgroup is torsion-free.
</document>

<document source="https://tamnd.github.io/math2/node/0DC3MPR7C9.md" tag="0DC3MPR7C9" kind="definition">
# Open ball

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $x \in X$ and let $r > 0$ be a real number.
The open ball with centre $x$ and radius $r$ is the set
$$B(x, r) = \{ y \in X : d(x, y) < r \}.$$

## Remarks

The centre is in the ball: $x \in B(x, r)$, because $d(x, x) = 0 < r$.
In $\mathbb{R}$ with the usual metric, $B(x, r)$ is the open interval $(x - r, x + r)$.
In the [[discrete metric|0NA4FK7B8S]], $B(x, r) = \{x\}$ for $r \le 1$ and $B(x, r) = X$ for $r > 1$.
Some authors also allow $r \le 0$. Then $B(x, r)$ is empty.

## Explanations

### L1 human

The open ball $B(x, r)$ is the set of points that are less than $r$ away from $x$.
In the plane with the Euclidean metric, it is the inside of the circle of radius $r$ about $x$, without the circle itself.
</document>

<document source="https://tamnd.github.io/math2/node/0DCT0QZVQ1.md" tag="0DCT0QZVQ1" kind="definition">
# Expected value

## Definition

Let $X$ be a real [[random variable|0DKRR22AEQ]] on a [[probability space|0ZYBZ34RT2]] $(\Omega, \mathcal{F}, P)$.
If $X \ge 0$, the expected value of $X$ is the Lebesgue integral
$$E[X] = \int_\Omega X \, dP \in [0, \infty].$$
In general, write $X = X^+ - X^-$ with $X^+ = \max(X, 0)$ and $X^- = \max(-X, 0)$. If $E[|X|] < \infty$, then $X$ is integrable and $E[X] = E[X^+] - E[X^-]$.

## Remarks

For a [[discrete random variable|0J6Z6H5VQJ]] with values $x_1, x_2, \dots$, $E[X] = \sum_i x_i \, P(X = x_i)$ when the series converges absolutely.
For a [[random variable with density|0C21RZG5V2]] $f$, $E[X] = \int_{-\infty}^{\infty} x f(x) \, dx$ when $\int |x| f(x) \, dx < \infty$.
Other notations are $\mathbb{E}[X]$, $\mathbb{E}X$ and $\mu_X$. The expected value depends only on the law of $X$.
If $E[X^+] = \infty$ and $E[X^-] < \infty$, one sets $E[X] = +\infty$. If both are infinite, $E[X]$ is not defined.

## Explanations

### L1 human

The expected value is the long-run average of many independent repetitions of the experiment.
For a fair die, it is $(1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5$. This value never appears on the die, but it is the center of mass of the distribution.
</document>

<document source="https://tamnd.github.io/math2/node/0DDA6CY7GJ.md" tag="0DDA6CY7GJ" kind="definition">
# Two-sided ideal

## Definition

Let $R$ be a [[ring|0894J980RK]].
A two-sided ideal of $R$ is a subset $I \subseteq R$ with these properties.

1. $I$ is a subgroup of the additive group $(R, +)$.
2. For all $r \in R$ and $a \in I$, $r a \in I$ and $a r \in I$.

So a two-sided ideal is a [[left ideal|0MYSA6MYNA]] that is also a right ideal.

## Remarks

In a [[commutative ring|0GYGHKX834]], left ideals, right ideals and two-sided ideals are the same.
The [[kernel|0AAHE7GEW6]] of every [[ring homomorphism|0GGE0Z0JQ1]] is a two-sided ideal.
The [[quotient ring|0VFW4Y69BT]] $R/I$ exists exactly when $I$ is a two-sided ideal.
In the ring of $n \times n$ matrices over a field, the only two-sided ideals are $\{0\}$ and the whole ring.

## Explanations

### L1 human

In a noncommutative ring, $r a$ and $a r$ can be different.
A two-sided ideal must absorb products from both sides.
This is the condition that makes the quotient ring work.
</document>

<document source="https://tamnd.github.io/math2/node/0DDPZAG5ER.md" tag="0DDPZAG5ER" kind="definition">
# Right inverse

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
A right inverse of $f$ is a function $h \colon B \to A$ such that $f \circ h = \mathrm{id}_B$, that is, $f(h(b)) = b$ for each $b \in B$.

## Remarks

A function has a right inverse exactly when it is [[surjective|0XKH881BZ7]]. The proof uses the [[axiom of choice|0HTGB13FE2]]. See [[surjective iff it has a right inverse|0TYQNHVB4C]].
A right inverse is in general not unique.
A right inverse is [[injective|0PVG1CRAJN]].

## Explanations

### L1 human

For $f(x) = x^2$ from $\mathbb{R}$ to $[0, \infty)$, the square root $h(y) = \sqrt{y}$ is a right inverse: $(\sqrt{y})^2 = y$.
The function $y \mapsto -\sqrt{y}$ is another right inverse.
</document>

<document source="https://tamnd.github.io/math2/node/0DE5R5EB4D.md" tag="0DE5R5EB4D" kind="definition">
# Elementary row operations

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $F$, with rows $R_1, \dots, R_m$.
An *elementary row operation* on $A$ is one of these three operations:

1. Row switching: exchange $R_i$ and $R_j$ for some $i \ne j$.
2. Row scaling: replace $R_i$ by $c R_i$ for some $c \in F$ with $c \ne 0$.
3. Row addition: replace $R_i$ by $R_i + c R_j$ for some $i \ne j$ and some $c \in F$.

## Remarks

Each elementary row operation has an inverse operation of the same type.
Each elementary row operation on $A$ is the same as the product $EA$ for an [[elementary matrix|01REXW3WWE]] $E$.
Elementary row operations on an [[augmented matrix|0T6MWGZSXA]] do not change the solution set of the system.
Elementary column operations have the same definition with columns in place of rows.

## Explanations

### L1 human

The three row operations are the moves that we make when we solve equations by hand.
We can swap two equations, multiply an equation by a nonzero number, or add a multiple of one equation to another.
None of these moves gains or loses a solution, and each move can be undone.
</document>

<document source="https://tamnd.github.io/math2/node/0DEB0JMN0W.md" tag="0DEB0JMN0W" kind="concept">
# Harmonic function

## Summary

A harmonic function is a twice continuously differentiable function on an open subset of $\mathbb{R}^n$ whose [[Laplacian|0E0B6RA8K0]] is zero at each point.
Harmonic functions describe steady states, such as equilibrium temperatures and electrostatic potentials in empty space.
</document>

<document source="https://tamnd.github.io/math2/node/0DEBXAJ5SA.md" tag="0DEBXAJ5SA" kind="concept">
# Conjugacy class

## Summary

The conjugacy class of an element $a$ of a group $G$ is the set of all elements $gag^{-1}$ with $g \in G$.
The conjugacy classes partition the group.
</document>

<document source="https://tamnd.github.io/math2/node/0DFT1GP1JP.md" tag="0DFT1GP1JP" kind="example">
# Gaussian integral

## Example

The improper integral of $e^{-x^2}$ over $\mathbb{R}$ converges, and
$$\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}.$$
Convergence follows from the [[comparison test|0FR1BDKK10]] with $e^{-|x|}$ for $|x| \ge 1$.
For the value, let $I$ be the integral. In polar coordinates, $I^2 = \int_0^{2\pi} \int_0^\infty e^{-r^2} r \, dr \, d\theta = \pi$.

## Remarks

The function $e^{-x^2}$ has no elementary antiderivative. So the second fundamental theorem does not give the value.
A change of variables gives $\int_{-\infty}^{\infty} e^{-a x^2} \, dx = \sqrt{\pi / a}$ for $a > 0$. Mathlib states this as `integral_gaussian`.
Abraham de Moivre found this type of integral in 1733. Carl Friedrich Gauss published the exact value in 1809.

## Explanations

### L1 human

The bell curve $e^{-x^2}$ has total area $\sqrt{\pi}$.
This number gives the constant in the normal distribution of probability theory.
</document>

<document source="https://tamnd.github.io/math2/node/0DFWHQ839T.md" tag="0DFWHQ839T" kind="statement">
# Strong law of large numbers

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with $E[|X_1|] < \infty$ and $\mu = E[X_1]$. Let $\bar{X}_n = (X_1 + \dots + X_n)/n$.
Then $\bar{X}_n \to \mu$ [[almost surely|04P7D63W8P]]:
$$P\left(\lim_{n \to \infty} \bar{X}_n = \mu\right) = 1.$$

## Remarks

The strong law implies the [[weak law|0BFDHF2H7N]], because almost sure convergence implies convergence in probability.
A short proof exists if $E[X_1^4] < \infty$: then $E[(\bar{X}_n - \mu)^4] \le C/n^2$, and the first Borel–Cantelli lemma applies.
Etemadi proved that the result holds when the $X_i$ are only pairwise independent and identically distributed.
Conversely, if $E[|X_1|] = \infty$, then almost surely $\bar{X}_n$ does not converge to a finite limit.

## Explanations

### L1 human

The strong law says more than the weak law. Almost every infinite sequence of trials has sample means that converge to the true mean.
So, with probability $1$, the fraction of heads in an infinite sequence of fair coin tosses tends to $1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/0DFYTNZ6AM.md" tag="0DFYTNZ6AM" kind="statement">
# Finite extensions are algebraic

## Statement

Let $L/K$ be a [[finite extension|0NRYPM9Y27]].
Then $L/K$ is [[algebraic|09EYMKMSB9]].

## Remarks

The proof: let $n = [L : K]$ and $a \in L$. The $n + 1$ elements $1, a, \dots, a^n$ are linearly dependent over $K$. A linear dependence $c_0 + c_1 a + \dots + c_n a^n = 0$ gives a nonzero polynomial with root $a$.
So every element of $L$ has degree at most $[L : K]$ over $K$.
The converse is false, as the algebraic numbers show.
An extension generated by finitely many algebraic elements is finite, by the [[tower law|0A7KJ68XHT]].

## Explanations

### L1 human

In a finite-dimensional extension, the powers of an element cannot all be independent.
A dependence between the powers is a polynomial equation for the element.
</document>

<document source="https://tamnd.github.io/math2/node/0DFZNHJ0S2.md" tag="0DFZNHJ0S2" kind="statement">
# Commutativity of addition

## Statement

For all natural numbers $a$ and $b$, $a + b = b + a$.
Here $+$ is the [[addition of natural numbers|0W5DSDFQ4G]].

## Remarks

The proof from the recursive definition of addition uses the [[principle of mathematical induction|0ASJVEFA9J]] two times.
An operation $\ast$ on a set is commutative when $x \ast y = y \ast x$ for all $x$ and $y$. Multiplication of natural numbers is also commutative.
Subtraction of integers is not commutative: $3 - 1 \neq 1 - 3$.

## Explanations

### L1 human

The order of two numbers in a sum does not change the result: $2 + 5$ and $5 + 2$ are both $7$.
Think of two piles of stones: you count the same total when you count the second pile first.
</document>

<document source="https://tamnd.github.io/math2/node/0DG0Z72CQD.md" tag="0DG0Z72CQD" kind="definition">
# Holomorphic function

## Definition

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be a function.
The function $f$ is holomorphic on $U$ if $f$ is [[complex differentiable|0D1C9Y1EXA]] at each point of $U$.
Let $z_0 \in \mathbb{C}$. A function $f$ is holomorphic at $z_0$ if $f$ is holomorphic on some open disc with center $z_0$.

## Remarks

Polynomials in $z$ are holomorphic on $\mathbb{C}$, and $z \mapsto 1/z$ is holomorphic on $\mathbb{C} \setminus \{0\}$.
Sums, products, quotients with a nonzero denominator and compositions of holomorphic functions are holomorphic.
Each holomorphic function is analytic, see [[holomorphic functions are analytic|0826SGV3R0]]. For this reason, many authors use the words holomorphic and analytic for the same functions.
A function that is holomorphic on all of $\mathbb{C}$ is an [[entire function|0419794R0T]].

## Explanations

### L1 human

A holomorphic function is a function of a complex variable that has a complex derivative at each point of an open set.
This one condition has strong consequences: the function then has derivatives of all orders and is equal to its Taylor series near each point.
</document>

<document source="https://tamnd.github.io/math2/node/0DG10TVYPN.md" tag="0DG10TVYPN" kind="statement">
# Cauchy's integral formula

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $c \in \mathbb{C}$ and $r > 0$ be such that the closed disc $\{z : |z - c| \le r\}$ is contained in $U$.
Then for each $a \in \mathbb{C}$ with $|a - c| < r$,
$$f(a) = \frac{1}{2 \pi i} \oint_{|z - c| = r} \frac{f(z)}{z - a} \, dz,$$
where the integral is the [[circle integral|0T7ABX4CWE]] over the circle $|z - c| = r$.

## Remarks

The values of $f$ on the circle determine the values of $f$ at each point inside the circle.
With $a = c$ and $z = c + r e^{i\theta}$, the formula gives $f(c) = \frac{1}{2\pi} \int_0^{2\pi} f(c + r e^{i\theta}) \, d\theta$. So $f(c)$ is the average of $f$ on the circle.
For $f = 1$, the formula is the [[integral of 1/(z - a)|0SXZ68HV6F]].
The formula gives [[the formula for the derivatives|0F0H5P5DED]], and it shows that [[holomorphic functions are analytic|0826SGV3R0]].

## Explanations

### L1 human

If you know a holomorphic function on a circle, you know it everywhere inside the circle.
The formula computes the value at an inside point $a$ as a weighted average of the values on the circle, with the weight $1/(z - a)$.
</document>

<document source="https://tamnd.github.io/math2/node/0DGDS5XY84.md" tag="0DGDS5XY84" kind="concept">
# Alternating bilinear form

## Summary

An alternating bilinear form is a bilinear form whose value on each pair of equal vectors is zero.
</document>

<document source="https://tamnd.github.io/math2/node/0DGZBQWZWN.md" tag="0DGZBQWZWN" kind="statement">
# Spectral theorem for Hermitian matrices

## Statement

Let $A$ be a [[Hermitian|0P17NGQQMQ]] $n \times n$ complex matrix.
Then there are a [[unitary matrix|0AR68JTMDT]] $U$ and a real diagonal matrix $D$ with
$$A = U D U^*.$$
If $A$ is a real [[symmetric matrix|0VR671PHE5]], then $U$ can be chosen real, that is, [[orthogonal|0W02QESPJQ]]: $A = Q D Q^T$.

## Remarks

The diagonal entries of $D$ are the eigenvalues of $A$, and the columns of $U$ are orthonormal eigenvectors.
This is the matrix form of the [[spectral theorem|06J4PXRG22]].
For example, $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = Q \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix} Q^T$ with $Q = \frac{1}{\sqrt 2} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$.

## Explanations

### L1 human

Every Hermitian matrix is diagonal after a unitary change of coordinates.
The new coordinate axes are perpendicular eigenvectors.
For real symmetric matrices, this is a rotation, possibly with a reflection, of the axes.
</document>

<document source="https://tamnd.github.io/math2/node/0DH7HMXP9F.md" tag="0DH7HMXP9F" kind="statement">
# Order of a permutation is the lcm of its cycle lengths

## Statement

Let $\sigma \in S_n$, and let $\sigma = c_1 c_2 \cdots c_r$ be its decomposition into disjoint cycles of lengths $k_1, \dots, k_r$.
Then the [[order|0WE2NHZJ7B]] of $\sigma$ is $\operatorname{lcm}(k_1, \dots, k_r)$.

## Remarks

A $k$-cycle has order $k$.
The disjoint cycles commute, so $\sigma^m = c_1^m \cdots c_r^m$, and this is the identity exactly when each $c_i^m$ is the identity.
For example, $(1\ 2)(3\ 4\ 5)$ has order $6$.

## Explanations

### L1 human

A permutation returns to the identity when each of its cycles is back at the start.
A cycle of length $k$ is back at the start after each multiple of $k$ steps.
So you need the first common multiple of all the cycle lengths.
</document>

<document source="https://tamnd.github.io/math2/node/0DJ8W3HP2D.md" tag="0DJ8W3HP2D" kind="definition">
# Covariance

## Definition

Let $X$ and $Y$ be real random variables on one probability space with $E[X^2] < \infty$ and $E[Y^2] < \infty$.
The covariance of $X$ and $Y$ is
$$\operatorname{Cov}(X, Y) = E\left[(X - E[X]) \, (Y - E[Y])\right].$$
$X$ and $Y$ are uncorrelated if $\operatorname{Cov}(X, Y) = 0$.

## Remarks

An equivalent formula is $\operatorname{Cov}(X, Y) = E[XY] - E[X] \, E[Y]$.
The covariance of $X$ with itself is the [[variance|007QED3E0Y]]: $\operatorname{Cov}(X, X) = \operatorname{Var}(X)$.
The covariance is symmetric and bilinear: $\operatorname{Cov}(aX + bY, Z) = a \operatorname{Cov}(X, Z) + b \operatorname{Cov}(Y, Z)$. It does not change when one adds constants to $X$ or $Y$.
The Cauchy–Schwarz inequality gives $|\operatorname{Cov}(X, Y)| \le \sigma_X \sigma_Y$.

## Explanations

### L1 human

Covariance says whether two quantities tend to move in the same direction.
Height and weight of people have positive covariance. The size of a covariance depends on the units, so one often uses the correlation coefficient to compare.
</document>

<document source="https://tamnd.github.io/math2/node/0DJ9Y9V610.md" tag="0DJ9Y9V610" kind="definition">
# Upper central series

## Definition

Let $G$ be a [[group|G5S1M9]].
The upper central series of $G$ is the sequence of normal subgroups $Z_0 = \{e\}$ and $Z_{i+1} = \{x \in G : [x, y] \in Z_i \text{ for all } y \in G\}$ for $i \geq 0$.

## Remarks

The first term $Z_1$ is the [[centre|0DZ93493VN]] $Z(G)$, and $Z_2$ is called the second centre.
For each $i$, $Z_{i+1}/Z_i$ is the centre of the [[quotient group|0KYX2N69J0]] $G/Z_i$.
For the quaternion group $Q_8$, the series is $\{1\} \subseteq \{1, -1\} \subseteq Q_8$.
A group with trivial centre, such as $S_3$, has $Z_i = \{e\}$ for each $i$.

## Explanations

### L1 human

The upper central series starts with the identity and first adds the centre.
Then it adds the elements that are central modulo the centre, and so on.
If this process reaches the whole group, the group is nilpotent.
</document>

<document source="https://tamnd.github.io/math2/node/0DJMGPN8K9.md" tag="0DJMGPN8K9" kind="statement">
# Estimation lemma

## Statement

Let $\gamma : [a, b] \to \mathbb{C}$ be a continuous and piecewise continuously differentiable path with length $L(\gamma) = \int_a^b |\gamma'(t)| \, dt$.
Let $f$ be a complex function that is continuous on the image of $\gamma$.
Let $M \ge 0$ be a real number with $|f(z)| \le M$ for each $z$ on the image of $\gamma$.
Then the [[contour integral|00FWNP9TXY]] of $f$ along $\gamma$ satisfies
$$\left| \int_\gamma f(z) \, dz \right| \le M \, L(\gamma).$$

## Remarks

For a [[circle integral|0T7ABX4CWE]] over $|z - c| = R$, the bound is $\left| \oint f(z) \, dz \right| \le 2 \pi R M$.
Proof: $\left| \int_a^b g(t) \, dt \right| \le \int_a^b |g(t)| \, dt$ for a complex function $g$, and $|f(\gamma(t)) \gamma'(t)| \le M |\gamma'(t)|$.
A typical use shows that the integral over a large half-circle goes to $0$ as the radius goes to $\infty$.

## Explanations

### L1 human

The integral cannot be larger than the largest size of the function times the length of the path.
For example, if $|f| \le 1$ on a circle of radius $3$, then the integral over the circle has size at most $6\pi$.
</document>

<document source="https://tamnd.github.io/math2/node/0DKCECSYCF.md" tag="0DKCECSYCF" kind="concept">
# Quadratic residue

## Summary

A quadratic residue modulo $n$ is an integer that is congruent to a perfect square modulo $n$.
The [[definition of a quadratic residue|0SQDB87Z3T]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0DKMBWTD7P.md" tag="0DKMBWTD7P" kind="statement">
# Cosets have the same cardinality as the subgroup

## Statement

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \in G$.
Then the map $h \mapsto gh$ is a bijection from $H$ to the [[left coset|03DEH43F4H]] $gH$. The map $h \mapsto hg$ is a bijection from $H$ to the [[right coset|0NE1DK7H4T]] $Hg$.
In particular, $|gH| = |Hg| = |H|$.

## Remarks

The inverse of $h \mapsto gh$ is $x \mapsto g^{-1}x$.
This fact and the partition of $G$ into left cosets give the proof of [[Lagrange's theorem|3C53TD]].

## Explanations

### L1 human

Multiplying by a fixed element never makes two different elements equal, by cancellation.
Thus a coset has exactly as many elements as the subgroup.
All cosets of a subgroup have the same size.
</document>

<document source="https://tamnd.github.io/math2/node/0DKRR22AEQ.md" tag="0DKRR22AEQ" kind="definition">
# Random variable

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $(E, \mathcal{E})$ be a set $E$ with a [[σ-algebra|04WSW8T3SY]] $\mathcal{E}$.
A random variable with values in $E$ is a [[function|0XHDTE5YMZ]] $X \colon \Omega \to E$ such that the [[preimage|0FGCFY1AZX]] $X^{-1}(B) = \{\omega \in \Omega : X(\omega) \in B\}$ is in $\mathcal{F}$ for each $B \in \mathcal{E}$.
A real random variable is a random variable with values in $\mathbb{R}$, with the [[Borel σ-algebra|0BVHZXWKCD]] on $\mathbb{R}$.

## Remarks

The condition says that $X$ is a measurable function. It makes $\{X \in B\}$ an event, so $P(X \in B)$ is defined.
A real function $X$ on $\Omega$ is a random variable if and only if $\{X \le x\} \in \mathcal{F}$ for each $x \in \mathbb{R}$.
For two rolls of a die, $\Omega = \{1, \dots, 6\}^2$ and the sum $X(a, b) = a + b$ is a random variable.
Sums, products and continuous functions of real random variables are random variables.

## Explanations

### L1 human

A random variable is a number that you read off from the result of the experiment.
For two rolls of a die, the sum of the two numbers is a random variable.
It is $7$ for the result $(3, 4)$ and $12$ for the result $(6, 6)$.
</document>

<document source="https://tamnd.github.io/math2/node/0DMY75D6FV.md" tag="0DMY75D6FV" kind="statement">
# Basel problem

## Statement

The series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, and
$$\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \cdots = \frac{\pi^2}{6}.$$

## Remarks

The convergence follows from the [[p-series test|0ZNSSK8ZK1]] with $p = 2$. The value is the hard part.
The value $\pi^2/6 \approx 1.6449$ is $\zeta(2)$, where $\zeta$ is the Riemann zeta function.
Mathlib states it as `hasSum_zeta_two`.

## History

Pietro Mengoli posed the problem in 1650.
Leonhard Euler solved it in 1734. The problem is named after Basel, the home town of Euler.

## Explanations

### L1 human

The sum of the reciprocals of the squares $1, 4, 9, 16, \dots$ is a finite number.
Surprisingly, this number involves $\pi$, which comes from circles.
</document>

<document source="https://tamnd.github.io/math2/node/0DNN9B6Q2F.md" tag="0DNN9B6Q2F" kind="concept">
# Correlation coefficient

## Summary

The correlation coefficient of two random variables is their covariance divided by the product of their standard deviations. It is a number in $[-1, 1]$ that measures the strength of a linear relation.
The node [[correlation coefficient|04CTY6XYWN]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0DNNT7DBFM.md" tag="0DNNT7DBFM" kind="definition">
# Linear equation

## Definition

Let $F$ be a field and let $n \ge 1$ be an integer.
A *linear equation* over $F$ in the unknowns $x_1, \dots, x_n$ is an equation of the form
$$a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b,$$
where the coefficients $a_1, \dots, a_n$ are elements of $F$. The constant term $b$ is also an element of the field.
A *solution* of the equation is a tuple $(s_1, \dots, s_n) \in F^n$ such that $a_1 s_1 + \cdots + a_n s_n = b$.

## Remarks

The equation $2x + 3y = 6$ over $\R$ is linear, and $(3, 0)$ is a solution.
The equation $xy = 1$ is not linear, because it has a product of two unknowns.
Over $\R$ with $n = 2$, the solutions of a linear equation with $(a_1, a_2) \ne (0, 0)$ form a line in the plane.

## Explanations

### L1 human

A linear equation adds up the unknowns, each one times a fixed number, and sets the sum equal to a fixed number.
No unknown is squared, and no two unknowns are multiplied together.
With two unknowns over the real numbers, the solutions usually draw a straight line, and this gives the name "linear".
</document>

<document source="https://tamnd.github.io/math2/node/0DPF6WRZW6.md" tag="0DPF6WRZW6" kind="definition">
# Degree of a polynomial

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $f = a_0 + a_1 x + \dots + a_n x^n$ be a [[polynomial|0Q9EZS90GY]] in $R[x]$.
If $f \neq 0$, the degree $\deg f$ of $f$ is the largest integer $d$ with $a_d \neq 0$.
The degree of the zero polynomial is $\deg 0 = -\infty$.

## Remarks

The degree of $3x^4 - x + 2$ in $\mathbb{Z}[x]$ is $4$. A nonzero constant polynomial has degree $0$.
Some authors leave $\deg 0$ undefined, or set it to $-1$. The convention $-\infty$ makes the formula $\deg(f g) = \deg f + \deg g$ hold also for $f = 0$ or $g = 0$.
For all $f, g \in R[x]$, $\deg(f + g) \le \max(\deg f, \deg g)$.
The coefficient $a_d$ with $d = \deg f$ is the [[leading coefficient|0P05RJC5XA]] of $f$.

## Explanations

### L1 human

The degree is the highest power of $x$ that really appears in the polynomial.
For example, $x^3 + 5x$ has degree $3$.
The zero polynomial has no nonzero coefficient, so it needs a special convention.
</document>

<document source="https://tamnd.github.io/math2/node/0DQ8ERJZNS.md" tag="0DQ8ERJZNS" kind="definition">
# Surface integral

## Definition

Let $r \colon D \to \mathbb{R}^3$ be a [[parametrised surface|05788J51WZ]] as in the definition of [[surface area|0WQE3G7B88]], let $S = r(D)$, and let $f \colon S \to \mathbb{R}$ be continuous.
The surface integral of $f$ over $S$ is
$$\iint_S f \, dS = \iint_D f\big( r(u, v) \big) \, \| r_u(u, v) \times r_v(u, v) \| \, du \, dv.$$

## Remarks

The expression $dS = \| r_u \times r_v \| \, du \, dv$ is the surface element. With $f = 1$, the integral is the area of $S$.
The value does not depend on the parametrisation of $S$. For a surface made of several pieces, add the integrals over the pieces.
For the graph $z = g(x, y)$ over $D$, $\iint_S f \, dS = \iint_D f(x, y, g(x, y)) \sqrt{1 + g_x^2 + g_y^2} \, dx \, dy$.
Example: on the unit sphere $S$, symmetry gives $\iint_S x^2 \, dS = \iint_S y^2 \, dS = \iint_S z^2 \, dS$. These add up to the area $4\pi$, so $\iint_S z^2 \, dS = 4\pi / 3$.

## Explanations

### L1 human

A surface integral adds up the values of a function over a curved surface, each weighted by a small piece of area.
If $f$ is a mass density per unit area, the surface integral is the total mass of the surface.
</document>

<document source="https://tamnd.github.io/math2/node/0DQEHHVRBX.md" tag="0DQEHHVRBX" kind="concept">
# Slope field

## Summary

The slope field of a first-order equation $y' = f(x, y)$ is a picture with a short segment of slope $f(x, y)$ at many points $(x, y)$ of the plane.
The graphs of the solutions are tangent to these segments. See the [[definition|0RKQQQACY7]].
</document>

<document source="https://tamnd.github.io/math2/node/0DQG39K2NH.md" tag="0DQG39K2NH" kind="statement">
# K[x] is a Euclidean domain

## Statement

Let $K$ be a [[field|0QA8CHH8MG]].
Then the polynomial ring $K[x]$ is a [[Euclidean domain|0GJ2HRG95T]], with the [[degree|0DPF6WRZW6]] as Euclidean function.
That is, for all $f, g \in K[x]$ with $g \neq 0$, there are $q, r \in K[x]$ with $f = q g + r$ and $r = 0$ or $\deg r < \deg g$.

## Remarks

The ring $K[x]$ is an integral domain because $K$ is one. The existence of $q$ and $r$ is the [[division algorithm|0MES3KS0ZC]] over a field.
A consequence is that $K[x]$ is a principal ideal domain and a unique factorization domain.
The ring $\mathbb{Z}[x]$ is not a Euclidean domain, because it is [[not a principal ideal domain|06E3AG9TM2]].

## Explanations

### L1 human

Over a field, you can divide any polynomial by any nonzero polynomial and get a remainder of smaller degree.
This is the same situation as division with remainder for integers.
So all the theory of the Euclidean algorithm works for polynomials over a field.
</document>

<document source="https://tamnd.github.io/math2/node/0DQMDSVSXJ.md" tag="0DQMDSVSXJ" kind="statement">
# Universal property of the polynomial ring

## Statement

Let $R$ and $S$ be [[commutative rings|0GYGHKX834]], let $\varphi \colon R \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]] and let $s \in S$.
Then there is exactly one ring homomorphism $\Phi \colon R[x] \to S$ such that $\Phi(r) = \varphi(r)$ for all $r \in R$ and $\Phi(x) = s$.
It is given by $\Phi(a_0 + a_1 x + \dots + a_n x^n) = \varphi(a_0) + \varphi(a_1) s + \dots + \varphi(a_n) s^n$.

## Remarks

With $S = R$, $\varphi$ the identity and $s = c$, the map $\Phi$ is the [[evaluation homomorphism|0125QVNCFQ]] at $c$.
In the language of algebras: for each commutative $R$-algebra $A$ and each $a \in A$, there is exactly one $R$-algebra homomorphism $R[x] \to A$ that sends $x$ to $a$.
This property characterizes $R[x]$ up to a unique isomorphism.

## Explanations

### L1 human

A ring homomorphism from $R[x]$ is fixed by two choices: what it does on the coefficients, and where it sends $x$.
Every pair of choices works, and it gives exactly one homomorphism.
Evaluation at a number is the main example.
</document>

<document source="https://tamnd.github.io/math2/node/0DQPNJNPYK.md" tag="0DQPNJNPYK" kind="concept">
# Well-founded relation

## Summary

A relation is well-founded when each nonempty subset has a minimal element for it.
Then there is no infinite descending chain.
</document>

<document source="https://tamnd.github.io/math2/node/0DQVHMYAGA.md" tag="0DQVHMYAGA" kind="statement">
# Bézout's identity

## Statement

Let $a$ and $b$ be integers, and let $d = \gcd(a, b)$ be their [[greatest common divisor|02WF878BH1]].
Then there are integers $x$ and $y$ such that $a x + b y = d$.

## Remarks

The integers $x$ and $y$ are not unique. If $(x, y)$ is a solution, then $(x + k b / d, \; y - k a / d)$ is also a solution for each integer $k$, when $d \neq 0$.
The set of all integers $a x + b y$ is the set of the multiples of $d$.
The [[extended Euclidean algorithm|0964YQFSA8]] computes one pair $(x, y)$.
For example, $\gcd(12, 42) = 6$ and $12 \cdot (-3) + 42 \cdot 1 = 6$.

## Explanations

### L1 human

With coins of value $a$ and $b$, and if you can give and take back coins, you can pay exactly the amount $\gcd(a, b)$.
For example, with coins of $4$ and $6$, you pay $2 = 2 \cdot 4 - 1 \cdot 6$.
You cannot pay $1$ this way, because each sum $4 x + 6 y$ is even.
</document>

<document source="https://tamnd.github.io/math2/node/0DR81R1F2B.md" tag="0DR81R1F2B" kind="concept">
# Joint probability distribution

## Summary

The joint distribution of several random variables on one probability space is the distribution of the vector that they form.
It gives the probability of each event that involves several of the variables together. The node [[joint probability distribution|0QAR0XDZH4]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0DRF0F3XC9.md" tag="0DRF0F3XC9" kind="statement">
# Inequality of arithmetic and geometric means

## Statement

Let $n \ge 1$ and let $x_1, \dots, x_n$ be nonnegative real numbers.
Then
$$\sqrt[n]{x_1 x_2 \cdots x_n} \le \frac{x_1 + x_2 + \dots + x_n}{n},$$
with equality if and only if $x_1 = x_2 = \dots = x_n$.

## Remarks

For $n = 2$, the inequality $\sqrt{x y} \le (x + y)/2$ follows from $(\sqrt{x} - \sqrt{y})^2 \ge 0$.
The weighted form: for weights $w_i \ge 0$ with $w_1 + \dots + w_n = 1$, $x_1^{w_1} \cdots x_n^{w_n} \le w_1 x_1 + \dots + w_n x_n$.

## Explanations

### L1 human

The geometric mean of some nonnegative numbers is never larger than their average.
For example, for $2$ and $8$, the geometric mean is $\sqrt{16} = 4$ and the average is $5$.
Among all rectangles with a fixed perimeter, the square has the largest area. This is the case $n = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/0DRKT1QJR7.md" tag="0DRKT1QJR7" kind="definition">
# Closure

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
The closure of $S$ is the [[intersection|0839XNQXDK]] of all [[closed sets|0G5AV9Q4RR]] $F$ of $X$ with $S \subseteq F$.
It is written $\overline{S}$ or $\operatorname{cl} S$.

## Remarks

The family of these closed sets is not empty, because it contains $X$.
A point $x$ is in $\overline{S}$ if and only if each open set that contains $x$ meets $S$.
In $\mathbb{R}$, the closure of $(0, 1)$ is $[0, 1]$, and the closure of $\mathbb{Q}$ is $\mathbb{R}$.
The closure and the [[interior|0C38S0KMJK]] are related by $X \setminus \overline{S} = \operatorname{int}(X \setminus S)$.

## Explanations

### L1 human

The closure of a set adds to the set all the points that you can approach from inside it.
For the open interval $(0, 1)$, you add the two end points $0$ and $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0DS2F7GK05.md" tag="0DS2F7GK05" kind="definition">
# Galois extension

## Definition

An [[algebraic extension|09EYMKMSB9]] $L/K$ is a Galois extension if it is [[normal|0CBRFQ7DVZ]] and [[separable|0N019F6RE9]].

## Remarks

For a finite extension $L/K$, these conditions are equivalent.

1. $L/K$ is Galois.
2. $L$ is the splitting field of a separable polynomial over $K$.
3. $|\operatorname{Aut}(L/K)| = [L : K]$.
4. $K$ is the [[fixed field|0KMJ6919WS]] of $\operatorname{Aut}(L/K)$.

Examples: $\mathbb{C}/\mathbb{R}$, $\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}$ and every extension of finite fields $\mathbb{F}_{p^n}/\mathbb{F}_p$. The extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not Galois.

## Explanations

### L1 human

A Galois extension has as many symmetries over the base field as its degree allows.
For these extensions, the subgroups of the Galois group describe all the intermediate fields.
</document>

<document source="https://tamnd.github.io/math2/node/0DS2QFYJHE.md" tag="0DS2QFYJHE" kind="concept">
# Fundamental system of solutions

## Summary

A fundamental system of solutions of a homogeneous linear equation of order $n$ is a basis of its space of solutions: $n$ linearly independent solutions.
See the [[definition|0H4KNNCHPY]].
</document>

<document source="https://tamnd.github.io/math2/node/0DS9KWFDK2.md" tag="0DS9KWFDK2" kind="statement">
# The rationals are countable

## Statement

The set $\mathbb{Q}$ of rational numbers is [[countably infinite|0SEZ3JGZP7]].

## Remarks

Proof: each rational number has a unique form $p/q$ with $p \in \mathbb{Z}$, $q \ge 1$ and $\gcd(p, q) = 1$. The map $p/q \mapsto (p, q)$ is an injection into $\mathbb{Z} \times \mathbb{N}$, which is countable. And $\mathbb{Q}$ is infinite.
The rationals are dense in $\mathbb{R}$, but they are countable, and $\mathbb{R}$ is [[not countable|0518MM6BE1]].

## Explanations

### L1 human

Write the fractions $p/q$ in a grid, with $p$ along one side and $q$ along the other.
List the grid diagonal by diagonal, and skip each fraction that is not in lowest terms.
Each rational number appears exactly one time.
</document>

<document source="https://tamnd.github.io/math2/node/0DSHAVH45F.md" tag="0DSHAVH45F" kind="concept">
# Definition by recursion

## Summary

A definition by recursion defines a function on the [[natural numbers|0MWKJ44V5A]] in two parts.
It gives the value at $0$, and a rule that gives the value at $n + 1$ from the value at $n$.
The recursion theorem says that such a definition always gives exactly one function.
Precisely, let $X$ be a set, let $a$ be an element of $X$ and let $g \colon \mathbb{N} \times X \to X$ be a function.
Then there is exactly one function $f \colon \mathbb{N} \to X$ with $f(0) = a$ and $f(n + 1) = g(n, f(n))$ for each $n$ in $\mathbb{N}$.

## Remarks

The factorial is an example: $0! = 1$ and $(n + 1)! = (n + 1) \cdot n!$.
The [[addition|0W5DSDFQ4G]] and the [[multiplication|0JRC32D6K2]] of natural numbers have definitions by recursion.
The proof of the recursion theorem uses the [[principle of mathematical induction|0ASJVEFA9J]].

## Explanations

### L1 human

A recursive definition tells you where to start and how to take one step.
For the factorial, you start with $0! = 1$, and each step multiplies by the next number.
Step by step, this gives $1! = 1$, $2! = 2$, $3! = 6$ and $4! = 24$.
</document>

<document source="https://tamnd.github.io/math2/node/0DT0TCV75N.md" tag="0DT0TCV75N" kind="definition">
# Set difference

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The set difference of $A$ and $B$ is the set
$$A \setminus B = \{x \in A \mid x \notin B\}.$$
It is also called the relative complement of $B$ in $A$.

## Remarks

Another notation is $A - B$.
Example: $\{1, 2, 3\} \setminus \{2, 4\} = \{1, 3\}$.
The set $B$ need not be a subset of $A$.
In general, $A \setminus B \neq B \setminus A$.

## Explanations

### L1 human

To compute $A \setminus B$, start with $A$ and remove each element that is also in $B$.
Elements of $B$ that are not in $A$ do not matter.
</document>

<document source="https://tamnd.github.io/math2/node/0DTCBCKEJP.md" tag="0DTCBCKEJP" kind="concept">
# Transvection

## Summary

A transvection is a linear map, or a matrix, of the form $I + cE_{ij}$ with $i \ne j$, that adds a multiple of one coordinate to another.
Geometrically it is a shear.
</document>

<document source="https://tamnd.github.io/math2/node/0DTXBQPA2S.md" tag="0DTXBQPA2S" kind="concept">
# Index of a subgroup

## Summary

The index of a subgroup $H$ in a group $G$ is the number of left cosets of $H$ in $G$.
For finite groups, it is $|G| / |H|$.
</document>

<document source="https://tamnd.github.io/math2/node/0DTY8B9HMS.md" tag="0DTY8B9HMS" kind="concept">
# Linear span

## Summary

The linear span of a set of vectors is the set of all linear combinations of the vectors.
It is the smallest subspace that contains the set.
</document>

<document source="https://tamnd.github.io/math2/node/0DVDCMA2DK.md" tag="0DVDCMA2DK" kind="concept">
# Probability distribution

## Summary

A probability distribution describes how probability is spread over the possible values of a random quantity.
Formally it is a probability measure on a space of values, such as the real line with its Borel sets. The node [[probability distribution|0QB913WM5E]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0DXD4QSZ8A.md" tag="0DXD4QSZ8A" kind="concept">
# Conjugate elements

## Summary

Two elements $a$ and $b$ of a group $G$ are conjugate when $b = gag^{-1}$ for some $g \in G$.
Conjugate elements play the same role in the group, up to a change of names given by $g$.
</document>

<document source="https://tamnd.github.io/math2/node/0DXKE10PHF.md" tag="0DXKE10PHF" kind="concept">
# Conjugate transpose

## Summary

The conjugate transpose of a complex matrix is the transpose of the matrix with each entry replaced by its complex conjugate.
</document>

<document source="https://tamnd.github.io/math2/node/0DYS9C3HH5.md" tag="0DYS9C3HH5" kind="statement">
# Stone's representation theorem for finite Boolean algebras

## Statement

An atom of a [[Boolean algebra|06GKBTAA5E]] $B$ is a minimal element of $B \setminus \{0\}$.
Let $B$ be a finite Boolean algebra, and let $\operatorname{At}(B)$ be its set of atoms.
Then the map $b \mapsto \{ x \in \operatorname{At}(B) : x \le b \}$ is an isomorphism from $B$ to the [[power set Boolean algebra|004J7ZEFAT]] of $\operatorname{At}(B)$.

## Remarks

A consequence: a finite Boolean algebra with $n$ atoms has $2^n$ elements.
The general theorem says that every Boolean algebra is isomorphic to the algebra of clopen subsets of its Stone space. For a finite Boolean algebra this space is finite and discrete, and its points correspond to the atoms.
The general theorem needs a weak form of the axiom of choice, but the finite case does not.

## History

Marshall Stone published the representation theorem for all Boolean algebras in 1936.

## Explanations

### L1 human

Each finite Boolean algebra is the algebra of all subsets of some finite set.
Each element is the set of atoms below it, and the atoms are the smallest nonzero elements.
</document>

<document source="https://tamnd.github.io/math2/node/0DZ93493VN.md" tag="0DZ93493VN" kind="concept">
# Centre of a group

## Summary

The centre of a group is the set of elements that commute with every element of the group.
It is a normal abelian subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/0DZ9DKAKZR.md" tag="0DZ9DKAKZR" kind="concept">
# Free module

## Summary

A free module is a module that has a basis, that is, a linearly independent generating set.
</document>

<document source="https://tamnd.github.io/math2/node/0DZGSDHD92.md" tag="0DZGSDHD92" kind="statement">
# Irrationality of the square root of 2

## Statement

The [[square root|0VB8BFRABH]] $\sqrt{2}$ of $2$ is an [[irrational number|080T2TXVQV]].
That is, there are no integers $a$ and $b$ with $b \neq 0$ and $(a / b)^2 = 2$.

## Remarks

A classical proof: suppose that $a / b = \sqrt{2}$ in [[lowest terms|0R64A2CVN4]]. Then $a^2 = 2 b^2$, thus $a$ is even. Then $b$ is also even, which is a contradiction.
The theorem is a special case of the [[irrationality of the square root of a prime|0YHFMXJJAZ]].

## History

The ancient Greeks knew this result. Tradition attributes its discovery to the Pythagoreans.

## Explanations

### L1 human

No fraction squared gives exactly $2$.
Fractions such as $7/5$ and $141/100$ come close, but there is always an error.
This means that the diagonal of a square with side $1$ has a length that is not a ratio of whole numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0DZPD2MG33.md" tag="0DZPD2MG33" kind="definition">
# Nilpotent group

## Definition

A [[group|G5S1M9]] $G$ is nilpotent when it has a central series. This is a finite sequence $\{e\} = N_0 \subseteq N_1 \subseteq \cdots \subseteq N_n = G$ of [[normal subgroups|0CB0DX5BVV]] of $G$ with $N_{i+1}/N_i \subseteq Z(G/N_i)$ for each $i < n$.
Here $Z(G/N_i)$ is the [[centre|0DZ93493VN]] of the [[quotient group|0KYX2N69J0]] $G/N_i$.

## Remarks

The condition $N_{i+1}/N_i \subseteq Z(G/N_i)$ means $[N_{i+1}, G] \subseteq N_i$.
Equivalently, the [[lower central series|0K98R7A4VC]] reaches $\{e\}$, or the [[upper central series|0XW54DH28V]] reaches $G$; the least such length is the nilpotency class.
The nilpotent groups of class at most $1$ are exactly the [[abelian groups|082GHNPJ53]].
The quaternion group $Q_8$ is nilpotent of class $2$, but $S_3$ is not nilpotent.

## Explanations

### L1 human

A nilpotent group is close to abelian in a strong sense.
Each layer of a central series commutes with the whole group, up to the lower layers.
Each nilpotent group is solvable, but the converse is false, for example for $S_3$.
</document>

<document source="https://tamnd.github.io/math2/node/0DZS7ZG0QY.md" tag="0DZS7ZG0QY" kind="definition">
# Length of a vector

## Definition

Let $n$ be a positive integer and let $v = (v_1, \dots, v_n)$ be a [[vector|0QN3D80AWP]] in $\mathbb{R}^n$.
The length of $v$ is the real number
$$\|v\| = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2} = \sqrt{v \cdot v},$$
where $v \cdot v = v_1^2 + \dots + v_n^2$ is the dot product of $v$ with itself.
A unit vector is a vector of length 1.

## Remarks

For each vector $v$, $\|v\| \geq 0$, and $\|v\| = 0$ if and only if $v$ is the zero vector.
For each real number $c$, $\|c v\| = |c| \, \|v\|$. Thus for $v \neq 0$ the vector $v / \|v\|$ is a unit vector.
Other names for the length are the magnitude and the Euclidean norm. The notation $|v|$ is also common.

## Explanations

### L1 human

The length of a vector is how far the vector moves you. In the plane, the vector $(3, 4)$ is the hypotenuse of a right triangle with legs 3 and 4. So its length is $\sqrt{3^2 + 4^2} = 5$.
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# Irrational number

## Summary

An irrational number is a [[real number|08J2M1JYAG]] that is not a [[rational number|0NCJ5WXN6Z]].
Examples are $\sqrt{2}$, $\pi$ and $e$.
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# Number of nth roots of unity

## Statement

Let $n$ be a positive integer.
Then there are exactly $n$ complex $n$th [[roots of unity|0S61N75H45]].

## Remarks

The [[list of the nth roots of unity|057BH3JC9S]] has $n$ numbers $e^{2 \pi i k / n}$ with $0 \le k < n$.
They are pairwise different, because their arguments are different numbers in $[0, 2\pi)$.
A polynomial of degree $n$ has at most $n$ roots in a field, so $X^n - 1$ has no other roots.
In other fields there can be fewer: in $\mathbb{R}$, the only $4$th roots of unity are $1$ and $-1$.

## Explanations

### L1 human

The equation $z^n = 1$ has exactly $n$ complex solutions.
Over the real numbers, it has only one or two solutions, so most of the solutions are not real.
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# Jordan–Hölder theorem

## Statement

Let $G$ be a [[group|G5S1M9]] and let $\{e\} = G_0 \trianglelefteq \cdots \trianglelefteq G_n = G$ and $\{e\} = H_0 \trianglelefteq \cdots \trianglelefteq H_m = G$ be two [[composition series|0XV2A49X6Q]] of $G$.
Then $n = m$, and there is a permutation $\sigma$ of $\{1, \dots, n\}$ with $G_i / G_{i-1} \cong H_{\sigma(i)} / H_{\sigma(i) - 1}$ for each $i$.

## Remarks

So the [[composition factors|0PHTC93M58]] and the composition length of $G$ are well defined.
For $G = \mathbb{Z}/n\mathbb{Z}$, the theorem gives the uniqueness of the prime factorisation of $n$.
One proof uses the Schreier refinement theorem, which says that two subnormal series have equivalent refinements.

## History

The theorem is named after Camille Jordan and Otto Hölder.

## Explanations

### L1 human

A group can be cut into simple layers in many ways.
The theorem says that the layers that you get are always the same, up to order.
This is the group version of the uniqueness of prime factorisation.
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# Laplacian

## Summary

The Laplacian of a twice differentiable function on $\mathbb{R}^n$ is the [[divergence|0MWV6ZTK6S]] of its [[gradient|0ABK9V9Y64]].
It measures how the value of the function at a point differs from its average value on small spheres around the point.
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# Primitive root theorem

## Statement

Let $n$ be an integer with $n \ge 2$.
There is a [[primitive root|0KM2CJ80G7]] modulo $n$ if and only if $n \in \{2, 4, p^k, 2p^k\}$.
Here $p$ is an odd prime and $k$ is a positive integer.

## Remarks

Equivalently, the group $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic exactly for these $n$.
The case $n = p$ is the [[existence of primitive roots modulo a prime|04XGKK8WQW]].
For example, there are primitive roots modulo $9$, $18$ and $25$, but not modulo $8$, $12$ or $15$.

## History

Carl Friedrich Gauss gave the first proof.
He defined primitive roots in his Disquisitiones Arithmeticae of 1801.

## Explanations

### L1 human

Modulo $12$, the residues coprime to $12$ are $1, 5, 7, 11$, and each of them squares to $1$.
So no single residue generates all four, and there is no primitive root modulo $12$.
The theorem lists exactly the moduli for which a generator exists.
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# Square root function

## Summary

The square root function sends a nonnegative real number to the unique nonnegative number whose square it is. It is the inverse of $t \mapsto t^2$ on $[0, \infty)$. The node [[square root function|0V7E9KP6Q2]] gives the precise definition.
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<document source="https://tamnd.github.io/math2/node/0E1MMCAY5B.md" tag="0E1MMCAY5B" kind="example">
# Euclidean metric

## Example

Let $n \ge 1$ be an integer.
For points $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let
$$d_2(x, y) = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}.$$
Then $d_2$ is a [[metric|063W5JMY85]] on $\mathbb{R}^n$, the Euclidean metric.

## Remarks

For $n = 1$, $d_2(x, y) = |x - y|$, the [[usual metric on the real line|0KXJTZSW9B]].
For $n = 2$, the formula is the theorem of Pythagoras for the segment from $x$ to $y$.
The triangle inequality for $d_2$ follows from the Cauchy–Schwarz inequality.

## Explanations

### L1 human

The Euclidean metric is the length of the straight segment between two points.
In the plane, the distance from $(0, 0)$ to $(3, 4)$ is $\sqrt{9 + 16} = 5$.
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# Riccati equation

## Definition

Let $I$ be an open interval and let $q_0, q_1, q_2 : I \to \mathbb{R}$ be continuous functions, with $q_0$ and $q_2$ not identically zero.
A Riccati equation is an [[ordinary differential equation|0P97V20RW7]] of the form
$$y' = q_0(x) + q_1(x)\, y + q_2(x)\, y^2.$$

## Remarks

If $q_2 = 0$, the equation is [[linear|0Z3YKRQZ48]]. If $q_0 = 0$, it is a [[Bernoulli equation|0ZB84QCDKR]] with $n = 2$.
If $y_1$ is one solution, then the substitution $y = y_1 + 1/v$ gives a linear first-order equation for $v$.
If $q_2$ has no zero and is differentiable, the substitution $y = -u' / (q_2 u)$ gives the second-order linear equation $u'' - \left(q_1 + q_2'/q_2\right) u' + q_0 q_2\, u = 0$.
Example: $y' = 1 + y^2$ has the solutions $y = \tan(x + C)$.

## Explanations

### L1 human

A Riccati equation is the next case after a linear equation: the right side is a quadratic polynomial in $y$.
In general, there is no formula for its solutions.
But if you know one solution, you can find all the others.
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# Complete residue system

## Definition

Let $n$ be a positive integer.
A complete residue system modulo $n$ is a set $S$ of integers that contains exactly one element of each [[residue class|0QRV8FPK5M]] modulo $n$.

## Remarks

Equivalently, $S$ has exactly $n$ elements, and no two distinct elements of $S$ are congruent modulo $n$.
The set $\{0, 1, \dots, n - 1\}$ is the set of the least nonnegative residues. It is a complete residue system modulo $n$.
For odd $n$, the set $\{ -\tfrac{n-1}{2}, \dots, -1, 0, 1, \dots, \tfrac{n-1}{2} \}$ of the least absolute residues is also a complete residue system.
If $S$ is a complete residue system modulo $n$ and $\gcd(a, n) = 1$, then $\{ a s + b : s \in S \}$ is also a complete residue system for each integer $b$.

## Explanations

### L1 human

A complete residue system picks one representative from each box of the integers modulo $n$.
For $n = 4$, both $\{0, 1, 2, 3\}$ and $\{4, -3, 10, 7\}$ are complete residue systems.
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# Orbit-stabiliser theorem

## Statement

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \in X$.
Then the map $gG_x \mapsto g \cdot x$ is a well defined bijection from the set of the [[left cosets|03DEH43F4H]] of the [[stabiliser|0AYMP41THN]] $G_x$ to the [[orbit|0YQPZB5Q1G]] $G \cdot x$.
In particular, $|G \cdot x| = [G : G_x]$.

## Remarks

The key equivalence is $g \cdot x = h \cdot x$ if and only if $g^{-1}h \in G_x$, if and only if $gG_x = hG_x$.
The bijection is compatible with the actions: it sends $k \cdot gG_x$ to $k \cdot (g \cdot x)$ for each $k \in G$.
So each transitive action of $G$ is isomorphic to the [[action on the left cosets|0AZG100DKH]] of a stabiliser.

## Explanations

### L1 human

Two group elements move $x$ to the same point exactly when they differ by an element that fixes $x$.
So the points of the orbit correspond to the cosets of the stabiliser.
A big stabiliser means a small orbit, and a small stabiliser means a big orbit.
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# Dimension of the orthogonal complement

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $W$ be a subspace of $V$.
Then
$$\dim W + \dim W^\perp = \dim V,$$
where $W^\perp$ is the [[orthogonal complement|0XJXTBCB2T]] of $W$.

## Remarks

It follows from $V = W \oplus W^\perp$.
For example, in $\R^3$, a line has a plane as orthogonal complement: $1 + 2 = 3$.
For a real matrix, it gives $\rank A + \dim \ker A = n$ once more, because $\ker A$ is the orthogonal complement of the row space.

## Explanations

### L1 human

A subspace and its orthogonal complement share the dimensions of the space.
If $W$ uses $k$ directions, then $W^\perp$ uses the remaining $n - k$ directions.
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# Differential form

## Summary

A differential form is an object that one can integrate over curves, surfaces, volumes and their higher-dimensional analogues.
A $k$-form assigns to each point an alternating $k$-linear function of tangent vectors. Forms give a coordinate-free approach to multivariable calculus.
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# Polynomial of an endomorphism

## Summary

Let $p$ be a polynomial and let $A$ be a square matrix or an endomorphism.
Then $p(A)$ is the result of substituting $A$ for the variable of $p$, with the identity in place of the constant $1$.
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# Algebra of endomorphisms

## Description

Let $F$ be a field and let $V$ be a vector space over $F$.
The *algebra of endomorphisms* $\End_F(V)$ is the set of [[endomorphisms|0J7H0YBFDJ]] of $V$, with pointwise addition, scalar multiplication, and composition as multiplication.
It is an associative algebra over $F$, with the identity map $\id_V$ as unit.
Thus $S(T + U) = ST + SU$, $(S + T)U = SU + TU$ and $a(ST) = (aS)T = S(aT)$ for all $S, T, U \in \End_F(V)$ and $a \in F$.

## Remarks

If $\dim V = n$, then the choice of a basis gives an algebra isomorphism $\End_F(V) \cong M_n(F)$.
For $\dim V \ge 2$, the algebra is not commutative.
Its group of units is the group $GL(V)$ of linear isomorphisms $V \to V$.

## Explanations

### L1 human

The endomorphisms of a space can be added, scaled and composed.
Composition plays the role of multiplication.
This makes them a ring that is also a vector space, like the square matrices.
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# Limit point

## Summary

A limit point of a set $S$ is a point $x$ such that each neighbourhood of $x$ contains a point of $S$ other than $x$.
Limit points are the points that the subset approaches without using the point itself.
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# Base rate fallacy

## Example

A disease affects $1\%$ of a population. A test is positive for $90\%$ of the sick people and for $9\%$ of the healthy people.
Let $S$ be the event "sick" and $T$ the event "positive test". Then $P(S) = 0.01$, $P(T \mid S) = 0.9$ and $P(T \mid S^c) = 0.09$.
By the [[law of total probability|0W62TKMW7V]], $P(T) = 0.9 \cdot 0.01 + 0.09 \cdot 0.99 = 0.0981$.
By [[Bayes' theorem|0NM6E9VW04]], $P(S \mid T) = 0.009 / 0.0981 \approx 0.092$.
So only about $9\%$ of the people with a positive test are sick.
This is true although the test is right for $90\%$ of the sick and for $91\%$ of the healthy.

## Remarks

The base rate fallacy is the error to ignore the base rate $P(S)$ and to take $P(S \mid T)$ to be close to $P(T \mid S)$.
When the condition is rare, most positive results are false positives, even for a good test.

## Explanations

### L1 human

Among $10000$ people, $100$ are sick and $90$ of them test positive. Of the $9900$ healthy people, $891$ also test positive.
So of the $981$ positive tests, only $90$ come from sick people.
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# Blow-up in finite time

## Example

Let $x_0 > 0$ and consider the [[initial value problem|075VBY4K4X]] $x'(t) = x(t)^2$, $x(0) = x_0$.
[[Separation of variables|0V4E623THY]] gives the solution
$$x(t) = \frac{x_0}{1 - x_0 t}, \qquad t \in \left(-\infty, \frac{1}{x_0}\right).$$
This is the [[maximal solution|019VSXSFMQ]]: $x(t) \to \infty$ as $t \to 1/x_0$ from below.
So the solution blows up at the finite time $1/x_0$, although the field $f(x) = x^2$ is smooth on all of $\mathbb{R}$.

## Remarks

The blow-up time $1/x_0$ depends on the initial value. Larger initial values blow up sooner.
For $x_0 < 0$, the maximal interval is $(1/x_0, \infty)$, and for $x_0 = 0$ the solution is $0$ on $\mathbb{R}$.
The field $x^2$ is not globally Lipschitz. A [[globally Lipschitz field|0BFD3A6CRH]] has no blow-up.
Near the blow-up time, $x(t)$ behaves like $(1/x_0 - t)^{-1}$. This is a finite-time singularity.

## Explanations

### L1 human

The rate of growth is the square of the size, so the growth feeds itself very fast.
The solution reaches infinity in finite time, unlike exponential growth.
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# Cayley table

## Definition

Let $S = \{a_1, \ldots, a_n\}$ be a finite set with $n$ elements, listed in a fixed order, and let $\ast$ be a [[binary operation|0M418FE7BB]] on $S$.
The Cayley table of $(S, \ast)$ is the $n \times n$ table whose rows and columns are labelled $a_1, \ldots, a_n$.
The entry in row $i$ and column $j$ is $a_i \ast a_j$.

## Remarks

The operation is [[commutative|099E72KTF8]] exactly when the table is symmetric about its main diagonal.
In the Cayley table of a finite [[group|G5S1M9]], each element occurs exactly one time in each row and one time in each column.
Thus the Cayley table of a finite group is a Latin square.

## Explanations

### L1 human

A Cayley table is like the multiplication table from school, but for any finite set with an operation.
For the integers modulo $2$ under addition, the table has the rows $0, 1$ and $1, 0$.
You can read many properties from the table, for example if the operation is commutative.
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# Directrix

## Summary

A directrix of a conic is a line with the following property. For each point of the conic, the distance to a focus divided by the distance to the line is the same constant. This constant is the eccentricity.
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# Law of total probability

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space.
Let $B_1, B_2, \dots$ be a finite or countable family of pairwise disjoint events with $\bigcup_i B_i = \Omega$ and $P(B_i) > 0$ for each $i$.
Then for each event $A$,
$$P(A) = \sum_{i} P(A \mid B_i) \, P(B_i).$$

## Remarks

The events $B_i$ form a [[partition|09CCE86A6D]] of $\Omega$ into events of positive probability. The proof uses $A = \bigcup_i (A \cap B_i)$, countable additivity and the [[multiplication rule|0BS3RMX8HM]].
The case of two events $B$ and $B^c$ is the [[law of total probability for an event and its complement|0W62TKMW7V]].
Example: urn $1$ has $2$ red balls of $5$ and urn $2$ has $3$ red balls of $4$. Choose an urn with a fair coin and draw one ball. The ball is red with probability $\frac12 \cdot \frac25 + \frac12 \cdot \frac34 = \frac{23}{40}$.

## Explanations

### L1 human

Split the world into cases that do not overlap.
The chance of $A$ is the average of its chances in the cases, weighted by the chances of the cases.
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# Convergents converge to the number

## Statement

Let $x$ be an irrational real number, and let $p_n / q_n$ be the [[convergents|0SRWNR42BQ]] of its [[continued fraction expansion|0SJ7Z6VKWK]].
Then
$$\left| x - \frac{p_n}{q_n} \right| < \frac{1}{q_n q_{n+1}} \le \frac{1}{q_n^2}$$
for each $n \ge 0$, and $p_n / q_n \to x$ as $n \to \infty$.

## Remarks

The convergents with even $n$ increase toward $x$, and the convergents with odd $n$ decrease toward $x$.
For a rational $x$, the last convergent is equal to $x$.
Conversely, each infinite simple continued fraction converges to an irrational number, and this number has that continued fraction as its expansion.
For example, $\frac{355}{113}$ is a convergent of $\pi$, and $\left| \pi - \frac{355}{113} \right| < 3 \cdot 10^{-7}$.

## Explanations

### L1 human

The convergents of $x$ get closer and closer to $x$, and the error is smaller than one over the square of the denominator.
This makes them very good rational approximations of $x$.
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# Fermat's Last Theorem for n = 4

## Statement

There are no positive integers $a$, $b$ and $c$ with $a^4 + b^4 = c^4$.

## Remarks

A stronger statement holds: $x^4 + y^4 = z^2$ has no solution in positive integers.
The theorem also follows from [[Fermat's right triangle theorem|086BE0J3FZ]], in the form that $x^4 - y^4 = z^2$ has no solution in positive integers.
The proof uses [[infinite descent|0EFYCFHYFG]] and the [[classification of Pythagorean triples|0KF3J2KAER]].
Since $4$ divides every power of $2$ above $2$, the case $n = 4$ settles [[Fermat's Last Theorem|0Y5X3SDBCZ]] for all such exponents.

## History

Fermat's proof of his right triangle theorem, by infinite descent, gives this case.

## Explanations

### L1 human

The equation $a^4 + b^4 = c^4$ has no whole-number solutions.
The proof assumes a solution and builds a smaller one from it, which leads to a contradiction.
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# Riemann rearrangement theorem

## Statement

Let $\sum a_n$ be a real series that [[converges conditionally|02YRQXZDCS]].
Let $M$ be a real number.
Then there is a [[rearrangement|0HRJ2Y74T3]] $\sum a_{\sigma(n)}$ that converges to $M$.
There is also a rearrangement that diverges to $+\infty$, and one that diverges to $-\infty$.

## Remarks

The positive terms and the negative terms each give a divergent series, and $a_n \to 0$. Take positive terms until the partial sum exceeds $M$, then negative terms until it falls below $M$, and repeat.
So a real series converges absolutely if and only if all its rearrangements converge to the same sum.
For the [[alternating harmonic series|0JQ5Y3WWVE]], take two positive terms, then one negative term, and repeat. The sum becomes $\frac{3}{2} \ln 2$.

## History

Augustin-Louis Cauchy is usually credited with the first remark that a new order can change a sum. He studied the alternating harmonic series in 1833.
The theorem is named after Bernhard Riemann.

## Explanations

### L1 human

A conditionally convergent series has positive terms with an infinite total and negative terms with an infinite total.
By a careful choice of order, we can steer the partial sums to any target.
</document>

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# Casting out nines

## Summary

Casting out nines is a method to check a computation with integers by the remainders modulo $9$.
It uses the fact that a natural number is [[congruent|0TPW25CBN6]] modulo $9$ to the sum of its decimal digits.
To check $a \cdot b = c$, one compares the digit sum of $c$ with the product of the digit sums of $a$ and $b$, all modulo $9$. The same works for sums and differences.

## Remarks

If the two results are not congruent modulo $9$, the computation is wrong. If they are congruent, the computation can still be wrong. For example, an exchange of two digits in the result is not detected.
The method works because of [[divisibility by 9|0ESHD7ZQDX]] and the compatibility of congruences with [[addition|00FWBN7F0Y]] and [[multiplication|072D5G4SK6]].

## Explanations

### L1 human

Check $46 \cdot 37 = 1702$. The digit sums are $10 \equiv 1$ and $10 \equiv 1$, so the product should be $1 \cdot 1 = 1$ modulo $9$.
The digit sum of $1702$ is $10 \equiv 1$, so the check passes.
A wrong result such as $1712$ has the digit sum $11 \equiv 2$, so the check finds the error.
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# Similar matrices

## Summary

Two square matrices are similar when they are the matrices of the same endomorphism in two bases.
Equivalently, $B = P^{-1} A P$ for an invertible matrix $P$.
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# Exponential of a sum

## Statement

For all real numbers $x$ and $y$,
$$\exp(x + y) = \exp(x) \exp(y).$$

## Remarks

Multiply the two series by the [[Cauchy product|0AN28XQ45Z]]. The binomial theorem gives $\sum_{k=0}^{n} \frac{x^k}{k!} \frac{y^{n-k}}{(n-k)!} = \frac{(x + y)^n}{n!}$.
In particular, $\exp(x) \exp(-x) = \exp(0) = 1$.
So $\exp$ is a group homomorphism from $(\mathbb{R}, +)$ to the multiplicative group $(0, \infty)$. Mathlib states the formula as `Real.exp_add`.

## Explanations

### L1 human

The exponential function turns addition into multiplication.
This is the rule $e^{x + y} = e^x e^y$ for powers.
</document>

<document source="https://tamnd.github.io/math2/node/0E9502NT92.md" tag="0E9502NT92" kind="definition">
# Independence of two random variables

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $X$ be a [[random variable|0DKRR22AEQ]] with values in $(E_1, \mathcal{E}_1)$ and $Y$ a random variable with values in $(E_2, \mathcal{E}_2)$.
$X$ and $Y$ are independent if for all $A \in \mathcal{E}_1$ and $B \in \mathcal{E}_2$,
$$P(X \in A, \, Y \in B) = P(X \in A) \, P(Y \in B).$$

## Remarks

In other words, the events $\{X \in A\}$ and $\{Y \in B\}$ are [[independent|0P0MZ4X682]] for all $A$ and $B$.
For real random variables, it is enough to check $P(X \le x, Y \le y) = P(X \le x) \, P(Y \le y)$ for all real $x$ and $y$.
For discrete random variables, independence is equivalent to $P(X = x, Y = y) = P(X = x) \, P(Y = y)$ for all $x$ and $y$.
If $X$ and $Y$ are independent and $g$, $h$ are measurable functions, then $g(X)$ and $h(Y)$ are independent.

## Explanations

### L1 human

Two random variables are independent when knowing the value of one tells you nothing about the other.
The results of two separate dice are independent. The result of one die and the sum of both dice are not.
</document>

<document source="https://tamnd.github.io/math2/node/0E99F51GVB.md" tag="0E99F51GVB" kind="concept">
# Exponential distribution

## Summary

The exponential distribution with rate $\lambda$ is the law of a waiting time with no memory.
An example is the time between two events of a Poisson process.
The node [[exponential distribution|028Z8WC5Y9]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0E9GTAV478.md" tag="0E9GTAV478" kind="definition">
# Range of a function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The range of $f$ is the set
$$\operatorname{ran}(f) = \{f(a) \mid a \in A\} = \{b \in B \mid \exists a \in A, \, f(a) = b\}.$$

## Remarks

The range is also called the image of $f$, and it is the [[image|0Q3F20XB0A]] $f(A)$ of the whole domain.
Some authors use "range" for the [[codomain|0F9JHP8AJN]]. This node uses the meaning above.
For $f \colon \mathbb{R} \to \mathbb{R}$ with $f(x) = x^2$, the range is $[0, \infty)$.

## Explanations

### L1 human

The range is the set of outputs that actually occur.
It can be smaller than the codomain.
</document>

<document source="https://tamnd.github.io/math2/node/0E9M24KS9J.md" tag="0E9M24KS9J" kind="statement">
# Harmonic series diverges

## Statement

The partial sums $H_N = \sum_{n=1}^{N} \frac{1}{n}$ of the [[harmonic series|0CXTMAY5PX]] tend to $+\infty$ as $N \to \infty$.
In particular, the harmonic series is a divergent series.

## Remarks

The classical proof groups the terms in blocks from $2^{k-1} + 1$ to $2^k$. Each block has $2^{k-1}$ terms, each at least $1/2^k$, so each block sums to at least $1/2$. Thus $H_{2^k} \ge 1 + k/2$.
A second proof uses the [[integral test|019QZ6RRVH]]: $H_N \ge \int_1^{N+1} dx/x = \ln(N + 1)$.
Mathlib states it as the limit of $\sum_{n < N} 1/(n+1)$, in `Real.tendsto_sum_range_one_div_nat_succ_atTop`.

## History

Nicole Oresme proved the divergence in the 14th century, with the grouping argument. Pietro Mengoli and Jacob Bernoulli gave other proofs in the 17th century.

## Explanations

### L1 human

Group the terms: $1/3 + 1/4 \ge 1/2$, then $1/5 + \cdots + 1/8 \ge 1/2$, then $1/9 + \cdots + 1/16 \ge 1/2$, and so on.
There are infinitely many such groups, so the sum exceeds each bound.
</document>

<document source="https://tamnd.github.io/math2/node/0EA4XFB0EY.md" tag="0EA4XFB0EY" kind="concept">
# Ceiling function

## Summary

The ceiling function sends a real number $x$ to the least integer that is greater than or equal to $x$. We write $\lceil x \rceil$ for it.
For example, $\lceil 2.3 \rceil = 3$ and $\lceil -2.3 \rceil = -2$.
</document>

<document source="https://tamnd.github.io/math2/node/0EA5BTA0CX.md" tag="0EA5BTA0CX" kind="concept">
# Division ring

## Summary

A division ring is a nonzero ring in which every nonzero element has a multiplicative inverse.
It is like a field, but the multiplication need not be commutative.
The [[definition of a division ring|0WZXK05ME7]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0EAAS2EHDR.md" tag="0EAAS2EHDR" kind="statement">
# Chain rule in several variables

## Statement

Let $U \subseteq \mathbb{R}^n$ and $W \subseteq \mathbb{R}^m$ be open sets, let $g \colon U \to \mathbb{R}^m$ with $g(U) \subseteq W$, let $f \colon W \to \mathbb{R}^k$, and let $a \in U$.
Suppose that $g$ is [[differentiable|0135XZZQ8M]] at $a$ and $f$ is differentiable at $g(a)$.
Then the [[composition|0THBK9V825]] $f \circ g \colon U \to \mathbb{R}^k$ is differentiable at $a$, and
$$D(f \circ g)(a) = Df(g(a)) \circ Dg(a).$$

## Remarks

In matrix form, the [[Jacobian matrix|047NDTEPCV]] of $f \circ g$ at $a$ is the product $J_f(g(a)) \, J_g(a)$. In terms of partial derivatives, $\dfrac{\partial (f \circ g)_i}{\partial x_j}(a) = \sum_{l=1}^m \dfrac{\partial f_i}{\partial y_l}(g(a)) \, \dfrac{\partial g_l}{\partial x_j}(a)$.
The same rule, with the same proof, holds for [[Fréchet derivatives|08EVNYZF42]] of maps between normed spaces.
A special case is the [[chain rule for a composition with a curve|0A8TJ0VHC7]].

## Explanations

### L1 human

The derivative of a composition is the composition of the derivatives: first the linear approximation of $g$, then the linear approximation of $f$.
In one variable this is $(f \circ g)'(a) = f'(g(a)) \, g'(a)$, a product of two numbers. In several variables the numbers become matrices.
</document>

<document source="https://tamnd.github.io/math2/node/0EAJ2XXADC.md" tag="0EAJ2XXADC" kind="concept">
# Degree of a field extension

## Summary

The degree of a field extension $L/K$ is the dimension of $L$ as a vector space over $K$.
It is written $[L : K]$.
The [[definition of the degree|0GS74FGFEK]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0EAR3KNRTN.md" tag="0EAR3KNRTN" kind="concept">
# Trigonometric substitution

## Summary

Trigonometric substitution is a [[substitution|0D37QFPEZE]] method for integrals that contain $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$. We set $x$ equal to $a \sin \theta$, $a \tan \theta$, or $a \sec \theta$. The [[Pythagorean identity|03H7WZ4T3K]] then removes the square root.

## Remarks

With $x = a \sin \theta$ for $\theta \in [-\pi/2, \pi/2]$, we get $\sqrt{a^2 - x^2} = a \cos \theta$, for $a > 0$.
For example, this substitution gives $\int_{-1}^{1} \sqrt{1 - x^2} \, dx = \pi/2$, the area of a half disc.
Hyperbolic functions can replace the trigonometric functions in the second and the third case.

## Explanations

### L1 human

A square root such as $\sqrt{1 - x^2}$ is the side of a right triangle.
Writing $x$ as the sine of an angle turns the root into a cosine.
</document>

<document source="https://tamnd.github.io/math2/node/0EB6BBMQMC.md" tag="0EB6BBMQMC" kind="statement">
# A composite number has a prime factor at most its square root

## Statement

Let $n$ be a [[composite number|0N3ADXHNTQ]].
Then $n$ has a prime factor $p$ with $p^2 \le n$, that is, $p \le \sqrt{n}$.

## Remarks

The [[smallest prime factor|0S2SYHTWPY]] $p$ of $n$ has this property. Write $n = p m$. Then $m > 1$, and each prime factor of $m$ is at least $p$, so $m \ge p$ and $n \ge p^2$.
This result justifies [[trial division|0Z962ECCXR]]: to test whether $n > 1$ is prime, it is enough to test the primes $p \le \sqrt{n}$.
For example, $\sqrt{97} < 10$ and none of $2, 3, 5, 7$ divides $97$, so $97$ is prime.

## Explanations

### L1 human

If $n = a b$ with both factors larger than $\sqrt{n}$, then $a b > n$, which is impossible.
So one of the two factors is at most $\sqrt{n}$, and a prime factor of that factor is also at most $\sqrt{n}$.
</document>

<document source="https://tamnd.github.io/math2/node/0EB9N0S33C.md" tag="0EB9N0S33C" kind="concept">
# Invertible matrix

## Summary

An invertible matrix is a square matrix that has a two-sided inverse for matrix multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/0EBACQFAZW.md" tag="0EBACQFAZW" kind="concept">
# Cardinal number

## Summary

Cardinal numbers measure the size of sets, finite or infinite.
They generalize the natural numbers as answers to the question "how many elements?".
</document>

<document source="https://tamnd.github.io/math2/node/0EBCWQ8BVE.md" tag="0EBCWQ8BVE" kind="concept">
# Probability-generating function

## Summary

The probability-generating function of a random variable with values in $\{0, 1, 2, \dots\}$ is the power series whose coefficients are the probabilities $P(X = k)$.
The node [[probability-generating function|0Z8PSSAVBA]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0EBE031B8E.md" tag="0EBE031B8E" kind="statement">
# A finite integral domain is a field

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]] that has a finite number of elements.
Then $R$ is a [[field|0QA8CHH8MG]].

## Remarks

The proof: for $a \neq 0$, the map $x \mapsto a x$ from $R$ to $R$ is injective by [[cancellation|0GJM24EXP9]]. A finite set has no injective map to itself that is not surjective, so $a x = 1$ for some $x$.
A consequence is that [[$\mathbb{Z}/p\mathbb{Z}$ is a field|0NV6E43P4A]] for every prime $p$.
The hypothesis of finiteness is necessary: $\mathbb{Z}$ is an infinite integral domain that is not a field.

## Explanations

### L1 human

In a finite integral domain, multiplication by a nonzero element $a$ never sends two elements to the same element.
Since there are only finitely many elements, every element is a product $a x$.
In particular, $1 = a x$ for some $x$, and $x$ is the inverse of $a$.
</document>

<document source="https://tamnd.github.io/math2/node/0EBZ3G455G.md" tag="0EBZ3G455G" kind="statement">
# The multiplicative group of a finite field is cyclic

## Statement

Let $F$ be a [[finite field|08RTDHX1SK]] of order $q$.
Then the multiplicative group $F^\times$ is cyclic of order $q - 1$.

## Remarks

This is the case $G = F^\times$ of the [[theorem on finite subgroups|0CKGBEJX6B]].
A generator of $F^\times$ is a [[primitive element|0YAA9CJQXF]] of $F$.
For $F = \mathbb{Z}/p\mathbb{Z}$, the statement says that there is a primitive root modulo $p$.
For example, $3$ generates $(\mathbb{Z}/7\mathbb{Z})^\times$, with powers $3, 2, 6, 4, 5, 1$.

## Explanations

### L1 human

All the nonzero elements of a finite field are powers of one element.
So multiplication in a finite field is like addition of exponents modulo $q - 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0EC7AGJ0PX.md" tag="0EC7AGJ0PX" kind="definition">
# Exact equation

## Definition

Let $D$ be an open subset of $\mathbb{R}^2$ and let $M, N : D \to \mathbb{R}$ be continuous functions.
The equation
$$M(x, y) + N(x, y)\, y' = 0,$$
also written $M\, dx + N\, dy = 0$, is exact on $D$ if there is a differentiable function $\psi : D \to \mathbb{R}$ with
$$\frac{\partial \psi}{\partial x} = M \quad \text{and} \quad \frac{\partial \psi}{\partial y} = N \quad \text{on } D.$$
Such a function $\psi$ is a potential function of the equation.

## Remarks

Let the equation be exact. Then a differentiable function $y$ on an interval, with graph in $D$, is a [[solution|04SW7WECS8]] if and only if $\psi(x, y(x))$ is constant. This follows from the chain rule: $\frac{d}{dx} \psi(x, y(x)) = M + N y'$.
So the solutions are given implicitly by $\psi(x, y) = C$.
Example: $2 x y + x^2 y' = 0$ is exact with $\psi = x^2 y$, and its solutions satisfy $x^2 y = C$.
The [[test for exactness|0MSCQNFTM8]] decides whether an equation is exact.

## Explanations

### L1 human

An exact equation is the total derivative of one function $\psi$ set to zero.
So $\psi$ stays constant along each solution, and the solution curves are the level curves of $\psi$.
To solve the equation, you find $\psi$ by integration.
</document>

<document source="https://tamnd.github.io/math2/node/0ECVK59Y03.md" tag="0ECVK59Y03" kind="object">
# Empty set

## Description

The empty set is the [[set|0SJ3RX9TXA]] that has no [[elements|0EZ758XE73]].
It is written $\varnothing$ or $\{\}$.
For each object $x$, $x \notin \varnothing$.
By the [[axiom of extensionality|0TB16T2C97]], there is only one empty set.

## Remarks

The empty set is a [[subset|0FE8Q5S42Y]] of every set. This is a case of [[vacuous truth|0V4QXPGJQK]].
Its [[cardinality|0JTGRV4WT4]] is $0$.
The set $\{\varnothing\}$ is not empty: it has one element.

## Explanations

### L1 human

The empty set is an empty bag.
For example, the set of real numbers $x$ with $x^2 = -1$ is empty.
It is useful in the same way that the number $0$ is useful.
</document>

<document source="https://tamnd.github.io/math2/node/0ED4HWNDGW.md" tag="0ED4HWNDGW" kind="concept">
# Interval of convergence

## Summary

The interval of convergence of a real power series is the set of points where the series converges. It is an interval with center $c$ and half length equal to the radius of convergence. The node [[interval of convergence|0Z5R9TVWQH]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ED72CTBNP.md" tag="0ED72CTBNP" kind="concept">
# Counterexample

## Summary

A counterexample to a universal statement $\forall x \in X, \, P(x)$ is an element $a \in X$ such that $P(a)$ is false.
One counterexample shows that the universal statement is false.

## Remarks

The number $2$ is a counterexample to "each prime number is odd".
This works because of the [[negation of a universal statement|0P3A06QNDP]]: $\neg \forall x \, P(x)$ is equivalent to $\exists x \, \neg P(x)$.

## Explanations

### L1 human

To show that a general claim is wrong, one bad case is enough.
"Each continuous function is differentiable" is false: $f(x) = |x|$ is continuous but not differentiable at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0EDH403M21.md" tag="0EDH403M21" kind="statement">
# Existence of the infimum

## Statement

Let $S$ be a nonempty subset of the real numbers that is [[bounded below|0B5H2V4JW8]].
Then $S$ has an [[infimum|04KFTA34C5]] in $\mathbb{R}$.

## Remarks

Proof: the set $-S = \{-s : s \in S\}$ is nonempty and bounded above. By the [[completeness axiom|0AXWET80YW]], it has a supremum $b$. Then $-b$ is the infimum of $S$.
Another proof: the set of lower bounds of $S$ is nonempty and bounded above by each element of $S$, and its supremum is $\inf S$.

## Explanations

### L1 human

If a set of real numbers has a floor, then it has a highest floor.
This follows from the completeness axiom by a reflection of the number line through $0$, which turns floors into ceilings.
</document>

<document source="https://tamnd.github.io/math2/node/0EDKGA2FAK.md" tag="0EDKGA2FAK" kind="concept">
# Hamiltonian system

## Summary

A Hamiltonian system is a dynamical system given by Hamilton's equations $q' = \partial H / \partial p$, $p' = -\partial H / \partial q$ for a function $H$ of positions and momenta.
Such systems model mechanical systems without friction, such as planetary systems. See the [[definition|0C2AKPDCD9]].

## Remarks

The name honours William Rowan Hamilton.
</document>

<document source="https://tamnd.github.io/math2/node/0EE19CFEEF.md" tag="0EE19CFEEF" kind="statement">
# Neyman–Pearson lemma

## Statement

Consider a [[hypothesis test|087P03NEND]] of the simple null hypothesis $H_0 \colon \theta = \theta_0$ against the simple alternative $H_1 \colon \theta = \theta_1$. Let $L(\theta \mid x)$ be the [[likelihood|05FSTXM92W]] of the data $x$.
For $\eta > 0$, let the test reject $H_0$ when
$$\Lambda(x) = \frac{L(\theta_0 \mid x)}{L(\theta_1 \mid x)} \le \eta,$$
and let $\alpha = P_{\theta_0}(\Lambda(X) \le \eta)$ be its size.
Then this likelihood-ratio test is a most powerful test of size $\alpha$. Each other test with size at most $\alpha$ has at most the same power at $\theta_1$.

## Remarks

Consider all tests with a probability of a [[type I error|0R41PVFA39]] at most $\alpha$. Among them, the likelihood-ratio test has the smallest probability of a type II error.
For a discrete law, an exact size $\alpha$ can need a randomized test on the boundary $\Lambda(x) = \eta$.
Example: for an i.i.d. $\mathcal{N}(\mu, 1)$ sample and $\mu_1 > \mu_0$, the test rejects $H_0$ when $\bar{x}$ is large.

## History

Jerzy Neyman and Egon Pearson introduced the lemma in a paper in 1933.

## Explanations

### L1 human

To choose between two exact hypotheses, the best test compares how well each hypothesis explains the data.
If the alternative explains the data much better than the null hypothesis, reject the null hypothesis. No other test with the same false-alarm rate finds more true effects.
</document>

<document source="https://tamnd.github.io/math2/node/0EE2V3CHFK.md" tag="0EE2V3CHFK" kind="statement">
# Rule of sum

## Statement

Let $A$ and $B$ be finite sets with $A \cap B = \varnothing$.
Then $A \cup B$ is finite and $|A \cup B| = |A| + |B|$.

More generally, let $A_1, \dots, A_k$ be finite sets that are pairwise disjoint.
Then $|A_1 \cup \dots \cup A_k| = |A_1| + \dots + |A_k|$.

## Remarks

For the disjoint union $A \sqcup B$ of two arbitrary finite sets, the same formula $|A \sqcup B| = |A| + |B|$ holds.
If $A$ and $B$ are not disjoint, the formula fails. The [[inclusion-exclusion formula for two sets|0C4YEQ9H5E]] gives the correct count.

## Explanations

### L1 human

If you choose one item from a first group or from a second group, and no item is in both groups, you add the sizes.
For example, with 3 soups and 4 salads on a menu, there are $3 + 4 = 7$ ways to choose one starter.
</document>

<document source="https://tamnd.github.io/math2/node/0EEE2CZB2A.md" tag="0EEE2CZB2A" kind="definition">
# Continuity at a point

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function, and let $a \in D$.
The function $f$ is continuous at $a$ if the following holds.
For each open interval $V$ containing $f(a)$ there is an open interval $U$ containing $a$ with $f(x) \in V$ for all $x \in U \cap D$.

## Remarks

If $a$ is a limit point of $D$, then $f$ is continuous at $a$ if and only if $\lim_{x \to a} f(x) = f(a)$. See [[limit of a function|0TXTSEMHR5]].
If $a$ is an isolated point of $D$, then each function on $D$ is continuous at $a$.
The [[epsilon-delta form|0W8YHHGCJG]] states the same condition with numbers.
Mathlib defines it for topological spaces, as `ContinuousAt`.

## Explanations

### L1 human

A function is continuous at $a$ if small changes of $x$ near $a$ give small changes of $f(x)$.
The graph has no jump or gap at $a$.
</document>

<document source="https://tamnd.github.io/math2/node/0EEKF7YFBW.md" tag="0EEKF7YFBW" kind="definition">
# Refinement of a partition

## Definition

Let $P$ and $Q$ be [[partitions|069P20KYQT]] of $[a, b]$.
The partition $Q$ is a refinement of $P$ if each point of $P$ is also a point of $Q$.

## Remarks

Any two partitions $P$ and $Q$ have a common refinement, made of the points of both.
Refining a partition lowers the upper [[Darboux sum|0QRQVH0CRW]] and raises the lower Darboux sum, or leaves them unchanged.

## Explanations

### L1 human

A refinement cuts the interval at more points.
Each old subinterval is cut into one or more new subintervals.
</document>

<document source="https://tamnd.github.io/math2/node/0EERDDPSVT.md" tag="0EERDDPSVT" kind="definition">
# Limit cycle

## Definition

Let $U$ be an open subset of $\mathbb{R}^2$, let $f : U \to \mathbb{R}^2$ be continuously differentiable, and let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$.
Let $y \in U$ be a point whose solution exists for all $t \ge 0$. Its $\omega$-limit set $\omega(y)$ is the set of points $z \in U$ such that $\varphi_{t_k}(y) \to z$ for some sequence $t_k \to \infty$.
For a point $y$ whose solution exists for all $t \le 0$, the $\alpha$-limit set $\alpha(y)$ is defined in the same way with $t_k \to -\infty$.
A limit cycle is a [[periodic orbit|01DSNP0B4R]] $\Gamma$ such that $\Gamma = \omega(y)$ or $\Gamma = \alpha(y)$ for some point $y \in U$ that is not on $\Gamma$.

## Remarks

A limit cycle is stable if all nearby trajectories approach it as $t \to \infty$, and unstable if they approach it as $t \to -\infty$. It can also be semi-stable.
A linear system has no limit cycles: its periodic orbits, if any, fill a region around the origin.
The van der Pol oscillator $x'' - \mu (1 - x^2) x' + x = 0$ with $\mu > 0$ has exactly one limit cycle, and it is stable.
The [[Poincaré–Bendixson theorem|0EW1X61ZT1]] is a main tool to prove that a limit cycle exists.

## Explanations

### L1 human

A limit cycle is an isolated loop in the phase plane that nearby motions spiral toward or away from.
It models a self-sustained oscillation, such as a heartbeat, whose size does not depend on the initial state.
</document>

<document source="https://tamnd.github.io/math2/node/0EF2VG8XKQ.md" tag="0EF2VG8XKQ" kind="definition">
# Concave function

## Definition

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be a function.
The function $f$ is concave if
$$f(t x + (1 - t) y) \ge t f(x) + (1 - t) f(y)$$
for all $x, y \in I$ and all $t \in [0, 1]$.

## Remarks

So $f$ is concave exactly when $-f$ is [[convex|0VV0HSW1ZS]].
For example, $\sqrt{x}$ and $\ln x$ are concave on $(0, \infty)$.
Mathlib defines concavity on a set as `ConcaveOn`.

## Explanations

### L1 human

A concave function bends downward, like a dome.
Each chord of the graph lies on or below the graph.
</document>

<document source="https://tamnd.github.io/math2/node/0EFDDXTVD6.md" tag="0EFDDXTVD6" kind="definition">
# Convergence in probability

## Definition

Let $X, X_1, X_2, \dots$ be real random variables on one probability space $(\Omega, \mathcal{F}, P)$.
The sequence $(X_n)$ converges in probability to $X$ if for each $\varepsilon > 0$,
$$\lim_{n \to \infty} P\left(|X_n - X| > \varepsilon\right) = 0.$$
One writes $X_n \xrightarrow{P} X$.

## Remarks

The limit is unique up to equality almost surely.
If $X_n \to X$ in probability, then a subsequence of $(X_n)$ converges to $X$ almost surely.
The weak law of large numbers is a result about convergence in probability.
Example: let $X_n$ be $1$ with probability $1/n$ and $0$ otherwise. Then $X_n \to 0$ in probability.

## Explanations

### L1 human

Convergence in probability means that, for large $n$, $X_n$ is very likely to be close to $X$.
It does not say that one fixed run of the experiment gives a convergent sequence.
</document>

<document source="https://tamnd.github.io/math2/node/0EFYCFHYFG.md" tag="0EFYCFHYFG" kind="concept">
# Proof by infinite descent

## Summary

A proof by infinite descent shows that a statement has no solution in positive integers.
It assumes a solution, and from it constructs a strictly smaller solution in positive integers.
This gives an infinite decreasing sequence of positive integers, which is impossible.

## Remarks

The method rests on the well-ordering of the natural numbers: each nonempty set of natural numbers has a least element.
A classic use shows that $\sqrt{2}$ is irrational: a solution of $a^2 = 2 b^2$ in positive integers gives a smaller one, $(2b - a, a - b)$.
Fermat used the method to prove [[Fermat's right triangle theorem|086BE0J3FZ]].

## Explanations

### L1 human

You cannot go down forever in the positive integers: you reach the bottom after finitely many steps.
So if each solution gives a smaller one, there can be no solution at all.
</document>

<document source="https://tamnd.github.io/math2/node/0EGJZHZ02C.md" tag="0EGJZHZ02C" kind="concept">
# Primality test

## Summary

A primality test is an algorithm that decides whether a given natural number is [[prime|0SN0MAEKYX]].
Unlike factorization, it does not need to find the factors of a composite number.

## Remarks

The simplest test is [[trial division|0Z962ECCXR]] by the primes up to $\sqrt{n}$, but it is slow for large $n$.
Probabilistic tests, such as the [[Fermat test|0N5ZSZMYH6]] and the [[Miller–Rabin test|0SZCSYHW60]], are fast but can accept a composite number with a small probability.
In 2002, Agrawal, Kayal and Saxena gave the first deterministic test that runs in polynomial time without an unproven hypothesis.

## Explanations

### L1 human

A primality test answers the question "is $n$ prime?".
For numbers with hundreds of digits, this is much easier than finding the factors of $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0EGN1E1JPB.md" tag="0EGN1E1JPB" kind="concept">
# Formal derivative

## Summary

The formal derivative of a polynomial is defined by the usual rule of calculus, $(x^n)' = n x^{n-1}$, with no limits.
It makes sense over every ring.
The [[definition of the formal derivative|0AD4VS02H3]] gives the formula.
</document>

<document source="https://tamnd.github.io/math2/node/0EGSEE9J9R.md" tag="0EGSEE9J9R" kind="statement">
# Boole's inequality

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A_1, A_2, A_3, \dots$ be a finite or infinite sequence of [[events|00SQGJNRTB]].
Then
$$P\left(\bigcup_{i} A_i\right) \le \sum_{i} P(A_i).$$

## Remarks

The inequality is also called the union bound. It needs no hypothesis on the events: they need not be disjoint or independent.
Equality holds when the events are pairwise disjoint.
The Bonferroni inequalities refine it with the terms of the [[inclusion–exclusion principle|0FF7726C7P]]. For example, $P\left(\bigcup_{i=1}^n A_i\right) \ge \sum_i P(A_i) - \sum_{i < j} P(A_i \cap A_j)$.

## History

The inequality is named after George Boole.

## Explanations

### L1 human

The chance that at least one of several bad things happens is at most the sum of their chances.
If each of $10$ parts fails with probability $0.01$, then some part fails with probability at most $0.1$.
</document>

<document source="https://tamnd.github.io/math2/node/0EHBY5Y84N.md" tag="0EHBY5Y84N" kind="statement">
# Prime number theorem

## Statement

Let $\pi(x)$ be the [[prime-counting function|0XRJB5SSNE]], and let $\ln$ be the natural logarithm.
Then
$$\lim_{x \to \infty} \frac{\pi(x)}{x / \ln x} = 1.$$

## Remarks

We write this as $\pi(x) \sim x / \ln x$.
An equivalent form: the $n$-th prime $p_n$ satisfies $p_n \sim n \ln n$.
The logarithmic integral $\operatorname{li}(x) = \int_2^x \frac{dt}{\ln t}$ gives a better approximation of $\pi(x)$, with the same limit.
For example, $\pi(10^6) = 78498$ and $10^6 / \ln 10^6 \approx 72382$.

## History

Legendre and Gauss conjectured the result from tables of primes, at the end of the 18th century.
Hadamard and de la Vallée Poussin proved it independently in 1896, with methods of complex analysis. Selberg and Erdős found elementary proofs in 1949.

## Explanations

### L1 human

Near a large number $x$, about one number in $\ln x$ is prime.
For numbers with $100$ digits, $\ln x \approx 230$, so about one number in $230$ is prime.
The theorem makes this rough rule precise.
</document>

<document source="https://tamnd.github.io/math2/node/0EHKVRF4J6.md" tag="0EHKVRF4J6" kind="statement">
# Liouville's theorem

## Statement

Let $f : \mathbb{C} \to \mathbb{C}$ be an [[entire function|0419794R0T]].
Assume that $f$ is bounded: there is a real number $M$ with $|f(z)| \le M$ for each $z \in \mathbb{C}$.
Then $f$ is constant.

## Remarks

Proof: let $c \in \mathbb{C}$. By [[Cauchy's estimates|0JJRENWMP6]], $|f'(c)| \le M / r$ for each $r > 0$. So $f'(c) = 0$ for each $c$, and $f$ is constant.
For example, $\sin$ is entire and not constant, so it is not bounded on $\mathbb{C}$. On $\mathbb{R}$ it is bounded, which shows that the theorem has no real analogue.
The theorem gives a short proof of the [[fundamental theorem of algebra|0JP0W07Y9A]].

## Explanations

### L1 human

A holomorphic function on the whole plane cannot stay inside a bounded region unless it does not move at all.
Each nonconstant entire function, for example $z$, $e^z$ or $\cos z$, takes values that are as large as you like.
</document>

<document source="https://tamnd.github.io/math2/node/0EHM5ZP5QH.md" tag="0EHM5ZP5QH" kind="concept">
# Primitive root of unity

## Summary

A primitive $n$th root of unity is an $n$th [[root of unity|0YV2CKP2AV]] that is not a $k$th root of unity for a smaller positive $k$.
Its powers give all the $n$th roots of unity.
</document>

<document source="https://tamnd.github.io/math2/node/0EHWPWGHB2.md" tag="0EHWPWGHB2" kind="statement">
# Binomial theorem

## Statement

Let $R$ be a commutative ring, let $x, y \in R$, and let $n$ be a natural number.
Then
$$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}.$$
Here $\binom{n}{k}$ is the [[binomial coefficient|092GEF38YD]], and the integer $\binom{n}{k}$ acts on $R$ by repeated addition.

## Remarks

For $n = 2$ it gives $(x + y)^2 = x^2 + 2xy + y^2$. For $n = 3$ it gives $x^3 + 3x^2 y + 3x y^2 + y^3$.
The formula stays true in a noncommutative ring when $xy = yx$.
With $x = y = 1$ it gives the [[sum of binomial coefficients|0KAMF19NV8]]. With $x = -1$ and $y = 1$ it gives the [[alternating sum|0PQQGFAWSH]].
The [[multinomial theorem|0XFC1VQYEK]] extends it to sums of more than two terms.

## Explanations

### L1 human

When you expand $(x + y)^n$, each term comes from choosing $x$ or $y$ in each of the $n$ factors.
The term $x^k y^{n-k}$ appears once for each choice of the $k$ factors that give $x$, so its coefficient is $\binom{n}{k}$.
</document>

<document source="https://tamnd.github.io/math2/node/0EHYZFZ4PK.md" tag="0EHYZFZ4PK" kind="definition">
# P-value

## Definition

Consider a [[hypothesis test|087P03NEND]] of a null hypothesis $H_0 \colon \theta \in \Theta_0$ with a test statistic $T(X)$, where large values of $T$ are evidence against $H_0$. Let $t$ be the observed value of $T(X)$.
The p-value of the observation is
$$p(t) = \sup_{\theta \in \Theta_0} P_\theta\big(T(X) \ge t\big).$$
If $H_0$ is simple, $\Theta_0 = \{\theta_0\}$, then $p(t) = P_{\theta_0}(T(X) \ge t)$.

## Remarks

For a two-sided test with a statistic that is symmetric about $0$ under $H_0$, one uses $P_{\theta_0}(|T(X)| \ge |t|)$.
The test that rejects $H_0$ when $p \le \alpha$ has significance level $\alpha$.
If $H_0$ is simple and $T(X)$ has a continuous law under $H_0$, then the random p-value $p(T(X))$ has the [[uniform distribution|0WGDPYR30Q]] on $[0, 1]$ under $H_0$.
The p-value is not the probability that $H_0$ is true.

## Explanations

### L1 human

The p-value tells how surprising your data would be if the null hypothesis were true.
Example: a coin shows $14$ heads in $20$ tosses. If the coin is fair, the probability of at least $14$ heads is about $0.058$. This is the one-sided p-value.
</document>

<document source="https://tamnd.github.io/math2/node/0EJ4PC8NJ6.md" tag="0EJ4PC8NJ6" kind="statement">
# Universal property of the abelianisation

## Statement

Let $G$ be a [[group|G5S1M9]], let $\pi \colon G \to G^{\mathrm{ab}}$ be the projection to its [[abelianisation|0AQZV0905N]], and let $A$ be an [[abelian group|082GHNPJ53]].
For each [[group homomorphism|0JYTG5K9Y4]] $\varphi \colon G \to A$, there is exactly one group homomorphism $\bar{\varphi} \colon G^{\mathrm{ab}} \to A$ with $\bar{\varphi} \circ \pi = \varphi$.

## Remarks

Each commutator goes to $e$ under $\varphi$, because $A$ is abelian, so $[G, G] \subseteq \ker \varphi$.
Then the [[universal property of the quotient group|0GJ83E78A5]] gives $\bar{\varphi}$.
Thus the homomorphisms from $G$ to $A$ correspond exactly to the homomorphisms from $G^{\mathrm{ab}}$ to $A$.

## Explanations

### L1 human

A homomorphism into an abelian group cannot see the commutators of $G$.
So it factors through the abelianisation, where the commutators are gone.
This makes the abelianisation the best abelian approximation of $G$.
</document>

<document source="https://tamnd.github.io/math2/node/0EJVD9K0K1.md" tag="0EJVD9K0K1" kind="concept">
# Normal convergence

## Summary

A series of functions converges normally when the series of the suprema of the absolute values of its terms converges. Normal convergence implies uniform and absolute convergence. The node [[normal convergence|0T0EJDGHMB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0EKGKY5TGH.md" tag="0EKGKY5TGH" kind="statement">
# Epsilon-delta continuity in metric spaces

## Statement

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]], let $f \colon X \to Y$ be a function and let $a \in X$.
Then $f$ is [[continuous at|07N7W5C29X]] $a$ if and only if for each real $\varepsilon > 0$ there is a real $\delta > 0$ such that
$$d_X(x, a) < \delta \implies d_Y\bigl(f(x), f(a)\bigr) < \varepsilon \quad \text{for all } x \in X.$$

## Remarks

In terms of [[open balls|0DC3MPR7C9]], the condition says: for each $\varepsilon > 0$ there is $\delta > 0$ with $f(B(a, \delta)) \subseteq B(f(a), \varepsilon)$.
The number $\delta$ can depend on $\varepsilon$ and on $a$. When one $\delta$ works for all $a$, the map is [[uniformly continuous|0FHSDCQACY]].
For $X = Y = \mathbb{R}$, this is the epsilon-delta definition of continuity from calculus.

## Explanations

### L1 human

You can make $f(x)$ as close to $f(a)$ as you want by taking $x$ close enough to $a$.
The tolerance $\varepsilon$ for the output comes first, and then you find a tolerance $\delta$ for the input.
</document>

<document source="https://tamnd.github.io/math2/node/0EKZCX4K3P.md" tag="0EKZCX4K3P" kind="definition">
# Hermitian matrix

## Definition

Let $A = (a_{ij})$ be an $n \times n$ complex [[square matrix|0JXZKX526R]].
The matrix $A$ is *Hermitian* when $A^* = A$, where $A^*$ is the [[conjugate transpose|0DXKE10PHF]] of $A$.
Equivalently, $a_{ij} = \overline{a_{ji}}$ for all $i$ and $j$.

## Remarks

The diagonal entries of a Hermitian matrix are real.
A real matrix is Hermitian if and only if it is [[symmetric|0VR671PHE5]].
The eigenvalues of a Hermitian matrix are real, and $\C^n$ has an orthonormal basis of eigenvectors, by the [[spectral theorem for Hermitian matrices|0DGZBQWZWN]].
The matrix $\begin{pmatrix} 2 & 1 - i \\ 1 + i & 3 \end{pmatrix}$ is Hermitian.

## Explanations

### L1 human

A Hermitian matrix is the complex version of a symmetric matrix.
The entry across the diagonal is the complex conjugate of the entry here.
In quantum mechanics, observable quantities are Hermitian matrices or operators.
</document>

<document source="https://tamnd.github.io/math2/node/0EKZW08FQ5.md" tag="0EKZW08FQ5" kind="definition">
# Orientable surface

## Definition

Let $S \subseteq \mathbb{R}^3$ be a surface that has a tangent plane at each point.
The surface $S$ is orientable if there is a continuous map $n \colon S \to \mathbb{R}^3$ with the following property. For each $p \in S$, $n(p)$ is a [[unit normal vector|0BJDQ2HBH1]] of $S$ at $p$.
Such a map $n$ is an orientation of $S$, and $S$ together with $n$ is an oriented surface.

## Remarks

If $S$ is connected and orientable, then it has exactly two orientations, $n$ and $-n$.
A regular injective [[parametrisation|05788J51WZ]] $r$ gives the orientation $n = (r_u \times r_v) / \| r_u \times r_v \|$. Swapping $u$ and $v$ gives the opposite orientation.
Spheres, planes and tori are orientable. The boundary of a bounded region in $\mathbb{R}^3$ has the outward normal as its standard orientation. The Möbius strip is not orientable.
Integrals of vector fields across a surface, such as flux, need an orientation. Reversing the orientation changes their sign. [[Integrals of scalar functions|0DQ8ERJZNS]] do not need one.

## Explanations

### L1 human

An orientable surface has two sides, an inside and an outside, or a top and a bottom.
On a Möbius strip, an ant can walk from one side to the other without crossing an edge. So there is no consistent choice of side.
</document>

<document source="https://tamnd.github.io/math2/node/0EM6C9SZN5.md" tag="0EM6C9SZN5" kind="statement">
# Existence of nontrivial solutions of Pell's equation

## Statement

Let $d$ be a positive integer that is not a perfect square.
Then [[Pell's equation|0AF3DANJ76]] $x^2 - d y^2 = 1$ has a solution in integers with $y \neq 0$.
In fact, it has infinitely many solutions in integers.

## Remarks

One proof uses [[Dirichlet's approximation theorem|04JAY9DGB7]] to find infinitely many $x/y$ with $|x^2 - d y^2|$ bounded.
Another proof uses the [[periodic continued fraction|0XKJPFS2W0]] of $\sqrt{d}$.
The smallest solution can be very large. For $d = 61$, it is $(1766319049, 226153980)$.

## History

Indian mathematicians, such as Brahmagupta, studied the equation many centuries before European mathematicians.
Lagrange proved that it always has infinitely many solutions.

## Explanations

### L1 human

For each non-square $d$, there are integers $x$ and $y \neq 0$ with $x^2 - d y^2 = 1$.
They may be hard to find by trial: for $d = 61$, the smallest $x$ has ten digits.
</document>

<document source="https://tamnd.github.io/math2/node/0EN0V3NY20.md" tag="0EN0V3NY20" kind="concept">
# Chi-squared test

## Summary

A chi-squared test is a [[hypothesis test|087P03NEND]] whose test statistic has approximately a [[chi-squared distribution|0RFMMBHFTV]] under the null hypothesis. The main example is Pearson's chi-squared test for counts in categories.
For observed counts $O_1, \dots, O_k$ and expected counts $E_1, \dots, E_k$ under $H_0$, Pearson's statistic is
$$\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}.$$
One rejects $H_0$ when $\chi^2$ is large.

## Remarks

In the goodness-of-fit test, $H_0$ fixes the probabilities of the $k$ categories. For a large sample, $\chi^2$ has approximately the law $\chi^2_{k - 1}$ under $H_0$.
In the test of independence for an $r \times c$ contingency table, $H_0$ states that the two categorical variables are [[independent|0XCE070V8B]]. Then $E_{ij}$ is (row total $\times$ column total) / $n$, and the law is approximately $\chi^2_{(r-1)(c-1)}$.
The approximation is poor when expected counts are small. A common rule asks for $E_i \ge 5$ in each cell. For small tables, Fisher's exact test is an alternative.
Karl Pearson introduced the test in 1900.

## Explanations

### L1 human

The chi-squared test compares the counts that you observe with the counts that a hypothesis predicts.
If the squared differences, scaled by the predictions, add up to a large number, the hypothesis does not fit the data.
</document>

<document source="https://tamnd.github.io/math2/node/0ENVM493VK.md" tag="0ENVM493VK" kind="statement">
# Inverse in terms of the adjugate

## Statement

Let $K$ be a commutative ring, let $n \ge 2$, and let $A$ be an $n \times n$ matrix over $K$ such that $\det A$ is a unit of $K$.
Then $A$ is [[invertible|0EB9N0S33C]], and
$$A^{-1} = (\det A)^{-1} \operatorname{adj}(A),$$
where $\operatorname{adj}(A)$ is the [[adjugate|07W9H8QF38]] of $A$.

## Remarks

The statement follows by dividing [[$A \operatorname{adj}(A) = \operatorname{adj}(A) A = \det(A) I_n$|0528HKWZ1Z]] by $\det A$.
For $n = 2$ and $ad - bc \ne 0$ over a field, $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.
For large $n$, row reduction computes the inverse much faster than this formula.

## Explanations

### L1 human

This formula gives every entry of the inverse as a quotient of two determinants.
It is useful for theory and for small matrices.
For $2 \times 2$ matrices, it is the formula that most students learn first.
</document>

<document source="https://tamnd.github.io/math2/node/0ENZN5G4CV.md" tag="0ENZN5G4CV" kind="definition">
# Normal operator

## Definition

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The map $T$ is *normal* when
$$T T^* = T^* T,$$
where $T^*$ is the [[adjoint|0R9NMSM1ZP]] of $T$.

## Remarks

Self-adjoint maps, skew-adjoint maps and unitary maps are normal.
$T$ is normal if and only if $\norm{T(v)} = \norm{T^*(v)}$ for all $v \in V$.
Over $\C$, normal maps are exactly the maps with an orthonormal basis of eigenvectors.

## Explanations

### L1 human

A normal map commutes with its adjoint.
This condition is weaker than self-adjoint, but it still gives perpendicular eigenvectors over $\C$.
Rotations of the plane are normal, but not self-adjoint.
</document>

<document source="https://tamnd.github.io/math2/node/0EP59ESQ94.md" tag="0EP59ESQ94" kind="definition">
# Invariant subspace

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
A [[subspace|00YC1D652K]] $W$ of $V$ is *invariant* under $T$, or *$T$-invariant*, when $T(w) \in W$ for all $w \in W$.

## Remarks

The subspaces $0$ and $V$, $\ker T$ and $\im T$, and each [[eigenspace|0G0RFPG8HR]] of $T$ are $T$-invariant.
A line spanned by $v \ne 0$ is $T$-invariant if and only if $v$ is an eigenvector of $T$.
If $V = W_1 \oplus W_2$ with both $W_i$ invariant, then the matrix of $T$ in an adapted basis is block diagonal.

## Explanations

### L1 human

An invariant subspace is a part of the space that the map never leaves.
Vectors inside it stay inside it.
We can then study the map on this smaller space alone.
</document>

<document source="https://tamnd.github.io/math2/node/0EP73HK4C4.md" tag="0EP73HK4C4" kind="definition">
# Row space

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$ with rows $r_1, \dots, r_m \in F^n$.
The *row space* of $A$ is
$$\operatorname{Row}(A) = \operatorname{span}(r_1, \dots, r_m) \subseteq F^n,$$
the [[linear span|0DTY8B9HMS]] of the rows.

## Remarks

The row space of $A$ is the [[column space|03AJKXH1AV]] of $A^T$.
[[Elementary row operations|04XWH28452]] do not change the row space, so [[row equivalent|006BHFWMAP]] matrices have the same row space.
The nonzero rows of a row echelon form of $A$ form a basis of $\operatorname{Row}(A)$.
Its dimension equals the dimension of the column space, see [[row rank equals column rank|0TZAG60ZYT]].

## Explanations

### L1 human

The row space collects all the combinations of the rows of a matrix.
Row operations only mix the rows, so they keep the row space.
After row reduction, the nonzero rows give a basis of it.
</document>

<document source="https://tamnd.github.io/math2/node/0EPAT51DXQ.md" tag="0EPAT51DXQ" kind="concept">
# Function

## Summary

A function from a set $A$ to a set $B$ assigns to each element of $A$ exactly one element of $B$.
</document>

<document source="https://tamnd.github.io/math2/node/0EPD87YVG8.md" tag="0EPD87YVG8" kind="statement">
# Continuous image of a connected set is connected

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], let $f \colon X \to Y$ be [[continuous|0SY9ECQ19Q]] and let $S \subseteq X$ be [[connected|0R37DFJKFW]].
Then $f(S)$ is connected.

## Remarks

Proof: if open sets $U$ and $V$ split $f(S)$, then the [[preimages|0FGCFY1AZX]] $f^{-1}(U)$ and $f^{-1}(V)$ are open and split $S$.
Thus connectedness is a [[topological property|02SMVGC690]].
For $Y = \mathbb{R}$, this gives the [[intermediate value theorem|098C0KZZAP]].
For example, the circle is connected, because it is the image of $[0, 1]$ under $t \mapsto (\cos 2\pi t, \sin 2\pi t)$.

## Explanations

### L1 human

A continuous map cannot break a space that is in one piece into separate pieces.
</document>

<document source="https://tamnd.github.io/math2/node/0EPTDZPJ83.md" tag="0EPTDZPJ83" kind="statement">
# Convergence in probability implies convergence in distribution

## Statement

Let $X, X_1, X_2, \dots$ be real random variables on one probability space.
If $X_n \to X$ [[in probability|0EFDDXTVD6]], then $X_n \to X$ [[in distribution|0JDAQHWA9N]].

## Remarks

Proof: for $\varepsilon > 0$, $F_n(x) \le F(x + \varepsilon) + P(|X_n - X| > \varepsilon)$ and $F(x - \varepsilon) \le F_n(x) + P(|X_n - X| > \varepsilon)$.
Let $n \to \infty$ and then $\varepsilon \to 0$. At a continuity point $x$ of $F$, this gives $F_n(x) \to F(x)$.
The converse is false. Let $X \sim \mathcal{N}(0, 1)$ and $X_n = -X$ for all $n$. Then $X_n \to X$ in distribution, but $P(|X_n - X| > 1) = P(|X| > 1/2)$ does not tend to $0$.
The converse holds when the limit is a constant: if $X_n \to c$ in distribution for a real $c$, then $X_n \to c$ in probability.

## Explanations

### L1 human

If $X_n$ is very likely to be close to $X$, then the probabilities for $X_n$ are close to those for $X$.
But two variables can have the same law and still be far apart, so the converse fails.
</document>

<document source="https://tamnd.github.io/math2/node/0EPTJA5QV4.md" tag="0EPTJA5QV4" kind="definition">
# Linear subspace

## Definition

Let $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
A *linear subspace* of $V$ is a subset $W \subseteq V$ with these three properties:

1. $0 \in W$.
2. If $u, v \in W$, then $u + v \in W$.
3. If $a \in F$ and $v \in W$, then $av \in W$.

## Remarks

With the operations of $V$, a subspace $W$ is itself a vector space over $F$.
The sets $\{0\}$ and $V$ are subspaces of $V$.
The lines through the origin are the one-dimensional subspaces of $\R^2$; a line that misses the origin is not a subspace.
The kernel and the image of a linear map are subspaces.

## Explanations

### L1 human

A subspace is a smaller vector space inside a bigger one.
We cannot leave it by adding its vectors or by scaling them.
In space, the subspaces are the origin, the lines and planes through the origin, and the whole space.
</document>

<document source="https://tamnd.github.io/math2/node/0EPW94MJ1P.md" tag="0EPW94MJ1P" kind="concept">
# Constrained optimisation

## Summary

Constrained optimisation looks for the maximum or minimum of an objective function $f$ of several variables over the set of points that satisfy given constraints.
Typical constraints are equations $g_i(x) = c_i$ and inequalities $h_j(x) \ge d_j$.

## Remarks

An example is to maximise $f(x, y) = x y$ on the circle $x^2 + y^2 = 2$. The maximum value is $1$, at $(1, 1)$ and $(-1, -1)$.
For equality constraints, one method solves the constraints for some variables and substitutes them into $f$. Another method uses [[Lagrange multipliers|0VKKP8X6Q6]].
A constrained extremum need not be a critical point of $f$. In the example above, $\nabla f(1, 1) = (1, 1) \ne 0$.

## Explanations

### L1 human

In many problems you cannot choose the variables freely.
For example, you want the largest area for a fixed length of fence.
Constrained optimisation finds the best value only among the choices that obey the rules.
</document>

<document source="https://tamnd.github.io/math2/node/0EPZC4Z1FC.md" tag="0EPZC4Z1FC" kind="statement">
# Central limit theorem

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with mean $\mu$ and variance $\sigma^2$, where $0 < \sigma^2 < \infty$. Let $\bar{X}_n = (X_1 + \dots + X_n)/n$.
Then $\sqrt{n} \, (\bar{X}_n - \mu) / \sigma$ converges [[in distribution|0JDAQHWA9N]] to $\mathcal{N}(0, 1)$. That is, for each $x \in \mathbb{R}$,
$$\lim_{n \to \infty} P\left(\frac{X_1 + \dots + X_n - n\mu}{\sigma \sqrt{n}} \le x\right) = \Phi(x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-u^2/2} \, du.$$

## Remarks

A standard proof uses [[characteristic functions|0MYDNRT7PP]]. With $Y_i = (X_i - \mu)/\sigma$, $\varphi_{Y_1}(t) = 1 - t^2/2 + o(t^2)$, so $\varphi_{Y_1}(t/\sqrt{n})^n \to e^{-t^2/2}$.
[[Lévy's continuity theorem|0FPWF7TTMB]] then gives the convergence in distribution.
The Lindeberg and Lyapunov forms remove the condition that the $X_i$ have the same law. The Berry–Esseen theorem bounds the rate of convergence when $E[|X_1|^3] < \infty$.
The finite variance is necessary. Sums of i.i.d. Cauchy variables, scaled by $1/n$, stay Cauchy.

## Explanations

### L1 human

The central limit theorem explains why the bell curve is everywhere.
A sum of many small independent effects is approximately normal, whatever the law of each effect, as long as the variance is finite.
</document>

<document source="https://tamnd.github.io/math2/node/0EQFWKJ37J.md" tag="0EQFWKJ37J" kind="definition">
# Hyperbolic cosine

## Definition

The hyperbolic cosine is the function $\cosh \colon \mathbb{R} \to \mathbb{R}$ defined by
$$\cosh x = \frac{e^x + e^{-x}}{2}.$$

## Remarks

It is even, and $\cosh x \ge 1$, with equality only at $x = 0$.
Its power series is $\cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}$.
The graph of $\cosh$ is the catenary, the shape of a hanging chain. Mathlib defines it as `Real.cosh`.

## Explanations

### L1 human

The hyperbolic cosine is the even part of $e^x$.
Its graph looks like a parabola near $0$, but it grows exponentially.
</document>

<document source="https://tamnd.github.io/math2/node/0EQMWQ4WMJ.md" tag="0EQMWQ4WMJ" kind="definition">
# Locally compact space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is locally compact when each point $x \in X$ has a [[neighbourhood|09CJ00H3NW]] $N$ that is [[compact|00R0X1WGPX]].

## Remarks

Authors use several stronger definitions, for example that each neighbourhood of $x$ contains a compact neighbourhood of $x$. For [[Hausdorff spaces|0MXB4CKHFW]], all the usual definitions agree.
Each compact space is locally compact. The space $\mathbb{R}^n$ is locally compact, because each closed ball is compact.
The rationals $\mathbb{Q}$ and each infinite-dimensional normed space are not locally compact.
A locally compact Hausdorff space has a Hausdorff [[one-point compactification|0MM45BT6HC]].

## Explanations

### L1 human

A locally compact space is compact in small pieces: each point has a compact set around it.
The real line is not compact, but each closed interval in it is.
</document>

<document source="https://tamnd.github.io/math2/node/0EQWVHF3W5.md" tag="0EQWVHF3W5" kind="concept">
# Field extension

## Summary

A field extension is a pair of fields $K \subseteq L$, where $K$ is a subfield of $L$.
One writes $L/K$ and says that $L$ is an extension of $K$.
The [[definition of a field extension|0WQ1A8SHTZ]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0EQZSAB9EC.md" tag="0EQZSAB9EC" kind="definition">
# Homogeneous linear equation

## Definition

Let $I$ be an open interval and let $a_0, \dots, a_{n-1} : I \to \mathbb{R}$ be continuous functions.
A homogeneous linear equation of order $n$ is a [[linear differential equation|0PB7K8M7FG]] with right side $0$:
$$y^{(n)} + a_{n-1}(x)\, y^{(n-1)} + \dots + a_1(x)\, y' + a_0(x)\, y = 0.$$
For a linear equation $L y = b$, the equation $L y = 0$ is the associated homogeneous equation.

## Remarks

The zero function is always a solution.
By the [[superposition principle|0F2VBPXNP9]], the solutions on $I$ form a vector space. It has [[dimension n|0GK68PZ6KH]].
Example: $y'' - y = 0$ is homogeneous, and its solutions are $C_1 e^{x} + C_2 e^{-x}$.

## Explanations

### L1 human

Homogeneous means that the equation has no term without $y$.
So if $y$ is a solution, then $2y$ and $-y$ are solutions, and so is the sum of two solutions.
</document>

<document source="https://tamnd.github.io/math2/node/0EREPBVG2G.md" tag="0EREPBVG2G" kind="concept">
# Induced subgraph

## Summary

An induced subgraph is the subgraph on a chosen set of vertices that keeps all edges between them. Induced subgraphs describe the structure of a graph on a part of its vertex set. The node [[induced subgraph|0V05FC4GKQ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ERWG4NXQ9.md" tag="0ERWG4NXQ9" kind="statement">
# QR decomposition

## Statement

Let $A$ be an $n \times n$ complex matrix.
Then there are a [[unitary matrix|0AR68JTMDT]] $Q$ and an [[upper triangular matrix|0N4XY7Z3NN]] $R$ with
$$A = QR.$$
If $A$ is real, then $Q$ and $R$ can be chosen real, with $Q$ orthogonal.
If $A$ is invertible and the diagonal entries of $R$ are required to be positive, then $Q$ and $R$ are unique.

## Remarks

For an invertible $A$, the [[Gram–Schmidt process|0VT6P0GNQR]] applied to the columns of $A$ gives the columns of $Q$.
A rectangular $m \times n$ matrix with $m \ge n$ has a decomposition $A = QR$ with $Q$ unitary $m \times m$ and $R$ upper triangular $m \times n$.
Householder reflections and Givens rotations compute it stably.
It is used to solve least squares problems and in the QR algorithm for eigenvalues.

## Explanations

### L1 human

The QR decomposition writes a matrix as a rotation, possibly with reflections, times a triangular matrix.
The columns of $Q$ are orthonormal, and $R$ records how to rebuild the columns of $A$ from them.
It is the matrix form of the Gram–Schmidt process.
</document>

<document source="https://tamnd.github.io/math2/node/0ESCX36MMC.md" tag="0ESCX36MMC" kind="object">
# Imaginary unit

## Description

The imaginary unit is the [[complex number|09FTAET5Z4]] $i = 0 + 1 i$. It satisfies $i^2 = -1$, see [[i squared is minus one|07KBQCG9KV]].
It is a solution of the equation $x^2 + 1 = 0$. The other solution is $-i$.
In the [[complex plane|0CTE7A31XQ]], $i$ is the point $(0, 1)$, at distance $1$ above $0$.

## Remarks

The powers of $i$ repeat with period $4$: $i^0 = 1$, $i^1 = i$, $i^2 = -1$, $i^3 = -i$ and $i^4 = 1$.
Multiplication by $i$ turns the complex plane by a quarter turn about $0$, counterclockwise.
In electrical engineering, people often write $j$ for $i$.

## Explanations

### L1 human

The imaginary unit is a new number whose square is $-1$.
No real number has this property, so $i$ is not on the real number line.
It sits one unit above $0$ in the plane of complex numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0ESHD7ZQDX.md" tag="0ESHD7ZQDX" kind="statement">
# Divisibility by 9

## Statement

Let $n$ be a natural number with decimal digits $d_k d_{k-1} \dots d_0$, so that $n = \sum_{i=0}^{k} d_i 10^i$.
Then $9 \mid n$ if and only if $9 \mid d_0 + d_1 + \dots + d_k$.

## Remarks

More precisely, $n \equiv d_0 + d_1 + \dots + d_k \pmod 9$, because $10^i \equiv 1 \pmod 9$ for each $i$.
For example, $9 \mid 81{,}243$, because $8 + 1 + 2 + 4 + 3 = 18$.
The congruence is the basis of [[casting out nines|0E7W0EM6J6]].

## Explanations

### L1 human

Add the digits. The number is a multiple of $9$ exactly when the sum is a multiple of $9$.
For example, $123{,}456$ has the digit sum $21$, which is not a multiple of $9$. So $9 \nmid 123{,}456$.
</document>

<document source="https://tamnd.github.io/math2/node/0ETCTBTJM1.md" tag="0ETCTBTJM1" kind="definition">
# Collinear points

## Definition

Let $K$ be a field and let $A$ be an [[affine space|09XNAPTJQ7]] over a vector space $V$ over $K$.
A set $S$ of points of $A$ is collinear if the linear span of the vectors $q - p$ for $p, q \in S$ has dimension at most 1.
Equivalently, $S$ is collinear if $S$ has at most one point or a [[line|0MP9BCF6DP]] of $A$ contains $S$.

## Remarks

The empty set and each set with one point are collinear. Each set of two points is collinear.
A set $S$ with at least two points is collinear if and only if its [[affine span|0YMB1676R1]] is a line.
In $\mathbb{R}^2$, the points $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$ are collinear if and only if $(x_2 - x_1)(y_3 - y_1) - (x_3 - x_1)(y_2 - y_1) = 0$.
Mathlib's `Collinear` uses the first form of the definition.

## Explanations

### L1 human

Points are collinear if you can draw one straight line through all of them. The points $(0, 0)$, $(1, 2)$ and $(2, 4)$ are collinear: they all lie on $y = 2x$. The point $(1, 1)$ is not on this line, so $(0, 0)$, $(1, 2)$ and $(1, 1)$ are not collinear.
</document>

<document source="https://tamnd.github.io/math2/node/0ETGBAK2CC.md" tag="0ETGBAK2CC" kind="statement">
# Existence of an algebraic closure

## Statement

Every [[field|0QA8CHH8MG]] $K$ has an [[algebraic closure|0MCXX29HWY]].

## Remarks

One proof: take a polynomial ring $R$ over $K$ with one variable for each root of each monic irreducible polynomial over $K$. A [[maximal ideal|02HJYT53F7]] of $R$ that contains the relations gives a field $K_1$ in which every polynomial over $K$ splits.
Repeating the step gives fields $K \subseteq K_1 \subseteq K_2 \subseteq \dots$, and their union is an algebraic closure of $K$.
The proof uses Zorn's lemma, through the existence of maximal ideals. The weaker ultrafilter lemma is enough.
For $K = \mathbb{Q}$ or $K = \mathbb{R}$, the complex numbers give a direct proof.

## Explanations

### L1 human

Every field sits inside an algebraically closed field that is algebraic over it.
The general proof needs a form of the axiom of choice, because a field can have very many polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/0ETGP0XVCJ.md" tag="0ETGP0XVCJ" kind="statement">
# Gershgorin circle theorem

## Statement

Let $A = (a_{ij})$ be an $n \times n$ matrix over $\C$.
For $1 \le i \le n$, let $R_i = \sum_{j \ne i} \abs{a_{ij}}$, and let $D_i = \{z \in \C : \abs{z - a_{ii}} \le R_i\}$ be the closed disc with centre $a_{ii}$ and radius $R_i$.
Then each [[eigenvalue|0BQAA2BK3Q]] of $A$ lies in at least one of the discs $D_1, \dots, D_n$.

## Remarks

The discs are called *Gershgorin discs*.
The proof takes an eigenvector $x$ and an index $i$ with $\abs{x_i}$ maximal, and estimates row $i$ of $Ax = \lambda x$.
Since $A$ and $A^T$ have the same eigenvalues, the same holds with column sums.
A strictly diagonally dominant matrix, with $\abs{a_{ii}} > R_i$ for all $i$, has no eigenvalue $0$, so it is invertible.

## Explanations

### L1 human

The eigenvalues of a matrix lie near its diagonal entries, when the other entries are small.
Each row gives a disc around its diagonal entry, with the radius equal to the sum of the other absolute values of the row.
Every eigenvalue lies in one of these discs.
</document>

<document source="https://tamnd.github.io/math2/node/0EVHJFTG9Z.md" tag="0EVHJFTG9Z" kind="definition">
# Linear independence

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A family $(v_1, \dots, v_k)$ of vectors of $V$ is *linearly independent* when the only [[linear combination|0V8F3E48E6]] of them that equals $0$ is the trivial one. This means that
$$a_1 v_1 + \cdots + a_k v_k = 0 \implies a_1 = \cdots = a_k = 0$$
for all $a_1, \dots, a_k \in F$.
An arbitrary family $(v_i)_{i \in I}$ is *linearly independent* when each finite subfamily is linearly independent.
A family that is not linearly independent is *linearly dependent*.

## Remarks

The vectors $(1, 0)$ and $(1, 1)$ in $\R^2$ are linearly independent; the vectors $(1, 2)$ and $(2, 4)$ are linearly dependent.
A family that contains the zero vector, or the same vector twice, is linearly dependent.
A family of at least two vectors is linearly dependent if and only if one of its vectors is a linear combination of the others.
In $F^n$, a family of more than $n$ vectors is linearly dependent.

## Explanations

### L1 human

Vectors are linearly independent when none of them is redundant.
No vector can be built from the others, so each one adds a new direction.
Two arrows in the plane are independent exactly when they do not lie on one line.
</document>

<document source="https://tamnd.github.io/math2/node/0EW1X61ZT1.md" tag="0EW1X61ZT1" kind="statement">
# Poincaré–Bendixson theorem

## Statement

Let $U$ be an open subset of $\mathbb{R}^2$, let $f : U \to \mathbb{R}^2$ be continuously differentiable, and let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$.
Let $y \in U$ be a point whose solution exists for all $t \ge 0$. Let $\omega(y)$ be its $\omega$-limit set, as in the definition of a [[limit cycle|0EERDDPSVT]].
Assume that $\omega(y)$ is non-empty and compact, and that it contains only finitely many [[equilibrium points|0G4X36AE0X]].
Then $\omega(y)$ is one of the following three kinds of set.
(1) It is a single equilibrium point.
(2) It is a [[periodic orbit|01DSNP0B4R]].
(3) It is a connected set of finitely many equilibrium points and of trajectories that tend to these points as $t \to \infty$ and as $t \to -\infty$.

## Remarks

A common corollary: let $K \subseteq U$ be compact, contain no equilibrium point, and contain the positive half-trajectory of $y$. Then $\omega(y)$ is a periodic orbit.
To apply it, one often finds an annulus that trajectories enter but cannot leave. Then the annulus contains a periodic orbit.
The theorem is special to the plane. It fails in $\mathbb{R}^3$, where chaotic behaviour is possible, as for the Lorenz system.
The trajectories in case (3) are homoclinic or heteroclinic orbits.

## History

The theorem is named after Henri Poincaré and Ivar Bendixson.

## Explanations

### L1 human

In the plane, a bounded motion has few options in the long run.
It settles at an equilibrium, approaches a closed loop, or follows a chain of paths between equilibria.
Trajectories in the plane cannot cross, and this is what limits the possibilities.
</document>

<document source="https://tamnd.github.io/math2/node/0EW824Z7G2.md" tag="0EW824Z7G2" kind="concept">
# Beta distribution

## Summary

The beta distribution is a two-parameter family of continuous distributions on the interval $[0, 1]$.
It is often used as a distribution for an unknown probability. The node [[beta distribution|0PP5ZAJPP7]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0EWMDD6C14.md" tag="0EWMDD6C14" kind="concept">
# Domain in the complex plane

## Summary

A domain in the complex plane is a nonempty open connected subset of $\mathbb{C}$, see the [[definition|097KNJFJ7V]].
Domains are the usual sets on which complex analysis studies holomorphic functions.
</document>

<document source="https://tamnd.github.io/math2/node/0EWQEYRVA8.md" tag="0EWQEYRVA8" kind="definition">
# Rank of a matrix

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$.
The *rank* of $A$ is
$$\rank A = \dim \operatorname{Col}(A),$$
the [[dimension|03GSZ8RTZJ]] of the [[column space|03AJKXH1AV]] of $A$.

## Remarks

The rank is the maximal number of linearly independent columns of $A$.
It also equals the dimension of the row space, by [[row rank equals column rank|0TZAG60ZYT]].
It equals the number of pivots in a row echelon form of $A$, and the largest order of a nonzero minor of $A$.
An $n \times n$ matrix is invertible if and only if its rank is $n$.

## Explanations

### L1 human

The rank of a matrix counts how many of its columns are really independent.
For $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$, the second column is twice the first, so the rank is $1$.
We compute the rank by row reduction and count the pivots.
</document>

<document source="https://tamnd.github.io/math2/node/0EWXPZCSZ0.md" tag="0EWXPZCSZ0" kind="statement">
# Impossibility of trisecting the angle

## Statement

The angle $\pi/3$, that is, $60^\circ$, cannot be [[trisected|00J3DCCEK5]] with straightedge and compass.
Equivalently, $\cos(\pi/9)$ is not a [[constructible number|0FMN2MN7TM]].

## Remarks

The proof: $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$ shows that $c = \cos(\pi/9)$ is a root of $8x^3 - 6x - 1$. Then $u = 2c$ is a root of $x^3 - 3x - 1$.
The polynomial $x^3 - 3x - 1$ has no rational root, by the [[rational root theorem|0819CQ1V8H]]. So it is [[irreducible|0VNE161QCK]], and $[\mathbb{Q}(c) : \mathbb{Q}] = 3$.
Since $3$ is not a power of $2$, the number $c$ is not constructible.
The angle $\pi/3$ itself is constructible, so no general trisection method exists.

## History

Pierre Wantzel published the proof in 1837.

## Explanations

### L1 human

If you could trisect every angle, you could trisect $60^\circ$ and construct an angle of $20^\circ$.
The cosine of $20^\circ$ has degree $3$ over the rationals, so it is not constructible.
</document>

<document source="https://tamnd.github.io/math2/node/0EX045E2BP.md" tag="0EX045E2BP" kind="definition">
# Conjugate transpose

## Definition

Let $A = (a_{ij})$ be an $m \times n$ matrix with complex entries.
The *conjugate transpose* of $A$ is the $n \times m$ matrix $A^*$ with entries
$$(A^*)_{ij} = \overline{a_{ji}}$$
for $1 \le i \le n$ and $1 \le j \le m$, where $\overline{z}$ is the complex conjugate of $z$.

## Remarks

Other notations are $A^H$ and $A^\dagger$.
For a real matrix, the conjugate transpose is the [[transpose|09DXWYRD43]].
The rules $(A^*)^* = A$ and $(AB)^* = B^* A^*$ hold.
For the standard inner product on $\C^n$, $\langle Ax, y \rangle = \langle x, A^* y \rangle$.

## Explanations

### L1 human

The conjugate transpose flips a complex matrix over its diagonal and conjugates each entry.
For example, the conjugate transpose of $\begin{pmatrix} 1 & i \\ 0 & 2 \end{pmatrix}$ is $\begin{pmatrix} 1 & 0 \\ -i & 2 \end{pmatrix}$.
For complex matrices it plays the role that the transpose plays for real matrices.
</document>

<document source="https://tamnd.github.io/math2/node/0EX2HPXPJ6.md" tag="0EX2HPXPJ6" kind="concept">
# Product of vector spaces

## Summary

The product of two vector spaces over the same field is the set of pairs, with addition and scalar multiplication done in each component.
</document>

<document source="https://tamnd.github.io/math2/node/0EX9GFJ0B3.md" tag="0EX9GFJ0B3" kind="statement">
# The exponential function is its own derivative

## Statement

The [[exponential function|0RMVS7RXZX]] is differentiable on $\mathbb{R}$, and $\exp'(x) = \exp(x)$ for all real $x$.

## Remarks

By [[the sum rule|0E8HNVBCSP]], $\frac{\exp(x + h) - \exp(x)}{h} = \exp(x) \cdot \frac{\exp(h) - 1}{h}$. The series gives $\frac{\exp(h) - 1}{h} \to 1$ as $h \to 0$.
Term-by-term differentiation of the power series gives the same result.
The exponential function is the only solution of $f' = f$ with $f(0) = 1$. Mathlib states the derivative as `Real.hasDerivAt_exp`.

## Explanations

### L1 human

The slope of the graph of $e^x$ at each point equals its height there.
No other function with value $1$ at $0$ has this property.
</document>

<document source="https://tamnd.github.io/math2/node/0EY05Z9RPN.md" tag="0EY05Z9RPN" kind="definition">
# Iterated derivative

## Definition

Let $U \subseteq \mathbb{R}^n$ be open and let $f \colon U \to \mathbb{R}^m$.
For an integer $k \ge 0$, let $L^k(\mathbb{R}^n; \mathbb{R}^m)$ be the space of $k$-linear maps from $(\mathbb{R}^n)^k$ to $\mathbb{R}^m$, with $L^0(\mathbb{R}^n; \mathbb{R}^m) = \mathbb{R}^m$.
The iterated derivatives are defined by recursion. First, $D^0 f = f$.
Suppose that $D^k f \colon U \to L^k(\mathbb{R}^n; \mathbb{R}^m)$ is defined on $U$ and is [[differentiable on U|08S7FDRS5Z]].
Then $D^{k+1} f(a)$, for $a \in U$, is the $(k+1)$-linear map
$$D^{k+1} f(a)(v_1, v_2, \dots, v_{k+1}) = \big(D(D^k f)(a)(v_1)\big)(v_2, \dots, v_{k+1}).$$
Here $D(D^k f)(a)$ is the [[Fréchet derivative|08EVNYZF42]] of $D^k f$ at $a$, a linear map from $\mathbb{R}^n$ to $L^k(\mathbb{R}^n; \mathbb{R}^m)$.

## Remarks

For $k = 1$, $D^1 f = Df$. For $k = 2$, $D^2 f(a)(u, v)$ is the derivative of $x \mapsto Df(x)(v)$ at $a$ in the direction $u$.
On basis vectors, $D^2 f(a)(e_i, e_j) = \dfrac{\partial}{\partial x_i}\left(\dfrac{\partial f}{\partial x_j}\right)(a)$. In general, $D^k f(a)(e_{i_1}, \dots, e_{i_k})$ is a [[partial derivative of order k|02RSXVM1ZN]].
For a real-valued $f$, the matrix of the bilinear map $D^2 f(a)$ is the [[Hessian matrix|0693Y2METT]].
If $f$ is of class $C^k$, then $D^k f(a)$ is a symmetric $k$-linear map, by the [[symmetry of second derivatives|01V9FPBBVK]].

## Explanations

### L1 human

The first derivative takes one direction and gives a rate of change.
The second derivative takes two directions $u$ and $v$. It tells how the rate of change in direction $v$ itself changes as you move in direction $u$.
</document>

<document source="https://tamnd.github.io/math2/node/0EYQ03XCR3.md" tag="0EYQ03XCR3" kind="example">
# 68–95–99.7 rule

## Example

Let $X \sim \mathcal{N}(\mu, \sigma^2)$ with $\sigma > 0$, and let $\Phi$ be the cumulative distribution function of $\mathcal{N}(0, 1)$.
Since $Z = (X - \mu)/\sigma \sim \mathcal{N}(0, 1)$, $P(|X - \mu| \le k\sigma) = 2\Phi(k) - 1$ for each $k > 0$. Numerical values of $\Phi$ give:
$$P(|X - \mu| \le \sigma) \approx 0.6827, \quad P(|X - \mu| \le 2\sigma) \approx 0.9545, \quad P(|X - \mu| \le 3\sigma) \approx 0.9973.$$
So about $68\%$, $95\%$ and $99.7\%$ of the mass of a normal distribution lie within one, two and three standard deviations of the mean.

## Remarks

The rule holds for normal distributions only. For an arbitrary distribution with finite variance, Chebyshev's inequality gives only $P(|X - \mu| \le k\sigma) \ge 1 - 1/k^2$, for example $3/4$ for $k = 2$.
The interval $\mu \pm 1.96\sigma$ contains $95\%$ of the mass. It is the base of the usual $95\%$ confidence intervals.

## Explanations

### L1 human

For data that follow a bell curve, almost all values are within three standard deviations of the mean.
If adult heights have mean $170$ cm and standard deviation $10$ cm, then about $95\%$ of the heights are between $150$ cm and $190$ cm.
</document>

<document source="https://tamnd.github.io/math2/node/0EZ758XE73.md" tag="0EZ758XE73" kind="definition">
# Element

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]] and let $x$ be an object.
The object $x$ is an element of $A$, written $x \in A$, when $A$ contains $x$.
Otherwise, $x$ is not an element of $A$, written $x \notin A$.
The proposition $x \notin A$ is the negation of $x \in A$.

## Remarks

Other phrases for $x \in A$ are "$x$ is a member of $A$", "$x$ belongs to $A$" and "$x$ is in $A$".
For $A = \{1, 2, 3\}$, $2 \in A$ and $4 \notin A$.
An element of a set can itself be a set: $\{1\} \in \{\{1\}, 2\}$, but $1 \notin \{\{1\}, 2\}$.

## Explanations

### L1 human

The symbol $\in$ asks one yes or no question: is this object in the set?
For the set of even numbers, $6 \in$ the set and $7 \notin$ the set.
</document>

<document source="https://tamnd.github.io/math2/node/0EZBG8SKJM.md" tag="0EZBG8SKJM" kind="statement">
# Cantor's intersection theorem

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]].
Let $(C_n)_{n \ge 1}$ be a sequence of nonempty [[compact|00R0X1WGPX]] [[closed|0G5AV9Q4RR]] subsets of $X$ with $C_1 \supseteq C_2 \supseteq C_3 \supseteq \cdots$.
Then the intersection $\bigcap_{n \ge 1} C_n$ is not empty.

## Remarks

The sets must be compact: in $\mathbb{R}$, the closed sets $[n, \infty)$ are nested and nonempty, but their intersection is empty.
For $X = \mathbb{R}$, the sets can be nested closed bounded intervals $[a_n, b_n]$.
There is a variant for metric spaces. A metric space is [[complete|08EWT7J0SC]] if and only if each nested sequence of nonempty closed sets whose [[diameters|0692CGQF00]] tend to $0$ has nonempty intersection. The intersection is then one point.

## Explanations

### L1 human

If you have a chain of nonempty compact sets, each inside the one before, at least one point is in all of them.
For example, the intervals $[0, 1/n]$ are nested, and $0$ is in all of them.
</document>

<document source="https://tamnd.github.io/math2/node/0EZMAY1FQD.md" tag="0EZMAY1FQD" kind="example">
# x sin(1/x) is continuous at 0

## Example

Let $f(x) = x \sin(1/x)$ for $x \ne 0$, and let $f(0) = 0$.
Then $f$ is [[continuous|0EEE2CZB2A]] at $0$, and so $f$ is continuous on $\mathbb{R}$.
Indeed, $|f(x)| \le |x|$ for all $x$, so $f(x) \to 0$ as $x \to 0$ by the [[squeeze theorem|0XMF00T4XQ]].

## Remarks

The factor $x$ damps the oscillation of $\sin(1/x)$.
The function is not differentiable at $0$. The difference quotient $(f(x) - f(0))/x = \sin(1/x)$ has no limit.
The function $x^2 \sin(1/x)$ is [[differentiable with a discontinuous derivative|0WDAKKN4NC]].

## Explanations

### L1 human

The graph of $x \sin(1/x)$ oscillates infinitely often near $0$.
But it stays between the lines $y = x$ and $y = -x$, which meet at the origin.
</document>

<document source="https://tamnd.github.io/math2/node/0EZSD6T9E0.md" tag="0EZSD6T9E0" kind="definition">
# Lower triangular matrix

## Definition

Let $R$ be a ring and let $n \ge 0$ be an integer.
A [[square matrix|0JXZKX526R]] $L = (l_{ij})$ of order $n$ over $R$ is *lower triangular* when $l_{ij} = 0$ for all $i < j$.
It is *strictly lower triangular* when $l_{ij} = 0$ for all $i \le j$.

## Remarks

A matrix is lower triangular if and only if its [[transpose|09DXWYRD43]] is [[upper triangular|0N4XY7Z3NN]].
Sums and products of lower triangular matrices of the same order are lower triangular.
A system $Lx = b$ with $L$ lower triangular and invertible is solved by forward substitution.
In an [[LU decomposition|0RKG5YW66Y]], the factor $L$ is lower triangular.

## Explanations

### L1 human

A lower triangular matrix has zeros everywhere above its diagonal.
Its nonzero entries fill a triangle at the bottom left.
For example, $\begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}$ is lower triangular.
</document>

<document source="https://tamnd.github.io/math2/node/0F0H5P5DED.md" tag="0F0H5P5DED" kind="statement">
# Cauchy's integral formula for derivatives

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $c \in \mathbb{C}$ and $r > 0$ be such that the closed disc $\{z : |z - c| \le r\}$ is contained in $U$.
Then for each integer $n \ge 0$ and each $a \in \mathbb{C}$ with $|a - c| < r$,
$$f^{(n)}(a) = \frac{n!}{2 \pi i} \oint_{|z - c| = r} \frac{f(z)}{(z - a)^{n+1}} \, dz,$$
where $f^{(n)}$ is the $n$-th complex derivative of $f$. This derivative exists by [[holomorphic functions are infinitely differentiable|0QEBN8DGVE]].

## Remarks

For $n = 0$, this is [[Cauchy's integral formula|0DG10TVYPN]].
Formally, the formula comes from differentiating $\frac{1}{z - a}$ $n$ times under the integral sign with respect to $a$.
For example, with $f(z) = e^z$, $c = a = 0$ and $r = 1$: $\oint_{|z| = 1} \frac{e^z}{z^{n+1}} \, dz = \frac{2 \pi i}{n!}$.
It gives [[Cauchy's estimates|0JJRENWMP6]] for the derivatives.

## Explanations

### L1 human

The values of a holomorphic function on a circle determine not only its values inside, but also all its derivatives inside.
To get the $n$-th derivative, you put a higher power of $z - a$ in the denominator and multiply by $n!$.
</document>

<document source="https://tamnd.github.io/math2/node/0F0J193T76.md" tag="0F0J193T76" kind="concept">
# Set bounded above

## Summary

A set of numbers is bounded above when one number is greater than or equal to every element of the set. Such a number is an upper bound of the set.
The same idea applies to a subset of any ordered set.
</document>

<document source="https://tamnd.github.io/math2/node/0F0K8DB9TD.md" tag="0F0K8DB9TD" kind="statement">
# Prime elements are irreducible in an integral domain

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]] and let $p \in R$ be a [[prime element|0MNHXBSX3M]].
Then $p$ is [[irreducible|052SDEPZH7]].

## Remarks

The proof: let $p = a b$. Then $p \mid a b$, so $p \mid a$, say. Write $a = p c$. Then $p = p c b$, and [[cancellation|0GJM24EXP9]] gives $c b = 1$, so $b$ is a unit.
The converse is false in general: in $\mathbb{Z}[\sqrt{-5}]$, [[$2$ is irreducible but not prime|0CHE3J7QMX]].
In a [[unique factorization domain|06YMXJMXYS]], [[irreducible elements are prime|0QRCFZKXH8]].

## Explanations

### L1 human

Suppose that a prime $p$ factors as $a b$.
Then $p$ divides one factor, so that factor is at least as large as $p$ for divisibility.
The other factor must then be a unit.
</document>

<document source="https://tamnd.github.io/math2/node/0F0RPPJ2TT.md" tag="0F0RPPJ2TT" kind="definition">
# Column space

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$ with columns $a_1, \dots, a_n \in F^m$.
The *column space* of $A$ is
$$\operatorname{Col}(A) = \operatorname{span}(a_1, \dots, a_n) \subseteq F^m,$$
the [[linear span|0DTY8B9HMS]] of the columns.

## Remarks

The column space is $\{Ax : x \in F^n\}$, the image of the map $x \mapsto Ax$.
The system $Ax = b$ is [[consistent|0X6V9B1V10]] if and only if $b \in \operatorname{Col}(A)$.
The dimension of the column space is the [[rank|0ZC9YTXZRY]] of $A$.
The pivot columns of $A$ form a basis of $\operatorname{Col}(A)$.

## Explanations

### L1 human

The column space collects all the combinations of the columns of a matrix.
These are exactly the vectors $b$ for which $Ax = b$ has a solution.
For the matrix $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$, the column space is the line spanned by $(1, 2)$.
</document>

<document source="https://tamnd.github.io/math2/node/0F1AVGX26N.md" tag="0F1AVGX26N" kind="definition">
# Closed and exact forms

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $\omega$ be a [[differential k-form|0QHTX7WQEY]] on $U$ with $C^1$ coefficients, where $k \ge 1$.
The form $\omega$ is closed if $d\omega = 0$, where $d$ is the [[exterior derivative|004FRCXH6M]].
The form $\omega$ is exact if $\omega = d\eta$ for some $(k - 1)$-form $\eta$ on $U$.

## Remarks

Each exact form $d\eta$ with $\eta$ of class $C^2$ is closed, because [[d d = 0|0X308R2DBE]].
A [[one-form|0MTENFBW3Q]] $\sum_i f_i \, dx_i$ is closed if and only if $\partial_j f_i = \partial_i f_j$ for all $i, j$. It is exact if and only if $(f_1, \dots, f_n)$ is a [[conservative|05R9NF2J49]] field. In $\mathbb{R}^3$, closed means that the [[curl|0GDNHS9J2X]] is zero.
The angle form $d\theta = \frac{-y \, dx + x \, dy}{x^2 + y^2}$ on $\mathbb{R}^2 \setminus \{0\}$ is closed but not exact. Its integral around the unit circle is $2\pi$, not $0$.
On a convex open set, each closed form is exact. This is the [[Poincaré lemma|05MC3HND79]].

## Explanations

### L1 human

Exact forms are derivatives of other forms, and closed forms only pass the local test for this: their derivative is zero.
The gap between the two notions is a measure of the holes of the domain.
</document>

<document source="https://tamnd.github.io/math2/node/0F1D6PK9RZ.md" tag="0F1D6PK9RZ" kind="definition">
# Well-order

## Definition

Let $A$ be a set.
A well-order on $A$ is a [[total order|0316AYH1SC]] $\le$ on $A$ such that each nonempty subset of $A$ has a [[least element|0T2NTJCV83]].
The pair $(A, \le)$ is a well-ordered set.

## Remarks

The usual order on $\mathbb{N}$ is a well-order.
The usual orders on $\mathbb{Z}$ and on $[0, 1]$ are not well-orders: $\mathbb{Z}$ and $(0, 1]$ have no least element.
A total order is a well-order exactly when its strict order $<$ is [[well-founded|0230ECX1Q5]].
The [[well-ordering theorem|0N109NG5ZX]] says that each set has a well-order.

## Explanations

### L1 human

In a well-ordered set, each nonempty part has a first element.
So you can always say "take the smallest element with this property".
</document>

<document source="https://tamnd.github.io/math2/node/0F1KBVY161.md" tag="0F1KBVY161" kind="definition">
# Transition matrix

## Definition

Let $S$ be a nonempty countable set.
A transition matrix, or stochastic matrix, on $S$ is a family $P = (p_{ij})_{i, j \in S}$ of real numbers with
$$p_{ij} \ge 0 \quad \text{for all } i, j \in S \quad \text{and} \quad \sum_{j \in S} p_{ij} = 1 \quad \text{for each } i \in S.$$
The transition matrix of a time-homogeneous [[Markov chain|033TJBZDG0]] $(X_n)$ on $S$ has $p_{ij} = P(X_{n+1} = j \mid X_n = i)$.

## Remarks

Each row of $P$ is a probability distribution on $S$: the law of the next state, given the present state $i$.
A product of two stochastic matrices is stochastic. The $n$-th power $P^n$ gives the $n$-step probabilities $P(X_{m+n} = j \mid X_m = i)$.
If the row vector $\lambda$ is the law of $X_0$, then $\lambda P^n$ is the law of $X_n$.
A finite stochastic matrix has the eigenvalue $1$, with the eigenvector $(1, \dots, 1)^{\mathsf{T}}$, and all its eigenvalues have absolute value at most $1$.

## Explanations

### L1 human

The transition matrix is a table of the chances to move from each state to each state in one step.
For weather with states "sun" and "rain", the row of "sun" gives the chances of sun and of rain tomorrow, if today is sunny.
</document>

<document source="https://tamnd.github.io/math2/node/0F1PR74BRX.md" tag="0F1PR74BRX" kind="concept">
# Subsequence

## Summary

A subsequence of a sequence keeps infinitely many of its terms, in their original order. Subsequences are the main tool of the Bolzano–Weierstrass theorem. The node [[subsequence|0R8J3ZQMJS]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0F1SN93KXE.md" tag="0F1SN93KXE" kind="definition">
# Conditional variance

## Definition

Let $X$ be a real random variable on $(\Omega, \mathcal{F}, P)$ with $E[X^2] < \infty$, and let $\mathcal{G} \subseteq \mathcal{F}$ be a σ-algebra.
The conditional variance of $X$ given $\mathcal{G}$ is the random variable
$$\operatorname{Var}(X \mid \mathcal{G}) = E\left[\left(X - E[X \mid \mathcal{G}]\right)^2 \,\middle|\, \mathcal{G}\right],$$
where $E[\cdot \mid \mathcal{G}]$ is the [[conditional expectation|0YAMJRK04E]]. For a random variable $Y$, $\operatorname{Var}(X \mid Y) = \operatorname{Var}(X \mid \sigma(Y))$.

## Remarks

As for the [[variance|007QED3E0Y]], $\operatorname{Var}(X \mid \mathcal{G}) = E[X^2 \mid \mathcal{G}] - (E[X \mid \mathcal{G}])^2$ almost surely.
For a discrete $Y$ and $y$ with $P(Y = y) > 0$, $\operatorname{Var}(X \mid Y)$ takes the value $\operatorname{Var}(X \mid Y = y)$ on $\{Y = y\}$. This is the variance of $X$ under $P(\cdot \mid Y = y)$.
$\operatorname{Var}(X \mid \mathcal{G}) = 0$ almost surely if and only if $X$ is equal almost surely to a $\mathcal{G}$-measurable random variable.

## Explanations

### L1 human

The conditional variance tells how uncertain $X$ still is after you learn some information.
For example, the variance of the height of a person given the sex of that person is the spread of heights within each group.
</document>

<document source="https://tamnd.github.io/math2/node/0F1WR9C83K.md" tag="0F1WR9C83K" kind="definition">
# Derivative

## Definition

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$ be a function, and let $a \in I$.
A real number $L$ is the derivative of $f$ at $a$ if
$$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h} = L.$$
The notation is $f'(a) = L$.

## Remarks

The quotient is the slope of the line through $(a, f(a))$ and $(a + h, f(a + h))$. The limit is a [[limit of a function|0TXTSEMHR5]] of $h$ at $0$.
The derivative is unique when it exists.
For example, the derivative of $x^2$ at $a$ is $\lim_{h \to 0} (2a + h) = 2a$.
Mathlib defines the relation "$f$ has derivative $L$ at $a$" as `HasDerivAt`.

## Explanations

### L1 human

The derivative of $f$ at $a$ is the instantaneous rate of change of $f$ at $a$.
Geometrically, it gives the steepness of the graph at the point $(a, f(a))$.
</document>

<document source="https://tamnd.github.io/math2/node/0F27PBRWD7.md" tag="0F27PBRWD7" kind="definition">
# Row echelon form

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $F$.
The *leading entry* of a nonzero row of $A$ is its first nonzero entry, from the left.
The matrix $A$ is in *row echelon form* when these two conditions hold:

1. Each zero row of $A$ is below each nonzero row of $A$.
2. For two consecutive nonzero rows, the leading entry of the lower row is strictly to the right of the leading entry of the upper row.

## Remarks

The matrix $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 4 \\ 0 & 0 & 0 \end{pmatrix}$ is in row echelon form.
Some authors also require that each leading entry is $1$.
In row echelon form, all the entries below a leading entry are zero.
[[Gaussian elimination|0JVJ2WCXZH]] changes each matrix into a row echelon form by elementary row operations.

## Explanations

### L1 human

Row echelon form looks like a staircase.
Each nonzero row starts further to the right than the row above it, and the zero rows sit at the bottom.
A system in this form is easy to solve from the bottom row up.
</document>

<document source="https://tamnd.github.io/math2/node/0F2PTGX1A2.md" tag="0F2PTGX1A2" kind="concept">
# Field of fractions

## Summary

The field of fractions of an integral domain is the smallest field that contains it.
Its elements are the fractions $a/b$ with $b \neq 0$.
The rational numbers are the field of fractions of the integers.
The [[definition of the field of fractions|0VF11ZTTMD]] gives the construction.
</document>

<document source="https://tamnd.github.io/math2/node/0F2QCWVFT7.md" tag="0F2QCWVFT7" kind="concept">
# Riemann integrable function

## Summary

A function on a closed bounded interval is Riemann integrable when its Riemann sums converge to one number as the mesh tends to zero. Continuous functions and monotone functions are Riemann integrable. The node [[Riemann integrable function|0N1JQXZXHC]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0F2SYCKDT2.md" tag="0F2SYCKDT2" kind="concept">
# Lyapunov stability

## Summary

An equilibrium point of a dynamical system is Lyapunov stable if all solutions that start near it stay near it for all future time.
The theory goes back to Aleksandr Lyapunov. See the [[definition|092S6VBECQ]] and the stronger notion of [[asymptotic stability|05Y1JAM3Y5]].
</document>

<document source="https://tamnd.github.io/math2/node/0F2VBPXNP9.md" tag="0F2VBPXNP9" kind="statement">
# Superposition principle

## Statement

Let $I$ be an open interval, and let $L y = y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y$ with continuous functions $a_0, \dots, a_{n-1}$ on $I$.
Let $y_1$ and $y_2$ be solutions of the [[homogeneous linear equation|0EQZSAB9EC]] $L y = 0$ on $I$, and let $c_1, c_2$ be real numbers. Then $c_1 y_1 + c_2 y_2$ is a solution of $L y = 0$ on $I$.

## Remarks

The proof is the linearity of $L$: $L(c_1 y_1 + c_2 y_2) = c_1 L y_1 + c_2 L y_2 = 0$.
The same argument gives a variant for right sides: if $L y_1 = b_1$ and $L y_2 = b_2$, then $L(y_1 + y_2) = b_1 + b_2$.
The principle fails for nonlinear equations. For $y' = y^2$, the function $y = -1/x$ is a solution on $x > 0$, but $2y$ is not.

## Explanations

### L1 human

For a linear equation, you can add solutions and scale them, and you get solutions again.
In physics this is superposition: two waves that solve a linear wave equation add up to a third solution.
</document>

<document source="https://tamnd.github.io/math2/node/0F32S73FJN.md" tag="0F32S73FJN" kind="concept">
# Iterated derivative

## Summary

The iterated derivative $D^k f$ of a map $f$ is the derivative of the derivative, taken $k$ times.
The [[Fréchet derivative|06EH97V2A7]] of $D^{k-1} f$ at a point is a linear map with values in $(k-1)$-linear maps, which is the same as a $k$-linear map.
So $D^k f(a)$ is a $k$-linear map, and for a function on $\mathbb{R}^n$ its values on basis vectors are the partial derivatives of order $k$.
</document>

<document source="https://tamnd.github.io/math2/node/0F33KX9Q6F.md" tag="0F33KX9Q6F" kind="concept">
# Primitive root modulo n

## Summary

A primitive root modulo $n$ is an integer whose powers give every residue class modulo $n$ that is coprime to $n$.
The [[definition of a primitive root modulo n|0KM2CJ80G7]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0F3K4C7XYW.md" tag="0F3K4C7XYW" kind="statement">
# Bounded linear maps are continuous

## Statement

Let $V$ and $W$ be normed spaces and let $T \colon V \to W$ be linear.
Then $T$ is [[continuous|0S3SMRH0Z5]] if and only if $T$ is bounded, that is, there is a real $C \ge 0$ with $\|Tx\|_W \le C \|x\|_V$ for all $x \in V$.

## Remarks

If $T$ is bounded, then $\|Tx - Ty\|_W = \|T(x - y)\|_W \le C \|x - y\|_V$. Thus $T$ is [[Lipschitz|0TMZ21WKCN]], and so [[uniformly continuous|07DCKDXRBP]].
Conversely, if $T$ is continuous at $0$, there is $\delta > 0$ with $\|Tx\|_W \le 1$ when $\|x\|_V \le \delta$. By linearity, $\|Tx\|_W \le \|x\|_V / \delta$ for all $x$.
Thus, for a linear map, continuity at one point already gives continuity everywhere.
The name bounded means that $T$ maps bounded sets to bounded sets. It does not mean that $T(V)$ is bounded.

## Explanations

### L1 human

For linear maps, continuity is the same as a simple estimate. The length of $Tx$ is at most a fixed multiple of the length of $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0F3Z22M2GT.md" tag="0F3Z22M2GT" kind="concept">
# Mersenne prime

## Summary

A Mersenne prime is a prime number of the form $2^n - 1$.
The [[definition of a Mersenne prime|06GPJF9V7M]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0F405477F8.md" tag="0F405477F8" kind="statement">
# Derivative vanishes at a local extremum

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $a \in U$ be a [[local extremum|04CPG9K1C4]] of $f$.
If the [[partial derivative|05AXG3QRKR]] $\partial f / \partial x_i (a)$ exists for some $i$, then $\partial f / \partial x_i (a) = 0$.
In particular, if $f$ is [[differentiable|0135XZZQ8M]] at $a$, then $\nabla f(a) = 0$ and the derivative $Df(a)$ is the zero linear map.

## Remarks

A point $a \in U$ where $f$ is differentiable and $\nabla f(a) = 0$ is a critical point, or stationary point, of $f$.
The proof applies the one-variable theorem to $t \mapsto f(a + t e_i)$, where $e_i$ is the $i$-th standard basis vector. This function has a local extremum at $t = 0$.
The converse is false. For $f(x, y) = x^2 - y^2$, $\nabla f(0, 0) = 0$, but $(0, 0)$ is a [[saddle point|00N3T18QNA]], not an extremum.
The hypothesis that $U$ is open matters. On $[0, 1]^2$, $f(x, y) = x + y$ has a maximum at $(1, 1)$, where $\nabla f = (1, 1)$.

## Explanations

### L1 human

At the top of a smooth hill the ground is flat in every direction.
So all partial derivatives are zero there, and the tangent plane to the graph is horizontal.
This turns the search for extrema into the solution of the equations $\nabla f = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0F462493EA.md" tag="0F462493EA" kind="concept">
# Transition matrix

## Summary

A transition matrix, or stochastic matrix, is a square matrix of nonnegative entries in which each row adds up to $1$. It gives the one-step probabilities of a Markov chain.
The node [[transition matrix|0F1KBVY161]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0F4JSQE2EK.md" tag="0F4JSQE2EK" kind="concept">
# Divisors of a natural number

## Summary

The divisors of a positive integer $n$ form the finite set of positive integers that divide $n$.
The [[definition of the set of divisors|0FCWA16FZE]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0F4WRWBZZC.md" tag="0F4WRWBZZC" kind="concept">
# Quadratic integer

## Summary

A quadratic integer is a root of a monic quadratic polynomial with integer coefficients, that is, an algebraic integer of degree at most $2$.
The [[definition of a quadratic integer|01VN79T2PR]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0F50N2P3FC.md" tag="0F50N2P3FC" kind="concept">
# Outcome

## Summary

An outcome is one possible result of a random experiment.
The outcomes of an experiment are the elements of its sample space. The node [[outcome|0Y5R7TPNT1]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0F56CMMC5P.md" tag="0F56CMMC5P" kind="statement">
# Divergence of a curl is zero

## Statement

Let $U \subseteq \mathbb{R}^3$ be open, and let $A \colon U \to \mathbb{R}^3$ be a vector field of class $C^2$.
Then
$$\nabla \cdot (\nabla \times A) = 0 \quad \text{on } U.$$

## Remarks

Proof: with $A = (A_1, A_2, A_3)$, the [[divergence|02NSMQVQWK]] of the [[curl|0GDNHS9J2X]] is $\partial_x (\partial_y A_3 - \partial_z A_2) + \partial_y (\partial_z A_1 - \partial_x A_3) + \partial_z (\partial_x A_2 - \partial_y A_1)$. The six terms cancel in pairs by [[Schwarz's theorem|0NNQY8WCW6]].
So a field that has a vector potential is divergence-free. On an open ball, the converse is also true: a $C^1$ divergence-free field is the curl of some field.
The identity is a special case of the rule [[d d = 0|0X308R2DBE]] for the exterior derivative.

## Explanations

### L1 human

The curl of a field describes rotation, and pure rotation does not create or destroy anything.
So the curl of any smooth field has no sources or sinks, which means its divergence is zero.
</document>

<document source="https://tamnd.github.io/math2/node/0F5QC101Q7.md" tag="0F5QC101Q7" kind="statement">
# g to the n equals 1 iff the order divides n

## Statement

Let $G$ be a [[group|G5S1M9]] with identity element $e$, and let $g \in G$ be an element of finite [[order|0WE2NHZJ7B]] $m$.
Let $n$ be an integer.
Then $g^n = e$ if and only if $m$ divides $n$.

## Remarks

The proof divides $n$ by $m$ with remainder: $n = qm + r$ with $0 \leq r < m$, so that $g^n = (g^m)^q g^r = g^r$.
By the minimality of $m$, the power $g^r$ is $e$ only when $r = 0$.
A consequence is that $g^a = g^b$ if and only if $a \equiv b \pmod m$.

## Explanations

### L1 human

An element of order $m$ comes back to the identity exactly after $m$, $2m$, $3m$ steps, and so on.
It is never at the identity after a number of steps that is not a multiple of $m$.
For example, a rotation by $90$ degrees is the identity after $8$ steps, but not after $6$ steps.
</document>

<document source="https://tamnd.github.io/math2/node/0F6ES3BRYP.md" tag="0F6ES3BRYP" kind="concept">
# Characteristic of a ring

## Summary

The characteristic of a ring is the least positive integer $n$ such that the sum of $n$ copies of $1$ is $0$.
If there is no such integer, the characteristic is $0$.
The [[definition of the characteristic|0P7CH1CND5]] states this precisely.
</document>

<document source="https://tamnd.github.io/math2/node/0F6FM9Y4Y6.md" tag="0F6FM9Y4Y6" kind="definition">
# Order on the integers

## Definition

Let $m$ and $n$ be [[integers|0C4JS9RF5M]].
The notation $m \le n$ means that $n - m$ is a [[natural number|0MWKJ44V5A]], that is, when $n - m \in \{0, 1, 2, \dots\}$.
The notation $m < n$ means that $m \le n$ and $m \neq n$.

## Remarks

This relation is a total order on $\mathbb{Z}$, and on $\mathbb{N}$ it agrees with the [[order on the natural numbers|0Y9H52BT03]].
If $m \le n$, then $m + k \le n + k$ for each integer $k$. If $m \le n$ and $0 \le k$, then $m k \le n k$.
Thus $\mathbb{Z}$ is an ordered ring. Unlike $\mathbb{N}$, it has no least element.

## Explanations

### L1 human

On the number line, $m \le n$ means that $m$ is at the same place as $n$ or to the left of it.
For example, $-5 \le -2$ because $-2 - (-5) = 3$ is a natural number.
</document>

<document source="https://tamnd.github.io/math2/node/0F72FK8EKB.md" tag="0F72FK8EKB" kind="concept">
# Disjunction

## Summary

Disjunction is the logical operation "or".
It joins two [[propositions|0ZGC7ZESYN]] into one proposition that is true when at least one part is true.
</document>

<document source="https://tamnd.github.io/math2/node/0F765KFC2J.md" tag="0F765KFC2J" kind="statement">
# Cayley–Hamilton theorem

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $A$ be an $n \times n$ matrix over $K$ with [[characteristic polynomial|0VSSPAFM2C]] $p_A(t) = \det(t I_n - A)$.
Then
$$p_A(A) = 0,$$
where $p_A(A)$ is the [[value of $p_A$ at $A$|0E3MV6E6MG]].

## Remarks

For $n = 2$, it says $A^2 - (\tr A) A + (\det A) I_2 = 0$.
Substituting $t = A$ into $\det(t I_n - A)$ is not a valid proof, because $p_A(A)$ is a matrix and $\det(A - A)$ is a scalar.
One proof uses the [[adjugate|07W9H8QF38]] identity $(t I_n - A) \operatorname{adj}(t I_n - A) = p_A(t) I_n$.
If $A$ is invertible, then the theorem expresses $A^{-1}$ as a polynomial in $A$.

## Explanations

### L1 human

Every square matrix satisfies its own characteristic polynomial.
For $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, the polynomial is $t^2 - 5t - 2$, so $A^2 = 5A + 2I$.
So every power of $A$ is a combination of $I$ and $A$.
</document>

<document source="https://tamnd.github.io/math2/node/0F7FQS6XY6.md" tag="0F7FQS6XY6" kind="concept">
# Square root

## Summary

A square root of a number $x$ is a number $y$ with $y^2 = x$.
Each nonnegative real number $x$ has exactly one nonnegative square root, the principal square root $\sqrt{x}$.
For example, $\sqrt{9} = 3$, and the square roots of $9$ are $3$ and $-3$.
</document>

<document source="https://tamnd.github.io/math2/node/0F7KNJ1ZYN.md" tag="0F7KNJ1ZYN" kind="definition">
# Greatest common divisor in an integral domain

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]] and let $a, b \in R$.
An element $d \in R$ is a greatest common divisor of $a$ and $b$ if it has these two properties.

1. $d \mid a$ and $d \mid b$.
2. For every $c \in R$ with $c \mid a$ and $c \mid b$, $c \mid d$.

## Remarks

If $d$ and $d'$ are two greatest common divisors of $a$ and $b$, then $d$ and $d'$ are [[associates|053E86NWVK]].
In $\mathbb{Z}$, the greatest common divisors of $12$ and $18$ are $6$ and $-6$.
A greatest common divisor need not exist. A GCD domain is an integral domain in which every pair of elements has one.
In a [[principal ideal domain|0RZYDVZ475]], $d$ is a greatest common divisor of $a$ and $b$ if and only if $(a, b) = (d)$.

## Explanations

### L1 human

In a general ring, "greatest" does not refer to size.
It means the largest for divisibility: every other common divisor divides it.
For the integers, this gives the usual gcd, up to the sign.
</document>

<document source="https://tamnd.github.io/math2/node/0F7QKGJS6B.md" tag="0F7QKGJS6B" kind="concept">
# Graph isomorphism

## Summary

A graph isomorphism is a bijection between the vertices of two graphs that preserves and reflects adjacency. Isomorphic graphs have the same structure and differ only in the names of their vertices. The node [[graph isomorphism|00CT0PTH74]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0F8GDKM0A7.md" tag="0F8GDKM0A7" kind="concept">
# Convolution of probability distributions

## Summary

The convolution of two probability distributions on $\mathbb{R}$ is the distribution of the sum of two independent random variables with these distributions.
The node [[convolution of probability distributions|07CHJ5YX28]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0F8X8ZRQ53.md" tag="0F8X8ZRQ53" kind="object">
# Group of units of Z/nZ

## Description

Let $n \geq 1$ be an integer.
The group of units of $\mathbb{Z}/n\mathbb{Z}$ is the [[group of units|0FJ90Y9926]] $(\mathbb{Z}/n\mathbb{Z})^\times$ of the ring $\mathbb{Z}/n\mathbb{Z}$ under multiplication.
Its elements are the residue classes $[a]$ with $\gcd(a, n) = 1$, and its identity is $[1]$.
It is a finite [[abelian group|082GHNPJ53]] of order $\varphi(n)$.

## Remarks

For $n = 8$, the group is $\{[1], [3], [5], [7]\}$, and the square of each element is $[1]$, so it is isomorphic to the [[Klein four-group|0PMRQJVTME]].
For $n = 16$, the group is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, so it is not cyclic.
The group is cyclic if and only if $n$ is $1$, $2$, $4$, $p^k$ or $2 p^k$, with $p$ an odd prime and $k \geq 1$. A generator is called a primitive root modulo $n$.
Other notations are $(\mathbb{Z}/n\mathbb{Z})^*$, $U(\mathbb{Z}/n\mathbb{Z})$ and $\mathbb{Z}_n^*$.

## Explanations

### L1 human

Multiplication modulo $n$ does not make all of $\mathbb{Z}/n\mathbb{Z}$ a group, because $[0]$ has no inverse.
The classes coprime to $n$ do form a group under multiplication.
This group is important in number theory and in cryptography.
</document>

<document source="https://tamnd.github.io/math2/node/0F8YBHF2W3.md" tag="0F8YBHF2W3" kind="concept">
# Double counting

## Summary

Double counting is a proof technique in combinatorics.
To prove that two expressions are equal, one shows that both count the elements of the same finite set.
Each expression comes from a different way to count that set.

A common form uses a relation $R \subseteq A \times B$ between finite sets $A$ and $B$.
For $a \in A$, let $d(a)$ be the number of $b \in B$ with $(a, b) \in R$.
For $b \in B$, let $e(b)$ be the number of $a \in A$ with $(a, b) \in R$.
Then both sums count the pairs in $R$, so
$$\sum_{a \in A} d(a) = |R| = \sum_{b \in B} e(b).$$

## Remarks

The [[handshaking lemma|0XAP8WHNES]] is double counting applied to the pairs of a vertex and an edge that contains it.
The [[symmetry of binomial coefficients|0R7JS3DP3W]] has a short proof by this technique.

## Explanations

### L1 human

Count the same thing in two ways, and the two answers must be equal.
For example, in a class each pair of friends shakes hands once. If you add the number of handshakes of each person, you count each handshake twice, so the sum is even.
</document>

<document source="https://tamnd.github.io/math2/node/0F98FZK14A.md" tag="0F98FZK14A" kind="concept">
# Circumcentre

## Summary

The circumcentre of a triangle is the centre of the circle through its three vertices. It is the point of the plane of the triangle at the same distance from all three vertices.
</document>

<document source="https://tamnd.github.io/math2/node/0F9JHP8AJN.md" tag="0F9JHP8AJN" kind="definition">
# Codomain

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The codomain of $f$ is the set $B$.

## Remarks

Each value $f(a)$ is in the codomain, but not every element of the codomain needs to be a value.
The set of values is the [[range|0E9GTAV478]] of $f$. It is a subset of the codomain.
The function $f$ is [[surjective|0XKH881BZ7]] when its range is the whole codomain.

## Explanations

### L1 human

For $f \colon \mathbb{R} \to \mathbb{R}$ with $f(x) = x^2$, the codomain is $\mathbb{R}$.
The value $-1$ is in the codomain, but no input gives it.
</document>

<document source="https://tamnd.github.io/math2/node/0F9VKJPWCA.md" tag="0F9VKJPWCA" kind="concept">
# Absolutely convergent improper integral

## Summary

An improper integral converges absolutely when the improper integral of the absolute value of the integrand converges. Absolute convergence implies convergence, but not conversely. The node [[absolutely convergent improper integral|03GP7KMBYM]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0FANE7P0DY.md" tag="0FANE7P0DY" kind="concept">
# Möbius function

## Summary

The Möbius function $\mu$ is an arithmetic function that is $0$ on integers with a square factor greater than $1$.
On a squarefree integer, its value $1$ or $-1$ depends on the parity of the number of prime factors.
The [[definition of the Möbius function|0C8QENHYZW]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0FBETJR0JG.md" tag="0FBETJR0JG" kind="concept">
# Local maximum

## Summary

A function has a local maximum at a point when its value there is at least as large as its values at all nearby points. Local maxima are the tops of the hills of the graph. The node [[local maximum|00999TYPXB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0FBXFBZBH8.md" tag="0FBXFBZBH8" kind="definition">
# Affine subspace

## Definition

Let $K$ be a field, let $V$ be a vector space over $K$, and let $A$ be an [[affine space|09XNAPTJQ7]] over $V$.
A nonempty subset $S$ of $A$ is an affine subspace if $S = p + W$ for a point $p$ of $A$ and a linear subspace $W$ of $V$. This means that
$$S = p + W = \{\, p + w \;:\; w \in W \,\}.$$
The subspace $W$ is the direction of $S$. It is unique: $W = \{\, q - r : q, r \in S \,\}$. The dimension of $S$ is the dimension of $W$.

## Remarks

For each point $p$ of $S$, $S = p + W$. Thus each point of $S$ can serve as the base point.
In $\mathbb{R}^n$, the affine subspaces of dimension 1 are the [[lines|0MP9BCF6DP]], and those of dimension 2 are the [[planes|0V3XAMF3C1]]. Those of dimension $n - 1$ are the hyperplanes.
Two affine subspaces are parallel if they have the same direction.
Mathlib's `AffineSubspace` also allows the empty set. A subset $S$ is an affine subspace in this sense if $c(q - r) + s \in S$ for all $c \in K$ and $q, r, s \in S$.

## Explanations

### L1 human

An affine subspace is a flat piece of space that need not pass through the origin. A line or a plane that has been moved away from the origin is an example. It is a linear subspace that has been shifted by a translation.
</document>

<document source="https://tamnd.github.io/math2/node/0FBXJZX3MK.md" tag="0FBXJZX3MK" kind="statement">
# Closed subsets of complete spaces are complete

## Statement

Let $(X, d)$ be a [[complete metric space|08EWT7J0SC]] and let $F$ be a [[closed subset|0G5AV9Q4RR]] of $X$.
Then $F$ with the [[subspace metric|0T2WYDTRP2]] is complete.

## Remarks

Proof: let $(x_n)$ be a Cauchy sequence in $F$. It is Cauchy in $X$, so it converges to some $x \in X$. Then $x \in F$, because [[closed sets contain their limits|09GQ3ZTQJ9]].
For example, $[0, 1]$, $\mathbb{Z}$ and each closed ball of $\mathbb{R}^n$ are complete.
The converse is [[complete subspaces are closed|0W41XG4W60]].

## Explanations

### L1 human

A Cauchy sequence in $F$ has a limit in the big complete space.
Because $F$ is closed, this limit cannot escape from $F$.
</document>

<document source="https://tamnd.github.io/math2/node/0FCCDXH97W.md" tag="0FCCDXH97W" kind="concept">
# Evaluation homomorphism

## Summary

The evaluation homomorphism at an element $c$ of a commutative ring $R$ is the map $R[x] \to R$ that sends each polynomial to its value at $c$.
The [[definition of the evaluation homomorphism|0125QVNCFQ]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0FCCTFA275.md" tag="0FCCTFA275" kind="definition">
# Exponent of a group

## Definition

Let $G$ be a [[group|G5S1M9]].
The exponent of $G$ is the least positive integer $n$ such that $g^n = e$ for each $g \in G$.
If no such integer exists, then $G$ has infinite exponent.

## Remarks

When the exponent is finite, it is the least common multiple of the [[orders of the elements|0WE2NHZJ7B]] of $G$.
The exponent of a finite group divides the order of the group, by [[Lagrange's theorem|3C53TD]].
The exponent of the cyclic group of order $n$ is $n$, and the exponent of the [[Klein four-group|0PMRQJVTME]] is $2$.
The exponent of $(\mathbb{Z}/n\mathbb{Z})^\times$ is the Carmichael function $\lambda(n)$.

## Explanations

### L1 human

The exponent is the smallest power that sends every element to the identity.
It is a common multiple of all the element orders.
A finite group always has a finite exponent, and the exponent divides the order.
</document>

<document source="https://tamnd.github.io/math2/node/0FCHE902D7.md" tag="0FCHE902D7" kind="object">
# General linear group

## Description

Let $R$ be a commutative ring and let $n \ge 1$ be an integer.
The *general linear group* $GL_n(R)$ is the set of [[invertible|0EB9N0S33C]] $n \times n$ matrices over $R$, with [[matrix multiplication|032Q9FB8MT]] as operation.
It is a [[group|G5S1M9]]: the product of invertible matrices is invertible, the identity element is $I_n$, and the inverse of $A$ is $A^{-1}$.
Equivalently, $GL_n(R)$ is the group of units of the [[matrix ring|0FHVPRAE60]] $M_n(R)$.

## Remarks

For a field $F$, $GL_n(F)$ is isomorphic to the group of linear isomorphisms of $F^n$ to itself.
For $n \ge 2$, the group $GL_n(F)$ is not abelian.
For a finite field with $q$ elements, $|GL_n(\F_q)| = (q^n - 1)(q^n - q) \cdots (q^n - q^{n-1})$.
The [[special linear group|0V9PMN0RBH]] is the subgroup of matrices with determinant $1$.

## Explanations

### L1 human

The general linear group collects all the invertible matrices of one size.
We can multiply two of them, and we can undo each of them, so they form a group.
Geometrically, its elements are all the ways to transform space linearly without flattening it.
</document>

<document source="https://tamnd.github.io/math2/node/0FCWA16FZE.md" tag="0FCWA16FZE" kind="definition">
# Divisors of a natural number

## Definition

Let $n$ be a positive integer.
The set of divisors of $n$ is the set of positive integers $d$ that [[divide|096PW7Q014]] $n$:
$$D(n) = \{ d \in \mathbb{Z}_{>0} : d \mid n \}.$$

## Remarks

The set $D(n)$ is finite, because each $d \in D(n)$ satisfies $1 \le d \le n$. It always contains $1$ and $n$.
For example, $D(12) = \{1, 2, 3, 4, 6, 12\}$.
The map $d \mapsto n/d$ is a bijection from $D(n)$ to itself.
Sums over $D(n)$ define the [[divisor function|0YEW47VJ0J]] and the [[Dirichlet convolution|0KH1N66QWW]].

## Explanations

### L1 human

The divisors of $n$ are the numbers that go into $n$ with no remainder.
They come in pairs $d$ and $n/d$. For $n = 12$, the pairs are $1$ and $12$, $2$ and $6$, and $3$ and $4$.
</document>

<document source="https://tamnd.github.io/math2/node/0FDCW77H44.md" tag="0FDCW77H44" kind="statement">
# The gamma function extends the factorial

## Statement

For each $s > 0$, the [[gamma function|0V3FZMVB4K]] satisfies $\Gamma(s + 1) = s \, \Gamma(s)$.
For each natural number $n$, $\Gamma(n + 1) = n!$.

## Remarks

[[Integration by parts|08Y1BBQTK9]] on $[\varepsilon, b]$, with $u = t^s$ and $v = -e^{-t}$, gives the recursion in the limit.
The [[integral of $e^{-t}$|039JAF7ZAN]] gives $\Gamma(1) = 1$. Induction on $n$ then gives $\Gamma(n + 1) = n!$.
Also $\Gamma(1/2) = \sqrt{\pi}$, by the substitution $t = x^2$ and the Gaussian integral.
Mathlib states it as `Real.Gamma_nat_eq_factorial`.

## History

Leonhard Euler announced the integral representation in a letter to Christian Goldbach, dated 8 January 1730.

## Explanations

### L1 human

The recursion $\Gamma(s + 1) = s \, \Gamma(s)$ is the same as the rule $(n + 1)! = (n + 1) \cdot n!$.
So the gamma function agrees with the factorial at the integers, shifted by one.
</document>

<document source="https://tamnd.github.io/math2/node/0FDKRX9M9Y.md" tag="0FDKRX9M9Y" kind="concept">
# Reachability

## Summary

Reachability says which vertices of a graph can be joined by a walk. It is an equivalence relation, and its classes are the connected components. The node [[reachability|0J8XE7W68F]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0FE2SB8WVJ.md" tag="0FE2SB8WVJ" kind="concept">
# Locally compact space

## Summary

A locally compact space is a topological space in which each point has a compact neighbourhood.
Such spaces look like compact spaces near each point, as $\mathbb{R}^n$ does.
</document>

<document source="https://tamnd.github.io/math2/node/0FE8Q5S42Y.md" tag="0FE8Q5S42Y" kind="definition">
# Subset

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The set $A$ is a subset of $B$, written $A \subseteq B$, when each [[element|0EZ758XE73]] of $A$ is an element of $B$:
$$\forall x \, (x \in A \to x \in B).$$
Then $B$ is a superset of $A$.
The set $A$ is a proper subset of $B$, written $A \subsetneq B$, when $A \subseteq B$ and $A \neq B$.

## Remarks

Some authors write $A \subset B$ for $A \subseteq B$, and others write it for a proper subset.
Two sets are equal if and only if each is a subset of the other. This follows from the [[axiom of extensionality|0TB16T2C97]].
The [[empty set|0ECVK59Y03]] is a subset of every set, and every set is a subset of itself.

## Explanations

### L1 human

$A \subseteq B$ means that $A$ fits inside $B$.
The set of even numbers is a subset of the set of integers.
To prove $A \subseteq B$, take an arbitrary $x \in A$ and show $x \in B$.
</document>

<document source="https://tamnd.github.io/math2/node/0FEBE9NC0E.md" tag="0FEBE9NC0E" kind="statement">
# A subgroup whose index is the smallest prime divisor of the order is normal

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be the smallest prime number that divides $|G|$. Let $H$ be a [[subgroup|0X1JXT47GA]] of $G$ of [[index|0DTXBQPA2S]] $p$.
Then $H$ is [[normal|0CB0DX5BVV]] in $G$.

## Remarks

For $p = 2$ this is the fact that [[subgroups of index two are normal|0V547YQRMV]].
The proof lets $G$ act on the $p$ left cosets of $H$. The kernel $K$ of this action lies in $H$, and $[G : K]$ divides both $|G|$ and $p!$.
Since $p$ is the smallest prime divisor of $|G|$, this forces $[G : K] = p$, so $H = K$ is normal.

## Explanations

### L1 human

Index $2$ is not the only small index that forces normality.
If the index is the smallest prime that divides the order of the group, the subgroup is also normal.
For example, a subgroup of index $3$ in a group of order $75 = 3 \cdot 5^2$ is normal.
</document>

<document source="https://tamnd.github.io/math2/node/0FEMF8JKAF.md" tag="0FEMF8JKAF" kind="statement">
# e as the limit of (1 + 1/n)^n

## Statement

Let $e = \exp(1)$, where $\exp$ is the [[exponential function|0RMVS7RXZX]].
Then
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e.$$
More generally, for each real number $x$,
$$\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x.$$

## Remarks

Some texts take the limit as the definition of [[the number e|0MVCFMW7S4]]. The sequence $(1 + 1/n)^n$ is increasing and bounded above by $3$, so the [[monotone convergence theorem|0FM6N4R0VB]] gives the limit.
The first values are $2$, $2.25$, about $2.370$ and about $2.441$. The limit is $e \approx 2.71828$.
Jacob Bernoulli found this limit when he studied compound interest.
Mathlib states the general form as `Real.tendsto_one_add_div_pow_exp`.

## Explanations

### L1 human

Put one unit of money in a bank at $100\%$ yearly interest, paid $n$ times a year.
After one year you have $(1 + 1/n)^n$ units. As $n$ grows, this approaches $e$, and it never exceeds $e$.
</document>

<document source="https://tamnd.github.io/math2/node/0FER36N8SB.md" tag="0FER36N8SB" kind="conjecture">
# Twin prime conjecture

## Statement

There are infinitely many [[prime numbers|0SN0MAEKYX]] $p$ such that $p + 2$ is also prime.

## Remarks

A pair of primes $(p, p + 2)$ is a pair of twin primes. The first pairs are $(3, 5)$, $(5, 7)$, $(11, 13)$, $(17, 19)$, $(29, 31)$.
The conjecture is the case $k = 1$ of the conjecture of de Polignac: for each positive integer $k$, there are infinitely many primes $p$ such that $p + 2k$ is prime.
The conjecture is open.

## History

De Polignac stated the general conjecture in 1849.
In 2013, Yitang Zhang proved that some gap $N < 7 \cdot 10^7$ occurs between infinitely many pairs of primes. Later work by Maynard and the Polymath project reduced the bound to $246$.

## Explanations

### L1 human

Twin primes are primes with only one number between them, like $41$ and $43$.
The conjecture says that such pairs never stop, even though primes become rarer.
</document>

<document source="https://tamnd.github.io/math2/node/0FF2FJ8CH2.md" tag="0FF2FJ8CH2" kind="statement">
# Universal property of the quotient topology

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]], let $\sim$ be an equivalence relation on $X$, and give $X / {\sim}$ the [[quotient topology|0V8046R0P9]], with quotient map $q \colon X \to X / {\sim}$.
Let $Y$ be a topological space and let $g \colon X / {\sim} \to Y$ be a function.
Then $g$ is [[continuous|0SY9ECQ19Q]] if and only if the [[composition|07HF9YCWPP]] $g \circ q \colon X \to Y$ is continuous.

## Remarks

Proof of the converse: for $V$ open in $Y$, the set $q^{-1}(g^{-1}(V)) = (g \circ q)^{-1}(V)$ is open in $X$. Thus $g^{-1}(V)$ is open in the quotient.
So each continuous map $f \colon X \to Y$ that is constant on each class gives a unique continuous map $g$ with $f = g \circ q$.
For example, $t \mapsto (\cos 2\pi t, \sin 2\pi t)$ is constant on the classes of $\mathbb{R} / \mathbb{Z}$, so it gives a continuous map from $\mathbb{R} / \mathbb{Z}$ to the circle.

## Explanations

### L1 human

To define a continuous map on a quotient space, define a continuous map on the original space that takes the same value on glued points.
The map then passes to the quotient.
</document>

<document source="https://tamnd.github.io/math2/node/0FF6CAQT2Y.md" tag="0FF6CAQT2Y" kind="definition">
# Ellipse

## Definition

Let $F_1$ and $F_2$ be points of the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $a$ be a real number with $2a > d(F_1, F_2)$.
The ellipse with the foci $F_1$, $F_2$ and the semi-major axis $a$ is the set
$$\{\, P \in \mathbb{R}^2 \;:\; d(P, F_1) + d(P, F_2) = 2a \,\}.$$

## Remarks

Let $c = \frac{1}{2} d(F_1, F_2)$ and $b = \sqrt{a^2 - c^2}$. For $F_1 = (-c, 0)$ and $F_2 = (c, 0)$, the ellipse is the set of the points with $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$.
If $F_1 = F_2$, the ellipse is the [[circle|060W5CBZP6]] with the centre $F_1$ and the radius $a$.
The [[eccentricity|0HVX863SAJ]] of the ellipse is $e = c / a$, with $0 \leq e < 1$. The ellipse bounds a region of area $\pi a b$.

## Explanations

### L1 human

Fix two pins in a board and tie a loose string to both. Pull the string tight with a pencil and move it round: the pencil draws an ellipse. The length of the string is the constant sum $2a$ of the distances to the two pins.
</document>

<document source="https://tamnd.github.io/math2/node/0FF7726C7P.md" tag="0FF7726C7P" kind="statement">
# Inclusion–exclusion principle

## Statement

Let $A_1, \dots, A_n$ be finite sets.
For a nonempty set $I \subseteq \{1, \dots, n\}$, let $A_I = \bigcap_{i \in I} A_i$.
Then
$$\left| \bigcup_{i=1}^{n} A_i \right| = \sum_{\varnothing \neq I \subseteq \{1, \dots, n\}} (-1)^{|I| + 1} |A_I|.$$

## Remarks

For $n = 2$ this is [[inclusion–exclusion for two sets|0C4YEQ9H5E]].
A common use counts the elements of a finite set $U$ that are in none of the sets $A_i \subseteq U$. This number is $\sum_{I} (-1)^{|I|} |A_I|$, over all $I$, with $A_\varnothing = U$.
The [[formula for the number of derangements|03RSJXF728]] follows in this way.

## Explanations

### L1 human

Add the sizes of the single sets, subtract the sizes of the pairwise intersections, add the sizes of the triple intersections, and so on.
Each element of the union is then counted exactly once in total.
</document>

<document source="https://tamnd.github.io/math2/node/0FFAE5P6R4.md" tag="0FFAE5P6R4" kind="definition">
# Affine span

## Definition

Let $K$ be a field, and let $A$ be an [[affine space|09XNAPTJQ7]] over a vector space $V$ over $K$. Let $S$ be a nonempty subset of $A$.
The affine span of $S$, written $\operatorname{aff}(S)$, is the intersection of all [[affine subspaces|0PYW26VNT6]] of $A$ that contain $S$.
It is the smallest affine subspace of $A$ that contains $S$.

## Remarks

The affine span of $S$ is the set of all [[affine combinations|0KJH6SRFZK]] of finitely many points of $S$.
In $\mathbb{R}^3$: the affine span of two distinct points is the line through them. The span of three points that are not on one line is the plane through them. The span of four points that are not in one plane is all of $\mathbb{R}^3$.
For each point $p$ of $S$, the affine span is $p + W$, where $W$ is the linear span of the vectors $q - p$ for $q \in S$.
Mathlib also defines the affine span of the empty set: it is the empty affine subspace.

## Explanations

### L1 human

The affine span is the flat object that the given points determine. Two points give a line, and three points in general position give a plane. Compare it with the linear span of vectors, which must always pass through the origin.
</document>

<document source="https://tamnd.github.io/math2/node/0FFAYNMAJZ.md" tag="0FFAYNMAJZ" kind="definition">
# Connected graph

## Definition

A [[graph|0JD5C82QMB]] $G$ is connected if its vertex set is nonempty and any two vertices of $G$ are [[reachable|0J8XE7W68F]] from each other.

## Remarks

The empty graph, with no vertices, is not connected under this convention. Some authors call it connected.
A graph is connected if and only if it has exactly one [[connected component|01VVP10MWJ]].
The [[cycle graphs|0QQ0F12962]], the [[path graphs|075WR6W3Y3]] and the [[trees|0MJ0JS72DZ]] are connected.

## Explanations

### L1 human

A graph is connected when it is in one piece.
From any vertex you can reach any other vertex along edges.
</document>

<document source="https://tamnd.github.io/math2/node/0FFH3HSRBA.md" tag="0FFH3HSRBA" kind="concept">
# Construction of the rationals as fractions

## Summary

The rational numbers can be constructed as classes of pairs of [[integers|0C4JS9RF5M]].
The idea is that the pair $(a, b)$ with $b \neq 0$ stands for the fraction $a / b$.
On the set of pairs $(a, b)$ with $a, b \in \mathbb{Z}$ and $b \neq 0$, define $(a, b) \sim (c, d)$ when $a d = b c$. This relation is an equivalence relation.
The set $\mathbb{Q}$ of [[rational numbers|0NCJ5WXN6Z]] is the set of the equivalence classes, and we write $a / b$ for the class of $(a, b)$.
The operations are $\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}$ and $\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}$. They do not depend on the choice of the pairs.

## Remarks

The same construction works for each integral domain $R$. It gives the field of fractions of $R$, the smallest field that contains $R$.
The map $a \mapsto a / 1$ embeds $\mathbb{Z}$ in $\mathbb{Q}$.

## Explanations

### L1 human

A fraction is a pair of a top number and a nonzero bottom number.
The pairs $(1, 2)$ and $(2, 4)$ are different pairs, but $1 \cdot 4 = 2 \cdot 2$, so they give the same rational number.
The construction makes this precise: a rational number is the collection of all the pairs that name it.
</document>

<document source="https://tamnd.github.io/math2/node/0FFXKTKKQC.md" tag="0FFXKTKKQC" kind="concept">
# Absolute value

## Summary

The absolute value of a number is its distance from $0$. We write $|x|$ for the absolute value of $x$.
For a real number, $|x|$ is $x$ without its sign, for example $|-3| = |3| = 3$.
For a complex number $a + b i$, the absolute value is the [[modulus|0KW2X0TFBX]] $\sqrt{a^2 + b^2}$.
</document>

<document source="https://tamnd.github.io/math2/node/0FGAJSPX8E.md" tag="0FGAJSPX8E" kind="concept">
# Kurtosis

## Summary

The kurtosis of a distribution measures the weight of its tails, compared with the center.
The node [[kurtosis|0CBQVVNBMP]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0FGCFY1AZX.md" tag="0FGCFY1AZX" kind="definition">
# Preimage of a set

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]] and let $T$ be a [[subset|0FE8Q5S42Y]] of $B$.
The preimage of $T$ under $f$ is the set
$$f^{-1}(T) = \{a \in A \mid f(a) \in T\}.$$

## Remarks

Another name is the inverse image. The notation $f^{-1}(T)$ makes sense for every $f$, also when $f$ has no [[inverse function|0W766JAPSN]].
Preimages respect all set operations: $f^{-1}(T_1 \cap T_2) = f^{-1}(T_1) \cap f^{-1}(T_2)$, and the same holds for unions and complements.
Example: for $f(x) = x^2$ on $\mathbb{R}$, $f^{-1}([1, 4]) = [-2, -1] \cup [1, 2]$.

## Explanations

### L1 human

The preimage answers the question: which inputs land in $T$?
For the function that sends a student to their grade, the preimage of $\{A\}$ is the set of students with grade $A$.
</document>

<document source="https://tamnd.github.io/math2/node/0FGENPMPJX.md" tag="0FGENPMPJX" kind="statement">
# The reals form a field

## Statement

The [[real numbers|08J2M1JYAG]] $\mathbb{R}$, with their addition and multiplication, form a field.
That is, $\mathbb{R}$ is a commutative ring with $0 \neq 1$, and each real number $x \neq 0$ has a multiplicative inverse $x^{-1}$ with $x \cdot x^{-1} = 1$.

## Remarks

The field $\mathbb{R}$ contains the field $\mathbb{Q}$ of rational numbers as a subfield.
In the axiomatic approach, this is part of the axioms. In a construction of $\mathbb{R}$, for example with [[Cauchy sequences|0D8HE681SC]], it is a theorem.

## Explanations

### L1 human

The usual rules of arithmetic hold for the real numbers.
You can add, subtract and multiply real numbers, and you can divide by each real number that is not zero.
</document>

<document source="https://tamnd.github.io/math2/node/0FHM8DB6PM.md" tag="0FHM8DB6PM" kind="statement">
# Characterization of bridges

## Statement

Let $G$ be a [[graph|0JD5C82QMB]] and let $e$ be an edge of $G$.
Then $e$ is a [[bridge|0D81KWH9ZM]] of $G$ if and only if $e$ is not an edge of any [[cycle|07CT9RDY7Q]] of $G$.

## Remarks

Proof idea: let $e = uv$. If $e$ lies on a cycle, the rest of the cycle is a path from $u$ to $v$ in $G - e$. Conversely, a path from $u$ to $v$ in $G - e$ together with $e$ gives a cycle.
A consequence: a graph is a forest if and only if every edge is a bridge.

## Explanations

### L1 human

An edge on a cycle is never a bridge: if you delete it, you can still go around the other way.
An edge on no cycle has no detour, so deleting it splits the graph.
</document>

<document source="https://tamnd.github.io/math2/node/0FHSDCQACY.md" tag="0FHSDCQACY" kind="statement">
# Epsilon-delta uniform continuity

## Statement

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be a function.
Then the following two conditions are equivalent:

1. For each real $\varepsilon > 0$ there is a real $\delta > 0$ such that $d_Y(f(x), f(y)) < \varepsilon$ for all $x, y \in X$ with $d_X(x, y) < \delta$.
2. For all sequences $(x_n)$ and $(y_n)$ in $X$ with $d_X(x_n, y_n) \to 0$, also $d_Y(f(x_n), f(y_n)) \to 0$.

The map $f$ is uniformly continuous when it satisfies these conditions.

## Remarks

Condition 1 is the epsilon-delta definition of uniform continuity. Here $\delta$ depends only on $\varepsilon$, not on the points.
Each uniformly continuous map is continuous. The converse fails: $x \mapsto x^2$ on $\mathbb{R}$ is continuous but not uniformly continuous. Take $x_n = n + 1/n$ and $y_n = n$; then $x_n^2 - y_n^2 > 2$.
On a compact space, each continuous map to a metric space is uniformly continuous: this is the [[Heine–Cantor theorem|06B8BNJCD6]].
Each [[Lipschitz map|0TMZ21WKCN]] is uniformly continuous.

## Explanations

### L1 human

For a uniformly continuous map, one input tolerance $\delta$ works in the whole space.
The map $x \mapsto x^2$ gets steeper and steeper, so near large $x$ you need a smaller and smaller $\delta$, and no single $\delta$ works.
</document>

<document source="https://tamnd.github.io/math2/node/0FHVPRAE60.md" tag="0FHVPRAE60" kind="object">
# Matrix ring

## Description

Let $R$ be a ring and let $n \ge 1$ be an integer.
The *matrix ring* $M_n(R)$ is the set of $n \times n$ [[matrices|0VD9ZSWWJG]] over $R$, with [[matrix addition|0Y7H3SAVA1]] and [[matrix multiplication|032Q9FB8MT]].
It is a ring, with the [[zero matrix|0ZTKJJDGR7]] as zero and the [[identity matrix|06DW9BAWB1]] $I_n$ as unit.
If $R$ is commutative, then $M_n(R)$ is also an algebra over $R$, with [[scalar multiplication|09V6DDACXR]].

## Remarks

For $n \ge 2$ and $R \ne 0$, the ring $M_n(R)$ is not commutative and has nonzero elements whose product is zero.
The units of $M_n(R)$ are the [[invertible matrices|0EB9N0S33C]], and they form the [[general linear group|0FCHE902D7]].
For a field $F$, the ring $M_n(F)$ is isomorphic to the ring of endomorphisms of $F^n$.

## Explanations

### L1 human

Square matrices of one fixed size can be added and multiplied, and the result is again a square matrix of that size.
They follow most of the rules of numbers.
The main differences are that $AB$ can differ from $BA$, and that $AB$ can be zero when $A$ and $B$ are not.
</document>

<document source="https://tamnd.github.io/math2/node/0FJ90Y9926.md" tag="0FJ90Y9926" kind="concept">
# Group of units of a monoid

## Summary

The units of a monoid are its invertible elements.
They form a group under the operation of the monoid.
</document>

<document source="https://tamnd.github.io/math2/node/0FJB06AD2P.md" tag="0FJB06AD2P" kind="definition">
# Conditional probability

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \in \mathcal{F}$ be [[events|00SQGJNRTB]] with $P(B) > 0$.
The conditional probability of $A$ given $B$ is
$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}.$$

## Remarks

For fixed $B$ with $P(B) > 0$, the map $A \mapsto P(A \mid B)$ is a [[probability measure|0VGMEBVW1M]] on $(\Omega, \mathcal{F})$. It gives all its mass to $B$.
For one roll of a fair die, $P(\{6\} \mid \text{even}) = (1/6) / (1/2) = 1/3$.
In general $P(A \mid B) \ne P(B \mid A)$. [[Bayes' theorem|0NM6E9VW04]] relates the two.
The definition does not apply when $P(B) = 0$. Conditioning on such events needs the [[conditional expectation|0YAMJRK04E]].

## Explanations

### L1 human

Once you know that $B$ happened, the outcomes outside $B$ are no longer possible.
So you keep only the part of $A$ inside $B$, and you rescale by $P(B)$ so that the total is $1$ again.
</document>

<document source="https://tamnd.github.io/math2/node/0FJBBCXYPS.md" tag="0FJBBCXYPS" kind="statement">
# Chromatic number of a complete bipartite graph

## Statement

Let $m, n \ge 1$ be integers.
Then the [[chromatic number|0C5M7AM6FH]] of the [[complete bipartite graph|0V0GC7BQAN]] $K_{m,n}$ is $2$.

## Remarks

Give one colour to each of the two parts. Each edge joins the two parts, so this is a colouring with $2$ colours.
One colour is not enough, because $K_{m,n}$ has an edge.
More generally, a [[bipartite|07DVHJ8N0A]] graph with at least one edge has chromatic number $2$.

## Explanations

### L1 human

Colour one side of $K_{m,n}$ red and the other side blue.
Every edge goes from red to blue, so two colours are enough, and one colour is too few.
</document>

<document source="https://tamnd.github.io/math2/node/0FJTGZA4GS.md" tag="0FJTGZA4GS" kind="statement">
# Bertrand's postulate

## Statement

Let $n$ be a positive integer.
Then there is a [[prime number|0SN0MAEKYX]] $p$ such that $n < p \le 2 n$.

## Remarks

For example, for $n = 10$ the primes $11, 13, 17, 19$ are in the range $10 < p \le 20$.
An equivalent form: if $p_k$ is the $k$-th prime, then $p_{k+1} < 2 p_k$ for each $k \ge 1$.
The [[prime number theorem|0EHBY5Y84N]] gives a stronger result for large $n$, but Bertrand's postulate holds for each $n \ge 1$.

## History

Joseph Bertrand stated the result as a conjecture in 1845. Chebyshev proved it in 1852.

## Explanations

### L1 human

Between each number and its double, there is always at least one prime.
So the gap after a prime $p$ to the next prime is always smaller than $p$.
</document>

<document source="https://tamnd.github.io/math2/node/0FJY6G7FNA.md" tag="0FJY6G7FNA" kind="concept">
# Maximum degree

## Summary

The maximum degree of a finite graph is the largest degree of its vertices. It bounds the chromatic number and many other parameters. The node [[maximum degree|0KNAN9F27P]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0FJZMBX0ZZ.md" tag="0FJZMBX0ZZ" kind="statement">
# Triangle inequality for norms

## Statement

Let $V$ be a vector space over $\R$ or $\C$ with a [[norm|0V2WP9B1MF]] $\norm{\cdot}$.
Then, for all $x, y \in V$,
$$\norm{x + y} \le \norm{x} + \norm{y} \qquad \text{and} \qquad \bigl| \norm{x} - \norm{y} \bigr| \le \norm{x - y}.$$

## Remarks

The first inequality is an axiom of a norm; the second, the *reverse triangle inequality*, follows from it.
For the norm induced by an inner product, the first inequality follows from the [[Cauchy–Schwarz inequality|0RFWW56Y22]].
In $\R^n$, it says that one side of a triangle is at most the sum of the other two sides.

## Explanations

### L1 human

The straight way from $0$ to $x + y$ is never longer than the way through $x$.
This is the triangle inequality.
The reverse form says that two vectors with a small difference have almost equal lengths.
</document>

<document source="https://tamnd.github.io/math2/node/0FK17N4DKA.md" tag="0FK17N4DKA" kind="statement">
# Simultaneous diagonalisation

## Statement

Let $F$ be a field and let $A$ and $B$ be [[diagonalisable|07SQCNV233]] $n \times n$ matrices over $F$.
Then there is one invertible $n \times n$ matrix $P$ such that both $P^{-1} A P$ and $P^{-1} B P$ are diagonal if and only if $AB = BA$.

## Remarks

Diagonal matrices commute, so the condition $AB = BA$ is necessary.
For sufficiency, $B$ maps each eigenspace of $A$ to itself, and it is diagonalisable on each of them.
The same holds for each family of pairwise commuting diagonalisable matrices.

## Explanations

### L1 human

Two diagonalisable matrices can be diagonalised by the same change of basis exactly when they commute.
Then they share a basis of eigenvectors.
This fact is important in quantum mechanics, where commuting observables can be measured together.
</document>

<document source="https://tamnd.github.io/math2/node/0FM6N4R0VB.md" tag="0FM6N4R0VB" kind="statement">
# Monotone convergence theorem for sequences

## Statement

Let $(a_n)$ be an increasing real sequence, in the sense of a [[monotone sequence|0KN22AAW5R]].
Assume that $(a_n)$ is bounded above.
Then $(a_n)$ converges, and its limit is the supremum $\sup \{a_n : n \in \mathbb{N}\}$.

## Remarks

The supremum $s$ exists by the completeness of $\mathbb{R}$. For $\varepsilon > 0$ there is $N$ with $a_N > s - \varepsilon$, and then $s - \varepsilon < a_n \le s$ for all $n \ge N$.
The statement fails in $\mathbb{Q}$. The decimal approximations $1, 1.4, 1.41, 1.414, \dots$ of $\sqrt{2}$ increase and are bounded, but they have no limit in $\mathbb{Q}$.
The node [[decreasing sequence bounded below converges|02V2GYAC8P]] gives the dual statement.
An increasing sequence that is not bounded above [[tends to infinity|09AZCV6GMC]].

## Explanations

### L1 human

An increasing sequence that cannot pass a ceiling must pile up somewhere.
It piles up at the lowest ceiling, which is the supremum of its terms.
</document>

<document source="https://tamnd.github.io/math2/node/0FMAQSB1SC.md" tag="0FMAQSB1SC" kind="statement">
# Burnside's p^a q^b theorem

## Statement

Let $p$ and $q$ be prime numbers, let $a, b \geq 0$ be integers, and let $G$ be a finite [[group|G5S1M9]] of order $p^a q^b$.
Then $G$ is [[solvable|0FXJ7YFMMX]].

## Remarks

A consequence is that the order of each finite [[simple group|0YMYMNQ0QZ]] that is not abelian has at least three distinct prime divisors.
For example, $A_5$ has order $60 = 2^2 \cdot 3 \cdot 5$.
For $b = 0$, the result says that finite $p$-groups are solvable, which follows from [[their nilpotency|081WTKYX6C]].

## History

William Burnside proved the theorem in 1904, with the representation theory of finite groups.
Burnside, Jordan and Frobenius had proved special cases before, in 1897, 1898 and 1902.
John G. Thompson later showed that his work on the N-group theorem gives a proof without representation theory.

## Explanations

### L1 human

If the order of a group has only two prime factors, then the group is solvable.
So a simple group that is not abelian needs at least three primes in its order.
The smallest such group, $A_5$, uses the primes $2$, $3$ and $5$.
</document>

<document source="https://tamnd.github.io/math2/node/0FMN2MN7TM.md" tag="0FMN2MN7TM" kind="definition">
# Constructible number

## Definition

Identify the plane with $\mathbb{C}$.
A complex number $z$ is constructible if the point $z$ can be obtained from $0$ and $1$ by finitely many [[straightedge and compass|0DAECMZVRT]] steps.
In each step, one adds an intersection point of two distinct lines or circles drawn through, or centred at, points already obtained.

## Remarks

A real number $x$ is constructible if and only if a segment of length $|x|$ can be constructed from a segment of length $1$.
A complex number $a + b i$ is constructible if and only if $a$ and $b$ are constructible.
The constructible numbers [[form a field closed under square roots|0PN9WMRWT7]].
The numbers $\sqrt{2}$, $\sqrt{1 + \sqrt{3}}$ and $\cos(2\pi/5)$ are constructible. The numbers $\sqrt[3]{2}$ and $\pi$ are not.

## Explanations

### L1 human

Start with two points, $0$ and $1$.
Draw lines through points you have, and circles centred at points you have, and mark the new intersections.
The constructible numbers are all the points you can reach in this way.
</document>

<document source="https://tamnd.github.io/math2/node/0FNKFT8H6E.md" tag="0FNKFT8H6E" kind="concept">
# Total derivative

## Summary

Let $f$ map an open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$. The total derivative of $f$ at a point $a$ is the linear map $Df(a) \colon \mathbb{R}^n \to \mathbb{R}^m$ that best approximates $f$ near $a$. That is, $f(a + h) = f(a) + Df(a)(h) + o(\|h\|)$ as $h \to 0$.
It is the [[Fréchet derivative|06EH97V2A7]] in the case of [[Euclidean spaces|0XPR1E85W0]].
Its matrix in the standard bases is the [[Jacobian matrix|047NDTEPCV]] of $f$ at $a$.

## Remarks

The total derivative accounts for changes in all variables at once. A partial derivative accounts for a change in one variable only.
When $f$ depends on $t$ both directly and through other functions of $t$, the total derivative $\dfrac{d}{dt} f(t, x(t))$ is the sum $\dfrac{\partial f}{\partial t} + \dfrac{\partial f}{\partial x} \dfrac{dx}{dt}$, by the [[chain rule|0EAAS2EHDR]].

## Explanations

### L1 human

For a function of one variable, the derivative at a point is a number: the slope of the tangent line.
For a function of several variables, the derivative at a point is a linear map: it tells how the output changes for each small step in any direction.
</document>

<document source="https://tamnd.github.io/math2/node/0FNN9B4W64.md" tag="0FNN9B4W64" kind="statement">
# Chinese remainder theorem

## Statement

Let $n_1, \dots, n_k$ be positive integers that are pairwise [[coprime|0GFKP8F8AG]], and let $N = n_1 n_2 \cdots n_k$.
Let $a_1, \dots, a_k$ be integers.
Then there is an integer $x$ such that $x \equiv a_i \pmod{n_i}$ for each $i$.
Two such integers are congruent modulo $N$.

## Remarks

In the language of rings, the map $\mathbb{Z}/N\mathbb{Z} \to \mathbb{Z}/n_1\mathbb{Z} \times \cdots \times \mathbb{Z}/n_k\mathbb{Z}$, $\overline{x} \mapsto (\overline{x}, \dots, \overline{x})$, is a ring isomorphism.
For two moduli $m$ and $n$, write $m u + n v = 1$ with [[Bézout's identity|0DQVHMYAGA]]. Then $x = a_2 m u + a_1 n v$ solves $x \equiv a_1 \pmod m$ and $x \equiv a_2 \pmod n$.
The [[Chinese remainder theorem for non-coprime moduli|04SJTX9P4N]] removes the hypothesis of coprimality.

## History

The Chinese book Sunzi Suanjing, from the 5th century, contains the oldest known example of the problem. Qin Jiushao gave a complete method in 1247.

## Explanations

### L1 human

Find a number that leaves the remainder $2$ when you divide by $3$, the remainder $3$ by $5$, and the remainder $2$ by $7$.
The theorem says that such a number exists, and that it is unique up to multiples of $3 \cdot 5 \cdot 7 = 105$.
The smallest positive answer is $23$.
</document>

<document source="https://tamnd.github.io/math2/node/0FP31SX5EA.md" tag="0FP31SX5EA" kind="concept">
# Even and odd functions

## Summary

A function is even when $f(-x) = f(x)$ and odd when $f(-x) = -f(x)$ for all $x$. Even functions have graphs symmetric about the vertical axis, and odd functions have graphs symmetric about the origin. The node [[even and odd functions|0XE690QF65]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0FP4BVDAAQ.md" tag="0FP4BVDAAQ" kind="definition">
# Scalar potential

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $F \colon U \to \mathbb{R}^n$ be a continuous [[vector field|0N67NBKQ5T]].
A scalar potential of $F$ is a function $\phi \colon U \to \mathbb{R}$ of class $C^1$ with $\nabla \phi = F$ on $U$.

## Remarks

In physics, a potential is often a function $V$ with $F = -\nabla V$, so that $V$ is a potential energy. The two conventions differ by a sign.
If $U$ is connected, two scalar potentials of $F$ differ by a constant. Indeed, their difference has zero gradient on $U$.
On a connected open set $U$ one finds a potential by fixing $p \in U$ and setting $\phi(x) = \int_\gamma F \cdot d\mathbf{r}$ for a path $\gamma$ from $p$ to $x$. This works when $F$ is [[conservative|05R9NF2J49]].
Example: the gravitational field $F(x) = -x / \|x\|^3$ on $\mathbb{R}^3 \setminus \{0\}$ has the scalar potential $\phi(x) = 1 / \|x\|$.

## Explanations

### L1 human

A scalar potential is a height function for a vector field.
The field points in the direction in which the potential grows fastest, and its length is the rate of growth.
</document>

<document source="https://tamnd.github.io/math2/node/0FPWF7TTMB.md" tag="0FPWF7TTMB" kind="statement">
# Lévy's continuity theorem

## Statement

Let $X, X_1, X_2, \dots$ be real random variables with [[characteristic functions|0MYDNRT7PP]] $\varphi, \varphi_1, \varphi_2, \dots$.
1. If $X_n \to X$ [[in distribution|0JDAQHWA9N]], then $\varphi_n(t) \to \varphi(t)$ for each $t \in \mathbb{R}$.
2. Conversely, suppose that $\varphi_n(t)$ converges to a limit $\psi(t)$ for each $t \in \mathbb{R}$, and that $\psi$ is continuous at $0$. Then $\psi$ is the characteristic function of a random variable $Y$, and $X_n \to Y$ in distribution.

## Remarks

Part 1 follows from the definition of convergence in distribution with the bounded continuous functions $\cos(tx)$ and $\sin(tx)$.
The continuity at $0$ in part 2 is necessary. For $X_n \sim \mathcal{N}(0, n)$, $\varphi_n(t) = e^{-nt^2/2}$ tends to $1_{\{t = 0\}}$, which is not continuous at $0$, and $X_n$ does not converge in distribution.
The theorem holds in the same form for random vectors in $\mathbb{R}^k$.

## History

The theorem is named after Paul Lévy.

## Explanations

### L1 human

To show that random variables converge in distribution, it is enough to show that their characteristic functions converge.
This is often much easier, because characteristic functions of sums of independent variables are products.
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# Reduced word

## Definition

Let $S$ be a set, and let $S^{-1} = \{s^{-1} : s \in S\}$ be a set of new symbols, one for each $s \in S$.
A word on $S$ is a finite string $t_1 t_2 \cdots t_n$ with $n \geq 0$ and each $t_i$ in $S \cup S^{-1}$.
A word is reduced when it contains no adjacent pair $s s^{-1}$ or $s^{-1} s$ with $s \in S$.

## Remarks

The empty word, with $n = 0$, is reduced.
Remove adjacent pairs $s s^{-1}$ and $s^{-1} s$, one at a time, until no such pair is left. The result is a reduced word.
For example, for $S = \{a, b, c\}$, the word $a b^3 c^{-1} c a^{-1} c$ reduces to $a b^3 a^{-1} c$.
A reduced word is cyclically reduced when its first and last letters are not inverse to each other.

## Explanations

### L1 human

A word is a string of letters and inverse letters.
It is reduced when you cannot cancel a letter against its inverse next to it.
Reduced words are the normal forms of the elements of a [[free group|0KTW26C80J]].
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# Spanning tree

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A spanning tree of $G$ is a [[subgraph|0P11X5HKAX]] $T$ of $G$ that contains every vertex of $G$ and is a [[tree|0MJ0JS72DZ]].

## Remarks

A graph has a spanning tree if and only if it is connected. The node [[every connected graph has a spanning tree|04XPQT2XTC]] gives one direction.
A spanning tree of a finite connected graph with $n$ vertices has $n - 1$ edges.
The complete graph $K_n$ has $n^{n-2}$ spanning trees, by [[Cayley's formula|0M2SSGSYPE]]. A [[minimum spanning tree|0Z367CGNRM]] has the least total weight.

## Explanations

### L1 human

A spanning tree keeps all the vertices of a graph and just enough edges to stay connected without any loop.
For a road network, it is a smallest set of roads that still links all towns.
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# Convergence of random variables

## Summary

A sequence of random variables can converge to a limit in several different senses.
The main modes are [[almost sure convergence|04P7D63W8P]], [[convergence in probability|0EFDDXTVD6]], [[convergence in mean|0TTA57SEJG]] and [[convergence in distribution|0JDAQHWA9N]].
Almost sure convergence and convergence in mean each imply convergence in probability, and convergence in probability implies convergence in distribution. The converse implications are false in general.

## Remarks

Convergence in distribution concerns only the laws of the random variables. So the random variables can be on different probability spaces.
The law of large numbers is a statement about convergence in probability or almost surely. The central limit theorem is a statement about convergence in distribution.

## Explanations

### L1 human

For numbers, there is one meaning of "$x_n \to x$". For random variables, there are several, because a random variable has many values at once.
Some modes ask that the values themselves come close, and the weakest mode asks only that the probabilities come close.
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<document source="https://tamnd.github.io/math2/node/0FQPYAY9PJ.md" tag="0FQPYAY9PJ" kind="definition">
# Meagre set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $A \subseteq X$.
The set $A$ is meagre in $X$ when there is a [[countable|0ZG61TFBES]] family $(A_n)_{n \in \mathbb{N}}$ of [[nowhere dense|0KSM23SN3H]] subsets of $X$ with $A = \bigcup_{n} A_n$.

## Remarks

Another name for a meagre set is a set of first category. A set that is not meagre is of second category.
Each subset of a meagre set is meagre, and each countable union of meagre sets is meagre.
The set $\mathbb{Q}$ is meagre in $\mathbb{R}$, because it is a countable union of one-point sets.
By the [[Baire category theorem|0A7KGAYHGV]], $\mathbb{R}$ itself is not meagre in $\mathbb{R}$.

## Explanations

### L1 human

A meagre set is made of countably many thin pieces.
It can still be dense, as $\mathbb{Q}$ is, but in a complete metric space it can never fill an open set.
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# Inverse of a complex number

## Statement

Let $z$ be a complex number with $z \neq 0$.
Then
$$z^{-1} = \frac{\bar{z}}{|z|^2}.$$
With $z = a + b i$ and $a, b$ real, this is $z^{-1} = \dfrac{a}{a^2 + b^2} - \dfrac{b}{a^2 + b^2} \, i$.

## Remarks

Proof: $z \bar{z} = |z|^2$ by the formula for [[a complex number times its conjugate|0ZB7J071WQ]], and $|z|^2 > 0$ because $z \neq 0$. Divide by $|z|^2$.
To compute a quotient $w / z$, multiply the numerator and the denominator by $\bar{z}$.

## Explanations

### L1 human

To divide by a complex number, multiply above and below by its conjugate. Then the denominator becomes a positive real number.
For example, $\dfrac{1}{1 + i} = \dfrac{1 - i}{(1 + i)(1 - i)} = \dfrac{1 - i}{2}$.
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<document source="https://tamnd.github.io/math2/node/0FQSDC2EQ2.md" tag="0FQSDC2EQ2" kind="concept">
# Convex hull

## Summary

The convex hull of a set of points is the smallest convex set that contains it. It is the set of all convex combinations of points of the set.
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<document source="https://tamnd.github.io/math2/node/0FR1BDKK10.md" tag="0FR1BDKK10" kind="statement">
# Comparison test for improper integrals

## Statement

Let $f, g \colon [a, \infty) \to \mathbb{R}$ be Riemann integrable on each $[a, b]$, with $0 \le f(x) \le g(x)$ for all $x \ge a$.
If $\int_a^\infty g(x) \, dx$ converges, then $\int_a^\infty f(x) \, dx$ converges, and
$$\int_a^\infty f(x) \, dx \le \int_a^\infty g(x) \, dx.$$
If $\int_a^\infty f(x) \, dx$ diverges, then $\int_a^\infty g(x) \, dx$ diverges.

## Remarks

The function $b \mapsto \int_a^b f$ is increasing, and it is bounded by $\int_a^\infty g$ by [[monotonicity|0MBWQKM8QR]]. A [[bounded monotone function has a limit|0KJXHN0TW0]].
For example, $0 \le e^{-x^2} \le e^{-x}$ for $x \ge 1$. So $\int_1^\infty e^{-x^2} \, dx$ converges.
The same test holds for integrals that are improper at a finite endpoint.

## Explanations

### L1 human

If a larger function has finite area, a smaller nonnegative function also has finite area.
This lets us prove convergence without computing the integral.
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# Heron's formula

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane, with the side lengths $a$, $b$ and $c$. Let $s = \frac{1}{2}(a + b + c)$ be its semiperimeter.
Then the [[area|065JWMZW86]] $T$ of the triangle is
$$T = \sqrt{s (s - a)(s - b)(s - c)}.$$

## Remarks

An equivalent form without $s$ is $16 T^2 = (a + b + c)(-a + b + c)(a - b + c)(a + b - c)$.
For example, the triangle with the sides $4$, $13$ and $15$ has $s = 16$ and $T = \sqrt{16 \cdot 12 \cdot 3 \cdot 1} = 24$.
One proof uses the [[law of cosines|060ZVZBV3E]] to get $\cos \gamma$ from the sides, and then $T = \frac{1}{2} a b \sin \gamma$.

## History

The formula carries the name of Heron of Alexandria, who gave a proof in his Metrica in the 1st century. It was possibly known earlier. Qin Jiushao published an equivalent formula in 1247.

## Explanations

### L1 human

Heron's formula gives the area of a triangle when you know only its three sides. You do not need a height or an angle. For the right triangle with the sides $3$, $4$, $5$: $s = 6$, and $\sqrt{6 \cdot 3 \cdot 2 \cdot 1} = 6$, which is also $\frac{1}{2} \cdot 3 \cdot 4$.
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<document source="https://tamnd.github.io/math2/node/0FREPFKNS7.md" tag="0FREPFKNS7" kind="concept">
# Maximum likelihood estimation

## Summary

Maximum likelihood estimation is a method to build [[estimators|0TTNRP5PFQ]] of unknown parameters.
For observed data $x$, the maximum likelihood estimate is a value $\hat{\theta}$ that maximizes the [[likelihood function|05FSTXM92W]] $\theta \mapsto L(\theta \mid x)$ over the parameter set $\Theta$.
In other words, it is the parameter value under which the observed data are most probable.

## Remarks

The maximizer of $L$ is also the maximizer of the log-likelihood $\ell = \ln L$. For a smooth $\ell$ and an interior maximum, $\hat{\theta}$ solves the likelihood equations $\partial \ell / \partial \theta_j = 0$.
Example: for $k$ successes in $n$ Bernoulli($p$) trials, $\ell(p) = k \ln p + (n - k) \ln(1 - p)$, so $\hat{p} = k/n$.
Example: for an i.i.d. $\mathcal{N}(\mu, \sigma^2)$ sample, $\hat{\mu} = \bar{x}$ and $\hat{\sigma}^2 = \frac{1}{n}\sum_i (x_i - \bar{x})^2$. This estimator of $\sigma^2$ is biased.
Under regularity conditions, the estimator is [[consistent|0XJVF81QF6]] and asymptotically normal. Also, for a function $g$, the maximum likelihood estimate of $g(\theta)$ is $g(\hat{\theta})$.

## Explanations

### L1 human

Maximum likelihood chooses the parameter that makes the observed data most probable.
If a coin shows $7$ heads in $10$ tosses, the maximum likelihood estimate of the probability of heads is $7/10$.
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# Scalar matrix

## Definition

Let $R$ be a ring, let $n \ge 0$ be an integer, and let $c \in R$.
The *scalar matrix* of order $n$ with value $c$ is the [[diagonal matrix|044JNC7ZAA]] $\operatorname{diag}(c, \dots, c) = c I_n$, where $I_n$ is the [[identity matrix|06DW9BAWB1]].
A square matrix is a *scalar matrix* when it equals $c I_n$ for some $c \in R$.

## Remarks

Over a commutative ring, a scalar matrix commutes with every square matrix of the same order.
Over a field, the scalar matrices are exactly the $n \times n$ matrices that commute with every $n \times n$ matrix.
As a linear map, $c I_n$ multiplies each vector by $c$.

## Explanations

### L1 human

A scalar matrix has the same number at every place on the diagonal and zeros elsewhere.
Multiplying by it is the same as multiplying by that number.
For example, $3 I_2 = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$ triples each vector.
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# Sequential compactness

## Summary

A space or a set is sequentially compact when each sequence in it has a subsequence that converges to a point of it.
In metric spaces, sequential compactness is the same as compactness.
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<document source="https://tamnd.github.io/math2/node/0FRSR10DWF.md" tag="0FRSR10DWF" kind="statement">
# The complex numbers form a field

## Statement

The [[complex numbers|09FTAET5Z4]] $\mathbb{C}$, with their addition and multiplication, form a field.
That is, $\mathbb{C}$ is a commutative ring with $0 \neq 1$, and each complex number $z \neq 0$ has a multiplicative inverse $z^{-1}$ with $z z^{-1} = 1$.

## Remarks

The [[inverse of a complex number|0FQRAV5SKH]] $a + b i \neq 0$ is $\dfrac{a - b i}{a^2 + b^2}$.
The field $\mathbb{C}$ contains $\mathbb{R}$ as a subfield. It has dimension $2$ as a vector space over $\mathbb{R}$, with basis $1, i$.
The field $\mathbb{C}$ is algebraically closed, but it is not an ordered field.

## Explanations

### L1 human

You can add, subtract, multiply and divide complex numbers, as with real numbers.
The only exception is division by $0$.
For example, $\dfrac{1}{i} = -i$, because $i \cdot (-i) = -i^2 = 1$.
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# Grönwall bound

## Definition

Let $\delta, K, \varepsilon$ be real numbers.
The Grönwall bound with these parameters is the function $G_{\delta, K, \varepsilon} : \mathbb{R} \to \mathbb{R}$ given by
$$G_{\delta, K, \varepsilon}(x) = \begin{cases} \delta\, e^{K x} + \dfrac{\varepsilon}{K} \left( e^{K x} - 1 \right) & \text{if } K \neq 0, \\ \delta + \varepsilon x & \text{if } K = 0. \end{cases}$$

## Remarks

The function $v = G_{\delta, K, \varepsilon}$ is the unique solution of $v' = K v + \varepsilon$, $v(0) = \delta$.
For $\varepsilon = 0$, it is $\delta e^{K x}$.
A form of [[Grönwall's inequality|07EJHQYR9A]] uses it: let $f : [a, b] \to \mathbb{R}^n$ be continuous and differentiable on $(a, b)$, with $\|f(a)\| \le \delta$ and $\|f'(x)\| \le K \|f(x)\| + \varepsilon$ on $(a, b)$. Then $\|f(x)\| \le G_{\delta, K, \varepsilon}(x - a)$ on $[a, b]$.
The [[estimate for approximate solutions|0JTPMCCZT2]] is stated with this bound.

## Explanations

### L1 human

Suppose that a quantity starts at most at $\delta$ and grows at most at rate $K$ times itself plus $\varepsilon$. The Grönwall bound is the largest value that it can reach.
It is the solution of the extreme case, where both inequalities are equalities.
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# Total order

## Summary

A total order is a partial order in which any two elements are comparable.
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<document source="https://tamnd.github.io/math2/node/0FRYPZ6D0X.md" tag="0FRYPZ6D0X" kind="statement">
# Trace is a coefficient of the characteristic polynomial

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $A$ be an $n \times n$ matrix over $K$.
Then the coefficient of $t^{n-1}$ in the [[characteristic polynomial|0VSSPAFM2C]] $p_A(t) = \det(t I_n - A)$ is $-\tr A$, where $\tr$ is the [[trace|0XC1RDBY7A]].
So $p_A(t) = t^n - (\tr A)\, t^{n-1} + \cdots$.

## Remarks

In the Leibniz expansion of $\det(t I_n - A)$, only the product of the diagonal entries $(t - a_{11}) \cdots (t - a_{nn})$ contributes to $t^{n-1}$.
For $n = 2$, $p_A(t) = t^2 - (\tr A) t + \det A$.
Together with [[the constant coefficient|0T0X8GTYYS]], it shows that $\tr$ and $\det$ are invariants of similarity.

## Explanations

### L1 human

The characteristic polynomial contains the trace as its second coefficient, with a minus sign.
For a $2 \times 2$ matrix, the full polynomial is $t^2 - (\text{trace})\,t + \text{determinant}$.
So the trace and the determinant are part of the information of the characteristic polynomial.
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<document source="https://tamnd.github.io/math2/node/0FS3RHF6KE.md" tag="0FS3RHF6KE" kind="statement">
# Infinite pigeonhole principle

## Statement

Let $A$ be an infinite set, let $B$ be a finite set, and let $f : A \to B$ be a function.
Then there is an element $b \in B$ such that the fibre $f^{-1}(b)$ is infinite.

## Remarks

The proof is short. If each of the finitely many fibres were finite, then $A$, their union, would be finite.
Example: in any infinite sequence of integers, infinitely many terms have the same remainder modulo $10$.
The statement is the starting point of the infinite form of [[Ramsey's theorem|00TQV3F5S4]].

## Explanations

### L1 human

If you put infinitely many objects into finitely many boxes, at least one box gets infinitely many of them.
For example, colour each natural number red or blue. Then one of the two colours is used infinitely often.
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# Determinant of a matrix with a zero row

## Statement

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$.
If one row of $A$ is zero, then $\det A = 0$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This follows from linearity in that row: the zero row is $0$ times itself, so $\det A = 0 \cdot \det A = 0$.
The same holds for a zero column.

## Explanations

### L1 human

A zero row means that one edge of the box spanned by the rows has length zero.
Such a box has no volume.
So the determinant is zero.
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<document source="https://tamnd.github.io/math2/node/0FS972T0ZH.md" tag="0FS972T0ZH" kind="statement">
# Bertrand's ballot theorem

## Statement

Let $p$ and $q$ be natural numbers with $p > q$.
In an election, candidate $A$ gets $p$ votes and candidate $B$ gets $q$ votes.
The votes are counted one at a time, in an order chosen uniformly at random among all orders.
Then the probability that $A$ is strictly ahead of $B$ after each vote is counted is
$$\frac{p - q}{p + q}.$$

## Remarks

Equivalently, the number of such counting orders, as words with $p$ letters $A$ and $q$ letters $B$, is $\frac{p - q}{p + q} \binom{p + q}{p}$.
Example: for $p = 3$ and $q = 1$, the probability is $\frac{1}{2}$. Of the $4$ orders, $AAAB$ and $AABA$ work.
A short proof uses the reflection principle for [[lattice paths|040WY0TT2G]].

## History

William Allen Whitworth published the result in 1878.
Joseph Bertrand found it again in 1887, and the theorem has his name.

## Explanations

### L1 human

If the winner gets many more votes than the loser, the winner is likely to lead the whole time.
The formula makes this precise: the chance of an unbroken lead is the margin divided by the total number of votes.
</document>

<document source="https://tamnd.github.io/math2/node/0FSCXYG856.md" tag="0FSCXYG856" kind="statement">
# Reversing a path changes the sign of a line integral

## Statement

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a piecewise $C^1$ curve with image $C$, and let $\gamma^- \colon [a, b] \to \mathbb{R}^n$, $\gamma^-(t) = \gamma(a + b - t)$, be the reversed curve.
For each continuous vector field $F$ on $C$ and each continuous function $f$ on $C$,
$$\int_{\gamma^-} F \cdot d\mathbf{r} = -\int_\gamma F \cdot d\mathbf{r}, \qquad \int_{\gamma^-} f \, ds = \int_\gamma f \, ds.$$

## Remarks

Proof: by the [[parametric formula|0W3W1VYWBP]], $(\gamma^-)'(t) = -\gamma'(a + b - t)$. The substitution $u = a + b - t$ then gives both identities.
The [[line integral of a vector field|0YS8RX87F4]] depends on the orientation of the curve. The [[line integral of a scalar field|0M50E4WYWJ]] does not, because it uses only $\|\gamma'\|$.
More generally, under a $C^1$ change of parameter $\varphi$ with $\varphi' > 0$, the vector line integral does not change. With $\varphi' < 0$ it changes sign.

## Explanations

### L1 human

If a force helps you along a path, it works against you on the way back.
So the work, a line integral of a vector field, changes sign when you reverse the path. The length of the path and the mass of a wire do not.
</document>

<document source="https://tamnd.github.io/math2/node/0FSJA9M43W.md" tag="0FSJA9M43W" kind="statement">
# Multiplicativity of the totient function

## Statement

Let $m$ and $n$ be positive integers that are [[coprime|0GFKP8F8AG]].
Then $\varphi(m n) = \varphi(m) \varphi(n)$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

The proof uses the [[Chinese remainder theorem|0FNN9B4W64]]. It gives a bijection between the residues modulo $m n$ and the pairs of residues modulo $m$ and modulo $n$. A residue is coprime to $m n$ if and only if both parts are coprime to their moduli.
The hypothesis is necessary: $\varphi(4) = 2$, but $\varphi(2) \varphi(2) = 1$.
So $\varphi$ is a [[multiplicative function|0AP80TT660]].
For example, $\varphi(36) = \varphi(4) \varphi(9) = 2 \cdot 6 = 12$.

## Explanations

### L1 human

When $m$ and $n$ have no common factor, a number is coprime to $m n$ exactly when it is coprime to $m$ and to $n$.
The two conditions are independent, so the counts multiply.
For example, $\varphi(15) = \varphi(3) \varphi(5) = 2 \cdot 4 = 8$.
</document>

<document source="https://tamnd.github.io/math2/node/0FSNQ0GNCC.md" tag="0FSNQ0GNCC" kind="definition">
# Identity function

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]].
The identity function of $A$ is the [[function|0XHDTE5YMZ]] $\mathrm{id}_A \colon A \to A$ with $\mathrm{id}_A(a) = a$ for each $a \in A$.

## Remarks

For each function $f \colon A \to B$, $f \circ \mathrm{id}_A = f$ and $\mathrm{id}_B \circ f = f$. See [[function composition|07HF9YCWPP]].
The identity function is a [[bijection|0SNQEF7Z53]], and it is its own inverse.
Its graph is the diagonal $\{(a, a) \mid a \in A\}$.

## Explanations

### L1 human

The identity function does nothing: each input comes out unchanged.
It plays the role for composition that the number $1$ plays for multiplication.
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# Complex conjugate

## Summary

The complex conjugate of a complex number has the same real part and the opposite imaginary part.
The conjugate of $a + b i$ is $a - b i$. In the [[complex plane|0CTE7A31XQ]], it is the reflection of the point in the real axis.
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<document source="https://tamnd.github.io/math2/node/0FWBQ53MPJ.md" tag="0FWBQ53MPJ" kind="concept">
# Choice function

## Summary

A choice function on a collection of nonempty sets picks one element from each set.
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<document source="https://tamnd.github.io/math2/node/0FWJ0VW561.md" tag="0FWJ0VW561" kind="definition">
# Orthonormal basis

## Definition

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$.
An *orthonormal basis* of $V$ is a [[basis|0KCQ9Y3QKA]] $(e_1, \dots, e_n)$ of $V$ that is an [[orthonormal family|0RW9TNNT87]].

## Remarks

An orthonormal family of $\dim V$ vectors is an orthonormal basis.
In an orthonormal basis, coordinates are inner products: $v = \sum_i \langle v, e_i \rangle e_i$.
The vectors $\frac{1}{\sqrt 2}(1, 1)$ and $\frac{1}{\sqrt 2}(1, -1)$ form an orthonormal basis of $\R^2$.
In infinite-dimensional Hilbert spaces, the term means a maximal orthonormal family, which is in general not a basis in the algebraic sense.

## Explanations

### L1 human

An orthonormal basis is a set of perpendicular unit vectors that spans the space.
Each coordinate of a vector is its inner product with one basis vector.
No linear system needs to be solved to find coordinates.
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<document source="https://tamnd.github.io/math2/node/0FWVRHBRWM.md" tag="0FWVRHBRWM" kind="concept">
# Transposition

## Summary

A transposition is a permutation that swaps two elements and fixes all other elements.
In cycle notation it is a $2$-cycle $(a\ b)$.
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<document source="https://tamnd.github.io/math2/node/0FXDW992DS.md" tag="0FXDW992DS" kind="concept">
# Initial value problem

## Summary

An initial value problem is an [[ordinary differential equation|0NT64JV96A]] together with a condition that fixes the value of the unknown function at one point.
A solution must satisfy both the equation and the condition. See the [[definition|075VBY4K4X]].
</document>

<document source="https://tamnd.github.io/math2/node/0FXJ7YFMMX.md" tag="0FXJ7YFMMX" kind="concept">
# Solvable group

## Summary

A solvable group is a group with a subnormal series whose factor groups are all abelian.
Equivalently, its derived series reaches the trivial subgroup.
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<document source="https://tamnd.github.io/math2/node/0FY42N02DV.md" tag="0FY42N02DV" kind="concept">
# Algebraic multiplicity

## Summary

The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/0FYQMKFE77.md" tag="0FYQMKFE77" kind="concept">
# Order of quantifiers

## Summary

In a statement with several quantifiers, the order of different quantifiers changes the meaning.
For a [[predicate|0T3DPBG981]] $P(x, y)$, the statement $\exists y \, \forall x \, P(x, y)$ implies $\forall x \, \exists y \, P(x, y)$, but the converse is false in general.
Two quantifiers of the same type commute: $\forall x \, \forall y \, P(x, y)$ is equivalent to $\forall y \, \forall x \, P(x, y)$, and the same holds for $\exists$.

## Remarks

On the integers, $\forall x \, \exists y \, (y > x)$ is true, but $\exists y \, \forall x \, (y > x)$ is false.
In $\forall x \, \exists y$, the choice of $y$ can depend on $x$.
The definitions of continuity and of uniform continuity differ only in the order of quantifiers.

## Explanations

### L1 human

"Every person has a mother" is $\forall x \, \exists y$: each person has their own mother.
"Some person is the mother of every person" is $\exists y \, \forall x$: one mother for all.
The first sentence is true, and the second is false.
</document>

<document source="https://tamnd.github.io/math2/node/0FYX44GP92.md" tag="0FYX44GP92" kind="concept">
# Recurrence relation

## Summary

A recurrence relation defines each term of a sequence from earlier terms. Recurrences are a main tool to describe counting sequences, and solving one means finding a closed formula for its terms. The node [[recurrence relation|07TZV87Y0T]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0FZFHK97P7.md" tag="0FZFHK97P7" kind="statement">
# Second derivative test

## Statement

Let $f$ be a real function differentiable on an open interval around $c$, with $f'(c) = 0$.
Assume that the [[second derivative|0Y9D2FJRQ4]] $f''(c)$ exists.
If $f''(c) > 0$, then $f$ has a strict [[local minimum|0NS8BP96QX]] at $c$. If $f''(c) < 0$, then $f$ has a strict local maximum at $c$.

## Remarks

If $f''(c) > 0$, then $f'(x)/(x - c) \to f''(c)$. So $f'$ is negative just left of $c$ and positive just right of $c$, and the [[first derivative test|01A2EGXJEM]] applies.
If $f''(c) = 0$, the test gives no answer. The functions $x^4$, $-x^4$, and $x^3$ at $0$ show all three cases.
Mathlib states the minimum case as `isLocalMin_of_deriv_deriv_pos`.

## Explanations

### L1 human

At a stationary point where the graph bends upward, the point is the bottom of a valley.
Where it bends downward, it is a hilltop.
</document>

<document source="https://tamnd.github.io/math2/node/0G01TS1RRP.md" tag="0G01TS1RRP" kind="definition">
# Deficient number

## Definition

A positive integer $n$ is deficient if the sum of its positive divisors other than $n$ is less than $n$.
Equivalently, $\sigma(n) < 2n$, where $\sigma$ is the [[sum of divisors|009QCP5T3Z]].

## Remarks

Each prime $p$ is deficient, since its only smaller divisor is $1$.
More generally, each prime power is deficient. For example, $1 + 2 + 4 = 7 < 8$.
Each divisor of a deficient number or of a [[perfect number|0SB7Q8SYSF]], other than the number itself, is deficient.

## Explanations

### L1 human

A deficient number has "too few" divisors: its smaller divisors add up to less than the number.
Most small numbers are deficient, for example $1, 2, 3, 4, 5, 7, 8, 9, 10$.
</document>

<document source="https://tamnd.github.io/math2/node/0G0ASKNA6Z.md" tag="0G0ASKNA6Z" kind="concept">
# Cardinality

## Summary

The cardinality of a set is a measure of its size, the number of its elements.
Two sets have the same cardinality exactly when they are equinumerous.
</document>

<document source="https://tamnd.github.io/math2/node/0G0B3DRG65.md" tag="0G0B3DRG65" kind="example">
# Cyclic group of order n

## Example

Let $n$ be a positive integer.
The integers modulo $n$, $\mathbb{Z}/n\mathbb{Z} = \{0, 1, \ldots, n - 1\}$, with addition modulo $n$, form a [[cyclic group|0KVGH0J5B3]] of [[order|0B85S2V2DR]] $n$.
The residue $1$ generates it, because each residue $k$ is the sum of $k$ copies of $1$.
Each cyclic group of order $n$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$, and this group is often written $C_n$.

## Remarks

A residue $k$ generates $\mathbb{Z}/n\mathbb{Z}$ exactly when $\gcd(k, n) = 1$.
The [[group of $n$th roots of unity|0WX11EA3F3]] is a cyclic group of order $n$ in multiplicative notation.
The rotations of a regular $n$-gon also form a cyclic group of order $n$.

## Explanations

### L1 human

Think of a clock with $n$ hours, where you add hours and wrap around after $n$.
This clock arithmetic gives the cyclic group of order $n$.
Every cyclic group of order $n$ looks the same as this clock, up to the names of the elements.
</document>

<document source="https://tamnd.github.io/math2/node/0G0F7FGCGP.md" tag="0G0F7FGCGP" kind="statement">
# Transcendence of pi

## Statement

The number $\pi$ is [[transcendental|00SW8B91E7]] over $\mathbb{Q}$.

## Remarks

The statement follows from the Lindemann–Weierstrass theorem: $e^\alpha$ is transcendental for every nonzero algebraic number $\alpha$.
Indeed, if $\pi$ were algebraic, then $i \pi$ would be algebraic and nonzero, but $e^{i \pi} = -1$ is algebraic.
So $\sqrt{\pi}$ is also transcendental, and it is not [[constructible|0FMN2MN7TM]].

## History

Charles Hermite proved the transcendence of $e$ in 1873.
Ferdinand von Lindemann proved in 1882 that $e^\alpha$ is transcendental for every nonzero algebraic $\alpha$, and so that $\pi$ is transcendental.
Karl Weierstrass proved a more general statement in 1885.

## Explanations

### L1 human

The number $\pi$ is not a root of any nonzero polynomial with rational coefficients.
This is much stronger than the irrationality of $\pi$, and the proof is much harder.
</document>

<document source="https://tamnd.github.io/math2/node/0G0RFPG8HR.md" tag="0G0RFPG8HR" kind="concept">
# Eigenspace

## Summary

The eigenspace of a linear map for a scalar $\lambda$ is the kernel of the map minus $\lambda$ times the identity.
It consists of the eigenvectors for $\lambda$ together with zero.
</document>

<document source="https://tamnd.github.io/math2/node/0G1022YXVQ.md" tag="0G1022YXVQ" kind="concept">
# Galton–Watson process

## Summary

A Galton–Watson process models a population. Each individual has a random number of children, independently of the others and with one common law.
The node [[Galton–Watson process|0H631CZP5K]] gives the precise definition.

## Remarks

The process comes from the study of the extinction of family names by Francis Galton.
</document>

<document source="https://tamnd.github.io/math2/node/0G24VQ1088.md" tag="0G24VQ1088" kind="statement">
# Sylvester's determinant theorem

## Statement

Let $K$ be a commutative ring, let $A$ be an $m \times n$ matrix and $B$ an $n \times m$ matrix over $K$.
Then
$$\det(I_m + AB) = \det(I_n + BA),$$
where $I_m$ and $I_n$ are [[identity matrices|06DW9BAWB1]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The two sides are determinants of different sizes, $m$ and $n$.
For a column $u$ and a row $v^T$, it gives $\det(I_m + u v^T) = 1 + v^T u$.
The statement is also called the Weinstein–Aronszajn identity.

## Explanations

### L1 human

The products $AB$ and $BA$ can have very different sizes.
Still, adding the identity to them gives the same determinant.
This can turn a large determinant into a small one.
</document>

<document source="https://tamnd.github.io/math2/node/0G2YJN0H5S.md" tag="0G2YJN0H5S" kind="statement">
# The complex exponential is entire

## Statement

The [[complex exponential|054VXHMCA3]] $\exp : \mathbb{C} \to \mathbb{C}$ is an [[entire function|0419794R0T]]: it is complex differentiable at each $z \in \mathbb{C}$.

## Remarks

Proof: $\exp$ is given by a power series with infinite radius of convergence, and [[power series are holomorphic in their disc of convergence|0YMP521ZR5]].
Its [[derivative|019H1Z6419]] is $\exp$ itself.

## Explanations

### L1 human

The exponential function has a complex derivative at every point of the plane.
So it is a holomorphic function without any exceptional points.
</document>

<document source="https://tamnd.github.io/math2/node/0G31VJ5YS5.md" tag="0G31VJ5YS5" kind="definition">
# Tensor product of vector spaces

## Definition

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$.
Let $L$ be the vector space over $F$ with basis the set $V \times W$; write $[v, w]$ for the basis vector of $(v, w)$.
Let $N$ be the subspace of $L$ spanned by all elements of these forms, for $v, v' \in V$, $w, w' \in W$ and $a \in F$:
$$[v + v', w] - [v, w] - [v', w], \qquad [v, w + w'] - [v, w] - [v, w'],$$
$$[av, w] - a[v, w], \qquad [v, aw] - a[v, w].$$
The *tensor product* of $V$ and $W$ is the quotient space $V \otimes W = L / N$.
For $v \in V$ and $w \in W$, the class of $[v, w]$ is written $v \otimes w$.

## Remarks

The map $(v, w) \mapsto v \otimes w$ is bilinear, and the vectors $v \otimes w$ span $V \otimes W$.
Not every vector of $V \otimes W$ is of the form $v \otimes w$; for example, $e_1 \otimes e_1 + e_2 \otimes e_2$ in $F^2 \otimes F^2$ is not.
The tensor product is characterised by its [[universal property|0YDRV01R8F]].
For finite-dimensional spaces, $V^* \otimes W$ is isomorphic to the space of linear maps from $V$ to $W$.

## Explanations

### L1 human

The tensor product combines two vector spaces into one space of formal products $v \otimes w$.
These products follow the rules of bilinearity and nothing more.
It turns bilinear maps into linear maps, which are easier to study.
</document>

<document source="https://tamnd.github.io/math2/node/0G3DBXRRQJ.md" tag="0G3DBXRRQJ" kind="statement">
# Continuous dependence on initial data

## Statement

Let $J$ be an interval, let $U$ be an open subset of $\mathbb{R}^n$, let $K \ge 0$, and let $f : J \times U \to \mathbb{R}^n$ be continuous.
Assume that $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ for all $t \in J$ and $x, y \in U$.
Let $a < b$ with $[a, b] \subseteq J$, and let $\gamma_1, \gamma_2 : [a, b] \to U$ be [[solutions|04SW7WECS8]] of $x' = f(t, x)$.
Then
$$\|\gamma_1(t) - \gamma_2(t)\| \le \|\gamma_1(a) - \gamma_2(a)\|\, e^{K (t - a)} \quad \text{for each } t \in [a, b].$$

## Remarks

So the solution at time $t$ is a Lipschitz function of the initial value, with constant $e^{K(t - a)}$.
The proof applies [[Grönwall's inequality|07EJHQYR9A]] to $u(t) = \|\gamma_1(t) - \gamma_2(t)\|$.
Existence, uniqueness and continuous dependence together make the initial value problem well posed, in the sense of Jacques Hadamard.
For equal initial values, the bound gives [[uniqueness|025PKW8AMR]] again.

## Explanations

### L1 human

If two solutions start close together, they stay close for a while.
Their distance can grow, but at most exponentially, with rate given by the Lipschitz constant.
So a small error in the initial value gives a small error in the solution over a bounded time.
</document>

<document source="https://tamnd.github.io/math2/node/0G3RRM41X9.md" tag="0G3RRM41X9" kind="concept">
# Dedekind cut

## Summary

A Dedekind cut splits the rational numbers into a lower part and an upper part.
Each real number corresponds to exactly one cut, so the cuts give a construction of the [[real numbers|08J2M1JYAG]] from the [[rational numbers|0NCJ5WXN6Z]].
</document>

<document source="https://tamnd.github.io/math2/node/0G48PYM96Q.md" tag="0G48PYM96Q" kind="concept">
# Markov chain

## Summary

A Markov chain is a random process in which the next state depends only on the present state, and not on the earlier history.
The node [[Markov chain|033TJBZDG0]] gives the precise definition in discrete time with a countable state space.
</document>

<document source="https://tamnd.github.io/math2/node/0G4C8SJ6PV.md" tag="0G4C8SJ6PV" kind="statement">
# Automorphism group of a cyclic group

## Statement

Let $n$ be a positive integer and let $G$ be a [[cyclic group|0KVGH0J5B3]] of order $n$.
For each integer $a$ with $\gcd(a, n) = 1$, the map $\sigma_a \colon x \mapsto x^a$ is an automorphism of $G$.
The map $a \bmod n \mapsto \sigma_a$ is an [[isomorphism|08MG7H8JW5]] from the [[group of units|0FJ90Y9926]] $(\mathbb{Z}/n\mathbb{Z})^\times$ to $\operatorname{Aut}(G)$.

## Remarks

An automorphism of $G = \langle g \rangle$ is determined by the image of $g$, and this image must be a generator $g^a$ with $\gcd(a, n) = 1$.
Thus $|\operatorname{Aut}(G)| = \varphi(n)$, where $\varphi$ is Euler's totient function.
For the infinite cyclic group, $\operatorname{Aut}(\mathbb{Z}) \cong \{1, -1\}$.

## Explanations

### L1 human

An automorphism of a cyclic group must send a generator to a generator.
Each choice of a generator gives exactly one automorphism.
For $\mathbb{Z}/5\mathbb{Z}$, the generators are $1, 2, 3, 4$, so there are four automorphisms, given by multiplication by $1, 2, 3, 4$.
</document>

<document source="https://tamnd.github.io/math2/node/0G4D3ZKTP8.md" tag="0G4D3ZKTP8" kind="example">
# Sum of the reciprocals of the factorials

## Example

The series
$$\sum_{n=0}^{\infty} \frac{1}{n!} = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \cdots$$
converges, and its sum is the number [[e|0MVCFMW7S4]] $\approx 2.71828$.

## Remarks

The series converges by the [[ratio test|0QNY6ZBXPN]], since the ratio of consecutive terms is $1/(n+1) \to 0$.
It is the value at $x = 1$ of the power series $\sum x^n / n!$ of the [[exponential function|0RMVS7RXZX]].
The series converges fast. The tail after the term $1/N!$ is less than $1/(N! \, N)$.

## Explanations

### L1 human

The terms shrink very fast, because $n!$ grows fast.
Only a few terms give a good value of $e$: the first five terms already give $2.708$.
</document>

<document source="https://tamnd.github.io/math2/node/0G4F0XB9X2.md" tag="0G4F0XB9X2" kind="concept">
# Entire function

## Summary

An entire function is a function that is holomorphic on the whole complex plane, see the [[definition|0419794R0T]].
Polynomials and the exponential function are entire.
</document>

<document source="https://tamnd.github.io/math2/node/0G4J7Z7WPJ.md" tag="0G4J7Z7WPJ" kind="definition">
# Completion of a metric space

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]].
A completion of $X$ is a pair $(\hat{X}, \iota)$ of a [[complete metric space|08EWT7J0SC]] $\hat{X}$ and an [[isometry|0878DN3WCS]] $\iota \colon X \to \hat{X}$ with [[dense|0D198GPMTJ]] image $\iota(X)$.

## Remarks

Each metric space has a completion. One construction takes the Cauchy sequences of $X$ and identifies two of them when the distance between their terms tends to $0$.
The completion is unique: for two completions $(\hat{X}, \iota)$ and $(\hat{X}', \iota')$, there is a unique isometric isomorphism $\varphi \colon \hat{X} \to \hat{X}'$ with $\varphi \circ \iota = \iota'$.
The completion of $\mathbb{Q}$ with the usual metric is $\mathbb{R}$. This is the [[construction of the reals from Cauchy sequences|0D8HE681SC]].
If $X$ is already complete, then $X$ with the identity map is a completion of $X$.

## Explanations

### L1 human

The completion fills the holes of a metric space with new points, and it adds no more points than it needs.
Each new point is a limit of a Cauchy sequence of old points.
</document>

<document source="https://tamnd.github.io/math2/node/0G4MFRQSGN.md" tag="0G4MFRQSGN" kind="definition">
# Cylindrical coordinates

## Definition

The cylindrical coordinate map is $C \colon [0, \infty) \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}^3$,
$$C(r, \theta, z) = (r \cos \theta, \ r \sin \theta, \ z).$$
A triple $(r, \theta, z)$ with $C(r, \theta, z) = (x, y, z)$ is a triple of cylindrical coordinates of $(x, y, z)$.
Here $r = \sqrt{x^2 + y^2}$ is the distance to the $z$-axis, $\theta$ is the azimuth, and $z$ is the height.

## Remarks

The pair $(r, \theta)$ is a pair of [[polar coordinates|0RJZV05MJA]] of $(x, y)$. So, away from the $z$-axis, the coordinates are unique if $r > 0$ and $\theta \in [0, 2\pi)$.
The Jacobian determinant of $C$ is $r$, so the volume element is $dV = r \, dr \, d\theta \, dz$.
Example: the solid cylinder $x^2 + y^2 \le a^2$, $0 \le z \le h$ has volume $\int_0^h \int_0^{2\pi} \int_0^a r \, dr \, d\theta \, dz = \pi a^2 h$.
Some texts write $\rho$ or $s$ for $r$, and $\varphi$ for $\theta$.

## Explanations

### L1 human

Cylindrical coordinates are polar coordinates in the floor plan, together with the height.
Surfaces $r = $ constant are cylinders around the $z$-axis, which gives the coordinates their name.
</document>

<document source="https://tamnd.github.io/math2/node/0G4W89YJRK.md" tag="0G4W89YJRK" kind="definition">
# Oriented angle

## Definition

Let $u = (u_1, u_2)$ and $v = (v_1, v_2)$ be nonzero vectors in the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$. Let $u \cdot v = u_1 v_1 + u_2 v_2$ and $\det(u, v) = u_1 v_2 - u_2 v_1$.
The oriented angle from $u$ to $v$ is the class modulo $2\pi$ of each real number $\theta$ with
$$\cos \theta = \frac{u \cdot v}{\|u\| \, \|v\|} \quad \text{and} \quad \sin \theta = \frac{\det(u, v)}{\|u\| \, \|v\|}.$$
For points $A$, $B$ and $C$ of $\mathbb{R}^2$ with $A \neq B$ and $C \neq B$, the oriented angle $\measuredangle ABC$ is the oriented angle from $A - B$ to $C - B$.

## Remarks

Such a $\theta$ exists because $(u \cdot v)^2 + \det(u, v)^2 = \|u\|^2 \|v\|^2$. It is unique modulo $2\pi$.
Swapping the vectors changes the sign: the oriented angle from $v$ to $u$ is minus the oriented angle from $u$ to $v$.
The [[unoriented angle|0ZEHZY85E6]] between $u$ and $v$ is the absolute value of the representative of the oriented angle in $(-\pi, \pi]$.
Mathlib's `EuclideanGeometry.oangle` works in each oriented Euclidean plane, with values in $\mathbb{R} / 2\pi\mathbb{Z}$. It gives $0$ when a vector is zero.

## Explanations

### L1 human

The oriented angle remembers the direction of the turn. Turning from $(1, 0)$ to $(0, 1)$ is a quarter turn anticlockwise, so the oriented angle is $\pi / 2$. Turning back, from $(0, 1)$ to $(1, 0)$, gives $-\pi / 2$.
</document>

<document source="https://tamnd.github.io/math2/node/0G4X36AE0X.md" tag="0G4X36AE0X" kind="definition">
# Equilibrium point

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be a function.
A point $x^* \in U$ is an equilibrium point of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ if $f(x^*) = 0$.

## Remarks

The point $x^*$ is an equilibrium point exactly when the constant function $x(t) = x^*$ is a [[solution|04SW7WECS8]].
For the [[logistic equation|0CC4VC7KR3]] $x' = r x (1 - x/K)$ with $r, K > 0$, the equilibrium points are $0$ and $K$.
The equilibrium points are the fixed points of the [[flow|02903SSA48]]: $\varphi_t(x^*) = x^*$ for each $t$.
The behaviour of solutions near an equilibrium point is the subject of [[Lyapunov stability|092S6VBECQ]].

## Explanations

### L1 human

An equilibrium point is a state where the velocity is zero.
A system that starts there stays there for all time.
</document>

<document source="https://tamnd.github.io/math2/node/0G52BPNPKG.md" tag="0G52BPNPKG" kind="definition">
# Clique

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A clique in $G$ is a set $S$ of vertices of $G$ such that any two distinct vertices of $S$ are adjacent.

## Remarks

A set $S$ is a clique if and only if the [[induced subgraph|0V05FC4GKQ]] on $S$ is a [[complete graph|03WKB2H474]].
A set is a clique in $G$ if and only if it is an [[independent set|0KZJ82WEPP]] in the [[complement graph|0RNMP6VCR2]].
To decide whether a graph has a clique of a given size is an NP-complete problem.

## Explanations

### L1 human

A clique is a group of vertices that are all joined to each other.
In a social network, it is a group of people who all know each other.
</document>

<document source="https://tamnd.github.io/math2/node/0G558GJQMW.md" tag="0G558GJQMW" kind="definition">
# Multivariate normal distribution

## Definition

Let $k \ge 1$ and let $X = (X_1, \dots, X_k)^{\mathsf{T}}$ be a random vector in $\mathbb{R}^k$.
Then $X$ is multivariate normal if there exist $\mu \in \mathbb{R}^k$, an integer $\ell \ge 1$ and a real $k \times \ell$ matrix $A$ with this property.
There is a vector $Z = (Z_1, \dots, Z_\ell)^{\mathsf{T}}$ of independent $\mathcal{N}(0, 1)$ random variables with
$$X = AZ + \mu.$$
Then the law of $X$ depends only on $\mu$ and on $\Sigma = AA^{\mathsf{T}}$. It is written $\mathcal{N}_k(\mu, \Sigma)$.

## Remarks

Here $\mu$ is the mean vector, with $\mu_i = E[X_i]$, and $\Sigma$ is the covariance matrix, with $\Sigma_{ij} = \operatorname{Cov}(X_i, X_j)$.
An equivalent condition: $X$ is multivariate normal if and only if $a^{\mathsf{T}} X$ has a [[normal distribution|07RXBVTVZB]] for each $a \in \mathbb{R}^k$. Here a constant counts as normal with variance $0$.
If $\Sigma$ is positive definite, $X$ has the density
$$f(x) = \frac{1}{\sqrt{(2\pi)^k \det \Sigma}} \exp\left(-\tfrac{1}{2} (x - \mu)^{\mathsf{T}} \Sigma^{-1} (x - \mu)\right), \quad x \in \mathbb{R}^k.$$
The components of a multivariate normal vector are independent if and only if they are uncorrelated, that is, if $\Sigma$ is diagonal.

## Explanations

### L1 human

A multivariate normal vector is a list of normal random variables that may be linked to each other.
The covariance matrix says how strongly each pair of components moves together. In two dimensions, the density looks like a hill with elliptical contour lines.
</document>

<document source="https://tamnd.github.io/math2/node/0G5AV9Q4RR.md" tag="0G5AV9Q4RR" kind="definition">
# Closed set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]].
A subset $F$ of $X$ is closed in $X$ when its [[complement|0B1FSS1VZW]] $X \setminus F$ is [[open|01ZQQDCERP]] in $X$.

## Remarks

The sets $\emptyset$ and $X$ are both open and closed.
A set can be neither open nor closed: $[0, 1)$ in $\mathbb{R}$ is an example.
In a metric space, $F$ is closed if and only if it [[contains the limits of its convergent sequences|09GQ3ZTQJ9]].

## Explanations

### L1 human

Closed is not the opposite of open. A closed set is a set whose outside is open.
The interval $[0, 1]$ is closed in $\mathbb{R}$, because its outside $(-\infty, 0) \cup (1, \infty)$ is open.
</document>

<document source="https://tamnd.github.io/math2/node/0G65751VSW.md" tag="0G65751VSW" kind="statement">
# Derivative of the inverse function

## Statement

Let $V, W \subseteq \mathbb{R}^n$ be open, and let $f \colon V \to W$ be a bijection.
Suppose that $f$ is differentiable at $a \in V$ and that $f^{-1}$ is differentiable at $b = f(a)$.
Then $Df(a)$ is invertible, and
$$D(f^{-1})(b) = \big( Df(a) \big)^{-1}.$$
In terms of [[Jacobian matrices|0YF0VXKXGB]], $J_{f^{-1}}(b) = J_f(a)^{-1}$.

## Remarks

Proof: apply the [[chain rule|0EAAS2EHDR]] to $f^{-1} \circ f = \mathrm{id}_V$ and $f \circ f^{-1} = \mathrm{id}_W$. This gives $D(f^{-1})(b) \, Df(a) = I$ and $Df(a) \, D(f^{-1})(b) = I$.
The hard part is the differentiability of $f^{-1}$. The [[inverse function theorem|0K9HSJFCMJ]] gives it when $f$ is of class $C^1$ and $Df(a)$ is invertible.
For polar coordinates $f(r, \theta) = (r \cos \theta, r \sin \theta)$ with $r > 0$, $J_f = \begin{pmatrix} \cos \theta & -r \sin \theta \\ \sin \theta & r \cos \theta \end{pmatrix}$, and the Jacobian matrix of a local inverse is the inverse of this matrix.

## Explanations

### L1 human

Undoing a map undoes its linear approximation.
So the derivative of the inverse is the inverse matrix of the derivative, taken at the matching point.
</document>

<document source="https://tamnd.github.io/math2/node/0G6D8E7CTA.md" tag="0G6D8E7CTA" kind="statement">
# Cauchy sequences are bounded

## Statement

Let $(a_n)$ be a [[Cauchy sequence|0QZMTN5AXX]] of real numbers.
Then $(a_n)$ is [[bounded|05FQ3VGAR9]].

## Remarks

Take $\varepsilon = 1$ to get $N$ with $|a_n - a_N| < 1$ for all $n \ge N$. Then $|a_n| \le \max(|a_0|, \dots, |a_{N-1}|, |a_N| + 1)$ for all $n$.
The statement holds in each metric space.
The proof of the [[Cauchy criterion|0ZA4AX4MW3]] uses this fact.

## Explanations

### L1 human

From some index on, all terms are within distance $1$ of one fixed term.
The finitely many terms before that index are bounded too, so the whole sequence is bounded.
</document>

<document source="https://tamnd.github.io/math2/node/0G77XFBV00.md" tag="0G77XFBV00" kind="definition">
# Lexicographic order

## Definition

Let $(A, \le_A)$ and $(B, \le_B)$ be [[partially ordered sets|011C29Q1S1]].
The lexicographic order on the [[Cartesian product|0C81YBVTWH]] $A \times B$ is the relation $\le$ with $(a, b) \le (a', b')$ when
$$a <_A a' \quad \text{or} \quad (a = a' \text{ and } b \le_B b').$$

## Remarks

It is a partial order. When $\le_A$ and $\le_B$ are [[total orders|0316AYH1SC]], it is a total order.
Example: in $\mathbb{N} \times \mathbb{N}$, $(1, 5) < (2, 0)$ and $(2, 0) < (2, 3)$.
A dictionary orders words in the same way, letter by letter.

## Explanations

### L1 human

The lexicographic order is the dictionary order.
"cat" comes before "cow", because the first letters agree and "a" comes before "o".
</document>

<document source="https://tamnd.github.io/math2/node/0G78GP5TF1.md" tag="0G78GP5TF1" kind="definition">
# Vector projection

## Definition

Let $n$ be a positive integer, let $a$ be a vector in $\mathbb{R}^n$, and let $b$ be a nonzero vector in $\mathbb{R}^n$. Let $a \cdot b = a_1 b_1 + \dots + a_n b_n$ be the dot product.
The vector projection of $a$ onto $b$ is the vector
$$\operatorname{proj}_b a = \frac{a \cdot b}{b \cdot b} \, b.$$
The scalar projection of $a$ onto $b$ is the real number $\dfrac{a \cdot b}{\|b\|}$, where $\|b\|$ is the [[length|0XKBKGTCPC]] of $b$.

## Remarks

The vector $a - \operatorname{proj}_b a$ is the vector rejection of $a$ from $b$. Its dot product with $b$ is $0$, so it is [[perpendicular to $b$|03NK6RSRQ1]].
Among all multiples $c b$ of $b$, the projection is the one that makes $\|a - c b\|$ smallest.
If $\theta$ is the angle between $a \neq 0$ and $b$, then the scalar projection is $\|a\| \cos \theta$.

## Explanations

### L1 human

The projection answers the question "how much of $a$ points along $b$?". For $a = (3, 4)$ and $b = (1, 0)$ the projection is $(3, 0)$. The rest, $(0, 4)$, is perpendicular to $b$.
</document>

<document source="https://tamnd.github.io/math2/node/0G7WM8DX4P.md" tag="0G7WM8DX4P" kind="concept">
# Parametrised curve

## Summary

A parametrised curve in $\mathbb{R}^n$ is a continuous [[vector-valued function|018GTRS7A4]] of one real variable, defined on an interval.
One can think of the parameter as time and of the curve as the trajectory of a moving point.
</document>

<document source="https://tamnd.github.io/math2/node/0G8G1TH5WK.md" tag="0G8G1TH5WK" kind="statement">
# Uniqueness of the complete ordered field

## Statement

Let $F$ and $K$ be [[complete ordered fields|0A9E6X0S3E]].
Then there is exactly one field isomorphism $\varphi \colon F \to K$ that preserves the order, that is, with $\varphi(a) \le \varphi(b)$ whenever $a \le b$.

## Remarks

The proof first maps the rational elements of $F$ to those of $K$. Each other element is the supremum of the rational elements below it, and $\varphi$ maps it to the supremum of their images.
In fact each field isomorphism between two complete ordered fields preserves the order, because the nonnegative elements are exactly the squares.

## Explanations

### L1 human

Two people may build the real numbers in different ways, for example with Dedekind cuts and with Cauchy sequences.
This theorem says that they get the same number system, with only different names for the numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0G8VEZ3WTK.md" tag="0G8VEZ3WTK" kind="concept">
# Equivalence relation

## Summary

An equivalence relation is a relation that behaves like equality: it is reflexive, symmetric and transitive.
</document>

<document source="https://tamnd.github.io/math2/node/0G9EKM0E15.md" tag="0G9EKM0E15" kind="concept">
# Sequence

## Summary

A sequence is an ordered list of elements of a set, with one term for each natural number. Sequences are the basic objects of the study of limits in analysis. The node [[sequence|0QE8STGJH2]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0GA1V8NN8C.md" tag="0GA1V8NN8C" kind="statement">
# Inscribed angle theorem

## Statement

Let $A$, $B$ and $V$ be distinct points of a [[circle|060W5CBZP6]] with the centre $O$ in the Euclidean plane $\mathbb{R}^2$. Let $\measuredangle$ be the [[oriented angle|03YN70NK0Z]], with values in $\mathbb{R} / 2\pi\mathbb{Z}$.
Then
$$\measuredangle AOB = 2 \, \measuredangle AVB.$$

## Remarks

For the unoriented [[angles|0GJAJC6TNZ]], with values in $[0, \pi]$: if $V$ and $O$ are not on opposite sides of the line $AB$, then $\angle AVB = \frac{1}{2} \angle AOB$. If they are on opposite sides, then $\angle AVB = \pi - \frac{1}{2} \angle AOB$.
Thus the inscribed angle does not change when $V$ moves on one arc from $A$ to $B$.
[[Thales's theorem|0GSDHYMA1T]] is the case where $AB$ is a diameter.
The oldest known version is in Propositions 20 and 21 of Book III of Euclid's Elements.

## Explanations

### L1 human

Fix two points $A$ and $B$ on a circle. The angle at the centre that looks at the arc $AB$ is twice the angle at any point of the circle on the other arc. The proof starts with the case where the line $VO$ passes through $B$. Then the triangle $VOA$ is isosceles, and its angle sum gives the result.
</document>

<document source="https://tamnd.github.io/math2/node/0GA6G43EA9.md" tag="0GA6G43EA9" kind="statement">
# Derivative of arcsine

## Statement

The arcsine is differentiable on $(-1, 1)$, and
$$\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \quad \text{for } -1 < x < 1.$$

## Remarks

Let $y = \arcsin x$. The [[inverse function rule|0M0RZ3HSB0]] gives the derivative $\frac{1}{\cos y}$. By the [[Pythagorean identity|03H7WZ4T3K]], $\cos y = \sqrt{1 - x^2}$, since $\cos y > 0$ for $|y| < \pi/2$.
The arcsine is not differentiable at $\pm 1$. Its graph has vertical tangents there.
Similarly, $\frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1 - x^2}}$. Mathlib states the main claim as `Real.hasDerivAt_arcsin`.

## Explanations

### L1 human

The derivative of the arcsine is an algebraic function, although the arcsine itself is not.
It becomes infinite at the ends of the interval $[-1, 1]$.
</document>

<document source="https://tamnd.github.io/math2/node/0GA9K57A39.md" tag="0GA9K57A39" kind="example">
# Fibonacci sequence

## Example

The Fibonacci sequence $(F_n)_{n \ge 0}$ is the sequence of natural numbers defined by
$$F_0 = 0, \qquad F_1 = 1, \qquad F_{n+2} = F_{n+1} + F_n.$$
It is an example of a sequence defined by a [[recurrence relation|07TZV87Y0T]], and in particular of a [[linear recurrence with constant coefficients|081THG13JZ]].
Its first terms are
$$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \dots$$

## Remarks

The ratios $F_{n+1} / F_n$ tend to the [[golden ratio|0SC8T7548D]] $\varphi = \frac{1 + \sqrt{5}}{2}$.
[[Binet's formula|0404D8TN7Y]] gives $F_n$ in closed form.
$F_{n+1}$ is the number of ways to tile a strip of length $n$ with squares of length $1$ and dominoes of length $2$.

## Explanations

### L1 human

Each Fibonacci number is the sum of the two before it: $5 + 8 = 13$, $8 + 13 = 21$, and so on.
The numbers grow by a factor close to $1.618$ at each step.
</document>

<document source="https://tamnd.github.io/math2/node/0GBG8B49AA.md" tag="0GBG8B49AA" kind="concept">
# Product topology

## Summary

The product topology is the natural topology on a Cartesian product of topological spaces.
It is the coarsest topology for which all projections onto the factors are continuous.
</document>

<document source="https://tamnd.github.io/math2/node/0GBJACNAP6.md" tag="0GBJACNAP6" kind="concept">
# Independence of two random variables

## Summary

Two random variables are independent when each event about the first one is independent of each event about the second one.
The node [[independence of two random variables|0E9502NT92]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0GBZDAFPX8.md" tag="0GBZDAFPX8" kind="statement">
# Epsilon-delta definition of a limit

## Statement

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, let $a$ be a limit point of $D$, and let $L$ be a real number.
Then $\lim_{x \to a} f(x) = L$ if and only if the following holds.
For each $\varepsilon > 0$ there is $\delta > 0$ such that $|f(x) - L| < \varepsilon$ for all $x \in D$ with $0 < |x - a| < \delta$.

## Remarks

Each open interval around a point contains an interval $(L - \varepsilon, L + \varepsilon)$, and each such interval is open. This gives the equivalence with the [[definition|0TXTSEMHR5]].
Many texts take this condition as the definition of the limit.
Mathlib states a form for metric spaces as `Metric.tendsto_nhds_nhds`.

## Explanations

### L1 human

For each error tolerance $\varepsilon$, we can force $f(x)$ within $\varepsilon$ of $L$.
We do this by taking $x$ within a suitable distance $\delta$ of $a$.
</document>

<document source="https://tamnd.github.io/math2/node/0GBZM0MT0F.md" tag="0GBZM0MT0F" kind="concept">
# Binomial coefficient

## Summary

The binomial coefficient $\binom{n}{k}$ is the number of ways to choose $k$ elements from an $n$-element set. These numbers are the coefficients in the expansion of $(x + y)^n$. The node [[binomial coefficient|092GEF38YD]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0GCM7B6PBK.md" tag="0GCM7B6PBK" kind="concept">
# Concave function

## Summary

A concave function on an interval is a function whose graph lies on or above each of its chords. A function is concave exactly when its negative is convex. The node [[concave function|0EF2VG8XKQ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0GCVEV6BJA.md" tag="0GCVEV6BJA" kind="definition">
# Mutual independence of events

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]], let $I$ be an index set, and let $A_i \in \mathcal{F}$ be an [[event|00SQGJNRTB]] for each $i \in I$.
The events $(A_i)_{i \in I}$ are mutually independent if for each finite nonempty subset $J \subseteq I$,
$$P\left(\bigcap_{j \in J} A_j\right) = \prod_{j \in J} P(A_j).$$

## Remarks

For two events this is the [[independence of two events|0P0MZ4X682]].
For three events $A, B, C$ the definition asks for four equations: one for each of the pairs $\{A, B\}$, $\{A, C\}$, $\{B, C\}$ and one for the triple $\{A, B, C\}$.
Mutual independence implies [[pairwise independence|0N57D1XY6T]]. The converse is false: see [[pairwise independent events that are not mutually independent|0AR4B3K5QD]].
If we replace some of the events $A_i$ by their complements, the family stays mutually independent.

## Explanations

### L1 human

For many events, independence must hold for every group of them, not only for every pair.
For three tosses of a fair coin, the events "toss $k$ gives heads" for $k = 1, 2, 3$ are mutually independent, and all three happen with probability $1/8$.
</document>

<document source="https://tamnd.github.io/math2/node/0GCWJS2EGB.md" tag="0GCWJS2EGB" kind="definition">
# Eccentricity

## Definition

Let $Q$ be a non-degenerate [[conic section|00HJCMC2CD]] in the Euclidean plane $\mathbb{R}^2$.
If $Q$ is a circle, its eccentricity is $e = 0$. Otherwise, let $F$ be a [[focus|0RBKKNH4GV]] of $Q$ and $\ell$ the [[directrix|0E5CXYNEPX]] for $F$. The eccentricity of $Q$ is the constant ratio
$$e = \frac{d(P, F)}{d(P, \ell)}, \qquad P \in Q.$$

## Remarks

An ellipse has $0 \leq e < 1$, a parabola has $e = 1$ and a hyperbola has $e > 1$.
For an ellipse or a hyperbola with the semi-major axis $a$ and the foci at the distance $c$ from the centre, $e = c / a$.
Two conic sections with the same eccentricity are similar.

## Explanations

### L1 human

The eccentricity tells how stretched a conic is. A circle has $e = 0$. The ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ has $c = 4$ and so $e = 4/5$. As $e$ gets near $1$, the ellipse becomes long and thin.
</document>

<document source="https://tamnd.github.io/math2/node/0GDNHS9J2X.md" tag="0GDNHS9J2X" kind="definition">
# Curl

## Definition

Let $U \subseteq \mathbb{R}^3$ be open, and let $F = (P, Q, R) \colon U \to \mathbb{R}^3$ be a [[vector field|0N67NBKQ5T]] whose first [[partial derivatives|05AXG3QRKR]] exist on $U$.
The curl of $F$ is the vector field $\operatorname{curl} F \colon U \to \mathbb{R}^3$,
$$\operatorname{curl} F = \nabla \times F = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \ \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right).$$
A field with $\operatorname{curl} F = 0$ on $U$ is irrotational, or curl-free.

## Remarks

As a mnemonic, $\nabla \times F$ is the formal determinant of the matrix with rows $(\mathbf{i}, \mathbf{j}, \mathbf{k})$, $(\partial_x, \partial_y, \partial_z)$ and $(P, Q, R)$.
For $F(x, y, z) = (-y, x, 0)$, $\operatorname{curl} F = (0, 0, 2)$. For $F(x, y, z) = (x, y, z)$, $\operatorname{curl} F = 0$.
For a plane field $(P, Q)$, one uses the scalar curl $\partial Q / \partial x - \partial P / \partial y$, the third component of the curl of $(P, Q, 0)$. It appears in [[Green's theorem|0HR3N02317]].
Take small loops orthogonal to a unit vector $n$. The component of $\operatorname{curl} F$ along $n$ is the limit of the circulation of $F$ around these loops, divided by the enclosed area. This follows from [[Stokes' theorem|0AKRKVHW94]].

## Explanations

### L1 human

Put a tiny paddle wheel into a fluid with velocity field $F$.
If it spins, the curl is not zero there. The curl points along the axis about which the wheel spins fastest.
</document>

<document source="https://tamnd.github.io/math2/node/0GDQRETR59.md" tag="0GDQRETR59" kind="concept">
# Prüfer sequence

## Summary

The Prüfer sequence encodes a labelled tree on $n$ vertices as a sequence of $n - 2$ labels. This encoding is a bijection, and it proves Cayley's formula. The node [[Prüfer sequence|09PV9JSY6E]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0GE4DVQKR8.md" tag="0GE4DVQKR8" kind="example">
# The quaternions

## Example

The quaternions $\mathbb{H}$ are the real vector space with basis $1, i, j, k$.
Each quaternion is $a + b i + c j + d k$ with $a, b, c, d \in \mathbb{R}$.
The multiplication is $\mathbb{R}$-bilinear, $1$ is the identity, and the basis elements obey
$$i^2 = j^2 = k^2 = i j k = -1.$$
These rules give $i j = k = -j i$, $j k = i = -k j$ and $k i = j = -i k$.
With this multiplication, $\mathbb{H}$ is a [[division ring|0WZXK05ME7]] that is not commutative.

## Remarks

The conjugate of $q = a + b i + c j + d k$ is $\bar{q} = a - b i - c j - d k$, and $q \bar{q} = a^2 + b^2 + c^2 + d^2$.
So a nonzero $q$ has the inverse $q^{-1} = \bar{q} / (a^2 + b^2 + c^2 + d^2)$.
The quaternions are not a field, because $i j \neq j i$.

## Explanations

### L1 human

The quaternions extend the complex numbers with two more square roots of $-1$, called $j$ and $k$.
You can divide by every nonzero quaternion.
But the order of multiplication matters: $i j = k$ and $j i = -k$.
</document>

<document source="https://tamnd.github.io/math2/node/0GEQ42PTGB.md" tag="0GEQ42PTGB" kind="definition">
# Critical point

## Definition

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$, and let $a \in I$.
The point $a$ is a critical point of $f$ if either $f$ is not differentiable at $a$, or $f$ is differentiable at $a$ with $f'(a) = 0$.
In the second case, $a$ is also called a stationary point.

## Remarks

By [[Fermat's theorem|0PCVBRV9VJ]], each local extremum of $f$ in $I$ is a critical point.
The converse fails: $x^3$ has a critical point at $0$, but no local extremum.
For $|x|$, the point $0$ is a critical point where $f$ is not differentiable.

## Explanations

### L1 human

Critical points are the candidates for local maxima and minima.
There the tangent is horizontal, or there is no tangent at all.
</document>

<document source="https://tamnd.github.io/math2/node/0GETX79CJN.md" tag="0GETX79CJN" kind="concept">
# Normal core

## Summary

The normal core of a subgroup $H$ of a group $G$ is the largest normal subgroup of $G$ that is contained in $H$.
It is the intersection of all conjugates of $H$.
</document>

<document source="https://tamnd.github.io/math2/node/0GF6AJEBZ0.md" tag="0GF6AJEBZ0" kind="statement">
# Induction from an arbitrary base case

## Statement

Let $m$ be a natural number and let $P(n)$ be a property of natural numbers $n$ with $n \ge m$.
Suppose that $P(m)$ holds.
Suppose also that for each natural number $n \ge m$, $P(n)$ implies $P(n + 1)$.
Then $P(n)$ holds for every natural number $n \ge m$.

## Remarks

The proof applies ordinary [[induction|0ASJVEFA9J]] to the property $Q(k)$: "$P(m + k)$ holds".
For example, induction from $m = 4$ shows that $2^n \ge n^2$ for every natural number $n \ge 4$.

## Explanations

### L1 human

Sometimes a property is false for small numbers and true from some point on.
Then you start the induction at that point instead of at $0$.
The dominoes start falling at domino number $m$, and the dominoes before it do not matter.
</document>

<document source="https://tamnd.github.io/math2/node/0GFGHRT2FJ.md" tag="0GFGHRT2FJ" kind="concept">
# Subfield

## Summary

A subfield of a field is a subset that is a field with the same addition and multiplication.
The [[definition of a subfield|07QY5RPFBR]] lists the closure conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0GFGK40582.md" tag="0GFGK40582" kind="concept">
# Pointwise convergence

## Summary

A sequence of functions converges pointwise when its values converge at every point of the domain. It is the weakest of the usual modes of convergence for functions. The node [[pointwise convergence|056FSKG32J]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0GFKP8F8AG.md" tag="0GFKP8F8AG" kind="definition">
# Coprime integers

## Definition

Two integers $a$ and $b$ are coprime when their [[greatest common divisor|02WF878BH1]] is $1$, that is, when $\gcd(a, b) = 1$.

## Remarks

Equivalently, $a$ and $b$ are coprime when the only integers that divide both of them are $1$ and $-1$.
For example, $8$ and $15$ are coprime, but $8$ and $12$ are not, because $4$ divides both.
The integers $1$ and $-1$ are coprime to each integer. The integer $0$ is coprime only to $1$ and $-1$.
By [[Bézout's identity|0DQVHMYAGA]], $a$ and $b$ are coprime if and only if $a x + b y = 1$ for some integers $x$ and $y$.

## Explanations

### L1 human

Two numbers are coprime when they have no common factor except $1$.
The numbers do not need to be prime themselves: $9$ and $10$ are not prime, but they are coprime.
A fraction $a / b$ is in lowest terms exactly when $a$ and $b$ are coprime.
</document>

<document source="https://tamnd.github.io/math2/node/0GG9TVW90M.md" tag="0GG9TVW90M" kind="concept">
# Bijective function

## Summary

A function is bijective when it is both injective and surjective, so each element of the codomain is the value at exactly one input.
</document>

<document source="https://tamnd.github.io/math2/node/0GGE0Z0JQ1.md" tag="0GGE0Z0JQ1" kind="definition">
# Ring homomorphism

## Definition

Let $R$ and $S$ be [[rings|0894J980RK]].
A ring homomorphism from $R$ to $S$ is a map $f \colon R \to S$ with these three properties.

1. $f(a + b) = f(a) + f(b)$ for all $a, b \in R$.
2. $f(a b) = f(a) f(b)$ for all $a, b \in R$.
3. $f(1) = 1$.

## Remarks

A ring homomorphism also satisfies $f(0) = 0$ and $f(-a) = -f(a)$.
For each integer $n \ge 1$, the map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ that sends $a$ to its class is a ring homomorphism.
The map $\mathbb{Z} \to \mathbb{Z}$, $a \mapsto 2a$, preserves sums but is not a ring homomorphism.
A bijective ring homomorphism is a [[ring isomorphism|0B6QC0C0G1]].

## Explanations

### L1 human

A ring homomorphism translates one ring into another and keeps the arithmetic.
It does not matter if you add or multiply first and then apply $f$, or apply $f$ first.
Reduction modulo $n$ is the main example.
</document>

<document source="https://tamnd.github.io/math2/node/0GGKJGZF6D.md" tag="0GGKJGZF6D" kind="concept">
# P-value

## Summary

The p-value of a test is the probability, under the null hypothesis, to get a result at least as extreme as the observed result. A small p-value is evidence against the null hypothesis.
The node [[p-value|0EHYZFZ4PK]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0GGXWDNKBR.md" tag="0GGXWDNKBR" kind="concept">
# Forest

## Summary

A forest is a graph without cycles, that is, a disjoint union of trees. The spanning forests of a graph generalize spanning trees to graphs that are not connected. The node [[forest|09JP6G9FTA]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0GHDDVAH8P.md" tag="0GHDDVAH8P" kind="definition">
# Frobenius endomorphism

## Definition

Let $p$ be a prime number and let $R$ be a [[commutative ring|0GYGHKX834]] of [[characteristic|0P7CH1CND5]] $p$.
The Frobenius endomorphism of $R$ is the map $F \colon R \to R$, $F(x) = x^p$.
It is a [[ring homomorphism|0GGE0Z0JQ1]] from $R$ to $R$.

## Remarks

The map $F$ preserves products because $R$ is commutative. It preserves sums by the [[freshman's dream|0396VGFA78]].
If $R$ is an [[integral domain|09S3PFBEPM]], then $F$ is injective.
If $R$ is a [[finite field|08RTDHX1SK]], then $F$ is an automorphism of $R$.
On $\mathbb{F}_p$, the map $F$ is the identity, by Fermat's little theorem.

## Explanations

### L1 human

In characteristic $p$, raising to the $p$-th power respects addition and multiplication.
So it is a homomorphism from the ring to itself.
This simple map is a basic tool in the study of finite fields.
</document>

<document source="https://tamnd.github.io/math2/node/0GHS47PDAQ.md" tag="0GHS47PDAQ" kind="concept">
# Universal quantifier

## Summary

The universal quantifier $\forall$ expresses "for all".
A universally quantified statement says that a [[predicate|0T3DPBG981]] holds for every element of its domain.
</document>

<document source="https://tamnd.github.io/math2/node/0GHWEH6H15.md" tag="0GHWEH6H15" kind="statement">
# General solution of an inhomogeneous linear equation

## Statement

Let $I$ be an open interval, let $a_0, \dots, a_{n-1}, b : I \to \mathbb{R}$ be continuous, and let $L y = y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y$.
Let $y_p$ be a solution of $L y = b$ on $I$, and let $(y_1, \dots, y_n)$ be a [[fundamental system|0H4KNNCHPY]] of $L y = 0$ on $I$.
Then a function $y : I \to \mathbb{R}$ is a solution of $L y = b$ on $I$ if and only if
$$y = y_p + c_1 y_1 + \dots + c_n y_n$$
for some real numbers $c_1, \dots, c_n$.

## Remarks

Proof: $y$ solves $L y = b$ if and only if $L(y - y_p) = 0$, by linearity. Then $y - y_p$ is a combination of the fundamental system.
The [[method of undetermined coefficients|0XRB469Z32]] and [[variation of parameters|0Y6TH5PQVS]] find a particular solution $y_p$.
Example: $y'' + y = 1$ has $y_p = 1$, so its solutions are $1 + c_1 \cos x + c_2 \sin x$.

## Explanations

### L1 human

To solve a linear equation with a right side, find one solution by any method.
Then add all the solutions of the equation without right side.
This gives every solution, and each only one time.
</document>

<document source="https://tamnd.github.io/math2/node/0GJ2HRG95T.md" tag="0GJ2HRG95T" kind="definition">
# Euclidean domain

## Definition

A Euclidean domain is an [[integral domain|09S3PFBEPM]] $R$ with a function $\delta \colon R \setminus \{0\} \to \mathbb{N}$ with this property.
For all $a, b \in R$ with $b \neq 0$, there are $q, r \in R$ such that $a = b q + r$, and $r = 0$ or $\delta(r) < \delta(b)$.
The function $\delta$ is a Euclidean function of $R$.

## Remarks

The absolute value is a Euclidean function on $\mathbb{Z}$. The degree is a Euclidean function on $K[x]$ for a field $K$.
Some authors also require $\delta(a) \le \delta(a b)$ for all nonzero $a, b$. This changes the class of Euclidean domains only in appearance, because a ring with a Euclidean function has one with this extra property.
Every Euclidean domain is a [[principal ideal domain|0RZYDVZ475]].

## Explanations

### L1 human

In a Euclidean domain, you can always divide with a remainder that is smaller than the divisor.
The function $\delta$ measures what smaller means.
This is all you need to run the Euclidean algorithm.
</document>

<document source="https://tamnd.github.io/math2/node/0GJ7T0NK49.md" tag="0GJ7T0NK49" kind="definition">
# Polynomial evaluation

## Definition

Let $R$ be a [[ring|0894J980RK]], let $f = a_0 + a_1 x + \dots + a_n x^n \in R[x]$ and let $c \in R$.
The value of $f$ at $c$ is the element
$$f(c) = a_0 + a_1 c + a_2 c^2 + \dots + a_n c^n \in R.$$

## Remarks

The value $f(c)$ does not depend on the choice of $n$, because the extra coefficients are $0$.
For $f = x^2 - 2$ in $\mathbb{Z}[x]$, $f(3) = 7$.
If $R$ is commutative, then $(f g)(c) = f(c) g(c)$ and $(f + g)(c) = f(c) + g(c)$. This gives the [[evaluation homomorphism|0125QVNCFQ]].
Horner's rule computes $f(c) = a_0 + c(a_1 + c(a_2 + \dots + c a_n))$ with $n$ multiplications.

## Explanations

### L1 human

To evaluate $f$ at $c$, you put $c$ in place of $x$ and compute.
For $f = x^2 + 1$ and $c = 2$, you get $f(2) = 5$.
</document>

<document source="https://tamnd.github.io/math2/node/0GJ83E78A5.md" tag="0GJ83E78A5" kind="statement">
# Universal property of the quotient group

## Statement

Let $G$ and $H$ be [[groups|G5S1M9]], let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$, and let $\pi \colon G \to G/N$ be the projection $\pi(g) = gN$ to the [[quotient group|0KYX2N69J0]].
Let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]] with $N \subseteq \ker \varphi$.
Then there is exactly one group homomorphism $\bar{\varphi} \colon G/N \to H$ with $\bar{\varphi} \circ \pi = \varphi$, and it is given by $\bar{\varphi}(gN) = \varphi(g)$.

## Remarks

The map $\bar{\varphi}$ is well defined because $\varphi$ is constant on each coset of $N$.
The image of $\bar{\varphi}$ is the image of $\varphi$, and the kernel of $\bar{\varphi}$ is $(\ker \varphi)/N$.
With $N = \ker \varphi$, this gives the [[first isomorphism theorem|0CXXJNG5B6]].

## Explanations

### L1 human

A homomorphism that sends all of $N$ to the identity cannot tell apart elements of the same coset of $N$.
So it already makes sense on the cosets, that is, on the quotient group.
This is the standard way to define a homomorphism out of a quotient group.
</document>

<document source="https://tamnd.github.io/math2/node/0GJAJC6TNZ.md" tag="0GJAJC6TNZ" kind="concept">
# Angle at a point

## Summary

The angle $\angle ABC$ at the point $B$ is the opening between the rays from $B$ through $A$ and from $B$ through $C$. It is a number from $0$ to $\pi$, and it does not depend on the order of the two rays.
</document>

<document source="https://tamnd.github.io/math2/node/0GJBJ19NXR.md" tag="0GJBJ19NXR" kind="definition">
# Mersenne number

## Definition

For a natural number $n$, the $n$-th Mersenne number is $M_n = 2^n - 1$.

## Remarks

The first values are $M_1 = 1$, $M_2 = 3$, $M_3 = 7$, $M_4 = 15$ and $M_5 = 31$.
In binary, $M_n$ is written with $n$ digits $1$.
Some texts call $M_n$ a Mersenne number only when $n$ is prime.
A Mersenne number that is prime is a [[Mersenne prime|06GPJF9V7M]].

## Explanations

### L1 human

A Mersenne number is one less than a power of $2$.
For example, $2^{10} - 1 = 1023$, which is $1111111111$ in binary.
</document>

<document source="https://tamnd.github.io/math2/node/0GJM24EXP9.md" tag="0GJM24EXP9" kind="statement">
# Cancellation in an integral domain

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]] and let $a, b, c \in R$ with $a \neq 0$.
If $a b = a c$, then $b = c$.

## Remarks

The proof is short: $a b = a c$ gives $a (b - c) = 0$, and $a \neq 0$ gives $b - c = 0$.
The converse also holds: a nonzero commutative ring with this cancellation property is an integral domain.
Cancellation fails in $\mathbb{Z}/6\mathbb{Z}$: $2 \cdot 1 = 2 \cdot 4$, but $1 \neq 4$.

## Explanations

### L1 human

In the integers, you can cancel a nonzero factor from both sides of an equation.
From $3 b = 3 c$ you get $b = c$.
This works in every integral domain, even when you cannot divide by $a$.
</document>

<document source="https://tamnd.github.io/math2/node/0GJWQZCPZM.md" tag="0GJWQZCPZM" kind="statement">
# Trace is linear

## Statement

Let $R$ be a commutative ring, let $A$ and $B$ be $n \times n$ matrices over $R$, and let $c \in R$.
Then $\tr(A + B) = \tr A + \tr B$ and $\tr(cA) = c \tr A$, where $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

Thus the trace is a linear map from $M_n(R)$ to $R$.
Both identities follow at once from the definition, because the diagonal of $A + B$ is the sum of the diagonals.
Over a field, the trace is a [[linear form|07YQ1C78G0]] on the vector space of $n \times n$ matrices.

## Explanations

### L1 human

The trace of a sum is the sum of the traces.
The trace of a multiple is the same multiple of the trace.
This holds because we only add diagonal entries, and addition and scaling act entry by entry.
</document>

<document source="https://tamnd.github.io/math2/node/0GJZCE87WP.md" tag="0GJZCE87WP" kind="statement">
# Addition formulas for hyperbolic functions

## Statement

For all real numbers $x$ and $y$,
$$\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y,$$
$$\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y.$$

## Remarks

Expand the right sides with the definitions and use [[$e^{x+y} = e^x e^y$|0E8HNVBCSP]].
With $y = x$, they give $\sinh 2x = 2 \sinh x \cosh x$ and $\cosh 2x = \cosh^2 x + \sinh^2 x$.
Mathlib states the first formula as `Real.sinh_add`.

## Explanations

### L1 human

These formulas look like the trigonometric addition formulas.
The only change is the plus sign in the formula for $\cosh$.
</document>

<document source="https://tamnd.github.io/math2/node/0GJZE2E4RX.md" tag="0GJZE2E4RX" kind="counterexample">
# Function for which the iterated integrals differ

## Counterexample

One could guess that the two [[iterated integrals|0P4WX7Y502]] of a function over a square are always equal when both exist.
This is false.
Let $f(x, y) = \dfrac{x^2 - y^2}{(x^2 + y^2)^2}$ on $(0, 1] \times (0, 1]$. Then
$$\int_0^1 \int_0^1 f(x, y) \, dy \, dx = \frac{\pi}{4}, \qquad \int_0^1 \int_0^1 f(x, y) \, dx \, dy = -\frac{\pi}{4}.$$

## Remarks

For fixed $x > 0$, $\partial_y \big( y / (x^2 + y^2) \big) = f(x, y)$. So the inner integral is $1/(1 + x^2)$, and its integral over $[0, 1]$ is $\pi / 4$. Since $f(y, x) = -f(x, y)$, the other order gives $-\pi / 4$.
The function is not integrable on the square: $\int_0^1 \int_0^1 |f| \, dy \, dx = \infty$. So [[Fubini's theorem|0R0T1VWH5X]] does not apply.
A discrete example is $a_{xy} = 1$ if $x = y$, $a_{xy} = -1$ if $x = y + 1$, and $a_{xy} = 0$ otherwise, on the positive integers. The two iterated sums are $0$ and $1$.

## Explanations

### L1 human

Near the corner $(0, 0)$ the function has large positive values and large negative values.
The two orders of integration cancel these parts in different ways, so they give different answers.
</document>

<document source="https://tamnd.github.io/math2/node/0GJZW9JFZQ.md" tag="0GJZW9JFZQ" kind="concept">
# Transitive relation

## Summary

A relation is transitive when it passes along chains.
If a first element is related to a second, and the second to a third, then the first is related to the third.
</document>

<document source="https://tamnd.github.io/math2/node/0GK1WB9R98.md" tag="0GK1WB9R98" kind="definition">
# Minimal polynomial of an algebraic element

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$ be [[algebraic|0NPW1DWECM]] over $K$.
The minimal polynomial of $a$ over $K$ is the [[monic polynomial|0BDX4MBKT4]] $m_a \in K[x]$ of smallest [[degree|0DPF6WRZW6]] such that $m_a(a) = 0$.

## Remarks

The minimal polynomial exists and is unique. If two monic polynomials of the smallest degree vanish at $a$, their difference has smaller degree and vanishes at $a$, so it is zero.
The minimal polynomial [[is irreducible|0NFEG48RDR]], and it [[divides each polynomial that vanishes at $a$|06K7SARZKR]].
The minimal polynomial of $\sqrt{2}$ over $\mathbb{Q}$ is $x^2 - 2$. Over $\mathbb{R}$, it is $x - \sqrt{2}$.
The minimal polynomial of $i$ over $\mathbb{R}$ is $x^2 + 1$.

## Explanations

### L1 human

Among all the polynomial equations that an algebraic element satisfies, the minimal polynomial is the simplest one.
It depends on the base field: a bigger base field can give a smaller minimal polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/0GK68PZ6KH.md" tag="0GK68PZ6KH" kind="statement">
# Solution space of a homogeneous linear equation has dimension n

## Statement

Let $n \ge 1$, let $I$ be an open interval, and let $a_0, \dots, a_{n-1} : I \to \mathbb{R}$ be continuous.
Let $S$ be the set of solutions $y : I \to \mathbb{R}$ of $y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y = 0$.
Then $S$ is a real vector space of dimension $n$.
More precisely, for each $x_0 \in I$, the map $S \to \mathbb{R}^n$, $y \mapsto \left(y(x_0), y'(x_0), \dots, y^{(n-1)}(x_0)\right)$ is a linear bijection.

## Remarks

The [[superposition principle|0F2VBPXNP9]] gives that $S$ is a vector space. The map is injective by the uniqueness of solutions of the initial value problem, and surjective by their existence on all of $I$.
A basis of $S$ is a [[fundamental system of solutions|0H4KNNCHPY]].
Example: the solutions of $y'' + y = 0$ on $\mathbb{R}$ are $C_1 \cos x + C_2 \sin x$, a space of dimension $2$.

## Explanations

### L1 human

A solution is fixed by its starting data: the value and the first $n - 1$ derivatives at one point.
These are $n$ numbers that you can choose freely.
So there are exactly $n$ independent solutions, and each solution is a combination of them.
</document>

<document source="https://tamnd.github.io/math2/node/0GKJ78948V.md" tag="0GKJ78948V" kind="definition">
# Separable space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is separable when there is a [[countable|0ZG61TFBES]] subset $D \subseteq X$ that is [[dense|0D198GPMTJ]] in $X$.

## Remarks

The real line is separable, because $\mathbb{Q}$ is countable and dense. In the same way $\mathbb{R}^n$ is separable, with the dense set $\mathbb{Q}^n$.
Each [[second-countable space|00RE1R4W7N]] is separable. For metric spaces, the converse is also true.
An uncountable set with the [[discrete metric|0NA4FK7B8S]] is not separable, because each dense subset is the whole set.

## Explanations

### L1 human

A list of points can approximate a separable space. Each point of the space has points of the list as close as you want.
For $\mathbb{R}$, the list of all fractions does the job.
</document>

<document source="https://tamnd.github.io/math2/node/0GKQKF0FXJ.md" tag="0GKQKF0FXJ" kind="example">
# The alternating group A5 is simple

## Example

The group $A_5$ has order $60$.
Its [[conjugacy classes|0DXD4QSZ8A]] have sizes $1$, $15$, $20$, $12$ and $12$.
The class of size $1$ is $\{e\}$, and the class of size $20$ contains the $3$-cycles.
The class of size $15$ contains the products of two disjoint transpositions.
The $24$ five-cycles split into two classes of size $12$.
A normal subgroup $N$ of $A_5$ is a union of conjugacy classes, and it contains the class $\{e\}$.
By [[Lagrange's theorem|3C53TD]], $|N|$ divides $60$.
No sum of $1$ and some of the numbers $15, 20, 12, 12$ divides $60$, except $1$ and $60$.
Thus $N = \{e\}$ or $N = A_5$, and $A_5$ is simple.
It is the smallest simple group that is not abelian.
</document>

<document source="https://tamnd.github.io/math2/node/0GMPYM0NSR.md" tag="0GMPYM0NSR" kind="definition">
# Adjugate matrix

## Definition

Let $K$ be a commutative ring, let $n \ge 2$, and let $A$ be an $n \times n$ matrix over $K$ with [[cofactors|0XPKR18APA]] $C_{ij}$.
The *adjugate* of $A$ is the $n \times n$ matrix $\operatorname{adj}(A)$ with entries
$$\operatorname{adj}(A)_{ij} = C_{ji}.$$
So $\operatorname{adj}(A)$ is the [[transpose|09DXWYRD43]] of the matrix of cofactors $(C_{ij})$.

## Remarks

For $n = 2$, $\operatorname{adj} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.
The adjugate satisfies [[$A \operatorname{adj}(A) = \det(A) I_n$|0528HKWZ1Z]].
Older texts call it the *classical adjoint*; it is not the adjoint of an inner product space.

## Explanations

### L1 human

To build the adjugate, replace each entry by its cofactor, and then transpose.
The adjugate is almost the inverse: it is the inverse multiplied by the determinant.
For $2 \times 2$ matrices, swap the diagonal entries and negate the other two.
</document>

<document source="https://tamnd.github.io/math2/node/0GMV7GP8W0.md" tag="0GMV7GP8W0" kind="statement">
# Absolute value is multiplicative

## Statement

For all real numbers $a$ and $b$, $|a b| = |a| \, |b|$.
Here $|\cdot|$ is the [[absolute value|00S7C68D51]].

## Remarks

Proof: check the four cases of the signs of $a$ and $b$.
In particular $|a^n| = |a|^n$ for each natural number $n$, and $|a / b| = |a| / |b|$ for $b \neq 0$.
The [[modulus of complex numbers|0HYNZE940T]] is also multiplicative.

## Explanations

### L1 human

You can take the absolute value before or after a product, and you get the same result.
For example, $|(-2) \cdot 3| = |-6| = 6$ and $|-2| \cdot |3| = 2 \cdot 3 = 6$.
</document>

<document source="https://tamnd.github.io/math2/node/0GNCRPYXYV.md" tag="0GNCRPYXYV" kind="concept">
# Subring

## Summary

A subring is a subset of a ring that contains $0$ and $1$ and is closed under the ring operations.
It is a ring with the operations of the larger ring.
The [[definition of a subring|0859D58WPD]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0GNG2KVPZJ.md" tag="0GNG2KVPZJ" kind="statement">
# Nilpotent groups are solvable

## Statement

Let $G$ be a [[nilpotent group|0Q7MMQHC1P]].
Then $G$ is [[solvable|0FXJ7YFMMX]].

## Remarks

A central series $\{e\} = N_0 \subseteq \cdots \subseteq N_n = G$ is a subnormal series, and each factor $N_{i+1}/N_i$ is central in $G/N_i$, so it is abelian.
The converse is false: $S_3$ is solvable, but it is not nilpotent.
So there are strict inclusions: abelian groups, then nilpotent groups, then solvable groups.

## Explanations

### L1 human

A central series has layers that commute with the whole group.
In particular, each layer is abelian.
So a nilpotent group is solvable.
</document>

<document source="https://tamnd.github.io/math2/node/0GNGPF7KS5.md" tag="0GNGPF7KS5" kind="definition">
# Antichain

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
A subset $A$ of $P$ is an antichain when, for all $a, b \in A$ with $a \neq b$, neither $a \le b$ nor $b \le a$.

## Remarks

In the power set of $\{1, 2, 3\}$, the set $\{\{1\}, \{2\}, \{3\}\}$ is an antichain.
A chain and an antichain have at most one common element.
The largest size of an antichain is the width of the poset.

## Explanations

### L1 human

An antichain is the opposite of a [[chain|0GP9ASVQ1X]]: its elements are side by side.
Under divisibility, the primes form an antichain, because no prime divides another prime.
</document>

<document source="https://tamnd.github.io/math2/node/0GNVN3CA2P.md" tag="0GNVN3CA2P" kind="statement">
# Dirichlet's theorem on arithmetic progressions

## Statement

Let $a$ and $d$ be positive integers that are [[coprime|0GFKP8F8AG]].
Then there are infinitely many [[prime numbers|0SN0MAEKYX]] of the form $a + n d$, with $n$ a natural number.
That is, there are infinitely many primes $p$ with $p \equiv a \pmod{d}$.

## Remarks

The hypothesis $\gcd(a, d) = 1$ is necessary. Each term of $a + n d$ is a multiple of $\gcd(a, d)$, so at most one term can be prime when $\gcd(a, d) > 1$.
For $d = 4$, the theorem gives infinitely many primes $p \equiv 1 \pmod 4$ and infinitely many primes $p \equiv 3 \pmod 4$.
A special case with an elementary proof is that there are [[infinitely many primes congruent to 1 modulo n|0T1BSD7H3M]].
The primes are in fact equally distributed among the $\varphi(d)$ possible classes, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## History

Dirichlet proved the theorem in 1837. His proof introduced Dirichlet characters and Dirichlet $L$-functions.

## Explanations

### L1 human

The sequence $3, 7, 11, 15, 19, 23, \dots$ adds $4$ at each step. It contains infinitely many primes.
In general, a sequence $a, a + d, a + 2d, \dots$ contains infinitely many primes, unless $a$ and $d$ have a common factor that blocks them all.
</document>

<document source="https://tamnd.github.io/math2/node/0GP9ASVQ1X.md" tag="0GP9ASVQ1X" kind="definition">
# Chain

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
A subset $C$ of $P$ is a chain when, for all $a, b \in C$, $a \le b$ or $b \le a$.

## Remarks

So a chain is a subset on which $\le$ is a [[total order|0316AYH1SC]].
In the power set of $\mathbb{N}$, the sets $\{0\} \subseteq \{0, 1\} \subseteq \{0, 1, 2\} \subseteq \dots$ form a chain.
Chains are the key notion in [[Zorn's lemma|04RXTQA5ZA]] and in the [[Hausdorff maximal principle|0KM4KVR9MV]].

## Explanations

### L1 human

A chain is a part of the order that lies on one line.
In the divisors of $12$, the set $\{1, 2, 4, 12\}$ is a chain, but $\{2, 3\}$ is not.
</document>

<document source="https://tamnd.github.io/math2/node/0GPBVKE31D.md" tag="0GPBVKE31D" kind="concept">
# Complete residue system

## Summary

A complete residue system modulo $n$ is a set of integers with exactly one element in each residue class modulo $n$.
The [[definition of a complete residue system|0E22JKS4W4]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0GPFKS96T7.md" tag="0GPFKS96T7" kind="statement">
# Cardinality of the power set

## Statement

Let $A$ be a [[set|0SJ3RX9TXA]].
Then the [[power set|0BC7KX51Z8]] of $A$ is equinumerous with the set $\{0, 1\}^A$ of functions from $A$ to $\{0, 1\}$, so
$$|\mathcal{P}(A)| = 2^{|A|}.$$

## Remarks

The bijection sends a subset $S$ to its [[indicator function|0VSB0XF4G7]] $\mathbf{1}_S$.
For a finite set with $n$ elements, the power set has $2^n$ elements.
For $A = \mathbb{N}$, $|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}$, the [[cardinality of the continuum|0WC28W9DNM]].

## Explanations

### L1 human

A subset is a choice of "in" or "out" for each element.
With $n$ elements, there are $2 \cdot 2 \cdots 2 = 2^n$ such choices.
</document>

<document source="https://tamnd.github.io/math2/node/0GPG24Q3GR.md" tag="0GPG24Q3GR" kind="concept">
# Cauchy distribution

## Summary

The Cauchy distribution is a continuous distribution on $\mathbb{R}$ with a bell-shaped density and heavy tails. It has no mean and no variance.
The node [[Cauchy distribution|0NA6Q82YR7]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0GPQ67XV7N.md" tag="0GPQ67XV7N" kind="statement">
# The medians meet at the centroid

## Statement

Let $ABC$ be a triangle in the Euclidean space $\mathbb{R}^n$, and let $G = \frac{1}{3}(A + B + C)$ be its [[centroid|0X7EEWES5G]].
Then $G$ lies on each of the three [[medians|0XW8HX8VEV]] of $ABC$.
More precisely, let $M_A$ be the midpoint of $B$ and $C$. Then $G = A + \frac{2}{3}(M_A - A)$, and the same holds for the vertices $B$ and $C$.

## Remarks

The formula shows that $G$ divides each median in the ratio $2 : 1$, counted from the vertex.
The proof is one line: $A + \frac{2}{3}\left(\frac{B + C}{2} - A\right) = \frac{1}{3}(A + B + C)$.
The statement is true in each affine space over a field in which $2$ and $3$ are invertible. Mathlib states it for each simplex.

## Explanations

### L1 human

Draw the three medians of a paper triangle. They always cross at one point. This point is two thirds of the way from each vertex to the middle of the opposite side.
</document>

<document source="https://tamnd.github.io/math2/node/0GQ3H7VW3V.md" tag="0GQ3H7VW3V" kind="definition">
# Skewness

## Definition

Let $X$ be a real random variable with $E[|X|^3] < \infty$, mean $\mu$ and standard deviation $\sigma > 0$.
The skewness of $X$ is
$$\gamma_1 = E\left[\left(\frac{X - \mu}{\sigma}\right)^3\right] = \frac{\mu_3}{\sigma^3},$$
where $\mu_3$ is the third [[central moment|0ZT24EC90K]].

## Remarks

A distribution that is symmetric about its mean has skewness $0$. The converse is false.
Positive skewness usually means a longer tail on the right. Negative skewness usually means a longer tail on the left.
The skewness does not change under $X \mapsto aX + b$ with $a > 0$. For $a < 0$, it changes sign.
The exponential distribution has skewness $2$.

## Explanations

### L1 human

Skewness tells to which side a distribution leans.
Incomes have positive skewness: most values are moderate, and a few are very large.
</document>

<document source="https://tamnd.github.io/math2/node/0GQ938GDE0.md" tag="0GQ938GDE0" kind="concept">
# Independence of random variables

## Summary

A family of random variables is independent, or mutually independent, when events about different members of the family are always mutually independent.
The node [[independence of random variables|0XCE070V8B]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0GR2APWTAC.md" tag="0GR2APWTAC" kind="statement">
# Quotient of continuous functions is continuous

## Statement

Let $D \subseteq \mathbb{R}$ and let $f, g \colon D \to \mathbb{R}$ be [[continuous|0P27W9RW0G]] functions with $g(x) \ne 0$ for all $x \in D$.
Then the quotient $f/g$, given by $(f/g)(x) = f(x)/g(x)$, is continuous.

## Remarks

The proof uses [[sequential continuity|0PAQM3H57X]] and the [[quotient rule for limits|03S6GWWKEZ]] of sequences.
If $g$ has zeros, then $f/g$ is continuous on the set where $g$ is not zero.
Mathlib states it as `Continuous.div`.

## Explanations

### L1 human

Division by a continuous function causes no problem away from its zeros.
Near a point where $g$ is not zero, $g$ stays away from zero.
</document>

<document source="https://tamnd.github.io/math2/node/0GRAATMGP1.md" tag="0GRAATMGP1" kind="concept">
# Wedge product of forms

## Summary

The wedge product combines a $k$-form and an $l$-form into a $(k + l)$-form.
It is bilinear and alternating, and it encodes signed areas and volumes. It makes the [[differential forms|0E3G0GRPZ6]] on a set into an algebra.
</document>

<document source="https://tamnd.github.io/math2/node/0GRM3BS0DD.md" tag="0GRM3BS0DD" kind="counterexample">
# Curl-free field that is not conservative

## Counterexample

One could guess that each $C^1$ vector field with zero [[curl|0GDNHS9J2X]] on an open set is [[conservative|05R9NF2J49]].
This is false.
Let $U = \{(x, y, z) \in \mathbb{R}^3 : (x, y) \ne (0, 0)\}$, which is $\mathbb{R}^3$ minus the $z$-axis, and let
$$F(x, y, z) = \left( \frac{-y}{x^2 + y^2}, \ \frac{x}{x^2 + y^2}, \ 0 \right).$$
Then $\nabla \times F = 0$ on $U$, but $F$ is not conservative on $U$.

## Remarks

The third component of $\nabla \times F$ is $\partial_x \big( x / (x^2 + y^2) \big) - \partial_y \big( -y / (x^2 + y^2) \big) = \frac{y^2 - x^2}{(x^2 + y^2)^2} - \frac{y^2 - x^2}{(x^2 + y^2)^2} = 0$, and the other two components are zero.
On the unit circle $\gamma(t) = (\cos t, \sin t, 0)$, $t \in [0, 2\pi]$, we have $F(\gamma(t)) \cdot \gamma'(t) = 1$. So the circulation is $2\pi \ne 0$, and [[path independence|0BFN21TF5F]] fails.
The domain $U$ is not [[simply connected|075C629XTH]], so this does not contradict the theorem for [[convex|0B22P010D6]] or simply connected sets.
Locally, $F = \nabla \theta$, where $\theta$ is the polar angle of $(x, y)$. But no continuous choice of $\theta$ exists on all of $U$.

## Explanations

### L1 human

The field circles around the $z$-axis, and its strength falls off with distance in just the right way to have no local swirl.
But a loop around the axis picks up work $2\pi$, so no potential can exist. The hole in the domain causes this.
</document>

<document source="https://tamnd.github.io/math2/node/0GS0AW6JRY.md" tag="0GS0AW6JRY" kind="statement">
# Angles in the same segment are equal

## Statement

Let $A$, $B$, $C$ and $D$ be distinct [[concyclic points|049DCJZ448]] of the Euclidean plane $\mathbb{R}^2$. Let $\measuredangle$ be the [[oriented angle|03YN70NK0Z]], with values in $\mathbb{R} / 2\pi\mathbb{Z}$.
Then
$$2 \, \measuredangle ACB = 2 \, \measuredangle ADB.$$
For the unoriented angles: if $C$ and $D$ are on the same side of the line $AB$, then $\angle ACB = \angle ADB$. If they are on opposite sides, then $\angle ACB + \angle ADB = \pi$.

## Remarks

The statement follows from the [[inscribed angle theorem|0GA1V8NN8C]]: both sides are equal to the oriented angle $\measuredangle AOB$ at the centre $O$.
The factor $2$ is necessary. It removes the difference of $\pi$ between points on the two arcs.
The case of opposite sides says that the opposite angles of a convex cyclic quadrilateral add up to $\pi$.

## Explanations

### L1 human

A segment of a circle is the region between a chord $AB$ and one of its arcs. From every point of that arc, the chord $AB$ is seen under the same angle. From the other arc, it is seen under the supplementary angle.
</document>

<document source="https://tamnd.github.io/math2/node/0GS27TKF1V.md" tag="0GS27TKF1V" kind="definition">
# Completely multiplicative function

## Definition

An [[arithmetic function|0H2EPRZC0D]] $f$ is completely multiplicative if $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n$.

## Remarks

Each completely multiplicative function is [[multiplicative|0AP80TT660]]. The converse fails: [[Euler's totient function|0VV04M41AN]] has $\varphi(4) = 2$ but $\varphi(2)^2 = 1$.
A completely multiplicative function is determined by its values at the primes.
Examples are $n \mapsto n^k$, the [[Liouville function|08RQAPXF1T]] and, for a fixed odd prime $p$, the [[Legendre symbol|01J8AMV9RS]] $n \mapsto \left(\frac{n}{p}\right)$.

## Explanations

### L1 human

A completely multiplicative function respects every product, not only products of coprime numbers.
So it is enough to know its values at the primes, as for the function $n \mapsto n^2$.
</document>

<document source="https://tamnd.github.io/math2/node/0GS74FGFEK.md" tag="0GS74FGFEK" kind="definition">
# Degree of a field extension

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
The field $L$ is a vector space over $K$, with the addition of $L$ and the multiplication $K \times L \to L$ of $L$.
The degree of $L/K$ is the dimension of this vector space, and it is written $[L : K]$.

## Remarks

The degree is a positive integer or an infinite cardinal.
One has $[L : K] = 1$ if and only if $L = K$.
Examples: $[\mathbb{C} : \mathbb{R}] = 2$, with the basis $1, i$. Also, $[\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2$, with the basis $1, \sqrt{2}$.
The field $\mathbb{R}$ has infinite degree over $\mathbb{Q}$, because $\mathbb{Q}$ is countable and $\mathbb{R}$ is not.

## Explanations

### L1 human

The degree counts how many elements of the big field you need, as a basis, to write every element with coefficients in the small field.
Every complex number is $a + b i$ with $a, b$ real, so the degree of $\mathbb{C}$ over $\mathbb{R}$ is $2$.
</document>

<document source="https://tamnd.github.io/math2/node/0GSDHYMA1T.md" tag="0GSDHYMA1T" kind="statement">
# Thales's theorem

## Statement

Let $A$, $B$ and $C$ be distinct points of a [[circle|060W5CBZP6]] in the Euclidean plane $\mathbb{R}^2$. Suppose that $AC$ is a diameter, so that the centre is the [[midpoint|0S706JBZBG]] of $A$ and $C$.
Then the [[angle|0GJAJC6TNZ]] $\angle ABC$ is a right angle: $\angle ABC = \pi / 2$.
Conversely, if $A$, $B$, $C$ are distinct points with $\angle ABC = \pi / 2$, then $B$ is on the circle with the diameter $AC$.

## Remarks

Thales's theorem is the special case of the inscribed angle theorem for a central angle of $\pi$.
An equivalent form: the circumcentre of a right triangle is the midpoint of its hypotenuse.
A proof: let $O$ be the centre. The triangles $OAB$ and $OBC$ are isosceles, and the angle sum of $ABC$ is $\pi$.

## History

The theorem is part of Proposition 31 of Book III of Euclid's Elements. Tradition gives it to Thales of Miletus, and sometimes to Pythagoras.

## Explanations

### L1 human

Take a diameter of a circle and a third point on the circle. Join this point to the two ends of the diameter. The angle at the third point is always a right angle.
</document>

<document source="https://tamnd.github.io/math2/node/0GSW76Q14N.md" tag="0GSW76Q14N" kind="concept">
# Ideal generated by a subset

## Summary

The ideal generated by a subset of a ring is the smallest ideal that contains the subset.
In a commutative ring, it is the set of the finite linear combinations of elements of the subset.
The [[definition of the generated ideal|030TZ0DQWT]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0GSZ7B0CGR.md" tag="0GSZ7B0CGR" kind="object">
# Extended real number line

## Description

The extended real number line is the set $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$.
It has two new elements, $-\infty$ and $+\infty$, that are not [[real numbers|08J2M1JYAG]].
Its order extends the order of $\mathbb{R}$, with $-\infty < x < +\infty$ for each real number $x$.

## Remarks

In $\overline{\mathbb{R}}$, each subset has a [[supremum|0BTMQ04556]] and an infimum. For example, $\sup \mathbb{N} = +\infty$ and $\sup \emptyset = -\infty$.
Some operations extend: $x + \infty = +\infty$ for each real $x$. Other expressions, such as $\infty - \infty$, stay undefined.
The set $\overline{\mathbb{R}}$ is not a field.

## Explanations

### L1 human

The extended real line adds two end points to the number line, one at each end.
This lets you write $\lim_{x \to \infty}$ and $\sup$ of unbounded sets with symbols instead of words.
</document>

<document source="https://tamnd.github.io/math2/node/0GT4WWGWH2.md" tag="0GT4WWGWH2" kind="concept">
# System of linear equations

## Summary

A system of linear equations is a finite collection of linear equations in the same unknowns.
A solution of the system is a solution of all the equations at the same time.
</document>

<document source="https://tamnd.github.io/math2/node/0GT85FBPTS.md" tag="0GT85FBPTS" kind="example">
# Decimal expansion as a series

## Example

Let $(d_n)_{n \ge 1}$ be a sequence of digits, with $d_n \in \{0, 1, \dots, 9\}$.
The decimal $0.d_1 d_2 d_3 \dots$ is the sum of the series
$$\sum_{n=1}^{\infty} \frac{d_n}{10^n}.$$
This series converges, and its sum lies in $[0, 1]$.

## Remarks

The series converges by the [[comparison test|04Z5G0THN6]] with the [[geometric series|01X5CVV1Q5]] $\sum 9/10^n = 1$.
Two different digit sequences can give the same number. For example, $0.4999\ldots = 0.5$.
Conversely, each real number in $[0, 1)$ has a decimal expansion.

## Explanations

### L1 human

A decimal like $0.314$ means $3/10 + 1/100 + 4/1000$.
An infinite decimal is the same idea with infinitely many terms, so it is a series.
</document>

<document source="https://tamnd.github.io/math2/node/0GTBGMZXV8.md" tag="0GTBGMZXV8" kind="definition">
# Triple integral

## Definition

Let $B = [a_1, b_1] \times [a_2, b_2] \times [a_3, b_3]$ with $a_i < b_i$, and let $f \colon B \to \mathbb{R}$ be bounded.
Partition each interval $[a_i, b_i]$, and let $B_1, \dots, B_N$ be the resulting closed sub-boxes, with volumes $\Delta V_1, \dots, \Delta V_N$.
For points $p_k \in B_k$, the sum $S = \sum_{k=1}^{N} f(p_k) \, \Delta V_k$ is a Riemann sum of $f$.
The function $f$ is Riemann integrable on $B$ if there is $I \in \mathbb{R}$ with this property. For each $\varepsilon > 0$, there is $\delta > 0$ with $|S - I| < \varepsilon$ for all Riemann sums whose sub-boxes have diameter less than $\delta$.
Then $I$ is the triple integral of $f$ over $B$, written $\iiint_B f(x, y, z) \, dV$.

## Remarks

This is the case $n = 3$ of the multiple integral over a box. The case $n = 2$ is the [[double integral|0NKBMFC18R]].
For a bounded solid $E \subseteq B$, the integral over $E$ is the integral over $B$ of $f$ times the indicator function of $E$. The volume of $E$ is $\iiint_E 1 \, dV$, when this integral exists.
By [[Fubini's theorem|0R0T1VWH5X]], for continuous $f$ the triple integral equals an iterated integral. For example, $\iiint_{[0,1]^3} x y z \, dV = (1/2)^3 = 1/8$.

## Explanations

### L1 human

A triple integral adds up the values of $f$ over a solid, each weighted by the volume of a small piece.
If $f$ is a density, the triple integral is the total mass of the solid.
</document>

<document source="https://tamnd.github.io/math2/node/0GTEHJQRFK.md" tag="0GTEHJQRFK" kind="definition">
# Gaussian prime

## Definition

A Gaussian prime is a prime element of the ring $\mathbb{Z}[i]$ of [[Gaussian integers|0B268GVX95]].
That is, it is a nonzero Gaussian integer $\pi$ that is not a unit, with the following property.
For all Gaussian integers $\alpha$ and $\beta$, if $\pi \mid \alpha \beta$, then $\pi \mid \alpha$ or $\pi \mid \beta$.

## Remarks

Since $\mathbb{Z}[i]$ is a [[unique factorization domain|0SPYEFH7H4]], the Gaussian primes are also the irreducible elements: the non-units that are not a product of two non-units.
A Gaussian integer $\alpha$ is a Gaussian prime if and only if one of two conditions holds.
Either $N(\alpha)$ is a prime number, or $\alpha$ is a unit times a prime $p \equiv 3 \pmod 4$.
For example, $1 + i$, $2 + i$ and $3$ are Gaussian primes. But $5 = (2 + i)(2 - i)$ is not.

## Explanations

### L1 human

Gaussian primes play the role of prime numbers in the Gaussian integers.
Some ordinary primes, such as $5$, split into two Gaussian primes. Others, such as $3$, stay prime.
</document>

<document source="https://tamnd.github.io/math2/node/0GTZ2F3PQY.md" tag="0GTZ2F3PQY" kind="statement">
# Cavalieri's principle

## Statement

Let $E \subseteq \mathbb{R}^3$ be a Jordan measurable solid that lies between the planes $z = a$ and $z = b$.
For $z \in [a, b]$, let $E_z = \{(x, y) : (x, y, z) \in E\}$ be its cross-section at height $z$, and suppose that each $E_z$ is Jordan measurable with area $A(z)$.
Then
$$\operatorname{vol}(E) = \int_a^b A(z) \, dz.$$
In particular, if two such solids have cross-sections of equal area at each height $z$, then they have equal volume.

## Remarks

This follows from [[Fubini's theorem|0R0T1VWH5X]] applied to the indicator function of $E$.
Example: the ball of radius $r$ has cross-sections of area $\pi (r^2 - z^2)$. A cylinder of radius $r$ and height $r$ with a cone removed has the same cross-sections, which gives the volume $\tfrac{2}{3} \pi r^3$ for a half ball.
The plane version says that two regions with equal chord lengths on each line parallel to a fixed line have equal area.
Archimedes used a similar method in the 3rd century BC. Cavalieri developed it as the method of indivisibles in 1635.

## Explanations

### L1 human

Cut two solids into thin horizontal slices.
Suppose that each slice of one has the same area as the slice of the other at the same height. Then the two solids have the same volume, like two stacks of coins of the same size.
</document>

<document source="https://tamnd.github.io/math2/node/0GV2573FP4.md" tag="0GV2573FP4" kind="statement">
# Bijection principle

## Statement

Let $A$ and $B$ be finite sets.
If there is a bijection $f : A \to B$, then $|A| = |B|$.
Conversely, if $|A| = |B|$, then there is a bijection from $A$ to $B$.

## Remarks

A bijective proof shows that two sets have the same size by a bijection between them. The proof of the [[symmetry of binomial coefficients|0R7JS3DP3W]] is a standard example. It uses the map $S \mapsto X \setminus S$ on the subsets of a finite set $X$.
[[Dyck words|0R9JR0465V]] and [[binary trees|02A8EMC83C]] are both counted by [[Catalan numbers|092DRBFKMG]], and a bijection explains this.

## Explanations

### L1 human

To show that two collections have the same size, you can pair each element of one with exactly one element of the other.
For example, the subsets of $\{1, 2, 3\}$ pair with the words of length 3 in the letters 0 and 1. A subset gives the word with 1 in the positions of its elements.
</document>

<document source="https://tamnd.github.io/math2/node/0GVDFAY4KW.md" tag="0GVDFAY4KW" kind="concept">
# Disjunctive normal form

## Summary

A formula in disjunctive normal form is a disjunction of conjunctions of literals. Every Boolean function has such a form, which can be read off from its truth table. The node [[disjunctive normal form|06S4WGSEQA]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0GVMR0C3S6.md" tag="0GVMR0C3S6" kind="definition">
# Moment of inertia

## Definition

Let $E \subseteq \mathbb{R}^3$ be a bounded Jordan measurable set with a continuous density $\delta \colon E \to [0, \infty)$, and let $L$ be a line in $\mathbb{R}^3$.
For $x \in E$, let $d_L(x)$ be the distance from $x$ to $L$.
The moment of inertia of $E$ about $L$ is
$$I_L = \iiint_E d_L(x)^2 \, \delta(x) \, dV.$$
In particular, $I_z = \iiint_E (x^2 + y^2) \, \delta(x, y, z) \, dV$ is the moment of inertia about the $z$-axis.

## Remarks

For a plane lamina $D \subseteq \mathbb{R}^2$, the moments of inertia about the coordinate axes are $I_x = \iint_D y^2 \delta \, dA$ and $I_y = \iint_D x^2 \delta \, dA$. These are [[double integrals|0NKBMFC18R]].
For a point mass $m$ at distance $r$ from the axis, the moment of inertia is $m r^2$.
Example: a solid cylinder of radius $a$, height $h$ and constant density has mass $M = \delta \pi a^2 h$ and $I_z = \delta \int_0^h \int_0^{2\pi} \int_0^a r^3 \, dr \, d\theta \, dz = \tfrac{1}{2} M a^2$.
For rotation about $L$ with angular velocity $\omega$, the kinetic energy is $\tfrac{1}{2} I_L \omega^2$.

## Explanations

### L1 human

Mass far from the axis is harder to set spinning than mass close to it.
The moment of inertia adds up all the mass, weighted by the square of its distance from the axis.
</document>

<document source="https://tamnd.github.io/math2/node/0GW1ZZH0R4.md" tag="0GW1ZZH0R4" kind="statement">
# Fundamental theorem of symmetric polynomials

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $f \in R[x_1, \dots, x_n]$ be [[symmetric|06CTFY2DEZ]].
Then there is a unique polynomial $g \in R[y_1, \dots, y_n]$ with
$$f = g(e_1, \dots, e_n),$$
where $e_1, \dots, e_n$ are the [[elementary symmetric polynomials|0ZMY8441GW]].

## Remarks

Equivalently, the map $R[y_1, \dots, y_n] \to R[x_1, \dots, x_n]$, $y_k \mapsto e_k$, is injective, and its image is the ring of symmetric polynomials.
One proof is by induction on the leading monomial in the lexicographic order. Each step subtracts a product of the $e_k$ with the same leading monomial.
Example: $x_1^2 + x_2^2 = e_1^2 - 2 e_2$.
So a symmetric expression in the roots of a polynomial over a field $K$ is an element of $K$.

## Explanations

### L1 human

Every symmetric polynomial can be written in exactly one way using the elementary symmetric polynomials.
The coefficients stay in the same ring.
</document>

<document source="https://tamnd.github.io/math2/node/0GWD21EWRW.md" tag="0GWD21EWRW" kind="definition">
# Hyperbolic sine

## Definition

The hyperbolic sine is the function $\sinh \colon \mathbb{R} \to \mathbb{R}$ defined by
$$\sinh x = \frac{e^x - e^{-x}}{2}.$$

## Remarks

It is odd and strictly increasing, and it is a bijection of $\mathbb{R}$ onto $\mathbb{R}$.
Its power series is $\sinh x = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}$, the sine series without the signs.
Mathlib defines it as `Real.sinh`.

## Explanations

### L1 human

The hyperbolic sine is the odd part of $e^x$.
For large $x$, it is close to $e^x / 2$.
</document>

<document source="https://tamnd.github.io/math2/node/0GWN3CMRK8.md" tag="0GWN3CMRK8" kind="concept">
# One-point compactification

## Summary

The one-point compactification of a topological space adds one point at infinity and makes the space compact.
For example, the plane with one added point is the sphere.
</document>

<document source="https://tamnd.github.io/math2/node/0GXA8Z205E.md" tag="0GXA8Z205E" kind="concept">
# Area under a curve

## Summary

For a nonnegative continuous function $f$ on $[a, b]$, the area under its graph is the [[integral|0QM4TXT3DR]] $\int_a^b f(x) \, dx$. The integral approximates the region by thin rectangles, through [[Riemann sums|0HBY48G8G1]]. For a function of any sign, the integral counts area below the $x$-axis as negative.

## Remarks

The area between the graphs of $f \le g$ over $[a, b]$ is $\int_a^b (g(x) - f(x)) \, dx$.
For example, the area under $x^2$ over $[0, 1]$ is $1/3$. In the 3rd century BC, Archimedes found the area under a parabola by the method of exhaustion.
The [[fundamental theorem of calculus|0NHGY0N0JB]] computes areas from antiderivatives.

## Explanations

### L1 human

The integral measures the area of the region under a graph.
We fill the region with thin rectangles and add up their areas.
</document>

<document source="https://tamnd.github.io/math2/node/0GXCXCX3QN.md" tag="0GXCXCX3QN" kind="statement">
# Integral of sine

## Statement

For all real numbers $a$ and $b$,
$$\int_a^b \sin x \, dx = \cos a - \cos b.$$

## Remarks

The [[derivative of cosine|0V2DV29EGT]] is $-\sin$, so $-\cos$ is an antiderivative of $\sin$.
For example, $\int_0^\pi \sin x \, dx = 2$.
Mathlib states it as `integral_sin`.

## Explanations

### L1 human

The area under one arch of the sine curve is exactly $2$.
This follows from the antiderivative $-\cos x$.
</document>

<document source="https://tamnd.github.io/math2/node/0GXKM1R7E4.md" tag="0GXKM1R7E4" kind="concept">
# Telescoping series

## Summary

A telescoping series is a series whose terms have the form $a_n = b_n - b_{n+1}$ for a sequence $(b_n)$. Its partial sums collapse: $\sum_{n=0}^{N} (b_n - b_{n+1}) = b_0 - b_{N+1}$. Thus the series converges if and only if $(b_n)$ converges, and then its sum is $b_0 - \lim b_n$.

## Remarks

For example, $\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$, so $\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = 1$.
The finite form $\sum_{k=0}^{n-1} (f(k+1) - f(k)) = f(n) - f(0)$ is a discrete analogue of the [[fundamental theorem of calculus|0NHGY0N0JB]]. Mathlib states it as `Finset.sum_range_sub`.

## Explanations

### L1 human

In a telescoping sum, each term cancels part of the next term, like the segments of a telescope that slide into each other.
Only the first piece and the last piece survive.
</document>

<document source="https://tamnd.github.io/math2/node/0GXMPDNYX3.md" tag="0GXMPDNYX3" kind="concept">
# Laurent series

## Summary

A Laurent series is a series $\sum_{n \in \mathbb{Z}} a_n (z - c)^n$ with powers of $z - c$ of all integer degrees, see the [[definition|0X3XD2ATB0]].
It represents holomorphic functions on annuli, also near isolated singularities where a Taylor series does not exist.
</document>

<document source="https://tamnd.github.io/math2/node/0GXNHEHQNA.md" tag="0GXNHEHQNA" kind="statement">
# Units of Z/nZ are the residues coprime to n

## Statement

Let $n \geq 1$ and $a$ be integers.
The residue class $[a]$ is a unit of the ring $\mathbb{Z}/n\mathbb{Z}$ if and only if $\gcd(a, n) = 1$.

## Remarks

If $\gcd(a, n) = 1$, then Bézout's lemma gives integers $x$ and $y$ with $a x + n y = 1$. So $[a][x] = [1]$.
If $[a][x] = [1]$, then $a x - 1$ is a multiple of $n$, so each common divisor of $a$ and $n$ divides $1$.
The condition does not depend on the representative, because $a \equiv b \pmod{n}$ implies $\gcd(a, n) = \gcd(b, n)$.
For $n = 12$, the units are $[1]$, $[5]$, $[7]$ and $[11]$.

## Explanations

### L1 human

A residue has a multiplicative inverse modulo $n$ exactly when it has no common factor with $n$.
Bézout's lemma gives the inverse.
So the [[group of units of Z/nZ|0F8X8ZRQ53]] is the set of residues coprime to $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0GYDV9FE98.md" tag="0GYDV9FE98" kind="statement">
# Second derivative test in several variables

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$ be of [[class C^2|0TCBZMEYGN]] on $U$, and let $a \in U$ with $\nabla f(a) = 0$.
Let $H = H_f(a)$ be the [[Hessian matrix|0SCSQWXRA4]] of $f$ at $a$.

1. If $H$ is [[positive definite|01RBSSGMY5]], then $f$ has a strict [[local minimum|04CPG9K1C4]] at $a$.
2. If $H$ is negative definite, then $f$ has a strict local maximum at $a$.
3. If $H$ has a positive and a negative eigenvalue, then $a$ is a [[saddle point|00N3T18QNA]] of $f$.

In the other cases the test gives no conclusion.

## Remarks

For $n = 2$, let $D = f_{xx}(a) f_{yy}(a) - f_{xy}(a)^2$ be the determinant of $H$. If $D > 0$ and $f_{xx}(a) > 0$, then $a$ is a local minimum. If $D > 0$ and $f_{xx}(a) < 0$, then $a$ is a local maximum. If $D < 0$, then $a$ is a saddle point.
The proof uses the [[second-order Taylor expansion|0P30YYZN85]]. If $H$ is positive definite, then $h^{\mathsf{T}} H h \ge m \|h\|^2$ for some $m > 0$, and this term dominates the remainder.
The functions $x^4 + y^4$, $-(x^4 + y^4)$ and $x^4 - y^4$ all have zero Hessian at the origin. The origin is a minimum, a maximum and a saddle point, respectively, so the test cannot decide when $H$ is singular and semidefinite.
By Sylvester's criterion, $H$ is positive definite if and only if all its leading principal minors are positive.

## Explanations

### L1 human

At a critical point the tangent plane is flat, so the second derivatives decide the shape of the graph.
If the graph bends upward in every direction, the point is a minimum. If it bends down in every direction, the point is a maximum.
If it bends up in some directions and down in others, the point is a saddle.
</document>

<document source="https://tamnd.github.io/math2/node/0GYGHKX834.md" tag="0GYGHKX834" kind="definition">
# Commutative ring

## Definition

A commutative ring is a [[ring|0894J980RK]] $R$ such that $a \cdot b = b \cdot a$ for all $a, b \in R$.

## Remarks

The integers $\mathbb{Z}$, the rational numbers $\mathbb{Q}$ and the polynomials in one variable over $\mathbb{Q}$ are commutative rings.
The ring of $2 \times 2$ real matrices is not commutative.
Every [[field|0QA8CHH8MG]] is a commutative ring.

## Explanations

### L1 human

In a commutative ring, the order of the factors in a product does not matter.
The integers are the model: $3 \cdot 5 = 5 \cdot 3$.
</document>

<document source="https://tamnd.github.io/math2/node/0GYQK9NDJF.md" tag="0GYQK9NDJF" kind="concept">
# Cayley graph

## Summary

The Cayley graph of a group $G$ with a generating set $S$ has the elements of $G$ as vertices. It has an edge from $g$ to $g s$ for each $g \in G$ and $s \in S$.
It encodes the structure of the group as a graph.
</document>

<document source="https://tamnd.github.io/math2/node/0GZ3D8F921.md" tag="0GZ3D8F921" kind="statement">
# Divergence theorem on a rectangle in the plane

## Statement

Let $a < b$ and $c < d$, and let $R = [a, b] \times [c, d]$.
Let $F = (P, Q)$ be a vector field of class $C^1$ on an open set that contains $R$.
Then
$$\iint_R \left( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \right) dA = \int_c^d \big( P(b, y) - P(a, y) \big) \, dy + \int_a^b \big( Q(x, d) - Q(x, c) \big) \, dx.$$

## Remarks

The right side is the outward flux $\oint_{\partial R} F \cdot n \, ds$. Here the outward unit normal $n$ is $(1, 0)$, $(-1, 0)$, $(0, 1)$ and $(0, -1)$ on the right, left, top and bottom sides.
Proof: by [[Fubini's theorem|0R0T1VWH5X]], $\iint_R \partial P / \partial x \, dA = \int_c^d \int_a^b \partial P / \partial x \, dx \, dy$. The fundamental theorem of calculus turns the inner integral into $P(b, y) - P(a, y)$. The $Q$ term is similar.
This is the simplest case of the [[divergence theorem|0JX86PSSA9]]. In the plane, the divergence theorem is equivalent to [[Green's theorem|0HR3N02317]].

## Explanations

### L1 human

The total source strength inside a rectangle equals the net flow out through its four sides.
In each coordinate direction, this is the fundamental theorem of calculus, applied along each line through the rectangle.
</document>

<document source="https://tamnd.github.io/math2/node/0GZ6Y1H4RN.md" tag="0GZ6Y1H4RN" kind="concept">
# Ascending chain condition

## Summary

A partially ordered set satisfies the ascending chain condition when every increasing sequence of its elements is eventually constant.
The [[definition of the ascending chain condition|0ZCAXY9563]] states this precisely.
</document>

<document source="https://tamnd.github.io/math2/node/0GZB2SN0PH.md" tag="0GZB2SN0PH" kind="example">
# Function space

## Example

Let $F$ be a field, let $X$ be a set, and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
The set $V^X$ of all functions $f \colon X \to V$ is a vector space over $F$ with the pointwise operations
$$(f + g)(x) = f(x) + g(x), \qquad (af)(x) = a\, f(x)$$
for $f, g \in V^X$, $a \in F$ and $x \in X$.
The zero vector is the function with value $0$ at each point.

## Remarks

For $X = \{1, \dots, n\}$ and $V = F$, this is the [[coordinate space|0XM8BZD76G]] $F^n$.
The continuous functions $[0, 1] \to \R$ form a subspace of $\R^{[0, 1]}$.
If $X$ is infinite and $V \ne 0$, then $V^X$ is infinite-dimensional.

## Explanations

### L1 human

We add two functions by adding their values at each point.
We scale a function by scaling its values.
With these rules, functions behave like vectors, and the zero function is the zero vector.
</document>

<document source="https://tamnd.github.io/math2/node/0GZNSHMG4Q.md" tag="0GZNSHMG4Q" kind="example">
# Dirichlet integral

## Example

The improper integral of $\frac{\sin x}{x}$ over $(0, \infty)$ converges, and
$$\int_0^\infty \frac{\sin x}{x} \, dx = \frac{\pi}{2}.$$
The integrand extends continuously to $x = 0$ with value $1$. So only the upper limit is improper.
[[Integration by parts|08Y1BBQTK9]] gives $\int_1^b \frac{\sin x}{x} \, dx = \cos 1 - \frac{\cos b}{b} - \int_1^b \frac{\cos x}{x^2} \, dx$. The last integral converges absolutely, so the limit exists.

## Remarks

The integral does not converge [[absolutely|03GP7KMBYM]]: $\int_0^\infty \frac{|\sin x|}{x} \, dx = \infty$.
So $\frac{\sin x}{x}$ is not Lebesgue integrable on $(0, \infty)$.
The value $\pi/2$ needs other methods, such as differentiation under the integral sign or contour integration.

## Explanations

### L1 human

The area under $\frac{\sin x}{x}$ alternates in sign, and the pieces shrink.
The positive and negative pieces cancel to give the value $\pi/2$.
</document>

<document source="https://tamnd.github.io/math2/node/0GZWXM2DKE.md" tag="0GZWXM2DKE" kind="concept">
# Characteristic equation

## Summary

The characteristic equation of a homogeneous linear differential equation with constant coefficients is the polynomial equation obtained from the trial solution $e^{r x}$.
Its roots determine the solutions of the differential equation. See the [[definition|0QDXMG46QW]].
</document>

<document source="https://tamnd.github.io/math2/node/0H11ZPX4ND.md" tag="0H11ZPX4ND" kind="concept">
# Conjugate subgroup

## Summary

A conjugate of a subgroup $H$ of a group $G$ is a subgroup $gHg^{-1}$ for some $g \in G$.
Conjugate subgroups are isomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/0H1GN6P9R5.md" tag="0H1GN6P9R5" kind="concept">
# Fundamental matrix

## Summary

A fundamental matrix of a homogeneous linear system $x' = A(t) x$ is a matrix function whose columns are $n$ linearly independent solutions.
It is the matrix form of a [[fundamental system of solutions|0DS2QFYJHE]]. See the [[definition|0MQ24RBG93]].
</document>

<document source="https://tamnd.github.io/math2/node/0H1JFKE7H4.md" tag="0H1JFKE7H4" kind="concept">
# Hyperbolic sine

## Summary

The hyperbolic sine is half the difference of $e^x$ and $e^{-x}$. It is an odd, strictly increasing function on $\mathbb{R}$. The node [[hyperbolic sine|0GWD21EWRW]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0H2EPRZC0D.md" tag="0H2EPRZC0D" kind="definition">
# Arithmetic function

## Definition

An arithmetic function is a function $f \colon \mathbb{Z}_{>0} \to \mathbb{C}$ from the positive integers to the complex numbers.

## Remarks

Examples are [[Euler's totient function|0VV04M41AN]] $\varphi$, the [[divisor function|0YEW47VJ0J]] $\sigma_k$ and the [[Möbius function|0C8QENHYZW]] $\mu$.
Some texts allow values in another commutative ring, or set $f(0) = 0$ and use all natural numbers as the domain.
The [[Dirichlet convolution|0KH1N66QWW]] makes the arithmetic functions into a commutative ring.

## Explanations

### L1 human

An arithmetic function is a sequence $f(1), f(2), f(3), \dots$ of numbers, seen as a function of $n$.
Number theory studies such functions when they describe arithmetic properties of $n$, such as its number of divisors.
</document>

<document source="https://tamnd.github.io/math2/node/0H3W614GEC.md" tag="0H3W614GEC" kind="definition">
# Pseudometric space

## Definition

Let $X$ be a set.
A pseudometric on $X$ is a function $d \colon X \times X \to \mathbb{R}$ such that, for all $x, y, z \in X$:

1. $d(x, x) = 0$.
2. $d(x, y) = d(y, x)$.
3. $d(x, z) \le d(x, y) + d(y, z)$.

A pseudometric space is a pair $(X, d)$, where $X$ is a set and $d$ is a pseudometric on $X$.

## Remarks

Each [[metric|063W5JMY85]] is a pseudometric. A pseudometric is a metric if and only if $d(x, y) = 0$ implies $x = y$.
On the set of functions $f \colon [0, 1] \to \mathbb{R}$, the rule $d(f, g) = |f(0) - g(0)|$ is a pseudometric, but not a metric.
The relation $d(x, y) = 0$ is an equivalence relation, and $d$ gives a metric on the set of its classes.

## Explanations

### L1 human

A pseudometric measures distance, but it can fail to see the difference between two points.
For example, measure the distance between two people by the difference of their ages. Then two people of the same age have distance zero.
</document>

<document source="https://tamnd.github.io/math2/node/0H3WZXCZNF.md" tag="0H3WZXCZNF" kind="concept">
# Semidirect product

## Summary

A semidirect product builds a group from a group $N$ and a group $H$ that acts on $N$ by automorphisms.
It generalises the direct product, and $N$ becomes a normal subgroup of the result.
</document>

<document source="https://tamnd.github.io/math2/node/0H3YY8CPKH.md" tag="0H3YY8CPKH" kind="object">
# Euclidean group

## Description

The Euclidean group $\mathrm{E}(n)$ is the set of all [[Euclidean isometries|0J1N3WTKM4]] of $\mathbb{R}^n$, with the composition of maps as the operation. It is a [[group|G5S1M9]]: the identity map is the neutral element, and the inverse of an isometry is an isometry.
Each element has the form $x \mapsto A x + b$, with $A$ in the orthogonal group $\mathrm{O}(n)$ and $b \in \mathbb{R}^n$.
The direct isometries, with $\det A = 1$, form a subgroup $\mathrm{SE}(n)$ of index $2$. Its elements are the rigid motions.

## Remarks

The [[translations|087VJM5Q77]] form an abelian normal subgroup $T \cong \mathbb{R}^n$, and $\mathrm{E}(n) / T \cong \mathrm{O}(n)$.
The group $\mathrm{E}(n)$ has $n(n+1)/2$ degrees of freedom: $3$ for $n = 2$ and $6$ for $n = 3$.
The group is not abelian for $n \geq 2$. For example, a rotation about the origin and a translation do not commute in general.
The symmetry group of each figure in $\mathbb{R}^n$ is a subgroup of $\mathrm{E}(n)$.

## Explanations

### L1 human

The Euclidean group collects all the ways to move space without changing distances. Two such moves done one after the other give again such a move, and each move can be undone. In the plane, three numbers fix a direct move: two for a shift and one for a turn.
</document>

<document source="https://tamnd.github.io/math2/node/0H47DMFN3S.md" tag="0H47DMFN3S" kind="definition">
# Inner automorphism

## Definition

Let $G$ be a [[group|G5S1M9]] and let $g \in G$.
The inner automorphism given by $g$ is the map $c_g \colon G \to G$, $c_g(x) = gxg^{-1}$.
The set of all inner automorphisms of $G$ is written $\operatorname{Inn}(G)$.

## Remarks

Each $c_g$ is an [[automorphism|0K1SCPBNKJ]] of $G$, with inverse $c_{g^{-1}}$.
Since $c_g \circ c_h = c_{gh}$, the map $g \mapsto c_g$ is a homomorphism from $G$ to $\operatorname{Aut}(G)$, and $\operatorname{Inn}(G)$ is its image.
The subgroup $\operatorname{Inn}(G)$ is normal in $\operatorname{Aut}(G)$, because $\alpha \circ c_g \circ \alpha^{-1} = c_{\alpha(g)}$ for each automorphism $\alpha$.
In an [[abelian group|082GHNPJ53]], each inner automorphism is the identity map.

## Explanations

### L1 human

Each element of a group gives a symmetry of the group: conjugate everything by this element.
These symmetries are called inner, because they come from inside the group.
In a group where all elements commute, conjugation changes nothing.
</document>

<document source="https://tamnd.github.io/math2/node/0H4KNNCHPY.md" tag="0H4KNNCHPY" kind="definition">
# Fundamental system of solutions

## Definition

Let $I$ be an open interval and let $a_0, \dots, a_{n-1} : I \to \mathbb{R}$ be continuous. Let $S$ be the vector space of solutions on $I$ of the [[homogeneous linear equation|0EQZSAB9EC]] $y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y = 0$.
A fundamental system of solutions of the equation is a basis $(y_1, \dots, y_n)$ of $S$.

## Remarks

Since $S$ has [[dimension n|0GK68PZ6KH]], a family of $n$ solutions is a fundamental system if and only if it is linearly independent. The [[Wronskian test|0MHE4B1QCQ]] checks this.
Each solution is then $c_1 y_1 + \dots + c_n y_n$ for unique constants $c_1, \dots, c_n$.
Example: $(\cos x, \sin x)$ is a fundamental system of $y'' + y = 0$ on $\mathbb{R}$.

## Explanations

### L1 human

A fundamental system is a set of building blocks.
Each solution of the homogeneous equation is a unique combination of these $n$ solutions.
</document>

<document source="https://tamnd.github.io/math2/node/0H4R3RESGV.md" tag="0H4R3RESGV" kind="definition">
# Limit inferior

## Definition

Let $(a_n)$ be a [[bounded sequence|05FQ3VGAR9]] of real numbers.
For each $n \in \mathbb{N}$, let $i_n = \inf \{a_k : k \ge n\}$.
The sequence $(i_n)$ is increasing and bounded above.
The limit inferior of $(a_n)$ is
$$\liminf_{n \to \infty} a_n = \lim_{n \to \infty} i_n = \sup_{n \in \mathbb{N}} \inf_{k \ge n} a_k.$$

## Remarks

The limit exists by the [[monotone convergence theorem|0FM6N4R0VB]].
It satisfies $\liminf a_n = -\limsup (-a_n)$, where $\limsup$ is the [[limit superior|0P8BA7T7S5]].
Always $\inf_n a_n \le \liminf a_n \le \limsup a_n \le \sup_n a_n$.
For $a_n = (-1)^n$, the limit inferior is $-1$ and the limit superior is $1$.

## Explanations

### L1 human

The limit inferior is the lowest value that the sequence keeps coming back to, in the long run.
Each number below it is undercut by only finitely many terms.
</document>

<document source="https://tamnd.github.io/math2/node/0H5GKDTB01.md" tag="0H5GKDTB01" kind="definition">
# Disjunction

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The disjunction of $p$ and $q$ is the proposition $p \lor q$, read "$p$ or $q$".
The proposition $p \lor q$ is false when $p$ is false and $q$ is false. In all other cases, $p \lor q$ is true.

| $p$ | $q$ | $p \lor q$ |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |

## Remarks

This "or" is inclusive: $p \lor q$ is true when both $p$ and $q$ are true.
The exclusive "or" is true when exactly one of $p$ and $q$ is true.
The [[union|07KVB4JSC7]] of two sets uses disjunction: $x \in A \cup B$ means $x \in A \lor x \in B$.

## Explanations

### L1 human

The sentence "$4$ is even or $4$ is odd" is a disjunction.
It is true, because one part is true.
In mathematics, "or" also allows both parts to be true.
</document>

<document source="https://tamnd.github.io/math2/node/0H5J02P8S9.md" tag="0H5J02P8S9" kind="definition">
# Upper bound

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.
An element $u \in P$ is an upper bound of $S$ when $s \le u$ for each $s \in S$.
The set $S$ is bounded above when it has an upper bound.

## Remarks

An upper bound need not be in $S$. The number $1$ is an upper bound of $[0, 1)$ in $\mathbb{R}$, and so is $5$.
A lower bound is defined in the same way with $u \le s$.
The least upper bound of $S$, when it exists, is the supremum of $S$.

## Explanations

### L1 human

An upper bound is a ceiling for the set: no element of the set is above it.
The set $\{1, 1/2, 1/3, \dots\}$ has upper bounds $1$, $2$, $100$ and many more.
</document>

<document source="https://tamnd.github.io/math2/node/0H5MR9B6GT.md" tag="0H5MR9B6GT" kind="concept">
# Line integral of a vector field

## Summary

The line integral of a [[vector field|0299VCW1TH]] along an oriented curve adds up the component of the field along the direction of motion.
In physics it is the work that a force field does on a particle that moves along the curve.
</document>

<document source="https://tamnd.github.io/math2/node/0H5SYBSED9.md" tag="0H5SYBSED9" kind="concept">
# Gamma function

## Summary

The gamma function extends the factorial to real arguments. For $s > 0$ it is an improper integral, and it satisfies $\Gamma(n + 1) = n!$ for natural numbers $n$. The node [[gamma function|0V3FZMVB4K]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0H5WWC8RZX.md" tag="0H5WWC8RZX" kind="statement">
# The order of a finite field is a prime power

## Statement

Let $F$ be a [[finite field|08RTDHX1SK]].
Then the [[characteristic|0P7CH1CND5]] of $F$ is a prime number $p$, and $|F| = p^n$ with $n = [F : \mathbb{F}_p]$.

## Remarks

The proof: a finite field has nonzero characteristic, and [[the characteristic of a domain is $0$ or prime|00758QM6JJ]]. The [[prime subfield|0VPFBAY7MW]] is $\mathbb{F}_p$, and $F$ is a vector space of dimension $n$ over it, so it has $p^n$ elements.
So there is no field with $6$ or $10$ elements.
The converse, that a field of order $p^n$ exists for each $n \ge 1$, is the [[existence theorem|0T8M3YC5RG]].

## Explanations

### L1 human

A finite field is a vector space over the field with $p$ elements.
A vector space of dimension $n$ over it has $p^n$ elements.
So the size of a finite field is always a power of a prime.
</document>

<document source="https://tamnd.github.io/math2/node/0H5ZAZCX2T.md" tag="0H5ZAZCX2T" kind="statement">
# Taylor's theorem with the Lagrange remainder

## Statement

Let $k \ge 0$ and let $a \ne x$ be real numbers. Let $J$ be the closed interval with endpoints $a$ and $x$.
Let $f$ be a real function with $f^{(k)}$ continuous on $J$ and $f^{(k+1)}$ existing on the interior of $J$.
Then there is $\xi$ strictly between $a$ and $x$ with
$$f(x) = P_k(x) + \frac{f^{(k+1)}(\xi)}{(k + 1)!} (x - a)^{k+1},$$
where $P_k$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.

## Remarks

For $k = 0$ this is the [[mean value theorem|0BARQY52N5]].
If $|f^{(k+1)}| \le M$ between $a$ and $x$, then $|f(x) - P_k(x)| \le M |x - a|^{k+1} / (k + 1)!$.
Mathlib states it as `taylor_mean_remainder_lagrange`.

## Explanations

### L1 human

The error of the Taylor polynomial looks like the next term of the polynomial.
Only the derivative is taken at an unknown point $\xi$ instead of at $a$.
</document>

<document source="https://tamnd.github.io/math2/node/0H60H1Q0SX.md" tag="0H60H1Q0SX" kind="statement">
# Changing the order of integration

## Statement

Let $D \subseteq \mathbb{R}^2$ be a region that has two descriptions,
$$D = \{(x, y) : a \le x \le b, \ \varphi_1(x) \le y \le \varphi_2(x)\} = \{(x, y) : c \le y \le d, \ \psi_1(y) \le x \le \psi_2(y)\},$$
where $\varphi_1 \le \varphi_2$ are continuous on $[a, b]$ and $\psi_1 \le \psi_2$ are continuous on $[c, d]$.
Let $f \colon D \to \mathbb{R}$ be continuous.
Then
$$\int_a^b \int_{\varphi_1(x)}^{\varphi_2(x)} f(x, y) \, dy \, dx = \int_c^d \int_{\psi_1(y)}^{\psi_2(y)} f(x, y) \, dx \, dy,$$
and both equal the double integral of $f$ over $D$.

## Remarks

The proof applies [[Fubini's theorem|0R0T1VWH5X]] on a rectangle that contains $D$ to $f$ times the indicator function of $D$.
The main work is to describe $D$ again in the other order. One way is to sketch the region.
Example: $\int_0^1 \int_x^1 e^{y^2} \, dy \, dx$ has no elementary inner antiderivative. The region is $0 \le x \le y \le 1$, so the integral equals $\int_0^1 \int_0^y e^{y^2} \, dx \, dy = \int_0^1 y e^{y^2} \, dy = (e - 1)/2$.
For unbounded regions or discontinuous integrands, a change of order needs extra hypotheses.

## Explanations

### L1 human

An iterated integral over a region can sometimes be hard in one order and easy in the other.
Changing the order means describing the same region by slicing it in the other direction, and then integrating in the new order.
</document>

<document source="https://tamnd.github.io/math2/node/0H628RSGWX.md" tag="0H628RSGWX" kind="concept">
# Algebraic element

## Summary

Let $L/K$ be a field extension.
An element of $L$ is algebraic over $K$ if it is a root of a nonzero polynomial with coefficients in $K$.
The [[definition of an algebraic element|0NPW1DWECM]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0H631CZP5K.md" tag="0H631CZP5K" kind="definition">
# Galton–Watson process

## Definition

Let $\mu$ be a probability distribution on $\{0, 1, 2, \dots\}$, the offspring distribution. Let $(\xi_{n, i})_{n \ge 0, \, i \ge 1}$ be [[i.i.d.|0AJN7WHBJZ]] random variables with law $\mu$.
The Galton–Watson process with offspring distribution $\mu$ is the [[stochastic process|0QH5ENKZM1]] $(Z_n)_{n \ge 0}$ with
$$Z_0 = 1 \quad \text{and} \quad Z_{n+1} = \sum_{i=1}^{Z_n} \xi_{n, i} \quad \text{for } n \ge 0.$$
Here $Z_n$ is the size of generation $n$, and an empty sum is $0$.

## Remarks

$(Z_n)$ is a [[Markov chain|033TJBZDG0]] on $\{0, 1, 2, \dots\}$, and $0$ is an absorbing state. Extinction is the event that $Z_n = 0$ for some $n$.
Let $m$ be the mean of $\mu$ and let $G$ be its [[probability-generating function|0Z8PSSAVBA]]. Then $E[Z_n] = m^n$, and the generating function of $Z_n$ is the $n$-fold composition $G \circ \dots \circ G$.
The extinction probability is the smallest solution $q \in [0, 1]$ of $G(q) = q$.
Assume $\mu(\{1\}) < 1$. Then $q = 1$ if $m \le 1$, and $q < 1$ if $m > 1$.



## Explanations

### L1 human

The Galton–Watson process follows a family tree in which each person has a random number of children.
If people have on average at most one child, the family dies out with probability $1$. If they have more, it can survive forever.
</document>

<document source="https://tamnd.github.io/math2/node/0H6DP8CHND.md" tag="0H6DP8CHND" kind="concept">
# Del operator

## Summary

The del operator $\nabla = \left( \frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n} \right)$ is a vector of partial derivative operators on $\mathbb{R}^n$.
It gives a short notation for the [[gradient|0ABK9V9Y64]] $\nabla f$, the [[divergence|0MWV6ZTK6S]] $\nabla \cdot F$, the [[curl|0NF6GQNXMY]] $\nabla \times F$ in $\mathbb{R}^3$, and the [[Laplacian|0E0B6RA8K0]] $\nabla^2 f = \nabla \cdot \nabla f$.

## Remarks

The notation treats $\nabla$ like a vector, but it is an operator. For example, $\nabla \cdot F$ and $F \cdot \nabla$ are different: the first is a function, and the second is the operator $\sum_i F_i \, \partial / \partial x_i$.
The directional derivative of $f$ along $v$ is $(v \cdot \nabla) f = v \cdot \nabla f$.
The symbol $\nabla$ is called nabla.

## Explanations

### L1 human

Del is a shorthand that makes vector calculus formulas look like vector algebra.
Multiply it by a function to get the gradient. Dot it with a field to get the divergence, and cross it with a field to get the curl.
</document>

<document source="https://tamnd.github.io/math2/node/0H6KWAYS87.md" tag="0H6KWAYS87" kind="statement">
# Product of power series

## Statement

Let $R$ be a commutative ring, let $f, g \in R[\![X]\!]$ be [[formal power series|072W67KFYK]], and let $n$ be a natural number.
Then
$$[X^n](f g) = \sum_{i + j = n} [X^i] f \cdot [X^j] g = \sum_{i=0}^{n} [X^i] f \cdot [X^{n-i}] g.$$

## Remarks

In many texts this formula is the definition of the product of formal power series.
The sequence of coefficients of $fg$ is the convolution of the sequences of $f$ and $g$. This is the Cauchy product.
Example: $[X^n] \frac{1}{(1 - X)^2} = \sum_{i=0}^{n} 1 \cdot 1 = n + 1$.

## Explanations

### L1 human

To find the coefficient of $X^n$ in a product, collect all ways to get $X^n$ as $X^i$ times $X^j$.
Add the products of the matching coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/0H7C3T1GJY.md" tag="0H7C3T1GJY" kind="concept">
# Scalar potential

## Summary

A scalar potential of a vector field is a scalar function whose [[gradient|0ABK9V9Y64]] is the field, or minus the field in the physics convention.
A field has a scalar potential exactly when it is [[conservative|0PKG97PX9Y]].
</document>

<document source="https://tamnd.github.io/math2/node/0H8H7ZTHW2.md" tag="0H8H7ZTHW2" kind="concept">
# Cumulative distribution function

## Summary

The cumulative distribution function of a real random variable $X$ gives, for each real $x$, the probability that $X$ is at most $x$.
It determines the law of $X$. The node [[cumulative distribution function|0MXMS3AE13]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0H8Q4MANBD.md" tag="0H8Q4MANBD" kind="statement">
# Every finite tree with at least two vertices has a leaf

## Statement

Let $T$ be a finite [[tree|0MJ0JS72DZ]] with at least two vertices.
Then $T$ has a vertex of [[degree|0T0F6CCS9W]] $1$.

## Remarks

In fact $T$ has at least two leaves: the two ends of a longest path are leaves.
Proof idea: take a longest path $v_0, \dots, v_k$. A neighbour of $v_k$ other than $v_{k-1}$ would extend the path or close a cycle, so $\deg(v_k) = 1$.
The lemma is the base of induction proofs on trees, as in the proof that [[a tree on n vertices has n − 1 edges|0VSNHTHK8T]].

## Explanations

### L1 human

A finite tree with at least one edge always has an end point.
Walk along the tree as far as you can without going back. You stop at a vertex with only one edge.
</document>

<document source="https://tamnd.github.io/math2/node/0H8SZFFDXJ.md" tag="0H8SZFFDXJ" kind="statement">
# Rule of Sarrus

## Statement

Let $K$ be a commutative ring and let $a, b, c, d, e, f, g, h, i \in K$.
Then
$$\det \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = aei + bfg + cdh - ceg - bdi - afh,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

Copy the first two columns to the right of the matrix.
Then the three positive terms are the products along the diagonals that go down to the right.
The three negative terms are products along the diagonals that go up to the right in the same picture.
This rule is the [[Leibniz formula|016PE4VZ8Q]] for $n = 3$; it has no analogue with diagonals for $n \ge 4$.

## Explanations

### L1 human

The rule of Sarrus is a memory aid for $3 \times 3$ determinants.
Write the first two columns again to the right of the matrix.
Add the three products down to the right, and subtract the three products up to the right.
</document>

<document source="https://tamnd.github.io/math2/node/0H97Q4Z2SF.md" tag="0H97Q4Z2SF" kind="definition">
# Plane

## Definition

Let $n \geq 3$ be an integer. A plane in the [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$ is a set of the form
$$P = \{\, p + s u + t v \;:\; s, t \in \mathbb{R} \,\},$$
where $p$ is a point of $\mathbb{R}^n$. Here $u$ and $v$ are linearly independent [[vectors|0QN3D80AWP]] of $\mathbb{R}^n$.

## Remarks

Three points that are not on one [[line|0MP9BCF6DP]] lie in exactly one plane.
In $\mathbb{R}^3$, the planes are also the solution sets of the linear equations $a x + b y + c z = d$ with $(a, b, c) \neq (0, 0, 0)$. See the [[equation of a plane|07SDJ5DNE4]].
In the language of affine geometry, a plane is an [[affine subspace|0PYW26VNT6]] of dimension 2.
The word plane also names the whole [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$.

## Explanations

### L1 human

A plane in space is like a sheet of glass that goes on forever. To describe it, pick one point on it and two directions in it that are not parallel. Each point of the plane is the start point plus some amount of the first direction plus some amount of the second.
</document>

<document source="https://tamnd.github.io/math2/node/0H9K5D3RDV.md" tag="0H9K5D3RDV" kind="definition">
# Abelianisation

## Definition

Let $G$ be a [[group|G5S1M9]] and let $[G, G]$ be its [[commutator subgroup|0ASGSTXSBW]].
The abelianisation of $G$ is the [[quotient group|0KYX2N69J0]] $G^{\mathrm{ab}} = G/[G, G]$.

## Remarks

The group $G^{\mathrm{ab}}$ is [[abelian|082GHNPJ53]], by [[this criterion|0MS8QGA6S5]].
If $G$ is abelian, then $G^{\mathrm{ab}} = G$.
The abelianisation of the symmetric group $S_n$ for $n \geq 2$ is cyclic of order $2$, and the sign of a permutation gives the isomorphism.

## Explanations

### L1 human

The abelianisation forces all elements of a group to commute, and changes nothing more than it must.
It keeps as much information about the group as an abelian group can keep.
For a group that is already abelian, it does nothing.
</document>

<document source="https://tamnd.github.io/math2/node/0HATW2VWAZ.md" tag="0HATW2VWAZ" kind="concept">
# Rectifiable curve

## Summary

A [[parametrised curve|0G7WM8DX4P]] is rectifiable when the lengths of the polygonal paths inscribed in it are bounded.
The least upper bound of these lengths is the length of the curve.
</document>

<document source="https://tamnd.github.io/math2/node/0HAZEFH137.md" tag="0HAZEFH137" kind="statement">
# Interior is the largest open subset

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then $\operatorname{int} S \subseteq S$, the set $\operatorname{int} S$ is open, and each open set $U$ of $X$ with $U \subseteq S$ satisfies $U \subseteq \operatorname{int} S$.

## Remarks

Here $\operatorname{int} S$ is the [[interior|0C38S0KMJK]] of $S$. The openness is the statement [[interior is open|0PNWCM559P]].
A consequence: $S$ is open if and only if $\operatorname{int} S = S$.
If $S \subseteq T$, then $\operatorname{int} S \subseteq \operatorname{int} T$.

## Explanations

### L1 human

Among all open sets inside $S$, the interior is the biggest one.
Each open set inside $S$ is already inside the interior.
</document>

<document source="https://tamnd.github.io/math2/node/0HB0XKDXHH.md" tag="0HB0XKDXHH" kind="example">
# Binomial series

## Example

Let $\alpha$ be a real number.
For each natural number $k$, let
$$\binom{\alpha}{k} = \frac{\alpha (\alpha - 1) \cdots (\alpha - k + 1)}{k!}.$$
The binomial series is the [[power series|0VP56B4EMX]] $\sum_{k=0}^{\infty} \binom{\alpha}{k} x^k$.
For $|x| < 1$ its sum is $(1 + x)^{\alpha}$.

## Remarks

If $\alpha$ is a natural number, then $\binom{\alpha}{k} = 0$ for $k > \alpha$. The series is then the binomial formula for $(1 + x)^{\alpha}$.
Otherwise the [[radius of convergence is 1|0KX1MXJ5MR]].
For $\alpha = -1$ it is the geometric series $\sum (-1)^k x^k = 1/(1 + x)$.
For $\alpha = 1/2$ it gives $\sqrt{1 + x} = 1 + x/2 - x^2/8 + \cdots$.

## Explanations

### L1 human

The binomial series extends the binomial formula to exponents that are not natural numbers.
The expansion then has infinitely many terms, and it holds only for small $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0HB8SXHPFA.md" tag="0HB8SXHPFA" kind="definition">
# Markov property

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space, let $T \subseteq [0, \infty)$, and let $(\mathcal{F}_t)_{t \in T}$ be a filtration: σ-algebras with $\mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F}$ for $s \le t$.
Let $(X_t)_{t \in T}$ be a [[stochastic process|0QH5ENKZM1]] with values in $(S, \mathcal{S})$ such that each $X_t$ is $\mathcal{F}_t$-measurable.
The process has the Markov property with respect to $(\mathcal{F}_t)$ if, for all $s \le t$ in $T$ and each $A \in \mathcal{S}$,
$$P(X_t \in A \mid \mathcal{F}_s) = P(X_t \in A \mid X_s) \quad \text{almost surely}.$$

## Remarks

Here $P(B \mid \mathcal{G}) = E[1_B \mid \mathcal{G}]$ is a [[conditional expectation|0YAMJRK04E]].
For a discrete-time process with a countable state space and the natural filtration, the property becomes $P(X_{n+1} = j \mid X_0, \dots, X_n) = P(X_{n+1} = j \mid X_n)$. This defines a Markov chain.
The property says that the future is [[conditionally independent|0ZJ87ZZ8PK]] of the past, given the present.
The strong Markov property is the same statement with a stopping time in place of the fixed time $s$.

## Explanations

### L1 human

A process with the Markov property forgets how it reached its present state.
To predict its future, the present state is all the information that you need.
</document>

<document source="https://tamnd.github.io/math2/node/0HBQC2GJXT.md" tag="0HBQC2GJXT" kind="definition">
# Limit point

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]], let $S \subseteq X$ and let $x \in X$.
The point $x$ is a limit point of $S$ when each [[open set|01ZQQDCERP]] $U$ with $x \in U$ contains a point of $S$ other than $x$.

## Remarks

The point $x$ need not be in $S$. In $\mathbb{R}$, the point $0$ is a limit point of $\{1/n : n \ge 1\}$.
In a metric space, $x$ is a limit point of $S$ if and only if each ball $B(x, \varepsilon)$ contains a point of $S \setminus \{x\}$.
The [[closure|0DRKT1QJR7]] of $S$ is the union of $S$ and the set of its limit points.
A point of $S$ that is not a limit point of $S$ is an [[isolated point|01PKQYTTMT]] of $S$.

## Explanations

### L1 human

A limit point is a point that other points of the set crowd around.
For the set $\{1, 1/2, 1/3, \ldots\}$, the points crowd around $0$, so $0$ is a limit point, but $1$ is not.
</document>

<document source="https://tamnd.github.io/math2/node/0HBRW4E6G1.md" tag="0HBRW4E6G1" kind="definition">
# Intersection

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The intersection of $A$ and $B$ is the set
$$A \cap B = \{x \in A \mid x \in B\}.$$
So an object $x$ is an [[element|0EZ758XE73]] of $A \cap B$ when $x \in A$ and $x \in B$.

## Remarks

Example: $\{1, 2\} \cap \{2, 3\} = \{2\}$.
Intersection is commutative and associative, and $A \cap B \subseteq A$.
When $A \cap B = \varnothing$, the sets are [[disjoint|0W4RPA8QH3]].
The intersection of a family of sets is in [[intersection of a family of sets|0839XNQXDK]].

## Explanations

### L1 human

The intersection keeps only what the two sets have in common.
The intersection of the even numbers and the multiples of $3$ is the set of multiples of $6$.
</document>

<document source="https://tamnd.github.io/math2/node/0HBS1F00DN.md" tag="0HBS1F00DN" kind="statement">
# Induction on the integers

## Statement

Let $P(z)$ be a property of [[integers|0C4JS9RF5M]] $z$.
Suppose that $P(0)$ holds.
Suppose that for each natural number $n$, $P(n)$ implies $P(n + 1)$.
Suppose that for each natural number $n$, $P(-n)$ implies $P(-n - 1)$.
Then $P(z)$ holds for every integer $z$.

## Remarks

The proof applies ordinary [[induction|0ASJVEFA9J]] two times: one time to $P(n)$ and one time to $P(-n)$, for natural numbers $n$.
Each integer is $n$ or $-n$ for some natural number $n$, thus the two inductions cover all of $\mathbb{Z}$.

## Explanations

### L1 human

You start at $0$ and you walk in both directions on the number line.
One induction goes up through $1, 2, 3, \dots$, and the other goes down through $-1, -2, -3, \dots$.
</document>

<document source="https://tamnd.github.io/math2/node/0HBY48G8G1.md" tag="0HBY48G8G1" kind="definition">
# Riemann sum

## Definition

Let $a < b$, let $f \colon [a, b] \to \mathbb{R}$, and let $P = (x_0, \dots, x_n)$ be a [[partition|069P20KYQT]] of $[a, b]$.
Let $t = (t_1, \dots, t_n)$ be tags with $t_i \in [x_{i-1}, x_i]$ for each $i$.
The Riemann sum of $f$ for the tagged partition $(P, t)$ is
$$S(f, P, t) = \sum_{i=1}^{n} f(t_i) (x_i - x_{i-1}).$$

## Remarks

The left rule takes $t_i = x_{i-1}$, the right rule takes $t_i = x_i$, and the midpoint rule takes the midpoints.
For bounded $f$, each Riemann sum lies between the lower and the upper [[Darboux sum|0QRQVH0CRW]] for $P$.
Mathlib has a general form for boxes, `BoxIntegral.integralSum`.

## Explanations

### L1 human

A Riemann sum adds up areas of rectangles.
Each rectangle stands on a piece of the partition, and its height is the value of $f$ at the chosen tag.
</document>

<document source="https://tamnd.github.io/math2/node/0HE5WDQ9FJ.md" tag="0HE5WDQ9FJ" kind="concept">
# Zermelo–Fraenkel set theory

## Summary

Zermelo–Fraenkel set theory, ZF, is an axiomatic theory of [[sets|0SJ3RX9TXA]] in [[first-order logic|086QKPF0A6]] with one binary relation $\in$.
Its axioms are extensionality, regularity, separation, pairing, union, replacement, infinity and power set. Separation and replacement are axiom schemas.
ZF with the [[axiom of choice|0HTGB13FE2]] is ZFC, the standard foundation of most of mathematics.

## Remarks

The [[axiom of extensionality|0TB16T2C97]] says that a set is determined by its elements.
Separation forms only subsets $\{x \in A \mid P(x)\}$ of a given set $A$. This avoids [[Russell's paradox|0HESMMSYEG]].
The [[continuum hypothesis|0MN2MER9WN]] is independent of ZFC, if ZFC is consistent.

## Explanations

### L1 human

ZF is a short list of rules that says which sets exist.
For example, for two sets there is a set that contains both, and for each set there is the set of its subsets.
From these rules, one can build the numbers, functions and most other objects of mathematics.
</document>

<document source="https://tamnd.github.io/math2/node/0HEF46RPEK.md" tag="0HEF46RPEK" kind="statement">
# Union of open sets is open

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $(U_i)_{i \in I}$ be a family of [[open subsets|01ZQQDCERP]] of $X$, indexed by any set $I$.
Then the [[union|05GK32H72H]] $\bigcup_{i \in I} U_i$ is open in $X$.

## Remarks

Proof: let $x$ be in the union. Then $x \in U_j$ for some $j \in I$, and there is $\varepsilon > 0$ with $B(x, \varepsilon) \subseteq U_j$. This ball is also in the union. Then use [[open sets of a metric space|0A26X79RTB]].
The family can be infinite, and even uncountable.
In a general [[topological space|0CNQXVBHCX]], this property is one of the axioms.

## Explanations

### L1 human

Each point of the union lies in one of the open sets, and that set already gives a small ball about the point.
For example, the union of the intervals $(1/n, 1)$ for $n \ge 2$ is the open interval $(0, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/0HESMMSYEG.md" tag="0HESMMSYEG" kind="statement">
# Russell's paradox

## Statement

There is no [[set|0SJ3RX9TXA]] $R$ such that, for each set $x$, $x \in R$ if and only if $x \notin x$.

## Remarks

Proof: if such an $R$ exists, apply the condition to $x = R$. Then $R \in R$ if and only if $R \notin R$, which is a contradiction.
So the collection $\{x \mid x \notin x\}$ is not a set. Unrestricted comprehension, which makes a set from each property, is inconsistent.
It follows that there is no set of all sets. Otherwise, separation would give $R$ as a subset of it.

## History

Bertrand Russell found the paradox in May 1901.
Ernst Zermelo found it independently, by 1902 and possibly in 1899, but he did not publish it.
In a 1902 letter, Russell told Gottlob Frege that the paradox arises in the logical system of Frege.
Axiomatic set theories, such as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], avoid it with a restricted form of comprehension.

## Explanations

### L1 human

Try to form "the set of all sets that do not contain themselves".
Ask whether this set contains itself: both answers lead to a contradiction.
So some collections are too large to be sets.
</document>

<document source="https://tamnd.github.io/math2/node/0HF30P75G4.md" tag="0HF30P75G4" kind="concept">
# Hamiltonian cycle

## Summary

A Hamiltonian cycle is a cycle that passes through every vertex of a graph. The search for such cycles is a classical hard problem, related to the travelling salesman problem. The node [[Hamiltonian cycle|0HKXNG2RS1]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0HFREYRSZP.md" tag="0HFREYRSZP" kind="concept">
# Rotation

## Summary

A rotation is a motion of a Euclidean space that keeps at least one point fixed and keeps the orientation. In the plane, a rotation turns each point about a centre through the same angle.
</document>

<document source="https://tamnd.github.io/math2/node/0HFSFVMWQB.md" tag="0HFSFVMWQB" kind="definition">
# Taylor polynomial

## Definition

Let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be $k$ times differentiable at $a$.
The Taylor polynomial of degree $k$ of $f$ at $a$ is
$$P_k(x) = \sum_{j=0}^{k} \frac{f^{(j)}(a)}{j!} (x - a)^j = f(a) + f'(a)(x - a) + \cdots + \frac{f^{(k)}(a)}{k!} (x - a)^k.$$

## Remarks

It is the unique polynomial $P$ of degree at most $k$ with $P^{(j)}(a) = f^{(j)}(a)$ for $j = 0, \dots, k$, using [[higher derivatives|084NGSTWM7]].
For $k = 1$, its graph is the [[tangent line|01R1HW3Y42]].
The difference $R_k = f - P_k$ is the remainder. [[Taylor's theorem|03QHBVW4TT]] says that it is small near $a$.
Mathlib has `taylorWithinEval` for Taylor polynomials within a set.

## Explanations

### L1 human

The Taylor polynomial copies the value and the first $k$ derivatives of $f$ at $a$.
It is the polynomial that best fits $f$ near $a$.
</document>

<document source="https://tamnd.github.io/math2/node/0HG11FW8MJ.md" tag="0HG11FW8MJ" kind="definition">
# First-countable space

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]].
The space $X$ is first-countable when each point $x \in X$ has a sequence $(N_n)_{n \ge 1}$ of [[neighbourhoods|09CJ00H3NW]] with this property: each neighbourhood of $x$ contains some $N_n$.

## Remarks

Such a sequence is a [[countable|0ZG61TFBES]] neighbourhood basis at $x$. One can choose the sets $N_n$ open, and also decreasing.
Each metric space is first-countable: take $N_n = B(x, 1/n)$, as in [[open balls form a basis|02EEX96DFS]].
Each [[second-countable space|00RE1R4W7N]] is first-countable.
In a first-countable space, a point is in the closure of a set $A$ exactly when it is a limit of a sequence in $A$.

## Explanations

### L1 human

At each point, you can shrink to the point along a countable list of neighbourhoods.
In a metric space, the balls of radius $1/n$ do this.
</document>

<document source="https://tamnd.github.io/math2/node/0HGVCWQGY3.md" tag="0HGVCWQGY3" kind="statement">
# Pigeonhole principle

## Statement

Let $A$ and $B$ be finite sets with $|A| > |B|$, and let $f : A \to B$ be a function.
Then there are elements $a \neq a'$ of $A$ with $f(a) = f(a')$.
In other words, $f$ is not injective.

## Remarks

In the usual image, if $n + 1$ pigeons sit in $n$ holes, then some hole has at least two pigeons.
Example: among any $13$ people, two have their birthdays in the same month.
The [[generalized pigeonhole principle|0BY7GN3449]] gives a fibre with more than $n$ elements. The [[infinite pigeonhole principle|0FS3RHF6KE]] is a version for infinite sets.

## Explanations

### L1 human

If there are more objects than boxes, some box has two objects.
For example, in any group of 367 people, two share a birthday, because a year has at most 366 days.
</document>

<document source="https://tamnd.github.io/math2/node/0HH4Y7HSRQ.md" tag="0HH4Y7HSRQ" kind="concept">
# Sum of a series

## Summary

The sum of a convergent series is the limit of its partial sums. It gives a meaning to an infinite sum of real numbers. The node [[sum of a series|0K0H5E3Z5N]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0HH6JX5WZZ.md" tag="0HH6JX5WZZ" kind="definition">
# Generating function

## Definition

Let $R$ be a commutative ring and let $(a_n)_{n \ge 0}$ be a sequence in $R$.
The ordinary generating function of $(a_n)$ is the [[formal power series|072W67KFYK]]
$$A(x) = \sum_{n \ge 0} a_n x^n \in R[\![x]\!].$$

## Remarks

Example: the sequence $1, 1, 1, \dots$ has the generating function $\sum_n x^n = \frac{1}{1 - x}$, the [[geometric series|0R8JR0ARJ4]].
Operations on sequences become operations on series. The [[product of power series|0H6KWAYS87]] corresponds to the convolution $c_n = \sum_{i+j=n} a_i b_j$.
The [[exponential generating function|0KE6NBTVY8]] is a variant with the weights $x^n / n!$.

## Explanations

### L1 human

A generating function stores a whole sequence as the coefficients of one power series.
Algebra with the series then gives facts about the sequence, such as recurrences or closed formulas.
</document>

<document source="https://tamnd.github.io/math2/node/0HHHDCXZV4.md" tag="0HHHDCXZV4" kind="concept">
# Differentiable function of several variables

## Summary

A function of several variables is differentiable at a point when, near that point, it is close to a linear function plus a constant. The error is small compared with the distance to the point.
The linear function is the [[total derivative|0FNKFT8H6E]] of the function at the point.
</document>

<document source="https://tamnd.github.io/math2/node/0HJ0HDA6W7.md" tag="0HJ0HDA6W7" kind="statement">
# Solutions for distinct real roots

## Statement

Let $a_0, \dots, a_{n-1}$ be real numbers, and suppose that the [[characteristic polynomial|0QDXMG46QW]] $p(r) = r^n + a_{n-1} r^{n-1} + \dots + a_0$ has $n$ distinct real roots $r_1, \dots, r_n$.
Then $\left(e^{r_1 x}, \dots, e^{r_n x}\right)$ is a [[fundamental system|0H4KNNCHPY]] of the equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = 0$ on $\mathbb{R}$.
So the solutions are exactly the functions $y = c_1 e^{r_1 x} + \dots + c_n e^{r_n x}$ with real constants $c_1, \dots, c_n$.

## Remarks

Each $e^{r_k x}$ is a solution because $p(r_k) = 0$. The Wronskian at $0$ is the Vandermonde determinant $\prod_{j < k} (r_k - r_j)$, which is not zero. So the [[Wronskian test|0MHE4B1QCQ]] gives independence.
Example: $y'' - 3 y' + 2 y = 0$ has the solutions $c_1 e^{x} + c_2 e^{2x}$.

## Explanations

### L1 human

Each root of the characteristic polynomial gives one exponential solution.
If the $n$ roots are different, these $n$ exponentials are independent, and their combinations are all the solutions.
</document>

<document source="https://tamnd.github.io/math2/node/0HJ28H9Y95.md" tag="0HJ28H9Y95" kind="example">
# Thomae's function

## Example

Define $f \colon \mathbb{R} \to \mathbb{R}$ as follows.
For irrational $x$, $f(x) = 0$. For rational $x = p/q$ in lowest terms with $q > 0$, $f(x) = 1/q$.
Then $f$ is [[continuous|0EEE2CZB2A]] at each irrational number and discontinuous at each rational number.

## Remarks

Let $a$ be irrational and let $\varepsilon > 0$. Only finitely many $p/q$ near $a$ have $1/q \ge \varepsilon$. So $f(x) < \varepsilon$ for all $x$ close enough to $a$.
At a rational $a$, irrational points near $a$ have value $0$, but $f(a) > 0$.
The function is periodic with period $1$.
No function on $\mathbb{R}$ is continuous exactly at the rationals.

## Explanations

### L1 human

The value $1/q$ is small when the denominator $q$ is large.
Near an irrational point, all rationals have large denominators, so the values there are small.
</document>

<document source="https://tamnd.github.io/math2/node/0HJ4DTW6FN.md" tag="0HJ4DTW6FN" kind="statement">
# Second isomorphism theorem for rings

## Statement

Let $R$ be a [[ring|0894J980RK]], let $S$ be a [[subring|0859D58WPD]] of $R$ and let $I$ be a two-sided ideal of $R$.
Then $S + I = \{s + x : s \in S, x \in I\}$ is a subring of $R$, and $I$ is a two-sided ideal of $S + I$.
The intersection $S \cap I$ is a two-sided ideal of $S$.
The map $S/(S \cap I) \to (S + I)/I$, $s + (S \cap I) \mapsto s + I$, is a [[ring isomorphism|0B6QC0C0G1]].

## Remarks

The proof applies the [[first isomorphism theorem|0ZMYX7GMH0]] to the map $S \to (S + I)/I$, $s \mapsto s + I$. It is surjective, and its kernel is $S \cap I$.
The theorem is also called the diamond theorem, after the picture of the four rings $S \cap I$, $S$, $I$ and $S + I$.

## Explanations

### L1 human

The theorem compares two quotients.
Divide the subring $S$ by its part inside $I$, or divide the larger ring $S + I$ by $I$.
The two quotients are the same ring.
</document>

<document source="https://tamnd.github.io/math2/node/0HJHYMGN3C.md" tag="0HJHYMGN3C" kind="concept">
# Discrete logarithm

## Summary

A discrete logarithm of an element $h$ of a cyclic group to a base $g$ is an exponent $k$ such that $g^k = h$.
The [[definition of the discrete logarithm|09B4X51J09]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0HKXNG2RS1.md" tag="0HKXNG2RS1" kind="definition">
# Hamiltonian cycle

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A Hamiltonian cycle of $G$ is a [[cycle|07CT9RDY7Q]] in $G$ that contains every vertex of $G$.

## Remarks

A Hamiltonian path is a [[path|0JDE56HWQ8]] that contains every vertex of $G$.
The names come from William Rowan Hamilton. His icosian game asks for a Hamiltonian cycle on the edges of a dodecahedron.
To decide whether a graph has a Hamiltonian cycle is an NP-complete problem.

## Explanations

### L1 human

A Hamiltonian cycle is a round trip that visits every vertex exactly once.
It is about visiting all vertices, while an Eulerian circuit is about using all edges.
</document>

<document source="https://tamnd.github.io/math2/node/0HM05N3FJD.md" tag="0HM05N3FJD" kind="concept">
# Random experiment

## Summary

A random experiment is a procedure that we can repeat as often as we like and that has a well-defined set of possible results.
The set of these results is the [[sample space|0W7NFW23RF]] of the experiment.
The experiment is random when it has more than one possible result, and deterministic when it has exactly one.

## Remarks

Each repetition of the experiment is a trial. Each trial gives exactly one [[outcome|0Y5R7TPNT1]].
A toss of a coin and a roll of a die are random experiments.

## Explanations

### L1 human

A random experiment is an action whose result you cannot predict, but whose possible results you know in advance.
When you roll a die, you do not know the number that you will get, but you know that it is one of $1, 2, 3, 4, 5, 6$.
</document>

<document source="https://tamnd.github.io/math2/node/0HM0HQCK6N.md" tag="0HM0HQCK6N" kind="concept">
# Distributive lattice

## Summary

A distributive lattice is a lattice in which meet and join distribute over each other. Lattices of sets and Boolean algebras are the main examples. The node [[distributive lattice|0T36P6TXQ6]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0HM584C465.md" tag="0HM584C465" kind="example">
# RSA cryptosystem

## Example

The RSA cryptosystem is an example of [[public-key cryptography|01E9MNJYEX]].
To make the keys, choose two distinct primes $p$ and $q$, and put $n = pq$ and $\varphi(n) = (p - 1)(q - 1)$, the value of [[Euler's totient function|0VV04M41AN]].
Choose an integer $e$ with $1 < e < \varphi(n)$ and $\gcd(e, \varphi(n)) = 1$. Let $d$ be the [[inverse|00GXTQMY8Z]] of $e$ modulo $\varphi(n)$.
The public key is $(n, e)$, and the private key is $d$.
To encrypt a message $m$ with $0 \le m < n$, compute $c = m^e \bmod n$. To decrypt, compute $c^d \bmod n$, which is equal to $m$.
A small case: $p = 5$, $q = 11$, $n = 55$, $\varphi(n) = 40$, $e = 3$ and $d = 27$, since $3 \cdot 27 = 81 \equiv 1 \pmod{40}$.
The message $m = 2$ gives $c = 2^3 = 8$, and $8^{27} \equiv 2 \pmod{55}$.

## Remarks

Decryption works by [[Euler's theorem|0VC4DVE7DQ]]: $ed = 1 + k \varphi(n)$, so $m^{ed} \equiv m \pmod n$ when $\gcd(m, n) = 1$. The general case follows from the [[Chinese remainder theorem|0FNN9B4W64]].
The security rests on the belief that it is hard to factor $n$ when $p$ and $q$ are large.
Ron Rivest, Adi Shamir and Leonard Adleman described the system publicly in 1977.

## Explanations

### L1 human

Anyone can raise a message to the power $e$ modulo $n$. To undo this, you need $d$, and to find $d$ you need the factors of $n$.
Multiplying two large primes is easy, but recovering them from the product is believed to be very hard.
</document>

<document source="https://tamnd.github.io/math2/node/0HM732ZNVC.md" tag="0HM732ZNVC" kind="statement">
# A sequence converges iff its liminf and limsup agree

## Statement

Let $(a_n)$ be a bounded real sequence.
Then $(a_n)$ converges if and only if $\liminf_{n \to \infty} a_n = \limsup_{n \to \infty} a_n$.
In this case, $\lim_{n \to \infty} a_n$ is equal to this common value.

## Remarks

The proof of the direction from right to left uses $\inf_{k \ge n} a_k \le a_n \le \sup_{k \ge n} a_k$ and the [[squeeze theorem|0XMF00T4XQ]].
In the extended real line, the statement holds for every sequence. Then convergence to $+\infty$ or to $-\infty$ is allowed.
For $(-1)^n$, the two values are $-1$ and $1$. They differ, so the sequence diverges.

## Explanations

### L1 human

The limit inferior and the limit superior are the lowest and the highest points where the sequence keeps returning.
If these two points are the same, the sequence has nowhere else to go, so it converges to that point.
</document>

<document source="https://tamnd.github.io/math2/node/0HNATXFEV5.md" tag="0HNATXFEV5" kind="concept">
# Order on the natural numbers

## Summary

The order on the natural numbers is the relation $\le$ on $\mathbb{N}$.
Here $a \le b$ means that $a$ comes at or before $b$ when we count.
It is a total order, and $0$ is its least element.
</document>

<document source="https://tamnd.github.io/math2/node/0HNK6KT05K.md" tag="0HNK6KT05K" kind="concept">
# Centre of mass

## Summary

The centre of mass of a body is the average position of its mass, with each point weighted by the density there.
For a body with a continuous density it is a quotient of [[triple integrals|0698M7C7AX]], or of double integrals for a plane lamina.
</document>

<document source="https://tamnd.github.io/math2/node/0HNMFG8F8K.md" tag="0HNMFG8F8K" kind="statement">
# Eigenvectors with distinct eigenvalues are linearly independent

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be a linear map.
Let $v_1, \dots, v_k$ be [[eigenvectors|0N6960W0BV]] of $T$ for eigenvalues $\lambda_1, \dots, \lambda_k$ that are pairwise distinct.
Then $v_1, \dots, v_k$ are [[linearly independent|0CRMGPAH83]].

## Remarks

The proof is by induction on $k$: apply $T - \lambda_k \id_V$ to a linear relation, and the term of $v_k$ disappears.
So the sum of the eigenspaces of $T$ for distinct eigenvalues is a direct sum.
In particular, an endomorphism of an $n$-dimensional space has at most $n$ distinct eigenvalues.

## Explanations

### L1 human

Eigenvectors for different eigenvalues point in genuinely different directions.
None of them is a combination of the others.
This is the key fact behind diagonalisation.
</document>

<document source="https://tamnd.github.io/math2/node/0HPHKCGEQ0.md" tag="0HPHKCGEQ0" kind="statement">
# Generalised Stokes theorem

## Statement

Let $\Omega$ be an oriented smooth manifold of dimension $n$ with boundary $\partial \Omega$, and give $\partial \Omega$ the induced orientation.
Let $\omega$ be a smooth [[differential form|0QHTX7WQEY]] of degree $n - 1$ on $\Omega$ with compact support, and let $d\omega$ be its [[exterior derivative|004FRCXH6M]]. Then
$$\int_\Omega d\omega = \int_{\partial \Omega} \omega.$$

## Remarks

For $\Omega = [a, b]$ and a function $f$, the theorem is the fundamental theorem of calculus, $\int_a^b f'(x) \, dx = f(b) - f(a)$. The boundary point $a$ gets the negative orientation.
For a curve and a 0-form, it is the [[gradient theorem|0SD237FEJD]]. For a plane region and a 1-form, it is [[Green's theorem|0HR3N02317]].
For a surface in $\mathbb{R}^3$ and a 1-form, it is the classical [[Stokes' theorem|0AKRKVHW94]]. For a solid in $\mathbb{R}^3$ and a 2-form, it is the [[divergence theorem|0JX86PSSA9]].
The orientation and the compact support are needed so that both integrals are defined.

## Explanations

### L1 human

The integral of a derivative over a region equals the integral of the original form over the boundary of the region.
One formula covers the fundamental theorem of calculus and the three integral theorems of vector calculus.
</document>

<document source="https://tamnd.github.io/math2/node/0HQARMKEV5.md" tag="0HQARMKEV5" kind="definition">
# Archimedean ordered field

## Definition

An [[ordered field|010JCV9J4E]] $F$ is Archimedean when, for all $x$ and $y$ in $F$ with $y > 0$, there is a natural number $n$ with $n y > x$.
Here $n y$ means $y + y + \dots + y$, with $n$ terms.

## Remarks

With $y = 1$, the definition gives the [[Archimedean property|0A74SX55QP]]: for each $x$, some natural number exceeds $x$. Conversely, this property implies the definition.
An equivalent condition: for each $\varepsilon > 0$ in $F$, there is a natural number $n \ge 1$ with $1/n < \varepsilon$.
The field of rational functions $\mathbb{R}(t)$, with the order in which $t$ is larger than every real number, is not Archimedean.

## Explanations

### L1 human

You can reach any distance with enough small steps of a fixed size.
In an Archimedean field there is no number so large that no multiple of a given positive number can pass it.
</document>

<document source="https://tamnd.github.io/math2/node/0HQFZAP3ND.md" tag="0HQFZAP3ND" kind="concept">
# Coefficient matrix

## Summary

The coefficient matrix of a system of linear equations is the matrix of the coefficients of the unknowns.
Its row $i$ holds the coefficients of equation $i$.
</document>

<document source="https://tamnd.github.io/math2/node/0HR3N02317.md" tag="0HR3N02317" kind="statement">
# Green's theorem

## Statement

Let $C$ be a positively oriented, piecewise smooth, simple [[closed curve|0D032NSZKY]] in $\mathbb{R}^2$, and let $D$ be the bounded region enclosed by $C$.
Let $L$ and $M$ be real functions with continuous [[partial derivatives|05AXG3QRKR]] on an open set that contains $D \cup C$.
Then
$$\oint_C (L \, dx + M \, dy) = \iint_D \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dA.$$

## Remarks

Positive orientation means that $C$ goes counterclockwise, so $D$ is on the left.
Proof sketch: first let $D$ be a region between the graphs of two functions of $x$. Write the [[double integral|0NKBMFC18R]] of $\partial L / \partial y$ as an iterated integral, and use the fundamental theorem of calculus. This gives $\oint_C L \, dx = -\iint_D \partial L / \partial y \, dA$. The $M$ term is similar, and general regions are cut into such pieces.
Example: with $L = -y$ and $M = x$, the right side is $2$ times the [[area of D|09CBQB6V67]].
Green's theorem is the planar case of [[Stokes' theorem|0AKRKVHW94]]. George Green stated a similar result in 1828, and Riemann gave a proof in 1851.

## Explanations

### L1 human

The theorem says that the total circulation of a plane field around a closed curve equals the sum of the small circulations inside.
The small circulations of neighbouring squares cancel along shared edges, so only the outer boundary is left.
</document>

<document source="https://tamnd.github.io/math2/node/0HR573CHNM.md" tag="0HR573CHNM" kind="definition">
# Linear span

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $S \subseteq V$.
The *linear span* of $S$ is the set $\operatorname{span}(S)$ of all [[linear combinations|0V8F3E48E6]] of finitely many vectors of $S$:
$$\operatorname{span}(S) = \{a_1 v_1 + \cdots + a_k v_k : k \ge 0, \ a_i \in F, \ v_i \in S\}.$$

## Remarks

The span is a [[linear subspace|00YC1D652K]] of $V$, and it is the intersection of all subspaces of $V$ that contain $S$.
The span of the empty set is $\{0\}$.
In $\R^3$, the span of one nonzero vector is a line through the origin.
The span of two non-parallel vectors is a plane through the origin.

## Explanations

### L1 human

The span of some vectors is everything we can reach from them by scaling and adding.
Two arrows in space that do not point along the same line span a whole plane.
The span is always a subspace.
</document>

<document source="https://tamnd.github.io/math2/node/0HRJ2Y74T3.md" tag="0HRJ2Y74T3" kind="definition">
# Rearrangement of a series

## Definition

Let $\sum_{n=0}^{\infty} a_n$ be a real [[series|0JH4MN4K1C]].
Let $\sigma \colon \mathbb{N} \to \mathbb{N}$ be a bijection.
The series $\sum_{n=0}^{\infty} a_{\sigma(n)}$ is a rearrangement of $\sum a_n$.

## Remarks

A rearrangement uses each term of the original series exactly once, in a new order.
[[Rearrangements of an absolutely convergent series have the same sum|0J76TP9FGD]].
For a [[conditionally convergent|02YRQXZDCS]] series, rearrangements can change the sum, by the [[Riemann rearrangement theorem|0E7VBYGFF7]].

## Explanations

### L1 human

A rearrangement adds up the same terms in a different order.
For finite sums the order does not matter, but for infinite series it can matter.
</document>

<document source="https://tamnd.github.io/math2/node/0HRVY67CNV.md" tag="0HRVY67CNV" kind="statement">
# Solutions for repeated roots

## Statement

Let $a_0, \dots, a_{n-1}$ be real numbers, and let $p(r) = r^n + a_{n-1} r^{n-1} + \dots + a_0$ be the [[characteristic polynomial|0QDXMG46QW]] of the equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = 0$.
Let $r$ be a real root of $p$ of multiplicity $m \ge 1$.
Then the $m$ functions $e^{r x}, x e^{r x}, \dots, x^{m-1} e^{r x}$ are linearly independent solutions of the equation on $\mathbb{R}$.
If all the roots of $p$ are real, the union of these families over the distinct roots of $p$ is a [[fundamental system|0H4KNNCHPY]].

## Remarks

For order $2$: if $r$ is a double root of $r^2 + a_1 r + a_0$, the solutions are $(c_1 + c_2 x) e^{r x}$.
Example: $y'' - 2 y' + y = 0$ has $p(r) = (r - 1)^2$, so its solutions are $(c_1 + c_2 x) e^{x}$.
The same result holds for complex roots, with complex solutions. For the real form, see [[complex conjugate roots|0BEZX8F2XW]].

## Explanations

### L1 human

A double root gives only one exponential, but we need two solutions.
The extra solution is the exponential times $x$.
A root of multiplicity $m$ gives the exponential times $1, x, \dots, x^{m-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/0HTGB13FE2.md" tag="0HTGB13FE2" kind="statement">
# Axiom of choice

## Statement

Let $X$ be a [[set|0SJ3RX9TXA]] of nonempty sets.
Then there is a [[choice function|0YXEQWRGP0]] on $X$, that is, a function $c \colon X \to \bigcup X$ such that $c(S) \in S$ for each $S \in X$.

## Remarks

An equivalent form: for each [[family|05Q6G4EH2G]] $(A_i)_{i \in I}$ of nonempty sets, the product $\prod_{i \in I} A_i$ is nonempty.
Over ZF, it is equivalent to [[Zorn's lemma|04RXTQA5ZA]], to the [[well-ordering theorem|0N109NG5ZX]] and to the [[comparability of cardinals|0YBHSJMN3J]].
It is independent of ZF, if ZF is consistent.
Kurt Gödel showed in 1938 that ZF does not refute it, and Paul Cohen showed in 1963 that ZF does not prove it.

## History

Ernst Zermelo formulated the axiom in 1904 for his proof of the [[well-ordering theorem|0N109NG5ZX]].

## Explanations

### L1 human

For finitely many nonempty boxes, you can pick one object from each box, one box at a time.
For infinitely many boxes without a rule to pick, the axiom of choice says that a selection still exists.
From infinitely many pairs of shoes, you can take the left shoe of each pair.
From infinitely many pairs of identical socks, there is no such rule, and you need the axiom.
</document>

<document source="https://tamnd.github.io/math2/node/0HV01NH20W.md" tag="0HV01NH20W" kind="concept">
# Reduced residue system

## Summary

A reduced residue system modulo $n$ is a set of integers with one element in each residue class of integers coprime to $n$.
The [[definition of a reduced residue system|0J0QEN1DZR]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0HV1SKWPMS.md" tag="0HV1SKWPMS" kind="statement">
# Rouché–Capelli theorem

## Statement

Let $F$ be a field, let $A$ be an $m \times n$ matrix over $F$, and let $b \in F^m$.
The [[system|0GT4WWGWH2]] $Ax = b$ has a solution if and only if $\rank A = \rank [A \mid b]$, where $[A \mid b]$ is the [[augmented matrix|0T6MWGZSXA]] and $\rank$ is the [[rank of a matrix|0ZC9YTXZRY]].
If a solution exists, then the solution set is an affine subspace of $F^n$ of dimension $n - \rank A$.
In particular, the solution is unique if and only if $\rank A = n$.

## Remarks

The theorem is also called the Kronecker–Capelli theorem.
If $\rank [A \mid b] = \rank A + 1$, then the system is [[inconsistent|0X6V9B1V10]].
The dimension $n - \rank A$ is the number of free variables.

## Explanations

### L1 human

The theorem says that a system has a solution exactly when the right side adds no new independent information.
If adding the column $b$ raises the rank, then $b$ is not a combination of the columns of $A$, and no solution exists.
When solutions exist, each unknown without a pivot is free, and it adds one dimension to the solution set.
</document>

<document source="https://tamnd.github.io/math2/node/0HV9Y7WG5R.md" tag="0HV9Y7WG5R" kind="definition">
# Group homomorphism

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]].
A group homomorphism from $G$ to $H$ is a map $\varphi \colon G \to H$ such that $\varphi(ab) = \varphi(a) \varphi(b)$ for all $a, b \in G$.
Here the product $ab$ is taken in $G$ and the product $\varphi(a) \varphi(b)$ is taken in $H$.

## Remarks

The map $n \mapsto 2^n$ is a homomorphism from $(\mathbb{Z}, +)$ to $(\mathbb{Q}^\times, \cdot)$.
The map $x \mapsto e^x$ is a homomorphism from $(\mathbb{R}, +)$ to the positive real numbers with multiplication.
For each pair of groups, the map that sends each element of $G$ to the identity of $H$ is a homomorphism, the trivial homomorphism.
The composite of two homomorphisms is a homomorphism.

## Explanations

### L1 human

A homomorphism translates one group into another so that products go to products.
It does not matter if you multiply first and then translate, or translate first and then multiply.
The logarithm turns multiplication of positive numbers into addition, and this is why it is a homomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/0HVX863SAJ.md" tag="0HVX863SAJ" kind="concept">
# Eccentricity

## Summary

The eccentricity of a conic section is a non-negative real number that measures how far the conic is from a circle. It is $0$ for a circle, between $0$ and $1$ for an ellipse, $1$ for a parabola and greater than $1$ for a hyperbola.
</document>

<document source="https://tamnd.github.io/math2/node/0HW7BNHHGF.md" tag="0HW7BNHHGF" kind="statement">
# Limit comparison test

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences with $a_n \ge 0$ and $b_n > 0$ for all $n$.
Assume that $a_n / b_n \to c$ as $n \to \infty$, for a real number $c$ with $0 < c < \infty$.
Then $\sum a_n$ converges if and only if $\sum b_n$ converges.

## Remarks

For large $n$ we have $\frac{c}{2} b_n \le a_n \le 2c\, b_n$. The [[comparison test|04Z5G0THN6]] then gives both directions.
For example, $\frac{n}{n^3 + 1} / \frac{1}{n^2} \to 1$, so $\sum n/(n^3 + 1)$ converges, like $\sum 1/n^2$.
If $c = 0$, convergence of $\sum b_n$ still implies convergence of $\sum a_n$. The converse can fail.

## Explanations

### L1 human

If two series of positive terms have terms of the same size in the long run, they behave the same way.
So you can replace a complicated term by a simple one with the same leading behaviour.
</document>

<document source="https://tamnd.github.io/math2/node/0HX8HZ1WTQ.md" tag="0HX8HZ1WTQ" kind="definition">
# Gram matrix

## Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $v_1, \dots, v_k \in V$.
The *Gram matrix* of $v_1, \dots, v_k$ is the $k \times k$ matrix $G$ with entries
$$G_{ij} = \langle v_i, v_j \rangle \qquad (1 \le i, j \le k),$$
where $\langle \cdot, \cdot \rangle$ is the [[inner product|0NB0AWGJBW]].

## Remarks

The Gram matrix is Hermitian and [[positive semidefinite|01MMVW8CKC]].
It is invertible if and only if $v_1, \dots, v_k$ are linearly independent.
For real column vectors $v_i \in \R^n$ with the dot product, $G = A^T A$, where $A$ has the columns $v_1, \dots, v_k$.
Its determinant is the square of the $k$-dimensional volume of the parallelepiped spanned by the $v_i$.

## Explanations

### L1 human

The Gram matrix records all inner products of a list of vectors.
It contains all lengths and all angles between them.
Two lists with the same Gram matrix have the same shape, up to an isometry.
</document>

<document source="https://tamnd.github.io/math2/node/0HXEX0DPRA.md" tag="0HXEX0DPRA" kind="definition">
# One-sided derivative

## Definition

Let $f$ be a real function defined on an interval $[a, a + r)$ with $r > 0$.
The right derivative of $f$ at $a$ is the [[one-sided limit|0909PWZR62]]
$$f'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h},$$
when it exists.
The left derivative $f'_-(a)$ is defined in the same way, for $f$ on $(a - r, a]$ and $h \to 0^-$.

## Remarks

A function on an open interval is differentiable at $a$ if and only if both one-sided derivatives exist and are equal. Their common value is the [[derivative|0F1WR9C83K]].
One-sided derivatives give a meaning to derivatives at the endpoints of a closed interval.
For $|x|$ at $0$, the right derivative is $1$ and the left derivative is $-1$.
Mathlib expresses them with `HasDerivWithinAt` on a half line.

## Explanations

### L1 human

A one-sided derivative only looks at one side of the point.
At a corner of a graph, the slopes from the left and from the right exist but differ.
</document>

<document source="https://tamnd.github.io/math2/node/0HY15D6WTX.md" tag="0HY15D6WTX" kind="definition">
# Finitely generated group

## Definition

A [[group|G5S1M9]] $G$ is finitely generated when there is a finite subset $S$ of $G$ with $\langle S \rangle = G$, where $\langle S \rangle$ is the [[subgroup generated by $S$|0RP7VWQ0YN]].

## Remarks

Each finite group is finitely generated, because it is generated by the finite set of all its elements.
The group $(\mathbb{Z}^n, +)$ is finitely generated by the $n$ standard basis vectors.
The group $(\mathbb{Q}, +)$ is not finitely generated: the subgroup generated by finitely many fractions has a common denominator, and it does not contain all fractions.

## Explanations

### L1 human

A group is finitely generated when a finite list of elements is enough to build all elements.
The group may still be infinite, as the integers show: one generator $1$ gives all integers.
The rational numbers need infinitely many generators.
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<document source="https://tamnd.github.io/math2/node/0HY1MXMEN9.md" tag="0HY1MXMEN9" kind="statement">
# Totient of a prime

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
Then $\varphi(p) = p - 1$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

The proof: the only divisors of $p$ are $1$ and $p$. So each of $1, \dots, p - 1$ is coprime to $p$, and $p$ is not.
The converse holds: if $n > 1$ and $\varphi(n) = n - 1$, then $n$ is prime.
The result is the case $k = 1$ of the [[totient of a prime power|0AVY8HKBGD]].

## Explanations

### L1 human

For a prime $p$, each number from $1$ to $p - 1$ has no common factor with $p$.
For example, $\varphi(7) = 6$, because $1, 2, 3, 4, 5, 6$ are all coprime to $7$.
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<document source="https://tamnd.github.io/math2/node/0HYC2RHD3M.md" tag="0HYC2RHD3M" kind="statement">
# Eigenvalues of a positive definite matrix are positive

## Statement

Let $A$ be a [[positive definite|0KX66CZMBB]] $n \times n$ complex Hermitian matrix.
Then each [[eigenvalue|0BQAA2BK3Q]] of $A$ is a positive real number.

## Remarks

If $Ax = \lambda x$ with $x \ne 0$, then $0 < x^* A x = \lambda\, x^* x$, and $x^* x > 0$.
The converse holds: a Hermitian matrix with only positive eigenvalues is positive definite, by the spectral theorem.
So $\det A > 0$ and $\tr A > 0$ for a positive definite $A$.

## Explanations

### L1 human

A positive definite matrix stretches each of its eigenvector directions by a positive factor.
It never flips or crushes a direction.
So its eigenvalues, its determinant and its trace are all positive.
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<document source="https://tamnd.github.io/math2/node/0HYFEEZAEM.md" tag="0HYFEEZAEM" kind="statement">
# Injective iff it has a left inverse

## Statement

Let $A$ be a nonempty set, let $B$ be a set and let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
Then $f$ is [[injective|0PVG1CRAJN]] if and only if $f$ has a [[left inverse|02486SD328]].

## Remarks

The hypothesis that $A$ is nonempty is necessary. The empty function from $\varnothing$ to a nonempty set is injective, but there is no function from a nonempty set to $\varnothing$.
To build a left inverse, fix $a_0 \in A$. Send each $b = f(a)$ to $a$, and send each $b \notin f(A)$ to $a_0$.

## Explanations

### L1 human

An injective function loses no information, so you can go back from each value to its unique input.
The values that $f$ never takes can go anywhere, and you need one point $a_0$ to send them to.
</document>

<document source="https://tamnd.github.io/math2/node/0HYFWKKY3C.md" tag="0HYFWKKY3C" kind="concept">
# Polar form of a quadratic form

## Summary

The polar form of a quadratic form $Q$ is the symmetric bilinear form $B(u, v) = Q(u + v) - Q(u) - Q(v)$.
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<document source="https://tamnd.github.io/math2/node/0HYNZE940T.md" tag="0HYNZE940T" kind="statement">
# Modulus is multiplicative

## Statement

For all complex numbers $z$ and $w$,
$$|z w| = |z| \, |w|,$$
where $|\cdot|$ is the [[modulus|0KW2X0TFBX]].

## Remarks

Proof: $|z w|^2 = z w \overline{z w} = (z \bar{z})(w \bar{w}) = |z|^2 |w|^2$, and both sides are nonnegative.
For $z = a + b i$ and $w = c + d i$, this is the identity $(a^2 + b^2)(c^2 + d^2) = (a c - b d)^2 + (a d + b c)^2$.
On the real numbers, it gives that the [[absolute value is multiplicative|0GMV7GP8W0]].

## Explanations

### L1 human

The length of a product is the product of the lengths.
For example, $|(1 + i)(1 - i)| = |2| = 2$, and $|1 + i| \cdot |1 - i| = \sqrt{2} \cdot \sqrt{2} = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/0HYP606RYF.md" tag="0HYP606RYF" kind="concept">
# Student's t-test

## Summary

A Student's t-test is a [[hypothesis test|087P03NEND]] whose test statistic has [[Student's t-distribution|0NQ5798SB1]] under the null hypothesis. It mostly tests hypotheses about the means of normal populations with unknown variance.
In the one-sample t-test, $X_1, \dots, X_n$ are i.i.d. $\mathcal{N}(\mu, \sigma^2)$ with unknown $\sigma$, and $H_0 \colon \mu = \mu_0$. The statistic is
$$t = \frac{\bar{X}_n - \mu_0}{S_n / \sqrt{n}},$$
with the [[sample mean|0QNRBHWV35]] $\bar{X}_n$ and the [[sample standard deviation|0JN9P4A944]] $S_n$. Under $H_0$, $t \sim t_{n - 1}$.

## Remarks

The two-sample t-test compares the means of two independent normal samples with sizes $n_1$, $n_2$ and equal variances. It uses the pooled standard deviation $s_p$ and $t = (\bar{X}_1 - \bar{X}_2) / \big(s_p \sqrt{1/n_1 + 1/n_2}\big)$, with $n_1 + n_2 - 2$ degrees of freedom.
For two samples with unequal variances, Welch's t-test uses an approximate number of degrees of freedom.
The paired t-test is the one-sample t-test applied to the differences of paired observations.
For large samples, the t-test gives almost the same results as a test with the normal law, because $t_\nu$ approaches $\mathcal{N}(0, 1)$.

## Explanations

### L1 human

A t-test checks whether an average differs from a given value, or whether two averages differ, more than chance alone can explain.
It divides the difference by its estimated standard error, and compares the result with the t-distribution.
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<document source="https://tamnd.github.io/math2/node/0HYPR64JM2.md" tag="0HYPR64JM2" kind="statement">
# Mean value property

## Statement

Let $U \subseteq \mathbb{C}$ be an open set, and identify $\mathbb{C}$ with $\mathbb{R}^2$ by $x + iy \leftrightarrow (x, y)$.
Let $u : U \to \mathbb{R}$ be harmonic: $u$ is twice continuously differentiable and $u_{xx} + u_{yy} = 0$ on $U$.
Let $c \in \mathbb{C}$ and $r > 0$ be such that the closed disc $\{z : |z - c| \le r\}$ is contained in $U$.
Then
$$u(c) = \frac{1}{2\pi} \int_0^{2\pi} u(c + r e^{i\theta}) \, d\theta.$$

## Remarks

The value of $u$ at the center of the circle is the average of $u$ on the circle.
For holomorphic functions, the same formula holds, and it follows from [[Cauchy's integral formula|0DG10TVYPN]] with $a = c$. On a disc, $u$ is the [[real part of a holomorphic function|031BG625Y3]], which gives the harmonic case.
The property also holds for averages over the solid disc. A continuous function with the mean value property on all small circles is harmonic.

## Explanations

### L1 human

If $u$ is harmonic, its value at a point is the average of its values on each circle around that point. This is true as long as the closed disc stays in the domain.
For example, $u = x^2 - y^2$ at $0$ is $0$, and on the unit circle it is $\cos 2\theta$, whose average is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0HYRJF4WNQ.md" tag="0HYRJF4WNQ" kind="concept">
# Norm of a Gaussian integer

## Summary

The norm of a Gaussian integer $a + bi$ is the integer $a^2 + b^2$. It is multiplicative.
The [[definition of the norm|00HPT88ZX0]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0HYV2NME07.md" tag="0HYV2NME07" kind="concept">
# Hamiltonian graph

## Summary

A Hamiltonian graph is a graph that has a cycle through all of its vertices. There is no simple test for this property, but degree conditions give sufficient criteria. The node [[Hamiltonian graph|090WPKFABX]] gives the precise definition.
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<document source="https://tamnd.github.io/math2/node/0HZNE5QZDJ.md" tag="0HZNE5QZDJ" kind="definition">
# Quadratic form

## Definition

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.
A *quadratic form* on $V$ is a map $Q \colon V \to F$ such that:

1. $Q(av) = a^2 Q(v)$ for all $a \in F$ and $v \in V$.
2. The map $(u, v) \mapsto Q(u + v) - Q(u) - Q(v)$ is a [[bilinear form|0SQZMXB6GP]] on $V$.

## Remarks

On $F^n$, the quadratic forms are the maps $Q(x) = \sum_{i,j} a_{ij} x_i x_j$, that is, homogeneous polynomials of degree $2$.
In matrix notation, $Q(x) = x^T A x$ for an $n \times n$ matrix $A$.
If the characteristic of $F$ is not $2$, the matrix $A$ can be chosen symmetric, and then it is unique.
For example, $Q(x, y) = x^2 + 3xy + 2y^2$ is a quadratic form on $\R^2$.

## Explanations

### L1 human

A quadratic form is a polynomial in which every term has degree two, such as $x^2 - 4xy + y^2$.
It is the simplest kind of function after linear functions.
The squared length $x_1^2 + \cdots + x_n^2$ is the main example.
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<document source="https://tamnd.github.io/math2/node/0J0BBCMJ5W.md" tag="0J0BBCMJ5W" kind="concept">
# Pivot

## Summary

A pivot of a matrix in row echelon form is the leading entry of a nonzero row.
In elimination, a pivot is the entry that we use to clear the other entries of its column.
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<document source="https://tamnd.github.io/math2/node/0J0FCWV32D.md" tag="0J0FCWV32D" kind="concept">
# Exponential function

## Summary

The exponential function $x \mapsto e^x$ is the function that equals its own derivative and takes the value $1$ at $0$. It turns sums into products, and it is the sum of the power series $\sum x^n / n!$. The node [[exponential function|0RMVS7RXZX]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0J0QEN1DZR.md" tag="0J0QEN1DZR" kind="definition">
# Reduced residue system

## Definition

Let $n$ be a positive integer.
A reduced residue system modulo $n$ is a set $R$ of integers with these properties.
Each element of $R$ is [[coprime|0GFKP8F8AG]] to $n$.
No two distinct elements of $R$ are congruent modulo $n$.
Each integer coprime to $n$ is congruent modulo $n$ to some element of $R$.

## Remarks

Thus $R$ contains exactly one element of each [[residue class|0QRV8FPK5M]] of integers coprime to $n$. It has $\varphi(n)$ elements, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].
For example, $\{1, 5, 7, 11\}$ is a reduced residue system modulo $12$.
If $R$ is a reduced residue system modulo $n$ and $\gcd(a, n) = 1$, then $\{ a r : r \in R \}$ is also a reduced residue system modulo $n$. This fact gives a proof of [[Euler's theorem|0VC4DVE7DQ]].

## Explanations

### L1 human

Take a [[complete residue system|0E22JKS4W4]] modulo $n$, and keep only the numbers with no common factor with $n$.
For $n = 10$, the numbers $0, 1, \dots, 9$ give the reduced residue system $\{1, 3, 7, 9\}$.
</document>

<document source="https://tamnd.github.io/math2/node/0J0SK5FF4M.md" tag="0J0SK5FF4M" kind="definition">
# Quotient map

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $W$ be a subspace of $V$.
The *quotient map* is the map
$$\pi \colon V \to V / W, \qquad \pi(v) = v + W,$$
from $V$ to the [[quotient space|0BR3FY5Q8M]] $V / W$.

## Remarks

The map $\pi$ is linear and surjective, and its kernel is $W$.
Each linear map $T \colon V \to U$ with $W \subseteq \ker T$ factors through $\pi$, see [[universal property of the quotient space|0ZERJX6PH6]].
In Mathlib, it is `Submodule.mkQ`.

## Explanations

### L1 human

The quotient map sends each vector to the class of all vectors that differ from it by an element of $W$.
It forgets exactly the information in $W$.
The vectors that it sends to zero are exactly the vectors of $W$.
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<document source="https://tamnd.github.io/math2/node/0J0XBKG64S.md" tag="0J0XBKG64S" kind="concept">
# Proof by contradiction

## Summary

A proof by contradiction proves a [[proposition|0ZGC7ZESYN]] $p$ in two steps.
It assumes $\neg p$, and it gets a contradiction from this assumption, that is, a statement $q$ together with $\neg q$.
Then $\neg p$ is false, so $p$ is true by the [[double negation law|0NFAGWH84K]].

## Remarks

Another name is reductio ad absurdum.
A classical example: $\sqrt{2}$ is irrational. Assume $\sqrt{2} = a/b$ in lowest terms. Then $a$ and $b$ are both even, which is a contradiction.
The last step uses classical logic. A proof of $\neg p$ from a contradiction with $p$ is valid also in intuitionistic logic.

## Explanations

### L1 human

To show that something is true, suppose that it is false and follow the consequences.
If you reach an impossible situation, the supposition was wrong.
So the statement must be true.
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<document source="https://tamnd.github.io/math2/node/0J1E5ZRZRM.md" tag="0J1E5ZRZRM" kind="concept">
# Fixed field

## Summary

The fixed field of a group of automorphisms of a field is the subfield of the elements that every automorphism in the group leaves unchanged.
The [[definition of the fixed field|0KMJ6919WS]] states it.
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<document source="https://tamnd.github.io/math2/node/0J1N3WTKM4.md" tag="0J1N3WTKM4" kind="concept">
# Euclidean isometry

## Summary

A Euclidean isometry is a map of a Euclidean space to itself that keeps the distance between each two points. Translations, rotations and reflections are examples. The Euclidean isometries of $\mathbb{R}^n$ form the Euclidean group.
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<document source="https://tamnd.github.io/math2/node/0J1R5NP36S.md" tag="0J1R5NP36S" kind="definition">
# Liouville number

## Definition

A real number $x$ is a Liouville number if for each positive integer $n$ there are integers $p$ and $q$ with $q > 1$ and
$$0 < \left| x - \frac{p}{q} \right| < \frac{1}{q^n}.$$

## Remarks

The standard example is Liouville's constant $\sum_{k=1}^{\infty} 10^{-k!} = 0.110001000\ldots$.
Each Liouville number is irrational, and in fact [[transcendental|079EGSD32Y]].
The numbers $e$ and $\pi$ are transcendental, but they are not Liouville numbers.
The Liouville numbers are uncountable, but they form a set of Lebesgue measure zero.

## Explanations

### L1 human

A Liouville number has rational approximations that are extremely good, better than any fixed power of $1/q$.
Algebraic numbers cannot be approximated this well, so a Liouville number is not algebraic.
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<document source="https://tamnd.github.io/math2/node/0J1TT091TN.md" tag="0J1TT091TN" kind="statement">
# Solutions of Pell's equation are powers of the fundamental solution

## Statement

Let $d$ be a positive integer that is not a perfect square, and let $(x_1, y_1)$ be the [[fundamental solution|0KCYJSB9FW]] of $x^2 - d y^2 = 1$.
Let $x$ and $y$ be integers with $x \ge 1$, $y \ge 0$ and $x^2 - d y^2 = 1$.
Then there is a unique natural number $n$ with $x + y \sqrt{d} = (x_1 + y_1 \sqrt{d})^n$.

## Remarks

Conversely, each power $(x_1 + y_1 \sqrt{d})^n = x_n + y_n \sqrt{d}$ gives a solution $(x_n, y_n)$.
The integer solutions are exactly the pairs with $x + y \sqrt{d} = \pm (x_1 + y_1 \sqrt{d})^n$ for $n \in \mathbb{Z}$.
For $d = 2$: $(3 + 2\sqrt{2})^2 = 17 + 12 \sqrt{2}$, so $(17, 12)$ is the next solution after $(3, 2)$.
In algebraic terms, the units of norm $1$ in $\mathbb{Z}[\sqrt{d}]$ form the group $\{\pm 1\} \times \mathbb{Z}$.

## Explanations

### L1 human

Once you know the smallest solution, you get all the others by taking powers.
Expand $(x_1 + y_1 \sqrt{d})^n$ and read the integer part and the coefficient of $\sqrt{d}$.
</document>

<document source="https://tamnd.github.io/math2/node/0J29GM30Z7.md" tag="0J29GM30Z7" kind="example">
# Existence of nth roots by the intermediate value theorem

## Example

Let $n \ge 1$ be a natural number and let $y \ge 0$ be real.
Then there is a unique real $x \ge 0$ with $x^n = y$.
For existence, let $b = \max(1, y)$. The function $p(x) = x^n$ is [[continuous|0C5TP6RYSV]], with $p(0) = 0 \le y$ and $p(b) \ge b \ge y$.
The [[intermediate value theorem|00KAFBR32E]] gives $x \in [0, b]$ with $x^n = y$.

## Remarks

Uniqueness holds because $x \mapsto x^n$ is strictly increasing on $[0, \infty)$.
The number $x$ is the $n$-th root $\sqrt[n]{y}$.
For example, it proves that $\sqrt{2}$ exists as a real number.

## Explanations

### L1 human

The graph of $x^n$ starts at $0$ and grows past $y$.
Since the graph has no gaps, it reaches the height $y$ at some point, which is the root.
</document>

<document source="https://tamnd.github.io/math2/node/0J2C7BDJQA.md" tag="0J2C7BDJQA" kind="statement">
# Polar decomposition

## Statement

Let $A$ be an $n \times n$ complex matrix.
Then there are a [[unitary matrix|0AR68JTMDT]] $U$ and a positive semidefinite Hermitian matrix $P$ with
$$A = UP.$$
The factor $P$ is unique: $P = (A^* A)^{1/2}$, the [[square root|0BJVDCF38Y]] of $A^* A$.
If $A$ is invertible, then $U$ is also unique.

## Remarks

It follows from the singular value decomposition $A = W \Sigma V^*$: take $U = W V^*$ and $P = V \Sigma V^*$.
For $n = 1$, it is the polar form $z = e^{i\theta} r$ of a complex number.
There is also a left polar decomposition $A = P' U$ with $P' = (A A^*)^{1/2}$.

## Explanations

### L1 human

Every square matrix is a stretch along perpendicular axes, followed by a rotation, possibly with a reflection.
This generalises writing a complex number as a positive length times a unit complex number.
The stretch part is always unique.
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<document source="https://tamnd.github.io/math2/node/0J2WAR83WP.md" tag="0J2WAR83WP" kind="concept">
# Inverse function

## Summary

The inverse function of a bijection reverses it: it sends each value back to the input that gives it.
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<document source="https://tamnd.github.io/math2/node/0J3578562S.md" tag="0J3578562S" kind="definition">
# Equivalent norms

## Definition

Let $V$ be a vector space over $\mathbb{R}$ or $\mathbb{C}$.
A norm on $V$ is a function $\|\cdot\| \colon V \to [0, \infty)$ such that, for all $x, y \in V$ and all scalars $\lambda$: $\|x\| = 0$ only for $x = 0$, $\|\lambda x\| = |\lambda| \, \|x\|$, and $\|x + y\| \le \|x\| + \|y\|$.
Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $V$ are equivalent when there are real numbers $c > 0$ and $C > 0$ with
$$c \, \|x\|_a \le \|x\|_b \le C \, \|x\|_a \quad \text{for all } x \in V.$$

## Remarks

Each norm gives a metric $d(x, y) = \|x - y\|$. Two norms are equivalent if and only if their metrics are strongly [[equivalent|0TWQ5PRKWS]].
For norms, equivalence is the same as giving the same [[metric topology|0A6YRNZB2R]].
On a finite-dimensional space, [[all norms are equivalent|03MC70ZRNR]].
On $C([0, 1])$, the sup norm and the norm $\int_0^1 |f(t)| \, dt$ are not equivalent.

## Explanations

### L1 human

Equivalent norms measure the same vectors as small or large, up to a fixed factor.
So they agree on all questions of convergence and continuity.
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<document source="https://tamnd.github.io/math2/node/0J3K4Y1KPD.md" tag="0J3K4Y1KPD" kind="concept">
# Cycle type

## Summary

The cycle type of a permutation of a finite set is the list of the lengths of the cycles in its disjoint cycle decomposition.
For a permutation in $S_n$ it is a partition of $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0J438GAC8K.md" tag="0J438GAC8K" kind="statement">
# Conjugacy classes of the symmetric group correspond to cycle types

## Statement

Let $\sigma, \tau \in S_n$.
Then $\sigma$ and $\tau$ are [[conjugate|0DXD4QSZ8A]] in $S_n$ if and only if they have the same [[cycle type|0J3K4Y1KPD]].

## Remarks

The key formula is $\pi (a_1\ a_2\ \dots\ a_k) \pi^{-1} = (\pi(a_1)\ \pi(a_2)\ \dots\ \pi(a_k))$ for each $\pi \in S_n$.
So conjugation keeps the cycle type, and a suitable $\pi$ sends the cycles of $\sigma$ onto the cycles of $\tau$.
Thus the number of conjugacy classes of $S_n$ is the number of partitions of $n$.
For example, $S_4$ has $5$ conjugacy classes.

## Explanations

### L1 human

Conjugation by $\pi$ renames the points, but it keeps the shape of each cycle.
Conversely, two permutations of the same shape differ only by a renaming of the points.
So in $S_n$, conjugate means "same shape".
</document>

<document source="https://tamnd.github.io/math2/node/0J53NE674D.md" tag="0J53NE674D" kind="statement">
# The symmetric group on five or more letters is not solvable

## Statement

Let $n \geq 5$.
Then the [[symmetric group|0515A6D2Y1]] $S_n$ is not [[solvable|0FXJ7YFMMX]].

## Remarks

The group $A_n$ is [[simple|0ARPRJQWZ2]] and not abelian, so its commutator subgroup is $A_n$, and the derived series of $A_n$ is constant.
So $A_n$ is not solvable, and then $S_n$ is not solvable, because [[subgroups of solvable groups are solvable|05Y0F9M0TS]].
The groups $S_1, S_2, S_3$ and $S_4$ are solvable.
By Galois theory, this is the reason why the general polynomial equation of degree $n \geq 5$ is not solvable by radicals.

## Explanations

### L1 human

From five letters on, the even permutations form a simple group that is not abelian.
Such a group cannot be cut into abelian layers.
So the full symmetric group cannot be cut into abelian layers either.
</document>

<document source="https://tamnd.github.io/math2/node/0J5EMJRRFM.md" tag="0J5EMJRRFM" kind="concept">
# Characteristic polynomial of a linear map

## Summary

The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of its matrix in any basis.
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<document source="https://tamnd.github.io/math2/node/0J5VGJYYC3.md" tag="0J5VGJYYC3" kind="statement">
# Congruence is compatible with powers

## Statement

Let $n$ be a positive integer, let $a$ and $b$ be integers, and let $k$ be a natural number.
If $a \equiv b \pmod n$, then $a^k \equiv b^k \pmod n$.

## Remarks

The proof is an induction on $k$ with [[compatibility with multiplication|072D5G4SK6]].
The exponent cannot be reduced modulo $n$ in general. For example, $2^5 = 32 \equiv 2 \pmod 3$, but $2^{5 \bmod 3} = 2^2 \equiv 1 \pmod 3$.
[[Euler's theorem|0VC4DVE7DQ]] tells when one can reduce the exponent, and by which modulus.

## Explanations

### L1 human

To find $10^{100} \bmod 9$, note that $10 \equiv 1 \pmod 9$.
So $10^{100} \equiv 1^{100} = 1 \pmod 9$, without a computation of the large number.
</document>

<document source="https://tamnd.github.io/math2/node/0J5XR7BW2S.md" tag="0J5XR7BW2S" kind="statement">
# Intersecting secants theorem

## Statement

Let $A$, $B$, $C$ and $D$ be [[concyclic points|049DCJZ448]] of the Euclidean plane $\mathbb{R}^2$, with $A \neq D$ and $B \neq C$. Let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
Suppose that the lines $AD$ and $BC$ meet at a point $P$ outside the circle.
Then
$$d(P, A)\, d(P, D) = d(P, B)\, d(P, C).$$

## Remarks

Both products are equal to the [[power|02E9VT1RVQ]] $d(P, O)^2 - r^2$ of $P$, where $O$ is the centre and $r$ the radius.
A proof: the triangles $PAC$ and $PBD$ are similar, because they have the same angle at $P$ and equal inscribed angles at $C$ and $D$.
If the second line becomes tangent, then $B = C$, and the statement becomes the [[tangent–secant theorem|0QX7E9KAAX]].

## Explanations

### L1 human

From a point outside a circle, draw two lines that cut the circle. On each line, multiply the distances from the point to the two crossing points. The two products are equal.
</document>

<document source="https://tamnd.github.io/math2/node/0J6A752DTZ.md" tag="0J6A752DTZ" kind="statement">
# Lyapunov's stability theorem

## Statement

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz, and let $x^* \in U$ be an equilibrium point of $x' = f(x)$.
Let $W \subseteq U$ be an open neighbourhood of $x^*$ and let $V : W \to \mathbb{R}$ be a [[Lyapunov function|04P7B5XC22]] for $x^*$.
(1) Then $x^*$ is [[Lyapunov stable|092S6VBECQ]].
(2) If $V$ is a strict Lyapunov function, then $x^*$ is [[asymptotically stable|05Y1JAM3Y5]].
(3) If $W = U = \mathbb{R}^n$, $V$ is strict, and $V(x) \to \infty$ as $\|x\| \to \infty$, then $x^*$ is globally asymptotically stable.

## Remarks

Proof idea for (1): choose $\varepsilon$ so small that the sphere $S$ of radius $\varepsilon$ around $x^*$ lies in $W$. Let $m > 0$ be the minimum of $V$ on $S$. A solution that starts where $V < m$ keeps $V < m$, so it cannot reach $S$.
For the [[damped harmonic oscillator|0RNXVVTJTJ]] $x' = y$, $y' = -x - c y$ with $c > 0$, the energy $V = \frac{1}{2}(x^2 + y^2)$ has $\dot{V} = -c y^2 \le 0$. This gives stability, and [[LaSalle's invariance principle|0659J5X7B5]] gives asymptotic stability.
The method needs no solution formula. The difficulty is to find a suitable $V$.

## History

The method is named after Aleksandr Lyapunov, who introduced it in his study of the stability of motion.

## Explanations

### L1 human

If some energy-like quantity decreases along all motions near an equilibrium, the motions stay near it.
If the quantity decreases strictly, the motions must also run down to the equilibrium.
</document>

<document source="https://tamnd.github.io/math2/node/0J6AKYZQT8.md" tag="0J6AKYZQT8" kind="statement">
# Associativity of composition

## Statement

Let $f \colon A \to B$, $g \colon B \to C$ and $h \colon C \to D$ be [[functions|0XHDTE5YMZ]].
Then
$$h \circ (g \circ f) = (h \circ g) \circ f.$$

## Remarks

Both sides send each $a \in A$ to $h(g(f(a)))$, so they are equal.
Because of this law, one can write $h \circ g \circ f$ without brackets.
With the [[identity functions|0FSNQ0GNCC]], the functions from a set to itself form a monoid under composition.

## Explanations

### L1 human

When three machines are chained, it does not matter which two you connect first.
The input passes through $f$, then $g$, then $h$ in both cases.
</document>

<document source="https://tamnd.github.io/math2/node/0J6G5NJC5R.md" tag="0J6G5NJC5R" kind="statement">
# Kuratowski's theorem

## Statement

A subdivision of a [[graph|0JD5C82QMB]] $H$ is a graph that $H$ gives when some edges are replaced by paths with new inner vertices.
Let $G$ be a finite graph.
Then $G$ is [[planar|029JQEK15B]] if and only if no [[subgraph|0P11X5HKAX]] of $G$ is a subdivision of $K_5$ or of $K_{3,3}$.

## Remarks

One direction is easy: a subdivision of a nonplanar graph is nonplanar, and [[$K_5$ and $K_{3,3}$ are not planar|030A3FHN14]].
Wagner's theorem is a related result: a finite graph is planar if and only if it has neither $K_5$ nor $K_{3,3}$ as a minor.

## History

Kazimierz Kuratowski published the theorem in 1930.
Orrin Frink and Paul Smith proved it independently in the same year, but they did not publish their proof.
Lev Pontryagin reportedly proved it around 1927, and in the Soviet Union it was also known under his name.

## Explanations

### L1 human

There are exactly two kinds of obstacle to drawing a graph without crossings.
A graph is planar unless it contains a copy of $K_5$ or $K_{3,3}$ in which some edges are stretched into paths.
</document>

<document source="https://tamnd.github.io/math2/node/0J6M03WSS6.md" tag="0J6M03WSS6" kind="definition">
# Hyperbolic functions

## Definition

The hyperbolic functions are the functions $\sinh$, $\cosh$ and $\tanh$ on $\mathbb{R}$, together with their reciprocals:
$$\coth x = \frac{\cosh x}{\sinh x} \ (x \ne 0), \qquad \operatorname{sech} x = \frac{1}{\cosh x}, \qquad \operatorname{csch} x = \frac{1}{\sinh x} \ (x \ne 0).$$
Here [[$\sinh$|0GWD21EWRW]], [[$\cosh$|0EQFWKJ37J]] and [[$\tanh$|0MWYQTZ07Z]] are the hyperbolic sine, cosine and tangent.

## Remarks

They satisfy identities similar to the trigonometric identities, often with a change of sign.
The point $(\cosh t, \sinh t)$ lies on the hyperbola $x^2 - y^2 = 1$, as $(\cos t, \sin t)$ lies on the circle.
Vincenzo Riccati introduced the hyperbolic functions in 1757.

## Explanations

### L1 human

The hyperbolic functions are built from $e^x$ and $e^{-x}$.
They play the role for the hyperbola that sine and cosine play for the circle.
</document>

<document source="https://tamnd.github.io/math2/node/0J6Z6H5VQJ.md" tag="0J6Z6H5VQJ" kind="definition">
# Discrete random variable

## Definition

Let $X$ be a real [[random variable|0DKRR22AEQ]] on a probability space $(\Omega, \mathcal{F}, P)$.
$X$ is discrete if there is a [[countable set|0ZG61TFBES]] $S \subseteq \mathbb{R}$ with $P(X \in S) = 1$.
A distribution $\mu$ on $\mathbb{R}$ is discrete if $\mu(S) = 1$ for some countable set $S$.

## Remarks

Then $P(X \in B) = \sum_{x \in B \cap S} P(X = x)$ for each Borel set $B$. So the [[probability mass function|0WF4EJP91W]] $x \mapsto P(X = x)$ determines the law of $X$.
The cumulative distribution function of a discrete random variable is a step function when the set of values has no limit points. It jumps by $P(X = x)$ at each value $x$.
The Bernoulli, binomial, geometric and Poisson distributions are discrete.

## Explanations

### L1 human

A discrete random variable takes values that you can list, such as $0, 1, 2, \dots$.
The number of heads in ten coin tosses and the number of calls in an hour are discrete random variables.
</document>

<document source="https://tamnd.github.io/math2/node/0J768YF1ZE.md" tag="0J768YF1ZE" kind="definition">
# Nine-point circle

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\mathbb{R}^2$. Let $M_A$, $M_B$ and $M_C$ be the [[midpoints|0S706JBZBG]] of the sides $BC$, $CA$ and $AB$.
These three points are not collinear. The nine-point circle of $ABC$ is the [[circumcircle|03VZJE6C8C]] of the triangle $M_A M_B M_C$.

## Remarks

The circle also contains the feet of the three [[altitudes|0C9MZY2RNA]], and the midpoints of the segments $AH$, $BH$ and $CH$, where $H$ is the [[orthocentre|0WB619CA0V]]. This gives the nine points of the name.
Its centre is the midpoint of $H$ and the circumcentre $O$ of $ABC$. Its radius is half of the circumradius of $ABC$.
Brianchon and Poncelet proved in 1821 that all nine points lie on one circle. In 1822, Feuerbach proved that the circle touches the incircle and the three excircles.

## Explanations

### L1 human

Join the midpoints of the sides of a triangle to get a smaller triangle. The circle through its corners is the nine-point circle. It also passes through six other special points of the big triangle.
</document>

<document source="https://tamnd.github.io/math2/node/0J76TP9FGD.md" tag="0J76TP9FGD" kind="statement">
# Rearrangements of an absolutely convergent series have the same sum

## Statement

Let $\sum a_n$ be a real series that [[converges absolutely|0W3K3QPEWT]], with sum $A$.
Let $\sigma \colon \mathbb{N} \to \mathbb{N}$ be a bijection.
Then the [[rearrangement|0HRJ2Y74T3]] $\sum a_{\sigma(n)}$ converges absolutely, and its sum is $A$.

## Remarks

Given $\varepsilon > 0$, choose $N$ with $\sum_{n > N} |a_n| < \varepsilon$. For large $m$, the first $m$ terms of the rearrangement include $a_0, \dots, a_N$. So the two partial sums differ by less than $\varepsilon$.
By the [[Riemann rearrangement theorem|0E7VBYGFF7]], the converse also holds for real series.
Mathlib states a general form for sums over a type, as `Equiv.tsum_eq`.

## Explanations

### L1 human

When the series converges absolutely, the far terms have a small total size, whatever their order.
So a new order changes the partial sums only by a small amount.
</document>

<document source="https://tamnd.github.io/math2/node/0J78KBPFZB.md" tag="0J78KBPFZB" kind="concept">
# Riemann sphere

## Summary

The Riemann sphere is the complex plane with one added point $\infty$, see the [[definition|0CDMXGPSES]].
Stereographic projection identifies it with the unit sphere in $\mathbb{R}^3$.
It is the natural domain and target of [[Möbius transformations|0SG74WBFA2]] and of meromorphic functions.
</document>

<document source="https://tamnd.github.io/math2/node/0J7H0YBFDJ.md" tag="0J7H0YBFDJ" kind="concept">
# Endomorphism

## Summary

An endomorphism of a vector space is a linear map from the space to itself.
</document>

<document source="https://tamnd.github.io/math2/node/0J7HF2PF1H.md" tag="0J7HF2PF1H" kind="definition">
# Picard–Lindelöf hypotheses

## Definition

Let $t_{\min} \le t_0 \le t_{\max}$ be real numbers, let $J = [t_{\min}, t_{\max}]$, let $x_0 \in \mathbb{R}^n$, and let $a, L, K \ge 0$.
Let $B = \{x \in \mathbb{R}^n : \|x - x_0\| \le a\}$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ be a function.
The function $f$ satisfies the Picard–Lindelöf hypotheses at $(t_0, x_0)$ with constants $a, L, K$ if these four conditions hold.
(1) For each $t \in J$, the map $x \mapsto f(t, x)$ is $K$-Lipschitz on $B$ ([[Lipschitz condition|055VVSN00V]]).
(2) For each $x \in B$, the map $t \mapsto f(t, x)$ is continuous on $J$.
(3) For each $t \in J$ and $x \in B$, $\|f(t, x)\| \le L$.
(4) $L \cdot \max(t_{\max} - t_0,\ t_0 - t_{\min}) \le a$.

## Remarks

Conditions (1) and (2) together imply that $f$ is continuous on $J \times B$.
Condition (4) makes the time interval short enough. A curve that starts at $x_0$ with speed at most $L$ cannot leave $B$ during the time interval $J$.
Under these hypotheses, the [[Picard–Lindelöf theorem|0P6KRD5B7A]] gives a unique solution of $x' = f(t, x)$, $x(t_0) = x_0$ on $J$.
A [[$C^1$ vector field satisfies these hypotheses|03Z9V0Y78A]] on a small time interval.

## Explanations

### L1 human


These are the conditions that make the proof of existence and uniqueness work.
The field must be continuous in time, Lipschitz in space and bounded near the start point.
The time interval must be so short that the solution stays in the region where we control the field.
</document>

<document source="https://tamnd.github.io/math2/node/0J7PHRBNNS.md" tag="0J7PHRBNNS" kind="statement">
# Logarithm of a quotient

## Statement

For all real $x > 0$ and $y > 0$,
$$\ln \frac{x}{y} = \ln x - \ln y.$$

## Remarks

Apply the [[product rule for logarithms|0K2FNY6DYV]] to $x = \frac{x}{y} \cdot y$.
With $x = 1$ it gives $\ln(1/y) = -\ln y$.
Mathlib states it as `Real.log_div`.

## Explanations

### L1 human

The logarithm of a quotient is the difference of the logarithms.
Division becomes subtraction.
</document>

<document source="https://tamnd.github.io/math2/node/0J817RTVG6.md" tag="0J817RTVG6" kind="definition">
# Primitive root of unity

## Definition

Let $n$ be a positive integer.
A complex number $\zeta$ is a primitive $n$th root of unity when $\zeta^n = 1$ and $\zeta^k \neq 1$ for each integer $k$ with $1 \le k < n$.

## Remarks

Equivalently, $\zeta$ has order exactly $n$ in the multiplicative group $\mathbb{C} \setminus \{0\}$.
If $\zeta$ is a primitive $n$th root of unity, then $1, \zeta, \zeta^2, \dots, \zeta^{n-1}$ are all the $n$th [[roots of unity|0S61N75H45]].
The number $e^{2 \pi i k / n}$ is primitive exactly when $\gcd(k, n) = 1$. Thus there are $\varphi(n)$ primitive $n$th roots of unity, where $\varphi$ is Euler's totient function.

## Explanations

### L1 human

The number $i$ is a primitive $4$th root of unity: $i, i^2 = -1, i^3 = -i$ are not $1$, and $i^4 = 1$.
The number $-1$ is a $4$th root of unity, but not a primitive one, because already $(-1)^2 = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0J87V9K1GE.md" tag="0J87V9K1GE" kind="statement">
# Sperner's theorem

## Statement

Let $E$ be a set with $n$ elements.
Let $\mathcal{S}$ be a family of subsets of $E$ in which no set is a proper subset of another.
Then $|\mathcal{S}| \le \binom{n}{\lfloor n/2 \rfloor}$.
Equality holds if and only if $\mathcal{S}$ is the family of all subsets of size $\lfloor n/2 \rfloor$, or of all subsets of size $\lceil n/2 \rceil$.

## Remarks

A family with this property is a Sperner family, or an antichain of sets.
For each $k$, the subsets of size $k$ form an antichain. The [[binomial coefficient|092GEF38YD]] $\binom{n}{k}$ is largest for $k = \lfloor n/2 \rfloor$.
Lubell gave a short proof in 1966, with the Lubell–Yamamoto–Meshalkin inequality.

## History

Emanuel Sperner published the theorem in 1928.

## Explanations

### L1 human

You want many subsets of a set, but no one of them may contain another.
The best you can do is to take all subsets of half the size.
</document>

<document source="https://tamnd.github.io/math2/node/0J8ANSYKAC.md" tag="0J8ANSYKAC" kind="definition">
# Composition of an integer

## Definition

Let $n$ be a natural number.
A composition of $n$ is a finite sequence $(a_1, \dots, a_k)$ of positive integers with $a_1 + \dots + a_k = n$.
The number $k$ is the number of parts of the composition.

## Remarks

Unlike a [[partition|0404R9P9J3]], a composition records the order of its parts.
The compositions of $3$ are $(3)$, $(2, 1)$, $(1, 2)$ and $(1, 1, 1)$.
The number $0$ has exactly one composition, the empty sequence.

## Explanations

### L1 human

A composition of $n$ writes $n$ as an ordered sum of positive integers.
So $2 + 1$ and $1 + 2$ are two different compositions of $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0J8T4K4YTE.md" tag="0J8T4K4YTE" kind="statement">
# Taylor's theorem in several variables

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $k \ge 1$ be an integer, and let $f \colon U \to \mathbb{R}$ be of [[class C^k|0TCBZMEYGN]] on $U$.
Let $a \in U$.
For a [[multi-index|0PJXBR6QA3]] $\alpha = (\alpha_1, \dots, \alpha_n)$, write $|\alpha| = \alpha_1 + \dots + \alpha_n$, $\alpha! = \alpha_1! \cdots \alpha_n!$, $h^\alpha = h_1^{\alpha_1} \cdots h_n^{\alpha_n}$, and $D^\alpha f = \partial^{|\alpha|} f / \partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}$.
Then
$$f(a + h) = \sum_{|\alpha| \le k} \frac{D^\alpha f(a)}{\alpha!} \, h^\alpha + R_k(h), \qquad \lim_{h \to 0} \frac{R_k(h)}{\|h\|^k} = 0.$$
If $f$ is of class $C^{k+1}$ and the segment from $a$ to $a + h$ lies in $U$, then
$$R_k(h) = \sum_{|\beta| = k + 1} \frac{k + 1}{\beta!} \, h^\beta \int_0^1 (1 - t)^k \, D^\beta f(a + t h) \, dt.$$

## Remarks

The sum over $|\alpha| \le k$ is the Taylor polynomial of order $k$ of $f$ at $a$.
For $k = 1$ the statement says that $f$ is differentiable at $a$, with $f(a + h) = f(a) + \nabla f(a) \cdot h + o(\|h\|)$.
One proof applies the one-variable Taylor theorem to $g(t) = f(a + t h)$ on $[0, 1]$, together with the chain rule.
The case $k = 2$ is the [[second-order Taylor expansion|0P30YYZN85]].

## Explanations

### L1 human

Near a point, a smooth function of several variables looks like a polynomial in the displacement $h$.
The coefficients of that polynomial are the partial derivatives at the point, divided by factorials.
The error is smaller than $\|h\|^k$, so the approximation gets better quickly as $h$ shrinks.
</document>

<document source="https://tamnd.github.io/math2/node/0J8XE7W68F.md" tag="0J8XE7W68F" kind="definition">
# Reachability

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $u, v$ be vertices of $G$.
The vertex $v$ is reachable from $u$ if there is a [[walk|0VJZM0F9YM]] in $G$ from $u$ to $v$.

## Remarks

Reachability is an equivalence relation on the vertices. It is reflexive by the walk of length $0$, symmetric by reversal of a walk, and transitive by concatenation.
Its equivalence classes are the [[connected components|01VVP10MWJ]] of $G$.
For a [[directed graph|0R5HTV7REV]], reachability along arcs is reflexive and transitive, but in general it is not symmetric.

## Explanations

### L1 human

A vertex is reachable from another one when you can go from the first to the second along edges.
On a map of islands with bridges, two places are reachable from each other when they are on islands linked by bridges.
</document>

<document source="https://tamnd.github.io/math2/node/0J8Y1NCY0K.md" tag="0J8Y1NCY0K" kind="concept">
# Limit of a function of several variables

## Summary

The limit of a function of several variables at a point $a$ is the value that $f(x)$ approaches when $x$ approaches $a$ in [[Euclidean space|0XPR1E85W0]]. The approach can be from any direction and along any route.
The distance in $\mathbb{R}^n$ measures how close $x$ is to $a$.
</document>

<document source="https://tamnd.github.io/math2/node/0J97P5KZY0.md" tag="0J97P5KZY0" kind="definition">
# Bernoulli distribution

## Definition

Let $p \in [0, 1]$.
The Bernoulli distribution with parameter $p$ is the distribution on $\{0, 1\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(1) = p, \qquad p(0) = 1 - p.$$
Equivalently, $p(k) = p^k (1 - p)^{1-k}$ for $k \in \{0, 1\}$. One writes $X \sim \operatorname{Bernoulli}(p)$.

## Remarks

The mean is $p$ and the variance is $p(1 - p)$.
The indicator $\mathbf{1}_A$ of an event $A$ has the Bernoulli distribution with parameter $P(A)$.
It is the distribution of the outcome of one [[Bernoulli trial|0PHC9FE2V5]], with $1$ for success. It is the [[binomial distribution|0VRWZDWW6B]] with $n = 1$.

## Explanations

### L1 human

The Bernoulli distribution describes one yes-or-no experiment: $1$ means yes and $0$ means no.
A fair coin, with $1$ for heads, has the Bernoulli distribution with $p = 1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/0J9TVTNB4K.md" tag="0J9TVTNB4K" kind="statement">
# Determinant of a block triangular matrix

## Statement

Let $K$ be a commutative ring, let $A$ be an $m \times m$ matrix, $B$ an $m \times n$ matrix and $D$ an $n \times n$ matrix over $K$.
Then the [[block matrix|0ASDFEQS6P]] satisfies
$$\det \begin{pmatrix} A & B \\ 0 & D \end{pmatrix} = \det A \cdot \det D,$$
where $0$ is the $n \times m$ zero matrix and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The same holds for the lower block triangular matrix $\begin{pmatrix} A & 0 \\ C & D \end{pmatrix}$.
It generalizes the rule for [[triangular matrices|0K1KKMFHYH]].
For a general block matrix, $\det \begin{pmatrix} A & B \\ C & D \end{pmatrix} \ne \det A \det D - \det B \det C$ in general.

## Explanations

### L1 human

When a matrix has a block of zeros in its lower left corner, its determinant splits.
It is the product of the determinants of the two diagonal blocks.
The block $B$ does not matter, just like the entries above the diagonal of a triangular matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0JAEHXTVV5.md" tag="0JAEHXTVV5" kind="concept">
# Riemann sum

## Summary

A Riemann sum approximates an integral by a finite sum of areas of rectangles. Each rectangle uses the value of the function at a chosen point of a subinterval. The node [[Riemann sum|0HBY48G8G1]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0JARTXX4XP.md" tag="0JARTXX4XP" kind="concept">
# Euler method

## Summary

The Euler method is a numerical method to approximate the solution of an [[initial value problem|075VBY4K4X]] $y' = f(t, y)$, $y(t_0) = y_0$.
With a step size $h > 0$, it sets $t_{k+1} = t_k + h$ and $y_{k+1} = y_k + h\, f(t_k, y_k)$ for $k = 0, 1, 2, \ldots$. Then $y_k$ approximates $y(t_k)$.
It follows the tangent line of the solution for one step at a time.

## Remarks

The error of one step, the local truncation error, is $\frac{h^2}{2} y''(\tau)$ for some $\tau$ between $t_k$ and $t_{k+1}$, when $y$ is twice continuously differentiable.
Let $f$ be $L$-[[Lipschitz|055VVSN00V]] in $y$ with $L > 0$, and let $|y''| \le M$ on the interval. Then the global error satisfies $|y(t_k) - y_k| \le \frac{h M}{2 L} \left( e^{L (t_k - t_0)} - 1 \right)$. The proof uses a [[discrete Grönwall inequality|0NTM3J5MFE]].
So the method is of first order: halving the step size roughly halves the error. It is the simplest Runge–Kutta method.
Leonhard Euler proposed it in his book Institutionum calculi integralis.

## Explanations

### L1 human

At each step, we compute the slope at the current point and walk a short distance $h$ along it.
The result is a broken line that stays close to the true solution when $h$ is small.
For $y' = y$, $y(0) = 1$, the method gives $y_k = (1 + h)^k$, and with $h = 1/k$ this tends to $e$ at time $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0JB228DCWB.md" tag="0JB228DCWB" kind="definition">
# Image of a linear map

## Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *image* of $T$, also called the *range*, is
$$\im T = \{T(v) : v \in V\} \subseteq W.$$

## Remarks

The image is a [[linear subspace|00YC1D652K]] of $W$.
For the map $x \mapsto Ax$ of a matrix $A$, the image is the [[column space|03AJKXH1AV]] of $A$.
The map $T$ is surjective if and only if $\im T = W$.
The dimension of the image is the [[rank|0ZZXSBTR4J]] of $T$.

## Explanations

### L1 human

The image of a linear map is everything that the map can produce.
If a map sends ordinary space onto a plane, then its image is that plane.
The image is always a subspace.
</document>

<document source="https://tamnd.github.io/math2/node/0JB3DP1BMF.md" tag="0JB3DP1BMF" kind="definition">
# Linear combination

## Definition

Let $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
Let $v_1, \dots, v_k \in V$.
A *linear combination* of $v_1, \dots, v_k$ is a vector of the form
$$a_1 v_1 + a_2 v_2 + \cdots + a_k v_k,$$
where $a_1, \dots, a_k \in F$ are the *coefficients*.
For an arbitrary family $(v_i)_{i \in I}$, a linear combination is a sum $\sum_{i \in I} a_i v_i$ in which only finitely many coefficients $a_i$ are nonzero.

## Remarks

The empty linear combination, with $k = 0$, is the zero vector.
The vector $(3, 5)$ is the linear combination $3 e_1 + 5 e_2$ of the standard basis vectors of $F^2$.
The set of all linear combinations of a family is its [[linear span|0DTY8B9HMS]].
The [[matrix–vector product|0PDVQEP0ZC]] $Ax$ is a linear combination of the columns of $A$, with the entries of $x$ as coefficients.

## Explanations

### L1 human

A linear combination mixes some vectors: scale each one by a number, then add the results.
With the vectors $(1, 0)$ and $(0, 1)$ we can make every vector of the plane in this way.
Infinite sums are not allowed: only finitely many coefficients can be nonzero.
</document>

<document source="https://tamnd.github.io/math2/node/0JBJYT69ZV.md" tag="0JBJYT69ZV" kind="concept">
# Graph of a function

## Summary

The graph of a function is the set of all pairs of an input and its value.
</document>

<document source="https://tamnd.github.io/math2/node/0JBMYAA9QH.md" tag="0JBMYAA9QH" kind="statement">
# Ratio formula for the radius of convergence

## Statement

Let $\sum a_n (x - c)^n$ be a real power series with $a_n \ne 0$ for all large $n$.
Assume that $|a_{n+1}| / |a_n| \to L$ as $n \to \infty$, for some $L$ in $[0, \infty]$.
Then the [[radius of convergence|0660QX1VPR]] of the series is $R = 1/L$, with the conventions $1/0 = \infty$ and $1/\infty = 0$.

## Remarks

The proof applies the [[ratio test|0QNY6ZBXPN]] to $\sum a_n (x - c)^n$. The ratio of consecutive terms tends to $L |x - c|$.
The formula needs the limit to exist. The [[Cauchy–Hadamard theorem|07E5MB1CA2]] works without this assumption.
For example, $\sum x^n / n!$ has $L = 0$ and $R = \infty$.
Mathlib states a form for nonzero $L$ as `FormalMultilinearSeries.ofScalars_radius_eq_inv_of_tendsto`.

## Explanations

### L1 human

When the ratio of consecutive coefficients settles to $L$, the coefficients behave like $L^n$.
So the series behaves like a geometric series with ratio $L |x - c|$.
</document>

<document source="https://tamnd.github.io/math2/node/0JC5X4HWR2.md" tag="0JC5X4HWR2" kind="statement">
# Vector calculus identities

## Statement

Let $U \subseteq \mathbb{R}^3$ be open, let $\phi, \psi \colon U \to \mathbb{R}$ be scalar fields, and let $A \colon U \to \mathbb{R}^3$ be a vector field.
If these fields are of class $C^1$, then
$$\nabla (\phi \psi) = \phi \nabla \psi + \psi \nabla \phi, \qquad \nabla \cdot (\psi A) = \psi \, \nabla \cdot A + \nabla \psi \cdot A, \qquad \nabla \times (\psi A) = \psi \, \nabla \times A + \nabla \psi \times A.$$
If these fields are of class $C^2$, then
$$\nabla \times (\nabla \phi) = 0, \qquad \nabla \cdot (\nabla \times A) = 0, \qquad \nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - \Delta A,$$
where $\Delta A$ applies the Laplacian to each component of $A$.

## Remarks

Here $\nabla \phi$ is the [[gradient|0NS2QYVC4Y]], $\nabla \cdot A$ the [[divergence|02NSMQVQWK]], $\nabla \times A$ the [[curl|0GDNHS9J2X]], and $\Delta$ the [[Laplacian|0XQ2ZVN1PZ]].
The product rules follow from the product rule for partial derivatives, applied to each component.
The identities [[curl of a gradient is zero|0M2JRW1J0N]] and [[divergence of a curl is zero|0F56CMMC5P]] follow from the symmetry of second derivatives.
Also, $\nabla \cdot (\nabla \phi) = \Delta \phi$ and $\Delta (\phi \psi) = \phi \Delta \psi + 2 \nabla \phi \cdot \nabla \psi + \psi \Delta \phi$.

## Explanations

### L1 human

These identities are the algebra rules of vector calculus.
They let you expand derivatives of products and simplify repeated derivatives, the way the ordinary product rule does in one variable.
</document>

<document source="https://tamnd.github.io/math2/node/0JCQGPY8Z4.md" tag="0JCQGPY8Z4" kind="concept">
# Stirling numbers of the second kind

## Summary

The Stirling numbers of the second kind $S(n, k)$ count the partitions of an $n$-element set into $k$ nonempty blocks. They connect ordinary powers with falling factorials and they sum to the Bell numbers. The node [[Stirling numbers of the second kind|0ZN07RQ3JS]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0JCSQNNP2M.md" tag="0JCSQNNP2M" kind="definition">
# Converse relation

## Definition

Let $R \subseteq A \times B$ be a [[binary relation|08EJC6Q25H]] from $A$ to $B$.
The converse of $R$ is the relation from $B$ to $A$
$$R^{-1} = \{(b, a) \in B \times A \mid (a, b) \in R\}.$$
So $b \mathrel{R^{-1}} a$ if and only if $a \mathrel{R} b$.

## Remarks

Other names are the inverse relation and the transpose. Another notation is $R^{T}$.
The converse of $<$ on $\mathbb{R}$ is $>$.
The converse of the converse of $R$ is $R$.
For a [[bijection|0SNQEF7Z53]] $f$, the converse of the graph of $f$ is the graph of the [[inverse function|0W766JAPSN]].

## Explanations

### L1 human

To get the converse, read each pair backwards.
The converse of "is a child of" is "is a parent of".
</document>

<document source="https://tamnd.github.io/math2/node/0JD5C82QMB.md" tag="0JD5C82QMB" kind="definition">
# Graph

## Definition

A graph, or simple graph, is a pair $G = (V, E)$ of a set $V$ and a set $E$ of $2$-element subsets of $V$.
The elements of $V$ are the vertices of $G$, and the elements of $E$ are its edges.
Two vertices $u$ and $v$ are adjacent if $\{u, v\} \in E$. The notation for this edge is $uv$, and the notation for adjacency is $u \sim v$.

Equivalently, a graph is a set $V$ with a relation $\sim$ on $V$ that is symmetric and irreflexive.
A graph is finite if $V$ is finite.

## Remarks

In a simple graph, no edge joins a vertex to itself, and two vertices are joined by at most one edge. A [[multigraph|01M5T0KDG7]] drops the second condition, and some variants allow loops.
In a [[directed graph|0R5HTV7REV]] the edges are ordered pairs.
The set $E$ is the [[edge set|0WNGQBPPQX]] of $G$.

## Explanations

### L1 human

A graph is a set of points with lines between some pairs of points.
For example, the people at a party are vertices, and two people are joined by an edge when they know each other.
</document>

<document source="https://tamnd.github.io/math2/node/0JD5ZNW0F4.md" tag="0JD5ZNW0F4" kind="definition">
# Analytic function

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$.
The function $f$ is analytic at $a \in I$ if there are $r > 0$ and a [[power series|0VP56B4EMX]] $\sum c_n (x - a)^n$ that converges to $f(x)$ for all $x$ with $|x - a| < r$.
It is analytic on $I$ if it is analytic at each point of $I$.

## Remarks

An analytic function is [[smooth|0BE3QDH73A]], and its power series at $a$ is its [[Taylor series|0225QPWRRE]] at $a$.
Polynomials, the exponential function, sine, and cosine are analytic on $\mathbb{R}$.
A smooth function need not be analytic. See the [[smooth non-analytic function|09RTYYHXKA]].
Mathlib defines analyticity at a point as `AnalyticAt`.

## Explanations

### L1 human

An analytic function equals a convergent power series near each point.
So it is determined near a point by its derivatives at that point.
</document>

<document source="https://tamnd.github.io/math2/node/0JDAQHWA9N.md" tag="0JDAQHWA9N" kind="definition">
# Convergence in distribution

## Definition

Let $X, X_1, X_2, \dots$ be real random variables, possibly on different probability spaces, with [[cumulative distribution functions|0MXMS3AE13]] $F, F_1, F_2, \dots$.
The sequence $(X_n)$ converges in distribution to $X$ if
$$\lim_{n \to \infty} F_n(x) = F(x)$$
for each $x \in \mathbb{R}$ at which $F$ is continuous. One writes $X_n \xrightarrow{d} X$.

## Remarks

By the portmanteau theorem, an equivalent condition is $E[g(X_n)] \to E[g(X)]$ for each bounded continuous $g \colon \mathbb{R} \to \mathbb{R}$. This form extends to random variables in metric spaces.
The condition depends only on the laws. So it is the weak convergence of the laws of the $X_n$ to the law of $X$.
The restriction to continuity points is necessary. For $X_n = 1/n$ and $X = 0$, $F_n(0) = 0$ for all $n$, but $F(0) = 1$.
The central limit theorem is a result about convergence in distribution.

## Explanations

### L1 human

Convergence in distribution means that the probabilities for $X_n$ come close to those for $X$. The values of $X_n$ themselves need not come close to the values of $X$.
For example, the number of heads in $n$ tosses, after standardization, converges in distribution to a standard normal variable.
</document>

<document source="https://tamnd.github.io/math2/node/0JDB8YJBSR.md" tag="0JDB8YJBSR" kind="definition">
# Triangle

## Definition

Let $K$ be a field and let $\mathcal{A}$ be an [[affine space|09XNAPTJQ7]] over a $K$-vector space.
A triangle in $\mathcal{A}$ is an ordered triple $(A, B, C)$ of points of $\mathcal{A}$ that are not [[collinear|0AFNNTRTJE]]. It is written $ABC$ or $\triangle ABC$.
The points $A$, $B$ and $C$ are its vertices. The segments $BC$, $CA$ and $AB$ are its sides.

## Remarks

Three points are not collinear if and only if they are affinely independent: the vectors $B - A$ and $C - A$ are linearly independent.
In a Euclidean space, the side lengths are often written $a = d(B, C)$, $b = d(C, A)$ and $c = d(A, B)$. The angles of the triangle are $\angle BAC$, $\angle CBA$ and $\angle ACB$.
A triangle is isosceles if two sides have the same length, equilateral if all three do, and scalene if no two do. It is a right triangle if one angle is $\pi / 2$.
Mathlib's `Affine.Triangle` is an affinely independent family of three points, indexed by $\{0, 1, 2\}$.

## Explanations

### L1 human

Pick three points that are not on one straight line and join each pair. The result is a triangle. If the three points were on one line, the "triangle" would be flat, and it would have no area.
</document>

<document source="https://tamnd.github.io/math2/node/0JDE56HWQ8.md" tag="0JDE56HWQ8" kind="definition">
# Path

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A path in $G$ is a [[walk|0VJZM0F9YM]] $(w_0, w_1, \dots, w_k)$ whose vertices $w_0, w_1, \dots, w_k$ are pairwise distinct.

## Remarks

A path uses each edge at most once, so every path is a [[trail|062MQGP8M5]].
A path of length $k$ has $k + 1$ vertices. Its vertices and edges form a subgraph isomorphic to the [[path graph|075WR6W3Y3]] $P_{k+1}$.
Each walk from $u$ to $v$ contains a path from $u$ to $v$. The node [[every walk contains a path|09VZBHB33V]] states this.

## Explanations

### L1 human

A path is a walk that never visits the same vertex twice.
On a map, it is a route that never passes through the same crossing again.
</document>

<document source="https://tamnd.github.io/math2/node/0JEJATBVD7.md" tag="0JEJATBVD7" kind="concept">
# Continuity in R^n

## Summary

Take a function between subsets of Euclidean spaces, and a point $a$. The function is continuous at this point when small changes of the input near $a$ give small changes of the value.
For a function of several variables, the input can change in all coordinates at once.
</document>

<document source="https://tamnd.github.io/math2/node/0JEJJCB0EX.md" tag="0JEJJCB0EX" kind="statement">
# Sign is a homomorphism

## Statement

Let $n \geq 2$.
The map $\operatorname{sgn} \colon S_n \to \{1, -1\}$ is a surjective [[group homomorphism|0JYTG5K9Y4]], where $\{1, -1\}$ is a group under multiplication.
That is, $\operatorname{sgn}(\sigma \tau) = \operatorname{sgn}(\sigma) \operatorname{sgn}(\tau)$ for all $\sigma, \tau \in S_n$.

## Remarks

If $\sigma$ is a product of $m$ transpositions and $\tau$ is a product of $l$ transpositions, then $\sigma \tau$ is a product of $m + l$ transpositions.
The map is surjective because $\operatorname{sgn}((1\ 2)) = -1$.
Its [[kernel|0W88CEGQP1]] is the [[alternating group|0Z5905Q73D]] $A_n$.

## Explanations

### L1 human

When you do two permutations one after the other, you join their chains of swaps.
The number of swaps adds, so the signs multiply.
This is exactly the rule of a homomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/0JEVZAFDY6.md" tag="0JEVZAFDY6" kind="statement">
# The cumulative distribution function determines the distribution

## Statement

Let $\mu$ and $\nu$ be [[probability distributions|0QB913WM5E]] on $\mathbb{R}$ with its Borel σ-algebra.
If $\mu((-\infty, x]) = \nu((-\infty, x])$ for each $x \in \mathbb{R}$, then $\mu(B) = \nu(B)$ for each Borel set $B \subseteq \mathbb{R}$.

## Remarks

In other words, two real random variables with the same [[cumulative distribution function|0MXMS3AE13]] are identically distributed.
The half-lines $(-\infty, x]$ are closed under finite intersections and generate the [[Borel σ-algebra|0BVHZXWKCD]]. The proof uses Dynkin's π-λ theorem to pass from these half-lines to all Borel sets.
So to describe a real distribution, it is enough to give its cumulative distribution function.

## Explanations

### L1 human

If you know $P(X \le x)$ for every $x$, then you know the probability of every reasonable event about $X$.
For example, $P(a < X \le b)$ is a difference of two values of the function, and more complex sets come from such intervals.
</document>

<document source="https://tamnd.github.io/math2/node/0JF0TVYGWT.md" tag="0JF0TVYGWT" kind="concept">
# Affine map

## Summary

An affine map is a map between [[affine spaces|09XNAPTJQ7]] that is a linear map followed by a translation. It sends lines to lines or points, and it keeps parallel lines parallel and ratios of lengths along a line.
</document>

<document source="https://tamnd.github.io/math2/node/0JF1XXGWB4.md" tag="0JF1XXGWB4" kind="definition">
# Parametrised curve

## Definition

Let $I \subseteq \mathbb{R}$ be an interval with more than one point.
A parametrised curve in $\mathbb{R}^n$ is a [[continuous|0S9EK5MQPE]] map $\gamma \colon I \to \mathbb{R}^n$, $\gamma(t) = (\gamma_1(t), \dots, \gamma_n(t))$.
Its image, or trace, is the set $\gamma(I) = \{\gamma(t) : t \in I\}$.
For $r \ge 1$, the curve is a $C^r$ curve if each component $\gamma_i$ is $r$ times continuously differentiable on $I$.
When $I = [a, b]$, the points $\gamma(a)$ and $\gamma(b)$ are the initial point and the end point of $\gamma$.

## Remarks

A curve is different from its image. The maps $t \mapsto (\cos t, \sin t)$ on $[0, 2\pi]$ and $t \mapsto (\cos 2t, \sin 2t)$ on $[0, 2\pi]$ have the same image, the unit circle, but the second goes around it twice.
The helix $\gamma(t) = (\cos t, \sin t, t)$, $t \in \mathbb{R}$, is a $C^\infty$ curve in $\mathbb{R}^3$.
The curve is simple if $\gamma$ is injective, and it is closed if $I = [a, b]$ and $\gamma(a) = \gamma(b)$.

## Explanations

### L1 human

A parametrised curve tells where a moving point is at each time $t$.
It records both the path and the way the point travels along it, including its speed and direction.
</document>

<document source="https://tamnd.github.io/math2/node/0JF9KW2TXB.md" tag="0JF9KW2TXB" kind="definition">
# Regular space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is regular when the property that follows holds.
For each [[closed set|0G5AV9Q4RR]] $F \subseteq X$ and each point $x \in X \setminus F$, there are [[open sets|01ZQQDCERP]] $U$ and $V$ with $x \in U$, $F \subseteq V$ and $U \cap V = \varnothing$.

## Remarks

Names vary. Some authors call a space T3 when it is regular and T0, and then it is also [[Hausdorff|0MXB4CKHFW]]. Other authors include this condition in the word regular.
Equivalently, $X$ is regular if and only if each neighbourhood of a point $x$ contains a closed neighbourhood of $x$.
Each metric space is regular. Each [[normal|0AJ646N339]] T1 space is regular.

## Explanations

### L1 human

In a regular space, you can separate a point from a closed set that does not contain it, by two disjoint open sets.
Hausdorff spaces only separate points from points.
</document>

<document source="https://tamnd.github.io/math2/node/0JFGKKWWFK.md" tag="0JFGKKWWFK" kind="definition">
# Residue

## Definition

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 < |z - c| < r$, so that $c$ is an [[isolated singularity|0TYB3HM9KE]] of $f$.
Let $f(z) = \sum_{n \in \mathbb{Z}} a_n (z - c)^n$ be the [[Laurent expansion|06WHRARZKY]] of $f$ on this punctured disc.
The residue of $f$ at $c$ is the coefficient
$$\operatorname{Res}(f, c) = a_{-1}.$$

## Remarks

For each $\rho$ with $0 < \rho < r$, $\operatorname{Res}(f, c) = \frac{1}{2 \pi i} \oint_{|z - c| = \rho} f(z) \, dz$. This follows from the coefficient formula for $n = -1$.
The residue is $0$ at a [[removable singularity|0P18JC72RE]]. It can be $0$ at a pole too: $\operatorname{Res}(1/z^2, 0) = 0$.
For example, $\operatorname{Res}(1/z, 0) = 1$ and $\operatorname{Res}(e^{1/z}, 0) = 1$.
Residues at simple poles are easy to compute, see [[residue at a simple pole|01991NCP7H]].

## Explanations

### L1 human

When you integrate a Laurent series term by term around a small circle, every term gives $0$ except the term $a_{-1}/(z - c)$, which gives $2 \pi i a_{-1}$.
So the residue is the one coefficient that the integral remembers.
</document>

<document source="https://tamnd.github.io/math2/node/0JG1KZSPFZ.md" tag="0JG1KZSPFZ" kind="statement">
# Turán's theorem

## Statement

Let $n \ge r \ge 1$ be integers, and let $G$ be a simple [[graph|0JD5C82QMB]] with $n$ vertices that has no [[clique|0G52BPNPKG]] with $r + 1$ vertices.
Then $G$ has at most as many edges as the [[Turán graph|0VTCABADBE]] $T(n, r)$.
In particular, $G$ has at most $\left(1 - \frac{1}{r}\right) \frac{n^2}{2}$ edges.

## Remarks

The bound is sharp, because $T(n, r)$ has no clique with $r + 1$ vertices.
The case $r = 2$ is [[Mantel's theorem|0MXEHSCYCS]] on graphs without triangles.
The theorem is a central result of extremal graph theory.

## History

Pál Turán described the theorem and the Turán graphs in 1941.

## Explanations

### L1 human

How many edges can a graph have if no $r + 1$ of its vertices are all joined to each other?
The answer is the Turán graph: split the vertices into $r$ balanced teams and join all pairs across teams.
</document>

<document source="https://tamnd.github.io/math2/node/0JG9W3KA6B.md" tag="0JG9W3KA6B" kind="concept">
# Connected component

## Summary

A connected component of a graph is a maximal piece in which all vertices are reachable from each other. The components split each graph into connected parts. The node [[connected component|01VVP10MWJ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0JGB7A6GTS.md" tag="0JGB7A6GTS" kind="definition">
# Rank of a linear map

## Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *rank* of $T$ is
$$\rank T = \dim \im T,$$
the [[dimension|03GSZ8RTZJ]] of the [[image|0WSFKKT6FF]] of $T$.

## Remarks

For the map $x \mapsto Ax$ of a matrix $A$, the rank is the [[rank of the matrix|0ZC9YTXZRY]].
The rank satisfies $\rank T \le \dim V$ and $\rank T \le \dim W$.
The [[rank–nullity theorem|0W4FSW1TVB]] relates the rank and the [[nullity|0APCFW7639]].
The rank of a composition satisfies $\rank(T \circ S) \le \min(\rank T, \rank S)$.

## Explanations

### L1 human

The rank of a linear map counts the dimensions of its output.
A map from space onto a plane has rank $2$.
The zero map has rank $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0JGEQXQ4AC.md" tag="0JGEQXQ4AC" kind="concept">
# Presentation of a group

## Summary

A presentation $\langle S \mid R \rangle$ describes a group by a set $S$ of generators and a set $R$ of relations.
The group is the quotient of the free group on $S$ by the normal subgroup generated by $R$.
</document>

<document source="https://tamnd.github.io/math2/node/0JGZCDT5DN.md" tag="0JGZCDT5DN" kind="example">
# Space of polynomials of bounded degree

## Example

Let $F$ be a field and let $n \ge 0$ be an integer.
Let $F[x]_{<n}$ be the set of polynomials in $x$ with coefficients in $F$ and of degree less than $n$, with the zero polynomial included.
It is a [[vector space|000AYPA9AJ]] over $F$, with the usual addition of polynomials and multiplication by constants.
The monomials $1, x, x^2, \dots, x^{n-1}$ form a [[basis|0KCQ9Y3QKA]], so $\dim F[x]_{<n} = n$.

## Remarks

The set of polynomials of degree exactly $n$ is not a vector space.
It does not contain $0$, and it is not closed under addition.
The whole polynomial ring $F[x]$ is an infinite-dimensional vector space with basis $1, x, x^2, \dots$.
[[Differentiation|0SS770NS8B]] is a linear map from $F[x]_{<n}$ to $F[x]_{<n-1}$ for $n \ge 1$.

## Explanations

### L1 human

Polynomials of small degree can be added and scaled, and the result stays a polynomial of small degree.
So they form a vector space.
A polynomial $a_0 + a_1 x + a_2 x^2$ is the same data as the list $(a_0, a_1, a_2)$.
</document>

<document source="https://tamnd.github.io/math2/node/0JH4MN4K1C.md" tag="0JH4MN4K1C" kind="definition">
# Series

## Definition

Let $(a_n)_{n \in \mathbb{N}}$ be a [[real sequence|0CTV783HYT]].
The series with terms $a_n$ is the sequence $(s_N)_{N \in \mathbb{N}}$ given by
$$s_N = a_0 + a_1 + \cdots + a_N = \sum_{n=0}^{N} a_n.$$
The notation for the series is $\sum_{n=0}^{\infty} a_n$, or $\sum a_n$.

## Remarks

The number $s_N$ is a [[partial sum|0NB643K60Q]] of the series.
The symbol $\sum_{n=0}^{\infty} a_n$ also denotes the [[sum of the series|0K0H5E3Z5N]], when the series is convergent. The context shows which meaning is intended.
A series can start at another index, for example $\sum_{n=1}^{\infty} 1/n$.

## Explanations

### L1 human

A series is an infinite sum $a_0 + a_1 + a_2 + \cdots$.
We cannot add infinitely many numbers directly, so we add the first terms and watch what happens as we add more.
</document>

<document source="https://tamnd.github.io/math2/node/0JHGAV9H25.md" tag="0JHGAV9H25" kind="statement">
# Eigenspaces of a self-adjoint operator are orthogonal

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be a [[self-adjoint|0NEGN5CRSC]] linear map.
Let $v$ and $w$ be eigenvectors of $T$ for distinct eigenvalues $\lambda \ne \mu$.
Then $v$ and $w$ are [[orthogonal|0Y2X7P2WB7]].
So the [[eigenspaces|0G0RFPG8HR]] of $T$ for distinct eigenvalues are orthogonal to each other.

## Remarks

The proof: $\lambda \langle v, w \rangle = \langle T(v), w \rangle = \langle v, T(w) \rangle = \mu \langle v, w \rangle$, since $\mu$ is real.
So $(\lambda - \mu) \langle v, w \rangle = 0$, and $\langle v, w \rangle = 0$.

## Explanations

### L1 human

For a symmetric matrix, eigenvectors for different eigenvalues are perpendicular.
So the eigenspaces meet at right angles.
This is the first step towards an orthonormal basis of eigenvectors.
</document>

<document source="https://tamnd.github.io/math2/node/0JHM6JK7FD.md" tag="0JHM6JK7FD" kind="example">
# The ring of integers

## Example

Let $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$ be the set of integers, with the usual addition and multiplication.
Then $\mathbb{Z}$ is a [[commutative ring|0GYGHKX834]].
Its zero element is $0$ and its identity element is $1$.

## Remarks

The only [[units|019NFGFPR5]] of $\mathbb{Z}$ are $1$ and $-1$.
The product of two nonzero integers is not zero, thus $\mathbb{Z}$ is an [[integral domain|09S3PFBEPM]].
The ring $\mathbb{Z}$ is not a [[field|0QA8CHH8MG]], because $2$ has no multiplicative inverse in $\mathbb{Z}$.

## Explanations

### L1 human

The integers are the first example of a ring.
You can add, subtract and multiply two integers, and the result is an integer.
You cannot always divide: $1 / 2$ is not an integer.
</document>

<document source="https://tamnd.github.io/math2/node/0JHXPVFKT0.md" tag="0JHXPVFKT0" kind="concept">
# Exponential generating function

## Summary

An exponential generating function encodes a sequence $(a_n)$ as the series $\sum a_n x^n / n!$. It is the natural tool to count labelled structures, because products give binomial convolutions. The node [[exponential generating function|0KE6NBTVY8]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0JJ6GDDE7Z.md" tag="0JJ6GDDE7Z" kind="concept">
# Underdamped

## Summary

A [[damped harmonic oscillator|0RNXVVTJTJ]] $x'' + 2 \zeta \omega_0 x' + \omega_0^2 x = 0$ with $\omega_0 > 0$ is underdamped when its damping ratio satisfies $0 < \zeta < 1$.
Then the characteristic roots are $-\zeta \omega_0 \pm i \omega_1$ with $\omega_1 = \omega_0 \sqrt{1 - \zeta^2}$.
Each nonzero solution has the form $x(t) = A e^{-\zeta \omega_0 t} \cos(\omega_1 t + \varphi)$ with $A > 0$: an oscillation whose amplitude decays exponentially.

## Remarks

The other cases are critical damping, $\zeta = 1$, and overdamping, $\zeta > 1$. In these cases the solution changes sign at most one time.
The damped frequency $\omega_1$ is smaller than the undamped frequency $\omega_0$.
The zeros of an underdamped solution are spaced $\pi / \omega_1$ apart.

## Explanations

### L1 human

An underdamped system overshoots its rest position again and again.
Each swing is smaller than the previous one, by the same factor.
A guitar string or a car on soft springs behaves like this.
</document>

<document source="https://tamnd.github.io/math2/node/0JJ9J1XZ8G.md" tag="0JJ9J1XZ8G" kind="definition">
# Boundary

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
The boundary of $S$ is the [[set difference|0DT0TCV75N]]
$$\partial S = \overline{S} \setminus \operatorname{int} S,$$
where $\overline{S}$ is the [[closure|0DRKT1QJR7]] and $\operatorname{int} S$ is the [[interior|0C38S0KMJK]] of $S$.

## Remarks

A point $x$ is in $\partial S$ if and only if each open set that contains $x$ meets both $S$ and $X \setminus S$.
The boundary is closed, and $\partial S = \partial (X \setminus S)$.
In $\mathbb{R}$, the boundary of $(0, 1)$ is $\{0, 1\}$, and the boundary of $\mathbb{Q}$ is $\mathbb{R}$.
Another name for the boundary is the frontier.

## Explanations

### L1 human

The boundary is the edge of a set. It has the points where you can step into the set and also out of it with a very small step.
For a disc in the plane, the boundary is its circle.
</document>

<document source="https://tamnd.github.io/math2/node/0JJRENWMP6.md" tag="0JJRENWMP6" kind="statement">
# Cauchy's estimates

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $c \in \mathbb{C}$ and $r > 0$ be such that the closed disc $\{z : |z - c| \le r\}$ is contained in $U$.
Let $M \ge 0$ be a real number with $|f(z)| \le M$ for each $z$ with $|z - c| = r$.
Then for each integer $n \ge 0$,
$$|f^{(n)}(c)| \le \frac{n! \, M}{r^n}.$$

## Remarks

Proof: apply the [[estimation lemma|0DJMGPN8K9]] to the [[formula for the derivatives|0F0H5P5DED]] with $a = c$. The circle has length $2 \pi r$, and $|f(z) / (z - c)^{n+1}| \le M / r^{n+1}$ on it.
For $n = 1$, the estimate is $|f'(c)| \le M / r$.
If $f(z) = \sum a_n (z - c)^n$ is the Taylor series, the estimate is $|a_n| \le M / r^n$.

## Explanations

### L1 human

If a holomorphic function is small on a circle, then its derivatives at the center cannot be large.
A larger circle with the same bound $M$ gives a better bound for the derivatives. This is the idea of [[Liouville's theorem|0EHKVRF4J6]].
</document>

<document source="https://tamnd.github.io/math2/node/0JJV80RXHD.md" tag="0JJV80RXHD" kind="statement">
# Abelian simple groups have prime order

## Statement

Let $G$ be an [[abelian|082GHNPJ53]] [[group|G5S1M9]].
Then $G$ is [[simple|0YMYMNQ0QZ]] if and only if $G$ is finite of prime order.
In that case, $G$ is [[cyclic|0KVGH0J5B3]], and $G \cong \mathbb{Z}/p\mathbb{Z}$ for a prime number $p$.

## Remarks

In an abelian group, each subgroup is normal, so a simple abelian group has no subgroup except $\{e\}$ and itself.
Then $G = \langle g \rangle$ for each $g \neq e$, so $G$ is cyclic; $\mathbb{Z}$ is not simple, because $2\mathbb{Z}$ is a proper nontrivial subgroup.
A finite cyclic group of order $n$ has a subgroup of order $d$ for each divisor $d$ of $n$, so $n$ must be prime.

## Explanations

### L1 human

In an abelian group, all subgroups are normal.
So a simple abelian group has no subgroups at all except the trivial ones.
Only the groups of prime order have this property.
</document>

<document source="https://tamnd.github.io/math2/node/0JK6M18QZH.md" tag="0JK6M18QZH" kind="definition">
# Kronecker product

## Definition

Let $F$ be a field, let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $F$, and let $B$ be a $p \times q$ matrix over $F$.
The *Kronecker product* $A \otimes B$ is the $mp \times nq$ [[block matrix|0ASDFEQS6P]]
$$A \otimes B = \begin{pmatrix} a_{11} B & \cdots & a_{1n} B \\ \vdots & \ddots & \vdots \\ a_{m1} B & \cdots & a_{mn} B \end{pmatrix}.$$

## Remarks

It satisfies $(A \otimes B)(C \otimes D) = (AC) \otimes (BD)$ when the products $AC$ and $BD$ are defined.
If $A$ and $B$ are invertible, then $(A \otimes B)^{-1} = A^{-1} \otimes B^{-1}$.
It is not commutative in general, but $A \otimes B$ and $B \otimes A$ are permutation equivalent.
It is the matrix of the tensor product of two linear maps, in the tensor product bases ordered suitably.

## Explanations

### L1 human

The Kronecker product replaces each entry of $A$ with that entry times the whole matrix $B$.
For example, $I_2 \otimes B$ is the block diagonal matrix with two copies of $B$.
It is the matrix form of the tensor product.
</document>

<document source="https://tamnd.github.io/math2/node/0JKW7W7XFR.md" tag="0JKW7W7XFR" kind="concept">
# Combinatorics

## Summary

Combinatorics is the part of mathematics that studies finite or countable discrete structures.
It asks three main types of question about such structures.
Enumerative questions ask how many structures of a given type exist.
Existence questions ask if a structure with given properties exists.
Extremal questions ask how large or small a structure with given properties can be.

Its basic objects are finite sets, arrangements of elements, subsets, partitions and graphs.
Its basic tools are counting rules, bijections, recurrences, generating functions and the pigeonhole principle.

## Remarks

The [[rule of sum|0EE2V3CHFK]] and the [[rule of product|0VPK4X73Q0]] are the first counting rules.
[[Binomial coefficients|092GEF38YD]] count the subsets of a given size of a finite set.
Graph theory is a large subfield of combinatorics. Its basic notion is a [[graph|0JD5C82QMB]].

## Explanations

### L1 human

Combinatorics is the mathematics of counting and arranging.
A typical question is: in how many ways can 5 people sit in a row?
The answer is $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$, and combinatorics gives general methods for questions of this type.
</document>

<document source="https://tamnd.github.io/math2/node/0JM7PA288B.md" tag="0JM7PA288B" kind="statement">
# Cauchy's theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number that divides $|G|$.
Then $G$ has an element of [[order|0WE2NHZJ7B]] $p$.

## Remarks

Equivalently, $G$ has a [[subgroup|0X1JXT47GA]] of order $p$, namely the cyclic subgroup generated by such an element.
It is a partial converse of [[Lagrange's theorem|3C53TD]]: a prime divisor of $|G|$ is always the order of a subgroup.
One proof lets $\mathbb{Z}/p\mathbb{Z}$ act by cyclic shifts on the set of tuples $(g_1, \dots, g_p)$ with $g_1 g_2 \cdots g_p = e$, and uses the [[fixed point lemma|098D2E7DEH]].
The [[first Sylow theorem|0V98WY94B8]] generalises the result to prime powers.

## History

Augustin-Louis Cauchy discovered the theorem in 1845.

## Explanations

### L1 human

Lagrange says that the order of an element divides the order of the group.
Cauchy says that, for prime numbers, each divisor really appears as the order of an element.
For example, a group of order $15$ has an element of order $3$ and an element of order $5$.
</document>

<document source="https://tamnd.github.io/math2/node/0JMR2HM2N2.md" tag="0JMR2HM2N2" kind="concept">
# Logical equivalence

## Summary

Two formulas are logically equivalent when they have the same truth value in every case.
</document>

<document source="https://tamnd.github.io/math2/node/0JMTY8MF54.md" tag="0JMTY8MF54" kind="concept">
# Sum of subspaces

## Summary

The sum of two subspaces is the set of all sums of a vector of the first subspace and a vector of the second subspace.
It is the smallest subspace that contains both.
</document>

<document source="https://tamnd.github.io/math2/node/0JN0SQG2JH.md" tag="0JN0SQG2JH" kind="concept">
# Kernel of a ring homomorphism

## Summary

The kernel of a ring homomorphism is the set of the elements that it sends to zero.
It is an ideal of the domain of the homomorphism.
The [[definition of the kernel|0AAHE7GEW6]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0JN9P4A944.md" tag="0JN9P4A944" kind="definition">
# Sample variance

## Definition

Let $n \ge 2$ and let $X_1, \dots, X_n$ be real random variables with [[sample mean|0QNRBHWV35]] $\bar{X}_n$.
The sample variance is the [[statistic|036756NFD6]]
$$S_n^2 = \frac{1}{n - 1} \sum_{i=1}^{n} \left(X_i - \bar{X}_n\right)^2.$$
The sample standard deviation is $S_n = \sqrt{S_n^2}$.

## Remarks

If the $X_i$ are i.i.d. with [[variance|007QED3E0Y]] $\sigma^2 < \infty$, then $E[S_n^2] = \sigma^2$. So $S_n^2$ is an unbiased estimator of $\sigma^2$.
The divisor $n - 1$ instead of $n$ is Bessel's correction. The version with divisor $n$ has expectation $\frac{n-1}{n} \sigma^2$, so it has the [[bias|01BD2TFFP4]] $-\sigma^2/n$.
The reason is that $\sum_i (X_i - \bar{X}_n)^2 \le \sum_i (X_i - \mu)^2$: the deviations from $\bar{X}_n$ are on average smaller than those from the true mean $\mu$.
For i.i.d. $\mathcal{N}(\mu, \sigma^2)$ variables, $(n - 1) S_n^2 / \sigma^2$ has the chi-squared distribution with $n - 1$ degrees of freedom, and it is independent of $\bar{X}_n$.



## Explanations

### L1 human

The sample variance is the average squared distance of the data from their mean, but with the divisor $n - 1$.
For the data $4$, $7$, $10$, the mean is $7$, the squared deviations add up to $18$, and the sample variance is $18/2 = 9$.
</document>

<document source="https://tamnd.github.io/math2/node/0JNCBFMPXR.md" tag="0JNCBFMPXR" kind="statement">
# Lindemann–Weierstrass theorem

## Statement

Let $\alpha_1, \dots, \alpha_n$ be distinct [[algebraic numbers|0MB4TGMXG6]].
Then $e^{\alpha_1}, \dots, e^{\alpha_n}$ are linearly independent over the field of algebraic numbers.
That is, if $\beta_1, \dots, \beta_n$ are algebraic numbers with $\beta_1 e^{\alpha_1} + \dots + \beta_n e^{\alpha_n} = 0$, then $\beta_1 = \dots = \beta_n = 0$.

## Remarks

A consequence: $e^{\alpha}$ is [[transcendental|079EGSD32Y]] for each nonzero algebraic number $\alpha$. In particular, $e$ is transcendental.
Another consequence: $\pi$ is transcendental. Otherwise $i\pi$ would be algebraic, and $e^{i\pi} + 1 = 0$ would contradict the theorem.
So the circle cannot be squared with ruler and compass.

## History

Ferdinand von Lindemann proved in 1882 that $e^{\alpha}$ is transcendental for each nonzero algebraic $\alpha$.
Karl Weierstrass proved the general statement in 1885.

## Explanations

### L1 human

The exponential function sends nonzero algebraic numbers to transcendental numbers.
With $\alpha = 1$ this shows that $e$ is transcendental, and with Euler's formula it shows the same for $\pi$.
</document>

<document source="https://tamnd.github.io/math2/node/0JNMJEDD93.md" tag="0JNMJEDD93" kind="statement">
# Convolution theorem for the Laplace transform

## Statement

Let $f, g : [0, \infty) \to \mathbb{C}$ be piecewise continuous, and let $M, a$ be real constants with $|f(t)| \le M e^{a t}$ and $|g(t)| \le M e^{a t}$ for all $t \ge 0$.
Let the convolution $f * g : [0, \infty) \to \mathbb{C}$ be
$$(f * g)(t) = \int_0^t f(\tau)\, g(t - \tau)\, d\tau.$$
Let $F$ and $G$ be the [[Laplace transforms|05XT5AVJBK]] of $f$ and $g$.
Then for each $s \in \mathbb{C}$ with $\operatorname{Re} s > a$, the Laplace transform of $f * g$ converges at $s$, and $\mathcal{L}\{f * g\}(s) = F(s)\, G(s)$.

## Remarks

The proof writes $F(s) G(s)$ as a double integral over the quadrant $\tau, u \ge 0$, and changes variables to $t = \tau + u$. The absolute convergence for $\operatorname{Re} s > a$ justifies the change of the order of integration.
The convolution is commutative: $f * g = g * f$.
Example: $\mathcal{L}^{-1}\left\{\frac{1}{s^2 (s - 1)}\right\}$ is the convolution of $t$ and $e^t$, that is $e^t - 1 - t$.
In [[solving initial value problems|0K3HFZJK5Q]], the solution with zero initial values is the convolution of the forcing term with the impulse response.

## Explanations

### L1 human

Multiplying two transforms corresponds to a special kind of product of the functions, the convolution.
The convolution mixes the past values of one function with the other, as a memory of earlier inputs.
</document>

<document source="https://tamnd.github.io/math2/node/0JNXSS9TJ9.md" tag="0JNXSS9TJ9" kind="concept">
# Universal set

## Summary

A universal set is a set that contains all the objects under discussion.
In most set theories, there is no set of all sets, so a universal set is always relative to a context.
</document>

<document source="https://tamnd.github.io/math2/node/0JP0W07Y9A.md" tag="0JP0W07Y9A" kind="statement">
# Fundamental theorem of algebra

## Statement

Let $n \ge 1$ be an integer and let $a_0, a_1, \dots, a_n$ be [[complex numbers|09FTAET5Z4]] with $a_n \neq 0$.
Let $p(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$.
Then there is a complex number $z_0$ with $p(z_0) = 0$.

## Remarks

Proof with [[Liouville's theorem|0EHKVRF4J6]]: if $p$ has no zero, then $1/p$ is entire. Since $|p(z)| \to \infty$ as $|z| \to \infty$, $1/p$ is bounded, so it is constant, which is false for $n \ge 1$.
By induction on $n$, it follows that $p(z) = a_n (z - z_1) \cdots (z - z_n)$ for some $z_1, \dots, z_n \in \mathbb{C}$. So $p$ has exactly $n$ roots, counted with multiplicity.
The theorem holds for polynomials with real coefficients too, but the roots can be non-real: $z^2 + 1$ has the roots $i$ and $-i$.

## Explanations

### L1 human

Each polynomial equation of degree at least $1$ has a solution in the complex numbers.
So we never need to add new numbers to $\mathbb{C}$ to solve polynomial equations, unlike the step from $\mathbb{R}$ to $\mathbb{C}$ for $x^2 = -1$.
</document>

<document source="https://tamnd.github.io/math2/node/0JP0ZF57PN.md" tag="0JP0ZF57PN" kind="statement">
# Kőnig's theorem

## Statement

Let $G$ be a finite [[bipartite graph|07DVHJ8N0A]].
Then the largest number of edges in a [[matching|0Z87BHAFTB]] of $G$ is equal to the smallest number of vertices in a [[vertex cover|0KDADE9RQ8]] of $G$.

## Remarks

In every graph, a vertex cover has at least as many vertices as a matching has edges. Kőnig's theorem says that equality holds for bipartite graphs.
The theorem fails for other graphs. The triangle $K_3$ has matchings with $1$ edge only, but each vertex cover has $2$ vertices.

## History

Dénes Kőnig published the theorem in 1931.
Jenő Egerváry found it independently in the same year, in a more general form with weights.

## Explanations

### L1 human

In a bipartite graph, the most pairs that you can match at once is the same as the fewest vertices that touch every edge.
Each of the two numbers gives a certificate that the other one is optimal.
</document>

<document source="https://tamnd.github.io/math2/node/0JPCHTXWG0.md" tag="0JPCHTXWG0" kind="concept">
# Function of several variables

## Summary

A function of several real variables is a [[function|0EPAT51DXQ]] $f \colon D \to \mathbb{R}$ whose domain $D$ is a subset of [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$ for some $n \ge 2$.
Its value at a point $x = (x_1, \dots, x_n)$ of $D$ is written $f(x)$ or $f(x_1, \dots, x_n)$.
Examples are $f(x, y) = x^2 + y^2$ on $\mathbb{R}^2$ and $g(x, y, z) = xyz$ on $\mathbb{R}^3$.

## Remarks

When the values lie in $\mathbb{R}^m$ with $m \ge 2$, the function is a [[vector-valued function|018GTRS7A4]] of several variables.
The domain is often an open subset of $\mathbb{R}^n$, so that each point of the domain has a whole ball around it in the domain.

## Explanations

### L1 human

A function of two variables takes two numbers and gives one number. An example is the height of a hill above the point $(x, y)$ of a map.
Each input is now a point of the plane or of space, not a point of a line.
</document>

<document source="https://tamnd.github.io/math2/node/0JPDTEDCFD.md" tag="0JPDTEDCFD" kind="statement">
# Subgroups of abelian groups are normal

## Statement

Let $G$ be an [[abelian group|082GHNPJ53]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $H$ is [[normal|0CB0DX5BVV]] in $G$.

## Remarks

For $g \in G$ and $h \in H$, commutativity gives $ghg^{-1} = hgg^{-1} = h \in H$.
The converse is false: in the quaternion group $Q_8$ each subgroup is normal, but $Q_8$ is not abelian.
More generally, each subgroup of the center of any group is normal.

## Explanations

### L1 human

In an abelian group, conjugation does nothing, because $ghg^{-1} = h$.
So a subgroup can never be moved by conjugation.
Thus all subgroups are normal.
</document>

<document source="https://tamnd.github.io/math2/node/0JPH2Y4DFV.md" tag="0JPH2Y4DFV" kind="concept">
# Central binomial coefficient

## Summary

The central binomial coefficients $\binom{2n}{n}$ are the middle entries of the even rows of Pascal's triangle. They count balanced lattice paths and appear in the formula for Catalan numbers. The node [[central binomial coefficient|01WHD9T0V6]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0JPYSD5E0B.md" tag="0JPYSD5E0B" kind="definition">
# Order of an element

## Definition

Let $G$ be a [[group|G5S1M9]] with identity element $e$, and let $g \in G$.
When there is a positive integer $n$ with $g^n = e$, the order of $g$ is the smallest such $n$.
When there is no such $n$, the element $g$ has infinite order.
The notation for the order of $g$ is $\operatorname{ord}(g)$ or $|g|$.

## Remarks

The identity element is the only element of order $1$.
In $(\mathbb{Z}, +)$, each nonzero element has infinite order.
In the multiplicative group $\{1, i, -1, -i\}$, the element $-1$ has order $2$ and the element $i$ has order $4$.
In a finite group, each element has finite order, and the order of $g$ equals the [[order|0B85S2V2DR]] of the subgroup that $g$ generates.

## Explanations

### L1 human

Take an element and multiply it by itself again and again.
The order is the number of steps that you need to come back to the identity for the first time.
For a rotation by $90$ degrees, you need $4$ steps, so its order is $4$.
</document>

<document source="https://tamnd.github.io/math2/node/0JQ1KNQW2K.md" tag="0JQ1KNQW2K" kind="statement">
# Order of the general linear group over a finite field

## Statement

Let $q$ be a prime power, let $\mathbb{F}_q$ be the finite field with $q$ elements, and let $n \geq 1$.
The group $\mathrm{GL}_n(\mathbb{F}_q)$ of the invertible $n \times n$ matrices over $\mathbb{F}_q$ has [[order|0B85S2V2DR]] $\prod_{k=0}^{n-1} (q^n - q^k) = (q^n - 1)(q^n - q) \cdots (q^n - q^{n-1})$.

## Remarks

A matrix is invertible exactly when its columns are linearly independent.
There are $q^n - 1$ choices for the first column. Column $k+1$ has $q^n - q^k$ choices, because it must avoid the $q^k$ vectors in the span of the first $k$ columns.
For example, $|\mathrm{GL}_2(\mathbb{F}_2)| = 3 \cdot 2 = 6$, and $\mathrm{GL}_2(\mathbb{F}_2) \cong S_3$.
Also, $|\mathrm{GL}_3(\mathbb{F}_2)| = 7 \cdot 6 \cdot 4 = 168$.

## History

Évariste Galois computed the order of the general linear group over a prime field in 1832, in his last letter to Chevalier.

## Explanations

### L1 human

To build an invertible matrix, choose its columns one after the other.
Each new column must not lie in the span of the columns that you already chose.
Counting the choices at each step gives the product formula.
</document>

<document source="https://tamnd.github.io/math2/node/0JQ5Y3WWVE.md" tag="0JQ5Y3WWVE" kind="example">
# Alternating harmonic series

## Example

The alternating harmonic series is
$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots.$$
It converges by the [[alternating series test|0PK8QH1W1V]], since $1/n$ decreases to $0$.
Its sum is $\ln 2 \approx 0.6931$.

## Remarks

The series of absolute values is the [[harmonic series|0CXTMAY5PX]], which diverges. So the convergence is [[conditional|02YRQXZDCS]].
The value $\ln 2$ comes from the [[Mercator series|0YZXNE3HFR]] $\ln(1 + x) = x - x^2/2 + x^3/3 - \cdots$ at $x = 1$, with [[Abel's theorem|0S4EMDQS2F]].
By the [[Riemann rearrangement theorem|0E7VBYGFF7]], a reordering of the terms can give any real sum.

## Explanations

### L1 human

The partial sums are $1, 0.5, 0.833, 0.583, 0.783, \dots$.
They swing around $\ln 2 \approx 0.693$ and close in on it slowly.
</document>

<document source="https://tamnd.github.io/math2/node/0JQE130DHW.md" tag="0JQE130DHW" kind="statement">
# Weierstrass approximation theorem

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous.
Then for every $\varepsilon > 0$ there is a polynomial $p$ with $|f(x) - p(x)| < \varepsilon$ for all $x \in [a, b]$.

## Remarks

Equivalently, there is a sequence of polynomials that converges [[uniformly|08FPABDQ4C]] to $f$ on $[a, b]$.
One proof uses the [[Bernstein polynomials|0C4VWBFY73]] on $[0, 1]$, together with [[uniform continuity|0W0TZDRF9C]] of $f$.
The Stone–Weierstrass theorem generalizes the result to compact spaces. Mathlib states the theorem as `polynomialFunctions_closure_eq_top`.

## History

Karl Weierstrass proved the theorem in 1885.

## Explanations

### L1 human

Every continuous function on a closed bounded interval is uniformly close to a polynomial.
So polynomials can imitate any continuous graph as closely as we like.
</document>

<document source="https://tamnd.github.io/math2/node/0JQWWFCS9R.md" tag="0JQWWFCS9R" kind="definition">
# Ordered pair

## Definition

An ordered pair $(a, b)$ has a first entry $a$ and a second entry $b$, with this property:
$$(a, b) = (c, d) \iff a = c \text{ and } b = d.$$
In set theory, the Kuratowski definition is $(a, b) = \{\{a\}, \{a, b\}\}$. It has the property above.

## Remarks

In general $(a, b) \neq (b, a)$, but the sets $\{a, b\}$ and $\{b, a\}$ are equal.
Ordered pairs are the elements of a [[Cartesian product|0C81YBVTWH]].
An ordered triple can be defined as $(a, b, c) = ((a, b), c)$.

## Explanations

### L1 human

A point in the plane is an ordered pair of coordinates.
The points $(1, 2)$ and $(2, 1)$ are different, so the order matters.
The Kuratowski definition only shows that sets can model pairs.
</document>

<document source="https://tamnd.github.io/math2/node/0JRC32D6K2.md" tag="0JRC32D6K2" kind="definition">
# Multiplication of natural numbers

## Definition

Let $\mathbb{N}$ be the set of [[natural numbers|0MWKJ44V5A]], let $S$ be the [[successor function|0832XVR512]] and let $+$ be the [[addition|0W5DSDFQ4G]] of $\mathbb{N}$.
Multiplication is the function $\cdot \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, $(a, b) \mapsto a \cdot b$, with these two properties for all $a$ and $b$ in $\mathbb{N}$:
$$a \cdot 0 = 0, \qquad a \cdot S(b) = a \cdot b + a.$$

## Remarks

For each $a$, these equations are a [[definition by recursion|0DSHAVH45F]] on $b$. Thus exactly one such function exists.
From the definition, $a \cdot 1 = a \cdot 0 + a = a$.
We often write $ab$ for $a \cdot b$.

## Explanations

### L1 human

Multiplication is repeated addition: $3 \cdot 4$ is $3 + 3 + 3 + 3 = 12$.
The definition says this in two rules: zero copies give $0$, and one more copy adds one more $a$.
</document>

<document source="https://tamnd.github.io/math2/node/0JRW4X1MDC.md" tag="0JRW4X1MDC" kind="definition">
# Inverse Laplace transform

## Definition

Let $a$ be a real number and let $F$ be a complex function on the half-plane $\{s \in \mathbb{C} : \operatorname{Re} s > a\}$.
An inverse Laplace transform of $F$ is a piecewise continuous function $f : [0, \infty) \to \mathbb{C}$ with two properties. First, there are constants $M, b$ with $|f(t)| \le M e^{b t}$ for $t \ge 0$. Second, the [[Laplace transform|05XT5AVJBK]] of $f$ equals $F(s)$ for each $s$ with $\operatorname{Re} s > \max(a, b)$.
The notation is $f = \mathcal{L}^{-1}\{F\}$.

## Remarks

By Lerch's theorem, two such functions with the same transform agree at each point where both are continuous. So $\mathcal{L}^{-1}\{F\}$ is unique in practice, and unique among continuous functions.
Let $\gamma$ be a real number larger than the real parts of all singularities of $F$. Under suitable conditions, the Bromwich integral recovers $f$: $f(t) = \frac{1}{2 \pi i} \lim_{T \to \infty} \int_{\gamma - iT}^{\gamma + iT} e^{s t} F(s)\, ds$.
In practice, one splits $F$ into partial fractions and reads the inverse of each term from a table. For example, $\mathcal{L}^{-1}\{1/(s - a)\} = e^{a t}$.

## Explanations

### L1 human

The inverse transform goes back from the $s$ side to the time side.
After we solve a problem in terms of $F(s)$, the inverse transform gives the solution as a function of time.
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<document source="https://tamnd.github.io/math2/node/0JT92VK307.md" tag="0JT92VK307" kind="concept">
# Bernstein polynomial

## Summary

A Bernstein polynomial of a function on $[0, 1]$ is a weighted average of its values at the points $k/n$. For a continuous function, these polynomials converge uniformly to the function. The node [[Bernstein polynomial|0C4VWBFY73]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0JTC045NNZ.md" tag="0JTC045NNZ" kind="concept">
# Legendre symbol

## Summary

The Legendre symbol $\left(\frac{a}{p}\right)$ takes the value $1$, $-1$ or $0$, and it tells whether $a$ is a square modulo an odd prime $p$.
The [[definition of the Legendre symbol|01J8AMV9RS]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0JTGRV4WT4.md" tag="0JTGRV4WT4" kind="definition">
# Cardinality

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]].
The cardinality of $A$, written $|A|$, is an object assigned to $A$ such that, for all sets $A$ and $B$,
$$|A| = |B| \iff A \text{ and } B \text{ are } \text{[[equinumerous|08EJHHMB5R]]}.$$
For a finite set $A$, $|A|$ is the number $n$ of its elements, that is, the unique $n \in \mathbb{N}$ with $A \approx \{1, \dots, n\}$.
For an infinite set, $|A|$ is a [[cardinal number|0QKPNZKNEA]].

## Remarks

Other notations are $\#A$, $\operatorname{card}(A)$ and $\overline{\overline{A}}$.
With the [[axiom of choice|0HTGB13FE2]], one can take $|A|$ to be the least [[ordinal|0RDJK7EQYY]] equinumerous with $A$.
$|A| \le |B|$ means that there is an [[injection|0PVG1CRAJN]] from $A$ to $B$. See [[order of cardinals|01GTP8M5RG]].

## Explanations

### L1 human

The cardinality of $\{a, b, c\}$ is $3$.
For infinite sets, cardinality is a new kind of number that records which sets can be put in bijection.
For example, $|\mathbb{N}| = |\mathbb{Z}|$, but $|\mathbb{N}| < |\mathbb{R}|$.
</document>

<document source="https://tamnd.github.io/math2/node/0JTPMCCZT2.md" tag="0JTPMCCZT2" kind="statement">
# Estimate for approximate solutions

## Statement

Let $J$ be an interval, let $U$ be an open subset of $\mathbb{R}^n$ and let $K \ge 0$. Let $f : J \times U \to \mathbb{R}^n$ satisfy $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ for all $t \in J$ and $x, y \in U$.
Let $a < b$ with $[a, b] \subseteq J$, and let $\delta, \varepsilon_1, \varepsilon_2 \ge 0$.
Let $\gamma_1, \gamma_2 : [a, b] \to U$ be continuous on $[a, b]$ and differentiable on $(a, b)$, with $\|\gamma_i'(t) - f(t, \gamma_i(t))\| \le \varepsilon_i$ for $i = 1, 2$ and $t \in (a, b)$.
If $\|\gamma_1(a) - \gamma_2(a)\| \le \delta$, then for each $t \in [a, b]$,
$$\|\gamma_1(t) - \gamma_2(t)\| \le G_{\delta, K, \varepsilon_1 + \varepsilon_2}(t - a),$$
where $G$ is the [[Grönwall bound|0FRX8JD7QH]].

## Remarks

A curve with $\|\gamma'(t) - f(t, \gamma(t))\| \le \varepsilon$ is an $\varepsilon$-approximate solution.
For $\varepsilon_1 = \varepsilon_2 = 0$, the result is [[continuous dependence on initial data|0G3DBXRRQJ]].
The proof applies the derivative form of [[Grönwall's inequality|07EJHQYR9A]] with the additional term $\varepsilon_1 + \varepsilon_2$ to $\|\gamma_1 - \gamma_2\|$.

## Explanations

### L1 human

A curve that almost solves the equation stays close to a true solution.
The error has two sources: the difference of initial values and the defects $\varepsilon_i$.
Both can grow at most exponentially in time.
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<document source="https://tamnd.github.io/math2/node/0JTRMBQ1SA.md" tag="0JTRMBQ1SA" kind="definition">
# System of linear equations

## Definition

Let $F$ be a field and let $m, n \ge 1$ be integers.
Consider a *system of linear equations* over $F$, with $m$ equations in the unknowns $x_1, \dots, x_n$. Such a system is a list of [[linear equations|0Q3ZT8J05R]]
$$\begin{aligned} a_{11} x_1 + a_{12} x_2 + \cdots + a_{1n} x_n &= b_1, \\ &\ \ \vdots \\ a_{m1} x_1 + a_{m2} x_2 + \cdots + a_{mn} x_n &= b_m, \end{aligned}$$
where all $a_{ij}$ and $b_i$ are elements of $F$.
A *solution* of the system is a tuple $(s_1, \dots, s_n) \in F^n$ that is a solution of each of the $m$ equations.
The *solution set* of the system is the set of all its solutions.

## Remarks

With the [[coefficient matrix|0HQFZAP3ND]] $A = (a_{ij})$, the vector $b = (b_1, \dots, b_m)^T$ and $x = (x_1, \dots, x_n)^T$, the system is the single equation $Ax = b$.
Over a field, a system has no solution, exactly one solution, or more than one solution.
When $F$ is infinite, "more than one" means infinitely many.

## Explanations

### L1 human

A system of linear equations asks for values of the unknowns that make several linear equations true together.
For example, $x + y = 3$ and $x - y = 1$ together have the single solution $x = 2$, $y = 1$.
In the plane, each equation is a line, and a solution is a point on all the lines.
</document>

<document source="https://tamnd.github.io/math2/node/0JTRSJKCFB.md" tag="0JTRSJKCFB" kind="statement">
# Tychonoff's theorem

## Statement

Let $(X_i)_{i \in I}$ be a family of [[compact spaces|0QPHW703PV]], indexed by any set $I$.
Then the product $\prod_{i \in I} X_i$, with the [[product topology|04S9HX3GAP]], is compact.

## Remarks

For finitely many factors, the statement follows from [[product of compact sets is compact|09ZKE5XQ74]].
For arbitrary products, the theorem is equivalent to the axiom of choice.
The statement fails for the [[box topology|0CTM1EHG9P]]: the box topology on $\{0, 1\}^{\mathbb{N}}$ is discrete and infinite.
For example, $[0, 1]^I$ is compact for each set $I$.

## History

Andrey Tikhonov proved the theorem in 1930 for powers of the closed unit interval. He stated the general theorem in 1935, with the remark that the same proof works.

## Explanations

### L1 human

Any product of compact spaces is compact, even with infinitely many factors.
It works because a basic open set of the product restricts only finitely many coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/0JV30XN73Z.md" tag="0JV30XN73Z" kind="concept">
# Integer partition

## Summary

An integer partition is a way to write a natural number as a sum of positive integers, without regard to order. Partitions are central objects of combinatorics, number theory and the representation theory of symmetric groups. The node [[integer partition|0404R9P9J3]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0JVJ2WCXZH.md" tag="0JVJ2WCXZH" kind="concept">
# Gaussian elimination

## Summary

Gaussian elimination is an algorithm that changes a matrix into a [[row echelon form|0VYCDGMJJJ]] by [[elementary row operations|04XWH28452]].
It works column by column, from left to right.
In each column, it chooses a nonzero entry as [[pivot|0J0BBCMJ5W]] and moves its row up.
Then it adds multiples of that row to the rows below, to make their entries in that column zero.
Applied to the [[augmented matrix|0T6MWGZSXA]] of a [[system of linear equations|0GT4WWGWH2]], it gives an equivalent system that [[back substitution|0ATFVE08MW]] solves.
For an $n \times n$ matrix, it uses on the order of $n^3$ arithmetic operations.

## Remarks

The algorithm also computes the [[rank|0ZC9YTXZRY]] of a matrix and, with a record of the row operations, its [[determinant|00KDZRW4C0]].
It also tests if a system is [[consistent|0X6V9B1V10]]: the system is inconsistent exactly when a row $(0, \dots, 0 \mid c)$ with $c \ne 0$ appears.
The method was known in China in the book *The Nine Chapters on the Mathematical Art*, long before Gauss.

## Explanations

### L1 human

Gaussian elimination is the method of solving equations by removing one unknown at a time.
We use the first equation to remove $x_1$ from all the other equations, then the second to remove $x_2$, and so on.
At the end, the last equation has the fewest unknowns, and we solve upwards.
</document>

<document source="https://tamnd.github.io/math2/node/0JVSNJCN68.md" tag="0JVSNJCN68" kind="concept">
# Split polynomial

## Summary

A polynomial splits over a field when it is a product of polynomials of degree one with coefficients in that field.
The [[definition of a split polynomial|0SHMC00RJV]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0JW13B48YP.md" tag="0JW13B48YP" kind="concept">
# Vector projection

## Summary

The vector projection of a vector $a$ onto a nonzero vector $b$ is the multiple of $b$ that is nearest to $a$. It is the shadow of $a$ on the line of $b$, under light that falls perpendicular to that line.
</document>

<document source="https://tamnd.github.io/math2/node/0JW2FHW7G3.md" tag="0JW2FHW7G3" kind="definition">
# Existential quantifier

## Definition

Let $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.
The existential quantification of $P$ over $X$ is the proposition $\exists x \in X, \, P(x)$, read "there exists $x$ in $X$ such that $P(x)$".
It is true when $P(a)$ is true for at least one $a \in X$, and false otherwise.

## Remarks

When $X$ is empty, $\exists x \in X, \, P(x)$ is false for each $P$.
For a finite set $X = \{a_1, \dots, a_n\}$, the statement is equivalent to $P(a_1) \lor \dots \lor P(a_n)$.
The [[uniqueness quantifier|0N3ADS4CHH]] $\exists!$ says that exactly one element exists.

## Explanations

### L1 human

"There is an even prime number" is $\exists n \in \mathbb{N}, \, (n \text{ is even} \land n \text{ is prime})$.
To prove it, one example is enough: $n = 2$.
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<document source="https://tamnd.github.io/math2/node/0JW6MDHAZD.md" tag="0JW6MDHAZD" kind="concept">
# Saddle point

## Summary

A saddle point of a differentiable function of several variables is a critical point that is not a [[local extremum|0WFBH1Q3WC]].
Near a saddle point, the function takes values both larger and smaller than its value at the point.
</document>

<document source="https://tamnd.github.io/math2/node/0JW7KT6PQW.md" tag="0JW7KT6PQW" kind="concept">
# Minimum spanning tree

## Summary

A minimum spanning tree of a weighted connected graph is a spanning tree of least total weight. It is a classical problem of combinatorial optimization, solved by greedy algorithms. The node [[minimum spanning tree|0Z367CGNRM]] gives the precise definition.
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<document source="https://tamnd.github.io/math2/node/0JWCX0TW2S.md" tag="0JWCX0TW2S" kind="definition">
# Concyclic points

## Definition

Let $S$ be a set of points of the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
The points of $S$ are concyclic if they lie on one [[circle|060W5CBZP6]]. That is, there are a [[plane|0V3XAMF3C1]] $P$ that contains $S$, a point $c \in P$ and a real number $r \geq 0$ with $d(x, c) = r$ for each $x \in S$.

## Remarks

Three points that are not collinear are always concyclic. Thus each triangle has a circumcircle.
Four or more points of the plane are not concyclic in general. Four concyclic points are the vertices of a cyclic quadrilateral.
The centre $c$ is on the perpendicular bisector of each two distinct points of $S$.
Mathlib's `EuclideanGeometry.Concyclic` asks for points that are cospherical and coplanar. This is the same condition.

## Explanations

### L1 human

Concyclic points are points on one common circle. The four corners of a rectangle are concyclic: the circle has its centre at the crossing of the diagonals. The four points $(0, 0)$, $(1, 0)$, $(2, 0)$ and $(0, 1)$ are not concyclic, because a circle meets a line in at most two points.
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<document source="https://tamnd.github.io/math2/node/0JWM6GJNQW.md" tag="0JWM6GJNQW" kind="statement">
# An orthonormal family is linearly independent

## Statement

Let $V$ be an inner product space over $\R$ or $\C$.
Then each [[orthonormal family|0RW9TNNT87]] in $V$ is [[linearly independent|0CRMGPAH83]].

## Remarks

If $\sum_i a_i e_i = 0$, then the inner product with $e_j$ gives $a_j = 0$.
More generally, a family of nonzero pairwise orthogonal vectors is linearly independent.
So an orthonormal family in an $n$-dimensional space has at most $n$ vectors.

## Explanations

### L1 human

Vectors at right angles cannot be combinations of each other.
To find the coefficient of $e_j$ in a combination, take the inner product with $e_j$.
If the combination is zero, then every coefficient is zero.
</document>

<document source="https://tamnd.github.io/math2/node/0JWMJSTBSG.md" tag="0JWMJSTBSG" kind="concept">
# Divergent series

## Summary

A divergent series is a series whose partial sums do not have a finite limit. Summation methods give values to some divergent series. The node [[divergent series|0NBRP6093V]] gives the precise definition.
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<document source="https://tamnd.github.io/math2/node/0JWRJX2RMP.md" tag="0JWRJX2RMP" kind="concept">
# Bayesian inference

## Summary

Bayesian inference is a method of statistical inference that treats an unknown parameter $\theta$ as a random variable. It uses [[Bayes' theorem|0NM6E9VW04]] to update a probability distribution for $\theta$ when data arrive.
One starts from a [[prior distribution|00NBBSXRQS]] with density $\pi(\theta)$ and a model with [[likelihood|05FSTXM92W]] $L(\theta \mid x) = f(x \mid \theta)$. The [[posterior distribution|0KJ7ZS8AJA]] has the density
$$\pi(\theta \mid x) = \frac{f(x \mid \theta)\, \pi(\theta)}{\int_\Theta f(x \mid \theta')\, \pi(\theta')\, d\theta'}.$$

## Remarks

In short, the posterior is proportional to the likelihood times the prior.
Example: with a $\operatorname{Beta}(a, b)$ prior for the success probability $p$, and $k$ successes in $n$ Bernoulli($p$) trials, the posterior is $\operatorname{Beta}(a + k, b + n - k)$ (see [[beta distribution|0PP5ZAJPP7]]). The prior is conjugate.
The posterior gives point estimates, such as the posterior mean, and credible intervals.
For new observations, the posterior predictive distribution is $f(\tilde{x} \mid x) = \int_\Theta f(\tilde{x} \mid \theta)\, \pi(\theta \mid x)\, d\theta$, if $\tilde{x}$ and $x$ are conditionally independent given $\theta$.

## Explanations

### L1 human

Bayesian inference starts with a belief about an unknown quantity, written as a probability distribution.
Each new piece of data updates this belief with Bayes' theorem. The result, the posterior, combines the prior belief with the evidence.
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<document source="https://tamnd.github.io/math2/node/0JWT4PPQ7B.md" tag="0JWT4PPQ7B" kind="statement">
# L'Hôpital's rule

## Statement

Let $I$ be an open interval and let $c \in I$.
Let $f, g$ be real functions differentiable on $I \setminus \{c\}$, with $g'(x) \ne 0$ for all $x \in I \setminus \{c\}$.
Assume that $f(x) \to 0$ and $g(x) \to 0$ as $x \to c$, and that $f'(x)/g'(x) \to L$ as $x \to c$.
Then $f(x)/g(x) \to L$ as $x \to c$.

## Remarks

The proof uses the [[Cauchy mean value theorem|02SYZAWM2K]] on intervals between $c$ and $x$.
Similar forms hold for the [[indeterminate form|0S7G92KKA7]] $\infty/\infty$, for one-sided limits, and for $x \to \pm\infty$.
For example, $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \lim_{x \to 0} \frac{\sin x}{2x} = \frac{1}{2}$.
Mathlib states a form as `deriv.lhopital_zero_nhds`.

## History

Johann Bernoulli found the rule and sent it to Guillaume de l'Hôpital in a letter of 1694.
L'Hôpital published it in his textbook of 1696.

## Explanations

### L1 human

Near $c$, both $f$ and $g$ are close to their tangent lines through $(c, 0)$.
So their ratio is close to the ratio of their slopes.
</document>

<document source="https://tamnd.github.io/math2/node/0JWZX6FKSH.md" tag="0JWZX6FKSH" kind="definition">
# Sylow p-subgroup

## Definition

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number.
Write $|G| = p^n m$ with $n \geq 0$ and $p \nmid m$.
A Sylow $p$-subgroup of $G$ is a [[subgroup|0X1JXT47GA]] of $G$ of order $p^n$.
The set of the Sylow $p$-subgroups of $G$ is written $\operatorname{Syl}_p(G)$, and its size is written $n_p$.

## Remarks

A Sylow $p$-subgroup is a maximal [[p-subgroup|06QYRTX9KQ]] of $G$.
If $p$ does not divide $|G|$, then the only Sylow $p$-subgroup is $\{e\}$.
In $S_3$, the Sylow $3$-subgroup is $A_3$, and there are three Sylow $2$-subgroups, each of order $2$.

## Explanations

### L1 human

A Sylow subgroup takes all the factors $p$ of the order of the group at once.
It is as large as a $p$-subgroup can be by Lagrange's theorem.
The Sylow theorems say that such subgroups exist and that they are all alike.
</document>

<document source="https://tamnd.github.io/math2/node/0JX86PSSA9.md" tag="0JX86PSSA9" kind="statement">
# Divergence theorem

## Statement

Let $V \subseteq \mathbb{R}^3$ be compact with a piecewise smooth boundary surface $S = \partial V$, oriented by the outward unit normal $n$.
If $F$ is a vector field of class $C^1$ on an open set that contains $V$, then
$$\iiint_V \nabla \cdot F \, dV = \iint_S F \cdot n \, dS.$$

## Remarks

The left side is the [[triple integral|0GTBGMZXV8]] of the [[divergence|02NSMQVQWK]], the total source strength in $V$. The right side is the outward [[flux|0YPRY7T4PK]] through the boundary.
Example: for $F(x, y, z) = (x, y, z)$ and the unit ball, $\nabla \cdot F = 3$. So both sides equal $3 \cdot \tfrac{4}{3} \pi = 4\pi$.
With $g F$ in place of $F$, for a $C^1$ function $g$, one gets $\iiint_V (F \cdot \nabla g + g \, \nabla \cdot F) \, dV = \iint_S g \, F \cdot n \, dS$. This is a form of integration by parts.
The same statement holds in $\mathbb{R}^n$. For a [[rectangle in the plane|0GZ3D8F921]], the proof is a direct use of the fundamental theorem of calculus.

## History

Lagrange discovered the theorem in 1762, and Gauss proved special cases from 1813.
Ostrogradsky gave the first proof of the general theorem in 1826, in his work on heat flow.

## Explanations

### L1 human

What flows out of a closed surface must come from sources inside it.
The divergence theorem makes this exact: the total divergence inside equals the net flux out through the boundary.
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<document source="https://tamnd.github.io/math2/node/0JXTQ392ZM.md" tag="0JXTQ392ZM" kind="statement">
# Legendre's theorem on continued fractions

## Statement

Let $\alpha$ be a real number, and let $p$ and $q$ be positive integers such that
$$\left| \alpha - \frac{p}{q} \right| < \frac{1}{2 q^2}.$$
Then $p / q$ is a [[convergent|0SRWNR42BQ]] of the [[continued fraction expansion|0SJ7Z6VKWK]] of $\alpha$.

## Remarks

So the very good rational approximations of $\alpha$ are all convergents.
The constant $\frac{1}{2}$ cannot be replaced by $1$: not every fraction with $\left| \alpha - \frac{p}{q} \right| < \frac{1}{q^2}$ is a convergent.
A use: it shows that the solutions of [[Pell's equation|0AF3DANJ76]] come from the convergents of $\sqrt{d}$.

## History

Adrien-Marie Legendre proved the theorem in his Essai sur la théorie des nombres of 1798.

## Explanations

### L1 human

If a fraction is extremely close to $\alpha$, then the continued fraction algorithm finds it.
So to look for the best approximations of a number, you only need to compute its convergents.
</document>

<document source="https://tamnd.github.io/math2/node/0JXZKX526R.md" tag="0JXZKX526R" kind="concept">
# Square matrix

## Summary

A square matrix is a matrix with the same number of rows and columns.
</document>

<document source="https://tamnd.github.io/math2/node/0JYF6PTKEW.md" tag="0JYF6PTKEW" kind="statement">
# Lifting-the-exponent lemma

## Statement

Let $p$ be an odd prime, let $x$ and $y$ be integers with $p \nmid x$, $p \nmid y$ and $p \mid x - y$, and let $n$ be a positive integer.
Then $\nu_p(x^n - y^n) = \nu_p(x - y) + \nu_p(n)$, where $\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].

## Remarks

For $p = 2$ the formula needs a stronger hypothesis. If $x$ and $y$ are odd and $4 \mid x - y$, then $\nu_2(x^n - y^n) = \nu_2(x - y) + \nu_2(n)$.
For $p = 2$, odd $x$ and $y$, and even $n$, the general formula is $\nu_2(x^n - y^n) = \nu_2(x - y) + \nu_2(x + y) + \nu_2(n) - 1$.
For example, $\nu_3(10^9 - 1) = \nu_3(9) + \nu_3(9) = 4$, and indeed $10^9 - 1 = 3^4 \cdot 37 \cdot 333667$.

## History

Édouard Lucas had some of the ideas by 1878. Mihai Manea published the modern form in 2006.

## Explanations

### L1 human

When $p$ divides $x - y$, the lemma tells you exactly how many factors $p$ are in $x^n - y^n$.
You add the factors $p$ of $x - y$ and the factors $p$ of $n$.
Olympiad problems often use the lemma.
</document>

<document source="https://tamnd.github.io/math2/node/0JYTG5K9Y4.md" tag="0JYTG5K9Y4" kind="concept">
# Group homomorphism

## Summary

A group homomorphism is a map between groups that preserves the group operation.
It is the basic way to compare two groups.
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<document source="https://tamnd.github.io/math2/node/0JZ27DMJ5D.md" tag="0JZ27DMJ5D" kind="concept">
# Dihedral group

## Summary

The dihedral group $D_n$ is the group of the symmetries of a regular polygon with $n$ sides, for $n \geq 3$.
It has $n$ rotations and $n$ reflections, so it has order $2n$.
</document>

<document source="https://tamnd.github.io/math2/node/0JZGW75C5Z.md" tag="0JZGW75C5Z" kind="statement">
# Power rule

## Statement

Let $n \ge 1$ be a natural number.
Then the function $x \mapsto x^n$ is differentiable on $\mathbb{R}$, and its derivative at $x$ is $n x^{n-1}$.

## Remarks

Induction on $n$ with the [[product rule|0V9A43H7WV]] gives $(x^{n+1})' = (x^n \cdot x)' = n x^{n-1} \cdot x + x^n = (n + 1) x^n$.
The same formula holds for negative integers $n$ when $x \ne 0$, and for [[real exponents|0KY08TWDMQ]] when $x > 0$.
Mathlib states it as `hasDerivAt_pow`.

## Explanations

### L1 human

To differentiate a power, bring the exponent down as a factor and lower the exponent by one.
For example, the derivative of $x^5$ is $5 x^4$.
</document>

<document source="https://tamnd.github.io/math2/node/0JZSRWRBZE.md" tag="0JZSRWRBZE" kind="definition">
# Combination

## Definition

Let $X$ be a set and let $k$ be a natural number.
A $k$-combination of $X$ is a subset $S \subseteq X$ with $|S| = k$.

The set of all $k$-combinations of $X$ is written $\binom{X}{k}$.

## Remarks

A combination does not record an order of its elements. Two lists with the same elements in a different order give the same combination.
For $X = \{a, b, c\}$, the 2-combinations are $\{a, b\}$, $\{a, c\}$ and $\{b, c\}$.
The [[number of k-element subsets|0BFMR2FQ8B]] of an $n$-element set is the [[binomial coefficient|092GEF38YD]] $\binom{n}{k}$.

## Explanations

### L1 human

A combination is a selection of $k$ objects where the order does not matter.
Choosing a team of 2 from Ann, Bob and Cy gives 3 teams, because the team $\{$Ann, Bob$\}$ is the same as $\{$Bob, Ann$\}$.
</document>

<document source="https://tamnd.github.io/math2/node/0JZY3R908H.md" tag="0JZY3R908H" kind="definition">
# Galois group of a polynomial

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be nonzero.
Let $L$ be a [[splitting field|02NZZ1J0FH]] of $f$ over $K$.
The Galois group of $f$ over $K$ is the group $\operatorname{Aut}(L/K)$ of the [[automorphisms of $L/K$|0XVR72M51M]].

## Remarks

By the [[uniqueness of splitting fields|069A3BC3EP]], the group does not depend on the choice of $L$, up to isomorphism.
Each automorphism permutes the distinct roots $r_1, \dots, r_m$ of $f$ in $L$, and it is determined by this permutation, because the roots generate $L$. So the group is isomorphic to a subgroup of the symmetric group $S_m$.
The Galois group of $x^2 - 2$ over $\mathbb{Q}$ has order $2$. The Galois group of $x^3 - 2$ over $\mathbb{Q}$ is isomorphic to $S_3$.
If $f$ is separable, then $L/K$ is a [[Galois extension|0DS2F7GK05]].

## Explanations

### L1 human

The Galois group of a polynomial lists the ways to permute its roots that respect all the algebraic relations between them.
It measures how complicated the polynomial is.
For example, a polynomial over $\mathbb{Q}$ can be solved by radicals exactly when its Galois group is solvable.
</document>

<document source="https://tamnd.github.io/math2/node/0K06X9QXDX.md" tag="0K06X9QXDX" kind="definition">
# Convex set

## Definition

A subset $S$ of $\mathbb{R}^n$ is convex if for all $x, y \in S$ and each real number $t$ with $0 \leq t \leq 1$,
$$(1 - t)\, x + t\, y \in S.$$
That is, $S$ contains the segment from $x$ to $y$.

## Remarks

The same definition works in each real vector space or real affine space. Mathlib's `Convex` is stated for a module over an ordered semiring.
A convex set contains each convex combination $\sum_i w_i x_i$ of its points, an [[affine combination|0KJH6SRFZK]] with $w_i \geq 0$.
The empty set and the whole space are convex. Each intersection of convex sets is convex, but a union of convex sets need not be convex.
Balls, half-spaces, lines and triangles with their inside are convex.

## Explanations

### L1 human

A set is convex if you can see each of its points from each other point, with the line of sight inside the set. A disc is convex. A ring is not: the segment between two opposite points of the ring passes through the hole.
</document>

<document source="https://tamnd.github.io/math2/node/0K090GVHVN.md" tag="0K090GVHVN" kind="statement">
# Laplace transform of a derivative

## Statement

Let $f : [0, \infty) \to \mathbb{C}$ be continuously differentiable, and let $M, a$ be real constants with $|f(t)| \le M e^{a t}$ for all $t \ge 0$.
Let $F$ be the [[Laplace transform|05XT5AVJBK]] of $f$.
Then for each $s \in \mathbb{C}$ with $\operatorname{Re} s > a$, the Laplace transform of $f'$ converges at $s$ and
$$\mathcal{L}\{f'\}(s) = s F(s) - f(0).$$

## Remarks

Proof: integration by parts gives $\int_0^T f'(t) e^{-st}\, dt = f(T) e^{-sT} - f(0) + s \int_0^T f(t) e^{-st}\, dt$. The term $f(T) e^{-sT}$ tends to $0$ for $\operatorname{Re} s > a$.
Applied twice, under the same hypotheses on $f$ and $f'$: $\mathcal{L}\{f''\}(s) = s^2 F(s) - s f(0) - f'(0)$.
This rule is the basis of [[solving initial value problems with the Laplace transform|0K3HFZJK5Q]]. The initial values enter the transformed equation directly.

## Explanations

### L1 human

Differentiation in time becomes multiplication by $s$ after the transform, with a correction by the initial value.
So a differential equation becomes an algebraic equation for $F$.
</document>

<document source="https://tamnd.github.io/math2/node/0K0A06X2MQ.md" tag="0K0A06X2MQ" kind="concept">
# Upper and lower Darboux sums

## Summary

The upper and lower Darboux sums of a bounded function for a partition use the supremum and the infimum of the function on each subinterval. They bound the area under the graph from above and from below. The node [[upper and lower Darboux sums|0QRQVH0CRW]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0K0GE6BCYW.md" tag="0K0GE6BCYW" kind="definition">
# Path-connected space

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]].
A path in $X$ from $x$ to $y$ is a [[continuous map|0SY9ECQ19Q]] $\gamma \colon [0, 1] \to X$ with $\gamma(0) = x$ and $\gamma(1) = y$.
The space $X$ is path-connected when $X$ is nonempty and for all $x, y \in X$ there is a path in $X$ from $x$ to $y$.
A subset of $X$ is path-connected when it is path-connected in the [[subspace topology|0D5YHQGDJY]].

## Remarks

Some authors do not require $X$ to be nonempty.
Each convex subset of $\mathbb{R}^n$ is path-connected: use the segment $\gamma(t) = (1 - t)x + ty$.
[[Path-connected spaces are connected|0ZMT1GP4KM]]. The [[topologist's sine curve|0K3T4A659R]] is connected but not path-connected.

## Explanations

### L1 human

A space is path-connected when you can walk from each point to each other point along a continuous path inside the space.
This is a concrete way to show that a space is in one piece.
</document>

<document source="https://tamnd.github.io/math2/node/0K0H5E3Z5N.md" tag="0K0H5E3Z5N" kind="definition">
# Sum of a series

## Definition

Let $\sum a_n$ be a [[convergent series|0CVVKY7GY2]] of real numbers with partial sums $s_N = \sum_{n=0}^{N} a_n$.
The sum of the series is the real number
$$\sum_{n=0}^{\infty} a_n = \lim_{N \to \infty} s_N.$$

## Remarks

The sum is well defined because a sequence has at most one limit.
The sum is linear: $\sum (a_n + b_n) = \sum a_n + \sum b_n$ and $\sum c\, a_n = c \sum a_n$ when $\sum a_n$ and $\sum b_n$ converge. This follows from the [[algebra of limits|07XBM3A9J5]].
For example, $\sum_{n=0}^{\infty} 1/2^n = 2$.

## Explanations

### L1 human

The sum of a series is the number that its running totals approach.
For $1 + 1/2 + 1/4 + \cdots$, the totals $1, 1.5, 1.75, 1.875, \dots$ approach $2$, so the sum is $2$.
</document>

<document source="https://tamnd.github.io/math2/node/0K1E2S6Z69.md" tag="0K1E2S6Z69" kind="statement">
# Distance from a point to a plane

## Statement

Let $a$, $b$, $c$ and $d$ be real numbers with $(a, b, c) \neq (0, 0, 0)$, and let $P = \{(x, y, z) \in \mathbb{R}^3 : a x + b y + c z = d\}$ be a [[plane|07SDJ5DNE4]].
Let $X = (x_0, y_0, z_0)$ be a point of $\mathbb{R}^3$. The distance from $X$ to $P$ is the smallest [[Euclidean distance|0NPPGHQNXQ]] $d(X, Q)$ for $Q \in P$. Then
$$\operatorname{dist}(X, P) = \frac{|a x_0 + b y_0 + c z_0 - d|}{\sqrt{a^2 + b^2 + c^2}}.$$

## Remarks

The smallest distance is reached at exactly one point of $P$, the orthogonal projection of $X$ on $P$. With $n = (a, b, c)$ this point is $X - \frac{a x_0 + b y_0 + c z_0 - d}{a^2 + b^2 + c^2} \, n$.
The number $(a x_0 + b y_0 + c z_0 - d) / \|n\|$ is the signed distance. Its sign tells on which side of the plane the point is.
The same formula holds for a hyperplane $\{x : a \cdot x = d\}$ with $a \neq 0$ in $\mathbb{R}^n$: the distance from $x_0$ is $|a \cdot x_0 - d| / \|a\|$.

## Explanations

### L1 human

The nearest point of a plane is straight below or above you, along the normal vector. For the floor plane $z = 0$, the formula gives $|z_0|$, your height above the floor. Tilted planes need the factor $\sqrt{a^2 + b^2 + c^2}$, because $(a, b, c)$ need not be a unit vector.
</document>

<document source="https://tamnd.github.io/math2/node/0K1EWAR6AF.md" tag="0K1EWAR6AF" kind="statement">
# Lipschitz functions are uniformly continuous

## Statement

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be [[Lipschitz|0QNVQ4VD9T]].
Then $f$ is [[uniformly continuous|0Q4DR37DVE]].

## Remarks

Let $K > 0$ be a Lipschitz constant. Given $\varepsilon > 0$, take $\delta = \varepsilon / K$.
The converse fails. The function $\sqrt{x}$ is uniformly continuous on $[0, 1]$ but not Lipschitz.
Mathlib states it as `LipschitzWith.uniformContinuous`.

## Explanations

### L1 human

A Lipschitz bound controls the change of $f$ in the same way at every point.
So one input tolerance works for the whole domain.
</document>

<document source="https://tamnd.github.io/math2/node/0K1KKMFHYH.md" tag="0K1KKMFHYH" kind="statement">
# Determinant of a triangular matrix

## Statement

Let $K$ be a commutative ring and let $A = (a_{ij})$ be an $n \times n$ upper or lower [[triangular matrix|0N4XY7Z3NN]] over $K$.
Then
$$\det A = a_{11} a_{22} \cdots a_{nn},$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

In the [[Leibniz formula|016PE4VZ8Q]], all terms except the term of the identity permutation contain an entry below, or above, the diagonal, so they are zero.
Row reduction to a triangular matrix, with this rule, is a practical way to compute determinants.
A triangular matrix over a field is invertible if and only if all its diagonal entries are nonzero.

## Explanations

### L1 human

For a triangular matrix, the determinant is the product of the diagonal entries.
The entries above the diagonal do not matter.
For example, $\det \begin{pmatrix} 2 & 7 \\ 0 & 3 \end{pmatrix} = 6$.
</document>

<document source="https://tamnd.github.io/math2/node/0K1N1Y4EGZ.md" tag="0K1N1Y4EGZ" kind="statement">
# Integral of 1/x

## Statement

Let $a, b$ be real numbers of the same sign, both nonzero.
Then
$$\int_a^b \frac{dx}{x} = \ln \frac{b}{a}.$$
In particular, $\int_1^x \frac{dt}{t} = \ln x$ for $x > 0$.

## Remarks

The function $\ln |x|$ is an antiderivative of $1/x$ on each of $(0, \infty)$ and $(-\infty, 0)$, by the [[derivative of the logarithm|07X1QA6BRW]].
Some texts use the second formula as the definition of the [[natural logarithm|0044CPHJBV]].
The integral over an interval that contains $0$ does not exist.
Mathlib states it as `integral_inv`, under the hypothesis $0 \notin [a, b]$.

## Explanations

### L1 human

The power rule fails for $x^{-1}$, since it would divide by $0$.
The logarithm fills this gap: its derivative is $1/x$.
</document>

<document source="https://tamnd.github.io/math2/node/0K1SCPBNKJ.md" tag="0K1SCPBNKJ" kind="concept">
# Automorphism group

## Summary

The automorphism group of a group $G$ is the group of all isomorphisms from $G$ to itself, with composition as the operation.
It describes the symmetries of the structure of $G$.
</document>

<document source="https://tamnd.github.io/math2/node/0K26YKNBSA.md" tag="0K26YKNBSA" kind="definition">
# Zero divisor

## Definition

Let $R$ be a [[ring|0894J980RK]].
An element $a \in R$ is a left zero divisor if there is a nonzero $x \in R$ with $a x = 0$.
An element $a \in R$ is a right zero divisor if there is a nonzero $y \in R$ with $y a = 0$.
An element is a zero divisor if it is a left or a right zero divisor.

## Remarks

In a [[commutative ring|0GYGHKX834]], left and right zero divisors are the same.
In $\mathbb{Z}/6\mathbb{Z}$, the classes of $2$ and $3$ are zero divisors, because $2 \cdot 3 = 0$.
Some authors do not count $0$ as a zero divisor. With the definition above, $0$ is a zero divisor of every nonzero ring.
A [[unit|019NFGFPR5]] is never a zero divisor.

## Explanations

### L1 human

In the integers, a product is zero only when one factor is zero.
In other rings this can fail.
In the integers modulo $6$, $2 \cdot 3 = 6 = 0$, but neither $2$ nor $3$ is zero.
</document>

<document source="https://tamnd.github.io/math2/node/0K28YXMS7Q.md" tag="0K28YXMS7Q" kind="definition">
# Scalar multiplication of matrices

## Definition

Let $R$ be a ring, let $c \in R$, and let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$.
The *scalar multiple* $cA$ is the $m \times n$ matrix with entries
$$(cA)_{ij} = c\, a_{ij}$$
for $1 \le i \le m$ and $1 \le j \le n$.

## Remarks

Over a field $F$, matrix addition and scalar multiplication make $M_{m \times n}(F)$ a [[vector space|000AYPA9AJ]] of dimension $mn$.
The rules $c(A + B) = cA + cB$, $(c + d)A = cA + dA$, $(cd)A = c(dA)$ and $1A = A$ hold.
For example, $2 \begin{pmatrix} 1 & -1 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -2 \\ 0 & 6 \end{pmatrix}$.

## Explanations

### L1 human

Multiplying a matrix by a number means multiplying every entry by that number.
The size of the matrix does not change.
Doubling a matrix doubles every number in the table.
</document>

<document source="https://tamnd.github.io/math2/node/0K2DMACEZ4.md" tag="0K2DMACEZ4" kind="statement">
# Brahmagupta–Fibonacci identity

## Statement

Let $a$, $b$, $c$ and $d$ be elements of a commutative ring. Then
$$(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2 = (ac + bd)^2 + (ad - bc)^2.$$

## Remarks

So the product of two sums of two squares is again a sum of two squares.
For example, $5 \cdot 13 = 65$, and the identity gives $65 = 1^2 + 8^2 = 7^2 + 4^2$.
For complex numbers, the identity says $|z|^2 |w|^2 = |zw|^2$ with $z = a + bi$ and $w = c + di$.
It is a step in the proof of the [[sum of two squares theorem|04NPWJMA6H]].

## History

The identity appears in the Arithmetica of Diophantus, of the third century.
Brahmagupta found it again in the seventh century, and Fibonacci published it in Europe in 1225.

## Explanations

### L1 human

If two numbers are sums of two squares, then so is their product.
You can check the identity by expanding both sides: the cross terms $2abcd$ cancel.
</document>

<document source="https://tamnd.github.io/math2/node/0K2FNY6DYV.md" tag="0K2FNY6DYV" kind="statement">
# Logarithm of a product

## Statement

For all real $x > 0$ and $y > 0$,
$$\ln(x y) = \ln x + \ln y.$$

## Remarks

By [[the sum rule for exp|0E8HNVBCSP]], $e^{\ln x + \ln y} = e^{\ln x} e^{\ln y} = x y$. Now [[take logarithms|0RKW2X63YR]].
So $\ln$ is a group isomorphism from $((0, \infty), \cdot)$ to $(\mathbb{R}, +)$.
Mathlib states it as `Real.log_mul`, for nonzero $x$ and $y$.

## Explanations

### L1 human

The logarithm turns multiplication into addition.
This property made logarithm tables useful for hand computation.
</document>

<document source="https://tamnd.github.io/math2/node/0K2TX12X58.md" tag="0K2TX12X58" kind="definition">
# Singular value

## Definition

Let $A$ be an $m \times n$ complex matrix.
The matrix $A^* A$ is positive semidefinite, so its [[eigenvalues|0BQAA2BK3Q]] $\lambda_1 \ge \cdots \ge \lambda_n \ge 0$ are real and nonnegative.
The *singular values* of $A$ are the numbers
$$\sigma_i = \sqrt{\lambda_i} \qquad (1 \le i \le n).$$

## Remarks

The number of nonzero singular values is the rank of $A$.
The largest singular value is the operator norm $\max_{\norm{x} = 1} \norm{Ax}$.
For a normal matrix, the singular values are the absolute values of the eigenvalues.
Some authors list only the first $\min(m, n)$ values.

## Explanations

### L1 human

The singular values of a matrix measure how much it stretches space in its main directions.
The unit sphere goes to an ellipsoid, and the singular values are the lengths of its semi-axes.
Unlike eigenvalues, they exist for every matrix, also rectangular ones.
</document>

<document source="https://tamnd.github.io/math2/node/0K387S2PKQ.md" tag="0K387S2PKQ" kind="statement">
# Minkowski's theorem

## Statement

Let $n$ be a positive integer, and let $L \subset \mathbb{R}^n$ be a lattice with determinant $d(L)$.
Let $S \subseteq \mathbb{R}^n$ be a convex set that is symmetric about the origin: $x \in S$ implies $-x \in S$.
If the volume of $S$ is greater than $2^n d(L)$, then $S$ contains a point of $L$ other than $0$.

## Remarks

The most common case is $L = \mathbb{Z}^n$, with $d(L) = 1$: a symmetric convex set of volume greater than $2^n$ contains a nonzero integer point.
The bound is sharp: the open cube $(-1, 1)^n$ has volume $2^n$ and contains no nonzero integer point.
Applications include proofs of [[Fermat's theorem on sums of two squares|043B3ENSTY]] and [[Lagrange's four-square theorem|0KYYSMEVHH]].

## History

Hermann Minkowski proved the theorem in 1889. It started the geometry of numbers.

## Explanations

### L1 human

A large, round, centered region in the plane must contain a lattice point other than the origin.
In the plane with $L = \mathbb{Z}^2$, the area must be more than $4$.
</document>

<document source="https://tamnd.github.io/math2/node/0K3HFZJK5Q.md" tag="0K3HFZJK5Q" kind="concept">
# Solving initial value problems with the Laplace transform

## Summary

To solve an [[initial value problem|075VBY4K4X]] for a [[linear equation with constant coefficients|0A6PGTTP2A]], one applies the [[Laplace transform|05XT5AVJBK]] to both sides of the equation.
The [[rule for derivatives|0K090GVHVN]] turns the equation and the initial values into one algebraic equation for the transform $Y(s)$ of the unknown $y(t)$.
One solves this equation for $Y(s)$ and then finds $y = \mathcal{L}^{-1}\{Y\}$ with the [[inverse Laplace transform|0JRW4X1MDC]].

## Remarks

Example: for $y'' + y = 0$, $y(0) = 0$, $y'(0) = 1$, the transform gives $s^2 Y(s) - 1 + Y(s) = 0$. So $Y(s) = 1/(s^2 + 1)$ and $y(t) = \sin t$.
For $a_n y^{(n)} + \cdots + a_0 y = g(t)$ with zero initial values, $Y(s) = G(s) / P(s)$, where $P(s) = a_n s^n + \cdots + a_0$ and $G$ is the transform of $g$. The [[convolution theorem|0JNMJEDD93]] gives $y = h * g$ with $h = \mathcal{L}^{-1}\{1/P\}$.
The method handles discontinuous forcing, such as a [[step function|09C4JP7Y0M]], and impulses, such as a [[Dirac delta|0KPHENZ5S7]].

## Explanations

### L1 human

The Laplace transform turns a differential equation into an algebra problem.
We solve the algebra problem, which is often easy, and then transform back to get the solution.
The initial values are built in from the start, so there are no constants to find at the end.
</document>

<document source="https://tamnd.github.io/math2/node/0K3MWA7DMC.md" tag="0K3MWA7DMC" kind="statement">
# Factor theorem

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]], let $f \in R[x]$ and let $c \in R$.
Then $x - c$ divides $f$ in $R[x]$ if and only if $c$ is a [[root|00EHX6FWC0]] of $f$.

## Remarks

The proof uses the [[remainder theorem|0TEDQH56RW]]: $f = (x - c) q + f(c)$. So $x - c$ divides $f$ exactly when $f(c) = 0$.
For example, $1$ is a root of $x^3 - 1$, and $x^3 - 1 = (x - 1)(x^2 + x + 1)$.
The theorem gives the bound on the [[number of roots|0KJVQV3E0F]] of a polynomial over a field.

## Explanations

### L1 human

Each root gives a linear factor.
If $f(c) = 0$, then you can write $f$ as $(x - c)$ times another polynomial.
Conversely, a factor $x - c$ makes $f(c) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0K3T4A659R.md" tag="0K3T4A659R" kind="counterexample">
# Topologist's sine curve

## Counterexample

Let
$$T = \left\{ \left( x, \sin \tfrac{1}{x} \right) : 0 < x \le 1 \right\} \cup \{ (0, 0) \} \subseteq \mathbb{R}^2,$$
with the [[subspace topology|0D5YHQGDJY]] of the plane.
Then $T$ is [[connected|0R37DFJKFW]] but not [[path-connected|0K0GE6BCYW]]. Thus connected spaces need not be path-connected.

## Remarks

The graph part $G$ is connected, as the continuous image of $(0, 1]$. The origin is in the closure of $G$, and a set between a connected set and its closure is connected. Thus $T$ is connected.
No path joins the origin to a point of $G$. Near $x = 0$, the curve goes up and down between $-1$ and $1$ infinitely often, and a path cannot follow it continuously.
Some texts use the closed version, which adds the whole segment $\{0\} \times [-1, 1]$. It is compact, connected and not path-connected.
The space $T$ is also not locally connected.

## Explanations

### L1 human

The curve $\sin(1/x)$ oscillates faster and faster as $x$ goes to $0$. The origin is a limit of the curve, so the space is in one piece.
But you cannot walk along the curve to the origin, because the curve never settles down.
</document>

<document source="https://tamnd.github.io/math2/node/0K4NMMJBG2.md" tag="0K4NMMJBG2" kind="definition">
# Confidence interval

## Definition

Let $X$ be a sample whose law $P_\theta$ depends on an unknown parameter $\theta \in \Theta$. Let $\tau \colon \Theta \to \mathbb{R}$ be a function and let $\gamma \in (0, 1)$.
A confidence interval for $\tau(\theta)$ with confidence level $\gamma$ is a random interval $[L(X), U(X)]$, where $L \le U$ are [[statistics|036756NFD6]], such that
$$P_\theta\big(L(X) \le \tau(\theta) \le U(X)\big) = \gamma \quad \text{for each } \theta \in \Theta.$$
If only "$\ge \gamma$" holds for each $\theta$, the interval is a conservative confidence interval.

## Remarks

Example: let $X_1, \dots, X_n$ be i.i.d. $\mathcal{N}(\mu, \sigma^2)$ with $\sigma$ known. Then $\bar{X}_n \pm 1.96\, \sigma / \sqrt{n}$ is a confidence interval for $\mu$ with level approximately $0.95$.
If $\sigma$ is unknown, one replaces $\sigma$ with the sample standard deviation and $1.96$ with a quantile of Student's t-distribution with $n - 1$ degrees of freedom.
The probability refers to the random interval, not to $\theta$. After one computes the interval from data, it either contains $\tau(\theta)$ or it does not.
The confidence level is often written as $1 - \alpha$, for example $95\%$ for $\alpha = 0.05$.

## Explanations

### L1 human

A $95\%$ confidence interval comes from a procedure that captures the true value in $95\%$ of repeated experiments.
It does not mean that the true value has a $95\%$ probability to be in one particular computed interval.
</document>

<document source="https://tamnd.github.io/math2/node/0K4VQB7HWR.md" tag="0K4VQB7HWR" kind="statement">
# Determinant of a matrix with two equal rows

## Statement

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$.
If two different rows of $A$ are equal, then $\det A = 0$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is property 2 of the definition of the determinant.
The same holds for two equal columns.
Over a field, more generally, $\det A = 0$ when the rows of $A$ are linearly dependent.

## Explanations

### L1 human

Two equal rows mean that the rows span less than the full space.
Then the box spanned by the rows is flat, and its volume is zero.
So the determinant is zero.
</document>

<document source="https://tamnd.github.io/math2/node/0K4WCNTMHG.md" tag="0K4WCNTMHG" kind="definition">
# Rayleigh quotient

## Definition

Let $A$ be a [[Hermitian|0P17NGQQMQ]] $n \times n$ complex matrix, and let $x \in \C^n$ with $x \ne 0$.
The *Rayleigh quotient* of $A$ and $x$ is the real number
$$R(A, x) = \frac{x^* A x}{x^* x}.$$

## Remarks

For each $x \ne 0$, $\lambda_{\min} \le R(A, x) \le \lambda_{\max}$, where $\lambda_{\min}$ and $\lambda_{\max}$ are the least and the greatest [[eigenvalues|0BQAA2BK3Q]] of $A$.
If $x$ is an eigenvector for $\lambda$, then $R(A, x) = \lambda$.
So $\lambda_{\max} = \max_{x \ne 0} R(A, x)$ and $\lambda_{\min} = \min_{x \ne 0} R(A, x)$.

## Explanations

### L1 human

The Rayleigh quotient measures how much a symmetric matrix stretches a given direction, on average.
Its largest value over all directions is the largest eigenvalue.
This gives a way to estimate eigenvalues without solving the characteristic equation.
</document>

<document source="https://tamnd.github.io/math2/node/0K4ZWTFSV5.md" tag="0K4ZWTFSV5" kind="definition">
# Homogeneous first-order equation

## Definition

Let $J$ be an open interval and let $F : J \to \mathbb{R}$ be a continuous function.
A homogeneous first-order equation is an [[ordinary differential equation|0P97V20RW7]] of the form
$$y' = F\left(\frac{y}{x}\right)$$
for an unknown function $y$ of $x$. Its domain is an interval that does not contain $0$, with $y(x)/x \in J$.

## Remarks

An equivalent form is $M(x, y)\, dx + N(x, y)\, dy = 0$, where $M$ and $N$ are homogeneous functions of the same degree $d$: $M(t x, t y) = t^d M(x, y)$ and $N(t x, t y) = t^d N(x, y)$ for $t \ne 0$.
The substitution $y = x v$ gives $x v' = F(v) - v$. This is a [[separable equation|0V4E623THY]] for $v$.
Example: $y' = (x + y)/x = 1 + y/x$ gives $x v' = 1$, so $v = \ln|x| + C$ and $y = x \ln|x| + C x$.
This use of the word homogeneous is different from its use for [[homogeneous linear equations|0EQZSAB9EC]].

## Explanations

### L1 human

In a homogeneous equation, the slope $y'$ depends only on the ratio $y/x$.
So the slope field is the same along each line through the origin.
The new unknown $v = y/x$ captures this, and the equation becomes separable.
</document>

<document source="https://tamnd.github.io/math2/node/0K57YG3BB8.md" tag="0K57YG3BB8" kind="concept">
# Differentiable function

## Summary

A differentiable function is a function that has a derivative at each point of its domain. Near each point, it is well approximated by a linear function. The node [[differentiable function|0A61B60B5R]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0K5TWJAGPD.md" tag="0K5TWJAGPD" kind="example">
# Action by left multiplication

## Example

Let $G$ be a [[group|G5S1M9]].
The map $G \times G \to G$, $(g, x) \mapsto gx$, is an [[action|0WCY42EV9V]] of $G$ on the set $G$.
The two axioms are $ex = x$ and $g(hx) = (gh)x$, and they follow from the group axioms.
This action is [[transitive|02GK6HZYBT]], because $(yx^{-1})x = y$ for all $x, y \in G$.
The [[stabiliser|0AYMP41THN]] of each point is $\{e\}$, because $gx = x$ implies $g = e$.
In particular, the action is [[faithful|0P7G9NGG2G]], and this gives [[Cayley's theorem|0151VYVX27]].
</document>

<document source="https://tamnd.github.io/math2/node/0K5W69AC0J.md" tag="0K5W69AC0J" kind="object">
# Gaussian integers

## Description

The ring of Gaussian integers is
$$\mathbb{Z}[i] = \{a + b i : a, b \in \mathbb{Z}\},$$
where $i \in \mathbb{C}$ satisfies $i^2 = -1$.
It is a [[subring|0859D58WPD]] of $\mathbb{C}$, with $(a + b i)(c + d i) = (a c - b d) + (a d + b c) i$.
As a subring of a field, it is an [[integral domain|09S3PFBEPM]].

## Remarks

The [[norm|03AN1V4ZFJ]] $N(a + b i) = a^2 + b^2$ is multiplicative.
The [[units of $\mathbb{Z}[i]$|06RH4XAGRY]] are $1, -1, i, -i$.
The ring $\mathbb{Z}[i]$ is a [[Euclidean domain|0PN13BB7BT]], so it has unique factorization.
A prime number $p$ is a sum of two squares if and only if $p$ is not prime in $\mathbb{Z}[i]$. This holds exactly for $p = 2$ and the primes $p \equiv 1 \pmod 4$.

## Explanations

### L1 human

The Gaussian integers are the points with integer coordinates in the complex plane.
You add and multiply them as complex numbers.
They form a ring with a theory of primes and factorization, like the integers.
</document>

<document source="https://tamnd.github.io/math2/node/0K665H5V1S.md" tag="0K665H5V1S" kind="definition">
# Matrix exponential

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $A$ be an $n \times n$ matrix over $\mathbb{K}$.
The *matrix exponential* of $A$ is the $n \times n$ matrix
$$e^A = \exp(A) = \sum_{k=0}^{\infty} \frac{1}{k!} A^k = I_n + A + \frac{1}{2} A^2 + \frac{1}{6} A^3 + \cdots,$$
with $A^0 = I_n$.

## Remarks

The series converges for each matrix $A$, entry by entry.
For a diagonal matrix $D$ with entries $d_i$, $e^D$ is diagonal with entries $e^{d_i}$.
If $A = P D P^{-1}$, then $e^A = P e^D P^{-1}$.
The function $x(t) = e^{tA} x_0$ solves the linear differential equation $x' = Ax$ with $x(0) = x_0$.

## Explanations

### L1 human

The matrix exponential uses the familiar series of $e^x$, with a matrix in place of the number $x$.
It is the main tool for solving systems of linear differential equations.
For $A = \begin{pmatrix} 0 & -t \\ t & 0 \end{pmatrix}$, $e^A$ is the rotation of the plane by the angle $t$.
</document>

<document source="https://tamnd.github.io/math2/node/0K67Q60Z8P.md" tag="0K67Q60Z8P" kind="statement">
# Complement of an open set is closed

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $U \subseteq X$.
Then $U$ is [[open|01ZQQDCERP]] in $X$ if and only if $X \setminus U$ is [[closed|0G5AV9Q4RR]] in $X$.

## Remarks

Proof: $X \setminus (X \setminus U) = U$, so the claim is the definition of a closed set applied to $X \setminus U$.
Thus the closed sets determine the open sets, and the open sets determine the closed sets.

## Explanations

### L1 human

Taking the complement turns open sets into closed sets and closed sets back into open sets.
For example, $(0, 1)$ is open in $\mathbb{R}$, so $(-\infty, 0] \cup [1, \infty)$ is closed.
</document>

<document source="https://tamnd.github.io/math2/node/0K6CH0MT0F.md" tag="0K6CH0MT0F" kind="concept">
# Alternating multilinear map

## Summary

An alternating multilinear map is a multilinear map that is zero whenever two of its arguments are equal.
</document>

<document source="https://tamnd.github.io/math2/node/0K78D81EF7.md" tag="0K78D81EF7" kind="definition">
# Young diagram

## Definition

A Young diagram is a finite set $D \subseteq \mathbb{N} \times \mathbb{N}$ with this property: if $(i, j) \in D$, $i' \le i$ and $j' \le j$, then $(i', j') \in D$.
The elements of $D$ are the cells, or boxes, of the diagram.
A cell $(i, j)$ is drawn as a unit square in row $i$ and column $j$, with rows numbered from the top.

## Remarks

The rows of a Young diagram have weakly decreasing lengths $\lambda_1 \ge \lambda_2 \ge \dots$. Thus a Young diagram with $n$ cells is the same as a [[partition|0404R9P9J3]] of $n$.
This drawing convention is the English one. The French convention draws row $0$ at the bottom.
Transposing a diagram, $(i, j) \mapsto (j, i)$, gives the diagram of the conjugate partition.

## Explanations

### L1 human

A Young diagram draws a partition as rows of boxes, left-aligned, with the longest row at the top.
The partition $4 + 2 + 1$ gives rows of $4$, $2$ and $1$ boxes.
</document>

<document source="https://tamnd.github.io/math2/node/0K7P6PR838.md" tag="0K7P6PR838" kind="concept">
# Maclaurin series

## Summary

The Maclaurin series of a smooth function is its Taylor series at the point $0$. It is a power series in $x$. The node [[Maclaurin series|043SJ5RF59]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0K7TD9BV2D.md" tag="0K7TD9BV2D" kind="definition">
# Idempotent matrix

## Definition

Let $R$ be a ring and let $P$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $P$ is *idempotent* when $P^2 = P$.

## Remarks

The matrices $0$, $I_n$ and $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ are idempotent.
Over a field, an idempotent matrix is the matrix of a [[projection|033DA1WE8Y]] onto its image along its kernel.
If $P$ is idempotent, then so is $I - P$.
Over a field, the eigenvalues of an idempotent matrix are in $\{0, 1\}$, and its rank equals its trace in characteristic $0$.

## Explanations

### L1 human

An idempotent matrix does nothing new when we apply it twice.
A projection onto a line is the main example: once a point is on the line, projecting again keeps it there.
So $P^2 = P$.
</document>

<document source="https://tamnd.github.io/math2/node/0K83KEJ43P.md" tag="0K83KEJ43P" kind="statement">
# Isolated zeros theorem

## Statement

Let $U \subseteq \mathbb{C}$ be an open set, let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$, and let $z_0 \in U$.
Then exactly one of these two cases holds.

1. There is $r > 0$ such that $f(z) = 0$ for each $z$ with $|z - z_0| < r$.
2. There is $r > 0$ such that $f(z) \neq 0$ for each $z$ with $0 < |z - z_0| < r$.

## Remarks

Proof: if case 1 does not hold, then $f(z) = (z - z_0)^m g(z)$ near $z_0$ with $g(z_0) \neq 0$, where $m$ is the [[order|064N002TX4]] of $f$ at $z_0$. By continuity, $g \neq 0$ near $z_0$.
On a [[domain|097KNJFJ7V]], it follows that the zeros of a holomorphic function that is not identically zero have no accumulation point in the domain.
The real function $x \mapsto e^{-1/x^2}$, with value $0$ at $0$, is infinitely differentiable and has all derivatives $0$ at $0$. The theorem has no analogue for such real functions.

## Explanations

### L1 human

Near a point, a holomorphic function is either zero everywhere or zero only at that point.
So the zeros of a nonzero holomorphic function cannot pile up inside its domain, unlike the zeros of $\sin(1/x)$ near $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0K8F7PVDMX.md" tag="0K8F7PVDMX" kind="concept">
# Associativity

## Summary

A binary operation is associative when the grouping of three elements does not change the result.
Then the parentheses in a product of several elements can be omitted.
</document>

<document source="https://tamnd.github.io/math2/node/0K8SGQSV7G.md" tag="0K8SGQSV7G" kind="statement">
# Compactness and sequential compactness agree in metric spaces

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $K \subseteq X$.
Then $K$ is [[compact|00R0X1WGPX]] if and only if $K$ is [[sequentially compact|0CARW59WD2]].

## Remarks

One proof of the hard direction uses the [[Lebesgue number lemma|040CFNJEAP]] and the fact that a sequentially compact metric space is [[totally bounded|0WE5BBHGM6]].
Both conditions are also equivalent to: $K$ is complete and totally bounded. This is the statement [[compact iff complete and totally bounded|054Q5PW5GX]].
Without a metric, the two notions differ. For example, the product $\{0, 1\}^{[0, 1]}$ is compact but not sequentially compact.

## Explanations

### L1 human

In a metric space, you can test compactness with sequences. A set is compact exactly when each sequence in it has a subsequence with a limit in the set.
This test is often easier than the test with open covers.
</document>

<document source="https://tamnd.github.io/math2/node/0K96Z2S699.md" tag="0K96Z2S699" kind="concept">
# Removable discontinuity

## Summary

A removable discontinuity is a point where the limit of a function exists but differs from the value of the function. Redefining the value at the point makes the function continuous there. The node [[removable discontinuity|0QG9C579SR]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0K98R7A4VC.md" tag="0K98R7A4VC" kind="concept">
# Lower central series

## Summary

The lower central series of a group $G$ is the sequence $G = \gamma_1(G) \supseteq \gamma_2(G) \supseteq \cdots$ with $\gamma_{i+1}(G) = [\gamma_i(G), G]$.
The group is nilpotent if and only if this series reaches the trivial subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/0K9AAJ8G51.md" tag="0K9AAJ8G51" kind="statement">
# Continuous image of a compact set is compact

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], let $f \colon X \to Y$ be a [[continuous map|0SY9ECQ19Q]] and let $K \subseteq X$ be [[compact|00R0X1WGPX]].
Then the [[image|0Q3F20XB0A]] $f(K)$ is compact in $Y$.

## Remarks

Proof: let $(V_i)$ be open sets of $Y$ that cover $f(K)$. The sets $f^{-1}(V_i)$ are open and cover $K$. A finite subfamily covers $K$, and the matching $V_i$ cover $f(K)$.
A consequence is that a [[continuous real function on a compact set attains its bounds|0V4DET59NW]].
Compactness is a [[topological property|02SMVGC690]].

## Explanations

### L1 human

Continuous maps keep compactness.
For example, the image of $[0, 1]$ under a continuous map into the plane is a compact curve, so it is closed and bounded.
</document>

<document source="https://tamnd.github.io/math2/node/0K9GBTZ1FX.md" tag="0K9GBTZ1FX" kind="definition">
# Parallelepiped

## Definition

Let $p$ be a point of $\mathbb{R}^3$, and let $a$, $b$ and $c$ be linearly independent [[vectors|0QN3D80AWP]] in $\mathbb{R}^3$.
The parallelepiped with the vertex $p$ and the edges $a$, $b$ and $c$ is the set
$$\{\, p + s a + t b + u c \;:\; s, t, u \in [0, 1] \,\}.$$

## Remarks

The parallelepiped has 8 vertices $p + s a + t b + u c$ with $s, t, u \in \{0, 1\}$, 12 edges and 6 faces. Opposite faces are congruent parallelograms in parallel planes.
Its volume is the absolute value of the [[scalar triple product|059Q2XWJDC]] $a \cdot (b \times c)$.
In Mathlib, the parallelepiped of a finite family of vectors $v_i$ is the set of the sums $\sum_i t_i v_i$ with each $t_i \in [0, 1]$. It does not ask for linear independence.
A cube and a rectangular cuboid are special parallelepipeds.

## Explanations

### L1 human

Take a cube and push it sideways so that its faces lean, but stay flat. The result is a parallelepiped. Each face is a parallelogram, and opposite faces are copies of each other.
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<document source="https://tamnd.github.io/math2/node/0K9HSJFCMJ.md" tag="0K9HSJFCMJ" kind="statement">
# Inverse function theorem

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $k \ge 1$, and let $f \colon U \to \mathbb{R}^n$ be of [[class C^k|0TCBZMEYGN]] on $U$.
Let $a \in U$, and suppose that the derivative $Df(a)$ is invertible, that is, the [[Jacobian determinant|03W6Q1MFR5]] of $f$ at $a$ is not zero.
Then there are open sets $V \subseteq U$ with $a \in V$ and $W \subseteq \mathbb{R}^n$ with $f(a) \in W$ such that $f$ restricted to $V$ is a bijection from $V$ onto $W$.
Its inverse $f^{-1} \colon W \to V$ is of class $C^k$, and $D(f^{-1})(f(x)) = Df(x)^{-1}$ for each $x \in V$.

## Remarks

The theorem is local. The map $f(x, y) = (e^x \cos y, e^x \sin y)$ has invertible derivative at each point of $\mathbb{R}^2$, but it is not injective, since $f(x, y + 2\pi) = f(x, y)$.
Continuity of the derivative matters. The function $g(x) = x + 2 x^2 \sin(1/x)$, $g(0) = 0$, has $g'(0) = 1$, but $g$ is not injective on any interval around $0$.
A standard proof applies the contraction mapping principle to $x \mapsto x + Df(a)^{-1}(y - f(x))$ near $a$.
The formula for the derivative of the inverse is the [[derivative of the inverse function|0G65751VSW]].

## Explanations

### L1 human

If the linear approximation of $f$ at $a$ is invertible, then $f$ itself is invertible near $a$.
So one can solve the system $f(x) = y$ for $x$ when $y$ is close to $f(a)$, and the solution depends smoothly on $y$.
</document>

<document source="https://tamnd.github.io/math2/node/0K9QCK6WJC.md" tag="0K9QCK6WJC" kind="statement">
# General Leibniz rule

## Statement

Let $n$ be a natural number, and let $f$ and $g$ be $n$ times differentiable real functions on an open interval $I$.
Then $f g$ is $n$ times differentiable on $I$, and
$$(f g)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} f^{(k)} g^{(n-k)}.$$

## Remarks

The proof is an induction on $n$ with the [[product rule|0V9A43H7WV]], like the proof of the binomial theorem.
For $n = 1$ it is the product rule. For $n = 2$ it is $(f g)'' = f'' g + 2 f' g' + f g''$.
Mathlib states it as `iteratedDeriv_mul`.

## Explanations

### L1 human

The rule for the $n$-th derivative of a product looks like the binomial formula for $(a + b)^n$.
Each term distributes $n$ differentiations between the two factors.
</document>

<document source="https://tamnd.github.io/math2/node/0K9QVPJYJF.md" tag="0K9QVPJYJF" kind="concept">
# Ramsey number

## Summary

The Ramsey number $R(r, s)$ is the least $n$ such that every two-colouring of the edges of $K_n$ has a blue $K_r$ or a red $K_s$. Ramsey numbers measure how large a structure must be to contain order. The node [[Ramsey number|01DSVCRF6X]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0KA4S8R39M.md" tag="0KA4S8R39M" kind="definition">
# Regular graph

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $k$ be a natural number.
The graph $G$ is $k$-regular, or regular of degree $k$, if each vertex of $G$ has [[degree|0T0F6CCS9W]] $k$.
A graph is regular if it is $k$-regular for some $k$.

## Remarks

A $0$-regular graph has no edges. A $1$-regular graph is a perfect matching. A finite $2$-regular graph is a disjoint union of cycles.
A $3$-regular graph is also called cubic. The [[Petersen graph|0NR95CTFCZ]] is cubic.
The complete graph $K_n$ is $(n-1)$-regular. By the [[handshaking lemma|0XAP8WHNES]], a finite $k$-regular graph on $n$ vertices has $kn/2$ edges.

## Explanations

### L1 human

A graph is regular when all vertices have the same number of neighbours.
The corners and edges of a cube form a $3$-regular graph.
</document>

<document source="https://tamnd.github.io/math2/node/0KA7A4ENVJ.md" tag="0KA7A4ENVJ" kind="statement">
# Determinant via the Schur complement

## Statement

Let $K$ be a commutative ring and let $A$ be an invertible $p \times p$ matrix over $K$.
Let $B$, $C$ and $D$ be matrices over $K$ of sizes $p \times q$, $q \times p$ and $q \times q$.
Then
$$\det \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \det A \cdot \det(D - C A^{-1} B),$$
where $\det$ is the [[determinant|00KDZRW4C0]].
The matrix $D - C A^{-1} B$ is the *Schur complement* of $A$ in the [[block matrix|0ASDFEQS6P]].

## Remarks

The proof factors the block matrix as $\begin{pmatrix} I & 0 \\ C A^{-1} & I \end{pmatrix} \begin{pmatrix} A & B \\ 0 & D - C A^{-1} B \end{pmatrix}$.
Then it uses the rule for [[block triangular matrices|0J9TVTNB4K]].
For $p = q = 1$ and $a \ne 0$, it gives $\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = a(d - cb/a) = ad - bc$.

## Explanations

### L1 human

Block elimination removes the block $C$, in the same way as Gaussian elimination removes an entry.
What remains in the lower right corner is the Schur complement.
The determinant is then the product of two smaller determinants.
</document>

<document source="https://tamnd.github.io/math2/node/0KABP2HWMQ.md" tag="0KABP2HWMQ" kind="concept">
# Upper and lower Darboux integrals

## Summary

The upper Darboux integral of a bounded function is the infimum of its upper Darboux sums. The lower one is the supremum of its lower Darboux sums. A function is integrable when the two agree. The node [[upper and lower Darboux integrals|09FQT057T7]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0KABPMRHM2.md" tag="0KABPMRHM2" kind="concept">
# Characteristic subgroup

## Summary

A characteristic subgroup of a group is a subgroup that each automorphism of the group maps onto itself.
Each characteristic subgroup is normal.
</document>

<document source="https://tamnd.github.io/math2/node/0KAMF19NV8.md" tag="0KAMF19NV8" kind="statement">
# Sum of binomial coefficients

## Statement

Let $n$ be a natural number.
Then
$$\sum_{k=0}^{n} \binom{n}{k} = 2^n.$$

## Remarks

For $n = 4$: $1 + 4 + 6 + 4 + 1 = 16$.
The identity is the [[binomial theorem|0EHWPWGHB2]] with $x = y = 1$.
Combinatorially, both sides count the subsets of an $n$-element set. The [[number of subsets of a finite set|0C5RDJE73X]] is $2^n$, and $\binom{n}{k}$ of them have size $k$.

## Explanations

### L1 human

The numbers in row $n$ of [[Pascal's triangle|05DME4TZ3K]] add up to $2^n$.
This is because each subset of an $n$-element set has some size $k$ from $0$ to $n$, and there are $2^n$ subsets in total.
</document>

<document source="https://tamnd.github.io/math2/node/0KAN330PAH.md" tag="0KAN330PAH" kind="definition">
# Nondegenerate bilinear form

## Definition

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.
A [[bilinear form|0SQZMXB6GP]] $B$ on $V$ is *nondegenerate* when, for each $u \in V$,
$$B(u, v) = 0 \text{ for all } v \in V \implies u = 0.$$
A bilinear form that is not nondegenerate is *degenerate*.

## Remarks

$B$ is nondegenerate if and only if its matrix in a basis has a nonzero determinant.
So, in finite dimensions, the condition in the first argument is equivalent to the same condition in the second argument.
$B$ is nondegenerate if and only if $u \mapsto B(u, \cdot)$ is an isomorphism from $V$ to its dual space.
Each inner product is nondegenerate; the form $B(x, y) = x_1 y_1$ on $\R^2$ is degenerate.

## Explanations

### L1 human

A form is nondegenerate when it can detect every nonzero vector.
For each nonzero $u$, some $v$ gives $B(u, v) \ne 0$.
In matrix terms, the matrix of the form is invertible.
</document>

<document source="https://tamnd.github.io/math2/node/0KB1ND7XDS.md" tag="0KB1ND7XDS" kind="statement">
# Subgroups of cyclic groups are cyclic

## Statement

Let $G$ be a [[cyclic group|0KVGH0J5B3]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $H$ is cyclic.

## Remarks

Let $g$ generate $G$.
If $H \neq \{e\}$, let $m$ be the smallest positive integer with $g^m \in H$; then division with remainder shows that $H = \langle g^m \rangle$.
For example, each subgroup of $(\mathbb{Z}, +)$ has the form $m\mathbb{Z}$, generated by $m$.

## Explanations

### L1 human

A subgroup of a cyclic group is generated by the smallest positive power of the generator that it contains.
Any other power in the subgroup is a multiple of this one, or the subgroup would contain a smaller positive power.
Thus the subgroup is again cyclic.
</document>

<document source="https://tamnd.github.io/math2/node/0KBEMAJAYA.md" tag="0KBEMAJAYA" kind="statement">
# Distributive law

## Statement

For all natural numbers $a$, $b$ and $c$, $a \cdot (b + c) = a \cdot b + a \cdot c$.
Here $+$ is the [[addition|0W5DSDFQ4G]] and $\cdot$ is the [[multiplication|0JRC32D6K2]] of natural numbers.

## Remarks

This is the left distributive law. With commutativity of multiplication, it gives the right law $(b + c) \cdot a = b \cdot a + c \cdot a$.
The proof from the recursive definitions uses [[induction|0ASJVEFA9J]] on $c$.
The same law holds in the integers, the rationals, the reals and the complex numbers.

## Explanations

### L1 human

To multiply a number by a sum, you can multiply it by each part and then add.
For example, $3 \cdot (10 + 2) = 30 + 6 = 36$, and this is how many people compute $3 \cdot 12$.
</document>

<document source="https://tamnd.github.io/math2/node/0KBFAWYRTE.md" tag="0KBFAWYRTE" kind="concept">
# Covariance

## Summary

The covariance of two random variables measures how they vary together. It is positive when large values of one go with large values of the other, and negative in the opposite case.
The node [[covariance|0DJ8W3HP2D]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0KBTP8Q2FS.md" tag="0KBTP8Q2FS" kind="concept">
# Reduced word

## Summary

A reduced word on a set $S$ is a finite string of letters $s$ and $s^{-1}$ with $s \in S$, with no letter next to its inverse.
The reduced words on $S$ are the elements of the free group on $S$.
</document>

<document source="https://tamnd.github.io/math2/node/0KC00K7KY4.md" tag="0KC00K7KY4" kind="concept">
# Bernoulli number

## Summary

The Bernoulli numbers are a sequence of rational numbers $B_n$, the coefficients of $t^n / n!$ in the expansion of $t / (e^t - 1)$.
The [[definition of the Bernoulli numbers|0RAS24E1R5]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0KC0F51T6A.md" tag="0KC0F51T6A" kind="statement">
# Stirling's formula

## Statement

As $n \to \infty$,
$$n! \sim \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n.$$
That is, the ratio $\frac{n!}{\sqrt{2 \pi n} \, (n/e)^n}$ tends to $1$.

## Remarks

The relation $\sim$ is [[asymptotic equivalence|00SB5EA3VW]]. The absolute error grows, but the relative error tends to $0$.
Comparing $\ln n! = \sum_{k \le n} \ln k$ with $\int_1^n \ln x \, dx$ gives the form $n! \sim C \sqrt{n} (n/e)^n$. The [[Wallis product|0XCYA4YERS]] gives $C = \sqrt{2\pi}$.
A sharper form is $n! = \sqrt{2 \pi n} (n/e)^n \big(1 + \frac{1}{12 n} + O(n^{-2})\big)$. Mathlib states the limit as `Stirling.tendsto_stirlingSeq_sqrt_pi`.

## History

Abraham de Moivre first stated a less precise form of the result.
In 1730, James Stirling showed that the constant is $\sqrt{2\pi}$.

## Explanations

### L1 human

Stirling's formula gives the size of $n!$ for large $n$.
For example, it gives $10! \approx 3.599 \times 10^6$, while $10! = 3628800$.
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<document source="https://tamnd.github.io/math2/node/0KC1NMG90B.md" tag="0KC1NMG90B" kind="concept">
# Wronskian

## Summary

The Wronskian of $n$ functions is the determinant of the matrix of the functions and their derivatives up to order $n - 1$.
It is a test for the linear independence of solutions of a linear differential equation. See the [[definition|0N5NF1RGBP]].

## Remarks

It is named after Józef Hoene-Wroński.
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<document source="https://tamnd.github.io/math2/node/0KCE8CG86P.md" tag="0KCE8CG86P" kind="concept">
# Idempotent matrix

## Summary

An idempotent matrix is a square matrix that is equal to its own square.
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<document source="https://tamnd.github.io/math2/node/0KCQ9Y3QKA.md" tag="0KCQ9Y3QKA" kind="concept">
# Basis

## Summary

A basis of a vector space is a linearly independent family of vectors that spans the space.
Each vector of the space is then a linear combination of the basis vectors in exactly one way.
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<document source="https://tamnd.github.io/math2/node/0KCT6DPXJK.md" tag="0KCT6DPXJK" kind="concept">
# Venn diagram

## Summary

A Venn diagram is a picture of the relations between a small number of sets.
Each set is a closed curve, often a circle, inside a rectangle for the [[universal set|0BPYRMHJ48]].
For $n$ sets, the curves cut the plane into $2^n$ regions, one for each combination of "in" and "out" of the sets.

## Remarks

John Venn made these diagrams popular in the 1880s.
The diagrams make laws such as [[De Morgan's laws for sets|0VK0W4D5Y5]] visible, but a picture is not a proof.
An Euler diagram is similar, but it shows only the regions that are not empty.

## Explanations

### L1 human

Draw two overlapping circles $A$ and $B$.
The overlap is $A \cap B$, both circles together are $A \cup B$, and the region outside both circles is the complement of $A \cup B$.
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<document source="https://tamnd.github.io/math2/node/0KCYJSB9FW.md" tag="0KCYJSB9FW" kind="definition">
# Fundamental solution of Pell's equation

## Definition

Let $d$ be a positive integer that is not a perfect square.
The fundamental solution of [[Pell's equation|0AF3DANJ76]] $x^2 - d y^2 = 1$ is the solution $(x_1, y_1)$ with $x_1 > 1$ and $y_1 > 0$ such that $x_1$ is as small as possible.

## Remarks

The fundamental solution exists by the [[existence of nontrivial solutions|0EM6C9SZN5]].
Among the solutions with $x > 1$ and $y > 0$, it also has the smallest $y$ and the smallest value of $x + y \sqrt{d}$.
For example, the fundamental solution for $d = 2$ is $(3, 2)$, and for $d = 7$ it is $(8, 3)$.
It appears among the convergents of the continued fraction of $\sqrt{d}$.

## Explanations

### L1 human

The fundamental solution is the smallest solution with positive entries.
All other solutions come from it by taking [[powers|0J1TT091TN]] of $x_1 + y_1 \sqrt{d}$.
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<document source="https://tamnd.github.io/math2/node/0KDADE9RQ8.md" tag="0KDADE9RQ8" kind="definition">
# Vertex cover

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A vertex cover of $G$ is a set $S$ of vertices of $G$ that contains at least one end of each edge.

## Remarks

A set $S$ is a vertex cover if and only if its complement is an [[independent set|0KZJ82WEPP]].
For the [[complete bipartite graph|0V0GC7BQAN]] $K_{m,n}$, the least size of a vertex cover is $\min\{m, n\}$.
Each vertex of a cover meets at most one edge of a [[matching|0Z87BHAFTB]]. Thus every vertex cover has at least as many vertices as any matching has edges.

## Explanations

### L1 human

A vertex cover is a set of vertices that touches every edge.
Think of guards at crossings: each street must have a guard at one of its two ends.
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<document source="https://tamnd.github.io/math2/node/0KDJD249DT.md" tag="0KDJD249DT" kind="statement">
# Bolzano–Weierstrass theorem

## Statement

Let $(a_n)$ be a [[bounded sequence|05FQ3VGAR9]] of real numbers.
Then $(a_n)$ has a [[subsequence|0R8J3ZQMJS]] that converges to a real number.

## Remarks

The proof takes a [[monotone subsequence|0MHGJRZMTM]]. It is bounded, so it converges by the [[monotone convergence theorem|0FM6N4R0VB]].
The limit of the subsequence lies in each closed interval $[m, M]$ that contains all terms, by [[the preservation of non-strict inequalities|0N32V0E3XJ]].
The theorem holds in $\mathbb{R}^k$ for each $k$. It says that each closed and bounded subset of $\mathbb{R}^k$ is sequentially compact.
The theorem fails in $\mathbb{Q}$. The decimal approximations of $\sqrt{2}$ are bounded, and no subsequence has a limit in $\mathbb{Q}$.

## History

Bernard Bolzano proved the result in 1817, as a lemma for the intermediate value theorem. Karl Weierstrass proved it again about fifty years later and made it a central tool of analysis.

## Explanations

### L1 human

A bounded sequence cannot run away, so its terms must crowd together somewhere.
The theorem picks infinitely many terms that approach one point of crowding.
For $(-1)^n$, the even terms form a subsequence that converges to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0KE6NBTVY8.md" tag="0KE6NBTVY8" kind="definition">
# Exponential generating function

## Definition

Let $K$ be a field of characteristic $0$, for example $\mathbb{Q}$ or $\mathbb{R}$, and let $(a_n)_{n \ge 0}$ be a sequence in $K$.
The exponential generating function of $(a_n)$ is the [[formal power series|072W67KFYK]]
$$\hat{A}(x) = \sum_{n \ge 0} a_n \frac{x^n}{n!} \in K[\![x]\!].$$

## Remarks

The constant sequence $1$ has the exponential generating function $e^x = \sum_n x^n / n!$.
For exponential generating functions, the product corresponds to the binomial convolution $c_n = \sum_{k=0}^{n} \binom{n}{k} a_k b_{n-k}$.
They suit labelled structures. For example, the [[derangement numbers|03WY4HHMT1]] have the exponential generating function $\frac{e^{-x}}{1 - x}$.

## Explanations

### L1 human

An exponential generating function divides the $n$-th term by $n!$ before it puts it in the series.
This makes the series of fast-growing sequences, such as $n!$, easy to handle: the sequence $n!$ gives $\frac{1}{1 - x}$.
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<document source="https://tamnd.github.io/math2/node/0KF3J2KAER.md" tag="0KF3J2KAER" kind="statement">
# Classification of Pythagorean triples

## Statement

Let $m$ and $n$ be [[coprime|0GFKP8F8AG]] positive integers with $m > n$ and $m - n$ odd.
Then $(m^2 - n^2, \, 2mn, \, m^2 + n^2)$ is a primitive [[Pythagorean triple|01G1WY1GVX]].
Conversely, each primitive Pythagorean triple $(a, b, c)$ with $b$ even arises in this way from exactly one such pair $(m, n)$.
Each Pythagorean triple is $(ka, kb, kc)$ for a positive integer $k$ and a primitive triple $(a, b, c)$, up to the order of the first two entries.

## Remarks

The formula is called Euclid's formula.
For example, $(m, n) = (2, 1)$ gives $(3, 4, 5)$, and $(m, n) = (3, 2)$ gives $(5, 12, 13)$.
One proof writes $b^2 = (c - a)(c + a)$ and uses unique factorization. Another uses the rational points on the unit circle.

## Explanations

### L1 human

Euclid's formula produces every right triangle with whole-number sides, from two numbers $m$ and $n$.
The identity $(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2$ shows that the formula always works.
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<document source="https://tamnd.github.io/math2/node/0KFYGSHHF5.md" tag="0KFYGSHHF5" kind="concept">
# Directed graph

## Summary

A directed graph is a set of vertices with arrows between some ordered pairs of vertices. Directed graphs model one-way relations such as links, flows and dependencies. The node [[directed graph|0R5HTV7REV]] gives the precise definition.
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# The alternating group is generated by three-cycles

## Statement

Let $n \geq 3$.
The [[alternating group|0Z5905Q73D]] $A_n$ is [[generated|0RP7VWQ0YN]] by the $3$-cycles in $S_n$.

## Remarks

Each $3$-cycle is even, so the $3$-cycles lie in $A_n$.
An even permutation is a product of an even number of transpositions, and each product of two transpositions is a product of $3$-cycles.
For distinct $a, b, c, d$, the formulas are $(a\ b)(a\ c) = (a\ c\ b)$ and $(a\ b)(c\ d) = (a\ c\ b)(a\ c\ d)$.

## Explanations

### L1 human

An even permutation is a chain of swaps, taken two at a time.
Each pair of swaps can be written with $3$-cycles.
So the $3$-cycles are enough to build all even permutations.
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<document source="https://tamnd.github.io/math2/node/0KG4D8AP6G.md" tag="0KG4D8AP6G" kind="definition">
# Norm induced by an inner product

## Definition

Let $V$ be an [[inner product space|0QVNCJXRY3]] over $\R$ or $\C$, with inner product $\langle \cdot, \cdot \rangle$.
The *norm induced by the inner product* is the map
$$\norm{x} = \sqrt{\langle x, x \rangle} \qquad (x \in V).$$

## Remarks

It is a [[norm|0V2WP9B1MF]]; the triangle inequality follows from the [[Cauchy–Schwarz inequality|0RFWW56Y22]].
For the dot product on $\R^n$, it is the Euclidean length $\sqrt{x_1^2 + \cdots + x_n^2}$.
It satisfies the [[parallelogram law|026HPGYGS4]], and the [[polarisation identity|05FKHRT0B4]] recovers the inner product from it.

## Explanations

### L1 human

The length of a vector is the square root of its inner product with itself.
In the plane, this is the Pythagorean formula $\sqrt{x^2 + y^2}$.
So every inner product gives a way to measure lengths.
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<document source="https://tamnd.github.io/math2/node/0KH1N66QWW.md" tag="0KH1N66QWW" kind="definition">
# Dirichlet convolution

## Definition

Let $f$ and $g$ be [[arithmetic functions|0H2EPRZC0D]].
Their Dirichlet convolution is the arithmetic function $f * g$ given by
$$(f * g)(n) = \sum_{d \mid n} f(d) \, g\!\left(\frac{n}{d}\right)$$
for each positive integer $n$, where $d$ runs over the [[positive divisors|0FCWA16FZE]] of $n$.

## Remarks

The convolution is commutative and associative, and it distributes over addition.
The identity element is $\varepsilon$, with $\varepsilon(1) = 1$ and $\varepsilon(n) = 0$ for $n > 1$.
An arithmetic function $f$ has an inverse for $*$ if and only if $f(1) \ne 0$.
For example, with the constant function $\mathbf{1}$, $(\mathbf{1} * \mathbf{1})(n)$ is the [[number of divisors|0BKJT6WG96]] of $n$.

## Explanations

### L1 human

The Dirichlet convolution combines two functions by a sum over the factorizations $n = d \cdot e$.
It is the natural product for functions defined by sums over divisors.
</document>

<document source="https://tamnd.github.io/math2/node/0KJ7ZS8AJA.md" tag="0KJ7ZS8AJA" kind="concept">
# Posterior probability

## Summary

The posterior probability of a hypothesis $H$ is its [[conditional probability|0FJB06AD2P]] $P(H \mid D)$ given the observed data $D$.
By [[Bayes' theorem|0NM6E9VW04]], $P(H \mid D) = P(D \mid H) \, P(H) / P(D)$. So the posterior is proportional to the likelihood $P(D \mid H)$ times the [[prior probability|00NBBSXRQS]] $P(H)$.
For an unknown parameter $\theta$ with prior density $\pi(\theta)$ and likelihood $L(\theta \mid x)$, the posterior density is $\pi(\theta \mid x) = L(\theta \mid x) \, \pi(\theta) / \int L(t \mid x) \, \pi(t) \, dt$.

## Remarks

The posterior of one experiment can be the prior of the next experiment. So Bayesian updates can be done one observation at a time.
A credible interval is an interval that has a given posterior probability, for example $95\%$.

## Explanations

### L1 human

The posterior is what you believe after you see the data.
It mixes what you believed before (the prior) with how well each hypothesis explains the data (the likelihood).
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<document source="https://tamnd.github.io/math2/node/0KJH6SRFZK.md" tag="0KJH6SRFZK" kind="concept">
# Affine combination

## Summary

An affine combination of points is a weighted sum of the points whose weights add up to 1. It makes sense in an [[affine space|09XNAPTJQ7]], where a sum of points with arbitrary weights does not.
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<document source="https://tamnd.github.io/math2/node/0KJVQV3E0F.md" tag="0KJVQV3E0F" kind="statement">
# A polynomial over a field has at most as many roots as its degree

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be a nonzero polynomial of [[degree|0DPF6WRZW6]] $n$.
Then $f$ has at most $n$ roots in $K$.
More precisely, the sum of the [[multiplicities|03XVP8Q4GY]] of the roots of $f$ in $K$ is at most $n$.

## Remarks

The proof is an induction on $n$ with the [[factor theorem|0K3MWA7DMC]]: a root $c$ gives $f = (x - c) g$ with $\deg g = n - 1$.
The same statement holds over every [[integral domain|09S3PFBEPM]].
It fails without this hypothesis: in $\mathbb{Z}/8\mathbb{Z}$, the polynomial $x^2 - 1$ has the four roots $1, 3, 5, 7$.

## Explanations

### L1 human

A polynomial of degree $2$ over the real numbers has at most two roots.
In general, each root takes away one linear factor, and you can do this at most $n$ times.
Strange rings like $\mathbb{Z}/8\mathbb{Z}$ can break this rule, because they have zero divisors.
</document>

<document source="https://tamnd.github.io/math2/node/0KJXHN0TW0.md" tag="0KJXHN0TW0" kind="statement">
# One-sided limits of a monotone function exist

## Statement

Let $a < c < b$ and let $f \colon (a, b) \to \mathbb{R}$ be increasing.
Then both [[one-sided limits|0909PWZR62]] of $f$ at $c$ exist, and
$$\lim_{x \to c^-} f(x) = \sup_{a < x < c} f(x) \le f(c) \le \inf_{c < x < b} f(x) = \lim_{x \to c^+} f(x).$$

## Remarks

The proof is the same as for the [[monotone convergence theorem|0FM6N4R0VB]]. The supremum exists because $f(c)$ is an upper bound.
A similar statement holds for decreasing functions, with the inequalities reversed.
So $f$ is continuous at $c$ exactly when the two one-sided limits are equal. Otherwise $c$ is a [[jump discontinuity|05B0FKFXQA]].
Mathlib states the left limit as `Monotone.tendsto_nhdsLT`.

## Explanations

### L1 human

An increasing function can only move up as $x$ approaches $c$ from the left.
Its values are bounded by $f(c)$, so they settle at a limit.
</document>

<document source="https://tamnd.github.io/math2/node/0KM2CJ80G7.md" tag="0KM2CJ80G7" kind="definition">
# Primitive root modulo n

## Definition

Let $n$ be a positive integer.
A primitive root modulo $n$ is an integer $g$ coprime to $n$ whose [[multiplicative order|0SBRX5CZ5R]] modulo $n$ is $\varphi(n)$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

Equivalently, $g$ is a primitive root modulo $n$ when the powers $g, g^2, \dots, g^{\varphi(n)}$ represent all the residue classes coprime to $n$.
So a primitive root exists exactly when the group $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic, and then $\overline{g}$ generates this group.
For example, $3$ is a primitive root modulo $7$: its powers are $3, 2, 6, 4, 5, 1$ modulo $7$.
There is no primitive root modulo $8$: each odd $a$ satisfies $a^2 \equiv 1 \pmod 8$, but $\varphi(8) = 4$.

## Explanations

### L1 human

A primitive root is a number whose powers run through all the residues coprime to $n$ before they come back to $1$.
Modulo $5$, the powers of $2$ are $2, 4, 3, 1$. So $2$ is a primitive root modulo $5$, but $4$ is not, since $4^2 \equiv 1$.
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<document source="https://tamnd.github.io/math2/node/0KM4KVR9MV.md" tag="0KM4KVR9MV" kind="statement">
# Hausdorff maximal principle

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
Then each [[chain|0GP9ASVQ1X]] of $P$ is a subset of a maximal chain.
A maximal chain is a chain that is not a proper subset of another chain of $P$.

## Remarks

Over ZF, the principle is equivalent to the axiom of choice and to [[Zorn's lemma|04RXTQA5ZA]].
With the empty chain, it says that each partially ordered set has a maximal chain.
Felix Hausdorff proved it in 1914.

## Explanations

### L1 human

Start with any chain in the order.
You can extend it until no element can be added without breaking the chain property.
</document>

<document source="https://tamnd.github.io/math2/node/0KMJ6919WS.md" tag="0KMJ6919WS" kind="definition">
# Fixed field

## Definition

Let $L$ be a [[field|0QA8CHH8MG]] and let $H$ be a group of ring automorphisms of $L$.
The fixed field of $H$ is
$$L^H = \{ a \in L : \sigma(a) = a \text{ for every } \sigma \in H \}.$$

## Remarks

The set $L^H$ is a [[subfield|07QY5RPFBR]] of $L$, because each $\sigma \in H$ respects the field operations.
For $L = \mathbb{C}$ and $H = \{\mathrm{id}, \sigma\}$, with $\sigma$ complex conjugation, $L^H = \mathbb{R}$.
If $H$ is finite, then $[L : L^H] = |H|$ and $L/L^H$ is a [[Galois extension|0DS2F7GK05]], by a theorem of Emil Artin.
More generally, a group acting on a ring has a fixed subring, the ring of invariants.

## Explanations

### L1 human

The fixed field is the part of the field that a group of symmetries does not move.
In the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], fixed fields turn subgroups into intermediate fields.
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<document source="https://tamnd.github.io/math2/node/0KMN32XGPF.md" tag="0KMN32XGPF" kind="statement">
# Determinant of a linear map is the determinant of its matrix

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$ with a basis $B$, and let $T \colon V \to V$ be a linear map.
Then the [[determinant of $T$|0X0HQYSZ0K]] equals the determinant of the [[matrix of $T$|0YQSGNYE3K]] in $B$:
$$\det T = \det [T]_B.$$

## Remarks

The key fact is $\det(P^{-1} A P) = \det A$, which follows from the product rule.
Over a commutative ring, the same statement holds for free modules of finite rank.

## Explanations

### L1 human

We can compute the determinant of a linear map in any basis.
We write its matrix in that basis, and we take the determinant of the matrix.
Every basis gives the same answer.
</document>

<document source="https://tamnd.github.io/math2/node/0KMVFSHFGZ.md" tag="0KMVFSHFGZ" kind="statement">
# Strong induction

## Statement

Let $P(n)$ be a property of natural numbers $n$.
Suppose that for each natural number $n$, if $P(k)$ holds for every natural number $k < n$, then $P(n)$ holds.
Then $P(n)$ holds for every natural number $n$.

## Remarks

The hypothesis for $n = 0$ asks for a proof of $P(0)$ with no assumption, because no natural number is less than $0$.
The proof applies ordinary [[induction|0ASJVEFA9J]] to the property "$P(k)$ holds for every $k < n$".
A standard use is the proof that each integer $n \ge 2$ is a product of primes.

## Explanations

### L1 human

In ordinary induction, you may use only the case just before $n$.
In strong induction, you may use all the cases before $n$.
This helps when $n$ breaks into smaller pieces that are not just $n - 1$, for example when $n = a b$ with $a$ and $b$ smaller than $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0KN22AAW5R.md" tag="0KN22AAW5R" kind="definition">
# Monotone sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]].
The sequence is increasing if $a_n \le a_{n+1}$ for all $n \in \mathbb{N}$.
The sequence is decreasing if $a_n \ge a_{n+1}$ for all $n \in \mathbb{N}$.
The sequence is monotone if it is increasing or decreasing.
The sequence is strictly increasing if $a_n < a_{n+1}$ for all $n$, and strictly decreasing if $a_n > a_{n+1}$ for all $n$.

## Remarks

Some texts say non-decreasing for increasing in this sense, and they keep the word increasing for strictly increasing.
By induction, $(a_n)$ is increasing if and only if $a_m \le a_n$ for all $m \le n$. This is the definition of a monotone function on $\mathbb{N}$.
The sequence $1/n$ is strictly decreasing. The sequence $(-1)^n$ is not monotone.

## Explanations

### L1 human

A monotone sequence always moves in one direction.
An increasing sequence never goes down, and a decreasing sequence never goes up.
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<document source="https://tamnd.github.io/math2/node/0KN2S81K72.md" tag="0KN2S81K72" kind="concept">
# Ring isomorphism

## Summary

A ring isomorphism is a bijective ring homomorphism.
Two rings are isomorphic when there is a ring isomorphism between them.
The [[definition of a ring isomorphism|0B6QC0C0G1]] gives the details.
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<document source="https://tamnd.github.io/math2/node/0KN2V7840E.md" tag="0KN2V7840E" kind="statement">
# The exponential function is strictly increasing

## Statement

The exponential function is strictly increasing on $\mathbb{R}$.
That is, $x < y$ implies $\exp(x) < \exp(y)$.

## Remarks

The derivative is [[$\exp$ itself|0EX9GFJ0B3]], and it is [[positive|0P9BNBW2W3]]. Then [[positive derivative|07J5X2T1ZX]] gives the claim.
Also $\exp(x) \to \infty$ as $x \to \infty$ and $\exp(x) \to 0$ as $x \to -\infty$. So $\exp$ is a bijection from $\mathbb{R}$ onto $(0, \infty)$.
Mathlib states it as `Real.exp_strictMono`.

## Explanations

### L1 human

The exponential function always grows as $x$ grows.
So it takes each positive value exactly once.
</document>

<document source="https://tamnd.github.io/math2/node/0KN7EPH7Z0.md" tag="0KN7EPH7Z0" kind="concept">
# Restriction of an endomorphism to an invariant subspace

## Summary

The restriction of an endomorphism to an invariant subspace is the endomorphism of the subspace that agrees with the original map on it.
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<document source="https://tamnd.github.io/math2/node/0KN93VDSXA.md" tag="0KN93VDSXA" kind="statement">
# Ceva's theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\mathbb{R}^2$. For [[collinear points|0AFNNTRTJE]] $X$, $Y$, $Z$ with $Z \neq Y$, let the signed ratio $\frac{XZ}{ZY}$ be the real number $\lambda$ with $Z - X = \lambda\,(Y - Z)$.
Let $O$ be a point that is not on the [[lines|0MP9BCF6DP]] $BC$, $CA$ and $AB$. Suppose that the lines $AO$, $BO$ and $CO$ meet the lines $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$.
Then
$$\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1.$$
Conversely, let $D$, $E$ and $F$ be points of the lines $BC$, $CA$ and $AB$, different from the vertices, with this product equal to $1$. Then the lines $AD$, $BE$ and $CF$ meet in one point or are all parallel.

## Remarks

The signed ratio $\frac{AF}{FB}$ is positive if $F$ is between $A$ and $B$, and negative otherwise.
The theorem uses only ratios of segments on one line. So it is true in each affine plane over each field.
For the three medians, each ratio is $1$. The converse then shows again that the medians meet in one point.

## History

Giovanni Ceva published the theorem in 1678, in his work De lineis rectis. Yusuf al-Mu'taman ibn Hud, an 11th century king of Zaragoza, had proved it much earlier.

## Explanations

### L1 human

Draw three lines from the corners of a triangle through one common inside point. Each line cuts the opposite side into two parts. Multiply the three ratios of the parts, going round the triangle in one direction: the product is always $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0KNAN9F27P.md" tag="0KNAN9F27P" kind="definition">
# Maximum degree

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]] with a nonempty vertex set $V$.
The maximum degree of $G$ is
$$\Delta(G) = \max_{v \in V} \deg(v),$$
the largest [[degree|0T0F6CCS9W]] of a vertex of $G$.

## Remarks

The complete graph $K_n$ has $\Delta = n - 1$.
A [[greedy colouring|04AAKZNXHD]] uses at most $\Delta(G) + 1$ colours. [[Brooks' theorem|0Y4TV2D5RQ]] improves this to $\Delta(G)$ in most cases.

## Explanations

### L1 human

The maximum degree is the largest number of edges at any vertex.
In a star with one centre and $5$ leaves, the maximum degree is $5$.
</document>

<document source="https://tamnd.github.io/math2/node/0KNJWEVSMG.md" tag="0KNJWEVSMG" kind="example">
# p-norm on R^n

## Example

Let $n \ge 1$ be an integer and let $p \ge 1$ be a real number.
For $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$, let
$$\|x\|_p = \left( \sum_{i=1}^{n} |x_i|^p \right)^{1/p}, \qquad \|x\|_\infty = \max_{1 \le i \le n} |x_i|.$$
Then $\|\cdot\|_p$ and $\|\cdot\|_\infty$ are norms on $\mathbb{R}^n$.
They are $0$ only at $x = 0$, they satisfy $\|\lambda x\| = |\lambda| \, \|x\|$ for real $\lambda$, and they satisfy the triangle inequality $\|x + y\| \le \|x\| + \|y\|$.

## Remarks

The triangle inequality for $\|\cdot\|_p$ is the Minkowski inequality.
The metric $\|x - y\|_p$ is the [[p-metric|0TFDZN6AXP]]. For $p = 2$, the norm is the Euclidean norm.
For each $x$, $\|x\|_p \to \|x\|_\infty$ as $p \to \infty$.
All these norms are [[equivalent|03MC70ZRNR]], and $\mathbb{R}^n$ is a [[Banach space|0BDTTM24DP]] with each of them.

## Explanations

### L1 human

The p-norms give different ways to measure the length of a vector from its coordinates.
For $p = 1$, you add the absolute values. For $p = 2$, you get the usual length. For $p = \infty$, you take the largest coordinate.
</document>

<document source="https://tamnd.github.io/math2/node/0KP0QNXPRT.md" tag="0KP0QNXPRT" kind="concept">
# Homogeneous linear equation

## Summary

A homogeneous linear equation is a linear differential equation $L y = 0$ with right side zero.
Its solutions form a vector space whose dimension is the order of the equation. See the [[definition|0EQZSAB9EC]].
</document>

<document source="https://tamnd.github.io/math2/node/0KPHENZ5S7.md" tag="0KPHENZ5S7" kind="object">
# Dirac delta function

## Description

Let $C_c^\infty(\mathbb{R})$ be the space of infinitely differentiable functions $\varphi : \mathbb{R} \to \mathbb{C}$ that vanish outside a bounded set.
The Dirac delta is the distribution $\delta$ that maps each test function $\varphi \in C_c^\infty(\mathbb{R})$ to $\delta[\varphi] = \varphi(0)$.
Informally, one writes $\int_{-\infty}^{\infty} \delta(t)\, \varphi(t)\, dt = \varphi(0)$, as if $\delta$ were a function that is $0$ away from $0$ and has total integral $1$.
No actual function has this property. The delta is a generalized function.

## Remarks

As a measure, $\delta$ gives the mass $1$ to each set that contains $0$ and the mass $0$ to the other sets.
The shifted delta $\delta(t - a)$ maps $\varphi$ to $\varphi(a)$. For $a > 0$, its Laplace transform is $e^{-a s}$, and for $a = 0$ the usual convention gives $1$.
In the sense of distributions, $\delta$ is the derivative of the [[Heaviside step function|09C4JP7Y0M]].
In differential equations, $\delta(t - a)$ models an impulse: a large force that acts for a very short time around $t = a$.

## Explanations

### L1 human

The delta function is an idealized spike at $0$, infinitely tall and infinitely thin, with area $1$.
It is not a real function. But it makes sense when we integrate it against a smooth function, and the result is the value of that function at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0KQZYGEJTF.md" tag="0KQZYGEJTF" kind="definition">
# Signature of a quadratic form

## Definition

Let $V$ be a real vector space of finite dimension $n$, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
Choose a basis in which $Q(x_1 e_1 + \cdots + x_n e_n) = a_1 x_1^2 + \cdots + a_n x_n^2$.
Let $p$, $q$ and $r$ be the numbers of indices $i$ with $a_i > 0$, $a_i < 0$ and $a_i = 0$.
The *signature* of $Q$ is the triple $(p, q, r)$.

## Remarks

By [[Sylvester's law of inertia|0WFED3K3DQ]], the triple does not depend on the choice of the basis.
For a symmetric matrix $A$ of $Q$, the numbers $p$, $q$ and $r$ count the positive, negative and zero eigenvalues of $A$, with multiplicity.
Conventions vary: some authors write only $(p, q)$, or the number $p - q$.
The Minkowski form $x_1^2 - x_2^2 - x_3^2 - x_4^2$ on $\R^4$ has $p = 1$, $q = 3$ and $r = 0$.

## Explanations

### L1 human

The signature counts the plus signs, the minus signs and the zeros in a real quadratic form, after diagonalisation.
It is the complete invariant of a real quadratic form.
Two real quadratic forms are equivalent exactly when they have the same signature.
</document>

<document source="https://tamnd.github.io/math2/node/0KRH5157WP.md" tag="0KRH5157WP" kind="concept">
# Stationary distribution

## Summary

A stationary distribution of a Markov chain is a distribution of the state that does not change in one step of the chain.
The node [[stationary distribution|02KYGSCTCN]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0KRMJRDRZT.md" tag="0KRMJRDRZT" kind="concept">
# Continuity at a point

## Summary

A function is continuous at a point when points near the point are sent to points near its image.
In topological spaces, nearness is expressed with neighbourhoods.
</document>

<document source="https://tamnd.github.io/math2/node/0KRVRV5V80.md" tag="0KRVRV5V80" kind="concept">
# Symmetric bilinear form associated with a quadratic form

## Summary

Let $Q$ be a quadratic form over a field of characteristic not two.
The symmetric bilinear form associated with $Q$ is the unique symmetric bilinear form $B$ with $B(v, v) = Q(v)$.
</document>

<document source="https://tamnd.github.io/math2/node/0KS0KRDV4Y.md" tag="0KS0KRDV4Y" kind="concept">
# Degree of a vertex

## Summary

The degree of a vertex is the number of edges that meet it. Degrees are the simplest numerical invariants of a graph. The node [[degree of a vertex|0T0F6CCS9W]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0KSM23SN3H.md" tag="0KSM23SN3H" kind="definition">
# Nowhere dense set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $A \subseteq X$.
The set $A$ is nowhere dense in $X$ when the [[interior|0C38S0KMJK]] of its [[closure|0DRKT1QJR7]] is empty, that is, $\operatorname{int} \overline{A} = \emptyset$.

## Remarks

The set $\mathbb{Z}$ is nowhere dense in $\mathbb{R}$. So is each finite subset of $\mathbb{R}$.
The [[Cantor set|0BXNDZFQME]] is nowhere dense in $\mathbb{R}$, though it is uncountable.
The set $\mathbb{Q}$ is not nowhere dense in $\mathbb{R}$, because its closure is $\mathbb{R}$.
A closed set is nowhere dense if and only if its complement is open and [[dense|0D198GPMTJ]].

## Explanations

### L1 human

A nowhere dense set is thin everywhere: inside each open set, you can find a smaller open set that misses it completely.
The integers on the real line are an example, since there are gaps between them.
</document>

<document source="https://tamnd.github.io/math2/node/0KT8TGXD8K.md" tag="0KT8TGXD8K" kind="concept">
# Equivalence relation induced by a function

## Summary

Each function induces an equivalence relation on its domain: two elements are equivalent when the function has the same value at both.
</document>

<document source="https://tamnd.github.io/math2/node/0KTGJW1ZTH.md" tag="0KTGJW1ZTH" kind="concept">
# Jordan measure

## Summary

The Jordan measure, or Jordan content, extends the volume of boxes in [[R^n|0XPR1E85W0]] to some bounded sets. Finite unions of boxes approximate these sets well from inside and from outside.
A bounded set has Jordan measure exactly when its indicator function is Riemann integrable, and then the measure is the integral of that function.
</document>

<document source="https://tamnd.github.io/math2/node/0KTW26C80J.md" tag="0KTW26C80J" kind="concept">
# Free group

## Summary

The free group on a set $S$ is the group of reduced words on $S$, with concatenation followed by reduction as the operation.
It is the group generated by $S$ with no relations other than those that the group axioms force.
</document>

<document source="https://tamnd.github.io/math2/node/0KVGH0J5B3.md" tag="0KVGH0J5B3" kind="concept">
# Cyclic group

## Summary

A cyclic group is a group that one element generates.
Up to isomorphism, the cyclic groups are the integers and the integers modulo $n$ under addition.
</document>

<document source="https://tamnd.github.io/math2/node/0KVQ9D86BM.md" tag="0KVQ9D86BM" kind="statement">
# Variance under scaling

## Statement

Let $X$ be a real random variable with finite [[variance|007QED3E0Y]] and let $a \in \mathbb{R}$. Then
$$\operatorname{Var}(aX) = a^2 \operatorname{Var}(X).$$

## Remarks

Proof: $E[aX] = a E[X]$ by [[linearity of expectation|0P7PTC8DDE]]. So $(aX - E[aX])^2 = a^2 (X - E[X])^2$, and one takes expectations.
For the standard deviation, the formula becomes $\sigma_{aX} = |a| \, \sigma_X$.
In particular, $\operatorname{Var}(-X) = \operatorname{Var}(X)$.

## Explanations

### L1 human

If you measure in centimeters instead of meters, every value is multiplied by $100$. The variance is then multiplied by $100^2 = 10000$, because it is measured in squared units.
</document>

<document source="https://tamnd.github.io/math2/node/0KW2X0TFBX.md" tag="0KW2X0TFBX" kind="definition">
# Modulus of a complex number

## Definition

Let $z$ be a [[complex number|09FTAET5Z4]] with real part $a = \operatorname{Re} z$ and imaginary part $b = \operatorname{Im} z$.
The modulus, or absolute value, of $z$ is the nonnegative real number
$$|z| = \sqrt{a^2 + b^2}.$$

## Remarks

For a real number $z = a$, the modulus is $\sqrt{a^2} = |a|$, the [[absolute value|00S7C68D51]] of $a$.
In the [[complex plane|0CTE7A31XQ]], $|z|$ is the distance from $z$ to $0$, by the Pythagorean theorem. The distance between $z$ and $w$ is $|z - w|$.
For example, $|3 + 4 i| = 5$.

## Explanations

### L1 human

The modulus of a complex number is its length, that is, its distance from $0$ in the plane.
The point $3 + 4 i$ is the corner of a right triangle with sides $3$ and $4$, so its distance from $0$ is $5$.
</document>

<document source="https://tamnd.github.io/math2/node/0KW3PN547K.md" tag="0KW3PN547K" kind="concept">
# Diameter of a set

## Summary

The diameter of a subset of a metric space is the supremum of the distances between pairs of its points.
It generalizes the diameter of a circle.
</document>

<document source="https://tamnd.github.io/math2/node/0KWFFMDF8T.md" tag="0KWFFMDF8T" kind="concept">
# Connected graph

## Summary

A connected graph is a graph in one piece, in which any two vertices are joined by a walk. Connectivity is a basic hypothesis of many results, such as the existence of spanning trees. The node [[connected graph|0FFAYNMAJZ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0KWWEXW0RN.md" tag="0KWWEXW0RN" kind="statement">
# Reduction modulo p test for irreducibility

## Statement

Let $f \in \mathbb{Z}[x]$ be a [[monic polynomial|0BDX4MBKT4]] and let $p$ be a prime number.
Let $\bar{f} \in \mathbb{F}_p[x]$ be the polynomial whose coefficients are those of $f$ modulo $p$.
If $\bar{f}$ is [[irreducible|00X2A9G9W3]] in $\mathbb{F}_p[x]$, then $f$ is irreducible in $\mathbb{Z}[x]$ and in $\mathbb{Q}[x]$.

## Remarks

The proof: a factorisation $f = g h$ into monic factors of positive degree gives $\bar{f} = \bar{g} \bar{h}$, with $\deg \bar{g} = \deg g$ and $\deg \bar{h} = \deg h$.
Irreducibility in $\mathbb{Q}[x]$ then follows by [[Gauss's lemma|0QFV5Y1Z5X]].
The converse is false: $x^4 + 1$ is irreducible over $\mathbb{Q}$, but it is reducible modulo every prime.
Example: $x^3 + x + 1$ has no root in $\mathbb{F}_2$, so it is irreducible over $\mathbb{F}_2$, and so over $\mathbb{Q}$.

## Explanations

### L1 human

A factorisation over the integers stays a factorisation after reduction modulo $p$.
So if the reduced polynomial does not factor, the original one does not factor either.
Over a small finite field, you can check irreducibility by a finite search.
</document>

<document source="https://tamnd.github.io/math2/node/0KWXYZPJKN.md" tag="0KWXYZPJKN" kind="definition">
# Change of basis matrix

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$ with two [[bases|0KCQ9Y3QKA]] $B = (b_1, \dots, b_n)$ and $B' = (b'_1, \dots, b'_n)$.
The *change of basis matrix* from $B$ to $B'$ is the $n \times n$ matrix $P = (p_{ij})$ defined by
$$b'_j = \sum_{i=1}^{n} p_{ij} b_i \qquad (1 \le j \le n).$$
Its column $j$ is the [[coordinate vector|04CYQHN6MC]] $[b'_j]_B$.

## Remarks

For each $v \in V$, $[v]_B = P [v]_{B'}$.
The matrix $P$ is invertible, and $P^{-1}$ is the change of basis matrix from $B'$ to $B$.
$P$ is the matrix of the identity map of $V$ in the bases $B'$ (domain) and $B$ (codomain).
Conventions differ between authors; some call $P^{-1}$ the change of basis matrix.

## Explanations

### L1 human

The change of basis matrix lists the new basis vectors in the old coordinates.
It translates coordinates in the new basis into coordinates in the old basis.
Its inverse translates in the other direction.
</document>

<document source="https://tamnd.github.io/math2/node/0KX1MXJ5MR.md" tag="0KX1MXJ5MR" kind="statement">
# Radius of convergence of the binomial series

## Statement

Let $\alpha$ be a real number that is not a natural number.
Then the [[binomial series|0HB0XKDXHH]] $\sum_{k=0}^{\infty} \binom{\alpha}{k} x^k$ has radius of convergence $1$.

## Remarks

The coefficients are nonzero, and $\binom{\alpha}{k+1} / \binom{\alpha}{k} = (\alpha - k)/(k + 1)$. Its absolute value tends to $1$, so the [[ratio formula|0JBMYAA9QH]] gives $R = 1$.
At $x = \pm 1$ the behavior depends on $\alpha$. For $\alpha > 0$ the series converges absolutely at both endpoints.
Mathlib states the result as `binomialSeries_radius_eq_one`.

## Explanations

### L1 human

Unless the series stops after finitely many terms, it converges exactly inside $(-1, 1)$, plus possibly some endpoints.
This matches the fact that $(1 + x)^{\alpha}$ has trouble at $x = -1$.
</document>

<document source="https://tamnd.github.io/math2/node/0KX66CZMBB.md" tag="0KX66CZMBB" kind="concept">
# Positive definite matrix

## Summary

A positive definite matrix is a Hermitian matrix $A$ with $x^* A x > 0$ for every nonzero vector $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0KXJTZSW9B.md" tag="0KXJTZSW9B" kind="example">
# Usual metric on the real line

## Example

On the set $\mathbb{R}$ of [[real numbers|08J2M1JYAG]], let
$$d(x, y) = |x - y|,$$
where $|\cdot|$ is the [[absolute value|00S7C68D51]].
Then $d$ is a [[metric|063W5JMY85]] on $\mathbb{R}$.
The [[triangle inequality|03HF5PMY5T]] for real numbers gives the triangle inequality for $d$.

## Remarks

The number $|x - y|$ is the absolute difference of $x$ and $y$.
This metric is the case $n = 1$ of the [[Euclidean metric|0E1MMCAY5B]].
The same formula with the [[modulus|0KW2X0TFBX]] gives a metric on the complex numbers.

## Explanations

### L1 human

The distance between two numbers is the length of the segment between them on the number line.
For example, the distance between $-2$ and $3$ is $|-2 - 3| = 5$.
</document>

<document source="https://tamnd.github.io/math2/node/0KXQKES075.md" tag="0KXQKES075" kind="concept">
# Number of prime factors counted with multiplicity

## Summary

The function $\Omega(n)$ counts the prime factors of a positive integer $n$, with each prime counted as often as it divides $n$.
The [[definition of the number of prime factors with multiplicity|08420828EY]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0KXQKX9P63.md" tag="0KXQKX9P63" kind="concept">
# Splitting field

## Summary

A splitting field of a polynomial over a field is a smallest extension field over which the polynomial splits into linear factors.
It is generated by the roots of the polynomial.
The [[definition of a splitting field|02NZZ1J0FH]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0KXSK8TEEB.md" tag="0KXSK8TEEB" kind="statement">
# Continuous partial derivatives imply differentiability

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$.
Suppose that each [[partial derivative|09EGC6F8PT]] $\partial f_j / \partial x_i$ exists at each point of $U$, for $1 \le i \le n$ and $1 \le j \le m$, and that each of them is continuous at $a$.
Then $f$ is [[differentiable|0135XZZQ8M]] at $a$.

## Remarks

If the partial derivatives exist on $U$ and are continuous on all of $U$, then $f$ is differentiable on $U$, and $Df$ is continuous. So $f$ is of class $C^1$ on $U$.
The converse is false. The function $f(x) = \|x\|^2 \sin(1/\|x\|)$ for $x \ne 0$, with $f(0) = 0$, is differentiable on $\mathbb{R}^n$, but its partial derivatives are not continuous at $0$.
Without continuity, the existence of the partial derivatives is not enough, see [[this counterexample|0TNZC55X3J]].

## Explanations

### L1 human

The partial derivatives give a candidate for the derivative: the matrix of all $\partial f_j / \partial x_i$ at $a$.
Continuity of the partial derivatives near $a$ is what makes this candidate a good linear approximation in all directions, and not only along the axes.
In practice, this theorem is how you check differentiability: compute the partial derivatives and see that they are continuous.
</document>

<document source="https://tamnd.github.io/math2/node/0KY08TWDMQ.md" tag="0KY08TWDMQ" kind="statement">
# Derivative of x^a

## Statement

Let $a$ be a real number. The function $x \mapsto x^a$ is differentiable on $(0, \infty)$, and
$$\frac{d}{dx} x^a = a x^{a-1} \quad \text{for } x > 0.$$

## Remarks

Write $x^a = e^{a \ln x}$. The [[chain rule|0RXX2PT1MQ]] with [[$\exp' = \exp$|0EX9GFJ0B3]] and [[$\ln' x = 1/x$|07X1QA6BRW]] gives $e^{a \ln x} \cdot \frac{a}{x} = a x^{a-1}$.
For example, $\frac{d}{dx} \sqrt{x} = \frac{1}{2 \sqrt{x}}$.
Similarly, $\frac{d}{dx} a^x = a^x \ln a$ for a fixed $a > 0$. Mathlib states the main claim as `Real.hasDerivAt_rpow_const`.

## Explanations

### L1 human

The power rule for whole exponents also works for real exponents, if $x > 0$.
Bring the exponent down and lower it by one.
</document>

<document source="https://tamnd.github.io/math2/node/0KY7GS695G.md" tag="0KY7GS695G" kind="concept">
# Stirling numbers of the first kind

## Summary

The Stirling numbers of the first kind count the permutations of $n$ elements with exactly $k$ cycles. They are the coefficients of the rising and falling factorials as polynomials. The node [[Stirling numbers of the first kind|0Z22VQH17V]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0KYP951QWH.md" tag="0KYP951QWH" kind="definition">
# Discrete uniform distribution

## Definition

Let $S$ be a nonempty [[finite set|0BBTH3YSZA]] with $n$ elements.
The discrete uniform distribution on $S$ is the distribution with the [[probability mass function|0WF4EJP91W]]
$$p(s) = \frac{1}{n} \quad \text{for } s \in S.$$
A random variable $X$ is uniform on $S$ if $P(X = s) = 1/n$ for each $s \in S$.

## Remarks

Then $P(X \in A) = |A| / n$ for each $A \subseteq S$. This is the [[classical definition of probability|0AKNPYQ04R]].
For $S = \{a, a + 1, \dots, b\}$ with integers $a \le b$ and $n = b - a + 1$, the mean is $(a + b)/2$ and the variance is $(n^2 - 1)/12$.
The result of a fair die is uniform on $\{1, \dots, 6\}$, with mean $7/2$.

## Explanations

### L1 human

The discrete uniform distribution is "all outcomes equally likely".
A fair die, a fair coin and a well-shuffled deck all give uniform distributions.
</document>

<document source="https://tamnd.github.io/math2/node/0KYX2N69J0.md" tag="0KYX2N69J0" kind="concept">
# Quotient group

## Summary

The quotient group $G/N$ of a group $G$ by a normal subgroup $N$ is the set of cosets of $N$, with the product $(aN)(bN) = abN$.
It collapses $N$ to the identity and keeps the rest of the structure of $G$.
</document>

<document source="https://tamnd.github.io/math2/node/0KYYSMEVHH.md" tag="0KYYSMEVHH" kind="statement">
# Lagrange's four-square theorem

## Statement

Each natural number $n$ is a sum of four squares of integers: $n = a^2 + b^2 + c^2 + d^2$ for some integers $a$, $b$, $c$ and $d$.

## Remarks

For example, $7 = 4 + 1 + 1 + 1$ and $15 = 9 + 4 + 1 + 1$.
Three squares are not enough: $7$ is not a sum of three squares, by [[Legendre's three-square theorem|0B64Q7XNE3]].
By [[Euler's four-square identity|08WTN6SCQ6]], it is enough to prove the theorem for primes.
Other proofs use [[Minkowski's theorem|0K387S2PKQ]] or the Hurwitz quaternions.

## History

Diophantus knew examples, and Bachet stated the theorem in his Latin translation of the Arithmetica of 1621.
Lagrange gave the first proof in 1770.

## Explanations

### L1 human

Every whole number is a sum of at most four perfect squares, if you allow $0$ as a square.
Some numbers, such as $7$, $15$ and $23$, need all four.
</document>

<document source="https://tamnd.github.io/math2/node/0KYZXYDGQZ.md" tag="0KYZXYDGQZ" kind="statement">
# Quadratic formula

## Statement

Let $a$, $b$ and $c$ be complex numbers with $a \neq 0$, and let $s$ be a complex number with $s^2 = b^2 - 4 a c$.
Then, for each complex number $x$,
$$a x^2 + b x + c = 0 \iff x = \frac{-b + s}{2 a} \text{ or } x = \frac{-b - s}{2 a}.$$

## Remarks

Such an $s$ exists, by the [[existence of nth roots|0YM6P6C206]] for $n = 2$.
The number $\Delta = b^2 - 4 a c$ is the discriminant. For real $a, b, c$, there are two real roots if $\Delta > 0$, one double real root if $\Delta = 0$, and two conjugate non-real roots if $\Delta < 0$.
The same formula holds in each field in which $2 \neq 0$.

## Explanations

### L1 human

The formula solves every quadratic equation.
For $x^2 + 2 x + 5 = 0$, the discriminant is $4 - 20 = -16$, so $s = 4 i$, and the roots are $-1 + 2 i$ and $-1 - 2 i$.
Completing the square proves the formula: the equation becomes $(2 a x + b)^2 = b^2 - 4 a c$.
</document>

<document source="https://tamnd.github.io/math2/node/0KZDNGA561.md" tag="0KZDNGA561" kind="concept">
# Trace of a linear map

## Summary

The trace of an endomorphism of a finite-dimensional vector space is the trace of its matrix in any basis.
</document>

<document source="https://tamnd.github.io/math2/node/0KZJ82WEPP.md" tag="0KZJ82WEPP" kind="definition">
# Independent set

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
An independent set in $G$ is a set $S$ of vertices of $G$ such that no two vertices of $S$ are adjacent.

## Remarks

A set is independent in $G$ if and only if it is a [[clique|0G52BPNPKG]] in the [[complement graph|0RNMP6VCR2]].
A set $S$ is independent if and only if its complement in the vertex set is a [[vertex cover|0KDADE9RQ8]].
The largest size of an independent set is the independence number $\alpha(G)$. Each colour class of a [[colouring|0874138YP6]] is an independent set.

## Explanations

### L1 human

An independent set is a set of vertices with no edge between any two of them.
In a conflict graph, it is a set of tasks that can all happen at the same time.
</document>

<document source="https://tamnd.github.io/math2/node/0KZZ70ZHXP.md" tag="0KZZ70ZHXP" kind="concept">
# Diffeomorphism

## Summary

A diffeomorphism is a bijection between open sets, or between manifolds, such that both the map and its inverse are [[differentiable|0HHHDCXZV4]].
Diffeomorphisms are the changes of coordinates of calculus, and they preserve all differentiable structure.
</document>

<document source="https://tamnd.github.io/math2/node/0M0H1ZH8TT.md" tag="0M0H1ZH8TT" kind="concept">
# Antisymmetric relation

## Summary

A relation is antisymmetric when two different elements are never related in both directions.
</document>

<document source="https://tamnd.github.io/math2/node/0M0RZ3HSB0.md" tag="0M0RZ3HSB0" kind="statement">
# Derivative of an inverse function

## Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be continuous and strictly monotone.
Assume that $f$ is differentiable at $a \in I$ with $f'(a) \ne 0$, and let $b = f(a)$.
Then $f^{-1}$ is differentiable at $b$, and
$$(f^{-1})'(b) = \frac{1}{f'(a)} = \frac{1}{f'(f^{-1}(b))}.$$

## Remarks

The inverse is [[continuous|07BY47HG5S]], so $f^{-1}(b + k) \to a$ as $k \to 0$. The difference quotient of $f^{-1}$ at $b$ is the reciprocal of a difference quotient of $f$ at $a$.
If $f'(a) = 0$, then $f^{-1}$ is not differentiable at $b$. An example is $f(x) = x^3$ at $a = 0$.
For example, it gives the [[derivative of the logarithm|07X1QA6BRW]] from the derivative of the exponential function.

## Explanations

### L1 human

The graph of $f^{-1}$ is the graph of $f$ reflected in the line $y = x$.
The reflection turns a slope $m$ into the slope $1/m$.
</document>

<document source="https://tamnd.github.io/math2/node/0M0SNKZ482.md" tag="0M0SNKZ482" kind="definition">
# Block diagonal matrix

## Definition

Let $R$ be a ring and let $A_1, \dots, A_k$ be square matrices over $R$, where $A_i$ has order $n_i$.
The *block diagonal matrix* $\operatorname{diag}(A_1, \dots, A_k)$ is the [[block matrix|0ASDFEQS6P]] of order $n_1 + \cdots + n_k$
$$\begin{pmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_k \end{pmatrix},$$
whose block in position $(i, i)$ is $A_i$. The other blocks are zero matrices.

## Remarks

This matrix is also called the *direct sum* $A_1 \oplus \cdots \oplus A_k$.
Products of block diagonal matrices with matching block sizes are computed block by block.
Over a commutative ring, $\det \operatorname{diag}(A_1, \dots, A_k) = \det A_1 \cdots \det A_k$.
The [[Jordan normal form|085HQ5G5TE]] is a block diagonal matrix of [[Jordan blocks|07PH4RX0X5]].

## Explanations

### L1 human

A block diagonal matrix has square blocks along the diagonal and zeros everywhere else.
Each block acts on its own group of coordinates and ignores the other coordinates.
So we can study each block separately.
</document>

<document source="https://tamnd.github.io/math2/node/0M0TKFRX2A.md" tag="0M0TKFRX2A" kind="statement">
# Linearity of differentiation

## Statement

Let $f$ and $g$ be real functions on an open interval $I$, both differentiable at $a \in I$.
Let $\alpha$ and $\beta$ be real numbers.
Then $\alpha f + \beta g$ is differentiable at $a$, and
$$(\alpha f + \beta g)'(a) = \alpha f'(a) + \beta g'(a).$$

## Remarks

The difference quotient of $\alpha f + \beta g$ is $\alpha$ times that of $f$ plus $\beta$ times that of $g$. The limit laws give the result.
The case $\alpha = \beta = 1$ is the sum rule $(f + g)' = f' + g'$. The case $\beta = 0$ is the [[constant multiple rule|0Z9JB71BS0]].
Mathlib states the sum rule as `deriv_add`.

## Explanations

### L1 human

Differentiation respects sums and constant factors.
So we can differentiate a sum term by term.
</document>

<document source="https://tamnd.github.io/math2/node/0M1GFS6PVA.md" tag="0M1GFS6PVA" kind="statement">
# Intercept theorem

## Statement

Let $S$ be a point of the Euclidean plane, and let $\ell_1$ and $\ell_2$ be two distinct [[lines|0MP9BCF6DP]] through $S$. Let $A$ and $B$ be points of $\ell_1$, and let $C$ and $D$ be points of $\ell_2$, all different from $S$. Let $|XY|$ denote the [[distance|0NPPGHQNXQ]] between the points $X$ and $Y$.
Assume that the line $AC$ is parallel to the line $BD$. Then
$$\frac{|SA|}{|SB|} = \frac{|SC|}{|SD|} = \frac{|AC|}{|BD|}.$$

## Remarks

If also $A \neq B$ and $A$ and $B$ are on the same ray from $S$, then the theorem also gives $\frac{|SA|}{|AB|} = \frac{|SC|}{|CD|}$.
A converse holds for two rays from $S$: if $A, B$ are on one ray, $C, D$ on the other, and $\frac{|SA|}{|AB|} = \frac{|SC|}{|CD|}$, then $AC$ is parallel to $BD$.
The theorem says that the triangles $SAC$ and $SBD$ are [[similar|0YF6QXXWB5]]. The [[homothety|0ZG1PNEP35]] with the centre $S$ that sends $A$ to $B$ also sends $C$ to $D$.

## History

Tradition attributes the theorem to Thales of Miletus. He may have used it to find the heights of the pyramids of Egypt.

## Explanations

### L1 human

Draw two lines from one point, and cut them with two parallel lines. The parallel lines cut both lines in the same proportions. This is how you can find the height of a tree from the length of its shadow and the shadow of a stick.
</document>

<document source="https://tamnd.github.io/math2/node/0M1GJSWZDA.md" tag="0M1GJSWZDA" kind="definition">
# Convex function on R^n

## Definition

A set $C \subseteq \mathbb{R}^n$ is convex if $t x + (1 - t) y \in C$ for all $x, y \in C$ and all $t \in [0, 1]$.
Let $C \subseteq \mathbb{R}^n$ be convex.
A function $f \colon C \to \mathbb{R}$ is convex if, for all $x, y \in C$ and all $t \in [0, 1]$,
$$f(t x + (1 - t) y) \le t f(x) + (1 - t) f(y).$$
The function $f$ is strictly convex if the inequality is strict for all $x \ne y$ and all $t \in (0, 1)$.

## Remarks

If $C$ is open and $f$ is differentiable on $C$, then $f$ is convex if and only if $f(y) \ge f(x) + \nabla f(x) \cdot (y - x)$ for all $x, y \in C$.
Let $C$ be open and let $f$ be of class $C^2$. Then $f$ is convex if and only if its [[Hessian matrix|0SCSQWXRA4]] is positive semidefinite at each point of $C$.
Each local minimum of a convex function is a global minimum, and a strictly convex function has at most one global minimum.
A function can be convex in each variable separately but not convex. The function $f(x, y) = x y$ is linear in each variable, but $f(t, -t) = -t^2$ is not convex.

## Explanations

### L1 human

A convex function is bowl-shaped: the straight segment between two points of its graph never goes below the graph.
Examples are $\|x\|^2$ and, on all of $\mathbb{R}^n$, each linear function.
</document>

<document source="https://tamnd.github.io/math2/node/0M1MX28VXK.md" tag="0M1MX28VXK" kind="statement">
# Bernoulli's inequality

## Statement

Let $x$ be a real number with $x \ge -1$, and let $n$ be a natural number.
Then
$$(1 + x)^n \ge 1 + n x.$$

## Remarks

When $x \neq 0$ and $n \ge 2$, the inequality is strict.
For natural numbers $n$, the inequality also holds when $x \ge -2$.
A proof by [[induction|0ASJVEFA9J]] on $n$ multiplies $(1 + x)^n \ge 1 + n x$ by $1 + x \ge 0$ and uses $n x^2 \ge 0$.

## Explanations

### L1 human

A small growth rate $x$, applied $n$ times, gives at least $n$ times that growth.
For example, $1.1^{10}$ is at least $1 + 10 \cdot 0.1 = 2$; its true value is about $2.59$.
The difference comes from the growth on the growth, which the right side ignores.
</document>

<document source="https://tamnd.github.io/math2/node/0M25PRN3WQ.md" tag="0M25PRN3WQ" kind="statement">
# Fermat's little theorem

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]] and let $a$ be an integer that $p$ does not divide.
Then $a^{p-1} \equiv 1 \pmod p$.

## Remarks

An equivalent form is [[Fermat's little theorem in the form a^p = a|0BMXWXDVJZ]]: $a^p \equiv a \pmod p$ for each integer $a$.
In the ring $\mathbb{Z}/p\mathbb{Z}$, the statement is $x^{p-1} = 1$ for each nonzero $x$. It follows from [[Lagrange's theorem|3C53TD]] for the multiplicative [[group|G5S1M9]] of $\mathbb{Z}/p\mathbb{Z}$, which has $p - 1$ elements.
[[Euler's theorem|0VC4DVE7DQ]] generalizes the result to each modulus $n$.
The converse is false. For example, $2^{340} \equiv 1 \pmod{341}$, but $341 = 11 \cdot 31$. See [[Fermat pseudoprime|02M2K4QH9P]].

## History

Fermat stated the theorem in a letter of 1640, without a proof. Euler published the first known proof in 1736. Leibniz had a proof earlier, in an unpublished manuscript.

## Explanations

### L1 human

Take $p = 5$ and $a = 2$: $2^4 = 16 = 3 \cdot 5 + 1$, so $2^4 \equiv 1 \pmod 5$.
The theorem says that this always happens for a prime $p$ and a number $a$ that $p$ does not divide.
It lets you reduce large powers: $2^{100} = (2^4)^{25} \equiv 1 \pmod 5$.
</document>

<document source="https://tamnd.github.io/math2/node/0M28ASWG8T.md" tag="0M28ASWG8T" kind="definition">
# Conjunction

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The conjunction of $p$ and $q$ is the proposition $p \land q$, read "$p$ and $q$".
The proposition $p \land q$ is true when $p$ is true and $q$ is true. In all other cases, $p \land q$ is false.

| $p$ | $q$ | $p \land q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |

## Remarks

Conjunction is commutative and associative: $p \land q$ has the same truth value as $q \land p$.
The [[intersection|0HBRW4E6G1]] of two sets uses conjunction: $x \in A \cap B$ means $x \in A \land x \in B$.

## Explanations

### L1 human

The sentence "$6$ is even and $6$ is divisible by $3$" is a conjunction.
It is true, because both parts are true.
If one part is false, the whole sentence is false.
</document>

<document source="https://tamnd.github.io/math2/node/0M2JRW1J0N.md" tag="0M2JRW1J0N" kind="statement">
# Curl of a gradient is zero

## Statement

Let $U \subseteq \mathbb{R}^3$ be open, and let $\phi \colon U \to \mathbb{R}$ be of class $C^2$.
Then
$$\nabla \times (\nabla \phi) = 0 \quad \text{on } U.$$

## Remarks

Proof: the first component of the [[curl|0GDNHS9J2X]] of $\nabla \phi$ is $\partial_y \partial_z \phi - \partial_z \partial_y \phi$, and this is zero by [[Schwarz's theorem|0NNQY8WCW6]]. The other components are similar.
So a [[conservative|05R9NF2J49]] $C^1$ vector field is irrotational. This gives a quick test: if $\operatorname{curl} F \ne 0$ at some point, then $F$ has no scalar potential.
In $\mathbb{R}^n$, the same argument shows $\partial_i F_j = \partial_j F_i$ for all $i, j$ when $F = \nabla \phi$ with $\phi$ of class $C^2$.
The converse needs a condition on $U$. It holds on [[convex sets|0B22P010D6]], but [[not on every open set|0GRM3BS0DD]].

## Explanations

### L1 human

A gradient field has no swirl.
Going around a small loop in a landscape, you return to the same height. So the field $\nabla \phi$ has no circulation, and its curl is zero.
</document>

<document source="https://tamnd.github.io/math2/node/0M2SSGSYPE.md" tag="0M2SSGSYPE" kind="statement">
# Cayley's formula

## Statement

Let $n \ge 2$ be an integer.
Then the number of [[trees|0MJ0JS72DZ]] with vertex set $\{1, 2, \dots, n\}$ is $n^{n-2}$.

## Remarks

The trees are labelled: two trees are the same only if they have the same edge set. Thus the formula counts the [[spanning trees|0FQ5B8S1ZC]] of $K_n$.
For $n = 3$ there are $3$ trees, and for $n = 4$ there are $16$.
A proof uses the [[Prüfer sequence|09PV9JSY6E]], a bijection between these trees and the sequences of length $n - 2$ over $\{1, \dots, n\}$.

## History

Carl Wilhelm Borchardt found the formula in 1860.
Arthur Cayley extended it in 1889, and the formula has his name.

## Explanations

### L1 human

Label $n$ points with the numbers $1$ to $n$ and count the ways to connect them by a tree.
For $4$ points there are $4^2 = 16$ ways: $12$ paths and $4$ stars.
</document>

<document source="https://tamnd.github.io/math2/node/0M3PYJ3TSE.md" tag="0M3PYJ3TSE" kind="statement">
# The number of Sylow subgroups divides the index

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a prime number, and let $P$ be a [[Sylow p-subgroup|0053CQE9Y2]] of $G$.
Then $n_p$ divides the [[index|0DTXBQPA2S]] $[G : P]$.

## Remarks

By [[this formula|09ZJ6CAK7S]], $n_p = [G : N_G(P)]$, and $[G : P] = [G : N_G(P)] \cdot [N_G(P) : P]$.
If $|G| = p^n m$ with $p \nmid m$, then $n_p$ divides $m$.
For example, in a group of order $15$, $n_3$ divides $5$ and $n_3 \equiv 1 \pmod 3$, so $n_3 = 1$.

## Explanations

### L1 human

The number of Sylow subgroups is the size of a conjugation orbit.
Orbit sizes divide the order of the group, and here even the part of the order without $p$.
This gives a strong limit on $n_p$.
</document>

<document source="https://tamnd.github.io/math2/node/0M3YQJ81NB.md" tag="0M3YQJ81NB" kind="concept">
# Analytic continuation

## Summary

Let $U \subseteq V$ be open subsets of $\mathbb{C}$, with $U$ nonempty and $V$ a [[domain|097KNJFJ7V]], and let $f$ be [[holomorphic|0DG0Z72CQD]] on $U$.
An analytic continuation of $f$ to $V$ is a function $F$ that is holomorphic on $V$ with $F(z) = f(z)$ for each $z \in U$.
By the [[identity theorem|0YBVKMYX0W]], $f$ has at most one analytic continuation to $V$.

## Remarks

The geometric series $\sum_{n \ge 0} z^n$ is holomorphic on the unit disc, and $1/(1 - z)$ is its analytic continuation to $\mathbb{C} \setminus \{1\}$.
The Riemann zeta function $\sum_{n \ge 1} n^{-s}$ on $\operatorname{Re} s > 1$ has an analytic continuation to $\mathbb{C} \setminus \{1\}$.
Continuation along a path is a more general process. Its result can depend on the path: the [[logarithm|0X2FY491PF]] continued once around $0$ changes by $2\pi i$.

## Explanations

### L1 human

A formula such as a power series often defines a function only on a small region.
Analytic continuation extends the function to a larger region in the only possible holomorphic way.
</document>

<document source="https://tamnd.github.io/math2/node/0M406JC8J1.md" tag="0M406JC8J1" kind="statement">
# Four colour theorem

## Statement

Let $G$ be a finite loopless [[planar graph|029JQEK15B]].
Then $\chi(G) \le 4$.

## Remarks

In terms of maps: four colours are enough for the regions of a map in the plane. Regions with a common boundary segment get different colours.
The bound is sharp, because $K_4$ is planar and $\chi(K_4) = 4$.

## History

Francis Guthrie stated the problem in 1852.
Kenneth Appel and Wolfgang Haken proved the theorem in 1976, with the help of a computer.
Robertson, Sanders, Seymour and Thomas gave a simpler proof in 1997. Georges Gonthier and Benjamin Werner checked a proof in the Coq proof assistant in 2005.

## Explanations

### L1 human

Four colours are enough to colour any map, so that neighbouring countries always have different colours.
It was the first major theorem whose proof needed a computer.
</document>

<document source="https://tamnd.github.io/math2/node/0M418FE7BB.md" tag="0M418FE7BB" kind="concept">
# Binary operation

## Summary

A binary operation on a set combines two elements of the set and gives one element of the same set.
Addition and multiplication of numbers are the basic examples.
</document>

<document source="https://tamnd.github.io/math2/node/0M4JJT780X.md" tag="0M4JJT780X" kind="concept">
# Uniform continuity

## Summary

A function is uniformly continuous when one input tolerance $\delta$ controls the output error $\varepsilon$ at all points of the domain at once. This is stronger than continuity at each point. The node [[uniform continuity|0Q4DR37DVE]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0M4SXE5WW3.md" tag="0M4SXE5WW3" kind="statement">
# Every linearly independent set extends to a basis

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $L \subseteq V$ be a [[linearly independent|0CRMGPAH83]] set.
Then there is a [[basis|0KCQ9Y3QKA]] $B$ of $V$ with $L \subseteq B$.

## Remarks

For finite-dimensional $V$, add to $L$, one at a time, a vector that is not in the span of the current set.
This stops after finitely many steps.
In general, the proof uses Zorn's lemma.
A stronger form: if $L \subseteq S$, with $L$ independent and $S$ spanning, then a basis $B$ exists with $L \subseteq B \subseteq S$.
For example, $\{(1, 1, 0)\}$ extends to the basis $\{(1, 1, 0), (0, 1, 0), (0, 0, 1)\}$ of $\R^3$.

## Explanations

### L1 human

We can always complete an independent set of vectors to a basis.
We keep adding vectors that point in new directions until the set reaches the whole space.
This shows that an independent set can never be larger than a basis.
</document>

<document source="https://tamnd.github.io/math2/node/0M4TCS0RRG.md" tag="0M4TCS0RRG" kind="statement">
# Classification of the Platonic solids

## Statement

Let $P$ be a [[Platonic solid|017E5PRS1Q]] whose faces have $p$ sides each, with $q$ faces at each vertex. Then $\{p, q\}$ is one of
$$\{3, 3\}, \quad \{4, 3\}, \quad \{3, 4\}, \quad \{5, 3\}, \quad \{3, 5\}.$$
Each of these five cases occurs, by the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron and the regular icosahedron. Up to similarity, there is exactly one Platonic solid for each case.

## Remarks

A proof: let $V$, $E$ and $F$ be the numbers of vertices, edges and faces. Then $pF = 2E = qV$, and [[Euler's formula|09HT3P35YP]] $V - E + F = 2$ gives $\frac{1}{p} + \frac{1}{q} = \frac{1}{2} + \frac{1}{E} > \frac{1}{2}$.
With $p, q \geq 3$, this inequality has exactly the five solutions above.
A geometric proof uses the angles: at each vertex, the $q$ face angles $\frac{(p - 2)\pi}{p}$ must add up to less than $2\pi$.

## History

Theaetetus, a contemporary of Plato, described all five solids. He may have given the first proof that no other convex regular polyhedra exist. Euclid's Elements contains a geometric proof.

## Explanations

### L1 human

Only five perfectly regular convex solids exist. At each corner, at least three faces meet, and their angles must add up to less than a full turn. This leaves room only for three, four or five triangles, three squares or three pentagons.
</document>

<document source="https://tamnd.github.io/math2/node/0M50E4WYWJ.md" tag="0M50E4WYWJ" kind="definition">
# Line integral of a scalar field

## Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a [[rectifiable curve|0C28RT5BG7]] with image $C$, and let $f \colon C \to \mathbb{R}$ be [[continuous|0S9EK5MQPE]].
For a partition $a = t_0 < t_1 < \dots < t_k = b$ and points $\tau_i \in [t_{i-1}, t_i]$, let $\Delta s_i$ be the length of the restriction of $\gamma$ to $[t_{i-1}, t_i]$.
The line integral of $f$ along $\gamma$ is the limit
$$\int_\gamma f \, ds = \lim \sum_{i=1}^{k} f(\gamma(\tau_i)) \, \Delta s_i,$$
taken as $\max_i (t_i - t_{i-1}) \to 0$.

## Remarks

The limit exists, and it does not depend on the choice of the points $\tau_i$.
For a $C^1$ curve, $\int_\gamma f \, ds = \int_a^b f(\gamma(t)) \, \|\gamma'(t)\| \, dt$. This is the [[parametric formula|0W3W1VYWBP]].
For $f = 1$ the line integral is the length of $\gamma$. If $\gamma$ is a wire with linear density $f$, the line integral is its mass.
The value does not change under a reparametrisation of $\gamma$, in either direction.

## Explanations

### L1 human

Walk along the curve, and at each small step multiply the length of the step by the value of $f$ there.
The sum of these products, in the limit of small steps, is the line integral with respect to arc length.
</document>

<document source="https://tamnd.github.io/math2/node/0M67FWZWPQ.md" tag="0M67FWZWPQ" kind="statement">
# Existence of inverses modulo n

## Statement

Let $n$ be a positive integer and let $a$ be an integer.
Then $a$ has a [[modular multiplicative inverse|00GXTQMY8Z]] modulo $n$ if and only if $a$ and $n$ are [[coprime|0GFKP8F8AG]].

## Remarks

If $\gcd(a, n) = 1$, [[Bézout's identity|0DQVHMYAGA]] gives integers $x$ and $y$ with $a x + n y = 1$. Then $a x \equiv 1 \pmod n$.
Conversely, if $a x \equiv 1 \pmod n$, then $a x + n y = 1$ for some integer $y$, so each common divisor of $a$ and $n$ divides $1$.
The [[extended Euclidean algorithm|0964YQFSA8]] computes the inverse.
For example, $4$ has no inverse modulo $6$, because $\gcd(4, 6) = 2$.

## Explanations

### L1 human

You can "divide by $a$" modulo $n$ exactly when $a$ and $n$ have no common factor.
Modulo $10$, the numbers $1, 3, 7, 9$ have inverses, since $1 \cdot 1$, $3 \cdot 7$ and $9 \cdot 9$ all end with the digit $1$.
The numbers $2, 4, 5, 6, 8$ have no inverse modulo $10$.
</document>

<document source="https://tamnd.github.io/math2/node/0M6G75E95E.md" tag="0M6G75E95E" kind="statement">
# The only ideals of a field are zero and the whole field

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $I$ be an [[ideal|0MYSA6MYNA]] of $K$.
Then $I = \{0\}$ or $I = K$.

## Remarks

The proof: if $a \in I$ and $a \neq 0$, then $1 = a^{-1} a \in I$, so $r = r \cdot 1 \in I$ for each $r \in K$.
The converse holds: a nonzero commutative ring whose only ideals are $\{0\}$ and the whole ring is a field.
For each nonzero $a$, the ideal $(a)$ is the whole ring, so $1 = b a$ for some $b$.

## Explanations

### L1 human

If an ideal of a field contains a nonzero element, then it contains $1$, because you can divide.
Once an ideal contains $1$, it contains every element.
So a field has only the two trivial ideals.
</document>

<document source="https://tamnd.github.io/math2/node/0M6W4XNST6.md" tag="0M6W4XNST6" kind="statement">
# Universal property of the product topology

## Statement

Let $(X_i)_{i \in I}$ be a family of [[topological spaces|0CNQXVBHCX]], let $X = \prod_{i \in I} X_i$ have the [[product topology|04S9HX3GAP]], and let $\pi_i \colon X \to X_i$ be the projections.
Let $Z$ be a topological space and let $f \colon Z \to X$ be a function.
Then $f$ is [[continuous|0SY9ECQ19Q]] if and only if the [[composition|07HF9YCWPP]] $\pi_i \circ f \colon Z \to X_i$ is continuous for each $i \in I$.

## Remarks

Proof of the converse: preimages of the subbasic sets $\pi_i^{-1}(U)$ under $f$ are $(\pi_i \circ f)^{-1}(U)$, which are open. Preimages respect unions and intersections, so all preimages of open sets are open.
Thus a map into a product is continuous if and only if all its coordinate functions are continuous. For example, $t \mapsto (\cos t, \sin t)$ is continuous from $\mathbb{R}$ to $\mathbb{R}^2$.
The statement fails for the [[box topology|0CTM1EHG9P]].

## Explanations

### L1 human

To check that a map into a product is continuous, check each coordinate separately.
This property characterises the product topology among all topologies on the product.
</document>

<document source="https://tamnd.github.io/math2/node/0M79HRJCRR.md" tag="0M79HRJCRR" kind="definition">
# Elementary matrix

## Definition

Let $F$ be a field and let $n \ge 1$ be an integer.
An $n \times n$ *elementary matrix* over $F$ is a matrix that results from the [[identity matrix|06DW9BAWB1]] $I_n$ by one [[elementary row operation|04XWH28452]].
There are three types:

1. $P_{ij}$, for $i \ne j$: exchange rows $i$ and $j$ of $I_n$.
2. $D_i(c)$, for $c \ne 0$: multiply row $i$ of $I_n$ by $c$.
3. $T_{ij}(c)$, for $i \ne j$ and $c \in F$: add $c$ times row $j$ of $I_n$ to row $i$.

## Remarks

For each $n \times p$ matrix $A$, the product $EA$ is the result of the same row operation on $A$.
The product $AE$ performs the matching column operation on $A$.
Each elementary matrix is invertible, and its inverse is an elementary matrix of the same type.
Each invertible matrix over a field is a product of elementary matrices.

## Explanations

### L1 human

An elementary matrix stores one row operation as a matrix.
To do the operation on $A$, we multiply $A$ on the left by this matrix.
Then a whole elimination becomes a product of elementary matrices.
</document>

<document source="https://tamnd.github.io/math2/node/0M7ARED5Q7.md" tag="0M7ARED5Q7" kind="concept">
# Pythagorean triple

## Summary

A Pythagorean triple is a triple of positive integers $a$, $b$, $c$ with $a^2 + b^2 = c^2$, the side lengths of a right triangle.
The [[definition of a Pythagorean triple|01G1WY1GVX]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0M7DEDKSYB.md" tag="0M7DEDKSYB" kind="statement">
# Euler's criterion

## Statement

Let $p$ be an odd prime and let $a$ be an integer.
Then
$$a^{(p-1)/2} \equiv \left(\frac{a}{p}\right) \pmod p,$$
where $\left(\frac{a}{p}\right)$ is the [[Legendre symbol|01J8AMV9RS]].

## Remarks

So for $p \nmid a$, the integer $a$ is a quadratic residue modulo $p$ if and only if $a^{(p-1)/2} \equiv 1 \pmod p$.
For example, $3^3 = 27 \equiv -1 \pmod 7$, so $3$ is a quadratic nonresidue modulo $7$.
The proof uses [[Fermat's little theorem|0M25PRN3WQ]]: $x = a^{(p-1)/2}$ satisfies $x^2 \equiv 1$, so $x \equiv \pm 1 \pmod p$.
A consequence is the [[multiplicativity of the Legendre symbol|0NW1JP7TBR]].

## History

Leonhard Euler found the criterion.

## Explanations

### L1 human

To test whether $a$ is a square modulo $p$, raise $a$ to the power $(p-1)/2$.
The result is $1$ for a square and $-1$ for a non-square, modulo $p$.
With [[fast modular exponentiation|04DR5PM25N]], this test is quick even for large $p$.
</document>

<document source="https://tamnd.github.io/math2/node/0M83V17KHX.md" tag="0M83V17KHX" kind="concept">
# Complementary subspaces

## Summary

Two subspaces of a vector space are complementary when the space is their direct sum.
</document>

<document source="https://tamnd.github.io/math2/node/0M856Q1AMZ.md" tag="0M856Q1AMZ" kind="definition">
# Dual space

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
The *dual space* of $V$ is the vector space
$$V^* = \Hom_F(V, F)$$
of all [[linear forms|07YQ1C78G0]] on $V$.
Its operations are pointwise: $(f + g)(v) = f(v) + g(v)$ and $(af)(v) = a f(v)$.

## Remarks

It is a special case of the [[space of linear maps|0X9MBPP2Z3]].
If $\dim V = n$ is finite, then $\dim V^* = n$, see [[the dimension of the dual space|00889YD15K]].
The elements of $F^n$ act on column vectors by the dot product, and this identifies $(F^n)^*$ with $F^n$.
For infinite-dimensional $V$, the space $V^*$ is not isomorphic to $V$.

## Explanations

### L1 human

The dual space collects all the ways to measure a vector by one linear number.
On $\R^3$, each linear form is a dot product with a fixed vector.
So, in finite dimensions, the dual space looks like the original space, but its elements play a different role.
</document>

<document source="https://tamnd.github.io/math2/node/0M85DXB5RM.md" tag="0M85DXB5RM" kind="concept">
# Equivalent norms

## Summary

Two norms on a vector space are equivalent when each is bounded by a constant multiple of the other.
Equivalent norms give the same open sets, the same convergent sequences and the same Cauchy sequences.
</document>

<document source="https://tamnd.github.io/math2/node/0M893X7YY7.md" tag="0M893X7YY7" kind="definition">
# Adjoining a root of a polynomial

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be [[irreducible|00X2A9G9W3]].
The field obtained by adjoining a root of $f$ to $K$ is the [[quotient ring|0VFW4Y69BT]] $K[x]/(f)$.
The map $K \to K[x]/(f)$, $c \mapsto c + (f)$, identifies $K$ with a subfield, and the class $\alpha = x + (f)$ is a root of $f$.

## Remarks

The quotient is a field, because [[$f$ is irreducible|0AWMPX35CW]].
Its degree over $K$ is $\deg f$, with the basis $1, \alpha, \dots, \alpha^{\deg f - 1}$.
For example, adjoining a root of $x^2 + 1$ to $\mathbb{R}$ gives a field isomorphic to $\mathbb{C}$.

## Explanations

### L1 human

You can create a root of a polynomial that has none.
Compute with polynomials in $x$, and replace $f(x)$ by zero each time it appears.
This is how $\mathbb{C}$ arises from $\mathbb{R}$ and $x^2 + 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0M8EWQJ1YV.md" tag="0M8EWQJ1YV" kind="statement">
# Vieta's formulas

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $f = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in R[x]$.
Suppose that $f = a_n (x - r_1)(x - r_2) \cdots (x - r_n)$ with $r_1, \dots, r_n \in R$.
Then for $1 \le k \le n$,
$$a_n \, e_k(r_1, \dots, r_n) = (-1)^k a_{n-k},$$
where $e_k$ is the $k$-th [[elementary symmetric polynomial|0ZMY8441GW]].

## Remarks

If $a_n$ is a unit, this says $e_k(r_1, \dots, r_n) = (-1)^k a_{n-k}/a_n$.
For $x^2 + b x + c = (x - r_1)(x - r_2)$, it gives $r_1 + r_2 = -b$ and $r_1 r_2 = c$.
The proof expands the product and compares coefficients.
Over a field, the hypothesis says that $f$ [[splits|0SHMC00RJV]] with roots $r_1, \dots, r_n$, counted with multiplicity.

## History

The formulas are named after the French mathematician François Viète, who derived them for positive roots.

## Explanations

### L1 human

Vieta's formulas say that the sum of the roots, the sum of their products in pairs, and so on, are given by the coefficients.
You can know these sums without finding the roots.
</document>

<document source="https://tamnd.github.io/math2/node/0M8MHYERTY.md" tag="0M8MHYERTY" kind="definition">
# Nilpotent endomorphism

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The map $T$ is *nilpotent* when $T^k = 0$ for some integer $k \ge 1$.
The least such $k$ is the *index* of $T$.

## Remarks

If $\dim V = n$ is finite and $T$ is nilpotent, then $T^n = 0$.
In finite dimensions, $T$ is nilpotent if and only if its characteristic polynomial is $t^n$.
Over an algebraically closed field, this means that $0$ is its only eigenvalue.
A matrix is nilpotent when its map is nilpotent, see [[nilpotent matrix|07BSPZPSMF]].
The derivative on the space of polynomials of degree less than $n$ is nilpotent.

## Explanations

### L1 human

A nilpotent map destroys everything after a few steps.
Each application pushes vectors further towards zero, until they vanish.
The shift $e_1 \mapsto 0$, $e_2 \mapsto e_1$, $e_3 \mapsto e_2$ in $F^3$ is nilpotent of index $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0M8TGGP9JP.md" tag="0M8TGGP9JP" kind="statement">
# Approximation property of the supremum

## Statement

Let $S$ be a nonempty subset of the real numbers that is bounded above, and let $b = \sup S$ be its [[supremum|0BTMQ04556]].
For each real number $c < b$, there is an element $s$ of $S$ with $c < s \le b$.

## Remarks

Proof: if no such $s$ exists, then $c$ is an upper bound of $S$ that is less than $b$. This contradicts the definition of the supremum.
An equivalent form: for each $\varepsilon > 0$, there is $s$ in $S$ with $b - \varepsilon < s$.
For an infimum $a = \inf S$ and each $c > a$, there is $s$ in $S$ with $a \le s < c$.

## Explanations

### L1 human

You can find elements of a set as close to its supremum as you want.
If you lower the ceiling even a little, then some element of the set sticks out above it.
</document>

<document source="https://tamnd.github.io/math2/node/0M98XYVX6S.md" tag="0M98XYVX6S" kind="definition">
# Coefficient of a power series

## Definition

Let $R$ be a commutative ring and let $n$ be a natural number.
The coefficient of $X^n$ is the map $[X^n] : R[\![X]\!] \to R$ that sends a [[formal power series|072W67KFYK]] $f = \sum_{m \ge 0} a_m X^m$ to $a_n$.

## Remarks

The map $[X^n]$ is $R$-linear: $[X^n](f + g) = [X^n] f + [X^n] g$ and $[X^n](r f) = r\, [X^n] f$.
Two series are equal if and only if all their coefficients are equal.
The coefficient of a product is given by the node [[product of power series|0H6KWAYS87]].

## Explanations

### L1 human

The coefficient $[X^n] f$ is the number in front of $X^n$ in the series $f$.
For $f = 1 + 2X + 3X^2 + \dots$, the coefficient $[X^2] f$ is $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0M9MX8YW0A.md" tag="0M9MX8YW0A" kind="concept">
# Incentre

## Summary

The incentre of a triangle is the point where the three internal angle bisectors meet. It is the point inside the triangle at the same distance from all three sides, and it is the centre of the incircle.
</document>

<document source="https://tamnd.github.io/math2/node/0M9XEDF1HJ.md" tag="0M9XEDF1HJ" kind="definition">
# Step function

## Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be a function.
The function $f$ is a step function if it is constant on each open subinterval of some [[partition|069P20KYQT]] $(x_0, \dots, x_n)$ of $[a, b]$.
That is, there are real numbers $c_1, \dots, c_n$ with $f(x) = c_i$ for all $x \in (x_{i-1}, x_i)$.

## Remarks

The values of $f$ at the points $x_i$ are arbitrary.
A step function is Riemann integrable, and $\int_a^b f = \sum_{i=1}^{n} c_i (x_i - x_{i-1})$.
The floor function $\lfloor x \rfloor$ restricted to $[a, b]$ is a step function.

## Explanations

### L1 human

The graph of a step function looks like a staircase.
It is flat on each piece of a partition, and it can jump at the cut points.
</document>

<document source="https://tamnd.github.io/math2/node/0MA5B744K3.md" tag="0MA5B744K3" kind="definition">
# Field generated by adjoining elements

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $S \subseteq L$.
The field $K(S)$ is the intersection of all the [[subfields|07QY5RPFBR]] of $L$ that contain $K$ and $S$.
It is the smallest [[intermediate field|02NBKVQFRQ]] of $L/K$ that contains $S$.

## Remarks

For $S = \{a_1, \dots, a_n\}$ one writes $K(a_1, \dots, a_n)$.
The elements of $K(S)$ are the quotients $f(s_1, \dots, s_n)/g(s_1, \dots, s_n)$ with $f, g$ polynomials over $K$, $s_i \in S$ and $g(s_1, \dots, s_n) \neq 0$.
One has $K(a, b) = K(a)(b)$.
For example, $\mathbb{Q}(\sqrt{2}) = \{ p + q \sqrt{2} : p, q \in \mathbb{Q} \}$ inside $\mathbb{R}$.

## Explanations

### L1 human

To adjoin elements to a field, you take the smallest field that contains the old field and the new elements.
It contains everything you can get from them by addition, subtraction, multiplication and division.
</document>

<document source="https://tamnd.github.io/math2/node/0MB48XDBY3.md" tag="0MB48XDBY3" kind="statement">
# Schwarz lemma

## Statement

Let $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$ be the open unit disc.
Let $f : \mathbb{D} \to \mathbb{D}$ be [[holomorphic|0DG0Z72CQD]] on $\mathbb{D}$ with $f(0) = 0$.
Then $|f(z)| \le |z|$ for each $z \in \mathbb{D}$, and $|f'(0)| \le 1$.
If $|f(z_0)| = |z_0|$ for some $z_0 \neq 0$ in $\mathbb{D}$, or if $|f'(0)| = 1$, then there is $a \in \mathbb{C}$ with $|a| = 1$ such that $f(z) = a z$ for each $z \in \mathbb{D}$.

## Remarks

Proof idea: $g(z) = f(z)/z$, with $g(0) = f'(0)$, is holomorphic on $\mathbb{D}$. The [[maximum modulus principle|048F2P90WR]] on the discs $|z| \le r$ gives $|g| \le 1/r$ for each $r < 1$, so $|g| \le 1$.
It follows that the bijective holomorphic maps $\mathbb{D} \to \mathbb{D}$ with $f(0) = 0$ are the rotations $z \mapsto az$ with $|a| = 1$.
The Schwarz–Pick theorem is the version without the condition $f(0) = 0$.

## Explanations

### L1 human

A holomorphic map from the unit disc to itself that keeps $0$ fixed cannot move any point farther away from $0$.
The only maps that keep the distance to $0$ of some point are the rotations.
</document>

<document source="https://tamnd.github.io/math2/node/0MB4TGMXG6.md" tag="0MB4TGMXG6" kind="definition">
# Algebraic number

## Definition

A complex number $\alpha$ is algebraic if there is a nonzero polynomial $f$ with rational coefficients such that $f(\alpha) = 0$.
The degree of $\alpha$ is the smallest degree of such a polynomial.

## Remarks

One may use integer coefficients instead, by clearing denominators.
Examples are each rational number (degree $1$), $\sqrt{2}$ and $i$ (degree $2$), and $\sqrt[3]{2}$ (degree $3$).
The algebraic numbers form a field, and this field is countable.
A complex number that is not algebraic is [[transcendental|079EGSD32Y]].

## Explanations

### L1 human

An algebraic number is a solution of a polynomial equation with integer coefficients.
For example, $\sqrt{2}$ is algebraic because it solves $x^2 - 2 = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0MB8HQW495.md" tag="0MB8HQW495" kind="definition">
# Self-adjoint operator

## Definition

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The map $T$ is *self-adjoint* when $T^* = T$, where $T^*$ is the [[adjoint|0R9NMSM1ZP]] of $T$.
Equivalently, $\langle T(v), w \rangle = \langle v, T(w) \rangle$ for all $v, w \in V$.

## Remarks

In an orthonormal basis, $T$ is self-adjoint if and only if its matrix is [[symmetric|0VR671PHE5]] (over $\R$) or [[Hermitian|0P17NGQQMQ]] (over $\C$).
Orthogonal projections are self-adjoint.
The eigenvalues of a self-adjoint map are real, see [[the reality of the eigenvalues|0C4NT4GT3X]].
Self-adjoint maps are the subject of the [[spectral theorem|06J4PXRG22]].

## Explanations

### L1 human

A self-adjoint map can move freely from one side of the inner product to the other.
For real matrices, these are the symmetric matrices.
They model observable quantities in quantum mechanics, because their eigenvalues are real.
</document>

<document source="https://tamnd.github.io/math2/node/0MBJVDSK1R.md" tag="0MBJVDSK1R" kind="concept">
# Sum of ideals

## Summary

Let $R$ be a [[ring|0894J980RK]] and let $I$ and $J$ be left [[ideals|0MYSA6MYNA]] of $R$.
The sum of $I$ and $J$ is $I + J = \{a + b : a \in I, b \in J\}$.
It is a left ideal, and it is the smallest left ideal that contains $I$ and $J$.

## Remarks

The sum $I + J$ is the ideal [[generated by|030TZ0DQWT]] $I \cup J$.
In $\mathbb{Z}$, $m\mathbb{Z} + n\mathbb{Z} = d\mathbb{Z}$, where $d$ is the greatest common divisor of $m$ and $n$.
The ideals $I$ and $J$ are [[coprime|0MDJP82TEH]] when $I + J = R$.

## Explanations

### L1 human

To add two ideals, take all the sums of one element from each.
In the integers, the sum of the multiples of $4$ and the multiples of $6$ is the set of the multiples of $2$.
</document>

<document source="https://tamnd.github.io/math2/node/0MBV2VHMXA.md" tag="0MBV2VHMXA" kind="concept">
# Generators of a group

## Summary

A subset $S$ of a [[group|G5S1M9]] $G$ generates $G$ when the [[subgroup generated by $S$|0RP7VWQ0YN]] is all of $G$.
Then each element of $G$ is a finite product of elements of $S$ and of their inverses. The elements of $S$ are called generators of $G$.

## Remarks

The set $\{1\}$ generates $(\mathbb{Z}, +)$, and so does the set $\{2, 3\}$.
Each group is generated by the set of all its elements.
A group generated by one element is a cyclic group, and a group generated by a finite set is a finitely generated group.

## Explanations

### L1 human

Generators are building blocks for a group: you can reach each element from them with products and inverses.
A small set of generators gives a short description of a big group.
For example, all symmetries of a square come from one rotation and one reflection.
</document>

<document source="https://tamnd.github.io/math2/node/0MBWQKM8QR.md" tag="0MBWQKM8QR" kind="statement">
# Monotonicity of the integral

## Statement

Let $f, g \colon [a, b] \to \mathbb{R}$ be Riemann integrable with $f(x) \le g(x)$ for all $x \in [a, b]$.
Then
$$\int_a^b f(x) \, dx \le \int_a^b g(x) \, dx.$$

## Remarks

Each [[Riemann sum|0HBY48G8G1]] of $f$ is at most the Riemann sum of $g$ for the same tagged partition.
In particular, if $m \le f \le M$ on $[a, b]$, then $m (b - a) \le \int_a^b f \le M (b - a)$.
Mathlib states it as `intervalIntegral.integral_mono_on`.

## Explanations

### L1 human

A larger function has a larger integral.
If one graph lies below another, the area under it is smaller.
</document>

<document source="https://tamnd.github.io/math2/node/0MC0AQZSHE.md" tag="0MC0AQZSHE" kind="statement">
# A finite abelian group is cyclic iff its exponent equals its order

## Statement

Let $G$ be a finite [[abelian group|082GHNPJ53]].
Then $G$ is [[cyclic|0KVGH0J5B3]] if and only if the [[exponent|0ZDW0H0VX4]] of $G$ equals $|G|$.

## Remarks

If $G$ is cyclic with generator $g$, then the order of $g$ is $|G|$, so the exponent is $|G|$.
For the converse, write $G \cong \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z}$ with $1 < k_1 \mid k_2 \mid \cdots \mid k_u$, by the [[classification|07CQR9BW3R]]. The exponent is $k_u$ and the order is $k_1 \cdots k_u$, so they are equal only when $u \leq 1$.
The abelian condition is necessary: $S_3$ has exponent $6$ and order $6$, but it is not cyclic.
For example, $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if $\lambda(n) = \varphi(n)$.

## Explanations

### L1 human

In a cyclic group, one element has order equal to the size of the group.
For a finite abelian group, it is enough that the exponent equals the size.
Then the group is cyclic.
</document>

<document source="https://tamnd.github.io/math2/node/0MCDDSCB9Z.md" tag="0MCDDSCB9Z" kind="statement">
# Orthogonal projection is the closest point

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, let $W$ be a finite-dimensional subspace of $V$, and let $v \in V$.
Let $P_W$ be the [[orthogonal projection|0B8KZRT28M]] onto $W$.
Then, for all $w \in W$,
$$\norm{v - P_W(v)} \le \norm{v - w},$$
with equality only for $w = P_W(v)$.

## Remarks

The proof: $v - w = (v - P_W(v)) + (P_W(v) - w)$ is a sum of orthogonal vectors, so the Pythagorean theorem applies.
The number $\norm{v - P_W(v)}$ is the distance from $v$ to $W$.
The Hilbert projection theorem gives the same result for closed convex sets in Hilbert spaces.
This is the basis of the method of [[least squares|0BYTQ4X1HC]].

## Explanations

### L1 human

The nearest point of a subspace to a given vector is the foot of the perpendicular.
Every other point of the subspace is farther away, by the Pythagorean theorem.
This turns problems of best approximation into problems of projection.
</document>

<document source="https://tamnd.github.io/math2/node/0MCK8K2D7Z.md" tag="0MCK8K2D7Z" kind="statement">
# Epsilon characterisation of Cauchy sequences

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $(x_n)_{n \ge 1}$ be a sequence in $X$.
For each $N \ge 1$, let $T_N = \{x_n : n \ge N\}$.
Then the following two conditions are equivalent:

1. For each real $\varepsilon > 0$ there is an integer $N$ such that $d(x_m, x_n) < \varepsilon$ for all $m, n \ge N$.
2. The [[diameters|0692CGQF00]] $\operatorname{diam}(T_N)$ tend to $0$ as $N \to \infty$.

The sequence $(x_n)$ is a Cauchy sequence when it satisfies these conditions.

## Remarks

Condition 1 is the usual definition of a Cauchy sequence.
Each convergent sequence is a Cauchy sequence: if $x_n \to x$, then $d(x_m, x_n) \le d(x_m, x) + d(x, x_n)$.
In $\mathbb{Q}$ with the usual metric, the decimal approximations $1, 1.4, 1.41, 1.414, \ldots$ of $\sqrt{2}$ form a Cauchy sequence that does not converge in $\mathbb{Q}$.
Each Cauchy sequence is [[bounded|0P953MG05E]].

## Explanations

### L1 human

The terms of a Cauchy sequence get closer and closer to each other.
You can check this without knowing a limit, and this is the point. A Cauchy sequence looks as if it converges. But the space can have no point for it to converge to.
</document>

<document source="https://tamnd.github.io/math2/node/0MCR8ZAXQD.md" tag="0MCR8ZAXQD" kind="definition">
# Minimal element

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.
An element $m \in S$ is a minimal element of $S$ when no $s \in S$ satisfies $s < m$.
Equivalently, for each $s \in S$, $s \le m$ implies $s = m$.

## Remarks

It is the [[maximal element|0TZEEFTNCC]] for the reversed order $\ge$.
In $\{2, 3, 4, 6, 9\}$ under divisibility, the minimal elements are $2$ and $3$.
A [[well-founded relation|0230ECX1Q5]] is one in which each nonempty subset has a minimal element.

## Explanations

### L1 human

A minimal element has nothing below it.
Among the integers greater than $1$ under divisibility, the minimal elements are the primes.
</document>

<document source="https://tamnd.github.io/math2/node/0MCXX29HWY.md" tag="0MCXX29HWY" kind="definition">
# Algebraic closure

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
An algebraic closure of $K$ is a [[field extension|0WQ1A8SHTZ]] $\overline{K}/K$ such that $\overline{K}/K$ is [[algebraic|09EYMKMSB9]] and $\overline{K}$ is [[algebraically closed|02SJG1HC32]].

## Remarks

The field $\mathbb{C}$ is an algebraic closure of $\mathbb{R}$, by the fundamental theorem of algebra.
The field $\overline{\mathbb{Q}}$ of [[algebraic numbers|0QFTETCHKB]] is an algebraic closure of $\mathbb{Q}$. The field $\mathbb{C}$ is not, because $\mathbb{C}/\mathbb{Q}$ is not algebraic.
If $K$ is infinite, then $\overline{K}$ has the same cardinality as $K$. If $K$ is finite, then $\overline{K}$ is countably infinite.
An algebraic closure is a largest algebraic extension of $K$, and a smallest algebraically closed field that contains $K$.

## Explanations

### L1 human

The algebraic closure adds to a field all the roots of all its polynomials, and nothing else.
For the rationals, it gives the algebraic numbers, not all the complex numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0MCYJ72VGA.md" tag="0MCYJ72VGA" kind="concept">
# Complement graph

## Summary

The complement of a graph has the same vertices and exactly the edges that the graph does not have. It swaps cliques and independent sets. The node [[complement graph|0RNMP6VCR2]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0MDJP82TEH.md" tag="0MDJP82TEH" kind="definition">
# Coprime ideals

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $I$ and $J$ be ideals of $R$.
The ideals $I$ and $J$ are coprime if $I + J = R$, where $I + J$ is the [[sum of the ideals|0MBJVDSK1R]].
Equivalently, there are $a \in I$ and $b \in J$ with $a + b = 1$.

## Remarks

In $\mathbb{Z}$, $m\mathbb{Z} + n\mathbb{Z} = \gcd(m, n)\mathbb{Z}$. So $m\mathbb{Z}$ and $n\mathbb{Z}$ are coprime if and only if $\gcd(m, n) = 1$.
If $I$ and $J$ are coprime, then $I \cap J = I J$.
Two distinct [[maximal ideals|09J6KGT65G]] are always coprime.

## Explanations

### L1 human

For integers, $3$ and $5$ are coprime, and $2 \cdot 3 - 1 \cdot 5 = 1$.
So $1$ is a multiple of $3$ plus a multiple of $5$.
Coprime ideals generalize this: $1$ is an element of $I$ plus an element of $J$.
</document>

<document source="https://tamnd.github.io/math2/node/0MDMZ965VE.md" tag="0MDMZ965VE" kind="concept">
# Exterior algebra

## Summary

The exterior algebra of a vector space is the associative algebra generated by the space in which the square of every vector is zero.
</document>

<document source="https://tamnd.github.io/math2/node/0ME21QN125.md" tag="0ME21QN125" kind="concept">
# Symmetric difference

## Summary

The symmetric difference of two sets is the set of the objects that are elements of exactly one of the two sets.
</document>

<document source="https://tamnd.github.io/math2/node/0MES3KS0ZC.md" tag="0MES3KS0ZC" kind="statement">
# Division algorithm for polynomials

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]], let $f \in R[x]$ and let $g \in R[x]$ be a [[monic polynomial|0BDX4MBKT4]].
Then there are unique polynomials $q, r \in R[x]$ such that $f = q g + r$ and $\deg r < \deg g$.

## Remarks

The polynomials $q$ and $r$ are the quotient and the remainder. Long division of polynomials computes them.
Over a [[field|0QA8CHH8MG]] $K$, every nonzero $g$ is a unit times a monic polynomial. So the statement holds for every nonzero $g \in K[x]$.
For example, $x^3 + 1 = (x^2 - x + 1)(x + 1) + 0$ and $x^2 + 1 = (x - 1)(x + 1) + 2$.
The statement fails for a divisor that is not monic: in $\mathbb{Z}[x]$, no $q$ and $r$ with $\deg r < 1$ give $x = q \cdot 2x + r$.

## Explanations

### L1 human

This is the polynomial version of division with remainder for integers.
You can divide by a monic polynomial and get a remainder of smaller degree.
The degree plays the role that the size of a number plays for integers.
</document>

<document source="https://tamnd.github.io/math2/node/0MFJJV97RJ.md" tag="0MFJJV97RJ" kind="concept">
# Norm induced by an inner product

## Summary

The norm induced by an inner product is the square root of the inner product of a vector with itself.
</document>

<document source="https://tamnd.github.io/math2/node/0MH1WK89PP.md" tag="0MH1WK89PP" kind="statement">
# Derivative of a polynomial

## Statement

Let $p(x) = \sum_{k=0}^{n} a_k x^k$ with real coefficients $a_k$.
Then $p$ is differentiable on $\mathbb{R}$, and
$$p'(x) = \sum_{k=1}^{n} k a_k x^{k-1}.$$

## Remarks

It follows from the [[power rule|0JZGW75C5Z]] and the [[linearity of differentiation|0M0TKFRX2A]].
So the derivative of a polynomial of degree $n \ge 1$ is a polynomial of degree $n - 1$.
Mathlib states it as `Polynomial.hasDerivAt`.

## Explanations

### L1 human

We differentiate a polynomial term by term with the power rule.
For example, the derivative of $3 x^2 + 5 x + 7$ is $6 x + 5$.
</document>

<document source="https://tamnd.github.io/math2/node/0MHE4B1QCQ.md" tag="0MHE4B1QCQ" kind="statement">
# Wronskian test for linear independence of solutions

## Statement

Let $I$ be an open interval, let $a_0, \dots, a_{n-1} : I \to \mathbb{R}$ be continuous, and let $y_1, \dots, y_n$ be solutions on $I$ of the [[homogeneous linear equation|0EQZSAB9EC]] $y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y = 0$.
Let $W = W(y_1, \dots, y_n)$ be their [[Wronskian|0N5NF1RGBP]].
Then these three conditions are equivalent.
(a) The functions $y_1, \dots, y_n$ are linearly independent on $I$.
(b) There is a point $x_0 \in I$ with $W(x_0) \ne 0$.
(c) For each $x \in I$, $W(x) \ne 0$.

## Remarks

So the Wronskian of $n$ solutions is either zero at each point of $I$ or zero at no point of $I$. [[Abel's identity|0VGWKX2JHB]] gives the same result for $n = 2$.
For arbitrary functions, $W(x_0) \ne 0$ at one point still implies linear independence. But the converse fails: $x^2$ and $x |x|$ are linearly independent on $\mathbb{R}$, and their Wronskian is $0$ at each point.
Example: $e^{x}$ and $e^{-x}$ solve $y'' - y = 0$, and their Wronskian is $-2$, so they are independent.

## Explanations

### L1 human

To check that $n$ solutions are independent, compute one determinant at one point.
If it is not zero, the solutions are independent and form a basis of all solutions.
For solutions of a linear equation, the value at one point already decides the question.
</document>

<document source="https://tamnd.github.io/math2/node/0MHGJRZMTM.md" tag="0MHGJRZMTM" kind="statement">
# Every sequence has a monotone subsequence

## Statement

Let $(a_n)$ be a real sequence.
Then $(a_n)$ has a [[subsequence|0R8J3ZQMJS]] that is [[monotone|0KN22AAW5R]].

## Remarks

The usual proof uses peaks. An index $n$ is a peak if $a_m \le a_n$ for all $m > n$.
If there are infinitely many peaks, the terms at the peaks form a decreasing subsequence.
If there are finitely many peaks, start after the last one. Each index then has a later index with a larger term, and this gives a strictly increasing subsequence.
With the [[monotone convergence theorem|0FM6N4R0VB]], the lemma gives the [[Bolzano–Weierstrass theorem|0KDJD249DT]].

## Explanations

### L1 human

Think of the terms as heights of hills seen from left to right.
Either infinitely many hills have no higher hill to their right, or from some point on you can always find a higher hill further right.
</document>

<document source="https://tamnd.github.io/math2/node/0MJ04A50QW.md" tag="0MJ04A50QW" kind="concept">
# Matching

## Summary

A matching in a graph is a set of edges without common ends. Matchings model pairing and assignment problems. The node [[matching|0Z87BHAFTB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0MJ0JS72DZ.md" tag="0MJ0JS72DZ" kind="definition">
# Tree

## Definition

A tree is a [[graph|0JD5C82QMB]] that is [[connected|0FFAYNMAJZ]] and [[acyclic|0XNN9Z8E5A]].

## Remarks

Let $T$ be a finite graph with $n \ge 1$ vertices. Then these conditions are equivalent: $T$ is a tree; $T$ is connected with $n - 1$ edges; $T$ is acyclic with $n - 1$ edges.
A graph is a tree if and only if any two vertices are joined by exactly one path. The node [[unique paths in trees|096B7Q5742]] states this.
The [[path graphs|075WR6W3Y3]] and the stars $K_{1,n}$ are trees.

## Explanations

### L1 human

A tree is a connected graph with no loops.
A family tree and the folders on a computer are pictures of trees.
</document>

<document source="https://tamnd.github.io/math2/node/0MKABWDRNC.md" tag="0MKABWDRNC" kind="definition">
# Upper triangular matrix

## Definition

Let $R$ be a ring and let $n \ge 0$ be an integer.
A [[square matrix|0JXZKX526R]] $U = (u_{ij})$ of order $n$ over $R$ is *upper triangular* when $u_{ij} = 0$ for all $i > j$.
It is *strictly upper triangular* when $u_{ij} = 0$ for all $i \ge j$.

## Remarks

Sums and products of upper triangular matrices of the same order are upper triangular.
The [[determinant of a triangular matrix|0K1KKMFHYH]] is the product of its diagonal entries.
The diagonal entries of an upper triangular matrix over a field are its [[eigenvalues|0BQAA2BK3Q]].
A system $Ux = b$ with $U$ upper triangular and invertible is solved by [[back substitution|0ATFVE08MW]].

## Explanations

### L1 human

An upper triangular matrix has zeros everywhere below its diagonal.
Its nonzero entries fill a triangle at the top right.
For example, $\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ is upper triangular.
</document>

<document source="https://tamnd.github.io/math2/node/0MKRHZB3WS.md" tag="0MKRHZB3WS" kind="concept">
# Change of basis matrix

## Summary

The change of basis matrix from one basis to another is a square matrix.
Its columns are the coordinate vectors of the new basis vectors in the old basis.
It converts new coordinates into old coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/0MM085RX00.md" tag="0MM085RX00" kind="definition">
# Product of vector spaces

## Definition

Let $F$ be a field and let $V$ and $W$ be [[vector spaces|000AYPA9AJ]] over $F$.
The *product* $V \times W$ is the set of pairs $(v, w)$ with $v \in V$ and $w \in W$, with the operations
$$(v, w) + (v', w') = (v + v', w + w'), \qquad a(v, w) = (av, aw)$$
for $v, v' \in V$, $w, w' \in W$ and $a \in F$.
It is a vector space over $F$ with zero vector $(0, 0)$.

## Remarks

For finitely many spaces, the product is also the [[external direct sum|0S7W6EG32V]], written $V \oplus W$.
The [[dimension of a direct product|06HN47VPTH]] is $\dim V + \dim W$.
For example, $F^m \times F^n$ is isomorphic to $F^{m+n}$.
The projections $(v, w) \mapsto v$ and $(v, w) \mapsto w$ are linear maps.

## Explanations

### L1 human

The product of two vector spaces has vectors that are pairs: one vector from each space.
We add pairs part by part and scale both parts.
The plane $\R^2$ is the product of two copies of the line $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/0MM45BT6HC.md" tag="0MM45BT6HC" kind="definition">
# One-point compactification

## Definition

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]] and let $\infty$ be a point not in $X$. Let $X^* = X \cup \{\infty\}$.
The one-point compactification, or Alexandroff extension, of $X$ is the set $X^*$ with the following topology.
The open sets are the sets $U \in \tau$ and the sets $X^* \setminus C$, where $C \subseteq X$ is [[closed|0G5AV9Q4RR]] and [[compact|00R0X1WGPX]] in $X$.

## Remarks

The space $X^*$ is compact, and $X$ is an open subspace of $X^*$.
The space $X^*$ is Hausdorff if and only if $X$ is [[locally compact|0EQMWQ4WMJ]] and [[Hausdorff|0MXB4CKHFW]].
The one-point compactification of $\mathbb{R}^n$ is homeomorphic to the sphere $S^n$, by stereographic projection. For $n = 1$, it is the circle.

## Explanations

### L1 human

You make a non-compact space compact by adding one point, and you declare that the new point is near everything outside large compact sets.
The real line becomes a circle: both ends meet at the new point.
</document>

<document source="https://tamnd.github.io/math2/node/0MMCB8RXGA.md" tag="0MMCB8RXGA" kind="statement">
# A ring homomorphism is injective iff its kernel is zero

## Statement

Let $f \colon R \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]].
Then $f$ is injective if and only if its [[kernel|0AAHE7GEW6]] is $\{0\}$.

## Remarks

The proof uses only the addition: $f(a) = f(b)$ holds if and only if $f(a - b) = 0$, that is, $a - b \in \ker f$.
The same criterion holds for group homomorphisms and for linear maps.

## Explanations

### L1 human

If two elements have the same image, then their difference goes to $0$.
So when only $0$ goes to $0$, no two different elements have the same image.
</document>

<document source="https://tamnd.github.io/math2/node/0MN2MER9WN.md" tag="0MN2MER9WN" kind="statement">
# Continuum hypothesis

## Statement

There is no [[set|0SJ3RX9TXA]] $S$ with $\aleph_0 < |S| < 2^{\aleph_0}$.
Equivalently, with the axiom of choice, $2^{\aleph_0} = \aleph_1$.

## Remarks

In words: each infinite subset of $\mathbb{R}$ is countable or equinumerous with $\mathbb{R}$.
The hypothesis is independent of ZFC: if ZFC is consistent, then ZFC proves neither the hypothesis nor its negation.
The generalized continuum hypothesis says that $2^{\aleph_\alpha} = \aleph_{\alpha + 1}$ for each ordinal $\alpha$.

## History

Georg Cantor proposed the hypothesis in 1878.
It was the first of the problems that David Hilbert presented in 1900.
Kurt Gödel showed in 1940 that ZFC does not refute it, and Paul Cohen showed in 1963 that ZFC does not prove it.

## Explanations

### L1 human

Cantor showed that $\mathbb{R}$ is larger than $\mathbb{N}$.
The continuum hypothesis says that no size lies strictly between them.
The usual axioms of set theory cannot decide whether this is true.
</document>

<document source="https://tamnd.github.io/math2/node/0MN882VJ4P.md" tag="0MN882VJ4P" kind="statement">
# Derivative as a little-o approximation

## Statement

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$, let $U \subseteq V$ be open, and let $f \colon U \to W$. Let $x \in U$, and let $A \colon V \to W$ be a bounded linear map.
Then $A$ is the [[Fréchet derivative|08EVNYZF42]] of $f$ at $x$ if and only if
$$f(x + h) = f(x) + A h + r(h), \qquad \text{where } r(h) = o(\|h\|) \text{ as } h \to 0.$$
Here $r(h) = o(\|h\|)$ means that for each $\varepsilon > 0$ there is a $\delta > 0$ such that $\|r(h)\| \le \varepsilon \|h\|$ for each $h$ with $\|h\| < \delta$ and $x + h \in U$.

## Remarks

The remainder $r(h)$ is defined by the equation, so the statement says that the error of the linear approximation tends to $0$ faster than $\|h\|$.
The little-o form includes $h = 0$, where $r(0) = 0$. This is why it uses $\le$ and does not divide by $\|h\|$.
This form is often the most convenient one for proofs, for example of the [[chain rule|0EAAS2EHDR]].

## Explanations

### L1 human

The equation splits $f(x + h)$ into three parts: the value at $x$, a linear part in $h$, and an error.
The derivative condition says only that the error is tiny compared with the step: halving the step more than halves the error.
</document>

<document source="https://tamnd.github.io/math2/node/0MNF0BYFQC.md" tag="0MNF0BYFQC" kind="concept">
# Algebraic number

## Summary

An algebraic number is a complex number that is a root of a nonzero polynomial with rational coefficients.
The [[definition of an algebraic number|0MB4TGMXG6]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0MNHXBSX3M.md" tag="0MNHXBSX3M" kind="definition">
# Prime element

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
An element $p \in R$ is prime if it has these three properties.

1. $p \neq 0$.
2. $p$ is not a [[unit|019NFGFPR5]].
3. For all $a, b \in R$, if $p \mid a b$, then $p \mid a$ or $p \mid b$.

## Remarks

A nonzero $p$ is prime if and only if the principal ideal $(p)$ is a [[prime ideal|06HC6HK0Z8]].
The prime elements of $\mathbb{Z}$ are the numbers $p$ and $-p$ with $p$ a prime number.
In an integral domain, [[prime elements are irreducible|0F0K8DB9TD]].

## Explanations

### L1 human

Euclid's lemma says that a prime number dividing a product divides one of the factors.
A prime element of a ring is an element with this property.
In the integers, $6$ is not prime: $6$ divides $2 \cdot 3$, but not $2$ or $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0MNM54WB99.md" tag="0MNM54WB99" kind="definition">
# Geometric sequence

## Definition

Let $a$ and $r$ be real numbers.
The geometric sequence with initial term $a$ and common ratio $r$ is the [[real sequence|0CTV783HYT]] $(a\,r^n)_{n \in \mathbb{N}}$.
Its terms are $a, a r, a r^2, a r^3, \dots$, and each term is $r$ times the term before it.

## Remarks

Many texts require $a \ne 0$ and $r \ne 0$, so that the quotient of two consecutive terms is always $r$.
The sequence $1, 2, 4, 8, \dots$ has $a = 1$ and $r = 2$. The sequence $1, 1/2, 1/4, \dots$ has $r = 1/2$.
For $|r| < 1$, the sequence [[converges to $0$|0Q7K4AXE53]]. For $r > 1$ and $a > 0$, it [[tends to infinity|0TZK91NQE3]].
The sum of the terms is the [[geometric series|01X5CVV1Q5]].

## Explanations

### L1 human

In a geometric sequence, you go from one term to the next by multiplication with the same number $r$.
If $r = 3$ and you start at $2$, you get $2, 6, 18, 54, \dots$.
</document>

<document source="https://tamnd.github.io/math2/node/0MNZCN4BQK.md" tag="0MNZCN4BQK" kind="definition">
# Product of rings

## Definition

Let $I$ be a set and let $(R_i)_{i \in I}$ be a family of [[rings|0894J980RK]].
The product $\prod_{i \in I} R_i$ is the set of families $(a_i)_{i \in I}$ with $a_i \in R_i$ for each $i \in I$.
Its addition is $(a_i)_{i \in I} + (b_i)_{i \in I} = (a_i + b_i)_{i \in I}$.
Its multiplication is $(a_i)_{i \in I} \cdot (b_i)_{i \in I} = (a_i \cdot b_i)_{i \in I}$.
With these operations, $\prod_{i \in I} R_i$ is a ring, with zero $(0)_{i \in I}$ and identity $(1)_{i \in I}$.

## Remarks

For two rings $R$ and $S$, the product is $R \times S = \{(r, s) : r \in R, s \in S\}$.
For each $j \in I$, the projection $\prod_{i \in I} R_i \to R_j$, $(a_i)_{i \in I} \mapsto a_j$, is a [[ring homomorphism|0GGE0Z0JQ1]].
The product of two nonzero rings is not an [[integral domain|09S3PFBEPM]]: in $R \times S$, $(1, 0) \cdot (0, 1) = (0, 0)$.

## Explanations

### L1 human

To multiply two pairs, you multiply the first entries and the second entries apart.
For example, in $\mathbb{Z} \times \mathbb{Z}$, $(2, 3) \cdot (4, 5) = (8, 15)$.
</document>

<document source="https://tamnd.github.io/math2/node/0MP9BCF6DP.md" tag="0MP9BCF6DP" kind="concept">
# Line

## Summary

A line is a straight figure of dimension one that extends without end in both directions. In a Euclidean space, through each two distinct points there is exactly one line.
</document>

<document source="https://tamnd.github.io/math2/node/0MPAJRB23T.md" tag="0MPAJRB23T" kind="statement">
# Diagonalisable iff the minimal polynomial has distinct roots

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$ with [[minimal polynomial|03NDVCSHSB]] $m_A$.
Then $A$ is [[diagonalisable|07SQCNV233]] over $F$ if and only if
$$m_A(t) = (t - \lambda_1)(t - \lambda_2) \cdots (t - \lambda_k)$$
for some pairwise distinct $\lambda_1, \dots, \lambda_k \in F$.

## Remarks

In words: $m_A$ splits over $F$ and has no repeated roots.
If $A^2 = A$, then $m_A$ divides $t^2 - t = t(t - 1)$, so each [[projection|033DA1WE8Y]] matrix is diagonalisable.
The matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ has $m_A = (t - 1)^2$, so it is not diagonalisable.

## Explanations

### L1 human

The minimal polynomial decides whether a matrix is diagonalisable.
The matrix is diagonalisable exactly when this polynomial is a product of distinct linear factors.
A repeated factor, like $(t - 1)^2$, signals a Jordan block of size at least $2$.
</document>

<document source="https://tamnd.github.io/math2/node/0MPZGAZ2GK.md" tag="0MPZGAZ2GK" kind="definition">
# Centre of mass

## Definition

Let $E \subseteq \mathbb{R}^n$ be a bounded Jordan measurable set, and let $\delta \colon E \to [0, \infty)$ be a continuous density with total mass $M = \int_E \delta(x) \, dx > 0$.
The centre of mass of $E$ with density $\delta$ is the point $\bar{x} = (\bar{x}_1, \dots, \bar{x}_n)$ with
$$\bar{x}_i = \frac{1}{M} \int_E x_i \, \delta(x) \, dx, \qquad i = 1, \dots, n.$$
If $\delta$ is constant, $\bar{x}$ is the centroid of $E$, and it does not depend on the value of the constant.

## Remarks

For $n = 3$ the integrals are [[triple integrals|0GTBGMZXV8]], and for a plane lamina, $n = 2$, they are [[double integrals|0NKBMFC18R]].
For point masses $m_1, \dots, m_k$ at $p_1, \dots, p_k$, the centre of mass is $\sum_j m_j p_j / \sum_j m_j$. The definition above replaces the sum by an integral.
Example: the half disc $\{x^2 + y^2 \le a^2, \ y \ge 0\}$ with constant density has centroid $(0, 4a / (3\pi))$.
If $E$ and $\delta$ are symmetric under a reflection, then $\bar{x}$ lies in the mirror of that reflection.

## Explanations

### L1 human

The centre of mass is the balance point of a body.
If the body is supported at that point, gravity does not turn it in any direction.
</document>

<document source="https://tamnd.github.io/math2/node/0MQ24RBG93.md" tag="0MQ24RBG93" kind="definition">
# Fundamental matrix

## Definition

Let $I$ be an open interval and let $A : I \to \mathbb{R}^{n \times n}$ be continuous.
A fundamental matrix of the homogeneous [[linear system|0A6QZ367GX]] $x' = A(t) x$ is a differentiable function $\Phi : I \to \mathbb{R}^{n \times n}$ with these two properties.
For each $t \in I$, $\Phi'(t) = A(t)\, \Phi(t)$.
For each $t \in I$, the matrix $\Phi(t)$ is invertible.

## Remarks

Equivalently, the columns of $\Phi$ are $n$ solutions that form a basis of the space of solutions.
Each solution is $x(t) = \Phi(t)\, c$ for a unique $c \in \mathbb{R}^n$, and the solution with $x(t_0) = x_0$ is $\Phi(t) \Phi(t_0)^{-1} x_0$.
If $\Phi'(t) = A(t) \Phi(t)$ and $\Phi(t_0)$ is invertible at one point $t_0$, then $\Phi(t)$ is invertible at each point, by [[Liouville's formula|09PS3S0EF6]].
For a constant matrix $A$, $\Phi(t) = e^{tA}$ is a fundamental matrix. See [[the matrix exponential as the solution|0YC8JT513J]].

## Explanations

### L1 human

A fundamental matrix puts $n$ independent solutions side by side as columns.
Then one matrix product gives each solution: choose the vector $c$ of coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/0MQQQ5NA2S.md" tag="0MQQQ5NA2S" kind="statement">
# Compact Hausdorff spaces are normal

## Statement

Let $X$ be a [[compact space|0QPHW703PV]] that is [[Hausdorff|0MXB4CKHFW]].
Then $X$ is [[normal|0AJ646N339]].

## Remarks

Proof sketch: closed subsets of $X$ are compact, by [[closed subsets of a compact set are compact|066HW6D9A7]]. In a Hausdorff space, a point and a disjoint compact set have disjoint open neighbourhoods. A second use of compactness separates two disjoint compact sets.
With [[Urysohn's lemma|01EY3VR7ZD]], this shows that compact Hausdorff spaces have many continuous real functions.

## Explanations

### L1 human

In a compact Hausdorff space, you first separate points by open sets, and then compactness lets you combine finitely many of these choices.
The result separates any two disjoint closed sets.
</document>

<document source="https://tamnd.github.io/math2/node/0MQXVZEBMT.md" tag="0MQXVZEBMT" kind="statement">
# Change of basis formula

## Statement

Let $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Let $B, B'$ be bases of $V$, and let $C, C'$ be bases of $W$.
Let $P$ be the [[change of basis matrix|0MKRHZB3WS]] from $B$ to $B'$, and $Q$ the change of basis matrix from $C$ to $C'$.
Then, for each linear map $T \colon V \to W$,
$$[T]_{C',B'} = Q^{-1} [T]_{C,B} P,$$
where $[T]$ denotes the [[matrix of $T$|0YQSGNYE3K]] in the given bases.
In particular, for an endomorphism $T$ of $V$ and $C = B$, $C' = B'$, one gets $[T]_{B'} = P^{-1} [T]_B P$.

## Remarks

The formula follows from [[the matrix of a composition|05Q1B7MH0M]], applied to $T = \id_W \circ T \circ \id_V$.
For endomorphisms, the two matrices are [[similar|0E89HQATM8]].
For linear maps between two spaces, the two matrices are [[equivalent|0CMHSGTWXB]].

## Explanations

### L1 human

When we change bases, the matrix of a linear map changes in a fixed way.
We translate the input from the new to the old coordinates, apply the old matrix, and translate the result back.
This gives the product $Q^{-1} A P$.
</document>

<document source="https://tamnd.github.io/math2/node/0MR0BYF0K2.md" tag="0MR0BYF0K2" kind="concept">
# Normal matrix

## Summary

A normal matrix is a complex square matrix that commutes with its conjugate transpose.
</document>

<document source="https://tamnd.github.io/math2/node/0MRE6XRE5V.md" tag="0MRE6XRE5V" kind="definition">
# Series of functions

## Definition

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ for $n \ge 0$.
The series of functions $\sum_{n=0}^{\infty} f_n$ is the sequence of partial sums $s_N = \sum_{n=0}^{N} f_n$.
The series converges pointwise, or uniformly, on $S$ if the sequence $(s_N)$ converges [[pointwise|056FSKG32J]], or [[uniformly|08FPABDQ4C]], on $S$.
The limit is the sum $\sum_{n=0}^{\infty} f_n(x)$.

## Remarks

Power series and Fourier series are series of functions.
The [[Weierstrass M-test|0NX5NWAFC9]] is the main tool to prove uniform convergence.
By the [[uniform limit theorem|0X9XT05JVC]], a uniformly convergent series of continuous functions has a continuous sum.

## Explanations

### L1 human

A series of functions adds infinitely many functions, point by point.
At each $x$ it is an ordinary series of numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0MS8QGA6S5.md" tag="0MS8QGA6S5" kind="statement">
# A quotient is abelian iff the subgroup contains the commutator subgroup

## Statement

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
Then the [[quotient group|0KYX2N69J0]] $G/N$ is [[abelian|082GHNPJ53]] if and only if $[G, G] \subseteq N$, where $[G, G]$ is the [[commutator subgroup|0ASGSTXSBW]].

## Remarks

The cosets $gN$ and $hN$ commute exactly when $g^{-1} h^{-1} g h \in N$.
Thus $[G, G]$ is the smallest normal subgroup of $G$ with an abelian quotient.
A subgroup that contains $[G, G]$ is automatically normal.

## Explanations

### L1 human

To make a group abelian by a quotient, you must send each commutator to the identity.
So the subgroup that you divide by must contain all commutators.
If it does, the quotient is abelian.
</document>

<document source="https://tamnd.github.io/math2/node/0MSCQNFTM8.md" tag="0MSCQNFTM8" kind="statement">
# Test for exactness

## Statement

Let $R = (\alpha, \beta) \times (\gamma, \delta)$ be an open rectangle in $\mathbb{R}^2$.
Let $M, N : R \to \mathbb{R}$ be continuous functions such that the partial derivatives $\partial M / \partial y$ and $\partial N / \partial x$ exist and are continuous on $R$.
Then the equation $M\, dx + N\, dy = 0$ is [[exact|0EC7AGJ0PX]] on $R$ if and only if
$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \quad \text{on } R.$$

## Remarks

The condition is necessary on each open set: if $\psi_x = M$ and $\psi_y = N$, the mixed partial derivatives of $\psi$ are equal, so $M_y = N_x$.
For the converse, fix $(x_0, y_0) \in R$ and set $\psi(x, y) = \int_{x_0}^{x} M(s, y_0)\, ds + \int_{y_0}^{y} N(x, t)\, dt$.
The converse holds on each simply connected open set. It can fail on other sets: $(-y\, dx + x\, dy)/(x^2 + y^2) = 0$ satisfies the condition on $\mathbb{R}^2 \setminus \{0\}$, but it is not exact there.

## Explanations

### L1 human

If $\psi$ exists, then $M$ and $N$ are its two partial derivatives.
Their cross derivatives must agree, because the order of differentiation does not matter for nice functions.
On a rectangle, this easy test is also enough.
</document>

<document source="https://tamnd.github.io/math2/node/0MSRQHCNN9.md" tag="0MSRQHCNN9" kind="concept">
# Existential quantifier

## Summary

The existential quantifier $\exists$ expresses "there exists".
An existentially quantified statement says that a [[predicate|0T3DPBG981]] holds for at least one element of its domain.
</document>

<document source="https://tamnd.github.io/math2/node/0MSSD1VCW9.md" tag="0MSSD1VCW9" kind="concept">
# Order on the reals

## Summary

The order on the reals is the relation $\le$ on $\mathbb{R}$ that compares two real numbers by their position on the number line.
It extends the order of $\mathbb{Q}$ and makes $\mathbb{R}$ an [[ordered field|010JCV9J4E]].
</document>

<document source="https://tamnd.github.io/math2/node/0MSWYWHYF5.md" tag="0MSWYWHYF5" kind="definition">
# Alternating multilinear map

## Definition

Let $F$ be a field, let $n \ge 1$, and let $V$ and $W$ be vector spaces over $F$.
A [[multilinear map|0443XNARDV]] $f \colon V^n \to W$ is *alternating* when $f(v_1, \dots, v_n) = 0$ for each family with $v_i = v_j$ for some $i \ne j$.

## Remarks

An alternating map changes sign when two arguments are exchanged.
If the characteristic of $F$ is not $2$, then this sign rule implies that the map is alternating.
An alternating map is zero on each linearly dependent family.
The [[determinant|00KDZRW4C0]] is an alternating multilinear map of the rows of a square matrix.

## Explanations

### L1 human

An alternating map gives zero as soon as two inputs are the same.
It follows that swapping two inputs changes the sign of the value.
The signed area of the parallelogram of two plane vectors is an alternating bilinear map.
</document>

<document source="https://tamnd.github.io/math2/node/0MT2S0ZX4Y.md" tag="0MT2S0ZX4Y" kind="definition">
# Contraction mapping

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]].
A map $f \colon X \to X$ is a contraction when there is a real number $k$ with $0 \le k < 1$ such that
$$d\bigl(f(x), f(y)\bigr) \le k \, d(x, y) \quad \text{for all } x, y \in X.$$

## Remarks

A contraction is a [[Lipschitz map|0TMZ21WKCN]] with a constant less than $1$. Thus it is uniformly continuous.
The map $x \mapsto x/2$ on $\mathbb{R}$ is a contraction with $k = 1/2$.
The condition $d(f(x), f(y)) < d(x, y)$ for all $x \neq y$ is weaker. The map $x \mapsto x + 1/x$ on $[1, \infty)$ satisfies it, but it is not a contraction and has no fixed point.

## Explanations

### L1 human

A contraction pulls the points of the space closer together, each time by at least the same factor.
If you apply it again and again, all points are pulled toward one point.
</document>

<document source="https://tamnd.github.io/math2/node/0MTBGVZ4XJ.md" tag="0MTBGVZ4XJ" kind="statement">
# First isomorphism theorem for linear maps

## Statement

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a linear map.
Then the map
$$\bar T \colon V / \ker T \to \im T, \qquad v + \ker T \mapsto T(v),$$
is well defined, and it is a linear isomorphism.
Here $V / \ker T$ is the [[quotient space|0BR3FY5Q8M]] by the [[kernel|0S23E6YE83]], and $\im T$ is the [[image|0WSFKKT6FF]] of $T$.

## Remarks

In finite dimensions, it gives the [[rank–nullity theorem|0W4FSW1TVB]]: $\dim V - \dim \ker T = \dim \im T$.
The same theorem holds for groups, rings and modules.
For a surjective map $T$, it says $V / \ker T \cong W$.

## Explanations

### L1 human

A linear map identifies vectors that differ by an element of the kernel.
After we glue these vectors together, the map becomes one to one.
So the quotient by the kernel is a copy of the image.
</document>

<document source="https://tamnd.github.io/math2/node/0MTENFBW3Q.md" tag="0MTENFBW3Q" kind="definition">
# One-form

## Definition

Let $U \subseteq \mathbb{R}^n$ be open.
A one-form on $U$ is a [[differential form|0QHTX7WQEY]] of degree $1$, that is, a sum $\omega = f_1 \, dx_1 + \dots + f_n \, dx_n$ with functions $f_i \colon U \to \mathbb{R}$.
At $x \in U$, it gives the linear map $\omega_x(v) = f_1(x) v_1 + \dots + f_n(x) v_n$ for $v \in \mathbb{R}^n$.

## Remarks

For a differentiable function $g$ on $U$, the differential of $g$ is the one-form $dg = \sum_i \frac{\partial g}{\partial x_i} \, dx_i$. So $dg_x(v)$ is the directional derivative of $g$ at $x$ in direction $v$.
For a continuous $\omega$ and a piecewise $C^1$ curve $\gamma \colon [a, b] \to U$, the integral is $\int_\gamma \omega = \int_a^b \omega_{\gamma(t)}\big( \gamma'(t) \big) \, dt$. This is the [[line integral|0YS8RX87F4]] of the [[vector field|0N67NBKQ5T]] $(f_1, \dots, f_n)$.
Under this correspondence, $dg$ corresponds to $\nabla g$, and $\omega$ is of the form $dg$ exactly when the field is [[conservative|05R9NF2J49]].
On $\mathbb{R}^2 \setminus \{0\}$, the angle form $d\theta = \frac{-y \, dx + x \, dy}{x^2 + y^2}$ is a one-form. Its integral around a closed curve is $2\pi$ times the winding number of the curve about the origin.

## Explanations

### L1 human

A one-form is a rule that, at each point, turns a direction vector into a number, in a linear way.
Integrating a one-form along a curve adds up these numbers for the velocity of the curve. It is the same thing as the work of a vector field.
</document>

<document source="https://tamnd.github.io/math2/node/0MTNR28K60.md" tag="0MTNR28K60" kind="concept">
# Poisson point process

## Summary

A Poisson point process is a random set of points in which the numbers of points in disjoint regions are independent Poisson random variables. On the time axis, it models events that occur at random times with a constant rate.
The node [[Poisson point process|05TG0SE4HT]] gives the precise definition of the homogeneous process on $[0, \infty)$.
</document>

<document source="https://tamnd.github.io/math2/node/0MVCFMW7S4.md" tag="0MVCFMW7S4" kind="object">
# e

## Description

The number $e$ is the value of the [[exponential function|0RMVS7RXZX]] at $1$:
$$e = \exp(1) = \sum_{n=0}^{\infty} \frac{1}{n!} \approx 2.718281828.$$
It is also the limit $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$.
It is the base of the natural logarithm, so $\ln e = 1$.

## Remarks

The series is the [[sum of the reciprocals of the factorials|0G4D3ZKTP8]].
The number $e$ is [[irrational|0W4GQSG7B6]]. Charles Hermite proved in 1873 that it is transcendental.
Jacob Bernoulli introduced the constant in 1683, in a problem of continuous compound interest. Leonhard Euler began to use the letter $e$ for it.

## Explanations

### L1 human

The number $e$ is about $2.718$. It is the natural base for exponential growth.
An amount that grows by $100\%$ per year, compounded continuously, becomes $e$ times larger in one year.
</document>

<document source="https://tamnd.github.io/math2/node/0MVDQEAB60.md" tag="0MVDQEAB60" kind="definition">
# Sequential continuity

## Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
The function $f$ is sequentially continuous if the following holds.
For each sequence $(x_n)$ in $D$ that [[converges|0XKWSSY8P2]] to a point $a \in D$, the sequence $(f(x_n))$ converges to $f(a)$.

## Remarks

For real functions, [[sequential continuity is the same as continuity|0PAQM3H57X]].
The definition is useful to prove that a function is not continuous. It is enough to find one sequence $x_n \to a$ with $f(x_n) \not\to f(a)$.
Mathlib defines it for topological spaces, as `SeqContinuous`.

## Explanations

### L1 human

A sequentially continuous function preserves limits of sequences.
The function of the limit is the limit of the function values.
</document>

<document source="https://tamnd.github.io/math2/node/0MVH4Y1CRZ.md" tag="0MVH4Y1CRZ" kind="definition">
# Noetherian ring

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
The ring $R$ is Noetherian if the set of its ideals, ordered by inclusion, satisfies the [[ascending chain condition|0ZCAXY9563]].
That is, for every chain $I_1 \subseteq I_2 \subseteq \dots$ of ideals of $R$, there is $N$ with $I_n = I_N$ for all $n \ge N$.

## Remarks

The ring $R$ is Noetherian if and only if every ideal of $R$ is [[finitely generated|030TZ0DQWT]].
Every field and every [[principal ideal domain|0RZYDVZ475]] is Noetherian.
By [[Hilbert's basis theorem|0YKA9J3XEE]], $R[x_1, \dots, x_n]$ is Noetherian when $R$ is.
The polynomial ring in infinitely many variables $x_1, x_2, \dots$ over a field is not Noetherian: $(x_1) \subset (x_1, x_2) \subset \dots$ never stops.
For a noncommutative ring, one defines left and right Noetherian rings with left and right ideals.

## Explanations

### L1 human

In a Noetherian ring, each ideal is generated by a finite list of elements.
This makes many arguments work by induction, because a growing chain of ideals must stop.
Rings met in practice, like $\mathbb{Z}[x, y]$, are Noetherian.
</document>

<document source="https://tamnd.github.io/math2/node/0MVTCFW42K.md" tag="0MVTCFW42K" kind="statement">
# Exponentials dominate powers in sequences

## Statement

Let $k$ be a natural number and let $r$ be a real number with $r > 1$.
Then
$$\lim_{n \to \infty} \frac{n^k}{r^n} = 0.$$

## Remarks

For $k = 1$ and $r = 2$, it says $n / 2^n \to 0$. The terms are $1/2, 2/4, 3/8, 4/16, \dots$.
One proof uses the ratio of consecutive terms. The ratio $\frac{(n+1)^k}{n^k} \cdot \frac{1}{r}$ tends to $1/r < 1$, so the terms eventually shrink by a fixed factor.
In the language of [[little o notation|04TSHB9H8D]], it says $n^k = o(r^n)$.
The node [[exponential growth beats polynomial growth|0CT6CBGTYY]] gives the version for real variables.

## Explanations

### L1 human

A power such as $n^{10}$ grows fast, but an exponential such as $1.01^n$ grows faster in the end.
For small $n$ the power can be much larger. For large $n$ the exponential always wins.
</document>

<document source="https://tamnd.github.io/math2/node/0MW2WXTHMC.md" tag="0MW2WXTHMC" kind="definition">
# Equicontinuity

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]] and let $\mathcal{F}$ be a set of functions from $X$ to $Y$.
The family $\mathcal{F}$ is equicontinuous at a point $a \in X$ when the property that follows holds.
For each real $\varepsilon > 0$ there is a real $\delta > 0$ such that $d_Y(f(x), f(a)) < \varepsilon$ for all $f \in \mathcal{F}$ and all $x \in X$ with $d_X(x, a) < \delta$.
The family $\mathcal{F}$ is equicontinuous when it is equicontinuous at each point of $X$.

## Remarks

The number $\delta$ depends on $\varepsilon$ and $a$, but not on $f$.
Each finite set of continuous functions is equicontinuous.
The family of all functions $f \colon \mathbb{R} \to \mathbb{R}$ that are [[Lipschitz|0TMZ21WKCN]] with constant $1$ is equicontinuous: take $\delta = \varepsilon$.
The family $f_n(x) = x^n$ on $[0, 1]$ is not equicontinuous at $x = 1$.

## Explanations

### L1 human

Each continuous function has its own $\delta$ for each $\varepsilon$. An equicontinuous family has one $\delta$ that works for all functions in the family.
The functions $x^n$ become steeper and steeper near $1$, so no common $\delta$ exists there.
</document>

<document source="https://tamnd.github.io/math2/node/0MWKJ44V5A.md" tag="0MWKJ44V5A" kind="object">
# Natural numbers

## Description

The natural numbers are the numbers $0, 1, 2, 3, \dots$ that count the elements of a finite set.
We write $\mathbb{N} = \{0, 1, 2, 3, \dots\}$ for the set of natural numbers.
Each natural number $n$ has a next natural number $n + 1$, its successor.
The [[Peano axioms|0QFT7Z76N5]] describe $\mathbb{N}$ with the number $0$ and the [[successor function|0AK9TGSNXY]].
The [[addition|0S13V32CPB]], the [[multiplication|06J5DE801P]] and the [[order|0HNATXFEV5]] of $\mathbb{N}$ come from $0$ and the successor function.

## Remarks

Some authors start the natural numbers at $1$. They then write $\mathbb{N}_0$ or $\mathbb{Z}_{\ge 0}$ for the set $\{0, 1, 2, \dots\}$.
Every nonempty set of natural numbers has a least element. This is the [[well-ordering principle|01BX15HSRR]].
The [[integers|0C4JS9RF5M]] contain the natural numbers as the integers that are not negative.

## Explanations

### L1 human

The natural numbers are the numbers that you use to count: zero, one, two, three, and so on.
There is no largest natural number, because you can always add one more.
All the other number systems of this course, such as the integers and the real numbers, start from the natural numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0MWV6ZTK6S.md" tag="0MWV6ZTK6S" kind="concept">
# Divergence

## Summary

The divergence of a differentiable [[vector field|0299VCW1TH]] is a scalar function. At each point, it measures the net rate of outward flow of the field per unit volume.
In Cartesian coordinates it is the sum of the [[partial derivatives|09EGC6F8PT]] $\partial F_i / \partial x_i$ of the components.
</document>

<document source="https://tamnd.github.io/math2/node/0MWYQTZ07Z.md" tag="0MWYQTZ07Z" kind="definition">
# Hyperbolic tangent

## Definition

The hyperbolic tangent is the function $\tanh \colon \mathbb{R} \to \mathbb{R}$ defined by
$$\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}.$$

## Remarks

The quotient is defined everywhere, since $\cosh x \ge 1$.
It is odd and strictly increasing, with range $(-1, 1)$. Also $\tanh x \to \pm 1$ as $x \to \pm \infty$.
Mathlib defines it as `Real.tanh`.

## Explanations

### L1 human

The hyperbolic tangent is an S-shaped curve between $-1$ and $1$.
It is close to $x$ near $0$ and close to $\pm 1$ far away.
</document>

<document source="https://tamnd.github.io/math2/node/0MX1HZBBXJ.md" tag="0MX1HZBBXJ" kind="statement">
# A series of nonnegative terms converges iff its partial sums are bounded

## Statement

Let $(a_n)$ be a real sequence with $a_n \ge 0$ for all $n$.
Then the series $\sum a_n$ converges if and only if its [[partial sums|0NB643K60Q]] $s_N = \sum_{n=0}^{N} a_n$ are bounded above.
In this case, $\sum_{n=0}^{\infty} a_n = \sup_N s_N$.

## Remarks

The partial sums increase, since $s_{N+1} - s_N = a_{N+1} \ge 0$. The result is then the [[monotone convergence theorem|0FM6N4R0VB]].
If the partial sums are not bounded, they tend to $+\infty$, and one writes $\sum a_n = +\infty$.
The [[comparison test|04Z5G0THN6]], the [[integral test|019QZ6RRVH]] and the [[Cauchy condensation test|0B96KMDDD1]] all rest on this fact.

## Explanations

### L1 human

When all terms are nonnegative, the running total can only go up.
So it either stays below some ceiling and converges, or it grows past every bound.
</document>

<document source="https://tamnd.github.io/math2/node/0MXB4CKHFW.md" tag="0MXB4CKHFW" kind="definition">
# Hausdorff space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is Hausdorff when for all points $x \neq y$ of $X$ there are [[open sets|01ZQQDCERP]] $U$ and $V$ with $x \in U$, $y \in V$ and $U \cap V = \varnothing$.

## Remarks

Each [[metric space is Hausdorff|0P9DCDES73]]. Subspaces and products of Hausdorff spaces are Hausdorff.
In a Hausdorff space, [[limits of sequences are unique|0D8T984XEE]] and [[compact subsets are closed|0MYFJVMN7M]].
Each Hausdorff space is [[T1|05GQJXBP3F]]. An infinite set with the [[cofinite topology|0B906ZBTMZ]] is T1 but not Hausdorff.
The Hausdorff condition is also called T2.

## Explanations

### L1 human

In a Hausdorff space, you can put any two distinct points into two open sets that do not meet.
Most spaces of analysis, such as $\mathbb{R}^n$, are Hausdorff.
</document>

<document source="https://tamnd.github.io/math2/node/0MXCP7MXWJ.md" tag="0MXCP7MXWJ" kind="problem">
# Coupon collector's problem

## Problem

There are $n \ge 1$ different types of coupons. Each draw gives a coupon of a uniform random type, independently of the other draws.
Let $T$ be the number of draws needed to collect all $n$ types. Find $E[T]$.

## Remarks

Solution: write $T = t_1 + \dots + t_n$, where $t_i$ is the number of draws to get a new type when $i - 1$ types are known.
Each $t_i$ counts trials up to the first success, with success probability $p_i = (n - i + 1)/n$. So $E[t_i] = 1/p_i$ by the [[mean of the geometric distribution|0DB0R4N5XX]].
By [[linearity of expectation|0P7PTC8DDE]], $E[T] = n \left(\frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{n}\right) = n H_n$, where $H_n$ is the $n$-th harmonic number. So $E[T] = n \ln n + \gamma n + \frac{1}{2} + o(1)$, with the Euler–Mascheroni constant $\gamma \approx 0.5772$.
The $t_i$ are independent, so $\operatorname{Var}(T) = \sum_i (1 - p_i)/p_i^2 < \pi^2 n^2 / 6$.

## Explanations

### L1 human

To collect all $n$ pictures of a series, you need about $n \ln n$ purchases, not $n$.
The last pictures are the slow ones: when only one type is missing, you wait on average $n$ draws for it. For $n = 50$, $E[T] = 50 H_{50} \approx 225$.
</document>

<document source="https://tamnd.github.io/math2/node/0MXEHSCYCS.md" tag="0MXEHSCYCS" kind="statement">
# Mantel's theorem

## Statement

Let $G$ be a finite simple [[graph|0JD5C82QMB]] with $n$ vertices and no cycle of length $3$.
Then $G$ has at most $\lfloor n^2/4 \rfloor$ edges.

## Remarks

The bound is sharp: the [[complete bipartite graph|0V0GC7BQAN]] $K_{\lfloor n/2 \rfloor, \lceil n/2 \rceil}$ has no triangle and has $\lfloor n^2/4 \rfloor$ edges.
The theorem is the case $r = 2$ of [[Turán's theorem|0JG1KZSPFZ]].

## History

Willem Mantel stated the theorem in 1907.

## Explanations

### L1 human

A graph on $n$ vertices without a triangle can have at most about a quarter of $n^2$ edges.
Split the vertices into two equal halves and join each pair across: this graph reaches the bound.
</document>

<document source="https://tamnd.github.io/math2/node/0MXFJ0J00R.md" tag="0MXFJ0J00R" kind="concept">
# Deficient number

## Summary

A deficient number is a positive integer whose positive divisors other than itself have a sum less than the number.
The [[definition of a deficient number|0G01TS1RRP]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0MXGYCH4SY.md" tag="0MXGYCH4SY" kind="concept">
# Perfect number

## Summary

A perfect number is a positive integer that is equal to the sum of its positive divisors other than itself.
The [[definition of a perfect number|0SB7Q8SYSF]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0MXMS3AE13.md" tag="0MXMS3AE13" kind="definition">
# Cumulative distribution function

## Definition

Let $X$ be a real [[random variable|0DKRR22AEQ]] on a probability space $(\Omega, \mathcal{F}, P)$.
The cumulative distribution function of $X$ is the function $F_X \colon \mathbb{R} \to [0, 1]$ with
$$F_X(x) = P(X \le x) \quad \text{for } x \in \mathbb{R}.$$
More generally, the cumulative distribution function of a [[probability distribution|0QB913WM5E]] $\mu$ on $\mathbb{R}$ is $F(x) = \mu((-\infty, x])$.

## Remarks

For $a < b$, $P(a < X \le b) = F_X(b) - F_X(a)$.
$F_X$ is nondecreasing and right-continuous, with limit $0$ at $-\infty$ and limit $1$ at $+\infty$. Conversely, each function with these properties is the cumulative distribution function of some distribution.
The jump of $F_X$ at $x$ is $P(X = x) = F_X(x) - \lim_{t \to x^-} F_X(t)$.
For a fair die, $F_X(x) = \lfloor x \rfloor / 6$ for $0 \le x < 6$, $F_X(x) = 0$ for $x < 1$, and $F_X(x) = 1$ for $x \ge 6$.

## Explanations

### L1 human

The cumulative distribution function adds up the probability from the left up to the point $x$.
Its graph starts at height $0$ on the far left, never goes down, and reaches height $1$ on the far right.
</document>

<document source="https://tamnd.github.io/math2/node/0MXN7AFBPE.md" tag="0MXN7AFBPE" kind="statement">
# Open connected subsets of R^n are path-connected

## Statement

Let $n \ge 1$ and let $U \subseteq \mathbb{R}^n$ be open and [[connected|0R37DFJKFW]], for the Euclidean topology.
Then $U$ is [[path-connected|0K0GE6BCYW]] or empty.

## Remarks

Proof: fix $x \in U$, and let $P$ be the set of points of $U$ that a path in $U$ joins to $x$. Each [[open ball|0DC3MPR7C9]] in $U$ is convex, so $P$ and $U \setminus P$ are both open. As $U$ is connected and $x \in P$, we get $P = U$.
The proof works in each locally path-connected space, for example in each open subset of a normed space.
In fact, any two points of $U$ can be joined by a path made of finitely many line segments.

## Explanations

### L1 human

For open sets in $\mathbb{R}^n$, being in one piece and being walkable are the same.
The bad behaviour of the topologist's sine curve cannot happen in an open set, because each point has a small ball around it.
</document>

<document source="https://tamnd.github.io/math2/node/0MXT6PRZZ6.md" tag="0MXT6PRZZ6" kind="concept">
# Marginal distribution

## Summary

A marginal distribution is the distribution of one variable, or of some of the variables, of a random vector, without regard to the other variables.
The node [[marginal distribution|0AXP2Z5N9H]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0MYDNRT7PP.md" tag="0MYDNRT7PP" kind="definition">
# Characteristic function

## Definition

Let $X$ be a real random variable with law $\mu_X$. Let $i$ be the imaginary unit in the [[complex numbers|09FTAET5Z4]].
The characteristic function of $X$ is the function $\varphi_X \colon \mathbb{R} \to \mathbb{C}$ with
$$\varphi_X(t) = E\left[e^{itX}\right] = E[\cos(tX)] + i \, E[\sin(tX)] = \int_{\mathbb{R}} e^{itx} \, \mu_X(dx).$$

## Remarks

The expectation exists for all $t$, because $|e^{itX}| = 1$. The second form uses [[Euler's formula|04R3CY41VJ]].
Always $\varphi_X(0) = 1$, $|\varphi_X(t)| \le 1$, and $\varphi_X$ is uniformly continuous on $\mathbb{R}$.
For $a, b \in \mathbb{R}$, $\varphi_{aX + b}(t) = e^{itb} \varphi_X(at)$.
If $E[|X|^k] < \infty$, then $\varphi_X$ is $k$ times continuously differentiable and $\varphi_X^{(k)}(0) = i^k E[X^k]$.

## Explanations

### L1 human

The characteristic function is like the moment-generating function, but with an imaginary argument.
This small change makes it finite for every distribution, so it is a tool that always works.
</document>

<document source="https://tamnd.github.io/math2/node/0MYFJVMN7M.md" tag="0MYFJVMN7M" kind="statement">
# Compact subsets of a Hausdorff space are closed

## Statement

Let $X$ be a [[Hausdorff space|0MXB4CKHFW]] and let $K$ be a [[compact subset|00R0X1WGPX]] of $X$.
Then $K$ is [[closed|0G5AV9Q4RR]] in $X$.

## Remarks

Proof: fix $y \notin K$. For each $x \in K$, choose disjoint open sets $U_x \ni x$ and $V_x \ni y$. Finitely many $U_{x_1}, \ldots, U_{x_n}$ cover $K$. Then $V_{x_1} \cap \cdots \cap V_{x_n}$ is an open set that contains $y$ and misses $K$.
The Hausdorff condition is needed. With the [[indiscrete topology|0XJ02JBD6S]] on a set with two points, each one-point set is compact but not closed.
Each metric space is Hausdorff, so compact subsets of metric spaces are closed.

## Explanations

### L1 human

In a Hausdorff space, you can separate a point outside $K$ from each point of $K$ by open sets.
Compactness lets you do this with finitely many open sets at once, which gives an open set about the point that misses $K$.
</document>

<document source="https://tamnd.github.io/math2/node/0MYSA6MYNA.md" tag="0MYSA6MYNA" kind="definition">
# Ideal

## Definition

Let $R$ be a [[ring|0894J980RK]].
A left ideal of $R$ is a subset $I \subseteq R$ with these properties.

1. $I$ is a subgroup of the additive [[group|G5S1M9]] $(R, +)$.
2. For all $r \in R$ and $a \in I$, $r a \in I$.

If $R$ is a [[commutative ring|0GYGHKX834]], a left ideal is simply called an ideal.

## Remarks

In $\mathbb{Z}$, the set $n\mathbb{Z}$ of the multiples of $n$ is an ideal for every integer $n$.
Every ring $R$ has the ideals $\{0\}$ and $R$.
An ideal contains $1$ if and only if it is all of $R$.
In a noncommutative ring, one also uses right ideals and [[two-sided ideals|0DDA6CY7GJ]].

## Explanations

### L1 human

An ideal is a set that absorbs multiplication.
The even integers are an example: an even number times any integer is even.
Ideals play the role in rings that normal subgroups play in groups.
</document>

<document source="https://tamnd.github.io/math2/node/0MZB06ST2B.md" tag="0MZB06ST2B" kind="concept">
# Power series

## Summary

A power series is an infinite sum of terms $a_n (x - c)^n$, with constant coefficients $a_n$ and center $c$. Inside its radius of convergence, it defines a function. The node [[power series|0VP56B4EMX]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0N019F6RE9.md" tag="0N019F6RE9" kind="definition">
# Separable extension

## Definition

Let $L/K$ be an [[algebraic extension|09EYMKMSB9]].
An element $a \in L$ is separable over $K$ if its [[minimal polynomial|0GK1WB9R98]] over $K$ is a [[separable polynomial|03J8BS7FMV]].
The extension $L/K$ is separable if every element of $L$ is separable over $K$.

## Remarks

An irreducible polynomial over $K$ is separable if and only if its derivative is not the zero polynomial.
So every algebraic extension of a field of characteristic $0$ is separable.
Every algebraic extension of a finite field is separable.
For a prime $p$ and the field $K = \mathbb{F}_p(t)$ of rational functions, the extension $K(t^{1/p})/K$ is not separable: the minimal polynomial $x^p - t$ equals $(x - t^{1/p})^p$.

## Explanations

### L1 human

In a separable extension, the roots of each minimal polynomial are all different.
Problems arise only in characteristic $p$, where a derivative like that of $x^p - t$ can vanish.
</document>

<document source="https://tamnd.github.io/math2/node/0N0PXCTYA7.md" tag="0N0PXCTYA7" kind="definition">
# Subgroup

## Definition

Let $G$ be a [[group|G5S1M9]] with identity element $e$.
A subgroup of $G$ is a subset $H \subseteq G$ with these three properties.
The identity $e$ is in $H$.
For all $a, b \in H$, the product $ab$ is in $H$.
For each $a \in H$, the inverse $a^{-1}$ is in $H$.
The notation $H \leq G$ means that $H$ is a subgroup of $G$.

## Remarks

A subgroup $H$ is a group with the operation of $G$ restricted to $H$, and its identity is the identity of $G$.
Each group $G$ has the subgroups $\{e\}$ and $G$.
The even integers $2\mathbb{Z}$ form a subgroup of $(\mathbb{Z}, +)$.
The natural numbers do not form a subgroup of $(\mathbb{Z}, +)$, because $-1$ is not a natural number.

## Explanations

### L1 human

A subgroup is a smaller group that lives inside a bigger group and uses the same operation.
To check that a subset is a subgroup, you check that it contains the identity and that it is closed under products and inverses.
For example, the rotations of a square form a subgroup of all the symmetries of the square.
</document>

<document source="https://tamnd.github.io/math2/node/0N109NG5ZX.md" tag="0N109NG5ZX" kind="statement">
# Well-ordering theorem

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Then each [[set|0SJ3RX9TXA]] $A$ has a [[well-order|0F1D6PK9RZ]].

## Remarks

Another name is Zermelo's theorem.
Over ZF, it is equivalent to the axiom of choice and to [[Zorn's lemma|04RXTQA5ZA]].
A well-order of $\mathbb{R}$ is hard or impossible to visualize, because its construction needs the axiom of choice.
With a well-order, each set allows proofs by [[transfinite induction|0ZHQMQ2C1N]].

## History

Ernst Zermelo proved the theorem in 1904 with the axiom of choice, which he formulated for this purpose.

## Explanations

### L1 human

The theorem says that each set has an order with a first element, a next element, and so on, possibly beyond infinity.
For $\mathbb{N}$ the usual order works.
For $\mathbb{R}$ such an order exists, but it is very hard to picture.
</document>

<document source="https://tamnd.github.io/math2/node/0N10YRNNWV.md" tag="0N10YRNNWV" kind="statement">
# Irrationality of square roots of non-squares

## Statement

Let $n$ be a natural number.
Then $\sqrt{n}$ is irrational if and only if $n$ is not a perfect square.

## Remarks

If $n = m^2$, then $\sqrt{n} = m$ is rational. The content is the other direction.
A proof: if $\sqrt{n} = a/b$, then $a^2 = n b^2$. By [[unique factorization|084HH1NDCB]], each prime has an even exponent in $a^2$ and in $b^2$, so also in $n$. Then $n$ is a square.
For example, $\sqrt{2}$, $\sqrt{3}$ and $\sqrt{12}$ are irrational.
The same argument shows that the $k$-th root of $n$ is irrational unless $n$ is a $k$-th power.

## Explanations

### L1 human

The square root of a whole number is either a whole number or an irrational number.
It is never a fraction such as $\frac{3}{2}$.
</document>

<document source="https://tamnd.github.io/math2/node/0N1JQXZXHC.md" tag="0N1JQXZXHC" kind="definition">
# Riemann integrable function

## Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be a function.
The function $f$ is Riemann integrable on $[a, b]$ if there is a real number $I$ with the following property.
For each $\varepsilon > 0$ there is $\delta > 0$ such that $|S(f, P, t) - I| < \varepsilon$ for each tagged partition $(P, t)$ of $[a, b]$ with [[mesh|04GM4NYXJB]] $\| P \| < \delta$.
Here $S(f, P, t)$ is the [[Riemann sum|0HBY48G8G1]] of $f$ for $(P, t)$.

## Remarks

The number $I$ is unique. It is the [[Riemann integral|0QM4TXT3DR]] of $f$.
A Riemann integrable function is bounded.
[[Continuous functions|06SESV7GQY]] and [[monotone functions|04YKAYDQPW]] on $[a, b]$ are Riemann integrable. The [[Dirichlet function|03Z4JC4YRJ]] is not.

## Explanations

### L1 human

A function is Riemann integrable if its Riemann sums settle down to one number as the partitions get finer.
It does not matter which tags we choose in the pieces.
</document>

<document source="https://tamnd.github.io/math2/node/0N1MY1T0W9.md" tag="0N1MY1T0W9" kind="definition">
# Matrix

## Definition

Let $R$ be a ring, for example a field, and let $m, n \ge 0$ be integers.
An $m \times n$ *matrix* over $R$ is a family $A = (a_{ij})$ of elements $a_{ij} \in R$, indexed by the pairs $(i, j)$ with $1 \le i \le m$ and $1 \le j \le n$.
The element $a_{ij}$ is the *entry* of $A$ in row $i$ and column $j$.
The notation for $A$ is a rectangular array with $m$ rows and $n$ columns:
$$A = \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}.$$
The set of all $m \times n$ matrices over $R$ is written $M_{m \times n}(R)$ or $R^{m \times n}$.

## Remarks

Two matrices are equal when they have the same size and the same entries in each position.
A $1 \times n$ matrix is a *row vector*, and an $m \times 1$ matrix is a *column vector*.
Mathlib uses arbitrary finite index types in place of $\{1, \dots, m\}$ and $\{1, \dots, n\}$.

## Explanations

### L1 human

A matrix is a table of numbers with a fixed number of rows and columns.
The matrix $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ has 2 rows and 3 columns, so it is a $2 \times 3$ matrix.
We name an entry by its row first and then its column.
</document>

<document source="https://tamnd.github.io/math2/node/0N1ZWFJDN5.md" tag="0N1ZWFJDN5" kind="statement">
# Completeness via Cauchy sequences

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]].
Then $X$ is [[complete|08EWT7J0SC]] if and only if each [[Cauchy sequence|0MCK8K2D7Z]] in $X$ has a [[convergent|06JYPJ4WHH]] subsequence.

## Remarks

The key fact: if a Cauchy sequence $(x_n)$ has a subsequence that converges to $x$, then $x_n \to x$. For $\varepsilon > 0$, choose $N$ for $\varepsilon / 2$ in the Cauchy condition and a term $x_{n_k}$ with $n_k \ge N$ and $d(x_{n_k}, x) < \varepsilon / 2$.
A consequence: a metric space in which each sequence has a convergent subsequence is complete. Thus each [[sequentially compact|0CARW59WD2]] metric space is complete.

## Explanations

### L1 human

To show that a space is complete, you need not find the limit of a whole Cauchy sequence.
It is enough to find a limit of some subsequence, and then the whole sequence follows it.
</document>

<document source="https://tamnd.github.io/math2/node/0N288P2TB5.md" tag="0N288P2TB5" kind="concept">
# Radius of convergence

## Summary

Each power series with center $c$ has a unique $R$ in $[0, \infty]$ such that it converges for $|x - c| < R$ and diverges for $|x - c| > R$. This $R$ is its radius of convergence. The node [[radius of convergence|0660QX1VPR]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0N28QN4CYA.md" tag="0N28QN4CYA" kind="definition">
# Symmetric matrix

## Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $A$ is *symmetric* when $A^T = A$, where $A^T$ is the [[transpose|09DXWYRD43]] of $A$.
Equivalently, $a_{ij} = a_{ji}$ for all $i$ and $j$.

## Remarks

The matrix $\begin{pmatrix} 1 & 7 \\ 7 & 2 \end{pmatrix}$ is symmetric.
For each $m \times n$ matrix $B$ over a commutative ring, $B^T B$ and $B B^T$ are symmetric.
A real symmetric matrix has real eigenvalues and an orthonormal basis of eigenvectors, by the [[spectral theorem|06J4PXRG22]].
Symmetric matrices represent [[symmetric bilinear forms|0AXZNE5XNH]] in a basis.

## Explanations

### L1 human

A symmetric matrix looks the same after we flip it over its main diagonal.
The entry in row $i$, column $j$ equals the entry in row $j$, column $i$.
Distance tables between cities are symmetric, because the distance from $A$ to $B$ equals the distance from $B$ to $A$.
</document>

<document source="https://tamnd.github.io/math2/node/0N2P0XRTDF.md" tag="0N2P0XRTDF" kind="statement">
# A homomorphism is injective iff its kernel is trivial

## Statement

Let $G$ and $H$ be [[groups|G5S1M9]], let $e_G$ be the identity element of $G$, and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
Then $\varphi$ is injective if and only if $\ker \varphi = \{e_G\}$.

## Remarks

If $\varphi(a) = \varphi(b)$, then $\varphi(a^{-1}b) = e_H$, so $a^{-1}b \in \ker \varphi$.
More generally, $\varphi(a) = \varphi(b)$ exactly when $a$ and $b$ lie in the same coset of $\ker \varphi$.

## Explanations

### L1 human

To check that a homomorphism is injective, you only need to check one fibre: the elements that go to the identity.
If only the identity goes to the identity, then no two elements can go to the same place.
This is much easier than comparing all pairs of elements.

### L2 agent

For a map of sets, injectivity needs a check on each pair of points. For a homomorphism, it is enough to look at one fibre, the kernel.

The proof uses the group structure to move any pair to the identity:

1. If $\varphi$ is injective, then only $e$ maps to $e$, because $\varphi(e) = e$.
2. Conversely, suppose that $\ker \varphi = \{e\}$ and $\varphi(a) = \varphi(b)$.
3. Then $\varphi(a^{-1}b) = \varphi(a)^{-1}\varphi(b) = e$.
4. Thus $a^{-1}b = e$, and $a = b$.

A worked example with $\varphi : \mathbb{Z} \to \mathbb{Z}$, $\varphi(n) = 3n$:

1. The map is a homomorphism, because $3(m + n) = 3m + 3n$.
2. If $3n = 0$, then $n = 0$. Thus the kernel is $\{0\}$.
3. By the result, $\varphi$ is injective. There is no need to compare pairs.

A non-example is reduction modulo $3$, $\mathbb{Z} \to \mathbb{Z}/3\mathbb{Z}$. Its kernel is $3\mathbb{Z}$, which is not trivial. Thus the map is not injective: for example, $0$ and $3$ have the same image.

More generally, $\varphi(a) = \varphi(b)$ exactly when $a^{-1}b \in \ker\varphi$, that is, when $a$ and $b$ are in the same coset of the kernel. Thus each fibre of $\varphi$ is a coset of the kernel, or it is empty.

A common error is to use the test for a map that is not a homomorphism. The map $x \mapsto x^2$ on $\mathbb{R}$ sends only $0$ to $0$, but it is not injective.
</document>

<document source="https://tamnd.github.io/math2/node/0N2SP6K9ZC.md" tag="0N2SP6K9ZC" kind="concept">
# Cumulant generating function

## Summary

The cumulant generating function of a real random variable is the logarithm of its moment-generating function. Its Taylor coefficients at $0$ are the cumulants.
The node [[cumulant generating function|0Q2QC81VTW]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0N2WPCC00T.md" tag="0N2WPCC00T" kind="statement">
# The rationals form a field

## Statement

The [[rational numbers|0NCJ5WXN6Z]] $\mathbb{Q}$, with their addition and multiplication, form a field.
That is, $\mathbb{Q}$ is a commutative ring with $0 \neq 1$, and each rational number $q \neq 0$ has a multiplicative inverse $q^{-1}$ with $q \cdot q^{-1} = 1$.

## Remarks

The inverse of $a / b$ with $a \neq 0$ is $b / a$.
The field $\mathbb{Q}$ is the smallest subfield of the real numbers. Each field of characteristic $0$ contains a copy of $\mathbb{Q}$.
The [[integers|0C4JS9RF5M]] are not a field, and $\mathbb{Q}$ is the field of fractions of $\mathbb{Z}$.

## Explanations

### L1 human

In the rational numbers, you can add, subtract, multiply, and divide by any number that is not zero.
The result is always a rational number again.
</document>

<document source="https://tamnd.github.io/math2/node/0N32V0E3XJ.md" tag="0N32V0E3XJ" kind="statement">
# Limits preserve non-strict inequalities

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences that converge to real numbers $a$ and $b$.
Assume that there is a natural number $N$ such that $a_n \le b_n$ for all $n \ge N$.
Then $a \le b$.

## Remarks

Strict inequalities do not pass to the limit. For example, $0 < 1/n$ for all $n \ge 1$, but $\lim 1/n = 0$.
With a constant sequence $b_n = c$, it says: if $a_n \le c$ for large $n$, then $\lim a_n \le c$.
The proof is by contradiction. If $a > b$, take $\varepsilon = (a - b)/2$. Then for large $n$ we get $b_n < a_n$.

## Explanations

### L1 human

If one sequence is always below another, its limit cannot be above the limit of the other.
The limits can be equal, even when the terms are never equal.
</document>

<document source="https://tamnd.github.io/math2/node/0N3ADS4CHH.md" tag="0N3ADS4CHH" kind="definition">
# Uniqueness quantifier

## Definition

Let $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.
The proposition $\exists! x \in X, \, P(x)$ is true when exactly one $a \in X$ makes $P(a)$ true.
In terms of the other quantifiers, it is the proposition
$$\exists x \in X, \, \bigl( P(x) \land \forall y \in X, \, (P(y) \to y = x) \bigr).$$

## Remarks

A proof of $\exists! x, \, P(x)$ usually has two parts: existence, an $x$ with $P(x)$, and uniqueness, if $P(x)$ and $P(y)$ then $x = y$.
For example, $\exists! x \in \mathbb{R}, \, 2x + 1 = 7$ is true, with $x = 3$.

## Explanations

### L1 human

"There is exactly one even prime" says two things.
There is an even prime, namely $2$, and no other number is an even prime.
</document>

<document source="https://tamnd.github.io/math2/node/0N3ADXHNTQ.md" tag="0N3ADXHNTQ" kind="definition">
# Composite number

## Definition

A composite number is a natural number $n$ such that $n > 1$ and $n$ is not a [[prime number|0SN0MAEKYX]].

## Remarks

Equivalently, $n$ is composite when $n = a b$ for some natural numbers $a$ and $b$ with $1 < a < n$ and $1 < b < n$.
The first composite numbers are $4, 6, 8, 9, 10, 12$.
The number $1$ is neither prime nor composite.

## Explanations

### L1 human

A composite number is a number that you can write as a product of two smaller numbers, each greater than $1$.
For example, $15 = 3 \cdot 5$ is composite.
You can arrange a composite number of dots as a rectangle with at least two rows and at least two columns.
</document>

<document source="https://tamnd.github.io/math2/node/0N3WFZEFPM.md" tag="0N3WFZEFPM" kind="statement">
# Liouville's theorem on Diophantine approximation

## Statement

Let $\alpha$ be a real [[algebraic number|0MB4TGMXG6]] of degree $d \ge 2$.
Then there is a constant $c > 0$ such that $\left| \alpha - \frac{p}{q} \right| > \frac{c}{q^d}$ for all integers $p$ and $q$ with $q > 0$.
As a consequence, each [[Liouville number|0J1R5NP36S]] is [[transcendental|079EGSD32Y]].

## Remarks

The proof uses the minimal polynomial $f$ of $\alpha$: $q^d f(p/q)$ is a nonzero integer, and the mean value theorem bounds $f(p/q)$ by $|\alpha - p/q|$.
For a Liouville number $x$, the approximations are better than $c / q^d$ for every $d$, so $x$ is not algebraic.
Roth's theorem improves the exponent $d$ to $2 + \varepsilon$ for each $\varepsilon > 0$.

## History

Joseph Liouville proved the bound in 1844. With it, he gave the first proof that transcendental numbers exist.

## Explanations

### L1 human

Algebraic numbers cannot be approximated by fractions too well: the error is at least a constant times $1/q^d$.
A number with better approximations of every order cannot be algebraic.
This gave the first explicit transcendental numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0N3Y8G340X.md" tag="0N3Y8G340X" kind="example">
# Hilbert's hotel

## Example

A hotel has rooms numbered $1, 2, 3, \dots$, one for each positive integer, and each room has a guest.
A new guest arrives. The manager moves the guest of room $n$ to room $n + 1$ for each $n$. Then room $1$ is free for the new guest.
Next, infinitely many new guests $g_1, g_2, \dots$ arrive. The manager moves the guest of room $n$ to room $2n$, and gives room $2k - 1$ to $g_k$.
In both cases, each guest has a room and no two guests share a room.

## Remarks

The maps $n \mapsto n + 1$ and $n \mapsto 2n$ are [[injections|0PVG1CRAJN]] of $\mathbb{N}_{>0}$ into a proper subset of itself. So the set of rooms is equinumerous with a proper subset.
This cannot happen for a [[finite set|0BBTH3YSZA]].
David Hilbert introduced the hotel in a lecture of 1924 to 1925, and George Gamow made it popular in a book of 1947.

## Explanations

### L1 human

A full finite hotel cannot take a new guest, but a full infinite hotel can.
The example shows that "as many as" for infinite sets does not behave like counting.
</document>

<document source="https://tamnd.github.io/math2/node/0N4F5FMQZW.md" tag="0N4F5FMQZW" kind="concept">
# Quotient map

## Summary

The quotient map of a vector space onto a quotient space sends each vector to its coset.
It is a surjective linear map whose kernel is the subspace.
</document>

<document source="https://tamnd.github.io/math2/node/0N4XY7Z3NN.md" tag="0N4XY7Z3NN" kind="concept">
# Upper triangular matrix

## Summary

An upper triangular matrix is a square matrix whose entries below the main diagonal are all zero.
</document>

<document source="https://tamnd.github.io/math2/node/0N57D1XY6T.md" tag="0N57D1XY6T" kind="definition">
# Pairwise independence

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $(A_i)_{i \in I}$ be a family of events.
The family is pairwise independent if $A_i$ and $A_j$ are [[independent|0P0MZ4X682]] for all $i, j \in I$ with $i \ne j$, that is, $P(A_i \cap A_j) = P(A_i) \, P(A_j)$.

## Remarks

[[Mutual independence|0GCVEV6BJA]] implies pairwise independence, but the converse is false. The node [[pairwise independent events that are not mutually independent|0AR4B3K5QD]] gives an example.
The same definition applies to random variables: any two of them are independent.
Pairwise independence is enough for some results.
Examples are the formula for the variance of a sum and the weak law of large numbers with finite variance.

## Explanations

### L1 human

Pairwise independence checks the events two at a time.
It can happen that each pair of events is independent, but a group of three events is not.
</document>

<document source="https://tamnd.github.io/math2/node/0N5FCV2YV6.md" tag="0N5FCV2YV6" kind="definition">
# Boolean ring

## Definition

A Boolean ring is a ring $R$ with identity such that $x^2 = x$ for every $x \in R$.

## Remarks

In a Boolean ring, $x + x = 0$ for every $x$, and the ring is commutative.
The ring $\mathbb{Z}/2\mathbb{Z}$ is a Boolean ring. The power set of a set is a Boolean ring, with symmetric difference as addition and intersection as multiplication.
Each Boolean ring is a [[Boolean algebra|06GKBTAA5E]] with $x \wedge y = xy$, $x \vee y = x + y + xy$ and $\neg x = 1 + x$. This gives a correspondence between Boolean rings and Boolean algebras.

## Explanations

### L1 human

A Boolean ring is a ring in which squaring changes nothing.
It is the same thing as a Boolean algebra, written with plus and times instead of or and and.
</document>

<document source="https://tamnd.github.io/math2/node/0N5GP3YT8T.md" tag="0N5GP3YT8T" kind="definition">
# Square root of a positive semidefinite matrix

## Definition

Let $A$ be a [[positive semidefinite|01MMVW8CKC]] $n \times n$ complex Hermitian matrix.
There is exactly one positive semidefinite matrix $B$ with $B^2 = A$.
This matrix is the *square root* of $A$, written $A^{1/2}$ or $\sqrt{A}$.

## Remarks

If $A = U D U^*$ with $U$ unitary and $D$ diagonal with entries $d_i \ge 0$, then $A^{1/2} = U D^{1/2} U^*$, with entries $\sqrt{d_i}$ in $D^{1/2}$.
$A^{1/2}$ commutes with each matrix that commutes with $A$.
Without the condition $B \succeq 0$, a matrix can have many square roots; for example, $I_2$ has infinitely many.

## Explanations

### L1 human

A positive semidefinite matrix has exactly one positive semidefinite square root.
In an orthonormal basis of eigenvectors, we take the square root of each eigenvalue.
This is like the positive square root of a nonnegative number.
</document>

<document source="https://tamnd.github.io/math2/node/0N5M924N3P.md" tag="0N5M924N3P" kind="definition">
# Falling factorial

## Definition

Let $x$ be an element of a commutative ring, for example a real number, and let $k$ be a natural number.
The falling factorial of $x$ of order $k$ is
$$(x)_k = x (x - 1) (x - 2) \cdots (x - k + 1) = \prod_{i=0}^{k-1} (x - i).$$
For $k = 0$ the product is empty, and $(x)_0 = 1$.

## Remarks

Other notations are $x^{\underline{k}}$ and $P(n, k)$.
For natural numbers $n \ge k$, $(n)_k = n! / (n - k)!$. For $k > n$, $(n)_k = 0$.
For natural numbers $n$ and $k$, $(n)_k$ is the [[number of injections|0X0N833FF8]] from a $k$-element set to an $n$-element set.

## Explanations

### L1 human

The falling factorial starts like the [[factorial|0WFCGVMRRC]] but stops after $k$ factors.
For example, $(5)_3 = 5 \cdot 4 \cdot 3 = 60$ is the number of ways to award gold, silver and bronze among 5 runners.
</document>

<document source="https://tamnd.github.io/math2/node/0N5N15HWZ7.md" tag="0N5N15HWZ7" kind="concept">
# Cross product

## Summary

The cross product is a binary operation on the vectors of three-dimensional space. It takes two vectors and gives a vector that is perpendicular to both. The length of the result is the area of the parallelogram with the two vectors as sides.
</document>

<document source="https://tamnd.github.io/math2/node/0N5NF1RGBP.md" tag="0N5NF1RGBP" kind="definition">
# Wronskian

## Definition

Let $I$ be an interval and let $f_1, \dots, f_n : I \to \mathbb{R}$ be $n - 1$ times differentiable functions.
The Wronskian of $f_1, \dots, f_n$ is the function $W(f_1, \dots, f_n) : I \to \mathbb{R}$ given by
$$W(f_1, \dots, f_n)(x) = \det \begin{pmatrix} f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & \cdots & f_n'(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \end{pmatrix}.$$

## Remarks

For two functions, $W(f, g) = f g' - f' g$.
Example: $W(\cos, \sin) = \cos^2 + \sin^2 = 1$.
The [[Wronskian test|0MHE4B1QCQ]] uses $W$ to decide whether solutions of a linear equation are linearly independent.

## Explanations

### L1 human

The rows of the matrix are the functions, then their first derivatives, and so on.
The determinant measures whether the vectors of starting data of the functions are independent at the point $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0N5NJZKKN3.md" tag="0N5NJZKKN3" kind="concept">
# Idempotent element

## Summary

An idempotent element of a ring is an element that is equal to its own square.
The [[definition of an idempotent element|0X4JKZXJ86]] states this precisely.
</document>

<document source="https://tamnd.github.io/math2/node/0N5NR32NQG.md" tag="0N5NR32NQG" kind="statement">
# Classification of conics by the discriminant

## Statement

Let $Q = \{\, (x, y) \in \mathbb{R}^2 : A x^2 + B x y + C y^2 + D x + E y + F = 0 \,\}$ be a non-degenerate [[conic section|00HJCMC2CD]]. Let $\Delta = B^2 - 4AC$ be the discriminant.
Then $Q$ is
- an [[ellipse|0VKCCEXBJ1]] if and only if $\Delta < 0$,
- a [[parabola|04S2GMHX82]] if and only if $\Delta = 0$,
- a [[hyperbola|0WN5G117YC]] if and only if $\Delta > 0$.

Moreover, $Q$ is a [[circle|060W5CBZP6]] if and only if $A = C$ and $B = 0$.

## Remarks

The discriminant is $-4$ times the determinant of $\begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix}$, the matrix of the quadratic part.
A rotation or a translation of the coordinates does not change $\Delta$ or $A + C$. So the type of a conic does not depend on the choice of Cartesian coordinates.
A hyperbola with $A + C = 0$ is rectangular: its asymptotes are perpendicular.

## Explanations

### L1 human

The terms of degree $2$ decide the shape of a conic. For $x^2 + y^2 = 1$, $\Delta = -4 < 0$ and we get a circle, a special ellipse. For $y = x^2$, $\Delta = 0$ and we get a parabola. For $x y = 1$, $\Delta = 1 > 0$ and we get a hyperbola.
</document>

<document source="https://tamnd.github.io/math2/node/0N5QW9K7MH.md" tag="0N5QW9K7MH" kind="statement">
# Degree of a simple extension is the degree of the minimal polynomial

## Statement

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$ be algebraic over $K$ with [[minimal polynomial|0GK1WB9R98]] $m_a$ of degree $n$.
Then $[K(a) : K] = n$, and $1, a, a^2, \dots, a^{n-1}$ is a basis of $K(a)$ over $K$.

## Remarks

The proof: by [[the structure theorem|0ZN39QDTP9]], $K(a) \cong K[x]/(m_a)$. By polynomial division, the classes of $1, x, \dots, x^{n-1}$ are a basis of $K[x]/(m_a)$.
For example, $[\mathbb{Q}(\sqrt[3]{2}) : \mathbb{Q}] = 3$, with the basis $1, \sqrt[3]{2}, \sqrt[3]{4}$.
With the [[tower law|0A7KJ68XHT]], the degree of every element of a finite extension $L/K$ divides $[L : K]$.

## Explanations

### L1 human

The degree of the field you get by adding one algebraic element is the degree of its minimal polynomial.
Each element of $K(a)$ is a polynomial in $a$ of degree less than $n$, in exactly one way.
</document>

<document source="https://tamnd.github.io/math2/node/0N5ZSZMYH6.md" tag="0N5ZSZMYH6" kind="concept">
# Fermat primality test

## Summary

The Fermat primality test is a probabilistic test of whether an integer $n > 1$ is prime.
It chooses a base $a$ with $1 < a < n - 1$ and computes $a^{n-1} \bmod n$ by [[modular exponentiation|04DR5PM25N]].
If the result is not $1$, then $n$ is composite, by [[Fermat's little theorem|0M25PRN3WQ]]. If the result is $1$, then $n$ is called a probable prime to the base $a$.

## Remarks

A probable prime can be composite. Such a number is a [[Fermat pseudoprime|02M2K4QH9P]].
The [[Carmichael numbers|0TA7WT68RK]] pass the test for each base coprime to them. So repeated tests with many bases cannot detect all composite numbers.
The [[Miller–Rabin primality test|0SZCSYHW60]] is a stronger test of the same kind, without this weakness.

## Explanations

### L1 human

To test $n = 15$ with $a = 2$, compute $2^{14} = 16384 \equiv 4 \pmod{15}$.
The result is not $1$, so $15$ is surely composite. The test does not find a factor; it only proves that one exists.
</document>

<document source="https://tamnd.github.io/math2/node/0N67NBKQ5T.md" tag="0N67NBKQ5T" kind="definition">
# Vector field

## Definition

Let $S \subseteq \mathbb{R}^n$.
A vector field on $S$ is a [[vector-valued function|018GTRS7A4]] $F \colon S \to \mathbb{R}^n$, $F(x) = (F_1(x), \dots, F_n(x))$.
The vector field is continuous, or of class $C^k$, if each component $F_i$ is [[continuous|0S9EK5MQPE]], or of [[class C^k|0TCBZMEYGN]], on $S$.

## Remarks

One pictures $F$ by an arrow $F(x)$ drawn with its tail at the point $x$.
For a differentiable function $f \colon U \to \mathbb{R}$ on an open set $U$, the gradient $\nabla f$ is a vector field on $U$. Such fields are gradient fields.
The field $F(x, y) = (-y, x)$ on $\mathbb{R}^2$ describes a counterclockwise rotation about the origin. The field $F(x) = x$ points away from the origin.
In $\mathbb{R}^3$ one often writes $F = P \, \mathbf{i} + Q \, \mathbf{j} + R \, \mathbf{k}$, with $P$, $Q$, $R$ the components of $F$.

## Explanations

### L1 human

A vector field attaches an arrow to each point.
Think of the wind on a weather map: at each place there is a wind direction and a wind speed.
</document>

<document source="https://tamnd.github.io/math2/node/0N681SSTHC.md" tag="0N681SSTHC" kind="concept">
# Simple extension

## Summary

A simple extension is a field extension that is generated by one element.
The [[definition of a simple extension|0P6SBR7YM2]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0N6960W0BV.md" tag="0N6960W0BV" kind="concept">
# Eigenvector

## Summary

An eigenvector of a linear map from a space to itself is a nonzero vector that the map sends to a scalar multiple of itself.
</document>

<document source="https://tamnd.github.io/math2/node/0N6R902FG1.md" tag="0N6R902FG1" kind="concept">
# Monoid

## Summary

A monoid is a semigroup that has an identity element.
Examples are the natural numbers with addition and the maps from a set to itself with composition.
</document>

<document source="https://tamnd.github.io/math2/node/0N7ATVF2X8.md" tag="0N7ATVF2X8" kind="concept">
# Classification of equilibria of planar linear systems

## Summary

Let $A$ be a real $2 \times 2$ matrix with $\det A \neq 0$, let $\tau = \operatorname{tr} A$ and $\Delta = \det A$, and consider $x' = A x$ in the [[phase plane|05VDGH129F]].
The origin is a saddle if $\Delta < 0$. It is a node if $\Delta > 0$ and $\tau^2 - 4 \Delta \ge 0$, and a spiral, or focus, if $\tau \neq 0$ and $\tau^2 - 4 \Delta < 0$. It is a centre if $\tau = 0$ and $\Delta > 0$.
A node or a spiral is stable for $\tau < 0$ and unstable for $\tau > 0$.

## Remarks

The eigenvalues are the roots of $\lambda^2 - \tau \lambda + \Delta = 0$. A saddle has real eigenvalues of opposite signs. A node has real eigenvalues of the same sign. A spiral or a centre has complex conjugate eigenvalues.
The case $\tau^2 = 4 \Delta$ gives a repeated eigenvalue. The origin is then a star node or a degenerate node.
The trace and the determinant do not change under a change of basis $A \mapsto C^{-1} A C$. So the classification depends only on the system, not on the coordinates.
The [[damped harmonic oscillator|0RNXVVTJTJ]] gives a stable spiral when it is underdamped and a stable node when it is overdamped.

## Explanations

### L1 human

For a linear system in the plane, two numbers, the trace and the determinant, tell the shape of the phase portrait near the origin.
For a node, trajectories run into or out of the origin along straight directions. They turn around it for a spiral, and they go around it in closed loops for a centre.
</document>

<document source="https://tamnd.github.io/math2/node/0N7ENQRCXN.md" tag="0N7ENQRCXN" kind="concept">
# Kronecker product

## Summary

The Kronecker product of two matrices is the block matrix whose blocks are the entries of the first matrix times the second matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0N8NGKDZDZ.md" tag="0N8NGKDZDZ" kind="concept">
# Equilibrium point

## Summary

An equilibrium point of a differential equation is a state that gives a constant solution.
For an autonomous system $x' = f(x)$, these are the zeros of $f$. See the [[definition|0G4X36AE0X]].
</document>

<document source="https://tamnd.github.io/math2/node/0N91N98EG5.md" tag="0N91N98EG5" kind="statement">
# Mean value theorem for integrals

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous.
Then there is $c \in [a, b]$ with
$$\int_a^b f(x) \, dx = f(c) \, (b - a).$$

## Remarks

Let $m$ and $M$ be the minimum and the maximum of $f$, from the [[extreme value theorem|0WS1A8E2P9]]. By [[monotonicity|0MBWQKM8QR]], the average $\frac{1}{b - a} \int_a^b f$ lies in $[m, M]$. The [[intermediate value theorem|00KAFBR32E]] gives $c$.
The number $\frac{1}{b - a} \int_a^b f$ is the average value of $f$ on $[a, b]$.
Mathlib has a form for averages, `exists_eq_interval_average`.

## Explanations

### L1 human

A continuous function takes its average value somewhere.
The area under the graph equals the area of a rectangle whose height is a value of $f$.
</document>

<document source="https://tamnd.github.io/math2/node/0N9E4G2P97.md" tag="0N9E4G2P97" kind="definition">
# Fermat prime

## Definition

A Fermat prime is a [[Fermat number|0R03B54V4Z]] $F_n = 2^{2^n} + 1$ that is a [[prime number|0SN0MAEKYX]].

## Remarks

The numbers $F_0, \dots, F_4$, that is $3, 5, 17, 257, 65537$, are prime. No other Fermat prime is known.
Fermat conjectured that all Fermat numbers are prime. Euler refuted this in 1732 with $F_5 = 641 \cdot 6700417$.
A regular $n$-gon is constructible with ruler and compass if and only if $n$ is a power of $2$ times distinct Fermat primes.
This is the Gauss–Wantzel theorem.

## Explanations

### L1 human

A Fermat prime is a prime of the form $2^{2^n} + 1$.
Only five of them are known, and it is an open problem whether there are more.
</document>

<document source="https://tamnd.github.io/math2/node/0N9R5NN6TW.md" tag="0N9R5NN6TW" kind="definition">
# Nilpotent element

## Definition

Let $R$ be a [[ring|0894J980RK]].
An element $x \in R$ is nilpotent if there is a positive integer $n$ such that $x^n = 0$.

## Remarks

In $\mathbb{Z}/8\mathbb{Z}$, the class of $2$ is nilpotent, because $2^3 = 8 = 0$.
The matrix $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ is nilpotent: its square is the zero matrix.
In a nonzero ring, a nilpotent element is a [[zero divisor|0K26YKNBSA]]: if $n$ is the least integer with $x^n = 0$, then $x \cdot x^{n-1} = 0$ and $x^{n-1} \neq 0$.
An [[integral domain|09S3PFBEPM]] has no nilpotent element except $0$.

## Explanations

### L1 human

A nilpotent element is not zero, but some power of it is zero.
Numbers do not behave like this.
In the integers modulo $8$, the element $2$ does: $2 \cdot 2 \cdot 2 = 8$, and $8$ is $0$ modulo $8$.
</document>

<document source="https://tamnd.github.io/math2/node/0N9SA25Z3V.md" tag="0N9SA25Z3V" kind="definition">
# Symmetric relation

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is symmetric when, for all $a, b \in A$, $a \mathrel{R} b$ implies $b \mathrel{R} a$.

## Remarks

Equality is symmetric. The relation $\le$ on $\mathbb{R}$ is not symmetric, because $1 \le 2$ but not $2 \le 1$.
A relation $R$ is symmetric exactly when it is equal to its [[converse|0JCSQNNP2M]].

## Explanations

### L1 human

"Is a sibling of" is symmetric: if Ann is a sibling of Bob, then Bob is a sibling of Ann.
"Is a parent of" is not symmetric.
</document>

<document source="https://tamnd.github.io/math2/node/0NA0732YAG.md" tag="0NA0732YAG" kind="concept">
# Basis of a topology

## Summary

A basis of a topology is a family of open sets such that every open set is a union of members of the family.
A basis lets you describe a topology by a small set of simple open sets, such as open intervals or open balls.
</document>

<document source="https://tamnd.github.io/math2/node/0NA4FK7B8S.md" tag="0NA4FK7B8S" kind="example">
# Discrete metric

## Example

Let $X$ be a set.
For $x, y \in X$, let $d(x, y) = 0$ if $x = y$ and $d(x, y) = 1$ if $x \neq y$.
Then $d$ is a [[metric|063W5JMY85]] on $X$, the discrete metric.

## Remarks

For each $x \in X$, the open ball of radius $1/2$ about $x$ is $\{x\}$. Thus each subset of $X$ is open, and the metric gives the [[discrete topology|0S2TEHX4E1]].
A sequence converges in this metric if and only if it is eventually constant.

## Explanations

### L1 human

In the discrete metric, two points are either equal or at distance $1$.
No point is close to another point, so the space has no interesting limits.
</document>

<document source="https://tamnd.github.io/math2/node/0NA6Q82YR7.md" tag="0NA6Q82YR7" kind="definition">
# Cauchy distribution

## Definition

Let $x_0 \in \mathbb{R}$ and $\gamma > 0$.
The Cauchy distribution with location $x_0$ and scale $\gamma$ is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$f(x) = \frac{1}{\pi \gamma \left(1 + \left(\frac{x - x_0}{\gamma}\right)^2\right)} = \frac{\gamma}{\pi \left((x - x_0)^2 + \gamma^2\right)}, \quad x \in \mathbb{R}.$$
The standard Cauchy distribution has $x_0 = 0$ and $\gamma = 1$, with density $\frac{1}{\pi(1 + x^2)}$.

## Remarks

The cumulative distribution function is $F(x) = \frac{1}{2} + \frac{1}{\pi} \arctan\left(\frac{x - x_0}{\gamma}\right)$. The median is $x_0$.
The integral $\int |x| f(x) \, dx$ is infinite, so the distribution has no mean. It also has no variance.
If $X_1, \dots, X_n$ are independent standard Cauchy, then their average $(X_1 + \dots + X_n)/n$ is again standard Cauchy. So the law of large numbers fails without a finite mean.
The ratio $X/Y$ of two independent standard normal random variables is standard Cauchy.

## Explanations

### L1 human

The Cauchy distribution looks like a bell curve, but its tails are much heavier: very large values are not rare.
Shine a lamp at a uniform random angle onto a wall. The point where the light hits the wall has a Cauchy distribution.
</document>

<document source="https://tamnd.github.io/math2/node/0NACHJYPKW.md" tag="0NACHJYPKW" kind="statement">
# Ore's theorem

## Statement

Let $G$ be a finite simple [[graph|0JD5C82QMB]] with $n \ge 3$ vertices.
Suppose that $\deg(u) + \deg(v) \ge n$ for any two distinct vertices $u$ and $v$ that are not adjacent.
Then $G$ is [[Hamiltonian|090WPKFABX]].

## Remarks

[[Dirac's theorem|0TC99BRSSB]] is a special case.
The Bondy–Chvátal theorem of 1976 extends it. It says that $G$ is Hamiltonian if and only if its closure is Hamiltonian.
The closure comes from $G$ when you add edges $uv$ with $\deg(u) + \deg(v) \ge n$ again and again.

## History

Øystein Ore proved the theorem in 1960.

## Explanations

### L1 human

Ore's theorem looks only at pairs of vertices that are not neighbours.
If each such pair has many edges together, the graph has a round trip through all vertices.
</document>

<document source="https://tamnd.github.io/math2/node/0NAHXR1FRH.md" tag="0NAHXR1FRH" kind="concept">
# Holomorphic function

## Summary

A holomorphic function is a complex function that is complex differentiable at each point of an open subset of $\mathbb{C}$, see the [[definition|0DG0Z72CQD]].
Holomorphic functions are the main objects of complex analysis.
They are infinitely differentiable and they are equal to their Taylor series near each point.
</document>

<document source="https://tamnd.github.io/math2/node/0NAKHQ9XYY.md" tag="0NAKHQ9XYY" kind="statement">
# Apollonius's theorem

## Statement

Let $A$, $B$ and $C$ be points of the Euclidean space $\mathbb{R}^n$, and let $D$ be the [[midpoint|0S706JBZBG]] of $B$ and $C$. Let $|XY|$ denote the [[distance|0NPPGHQNXQ]] between the points $X$ and $Y$.
Then
$$|AB|^2 + |AC|^2 = 2 \left( |AD|^2 + |BD|^2 \right).$$

## Remarks

For a triangle $ABC$ with the side lengths $a = |BC|$, $b = |CA|$, $c = |AB|$, the [[median|0XW8HX8VEV]] from $A$ thus has the length $m_a = \frac{1}{2} \sqrt{2 b^2 + 2 c^2 - a^2}$.
In vector form, with $u = B - A$ and $v = C - A$, the statement is the parallelogram law $\|u + v\|^2 + \|u - v\|^2 = 2\|u\|^2 + 2\|v\|^2$.
For an isosceles triangle with $|AB| = |AC|$, the median $AD$ is perpendicular to $BC$, and the theorem reduces to the [[Pythagorean theorem|0VP3A1290K]].

## History

The theorem is Proposition 122 of Book VII of the Collection of Pappus of Alexandria. It was possibly in the lost work Plane Loci of Apollonius of Perga.

## Explanations

### L1 human

The theorem relates the two sides from $A$ to the median from $A$. If you know the three sides of a triangle, it gives the length of each median. For the right triangle with legs $3$, $4$ and hypotenuse $5$, the median to the hypotenuse has length $\frac{1}{2}\sqrt{18 + 32 - 25} = \frac{5}{2}$.
</document>

<document source="https://tamnd.github.io/math2/node/0NANDX7V02.md" tag="0NANDX7V02" kind="concept">
# Harmonic number

## Summary

The harmonic numbers are the sums $1 + 1/2 + \cdots + 1/n$ of the reciprocals of the first positive integers. They grow like the natural logarithm of $n$. The node [[harmonic number|0D2Y6G4C4S]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0NB0AWGJBW.md" tag="0NB0AWGJBW" kind="concept">
# Inner product

## Summary

An inner product on a real or complex vector space is a function of two vectors.
It is linear in the first vector, conjugate symmetric, and positive on nonzero vectors.
</document>

<document source="https://tamnd.github.io/math2/node/0NB31CC4SN.md" tag="0NB31CC4SN" kind="example">
# exp(1/z) has an essential singularity at 0

## Example

Let $f(z) = e^{1/z}$ for $z \in \mathbb{C} \setminus \{0\}$, where $e^w$ is the [[complex exponential|054VXHMCA3]].
The function $f$ is holomorphic on $\mathbb{C} \setminus \{0\}$, so $0$ is an isolated singularity of $f$.
Its [[Laurent series|0X3XD2ATB0]] at $0$ is
$$e^{1/z} = \sum_{n=0}^{\infty} \frac{1}{n!} z^{-n} = 1 + \frac{1}{z} + \frac{1}{2! \, z^2} + \cdots \qquad (z \neq 0).$$
It has infinitely many nonzero terms with negative powers, so $0$ is an [[essential singularity|0YKDJBV26S]] of $f$.

## Remarks

A direct check: for real $x \to 0^+$, $e^{1/x} \to \infty$, and for real $x \to 0^-$, $e^{1/x} \to 0$. So $f$ has no limit at $0$ and $|f|$ does not tend to $\infty$.
In each punctured disc $0 < |z| < \delta$, $f$ takes each value $w \neq 0$ infinitely often: the solutions of $e^{1/z} = w$ are $z = 1 / (\operatorname{Log} w + 2\pi i k)$ with $k \in \mathbb{Z}$ and $\operatorname{Log} w + 2\pi i k \neq 0$, and these points tend to $0$ as $|k| \to \infty$.
The value $0$ is never taken. This agrees with [[Picard's great theorem|08QA8QTM71]], which allows one exception.
The residue of $f$ at $0$ is the coefficient of $z^{-1}$, which is $1$.

## Explanations

### L1 human

Near $0$, the number $1/z$ is huge and can point in any direction, so $e^{1/z}$ can be huge, tiny, or anything in between.
That is why no single value or infinity describes $e^{1/z}$ near $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0NB643K60Q.md" tag="0NB643K60Q" kind="definition">
# Partial sum

## Definition

Let $\sum_{n=0}^{\infty} a_n$ be a [[series|0JH4MN4K1C]] of real numbers and let $N$ be a natural number.
The $N$-th partial sum of the series is the finite sum
$$s_N = \sum_{n=0}^{N} a_n.$$

## Remarks

The partial sums satisfy $s_0 = a_0$ and $s_N = s_{N-1} + a_N$ for $N \ge 1$.
The terms are the differences of the partial sums: $a_N = s_N - s_{N-1}$ for $N \ge 1$.
For the series $\sum 1/2^n$, the partial sums are $1, 3/2, 7/4, 15/8, \dots$, and $s_N = 2 - 1/2^N$.

## Explanations

### L1 human

The partial sums are the running totals of the series.
You add one more term at each step and record the total.
</document>

<document source="https://tamnd.github.io/math2/node/0NB689782W.md" tag="0NB689782W" kind="concept">
# Asymptotic stability

## Summary

An equilibrium point is asymptotically stable if it is Lyapunov stable and nearby solutions converge to it as time tends to infinity.
See the [[definition|05Y1JAM3Y5]].
</document>

<document source="https://tamnd.github.io/math2/node/0NB8VPB93V.md" tag="0NB8VPB93V" kind="definition">
# Composition series

## Definition

Let $G$ be a [[group|G5S1M9]].
A composition series of $G$ is a [[subnormal series|0ZZR8D7Y1Z]] $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$ in which each factor $G_i / G_{i-1}$ is a [[simple group|0YMYMNQ0QZ]].

## Remarks

Equivalently, the inclusions are strict and each $G_{i-1}$ is a maximal proper normal subgroup of $G_i$.
Each finite group has a composition series, but the group $\mathbb{Z}$ has none.
For example, $\{0\} \trianglelefteq \langle 6 \rangle \trianglelefteq \langle 3 \rangle \trianglelefteq \mathbb{Z}/12\mathbb{Z}$ and $\{0\} \trianglelefteq \langle 6 \rangle \trianglelefteq \langle 2 \rangle \trianglelefteq \mathbb{Z}/12\mathbb{Z}$ are composition series.

## Explanations

### L1 human

A composition series cuts a group into layers that cannot be cut further.
Each layer is a simple group.
It is like the factorisation of an integer into prime numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0NBRP6093V.md" tag="0NBRP6093V" kind="definition">
# Divergent series

## Definition

A [[series|0JH4MN4K1C]] $\sum a_n$ of real numbers is divergent if it is not a [[convergent series|0CVVKY7GY2]].
Thus its sequence of partial sums has no limit in $\mathbb{R}$.

## Remarks

The series $1 + 1 + 1 + \cdots$ diverges, and its partial sums tend to $+\infty$.
The series $1 - 1 + 1 - 1 + \cdots$ diverges, and its partial sums oscillate between $1$ and $0$.
The [[harmonic series|0CXTMAY5PX]] diverges, although its terms tend to $0$.
Summation methods, such as Cesàro summation, give values to some divergent series.

## Explanations

### L1 human

A divergent series has no sum in the usual sense.
Its running totals grow without bound or keep jumping around.
</document>

<document source="https://tamnd.github.io/math2/node/0NC2A3HWGT.md" tag="0NC2A3HWGT" kind="statement">
# The integral of a continuous function is an antiderivative

## Statement

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$ be continuous, and let $a \in I$.
Then $F(x) = \int_a^x f(t) \, dt$ is an [[antiderivative|0BW9N5CDSB]] of $f$ on $I$.

## Remarks

The integral exists, since [[continuous functions are integrable|06SESV7GQY]]. The [[first fundamental theorem|063H5KJC69]] applies at each point of $I$.
So every continuous function on an open interval has an antiderivative.
Mathlib states the derivative as `Continuous.deriv_integral`.

## Explanations

### L1 human

Every continuous function has an antiderivative.
The area function from a fixed starting point is one.
</document>

<document source="https://tamnd.github.io/math2/node/0NC9R64TXP.md" tag="0NC9R64TXP" kind="concept">
# Planar graph

## Summary

A planar graph is a graph that can be drawn in the plane with no crossing edges. Planar graphs are the subject of Euler's formula and of the four colour theorem. The node [[planar graph|029JQEK15B]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0NCJ5WXN6Z.md" tag="0NCJ5WXN6Z" kind="object">
# Rational numbers

## Description

A rational number is a number of the form $a / b$, where $a$ and $b$ are [[integers|0C4JS9RF5M]] and $b \neq 0$.
We write $\mathbb{Q}$ for the set of rational numbers.
Two fractions $a / b$ and $c / d$ are the same rational number when $a d = b c$.
With addition and multiplication, $\mathbb{Q}$ is a [[field|0N2WPCC00T]], and with its usual order it is an [[ordered field|09RFKMBQ63]].
The set $\mathbb{Q}$ contains $\mathbb{Z}$, because each integer $a$ is the fraction $a / 1$.

## Remarks

Each rational number has exactly one [[representation in lowest terms|0R64A2CVN4]] $a / b$ with $b > 0$.
The set $\mathbb{Q}$ is countable.
Not every real number is rational: $\sqrt{2}$ is [[irrational|0DZGSDHD92]].

## Explanations

### L1 human

The rational numbers are the fractions, such as $1/2$, $-3/4$ and $7 = 7/1$.
Many fractions name the same number: $1/2$, $2/4$ and $3/6$ are all equal.
In $\mathbb{Q}$, you can divide by any number except $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0NCPXH194Y.md" tag="0NCPXH194Y" kind="concept">
# Graph colouring

## Summary

A colouring of a graph assigns colours to the vertices so that adjacent vertices get different colours. Colourings model scheduling and other conflict problems. The node [[graph colouring|0874138YP6]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0NDV5R6Z6X.md" tag="0NDV5R6Z6X" kind="definition">
# Principal part

## Definition

Let $c \in \mathbb{C}$, let $R \in (0, \infty]$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 < |z - c| < R$.
Let $f(z) = \sum_{n \in \mathbb{Z}} a_n (z - c)^n$ be the [[Laurent expansion|06WHRARZKY]] of $f$ on this punctured disc.
The principal part of $f$ at $c$ is the series
$$\sum_{n = 1}^{\infty} a_{-n} (z - c)^{-n},$$
the part of the [[Laurent series|0X3XD2ATB0]] with the negative powers of $z - c$.

## Remarks

The principal part converges for each $z \neq c$. The difference $f(z) - \sum_{n \ge 1} a_{-n} (z - c)^{-n}$ extends to a holomorphic function on $|z - c| < R$.
The principal part is $0$ at a [[removable singularity|0P18JC72RE]], it has finitely many nonzero terms at a [[pole|0WPQYW8MTE]], and it has infinitely many at an [[essential singularity|0YKDJBV26S]].
For example, the principal part of $\frac{e^z}{z^2}$ at $0$ is $\frac{1}{z^2} + \frac{1}{z}$.

## Explanations

### L1 human

The principal part is the piece of the function that is responsible for its bad behavior at $c$.
If you subtract it, what is left is holomorphic at $c$.
</document>

<document source="https://tamnd.github.io/math2/node/0NDVHBVBXB.md" tag="0NDVHBVBXB" kind="concept">
# Linear recurrence with constant coefficients

## Summary

A linear recurrence with constant coefficients makes each term a fixed linear combination of the previous terms. Its solutions have closed forms in terms of the roots of its characteristic polynomial. The node [[linear recurrence with constant coefficients|081THG13JZ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0NE1DK7H4T.md" tag="0NE1DK7H4T" kind="concept">
# Right coset

## Summary

A right coset of a subgroup $H$ of a group $G$ is a set $Hg$, with $g \in G$. You get it when you multiply each element of $H$ on the right by $g$.
The right cosets of $H$ partition $G$.
</document>

<document source="https://tamnd.github.io/math2/node/0NE233WGD1.md" tag="0NE233WGD1" kind="concept">
# Conditional convergence

## Summary

A series converges conditionally when it converges but the series of absolute values diverges. The sum of such a series depends on the order of its terms. The node [[conditional convergence|02YRQXZDCS]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0NE2W2XPQJ.md" tag="0NE2W2XPQJ" kind="statement">
# Implicit function theorem for a surjective derivative

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $m \le n$, and let $F \colon U \to \mathbb{R}^m$ be of class $C^1$.
Let $p \in U$ with $F(p) = 0$, and suppose that $DF(p) \colon \mathbb{R}^n \to \mathbb{R}^m$ is surjective.
Then there are $m$ coordinates $x_{i_1}, \dots, x_{i_m}$ and an open neighbourhood $W$ of $p$ with this property. The set $\{x \in W : F(x) = 0\}$ is the graph of a $C^1$ function of the other $n - m$ coordinates.

## Remarks

The derivative is surjective if and only if the [[Jacobian matrix|0YF0VXKXGB]] $J_F(p)$ has rank $m$. Then some $m$ columns of $J_F(p)$ are linearly independent, and these columns choose the coordinates.
After a renumbering of coordinates, the statement is the [[implicit function theorem|0BXBADR94C]] with $y = (x_{i_1}, \dots, x_{i_m})$.
If $DF(x)$ is surjective at each point of the [[level set|0Y5Q8Q9WJ7]] $F^{-1}(0)$, this set is locally a graph near each of its points. So it is a $C^1$ submanifold of dimension $n - m$.
Example: for $F(x, y, z) = x^2 + y^2 + z^2 - 1$, $DF$ is surjective on the unit sphere. So the sphere is locally the graph of a function of two coordinates.

## Explanations

### L1 human

The condition that the derivative is onto means the $m$ equations are independent at the point.
Then $m$ of the variables are determined by the others, and the solution set has $n - m$ free directions.
</document>

<document source="https://tamnd.github.io/math2/node/0NEGN5CRSC.md" tag="0NEGN5CRSC" kind="concept">
# Self-adjoint operator

## Summary

A self-adjoint operator on a finite-dimensional inner product space is a linear map from the space to itself that equals its adjoint.
</document>

<document source="https://tamnd.github.io/math2/node/0NF1KGDXX3.md" tag="0NF1KGDXX3" kind="concept">
# Restriction of a function

## Summary

The restriction of a function to a subset of its domain is the function that has the same values but only on that subset.
</document>

<document source="https://tamnd.github.io/math2/node/0NF6GQNXMY.md" tag="0NF6GQNXMY" kind="concept">
# Curl

## Summary

The curl of a differentiable [[vector field|0299VCW1TH]] in $\mathbb{R}^3$ is a vector field that measures the rotation of the field at each point.
Its direction is the axis of the strongest rotation, by the right-hand rule, and its length measures the rate of rotation.
</document>

<document source="https://tamnd.github.io/math2/node/0NFAGWH84K.md" tag="0NFAGWH84K" kind="statement">
# Double negation law

## Statement

In classical logic, for each [[proposition|0ZGC7ZESYN]] $p$, the proposition $\neg \neg p$ is [[logically equivalent|06GYGS41JK]] to $p$.
$$\neg \neg p \equiv p$$

## Remarks

The implication $p \to \neg \neg p$ is also valid in intuitionistic logic.
The other direction, $\neg \neg p \to p$, needs classical logic.
This direction is the basis of [[proof by contradiction|0J0XBKG64S]].

## Explanations

### L1 human

"It is not true that $5$ is not prime" says the same as "$5$ is prime".
Two negations cancel each other.
</document>

<document source="https://tamnd.github.io/math2/node/0NFEG48RDR.md" tag="0NFEG48RDR" kind="statement">
# The minimal polynomial is irreducible

## Statement

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$ be algebraic over $K$.
Then the [[minimal polynomial|0GK1WB9R98]] $m_a$ of $a$ over $K$ is [[irreducible|00X2A9G9W3]] over $K$.

## Remarks

The proof: if $m_a = g h$ with $g, h \in K[x]$ of smaller degree, then $g(a) h(a) = 0$ in the field $L$. So $g(a) = 0$ or $h(a) = 0$, and this contradicts the minimality of $\deg m_a$.
Conversely, a monic irreducible polynomial $f \in K[x]$ with $f(a) = 0$ is the minimal polynomial of $a$.
For example, $x^3 - 2$ is irreducible over $\mathbb{Q}$, so it is the minimal polynomial of $\sqrt[3]{2}$ over $\mathbb{Q}$.

## Explanations

### L1 human

The minimal polynomial cannot be factored over the base field.
If it could, one of the factors would already vanish at the element, and that factor would be a smaller polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/0NFPJVJJHX.md" tag="0NFPJVJJHX" kind="concept">
# Probability space

## Summary

A probability space is the mathematical model of a random experiment: a sample space, a σ-algebra of events and a probability measure on the events.
The node [[probability space|0ZYBZ34RT2]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0NHGY0N0JB.md" tag="0NHGY0N0JB" kind="statement">
# Fundamental theorem of calculus

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous.
Then the function $F(x) = \int_a^x f(t) \, dt$ is differentiable on $(a, b)$, with $F'(x) = f(x)$.
If $G$ is continuous on $[a, b]$ and $G' = f$ on $(a, b)$, then
$$\int_a^b f(x) \, dx = G(b) - G(a).$$

## Remarks

The first part is the [[first fundamental theorem|063H5KJC69]]. The second part is the [[second fundamental theorem|0XYAFCN0PT]].
Together they say that differentiation and integration are inverse operations.
The theorem reduces the computation of integrals to the search for [[antiderivatives|0BW9N5CDSB]].

## History

James Gregory published the first statement and proof of a geometric form of the theorem.
Isaac Barrow proved a more general version. Isaac Newton and Gottfried Leibniz then built the calculus around it.

## Explanations

### L1 human

The fundamental theorem connects the two main ideas of calculus.
The derivative of an area function is the height of the graph, and an integral is a change in an antiderivative.
</document>

<document source="https://tamnd.github.io/math2/node/0NHK9MFFW9.md" tag="0NHK9MFFW9" kind="concept">
# Perfect matching

## Summary

A perfect matching is a matching that covers every vertex of a graph. The theorems of Hall and Tutte say when a graph has one. The node [[perfect matching|05ZPH8KF9K]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0NHXGY2C5J.md" tag="0NHXGY2C5J" kind="concept">
# Sample variance

## Summary

The sample variance of a sample measures the spread of its values around the sample mean. With the divisor $n - 1$, it is an unbiased estimator of the variance of the population.
The node [[sample variance|0JN9P4A944]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0NK8JTE1NN.md" tag="0NK8JTE1NN" kind="concept">
# von Mangoldt function

## Summary

The von Mangoldt function $\Lambda$ takes the value $\log p$ at each power of a prime $p$, and $0$ at all other positive integers.
The [[definition of the von Mangoldt function|087EMZTHWN]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0NKBMFC18R.md" tag="0NKBMFC18R" kind="definition">
# Double integral

## Definition

Let $R = [a, b] \times [c, d]$ with $a < b$ and $c < d$, and let $f \colon R \to \mathbb{R}$ be bounded.
Choose partitions $a = x_0 < \dots < x_k = b$ and $c = y_0 < \dots < y_l = d$, and let $R_{ij} = [x_{i-1}, x_i] \times [y_{j-1}, y_j]$ with area $\Delta A_{ij} = (x_i - x_{i-1})(y_j - y_{j-1})$.
For points $p_{ij} \in R_{ij}$, the sum $S = \sum_{i, j} f(p_{ij}) \, \Delta A_{ij}$ is a Riemann sum of $f$.
The function $f$ is Riemann integrable on $R$ if there is $I \in \mathbb{R}$ with the following property. For each $\varepsilon > 0$ there is $\delta > 0$ such that $|S - I| < \varepsilon$ for all such sums whose rectangles have diameter less than $\delta$.
Then $I$ is the double integral of $f$ over $R$, written $\iint_R f(x, y) \, dA$ or $\iint_R f(x, y) \, dx \, dy$.

## Remarks

Each [[continuous|0S9EK5MQPE]] function on $R$ is Riemann integrable on $R$.
For a bounded region $D \subseteq R$, one integrates $f \mathbf{1}_D$ over $R$, where $\mathbf{1}_D$ is $1$ on $D$ and $0$ elsewhere. This gives [[integrals over non-rectangular regions|0TXCQ9CST1]].
In practice one computes double integrals as [[iterated integrals|0P4WX7Y502]], by [[Fubini's theorem|0R0T1VWH5X]]. For example, $\iint_{[0, 1]^2} x y \, dA = \int_0^1 \int_0^1 x y \, dy \, dx = 1/4$.
The integral is linear in $f$, and $\iint_R 1 \, dA = (b - a)(d - c)$ is the area of $R$.

## Explanations

### L1 human

Cut the rectangle into small pieces, multiply the area of each piece by a value of $f$ in it, and add.
As the pieces get smaller, these sums approach the double integral.
For $f \ge 0$ this is the volume under the graph of $f$.
</document>

<document source="https://tamnd.github.io/math2/node/0NM626FH6Q.md" tag="0NM626FH6Q" kind="concept">
# Series

## Summary

A series is the infinite sum of the terms of a sequence. It is studied through its partial sums, the sums of finitely many first terms. The node [[series|0JH4MN4K1C]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0NM6E9VW04.md" tag="0NM6E9VW04" kind="statement">
# Bayes' theorem

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A, B \in \mathcal{F}$ with $P(A) > 0$ and $P(B) > 0$.
Then
$$P(A \mid B) = \frac{P(B \mid A) \, P(A)}{P(B)}.$$

## Remarks

Let $A_1, A_2, \dots$ be a finite or countable partition of $\Omega$ into events of positive probability. With the [[law of total probability|0E5G5FMT2R]], $P(A_j \mid B) = \dfrac{P(B \mid A_j) \, P(A_j)}{\sum_i P(B \mid A_i) \, P(A_i)}$.
In Bayesian language, $P(A)$ is the [[prior probability|00NBBSXRQS]], $P(B \mid A)$ is the likelihood and $P(A \mid B)$ is the [[posterior probability|0KJ7ZS8AJA]].
The proof: both $P(A \mid B) \, P(B)$ and $P(B \mid A) \, P(A)$ are equal to $P(A \cap B)$.

## History

The theorem is named after Thomas Bayes. Pierre-Simon Laplace found it independently and developed it further.

## Explanations

### L1 human

Bayes' theorem turns "the chance of the evidence given the cause" into "the chance of the cause given the evidence".
A medical test tells you how often sick people test positive. Bayes' theorem tells you how likely a person with a positive test is to be sick.
</document>

<document source="https://tamnd.github.io/math2/node/0NMF95YK8W.md" tag="0NMF95YK8W" kind="statement">
# Variance as the mean of the square minus the square of the mean

## Statement

Let $X$ be a real random variable with $E[X^2] < \infty$. Then
$$\operatorname{Var}(X) = E[X^2] - (E[X])^2.$$

## Remarks

Proof: with $\mu = E[X]$, $(X - \mu)^2 = X^2 - 2\mu X + \mu^2$. By [[linearity of expectation|0P7PTC8DDE]], $E[(X - \mu)^2] = E[X^2] - 2\mu^2 + \mu^2 = E[X^2] - \mu^2$.
As a corollary, $(E[X])^2 \le E[X^2]$.
For a fair die, $E[X^2] = 91/6$ and $E[X] = 7/2$, so $\operatorname{Var}(X) = 91/6 - 49/4 = 35/12$.
In floating-point arithmetic, this formula can lose much precision when $E[X^2]$ and $(E[X])^2$ are close.

## Explanations

### L1 human

This formula is often the fastest way to compute a variance: find the average of the squares and subtract the square of the average.
</document>

<document source="https://tamnd.github.io/math2/node/0NMG7W7Q11.md" tag="0NMG7W7Q11" kind="concept">
# Implication

## Summary

Implication is the logical operation "if ... then".
The implication from $p$ to $q$ is false only when $p$ is true and $q$ is false.
</document>

<document source="https://tamnd.github.io/math2/node/0NMGJJ19HD.md" tag="0NMGJJ19HD" kind="statement">
# Epsilon-net characterisation of total boundedness

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $S \subseteq X$.
Then $S$ is [[totally bounded|0WE5BBHGM6]] if and only if for each real $\varepsilon > 0$ there is a finite set $F \subseteq S$ with
$$S \subseteq \bigcup_{y \in F} B(y, \varepsilon).$$

## Remarks

Such a finite set $F$ is an $\varepsilon$-net of $S$. Here $B(y, \varepsilon)$ is an [[open ball|0DC3MPR7C9]].
Proof: an $\varepsilon$-net gives a cover by balls of diameter at most $2\varepsilon$. Conversely, from a cover by sets $A_k$ of diameter less than $\varepsilon$, choose one point of $S$ in each $A_k$ that meets $S$.
The centres can be taken in $S$ or in $X$. Both choices give the same notion.

## Explanations

### L1 human

For each $\varepsilon$, you can place finitely many points in $S$ such that each point of $S$ is within $\varepsilon$ of one of them.
For $[0, 1]$ and $\varepsilon = 0.1$, the points $0, 0.1, 0.2, \ldots, 1$ do the job.
</document>

<document source="https://tamnd.github.io/math2/node/0NMN00FNW1.md" tag="0NMN00FNW1" kind="definition">
# Transvection

## Definition

Let $R$ be a commutative ring, let $n \ge 2$, let $i \ne j$ be indices in $\{1, \dots, n\}$ and let $c \in R$.
Let $E_{ij}$ be the $n \times n$ matrix with entry $1$ in position $(i, j)$ and $0$ elsewhere.
The *transvection matrix* is
$$T_{ij}(c) = I_n + c E_{ij}.$$

## Remarks

The product $T_{ij}(c) A$ adds $c$ times row $j$ of $A$ to row $i$; this is an [[elementary matrix|01REXW3WWE]] of the row-addition type.
Its determinant is $1$, and its inverse is $T_{ij}(-c)$.
Over a field, the transvections generate the [[special linear group|0V9PMN0RBH]].
In the plane, $\begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix}$ is a horizontal shear.

## Explanations

### L1 human

A transvection slides each point parallel to one axis, by an amount that depends on another coordinate.
A deck of cards that we push sideways is a picture of a shear.
Areas and volumes do not change, which matches the determinant $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0NMWG2PHJ0.md" tag="0NMWG2PHJ0" kind="example">
# Birthday problem

## Example

Assume that each person has a birthday on one of $365$ days, and that each day is equally likely.
Assume also that the birthdays of different people are independent.
Let $p(n)$ be the probability that among $n$ people, at least two have the same birthday.
By the [[rule for the complement|0PQEBTYDZM]], for $1 \le n \le 365$,
$$p(n) = 1 - \frac{365 \cdot 364 \cdots (365 - n + 1)}{365^n}.$$
Then $p(22) \approx 0.476$ and $p(23) \approx 0.507$. So $23$ people are enough for a probability above $1/2$.

## Remarks

The answer is surprising because there are many pairs: $23$ people form $\binom{23}{2} = 253$ pairs.
For $n$ much smaller than $365$, $p(n) \approx 1 - e^{-n(n-1)/730}$.
For $n = 70$, $p(n) > 0.999$. For $n \ge 366$, $p(n) = 1$ by the pigeonhole principle.

## Explanations

### L1 human

It is easier to compute the chance that all birthdays are different.
The second person must avoid $1$ day, the third person must avoid $2$ days, and so on. These small losses multiply quickly.
</document>

<document source="https://tamnd.github.io/math2/node/0NMXDCNNKR.md" tag="0NMXDCNNKR" kind="statement">
# Trace is the sum of the eigenvalues

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Assume that the characteristic polynomial of $A$ splits over $F$ as $p_A(t) = (t - \lambda_1) \cdots (t - \lambda_n)$.
Then
$$\tr A = \lambda_1 + \lambda_2 + \cdots + \lambda_n.$$

## Remarks

The $\lambda_i$ are the [[eigenvalues|0BQAA2BK3Q]] of $A$, each repeated according to its [[algebraic multiplicity|0FY42N02DV]].
The hypothesis always holds over $\C$.
The statement follows by comparing the coefficients of $t^{n-1}$, see [[the trace as a coefficient|0FRYPZ6D0X]].
For example, $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ has eigenvalues $1$ and $3$, and trace $4$.

## Explanations

### L1 human

The trace of a matrix is the sum of its eigenvalues, counted with multiplicity.
For a diagonal matrix this is clear, because the eigenvalues are the diagonal entries.
The statement says that it stays true for every complex matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0NN5JEMTVE.md" tag="0NN5JEMTVE" kind="example">
# 0.999... equals 1

## Example

The [[decimal expansion|06XY4X9430]] $0.999\dots$, with the digit $9$ at every place after the point, represents the real number $1$.
By definition, $0.999\dots = \sum_{k=1}^{\infty} 9 \cdot 10^{-k}$.
This is a geometric series with first term $9/10$ and ratio $1/10$, thus its sum is $\frac{9/10}{1 - 1/10} = 1$.
Thus the number $1$ has two decimal expansions, $1.000\dots$ and $0.999\dots$.

## Remarks

Another proof: the $n$th truncation is $1 - 10^{-n}$. The supremum of these numbers is $1$, by the [[Archimedean property|0A3GWZ86AY]].
The same happens for each number with a finite expansion. For example, $0.25 = 0.24999\dots$.

## Explanations

### L1 human

Many people feel that $0.999\dots$ is a little less than $1$.
But the difference $1 - 0.999\dots$ is smaller than $10^{-n}$ for every $n$.
The only nonnegative real number with this property is $0$.
Another view: $1/3 = 0.333\dots$, and three times this gives $1 = 0.999\dots$.
</document>

<document source="https://tamnd.github.io/math2/node/0NNDQVJ54Q.md" tag="0NNDQVJ54Q" kind="statement">
# Closed formula for Catalan numbers

## Statement

Let $n$ be a natural number, and let $C_n$ be the $n$-th [[Catalan number|092DRBFKMG]].
Then
$$C_n = \frac{1}{n + 1} \binom{2n}{n} = \frac{(2n)!}{(n + 1)!\, n!}.$$

## Remarks

The fraction is an integer, because $C_n = \binom{2n}{n} - \binom{2n}{n+1}$.
The numerator $\binom{2n}{n}$ is the [[central binomial coefficient|01WHD9T0V6]].
Example: $C_4 = \frac{1}{5} \binom{8}{4} = \frac{70}{5} = 14$.

## Explanations

### L1 human

Among the $\binom{2n}{n}$ grid paths from $(0, 0)$ to $(n, n)$, exactly one in $n + 1$ stays below the diagonal.
So the Catalan number is the central binomial coefficient divided by $n + 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0NNQ4KFEX7.md" tag="0NNQ4KFEX7" kind="definition">
# Associativity

## Definition

Let $S$ be a set and let $\ast$ be a [[binary operation|0M418FE7BB]] on $S$.
The operation $\ast$ is associative when $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a, b, c \in S$.

## Remarks

Addition and multiplication of real numbers are associative.
Subtraction of integers is not associative: $(1 - 1) - 1 = -1$, but $1 - (1 - 1) = 1$.
Composition of maps from a set to itself is associative.
For an associative operation, each way to put parentheses in $a_1 \ast a_2 \ast \cdots \ast a_n$ gives the same result.

## Explanations

### L1 human

Associativity says that you can do the operations in a chain in any grouping.
For example, $(2 + 3) + 4$ and $2 + (3 + 4)$ are both $9$.
Thus we can write $2 + 3 + 4$ without parentheses.
</document>

<document source="https://tamnd.github.io/math2/node/0NNQY8WCW6.md" tag="0NNQY8WCW6" kind="statement">
# Schwarz's theorem for C^2 functions

## Statement

Let $U \subseteq \mathbb{R}^n$ be open and let $f \colon U \to \mathbb{R}$ be of [[class C^2|0TCBZMEYGN]] on $U$.
Then for all $i, j \in \{1, \dots, n\}$ and each $a \in U$,
$$\frac{\partial^2 f}{\partial x_i \, \partial x_j}(a) = \frac{\partial^2 f}{\partial x_j \, \partial x_i}(a).$$

## Remarks

A function of class $C^2$ is twice differentiable at each point, so this is a case of the [[symmetry of second derivatives|01V9FPBBVK]].
The result is also called Clairaut's theorem. By induction, for a function of class $C^k$ the [[partial derivatives|02RSXVM1ZN]] of order at most $k$ do not depend on the order of differentiation.
So for $C^k$ functions one can use [[multi-index notation|0PJXBR6QA3]] $\partial^\alpha f$ without ambiguity.

## Explanations

### L1 human

If the second partial derivatives are continuous, then $f_{xy} = f_{yx}$.
This saves work: in a $C^2$ function of $n$ variables, only $n(n+1)/2$ of the $n^2$ second partial derivatives can differ.
</document>

<document source="https://tamnd.github.io/math2/node/0NNR68BFRY.md" tag="0NNR68BFRY" kind="definition">
# Reduced row echelon form

## Definition

Let $F$ be a field and let $A$ be a matrix over $F$.
The matrix $A$ is in *reduced row echelon form* when these three conditions hold:

1. $A$ is in [[row echelon form|0VYCDGMJJJ]].
2. The leading entry of each nonzero row is $1$.
3. In each column that has a leading entry, all the other entries are $0$.

## Remarks

The matrix $\begin{pmatrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$ is in reduced row echelon form.
Each matrix is [[row equivalent|006BHFWMAP]] to exactly one matrix in reduced row echelon form.
[[Gauss–Jordan elimination|098D6FQN3N]] computes this matrix.
The number of nonzero rows of the reduced row echelon form of $A$ is the [[rank|0ZC9YTXZRY]] of $A$.

## Explanations

### L1 human

Reduced row echelon form is the simplest form that row operations can reach.
Each pivot is a $1$, and it has zeros above it and below it.
From this form we can read the solutions of the system directly.
</document>

<document source="https://tamnd.github.io/math2/node/0NNRDW10RZ.md" tag="0NNRDW10RZ" kind="concept">
# Zero divisor

## Summary

A zero divisor of a ring is an element that gives zero when you multiply it by some nonzero element.
The [[definition of a zero divisor|0K26YKNBSA]] gives the precise condition.
</document>

<document source="https://tamnd.github.io/math2/node/0NNRNGKMFM.md" tag="0NNRNGKMFM" kind="definition">
# Moment-generating function

## Definition

Let $X$ be a real random variable.
The moment-generating function of $X$ is the function
$$M_X(t) = E\left[e^{tX}\right], \quad t \in \mathbb{R},$$
with values in $(0, \infty]$.

## Remarks

Always $M_X(0) = 1$. The set of $t$ with $M_X(t) < \infty$ is an interval that contains $0$.
If $M_X$ is finite on an open interval around $0$, then all [[moments|0PH0VHTQS0]] of $X$ are finite and $M_X$ determines the law of $X$.
Some random variables have $M_X(t) = \infty$ for all $t \ne 0$. An example is a Cauchy random variable.
For $X \sim \operatorname{Exp}(\lambda)$, $M_X(t) = \lambda / (\lambda - t)$ for $t < \lambda$, and $M_X(t) = \infty$ for $t \ge \lambda$.

## Explanations

### L1 human

The moment-generating function is a tool that turns a distribution into a function of one real variable.
It is useful because sums of independent variables become products, and moments become derivatives.
</document>

<document source="https://tamnd.github.io/math2/node/0NNYDD1AVR.md" tag="0NNYDD1AVR" kind="definition">
# Eulerian trail

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] or a [[multigraph|01M5T0KDG7]].
An Eulerian trail of $G$ is a [[trail|062MQGP8M5]] in $G$ that uses every edge of $G$ exactly once.
An Eulerian circuit is an Eulerian trail that starts and ends at the same vertex.

## Remarks

A graph with an Eulerian circuit is called Eulerian.
[[Euler's theorem|020422MDQF]] characterizes the connected graphs with an Eulerian circuit by the parity of the degrees.
The [[Seven Bridges of Königsberg|0YCW781Q9D]] form a multigraph with no Eulerian trail.

## Explanations

### L1 human

An Eulerian trail draws the whole graph in one stroke, without lifting the pen and without going over any line twice.
An Eulerian circuit also ends where it starts.
</document>

<document source="https://tamnd.github.io/math2/node/0NP5N5PYW5.md" tag="0NP5N5PYW5" kind="example">
# Cantor pairing function

## Example

The Cantor pairing function is the function $\pi \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ with
$$\pi(x, y) = \frac{(x + y)(x + y + 1)}{2} + y.$$
It is a [[bijection|0SNQEF7Z53]].
It lists the pairs along the diagonals $x + y = 0, 1, 2, \dots$: $\pi(0, 0) = 0$, $\pi(1, 0) = 1$, $\pi(0, 1) = 2$, $\pi(2, 0) = 3$.

## Remarks

So $\mathbb{N} \times \mathbb{N}$ is [[countably infinite|0SEZ3JGZP7]]. See [[product of countable sets is countable|08G4XH86JC]].
The value $(x + y)(x + y + 1)/2$ is the number of pairs on the earlier diagonals, and $y$ is the position on the diagonal.

## Explanations

### L1 human

Write the pairs $(x, y)$ in an infinite grid.
Walk through the grid along the short diagonals, one after the other, and number the pairs as you go.
Each pair gets exactly one number, and each number gets exactly one pair.
</document>

<document source="https://tamnd.github.io/math2/node/0NPPGHQNXQ.md" tag="0NPPGHQNXQ" kind="concept">
# Euclidean distance

## Summary

The Euclidean distance between two points of a Euclidean space is the length of the line segment that joins them. In Cartesian coordinates, the Pythagorean theorem gives a formula for it.
</document>

<document source="https://tamnd.github.io/math2/node/0NPW1DWECM.md" tag="0NPW1DWECM" kind="definition">
# Algebraic element

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$.
The element $a$ is algebraic over $K$ if there is a nonzero polynomial $f \in K[x]$ with $f(a) = 0$.

## Remarks

The number $\sqrt{2}$ is algebraic over $\mathbb{Q}$, because it is a [[root|00EHX6FWC0]] of $x^2 - 2$.
Every element $a \in K$ is algebraic over $K$, as a root of $x - a$.
The element $a$ is algebraic over $K$ if and only if $[K(a) : K]$ is finite.
An element that is not algebraic is [[transcendental|00SW8B91E7]].

## Explanations

### L1 human

An algebraic element solves a polynomial equation with coefficients in the smaller field.
The algebraic numbers are the complex numbers that are algebraic over $\mathbb{Q}$.
</document>

<document source="https://tamnd.github.io/math2/node/0NQ1H2X495.md" tag="0NQ1H2X495" kind="concept">
# Quotient topology

## Summary

The quotient topology is the topology on a set of equivalence classes that comes from a topological space.
It formalises the idea of gluing points of a space together.
</document>

<document source="https://tamnd.github.io/math2/node/0NQ5798SB1.md" tag="0NQ5798SB1" kind="definition">
# Student's t-distribution

## Definition

Let $\nu > 0$ be a real number.
Student's t-distribution with $\nu$ degrees of freedom, written $t_\nu$, is the law on $\mathbb{R}$ with the density
$$f(t) = \frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu \pi}\, \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{t^2}{\nu}\right)^{-(\nu + 1)/2}, \qquad t \in \mathbb{R}.$$

## Remarks

Let $Z$ be standard normal and $V \sim \chi^2_\nu$ (see [[chi-squared distribution|0RFMMBHFTV]]), [[independent|0XCE070V8B]]. Then $Z / \sqrt{V / \nu} \sim t_\nu$.
For an i.i.d. $\mathcal{N}(\mu, \sigma^2)$ sample of size $n \ge 2$, $\frac{\bar{X}_n - \mu}{S_n / \sqrt{n}} \sim t_{n - 1}$, with the [[sample mean|0QNRBHWV35]] $\bar{X}_n$ and the [[sample standard deviation|0JN9P4A944]] $S_n$.
The mean is $0$ for $\nu > 1$, and the variance is $\nu / (\nu - 2)$ for $\nu > 2$. For $\nu = 1$, $t_1$ is the standard [[Cauchy distribution|0NA6Q82YR7]].
As $\nu \to \infty$, the density of $t_\nu$ converges to the standard normal density.



## Explanations

### L1 human

The t-distribution looks like the bell curve, but it gives more probability to values far from $0$.
It takes into account the extra uncertainty when the standard deviation comes from a small sample. With many degrees of freedom, it is almost the normal law.
</document>

<document source="https://tamnd.github.io/math2/node/0NQMZ89X4A.md" tag="0NQMZ89X4A" kind="concept">
# Transcendental element

## Summary

An element of a field extension is transcendental over the smaller field if it is not algebraic over it.
The [[definition of a transcendental element|00SW8B91E7]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0NQSV2EWDR.md" tag="0NQSV2EWDR" kind="concept">
# Winding number

## Summary

The winding number of a closed curve around a point is the integer that counts the counterclockwise turns of the curve around the point. See the [[definition|0X7SBS8E0F]].
</document>

<document source="https://tamnd.github.io/math2/node/0NR82KMYKD.md" tag="0NR82KMYKD" kind="statement">
# Triangle inequality for integrals

## Statement

Let $f \colon [a, b] \to \mathbb{R}$ be Riemann integrable.
Then $|f|$ is Riemann integrable on $[a, b]$, and
$$\Big| \int_a^b f(x) \, dx \Big| \le \int_a^b |f(x)| \, dx.$$

## Remarks

The inequality follows from $-|f| \le f \le |f|$ and the [[monotonicity of the integral|0MBWQKM8QR]].
The converse of the first claim fails. On $[0, 1]$, let $f = 1$ on the rationals and $f = -1$ elsewhere. Then $|f| = 1$ is integrable, but $f$ is not.
Mathlib states the inequality as `intervalIntegral.norm_integral_le_integral_norm`.

## Explanations

### L1 human

Cancellation can only make an integral smaller in size.
The size of the integral is at most the integral of the size of the function.
</document>

<document source="https://tamnd.github.io/math2/node/0NR95CTFCZ.md" tag="0NR95CTFCZ" kind="example">
# Petersen graph

## Example

The Petersen graph is the [[graph|0JD5C82QMB]] whose vertices are the $2$-element subsets of $\{1, 2, 3, 4, 5\}$.
Two vertices are adjacent if and only if they are disjoint subsets.

It has $\binom{5}{2} = 10$ vertices and $15$ edges, and it is $3$-[[regular|0KA4S8R39M]].

## Remarks

Its shortest cycle has length $5$. It is not [[Hamiltonian|090WPKFABX]], and it is not [[planar|029JQEK15B]].
Its [[chromatic number|0C5M7AM6FH]] is $3$, and its edges cannot be coloured with $3$ colours.
A common drawing shows it as a pentagon outside, a pentagram inside, and five spokes.

## Explanations

### L1 human

The Petersen graph is a small graph with $10$ vertices and $15$ edges.
It is a standard counterexample: many statements that seem true for all cubic graphs fail for it.
</document>

<document source="https://tamnd.github.io/math2/node/0NRJGW864D.md" tag="0NRJGW864D" kind="object">
# Quaternion group

## Description

The quaternion group $Q_8$ is the set $\{1, -1, i, -i, j, -j, k, -k\}$ of quaternions, with quaternion multiplication as the operation.
The multiplication follows from the rules $i^2 = j^2 = k^2 = ijk = -1$, where $-1$ commutes with each element.
These rules give $ij = k$, $jk = i$, $ki = j$ and $ji = -k$, $kj = -i$, $ik = -j$.
The group has order $8$, and it is not [[abelian|082GHNPJ53]].

## Remarks

The element $-1$ is the only element of order $2$, and the six elements $\pm i, \pm j, \pm k$ have order $4$.
The [[centre|0DZ93493VN]] and the [[commutator subgroup|0ASGSTXSBW]] of $Q_8$ are both $\{1, -1\}$, and $Q_8 / \{1, -1\}$ is the [[Klein four-group|0PMRQJVTME]].
Each subgroup of $Q_8$ is [[normal|0CB0DX5BVV]], although $Q_8$ is not abelian.
The groups $Q_8$ and $D_4$ are not isomorphic, because $D_4$ has five elements of order $2$.

## Explanations

### L1 human

The quaternion group is a small group of order $8$ in which the elements do not commute.
You can think of $i$, $j$, $k$ as three square roots of $-1$ whose products go around in a circle.
It is a good test case, because many simple guesses about groups fail for it.
</document>

<document source="https://tamnd.github.io/math2/node/0NRQPQX3VA.md" tag="0NRQPQX3VA" kind="statement">
# Composition of continuous functions is continuous

## Statement

Let $D, E \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function with $f(D) \subseteq E$, and let $g \colon E \to \mathbb{R}$.
If $f$ is continuous at $a \in D$ and $g$ is continuous at $f(a)$, then $g \circ f$ is continuous at $a$.
In particular, if $f$ and $g$ are [[continuous|0P27W9RW0G]], then $g \circ f$ is continuous.

## Remarks

Given $\varepsilon > 0$, choose $\eta > 0$ for $g$ at $f(a)$, and then $\delta > 0$ for $f$ at $a$ and the tolerance $\eta$. See the [[epsilon-delta form|0W8YHHGCJG]].
For example, $x \mapsto \sin(x^2)$ is continuous.
Mathlib states it for topological spaces, as `Continuous.comp`.

## Explanations

### L1 human

A small change of $x$ gives a small change of $f(x)$.
This in turn gives a small change of $g(f(x))$.
</document>

<document source="https://tamnd.github.io/math2/node/0NRT0E2QT2.md" tag="0NRT0E2QT2" kind="statement">
# Connected components are closed

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $x \in X$.
The connected component of $x$ is the [[union|05GK32H72H]] $C(x)$ of all [[connected subsets|0R37DFJKFW]] of $X$ that contain $x$.
Then $C(x)$ is connected and [[closed|0G5AV9Q4RR]] in $X$.

## Remarks

Proof: a union of connected sets with a common point is connected, so $C(x)$ is the largest connected set that contains $x$. The [[closure|0DRKT1QJR7]] of $C(x)$ is also connected, so it equals $C(x)$.
The components of $X$ form a partition of $X$.
Components need not be open. In $\mathbb{Q}$, the components are the single points, which are not open.
If $X$ has only finitely many components, then each of them is also open.

## Explanations

### L1 human

The component of a point is the largest piece of the space that contains the point and is in one piece.
Taking the closure cannot make the piece larger, so each component is closed.
</document>

<document source="https://tamnd.github.io/math2/node/0NRYHJPQG2.md" tag="0NRYHJPQG2" kind="concept">
# Linear map

## Summary

A linear map is a map between vector spaces over the same field that preserves addition and scalar multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/0NRYPM9Y27.md" tag="0NRYPM9Y27" kind="definition">
# Finite extension

## Definition

A [[field extension|0WQ1A8SHTZ]] $L/K$ is finite if its [[degree|0GS74FGFEK]] $[L : K]$ is finite.
Otherwise $L/K$ is an infinite extension.

## Remarks

A finite extension is not the same as an extension of finite fields. For example, $\mathbb{C}/\mathbb{R}$ is finite, and both fields are infinite.
The extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is finite of degree $3$.
The field $K(x)$ of rational functions is an infinite extension of $K$, because $1, x, x^2, \dots$ are linearly independent over $K$.
[[Finite extensions are algebraic|0DFYTNZ6AM]].

## Explanations

### L1 human

A finite extension is one that is a finite-dimensional vector space over the smaller field.
Each element of it is a combination of finitely many fixed elements, with coefficients in the smaller field.
</document>

<document source="https://tamnd.github.io/math2/node/0NS0TDA4E2.md" tag="0NS0TDA4E2" kind="statement">
# Characteristic function of the normal distribution

## Statement

Let $\mu \in \mathbb{R}$, let $\sigma^2 > 0$ and let $X \sim \mathcal{N}(\mu, \sigma^2)$. Then for each $t \in \mathbb{R}$,
$$\varphi_X(t) = E\left[e^{itX}\right] = \exp\left(i \mu t - \tfrac{1}{2} \sigma^2 t^2\right).$$
In particular, the standard normal distribution has $\varphi(t) = e^{-t^2/2}$.

## Remarks

One proof: the [[moment-generating function|0B1G4E21G7]] $E[e^{zX}] = \exp(\mu z + \sigma^2 z^2/2)$ holds for real $z$. Both sides are entire functions of $z$, so the identity extends to all complex $z$, and one puts $z = it$.
Another proof: $\varphi(t) = e^{-t^2/2}$ for $\mathcal{N}(0, 1)$ is the solution of $\varphi'(t) = -t \varphi(t)$ with $\varphi(0) = 1$. Integration by parts gives this differential equation.
So the standard normal density is, up to the factor $\sqrt{2\pi}$, its own Fourier transform.

## Explanations

### L1 human

The characteristic function of a normal variable is again a bell-shaped function.
A narrow normal distribution, with small $\sigma$, has a wide characteristic function, and conversely.
</document>

<document source="https://tamnd.github.io/math2/node/0NS0X75RS1.md" tag="0NS0X75RS1" kind="statement">
# Derivative of a map into R^n by components

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f = (f_1, \dots, f_m) \colon U \to \mathbb{R}^m$, let $a \in U$, and let $L \colon \mathbb{R}^n \to \mathbb{R}^m$ be linear with components $L_1, \dots, L_m$.
Then $f$ is [[differentiable|0135XZZQ8M]] at $a$ with $Df(a) = L$ if and only if, for each $j \in \{1, \dots, m\}$, the component $f_j$ is differentiable at $a$ with $Df_j(a) = L_j$.

## Remarks

So you can differentiate a vector-valued map one component at a time. The derivative of $f_j$ is row $j$ of the [[Jacobian matrix|047NDTEPCV]] of $f$.
The proof compares norms: each $|v_j| \le \|v\|$, and $\|v\| \le \sqrt{m} \max_j |v_j|$, for $v \in \mathbb{R}^m$.
This is the case of [[a map into a product|0QQMXFTX0A]] with $m$ factors equal to $\mathbb{R}$.

## Explanations

### L1 human

The map $f(t) = (\cos t, \sin t)$ has the derivative $h \mapsto (-h \sin t, \, h \cos t)$ at $t$.
You get it by differentiating $\cos t$ and $\sin t$ separately.
</document>

<document source="https://tamnd.github.io/math2/node/0NS1YTGM19.md" tag="0NS1YTGM19" kind="definition">
# Product of two subgroups

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ and $K$ be [[subgroups|0X1JXT47GA]] of $G$.
The product of $H$ and $K$ is the subset $HK = \{hk : h \in H, k \in K\}$ of $G$.

## Remarks

The set $HK$ is a subgroup of $G$ if and only if $HK = KH$.
It need not be a subgroup. In the symmetric group $S_3$, two distinct subgroups of order $2$ have a product with $4$ elements, and $4$ does not divide $6$.
When $HK$ is a subgroup, it is the [[subgroup generated by|0RP7VWQ0YN]] $H \cup K$.

## Explanations

### L1 human

To multiply two subgroups, multiply each element of the first by each element of the second.
The result contains both subgroups, but it may fail to be a subgroup, because a product $hk h'k'$ need not have the form $h''k''$.
In an abelian group, the product of two subgroups is always a subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/0NS2QYVC4Y.md" tag="0NS2QYVC4Y" kind="definition">
# Gradient

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $a \in U$ be a point at which $f$ is [[differentiable|0135XZZQ8M]].
The gradient of $f$ at $a$ is the unique vector $\nabla f(a) \in \mathbb{R}^n$ such that
$$Df(a)(v) = \nabla f(a) \cdot v \quad \text{for each } v \in \mathbb{R}^n,$$
where $\cdot$ is the dot product of [[Euclidean space|0XPR1E85W0]].
In coordinates, it is the vector of [[partial derivatives|09EGC6F8PT]]
$$\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \dots, \frac{\partial f}{\partial x_n}(a)\right).$$
Other notations are $\operatorname{grad} f(a)$ and $\nabla f(a)$, where $\nabla$ is the [[del operator|0H6DP8CHND]].

## Remarks

The vector exists and is unique because each linear map $\mathbb{R}^n \to \mathbb{R}$ is $v \mapsto w \cdot v$ for exactly one $w \in \mathbb{R}^n$.
For $f(x, y) = x^2 + 3 x y$, the gradient is $\nabla f(x, y) = (2x + 3y, \, 3x)$.
A gradient is a column vector, and the derivative $Df(a)$ is the row vector of the same entries. Some books do not make this distinction.
The map $a \mapsto \nabla f(a)$ is a [[vector field|0299VCW1TH]] on $U$, the gradient field of $f$.

## Explanations

### L1 human

The gradient collects all slopes of $f$ along the axes into one arrow.
On a map of a hill, the gradient at a point is an arrow that points straight uphill. The arrow is longer where the hill is steeper.
</document>

<document source="https://tamnd.github.io/math2/node/0NS58JP68E.md" tag="0NS58JP68E" kind="concept">
# Construction of the integers from the natural numbers

## Summary

The integers can be constructed as classes of pairs of [[natural numbers|0MWKJ44V5A]].
The idea is that the pair $(a, b)$ stands for the difference $a - b$.
On the set $\mathbb{N} \times \mathbb{N}$, define $(a, b) \sim (c, d)$ when $a + d = b + c$. This relation is an equivalence relation.
The set $\mathbb{Z}$ of [[integers|0C4JS9RF5M]] is the set of the equivalence classes $[(a, b)]$.
The operations on classes are $[(a, b)] + [(c, d)] = [(a + c, b + d)]$ and $[(a, b)] \cdot [(c, d)] = [(a c + b d, a d + b c)]$.
These operations do not depend on the choice of the pairs in the classes.

## Remarks

The negative of $[(a, b)]$ is $[(b, a)]$.
The map $n \mapsto [(n, 0)]$ embeds $\mathbb{N}$ in $\mathbb{Z}$, and it respects addition and multiplication.
Each class contains a pair of the form $(n, 0)$ or $(0, n)$. These classes are the integers $n$ and $-n$.

## Explanations

### L1 human

The pairs $(5, 2)$, $(4, 1)$ and $(3, 0)$ all stand for the integer $3$, because each first number is $3$ more than the second.
The pairs $(2, 5)$ and $(0, 3)$ stand for $-3$.
The construction makes new numbers from the natural numbers only, without the assumption that negative numbers already exist.
</document>

<document source="https://tamnd.github.io/math2/node/0NS8BP96QX.md" tag="0NS8BP96QX" kind="definition">
# Local minimum

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$.
The function $f$ has a local minimum at $a$ if there is $\delta > 0$ with $f(x) \ge f(a)$ for all $x \in D$ with $|x - a| < \delta$.
It is a strict local minimum if $f(x) > f(a)$ for all such $x \ne a$.

## Remarks

So $f$ has a local minimum at $a$ exactly when $-f$ has a [[local maximum|00999TYPXB]] at $a$.
For example, $x^2$ has a strict local minimum at $0$.
Mathlib defines it as `IsLocalMin`.

## Explanations

### L1 human

A local minimum is the bottom of a valley of the graph.
Nearby values are not lower.
</document>

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# Meromorphic function

## Definition

Let $U \subseteq \mathbb{C}$ be an open set.
A function $f$ is meromorphic on $U$ if there is a set $P \subseteq U$ with these properties.

1. $P$ has no accumulation point in $U$.
2. $f$ is [[holomorphic|0DG0Z72CQD]] on $U \setminus P$.
3. Each point of $P$ is a [[pole|0WPQYW8MTE]] of $f$.

## Remarks

Rational functions $p/q$, with polynomials $p$ and $q \neq 0$, are meromorphic on $\mathbb{C}$. So are $\tan z$, $\frac{1}{\sin z}$ and the Gamma function.
If $U$ is a [[domain|097KNJFJ7V]], $g$ and $h$ are holomorphic on $U$, and $h$ is not identically zero, then $g/h$ is meromorphic on $U$. The meromorphic functions on a domain form a field.
The function $e^{1/z}$ is not meromorphic on $\mathbb{C}$, because its singularity at $0$ is [[essential|0YKDJBV26S]].
The set $P$ can be infinite. It is countable, and each compact subset of $U$ contains only finitely many points of $P$.

## Explanations

### L1 human

A meromorphic function is holomorphic except at some isolated points, where it has poles.
For example, $\frac{1}{z^2 + 1}$ is holomorphic on $\mathbb{C}$ except at $i$ and $-i$, where it goes to infinity.
</document>

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# Normal space

## Summary

A normal space is a topological space in which any two disjoint closed sets can be put into disjoint open sets.
In normal spaces, there are many continuous real functions, by Urysohn's lemma.
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<document source="https://tamnd.github.io/math2/node/0NT64JV96A.md" tag="0NT64JV96A" kind="concept">
# Ordinary differential equation

## Summary

An ordinary differential equation is a [[differential equation|003R0V5M7V]] for an unknown function of one variable.
It relates the function, its derivatives and the variable. See the [[definition|0P97V20RW7]].
</document>

<document source="https://tamnd.github.io/math2/node/0NTF6DMVAP.md" tag="0NTF6DMVAP" kind="concept">
# Direct proof

## Summary

A direct proof of an [[implication|0ZNX6DQ3EJ]] $p \to q$ assumes $p$ and then gets $q$ by a chain of valid steps.
The steps use axioms, definitions and theorems that are already proved.

## Remarks

Example: if $n$ is odd, then $n^2$ is odd. Write $n = 2k + 1$. Then $n^2 = 2(2k^2 + 2k) + 1$, which is odd.
When a direct proof is hard, a [[proof by contrapositive|0BB5TPGQ1N]] or a [[proof by contradiction|0J0XBKG64S]] can help.

## Explanations

### L1 human

In a direct proof you go straight from the hypothesis to the conclusion.
You write down what the hypothesis means, then you compute or reason until you reach the conclusion.
</document>

<document source="https://tamnd.github.io/math2/node/0NTM3J5MFE.md" tag="0NTM3J5MFE" kind="statement">
# Discrete Grönwall inequality

## Statement

Let $(u_k)_{k \ge 0}$ be a sequence of real numbers with $u_0 \ge 0$, and let $(b_k)_{k \ge 0}$ and $(c_k)_{k \ge 0}$ be sequences of non-negative real numbers.
Assume that $u_{k+1} \le (1 + c_k)\, u_k + b_k$ for each $k \ge 0$.
Then for each $n \ge 0$,
$$u_n \le \left( u_0 + \sum_{k=0}^{n-1} b_k \right) \exp\left( \sum_{k=0}^{n-1} c_k \right).$$

## Remarks

Proof idea: by induction, $u_n \le u_0 \prod_{k<n} (1 + c_k) + \sum_{k<n} b_k \prod_{k<j<n} (1 + c_j)$. Then use $1 + c \le e^{c}$.
It is the discrete analogue of [[Grönwall's inequality|07EJHQYR9A]], with sums in place of integrals.
It bounds the growth of errors in numerical methods such as the [[Euler method|0JARTXX4XP]].

## Explanations

### L1 human

At each step, the quantity grows at most by a factor $1 + c_k$ and an extra amount $b_k$.
The inequality says that the total growth is at most exponential in the sum of the rates.
</document>

<document source="https://tamnd.github.io/math2/node/0NTSE7EREA.md" tag="0NTSE7EREA" kind="statement">
# Exterior angle theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in a Euclidean space $\mathbb{R}^n$, and let $\angle$ denote the [[angle at a point|0GJAJC6TNZ]]. Let $D$ be a point such that $C$ lies strictly between $B$ and $D$ on the line $BC$.
Then the exterior angle at $C$ is the sum of the two interior angles at the other vertices:
$$\angle ACD = \angle BAC + \angle CBA.$$

## Remarks

The angles $\angle ACD$ and $\angle ACB$ add up to $\pi$. Thus the statement follows from the [[angle sum of a triangle|05MDKQBEAF]].
A consequence is that the exterior angle is greater than each of the two remote interior angles. This weaker form does not need the parallel postulate.

## History

In Book I of Euclid's Elements, the weaker form is Proposition 16. The form with the sum is part of Proposition 32 of the same book. Many school books call the form with the sum the exterior angle theorem.

## Explanations

### L1 human

Extend one side of a triangle past a corner. The angle outside the triangle at that corner is the exterior angle. It is exactly as large as the two far angles of the triangle together.
</document>

<document source="https://tamnd.github.io/math2/node/0NTWN57GM3.md" tag="0NTWN57GM3" kind="concept">
# Multiplicative function

## Summary

A multiplicative function is an arithmetic function with $f(1) = 1$ that takes products of coprime integers to products of values.
The [[definition of a multiplicative function|0AP80TT660]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0NV6E43P4A.md" tag="0NV6E43P4A" kind="statement">
# Z/pZ is a field for p prime

## Statement

Let $n \ge 2$ be an integer.
The ring $\mathbb{Z}/n\mathbb{Z}$ of integers modulo $n$ is a [[field|0QA8CHH8MG]] if and only if $n$ is a prime number.

## Remarks

If $n = a b$ with $1 < a, b < n$, then the classes of $a$ and $b$ are nonzero and their product is $0$. So $\mathbb{Z}/n\mathbb{Z}$ is not an integral domain.
If $n = p$ is prime and $p \nmid a$, then Bézout's identity gives $a x + p y = 1$ for some integers $x, y$. So $x$ is an inverse of $a$ modulo $p$.
The field $\mathbb{Z}/p\mathbb{Z}$ is often written $\mathbb{F}_p$. It is the smallest [[finite field|08RTDHX1SK]] of characteristic $p$.

## Explanations

### L1 human

Modulo $5$, every nonzero class has an inverse: $1 \cdot 1 = 1$, $2 \cdot 3 = 6 \equiv 1$ and $4 \cdot 4 = 16 \equiv 1$.
Modulo $6$, the class of $2$ has no inverse, because $2 x$ is always even.
Division works modulo $n$ exactly when $n$ is prime.
</document>

<document source="https://tamnd.github.io/math2/node/0NVPNC12JY.md" tag="0NVPNC12JY" kind="definition">
# Open cover

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $A \subseteq X$.
An open cover of $A$ is a family $(U_i)_{i \in I}$ of [[open subsets|01ZQQDCERP]] of $X$ with $A \subseteq \bigcup_{i \in I} U_i$.
A subcover is a subfamily $(U_j)_{j \in J}$, with $J \subseteq I$, that is still an open cover of $A$.
A subcover is finite when $J$ is a finite set.

## Remarks

The intervals $(-n, n)$ for $n \ge 1$ form an open cover of $\mathbb{R}$. It has no finite subcover.
The intervals $(1/n, 1)$ for $n \ge 2$ form an open cover of $(0, 1)$ with no finite subcover.
A set $K$ is [[compact|00R0X1WGPX]] when each open cover of $K$ has a finite subcover.

## Explanations

### L1 human

An open cover is a collection of open sets that together cover the whole set.
You can think of them as patches, and a finite subcover uses only finitely many patches to cover the set.
</document>

<document source="https://tamnd.github.io/math2/node/0NW1JP7TBR.md" tag="0NW1JP7TBR" kind="statement">
# Multiplicativity of the Legendre symbol

## Statement

Let $p$ be an odd prime, and let $a$ and $b$ be integers.
Then
$$\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right) \left(\frac{b}{p}\right),$$
where the symbols are [[Legendre symbols|01J8AMV9RS]].

## Remarks

The proof follows from [[Euler's criterion|0M7DEDKSYB]], because $(ab)^{(p-1)/2} = a^{(p-1)/2} b^{(p-1)/2}$.
In words: modulo $p$, the product of two nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue.
For example, $3$ and $5$ are nonresidues modulo $7$, and $15 \equiv 1 \pmod 7$ is a residue.
So the map $a \mapsto \left(\frac{a}{p}\right)$ is a completely multiplicative function of $a$.

## Explanations

### L1 human

To decide whether a product is a square modulo $p$, you can test each factor.
Two "no" answers give "yes", just as two negative signs give a positive sign.
</document>

<document source="https://tamnd.github.io/math2/node/0NW6CH38YR.md" tag="0NW6CH38YR" kind="concept">
# T1 space

## Summary

A T1 space is a topological space in which, for any two distinct points, each one has an open set that misses the other.
Equivalently, every one-point set is closed.
</document>

<document source="https://tamnd.github.io/math2/node/0NWXRAMEE7.md" tag="0NWXRAMEE7" kind="statement">
# Second Borel–Cantelli lemma

## Statement

Let $A_1, A_2, \dots$ be [[mutually independent|0GCVEV6BJA]] events in a probability space $(\Omega, \mathcal{F}, P)$ with $\sum_{n=1}^{\infty} P(A_n) = \infty$.
Then
$$P\left(\limsup_{n \to \infty} A_n\right) = 1.$$
So almost surely, infinitely many of the events $A_n$ occur.

## Remarks

Proof: the complement of $\limsup_n A_n$ is $\bigcup_n \bigcap_{m \ge n} A_m^c$. By independence and $1 - x \le e^{-x}$, $P\left(\bigcap_{m = n}^{N} A_m^c\right) = \prod_{m=n}^{N} (1 - P(A_m)) \le \exp\left(-\sum_{m=n}^{N} P(A_m)\right)$.
The right side tends to $0$ as $N \to \infty$. So each $\bigcap_{m \ge n} A_m^c$ has probability $0$, and so does their union.
Independence is necessary. For $A_n = A$ with $0 < P(A) < 1$ for all $n$, the sum diverges but $P(\limsup_n A_n) = P(A) < 1$.
With the [[first lemma|0T6WS7TT77]], it follows that for independent events, $P(\limsup_n A_n)$ is $0$ or $1$.

## Explanations

### L1 human

If independent events have chances that add up to infinity, then infinitely many of them happen, with probability $1$.
For example, in an infinite sequence of tosses of a fair coin, the pattern "ten heads in a row" appears infinitely often.
</document>

<document source="https://tamnd.github.io/math2/node/0NX5NWAFC9.md" tag="0NX5NWAFC9" kind="statement">
# Weierstrass M-test

## Statement

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ for $n \ge 0$.
Assume that there are real numbers $M_n \ge 0$ with $|f_n(x)| \le M_n$ for all $n$ and all $x \in S$, and $\sum_{n=0}^{\infty} M_n < \infty$.
Then the [[series of functions|0MRE6XRE5V]] $\sum_{n=0}^{\infty} f_n$ converges absolutely and uniformly on $S$.

## Remarks

Absolute convergence at each $x$ follows from the [[comparison test|04Z5G0THN6]]. For $m < n$, $|s_n(x) - s_m(x)| \le \sum_{k=m+1}^{n} M_k$. So the partial sums are [[uniformly Cauchy|01XAY17QA3]].
For example, $\sum \frac{\sin(n x)}{n^2}$ converges uniformly on $\mathbb{R}$, with $M_n = 1/n^2$.
Mathlib states it as `tendstoUniformlyOn_tsum`.

## Explanations

### L1 human

If each function is bounded by a number, and these numbers have a finite sum, then the series of functions converges uniformly.
The bounds do not depend on $x$, so the convergence is the same everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/0NXGPJWNFN.md" tag="0NXGPJWNFN" kind="statement">
# Sum of independent normal random variables

## Statement

Let $X \sim \mathcal{N}(\mu_1, \sigma_1^2)$ and $Y \sim \mathcal{N}(\mu_2, \sigma_2^2)$ be [[independent|0E9502NT92]], with $\sigma_1^2, \sigma_2^2 > 0$. Then
$$X + Y \sim \mathcal{N}(\mu_1 + \mu_2, \, \sigma_1^2 + \sigma_2^2).$$

## Remarks

Proof with [[characteristic functions|0NS0TDA4E2]]: $\varphi_{X+Y}(t) = \varphi_X(t) \varphi_Y(t) = \exp(i(\mu_1 + \mu_2)t - (\sigma_1^2 + \sigma_2^2) t^2 / 2)$. By the [[uniqueness theorem|0T8BYAFEXX]], this fixes the law.
By induction, a sum of $n$ independent normal variables is normal. Its mean is the sum of the means and its variance is the sum of the variances.
With the rule for $aX + b$, every linear combination $a_1 X_1 + \dots + a_n X_n$ of independent normal variables is normal, if some $a_i \ne 0$.
The independence condition is necessary. Two normal variables that are not jointly normal can have a sum that is not normal.

## Explanations

### L1 human

Adding independent normal errors gives another normal error. The means add, and the variances add, but the standard deviations do not add.
</document>

<document source="https://tamnd.github.io/math2/node/0NXJM01QX1.md" tag="0NXJM01QX1" kind="definition">
# Euclidean isometry

## Definition

Let $d$ be the [[Euclidean distance|0NPPGHQNXQ]] on the [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$.
A Euclidean isometry of $\mathbb{R}^n$ is a map $f \colon \mathbb{R}^n \to \mathbb{R}^n$ with
$$d(f(p), f(q)) = d(p, q) \quad \text{for all } p, q \in \mathbb{R}^n.$$

## Remarks

Each Euclidean isometry is an [[affine map|0JF0TVYGWT]] $f(x) = A x + b$, with an orthogonal $n \times n$ matrix $A$ and a vector $b \in \mathbb{R}^n$. Thus $f$ is a bijection, and its inverse is also an isometry.
The isometry is direct if $\det A = 1$ and indirect if $\det A = -1$. The direct isometries keep the orientation of figures.
Mathlib's `AffineIsometryEquiv` is an affine bijection that keeps distances, between normed affine spaces.

## Explanations

### L1 human

An isometry moves the plane or the space without stretching, bending or tearing it. Lengths and angles do not change. In the plane, examples are a shift, a turn about a point and a mirror image in a line.
</document>

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# Adjoining a root of a polynomial

## Summary

Let $f$ be an irreducible polynomial over a field $K$.
To adjoin a root of $f$ to $K$, one forms the field $K[x]/(f)$, in which the class of $x$ is a root of $f$.
The [[definition of this construction|0M893X7YY7]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0NYM0ZGAK9.md" tag="0NYM0ZGAK9" kind="statement">
# Theorem on friends and strangers

## Statement

Colour each edge of the [[complete graph|03WKB2H474]] $K_6$ red or blue.
Then there are three vertices such that the three edges between them have the same colour.

## Remarks

Proof: a vertex $v$ has $5$ edges, so at least $3$ of them have one colour, say blue, by the [[pigeonhole principle|0HGVCWQGY3]]. If an edge between their ends is blue, it closes a blue triangle with $v$. Otherwise these three ends form a red triangle.
The number $6$ is the least possible: $K_5$ has a colouring with no triangle of one colour.
Thus the [[Ramsey number|01DSVCRF6X]] $R(3, 3)$ is $6$.

## Explanations

### L1 human

At any party of six people, there are three people who all know each other, or three who are all strangers to each other.
With five people this can fail.
</document>

<document source="https://tamnd.github.io/math2/node/0NYZ2M45G9.md" tag="0NYZ2M45G9" kind="definition">
# Closed map

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]].
A function $f \colon X \to Y$ is a closed map when for each [[closed set|0G5AV9Q4RR]] $F$ of $X$, the [[image|0Q3F20XB0A]] $f(F)$ is closed in $Y$.

## Remarks

A continuous map from a [[compact space|0QPHW703PV]] to a [[Hausdorff space|0MXB4CKHFW]] is closed. This gives [[continuous bijections from compact to Hausdorff spaces are homeomorphisms|0430ETCMKD]].
The projection $\mathbb{R}^2 \to \mathbb{R}$, $(x, y) \mapsto x$, is open but not closed: it maps the closed set $\{(x, y) : xy = 1\}$ to $\mathbb{R} \setminus \{0\}$.
A map can be closed but not open. A constant map $\mathbb{R} \to \mathbb{R}$ is closed, but the image of $\mathbb{R}$ is a point, which is not open.

## Explanations

### L1 human

A closed map keeps closed sets closed when you push them forward.
Open maps and closed maps are different: a map can have one property without the other.
</document>

<document source="https://tamnd.github.io/math2/node/0NZ8JA25WK.md" tag="0NZ8JA25WK" kind="concept">
# Module over a ring

## Summary

A module over a ring is an abelian group with a multiplication by the elements of the ring.
This multiplication satisfies the same axioms as for a vector space.
</document>

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# Adjacency matrix

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]] with vertex set $\{v_1, \dots, v_n\}$.
The adjacency matrix of $G$ is the $n \times n$ matrix $A = (a_{ij})$ with
$$a_{ij} = \begin{cases} 1 & \text{if } v_i \sim v_j, \\ 0 & \text{otherwise.} \end{cases}$$

## Remarks

The matrix $A$ is symmetric and has zeros on the diagonal. The sum of row $i$ is the [[degree|0T0F6CCS9W]] of $v_i$.
The entry $(A^k)_{ij}$ is the number of [[walks|0VJZM0F9YM]] of length $k$ from $v_i$ to $v_j$.
Another order of the vertices gives the matrix $P^{T} A P$ for a permutation matrix $P$.

## Explanations

### L1 human

The adjacency matrix is a table of zeros and ones. A $1$ in row $i$ and column $j$ means that vertices $i$ and $j$ are joined.
For a triangle, every entry off the diagonal is $1$.
</document>

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# Angle-angle similarity

## Statement

Let $ABC$ and $A'B'C'$ be triangles in the Euclidean space $\mathbb{R}^n$, and let $\angle$ denote the [[angle at a point|0GJAJC6TNZ]].
If $\angle BAC = \angle B'A'C'$ and $\angle CBA = \angle C'B'A'$, then $ABC$ and $A'B'C'$ are [[similar|0YF6QXXWB5]].

## Remarks

By the [[angle sum of a triangle|05MDKQBEAF]], the third angles are then equal too. So the criterion is also called angle-angle-angle similarity.
Another criterion for similarity is side-angle-side: two equal ratios of sides, and equal angles between those sides.
In hyperbolic geometry, triangles with equal angles are congruent, so the criterion has a different meaning there.

## Explanations

### L1 human

The shape of a triangle depends only on its angles. If two angles match, the third one matches too, since the three always add up to $\pi$. So two triangles with matching angles are scaled copies of each other.
</document>

<document source="https://tamnd.github.io/math2/node/0P05RJC5XA.md" tag="0P05RJC5XA" kind="definition">
# Leading coefficient

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $f = a_0 + a_1 x + \dots + a_n x^n$ be a nonzero polynomial in $R[x]$ of [[degree|0DPF6WRZW6]] $d$.
The leading coefficient of $f$ is $a_d$.
The leading coefficient of the zero polynomial is $0$.

## Remarks

The leading coefficient of $3x^4 - x + 2$ is $3$.
A polynomial with leading coefficient $1$ is [[monic|0BDX4MBKT4]].
In an [[integral domain|09S3PFBEPM]], the leading coefficient of $f g$ is the product of the leading coefficients of $f$ and $g$.

## Explanations

### L1 human

The leading coefficient is the number in front of the highest power of $x$.
For large values of $x$, the leading term $a_d x^d$ decides the size of a real polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/0P0DJHBGBB.md" tag="0P0DJHBGBB" kind="definition">
# Cesàro mean

## Definition

Let $(a_n)_{n \ge 1}$ be a [[real sequence|0CTV783HYT]].
For each integer $n \ge 1$, the $n$-th Cesàro mean of $(a_n)$ is the arithmetic mean of its first $n$ terms,
$$\sigma_n = \frac{a_1 + a_2 + \cdots + a_n}{n}.$$

## Remarks

The sequence $(-1)^{n+1}$ diverges, but its Cesàro means are $1, 0, 1/3, 0, 1/5, \dots$, and they tend to $0$.
If $(a_n)$ converges, the [[Cesàro means converge to the same limit|0RDRPBMNAC]].
Cesàro summation applies the Cesàro means to the partial sums of a series. A series is Cesàro summable with sum $A$ when the means of its partial sums tend to $A$.

## Explanations

### L1 human

The Cesàro mean is the running average of the sequence.
Averaging smooths out oscillations. The sequence $1, 0, 1, 0, \dots$ has no limit, but its running averages tend to $1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/0P0MZ4X682.md" tag="0P0MZ4X682" kind="definition">
# Independence of two events

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \in \mathcal{F}$ be [[events|00SQGJNRTB]].
The events $A$ and $B$ are independent if
$$P(A \cap B) = P(A) \, P(B).$$

## Remarks

If $P(B) > 0$, then $A$ and $B$ are independent if and only if $P(A \mid B) = P(A)$, with the [[conditional probability|0FJB06AD2P]].
If $A$ and $B$ are independent, then so are $A$ and $B^c$, $A^c$ and $B$, and $A^c$ and $B^c$.
An event of probability $0$ or $1$ is independent of each event. An event $A$ is independent of itself if and only if $P(A) \in \{0, 1\}$.
Independence is not the same as disjointness. Disjoint events $A$ and $B$ with $P(A) > 0$ and $P(B) > 0$ are never independent.

## Explanations

### L1 human

Two events are independent when knowing that one happened tells you nothing about the other.
Roll a fair die two times. The events "the first roll is $6$" and "the second roll is $6$" are independent.
Both happen with probability $\frac16 \cdot \frac16 = \frac{1}{36}$.
</document>

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# Classification of isometries of the plane

## Statement

Let $f$ be a [[Euclidean isometry|0J1N3WTKM4]] of the plane $\mathbb{R}^2$. Then $f$ is of exactly one of these five types:
- the identity,
- a [[translation|087VJM5Q77]] by a nonzero vector,
- a [[rotation|0HFREYRSZP]] about a point through an angle $\theta \notin 2\pi\mathbb{Z}$,
- a [[reflection|06BRDF3F0E]] in a line,
- a [[glide reflection|0167F9B344]] with a nonzero glide vector.

## Remarks

Write $f(x) = A x + b$ with $A$ orthogonal. If $\det A = 1$, then $f$ is the identity, a translation or a rotation. If $\det A = -1$, then $f$ is a reflection or a glide reflection.
The fixed points tell the types apart. A translation and a glide reflection have no fixed point. A rotation has one, and a reflection fixes a whole line.
Each isometry of the plane is the composite of at most three reflections in lines.

## Explanations

### L1 human

Each rigid motion of the plane is a shift, a turn, a mirror image, or a mirror image with a shift along the mirror. A turn and a shift keep the clockwise order of points. A mirror image and a glide reverse it.
</document>

<document source="https://tamnd.github.io/math2/node/0P11X5HKAX.md" tag="0P11X5HKAX" kind="definition">
# Subgraph

## Definition

Let $G = (V, E)$ be a [[graph|0JD5C82QMB]].
A subgraph of $G$ is a graph $H = (V', E')$ with $V' \subseteq V$ and $E' \subseteq E$.
Each edge of $E'$ has both ends in $V'$, because $H$ is a graph.

## Remarks

A subgraph is spanning if $V' = V$.
An [[induced subgraph|0V05FC4GKQ]] keeps all edges of $G$ between the chosen vertices.
In the cycle $C_4$ with vertices $1, 2, 3, 4$, the path $1 - 2 - 3$ is a subgraph.

## Explanations

### L1 human

To get a subgraph, delete some vertices and some edges of a graph.
When you delete a vertex, you must also delete the edges at that vertex.
</document>

<document source="https://tamnd.github.io/math2/node/0P17NGQQMQ.md" tag="0P17NGQQMQ" kind="concept">
# Hermitian matrix

## Summary

A Hermitian matrix is a complex square matrix that is equal to its conjugate transpose.
</document>

<document source="https://tamnd.github.io/math2/node/0P18JC72RE.md" tag="0P18JC72RE" kind="definition">
# Removable singularity

## Definition

Let $c$ be an [[isolated singularity|0TYB3HM9KE]] of $f$, so that $f$ is holomorphic on $0 < |z - c| < r$ for some $r > 0$.
The singularity $c$ is removable if there is a function $g$ that is [[holomorphic|0DG0Z72CQD]] on the disc $|z - c| < r$ with $g(z) = f(z)$ for each $z$ with $0 < |z - c| < r$.

## Remarks

The standard example is $f(z) = \frac{\sin z}{z}$ at $0$. With $g(0) = 1$, the function $g$ is the sum of $1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \cdots$, which is holomorphic.
The singularity is removable if and only if the [[principal part|0NDV5R6Z6X]] of $f$ at $c$ is $0$.
[[Riemann's theorem|0BSQDZWF34]] says that the singularity is removable if $f$ is bounded near $c$.

## Explanations

### L1 human

A removable singularity is a hole in the domain of the function that you can fill with one value.
After you fill it, the function is holomorphic at the point too.
</document>

<document source="https://tamnd.github.io/math2/node/0P21AVFFSP.md" tag="0P21AVFFSP" kind="definition">
# Equivalent matrices

## Definition

Let $F$ be a field and let $A$ and $B$ be $m \times n$ matrices over $F$.
The matrix $B$ is *equivalent* to $A$ when there are an [[invertible|0EB9N0S33C]] $n \times n$ matrix $P$ and an invertible $m \times m$ matrix $Q$ over $F$ with
$$B = Q^{-1} A P.$$

## Remarks

Two matrices are equivalent if and only if they represent the same linear map $F^n \to F^m$ in two pairs of bases.
Two $m \times n$ matrices over a field are equivalent if and only if they have the same [[rank|0ZC9YTXZRY]].
A matrix of rank $r$ is equivalent to the block matrix $\begin{pmatrix} I_r & 0 \\ 0 & 0 \end{pmatrix}$.
[[Similarity|0E89HQATM8]] is a finer relation: similar matrices are equivalent, but equivalent square matrices need not be similar.

## Explanations

### L1 human

Matrix equivalence allows independent changes of basis on the input side and on the output side.
With so much freedom, only the rank survives.
Every matrix of rank $r$ becomes the identity of size $r$, padded with zeros.
</document>

<document source="https://tamnd.github.io/math2/node/0P234N3M28.md" tag="0P234N3M28" kind="definition">
# Hyperbola

## Definition

Let $F_1$ and $F_2$ be distinct points of the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $a$ be a real number with $0 < 2a < d(F_1, F_2)$.
The hyperbola with the foci $F_1$, $F_2$ and the semi-major axis $a$ is the set
$$\{\, P \in \mathbb{R}^2 \;:\; |d(P, F_1) - d(P, F_2)| = 2a \,\}.$$

## Remarks

Let $c = \frac{1}{2} d(F_1, F_2)$ and $b = \sqrt{c^2 - a^2}$. For $F_1 = (-c, 0)$ and $F_2 = (c, 0)$, the hyperbola is the set of the points with $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$.
This hyperbola has the two asymptotes $y = \pm \frac{b}{a} x$. Its [[eccentricity|0HVX863SAJ]] is $e = c / a > 1$.
For $a = b$, the asymptotes are perpendicular, and the hyperbola is rectangular. The graph of $y = 1/x$ is a rectangular hyperbola.

## Explanations

### L1 human

A hyperbola has two separate branches, like two opposite bows. On one branch, the points are nearer to one focus; on the other branch, they are nearer to the other focus. The difference of the two distances is always $2a$.
</document>

<document source="https://tamnd.github.io/math2/node/0P24YW565F.md" tag="0P24YW565F" kind="definition">
# Matrix of a bilinear form

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$ with a [[basis|0KCQ9Y3QKA]] $(e_1, \dots, e_n)$.
Let $B$ be a [[bilinear form|0SQZMXB6GP]] on $V$.
The *matrix* of $B$ in this basis is the $n \times n$ matrix $M$ with entries
$$M_{ij} = B(e_i, e_j) \qquad (1 \le i, j \le n).$$

## Remarks

If $x$ and $y$ are the coordinate vectors of $u$ and $v$, then $B(u, v) = x^T M y$.
The map $B \mapsto M$ is a linear bijection from the bilinear forms on $V$ to the $n \times n$ matrices over $F$.
For the dot product on $\R^n$ and the standard basis, $M$ is the identity matrix.

## Explanations

### L1 human

A bilinear form is fixed by its values on pairs of basis vectors.
We store these $n^2$ values in a matrix.
Then the form becomes a product of a row vector, the matrix and a column vector.
</document>

<document source="https://tamnd.github.io/math2/node/0P27W9RW0G.md" tag="0P27W9RW0G" kind="definition">
# Continuous function

## Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
The function $f$ is continuous if it is [[continuous at|0EEE2CZB2A]] each point of $D$.
For a subset $S \subseteq D$, $f$ is continuous on $S$ if its restriction to $S$ is continuous.

## Remarks

Sums, [[products|0VGMHCF4SC]], [[quotients|0GR2APWTAC]], and [[compositions|0NRQPQX3VA]] of continuous functions are continuous.
[[Polynomial functions are continuous|0C5TP6RYSV]]. So are the exponential function, the sine, and the cosine.
On a closed bounded interval, a continuous function has the [[intermediate value property|00KAFBR32E]] and [[attains its extreme values|0WS1A8E2P9]].
Mathlib defines it for topological spaces, as `Continuous`.

## Explanations

### L1 human

A continuous function has no jumps or breaks at any point of its domain.
On an interval, we can draw its graph without lifting the pen.
</document>

<document source="https://tamnd.github.io/math2/node/0P30YYZN85.md" tag="0P30YYZN85" kind="statement">
# Second-order Taylor expansion

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$ be of [[class C^2|0TCBZMEYGN]] on $U$, and let $a \in U$.
Let $\nabla f(a)$ be the [[gradient|0NS2QYVC4Y]] of $f$ at $a$, and let $H_f(a)$ be the [[Hessian matrix|0SCSQWXRA4]] of $f$ at $a$.
Then
$$f(a + h) = f(a) + \nabla f(a) \cdot h + \tfrac{1}{2} \, h^{\mathsf{T}} H_f(a) \, h + R(h), \qquad \lim_{h \to 0} \frac{R(h)}{\|h\|^2} = 0.$$

## Remarks

This is the case $k = 2$ of [[Taylor's theorem in several variables|0J8T4K4YTE]], written with matrices. The quadratic term collects the terms with $|\alpha| = 2$.
For $n = 2$ and $h = (u, v)$, the quadratic term is $\tfrac{1}{2} \left( f_{xx}(a) u^2 + 2 f_{xy}(a) u v + f_{yy}(a) v^2 \right)$.
At a critical point the linear term is zero, so the sign of $h^{\mathsf{T}} H_f(a) h$ controls the behaviour of $f$ near $a$. This is the basis of the [[second derivative test|0GYDV9FE98]].

## Explanations

### L1 human

A function of class $C^2$ is close to a quadratic polynomial near each point.
The gradient gives the linear part and the Hessian gives the quadratic part.
The error goes to zero faster than $\|h\|^2$.
</document>

<document source="https://tamnd.github.io/math2/node/0P3A06QNDP.md" tag="0P3A06QNDP" kind="statement">
# Negation of a universal statement

## Statement

Let $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.
In classical logic, the negation of $\forall x \in X, \, P(x)$ is logically equivalent to $\exists x \in X, \, \neg P(x)$.
$$\neg \bigl( \forall x \in X, \, P(x) \bigr) \equiv \exists x \in X, \, \neg P(x)$$

## Remarks

The direction from $\exists x, \, \neg P(x)$ to $\neg \forall x, \, P(x)$ is valid in intuitionistic logic. The other direction needs classical logic.
This law explains why one [[counterexample|0ED72CTBNP]] refutes a universal statement.
For a finite $X$, it is a form of [[De Morgan's laws|09PKJKA6J0]].

## Explanations

### L1 human

"Not every student passed" means "some student did not pass".
To deny that all elements have a property, you show one element without it.
</document>

<document source="https://tamnd.github.io/math2/node/0P3KZK0FTV.md" tag="0P3KZK0FTV" kind="definition">
# Solvable group

## Definition

A [[group|G5S1M9]] $G$ is solvable when it has a [[subnormal series|0ZZR8D7Y1Z]] $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$ in which each factor $G_i / G_{i-1}$ is [[abelian|082GHNPJ53]].

## Remarks

Equivalently, the [[derived series|0Q5G5DJ4P2]] of $G$ reaches $\{e\}$ after finitely many steps.
A finite group is solvable if and only if its [[composition factors|0PHTC93M58]] are all cyclic of prime order.
The group $S_4$ is solvable, by the series $\{e\} \trianglelefteq V \trianglelefteq A_4 \trianglelefteq S_4$.
The name comes from Galois theory: a polynomial equation over $\mathbb{Q}$ is solvable by radicals if and only if its Galois group is solvable.

## Explanations

### L1 human

A solvable group can be built from abelian groups in finitely many layers.
Abelian groups are easy to understand, so solvable groups are close to easy.
The smallest group that is not solvable is $A_5$, of order $60$.
</document>

<document source="https://tamnd.github.io/math2/node/0P3M4ZY9X8.md" tag="0P3M4ZY9X8" kind="concept">
# Divergent sequence

## Summary

A divergent sequence is a sequence that has no limit. Its terms can grow without bound or oscillate. The node [[divergent sequence|01HR2AH0BX]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0P3VCBDEE2.md" tag="0P3VCBDEE2" kind="concept">
# Ordered field

## Summary

An ordered field is a field with a total order that is compatible with addition and with multiplication by positive elements.
The [[rational numbers|0NCJ5WXN6Z]] and the [[real numbers|08J2M1JYAG]] are ordered fields.
</document>

<document source="https://tamnd.github.io/math2/node/0P42JB4E3M.md" tag="0P42JB4E3M" kind="definition">
# Glide reflection

## Definition

Let $H = \{\, x \in \mathbb{R}^n : x \cdot a = c \,\}$ be a hyperplane, with $a \neq 0$, and let $\sigma_H$ be the [[reflection|06BRDF3F0E]] in $H$. Let $w \in \mathbb{R}^n$ be a nonzero vector with $w \cdot a = 0$, and let $T_w$ be the [[translation|087VJM5Q77]] by $w$.
The glide reflection with the axis $H$ and the glide vector $w$ is the map $T_w \circ \sigma_H$.

## Remarks

Because $w$ is parallel to $H$, the two maps commute: $T_w \circ \sigma_H = \sigma_H \circ T_w$.
A glide reflection is an indirect isometry with no fixed point. Its square is the translation by $2w$.
If the translation vector is perpendicular to $H$ in place of parallel, then $T_w \circ \sigma_H$ is the reflection in a hyperplane parallel to $H$.
In the plane, the axis is the glide line. Glide reflections are symmetries of many frieze and wallpaper patterns.

## Explanations

### L1 human

First reflect the plane in a line, then slide it along that line. Neither step alone gives the result. The map $(x, y) \mapsto (x + 1, -y)$ is the glide reflection with the axis $y = 0$ and the glide vector $(1, 0)$.
</document>

<document source="https://tamnd.github.io/math2/node/0P4FTHHMYS.md" tag="0P4FTHHMYS" kind="concept">
# Standard deviation

## Summary

The standard deviation of a random variable is the square root of its variance. It measures the spread of the values in the same units as the variable.
The node [[standard deviation|0VTN5RKNC5]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0P4JRMSWNH.md" tag="0P4JRMSWNH" kind="concept">
# Degree of a polynomial

## Summary

The degree of a nonzero polynomial is the largest exponent of $x$ with a nonzero coefficient.
The [[definition of the degree|0DPF6WRZW6]] gives the details and the convention for the zero polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/0P4NQD746K.md" tag="0P4NQD746K" kind="statement">
# Sine is 2π-periodic

## Statement

For every real $x$,
$$\sin(x + 2\pi) = \sin x, \qquad \cos(x + 2\pi) = \cos x.$$
The number $2\pi$ is the smallest positive period of $\sin$ and of $\cos$.

## Remarks

From [[$\cos \pi = -1$ and $\sin \pi = 0$|01A8JGR4JM]], the [[addition formulas|028XEQCE88]] give $\cos 2\pi = 1$ and $\sin 2\pi = 0$. Apply the addition formulas again with $y = 2\pi$.
So $\sin$ and $\cos$ are [[periodic|089KWVYD69]] functions.
Also $\sin(x + \pi) = -\sin x$ and $\sin(x + \pi/2) = \cos x$. Mathlib states the periodicity as `Real.sin_periodic`.

## Explanations

### L1 human

A full turn around the unit circle has length $2\pi$, and it brings the point back to where it started.
So the graphs of sine and cosine repeat after each interval of length $2\pi$.
</document>

<document source="https://tamnd.github.io/math2/node/0P4WX7Y502.md" tag="0P4WX7Y502" kind="definition">
# Iterated integral

## Definition

Let $R = [a, b] \times [c, d]$ and let $f \colon R \to \mathbb{R}$.
Suppose that, for each $x \in [a, b]$, the function $y \mapsto f(x, y)$ is integrable on $[c, d]$, and that $F(x) = \int_c^d f(x, y) \, dy$ is integrable on $[a, b]$.
Then the iterated integral of $f$, first in $y$ and then in $x$, is
$$\int_a^b \int_c^d f(x, y) \, dy \, dx = \int_a^b \left( \int_c^d f(x, y) \, dy \right) dx = \int_a^b F(x) \, dx.$$
The iterated integral in the other order, $\int_c^d \int_a^b f(x, y) \, dx \, dy$, is defined in the same way.

## Remarks

Iterated integrals in three or more variables are defined by repeating this step.
The inner bounds may depend on the outer variable, as in $\int_0^1 \int_0^x f(x, y) \, dy \, dx$. This integrates over the triangle $0 \le y \le x \le 1$.
An iterated integral is different in principle from the [[double integral|0NKBMFC18R]], but [[Fubini's theorem|0R0T1VWH5X]] says that they are equal under suitable hypotheses.
The two orders of integration can give [[different values|0GJZE2E4RX]] when the hypotheses fail.

## Explanations

### L1 human

To compute an iterated integral, work from the inside out.
First integrate in $y$ as if $x$ were a constant; the result is a function of $x$ only. Then integrate this function in $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0P506SPZP2.md" tag="0P506SPZP2" kind="counterexample">
# Non-commutativity of matrix multiplication

## Counterexample

Let $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, as real $2 \times 2$ matrices.
Then
$$AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \qquad BA = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.$$
Thus $AB \ne BA$.
This shows that [[matrix multiplication|032Q9FB8MT]] of $n \times n$ matrices is not commutative for $n = 2$.

## Remarks

The same example, extended with zero rows and zero columns, shows that $M_n(R)$ is not commutative for each $n \ge 2$ and each nonzero ring $R$.
Two matrices $A$ and $B$ with $AB = BA$ are said to *commute*.
Some pairs commute, for example a matrix and the [[identity matrix|06DW9BAWB1]], or two [[diagonal matrices|044JNC7ZAA]].

## Explanations

### L1 human

For numbers, $2 \cdot 3 = 3 \cdot 2$, but for matrices the order of the factors matters.
Here $AB$ and $BA$ are different matrices.
A reason is that $AB$ means "first apply $B$, then $A$", and two actions done in a different order can give different results.
</document>

<document source="https://tamnd.github.io/math2/node/0P64VE79X3.md" tag="0P64VE79X3" kind="statement">
# A function with bounded derivative is Lipschitz

## Statement

Let $I$ be an interval, let $f \colon I \to \mathbb{R}$ be continuous and differentiable at each interior point of $I$, and let $K \ge 0$.
If $|f'(x)| \le K$ for each interior point $x$ of $I$, then $f$ is [[Lipschitz|0QNVQ4VD9T]] with constant $K$.

## Remarks

The [[mean value theorem|0BARQY52N5]] gives $|f(y) - f(x)| = |f'(c)| \, |y - x| \le K |y - x|$.
For example, $\sin$ is Lipschitz with constant $1$ on $\mathbb{R}$.
Mathlib states a form for convex sets as `Convex.lipschitzOnWith_of_nnnorm_deriv_le`.

## Explanations

### L1 human

If the slope of the graph never exceeds $K$ in size, no chord can be steeper than $K$.
</document>

<document source="https://tamnd.github.io/math2/node/0P6KRD5B7A.md" tag="0P6KRD5B7A" kind="statement">
# Picard–Lindelöf theorem

## Statement

Let $J = [t_{\min}, t_{\max}]$ and $t_0 \in J$, let $x_0 \in \mathbb{R}^n$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ satisfy the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] at $(t_0, x_0)$ with constants $a, L, K$.
Let $B$ be the closed ball of radius $a$ around $x_0$.
Then there is exactly one continuous curve $\alpha : J \to B$ with $\alpha(t_0) = x_0$ and with this property. For each $t \in J$, $\alpha$ has the derivative $f(t, \alpha(t))$ at $t$ within $J$.

## Remarks

The proof uses the [[integral equation form|015KK993C5]] and the fact that [[an iterate of the Picard operator is a contraction|08XDXP86XE]]. The limit of [[Picard iteration|0RJ5NHWYTP]] is the solution.
A common form: let $f$ be continuous on a neighbourhood of $(t_0, x_0)$ and locally Lipschitz in $x$. Then the [[initial value problem|075VBY4K4X]] has a unique solution on $[t_0 - \varepsilon, t_0 + \varepsilon]$ for some $\varepsilon > 0$.
The theorem is local. For $x' = 1 + x^2$, $x(0) = 0$, the solution $\tan t$ exists only for $|t| < \pi/2$.
Without the Lipschitz condition, solutions exist by the [[Peano existence theorem|098YP831QY]], but they need not be unique. See the [[non-uniqueness example|00K7ASZ1PR]].

## History

The theorem is named after Émile Picard and Ernst Lindelöf. It is also called the Cauchy–Lipschitz theorem, after Augustin-Louis Cauchy and Rudolf Lipschitz.

## Explanations

### L1 human

If the vector field is regular enough, the present state determines the whole motion near the start time.
Regular enough means continuous in time and Lipschitz in the state.
Then there is one and only one solution through the initial point, at least for a short time.
</document>

<document source="https://tamnd.github.io/math2/node/0P6PMJB4MA.md" tag="0P6PMJB4MA" kind="statement">
# Compactness via open covers

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $K \subseteq X$.
Then $K$ is [[compact|00R0X1WGPX]] if and only if each family $(U_i)_{i \in I}$ of open subsets of $X$ with $K \subseteq \bigcup_{i \in I} U_i$ has a finite subfamily $U_{i_1}, \ldots, U_{i_n}$ with $K \subseteq U_{i_1} \cup \cdots \cup U_{i_n}$.

## Remarks

Proof: the open sets of the [[subspace topology|0D5YHQGDJY]] on $K$ are the sets $U \cap K$ with $U$ open in $X$. So covers of $K$ by open sets of $K$ and covers by open sets of $X$ match each other.
For $K = X$, the statement is the definition of a [[compact space|0QPHW703PV]].
Thus compactness of a subset does not depend on the space around it. If $K \subseteq Y \subseteq X$, then $K$ is compact in $Y$ if and only if it is compact in $X$.

## Explanations

### L1 human

To test if a subset is compact, you can use open sets of the big space.
Each time you cover the subset with open sets, finitely many of them must already be enough.
</document>

<document source="https://tamnd.github.io/math2/node/0P6SBR7YM2.md" tag="0P6SBR7YM2" kind="definition">
# Simple extension

## Definition

A [[field extension|0WQ1A8SHTZ]] $L/K$ is simple if there is an element $\theta \in L$ with $L = K(\theta)$.
Such an element $\theta$ is a primitive element, or a generator, of $L/K$.

## Remarks

The extensions $\mathbb{C}/\mathbb{R}$, generated by $i$, and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$, generated by $\sqrt{2}$, are simple.
The extension $\mathbb{Q}(\sqrt{3}, \sqrt{7})/\mathbb{Q}$ is simple, with the generator $\sqrt{3} + \sqrt{7}$.
By the [[primitive element theorem|0SVX98HYV1]], every finite separable extension is simple.
For finite fields, the term "primitive element" usually means a [[generator of the multiplicative group|0YAA9CJQXF]], which is a stronger condition.

## Explanations

### L1 human

A simple extension is built by adding one new element to a field.
Simple extensions are well understood: there is one kind for an algebraic generator and one kind for a transcendental generator.
</document>

<document source="https://tamnd.github.io/math2/node/0P799D6AMN.md" tag="0P799D6AMN" kind="definition">
# Bilinear form

## Definition

Let $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
A *bilinear form* on $V$ is a map $B \colon V \times V \to F$ such that, for all $u, v, w \in V$ and $a \in F$:

1. $B(u + v, w) = B(u, w) + B(v, w)$ and $B(au, w) = a B(u, w)$.
2. $B(u, v + w) = B(u, v) + B(u, w)$ and $B(u, aw) = a B(u, w)$.

## Remarks

The [[dot product|0S52TNNTH9]] on $\R^n$ is a bilinear form.
For each $n \times n$ matrix $M$ over $F$, the map $B(x, y) = x^T M y$ is a bilinear form on $F^n$.
The bilinear forms on $V$ form a vector space over $F$; if $\dim V = n$, it has dimension $n^2$.
More generally, a bilinear map $V \times W \to F$ uses two different spaces.

## Explanations

### L1 human

A bilinear form takes two vectors and gives a number.
It is linear in the first vector and also linear in the second vector.
The dot product is the main example, but bilinear forms need not be symmetric or positive.
</document>

<document source="https://tamnd.github.io/math2/node/0P7A2MJD24.md" tag="0P7A2MJD24" kind="definition">
# Coefficient matrix

## Definition

Let $F$ be a field.
Take a [[system of linear equations|0GT4WWGWH2]] over $F$ with $m$ equations in $n$ unknowns, where equation $i$ is $a_{i1} x_1 + \cdots + a_{in} x_n = b_i$.
The *coefficient matrix* of the system is the $m \times n$ [[matrix|0VD9ZSWWJG]] $A = (a_{ij})$.
Its entry in row $i$ and column $j$ is the coefficient of $x_j$ in equation $i$.

## Remarks

The system $2x + y = 5$, $x - 3y = 0$ has the coefficient matrix $\begin{pmatrix} 2 & 1 \\ 1 & -3 \end{pmatrix}$.
With $x = (x_1, \dots, x_n)^T$ and $b = (b_1, \dots, b_m)^T$, the system is $Ax = b$.
If $A$ is square and [[invertible|0EB9N0S33C]], the system has exactly one solution, $x = A^{-1} b$.

## Explanations

### L1 human

The coefficient matrix keeps only the numbers in front of the unknowns, in a rectangular table.
Each row is one equation, and each column is one unknown.
This table lets us work with the whole system as one object.
</document>

<document source="https://tamnd.github.io/math2/node/0P7CH1CND5.md" tag="0P7CH1CND5" kind="definition">
# Characteristic of a ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
For a positive integer $n$, write $n \cdot 1 = 1 + 1 + \dots + 1$, with $n$ terms.
If there is a positive integer $n$ with $n \cdot 1 = 0$ in $R$, then the characteristic of $R$ is the least such $n$.
If there is no such $n$, then the characteristic of $R$ is $0$.

## Remarks

The rings $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ have characteristic $0$.
For $n \ge 1$, the ring $\mathbb{Z}/n\mathbb{Z}$ has characteristic $n$.
The characteristic of $R$ is the nonnegative generator of the kernel of the unique ring homomorphism $\mathbb{Z} \to R$.
The [[characteristic of an integral domain is zero or prime|00758QM6JJ]].

## Explanations

### L1 human

Start with $1$ and keep adding $1$.
In the integers you never come back to $0$, so the characteristic is $0$.
Modulo $5$, you get $0$ after five steps, so the characteristic is $5$.
</document>

<document source="https://tamnd.github.io/math2/node/0P7FZHXZ5S.md" tag="0P7FZHXZ5S" kind="concept">
# Composition of an integer

## Summary

A composition of an integer is a way to write it as an ordered sum of positive integers. Compositions are the ordered analogue of partitions, and they are easy to count. The node [[composition of an integer|0J8ANSYKAC]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0P7G9NGG2G.md" tag="0P7G9NGG2G" kind="concept">
# Faithful action

## Summary

A group action is faithful when only the identity element of the group fixes each point.
Then different group elements move the points in different ways.
</document>

<document source="https://tamnd.github.io/math2/node/0P7PTC8DDE.md" tag="0P7PTC8DDE" kind="statement">
# Linearity of expectation

## Statement

Let $X$ and $Y$ be integrable real random variables on one probability space, and let $a, b \in \mathbb{R}$.
Then $aX + bY$ is integrable and
$$E[aX + bY] = a \, E[X] + b \, E[Y].$$

## Remarks

The statement is the linearity of the Lebesgue integral. It holds for all integrable $X$ and $Y$, also when they are not independent.
By induction, $E[X_1 + \dots + X_n] = E[X_1] + \dots + E[X_n]$.
A standard use is the method of indicators. If $N$ counts how many of the events $A_1, \dots, A_n$ occur, then $N = \sum_i 1_{A_i}$ and $E[N] = \sum_i P(A_i)$.
For nonnegative random variables, additivity $E[X + Y] = E[X] + E[Y]$ holds without integrability, in $[0, \infty]$.

## Explanations

### L1 human

The average of a sum is the sum of the averages, always.
If a random permutation of $n$ letters goes into $n$ envelopes, each letter is in its own envelope with probability $1/n$. So the expected number of correct letters is $n \cdot 1/n = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0P84B554AS.md" tag="0P84B554AS" kind="statement">
# Vandermonde determinant

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $x_1, \dots, x_n \in K$.
Let $V$ be the $n \times n$ *Vandermonde matrix* with entries $V_{ij} = x_i^{\,j-1}$.
Then
$$\det V = \prod_{1 \le i < j \le n} (x_j - x_i),$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

Over a field, $\det V \ne 0$ if and only if the $x_i$ are pairwise distinct.
Then, for all $y_1, \dots, y_n$, there is exactly one polynomial $p$ of degree less than $n$ with $p(x_i) = y_i$ for all $i$.
For $n = 2$, the matrix is $\begin{pmatrix} 1 & x_1 \\ 1 & x_2 \end{pmatrix}$ with determinant $x_2 - x_1$.

## Explanations

### L1 human

A Vandermonde matrix lists the powers $1, x, x^2, \dots$ of some numbers, one row for each number.
Its determinant is the product of all differences of the numbers.
So it is nonzero exactly when the numbers are distinct.
</document>

<document source="https://tamnd.github.io/math2/node/0P8BA7T7S5.md" tag="0P8BA7T7S5" kind="definition">
# Limit superior

## Definition

Let $(a_n)$ be a [[bounded sequence|05FQ3VGAR9]] of real numbers.
For each $n \in \mathbb{N}$, let $s_n = \sup \{a_k : k \ge n\}$.
The sequence $(s_n)$ is decreasing and bounded below.
The limit superior of $(a_n)$ is
$$\limsup_{n \to \infty} a_n = \lim_{n \to \infty} s_n = \inf_{n \in \mathbb{N}} \sup_{k \ge n} a_k.$$

## Remarks

The limit exists because a [[decreasing sequence bounded below converges|02V2GYAC8P]].
For a sequence that is not bounded above, one sets $\limsup a_n = +\infty$. For a sequence that is bounded above and not below, the same formula is used in the extended real line, and it can give $-\infty$.
For $a_n = (-1)^n (1 + 1/(n+1))$, the limit superior is $1$, although no term equals $1$.
For a bounded sequence, it is the largest [[cluster point|0RCKQN1E09]].

## Explanations

### L1 human

The limit superior is the highest value that the sequence keeps coming back to, in the long run.
Each number above it is exceeded by only finitely many terms.
</document>

<document source="https://tamnd.github.io/math2/node/0P8CY9V437.md" tag="0P8CY9V437" kind="example">
# Action by conjugation

## Example

Let $G$ be a [[group|G5S1M9]].
The map $G \times G \to G$, $(g, x) \mapsto gxg^{-1}$, is an [[action|0WCY42EV9V]] of $G$ on the set $G$.
Indeed, $exe^{-1} = x$ and $g(hxh^{-1})g^{-1} = (gh)x(gh)^{-1}$.
The [[orbits|0YQPZB5Q1G]] of this action are the [[conjugacy classes|0DEBXAJ5SA]] of $G$.
The [[stabiliser|0AYMP41THN]] of $x$ is the [[centraliser|02YYV36BFM]] $C_G(x)$.
The [[fixed points|0RRWNMYRXC]] are the elements of the [[centre|0DZ93493VN]] $Z(G)$.
For an [[abelian group|082GHNPJ53]], each element is a fixed point, and each orbit has one element.
</document>

<document source="https://tamnd.github.io/math2/node/0P953MG05E.md" tag="0P953MG05E" kind="definition">
# Bounded set

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $S \subseteq X$.
The set $S$ is bounded when there is a real number $M$ such that $d(x, y) \le M$ for all $x, y \in S$.

## Remarks

The empty set and each finite set are bounded.
If $X$ is not empty, then $S$ is bounded if and only if $S$ lies in some [[closed ball|05DSV40EB9]] $\bar{B}(x, r)$ with $x \in X$ and $r \ge 0$.
In $\mathbb{R}$ with the usual metric, $S$ is bounded in this sense if and only if $S$ is bounded above and bounded below.
Boundedness depends on the metric: $\mathbb{R}$ is not bounded for $|x - y|$, but it is bounded for $\min(|x - y|, 1)$.

## Explanations

### L1 human

A set is bounded when you can put all of it inside one large ball.
For example, the interval $(0, 1)$ is bounded, but the set $\mathbb{N}$ of natural numbers is not bounded in $\mathbb{R}$.
</document>

<document source="https://tamnd.github.io/math2/node/0P97V20RW7.md" tag="0P97V20RW7" kind="definition">
# Ordinary differential equation

## Definition

Let $n \ge 1$ be an integer, let $U$ be a subset of $\mathbb{R}^{n+2}$, and let $F : U \to \mathbb{R}$ be a function.
An ordinary differential equation of [[order|0XK0G19SK8]] $n$ is an equation
$$F\left(x, y, y', \dots, y^{(n)}\right) = 0$$
for an unknown real function $y$ of one real variable $x$. Here $y^{(k)}$ is the $k$-th derivative of $y$.
The equation must depend on $y^{(n)}$.
If $y$ takes values in $\mathbb{R}^m$ and $F$ takes values in $\mathbb{R}^m$, the equation is a system of $m$ ordinary differential equations.

## Remarks

The word ordinary means that the unknown function depends on one variable only. An equation with partial derivatives of a function of several variables is a partial differential equation.
The equation is in [[explicit form|08WWTN738R]] when it is solved for the highest derivative: $y^{(n)} = f\left(x, y, y', \dots, y^{(n-1)}\right)$.
A [[solution|07VX34QXAV]] is a function on an interval that satisfies the equation at each point of the interval.

## Explanations

### L1 human

An ordinary differential equation is a rule that links a function of one variable with its derivatives.
For example, $y'' + y = 0$ links $y$ with its second derivative. The functions $\sin x$ and $\cos x$ satisfy it.
</document>

<document source="https://tamnd.github.io/math2/node/0P9BNBW2W3.md" tag="0P9BNBW2W3" kind="statement">
# The exponential function is positive

## Statement

For every real $x$, $\exp(x) > 0$.

## Remarks

By [[the sum rule|0E8HNVBCSP]], $\exp(x) = \exp(x/2)^2 \ge 0$. Also $\exp(x) \exp(-x) = 1$, so $\exp(x) \ne 0$.
For $x \ge 0$, it also follows from the series, since all terms are nonnegative and the first is $1$.
Mathlib states it as `Real.exp_pos`.

## Explanations

### L1 human

The graph of $e^x$ stays above the $x$-axis.
It comes close to $0$ as $x \to -\infty$, but it never reaches $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0P9DCDES73.md" tag="0P9DCDES73" kind="statement">
# Metric spaces are Hausdorff

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], with the [[metric topology|0A6YRNZB2R]].
Then $X$ is a [[Hausdorff space|0MXB4CKHFW]].

## Remarks

Proof: for $x \neq y$, let $r = d(x, y) / 2 > 0$. The [[open balls|0DC3MPR7C9]] $B(x, r)$ and $B(y, r)$ are open. They are disjoint: a point $z$ in both would give $d(x, y) \le d(x, z) + d(z, y) < 2r$, by the [[triangle inequality|01M6CW9VM6]].
Thus each [[metrizable space|09KDHB82SG]] is Hausdorff. A non-Hausdorff space, such as an infinite set with the cofinite topology, is not metrizable.
For a pseudometric, the statement fails: points at distance $0$ cannot be separated.

## Explanations

### L1 human

Two distinct points of a metric space are at a positive distance. Balls of half that radius about them do not meet.
</document>

<document source="https://tamnd.github.io/math2/node/0P9T8WHNMH.md" tag="0P9T8WHNMH" kind="statement">
# Existence of a complementary subspace

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ be a subspace of $V$.
Then $U$ has a [[complement|0M83V17KHX]] in $V$: there is a subspace $W$ of $V$ with $V = U \oplus W$.

## Remarks

Proof: extend a basis $B_U$ of $U$ to a basis $B$ of $V$, and let $W$ be the span of $B \setminus B_U$.
For infinite-dimensional $V$, this uses the extension of independent sets to bases, and hence Zorn's lemma.
The statement fails for modules over rings: the submodule $2\Z$ of $\Z$ has no complement.
For closed subspaces of topological vector spaces, a closed complement need not exist.

## Explanations

### L1 human

Every subspace has a partner subspace that fills up the rest of the space.
We find it by completing a basis of the subspace and taking the span of the new vectors.
The partner is not unique, but its dimension is fixed.
</document>

<document source="https://tamnd.github.io/math2/node/0PAQM3H57X.md" tag="0PAQM3H57X" kind="statement">
# Continuity is equivalent to sequential continuity

## Statement

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
Then $f$ is [[continuous|0P27W9RW0G]] if and only if $f$ is [[sequentially continuous|0MVDQEAB60]].

## Remarks

The forward direction follows from the [[epsilon-delta form|0W8YHHGCJG]] of continuity.
For the converse, suppose that $f$ is not continuous at $a$. Then there is $\varepsilon > 0$ and for each $n$ a point $x_n$ with $|x_n - a| < 1/n$ and $|f(x_n) - f(a)| \ge \varepsilon$.
The equivalence holds in all metric spaces. Mathlib states it for sequential spaces, as `continuous_iff_seqContinuous`.

## Explanations

### L1 human

We can test continuity with sequences.
A function is continuous exactly when it sends every convergent sequence to a convergent sequence with the right limit.
</document>

<document source="https://tamnd.github.io/math2/node/0PARCKE8WP.md" tag="0PARCKE8WP" kind="example">
# Sup norm on bounded continuous functions

## Example

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $C_b(X)$ be the real vector space of bounded [[continuous maps|0SY9ECQ19Q]] $f \colon X \to \mathbb{R}$.
For $f \in C_b(X)$, let $\|f\|_\infty = \sup_{x \in X} |f(x)|$, with $\|f\|_\infty = 0$ when $X$ is empty.
Then $\|\cdot\|_\infty$ is a norm on $C_b(X)$, the sup norm, and $C_b(X)$ is a [[Banach space|0BDTTM24DP]] with this norm.

## Remarks

The [[supremum|0BTMQ04556]] is finite because $f$ is bounded.
The metric $\|f - g\|_\infty$ is the [[uniform metric|06HE9J7DXF]]. Its completeness is [[completeness of the space of bounded continuous functions|09GRCZVR5E]].
A sequence converges in the sup norm if and only if it converges uniformly.
For a compact space $X$, each continuous real function is bounded, so $C_b(X) = C(X)$.

## Explanations

### L1 human

The sup norm of a function is the height of the highest point of its graph, in absolute value.
Two functions are close in this norm when their graphs are close everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/0PASQ34GHG.md" tag="0PASQ34GHG" kind="statement">
# Law of sines

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$. Let $a = d(B, C)$, $b = d(C, A)$ and $c = d(A, B)$ be its side lengths, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]]. Let $\alpha = \angle BAC$, $\beta = \angle CBA$ and $\gamma = \angle ACB$ be its [[angles|0GJAJC6TNZ]].
Then
$$\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}.$$

## Remarks

The common value of the three quotients is $2R$, where $R$ is the radius of the [[circumcircle|03VZJE6C8C]] of the triangle.
The angles of a triangle are in $(0, \pi)$, so their sines are positive and the quotients exist.
From two angles and one side, the law of sines gives the other sides. From two sides and an angle that is not between them, it can give two different triangles. This is the ambiguous case.

## History

Ptolemy knew an equivalent form, with chords in place of sines, in the 2nd century.

## Explanations

### L1 human

In a triangle, a larger angle sits opposite a longer side. The law of sines makes this exact: each side divided by the sine of the opposite angle gives the same number. For an equilateral triangle with side $1$, each quotient is $1 / \sin(\pi/3) = 2/\sqrt{3}$.
</document>

<document source="https://tamnd.github.io/math2/node/0PB7K8M7FG.md" tag="0PB7K8M7FG" kind="definition">
# Linear differential equation of order n

## Definition

Let $n \ge 1$, let $I$ be an open [[interval|0Q1D85B8G6]], and let $a_0, \dots, a_{n-1}, b : I \to \mathbb{R}$ be continuous functions.
The linear differential equation of order $n$ with these coefficients and right side $b$ is
$$y^{(n)} + a_{n-1}(x)\, y^{(n-1)} + \dots + a_1(x)\, y' + a_0(x)\, y = b(x)$$
for an unknown $n$ times differentiable function $y : I \to \mathbb{R}$.
With the operator $L y = y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y$, the equation is $L y = b$.

## Remarks

The map $L$ is linear: $L(c_1 y_1 + c_2 y_2) = c_1 L y_1 + c_2 L y_2$ for constants $c_1, c_2$.
An equation $a_n(x) y^{(n)} + \dots + a_0(x) y = b(x)$ with continuous $a_n$ and no zero of $a_n$ on $I$ takes this form after division by $a_n$.
The equation is [[homogeneous|0EQZSAB9EC]] if $b = 0$. The case $n = 1$ is the [[linear first-order equation|0Z3YKRQZ48]].
For each $x_0 \in I$ and each $(c_0, \dots, c_{n-1}) \in \mathbb{R}^n$, there is exactly one solution on all of $I$ with $y^{(k)}(x_0) = c_k$ for $k = 0, \dots, n-1$.

## Explanations

### L1 human

In a linear equation, the unknown $y$ and its derivatives appear only to the first power, each multiplied by a function of $x$.
For example, $y'' + x y' + y = \cos x$ is linear, but $y'' + y^2 = 0$ is not.
Linear equations have a rich theory, because sums and multiples of solutions behave well.
</document>

<document source="https://tamnd.github.io/math2/node/0PBFS92AJP.md" tag="0PBFS92AJP" kind="definition">
# Jordan block

## Definition

Let $F$ be a field, let $\lambda \in F$, and let $k \ge 1$.
The *Jordan block* $J_k(\lambda)$ is the $k \times k$ matrix with entries
$$(J_k(\lambda))_{ij} = \begin{cases} \lambda & \text{if } j = i, \\ 1 & \text{if } j = i + 1, \\ 0 & \text{otherwise}. \end{cases}$$

## Remarks

For example, $J_3(\lambda) = \begin{pmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{pmatrix}$.
$J_k(\lambda) - \lambda I_k$ is nilpotent of index $k$.
Its only eigenvalue is $\lambda$, with algebraic multiplicity $k$ and geometric multiplicity $1$.
Some authors put the $1$ entries below the diagonal.

## Explanations

### L1 human

A Jordan block is the simplest matrix that is not diagonalisable, when its size is at least $2$.
It has one eigenvalue and only one line of eigenvectors.
Every complex matrix is built from such blocks, by the Jordan normal form.
</document>

<document source="https://tamnd.github.io/math2/node/0PBN8VEKY8.md" tag="0PBN8VEKY8" kind="concept">
# Annihilating polynomial

## Summary

An annihilating polynomial of an endomorphism $T$ is a polynomial $p$ with $p(T) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0PBWWQD9KY.md" tag="0PBWWQD9KY" kind="problem">
# Monty Hall problem

## Problem

A game show has three closed doors. A car is behind one door, chosen uniformly at random, and a goat is behind each of the two other doors.
The player chooses one door. Then the host, who knows where the car is, opens one of the two other doors and shows a goat.
If both other doors hide goats, the host chooses one of them with probability $1/2$ each. The host always opens a door and always offers a switch.
The player can keep the first door or switch to the other closed door. Which choice gives the larger probability of the car?

## Remarks

Under these assumptions, switching wins with probability $2/3$ and staying wins with probability $1/3$. The player's first door hides the car with probability $1/3$, and the host's action does not change this.
The conditional probability that switching wins, given the door that the host opens, is also $2/3$. [[Bayes' theorem|0NM6E9VW04]] gives this.
The answer depends on the rules of the host. If the host opens a random other door and it shows a goat by chance, both choices win with probability $1/2$.

## History

Steve Selvin posed the problem in a letter to The American Statistician in 1975. It became famous through the column of Marilyn vos Savant in Parade magazine.

## Explanations

### L1 human

Staying wins only when your first choice was right, which happens one time in three.
Switching wins every time your first choice was wrong, which happens two times in three.
</document>

<document source="https://tamnd.github.io/math2/node/0PBY4E9GQF.md" tag="0PBY4E9GQF" kind="concept">
# Skew-symmetric matrix

## Summary

A skew-symmetric matrix is a square matrix whose transpose is its negative.
</document>

<document source="https://tamnd.github.io/math2/node/0PCKBBMEXC.md" tag="0PCKBBMEXC" kind="statement">
# Order of the symmetric group

## Statement

Let $n$ be a positive integer.
The [[symmetric group|0515A6D2Y1]] $S_n$ has [[order|0B85S2V2DR]] $n!$.

## Remarks

A permutation $\sigma$ has $n$ choices for $\sigma(1)$, then $n - 1$ choices for $\sigma(2)$, and so on, down to $1$ choice for $\sigma(n)$.
For example, $|S_3| = 6$ and $|S_4| = 24$.

## Explanations

### L1 human

To make a permutation, you choose where each symbol goes, one symbol after the other.
Each choice removes one place from the next choices.
So the number of permutations is $n \cdot (n-1) \cdots 1 = n!$.
</document>

<document source="https://tamnd.github.io/math2/node/0PCV9EDJM2.md" tag="0PCV9EDJM2" kind="concept">
# Multinomial coefficient

## Summary

A multinomial coefficient counts the ways to distribute $n$ distinct objects into labelled boxes of given sizes. It extends the binomial coefficient from two boxes to any number of boxes. The node [[multinomial coefficient|00K0RF9BFY]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0PCVBRV9VJ.md" tag="0PCVBRV9VJ" kind="statement">
# Fermat's theorem on stationary points

## Statement

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$, and let $a \in I$.
Assume that $f$ has a [[local maximum|00999TYPXB]] or a [[local minimum|0NS8BP96QX]] at $a$, and that $f$ is differentiable at $a$.
Then $f'(a) = 0$.

## Remarks

At a local maximum, the difference quotient $(f(a + h) - f(a))/h$ is $\le 0$ for small $h > 0$ and $\ge 0$ for small $h < 0$. So the limit is both $\le 0$ and $\ge 0$.
The point must be interior. On $[0, 1]$, the function $x$ has a maximum at $1$, where its derivative is $1$.
Mathlib states it as `IsLocalExtr.deriv_eq_zero`.

## Explanations

### L1 human

At the top of a smooth hill, the tangent line is horizontal.
If the slope were positive or negative, moving a little would give a higher value.
</document>

<document source="https://tamnd.github.io/math2/node/0PD57PSHNR.md" tag="0PD57PSHNR" kind="definition">
# Circuit

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A circuit in $G$ is a closed [[trail|062MQGP8M5]] of positive length.
That is a walk $(w_0, w_1, \dots, w_k)$ with $k \ge 1$, $w_k = w_0$, and pairwise distinct edges.

## Remarks

A circuit may visit a vertex more than once. A [[cycle|07CT9RDY7Q]] is a circuit with no repeated vertex other than the start.
In a simple graph each circuit has length at least $3$.
Some authors use the word circuit for a cycle. Others use the word tour.

## Explanations

### L1 human

A circuit is a trail that ends where it starts.
It is a round trip that never uses the same edge twice.
</document>

<document source="https://tamnd.github.io/math2/node/0PDBVPP233.md" tag="0PDBVPP233" kind="counterexample">
# Converse of Lagrange's theorem fails for the alternating group A4

## Counterexample

The converse of [[Lagrange's theorem|3C53TD]] would say this. If $G$ is a finite group and $d$ is a positive divisor of $|G|$, then $G$ has a subgroup of order $d$.
This is false.
Let $A_4$ be the group of the even permutations of $\{1, 2, 3, 4\}$, under composition.
It has order $12$, and $6$ divides $12$, but $A_4$ has no subgroup of order $6$.

## Remarks

The elements of $A_4$ are the identity, the three products of two disjoint transpositions, and the eight $3$-cycles.
A subgroup $H$ of order $6$ would have index $2$, and thus it would contain the square of each element of $A_4$.
The square of each $3$-cycle is a $3$-cycle, and each $3$-cycle is such a square. So $H$ would contain all eight $3$-cycles, which is too many.
The group $A_4$ is the smallest group for which the converse fails.

## Explanations

### L1 human

Lagrange's theorem says which sizes a subgroup can have, but it does not promise that each allowed size occurs.
The group $A_4$ has $12$ elements, and $6$ is an allowed size, but no subgroup has $6$ elements.
For cyclic groups the converse is true, so a counterexample must be a group that is not cyclic.
</document>

<document source="https://tamnd.github.io/math2/node/0PDVQEP0ZC.md" tag="0PDVQEP0ZC" kind="concept">
# Matrix–vector product

## Summary

The product of an $m \times n$ matrix and a column vector with $n$ entries is a column vector with $m$ entries.
Its entry $i$ is the dot product of row $i$ of the matrix with the vector.
</document>

<document source="https://tamnd.github.io/math2/node/0PEJH7AC74.md" tag="0PEJH7AC74" kind="definition">
# Transitive closure

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The transitive closure of $R$ is the relation $R^{+}$ on $A$ such that $a \mathrel{R^{+}} b$ when there are $n \ge 1$ and $x_0, \dots, x_n \in A$ with
$$a = x_0 \mathrel{R} x_1 \mathrel{R} x_2 \cdots x_{n-1} \mathrel{R} x_n = b.$$

## Remarks

The relation $R^{+}$ is [[transitive|0CYRECG2YQ]], it contains $R$, and it is a subset of each transitive relation that contains $R$.
It is also the intersection of all transitive relations on $A$ that contain $R$.
Example: the transitive closure of "is a parent of" is "is an ancestor of".

## Explanations

### L1 human

The transitive closure adds all the shortcuts.
In a graph, $a \mathrel{R^{+}} b$ means that there is a path of one or more arrows from $a$ to $b$.
</document>

<document source="https://tamnd.github.io/math2/node/0PF6TM8TWT.md" tag="0PF6TM8TWT" kind="concept">
# Algebraically closed field

## Summary

An algebraically closed field is a field in which every nonconstant polynomial has a root.
The complex numbers are the main example.
The [[definition of an algebraically closed field|02SJG1HC32]] gives equivalent conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0PFPWKBDTJ.md" tag="0PFPWKBDTJ" kind="definition">
# Restriction of an endomorphism to an invariant subspace

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an endomorphism.
Let $W$ be a $T$-[[invariant subspace|0ACX5QT8TE]] of $V$.
The *restriction* of $T$ to $W$ is the endomorphism
$$T|_W \colon W \to W, \qquad T|_W(w) = T(w).$$

## Remarks

The map is well defined because $T(W) \subseteq W$, and it is linear.
If $V$ is finite-dimensional, then the characteristic polynomial of $T|_W$ divides that of $T$.
Take a basis of $V$ that extends a basis of $W$.
In this basis, the matrix of $T$ is block upper triangular, and its first diagonal block is the matrix of $T|_W$.

## Explanations

### L1 human

If a map keeps a subspace inside itself, then it is also a map of that subspace.
This smaller map is the restriction.
Its eigenvalues are some of the eigenvalues of the original map.
</document>

<document source="https://tamnd.github.io/math2/node/0PG2H7WKGG.md" tag="0PG2H7WKGG" kind="concept">
# Divisibility

## Summary

Divisibility is the relation between two integers $a$ and $b$ that holds when $b$ is an integer multiple of $a$.
Then $a$ is a divisor of $b$, and we write $a \mid b$.
The [[definition of divisibility|096PW7Q014]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0PH0VHTQS0.md" tag="0PH0VHTQS0" kind="definition">
# Moment

## Definition

Let $X$ be a real random variable and let $k \ge 1$ be an integer with $E[|X|^k] < \infty$.
The $k$-th moment of $X$ is
$$m_k = E[X^k].$$
The $k$-th absolute moment is $E[|X|^k]$.

## Remarks

By the [[law of the unconscious statistician|01ZWX875AT]], $m_k = \int x^k \, \mu_X(dx)$. So the moments depend only on the law $\mu_X$ of $X$.
The first moment is the mean. The variance is $m_2 - m_1^2$.
If $E[|X|^k] < \infty$, then $E[|X|^j] < \infty$ for each $j \le k$.
The [[Cauchy distribution|0NA6Q82YR7]] has no moments of order $k \ge 1$.

## Explanations

### L1 human

Moments are numbers that sum up the shape of a distribution.
The first gives the center, the second with the first gives the spread, and the higher ones describe asymmetry and the weight of the tails.
</document>

<document source="https://tamnd.github.io/math2/node/0PHAC833JF.md" tag="0PHAC833JF" kind="concept">
# Generating function

## Summary

A generating function encodes a sequence as the coefficients of a formal power series. It turns counting problems into algebra with series. The node [[generating function|0HH6JX5WZZ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0PHBS78HKT.md" tag="0PHBS78HKT" kind="statement">
# First supplement to quadratic reciprocity

## Statement

Let $p$ be an odd prime.
Then
$$\left(\frac{-1}{p}\right) = (-1)^{(p-1)/2},$$
where the left side is a [[Legendre symbol|01J8AMV9RS]].

## Remarks

So $\left(\frac{-1}{p}\right) = 1$ if $p \equiv 1 \pmod 4$, and $\left(\frac{-1}{p}\right) = -1$ if $p \equiv 3 \pmod 4$.
The proof is [[Euler's criterion|0M7DEDKSYB]] with $a = -1$.
For example, $2^2 = 4 \equiv -1 \pmod 5$, but $-1$ is not a square modulo $7$.

## Explanations

### L1 human

The number $-1$ is a square modulo an odd prime $p$ exactly when $p$ leaves remainder $1$ on division by $4$.
So $-1$ is a square modulo $5$, $13$ and $17$, but not modulo $3$, $7$ and $11$.
</document>

<document source="https://tamnd.github.io/math2/node/0PHC9FE2V5.md" tag="0PHC9FE2V5" kind="concept">
# Bernoulli trial

## Summary

A Bernoulli trial is a [[random experiment|0HM05N3FJD]] with exactly two possible outcomes, called success and failure.
The probability $p$ of success is the same each time the trial is repeated, and the failure has probability $q = 1 - p$.
A sequence of Bernoulli trials usually means independent repetitions of one trial. The number of successes in $n$ such trials has the [[binomial distribution|0VRWZDWW6B]].

## Remarks

The outcome of one trial, coded as $1$ for success and $0$ for failure, has the [[Bernoulli distribution|0J97P5KZY0]].
The trials are named after Jacob Bernoulli, who studied them in his book Ars Conjectandi of 1713.
Examples are a coin toss, with heads as success, and a roll of a die, with a six as success and $p = 1/6$.

## Explanations

### L1 human

A Bernoulli trial is a yes-or-no experiment.
Many questions have this form: does the coin show heads, does the patient recover, does the part fail.
</document>

<document source="https://tamnd.github.io/math2/node/0PHDYNR24D.md" tag="0PHDYNR24D" kind="concept">
# Null space

## Summary

The null space of a matrix is the set of vectors that the matrix sends to zero.
It is the solution set of the homogeneous system with this matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0PHN07KVQT.md" tag="0PHN07KVQT" kind="concept">
# Residue

## Summary

The residue of a function at an isolated singularity $c$ is the coefficient of $(z - c)^{-1}$ in its Laurent series, see the [[definition|0JFGKKWWFK]].
It is $\frac{1}{2\pi i}$ times the integral of the function around a small circle around $c$.
</document>

<document source="https://tamnd.github.io/math2/node/0PHT6NBYPB.md" tag="0PHT6NBYPB" kind="statement">
# Heine–Borel theorem in finite-dimensional normed spaces

## Statement

Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, with a norm $\|\cdot\|$ and the metric $d(x, y) = \|x - y\|$.
Then a subset $K \subseteq V$ is [[compact|00R0X1WGPX]] if and only if $K$ is [[closed|0G5AV9Q4RR]] and [[bounded|0P953MG05E]].

## Remarks

Proof: choose a basis. By [[equivalence of norms in finite dimension|03MC70ZRNR]], the norm is equivalent to the Euclidean norm of the coordinates. Equivalent norms have the same closed sets, bounded sets and compact sets. Then use the [[Heine–Borel theorem|07PFYV65MB]] in $\mathbb{R}^n$ or $\mathbb{C}^n = \mathbb{R}^{2n}$.
In other words, each finite-dimensional normed space is a [[proper metric space|0Z4V98VN5E]].
The converse also holds: if the closed unit ball is compact, the space is [[finite-dimensional|0VZVV8VME5]].

## Explanations

### L1 human

In each finite-dimensional normed space, the compact sets are the closed bounded sets, just as in $\mathbb{R}^n$.
</document>

<document source="https://tamnd.github.io/math2/node/0PHTC93M58.md" tag="0PHTC93M58" kind="concept">
# Composition factors

## Summary

The composition factors of a group are the factor groups of a composition series.
By the Jordan–Hölder theorem, they do not depend on the choice of the series, up to order and isomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/0PJXBR6QA3.md" tag="0PJXBR6QA3" kind="concept">
# Multi-index notation

## Summary

A multi-index of dimension $n$ is an $n$-tuple $\alpha = (\alpha_1, \dots, \alpha_n)$ of non-negative integers.
With $|\alpha| = \alpha_1 + \dots + \alpha_n$, $\alpha! = \alpha_1! \cdots \alpha_n!$, $x^\alpha = x_1^{\alpha_1} \cdots x_n^{\alpha_n}$ and $\partial^\alpha = \partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n}$, formulas of several variables take the short form of their one-variable versions.

## Remarks

Here $\partial_i$ is the [[partial derivative|02RSXVM1ZN]] with respect to $x_i$, and $\partial^\alpha f$ is a partial derivative of order $|\alpha|$.
For $\alpha = (2, 0, 1)$, $|\alpha| = 3$, $\alpha! = 2$, $x^\alpha = x_1^2 x_3$ and $\partial^\alpha f = \partial^3 f / \partial x_1^2 \, \partial x_3$.
The notation $\partial^\alpha f$ does not show the order of differentiation. It is unambiguous when $f$ is of class $C^{|\alpha|}$, by [[Schwarz's theorem|0NNQY8WCW6]].
Examples of use are [[Taylor's theorem in several variables|0J8T4K4YTE]] and the multinomial theorem $(x_1 + \dots + x_n)^k = \sum_{|\alpha| = k} \frac{k!}{\alpha!} x^\alpha$.

## Explanations

### L1 human

Multi-index notation is a bookkeeping device.
One symbol $\alpha$ records how many times each variable appears, so one formula covers all mixed derivatives or monomials of a given order.
</document>

<document source="https://tamnd.github.io/math2/node/0PK8QH1W1V.md" tag="0PK8QH1W1V" kind="statement">
# Alternating series test

## Statement

Let $(b_n)$ be a decreasing real sequence with $b_n \to 0$ as $n \to \infty$.
Then the [[alternating series|017BR1M98J]] $\sum_{n=0}^{\infty} (-1)^n b_n$ converges.
Moreover, its sum $S$ and its partial sums $s_N = \sum_{n=0}^{N} (-1)^n b_n$ satisfy $|S - s_N| \le b_{N+1}$ for each $N$.

## Remarks

The even partial sums $s_0, s_2, s_4, \dots$ decrease, and the odd partial sums $s_1, s_3, \dots$ increase. Each odd partial sum is below each even one, so both sequences converge by the [[monotone convergence theorem|0FM6N4R0VB]]. Their limits agree, since $s_{2k} - s_{2k+1} = b_{2k+1} \to 0$.
The hypothesis that $(b_n)$ decreases is needed. Without it, an alternating series with terms tending to $0$ can diverge.
Mathlib states the convergence for an antitone sequence, in `Antitone.tendsto_alternating_series_of_tendsto_zero`.

## History

Leibniz discussed the criterion in an unpublished work of 1676. For this reason it is also called the Leibniz test.

## Explanations

### L1 human

Each step goes in the other direction and is shorter than the step before.
So the partial sums swing back and forth with smaller and smaller swings, and they settle down.
</document>

<document source="https://tamnd.github.io/math2/node/0PKG97PX9Y.md" tag="0PKG97PX9Y" kind="concept">
# Conservative vector field

## Summary

A conservative vector field is a [[vector field|0299VCW1TH]] that is the [[gradient|0ABK9V9Y64]] of a scalar function, called a [[scalar potential|0H7C3T1GJY]] of the field.
Its line integrals depend only on the end points of the path, and they vanish around each closed path.
</document>

<document source="https://tamnd.github.io/math2/node/0PKRYB8ATM.md" tag="0PKRYB8ATM" kind="concept">
# Limit of a function

## Summary

The limit of a function at a point is the value that the function approaches as its argument approaches the point. It does not depend on the value at the point itself. The node [[limit of a function|0TXTSEMHR5]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0PM88Z9W4M.md" tag="0PM88Z9W4M" kind="statement">
# Projections are continuous

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]] and give $X \times Y$ the [[product topology|0Z57EC6YX3]].
Then the projections $\pi_1 \colon X \times Y \to X$, $(x, y) \mapsto x$, and $\pi_2 \colon X \times Y \to Y$, $(x, y) \mapsto y$, are [[continuous|0SY9ECQ19Q]].

## Remarks

Proof: for $U$ open in $X$, we have $\pi_1^{-1}(U) = U \times Y$, which is a basic open set. The proof for $\pi_2$ is the same.
The projections are also [[open maps|0SQ2PC596Y]], but in general they are not closed maps.
The product topology is the coarsest topology on $X \times Y$ that makes both projections continuous.

## Explanations

### L1 human

Taking one coordinate of a point is a continuous operation.
For example, the map $\mathbb{R}^2 \to \mathbb{R}$, $(x, y) \mapsto x$, is continuous.
</document>

<document source="https://tamnd.github.io/math2/node/0PMRQJVTME.md" tag="0PMRQJVTME" kind="object">
# Klein four-group

## Description

The Klein four-group $V$ is the group with four elements $e, a, b, c$, in which $e$ is the identity and each element is its own inverse.
The product of two distinct elements of $\{a, b, c\}$ is the third one.
It is [[abelian|082GHNPJ53]], and it is [[isomorphic|08MG7H8JW5]] to the [[direct product|00HF4RJ5QN]] $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

## Remarks

The Klein four-group is the smallest group that is not [[cyclic|0KVGH0J5B3]], because it has no element of order $4$.
It is the symmetry group of a rectangle that is not a square.
In $S_4$, the set $\{e, (1\ 2)(3\ 4), (1\ 3)(2\ 4), (1\ 4)(2\ 3)\}$ is a normal subgroup isomorphic to $V$.
Each group of order $4$ is isomorphic to $\mathbb{Z}/4\mathbb{Z}$ or to $V$.

## Explanations

### L1 human

The Klein four-group is like two light switches that you can flip separately.
Each switch flipped two times is back to the start.
Flipping both switches is the third element that is not the identity.
</document>

<document source="https://tamnd.github.io/math2/node/0PMT9YJZVW.md" tag="0PMT9YJZVW" kind="statement">
# Order of an inverse

## Statement

Let $G$ be a [[group|G5S1M9]] and let $g \in G$.
Then $g$ and its inverse $g^{-1}$ have the same [[order|0WE2NHZJ7B]].
In particular, $g$ has infinite order if and only if $g^{-1}$ has infinite order.

## Remarks

For each positive integer $n$, $(g^{-1})^n = (g^n)^{-1}$, and $g^n = e$ if and only if $(g^n)^{-1} = e$.
For example, in $\{1, i, -1, -i\}$, both $i$ and $i^{-1} = -i$ have order $4$.

## Explanations

### L1 human

The inverse of an element walks the same path back to the identity, in the opposite direction.
It needs the same number of steps.
A rotation by $90$ degrees and a rotation by $-90$ degrees both have order $4$.
</document>

<document source="https://tamnd.github.io/math2/node/0PMY0YQYXX.md" tag="0PMY0YQYXX" kind="statement">
# The continued fraction of a rational number terminates

## Statement

Let $x$ be a real number.
The [[continued fraction expansion|0SJ7Z6VKWK]] of $x$ is finite if and only if $x$ is rational.

## Remarks

If $x = a/b$ with $b > 0$, the expansion follows the [[Euclidean algorithm|0TRNT4X7J0]] on $a$ and $b$. So it stops after finitely many steps.
Conversely, a finite simple continued fraction is a rational number.
For example, $\frac{43}{19}$: $43 = 2 \cdot 19 + 5$, $19 = 3 \cdot 5 + 4$, $5 = 1 \cdot 4 + 1$, $4 = 4 \cdot 1$. So $\frac{43}{19} = [2; 3, 1, 4]$.
A rational number has exactly two finite expansions, since $[a_0; \dots, a_n] = [a_0; \dots, a_n - 1, 1]$ when $a_n > 1$.

## Explanations

### L1 human

For a fraction, the continued fraction algorithm is the Euclidean algorithm in another form.
The remainders get smaller, so the process must end.
An irrational number gives an infinite continued fraction.
</document>

<document source="https://tamnd.github.io/math2/node/0PN13BB7BT.md" tag="0PN13BB7BT" kind="statement">
# The Gaussian integers are a Euclidean domain

## Statement

The ring of [[Gaussian integers|0K5W69AC0J]] $\mathbb{Z}[i]$ is a [[Euclidean domain|0GJ2HRG95T]], with the [[norm|03AN1V4ZFJ]] $N(a + b i) = a^2 + b^2$ as Euclidean function.
That is, for all $z, w \in \mathbb{Z}[i]$ with $w \neq 0$, there are $q, r \in \mathbb{Z}[i]$ with $z = q w + r$ and $N(r) < N(w)$.

## Remarks

The proof: write $z / w = x + y i$ in $\mathbb{C}$. Let $q = m + n i$, where $m$ and $n$ are integers nearest to $x$ and $y$.
Then $|z/w - q|^2 \le \tfrac{1}{4} + \tfrac{1}{4} = \tfrac{1}{2}$, so $r = z - q w$ has $N(r) \le \tfrac{1}{2} N(w) < N(w)$.
A consequence is that $\mathbb{Z}[i]$ is a principal ideal domain and a unique factorization domain.

## Explanations

### L1 human

To divide $z$ by $w$, compute the complex quotient and round both coordinates to the nearest integer.
The rounding error is small, so the remainder is smaller than $w$ in norm.
This makes the Euclidean algorithm work for Gaussian integers.
</document>

<document source="https://tamnd.github.io/math2/node/0PN2BXRS8G.md" tag="0PN2BXRS8G" kind="statement">
# Argument principle

## Statement

Let $c \in \mathbb{C}$ and $r > 0$.
Let $f$ be a [[meromorphic function|0NSD1TPHR0]] on an open set that contains the closed disc $\{ z : |z - c| \le r \}$.
Suppose that $f$ has no zeros and no poles on the circle $|z - c| = r$.
Let $Z$ be the number of zeros of $f$ in the open disc $|z - c| < r$, each counted as many times as its [[order|064N002TX4]].
Let $P$ be the number of poles of $f$ in the open disc, each counted as many times as its [[order|0YK27TJ4WF]].
Then
$$\frac{1}{2\pi i} \oint_{|z - c| = r} \frac{f'(z)}{f(z)} \, dz = Z - P,$$
where the [[circle integral|0T7ABX4CWE]] goes once around the circle counterclockwise.

## Remarks

The function $f'/f$ is the logarithmic derivative of $f$.
At a zero of order $m$, it has a simple pole with residue $m$; at a pole of order $m$, it has a simple pole with residue $-m$.
So the theorem follows from the [[residue theorem|032HKYS329]].
The left side is also the [[winding number|0X7SBS8E0F]] around $0$ of the closed curve $\theta \mapsto f(c + r e^{i\theta})$, $\theta \in [0, 2\pi]$.

## Explanations

### L1 human

Walk once around the circle and watch the point $f(z)$.
Each zero inside the circle makes $f(z)$ turn once around $0$ in the positive direction, and each pole makes it turn once in the negative direction.
For example, $f(z) = z^3$ on the unit circle turns three times around $0$, and $f$ has a zero of order $3$ at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0PN4QJE969.md" tag="0PN4QJE969" kind="definition">
# Farey sequence

## Definition

Let $n$ be a positive integer.
The Farey sequence $F_n$ of order $n$ is the increasing sequence of all fractions $\frac{a}{b}$ with $0 \le a \le b \le n$ and $\gcd(a, b) = 1$.
Each fraction is in lowest terms, and $F_n$ starts with $\frac{0}{1}$ and ends with $\frac{1}{1}$.

## Remarks

For example, $F_5 = \left( \frac{0}{1}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{1}{1} \right)$.
If $\frac{a}{b} < \frac{c}{d}$ are neighbours in $F_n$, then $bc - ad = 1$.
The length of $F_n$ is $1 + \sum_{k=1}^{n} \varphi(k)$, with $\varphi$ [[Euler's totient function|0VV04M41AN]].
The sequences are named after John Farey, whose letter about them appeared in 1816.

## Explanations

### L1 human

Write all fractions between $0$ and $1$ with denominators up to $n$, in lowest terms and in increasing order.
Neighbouring fractions in this list are always very close in a precise sense: their cross difference is $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0PN9WMRWT7.md" tag="0PN9WMRWT7" kind="statement">
# The constructible numbers form a field closed under square roots

## Statement

The set of the [[constructible numbers|0FMN2MN7TM]] is a [[subfield|07QY5RPFBR]] of $\mathbb{C}$.
If $z$ is constructible, then the square roots of $z$ are constructible.
Moreover, the field of constructible numbers is the smallest subfield of $\mathbb{C}$ that is closed under square roots.

## Remarks

The proof of closure uses classical constructions: parallels and similar triangles give products and quotients, and a circle on a diameter of length $1 + x$ gives $\sqrt{x}$.
Conversely, the intersection points of lines and circles have coordinates that come from the old ones by field operations and square roots.
So a number is constructible if and only if it is obtained from $\mathbb{Q}$ by finitely many field operations and square roots.
The constructible real numbers form an ordered field in which every positive element has a square root.

## Explanations

### L1 human

You can add, subtract, multiply and divide lengths with straightedge and compass, and you can take square roots.
Nothing else is possible, because lines and circles only lead to equations of degree one or two.
</document>

<document source="https://tamnd.github.io/math2/node/0PNPG2PP9H.md" tag="0PNPG2PP9H" kind="statement">
# Jordan's lemma

## Statement

Let $a > 0$ and $R > 0$ be real numbers.
Let $C_R$ be the semicircle $\{ R e^{i\theta} : \theta \in [0, \pi] \}$ in the upper half-plane, with the parametrization $\theta \mapsto R e^{i\theta}$.
Let $g$ be a continuous complex function on $C_R$, and let $f(z) = e^{iaz} g(z)$ for $z \in C_R$, where $e^w$ is the [[complex exponential|054VXHMCA3]].
Let $M_R = \max_{\theta \in [0, \pi]} |g(R e^{i\theta})|$.
Then the [[contour integral|00FWNP9TXY]] of $f$ along $C_R$ satisfies
$$\left| \int_{C_R} f(z) \, dz \right| \le \frac{\pi}{a} M_R.$$

## Remarks

If $M_R \to 0$ as $R \to \infty$, then $\int_{C_R} f(z) \, dz \to 0$ as $R \to \infty$.
The bound does not contain the length $\pi R$ of $C_R$, so it is stronger than the [[estimation lemma|0DJMGPN8K9]] for this type of integrand.
A similar result holds for the semicircle in the lower half-plane when $a < 0$.
The lemma is a main step in the [[evaluation of real integrals|0476QH9846]] such as $\int_{-\infty}^{\infty} \frac{e^{iax}}{x^2 + 1} \, dx$.

## Explanations

### L1 human

On the semicircle, $|e^{iaz}| = e^{-aR \sin \theta}$, and this number is very small except near the two ends of the semicircle.
So the factor $e^{iaz}$ kills most of the long path.
This is why the bound stays small even when the radius $R$ is large.
</document>

<document source="https://tamnd.github.io/math2/node/0PNRKMRC55.md" tag="0PNRKMRC55" kind="concept">
# Random variable

## Summary

A random variable is a quantity whose value depends on the outcome of a random experiment.
Formally it is a measurable function on the sample space of a probability space. The node [[random variable|0DKRR22AEQ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0PNWCM559P.md" tag="0PNWCM559P" kind="statement">
# Interior is open

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then the [[interior|0C38S0KMJK]] $\operatorname{int} S$ is an [[open set|01ZQQDCERP]] of $X$.

## Remarks

Proof: $\operatorname{int} S$ is a union of open sets, and each union of open sets is open by the axioms of a topology.
It follows that $\operatorname{int}(\operatorname{int} S) = \operatorname{int} S$.

## Explanations

### L1 human

The interior is made by putting together open sets, and open sets put together give an open set.
</document>

<document source="https://tamnd.github.io/math2/node/0PP5ZAJPP7.md" tag="0PP5ZAJPP7" kind="definition">
# Beta distribution

## Definition

Let $\alpha > 0$ and $\beta > 0$.
The beta distribution with parameters $\alpha$ and $\beta$ is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$f(x) = \begin{cases} \dfrac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)} & \text{if } 0 < x < 1, \\ 0 & \text{otherwise,} \end{cases}$$
where $B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} \, dt$ is the beta function.

## Remarks

The mean is $\alpha / (\alpha + \beta)$ and the variance is $\dfrac{\alpha\beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$.
$\operatorname{Beta}(1, 1)$ is the [[uniform distribution|0WGDPYR30Q]] on $[0, 1]$.
The beta function satisfies $B(\alpha, \beta) = \Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha + \beta)$.
In Bayesian statistics, the beta distribution is the conjugate prior of the success probability $p$ of binomial data. A $\operatorname{Beta}(\alpha, \beta)$ prior and $k$ successes in $n$ trials give a $\operatorname{Beta}(\alpha + k, \beta + n - k)$ posterior.

## Explanations

### L1 human

The beta distribution describes a random proportion, a number between $0$ and $1$.
Large $\alpha$ pushes the mass toward $1$, large $\beta$ pushes it toward $0$, and $\alpha = \beta$ gives a shape that is symmetric about $1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/0PPG793642.md" tag="0PPG793642" kind="statement">
# Determinant is multilinear in the rows

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $1 \le i \le n$.
Let $A$, $B$, $C$ be $n \times n$ matrices over $K$ that agree outside row $i$.
Let $a$, $b$, $c$ be row $i$ of $A$, $B$, $C$, and assume $c = sa + tb$ with $s, t \in K$.
Then
$$\det C = s \det A + t \det B,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is property 1 of the definition of the determinant.
The determinant is not additive in the whole matrix: in general $\det(A + B) \ne \det A + \det B$.
By $\det A^T = \det A$, the determinant is also multilinear in the columns.

## Explanations

### L1 human

If we fix all rows but one, then the determinant is a linear function of the remaining row.
So a sum in one row splits the determinant into a sum of two determinants.
A factor in one row comes out of the determinant.
</document>

<document source="https://tamnd.github.io/math2/node/0PPP2J2DJV.md" tag="0PPP2J2DJV" kind="statement">
# Second isomorphism theorem

## Statement

Let $G$ be a [[group|G5S1M9]], let $S$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
Then $SN$ is a subgroup of $G$, $N$ is normal in $SN$, and $S \cap N$ is normal in $S$.
Moreover, the map $s(S \cap N) \mapsto sN$ is an [[isomorphism|08MG7H8JW5]] from $S/(S \cap N)$ to $SN/N$.

## Remarks

The proof applies the [[first isomorphism theorem|0CXXJNG5B6]] to the homomorphism $S \to SN/N$, $s \mapsto sN$, which is surjective with kernel $S \cap N$.
For finite groups, it gives $|SN| = |S| \, |N| / |S \cap N|$.
It is enough that $S$ normalizes $N$, that is, $sNs^{-1} = N$ for each $s \in S$.

## Explanations

### L1 human

Picture a diamond with $SN$ at the top, $S$ and $N$ in the middle, and $S \cap N$ at the bottom.
The theorem says that the two slanted sides opposite each other give the same quotient group.
For integers, with $S = 4\mathbb{Z}$ and $N = 6\mathbb{Z}$, it gives $4\mathbb{Z}/12\mathbb{Z} \cong 2\mathbb{Z}/6\mathbb{Z}$, both cyclic of order $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0PQ2GGY1QV.md" tag="0PQ2GGY1QV" kind="definition">
# Null space

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$.
The *null space* of $A$ is
$$\operatorname{Null}(A) = \{x \in F^n : Ax = 0\}.$$
It is the [[kernel|0S23E6YE83]] of the linear map $x \mapsto Ax$.

## Remarks

The null space is a subspace of $F^n$, of dimension $n - \rank A$, the [[nullity|0APCFW7639]] of $A$.
Row equivalent matrices have the same null space.
The null space of $\begin{pmatrix} 1 & 1 \end{pmatrix}$ is the line spanned by $(1, -1)^T$.
Each vector of the null space is orthogonal to each row of $A$, for the dot product.

## Explanations

### L1 human

The null space of a matrix is the set of solutions of $Ax = 0$.
We find a basis of it by row reduction: each free variable gives one basis vector.
It is the kernel of the matrix, viewed as a linear map.
</document>

<document source="https://tamnd.github.io/math2/node/0PQEBTYDZM.md" tag="0PQEBTYDZM" kind="statement">
# Probability of the complement

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A \in \mathcal{F}$ be an [[event|00SQGJNRTB]].
Let $A^c = \Omega \setminus A$ be the [[complement|0B1FSS1VZW]] of $A$.
Then
$$P(A^c) = 1 - P(A).$$

## Remarks

The proof: $A$ and $A^c$ are disjoint and $A \cup A^c = \Omega$, so $P(A) + P(A^c) = P(\Omega) = 1$ by additivity.
With $A = \Omega$, the rule gives $P(\varnothing) = 0$.
The rule is useful when $A^c$ is simpler than $A$. For example, the probability of at least one six in four rolls of a fair die is $1 - (5/6)^4 \approx 0.518$.

## Explanations

### L1 human

Either $A$ happens or it does not, and the two chances add up to $1$.
If the chance of rain is $0.3$, then the chance of no rain is $0.7$.
</document>

<document source="https://tamnd.github.io/math2/node/0PQQGFAWSH.md" tag="0PQQGFAWSH" kind="statement">
# Alternating sum of binomial coefficients

## Statement

Let $n$ be a positive integer.
Then
$$\sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0.$$

## Remarks

For $n = 0$ the sum is $1$, so the hypothesis $n \ge 1$ is necessary.
For $n = 4$: $1 - 4 + 6 - 4 + 1 = 0$.
The identity is the [[binomial theorem|0EHWPWGHB2]] with $x = -1$ and $y = 1$. It says that a nonempty finite set has as many subsets of even size as of odd size.

## Explanations

### L1 human

Add and subtract the numbers of a row of [[Pascal's triangle|05DME4TZ3K]] in turn, and you get $0$.
For a nonempty set, add or remove one fixed element. This pairs the even subsets with the odd subsets.
</document>

<document source="https://tamnd.github.io/math2/node/0PR5P9Z4KM.md" tag="0PR5P9Z4KM" kind="statement">
# Principal axis theorem

## Statement

Let $A$ be a real symmetric $n \times n$ matrix, and let $Q(x) = x^T A x$ for $x \in \R^n$.
Then there are an [[orthogonal matrix|0W02QESPJQ]] $P$ and real numbers $\lambda_1, \dots, \lambda_n$ such that, for $x = P y$,
$$Q(x) = \lambda_1 y_1^2 + \lambda_2 y_2^2 + \cdots + \lambda_n y_n^2.$$
The numbers $\lambda_i$ are the eigenvalues of $A$, and the columns of $P$ are orthonormal eigenvectors of $A$.

## Remarks

It follows from the [[spectral theorem|0DGZBQWZWN]] $A = P D P^T$ with $D$ diagonal.
The lines spanned by the columns of $P$ are the *principal axes* of $Q$.
For example, $Q(x) = 5x_1^2 + 4x_1 x_2 + 2x_2^2$ has the eigenvalues $6$ and $1$, so it becomes $6y_1^2 + y_2^2$.
So the curve $Q(x) = 1$ is an ellipse, with semi-axes $1/\sqrt 6$ and $1$.

## Explanations

### L1 human

Each real quadratic form becomes a sum of squares after a rotation of the axes, possibly with a reflection.
The new axes are the principal axes of the form.
This is how we find the shape of an ellipse or a hyperbola from its equation.
</document>

<document source="https://tamnd.github.io/math2/node/0PR88XHDMB.md" tag="0PR88XHDMB" kind="statement">
# Dirichlet convolution of multiplicative functions is multiplicative

## Statement

Let $f$ and $g$ be [[multiplicative functions|0AP80TT660]].
Then their [[Dirichlet convolution|0KH1N66QWW]] $f * g$ is a multiplicative function.

## Remarks

Let $m$ and $n$ be coprime. The proof uses that each divisor of $mn$ is a unique product of a divisor of $m$ and a divisor of $n$.
With $f = \mathrm{id}_k \colon n \mapsto n^k$ and $g = \mathbf{1}$, the theorem gives the [[multiplicativity of the divisor function|04WC36XKV6]], since $\sigma_k = \mathrm{id}_k * \mathbf{1}$.
The Dirichlet inverse of a multiplicative function is also multiplicative.

## Explanations

### L1 human

Multiplicative functions stay multiplicative when you convolve them.
So a new function built from known ones by divisor sums keeps the property, and you can compute it from prime powers.
</document>

<document source="https://tamnd.github.io/math2/node/0PRAS55318.md" tag="0PRAS55318" kind="definition">
# Angle between two vectors

## Definition

Let $n$ be a positive integer, and let $u$ and $v$ be nonzero [[vectors|0QN3D80AWP]] in $\mathbb{R}^n$.
Let $u \cdot v = u_1 v_1 + \dots + u_n v_n$ be their dot product, and let $\|u\|$ and $\|v\|$ be their [[lengths|0XKBKGTCPC]].
The angle between $u$ and $v$ is the unique real number $\theta$ with $0 \leq \theta \leq \pi$ such that
$$\cos \theta = \frac{u \cdot v}{\|u\| \, \|v\|}.$$
The Cauchy–Schwarz inequality $|u \cdot v| \leq \|u\| \, \|v\|$ makes sure that the quotient is in $[-1, 1]$. Thus $\theta = \arccos\left(\frac{u \cdot v}{\|u\| \, \|v\|}\right)$ exists.

## Remarks

The angle is $0$ if and only if $v = c u$ for a real number $c > 0$. It is $\pi$ if and only if $v = c u$ for a real number $c < 0$.
The angle does not depend on the order of the vectors. It does not change when you multiply $u$ or $v$ by a positive number.
Mathlib uses the same formula for all vectors. With the convention $0 / 0 = 0$, the angle between the zero vector and any vector is then $\pi / 2$.

## Explanations

### L1 human

Put the two arrows at the same start point. The angle between them is the opening between the arrows, measured in radians from $0$ to $\pi$. For example, $(1, 0)$ and $(0, 1)$ have dot product $0$, so the angle between them is $\arccos 0 = \pi / 2$, a right angle.
</document>

<document source="https://tamnd.github.io/math2/node/0PSKT5W817.md" tag="0PSKT5W817" kind="concept">
# Identity function

## Summary

The identity function of a set sends each element to itself.
</document>

<document source="https://tamnd.github.io/math2/node/0PSRCMCM5X.md" tag="0PSRCMCM5X" kind="definition">
# Moore–Penrose inverse

## Definition

Let $A$ be an $m \times n$ complex matrix.
There is exactly one $n \times m$ complex matrix $A^+$ with these four properties:

1. $A A^+ A = A$.
2. $A^+ A A^+ = A^+$.
3. $(A A^+)^* = A A^+$.
4. $(A^+ A)^* = A^+ A$.

This matrix is the *Moore–Penrose inverse*, or *pseudoinverse*, of $A$, where $^*$ is the [[conjugate transpose|0DXKE10PHF]].

## Remarks

If $A$ is invertible, then $A^+ = A^{-1}$.
If the columns of $A$ are linearly independent, then $A^+ = (A^* A)^{-1} A^*$.
For $b \in \C^m$, the vector $A^+ b$ is the least squares solution of $Ax = b$ with the least norm.
If $A = U \Sigma V^*$ is a singular value decomposition, then $A^+ = V \Sigma^+ U^*$, where $\Sigma^+$ inverts the nonzero singular values and transposes.

## Explanations

### L1 human

The pseudoinverse extends the inverse to all matrices, also singular and rectangular ones.
It gives the best possible solution of $Ax = b$ when no exact solution exists or when there are many.
For the zero matrix, the pseudoinverse is the zero matrix of the transposed size.
</document>

<document source="https://tamnd.github.io/math2/node/0PT52XQZD7.md" tag="0PT52XQZD7" kind="concept">
# Product of two subgroups

## Summary

The product of two subgroups $H$ and $K$ of a group is the set $HK$ of all products $hk$ with $h \in H$ and $k \in K$.
It is a subgroup exactly when $HK = KH$.
</document>

<document source="https://tamnd.github.io/math2/node/0PTBM7K0R2.md" tag="0PTBM7K0R2" kind="concept">
# Algebraic extension

## Summary

An algebraic extension is a field extension in which every element of the larger field is algebraic over the smaller field.
The [[definition of an algebraic extension|09EYMKMSB9]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0PTF1PTJK0.md" tag="0PTF1PTJK0" kind="definition">
# Endomorphism

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
An *endomorphism* of $V$ is a [[linear map|0NRYHJPQG2]] $T \colon V \to V$.
The set of all endomorphisms of $V$ is written $\End_F(V)$.

## Remarks

An endomorphism is also called a *linear operator* on $V$.
Endomorphisms can be composed with each other, so their powers $T^k$ and polynomials $p(T)$ are defined.
Under composition and addition, $\End_F(V)$ is the [[algebra of endomorphisms|0E3X2G1ZQB]].
Eigenvalues, the trace and the determinant are defined for endomorphisms of finite-dimensional spaces.

## Explanations

### L1 human

An endomorphism maps a space into itself.
So we can apply it again and again, and compare a vector with its image.
In a basis, an endomorphism of an $n$-dimensional space is a square $n \times n$ matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0PTVNH4GPS.md" tag="0PTVNH4GPS" kind="concept">
# Linear congruence

## Summary

A linear congruence is a [[congruence|0TPW25CBN6]] $a x \equiv b \pmod n$, with given integers $a$ and $b$ and a positive integer $n$.
Its solutions are the integers $x$ that satisfy it.
Let $d = \gcd(a, n)$. The congruence has a solution if and only if $d \mid b$.
In this case, the solutions form exactly $d$ residue classes modulo $n$, and one residue class modulo $n / d$.

## Remarks

When $\gcd(a, n) = 1$, the unique solution modulo $n$ is $x \equiv a^{-1} b \pmod n$, where $a^{-1}$ is the [[modular multiplicative inverse|00GXTQMY8Z]] of $a$.
The congruence $a x \equiv b \pmod n$ is the same as the [[linear Diophantine equation|0R9GF3JD0K]] $a x - n y = b$.
For example, $6 x \equiv 4 \pmod{10}$ has $d = 2$. Its solutions are $x \equiv 4 \pmod 5$, that is, $x \equiv 4$ or $x \equiv 9 \pmod{10}$.

## Explanations

### L1 human

A linear congruence is a linear equation in modular arithmetic.
To solve $2 x \equiv 3 \pmod 5$, find the inverse $3$ of $2$ modulo $5$, and multiply: $x \equiv 9 \equiv 4 \pmod 5$.
Check: $2 \cdot 4 = 8 \equiv 3 \pmod 5$.
</document>

<document source="https://tamnd.github.io/math2/node/0PV74TR5FP.md" tag="0PV74TR5FP" kind="concept">
# Maximal ideal

## Summary

A maximal ideal of a ring is a proper ideal that is not contained in a larger proper ideal.
The maximal ideals of $\mathbb{Z}$ are the ideals $p\mathbb{Z}$ for $p$ prime.
The [[definition of a maximal ideal|09J6KGT65G]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0PVCTCKD3C.md" tag="0PVCTCKD3C" kind="statement">
# Expectation with respect to the binomial distribution

## Statement

Let $n \ge 1$ be an integer, let $p \in [0, 1]$, and let $X$ have the [[binomial distribution|0VRWZDWW6B]] $\operatorname{Bin}(n, p)$. Then
$$E[X] = \sum_{k=0}^{n} k \binom{n}{k} p^k (1 - p)^{n - k} = np.$$

## Remarks

A short proof: $X$ has the same law as $Y_1 + \dots + Y_n$ with independent [[Bernoulli|0J97P5KZY0]] $Y_i$ with $E[Y_i] = p$. Then [[linearity of expectation|0P7PTC8DDE]] gives $np$.
A direct proof uses $k \binom{n}{k} = n \binom{n-1}{k-1}$ and the binomial theorem.
The variance of $X$ is $np(1 - p)$.

## Explanations

### L1 human

In $n$ trials with success probability $p$, you expect $np$ successes.
For $100$ tosses of a fair coin, the expected number of heads is $50$.
</document>

<document source="https://tamnd.github.io/math2/node/0PVG1CRAJN.md" tag="0PVG1CRAJN" kind="definition">
# Injective function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The function $f$ is injective when, for all $a, a' \in A$, $f(a) = f(a')$ implies $a = a'$.

## Remarks

Other names are one-to-one function and injection.
The function $x \mapsto 2x$ on $\mathbb{R}$ is injective, and $x \mapsto x^2$ on $\mathbb{R}$ is not, because $f(1) = f(-1)$.
An injective function with a nonempty domain has a [[left inverse|02486SD328]]. See [[injective iff it has a left inverse|0HYFEEZAEM]].

## Explanations

### L1 human

An injective function never sends two inputs to the same place.
To prove injectivity, assume $f(a) = f(a')$ and deduce $a = a'$.
</document>

<document source="https://tamnd.github.io/math2/node/0PVJRACB9H.md" tag="0PVJRACB9H" kind="statement">
# The exponential series has infinite radius of convergence

## Statement

The [[radius of convergence|0660QX1VPR]] of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $\infty$.
So the series converges absolutely for every real $x$.

## Remarks

The ratio of consecutive coefficients is $\frac{1/(n+1)!}{1/n!} = \frac{1}{n + 1} \to 0$. The [[ratio formula|0JBMYAA9QH]] then gives radius $\infty$.
The same holds for the series of sine, cosine, $\sinh$ and $\cosh$.
Mathlib states it as `NormedSpace.expSeries_radius_eq_top`.

## Explanations

### L1 human

The factorials in the denominators grow faster than every power $x^n$.
So the terms of the series go to $0$ fast, for each fixed $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0PWE3BCHVJ.md" tag="0PWE3BCHVJ" kind="concept">
# Infinite set

## Summary

An infinite set is a set that is not finite.
</document>

<document source="https://tamnd.github.io/math2/node/0PWX18FM4G.md" tag="0PWX18FM4G" kind="concept">
# Cauchy sequence

## Summary

In a Cauchy sequence, any two terms with large enough indices are as near to each other as we wish. A metric space is complete when each Cauchy sequence in it converges, and the real numbers form a complete space. The node [[Cauchy sequence|0QZMTN5AXX]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0PXN9G4MX1.md" tag="0PXN9G4MX1" kind="statement">
# Orthogonal complement of the orthogonal complement

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $W$ be a subspace of $V$.
Then
$$(W^\perp)^\perp = W,$$
where $W^\perp$ is the [[orthogonal complement|0XJXTBCB2T]] of $W$.

## Remarks

Each vector of the subspace is orthogonal to each vector of its orthogonal complement, so $W \subseteq (W^\perp)^\perp$. Both have dimension $\dim V - \dim W^\perp = \dim W$.
In an infinite-dimensional Hilbert space, $(W^\perp)^\perp$ is the closure of $W$.

## Explanations

### L1 human

Taking the orthogonal complement twice gives back the original subspace.
The complement of the complement of a line is the same line.
</document>

<document source="https://tamnd.github.io/math2/node/0PXW0BN7DW.md" tag="0PXW0BN7DW" kind="concept">
# Commutator of two elements

## Summary

The commutator of two elements $g$ and $h$ of a group is $[g, h] = g^{-1} h^{-1} g h$.
It is the identity exactly when $g$ and $h$ commute.
</document>

<document source="https://tamnd.github.io/math2/node/0PYEZM6CN9.md" tag="0PYEZM6CN9" kind="concept">
# Liouville number

## Summary

A Liouville number is a real number that has rational approximations $p/q$ with error smaller than $1/q^n$ for each $n$.
The [[definition of a Liouville number|0J1R5NP36S]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0PYSR20HGF.md" tag="0PYSR20HGF" kind="concept">
# Composition of relations

## Summary

The composition of two relations relates $a$ to $c$ when there is an intermediate element $b$.
The first relation relates $a$ to $b$, and the second relation relates $b$ to $c$.
</document>

<document source="https://tamnd.github.io/math2/node/0PYW26VNT6.md" tag="0PYW26VNT6" kind="concept">
# Affine subspace

## Summary

An affine subspace of an [[affine space|09XNAPTJQ7]] is a subset that is itself an affine space, for the same action of the vectors. Points, lines and planes are the affine subspaces of dimension 0, 1 and 2.
</document>

<document source="https://tamnd.github.io/math2/node/0Q0H13G8ZB.md" tag="0Q0H13G8ZB" kind="concept">
# Closed ball

## Summary

A closed ball in a metric space is the set of all points at distance at most a given radius from a given centre.
</document>

<document source="https://tamnd.github.io/math2/node/0Q0Z1BEXTF.md" tag="0Q0Z1BEXTF" kind="definition">
# Centraliser

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$.
The centraliser of $S$ in $G$ is the set $C_G(S) = \{g \in G : gs = sg \text{ for all } s \in S\}$.
For an element $a \in G$, the notation $C_G(a)$ means $C_G(\{a\})$.

## Remarks

The centraliser $C_G(S)$ is a [[subgroup|0X1JXT47GA]] of $G$.
The centraliser of $G$ itself is the [[centre|0DZ93493VN]] $Z(G)$.
The element $a$ lies in its own centraliser $C_G(a)$, and so does each power of $a$.

## Explanations

### L1 human

The centraliser of an element is the set of elements that commute with it.
If the element commutes with everything, its centraliser is the whole group.
The size of the centraliser tells you how many conjugates the element has.
</document>

<document source="https://tamnd.github.io/math2/node/0Q1D85B8G6.md" tag="0Q1D85B8G6" kind="concept">
# Interval

## Summary

An interval is a set of real numbers that contains each real number between two of its elements.
Examples are $[0, 1]$, $(0, 1)$, $[0, \infty)$ and $\mathbb{R}$ itself.
</document>

<document source="https://tamnd.github.io/math2/node/0Q1FKQKTKJ.md" tag="0Q1FKQKTKJ" kind="concept">
# Exact equation

## Summary

An exact equation is a first-order equation $M\, dx + N\, dy = 0$ whose left side is the total differential of a function $\psi(x, y)$.
Its solutions lie on the level curves $\psi(x, y) = C$. See the [[definition|0EC7AGJ0PX]].
</document>

<document source="https://tamnd.github.io/math2/node/0Q2MDD13G4.md" tag="0Q2MDD13G4" kind="concept">
# Singular solution

## Summary

A singular solution of a differential equation is a solution along which uniqueness of solutions fails.
It is often the envelope of a family of solutions, and the general solution does not contain it. See the [[definition|0CZA0NCFWP]].
</document>

<document source="https://tamnd.github.io/math2/node/0Q2MVQ7JMT.md" tag="0Q2MVQ7JMT" kind="definition">
# Internal direct product

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ and $K$ be [[normal subgroups|0CB0DX5BVV]] of $G$.
The group $G$ is the internal direct product of $H$ and $K$ when $H \cap K = \{e\}$ and $HK = G$.

## Remarks

In this case, $hk = kh$ for all $h \in H$ and $k \in K$, because $hkh^{-1}k^{-1}$ lies in $H \cap K$.
Then the map $(h, k) \mapsto hk$ is an [[isomorphism|08MG7H8JW5]] from the [[direct product|00HF4RJ5QN]] $H \times K$ to $G$.
Also, each element of $G$ is a product $hk$ with $h \in H$ and $k \in K$ in exactly one way.
For example, $\mathbb{Z}/6\mathbb{Z}$ is the internal direct product of $\{0, 3\}$ and $\{0, 2, 4\}$.

## Explanations

### L1 human

Sometimes a group already contains two subgroups that behave like the two factors of a direct product.
They meet only in the identity, they fill the group together, and they are both normal.
Then the group is the direct product of these subgroups, up to isomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/0Q2QC81VTW.md" tag="0Q2QC81VTW" kind="definition">
# Cumulant generating function

## Definition

Let $X$ be a real random variable with [[moment-generating function|0NNRNGKMFM]] $M_X$.
The cumulant generating function of $X$ is
$$K_X(t) = \ln M_X(t) = \ln E\left[e^{tX}\right],$$
for each $t \in \mathbb{R}$ with $M_X(t) < \infty$.
If $M_X$ is finite on an open interval around $0$, the cumulants $\kappa_n$ of $X$ are the coefficients in $K_X(t) = \sum_{n=1}^{\infty} \kappa_n t^n / n!$, so $\kappa_n = K_X^{(n)}(0)$.

## Remarks

The first cumulants are $\kappa_1 = E[X]$, $\kappa_2 = \operatorname{Var}(X)$ and $\kappa_3 = E[(X - E[X])^3]$.
For independent $X$ and $Y$, $K_{X+Y} = K_X + K_Y$ by the [[product rule for moment-generating functions|0WRZ170YEH]]. So cumulants of independent sums add.
For $X \sim \mathcal{N}(\mu, \sigma^2)$, $K_X(t) = \mu t + \sigma^2 t^2 / 2$. For $X \sim \operatorname{Poisson}(\lambda)$, $K_X(t) = \lambda(e^t - 1)$, so all cumulants equal $\lambda$.
$K_X$ is convex on the interval where it is finite.

## Explanations

### L1 human

Taking the logarithm turns products into sums. So the cumulant generating function of a sum of independent variables is the sum of their cumulant generating functions.
This makes cumulants convenient for sums, for example in the central limit theorem.
</document>

<document source="https://tamnd.github.io/math2/node/0Q3AKZV74D.md" tag="0Q3AKZV74D" kind="statement">
# Stolz–Cesàro theorem

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences and let $L$ be a real number.
Assume that $(b_n)$ is strictly increasing and that $b_n \to +\infty$ as $n \to \infty$.
Assume that
$$\lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L.$$
Then $a_n / b_n \to L$ as $n \to \infty$.

## Remarks

There is also a $0/0$ form. If $a_n \to 0$, $b_n \to 0$, $(b_n)$ is strictly decreasing and the quotient of the differences tends to $L$, then $a_n / b_n \to L$.
With $b_n = n$ it gives the [[convergence of the Cesàro means|0RDRPBMNAC]].
For example, with $a_n = 1 + 2 + \cdots + n$ and $b_n = n^2$, the quotient of differences is $(n+1)/(2n+1) \to 1/2$, so $a_n / n^2 \to 1/2$.
It is a discrete analogue of [[L'Hôpital's rule|0JWT4PPQ7B]].

## History

Otto Stolz proved the $\infty/\infty$ case in a book of 1885. Ernesto Cesàro gave it in an article of 1888.

## Explanations

### L1 human

To compare two sequences that grow without bound, it is enough to compare their steps.
If each step of $a_n$ is about $L$ times the step of $b_n$, then $a_n$ is about $L$ times $b_n$ in the long run.
</document>

<document source="https://tamnd.github.io/math2/node/0Q3F20XB0A.md" tag="0Q3F20XB0A" kind="definition">
# Image of a set

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $A$.
The image of $S$ under $f$ is the set
$$f(S) = \{f(s) \mid s \in S\} = \{b \in B \mid \exists s \in S, \, f(s) = b\}.$$

## Remarks

Another notation is $f[S]$. It avoids a confusion with $f(a)$ for an element $a$.
The image $f(A)$ of the whole domain is the [[range|0E9GTAV478]] of $f$.
Example: for $f(x) = x^2$ on $\mathbb{R}$, $f([-1, 2]) = [0, 4]$.
The image of a union is the union of the images, but the image of an intersection can be smaller. See [[image of an intersection|0RX7DDBYCK]].

## Explanations

### L1 human

To compute $f(S)$, apply $f$ to each element of $S$ and collect the results.
For $f(x) = x + 1$ and $S = \{1, 2, 3\}$, the image is $\{2, 3, 4\}$.
</document>

<document source="https://tamnd.github.io/math2/node/0Q3HYFT9RQ.md" tag="0Q3HYFT9RQ" kind="statement">
# Integral of a power

## Statement

Let $n \ge 0$ be an integer and let $a, b \in \mathbb{R}$.
Then
$$\int_a^b x^n \, dx = \frac{b^{n+1} - a^{n+1}}{n + 1}.$$

## Remarks

By the [[power rule|0JZGW75C5Z]], $x^{n+1}/(n + 1)$ is an antiderivative of $x^n$. Apply the [[second fundamental theorem|0XYAFCN0PT]].
The same formula holds for each real exponent $r \ne -1$, on intervals inside $(0, \infty)$. The case $r = -1$ gives the [[logarithm|0K1N1Y4EGZ]].
Mathlib states it as `integral_pow`.

## Explanations

### L1 human

To integrate $x^n$, raise the power by one and divide by the new power.
For example, $\int_0^1 x^2 \, dx = 1/3$.
</document>

<document source="https://tamnd.github.io/math2/node/0Q3ZT8J05R.md" tag="0Q3ZT8J05R" kind="concept">
# Linear equation

## Summary

A linear equation is an equation in which each unknown appears only to the first power and is multiplied only by a constant.
Linear equations are the basic objects of linear algebra.
</document>

<document source="https://tamnd.github.io/math2/node/0Q4CBD3HX0.md" tag="0Q4CBD3HX0" kind="concept">
# Domain

## Summary

The domain of a function is the set of its inputs.
</document>

<document source="https://tamnd.github.io/math2/node/0Q4CJ5NAT1.md" tag="0Q4CJ5NAT1" kind="concept">
# Arc length parametrisation

## Summary

An arc length parametrisation of a [[regular curve|0V7090SSPC]] uses the distance travelled along the curve as the parameter.
In this parametrisation the curve has speed $1$ at each point.
</document>

<document source="https://tamnd.github.io/math2/node/0Q4DR37DVE.md" tag="0Q4DR37DVE" kind="definition">
# Uniform continuity

## Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
The function $f$ is uniformly continuous if the following holds.
For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - f(y)| < \varepsilon$ for all $x, y \in D$ with $|x - y| < \delta$.

## Remarks

Each uniformly continuous function is continuous. In the [[epsilon-delta form|0W8YHHGCJG]] of continuity, $\delta$ may depend on the point. Here one $\delta$ works for all points.
The function $x^2$ is continuous on $\mathbb{R}$ but [[not uniformly continuous|0A8JXHYTJC]].
Mathlib defines it for uniform spaces, as `UniformContinuous`.
Eduard Heine published the first definition of uniform continuity, in 1870.

## Explanations

### L1 human

For a uniformly continuous function, one input tolerance works everywhere in the domain.
The graph has no places where it becomes steeper and steeper without bound.
</document>

<document source="https://tamnd.github.io/math2/node/0Q4QJWEHW3.md" tag="0Q4QJWEHW3" kind="concept">
# Second derivative

## Summary

The second derivative of a function is the derivative of its derivative. It measures how the slope changes, and so how the graph bends. The node [[second derivative|0Y9D2FJRQ4]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0Q5A47J48D.md" tag="0Q5A47J48D" kind="statement">
# Darboux's theorem

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be differentiable on $[a, b]$, with [[one-sided derivatives|0HXEX0DPRA]] at the endpoints.
Let $y$ be a real number strictly between $f'(a)$ and $f'(b)$.
Then there is $c \in (a, b)$ with $f'(c) = y$.

## Remarks

Assume $f'(a) < y < f'(b)$. The function $g(x) = f(x) - y x$ has $g'(a) < 0 < g'(b)$. So its minimum on $[a, b]$, given by the [[extreme value theorem|0WS1A8E2P9]], is at an interior point $c$. [[Fermat's theorem|0PCVBRV9VJ]] gives $g'(c) = 0$.
So a derivative has the intermediate value property, even when it is not continuous.
In particular, a derivative has no jump discontinuities. Mathlib states it as `exists_hasDerivWithinAt_eq_of_gt_of_lt`.

## Explanations

### L1 human

The derivative of a function cannot skip values, just as a continuous function cannot.
For example, no function has derivative $0$ on the left of a point and $1$ on the right.
</document>

<document source="https://tamnd.github.io/math2/node/0Q5CG2NEQ8.md" tag="0Q5CG2NEQ8" kind="concept">
# Statistic

## Summary

A statistic is a quantity computed from a sample, by a fixed rule that does not depend on unknown parameters. Examples are the sample mean and the sample maximum.
The node [[statistic|036756NFD6]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0Q5G5DJ4P2.md" tag="0Q5G5DJ4P2" kind="concept">
# Derived series

## Summary

The derived series of a group is the sequence of iterated commutator subgroups $G \supseteq G' \supseteq G'' \supseteq \cdots$.
A group is solvable if and only if this series reaches the trivial subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/0Q5NA8Y362.md" tag="0Q5NA8Y362" kind="concept">
# Galois group of a polynomial

## Summary

The Galois group of a polynomial over a field is the Galois group of its splitting field.
It acts on the roots of the polynomial by permutations.
The [[definition of the Galois group of a polynomial|0JZY3R908H]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0Q5SM60HTH.md" tag="0Q5SM60HTH" kind="definition">
# Permutation matrix

## Definition

Let $n \ge 1$ and let $\sigma$ be a permutation of $\{1, \dots, n\}$.
The *permutation matrix* of $\sigma$ is the $n \times n$ matrix $P_\sigma$ with entries
$$(P_\sigma)_{ij} = \begin{cases} 1 & \text{if } i = \sigma(j), \\ 0 & \text{otherwise.} \end{cases}$$
A *permutation matrix* is a matrix of the form $P_\sigma$.

## Remarks

The matrix $P_\sigma$ sends the standard basis vector $e_j$ to $e_{\sigma(j)}$.
With this convention, $P_\sigma P_\tau = P_{\sigma \tau}$; some authors use the transposed convention.
Each permutation matrix is invertible with $P_\sigma^{-1} = P_\sigma^T$, and $\det P_\sigma = \sgn \sigma$.
The matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is the permutation matrix that exchanges two coordinates.

## Explanations

### L1 human

A permutation matrix is the identity matrix with its columns shuffled.
Multiplying a vector by it shuffles the entries of the vector in the same way.
Each row and each column has exactly one $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0Q70YFB6ZM.md" tag="0Q70YFB6ZM" kind="concept">
# Regular graph

## Summary

A regular graph is a graph in which all vertices have the same degree. Regular graphs are a basic family with strong symmetry and counting properties. The node [[regular graph|0KA4S8R39M]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0Q7K4AXE53.md" tag="0Q7K4AXE53" kind="example">
# The sequence r^n tends to zero for |r| < 1

## Example

Let $r$ be a real number with $|r| < 1$.
Then $r^n \to 0$ as $n \to \infty$.
For $r = 0$ this is clear. For $0 < |r| < 1$, write $1/|r| = 1 + h$ with $h > 0$.
By Bernoulli's inequality, $(1 + h)^n \ge 1 + n h > n h$ for each $n \ge 1$.
Thus $0 \le |r^n| < 1/(n h)$, and the [[squeeze theorem|0XMF00T4XQ]] gives $|r^n| \to 0$, so $r^n \to 0$.

## Remarks

The [[geometric sequence|0MNM54WB99]] $a r^n$ then tends to $0$ for each real $a$.
For $r = 1$ the sequence is constant. For $r = -1$ it is $1, -1, 1, \dots$, which diverges.
For $|r| > 1$ the sequence $|r^n|$ [[tends to infinity|0TZK91NQE3]].

## Explanations

### L1 human

Multiply a number by $1/2$ again and again: $1, 1/2, 1/4, 1/8, \dots$.
The result gets closer and closer to $0$. The same happens for each factor between $-1$ and $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0Q7M6K6VD7.md" tag="0Q7M6K6VD7" kind="statement">
# Holomorphic functions with nonzero derivative are conformal

## Statement

Let $U \subseteq \mathbb{C}$ be an open set, let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]], and let $z_0 \in U$.
If $f'(z_0) \neq 0$, then $f$ is [[conformal|0VHNKP1N3K]] at $z_0$.
In particular, if $f'(z) \neq 0$ for each $z \in U$, then $f$ is conformal on $U$.

## Remarks

Write $f'(z_0) = r e^{i\varphi}$ with $r > 0$; then the real derivative of $f$ at $z_0$ is $h \mapsto r e^{i\varphi} h$, a rotation by $\varphi$ followed by a scaling by $r$.
The converse also holds: a map that is conformal at each point of an open set $U$ is holomorphic on $U$ with a nonzero derivative.
The condition $f'(z_0) \neq 0$ is necessary: $z \mapsto z^2$ doubles the angles between curves through $0$.
The exponential function is conformal at each point but not injective, so a conformal map need not be injective.

## Explanations

### L1 human

Near $z_0$, a [[complex differentiable|0D1C9Y1EXA]] function acts almost like $z \mapsto f(z_0) + f'(z_0)(z - z_0)$.
Multiplication by the complex number $f'(z_0)$ turns the plane and scales it.
Turning and scaling keep all angles, so $f$ keeps the angles between curves through $z_0$.
</document>

<document source="https://tamnd.github.io/math2/node/0Q7MMQHC1P.md" tag="0Q7MMQHC1P" kind="concept">
# Nilpotent group

## Summary

A nilpotent group is a group that has a central series of finite length.
Equivalently, its lower central series reaches the trivial subgroup, or its upper central series reaches the whole group.
</document>

<document source="https://tamnd.github.io/math2/node/0Q88KQD17A.md" tag="0Q88KQD17A" kind="definition">
# Conjugacy class

## Definition

Let $G$ be a [[group|G5S1M9]] and let $a \in G$.
The conjugacy class of $a$ is the set $\operatorname{Cl}(a) = \{gag^{-1} : g \in G\}$.

## Remarks

The conjugacy classes are the equivalence classes of the relation "is [[conjugate to|0DXD4QSZ8A]]", so they partition $G$.
They are the [[orbits|0YQPZB5Q1G]] of the [[action by conjugation|0P8CY9V437]].
The class of $a$ is $\{a\}$ if and only if $a$ lies in the [[centre|0DZ93493VN]] of $G$.
In the symmetric group $S_n$, the conjugacy classes are given by the [[cycle types|0J438GAC8K]].

## Explanations

### L1 human

The conjugacy class of an element collects all the elements that look like it after a renaming by conjugation.
In an abelian group, each element is alone in its class.
In $S_3$, the three transpositions form one class, and the two $3$-cycles form another class.
</document>

<document source="https://tamnd.github.io/math2/node/0Q8BZXVT1Z.md" tag="0Q8BZXVT1Z" kind="definition">
# Orientation of a real vector space

## Definition

Let $V$ be a real vector space of finite dimension $n \ge 1$.
Two ordered [[bases|0KCQ9Y3QKA]] $B$ and $B'$ of $V$ have the *same orientation* when the [[change of basis matrix|0MKRHZB3WS]] from $B$ to $B'$ has positive [[determinant|00KDZRW4C0]].
This is an equivalence relation with exactly two classes.
An *orientation* of $V$ is a choice of one of these two classes.

## Remarks

A vector space with a chosen orientation is *oriented*, and the bases in the chosen class are *positively oriented*.
The standard orientation of $\R^n$ is the class of the standard basis.
In $\R^2$, exchanging the two basis vectors reverses the orientation.
An invertible linear map keeps the orientation if and only if its determinant is positive.

## Explanations

### L1 human

An orientation fixes what counts as turning to the left, or as a right-handed system.
In the plane, a counterclockwise pair of basis vectors and a clockwise pair have opposite orientations.
A mirror reflection changes the orientation, and a rotation keeps it.
</document>

<document source="https://tamnd.github.io/math2/node/0Q8DGAZSJ4.md" tag="0Q8DGAZSJ4" kind="statement">
# Third Sylow theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number.
Then the number $n_p$ of [[Sylow p-subgroups|0053CQE9Y2]] of $G$ satisfies $n_p \equiv 1 \pmod p$.

## Remarks

The proof lets a Sylow $p$-subgroup $P$ act by conjugation on $\operatorname{Syl}_p(G)$, and $P$ is the only fixed point.
Together with [[the divisibility condition|0M3PYJ3TSE]], this often forces $n_p = 1$.
In $S_4$, $n_2 = 3$ and $n_3 = 4$, and both numbers are $\equiv 1$ modulo the prime.

## Explanations

### L1 human

The number of Sylow $p$-subgroups is one more than a multiple of $p$.
This simple fact, together with a divisibility condition, leaves only a few possible values.
Often the only possible value is $1$, and then the Sylow subgroup is normal.
</document>

<document source="https://tamnd.github.io/math2/node/0Q8HZTJC7G.md" tag="0Q8HZTJC7G" kind="definition">
# Orthogonal projection

## Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $W$ be a finite-dimensional subspace of $V$.
By the [[orthogonal decomposition|0SP7P0G7KQ]], each $v \in V$ is $v = w + w'$ with unique $w \in W$ and $w' \in W^\perp$.
The *orthogonal projection* onto $W$ is the map $P_W \colon V \to V$, $P_W(v) = w$.

## Remarks

$P_W$ is a linear [[projection|033DA1WE8Y]] with image $W$ and kernel $W^\perp$, and it is self-adjoint.
For an orthonormal basis $e_1, \dots, e_k$ of $W$, $P_W(v) = \sum_{i=1}^{k} \langle v, e_i \rangle e_i$.
For the line spanned by $u \ne 0$, $P(v) = \frac{\langle v, u \rangle}{\langle u, u \rangle} u$.
$P_W(v)$ is the point of $W$ [[closest to $v$|0MCDDSCB9Z]].

## Explanations

### L1 human

The orthogonal projection drops a perpendicular from a vector onto a subspace.
The foot of the perpendicular is the projection.
In $\R^3$, projecting onto the $xy$-plane sets the $z$-coordinate to zero.
</document>

<document source="https://tamnd.github.io/math2/node/0Q8Y862DTZ.md" tag="0Q8Y862DTZ" kind="concept">
# Outer automorphism group

## Summary

The outer automorphism group of a group is the quotient of its automorphism group by the inner automorphisms.
It measures how many automorphisms do not come from conjugation.
</document>

<document source="https://tamnd.github.io/math2/node/0Q9EZS90GY.md" tag="0Q9EZS90GY" kind="definition">
# Polynomial ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
A polynomial over $R$ is a sequence $(a_0, a_1, a_2, \dots)$ of elements of $R$ such that $a_n = 0$ for all but finitely many $n$.
The element $a_n$ is the coefficient of degree $n$.
The polynomial ring $R[x]$ is the set of these polynomials, with these operations:
$$(a_n)_n + (b_n)_n = (a_n + b_n)_n, \qquad (a_n)_n \cdot (b_n)_n = (c_n)_n, \quad c_n = \sum_{i + j = n} a_i b_j.$$
With these operations, $R[x]$ is a ring.

## Remarks

Let $x$ be the polynomial $(0, 1, 0, 0, \dots)$, and identify $r \in R$ with $(r, 0, 0, \dots)$.
Then each polynomial with $a_n = 0$ for $n > d$ is $a_0 + a_1 x + \dots + a_d x^d$, in exactly one way.
If $R$ is a [[commutative ring|0GYGHKX834]], then $R[x]$ is commutative.
A polynomial is a formal expression and not a function: over $\mathbb{F}_2$, $x^2 + x$ is a nonzero polynomial, but it is zero at both elements of $\mathbb{F}_2$.

## Explanations

### L1 human

A polynomial is a finite list of coefficients.
You add polynomials coefficient by coefficient.
You multiply them with the rule $x^i \cdot x^j = x^{i + j}$ and the distributive law.
</document>

<document source="https://tamnd.github.io/math2/node/0Q9PGXD9E4.md" tag="0Q9PGXD9E4" kind="definition">
# Diagonal matrix

## Definition

Let $R$ be a ring and let $n \ge 0$ be an integer.
A [[square matrix|0JXZKX526R]] $D = (d_{ij})$ of order $n$ over $R$ is *diagonal* when $d_{ij} = 0$ for all $i \ne j$.
For $d_1, \dots, d_n \in R$, the matrix $\operatorname{diag}(d_1, \dots, d_n)$ is the diagonal matrix with $d_{ii} = d_i$ for each $i$.

## Remarks

The product of two diagonal matrices is diagonal: $\operatorname{diag}(a_1, \dots, a_n) \operatorname{diag}(b_1, \dots, b_n) = \operatorname{diag}(a_1 b_1, \dots, a_n b_n)$.
Over a commutative ring, diagonal matrices commute with each other.
The [[determinant|00KDZRW4C0]] of a diagonal matrix is the product of its diagonal entries.
A matrix that is similar to a diagonal matrix is [[diagonalisable|07SQCNV233]].

## Explanations

### L1 human

A diagonal matrix has numbers only on its main diagonal, from the top left to the bottom right.
As a map, it stretches each coordinate axis by its own factor.
Diagonal matrices are easy to add, multiply and raise to powers.
</document>

<document source="https://tamnd.github.io/math2/node/0QA1HBD9Z4.md" tag="0QA1HBD9Z4" kind="concept">
# Convergence in probability

## Summary

A sequence of random variables converges in probability when, for each tolerance, the probability of a deviation larger than this tolerance tends to $0$.
The node [[convergence in probability|0EFDDXTVD6]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0QA8CHH8MG.md" tag="0QA8CHH8MG" kind="definition">
# Field

## Definition

A field is a [[commutative ring|0GYGHKX834]] $K$ with these two properties.

1. $1 \neq 0$ in $K$.
2. Every nonzero element of $K$ is a [[unit|019NFGFPR5]]: for each $a \in K$ with $a \neq 0$, there is $b \in K$ with $a b = 1$.

## Remarks

The rings $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ are fields. The ring $\mathbb{Z}$ is not a field.
For a prime number $p$, the ring [[$\mathbb{Z}/p\mathbb{Z}$ is a field|0NV6E43P4A]].
Every field is an [[integral domain|09S3PFBEPM]].
A [[division ring|0WZXK05ME7]] has the same axioms without the commutativity of the multiplication.

## Explanations

### L1 human

A field is a number system where the four operations of arithmetic work.
You can add, subtract, multiply, and divide by any number that is not zero.
The rational numbers are the simplest example.
</document>

<document source="https://tamnd.github.io/math2/node/0QAJHFGZZJ.md" tag="0QAJHFGZZJ" kind="concept">
# Uniqueness quantifier

## Summary

The uniqueness quantifier $\exists!$ expresses "there exists exactly one".
</document>

<document source="https://tamnd.github.io/math2/node/0QAR0XDZH4.md" tag="0QAR0XDZH4" kind="definition">
# Joint probability distribution

## Definition

Let $X_1, \dots, X_n$ be real [[random variables|0DKRR22AEQ]] on one probability space $(\Omega, \mathcal{F}, P)$.
Their joint distribution is the [[law|0TST4MHA3A]] of the random vector $(X_1, \dots, X_n) \colon \Omega \to \mathbb{R}^n$. It is the probability measure on the Borel sets of $\mathbb{R}^n$ with
$$\mu(B) = P((X_1, \dots, X_n) \in B).$$
Their joint cumulative distribution function is $F(x_1, \dots, x_n) = P(X_1 \le x_1, \dots, X_n \le x_n)$.

## Remarks

For discrete variables, the joint probability mass function $p(x_1, \dots, x_n) = P(X_1 = x_1, \dots, X_n = x_n)$ determines the joint distribution.
The variables have a joint density $f$ if $\mu(B) = \int_B f(x) \, dx$ for each Borel set $B \subseteq \mathbb{R}^n$.
The joint distribution determines the [[marginal distributions|0AXP2Z5N9H]], but the marginal distributions do not determine the joint distribution.
For two rolls of a fair die, the joint mass function of the two results is $1/36$ at each pair $(i, j)$ with $1 \le i, j \le 6$.

## Explanations

### L1 human

The joint distribution describes several random quantities together, not one by one.
For example, it says how likely it is that a person is tall and also heavy.
It gives more than the chance of each of the two alone.
</document>

<document source="https://tamnd.github.io/math2/node/0QB913WM5E.md" tag="0QB913WM5E" kind="definition">
# Probability distribution

## Definition

Let $E$ be a set with a σ-algebra $\mathcal{E}$.
A probability distribution on $(E, \mathcal{E})$ is a [[probability measure|0VGMEBVW1M]] $\mu$ on $(E, \mathcal{E})$.
A probability distribution on $\mathbb{R}$ is a probability measure on $\mathbb{R}$ with its [[Borel σ-algebra|0BVHZXWKCD]] $\mathcal{B}(\mathbb{R})$.

## Remarks

Each random variable $X$ with values in $E$ gives a distribution, its [[law|0TST4MHA3A]] $B \mapsto P(X \in B)$. Conversely, each distribution $\mu$ is the law of some random variable, for example the identity map on $(E, \mathcal{E}, \mu)$.
A distribution on $\mathbb{R}$ is determined by its [[cumulative distribution function|0MXMS3AE13]].
Important types are discrete distributions, given by a [[probability mass function|0WF4EJP91W]], and absolutely continuous distributions, given by a [[probability density function|04YG690N5V]].

## Explanations

### L1 human

A distribution tells you which values are possible and how likely each range of values is.
For a fair die, the distribution gives each of the values $1, \dots, 6$ the probability $1/6$.
</document>

<document source="https://tamnd.github.io/math2/node/0QBYFZV91P.md" tag="0QBYFZV91P" kind="concept">
# Subspace metric

## Summary

The subspace metric on a subset of a metric space is the restriction of the metric to pairs of points of the subset.
It makes each subset of a metric space a metric space.
</document>

<document source="https://tamnd.github.io/math2/node/0QD4E5JYPT.md" tag="0QD4E5JYPT" kind="statement">
# Reciprocal rule

## Statement

Let $g$ be a real function on an open interval $I$, differentiable at $a \in I$, with $g(a) \ne 0$.
Then $1/g$ is differentiable at $a$, and
$$\left( \frac{1}{g} \right)'(a) = -\frac{g'(a)}{g(a)^2}.$$

## Remarks

The difference quotient is $-\frac{g(a + h) - g(a)}{h} \cdot \frac{1}{g(a + h) g(a)}$. Continuity of $g$ at $a$ gives the limit.
For example, $(1/x)' = -1/x^2$ for $x \ne 0$.
Mathlib states it as `HasDerivAt.inv`.

## Explanations

### L1 human

When $g$ grows, $1/g$ shrinks, so the two derivatives have opposite signs.
The factor $1/g^2$ shows that $1/g$ changes fast where $g$ is small.
</document>

<document source="https://tamnd.github.io/math2/node/0QD83RAN8G.md" tag="0QD83RAN8G" kind="concept">
# Contour line

## Summary

A contour line of a function $f$ of two real variables is a curve in the plane along which $f$ has a constant value $c$.
It is a [[level set|0B2RZ0778N]] $\{(x, y) : f(x, y) = c\}$ of $f$, drawn in the $(x, y)$-plane.
A contour map draws the contour lines for several equally spaced values of $c$. It shows the shape of the [[graph|0JBJYT69ZV]] of $f$ in the plane.

## Remarks

On a topographic map, $f$ is the height above sea level, and the contour lines join points of equal height.
Where the contour lines of equally spaced values are close together, the function changes fast.

## Explanations

### L1 human

Imagine that you cut a hill with horizontal planes at heights $10$, $20$, $30$ and so on, and you look at the cuts from above.
Each cut gives a closed curve on the map, and these curves are the contour lines.
</document>

<document source="https://tamnd.github.io/math2/node/0QDXMG46QW.md" tag="0QDXMG46QW" kind="definition">
# Characteristic equation

## Definition

Let $a_0, \dots, a_{n-1}$ be real numbers, and consider the homogeneous [[linear equation with constant coefficients|0A6PGTTP2A]]
$$y^{(n)} + a_{n-1}\, y^{(n-1)} + \dots + a_1\, y' + a_0\, y = 0.$$
Its characteristic polynomial is $p(r) = r^n + a_{n-1} r^{n-1} + \dots + a_1 r + a_0$.
Its characteristic equation is the polynomial equation $p(r) = 0$ for an unknown complex number $r$.

## Remarks

For each complex $r$, the function $y = e^{r x}$ satisfies $y^{(n)} + \dots + a_0 y = p(r)\, e^{r x}$. So $e^{rx}$ is a solution if and only if $p(r) = 0$.
Example: the equation $y'' - 3 y' + 2 y = 0$ has the characteristic equation $r^2 - 3 r + 2 = 0$, with roots $1$ and $2$.
The [[characteristic polynomial of a linear recurrence|0BWDJGQYST]] plays the same role for recurrences, with $r^k$ in place of $e^{r x}$.

## Explanations

### L1 human

Try a solution of the form $e^{r x}$.
Each derivative multiplies it by $r$, so the equation becomes $p(r) e^{rx} = 0$.
Since $e^{rx}$ is never zero, $r$ must be a root of $p$.
</document>

<document source="https://tamnd.github.io/math2/node/0QE5PFJY7V.md" tag="0QE5PFJY7V" kind="definition">
# Equivalence relation induced by a function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The equivalence relation induced by $f$ is the relation $\sim_f$ on $A$ with
$$a \sim_f a' \iff f(a) = f(a')$$
for all $a, a' \in A$.

## Remarks

Another name is the kernel of $f$.
The relation $\sim_f$ is an [[equivalence relation|0SGTMBEWRF]], because equality is one.
Its classes are the nonempty preimages $f^{-1}(\{b\})$ of the points $b \in B$.
The function $f$ is [[injective|0PVG1CRAJN]] exactly when $\sim_f$ is equality.

## Explanations

### L1 human

Take the function that sends each person to their birthday.
Two people are equivalent under it when they have the same birthday.
The function forgets everything except its value, and the relation records what it forgets.
</document>

<document source="https://tamnd.github.io/math2/node/0QE8STGJH2.md" tag="0QE8STGJH2" kind="definition">
# Sequence

## Definition

Let $X$ be a set.
A sequence in $X$ is a function $a \colon \mathbb{N} \to X$, where $\mathbb{N} = \{0, 1, 2, \dots\}$.
The value $a(n)$ is the $n$-th term of the sequence, with the notation $a_n$.
The notation for the sequence is $(a_n)_{n \in \mathbb{N}}$, or $(a_n)$ when the index set is clear.

## Remarks

Many texts start the index at $1$, and they use a function on $\{1, 2, 3, \dots\}$. The theory does not change.
The terms of a sequence can repeat. For example, $(1, 0, 1, 0, \dots)$ is the sequence with $a_n = 1$ for even $n$ and $a_n = 0$ for odd $n$.
A sequence of real numbers is a [[real sequence|0CTV783HYT]].

## Explanations

### L1 human

A sequence is an infinite list $a_0, a_1, a_2, \dots$ of elements of a set.
The order matters, and the same element can occur more than once.
For example, the squares $0, 1, 4, 9, 16, \dots$ form the sequence $a_n = n^2$.
</document>

<document source="https://tamnd.github.io/math2/node/0QEAJB91GZ.md" tag="0QEAJB91GZ" kind="concept">
# Closure

## Summary

The closure of a subset of a topological space is the intersection of all closed sets that contain it.
It is the smallest closed set that contains the subset.
In a metric space, it is the set of limits of convergent sequences of points of the subset.
</document>

<document source="https://tamnd.github.io/math2/node/0QEBN8DGVE.md" tag="0QEBN8DGVE" kind="statement">
# Holomorphic functions are infinitely differentiable

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Then the complex derivative $f'$ is holomorphic on $U$.
Thus, for each $n \ge 1$, the $n$-th complex derivative $f^{(n)}$ exists and is holomorphic on $U$.
As a function of two real variables $(x, y)$ with $z = x + iy$, $f$ has continuous partial derivatives of all orders on $U$.

## Remarks

Proof: near each point, $f$ is the sum of a power series by [[holomorphic functions are analytic|0826SGV3R0]]. A [[power series can be differentiated term by term|0YMP521ZR5]], and the result is again a power series.
There is no real analogue: $x \mapsto x |x|$ is differentiable on $\mathbb{R}$, but its derivative $2|x|$ is not differentiable at $0$.
The [[formula for the derivatives|0F0H5P5DED]] gives each $f^{(n)}$ as a circle integral.

## Explanations

### L1 human

Once a complex function has one derivative on an open set, it has a second one, a third one, and so on forever.
So in complex analysis there are no functions that are differentiable only once.
</document>

<document source="https://tamnd.github.io/math2/node/0QED0GPYKY.md" tag="0QED0GPYKY" kind="concept">
# Alternating series

## Summary

An alternating series is a series whose terms alternate in sign. Such a series converges when the absolute values of its terms decrease to zero. The node [[alternating series|017BR1M98J]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0QEM68Y0DE.md" tag="0QEM68Y0DE" kind="statement">
# Cayley–Hamilton theorem for linear maps

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map with [[characteristic polynomial|0J5EMJRRFM]] $p_T$.
Then $p_T(T) = 0$, where $p_T(T)$ is the [[value of $p_T$ at $T$|0E3MV6E6MG]].

## Remarks

It follows from the [[Cayley–Hamilton theorem for matrices|0F765KFC2J]] applied to the matrix of $T$ in a basis.
The map $p \mapsto p([T]_B)$ corresponds to $p \mapsto p(T)$ under $T \mapsto [T]_B$.
The statement also holds for endomorphisms of finitely generated modules over a commutative ring, with a suitable characteristic polynomial.

## Explanations

### L1 human

The theorem does not depend on a basis.
A linear map satisfies its characteristic polynomial, because each of its matrices does.
</document>

<document source="https://tamnd.github.io/math2/node/0QF1MZ9FRG.md" tag="0QF1MZ9FRG" kind="concept">
# Infimum

## Summary

The infimum of a set is its greatest lower bound.
It is a lower bound of the set that is greater than or equal to every other lower bound.
It is the dual notion of the [[supremum|0R3H5PANRQ]].
</document>

<document source="https://tamnd.github.io/math2/node/0QF4ZEFG52.md" tag="0QF4ZEFG52" kind="statement">
# Eigenvalues are the roots of the minimal polynomial

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $\lambda \in F$.
Let $T \colon V \to V$ be an endomorphism with [[minimal polynomial|03NDVCSHSB]] $m_T$.
Then $\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $T$ if and only if $m_T(\lambda) = 0$.

## Remarks

If $T(v) = \lambda v$ with $v \ne 0$, then $0 = m_T(T) v = m_T(\lambda) v$, so $m_T(\lambda) = 0$.
Conversely, if $m_T = (x - \lambda) q$, then $q(T) \ne 0$ and $T - \lambda \id_V$ kills the image of $q(T)$.
So the minimal and the characteristic polynomial have the same roots, possibly with different multiplicities.

## Explanations

### L1 human

The minimal polynomial and the characteristic polynomial have the same roots: the eigenvalues.
Only the multiplicities of the roots can differ.
So the minimal polynomial is a second way to find the eigenvalues.
</document>

<document source="https://tamnd.github.io/math2/node/0QFT7Z76N5.md" tag="0QFT7Z76N5" kind="concept">
# Peano axioms

## Summary

The Peano axioms are axioms for the [[natural numbers|0MWKJ44V5A]].
They use a set $\mathbb{N}$, an element $0$ of $\mathbb{N}$ and a function $S \colon \mathbb{N} \to \mathbb{N}$, the [[successor function|0AK9TGSNXY]].
The axioms state three properties of these data.
First, $S(n) \neq 0$ for each $n$ in $\mathbb{N}$.
Second, $S$ is injective: if $S(m) = S(n)$, then $m = n$.
Third, the induction axiom holds: if a subset $A$ of $\mathbb{N}$ contains $0$ and contains $S(n)$ for each $n$ in $A$, then $A = \mathbb{N}$.

## Remarks

Giuseppe Peano published the axioms in 1889. Richard Dedekind gave a similar system in 1888.
The induction axiom speaks about all subsets of $\mathbb{N}$, thus it is a second-order axiom. First-order Peano arithmetic replaces it with an axiom schema.
The axioms are the base for the [[principle of mathematical induction|0ASJVEFA9J]] and for [[definition by recursion|0DSHAVH45F]].

## Explanations

### L1 human

The Peano axioms say what is special about counting: you start at $0$ and you always go to the next number.
Two different numbers never have the same next number, and you never come back to $0$.
The induction axiom says that you reach every natural number in this way.
</document>

<document source="https://tamnd.github.io/math2/node/0QFTETCHKB.md" tag="0QFTETCHKB" kind="statement">
# The algebraic numbers form a field

## Statement

Let $\overline{\mathbb{Q}}$ be the set of the complex numbers that are [[algebraic|0NPW1DWECM]] over $\mathbb{Q}$.
Then $\overline{\mathbb{Q}}$ is a [[subfield|07QY5RPFBR]] of $\mathbb{C}$.

## Remarks

The proof: if $a, b$ are algebraic over $\mathbb{Q}$, then $\mathbb{Q}(a, b)$ is a finite extension of $\mathbb{Q}$, by the [[tower law|0A7KJ68XHT]]. So $a + b$, $a - b$, $a b$ and $a/b$ for $b \neq 0$ are algebraic, because [[finite extensions are algebraic|0DFYTNZ6AM]].
More generally, for every extension $L/K$, the elements of $L$ that are algebraic over $K$ form a subfield of $L$.
The field $\overline{\mathbb{Q}}$ is countable and algebraically closed. It is an [[algebraic closure|0MCXX29HWY]] of $\mathbb{Q}$.

## Explanations

### L1 human

The sum and the product of two algebraic numbers are algebraic.
For example, $\sqrt{2} + \sqrt{3}$ is a root of $x^4 - 10x^2 + 1$.
The proof does not need to find such a polynomial: it uses only dimensions of vector spaces.
</document>

<document source="https://tamnd.github.io/math2/node/0QFV5Y1Z5X.md" tag="0QFV5Y1Z5X" kind="statement">
# Gauss's lemma

## Statement

Let $R$ be a [[unique factorisation domain|06YMXJMXYS]] with [[field of fractions|0VF11ZTTMD]] $K$ and let $f \in R[x]$ be a [[primitive polynomial|0SP14DZN0G]].
Then $f$ is [[irreducible|00X2A9G9W3]] in $R[x]$ if and only if $f$ is irreducible in $K[x]$.

## Remarks

The key step: if $f = g h$ in $K[x]$, then one can scale $g$ and $h$ by elements of $K$ so that they are in $R[x]$ and $f = g h$ still holds. This uses that the [[product of primitive polynomials is primitive|08C1NWBNHA]].
For $R = \mathbb{Z}$: a primitive integer polynomial that factors over $\mathbb{Q}$ already factors over $\mathbb{Z}$, into factors of the same degrees.
The hypothesis that $f$ is primitive is necessary: $2x$ is irreducible in $\mathbb{Q}[x]$, but $2x = 2 \cdot x$ is a product of two nonunits in $\mathbb{Z}[x]$.

## Explanations

### L1 human

To show that an integer polynomial has no factorisation over the rationals, it is enough to check factorisations over the integers.
Fractions do not help to factor a primitive polynomial.
This is why criteria such as Eisenstein's work with integer coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/0QG0Q3NDT9.md" tag="0QG0Q3NDT9" kind="concept">
# Fundamental solution of Pell's equation

## Summary

The fundamental solution of Pell's equation $x^2 - d y^2 = 1$ is its smallest solution in positive integers.
The [[definition of the fundamental solution|0KCYJSB9FW]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0QG9C579SR.md" tag="0QG9C579SR" kind="definition">
# Removable discontinuity

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$ be a limit point of $D$.
The point $a$ is a removable discontinuity of $f$ if the [[limit|0TXTSEMHR5]] $L = \lim_{x \to a} f(x)$ exists and $L \ne f(a)$.

## Remarks

Change the value of $f$ at $a$ to $L$. The new function is [[continuous at|0EEE2CZB2A]] $a$.
For example, let $f(x) = 1$ for $x \ne 0$ and $f(0) = 0$. Then $0$ is a removable discontinuity.
Some texts also call $a$ removable when $a \notin D$ and the limit exists. An example is $\sin(x)/x$ at $0$.

## Explanations

### L1 human

At a removable discontinuity, the graph has a single misplaced point.
Moving this one point repairs the continuity.
</document>

<document source="https://tamnd.github.io/math2/node/0QH5ENKZM1.md" tag="0QH5ENKZM1" kind="concept">
# Stochastic process

## Summary

A stochastic process is a family $(X_t)_{t \in T}$ of [[random variables|0DKRR22AEQ]] on one [[probability space|0ZYBZ34RT2]] $(\Omega, \mathcal{F}, P)$, all with values in the same measurable space $(S, \mathcal{S})$.
The set $T$ is the index set, often a set of times such as $\{0, 1, 2, \dots\}$ or $[0, \infty)$. The set $S$ is the state space.
For each outcome $\omega \in \Omega$, the function $t \mapsto X_t(\omega)$ is a sample path of the process.

## Remarks

The process is in discrete time if $T$ is countable, such as $\{0, 1, 2, \dots\}$, and in continuous time if $T$ is an interval.
The finite-dimensional distributions are the laws of the vectors $(X_{t_1}, \dots, X_{t_n})$ for finite sets $\{t_1, \dots, t_n\} \subseteq T$. The Kolmogorov extension theorem builds a process from a consistent family of such laws.
Important examples are the Bernoulli process, random walks, Markov chains, the Poisson process and the Wiener process.

## Explanations

### L1 human

A stochastic process is a random quantity that changes with time.
Examples are the price of a share day by day and the number of customers in a queue.
At each time it is a random variable, and one run of the whole process gives a random path.
</document>

<document source="https://tamnd.github.io/math2/node/0QHEWQD30B.md" tag="0QHEWQD30B" kind="concept">
# Predicate

## Summary

A predicate is a statement with variables that becomes a [[proposition|0ZGC7ZESYN]] when each variable gets a value.
</document>

<document source="https://tamnd.github.io/math2/node/0QHTX7WQEY.md" tag="0QHTX7WQEY" kind="definition">
# Differential form

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $0 \le k \le n$.
Take a multi-index $I = (i_1, \dots, i_k)$ with $1 \le i_1 < \dots < i_k \le n$. Let $dx_I$ be the function of $k$ vectors $v_1, \dots, v_k \in \mathbb{R}^n$ given by
$$dx_I(v_1, \dots, v_k) = \det \big[ (v_j)_{i_l} \big]_{l, j = 1}^{k},$$
the determinant of a $k \times k$ matrix. The entry of this matrix in row $l$ and column $j$ is the $i_l$-th component of $v_j$.
A differential $k$-form on $U$ is a sum $\omega = \sum_I f_I \, dx_I$ over these multi-indices, where each $f_I \colon U \to \mathbb{R}$ is a function. At $x \in U$, it gives the function $\omega_x(v_1, \dots, v_k) = \sum_I f_I(x) \, dx_I(v_1, \dots, v_k)$.

## Remarks

The form is of class $C^r$ if all $f_I$ are of [[class C^r|0TCBZMEYGN]]. A $0$-form is a function, and a [[1-form|0MTENFBW3Q]] is $\sum_i f_i \, dx_i$.
Each $\omega_x$ is multilinear and alternating: it changes sign when two of the vectors are swapped. The forms $dx_I$ give a basis of such functions, so there are $\binom{n}{k}$ of them, and no nonzero forms of degree $k > n$.
In the [[wedge product|0S166D363M]] notation, $dx_I = dx_{i_1} \wedge \dots \wedge dx_{i_k}$. On $\mathbb{R}^3$, $dx \wedge dy \wedge dz$ gives the signed volume of the parallelepiped spanned by three vectors.
In $\mathbb{R}^3$, 1-forms and 2-forms correspond to vector fields, and 0-forms and 3-forms to functions. The [[exterior derivative|004FRCXH6M]] then corresponds to the gradient, curl and divergence.

## Explanations

### L1 human

A $k$-form measures small $k$-dimensional pieces: a 1-form measures small arrows, a 2-form small parallelograms, and so on.
Integrating a $k$-form over a $k$-dimensional surface adds up these measurements over the whole surface.
</document>

<document source="https://tamnd.github.io/math2/node/0QHX5GGABE.md" tag="0QHX5GGABE" kind="definition">
# Affine map

## Definition

Let $K$ be a field. Let $A$ be an [[affine space|09XNAPTJQ7]] over a $K$-vector space $V$, and let $B$ be an affine space over a $K$-vector space $W$.
A map $f : A \to B$ is an affine map if there is a linear map $L : V \to W$ such that
$$f(p + v) = f(p) + L(v) \quad \text{for each point } p \text{ of } A \text{ and each vector } v \text{ of } V.$$
The map $L$ is unique. It is the linear part of $f$, and it satisfies $L(q - p) = f(q) - f(p)$ for all points $p$ and $q$ of $A$.

## Remarks

The affine maps from $\mathbb{R}^n$ to $\mathbb{R}^m$ are the maps $x \mapsto M x + b$, with $M$ a real $m \times n$ matrix and $b$ a vector of $\mathbb{R}^m$.
An affine map keeps [[affine combinations|0KJH6SRFZK]]. In particular it sends [[midpoints|0S706JBZBG]] to midpoints.
An affine transformation is a bijective affine map from $A$ to itself. The affine transformations of $A$ form a group under composition.
Mathlib's `AffineMap` is this notion, with the linear part as a field of the structure.

## Explanations

### L1 human

An affine map is a linear map that may also shift the origin. In the plane, examples are rotations, stretches, shears and translations, and the compositions of these. A grid of squares goes to a grid of parallelograms.
</document>

<document source="https://tamnd.github.io/math2/node/0QHZX1PK8V.md" tag="0QHZX1PK8V" kind="statement">
# Irrationality of pi

## Statement

The number [[$\pi$|0C1VYVNV2N]] is irrational.

## Remarks

Ivan Niven gave a short proof by calculus. Assume $\pi = a/b$, and let $f(x) = \frac{x^n (a - b x)^n}{n!}$.
Then $\int_0^\pi f(x) \sin x \, dx$ is a positive integer for every $n$. But it tends to $0$ as $n \to \infty$, a contradiction.
In 1882, Ferdinand von Lindemann proved the stronger fact that $\pi$ is transcendental. Mathlib states irrationality as `irrational_pi`.

## History

Johann Heinrich Lambert gave the first proof in the 1760s. He used a continued fraction for $\tan x$.

## Explanations

### L1 human

The number $\pi$ is not a fraction of two integers.
So its decimal expansion never ends and never repeats.
</document>

<document source="https://tamnd.github.io/math2/node/0QJJAQ6ZYZ.md" tag="0QJJAQ6ZYZ" kind="concept">
# Orientation of a real vector space

## Summary

An orientation of a finite-dimensional real vector space is a choice of one of the two classes of ordered bases.
Two bases are in the same class when the change of basis matrix has positive determinant.
</document>

<document source="https://tamnd.github.io/math2/node/0QK6K857HS.md" tag="0QK6K857HS" kind="concept">
# Matrix of a bilinear form

## Summary

The matrix of a bilinear form in a basis is the square matrix of the values of the form on all pairs of basis vectors.
</document>

<document source="https://tamnd.github.io/math2/node/0QK7FPE6VE.md" tag="0QK7FPE6VE" kind="statement">
# Dimension of the coordinate space

## Statement

Let $F$ be a field and let $n \ge 0$ be an integer.
The [[coordinate space|0XM8BZD76G]] $F^n$ has [[dimension|03GSZ8RTZJ]] $n$ over $F$.

## Remarks

The [[standard basis|0V7ZD6P7RK]] $e_1, \dots, e_n$ is a basis of $F^n$ with $n$ vectors.
A consequence: each $n$-dimensional space over $F$ is isomorphic to $F^n$.
In Mathlib, the index type is `Fin n`, and the space is the type of functions `Fin n → F`.

## Explanations

### L1 human

The space of lists of $n$ numbers has dimension $n$.
The $n$ standard unit vectors form a basis.
For example, $\R^3$ has dimension $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0QKA345AW1.md" tag="0QKA345AW1" kind="concept">
# Implicit differentiation

## Summary

Implicit differentiation finds the derivative of a function that an equation $F(x, y) = 0$ defines, without solving the equation for $y$.
One differentiates both sides of $F(x, y(x)) = 0$ with the [[chain rule|0EAAS2EHDR]] and solves the result for $y'(x)$.

## Remarks

If $F$ is of class $C^1$ and $\partial F / \partial y \ne 0$ at the point, then $y'(x) = -\dfrac{\partial F / \partial x}{\partial F / \partial y}$, evaluated at $(x, y(x))$. The [[implicit function theorem|0BXBADR94C]] guarantees that $y(x)$ exists and is differentiable.
For the circle $x^2 + y^2 = 1$, differentiation gives $2 x + 2 y y' = 0$. So $y' = -x / y$ at each point with $y \ne 0$, on both the upper and the lower arc.
With several variables, $F(x, y, z) = 0$ gives $\partial z / \partial x = -F_x / F_z$ and $\partial z / \partial y = -F_y / F_z$ where $F_z \ne 0$. Here $F_x$, $F_y$, $F_z$ are the [[partial derivatives|05AXG3QRKR]] of $F$.

## Explanations

### L1 human

Sometimes a curve is given by an equation and not by a formula $y = f(x)$.
You can still find its slope: treat $y$ as a function of $x$, differentiate the equation term by term, and solve for $y'$.
</document>

<document source="https://tamnd.github.io/math2/node/0QKJ6NT0R9.md" tag="0QKJ6NT0R9" kind="concept">
# Derivative function

## Summary

The derivative function of a differentiable function $f$ is the function $f'$ that gives the derivative of $f$ at each point. Repeating the process gives higher derivatives. The node [[derivative function|027K9RMZ43]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0QKPNZKNEA.md" tag="0QKPNZKNEA" kind="definition">
# Cardinal number

## Definition

A cardinal number is the [[cardinality|0JTGRV4WT4]] $|A|$ of some set $A$.
In ZFC, the von Neumann definition takes $|A|$ to be the least [[ordinal|0RDJK7EQYY]] $\kappa$ such that $\kappa$ is [[equinumerous|08EJHHMB5R]] with $A$. A cardinal number is then an ordinal that is not equinumerous with any smaller ordinal.

## Remarks

The finite cardinals are the natural numbers $0, 1, 2, \dots$.
The least infinite cardinal is [[aleph-null|0V2H5PGJ9B]], the cardinality of $\mathbb{N}$.
Cardinals have an order, an addition, a multiplication and an exponentiation.
The sum $|A| + |B|$ is the cardinality of a disjoint union, and $|A| \cdot |B| = |A \times B|$.
The power $|B|^{|A|}$ is the cardinality of the set of functions from $A$ to $B$.

## Explanations

### L1 human

A cardinal number answers "how many?".
For finite sets, the answers are $0, 1, 2, \dots$.
For infinite sets, there are many different answers, such as $\aleph_0$ for $\mathbb{N}$ and $\mathfrak{c}$ for $\mathbb{R}$.
</document>

<document source="https://tamnd.github.io/math2/node/0QKVE1GD9A.md" tag="0QKVE1GD9A" kind="statement">
# Euclidean domains are principal ideal domains

## Statement

Let $R$ be a [[Euclidean domain|0GJ2HRG95T]].
Then $R$ is a [[principal ideal domain|0RZYDVZ475]].

## Remarks

The proof: let $I \neq 0$ be an ideal and let $b \in I$ be a nonzero element with $\delta(b)$ minimal. For $a \in I$, write $a = b q + r$. Then $r \in I$ and $\delta(r) < \delta(b)$ is impossible, so $r = 0$ and $I = (b)$.
The converse is false: $\mathbb{Z}[\tfrac{1 + \sqrt{-19}}{2}]$ is a principal ideal domain that is not Euclidean.

## Explanations

### L1 human

Take a nonzero element of the ideal that is as small as possible.
Every other element of the ideal leaves remainder zero when you divide it by this one.
So the smallest element generates the ideal.
</document>

<document source="https://tamnd.github.io/math2/node/0QM0T7T3PZ.md" tag="0QM0T7T3PZ" kind="concept">
# Element

## Summary

An element of a set is one of the objects that the set contains.
The relation "is an element of" is written with the symbol $\in$.
</document>

<document source="https://tamnd.github.io/math2/node/0QM2G72DXA.md" tag="0QM2G72DXA" kind="concept">
# Product of rings

## Summary

The product of a family of rings is the set of tuples of their elements, with addition and multiplication in each coordinate.
The [[definition of the product of rings|0MNZCN4BQK]] gives the operations.
</document>

<document source="https://tamnd.github.io/math2/node/0QM4TXT3DR.md" tag="0QM4TXT3DR" kind="definition">
# Riemann integral

## Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be [[Riemann integrable|0N1JQXZXHC]].
The Riemann integral of $f$ over $[a, b]$ is the unique number $I$ in the definition of Riemann integrability.
It is written
$$\int_a^b f(x) \, dx.$$

## Remarks

For $f \ge 0$, the integral equals the area under the graph of $f$ and above the $x$-axis.
By convention, $\int_a^a f = 0$ and $\int_b^a f = -\int_a^b f$.
The [[fundamental theorem of calculus|0NHGY0N0JB]] computes the integral from an antiderivative.
The integral is named after Bernhard Riemann.

## Explanations

### L1 human

The Riemann integral is the limit of sums of areas of thin rectangles under the graph.
It gives the signed area between the graph and the $x$-axis.
</document>

<document source="https://tamnd.github.io/math2/node/0QM98DNWVW.md" tag="0QM98DNWVW" kind="statement">
# The Galois group has order at most the degree

## Statement

Let $L/K$ be a [[finite extension|0NRYPM9Y27]].
Then the group $\operatorname{Aut}(L/K)$ of [[automorphisms of $L/K$|0XVR72M51M]] is finite, and
$$|\operatorname{Aut}(L/K)| \le [L : K].$$

## Remarks

For a simple extension $L = K(a)$, an automorphism is determined by $\sigma(a)$, which is a root in $L$ of the minimal polynomial of $a$. There are at most $[L : K]$ such roots.
Equality holds if and only if $L/K$ is a [[Galois extension|0DS2F7GK05]].
For $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$, the inequality is strict: $1 < 3$.

## Explanations

### L1 human

A finite extension has only finitely many symmetries over the base field, and at most as many as its degree.
When there are exactly as many as the degree, the extension is Galois.
</document>

<document source="https://tamnd.github.io/math2/node/0QMYQ1X03H.md" tag="0QMYQ1X03H" kind="definition">
# Orthocentre

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$. The three [[lines|0MP9BCF6DP]] that contain the [[altitudes|0C9MZY2RNA]] of $ABC$ meet in exactly one point $H$.
This point $H$ is the orthocentre of $ABC$.

## Remarks

The orthocentre is inside the triangle if the triangle is acute. In a right triangle, it is the vertex at the right angle. In an obtuse triangle, it is outside.
In an equilateral triangle, the orthocentre is the same point as the centroid, the circumcentre and the incentre.
The Book of Lemmas, attributed to Archimedes, uses the fact that the three altitudes meet. Al-Nasawi stated and proved it explicitly in the 11th century.
Mathlib's `Affine.Triangle.orthocenter` defines it as the Monge point of the triangle.

## Explanations

### L1 human

Draw the three altitudes of a triangle, each from a corner at a right angle to the opposite side. The three lines always pass through one common point. For the right triangle with the vertices $(0, 0)$, $(3, 0)$ and $(0, 4)$, this point is the corner $(0, 0)$.
</document>

<document source="https://tamnd.github.io/math2/node/0QMZAQ6QQ4.md" tag="0QMZAQ6QQ4" kind="concept">
# Galois extension

## Summary

A Galois extension is an algebraic field extension that is normal and separable.
Galois extensions are the extensions to which the fundamental theorem of Galois theory applies.
The [[definition of a Galois extension|0DS2F7GK05]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0QN3D80AWP.md" tag="0QN3D80AWP" kind="concept">
# Euclidean vector

## Summary

A Euclidean vector is a geometric quantity with a length and a direction, often drawn as an arrow from one point to another.
Vectors can be added and multiplied by real numbers, and with these operations they form a real vector space.
</document>

<document source="https://tamnd.github.io/math2/node/0QN69YA357.md" tag="0QN69YA357" kind="concept">
# Contraction mapping

## Summary

A contraction mapping is a map from a metric space to itself that shrinks all distances by at least a fixed factor less than one.
</document>

<document source="https://tamnd.github.io/math2/node/0QNFRBJF4Q.md" tag="0QNFRBJF4Q" kind="concept">
# Constructible number

## Summary

A constructible number is a complex number that is a point of the plane reachable from $0$ and $1$ by a straightedge and compass construction.
The [[definition of a constructible number|0FMN2MN7TM]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0QNRBHWV35.md" tag="0QNRBHWV35" kind="definition">
# Sample mean

## Definition

Let $X_1, \dots, X_n$ be real random variables, for example an [[i.i.d.|0AJN7WHBJZ]] sample.
The sample mean is the [[statistic|036756NFD6]]
$$\bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i.$$
For observed values $x_1, \dots, x_n$, the sample mean is the number $\bar{x} = (x_1 + \dots + x_n)/n$.

## Remarks

If the $X_i$ are i.i.d. with mean $\mu$, then $E[\bar{X}_n] = \mu$ by [[linearity of expectation|0P7PTC8DDE]]. So $\bar{X}_n$ is an unbiased estimator of $\mu$.
If also $\operatorname{Var}(X_i) = \sigma^2 < \infty$, then $\operatorname{Var}(\bar{X}_n) = \sigma^2 / n$.
For i.i.d. $\mathcal{N}(\mu, \sigma^2)$ variables, $\bar{X}_n \sim \mathcal{N}(\mu, \sigma^2/n)$.
By the law of large numbers, $\bar{X}_n \to \mu$ almost surely.

## Explanations

### L1 human

The sample mean is the ordinary average of the data.
For the measurements $4$, $7$ and $10$, the sample mean is $21/3 = 7$.
</document>

<document source="https://tamnd.github.io/math2/node/0QNVQ4VD9T.md" tag="0QNVQ4VD9T" kind="definition">
# Lipschitz continuity

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function, and let $K \ge 0$ be real.
The function $f$ is Lipschitz with constant $K$ if
$$|f(x) - f(y)| \le K |x - y| \quad \text{for all } x, y \in D.$$
It is Lipschitz if it is Lipschitz with some constant $K$.

## Remarks

[[Lipschitz functions are uniformly continuous|0K1EWAR6AF]].
A differentiable function on an interval with $|f'| \le K$ is [[Lipschitz with constant $K$|0P64VE79X3]].
For example, $|x|$ and $\sin x$ are Lipschitz with constant $1$. The function $\sqrt{x}$ on $[0, 1]$ is not Lipschitz.
A Lipschitz map with constant $K < 1$ from a set to itself is a [[contraction|0CZRT49QNG]]. Mathlib defines it as `LipschitzWith`.

## Explanations

### L1 human

The slope of each chord of the graph of a Lipschitz function is at most $K$ in absolute value.
So the function never changes faster than a fixed rate.
</document>

<document source="https://tamnd.github.io/math2/node/0QNY6ZBXPN.md" tag="0QNY6ZBXPN" kind="statement">
# Ratio test

## Statement

Let $(a_n)$ be a real sequence with $a_n \ne 0$ for all $n$.
Assume that $|a_{n+1}| / |a_n| \to L$ as $n \to \infty$, for a real number $L$ with $L < 1$.
Then the series $\sum a_n$ converges [[absolutely|0W3K3QPEWT]].

## Remarks

Choose $q$ with $L < q < 1$. For $n \ge N$ we get $|a_{n+1}| \le q |a_n|$, so $|a_n| \le |a_N| q^{n-N}$. The [[comparison test|04Z5G0THN6]] with a [[geometric series|01X5CVV1Q5]] gives the result.
For $L > 1$ the series diverges, as the [[divergence part of the ratio test|08AM0KRCZ1]] states.
For $L = 1$ the test gives no answer. The series $\sum 1/n$ diverges and $\sum 1/n^2$ converges, and both have $L = 1$.
For example, $\sum x^n / n!$ converges for each real $x$, since the ratio is $|x|/(n+1) \to 0$.

## Explanations

### L1 human

If each term is eventually less than a fixed fraction $q < 1$ of the term before, the terms shrink at least as fast as $q^n$.
So the series behaves at least as well as a convergent geometric series.
</document>

<document source="https://tamnd.github.io/math2/node/0QP2YYR79N.md" tag="0QP2YYR79N" kind="definition">
# Transitive action

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a nonempty set $X$.
The action is transitive when for all $x, y \in X$ there is a $g \in G$ with $g \cdot x = y$.

## Remarks

The action is transitive if and only if it has exactly one [[orbit|0YQPZB5Q1G]].
The action of $S_n$ on $\{1, \dots, n\}$ is transitive.
The action of $G$ on the left cosets of a subgroup $H$ is transitive. Each transitive action of $G$ is isomorphic to such an action.

## Explanations

### L1 human

A transitive action can move any point to any other point.
So the set does not split into separate pieces under the action.
The rotations of a square act transitively on its four corners.
</document>

<document source="https://tamnd.github.io/math2/node/0QPHW703PV.md" tag="0QPHW703PV" kind="definition">
# Compact space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is compact when each [[open cover|0NVPNC12JY]] of $X$ has a finite subcover.

## Remarks

Some authors, mainly in the French tradition, also require a compact space to be Hausdorff, and they call the general notion quasi-compact.
Each finite topological space is compact. An infinite set with the [[discrete topology|0S2TEHX4E1]] is not compact.
For metric spaces, compactness is the same as [[sequential compactness|0CARW59WD2]], by [[this theorem|0K8SGQSV7G]].
The [[continuous image of a compact space is compact|0K9AAJ8G51]].

## Explanations

### L1 human

In a compact space, each way to cover the space by open sets can be reduced to finitely many of them.
This finiteness lets you turn local facts, which hold near each point, into global facts with one bound for the whole space.
</document>

<document source="https://tamnd.github.io/math2/node/0QPYHHNRXC.md" tag="0QPYHHNRXC" kind="statement">
# Von Dyck's theorem

## Statement

Let $G = \langle S \mid R \rangle$, let $H$ be a [[group|G5S1M9]], and let $f \colon S \to H$ be a function.
Let $\varphi \colon F_S \to H$ be the homomorphism with $\varphi(s) = f(s)$ for each $s \in S$.
If $\varphi(r) = e$ for each $r \in R$, then there is exactly one [[group homomorphism|0JYTG5K9Y4]] $\psi \colon G \to H$ with $\psi(s N) = f(s)$ for each $s \in S$.

## Remarks

Here $N$ is the normal closure of $R$ in $F_S$, so $G = F_S / N$.
The proof combines the [[universal property of the free group|0CDWKDYWN6]] with the [[universal property of the quotient group|0GJ83E78A5]], because $N \subseteq \ker \varphi$.
If the elements $f(s)$ generate $H$, then $\psi$ is surjective, so $H$ is a quotient of $G$.
The name refers to Walther von Dyck, who gave the first systematic study of group presentations in the early 1880s.

## Explanations

### L1 human

To map a presented group to another group, choose the images of the generators.
If these images obey all the relations, then the choice gives exactly one homomorphism.
You do not need to check anything else.
</document>

<document source="https://tamnd.github.io/math2/node/0QQ0F12962.md" tag="0QQ0F12962" kind="example">
# Cycle graph

## Example

Let $n \ge 3$ be an integer.
The cycle graph $C_n$ is the [[graph|0JD5C82QMB]] with vertex set $\mathbb{Z}/n\mathbb{Z}$ in which $i$ and $j$ are adjacent if and only if $j - i = \pm 1$.
Its edges are $\{0, 1\}, \{1, 2\}, \dots, \{n-2, n-1\}$ and $\{n-1, 0\}$.

## Remarks

$C_n$ has $n$ vertices and $n$ edges, and it is $2$-[[regular|0KA4S8R39M]] and connected.
$C_3$ is the complete graph $K_3$.
The chromatic number of $C_n$ is $2$ for even $n$ and $3$ for odd $n$. The odd cycles are the obstruction in the theorem that [[a graph is bipartite iff it has no odd cycle|01R1HPEK6S]].

## Explanations

### L1 human

A cycle graph is a ring of $n$ vertices, each joined to the next one and the last joined to the first.
It looks like a polygon with $n$ corners.
</document>

<document source="https://tamnd.github.io/math2/node/0QQCG3JWJ4.md" tag="0QQCG3JWJ4" kind="definition">
# Positive definite matrix

## Definition

Let $A$ be an $n \times n$ complex [[Hermitian matrix|0P17NGQQMQ]].
The matrix $A$ is *positive definite* when
$$x^* A x > 0 \qquad \text{for all } x \in \C^n \text{ with } x \ne 0.$$
A real [[symmetric matrix|0VR671PHE5]] $A$ is positive definite when $x^T A x > 0$ for all $x \in \R^n$ with $x \ne 0$.

## Remarks

For a Hermitian $A$, the number $x^* A x$ is always real.
The identity matrix and $\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$ are positive definite.
$A$ is positive definite if and only if all its eigenvalues are positive.
$A$ is positive definite if and only if $\langle x, y \rangle = y^* A x$ is an inner product on $\C^n$.

## Explanations

### L1 human

A positive definite matrix gives a positive number $x^T A x$ for every nonzero vector $x$.
Such matrices behave like positive numbers in many ways: they have square roots and positive determinants.
They appear as Hessians at minima and as covariance matrices.
</document>

<document source="https://tamnd.github.io/math2/node/0QQMXFTX0A.md" tag="0QQMXFTX0A" kind="statement">
# Derivative of a map into a product

## Statement

Let $V$, $W_1$ and $W_2$ be normed vector spaces over $\mathbb{R}$, let $U \subseteq V$ be open, and let $x \in U$.
Let $f_1 \colon U \to W_1$ and $f_2 \colon U \to W_2$ be Fréchet differentiable at $x$.
Then the map $F \colon U \to W_1 \times W_2$, $F(y) = (f_1(y), f_2(y))$, is Fréchet differentiable at $x$, and
$$DF(x)(h) = \big(Df_1(x)(h), \, Df_2(x)(h)\big) \quad \text{for each } h \in V.$$
Here $W_1 \times W_2$ has the norm $\|(w_1, w_2)\| = \max(\|w_1\|, \|w_2\|)$.

## Remarks

The converse also holds: if $F$ is differentiable at $x$, then so are its two components, because each projection $W_1 \times W_2 \to W_i$ is a bounded linear map.
Any norm on $W_1 \times W_2$ that is equivalent to the max norm gives the same derivative.
Applied $m$ times with $W_i = \mathbb{R}$, this gives the [[derivative of a map into R^n by components|0NS0X75RS1]].

## Explanations

### L1 human

To differentiate a map with two outputs, differentiate each output and put the two results side by side.
The remainder of the pair is small exactly when both remainders are small.
</document>

<document source="https://tamnd.github.io/math2/node/0QQQQCPZA8.md" tag="0QQQQCPZA8" kind="statement">
# Number of odd-degree vertices is even

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]].
Then the number of vertices of $G$ with odd [[degree|0T0F6CCS9W]] is even.

## Remarks

It follows from the [[handshaking lemma|0XAP8WHNES]]: the sum of all degrees is even, so it has an even number of odd terms.
Thus no graph has exactly one vertex of odd degree. For example, no graph has the degrees $3, 2, 2, 2$.

## Explanations

### L1 human

The total of all degrees is even, because it is twice the number of edges.
A sum of whole numbers is even only if it has an even number of odd terms.
</document>

<document source="https://tamnd.github.io/math2/node/0QQRP17Z7S.md" tag="0QQRP17Z7S" kind="concept">
# Linear differential equation of order n

## Summary

A linear differential equation is an equation in which the unknown function and its derivatives occur linearly, with coefficients that are functions of the variable.
Its solutions are a particular solution plus the solutions of the associated homogeneous equation, which form a vector space. See the [[definition|0PB7K8M7FG]].
</document>

<document source="https://tamnd.github.io/math2/node/0QR92XBV6H.md" tag="0QR92XBV6H" kind="concept">
# Asymptotic equivalence

## Summary

Two functions are asymptotically equivalent at a point when their ratio tends to $1$ there. The notation is $f \sim g$. The node [[asymptotic equivalence|00SB5EA3VW]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0QRBVFJ50X.md" tag="0QRBVFJ50X" kind="definition">
# Orthogonal matrix

## Definition

Let $n \ge 1$ and let $Q$ be a real $n \times n$ matrix.
The matrix $Q$ is *orthogonal* when
$$Q^T Q = I_n,$$
where $Q^T$ is the [[transpose|09DXWYRD43]] and $I_n$ is the [[identity matrix|06DW9BAWB1]].

## Remarks

Equivalently, $Q$ is invertible with $Q^{-1} = Q^T$, and then also $Q Q^T = I_n$.
Equivalently, the columns of $Q$ form an orthonormal basis of $\R^n$; the same holds for the rows.
The map $x \mapsto Qx$ keeps the dot product, so it is a linear isometry of $\R^n$.
The determinant of an orthogonal matrix is $1$ or $-1$.

## Explanations

### L1 human

Orthogonal matrices describe the rigid motions of space that fix the origin: rotations and reflections.
They keep all lengths and angles.
To invert an orthogonal matrix, we only need to transpose it.
</document>

<document source="https://tamnd.github.io/math2/node/0QRCFZKXH8.md" tag="0QRCFZKXH8" kind="statement">
# In a UFD irreducible elements are prime

## Statement

Let $R$ be a [[unique factorisation domain|06YMXJMXYS]] and let $p \in R$.
Then $p$ is [[irreducible|052SDEPZH7]] if and only if $p$ is [[prime|0MNHXBSX3M]].

## Remarks

[[Prime elements are irreducible|0F0K8DB9TD]] in every integral domain. The other direction needs unique factorisation.
The proof: suppose $p$ is irreducible and $p c = a b$. Factor $a$, $b$ and $c$ into irreducibles. By uniqueness, $p$ is an associate of an irreducible factor of $a$ or of $b$.
Conversely, an integral domain in which every nonzero nonunit is a product of irreducibles, and every irreducible is prime, is a unique factorisation domain.

## Explanations

### L1 human

In the integers, Euclid's lemma says that a prime that divides a product divides a factor.
In a unique factorisation domain, the same thing is true for every irreducible element.
In $\mathbb{Z}[\sqrt{-5}]$ this fails, and so factorisation is not unique there.
</document>

<document source="https://tamnd.github.io/math2/node/0QRQVH0CRW.md" tag="0QRQVH0CRW" kind="definition">
# Upper and lower Darboux sums

## Definition

Let $a < b$, let $f \colon [a, b] \to \mathbb{R}$ be bounded, and let $P = (x_0, \dots, x_n)$ be a [[partition|069P20KYQT]] of $[a, b]$.
For each $i$, let $M_i = \sup f([x_{i-1}, x_i])$ and $m_i = \inf f([x_{i-1}, x_i])$.
The upper and lower Darboux sums of $f$ for $P$ are
$$U(f, P) = \sum_{i=1}^{n} M_i (x_i - x_{i-1}), \qquad L(f, P) = \sum_{i=1}^{n} m_i (x_i - x_{i-1}).$$

## Remarks

Always $(b - a) \inf f \le L(f, P) \le U(f, P) \le (b - a) \sup f$.
For a [[refinement|0EEKF7YFBW]] $Q$ of $P$, we have $L(f, P) \le L(f, Q)$ and $U(f, Q) \le U(f, P)$.
So $L(f, P) \le U(f, Q)$ for any two partitions $P$ and $Q$.

## Explanations

### L1 human

The upper sum covers the graph with rectangles that reach its highest point on each piece.
The lower sum uses rectangles that reach its lowest point on each piece.
</document>

<document source="https://tamnd.github.io/math2/node/0QRRYXG7V7.md" tag="0QRRYXG7V7" kind="statement">
# Recurrence for derangements

## Statement

Let $n$ be a natural number, and let $D_m$ be the number of [[derangements|0TH61JR04G]] of an $m$-element set.
Then
$$D_{n+2} = (n + 1)\,(D_{n+1} + D_n).$$

## Remarks

Together with $D_0 = 1$ and $D_1 = 0$, the recurrence gives all values: $1, 0, 1, 2, 9, 44, 265, \dots$
A proof looks at a derangement $\sigma$ of $\{1, \dots, n + 2\}$ and the value $i = \sigma(n+2)$. There are $n + 1$ choices of $i$. The case $\sigma(i) = n + 2$ gives $D_n$ derangements, and the other case gives $D_{n+1}$.
Another recurrence is $D_{n+1} = (n + 1) D_n + (-1)^{n+1}$.

## Explanations

### L1 human

Look at where the last element goes, say to position $i$.
Either $i$ goes back to the last position, and the other $n$ elements form a derangement. Or it does not, and this acts like a derangement of $n + 1$ elements.
</document>

<document source="https://tamnd.github.io/math2/node/0QRV8FPK5M.md" tag="0QRV8FPK5M" kind="definition">
# Residue class

## Definition

Let $n$ be a positive integer and let $a$ be an integer.
The residue class of $a$ modulo $n$ is the set $\overline{a} = \{ b \in \mathbb{Z} : b \equiv a \pmod n \}$ of the integers [[congruent|0TPW25CBN6]] to $a$ modulo $n$.
Thus $\overline{a} = \{ a + k n : k \in \mathbb{Z} \} = a + n \mathbb{Z}$.

## Remarks

Two residue classes $\overline{a}$ and $\overline{b}$ are equal if $a \equiv b \pmod n$, and they have no common element otherwise.
There are exactly $n$ residue classes modulo $n$: $\overline{0}, \overline{1}, \dots, \overline{n - 1}$.
Other notations are $[a]$, $[a]_n$ and $a \bmod n$.
The residue classes are the elements of the ring of [[integers modulo n|01N3AYAA6N]].

## Explanations

### L1 human

Modulo $3$, each integer falls in one of three boxes: the multiples of $3$, the numbers $1$ more than a multiple of $3$, and the numbers $2$ more.
Each box is a residue class. For example, $\overline{1} = \{ \dots, -5, -2, 1, 4, 7, \dots \}$.
</document>

<document source="https://tamnd.github.io/math2/node/0QS8P74F2Z.md" tag="0QS8P74F2Z" kind="definition">
# Semisimple endomorphism

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an endomorphism.
The map $T$ is *semisimple* when, for each $T$-[[invariant subspace|0ACX5QT8TE]] $W$ of $V$, there is a $T$-invariant subspace $W'$ with $V = W \oplus W'$.

## Remarks

The map $T$ is semisimple if and only if its minimal polynomial has no repeated irreducible factor.
Over an algebraically closed field, $T$ is semisimple if and only if it is [[diagonalisable|07SQCNV233]].
The rotation of $\R^2$ by $90$ degrees is semisimple, but it is not diagonalisable over $\R$.

## Explanations

### L1 human

Semisimple maps generalise diagonalisable maps to fields where eigenvalues can be missing.
They have no hidden nilpotent part.
Over $\C$, semisimple simply means diagonalisable.
</document>

<document source="https://tamnd.github.io/math2/node/0QSPNASE50.md" tag="0QSPNASE50" kind="statement">
# The reals form an ordered field

## Statement

The [[real numbers|08J2M1JYAG]] $\mathbb{R}$, with their addition, multiplication and [[order|0337CYB0VD]] $\le$, form an [[ordered field|010JCV9J4E]].
That is, $\le$ is a total order with two properties for all real numbers $a$, $b$ and $c$.
If $a \le b$, then $a + c \le b + c$. If $0 \le a$ and $0 \le b$, then $0 \le a b$.

## Remarks

The ordered field $\mathbb{R}$ contains $\mathbb{Q}$ as an ordered subfield.
The order of $\mathbb{R}$ is the only order that makes $\mathbb{R}$ an ordered field. The nonnegative reals are exactly the squares.

## Explanations

### L1 human

All the usual rules for inequalities hold for the real numbers.
For example, from $a \le b$ you get $a + 5 \le b + 5$, and from $a \ge 0$ and $b \ge 0$ you get $a b \ge 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0QT00PB5G2.md" tag="0QT00PB5G2" kind="concept">
# Homeomorphism

## Summary

A homeomorphism is a continuous bijection between topological spaces whose inverse is also continuous.
Homeomorphic spaces are the same from the point of view of topology.
</document>

<document source="https://tamnd.github.io/math2/node/0QTHB6PE52.md" tag="0QTHB6PE52" kind="statement">
# The group of units of Z/pZ is cyclic

## Statement

Let $p$ be a prime number.
Then the group $(\mathbb{Z}/p\mathbb{Z})^\times$ is [[cyclic|0KVGH0J5B3]] of order $p - 1$.

## Remarks

So $(\mathbb{Z}/p\mathbb{Z})^\times$ is isomorphic to the additive group $\mathbb{Z}/(p-1)\mathbb{Z}$, but no simple general formula for a generator is known.
A generator is a primitive root modulo $p$, and there are $\varphi(p - 1)$ of them.
For $p = 7$, the powers of $3$ modulo $7$ are $1, 3, 2, 6, 4, 5$, so $3$ is a primitive root modulo $7$.
More generally, $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if $n$ is $1$, $2$, $4$, $p^k$ or $2 p^k$ for an odd prime $p$. Gauss first proved this.

## Explanations

### L1 human

For a prime $p$, one residue modulo $p$ has powers that give all the nonzero residues.
Such a residue is a primitive root.
So the multiplicative group modulo $p$ is cyclic.
</document>

<document source="https://tamnd.github.io/math2/node/0QTZDKJEDV.md" tag="0QTZDKJEDV" kind="statement">
# Frattini argument

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $H$ be a [[normal subgroup|0CB0DX5BVV]] of $G$, and let $P$ be a [[Sylow p-subgroup|0053CQE9Y2]] of $H$.
Then $G = N_G(P) H$, where $N_G(P)$ is the [[normaliser|0YGN3VAA0D]] of $P$ in $G$.

## Remarks

For $g \in G$, the subgroup $gPg^{-1}$ lies in $H$, because $H$ is normal, and it is a Sylow $p$-subgroup of $H$.
By the [[second Sylow theorem|09GCQ5STVJ]] in $H$, there is $h \in H$ with $gPg^{-1} = hPh^{-1}$, so $h^{-1}g \in N_G(P)$.
The argument is used in the proof that a finite [[nilpotent group|0Q7MMQHC1P]] is the [[direct product of its Sylow subgroups|00Y9N80VSS]].

## History

Giovanni Frattini used the argument in a paper from 1885.
He took it, as he said himself, from a paper of Alfredo Capelli from 1884.

## Explanations

### L1 human

Conjugation by an element of $G$ moves $P$ to another Sylow subgroup of $H$.
An element of $H$ can move it back.
So each element of $G$ is an element of $H$ times an element that normalises $P$.
</document>

<document source="https://tamnd.github.io/math2/node/0QV2JS9RJ3.md" tag="0QV2JS9RJ3" kind="concept">
# Partial order

## Summary

A partial order is a relation that is reflexive, antisymmetric and transitive.
A set with a partial order is a partially ordered set, or poset.
</document>

<document source="https://tamnd.github.io/math2/node/0QV9VNNB4Z.md" tag="0QV9VNNB4Z" kind="concept">
# Distance in a graph

## Summary

The distance between two vertices of a graph is the length of a shortest walk between them. It makes a connected graph into a metric space. The node [[distance in a graph|0WPZ5YPAGR]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0QVD7WBFGN.md" tag="0QVD7WBFGN" kind="concept">
# Integral domain

## Summary

An integral domain is a nonzero commutative ring in which the product of two nonzero elements is never zero.
The integers are the model example.
The [[definition of an integral domain|09S3PFBEPM]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0QVDWMKQ0T.md" tag="0QVDWMKQ0T" kind="definition">
# Homothety

## Definition

Let $K$ be a field, and let $A$ be an [[affine space|09XNAPTJQ7]] over a $K$-vector space $V$. Let $c$ be a point of $A$ and let $k$ be a nonzero element of $K$.
The homothety with the centre $c$ and the ratio $k$ is the map $h : A \to A$ with
$$h(p) = c + k (p - c) \quad \text{for each point } p \text{ of } A.$$

## Remarks

A homothety is an [[affine map|0JF0TVYGWT]] with the linear part $v \mapsto k v$. It fixes its centre, and it sends each line to a parallel line.
The ratio $k = 1$ gives the identity. The ratio $k = -1$ gives the point reflection in $c$.
In a Euclidean space, a homothety with the ratio $k$ multiplies each distance by $|k|$, each area by $k^2$ and each volume by $|k|^3$.
Mathlib's `AffineMap.homothety` allows each $k$, also $k = 0$, which gives the constant map with the value $c$.

## Explanations

### L1 human

A homothety is a zoom centred at a point. With $k = 2$, each point moves to twice its distance from the centre, on the same ray. A triangle becomes a triangle with sides twice as long, and the same angles.
</document>

<document source="https://tamnd.github.io/math2/node/0QVNCJXRY3.md" tag="0QVNCJXRY3" kind="concept">
# Inner product space

## Summary

An inner product space is a real or complex vector space together with an inner product.
</document>

<document source="https://tamnd.github.io/math2/node/0QW084460E.md" tag="0QW084460E" kind="statement">
# Dimension of a subspace

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a [[linear subspace|00YC1D652K]] of $V$.
Then $W$ is finite-dimensional, and $\dim W \le \dim V$, where $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

A basis of $W$ is a linearly independent family in $V$, so it has at most $\dim V$ vectors.
Equality holds only when $W = V$, see [[a subspace of full dimension is the whole space|0WGAXEK99E]].
For infinite-dimensional $V$, the inequality $\dim W \le \dim V$ of cardinals still holds.

## Explanations

### L1 human

A subspace cannot have more independent directions than the space around it.
A plane in ordinary space has dimension $2$, which is at most $3$.
The only subspace with the same dimension is the whole space.
</document>

<document source="https://tamnd.github.io/math2/node/0QW64ARSHB.md" tag="0QW64ARSHB" kind="concept">
# Simply connected domain

## Summary

A simply connected domain is a path-connected set in which each closed loop can be shrunk continuously to a point without leaving the set.
Informally, the set is in one piece and has no hole that a loop can go around.
On such domains, curl-free vector fields are conservative.
</document>

<document source="https://tamnd.github.io/math2/node/0QWR8WTZEW.md" tag="0QWR8WTZEW" kind="statement">
# Every element of a finite field of order q satisfies x^q = x

## Statement

Let $F$ be a [[finite field|08RTDHX1SK]] of order $q$.
Then $a^q = a$ for every $a \in F$, and
$$x^q - x = \prod_{a \in F} (x - a) \quad \text{in } F[x].$$

## Remarks

The proof: the [[group of units|05THZKR9AN]] $F^\times$ has order $q - 1$, so $a^{q-1} = 1$ for $a \neq 0$, by [[Lagrange's theorem|3C53TD]]. Then $a^q = a$ for all $a$, also for $a = 0$.
So the $q$ elements of $F$ are $q$ distinct roots of $x^q - x$, which has degree $q$. This gives the factorisation.
For $q = p$ prime, $a^p = a$ in $\mathbb{Z}/p\mathbb{Z}$ is Fermat's little theorem.

## Explanations

### L1 human

In a field with $q$ elements, raising to the power $q$ changes nothing.
So the polynomial $x^q - x$ has every element of the field as a root.
</document>

<document source="https://tamnd.github.io/math2/node/0QX19C7K1C.md" tag="0QX19C7K1C" kind="concept">
# Order of a zero

## Summary

The order of a zero $z_0$ of a holomorphic function $f$ is the smallest $m$ with $f^{(m)}(z_0) \neq 0$, see the [[definition|064N002TX4]].
Near such a zero, $f(z) = (z - z_0)^m g(z)$ with $g(z_0) \neq 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0QX7E9KAAX.md" tag="0QX7E9KAAX" kind="statement">
# Tangent–secant theorem

## Statement

Let $C$ be a [[circle|060W5CBZP6]] in the Euclidean plane $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $P$ be a point outside $C$.
Let a [[tangent line|0T7DX8NY1X]] through $P$ touch $C$ at $T$. Let a second line through $P$ meet $C$ at the two points $G_1$ and $G_2$.
Then
$$d(P, T)^2 = d(P, G_1)\, d(P, G_2).$$

## Remarks

Both sides are equal to the [[power|02E9VT1RVQ]] of $P$ with respect to $C$.
It is the limit case of the [[intersecting secants theorem|0J5XR7BW2S]] where one secant becomes a tangent.
The theorem is Proposition 36 of Book III of Euclid's Elements.

## Explanations

### L1 human

From a point outside a circle, the square of the tangent length equals the product of the two distances along any secant. For the unit circle and $P = (3, 0)$, the secant along the $x$-axis meets the circle at distances $2$ and $4$. So the tangent length is $\sqrt{8}$.
</document>

<document source="https://tamnd.github.io/math2/node/0QXM0A9878.md" tag="0QXM0A9878" kind="concept">
# Picard–Lindelöf hypotheses

## Summary

The Picard–Lindelöf hypotheses on a vector field are continuity in time, a Lipschitz condition in space and a bound near the initial point. They also ask for a short enough time interval.
They are the conditions of the [[Picard–Lindelöf theorem|0P6KRD5B7A]]. See the [[definition|0J7HF2PF1H]].
</document>

<document source="https://tamnd.github.io/math2/node/0QY87R9CY3.md" tag="0QY87R9CY3" kind="statement">
# Convergent sequences are Cauchy

## Statement

Let $(a_n)$ be a [[convergent sequence|0AZ3R52593]] of real numbers.
Then $(a_n)$ is a [[Cauchy sequence|0QZMTN5AXX]].

## Remarks

Let $L$ be the limit and let $\varepsilon > 0$. Take $N$ with $|a_n - L| < \varepsilon/2$ for $n \ge N$. Then $|a_m - a_n| \le |a_m - L| + |L - a_n| < \varepsilon$ for $m, n \ge N$.
The statement holds in each metric space.
The converse holds in $\mathbb{R}$, but it fails in $\mathbb{Q}$ and in other spaces that are not complete.

## Explanations

### L1 human

If all late terms are close to one number $L$, then they are also close to each other.
Two terms that are each within $\varepsilon/2$ of $L$ are within $\varepsilon$ of each other.
</document>

<document source="https://tamnd.github.io/math2/node/0QZCY9PE3Q.md" tag="0QZCY9PE3Q" kind="definition">
# Presentation of a group

## Definition

Let $S$ be a set, let $F_S$ be the [[free group|0KTW26C80J]] on $S$, and let $R \subseteq F_S$.
Let $N$ be the smallest [[normal subgroup|0CB0DX5BVV]] of $F_S$ that contains $R$, the normal closure of $R$.
The group with presentation $\langle S \mid R \rangle$ is the [[quotient group|0KYX2N69J0]] $F_S / N$.
A group $G$ has the presentation $\langle S \mid R \rangle$ when $G$ is [[isomorphic|08MG7H8JW5]] to $F_S / N$.

## Remarks

The elements of $S$ are the generators and the elements of $R$ are the relators; a relation $x = y$ stands for the relator $y^{-1} x$.
The cyclic group of order $n$ has the presentation $\langle a \mid a^n \rangle$, and the dihedral group of order $16$ has the presentation $\langle r, f \mid r^8, f^2, (r f)^2 \rangle$.
Each group has a presentation, and each finite group has a presentation with $S$ and $R$ finite.
A group is finitely presented when it has a presentation with $S$ and $R$ finite.

## Explanations

### L1 human

A presentation lists generators and the rules that they obey.
The group is the largest group that these generators make, subject only to these rules.
Presentations are often the shortest way to describe a group.
</document>

<document source="https://tamnd.github.io/math2/node/0QZMTN5AXX.md" tag="0QZMTN5AXX" kind="definition">
# Cauchy sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]].
The sequence is a Cauchy sequence if the following holds for each real number $\varepsilon > 0$.
There is a natural number $N$ such that $|a_m - a_n| < \varepsilon$ for all natural numbers $m, n \ge N$.

## Remarks

In a metric space $(X, d)$, the definition is the same with $d(a_m, a_n)$ in place of $|a_m - a_n|$.
The definition does not mention a limit. In $\mathbb{R}$, the [[Cauchy criterion|0ZA4AX4MW3]] shows that the Cauchy sequences are exactly the convergent sequences.
It is not enough that $|a_{n+1} - a_n| \to 0$. The sequence $a_n = \sqrt{n}$ has this property, but it is not Cauchy.

## Explanations

### L1 human

In a Cauchy sequence, the terms become close to each other, not only to their neighbours.
From some index on, any two terms differ by less than the error that you allow.
</document>

<document source="https://tamnd.github.io/math2/node/0R03B54V4Z.md" tag="0R03B54V4Z" kind="definition">
# Fermat number

## Definition

For a natural number $n$, the $n$-th Fermat number is $F_n = 2^{2^n} + 1$.

## Remarks

The first values are $F_0 = 3$, $F_1 = 5$, $F_2 = 17$, $F_3 = 257$ and $F_4 = 65537$.
The Fermat numbers satisfy $F_0 F_1 \cdots F_{n-1} = F_n - 2$ for $n \ge 1$.
A Fermat number that is prime is a [[Fermat prime|0N9E4G2P97]].

## Explanations

### L1 human

The Fermat numbers grow very fast, because the exponent itself is a power of $2$.
For example, $F_5 = 2^{32} + 1 = 4294967297$.
</document>

<document source="https://tamnd.github.io/math2/node/0R0D0XY4HT.md" tag="0R0D0XY4HT" kind="concept">
# Coefficient of a power series

## Summary

The coefficient of $X^n$ in a formal power series is the $n$-th term of its sequence. Coefficient extraction is how a generating function gives back its counting numbers. The node [[coefficient of a power series|0M98XYVX6S]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0R0MYT0BJD.md" tag="0R0MYT0BJD" kind="statement">
# Continuity of the sum of a uniformly convergent series of functions

## Statement

Let $S \subseteq \mathbb{R}$, and let $f_n \colon S \to \mathbb{R}$ be continuous for each $n \ge 0$.
If the [[series|0MRE6XRE5V]] $\sum_{n=0}^{\infty} f_n$ converges uniformly on $S$, then its sum is continuous on $S$.

## Remarks

The partial sums are continuous, and they converge uniformly. Apply the [[uniform limit theorem|0X9XT05JVC]].
Uniform convergence often comes from the [[M-test|0NX5NWAFC9]]. Mathlib states the result in this form as `continuous_tsum`.
For example, $\sum_{n=1}^{\infty} \frac{\cos(n x)}{n^2}$ is continuous on $\mathbb{R}$.

## Explanations

### L1 human

Adding infinitely many continuous functions gives a continuous function, if the sum converges uniformly.
</document>

<document source="https://tamnd.github.io/math2/node/0R0T1VWH5X.md" tag="0R0T1VWH5X" kind="statement">
# Fubini's theorem

## Statement

Let $R = [a, b] \times [c, d]$, and let $f \colon R \to \mathbb{R}$ be Riemann integrable on $R$.
Suppose that $y \mapsto f(x, y)$ is integrable on $[c, d]$ for each $x \in [a, b]$, and that $x \mapsto f(x, y)$ is integrable on $[a, b]$ for each $y \in [c, d]$.
Then
$$\iint_R f \, dA = \int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy.$$
In particular, this holds for each [[continuous|0S9EK5MQPE]] function $f$ on $R$.

## Remarks

The left side is the [[double integral|0NKBMFC18R]], and the other two are [[iterated integrals|0P4WX7Y502]].
The Lebesgue version: let $f$ be Lebesgue measurable on $\mathbb{R}^n \times \mathbb{R}^m$ with $\int |f| < \infty$. Then the double integral equals both iterated integrals, where the inner integrals exist for almost all values of the outer variable.
Without integrability of $|f|$, the [[iterated integrals can differ|0GJZE2E4RX]]. For non-negative functions, [[Tonelli's theorem|067SYD2J4B]] removes the integrability hypothesis.
Euler knew the case of continuous functions. Lebesgue proved a version for bounded measurable functions in 1904, and Fubini proved the version for integrable functions in 1907.

## Explanations

### L1 human

To integrate over a rectangle, you can slice it into lines, integrate along each line, and then integrate the results.
Fubini's theorem says that both slicing directions give the same answer as the double integral, for well-behaved functions.
</document>

<document source="https://tamnd.github.io/math2/node/0R0X73XHAT.md" tag="0R0X73XHAT" kind="statement">
# Abelian groups are solvable

## Statement

Let $G$ be an [[abelian group|082GHNPJ53]].
Then $G$ is [[solvable|0FXJ7YFMMX]].

## Remarks

The series $\{e\} \trianglelefteq G$ has one factor, $G$ itself, and this factor is abelian.
Equivalently, the commutator subgroup $G^{(1)}$ is already $\{e\}$.
The converse is false: $S_3$ is solvable, by the series $\{e\} \trianglelefteq A_3 \trianglelefteq S_3$, but it is not abelian.

## Explanations

### L1 human

An abelian group is already one abelian layer.
So it is solvable with a series of length one.
Solvable groups generalise abelian groups.
</document>

<document source="https://tamnd.github.io/math2/node/0R2X8K9QA7.md" tag="0R2X8K9QA7" kind="concept">
# Content of a polynomial

## Summary

The content of a polynomial over a unique factorisation domain is a greatest common divisor of its coefficients.
The [[definition of the content|08VV7YJRB7]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0R37DFJKFW.md" tag="0R37DFJKFW" kind="definition">
# Connected set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
The set $S$ is connected when $S$, with the [[subspace topology|0D5YHQGDJY]], is a [[connected space|0D939CMME7]].

## Remarks

Equivalently, there are no open sets $U, V$ of $X$ with $S \subseteq U \cup V$, $S \cap U \cap V = \varnothing$, $S \cap U \neq \varnothing$ and $S \cap V \neq \varnothing$.
The [[connected subsets of the real line are the intervals|0B9ZQFMQNV]].
The closure of a connected set is connected. A union of connected sets with a common point is connected.

## Explanations

### L1 human

A subset is connected when it is in one piece, as a space of its own.
For example, the set $\mathbb{Q}$ of rationals is not connected in $\mathbb{R}$: the sets $(-\infty, \sqrt{2})$ and $(\sqrt{2}, \infty)$ split it.
</document>

<document source="https://tamnd.github.io/math2/node/0R3B7DX55J.md" tag="0R3B7DX55J" kind="definition">
# T0 space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is T0 when for all $x \neq y$ in $X$, some [[open set|01ZQQDCERP]] contains exactly one of $x$ and $y$.

## Remarks

Each [[T1 space|05GQJXBP3F]] is T0.
The Sierpiński space $\{0, 1\}$, with open sets $\varnothing$, $\{1\}$ and $\{0, 1\}$, is T0 but not T1.
An [[indiscrete space|0XJ02JBD6S]] with at least two points is not T0.

## Explanations

### L1 human

In a T0 space, no two distinct points have exactly the same open sets around them.
The open sets may still fail to separate the points in both directions.
</document>

<document source="https://tamnd.github.io/math2/node/0R3H35G96G.md" tag="0R3H35G96G" kind="definition">
# Derived series

## Definition

Let $G$ be a [[group|G5S1M9]].
The derived series of $G$ is the sequence of subgroups $G^{(0)} = G$ and $G^{(i+1)} = [G^{(i)}, G^{(i)}]$ for $i \geq 0$.
Here $[H, H]$ is the [[commutator subgroup|0ASGSTXSBW]] of $H$.

## Remarks

Each $G^{(i)}$ is a [[characteristic subgroup|0KABPMRHM2]] of $G$, and $G^{(i)} / G^{(i+1)}$ is abelian.
The group $G$ is [[solvable|0FXJ7YFMMX]] if and only if $G^{(n)} = \{e\}$ for some $n$, and the least such $n$ is the derived length.
For $S_4$, the derived series is $S_4 \supseteq A_4 \supseteq V \supseteq \{e\}$.
For $n \geq 5$, the derived series of $S_n$ is $S_n \supseteq A_n \supseteq A_n \supseteq \cdots$, so it never reaches $\{e\}$.

## Explanations

### L1 human

Each step of the derived series takes the commutator subgroup of the last step.
So each step removes an abelian layer from the group.
If you reach the identity, the group is solvable.
</document>

<document source="https://tamnd.github.io/math2/node/0R3H5PANRQ.md" tag="0R3H5PANRQ" kind="concept">
# Supremum

## Summary

The supremum of a set is its least upper bound.
It is an upper bound of the set that is less than or equal to every other upper bound.
A set can have a supremum that is not an element of the set.
</document>

<document source="https://tamnd.github.io/math2/node/0R41PVFA39.md" tag="0R41PVFA39" kind="definition">
# Type I and type II errors

## Definition

Consider a [[hypothesis test|087P03NEND]] of $H_0 \colon \theta \in \Theta_0$ against $H_1 \colon \theta \in \Theta_1$ that rejects $H_0$ when the data $X$ are in a rejection region $R$.
A type I error occurs when $H_0$ is true and the test rejects $H_0$. Its probability at $\theta \in \Theta_0$ is $P_\theta(X \in R)$.
A type II error occurs when $H_0$ is false and the test does not reject $H_0$. Its probability at $\theta \in \Theta_1$ is $\beta(\theta) = P_\theta(X \notin R)$.

## Remarks

The size of the test is $\alpha = \sup_{\theta \in \Theta_0} P_\theta(X \in R)$.
The power of the test at $\theta \in \Theta_1$ is $1 - \beta(\theta)$.
For a fixed sample size, a smaller rejection region lowers the probability of a type I error. It raises the probability of a type II error.
In the frequentist practice, one fixes a bound $\alpha$ on the type I error, and then chooses a test with a small type II error.

## Explanations

### L1 human

A type I error is a false alarm: the test says "effect" when there is none. A type II error is a miss: the test says "no effect" when there is one.
For a medical test, a positive result for a healthy person is a type I error. A negative result for an ill person is a type II error.
</document>

<document source="https://tamnd.github.io/math2/node/0R46A1XY60.md" tag="0R46A1XY60" kind="definition">
# Almost surely

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]].
An [[event|00SQGJNRTB]] $E \in \mathcal{F}$ happens almost surely if $P(E) = 1$.
More generally, a property of outcomes holds almost surely (a.s.) if an event of probability $0$ contains all outcomes where it fails.

## Remarks

The property need not hold for all outcomes. In this sense, "almost surely" is weaker than "surely".
Two random variables $X$ and $Y$ are equal almost surely if $P(X = Y) = 1$. They then have the same law.
A countable intersection of events that happen almost surely also happens almost surely.
It is the probability form of "almost everywhere" in measure theory.

## Explanations

### L1 human

If you choose a uniform random number in $[0, 1]$, it is almost surely irrational. Rational outcomes are possible, but together they have probability $0$.
If you toss a fair coin forever, you get heads at least once almost surely.
</document>

<document source="https://tamnd.github.io/math2/node/0R47MAD57S.md" tag="0R47MAD57S" kind="definition">
# Monoid

## Definition

A monoid is a triple $(M, \ast, e)$, where $(M, \ast)$ is a [[semigroup|0XMEM406JM]] and $e \in M$ is an [[identity element|07WYQPB8E3]] for $\ast$.
Thus $(a \ast b) \ast c = a \ast (b \ast c)$ and $e \ast a = a \ast e = a$ for all $a, b, c \in M$.

## Remarks

The natural numbers $\mathbb{N} = \{0, 1, 2, \ldots\}$ with addition form a monoid with identity $0$.
The integers with multiplication form a monoid with identity $1$.
The identity element of a monoid is unique, thus some authors write a monoid as the pair $(M, \ast)$.
Each [[group|G5S1M9]] is a monoid in which each element has an [[inverse|0ZNZH0EXFX]].

## Explanations

### L1 human

A monoid is a set with an associative operation and an element that does nothing.
The strings of letters with concatenation form a monoid, and the empty string is the identity.
In a monoid, an element need not have an inverse: you cannot undo the addition of a letter.
</document>

<document source="https://tamnd.github.io/math2/node/0R5HTV7REV.md" tag="0R5HTV7REV" kind="definition">
# Directed graph

## Definition

A directed graph, or digraph, is a pair $D = (V, A)$ of a set $V$ and a relation $A \subseteq V \times V$.
The elements of $V$ are the vertices of $D$.
The elements of $A$ are the arcs, or directed edges, of $D$.
An arc $(u, v)$ goes from its tail $u$ to its head $v$.

## Remarks

With this definition a digraph can have loops $(v, v)$, and it has at most one arc from $u$ to $v$. Some authors exclude loops.
A [[graph|0JD5C82QMB]] is the same as a digraph whose relation is symmetric and irreflexive.
The out-degree of $v$ is the number of arcs with tail $v$, and the in-degree is the number of arcs with head $v$.

## Explanations

### L1 human

A directed graph is a graph whose lines are arrows.
One-way streets between crossings form a directed graph.
</document>

<document source="https://tamnd.github.io/math2/node/0R5QPQ3AGJ.md" tag="0R5QPQ3AGJ" kind="definition">
# Translation

## Definition

Let $v$ be a [[vector|0QN3D80AWP]] of $\mathbb{R}^n$. The translation by $v$ is the map
$$T_v \colon \mathbb{R}^n \to \mathbb{R}^n, \qquad T_v(p) = p + v.$$

## Remarks

Each translation is a [[Euclidean isometry|0J1N3WTKM4]], and it is direct. For $v \neq 0$, it has no fixed point.
$T_v \circ T_w = T_{v + w}$ and $T_v^{-1} = T_{-v}$. So the translations form an abelian group, isomorphic to $\mathbb{R}^n$.
In the plane, the composite of the reflections in two parallel lines is a translation.
Mathlib's `AffineEquiv.constVAdd` is the translation by a vector in an affine space.

## Explanations

### L1 human

A translation slides every point by the same arrow. Shapes keep their size, their form and their direction. The translation by $(2, 1)$ sends the point $(0, 0)$ to $(2, 1)$ and the point $(3, -1)$ to $(5, 0)$.
</document>

<document source="https://tamnd.github.io/math2/node/0R64A2CVN4.md" tag="0R64A2CVN4" kind="statement">
# Fraction in lowest terms

## Statement

Let $q$ be a [[rational number|0NCJ5WXN6Z]].
Then there is exactly one pair of [[integers|0C4JS9RF5M]] $a$ and $b$ with $b > 0$, $\gcd(a, b) = 1$ and $q = a / b$.

## Remarks

The fraction $a / b$ is the irreducible fraction, or the fraction in lowest terms, of $q$.
To find it from any fraction $c / d$ with $d \neq 0$, divide $c$ and $d$ by $\pm \gcd(c, d)$, with the sign that makes the denominator positive.
For example, $-6 / -4$ has lowest terms $3 / 2$.

## Explanations

### L1 human

You can simplify each fraction until the top and the bottom have no common factor larger than $1$.
For example, $12 / 18$ becomes $2 / 3$ when you divide both numbers by $6$.
The result does not depend on the order of the simplification steps.
</document>

<document source="https://tamnd.github.io/math2/node/0R6EWFKRCF.md" tag="0R6EWFKRCF" kind="concept">
# Compact set

## Summary

A compact subset of a topological space is a subset that is a compact space with the subspace topology.
Equivalently, each cover of the subset by open sets of the space has a finite subcover.
</document>

<document source="https://tamnd.github.io/math2/node/0R6M158SQ9.md" tag="0R6M158SQ9" kind="definition">
# Annihilator of a subspace

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $W$ be a [[subspace|00YC1D652K]] of $V$.
The *annihilator* of $W$ is the subset of the [[dual space|07JBSZE0V4]]
$$W^0 = \{f \in V^* : f(w) = 0 \text{ for all } w \in W\}.$$

## Remarks

The annihilator $W^0$ is a subspace of $V^*$.
If $W_1 \subseteq W_2$, then $W_2^0 \subseteq W_1^0$.
In finite dimensions, $\dim W + \dim W^0 = \dim V$, see [[the dimension of the annihilator|06Q1EFVH0Q]].
The annihilator of the line spanned by $(1, 1)$ in $F^2$ is spanned by the form $(x, y) \mapsto x - y$.

## Explanations

### L1 human

The annihilator of a subspace collects the linear forms that are zero on all of it.
In $\R^3$, the forms that vanish on a plane through the origin are the multiples of one form, whose kernel is that plane.
A larger subspace has a smaller annihilator.
</document>

<document source="https://tamnd.github.io/math2/node/0R74MZYNRB.md" tag="0R74MZYNRB" kind="statement">
# Universal property of the quotient ring

## Statement

Let $R$ be a [[ring|0894J980RK]], let $I$ be a two-sided ideal of $R$ and let $\pi \colon R \to R/I$ be the quotient map.
Let $f \colon R \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]] with $I \subseteq \ker f$.
Then there is exactly one ring homomorphism $\bar{f} \colon R/I \to S$ such that $\bar{f} \circ \pi = f$.
It is given by $\bar{f}(a + I) = f(a)$.

## Remarks

The map $\bar{f}$ is well defined: if $a + I = b + I$, then $a - b \in I \subseteq \ker f$, so $f(a) = f(b)$.
The image of $\bar{f}$ is the image of $f$, and $\ker \bar{f} = (\ker f)/I$.
With $I = \ker f$, this gives the [[first isomorphism theorem|0ZMYX7GMH0]].

## Explanations

### L1 human

A homomorphism that sends all of $I$ to zero does not see the difference between $a$ and $a + x$ for $x \in I$.
So it already makes sense on the cosets of $I$.
This gives a homomorphism from the quotient ring.
</document>

<document source="https://tamnd.github.io/math2/node/0R7BTEWCJ2.md" tag="0R7BTEWCJ2" kind="statement">
# Compact subsets of a metric space are bounded

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $K$ be a [[compact subset|00R0X1WGPX]] of $X$.
Then $K$ is [[bounded|0P953MG05E]].

## Remarks

Proof: if $K$ is empty, it is bounded. Otherwise fix $x \in K$. The [[open balls|0DC3MPR7C9]] $B(x, n)$ for $n \ge 1$ cover $K$, so finitely many of them do. The largest of these balls contains $K$.
A compact subset of a metric space is also [[closed|0G5AV9Q4RR]], because metric spaces are Hausdorff.
The converse fails in general: a closed bounded set need not be compact. With the metric $\min(|x - y|, 1)$, the real line is closed and bounded, but not compact.

## Explanations

### L1 human

A compact set can be covered by finitely many balls about one point, and then it fits in the largest of them.
</document>

<document source="https://tamnd.github.io/math2/node/0R7JS3DP3W.md" tag="0R7JS3DP3W" kind="statement">
# Symmetry of binomial coefficients

## Statement

Let $n$ and $k$ be natural numbers with $k \le n$.
Then
$$\binom{n}{k} = \binom{n}{n - k}.$$

## Remarks

For a set $X$ with $n$ elements, the map $S \mapsto X \setminus S$ is a bijection from the $k$-subsets of $X$ to the $(n-k)$-subsets. This is a [[bijective proof|0GV2573FP4]] of the identity.
The identity makes each row of [[Pascal's triangle|05DME4TZ3K]] read the same from left to right and from right to left.

## Explanations

### L1 human

To choose $k$ people for a team is the same as to choose the $n - k$ people who stay out.
For example, $\binom{5}{2} = \binom{5}{3} = 10$.
</document>

<document source="https://tamnd.github.io/math2/node/0R7SVNYTJS.md" tag="0R7SVNYTJS" kind="statement">
# pi is greater than 3

## Statement

$$\pi > 3.$$

## Remarks

The [[cosine series|08W5RN375X]] alternates. At $x = 3/2$, its terms decrease in size from the term $x^4/4!$ on. So $\cos(3/2) \ge 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}$ at $x = 3/2$. This is about $0.070 > 0$.
Since $\cos 2 < 0$, the zero of $\cos$ in $[0, 2]$ lies in $(3/2, 2)$. So $\pi/2 > 3/2$.
Archimedes proved the sharper bounds $\frac{223}{71} < \pi < \frac{22}{7}$. Mathlib states the claim as `Real.pi_gt_three`.

## Explanations

### L1 human

We show that $\cos x$ is still positive at $x = 3/2$.
So its first zero $\pi/2$ comes later, and $\pi > 3$.
</document>

<document source="https://tamnd.github.io/math2/node/0R7VXTYHYZ.md" tag="0R7VXTYHYZ" kind="statement">
# A linear isometry preserves the inner product

## Statement

Let $V$ and $W$ be inner product spaces over $\R$ or $\C$, with the induced norms, and let $T \colon V \to W$ be a [[linear isometry|07SJXTWVQW]].
Then, for all $u, v \in V$,
$$\langle T(u), T(v) \rangle = \langle u, v \rangle.$$

## Remarks

The proof applies the [[polarisation identity|05FKHRT0B4]], which writes inner products in terms of norms.
Conversely, a linear map that keeps inner products keeps norms.
So a linear isometry also keeps angles and orthogonality.

## Explanations

### L1 human

If a linear map keeps lengths, then it also keeps angles.
The reason is that lengths determine inner products, by polarisation.
So rigid linear motions keep the whole geometry.
</document>

<document source="https://tamnd.github.io/math2/node/0R84PF4JV7.md" tag="0R84PF4JV7" kind="statement">
# Dimension of a tensor product

## Statement

Let $F$ be a field, and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Then
$$\dim (V \otimes W) = \dim V \cdot \dim W,$$
where $\dim$ is the [[dimension|03GSZ8RTZJ]] over $F$.

## Remarks

It follows from the [[basis of a tensor product|0SH5VT8QYA]], which has $\dim V \cdot \dim W$ vectors.
For example, $F^m \otimes F^n$ has dimension $mn$, like the space of $m \times n$ matrices.
Compare this with the direct sum: $\dim (V \oplus W) = \dim V + \dim W$.

## Explanations

### L1 human

The dimension of a tensor product is the product of the dimensions.
The dimension of a direct sum is their sum instead.
So $\R^2 \otimes \R^3$ has dimension $6$.
</document>

<document source="https://tamnd.github.io/math2/node/0R8J3ZQMJS.md" tag="0R8J3ZQMJS" kind="definition">
# Subsequence

## Definition

Let $(a_n)_{n \in \mathbb{N}}$ be a [[sequence|0QE8STGJH2]] in a set $X$.
Let $(n_k)_{k \in \mathbb{N}}$ be a strictly increasing sequence of natural numbers: $n_0 < n_1 < n_2 < \cdots$.
The sequence $(a_{n_k})_{k \in \mathbb{N}}$ is a subsequence of $(a_n)$.

## Remarks

Equivalently, a subsequence is the composite $a \circ \varphi$ of the sequence $a \colon \mathbb{N} \to X$ with a strictly increasing map $\varphi \colon \mathbb{N} \to \mathbb{N}$.
By induction, $n_k \ge k$ for each $k$.
The even terms $a_0, a_2, a_4, \dots$ form a subsequence, with $n_k = 2k$.

## Explanations

### L1 human

You get a subsequence when you delete some terms of a sequence and keep infinitely many terms in their old order.
From $1, 2, 3, 4, \dots$ you can keep the even numbers $2, 4, 6, \dots$.
</document>

<document source="https://tamnd.github.io/math2/node/0R8JR0ARJ4.md" tag="0R8JR0ARJ4" kind="example">
# Geometric series as a power series

## Example

Let $R$ be a commutative ring and let $a \in R$.
In the ring $R[\![X]\!]$ of [[formal power series|072W67KFYK]],
$$(1 - aX) \sum_{n \ge 0} a^n X^n = 1.$$
Thus $1 - aX$ is a unit, and
$$\frac{1}{1 - aX} = \sum_{n \ge 0} a^n X^n.$$

## Remarks

The proof is a telescoping product. The coefficient of $X^n$ in the product is $a^n - a \cdot a^{n-1} = 0$ for $n \ge 1$, and it is $1$ for $n = 0$.
With $a = 1$, it gives $\frac{1}{1 - X} = 1 + X + X^2 + \dots$, the [[generating function|0HH6JX5WZZ]] of the constant sequence $1$.
Over $\mathbb{R}$ the same identity holds as a convergent series for $|aX| < 1$, but the formal identity needs no convergence.

## Explanations

### L1 human

Multiply $1 + aX + a^2 X^2 + \dots$ by $1 - aX$. Every term cancels with the next one, and only $1$ is left.
So the geometric series is the inverse of $1 - aX$.
</document>

<document source="https://tamnd.github.io/math2/node/0R8X9XA83A.md" tag="0R8X9XA83A" kind="statement">
# A linear map is determined by its values on a basis

## Statement

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $(b_i)_{i \in I}$ be a [[basis|0KCQ9Y3QKA]] of $V$.
If $S, T \colon V \to W$ are [[linear maps|0NRYHJPQG2]] with $S(b_i) = T(b_i)$ for all $i \in I$, then $S = T$.
Moreover, for each family $(w_i)_{i \in I}$ of vectors of $W$, there is exactly one linear map $T \colon V \to W$ with $T(b_i) = w_i$ for all $i \in I$.

## Remarks

The map is $T(\sum_i a_i b_i) = \sum_i a_i w_i$.
The values on a basis can be chosen freely; the values on a merely spanning family cannot.
This statement is the reason why a linear map between finite-dimensional spaces is described by a matrix.

## Explanations

### L1 human

A linear map is known as soon as we know where it sends the basis vectors.
Every other vector is a combination of basis vectors, and linearity tells us where it goes.
Conversely, we may send the basis vectors anywhere we like.
</document>

<document source="https://tamnd.github.io/math2/node/0R8YXRC9AJ.md" tag="0R8YXRC9AJ" kind="statement">
# Integers modulo a prime form a field

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
Then the ring $\mathbb{Z}/p\mathbb{Z}$ of the [[integers modulo p|01N3AYAA6N]] is a field.
That is, each nonzero element of $\mathbb{Z}/p\mathbb{Z}$ has a multiplicative inverse.

## Remarks

The proof: if $\overline{a} \neq \overline{0}$, then $p \nmid a$, so $\gcd(a, p) = 1$. [[Bézout's identity|0DQVHMYAGA]] gives $a x + p y = 1$, and then $\overline{a} \cdot \overline{x} = \overline{1}$.
The converse holds: if $n > 1$ is composite, with $n = a b$ and $1 < a, b < n$, then $\overline{a} \cdot \overline{b} = \overline{0}$, so $\mathbb{Z}/n\mathbb{Z}$ is not a field.
The field $\mathbb{Z}/p\mathbb{Z}$ is often written $\mathbb{F}_p$ or $\mathrm{GF}(p)$. It is the finite field with $p$ elements.

## Explanations

### L1 human

Modulo $7$, you can divide by each nonzero number. For example, $3 \cdot 5 = 15 \equiv 1 \pmod 7$, so $5$ plays the part of $1/3$.
Modulo $6$ this fails: no number $x$ gives $2 x \equiv 1 \pmod 6$, because $2 x$ is always even.
</document>

<document source="https://tamnd.github.io/math2/node/0R9B0B5458.md" tag="0R9B0B5458" kind="statement">
# Characterization of the floor

## Statement

Let $x$ be a real number and let $n$ be an integer.
Then $n = \lfloor x \rfloor$ if and only if $n \le x < n + 1$.
In particular, $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$.

## Remarks

Thus the [[floor|0YVRNEDSVG]] of $x$ is the unique integer $n$ with $n \le x < n + 1$.
The fractional part of $x$ is $x - \lfloor x \rfloor$. It lies in the interval $[0, 1)$.
An equivalent form: $x - 1 < \lfloor x \rfloor \le x$.

## Explanations

### L1 human

The intervals $[n, n + 1)$ for integers $n$ cover the number line without overlap.
Each real number lies in exactly one of them, and the floor of the number is the left end of that interval.
</document>

<document source="https://tamnd.github.io/math2/node/0R9D4CEJEB.md" tag="0R9D4CEJEB" kind="statement">
# Divisibility by 3

## Statement

Let $n$ be a natural number with decimal digits $d_k d_{k-1} \dots d_0$, so that $n = \sum_{i=0}^{k} d_i 10^i$.
Then $3 \mid n$ if and only if $3 \mid d_0 + d_1 + \dots + d_k$.

## Remarks

More precisely, $n \equiv d_0 + d_1 + \dots + d_k \pmod 3$. This holds because $10 \equiv 1 \pmod 3$, so $10^i \equiv 1 \pmod 3$ by [[compatibility with powers|0J5VGJYYC3]].
For example, $3 \mid 7{,}341$, because $7 + 3 + 4 + 1 = 15$ and $3 \mid 15$.
The rule for [[divisibility by 9|0ESHD7ZQDX]] has the same proof.

## Explanations

### L1 human

Add the digits of the number. If the sum is a multiple of $3$, the number is a multiple of $3$, and if not, it is not.
You can repeat: for $987{,}654$ the sum is $39$, and $3 + 9 = 12$ is a multiple of $3$. So $3 \mid 987{,}654$.
</document>

<document source="https://tamnd.github.io/math2/node/0R9F850CTH.md" tag="0R9F850CTH" kind="statement">
# Gcd of Fibonacci numbers

## Statement

Let $m$ and $n$ be natural numbers, and let $F_k$ be the [[Fibonacci numbers|0GA9K57A39]].
Then
$$\gcd(F_m, F_n) = F_{\gcd(m, n)}.$$

## Remarks

Example: $\gcd(F_{12}, F_{18}) = \gcd(144, 2584) = 8 = F_6$.
In particular, $F_m$ divides $F_n$ when $m$ divides $n$. Consecutive Fibonacci numbers are coprime, because $\gcd(n, n+1) = 1$ and $F_1 = 1$.
The proof follows the Euclidean algorithm and uses the [[addition formula|00CN2F997Q]].

## Explanations

### L1 human

The greatest common divisor of two Fibonacci numbers is again a Fibonacci number.
To find which one, take the greatest common divisor of the two indices.
</document>

<document source="https://tamnd.github.io/math2/node/0R9GF3JD0K.md" tag="0R9GF3JD0K" kind="concept">
# Linear Diophantine equation

## Summary

A linear Diophantine equation in two unknowns is an equation $a x + b y = c$ with given integers $a$, $b$ and $c$.
Its solutions are the pairs of integers $(x, y)$ that satisfy it.
When $a$ and $b$ are not both $0$, the equation has an integer solution if and only if $\gcd(a, b)$ divides $c$.
In this case, let $d = \gcd(a, b)$ and let $(x_0, y_0)$ be one solution. Then the solutions are the pairs $(x_0 + k b / d, \; y_0 - k a / d)$ with $k$ an integer.

## Remarks

[[Bézout's identity|0DQVHMYAGA]] gives the existence of a solution, and the [[extended Euclidean algorithm|0964YQFSA8]] computes one.
For example, $6 x + 10 y = 7$ has no integer solution, because $\gcd(6, 10) = 2$ does not divide $7$.
The equation is a special case of a [[Diophantine equation|08B1RTW7VZ]].

## Explanations

### L1 human

Ask whether you can pay $c$ with coins of value $a$ and $b$, when you can also get coins back as change.
Each amount that you can pay is a multiple of $\gcd(a, b)$, and each such multiple is possible.
For example, $3 x + 5 y = 1$ has the solution $x = 2$, $y = -1$.
</document>

<document source="https://tamnd.github.io/math2/node/0R9JR0465V.md" tag="0R9JR0465V" kind="definition">
# Dyck word

## Definition

A Dyck word is a finite word $w$ over the two-letter alphabet $\{U, D\}$ with these two properties.
The word $w$ has as many letters $U$ as letters $D$.
Each prefix of $w$ has at least as many letters $U$ as letters $D$.

The semilength of a Dyck word is its number of letters $U$, which is half its length.

## Remarks

Other common alphabets are $\{(, )\}$ and $\{[, ]\}$. Then the Dyck words are the balanced strings of brackets.
The Dyck words of semilength $2$ are $UUDD$ and $UDUD$.
Read $U$ as a step $(1, 1)$ and $D$ as a step $(1, -1)$. A Dyck word is then a [[lattice path|040WY0TT2G]] from $(0, 0)$ to $(2n, 0)$ that never goes below the horizontal axis.
The Dyck words of semilength $n$ are [[counted by the Catalan number|017SRZ7NE9]] $C_n$.

## Explanations

### L1 human

A Dyck word is a correct sequence of opening and closing brackets.
The sequence $(()())$ is a Dyck word, but $())($ is not, because after three symbols there are more closing than opening brackets.
</document>

<document source="https://tamnd.github.io/math2/node/0R9NMSM1ZP.md" tag="0R9NMSM1ZP" kind="concept">
# Adjoint of a linear map

## Summary

The adjoint of a linear map between finite-dimensional inner product spaces is a linear map in the opposite direction.
It moves the map from one side of the inner product to the other.
</document>

<document source="https://tamnd.github.io/math2/node/0R9PHTGVNW.md" tag="0R9PHTGVNW" kind="concept">
# One-sided derivative

## Summary

A one-sided derivative is the limit of difference quotients from one side of a point only. A function is differentiable at an interior point exactly when its left and right derivatives exist and agree. The node [[one-sided derivative|0HXEX0DPRA]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0R9R0Y1HDK.md" tag="0R9R0Y1HDK" kind="object">
# Unitary group

## Description

Let $n \ge 1$.
The *unitary group* is the set
$$U(n) = \{U \in M_n(\C) : U^* U = I_n\}$$
of complex $n \times n$ [[unitary matrices|0AR68JTMDT]], with matrix multiplication.
It is a subgroup of the general linear group $GL_n(\C)$.

## Remarks

It is the group of linear isometries of $\C^n$ with the standard inner product.
$U(1)$ is the circle group $\{z \in \C : \abs{z} = 1\}$.
The unitary matrices with determinant $1$ form the special unitary group $SU(n)$.
$U(n)$ is a compact connected Lie group of real dimension $n^2$.

## Explanations

### L1 human

The unitary group collects all complex matrices that keep the length of every vector.
For $n = 1$, these are multiplications by complex numbers of absolute value $1$, that is, rotations of the complex plane.
The group appears throughout physics.
</document>

<document source="https://tamnd.github.io/math2/node/0RAM4E07NR.md" tag="0RAM4E07NR" kind="statement">
# Limit of a sum of sequences

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers.
If $a_n \to a$ and $b_n \to b$ as $n \to \infty$, then $a_n + b_n \to a + b$ as $n \to \infty$.

## Remarks

Given $\varepsilon > 0$, choose $N$ so that $|a_n - a| < \varepsilon/2$ and $|b_n - b| < \varepsilon/2$ for all $n \ge N$. The triangle inequality then gives $|(a_n + b_n) - (a + b)| < \varepsilon$.
With the constant sequence $-1$ and the [[product rule for limits|0734RSVF2V]], it also gives $a_n - b_n \to a - b$.
It is one of the rules of the [[algebra of limits|07XBM3A9J5]].

## Explanations

### L1 human

If $a_n$ is close to $a$ and $b_n$ is close to $b$, then $a_n + b_n$ is close to $a + b$.
The two errors add, so we make each error smaller than half of the allowed error.
</document>

<document source="https://tamnd.github.io/math2/node/0RAS24E1R5.md" tag="0RAS24E1R5" kind="definition">
# Bernoulli number

## Definition

The Bernoulli numbers $B_0, B_1, B_2, \dots$ are the rational numbers given by the power series expansion
$$\frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}.$$
Equivalently, $B_0 = 1$ and $\sum_{k=0}^{n} \binom{n+1}{k} B_k = 0$ for each $n \ge 1$.

## Remarks

The first values are $B_0 = 1$, $B_1 = -\frac{1}{2}$, $B_2 = \frac{1}{6}$, $B_4 = -\frac{1}{30}$ and $B_6 = \frac{1}{42}$. Also $B_n = 0$ for each odd $n \ge 3$.
Some authors use the convention $B_1 = +\frac{1}{2}$, from the series $\frac{t e^t}{e^t - 1}$.
They appear in the formulas for the sums of powers $1^k + 2^k + \dots + n^k$, and in the values $\zeta(2n)$ of the Riemann zeta function.
They are named after Jacob Bernoulli. Seki Takakazu tabulated them in his Katsuyō Sanpō of 1712.

## Explanations

### L1 human

The Bernoulli numbers are a sequence of fractions that appear in many formulas, for example in the formula for $1^k + 2^k + \dots + n^k$.
You can compute them one after another from a simple recurrence.
</document>

<document source="https://tamnd.github.io/math2/node/0RBKKNH4GV.md" tag="0RBKKNH4GV" kind="concept">
# Focus of a conic

## Summary

A focus of a conic is a special point that is used to define the conic. An ellipse and a hyperbola have two foci, a parabola has one, and the focus of a circle is its centre.
</document>

<document source="https://tamnd.github.io/math2/node/0RBQS1JJCR.md" tag="0RBQS1JJCR" kind="concept">
# Quadratic form

## Summary

A quadratic form on a vector space is a map $Q$ to the scalars with $Q(av) = a^2 Q(v)$.
Also, $Q(u + v) - Q(u) - Q(v)$ must be bilinear in $u$ and $v$.
In coordinates, it is a homogeneous polynomial of degree two.
</document>

<document source="https://tamnd.github.io/math2/node/0RBT5PB74K.md" tag="0RBT5PB74K" kind="statement">
# Bessel's inequality

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, with the [[induced norm|0MFJJV97RJ]] $\norm{\cdot}$.
Let $e_1, \dots, e_k$ be an [[orthonormal family|0RW9TNNT87]] in $V$.
Then, for each $x \in V$,
$$\sum_{i=1}^{k} \abs{\langle x, e_i \rangle}^2 \le \norm{x}^2.$$

## Remarks

The proof applies the Pythagorean theorem to $x = \sum_i \langle x, e_i \rangle e_i + r$, where $r$ is orthogonal to each $e_i$.
Equality holds if and only if $x$ lies in the span of $e_1, \dots, e_k$.
The inequality also holds for infinite orthonormal families, with a series on the left.

## Explanations

### L1 human

The squared lengths of the projections of a vector onto perpendicular directions add up to at most its squared length.
The missing part is the squared length of the component outside these directions.
If the directions span the whole space, then nothing is missing.
</document>

<document source="https://tamnd.github.io/math2/node/0RCKQN1E09.md" tag="0RCKQN1E09" kind="definition">
# Cluster point of a sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]] and let $c$ be a real number.
The number $c$ is a cluster point of $(a_n)$ if the following holds for each $\varepsilon > 0$ and each natural number $N$.
There is a natural number $n \ge N$ such that $|a_n - c| < \varepsilon$.

## Remarks

Equivalently, for each $\varepsilon > 0$ there are infinitely many indices $n$ with $|a_n - c| < \varepsilon$.
In $\mathbb{R}$, the number $c$ is a cluster point of $(a_n)$ if and only if some [[subsequence|0R8J3ZQMJS]] of $(a_n)$ converges to $c$.
The sequence $(-1)^n$ has the two cluster points $1$ and $-1$, and no limit.
For a bounded sequence, the [[limit superior|0P8BA7T7S5]] is the largest cluster point and the [[limit inferior|0H4R3RESGV]] is the smallest.

## Explanations

### L1 human

A cluster point is a number that the sequence visits again and again, as closely as you like.
A limit is a cluster point that the sequence approaches with all its late terms, not only with some of them.
</document>

<document source="https://tamnd.github.io/math2/node/0RCP929Y1M.md" tag="0RCP929Y1M" kind="concept">
# Cyclic quadrilateral

## Summary

A cyclic quadrilateral is a quadrilateral whose four vertices lie on one circle. Its sides are chords of that circle.
</document>

<document source="https://tamnd.github.io/math2/node/0RCQZ36MME.md" tag="0RCQZ36MME" kind="concept">
# Indicator function

## Summary

The indicator function of a subset takes the value $1$ on the subset and the value $0$ outside it.
</document>

<document source="https://tamnd.github.io/math2/node/0RCV1ZFP8G.md" tag="0RCV1ZFP8G" kind="concept">
# Boolean ring

## Summary

A Boolean ring is a ring in which every element is idempotent. Boolean rings are Boolean algebras written in the language of rings, with exclusive or as addition. The node [[Boolean ring|0N5FCV2YV6]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0RDJK7EQYY.md" tag="0RDJK7EQYY" kind="definition">
# Ordinal number

## Definition

A set $X$ is transitive when each element of an element of $X$ is an element of $X$.
An ordinal number, in the definition of von Neumann, is a transitive set $\alpha$ that is [[well-ordered|0F1D6PK9RZ]] by the relation $\in$, with $\beta < \gamma$ when $\beta \in \gamma$.

## Remarks

The first ordinals are $0 = \varnothing$, $1 = \{0\}$, $2 = \{0, 1\}$, and in general $n + 1 = n \cup \{n\}$.
The set $\omega = \{0, 1, 2, \dots\}$ of all finite ordinals is the first infinite ordinal.
Each well-ordered set is [[order isomorphic|0Y74KN99A5]] to exactly one ordinal, its order type.
Each ordinal $\alpha$ is the set of all ordinals smaller than $\alpha$.

## Explanations

### L1 human

Ordinals count positions: first, second, third, and then beyond all finite positions.
After $0, 1, 2, \dots$ comes $\omega$, then $\omega + 1$, and so on.
They let you continue a step-by-step construction past infinity.
</document>

<document source="https://tamnd.github.io/math2/node/0RDP2HEFEW.md" tag="0RDP2HEFEW" kind="definition">
# Arccosine

## Definition

The arccosine is the function $\arccos \colon [-1, 1] \to [0, \pi]$ such that, for each $x \in [-1, 1]$, $\arccos x$ is the unique $y \in [0, \pi]$ with $\cos y = x$.

## Remarks

The cosine is strictly decreasing and continuous on $[0, \pi]$, with range $[-1, 1]$. So $\arccos$ is well defined and [[continuous|07BY47HG5S]].
For all $x \in [-1, 1]$, $\arccos x = \frac{\pi}{2} - \arcsin x$.
For example, $\arccos 1 = 0$ and $\arccos(-1) = \pi$. Mathlib defines `Real.arccos` by the identity above.

## Explanations

### L1 human

The arccosine of $x$ is the angle between $0$ and $\pi$ whose cosine is $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0RDRPBMNAC.md" tag="0RDRPBMNAC" kind="statement">
# Cesàro means of a convergent sequence converge

## Statement

Let $(a_n)_{n \ge 1}$ be a real sequence that converges to a real number $L$.
For $n \ge 1$, let $\sigma_n = (a_1 + \cdots + a_n)/n$ be its [[Cesàro mean|0P0DJHBGBB]].
Then $\sigma_n \to L$ as $n \to \infty$.

## Remarks

For the proof, let $\varepsilon > 0$ and take $N$ with $|a_k - L| < \varepsilon$ for $k > N$. For $n > N$,
$$|\sigma_n - L| \le \frac{1}{n} \sum_{k=1}^{N} |a_k - L| + \frac{n - N}{n} \varepsilon.$$
The first term tends to $0$, so $|\sigma_n - L| < 2\varepsilon$ for large $n$.
The converse fails. The means of $(-1)^n$ tend to $0$, but $(-1)^n$ diverges.
The statement is a special case of the [[Stolz–Cesàro theorem|0Q3AKZV74D]] with $b_n = n$.

## Explanations

### L1 human

If the terms settle near $L$, then the running average also settles near $L$.
The early terms that are far from $L$ count less and less, because we divide by $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0RE6X3BGPN.md" tag="0RE6X3BGPN" kind="example">
# Space of matrices

## Example

Let $F$ be a field and let $m, n \ge 1$.
The set $M_{m \times n}(F)$ of $m \times n$ [[matrices|0VD9ZSWWJG]] over $F$ is a [[vector space|000AYPA9AJ]] over $F$, with [[matrix addition|0Y7H3SAVA1]] and [[scalar multiplication|09V6DDACXR]].
Its zero vector is the [[zero matrix|0ZTKJJDGR7]].
The matrices $E_{ij}$, with entry $1$ in position $(i, j)$ and $0$ elsewhere, form a basis, so $\dim M_{m \times n}(F) = mn$.

## Remarks

The map that lists the entries of a matrix row by row is a linear isomorphism $M_{m \times n}(F) \to F^{mn}$.
The space $M_{m \times n}(F)$ is isomorphic to the [[space of linear maps|0X9MBPP2Z3]] from $F^n$ to $F^m$.
The symmetric $n \times n$ matrices form a subspace of dimension $n(n+1)/2$.

## Explanations

### L1 human

Matrices of one fixed size can be added and scaled, so they form a vector space.
A $2 \times 2$ matrix holds four numbers, so this space has dimension $4$.
It is $F^4$ written as a square instead of a row.
</document>

<document source="https://tamnd.github.io/math2/node/0REMT01N36.md" tag="0REMT01N36" kind="statement">
# Generating function of a sum of independent random variables

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] random variables with values in $\{0, 1, 2, \dots\}$, and let $S = X_1 + \dots + X_n$.
Then for each complex $z$ with $|z| \le 1$, the [[probability-generating functions|0Z8PSSAVBA]] satisfy
$$G_S(z) = G_{X_1}(z) \, G_{X_2}(z) \cdots G_{X_n}(z).$$

## Remarks

Proof: $z^S = z^{X_1} \cdots z^{X_n}$. The factors are independent, so the [[product rule for expectations|05AW1G0WE1]] applies.
Example: a sum of $n$ independent Bernoulli($p$) variables has the generating function $(1 - p + pz)^n$. This is the generating function of $\operatorname{Bin}(n, p)$.
For a random number $N$ of i.i.d. terms $X_i$ with $N$ independent of the $X_i$, the sum $S_N = X_1 + \dots + X_N$ has $G_{S_N}(z) = G_N(G_{X_1}(z))$.

## Explanations

### L1 human

Adding independent counts corresponds to multiplying their generating functions.
This turns the hard computation of the law of a sum into a product of power series.
</document>

<document source="https://tamnd.github.io/math2/node/0RFMMBHFTV.md" tag="0RFMMBHFTV" kind="definition">
# Chi-squared distribution

## Definition

Let $k \ge 1$ be an integer and let $Z_1, \dots, Z_k$ be [[i.i.d.|0AJN7WHBJZ]] standard normal random variables.
The chi-squared distribution with $k$ degrees of freedom, written $\chi^2_k$, is the law of
$$Q = Z_1^2 + Z_2^2 + \dots + Z_k^2.$$

## Remarks

It has the [[density|04YG690N5V]] $f(x) = \frac{x^{k/2 - 1} e^{-x/2}}{2^{k/2}\, \Gamma(k/2)}$ for $x > 0$, and $f(x) = 0$ for $x \le 0$. So $\chi^2_k$ is the [[gamma distribution|06395GBNWD]] with shape $k/2$ and rate $1/2$.
If $Q \sim \chi^2_k$, then $E[Q] = k$ and $\operatorname{Var}(Q) = 2k$.
If $Q_1 \sim \chi^2_{k_1}$ and $Q_2 \sim \chi^2_{k_2}$ are independent, then $Q_1 + Q_2 \sim \chi^2_{k_1 + k_2}$.
For an i.i.d. $\mathcal{N}(\mu, \sigma^2)$ sample of size $n$ with [[sample variance|0JN9P4A944]] $S_n^2$, $(n - 1) S_n^2 / \sigma^2 \sim \chi^2_{n - 1}$.

## Explanations

### L1 human

Square $k$ independent standard normal numbers and add them: the result has the chi-squared distribution with $k$ degrees of freedom.
Its mean is $k$, and it has values in $[0, \infty)$ only.
</document>

<document source="https://tamnd.github.io/math2/node/0RFP7F3SEW.md" tag="0RFP7F3SEW" kind="definition">
# Composition factors

## Definition

Let $G$ be a [[group|G5S1M9]] with a [[composition series|0XV2A49X6Q]] $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$.
The composition factors of $G$ are the simple groups $G_i / G_{i-1}$ for $i = 1, \dots, n$, counted with multiplicity.
The number $n$ is the composition length of $G$.

## Remarks

The [[Jordan–Hölder theorem|0E088ZP533]] shows that the composition factors do not depend on the series, up to order and isomorphism.
The composition factors of $\mathbb{Z}/12\mathbb{Z}$ are $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$.
The composition factors of $S_4$ are $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$.
Groups that are not isomorphic can have the same composition factors, for example $\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

## Explanations

### L1 human

The composition factors are the simple pieces of a group.
They are like the prime factors of an integer, counted with multiplicity.
But unlike for integers, the pieces do not determine the group.
</document>

<document source="https://tamnd.github.io/math2/node/0RFWW56Y22.md" tag="0RFWW56Y22" kind="statement">
# Cauchy–Schwarz inequality

## Statement

Let $V$ be an [[inner product space|0QVNCJXRY3]] over $\R$ or $\C$, with the [[induced norm|0MFJJV97RJ]] $\norm{\cdot}$.
Then, for all $x, y \in V$,
$$\abs{\langle x, y \rangle} \le \norm{x} \norm{y}.$$
Equality holds if and only if $x$ and $y$ are linearly dependent.

## Remarks

For the dot product on $\R^n$, it says $\left(\sum_i x_i y_i\right)^2 \le \left(\sum_i x_i^2\right)\left(\sum_i y_i^2\right)$.
In a real inner product space, it allows the definition of the angle $\theta \in [0, \pi]$ with $\cos\theta = \langle x, y \rangle / (\norm{x} \norm{y})$ for nonzero $x$ and $y$.
One proof uses that $\norm{x - ty}^2 \ge 0$ for all real $t$, so a quadratic has a nonpositive discriminant.
It implies the [[triangle inequality|0FJZMBX0ZZ]] for the induced norm.

## Explanations

### L1 human

The inner product of two vectors is at most the product of their lengths, in absolute value.
In the plane, this is the fact that $\abs{\cos\theta} \le 1$.
Equality happens only when the vectors lie on one line.
</document>

<document source="https://tamnd.github.io/math2/node/0RG9SPQSWM.md" tag="0RG9SPQSWM" kind="definition">
# Subgroup generated by a subset

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$.
The subgroup generated by $S$, written $\langle S \rangle$, is the intersection of all [[subgroups|0X1JXT47GA]] of $G$ that contain $S$.

## Remarks

By [[the intersection property|0ZTGMDNVTB]], $\langle S \rangle$ is a subgroup, and it is contained in each subgroup that contains $S$.
Its elements are the products $s_1^{\varepsilon_1} s_2^{\varepsilon_2} \cdots s_k^{\varepsilon_k}$ with $k \geq 0$, $s_i \in S$ and $\varepsilon_i \in \{1, -1\}$, where the empty product ($k = 0$) is the identity.
In particular, $\langle \emptyset \rangle = \{e\}$.

## Explanations

### L1 human

Start with some elements of a group, and add everything that you can make from them with products and inverses.
The result is the subgroup generated by these elements.
For example, in $(\mathbb{Z}, +)$, the subgroup generated by $\{4, 6\}$ is the set $2\mathbb{Z}$ of even integers.
</document>

<document source="https://tamnd.github.io/math2/node/0RGGF8ZS6D.md" tag="0RGGF8ZS6D" kind="definition">
# Stabiliser

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \in X$.
The stabiliser of $x$ is the set $G_x = \{g \in G : g \cdot x = x\}$.

## Remarks

The stabiliser $G_x$ is a [[subgroup|0X1JXT47GA]] of $G$, but in general it is not normal.
For the action of $S_n$ on $\{1, \dots, n\}$, the stabiliser of $n$ is a subgroup isomorphic to $S_{n-1}$.
For the action of $G$ on itself by conjugation, the stabiliser of $a$ is the [[centraliser|02YYV36BFM]] $C_G(a)$.

## Explanations

### L1 human

The stabiliser of a point collects the group elements that keep this point in place.
They may move other points.
For the rotations of a square about its centre, only the identity keeps a corner in place.
</document>

<document source="https://tamnd.github.io/math2/node/0RGWYWWBM5.md" tag="0RGWYWWBM5" kind="definition">
# Incircle

## Definition

Let $ABC$ be a triangle in the Euclidean plane $\mathbb{R}^2$, with [[incentre|0M9MX8YW0A]] $I$ and inradius $r$. Let $d$ be the Euclidean distance.
The incircle of $ABC$ is the [[circle|060W5CBZP6]] with the centre $I$ and the radius $r$:
$$\{\, X \in \mathbb{R}^2 \;:\; d(X, I) = r \,\}.$$

## Remarks

The incircle touches each side of the triangle at exactly one point, the foot of the perpendicular from $I$ to that side.
Let $T$ be the area of $ABC$ and $s$ half of its perimeter. Then $T = r s$, so $r = T / s$.
An excircle is a circle outside the triangle that touches one side and the extensions of the other two sides. Each triangle has three excircles.

## Explanations

### L1 human

The incircle is the biggest circle that fits inside a triangle. It touches each side once. Cut the triangle into three pieces from $I$ to the corners: each piece has the height $r$, and this gives $T = r s$.
</document>

<document source="https://tamnd.github.io/math2/node/0RGZ10DFPG.md" tag="0RGZ10DFPG" kind="definition">
# Submodule

## Definition

Let $R$ be a ring with identity element and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.
A *submodule* of $M$ is a subset $N \subseteq M$ such that:

1. $N$ is a subgroup of the abelian group $(M, +)$.
2. $rx \in N$ for all $r \in R$ and $x \in N$.

## Remarks

A submodule is a module over $R$ with the restricted operations.
The submodules of $R$, as a left module over itself, are the left ideals of $R$.
The $\Z$-submodules of an abelian group are its subgroups, and over a field the submodules are the [[subspaces|00YC1D652K]].
Intersections and sums of submodules are submodules.

## Explanations

### L1 human

A submodule is a module inside a bigger module.
It contains the sums and the scalar multiples of its elements.
For example, the even integers are a submodule of $\Z$ over $\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/0RH1TPDGCQ.md" tag="0RH1TPDGCQ" kind="definition">
# Torsion subgroup

## Definition

Let $A$ be an [[abelian group|082GHNPJ53]].
The torsion subgroup of $A$ is $A_T = \{a \in A : a \text{ has finite order}\}$.
The group $A$ is a torsion group when $A_T = A$, and it is torsion-free when $A_T = \{e\}$.

## Remarks

The set $A_T$ is a [[subgroup|0X1JXT47GA]] because $A$ is commutative: if $a^m = e$ and $b^n = e$, then $(a b)^{m n} = e$.
In a nonabelian group, the elements of finite order need not form a subgroup. In $\langle x, y \mid x^2, y^2 \rangle$, the product $x y$ has infinite order.
The [[quotient group|0KYX2N69J0]] $A / A_T$ is torsion-free, and $A_T$ is a characteristic subgroup of $A$.
For $A = \mathbb{Q}/\mathbb{Z}$, each element has finite order, so $A_T = A$, but the orders have no upper bound.

## Explanations

### L1 human

The torsion subgroup collects the elements that return to the identity after finitely many steps.
In an abelian group, these elements form a subgroup.
Removing them leaves a group in which only the identity has finite order.
</document>

<document source="https://tamnd.github.io/math2/node/0RH4E3CAV9.md" tag="0RH4E3CAV9" kind="definition">
# Projection

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A *projection* on $V$ is an [[endomorphism|0J7H0YBFDJ]] $P \colon V \to V$ with $P \circ P = P$.

## Remarks

If $P$ is a projection, then $V = \im P \oplus \ker P$, and $P$ acts as the identity on $\im P$.
Conversely, for each [[direct sum|0XYYT5KNTX]] $V = U \oplus W$, the map $u + w \mapsto u$ is a projection with image $U$ and kernel $W$.
If $P$ is a projection, then $I - P$ is the projection with image $\ker P$ and kernel $\im P$.
In an inner product space, an [[orthogonal projection|0B8KZRT28M]] is a projection whose kernel is orthogonal to its image.

## Explanations

### L1 human

A projection maps space onto a subspace and leaves the vectors of that subspace where they are.
Projecting a second time changes nothing.
The shadow of an object on the floor, in light from a fixed direction, is a picture of a projection.
</document>

<document source="https://tamnd.github.io/math2/node/0RJ5NHWYTP.md" tag="0RJ5NHWYTP" kind="concept">
# Picard iteration

## Summary

Picard iteration is the construction of a solution of $x' = f(t, x)$, $x(t_0) = x_0$ as a limit of successive approximations.
One starts with the constant curve $\varphi_0(t) = x_0$ and sets $\varphi_{k+1} = P \varphi_k$, where $P$ is the [[Picard operator|00WAVB68F9]]: $\varphi_{k+1}(t) = x_0 + \int_{t_0}^{t} f(s, \varphi_k(s))\, ds$.
Under the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]], the curves $\varphi_k$ converge uniformly on the time interval to the unique solution.

## Remarks

Example: for $x' = x$, $x(0) = 1$, the iterates are $\varphi_k(t) = \sum_{j=0}^{k} t^j / j!$. They converge to $e^t$.
Let $K$ be the Lipschitz constant and $T$ the length of the time interval. The error after $k$ steps is at most $C (K T)^{k} / k!$ for a constant $C$. See [[an iterate of the Picard operator is a contraction|08XDXP86XE]].

## Explanations

### L1 human

Start with a rough guess, the constant $x_0$.
Put the guess into the right side of the integral equation to get a better guess, and repeat.
Each step improves the approximation, and the guesses converge to the solution.
</document>

<document source="https://tamnd.github.io/math2/node/0RJ5TZH6G0.md" tag="0RJ5TZH6G0" kind="statement">
# Limit of the cumulative distribution function at infinity

## Statement

Let $X$ be a real random variable with [[cumulative distribution function|0MXMS3AE13]] $F_X$.
Then
$$\lim_{x \to +\infty} F_X(x) = 1.$$

## Remarks

The proof: the events $\{X \le n\}$ increase with $n$ and their union is $\Omega$. So [[continuity from below|0X8HBKCSYB]] gives $F_X(n) \to 1$, and $F_X$ is nondecreasing.
In the same way, $\lim_{x \to -\infty} F_X(x) = 0$, because the events $\{X \le -n\}$ decrease to $\varnothing$.
The proof uses that $X$ takes only real values. For a random variable with values in $[-\infty, +\infty]$, the limit is $1 - P(X = +\infty)$.

## Explanations

### L1 human

For a very large $x$, it is almost sure that $X \le x$, because $X$ is some real number.
So the probability $F_X(x)$ gets as close to $1$ as we like.
</document>

<document source="https://tamnd.github.io/math2/node/0RJF668P1T.md" tag="0RJF668P1T" kind="definition">
# Linear map

## Definition

Let $F$ be a field and let $V$ and $W$ be [[vector spaces|000AYPA9AJ]] over $F$.
A map $T \colon V \to W$ is *linear* when, for all $u, v \in V$ and $a \in F$,
$$T(u + v) = T(u) + T(v), \qquad T(av) = a\, T(v).$$
A linear map is also called a *linear transformation*.

## Remarks

A linear map sends $0$ to $0$ and linear combinations to linear combinations: $T(\sum a_i v_i) = \sum a_i T(v_i)$.
For an $m \times n$ matrix $A$, the map $F^n \to F^m$, $x \mapsto Ax$, is linear.
Rotations of $\R^2$ about the origin are linear; translations by a nonzero vector are not linear.
Each linear map between finite-dimensional spaces has a [[matrix|0YQSGNYE3K]] in a choice of bases.

## Explanations

### L1 human

A linear map respects the two operations of a vector space.
The image of a sum is the sum of the images, and the image of a scaled vector is the scaled image.
In the plane, linear maps send lines through the origin to lines through the origin, or to the origin, and keep parallel lines parallel.
</document>

<document source="https://tamnd.github.io/math2/node/0RJMNEN3BJ.md" tag="0RJMNEN3BJ" kind="definition">
# Symmetric group

## Definition

Let $X$ be a set.
The symmetric group $\operatorname{Sym}(X)$ is the set of all bijections $\sigma \colon X \to X$, with composition of maps as the operation.
For a positive integer $n$, the symmetric group $S_n$ is $\operatorname{Sym}(\{1, 2, \dots, n\})$.
An element of $S_n$ is a permutation of degree $n$.

## Remarks

The identity map is the identity element, and the inverse of $\sigma$ is the inverse map $\sigma^{-1}$.
In this node, the product $\sigma \tau$ means first $\tau$, then $\sigma$.
The group $S_n$ is not [[abelian|082GHNPJ53]] for $n \geq 3$, because $(1\ 2)(2\ 3) \neq (2\ 3)(1\ 2)$.
If $X$ and $Y$ have the same cardinality, then $\operatorname{Sym}(X)$ and $\operatorname{Sym}(Y)$ are [[isomorphic|08MG7H8JW5]].

## Explanations

### L1 human

The symmetric group collects all the ways to shuffle the elements of a set.
Two shuffles combine into one shuffle when you do them one after the other.
Each shuffle can be undone, so the shuffles form a group.
</document>

<document source="https://tamnd.github.io/math2/node/0RJSBR9GW1.md" tag="0RJSBR9GW1" kind="concept">
# Totally disconnected space

## Summary

A totally disconnected space is a topological space whose only connected subsets are the empty set and the single points.
</document>

<document source="https://tamnd.github.io/math2/node/0RJZV05MJA.md" tag="0RJZV05MJA" kind="definition">
# Polar coordinates

## Definition

The polar coordinate map is $P \colon [0, \infty) \times \mathbb{R} \to \mathbb{R}^2$, $P(r, \theta) = (r \cos \theta, r \sin \theta)$.
A pair $(r, \theta)$ with $P(r, \theta) = (x, y)$ is a pair of polar coordinates of the point $(x, y)$. Here $r = \sqrt{x^2 + y^2}$ is the radial coordinate and $\theta$ is the angular coordinate.

## Remarks

Each point other than the origin has exactly one pair of polar coordinates with $r > 0$ and $\theta \in [0, 2\pi)$. The origin has $r = 0$ and any $\theta$.
For $x > 0$, $\theta = \arctan(y / x)$ is one valid angle. In general one uses the two-argument function $\operatorname{atan2}(y, x)$, with values in $(-\pi, \pi]$.
The restriction of $P$ to $(0, \infty) \times (0, 2\pi)$ is a $C^\infty$ [[diffeomorphism|0Z1Y7JHGBR]] onto the plane without the half-line $\{(x, 0) : x \ge 0\}$. Its Jacobian determinant is $r$.
The circle of radius $a$ about the origin is $r = a$, and the ray at angle $\alpha$ is $\theta = \alpha$.

## Explanations

### L1 human

Instead of moving $x$ units right and $y$ units up, you turn by the angle $\theta$ and then walk the distance $r$.
Both descriptions name the same point.
</document>

<document source="https://tamnd.github.io/math2/node/0RKB5MKC05.md" tag="0RKB5MKC05" kind="definition">
# Circumcircle

## Definition

Let $ABC$ be a triangle in the Euclidean plane $\mathbb{R}^2$, with [[circumcentre|0F98FZK14A]] $O$ and circumradius $R = d(O, A)$, where $d$ is the Euclidean distance.
The circumcircle of $ABC$ is the [[circle|060W5CBZP6]] with the centre $O$ and the radius $R$:
$$\{\, X \in \mathbb{R}^2 \;:\; d(X, O) = R \,\}.$$
It is the unique circle that contains $A$, $B$ and $C$.

## Remarks

For a triangle in $\mathbb{R}^n$, the circumcircle is the circle in the plane of the triangle with the centre $O$ and the radius $R$. Mathlib's `Affine.Simplex.circumsphere` is the sphere with this centre and radius in the ambient space.
The [[law of sines|0PASQ34GHG]] gives $2R = a / \sin \alpha$, with the side $a$ opposite the angle $\alpha$.
A polygon whose vertices all lie on one circle is a cyclic polygon. Each triangle is cyclic, but not each quadrilateral is.

## Explanations

### L1 human

Through any three points that are not on one line, there is exactly one circle. For a triangle, this is the circumcircle. For the right triangle with the vertices $(0, 0)$, $(4, 0)$ and $(0, 3)$, the circumcircle has the centre $(2, 1.5)$ and the radius $2.5$.
</document>

<document source="https://tamnd.github.io/math2/node/0RKF1ZT5JJ.md" tag="0RKF1ZT5JJ" kind="concept">
# Complement

## Summary

The complement of a set is the set of all objects of the universe that are not elements of the set.
</document>

<document source="https://tamnd.github.io/math2/node/0RKG5YW66Y.md" tag="0RKG5YW66Y" kind="concept">
# LU decomposition

## Summary

An *LU decomposition* of a square matrix $A$ is a factorization $A = LU$.
Here $L$ is a [[lower triangular matrix|00J3KZP8Y0]] and $U$ is an [[upper triangular matrix|0N4XY7Z3NN]].
[[Gaussian elimination|0JVJ2WCXZH]] without row exchanges produces such a factorization: $U$ is the row echelon form, and $L$ stores the multipliers of the row operations.
With row exchanges, one gets $PA = LU$ for a [[permutation matrix|08SP3DF8NB]] $P$, and each square matrix over a field has such a factorization.

## Remarks

An invertible matrix has an LU decomposition with $L$ unit lower triangular if and only if all its leading principal minors are nonzero.
This decomposition is then unique.
The matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ has no LU decomposition, but it has one after a row exchange.
To solve $Ax = b$, solve $Ly = b$ by forward substitution and $Ux = y$ by [[back substitution|0ATFVE08MW]].
The decomposition also gives $\det A = \det L \det U$.

## Explanations

### L1 human

An LU decomposition writes a matrix as a product of two triangular matrices.
Triangular systems are easy to solve, so we solve $Ax = b$ in two easy steps.
It is useful when we solve many systems with the same $A$ and different right sides.
</document>

<document source="https://tamnd.github.io/math2/node/0RKQQQACY7.md" tag="0RKQQQACY7" kind="definition">
# Slope field

## Definition

Let $\Omega$ be an open subset of $\mathbb{R}^2$ and let $f : \Omega \to \mathbb{R}$ be a function.
The slope field of the first-order equation $y' = f(x, y)$ is the map that assigns to each point $(x, y) \in \Omega$ the line through $(x, y)$ with slope $f(x, y)$.
In a picture, one draws a short segment of this line at each point of a grid in $\Omega$.

## Remarks

Let $u$ be a differentiable function on an interval. It is a [[solution|04SW7WECS8]] of $y' = f(x, y)$ if and only if its graph is tangent to the slope field at each of its points.
For $y' = y$, the segments are horizontal on the $x$-axis and get steeper as $|y|$ grows.
For a system $x' = F(x)$ in the plane, the same picture with arrows is a picture of the vector field $F$.

## Explanations

### L1 human

A slope field shows the direction that a solution must take at each point.
You can sketch a solution without a formula: start at a point and follow the small segments.
This gives a quick picture of how all the solutions behave.
</document>

<document source="https://tamnd.github.io/math2/node/0RKW2X63YR.md" tag="0RKW2X63YR" kind="statement">
# Logarithm is the inverse of the exponential

## Statement

For every real $x > 0$, $e^{\ln x} = x$. For every real $y$, $\ln(e^y) = y$.
The function $\ln$ is continuous and strictly increasing on $(0, \infty)$, and its range is $\mathbb{R}$.

## Remarks

The two identities say that $\ln$ and $\exp$ are inverse bijections between $(0, \infty)$ and $\mathbb{R}$.
Continuity follows from the [[continuity of inverse functions|07BY47HG5S]], since $\exp$ is [[strictly increasing|0KN2V7840E]] and continuous.
Also $\ln x \to \infty$ as $x \to \infty$, and $\ln x \to -\infty$ as $x \to 0^+$. Mathlib states the first identity as `Real.exp_log`.

## Explanations

### L1 human

Taking $\ln$ undoes taking $\exp$, and the other way round.
The graph of $\ln$ is the mirror image of the graph of $e^x$ in the line $y = x$.
</document>

<document source="https://tamnd.github.io/math2/node/0RMKABS2S5.md" tag="0RMKABS2S5" kind="definition">
# Sum of subspaces

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be [[linear subspaces|00YC1D652K]] of $V$.
The *sum* of $U$ and $W$ is
$$U + W = \{u + w : u \in U, \ w \in W\}.$$
For subspaces $U_1, \dots, U_k$, the sum is $U_1 + \cdots + U_k = \{u_1 + \cdots + u_k : u_i \in U_i\}$.

## Remarks

The sum $U + W$ is a subspace, and it is the smallest subspace of $V$ that contains $U \cup W$.
The sum is the [[span|0DTY8B9HMS]] of $U \cup W$.
When $U \cap W = \{0\}$, the sum is a [[direct sum|0XYYT5KNTX]].
For finite-dimensional subspaces, [[Grassmann's formula|06HTJNCMPA]] gives $\dim(U + W) = \dim U + \dim W - \dim(U \cap W)$.

## Explanations

### L1 human

The sum of two subspaces collects every vector that we can build from one piece in each subspace.
In space, the sum of two different lines through the origin is the plane that contains both lines.
It is the smallest subspace that holds both subspaces.
</document>

<document source="https://tamnd.github.io/math2/node/0RMVS7RXZX.md" tag="0RMVS7RXZX" kind="definition">
# Exponential function

## Definition

The exponential function $\exp \colon \mathbb{R} \to \mathbb{R}$ is defined by
$$\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots.$$

## Remarks

The [[power series|0VP56B4EMX]] converges absolutely for every real $x$, because [[its radius of convergence is infinite|0PVJRACB9H]].
One also writes $e^x$ for $\exp(x)$, where $e = \exp(1)$ is the number [[e|0MVCFMW7S4]].
Other common definitions use the limit $\lim_{n \to \infty} (1 + x/n)^n$, or the equation $f' = f$ with $f(0) = 1$. They give the same function.
Mathlib defines `Real.exp` as the real part of the complex exponential series.

## Explanations

### L1 human

The exponential function is an infinite polynomial with the coefficients $1/n!$.
It grows faster than every polynomial, and its derivative is itself.
</document>

<document source="https://tamnd.github.io/math2/node/0RN00NTM3S.md" tag="0RN00NTM3S" kind="concept">
# Inner automorphism

## Summary

An inner automorphism of a group is the automorphism given by conjugation by a fixed element.
The inner automorphisms form a normal subgroup of the automorphism group.
</document>

<document source="https://tamnd.github.io/math2/node/0RN765S7DH.md" tag="0RN765S7DH" kind="concept">
# Proper metric space

## Summary

A proper metric space is a metric space in which each closed ball is compact.
In such a space, a set is compact if and only if it is closed and bounded.
</document>

<document source="https://tamnd.github.io/math2/node/0RN88KZBJ3.md" tag="0RN88KZBJ3" kind="concept">
# Homogeneous first-order equation

## Summary

A homogeneous first-order equation is an equation $y' = F(y/x)$, in which the slope depends only on the ratio $y/x$.
The substitution $y = x v$ changes it into a separable equation. See the [[definition|0K4ZWTFSV5]].
</document>

<document source="https://tamnd.github.io/math2/node/0RNGDMHT7M.md" tag="0RNGDMHT7M" kind="concept">
# Complex power

## Summary

A complex power $z^w$ with $z \neq 0$ is a number $e^{w \ell}$ where $\ell$ is a logarithm of $z$, see the [[definition|0ZV9A7EV0F]].
The principal value is $e^{w \operatorname{Log} z}$.
</document>

<document source="https://tamnd.github.io/math2/node/0RNMP6VCR2.md" tag="0RNMP6VCR2" kind="definition">
# Complement graph

## Definition

Let $G = (V, E)$ be a [[graph|0JD5C82QMB]].
The complement of $G$ is the graph $\overline{G}$ with vertex set $V$.
Two distinct vertices are adjacent in $\overline{G}$ if and only if they are not adjacent in $G$.

## Remarks

The complement of the complement of $G$ is $G$.
The complement of the complete graph $K_n$ is the graph on $n$ vertices with no edges.
A [[clique|0G52BPNPKG]] of $G$ is an [[independent set|0KZJ82WEPP]] of $\overline{G}$. This links Ramsey numbers to colourings of $K_n$.

## Explanations

### L1 human

To get the complement, remove all edges of the graph and draw all the missing ones.
If $G$ records which people know each other, then $\overline{G}$ records which people are strangers.
</document>

<document source="https://tamnd.github.io/math2/node/0RNXVVTJTJ.md" tag="0RNXVVTJTJ" kind="example">
# Damped harmonic oscillator

## Example

Let $m > 0$, $k > 0$ and $c > 0$ be constants. The damped harmonic oscillator is the equation
$$m\, x'' + c\, x' + k\, x = 0, \quad \text{that is} \quad x'' + 2 \zeta \omega_0\, x' + \omega_0^2\, x = 0,$$
with $\omega_0 = \sqrt{k/m}$ and damping ratio $\zeta = c / \left(2 \sqrt{m k}\right)$.
The characteristic roots are $r = \omega_0 \left(-\zeta \pm \sqrt{\zeta^2 - 1}\right)$, and there are three cases.
If $\zeta > 1$ (overdamped), the roots are real, distinct and negative, and $x = c_1 e^{r_1 t} + c_2 e^{r_2 t}$.
If $\zeta = 1$ (critically damped), $r = -\omega_0$ is a double root, and $x = (c_1 + c_2 t) e^{-\omega_0 t}$.
If $0 < \zeta < 1$ ([[underdamped|0JJ6GDDE7Z]]), $x = e^{-\zeta \omega_0 t} \left(c_1 \cos \omega_1 t + c_2 \sin \omega_1 t\right)$ with $\omega_1 = \omega_0 \sqrt{1 - \zeta^2}$.

## Remarks

In each case, each solution tends to $0$ as $t \to \infty$. The origin is an asymptotically stable [[equilibrium point|0N8NGKDZDZ]] of the system for $(x, x')$.
The term $c x'$ models a friction force proportional to the velocity.
The energy $E = \frac{1}{2} m (x')^2 + \frac{1}{2} k x^2$ satisfies $E' = -c (x')^2 \le 0$. So the energy decreases along each solution.

## Explanations

### L1 human

Friction takes energy from the oscillating mass.
With little friction, the mass still oscillates, but the swings get smaller.
With much friction, the mass creeps back to rest without any swing.
</document>

<document source="https://tamnd.github.io/math2/node/0RP20HFYM3.md" tag="0RP20HFYM3" kind="concept">
# Positive definite quadratic form

## Summary

A positive definite quadratic form is a real quadratic form that takes a positive value at every nonzero vector.
</document>

<document source="https://tamnd.github.io/math2/node/0RP7VWQ0YN.md" tag="0RP7VWQ0YN" kind="concept">
# Subgroup generated by a subset

## Summary

The subgroup generated by a subset of a group is the smallest subgroup that contains the subset.
Its elements are the finite products of elements of the subset and their inverses.
</document>

<document source="https://tamnd.github.io/math2/node/0RPDTWY2JS.md" tag="0RPDTWY2JS" kind="definition">
# Trace of a linear map

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The *trace* of $T$ is
$$\tr T = \tr [T]_B,$$
the [[trace|0XC1RDBY7A]] of the matrix of $T$ in a basis $B$ of $V$.

## Remarks

The value does not depend on $B$: by the [[change of basis formula|0MQXVZEBMT]], another basis gives $P^{-1} [T]_B P$, and $\tr(P^{-1} A P) = \tr A$.
The trace is a linear map $\End_F(V) \to F$, and $\tr(ST) = \tr(TS)$.
The trace of the identity map of $V$ is $\dim V \cdot 1$.
A [[projection|033DA1WE8Y]] on a real or complex space has trace equal to its rank.

## Explanations

### L1 human

We define the trace of a linear map by its matrix in some basis.
Another basis gives a different matrix, but the trace stays the same.
So the trace belongs to the map, not to the basis.
</document>

<document source="https://tamnd.github.io/math2/node/0RQ8DHHR8V.md" tag="0RQ8DHHR8V" kind="statement">
# Cantor set is homeomorphic to the space of binary sequences

## Statement

Let $\{0, 1\}$ have the [[discrete topology|0S2TEHX4E1]] and let $\{0, 1\}^{\mathbb{N}}$ have the [[product topology|04S9HX3GAP]].
Then the map
$$\varphi \colon \{0, 1\}^{\mathbb{N}} \to C, \qquad \varphi((a_n)_{n \ge 1}) = \sum_{n \ge 1} \frac{2 a_n}{3^n}$$
is a [[homeomorphism|05GZ11PWRY]] onto the [[Cantor set|0BXNDZFQME]] $C$.

## Remarks

The map is a bijection, because each point of $C$ has exactly one base-3 expansion with digits in $\{0, 2\}$.
It is continuous: if two sequences agree in their first $N$ terms, their images differ by at most $3^{-N}$.
The space $\{0, 1\}^{\mathbb{N}}$ is compact by [[Tychonoff's theorem|0JTRSJKCFB]], and $C$ is Hausdorff. Thus $\varphi$ is a homeomorphism, by [[this theorem|0430ETCMKD]].
A space homeomorphic to $C$ is a Cantor space. By a theorem of Brouwer, these are exactly the nonempty compact metrizable spaces that are totally disconnected and have no isolated points.

## Explanations

### L1 human

Each point of the Cantor set is described by an infinite list of choices: left or right third, at each step.
These lists are the binary sequences, and the topology of the Cantor set matches the product topology on them.
</document>

<document source="https://tamnd.github.io/math2/node/0RQPZZ44T4.md" tag="0RQPZZ44T4" kind="definition">
# Identity element

## Definition

Let $S$ be a set and let $\ast$ be a [[binary operation|0M418FE7BB]] on $S$.
An element $e \in S$ is a left identity when $e \ast a = a$ for all $a \in S$.
An element $e \in S$ is a right identity when $a \ast e = a$ for all $a \in S$.
An element $e \in S$ is an identity element (or two-sided identity) when it is both a left identity and a right identity.

## Remarks

The number $0$ is the identity element for addition on $\mathbb{Z}$, and $1$ is the identity element for multiplication.
If $e$ is a left identity and $f$ is a right identity, then $e = e \ast f = f$.
Thus an operation has at most one two-sided identity element.

## Explanations

### L1 human

An identity element is an element that does nothing when you combine it with another element.
Adding $0$ to a number gives the same number, and multiplying a number by $1$ gives the same number.
Some operations have no identity element, for example addition on the positive integers.
</document>

<document source="https://tamnd.github.io/math2/node/0RQSNVTFVZ.md" tag="0RQSNVTFVZ" kind="concept">
# Prime-counting function

## Summary

The prime-counting function $\pi(x)$ gives the number of primes that are less than or equal to a real number $x$.
The [[definition of the prime-counting function|0XRJB5SSNE]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0RRNVX1VKA.md" tag="0RRNVX1VKA" kind="statement">
# Wedderburn's little theorem

## Statement

Let $D$ be a [[division ring|0WZXK05ME7]] with a finite number of elements.
Then the multiplication of $D$ is commutative, so $D$ is a [[field|0QA8CHH8MG]].

## Remarks

The hypothesis of finiteness is necessary: the [[quaternions|0GE4DVQKR8]] are an infinite division ring that is not commutative.
A consequence is that every finite domain, commutative or not, is a field.

## History

Joseph Wedderburn gave the first proof in 1905.
Leonard Eugene Dickson gave another proof soon after.
The first proof of Wedderburn had a gap, and for this reason some historians credit Dickson with the first correct proof.

## Explanations

### L1 human

In an infinite division ring, $a b$ and $b a$ can be different.
The theorem says that this cannot happen when the ring is finite.
So there are no finite analogues of the quaternions.
</document>

<document source="https://tamnd.github.io/math2/node/0RRWNMYRXC.md" tag="0RRWNMYRXC" kind="concept">
# Fixed points of an action

## Summary

A fixed point of a group action is a point that each element of the group keeps in place.
The fixed points are exactly the points whose orbit has one element.
</document>

<document source="https://tamnd.github.io/math2/node/0RT96J4D3H.md" tag="0RT96J4D3H" kind="definition">
# Linear form

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A *linear form* on $V$ is a [[linear map|0NRYHJPQG2]] $f \colon V \to F$, where $F$ is a vector space over itself.
Thus $f(u + v) = f(u) + f(v)$ and $f(av) = a f(v)$ for all $u, v \in V$ and $a \in F$.

## Remarks

A linear form is also called a *linear functional* or a *covector*.
On $F^n$, each linear form is $x \mapsto c_1 x_1 + \cdots + c_n x_n$ for unique scalars $c_1, \dots, c_n$.
The [[trace|0XC1RDBY7A]] is a linear form on the space of $n \times n$ matrices.
The linear forms on $V$ form the [[dual space|07JBSZE0V4]] of $V$.

## Explanations

### L1 human

A linear form takes a vector and returns one number, in a linear way.
On the plane, the map $(x, y) \mapsto 2x - 3y$ is a linear form.
Its kernel is the line $2x = 3y$, unless the form is zero.
</document>

<document source="https://tamnd.github.io/math2/node/0RV3AEXPT6.md" tag="0RV3AEXPT6" kind="definition">
# Focus of a conic

## Definition

Let $Q$ be a non-degenerate conic section in the Euclidean plane $\mathbb{R}^2$.
Suppose that $Q$ is an [[ellipse|0VKCCEXBJ1]] or a [[hyperbola|0WN5G117YC]] with the foci $F_1$ and $F_2$, as in their definitions. Then $F_1$ and $F_2$ are the foci of $Q$. If $Q$ is a [[circle|060W5CBZP6]], its centre is its only focus.
If $Q$ is a [[parabola|04S2GMHX82]] with the focus $F$ and a directrix, as in its definition, then $F$ is the only focus of $Q$.

## Remarks

The foci are determined by the conic. For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b > 0$, the foci are $(\pm c, 0)$ with $c^2 = a^2 - b^2$.
For the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the foci are $(\pm c, 0)$ with $c^2 = a^2 + b^2$.
Let $Q$ be a conic that is not a circle, and let $F$ be a focus of $Q$. Then $Q$ is the set of the points whose distances to $F$ and to a line, the [[directrix|0E5CXYNEPX]], have a constant ratio $e > 0$.

## Explanations

### L1 human

The foci are the two pins in the string construction of an ellipse. A parabolic mirror sends the light of a lamp at its focus out in parallel rays. For the parabola $y = x^2/4$, the focus is $(0, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/0RVD9R2V99.md" tag="0RVD9R2V99" kind="concept">
# Cauchy product

## Summary

The Cauchy product of two series is the series whose $n$-th term is the sum of the products $a_k b_{n-k}$. It is the discrete convolution of the two sequences of terms. The node [[Cauchy product|0B1K9PVWE9]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0RVRNR7KA1.md" tag="0RVRNR7KA1" kind="concept">
# Gaussian prime

## Summary

A Gaussian prime is a prime element of the ring of Gaussian integers.
The [[definition of a Gaussian prime|0GTEHJQRFK]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0RVYE75E2M.md" tag="0RVYE75E2M" kind="definition">
# Converse

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The converse of the [[implication|0ZNX6DQ3EJ]] $p \to q$ is the implication $q \to p$.

## Remarks

An implication and its converse are not [[logically equivalent|06GYGS41JK]].
For example, "if $n = 2$, then $n$ is prime" is true, but its converse "if $n$ is prime, then $n = 2$" is false.
The inverse of $p \to q$ is $\neg p \to \neg q$. It is the [[contrapositive|04T1CT7212]] of the converse.

## Explanations

### L1 human

The converse reads the arrow backwards.
"If an animal is a dog, then it is a mammal" is true.
Its converse, "if an animal is a mammal, then it is a dog", is false.
</document>

<document source="https://tamnd.github.io/math2/node/0RW8KPMXF6.md" tag="0RW8KPMXF6" kind="definition">
# Spherical coordinates

## Definition

The spherical coordinate map is $S \colon [0, \infty) \times [0, \pi] \times \mathbb{R} \to \mathbb{R}^3$,
$$S(\rho, \varphi, \theta) = (\rho \sin \varphi \cos \theta, \ \rho \sin \varphi \sin \theta, \ \rho \cos \varphi).$$
A triple $(\rho, \varphi, \theta)$ with $S(\rho, \varphi, \theta) = (x, y, z)$ is a triple of spherical coordinates of $(x, y, z)$.
Here $\rho = \sqrt{x^2 + y^2 + z^2}$ is the distance to the origin, $\varphi$ is the polar angle from the positive $z$-axis, and $\theta$ is the azimuth.

## Remarks

The pair $(\rho \sin \varphi, \theta)$ is a pair of [[polar coordinates|0RJZV05MJA]] of $(x, y)$, and $\cos \varphi = z / \rho$ for $\rho > 0$.
The coordinates are unique for points off the $z$-axis if $\rho > 0$, $0 < \varphi < \pi$ and $\theta \in [0, 2\pi)$.
The sphere of radius $a$ is $\rho = a$, and $\varphi = \pi / 4$ is a cone around the positive $z$-axis.
This is a common convention in mathematics. Physics texts often write $(r, \theta, \varphi)$, with $\theta$ the polar angle and $\varphi$ the azimuth.

## Explanations

### L1 human

To locate a point in space, give its distance from the origin, how far it tilts away from the north pole, and its longitude.
This is like latitude and longitude on the Earth, with the polar angle measured from the north pole.
</document>

<document source="https://tamnd.github.io/math2/node/0RW9TNNT87.md" tag="0RW9TNNT87" kind="concept">
# Orthonormal family

## Summary

An orthonormal family is a family of unit vectors that are pairwise orthogonal.
</document>

<document source="https://tamnd.github.io/math2/node/0RWE7CPJ32.md" tag="0RWE7CPJ32" kind="concept">
# Probability theory

## Summary

Probability theory is the branch of mathematics that studies chance in a precise way.
It models a random experiment by a [[probability space|0ZYBZ34RT2]]: a sample space, a family of events and a probability measure that obeys the [[probability axioms|0DAX8PMAJ4]].
Its main objects are events, random variables, their distributions and their expected values.
Its main results include limit theorems, such as the law of large numbers and the central limit theorem.

## Remarks

The modern axioms are due to Andrey Kolmogorov, who based them on measure theory in 1933.
Statistics uses probability theory to draw conclusions from data.

## Explanations

### L1 human

Probability theory gives numbers to uncertain things, such as the chance of rain or the chance of a six on a die.
The rules for these numbers are few and simple, but they give strong results.
For example, the average of many fair coin tosses is close to one half.
</document>

<document source="https://tamnd.github.io/math2/node/0RWKYX8339.md" tag="0RWKYX8339" kind="concept">
# Cartesian coordinate system

## Summary

A Cartesian coordinate system in the [[Euclidean plane|0XW7V9ZZ05]] is a choice of a unit of length and of two perpendicular directed lines, the axes. The axes meet at a point $O$, the origin.
Each point $P$ then has a unique pair $(x, y)$ of real numbers, its coordinates. The numbers $x$ and $y$ are the signed distances from the origin to the perpendicular projections of $P$ on the two axes.
In a [[Euclidean space|0XPR1E85W0]] of dimension $n$, a Cartesian coordinate system has $n$ mutually perpendicular axes through one origin, and each point has $n$ coordinates.
The map from a point to its coordinates is a bijection onto $\mathbb{R}^n$ that keeps distances.

## Remarks

In three dimensions, the usual names of the coordinates are $x$, $y$ and $z$.
The two axes of the plane divide it into four quadrants. The three coordinate planes of space divide it into eight octants.
Descartes and Fermat developed the idea in the 17th century.

## Explanations

### L1 human

A Cartesian coordinate system is like the grid of a map. You choose a starting point and two perpendicular directions. Then you describe each point by how far you go along each direction. For example, "3 to the right and 2 up" is the point $(3, 2)$.
</document>

<document source="https://tamnd.github.io/math2/node/0RWMRTPC87.md" tag="0RWMRTPC87" kind="statement">
# Jensen's inequality

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be [[convex|0VV0HSW1ZS]].
Let $x_1, \dots, x_n \in I$, and let $w_1, \dots, w_n \ge 0$ with $w_1 + \cdots + w_n = 1$.
Then
$$f\Big( \sum_{i=1}^{n} w_i x_i \Big) \le \sum_{i=1}^{n} w_i f(x_i).$$

## Remarks

The case $n = 2$ is the definition of convexity. Induction on $n$ gives the general case.
For a concave function, the inequality is reversed. With $f = \ln$ this gives the [[AM–GM inequality|0DRF0F3XC9]].
Mathlib states it as `ConvexOn.map_sum_le`.

## History

Johan Jensen proved the inequality in 1906.

## Explanations

### L1 human

The value of a convex function at an average is at most the average of its values.
The weights $w_i$ say how much each point counts in the average.
</document>

<document source="https://tamnd.github.io/math2/node/0RWQ5VZ6VQ.md" tag="0RWQ5VZ6VQ" kind="statement">
# Every infinite set has a countably infinite subset

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Let $A$ be an [[infinite set|01MKW4R283]].
Then there is an [[injective function|0PVG1CRAJN]] $f \colon \mathbb{N} \to A$, so $A$ has a [[countably infinite|0SEZ3JGZP7]] subset $f(\mathbb{N})$.

## Remarks

The proof chooses $a_0 \in A$, then $a_1 \in A \setminus \{a_0\}$, and so on. Each set $A \setminus \{a_0, \dots, a_n\}$ is nonempty because $A$ is infinite.
The axiom of countable choice is enough, but some choice is needed: in ZF alone, the statement cannot be proved.
It follows that $\aleph_0$ is the least infinite cardinal.

## Explanations

### L1 human

In an infinite set you can always pick one more element.
Picking forever gives a sequence of different elements, which is a copy of $\mathbb{N}$ inside the set.
</document>

<document source="https://tamnd.github.io/math2/node/0RX7DDBYCK.md" tag="0RX7DDBYCK" kind="statement">
# Image of an intersection

## Statement

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]] and let $S$ and $T$ be [[subsets|0FE8Q5S42Y]] of $A$.
Then the [[image|0Q3F20XB0A]] of $S \cap T$ is a subset of the intersection of the images:
$$f(S \cap T) \subseteq f(S) \cap f(T).$$

## Remarks

The inclusion can be strict. For the constant function $f \colon \{1, 2\} \to \{0\}$, $S = \{1\}$ and $T = \{2\}$, the left side is $\varnothing$ and the right side is $\{0\}$.
Equality holds for all $S$ and $T$ if and only if $f$ is [[injective|0PVG1CRAJN]].
For unions, equality always holds: $f(S \cup T) = f(S) \cup f(T)$.

## Explanations

### L1 human

If $b = f(x)$ with $x$ in both $S$ and $T$, then $b$ is in both images.
But $b$ can be in both images for another reason: $b = f(s) = f(t)$ with $s \in S$, $t \in T$ and $s \neq t$.
</document>

<document source="https://tamnd.github.io/math2/node/0RXD6EZYBK.md" tag="0RXD6EZYBK" kind="statement">
# Distributivity of matrix multiplication

## Statement

Let $R$ be a ring.
Let $A$ be an $m \times n$ matrix over $R$, and let $B$ and $C$ be $n \times p$ matrices over $R$.
Then $A(B + C) = AB + AC$.
Let $D$ and $E$ be $m \times n$ matrices over $R$, and let $G$ be an $n \times p$ matrix over $R$.
Then $(D + E)G = DG + EG$.

## Remarks

Both identities follow entry by entry from the distributive law in $R$.
We need both identities, because [[matrix multiplication|032Q9FB8MT]] is not commutative.
Together with [[associativity|073E4XC016]], they make the square matrices a [[ring|0FHVPRAE60]].

## Explanations

### L1 human

Multiplication of matrices spreads over addition, as for numbers.
For example, $A(B + C)$ is the same as $AB + AC$.
We must keep the order: $A$ stays on the left in both terms.
</document>

<document source="https://tamnd.github.io/math2/node/0RXK2Z808Q.md" tag="0RXK2Z808Q" kind="counterexample">
# A continuous nowhere differentiable function

## Counterexample

Let $\varphi \colon \mathbb{R} \to \mathbb{R}$ be the zigzag function with $\varphi(x) = |x|$ for $-1 \le x \le 1$ and $\varphi(x + 2) = \varphi(x)$ for all $x$.
Define
$$f(x) = \sum_{n=0}^{\infty} \Big( \frac{3}{4} \Big)^n \varphi(4^n x).$$
The function $f$ is continuous on $\mathbb{R}$, but it is not [[differentiable|0A61B60B5R]] at any point.

## Remarks

Since $0 \le \varphi \le 1$, the [[M-test|0NX5NWAFC9]] with $M_n = (3/4)^n$ and the [[continuity of uniform sums|0R0MYT0BJD]] give continuity.
Fix $x$. For each $m$, choose $\delta_m = \pm \frac{1}{2} 4^{-m}$ so that no integer lies strictly between $4^m x$ and $4^m (x + \delta_m)$.
Then $\Big| \frac{f(x + \delta_m) - f(x)}{\delta_m} \Big| \ge \frac{3^m + 1}{2}$. This tends to infinity while $\delta_m \to 0$, so $f'(x)$ does not exist.
This function replaces the cosine in the [[Weierstrass function|01A0MK1730]] by a piecewise linear zigzag.

## Explanations

### L1 human

Each term adds smaller and steeper zigzags to the graph.
The sum is continuous, but its graph has a corner at every scale near every point.
</document>

<document source="https://tamnd.github.io/math2/node/0RXKW044R4.md" tag="0RXKW044R4" kind="concept">
# Completely multiplicative function

## Summary

A completely multiplicative function is an arithmetic function with $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n$.
The [[definition of a completely multiplicative function|0GS27TKF1V]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0RXS8R6K37.md" tag="0RXS8R6K37" kind="definition">
# Preorder

## Definition

Let $A$ be a set.
A preorder on $A$ is a [[binary relation|08EJC6Q25H]] $\le$ on $A$ that is [[reflexive|0XEGNW851S]] and [[transitive|0CYRECG2YQ]]. For all $a, b, c \in A$:

1. $a \le a$;
2. $a \le b$ and $b \le c$ imply $a \le c$.

## Remarks

Divisibility on $\mathbb{Z}$ is a preorder. It is not antisymmetric, because $2 \mid -2$ and $-2 \mid 2$.
An antisymmetric preorder is a [[partial order|011C29Q1S1]].
For a preorder, $a \le b$ and $b \le a$ together define an [[equivalence relation|0SGTMBEWRF]]. The preorder gives a partial order on the [[quotient set|0ZVHXDSZC7]].

## Explanations

### L1 human

Compare people by height with "is at most as tall as".
This is a preorder: two different people can have the same height.
</document>

<document source="https://tamnd.github.io/math2/node/0RXX2PT1MQ.md" tag="0RXX2PT1MQ" kind="statement">
# Chain rule

## Statement

Let $f$ be a real function on an open interval $I$ that is differentiable at $a \in I$.
Let $g$ be a real function on an open interval $J$ with $f(I) \subseteq J$, and assume that $g$ is differentiable at $f(a)$.
Then $g \circ f$ is differentiable at $a$, and
$$(g \circ f)'(a) = g'(f(a)) \, f'(a).$$

## Remarks

A naive proof multiplies and divides by $f(a + h) - f(a)$. This fails when $f(a + h) = f(a)$ for small $h$. A correct proof uses the linear approximation of $g$ at $f(a)$.
For example, $(\sin(x^2))' = 2 x \cos(x^2)$.
In Leibniz notation, $\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx}$. Mathlib states the rule as `HasDerivAt.comp`.

## History

Gottfried Wilhelm Leibniz seems to have used the rule first, in a memoir of 1676.

## Explanations

### L1 human

Suppose $y$ changes $f'(a)$ times as fast as $x$, and $z$ changes $g'(y)$ times as fast as $y$.
Then $z$ changes $g'(y) f'(a)$ times as fast as $x$.
The rates multiply.
</document>

<document source="https://tamnd.github.io/math2/node/0RY0B72XAS.md" tag="0RY0B72XAS" kind="statement">
# Structure theorem for finitely generated abelian groups

## Statement

Let $G$ be a [[finitely generated|082DWWXMRJ]] [[abelian group|082GHNPJ53]].
Then $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z}$ for an integer $n \geq 0$ and prime powers $q_1, \ldots, q_t$.
Also $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z}$ with $1 < k_1 \mid k_2 \mid \cdots \mid k_u$.
The rank $n$, the multiset of the $q_i$ and the sequence of the $k_i$ are unique.

## Remarks

The finite part is the [[torsion subgroup|0DBWXQRXQJ]] of $G$, and $\mathbb{Z}^n$ is a [[free abelian group|0WJX8MGG37]] of rank $n$.
So $G$ is finite if and only if $n = 0$, and then the theorem is the [[classification of finite abelian groups|07CQR9BW3R]].
Each finitely generated torsion-free abelian group is free abelian. The group $(\mathbb{Q}, +)$ shows that the condition "finitely generated" is necessary.
The theorem is a special case of the structure theorem for finitely generated modules over a principal ideal domain.

## History

Henry John Stephen Smith solved the case of finitely presented abelian groups in 1861, with the Smith normal form of integer matrices.
Henri Poincaré proved the theorem for finitely generated abelian groups in 1900, with a matrix proof.
Emmy Noether extended Kronecker's proof to finitely generated abelian groups in 1926.

## Explanations

### L1 human

A finitely generated abelian group is a free part $\mathbb{Z}^n$ plus a finite part.
The finite part is a sum of cyclic groups.
So a short list of integers describes the group completely.
</document>

<document source="https://tamnd.github.io/math2/node/0RY4GGMEQG.md" tag="0RY4GGMEQG" kind="statement">
# Lucas's theorem

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]], and let $m$ and $n$ be natural numbers with base-$p$ digits $m = \sum_{i=0}^{k} m_i p^i$ and $n = \sum_{i=0}^{k} n_i p^i$, where $0 \le m_i, n_i < p$.
Then
$$\binom{m}{n} \equiv \prod_{i=0}^{k} \binom{m_i}{n_i} \pmod p,$$
with the convention $\binom{a}{b} = 0$ when $b > a$.

## Remarks

A consequence: $p \mid \binom{m}{n}$ if and only if $n_i > m_i$ for some digit $i$.
For example, with $p = 3$: $10 = 101_3$ and $4 = 011_3$. The theorem gives $\binom{10}{4} \equiv \binom{1}{0} \binom{0}{1} \binom{1}{1} = 0 \pmod 3$, and indeed $\binom{10}{4} = 210$.
For $p = 2$, the binomial coefficient $\binom{m}{n}$ is odd if and only if $m$ has a binary digit $1$ at each place where $n$ has one.

## History

Édouard Lucas published the theorem in 1878.

## Explanations

### L1 human

To find a binomial coefficient modulo a prime $p$, write both numbers in base $p$.
Then compute small binomial coefficients digit by digit and multiply them.
This explains the pattern of odd numbers in Pascal's triangle, which looks like the Sierpiński triangle.
</document>

<document source="https://tamnd.github.io/math2/node/0RYTGBJAGN.md" tag="0RYTGBJAGN" kind="definition">
# Big O notation

## Definition

Let $f$ and $g$ be real functions defined on an interval $[x_0, \infty)$.
The notation $f(x) = O(g(x))$ as $x \to \infty$ means that there are real numbers $C > 0$ and $x_1 \ge x_0$ with
$$|f(x)| \le C |g(x)| \quad \text{for all } x \ge x_1.$$

## Remarks

The same definition works as $x \to a$, with $|x - a| < \delta$ in place of $x \ge x_1$.
If $g(x) \ne 0$ for large $x$, then $f = O(g)$ means that $f/g$ is bounded for large $x$.
For example, $3x^2 + 5x = O(x^2)$ as $x \to \infty$.
Mathlib defines it for filters, as `Asymptotics.IsBigO`.

## Explanations

### L1 human

The statement $f = O(g)$ says that $f$ grows at most as fast as $g$, up to a constant factor.
It ignores constant factors and small values of $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0RYYK8PG4S.md" tag="0RYYK8PG4S" kind="statement">
# Law of excluded middle

## Statement

In classical logic, for each [[proposition|0ZGC7ZESYN]] $p$, the proposition $p \lor \neg p$ is true.

## Remarks

The formula $p \lor \neg p$ is a [[tautology|0D1AHYPM8N]].
Some systems take the law as an axiom.
Intuitionistic logic does not accept the law in general.
The law is the basis of [[proof by cases|0WCP0PVET6]] on $p$ and $\neg p$.

## Explanations

### L1 human

The law says that there is no third option: a statement is true, or its negation is true.
For example, each natural number $n$ is even or not even.
</document>

<document source="https://tamnd.github.io/math2/node/0RZ89QWTSJ.md" tag="0RZ89QWTSJ" kind="concept">
# Separable polynomial

## Summary

A separable polynomial is a polynomial that is coprime to its formal derivative.
Over a field, these are the polynomials with no repeated root in an algebraic closure.
The [[definition of a separable polynomial|03J8BS7FMV]] gives the condition.
</document>

<document source="https://tamnd.github.io/math2/node/0RZCTT30H5.md" tag="0RZCTT30H5" kind="statement">
# A nontrivial finite p-group has nontrivial centre

## Statement

Let $p$ be a prime number and let $G$ be a finite [[p-group|06QYRTX9KQ]] with $G \neq \{e\}$.
Then the [[centre|0DZ93493VN]] $Z(G)$ is not $\{e\}$.

## Remarks

In the [[class equation|08BPSC6YTE]] $|G| = |Z(G)| + \sum [G : C_G(a_i)]$, the left side and each term of the sum are divisible by $p$.
So $p$ divides $|Z(G)|$, and $|Z(G)| \geq p$.
Equivalently, apply the [[fixed point lemma|098D2E7DEH]] to the action of $G$ on itself by conjugation.
A consequence is that each finite $p$-group is [[nilpotent|0Q7MMQHC1P]].

## Explanations

### L1 human

The conjugacy classes of a $p$-group that have more than one element all have sizes divisible by $p$.
The order of the group is also divisible by $p$.
So the classes with one element, which form the centre, cannot be just the identity.
</document>

<document source="https://tamnd.github.io/math2/node/0RZH6W98HK.md" tag="0RZH6W98HK" kind="concept">
# Interior

## Summary

The interior of a subset of a topological space is the union of all open sets inside it.
It is the largest open set contained in the subset.
</document>

<document source="https://tamnd.github.io/math2/node/0RZYDVZ475.md" tag="0RZYDVZ475" kind="definition">
# Principal ideal domain

## Definition

A principal ideal domain is an [[integral domain|09S3PFBEPM]] $R$ in which every ideal is [[principal|00A2EDFT9Q]].
That is, for every ideal $I$ of $R$, there is $a \in R$ with $I = (a)$.

## Remarks

Every [[Euclidean domain is a principal ideal domain|0QKVE1GD9A]]. So $\mathbb{Z}$, $K[x]$ for a field $K$, and $\mathbb{Z}[i]$ are principal ideal domains.
The ring $\mathbb{Z}[x]$ is [[not a principal ideal domain|06E3AG9TM2]].
Every principal ideal domain is a [[unique factorization domain|06YMXJMXYS]].
The ring $\mathbb{Z}[\tfrac{1 + \sqrt{-19}}{2}]$ is a principal ideal domain that is not a Euclidean domain.

## Explanations

### L1 human

In a principal ideal domain, every ideal is the set of the multiples of one element.
For the integers, every ideal is $n\mathbb{Z}$ for some $n$.
This makes divisibility in a principal ideal domain behave much like divisibility of integers.
</document>

<document source="https://tamnd.github.io/math2/node/0S049HG1W5.md" tag="0S049HG1W5" kind="statement">
# Power-reduction formula for cosine

## Statement

For every real $x$,
$$\cos^2 x = \frac{1 + \cos 2x}{2}, \qquad \sin^2 x = \frac{1 - \cos 2x}{2}.$$

## Remarks

The [[addition formula|028XEQCE88]] gives $\cos 2x = \cos^2 x - \sin^2 x$. With the [[Pythagorean identity|03H7WZ4T3K]], $\cos 2x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x$.
These formulas give $\int_0^{\pi} \cos^2 x \, dx = \frac{\pi}{2}$.
Mathlib states the first one as `Real.cos_sq`.

## Explanations

### L1 human

A square of a cosine is a cosine of the double angle, shifted and scaled.
This turns squares into first powers, which are easy to integrate.
</document>

<document source="https://tamnd.github.io/math2/node/0S13V32CPB.md" tag="0S13V32CPB" kind="concept">
# Addition of natural numbers

## Summary

Addition of natural numbers is the operation that sends two [[natural numbers|0MWKJ44V5A]] $a$ and $b$ to their sum $a + b$.
The sum $a + b$ is the number that you get when you start at $a$ and apply the [[successor function|0AK9TGSNXY]] $b$ times.
</document>

<document source="https://tamnd.github.io/math2/node/0S166D363M.md" tag="0S166D363M" kind="definition">
# Wedge product of forms

## Definition

Let $U \subseteq \mathbb{R}^n$ be open. For indices $j_1, \dots, j_m \in \{1, \dots, n\}$, let $dx_{j_1} \wedge \dots \wedge dx_{j_m}$ be $0$ if two indices are equal. Otherwise it is $\operatorname{sgn}(\sigma) \, dx_I$, where $I$ is the increasing rearrangement of the indices and $\sigma$ is the permutation that sorts them.
For [[differential forms|0QHTX7WQEY]] $\alpha = \sum_I f_I \, dx_I$ of degree $k$ and $\beta = \sum_J g_J \, dx_J$ of degree $l$ on $U$, the wedge product is the $(k + l)$-form
$$\alpha \wedge \beta = \sum_{I, J} f_I \, g_J \; dx_{i_1} \wedge \dots \wedge dx_{i_k} \wedge dx_{j_1} \wedge \dots \wedge dx_{j_l}.$$

## Remarks

The product is bilinear and associative. It is graded commutative: $\beta \wedge \alpha = (-1)^{k l} \, \alpha \wedge \beta$. In particular, $dx_i \wedge dx_j = -dx_j \wedge dx_i$ and $dx_i \wedge dx_i = 0$.
For a $0$-form $f$, the product $f \wedge \beta$ is the form $f \beta$.
In the plane, $(a \, dx + b \, dy) \wedge (c \, dx + d \, dy) = (a d - b c) \, dx \wedge dy$. The coefficient is the determinant, which is the signed area of the parallelogram spanned by $(a, b)$ and $(c, d)$.
For 1-forms $\alpha$ and $\beta$ and vectors $v, w$, $(\alpha \wedge \beta)(v, w) = \alpha(v) \beta(w) - \alpha(w) \beta(v)$.

## Explanations

### L1 human

The wedge product is a multiplication of forms in which order matters up to sign, and a form times itself is zero.
It is the algebra behind determinants, signed areas and signed volumes.
</document>

<document source="https://tamnd.github.io/math2/node/0S19CWFZM2.md" tag="0S19CWFZM2" kind="statement">
# Size of an orbit times the size of the stabiliser

## Statement

Let a finite [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \in X$.
Then $|G \cdot x| \cdot |G_x| = |G|$.
In particular, the size of each [[orbit|0YQPZB5Q1G]] divides $|G|$.

## Remarks

This follows from the [[orbit-stabiliser theorem|0E2HSCKZ6C]] and from $[G : G_x] \cdot |G_x| = |G|$, which is [[Lagrange's theorem|3C53TD]].
For example, the rotation group of a cube acts transitively on the $6$ faces, and the stabiliser of a face has $4$ elements. So the group has order $24$.

## Explanations

### L1 human

Each point of the orbit is reached by exactly $|G_x|$ group elements.
So the group elements are counted by the orbit size times the stabiliser size.
This is a good way to compute the order of a symmetry group.
</document>

<document source="https://tamnd.github.io/math2/node/0S1FJ8VEX9.md" tag="0S1FJ8VEX9" kind="concept">
# Hausdorff space

## Summary

A Hausdorff space is a topological space in which any two distinct points have disjoint open neighbourhoods.
It is the most used separation axiom, and it makes limits of sequences unique.
</document>

<document source="https://tamnd.github.io/math2/node/0S1KG0XE40.md" tag="0S1KG0XE40" kind="statement">
# Variation of constants formula for linear systems

## Statement

Let $I$ be an open interval, let $A : I \to \mathbb{R}^{n \times n}$ and $b : I \to \mathbb{R}^n$ be continuous, and let $\Phi$ be a [[fundamental matrix|0MQ24RBG93]] of $x' = A(t) x$ on $I$.
Let $t_0 \in I$ and $x_0 \in \mathbb{R}^n$.
Then the unique solution on $I$ of $x' = A(t) x + b(t)$, $x(t_0) = x_0$ is
$$x(t) = \Phi(t)\, \Phi(t_0)^{-1} x_0 + \Phi(t) \int_{t_0}^{t} \Phi(s)^{-1}\, b(s)\, ds, \qquad t \in I.$$

## Remarks

For a constant matrix $A$, take $\Phi(t) = e^{tA}$. The formula becomes $x(t) = e^{(t - t_0) A} x_0 + \int_{t_0}^{t} e^{(t - s) A}\, b(s)\, ds$.
The formula comes from the trial solution $x(t) = \Phi(t)\, c(t)$: the equation gives $\Phi(t)\, c'(t) = b(t)$. This is [[variation of parameters|0Y6TH5PQVS]] for systems.
For $n = 1$, it is the [[solution formula for linear first-order equations|06B9W45XV0]].

## Explanations

### L1 human


The first term is the free motion from the initial state.
The integral adds up the effect of the force $b(s)$ at each earlier time $s$, carried forward to time $t$ by the homogeneous system.
</document>

<document source="https://tamnd.github.io/math2/node/0S21GNW9ZB.md" tag="0S21GNW9ZB" kind="concept">
# Image of a homomorphism

## Summary

The image of a group homomorphism $\varphi \colon G \to H$ is the set of the values $\varphi(g)$ for $g \in G$.
It is a subgroup of $H$.
</document>

<document source="https://tamnd.github.io/math2/node/0S23E6YE83.md" tag="0S23E6YE83" kind="concept">
# Kernel of a linear map

## Summary

The kernel of a linear map is the set of vectors that the map sends to zero.
It is a subspace of the domain.
</document>

<document source="https://tamnd.github.io/math2/node/0S29Y4ZW0Y.md" tag="0S29Y4ZW0Y" kind="concept">
# Surface area

## Summary

The surface area of a surface is a measure of its two-dimensional size.
For a [[parametrised surface|0ZG9QF40HE]], it is the double integral of the length of the normal vector over the parameter domain.
</document>

<document source="https://tamnd.github.io/math2/node/0S2SYHTWPY.md" tag="0S2SYHTWPY" kind="definition">
# Smallest prime factor

## Definition

Let $n$ be a natural number with $n \ge 2$.
The smallest prime factor of $n$ is the smallest [[prime number|0SN0MAEKYX]] $p$ such that $p \mid n$.
It exists by the [[existence of a prime divisor|0SPBEDVNW1]].

## Remarks

The smallest prime factor of $n$ is also the smallest divisor $d$ of $n$ with $d > 1$.
The number $n$ is prime if and only if its smallest prime factor is $n$.
If $n$ is composite, its smallest prime factor $p$ satisfies $p^2 \le n$. See [[a composite number has a prime factor at most its square root|0EB6BBMQMC]].
For example, the smallest prime factor of $91 = 7 \cdot 13$ is $7$.

## Explanations

### L1 human

Divide $n$ by $2$, then by $3$, then by $4$, and so on. The first number that divides $n$ exactly is the smallest prime factor.
It is always prime, because a smaller factor of it would also divide $n$ and would come earlier.
</document>

<document source="https://tamnd.github.io/math2/node/0S2TEHX4E1.md" tag="0S2TEHX4E1" kind="example">
# Discrete topology

## Example

Let $X$ be a set and let $\tau = \mathcal{P}(X)$ be the set of all subsets of $X$.
Then $\tau$ is a [[topology|0CNQXVBHCX]] on $X$, the discrete topology. The space $(X, \tau)$ is a discrete space.

## Remarks

Each subset of $X$ is both open and closed. Each point $x$ gives an open set $\{x\}$.
It is the finest topology on $X$. It is the [[metric topology|0A6YRNZB2R]] of the [[discrete metric|0NA4FK7B8S]].
Each function from a discrete space to any topological space is [[continuous|0SY9ECQ19Q]].
A discrete space is [[compact|0QPHW703PV]] if and only if it is finite.

## Explanations

### L1 human

In the discrete topology, every set is open, so each point is isolated from all other points.
The integers $\mathbb{Z}$, as a subspace of $\mathbb{R}$, carry the discrete topology.
</document>

<document source="https://tamnd.github.io/math2/node/0S3SMRH0Z5.md" tag="0S3SMRH0Z5" kind="definition">
# Continuous linear map

## Definition

Let $V$ and $W$ be vector spaces over $\mathbb{R}$ or $\mathbb{C}$, with [[norms|0J3578562S]] $\|\cdot\|_V$ and $\|\cdot\|_W$.
A map $T \colon V \to W$ is a continuous linear map when $T$ is linear and [[continuous|0SY9ECQ19Q]] for the metrics $\|x - y\|_V$ and $\|x - y\|_W$.

## Remarks

A linear map is continuous if and only if it is bounded, that is, $\|Tx\|_W \le C \|x\|_V$ for some constant $C$ and all $x$. This is [[bounded linear maps are continuous|0F3K4C7XYW]].
The smallest such $C$ is the [[operator norm|0ZTEH055YF]] of $T$.
On a finite-dimensional space, [[each linear map is continuous|0C13E623WD]].
The derivative $p \mapsto p'$ on polynomials, with the sup norm on $[0, 1]$, is linear but not continuous: $\|x^n\|_\infty = 1$ but $\|n x^{n-1}\|_\infty = n$.

## Explanations

### L1 human

A continuous linear map respects the vector space structure and also the notion of closeness.
In infinite dimensions, a linear map can fail to be continuous, so this is a real condition.
</document>

<document source="https://tamnd.github.io/math2/node/0S4EMDQS2F.md" tag="0S4EMDQS2F" kind="statement">
# Abel's theorem

## Statement

Let $(a_n)$ be a real sequence such that the series $\sum_{n=0}^{\infty} a_n$ [[converges|0CVVKY7GY2]].
Let $G(x) = \sum_{n=0}^{\infty} a_n x^n$ for $-1 < x < 1$.
Then $G(x) \to \sum_{n=0}^{\infty} a_n$ as $x \to 1$ from the left.

## Remarks

The [[power series|0VP56B4EMX]] converges on $(-1, 1)$, since its radius of convergence is at least $1$.
The theorem gives the sum of the [[alternating harmonic series|0JQ5Y3WWVE]]. Apply it to the [[Mercator series|0YZXNE3HFR]] for $\ln(1 + x)$.
The converse fails. For $a_n = (-1)^n$, $G(x) = 1/(1 + x) \to 1/2$, but $\sum (-1)^n$ diverges.
Mathlib states it as `Real.tendsto_tsum_powerSeries_nhdsWithin_lt`.

## Explanations

### L1 human

A power series that converges at the endpoint $x = 1$ is continuous there from the left.
So its value at $1$ is the limit of its values inside the interval.
</document>

<document source="https://tamnd.github.io/math2/node/0S4JF619V7.md" tag="0S4JF619V7" kind="statement">
# Convergent sequences are bounded

## Statement

Let $(a_n)$ be a [[convergent sequence|0AZ3R52593]] of real numbers.
Then $(a_n)$ is [[bounded|05FQ3VGAR9]]: there is a real number $K \ge 0$ such that $|a_n| \le K$ for all $n \in \mathbb{N}$.

## Remarks

Let $L$ be the limit. Take $\varepsilon = 1$ in the [[epsilon-N definition|09G6KHS6GY]] to get $N$. Then $K = \max(|a_0|, \dots, |a_{N-1}|, |L| + 1)$ works.
The converse is false. The sequence $(-1)^n$ is bounded and divergent.
The statement holds in each metric space, where bounded means contained in some ball.

## Explanations

### L1 human

After some index, all terms are close to the limit, so they lie in a small interval.
Only finitely many terms come before that index, and finitely many numbers are always bounded.
</document>

<document source="https://tamnd.github.io/math2/node/0S4WTRE9GG.md" tag="0S4WTRE9GG" kind="statement">
# Inverse of a product

## Statement

Let $R$ be a commutative ring and let $A$ and $B$ be [[invertible|0EB9N0S33C]] $n \times n$ matrices over $R$.
Then $AB$ is invertible, and
$$(AB)^{-1} = B^{-1} A^{-1}.$$

## Remarks

The proof is the computation $(AB)(B^{-1} A^{-1}) = A (B B^{-1}) A^{-1} = A A^{-1} = I_n$, and the same in the other order.
By induction, $(A_1 \cdots A_k)^{-1} = A_k^{-1} \cdots A_1^{-1}$.
Thus the invertible $n \times n$ matrices form a group, the [[general linear group|0FCHE902D7]].

## Explanations

### L1 human

To undo two steps, we undo the last step first.
If we first apply $B$ and then $A$, then we must first undo $A$ and then undo $B$.
So the inverse of $AB$ is $B^{-1} A^{-1}$, with the order reversed.
</document>

<document source="https://tamnd.github.io/math2/node/0S4ZEZM7KE.md" tag="0S4ZEZM7KE" kind="concept">
# Spanning tree

## Summary

A spanning tree of a graph is a tree inside the graph that reaches every vertex. Spanning trees are the minimal connected spanning subgraphs, with uses in network design and in counting. The node [[spanning tree|0FQ5B8S1ZC]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0S52TNNTH9.md" tag="0S52TNNTH9" kind="concept">
# Dot product

## Summary

The dot product of two vectors of $\R^n$ is the sum of the products of their corresponding coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/0S60DVMVPN.md" tag="0S60DVMVPN" kind="concept">
# Upper bound

## Summary

Let $S$ be a subset of a partially ordered set.
An upper bound of $S$ is an element that is at least as large as each element of $S$.
</document>

<document source="https://tamnd.github.io/math2/node/0S61N75H45.md" tag="0S61N75H45" kind="definition">
# Root of unity

## Definition

Let $n$ be a positive integer.
A complex number $z$ is an $n$th root of unity when $z^n = 1$.
A complex number $z$ is a root of unity when it is an $n$th root of unity for some positive integer $n$.

## Remarks

The same definition works in any field, or in any commutative ring.
The $n$th roots of unity in $\mathbb{C}$ form a group under multiplication, written $\mu_n$.
For example, the $4$th roots of unity are $1$, $i$, $-1$ and $-i$.
Each root of unity $z$ satisfies $|z| = 1$, because $|z|^n = |z^n| = 1$.

## Explanations

### L1 human

A root of unity is a number that gives $1$ when you multiply it by itself often enough.
The number $-1$ is one, because $(-1)^2 = 1$, and so is $i$, because $i^4 = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0S6GMNJ62Y.md" tag="0S6GMNJ62Y" kind="concept">
# Biconditional

## Summary

The biconditional is the logical operation "if and only if".
It joins two [[propositions|0ZGC7ZESYN]] into one proposition that is true when both parts have the same truth value.
</document>

<document source="https://tamnd.github.io/math2/node/0S6T9KQNC0.md" tag="0S6T9KQNC0" kind="concept">
# Confidence interval

## Summary

A confidence interval is an interval computed from the data. Over repetitions of the experiment, it contains an unknown parameter with a fixed probability, such as $95\%$.
The node [[confidence interval|0K4NMMJBG2]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0S706JBZBG.md" tag="0S706JBZBG" kind="concept">
# Midpoint

## Summary

The midpoint of a line segment is the point of the segment that is at the same distance from both endpoints. It divides the segment into two halves.
</document>

<document source="https://tamnd.github.io/math2/node/0S7495H4HM.md" tag="0S7495H4HM" kind="statement">
# Order of the group of units of Z/nZ is Euler's totient

## Statement

Let $n \geq 1$ be an integer.
Then the [[order|0B85S2V2DR]] of the group $(\mathbb{Z}/n\mathbb{Z})^\times$ is $\varphi(n)$.
Here $\varphi(n)$ is the number of integers $a$ with $0 \leq a \leq n - 1$ and $\gcd(a, n) = 1$.

## Remarks

The proof is a count, because the [[units of Z/nZ are the residues coprime to n|0GXNHEHQNA]].
For a prime $p$, the order is $\varphi(p) = p - 1$.
For example, $(\mathbb{Z}/9\mathbb{Z})^\times = \{1, 2, 4, 5, 7, 8\}$ has order $\varphi(9) = 6$.
With [[Lagrange's theorem|3C53TD]], this gives Euler's theorem: $a^{\varphi(n)} \equiv 1 \pmod{n}$ for each $a$ coprime to $n$.

## Explanations

### L1 human

The size of the group of units modulo $n$ is the number of residues coprime to $n$.
Euler's totient function counts exactly these residues.
So the group has $\varphi(n)$ elements.
</document>

<document source="https://tamnd.github.io/math2/node/0S7A0EMPF4.md" tag="0S7A0EMPF4" kind="definition">
# Algebraic multiplicity

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Let $\lambda$ be an [[eigenvalue|0BQAA2BK3Q]] of $T$.
The *algebraic multiplicity* of $\lambda$ is the largest integer $m \ge 1$ such that $(t - \lambda)^m$ divides the [[characteristic polynomial|0J5EMJRRFM]] $p_T(t)$.

## Remarks

The sum of the algebraic multiplicities of all eigenvalues is at most $\dim V$, with equality when $p_T$ splits over $F$.
For $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the eigenvalue $1$ has algebraic multiplicity $2$.
The algebraic multiplicity is at least the [[geometric multiplicity|099TWVJ909]].

## Explanations

### L1 human

An eigenvalue can be a root of the characteristic polynomial several times.
The algebraic multiplicity counts how many times.
For the identity matrix of size $n$, the eigenvalue $1$ has algebraic multiplicity $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0S7G92KKA7.md" tag="0S7G92KKA7" kind="concept">
# Indeterminate form

## Summary

An indeterminate form is a type of [[limit|0TXTSEMHR5]] where the limits of the parts do not determine the limit of the whole.
The seven common forms are $0/0$, $\infty/\infty$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^{\infty}$, and $\infty^0$.

## Remarks

For example, $x \to 0$ and $x^2 \to 0$, but $x^2/x \to 0$, $x/x \to 1$, and $x/x^2$ has no finite limit. All three are of the form $0/0$.
[[L'Hôpital's rule|0JWT4PPQ7B]] treats the forms $0/0$ and $\infty/\infty$. Logarithms turn the forms $0^0$, $1^{\infty}$, and $\infty^0$ into the form $0 \cdot \infty$.
Forms such as $1/0^+$ and $0^{\infty}$ are not indeterminate.

## Explanations

### L1 human

Some limits cannot be found by plugging in the limits of the parts.
Then a finer analysis is needed, because many different answers are possible.
</document>

<document source="https://tamnd.github.io/math2/node/0S7W6EG32V.md" tag="0S7W6EG32V" kind="concept">
# External direct sum

## Summary

The external direct sum of a family of vector spaces is the space of families of vectors, one vector from each space.
Only finitely many entries of such a family are nonzero.
For finitely many spaces it is the product of the spaces.
</document>

<document source="https://tamnd.github.io/math2/node/0S7Y9Q990Q.md" tag="0S7Y9Q990Q" kind="definition">
# Centre of a group

## Definition

Let $G$ be a [[group|G5S1M9]].
The centre of $G$ is the set $Z(G) = \{z \in G : zg = gz \text{ for all } g \in G\}$.

## Remarks

The centre is an [[abelian|082GHNPJ53]] [[normal subgroup|0CB0DX5BVV]] of $G$, and it is [[characteristic|0KABPMRHM2]].
The group $G$ is abelian if and only if $Z(G) = G$.
The centre of the symmetric group $S_n$ is trivial for $n \geq 3$.
The centre of $\mathrm{GL}_n(\mathbb{R})$ is the set of the nonzero scalar matrices $\lambda I$.

## Explanations

### L1 human

The centre collects the elements that commute with everybody.
The identity is always in the centre.
In an abelian group the centre is everything, and in many nonabelian groups it is only the identity.
</document>

<document source="https://tamnd.github.io/math2/node/0S8FE7ZMFM.md" tag="0S8FE7ZMFM" kind="concept">
# Chain

## Summary

A chain in a partially ordered set is a subset in which any two elements are comparable.
</document>

<document source="https://tamnd.github.io/math2/node/0S8WX8A4V9.md" tag="0S8WX8A4V9" kind="statement">
# Hartman–Grobman theorem

## Statement

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be continuously differentiable. Let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$, and let $x^* \in U$ be a [[hyperbolic equilibrium point|0C5PREPZWG]].
Let $A = Df(x^*)$.
Then there are open neighbourhoods $V \subseteq U$ of $x^*$ and $W \subseteq \mathbb{R}^n$ of $0$, and a homeomorphism $h : V \to W$ with $h(x^*) = 0$, as follows.
For each $x \in V$ and each $t \in \mathbb{R}$ such that $\varphi_s(x) \in V$ for all $s$ between $0$ and $t$, $h(\varphi_t(x)) = e^{tA} h(x)$.

## Remarks

So near $x^*$ the flow is topologically conjugate to the flow of the [[linearisation|09ZWYRQ41J]] $y' = A y$, given by the [[matrix exponential|0YC8JT513J]]. The map $h$ sends trajectories to trajectories and keeps the direction of time.
The map $h$ is in general only continuous, not differentiable.
The hypothesis of hyperbolicity is necessary. For $x' = -x^3$ and $x' = x^3$ on $\mathbb{R}$, the linearisation at $0$ is the same, but the two phase portraits are not conjugate.

## History

The theorem is named after Philip Hartman and David M. Grobman. It is also called the linearization theorem.

## Explanations

### L1 human

Near a hyperbolic equilibrium, a nonlinear system looks like its linear approximation, after a continuous change of coordinates.
So the pattern of trajectories of the linear system, such as a saddle or a spiral, also appears in the nonlinear system.
</document>

<document source="https://tamnd.github.io/math2/node/0S9EK5MQPE.md" tag="0S9EK5MQPE" kind="definition">
# Continuity in R^n

## Definition

Let $D \subseteq \mathbb{R}^n$, let $f \colon D \to \mathbb{R}^m$, and let $a \in D$.
The function $f$ is continuous at $a$ when, for each $\varepsilon > 0$, there is a $\delta > 0$ such that $\|f(x) - f(a)\| < \varepsilon$ for each $x \in D$ with $\|x - a\| < \delta$.
Here $\|\cdot\|$ is the Euclidean norm of [[Euclidean space|0XPR1E85W0]].
The function $f$ is continuous on $D$ when it is continuous at each point of $D$.

## Remarks

Let $a$ be a limit point of $D$. Then $f$ is continuous at $a$ if and only if $\lim_{x \to a} f(x) = f(a)$, with the [[limit of a function of several variables|0J8Y1NCY0K]].
A function into $\mathbb{R}^m$ is continuous if and only if each component function is continuous.
Sums, products and compositions of continuous functions are continuous. So each polynomial in $x_1, \dots, x_n$ is continuous on $\mathbb{R}^n$.
Continuity in each variable separately is weaker: see the [[separately continuous function that is not continuous|0C0KPAKK9Y]].

## Explanations

### L1 human

A continuous function has no jumps: if you move the input a little, the output moves only a little.
In the plane, "a little" means inside a small disc, so the condition must hold for moves in every direction.
</document>

<document source="https://tamnd.github.io/math2/node/0S9GGTR0EX.md" tag="0S9GGTR0EX" kind="concept">
# Orthonormal basis

## Summary

An orthonormal basis of a finite-dimensional inner product space is a basis whose vectors have length one and are pairwise orthogonal.
</document>

<document source="https://tamnd.github.io/math2/node/0S9ZVZ34MD.md" tag="0S9ZVZ34MD" kind="statement">
# Correspondence theorem

## Statement

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
Then the map $K \mapsto K/N$ is a bijection from the set of [[subgroups|0X1JXT47GA]] $K$ of $G$ with $N \subseteq K$ to the set of all subgroups of $G/N$. Here $G/N$ is the [[quotient group|0KYX2N69J0]].
Let $K$ and $L$ be subgroups of $G$ that contain $N$. Then $K \subseteq L$ if and only if $K/N \subseteq L/N$. Also, $K$ is normal in $G$ if and only if $K/N$ is normal in $G/N$.

## Remarks

The inverse map sends a subgroup $\bar{K}$ of $G/N$ to its preimage $\{g \in G : gN \in \bar{K}\}$.
The bijection is an isomorphism of [[lattices of subgroups|0B8BW4YWTF]], and it keeps indices: $[L : K] = [L/N : K/N]$ when $N \subseteq K \subseteq L$.
For example, the subgroups of $\mathbb{Z}/12\mathbb{Z}$ correspond to the subgroups $d\mathbb{Z}$ of $\mathbb{Z}$ with $d$ dividing $12$.

## Explanations

### L1 human

The subgroups of a quotient group $G/N$ are the same as the subgroups of $G$ that contain $N$.
Passing to the quotient does not create new subgroups or lose old ones above $N$.
So the picture of the subgroups above $N$ in $G$ is the full picture of the subgroups of $G/N$.
</document>

<document source="https://tamnd.github.io/math2/node/0SAAHC36J9.md" tag="0SAAHC36J9" kind="definition">
# Generalised eigenvector

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$.
A *generalised eigenvector* of $T$ for $\lambda$ is a vector $v \in V$ with $v \ne 0$ and
$$(T - \lambda \id_V)^k (v) = 0$$
for some integer $k \ge 1$.
The least such $k$ is the *rank* of the generalised eigenvector.

## Remarks

The generalised eigenvectors of rank $1$ are the [[eigenvectors|0N6960W0BV]].
If a generalised eigenvector for $\lambda$ exists, then $\lambda$ is an eigenvalue.
For $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the vector $e_2$ is a generalised eigenvector for $1$ of rank $2$, and it is not an eigenvector.

## Explanations

### L1 human

Some matrices do not have enough eigenvectors to make a basis.
Generalised eigenvectors fill the gap: they reach zero after several applications of $T - \lambda \id$ instead of one.
They are the building blocks of the Jordan normal form.
</document>

<document source="https://tamnd.github.io/math2/node/0SAMR34WXC.md" tag="0SAMR34WXC" kind="statement">
# Derivative of tangent

## Statement

The tangent function is differentiable at every $x$ with $\cos x \ne 0$, and
$$\tan'(x) = \frac{1}{\cos^2 x} = 1 + \tan^2 x.$$

## Remarks

The [[quotient rule|0BS0NJPKVE]] gives $\tan' = \frac{\cos \cdot \cos - \sin \cdot (-\sin)}{\cos^2}$. The [[Pythagorean identity|03H7WZ4T3K]] turns the numerator into $1$.
The derivative is positive. So $\tan$ is strictly increasing on $(-\pi/2, \pi/2)$.
Mathlib states it as `Real.hasDerivAt_tan`.

## Explanations

### L1 human

The tangent function always rises, and it rises faster and faster near its asymptotes.
Its slope at $0$ is $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0SAR26CKPS.md" tag="0SAR26CKPS" kind="statement">
# Dirichlet convolution formula

## Statement

Let $f$ and $g$ be arithmetic functions, and let $n$ be a positive integer.
Then
$$(f * g)(n) = \sum_{\substack{a, b \ge 1 \\ ab = n}} f(a) \, g(b),$$
where the sum runs over the ordered pairs of positive integers $(a, b)$ with $ab = n$, and $*$ is the [[Dirichlet convolution|0KH1N66QWW]].

## Remarks

The map $d \mapsto (d, n/d)$ is a bijection from the divisors of $n$ to these pairs. This gives the formula.
The symmetric form shows at once that $f * g = g * f$.
For example, $(f * g)(6) = f(1)g(6) + f(2)g(3) + f(3)g(2) + f(6)g(1)$.

## Explanations

### L1 human

Each divisor $d$ of $n$ comes with its partner $n/d$.
So a sum over divisors is the same as a sum over the ways to write $n$ as an ordered product of two factors.
</document>

<document source="https://tamnd.github.io/math2/node/0SB7Q8SYSF.md" tag="0SB7Q8SYSF" kind="definition">
# Perfect number

## Definition

A positive integer $n$ is perfect if it is equal to the sum of its positive divisors other than $n$.
Equivalently, $\sigma(n) = 2n$, where $\sigma$ is the [[sum of divisors|009QCP5T3Z]].

## Remarks

The first perfect numbers are $6$, $28$, $496$ and $8128$. For example, $28 = 1 + 2 + 4 + 7 + 14$.
The [[Euclid–Euler theorem|0WETJCKNRG]] describes all even perfect numbers by [[Mersenne primes|06GPJF9V7M]].
No odd perfect number is known, and it is an open problem whether one exists.

## Explanations

### L1 human

A perfect number is the sum of its smaller divisors.
The divisors of $6$ other than $6$ are $1$, $2$ and $3$, and $1 + 2 + 3 = 6$.
</document>

<document source="https://tamnd.github.io/math2/node/0SB9FNZECG.md" tag="0SB9FNZECG" kind="definition">
# Path component

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]].
For $x, y \in X$, write $x \simeq y$ when there is a [[path|0K0GE6BCYW]] in $X$ from $x$ to $y$.
This is an [[equivalence relation|0SGTMBEWRF]] on $X$. Its classes are the path components of $X$.

## Remarks

The relation is reflexive by constant paths, symmetric by $t \mapsto \gamma(1 - t)$, and transitive by joining two paths one after the other.
The path component of $x$ is the largest path-connected subset of $X$ that contains $x$.
Each path component lies in a [[connected component|0NRT0E2QT2]]. The [[topologist's sine curve|0K3T4A659R]] is one connected component but has two path components.
Path components need not be closed.

## Explanations

### L1 human

The path component of a point is the set of all points that you can reach from it along a path.
</document>

<document source="https://tamnd.github.io/math2/node/0SBRX5CZ5R.md" tag="0SBRX5CZ5R" kind="definition">
# Multiplicative order

## Definition

Let $n$ be a positive integer and let $a$ be an integer [[coprime|0GFKP8F8AG]] to $n$.
The multiplicative order of $a$ modulo $n$ is the smallest positive integer $k$ such that $a^k \equiv 1 \pmod n$.
The notation for it is $\operatorname{ord}_n(a)$.

## Remarks

The order exists, because $a^{\varphi(n)} \equiv 1 \pmod n$ by [[Euler's theorem|0VC4DVE7DQ]].
For each integer $m \ge 0$, $a^m \equiv 1 \pmod n$ if and only if $\operatorname{ord}_n(a) \mid m$. In particular, $\operatorname{ord}_n(a) \mid \varphi(n)$.
The order of $a$ modulo $n$ is the order of $\overline{a}$ in the group $(\mathbb{Z}/n\mathbb{Z})^\times$.
For example, $\operatorname{ord}_7(2) = 3$, because $2^1 = 2$, $2^2 = 4$ and $2^3 = 8 \equiv 1 \pmod 7$.

## Explanations

### L1 human

Compute the powers $a, a^2, a^3, \dots$ modulo $n$, until you reach $1$. The number of steps is the order.
Modulo $10$, the powers of $3$ are $3, 9, 7, 1$, so the order of $3$ modulo $10$ is $4$.
</document>

<document source="https://tamnd.github.io/math2/node/0SC8T7548D.md" tag="0SC8T7548D" kind="object">
# Golden ratio

## Description

The golden ratio is the real number
$$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887.$$
It is the positive root of the polynomial $x^2 - x - 1$, so $\varphi^2 = \varphi + 1$.
The other root is $\psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = -\frac{1}{\varphi}$.

## Remarks

Two positive numbers $a > b$ are in the golden ratio when $\frac{a + b}{a} = \frac{a}{b}$. Then $\frac{a}{b} = \varphi$.
The number $\varphi$ is irrational, because $\sqrt{5}$ is irrational.
The ratios of consecutive [[Fibonacci numbers|0GA9K57A39]] tend to $\varphi$. [[Binet's formula|0404D8TN7Y]] uses $\varphi$ and $\psi$.

## Explanations

### L1 human

Cut a stick into a long part and a short part. Do it so that the whole stick relates to the long part as the long part relates to the short part.
The ratio of long to short is then $\varphi$, about $1.618$.
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<document source="https://tamnd.github.io/math2/node/0SCSQWXRA4.md" tag="0SCSQWXRA4" kind="definition">
# Hessian matrix

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $a \in U$ be a point at which all second [[partial derivatives|02RSXVM1ZN]] of $f$ exist.
The Hessian matrix of $f$ at $a$ is the $n \times n$ matrix
$$H_f(a) = \left( \frac{\partial^2 f}{\partial x_i \, \partial x_j}(a) \right)_{1 \le i, j \le n},$$
whose entry in row $i$ and column $j$ is $\partial^2 f / \partial x_i \, \partial x_j$ at $a$.
It is also written $\nabla^2 f(a)$ or $D^2 f(a)$.

## Remarks

If the second partial derivatives of $f$ are continuous near $a$, then $H_f(a)$ is a symmetric matrix, by [[Schwarz's theorem|0NNQY8WCW6]].
For $f(x, y) = x^2 + 3 x y - y^2$, the Hessian matrix at each point is $\begin{pmatrix} 2 & 3 \\ 3 & -2 \end{pmatrix}$.
The Hessian matrix is the Jacobian matrix of the gradient map $x \mapsto \nabla f(x)$, when this map is differentiable.
At a critical point, the signs of the eigenvalues of $H_f(a)$ decide the [[second derivative test|0GYDV9FE98]].

## Explanations

### L1 human

The gradient tells how a function tilts; the Hessian tells how it bends.
For a bowl like $x^2 + y^2$ the Hessian is $2 I$, and both directions bend upward. For a saddle like $x^2 - y^2$ one direction bends up and the other bends down.
</document>

<document source="https://tamnd.github.io/math2/node/0SD237FEJD.md" tag="0SD237FEJD" kind="statement">
# Gradient theorem

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $\phi \colon U \to \mathbb{R}$ be of class $C^1$, and let $\gamma \colon [a, b] \to U$ be a piecewise $C^1$ curve.
Then
$$\int_\gamma \nabla \phi \cdot d\mathbf{r} = \phi(\gamma(b)) - \phi(\gamma(a)).$$

## Remarks

Proof for a $C^1$ curve: by the [[chain rule along a curve|0A8TJ0VHC7]], $(\phi \circ \gamma)'(t) = \nabla \phi(\gamma(t)) \cdot \gamma'(t)$. Then the [[parametric formula|0W3W1VYWBP]] and the fundamental theorem of calculus give the result. For a piecewise $C^1$ curve, add the pieces.
So the line integral of a gradient field depends only on the end points. Around a closed curve it is $0$.
Example: for $\phi(x, y) = x^2 y$ and any path from $(0, 0)$ to $(1, 2)$, $\int_\gamma (2 x y, x^2) \cdot d\mathbf{r} = 2$.
This is the case of curves in the [[generalised Stokes theorem|0HPHKCGEQ0]]: the boundary of a curve consists of its two end points.

## Explanations

### L1 human

This is the fundamental theorem of calculus for curves.
To integrate the gradient of $\phi$ along a path, you only need the values of $\phi$ at the end and at the start.
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<document source="https://tamnd.github.io/math2/node/0SD528P0EK.md" tag="0SD528P0EK" kind="definition">
# Minor

## Definition

Let $K$ be a commutative ring, let $A$ be an $m \times n$ matrix over $K$, and let $1 \le k \le \min(m, n)$.
A *minor of order $k$* of $A$ is the [[determinant|00KDZRW4C0]] of a $k \times k$ submatrix of $A$.
Such a submatrix consists of the entries in $k$ chosen rows and $k$ chosen columns of $A$.
For an $n \times n$ matrix $A$, delete row $i$ and column $j$ to get an $(n-1) \times (n-1)$ matrix.
The minor $M_{ij}$ is the determinant of this matrix.

## Remarks

The minors $M_{ij}$ are called *first minors*.
Over a field, the [[rank|0ZC9YTXZRY]] of $A$ is the largest $k$ such that $A$ has a nonzero minor of order $k$.
The minors with the same chosen rows and columns are *principal minors*.

## Explanations

### L1 human

A minor is the determinant of a smaller square piece of a matrix.
For a $3 \times 3$ matrix, we delete one row and one column and take the $2 \times 2$ determinant of what is left.
Minors appear in the expansion of determinants and in the formula for the inverse.
</document>

<document source="https://tamnd.github.io/math2/node/0SDC934R52.md" tag="0SDC934R52" kind="concept">
# Polynomial ring

## Summary

The polynomial ring $R[x]$ over a ring $R$ is the ring of the polynomials in one variable $x$ with coefficients in $R$.
The [[definition of the polynomial ring|0Q9EZS90GY]] gives the construction.
</document>

<document source="https://tamnd.github.io/math2/node/0SDFXN1VDQ.md" tag="0SDFXN1VDQ" kind="definition">
# Dihedral angle

## Definition

Let $P_1$ and $P_2$ be two [[planes|0V3XAMF3C1]] in $\mathbb{R}^3$ that meet in a line. Let $n_1$ and $n_2$ be [[normal vectors|0WQJJCCTC8]] of $P_1$ and $P_2$, and let $\cdot$ be the dot product.
The dihedral angle between $P_1$ and $P_2$ is the unique real number $\varphi$ with $0 \leq \varphi \leq \pi / 2$ such that
$$\cos \varphi = \frac{|n_1 \cdot n_2|}{\|n_1\| \, \|n_2\|}.$$

## Remarks

The value does not depend on the choice of the normal vectors. The absolute value is necessary because $-n_1$ is also a normal vector of $P_1$.
For two half-planes with a common edge line, the dihedral angle is in $[0, \pi]$. Take a point $O$ on the edge. Let $u_1$ and $u_2$ be vectors that are perpendicular to the edge and point from $O$ into the two half-planes. The dihedral angle is the [[angle between $u_1$ and $u_2$|0ZEHZY85E6]].
For the planes $z = 0$ and $x = 0$, the normal vectors $(0, 0, 1)$ and $(1, 0, 0)$ have dot product $0$, so the dihedral angle is $\pi / 2$.
The dihedral angles of a polyhedron are the angles between the planes of two faces with a common edge, measured inside the solid.

## Explanations

### L1 human

Open a book partly. The two covers are half-planes that meet along the spine. The dihedral angle is how far the book is open: $0$ when it is closed and $\pi$ when it lies flat.
</document>

<document source="https://tamnd.github.io/math2/node/0SDMPMCN43.md" tag="0SDMPMCN43" kind="definition">
# Multilinear map

## Definition

Let $F$ be a field, let $n \ge 1$, and let $V_1, \dots, V_n$ and $W$ be vector spaces over $F$.
A map $f \colon V_1 \times \cdots \times V_n \to W$ is *multilinear* when it is linear in each variable separately.
That is, for each $i$ and each fixed choice of $v_j \in V_j$ for $j \ne i$, the map $v \mapsto f(v_1, \dots, v_{i-1}, v, v_{i+1}, \dots, v_n)$ is a [[linear map|0NRYHJPQG2]] $V_i \to W$.

## Remarks

For $n = 1$, a multilinear map is a linear map, and for $n = 2$ it is a *bilinear map*.
The dot product on $\R^n$ and the matrix product $M_n(F) \times M_n(F) \to M_n(F)$ are bilinear.
The [[determinant|00KDZRW4C0]] is multilinear in the $n$ columns of a matrix.
A multilinear map is in general not linear on the product space: $f(2v_1, 2v_2) = 4 f(v_1, v_2)$ for a bilinear $f$.

## Explanations

### L1 human

A multilinear map takes several vectors and is linear in each one, while the others stay fixed.
The product $xy$ of two real numbers is a simple example: it is linear in $x$ and linear in $y$.
It is not linear in the pair $(x, y)$.
</document>

<document source="https://tamnd.github.io/math2/node/0SEDGXG9S3.md" tag="0SEDGXG9S3" kind="definition">
# Cross product

## Definition

Let $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ be [[vectors|0QN3D80AWP]] in $\mathbb{R}^3$.
The cross product of $a$ and $b$ is the vector
$$a \times b = (a_2 b_3 - a_3 b_2, \; a_3 b_1 - a_1 b_3, \; a_1 b_2 - a_2 b_1).$$

## Remarks

For the standard basis vectors $e_1$, $e_2$, $e_3$ of $\mathbb{R}^3$: $e_1 \times e_2 = e_3$, $e_2 \times e_3 = e_1$ and $e_3 \times e_1 = e_2$.
If $\theta$ is the angle between nonzero vectors $a$ and $b$, then $\|a \times b\| = \|a\| \, \|b\| \sin \theta$. This is the area of the parallelogram with the sides $a$ and $b$.
The cross product is [[perpendicular to its factors|0T4QHJXSD4]] and [[anticommutative|07ZS685JJC]]. It is bilinear, but it is not associative.
In the usual right-handed coordinates, $a$, $b$ and $a \times b$ follow the right-hand rule. The physics notation $a \wedge b$ is also used.

## Explanations

### L1 human

The cross product of two arrows in space gives a third arrow that sticks out at right angles to both. For example, $(1, 0, 0) \times (0, 1, 0) = (0, 0, 1)$. The longer and the more perpendicular the two arrows are, the longer their cross product.
</document>

<document source="https://tamnd.github.io/math2/node/0SEX9F9J5Z.md" tag="0SEX9F9J5Z" kind="definition">
# Order of a group

## Definition

Let $G$ be a [[group|G5S1M9]].
The order of $G$, written $|G|$, is the cardinality of the underlying set of $G$.
When $G$ is finite, $|G|$ is the number of elements of $G$.
When $G$ is infinite, $G$ has infinite order.

## Remarks

The [[trivial group|0BNPC9850Y]] has order $1$, and the [[group of $n$th roots of unity|0WX11EA3F3]] has order $n$.
The group $(\mathbb{Z}, +)$ has infinite order.
[[Lagrange's theorem|3C53TD]] says that the order of a subgroup of a finite group divides the order of the group.

## Explanations

### L1 human

The order of a group tells you how big the group is.
For example, the rotations of a square that map the square to itself form a group of order $4$.
The word "order" has a second meaning in group theory: the order of an element.
</document>

<document source="https://tamnd.github.io/math2/node/0SEZ3JGZP7.md" tag="0SEZ3JGZP7" kind="definition">
# Countably infinite set

## Definition

A [[set|0SJ3RX9TXA]] $A$ is countably infinite when there is a [[bijective function|0SNQEF7Z53]] $f \colon \mathbb{N} \to A$.

## Remarks

Another name is denumerable. Its cardinality is [[aleph-null|0V2H5PGJ9B]].
Examples are $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$, and also the set of primes.
A set is countably infinite exactly when it is [[countable|0ZG61TFBES]] and [[infinite|01MKW4R283]].

## Explanations

### L1 human

A countably infinite set can be listed as $a_0, a_1, a_2, \dots$ without end, and each element appears exactly one time.
The integers have such a list: $0, 1, -1, 2, -2, \dots$.
</document>

<document source="https://tamnd.github.io/math2/node/0SF85PY0QN.md" tag="0SF85PY0QN" kind="statement">
# Product rule in several variables

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f, g \colon U \to \mathbb{R}$, and let $a \in U$.
Suppose that $f$ and $g$ are [[differentiable|0135XZZQ8M]] at $a$.
Then the product $f g$ is differentiable at $a$, and its derivative is
$$D(f g)(a) = f(a) \, Dg(a) + g(a) \, Df(a).$$
That is, $D(f g)(a)(h) = f(a) \, Dg(a)(h) + g(a) \, Df(a)(h)$ for each $h \in \mathbb{R}^n$.

## Remarks

In terms of the [[gradient|0ABK9V9Y64]], the rule reads $\nabla (f g) = f \nabla g + g \nabla f$.
The same rule holds for each partial derivative: $\partial_i (f g) = f \, \partial_i g + g \, \partial_i f$.
More generally, let $B$ be a bounded bilinear map. If $f$ and $g$ are [[Fréchet differentiable|08EVNYZF42]] at $a$, then $D(B(f, g))(a)(h) = B(Df(a) h, g(a)) + B(f(a), Dg(a) h)$. This gives product rules for the dot product and the cross product.

## Explanations

### L1 human

This is the familiar rule $(f g)' = f' g + f g'$, applied to the change in each direction at once.
For $f = x$ and $g = y$ on $\mathbb{R}^2$, the product $x y$ has the derivative $h \mapsto y h_1 + x h_2$ at the point $(x, y)$.
</document>

<document source="https://tamnd.github.io/math2/node/0SG74WBFA2.md" tag="0SG74WBFA2" kind="concept">
# Möbius transformation

## Summary

A Möbius transformation is a map $z \mapsto \frac{az + b}{cz + d}$ of the Riemann sphere to itself, with $ad - bc \neq 0$, see the [[definition|0B346DDAM5]].
These maps are bijective and holomorphic, they preserve angles, and they [[map circles and lines to circles and lines|0SXDHQM2MH]].
</document>

<document source="https://tamnd.github.io/math2/node/0SGTMBEWRF.md" tag="0SGTMBEWRF" kind="definition">
# Equivalence relation

## Definition

Let $A$ be a set.
An equivalence relation on $A$ is a [[binary relation|08EJC6Q25H]] $\sim$ on $A$ that is [[reflexive|0XEGNW851S]], [[symmetric|0N9SA25Z3V]] and [[transitive|0CYRECG2YQ]]. For all $a, b, c \in A$:

1. $a \sim a$;
2. $a \sim b$ implies $b \sim a$;
3. $a \sim b$ and $b \sim c$ imply $a \sim c$.

## Remarks

For a fixed integer $n \ge 1$, congruence modulo $n$ on $\mathbb{Z}$ is an equivalence relation: $a \sim b$ when $n$ divides $a - b$.
An equivalence relation cuts the set into [[equivalence classes|082RMRSY26]], and these form a [[partition|09CCE86A6D]].
Each [[function|0XHDTE5YMZ]] gives an equivalence relation. See [[equivalence relation induced by a function|0QE5PFJY7V]].

## Explanations

### L1 human

An equivalence relation says when two objects are "the same for our purpose".
Two integers are congruent modulo $2$ when they have the same parity.
Then all even numbers count as one kind, and all odd numbers as another.
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<document source="https://tamnd.github.io/math2/node/0SH5VT8QYA.md" tag="0SH5VT8QYA" kind="statement">
# Basis of a tensor product

## Statement

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$.
Let $(e_i)_{i \in I}$ be a [[basis|0KCQ9Y3QKA]] of $V$ and let $(f_j)_{j \in J}$ be a basis of $W$.
Then the family $(e_i \otimes f_j)_{(i, j) \in I \times J}$ is a basis of the [[tensor product|0BF2DE2SYK]] $V \otimes W$.

## Remarks

So each vector of $V \otimes W$ is uniquely $\sum_{i,j} c_{ij}\, e_i \otimes f_j$ with only finitely many nonzero $c_{ij} \in F$.
For $v = \sum_i a_i e_i$ and $w = \sum_j b_j f_j$, $v \otimes w = \sum_{i,j} a_i b_j\, e_i \otimes f_j$.
Thus the coefficients of a vector $v \otimes w$ form a matrix of rank at most $1$.

## Explanations

### L1 human

From a basis of each factor, we get a basis of the tensor product.
Its vectors are all the products of a basis vector of $V$ and a basis vector of $W$.
A vector of $V \otimes W$ is like a matrix of coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/0SHC6Z304Z.md" tag="0SHC6Z304Z" kind="statement">
# Closure is the smallest closed superset

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then $S \subseteq \overline{S}$, the set $\overline{S}$ is closed, and each closed set $F$ of $X$ with $S \subseteq F$ satisfies $\overline{S} \subseteq F$.

## Remarks

Here $\overline{S}$ is the [[closure|0DRKT1QJR7]] of $S$. That it is closed is the statement [[closure is closed|02TZDKG50Q]].
This is the dual of the statement that the [[interior is the largest open subset|0HAZEFH137]].
If $S \subseteq T$, then $\overline{S} \subseteq \overline{T}$.

## Explanations

### L1 human

Among all closed sets that contain $S$, the closure is the smallest one.
For example, each closed set that contains $(0, 1)$ in $\mathbb{R}$ also contains $[0, 1]$.
</document>

<document source="https://tamnd.github.io/math2/node/0SHMC00RJV.md" tag="0SHMC00RJV" kind="definition">
# Split polynomial

## Definition

Let $L$ be a [[field|0QA8CHH8MG]] and let $f \in L[x]$ be nonzero.
The polynomial $f$ splits over $L$ if there are $c, a_1, \dots, a_n \in L$ with
$$f = c \, (x - a_1)(x - a_2) \cdots (x - a_n).$$

## Remarks

Then $c$ is the leading coefficient of $f$, $n = \deg f$, and $a_1, \dots, a_n$ are the [[roots|00EHX6FWC0]] of $f$, not necessarily distinct.
The polynomial $x^2 - 2$ splits over $\mathbb{R}$, but not over $\mathbb{Q}$.
The polynomial $x^2 + 1$ splits over $\mathbb{C}$, but not over $\mathbb{R}$.
Over an [[algebraically closed field|02SJG1HC32]], every nonzero polynomial splits.

## Explanations

### L1 human

A polynomial splits when you can factor it completely into linear factors.
Then you can read all its roots from the factors.
</document>

<document source="https://tamnd.github.io/math2/node/0SHPXM2HWH.md" tag="0SHPXM2HWH" kind="example">
# Pendulum equation

## Example

Let $g > 0$ be the acceleration of gravity and $\ell > 0$ the length of a rigid massless rod, with a point mass at its end. Let $\theta(t)$ be the angle of the rod from the downward vertical at time $t$.
Without friction, the motion satisfies the pendulum equation
$$\theta''(t) + \frac{g}{\ell} \sin \theta(t) = 0.$$
As a first-order system, it is $\theta' = \omega$, $\omega' = -\frac{g}{\ell} \sin \theta$. The function $E(\theta, \omega) = \frac{1}{2} \omega^2 - \frac{g}{\ell} \cos \theta$ is a [[first integral|0SS1P1YC9A]].

## Remarks

For small angles, $\sin \theta \approx \theta$. This gives the [[harmonic oscillator|0YG0ZJPTWG]] $\theta'' + \frac{g}{\ell} \theta = 0$, with the solution $\theta(t) = \theta_0 \cos\left(\sqrt{g/\ell}\, t\right)$ for $\theta(0) = \theta_0$, $\theta'(0) = 0$, and the period $2 \pi \sqrt{\ell / g}$.
For the nonlinear equation, the period depends on the amplitude. It is given by an elliptic integral and it is longer than $2 \pi \sqrt{\ell / g}$.
The equilibrium $(0, 0)$, hanging down, is stable. The equilibrium $(\pi, 0)$, standing up, is a [[hyperbolic|0C5PREPZWG]] saddle, so it is unstable.

## Explanations

### L1 human

Gravity pulls the mass back toward the lowest point, with a force proportional to $\sin \theta$.
For small swings, this is nearly proportional to $\theta$, and the pendulum behaves like a spring.
For large swings, the motion is slower, and with enough energy the pendulum turns over the top.
</document>

<document source="https://tamnd.github.io/math2/node/0SJ3RX9TXA.md" tag="0SJ3RX9TXA" kind="definition">
# Set

## Definition

A set is a collection of objects, called the elements of the set.
For an object $x$ and a set $A$, either $x$ is an [[element|0EZ758XE73]] of $A$, written $x \in A$, or it is not, written $x \notin A$.
Two sets $A$ and $B$ are equal when they have the same elements: for each $x$, $x \in A$ if and only if $x \in B$.
A finite set can be written by a list of its elements in braces, for example $\{1, 2, 3\}$.

## Remarks

The sets $\{1, 2, 3\}$, $\{3, 2, 1\}$ and $\{1, 1, 2, 3\}$ are the same set.
In [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], "set" and "$\in$" are primitive notions, and the axioms say how they behave.
Not every collection can be a set. [[Russell's paradox|0HESMMSYEG]] shows this.

## Explanations

### L1 human

A set is like a bag of objects, where only membership counts.
The bag $\{1, 2, 3\}$ contains $1$, $2$ and $3$, and nothing else.
You cannot put the same object in twice, and the objects have no order.
</document>

<document source="https://tamnd.github.io/math2/node/0SJ7Z6VKWK.md" tag="0SJ7Z6VKWK" kind="definition">
# Continued fraction expansion of a real number

## Definition

Let $x$ be a real number.
Put $x_0 = x$ and $a_0 = \lfloor x_0 \rfloor$.
For $n \ge 0$, if $x_n \neq a_n$, put $x_{n+1} = \dfrac{1}{x_n - a_n}$ and $a_{n+1} = \lfloor x_{n+1} \rfloor$. If $x_n = a_n$, the process stops.
The continued fraction expansion of $x$ is the [[simple continued fraction|06XF7BCFFJ]] $[a_0; a_1, a_2, \dots]$, finite or infinite.

## Remarks

Here $\lfloor y \rfloor$ is the largest integer that is at most $y$. Since $0 < x_n - a_n < 1$, each $a_{n+1}$ with $n \ge 0$ is a positive integer.
For example, $\frac{43}{19} = [2; 3, 1, 4]$, and $\sqrt{2} = [1; 2, 2, 2, \dots]$.
The process stops if and only if $x$ is rational, by the [[termination theorem|0PMY0YQYXX]].
For a rational $x$, the steps are the steps of the [[Euclidean algorithm|0TRNT4X7J0]] on its numerator and denominator.

## Explanations

### L1 human

Take the integer part of $x$, then flip the remaining fractional part, and repeat.
The integer parts that you collect are the terms of the continued fraction of $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0SJNCJCCQG.md" tag="0SJNCJCCQG" kind="concept">
# Falling factorial

## Summary

The falling factorial $(x)_k$ is the product of $k$ consecutive descending factors that start at $x$. It counts ordered selections without repetition, and it is a basis of the polynomials that is useful in finite difference calculus. The node [[falling factorial|0N5M924N3P]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0SK1VHZ5R2.md" tag="0SK1VHZ5R2" kind="statement">
# Irreducibility of the cyclotomic polynomials over the rationals

## Statement

For every integer $n \ge 1$, the [[cyclotomic polynomial|0W0KVBM54S]] $\Phi_n$ is [[irreducible|00X2A9G9W3]] in $\mathbb{Q}[x]$.

## Remarks

For $n = p$ prime, [[Eisenstein's criterion|02BPKD7TMD]] applied to $\Phi_p(x + 1)$ with the prime $p$ proves the statement.
The general case is harder. One proof shows that if $\zeta$ is a root of an irreducible factor of $\Phi_n$, then so is $\zeta^p$ for each prime $p \nmid n$.
A consequence: $\Phi_n$ is the [[minimal polynomial|0GK1WB9R98]] over $\mathbb{Q}$ of each primitive $n$-th root of unity. So the degree of $\mathbb{Q}(e^{2\pi i/n})$ over $\mathbb{Q}$ is $\varphi(n)$.

## Explanations

### L1 human

The polynomial $\Phi_n$ cannot be split into smaller polynomials with rational coefficients.
So all the primitive $n$-th roots of unity are conjugate over the rationals: no rational polynomial equation separates them.
</document>

<document source="https://tamnd.github.io/math2/node/0SK90238BH.md" tag="0SK90238BH" kind="definition">
# Sign of a permutation

## Definition

Let $n \geq 2$ and let $\sigma \in S_n$.
Write $\sigma = \tau_1 \tau_2 \cdots \tau_m$ as a product of [[transpositions|0FWVRHBRWM]].
The sign of $\sigma$ is $\operatorname{sgn}(\sigma) = (-1)^m$.

## Remarks

The parity of $m$ does not depend on the decomposition, so the sign is well defined.
Equivalently, $\operatorname{sgn}(\sigma) = (-1)^{N(\sigma)}$, where $N(\sigma)$ is the number of pairs $i < j$ with $\sigma(i) > \sigma(j)$.
The sign of a permutation is the determinant of its permutation matrix.

## Explanations

### L1 human

Each permutation is a product of swaps, and the number of swaps is always even or always odd.
The sign records this: $+1$ for an even number and $-1$ for an odd number.
A single swap has sign $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/0SKVEMXJWF.md" tag="0SKVEMXJWF" kind="statement">
# Stability of linear systems via eigenvalues

## Statement

Let $A$ be a real $n \times n$ matrix and consider the [[linear system with constant coefficients|0VJF99CZ0Z]] $x' = A x$ on $\mathbb{R}^n$, with the equilibrium point $0$.
(1) The point $0$ is [[asymptotically stable|05Y1JAM3Y5]] if and only if each eigenvalue $\lambda$ of $A$ has $\operatorname{Re} \lambda < 0$.
(2) If some eigenvalue $\lambda$ of $A$ has $\operatorname{Re} \lambda > 0$, then $0$ is unstable.
(3) The point $0$ is [[Lyapunov stable|092S6VBECQ]] if and only if two conditions hold. Each eigenvalue has $\operatorname{Re} \lambda \le 0$. Each eigenvalue with $\operatorname{Re} \lambda = 0$ has equal algebraic and geometric multiplicities.

## Remarks

The proof uses the [[matrix exponential|0YC8JT513J]]. The entries of $e^{tA}$ are sums of terms $t^k e^{\lambda t}$. Here $\lambda$ is an eigenvalue and $k$ is less than the size of its largest Jordan block.
In case (1), there are constants $C, \alpha > 0$ with $\|e^{tA}\| \le C e^{-\alpha t}$ for $t \ge 0$. So the origin is globally and exponentially stable.
For $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, both eigenvalues are $0$ but the solution $(t, 1)$ is unbounded. So $0$ is unstable, in accordance with (3).
The Routh–Hurwitz criterion tests the condition of (1) from the coefficients of the characteristic polynomial, without computing the roots.

## Explanations

### L1 human

For a linear system, the eigenvalues decide everything.
Eigenvalues with negative real part give decaying solutions, and an eigenvalue with positive real part gives a growing solution.
</document>

<document source="https://tamnd.github.io/math2/node/0SKYFQAFYD.md" tag="0SKYFQAFYD" kind="concept">
# General solution and particular solution

## Summary

The general solution of an [[ordinary differential equation|0P97V20RW7]] of order $n$ is a family of [[solutions|04SW7WECS8]] that depends on $n$ arbitrary constants $C_1, \dots, C_n$.
In the usual case, each solution of the equation is a member of the family, with the possible exception of [[singular solutions|0Q2MDD13G4]].
A particular solution is one solution of the equation. It is often the member of the general solution for one choice of the constants.

## Remarks

The general solution of $y'' + y = 0$ is $y = C_1 \cos x + C_2 \sin x$. The choice $C_1 = 1$, $C_2 = 0$ gives the particular solution $y = \cos x$.
An [[initial value problem|075VBY4K4X]] fixes the constants. For an equation of order $n$, one needs $n$ initial conditions.
For a linear equation, the [[general solution of the inhomogeneous equation|0GHWEH6H15]] is one particular solution plus the general solution of the homogeneous equation.

## Explanations

### L1 human

The general solution is the list of all answers, written with free constants.
A particular solution is one answer from the list.
For $y' = 2x$, the general solution is $y = x^2 + C$, and $y = x^2 + 5$ is a particular solution.
</document>

<document source="https://tamnd.github.io/math2/node/0SM1REAVYJ.md" tag="0SM1REAVYJ" kind="statement">
# Edge bound for planar graphs

## Statement

Let $G$ be a finite simple connected [[planar graph|029JQEK15B]] with $v \ge 3$ vertices and $e$ edges.
Then $e \le 3v - 6$.

## Remarks

Proof idea: each face is bounded by at least $3$ edges, and each edge touches at most $2$ faces. Thus $3f \le 2e$, and [[Euler's formula|0W1C598HMS]] gives the bound.
If $G$ also has no cycle of length $3$, then $e \le 2v - 4$.
A consequence: each finite simple planar graph has a vertex of [[degree|0T0F6CCS9W]] at most $5$.

## Explanations

### L1 human

A planar graph cannot have too many edges: at most about three times its number of vertices.
This simple count already shows that some graphs, such as $K_5$, are not planar.
</document>

<document source="https://tamnd.github.io/math2/node/0SN0MAEKYX.md" tag="0SN0MAEKYX" kind="definition">
# Prime number

## Definition

A prime number is a natural number $p$ such that $p > 1$ and the only positive divisors of $p$ are $1$ and $p$.

## Remarks

The first prime numbers are $2, 3, 5, 7, 11, 13$. The number $2$ is the only even prime.
The number $1$ is not prime by this definition. This choice makes the [[prime factorization|0AEKDFCYFA]] of each integer unique.
A natural number $n > 1$ that is not prime is a [[composite number|0N3ADXHNTQ]].
An equivalent property for $p > 1$ is [[Euclid's lemma|051VYKA1EX]]: if $p \mid a b$, then $p \mid a$ or $p \mid b$.

## Explanations

### L1 human

A prime number is a number greater than $1$ that you cannot split as a product of two smaller natural numbers.
For example, $7$ is prime, but $6 = 2 \cdot 3$ is not.
The primes are the building blocks of the natural numbers under multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/0SNJM8HSD3.md" tag="0SNJM8HSD3" kind="statement">
# Erdős–Szekeres theorem

## Statement

Let $r$ and $s$ be positive integers.
Let $a_1, a_2, \dots, a_N$ be a sequence of $N = (r - 1)(s - 1) + 1$ distinct real numbers.
Then the sequence has an increasing subsequence of length $r$ or a decreasing subsequence of length $s$.

## Remarks

The bound is sharp: some sequence of $(r - 1)(s - 1)$ distinct numbers has neither type of subsequence.
For $r = s = 3$: each sequence of $5$ distinct numbers has a monotone subsequence of length $3$.
A short proof labels each term with the lengths of the longest increasing and decreasing subsequences that end at it. The labels are distinct, and the [[pigeonhole principle|0HGVCWQGY3]] finishes the proof.

## History

Paul Erdős and George Szekeres proved the theorem in 1935.

## Explanations

### L1 human

A long enough list of different numbers always contains a long subsequence that goes up or one that goes down. The terms need not be adjacent.
For example, $3, 1, 4, 5, 2$ has the increasing subsequence $3, 4, 5$.
</document>

<document source="https://tamnd.github.io/math2/node/0SNQEF7Z53.md" tag="0SNQEF7Z53" kind="definition">
# Bijective function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The function $f$ is bijective when it is [[injective|0PVG1CRAJN]] and [[surjective|0XKH881BZ7]].
Equivalently, for each $b \in B$, there is exactly one $a \in A$ with $f(a) = b$.

## Remarks

Other names are bijection and one-to-one correspondence.
The function $x \mapsto 2x + 1$ on $\mathbb{R}$ is bijective.
A function is bijective if and only if it has an [[inverse function|0W766JAPSN]]. See [[bijective iff it has a two-sided inverse|01VG2A0BHW]].
Two finite sets have the same number of elements exactly when there is a bijection between them.

## Explanations

### L1 human

A bijection pairs the elements of $A$ and the elements of $B$ perfectly.
Each element of $A$ has one partner in $B$, and each element of $B$ has one partner in $A$.
</document>

<document source="https://tamnd.github.io/math2/node/0SP14DZN0G.md" tag="0SP14DZN0G" kind="definition">
# Primitive polynomial

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
A polynomial $f \in R[x]$ is primitive if every $r \in R$ that divides $f$ in $R[x]$ is a unit.
Here $r$ divides $f$ when $r$ divides each coefficient of $f$.

## Remarks

If $R$ is a [[unique factorisation domain|06YMXJMXYS]], then $f$ is primitive if and only if its [[content|08VV7YJRB7]] is a unit.
In $\mathbb{Z}[x]$, $3x^2 + 2x - 1$ is primitive and $6x^2 + 4x - 2$ is not.
Every [[monic polynomial|0BDX4MBKT4]] is primitive.
The word primitive polynomial has another meaning over a finite field: a polynomial whose root is a [[primitive element|0YAA9CJQXF]].

## Explanations

### L1 human

A primitive polynomial has coefficients with no common factor.
Over the integers, this means that the gcd of the coefficients is $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0SP7P0G7KQ.md" tag="0SP7P0G7KQ" kind="statement">
# A space is the direct sum of a subspace and its orthogonal complement

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, and let $W$ be a finite-dimensional subspace of $V$.
Then
$$V = W \oplus W^\perp,$$
where $W^\perp$ is the [[orthogonal complement|0XJXTBCB2T]] of $W$ and $\oplus$ is the [[direct sum|0XYYT5KNTX]].

## Remarks

For an orthonormal basis $e_1, \dots, e_k$ of $W$, each $v \in V$ is $w + (v - w)$ with $w = \sum_i \langle v, e_i \rangle e_i \in W$ and $v - w \in W^\perp$.
The statement holds for each closed subspace of a Hilbert space.
It can fail for a subspace that is not closed in an infinite-dimensional space.

## Explanations

### L1 human

Each vector splits in exactly one way into a part in $W$ and a part perpendicular to $W$.
The part in $W$ is the shadow of the vector on $W$.
The part perpendicular to $W$ is what remains.
</document>

<document source="https://tamnd.github.io/math2/node/0SPBEDVNW1.md" tag="0SPBEDVNW1" kind="statement">
# Existence of a prime divisor

## Statement

Let $n$ be a natural number with $n \ge 2$.
Then there is a [[prime number|0SN0MAEKYX]] $p$ such that $p \mid n$.

## Remarks

One proof takes the smallest divisor $d$ of $n$ with $d > 1$. Each divisor $e$ of $d$ with $e > 1$ also divides $n$, so $e = d$. Thus $d$ is prime.
This smallest divisor is the [[smallest prime factor|0S2SYHTWPY]] of $n$.
The result is also true for $n = 0$, since each prime divides $0$. It is false for $n = 1$.

## Explanations

### L1 human

Split $n$ into two smaller factors when you can, and repeat with one of the factors.
The numbers get smaller each time, so the process stops at a number that you cannot split. This number is a prime, and it divides $n$.
For example, $60 = 2 \cdot 30$, so $2$ is a prime divisor of $60$.
</document>

<document source="https://tamnd.github.io/math2/node/0SPBPW3K61.md" tag="0SPBPW3K61" kind="definition">
# Hypergeometric distribution

## Definition

Let $N$, $K$ and $n$ be integers with $N \ge 1$, $0 \le K \le N$ and $0 \le n \le N$.
The hypergeometric distribution with parameters $N$, $K$ and $n$ has the [[probability mass function|0WF4EJP91W]]
$$p(k) = \frac{\binom{K}{k} \binom{N - K}{n - k}}{\binom{N}{n}} \quad \text{for } \max(0, n + K - N) \le k \le \min(n, K).$$

## Remarks

A population has $N$ objects, and $K$ of them are successes. Draw $n$ objects without replacement, each subset of size $n$ equally likely. Then the number of successes in the sample has this distribution.
The mean is $nK/N$. The variance is $n \frac{K}{N} \frac{N - K}{N} \frac{N - n}{N - 1}$ for $N \ge 2$.
When $N$ is large compared with $n$, the distribution is close to the [[binomial distribution|0VRWZDWW6B]] with parameters $n$ and $K/N$.
The probability of exactly $2$ aces in a hand of $5$ cards is $\binom{4}{2} \binom{48}{3} / \binom{52}{5}$.

## Explanations

### L1 human

An urn has red and blue balls. Draw a few balls without putting them back, and count the red ones.
The count has the hypergeometric distribution. Each draw changes the urn, so the draws are not independent.
</document>

<document source="https://tamnd.github.io/math2/node/0SPE8WNWFV.md" tag="0SPE8WNWFV" kind="concept">
# Connected set

## Summary

A connected set is a subset of a topological space that is connected in the subspace topology.
</document>

<document source="https://tamnd.github.io/math2/node/0SPYAGE7T0.md" tag="0SPYAGE7T0" kind="statement">
# Change of basis for a bilinear form

## Statement

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$, with two bases $\mathcal{B}$ and $\mathcal{B}'$.
Let $P$ be the [[change of basis matrix|0MKRHZB3WS]] whose columns are the coordinate vectors of the vectors of $\mathcal{B}'$ in $\mathcal{B}$.
Let $B$ be a bilinear form on $V$, with [[matrix|0QK6K857HS]] $M$ in $\mathcal{B}$ and $M'$ in $\mathcal{B}'$.
Then
$$M' = P^T M P.$$

## Remarks

Two square matrices $M$ and $M'$ are *congruent* when $M' = P^T M P$ for an invertible matrix $P$.
So two matrices represent the same bilinear form in two bases if and only if they are congruent.
Congruence keeps the rank and the symmetry of a matrix, but in general not its eigenvalues.
Compare this with the rule $P^{-1} A P$ for the matrix of a linear map.

## Explanations

### L1 human

When we change the basis, the matrix of a bilinear form changes by $P^T M P$.
Both vectors change coordinates, so $P$ appears twice.
For a linear map, the rule is $P^{-1} A P$ instead.
</document>

<document source="https://tamnd.github.io/math2/node/0SPYEFH7H4.md" tag="0SPYEFH7H4" kind="statement">
# The Gaussian integers form a Euclidean domain

## Statement

Let $\alpha$ and $\beta$ be [[Gaussian integers|0B268GVX95]] with $\beta \neq 0$.
Then there are Gaussian integers $q$ and $r$ with $\alpha = q \beta + r$ and $N(r) < N(\beta)$, where $N$ is the [[norm|00HPT88ZX0]].
So $\mathbb{Z}[i]$, with the norm as Euclidean function, is a Euclidean domain.

## Remarks

To find $q$, round the real part and the imaginary part of the complex number $\alpha / \beta$ to nearest integers. This gives even $N(r) \le \frac{1}{2} N(\beta)$.
As a consequence, $\mathbb{Z}[i]$ is a principal ideal domain and a unique factorization domain.
The quotient and the remainder are not always unique.
This is the analogue of the [[division algorithm|0CHQKZFK5S]] for integers.

## History

Carl Friedrich Gauss introduced the Gaussian integers in 1832, in his second paper on biquadratic reciprocity. In it, he proved unique factorization.

## Explanations

### L1 human

You can divide one Gaussian integer by another and get a remainder that is smaller in norm than the divisor.
So the Euclidean algorithm works in $\mathbb{Z}[i]$, and factorization into primes is unique, as for the integers.
</document>

<document source="https://tamnd.github.io/math2/node/0SPZ1H3N9V.md" tag="0SPZ1H3N9V" kind="concept">
# Refinement of a partition

## Summary

A refinement of a partition is a partition that contains all the points of the given one, and possibly more. Two partitions always have a common refinement. The node [[refinement of a partition|0EEKF7YFBW]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0SQ1F2BEDS.md" tag="0SQ1F2BEDS" kind="definition">
# Branch of the logarithm

## Definition

Let $U$ be an open subset of $\mathbb{C} \setminus \{0\}$.
A branch of the logarithm on $U$ is a continuous function $L : U \to \mathbb{C}$ such that $e^{L(z)} = z$ for each $z \in U$.
Thus $L(z)$ is a [[complex logarithm|0X2FY491PF]] of $z$ for each $z \in U$.

## Remarks

Each branch $L$ of the logarithm on $U$ is holomorphic on $U$, with $L'(z) = 1/z$.
If $U$ is a [[domain|097KNJFJ7V]] and $L_1, L_2$ are two branches on $U$, then $L_1 - L_2$ is the constant $2 \pi i k$ for one $k \in \mathbb{Z}$.
There is no branch of the logarithm on $\mathbb{C} \setminus \{0\}$, or on any open set that contains a circle with center $0$.
The [[principal branch|0YKY4M3KRQ]] $\operatorname{Log}$ is a branch on $\mathbb{C} \setminus (-\infty, 0]$. The functions $\operatorname{Log} z + 2 \pi i k$ with $k \in \mathbb{Z}$ are the other branches on this set.

## Explanations

### L1 human

The logarithm has many values at each point, so to get a function we must choose one value at each point, and the choice must change continuously.
Such a choice is a branch.
If you go once around $0$, the argument grows by $2\pi$, so no continuous choice is possible on a full circle around $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0SQ2PC596Y.md" tag="0SQ2PC596Y" kind="definition">
# Open map

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]].
A function $f \colon X \to Y$ is an open map when for each [[open set|01ZQQDCERP]] $U$ of $X$, the [[image|0Q3F20XB0A]] $f(U)$ is open in $Y$.

## Remarks

The [[projections of a product|0PM88Z9W4M]] $X \times Y \to X$ are open maps.
A continuous bijection is a [[homeomorphism|05GZ11PWRY]] if and only if it is an open map.
The map $\mathbb{R} \to \mathbb{R}$, $x \mapsto x^2$, is continuous but not open: the image of $(-1, 1)$ is $[0, 1)$.

## Explanations

### L1 human

An open map keeps open sets open when you push them forward.
Continuity is about pulling open sets back, so the two notions are different.
</document>

<document source="https://tamnd.github.io/math2/node/0SQDB87Z3T.md" tag="0SQDB87Z3T" kind="definition">
# Quadratic residue

## Definition

Let $n$ be a positive integer and let $q$ be an integer.
Then $q$ is a quadratic residue modulo $n$ if there is an integer $x$ with $x^2 \equiv q \pmod n$.
Otherwise $q$ is a quadratic nonresidue modulo $n$.

## Remarks

Many texts also require $\gcd(q, n) = 1$, and so they do not count $0$ as a quadratic residue.
Modulo $7$, the squares of $1, \dots, 6$ are $1, 4, 2, 2, 4, 1$. So the nonzero quadratic residues modulo $7$ are $1$, $2$ and $4$.
Modulo an odd prime $p$, exactly $(p - 1)/2$ of the classes $1, \dots, p - 1$ are quadratic residues.
The product of two quadratic residues is a quadratic residue.

## Explanations

### L1 human

A quadratic residue modulo $n$ is a number that is a perfect square in the arithmetic modulo $n$.
For example, $2$ is not a square in the integers, but $3^2 = 9 \equiv 2 \pmod 7$. So $2$ is a quadratic residue modulo $7$.
</document>

<document source="https://tamnd.github.io/math2/node/0SQSQP748R.md" tag="0SQSQP748R" kind="statement">
# Principal value of the argument

## Statement

Let $z$ be a complex number with $z \neq 0$.
Then $z$ has exactly one [[argument|0ZZRSX91QR]] $\theta$ with $-\pi < \theta \le \pi$.
This argument is the principal value of the argument, written $\operatorname{Arg} z$.

## Remarks

Proof: the arguments of $z$ form a set $\theta_0 + 2 \pi \mathbb{Z}$, and each interval $(-\pi, \pi]$ of length $2\pi$ contains exactly one element of such a set.
For example, $\operatorname{Arg}(-1) = \pi$, $\operatorname{Arg}(-i) = -\pi/2$ and $\operatorname{Arg} x = 0$ for real $x > 0$.
Some authors use the interval $[0, 2\pi)$ instead.

## Explanations

### L1 human

Many angles describe the same direction, because full turns do not change it.
The principal value picks one of them, the one between $-180$ and $180$ degrees, with $180$ allowed and $-180$ not.
</document>

<document source="https://tamnd.github.io/math2/node/0SQTMQTSRZ.md" tag="0SQTMQTSRZ" kind="concept">
# Even and odd permutations

## Summary

A permutation of a finite set is even when it is a product of an even number of transpositions, and odd otherwise.
The even permutations form the alternating group.
</document>

<document source="https://tamnd.github.io/math2/node/0SQV2VJTGY.md" tag="0SQV2VJTGY" kind="concept">
# Parallelepiped

## Summary

A parallelepiped is a solid in three-dimensional space with six faces, each of which is a parallelogram. It relates to a parallelogram as a cube relates to a square.
</document>

<document source="https://tamnd.github.io/math2/node/0SQZMXB6GP.md" tag="0SQZMXB6GP" kind="concept">
# Bilinear form

## Summary

A bilinear form on a vector space is a map of two vectors to the scalars.
It is linear in each vector when the other vector is fixed.
</document>

<document source="https://tamnd.github.io/math2/node/0SRATCFF1W.md" tag="0SRATCFF1W" kind="statement">
# Subgroups of the integers

## Statement

Let $H$ be a [[subgroup|0X1JXT47GA]] of the [[additive group of integers|0BSCJA5G53]] $(\mathbb{Z}, +)$.
Then there is exactly one integer $n \geq 0$ with $H = n\mathbb{Z} = \{nk : k \in \mathbb{Z}\}$.

## Remarks

When $H \neq \{0\}$, the integer $n$ is the smallest positive element of $H$.
For $m, n \geq 0$, $m\mathbb{Z} \subseteq n\mathbb{Z}$ holds exactly when $n$ divides $m$.
A consequence is that, for integers $a$ and $b$, the subgroup $a\mathbb{Z} + b\mathbb{Z}$ is $d\mathbb{Z}$ with $d = \gcd(a, b)$, which gives Bezout's identity.

## Explanations

### L1 human

Each subgroup of the integers is the set of multiples of one number.
For example, the set of all sums $4x + 6y$ with integers $x, y$ is the set of multiples of $2$.
The number $0$ gives the trivial subgroup $\{0\}$.
</document>

<document source="https://tamnd.github.io/math2/node/0SRE7K4E49.md" tag="0SRE7K4E49" kind="statement">
# Characteristic subgroups are normal

## Statement

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[characteristic subgroup|0KABPMRHM2]] of $G$.
Then $H$ is a [[normal subgroup|0CB0DX5BVV]] of $G$.

## Remarks

Each [[inner automorphism|0RN00NTM3S]] $c_g$ is an automorphism, so $gHg^{-1} = c_g(H) = H$ for each $g \in G$.
The converse is false: in the Klein four-group $V = \{e, a, b, ab\}$, each subgroup is normal, but an automorphism can swap $\{e, a\}$ and $\{e, b\}$.

## Explanations

### L1 human

Normal means fixed by the inner automorphisms, and characteristic means fixed by all automorphisms.
All automorphisms include the inner ones.
So a characteristic subgroup is normal.
</document>

<document source="https://tamnd.github.io/math2/node/0SRWNR42BQ.md" tag="0SRWNR42BQ" kind="definition">
# Convergents of a continued fraction

## Definition

Let $[a_0; a_1, a_2, \dots]$ be a [[simple continued fraction|06XF7BCFFJ]].
For each $n$ such that $a_n$ exists, the $n$-th convergent is the rational number $\dfrac{p_n}{q_n} = [a_0; a_1, \dots, a_n]$.
Here $p_n$ and $q_n$ are given by $p_{-2} = 0$, $p_{-1} = 1$, $q_{-2} = 1$, $q_{-1} = 0$ and
$$p_n = a_n p_{n-1} + p_{n-2}, \qquad q_n = a_n q_{n-1} + q_{n-2} \quad (n \ge 0).$$

## Remarks

The recurrences give $p_n q_{n-1} - p_{n-1} q_n = (-1)^{n-1}$. So $\gcd(p_n, q_n) = 1$, and the fraction $p_n / q_n$ is in lowest terms.
For $\sqrt{2} = [1; 2, 2, \dots]$, the convergents are $1, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \frac{41}{29}, \dots$.
For a general [[continued fraction|0BK6JBK58W]], the same recurrences hold with $p_n = b_n p_{n-1} + a_n p_{n-2}$ and $q_n = b_n q_{n-1} + a_n q_{n-2}$.

## Explanations

### L1 human

The convergents are the fractions that you get when you cut a continued fraction off after finitely many terms.
They approach the value of the continued fraction, alternately from below and from above.
</document>

<document source="https://tamnd.github.io/math2/node/0SRYJGXKR4.md" tag="0SRYJGXKR4" kind="concept">
# Convergence in distribution

## Summary

A sequence of random variables converges in distribution when their cumulative distribution functions converge to a limit at each point where the limit is continuous.
The node [[convergence in distribution|0JDAQHWA9N]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0SS1P1YC9A.md" tag="0SS1P1YC9A" kind="definition">
# First integral

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be a function.
Let $H : U \to \mathbb{R}$ be a continuously differentiable function. It is a first integral of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ if
$$\nabla H(x) \cdot f(x) = 0 \quad \text{for each } x \in U,$$
and if $H$ is not constant on any non-empty open subset of $U$.

## Remarks

The condition means that $\frac{d}{dt} H(\gamma(t)) = 0$ for each [[solution|04SW7WECS8]] $\gamma$. So $H$ is constant along each solution.
Each [[trajectory|0BVT6VSB3S]] lies in a level set $\{x : H(x) = c\}$. In the plane, the level curves of $H$ give the phase portrait.
For the [[harmonic oscillator|0YG0ZJPTWG]] $x' = y$, $y' = -x$, the function $H(x, y) = x^2 + y^2$ is a first integral.
The Hamiltonian of a time-independent [[Hamiltonian system|0C2AKPDCD9]] is a first integral.

## Explanations

### L1 human

A first integral is a quantity that does not change while the system moves, such as the energy of a system without friction.
Knowing it tells us that each motion stays on one level set of this quantity.
</document>

<document source="https://tamnd.github.io/math2/node/0SS770NS8B.md" tag="0SS770NS8B" kind="example">
# Differentiation as a linear map

## Example

Let $F$ be a field and let $F[x]$ be the vector space of polynomials over $F$.
The *formal derivative* is the map $D \colon F[x] \to F[x]$ with
$$D\left(\sum_{k=0}^{n} a_k x^k\right) = \sum_{k=1}^{n} k\, a_k x^{k-1}.$$
It is a [[linear map|0NRYHJPQG2]]: $D(p + q) = D(p) + D(q)$ and $D(cp) = c\, D(p)$ for all polynomials $p, q$ and all $c \in F$.

## Remarks

Over a field of characteristic $0$, the kernel of $D$ is the space of constant polynomials.
On the space $F[x]_{<n}$ of polynomials of degree less than $n$, $D$ is nilpotent: $D^n = 0$.
In the basis $1, x, x^2$ of $F[x]_{<3}$, the matrix of $D$ is $\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{pmatrix}$.
Over $\R$, differentiation is also a linear map on the space of differentiable functions.

## Explanations

### L1 human

Differentiation is linear: the derivative of a sum is the sum of the derivatives, and constants come out.
So calculus gives important examples of linear maps.
On polynomials of bounded degree, we can even write differentiation as a matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0SSKBQN58M.md" tag="0SSKBQN58M" kind="statement">
# Hurwitz's theorem

## Statement

Let $G$ be a [[domain|097KNJFJ7V]] in $\mathbb{C}$, that is, a connected open set.
Let $(f_k)$ be a sequence of [[holomorphic|0DG0Z72CQD]] functions on $G$ that converges uniformly on each compact subset of $G$ to a function $f$.
Suppose that $f$ is not identically zero on $G$, and that $f$ has a zero of [[order|064N002TX4]] $m$ at a point $z_0 \in G$.
Then for each sufficiently small $\rho > 0$ there is an integer $N$ with the following property.
For each $k \ge N$, the function $f_k$ has exactly $m$ zeros in the disc $|z - z_0| < \rho$, each counted as many times as its order.

## Remarks

The limit $f$ is holomorphic on $G$ by the [[theorem on uniform limits|0431RN0R1N]].
A consequence: if no $f_k$ has a zero in $G$, then $f$ either has no zero in $G$ or is identically zero.
A second consequence: if each $f_k$ is injective on $G$, then $f$ is injective or constant.
The radius $\rho$ must be small: $f_n(z) = z - 1 + \frac{1}{n}$ has one zero in the unit disc, but its limit $z - 1$ has none there.

## Explanations

### L1 human

Zeros of holomorphic functions do not suddenly appear or disappear under uniform limits.
Near a zero of the limit, each function far along in the sequence has the same number of zeros.
The proof compares $f_k$ with $f$ on a small circle, as in [[Rouché's theorem|0D430QR02W]].
</document>

<document source="https://tamnd.github.io/math2/node/0SSS6ERQ0S.md" tag="0SSS6ERQ0S" kind="concept">
# Estimator

## Summary

An estimator is a rule that computes a guess of an unknown parameter from the data. It is a statistic that one uses for this purpose.
The node [[estimator|0TTNRP5PFQ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0SSSM1D475.md" tag="0SSSM1D475" kind="statement">
# The degree of a constructible number is a power of two

## Statement

Let $z \in \mathbb{C}$ be a [[constructible number|0FMN2MN7TM]].
Then $z$ is algebraic over $\mathbb{Q}$, and $[\mathbb{Q}(z) : \mathbb{Q}] = 2^r$ for some integer $r \ge 0$.

## Remarks

The proof: there is a tower $\mathbb{Q} = F_0 \subseteq F_1 \subseteq \dots \subseteq F_n$ with $z \in F_n$ and $[F_j : F_{j-1}] = 2$ for each $j$. By the [[tower law|0A7KJ68XHT]], $[F_n : \mathbb{Q}] = 2^n$, and $[\mathbb{Q}(z) : \mathbb{Q}]$ divides it.
So the [[minimal polynomial|0GK1WB9R98]] of $z$ over $\mathbb{Q}$ has degree a power of two.
The converse is false: some algebraic numbers whose degree is a power of two are not constructible.
A sufficient condition: the splitting field of the minimal polynomial of $z$ has degree a power of two over $\mathbb{Q}$.

## Explanations

### L1 human

Each construction step at most doubles the degree of the field.
So a number whose minimal polynomial has degree $3$, like $\sqrt[3]{2}$, can never be constructed.
This simple test settles several problems that were open for two thousand years.
</document>

<document source="https://tamnd.github.io/math2/node/0ST21PGDAP.md" tag="0ST21PGDAP" kind="concept">
# Almost surely

## Summary

An event happens almost surely when its probability is $1$. The event can still fail for some outcomes, but these outcomes together have probability $0$.
The node [[almost surely|0R46A1XY60]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ST8QF9T3T.md" tag="0ST8QF9T3T" kind="definition">
# Adjoint of a linear map

## Definition

Let $V$ and $W$ be finite-dimensional inner product spaces over $\R$ or $\C$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *adjoint* of $T$ is the unique linear map $T^* \colon W \to V$ such that
$$\langle T(v), w \rangle = \langle v, T^*(w) \rangle \qquad \text{for all } v \in V \text{ and } w \in W.$$

## Remarks

The existence and uniqueness follow from the [[Riesz representation theorem|04GYPXDXNH]].
It satisfies $(S + T)^* = S^* + T^*$, $(aT)^* = \bar a T^*$, $(ST)^* = T^* S^*$ and $T^{**} = T$.
In orthonormal bases, the matrix of $T^*$ is the [[conjugate transpose|0DXKE10PHF]] of the matrix of $T$.
It satisfies $\ker T^* = (\im T)^\perp$.

## Explanations

### L1 human

The adjoint lets us move a map from one side of an inner product to the other.
For real matrices with the dot product, the adjoint is the transpose: $(Ax) \cdot y = x \cdot (A^T y)$.
For complex matrices, it is the conjugate transpose.
</document>

<document source="https://tamnd.github.io/math2/node/0STK8KEY9G.md" tag="0STK8KEY9G" kind="statement">
# Moments are derivatives of the moment-generating function at zero

## Statement

Let $X$ be a real random variable, and let $h > 0$ with $M_X(t) = E[e^{tX}] < \infty$ for all $t \in (-h, h)$.
Then $X$ has finite [[moments|0PH0VHTQS0]] of all orders, the [[moment-generating function|0NNRNGKMFM]] $M_X$ is infinitely differentiable on $(-h, h)$, and for each integer $k \ge 0$,
$$M_X^{(k)}(0) = E\left[X^k\right].$$
Moreover, $M_X(t) = \sum_{k=0}^{\infty} E[X^k] \, t^k / k!$ for $|t| < h$.

## Remarks

The proof differentiates under the expectation sign. The bound $|x|^k e^{t x} \le C(e^{a x} + e^{-a x})$ for $|t| < a < h$ justifies this by dominated convergence.
Example: for $X \sim \operatorname{Exp}(\lambda)$, $M_X(t) = \lambda/(\lambda - t)$, so $M_X'(0) = 1/\lambda$ and $M_X''(0) = 2/\lambda^2$. So $\operatorname{Var}(X) = 1/\lambda^2$.

## Explanations

### L1 human

This result is the reason for the name: the function "generates" the moments.
To find $E[X^k]$, you take the $k$-th derivative of $M_X$ and put $t = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0STPPDXGME.md" tag="0STPPDXGME" kind="statement">
# C^1 functions are strictly differentiable

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}^m$ be of [[class C^1|0TCBZMEYGN]] on $U$.
Then $f$ is [[strictly differentiable|0146MBQNNE]] at each point $a \in U$, with linear map $Df(a)$.

## Remarks

Proof: let $\varepsilon > 0$ and choose $r > 0$ with $\|Df(z) - Df(a)\| \le \varepsilon$ for $\|z - a\| < r$. Apply the [[mean value inequality|02CYB184RC]] to $g(z) = f(z) - Df(a) z$ on the ball of radius $r$. This gives $\|f(x) - f(y) - Df(a)(x - y)\| \le \varepsilon \|x - y\|$ for $x, y$ in that ball.
For a function of one variable on an open interval, strict differentiability at each point is equivalent to being of class $C^1$.

## Explanations

### L1 human

If the derivative changes continuously, then near $a$ it stays close to $Df(a)$.
So every chord of the graph near $a$ has a slope close to $Df(a)$, which is what strict differentiability asks.
</document>

<document source="https://tamnd.github.io/math2/node/0STX0WRDNG.md" tag="0STX0WRDNG" kind="statement">
# Density of the rationals

## Statement

Let $x$ and $y$ be real numbers with $x < y$.
Then there is a [[rational number|0NCJ5WXN6Z]] $q$ with $x < q < y$.

## Remarks

Proof idea: choose $n$ with $1/n < y - x$ by the [[Archimedean property|0WH916TJNY]]. Then some fraction $m/n$ with $m \in \mathbb{Z}$ lies between $x$ and $y$.
Thus between two real numbers there are infinitely many rational numbers.
The [[irrationals are also dense|04J6G4H78J]].

## Explanations

### L1 human

Between any two different real numbers, however close, there is a fraction.
For example, between $\sqrt{2} = 1.41421\dots$ and $1.4143$ lies the fraction $1.41425 = 28285/20000$.
</document>

<document source="https://tamnd.github.io/math2/node/0SV8K0TDGY.md" tag="0SV8K0TDGY" kind="concept">
# Sigma-algebra

## Summary

A σ-algebra on a set is a family of subsets that contains the whole set and is closed under complements and countable unions.
In probability theory, the σ-algebra is the family of events, the sets to which we give a probability. The node [[sigma-algebra|04WSW8T3SY]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0SVN6X695K.md" tag="0SVN6X695K" kind="statement">
# Almost sure convergence implies convergence in probability

## Statement

Let $X, X_1, X_2, \dots$ be real random variables on one probability space.
If $X_n \to X$ [[almost surely|04P7D63W8P]], then $X_n \to X$ [[in probability|0EFDDXTVD6]].

## Remarks

Proof: fix $\varepsilon > 0$ and let $A_n = \bigcup_{m \ge n} \{|X_m - X| > \varepsilon\}$. The events $A_n$ decrease, and their intersection has probability $0$ by almost sure convergence.
By the [[continuity of probability|0X8HBKCSYB]], $P(A_n) \to 0$. Since $\{|X_n - X| > \varepsilon\} \subseteq A_n$, the claim follows.
The converse is false. Let $X_n$ be independent with $P(X_n = 1) = 1/n$ and $P(X_n = 0) = 1 - 1/n$. Then $X_n \to 0$ in probability.
But by the second Borel–Cantelli lemma, $X_n = 1$ for infinitely many $n$ almost surely, so $X_n$ does not converge to $0$ almost surely.

## Explanations

### L1 human

If almost every run of the experiment gives a sequence that settles down at $X$, then for large $n$ a large deviation becomes unlikely.
</document>

<document source="https://tamnd.github.io/math2/node/0SVR29SV1V.md" tag="0SVR29SV1V" kind="definition">
# Quotient space

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $W$ be a [[linear subspace|00YC1D652K]] of $V$.
For $v \in V$, the *coset* of $v$ is $v + W = \{v + w : w \in W\}$.
The *quotient space* $V / W$ is the set of all cosets $v + W$ with $v \in V$, with the operations
$$(u + W) + (v + W) = (u + v) + W, \qquad a(v + W) = av + W$$
for $u, v \in V$ and $a \in F$.

## Remarks

The operations are well defined, because $W$ is a subspace: the result does not depend on the choice of $u$ and $v$ in their cosets.
Two cosets $u + W$ and $v + W$ are equal if and only if $u - v \in W$.
The zero vector of $V / W$ is the coset $0 + W = W$.
In $\R^2$ with $W$ the $x$-axis, the cosets are the horizontal lines, and $V / W$ is isomorphic to $\R$.

## Explanations

### L1 human

In the quotient space $V / W$, we treat two vectors as the same when they differ by a vector of $W$.
Each element of $V / W$ is a copy of $W$ shifted by some vector.
Dividing the plane by a line gives the set of all lines parallel to it, which behaves like a single line.
</document>

<document source="https://tamnd.github.io/math2/node/0SVX98HYV1.md" tag="0SVX98HYV1" kind="statement">
# Primitive element theorem

## Statement

Let $L/K$ be a [[finite|0NRYPM9Y27]] [[separable extension|0N019F6RE9]].
Then $L/K$ is a [[simple extension|0P6SBR7YM2]]: there is $\theta \in L$ with $L = K(\theta)$.

## Remarks

Every finite extension of a field of characteristic $0$ is separable, so it is simple. For example, $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2} + \sqrt{3})$.
For an infinite field $K$, the proof shows that $\theta = a + c b$ works for all but finitely many $c \in K$.
For finite fields, a [[primitive element|0YAA9CJQXF]] of $L$ generates $L$ over $K$.
Without separability the statement can fail: there are finite extensions of degree $p^2$ in characteristic $p$ with no primitive element.

## History

Évariste Galois sketched a proof for splitting fields over the rationals in his First Memoir of 1831, published in 1846.
Ernst Steinitz proved the theorem in its modern form in 1910.
Emil Artin later developed Galois theory without primitive elements.

## Explanations

### L1 human

A finite separable extension, even one built from many elements, can be generated by one element.
So many statements about finite extensions reduce to the case of one algebraic element and its minimal polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/0SVYSP8HCG.md" tag="0SVYSP8HCG" kind="concept">
# Principal ideal

## Summary

A principal ideal is an ideal generated by a single element.
In a commutative ring, it is the set of the multiples of that element.
The [[definition of a principal ideal|00A2EDFT9Q]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0SWTJ5VC07.md" tag="0SWTJ5VC07" kind="concept">
# Linear equation with constant coefficients

## Summary

A linear equation with constant coefficients is a linear differential equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = b(x)$ with real constants $a_0, \dots, a_{n-1}$.
The roots of its characteristic polynomial give all solutions of the homogeneous equation. See the [[definition|0A6PGTTP2A]].
</document>

<document source="https://tamnd.github.io/math2/node/0SWXJHKJ02.md" tag="0SWXJHKJ02" kind="concept">
# Sample space

## Summary

The sample space of a random experiment is the set of all its possible outcomes.
The node [[sample space|0W7NFW23RF]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0SXDHQM2MH.md" tag="0SXDHQM2MH" kind="statement">
# Möbius transformations map circles and lines to circles and lines

## Statement

A generalized circle in the [[Riemann sphere|0CDMXGPSES]] $\widehat{\mathbb{C}}$ is either a circle in $\mathbb{C}$, or a set $L \cup \{\infty\}$ where $L$ is a line in $\mathbb{C}$.
Let $f$ be a [[Möbius transformation|0B346DDAM5]].
Then for each generalized circle $C$, the image $f(C)$ is a generalized circle.

## Remarks

A circle can go to a line: the map $z \mapsto \frac{1}{z}$ sends the circle $|z - 1| = 1$, which contains $0$, to the line $\operatorname{Re} w = \frac{1}{2}$ together with $\infty$.
The image of a generalized circle is a line together with $\infty$ exactly when the generalized circle contains the point that $f$ sends to $\infty$.
The proof checks the result for $z \mapsto z + b$, $z \mapsto az$ and $z \mapsto \frac{1}{z}$, because each Möbius transformation is a composition of such maps.

## Explanations

### L1 human

For a Möbius map, a line is only a circle through the point $\infty$.
So the theorem says that circles go to circles, if we count lines as circles.
For example, $z \mapsto 2z + 3$ sends each circle to a circle with twice the radius.
</document>

<document source="https://tamnd.github.io/math2/node/0SXDQ6TN00.md" tag="0SXDQ6TN00" kind="statement">
# Dimension of a quotient space

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a subspace of $V$.
Then the [[quotient space|0BR3FY5Q8M]] $V / W$ is finite-dimensional, and
$$\dim(V / W) + \dim W = \dim V,$$
where $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

The number $\dim(V / W) = \dim V - \dim W$ is the *codimension* of $W$ in $V$.
The formula is the [[rank–nullity theorem|0W4FSW1TVB]] for the [[quotient map|0N4F5FMQZW]] $V \to V / W$.
For example, a plane $W$ in $\R^3$ gives $\dim(\R^3 / W) = 1$.

## Explanations

### L1 human

Dividing by a subspace removes its dimensions.
What remains has the dimension of the space minus the dimension of the subspace.
Ordinary space divided by a plane leaves one dimension.
</document>

<document source="https://tamnd.github.io/math2/node/0SXHBTNM9T.md" tag="0SXHBTNM9T" kind="concept">
# Acyclic graph

## Summary

An acyclic graph is a graph without cycles. Acyclic graphs are the forests, and their connected components are trees. The node [[acyclic graph|0XNN9Z8E5A]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0SXZ68HV6F.md" tag="0SXZ68HV6F" kind="example">
# Integral of 1/z around the unit circle

## Example

The [[circle integral|0T7ABX4CWE]] of $f(z) = 1/z$ over the unit circle $|z| = 1$ is
$$\oint_{|z| = 1} \frac{dz}{z} = 2 \pi i.$$
Computation: the path is $\gamma(\theta) = e^{i\theta}$ with $\theta \in [0, 2\pi]$, so $\gamma'(\theta) = i e^{i\theta}$. Then
$$\oint_{|z| = 1} \frac{dz}{z} = \int_0^{2\pi} \frac{i e^{i\theta}}{e^{i\theta}} \, d\theta = \int_0^{2\pi} i \, d\theta = 2 \pi i.$$
More generally, let $c, w \in \mathbb{C}$ and $R > 0$. If $|w - c| < R$, then $\oint_{|z - c| = R} \frac{dz}{z - w} = 2 \pi i$. If $|w - c| > R$, the integral is $0$.

## Remarks

For each integer $n \neq -1$, $\oint_{|z| = 1} z^n \, dz = 0$, because $z^n$ has a primitive on $\mathbb{C} \setminus \{0\}$.
The value is not $0$, so $1/z$ has no primitive on $\mathbb{C} \setminus \{0\}$, and [[Cauchy's integral theorem|01MPZXYVYF]] does not apply on $\mathbb{C} \setminus \{0\}$.
This integral is the base of [[Cauchy's integral formula|0DG10TVYPN]], of the [[residue|0JFGKKWWFK]] and of the [[winding number|0X7SBS8E0F]].

## Explanations

### L1 human

On the unit circle, $dz$ is $i z \, d\theta$, so $dz / z$ is just $i \, d\theta$.
Going once around adds up $i \, d\theta$ over a full turn of $2\pi$, which gives $2 \pi i$.
</document>

<document source="https://tamnd.github.io/math2/node/0SY9ECQ19Q.md" tag="0SY9ECQ19Q" kind="definition">
# Continuous map of topological spaces

## Definition

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be [[topological spaces|0CNQXVBHCX]] and let $f \colon X \to Y$ be a function.
The function $f$ is continuous when for each [[open set|01ZQQDCERP]] $V \in \tau_Y$, the [[preimage|0FGCFY1AZX]] $f^{-1}(V)$ is in $\tau_X$.

## Remarks

Equivalently, $f$ is continuous if and only if preimages of closed sets are closed, as in [[continuity via closed sets|0B8GPXFSP7]].
For metric spaces, this is the same as the epsilon-delta condition, by [[continuity via open sets|0YF82GBTND]].
A function is continuous if and only if it is [[continuous at each point|07N7W5C29X]].
Constant maps and identity maps are continuous, and the [[composition of continuous maps is continuous|05NK26G0DR]].

## Explanations

### L1 human

A continuous map does not tear the space apart: the points that go into an open set come from an open set.
This definition uses only open sets, so it works in each topological space.
</document>

<document source="https://tamnd.github.io/math2/node/0SYR10HD4R.md" tag="0SYR10HD4R" kind="concept">
# Cauchy principal value

## Summary

The Cauchy principal value gives a value to some improper integrals that do not converge in the usual sense, see the [[definition|0V3RDQV3ZX]].
It removes a symmetric neighborhood of the singular point, or integrates over a symmetric interval $[-R, R]$, and then takes the limit.
</document>

<document source="https://tamnd.github.io/math2/node/0SZ1KMNWVE.md" tag="0SZ1KMNWVE" kind="statement">
# Order of the alternating group

## Statement

Let $n \geq 2$.
The [[alternating group|0Z5905Q73D]] $A_n$ has [[order|0B85S2V2DR]] $n!/2$, and $[S_n : A_n] = 2$.

## Remarks

The sign map is a surjective homomorphism onto a group of order $2$ with kernel $A_n$, so $S_n / A_n$ has order $2$.
Another proof: the map $\sigma \mapsto (1\ 2) \sigma$ is a bijection from the even permutations to the odd permutations.
For example, $|A_4| = 12$ and $|A_5| = 60$.

## Explanations

### L1 human

Half of all permutations are even.
To see this, multiply by one fixed swap: this changes each even permutation into an odd one, and back.
So $S_n$ splits into two halves of the same size.
</document>

<document source="https://tamnd.github.io/math2/node/0SZCSYHW60.md" tag="0SZCSYHW60" kind="concept">
# Miller–Rabin primality test

## Summary

The Miller–Rabin test is a probabilistic [[primality test|0EGJZHZ02C]] for an odd integer $n > 2$.
Write $n - 1 = 2^s d$ with $d$ odd, and choose a base $a$ with $1 < a < n - 1$.
The number $n$ passes for base $a$ if $a^d \equiv 1 \pmod n$, or if $a^{2^r d} \equiv -1 \pmod n$ for some $r$ with $0 \le r < s$.
Each odd prime $n$ passes for each base. If $n$ fails for some base, then $n$ is composite.

## Remarks

The prime case follows from [[Fermat's little theorem|0M25PRN3WQ]] and the fact that $\pm 1$ are the only square roots of $1$ modulo a prime.
If $n$ is an odd composite number, then at most $\frac{1}{4}$ of the bases let it pass. So $k$ random bases give an error probability of at most $4^{-k}$.
Gary Miller found the test in 1976, in a deterministic form that depends on the extended Riemann hypothesis. Michael Rabin made it an unconditional probabilistic test in 1980.

## Explanations

### L1 human

The test checks a property that every prime has, for a few random bases.
A composite number fails for most bases, so a few rounds expose it with high probability.
The powers are fast to compute by [[modular exponentiation|04DR5PM25N]].
</document>

<document source="https://tamnd.github.io/math2/node/0T0EJDGHMB.md" tag="0T0EJDGHMB" kind="definition">
# Normal convergence

## Definition

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ be bounded functions for $n \ge 0$.
The [[series of functions|0MRE6XRE5V]] $\sum_{n=0}^{\infty} f_n$ converges normally on $S$ if
$$\sum_{n=0}^{\infty} \sup_{x \in S} |f_n(x)| < \infty.$$

## Remarks

By the [[Weierstrass M-test|0NX5NWAFC9]], a normally convergent series converges absolutely and uniformly on $S$.
The converse fails. For constant functions $f_n = (-1)^n / (n + 1)$, the series converges uniformly but not normally.
René Baire introduced the concept in 1908.

## Explanations

### L1 human

Normal convergence means that the sizes of the functions, measured by their suprema, form a convergent series.
It is a simple condition that is stronger than uniform convergence.
</document>

<document source="https://tamnd.github.io/math2/node/0T0F6CCS9W.md" tag="0T0F6CCS9W" kind="definition">
# Degree of a vertex

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $v$ be a vertex of $G$ with a finite [[neighbourhood|01T3TMGTTE]] $N(v)$.
The degree of $v$ is the number
$$\deg(v) = |N(v)|$$
of vertices adjacent to $v$. It is also the number of edges that contain $v$.

## Remarks

A vertex of degree $0$ is isolated, and a vertex of degree $1$ is a leaf.
A graph is locally finite when each vertex has finite degree. Each finite graph is locally finite.
In a [[multigraph|01M5T0KDG7]], the degree counts edges with multiplicity, and a loop counts twice.

## Explanations

### L1 human

The degree of a vertex is the number of edges at that vertex.
In a friendship graph it is the number of friends of a person.
</document>

<document source="https://tamnd.github.io/math2/node/0T0JWJ3M47.md" tag="0T0JWJ3M47" kind="example">
# Bertrand paradox

## Example

Take a circle and an equilateral triangle inscribed in it. Choose a chord of the circle "at random". What is the probability that the chord is longer than a side of the triangle?
Three natural ways to choose the chord give three different answers.
Choose two independent uniform points on the circle and join them. The probability is $1/3$.
Choose a uniform radius, then a uniform point on it, and take the chord perpendicular to the radius through that point. The probability is $1/2$.
Choose a uniform point in the disc and take the chord with that point as midpoint. The probability is $1/4$.

## Remarks

The paradox shows that "at random" has no meaning until one fixes a probability measure. The three methods give three different distributions on the set of chords.
Joseph Bertrand gave the example in his book Calcul des probabilités of 1889, as a criticism of the principle of indifference.

## Explanations

### L1 human

The question sounds complete, but it is not: it does not say how to choose the chord.
Each method is fair in its own sense, and each one gives a different answer. In probability you must always say exactly how the random choice is made.
</document>

<document source="https://tamnd.github.io/math2/node/0T0KMZPYV9.md" tag="0T0KMZPYV9" kind="definition">
# Normal core

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
The normal core of $H$ in $G$ is the subgroup $\operatorname{Core}_G(H) = \bigcap_{g \in G} gHg^{-1}$.

## Remarks

The core is a [[normal subgroup|0CB0DX5BVV]] of $G$, and it contains each normal subgroup of $G$ that lies in $H$.
The subgroup $H$ is normal exactly when $\operatorname{Core}_G(H) = H$.
The core is the kernel of the action of $G$ on the left cosets of $H$ by left multiplication.

## Explanations

### L1 human

Some subgroups are not normal, but they still contain normal subgroups.
The normal core is the biggest of these.
You find it by keeping only the elements that lie in every conjugate of the subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/0T0X8GTYYS.md" tag="0T0X8GTYYS" kind="statement">
# Determinant is a coefficient of the characteristic polynomial

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $A$ be an $n \times n$ matrix over $K$.
Then the constant coefficient of the [[characteristic polynomial|0VSSPAFM2C]] $p_A(t) = \det(t I_n - A)$ is
$$p_A(0) = (-1)^n \det A,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The proof sets $t = 0$: $p_A(0) = \det(-A) = (-1)^n \det A$.
So, over a field, $0$ is an eigenvalue of $A$ if and only if $\det A = 0$.

## Explanations

### L1 human

The constant term of the characteristic polynomial is the determinant, up to a sign.
This follows when we put $t = 0$.
So a matrix with determinant zero has the eigenvalue $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0T13HM377F.md" tag="0T13HM377F" kind="definition">
# Alternating group

## Definition

Let $n$ be a positive integer.
The alternating group $A_n$ is the set of the [[even permutations|0SQTMQTSRZ]] in $S_n$, with composition as the operation.

## Remarks

For $n \geq 2$, $A_n$ is the [[kernel|0W88CEGQP1]] of the [[sign homomorphism|0JEJJCB0EX]], so it is a [[normal subgroup|0CB0DX5BVV]] of $S_n$.
For $n \geq 2$, $A_n$ is also the [[commutator subgroup|0ASGSTXSBW]] of $S_n$.
The group $A_3$ is cyclic of order $3$, and $A_4$ has order $12$.
The group $A_n$ is abelian if and only if $n \leq 3$.

## Explanations

### L1 human

The alternating group keeps only the even permutations.
The product of two even permutations is even, so these form a group.
For $n \geq 5$ it is one of the most important examples of a simple group.
</document>

<document source="https://tamnd.github.io/math2/node/0T1BSD7H3M.md" tag="0T1BSD7H3M" kind="statement">
# Infinitely many primes congruent to 1 modulo n

## Statement

Let $n$ be a positive integer.
Then there are infinitely many [[prime numbers|0SN0MAEKYX]] $p$ such that $p \equiv 1 \pmod{n}$.

## Remarks

This is the case $a = 1$ of [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]].
The case $a = 1$ has an elementary proof with cyclotomic polynomials. A prime factor $p$ of $\Phi_n(m)$ that does not divide $n$ satisfies $p \equiv 1 \pmod n$.
For example, the primes $p \equiv 1 \pmod{10}$ start with $11, 31, 41, 61, 71$.

## Explanations

### L1 human

Fix a number $n$, for example $n = 6$. Then infinitely many primes leave the remainder $1$ when you divide them by $6$.
The first ones are $7, 13, 19, 31, 37$.
</document>

<document source="https://tamnd.github.io/math2/node/0T1Q8D0FW8.md" tag="0T1Q8D0FW8" kind="statement">
# Fundamental theorem of Galois theory

## Statement

Let $L/K$ be a finite [[Galois extension|0DS2F7GK05]] with [[Galois group|0XVR72M51M]] $G = \operatorname{Gal}(L/K)$.

1. The maps $H \mapsto L^H$ and $E \mapsto \operatorname{Gal}(L/E)$ are inverse bijections between the subgroups $H$ of $G$ and the [[intermediate fields|02NBKVQFRQ]] $E$ of $L/K$.
2. These bijections reverse inclusions: $H_1 \subseteq H_2$ if and only if $L^{H_1} \supseteq L^{H_2}$.
3. For every subgroup $H$, $[L : L^H] = |H|$ and $[L^H : K] = |G|/|H|$.
4. The extension $L^H/K$ is normal if and only if $H$ is a normal subgroup of $G$. Then $\operatorname{Gal}(L^H/K) \cong G/H$.

Here $L^H$ is the [[fixed field|0KMJ6919WS]] of $H$.

## Remarks

For $L = \mathbb{Q}(\sqrt{2}, \sqrt{3})$ and $K = \mathbb{Q}$, the group $G$ has order $4$ and three subgroups of order $2$. They match the three intermediate fields $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{3})$ and $\mathbb{Q}(\sqrt{6})$.
The top field $L$ matches the trivial subgroup, and the base field $K$ matches $G$.
A finite Galois extension has only finitely many intermediate fields, because a finite group has finitely many subgroups.
The theorem is used to prove that the general equation of degree $5$ is not solvable by radicals.

## History

Évariste Galois proved the theorem in his development of Galois theory.

## Explanations

### L1 human

The theorem turns questions about fields into questions about groups.
Each intermediate field matches one subgroup of the Galois group, and the bigger the field, the smaller the subgroup.
Finite groups are often easier to study than fields.
</document>

<document source="https://tamnd.github.io/math2/node/0T233AZ1GE.md" tag="0T233AZ1GE" kind="statement">
# Degree of a product

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]] and let $f, g \in R[x]$.
Then $\deg(f g) = \deg f + \deg g$, with the [[convention|0DPF6WRZW6]] $\deg 0 = -\infty$.

## Remarks

The proof: if $f$ and $g$ are nonzero with [[leading coefficients|0P05RJC5XA]] $a$ and $b$, then the coefficient of $x^{\deg f + \deg g}$ in $f g$ is $a b \neq 0$.
Over every ring, $\deg(f g) \le \deg f + \deg g$.
The equality fails in $(\mathbb{Z}/4\mathbb{Z})[x]$: $(2x + 1)(2x + 1) = 4x^2 + 4x + 1 = 1$.

## Explanations

### L1 human

When you multiply two polynomials, the highest terms multiply to give the highest term of the product.
For example, $(x^2 + 1)(x^3 - x)$ has degree $2 + 3 = 5$.
This needs the product of the leading coefficients to be nonzero.
</document>

<document source="https://tamnd.github.io/math2/node/0T2NTJCV83.md" tag="0T2NTJCV83" kind="definition">
# Least element

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.
An element $l \in S$ is the least element of $S$ when $l \le s$ for each $s \in S$.

## Remarks

Another name is minimum. By antisymmetry, a set has at most one least element.
Each nonempty subset of $\mathbb{N}$ has a least element. This is the well-ordering principle of $\mathbb{N}$. See [[well-order|0F1D6PK9RZ]].
The interval $(0, 1]$ in $\mathbb{R}$ has no least element.

## Explanations

### L1 human

The least element is below all others.
In the power set of a set under $\subseteq$, the least element is the empty set.
</document>

<document source="https://tamnd.github.io/math2/node/0T2WYDTRP2.md" tag="0T2WYDTRP2" kind="definition">
# Subspace metric

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $A$ be a [[subset|0FE8Q5S42Y]] of $X$.
The subspace metric on $A$ is the function $d_A \colon A \times A \to \mathbb{R}$ with $d_A(a, b) = d(a, b)$ for all $a, b \in A$.
It is the [[restriction|0D2MXBV21T]] of $d$ to $A \times A$.
The metric space $(A, d_A)$ is a subspace of $(X, d)$.

## Remarks

The function $d_A$ is a metric, because each axiom for $d_A$ is a special case of the same axiom for $d$.
For example, the interval $[0, 1]$ with $d(x, y) = |x - y|$ is a subspace of the [[real line|0KXJTZSW9B]].
The open sets of $(A, d_A)$ are the sets $U \cap A$ with $U$ open in $X$. Thus the topology of $d_A$ is the [[subspace topology|0D5YHQGDJY]].

## Explanations

### L1 human

A subspace keeps the distances of the big space and forgets the other points.
For example, the unit circle in the plane, with the Euclidean distance, is a subspace of $\mathbb{R}^2$. The distance between two points of the circle is the length of the straight chord, not of the arc.
</document>

<document source="https://tamnd.github.io/math2/node/0T32ME0A6H.md" tag="0T32ME0A6H" kind="statement">
# Power series of arctangent

## Statement

For every real $x$ with $|x| \le 1$,
$$\arctan x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n + 1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots.$$

## Remarks

For $|x| < 1$, integrate the [[geometric series|01X5CVV1Q5]] $\frac{1}{1 + t^2} = \sum (-1)^n t^{2n}$ term by term from $0$ to $x$. Its sum is the [[derivative of arctangent|05P4B4Q22V]].
At $x = \pm 1$, the series converges by the alternating series test, and [[Abel's theorem|0S4EMDQS2F]] gives the equality. At $x = 1$ this is the [[Leibniz formula|037FD5NBRT]] $\pi/4 = 1 - 1/3 + 1/5 - \cdots$.
The radius of convergence is $1$. Mathlib states the series for $|x| < 1$ as `Real.hasSum_arctan`.

## Explanations

### L1 human

The arctangent is a power series with odd powers and the simple coefficients $\pm 1/(2n+1)$.
At $x = 1$, it gives a famous series for $\pi/4$.
</document>

<document source="https://tamnd.github.io/math2/node/0T36P6TXQ6.md" tag="0T36P6TXQ6" kind="definition">
# Distributive lattice

## Definition

A lattice $(L, \vee, \wedge)$ is distributive if $x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$ for all $x, y, z \in L$.

## Remarks

The condition is equivalent to its dual: $x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)$ for all $x, y, z$.
Every totally ordered set is a distributive lattice, with $\max$ as join and $\min$ as meet. Every [[Boolean algebra|06GKBTAA5E]] is a distributive lattice.
The diamond lattice $M_3$ and the pentagon lattice $N_5$ are the basic lattices that are not distributive.

## Explanations

### L1 human

A lattice is distributive when meet distributes over join, like intersection over union for sets.
Then join also distributes over meet.
</document>

<document source="https://tamnd.github.io/math2/node/0T3DPBG981.md" tag="0T3DPBG981" kind="definition">
# Predicate

## Definition

Let $X$ be a set.
A predicate on $X$ is a statement $P(x)$ with a variable $x$ that ranges over $X$, such that $P(a)$ is a [[proposition|0ZGC7ZESYN]] for each $a \in X$.
More generally, a predicate $P(x_1, \dots, x_n)$ has $n$ variables, and $n$ is its arity.

## Remarks

The predicate "$x > 3$" on the integers gives the true proposition "$5 > 3$" and the false proposition "$1 > 3$".
A predicate on $X$ determines the set $\{x \in X \mid P(x)\}$ of the values that make it true. See [[set-builder notation|0TMZ89Q5KA]].
A predicate with two variables is a [[binary relation|08EJC6Q25H]].

## Explanations

### L1 human

A predicate is a sentence with a blank, such as "___ is prime".
It is not true or false yet.
When you fill the blank with a number, you get a proposition.
</document>

<document source="https://tamnd.github.io/math2/node/0T3GK0XCBC.md" tag="0T3GK0XCBC" kind="concept">
# Phase portrait

## Summary

A phase portrait of an autonomous system is a picture of a representative set of its [[trajectories|0BVT6VSB3S]] in the phase space, usually the [[phase plane|05VDGH129F]].
Arrows on the curves show the direction of time.
It shows the [[equilibrium points|0G4X36AE0X]], the periodic orbits and the long-time behaviour of the solutions.

## Remarks

The [[phase line|081QSEAGEY]] is the phase portrait of a scalar autonomous equation.
For the undamped [[harmonic oscillator|0YG0ZJPTWG]] $x'' = -x$, the phase portrait in the $(x, x')$ plane consists of the origin and the circles around it.

## Explanations

### L1 human

A phase portrait is a map of all the possible motions of a system.
One look shows where the system rests, where it cycles and where it goes in the long run.
</document>

<document source="https://tamnd.github.io/math2/node/0T3RZQXDYD.md" tag="0T3RZQXDYD" kind="concept">
# Limit superior

## Summary

The limit superior of a sequence is the limit of the suprema of its tails. It describes the largest value near which the sequence lies infinitely often. The node [[limit superior|0P8BA7T7S5]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0T3WVTD675.md" tag="0T3WVTD675" kind="concept">
# Frobenius endomorphism

## Summary

The Frobenius endomorphism of a commutative ring of prime characteristic $p$ is the map that sends each element to its $p$-th power.
It is a ring homomorphism.
The [[definition of the Frobenius endomorphism|0GHDDVAH8P]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0T4EJ8ZJM9.md" tag="0T4EJ8ZJM9" kind="counterexample">
# Dirichlet function

## Counterexample

Let $f \colon \mathbb{R} \to \mathbb{R}$ be given by $f(x) = 1$ for rational $x$ and $f(x) = 0$ for irrational $x$.
Then $f$ is not [[continuous at|0EEE2CZB2A]] any point.
Indeed, each real $a$ is the limit of a sequence of rationals and of a sequence of irrationals. Their images tend to $1$ and to $0$, so one of them does not tend to $f(a)$.

## Remarks

The function $f$ is the indicator function of $\mathbb{Q}$.
Every rational number is a period of $f$.
The function is [[not Riemann integrable|03Z4JC4YRJ]] on any interval with more than one point.

## Explanations

### L1 human

Rational and irrational numbers lie mixed together in every small interval.
So the function jumps between $0$ and $1$ near every point, and it has no limit anywhere.
</document>

<document source="https://tamnd.github.io/math2/node/0T4QHJXSD4.md" tag="0T4QHJXSD4" kind="statement">
# The cross product is perpendicular to its factors

## Statement

Let $a$ and $b$ be vectors in $\mathbb{R}^3$, and let $a \times b$ be their [[cross product|0N5N15HWZ7]]. Let $u \cdot v = u_1 v_1 + u_2 v_2 + u_3 v_3$ be the dot product.
Then
$$a \cdot (a \times b) = 0 \quad \text{and} \quad b \cdot (a \times b) = 0.$$

## Remarks

By the [[test with the dot product|03NK6RSRQ1]], $a \times b$ is perpendicular to $a$ and to $b$ when all three vectors are nonzero.
The proof is a direct computation. For example, $a \cdot (a \times b) = a_1(a_2 b_3 - a_3 b_2) + a_2(a_3 b_1 - a_1 b_3) + a_3(a_1 b_2 - a_2 b_1)$, and the six terms cancel in pairs.
If $a$ and $b$ are linearly independent, then $a \times b$ is a normal vector of the plane through $0$ that contains $a$ and $b$.

## Explanations

### L1 human

Lay two pencils flat on a table, starting at the same point. Their cross product points straight up from the table, or straight down. So it makes a right angle with each pencil.
</document>

<document source="https://tamnd.github.io/math2/node/0T6D4PE51C.md" tag="0T6D4PE51C" kind="example">
# Clairaut's equation

## Example

Let $f$ be a twice continuously differentiable function on an open interval $J$.
Clairaut's equation is the implicit first-order equation
$$y = x\, y' + f(y'),$$
for an unknown $y$ with $y' \in J$.
For each constant $C \in J$, the line $y = C x + f(C)$ is a solution. These lines form the [[general solution|0SKYFQAFYD]].
If $f'' \ne 0$ on $J$, the curve with the parametric form $x = -f'(p)$, $y = f(p) - p f'(p)$ for $p \in J$ is also a solution. It is the envelope of the lines, and it is a [[singular solution|0CZA0NCFWP]].

## Remarks

Differentiation of the equation gives $y''\left(x + f'(y')\right) = 0$. The case $y'' = 0$ gives the lines, and the case $x + f'(y') = 0$ gives the envelope.
Example: for $f(p) = p^2$, the lines are $y = C x + C^2$, and the singular solution is the parabola $y = -x^2/4$.
Each line of the general solution is tangent to the singular solution at one point.

## Explanations

### L1 human

Clairaut's equation has a family of straight lines as solutions, one for each slope $C$.
These lines wrap around a curve, their envelope, and this curve is also a solution.
No choice of $C$ gives the envelope, so it is a new, singular solution.
</document>

<document source="https://tamnd.github.io/math2/node/0T6MWGZSXA.md" tag="0T6MWGZSXA" kind="concept">
# Augmented matrix

## Summary

The augmented matrix of a system of linear equations is the coefficient matrix with the column of constant terms added on the right.
</document>

<document source="https://tamnd.github.io/math2/node/0T6WS7TT77.md" tag="0T6WS7TT77" kind="statement">
# First Borel–Cantelli lemma

## Statement

Let $A_1, A_2, \dots$ be [[events|00SQGJNRTB]] in a probability space $(\Omega, \mathcal{F}, P)$ with $\sum_{n=1}^{\infty} P(A_n) < \infty$.
Then
$$P\left(\limsup_{n \to \infty} A_n\right) = 0, \quad \text{where} \quad \limsup_{n \to \infty} A_n = \bigcap_{n=1}^{\infty} \bigcup_{m=n}^{\infty} A_m.$$
So almost surely, only finitely many of the events $A_n$ occur.

## Remarks

The event $\limsup_n A_n$ is the set of outcomes that lie in infinitely many $A_n$. It is written $\{A_n \text{ i.o.}\}$, for "infinitely often".
Proof: for each $n$, $P(\limsup_m A_m) \le P\left(\bigcup_{m \ge n} A_m\right) \le \sum_{m \ge n} P(A_m)$ by [[Boole's inequality|0EGSEE9J9R]]. The right side tends to $0$ as $n \to \infty$.
The lemma needs no independence. The [[second Borel–Cantelli lemma|0NWXRAMEE7]] gives a partial converse for independent events.
The lemma holds in each measure space: if $\sum_n \mu(A_n) < \infty$, then $\mu(\limsup_n A_n) = 0$.

## History

The lemma is named after Émile Borel and Francesco Paolo Cantelli.

## Explanations

### L1 human

If the chances of the events add up to a finite number, then after some time none of them happens any more, with probability $1$.
For example, if $P(A_n) = 1/n^2$, then only finitely many $A_n$ occur.
</document>

<document source="https://tamnd.github.io/math2/node/0T71SSYZ39.md" tag="0T71SSYZ39" kind="statement">
# Cauchy–Riemann equations

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be a function.
Identify $\mathbb{C}$ with $\mathbb{R}^2$ by $x + i y \leftrightarrow (x, y)$.
Let $u = \operatorname{Re} f$ and $v = \operatorname{Im} f$ be the [[real and imaginary parts|02ZE6JHJ7G]] of $f$, as real functions of $(x, y)$ on $U$.
Let $z_0 = x_0 + i y_0 \in U$.
Then $f$ is [[complex differentiable|0D1C9Y1EXA]] at $z_0$ if and only if the following two conditions hold. First, $u$ and $v$ are differentiable at $(x_0, y_0)$ as functions of two real variables. Second, they satisfy
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$$
at $(x_0, y_0)$.
In this case $f'(z_0) = \frac{\partial u}{\partial x}(x_0, y_0) + i \frac{\partial v}{\partial x}(x_0, y_0)$.

## Remarks

For $f(z) = z^2$ we have $u = x^2 - y^2$ and $v = 2xy$. Then $u_x = 2x = v_y$ and $u_y = -2y = -v_x$.
The real differentiability of $u$ and $v$ is necessary. The function $u(x, y) = \sqrt{|xy|}$, $v = 0$ satisfies the two equations at $(0, 0)$, but $u + iv$ is not complex differentiable at $0$.
The two equations together are the one complex equation $\frac{\partial f}{\partial y} = i \frac{\partial f}{\partial x}$.

## Explanations

### L1 human

A complex derivative must be the same in each direction. If you compare the derivative along the real axis with the derivative along the imaginary axis, you get the two equations.
So the equations are a test with partial derivatives for a complex derivative.
For example, $\bar z = x - i y$ has $u_x = 1$ and $v_y = -1$, so it fails the test.
</document>

<document source="https://tamnd.github.io/math2/node/0T7ABX4CWE.md" tag="0T7ABX4CWE" kind="definition">
# Circle integral

## Definition

Let $c \in \mathbb{C}$ and let $R > 0$ be a real number.
Let $f$ be a complex function that is continuous on the circle $\{z : |z - c| = R\}$.
The circle integral of $f$ over this circle is the [[contour integral|00FWNP9TXY]] along the path $\gamma(\theta) = c + R e^{i\theta}$, $\theta \in [0, 2\pi]$:
$$\oint_{|z - c| = R} f(z) \, dz = \int_0^{2\pi} f(c + R e^{i\theta}) \, i R e^{i\theta} \, d\theta.$$

## Remarks

The path goes once around the circle, in the positive (counterclockwise) direction.
The length of the path is $2 \pi R$.
The basic example is the [[integral of 1/z|0SXZ68HV6F]]: $\oint_{|z| = 1} \frac{dz}{z} = 2 \pi i$.

## Explanations

### L1 human

A circle integral is a contour integral where the path is a circle, traveled once counterclockwise.
Circles are the paths that appear most in complex analysis, for example in [[Cauchy's integral formula|0DG10TVYPN]].
</document>

<document source="https://tamnd.github.io/math2/node/0T7DX8NY1X.md" tag="0T7DX8NY1X" kind="concept">
# Tangent line to a circle

## Summary

A tangent line to a circle is a line that meets the circle in exactly one point, the point of tangency. It does not enter the inside of the circle, and it is perpendicular to the radius at the point of tangency.
</document>

<document source="https://tamnd.github.io/math2/node/0T89W1QZ9K.md" tag="0T89W1QZ9K" kind="statement">
# Integration in polar coordinates

## Statement

Let $0 \le r_1 < r_2$ and $\alpha < \beta \le \alpha + 2\pi$, and let $D = \{(r \cos \theta, r \sin \theta) : r_1 \le r \le r_2, \ \alpha \le \theta \le \beta\}$.
Let $f \colon D \to \mathbb{R}$ be continuous.
Then
$$\iint_D f(x, y) \, dx \, dy = \int_\alpha^\beta \int_{r_1}^{r_2} f(r \cos \theta, r \sin \theta) \, r \, dr \, d\theta.$$

## Remarks

This is the [[change of variables formula|00N5FHP1CV]] for the [[polar coordinate map|0RJZV05MJA]], whose Jacobian determinant is $r$. The boundary rays and the origin have area zero and do not change the integral.
Example: the area of the disc of radius $a$ is $\int_0^{2\pi} \int_0^a r \, dr \, d\theta = \pi a^2$.
For a region $\alpha \le \theta \le \beta$, $0 \le r \le g(\theta)$ with $g$ continuous, the formula gives the area $\tfrac{1}{2} \int_\alpha^\beta g(\theta)^2 \, d\theta$.
The Gaussian integral follows from $\iint_{\mathbb{R}^2} e^{-(x^2 + y^2)} \, dx \, dy = \int_0^{2\pi} \int_0^\infty e^{-r^2} r \, dr \, d\theta = \pi$. Hence $\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$.

## Explanations

### L1 human

In polar coordinates a small piece of the plane is almost a rectangle with sides $dr$ and $r \, d\theta$.
So its area is $r \, dr \, d\theta$, and the factor $r$ appears in the integral.
</document>

<document source="https://tamnd.github.io/math2/node/0T8BYAFEXX.md" tag="0T8BYAFEXX" kind="statement">
# The characteristic function determines the distribution

## Statement

Let $X$ and $Y$ be real random variables with [[characteristic functions|0MYDNRT7PP]] $\varphi_X$ and $\varphi_Y$.
If $\varphi_X(t) = \varphi_Y(t)$ for all $t \in \mathbb{R}$, then $X$ and $Y$ have the same [[law|0TST4MHA3A]].

## Remarks

The proof uses the Lévy inversion formula. For real $a < b$ that are continuity points of the [[cumulative distribution function|0MXMS3AE13]] $F_X$,
$$F_X(b) - F_X(a) = \lim_{T \to \infty} \frac{1}{2\pi} \int_{-T}^{T} \frac{e^{-ita} - e^{-itb}}{it} \, \varphi_X(t) \, dt.$$
If $\int |\varphi_X(t)| \, dt < \infty$, then $X$ has the continuous density $f_X(x) = \frac{1}{2\pi} \int e^{-itx} \varphi_X(t) \, dt$.
The same result holds for random vectors in $\mathbb{R}^n$, with $\varphi_X(t) = E[e^{i \langle t, X \rangle}]$.

## Explanations

### L1 human

Two different distributions cannot have the same characteristic function.
So, to identify a distribution, it is enough to compute its characteristic function and recognize it.
</document>

<document source="https://tamnd.github.io/math2/node/0T8M3YC5RG.md" tag="0T8M3YC5RG" kind="statement">
# Existence of finite fields of every prime power order

## Statement

Let $p$ be a prime number and let $n \ge 1$.
Then there is a [[finite field|08RTDHX1SK]] of order $q = p^n$.

## Remarks

The proof: let $L$ be a [[splitting field|01441KDGBY]] of $f = x^q - x$ over $\mathbb{F}_p$. The [[derivative|0AD4VS02H3]] $f' = q x^{q-1} - 1 = -1$ has no common root with $f$, so $f$ has $q$ distinct roots in $L$, by the [[criterion for repeated roots|067F8YVYMH]].
The roots of $f$ form a subfield of $L$, because $(a + b)^q = a^q + b^q$ and $(a b)^q = a^q b^q$. This subfield has $q$ elements.
So for each $n$ there is an irreducible polynomial of degree $n$ over $\mathbb{F}_p$.

## History

E. H. Moore first proved the classification of finite fields in 1893: there is a field of each prime power order, and all fields of a given order are isomorphic.

## Explanations

### L1 human

The field with $p^n$ elements is made of the roots of $x^{p^n} - x$.
The proof checks that these roots are distinct and closed under the field operations.
</document>

<document source="https://tamnd.github.io/math2/node/0T92VGGVWP.md" tag="0T92VGGVWP" kind="concept">
# Bernoulli equation

## Summary

A Bernoulli equation is a first-order equation $y' + P(x) y = Q(x) y^n$.
The substitution $v = y^{1-n}$ changes it into a linear equation. See the [[definition|0ZB84QCDKR]].

## Remarks

Jacob Bernoulli discussed the equation in 1695. Leibniz published a solution in the same year.
</document>

<document source="https://tamnd.github.io/math2/node/0TA7WT68RK.md" tag="0TA7WT68RK" kind="definition">
# Carmichael number

## Definition

A Carmichael number is a [[composite number|0N3ADXHNTQ]] $n$ such that $a^{n-1} \equiv 1 \pmod n$ for each integer $a$ [[coprime|0GFKP8F8AG]] to $n$.

## Remarks

Thus a Carmichael number is a [[Fermat pseudoprime|02M2K4QH9P]] to each base coprime to it.
Korselt's criterion: a composite $n$ is a Carmichael number if and only if $n$ is square-free and $p - 1 \mid n - 1$ for each prime $p \mid n$.
The smallest Carmichael number is $561 = 3 \cdot 11 \cdot 17$. Here $2$, $10$ and $16$ divide $560$.
Alford, Granville and Pomerance proved in 1994 that there are infinitely many Carmichael numbers.

## Explanations

### L1 human

A Carmichael number behaves like a prime for the Fermat test, for each base that has no common factor with it.
So the [[Fermat primality test|0N5ZSZMYH6]] cannot detect it, unless the base shares a factor with it.
</document>

<document source="https://tamnd.github.io/math2/node/0TB16T2C97.md" tag="0TB16T2C97" kind="statement">
# Axiom of extensionality

## Statement

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
If $A$ and $B$ have the same [[elements|0EZ758XE73]], then $A = B$:
$$\forall A \, \forall B \, \bigl( \forall x \, (x \in A \leftrightarrow x \in B) \to A = B \bigr).$$

## Remarks

It is one of the axioms of [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].
The converse is a rule of logic: equal sets have the same elements.
In practice, $A = B$ is often proved with $A \subseteq B$ and $B \subseteq A$. See [[subset|0FE8Q5S42Y]].

## Explanations

### L1 human

A set is nothing more than its elements.
If two sets contain exactly the same things, they are the same set, however you describe them.
For example, $\{x \in \mathbb{R} \mid x^2 = 1\}$ and $\{-1, 1\}$ are equal.
</document>

<document source="https://tamnd.github.io/math2/node/0TBYMWPWWB.md" tag="0TBYMWPWWB" kind="statement">
# The successor function is injective

## Statement

Let $S \colon \mathbb{N} \to \mathbb{N}$ be the [[successor function|0832XVR512]] of the natural numbers.
For all natural numbers $m$ and $n$, if $S(m) = S(n)$, then $m = n$.

## Remarks

This is one of the [[Peano axioms|0QFT7Z76N5]]. In a construction of $\mathbb{N}$, for example in set theory, it is a theorem.
An equivalent form is: if $m \neq n$, then $m + 1 \neq n + 1$.

## Explanations

### L1 human

Two different numbers never have the same next number.
If you know the number after $n$, then you know $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0TC99BRSSB.md" tag="0TC99BRSSB" kind="statement">
# Dirac's theorem on Hamiltonian cycles

## Statement

Let $G$ be a finite simple [[graph|0JD5C82QMB]] with $n \ge 3$ vertices.
Suppose that every vertex of $G$ has [[degree|0T0F6CCS9W]] at least $n/2$.
Then $G$ is [[Hamiltonian|090WPKFABX]].

## Remarks

The hypothesis says that the [[minimum degree|0TKZCSGCCA]] is at least $n/2$.
The theorem follows from [[Ore's theorem|0NACHJYPKW]]: if all degrees are at least $n/2$, then each sum of two degrees is at least $n$.
The bound is sharp. The complete bipartite graph $K_{k, k+1}$ has minimum degree $k < (2k+1)/2$, and it is not Hamiltonian.

## History

G. A. Dirac published the theorem in 1952.

## Explanations

### L1 human

If each vertex is joined to at least half of the other vertices, there is a round trip through all vertices.
A graph with this many edges cannot avoid a Hamiltonian cycle.
</document>

<document source="https://tamnd.github.io/math2/node/0TCBZMEYGN.md" tag="0TCBZMEYGN" kind="statement">
# C^k function characterised by continuous derivatives

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $k \ge 1$ be an integer.
Say that $f$ is of class $C^k$ on $U$ when the following is true for each component $f_j$ and each $r \le k$. Each [[partial derivative|02RSXVM1ZN]] $\partial^r f_j / \partial x_{i_r} \cdots \partial x_{i_1}$ of order $r$ exists on $U$ and is continuous on $U$.
Then $f$ is of class $C^k$ on $U$ if and only if these two conditions hold.

1. For each $r$ with $0 \le r \le k$, the [[iterated derivative|0EY05Z9RPN]] $D^r f$ exists on $U$ and is continuous on $U$.
2. For each $r$ with $0 \le r < k$, the map $D^r f$ is differentiable on $U$.

## Remarks

So in finite dimensions, $C^k$ means $k$ times continuously differentiable, in either the sense of partial derivatives or the sense of Fréchet derivatives.
A function is of class $C^\infty$, or smooth, when it is of class $C^k$ for each $k$.
Each polynomial in $x_1, \dots, x_n$ is of class $C^\infty$ on $\mathbb{R}^n$. The function $x \mapsto \|x\|$ is of class $C^\infty$ on $\mathbb{R}^n \setminus \{0\}$, but it is not differentiable at $0$.

## Explanations

### L1 human

There are two ways to count derivatives: one variable at a time with partial derivatives, or all variables at once with total derivatives.
For functions on open subsets of $\mathbb{R}^n$, both ways give the same classes $C^1, C^2, \dots$, as long as you also ask for continuity.
</document>

<document source="https://tamnd.github.io/math2/node/0TCHAS04WK.md" tag="0TCHAS04WK" kind="definition">
# Convex hull

## Definition

Let $X$ be a subset of $\mathbb{R}^n$. The convex hull of $X$ is the intersection of all [[convex sets|02AS45T3H9]] of $\mathbb{R}^n$ that contain $X$:
$$\operatorname{conv}(X) = \bigcap \{\, S \subseteq \mathbb{R}^n \;:\; S \supseteq X \text{ and } S \text{ is convex} \,\}.$$

## Remarks

$\operatorname{conv}(X)$ is convex, and it is the smallest convex set that contains $X$.
It is also the set of all convex combinations $\sum_{i=1}^{k} w_i x_i$, with $k \geq 1$, $x_i \in X$, $w_i \geq 0$ and $\sum_i w_i = 1$. These are the [[affine combinations|0KJH6SRFZK]] with nonnegative weights.
The convex hull of a compact set is compact, and the convex hull of an open set is open.
The convex hull of a finite set is a convex polytope. For three points not on a line, it is the triangle with its inside.

## Explanations

### L1 human

For a finite set of points in the plane, put a rubber band round all the points and let it go. The band takes the shape of the boundary of the convex hull. The convex hull of the four corners of a square is the full square.
</document>

<document source="https://tamnd.github.io/math2/node/0TD904MG06.md" tag="0TD904MG06" kind="concept">
# Orientable surface

## Summary

A surface in space is orientable if one can choose a [[unit normal vector|073JKP99K2]] at each point in a continuous way. The choice must be continuous over the whole surface.
Such a choice is an orientation. The Möbius strip is the standard surface that has no orientation.
</document>

<document source="https://tamnd.github.io/math2/node/0TDY6JZHKS.md" tag="0TDY6JZHKS" kind="statement">
# Product of the gcd and the lcm

## Statement

Let $a$ and $b$ be natural numbers.
Then $\gcd(a, b) \cdot \operatorname{lcm}(a, b) = a b$, where $\gcd$ is the [[greatest common divisor|02WF878BH1]] and $\operatorname{lcm}$ is the [[least common multiple|09S5SN35DJ]].

## Remarks

For integers $a$ and $b$ of any sign, the same argument gives $\gcd(a, b) \cdot \operatorname{lcm}(a, b) = |a b|$.
Thus $\operatorname{lcm}(a, b) = a b / \gcd(a, b)$ when $a$ and $b$ are not both $0$. This is the usual way to compute the lcm, with the [[Euclidean algorithm|0TRNT4X7J0]].
For example, $\gcd(4, 6) = 2$, $\operatorname{lcm}(4, 6) = 12$ and $2 \cdot 12 = 24 = 4 \cdot 6$.

## Explanations

### L1 human

Think of the prime factors of $a$ and $b$. The gcd takes, for each prime, the smaller of the two exponents, and the lcm takes the larger one.
The smaller exponent plus the larger exponent is the sum of the two exponents.
So the product of the gcd and the lcm has the same prime factors as $a b$.
</document>

<document source="https://tamnd.github.io/math2/node/0TEDQH56RW.md" tag="0TEDQH56RW" kind="statement">
# Remainder theorem

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]], let $f \in R[x]$ and let $c \in R$.
Then the remainder of the [[division|0MES3KS0ZC]] of $f$ by $x - c$ is the constant polynomial $f(c)$.
That is, there is $q \in R[x]$ with $f = (x - c) q + f(c)$.

## Remarks

The proof: division by the monic polynomial $x - c$ gives $f = (x - c) q + r$ with $r$ a constant. [[Evaluation|0GJ7T0NK49]] at $c$ gives $r = f(c)$.
For example, the remainder of $x^3 - 2x + 5$ divided by $x - 2$ is $2^3 - 4 + 5 = 9$.

## Explanations

### L1 human

You do not need long division to find the remainder of $f$ divided by $x - c$.
You only compute $f(c)$.
</document>

<document source="https://tamnd.github.io/math2/node/0TEJ6QF7K9.md" tag="0TEJ6QF7K9" kind="definition">
# Affine space

## Definition

Let $K$ be a field and let $V$ be a vector space over $K$.
An affine space over $V$ is a nonempty set $A$ with a map $A \times V \to A$, written $(p, v) \mapsto p + v$, that has these properties.
For each point $p$ of $A$: $p + 0 = p$.
For each point $p$ of $A$ and all vectors $u$ and $v$ of $V$: $(p + u) + v = p + (u + v)$.
For all points $p$ and $q$ of $A$, there is exactly one vector $v$ of $V$ with $p + v = q$. This vector is written $q - p$ or $\overrightarrow{pq}$.
The elements of $A$ are the points, and $V$ is the space of translations of $A$. The dimension of $A$ is the dimension of $V$.

## Remarks

Each vector space $V$ is an affine space over itself, with the addition of $V$ as the action. In particular $\mathbb{R}^n$ is an affine space over $\mathbb{R}^n$.
For all points $p$, $q$ and $r$, Chasles's relation holds: $(q - p) + (r - q) = r - p$.
Mathlib calls this structure an additive torsor, `AddTorsor`. Its definition works for an additive group $V$ in place of a vector space.
A [[Euclidean space|0XPR1E85W0]] is a real affine space of finite dimension whose space of translations has an inner product.

## Explanations

### L1 human

In an affine space you can subtract two points to get the arrow from one to the other. You can also add an arrow to a point. But you cannot add two points: there is no special origin. A good picture is the plane without its coordinate axes.
</document>

<document source="https://tamnd.github.io/math2/node/0TFDZN6AXP.md" tag="0TFDZN6AXP" kind="example">
# p-metric on R^n

## Example

Let $n \ge 1$ be an integer and let $p \ge 1$ be a real number.
For $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let
$$d_p(x, y) = \left( \sum_{i=1}^{n} |x_i - y_i|^p \right)^{1/p}.$$
Then $d_p$ is a [[metric|063W5JMY85]] on $\mathbb{R}^n$.

## Remarks

The triangle inequality for $d_p$ is the Minkowski inequality.
For $p = 1$ it is the [[taxicab metric|07CJNE6ZAQ]], and for $p = 2$ it is the [[Euclidean metric|0E1MMCAY5B]].
For $0 < p < 1$, the same formula does not give a metric when $n \ge 2$: the triangle inequality fails.
The metric $d_p$ comes from the [[p-norm|0KNJWEVSMG]]: $d_p(x, y) = \|x - y\|_p$.

## Explanations

### L1 human

The p-metrics form a family of ways to combine the differences of the coordinates into one distance.
A larger $p$ gives more weight to the largest difference, and for very large $p$ the distance is close to the [[sup metric|0VTAXQEWQD]].
</document>

<document source="https://tamnd.github.io/math2/node/0TGV80E6A4.md" tag="0TGV80E6A4" kind="concept">
# Big O notation

## Summary

Big O notation compares the growth of two functions. The statement $f = O(g)$ means that $|f|$ is at most a constant times $|g|$ near a given point or at infinity. The node [[big O notation|0RYTGBJAGN]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0TH61JR04G.md" tag="0TH61JR04G" kind="definition">
# Derangement

## Definition

Let $X$ be a set.
A derangement of $X$ is a [[permutation|0BQMWCNS8S]] $\sigma$ of $X$ such that $\sigma(x) \neq x$ for all $x \in X$.
In other words, it is a permutation with no fixed point.

## Remarks

The derangements of $\{1, 2, 3\}$ are $231$ and $312$ in one-line notation.
The [[number of derangements|03WY4HHMT1]] of an $n$-element set is written $D_n$ or $!n$.

## Explanations

### L1 human

A derangement is a shuffle in which nothing stays in its place.
Suppose that $n$ people leave their hats, and each person later takes a hat at random. A derangement is an outcome in which nobody gets their own hat.
</document>

<document source="https://tamnd.github.io/math2/node/0TH78S7GBZ.md" tag="0TH78S7GBZ" kind="concept">
# Arc length

## Summary

The arc length of a curve is its length, measured along the curve. It is the supremum of the lengths of inscribed polygonal paths, and an integral of the speed for smooth curves. The node [[arc length|0X7G61297X]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0TH9CPW3VS.md" tag="0TH9CPW3VS" kind="statement">
# Reduction of a higher-order equation to a first-order system

## Statement

Let $n \ge 1$, let $U$ be a subset of $\mathbb{R} \times \mathbb{R}^n$, and let $f : U \to \mathbb{R}$ be a function.
Consider the equation $y^{(n)} = f\left(x, y, y', \dots, y^{(n-1)}\right)$ and the first-order system for $z = (z_1, \dots, z_n)$:
$$z_1' = z_2, \quad z_2' = z_3, \quad \dots, \quad z_{n-1}' = z_n, \quad z_n' = f(x, z_1, \dots, z_n).$$
Let $I$ be an open interval.
If $y$ is a [[solution|04SW7WECS8]] of the equation on $I$, then $z = \left(y, y', \dots, y^{(n-1)}\right)$ is a solution of the system on $I$.
Conversely, let $z$ be a solution of the system on $I$. Then $y = z_1$ is a solution of the equation on $I$, and $z_k = y^{(k-1)}$ for $k = 1, \dots, n$.

## Remarks

The initial conditions $y^{(k)}(x_0) = c_k$ for $k = 0, \dots, n-1$ become the one condition $z(x_0) = (c_0, \dots, c_{n-1})$. So results on first-order [[initial value problems|075VBY4K4X]] apply to equations of each order.
For $y'' = -y$, the system is $z_1' = z_2$, $z_2' = -z_1$.
The same method changes a system of equations of order $n$ into a larger first-order system.

## Explanations

### L1 human

The trick is to give new names to the derivatives of $y$.
Each new unknown is the derivative of the one before it, and the last one is given by the equation.
So we only need a theory of first-order systems.
</document>

<document source="https://tamnd.github.io/math2/node/0THBK9V825.md" tag="0THBK9V825" kind="concept">
# Function composition

## Summary

The composition of two functions applies the first function and then the second.
</document>

<document source="https://tamnd.github.io/math2/node/0THJ7416DX.md" tag="0THJ7416DX" kind="concept">
# Conditional independence

## Summary

Two events are conditionally independent given a third event when they are independent for the conditional probability given the third event.
The node [[conditional independence|0ZJ87ZZ8PK]] gives the precise definition for events.
</document>

<document source="https://tamnd.github.io/math2/node/0TJ1V9DDPC.md" tag="0TJ1V9DDPC" kind="statement">
# Third isomorphism theorem

## Statement

Let $G$ be a [[group|G5S1M9]], and let $N$ and $K$ be [[normal subgroups|0CB0DX5BVV]] of $G$ with $N \subseteq K$.
Then $K/N$ is a normal subgroup of $G/N$, and the map $(gN)(K/N) \mapsto gK$ is an [[isomorphism|08MG7H8JW5]] from $(G/N)/(K/N)$ to $G/K$.

## Remarks

The proof applies the [[first isomorphism theorem|0CXXJNG5B6]] to the surjective homomorphism $G/N \to G/K$, $gN \mapsto gK$, whose kernel is $K/N$.
For example, $(\mathbb{Z}/12\mathbb{Z})/(4\mathbb{Z}/12\mathbb{Z}) \cong \mathbb{Z}/4\mathbb{Z}$.
For finite groups, it gives $[G : K] = [G/N : K/N]$.

## Explanations

### L1 human

If you divide by $N$ and then by the image of $K$, you get the same result as dividing by $K$ in one step.
It looks like cancelling fractions: $(G/N)/(K/N) = G/K$.
The theorem makes this cancellation precise.
</document>

<document source="https://tamnd.github.io/math2/node/0TJ8QBJX5K.md" tag="0TJ8QBJX5K" kind="statement">
# Term-by-term differentiation

## Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be differentiable for each $n \ge 0$.
Assume that $\sum_{n} f_n(x_0)$ converges for some $x_0 \in [a, b]$, and that $\sum_{n} f_n'$ converges uniformly on $[a, b]$.
Then $\sum_{n} f_n$ converges uniformly on $[a, b]$, its sum is differentiable, and
$$\Big( \sum_{n=0}^{\infty} f_n \Big)' = \sum_{n=0}^{\infty} f_n'.$$

## Remarks

Apply [[the theorem on sequences|0CS42KEM5C]] to the partial sums.
The derivative series often converges uniformly by the M-test. Mathlib states that form as `hasDerivAt_tsum`.
Power series satisfy the hypotheses on each closed interval inside the interval of convergence.

## Explanations

### L1 human

We may differentiate a series term by term, if the series of derivatives converges uniformly.
Then the derivative of the sum is the sum of the derivatives.
</document>

<document source="https://tamnd.github.io/math2/node/0TK8Y3J5RW.md" tag="0TK8Y3J5RW" kind="definition">
# Interval

## Definition

A subset $I$ of $\mathbb{R}$ is an interval when, for all $x$ and $y$ in $I$ and each real $z$ with $x \le z \le y$, also $z \in I$.
For real numbers $a \le b$, the notation is:
$$[a, b] = \{x \in \mathbb{R} : a \le x \le b\}, \qquad (a, b) = \{x \in \mathbb{R} : a < x < b\},$$
$$[a, b) = \{x \in \mathbb{R} : a \le x < b\}, \qquad (a, b] = \{x \in \mathbb{R} : a < x \le b\}.$$
The set $[a, b]$ is a closed interval and $(a, b)$ is an open interval.
For each real $a$, the notation is $[a, \infty) = \{x : x \ge a\}$, $(a, \infty) = \{x : x > a\}$, $(-\infty, a] = \{x : x \le a\}$ and $(-\infty, a) = \{x : x < a\}$.

## Remarks

By the [[completeness axiom|0AXWET80YW]], each interval is one of the sets above, or $\mathbb{R}$, or $\emptyset$.
In some countries, the notation $]a, b[$ replaces $(a, b)$.
The symbols $\infty$ and $-\infty$ are not real numbers. They show only that the interval has no bound on that side.

## Explanations

### L1 human

An interval is a piece of the number line without holes.
A square bracket means that the end point belongs to the interval, and a round bracket means that it does not.
For example, $[0, 1)$ contains $0$ and $0.999$, but not $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0TK9MCFHBW.md" tag="0TK9MCFHBW" kind="concept">
# Cylindrical coordinates

## Summary

Cylindrical coordinates describe a point of space by the [[polar coordinates|0BSW4T7THS]] of its projection to the $xy$-plane and by its height $z$ above that plane.
They simplify problems with symmetry about an axis, such as integrals over cylinders and cones.
</document>

<document source="https://tamnd.github.io/math2/node/0TKG7T3K0E.md" tag="0TKG7T3K0E" kind="statement">
# Bounded derivative on a convex set implies Lipschitz

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$ be [[differentiable|0135XZZQ8M]] on $U$, and let $C \subseteq U$ be a convex set.
Suppose that $M \ge 0$ and $\|Df(z)\| \le M$ for each $z \in C$, with the operator norm.
Then $f$ is Lipschitz on $C$ with constant $M$: $\|f(y) - f(x)\| \le M \, \|y - x\|$ for all $x, y \in C$.

## Remarks

Because $C$ is convex, the segment between two points of $C$ lies in $C$. So this follows from the [[mean value inequality|02CYB184RC]].
Convexity matters. Take the annulus $1 < \|x\| < 2$ in $\mathbb{R}^2$ without the points on the negative $x$-axis, and the polar angle $\theta \in (-\pi, \pi)$ on it. Then $\|\nabla \theta\| = 1/\|x\| \le 1$, but two points just above and just below the slit are close, and their angles differ by almost $2\pi$.
A function of class $C^1$ on $U$ is Lipschitz on each closed ball in $U$, because $\|Df\|$ is bounded on the compact ball.

## Explanations

### L1 human

A Lipschitz function cannot change faster than a fixed rate $M$ per unit of distance.
A bound on the derivative gives such a rate. For this, you must be able to walk between any two points along a straight line inside the domain.
</document>

<document source="https://tamnd.github.io/math2/node/0TKP3TDD02.md" tag="0TKP3TDD02" kind="concept">
# Bell numbers

## Summary

The Bell numbers $B_n$ count all the partitions of an $n$-element set. They are the row sums of the Stirling numbers of the second kind. The node [[Bell numbers|0WW85SQVS3]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0TKYAEJ8N7.md" tag="0TKYAEJ8N7" kind="object">
# Orthogonal group

## Description

Let $n \ge 1$.
The *orthogonal group* is the set
$$O(n) = \{Q \in M_n(\R) : Q^T Q = I_n\}$$
of real $n \times n$ [[orthogonal matrices|0W02QESPJQ]], with matrix multiplication.
It is a subgroup of the general linear group $GL_n(\R)$.

## Remarks

It is the group of linear isometries of $\R^n$ with the dot product.
Each element has determinant $1$ or $-1$, and $\det \colon O(n) \to \{1, -1\}$ is a surjective homomorphism.
Its kernel is the [[special orthogonal group|0CGYNDCWRR]].
$O(n)$ is a compact Lie group of dimension $n(n-1)/2$.

## Explanations

### L1 human

The orthogonal group collects all rotations and reflections of $n$-dimensional space that fix the origin.
Composing two of them gives another one, and each one can be undone.
In the plane, it consists of the rotations about the origin and the reflections in lines through the origin.
</document>

<document source="https://tamnd.github.io/math2/node/0TKZCSGCCA.md" tag="0TKZCSGCCA" kind="definition">
# Minimum degree

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]] with a nonempty vertex set $V$.
The minimum degree of $G$ is
$$\delta(G) = \min_{v \in V} \deg(v),$$
the smallest [[degree|0T0F6CCS9W]] of a vertex of $G$.

## Remarks

The [[maximum degree|0KNAN9F27P]] $\Delta(G)$ is the largest degree. Always $\delta(G) \le \Delta(G)$.
Each [[cycle graph|0QQ0F12962]] has $\delta = 2$, and each tree with at least two vertices has $\delta = 1$.
A condition on $\delta(G)$ gives a Hamiltonian cycle in [[Dirac's theorem|0TC99BRSSB]].

## Explanations

### L1 human

The minimum degree is the smallest number of edges at any vertex.
If $\delta(G) = 3$, then each vertex has at least three neighbours.
</document>

<document source="https://tamnd.github.io/math2/node/0TMZ21WKCN.md" tag="0TMZ21WKCN" kind="definition">
# Lipschitz map

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]], let $f \colon X \to Y$ be a function and let $K \ge 0$ be a real number.
The map $f$ is Lipschitz with constant $K$ when
$$d_Y\bigl(f(x), f(y)\bigr) \le K \, d_X(x, y) \quad \text{for all } x, y \in X.$$
The map $f$ is Lipschitz when it is Lipschitz with some constant $K \ge 0$.

## Remarks

The smallest such $K$ is the Lipschitz constant of $f$.
The map $x \mapsto |x|$ on $\mathbb{R}$ is Lipschitz with constant $1$. The map $x \mapsto \sqrt{x}$ on $[0, \infty)$ is not Lipschitz, because $\sqrt{x} / x$ is not bounded near $0$.
A Lipschitz map with constant $K < 1$ is a [[contraction|0MT2S0ZX4Y]].
A differentiable $f \colon \mathbb{R} \to \mathbb{R}$ with $|f'(x)| \le K$ for all $x$ is Lipschitz with constant $K$, by the mean value theorem.

## Explanations

### L1 human

A Lipschitz map cannot stretch distances by more than the factor $K$.
For a real function, this means that the slopes of all its chords are between $-K$ and $K$.
</document>

<document source="https://tamnd.github.io/math2/node/0TMZ89Q5KA.md" tag="0TMZ89Q5KA" kind="concept">
# Set-builder notation

## Summary

Set-builder notation describes a [[set|0SJ3RX9TXA]] by a property of its [[elements|0EZ758XE73]].
For a set $X$ and a [[predicate|0T3DPBG981]] $P(x)$ on $X$, the set $\{x \in X \mid P(x)\}$ has as elements exactly the $a \in X$ such that $P(a)$ is true.

## Remarks

A colon is a common alternative to the bar: $\{x \in X : P(x)\}$.
Example: $\{n \in \mathbb{Z} \mid n \text{ is even}\}$ is the set of even integers.
The form $\{f(x) \mid x \in X\}$ denotes the set of the values $f(x)$. This is the [[image|0Q3F20XB0A]] of $X$ under $f$.
Without the bound $x \in X$, the notation $\{x \mid P(x)\}$ can lead to [[Russell's paradox|0HESMMSYEG]].

## Explanations

### L1 human

Instead of a list of elements, you give a test.
$\{x \in \mathbb{R} \mid x^2 = 4\}$ means "all real numbers whose square is $4$".
This set is $\{-2, 2\}$.
</document>

<document source="https://tamnd.github.io/math2/node/0TNR9BQ871.md" tag="0TNR9BQ871" kind="concept">
# Powers of a diagonalisable matrix

## Summary

Let $F$ be a field, and let $A$ be an $n \times n$ matrix over $F$ with $A = P D P^{-1}$.
Here $P$ is invertible and $D$ is diagonal with entries $\lambda_1, \dots, \lambda_n$.
Then, for each integer $k \ge 0$,
$$A^k = P D^k P^{-1},$$
where $D^k$ is the diagonal matrix with entries $\lambda_1^k, \dots, \lambda_n^k$.
So the powers of a [[diagonalisable|07SQCNV233]] matrix are easy to compute.

## Remarks

The factors $P^{-1} P$ in $(P D P^{-1})^k$ cancel in pairs.
The same idea gives $p(A) = P\, p(D)\, P^{-1}$ for each polynomial $p$.
Over $\R$ or $\C$, it also gives the matrix exponential $e^A = P e^D P^{-1}$.
It is the standard method to solve linear recurrences, such as the Fibonacci recurrence.

## Explanations

### L1 human

To raise a diagonalisable matrix to a power, change to the basis of eigenvectors.
There the matrix is diagonal, and its power is the diagonal of the powers of the eigenvalues.
Then change back to the old basis.
</document>

<document source="https://tamnd.github.io/math2/node/0TNS65QB87.md" tag="0TNS65QB87" kind="concept">
# Prime subfield

## Summary

The prime subfield of a field is its smallest subfield.
It is $\mathbb{Q}$ or $\mathbb{F}_p$, up to isomorphism, by the characteristic.
The [[definition of the prime subfield|0VPFBAY7MW]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0TNZC55X3J.md" tag="0TNZC55X3J" kind="counterexample">
# Partial derivatives do not imply differentiability

## Counterexample

One could guess that a function of several variables is differentiable at a point when all its partial derivatives exist there.
This is false.
Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be given by $f(x, y) = \dfrac{xy}{x^2 + y^2}$ for $(x, y) \ne (0, 0)$ and $f(0, 0) = 0$.
Both [[partial derivatives|09EGC6F8PT]] of $f$ exist at $(0, 0)$, and both are $0$.
But $f$ is not continuous at $(0, 0)$, so $f$ is not [[differentiable|0HHHDCXZV4]] at $(0, 0)$.

## Remarks

The function is $0$ on both axes, so $f(h, 0) = f(0, h) = 0$ for each $h$, and both difference quotients at the origin are $0$.
On the line $y = x$ the value is $1/2$, so $f$ is not continuous at the origin, as in the [[separately continuous function|0C0KPAKK9Y]]. A function that is differentiable at a point is continuous there.
Even all directional derivatives are not enough. For $g(x, y) = x^2 y / (x^4 + y^2)$ off the origin and $g(0, 0) = 0$, each directional derivative at the origin exists, but $g = 1/2$ on the parabola $y = x^2$ minus the origin.
If the partial derivatives exist near the point and are continuous at it, then the function is differentiable, by the [[criterion with continuous partial derivatives|0KXSK8TEEB]].

## Explanations

### L1 human

The partial derivatives look only along the two axes.
A function can be flat on both axes and still do something wild between them, here on the diagonal.
Differentiability asks for a good linear approximation in all directions at once, which is much more.
</document>

<document source="https://tamnd.github.io/math2/node/0TP2QKJ8DE.md" tag="0TP2QKJ8DE" kind="definition">
# Circumcentre

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $P$ be the plane through $A$, $B$ and $C$, their [[affine span|0YMB1676R1]].
The circumcentre of $ABC$ is the unique point $O$ of $P$ with
$$d(O, A) = d(O, B) = d(O, C).$$
This common distance $R$ is the circumradius of $ABC$.

## Remarks

The point $O$ exists and is unique. In the plane, it is the common point of the three [[perpendicular bisectors|055VNYT0PB]] of the sides. Two of them meet in one point because the sides are not parallel.
The circumcentre is inside the triangle if the triangle is acute, on the longest side if it is right, and outside if it is obtuse.
In a right triangle, the circumcentre is the midpoint of the hypotenuse, by [[Thales's theorem|0GSDHYMA1T]].
Mathlib defines the circumcentre of each simplex in the same way, inside the affine span of the vertices.

## Explanations

### L1 human

The circumcentre is the point that is equally far from the three corners of a triangle. To find it, draw the perpendicular bisectors of two sides; they cross at the circumcentre. A circle with this centre through one corner passes through all three.
</document>

<document source="https://tamnd.github.io/math2/node/0TPW25CBN6.md" tag="0TPW25CBN6" kind="definition">
# Congruence modulo n

## Definition

Let $n$ be a positive integer, and let $a$ and $b$ be integers.
The integer $a$ is congruent to $b$ modulo $n$ when $n$ [[divides|096PW7Q014]] $a - b$.
The notation is $a \equiv b \pmod{n}$. The integer $n$ is the modulus of the congruence.

## Remarks

Equivalently, $a \equiv b \pmod n$ if and only if $a$ and $b$ have the same remainder on division by $n$, that is, $a \bmod n = b \bmod n$.
Congruence modulo $n$ is an equivalence relation on the integers. Its classes are the [[residue classes|0QRV8FPK5M]] modulo $n$.
For example, $38 \equiv 14 \pmod{12}$, because $12 \mid 24$.
Some texts allow each integer $n$ as a modulus. Then congruence modulo $0$ is equality, and congruence modulo $-n$ is congruence modulo $n$.

## Explanations

### L1 human

Two integers are congruent modulo $n$ when they differ by a multiple of $n$.
On a $12$-hour clock, $14$ o'clock and $2$ o'clock show the same time, because $14 \equiv 2 \pmod{12}$.
</document>

<document source="https://tamnd.github.io/math2/node/0TPWS0K1BC.md" tag="0TPWS0K1BC" kind="statement">
# An algebraic extension of an algebraic extension is algebraic

## Statement

Let $K \subseteq L \subseteq M$ be fields.
If $M/L$ and $L/K$ are [[algebraic|09EYMKMSB9]], then $M/K$ is algebraic.

## Remarks

The proof: let $a \in M$ have minimal polynomial $x^n + c_{n-1} x^{n-1} + \dots + c_0$ over $L$. The field $E = K(c_0, \dots, c_{n-1})$ is finite over $K$, and $E(a)$ is finite over $E$. By the [[tower law|0A7KJ68XHT]], $E(a)/K$ is finite, so $a$ is algebraic over $K$.
Also, if $M/K$ is algebraic, then $M/L$ and $L/K$ are algebraic.

## Explanations

### L1 human

Algebraic extensions can be stacked.
If each step adds only algebraic elements, then the whole tower adds only algebraic elements.
</document>

<document source="https://tamnd.github.io/math2/node/0TR9SR9FTB.md" tag="0TR9SR9FTB" kind="definition">
# Strictly monotone function

## Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
The function $f$ is strictly increasing if $f(x) < f(y)$ for all $x, y \in D$ with $x < y$.
It is strictly decreasing if $f(x) > f(y)$ for all $x, y \in D$ with $x < y$.
It is strictly monotone if it is strictly increasing or strictly decreasing.

## Remarks

With $\le$ and $\ge$ in place of $<$ and $>$, we get increasing, decreasing, and monotone functions.
A strictly monotone function is injective.
For example, $x^3$ is strictly increasing on $\mathbb{R}$, and $x^2$ is strictly increasing on $[0, \infty)$.
Mathlib defines strictly increasing maps as `StrictMono`.

## Explanations

### L1 human

A strictly increasing function always goes up: a larger input gives a larger output.
Its graph rises from left to right, with no flat parts.
</document>

<document source="https://tamnd.github.io/math2/node/0TRNT4X7J0.md" tag="0TRNT4X7J0" kind="concept">
# Euclidean algorithm

## Summary

The Euclidean algorithm computes the [[greatest common divisor|02WF878BH1]] of two natural numbers $a$ and $b$.
It uses two rules: $\gcd(a, 0) = a$, and $\gcd(a, b) = \gcd(b, a \bmod b)$ when $b \neq 0$.
It applies the second rule until the second number is $0$. Then the first number is the result.
The algorithm stops, because the second number gets smaller at each step.

## Remarks

The second rule is true because $a$ and $b$ have the same common divisors as $b$ and $a \bmod b$.
The number of steps is at most a constant times the number of digits of the smaller input. Lamé's theorem gives this bound.
The algorithm also works in each Euclidean domain, for example in the ring of polynomials over a field.

## Explanations

### L1 human

To find $\gcd(48, 18)$, divide again and again, and keep the divisor and the remainder.
$48 = 2 \cdot 18 + 12$, then $18 = 1 \cdot 12 + 6$, then $12 = 2 \cdot 6 + 0$.
The last nonzero remainder is $6$, so $\gcd(48, 18) = 6$.
</document>

<document source="https://tamnd.github.io/math2/node/0TS7VZJBWD.md" tag="0TS7VZJBWD" kind="definition">
# Orthogonal vectors

## Definition

Let $V$ be an inner product space over $\R$ or $\C$.
Two vectors $x, y \in V$ are *orthogonal*, written $x \perp y$, when $\langle x, y \rangle = 0$, where $\langle \cdot, \cdot \rangle$ is the [[inner product|0NB0AWGJBW]].

## Remarks

The relation is symmetric, because $\langle y, x \rangle = \overline{\langle x, y \rangle}$.
The zero vector is orthogonal to every vector, and it is the only vector orthogonal to itself.
In $\R^2$ with the dot product, $(1, 2)$ and $(-2, 1)$ are orthogonal.

## Explanations

### L1 human

Orthogonal is the general word for perpendicular.
Two vectors are orthogonal when their inner product is zero.
In the plane and in space, this means that they meet at a right angle.
</document>

<document source="https://tamnd.github.io/math2/node/0TST4MHA3A.md" tag="0TST4MHA3A" kind="definition">
# Law of a random variable

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space, let $(E, \mathcal{E})$ be a set with a σ-algebra, and let $X \colon \Omega \to E$ be a [[random variable|0DKRR22AEQ]].
The law of $X$, also called the distribution of $X$, is the function $P_X \colon \mathcal{E} \to [0, 1]$ with
$$P_X(B) = P(X^{-1}(B)) = P(X \in B) \quad \text{for } B \in \mathcal{E}.$$

## Remarks

$P_X$ is a [[probability distribution|0QB913WM5E]] on $(E, \mathcal{E})$. It is the image measure, or pushforward, of $P$ under $X$, and one also writes $P \circ X^{-1}$ or $X_* P$.
Notation: $X \sim \mu$ means that the law of $X$ is $\mu$.
Two random variables can have the same law but be different functions. For a fair coin toss, the indicator of heads and the indicator of tails both have the Bernoulli law with parameter $1/2$.

## Explanations

### L1 human

The law of $X$ forgets the sample space and keeps only the chances of the values of $X$.
For the sum of two dice, the law says that $7$ has probability $6/36$, $2$ has probability $1/36$, and so on.
</document>

<document source="https://tamnd.github.io/math2/node/0TTA57SEJG.md" tag="0TTA57SEJG" kind="definition">
# Convergence in mean

## Definition

Let $r \ge 1$ be real. Let $X, X_1, X_2, \dots$ be real random variables on one probability space with $E[|X|^r] < \infty$ and $E[|X_n|^r] < \infty$ for each $n$.
The sequence $(X_n)$ converges in $r$-th mean, or in $L^r$, to $X$ if
$$\lim_{n \to \infty} E\left[|X_n - X|^r\right] = 0.$$
For $r = 1$ one says convergence in mean, and for $r = 2$ convergence in mean square.

## Remarks

By [[Markov's inequality|05PWWHYBX5]], $P(|X_n - X| > \varepsilon) \le E[|X_n - X|^r] / \varepsilon^r$. So convergence in $r$-th mean implies convergence in probability.
For $s \ge r \ge 1$, convergence in $s$-th mean implies convergence in $r$-th mean.
Convergence in mean does not imply almost sure convergence, and almost sure convergence does not imply convergence in mean.
Example: on $[0, 1]$ with the uniform measure, $X_n = n \, 1_{[0, 1/n]}$ converges to $0$ almost surely but $E[X_n] = 1$ for all $n$.

## Explanations

### L1 human

Convergence in mean asks that the average size of the error $|X_n - X|^r$ goes to $0$.
Mean square convergence, with $r = 2$, is common in statistics, where it says that the mean squared error of an estimator goes to $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0TTDGTEFMN.md" tag="0TTDGTEFMN" kind="concept">
# Conditional expectation

## Summary

The conditional expectation of a random variable given some information is its best estimate when one knows that information. It is itself a random variable.
The node [[conditional expectation|0YAMJRK04E]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0TTNRP5PFQ.md" tag="0TTNRP5PFQ" kind="definition">
# Estimator

## Definition

Let $\Theta$ be a set of parameters, and for each $\theta \in \Theta$ let $P_\theta$ be a probability distribution. Let $X_1, \dots, X_n$ be a sample whose joint law is determined by an unknown $\theta \in \Theta$. Let $\tau \colon \Theta \to \mathbb{R}^k$ be a function.
An estimator of $\tau(\theta)$ is a [[statistic|036756NFD6]] $\hat{\tau} = g(X_1, \dots, X_n)$ with values in $\mathbb{R}^k$, where $g$ does not depend on $\theta$.
The value of $\hat{\tau}$ for observed data is an estimate of $\tau(\theta)$.

## Remarks

The definition does not ask that $\hat{\tau}$ is close to $\tau(\theta)$. One measures the quality of an estimator with its [[bias|01BD2TFFP4]], its variance and its [[mean squared error|0V4VSCBXP3]].
For an i.i.d. sample, the sample mean is an estimator of the mean, and the sample variance is an estimator of the variance.
One often writes $\hat{\theta}$ for an estimator of $\theta$ itself.

## Explanations

### L1 human

An estimator is a recipe that turns data into a guess of an unknown quantity.
For example, the fraction of "yes" answers in a poll is an estimator of the fraction of "yes" voters in the whole population.
</document>

<document source="https://tamnd.github.io/math2/node/0TV07AQT62.md" tag="0TV07AQT62" kind="definition">
# Real exponentiation

## Definition

Let $a > 0$ and let $x$ be a real number.
The power $a^x$ is
$$a^x = e^{x \ln a}.$$

## Remarks

For natural numbers $x = n$, this agrees with the product of $n$ factors $a$. For $x = p/q$, it agrees with $\sqrt[q]{a^p}$.
The usual rules hold: $a^{x + y} = a^x a^y$, $(a^x)^y = a^{x y}$, and $(a b)^x = a^x b^x$.
For $a = e$, it gives $e^x = \exp(x)$, since $\ln e = 1$. Mathlib defines `Real.rpow` with extra conventions for $a \le 0$.

## Explanations

### L1 human

To raise $a$ to a real power $x$, multiply the logarithm of $a$ by $x$ and apply the exponential function.
This fills in all powers between the rational ones continuously.
</document>

<document source="https://tamnd.github.io/math2/node/0TVJZYB6SH.md" tag="0TVJZYB6SH" kind="concept">
# Compact space

## Summary

A compact space is a topological space in which each open cover has a finite subcover.
Closed bounded intervals of the real line are the basic examples.
</document>

<document source="https://tamnd.github.io/math2/node/0TWPTQ0AD7.md" tag="0TWPTQ0AD7" kind="definition">
# Incentre

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\mathbb{R}^2$. For a point $X$ and a line $\ell$, let $d(X, \ell)$ be the [[distance from the point to the line|01AAYNSBN6]].
The incentre of $ABC$ is the unique point $I$ in the interior of the triangle with
$$d(I, BC) = d(I, CA) = d(I, AB).$$
This common distance $r$ is the inradius of $ABC$.

## Remarks

The point $I$ lies on each of the three internal angle bisectors of the triangle. Thus the three bisectors meet in one point.
Let $a = d(B, C)$, $b = d(C, A)$ and $c = d(A, B)$ be the side lengths, with $d$ the [[Euclidean distance|0NPPGHQNXQ]]. Then $I = \dfrac{aA + bB + cC}{a + b + c}$.
Outside the triangle there are three more points at the same distance from the three side lines. They are the excentres.
Mathlib's `Affine.Simplex.incenter` defines the incentre of each simplex in a Euclidean affine space.

## Explanations

### L1 human

The incentre is the point inside a triangle that is equally far from the three sides. You find it as the crossing point of two angle bisectors. For the right triangle with the vertices $(0, 0)$, $(3, 0)$ and $(0, 4)$, the incentre is $(1, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/0TWQ5PRKWS.md" tag="0TWQ5PRKWS" kind="definition">
# Equivalent metrics

## Definition

Let $X$ be a set and let $d_1$ and $d_2$ be metrics on $X$.
The metrics $d_1$ and $d_2$ are equivalent when they have the same [[open sets|01ZQQDCERP]], that is, when they give the same [[metric topology|0A6YRNZB2R]] on $X$.
The metrics are strongly equivalent when there are real numbers $a > 0$ and $b > 0$ with
$$a \, d_1(x, y) \le d_2(x, y) \le b \, d_1(x, y) \quad \text{for all } x, y \in X.$$

## Remarks

Strongly equivalent metrics are equivalent: each ball for one metric contains a ball with the same centre for the other metric.
On $\mathbb{R}^n$, the [[taxicab metric|07CJNE6ZAQ]], the [[Euclidean metric|0E1MMCAY5B]] and the [[sup metric|0VTAXQEWQD]] are strongly equivalent.
On $\mathbb{R}$, the metrics $|x - y|$ and $\min(|x - y|, 1)$ are equivalent but not strongly equivalent, because only the second one is bounded.
Equivalent metrics have the same convergent sequences and the same continuous maps. Completeness can change: see the Remarks of [[complete metric space|08EWT7J0SC]].

## Explanations

### L1 human

Equivalent metrics can give different numbers for distances, but they agree on which points are near each other.
For example, on the plane, the taxicab distance is never more than $\sqrt{2}$ times the straight distance, and never less than it.
</document>

<document source="https://tamnd.github.io/math2/node/0TWTS88TWN.md" tag="0TWTS88TWN" kind="definition">
# Invertible matrix

## Definition

Let $R$ be a commutative ring and let $A$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $A$ is *invertible* when there is an $n \times n$ matrix $B$ over $R$ such that
$$AB = BA = I_n,$$
where $I_n$ is the [[identity matrix|06DW9BAWB1]].
The matrix $B$ is then unique; it is the *inverse* of $A$, written $A^{-1}$.

## Remarks

Over a field, other names are *nonsingular* and *nondegenerate*.
A matrix over $R$ is invertible if and only if its [[determinant is a unit|0783S9D0ZH]] of $R$.
Over a field, a [[one-sided inverse is two-sided|0753Y2X5K6]], and $A$ is invertible if and only if $Ax = 0$ has only the solution $x = 0$.
The matrix $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is not invertible over $\R$, because its determinant is $0$.

## Explanations

### L1 human

A matrix is invertible when another matrix undoes it.
If $A$ sends $x$ to $y$, then $A^{-1}$ sends $y$ back to $x$.
For $2 \times 2$ matrices, $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ when $ad - bc \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0TX2YH2FQS.md" tag="0TX2YH2FQS" kind="definition">
# Symmetric bilinear form associated with a quadratic form

## Definition

Let $F$ be a field of characteristic not $2$, and let $V$ be a finite-dimensional vector space over $F$.
Let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
The *symmetric bilinear form associated with* $Q$ is
$$B(u, v) = \tfrac{1}{2} \bigl( Q(u + v) - Q(u) - Q(v) \bigr) \qquad (u, v \in V).$$

## Remarks

$B$ is a [[symmetric bilinear form|0AXZNE5XNH]] with $B(v, v) = Q(v)$, and it is the only one.
Conversely, each symmetric bilinear form $B$ gives the quadratic form $v \mapsto B(v, v)$; the two constructions are inverse to each other.
For $Q(x) = x^T A x$ with $A$ symmetric, $B(x, y) = x^T A y$.
It is half of the [[polar form|0HYFWKKY3C]] of $Q$.

## Explanations

### L1 human

Outside characteristic $2$, quadratic forms and symmetric bilinear forms are the same thing in two forms.
From $B$, we get $Q$ by putting the same vector twice.
From $Q$, we get $B$ by this formula.
</document>

<document source="https://tamnd.github.io/math2/node/0TXCQ9CST1.md" tag="0TXCQ9CST1" kind="concept">
# Integration over a non-rectangular region

## Summary

To integrate a function $f$ over a bounded region $D$ that is not a rectangle, take a rectangle $R \supseteq D$. Then integrate over $R$ the function equal to $f$ on $D$ and to $0$ on $R \setminus D$.
For a region between two graphs, $D = \{(x, y) : a \le x \le b, \ \varphi_1(x) \le y \le \varphi_2(x)\}$ with $\varphi_1 \le \varphi_2$ continuous, a continuous $f$ gives $\iint_D f \, dA = \int_a^b \int_{\varphi_1(x)}^{\varphi_2(x)} f(x, y) \, dy \, dx$.

## Remarks

Such a region is normal with respect to the $x$-axis. A region of the form $\{c \le y \le d, \ \psi_1(y) \le x \le \psi_2(y)\}$ is normal with respect to the $y$-axis.
The definition does not depend on the choice of $R$. For continuous $f$, the integral exists when the boundary of $D$ has [[Jordan content|0KTGJW1ZTH]] zero.
Example: for $D = \{0 \le x \le 1, \ x^2 \le y \le 1\}$, $\iint_D x \, dA = \int_0^1 x (1 - x^2) \, dx = 1/4$.
The [[double integral|0WAR4F23B4]] over $D$ is computed as an [[iterated integral|0WZCHE1ZF9]] with variable bounds.

## Explanations

### L1 human

Most regions in practice are not rectangles, but many lie between two curves.
Then the inner integral runs from the lower curve to the upper curve, and the outer integral runs over the range of the other variable.
</document>

<document source="https://tamnd.github.io/math2/node/0TXNRQZY0H.md" tag="0TXNRQZY0H" kind="concept">
# Primitive polynomial

## Summary

A primitive polynomial is a polynomial whose coefficients have no common factor except units.
The [[definition of a primitive polynomial|0SP14DZN0G]] gives the condition.
</document>

<document source="https://tamnd.github.io/math2/node/0TXTSEMHR5.md" tag="0TXTSEMHR5" kind="definition">
# Limit of a function

## Definition

Let $D$ be a subset of $\mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function, and let $a$ be a [[limit point|07EGCNHAVW]] of $D$.
A real number $L$ is the limit of $f$ at $a$ if the following holds.
For each open interval $V$ containing $L$ there is an open interval $U$ containing $a$ with $f(x) \in V$ for all $x \in U \cap D$ with $x \ne a$.
The notation is $\lim_{x \to a} f(x) = L$, or $f(x) \to L$ as $x \to a$.

## Remarks

The value of $f$ at $a$ plays no role, and $a$ need not lie in $D$.
The [[epsilon-delta definition|0GBZDAFPX8]] states the same condition with numbers. The [[sequential characterization|000MXHEY22]] states it with sequences.
The limit is unique when it exists, since $a$ is a limit point of $D$.
Mathlib expresses limits with filters, as `Filter.Tendsto`.

## Explanations

### L1 human

The limit of $f$ at $a$ is the value that $f(x)$ approaches as $x$ comes close to $a$.
Only the values near $a$ matter, not the value at $a$ itself.
</document>

<document source="https://tamnd.github.io/math2/node/0TXWV1JGG1.md" tag="0TXWV1JGG1" kind="statement">
# Product of a subgroup and a normal subgroup is a subgroup

## Statement

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
Then the [[product|0PT52XQZD7]] $HN$ is a subgroup of $G$, and $HN = NH$.

## Remarks

If $H$ is also normal, then $HN$ is normal in $G$.
The proof uses $hN = Nh$ for each $h \in H$, so that $HN = \bigcup_{h \in H} hN = \bigcup_{h \in H} Nh = NH$.
This product appears in the second isomorphism theorem.

## Explanations

### L1 human

The product of two subgroups can fail to be a subgroup, but not when one of them is normal.
A normal subgroup lets you move its elements past other elements, at the cost of changing them inside the subgroup.
So products of the form $hn$ stay of that form when you multiply them.
</document>

<document source="https://tamnd.github.io/math2/node/0TYB3HM9KE.md" tag="0TYB3HM9KE" kind="definition">
# Isolated singularity

## Definition

Let $c \in \mathbb{C}$ and let $f$ be a complex function.
The point $c$ is an isolated singularity of $f$ if there is $r > 0$ such that $f$ is defined and [[holomorphic|0DG0Z72CQD]] on the punctured disc $\{z : 0 < |z - c| < r\}$.
The value of $f$ at $c$, if there is one, plays no role.

## Remarks

Each isolated singularity is of exactly one of three types: [[removable|0P18JC72RE]], a [[pole|0WPQYW8MTE]], or [[essential|0YKDJBV26S]].
For example, $0$ is an isolated singularity of $1/z$, of $\frac{\sin z}{z}$ and of $e^{1/z}$.
The point $0$ is not an isolated singularity of $\frac{1}{\sin(1/z)}$, because the poles $\frac{1}{k\pi}$ accumulate at $0$. It is not an isolated singularity of $\operatorname{Log} z$ either, because $\operatorname{Log}$ is not holomorphic on any punctured disc around $0$.

## Explanations

### L1 human

An isolated singularity is a single bad point of a function, with only good points around it.
The function is holomorphic on a small disc around the point, with only the center removed.
</document>

<document source="https://tamnd.github.io/math2/node/0TYMASG33Y.md" tag="0TYMASG33Y" kind="statement">
# Characteristic polynomial does not depend on the basis

## Statement

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$, and let $P$ be an invertible $n \times n$ matrix over $K$.
Then the [[characteristic polynomials|0VSSPAFM2C]] of $A$ and $P^{-1} A P$ are equal:
$$p_{P^{-1} A P}(t) = p_A(t).$$
So [[similar matrices|0E89HQATM8]] have the same characteristic polynomial.

## Remarks

The proof is $\det(t I - P^{-1} A P) = \det(P^{-1} (t I - A) P) = \det(t I - A)$.
Hence the characteristic polynomial of a linear map does not depend on the basis.
The converse fails: $I_2$ and $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ both have $p(t) = (t - 1)^2$, but they are not similar.

## Explanations

### L1 human

A change of basis does not change the characteristic polynomial.
So the eigenvalues, the trace and the determinant, which we can read from it, are the same in every basis.
They are properties of the linear map, not of one matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0TYQNHVB4C.md" tag="0TYQNHVB4C" kind="statement">
# Surjective iff it has a right inverse

## Statement

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
Assume the [[axiom of choice|0HTGB13FE2]].
Then $f$ is [[surjective|0XKH881BZ7]] if and only if $f$ has a [[right inverse|0DDPZAG5ER]].

## Remarks

If $f \circ h = \mathrm{id}_B$, then each $b$ equals $f(h(b))$, so $f$ is surjective. This direction does not use choice.
For the other direction, choose for each $b \in B$ an element $h(b)$ of the nonempty set $f^{-1}(\{b\})$.
The statement "each surjection has a right inverse" is equivalent to the axiom of choice.

## Explanations

### L1 human

For a surjection, each $b$ has at least one input that maps to it.
A right inverse picks one such input for each $b$.
For infinitely many $b$ with no rule to pick, you need the axiom of choice.
</document>

<document source="https://tamnd.github.io/math2/node/0TZ2DPT0JV.md" tag="0TZ2DPT0JV" kind="concept">
# Principal part

## Summary

The principal part of a function at an isolated singularity is the part of its Laurent series with the negative powers, see the [[definition|0NDV5R6Z6X]].
</document>

<document source="https://tamnd.github.io/math2/node/0TZAG60ZYT.md" tag="0TZAG60ZYT" kind="statement">
# Row rank equals column rank

## Statement

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$.
Then the dimension of the [[row space|069MCGX635]] of $A$ equals the dimension of the [[column space|03AJKXH1AV]] of $A$.
Equivalently, $\rank A^T = \rank A$, where $A^T$ is the [[transpose|09DXWYRD43]].

## Remarks

One proof uses row reduction: row operations keep the row space and the dimension of the column space.
In reduced row echelon form, both dimensions equal the number of pivots.
Another proof uses the [[annihilator|0ASHNYF7AG]]: the dual map has the transposed matrix and the same rank.
Over $\C$, also $\rank A^* = \rank A$.

## Explanations

### L1 human

The number of independent rows of a matrix always equals the number of independent columns.
This is surprising, because rows and columns live in different spaces.
For a $3 \times 5$ matrix, it shows at once that the rank is at most $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0TZEEFTNCC.md" tag="0TZEEFTNCC" kind="definition">
# Maximal element

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.
An element $m \in S$ is a maximal element of $S$ when no $s \in S$ satisfies $m < s$.
Equivalently, for each $s \in S$, $m \le s$ implies $s = m$.

## Remarks

A maximal element need not be the [[greatest element|0Z7HDDXX3X]], and a set can have several maximal elements.
In $\{\{1\}, \{2\}, \{1, 2, 3\}, \{4\}\}$ under $\subseteq$, the maximal elements are $\{1, 2, 3\}$ and $\{4\}$.
[[Zorn's lemma|04RXTQA5ZA]] gives maximal elements in many infinite posets.

## Explanations

### L1 human

A maximal element has nothing above it.
It may still be incomparable with other elements, so it is not necessarily above everything.
</document>

<document source="https://tamnd.github.io/math2/node/0TZK91NQE3.md" tag="0TZK91NQE3" kind="example">
# The sequence r^n tends to infinity for r > 1

## Example

Let $r$ be a real number with $r > 1$.
Then $r^n \to +\infty$ as $n \to \infty$, in the sense of a [[sequence tending to infinity|09AZCV6GMC]].
Write $r = 1 + h$ with $h > 0$. By Bernoulli's inequality, $r^n \ge 1 + n h$ for each natural number $n$.
Let $M$ be a real number, and choose a natural number $N$ with $N h > M$. Then $r^n > M$ for all $n \ge N$.

## Remarks

For $r < -1$ the sequence $r^n$ does not tend to $+\infty$ or to $-\infty$. Its terms change sign, but $|r^n| \to +\infty$.
Exponential growth is faster than polynomial growth. The node [[exponentials dominate powers in sequences|0MVTCFW42K]] makes this precise.

## Explanations

### L1 human

Doubling again and again gives $1, 2, 4, 8, 16, \dots$, and these numbers grow without bound.
Each factor $r > 1$ does the same, but a factor close to $1$ needs more steps.
</document>

<document source="https://tamnd.github.io/math2/node/0V032VWTHD.md" tag="0V032VWTHD" kind="concept">
# Connected space

## Summary

A topological space is connected when it is not the union of two disjoint nonempty open sets.
A connected space is in one piece.
</document>

<document source="https://tamnd.github.io/math2/node/0V05FC4GKQ.md" tag="0V05FC4GKQ" kind="definition">
# Induced subgraph

## Definition

Let $G = (V, E)$ be a [[graph|0JD5C82QMB]] and let $S \subseteq V$.
The subgraph induced by $S$ is the graph $G[S]$ with vertex set $S$ and edge set
$$\{ \{u, v\} \in E : u, v \in S \}.$$
A subgraph of this form is an induced subgraph.

## Remarks

An induced subgraph is a [[subgraph|0P11X5HKAX]] that keeps every edge of $G$ between its vertices.
In $K_4$, every subgraph induced by $3$ vertices is a triangle. A path on these $3$ vertices is a subgraph, but it is not induced.
A [[clique|0G52BPNPKG]] induces a complete graph, and an [[independent set|0KZJ82WEPP]] induces a graph with no edges.

## Explanations

### L1 human

Choose some vertices and keep all the edges between them.
The result is the induced subgraph on those vertices.
</document>

<document source="https://tamnd.github.io/math2/node/0V0778DCW9.md" tag="0V0778DCW9" kind="statement">
# Variance of a sum of independent random variables

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] real random variables with $E[X_i^2] < \infty$ for each $i$. Then
$$\operatorname{Var}(X_1 + \dots + X_n) = \operatorname{Var}(X_1) + \dots + \operatorname{Var}(X_n).$$

## Remarks

The proof uses the [[variance of a sum|0D86GJEZ39]]: independent variables are [[uncorrelated|077GN9HDR6]], so each covariance term is $0$.
So the formula holds more generally for pairwise uncorrelated $X_i$.
For i.i.d. $X_i$ with variance $\sigma^2$, the sample mean $\bar{X}_n = (X_1 + \dots + X_n)/n$ has variance $\sigma^2 / n$.
For $X \sim \operatorname{Bin}(n, p)$, the formula gives $\operatorname{Var}(X) = np(1 - p)$, as $X$ is a sum of $n$ independent Bernoulli variables.

## History

The formula is also called the Bienaymé formula, after Irénée-Jules Bienaymé.

## Explanations

### L1 human

For independent variables, variances simply add.
This is why an average of many independent measurements is more precise than one measurement: its variance gets smaller like $1/n$.
</document>

<document source="https://tamnd.github.io/math2/node/0V0E04DWX2.md" tag="0V0E04DWX2" kind="statement">
# Sign of a cycle

## Statement

Let $k \geq 1$ and let $\sigma$ be a $k$-cycle in $S_n$.
Then $\operatorname{sgn}(\sigma) = (-1)^{k-1}$.

## Remarks

A $k$-cycle is a product of $k - 1$ transpositions, which gives the formula.
Thus a cycle is an [[even permutation|0SQTMQTSRZ]] exactly when its length is odd.
With the [[disjoint cycle decomposition|0YWDAZPRPQ]], the sign of $\sigma \in S_n$ is $(-1)^{n - c}$, where $c$ is the number of cycles, with the fixed points counted as $1$-cycles.

## Explanations

### L1 human

A cycle of length $k$ is a chain of $k - 1$ swaps.
So a $3$-cycle is even and a transposition is odd.
The length and the parity of a cycle are opposite.
</document>

<document source="https://tamnd.github.io/math2/node/0V0GC7BQAN.md" tag="0V0GC7BQAN" kind="example">
# Complete bipartite graph

## Example

Let $m$ and $n$ be natural numbers.
The complete bipartite graph $K_{m,n}$ is the [[graph|0JD5C82QMB]] whose vertex set is a disjoint union $A \sqcup B$ with $|A| = m$ and $|B| = n$.
Two vertices are adjacent if and only if one is in $A$ and the other is in $B$.

## Remarks

$K_{m,n}$ has $m + n$ vertices and $mn$ edges.
It is [[bipartite|07DVHJ8N0A]] with parts $A$ and $B$. For $m, n \ge 1$ its [[chromatic number|0FJBBCXYPS]] is $2$.
$K_{1,n}$ is a star. [[$K_{3,3}$ is not planar|030A3FHN14]].

## Explanations

### L1 human

Split the vertices into two groups and join every vertex of one group to every vertex of the other group.
There are no edges inside a group.
</document>

<document source="https://tamnd.github.io/math2/node/0V11FEXK08.md" tag="0V11FEXK08" kind="statement">
# Frobenius coin problem

## Statement

Let $a$ and $b$ be [[coprime|0GFKP8F8AG]] integers with $a > 1$ and $b > 1$.
Then $ab - a - b$ is not of the form $ax + by$ with natural numbers $x$ and $y$.
Every integer $n > ab - a - b$ is of this form.

## Remarks

The number $ab - a - b$ is the Frobenius number of $a$ and $b$: the largest amount that one cannot pay with coins of values $a$ and $b$.
For example, with coins of $3$ and $5$, the amount $7$ is impossible, but $8$, $9$, $10$ and all larger amounts are possible.
Exactly $(a - 1)(b - 1)/2$ positive integers are not of the form $ax + by$.
For three or more coin values, no such simple formula is known.

## History

James Joseph Sylvester found the formula for two coins in 1882.

## Explanations

### L1 human

With coins of values $a$ and $b$, you can pay all large amounts when $a$ and $b$ are coprime.
The theorem gives the largest amount that you cannot pay: $ab - a - b$.
</document>

<document source="https://tamnd.github.io/math2/node/0V1EJCT8G6.md" tag="0V1EJCT8G6" kind="concept">
# Derivative

## Summary

The derivative of a function at a point is the limit of the slopes of secant lines through that point. It measures the rate of change of the function, and it is the slope of the tangent line. The node [[derivative|0F1WR9C83K]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0V2DV29EGT.md" tag="0V2DV29EGT" kind="statement">
# Derivative of cosine

## Statement

The cosine function is differentiable on $\mathbb{R}$, and $\cos'(x) = -\sin x$ for all real $x$.

## Remarks

[[Differentiate the power series|002X7EK91H]] term by term. The derivative of $\frac{(-1)^n x^{2n}}{(2n)!}$ is $-\frac{(-1)^{n-1} x^{2n-1}}{(2n-1)!}$ for $n \ge 1$.
So $\sin'' = -\sin$ and $\cos'' = -\cos$. Both functions solve $f'' + f = 0$.
Mathlib states it as `Real.hasDerivAt_cos`.

## Explanations

### L1 human

The slope of the cosine curve at $x$ is $-\sin x$.
The cosine curve falls where the sine is positive.
</document>

<document source="https://tamnd.github.io/math2/node/0V2H5PGJ9B.md" tag="0V2H5PGJ9B" kind="object">
# Aleph-null

## Description

Aleph-null, written $\aleph_0$, is the [[cardinality|0JTGRV4WT4]] of the set $\mathbb{N}$ of natural numbers.
It is the cardinality of each [[countably infinite set|0SEZ3JGZP7]].
It is the least infinite [[cardinal number|0QKPNZKNEA]]: each infinite set has cardinality at least $\aleph_0$, with the axiom of choice.

## Remarks

Other names are aleph-zero and aleph-nought.
$\aleph_0 + \aleph_0 = \aleph_0 \cdot \aleph_0 = \aleph_0$, but $2^{\aleph_0} > \aleph_0$ by [[Cantor's theorem|0WTEMGYV2C]].
As an ordinal, $\aleph_0$ is $\omega$, the set of the finite ordinals.

## Explanations

### L1 human

$\aleph_0$ is the size of the infinity of counting.
The sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ all have this size.
</document>

<document source="https://tamnd.github.io/math2/node/0V2WP9B1MF.md" tag="0V2WP9B1MF" kind="concept">
# Norm

## Summary

A norm on a real or complex vector space is a function that gives each vector a length.
It satisfies the triangle inequality and absolute homogeneity, and only the zero vector has length zero.
</document>

<document source="https://tamnd.github.io/math2/node/0V33BNKZ9S.md" tag="0V33BNKZ9S" kind="concept">
# Smooth function

## Summary

A smooth function is a function with continuous derivatives of all orders. Smoothness is the top of the scale of differentiability classes $C^k$. The node [[smooth function|0BE3QDH73A]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0V33SPH7RR.md" tag="0V33SPH7RR" kind="statement">
# Intersection of subspaces

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be [[linear subspaces|00YC1D652K]] of $V$.
Then $U \cap W$ is a linear subspace of $V$.
It is the largest subspace of $V$ that is contained in both $U$ and $W$.
More generally, the intersection of each nonempty family of subspaces of $V$ is a subspace of $V$.

## Remarks

The union $U \cup W$ is a subspace only when $U \subseteq W$ or $W \subseteq U$.
The smallest subspace that contains $U \cup W$ is the [[sum|0JMTY8MF54]] $U + W$.
In $\R^3$, two different planes through the origin meet in a line through the origin.

## Explanations

### L1 human

A vector that lies in two subspaces stays in both of them when we add or scale.
So the common part of two subspaces is again a subspace.
Two planes through the origin in space cross in a line, which is a subspace.
</document>

<document source="https://tamnd.github.io/math2/node/0V3CFG1KK9.md" tag="0V3CFG1KK9" kind="definition">
# Semigroup

## Definition

A semigroup is a pair $(S, \ast)$, where $S$ is a set and $\ast$ is a [[binary operation|0M418FE7BB]] on $S$ that is [[associative|0K8F7PVDMX]].
Thus $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a, b, c \in S$.

## Remarks

The positive integers with addition form a semigroup that has no identity element.
Each [[monoid|0N6R902FG1]] and each [[group|G5S1M9]] is a semigroup.
Some authors allow the empty set as a semigroup, and others do not.

## Explanations

### L1 human

A semigroup is the simplest structure with an associative operation.
You can combine elements and the grouping does not matter, but maybe nothing acts as "do nothing" and no step can be undone.
An example is the set of nonempty words over an alphabet, with the operation that writes one word after the other.
</document>

<document source="https://tamnd.github.io/math2/node/0V3FZMVB4K.md" tag="0V3FZMVB4K" kind="definition">
# Gamma function

## Definition

For a real number $s > 0$, the gamma function is
$$\Gamma(s) = \int_0^\infty t^{s-1} e^{-t} \, dt.$$

## Remarks

The [[improper integral|0VCTBJMEJF]] converges at $0$ because $s > 0$, and at $\infty$ because $e^{-t}$ decays fast. Both claims follow from the [[comparison test|0FR1BDKK10]].
Integration by parts gives $\Gamma(s + 1) = s \, \Gamma(s)$. Also $\Gamma(1) = 1$, so [[$\Gamma$ extends the factorial|0FDCW77H44]].
The function extends to all real numbers except $0, -1, -2, \dots$, by the recursion. Mathlib defines this extension as `Real.Gamma`.
The notation $\Gamma$ is due to Adrien-Marie Legendre.

## Explanations

### L1 human

The gamma function is an integral that depends on a parameter $s$.
At positive integers it gives factorials, so it interpolates the factorial.
</document>

<document source="https://tamnd.github.io/math2/node/0V3RDQV3ZX.md" tag="0V3RDQV3ZX" kind="definition">
# Cauchy principal value

## Definition

Let $a < b < c$ be [[real numbers|08J2M1JYAG]].
Let $f$ be a function on $[a, c] \setminus \{b\}$ that is integrable on $[a, b - \varepsilon]$ and on $[b + \varepsilon, c]$ for each $\varepsilon$ with $0 < \varepsilon < \min(b - a, c - b)$.
If the limit
$$\operatorname{p.v.} \int_a^c f(x) \, dx = \lim_{\varepsilon \to 0^+} \left( \int_a^{b - \varepsilon} f(x) \, dx + \int_{b + \varepsilon}^{c} f(x) \, dx \right)$$
exists, it is the Cauchy principal value of the integral of $f$ over $[a, c]$.
Let $f$ be a function on $\mathbb{R}$ that is integrable on each interval $[-R, R]$.
If the limit $\operatorname{p.v.} \int_{-\infty}^{\infty} f(x) \, dx = \lim_{R \to \infty} \int_{-R}^{R} f(x) \, dx$ exists, it is the Cauchy principal value of the integral of $f$ over $\mathbb{R}$.

## Remarks

The integral $\int_{-1}^{1} \frac{dx}{x}$ does not exist as an improper integral, but $\operatorname{p.v.} \int_{-1}^{1} \frac{dx}{x} = 0$, because the two parts cancel.
Similarly, $\operatorname{p.v.} \int_{-\infty}^{\infty} x \, dx = 0$, because $\int_{-R}^{R} x \, dx = 0$ for each $R$.
When the improper integral exists, the principal value exists and has the same value.
Principal values occur in the [[evaluation of real integrals by residues|0476QH9846]] when the integrand has a pole on the real axis.

## Explanations

### L1 human

Sometimes an integral is infinite on the left of a bad point and infinite on the right, with opposite signs.
The principal value cuts out a small symmetric interval around the bad point, so that the two sides can cancel.
For $\frac{1}{x}$ on $[-1, 1]$, the areas on the two sides cancel exactly.
</document>

<document source="https://tamnd.github.io/math2/node/0V3XAMF3C1.md" tag="0V3XAMF3C1" kind="concept">
# Plane

## Summary

A plane is a flat surface of dimension two that extends without end. It is the two-dimensional analogue of a point, which has dimension zero, and of a line, which has dimension one.
</document>

<document source="https://tamnd.github.io/math2/node/0V49JEZW06.md" tag="0V49JEZW06" kind="concept">
# Laplace transform

## Summary

The Laplace transform is an integral transform. It sends a function $f$ of a real variable $t \ge 0$ to the function $F(s) = \int_0^\infty f(t) e^{-st}\, dt$ of a complex variable $s$.
It is named after Pierre-Simon Laplace. See the [[definition|05XT5AVJBK]].
</document>

<document source="https://tamnd.github.io/math2/node/0V4DET59NW.md" tag="0V4DET59NW" kind="statement">
# Continuous real function on a compact set attains its bounds

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]], let $K \subseteq X$ be a nonempty [[compact set|00R0X1WGPX]] and let $f \colon X \to \mathbb{R}$ be a [[continuous map|0SY9ECQ19Q]].
Then there are points $a, b \in K$ with $f(a) \le f(x) \le f(b)$ for all $x \in K$.

## Remarks

Proof: the set $f(K)$ is compact in $\mathbb{R}$, because the [[continuous image of a compact set is compact|0K9AAJ8G51]]. By the [[Heine–Borel theorem|07PFYV65MB]], it is closed and bounded. Thus it contains its [[supremum|0BTMQ04556]] and its infimum.
For $K = [a, b]$, this is the extreme value theorem of calculus.
Compactness is needed: $f(x) = x$ on $(0, 1)$ has no maximum, and $f(x) = x$ on $\mathbb{R}$ is not bounded.

## Explanations

### L1 human

A continuous function on a compact set has a largest value and a smallest value, and it reaches them at some points.
On an open interval, the function can come closer and closer to a value without reaching it.
</document>

<document source="https://tamnd.github.io/math2/node/0V4E623THY.md" tag="0V4E623THY" kind="definition">
# Separable equation

## Definition

Let $I$ and $J$ be open [[intervals|0Q1D85B8G6]] of $\mathbb{R}$, and let $g : I \to \mathbb{R}$ and $h : J \to \mathbb{R}$ be continuous functions.
A separable equation is a first-order [[ordinary differential equation|0P97V20RW7]] of the form
$$y' = g(x)\, h(y),$$
for an unknown function $y$ of $x$ with values in $J$.

## Remarks

Method: if $h(y) \ne 0$, write the equation as $y'/h(y) = g(x)$ and integrate. With antiderivatives $H$ of $1/h$ and $G$ of $g$, the solutions satisfy $H(y(x)) = G(x) + C$ for a constant $C$.
For each zero $c$ of $h$, the constant function $y = c$ is a solution. The method above does not find these solutions.
Example: $y' = x y$ has the solutions $y = C e^{x^2/2}$ with a real constant $C$.
[[Exponential growth|00SVP6W3CB]] and the [[logistic equation|0CC4VC7KR3]] are separable.

## Explanations

### L1 human

In a separable equation, the right side is a product: one factor depends only on $x$ and the other only on $y$.
Put everything with $y$ on one side and everything with $x$ on the other side, then integrate both sides.
</document>

<document source="https://tamnd.github.io/math2/node/0V4QXPGJQK.md" tag="0V4QXPGJQK" kind="concept">
# Vacuous truth

## Summary

A vacuous truth is an [[implication|0ZNX6DQ3EJ]] or a universal statement that is true because its hypothesis is never true.
An implication $p \to q$ with a false $p$ is true, and a statement $\forall x \in X, \, P(x)$ with an empty $X$ is true.

## Remarks

Example: "each element of the [[empty set|0ECVK59Y03]] is an even number" is vacuously true.
Vacuous truth explains why the empty set is a [[subset|0FE8Q5S42Y]] of every set.

## Explanations

### L1 human

"All unicorns in this room are blue" is true when the room has no unicorns.
There is nothing that can break the statement.
This convention keeps the rules of logic simple and regular.
</document>

<document source="https://tamnd.github.io/math2/node/0V4VB6MQDC.md" tag="0V4VB6MQDC" kind="statement">
# Fibonacci numbers as sums of binomial coefficients

## Statement

Let $n$ be a natural number, and let $F_m$ be the [[Fibonacci numbers|0GA9K57A39]].
Then
$$F_{n+1} = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n - k}{k}.$$

## Remarks

Example: $F_6 = \binom{5}{0} + \binom{4}{1} + \binom{3}{2} = 1 + 4 + 3 = 8$.
Combinatorial proof: $F_{n+1}$ counts the tilings of a strip of length $n$ by squares and dominoes. A tiling with $k$ dominoes has $n - k$ tiles, and $\binom{n-k}{k}$ choices fix the positions of the dominoes.
In [[Pascal's triangle|05DME4TZ3K]], the sum runs along a shallow diagonal.

## Explanations

### L1 human

Add the numbers of Pascal's triangle along the shallow diagonals, and you get the Fibonacci numbers.
Each term counts the ways to tile a strip with a fixed number of dominoes.
</document>

<document source="https://tamnd.github.io/math2/node/0V4VSCBXP3.md" tag="0V4VSCBXP3" kind="definition">
# Mean squared error

## Definition

Let $\hat{\theta}$ be an [[estimator|0TTNRP5PFQ]] of a real parameter $\theta$, with $E_\theta[\hat{\theta}^2] < \infty$, where $E_\theta$ is the expectation when $\theta$ is the true parameter.
The mean squared error of $\hat{\theta}$ at $\theta$ is
$$\operatorname{MSE}_\theta(\hat{\theta}) = E_\theta\left[(\hat{\theta} - \theta)^2\right].$$

## Remarks

It decomposes as $\operatorname{MSE}_\theta(\hat{\theta}) = \operatorname{Var}_\theta(\hat{\theta}) + \operatorname{Bias}_\theta(\hat{\theta})^2$, with the [[bias|01BD2TFFP4]] $\operatorname{Bias}_\theta(\hat{\theta}) = E_\theta[\hat{\theta}] - \theta$.
For an unbiased estimator, the mean squared error is the [[variance|007QED3E0Y]].
For the sample mean of $n$ i.i.d. variables with variance $\sigma^2$, the mean squared error is $\sigma^2/n$.
For $n$ predictions $\hat{y}_i$ of observed values $y_i$, one also calls $\frac{1}{n}\sum_i (y_i - \hat{y}_i)^2$ the mean squared error.

## Explanations

### L1 human

The mean squared error measures how far, on average, an estimator is from the truth.
It is large if the estimator spreads widely, or if it is systematically off, or both.
</document>

<document source="https://tamnd.github.io/math2/node/0V547YQRMV.md" tag="0V547YQRMV" kind="statement">
# Subgroups of index two are normal

## Statement

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$ of [[index|0DTXBQPA2S]] $2$.
Then $H$ is [[normal|0CB0DX5BVV]] in $G$.

## Remarks

For $g \notin H$, the left cosets are $H$ and $gH = G \setminus H$, and the right cosets are $H$ and $Hg = G \setminus H$.
Thus $gH = Hg$ for each $g \in G$, and [[the coset criterion|01E7SWYJX3]] applies.
For example, the alternating group $A_n$ has index $2$ in the symmetric group $S_n$, and thus it is normal.

## Explanations

### L1 human

When a subgroup has only two cosets, one coset is the subgroup and the other is everything else.
This is true for the left cosets and for the right cosets.
So the left and right cosets agree, and the subgroup is normal.

### L2 agent

A subgroup $H$ is normal when $gH = Hg$ for each $g \in G$. In general this is a real condition. For a subgroup of index $2$ it is automatic.

The reason is that two cosets leave no choice:

1. The left cosets are $H$ and one other coset. Since the cosets partition $G$, the other coset is $G \setminus H$.
2. In the same way, the right cosets are $H$ and $G \setminus H$.
3. If $g \in H$, then $gH = H = Hg$.
4. If $g \notin H$, then $gH = G \setminus H = Hg$.

A worked example with $G = S_3$ and $H = A_3 = \{e, (1\,2\,3), (1\,3\,2)\}$:

1. $|S_3| = 6$ and $|A_3| = 3$, so the index is $2$.
2. For $g = (1\,2)$, the left coset $gA_3$ is the set of the three transpositions.
3. The right coset $A_3g$ is the same set. Thus $gA_3 = A_3g$.

The index two condition is sharp. The subgroup $\{e, (1\,2)\}$ of $S_3$ has index $3$, and it is not normal: $(1\,2\,3)H = \{(1\,2\,3), (1\,3)\}$, but $H(1\,2\,3) = \{(1\,2\,3), (2\,3)\}$.

Standard examples are $A_n$ in $S_n$, the rotations in a dihedral group, and $2\mathbb{Z}$ in $\mathbb{Z}$. A related fact is that a subgroup whose index is the smallest prime divisor of $|G|$ is also normal. The proof of that fact needs a group action.
</document>

<document source="https://tamnd.github.io/math2/node/0V58FPM3EQ.md" tag="0V58FPM3EQ" kind="definition">
# Minimal polynomial

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an endomorphism.
The *minimal polynomial* of $T$ is the monic polynomial $m_T \in F[x]$ of least degree with $m_T(T) = 0$.
It is the monic generator of the ideal of [[annihilating polynomials|0PBN8VEKY8]] of $T$.
The minimal polynomial of a square matrix is defined in the same way.

## Remarks

The minimal polynomial exists and is unique.
It [[divides each annihilating polynomial|0VTAD25A3G]], in particular the [[characteristic polynomial|0VSSPAFM2C]].
For $I_2$, the minimal polynomial is $x - 1$; for $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, it is $(x - 1)^2$.
Similar matrices have the same minimal polynomial.

## Explanations

### L1 human

The minimal polynomial is the shortest polynomial relation that a matrix satisfies.
All other relations are multiples of it.
Its roots are exactly the eigenvalues of the matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0V5B6D6BQH.md" tag="0V5B6D6BQH" kind="concept">
# Characteristic function

## Summary

The characteristic function of a real random variable $X$ is the function $t \mapsto E[e^{itX}]$. It is the Fourier transform of the law of $X$, and it exists for every random variable.
The node [[characteristic function|0MYDNRT7PP]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0V5GYWR8R8.md" tag="0V5GYWR8R8" kind="concept">
# Leading coefficient

## Summary

The leading coefficient of a nonzero polynomial is its coefficient of highest degree.
The [[definition of the leading coefficient|0P05RJC5XA]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0V63D9V0FY.md" tag="0V63D9V0FY" kind="definition">
# Regular polygon

## Definition

Let $n \geq 3$, and let $P_1 P_2 \cdots P_n$ be a convex polygon in the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$. Write $P_0 = P_n$ and $P_{n+1} = P_1$.
The polygon is regular if all sides have the same length, $d(P_i, P_{i+1}) = d(P_1, P_2)$, and all [[angles|0GJAJC6TNZ]] are equal, $\angle P_{i-1} P_i P_{i+1} = \angle P_n P_1 P_2$, for $i = 1, \dots, n$.

## Remarks

Each interior angle of a regular convex $n$-gon is $\frac{(n - 2)\pi}{n}$. Each exterior angle is $\frac{2\pi}{n}$.
The vertices are [[concyclic|049DCJZ448]], and the polygon also has an incircle. The vertices of a regular $n$-gon with the circumradius $1$ can be $\left(\cos \frac{2\pi k}{n}, \sin \frac{2\pi k}{n}\right)$, for $k = 0, \dots, n - 1$.
With the side $s$, the area is $\frac{1}{4} n s^2 \cot \frac{\pi}{n}$.
Star polygons, such as the pentagram, are also called regular polygons. They are not convex.

## Explanations

### L1 human

A regular polygon looks the same from each corner. Equal sides alone are not enough: a rhombus has four equal sides but is regular only when it is a square. Equal angles alone are not enough either, as a rectangle shows.
</document>

<document source="https://tamnd.github.io/math2/node/0V6GAWTBYW.md" tag="0V6GAWTBYW" kind="definition">
# Biconditional

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The biconditional of $p$ and $q$ is the proposition $p \leftrightarrow q$, read "$p$ if and only if $q$".
The proposition $p \leftrightarrow q$ is true when $p$ and $q$ are both true or both false. Otherwise, it is false.

| $p$ | $q$ | $p \leftrightarrow q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |

## Remarks

Other notations are $p \Leftrightarrow q$ and "$p$ iff $q$".
The proposition $p \leftrightarrow q$ has the same truth value as $(p \to q) \land (q \to p)$.

## Explanations

### L1 human

"A triangle has three equal sides if and only if it has three equal angles" is a biconditional.
It says two things: each property gives the other.
To prove a biconditional, you usually prove the two [[implications|0ZNX6DQ3EJ]] separately.
</document>

<document source="https://tamnd.github.io/math2/node/0V7090SSPC.md" tag="0V7090SSPC" kind="concept">
# Regular curve

## Summary

A regular curve is a continuously differentiable [[parametrised curve|0G7WM8DX4P]] whose [[tangent vector|0ZCZK72B45]] is never zero.
At each point a regular curve has a well-defined tangent line, and it can be reparametrised by arc length.
</document>

<document source="https://tamnd.github.io/math2/node/0V7E9KP6Q2.md" tag="0V7E9KP6Q2" kind="definition">
# Square root function

## Definition

For a real number $x \ge 0$, the square root $\sqrt{x}$ is the unique real number $y \ge 0$ with $y^2 = x$.
The square root function is the function $x \mapsto \sqrt{x}$ on $[0, \infty)$.

## Remarks

The number $y$ exists by the [[intermediate value theorem|00KAFBR32E]] for $t \mapsto t^2$. It is unique because $t^2$ is [[strictly increasing|0TR9SR9FTB]] on $[0, \infty)$.
The function is continuous and strictly increasing. It is differentiable on $(0, \infty)$, but not at $0$.
For $x > 0$, $\sqrt{x} = x^{1/2}$. Also $\sqrt{x^2} = |x|$ for every real $x$. Mathlib defines `Real.sqrt`, with $\sqrt{x} = 0$ for $x < 0$.

## Explanations

### L1 human

The square root of $x$ is the nonnegative number whose square is $x$.
For example, $\sqrt{9} = 3$, not $-3$.
</document>

<document source="https://tamnd.github.io/math2/node/0V7G6WMFK0.md" tag="0V7G6WMFK0" kind="concept">
# Covariance matrix

## Summary

The covariance matrix of a random vector is the square matrix of the covariances of all pairs of its components.
The node [[covariance matrix|02CBHV4ZZQ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0V7ZD6P7RK.md" tag="0V7ZD6P7RK" kind="object">
# Standard basis

## Description

Let $F$ be a field and let $n \ge 1$ be an integer.
For $i = 1, \dots, n$, let $e_i \in F^n$ be the vector with entry $1$ in position $i$ and $0$ in the other positions.
The *standard basis* of the coordinate space $F^n$ is the family $(e_1, \dots, e_n)$.
It is a [[basis|0KCQ9Y3QKA]] of $F^n$, and each $x = (x_1, \dots, x_n)$ is equal to $x_1 e_1 + \cdots + x_n e_n$.

## Remarks

The coordinates of a vector in the standard basis are its entries.
In $\R^3$, the standard basis vectors are often written $\mathbf{i}, \mathbf{j}, \mathbf{k}$.
The column $j$ of an $m \times n$ matrix $A$ is $A e_j$.
The standard basis of $\R^n$ is an orthonormal basis for the dot product.

## Explanations

### L1 human

The standard basis of $F^n$ consists of the vectors with one entry $1$ and all other entries $0$.
In the plane, these are the unit arrows along the $x$-axis and the $y$-axis.
The coordinates of a vector in this basis are just its entries.
</document>

<document source="https://tamnd.github.io/math2/node/0V8046R0P9.md" tag="0V8046R0P9" kind="definition">
# Quotient topology

## Definition

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]] and let $\sim$ be an [[equivalence relation|0SGTMBEWRF]] on $X$.
Let $q \colon X \to X / {\sim}$ be the map to the [[quotient set|0ZVHXDSZC7]] that sends $x$ to its class.
The quotient topology on $X / {\sim}$ is
$$\{ V \subseteq X / {\sim} : q^{-1}(V) \in \tau \}.$$

## Remarks

It is the finest topology on $X / {\sim}$ for which $q$ is continuous.
A [[preimage|0FGCFY1AZX]] $q^{-1}(V)$ is a union of classes. Thus $V$ is open if and only if the union of the classes in $V$ is open in $X$.
Quotients need not keep good properties: a quotient of a Hausdorff space need not be [[Hausdorff|0MXB4CKHFW]].
Examples are the [[circle|0WH1GVBJNH]], the [[torus|03NM7DZXXC]] and the [[Möbius strip|04AJE0M271]].

## Explanations

### L1 human

You make a quotient space by gluing the points of each class into one point.
For example, gluing the two ends of $[0, 1]$ gives a circle.
</document>

<document source="https://tamnd.github.io/math2/node/0V8F3E48E6.md" tag="0V8F3E48E6" kind="concept">
# Linear combination

## Summary

A linear combination of vectors is a sum of scalar multiples of the vectors.
</document>

<document source="https://tamnd.github.io/math2/node/0V948DE5KA.md" tag="0V948DE5KA" kind="statement">
# Logarithm of a power

## Statement

For every real $x > 0$ and every natural number $n$,
$$\ln(x^n) = n \ln x.$$

## Remarks

Induction on $n$ with the [[product rule for logarithms|0K2FNY6DYV]] gives the claim.
For [[real exponents|0TV07AQT62]], $\ln(x^a) = a \ln x$ holds by the definition $x^a = e^{a \ln x}$.
Mathlib states it as `Real.log_pow`.

## Explanations

### L1 human

The logarithm brings an exponent down as a factor.
For example, $\ln(x^3) = 3 \ln x$.
</document>

<document source="https://tamnd.github.io/math2/node/0V94H674MB.md" tag="0V94H674MB" kind="definition">
# Determinant of a family of vectors in a basis

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \dots, b_n)$ be a [[basis|0KCQ9Y3QKA]] of $V$.
For vectors $v_1, \dots, v_n \in V$, let $M$ be the $n \times n$ matrix whose column $j$ is the [[coordinate vector|04CYQHN6MC]] $[v_j]_B$.
The *determinant of $(v_1, \dots, v_n)$ in the basis $B$* is
$$\det_B(v_1, \dots, v_n) = \det M.$$

## Remarks

The map $\det_B \colon V^n \to F$ is the unique alternating multilinear form with $\det_B(b_1, \dots, b_n) = 1$.
The vectors $v_1, \dots, v_n$ form a basis of $V$ if and only if $\det_B(v_1, \dots, v_n) \ne 0$.
For another basis $B'$, $\det_{B'} = \det_{B'}(b_1, \dots, b_n) \cdot \det_B$.

## Explanations

### L1 human

Write $n$ vectors in the coordinates of a fixed basis, and put the coordinates side by side as columns.
The determinant of this square matrix measures the signed volume of the box that the vectors span.
The unit is the box of the basis vectors.
</document>

<document source="https://tamnd.github.io/math2/node/0V98WY94B8.md" tag="0V98WY94B8" kind="statement">
# First Sylow theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a prime number, and let $k \geq 0$ with $p^k \mid |G|$.
Then $G$ has a [[subgroup|0X1JXT47GA]] of order $p^k$.
In particular, $G$ has a [[Sylow p-subgroup|0053CQE9Y2]].

## Remarks

For $k = 1$, this is [[Cauchy's theorem|0JM7PA288B]].
Let $p^n$ be the largest power of $p$ that divides $|G|$. A proof by Wielandt lets $G$ act by left multiplication on the subsets of $G$ of size $p^n$.
For example, a group of order $12$ has a subgroup of order $4$ and a subgroup of order $3$.

## History

Ludwig Sylow proved the Sylow theorems in 1872, and he published them in Mathematische Annalen.

## Explanations

### L1 human

The converse of Lagrange's theorem is false in general.
But it is true for prime powers: each prime power that divides the order of the group is the order of a subgroup.
The largest such subgroups are the Sylow subgroups.
</document>

<document source="https://tamnd.github.io/math2/node/0V9A43H7WV.md" tag="0V9A43H7WV" kind="statement">
# Product rule

## Statement

Let $f$ and $g$ be real functions on an open interval $I$, both differentiable at $a \in I$.
Then $f g$ is differentiable at $a$, and
$$(f g)'(a) = f'(a) g(a) + f(a) g'(a).$$

## Remarks

Write $f(a + h) g(a + h) - f(a) g(a) = (f(a + h) - f(a)) g(a + h) + f(a) (g(a + h) - g(a))$. Divide by $h$ and use the [[continuity of $g$ at $a$|07ZMX4ANJS]].
For example, $(x \sin x)' = \sin x + x \cos x$.
Mathlib states it as `HasDerivAt.mul`.

## History

The discovery of the rule is credited to Gottfried Leibniz.

## Explanations

### L1 human

Think of $f g$ as the area of a rectangle with sides $f$ and $g$.
When both sides grow a little, the area grows by $f' g + f g'$, up to a tiny corner piece.
</document>

<document source="https://tamnd.github.io/math2/node/0V9ME8GQE7.md" tag="0V9ME8GQE7" kind="statement">
# Dirichlet's test

## Statement

Let $(a_n)$ be a [[monotone|0KN22AAW5R]] real sequence with $a_n \to 0$.
Let $(b_n)$ be a real sequence whose [[partial sums|0NB643K60Q]] are bounded. That is, there is $M$ with $\left| \sum_{k=1}^{n} b_k \right| \le M$ for all $n$.
Then the series $\sum_{n=1}^{\infty} a_n b_n$ converges.

## Remarks

The proof uses summation by parts and the [[Cauchy criterion for series|06PVXMY0TH]].
With $b_n = (-1)^n$ it gives the [[alternating series test|0PK8QH1W1V]].
It also shows that $\sum \sin(n x) / n$ converges for each real $x$.
Mathlib states a form with a Cauchy sequence of partial sums, as `Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded`.

## History

The test is named after Peter Gustav Lejeune Dirichlet. It was published after his death, in 1862.

## Explanations

### L1 human

The partial sums of $\sum b_n$ do not grow, and the weights $a_n$ shrink steadily to $0$.
Together these give enough cancellation for $\sum a_n b_n$ to converge.
</document>

<document source="https://tamnd.github.io/math2/node/0V9PMN0RBH.md" tag="0V9PMN0RBH" kind="object">
# Special linear group

## Description

Let $F$ be a field and let $n \ge 1$.
The *special linear group* is the set
$$SL_n(F) = \{A \in GL_n(F) : \det A = 1\}$$
of $n \times n$ matrices over $F$ with [[determinant|00KDZRW4C0]] $1$, with matrix multiplication.
It is a subgroup of the [[general linear group|0FCHE902D7]] $GL_n(F)$.

## Remarks

It is the kernel of the group homomorphism $\det \colon GL_n(F) \to F^\times$, so it is a normal subgroup.
The quotient $GL_n(F) / SL_n(F)$ is isomorphic to $F^\times$.
Over $\R$, its elements are the linear maps that keep volume and orientation.
$SL_2(\Z)$ is the analogous group of integer matrices with determinant $1$.

## Explanations

### L1 human

The special linear group collects the matrices with determinant $1$.
In the real case, these maps keep the volume of every region, and they do not flip orientation.
Shears, such as $\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}$, are typical elements.
</document>

<document source="https://tamnd.github.io/math2/node/0VABYWJWH9.md" tag="0VABYWJWH9" kind="concept">
# Preorder

## Summary

A preorder is a relation that is reflexive and transitive.
It compares elements, but two different elements can be equivalent under it.
</document>

<document source="https://tamnd.github.io/math2/node/0VARVQ9ZW1.md" tag="0VARVQ9ZW1" kind="definition">
# Power series in the complex plane

## Definition

Let $c \in \mathbb{C}$ and let $(a_n)_{n \ge 0}$ be a sequence of [[complex numbers|09FTAET5Z4]].
The power series with center $c$ and coefficients $a_n$ is the series
$$\sum_{n=0}^{\infty} a_n (z - c)^n$$
in the complex variable $z$.
The series converges at a point $z \in \mathbb{C}$ if the partial sums $\sum_{n=0}^{N} a_n (z - c)^n$ have a limit in $\mathbb{C}$ as $N \to \infty$. The sum of the series is the function that maps each such $z$ to this limit.

## Remarks

Each power series converges at $z = c$, with sum $a_0$.
The geometric series $\sum_{n \ge 0} z^n$ converges if and only if $|z| < 1$, and then its sum is $1/(1 - z)$.
The set where the series converges contains an open disc and is contained in its closure, see [[disc of convergence|045EWAXY9P]].
A [[formal power series|033WGN00BF]] is the same expression without any question of convergence.

## Explanations

### L1 human

A power series is a polynomial with infinitely many terms.
For some values of $z$ the terms get small fast enough and the series has a sum, and for other values it does not.
For example, $\sum z^n$ has a sum when $|z| < 1$, but not when $|z| \ge 1$, because then the terms do not go to $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0VB8BFRABH.md" tag="0VB8BFRABH" kind="definition">
# Square root

## Definition

Let $x$ be a real number with $x \ge 0$.
The square root of $x$, written $\sqrt{x}$, is the unique real number $y$ with $y \ge 0$ and $y^2 = x$.

## Remarks

The [[existence of square roots|068BZBXE0C]] shows that exactly one such $y$ exists.
The function $x \mapsto \sqrt{x}$ on $[0, \infty)$ is increasing, and $\sqrt{a b} = \sqrt{a} \sqrt{b}$ for $a, b \ge 0$.
For each real $x$, $\sqrt{x^2} = |x|$, the [[absolute value|00S7C68D51]] of $x$.

## Explanations

### L1 human

The square root of $x$ is the side length of a square with area $x$.
The equation $y^2 = 25$ has two solutions, $5$ and $-5$. The symbol $\sqrt{25}$ means the positive one, $5$.
</document>

<document source="https://tamnd.github.io/math2/node/0VB9G5B034.md" tag="0VB9G5B034" kind="statement">
# Multiplicativity of the Möbius function

## Statement

The [[Möbius function|0C8QENHYZW]] $\mu$ is a [[multiplicative function|0AP80TT660]].
That is, $\mu(1) = 1$ and $\mu(mn) = \mu(m) \mu(n)$ for all coprime positive integers $m$ and $n$.

## Remarks

The function $\mu$ is not completely multiplicative: $\mu(4) = 0$, but $\mu(2)^2 = 1$.
The proof: for coprime $m$ and $n$, the product $mn$ is squarefree exactly when $m$ and $n$ are both squarefree. Then $\omega(mn) = \omega(m) + \omega(n)$.

## Explanations

### L1 human

For example, $\mu(5) = -1$ and $\mu(6) = 1$, and $\mu(30) = -1 = \mu(5) \mu(6)$.
When $m$ and $n$ have no common prime, their prime factors just combine, so the signs multiply.
</document>

<document source="https://tamnd.github.io/math2/node/0VBWK9XYDG.md" tag="0VBWK9XYDG" kind="statement">
# Kummer's theorem

## Statement

Let $p$ be a prime number, and let $m$ and $n$ be natural numbers.
Then $\nu_p \binom{m + n}{m}$ is the number of carries when one adds $m$ and $n$ in base $p$. Here $\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].

## Remarks

For example, in base $2$, $3 + 1 = 11_2 + 1_2 = 100_2$ has $2$ carries, and $\binom{4}{1} = 4 = 2^2$.
The theorem follows from [[Legendre's formula|01WPV9PHN7]] in the form with the digit sums $s_p$.
A consequence: $p \mid \binom{m+n}{m}$ if and only if there is at least one carry.

## History

Ernst Kummer proved the theorem in 1852.

## Explanations

### L1 human

Write $m$ and $n$ in base $p$ and add them by hand. Count how many times you carry a $1$.
That count is the exponent of $p$ in the binomial coefficient $\binom{m+n}{m}$.
For $p = 10$ the rule fails, because $10$ is not prime. The theorem needs a prime base.
</document>

<document source="https://tamnd.github.io/math2/node/0VBZH90R1A.md" tag="0VBZH90R1A" kind="concept">
# Internal direct product

## Summary

A group $G$ is the internal direct product of two normal subgroups $H$ and $K$ when $H \cap K = \{e\}$ and $HK = G$.
Then $G$ is isomorphic to the direct product $H \times K$.
</document>

<document source="https://tamnd.github.io/math2/node/0VC4DVE7DQ.md" tag="0VC4DVE7DQ" kind="statement">
# Euler's theorem

## Statement

Let $n$ be a positive integer and let $a$ be an integer [[coprime|0GFKP8F8AG]] to $n$.
Then $a^{\varphi(n)} \equiv 1 \pmod n$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

When $n = p$ is prime, $\varphi(p) = p - 1$, and the result is [[Fermat's little theorem|0M25PRN3WQ]].
A proof uses [[Lagrange's theorem|3C53TD]]: the group $(\mathbb{Z}/n\mathbb{Z})^\times$ has order $\varphi(n)$, by the [[number of invertible residues modulo n|06RJWZCG95]].
The result lets us reduce exponents. If $\gcd(a, n) = 1$ and $k \equiv l \pmod{\varphi(n)}$, then $a^k \equiv a^l \pmod n$.
For example, $\varphi(10) = 4$, so $7^{4} \equiv 1 \pmod{10}$, and the last digit of $7^{222}$ is the last digit of $7^2 = 49$, that is, $9$.

## History

Euler proved the theorem in 1763.

## Explanations

### L1 human

Powers of $a$ modulo $n$ repeat in a cycle, and the cycle comes back to $1$.
Euler's theorem gives one exponent that always brings $a$ back to $1$, namely $\varphi(n)$.
For $n = 12$ and $a = 5$: $\varphi(12) = 4$ and $5^4 = 625 = 52 \cdot 12 + 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0VCTBJMEJF.md" tag="0VCTBJMEJF" kind="definition">
# Improper integral

## Definition

Let $f \colon [a, \infty) \to \mathbb{R}$ be Riemann integrable on $[a, b]$ for each $b > a$.
The improper integral of $f$ over $[a, \infty)$ converges if the following [[limit|0Y12QYTJK3]] exists in $\mathbb{R}$:
$$\int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx.$$
Similarly, let $f \colon (a, b] \to \mathbb{R}$ be Riemann integrable on $[c, b]$ for each $c \in (a, b)$.
Then $\int_a^b f(x) \, dx = \lim_{c \to a^+} \int_c^b f(x) \, dx$, if this [[one-sided limit|0909PWZR62]] exists.

## Remarks

If the limit does not exist, the improper integral diverges.
For an integral over $(-\infty, \infty)$, or with several bad points, we split the interval. Each piece must converge separately.
For example, $\int_1^\infty \frac{dx}{x^2} = 1$, and $\int_0^1 \frac{dx}{\sqrt{x}} = 2$. But $\int_1^\infty \frac{dx}{x}$ diverges.

## Explanations

### L1 human

The Riemann integral needs a bounded interval and a bounded function.
An improper integral removes these limits by a limit of ordinary integrals.
</document>

<document source="https://tamnd.github.io/math2/node/0VD9ZSWWJG.md" tag="0VD9ZSWWJG" kind="concept">
# Matrix

## Summary

A matrix is a rectangular array of elements of a ring or a field, arranged in rows and columns.
Matrices represent linear maps and systems of linear equations.
</document>

<document source="https://tamnd.github.io/math2/node/0VDD7W0EKZ.md" tag="0VDD7W0EKZ" kind="statement">
# Morera's theorem

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be continuous.
Assume that $\oint_{\partial T} f(z) \, dz = 0$ for each closed triangle $T \subseteq U$, where $\partial T$ is the boundary of $T$ traveled once and the integral is a [[contour integral|00FWNP9TXY]].
Then $f$ is [[holomorphic|0DG0Z72CQD]] on $U$.

## Remarks

It is enough to know that $\oint_\gamma f(z) \, dz = 0$ for each closed piecewise continuously differentiable path $\gamma$ in $U$, because the boundary of a triangle is such a path.
Proof idea: on each disc in $U$, $F(z) = \int_{[c, z]} f(w) \, dw$ is a primitive of $f$. So $F$ is holomorphic, and $f = F'$ is holomorphic because [[derivatives of holomorphic functions are holomorphic|0QEBN8DGVE]].
Together with [[Goursat's theorem|0BRG85WXYC]], this shows: a continuous $f$ is holomorphic on $U$ if and only if its integrals over all triangles in $U$ are zero.
A main use is the theorem on [[uniform limits of holomorphic functions|0431RN0R1N]].

## Explanations

### L1 human

Morera's theorem is a converse of Cauchy's theorem.
If the integral of a continuous function around each small triangle is zero, then the function is holomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/0VEFSSKX18.md" tag="0VEFSSKX18" kind="definition">
# Poisson distribution

## Definition

Let $\lambda > 0$.
The Poisson distribution with parameter $\lambda$ is the distribution on $\{0, 1, 2, \dots\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(k) = \frac{\lambda^k e^{-\lambda}}{k!} \quad \text{for } k = 0, 1, 2, \dots$$
Here $k!$ is the [[factorial|0WFCGVMRRC]] of $k$. One writes $X \sim \operatorname{Poisson}(\lambda)$.

## Remarks

The values add up to $1$ because $\sum_{k \ge 0} \lambda^k / k! = e^{\lambda}$.
The mean and the variance are both $\lambda$.
It is the limit of the [[binomial distribution|0VRWZDWW6B]] with parameters $n$ and $\lambda / n$ as $n \to \infty$. This is the [[Poisson limit theorem|0XKTZRWTXX]].
Siméon Denis Poisson introduced the distribution in 1837.

## Explanations

### L1 human

The Poisson distribution counts rare events that happen at random, such as calls to a help desk in one hour.
If $\lambda = 2$ calls come per hour on average, the chance of no call in an hour is $e^{-2} \approx 0.135$.
</document>

<document source="https://tamnd.github.io/math2/node/0VEKNG02W0.md" tag="0VEKNG02W0" kind="statement">
# Volume of a box

## Statement

Let $a_i < b_i$ for $i = 1, \dots, n$, and let $Q = [a_1, b_1] \times \dots \times [a_n, b_n] \subseteq \mathbb{R}^n$.
Then $Q$ is Jordan measurable, and its volume is
$$\operatorname{vol}(Q) = \int_Q 1 \, dV = (b_1 - a_1)(b_2 - a_2) \cdots (b_n - a_n).$$

## Remarks

For a box, the [[Jordan measure|0565DC8RMK]] and the Riemann integral of the constant function $1$ give the same value.
The integral can be computed with [[Fubini's theorem|0R0T1VWH5X]] as an iterated integral of $1$, one factor $b_i - a_i$ at a time.
Open and half-open boxes with the same edges have the same volume, because the faces of $Q$ have volume zero.
For $n = 3$ this is the formula length times width times height, a case of base area times height. The volume of a general solid $E$ is the [[triple integral|0GTBGMZXV8]] $\iiint_E 1 \, dV$.

## Explanations

### L1 human

The volume of a box is the product of its side lengths.
In the theory of multiple integrals this is the starting point: Riemann sums and Jordan measure are both built from volumes of small boxes.
</document>

<document source="https://tamnd.github.io/math2/node/0VF11ZTTMD.md" tag="0VF11ZTTMD" kind="definition">
# Field of fractions

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]].
On the set of pairs $(a, b)$ with $a, b \in R$ and $b \neq 0$, let $(a, b) \sim (c, d)$ if $a d = b c$.
This is an equivalence relation. Write $a/b$ for the class of $(a, b)$.
The field of fractions $\operatorname{Frac}(R)$ is the set of these classes, with the operations
$$\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}, \qquad \frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}.$$
It is a [[field|0QA8CHH8MG]], and the map $R \to \operatorname{Frac}(R)$, $a \mapsto a/1$, is an injective ring homomorphism.

## Remarks

The product $b d$ is not zero because $R$ is an integral domain. This makes the operations well defined.
The inverse of a nonzero $a/b$ is $b/a$.
Universal property: every injective ring homomorphism from $R$ to a field $K$ extends to exactly one ring homomorphism from $\operatorname{Frac}(R)$ to $K$.
The field of fractions of the polynomial ring $K[x]$ over a field $K$ is the field $K(x)$ of rational functions.

## Explanations

### L1 human

This is how the rational numbers come from the integers.
A fraction is a pair of a numerator and a nonzero denominator.
Two fractions are equal when cross-multiplication gives the same result, as with $1/2 = 2/4$.
</document>

<document source="https://tamnd.github.io/math2/node/0VFW4Y69BT.md" tag="0VFW4Y69BT" kind="definition">
# Quotient ring

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $I$ be a [[two-sided ideal|0DDA6CY7GJ]] of $R$.
For $a \in R$, the coset of $a$ is $a + I = \{a + x : x \in I\}$.
The quotient ring $R/I$ is the set of the cosets of $I$, with these operations:
$$(a + I) + (b + I) = (a + b) + I, \qquad (a + I)(b + I) = a b + I.$$
These operations do not depend on the choice of $a$ and $b$ in their cosets.
With them, $R/I$ is a ring with zero $0 + I = I$ and identity $1 + I$.

## Remarks

Two elements $a, b \in R$ have the same coset if and only if $a - b \in I$.
The map $\pi \colon R \to R/I$, $\pi(a) = a + I$, is a surjective [[ring homomorphism|0GGE0Z0JQ1]] with [[kernel|0AAHE7GEW6]] $I$.
For $n \ge 1$, the quotient $\mathbb{Z}/n\mathbb{Z}$ is the ring of integers modulo $n$.
The quotient $R/R$ is the [[zero ring|008TBCYDPX]].

## Explanations

### L1 human

To make the quotient ring, you declare all the elements of $I$ to be zero.
Two elements become equal when their difference is in $I$.
For $\mathbb{Z}/5\mathbb{Z}$, the numbers $2$ and $7$ become the same element.
</document>

<document source="https://tamnd.github.io/math2/node/0VG02M1FEZ.md" tag="0VG02M1FEZ" kind="statement">
# Every polynomial has a root in some extension field

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ with $\deg f \ge 1$.
Then there is a [[field extension|0WQ1A8SHTZ]] $L/K$ with $[L : K] \le \deg f$ such that $f$ has a [[root|00EHX6FWC0]] in $L$.

## Remarks

The proof: let $g$ be an irreducible factor of $f$ in $K[x]$. [[Adjoining a root of $g$|0M893X7YY7]] gives the field $L = K[x]/(g)$ of degree $\deg g$ over $K$.
The class of $x$ in $L$ is a root of $g$, and so of $f$.
The irreducible factor $g$ exists because $K[x]$ is a [[unique factorisation domain|0D6KH8Y59N]].
Repeating the step gives a [[splitting field|02NZZ1J0FH]] of $f$.

## Explanations

### L1 human

A polynomial with no root in a field gets a root in a bigger field.
The bigger field is built from the polynomial itself, without any reference to the complex numbers.
</document>

<document source="https://tamnd.github.io/math2/node/0VG7VNEWMG.md" tag="0VG7VNEWMG" kind="concept">
# Abundant number

## Summary

An abundant number is a positive integer whose positive divisors other than itself have a sum greater than the number.
The [[definition of an abundant number|0VS0MP1P5H]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0VGFHTM75Y.md" tag="0VGFHTM75Y" kind="concept">
# Continuous map

## Summary

A continuous map is a function that sends nearby points to nearby points.
Between topological spaces, this means that preimages of open sets are open. Between metric spaces, it is the same as the epsilon-delta condition.
</document>

<document source="https://tamnd.github.io/math2/node/0VGKB2NC4T.md" tag="0VGKB2NC4T" kind="example">
# Improper integral of x^p on an unbounded interval

## Example

Let $p$ be a real number. The [[improper integral|0VCTBJMEJF]] $\int_1^\infty x^p \, dx$ converges if and only if $p < -1$.
In this case,
$$\int_1^\infty x^p \, dx = -\frac{1}{p + 1}.$$
For $p \ne -1$, [[the power formula|0Q3HYFT9RQ]] gives $\int_1^b x^p \, dx = \frac{b^{p+1} - 1}{p + 1}$. This has a finite limit as $b \to \infty$ exactly when $p + 1 < 0$.
For $p = -1$, $\int_1^b \frac{dx}{x} = \ln b$, [[which tends to infinity|0K1N1Y4EGZ]].

## Remarks

This example parallels the [[$p$-series test|0ZNSSK8ZK1]] through the [[integral test|019QZ6RRVH]].
Mathlib states a form as `integrableOn_Ioi_rpow_iff`.

## Explanations

### L1 human

The function $x^p$ must decay faster than $1/x$ for the area over $[1, \infty)$ to be finite.
The function $1/x$ itself is just too slow.
</document>

<document source="https://tamnd.github.io/math2/node/0VGMEBVW1M.md" tag="0VGMEBVW1M" kind="definition">
# Probability measure

## Definition

Let $\Omega$ be a set and let $\mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] on $\Omega$.
A probability measure on $(\Omega, \mathcal{F})$ is a function $P \colon \mathcal{F} \to [0, 1]$ with these two properties.
$P(\Omega) = 1$.
For each sequence $A_1, A_2, \dots$ of elements of $\mathcal{F}$ with $A_i \cap A_j = \varnothing$ for all $i \ne j$,
$$P\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} P(A_n).$$

## Remarks

These are the [[probability axioms|0DAX8PMAJ4]]. In other words, a probability measure is a measure of total mass $1$.
Countable additivity implies $P(\varnothing) = 0$ and finite additivity: $P(A \cup B) = P(A) + P(B)$ for disjoint $A$ and $B$.
On a finite set $\Omega$ with the power set as $\mathcal{F}$, each probability measure has the form $P(A) = \sum_{\omega \in A} p(\omega)$ with $p(\omega) \ge 0$ and $\sum_{\omega \in \Omega} p(\omega) = 1$.

## Explanations

### L1 human

A probability measure is a rule that gives each event a number between $0$ and $1$.
The whole sample space gets $1$, and the number of a union of separate pieces is the sum of the numbers of the pieces.
For a fair die, each face gets $1/6$, and the event "even" gets $3/6 = 1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/0VGMHCF4SC.md" tag="0VGMHCF4SC" kind="statement">
# Product of continuous functions is continuous

## Statement

Let $D \subseteq \mathbb{R}$ and let $f, g \colon D \to \mathbb{R}$ be [[continuous|0P27W9RW0G]] functions.
Then the sum $f + g$ and the product $f g$, given by $(f g)(x) = f(x) g(x)$, are continuous.

## Remarks

The proof uses [[sequential continuity|0PAQM3H57X]] and the [[product rule for limits|0734RSVF2V]] of sequences.
In particular, $c f$ is continuous for each constant $c$.
Mathlib states it for topological monoids, as `Continuous.mul`.

## Explanations

### L1 human

If $f(x)$ and $g(x)$ change only a little when $x$ changes a little, so does $f(x) g(x)$.
</document>

<document source="https://tamnd.github.io/math2/node/0VGPKZ8MRK.md" tag="0VGPKZ8MRK" kind="statement">
# Continuity is equivalent to sequential continuity in metric spaces

## Statement

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be a function.
Then $f$ is [[continuous|0SY9ECQ19Q]] if and only if $f(x_n) \to f(x)$ for each sequence $(x_n)$ in $X$ that [[converges|06JYPJ4WHH]] to a point $x$.

## Remarks

The condition on sequences is sequential continuity.
Proof of the converse: if $f$ is not continuous at $x$, then by [[epsilon-delta continuity|0EKGKY5TGH]] there are $\varepsilon > 0$ and points $x_n$ with $d(x_n, x) < 1/n$ and $d(f(x_n), f(x)) \ge \varepsilon$. Then $x_n \to x$, but $f(x_n) \not\to f(x)$.
The same equivalence holds at each single point.
In general topological spaces, continuity implies sequential continuity, but the converse can fail.

## Explanations

### L1 human

A continuous map sends convergent sequences to convergent sequences, with the right limit.
To show that a map is not continuous, it is often enough to find one sequence $x_n \to x$ with $f(x_n) \not\to f(x)$.
</document>

<document source="https://tamnd.github.io/math2/node/0VGPX25NXQ.md" tag="0VGPX25NXQ" kind="concept">
# Conditional variance

## Summary

The conditional variance of $X$ given some information measures the spread of $X$ around its conditional expectation. It is itself a random variable.
The node [[conditional variance|0F1SN93KXE]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0VGSYBWHV2.md" tag="0VGSYBWHV2" kind="definition">
# Inner product

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $V$ be a vector space over $\mathbb{K}$.
An *inner product* on $V$ is a map $\langle \cdot, \cdot \rangle \colon V \times V \to \mathbb{K}$ such that, for all $x, y, z \in V$ and $a, b \in \mathbb{K}$:

1. $\langle ax + by, z \rangle = a \langle x, z \rangle + b \langle y, z \rangle$.
2. $\langle y, x \rangle = \overline{\langle x, y \rangle}$.
3. $\langle x, x \rangle > 0$ if $x \ne 0$.

## Remarks

Over $\R$, the conjugation does nothing, so an inner product is a symmetric positive definite bilinear form.
Over $\C$, conditions 1 and 2 give $\langle x, by \rangle = \bar b \langle x, y \rangle$, so the map is conjugate linear in the second argument.
Many physicists, and Mathlib, use the opposite convention: linear in the second argument.
The [[dot product|0S52TNNTH9]] on $\R^n$ and $\langle x, y \rangle = \sum_i x_i \overline{y_i}$ on $\C^n$ are inner products.

## Explanations

### L1 human

An inner product generalises the dot product to abstract vector spaces.
It lets us talk about lengths, angles and perpendicular vectors.
On the space of continuous functions on $[0, 1]$, the integral $\int_0^1 f(t) g(t)\, dt$ is an inner product.
</document>

<document source="https://tamnd.github.io/math2/node/0VGWKX2JHB.md" tag="0VGWKX2JHB" kind="statement">
# Abel's identity

## Statement

Let $I$ be an open interval, let $p, q : I \to \mathbb{R}$ be continuous, and let $y_1, y_2$ be solutions on $I$ of
$$y'' + p(x)\, y' + q(x)\, y = 0.$$
Let $W = y_1 y_2' - y_1' y_2$ be their [[Wronskian|0N5NF1RGBP]], and let $x_0 \in I$.
Then for each $x \in I$,
$$W(x) = W(x_0) \exp\left( -\int_{x_0}^{x} p(t)\, dt \right).$$

## Remarks

Proof idea: a computation with the equation gives $W' = -p W$, a linear first-order equation for $W$.
So $W$ is zero at each point or at no point of $I$.
For an equation of order $n$ with coefficient $a_{n-1}$ of $y^{(n-1)}$, the same formula holds with $p$ replaced by $a_{n-1}$. [[Liouville's formula|09PS3S0EF6]] extends it to linear systems.
Niels Henrik Abel found the identity.

## Explanations

### L1 human

You can find the Wronskian of two solutions without knowing the solutions.
Only the coefficient $p$ of $y'$ matters.
For $y'' + y = 0$, we have $p = 0$, so the Wronskian of two solutions is constant.
</document>

<document source="https://tamnd.github.io/math2/node/0VH1TC1PWF.md" tag="0VH1TC1PWF" kind="definition">
# Companion matrix

## Definition

Let $F$ be a field, let $n \ge 1$, and let $p(t) = t^n + c_{n-1} t^{n-1} + \cdots + c_1 t + c_0$ be a monic polynomial over $F$.
The *companion matrix* of $p$ is the $n \times n$ matrix
$$C(p) = \begin{pmatrix} 0 & 0 & \cdots & 0 & -c_0 \\ 1 & 0 & \cdots & 0 & -c_1 \\ 0 & 1 & \cdots & 0 & -c_2 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & 1 & -c_{n-1} \end{pmatrix}.$$

## Remarks

The [[characteristic polynomial|0VSSPAFM2C]] and the [[minimal polynomial|03NDVCSHSB]] of $C(p)$ both equal $p$.
In the basis $e_1, \dots, e_n$, the map of $C(p)$ sends $e_i$ to $e_{i+1}$ for $i < n$.
Some authors use the transpose of this matrix as the companion matrix.
Companion matrices are the blocks of the [[rational canonical form|042JWAR3Q0]].

## Explanations

### L1 human

For each monic polynomial, the companion matrix is a matrix with exactly that characteristic polynomial.
It shifts each basis vector to the next one, and the polynomial fixes the image of the last one.
So questions about roots of polynomials become questions about eigenvalues of matrices.
</document>

<document source="https://tamnd.github.io/math2/node/0VHAVXW08V.md" tag="0VHAVXW08V" kind="statement">
# Subsequences of a convergent sequence converge to the same limit

## Statement

Let $(a_n)$ be a real sequence that converges to a real number $L$.
Let $(a_{n_k})_{k \in \mathbb{N}}$ be a [[subsequence|0R8J3ZQMJS]] of $(a_n)$.
Then $a_{n_k} \to L$ as $k \to \infty$.

## Remarks

For the proof, let $\varepsilon > 0$ and take $N$ with $|a_n - L| < \varepsilon$ for $n \ge N$. Since $n_k \ge k$, we get $|a_{n_k} - L| < \varepsilon$ for all $k \ge N$.
It gives a test for divergence. If two subsequences have different limits, the sequence diverges. For $(-1)^n$ the even terms tend to $1$ and the odd terms tend to $-1$.

## Explanations

### L1 human

If all terms eventually stay close to $L$, then the terms that you keep also stay close to $L$.
Deleting terms cannot change where the sequence is going.
</document>

<document source="https://tamnd.github.io/math2/node/0VHJRXNAAS.md" tag="0VHJRXNAAS" kind="concept">
# Chromatic number

## Summary

The chromatic number of a graph is the least number of colours in a proper colouring. It is a central invariant of graph theory, and the four colour theorem bounds it for planar graphs. The node [[chromatic number|0C5M7AM6FH]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0VHNKP1N3K.md" tag="0VHNKP1N3K" kind="definition">
# Conformal map

## Definition

Let $U$ be an open subset of the [[complex plane|0CTE7A31XQ]], identified with $\mathbb{R}^2$, and let $z_0 \in U$.
Let $f : U \to \mathbb{C}$ be differentiable at $z_0$ as a map from an open subset of $\mathbb{R}^2$ to $\mathbb{R}^2$, with Jacobian matrix $J$ at $z_0$.
The map $f$ is conformal at $z_0$ if $J = \lambda Q$ for a real number $\lambda > 0$ and a rotation matrix $Q$. A rotation matrix is an orthogonal $2 \times 2$ matrix with determinant $1$.
The map $f$ is conformal on $U$ if it is conformal at each point of $U$.

## Remarks

Equivalently, $f$ is conformal at $z_0$ when its real derivative at $z_0$ is the map $h \mapsto a h$ for a complex number $a \neq 0$.
A map that is conformal at $z_0$ preserves the angles between curves through $z_0$, and their orientation.
Some authors also allow orientation-reversing maps, such as $z \mapsto \bar{z}$, whose Jacobian is a positive number times an orthogonal matrix.
Other authors use "conformal map" for an injective holomorphic map on an open set.

## Explanations

### L1 human

A conformal map can stretch and turn a small figure, but it does not distort its shape.
If two curves cross at an angle of $30$ degrees, then their images cross at the same angle, in the same direction.
For example, $z \mapsto 2z$ and $z \mapsto iz$ are conformal at each point.
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# Symmetric polynomial

## Summary

A symmetric polynomial is a polynomial in several variables that does not change when you permute the variables.
The [[definition of a symmetric polynomial|06CTFY2DEZ]] states it.
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<document source="https://tamnd.github.io/math2/node/0VJBSK15SN.md" tag="0VJBSK15SN" kind="statement">
# Finite-dimensional spaces are isomorphic iff they have equal dimension

## Statement

Let $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Then $V$ and $W$ are [[isomorphic|0B8T10CE7N]] if and only if $\dim V = \dim W$, where $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

An isomorphism sends a basis to a basis, so it keeps the dimension.
Conversely, a bijection between bases of $V$ and $W$ extends to a linear isomorphism.
The statement holds for all vector spaces, with dimension as a cardinal number.
For example, $M_{2 \times 3}(F)$, $F^6$ and $F[x]_{<6}$ are pairwise isomorphic.

## Explanations

### L1 human

Up to isomorphism, a finite-dimensional vector space is determined by one number: its dimension.
All three-dimensional real spaces behave like $\R^3$.
Only the names of the vectors differ.
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# Homogeneous linear system with constant coefficients

## Definition

Let $n \ge 1$ and let $A$ be a real $n \times n$ matrix.
The homogeneous linear system with constant coefficients given by $A$ is the [[system of linear differential equations|0A6QZ367GX]]
$$x'(t) = A\, x(t)$$
for an unknown differentiable function $x : \mathbb{R} \to \mathbb{R}^n$.

## Remarks

Its solutions are $x(t) = e^{t A} x(0)$. See [[the matrix exponential as the solution|0YC8JT513J]].
If $A v = \lambda v$, then $x(t) = e^{\lambda t} v$ is a solution. The [[eigenvalue method|08ZWFDJP3H]] builds all solutions from such solutions.
Example: for $A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, the solutions are rotations: $x(t) = \left(a \cos t + b \sin t, -a \sin t + b \cos t\right)$.

## Explanations

### L1 human

The rate of change of the vector $x$ is the matrix $A$ applied to $x$, and $A$ does not change with time.
This is the vector version of $y' = a y$, whose solutions are $e^{a t} y(0)$.
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<document source="https://tamnd.github.io/math2/node/0VJZM0F9YM.md" tag="0VJZM0F9YM" kind="definition">
# Walk

## Definition

Let $G$ be a [[graph|0JD5C82QMB]], let $k$ be a natural number, and let $u, v$ be vertices of $G$.
A walk of length $k$ from $u$ to $v$ is a sequence of vertices $(w_0, w_1, \dots, w_k)$ with $w_0 = u$, $w_k = v$, and $w_{i-1} \sim w_i$ for $1 \le i \le k$.
Its edges are the edges $w_{i-1} w_i$.

## Remarks

A walk may repeat vertices and edges. A walk of length $0$ is a single vertex.
A [[trail|062MQGP8M5]] has no repeated edge, and a [[path|0JDE56HWQ8]] has no repeated vertex.
A walk is closed if $u = v$.

## Explanations

### L1 human

A walk is a route through a graph that goes from vertex to vertex along edges.
You may visit the same vertex or use the same edge more than once.
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<document source="https://tamnd.github.io/math2/node/0VK0W4D5Y5.md" tag="0VK0W4D5Y5" kind="statement">
# De Morgan's laws for sets

## Statement

Let $U$ be a set and let $A$ and $B$ be [[subsets|0FE8Q5S42Y]] of $U$. Write $X^c = U \setminus X$ for the [[complement|0B1FSS1VZW]] of a subset $X$ in $U$.
Then
$$(A \cup B)^{c} = A^{c} \cap B^{c},$$
$$(A \cap B)^{c} = A^{c} \cup B^{c}.$$

## Remarks

The laws come from [[De Morgan's laws|09PKJKA6J0]] of logic, applied to $x \in A$ and $x \in B$.
They hold also for arbitrary families: the complement of $\bigcup_i A_i$ is $\bigcap_i A_i^c$, and the complement of $\bigcap_i A_i$ is $\bigcup_i A_i^c$.
With set differences, $C \setminus (A \cup B) = (C \setminus A) \cap (C \setminus B)$ for each set $C$.

## Explanations

### L1 human

An element is outside $A \cup B$ exactly when it is outside $A$ and outside $B$.
An element is outside $A \cap B$ exactly when it is outside $A$ or outside $B$.
A [[Venn diagram|0KCT6DPXJK]] with two circles shows both laws.
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<document source="https://tamnd.github.io/math2/node/0VK5V9P8G3.md" tag="0VK5V9P8G3" kind="statement">
# Size of a conjugacy class is the index of the centraliser

## Statement

Let $G$ be a [[group|G5S1M9]] and let $a \in G$.
Then $|\operatorname{Cl}(a)| = [G : C_G(a)]$, where $\operatorname{Cl}(a)$ is the [[conjugacy class|0DEBXAJ5SA]] of $a$ and $C_G(a)$ is its [[centraliser|02YYV36BFM]].
In particular, if $G$ is finite, then $|\operatorname{Cl}(a)|$ divides $|G|$.

## Remarks

This is the [[orbit-stabiliser theorem|0E2HSCKZ6C]] for the [[action by conjugation|0P8CY9V437]].
Directly, $gag^{-1} = hah^{-1}$ if and only if $h^{-1}g \in C_G(a)$.
For example, in $S_3$ the centraliser of $(1\ 2\ 3)$ is $A_3$, of index $2$, and the class of $(1\ 2\ 3)$ has $2$ elements.

## Explanations

### L1 human

Conjugates of $a$ by two elements are equal exactly when the two elements differ by something that commutes with $a$.
So the conjugates of $a$ correspond to the cosets of the centraliser.
An element that commutes with many elements has few conjugates.
</document>

<document source="https://tamnd.github.io/math2/node/0VK6VEH345.md" tag="0VK6VEH345" kind="statement">
# Banach fixed point theorem

## Statement

Let $(X, d)$ be a nonempty [[complete metric space|08EWT7J0SC]] and let $f \colon X \to X$ be a [[contraction|0MT2S0ZX4Y]] with constant $k \in [0, 1)$.
Then $f$ has exactly one fixed point $x^*$, that is, a point with $f(x^*) = x^*$.
Moreover, for each $x_0 \in X$, the sequence $x_{n+1} = f(x_n)$ converges to $x^*$, and
$$d(x_n, x^*) \le \frac{k^n}{1 - k} \, d(x_1, x_0) \quad \text{for all } n \ge 0.$$

## Remarks

Proof idea: $d(x_{n+1}, x_n) \le k^n d(x_1, x_0)$, so the sequence $(x_n)$ is a [[Cauchy sequence|0MCK8K2D7Z]] by a geometric series bound. Its limit is a fixed point because $f$ is continuous. Two fixed points $x, y$ satisfy $d(x, y) \le k \, d(x, y)$, so $x = y$.
Completeness is needed: $x \mapsto x/2$ on $(0, 1]$ is a contraction without a fixed point.
The theorem gives the existence of solutions of ordinary differential equations in the Picard–Lindelöf theorem.

## History

Stefan Banach stated the theorem in 1922.

## Explanations

### L1 human

Apply the contraction again and again, starting from any point. The points get closer to each other, and in a complete space they converge.
The limit does not move under $f$, and it is the only point that does not move.
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<document source="https://tamnd.github.io/math2/node/0VKCCEXBJ1.md" tag="0VKCCEXBJ1" kind="concept">
# Ellipse

## Summary

An ellipse is a closed plane curve with two foci. For each point of the curve, the sum of the distances to the two foci is the same constant. A circle is the special case of an ellipse where the two foci are the same point.
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<document source="https://tamnd.github.io/math2/node/0VKD1TP7F3.md" tag="0VKD1TP7F3" kind="concept">
# Metric space

## Summary

A metric space is a set with a distance function between its points.
The distance is zero only from a point to itself, it is symmetric, and it satisfies the triangle inequality.
Metric spaces are the setting for limits, continuity and completeness beyond the real line.
</document>

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# Lagrange multipliers

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $m < n$, and let $f \colon U \to \mathbb{R}$ and $g = (g_1, \dots, g_m) \colon U \to \mathbb{R}^m$ be of [[class C^1|0TCBZMEYGN]].
Let $M = \{x \in U : g(x) = 0\}$, and let $a \in M$ be a [[local extremum|04CPG9K1C4]] of the restriction of $f$ to $M$.
Suppose that the [[gradients|0NS2QYVC4Y]] $\nabla g_1(a), \dots, \nabla g_m(a)$ are linearly independent.
Then there is a unique $\lambda = (\lambda_1, \dots, \lambda_m) \in \mathbb{R}^m$ such that
$$\nabla f(a) = \lambda_1 \nabla g_1(a) + \dots + \lambda_m \nabla g_m(a).$$

## Remarks

The numbers $\lambda_i$ are the Lagrange multipliers. Equivalently, $(a, \lambda)$ is a critical point of the Lagrangian $L(x, \lambda) = f(x) - \lambda \cdot g(x)$.
The hypothesis on the gradients is the constraint qualification. It says that the [[Jacobian matrix|0YF0VXKXGB]] $Dg(a)$ has rank $m$.
The proof uses the [[implicit function theorem|0BXBADR94C]]. Near $a$, the set $M$ is the graph of a $C^1$ function of $n - m$ variables. The gradient of $f$ is orthogonal to the tangent space of $M$ at $a$.
The condition is necessary, not sufficient. Each candidate point needs a further check, for example a comparison of values on a compact set $M$.

## Explanations

### L1 human

At a constrained extremum, a small move along the constraint set does not change $f$ to first order.
So $\nabla f(a)$ has no component along the constraint set, and it is a combination of the normals $\nabla g_i(a)$.
This turns a constrained problem into a system of $n + m$ equations in $n + m$ unknowns.
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<document source="https://tamnd.github.io/math2/node/0VM8B2SB4P.md" tag="0VM8B2SB4P" kind="concept">
# Decimal expansion

## Summary

A decimal expansion writes a nonnegative real number as an integer part and an infinite sequence of digits after the decimal point.
It represents the number as a sum of multiples of powers of $10$.
For example, $1/4 = 0.25000\dots$ and $1/3 = 0.333\dots$.
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<document source="https://tamnd.github.io/math2/node/0VMG8TA8VF.md" tag="0VMG8TA8VF" kind="definition">
# Limit of a function of several variables

## Definition

Let $D \subseteq \mathbb{R}^n$ and let $f \colon D \to \mathbb{R}^m$.
Let $a \in \mathbb{R}^n$ be a limit point of $D$: each open ball about $a$ contains a point of $D$ other than $a$.
Let $L \in \mathbb{R}^m$.
When this holds, $f(x)$ tends to $L$ as $x$ tends to $a$, with the notation $\lim_{x \to a} f(x) = L$.
For each $\varepsilon > 0$ there is a $\delta > 0$ such that $\|f(x) - L\| < \varepsilon$ for each $x \in D$ with $0 < \|x - a\| < \delta$.
Here $\|\cdot\|$ is the Euclidean norm of [[Euclidean space|0XPR1E85W0]].

## Remarks

A limit is unique when it exists, because $a$ is a limit point of $D$.
The value $f(a)$, and even whether $a$ is in $D$, has no effect on the limit.
A function into $\mathbb{R}^m$ tends to $L$ if and only if each component $f_i$ tends to $L_i$.
To show that a limit does not exist, it is enough to find two [[paths with different limits|0BJ6H85DRQ]].

## Explanations

### L1 human

The definition says that you can make $f(x)$ as close to $L$ as you want. For this, take $x$ close enough to $a$ but not equal to $a$.
In the plane, "close to $a$" means inside a small disc about $a$, so $x$ can come from every direction at once.
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<document source="https://tamnd.github.io/math2/node/0VMWGYY35Q.md" tag="0VMWGYY35Q" kind="statement">
# Cartan–Dieudonné theorem

## Statement

Let $V$ be a real inner product space of finite dimension $n$.
Then each linear isometry $T \colon V \to V$ is a composition of at most $n$ [[reflections|06BRDF3F0E]] in hyperplanes of $V$.

## Remarks

The identity map is the composition of zero reflections.
In $\R^2$, each rotation is the composition of two reflections in lines through the origin.
The theorem holds more generally for nondegenerate symmetric bilinear forms on spaces over fields of characteristic not $2$.

## Explanations

### L1 human

Every rigid linear motion of $n$-dimensional space is a sequence of at most $n$ mirror reflections.
So reflections are the building blocks of all isometries.
In the plane, a rotation by $\theta$ is the product of two reflections in lines at the angle $\theta/2$.
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<document source="https://tamnd.github.io/math2/node/0VMWPGGG3C.md" tag="0VMWPGGG3C" kind="concept">
# Smallest prime factor

## Summary

The smallest prime factor of a natural number $n \ge 2$ is the least prime that divides $n$.
The [[definition of the smallest prime factor|0S2SYHTWPY]] gives the precise form.
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# If 2^n + 1 is prime then n is a power of 2

## Statement

Let $n$ be a positive integer.
If $2^n + 1$ is a [[prime number|0SN0MAEKYX]], then $n = 2^k$ for a natural number $k$.
So every prime of the form $2^n + 1$ with $n \ge 1$ is a [[Fermat number|0R03B54V4Z]].

## Remarks

The proof: if $n = k m$ with $m > 1$ odd, then $x + 1$ divides $x^m + 1$ for $x = 2^k$. So $2^k + 1$ is a proper divisor of $2^n + 1$.
For example, $2^6 + 1 = 65 = 5 \cdot 13$, and $5 = 2^2 + 1$.
The converse is false, because $F_5 = 2^{32} + 1$ is composite.

## Explanations

### L1 human

An odd factor of the exponent gives a factorization of $2^n + 1$.
So the only exponents that can give primes are the powers of $2$, which have no odd factor except $1$.
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<document source="https://tamnd.github.io/math2/node/0VNE161QCK.md" tag="0VNE161QCK" kind="statement">
# Polynomials of degree two or three are irreducible iff they have no roots

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ with $\deg f = 2$ or $\deg f = 3$.
Then $f$ is [[irreducible|00X2A9G9W3]] over $K$ if and only if $f$ has no [[root|00EHX6FWC0]] in $K$.

## Remarks

The proof: if $f = g h$ with $\deg g, \deg h \ge 1$, then $\deg g + \deg h \le 3$, so one factor has degree $1$. A factor $a x + b$ of degree $1$ gives the root $-b/a$. The [[factor theorem|0K3MWA7DMC]] gives the converse.
For example, $x^3 - 2$ has no root in $\mathbb{Q}$, so it is irreducible over $\mathbb{Q}$.
The statement fails in degree $4$: $(x^2 + 1)^2$ has no real root, but it is not irreducible over $\mathbb{R}$.

## Explanations

### L1 human

A factorisation of a polynomial of degree $2$ or $3$ must have a factor of degree $1$.
A factor of degree $1$ is the same as a root.
So to test irreducibility in low degree, you only look for roots.
</document>

<document source="https://tamnd.github.io/math2/node/0VP3A1290K.md" tag="0VP3A1290K" kind="statement">
# Pythagorean theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$. Let $d$ be the [[Euclidean distance|0NPPGHQNXQ]], and let $\angle ABC$ be the [[angle at $B$|0GJAJC6TNZ]].
Then
$$d(A, C)^2 = d(A, B)^2 + d(B, C)^2$$
if and only if $\angle ABC = \pi / 2$.

## Remarks

The direction "if" is the classical theorem: in a right triangle, the square of the hypotenuse is the sum of the squares of the legs, $a^2 + b^2 = c^2$. The direction "only if" is its converse.
In vector form: for vectors $u$ and $v$ with $u \cdot v = 0$, $\|u + v\|^2 = \|u\|^2 + \|v\|^2$.
The theorem is the special case $\gamma = \pi / 2$ of the [[law of cosines|060ZVZBV3E]].
Mathlib states the equivalence for all points $A$, $B$ and $C$, with its convention for the angle at a repeated point.

## History

The theorem carries the name of the Greek philosopher Pythagoras. Euclid's Elements proves it as Proposition 47 of Book I, and the converse as Proposition 48.

## Explanations

### L1 human

Build a square on each side of a right triangle. The two small squares together have the same area as the big square on the longest side. For the sides $3$, $4$ and $5$: $9 + 16 = 25$.
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<document source="https://tamnd.github.io/math2/node/0VP4TBVJ19.md" tag="0VP4TBVJ19" kind="definition">
# Frobenius automorphism of a finite field

## Definition

Let $F$ be a [[finite field|08RTDHX1SK]] of characteristic $p$.
The Frobenius automorphism of $F$ is the map $\varphi \colon F \to F$, $\varphi(x) = x^p$.

## Remarks

The map $\varphi$ is the [[Frobenius endomorphism|0GHDDVAH8P]] of $F$. It is injective, as is every field homomorphism, so it is bijective, because $F$ is finite.
The map $\varphi$ fixes every element of the prime subfield $\mathbb{F}_p$, by Fermat's little theorem.
If $|F| = p^n$, then $\varphi^n$ is the identity, and the automorphisms of $F$ are exactly $\mathrm{id}, \varphi, \dots, \varphi^{n-1}$.
So $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is cyclic of order $n$, generated by $\varphi$.

## Explanations

### L1 human

Raising to the power $p$ respects addition in characteristic $p$, by the freshman's dream.
In a finite field, this map permutes the elements, and every symmetry of the field is a power of it.
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<document source="https://tamnd.github.io/math2/node/0VP56B4EMX.md" tag="0VP56B4EMX" kind="definition">
# Power series

## Definition

Let $c$ be a real number and let $(a_n)_{n \ge 0}$ be a real sequence.
The power series with center $c$ and coefficients $a_n$ is the [[series|0JH4MN4K1C]]
$$\sum_{n=0}^{\infty} a_n (x - c)^n,$$
taken as a function of the real variable $x$.

## Remarks

The series always converges at $x = c$, with sum $a_0$.
The set of $x$ where it converges is an interval around $c$. Its half length is the [[radius of convergence|0660QX1VPR]].
Polynomials are power series with only finitely many nonzero coefficients.
Complex power series use the same definition, with a complex variable $z$.

## Explanations

### L1 human

A power series is a polynomial with infinitely many terms.
For each $x$, we put $x$ into every term and add up the resulting series of numbers.
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<document source="https://tamnd.github.io/math2/node/0VPCJWGMP9.md" tag="0VPCJWGMP9" kind="definition">
# Space of linear maps

## Definition

Let $F$ be a field and let $V$ and $W$ be vector spaces over $F$.
The *space of linear maps* $\Hom_F(V, W)$, also written $L(V, W)$, is the set of all [[linear maps|0NRYHJPQG2]] $V \to W$. It has the operations
$$(S + T)(v) = S(v) + T(v), \qquad (aT)(v) = a\, T(v)$$
for $S, T \in \Hom_F(V, W)$, $a \in F$ and $v \in V$.
It is a vector space over $F$, with the zero map as zero vector.

## Remarks

For $V = F^n$ and $W = F^m$, the matrix map is an isomorphism $\Hom_F(F^n, F^m) \cong M_{m \times n}(F)$.
The [[dimension of the space of linear maps|0Z9N1ER0W0]] is $\dim V \cdot \dim W$ in finite dimensions.
For $W = F$, it is the [[dual space|07JBSZE0V4]] of $V$, and for $W = V$, it is the space of [[endomorphisms|0J7H0YBFDJ]] of $V$.

## Explanations

### L1 human

Linear maps can be added and scaled, value by value, and the result is again linear.
So the linear maps from $V$ to $W$ form a vector space themselves.
In coordinates, this is the vector space of matrices of a fixed size.
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<document source="https://tamnd.github.io/math2/node/0VPFBAY7MW.md" tag="0VPFBAY7MW" kind="definition">
# Prime subfield

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
The prime subfield of $K$ is the intersection of all the [[subfields|07QY5RPFBR]] of $K$.
It is the smallest subfield of $K$.

## Remarks

If the [[characteristic|0P7CH1CND5]] of $K$ is a prime $p$, then the prime subfield is isomorphic to $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.
If the characteristic of $K$ is $0$, then the prime subfield is isomorphic to $\mathbb{Q}$.
The prime subfield of $\mathbb{R}$ is $\mathbb{Q}$.

## Explanations

### L1 human

Every subfield contains $0$ and $1$, and so it contains all the sums $1 + 1 + \dots + 1$ and their quotients.
These elements form the prime subfield.
In the real numbers, they are exactly the rational numbers.
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<document source="https://tamnd.github.io/math2/node/0VPK4X73Q0.md" tag="0VPK4X73Q0" kind="statement">
# Rule of product

## Statement

Let $A$ and $B$ be finite sets.
Then the Cartesian product $A \times B = \{(a, b) : a \in A, b \in B\}$ is finite and $|A \times B| = |A| \cdot |B|$.

More generally, for finite sets $A_1, \dots, A_k$, the product $A_1 \times \dots \times A_k$ has $|A_1| \cdots |A_k|$ elements.

## Remarks

A common form of the rule: a choice is made in $k$ steps, and step $i$ has $n_i$ options, whatever the earlier steps are. Then there are $n_1 n_2 \cdots n_k$ choices in total.
The [[number of functions between finite sets|08QQH9F7G0]] is a direct consequence.

## Explanations

### L1 human

If you choose one item from a first group and then one item from a second group, you multiply the sizes.
For example, with 3 shirts and 4 trousers there are $3 \cdot 4 = 12$ outfits.
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<document source="https://tamnd.github.io/math2/node/0VQA7F720Z.md" tag="0VQA7F720Z" kind="concept">
# Sequence tending to infinity

## Summary

A real sequence tends to infinity when its terms eventually exceed each given bound. Such a sequence diverges, and we say that its limit is $+\infty$. The node [[sequence tending to infinity|09AZCV6GMC]] gives the precise definition.
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<document source="https://tamnd.github.io/math2/node/0VQQCVG8WA.md" tag="0VQQCVG8WA" kind="concept">
# Combinations with repetition

## Summary

A combination with repetition is a selection of $k$ elements from a set in which the order does not matter and elements can repeat. Such selections are multisets of size $k$, and stars and bars counts them. The node [[combinations with repetition|0YEQG3VBYC]] gives the precise definition.
</document>

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# Nowhere dense set

## Summary

A nowhere dense subset of a topological space is a set whose closure has empty interior.
Such a set is not dense in any nonempty open set.
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<document source="https://tamnd.github.io/math2/node/0VR671PHE5.md" tag="0VR671PHE5" kind="concept">
# Symmetric matrix

## Summary

A symmetric matrix is a square matrix that is equal to its transpose.
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<document source="https://tamnd.github.io/math2/node/0VRD8GC5BM.md" tag="0VRD8GC5BM" kind="definition">
# Isometric isomorphism

## Definition

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]].
An isometric isomorphism from $X$ to $Y$ is a [[bijective|0SNQEF7Z53]] [[isometry|0878DN3WCS]] $f \colon X \to Y$.
The spaces $X$ and $Y$ are isometric when there is an isometric isomorphism from $X$ to $Y$.

## Remarks

Because each isometry is injective, an isometry is an isometric isomorphism if and only if it is surjective.
The inverse of an isometric isomorphism is again an isometric isomorphism, and the composition of two of them is one.
Each isometric isomorphism is a [[homeomorphism|05GZ11PWRY]].
The map $x \mapsto x + 1$ is an isometric isomorphism from $[0, 1]$ to $[1, 2]$ with the usual metric.

## Explanations

### L1 human

Isometric spaces are copies of each other: a bijection matches their points, and it keeps all the distances.
Each fact about distances in one space is also true in the other space.
</document>

<document source="https://tamnd.github.io/math2/node/0VRWZDWW6B.md" tag="0VRWZDWW6B" kind="definition">
# Binomial distribution

## Definition

Let $n \ge 0$ be an integer and let $p \in [0, 1]$.
The binomial distribution with parameters $n$ and $p$ is the distribution on $\{0, 1, \dots, n\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(k) = \binom{n}{k} p^k (1 - p)^{n-k} \quad \text{for } k = 0, 1, \dots, n.$$
Here $\binom{n}{k}$ is the [[binomial coefficient|092GEF38YD]]. One writes $X \sim \operatorname{B}(n, p)$ or $X \sim \operatorname{Bin}(n, p)$.

## Remarks

The values add up to $1$ by the [[binomial theorem|0EHWPWGHB2]]: $\sum_{k} \binom{n}{k} p^k (1 - p)^{n-k} = (p + 1 - p)^n = 1$.
The number of successes in $n$ independent [[Bernoulli trials|0PHC9FE2V5]] with success probability $p$ has this distribution. The factor $\binom{n}{k}$ counts the positions of the $k$ successes.
The mean is $np$ and the variance is $np(1 - p)$.
The probability of exactly $2$ heads in $4$ tosses of a fair coin is $\binom{4}{2} / 2^4 = 6/16$.

## Explanations

### L1 human

Repeat a yes-or-no experiment $n$ times, independently, and count the yes answers.
The count has the binomial distribution. Each sequence with $k$ yes answers has probability $p^k (1 - p)^{n-k}$, and there are $\binom{n}{k}$ such sequences.
</document>

<document source="https://tamnd.github.io/math2/node/0VS0MP1P5H.md" tag="0VS0MP1P5H" kind="definition">
# Abundant number

## Definition

A positive integer $n$ is abundant if the sum of its positive divisors other than $n$ is greater than $n$.
Equivalently, $\sigma(n) > 2n$, where $\sigma$ is the [[sum of divisors|009QCP5T3Z]].

## Remarks

The smallest abundant number is $12$, since $1 + 2 + 3 + 4 + 6 = 16 > 12$.
The smallest odd abundant number is $945$.
Each multiple $kn$ with $k \ge 2$ of a [[perfect number|0SB7Q8SYSF]] $n$ is abundant.

## Explanations

### L1 human

An abundant number has "too many" divisors: its smaller divisors add up to more than the number.
Compare with [[perfect numbers|0SB7Q8SYSF]], where the sum is equal, and [[deficient numbers|0G01TS1RRP]], where it is smaller.
</document>

<document source="https://tamnd.github.io/math2/node/0VSB0XF4G7.md" tag="0VSB0XF4G7" kind="definition">
# Indicator function

## Definition

Let $X$ be a set and let $A$ be a [[subset|0FE8Q5S42Y]] of $X$.
The indicator function of $A$ is the [[function|0XHDTE5YMZ]] $\mathbf{1}_A \colon X \to \{0, 1\}$ with
$$\mathbf{1}_A(x) = \begin{cases} 1 & \text{if } x \in A, \\ 0 & \text{if } x \notin A. \end{cases}$$

## Remarks

Other names are characteristic function and the notation $\chi_A$.
For subsets $A$ and $B$ of $X$, $\mathbf{1}_{A \cap B} = \mathbf{1}_A \mathbf{1}_B$ and $\mathbf{1}_{A^c} = 1 - \mathbf{1}_A$.
The map $A \mapsto \mathbf{1}_A$ is a [[bijection|0SNQEF7Z53]] from the [[power set|0BC7KX51Z8]] of $X$ to the set of functions from $X$ to $\{0, 1\}$.

## Explanations

### L1 human

The indicator function answers "is $x$ in $A$?" with $1$ for yes and $0$ for no.
For $A$ the even integers, $\mathbf{1}_A(4) = 1$ and $\mathbf{1}_A(5) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0VSNHTHK8T.md" tag="0VSNHTHK8T" kind="statement">
# A tree on n vertices has n − 1 edges

## Statement

Let $T$ be a finite [[tree|0MJ0JS72DZ]] with $n$ vertices.
Then $T$ has exactly $n - 1$ edges.

## Remarks

A tree has at least one vertex, because it is connected. Thus $n \ge 1$.
Proof by induction on $n$: for $n \ge 2$ delete a leaf, which exists by the [[leaf lemma|0H8Q4MANBD]]. The rest is a tree with $n - 1$ vertices.
A consequence: the sum of the degrees in a tree is $2n - 2$.

## Explanations

### L1 human

Build a tree by adding one vertex at a time, each joined to the tree by one new edge.
You start with $1$ vertex and $0$ edges, so at the end there is always one edge fewer than vertices.
</document>

<document source="https://tamnd.github.io/math2/node/0VSNX4P2Q0.md" tag="0VSNX4P2Q0" kind="statement">
# Cancellation laws

## Statement

Let $G$ be a [[group|G5S1M9]] and let $a, b, c \in G$.
If $ab = ac$, then $b = c$.
If $ba = ca$, then $b = c$.

## Remarks

The first law is left cancellation and the second law is right cancellation.
To prove the first law, multiply $ab = ac$ on the left by $a^{-1}$.
A consequence is that in the [[Cayley table|0YB2NRP0T5]] of a finite group, no element occurs two times in one row or in one column.
The laws can fail in a [[monoid|0N6R902FG1]]: in $(\mathbb{Z}, \cdot)$, $0 \cdot 1 = 0 \cdot 2$ but $1 \neq 2$.

## Explanations

### L1 human

In a group, you can remove the same element from both sides of an equation, if it is on the same side of each product.
This works because each element has an inverse that undoes it.
Take care with the side: $ab = ca$ does not give $b = c$ in general.

### L2 agent

The cancellation laws let us remove a common factor from both sides of an equation. The factor must be on the same side in both products.

The proof of left cancellation has three steps:

1. Suppose that $ab = ac$.
2. Multiply both sides on the left by $a^{-1}$.
3. Associativity and $a^{-1}a = e$ give $b = c$.

A worked example in $S_3$, with composition from right to left:

1. Let $a = (1\,2)$. Suppose that $ax = (1\,2\,3)$ and $ay = (1\,2\,3)$.
2. Then $ax = ay$, so $x = y$ by left cancellation.
3. Thus $x = a^{-1}(1\,2\,3) = (1\,2)(1\,2\,3) = (2\,3)$ is the only solution.

This shows a useful consequence. For fixed $a$ and $c$, the equation $ax = c$ has exactly one solution, $x = a^{-1}c$. Thus each row of the Cayley table of a finite group contains each element exactly one time. The same is true for each column.

A common error is to cancel across sides. From $ab = ca$ we cannot conclude $b = c$. In $S_3$, let $a = (1\,2)$, $b = (1\,3)$ and $c = (2\,3)$. Then $ab = (1\,3\,2)$ and $ca = (1\,3\,2)$, but $b \ne c$.

The laws need inverses. In the monoid $(\mathbb{Z}, \cdot)$, $0 \cdot 1 = 0 \cdot 2$, but $1 \ne 2$, because $0$ has no inverse.
</document>

<document source="https://tamnd.github.io/math2/node/0VSSPAFM2C.md" tag="0VSSPAFM2C" kind="concept">
# Characteristic polynomial

## Summary

The characteristic polynomial of a square matrix $A$ is the polynomial $\det(t I - A)$ in the variable $t$.
Its roots in the field are the eigenvalues of $A$.
</document>

<document source="https://tamnd.github.io/math2/node/0VT2QK6X6Q.md" tag="0VT2QK6X6Q" kind="concept">
# Riccati equation

## Summary

A Riccati equation is a first-order equation $y' = q_0(x) + q_1(x) y + q_2(x) y^2$, with a right side quadratic in $y$.
It can be changed into a second-order linear equation. See the [[definition|0E1ZZBHZ3Z]].
</document>

<document source="https://tamnd.github.io/math2/node/0VT6P0GNQR.md" tag="0VT6P0GNQR" kind="concept">
# Gram–Schmidt process

## Summary

Let $V$ be an inner product space over $\R$ or $\C$, and let $v_1, \dots, v_k \in V$ be [[linearly independent|0CRMGPAH83]].
The *Gram–Schmidt process* defines vectors $u_1, \dots, u_k$ by $u_1 = v_1$ and, for $2 \le j \le k$,
$$u_j = v_j - \sum_{i=1}^{j-1} \frac{\langle v_j, u_i \rangle}{\langle u_i, u_i \rangle} u_i.$$
Then the vectors $e_j = u_j / \norm{u_j}$ form an [[orthonormal family|0RW9TNNT87]] with the same [[span|0DTY8B9HMS]] as $v_1, \dots, v_j$ for each $j$.

## Remarks

Each step subtracts from $v_j$ its projections on the earlier vectors $u_i$.
For $v_1 = (1, 1)$ and $v_2 = (1, 0)$ in $\R^2$, it gives $u_1 = (1, 1)$ and $u_2 = (1/2, -1/2)$.
In floating point arithmetic, the modified Gram–Schmidt process is more stable.
In matrix form, the process gives the [[QR decomposition|0ERWG4NXQ9]].

## Explanations

### L1 human

The Gram–Schmidt process turns any independent list of vectors into a list of perpendicular unit vectors.
It treats the vectors one at a time, and it removes from each one its components along the earlier ones.
Then it scales each result to length one.
</document>

<document source="https://tamnd.github.io/math2/node/0VTAD25A3G.md" tag="0VTAD25A3G" kind="statement">
# Minimal polynomial divides every annihilating polynomial

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an endomorphism with [[minimal polynomial|03NDVCSHSB]] $m_T$.
If $p \in F[x]$ and $p(T) = 0$, then $m_T$ divides $p$.

## Remarks

The proof divides with remainder: $p = q m_T + r$ with $\deg r < \deg m_T$.
Then $r(T) = 0$, so $r = 0$ by the minimality of $\deg m_T$.

## Explanations

### L1 human

Every polynomial that kills $T$ is a multiple of the minimal polynomial.
If a matrix satisfies $A^2 = A$, then its minimal polynomial divides $x^2 - x$.
So it is $x$, $x - 1$ or $x^2 - x$.
</document>

<document source="https://tamnd.github.io/math2/node/0VTAXQEWQD.md" tag="0VTAXQEWQD" kind="example">
# Sup metric on R^n

## Example

Let $n \ge 1$ be an integer.
For $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let
$$d_\infty(x, y) = \max_{1 \le i \le n} |x_i - y_i|.$$
Then $d_\infty$ is a [[metric|063W5JMY85]] on $\mathbb{R}^n$, the sup metric.

## Remarks

It is also called the Chebyshev distance or the maximum metric.
For all $x, y$ in $\mathbb{R}^n$, $d_\infty(x, y) \le d_2(x, y) \le d_1(x, y) \le n \, d_\infty(x, y)$, where $d_2$ is the [[Euclidean metric|0E1MMCAY5B]] and $d_1$ is the [[taxicab metric|07CJNE6ZAQ]].
It is the limit of the [[p-metric|0TFDZN6AXP]] $d_p$ as $p \to \infty$.

## Explanations

### L1 human

The sup metric takes only the largest difference of the coordinates.
On a chessboard, it is the number of moves that a king needs to go from one square to another.
</document>

<document source="https://tamnd.github.io/math2/node/0VTCABADBE.md" tag="0VTCABADBE" kind="example">
# Turán graph

## Example

Let $n \ge r \ge 1$ be integers.
The Turán graph $T(n, r)$ has $n$ vertices, split into $r$ parts whose sizes differ by at most $1$.
Two vertices are adjacent if and only if they are in different parts.

## Remarks

Two vertices in the same part are not adjacent. Among any $r + 1$ vertices, two are in the same part, so $T(n, r)$ has no [[clique|0G52BPNPKG]] with $r + 1$ vertices.
The number of edges of $T(n, r)$ is $\left\lfloor \left(1 - \frac{1}{r}\right) \frac{n^2}{2} \right\rfloor$.
$T(n, 2)$ is the [[complete bipartite graph|0V0GC7BQAN]] $K_{\lfloor n/2 \rfloor, \lceil n/2 \rceil}$, and $T(n, n)$ is the [[complete graph|03WKB2H474]] $K_n$.

## Explanations

### L1 human

To build $T(n, r)$, split $n$ people into $r$ teams of almost equal size, and join two people when they are in different teams.
It is the graph with the most edges that has no clique of size $r + 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0VTH8KPCG2.md" tag="0VTH8KPCG2" kind="concept">
# Ring homomorphism

## Summary

A ring homomorphism is a map between two rings that preserves addition, multiplication and the identity element.
The [[definition of a ring homomorphism|0GGE0Z0JQ1]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0VTN5RKNC5.md" tag="0VTN5RKNC5" kind="definition">
# Standard deviation

## Definition

Let $X$ be a real random variable with finite [[variance|007QED3E0Y]].
The standard deviation of $X$ is the [[square root|0VB8BFRABH]]
$$\sigma_X = \sqrt{\operatorname{Var}(X)} = \sqrt{E\left[(X - E[X])^2\right]}.$$

## Remarks

The standard deviation is in the same units as $X$, but the variance is in squared units.
For $a, b \in \mathbb{R}$, $\sigma_{aX + b} = |a| \, \sigma_X$.
For the [[normal distribution|07RXBVTVZB]] $\mathcal{N}(\mu, \sigma^2)$, the standard deviation is $\sigma$. About $68\%$ of the mass lies within one standard deviation of the mean.

## Explanations

### L1 human

The standard deviation is a typical size of the distance between a value and the mean.
A small standard deviation means that the values stay close to the mean. A large one means that they spread out.
</document>

<document source="https://tamnd.github.io/math2/node/0VTQTPQGAV.md" tag="0VTQTPQGAV" kind="definition">
# Basis

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A *basis* of $V$ is a family $(b_i)_{i \in I}$ of vectors of $V$ that is [[linearly independent|0CRMGPAH83]] and [[spans|088RYKAWK9]] $V$.

## Remarks

A family $(b_i)_{i \in I}$ is a basis if and only if each $v \in V$ is equal to $\sum_{i \in I} a_i b_i$ for exactly one finitely supported family of scalars $(a_i)_{i \in I}$.
The scalars $a_i$ are the [[coordinates|04CYQHN6MC]] of $v$ in the basis.
The [[standard basis|0V7ZD6P7RK]] $e_1, \dots, e_n$ is a basis of $F^n$; the vectors $(1, 1)$ and $(1, -1)$ form another basis of $\R^2$.
[[Every vector space has a basis|00MS9VMQ9Z]], and all bases of a space have the same cardinality.

## Explanations

### L1 human

A basis is a set of directions that is just right: enough to reach every vector, and none of them redundant.
Once we fix a basis, each vector gets a unique list of coordinates.
A vector space has many bases, but all of them have the same number of vectors.
</document>

<document source="https://tamnd.github.io/math2/node/0VTQXTCR2G.md" tag="0VTQXTCR2G" kind="concept">
# Transcendental number

## Summary

A transcendental number is a complex number that is not algebraic, such as $e$ or $\pi$.
The [[definition of a transcendental number|079EGSD32Y]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0VV04M41AN.md" tag="0VV04M41AN" kind="definition">
# Euler's totient function

## Definition

Let $n$ be a positive integer.
Euler's totient function $\varphi(n)$ is the number of integers $k$ with $1 \le k \le n$ such that $k$ and $n$ are [[coprime|0GFKP8F8AG]].

## Remarks

For example, $\varphi(1) = 1$, $\varphi(9) = 6$ and $\varphi(12) = 4$. The integers coprime to $12$ in $\{1, \dots, 12\}$ are $1, 5, 7, 11$.
The number $\varphi(n)$ is the number of elements of a [[reduced residue system|0J0QEN1DZR]] modulo $n$.
If $n = p_1^{e_1} \cdots p_r^{e_r}$ with distinct primes $p_i$, then $\varphi(n) = n \prod_{i=1}^{r} \left(1 - \frac{1}{p_i}\right)$.
The Lean library Mathlib sets $\varphi(0) = 0$.

## Explanations

### L1 human

Count the numbers from $1$ to $n$ that have no common factor with $n$.
For $n = 10$, these are $1, 3, 7, 9$, so $\varphi(10) = 4$.
</document>

<document source="https://tamnd.github.io/math2/node/0VV0HSW1ZS.md" tag="0VV0HSW1ZS" kind="definition">
# Convex function

## Definition

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be a function.
The function $f$ is convex if
$$f(t x + (1 - t) y) \le t f(x) + (1 - t) f(y)$$
for all $x, y \in I$ and all $t \in [0, 1]$.

## Remarks

Geometrically, each chord of the graph lies on or above the graph.
For example, $x^2$, $|x|$, and $e^x$ are convex on $\mathbb{R}$. Linear functions are convex.
A twice differentiable function with [[$f'' \ge 0$ is convex|0Z41C17TGP]].
Mathlib defines convexity on a set as `ConvexOn`.

## Explanations

### L1 human

A convex function bends upward, like a bowl.
The straight segment between two points of the graph never goes below the graph.
</document>

<document source="https://tamnd.github.io/math2/node/0VV3ZMRF88.md" tag="0VV3ZMRF88" kind="concept">
# Number of divisors

## Summary

The number of divisors $d(n)$ of a positive integer $n$ is the count of its positive divisors.
The [[definition of the number of divisors|0BKJT6WG96]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0VVK64Q4JK.md" tag="0VVK64Q4JK" kind="concept">
# Discriminant of a polynomial

## Summary

The discriminant of a polynomial is a quantity computed from its coefficients that is zero exactly when the polynomial has a repeated root.
The [[definition of the discriminant|0CCPN09EMJ]] gives the formula in terms of the roots.
</document>

<document source="https://tamnd.github.io/math2/node/0VVPQNVM8S.md" tag="0VVPQNVM8S" kind="statement">
# Order of a power

## Statement

Let $G$ be a [[group|G5S1M9]] and let $g \in G$ be an element of finite [[order|0WE2NHZJ7B]] $n$.
Let $k$ be an integer.
Then the order of $g^k$ is $n / \gcd(n, k)$.

## Remarks

Here $\gcd(n, 0) = n$, so that $g^0 = e$ has order $1$.
In particular, $g^k$ has order $n$ if and only if $\gcd(n, k) = 1$.
For example, if $g$ has order $12$, then $g^8$ has order $12 / 4 = 3$.

## Explanations

### L1 human

A power of an element returns to the identity faster than the element itself, or at the same time.
If $g$ has order $12$, then $g^8$ is at the identity after $k$ steps exactly when $12$ divides $8k$, that is, when $3$ divides $k$.
Thus $g^8$ has order $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0VVY4H92G2.md" tag="0VVY4H92G2" kind="statement">
# Term test

## Statement

Let $\sum a_n$ be a [[convergent series|0CVVKY7GY2]] of real numbers.
Then $a_n \to 0$ as $n \to \infty$.
Equivalently, if $(a_n)$ does not converge to $0$, then the series $\sum a_n$ diverges.

## Remarks

For the proof, let $s$ be the sum. Then $a_n = s_n - s_{n-1} \to s - s = 0$.
The converse is false. The terms of the [[harmonic series|0CXTMAY5PX]] tend to $0$, and the series diverges.
For example, $\sum (-1)^n$ and $\sum n/(n+1)$ diverge by this test.

## Explanations

### L1 human

If you keep adding terms that do not get small, the total cannot settle down.
So small terms are necessary for convergence, but they are not enough.
</document>

<document source="https://tamnd.github.io/math2/node/0VW90BBSJ4.md" tag="0VW90BBSJ4" kind="concept">
# Continuously differentiable function

## Summary

A continuously differentiable function is a differentiable function whose derivative is continuous. The classes $C^k$ ask for $k$ derivatives, the last one continuous. The node [[continuously differentiable function|06SKCHDQT5]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0VW9E4QXR6.md" tag="0VW9E4QXR6" kind="concept">
# Likelihood function

## Summary

The likelihood function gives, for each value of an unknown parameter, the probability or the density of the observed data under that value. It is a function of the parameter, with the data fixed.
The node [[likelihood function|05FSTXM92W]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0VX0CPRQTZ.md" tag="0VX0CPRQTZ" kind="statement">
# Euler line

## Statement

Let $ABC$ be a triangle in the Euclidean space $\mathbb{R}^n$. Let $O$ be its [[circumcentre|0F98FZK14A]], $G = \frac{1}{3}(A + B + C)$ its [[centroid|0X7EEWES5G]] and $H$ its [[orthocentre|0WB619CA0V]].
Then
$$H = O + 3\,(G - O).$$
In particular, the points $O$, $G$ and $H$ are [[collinear|0AFNNTRTJE]].
If the triangle is not equilateral, then $O \neq G$, and the line through $O$ and $G$ is the Euler line of the triangle.

## Remarks

The formula says that $G$ lies between $O$ and $H$, with $d(G, H) = 2\, d(O, G)$.
An equivalent form is $H - O = (A - O) + (B - O) + (C - O)$.
The centre of the nine-point circle is the midpoint of $O$ and $H$, so it is also on the Euler line.
In an equilateral triangle, the points $O$, $G$ and $H$ are the same point.

## History

Euler showed in 1765 that the orthocentre, the circumcentre and the centroid of a triangle are collinear.

## Explanations

### L1 human

Three famous centres of a triangle always lie on one straight line. The centroid is on this line between the other two. It is twice as far from the orthocentre as from the circumcentre.
</document>

<document source="https://tamnd.github.io/math2/node/0VX9EP2GS8.md" tag="0VX9EP2GS8" kind="concept">
# Minimal polynomial (field theory)

## Summary

Let $a$ be algebraic over a field $K$.
The minimal polynomial of $a$ over $K$ is the monic polynomial of smallest degree in $K[x]$ that has $a$ as a root.
The [[definition of the minimal polynomial|0GK1WB9R98]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0VXJHZ1G9J.md" tag="0VXJHZ1G9J" kind="concept">
# Standard score

## Summary

The standard score of a value is the number of standard deviations by which it lies above or below the mean.
The node [[standard score|05RZ4YZG4E]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0VXYV3NQQ1.md" tag="0VXYV3NQQ1" kind="statement">
# Gradient is orthogonal to level sets

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $c \in \mathbb{R}$.
Let $I \subseteq \mathbb{R}$ be an open interval and let $\gamma \colon I \to U$ be a curve that lies in the [[level set|0Y5Q8Q9WJ7]] $L_c(f)$, so that $f(\gamma(t)) = c$ for each $t \in I$.
Let $t_0 \in I$, and suppose that $\gamma$ is differentiable at $t_0$ and $f$ is differentiable at $\gamma(t_0)$.
Then $\nabla f(\gamma(t_0)) \cdot \gamma'(t_0) = 0$.

## Remarks

The proof is one step: $f \circ \gamma$ is constant, so its derivative is $0$, and by the [[chain rule for a curve|0A8TJ0VHC7]] that derivative is $\nabla f(\gamma(t_0)) \cdot \gamma'(t_0)$.
So the [[gradient|0NS2QYVC4Y]] is orthogonal to each tangent vector of the level set at the point. Let $\nabla f \ne 0$ and let $f$ be of class $C^1$. Then the level set near the point is a smooth hypersurface, and $\nabla f$ is a normal vector to it.
For $f(x, y, z) = x^2 + y^2 + z^2$, the level set at $c = 1$ is the unit sphere, and $\nabla f(p) = 2p$ is normal to the sphere at $p$.

## Explanations

### L1 human

Along a contour line the height does not change, so the slope along the line is $0$.
The gradient gives the slope in each direction by a dot product, so it must be at right angles to the contour line.
</document>

<document source="https://tamnd.github.io/math2/node/0VY3WPP9Y3.md" tag="0VY3WPP9Y3" kind="statement">
# Bézout's identity in a Euclidean domain

## Statement

Let $R$ be a [[Euclidean domain|0GJ2HRG95T]] and let $a, b \in R$.
Then $a$ and $b$ have a [[greatest common divisor|0F7KNJ1ZYN]] $d$, and there are $s, t \in R$ with
$$d = s a + t b.$$

## Remarks

The [[Euclidean algorithm|023VY493K0]] computes $d$. The extended Euclidean algorithm also computes $s$ and $t$, by back substitution.
In $\mathbb{Z}$: $\gcd(252, 198) = 18 = 4 \cdot 252 - 5 \cdot 198$.
The same identity holds in every [[principal ideal domain|0RZYDVZ475]], because the ideal $(a, b)$ is principal.
If $d \neq 0$, the coefficients are not unique: $s + k b/d$ and $t - k a/d$ also work for each $k \in R$.

## Explanations

### L1 human

The gcd of two elements is a combination of them with coefficients in the ring.
For example, $\gcd(4, 6) = 2 = 2 \cdot 4 - 1 \cdot 6$.
This is the main tool to solve linear equations $s a + t b = c$ in a Euclidean domain.
</document>

<document source="https://tamnd.github.io/math2/node/0VYCDGMJJJ.md" tag="0VYCDGMJJJ" kind="concept">
# Row echelon form

## Summary

A matrix is in row echelon form when its zero rows are at the bottom.
Also, the first nonzero entry of each row is to the right of the first nonzero entry of the row above.
</document>

<document source="https://tamnd.github.io/math2/node/0VZ48N083F.md" tag="0VZ48N083F" kind="concept">
# Coprime integers

## Summary

Two integers are coprime when no integer other than $1$ and $-1$ divides both of them.
The [[definition of coprime integers|0GFKP8F8AG]] states this with the greatest common divisor.
</document>

<document source="https://tamnd.github.io/math2/node/0VZVV8VME5.md" tag="0VZVV8VME5" kind="statement">
# Compact closed unit ball implies finite dimension

## Statement

Let $X$ be a normed space over $\mathbb{R}$ or $\mathbb{C}$.
If the closed unit ball $\{ x \in X : \|x\| \le 1 \}$ is [[compact|00R0X1WGPX]], then $X$ is finite-dimensional.

## Remarks

Proof: if $X$ is infinite-dimensional, build unit vectors $x_1, x_2, \ldots$ by [[Riesz's lemma|097C7KCAT5]], with $x_{n+1}$ at distance at least $\tfrac{1}{2}$ from the span of $x_1, \ldots, x_n$. That span is finite-dimensional, so it is [[complete|0ABP2F2ZP7]] and closed. Then $\|x_m - x_n\| \ge \tfrac{1}{2}$ for $m \neq n$, so no subsequence converges, and the ball is not [[sequentially compact|0K8SGQSV7G]].
With the [[Heine–Borel theorem in finite dimension|0PHT6NBYPB]], we get: a normed space is finite-dimensional if and only if its closed unit ball is compact.
Equivalently, a normed space is locally compact if and only if it is finite-dimensional.

## Explanations

### L1 human

In infinite dimensions, the closed unit ball is never compact: there is always room for infinitely many unit vectors that stay far apart.
This is one of the main differences between finite and infinite dimensions.
</document>

<document source="https://tamnd.github.io/math2/node/0W02QESPJQ.md" tag="0W02QESPJQ" kind="concept">
# Orthogonal matrix

## Summary

An orthogonal matrix is a real square matrix whose transpose is its inverse, so that its columns form an orthonormal basis.
</document>

<document source="https://tamnd.github.io/math2/node/0W0KVBM54S.md" tag="0W0KVBM54S" kind="definition">
# Cyclotomic polynomial

## Definition

Let $n \ge 1$ be an integer.
The $n$-th cyclotomic polynomial is
$$\Phi_n(x) = \prod_{\substack{1 \le k \le n \\ \gcd(k, n) = 1}} \left(x - e^{2 \pi i k / n}\right) \in \mathbb{C}[x].$$
Its roots are the primitive $n$-th roots of unity: the complex numbers $\zeta$ with $\zeta^n = 1$ and $\zeta^m \neq 1$ for $0 < m < n$.

## Remarks

The polynomial $\Phi_n$ is [[monic|0BDX4MBKT4]] of degree $\varphi(n)$, where $\varphi$ is Euler's totient function.
For each $n \ge 1$, $x^n - 1 = \prod_{d \mid n} \Phi_d(x)$. By induction on $n$, this shows that $\Phi_n$ has integer coefficients.
The first values are $\Phi_1 = x - 1$, $\Phi_2 = x + 1$, $\Phi_3 = x^2 + x + 1$, $\Phi_4 = x^2 + 1$ and $\Phi_6 = x^2 - x + 1$.
For a prime $p$, $\Phi_p = x^{p-1} + x^{p-2} + \dots + x + 1$.

## Explanations

### L1 human

The $n$-th roots of unity are the corners of a regular $n$-gon on the unit circle.
The cyclotomic polynomial $\Phi_n$ collects only the corners that are not roots of unity of a smaller order.
The name means "circle dividing".
</document>

<document source="https://tamnd.github.io/math2/node/0W0TZDRF9C.md" tag="0W0TZDRF9C" kind="statement">
# Continuous functions on a closed bounded interval are uniformly continuous

## Statement

Let $a \le b$ and let $f \colon [a, b] \to \mathbb{R}$ be [[continuous|0P27W9RW0G]].
Then $f$ is [[uniformly continuous|0A3CK6DVVB]] on $[a, b]$.

## Remarks

Suppose not. Then there is $\varepsilon > 0$ and points $x_n, y_n$ with $|x_n - y_n| < 1/n$ and $|f(x_n) - f(y_n)| \ge \varepsilon$. A [[convergent subsequence|0KDJD249DT]] of $(x_n)$ leads to a contradiction with continuity.
The interval must be closed and bounded. The function $1/x$ is continuous on $(0, 1]$ but not uniformly continuous there.
The name honors Eduard Heine and Georg Cantor. Mathlib states it for compact sets, as `IsCompact.uniformContinuousOn_of_continuous`.

## Explanations

### L1 human

On a closed bounded interval, continuity at each point upgrades to uniform continuity.
There is no room for the function to become steeper and steeper.
</document>

<document source="https://tamnd.github.io/math2/node/0W1C598HMS.md" tag="0W1C598HMS" kind="statement">
# Euler's formula for planar graphs

## Statement

Let $G$ be a finite [[connected|0FFAYNMAJZ]] [[planar graph|029JQEK15B]], drawn in the plane with no crossing edges.
Let $v$, $e$ and $f$ be the numbers of vertices, edges and faces of the drawing, where the unbounded face counts.
Then $v - e + f = 2$.

## Remarks

Proof idea: if the graph has a cycle, delete an edge of the cycle. This removes one edge and one face.
At the end you have a [[spanning tree|0FQ5B8S1ZC]], with $e = v - 1$ and $f = 1$.
The same formula holds for the vertices, edges and faces of a convex polyhedron.

## Explanations

### L1 human

Draw a connected graph without crossings and count its vertices, edges and regions.
Vertices minus edges plus regions is always $2$. For a square, this is $4 - 4 + 2$.
</document>

<document source="https://tamnd.github.io/math2/node/0W1K48V5SG.md" tag="0W1K48V5SG" kind="concept">
# Rearrangement of a series

## Summary

A rearrangement of a series is the series of the same terms taken in a new order, given by a bijection of the indices. Rearrangements keep the sum of an absolutely convergent series. The node [[rearrangement of a series|0HRJ2Y74T3]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0W1QMKJ9W8.md" tag="0W1QMKJ9W8" kind="concept">
# Uniform convergence on a set

## Summary

Uniform convergence on a subset of the domain asks for uniform closeness only at the points of that subset. A sequence can converge uniformly on some subsets and not on others. The node [[uniform convergence on a set|08HJRTG5N7]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0W276SSSCG.md" tag="0W276SSSCG" kind="concept">
# Linear approximation

## Summary

A linear approximation of a function $f$ near a point $a$ replaces $f$ by an affine function. This function has the same value and the same derivative at $a$.
For a differentiable map $f$ from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$, it is $f(x) \approx f(a) + Df(a)(x - a)$, where $Df(a)$ is the [[total derivative|0FNKFT8H6E]].
The error is small compared with $\|x - a\|$ as $x \to a$.

## Remarks

For a function of one variable, the graph of the linear approximation is the tangent line. For a function of two variables, it is the [[tangent plane|035V802J58]].
For $f(x) = \sqrt{x}$ near $a = 4$, the approximation $\sqrt{x} \approx 2 + (x - 4)/4$ gives $\sqrt{4.1} \approx 2.025$, and the true value is $2.0248\ldots$.

## Explanations

### L1 human

Near a point, a smooth function looks like a straight line, or a flat plane in more variables.
So you can estimate values of the function near the point with simple linear arithmetic.
</document>

<document source="https://tamnd.github.io/math2/node/0W29TBTY12.md" tag="0W29TBTY12" kind="concept">
# Riemann integral

## Summary

The Riemann integral of a function on a closed bounded interval is the limit of its Riemann sums as the partitions get finer. For a nonnegative function, it is the area under the graph. The node [[Riemann integral|0QM4TXT3DR]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0W2QAVJQ1W.md" tag="0W2QAVJQ1W" kind="definition">
# Cofactor

## Definition

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$, with $n \ge 2$.
The *cofactor* of $A$ in position $(i, j)$ is
$$C_{ij} = (-1)^{i+j} M_{ij},$$
where $M_{ij}$ is a first [[minor|0BT9BYWVRY]] of $A$. It is the minor that results when row $i$ and column $j$ are deleted.

## Remarks

The signs $(-1)^{i+j}$ form a checkerboard pattern, with $+$ in the top left corner.
The cofactors give the [[Laplace expansion|0C08BQ3M05]] of the determinant.
The transpose of the matrix of cofactors is the [[adjugate matrix|07W9H8QF38]].

## Explanations

### L1 human

A cofactor is a minor with a sign.
The sign is $+$ or $-$, according to a checkerboard pattern.
For the matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the cofactors are $C_{11} = d$, $C_{12} = -c$, $C_{21} = -b$ and $C_{22} = a$.
</document>

<document source="https://tamnd.github.io/math2/node/0W3K3QPEWT.md" tag="0W3K3QPEWT" kind="definition">
# Absolute convergence

## Definition

Let $(a_n)$ be a real sequence.
The series $\sum a_n$ converges absolutely if the series $\sum |a_n|$ of absolute values [[converges|0CVVKY7GY2]].

## Remarks

The series $\sum |a_n|$ has nonnegative terms. So it converges exactly when its partial sums are bounded.
[[Absolutely convergent series converge|0Y123X48GA]], but the converse fails. The [[alternating harmonic series|0JQ5Y3WWVE]] converges, but not absolutely.
A convergent series that does not converge absolutely is [[conditionally convergent|02YRQXZDCS]].
The same definition works for complex terms, with the modulus in place of the absolute value.

## Explanations

### L1 human

A series converges absolutely if it still converges after we make all its terms nonnegative.
Then cancellation between positive and negative terms is not needed for convergence.
</document>

<document source="https://tamnd.github.io/math2/node/0W3W1VYWBP.md" tag="0W3W1VYWBP" kind="statement">
# Line integral as an integral over the parameter interval

## Statement

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a $C^1$ curve with image $C$.
Let $f \colon C \to \mathbb{R}$ be continuous, and let $F \colon C \to \mathbb{R}^n$ be a continuous vector field.
Then
$$\int_\gamma f \, ds = \int_a^b f(\gamma(t)) \, \|\gamma'(t)\| \, dt, \qquad \int_\gamma F \cdot d\mathbf{r} = \int_a^b F(\gamma(t)) \cdot \gamma'(t) \, dt.$$

## Remarks

The left sides are the [[line integral of a scalar field|0M50E4WYWJ]] and the [[line integral of a vector field|0YS8RX87F4]]. The right sides are ordinary integrals in one variable.
The formula follows from $\Delta s_i = \int_{t_{i-1}}^{t_i} \|\gamma'(t)\| \, dt$, the mean value theorem for each component of $\gamma$, and the uniform continuity of $f \circ \gamma$ and $\gamma'$.
The formulas extend to piecewise $C^1$ curves, by summing over the pieces.
Example: for $F(x, y) = (-y, x)$ and $\gamma(t) = (\cos t, \sin t)$, $t \in [0, 2\pi]$, the integrand is $\sin^2 t + \cos^2 t = 1$, so $\int_\gamma F \cdot d\mathbf{r} = 2\pi$.

## Explanations

### L1 human

To compute a line integral, write the curve with a parameter $t$.
Then $ds$ becomes $\|\gamma'(t)\| \, dt$ and $d\mathbf{r}$ becomes $\gamma'(t) \, dt$, and the line integral is an ordinary integral in $t$.
</document>

<document source="https://tamnd.github.io/math2/node/0W41XG4W60.md" tag="0W41XG4W60" kind="statement">
# Complete subspaces are closed

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $A \subseteq X$.
If $A$ with the [[subspace metric|0T2WYDTRP2]] is [[complete|08EWT7J0SC]], then $A$ is [[closed|0G5AV9Q4RR]] in $X$.

## Remarks

Proof: let $x \in \overline{A}$. By [[closure is the set of limits of sequences|003QP7A8HG]], there is a sequence in $A$ with limit $x$. It is a Cauchy sequence in $A$, so it has a limit in $A$. Limits are unique in $X$, so $x \in A$.
The space $X$ need not be complete here.
For example, $\mathbb{Q}$ is not closed in $\mathbb{R}$, so $\mathbb{Q}$ is not complete.
In a complete space, a subset is complete if and only if it is closed, by [[closed subsets of complete spaces are complete|0FBXJZX3MK]].

## Explanations

### L1 human

A complete subset contains the limits of its Cauchy sequences, and thus of its convergent sequences.
A set that keeps all its limits is closed.
</document>

<document source="https://tamnd.github.io/math2/node/0W44K2VNBD.md" tag="0W44K2VNBD" kind="definition">
# Sesquilinear form

## Definition

Let $V$ be a vector space over $\C$.
A *sesquilinear form* on $V$ is a map $\varphi \colon V \times V \to \C$ such that, for all $u, v, w \in V$ and $a \in \C$:

1. $\varphi(u + v, w) = \varphi(u, w) + \varphi(v, w)$ and $\varphi(au, w) = a\, \varphi(u, w)$.
2. $\varphi(u, v + w) = \varphi(u, v) + \varphi(u, w)$ and $\varphi(u, aw) = \bar a\, \varphi(u, w)$.

## Remarks

Here the form is linear in the first argument; many authors, especially in physics, use linearity in the second argument instead.
A sesquilinear form is *Hermitian* when $\varphi(v, u) = \overline{\varphi(u, v)}$ for all $u, v$.
On $\C^n$, each sesquilinear form is $\varphi(x, y) = y^* M x$ for one $n \times n$ complex matrix $M$.
A complex [[inner product|0NB0AWGJBW]] is a positive definite Hermitian sesquilinear form.

## Explanations

### L1 human

Over the complex numbers, bilinear forms cannot be positive, because $\varphi(iv, iv) = -\varphi(v, v)$.
Sesquilinear forms fix this by conjugating the scalar in one argument.
The standard inner product $\sum_i x_i \overline{y_i}$ on $\C^n$ is the main example.
</document>

<document source="https://tamnd.github.io/math2/node/0W4CN9E2W5.md" tag="0W4CN9E2W5" kind="concept">
# Convergent sequence

## Summary

A convergent sequence is a sequence that has a limit. Convergent sequences are bounded, and their limits obey the algebra of limits. The node [[convergent sequence|0AZ3R52593]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0W4FSW1TVB.md" tag="0W4FSW1TVB" kind="statement">
# Rank–nullity theorem

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a vector space over $F$.
Let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
Then
$$\dim \im T + \dim \ker T = \dim V,$$
that is, $\rank T + \operatorname{nullity} T = \dim V$, with the [[rank|0ZZXSBTR4J]] and the [[nullity|0APCFW7639]] of $T$.

## Remarks

For an $m \times n$ matrix $A$, this says $\rank A + \dim \{x : Ax = 0\} = n$.
A proof extends a basis of $\ker T$ to a basis of $V$; the images of the new vectors form a basis of $\im T$.
The theorem also follows from the [[first isomorphism theorem|0MTBGVZ4XJ]] $V / \ker T \cong \im T$.

## Explanations

### L1 human

Each dimension of the domain either survives in the image or is crushed into the kernel.
So the dimensions of image and kernel add up to the dimension of the domain.
A map from $\R^5$ to $\R^3$ has a kernel of dimension at least $2$.
</document>

<document source="https://tamnd.github.io/math2/node/0W4GQSG7B6.md" tag="0W4GQSG7B6" kind="statement">
# Irrationality of e

## Statement

The number [[$e$|0MVCFMW7S4]] is irrational.

## Remarks

Assume $e = a/b$ with positive integers $a, b$. Then $x = b! \big( e - \sum_{n=0}^{b} \frac{1}{n!} \big)$ is an integer.
But the [[series for $e$|0G4D3ZKTP8]] gives $0 < x < \sum_{k=1}^{\infty} (b + 1)^{-k} = \frac{1}{b} \le 1$. No integer lies strictly between $0$ and $1$.
This short proof is due to Joseph Fourier.

## History

Leonhard Euler wrote the first proof in 1737. It was published seven years later.

## Explanations

### L1 human

The series for $e$ converges so fast that $e$ cannot be a fraction.
If $e = a/b$, a suitable multiple of the tail of the series would be an integer between $0$ and $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0W4RPA8QH3.md" tag="0W4RPA8QH3" kind="definition">
# Disjoint sets

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The sets $A$ and $B$ are disjoint when $A \cap B = \varnothing$.
A family of sets $(A_i)_{i \in I}$ is pairwise disjoint when $A_i \cap A_j = \varnothing$ for all $i, j \in I$ with $i \neq j$.

## Remarks

Example: $\{1, 2\}$ and $\{3, 4\}$ are disjoint, but $\{1, 2\}$ and $\{2, 3\}$ are not.
The blocks of a [[partition|09CCE86A6D]] are pairwise disjoint.
For finite disjoint sets, the number of elements of $A \cup B$ is the sum of the numbers of elements of $A$ and $B$.

## Explanations

### L1 human

Disjoint sets do not overlap.
The even integers and the odd integers are disjoint, because no integer is both.
</document>

<document source="https://tamnd.github.io/math2/node/0W4X9G6263.md" tag="0W4X9G6263" kind="concept">
# Unique factorisation domain

## Summary

A unique factorisation domain is an integral domain in which every nonzero nonunit is a product of irreducible elements, in an essentially unique way.
The fundamental theorem of arithmetic says that $\mathbb{Z}$ is one.
The [[definition of a unique factorisation domain|06YMXJMXYS]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0W50MT1C6X.md" tag="0W50MT1C6X" kind="concept">
# Trail

## Summary

A trail is a walk in a graph that uses each edge at most once. Trails are the setting of Eulerian problems. The node [[trail|062MQGP8M5]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0W5DSDFQ4G.md" tag="0W5DSDFQ4G" kind="definition">
# Addition of natural numbers

## Definition

Let $\mathbb{N}$ be the set of [[natural numbers|0MWKJ44V5A]] and let $S$ be the [[successor function|0832XVR512]].
Addition is the function $+ \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, $(a, b) \mapsto a + b$, with these two properties for all $a$ and $b$ in $\mathbb{N}$:
$$a + 0 = a, \qquad a + S(b) = S(a + b).$$

## Remarks

For each $a$, these equations are a [[definition by recursion|0DSHAVH45F]] on $b$. Thus exactly one such function exists.
With $1 = S(0)$, the second equation gives $a + 1 = S(a)$.
For example, $2 + 2 = S(2 + 1) = S(S(2 + 0)) = S(S(2)) = 4$.

## Explanations

### L1 human

To add $b$ to $a$, you start at $a$ and count up $b$ steps.
The definition says this in two rules: adding $0$ changes nothing, and adding one more step gives the next number.
</document>

<document source="https://tamnd.github.io/math2/node/0W5NJQ8MKY.md" tag="0W5NJQ8MKY" kind="definition">
# Harmonic function

## Definition

Let $U \subseteq \mathbb{R}^n$ be open.
A function $u \colon U \to \mathbb{R}$ is harmonic on $U$ if $u$ is of [[class C^2|0TCBZMEYGN]] on $U$ and its [[Laplacian|0XQ2ZVN1PZ]] vanishes on $U$:
$$\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \dots + \frac{\partial^2 u}{\partial x_n^2} = 0.$$

## Remarks

Each affine function is harmonic. In the plane, $x^2 - y^2$, $x y$ and $e^x \sin y$ are harmonic, and so is $\ln(x^2 + y^2)$ on $\mathbb{R}^2 \setminus \{0\}$.
On $\mathbb{R}^3 \setminus \{0\}$, $1 / \|x\|$ is harmonic. It is the potential of a unit point charge at the origin.
The real and imaginary parts of a holomorphic function of $z = x + i y$ are harmonic functions of $(x, y)$.
Mean value property: let the closed ball of centre $a$ and radius $r$ lie in $U$. Then $u(a)$ is the average of $u$ on the sphere of radius $r$ about $a$, and also on the ball.

## Explanations

### L1 human

A harmonic function has no local bumps or dips: its value at each point is the average of its values around the point.
A stretched rubber sheet held at fixed heights on its rim has a harmonic height function inside.
</document>

<document source="https://tamnd.github.io/math2/node/0W62TKMW7V.md" tag="0W62TKMW7V" kind="statement">
# Law of total probability for an event and its complement

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A, B \in \mathcal{F}$ with $0 < P(B) < 1$. Let $B^c = \Omega \setminus B$.
Then
$$P(A) = P(A \mid B) \, P(B) + P(A \mid B^c) \, (1 - P(B)).$$

## Remarks

This is the [[law of total probability|0E5G5FMT2R]] for the partition $\{B, B^c\}$, together with $P(B^c) = 1 - P(B)$.
The formula is the denominator of [[Bayes' theorem|0NM6E9VW04]] in the case of two hypotheses.

## Explanations

### L1 human

Either $B$ happens or it does not.
So the chance of $A$ is its chance when $B$ happens times $P(B)$, plus its chance when $B$ does not happen times $1 - P(B)$.
</document>

<document source="https://tamnd.github.io/math2/node/0W6M3PSD08.md" tag="0W6M3PSD08" kind="definition">
# Dual basis

## Definition

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.
Let $B = (b_1, \dots, b_n)$ be a [[basis|0KCQ9Y3QKA]] of $V$.
The *dual basis* of $B$ is the family $(b_1^*, \dots, b_n^*)$ of elements of the [[dual space|07JBSZE0V4]] $V^*$. These elements are defined by
$$b_i^*(b_j) = \delta_{ij} \qquad (1 \le i, j \le n),$$
where $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ if $i \ne j$.

## Remarks

The dual basis is a basis of $V^*$.
The form $b_i^*$ returns coordinate $i$: $b_i^*(a_1 b_1 + \cdots + a_n b_n) = a_i$.
Each $f \in V^*$ is $f = \sum_i f(b_i)\, b_i^*$.
For the standard basis of $F^n$, $e_i^*$ is the map $x \mapsto x_i$.

## Explanations

### L1 human

Each vector of the dual basis reads off one coordinate.
The form $b_1^*$ gives the first coordinate in the basis $B$, the form $b_2^*$ gives the second, and so on.
The condition $b_i^*(b_j) = \delta_{ij}$ says exactly this.
</document>

<document source="https://tamnd.github.io/math2/node/0W766JAPSN.md" tag="0W766JAPSN" kind="definition">
# Inverse function

## Definition

Let $f \colon A \to B$ be a [[bijective function|0SNQEF7Z53]].
The inverse function of $f$ is the function $f^{-1} \colon B \to A$ such that, for each $b \in B$, $f^{-1}(b)$ is the unique $a \in A$ with $f(a) = b$.

## Remarks

Then $f^{-1} \circ f = \mathrm{id}_A$ and $f \circ f^{-1} = \mathrm{id}_B$. See [[bijective iff it has a two-sided inverse|01VG2A0BHW]].
The inverse $f^{-1}$ is a bijection, and its inverse is $f$.
Example: the inverse of $x \mapsto 2x + 1$ on $\mathbb{R}$ is $y \mapsto (y - 1)/2$.
Do not confuse $f^{-1}(b)$ with the [[preimage|0FGCFY1AZX]] $f^{-1}(\{b\})$, which exists for every function.

## Explanations

### L1 human

To find the inverse of $y = 2x + 1$, solve for $x$: $x = (y - 1)/2$.
The inverse runs the function backwards.
Its graph is the graph of $f$ reflected in the line $y = x$.
</document>

<document source="https://tamnd.github.io/math2/node/0W77CVBCHQ.md" tag="0W77CVBCHQ" kind="definition">
# Normal subgroup

## Definition

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[subgroup|0X1JXT47GA]] of $G$.
The subgroup $N$ is normal in $G$ when $gng^{-1} \in N$ for all $g \in G$ and all $n \in N$.
The notation is $N \trianglelefteq G$.

## Remarks

Equivalently, $gNg^{-1} = N$ for each $g \in G$, that is, $N$ equals each of its [[conjugates|0H11ZPX4ND]].
Each group $G$ has the normal subgroups $\{e\}$ and $G$.
In the symmetric group $S_3$, the subgroup $\{e, (1\,2\,3), (1\,3\,2)\}$ is normal, but the subgroup $\{e, (1\,2)\}$ is not.

## Explanations

### L1 human

A normal subgroup does not move when you conjugate it by any element of the group.
This is the condition that lets you divide the group by the subgroup and get a new group.
In an abelian group, every subgroup is normal.
</document>

<document source="https://tamnd.github.io/math2/node/0W7FBYS1F1.md" tag="0W7FBYS1F1" kind="concept">
# Uncountable set

## Summary

An uncountable set is a set that is not countable.
It is too large to be listed in a sequence.
</document>

<document source="https://tamnd.github.io/math2/node/0W7NFW23RF.md" tag="0W7NFW23RF" kind="definition">
# Sample space

## Definition

Let $E$ be a [[random experiment|0HM05N3FJD]].
A sample space of $E$ is a nonempty [[set|0SJ3RX9TXA]] $\Omega$ with two properties.
Each performance of $E$ gives exactly one element of $\Omega$.
Each element of $\Omega$ is a possible result of $E$.
The usual notation for a sample space is $\Omega$ or $S$.

## Remarks

The elements of $\Omega$ are mutually exclusive: two different elements cannot both be the result of one trial. They are also collectively exhaustive: some element is always the result.
For one toss of a coin, $\Omega = \{H, T\}$. For two rolls of a die, $\Omega = \{1, \dots, 6\}^2$ has $36$ elements.
One experiment can have more than one sample space. For a card drawn from a deck, $\Omega$ can be the set of $52$ cards or the set of the $4$ suits.
A sample space can be finite, countably infinite or uncountable. The time until a lamp fails has the sample space $[0, \infty)$.

## Explanations

### L1 human

The sample space is the list of everything that can happen in the experiment.
For a die, the list is $1, 2, 3, 4, 5, 6$. Every question about the die is then a question about this list.
</document>

<document source="https://tamnd.github.io/math2/node/0W809K6AW0.md" tag="0W809K6AW0" kind="definition">
# Transpose

## Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$.
The *transpose* of $A$ is the $n \times m$ matrix $A^T$ with entries
$$(A^T)_{ij} = a_{ji}$$
for $1 \le i \le n$ and $1 \le j \le m$.

## Remarks

The transpose of $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ is $\begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$.
The rules $(A^T)^T = A$, $(A + B)^T = A^T + B^T$ and $(cA)^T = c A^T$ hold.
Over a commutative ring, $(AB)^T = B^T A^T$, see [[transpose of a product|09GK5PS824]].
The [[matrix of the dual map is the transpose|0ZPB3180AY]] of the matrix of the map.

## Explanations

### L1 human

To transpose a matrix, flip it over its main diagonal.
Row $1$ becomes column $1$, row $2$ becomes column $2$, and so on.
A $2 \times 3$ matrix becomes a $3 \times 2$ matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0W88CEGQP1.md" tag="0W88CEGQP1" kind="concept">
# Kernel of a homomorphism

## Summary

The kernel of a group homomorphism is the set of elements that it sends to the identity.
It is a normal subgroup, and it measures how far the homomorphism is from being injective.
</document>

<document source="https://tamnd.github.io/math2/node/0W8ETY2AZ6.md" tag="0W8ETY2AZ6" kind="concept">
# Tangent half-angle substitution

## Summary

The tangent half-angle substitution sets $t = \tan(x/2)$ in an integral. It turns each rational expression in $\sin x$ and $\cos x$ into a rational function of $t$. Partial fractions then finish the computation.

## Remarks

The substitution gives $\sin x = \frac{2t}{1 + t^2}$, $\cos x = \frac{1 - t^2}{1 + t^2}$ and $dx = \frac{2 \, dt}{1 + t^2}$.
It works on intervals where $x/2$ stays inside $(-\pi/2, \pi/2)$.
For example, it gives $\int \frac{dx}{\sin x} = \ln |\tan(x/2)| + C$.

## Explanations

### L1 human

This substitution turns trigonometric integrals into integrals of fractions of polynomials.
Those integrals are always solvable by partial fractions.
</document>

<document source="https://tamnd.github.io/math2/node/0W8QHVSN95.md" tag="0W8QHVSN95" kind="concept">
# Branch cut

## Summary

A branch cut is a curve that is removed from the plane so that a multivalued function has a continuous single-valued branch on the rest. See the [[definition|06JYC06JHY]].
The usual branch cut for the logarithm is the negative real axis.
</document>

<document source="https://tamnd.github.io/math2/node/0W8YHHGCJG.md" tag="0W8YHHGCJG" kind="statement">
# Epsilon-delta definition of continuity

## Statement

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$.
Then $f$ is [[continuous at|0EEE2CZB2A]] $a$ if and only if the following holds.
For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - f(a)| < \varepsilon$ for all $x \in D$ with $|x - a| < \delta$.

## Remarks

Each open interval around a point contains an interval of the form $(c - r, c + r)$ with $r > 0$. This gives the equivalence.
Many texts take this condition as the definition of continuity.
Mathlib states a form for metric spaces as `Metric.continuousAt_iff`.

## Explanations

### L1 human

For each allowed error $\varepsilon$ in the output, there is a tolerance $\delta$ in the input.
Inputs within $\delta$ of $a$ give outputs within $\varepsilon$ of $f(a)$.
</document>

<document source="https://tamnd.github.io/math2/node/0W93FJAC8F.md" tag="0W93FJAC8F" kind="concept">
# Fermat pseudoprime

## Summary

A Fermat pseudoprime to a base $a$ is a composite number $n$ that satisfies the congruence $a^{n-1} \equiv 1 \pmod n$ of Fermat's little theorem.
The [[definition of a Fermat pseudoprime|02M2K4QH9P]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0W95W33CMP.md" tag="0W95W33CMP" kind="concept">
# Equivalence class

## Summary

The equivalence class of an element is the set of all elements that are equivalent to it.
</document>

<document source="https://tamnd.github.io/math2/node/0W97J8QED7.md" tag="0W97J8QED7" kind="statement">
# Chinese remainder theorem for rings

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $I_1, \dots, I_n$ be ideals of $R$, with $n \ge 1$.
Suppose that $I_j$ and $I_k$ are [[coprime|0MDJP82TEH]] for all $j \neq k$.
Then the map
$$R/(I_1 \cap \dots \cap I_n) \to R/I_1 \times \dots \times R/I_n, \qquad a + (I_1 \cap \dots \cap I_n) \mapsto (a + I_1, \dots, a + I_n)$$
is a [[ring isomorphism|0B6QC0C0G1]].

## Remarks

Under the same hypothesis, $I_1 \cap \dots \cap I_n = I_1 I_2 \cdots I_n$.
For $R = \mathbb{Z}$ and pairwise coprime integers $m_1, \dots, m_n$, the theorem gives $\mathbb{Z}/(m_1 \cdots m_n)\mathbb{Z} \cong \mathbb{Z}/m_1\mathbb{Z} \times \dots \times \mathbb{Z}/m_n\mathbb{Z}$.
For example, $\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$.

## Explanations

### L1 human

For integers, the theorem says that a number modulo $15$ is the same information as the number modulo $3$ and modulo $5$.
For each pair of remainders, exactly one class modulo $15$ has them.
The ring version says the same for any family of pairwise coprime ideals.
</document>

<document source="https://tamnd.github.io/math2/node/0WA7ZR95V4.md" tag="0WA7ZR95V4" kind="statement">
# Time translation of solutions of an autonomous system

## Statement

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be a function. Let $\gamma : I \to U$ be a solution of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ on an interval $I$.
Let $c \in \mathbb{R}$ and let $I - c = \{t - c : t \in I\}$.
Then the function $\delta : I - c \to U$ given by $\delta(t) = \gamma(t + c)$ is also a solution of $x' = f(x)$.

## Remarks

Proof: by the chain rule, $\delta'(t) = \gamma'(t + c) = f(\gamma(t + c)) = f(\delta(t))$.
So it is enough to solve an initial value problem with initial time $0$. The solution with $x(t_0) = x_0$ is $t \mapsto \varphi(t - t_0)$, where $\varphi$ solves $x(0) = x_0$.
The result fails for non-autonomous equations. For $x' = t$, the solution $t^2/2$ is a solution, but $(t+1)^2/2$ is not.

## Explanations

### L1 human

If the law of motion does not change with time, a motion that starts later is the same motion, only delayed.
So shifting a solution in time gives another solution.
</document>

<document source="https://tamnd.github.io/math2/node/0WAR4F23B4.md" tag="0WAR4F23B4" kind="concept">
# Double integral

## Summary

The double integral of a function of two variables over a region of the plane is a limit of Riemann sums. The partitions of the region into small rectangles become finer and finer.
For a non-negative function it is the volume of the solid between the region and the graph.
</document>

<document source="https://tamnd.github.io/math2/node/0WAY5WX8NY.md" tag="0WAY5WX8NY" kind="concept">
# Congruence modulo n

## Summary

Two integers are congruent modulo a positive integer $n$ when their difference is a multiple of $n$.
Modular arithmetic is the arithmetic of integers up to this relation.
The [[definition of congruence modulo n|0TPW25CBN6]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0WB57D9488.md" tag="0WB57D9488" kind="definition">
# Sine and cosine

## Definition

The sine and cosine functions $\sin, \cos \colon \mathbb{R} \to \mathbb{R}$ are defined by the power series
$$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}, \qquad \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}.$$

## Remarks

Both [[series|02KTRHQMHN]] [[converge|08W5RN375X]] absolutely for every real $x$.
Sine is odd and cosine is even. Also $\sin 0 = 0$ and $\cos 0 = 1$.
Geometrically, $(\cos x, \sin x)$ is the point of the unit circle at arc length $x$ from $(1, 0)$, counterclockwise.
Mathlib defines `Real.sin` and `Real.cos` from the complex exponential, with $\cos x + i \sin x = e^{i x}$.

## Explanations

### L1 human

Sine and cosine describe motion around the unit circle.
In analysis, we define them by power series, and we then prove their geometric properties.
</document>

<document source="https://tamnd.github.io/math2/node/0WB619CA0V.md" tag="0WB619CA0V" kind="concept">
# Orthocentre

## Summary

The orthocentre of a triangle is the point where the three lines that contain its altitudes meet. It is inside the triangle exactly when all angles are acute.
</document>

<document source="https://tamnd.github.io/math2/node/0WB6DH499T.md" tag="0WB6DH499T" kind="concept">
# Path-connected space

## Summary

A topological space is path-connected when any two of its points can be joined by a continuous path.
Path-connected spaces are connected, but the converse is false.
</document>

<document source="https://tamnd.github.io/math2/node/0WBFFC8XZ9.md" tag="0WBFFC8XZ9" kind="concept">
# Subgraph

## Summary

A subgraph of a graph is a graph formed by some of its vertices and some of its edges. Subgraphs are how one speaks of a structure, such as a cycle or a tree, inside a graph. The node [[subgraph|0P11X5HKAX]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0WBRJXXDPZ.md" tag="0WBRJXXDPZ" kind="concept">
# Norm on the Gaussian integers

## Summary

The norm of a Gaussian integer $a + b i$ is $a^2 + b^2$, the square of its absolute value.
The [[definition of the norm|03AN1V4ZFJ]] gives its properties.
</document>

<document source="https://tamnd.github.io/math2/node/0WBTWFXJQG.md" tag="0WBTWFXJQG" kind="concept">
# Integration by partial fractions

## Summary

Integration by partial fractions is a method to integrate [[rational functions|0WCZH4AFMT]]. We write the rational function as a polynomial plus a sum of simple fractions. Each simple fraction has a known antiderivative.

## Remarks

Over $\mathbb{R}$, the simple fractions have the forms $\frac{A}{(x - r)^k}$ and $\frac{B x + C}{(x^2 + p x + q)^k}$, with $p^2 < 4 q$.
Their antiderivatives use powers, the [[logarithm|0K1N1Y4EGZ]], and the [[arctangent|065XW2V6C3]].
For example, $\frac{1}{x^2 - 1} = \frac{1}{2} \Big( \frac{1}{x - 1} - \frac{1}{x + 1} \Big)$.
So every rational function has an elementary antiderivative.

## Explanations

### L1 human

A complicated fraction of polynomials splits into a sum of simple fractions.
We can integrate each simple fraction by a standard formula.
</document>

<document source="https://tamnd.github.io/math2/node/0WC28W9DNM.md" tag="0WC28W9DNM" kind="object">
# Cardinality of the continuum

## Description

The cardinality of the continuum, written $\mathfrak{c}$, is the [[cardinal number|0QKPNZKNEA]] $2^{\aleph_0}$.
It is the cardinality of the [[power set|0BC7KX51Z8]] of $\mathbb{N}$, and of the set of all sequences of $0$ and $1$.
It is also the cardinality of $\mathbb{R}$. See [[the reals have the cardinality of the continuum|075PM9FH7A]].

## Remarks

By [[Cantor's theorem|0WTEMGYV2C]], $\mathfrak{c} > \aleph_0$.
Other sets of cardinality $\mathfrak{c}$ are each nonempty open interval of $\mathbb{R}$, the plane $\mathbb{R}^2$ and the set of irrational numbers.
The [[continuum hypothesis|0MN2MER9WN]] says that $\mathfrak{c} = \aleph_1$.

## Explanations

### L1 human

$\mathfrak{c}$ is the size of the real line.
It is a larger infinity than $\aleph_0$.
Surprisingly, a short interval, a whole line and a whole plane all have this same size.
</document>

<document source="https://tamnd.github.io/math2/node/0WCKRBHFFD.md" tag="0WCKRBHFFD" kind="definition">
# Clique number

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]].
The clique number $\omega(G)$ is the largest number of vertices of a [[clique|0G52BPNPKG]] in $G$.

## Remarks

The vertices of a clique need different colours. Thus $\omega(G) \le \chi(G)$, where $\chi(G)$ is the [[chromatic number|0C5M7AM6FH]].
For the [[complete graph|03WKB2H474]], $\omega(K_n) = n$. For the [[cycle graph|0QQ0F12962]] $C_n$ with $n \ge 4$, $\omega(C_n) = 2$.
The inequality $\omega(G) \le \chi(G)$ can be strict: for $C_5$, $\omega = 2$ and $\chi = 3$.

## Explanations

### L1 human

The clique number is the size of the largest group of vertices that are all joined to each other.
It gives a lower bound for the number of colours that a graph needs.
</document>

<document source="https://tamnd.github.io/math2/node/0WCP0PVET6.md" tag="0WCP0PVET6" kind="concept">
# Proof by cases

## Summary

A proof by cases splits the situation into a finite number of cases that cover all possibilities.
Then it proves the statement in each case.
The basic form is this rule: from $p \lor q$, $p \to r$ and $q \to r$, conclude $r$.

## Remarks

Another name is proof by exhaustion, mainly when the cases are many.
Example: $n^2 + n$ is even for each integer $n$. If $n$ is even, $n^2$ and $n$ are even. If $n$ is odd, $n^2$ and $n$ are odd. In both cases, the sum is even.
A split into $p$ and $\neg p$ uses the [[law of excluded middle|0RYYK8PG4S]].

## Explanations

### L1 human

When one argument does not work for all cases, split the cases.
Integers are even or odd, so a proof for even integers and a proof for odd integers together cover all integers.
Make sure that no case is missing.
</document>

<document source="https://tamnd.github.io/math2/node/0WCPH5Z7FA.md" tag="0WCPH5Z7FA" kind="statement">
# Trace of a linear map is the trace of its matrix

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$ with a basis $B$, and let $T \colon V \to V$ be a linear map.
Then the [[trace of $T$|0KZDNGA561]] equals the trace of the [[matrix of $T$|0YQSGNYE3K]] in $B$:
$$\tr T = \tr [T]_B.$$

## Remarks

The key fact is that similar matrices have the same trace, by $\tr(AB) = \tr(BA)$.
With a basis-free definition, the trace is the map $\End_F(V) \cong V^* \otimes V \to F$ induced by evaluation $f \otimes v \mapsto f(v)$.

## Explanations

### L1 human

We can compute the trace of a linear map in any basis that we like.
We write the matrix in that basis and add its diagonal entries.
Every basis gives the same number.
</document>

<document source="https://tamnd.github.io/math2/node/0WCWA1KT3M.md" tag="0WCWA1KT3M" kind="concept">
# Binomial distribution

## Summary

The binomial distribution with parameters $n$ and $p$ is the distribution of the number of successes in $n$ independent Bernoulli trials.
Each trial has the success probability $p$.
The node [[binomial distribution|0VRWZDWW6B]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0WCX7NB4N7.md" tag="0WCX7NB4N7" kind="concept">
# Separable extension

## Summary

A separable extension is an algebraic extension in which the minimal polynomial of every element has no repeated roots.
The [[definition of a separable extension|0N019F6RE9]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0WCY42EV9V.md" tag="0WCY42EV9V" kind="concept">
# Group action

## Summary

An action of a group $G$ on a set $X$ lets each element of $G$ move the points of $X$. The moves must respect the group operation.
Equivalently, it is a group homomorphism from $G$ to the symmetric group of $X$.
</document>

<document source="https://tamnd.github.io/math2/node/0WCZH4AFMT.md" tag="0WCZH4AFMT" kind="definition">
# Rational function

## Definition

Let $p$ and $q$ be real polynomials, with $q$ not the zero polynomial.
Let $D = \{ x \in \mathbb{R} : q(x) \ne 0 \}$.
The function $f \colon D \to \mathbb{R}$ given by $f(x) = p(x)/q(x)$ is a rational function.

## Remarks

A nonzero polynomial has only finitely many roots, so $D$ is $\mathbb{R}$ minus a finite set.
Each rational function is continuous on $D$, since [[polynomials are continuous|0C5TP6RYSV]] and [[quotients of continuous functions are continuous|0GR2APWTAC]].
For example, $1/x$ and $(x^2 + 1)/(x - 2)$ are rational functions.

## Explanations

### L1 human

A rational function is a quotient of two polynomials.
It is defined everywhere except at the zeros of the denominator.
</document>

<document source="https://tamnd.github.io/math2/node/0WD3SE7S1X.md" tag="0WD3SE7S1X" kind="statement">
# R/M is a field iff M is maximal

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $M$ be an ideal of $R$.
Then the [[quotient ring|0VFW4Y69BT]] $R/M$ is a [[field|0QA8CHH8MG]] if and only if $M$ is a [[maximal ideal|09J6KGT65G]].

## Remarks

The proof uses the [[correspondence theorem|0D1R4SVMBQ]]: the ideals of $R/M$ are the images of the ideals of $R$ that contain $M$.
So $M$ is maximal if and only if $R/M$ is nonzero and has only the two ideals $\{0\}$ and $R/M$.
A nonzero commutative ring with only these two ideals is a field, by the [[converse for fields|0M6G75E95E]].
For example, $\mathbb{Z}/p\mathbb{Z}$ is a field because $p\mathbb{Z}$ is maximal for $p$ prime.

## Explanations

### L1 human

The quotient by an ideal is a field exactly when the ideal is as large as possible.
In the integers, $(5)$ is maximal, and $\mathbb{Z}/5\mathbb{Z}$ is a field.
The ideal $(6)$ is not maximal, and $\mathbb{Z}/6\mathbb{Z}$ is not a field.
</document>

<document source="https://tamnd.github.io/math2/node/0WDAKKN4NC.md" tag="0WDAKKN4NC" kind="counterexample">
# x^2 sin(1/x) is differentiable with discontinuous derivative

## Counterexample

Let $f(x) = x^2 \sin(1/x)$ for $x \ne 0$, and let $f(0) = 0$.
Then $f$ is [[differentiable|0A61B60B5R]] on $\mathbb{R}$, but $f'$ is not continuous at $0$.
At $0$, the difference quotient is $h \sin(1/h)$, which tends to $0$. So $f'(0) = 0$.
For $x \ne 0$, $f'(x) = 2 x \sin(1/x) - \cos(1/x)$. This has no limit as $x \to 0$, because $\cos(1/x)$ has none.

## Remarks

So $f$ is differentiable but not [[of class $C^1$|06SKCHDQT5]].
The discontinuity of $f'$ at $0$ is essential. By [[Darboux's theorem|0Q5A47J48D]], a derivative can have no jump discontinuities.

## Explanations

### L1 human

The factor $x^2$ flattens the oscillation of $\sin(1/x)$ enough to give a slope $0$ at the origin.
But near the origin the slopes keep swinging between about $-1$ and $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0WDNRBXX7K.md" tag="0WDNRBXX7K" kind="definition">
# Cauchy–Euler equation

## Definition

Let $n \ge 1$ and let $a_0, \dots, a_n$ be real numbers with $a_n \ne 0$.
The Cauchy–Euler equation with these coefficients is the [[linear differential equation|0PB7K8M7FG]]
$$a_n x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \dots + a_1 x\, y' + a_0\, y = 0$$
for an unknown function $y$ on the interval $x > 0$.

## Remarks

The substitution $x = e^{t}$, $u(t) = y(e^t)$ changes it into a [[linear equation with constant coefficients|0A6PGTTP2A]] for $u$.
For $n = 2$, the trial solution $y = x^m$ gives the indicial equation $a_2 m (m - 1) + a_1 m + a_0 = 0$.
For order $2$: a double root $m$ gives the solutions $x^m$ and $x^m \ln x$. Roots $\alpha \pm i \beta$ with $\beta \ne 0$ give $x^\alpha \cos(\beta \ln x)$ and $x^\alpha \sin(\beta \ln x)$.
Example: $x^2 y'' - 2 y = 0$ gives $m(m-1) - 2 = 0$, so $m = 2$ or $m = -1$, and the solutions are $c_1 x^2 + c_2 x^{-1}$.

## Explanations

### L1 human

In this equation, each derivative $y^{(k)}$ is multiplied by $x^k$.
So powers $x^m$ behave like exponentials do for constant coefficients: the equation becomes a polynomial equation in $m$.
</document>

<document source="https://tamnd.github.io/math2/node/0WE2NHZJ7B.md" tag="0WE2NHZJ7B" kind="concept">
# Order of an element

## Summary

The order of an element of a group is the smallest positive integer $n$ such that the $n$th power of the element is the identity.
When no such $n$ exists, the element has infinite order.
</document>

<document source="https://tamnd.github.io/math2/node/0WE5BBHGM6.md" tag="0WE5BBHGM6" kind="definition">
# Totally bounded set

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $S \subseteq X$.
The set $S$ is totally bounded when for each real $\varepsilon > 0$ there are finitely many sets $A_1, \ldots, A_n \subseteq X$ with $S \subseteq A_1 \cup \cdots \cup A_n$ and $\operatorname{diam}(A_k) < \varepsilon$ for each $k$.
The space $X$ is totally bounded when $X$ itself is a totally bounded set.

## Remarks

Here $\operatorname{diam}$ is the [[diameter|0692CGQF00]].
Each totally bounded set is [[bounded|0P953MG05E]]. The converse is false: an infinite set with the [[discrete metric|0NA4FK7B8S]] is bounded but not totally bounded.
In $\mathbb{R}^n$ with the Euclidean metric, a set is totally bounded if and only if it is bounded.
The [[epsilon-net characterisation|0NMGJJ19HD]] gives an equivalent form with balls.

## Explanations

### L1 human

A totally bounded set can be cut into finitely many small pieces, for each size of piece that you choose.
A bounded set only fits in one big ball, and this is not enough in infinite dimensions.
</document>

<document source="https://tamnd.github.io/math2/node/0WETJCKNRG.md" tag="0WETJCKNRG" kind="statement">
# Euclid–Euler theorem

## Statement

Let $n$ be an even positive integer.
Then $n$ is a [[perfect number|0SB7Q8SYSF]] if and only if $n = 2^{p-1} (2^p - 1)$, where $2^p - 1$ is a [[Mersenne prime|06GPJF9V7M]].

## Remarks

The primes $2^p - 1$ for $p = 2, 3, 5, 7$ give the perfect numbers $6$, $28$, $496$ and $8128$.
So the even perfect numbers correspond one to one to the Mersenne primes.
The proof uses the multiplicativity of the [[sum of divisors|009QCP5T3Z]]: $\sigma(2^{p-1} q) = (2^p - 1)(q + 1)$ for an odd prime $q$.

## History

Euclid proved in the Elements that $2^{p-1}(2^p - 1)$ is perfect when $2^p - 1$ is prime.
Ibn al-Haytham conjectured the converse around 1000 CE.
Euler proved the converse in the 18th century, in a paper published after his death.

## Explanations

### L1 human

Euclid found a recipe for perfect numbers: take a prime of the form $2^p - 1$ and multiply it by $2^{p-1}$.
Euler showed that the recipe gives every even perfect number.
Whether an odd perfect number exists is still unknown.
</document>

<document source="https://tamnd.github.io/math2/node/0WF4EJP91W.md" tag="0WF4EJP91W" kind="definition">
# Probability mass function

## Definition

Let $X$ be a [[discrete random variable|0J6Z6H5VQJ]] on a probability space $(\Omega, \mathcal{F}, P)$.
The probability mass function of $X$ is the function $p_X \colon \mathbb{R} \to [0, 1]$ with
$$p_X(x) = P(X = x) \quad \text{for } x \in \mathbb{R}.$$
More generally, a probability mass function on a countable set $S$ is a function $p \colon S \to [0, 1]$ with $\sum_{s \in S} p(s) = 1$.

## Remarks

The set $\{x : p_X(x) > 0\}$ is countable, and $\sum_x p_X(x) = 1$.
For each Borel set $B$, $P(X \in B) = \sum_{x \in B} p_X(x)$.
For a fair die, $p_X(k) = 1/6$ for $k = 1, \dots, 6$ and $p_X(x) = 0$ for other $x$.

## Explanations

### L1 human

The probability mass function is a table: each possible value with its probability.
The entries are never negative and they add up to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0WF5AQYTW6.md" tag="0WF5AQYTW6" kind="concept">
# Reduced row echelon form

## Summary

A matrix is in reduced row echelon form when it is in row echelon form and each leading entry is $1$.
Also, each leading entry is the only nonzero entry in its column.
</document>

<document source="https://tamnd.github.io/math2/node/0WFBH1Q3WC.md" tag="0WFBH1Q3WC" kind="concept">
# Local maximum and minimum

## Summary

A [[function of several real variables|0JPCHTXWG0]] has a local maximum at a point when no nearby point of the domain gives a larger value.
It has a local minimum when no nearby point gives a smaller value, and a local extremum is either of the two.
</document>

<document source="https://tamnd.github.io/math2/node/0WFCGVMRRC.md" tag="0WFCGVMRRC" kind="definition">
# Factorial

## Definition

Let $n$ be a natural number.
The factorial of $n$ is the natural number $n!$ defined by the recursion
$$0! = 1, \qquad (n+1)! = (n+1) \cdot n!.$$
For $n \ge 1$ this gives $n! = 1 \cdot 2 \cdots n$, the product of the positive integers up to $n$.

## Remarks

The first values are $0! = 1$, $1! = 1$, $2! = 2$, $3! = 6$, $4! = 24$ and $5! = 120$.
The value $n!$ is the [[number of permutations|0CEBCS0E0E]] of an $n$-element set.
The convention $0! = 1$ agrees with the empty product, and it makes the [[factorial formula for binomial coefficients|033KTKKRV6]] true for $k = 0$ and $k = n$.

## Explanations

### L1 human

The factorial $n!$ multiplies all whole numbers from $1$ to $n$.
It counts the ways to put $n$ different books in a row: $n$ choices for the first place, $n - 1$ for the second, and so on.
</document>

<document source="https://tamnd.github.io/math2/node/0WFCNG0MZC.md" tag="0WFCNG0MZC" kind="concept">
# Ordered pair

## Summary

An ordered pair is a pair of objects in which the order matters.
Two ordered pairs are equal exactly when their first entries are equal and their second entries are equal.
</document>

<document source="https://tamnd.github.io/math2/node/0WFED3K3DQ.md" tag="0WFED3K3DQ" kind="statement">
# Sylvester's law of inertia

## Statement

Let $A$ be a real symmetric $n \times n$ matrix.
Let $P$ and $P'$ be invertible real $n \times n$ matrices such that $D = P^T A P$ and $D' = P'^T A P'$ are diagonal.
Then $D$ and $D'$ have the same number of positive diagonal entries and the same number of negative diagonal entries.
They also have the same number of zero diagonal entries.

## Remarks

So these three numbers are invariants of the quadratic form $x \mapsto x^T A x$; they give its [[signature|07EZA80NEH]].
They are also the numbers of positive, negative and zero eigenvalues of $A$, counted with multiplicity.
Equivalently, two real symmetric matrices are congruent if and only if they have the same numbers of positive, negative and zero eigenvalues.

## History

James Joseph Sylvester published the theorem in 1852.

## Explanations

### L1 human

A real quadratic form can be written as a sum of squares in many ways.
The numbers of plus signs and minus signs are always the same.
For example, $x^2 + 2xy$ always has one plus sign and one minus sign.
</document>

<document source="https://tamnd.github.io/math2/node/0WFF6AZPBP.md" tag="0WFF6AZPBP" kind="definition">
# Monotone function

## Definition

Let $(P, \le_P)$ and $(Q, \le_Q)$ be [[preorders|0RXS8R6K37]] and let $f \colon P \to Q$ be a [[function|0XHDTE5YMZ]].
The function $f$ is monotone when, for all $a, b \in P$, $a \le_P b$ implies $f(a) \le_Q f(b)$.

## Remarks

Other names are order-preserving and, for real functions, monotonically increasing or non-decreasing.
The function $f$ is antitone when $a \le b$ implies $f(b) \le f(a)$.
The map $S \mapsto f(S)$ on [[power sets|0BC7KX51Z8]] is monotone for $\subseteq$.

## Explanations

### L1 human

A monotone function never reverses the order of two inputs.
On $\mathbb{R}$, $x \mapsto x^3$ is monotone, and $x \mapsto x^2$ is not, because $-2 \le 1$ but $4 > 1$.
</document>

<document source="https://tamnd.github.io/math2/node/0WFMKK7DFZ.md" tag="0WFMKK7DFZ" kind="problem">
# Gambler's ruin

## Problem

A gambler starts with $k$ units of money, where $0 \le k \le N$ for an integer $N \ge 1$. In each round, the gambler wins one unit with probability $p \in (0, 1)$ and loses one unit with probability $q = 1 - p$. The rounds are independent.
The game stops when the fortune reaches $0$ (ruin) or $N$ (the target).
Find the probability $P_k$ that the gambler reaches $N$ before $0$.

## Remarks

The fortune is a [[random walk|0C08YMV95H]] on $\{0, 1, \dots, N\}$ that stops at $0$ and $N$. By the [[law of total probability|0E5G5FMT2R]] for the first round, $P_k = p P_{k+1} + q P_{k-1}$ for $0 < k < N$, with $P_0 = 0$ and $P_N = 1$.
For a fair game, $p = q = 1/2$, the solution is $P_k = k/N$.
For $p \ne q$, the solution is $P_k = \dfrac{1 - (q/p)^k}{1 - (q/p)^N}$.
In a fair game against an opponent with unlimited money ($N \to \infty$), the gambler is ruined with probability $1$.

## History

An early form of the problem appears in a letter of Blaise Pascal to Pierre de Fermat in 1656.

## Explanations

### L1 human

A gambler who plays a fair game against a much richer opponent will almost surely lose everything in the end.
With $10$ units and a target of $100$ in a fair game, the chance to reach the target is only $10/100 = 0.1$.
</document>

<document source="https://tamnd.github.io/math2/node/0WFZX06698.md" tag="0WFZX06698" kind="concept">
# Inverse hyperbolic functions

## Summary

The inverse hyperbolic functions are the inverses of the hyperbolic sine, cosine and tangent, with a restricted domain for the cosine. They have explicit formulas in terms of the logarithm and the square root. The node [[inverse hyperbolic functions|04D7CHT5V1]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0WG2A3HE0S.md" tag="0WG2A3HE0S" kind="statement">
# Newton's identities

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $n \ge 1$.
Let $e_k$ be the [[elementary symmetric polynomials|0ZMY8441GW]] and $p_k$ the [[power sums|0AS1FZT6BM]] in $x_1, \dots, x_n$, with $e_0 = 1$ and $e_k = 0$ for $k > n$.
Then for every $k \ge 1$,
$$k \, e_k = \sum_{i=1}^{k} (-1)^{i-1} e_{k-i} \, p_i.$$

## Remarks

The first cases are $e_1 = p_1$, $2 e_2 = e_1 p_1 - p_2$ and $3 e_3 = e_2 p_1 - e_1 p_2 + p_3$.
For $k > n$, the left side is $0$, and the identity expresses $p_k$ through $p_{k-1}, \dots, p_{k-n}$.
Applied to the roots of a polynomial, the identities compute the power sums of the roots from the coefficients.
Applied to the eigenvalues of a matrix $A$, they relate the traces of the powers of $A$ to the coefficients of its characteristic polynomial.

## History

Isaac Newton found the identities around 1666.
Albert Girard had already found the first four formulas in 1629.

## Explanations

### L1 human

Newton's identities convert between two ways of describing the roots of a polynomial: the sums of their powers, and the coefficients.
Each identity gives the next quantity from the earlier ones.
</document>

<document source="https://tamnd.github.io/math2/node/0WG43AC0EV.md" tag="0WG43AC0EV" kind="example">
# Two-element Boolean algebra

## Example

The two-element Boolean algebra is the set $\{0, 1\}$ with these operations.
$a \wedge b = 1$ if and only if $a = b = 1$.
$a \vee b = 0$ if and only if $a = b = 0$.
$\neg 0 = 1$ and $\neg 1 = 0$.
It satisfies the axioms of a [[Boolean algebra|06GKBTAA5E]], with $0$ as least element and $1$ as greatest element.

## Remarks

If $1$ means true and $0$ means false, then $\wedge$, $\vee$ and $\neg$ are the connectives and, or and not.
An equation in the Boolean operations holds in all Boolean algebras if and only if it holds in the two-element Boolean algebra.
It is isomorphic to the power set of a set with one element.

## Explanations

### L1 human

The two-element Boolean algebra is the algebra of true and false.
It is also the algebra of one bit in a digital circuit.
</document>

<document source="https://tamnd.github.io/math2/node/0WGAXEK99E.md" tag="0WGAXEK99E" kind="statement">
# A subspace of full dimension is the whole space

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a [[linear subspace|00YC1D652K]] of $V$.
If $\dim W = \dim V$, then $W = V$.

## Remarks

A basis of $W$ is a linearly independent family of $\dim V$ vectors in $V$.
So it is a basis of $V$, see [[linearly independent set of the right size is a basis|096V7H13H9]].
The statement fails for infinite-dimensional spaces.
In $F[x]$, the subspace of polynomials with $p(0) = 0$ has the same dimension as $F[x]$, but it is not equal to $F[x]$.

## Explanations

### L1 human

If a subspace is as large as the whole space, measured by dimension, then it is the whole space.
For example, a two-dimensional subspace of the plane is the plane.
This lets us prove that two subspaces are equal by an inclusion and a count.
</document>

<document source="https://tamnd.github.io/math2/node/0WGDPYR30Q.md" tag="0WGDPYR30Q" kind="definition">
# Continuous uniform distribution

## Definition

Let $a < b$ be real numbers.
The continuous uniform distribution on the [[interval|0TK8Y3J5RW]] $[a, b]$ is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$f(x) = \begin{cases} \dfrac{1}{b - a} & \text{if } a \le x \le b, \\ 0 & \text{otherwise.} \end{cases}$$
One writes $X \sim \operatorname{U}(a, b)$.

## Remarks

For $a \le c \le d \le b$, $P(c \le X \le d) = (d - c)/(b - a)$.
The cumulative distribution function is $F(x) = (x - a)/(b - a)$ for $a \le x \le b$, with $F(x) = 0$ for $x < a$ and $F(x) = 1$ for $x > b$.
The mean is $(a + b)/2$ and the variance is $(b - a)^2 / 12$.
If $U \sim \operatorname{U}(0, 1)$ and $F$ is a cumulative distribution function with [[quantile function|0B3Q3XDM5X]] $Q$, then $Q(U)$ has the cumulative distribution function $F$.

## Explanations

### L1 human

The uniform distribution is "a point chosen at random in an interval, with no preference".
For $\operatorname{U}(0, 10)$, the chance to land between $2$ and $5$ is $3/10$, the length of the piece over the length of the interval.
</document>

<document source="https://tamnd.github.io/math2/node/0WH1GVBJNH.md" tag="0WH1GVBJNH" kind="example">
# Circle as a quotient of the real line

## Example

On $\mathbb{R}$, let $s \sim t$ when $s - t \in \mathbb{Z}$. Let $\mathbb{R} / \mathbb{Z}$ be the set of classes, with the [[quotient topology|0V8046R0P9]].
Let $S^1 = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1 \}$ be the unit circle, with the [[subspace topology|0D5YHQGDJY]] of $\mathbb{R}^2$.
Then the map $[t] \mapsto (\cos 2\pi t, \sin 2\pi t)$ is a [[homeomorphism|05GZ11PWRY]] from $\mathbb{R} / \mathbb{Z}$ to $S^1$.

## Remarks

The map is well defined and continuous by the [[universal property of the quotient topology|0FF2FJ8CH2]], and it is bijective.
The space $\mathbb{R} / \mathbb{Z}$ is compact, because it is the image of $[0, 1]$ under the quotient map. The circle is Hausdorff. Thus the map is a homeomorphism, by [[this theorem|0430ETCMKD]].
Equivalently, $S^1$ is homeomorphic to $[0, 1]$ with the two endpoints $0$ and $1$ glued together.
Addition of classes makes $\mathbb{R} / \mathbb{Z}$ a group, the circle group.

## Explanations

### L1 human

Wrap the real line around a circle of length $1$. Points that differ by an integer land at the same place.
The quotient $\mathbb{R} / \mathbb{Z}$ records exactly this wrapping.
</document>

<document source="https://tamnd.github.io/math2/node/0WH916TJNY.md" tag="0WH916TJNY" kind="statement">
# There is a natural number n with 1/n below any positive epsilon

## Statement

Let $\varepsilon$ be a real number with $\varepsilon > 0$.
Then there is a natural number $n \ge 1$ with $\dfrac{1}{n} < \varepsilon$.

## Remarks

Proof: since the [[reals are Archimedean|0A3GWZ86AY]], there is a natural number $n > 1/\varepsilon$. Then $n \ge 1$ and $1/n < \varepsilon$.
Thus $\inf \{1/n : n \ge 1\} = 0$. There are no positive infinitely small real numbers.
The same holds in each Archimedean ordered field.

## Explanations

### L1 human

The fractions $1, 1/2, 1/3, \dots$ get as small as you want.
For $\varepsilon = 0.001$, the number $n = 1001$ works, because $1/1001 < 0.001$.
</document>

<document source="https://tamnd.github.io/math2/node/0WHCA4SVP2.md" tag="0WHCA4SVP2" kind="concept">
# Invariant factors and elementary divisors

## Summary

Let $G$ be a finitely generated [[abelian group|082GHNPJ53]], and write $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z}$ with $1 < k_1 \mid k_2 \mid \cdots \mid k_u$.
The integers $k_1, \ldots, k_u$ are the invariant factors of $G$.
In the other form $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z}$, the prime powers $q_1, \ldots, q_t$ are the elementary divisors of $G$.

## Remarks

The [[structure theorem|0RY0B72XAS]] says that the rank $n$, the invariant factors and the elementary divisors are unique, and that they determine $G$ up to isomorphism.
The elementary divisors are the prime power factors of the invariant factors. For example, the invariant factors $2, 12$ give the elementary divisors $2, 4, 3$.
For a finite group $G$, the product of the invariant factors is $|G|$, and the largest invariant factor $k_u$ is the [[exponent|0ZDW0H0VX4]] of $G$.
The same terms are used for finitely generated modules over a principal ideal domain.

## Explanations

### L1 human

There are two standard ways to write a finitely generated abelian group as a sum of cyclic groups.
The invariant factors are orders that divide each other.
The elementary divisors are prime power orders.
</document>

<document source="https://tamnd.github.io/math2/node/0WJGR81HBD.md" tag="0WJGR81HBD" kind="statement">
# Taylor's theorem with the integral remainder

## Statement

Let $k \ge 0$, let $a \ne x$ be real numbers, and let $J$ be the closed interval with endpoints $a$ and $x$.
Let $f$ be a real function on $J$ such that $f^{(k+1)}$ exists and is continuous on $J$.
Then
$$f(x) = P_k(x) + \int_a^x \frac{f^{(k+1)}(t)}{k!} (x - t)^k \, dt,$$
where $P_k$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.

## Remarks

For $k = 0$ it is the [[fundamental theorem of calculus|0NHGY0N0JB]], $f(x) = f(a) + \int_a^x f'(t)\, dt$. Repeated [[integration by parts|08Y1BBQTK9]] gives the general case.
The hypotheses can be weakened to absolute continuity of $f^{(k)}$.
Mathlib states it as `taylor_integral_remainder_of_absolutelyContinuous`.

## Explanations

### L1 human

The integral form gives the error exactly, as an average of the next derivative with weights.
It needs no unknown point $\xi$.
</document>

<document source="https://tamnd.github.io/math2/node/0WJX8MGG37.md" tag="0WJX8MGG37" kind="concept">
# Free abelian group

## Summary

A free abelian group is an abelian group with a basis. Each element is an integer combination of finitely many basis elements in exactly one way.
The free abelian group of finite rank $n$ is isomorphic to $\mathbb{Z}^n$.
</document>

<document source="https://tamnd.github.io/math2/node/0WK0Y8RH1M.md" tag="0WK0Y8RH1M" kind="statement">
# Dimension theorem for vector spaces

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$.
Any two [[bases|0KCQ9Y3QKA]] of $V$ have the same cardinality.

## Remarks

For finite bases, this follows from the [[Steinitz exchange lemma|08ZDRN6H41]].
For infinite bases, the proof uses cardinal arithmetic: each vector of one basis is a combination of finitely many vectors of the other basis.
The common cardinality is the [[dimension|03GSZ8RTZJ]] of $V$.
The theorem holds more generally for free modules over a commutative ring that is not the zero ring.

## Explanations

### L1 human

However we choose a basis of a space, it always has the same number of vectors.
So "the number of vectors in a basis" is a property of the space itself.
We call this number the dimension.
</document>

<document source="https://tamnd.github.io/math2/node/0WKEF78PMV.md" tag="0WKEF78PMV" kind="statement">
# Sylvester's criterion

## Statement

Let $A$ be an $n \times n$ complex Hermitian matrix.
For $1 \le k \le n$, let $\Delta_k$ be the determinant of the upper left $k \times k$ submatrix of $A$.
Then $A$ is [[positive definite|0KX66CZMBB]] if and only if $\Delta_k > 0$ for all $k = 1, \dots, n$.

## Remarks

The numbers $\Delta_k$ are the *leading principal [[minors|0BT9BYWVRY]]* of $A$.
For $\begin{pmatrix} a & b \\ b & c \end{pmatrix}$ real, the test is $a > 0$ and $ac - b^2 > 0$.
For positive semidefinite matrices, the analogous test needs all principal minors, not only the leading ones, to be nonnegative.

## Explanations

### L1 human

Sylvester's criterion tests positive definiteness with $n$ determinants.
We check the top left $1 \times 1$, $2 \times 2$, and so on, up to the full matrix.
All of them must be positive.
</document>

<document source="https://tamnd.github.io/math2/node/0WKG15TV3G.md" tag="0WKG15TV3G" kind="concept">
# Fermat number

## Summary

A Fermat number is an integer of the form $2^{2^n} + 1$ for a natural number $n$.
The [[definition of a Fermat number|0R03B54V4Z]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0WKG8CYY6A.md" tag="0WKG8CYY6A" kind="statement">
# Cholesky decomposition

## Statement

Let $A$ be a [[positive definite|0KX66CZMBB]] $n \times n$ complex Hermitian matrix.
Then there is exactly one [[lower triangular matrix|00J3KZP8Y0]] $L$ with positive real diagonal entries such that
$$A = L L^*.$$
If $A$ is real, then $L$ is real.

## Remarks

The factor $L$ is the *Cholesky factor* of $A$.
For example, $\begin{pmatrix} 4 & 2 \\ 2 & 5 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$.
It needs about half the work of an LU decomposition, and it solves systems $Ax = b$ with positive definite $A$.
A positive semidefinite matrix also has a decomposition $A = L L^*$, but $L$ need not be unique.

## Explanations

### L1 human

A positive definite matrix is the product of a lower triangular matrix and its conjugate transpose.
This is like writing a positive number as the square of its positive square root.
It is the fastest standard way to solve linear systems with such a matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0WM7DJCM6G.md" tag="0WM7DJCM6G" kind="statement">
# Product of exponentials

## Statement

For all complex numbers $z$ and $w$,
$$e^{z + w} = e^{z} e^{w},$$
where $e^{z} = \sum_{n=0}^{\infty} z^n / n!$ is the complex exponential function.

## Remarks

The proof multiplies the two series and uses the binomial theorem.
It follows that $e^{z} \neq 0$ and $e^{-z} = 1 / e^{z}$ for each $z$.
With [[Euler's formula|04R3CY41VJ]], it gives the addition formulas for $\cos$ and $\sin$.

## Explanations

### L1 human

The exponential function turns sums into products, for complex numbers as for real numbers.
For example, $e^{i \theta} e^{i \varphi} = e^{i (\theta + \varphi)}$: turning by $\theta$ and then by $\varphi$ is turning by $\theta + \varphi$.
</document>

<document source="https://tamnd.github.io/math2/node/0WMDV48VSN.md" tag="0WMDV48VSN" kind="statement">
# Volume of a ball in R^3

## Statement

Let $R > 0$, and let $B = \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \le R^2\}$.
Then
$$\operatorname{vol}(B) = \frac{4}{3} \pi R^3.$$

## Remarks

In [[spherical coordinates|05WKRC7HWT]], $\operatorname{vol}(B) = \int_0^{2\pi} \int_0^\pi \int_0^R \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta = 2\pi \cdot 2 \cdot R^3 / 3$.
By slices, the cross-section at height $z$ is a disc of area $\pi (R^2 - z^2)$, and $\int_{-R}^{R} \pi (R^2 - z^2) \, dz = \tfrac{4}{3} \pi R^3$. This is [[Cavalieri's principle|0GTZ2F3PQY]].
This is the case $n = 3$ of the [[volume of a ball in R^n|00YH28R457]], since $\Gamma(5/2) = \tfrac{3}{4} \sqrt{\pi}$.
Archimedes found this volume, by comparison with a cone and a cylinder.

## Explanations

### L1 human

The ball of radius $R$ fills two thirds of the cylinder of radius $R$ and height $2R$ around it, since that cylinder has volume $2 \pi R^3$.
Any of the standard methods of multiple integration gives the same value.
</document>

<document source="https://tamnd.github.io/math2/node/0WMJSEFYFE.md" tag="0WMJSEFYFE" kind="statement">
# A prime is a Gaussian prime if and only if it is 3 modulo 4

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
Then $p$ is a [[Gaussian prime|0GTEHJQRFK]] if and only if $p \equiv 3 \pmod 4$.

## Remarks

If $p = 2$, then $2 = (1 + i)(1 - i)$, and $1 + i$ is a Gaussian prime.
If $p \equiv 1 \pmod 4$, then $p = a^2 + b^2 = (a + bi)(a - bi)$ by [[Fermat's theorem on sums of two squares|043B3ENSTY]], and $a \pm bi$ are Gaussian primes.
If $p \equiv 3 \pmod 4$, then $p$ is not a sum of two squares. So no Gaussian integer has norm $p$, and $p$ cannot split.
For example, $3$ and $7$ are Gaussian primes, but $13 = (3 + 2i)(3 - 2i)$.

## Explanations

### L1 human

An ordinary prime stays prime among the Gaussian integers exactly when it leaves remainder $3$ on division by $4$.
The other primes factor, because they are sums of two squares.
</document>

<document source="https://tamnd.github.io/math2/node/0WN1QXR7R3.md" tag="0WN1QXR7R3" kind="concept">
# Indefinite integral

## Summary

The indefinite integral of a function is the family of all its antiderivatives on an interval. It is written as one antiderivative plus an arbitrary constant. The node [[indefinite integral|03VHANA6ZK]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0WN5G117YC.md" tag="0WN5G117YC" kind="concept">
# Hyperbola

## Summary

A hyperbola is a plane curve with two foci. For each point of the curve, the difference of the distances to the two foci has the same absolute value. It has two branches that are mirror images of each other.
</document>

<document source="https://tamnd.github.io/math2/node/0WNGQBPPQX.md" tag="0WNGQBPPQX" kind="definition">
# Edge set

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] with vertex set $V$ and adjacency relation $\sim$.
The edge set of $G$ is the set
$$E(G) = \{ \{u, v\} : u, v \in V,\ u \sim v \}$$
of unordered pairs of adjacent vertices.

## Remarks

Each edge is a $2$-element subset of $V$, because $\sim$ is irreflexive.
The number of edges $|E(G)|$ is the size of $G$, and the number of vertices $|V|$ is its order.
The [[handshaking lemma|0XAP8WHNES]] relates $|E(G)|$ to the degrees of the vertices.

## Explanations

### L1 human

The edge set is the list of all lines of the graph, with each line given by its two end points.
A triangle on the vertices $a, b, c$ has the edge set $\{\{a, b\}, \{b, c\}, \{a, c\}\}$.
</document>

<document source="https://tamnd.github.io/math2/node/0WPDNXGSSN.md" tag="0WPDNXGSSN" kind="statement">
# Hoeffding's inequality

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] real random variables with $a_i \le X_i \le b_i$ almost surely, where $a_i < b_i$ are reals. Let $S_n = X_1 + \dots + X_n$.
Then for each $t > 0$,
$$P\left(S_n - E[S_n] \ge t\right) \le \exp\left(-\frac{2t^2}{\sum_{i=1}^n (b_i - a_i)^2}\right).$$

## Remarks

The same bound holds for $P(S_n - E[S_n] \le -t)$. So $P(|S_n - E[S_n]| \ge t) \le 2 \exp\left(-2t^2 / \sum_i (b_i - a_i)^2\right)$.
The proof uses the [[Chernoff bound|014PEBGDYF]] together with Hoeffding's lemma: $E[e^{s(X - E[X])}] \le e^{s^2 (b - a)^2 / 8}$ if $a \le X \le b$.
For $n$ independent tosses of a coin with $P(\text{heads}) = p$, the fraction $H$ of heads satisfies $P(|H - p| \ge \varepsilon) \le 2 e^{-2n\varepsilon^2}$.

## History

The inequality is named after Wassily Hoeffding.

## Explanations

### L1 human

A sum of many independent bounded terms is very unlikely to be far from its mean.
The probability of a large deviation decreases like $e^{-c t^2}$. This is why polls with a few thousand people give reliable results.
</document>

<document source="https://tamnd.github.io/math2/node/0WPQYW8MTE.md" tag="0WPQYW8MTE" kind="definition">
# Pole

## Definition

Let $c$ be an [[isolated singularity|0TYB3HM9KE]] of $f$, so that $f$ is holomorphic on $0 < |z - c| < r$ for some $r > 0$.
The singularity $c$ is a pole of $f$ if $|f(z)| \to \infty$ as $z \to c$. This means that for each $K > 0$ there is $\delta > 0$ such that $|f(z)| > K$ for each $z$ with $0 < |z - c| < \delta$.

## Remarks

The point $c$ is a pole of $f$ if and only if $f(z) = \frac{g(z)}{(z - c)^m}$ near $c$, for an integer $m \ge 1$ and a function $g$ that is holomorphic near $c$ with $g(c) \neq 0$. The integer $m$ is the [[order of the pole|0YK27TJ4WF]].
Equivalently, $1/f$ extends to a holomorphic function near $c$ with a [[zero|0CEGEASY49]] at $c$.
For example, $\frac{1}{z^2 (z - 1)}$ has a pole of order $2$ at $0$ and a pole of order $1$, a simple pole, at $1$.

## Explanations

### L1 human

A pole is a point where the function goes to infinity, in the way $1/z$ or $1/z^3$ does at $0$.
Near a pole, $1/f$ is a holomorphic function with a zero there.
</document>

<document source="https://tamnd.github.io/math2/node/0WPZ5YPAGR.md" tag="0WPZ5YPAGR" kind="definition">
# Distance in a graph

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $u, v$ be vertices of $G$.
If $v$ is [[reachable|0J8XE7W68F]] from $u$, the distance $d(u, v)$ is the smallest length of a [[walk|0VJZM0F9YM]] from $u$ to $v$.
If $v$ is not reachable from $u$, then $d(u, v) = \infty$.

## Remarks

A shortest walk is a [[path|0JDE56HWQ8]], so $d(u, v)$ is also the smallest length of a path. Such a path is a geodesic.
On a [[connected|0FFAYNMAJZ]] graph, $d$ is a metric: $d(u, v) = 0$ only for $u = v$, it is symmetric, and it satisfies the triangle inequality.

## Explanations

### L1 human

The distance between two vertices is the smallest number of edges that you must cross to go from one to the other.
In the cycle $C_6$, opposite vertices have distance $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0WQ19TJV24.md" tag="0WQ19TJV24" kind="concept">
# Distance from a point to a set

## Summary

In a metric space, the distance from a point $x$ to a nonempty set $A$ is the infimum of the distances $d(x, a)$ with $a \in A$.
</document>

<document source="https://tamnd.github.io/math2/node/0WQ1A8SHTZ.md" tag="0WQ1A8SHTZ" kind="definition">
# Field extension

## Definition

Let $K$ and $L$ be [[fields|0QA8CHH8MG]].
The field $L$ is an extension of $K$ if $K$ is a [[subfield|07QY5RPFBR]] of $L$.
One writes $L/K$ for this field extension.

## Remarks

Often one also calls $L$ an extension of $K$ when there is an injective [[ring homomorphism|0GGE0Z0JQ1]] $K \to L$. Then one identifies $K$ with its image.
An extension $L/K$ makes $L$ a vector space over $K$. Its dimension is the [[degree of the extension|0GS74FGFEK]].
Examples: $\mathbb{C}/\mathbb{R}$, $\mathbb{R}/\mathbb{Q}$ and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$.
The notation $L/K$ does not mean a quotient.

## Explanations

### L1 human

A field extension is a bigger field that contains a given field.
For example, the complex numbers extend the real numbers by an element $i$ with $i^2 = -1$.
Field theory studies how the bigger field is built from the smaller one.
</document>

<document source="https://tamnd.github.io/math2/node/0WQE3G7B88.md" tag="0WQE3G7B88" kind="definition">
# Surface area

## Definition

Let $D \subseteq \mathbb{R}^2$ be a bounded [[Jordan measurable|0565DC8RMK]] closed set, and let $r \colon D \to \mathbb{R}^3$ be a [[parametrised surface|05788J51WZ]] that is injective on the interior of $D$.
The surface area of $S = r(D)$ is
$$\operatorname{area}(S) = \iint_D \| r_u(u, v) \times r_v(u, v) \| \, du \, dv.$$

## Remarks

The integrand is the length of the [[normal vector|0BJDQ2HBH1]]. It is the area of the parallelogram spanned by $r_u$ and $r_v$, which approximates the image of a small parameter rectangle.
For the graph $z = f(x, y)$ over $D$, the area is $\iint_D \sqrt{1 + f_x^2 + f_y^2} \, dx \, dy$.
For the unit sphere, $\| r_\varphi \times r_\theta \| = \sin \varphi$, and the area is $\int_0^{2\pi} \int_0^\pi \sin \varphi \, d\varphi \, d\theta = 4\pi$.
The value does not depend on the parametrisation. A piecewise smooth surface gets the sum of the areas of its pieces. Inscribed polyhedra do not give a good definition: the Schwarz lantern is a counterexample.

## Explanations

### L1 human

To find the area of a curved surface, cut the parameter domain into small rectangles.
Each one maps to a small curved patch, which is nearly a parallelogram, and you add up the areas of these parallelograms.
</document>

<document source="https://tamnd.github.io/math2/node/0WQJJCCTC8.md" tag="0WQJJCCTC8" kind="concept">
# Normal vector

## Summary

A normal vector of a line, a plane or a surface is a nonzero vector that is perpendicular to it. For a curved surface, a normal vector at a point is perpendicular to the tangent plane at that point.
</document>

<document source="https://tamnd.github.io/math2/node/0WQNRCNY3M.md" tag="0WQNRCNY3M" kind="concept">
# Expected value

## Summary

The expected value of a random variable is its average value, where each value has a weight equal to its probability.
The node [[expected value|0DCT0QZVQ1]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0WQVXP7PDB.md" tag="0WQVXP7PDB" kind="concept">
# Moment-generating function

## Summary

The moment-generating function of a real random variable $X$ is the function $t \mapsto E[e^{tX}]$. When it is finite near $0$, its derivatives at $0$ are the moments of $X$.
The node [[moment-generating function|0NNRNGKMFM]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0WQXY32Q8D.md" tag="0WQXY32Q8D" kind="concept">
# Borel sigma-algebra

## Summary

The Borel σ-algebra of a topological space is the smallest σ-algebra that contains all open sets. Its elements are the Borel sets.
On the real line it is the usual family of events for real random variables. The node [[Borel sigma-algebra|0BVHZXWKCD]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0WRZ170YEH.md" tag="0WRZ170YEH" kind="statement">
# Moment-generating function of a sum of independent random variables

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] real random variables, and let $S = X_1 + \dots + X_n$.
Then for each $t \in \mathbb{R}$, the [[moment-generating functions|0NNRNGKMFM]] satisfy
$$M_S(t) = M_{X_1}(t) \, M_{X_2}(t) \cdots M_{X_n}(t),$$
in $(0, \infty]$.

## Remarks

Proof: $e^{tS} = e^{tX_1} \cdots e^{tX_n}$. The factors are nonnegative and independent, so the [[product rule for expectations|05AW1G0WE1]] applies.
For i.i.d. $X_i$, $M_S(t) = M_{X_1}(t)^n$.
Example: if $X_i \sim \operatorname{Exp}(\lambda)$ are independent, then $M_S(t) = (\lambda/(\lambda - t))^n$ for $t < \lambda$. This is the moment-generating function of the gamma distribution $\operatorname{Gamma}(n, \lambda)$.

## Explanations

### L1 human

When you add independent random variables, their moment-generating functions multiply.
This makes it easy to identify the law of a sum, if you know which law has the product as its moment-generating function.
</document>

<document source="https://tamnd.github.io/math2/node/0WS1A8E2P9.md" tag="0WS1A8E2P9" kind="statement">
# Extreme value theorem

## Statement

Let $a \le b$ and let $f \colon [a, b] \to \mathbb{R}$ be [[continuous|0P27W9RW0G]].
Then there are $c, d \in [a, b]$ with $f(c) \le f(x) \le f(d)$ for all $x \in [a, b]$.

## Remarks

By the [[boundedness theorem|0ATVXC3NCC]], $M = \sup f([a, b])$ is finite. If $f(x) < M$ for all $x$, then $1/(M - f(x))$ is continuous and unbounded, which is a contradiction.
Each hypothesis is needed. The function $x$ on $[0, 1)$ has no maximum, and $x$ on $[0, \infty)$ is not bounded above.
Mathlib states it for compact sets, as `IsCompact.exists_isMinOn` and `IsCompact.exists_isMaxOn`.

## History

Bernard Bolzano proved the theorem in the 1830s. His proof stayed unpublished until 1930.

## Explanations

### L1 human

On a closed bounded interval, a continuous function has a highest value and a lowest value.
Its graph has a top point and a bottom point.
</document>

<document source="https://tamnd.github.io/math2/node/0WS9P12FQR.md" tag="0WS9P12FQR" kind="concept">
# Subspace topology

## Summary

The subspace topology is the topology that a subset of a topological space gets from the whole space.
Its open sets are the intersections of the subset with the open sets of the whole space.
</document>

<document source="https://tamnd.github.io/math2/node/0WSFKKT6FF.md" tag="0WSFKKT6FF" kind="concept">
# Image of a linear map

## Summary

The image of a linear map is the set of all values of the map.
It is a subspace of the codomain.
</document>

<document source="https://tamnd.github.io/math2/node/0WTEMGYV2C.md" tag="0WTEMGYV2C" kind="statement">
# Cantor's theorem

## Statement

Let $A$ be a [[set|0SJ3RX9TXA]] and let $f \colon A \to \mathcal{P}(A)$ be a [[function|0XHDTE5YMZ]] to the [[power set|0BC7KX51Z8]] of $A$.
Then $f$ is not [[surjective|0XKH881BZ7]].
So $|A| < |\mathcal{P}(A)|$.

## Remarks

Proof: let $D = \{a \in A \mid a \notin f(a)\}$. If $D = f(d)$ for some $d \in A$, then $d \in D$ if and only if $d \notin D$, a contradiction.
The map $a \mapsto \{a\}$ is an injection, so $|A| \le |\mathcal{P}(A)|$, and the inequality is strict.
It follows that there is no largest cardinal. The proof is a form of the [[diagonal argument|0XCH8QVDCQ]], and it is close to [[Russell's paradox|0HESMMSYEG]].

## Explanations

### L1 human

For each set, its set of subsets is strictly larger.
For a set with $3$ elements, this is $3 < 8$.
For infinite sets, it gives an endless tower of larger and larger infinities.
</document>

<document source="https://tamnd.github.io/math2/node/0WW060JR6P.md" tag="0WW060JR6P" kind="definition">
# Nullcline

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f = (f_1, \ldots, f_n) : U \to \mathbb{R}^n$ be a function, and consider the [[autonomous system|07BNRFQ0PJ]] $x_j' = f_j(x_1, \ldots, x_n)$ for $j = 1, \ldots, n$.
For $j \in \{1, \ldots, n\}$, the $j$-th nullcline of the system is the set
$$N_j = \{x \in U : f_j(x) = 0\}.$$

## Remarks

On the $j$-th nullcline, the velocity has no component in the direction $x_j$. In the plane, trajectories cross the $x$-nullcline vertically and the $y$-nullcline horizontally.
The [[equilibrium points|0G4X36AE0X]] are the points that lie on all the nullclines.
The nullclines divide the [[phase plane|05VDGH129F]] into regions where each component of the velocity has a constant sign.
For a planar linear system, the nullclines are lines through the origin, or the whole plane when a row of the matrix is zero.

## Explanations

### L1 human

A nullcline is the set of states where one of the variables stops changing for an instant.
Drawing the nullclines is a quick first step to sketch a phase portrait.
</document>

<document source="https://tamnd.github.io/math2/node/0WW85SQVS3.md" tag="0WW85SQVS3" kind="definition">
# Bell numbers

## Definition

Let $n$ be a natural number.
The Bell number $B_n$ is the number of partitions of an $n$-element set into nonempty blocks.
Equivalently, it is the sum of [[Stirling numbers of the second kind|0ZN07RQ3JS]]:
$$B_n = \sum_{k=0}^{n} S(n, k).$$

## Remarks

The first values, for $n = 0, 1, \dots, 6$, are $1, 1, 2, 5, 15, 52, 203$.
$B_0 = 1$, because the empty set has exactly one partition, the partition with no blocks.
The values satisfy a [[recurrence|0110GBHPPB]] with binomial coefficients.

## Explanations

### L1 human

The Bell number $B_n$ counts all ways to split $n$ labelled objects into groups, with any number of groups.
For $\{1, 2, 3\}$ there are $5$ ways: $123$, $1|23$, $2|13$, $3|12$ and $1|2|3$.
</document>

<document source="https://tamnd.github.io/math2/node/0WWRD82HD2.md" tag="0WWRD82HD2" kind="concept">
# Right inverse

## Summary

A right inverse of a function chooses, for each target value, an input that the function sends to it.
</document>

<document source="https://tamnd.github.io/math2/node/0WX11EA3F3.md" tag="0WX11EA3F3" kind="object">
# Group of nth roots of unity

## Description

Let $n$ be a positive integer.
The group of $n$th roots of unity is the set $\mu_n = \{z \in \mathbb{C} : z^n = 1\}$ with complex multiplication.
It is a finite [[abelian group|082GHNPJ53]] of order $n$, and its elements are $e^{2\pi i k / n}$ for $k = 0, 1, \ldots, n - 1$.
It is cyclic, and the element $e^{2\pi i / n}$ generates it.

## Remarks

The group $\mu_n$ is a subgroup of the [[circle group|04EAFVFTF7]].
For $n = 4$, the group is $\{1, i, -1, -i\}$.
A generator of $\mu_n$ is called a primitive $n$th root of unity, and there are $\varphi(n)$ of them, where $\varphi$ is Euler's totient function.

## Explanations

### L1 human

The $n$th roots of unity are $n$ points on the unit circle that cut it into $n$ equal arcs.
Multiplying two of them adds their angles, so the product is again one of these points.
They behave like the rotations of a regular polygon with $n$ sides.
</document>

<document source="https://tamnd.github.io/math2/node/0WXBGBTA3D.md" tag="0WXBGBTA3D" kind="statement">
# Gram–Schmidt vectors are orthogonal

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, and let $v_1, \dots, v_k \in V$ be linearly independent.
Let $u_1, \dots, u_k$ be the vectors of the [[Gram–Schmidt process|0VT6P0GNQR]].
Then $u_1, \dots, u_k$ are nonzero and pairwise [[orthogonal|0Y2X7P2WB7]], and
$$\operatorname{span}(u_1, \dots, u_j) = \operatorname{span}(v_1, \dots, v_j) \qquad (1 \le j \le k).$$

## Remarks

The proof is by induction on $j$: by construction, $\langle u_j, u_i \rangle = 0$ for $i < j$.
The vector $u_j$ is nonzero, because $v_j$ is not in $\operatorname{span}(v_1, \dots, v_{j-1})$.
If the $v_i$ are only spanning, then the process produces some zero vectors, and dropping them still gives an orthogonal family with the same span.

## Explanations

### L1 human

The vectors made by Gram–Schmidt are really perpendicular to each other.
Also, the first $j$ new vectors span the same space as the first $j$ old vectors.
So no information is lost at any step.
</document>

<document source="https://tamnd.github.io/math2/node/0WYTM4C9QG.md" tag="0WYTM4C9QG" kind="statement">
# A monotone function has countably many discontinuities

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be monotone.
Then the set of points of $I$ where $f$ is not continuous is countable.

## Remarks

Assume $f$ increasing. By [[the existence of one-sided limits|0KJXHN0TW0]], each discontinuity $c$ inside $I$ gives an open interval $(f(c^-), f(c^+))$.
These intervals are disjoint. Each contains a rational number, so there are only countably many of them.
All these discontinuities are [[jumps|05B0FKFXQA]].
Mathlib states it as `Monotone.countable_not_continuousAt`.

## Explanations

### L1 human

Each discontinuity of an increasing function is a jump up.
The jumps take disjoint pieces of the vertical axis, so there can be only countably many.
</document>

<document source="https://tamnd.github.io/math2/node/0WZCHE1ZF9.md" tag="0WZCHE1ZF9" kind="concept">
# Iterated integral

## Summary

An iterated integral integrates a function of several variables with respect to one variable at a time. At each step, it keeps the other variables fixed.
It reduces a multiple integral to a sequence of integrals in one variable.
</document>

<document source="https://tamnd.github.io/math2/node/0WZTM0EFR6.md" tag="0WZTM0EFR6" kind="definition">
# Hölder continuity

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]], let $f \colon X \to Y$ be a function, and let $C \ge 0$ and $\alpha > 0$ be real numbers.
The map $f$ is Hölder continuous with constant $C$ and exponent $\alpha$ when
$$d_Y\bigl(f(x), f(y)\bigr) \le C \, d_X(x, y)^{\alpha} \quad \text{for all } x, y \in X.$$

## Remarks

For $\alpha = 1$, this is the condition for a [[Lipschitz map|0TMZ21WKCN]] with constant $C$.
The map $x \mapsto \sqrt{x}$ on $[0, \infty)$ is Hölder continuous with $C = 1$ and $\alpha = 1/2$.
A Hölder continuous map is uniformly continuous: for $\varepsilon > 0$ and $C > 0$, take $\delta = (\varepsilon / C)^{1/\alpha}$.
For a real function on an interval, Hölder continuity with an exponent $\alpha > 1$ forces the function to be constant.

## Explanations

### L1 human

The Hölder condition is a weaker form of the Lipschitz condition.
It allows a map to be steep, as $\sqrt{x}$ is near $0$, but it still controls how fast the values can change.
</document>

<document source="https://tamnd.github.io/math2/node/0WZXK05ME7.md" tag="0WZXK05ME7" kind="definition">
# Division ring

## Definition

A division ring is a [[ring|0894J980RK]] $D$ with these two properties.

1. $1 \neq 0$ in $D$.
2. Every nonzero element of $D$ is a [[unit|019NFGFPR5]]: for each $a \neq 0$ in $D$, there is $b \in D$ with $a b = b a = 1$.

## Remarks

A commutative division ring is a [[field|0QA8CHH8MG]].
The [[quaternions|0GE4DVQKR8]] are a division ring that is not commutative.
By [[Wedderburn's little theorem|0RRNVX1VKA]], every finite division ring is a field.

## Explanations

### L1 human

In a division ring, you can divide by every nonzero element, as in a field.
But $a b$ and $b a$ can be different.
The quaternions of Hamilton are the standard example.
</document>

<document source="https://tamnd.github.io/math2/node/0X05GS525E.md" tag="0X05GS525E" kind="concept">
# Boolean function

## Summary

A Boolean function is a function from $n$-tuples of bits to bits. Boolean functions are the objects that logic formulas and digital circuits compute. The node [[Boolean function|0A58THH5XV]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0X0HQYSZ0K.md" tag="0X0HQYSZ0K" kind="concept">
# Determinant of a linear map

## Summary

The determinant of an endomorphism of a finite-dimensional vector space is the determinant of its matrix in any basis.
</document>

<document source="https://tamnd.github.io/math2/node/0X0N833FF8.md" tag="0X0N833FF8" kind="statement">
# Number of injections

## Statement

Let $A$ and $B$ be finite sets with $|A| = k$ and $|B| = n$.
Then the number of injective functions from $A$ to $B$ is the [[falling factorial|0N5M924N3P]]
$$(n)_k = n (n - 1) \cdots (n - k + 1).$$

## Remarks

If $k > n$, then $(n)_k = 0$, and there is no injection. This is the [[pigeonhole principle|0HGVCWQGY3]].
An injection from $\{1, \dots, k\}$ to $B$ is a $k$-permutation of $B$: an ordered list of $k$ distinct elements of $B$.
For $k = n$ the count is $n!$, the [[number of permutations|0CEBCS0E0E]].

## Explanations

### L1 human

List the elements of $A$ and give each one an image in $B$ that is not used yet.
The first has $n$ choices, the second $n - 1$, and so on, for $k$ steps.
</document>

<document source="https://tamnd.github.io/math2/node/0X1GSWNBN1.md" tag="0X1GSWNBN1" kind="concept">
# Union

## Summary

The union of sets is the set of all objects that are elements of at least one of the sets.
</document>

<document source="https://tamnd.github.io/math2/node/0X1JXT47GA.md" tag="0X1JXT47GA" kind="concept">
# Subgroup

## Summary

A subgroup of a group is a subset that is itself a group under the same operation.
</document>

<document source="https://tamnd.github.io/math2/node/0X2FY491PF.md" tag="0X2FY491PF" kind="definition">
# Complex logarithm

## Definition

Let $z$ be a complex number with $z \neq 0$.
A complex logarithm of $z$ is a complex number $w$ with $e^w = z$, where $e^w$ is the [[complex exponential|054VXHMCA3]] of $w$.
The notation $\log z$ means the set of all complex logarithms of $z$, or one chosen element of this set.

## Remarks

Let $|z|$ be the [[modulus|0KW2X0TFBX]] of $z$, let $\ln$ be the real natural logarithm, and let $\theta$ be an [[argument|0ZZRSX91QR]] of $z$. Then the logarithms of $z$ are exactly the numbers $\ln |z| + i (\theta + 2 \pi k)$ with $k \in \mathbb{Z}$.
The number $0$ has no logarithm, because $e^w \neq 0$ for each $w$.
So $\log$ is a [[multivalued function|03CPD1HGF3]] on $\mathbb{C} \setminus \{0\}$. Its [[principal branch|0YKY4M3KRQ]] is $\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$, with $-\pi < \operatorname{Arg} z \le \pi$.
For example, the logarithms of $-1$ are $i \pi (2k + 1)$ with $k \in \mathbb{Z}$, and $\operatorname{Log}(-1) = i \pi$.

## Explanations

### L1 human

A logarithm of $z$ is an answer to the question: $e$ to which power gives $z$?
Because $e^{w + 2\pi i} = e^w$, each answer gives infinitely many other answers, spaced by $2\pi i$ on a vertical line.
The real part of each answer is $\ln |z|$, and the imaginary part is an angle of $z$.
</document>

<document source="https://tamnd.github.io/math2/node/0X2GJPNPFN.md" tag="0X2GJPNPFN" kind="statement">
# Hall's theorem for bipartite graphs

## Statement

Let $G$ be a finite [[bipartite graph|07DVHJ8N0A]] with parts $X$ and $Y$.
For $W \subseteq X$, let $N(W)$ be the set of vertices of $Y$ that are adjacent to some vertex of $W$.
Then $G$ has a [[matching|0Z87BHAFTB]] that covers every vertex of $X$ if and only if $|N(W)| \ge |W|$ for every $W \subseteq X$.

## Remarks

If $|X| = |Y|$, such a matching is a [[perfect matching|05ZPH8KF9K]]. Thus the condition then characterizes the perfect matchings.
A consequence: a finite regular bipartite graph with at least one edge has a perfect matching.
The theorem is the graph form of [[Hall's marriage theorem|08WCB90WND]]: let $X$ index the sets and let $Y$ be their union.

## Explanations

### L1 human

Think of $X$ as workers and $Y$ as jobs, with an edge when a worker can do a job.
Every worker can get a different job exactly when this holds.
Each group of workers can do, together, at least as many jobs as there are workers in the group.
</document>

<document source="https://tamnd.github.io/math2/node/0X2J0VY6DN.md" tag="0X2J0VY6DN" kind="concept">
# Discrete random variable

## Summary

A discrete random variable is a random variable whose values lie, with probability one, in a countable set.
Its law is described by the probabilities of the single values. The node [[discrete random variable|0J6Z6H5VQJ]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0X2T63GFQQ.md" tag="0X2T63GFQQ" kind="statement">
# Countable intersection of dense open sets is dense

## Statement

Let $X$ be a [[Baire space|0D5KR33H7X]] and let $(U_n)_{n \in \mathbb{N}}$ be a sequence of open subsets of $X$, each [[dense|0D198GPMTJ]] in $X$.
Then $\bigcap_{n} U_n$ is dense in $X$.

## Remarks

Proof: let $F_n = X \setminus U_n$. Each $F_n$ is closed, and $\operatorname{int} F_n = X \setminus \overline{U_n} = \emptyset$. Thus $\operatorname{int} \bigcup_n F_n = \emptyset$. Its complement $\bigcap_n U_n$ therefore has closure $X$.
The converse also holds, so this property characterises Baire spaces.
In $\mathbb{R}$, for an enumeration $q_1, q_2, \ldots$ of $\mathbb{Q}$, the sets $U_n = \mathbb{R} \setminus \{q_n\}$ are open and dense. Their intersection is the set of irrational numbers, which is dense.
The intersection need not be open.

## Explanations

### L1 human

Each dense open set leaves out only a thin closed set.
In a Baire space, removing countably many thin sets still leaves a set that is everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/0X308R2DBE.md" tag="0X308R2DBE" kind="statement">
# The exterior derivative squares to zero

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $\omega$ be a differential $k$-form on $U$ with coefficients of class $C^2$.
Then
$$d(d\omega) = 0.$$

## Remarks

Proof: for $\omega = f \, dx_I$, the [[exterior derivative|004FRCXH6M]] gives $d(d\omega) = \sum_{j, l} \partial_l \partial_j f \, dx_l \wedge dx_j \wedge dx_I$.
By [[Schwarz's theorem|0NNQY8WCW6]], $\partial_l \partial_j f = \partial_j \partial_l f$, while $dx_l \wedge dx_j = -dx_j \wedge dx_l$. So the terms cancel in pairs, and the terms with $j = l$ are zero.
In $\mathbb{R}^3$, the cases $k = 0$ and $k = 1$ are the identities [[curl of a gradient is zero|0M2JRW1J0N]] and [[divergence of a curl is zero|0F56CMMC5P]].
It follows that each [[exact form|0F1AVGX26N]] with $C^2$ coefficients is closed.

## Explanations

### L1 human

Taking the exterior derivative twice always gives zero, because mixed second partial derivatives are equal while the wedge product changes sign.
This one rule contains the facts that gradients have no curl and curls have no divergence.
</document>

<document source="https://tamnd.github.io/math2/node/0X3EG7Y65T.md" tag="0X3EG7Y65T" kind="concept">
# Path component

## Summary

A path component of a topological space is a maximal path-connected subset.
The path components form a partition of the space.
</document>

<document source="https://tamnd.github.io/math2/node/0X3TEM69AA.md" tag="0X3TEM69AA" kind="definition">
# Congruent triangles

## Definition

Let $n$ be a positive integer, and let $ABC$ and $A'B'C'$ be [[triangles|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$.
The triangles are congruent, written $ABC \cong A'B'C'$, if there is an isometry $f : \mathbb{R}^n \to \mathbb{R}^n$ with $f(A) = A'$, $f(B) = B'$ and $f(C) = C'$.
An isometry is a map $f$ with $d(f(x), f(y)) = d(x, y)$ for all points $x$ and $y$, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]].

## Remarks

Congruence is an equivalence relation on triangles.
Congruent triangles have equal corresponding sides and equal corresponding angles. The [[side-side-side criterion|0BRBYRK6PA]] gives the converse.
The definition depends on the order of the vertices. For example, $ABC \cong ACB$ holds only when $d(A, B) = d(A, C)$.
Mathlib's `Congruent` asks for equal distances between all pairs of corresponding points of two indexed families. In a Euclidean space this is equivalent, by the side-side-side criterion.

## Explanations

### L1 human

Two triangles are congruent if you can cut one out of paper and lay it exactly on the other. You may slide it, turn it, and flip it over. Then corresponding sides and corresponding angles match.
</document>

<document source="https://tamnd.github.io/math2/node/0X3WGX95FE.md" tag="0X3WGX95FE" kind="concept">
# Continuous function

## Summary

A continuous function is a function that is continuous at each point of its domain. Small changes of the argument then give small changes of the value. The node [[continuous function|0P27W9RW0G]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0X3XD2ATB0.md" tag="0X3XD2ATB0" kind="definition">
# Laurent series

## Definition

Let $c \in \mathbb{C}$ and let $(a_n)_{n \in \mathbb{Z}}$ be complex numbers.
The Laurent series with center $c$ and coefficients $a_n$ is the series
$$\sum_{n = -\infty}^{\infty} a_n (z - c)^n$$
in the complex variable $z$.
It converges at a point $z \neq c$ if both series $\sum_{n \ge 0} a_n (z - c)^n$ and $\sum_{n \ge 1} a_{-n} (z - c)^{-n}$ converge. Its sum at $z$ is then the sum of the two limits.

## Remarks

A [[power series|0VARVQ9ZW1]] is a Laurent series with $a_n = 0$ for each $n < 0$.
If the series converges on a nonempty open set, then it converges absolutely on an [[annulus|0AJ0YVK743]] $r < |z - c| < R$, and uniformly on each smaller closed annulus.
For example, $e^{1/z} = \sum_{n \ge 0} \frac{1}{n!} z^{-n}$ for each $z \neq 0$.
The part with negative powers is the [[principal part|0NDV5R6Z6X]].

## Explanations

### L1 human

A Laurent series is like a power series, but it can also have negative powers of $z - c$.
The negative powers allow it to describe functions that blow up at $c$, such as $1/z$ or $e^{1/z}$.
</document>

<document source="https://tamnd.github.io/math2/node/0X3Y0ZSKNF.md" tag="0X3Y0ZSKNF" kind="concept">
# Hypergeometric distribution

## Summary

The hypergeometric distribution describes the number of successes in a sample drawn without replacement from a finite population.
The node [[hypergeometric distribution|0SPBPW3K61]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0X45ZM731G.md" tag="0X45ZM731G" kind="example">
# Cayley transform

## Example

Let $\mathbb{H} = \{ z \in \mathbb{C} : \operatorname{Im} z > 0 \}$ be the upper half-plane and $\mathbb{D} = \{ w \in \mathbb{C} : |w| < 1 \}$ the open unit disc.
The Cayley transform is
$$W(z) = \frac{z - i}{z + i}.$$
It is a [[Möbius transformation|0B346DDAM5]] with $a = 1$, $b = -i$, $c = 1$, $d = i$, and $ad - bc = 2i \neq 0$.
For $z \in \mathbb{C}$, $|W(z)| < 1$ holds exactly when $z$ is nearer to $i$ than to $-i$, that is, when $\operatorname{Im} z > 0$.
The map $W$ is a bijection from $\mathbb{H}$ onto $\mathbb{D}$, with inverse $w \mapsto i \frac{1 + w}{1 - w}$.

## Remarks

Some values are $W(i) = 0$, $W(0) = -1$, $W(1) = -i$, $W(-1) = i$ and $W(\infty) = 1$.
The map $W$ sends the real line together with $\infty$ onto the unit circle.
It is an explicit case of the [[Riemann mapping theorem|02WHMEG5ZR]] for $\mathbb{H}$.
In hyperbolic geometry, it connects the half-plane model and the disc model.

## Explanations

### L1 human

The Cayley transform turns the upper half-plane into the unit disc.
A point of $\mathbb{H}$ is nearer to $i$ than to $-i$, so the quotient of the two distances is less than $1$.
It lets you move a problem on the half-plane to the disc, where it can be easier.
</document>

<document source="https://tamnd.github.io/math2/node/0X4JKZXJ86.md" tag="0X4JKZXJ86" kind="definition">
# Idempotent element

## Definition

Let $R$ be a [[ring|0894J980RK]].
An element $e \in R$ is idempotent if $e^2 = e$.

## Remarks

In every ring, $0$ and $1$ are idempotent.
In $\mathbb{Z}/6\mathbb{Z}$, the classes of $3$ and $4$ are idempotent, because $9 \equiv 3$ and $16 \equiv 4 \pmod 6$.
If $e$ is idempotent, then $1 - e$ is idempotent, and $e (1 - e) = 0$.
In the [[product|0MNZCN4BQK]] $R \times S$, the elements $(1, 0)$ and $(0, 1)$ are idempotent.

## Explanations

### L1 human

An idempotent does not change when you square it.
Among the integers, only $0$ and $1$ have this property.
Other rings can have more idempotents, for example the projection matrices.
</document>

<document source="https://tamnd.github.io/math2/node/0X4NT20S3G.md" tag="0X4NT20S3G" kind="example">
# Forced oscillation and resonance

## Example

Let $\omega_0 > 0$, $\zeta \ge 0$, $F > 0$ and $\omega > 0$ be constants. The forced oscillator is
$$x'' + 2 \zeta \omega_0\, x' + \omega_0^2\, x = F \cos(\omega t).$$
If $\zeta > 0$, it has the particular solution $x_p(t) = a(\omega) \cos(\omega t - \delta)$ for a phase $\delta$, with amplitude
$$a(\omega) = \frac{F}{\sqrt{\left(\omega_0^2 - \omega^2\right)^2 + \left(2 \zeta \omega_0 \omega\right)^2}}.$$
Each solution is $x_p$ plus a solution of the [[damped equation|0RNXVVTJTJ]], which tends to $0$. So $x_p$ is the steady state.
If $\zeta = 0$ and $\omega = \omega_0$, a particular solution is $x_p(t) = \frac{F}{2 \omega_0}\, t \sin(\omega_0 t)$, which is unbounded. This is resonance.

## Remarks

For $0 < \zeta < 1/\sqrt{2}$, the amplitude $a(\omega)$ is largest at $\omega = \omega_0 \sqrt{1 - 2 \zeta^2}$. For small $\zeta$, the maximum is close to $F / (2 \zeta \omega_0^2)$, which is large.
For $\zeta = 0$ and $\omega \ne \omega_0$, all solutions are bounded: $x_p = F \cos(\omega t) / (\omega_0^2 - \omega^2)$.
The factor $t$ in the resonant solution comes from the rule of the [[method of undetermined coefficients|0XRB469Z32]]: $i \omega_0$ is a root of the characteristic polynomial.

## Explanations

### L1 human

If you push a swing at the right rhythm, each push adds to the motion, and the swing goes higher and higher.
This is resonance: the driving frequency matches the natural frequency $\omega_0$.
Friction limits the growth, but the response is still very large near $\omega_0$.
</document>

<document source="https://tamnd.github.io/math2/node/0X52TNMTCJ.md" tag="0X52TNMTCJ" kind="statement">
# Tietze extension theorem

## Statement

Let $X$ be a [[normal space|0AJ646N339]], let $A \subseteq X$ be [[closed|0G5AV9Q4RR]], and let $f \colon A \to \mathbb{R}$ be [[continuous|0SY9ECQ19Q]] for the subspace topology on $A$.
Then there is a continuous $F \colon X \to \mathbb{R}$ whose [[restriction|0D2MXBV21T]] to $A$ is $f$.
If also $|f(a)| \le M$ for all $a \in A$, then one can choose $F$ with $|F(x)| \le M$ for all $x \in X$.

## Remarks

The proof applies [[Urysohn's lemma|01EY3VR7ZD]] many times and sums a uniformly convergent series of functions.
The converse holds: if each continuous function from each closed subset to $\mathbb{R}$ extends, then $X$ is normal.
The closedness of $A$ is needed: the function $\sin(1/x)$ on $(0, 1]$ has no continuous extension to $[0, 1]$.

## Explanations

### L1 human

On a normal space, each continuous real function on a closed set can be extended to the whole space, without losing continuity.
You can also keep the same bound on the function.
</document>

<document source="https://tamnd.github.io/math2/node/0X55M9XET4.md" tag="0X55M9XET4" kind="definition">
# Uncountable set

## Definition

A [[set|0SJ3RX9TXA]] $A$ is uncountable when it is not [[countable|0ZG61TFBES]], that is, when there is no [[injective function|0PVG1CRAJN]] from $A$ to $\mathbb{N}$.

## Remarks

The real numbers and the [[power set|0BC7KX51Z8]] of $\mathbb{N}$ are uncountable.
With the axiom of choice, a set is uncountable exactly when its cardinality is at least $\aleph_1$, the least cardinal larger than $\aleph_0$.
A superset of an uncountable set is uncountable.

## Explanations

### L1 human

For an uncountable set, every attempt to list its elements misses some of them.
[[Cantor's diagonal argument|0XCH8QVDCQ]] shows how to find a missing element.
</document>

<document source="https://tamnd.github.io/math2/node/0X5QVCF06N.md" tag="0X5QVCF06N" kind="concept">
# Elementary symmetric polynomial

## Summary

The elementary symmetric polynomials in $n$ variables are the sums of all products of $k$ distinct variables, for $k = 0, \dots, n$.
They generate all symmetric polynomials.
The [[definition of the elementary symmetric polynomials|0ZMY8441GW]] gives the formula.
</document>

<document source="https://tamnd.github.io/math2/node/0X60VE98HN.md" tag="0X60VE98HN" kind="concept">
# Quantile function

## Summary

The quantile function of a real distribution is a generalized inverse of its cumulative distribution function.
For a level $p$ between $0$ and $1$, it gives the smallest value $x$ with probability at least $p$ of not being exceeded. The node [[quantile function|0B3Q3XDM5X]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0X6AB5P3G1.md" tag="0X6AB5P3G1" kind="concept">
# Algebraic closure

## Summary

An algebraic closure of a field $K$ is an algebraic extension of $K$ that is algebraically closed.
It exists and is unique up to an isomorphism that fixes $K$.
The [[definition of an algebraic closure|0MCXX29HWY]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0X6C0MAF84.md" tag="0X6C0MAF84" kind="concept">
# Hyperbolic tangent

## Summary

The hyperbolic tangent is the quotient of the hyperbolic sine by the hyperbolic cosine. It is a bounded, odd, strictly increasing function with range $(-1, 1)$. The node [[hyperbolic tangent|0MWYQTZ07Z]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0X6V9B1V10.md" tag="0X6V9B1V10" kind="concept">
# Consistent and inconsistent systems

## Summary

A system of equations is consistent when it has at least one solution, and inconsistent when it has no solution.
</document>

<document source="https://tamnd.github.io/math2/node/0X7CVDE3BG.md" tag="0X7CVDE3BG" kind="statement">
# Impossibility of squaring the circle

## Statement

The number $\sqrt{\pi}$ is not a [[constructible number|0FMN2MN7TM]].
So the circle cannot be [[squared|0BSXRER81B]] with straightedge and compass.

## Remarks

The proof: every constructible number is [[algebraic over $\mathbb{Q}$|0SSSM1D475]]. If $\sqrt{\pi}$ were constructible, then $\pi = (\sqrt{\pi})^2$ would be constructible and so algebraic.
This contradicts the [[transcendence of $\pi$|0G0F7FGCGP]].

## History

The impossibility was proved in 1882, as a consequence of Lindemann's proof that $\pi$ is transcendental.
It had been known for decades that the transcendence of $\pi$ would imply the impossibility.

## Explanations

### L1 human

A square with the area of the unit circle has side $\sqrt{\pi}$.
Constructible numbers are algebraic, and $\pi$ is not, so this side cannot be constructed.
</document>

<document source="https://tamnd.github.io/math2/node/0X7EEWES5G.md" tag="0X7EEWES5G" kind="concept">
# Centroid

## Summary

The centroid of a finite set of points is their average, the arithmetic mean of their positions. The centroid of a triangle is the centroid of its three vertices, and it is the point where the three medians meet.
</document>

<document source="https://tamnd.github.io/math2/node/0X7G61297X.md" tag="0X7G61297X" kind="definition">
# Arc length

## Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be continuous.
The arc length of $\gamma$ is
$$L(\gamma) = \sup_{P} \sum_{i=1}^{k} \| \gamma(t_i) - \gamma(t_{i-1}) \|,$$
where the supremum runs over all [[partitions|069P20KYQT]] $P = (t_0, \dots, t_k)$ of $[a, b]$.
The curve $\gamma$ is rectifiable if $L(\gamma) < \infty$.

## Remarks

Each sum is the length of a polygonal path through points of the curve. Adding points does not decrease this length.
If $\gamma$ is continuously differentiable, then $L(\gamma) = \int_a^b \| \gamma'(t) \| \, dt$.
For the graph of a $C^1$ function $f$ on $[a, b]$, this gives $L = \int_a^b \sqrt{1 + f'(x)^2} \, dx$.
Some continuous curves are not rectifiable, for example the graph of the [[Weierstrass function|01A0MK1730]].

## Explanations

### L1 human

The arc length of a curve is the length of a string laid along it.
We approximate the curve by straight segments and take the best approximation.
</document>

<document source="https://tamnd.github.io/math2/node/0X7SBS8E0F.md" tag="0X7SBS8E0F" kind="definition">
# Winding number

## Definition

Let $\gamma : [a, b] \to \mathbb{C}$ be a closed path that is continuous and piecewise continuously differentiable. Let $w \in \mathbb{C}$ be a point that is not on the image of $\gamma$.
The winding number of $\gamma$ around $w$ is
$$n(\gamma, w) = \frac{1}{2 \pi i} \oint_\gamma \frac{dz}{z - w},$$
where the integral is a [[contour integral|00FWNP9TXY]].

## Remarks

The number $n(\gamma, w)$ is an integer. It counts how many times $\gamma$ goes around $w$ counterclockwise, minus the times it goes around $w$ clockwise.
As a function of $w$, it is constant on each connected component of the complement of the image of $\gamma$. It is $0$ on the unbounded component.
For the circle $|z - c| = R$, traveled once counterclockwise, $n = 1$ for $|w - c| < R$ and $n = 0$ for $|w - c| > R$, by the [[integral of 1/(z - w)|0SXZ68HV6F]].

## Explanations

### L1 human

Stand at the point $w$ and watch a point that moves along the closed curve. The winding number is the number of full turns that you make.
A circle around you gives $1$, the same circle traveled twice gives $2$, and a circle that does not go around you gives $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0X89GC65N1.md" tag="0X89GC65N1" kind="definition">
# Altitude of a triangle

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$, and let $\cdot$ be the dot product. Let $H_A$ be the orthogonal projection of $A$ onto the [[line|0MP9BCF6DP]] $BC$:
$$H_A = B + \frac{(A - B) \cdot (C - B)}{(C - B) \cdot (C - B)} \, (C - B).$$
The altitude of $ABC$ from $A$ is the segment from $A$ to $H_A$. The point $H_A$ is the foot of the altitude, and $h_a = \|A - H_A\|$ is its length.
The altitudes from $B$ and from $C$ are defined in the same way.

## Remarks

The formula uses the [[vector projection|0JW13B48YP]] of $A - B$ onto $C - B$. The length $h_a$ is the [[distance from $A$ to the line $BC$|01AAYNSBN6]].
The word altitude also names the whole line through $A$ and $H_A$, or the length $h_a$. Mathlib's `Affine.Simplex.altitude` is the line.
With $a = \|C - B\|$, the area of the triangle is $\frac{1}{2} a h_a$.
Let the right angle of a right triangle be at $C$. Then the leg $AC$ is the altitude from $A$, and the leg $BC$ is the altitude from $B$.

## Explanations

### L1 human

An altitude is the height of a triangle, measured from one corner. Drop a plumb line from the corner to the opposite side, or to its extension. The foot of the altitude can be outside the triangle when the triangle has an obtuse angle.
</document>

<document source="https://tamnd.github.io/math2/node/0X8HBKCSYB.md" tag="0X8HBKCSYB" kind="statement">
# Continuity of probability measures

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]].
If $A_1 \subseteq A_2 \subseteq \cdots$ are events and $A = \bigcup_{n=1}^{\infty} A_n$, then $\lim_{n \to \infty} P(A_n) = P(A)$.
If $B_1 \supseteq B_2 \supseteq \cdots$ are events and $B = \bigcap_{n=1}^{\infty} B_n$, then $\lim_{n \to \infty} P(B_n) = P(B)$.

## Remarks

The first part is continuity from below. Its proof writes $A$ as the disjoint union of $A_1$ and the sets $A_{n+1} \setminus A_n$, and uses countable additivity.
The second part, continuity from above, follows from the first part with the complements $B_n^c$.
For a general measure $\mu$, continuity from above needs $\mu(B_1) < \infty$. A [[probability measure|0VGMEBVW1M]] is finite, so this holds.
For example, $P(X \le x + 1/n) \to P(X \le x)$ for a real random variable $X$. This gives the right continuity of the cumulative distribution function.

## Explanations

### L1 human

If events grow step by step, their probabilities grow toward the probability of the limit event.
For example, the chance that a fair coin shows heads at least once in the first $n$ tosses is $1 - 2^{-n}$. It tends to the chance $1$ of heads at some toss.
</document>

<document source="https://tamnd.github.io/math2/node/0X8NMST0J2.md" tag="0X8NMST0J2" kind="concept">
# Reflexive relation

## Summary

A relation on a set is reflexive when each element is related to itself.
</document>

<document source="https://tamnd.github.io/math2/node/0X9MBPP2Z3.md" tag="0X9MBPP2Z3" kind="concept">
# Space of linear maps

## Summary

The space of linear maps between two vector spaces is the set of all linear maps from the first space to the second.
Its operations are pointwise addition and pointwise scalar multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/0X9XT05JVC.md" tag="0X9XT05JVC" kind="statement">
# Uniform limit theorem

## Statement

Let $S \subseteq \mathbb{R}$, and let $f_n \colon S \to \mathbb{R}$ be continuous for each $n$.
If $f_n \to f$ [[uniformly on $S$|08HJRTG5N7]], then $f$ is continuous on $S$.

## Remarks

Fix $c \in S$ and $\varepsilon > 0$. Choose $n$ with $|f_n - f| < \varepsilon/3$ on $S$, and $\delta$ for the [[continuity|0EEE2CZB2A]] of $f_n$ at $c$.
Then $|f(x) - f(c)| \le |f(x) - f_n(x)| + |f_n(x) - f_n(c)| + |f_n(c) - f(c)| < \varepsilon$ for $|x - c| < \delta$.
The [[sequence $x^n$ on $[0, 1]$|0YSRR1JTKZ]] shows that pointwise convergence is not enough. Mathlib states the theorem as `TendstoUniformly.continuous`.

## History

Augustin-Louis Cauchy claimed in 1821 that a series of continuous functions has a continuous sum, whenever it converges.
In 1826, Niels Henrik Abel found counterexamples among Fourier series. In modern terms, Cauchy's argument needs uniform convergence.

## Explanations

### L1 human

If continuous graphs approach a limit graph uniformly, the limit graph has no jumps.
A jump in the limit would force some $f_n$ to jump as well.
</document>

<document source="https://tamnd.github.io/math2/node/0XADJKM699.md" tag="0XADJKM699" kind="concept">
# Boolean algebra

## Summary

A Boolean algebra is a structure with meet, join and complement that satisfies the laws of classical logic and of set algebra. It is the algebra of propositional logic and of digital circuits. The node [[Boolean algebra|06GKBTAA5E]] gives the precise definition.
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<document source="https://tamnd.github.io/math2/node/0XAP8WHNES.md" tag="0XAP8WHNES" kind="statement">
# Handshaking lemma

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]] with vertex set $V$ and [[edge set|0WNGQBPPQX]] $E$.
Then
$$\sum_{v \in V} \deg(v) = 2\, |E|.$$

## Remarks

Proof: count the pairs $(v, e)$ with $v \in e$. Each edge has two ends, and each vertex $v$ lies in $\deg(v)$ edges. This is [[double counting|0F8YBHF2W3]].
A direct consequence is that the [[number of odd-degree vertices is even|0QQQQCPZA8]].
For a $3$-regular graph on $n$ vertices, it gives $3n = 2 |E|$, so $n$ is even.

## Explanations

### L1 human

At a party, add up the number of hands each person shakes.
Each handshake is counted twice, once for each of the two people, so the total is twice the number of handshakes.
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<document source="https://tamnd.github.io/math2/node/0XB0J0P64D.md" tag="0XB0J0P64D" kind="concept">
# Simple continued fraction

## Summary

A simple continued fraction is a continued fraction with all partial numerators equal to $1$ and positive integer partial denominators after the first term.
The [[definition of a simple continued fraction|06XF7BCFFJ]] gives the precise form.
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<document source="https://tamnd.github.io/math2/node/0XB9GQTN9G.md" tag="0XB9GQTN9G" kind="concept">
# Poisson distribution

## Summary

The Poisson distribution with parameter $\lambda$ describes the number of events in a fixed interval of time or space.
It applies when the events occur independently at a constant mean rate.
The node [[Poisson distribution|0VEFSSKX18]] gives the precise definition.
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<document source="https://tamnd.github.io/math2/node/0XBRG3965F.md" tag="0XBRG3965F" kind="concept">
# Negative binomial distribution

## Summary

The negative binomial distribution describes the number of failures before the $r$-th success in a sequence of independent Bernoulli trials.
The node [[negative binomial distribution|0XN7KEVSSS]] gives the precise definition.
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<document source="https://tamnd.github.io/math2/node/0XC1RDBY7A.md" tag="0XC1RDBY7A" kind="concept">
# Trace of a matrix

## Summary

The trace of a square matrix is the sum of the entries on its main diagonal.
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<document source="https://tamnd.github.io/math2/node/0XCE070V8B.md" tag="0XCE070V8B" kind="definition">
# Independence of random variables

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $I$ be an index set. For each $i \in I$, let $X_i$ be a [[random variable|0DKRR22AEQ]] with values in $(E_i, \mathcal{E}_i)$.
The family $(X_i)_{i \in I}$ is independent if for each finite nonempty set $J \subseteq I$ and each choice of $B_j \in \mathcal{E}_j$ for $j \in J$,
$$P\left(\bigcap_{j \in J} \{X_j \in B_j\}\right) = \prod_{j \in J} P(X_j \in B_j).$$

## Remarks

Equivalently, the σ-algebras $\sigma(X_i) = \{X_i^{-1}(B) : B \in \mathcal{E}_i\}$ are [[independent|09T8ZHST1V]].
For two random variables this is the [[independence of two random variables|0E9502NT92]].
Mutual independence implies pairwise independence, but the converse is false, as for [[events|0AR4B3K5QD]].
For real random variables $X_1, \dots, X_n$, independence holds if and only if the joint cumulative distribution function is the product of the marginal ones.

## Explanations

### L1 human

Independent random variables come from separate parts of an experiment that do not influence each other.
For $n$ tosses of a coin, the results of the tosses are independent random variables.
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<document source="https://tamnd.github.io/math2/node/0XCH8QVDCQ.md" tag="0XCH8QVDCQ" kind="concept">
# Cantor's diagonal argument

## Summary

Cantor's diagonal argument shows that some infinite sets are [[uncountable|0X55M9XET4]].
Given a list $s_0, s_1, s_2, \dots$ of infinite sequences of $0$ and $1$, define a sequence $t$ by $t(n) = 1 - s_n(n)$.
Then $t$ differs from each $s_n$ at position $n$, so $t$ is not in the list.
So the set of all such sequences is not countable.

## Remarks

Georg Cantor published the argument in 1891. His first proof that $\mathbb{R}$ is uncountable, from 1874, used another method.
The same idea proves [[Cantor's theorem|0WTEMGYV2C]] for every set.
Diagonal arguments also appear in the proofs of the halting problem and of Gödel's incompleteness theorems.

## Explanations

### L1 human

Write the list of sequences as the rows of a table.
Go down the diagonal and change each entry.
The new row differs from row $n$ at column $n$, so it cannot be any row of the table.
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<document source="https://tamnd.github.io/math2/node/0XCVV5Q500.md" tag="0XCVV5Q500" kind="definition">
# Geometric distribution

## Definition

Let $p \in (0, 1]$.
The geometric distribution with parameter $p$ is the distribution on $\{0, 1, 2, \dots\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(k) = (1 - p)^k \, p \quad \text{for } k = 0, 1, 2, \dots$$

## Remarks

It is the distribution of the number of failures before the first success in independent [[Bernoulli trials|0PHC9FE2V5]] with success probability $p$.
The other common convention counts the trials up to and including the first success. That number $Y = X + 1$ has $P(Y = k) = (1 - p)^{k-1} p$ for $k \ge 1$.
For $X$ as in the definition, $P(X \ge k) = (1 - p)^k$, the mean is $(1 - p)/p$ and the variance is $(1 - p)/p^2$.
The geometric distribution is the only distribution on $\{0, 1, 2, \dots\}$ with the [[memoryless property|0ZAZZ34WQS]].

## Explanations

### L1 human

Toss a coin until the first head. The number of tails before it has the geometric distribution.
Each extra tail multiplies the probability by $1 - p$, so the probabilities form a geometric sequence.
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<document source="https://tamnd.github.io/math2/node/0XCYA4YERS.md" tag="0XCYA4YERS" kind="statement">
# Wallis product

## Statement

The partial products of the following infinite product converge to $\pi/2$:
$$\frac{\pi}{2} = \prod_{n=1}^{\infty} \frac{4 n^2}{4 n^2 - 1} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots.$$

## Remarks

Let $I_n = \int_0^{\pi/2} \sin^n x \, dx$. The [[reduction formula|012RYET5E2]] gives $I_{2n}$ and $I_{2n+1}$ as explicit products.
The sequence $I_n$ is decreasing, and $I_{2n+1} / I_{2n} \to 1$ by the [[squeeze theorem|0XMF00T4XQ]]. This ratio is a partial product times $2/\pi$.
Mathlib states it as `Real.tendsto_prod_pi_div_two`.

## History

John Wallis published the product in 1656.

## Explanations

### L1 human

The Wallis product writes $\pi/2$ as an infinite product of simple fractions.
The partial products converge slowly to $\pi/2$.
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<document source="https://tamnd.github.io/math2/node/0XCYGCCX8V.md" tag="0XCYGCCX8V" kind="statement">
# Number of elements of each order in a cyclic group

## Statement

Let $G$ be a [[cyclic group|0KVGH0J5B3]] of finite order $n$, and let $d$ be a positive divisor of $n$.
Then $G$ has exactly $\varphi(d)$ elements of [[order|0WE2NHZJ7B]] $d$, where $\varphi$ is Euler's totient function.

## Remarks

Here $\varphi(d)$ is the number of integers $k$ with $1 \leq k \leq d$ and $\gcd(k, d) = 1$.
The elements of order $d$ are the generators of the unique subgroup of order $d$.
Counting all elements by their orders gives the identity $\sum_{d \mid n} \varphi(d) = n$.
In particular, $G$ has exactly $\varphi(n)$ generators.

## Explanations

### L1 human

In $\mathbb{Z}/6\mathbb{Z}$, the element $0$ has order $1$ and the element $3$ has order $2$. The elements $2$ and $4$ have order $3$, and $1$ and $5$ have order $6$.
These counts are $1, 1, 2, 2$, which are $\varphi(1), \varphi(2), \varphi(3), \varphi(6)$.
The totient function counts the elements of each order.
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<document source="https://tamnd.github.io/math2/node/0XD22PBMWA.md" tag="0XD22PBMWA" kind="statement">
# Monotonicity of the cumulative distribution function

## Statement

Let $X$ be a real random variable with [[cumulative distribution function|0MXMS3AE13]] $F_X$.
Then $F_X$ is nondecreasing: if $x \le y$, then $F_X(x) \le F_X(y)$.

## Remarks

The proof: $\{X \le x\} \subseteq \{X \le y\}$, so the claim follows from the [[monotonicity of probability|0464Z0JP2E]].
More precisely, $F_X(y) - F_X(x) = P(x < X \le y) \ge 0$.
A nondecreasing function has at most countably many points of discontinuity. So $P(X = x) > 0$ for at most countably many $x$.

## Explanations

### L1 human

As $x$ moves to the right, the event "$X \le x$" contains more outcomes, so its probability cannot go down.
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<document source="https://tamnd.github.io/math2/node/0XDB5AW7M1.md" tag="0XDB5AW7M1" kind="definition">
# Jacobi symbol

## Definition

Let $n$ be a positive odd integer with [[prime factorization|0AEKDFCYFA]] $n = p_1^{e_1} \cdots p_k^{e_k}$, and let $a$ be an integer.
The Jacobi symbol is
$$\left(\frac{a}{n}\right) = \left(\frac{a}{p_1}\right)^{e_1} \cdots \left(\frac{a}{p_k}\right)^{e_k},$$
where each factor on the right is a [[Legendre symbol|01J8AMV9RS]].
For $n = 1$, the empty product gives $\left(\frac{a}{1}\right) = 1$.

## Remarks

For a prime $n$, the Jacobi symbol equals the Legendre symbol.
If $\left(\frac{a}{n}\right) = -1$, then $a$ is not a quadratic residue modulo $n$.
The converse fails: $\left(\frac{2}{15}\right) = \left(\frac{2}{3}\right) \left(\frac{2}{5}\right) = 1$, but $2$ is not a square modulo $15$.
The symbol is $0$ exactly when $\gcd(a, n) > 1$.

## Explanations

### L1 human

The Jacobi symbol extends the Legendre symbol from odd primes to all odd positive integers.
It multiplies the Legendre symbols for the prime factors of $n$.
Its main use is fast computation, because it obeys [[quadratic reciprocity|0YY06KP3ZQ]] without the need to factor numbers.
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<document source="https://tamnd.github.io/math2/node/0XDK9CGYKJ.md" tag="0XDK9CGYKJ" kind="concept">
# Rational function

## Summary

A rational function is a function given by a quotient of two polynomials. It is defined and continuous wherever the denominator is not zero. The node [[rational function|0WCZH4AFMT]] gives the precise definition.
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<document source="https://tamnd.github.io/math2/node/0XE690QF65.md" tag="0XE690QF65" kind="definition">
# Even and odd functions

## Definition

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a function.
The function $f$ is even if $f(-x) = f(x)$ for all $x \in \mathbb{R}$.
It is odd if $f(-x) = -f(x)$ for all $x \in \mathbb{R}$.

## Remarks

The graph of an even function is symmetric about the $y$-axis. The graph of an odd function is symmetric about the origin.
For example, $x^2$ and $\cos x$ are even, and $x^3$ and $\sin x$ are odd.
Each function splits in a unique way into an even part and an odd part: $f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2}$.
Leonhard Euler introduced the two notions in 1727. Mathlib defines even functions as `Function.Even`.

## Explanations

### L1 human

An even function takes the same value at $x$ and at $-x$.
An odd function takes opposite values at $x$ and at $-x$.
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<document source="https://tamnd.github.io/math2/node/0XEGNW851S.md" tag="0XEGNW851S" kind="definition">
# Reflexive relation

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is reflexive when $a \mathrel{R} a$ for each $a \in A$.

## Remarks

On $\mathbb{R}$, the relation $\le$ is reflexive and the relation $<$ is not.
A relation is irreflexive when no element is related to itself.
Reflexivity is one of the conditions for an [[equivalence relation|0SGTMBEWRF]] and for a [[partial order|011C29Q1S1]].

## Explanations

### L1 human

In the graph of a reflexive relation, each point has a loop to itself.
"Has the same birthday as" is reflexive, because each person has the same birthday as themselves.
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<document source="https://tamnd.github.io/math2/node/0XFC1VQYEK.md" tag="0XFC1VQYEK" kind="statement">
# Multinomial theorem

## Statement

Let $R$ be a commutative ring, let $x_1, \dots, x_m \in R$, and let $n$ be a natural number.
Then
$$(x_1 + \dots + x_m)^n = \sum_{k_1 + \dots + k_m = n} \binom{n}{k_1, \dots, k_m} x_1^{k_1} \cdots x_m^{k_m}.$$
The sum runs over all $m$-tuples $(k_1, \dots, k_m)$ of natural numbers with sum $n$, and the coefficients are [[multinomial coefficients|00K0RF9BFY]].

## Remarks

For $m = 2$ it is the [[binomial theorem|0EHWPWGHB2]].
Example: $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2xz$.
With all $x_i = 1$ it gives $\sum \binom{n}{k_1, \dots, k_m} = m^n$.

## Explanations

### L1 human

To expand a power of a sum of $m$ terms, pick one term from each of the $n$ factors.
The monomial $x_1^{k_1} \cdots x_m^{k_m}$ appears once for each way to decide which factors give which term.
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<document source="https://tamnd.github.io/math2/node/0XFZ1BMF5Q.md" tag="0XFZ1BMF5Q" kind="statement">
# Euler's partition theorem

## Statement

Let $n$ be a natural number.
Then the number of [[partitions|0404R9P9J3]] of $n$ into odd parts equals the number of partitions of $n$ into distinct parts.

## Remarks

For $n = 5$, the partitions into odd parts are $5$, $3 + 1 + 1$ and $1 + 1 + 1 + 1 + 1$. The partitions into distinct parts are $5$, $4 + 1$ and $3 + 2$.
A proof with generating functions: $\prod_{k \ge 1} (1 + x^k) = \prod_{k \ge 1} \frac{1 - x^{2k}}{1 - x^k} = \prod_{k \text{ odd}} \frac{1}{1 - x^k}$.
Glaisher gave a bijection. It merges equal parts in pairs, again and again, until all parts are distinct.

## Explanations

### L1 human

Count the ways to write $n$ with only odd summands, and the ways to write it with summands that are all different.
The two counts are always equal, although the partitions themselves are not the same.
</document>

<document source="https://tamnd.github.io/math2/node/0XGCZ4GDZ6.md" tag="0XGCZ4GDZ6" kind="concept">
# Meagre set

## Summary

A meagre subset of a topological space is a countable union of nowhere dense sets.
Meagre sets are small in the sense of Baire category.
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<document source="https://tamnd.github.io/math2/node/0XH79S06J3.md" tag="0XH79S06J3" kind="concept">
# Archimedean ordered field

## Summary

An Archimedean ordered field is an [[ordered field|0P3VCBDEE2]] that has no infinitely large and no infinitely small positive elements.
In such a field, the natural numbers are not bounded above.
The fields $\mathbb{Q}$ and $\mathbb{R}$ are Archimedean.
</document>

<document source="https://tamnd.github.io/math2/node/0XHDTE5YMZ.md" tag="0XHDTE5YMZ" kind="definition">
# Function

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
A function from $A$ to $B$ is a [[binary relation|08EJC6Q25H]] $f \subseteq A \times B$ such that, for each $a \in A$, there is exactly one $b \in B$ with $(a, b) \in f$.
This $b$ is the value of $f$ at $a$, written $f(a)$.
The notation $f \colon A \to B$ says that $f$ is a function from $A$ to $B$.

## Remarks

The set $A$ is the [[domain|093CM69AGS]] and the set $B$ is the [[codomain|0F9JHP8AJN]] of $f$.
Many authors take the triple $(A, B, f)$ as the function. Then two functions with the same pairs but different codomains are different.
Two functions $f, g \colon A \to B$ are equal when $f(a) = g(a)$ for each $a \in A$.
Other names are map and mapping.

## Explanations

### L1 human

A function is a machine: you put in an element $a$ of $A$, and it gives back one element $f(a)$ of $B$.
Each input gives exactly one output, but two inputs can give the same output.
For example, $f(x) = x^2$ on $\mathbb{R}$ sends both $2$ and $-2$ to $4$.
</document>

<document source="https://tamnd.github.io/math2/node/0XHM30WBVB.md" tag="0XHM30WBVB" kind="definition">
# Finitely generated module

## Definition

Let $R$ be a ring with identity element and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.
The module $M$ is *finitely generated* when there are $x_1, \dots, x_n \in M$ such that each $x \in M$ is
$$x = r_1 x_1 + \cdots + r_n x_n \qquad \text{for some } r_1, \dots, r_n \in R.$$

## Remarks

Over a field, the finitely generated modules are the finite-dimensional vector spaces.
$\Z^n$ and $\Z / n\Z$ are finitely generated over $\Z$; the group $\Q$ is not.
Over a Noetherian ring, each submodule of a finitely generated module is finitely generated.
The finitely generated modules over a principal ideal domain are classified by the structure theorem.

## Explanations

### L1 human

A module is finitely generated when finitely many elements produce all others by linear combinations.
This is the module analogue of a finite-dimensional vector space.
Unlike in vector spaces, the generators need not be independent.
</document>

<document source="https://tamnd.github.io/math2/node/0XHWTD3MPV.md" tag="0XHWTD3MPV" kind="concept">
# Derangement

## Summary

A derangement is a permutation that moves every element. Derangements answer the classical hat-check problem, and their number is a standard application of inclusion and exclusion. The node [[derangement|0TH61JR04G]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0XHZW0VCPR.md" tag="0XHZW0VCPR" kind="concept">
# K-colourable graph

## Summary

A graph is $k$-colourable when it has a proper colouring with $k$ colours. The least such $k$ is the chromatic number. The node [[k-colourable graph|0Y7VDAX8Q3]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0XJ02JBD6S.md" tag="0XJ02JBD6S" kind="example">
# Indiscrete topology

## Example

Let $X$ be a set and let $\tau = \{\varnothing, X\}$.
Then $\tau$ is a [[topology|0CNQXVBHCX]] on $X$, the indiscrete topology, also called the trivial topology.

## Remarks

It is the coarsest topology on $X$.
Each function from any topological space to an indiscrete space is [[continuous|0SY9ECQ19Q]].
In an indiscrete space, each sequence converges to each point. Thus, if $X$ has at least two points, the space is not [[Hausdorff|0MXB4CKHFW]] and not [[metrizable|09KDHB82SG]].
Each indiscrete space is [[compact|0QPHW703PV]] and [[connected|0D939CMME7]].

## Explanations

### L1 human

In the indiscrete topology, the only open sets are the empty set and the whole space.
The topology cannot tell any two points apart.
</document>

<document source="https://tamnd.github.io/math2/node/0XJVF81QF6.md" tag="0XJVF81QF6" kind="definition">
# Consistent estimator

## Definition

For each $n \ge 1$, let $\hat{\theta}_n$ be an [[estimator|0TTNRP5PFQ]] of a real parameter $\theta$ that uses the first $n$ observations of a sample.
The sequence $(\hat{\theta}_n)$ is consistent if, for each $\theta \in \Theta$, $\hat{\theta}_n \to \theta$ [[in probability|0EFDDXTVD6]] when $\theta$ is the true parameter:
$$\lim_{n \to \infty} P_\theta\left(|\hat{\theta}_n - \theta| > \varepsilon\right) = 0 \quad \text{for each } \varepsilon > 0.$$

## Remarks

If the convergence is almost sure, the sequence is strongly consistent.
By the weak law of large numbers, the sample mean of an i.i.d. sample is a consistent estimator of the mean.
If the mean squared error of $\hat{\theta}_n$ tends to $0$, then $(\hat{\theta}_n)$ is consistent. This follows from Markov's inequality.
Consistent estimators can be biased, and unbiased estimators can fail to be consistent. For example, the estimator $X_1$ of the mean uses only one observation: it is unbiased but not consistent, unless $X_1$ is almost surely constant.

## Explanations

### L1 human

A consistent estimator gets closer and closer to the truth when you collect more and more data.
This is a minimal requirement for a reasonable estimator.
</document>

<document source="https://tamnd.github.io/math2/node/0XJXTBCB2T.md" tag="0XJXTBCB2T" kind="concept">
# Orthogonal complement

## Summary

The orthogonal complement of a subspace of an inner product space is the set of all vectors orthogonal to every vector of the subspace.
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<document source="https://tamnd.github.io/math2/node/0XK0G19SK8.md" tag="0XK0G19SK8" kind="concept">
# Order of a differential equation

## Summary

The order of a [[differential equation|003R0V5M7V]] is the highest order of a derivative of the unknown function in it.
See the [[definition|09M2RDB61G]].
</document>

<document source="https://tamnd.github.io/math2/node/0XK6FECMW5.md" tag="0XK6FECMW5" kind="statement">
# Decomposition of a permutation into transpositions

## Statement

Let $n \geq 2$.
Each permutation in $S_n$ is a product of [[transpositions|0FWVRHBRWM]].

## Remarks

A $k$-cycle is a product of $k - 1$ transpositions, for example $(a_1\ a_2\ \dots\ a_k) = (a_1\ a_k)(a_1\ a_{k-1}) \cdots (a_1\ a_2)$.
With the [[disjoint cycle decomposition|0YWDAZPRPQ]], this gives the result, and the identity is $(1\ 2)(1\ 2)$.
The decomposition is not unique, but the parity of the number of factors is unique.

## Explanations

### L1 human

You can sort any arrangement of objects by swapping two objects at a time.
So each permutation is a chain of swaps.
Different chains can give the same permutation, but they are all of even length or all of odd length.
</document>

<document source="https://tamnd.github.io/math2/node/0XKBKGTCPC.md" tag="0XKBKGTCPC" kind="concept">
# Length of a vector

## Summary

The length of a [[Euclidean vector|0QN3D80AWP]] is the nonnegative real number that measures its size. It is the distance between the start point and the end point of an arrow that represents the vector.
</document>

<document source="https://tamnd.github.io/math2/node/0XKH881BZ7.md" tag="0XKH881BZ7" kind="definition">
# Surjective function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The function $f$ is surjective when, for each $b \in B$, there is an $a \in A$ with $f(a) = b$.

## Remarks

Other names are onto function and surjection.
Equivalently, the [[range|0E9GTAV478]] of $f$ is the whole [[codomain|0F9JHP8AJN]] $B$.
The function $x \mapsto x^3$ on $\mathbb{R}$ is surjective, and $x \mapsto x^2$ on $\mathbb{R}$ is not, because $-1$ is not a value.

## Explanations

### L1 human

A surjective function hits every element of the codomain at least once.
To prove surjectivity, take an arbitrary $b \in B$ and find an input $a$ with $f(a) = b$.
</document>

<document source="https://tamnd.github.io/math2/node/0XKJPFS2W0.md" tag="0XKJPFS2W0" kind="definition">
# Periodic continued fraction

## Definition

An infinite [[simple continued fraction|06XF7BCFFJ]] $[a_0; a_1, a_2, \dots]$ is periodic if there are integers $N \ge 0$ and $k \ge 1$ with $a_{n+k} = a_n$ for all $n \ge N$.
It is purely periodic if this holds with $N = 0$.

## Remarks

One writes the repeating block with a bar, for example $\sqrt{2} = [1; \overline{2}]$ and $\sqrt{3} = [1; \overline{1, 2}]$.
By theorems of Euler and Lagrange, a real number has a periodic continued fraction if and only if it is a quadratic irrational. That is, it is an irrational root of a quadratic equation with integer coefficients.
The period of $\sqrt{d}$ gives the [[fundamental solution of Pell's equation|0KCYJSB9FW]].

## Explanations

### L1 human

A periodic continued fraction repeats the same block of terms forever.
The golden ratio $[1; 1, 1, 1, \dots]$ is the simplest example.
</document>

<document source="https://tamnd.github.io/math2/node/0XKTZRWTXX.md" tag="0XKTZRWTXX" kind="statement">
# Poisson limit theorem

## Statement

Let $\lambda > 0$, and let $p_n \in [0, 1]$ satisfy $n p_n \to \lambda$ as $n \to \infty$. Let $S_n \sim \operatorname{Bin}(n, p_n)$.
Then for each integer $k \ge 0$,
$$\lim_{n \to \infty} P(S_n = k) = \lim_{n \to \infty} \binom{n}{k} p_n^k (1 - p_n)^{n - k} = e^{-\lambda} \frac{\lambda^k}{k!}.$$
So $S_n$ converges [[in distribution|0JDAQHWA9N]] to the [[Poisson distribution|0VEFSSKX18]] with parameter $\lambda$.

## Remarks

Proof: write $\binom{n}{k} p_n^k = \frac{n(n-1)\cdots(n-k+1)}{n^k} \cdot \frac{(n p_n)^k}{k!}$. The first factor tends to $1$, and $(1 - p_n)^{n-k} \to e^{-\lambda}$.
In practice, $\operatorname{Bin}(n, p)$ is close to $\operatorname{Poisson}(np)$ when $n$ is large and $p$ is small.
Le Cam's inequality gives an explicit bound on the error of this approximation.

## Explanations

### L1 human

The number of rare events among many independent chances is approximately Poisson.
Examples are the number of misprints on a page, or the number of large insurance claims in a year.
</document>

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# Limit of a sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]] and let $L$ be a real number.
The sequence $(a_n)$ converges to $L$ if the following holds for each open interval $U$ that contains $L$.
There is a natural number $N$ such that $a_n \in U$ for all $n \ge N$.
Then $L$ is a limit of $(a_n)$, with the notation $\lim_{n \to \infty} a_n = L$ or $a_n \to L$ as $n \to \infty$.

## Remarks

The [[epsilon-N definition of the limit of a sequence|09G6KHS6GY]] is an equivalent form, with intervals $(L - \varepsilon, L + \varepsilon)$.
A sequence has at most one limit, by the [[uniqueness of limits|0ZXJ2455NJ]]. So the notation $\lim_{n \to \infty} a_n$ is well defined.
The same definition works in each topological space, with open neighbourhoods of $L$ in place of open intervals.
In Mathlib, the limit is the statement `Filter.Tendsto a Filter.atTop (nhds L)`.

## Explanations

### L1 human

The sequence $(a_n)$ tends to $L$ when its terms come close to $L$ and stay close.
However small a window around $L$ you choose, from some index on, all terms are inside the window.
For example, $1/(n+1)$ tends to $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0XM8BZD76G.md" tag="0XM8BZD76G" kind="object">
# Coordinate space

## Description

Let $F$ be a field and let $n \ge 0$ be an integer.
The *coordinate space* $F^n$ is the set of $n$-tuples $(x_1, \dots, x_n)$ with $x_i \in F$.
It is a [[vector space|000AYPA9AJ]] over $F$ with the operations
$$(x_1, \dots, x_n) + (y_1, \dots, y_n) = (x_1 + y_1, \dots, x_n + y_n), \qquad a(x_1, \dots, x_n) = (ax_1, \dots, ax_n).$$
Its zero vector is $(0, \dots, 0)$.

## Remarks

The space $F^n$ has [[dimension|03GSZ8RTZJ]] $n$, with the [[standard basis|0V7ZD6P7RK]] $e_1, \dots, e_n$.
More generally, for a set $I$, the set $F^I$ of all functions $I \to F$ is a vector space with pointwise operations.
Each $n$-dimensional vector space over $F$ is isomorphic to $F^n$.
The real spaces $\R^2$ and $\R^3$ model the plane and ordinary space.

## Explanations

### L1 human

The coordinate space $F^n$ is the space of lists of $n$ numbers.
We add two lists position by position, and we scale a list by scaling each entry.
Every finite-dimensional vector space looks like one of these spaces after we choose a basis.
</document>

<document source="https://tamnd.github.io/math2/node/0XM8ZA9RNM.md" tag="0XM8ZA9RNM" kind="concept">
# Sieve of Eratosthenes

## Summary

The sieve of Eratosthenes is an algorithm that finds all the [[prime numbers|0SN0MAEKYX]] up to a given bound $N$.
It writes the integers from $2$ to $N$. It takes the smallest number that is not crossed out, which is a prime $p$, and crosses out the multiples $p^2, p^2 + p, p^2 + 2p, \dots$ up to $N$.
It repeats this step while $p^2 \le N$. The numbers that are not crossed out at the end are the primes up to $N$.

## Remarks

It is enough to start at $p^2$, because a smaller multiple of $p$ has a smaller prime factor, and an earlier step crossed it out.
The algorithm needs about $N \log \log N$ operations.
The idea of the sieve is the basis of sieve methods in analytic number theory.

## Explanations

### L1 human

For $N = 30$, cross out the multiples of $2$ from $4$, then the multiples of $3$ from $9$, then the multiples of $5$ from $25$.
Then $7^2 = 49 > 30$, so you stop.
The numbers left are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.
</document>

<document source="https://tamnd.github.io/math2/node/0XMEM406JM.md" tag="0XMEM406JM" kind="concept">
# Semigroup

## Summary

A semigroup is a set with an associative binary operation.
It need not have an identity element or inverses.
</document>

<document source="https://tamnd.github.io/math2/node/0XMF00T4XQ.md" tag="0XMF00T4XQ" kind="statement">
# Squeeze theorem

## Statement

Let $(a_n)$, $(b_n)$ and $(c_n)$ be real sequences and let $L$ be a real number.
Assume that there is a natural number $N$ such that $a_n \le c_n \le b_n$ for all $n \ge N$.
Assume that $a_n \to L$ and $b_n \to L$ as $n \to \infty$.
Then $c_n \to L$ as $n \to \infty$.

## Remarks

The theorem gives the existence of the limit of $(c_n)$. It does not assume it.
For example, $-1/n \le \sin(n)/n \le 1/n$, so $\sin(n)/n \to 0$.
The same statement holds for the [[limit of a function|0TXTSEMHR5]] at a point.
Other names are the sandwich theorem and, in some countries, the two policemen theorem.

## Explanations

### L1 human

If a sequence is trapped between two sequences that go to the same limit, it must go there too.
For large $n$, the upper and lower sequences are both close to $L$, and $c_n$ lies between them.
</document>

<document source="https://tamnd.github.io/math2/node/0XN3K7RSZY.md" tag="0XN3K7RSZY" kind="concept">
# Multivariate normal distribution

## Summary

The multivariate normal distribution is the extension of the normal distribution to random vectors in $\mathbb{R}^k$. A mean vector and a covariance matrix determine it.
The node [[multivariate normal distribution|0G558GJQMW]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0XN66MZN0Y.md" tag="0XN66MZN0Y" kind="concept">
# Equicontinuity

## Summary

A family of functions is equicontinuous when all its members are continuous in the same way. For each tolerance, one neighbourhood works for all of them at the same time.
</document>

<document source="https://tamnd.github.io/math2/node/0XN7KEVSSS.md" tag="0XN7KEVSSS" kind="definition">
# Negative binomial distribution

## Definition

Let $r \ge 1$ be an integer and let $p \in (0, 1]$.
The negative binomial distribution with parameters $r$ and $p$ is the distribution on $\{0, 1, 2, \dots\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(k) = \binom{k + r - 1}{k} p^r (1 - p)^k \quad \text{for } k = 0, 1, 2, \dots$$

## Remarks

It is the distribution of the number of failures before the $r$-th success in independent Bernoulli trials with success probability $p$. The last trial is a success, and the $k$ failures are among the other $k + r - 1$ trials.
For $r = 1$ it is the [[geometric distribution|0XCVV5Q500]]. The sum of $r$ independent geometric random variables with parameter $p$ has the negative binomial distribution with parameters $r$ and $p$.
The mean is $r(1 - p)/p$ and the variance is $r(1 - p)/p^2$.
Some authors count the trials, or the successes before a fixed number of failures. Check the convention of each source.

## Explanations

### L1 human

Toss a coin until you have seen $r$ heads, and count the tails on the way.
This count has the negative binomial distribution. For $r = 1$ you wait only for the first head.
</document>

<document source="https://tamnd.github.io/math2/node/0XNN9Z8E5A.md" tag="0XNN9Z8E5A" kind="definition">
# Acyclic graph

## Definition

A [[graph|0JD5C82QMB]] is acyclic if it has no [[cycle|07CT9RDY7Q]].

## Remarks

An acyclic graph is also called a [[forest|09JP6G9FTA]]. A connected acyclic graph is a [[tree|0MJ0JS72DZ]].
The [[path graphs|075WR6W3Y3]] and the graphs with no edges are acyclic. The [[cycle graphs|0QQ0F12962]] are not.
A finite acyclic graph with $n$ vertices and $c$ components has $n - c$ edges.

## Explanations

### L1 human

A graph is acyclic when you cannot go around any loop in it.
Each edge then splits its piece of the graph in two when you delete it.
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<document source="https://tamnd.github.io/math2/node/0XNZWJ96HK.md" tag="0XNZWJ96HK" kind="definition">
# Rotation

## Definition

Let $c$ be a point of $\mathbb{R}^n$, and let $A$ be an orthogonal $n \times n$ matrix with $\det A = 1$. The rotation about $c$ with the matrix $A$ is the map
$$R \colon \mathbb{R}^n \to \mathbb{R}^n, \qquad R(p) = c + A\,(p - c).$$
In the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$, the rotation about $c$ through the angle $\theta \in \mathbb{R}$ is the case
$$A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}.$$

## Remarks

Each rotation is a direct [[Euclidean isometry|0J1N3WTKM4]] that fixes $c$. In the plane, each orthogonal matrix with determinant $1$ has this form, so each plane rotation has an angle.
In the plane, rotations about one centre commute, and their angles add modulo $2\pi$. Rotations about different centres do not commute in general.
In the plane, the composite of the reflections in two lines that meet at $c$ is a rotation about $c$.
Mathlib's `Orientation.rotation` is the rotation through an angle of an oriented Euclidean plane, as a linear map.

## Explanations

### L1 human

A plane rotation turns the whole plane about one fixed point through one angle. With $\theta > 0$, the turn is anticlockwise. The rotation about $(0, 0)$ through $\pi/2$ sends $(1, 0)$ to $(0, 1)$ and $(0, 1)$ to $(-1, 0)$.
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<document source="https://tamnd.github.io/math2/node/0XPKR18APA.md" tag="0XPKR18APA" kind="concept">
# Cofactor

## Summary

The cofactor in position $(i, j)$ of a square matrix is the first minor $M_{ij}$ with the sign $(-1)^{i+j}$.
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# Euclidean space

## Description

Let $n$ be a positive integer.
The Euclidean $n$-space is the set $\mathbb{R}^n$ of the $n$-tuples $x = (x_1, \dots, x_n)$ of real numbers, with the dot product
$$x \cdot y = x_1 y_1 + x_2 y_2 + \dots + x_n y_n.$$
The elements of $\mathbb{R}^n$ are its points, and also its vectors.
The dot product gives the length $\|x\| = \sqrt{x \cdot x}$ of a vector $x$.
It also gives the distance $\|x - y\|$ between two points $x$ and $y$, and the angle between two nonzero vectors.
For $n = 2$ this is the [[Euclidean plane|0XW7V9ZZ05]], and for $n = 3$ it is the space of elementary solid geometry.

## Remarks

In a more abstract form, a Euclidean space is a real [[affine space|09XNAPTJQ7]] of finite dimension whose space of translations has an inner product. Each such space of dimension $n$ is isometric to $\mathbb{R}^n$ with the dot product.
The notation $\mathbb{E}^n$ is also common for the Euclidean $n$-space.

## Explanations

### L1 human

The Euclidean $n$-space is the space of ordinary geometry in $n$ dimensions. A point is a list of $n$ coordinates. The dot product lets you measure lengths and angles, so the familiar facts of school geometry hold in each dimension.
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<document source="https://tamnd.github.io/math2/node/0XPX5NC67E.md" tag="0XPX5NC67E" kind="concept">
# Spectrum of a matrix

## Summary

The spectrum of a square matrix is the set of its eigenvalues.
</document>

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# Laplacian

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}$ be twice differentiable on $U$.
The Laplacian of $f$ is the function $\Delta f \colon U \to \mathbb{R}$,
$$\Delta f = \operatorname{div}(\nabla f) = \nabla \cdot \nabla f,$$
the [[divergence|02NSMQVQWK]] of the [[gradient|0NS2QYVC4Y]] of $f$. It is also written $\nabla^2 f$.

## Remarks

In Cartesian coordinates, $\Delta f$ is the [[sum of the unmixed second partial derivatives|05RCT1522R]] of $f$.
The operator $\Delta$ is linear, and it maps functions of [[class C^k|0TCBZMEYGN]] to functions of class $C^{k-2}$ for $k \ge 2$.
Let $f$ be of class $C^2$, and let $\bar{f}_r$ be the average of $f$ on the sphere of radius $r$ about $x$. Then $\Delta f(x)$ is $2n$ times the limit of $(\bar{f}_r - f(x)) / r^2$ as $r \to 0$. So $\Delta f(x) > 0$ means that $f(x)$ is below its nearby averages.
The Laplacian appears in the heat equation $\partial_t u = \Delta u$, the wave equation and Laplace's equation $\Delta u = 0$.

## Explanations

### L1 human

The Laplacian compares the value of a function at a point with its values nearby.
If $f$ at the point is lower than the average around it, the Laplacian is positive. If it is higher, the Laplacian is negative.
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<document source="https://tamnd.github.io/math2/node/0XR2QK49V6.md" tag="0XR2QK49V6" kind="concept">
# Greatest element

## Summary

The greatest element of a subset of a partially ordered set is an element of the subset that is above every element of the subset.
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# Negation

## Definition

Let $p$ be a [[proposition|0ZGC7ZESYN]].
The negation of $p$ is the proposition $\neg p$, read "not $p$".
The proposition $\neg p$ is true when $p$ is false, and $\neg p$ is false when $p$ is true.

| $p$ | $\neg p$ |
|---|---|
| T | F |
| F | T |

## Remarks

Other notations for $\neg p$ are $\sim p$ and $\overline{p}$.
In classical logic, $\neg \neg p$ has the same truth value as $p$. This is the [[double negation law|0NFAGWH84K]].

## Explanations

### L1 human

To negate a sentence, say that it is not the case.
The negation of "$5$ is prime" is "$5$ is not prime".
The first sentence is true, so its negation is false.
</document>

<document source="https://tamnd.github.io/math2/node/0XRB469Z32.md" tag="0XRB469Z32" kind="concept">
# Method of undetermined coefficients

## Summary

The method of undetermined coefficients finds a particular solution of a [[linear equation with constant coefficients|0A6PGTTP2A]] $L y = b(x)$ when $b$ has a special form.
The right side must be a sum of terms $x^k e^{\lambda x}$, $x^k e^{\lambda x} \cos(\omega x)$ and $x^k e^{\lambda x} \sin(\omega x)$.
For such a term, one tries a function of the same type, with a general polynomial of degree $k$ and unknown coefficients. If $\lambda$ (or $\lambda + i\omega$) is a root of multiplicity $s$ of the [[characteristic polynomial|0QDXMG46QW]], one multiplies the trial function by $x^s$.
Substitution into the equation gives linear equations for the unknown coefficients.

## Remarks

Example: for $y'' + y = x$, try $y_p = A x + B$. Then $A x + B = x$, so $y_p = x$.
Example with resonance: for $y'' + y = \cos x$, the number $i$ is a simple root of $r^2 + 1$. So try $y_p = x (A \cos x + B \sin x)$, which gives $y_p = \frac{1}{2} x \sin x$.
The method works because the derivatives of these functions stay in a finite-dimensional space. For other right sides, use [[variation of parameters|0Y6TH5PQVS]].

## Explanations

### L1 human

Guess that the answer looks like the right side, with unknown numbers in it.
Put the guess into the equation and match the coefficients.
If the guess already solves the homogeneous equation, multiply it by $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0XRDYJQES6.md" tag="0XRDYJQES6" kind="concept">
# Farey sequence

## Summary

The Farey sequence of order $n$ is the increasing list of reduced fractions between $0$ and $1$ with denominators at most $n$.
The [[definition of a Farey sequence|0PN4QJE969]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0XRJB5SSNE.md" tag="0XRJB5SSNE" kind="definition">
# Prime-counting function

## Definition

For a real number $x$, the prime-counting function $\pi(x)$ is the number of [[prime numbers|0SN0MAEKYX]] $p$ with $p \le x$.

## Remarks

For example, $\pi(10) = 4$, because the primes up to $10$ are $2, 3, 5, 7$. Also $\pi(100) = 25$.
The function is a step function. It increases by $1$ at each prime.
The [[prime number theorem|0EHBY5Y84N]] says that $\pi(x)$ grows like $x / \ln x$.
The symbol $\pi$ here has no relation to the number $3.14159\dots$.

## Explanations

### L1 human

The function $\pi(x)$ answers the question: how many primes are there up to $x$?
The answer grows without bound, by [[Euclid's theorem|0292EPPPT0]], but the primes become rarer as $x$ grows.
</document>

<document source="https://tamnd.github.io/math2/node/0XRRYF1HPE.md" tag="0XRRYF1HPE" kind="statement">
# Dini's theorem

## Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be continuous functions with $f_n(x) \le f_{n+1}(x)$ for all $n$ and all $x$.
Assume that $(f_n)$ converges [[pointwise|056FSKG32J]] to a continuous function $f$ on $[a, b]$.
Then $(f_n)$ converges [[uniformly|08FPABDQ4C]] to $f$ on $[a, b]$.

## Remarks

The same holds for decreasing sequences, and for any compact space in place of $[a, b]$.
One proof uses the [[Bolzano–Weierstrass theorem|0KDJD249DT]]. Another uses the open sets $\{x : f(x) - f_n(x) < \varepsilon\}$, which increase and cover $[a, b]$.
All hypotheses are needed. The sequence $x^n$ on $[0, 1]$ is monotone, but its limit is not continuous. Mathlib states a version as `Monotone.tendstoLocallyUniformly_of_forall_tendsto`.

## History

The theorem is named after Ulisse Dini. He gave the first version in an 1878 book on real functions, printed in Pisa.

## Explanations

### L1 human

If continuous functions increase to a continuous limit on a closed bounded interval, they get close to it everywhere at once.
Monotonicity and compactness rule out the bad behavior of examples such as $x^n$.
</document>

<document source="https://tamnd.github.io/math2/node/0XSDBCP4RZ.md" tag="0XSDBCP4RZ" kind="statement">
# Eisenstein's lemma

## Statement

Let $p$ be an odd prime and let $a$ be a positive integer with $p \nmid a$.
Then
$$\left(\frac{a}{p}\right) = (-1)^{\sum_{u} \lfloor a u / p \rfloor},$$
where the sum runs over the even integers $u = 2, 4, \dots, p - 1$, and $\left(\frac{a}{p}\right)$ is the [[Legendre symbol|01J8AMV9RS]].

## Remarks

For example, take $p = 7$ and $a = 3$. The sum is $\lfloor 6/7 \rfloor + \lfloor 12/7 \rfloor + \lfloor 18/7 \rfloor = 0 + 1 + 2 = 3$, so $\left(\frac{3}{7}\right) = -1$.
A common variant holds for odd $a$: then $\left(\frac{a}{p}\right) = (-1)^{\sum_{k=1}^{(p-1)/2} \lfloor k a / p \rfloor}$.
The proof is similar to the proof of [[Gauss's lemma|037ACYXB5Z]].
Eisenstein used the lemma in a proof of [[quadratic reciprocity|085MJS6907]] that counts lattice points in a triangle.

## Explanations

### L1 human

The lemma turns the question "is $a$ a square modulo $p$?" into a count of integer parts of fractions.
The parity of the count gives the answer.
</document>

<document source="https://tamnd.github.io/math2/node/0XSJ25Z9M4.md" tag="0XSJ25Z9M4" kind="definition">
# Linear isomorphism

## Definition

Let $F$ be a field and let $V$ and $W$ be vector spaces over $F$.
A *linear isomorphism* from $V$ to $W$ is a bijective [[linear map|0NRYHJPQG2]] $T \colon V \to W$.
The spaces $V$ and $W$ are *isomorphic*, written $V \cong W$, when a linear isomorphism $V \to W$ exists.

## Remarks

The inverse map $T^{-1} \colon W \to V$ of a linear isomorphism is linear.
A linear map is an isomorphism if and only if it sends some basis, or each basis, of $V$ to a basis of $W$.
For an $n \times n$ matrix $A$ over $F$, the map $x \mapsto Ax$ is an isomorphism of $F^n$ if and only if $A$ is [[invertible|0EB9N0S33C]].
[[Finite-dimensional spaces are isomorphic if and only if they have equal dimension|0VJBSK15SN]].

## Explanations

### L1 human

A linear isomorphism is a perfect translation between two vector spaces.
It matches the vectors one to one and respects addition and scaling.
Isomorphic spaces have the same linear algebra, even when their vectors look different.
</document>

<document source="https://tamnd.github.io/math2/node/0XTNJ6SVAC.md" tag="0XTNJ6SVAC" kind="concept">
# Cluster point of a sequence

## Summary

A cluster point of a sequence is a point near which infinitely many terms of the sequence lie. In the real numbers, the cluster points are exactly the limits of the convergent subsequences. The node [[cluster point of a sequence|0RCKQN1E09]] gives the precise definition.
</document>

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# Galois group

## Summary

The Galois group of a field extension $L/K$ is the group of the automorphisms of $L$ that fix every element of $K$.
The [[definition of the Galois group|0XVR72M51M]] gives the details.
</document>

<document source="https://tamnd.github.io/math2/node/0XV2A49X6Q.md" tag="0XV2A49X6Q" kind="concept">
# Composition series

## Summary

A composition series of a group is a subnormal series in which each factor group is simple.
It cannot be refined further, and its factors are unique up to order by the Jordan–Hölder theorem.
</document>

<document source="https://tamnd.github.io/math2/node/0XVEZDE9J8.md" tag="0XVEZDE9J8" kind="concept">
# Unit of a ring

## Summary

A unit of a ring is an element that has a multiplicative inverse in the ring.
The [[definition of a unit|019NFGFPR5]] states this precisely.
</document>

<document source="https://tamnd.github.io/math2/node/0XVGJE7Q04.md" tag="0XVGJE7Q04" kind="statement">
# The principal logarithm is holomorphic off the branch cut

## Statement

Let $S = \mathbb{C} \setminus (-\infty, 0]$ be the slit plane, the complement of the standard [[branch cut|06JYC06JHY]].
Let $\operatorname{Log}$ be the [[principal branch|0YKY4M3KRQ]] of the logarithm.
Then $\operatorname{Log}$ is [[holomorphic|0DG0Z72CQD]] on $S$, and
$$\operatorname{Log}'(z) = \frac{1}{z}$$
for each $z \in S$.

## Remarks

Proof idea: on $S$, $\operatorname{Log}$ is continuous and it is the inverse of the [[complex exponential|054VXHMCA3]] on the strip $\{w : -\pi < \operatorname{Im} w < \pi\}$. The inverse function rule gives $\operatorname{Log}'(z) = 1 / e^{\operatorname{Log} z} = 1/z$.
$\operatorname{Log}$ is not continuous at the points of $(-\infty, 0)$. For $x < 0$, its values tend to $\ln|x| + i\pi$ from above the axis and to $\ln|x| - i\pi$ from below.

## Explanations

### L1 human

Away from the cut along the negative real axis, the principal logarithm behaves like the real logarithm: its derivative is $1/z$.
On the cut itself, the angle jumps from about $\pi$ to about $-\pi$, so the function is not even continuous there.
</document>

<document source="https://tamnd.github.io/math2/node/0XVR72M51M.md" tag="0XVR72M51M" kind="definition">
# Galois group

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
An automorphism of $L/K$ is a ring isomorphism $\sigma \colon L \to L$ with $\sigma(c) = c$ for every $c \in K$.
These automorphisms form a [[group|G5S1M9]] $\operatorname{Aut}(L/K)$ under composition.
If $L/K$ is a [[Galois extension|0DS2F7GK05]], this group is the Galois group of $L/K$, written $\operatorname{Gal}(L/K)$.

## Remarks

Some authors call $\operatorname{Aut}(L/K)$ the Galois group for every extension $L/K$.
The group $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$ has two elements: the identity and complex conjugation.
The group $\operatorname{Aut}(\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q})$ is trivial, because an automorphism must send $\sqrt[3]{2}$ to a real cube root of $2$.
For a finite extension, $|\operatorname{Aut}(L/K)| \le [L : K]$, by the [[bound on the order|0QM98DNWVW]].

## Explanations

### L1 human

The Galois group collects the symmetries of a bigger field that leave the smaller field unchanged.
Each symmetry permutes the roots of every polynomial over the smaller field.
</document>

<document source="https://tamnd.github.io/math2/node/0XW54DH28V.md" tag="0XW54DH28V" kind="concept">
# Upper central series

## Summary

The upper central series of a group $G$ is the sequence $\{e\} = Z_0 \subseteq Z_1 \subseteq Z_2 \subseteq \cdots$, where $Z_1$ is the centre and $Z_{i+1}/Z_i$ is the centre of $G/Z_i$.
The group is nilpotent if and only if this series reaches $G$.
</document>

<document source="https://tamnd.github.io/math2/node/0XW7V9ZZ05.md" tag="0XW7V9ZZ05" kind="object">
# Euclidean plane

## Description

The Euclidean plane is the set $\mathbb{R}^2$ of the ordered pairs $(x, y)$ of real numbers, with the dot product $(x_1, y_1) \cdot (x_2, y_2) = x_1 x_2 + y_1 y_2$.
The elements of $\mathbb{R}^2$ are the points of the plane.
The dot product gives the length of a vector, the distance between two points and the angle between two vectors.
The distance between the points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ is $\sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2}$.
The Euclidean plane is the [[Euclidean space|0XPR1E85W0]] of dimension 2.

## Remarks

Each Euclidean space of dimension 2 is isometric to $\mathbb{R}^2$. A choice of a [[Cartesian coordinate system|0RWKYX8339]] gives such an isometry.
The plane with a fixed Cartesian coordinate system is also called the Cartesian plane.
Books I to IV and VI of Euclid's Elements study the geometry of the plane.

## Explanations

### L1 human

Think of the Euclidean plane as an infinite flat sheet of paper. Two numbers, the coordinates $x$ and $y$, fix the position of each point. For example, the points $(0, 0)$ and $(3, 4)$ are at distance $\sqrt{9 + 16} = 5$.
</document>

<document source="https://tamnd.github.io/math2/node/0XW8HX8VEV.md" tag="0XW8HX8VEV" kind="concept">
# Median of a triangle

## Summary

A median of a triangle is the segment from a vertex to the midpoint of the opposite side. Each triangle has three medians, and they meet at one point, the centroid.
</document>

<document source="https://tamnd.github.io/math2/node/0XWCKJZRYH.md" tag="0XWCKJZRYH" kind="statement">
# A root of an integer is an integer or irrational

## Statement

Let $x$ be a real number, let $n$ be a positive integer and let $m$ be an [[integer|0C4JS9RF5M]] with $x^n = m$.
If $x$ is not an integer, then $x$ is [[irrational|080T2TXVQV]].

## Remarks

Thus, a rational number $x$ with $x^n \in \mathbb{Z}$ is an integer.
For example, $\sqrt[3]{2}$ is irrational, because $1^3 < 2 < 2^3$ shows that it is not an integer.
The statement follows from the rational root theorem for the polynomial $X^n - m$.

## Explanations

### L1 human

Take a root of a whole number, for example $\sqrt{10}$ or $\sqrt[3]{20}$.
Either the root is a whole number, as for $\sqrt{9} = 3$, or it is not a fraction at all.
A root of a whole number is never a fraction like $5/2$.
</document>

<document source="https://tamnd.github.io/math2/node/0XWG0JGDFW.md" tag="0XWG0JGDFW" kind="concept">
# Factorial

## Summary

The factorial of a natural number $n$ is the product of the positive integers up to $n$. It counts the orderings of $n$ distinct objects and appears in most counting formulas. The node [[factorial|0WFCGVMRRC]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0XYAFCN0PT.md" tag="0XYAFCN0PT" kind="statement">
# Second fundamental theorem of calculus

## Statement

Let $a < b$, and let $F \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$.
Assume that $F'$ extends to a [[Riemann integrable|0N1JQXZXHC]] function $f$ on $[a, b]$.
Then
$$\int_a^b f(x) \, dx = F(b) - F(a).$$

## Remarks

For a partition $P$, the [[mean value theorem|0BARQY52N5]] gives tags $t_i$ with $F(x_i) - F(x_{i-1}) = f(t_i)(x_i - x_{i-1})$. So $F(b) - F(a)$ is a [[Riemann sum|0HBY48G8G1]] of $f$ for each partition.
One writes $F(b) - F(a) = \big[ F(x) \big]_a^b$.
The hypothesis that $f$ is integrable is needed. Some differentiable functions have derivatives that are not Riemann integrable.
Mathlib states it as `intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le`.

## Explanations

### L1 human

To integrate $f$, find a function $F$ with derivative $f$.
The integral is the change of $F$ from $a$ to $b$.
</document>

<document source="https://tamnd.github.io/math2/node/0XYKE8BE7X.md" tag="0XYKE8BE7X" kind="definition">
# Normal matrix

## Definition

Let $A$ be an $n \times n$ complex matrix.
The matrix $A$ is *normal* when
$$A A^* = A^* A,$$
where $A^*$ is the [[conjugate transpose|0DXKE10PHF]] of $A$.

## Remarks

Hermitian, skew-Hermitian and unitary matrices are normal.
A real matrix is normal when $A A^T = A^T A$; real symmetric and orthogonal matrices are examples.
The matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is not normal.
A matrix is normal if and only if its map on $\C^n$ is a [[normal operator|0C8BR8XRTR]] for the standard inner product.

## Explanations

### L1 human

A matrix is normal when it commutes with its conjugate transpose.
Normal matrices are exactly the matrices that a unitary change of basis makes diagonal.
They include the symmetric, the Hermitian and the unitary matrices.
</document>

<document source="https://tamnd.github.io/math2/node/0XYP2KKSPC.md" tag="0XYP2KKSPC" kind="definition">
# Normal vector

## Definition

Let $H$ be a [[line|0MP9BCF6DP]] in $\mathbb{R}^2$ or a [[plane|0V3XAMF3C1]] in $\mathbb{R}^3$. Let $\cdot$ be the dot product.
A normal vector of $H$ is a nonzero vector $n$ such that $n \cdot (x - y) = 0$ for all points $x$ and $y$ of $H$.

## Remarks

If $n$ is a normal vector of $H$, then the normal vectors of $H$ are exactly the vectors $c n$ with $c \neq 0$.
The plane $\{(x, y, z) : a x + b y + c z = d\}$ with $(a, b, c) \neq (0, 0, 0)$ has the normal vector $(a, b, c)$. The line $\{(x, y) : a x + b y = c\}$ with $(a, b) \neq (0, 0)$ has the normal vector $(a, b)$.
For the plane $\{p + s u + t v : s, t \in \mathbb{R}\}$, the [[cross product|0N5N15HWZ7]] $u \times v$ is a normal vector.
A unit normal vector is a normal vector of length 1.

## Explanations

### L1 human

A normal vector sticks straight out of a flat object. Think of a pencil standing upright on a table: it is normal to the table. Each arrow that lies in the table makes a right angle with the pencil, so its dot product with the pencil is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/0XYVDWGQX2.md" tag="0XYVDWGQX2" kind="statement">
# De Morgan's laws in a Boolean algebra

## Statement

Let $A$ be a [[Boolean algebra|06GKBTAA5E]].
Then $\neg(a \vee b) = \neg a \wedge \neg b$ and $\neg(a \wedge b) = \neg a \vee \neg b$ for all $a, b \in A$.

## Remarks

Proof idea: in a Boolean algebra each element has a unique complement. Check that $\neg a \wedge \neg b$ satisfies the two complement equations for $a \vee b$.
In the [[power set Boolean algebra|004J7ZEFAT]], the laws become $X \setminus (A \cup B) = (X \setminus A) \cap (X \setminus B)$ and its dual.
In the [[two-element Boolean algebra|0WG43AC0EV]], they are the laws of logic for not, and, and or.

## History

The laws are named after Augustus De Morgan. His book Formal Logic of 1847 states them.

## Explanations

### L1 human

The complement of an or is the and of the complements, and the complement of an and is the or of the complements.
Not (rain or snow) means no rain and no snow.
</document>

<document source="https://tamnd.github.io/math2/node/0XYYT5KNTX.md" tag="0XYYT5KNTX" kind="concept">
# Direct sum of subspaces

## Summary

A vector space is the direct sum of a family of subspaces when each vector is a sum of vectors from the subspaces.
This sum must be unique.
</document>

<document source="https://tamnd.github.io/math2/node/0XZ37KAB1Z.md" tag="0XZ37KAB1Z" kind="concept">
# Disjoint sets

## Summary

Two sets are disjoint when they have no element in common.
</document>

<document source="https://tamnd.github.io/math2/node/0Y0JGC2K9T.md" tag="0Y0JGC2K9T" kind="concept">
# Rooted tree

## Summary

A rooted tree is a tree with a chosen vertex, the root, which gives each edge a direction away from it. Rooted trees model hierarchies and recursive data. The node [[rooted tree|0C9J6W2A8A]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0Y0KFHF47G.md" tag="0Y0KFHF47G" kind="definition">
# Graph of a function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The graph of $f$ is the set
$$\Gamma(f) = \{(a, f(a)) \mid a \in A\} \subseteq A \times B.$$

## Remarks

For a real function $f \colon \mathbb{R} \to \mathbb{R}$, the graph is a curve in the plane $\mathbb{R}^2$.
Let $G \subseteq A \times B$.
$G$ is the graph of a function from $A$ to $B$ if and only if each $a \in A$ is the first entry of exactly one pair.
In the set-theoretic [[definition of a function|0XHDTE5YMZ]], the function and its graph are the same set.

## Explanations

### L1 human

The graph of $f(x) = x^2$ is the parabola $\{(x, x^2) \mid x \in \mathbb{R}\}$.
Each vertical line meets it in exactly one point. This is the vertical line test.
</document>

<document source="https://tamnd.github.io/math2/node/0Y0MQYST0M.md" tag="0Y0MQYST0M" kind="concept">
# Modus ponens

## Summary

Modus ponens is the rule of inference that gets $q$ from the two premises $p$ and $p \to q$.
$$\frac{p \qquad p \to q}{q}$$
It is valid: when the two premises are true, the conclusion $q$ is true.

## Remarks

Most proof systems for [[propositional logic|09K9MWD2QQ]] use modus ponens as a rule.
Its partner modus tollens gets $\neg p$ from $\neg q$ and $p \to q$.

## Explanations

### L1 human

Modus ponens is the most common step in a proof.
If you know that $n$ is divisible by $4$, and that each number divisible by $4$ is even, then you know that $n$ is even.
</document>

<document source="https://tamnd.github.io/math2/node/0Y0WS5BCJ7.md" tag="0Y0WS5BCJ7" kind="definition">
# Inflection point

## Definition

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$ be continuous, and let $c \in I$.
The point $c$ is an inflection point of $f$ if there is $\delta > 0$ with one of the following properties.
The function $f$ is [[convex|0VV0HSW1ZS]] on $(c - \delta, c]$ and [[concave|0EF2VG8XKQ]] on $[c, c + \delta)$.
Or $f$ is concave on $(c - \delta, c]$ and convex on $[c, c + \delta)$.

## Remarks

If $f$ is twice differentiable at an inflection point $c$, then $f''(c) = 0$.
The converse fails: $f(x) = x^4$ has $f''(0) = 0$, but $0$ is not an inflection point.
The function $x^3$ has an inflection point at $0$. So does $\sin x$, at each multiple of $\pi$.
Some texts also ask the graph to have a tangent line at $c$.

## Explanations

### L1 human

At an inflection point, the graph changes from bending up to bending down, or the reverse.
The graph crosses its tangent line there.
</document>

<document source="https://tamnd.github.io/math2/node/0Y123X48GA.md" tag="0Y123X48GA" kind="statement">
# Absolutely convergent series converge

## Statement

Let $(a_n)$ be a real sequence.
If the series $\sum a_n$ [[converges absolutely|0W3K3QPEWT]], then $\sum a_n$ converges.
Moreover, $\left| \sum_{n=0}^{\infty} a_n \right| \le \sum_{n=0}^{\infty} |a_n|$.

## Remarks

The proof uses the [[Cauchy criterion for series|06PVXMY0TH]]. For $n > m$ the triangle inequality gives $\left| \sum_{k=m+1}^{n} a_k \right| \le \sum_{k=m+1}^{n} |a_k|$.
The converse fails: the [[alternating harmonic series|0JQ5Y3WWVE]] converges, but $\sum 1/n$ diverges.
The result holds in every complete normed space. Mathlib states it as `Summable.of_norm`.

## Explanations

### L1 human

Cancellation between terms of opposite signs can only make partial sums smaller in absolute value.
So if the series converges without any cancellation, it also converges with it.
</document>

<document source="https://tamnd.github.io/math2/node/0Y12QYTJK3.md" tag="0Y12QYTJK3" kind="definition">
# Limit at infinity

## Definition

Let $D \subseteq \mathbb{R}$ be a set that is not bounded above, and let $f \colon D \to \mathbb{R}$ be a function.
A real number $L$ is the limit of $f$ at $+\infty$ if for each $\varepsilon > 0$ there is $M$ with $|f(x) - L| < \varepsilon$ for all $x \in D$ with $x > M$.
The notation is $\lim_{x \to \infty} f(x) = L$.
The limit at $-\infty$ is defined in the same way, with $x < M$.

## Remarks

A sequence is a function on $\mathbb{N}$. For sequences this definition gives the [[limit of a sequence|0XKWSSY8P2]].
For example, $\lim_{x \to \infty} 1/x = 0$.
Mathlib expresses $x \to \infty$ by the filter `Filter.atTop`.

## Explanations

### L1 human

The limit at infinity is the value that $f(x)$ approaches as $x$ grows without bound.
The graph then gets close to a horizontal line $y = L$.
</document>

<document source="https://tamnd.github.io/math2/node/0Y17PEKF24.md" tag="0Y17PEKF24" kind="definition">
# Diagonalisable matrix

## Definition

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
The matrix $A$ is *diagonalisable* over $F$ when there is an invertible $n \times n$ matrix $P$ over $F$ such that $P^{-1} A P$ is a [[diagonal matrix|044JNC7ZAA]].
An endomorphism of a finite-dimensional space is diagonalisable when its matrix in some basis is diagonal.

## Remarks

So $A$ is diagonalisable if and only if it is [[similar|0E89HQATM8]] to a diagonal matrix.
The diagonal entries of $P^{-1} A P$ are the eigenvalues of $A$, and the columns of $P$ are eigenvectors.
The matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is not diagonalisable over any field.
Over $\R$, the rotation matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ is not diagonalisable; over $\C$, it is.

## Explanations

### L1 human

A diagonalisable matrix becomes a diagonal matrix in a suitable basis.
In that basis, the map just stretches each coordinate axis by a factor.
This makes powers, exponentials and other functions of the matrix easy to compute.
</document>

<document source="https://tamnd.github.io/math2/node/0Y1MG9R5VN.md" tag="0Y1MG9R5VN" kind="statement">
# Catalan recurrence

## Statement

Let $n$ be a natural number, and let $C_m$ be the [[Catalan numbers|092DRBFKMG]].
Then
$$C_{n+1} = \sum_{i=0}^{n} C_i\, C_{n-i}.$$

## Remarks

With the definition of the node [[Catalan numbers|092DRBFKMG]], the statement holds by definition. Its content is in the combinatorial families that satisfy it.
Example: $C_4 = C_0 C_3 + C_1 C_2 + C_2 C_1 + C_3 C_0 = 5 + 2 + 2 + 5 = 14$.
For [[Dyck words|0R9JR0465V]], the recurrence splits a word at the first return to level $0$.

## Explanations

### L1 human

A structure of size $n + 1$ often splits into a left part of size $i$ and a right part of size $n - i$.
Add over all $i$ the number of ways to build the two parts, and you get the next Catalan number.
</document>

<document source="https://tamnd.github.io/math2/node/0Y2X7P2WB7.md" tag="0Y2X7P2WB7" kind="concept">
# Orthogonal vectors

## Summary

Two vectors of an inner product space are orthogonal when their inner product is zero.
</document>

<document source="https://tamnd.github.io/math2/node/0Y34JMJNE9.md" tag="0Y34JMJNE9" kind="statement">
# Gauss–Wantzel theorem

## Statement

Let $n \ge 3$.
The regular $n$-gon can be constructed with straightedge and compass if and only if $n = 2^k p_1 p_2 \cdots p_t$, with $k, t \ge 0$ and $p_1, \dots, p_t$ distinct Fermat primes.

## Remarks

A Fermat prime is a prime number of the form $2^{2^m} + 1$. The known Fermat primes are $3, 5, 17, 257$ and $65537$.
The regular $n$-gon is constructible if and only if $e^{2\pi i/n}$ is a [[constructible number|0FMN2MN7TM]]. Its degree over $\mathbb{Q}$ is $\varphi(n)$, by the [[irreducibility of $\Phi_n$|0SK1VHZ5R2]].
The condition on $n$ holds if and only if $\varphi(n)$ is a power of $2$.
So the $17$-gon and the $15$-gon are constructible, but the $7$-gon and the $9$-gon are not.

## History

Carl Friedrich Gauss proved in 1796 that the regular $17$-gon is constructible.
In his 1801 book Disquisitiones Arithmeticae, he gave the sufficient condition.
Pierre Wantzel gave a full proof of the necessity in 1837.

## Explanations

### L1 human

The theorem tells exactly which regular polygons you can draw with straightedge and compass.
The answer depends on the prime factors of the number of sides.
Only Fermat primes, each used at most once, and powers of $2$ are allowed.
</document>

<document source="https://tamnd.github.io/math2/node/0Y3KSN2E9X.md" tag="0Y3KSN2E9X" kind="concept">
# Series of functions

## Summary

A series of functions is an infinite sum of functions defined on a common set. Its convergence is the convergence of its partial sums, pointwise or uniformly. The node [[series of functions|0MRE6XRE5V]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0Y4B7P0GS8.md" tag="0Y4B7P0GS8" kind="statement">
# The order modulo p divides p − 1

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]], and let $a$ be an integer with $p \nmid a$.
Then the [[multiplicative order|0SBRX5CZ5R]] of $a$ modulo $p$ divides $p - 1$.

## Remarks

The proof: $a^{p-1} \equiv 1 \pmod p$ by [[Fermat's little theorem|0M25PRN3WQ]], and each exponent $m$ with $a^m \equiv 1$ is a multiple of the order.
The order is equal to $p - 1$ exactly when $a$ is a [[primitive root|0KM2CJ80G7]] modulo $p$.
For example, modulo $13$, the order of $3$ is $3$ and the order of $2$ is $12$. Both divide $12$.

## Explanations

### L1 human

The powers of $a$ modulo $p$ go round a cycle. The length of the cycle always divides $p - 1$.
Modulo $11$, the powers of $5$ are $5, 3, 4, 9, 1$. The cycle has length $5$, and $5 \mid 10$.
</document>

<document source="https://tamnd.github.io/math2/node/0Y4NF1DRKD.md" tag="0Y4NF1DRKD" kind="statement">
# Trace of a product is symmetric

## Statement

Let $R$ be a commutative ring.
Let $A$ be an $m \times n$ matrix and $B$ an $n \times m$ matrix over $R$.
Then $\tr(AB) = \tr(BA)$, where $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

Both sides equal $\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} b_{ji}$.
The matrices $AB$ and $BA$ can have different sizes, and the identity still holds.
A consequence is $\tr(P^{-1} A P) = \tr A$, so similar matrices have the same trace.
For three matrices, only cyclic permutations are allowed in general: $\tr(ABC) = \tr(BCA)$, but $\tr(ABC)$ can differ from $\tr(BAC)$.

## Explanations

### L1 human

Although $AB$ and $BA$ are usually different matrices, they have the same trace.
Each diagonal sum adds the same products $a_{ij} b_{ji}$, only in another order.
This fact makes the trace a property of a linear map, not of the basis that we choose.
</document>

<document source="https://tamnd.github.io/math2/node/0Y4TV2D5RQ.md" tag="0Y4TV2D5RQ" kind="statement">
# Brooks' theorem

## Statement

Let $G$ be a finite connected simple [[graph|0JD5C82QMB]] with [[maximum degree|0KNAN9F27P]] $\Delta$.
Suppose that $G$ is not a [[complete graph|03WKB2H474]] and not a [[cycle graph|0QQ0F12962]] of odd length.
Then $\chi(G) \le \Delta$.

## Remarks

A [[greedy colouring|04AAKZNXHD]] gives $\chi(G) \le \Delta + 1$ for every graph. Brooks' theorem improves this bound by $1$ in all other cases.
The two exceptions need $\Delta + 1$ colours: $\chi(K_n) = n$ with $\Delta = n - 1$, and an odd cycle has $\chi = 3$ with $\Delta = 2$.

## History

R. Leonard Brooks published the theorem in 1941.

## Explanations

### L1 human

If each vertex has at most $\Delta$ neighbours, then $\Delta + 1$ colours are always enough.
Brooks' theorem says that $\Delta$ colours are enough, except for complete graphs and odd cycles.
</document>

<document source="https://tamnd.github.io/math2/node/0Y5Q8Q9WJ7.md" tag="0Y5Q8Q9WJ7" kind="definition">
# Level set

## Definition

Let $D \subseteq \mathbb{R}^n$, let $f \colon D \to \mathbb{R}$ be a [[function of several variables|0JPCHTXWG0]], and let $c \in \mathbb{R}$.
The level set of $f$ at the value $c$ is the set
$$L_c(f) = \{x \in D : f(x) = c\}.$$
It is the [[preimage|01AHG4G9RT]] $f^{-1}(\{c\})$ of the one-point set $\{c\}$.
For $n = 2$ a level set is also called a level curve, and for $n = 3$ it is also called a level surface.

## Remarks

For $f(x, y) = x^2 + y^2$, the level set at $c > 0$ is the circle of radius $\sqrt{c}$ about the origin. The level set at $c = 0$ is the point $(0, 0)$, and the level set at $c < 0$ is empty.
The level sets of $f$ for different values of $c$ are disjoint, and their union is the domain $D$.
At a point where $f$ is differentiable with a nonzero [[gradient|0ABK9V9Y64]], the gradient is [[orthogonal to the level set|0VXYV3NQQ1]].
A level set of a function of two variables is a [[contour line|0QD83RAN8G]] of its graph.

## Explanations

### L1 human

On a map of a hill, a level set is the set of all places at one fixed height.
For the bowl $f(x, y) = x^2 + y^2$, each level set at a positive height is a circle, and the circles grow as the height grows.
</document>

<document source="https://tamnd.github.io/math2/node/0Y5R7TPNT1.md" tag="0Y5R7TPNT1" kind="definition">
# Outcome

## Definition

Let $\Omega$ be a [[sample space|0W7NFW23RF]] of a random experiment.
An outcome of the experiment is an [[element|0EZ758XE73]] $\omega$ of $\Omega$.

## Remarks

Two different outcomes cannot both happen in one trial.
An outcome $\omega$ gives the event $\{\omega\}$, an elementary event. An outcome and an event are different things: an [[event|00SQGJNRTB]] is a set of outcomes.
For a roll of a die, the outcomes are $1, 2, 3, 4, 5, 6$, and "the result is even" is the event $\{2, 4, 6\}$, not an outcome.

## Explanations

### L1 human

An outcome is one thing that can happen.
When you toss two coins, "heads then tails" is one outcome, and there are four outcomes in total.
</document>

<document source="https://tamnd.github.io/math2/node/0Y5RHNRKJZ.md" tag="0Y5RHNRKJZ" kind="definition">
# Centroid

## Definition

Let $k$ be a positive integer, and let $K$ be a field in which $k \neq 0$. Let $A$ be an [[affine space|09XNAPTJQ7]] over a $K$-vector space, and let $p_1, \dots, p_k$ be points of $A$.
The centroid of $p_1, \dots, p_k$ is the [[affine combination|0KJH6SRFZK]] with all weights equal to $\frac{1}{k}$:
$$G = \frac{1}{k} p_1 + \frac{1}{k} p_2 + \dots + \frac{1}{k} p_k.$$
The centroid of a triangle $ABC$ is the centroid $G = \frac{1}{3}(A + B + C)$ of its vertices.

## Remarks

In $\mathbb{R}^n$, the coordinates of the centroid are the means of the coordinates of the points.
In $\mathbb{R}^n$, the centroid is the point $x$ that makes the sum $\sum_i \|x - p_i\|^2$ smallest.
The centroid of two points is their [[midpoint|0S706JBZBG]].
The centroid of a triangle is also the centre of mass of a uniform triangular plate.

## Explanations

### L1 human

The centroid is the average of the points. For the triangle with the vertices $(0, 0)$, $(6, 0)$ and $(0, 3)$, the centroid is $\left(\frac{0 + 6 + 0}{3}, \frac{0 + 0 + 3}{3}\right) = (2, 1)$. A cardboard triangle balances on a pin at this point.
</document>

<document source="https://tamnd.github.io/math2/node/0Y5X3SDBCZ.md" tag="0Y5X3SDBCZ" kind="statement">
# Fermat's Last Theorem

## Statement

Let $n$ be an integer with $n > 2$.
Then there are no positive integers $a$, $b$ and $c$ with $a^n + b^n = c^n$.

## Remarks

For $n = 2$ there are infinitely many solutions, the [[Pythagorean triples|01G1WY1GVX]].
It is enough to prove the theorem for $n = 4$ and for each odd prime $n$, because each $n > 2$ has one of these as a divisor.
The cases [[n = 4|0E741HSMJ1]] and [[n = 3|07NKG38VEG]] have elementary proofs.

## History

Pierre de Fermat stated the theorem around 1637 and wrote that he had a proof.
Andrew Wiles gave the first proof in 1994, with a step completed jointly with Richard Taylor. The proof was published in 1995.
The proof uses the modularity of semistable elliptic curves.

## Explanations

### L1 human

The equation $a^2 + b^2 = c^2$ has many whole-number solutions, such as $3^2 + 4^2 = 5^2$.
For cubes, fourth powers and all higher powers, the theorem says that there are none.
The statement is simple, but the proof took more than 350 years.
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# Tail sum formula for expectation

## Statement

Let $X$ be a random variable with values in $[0, \infty)$. Then
$$E[X] = \int_0^\infty P(X > t) \, dt,$$
with both sides in $[0, \infty]$.
If $X$ takes values in $\{0, 1, 2, \dots\}$, then
$$E[X] = \sum_{n=1}^{\infty} P(X \ge n).$$

## Remarks

The proof writes $X = \int_0^\infty 1_{\{X > t\}} \, dt$ and exchanges expectation and integral by Tonelli's theorem.
In terms of the [[cumulative distribution function|0MXMS3AE13]] $F$, the first formula reads $E[X] = \int_0^\infty (1 - F(t)) \, dt$.
For $X \sim \operatorname{Exp}(\lambda)$, the formula gives $E[X] = \int_0^\infty e^{-\lambda t} \, dt = 1/\lambda$.

## Explanations

### L1 human

You can compute an average by adding the chances that $X$ passes each level.
For a count $X$, each $n \le X$ adds $1$ to $X$, and it does so with probability $P(X \ge n)$.
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# Variation of parameters

## Summary

Variation of parameters is a method to find a particular solution of an inhomogeneous linear equation from a [[fundamental system|0H4KNNCHPY]] of the homogeneous equation.
Let $p, q, f$ be continuous on an open interval $I$. Let $(y_1, y_2)$ be a fundamental system of $y'' + p y' + q y = 0$ on $I$, with [[Wronskian|0N5NF1RGBP]] $W$.
One looks for $y = u_1 y_1 + u_2 y_2$ with the extra condition $u_1' y_1 + u_2' y_2 = 0$. Then the equation $y'' + p y' + q y = f$ becomes $u_1' y_1' + u_2' y_2' = f$.
This linear system gives $u_1' = -y_2 f / W$ and $u_2' = y_1 f / W$, so a particular solution is
$$y_p = -y_1 \int \frac{y_2 f}{W}\, dx + y_2 \int \frac{y_1 f}{W}\, dx.$$

## Remarks

The method works for each continuous right side, unlike the [[method of undetermined coefficients|0XRB469Z32]].
Example: for $y'' + y = \tan x$ on $(-\pi/2, \pi/2)$, with $y_1 = \cos x$, $y_2 = \sin x$ and $W = 1$, one gets $y_p = -\cos x \ln\left(\sec x + \tan x\right)$.
For first-order systems, the same idea gives the [[variation of constants formula|0S1KG0XE40]].

## Explanations

### L1 human

The general solution of the homogeneous equation is $c_1 y_1 + c_2 y_2$ with constants.
The idea is to let the constants vary with $x$.
A smart extra condition makes the equations for $u_1'$ and $u_2'$ simple.
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# Order isomorphism

## Definition

Let $(P, \le_P)$ and $(Q, \le_Q)$ be [[partially ordered sets|011C29Q1S1]].
An order isomorphism from $P$ to $Q$ is a [[bijective function|0SNQEF7Z53]] $f \colon P \to Q$ such that, for all $a, b \in P$,
$$a \le_P b \iff f(a) \le_Q f(b).$$

## Remarks

The inverse of an order isomorphism is an order isomorphism.
The map $x \mapsto 2x$ is an order isomorphism from $\mathbb{Z}$ to the even integers.
The ordered sets $\mathbb{N}$ and $\mathbb{Z}$ are not order isomorphic, because $\mathbb{N}$ has a least element and $\mathbb{Z}$ has none.
A [[monotone|0WFF6AZPBP]] bijection need not be an order isomorphism, when the orders are not total.

## Explanations

### L1 human

Order isomorphic sets have the same shape as ordered sets.
Only the names of the elements differ.
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# Matrix addition

## Summary

Matrix addition is the operation that adds two matrices of the same size entry by entry.
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# Directional derivative

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, let $a \in U$, and let $v \in \mathbb{R}^n$.
The directional derivative of $f$ at $a$ in the direction $v$ is the limit
$$D_v f(a) = \lim_{t \to 0} \frac{f(a + t v) - f(a)}{t},$$
when this limit exists in $\mathbb{R}^m$.
Other notations for it are $\partial_v f(a)$ and $\nabla_v f(a)$.

## Remarks

For $v = e_i$, the directional derivative is the [[partial derivative|09EGC6F8PT]] $\partial f / \partial x_i$ at $a$.
Some books require that $v$ is a unit vector, so that $D_v f(a)$ is a rate of change per unit of distance.
If $D_v f(a)$ exists and $c \in \mathbb{R}$, then $D_{cv} f(a)$ exists and equals $c \, D_v f(a)$.
When $f$ is [[differentiable|0HHHDCXZV4]] at $a$, the directional derivatives are the values of the derivative: $D_v f(a) = Df(a)(v)$, see [[this statement|0A6PMR20N4]].

## Explanations

### L1 human

Stand at the point $a$ on the graph of $f$ and walk in the direction $v$.
The directional derivative is the slope that you feel at the first step.
For $f(x, y) = x^2 + y^2$ at $a = (1, 0)$ and $v = (0, 1)$, you get $f(1, t) = 1 + t^2$, so the slope is $0$. You start along the tangent of the circle $x^2 + y^2 = 1$, where $f$ is constant.
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# K-colourable graph

## Definition

Let $k$ be a natural number.
A [[graph|0JD5C82QMB]] $G$ is $k$-colourable if it has a [[colouring|0874138YP6]] with the set of colours $\{1, 2, \dots, k\}$.

## Remarks

A $k$-colourable graph is also $(k+1)$-colourable.
A graph is $2$-colourable if and only if it is [[bipartite|07DVHJ8N0A]].
The [[complete graph|03WKB2H474]] $K_n$ is $k$-colourable if and only if $k \ge n$.

## Explanations

### L1 human

A graph is $k$-colourable when $k$ colours are enough to colour it properly.
A triangle needs three colours, so it is $3$-colourable but not $2$-colourable.
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# Existence of an orthonormal basis

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$.
Then $V$ has an [[orthonormal basis|0S9GGTR0EX]].
Moreover, each orthonormal family in $V$ extends to an orthonormal basis of $V$.

## Remarks

The proof applies the [[Gram–Schmidt process|0VT6P0GNQR]] to any basis of $V$.
For the extension, first extend the orthonormal family to a basis, and then apply Gram–Schmidt; it keeps the first vectors.
Hence each finite-dimensional real inner product space is isometric to $\R^n$ with the dot product.

## Explanations

### L1 human

Every finite-dimensional inner product space has a basis of perpendicular unit vectors.
We construct one with the Gram–Schmidt process.
So every such space looks like $\R^n$ or $\C^n$ with the standard inner product.
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# Sequence of functions

## Summary

A sequence of functions is a sequence $(f_n)$ of functions $f_n \colon S \to \mathbb{R}$ on a common set $S$. For each fixed $x \in S$, the values $f_n(x)$ form a [[sequence of real numbers|0QE8STGJH2]]. Such sequences can converge in several ways, mainly [[pointwise|056FSKG32J]] or [[uniformly|08FPABDQ4C]].

## Remarks

The two modes give different properties of the limit. A uniform limit of continuous functions is continuous, but a pointwise limit need not be.
Examples are the Taylor polynomials of a function, and the partial sums of a power series.

## Explanations

### L1 human

A sequence of functions is a list of graphs, one for each $n$.
We ask whether the graphs approach the graph of a limit function, and in what sense.
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# Distinct eigenvalues imply diagonalisable

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
If $A$ has $n$ pairwise distinct eigenvalues in $F$, then $A$ is [[diagonalisable|07SQCNV233]] over $F$.

## Remarks

Eigenvectors for the $n$ distinct eigenvalues are [[linearly independent|0HNMFG8F8K]], so they form a basis of $F^n$.
The converse fails: $I_n$ is diagonal, but it has only one eigenvalue.
For example, $\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ has eigenvalues $1$ and $3$, so it is diagonalisable.

## Explanations

### L1 human

If an $n \times n$ matrix has $n$ different eigenvalues, then it has $n$ independent eigenvectors.
So it is diagonalisable.
This sufficient condition is easy to check with the characteristic polynomial.
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# Binary trees are counted by Catalan numbers

## Statement

Let $n$ be a natural number.
Then the number of [[binary trees|02A8EMC83C]] with exactly $n$ nodes is the [[Catalan number|092DRBFKMG]] $C_n$.

## Remarks

Here a binary tree is either empty or a root node with an ordered pair of binary trees as its left and right subtrees.
For $n = 3$ there are $C_3 = 5$ binary trees.
Proof idea: a nonempty tree with $n + 1$ nodes has a left subtree with $i$ nodes and a right subtree with $n - i$ nodes. This gives the [[Catalan recurrence|0Y1MG9R5VN]].

## Explanations

### L1 human

Remove the root of a binary tree, and you get a left tree and a right tree.
The sizes of the two trees add up to one less than the size of the original tree, and this gives the Catalan recurrence.
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# Second derivative

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be differentiable on $I$.
If $f'$ is differentiable at $a \in I$, the second derivative of $f$ at $a$ is $f''(a) = (f')'(a)$.

## Remarks

It is the [[higher derivative|084NGSTWM7]] $f^{(2)}$. Another notation is $\frac{d^2 f}{dx^2}$.
If $f'' > 0$ on $I$, then $f$ is [[convex|0VV0HSW1ZS]] on $I$.
In mechanics, if $f(t)$ is the position at time $t$, then $f''(t)$ is the acceleration.

## Explanations

### L1 human

The second derivative is the rate of change of the slope.
It tells how the graph bends: upward where $f'' > 0$ and downward where $f'' < 0$.
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# Order on the natural numbers

## Definition

Let $a$ and $b$ be [[natural numbers|0MWKJ44V5A]].
The notation $a \le b$ means that there is a natural number $c$ with $a + c = b$, where $+$ is the [[addition|0W5DSDFQ4G]] of $\mathbb{N}$.
The notation $a < b$ means that $a \le b$ and $a \neq b$.

## Remarks

The relation $\le$ is a total order on $\mathbb{N}$: it is reflexive, antisymmetric and transitive, and $a \le b$ or $b \le a$ for all $a$ and $b$.
The order agrees with the operations: if $a \le b$, then $a + c \le b + c$ and $a c \le b c$ for each natural number $c$.
Each nonempty set of natural numbers has a least element for this order. This is the [[well-ordering principle|01BX15HSRR]].

## Explanations

### L1 human

The number $a$ is at most $b$ when you can get from $a$ to $b$ by adding something.
For example, $3 \le 7$ because $3 + 4 = 7$, but $7 \le 3$ is false because no natural number added to $7$ gives $3$.
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# Characteristic polynomial of a linear recurrence

## Summary

The characteristic polynomial of a linear recurrence encodes its coefficients as a polynomial. Its roots give the geometric solutions, and they lead to closed formulas for all solutions. The node [[characteristic polynomial of a linear recurrence|0BWDJGQYST]] gives the precise definition.
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# Primitive element of a finite field

## Definition

Let $F$ be a [[finite field|08RTDHX1SK]] of order $q$.
An element $\alpha \in F$ is a primitive element of $F$ if $\alpha$ generates the multiplicative group $F^\times$.
That is, $F^\times = \{ \alpha, \alpha^2, \dots, \alpha^{q-1} = 1 \}$.

## Remarks

Primitive elements exist, because [[$F^\times$ is cyclic|0EBZ3G455G]]. There are exactly $\varphi(q - 1)$ of them, where $\varphi$ is Euler's totient function.
The element $2$ is a primitive element of $\mathbb{F}_3$ and of $\mathbb{F}_5$, but not of $\mathbb{F}_7$. The element $3$ is a primitive element of $\mathbb{F}_7$.
A primitive element $\alpha$ of $F$ generates $F$ as a [[simple extension|0P6SBR7YM2]] of its prime subfield. The converse is false in general.

## Explanations

### L1 human

A primitive element is one whose powers give every nonzero element of the field.
It works like a base for logarithms in the finite field.
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# Mersenne number

## Summary

A Mersenne number is an integer of the form $2^n - 1$, one less than a power of two.
The [[definition of a Mersenne number|0GJBJ19NXR]] gives the precise form.
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# Conditional expectation

## Definition

Let $X$ be an integrable real random variable on a probability space $(\Omega, \mathcal{F}, P)$, and let $\mathcal{G} \subseteq \mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]].
A conditional expectation of $X$ given $\mathcal{G}$ is an integrable random variable $Z$ with these two properties:
1. $Z$ is $\mathcal{G}$-measurable.
2. $E[Z \, 1_G] = E[X \, 1_G]$ for each $G \in \mathcal{G}$.
Such a $Z$ exists, and two such random variables are equal almost surely. One writes $Z = E[X \mid \mathcal{G}]$.
For a random variable $Y$, $E[X \mid Y] = E[X \mid \sigma(Y)]$, where $\sigma(Y)$ is the σ-algebra generated by $Y$.

## Remarks

The existence follows from the Radon–Nikodym theorem. For $E[X^2] < \infty$, $E[X \mid \mathcal{G}]$ is the orthogonal projection of $X$ onto $L^2(\mathcal{G})$.
For a discrete $Y$, $E[X \mid Y] = g(Y)$ with $g(y) = E[X \, 1_{\{Y = y\}}] / P(Y = y)$ when $P(Y = y) > 0$. This $g(y)$ is the expectation of $X$ under the [[conditional probability|0FJB06AD2P]] $P(\cdot \mid Y = y)$.
The conditional expectation is linear and monotone. If $W$ is $\mathcal{G}$-measurable and $WX$ is integrable, then $E[WX \mid \mathcal{G}] = W \, E[X \mid \mathcal{G}]$.
If $X$ is independent of $\mathcal{G}$, then $E[X \mid \mathcal{G}] = E[X]$.

## Explanations

### L1 human

$E[X \mid Y]$ is the average value of $X$ when you know $Y$. It changes with $Y$, so it is a random variable.
For two dice with results $D_1$ and $D_2$, $E[D_1 + D_2 \mid D_1] = D_1 + 3.5$.
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# Cayley table

## Summary

A Cayley table of a finite set with a binary operation is a square table. It shows the result of the operation for each pair of elements.
For a finite group, it shows the full multiplication of the group.
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# Absorbing Markov chain

## Definition

Let $(X_n)$ be a time-homogeneous [[Markov chain|033TJBZDG0]] with [[transition matrix|0F1KBVY161]] $P = (p_{ij})$ on a state space $S$.
A state $i \in S$ is absorbing if $p_{ii} = 1$. A state that is not absorbing is transient.
The chain is an absorbing Markov chain if it has at least one absorbing state and each state can reach an absorbing state.
That is, for each $i \in S$ there are an absorbing state $j$ and an $n \ge 0$ with $p^{(n)}_{ij} > 0$.

## Remarks

For a finite absorbing chain, write $P$ in the canonical form with the transient states first: $P = \begin{pmatrix} Q & R \\ 0 & I \end{pmatrix}$.
The fundamental matrix is $N = (I - Q)^{-1} = \sum_{n=0}^{\infty} Q^n$. The entry $N_{ij}$ is the expected number of visits to the transient state $j$, starting from the transient state $i$.
The expected number of steps before absorption from $i$ is the $i$-th entry of $N \mathbf{1}$, where $\mathbf{1}$ is the column vector of ones. The absorption probabilities are the entries of $NR$.
A finite absorbing chain reaches an absorbing state with probability $1$.

## Explanations

### L1 human

An absorbing chain wanders for a while and then gets stuck in a trap state forever.
The fortune in the gambler's ruin problem is an example: the states $0$ and $N$ are traps.
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# Absolute convergence

## Summary

A series converges absolutely when the series of the absolute values of its terms converges. Absolutely convergent series converge, and their sums do not depend on the order of the terms. The node [[absolute convergence|0W3K3QPEWT]] gives the precise definition.
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# Comparability of cardinals

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
Then there is an [[injective function|0PVG1CRAJN]] from $A$ to $B$ or an injective function from $B$ to $A$.
So $|A| \le |B|$ or $|B| \le |A|$.

## Remarks

With the [[Schröder–Bernstein theorem|0A9V6RKQG6]], it follows that the [[order of cardinals|01GTP8M5RG]] is a total order.
The statement is equivalent to the axiom of choice over ZF.
One proof uses [[Zorn's lemma|04RXTQA5ZA]] on the set of injective partial functions from $A$ to $B$.

## Explanations

### L1 human

Any two sets can be compared in size: one of them fits into the other.
Without the axiom of choice, two sets can exist such that neither fits into the other.
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<document source="https://tamnd.github.io/math2/node/0YBR6QPY4Y.md" tag="0YBR6QPY4Y" kind="example">
# Groups of order pq

## Example

Let $p < q$ be prime numbers such that $p$ does not divide $q - 1$, and let $G$ be a group of order $pq$.
The number $n_q$ divides $p$ and satisfies $n_q \equiv 1 \pmod q$, so $n_q = 1$, because $p < q$.
The number $n_p$ divides $q$ and satisfies $n_p \equiv 1 \pmod p$; since $p \nmid q - 1$, $n_p = 1$.
So $G$ has a [[normal|0CB0DX5BVV]] subgroup $P$ of order $p$ and a normal subgroup $Q$ of order $q$.
They meet only in $e$, so $G$ is the [[internal direct product|0VBZH90R1A]] of $P$ and $Q$.
Thus $G \cong \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z} \cong \mathbb{Z}/pq\mathbb{Z}$, and $G$ is cyclic.
For example, each group of order $15$ is cyclic.
If $p$ divides $q - 1$, there is also a group of order $pq$ that is not cyclic, for example the dihedral group $D_3$ of order $6$.
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# Identity theorem

## Statement

Let $D \subseteq \mathbb{C}$ be a [[domain|097KNJFJ7V]] and let $f, g : D \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.
Let $S = \{z \in D : f(z) = g(z)\}$.
Assume that $S$ has an accumulation point in $D$. This means that there is $z_0 \in D$ such that each disc with center $z_0$ contains a point of $S$ other than $z_0$.
Then $f(z) = g(z)$ for each $z \in D$.

## Remarks

Proof idea: apply the [[isolated zeros theorem|0K83KEJ43P]] to $f - g$. The set of points near which $f - g = 0$ is open, closed in $D$ and not empty, so it is $D$ because $D$ is connected.
For example, if $f$ is entire and $f(1/n) = 0$ for each $n \ge 1$, then $f = 0$.
The connectedness of $D$ is necessary: on the union of two disjoint discs, a function can be $0$ on one disc and $1$ on the other.
The theorem makes [[analytic continuation|0M3YQJ81NB]] unique.

## Explanations

### L1 human

Two holomorphic functions on a domain that agree on a small piece, even on a convergent sequence of points with its limit, agree everywhere.
So a holomorphic function is fixed everywhere by its values on a small disc.
</document>

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# Term-by-term integration

## Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be Riemann integrable for each $n \ge 0$.
If $\sum_{n} f_n$ converges uniformly on $[a, b]$, then its sum is Riemann integrable, and
$$\int_a^b \sum_{n=0}^{\infty} f_n(x) \, dx = \sum_{n=0}^{\infty} \int_a^b f_n(x) \, dx.$$

## Remarks

Apply [[the theorem on sequences|0BWGW68B66]] to the partial sums.
For example, integrating $\frac{1}{1 + t} = \sum (-t)^n$ term by term over $[0, x]$, with $|x| < 1$, gives the Mercator series for $\ln(1 + x)$.
Mathlib states a version with summable sup norms as `intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm`.

## Explanations

### L1 human

We may integrate a uniformly convergent series term by term.
The integral of the sum is the sum of the integrals.
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# Matrix exponential as the solution of a linear system

## Statement

Let $A$ be a real $n \times n$ matrix, and for $t \in \mathbb{R}$ let $e^{t A} = \sum_{k=0}^{\infty} \frac{t^k}{k!} A^k$, a series that converges for each $t$.
Then the map $t \mapsto e^{t A}$ is differentiable on $\mathbb{R}$, and
$$\frac{d}{dt}\, e^{t A} = A\, e^{t A} = e^{t A} A.$$
Consequently, for each $x_0 \in \mathbb{R}^n$, the function $x(t) = e^{t A} x_0$ is the unique solution on $\mathbb{R}$ of the [[initial value problem|075VBY4K4X]] $x' = A x$, $x(0) = x_0$.

## Remarks

Uniqueness: if $x$ is a solution, then $\frac{d}{dt}\left(e^{-tA} x(t)\right) = 0$, so $e^{-tA} x(t) = x_0$.
If $A = P D P^{-1}$ with $D$ diagonal, then $e^{tA} = P e^{tD} P^{-1}$, and $e^{tD}$ is diagonal with the entries $e^{t \lambda_k}$.
The matrix $e^{tA}$ is the [[fundamental matrix|0MQ24RBG93]] of $x' = A x$ with value $I$ at $t = 0$.

## Explanations

### L1 human

For one equation $y' = a y$, the solution is $e^{a t} y_0$.
The matrix exponential is the same formula for a system, with the matrix $A$ in place of the number $a$.
The series is the same as the series of $e^x$.
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# Augmented matrix

## Definition

Let $F$ be a field.
Take a [[system of linear equations|0GT4WWGWH2]] $Ax = b$ over $F$, with [[coefficient matrix|0HQFZAP3ND]] $A$ of size $m \times n$ and constant vector $b \in F^m$.
The *augmented matrix* of the system is the $m \times (n+1)$ matrix $[A \mid b]$.
Its first $n$ columns are the columns of $A$, and its last column is $b$.

## Remarks

The system $2x + y = 5$, $x - 3y = 0$ has the augmented matrix $\left(\begin{array}{cc|c} 2 & 1 & 5 \\ 1 & -3 & 0 \end{array}\right)$.
[[Elementary row operations|04XWH28452]] on the augmented matrix do not change the solution set of the system.
More generally, $[A \mid B]$ denotes the matrix $A$ with the columns of a matrix $B$ added on the right.

## Explanations

### L1 human

The augmented matrix is the full system written as a table of numbers.
The vertical bar reminds us where the equals signs were.
We do row reduction on this table instead of rewriting the equations each time.
</document>

<document source="https://tamnd.github.io/math2/node/0YCG648KDD.md" tag="0YCG648KDD" kind="concept">
# Open map

## Summary

An open map is a function between topological spaces that sends open sets to open sets.
Open maps need not be continuous, and continuous maps need not be open.
</document>

<document source="https://tamnd.github.io/math2/node/0YCW781Q9D.md" tag="0YCW781Q9D" kind="example">
# Seven Bridges of Königsberg

## Example

The city of Königsberg had four land areas, joined by seven bridges over the river Pregel.
The problem asked for a walk that crosses each bridge exactly once.

Model each land area as a vertex and each bridge as an edge. This gives a connected [[multigraph|01M5T0KDG7]] with $4$ vertices and $7$ edges.
The degrees of the vertices are $5$, $3$, $3$ and $3$.
An [[Eulerian trail|0NNYDD1AVR]] can have at most two vertices of odd degree, by the theorem on [[degrees along an Eulerian trail|0CDVDGWGEG]].
Here all four degrees are odd, so the walk does not exist.

## Remarks

Leonhard Euler solved the problem in 1736. His analysis is often seen as the start of graph theory.
The problem has no solution even if the walk does not have to return to its start.

## Explanations

### L1 human

Each time a walk passes through a land area, it uses two bridges there, one to enter and one to leave.
So a land area in the middle of the walk needs an even number of bridges.
Königsberg had four land areas with an odd number of bridges.
</document>

<document source="https://tamnd.github.io/math2/node/0YDRV01R8F.md" tag="0YDRV01R8F" kind="statement">
# Universal property of the tensor product

## Statement

Let $F$ be a field, and let $V$, $W$ and $U$ be vector spaces over $F$.
Let $h \colon V \times W \to U$ be a bilinear map.
Then there is exactly one linear map $\tilde h \colon V \otimes W \to U$ such that
$$\tilde h(v \otimes w) = h(v, w) \qquad \text{for all } v \in V \text{ and } w \in W,$$
where $V \otimes W$ is the [[tensor product|0BF2DE2SYK]].

## Remarks

So the bilinear maps $V \times W \to U$ correspond one to one with the [[linear maps|0NRYHJPQG2]] $V \otimes W \to U$.
This property determines $V \otimes W$ up to a unique isomorphism that fixes the vectors $v \otimes w$.
Uniqueness holds because the vectors $v \otimes w$ span $V \otimes W$.

## Explanations

### L1 human

Every bilinear map out of $V \times W$ factors through the tensor product in exactly one way.
So the study of bilinear maps reduces to the study of linear maps.
This property is the usual definition of the tensor product in more abstract settings.
</document>

<document source="https://tamnd.github.io/math2/node/0YE5868HZB.md" tag="0YE5868HZB" kind="concept">
# Open ball

## Summary

An open ball in a metric space is the set of all points at distance strictly less than a given radius from a given centre.
Open balls are the basic open sets of a metric space.
</document>

<document source="https://tamnd.github.io/math2/node/0YE597NHGB.md" tag="0YE597NHGB" kind="definition">
# Cayley graph

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a [[generating set|0MBV2VHMXA]] of $G$.
The Cayley graph $\Gamma(G, S)$ is the directed graph with vertex set $G$.
For each $g \in G$ and $s \in S$, it has an edge with colour $s$ from $g$ to $g s$.

## Remarks

For $G = \mathbb{Z}$ and $S = \{1, -1\}$, the Cayley graph is an infinite path. For $\mathbb{Z}/n\mathbb{Z}$ with a generator and its inverse, it is the cycle $C_n$.
The Cayley graph of a free group with respect to a basis is a tree.
The [[left multiplication action|0K5TWJAGPD]] of $G$ on itself acts on $\Gamma(G, S)$, so each Cayley graph is vertex-transitive.
Arthur Cayley first considered these graphs for finite groups in 1878.

## Explanations

### L1 human

A Cayley graph draws a group as a network.
Each element is a point, and each generator gives an arrow from $g$ to $g s$.
Paths in the graph correspond to words in the generators.
</document>

<document source="https://tamnd.github.io/math2/node/0YEQG3VBYC.md" tag="0YEQG3VBYC" kind="definition">
# Combinations with repetition

## Definition

Let $X$ be a set and let $k$ be a natural number.
A $k$-combination with repetition of $X$ is a multiset of size $k$ with elements in $X$.
That is a function $m : X \to \mathbb{N}$ with $\sum_{x \in X} m(x) = k$, where $m(x)$ is the multiplicity of $x$.

For natural numbers $n$ and $k$, the multiset coefficient $\left(\!\binom{n}{k}\!\right)$ is the number of $k$-combinations with repetition of an $n$-element set.

## Remarks

The value is $\left(\!\binom{n}{k}\!\right) = \binom{n + k - 1}{k}$. This is [[stars and bars|009TEMTQKH]].
For $X = \{a, b\}$ and $k = 2$ there are $3$ multisets: $\{a, a\}$, $\{a, b\}$ and $\{b, b\}$.
Unlike a [[combination|0JZSRWRBZE]], a combination with repetition can use an element more than once.

## Explanations

### L1 human

A combination with repetition is a choice of $k$ items from $n$ kinds. You can take the same kind more than once, and the order does not matter.
An example is a bag of 3 fruits from apples, pears and plums.
</document>

<document source="https://tamnd.github.io/math2/node/0YEW47VJ0J.md" tag="0YEW47VJ0J" kind="definition">
# Divisor function

## Definition

Let $k$ be a natural number.
The divisor function $\sigma_k$ is the [[arithmetic function|0H2EPRZC0D]] given by
$$\sigma_k(n) = \sum_{d \mid n} d^k,$$
where the sum runs over the [[positive divisors|0FCWA16FZE]] $d$ of $n$.

## Remarks

The case $k = 0$ is the [[number of divisors|0BKJT6WG96]], and the case $k = 1$ is the [[sum of divisors|009QCP5T3Z]].
For example, $\sigma_2(6) = 1 + 4 + 9 + 36 = 50$.
The same formula defines $\sigma_k$ for each real or complex number $k$.
The function $\sigma_k$ is [[multiplicative|04WC36XKV6]].

## Explanations

### L1 human

To compute $\sigma_k(n)$, list the divisors of $n$, raise each one to the power $k$, and add the results.
</document>

<document source="https://tamnd.github.io/math2/node/0YF0VXKXGB.md" tag="0YF0VXKXGB" kind="definition">
# Jacobian matrix

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f = (f_1, \dots, f_m) \colon U \to \mathbb{R}^m$, and let $a \in U$.
Suppose that each [[partial derivative|09EGC6F8PT]] $\partial f_i / \partial x_j$ exists at $a$, for $1 \le i \le m$ and $1 \le j \le n$.
The Jacobian matrix of $f$ at $a$ is the $m \times n$ matrix
$$J_f(a) = \begin{pmatrix} \dfrac{\partial f_1}{\partial x_1}(a) & \cdots & \dfrac{\partial f_1}{\partial x_n}(a) \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_m}{\partial x_1}(a) & \cdots & \dfrac{\partial f_m}{\partial x_n}(a) \end{pmatrix},$$
whose entry in row $i$ and column $j$ is $\partial f_i / \partial x_j (a)$.
Other notations are $Df(a)$, $f'(a)$ and $\dfrac{\partial (f_1, \dots, f_m)}{\partial (x_1, \dots, x_n)}$.

## Remarks

If $f$ is [[differentiable|0135XZZQ8M]] at $a$, then $Df(a)(h) = J_f(a) \, h$ for each column vector $h \in \mathbb{R}^n$.
Row $i$ of $J_f(a)$ is the transpose of the [[gradient|0ABK9V9Y64]] of $f_i$ at $a$. For $m = 1$, the Jacobian matrix is the gradient written as a row.
For the polar map $f(r, \theta) = (r \cos \theta, r \sin \theta)$, the Jacobian matrix is $\begin{pmatrix} \cos \theta & -r \sin \theta \\ \sin \theta & r \cos \theta \end{pmatrix}$.
For $m = n$, its determinant is the [[Jacobian determinant|0ZCSMYHTBP]].

## Explanations

### L1 human

The Jacobian matrix lists how each output changes with each input: one row for each output, one column for each input.
It is the multivariable form of the derivative. It turns a small step $h$ in the input into the approximate step $J_f(a) h$ in the output.
</document>

<document source="https://tamnd.github.io/math2/node/0YF6QXXWB5.md" tag="0YF6QXXWB5" kind="concept">
# Similar triangles

## Summary

Two triangles are similar when they have the same shape, but possibly different sizes. Then one is a scaled copy of the other, after a rigid motion and possibly a reflection.
</document>

<document source="https://tamnd.github.io/math2/node/0YF82GBTND.md" tag="0YF82GBTND" kind="statement">
# Continuity via open sets

## Statement

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be a function.
Then the following two conditions are equivalent:

1. For each $a \in X$ and each $\varepsilon > 0$ there is $\delta > 0$ such that $d_X(x, a) < \delta$ implies $d_Y(f(x), f(a)) < \varepsilon$ for all $x \in X$.
2. For each [[open set|01ZQQDCERP]] $V$ of $Y$, the [[preimage|0FGCFY1AZX]] $f^{-1}(V)$ is open in $X$.

## Remarks

Condition 2 is the definition of a [[continuous map|0SY9ECQ19Q]] between topological spaces. Thus, for metric spaces, the epsilon-delta definition and the topological definition agree.
Proof that 1 implies 2: let $a \in f^{-1}(V)$. Choose $\varepsilon$ with $B(f(a), \varepsilon) \subseteq V$, then $\delta$ from condition 1. Then $B(a, \delta) \subseteq f^{-1}(V)$.
Images of open sets need not be open: the constant map $\mathbb{R} \to \mathbb{R}$, $x \mapsto 0$, is continuous, but the image of $\mathbb{R}$ is $\{0\}$.

## Explanations

### L1 human

The topological form of continuity uses only open sets and does not mention distances.
This form lets us define continuity in spaces that have no metric at all.
</document>

<document source="https://tamnd.github.io/math2/node/0YG0ZJPTWG.md" tag="0YG0ZJPTWG" kind="example">
# Harmonic oscillator

## Example

Let $m > 0$ and $k > 0$ be constants, and let $\omega_0 = \sqrt{k/m}$.
The equation of the harmonic oscillator is
$$m\, x''(t) + k\, x(t) = 0, \quad \text{that is} \quad x'' + \omega_0^2\, x = 0.$$
It models a mass $m$ on a spring with stiffness $k$, where $x$ is the displacement from the rest position.
Its characteristic roots are $\pm i \omega_0$, so its solutions are $x(t) = c_1 \cos(\omega_0 t) + c_2 \sin(\omega_0 t) = A \cos(\omega_0 t + \varphi)$ with constants $A \ge 0$ and $\varphi$.

## Remarks

Each nonzero solution is periodic with period $2 \pi / \omega_0$. The period does not depend on the amplitude $A$.
The energy $E = \frac{1}{2} m (x')^2 + \frac{1}{2} k x^2$ is constant along each solution. It is a [[first integral|0SS1P1YC9A]].
With friction, the equation becomes the [[damped harmonic oscillator|0RNXVVTJTJ]].

## Explanations

### L1 human

The spring pulls the mass back with a force proportional to the displacement.
The mass overshoots the rest position, and the motion repeats forever as a sine wave.
A stiffer spring or a lighter mass gives a faster oscillation.
</document>

<document source="https://tamnd.github.io/math2/node/0YGN3VAA0D.md" tag="0YGN3VAA0D" kind="concept">
# Normaliser

## Summary

The normaliser of a subset $S$ of a group $G$ is the set of elements $g$ with $gSg^{-1} = S$.
It is the largest subgroup in which a given subgroup is normal.
</document>

<document source="https://tamnd.github.io/math2/node/0YH1M56SBN.md" tag="0YH1M56SBN" kind="statement">
# Sum of independent Poisson random variables

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] random variables with $X_i \sim \operatorname{Poisson}(\lambda_i)$ and $\lambda_i > 0$. Then
$$X_1 + \dots + X_n \sim \operatorname{Poisson}(\lambda_1 + \dots + \lambda_n).$$

## Remarks

Proof with [[generating functions|0REMT01N36]]: $G_{X_i}(z) = e^{\lambda_i (z - 1)}$, so the sum has $G(z) = e^{(\lambda_1 + \dots + \lambda_n)(z - 1)}$. This is the generating function of $\operatorname{Poisson}(\lambda_1 + \dots + \lambda_n)$.
A direct proof for $n = 2$ uses the binomial theorem: $\sum_{k=0}^{m} e^{-\lambda_1} \frac{\lambda_1^k}{k!} e^{-\lambda_2} \frac{\lambda_2^{m-k}}{(m-k)!} = e^{-(\lambda_1 + \lambda_2)} \frac{(\lambda_1 + \lambda_2)^m}{m!}$.
Conversely, given $X_1 + X_2 = m$, the conditional law of $X_1$ is $\operatorname{Bin}(m, \lambda_1 / (\lambda_1 + \lambda_2))$.

## Explanations

### L1 human

If calls come to two help desks as independent Poisson counts, the total number of calls is again a Poisson count. Its mean is the sum of the two means.
</document>

<document source="https://tamnd.github.io/math2/node/0YHFMXJJAZ.md" tag="0YHFMXJJAZ" kind="statement">
# Irrationality of the square root of a prime

## Statement

Let $p$ be a prime number.
Then $\sqrt{p}$ is an [[irrational number|080T2TXVQV]].

## Remarks

This generalizes the [[irrationality of the square root of 2|0DZGSDHD92]].
More generally, the square root of a natural number $n$ is irrational unless $n$ is a perfect square. See [[the roots of integers|0XWCKJZRYH]].

## Explanations

### L1 human

The numbers $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{7}$ are all irrational.
The proof for $\sqrt{2}$ works for each prime: if $a^2 = p b^2$, then $p$ divides $a$, and then $p$ also divides $b$.
</document>

<document source="https://tamnd.github.io/math2/node/0YJ3WRR9RY.md" tag="0YJ3WRR9RY" kind="example">
# Integers modulo n as a quotient group

## Example

Let $n$ be a positive integer.
The subgroup $n\mathbb{Z}$ of the [[additive group of integers|0BSCJA5G53]] is normal, because $\mathbb{Z}$ is abelian.
The [[quotient group|0KYX2N69J0]] $\mathbb{Z}/n\mathbb{Z}$ has the $n$ cosets $0 + n\mathbb{Z}, 1 + n\mathbb{Z}, \ldots, (n-1) + n\mathbb{Z}$.
The operation is $(a + n\mathbb{Z}) + (b + n\mathbb{Z}) = (a + b) + n\mathbb{Z}$, which is addition modulo $n$.

## Remarks

The coset $a + n\mathbb{Z}$ is the set of integers with the same remainder as $a$ on division by $n$.
Thus $\mathbb{Z}/n\mathbb{Z}$ is the cyclic group of order $n$ of clock arithmetic.
For $n = 0$, the quotient $\mathbb{Z}/0\mathbb{Z}$ is isomorphic to $\mathbb{Z}$.

## Explanations

### L1 human

Two integers are in the same coset of $n\mathbb{Z}$ when their difference is a multiple of $n$.
So the quotient group keeps only the remainder modulo $n$.
Adding cosets is the same as adding remainders and reducing modulo $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0YK27TJ4WF.md" tag="0YK27TJ4WF" kind="definition">
# Order of a pole

## Definition

Let $c$ be a [[pole|0WPQYW8MTE]] of $f$.
The order of the pole $c$ is the unique integer $m \ge 1$ with the following property. On some punctured disc $0 < |z - c| < \delta$,
$$f(z) = \frac{g(z)}{(z - c)^m},$$
where $g$ is [[holomorphic|0DG0Z72CQD]] on the disc $|z - c| < \delta$ and $g(c) \neq 0$.
A pole of order $1$ is a simple pole.

## Remarks

The order of the pole of $f$ at $c$ is the [[order of the zero|064N002TX4]] of $1/f$ at $c$.
In the Laurent series $\sum a_n (z - c)^n$ of $f$ at $c$, the order is $m$ if and only if $a_{-m} \neq 0$ and $a_n = 0$ for each $n < -m$.
Many authors give a pole of order $m$ the signed order $-m$. Then zeros have positive order and poles have negative order.
For example, $\frac{1}{\sin^2 z}$ has a pole of order $2$ at each point $k\pi$ with $k \in \mathbb{Z}$.

## Explanations

### L1 human

The order of a pole says how fast the function goes to infinity at the point.
Near a pole of order $m$, the function behaves like a constant times $1/(z - c)^m$.
</document>

<document source="https://tamnd.github.io/math2/node/0YKA9J3XEE.md" tag="0YKA9J3XEE" kind="statement">
# Hilbert's basis theorem

## Statement

Let $R$ be a [[Noetherian|0MVH4Y1CRZ]] [[commutative ring|0GYGHKX834]].
Then the [[polynomial ring|0Q9EZS90GY]] $R[x]$ is Noetherian.

## Remarks

By induction, $R[x_1, \dots, x_n]$ is Noetherian for every $n$. In particular, every ideal of $K[x_1, \dots, x_n]$ for a field $K$ is finitely generated.
The proof studies the ideals of the leading coefficients of the polynomials of each degree in an ideal of $R[x]$.
In algebraic geometry, the theorem says that every set of polynomial equations has the same solutions as a finite subset of it.

## History

David Hilbert proved the theorem in 1890, for polynomials in several variables over a field, in an article on invariant theory.
His proof was not constructive. It did not give a method to find the finite set of generators.

## Explanations

### L1 human

An ideal of polynomials can contain infinitely many polynomials.
The theorem says that a finite list of them always generates the whole ideal.
So every system of polynomial equations can be replaced by a finite one.
</document>

<document source="https://tamnd.github.io/math2/node/0YKDJBV26S.md" tag="0YKDJBV26S" kind="definition">
# Essential singularity

## Definition

Let $c$ be an [[isolated singularity|0TYB3HM9KE]] of $f$.
The singularity $c$ is essential if it is neither a [[removable singularity|0P18JC72RE]] nor a [[pole|0WPQYW8MTE]] of $f$.

## Remarks

The singularity $c$ is essential if and only if the [[principal part|0NDV5R6Z6X]] of $f$ at $c$ has infinitely many nonzero terms.
Equivalently, $f(z)$ has no limit in $\mathbb{C}$ as $z \to c$, and $|f(z)|$ does not tend to $\infty$.
The standard example is [[exp(1/z) at 0|0NB31CC4SN]]. The function $\sin(1/z)$ also has an essential singularity at $0$.
Near an essential singularity, $f$ comes close to each complex value, by the [[Casorati–Weierstrass theorem|02HHNPC8ED]].

## Explanations

### L1 human

An essential singularity is a bad point that is neither a removable hole nor a point where the function simply goes to infinity.
Near such a point, the function behaves wildly: along different paths it can go to $0$, to infinity, or to any other value.
</document>

<document source="https://tamnd.github.io/math2/node/0YKY4M3KRQ.md" tag="0YKY4M3KRQ" kind="definition">
# Principal branch

## Definition

Let $F$ be a [[multivalued function|03CPD1HGF3]] on a set $X \subseteq \mathbb{C}$, so that $F(z)$ is a set of complex numbers for each $z \in X$.
A principal branch of $F$ is a function $f$ on $X$ with $f(z) \in F(z)$ for each $z \in X$, chosen by a standard convention. Its value $f(z)$ is the principal value of $F$ at $z$.
For the main functions of complex analysis, the conventions are these. Let $z \neq 0$, and let $\operatorname{Arg} z \in (-\pi, \pi]$ be the [[principal value of the argument|0SQSQP748R]].

- The principal [[logarithm|0X2FY491PF]] is $\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$, where $\ln$ is the real natural logarithm.
- The principal square root is $\sqrt{z} = e^{\frac{1}{2} \operatorname{Log} z}$.
- For $w \in \mathbb{C}$, the principal [[power|0ZV9A7EV0F]] is $z^w = e^{w \operatorname{Log} z}$.

## Remarks

The imaginary part of $\operatorname{Log} z$ is in $(-\pi, \pi]$. For example, $\operatorname{Log} i = i\pi/2$ and $\operatorname{Log}(-1) = i\pi$.
These principal branches are continuous and holomorphic on $\mathbb{C} \setminus (-\infty, 0]$, see [[the principal logarithm is holomorphic off the branch cut|0XVGJE7Q04]]. They are not continuous at the points of $(-\infty, 0)$.
Some authors use $\operatorname{Arg} z \in [0, 2\pi)$ instead. This gives a different principal branch.

## Explanations

### L1 human

A multivalued function gives many values at a point. The principal branch is the rule that everybody agrees on for picking one value.
For the logarithm, the rule is to take the angle between $-180$ and $180$ degrees.
</document>

<document source="https://tamnd.github.io/math2/node/0YM6P6C206.md" tag="0YM6P6C206" kind="statement">
# Existence of nth roots of complex numbers

## Statement

Let $z$ be a complex number and let $n$ be a positive integer.
Then there is a complex number $w$ with $w^n = z$.

## Remarks

For $z = r e^{i \theta}$ with $r > 0$, one solution is $w = r^{1/n} e^{i \theta / n}$, by the [[polar form|0BRXFNQ5JV]].
For $z \neq 0$, there are exactly $n$ solutions. They are $w$ times the $n$th [[roots of unity|0E021GF2B8]].
This fact is a special case of the fundamental theorem of algebra.

## Explanations

### L1 human

Every complex number has a square root, a cube root, and so on.
For example, $-1$ has the square roots $i$ and $-i$, and $i$ has the square roots $\pm \frac{\sqrt{2}}{2}(1 + i)$.
To take an $n$th root, take the $n$th root of the length and divide the angle by $n$.
</document>

<document source="https://tamnd.github.io/math2/node/0YMB1676R1.md" tag="0YMB1676R1" kind="concept">
# Affine span

## Summary

The affine span of a set of points is the smallest [[affine subspace|0PYW26VNT6]] that contains them. For two distinct points it is the line through them.
</document>

<document source="https://tamnd.github.io/math2/node/0YMP521ZR5.md" tag="0YMP521ZR5" kind="statement">
# Power series are holomorphic in the disc of convergence

## Statement

Let $\sum_{n \ge 0} a_n (z - c)^n$ be a [[power series|0VARVQ9ZW1]] with radius of convergence $R > 0$, and let $D = \{z : |z - c| < R\}$ be its [[disc of convergence|045EWAXY9P]].
Let $f(z) = \sum_{n=0}^{\infty} a_n (z - c)^n$ for $z \in D$.
Then $f$ is [[holomorphic|0DG0Z72CQD]] on $D$, and for each $z \in D$,
$$f'(z) = \sum_{n=1}^{\infty} n a_n (z - c)^{n-1}.$$
The series for $f'$ has the same radius of convergence $R$.

## Remarks

By induction, $f$ has complex derivatives of all orders on $D$, and $a_n = f^{(n)}(c) / n!$. So the power series is the Taylor series of $f$ at $c$.
For example, the [[complex exponential|054VXHMCA3]] has $R = \infty$, so it is [[entire|0G2YJN0H5S]].
The converse is also true: [[holomorphic functions are analytic|0826SGV3R0]].

## Explanations

### L1 human

Inside its disc of convergence, a power series can be differentiated term by term, like a polynomial.
The result is again a power series with the same disc, so you can do it again and again.
</document>

<document source="https://tamnd.github.io/math2/node/0YMSEZE1VH.md" tag="0YMSEZE1VH" kind="definition">
# Free abelian group

## Definition

Let $A$ be an [[abelian group|082GHNPJ53]], written additively.
A basis of $A$ is a subset $B \subseteq A$ such that each $a \in A$ has exactly one expression $a = k_1 b_1 + \cdots + k_n b_n$ with distinct $b_i \in B$ and nonzero integers $k_i$.
The group $A$ is a free abelian group when it has a basis.

## Remarks

The additive group $\mathbb{Z}$ is free abelian with basis $\{1\}$, and $\mathbb{Z}^2$ is free abelian with basis $\{(1, 0), (0, 1)\}$.
All bases of $A$ have the same cardinality, the rank, and a free abelian group of finite rank $n$ is [[isomorphic|08MG7H8JW5]] to $\mathbb{Z}^n$.
The free abelian group on a set $S$ is the [[abelianisation|0AQZV0905N]] of the [[free group|0KTW26C80J]] $F_S$.
Each subgroup of a free abelian group is free abelian.

## Explanations

### L1 human

A free abelian group is like a vector space over the integers with a basis.
Each element is an integer combination of basis elements in exactly one way.
The standard examples are $\mathbb{Z}^n$ for $n \geq 0$.
</document>

<document source="https://tamnd.github.io/math2/node/0YMT0D2442.md" tag="0YMT0D2442" kind="concept">
# Irreducible element

## Summary

An irreducible element of an integral domain is a nonzero nonunit that is not a product of two nonunits.
The [[definition of an irreducible element|052SDEPZH7]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0YMYMNQ0QZ.md" tag="0YMYMNQ0QZ" kind="concept">
# Simple group

## Summary

A simple group is a nontrivial group whose only normal subgroups are the trivial subgroup and the group itself.
The finite simple groups are the building blocks of all finite groups.
</document>

<document source="https://tamnd.github.io/math2/node/0YP7WYXF9V.md" tag="0YP7WYXF9V" kind="definition">
# Direct product of groups

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]].
The direct product $G \times H$ is the set of ordered pairs $(g, h)$ with $g \in G$ and $h \in H$, with the operation $(g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2)$.

## Remarks

The identity is $(e_G, e_H)$, and the inverse of $(g, h)$ is $(g^{-1}, h^{-1})$.
If $G$ and $H$ are finite, then $|G \times H| = |G| \cdot |H|$.
The [[order|0WE2NHZJ7B]] of $(g, h)$ is the least common multiple of the orders of $g$ and $h$, when both are finite.
For example, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ is cyclic of order $6$, but $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ is not cyclic.

## Explanations

### L1 human

The direct product runs two groups side by side.
An element is a pair, and you multiply the first parts in $G$ and the second parts in $H$.
The two parts never interact.
</document>

<document source="https://tamnd.github.io/math2/node/0YPRY7T4PK.md" tag="0YPRY7T4PK" kind="definition">
# Flux through a surface

## Definition

Let $S \subseteq \mathbb{R}^3$ be an [[oriented surface|0EKZW08FQ5]] with unit normal field $n$, and let $F$ be a continuous [[vector field|0N67NBKQ5T]] on $S$.
The flux of $F$ through $S$ is the [[surface integral|0DQ8ERJZNS]]
$$\iint_S F \cdot n \, dS.$$

## Remarks

If $r \colon D \to \mathbb{R}^3$ is a regular [[parametrisation|05788J51WZ]] of $S$ with $r_u \times r_v$ pointing in the direction of $n$, then the flux is $\iint_D F\big( r(u, v) \big) \cdot \big( r_u \times r_v \big) \, du \, dv$.
Reversing the orientation of $S$ changes the sign of the flux. The notation $\iint_S F \cdot d\mathbf{S}$, with $d\mathbf{S} = n \, dS$, is common.
Example: for $F(x, y, z) = (x, y, z)$ and the unit sphere with the outward normal $n(p) = p$, we get $F \cdot n = 1$, so the flux is the area $4\pi$.
If $F$ is the velocity field of a fluid, the flux is the volume of fluid that crosses $S$ per unit time. Flow in the direction of $n$ counts as positive.

## Explanations

### L1 human

Flux measures how much of a field goes through a surface.
Only the part of the field perpendicular to the surface counts. Flow along the surface does not cross it.
</document>

<document source="https://tamnd.github.io/math2/node/0YQ458D7PF.md" tag="0YQ458D7PF" kind="statement">
# Fundamental theorem of contour integration

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be continuous.
Let $F : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] with $F'(z) = f(z)$ for each $z \in U$.
Let $\gamma : [a, b] \to U$ be a continuous and piecewise continuously differentiable path.
Then the [[contour integral|00FWNP9TXY]] of $f$ along $\gamma$ is
$$\int_\gamma f(z) \, dz = F(\gamma(b)) - F(\gamma(a)).$$
In particular, $\int_\gamma f(z) \, dz = 0$ if $\gamma$ is closed.

## Remarks

The function $F$ is a primitive of $f$ on $U$. The integral depends only on the end points of $\gamma$.
For each integer $n \neq -1$, $z^n$ has the primitive $z^{n+1}/(n+1)$ on $\mathbb{C} \setminus \{0\}$. So $\oint_\gamma z^n \, dz = 0$ for each closed path $\gamma$ that does not pass through $0$.
The function $1/z$ has no primitive on $\mathbb{C} \setminus \{0\}$, because its [[integral over the unit circle|0SXZ68HV6F]] is $2 \pi i \neq 0$.

## Explanations

### L1 human

This is the complex version of $\int_a^b F'(t) \, dt = F(b) - F(a)$.
If you know a function whose derivative is $f$, you get the integral along any path from its values at the two ends.
</document>

<document source="https://tamnd.github.io/math2/node/0YQPZB5Q1G.md" tag="0YQPZB5Q1G" kind="concept">
# Orbit

## Summary

The orbit of a point $x$ under a group action is the set of all points to which elements of the group move $x$.
</document>

<document source="https://tamnd.github.io/math2/node/0YQQYET8YQ.md" tag="0YQQYET8YQ" kind="statement">
# Intersecting chords theorem

## Statement

Let $A$, $B$, $C$ and $D$ be [[concyclic points|049DCJZ448]] of the Euclidean plane $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Suppose that the chords $AC$ and $BD$ meet at a point $S$ that is inside both segments.
Then
$$d(A, S)\, d(S, C) = d(B, S)\, d(S, D).$$

## Remarks

Both products are equal to $r^2 - d(S, O)^2$, where $O$ is the centre and $r$ the radius. This is minus the [[power|02E9VT1RVQ]] of $S$.
A proof: the triangles $ASD$ and $BSC$ are similar, by the inscribed angle theorem.
The theorem is Proposition 35 of Book III of Euclid's Elements.

## Explanations

### L1 human

Draw two chords of a circle that cross each other. Each chord is cut into two pieces. The product of the two pieces is the same for both chords.
</document>

<document source="https://tamnd.github.io/math2/node/0YQSGNYE3K.md" tag="0YQSGNYE3K" kind="concept">
# Matrix of a linear map

## Summary

The matrix of a linear map in two bases is a matrix with one column for each basis vector of the domain.
Its column $j$ is the coordinate vector of the image of basis vector $j$.
</document>

<document source="https://tamnd.github.io/math2/node/0YRG241HQQ.md" tag="0YRG241HQQ" kind="statement">
# Pascal's rule

## Statement

Let $n$ and $k$ be natural numbers.
Then
$$\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}.$$

## Remarks

For $n = 4$ and $k = 1$ it gives $\binom{5}{2} = \binom{4}{1} + \binom{4}{2} = 4 + 6 = 10$.
Combinatorial proof: fix one element $x$ of an $(n+1)$-element set. A $(k+1)$-subset contains $x$, and then it has $k$ other elements, or it omits $x$.
The rule builds [[Pascal's triangle|05DME4TZ3K]] row by row.

## Explanations

### L1 human

Each number in Pascal's triangle is the sum of the two numbers above it.
This rule is the same fact: a team of $k+1$ from $n+1$ people either includes a fixed person or does not.
</document>

<document source="https://tamnd.github.io/math2/node/0YS40B38JD.md" tag="0YS40B38JD" kind="statement">
# Closed balls are closed

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $x \in X$ and let $r \ge 0$.
Then the [[closed ball|05DSV40EB9]] $\bar{B}(x, r)$ is a [[closed set|0G5AV9Q4RR]] of $X$.

## Remarks

Proof: let $y \notin \bar{B}(x, r)$ and let $\varepsilon = d(x, y) - r > 0$. If $d(y, z) < \varepsilon$, then $d(x, z) \ge d(x, y) - d(y, z) > r$ by the [[triangle inequality|01M6CW9VM6]]. Thus the complement of $\bar{B}(x, r)$ is open.
For $r = 0$, this shows that each one-point set $\{x\}$ is closed in a metric space.

## Explanations

### L1 human

A point outside the closed ball is at some positive distance beyond its edge.
A small ball about that point stays outside, so the outside is open.
</document>

<document source="https://tamnd.github.io/math2/node/0YS8RX87F4.md" tag="0YS8RX87F4" kind="definition">
# Line integral of a vector field

## Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a [[rectifiable curve|0C28RT5BG7]] with image $C$, and let $F \colon C \to \mathbb{R}^n$ be a continuous [[vector field|0N67NBKQ5T]].
For a partition $a = t_0 < t_1 < \dots < t_k = b$ and points $\tau_i \in [t_{i-1}, t_i]$, consider the sum $\sum_{i=1}^{k} F(\gamma(\tau_i)) \cdot \big( \gamma(t_i) - \gamma(t_{i-1}) \big)$.
The line integral of $F$ along $\gamma$ is the limit of these sums as $\max_i (t_i - t_{i-1}) \to 0$. It is written
$$\int_\gamma F \cdot d\mathbf{r} \quad \text{or} \quad \int_\gamma F_1 \, dx_1 + \dots + F_n \, dx_n.$$

## Remarks

The limit exists, and it does not depend on the choice of the points $\tau_i$.
For a $C^1$ curve, $\int_\gamma F \cdot d\mathbf{r} = \int_a^b F(\gamma(t)) \cdot \gamma'(t) \, dt$, by the [[parametric formula|0W3W1VYWBP]].
For a regular curve with unit tangent vector $T$, $\int_\gamma F \cdot d\mathbf{r} = \int_\gamma (F \cdot T) \, ds$. So it is the line integral of the scalar field $F \cdot T$.
The value depends on the direction of travel. [[Reversing the path|0FSCXYG856]] changes its sign.

## Explanations

### L1 human

At each small step along the curve, take the dot product of the field with the step.
This counts only the part of the field that pushes along the curve, and the line integral is the total of these contributions.
</document>

<document source="https://tamnd.github.io/math2/node/0YSCSX5YZC.md" tag="0YSCSX5YZC" kind="concept">
# Ordinal number

## Summary

Ordinal numbers extend the natural numbers to measure the length of well-ordered sets, including infinite ones.
Each well-ordered set is order isomorphic to exactly one ordinal.
</document>

<document source="https://tamnd.github.io/math2/node/0YSE802YCQ.md" tag="0YSE802YCQ" kind="definition">
# Group action

## Definition

Let $G$ be a [[group|G5S1M9]] with identity $e$, and let $X$ be a set.
A (left) action of $G$ on $X$ is a map $G \times X \to X$, $(g, x) \mapsto g \cdot x$, with these two properties:

1. $e \cdot x = x$ for each $x \in X$.
2. $g \cdot (h \cdot x) = (gh) \cdot x$ for all $g, h \in G$ and $x \in X$.

## Remarks

For each $g \in G$, the map $x \mapsto g \cdot x$ is a bijection of $X$, with inverse $x \mapsto g^{-1} \cdot x$.
The map that sends $g$ to this bijection is a [[group homomorphism|0JYTG5K9Y4]] from $G$ to the [[symmetric group|0515A6D2Y1]] $\operatorname{Sym}(X)$.
Conversely, each homomorphism $G \to \operatorname{Sym}(X)$ gives an action of $G$ on $X$.
A right action is a map $X \times G \to X$ with $x \cdot e = x$ and $(x \cdot g) \cdot h = x \cdot (gh)$.

## Explanations

### L1 human

A group action lets the elements of a group move the points of a set.
The identity does not move any point, and moving by $h$ and then by $g$ is the same as moving by $gh$.
For example, the rotations of a square move the four corners of the square.
</document>

<document source="https://tamnd.github.io/math2/node/0YSRR1JTKZ.md" tag="0YSRR1JTKZ" kind="counterexample">
# x^n converges pointwise but not uniformly on [0, 1]

## Counterexample

Let $f_n(x) = x^n$ on $[0, 1]$. Let $f(x) = 0$ for $0 \le x < 1$ and $f(1) = 1$.
For $0 \le x < 1$, $x^n \to 0$ by the [[geometric sequence limit|0MNM54WB99]]. So $f_n \to f$ [[pointwise|056FSKG32J]] on $[0, 1]$.
But $\sup_{0 \le x < 1} |x^n - 0| = 1$ for every $n$. So the convergence is not [[uniform|08FPABDQ4C]].

## Remarks

Each $f_n$ is continuous, but $f$ is not. This also shows the failure, by the [[uniform limit theorem|0X9XT05JVC]].
On $[0, r]$ with $0 < r < 1$, the convergence is uniform, since $\sup_{[0, r]} x^n = r^n \to 0$.

## Explanations

### L1 human

Near $x = 1$, the values $x^n$ stay close to $1$ for a long time before they drop to $0$.
So no single $n$ makes all the values small at once.
</document>

<document source="https://tamnd.github.io/math2/node/0YT586TKXF.md" tag="0YT586TKXF" kind="definition">
# Cyclic subgroup generated by an element

## Definition

Let $G$ be a [[group|G5S1M9]] and let $g \in G$.
The cyclic subgroup generated by $g$ is the set $\langle g \rangle = \{g^k : k \in \mathbb{Z}\}$ of all integer powers of $g$.

## Remarks

By [[the laws of exponents|00DRK90Z6K]], $g^j (g^k)^{-1} = g^{j-k}$, thus $\langle g \rangle$ is a [[subgroup|0X1JXT47GA]] of $G$.
It is the [[subgroup generated by|0RP7VWQ0YN]] the set $\{g\}$.
In additive notation, $\langle g \rangle = \{kg : k \in \mathbb{Z}\}$, for example $\langle 3 \rangle = 3\mathbb{Z}$ in $(\mathbb{Z}, +)$.

## Explanations

### L1 human

Take one element and all its powers, positive and negative, and the identity.
These powers form a subgroup, the smallest one that contains the element.
In the integers with addition, the cyclic subgroup generated by $3$ is the set of multiples of $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0YTCMFB4HC.md" tag="0YTCMFB4HC" kind="concept">
# Critical point

## Summary

A critical point of a real function is a point where the derivative is zero or does not exist. All interior local extrema of the function occur at critical points. The node [[critical point|0GEQ42PTGB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0YTJ3RN3GE.md" tag="0YTJ3RN3GE" kind="concept">
# Dyck word

## Summary

A Dyck word is a balanced word of opening and closing brackets. Dyck words form the Dyck language of formal language theory, and those of semilength $n$ are counted by the Catalan number $C_n$. The node [[Dyck word|0R9JR0465V]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0YV2CKP2AV.md" tag="0YV2CKP2AV" kind="concept">
# Root of unity

## Summary

A root of unity is a number whose power with some positive integer exponent is $1$.
In $\mathbb{C}$, the $n$th roots of unity are the $n$ points $e^{2 \pi i k / n}$ on the unit circle. They are the vertices of a regular $n$-gon.
</document>

<document source="https://tamnd.github.io/math2/node/0YVDC5PYDW.md" tag="0YVDC5PYDW" kind="definition">
# Conditional probability distribution

## Definition

Let $X$ and $Y$ be random variables on one probability space.
Discrete case: let $X$ and $Y$ be discrete and let $x$ satisfy $P(X = x) > 0$. The conditional distribution of $Y$ given $X = x$ has the [[probability mass function|0WF4EJP91W]]
$$p_{Y \mid X}(y \mid x) = P(Y = y \mid X = x) = \frac{P(X = x, \, Y = y)}{P(X = x)}.$$
Continuous case: let $(X, Y)$ have a joint density $f_{X, Y}$, and let $f_X$ be the [[marginal|0AXP2Z5N9H]] density of $X$. For each $x$ with $f_X(x) > 0$, the conditional distribution of $Y$ given $X = x$ has the [[density|04YG690N5V]]
$$f_{Y \mid X}(y \mid x) = \frac{f_{X, Y}(x, y)}{f_X(x)}.$$

## Remarks

In the discrete case, $p_{Y \mid X}(\cdot \mid x)$ is the law of $Y$ under the [[conditional probability|0FJB06AD2P]] $P(\cdot \mid X = x)$.
In both cases, the joint law is the marginal times the conditional: $f_{X, Y}(x, y) = f_X(x) \, f_{Y \mid X}(y \mid x)$.
If $X$ and $Y$ are independent, the conditional distribution of $Y$ given $X = x$ is the law of $Y$, for each such $x$.
In general measure theory, one defines it as a regular conditional distribution. This is a probability kernel $\kappa$ with $P(X \in A, Y \in B) = \int_A \kappa(x, B) \, \mu_X(dx)$.

## Explanations

### L1 human

The conditional distribution answers the question: "If I know $X$, how is $Y$ distributed?"
For a person of height $180$ cm, it describes the possible weights of that person and their chances.
</document>

<document source="https://tamnd.github.io/math2/node/0YVKNB7V5F.md" tag="0YVKNB7V5F" kind="concept">
# Metric topology

## Summary

In the metric topology of a metric space, the open sets are the sets that contain a small ball about each of their points.
It turns each metric space into a topological space.
</document>

<document source="https://tamnd.github.io/math2/node/0YVQKBNV2R.md" tag="0YVQKBNV2R" kind="concept">
# Skewness

## Summary

The skewness of a distribution measures its asymmetry about the mean.
The node [[skewness|0GQ3H7VW3V]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0YVRNEDSVG.md" tag="0YVRNEDSVG" kind="definition">
# Floor function

## Definition

Let $x$ be a real number.
The floor of $x$, written $\lfloor x \rfloor$, is the greatest [[integer|0C4JS9RF5M]] $n$ with $n \le x$.
The floor function is the function $\mathbb{R} \to \mathbb{Z}$, $x \mapsto \lfloor x \rfloor$.

## Remarks

The greatest such integer exists because the [[reals are Archimedean|0A3GWZ86AY]] and because of the [[well-ordering principle|01BX15HSRR]].
For each integer $n$, $\lfloor n \rfloor = n$. For example, $\lfloor \pi \rfloor = 3$ and $\lfloor -0.5 \rfloor = -1$.
The [[characterization of the floor|0R9B0B5458]] gives an equivalent definition.

## Explanations

### L1 human

The floor of a number rounds it down to a whole number.
On the number line, it is the nearest integer at the number or to its left.
For negative numbers, take care: the floor of $-1.2$ is $-2$, not $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/0YVVN6R715.md" tag="0YVVN6R715" kind="statement">
# Homotopy invariance of line integrals of closed forms

## Statement

Let $U \subseteq \mathbb{R}^3$ be open, and let $F \colon U \to \mathbb{R}^3$ be a vector field of class $C^1$ with $\nabla \times F = 0$ on $U$.
Let $c_0, c_1 \colon [0, 1] \to U$ be piecewise smooth [[closed curves|0D032NSZKY]].
Suppose that there is a piecewise smooth map $H \colon [0, 1] \times [0, 1] \to U$ with $H(t, 0) = c_0(t)$, $H(t, 1) = c_1(t)$ and $H(0, s) = H(1, s)$ for all $s, t \in [0, 1]$.
Then
$$\oint_{c_0} F \cdot d\mathbf{r} = \oint_{c_1} F \cdot d\mathbf{r}.$$

## Remarks

Proof: apply [[Stokes' theorem|0AKRKVHW94]] to the surface $H([0, 1]^2)$. The curl term is zero, and the integrals along the curves $s \mapsto H(0, s)$ and $s \mapsto H(1, s)$ cancel.
If $c_1$ is a constant loop, then $\oint_{c_0} F \cdot d\mathbf{r} = 0$.
On a [[simply connected|075C629XTH]] domain, each loop deforms to a point, and a smoothing argument gives a piecewise smooth $H$. So curl-free $C^1$ fields on such domains are [[conservative|05R9NF2J49]].
The condition $\nabla \times F = 0$ means that the 1-form $F_1 \, dx + F_2 \, dy + F_3 \, dz$ is closed. This explains the name of the theorem.

## Explanations

### L1 human

For a field with no swirl, the work around a loop does not change when you slide and bend the loop continuously inside the domain.
A loop that can shrink to a point therefore has zero work.
</document>

<document source="https://tamnd.github.io/math2/node/0YVVXT3A0B.md" tag="0YVVXT3A0B" kind="definition">
# Number of distinct prime factors

## Definition

For a positive integer $n$, $\omega(n)$ is the number of distinct [[prime numbers|0SN0MAEKYX]] that divide $n$.
So if $n = p_1^{e_1} \cdots p_k^{e_k}$ with distinct primes $p_i$ and $e_i \ge 1$, then $\omega(n) = k$.

## Remarks

For example, $\omega(1) = 0$, $\omega(12) = 2$ and $\omega(30) = 3$.
The function $\omega$ is additive: $\omega(mn) = \omega(m) + \omega(n)$ for coprime $m$ and $n$.
Compare with the [[number of prime factors with multiplicity|08420828EY]] $\Omega(n)$. We have $\omega(n) \le \Omega(n)$, with equality exactly for squarefree $n$.

## Explanations

### L1 human

The function $\omega(n)$ counts the different primes in $n$ and ignores repeats.
For $360 = 2^3 \cdot 3^2 \cdot 5$, the different primes are $2$, $3$ and $5$, so $\omega(360) = 3$.
</document>

<document source="https://tamnd.github.io/math2/node/0YWDAZPRPQ.md" tag="0YWDAZPRPQ" kind="statement">
# Decomposition of a permutation into disjoint cycles

## Statement

Let $X$ be a finite set and let $\sigma \in \operatorname{Sym}(X)$.
Then $\sigma$ is a product of cycles of length at least $2$ whose sets of moved points are pairwise disjoint.
This product is unique up to the order of the factors.

## Remarks

The cycles come from the orbits of $\langle \sigma \rangle$ on $X$: each orbit with more than one point gives one cycle.
The order of the factors does not matter, because [[disjoint cycles commute|07E6D8R4D6]].
For example, the permutation $1 \mapsto 2$, $2 \mapsto 1$, $3 \mapsto 4$, $4 \mapsto 5$, $5 \mapsto 3$ in $S_5$ is $(1\ 2)(3\ 4\ 5)$.

## Explanations

### L1 human

Start at a point and follow the permutation until you return.
This gives one cycle; then start again at a point that you did not visit yet.
When all points are visited, the permutation is the product of the cycles that you found.
</document>

<document source="https://tamnd.github.io/math2/node/0YWZJEABB2.md" tag="0YWZJEABB2" kind="concept">
# Catalan numbers

## Summary

The Catalan numbers $C_n$ form a sequence that counts many recursive structures, such as Dyck words, binary trees and bracketings. They satisfy a convolution recurrence and the formula $C_n = \binom{2n}{n}/(n+1)$. The node [[Catalan numbers|092DRBFKMG]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0YXEHVA93R.md" tag="0YXEHVA93R" kind="definition">
# Tangent plane to a graph

## Definition

Let $U \subseteq \mathbb{R}^2$ be open, let $f \colon U \to \mathbb{R}$, and let $(a, b) \in U$ be a point at which $f$ is [[differentiable|0135XZZQ8M]].
The tangent plane to the [[graph|0JBJYT69ZV]] $z = f(x, y)$ at the point $(a, b, f(a, b))$ is the plane in $\mathbb{R}^3$ with the equation
$$z = f(a, b) + \frac{\partial f}{\partial x}(a, b)\,(x - a) + \frac{\partial f}{\partial y}(a, b)\,(y - b).$$
More generally, let $U \subseteq \mathbb{R}^n$ be open and let $f$ be differentiable at $a \in U$. The tangent hyperplane to the graph at $(a, f(a))$ is the set of points $(x, z) \in \mathbb{R}^n \times \mathbb{R}$ with $z = f(a) + \nabla f(a) \cdot (x - a)$.

## Remarks

The vector $\big(\partial f / \partial x(a, b), \, \partial f / \partial y(a, b), \, -1\big)$ is normal to the tangent plane.
For $f(x, y) = x^2 + y^2$ at $(1, 1)$, the tangent plane is $z = 2 + 2(x - 1) + 2(y - 1)$, that is, $z = 2x + 2y - 2$.
The tangent plane exists only where $f$ is differentiable. The existence of the [[partial derivatives|09EGC6F8PT]] alone gives a plane, but this plane need not touch the graph well.

## Explanations

### L1 human

Put a flat sheet on top of a smooth hill so that it touches the hill at one point and does not tilt against it.
That sheet is the tangent plane. Near the point of contact, the hill and the sheet are almost the same.
</document>

<document source="https://tamnd.github.io/math2/node/0YXEQWRGP0.md" tag="0YXEQWRGP0" kind="definition">
# Choice function

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]] of nonempty sets.
A choice function on $X$ is a [[function|0XHDTE5YMZ]] $c \colon X \to \bigcup X$ such that $c(S) \in S$ for each $S \in X$.

## Remarks

For a finite set $X$, a choice function exists without any axiom of choice.
For the set of nonempty subsets of $\mathbb{N}$, the function that sends each subset to its least element is a choice function.
The [[axiom of choice|0HTGB13FE2]] says that each set of nonempty sets has a choice function.

## Explanations

### L1 human

Think of $X$ as a collection of boxes, none empty.
A choice function names one object in each box.
Sometimes a rule gives it, such as "take the smallest number".
</document>

<document source="https://tamnd.github.io/math2/node/0YY06KP3ZQ.md" tag="0YY06KP3ZQ" kind="statement">
# Quadratic reciprocity for the Jacobi symbol

## Statement

Let $m$ and $n$ be coprime positive odd integers.
Then
$$\left(\frac{m}{n}\right) \left(\frac{n}{m}\right) = (-1)^{\frac{m-1}{2} \cdot \frac{n-1}{2}},$$
where the symbols are [[Jacobi symbols|0XDB5AW7M1]].

## Remarks

The supplements also extend: for a positive odd $n$, $\left(\frac{-1}{n}\right) = (-1)^{(n-1)/2}$ and $\left(\frac{2}{n}\right) = (-1)^{(n^2-1)/8}$.
The proof uses the [[law of quadratic reciprocity|085MJS6907]] for each pair of prime factors.
With this law, one computes $\left(\frac{a}{n}\right)$ by an algorithm like the [[Euclidean algorithm|0TRNT4X7J0]], with no factorization.

## Explanations

### L1 human

The Jacobi symbol obeys the same reciprocity law as the Legendre symbol.
So you can flip the symbol, reduce the top number modulo the bottom number, and repeat.
This gives a fast way to compute Legendre symbols for large primes.
</document>

<document source="https://tamnd.github.io/math2/node/0YY8VX28JR.md" tag="0YY8VX28JR" kind="concept">
# Moore–Penrose inverse

## Summary

The Moore–Penrose inverse of a matrix is the unique matrix that satisfies four equations of Penrose.
It gives the least squares solution of least norm of a linear system.
</document>

<document source="https://tamnd.github.io/math2/node/0YYMY95FWS.md" tag="0YYMY95FWS" kind="definition">
# Matrix of a linear map

## Definition

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$ with [[bases|0KCQ9Y3QKA]] $B = (b_1, \dots, b_n)$ and $C = (c_1, \dots, c_m)$.
Let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *matrix of $T$* in the bases $B$ and $C$ is the $m \times n$ matrix $[T]_{C,B} = (a_{ij})$ defined by
$$T(b_j) = \sum_{i=1}^{m} a_{ij} c_i \qquad (1 \le j \le n).$$
Equivalently, column $j$ of $[T]_{C,B}$ is the [[coordinate vector|04CYQHN6MC]] $[T(b_j)]_C$.

## Remarks

For each $v \in V$, $[T(v)]_C = [T]_{C,B} [v]_B$.
The map $T \mapsto [T]_{C,B}$ is a linear isomorphism $\Hom_F(V, W) \to M_{m \times n}(F)$.
For the rotation of $\R^2$ by an angle $\theta$, in the standard basis, the matrix is $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$.
Mathlib writes this map as `LinearMap.toMatrix`.

## Explanations

### L1 human

To write a linear map as a matrix, apply the map to each basis vector.
Write the coordinates of each image as a column.
These columns, side by side, are the matrix.
</document>

<document source="https://tamnd.github.io/math2/node/0YYT36R4YD.md" tag="0YYT36R4YD" kind="concept">
# Conformal map

## Summary

A conformal map is a map that preserves angles and their orientation at each point, but not necessarily lengths, see the [[definition|0VHNKP1N3K]].
In the plane, the conformal maps on an open set are the [[holomorphic|0DG0Z72CQD]] functions with a derivative that is never zero.
</document>

<document source="https://tamnd.github.io/math2/node/0YYWCNTPWR.md" tag="0YYWCNTPWR" kind="definition">
# Quotient group

## Definition

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
The quotient group $G/N$ is the set of [[left cosets|03DEH43F4H]] $\{gN : g \in G\}$ with the operation $(aN)(bN) = (ab)N$.

## Remarks

The operation is well defined because $N$ is normal: if $aN = a'N$ and $bN = b'N$, then $abN = a'b'N$.
The identity of $G/N$ is $N = eN$, and the inverse of $aN$ is $a^{-1}N$.
When $G$ is finite, $|G/N| = [G : N] = |G| / |N|$.
The map $g \mapsto gN$ is a surjective homomorphism from $G$ to $G/N$ with kernel $N$.

## Explanations

### L1 human

A quotient group treats all elements of one coset as one element.
For example, the integers modulo $2$ come from the integers when you treat all even numbers as one element and all odd numbers as another.
The normality of $N$ makes sure that the product of two cosets does not depend on the chosen elements.
</document>

<document source="https://tamnd.github.io/math2/node/0YZ3T5Q7XN.md" tag="0YZ3T5Q7XN" kind="example">
# Determinant as a group homomorphism

## Example

Let $n$ be a positive integer and let $\mathrm{GL}_n(\mathbb{R})$ be the group of invertible $n \times n$ real matrices under multiplication.
The determinant is a map $\det \colon \mathrm{GL}_n(\mathbb{R}) \to \mathbb{R}^\times$ to the group of nonzero real numbers under multiplication.
It is a [[group homomorphism|0JYTG5K9Y4]], because $\det(AB) = \det(A) \det(B)$ for all $A, B$.

## Remarks

The [[kernel|0W88CEGQP1]] of $\det$ is the special linear group $\mathrm{SL}_n(\mathbb{R})$ of matrices of determinant $1$, so this group is normal in $\mathrm{GL}_n(\mathbb{R})$.
The determinant is surjective onto $\mathbb{R}^\times$, so $\mathrm{GL}_n(\mathbb{R}) / \mathrm{SL}_n(\mathbb{R}) \cong \mathbb{R}^\times$.
The same holds over each field $F$, with $\mathrm{GL}_n(F)$, $\mathrm{SL}_n(F)$ and $F^\times$.

## Explanations

### L1 human

The determinant turns a product of matrices into a product of numbers.
That is exactly what a homomorphism does.
The matrices that it sends to $1$ form an important subgroup, the special linear group.
</document>

<document source="https://tamnd.github.io/math2/node/0YZXNE3HFR.md" tag="0YZXNE3HFR" kind="statement">
# Mercator series

## Statement

For every real $x$ with $-1 < x \le 1$,
$$\ln(1 + x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} x^n = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots.$$

## Remarks

For $|x| < 1$, integrate the [[geometric series|01X5CVV1Q5]] $\frac{1}{1 + t} = \sum (-t)^n$ term by term from $0$ to $x$.
At $x = 1$ the series converges by the alternating series test. [[Abel's theorem|0S4EMDQS2F]] gives $\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \cdots$.
Mathlib states the form $\sum_{n \ge 0} \frac{x^{n+1}}{n+1} = -\ln(1 - x)$ as `Real.hasSum_pow_div_log_of_abs_lt_one`.

## History

Johannes Hudde found the series in 1656 and Isaac Newton in 1665, but neither of them published it.
Nicholas Mercator found it independently and used it in his book Logarithmotechnia of 1668.

## Explanations

### L1 human

The logarithm of $1 + x$ is a power series in $x$, for $x$ in $(-1, 1]$.
At $x = 1$ it gives the alternating harmonic series, with sum $\ln 2$.
</document>

<document source="https://tamnd.github.io/math2/node/0Z05VCBQ7Y.md" tag="0Z05VCBQ7Y" kind="concept">
# Cycle

## Summary

A cycle in a graph is a closed walk that visits each of its vertices once and returns to its start. Cycles are central in graph theory, for example in the definitions of trees and of bipartite graphs. The node [[cycle|07CT9RDY7Q]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0Z0NE3J6JX.md" tag="0Z0NE3J6JX" kind="concept">
# Pell's equation

## Summary

Pell's equation is the Diophantine equation $x^2 - d y^2 = 1$, where $d$ is a positive integer that is not a perfect square.
The [[definition of Pell's equation|0AF3DANJ76]] gives the precise form.
</document>

<document source="https://tamnd.github.io/math2/node/0Z1G2F8DGY.md" tag="0Z1G2F8DGY" kind="statement">
# Power series of complex cosine

## Statement

For each $z \in \mathbb{C}$, the series $\sum_{n \ge 0} (-1)^n z^{2n} / (2n)!$ converges absolutely, and
$$\cos z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \cdots,$$
where $\cos$ is the [[complex cosine|0A7YDKRNR6]].

## Remarks

Proof: put $iz$ and $-iz$ in the series of the [[complex exponential|054VXHMCA3]] and add. The terms with odd powers cancel, and $i^{2n} = (-1)^n$.
In the same way, $\sin z = \sum_{n \ge 0} (-1)^n z^{2n+1} / (2n+1)!$ for each $z \in \mathbb{C}$.
The series shows that $\cos$ is an even function: $\cos(-z) = \cos z$.

## Explanations

### L1 human

The series of $\cos$ has only the even powers of $e^z$'s series, with signs that alternate.
For example, $\cos(0.1) \approx 1 - 0.005 = 0.995$ from the first two terms.
</document>

<document source="https://tamnd.github.io/math2/node/0Z1Y7JHGBR.md" tag="0Z1Y7JHGBR" kind="definition">
# Diffeomorphism

## Definition

Let $U, V \subseteq \mathbb{R}^n$ be open, and let $k \ge 1$ be an integer or $k = \infty$.
A map $f \colon U \to V$ is a $C^k$ diffeomorphism if $f$ is a bijection of [[class C^k|0TCBZMEYGN]] on $U$. In addition, $f^{-1} \colon V \to U$ must be of class $C^k$ on $V$.
The sets $U$ and $V$ are $C^k$ diffeomorphic if there is a $C^k$ diffeomorphism from $U$ to $V$.

## Remarks

If $f$ is a diffeomorphism, the chain rule applied to $f^{-1} \circ f = \mathrm{id}$ shows that $Df(x)$ is invertible for each $x \in U$.
Conversely, a $C^k$ bijection $f \colon U \to V$ with $Df(x)$ invertible at each point is a $C^k$ diffeomorphism, by the [[inverse function theorem|0K9HSJFCMJ]].
The map $x \mapsto x^3$ is a $C^\infty$ bijection of $\mathbb{R}$, but it is not a diffeomorphism, because its inverse $x \mapsto x^{1/3}$ is not differentiable at $0$.
Polar coordinates $(r, \theta) \mapsto (r \cos \theta, r \sin \theta)$ give a $C^\infty$ diffeomorphism from $(0, \infty) \times (0, 2\pi)$ onto the plane without the closed half-line $\{(x, 0) : x \ge 0\}$.

## Explanations

### L1 human

A diffeomorphism is a smooth change of coordinates that can be undone smoothly.
It may stretch and bend space, but it does not tear, fold or crush it.
</document>

<document source="https://tamnd.github.io/math2/node/0Z22VQH17V.md" tag="0Z22VQH17V" kind="definition">
# Stirling numbers of the first kind

## Definition

Let $n$ and $k$ be natural numbers.
The unsigned Stirling number of the first kind $c(n, k)$ is the number of [[permutations|0BQMWCNS8S]] of an $n$-element set that have exactly $k$ disjoint cycles.
Fixed points count as cycles of length $1$.

## Remarks

Other notations are $\left[ {n \atop k} \right]$ and $|s(n, k)|$. The signed numbers are $s(n, k) = (-1)^{n-k} c(n, k)$.
Example: $c(3, 2) = 3$. The permutations of $\{1, 2, 3\}$ with two cycles are the three transpositions.
The numbers satisfy $\sum_k c(n, k) = n!$ and $x (x + 1) \cdots (x + n - 1) = \sum_k c(n, k)\, x^k$.
The values satisfy a [[recurrence|0C5BFX94J9]] in $n$ and $k$.

## Explanations

### L1 human

Write a permutation as a set of cycles, for example $(1\,2)(3)$.
$c(n, k)$ counts the permutations of $n$ elements that split into exactly $k$ cycles.
</document>

<document source="https://tamnd.github.io/math2/node/0Z2P41SVDP.md" tag="0Z2P41SVDP" kind="concept">
# Separable space

## Summary

A separable space is a topological space with a countable dense subset.
</document>

<document source="https://tamnd.github.io/math2/node/0Z367CGNRM.md" tag="0Z367CGNRM" kind="definition">
# Minimum spanning tree

## Definition

Let $G$ be a finite [[connected|0FFAYNMAJZ]] [[graph|0JD5C82QMB]] with edge set $E$, and let $w : E \to \mathbb{R}$ be a weight function.
The weight of a subgraph is the sum of the weights of its edges.
A minimum spanning tree of $(G, w)$ is a [[spanning tree|0FQ5B8S1ZC]] of $G$ whose weight is the smallest among all spanning trees of $G$.

## Remarks

A minimum spanning tree exists, because a finite graph has finitely many spanning trees. It is unique when all weights are distinct.
Kruskal's algorithm and Prim's algorithm find a minimum spanning tree with greedy choices.

## Explanations

### L1 human

Suppose that each road between towns has a cost to build.
A minimum spanning tree is the cheapest set of roads that links all towns.
</document>

<document source="https://tamnd.github.io/math2/node/0Z3FEWZ8HA.md" tag="0Z3FEWZ8HA" kind="statement">
# Diagonalisation of a quadratic form

## Statement

Let $F$ be a field of characteristic not $2$, and let $V$ be a vector space over $F$ of finite dimension $n$.
Let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
Then $V$ has a basis $(e_1, \dots, e_n)$ and there are $a_1, \dots, a_n \in F$ such that, for all $x_1, \dots, x_n \in F$,
$$Q(x_1 e_1 + \cdots + x_n e_n) = a_1 x_1^2 + a_2 x_2^2 + \cdots + a_n x_n^2.$$

## Remarks

The basis is [[orthogonal|01TXN7ETH2]] for the associated symmetric bilinear form, and $a_i = Q(e_i)$.
In matrix terms, each symmetric matrix $A$ over $F$ has an invertible $P$ such that $P^T A P$ is diagonal.
The numbers $a_i$ are not unique; for example, each $a_i$ can be multiplied by a nonzero square.
Over $\R$, the orthogonal diagonalisation of the [[principal axis theorem|0PR5P9Z4KM]] is a special case.

## Explanations

### L1 human

Every quadratic form becomes a sum of squares with coefficients after a good change of variables.
For example, $x^2 + 2xy = (x + y)^2 - y^2$.
With $u = x + y$ and $v = y$, the form becomes $u^2 - v^2$.
</document>

<document source="https://tamnd.github.io/math2/node/0Z3GDY89T5.md" tag="0Z3GDY89T5" kind="statement">
# Decomposition into generalised eigenspaces

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Assume that the characteristic polynomial of $T$ splits over $F$ as $p_T(t) = (t - \lambda_1)^{m_1} \cdots (t - \lambda_k)^{m_k}$ with pairwise distinct $\lambda_i$.
Then
$$V = V_{\lambda_1} \oplus \cdots \oplus V_{\lambda_k},$$
where $V_{\lambda_i}$ is the [[generalised eigenspace|0AJ035ZTZE]] of $\lambda_i$, and $\dim V_{\lambda_i} = m_i$.

## Remarks

The hypothesis holds for every $T$ when $F$ is algebraically closed, for example $F = \C$.
Each $V_{\lambda_i}$ is $T$-invariant, and $T - \lambda_i \id$ is nilpotent on it.
A proof uses the coprime polynomials $(t - \lambda_i)^{m_i}$ and the [[Cayley–Hamilton theorem|0F765KFC2J]].
This [[direct sum|0XYYT5KNTX]] decomposition is the first step towards the [[Jordan normal form|085HQ5G5TE]].

## Explanations

### L1 human

Over $\C$, the space splits into pieces, one for each eigenvalue.
On each piece, the map is $\lambda$ times the identity plus a nilpotent part.
So a hard problem breaks into simpler problems, one for each eigenvalue.
</document>

<document source="https://tamnd.github.io/math2/node/0Z3YKRQZ48.md" tag="0Z3YKRQZ48" kind="definition">
# Linear first-order equation

## Definition

Let $I$ be an open [[interval|0Q1D85B8G6]] and let $p, q : I \to \mathbb{R}$ be continuous functions.
The linear first-order equation with coefficient $p$ and right side $q$ is the equation
$$y' + p(x)\, y = q(x)$$
for an unknown differentiable function $y : I \to \mathbb{R}$.
It is homogeneous if $q = 0$ on $I$, and inhomogeneous otherwise.

## Remarks

The [[solution formula|06B9W45XV0]] gives each solution explicitly, with an [[integrating factor|01THH3Y3YP]].
The homogeneous equation $y' + p(x) y = 0$ has the solutions $y = C e^{-P(x)}$, where $P$ is an antiderivative of $p$ and $C$ is a constant.
Many authors write the equation as $y' = a(x) y + b(x)$, with $a = -p$ and $b = q$.

## Explanations

### L1 human

The equation is linear because $y$ and $y'$ appear only to the first power, with coefficients that depend on $x$.
There is no product such as $y\, y'$ and no function such as $\sin y$.
This structure is the reason why there is a formula for all the solutions.
</document>

<document source="https://tamnd.github.io/math2/node/0Z409N8RDG.md" tag="0Z409N8RDG" kind="concept">
# Intermediate field

## Summary

An intermediate field of an extension $L/K$ is a subfield of $L$ that contains $K$.
The [[definition of an intermediate field|02NBKVQFRQ]] states it.
</document>

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# A function with nonnegative second derivative is convex

## Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be twice differentiable.
If $f''(x) \ge 0$ for all $x \in I$, then $f$ is [[convex|0VV0HSW1ZS]].

## Remarks

The derivative $f'$ has a nonnegative derivative, so $f'$ is [[increasing|026BJBQNGJ]]. Then [[convexity follows|005JT76HCK]].
If $f'' > 0$ on $I$, then $f$ is strictly convex. The converse fails: $x^4$ is strictly convex, but its second derivative at $0$ is $0$.
Mathlib states it as `convexOn_of_deriv2_nonneg`.

## Explanations

### L1 human

A nonnegative [[second derivative|0Y9D2FJRQ4]] means that the slope never decreases.
So the graph bends upward everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/0Z424HK8CQ.md" tag="0Z424HK8CQ" kind="definition">
# Kernel of a homomorphism

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]], let $e_H$ be the identity element of $H$, and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
The kernel of $\varphi$ is the set $\ker \varphi = \{g \in G : \varphi(g) = e_H\}$.

## Remarks

The kernel is a [[normal subgroup|0CB0DX5BVV]] of $G$.
The kernel of the map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$, $k \mapsto k + n\mathbb{Z}$, is $n\mathbb{Z}$.
The kernel of the determinant map from $\mathrm{GL}_n(\mathbb{R})$ to $\mathbb{R}^\times$ is the special linear group $\mathrm{SL}_n(\mathbb{R})$.

## Explanations

### L1 human

The kernel is the part of the group that the homomorphism forgets: everything in it goes to the identity.
If the kernel is only the identity, the homomorphism forgets nothing, and it is injective.
For the map that sends an integer to its remainder modulo $3$, the kernel is the set of multiples of $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0Z47M1B2VS.md" tag="0Z47M1B2VS" kind="statement">
# Burnside's lemma

## Statement

Let a finite [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a finite set $X$.
For $g \in G$, let $X^g = \{x \in X : g \cdot x = x\}$.
Then the number of [[orbits|0YQPZB5Q1G]] is $|X/G| = \frac{1}{|G|} \sum_{g \in G} |X^g|$.

## Remarks

The proof counts the pairs $(g, x)$ with $g \cdot x = x$ in two ways, and uses $|G_x| = |G| / |G \cdot x|$.
For example, the rotation group $\mathbb{Z}/4\mathbb{Z}$ of a square acts on the $2^4 = 16$ colourings of the corners with two colours.
The fixed colourings number $16$, $2$, $4$ and $2$, so there are $(16 + 2 + 4 + 2)/4 = 6$ colourings up to rotation.

## History

William Burnside stated and proved the lemma in his 1897 book on finite groups, and he attributed it to Frobenius, 1887.
Cauchy already knew the formula in 1845.
For this reason, the lemma is sometimes called "the lemma that is not Burnside's".

## Explanations

### L1 human

The number of orbits is the average number of points that a group element fixes.
This lets you count objects up to symmetry without listing them.
You count the fixed objects for each symmetry, add, and divide by the number of symmetries.
</document>

<document source="https://tamnd.github.io/math2/node/0Z49NTR1DZ.md" tag="0Z49NTR1DZ" kind="definition">
# Symmetric difference

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The symmetric difference of $A$ and $B$ is the set
$$A \mathbin{\triangle} B = (A \setminus B) \cup (B \setminus A).$$
An object $x$ is an element of $A \mathbin{\triangle} B$ when $x$ is an element of exactly one of $A$ and $B$.

## Remarks

Another notation is $A \ominus B$.
Example: $\{1, 2, 3\} \mathbin{\triangle} \{2, 3, 4\} = \{1, 4\}$.
Also $A \mathbin{\triangle} B = (A \cup B) \setminus (A \cap B)$.
The [[power set|0BC7KX51Z8]] of a set is a [[group|G5S1M9]] under symmetric difference, with identity $\varnothing$.

## Explanations

### L1 human

The symmetric difference is the set version of "exclusive or".
It keeps what is in one set or the other, but not what is in both.
</document>

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# Taylor series

## Summary

The Taylor series of a smooth function at a point is the power series built from all the derivatives of the function at that point. For analytic functions, it converges to the function near the point. The node [[Taylor series|0225QPWRRE]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0Z4BWARC38.md" tag="0Z4BWARC38" kind="statement">
# Determinant of the identity matrix

## Statement

Let $K$ be a commutative ring and let $n \ge 1$.
Then $\det I_n = 1$, where $I_n$ is the [[identity matrix|06DW9BAWB1]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is property 3 of the definition of the determinant.
It fixes the scale: the unit cube has volume $1$.
More generally, the determinant of a diagonal matrix is the product of its diagonal entries.

## Explanations

### L1 human

The identity matrix leaves every vector where it is.
It does not change any volume.
So its determinant is $1$.
</document>

<document source="https://tamnd.github.io/math2/node/0Z4V98VN5E.md" tag="0Z4V98VN5E" kind="definition">
# Proper metric space

## Definition

A [[metric space|063W5JMY85]] $(X, d)$ is proper when each [[closed ball|05DSV40EB9]] $\bar{B}(x, r)$, with $x \in X$ and $r \ge 0$, is [[compact|00R0X1WGPX]].

## Remarks

In a proper metric space, a subset is compact if and only if it is closed and bounded. Thus the [[Heine–Borel theorem|07PFYV65MB]] says that $\mathbb{R}^n$ is proper.
Each proper metric space is complete and [[locally compact|0EQMWQ4WMJ]].
An infinite set with the [[discrete metric|0NA4FK7B8S]] is not proper: the closed ball of radius $1$ is the whole set, which is not compact.

## Explanations

### L1 human

A proper metric space is a space where the Heine–Borel test for compactness works: closed and bounded is enough.
Euclidean spaces are proper, but many infinite-dimensional spaces are not.
</document>

<document source="https://tamnd.github.io/math2/node/0Z57EC6YX3.md" tag="0Z57EC6YX3" kind="definition">
# Product topology

## Definition

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be [[topological spaces|0CNQXVBHCX]].
The product topology on the [[Cartesian product|0C81YBVTWH]] $X \times Y$ is the topology with [[basis|06EX0FHBTF]]
$$\{ U \times V : U \in \tau_X, \ V \in \tau_Y \}.$$

## Remarks

This family is a basis of a topology, because $(U_1 \times V_1) \cap (U_2 \times V_2) = (U_1 \cap U_2) \times (V_1 \cap V_2)$.
A set $W \subseteq X \times Y$ is open if and only if for each $(x, y) \in W$ there are open $U \ni x$ and $V \ni y$ with $U \times V \subseteq W$.
For metric spaces, it is the metric topology of the [[product metric|044D4AQV8V]]. The product topology on $\mathbb{R} \times \mathbb{R}$ is the usual topology of $\mathbb{R}^2$.
The [[projections are continuous|0PM88Z9W4M]]. For products of any family of spaces, see the [[product topology on an arbitrary product|04S9HX3GAP]].

## Explanations

### L1 human

The open sets of $X \times Y$ are built from open rectangles $U \times V$.
In the plane, open rectangles and open discs give the same topology.
</document>

<document source="https://tamnd.github.io/math2/node/0Z5905Q73D.md" tag="0Z5905Q73D" kind="concept">
# Alternating group

## Summary

The alternating group $A_n$ is the group of the even permutations of $n$ symbols.
It is a normal subgroup of index $2$ in $S_n$ for $n \geq 2$.
</document>

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# Interval of convergence

## Definition

Let $\sum a_n (x - c)^n$ be a real [[power series|0VP56B4EMX]].
Its interval of convergence is the set of real numbers $x$ for which the series converges.

## Remarks

Let $R$ be the [[radius of convergence|0660QX1VPR]]. The interval of convergence contains $(c - R, c + R)$ and lies in $[c - R, c + R]$.
Each endpoint must be checked on its own. For $\sum x^n / n$, the interval is $[-1, 1)$.
For $\sum x^n$ it is $(-1, 1)$, and for $\sum x^n / n^2$ it is $[-1, 1]$.

## Explanations

### L1 human

The radius of convergence fixes the interval up to its two endpoints.
At each endpoint the series may converge or diverge, so the interval may be open, closed, or half open.
</document>

<document source="https://tamnd.github.io/math2/node/0Z7HDDXX3X.md" tag="0Z7HDDXX3X" kind="definition">
# Greatest element

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.
An element $g \in S$ is the greatest element of $S$ when $s \le g$ for each $s \in S$.

## Remarks

Another name is maximum. By antisymmetry, a set has at most one greatest element.
A greatest element is a [[maximal element|0TZEEFTNCC]] and an [[upper bound|0H5J02P8S9]] of $S$ that lies in $S$.
The interval $[0, 1)$ in $\mathbb{R}$ has no greatest element.

## Explanations

### L1 human

The greatest element beats every other element of the set.
In $\{1, 2, 3, 6\}$ under divisibility, $6$ is the greatest element, because each element divides $6$.
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<document source="https://tamnd.github.io/math2/node/0Z83DHFMYM.md" tag="0Z83DHFMYM" kind="concept">
# Computing the inverse by row reduction

## Summary

Row reduction computes the inverse of a square matrix, or shows that no inverse exists.
Let $A$ be an $n \times n$ matrix over a field $F$, and form the $n \times 2n$ [[augmented matrix|0T6MWGZSXA]] $[A \mid I_n]$.
Apply [[Gauss–Jordan elimination|098D6FQN3N]] to it.
If the left block becomes $I_n$, then the result is $[I_n \mid A^{-1}]$, and the right block is the inverse.
If the left block gets a zero row, then $A$ is not [[invertible|0EB9N0S33C]].

## Remarks

The method works because each row operation is left multiplication by an [[elementary matrix|01REXW3WWE]].
If $E_k \cdots E_1 A = I_n$, then $E_k \cdots E_1 = A^{-1}$, and the same product changes $I_n$ into $A^{-1}$.
For $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$, the method gives $A^{-1} = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$.

## Explanations

### L1 human

We write the identity matrix next to $A$ and row reduce the whole wide matrix.
When the left half becomes the identity, the right half has become $A^{-1}$.
The same row operations that turn $A$ into $I$ turn $I$ into $A^{-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/0Z87BHAFTB.md" tag="0Z87BHAFTB" kind="definition">
# Matching

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A matching in $G$ is a set $M$ of edges of $G$ such that no two distinct edges of $M$ have a common end.

## Remarks

A vertex is matched if it is an end of an edge of $M$. Otherwise it is unmatched.
A maximal matching is not contained in a larger matching. A maximum matching has the largest number of edges, the matching number $\nu(G)$.
Every maximum matching is maximal, but the converse is false: in the [[path graph|075WR6W3Y3]] $P_4$, the middle edge alone is a maximal matching.

## Explanations

### L1 human

A matching pairs some vertices along edges, and no vertex is in two pairs.
For example, it can pair workers with jobs that they can do, one job per worker.
</document>

<document source="https://tamnd.github.io/math2/node/0Z8ATBPM5W.md" tag="0Z8ATBPM5W" kind="counterexample">
# The complex conjugate is not holomorphic

## Counterexample

Let $f : \mathbb{C} \to \mathbb{C}$ be the [[complex conjugate|0BJV24C4SY]] map, $f(z) = \bar z$.
The function $f$ is continuous, but it is not [[complex differentiable|0D1C9Y1EXA]] at any point $z_0 \in \mathbb{C}$.
Thus $f$ is not [[holomorphic|0DG0Z72CQD]] on any nonempty open set.
Proof: the difference quotient at $z_0$ is
$$\frac{f(z_0 + h) - f(z_0)}{h} = \frac{\bar h}{h}.$$
It is $1$ for each real $h \neq 0$ and $-1$ for each purely imaginary $h \neq 0$. So it has no limit as $h \to 0$.

## Remarks

The real and imaginary parts $u = x$ and $v = -y$ are differentiable, but $u_x = 1$ and $v_y = -1$. So the [[Cauchy–Riemann equations|0T71SSYZ39]] fail at each point.
As a map of $\mathbb{R}^2$, $f$ is the reflection in the real axis. It is real linear but not complex linear.
For the same reason, $\operatorname{Re} z$, $\operatorname{Im} z$ and $|z|$ are not holomorphic on any nonempty open set.

## Explanations

### L1 human

As a map of the plane, conjugation is a mirror, and it is very smooth.
But a complex derivative must be the same in each direction, and conjugation fixes the real direction and reverses the imaginary direction.
So the quotient gives $1$ in one direction and $-1$ in the other, and there is no complex derivative.
</document>

<document source="https://tamnd.github.io/math2/node/0Z8CSYVHDJ.md" tag="0Z8CSYVHDJ" kind="statement">
# Local homeomorphism from the inverse function theorem

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}^n$ be of class $C^1$ with $Df(x)$ invertible for each $x \in U$.
Then each $a \in U$ has an open neighbourhood $V \subseteq U$ such that $f(V)$ is open and $f$ restricted to $V$ is a homeomorphism onto $f(V)$.
In particular, $f$ is an open map: $f(O)$ is open for each open set $O \subseteq U$.

## Remarks

This follows from the [[inverse function theorem|0K9HSJFCMJ]], since a $C^1$ inverse is [[continuous|0S9EK5MQPE]].
For each $a$, the hypothesis is needed only at $a$: if $Df(a)$ is invertible, $f$ is a homeomorphism from a neighbourhood of $a$ onto an open set.
A local homeomorphism need not be injective. The map $(x, y) \mapsto (e^x \cos y, e^x \sin y)$ is a local homeomorphism from $\mathbb{R}^2$ onto $\mathbb{R}^2 \setminus \{0\}$, but it is not injective.

## Explanations

### L1 human

Where the derivative is invertible, the map neither folds nor crushes space in a small region.
So small open sets go to open sets, and the map can be undone continuously on each small region.
</document>

<document source="https://tamnd.github.io/math2/node/0Z8FM54P86.md" tag="0Z8FM54P86" kind="statement">
# Countable union of countable sets is countable

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Let $I$ be a [[countable|0ZG61TFBES]] set and let $(A_i)_{i \in I}$ be a [[family of sets|05Q6G4EH2G]] such that each $A_i$ is countable.
Then the [[union|05GK32H72H]] $\bigcup_{i \in I} A_i$ is countable.

## Remarks

Proof: choose for each $i$ a surjection from $\mathbb{N}$ onto $A_i$, when $A_i$ is nonempty. Together they give a surjection from a subset of $\mathbb{N} \times \mathbb{N}$ onto the union, and $\mathbb{N} \times \mathbb{N}$ is countable.
The proof uses the axiom of countable choice to choose the surjections. Without any choice, the statement cannot be proved in ZF.
When the enumerations of the $A_i$ are given, no choice is needed.

## Explanations

### L1 human

List each set $A_i$ as a row of a grid.
Then list the grid diagonal by diagonal.
Each element of the union appears in the list.
</document>

<document source="https://tamnd.github.io/math2/node/0Z8PSSAVBA.md" tag="0Z8PSSAVBA" kind="definition">
# Probability-generating function

## Definition

Let $X$ be a random variable with values in $\{0, 1, 2, \dots\}$ and with [[mass function|0WF4EJP91W]] $p(k) = P(X = k)$.
The probability-generating function of $X$ is
$$G_X(z) = E\left[z^X\right] = \sum_{k=0}^{\infty} p(k) \, z^k,$$
for each complex $z$ with $|z| \le 1$.

## Remarks

$G_X$ is the [[generating function|0HH6JX5WZZ]] of the sequence $p(0), p(1), \dots$. The series converges absolutely for $|z| \le 1$, and $G_X(1) = 1$.
$G_X$ determines the law of $X$: $P(X = k) = G_X^{(k)}(0)/k!$.
If $E[X] < \infty$, then $E[X] = G_X'(1)$, with the left derivative at $1$. Also $E[X(X - 1)] = G_X''(1)$ when it is finite.
For $X \sim \operatorname{Poisson}(\lambda)$, $G_X(z) = e^{\lambda(z - 1)}$. For $X \sim \operatorname{Bin}(n, p)$, $G_X(z) = (1 - p + pz)^n$.

## Explanations

### L1 human

The probability-generating function packs all the probabilities of a count into one function.
Derivatives of this function at $0$ give back the probabilities, and derivatives at $1$ give the moments.
</document>

<document source="https://tamnd.github.io/math2/node/0Z962ECCXR.md" tag="0Z962ECCXR" kind="concept">
# Trial division

## Summary

Trial division is the simplest method to factor an integer $n > 1$ or to test whether it is prime.
It divides $n$ by the integers $d = 2, 3, 4, \dots$ in turn, and it stops at the first $d$ that divides $n$, or when $d^2 > n$.
If no $d$ with $d^2 \le n$ divides $n$, then $n$ is prime.

## Remarks

The method is correct because [[a composite number has a prime factor at most its square root|0EB6BBMQMC]].
It is enough to test the primes $d$, but a test of all integers $d$ is also correct.
For a number $n$ with $k$ digits, the method can need about $\sqrt{n} \approx 10^{k/2}$ divisions. Thus it is slow for large $n$.

## Explanations

### L1 human

To test whether $91$ is prime, try to divide it by $2$, $3$, $5$ and $7$. You can stop after $9$, because $10^2 > 91$.
The division by $7$ works: $91 = 7 \cdot 13$. So $91$ is not prime.
</document>

<document source="https://tamnd.github.io/math2/node/0Z9JB71BS0.md" tag="0Z9JB71BS0" kind="statement">
# Constant multiple rule

## Statement

Let $f$ be a real function on an open interval $I$, differentiable at $a \in I$, and let $c$ be a real number.
Then $c f$ is differentiable at $a$, and $(c f)'(a) = c f'(a)$.

## Remarks

It is the case $\beta = 0$ of the [[linearity of differentiation|0M0TKFRX2A]].
For example, $(5 x^2)' = 10 x$.
Mathlib states it as `deriv_const_mul`.

## Explanations

### L1 human

Stretching a graph vertically by a factor $c$ multiplies all its slopes by $c$.
</document>

<document source="https://tamnd.github.io/math2/node/0Z9N1ER0W0.md" tag="0Z9N1ER0W0" kind="statement">
# Dimension of the space of linear maps

## Statement

Let $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Then the [[space of linear maps|0X9MBPP2Z3]] $\Hom_F(V, W)$ is finite-dimensional, and
$$\dim \Hom_F(V, W) = \dim V \cdot \dim W.$$

## Remarks

With bases of $V$ and $W$, the map that sends $T$ to its matrix is an isomorphism $\Hom_F(V, W) \cong M_{m \times n}(F)$, with $n = \dim V$ and $m = \dim W$.
The matrices $E_{ij}$ form a basis of $M_{m \times n}(F)$, so its dimension is $mn$.
In particular, the [[dual space|07JBSZE0V4]] of $V$ has dimension $\dim V$.

## Explanations

### L1 human

A linear map from an $n$-dimensional space to an $m$-dimensional space is an $m \times n$ matrix.
A matrix of this size has $mn$ free entries.
So the space of these maps has dimension $mn$.
</document>

<document source="https://tamnd.github.io/math2/node/0ZA4AX4MW3.md" tag="0ZA4AX4MW3" kind="statement">
# Cauchy criterion for sequences

## Statement

Let $(a_n)$ be a real sequence.
Then $(a_n)$ converges if and only if $(a_n)$ is a [[Cauchy sequence|0QZMTN5AXX]].

## Remarks

One direction is [[convergent sequences are Cauchy|0QY87R9CY3]].
For the other direction, a Cauchy sequence is [[bounded|0G6D8E7CTA]], so it has a convergent subsequence by the [[Bolzano–Weierstrass theorem|0KDJD249DT]]. A Cauchy sequence with a convergent subsequence converges to the same limit.
The criterion says that $\mathbb{R}$ is a complete metric space. It fails in $\mathbb{Q}$.
Its value is that it proves convergence without a candidate for the limit.

## Explanations

### L1 human

To show that a sequence has a limit, you often need to know the limit first.
The Cauchy criterion avoids this. It is enough to show that the terms become close to each other.
</document>

<document source="https://tamnd.github.io/math2/node/0ZAF9KHDDF.md" tag="0ZAF9KHDDF" kind="concept">
# Greatest common divisor in an integral domain

## Summary

A greatest common divisor of two elements of an integral domain is a common divisor that every common divisor divides.
It is unique up to associates when it exists.
The [[definition of a greatest common divisor|0F7KNJ1ZYN]] gives the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/0ZAZZ34WQS.md" tag="0ZAZZ34WQS" kind="concept">
# Memorylessness

## Summary

A random waiting time $X \ge 0$ is memoryless if, for all $s, t \ge 0$ with $P(X > s) > 0$,
$$P(X > s + t \mid X > s) = P(X > t).$$
So the time already waited does not change the law of the remaining waiting time.
Among the distributions on $[0, \infty)$ with $P(X > t) > 0$ for each $t$, only the [[exponential distributions|028Z8WC5Y9]] are memoryless. Among the distributions on $\{0, 1, 2, \dots\}$, with $s$ and $t$ integers, only the [[geometric distributions|0XCVV5Q500]] are memoryless.

## Remarks

For $X \sim \operatorname{Exp}(\lambda)$, $P(X > s + t) = e^{-\lambda(s + t)} = P(X > s) \, P(X > t)$. This is the memoryless property.
The proof of the characterization uses that $G(t) = P(X > t)$ satisfies $G(s + t) = G(s) \, G(t)$ and is right-continuous. The only such functions with values in $(0, 1]$ are $G(t) = e^{-\lambda t}$ with $\lambda \ge 0$.

## Explanations

### L1 human

A memoryless waiting time does not "age".
If a bus comes at exponential times and you have already waited ten minutes, your expected further wait is the same as when you arrived.
</document>

<document source="https://tamnd.github.io/math2/node/0ZB2JG1M9C.md" tag="0ZB2JG1M9C" kind="concept">
# Nilpotent element

## Summary

A nilpotent element of a ring is an element that has a power equal to zero.
The [[definition of a nilpotent element|0N9R5NN6TW]] states this precisely.
</document>

<document source="https://tamnd.github.io/math2/node/0ZB3M68NYQ.md" tag="0ZB3M68NYQ" kind="concept">
# Pairwise independence

## Summary

A family of events, or of random variables, is pairwise independent when any two of its members are independent.
It is weaker than mutual independence. The node [[pairwise independence|0N57D1XY6T]] gives the precise definition for events.
</document>

<document source="https://tamnd.github.io/math2/node/0ZB7J071WQ.md" tag="0ZB7J071WQ" kind="statement">
# A complex number times its conjugate

## Statement

Let $z$ be a complex number, let $\bar{z}$ be its [[conjugate|0BJV24C4SY]] and let $|z|$ be its [[modulus|0KW2X0TFBX]].
Then
$$z \bar{z} = |z|^2.$$
In particular, $z \bar{z}$ is a nonnegative real number.

## Remarks

With $z = a + b i$: $(a + b i)(a - b i) = a^2 - (b i)^2 = a^2 + b^2$.
This identity gives the formula for the [[inverse of a complex number|0FQRAV5SKH]].
It also gives a quick proof that the [[modulus is multiplicative|0HYNZE940T]].

## Explanations

### L1 human

When you multiply a complex number by its conjugate, the imaginary parts cancel.
What remains is the square of the length of the number, for example $(3 + 4 i)(3 - 4 i) = 9 + 16 = 25$.
</document>

<document source="https://tamnd.github.io/math2/node/0ZB84QCDKR.md" tag="0ZB84QCDKR" kind="definition">
# Bernoulli equation

## Definition

Let $I$ be an open interval, let $P, Q : I \to \mathbb{R}$ be continuous functions, and let $n$ be a real number.
A Bernoulli equation is an [[ordinary differential equation|0P97V20RW7]] of the form
$$y' + P(x)\, y = Q(x)\, y^{n}$$
for an unknown function $y$ on a subinterval of $I$. If $n$ is not an integer, the function must satisfy $y > 0$.

## Remarks

For $n = 0$ and $n = 1$ the equation is [[linear|0Z3YKRQZ48]]. Some authors exclude these two values.
For $n \ne 1$, the substitution $v = y^{1 - n}$ changes the equation, on the set where $y \ne 0$, into the linear equation $v' + (1 - n) P(x)\, v = (1 - n) Q(x)$.
For $n > 0$, the function $y = 0$ is also a solution.
The [[logistic equation|0CC4VC7KR3]] is a Bernoulli equation with $n = 2$.

## Explanations

### L1 human

A Bernoulli equation is a linear equation with an extra power of $y$ on the right side.
It is not linear, but a change of the unknown makes it linear.
Then the [[solution formula|06B9W45XV0]] for linear equations solves it.
</document>

<document source="https://tamnd.github.io/math2/node/0ZC78099PA.md" tag="0ZC78099PA" kind="statement">
# Gradient points in the direction of steepest ascent

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$ be differentiable at $a \in U$, and suppose that $\nabla f(a) \ne 0$.
Then for each unit vector $u \in \mathbb{R}^n$, the [[directional derivative|0Y7PVW4Q44]] satisfies $-\|\nabla f(a)\| \le D_u f(a) \le \|\nabla f(a)\|$.
The upper bound is attained if and only if $u = \nabla f(a) / \|\nabla f(a)\|$, and the lower bound if and only if $u = -\nabla f(a) / \|\nabla f(a)\|$.

## Remarks

The proof combines $D_u f(a) = \nabla f(a) \cdot u$, from the statement that [[the derivative gives the directional derivatives|0A6PMR20N4]], with the Cauchy–Schwarz inequality $|w \cdot u| \le \|w\| \, \|u\|$ and its case of equality.
So the [[gradient|0NS2QYVC4Y]] gives the direction of fastest increase, and its norm is the fastest rate of increase per unit of distance.
The method of gradient descent uses the opposite direction $-\nabla f(a)$ to look for a minimum of $f$.

## Explanations

### L1 human

If you stand on a hillside and want to climb as fast as possible, walk in the direction of the gradient.
If you walk at right angles to the gradient, your height does not change at first: you follow a contour line.
</document>

<document source="https://tamnd.github.io/math2/node/0ZC7FYE0NZ.md" tag="0ZC7FYE0NZ" kind="statement">
# Orbits partition the set

## Statement

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$.
The relation $x \sim y$ if and only if $y = g \cdot x$ for some $g \in G$ is an equivalence relation on $X$.
Its equivalence classes are the [[orbits|0YQPZB5Q1G]], so each point of $X$ lies in exactly one orbit.

## Remarks

Reflexivity uses $e \cdot x = x$, symmetry uses $x = g^{-1} \cdot (g \cdot x)$, and transitivity uses $h \cdot (g \cdot x) = (hg) \cdot x$.
If $X$ is finite, then $|X|$ is the sum of the sizes of the distinct orbits.
The [[cosets|03DEH43F4H]] of a subgroup and the [[conjugacy classes|0DEBXAJ5SA]] of a group are examples of orbits.

## Explanations

### L1 human

The group action cuts the set into separate pieces, the orbits.
Inside one piece, each point can reach each other point.
No point can reach a point of another piece.
</document>

<document source="https://tamnd.github.io/math2/node/0ZC9YTXZRY.md" tag="0ZC9YTXZRY" kind="concept">
# Rank of a matrix

## Summary

The rank of a matrix is the dimension of the span of its columns, which equals the dimension of the span of its rows.
</document>

<document source="https://tamnd.github.io/math2/node/0ZCAXY9563.md" tag="0ZCAXY9563" kind="definition">
# Ascending chain condition

## Definition

Let $(P, \le)$ be a partially ordered set.
The set $P$ satisfies the ascending chain condition if every sequence $a_1 \le a_2 \le a_3 \le \dots$ of elements of $P$ is eventually constant.
That is, there is an integer $N$ with $a_n = a_N$ for all $n \ge N$.

## Remarks

Equivalently, every nonempty subset of $P$ has a maximal element. This equivalence uses the axiom of dependent choice.
The set $\mathbb{N}$ with its usual order does not satisfy the condition. With the reverse order, it does.
For the set of the ideals of a ring, ordered by inclusion, the condition defines [[Noetherian rings|0MVH4Y1CRZ]].
The ideals of $\mathbb{Z}$ satisfy the condition: a chain $(a_1) \subseteq (a_2) \subseteq \dots$ means $a_2 \mid a_1$, $a_3 \mid a_2$, and so on.

## Explanations

### L1 human

The condition says that you cannot climb forever.
Every increasing chain stops growing after a finite number of steps.
For the ideals of the integers, a strictly increasing chain like $(12) \subset (6) \subset (3) \subset (1)$ must end.
</document>

<document source="https://tamnd.github.io/math2/node/0ZCK8DQYN1.md" tag="0ZCK8DQYN1" kind="concept">
# Closed set

## Summary

A closed set in a topological space is a set whose complement is open.
In a metric space, the closed sets are the sets that contain the limits of all their convergent sequences.
</document>

<document source="https://tamnd.github.io/math2/node/0ZCSMYHTBP.md" tag="0ZCSMYHTBP" kind="concept">
# Jacobian determinant

## Summary

The Jacobian determinant of a map from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^n$ at a point is the determinant of its [[Jacobian matrix|047NDTEPCV]] there.
Its absolute value is the factor by which the map scales small volumes near the point. Its sign tells whether the map keeps or reverses orientation.
</document>

<document source="https://tamnd.github.io/math2/node/0ZCWPNZTQ3.md" tag="0ZCWPNZTQ3" kind="concept">
# Companion matrix

## Summary

The companion matrix of a monic polynomial is a square matrix with ones just below the diagonal.
Its last column holds the negated coefficients, and all other entries are zero.
Its characteristic polynomial is the given polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/0ZCY0Z80B6.md" tag="0ZCY0Z80B6" kind="concept">
# Neighbourhood

## Summary

The neighbourhood of a vertex in a graph is the set of vertices adjacent to it. Many local properties of graphs, such as degrees, are defined through neighbourhoods. The node [[neighbourhood|01T3TMGTTE]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ZCZK72B45.md" tag="0ZCZK72B45" kind="concept">
# Tangent vector of a curve

## Summary

The tangent vector of a differentiable [[parametrised curve|0G7WM8DX4P]] at a parameter value is the derivative of the curve there.
If the curve is the path of a moving point, the tangent vector is its velocity, and its length is the speed.
</document>

<document source="https://tamnd.github.io/math2/node/0ZD2CF6ZS0.md" tag="0ZD2CF6ZS0" kind="concept">
# Dual basis

## Summary

The dual basis of a basis $b_1, \dots, b_n$ is the family of linear forms $f_1, \dots, f_n$ with $f_i(b_j) = 1$ when $i = j$ and $0$ otherwise.
</document>

<document source="https://tamnd.github.io/math2/node/0ZDDNBQSC4.md" tag="0ZDDNBQSC4" kind="definition">
# Angle at a point

## Definition

Let $A$, $B$ and $C$ be points of a [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$ with $A \neq B$ and $C \neq B$.
The angle $\angle ABC$ is the [[angle between the vectors|0ZEHZY85E6]] $A - B$ and $C - B$. Thus it is the unique real number $\theta$ with $0 \leq \theta \leq \pi$ such that
$$\cos \theta = \frac{(A - B) \cdot (C - B)}{\|A - B\| \, \|C - B\|}.$$
The point $B$ is the vertex of the angle.

## Remarks

The angle is symmetric: $\angle ABC = \angle CBA$.
The angle is $\pi$ if and only if $B$ lies strictly between $A$ and $C$ on a line. It is $0$ if and only if $A$ and $C$ are on the same ray from $B$.
Mathlib's `EuclideanGeometry.angle` uses the same formula for all points. It gives $\pi / 2$ when $A = B$ or $C = B$.
The angle is a right angle when $\theta = \pi / 2$, acute when $\theta < \pi / 2$ and obtuse when $\theta > \pi / 2$.

## Explanations

### L1 human

To measure the angle at $B$, stand at $B$ and look at $A$, then turn to look at $C$. The angle is how far you turn, using the short way round. The order of $A$ and $C$ does not matter.
</document>

<document source="https://tamnd.github.io/math2/node/0ZDW0H0VX4.md" tag="0ZDW0H0VX4" kind="concept">
# Exponent of a group

## Summary

The exponent of a group is the least positive integer $n$ with $g^n = e$ for each element $g$, when such an integer exists.
It is the least common multiple of the orders of the elements.
</document>

<document source="https://tamnd.github.io/math2/node/0ZEHZY85E6.md" tag="0ZEHZY85E6" kind="concept">
# Angle between two vectors

## Summary

The angle between two nonzero [[vectors|0QN3D80AWP]] measures the turn from the direction of one to the direction of the other. It is a number from $0$ to $\pi$. The dot product of the two vectors and their lengths give its cosine.
</document>

<document source="https://tamnd.github.io/math2/node/0ZEPNS9BXK.md" tag="0ZEPNS9BXK" kind="statement">
# g to the order of the group is the identity

## Statement

Let $G$ be a finite [[group|G5S1M9]] with identity element $e$, and let $g \in G$.
Then $g^{|G|} = e$.

## Remarks

The [[order|0WE2NHZJ7B]] of $g$ divides $|G|$ by [[this corollary|08V7JHS3QZ]], and [[a power is $e$ when the order divides the exponent|0F5QC101Q7]].
For the group $(\mathbb{Z}/p\mathbb{Z})^\times$ of order $p - 1$, with $p$ prime, this gives Fermat's little theorem $a^{p-1} \equiv 1 \pmod p$ for $p \nmid a$.
For the group $(\mathbb{Z}/n\mathbb{Z})^\times$ of order $\varphi(n)$, it gives Euler's theorem.

## Explanations

### L1 human

If a group has $n$ elements, then the $n$th power of each element is the identity.
For example, in a group of order $6$, each element $g$ satisfies $g^6 = e$.
This simple fact proves Fermat's little theorem in number theory.
</document>

<document source="https://tamnd.github.io/math2/node/0ZERJX6PH6.md" tag="0ZERJX6PH6" kind="statement">
# Universal property of the quotient space

## Statement

Let $F$ be a field, let $V$ and $U$ be vector spaces over $F$, and let $W$ be a subspace of $V$.
Let $\pi \colon V \to V / W$ be the [[quotient map|0N4F5FMQZW]].
Let $T \colon V \to U$ be a [[linear map|0NRYHJPQG2]] with $W \subseteq \ker T$.
Then there is exactly one linear map $\bar T \colon V / W \to U$ with $\bar T \circ \pi = T$.
It is given by $\bar T(v + W) = T(v)$.

## Remarks

The condition $W \subseteq \ker T$ makes $\bar T$ well defined.
The kernel of $\bar T$ is $\ker T / W$, and its image is $\im T$.
For $W = \ker T$, the map $\bar T$ is injective, which gives the [[first isomorphism theorem|0MTBGVZ4XJ]].

## Explanations

### L1 human

A linear map that kills every vector of $W$ cannot tell apart two vectors that differ by an element of $W$.
So it makes sense on the cosets, and it becomes a map from the quotient space.
This property describes the quotient space completely.
</document>

<document source="https://tamnd.github.io/math2/node/0ZET0ZH16W.md" tag="0ZET0ZH16W" kind="definition">
# Median of a triangle

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$. Let $M_A$ be the [[midpoint|0S706JBZBG]] of $B$ and $C$.
The median from $A$ is the segment $\{\, A + t (M_A - A) : t \in [0, 1] \,\}$ from $A$ to $M_A$. The medians from $B$ and from $C$ are defined in the same way, with the midpoints of $CA$ and of $AB$.

## Remarks

Some texts call the whole line through $A$ and $M_A$ the median.
The three medians meet at the [[centroid|0X7EEWES5G]]. See [[the medians meet at the centroid|0GPQ67XV7N]].
Each median divides the triangle into two triangles of equal area.
[[Apollonius's theorem|0NAKHQ9XYY]] gives the length of a median from the side lengths.

## Explanations

### L1 human

Join a corner of a triangle to the middle of the opposite side. The segment is a median. If the triangle is a thin plate of uniform material, it balances on each median.
</document>

<document source="https://tamnd.github.io/math2/node/0ZFZ8ZG3EN.md" tag="0ZFZ8ZG3EN" kind="concept">
# Minimum degree

## Summary

The minimum degree of a finite graph is the smallest degree of its vertices. Many theorems guarantee a structure, such as a long cycle, from a lower bound on the minimum degree. The node [[minimum degree|0TKZCSGCCA]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ZG1PNEP35.md" tag="0ZG1PNEP35" kind="concept">
# Homothety

## Summary

A homothety is a transformation that scales the space by a fixed factor from a fixed point, its centre. It sends each figure to a figure of the same shape.
</document>

<document source="https://tamnd.github.io/math2/node/0ZG3Z5JZ85.md" tag="0ZG3Z5JZ85" kind="concept">
# Symmetry group

## Summary

The symmetry group of an object is the group of all transformations of the ambient space that map the object onto itself. These transformations must keep the relevant structure of the object.
The operation is composition of transformations.
It is often written $\operatorname{Sym}(X)$ for an object $X$.

## Remarks

The symmetry group of a regular $n$-gon in the plane is the [[dihedral group|0JZ27DMJ5D]] of order $2n$.
The rotations of a regular tetrahedron form a group [[isomorphic|08MG7H8JW5]] to the [[alternating group|0Z5905Q73D]] $A_4$.
The symmetry group of an object is a [[subgroup|0X1JXT47GA]] of the group of all transformations of the ambient space.

## Explanations

### L1 human

A symmetry of an object is a motion that leaves the object looking the same.
Two symmetries done one after the other give another symmetry, and each symmetry can be undone.
So the symmetries of an object form a group, and this group measures how symmetric the object is.
</document>

<document source="https://tamnd.github.io/math2/node/0ZG61TFBES.md" tag="0ZG61TFBES" kind="definition">
# Countable set

## Definition

A [[set|0SJ3RX9TXA]] $A$ is countable when there is an [[injective function|0PVG1CRAJN]] $f \colon A \to \mathbb{N}$.

## Remarks

Equivalently, $A$ is [[finite|0BBTH3YSZA]] or [[countably infinite|0SEZ3JGZP7]].
A nonempty set is countable exactly when there is a [[surjection|0XKH881BZ7]] from $\mathbb{N}$ to it.
Some authors use "countable" only for countably infinite sets, and "at most countable" for this notion.
Subsets of countable sets are countable.

## Explanations

### L1 human

A countable set is one that you can list: first element, second element, and so on.
The list may stop, or it may go on forever, but each element appears at some finite position.
</document>

<document source="https://tamnd.github.io/math2/node/0ZG9QF40HE.md" tag="0ZG9QF40HE" kind="concept">
# Parametrised surface

## Summary

A parametrised surface is a surface in $\mathbb{R}^3$ given by a map of two parameters into space.
Surfaces in Stokes' theorem and the divergence theorem are often given in this form.
From a parametrisation one can compute the surface area, the normal vectors and the curvature of the surface.
</document>

<document source="https://tamnd.github.io/math2/node/0ZGC7ZESYN.md" tag="0ZGC7ZESYN" kind="concept">
# Proposition

## Summary

A proposition is a statement that is either true or false.
In classical logic, each proposition has exactly one of the two truth values, true or false.
For example, "$2 + 2 = 4$" is a true proposition, and "$7$ is even" is a false proposition.
A question or a command is not a proposition, because it has no truth value.

## Remarks

Letters such as $p$, $q$ and $r$ usually denote propositions.
An open sentence such as "$x > 3$" is not a proposition until $x$ has a value. It is a [[predicate|0T3DPBG981]].
The connectives [[negation|0XR7A7RSXB]], [[conjunction|0M28ASWG8T]] and [[disjunction|0H5GKDTB01]] make new propositions from given propositions.

## Explanations

### L1 human

A proposition is a sentence that you can mark with "true" or "false".
"It rains in Paris now" is a proposition, because it is true or false at a given time.
"Close the door" is not a proposition, because it is neither true nor false.
</document>

<document source="https://tamnd.github.io/math2/node/0ZH5AJ2ED7.md" tag="0ZH5AJ2ED7" kind="concept">
# Baire space

## Summary

A Baire space is a topological space in which countable unions of closed sets with empty interior have empty interior.
Equivalently, countable intersections of dense open sets are dense.
</document>

<document source="https://tamnd.github.io/math2/node/0ZHQMQ2C1N.md" tag="0ZHQMQ2C1N" kind="statement">
# Transfinite induction

## Statement

Let $R$ be a [[well-founded relation|0230ECX1Q5]] on a set $A$ and let $P(x)$ be a [[predicate|0T3DPBG981]] on $A$.
Assume that, for each $x \in A$, if $P(y)$ holds for each $y \in A$ with $y \mathrel{R} x$, then $P(x)$ holds.
Then $P(x)$ holds for each $x \in A$.

## Remarks

Proof: if the set of $x$ with $\neg P(x)$ is nonempty, it has an $R$-minimal element $m$. Then $P(y)$ holds for each $y \mathrel{R} m$, so $P(m)$ holds, a contradiction.
For $<$ on $\mathbb{N}$, this is strong induction.
For [[ordinal numbers|0RDJK7EQYY]], one often splits the step into three cases: zero, successor ordinals and limit ordinals.

## Explanations

### L1 human

Transfinite induction extends induction on $\mathbb{N}$ to every well-founded order.
You show: if the statement holds for everything below $x$, then it holds for $x$.
Since you cannot go down forever, the statement holds everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/0ZHSTAG8ZY.md" tag="0ZHSTAG8ZY" kind="concept">
# Exterior derivative

## Summary

The exterior derivative extends the differential of a function to [[differential forms|0E3G0GRPZ6]] of higher degree. It maps $k$-forms to $(k + 1)$-forms.
In $\mathbb{R}^3$ it unifies the gradient, curl and divergence, and it gives a common form of the theorems of Green, Stokes and Gauss. Élie Cartan described it in its current form in 1899.
</document>

<document source="https://tamnd.github.io/math2/node/0ZHWKQMVX5.md" tag="0ZHWKQMVX5" kind="concept">
# Finite-dimensional vector space

## Summary

A finite-dimensional vector space is a vector space that has a finite basis, or equivalently a finite spanning set.
</document>

<document source="https://tamnd.github.io/math2/node/0ZJ87ZZ8PK.md" tag="0ZJ87ZZ8PK" kind="definition">
# Conditional independence

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A, B, C \in \mathcal{F}$ with $P(C) > 0$.
The events $A$ and $B$ are conditionally independent given $C$ if
$$P(A \cap B \mid C) = P(A \mid C) \, P(B \mid C),$$
where $P(\cdot \mid C)$ is the [[conditional probability|0FJB06AD2P]] given $C$.

## Remarks

If also $P(B \cap C) > 0$, the condition is equivalent to $P(A \mid B \cap C) = P(A \mid C)$.
Conditional independence does not imply independence, and independence does not imply conditional independence.
Example: choose one of two coins at random, a fair coin or a coin with two heads, and toss it two times. Given the coin, the two tosses are independent. Without this knowledge, they are dependent.
For random variables $X$ and $Y$ given $Z$, the definition uses the conditional distributions given $Z$.

## Explanations

### L1 human

Two results can be linked only through a hidden common cause.
When you know the cause, the link disappears. This is conditional independence given the cause.
</document>

<document source="https://tamnd.github.io/math2/node/0ZJM9D7NPS.md" tag="0ZJM9D7NPS" kind="statement">
# Canonical isomorphism with the double dual

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$.
Let $V^{**} = (V^*)^*$ be the dual of the [[dual space|07JBSZE0V4]].
The *evaluation map* $\operatorname{ev} \colon V \to V^{**}$ is defined by $\operatorname{ev}(v)(f) = f(v)$ for $v \in V$ and $f \in V^*$.
This map is linear and injective.
If $V$ is finite-dimensional, then $\operatorname{ev}$ is an isomorphism.

## Remarks

The evaluation map does not depend on a choice of basis, so it is called *canonical* or *natural*.
In finite dimensions, it is an isomorphism because it is injective and $\dim V^{**} = \dim V^* = \dim V$.
For infinite-dimensional $V$, the evaluation map is not surjective.

## Explanations

### L1 human

A vector $v$ can act on linear forms: it sends each form $f$ to the number $f(v)$.
This turns each vector into an element of the double dual.
In finite dimensions, every element of the double dual comes from exactly one vector in this way.
</document>

<document source="https://tamnd.github.io/math2/node/0ZKPH2BD1T.md" tag="0ZKPH2BD1T" kind="concept">
# Cycle notation

## Summary

Cycle notation writes a permutation as a list of cycles.
The cycle $(a_1\ a_2\ \dots\ a_k)$ sends $a_i$ to $a_{i+1}$ for $i < k$, sends $a_k$ to $a_1$, and fixes all other points.
A cycle with $k$ entries is a $k$-cycle, and a $1$-cycle is the identity, so it is usually not written.

## Remarks

The same cycle has $k$ different notations, because you can start the list at any entry: $(1\ 2\ 3) = (2\ 3\ 1) = (3\ 1\ 2)$.
The inverse of $(a_1\ a_2\ \dots\ a_k)$ is $(a_k\ \dots\ a_2\ a_1)$.
Each permutation of a finite set is a product of [[disjoint cycles|0YWDAZPRPQ]], so cycle notation can show each permutation.
For example, in $S_5$ the permutation with $1 \mapsto 3$, $3 \mapsto 1$, $2 \mapsto 4$, $4 \mapsto 5$, $5 \mapsto 2$ is $(1\ 3)(2\ 4\ 5)$.

## Explanations

### L1 human

Cycle notation follows each symbol to where it goes, until you come back to the start.
Each round trip is a cycle, and you write it in parentheses.
This is shorter than a table, and it shows the structure of the permutation directly.
</document>

<document source="https://tamnd.github.io/math2/node/0ZKZBJPG2J.md" tag="0ZKZBJPG2J" kind="statement">
# Absorption law

## Statement

Let $L$ be a lattice with join $\vee$ and meet $\wedge$.
Then $a \vee (a \wedge b) = a$ and $a \wedge (a \vee b) = a$ for all $a, b \in L$.

## Remarks

In a lattice defined as a partially ordered set, the proof is short. Since $a \wedge b \le a$, the join of $a$ and $a \wedge b$ is $a$. The second law is dual.
In the algebraic definition of a lattice, the two absorption laws are axioms, together with associativity and commutativity.
The absorption laws imply idempotence: $a \vee a = a \vee (a \wedge (a \vee a)) = a$.

## Explanations

### L1 human

Absorption says that combining $a$ with something made from $a$ gives back $a$.
For sets: $A \cup (A \cap B) = A$, because $A \cap B$ is already inside $A$.
</document>

<document source="https://tamnd.github.io/math2/node/0ZMT1GP4KM.md" tag="0ZMT1GP4KM" kind="statement">
# Path-connected spaces are connected

## Statement

Let $X$ be a [[path-connected space|0K0GE6BCYW]].
Then $X$ is [[connected|0D939CMME7]].

## Remarks

Proof: suppose open sets $U$ and $V$ split $X$, with $x \in U$ and $y \in V$. Take a path $\gamma$ from $x$ to $y$. Then $\gamma([0, 1])$ is connected, by [[continuous image of a connected set is connected|0EPD87YVG8]] and [[connectedness of intervals|0B9ZQFMQNV]]. But $U$ and $V$ split it, a contradiction.
The converse fails, as the [[topologist's sine curve|0K3T4A659R]] shows. It holds for [[open subsets of R^n|0MXN7AFBPE]].

## Explanations

### L1 human

If you can walk between any two points, the space cannot be split into two separate open parts.
</document>

<document source="https://tamnd.github.io/math2/node/0ZMY8441GW.md" tag="0ZMY8441GW" kind="definition">
# Elementary symmetric polynomials

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $n \ge 1$.
For $0 \le k \le n$, the $k$-th elementary symmetric polynomial in $x_1, \dots, x_n$ is
$$e_k = \sum_{1 \le i_1 < i_2 < \dots < i_k \le n} x_{i_1} x_{i_2} \cdots x_{i_k} \in R[x_1, \dots, x_n].$$

## Remarks

So $e_0 = 1$, $e_1 = x_1 + \dots + x_n$ and $e_n = x_1 x_2 \cdots x_n$. One sets $e_k = 0$ for $k > n$.
In three variables, $e_2 = x_1 x_2 + x_1 x_3 + x_2 x_3$.
The generating identity is $\prod_{i=1}^n (t + x_i) = \sum_{k=0}^n e_k t^{n-k}$.
Each $e_k$ is [[symmetric|06CTFY2DEZ]] and homogeneous of degree $k$.

## Explanations

### L1 human

The elementary symmetric polynomials are the simplest symmetric polynomials: the sum of the variables, the sum of their products in pairs, and so on.
Up to sign, they are the coefficients of the polynomial whose roots are the variables.
</document>

<document source="https://tamnd.github.io/math2/node/0ZMYX7GMH0.md" tag="0ZMYX7GMH0" kind="statement">
# First isomorphism theorem for rings

## Statement

Let $f \colon R \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]].
Then the [[kernel|0AAHE7GEW6]] $\ker f$ is a two-sided ideal of $R$, and the image $f(R)$ is a subring of $S$.
The map $R/\ker f \to f(R)$, $a + \ker f \mapsto f(a)$, is a [[ring isomorphism|0B6QC0C0G1]].
In particular, if $f$ is surjective, then $R/\ker f \cong S$.

## Remarks

The reduction map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ has kernel $n\mathbb{Z}$.
The evaluation map $\mathbb{R}[x] \to \mathbb{C}$, $p \mapsto p(i)$, is surjective with kernel $(x^2 + 1)$. So $\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}$.
The theorem has the same form for groups and for modules.

## Explanations

### L1 human

A homomorphism forgets exactly the information in its kernel.
If you divide out the kernel, the homomorphism becomes a bijection onto its image.
So every image of a ring is a quotient of that ring.
</document>

<document source="https://tamnd.github.io/math2/node/0ZN07RQ3JS.md" tag="0ZN07RQ3JS" kind="definition">
# Stirling numbers of the second kind

## Definition

Let $n$ and $k$ be natural numbers.
The Stirling number of the second kind $S(n, k)$ is the number of partitions of an $n$-element set into $k$ nonempty blocks.
A partition into blocks is a set of pairwise disjoint nonempty subsets whose union is the whole set.

## Remarks

Other notations are $\left\{ {n \atop k} \right\}$ and $S_n^{(k)}$.
Boundary values: $S(0, 0) = 1$, $S(n, 0) = 0$ for $n \ge 1$, and $S(n, k) = 0$ for $k > n$.
Example: $S(4, 2) = 7$. The partitions of $\{1, 2, 3, 4\}$ into two blocks are $1|234$, $2|134$, $3|124$, $4|123$, $12|34$, $13|24$ and $14|23$.
The values satisfy a [[recurrence|0C019SDTER]], and the sum $\sum_k S(n, k)$ is the [[Bell number|0WW85SQVS3]] $B_n$.

## Explanations

### L1 human

$S(n, k)$ counts the ways to split $n$ labelled objects into $k$ groups, when the groups have no labels and none is empty.
For example, there are $S(3, 2) = 3$ ways to split three people into two teams.
</document>

<document source="https://tamnd.github.io/math2/node/0ZN2J42SK7.md" tag="0ZN2J42SK7" kind="statement">
# Determinant of a 2 by 2 matrix

## Statement

Let $K$ be a commutative ring and let $a, b, c, d \in K$.
Then
$$\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is the [[Leibniz formula|016PE4VZ8Q]] for $n = 2$, with the identity and one transposition.
Over $\R$, $\abs{ad - bc}$ is the area of the parallelogram spanned by the columns $(a, c)$ and $(b, d)$.
For example, $\det \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = 4 - 6 = -2$.

## Explanations

### L1 human

The determinant of a $2 \times 2$ matrix is the product of the main diagonal minus the product of the other diagonal.
It is zero exactly when one column is a multiple of the other, over a field.
Then the matrix squeezes the plane onto a line or a point.
</document>

<document source="https://tamnd.github.io/math2/node/0ZN39QDTP9.md" tag="0ZN39QDTP9" kind="statement">
# K(a) is isomorphic to K[x] modulo the minimal polynomial

## Statement

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$ be algebraic over $K$ with [[minimal polynomial|0GK1WB9R98]] $m_a$.
Then the [[evaluation map|0125QVNCFQ]] $K[x] \to L$, $f \mapsto f(a)$, induces a ring isomorphism
$$K[x]/(m_a) \cong K(a).$$
In particular, $K(a) = K[a] = \{ f(a) : f \in K[x] \}$.

## Remarks

The proof: the kernel of the evaluation map is $(m_a)$, by [[the divisibility property|06K7SARZKR]]. The [[first isomorphism theorem|0ZMYX7GMH0]] gives $K[x]/(m_a) \cong K[a]$.
Since $m_a$ is irreducible, $K[x]/(m_a)$ is a [[field|0AWMPX35CW]]. So $K[a]$ is a field, and $K[a] = K(a)$.
If $a$ is transcendental over $K$, then the evaluation map is injective, and it extends to an isomorphism $K(x) \cong K(a)$.

## Explanations

### L1 human

An algebraic element adds nothing to the field beyond the relation given by its minimal polynomial.
So $K(a)$ is the same as polynomials in a variable, computed modulo the minimal polynomial.
For example, $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}[x]/(x^2 - 2)$.
</document>

<document source="https://tamnd.github.io/math2/node/0ZN9EZTEM6.md" tag="0ZN9EZTEM6" kind="concept">
# Conditional probability distribution

## Summary

The conditional distribution of $Y$ given $X = x$ is the distribution of $Y$ when one knows that $X$ has the value $x$.
The node [[conditional probability distribution|0YVDC5PYDW]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ZND7NQ970.md" tag="0ZND7NQ970" kind="definition">
# Eigenvalue

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
A scalar $\lambda \in F$ is an *eigenvalue* of $T$ when there is a vector $v \in V$ with $v \ne 0$ and
$$T(v) = \lambda v.$$
An eigenvalue of an $n \times n$ matrix $A$ over $F$ is an eigenvalue of the map $F^n \to F^n$, $x \mapsto Ax$.

## Remarks

So $\lambda$ is an eigenvalue of $T$ if and only if $T - \lambda \id_V$ is not injective.
In finite dimensions, this holds if and only if $\det(T - \lambda \id_V) = 0$.
The eigenvalues of the diagonal matrix $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ are $2$ and $3$.
The rotation of $\R^2$ by $90$ degrees has no real eigenvalue.

## Explanations

### L1 human

Most vectors change direction when a linear map acts on them.
An eigenvalue is a scaling factor $\lambda$ such that some nonzero vector only gets stretched by $\lambda$, without a change of its line.
These special directions often reveal the structure of the map.
</document>

<document source="https://tamnd.github.io/math2/node/0ZNMMA9TF5.md" tag="0ZNMMA9TF5" kind="concept">
# Intersection

## Summary

The intersection of sets is the set of all objects that are elements of each of the sets.
</document>

<document source="https://tamnd.github.io/math2/node/0ZNSSK8ZK1.md" tag="0ZNSSK8ZK1" kind="statement">
# p-series test

## Statement

Let $p$ be a real number.
The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if and only if $p > 1$.

## Remarks

For $p = 1$ it is the [[harmonic series|0CXTMAY5PX]], which diverges. For $p \le 0$ the terms do not tend to $0$.
One proof uses the [[Cauchy condensation test|0B96KMDDD1]]. The condensed series is $\sum 2^n \cdot 2^{-np} = \sum (2^{1-p})^n$, a geometric series with ratio $2^{1-p}$.
Another proof uses the [[integral test|019QZ6RRVH]] with $f(x) = x^{-p}$.
For $p > 1$ the sum is the value $\zeta(p)$ of the Riemann zeta function. For example, $\zeta(2) = \pi^2/6$, which is the [[Basel problem|0DMY75D6FV]].

## Explanations

### L1 human

The terms $1/n^p$ shrink faster when $p$ is larger.
The value $p = 1$ is the border. Above it the sum is finite, and at it or below it the sum is infinite.
</document>

<document source="https://tamnd.github.io/math2/node/0ZNX6DQ3EJ.md" tag="0ZNX6DQ3EJ" kind="definition">
# Implication

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The implication from $p$ to $q$ is the proposition $p \to q$, read "if $p$, then $q$".
The proposition $p$ is the hypothesis, and the proposition $q$ is the conclusion.
The proposition $p \to q$ is false when $p$ is true and $q$ is false. In all other cases, $p \to q$ is true.

| $p$ | $q$ | $p \to q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |

## Remarks

Another notation for $p \to q$ is $p \Rightarrow q$.
The proposition $p \to q$ has the same truth value as $\neg p \lor q$.
When $p$ is false, $p \to q$ is true. This is [[vacuous truth|0V4QXPGJQK]].
The [[converse|0RVYE75E2M]] of $p \to q$ is $q \to p$. It can have a different truth value.

## Explanations

### L1 human

Think of $p \to q$ as a promise: "if it rains, I take an umbrella".
You break the promise only when it rains and you take no umbrella.
When it does not rain, you keep the promise, whatever you do.
</document>

<document source="https://tamnd.github.io/math2/node/0ZNXZB4HNB.md" tag="0ZNXZB4HNB" kind="concept">
# Power sum symmetric polynomial

## Summary

The $k$-th power sum in $n$ variables is the sum of the $k$-th powers of the variables.
The [[definition of the power sums|0AS1FZT6BM]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0ZNZH0EXFX.md" tag="0ZNZH0EXFX" kind="concept">
# Inverse element

## Summary

An inverse of an element undoes the element: their combination gives the identity element.
For example, $-a$ is the inverse of $a$ for addition.
</document>

<document source="https://tamnd.github.io/math2/node/0ZPB3180AY.md" tag="0ZPB3180AY" kind="statement">
# Matrix of the dual map is the transpose

## Statement

Let $F$ be a field, and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Let $B$ be a basis of $V$ and $C$ a basis of $W$, with [[dual bases|0ZD2CF6ZS0]] $B^*$ and $C^*$.
Let $T \colon V \to W$ be a linear map with [[dual map|0AZ5ARP287]] $T^*$.
Then
$$[T^*]_{B^*, C^*} = \left([T]_{C, B}\right)^T,$$
where $[\,\cdot\,]$ is the [[matrix of a linear map|0YQSGNYE3K]] and the exponent $T$ is the [[transpose|09DXWYRD43]].

## Remarks

The proof computes entry $(j, i)$ of $[T^*]$ as $T^*(c_i^*)(b_j) = c_i^*(T(b_j))$, which is entry $(i, j)$ of $[T]$.
With the [[annihilator|0ASHNYF7AG]], this gives a proof that [[row rank equals column rank|0TZAG60ZYT]].

## Explanations

### L1 human

The dual map is the linear algebra form of the transpose of a matrix.
In dual bases, its matrix is the transpose of the matrix of the original map.
This explains why rows and columns play symmetric roles.
</document>

<document source="https://tamnd.github.io/math2/node/0ZPQHH37PF.md" tag="0ZPQHH37PF" kind="concept">
# Grönwall bound

## Summary

The Grönwall bound is the solution $x \mapsto \delta e^{Kx} + \frac{\varepsilon}{K}(e^{Kx} - 1)$ of $v' = K v + \varepsilon$, $v(0) = \delta$, with the value $\delta + \varepsilon x$ when $K = 0$.
It is the upper bound in a form of [[Grönwall's inequality|07EJHQYR9A]]. See the [[definition|0FRX8JD7QH]].
</document>

<document source="https://tamnd.github.io/math2/node/0ZPV68RN1X.md" tag="0ZPV68RN1X" kind="concept">
# Recursively defined sequence

## Summary

A recursively defined sequence is a [[sequence|0QE8STGJH2]] given by its first terms and a rule that computes each later term from the terms before it. A typical form is $a_0 = c$ and $a_{n+1} = f(n, a_n)$ for a given value $c$ and a given function $f$. The recursion theorem shows that exactly one sequence satisfies such a rule.

## Remarks

The factorials satisfy $0! = 1$ and $(n+1)! = (n+1) \cdot n!$.
The Fibonacci numbers satisfy $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$. Here each term needs the two terms before it.
In analysis, the sequence $a_0 = 1$, $a_{n+1} = \frac{1}{2}(a_n + 2/a_n)$ converges to $\sqrt{2}$. It is [[Newton's method|07EXVNF9VP]] for $x^2 - 2$.
The rule that links the terms is a recurrence relation.

## Explanations

### L1 human

A recursive rule tells you how to get the next term from the terms you already have.
Start with $a_0 = 1$ and the rule $a_{n+1} = 2 a_n$, and you get $1, 2, 4, 8, \dots$.
</document>

<document source="https://tamnd.github.io/math2/node/0ZQJ7KEFQ5.md" tag="0ZQJ7KEFQ5" kind="statement">
# Tower law for indices

## Statement

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $K$ be a subgroup of $H$.
Then $[G : K] = [G : H] \cdot [H : K]$, where $[\cdot : \cdot]$ is the [[index|0DTXBQPA2S]].

## Remarks

The equation holds as an equation of cardinal numbers; in particular $[G : K]$ is finite if and only if $[G : H]$ and $[H : K]$ are both finite.
For example, in $(\mathbb{Z}, +)$ with $H = 2\mathbb{Z}$ and $K = 6\mathbb{Z}$, the indices are $[\mathbb{Z} : 6\mathbb{Z}] = 6 = 2 \cdot 3$.

## Explanations

### L1 human

Each coset of $H$ in $G$ splits into $[H : K]$ cosets of $K$.
So the number of cosets of $K$ is the number of cosets of $H$ times $[H : K]$.
It is like a box with several drawers, each with the same number of compartments.
</document>

<document source="https://tamnd.github.io/math2/node/0ZS54B9GJF.md" tag="0ZS54B9GJF" kind="statement">
# The logarithm is strictly concave

## Statement

The [[natural logarithm|0044CPHJBV]] is strictly [[concave|0EF2VG8XKQ]] on $(0, \infty)$.
That is, $\ln(t x + (1 - t) y) > t \ln x + (1 - t) \ln y$ for all $x \ne y$ in $(0, \infty)$ and $t \in (0, 1)$.

## Remarks

The [[derivative of $\ln x$|07X1QA6BRW]] is $1/x$, and its second derivative is $-1/x^2 < 0$.
The concavity gives the [[AM–GM inequality|0DRF0F3XC9]] through Jensen's inequality.
Mathlib states it as `strictConcaveOn_log_Ioi`.

## Explanations

### L1 human

The graph of $\ln x$ bends downward everywhere.
Every chord of the graph lies below the graph.
</document>

<document source="https://tamnd.github.io/math2/node/0ZT24EC90K.md" tag="0ZT24EC90K" kind="definition">
# Central moment

## Definition

Let $X$ be a real random variable and let $k \ge 1$ be an integer with $E[|X|^k] < \infty$. Let $\mu = E[X]$.
The $k$-th central moment of $X$ is
$$\mu_k = E\left[(X - \mu)^k\right].$$

## Remarks

Always $\mu_1 = 0$, and $\mu_2 = \operatorname{Var}(X)$ is the [[variance|007QED3E0Y]].
Central moments do not change when one adds a constant to $X$. For $a \in \mathbb{R}$, the central moments of $aX$ are $a^k \mu_k$.
The binomial theorem gives $\mu_k = \sum_{j=0}^{k} \binom{k}{j} (-\mu)^{k - j} m_j$ in terms of the [[moments|0PH0VHTQS0]] $m_j = E[X^j]$, with $m_0 = 1$.
For a distribution that is symmetric about its mean, all odd central moments are $0$.

## Explanations

### L1 human

Central moments measure the shape of a distribution around its center, and not its position.
The third gives asymmetry and the fourth gives the weight of the tails.
</document>

<document source="https://tamnd.github.io/math2/node/0ZTEH055YF.md" tag="0ZTEH055YF" kind="definition">
# Operator norm

## Definition

Let $V$ and $W$ be normed spaces and let $T \colon V \to W$ be a [[continuous linear map|0S3SMRH0Z5]].
The operator norm of $T$ is
$$\|T\| = \sup \{ \|Tx\|_W : x \in V, \ \|x\|_V \le 1 \}.$$

## Remarks

The [[supremum|0BTMQ04556]] is finite because $T$ is [[bounded|0F3K4C7XYW]]. If $V \neq \{0\}$, then $\|T\| = \sup_{x \neq 0} \|Tx\|_W / \|x\|_V$.
It is the smallest $C \ge 0$ with $\|Tx\|_W \le C \|x\|_V$ for all $x \in V$.
The operator norm is a norm on the space $L(V, W)$ of continuous linear maps. It satisfies $\|ST\| \le \|S\| \, \|T\|$. If $W$ is a [[Banach space|0BDTTM24DP]], so is $L(V, W)$.
For a diagonal matrix on $\mathbb{R}^n$ with the Euclidean norm, the operator norm is the largest absolute value of a diagonal entry.

## Explanations

### L1 human

The operator norm tells you how much $T$ can stretch a vector at most.
For example, the map $x \mapsto 3x$ on $\mathbb{R}$ has operator norm $3$.
</document>

<document source="https://tamnd.github.io/math2/node/0ZTGMDNVTB.md" tag="0ZTGMDNVTB" kind="statement">
# Intersection of subgroups is a subgroup

## Statement

Let $G$ be a [[group|G5S1M9]] and let $(H_i)_{i \in I}$ be a family of [[subgroups|0X1JXT47GA]] of $G$, indexed by a nonempty set $I$.
Then the intersection $\bigcap_{i \in I} H_i$ is a subgroup of $G$.

## Remarks

The identity is in each $H_i$, and if $a, b$ are in each $H_i$, then $ab^{-1}$ is in each $H_i$.
The union of two subgroups is in general not a subgroup: in $(\mathbb{Z}, +)$, $2 \in 2\mathbb{Z}$ and $3 \in 3\mathbb{Z}$, but $5$ is in neither.
The union of two subgroups $H$ and $K$ is a subgroup only when $H \subseteq K$ or $K \subseteq H$.

## Explanations

### L1 human

If several subgroups each pass the subgroup test, then their common part also passes it.
For example, $2\mathbb{Z} \cap 3\mathbb{Z} = 6\mathbb{Z}$ in the integers.
This fact lets us define the smallest subgroup that contains a given subset.
</document>

<document source="https://tamnd.github.io/math2/node/0ZTKJJDGR7.md" tag="0ZTKJJDGR7" kind="object">
# Zero matrix

## Description

Let $R$ be a ring and let $m, n \ge 0$ be integers.
The $m \times n$ *zero matrix* over $R$ is the [[matrix|0VD9ZSWWJG]] $0_{m \times n}$ whose entries are all $0$.
It is the identity element for [[matrix addition|0Y7H3SAVA1]]: $A + 0 = 0 + A = A$ for each $m \times n$ matrix $A$.
For each matrix $B$ with $n$ rows, the product $0_{m \times n} B$ is a zero matrix.

## Remarks

The $2 \times 2$ zero matrix is $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$.
The zero matrix is the matrix of the zero linear map in any pair of bases.
A product of two nonzero matrices can be a zero matrix, for example $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}^2 = 0$.

## Explanations

### L1 human

The zero matrix is the table with $0$ in every position.
It plays the role of the number $0$: adding it to a matrix changes nothing.
There is one zero matrix for each size.
</document>

<document source="https://tamnd.github.io/math2/node/0ZV14XRNF0.md" tag="0ZV14XRNF0" kind="concept">
# Closed and exact forms

## Summary

A [[differential form|0E3G0GRPZ6]] is closed if its [[exterior derivative|0ZHSTAG8ZY]] is zero, and exact if it is the exterior derivative of another form.
Each exact form is closed. Whether each closed form is exact depends on the shape of the domain, so these notions detect holes in the domain.
</document>

<document source="https://tamnd.github.io/math2/node/0ZV2YKKZ58.md" tag="0ZV2YKKZ58" kind="definition">
# Solution of a linear recurrence

## Definition

Let $R$ be a commutative ring, and let $E$ be a [[linear recurrence with constant coefficients|081THG13JZ]] over $R$, of order $d$ with coefficients $c_0, \dots, c_{d-1}$.
A solution of $E$ is a sequence $u : \mathbb{N} \to R$ such that
$$u_{n+d} = \sum_{i=0}^{d-1} c_i\, u_{n+i} \quad \text{for all } n \in \mathbb{N}.$$

## Remarks

Each choice of the initial values $u_0, \dots, u_{d-1}$ gives exactly one solution.
The sum of two solutions is a solution, and a multiple of a solution is a solution. So the solutions form the [[space of solutions|05BDYVQQ0T]].
For $u_{n+2} = u_{n+1} + u_n$, the Lucas numbers $2, 1, 3, 4, 7, 11, \dots$ form a solution.

## Explanations

### L1 human

A solution is a sequence that obeys the rule of the recurrence at every step.
The rule alone does not fix the sequence: you also choose the first $d$ terms.
</document>

<document source="https://tamnd.github.io/math2/node/0ZV9A7EV0F.md" tag="0ZV9A7EV0F" kind="definition">
# Complex power

## Definition

Let $z, w \in \mathbb{C}$ with $z \neq 0$.
The complex powers of $z$ with exponent $w$ are the numbers $e^{w \ell}$, where $\ell$ runs through the [[complex logarithms|0X2FY491PF]] of $z$.
The principal value of $z^w$ is
$$z^w = e^{w \operatorname{Log} z},$$
where $\operatorname{Log}$ is the [[principal branch|0YKY4M3KRQ]] of the logarithm and $e^u$ is the [[complex exponential|054VXHMCA3]] of $u$.

## Remarks

The powers of $z$ with exponent $w$ are the numbers $e^{w \operatorname{Log} z} e^{2 \pi i k w}$ with $k \in \mathbb{Z}$. There is one value if $w$ is an integer, there are $q$ values if $w = p/q$ in lowest terms with $q \ge 2$, and there are infinitely many values otherwise.
For an integer $w = n$, $z^n$ is the usual product of $n$ factors $z$, or of $-n$ factors $1/z$.
For example, $i^i = e^{i \cdot i \pi/2} = e^{-\pi/2}$, a positive real number.
The usual rules can fail: with principal values, $(z^a)^b \neq z^{ab}$ for some $z, a, b$. A common convention for $z = 0$ is $0^0 = 1$ and $0^w = 0$ for $w \neq 0$.

## Explanations

### L1 human

For real numbers, $a^b = e^{b \ln a}$ when $a > 0$. The complex power uses the same formula with a complex logarithm.
Because the logarithm has many values, the power can have many values too, and the principal value picks one of them.
</document>

<document source="https://tamnd.github.io/math2/node/0ZVEAYVRC3.md" tag="0ZVEAYVRC3" kind="concept">
# Law of large numbers

## Summary

The law of large numbers says that the average of many independent repetitions of a random experiment tends to the expected value.
For an [[i.i.d.|0AJN7WHBJZ]] sequence $X_1, X_2, \dots$ with finite mean $\mu$, the sample mean $\bar{X}_n = (X_1 + \dots + X_n)/n$ tends to $\mu$.
The [[weak law|0BFDHF2H7N]] gives convergence in probability, and the [[strong law|0DFWHQ839T]] gives almost sure convergence.

## Remarks

The law fails without a finite mean. For i.i.d. standard Cauchy variables, $\bar{X}_n$ is standard Cauchy for each $n$ and does not converge.
For indicator variables of an event $A$, the law says that the relative frequency of $A$ tends to $P(A)$. This links probability with long-run frequency.

## Explanations

### L1 human

If you toss a fair coin many times, the fraction of heads comes close to $1/2$.
The law of large numbers is the precise form of this experience. It is the reason why averages of many measurements are reliable.
</document>

<document source="https://tamnd.github.io/math2/node/0ZVHXDSZC7.md" tag="0ZVHXDSZC7" kind="definition">
# Quotient set

## Definition

Let $\sim$ be an [[equivalence relation|0SGTMBEWRF]] on a set $A$.
The quotient set of $A$ by $\sim$ is the set
$$A/{\sim} = \{[a] \mid a \in A\}$$
of all [[equivalence classes|082RMRSY26]].
The canonical projection is the [[function|0XHDTE5YMZ]] $\pi \colon A \to A/{\sim}$ with $\pi(a) = [a]$.

## Remarks

For congruence modulo $n$ on $\mathbb{Z}$, the quotient set $\mathbb{Z}/n\mathbb{Z}$ has the $n$ elements $[0], [1], \dots, [n-1]$.
The canonical projection is [[surjective|0XKH881BZ7]].
A function on $A/{\sim}$ is often defined by a formula on representatives. The formula must give the same value for equivalent representatives.

## Explanations

### L1 human

In the quotient set, each class becomes a single point.
From $\mathbb{Z}$ with congruence modulo $2$ you get a set with two elements: "even" and "odd".
</document>

<document source="https://tamnd.github.io/math2/node/0ZVQKTRKVQ.md" tag="0ZVQKTRKVQ" kind="concept">
# Bipartite graph

## Summary

A bipartite graph is a graph whose vertices split into two parts with every edge between the parts. Bipartite graphs model relations between two kinds of object, and they are the setting of matching theory. The node [[bipartite graph|07DVHJ8N0A]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ZXJ2455NJ.md" tag="0ZXJ2455NJ" kind="statement">
# Uniqueness of limits

## Statement

Let $(a_n)$ be a real sequence and let $L$ and $L'$ be real numbers.
If $(a_n)$ converges to $L$ and $(a_n)$ converges to $L'$, then $L = L'$.

## Remarks

The proof takes $\varepsilon = |L - L'|/2$ in the [[epsilon-N definition|09G6KHS6GY]]. If $L \ne L'$, then for large $n$ the term $a_n$ would be closer than $\varepsilon$ to both numbers, which is impossible.
The statement holds in each metric space and, more generally, in each Hausdorff topological space.
The statement fails in some spaces that are not Hausdorff. In a set with the trivial topology, each sequence converges to each point.

## Explanations

### L1 human

A sequence cannot approach two different numbers at the same time.
If the two numbers are a distance $d$ apart, the terms cannot be within $d/2$ of both of them.
</document>

<document source="https://tamnd.github.io/math2/node/0ZXSF2YQV2.md" tag="0ZXSF2YQV2" kind="concept">
# Isometric isomorphism

## Summary

An isometric isomorphism is a bijective isometry between metric spaces.
Two metric spaces with an isometric isomorphism between them have the same metric properties.
</document>

<document source="https://tamnd.github.io/math2/node/0ZXW95S44M.md" tag="0ZXW95S44M" kind="statement">
# Second supplement to quadratic reciprocity

## Statement

Let $p$ be an odd prime.
Then
$$\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8},$$
where the left side is a [[Legendre symbol|01J8AMV9RS]].

## Remarks

So $2$ is a quadratic residue modulo $p$ if $p \equiv \pm 1 \pmod 8$, and a nonresidue if $p \equiv \pm 3 \pmod 8$.
For example, $3^2 \equiv 2 \pmod 7$, but $2$ is not a square modulo $5$.
One proof uses [[Gauss's lemma|037ACYXB5Z]] with $a = 2$.

## History

Euler proved the second supplement.

## Explanations

### L1 human

Whether $2$ is a square modulo $p$ depends only on the remainder of $p$ on division by $8$.
The remainders $1$ and $7$ give a square, and the remainders $3$ and $5$ do not.
</document>

<document source="https://tamnd.github.io/math2/node/0ZXZ82FTX2.md" tag="0ZXZ82FTX2" kind="concept">
# Finite field

## Summary

A finite field is a field with a finite number of elements.
For each prime power $q$ there is exactly one finite field of order $q$, up to isomorphism, and there are no other finite fields.
The [[definition of a finite field|08RTDHX1SK]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/0ZY9DX8FBF.md" tag="0ZY9DX8FBF" kind="concept">
# Probability density function

## Summary

A probability density function of a real random variable $X$ is a nonnegative function $f$ with this property.
For each set $B$ of values, the integral of $f$ over $B$ is the probability that $X$ lies in $B$.
The node [[probability density function|04YG690N5V]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ZYBZ34RT2.md" tag="0ZYBZ34RT2" kind="definition">
# Probability space

## Definition

A probability space is a triple $(\Omega, \mathcal{F}, P)$ with these parts.
$\Omega$ is a nonempty set, the [[sample space|0W7NFW23RF]].
$\mathcal{F}$ is a [[σ-algebra|04WSW8T3SY]] on $\Omega$. Its elements are the events.
$P$ is a [[probability measure|0VGMEBVW1M]] on $(\Omega, \mathcal{F})$.

## Remarks

For one roll of a fair die: $\Omega = \{1, \dots, 6\}$, $\mathcal{F}$ is the power set of $\Omega$, and $P(A) = |A| / 6$.
A discrete probability space has a countable $\Omega$, the power set as $\mathcal{F}$, and $P(A) = \sum_{\omega \in A} p(\omega)$ for a function $p \ge 0$ with total sum $1$.
The interval $[0, 1]$ with its Borel σ-algebra and the length of intervals as $P$ is a probability space. It models a point chosen uniformly at random in $[0, 1]$.

## Explanations

### L1 human

A probability space puts three things together: what can happen, which questions we can ask about it, and how likely each answer is.
Every problem in probability starts, at least in the background, with one probability space.
</document>

<document source="https://tamnd.github.io/math2/node/0ZZ4RHTR9X.md" tag="0ZZ4RHTR9X" kind="concept">
# Taylor polynomial

## Summary

The Taylor polynomial of degree $k$ of a function at a point has the same value and the same first $k$ derivatives there. It approximates the function near the point. The node [[Taylor polynomial|0HFSFVMWQB]] gives the precise definition.
</document>

<document source="https://tamnd.github.io/math2/node/0ZZ7ZYDR8W.md" tag="0ZZ7ZYDR8W" kind="concept">
# Meromorphic function

## Summary

A meromorphic function on an open set is a function that is holomorphic on the set except at isolated points, which are poles. See the [[definition|0NSD1TPHR0]].
Quotients of holomorphic functions are meromorphic.
</document>

<document source="https://tamnd.github.io/math2/node/0ZZQ2E1MTE.md" tag="0ZZQ2E1MTE" kind="statement">
# Determinant of a transpose

## Statement

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$.
Then $\det A^T = \det A$, where $A^T$ is the [[transpose|09DXWYRD43]] of $A$ and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

In the [[Leibniz formula|016PE4VZ8Q]], the term of $\sigma$ for $A^T$ equals the term of $\sigma^{-1}$ for $A$, and $\sgn(\sigma^{-1}) = \sgn(\sigma)$.
So each property of the determinant for rows also holds for columns.

## Explanations

### L1 human

Exchanging rows and columns does not change the determinant.
So we can expand, reduce or test a determinant by columns as well as by rows.
Every rule for rows has a twin rule for columns.
</document>

<document source="https://tamnd.github.io/math2/node/0ZZR8D7Y1Z.md" tag="0ZZR8D7Y1Z" kind="concept">
# Subnormal series

## Summary

A subnormal series of a group is a finite chain of subgroups from the trivial subgroup to the whole group.
Each subgroup in the chain is normal in the next one.
Its factor groups are the quotients of consecutive terms.
</document>

<document source="https://tamnd.github.io/math2/node/0ZZRSX91QR.md" tag="0ZZRSX91QR" kind="definition">
# Argument of a complex number

## Definition

Let $z$ be a [[complex number|09FTAET5Z4]] with $z \neq 0$, and let $|z|$ be its [[modulus|0KW2X0TFBX]].
An argument of $z$ is a real number $\theta$ with
$$z = |z| (\cos \theta + i \sin \theta).$$
The notation $\arg z$ means the set of all arguments of $z$, or one chosen argument.

## Remarks

If $\theta$ is an argument of $z$, then the arguments of $z$ are exactly the numbers $\theta + 2 \pi k$ with $k \in \mathbb{Z}$.
The number $0$ has no argument in this sense. Some conventions set $\arg 0 = 0$.
For example, the arguments of $i$ are $\pi/2 + 2 \pi k$, and those of $-1$ are $\pi + 2 \pi k$.

## Explanations

### L1 human

The argument says in which direction a complex number points from $0$.
The number $1 + i$ points at $45$ degrees, so $\pi/4$ is an argument of it.
A full turn brings you back to the same direction, so adding $2\pi$ gives another argument.
</document>

<document source="https://tamnd.github.io/math2/node/0ZZXSBTR4J.md" tag="0ZZXSBTR4J" kind="concept">
# Rank of a linear map

## Summary

The rank of a linear map is the dimension of its image.
</document>

<document source="https://tamnd.github.io/math2/node/101H2VSX2R.md" tag="101H2VSX2R" kind="conjecture">
# Sendov's conjecture

## Statement

Let $n \ge 2$ and let $p$ be a polynomial of degree $n$ with complex coefficients whose [[roots|019MA1VTR2]] all lie in the closed unit disc $\set{z \in \C : \abs{z} \le 1}$.
Then for each root $a$ of $p$ there is a root $b$ of the derivative $p'$ with $\abs{a - b} \le 1$.

## Remarks

The roots of $p'$ are the [[critical points|0YTCMFB4HC]] of $p$.
The Gauss–Lucas theorem says that the critical points lie in the [[convex hull|0FQSDC2EQ2]] of the roots; this only gives the distance bound $2$.
The bound $1$ would be sharp: for $p(z) = z^n - 1$ and the root $a = 1$, the only critical point is $0$, at distance exactly $1$.
The conjecture has been proved for polynomials of degree at most $8$, by Brown and Xiang.
Terence Tao proved it for all polynomials of sufficiently large degree $n$.
The general case is open.

## Explanations

### L1 human

Place the roots of a polynomial inside the unit disc.
The derivative of the polynomial has its own roots, which tend to sit among the roots of the polynomial.
The conjecture says that every root of the polynomial has a root of the derivative within distance $1$.
It is simple to state, yet it has resisted proof for decades.
</document>

<document source="https://tamnd.github.io/math2/node/105G7PWKS1.md" tag="105G7PWKS1" kind="counterexample">
# Function with no Laplace transform

## Counterexample

The function $f(t) = e^{t^2}$ on $[0, \infty)$ is continuous, but its [[Laplace integral|05XT5AVJBK]] $\int_0^\infty e^{t^2} e^{-st}\, dt$ converges for no $s \in \C$.
Check: suppose it converged at some $s_0 \in \C$.
By [[the half-plane theorem|6X05N89XDP]] it would then converge at every real $s > \operatorname{Re} s_0$.
But for real $s$ and $t \ge \max(s, 0)$, the integrand satisfies $e^{t^2 - st} = e^{t(t - s)} \ge 1$, so $\int_0^T e^{t^2 - st}\, dt \to \infty$ as $T \to \infty$.
This is a contradiction, so the abscissa of convergence of $f$ is $+\infty$.
The function is not of [[exponential order|TKZAV3KR86]]: $e^{t^2} / e^{at} \to \infty$ for every real $a$.

## Remarks

Exponential order is sufficient, not necessary, for a Laplace transform to exist.
For example, $f(t) = 2t e^{t^2} \cos(e^{t^2})$ is not of exponential order, yet $\int_0^T f(t)\, dt = \sin(e^{T^2}) - \sin 1$ stays bounded, and integration by parts shows that its Laplace integral converges for every $s$ with $\operatorname{Re} s > 0$.
A different obstruction is local: $f(t) = 1/t$ has no Laplace transform because $\int_0^1 dt/t$ diverges, whatever $s$ is.

## Explanations

### L1 human

The Laplace transform weighs $f(t)$ by $e^{-st}$, which shrinks at most exponentially fast.
The function $e^{t^2}$ grows faster than any exponential, so no choice of $s$ can tame it and the integral always blows up.
Fast growth is not the only way to fail: a function such as $1/t$ is too large near $0$.
</document>

<document source="https://tamnd.github.io/math2/node/10B1TS96VA.md" tag="10B1TS96VA" kind="definition">
# Morse–Smale system

## Definition

Let $M$ be a compact smooth manifold and let $f \colon M \to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]].
A [[periodic point|T75CZ5H47N]] $p$ of period $n$ is hyperbolic if no eigenvalue of $Df^n_p$ has modulus $1$.
Such a point has a stable manifold $W^s(p)$ and an unstable manifold $W^u(p)$.
The diffeomorphism $f$ is *Morse–Smale* if the following hold.

1. The [[non-wandering set|EY2W34JZA4]] $\Omega(f)$ is finite, so it consists of finitely many periodic orbits.
2. Every periodic point of $f$ is hyperbolic.
3. For all periodic points $p$ and $q$, the manifolds $W^s(p)$ and $W^u(q)$ intersect [[transversally|EWVATW6EYZ]].

## Remarks

For a flow $\varphi_t$ generated by a vector field on $M$, the non-wandering set must be the union of finitely many equilibria and closed orbits.
All of them must be hyperbolic, with transverse stable and unstable manifolds.
Example: consider the flow on the round sphere $S^2$ that moves every point along its meridian from the north pole $N$ to the south pole $S$.
Let $f$ be its time-one map.
Then $\Omega(f) = \set{N, S}$, the point $N$ is a hyperbolic source and $S$ a hyperbolic sink.
Here $W^s(S) = S^2 \setminus \set{N}$ is open, so it meets $W^u(N)$ transversally, and the other intersections are single points with full-dimensional manifolds or empty.
Non-example: an irrational rotation of the circle has no periodic points, and its non-wandering set is the whole circle, so clause 1 fails.
By the stable manifold theorem for hyperbolic periodic points, $W^s(p)$ and $W^u(p)$ are injectively immersed submanifolds of $M$.
Smale showed that the gradient flow of a Morse function satisfies clause 3 after a small change of the metric.
</document>

<document source="https://tamnd.github.io/math2/node/10CQ5007PH.md" tag="10CQ5007PH" kind="concept">
# Cotton tensor

## Summary

The Cotton tensor is defined on a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] of dimension at least $3$.
It is a $3$-index tensor built from the covariant derivatives of the Schouten tensor, a modified [[Ricci curvature|4GTNM0WHDH]].
In dimension $3$ the [[Weyl tensor|SBHY31HESP]] is always zero, and the Cotton tensor takes over its role.
A $3$-dimensional metric is locally conformally flat if and only if its Cotton tensor vanishes.
In higher dimensions it is, up to a constant factor, the divergence of the Weyl tensor.

## Explanations

### L1 human

In three dimensions all of the curvature is determined by the Ricci curvature, so the usual test for being conformally flat, the Weyl tensor, says nothing.
The Cotton tensor is the replacement test.
It looks at how the Ricci curvature changes from point to point, through its derivatives.
A space where the Ricci curvature is a constant multiple of the metric, such as the round $3$-sphere, has zero Cotton tensor.
</document>

<document source="https://tamnd.github.io/math2/node/10FGGCQ6TS.md" tag="10FGGCQ6TS" kind="definition">
# Power iteration

## Definition

Let $A$ be an $n \times n$ complex [[matrix|0VD9ZSWWJG]] and let $x_0 \in \C^n$ be a nonzero starting vector.
*Power iteration* is the sequence defined by
$$x_{k+1} = \frac{A x_k}{\norm{A x_k}}, \qquad k \ge 0,$$
as long as $A x_k \ne 0$, together with the eigenvalue estimates $\mu_k = x_k^* A x_k / (x_k^* x_k)$.

## Remarks

The numbers $\mu_k$ are [[Rayleigh quotients|06WHP8SE98]].
Convergence: suppose $A$ is diagonalizable with [[eigenvalues|0BQAA2BK3Q]] $\abs{\lambda_1} > \abs{\lambda_2} \ge \cdots \ge \abs{\lambda_n}$, and $x_0$ has a nonzero component along an [[eigenvector|0N6960W0BV]] $v_1$ for $\lambda_1$ in the eigenbasis.
Then the direction of $x_k$ tends to that of $v_1$, up to a scalar factor of modulus $1$, with error of order $\abs{\lambda_2 / \lambda_1}^k$, and $\mu_k \to \lambda_1$.
Example: $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ has eigenvalues $3$ and $1$ with eigenvectors $(1, 1)$ and $(1, -1)$; for $x_0 = (1, 0) = \tfrac12 (1, 1) + \tfrac12 (1, -1)$, one gets $A^k x_0 = \tfrac12 (3^k + 1, 3^k - 1)$, whose direction tends to $(1, 1)$ with error of order $3^{-k}$.
Non-example of convergence: for $A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, the condition $\abs{\lambda_1} > \abs{\lambda_2}$ fails, and from $x_0 = (1, 1)$ the iterates alternate between the directions of $(1, -1)$ and $(1, 1)$.
Applying the method to $(A - \sigma I)^{-1}$ gives inverse iteration, which finds the eigenvalue closest to a shift $\sigma$.
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<document source="https://tamnd.github.io/math2/node/10HFW1GRV2.md" tag="10HFW1GRV2" kind="statement">
# Doob's upcrossing inequality

## Statement

Let $n \ge 1$, let $(X_k)_{0 \le k \le n}$ be a [[submartingale|EQ055QMACW]] with respect to a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_k)_{0 \le k \le n}$, and let $a < b$ be real numbers.
Let $U_n[a, b]$ be the number of upcrossings of $[a, b]$ by $X_0, \dots, X_n$.
This is the largest $m \ge 0$ with times $0 \le s_1 < t_1 < \dots < s_m < t_m \le n$ such that $X_{s_i} \le a$ and $X_{t_i} \ge b$ for each $i$.
Then
$$(b - a)\, E\bigl[U_n[a, b]\bigr] \le E\bigl[(X_n - a)^+\bigr] - E\bigl[(X_0 - a)^+\bigr],$$
where $y^+ = \max(y, 0)$.

## Remarks

The proof uses a betting strategy: buy one unit when the process is at most $a$, and sell when it reaches at least $b$.
With $Y_k = (X_k - a)^+$, which is again a submartingale, each completed upcrossing earns at least $b - a$, so the gain is at least $(b - a) U_n[a, b]$.
The opposite strategy holds exactly when the first one does not.
It is predictable and bounded, so its expected gain on the submartingale $Y$ is non-negative.
The two gains add up to $Y_n - Y_0$.
As a consequence, if $\sup_n E[X_n^+] < \infty$, then the total number of upcrossings of every interval $[a, b]$ is finite almost surely.
This is the key step of [[Doob's martingale convergence theorems|MRXBC4ZNRR]]: a sequence that crosses no rational interval infinitely often has a limit in $[-\infty, \infty]$.

## Explanations

### L1 human

An upcrossing is a trip of the process from below $a$ to above $b$.
The inequality says that a process with an upward drift cannot make many such trips, on average, unless it ends up high.
In other words, a submartingale with controlled size cannot oscillate forever.

### L2 human

The inequality turns a question about convergence into a counting question.
A real sequence fails to converge in $[-\infty, \infty]$ exactly when it crosses some interval with rational endpoints infinitely often.
The upcrossing bound and monotone convergence show this happens with probability zero for $L^1$-bounded submartingales.
For a [[supermartingale|KYW1PV0SX5]] the analogous bound is $(b - a)\, E[U_n[a, b]] \le E[(X_n - a)^-]$, where $y^- = \max(-y, 0)$.
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<document source="https://tamnd.github.io/math2/node/10P0C0Y5V0.md" tag="10P0C0Y5V0" kind="definition">
# Completely normal space

## Definition

A [[topological space|0CQHFPGJ45]] $X$ is *completely normal* when every subset of $X$, with the [[subspace topology|0WS9P12FQR]], is a [[normal space|0NSR9V8TED]].
A completely normal space that is also [[T1|0NW6CH38YR]] is called a $T_5$ space.

## Remarks

Equivalently, $X$ is completely normal if and only if the following holds.
For all $A, B \subseteq X$ with $A \cap \overline{B} = \varnothing$ and $\overline{A} \cap B = \varnothing$, there are disjoint [[open sets|02BH3MCBQB]] $U \supseteq A$ and $V \supseteq B$.
It is enough to check that every open subspace is normal.

Example: every metric space $(X, d)$ is completely normal.
Each subspace is again a metric space, and every metric space is [[normal|0AD0BZ49DW]].

Non-example: the [[Tychonoff plank|D440JPY3QG]] is compact Hausdorff, hence normal, but it is not completely normal.
Its subspace obtained by removing one corner point is not normal.

Every [[perfectly normal space|VK3ZMD1BCP]] is completely normal.
</document>

<document source="https://tamnd.github.io/math2/node/10TJ890QK0.md" tag="10TJ890QK0" kind="definition">
# Quotient algebra

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$ and let $\theta$ be a [[congruence|DCSXXT4SER]] on $\mathbf{A}$.
For $a \in A$, let $a/\theta$ be the class of $a$, and let $A/\theta$ be the [[quotient set|0011MTQMR6]] of all classes.
The *quotient algebra* $\mathbf{A}/\theta$ has universe $A/\theta$ and the same signature as $\mathbf{A}$.
Each $n$-ary fundamental operation $f$ is interpreted by
$$f^{\mathbf{A}/\theta}(a_1/\theta, \dots, a_n/\theta) = f^{\mathbf{A}}(a_1, \dots, a_n)/\theta.$$

## Remarks

The operations are well defined because $\theta$ is compatible with each $f$.
The map $\nu_\theta \colon a \mapsto a/\theta$ is a surjective homomorphism $\mathbf{A} \to \mathbf{A}/\theta$, the natural map.

Example: for the ring $\Z$ and congruence modulo $n$, the quotient $\Z/\theta$ is the ring $\Z/n\Z$.
For instance $(3/\theta) \cdot (4/\theta) = 12/\theta = 2/\theta$ when $n = 5$.

Non-example: for an equivalence relation that is not compatible, the formula gives no operation.
Take $(\Z, +)$ with the classes "negative", "zero" and "positive".
Computing with $1$ and $-1$ gives the class of $0$, but computing with $2$ and $-1$ gives the class of $1$.

For a group $G$ and the congruence of a normal subgroup $N$, the quotient algebra is the [[quotient group|0KYX2N69J0]] $G/N$.
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<document source="https://tamnd.github.io/math2/node/10WTA1QZXD.md" tag="10WTA1QZXD" kind="definition">
# Fractional factorial design

## Definition

Let $k > p \ge 1$, and code the levels of $k$ two-level factors as $\pm 1$.
Then the runs of the full $2^k$ [[factorial experiment|EHJ3R589ZX]] are the points of $\set{-1, 1}^k$.
For a subset $w \subseteq \set{1, \dots, k}$, called a *word*, write $x^w = \prod_{j \in w} x_j$ for $x \in \set{-1, 1}^k$.
Identify words with vectors of $\F_2^k$, with the symmetric difference as addition.
A *regular $2^{k-p}$ fractional factorial design* is a set of runs
$$F = \set{x \in \set{-1, 1}^k : x^{w} = \epsilon_w \text{ for all } w \in H},$$
where:

1. $H$ is a [[subgroup|0X1JXT47GA]] of $\F_2^k$ of order $2^p$, the *defining relation*, with no word of size $1$;
2. the signs $\epsilon_w \in \set{-1, 1}$ satisfy $\epsilon_{\emptyset} = 1$ and $\epsilon_{w \triangle w'} = \epsilon_w \epsilon_{w'}$, so that $F$ is not empty.

Then $F$ has $2^{k-p}$ runs.
Two effects with words $u$ and $v$ are *aliased* if $u \triangle v \in H$.
The *resolution* of the design is the smallest size of a nonempty word of $H$.

## Remarks

On $F$, the contrast columns $x^u$ and $x^v$ of aliased effects agree up to the sign $\epsilon_{u \triangle v}$, so their effects cannot be separated.
Here $\F_2$ is the [[field with two elements|0ZXZ82FTX2]].

Example: $k = 3$, $p = 1$, $H = \set{\emptyset, \set{1,2,3}}$ with $\epsilon = +1$, that is, the generator $C = AB$ and the defining relation $I = ABC$.
The runs are $(-1,-1,+1)$, $(+1,-1,-1)$, $(-1,+1,-1)$, $(+1,+1,+1)$, four of the eight.
The main effect $A$, with word $\set{1}$, is aliased with $BC$, since $\set{1} \triangle \set{1,2,3} = \set{2,3}$.
The design has resolution III.

Non-example: the four runs with $x_1 = -1$ form the set $\set{x : x^{\set{1}} = -1}$, which uses the word $\set{1}$ of size $1$, so clause 1 fails.
In these runs factor $A$ never changes, so its main effect cannot be estimated at all.
</document>

<document source="https://tamnd.github.io/math2/node/114KYR2JGP.md" tag="114KYR2JGP" kind="definition">
# Median algebra

## Definition

Let $X$ be a set and let $\langle \cdot, \cdot, \cdot \rangle \colon X \times X \times X \to X$ be a ternary operation.
The pair $(X, \langle \cdot, \cdot, \cdot \rangle)$ is a *median algebra* if for all $x, y, z, w \in X$:

1. $\langle x, y, y \rangle = y$;
2. $\langle x, y, z \rangle = \langle z, x, y \rangle$;
3. $\langle x, y, z \rangle = \langle x, z, y \rangle$;
4. $\langle \langle x, w, y \rangle, w, z \rangle = \langle x, w, \langle y, w, z \rangle \rangle$.

## Remarks

Clauses 2 and 3 together say that $\langle x, y, z \rangle$ does not change under any permutation of $x, y, z$.
Clause 4 says that for fixed $w$, the binary operation $(x, y) \mapsto \langle x, w, y \rangle$ is associative.

Example: $X = \R$ with $\langle x, y, z \rangle$ the median of $x, y, z$, which equals $\max(\min(x, y), \min(y, z), \min(z, x))$.
More generally, every [[distributive lattice|0HM0HQCK6N]] is a median algebra with $\langle x, y, z \rangle = (x \wedge y) \vee (y \wedge z) \vee (z \wedge x)$.
For clause 1, $(x \wedge y) \vee y \vee (y \wedge x) = y$.

Non-example: a set $X$ with at least two elements and $\langle x, y, z \rangle = x$.
Clause 1 fails: for $x \ne y$, $\langle x, y, y \rangle = x \ne y$.
</document>

<document source="https://tamnd.github.io/math2/node/116TPXR5FS.md" tag="116TPXR5FS" kind="concept">
# Smooth dynamical system

## Summary

A smooth dynamical system is a [[dynamical system|BBZP1R872N]] whose phase space is a [[smooth manifold|F45VQ47GPV]] and whose time evolution is differentiable.
In discrete time it is a [[diffeomorphism|0KZZ70ZHXP]] iterated forward and backward; in continuous time it is the [[flow|06738GGWWK]] of a smooth vector field.
Differentiability allows one to linearise the motion along orbits, which leads to hyperbolicity, invariant manifolds and Lyapunov exponents.
Most dynamical systems of mechanics and of the sciences are smooth.

## Explanations

### L1 human

A smooth dynamical system is a rule that moves the points of a smooth space, such as a surface, without tearing or creasing it.
Either the rule is applied in steps, or the points flow continuously along arrows drawn on the space.
Near each orbit the motion looks, to first order, like a linear map, so tools from linear algebra apply.
For example, rotating the circle by a fixed angle, or letting a pendulum swing, are smooth dynamical systems.
See the [[definition|HZKFNZTHZS]].
</document>

<document source="https://tamnd.github.io/math2/node/119J1Q1MG9.md" tag="119J1Q1MG9" kind="statement">
# Stationary phase approximation

## Statement

Let $g \colon \R \to \C$ be a [[smooth function|0V33BNKZ9S]] with compact support, and let $f \colon \R \to \R$ be smooth.
Assume that on the support of $g$ the function $f$ has exactly one [[critical point|0YTCMFB4HC]] $x_0$, and that $f''(x_0) \ne 0$.
Then, as $k \to +\infty$,
$$\int_{\R} g(x)\, e^{i k f(x)}\,dx = \sqrt{\frac{2\pi}{k \abs{f''(x_0)}}}\; g(x_0)\, e^{i k f(x_0) + i \frac{\pi}{4} \sgn f''(x_0)} + O\big(k^{-3/2}\big).$$

## Remarks

Proof idea: if $f' \ne 0$ on the support of $g$, repeated integration by parts with $e^{ikf} = \frac{1}{ikf'} \frac{d}{dx} e^{ikf}$ shows that the integral is $O(k^{-N})$ for every $N$.
A partition of unity therefore reduces the problem to a small neighbourhood of $x_0$.
There the Morse lemma gives a change of variable with $f(x) = f(x_0) + \frac12 f''(x_0)\, y^2$, and the Fresnel integral $\int_{\R} e^{i a y^2 / 2}\,dy = \sqrt{2\pi / \abs{a}}\, e^{i \frac{\pi}{4} \sgn a}$ produces the main term.

Example: for $f(x) = x^2$ and $g$ equal to $1$ near $0$, the main term is $\sqrt{\pi / k}\, e^{i\pi/4}$, which matches $\int_{\R} e^{i k x^2}\,dx = \sqrt{\pi / k}\, e^{i\pi/4}$.

The hypothesis $f''(x_0) \ne 0$ is needed for this rate: for $f(x) = x^3$ the critical point at $0$ is degenerate, and the integral decays only like $k^{-1/3}$.

## Explanations

### L1 human

When $k$ is large, the factor $e^{ikf(x)}$ spins around the unit circle very fast as $x$ moves.
The positive and negative parts then cancel almost completely.
The cancellation fails only near points where the phase $f$ is stationary, that is, where $f'(x) = 0$.
So the whole integral is decided by a small neighbourhood of these points, and the theorem gives the exact leading term.

### L2 human

The stationary phase approximation is the oscillatory counterpart of [[Laplace's method|9G7K8DBT9Y]], which treats real exponents $e^{k f}$.
Both are special cases of the [[method of steepest descent|Q5H9VYFK98]] when the functions extend holomorphically.
With several nondegenerate critical points the contributions add up, and there is a full asymptotic expansion in powers of $k^{-1/2 - j}$.
In $\R^n$ the factor becomes $(2\pi/k)^{n/2} \abs{\det f''(x_0)}^{-1/2} e^{i \frac{\pi}{4} \operatorname{sgn} f''(x_0)}$, where $\operatorname{sgn}$ of the Hessian is its signature.
It gives the asymptotics of [[Bessel functions|CZCMRDWDYW]] and Airy functions and the decay of Fourier transforms of surface measures.
In physics it describes the passage from wave optics to geometric optics.
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<document source="https://tamnd.github.io/math2/node/11FFYBNF0E.md" tag="11FFYBNF0E" kind="statement">
# Homotopy lifting for covering maps

## Statement

Let $p \colon E \to B$ be a [[covering map|3XZRRE165N]], and let $X$ be a [[topological space|0CQHFPGJ45]].
Let $H \colon X \times [0, 1] \to B$ be a [[homotopy|9YA6S6HD9V]], and let $\tilde f \colon X \to E$ be a continuous map with $p \circ \tilde f = H(\cdot, 0)$.
Then there is a unique homotopy $\tilde H \colon X \times [0, 1] \to E$ such that $p \circ \tilde H = H$ and $\tilde H(\cdot, 0) = \tilde f$.

## Remarks

For each $x \in X$, cover $\{x\} \times [0, 1]$ by sets of the form $N \times [t_i, t_{i+1}]$ that $H$ maps into evenly covered open sets of $B$.
On each such piece the lift is forced: it is the composite with the inverse of $p$ on the sheet that contains the lift already built.
Uniqueness of lifts on the connected sets $\{x\} \times [0, 1]$ makes these local lifts agree, so they glue to a continuous $\tilde H$.
When $X$ is a point, the statement is the path lifting property: each path in $B$ lifts uniquely once the start point of the lift is chosen.
Applied to $X = [0, 1]$ and a homotopy of loops rel endpoints, it shows that $p_* \colon \pi_1(E, e_0) \to \pi_1(B, p(e_0))$ is injective, where $\pi_1$ is the [[fundamental group|S1XK3HXHFX]].
For example, take $p \colon \R \to S^1$, $p(t) = (\cos 2\pi t, \sin 2\pi t)$, and a loop at $(1, 0)$ whose lift runs from $0$ to $n \ne 0$.
This loop is not null-homotopic, because a null-homotopy would lift to a homotopy rel endpoints in $\R$.

## Explanations

### L1 human

A covering map looks locally like several stacked copies of the base.
If a movie of maps into the base is given, and the first frame is already lifted to the cover, then the whole movie lifts, and in only one way.
The picture is a helix over a circle: a motion on the circle can be followed upstairs on the helix without any choice once the starting level is fixed.

### L2 human

This property is the engine of covering space theory.
It gives the injectivity of $p_*$ on fundamental groups, the computation $\pi_1(S^1) \cong \Z$, and the lifting criterion behind the classification of covering spaces.
It also says that a covering map is a fibration with discrete fibers, so the long exact sequence of homotopy groups applies and gives $\pi_n(E) \cong \pi_n(B)$ for $n \ge 2$.
</document>

<document source="https://tamnd.github.io/math2/node/11GMR0F2Q8.md" tag="11GMR0F2Q8" kind="definition">
# Rigged Hilbert space

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] whose inner product $\ip{\cdot}{\cdot}$ is linear in the first argument.
Let $\Phi$ be a subspace of $H$ with a locally convex topology such that these conditions hold:

1. The inclusion $\Phi \to H$ is continuous.
2. $\Phi$ is dense in $H$.

Let $\Phi^\times$ be the space of continuous conjugate-linear maps $\Phi \to \C$.
Each $h \in H$ gives the element $\varphi \mapsto \ip{h}{\varphi}$ of $\Phi^\times$, and the resulting map $H \to \Phi^\times$ is linear and injective.
The chain $\Phi \subset H \subset \Phi^\times$ is a *rigged Hilbert space*, or *Gelfand triple*.
In the theory of generalized eigenvectors one also asks $\Phi$ to be a [[nuclear space|XC6XPMC2FQ]].

## Remarks

Injectivity of $H \to \Phi^\times$ comes from condition 2: if $\ip{h}{\varphi} = 0$ for all $\varphi$ in a dense subspace, then $h = 0$.
Many authors use the [[continuous dual|4A82S5FVRT]] $\Phi'$ of linear forms instead of $\Phi^\times$; then the map $H \to \Phi'$ is conjugate-linear.

Example: $\Phi = \mathcal{S}(\R)$, the Schwartz space with its Fréchet topology, $H = L^2(\R)$, and $\Phi^\times$ the conjugate-linear tempered distributions.
The inclusion is continuous because $\norm{\varphi}_{L^2} \le \sqrt{\pi}\, \sup_x \abs{(1 + x^2)^{1/2}\varphi(x)}$, and $\mathcal{S}(\R)$ is dense in $L^2(\R)$.
The delta $\varphi \mapsto \overline{\varphi(a)}$ lies in $\Phi^\times$ but not in $H$.

Non-example: let $\Phi$ be the Schwartz functions that vanish on $[0,1]$.
Its closure in $L^2(\R)$ consists of functions that vanish almost everywhere on $[0,1]$, so condition 2 fails.
Then every $h \in L^2(\R)$ supported in $[0,1]$ gives the zero functional on $\Phi$, and $H \to \Phi^\times$ is not injective.
</document>

<document source="https://tamnd.github.io/math2/node/11KYREAH1J.md" tag="11KYREAH1J" kind="concept">
# Free probability

## Summary

Free probability is a theory of noncommuting random variables in which the classical notion of independence is replaced by free independence.
It was introduced by Dan Voiculescu in the 1980s to study the von Neumann algebras of [[free groups|0KTW26C80J]].
It has analogues of the central limit theorem, of convolution and of cumulants.
In it the [[Wigner semicircle law|3QTT95DMFH]] plays the role of the normal distribution.
It also describes the limit of large random matrices, which become asymptotically free.

## Explanations

### L1 human

In ordinary probability, random numbers commute, and independence means that knowing one tells nothing about the other.
Matrices and operators do not commute, and for them there is a different and very useful rule of independence, called freeness.
It is modelled on free groups, in which products of letters from different generators never cancel.
The sum of many free variables, suitably scaled, has a semicircle distribution rather than a bell curve.
Large random matrices chosen independently behave almost exactly like free variables, which makes the theory useful in physics and statistics.
</document>

<document source="https://tamnd.github.io/math2/node/11MVWQ5R31.md" tag="11MVWQ5R31" kind="definition">
# Graph of groups

## Definition

A *graph* in the sense of Serre consists of a set $V$ of vertices and a set $E$ of oriented edges.
It comes with a map $e \mapsto \bar e$ from $E$ to $E$ with $\bar{\bar e} = e$ and $\bar e \neq e$, and maps $o, t \colon E \to V$ with $o(\bar e) = t(e)$.
A *graph of groups* $\mathbb{G}$ consists of:

1. a connected [[graph|00C0BR7D7H]] $Y$ in this sense;
2. a [[group|G5S1M9]] $G_v$ for each vertex $v$;
3. a group $G_e$ for each edge $e$, with $G_{\bar e} = G_e$;
4. an injective [[homomorphism|0JYTG5K9Y4]] $\alpha_e \colon G_e \to G_{t(e)}$ for each edge $e$.

Let $T$ be a maximal [[tree|062D9KXJB1]] in $Y$.
Form the [[free product|TM7QS7P7X9]] of the groups $G_v$ and of the free group on symbols $t_e$, $e \in E$.
The *fundamental group* $\pi_1(\mathbb{G}, T)$ is its quotient by the relations

- $t_{\bar e} = t_e^{-1}$ for each edge $e$;
- $t_e \, \alpha_e(g) \, t_e^{-1} = \alpha_{\bar e}(g)$ for each edge $e$ and each $g \in G_e$;
- $t_e = 1$ for each edge $e$ of $T$.

## Remarks

Up to isomorphism, $\pi_1(\mathbb{G}, T)$ does not depend on the choice of $T$, and each $G_v$ embeds in it.

Example: $Y$ has two vertices $u, v$ and one pair of edges $e, \bar e$ from $u$ to $v$, with $G_u = A$, $G_v = B$ and $G_e = C$.
Then $T = Y$, all $t_e$ are trivial, and the relations identify $\alpha_{\bar e}(c) \in A$ with $\alpha_e(c) \in B$; so $\pi_1(\mathbb{G}, T)$ is the [[amalgamated free product|M848AHBRCM]] $A *_C B$.

Example: $Y$ has one vertex $v$ and a loop $e$, with $G_v = A$ and $G_e = C$.
Then $T = \set{v}$, and $\pi_1(\mathbb{G}, T)$ is generated by $A$ and $t = t_e$ with $t \, \alpha_e(c) \, t^{-1} = \alpha_{\bar e}(c)$; this is the [[HNN extension|YEYERRY37K]] of $A$ along the isomorphism $\alpha_e(C) \to \alpha_{\bar e}(C)$.

Non-example: if some map $\alpha_e$ is not injective, clause 4 fails, and the data is not a graph of groups; the vertex groups need not embed in the resulting quotient.

When all vertex and edge groups are trivial, $\pi_1(\mathbb{G}, T)$ is the free group on the edges of $Y$ outside $T$, one generator for each pair $e, \bar e$.
This is the fundamental group of $Y$ as a topological graph.
</document>

<document source="https://tamnd.github.io/math2/node/11MYA5W3KT.md" tag="11MYA5W3KT" kind="statement">
# Willmore conjecture

## Statement

Let $T$ be a torus and let $f \colon T \to \R^3$ be a smooth [[immersion|6RW68JGBHC]].
Then the [[Willmore energy|R7WFJGJBV4]] of $f$ satisfies
$$\mathcal{W}(f) = \int_T H^2 \, dA \ge 2\pi^2.$$
Let $T_0$ be the torus of revolution whose generating circle has radius $1$ and center at distance $\sqrt{2}$ from the axis.
Equality holds if and only if $f(T)$ is the image of $T_0$ under a Möbius transformation of $\R^3 \cup \set{\infty}$.

## Remarks

Willmore stated the conjecture in 1965, and Marques and Neves proved it; their proof appeared in 2014.
The torus in the equality case is the stereographic image of the Clifford torus in the three-sphere.
If $f$ is not an embedding, a theorem of Li and Yau gives $\mathcal{W}(f) \ge 8\pi > 2\pi^2$, so only embedded tori need work.
For those, Marques and Neves used conformal invariance to move the problem to the three-sphere and applied Almgren–Pitts min-max theory to a five-parameter family of surfaces built from the torus.
A key step is that every closed embedded minimal surface of positive genus in the three-sphere has area at least $2\pi^2$.
Equality holds there only for the Clifford torus, up to isometry.

For every closed surface $\mathcal{W} \ge 4\pi$, so the conjecture says that being a torus costs at least $2\pi^2 - 4\pi$ extra energy.

## Explanations

### L1 human

The Willmore energy measures how much a closed surface bends overall.
Among all surfaces it is smallest for a round sphere.
The conjecture, now a theorem, finds the least possible energy for a doughnut-shaped surface: $2\pi^2$, reached by one particular fat doughnut and its Möbius images.

### L2 human

Because $\mathcal{W}$ is invariant under Möbius transformations, the minimizers come in whole families, and the natural setting is conformal geometry.
The proof is one of the main applications of min-max theory for minimal surfaces.
It belongs to a larger program of finding the least Willmore energy for each genus.
</document>

<document source="https://tamnd.github.io/math2/node/11N1AXSXYK.md" tag="11N1AXSXYK" kind="concept">
# Expectation–maximization algorithm

## Summary

The expectation–maximization (EM) algorithm is an iterative method to compute maximum likelihood estimates when the data are incomplete or the model has latent variables.
Each iteration has two steps.
The expectation step averages the complete-data log-likelihood over the missing part, given the current parameter; the maximization step maximizes this average.
Each iteration does not decrease the likelihood of the observed data.
It is the standard method for fitting mixture models and hidden Markov models; Dempster, Laird and Rubin gave its general formulation in 1977.

## Explanations

### L1 human

Suppose heights come from a mix of two groups, but nobody recorded who belongs to which group.
If you knew the groups, estimating each group's average would be easy; if you knew the averages, guessing the groups would be easy.
EM goes back and forth: it guesses soft group memberships from the current averages, then recomputes the averages from these memberships, and repeats.
Each round makes the fit to the data at least as good as before.
</document>

<document source="https://tamnd.github.io/math2/node/11PJJQRECB.md" tag="11PJJQRECB" kind="definition">
# Relative CW complex

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $A \subseteq X$ be a subspace.
For $n \ge 0$ let $D^n$ be the closed unit [[ball|05DSV40EB9]] in $\R^n$ with boundary $S^{n-1}$, where $S^{-1} = \emptyset$.
A *relative CW structure* on $(X, A)$ is a sequence of subspaces $A = X^{-1} \subseteq X^0 \subseteq X^1 \subseteq \cdots$ of $X$ such that:

1. For each $n \ge 0$ there are a set $J_n$ and maps $\varphi_j \colon S^{n-1} \to X^{n-1}$, $j \in J_n$, with the following property.
   The space $X^n$ is the pushout of $X^{n-1} \leftarrow \bigsqcup_{j \in J_n} S^{n-1} \to \bigsqcup_{j \in J_n} D^n$, where the left map is given by the $\varphi_j$.
   That is, $X^n$ is the [[quotient|0V8046R0P9]] of the [[disjoint union|A486VR787Z]] $X^{n-1} \sqcup \bigsqcup_{j} D^n$ that identifies each $x \in S^{n-1}$ in the $j$-th disk with $\varphi_j(x)$.
2. $X = \bigcup_{n \ge -1} X^n$.
3. A subset $C \subseteq X$ is closed if and only if $C \cap X^n$ is closed in $X^n$ for every $n$.

A *relative CW complex* is a pair $(X, A)$ with a relative CW structure.
The images of the open disks $D^n \setminus S^{n-1}$ are the $n$-cells of $X$ relative to $A$.

## Remarks

When $A = \emptyset$, a relative CW complex is the same as a [[CW complex|5W95QFHXPR]].
If $A$ is a subcomplex of a CW complex $X$, then $(X, A)$ is a relative CW complex.

Example: let $A = S^1$ and $X = D^2$.
Take $X^0 = X^1 = S^1$ with $J_0 = J_1 = \emptyset$, and $X^2 = D^2$ with one $2$-cell attached by the identity $S^1 \to S^1$.
Clause 1 holds since $D^2$ is the pushout of $S^1 \leftarrow S^1 \to D^2$, and clauses 2 and 3 hold since the filtration stops at $X^2 = X$.

Non-example: let $A = \set{0}$ and $X = \set{0} \cup \set{1/m : m \ge 1} \subseteq \R$.
The only cells available are $0$-cells, so $X = X^0$.
Clause 1 then forces $X^0$ to be the disjoint union of $A$ and a discrete set, but in $X$ the points $1/m$ converge to $0$, so clause 1 fails.
</document>

<document source="https://tamnd.github.io/math2/node/11VC51YHQJ.md" tag="11VC51YHQJ" kind="statement">
# Weierstrass factorization theorem

## Statement

For an integer $p \ge 0$, let $E_p$ be the elementary factor
$$E_0(z) = 1 - z, \qquad E_p(z) = (1 - z) \exp\left(z + \frac{z^2}{2} + \cdots + \frac{z^p}{p}\right) \quad (p \ge 1).$$

1. Let $(a_n)$ be a finite or infinite sequence of nonzero complex numbers with $\abs{a_n} \to \infty$ if it is infinite. Then there is an [[entire function|0419794R0T]] whose [[zeros|0CEGEASY49]] are exactly the points $a_n$. The multiplicity of each zero is the number of times it occurs in the sequence.
2. Let $f$ be an entire function that is not identically zero. Let $m \ge 0$ be the order of the zero of $f$ at $0$. Let $(a_n)$ be the nonzero zeros of $f$, repeated according to multiplicity. Then there are an entire function $g$ and integers $p_n \ge 0$ such that
$$f(z) = z^m e^{g(z)} \prod_{n} E_{p_n}\!\left(\frac{z}{a_n}\right),$$
where the product converges uniformly on compact subsets of $\C$.

## Remarks

The key estimate is $\abs{1 - E_p(z)} \le \abs{z}^{p+1}$ for $\abs{z} \le 1$.
Choose the $p_n$ so that $\sum_n (r / \abs{a_n})^{p_n + 1} < \infty$ for every $r > 0$, for example $p_n = n - 1$.
Then the [[Weierstrass product|ACSF58HYDH]] converges and has the prescribed zeros.
For part 2, the quotient of $f$ by $z^m$ times this product is entire and has no zeros.
So it is $e^{g}$ for an entire $g$, because $\C$ is simply connected.
A classical example is [[Euler's sine product|DQEHD56649]] $\sin(\pi z) = \pi z \prod_{n \ge 1} (1 - z^2/n^2)$.
The convergence factors are needed: for $a_n = n$ the product $\prod_n (1 - z/n)$ diverges, while $\prod_n (1 - z/n) e^{z/n}$ converges.

## Explanations

### L1 human

A polynomial is determined, up to a constant factor, by its roots: it is a constant times a product of factors $z - a$.
The theorem extends this to entire functions, which can have infinitely many zeros.
The plain factors $1 - z/a_n$ may give a product that does not converge, so each factor gets an exponential correction that is close to $1$ near $0$.
What is left after removing all zeros is a function without zeros, which is the exponential of an entire function.

### L2 human

Part 1 shows that any sequence tending to infinity is the zero set of some entire function.
As a consequence, every [[meromorphic function|0NSD1TPHR0]] on $\C$ is a quotient of two entire functions: multiply it by an entire function with zeros at its poles.
The theorem leaves $g$ and the $p_n$ very free.
For functions of finite order the [[Hadamard factorization theorem|KYSR7NHNFQ]] makes both choices precise.
The companion result for poles and principal parts is [[Mittag-Leffler's theorem|X4R2GY1F6J]].
</document>

<document source="https://tamnd.github.io/math2/node/11YRFWD951.md" tag="11YRFWD951" kind="concept">
# Partial autocorrelation function

## Summary

The partial autocorrelation function of a weakly stationary process gives, for each lag $h$, the correlation between $X_{t+h}$ and $X_t$.
It first removes the linear effect of the values in between.
It complements the [[autocorrelation function|YCZHYY8VE5]].
For an autoregressive model of order $p$ it vanishes beyond lag $p$.
This makes it the main tool for choosing the order of such a model.

## Explanations

### L1 human

Today's value may be correlated with the value two days ago only because both are linked to yesterday's value.
The partial autocorrelation at lag $2$ asks what correlation is left once yesterday's value has been taken into account.
In a process where each value depends only on the previous one plus fresh noise, nothing is left: the partial autocorrelation at lag $2$ is $0$.
The ordinary autocorrelation at lag $2$ of the same process is not $0$, because the influence passes through the middle day.
</document>

<document source="https://tamnd.github.io/math2/node/125Z033VCQ.md" tag="125Z033VCQ" kind="concept">
# Auslander–Reiten quiver

## Summary

The Auslander–Reiten quiver of a finite-dimensional algebra is a [[quiver|RE9YFFS3TS]] that maps out its module category.
Its vertices are the isomorphism classes of [[indecomposable|E2P29GCD9T]] finitely generated modules, and its arrows record the irreducible morphisms between them.
It also carries the Auslander–Reiten translation $\tau$, which makes it a translation quiver.
For algebras of finite representation type, it describes the whole module category.
Its meshes come from the [[almost split sequences|M5D95FN7G3]].

## Explanations

### L1 human

The Auslander–Reiten quiver is a map of all the indecomposable modules of an algebra.
Each module is a dot, and an arrow goes from one dot to another when there is a basic map between them that cannot be broken down further.
The almost split sequences show up as small diamond or triangle shapes, called meshes.
For the path algebra of $1 \to 2$ the map has three dots, $S_2 \to P_1 \to S_1$.
For an algebra with infinitely many indecomposables, the quiver is infinite, but it still has a regular structure in many cases.
</document>

<document source="https://tamnd.github.io/math2/node/128WX7HXAT.md" tag="128WX7HXAT" kind="definition">
# Additive category

## Definition

An *additive category* is a [[preadditive category|SC6FD5F4J0]] $\mathcal{C}$ such that:

1. $\mathcal{C}$ has a [[zero object|AQET9YHH6F]];
2. every two objects $X_1, X_2$ of $\mathcal{C}$ have a [[biproduct|BZEJKTQ76T]] $X_1 \oplus X_2$.

## Remarks

Then every finite family of objects has a biproduct, and a morphism $\bigoplus_{j} X_j \to \bigoplus_{i} Y_i$ is the same as a matrix $(f_{ij})$ with $f_{ij} \colon X_j \to Y_i$, composed by matrix multiplication.
Being additive is a property of the underlying category: the addition of morphisms is determined by the biproducts, through $f + g = \nabla \circ (f \oplus g) \circ \Delta$, where $\Delta$ is the diagonal and $\nabla$ the codiagonal.

Example: the category of left [[modules|0NZ8JA25WK]] over a ring $R$ is additive.
The zero module is a zero object, and the direct sum $M_1 \oplus M_2$ is a biproduct.

Non-example: the ring $\Z$, as a preadditive category with one object $\ast$, is not additive.
It has no zero object, since a zero object $Z$ has $\Hom(Z, Z) = \set{0}$, while $\Hom(\ast, \ast) = \Z$.
</document>

<document source="https://tamnd.github.io/math2/node/12BXVH5W5H.md" tag="12BXVH5W5H" kind="statement">
# Rokhlin's theorem

## Statement

Let $M$ be a closed, oriented, [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]] that admits a [[spin structure|RSPZ9TCFQ8]].
Then the [[signature|07EZA80NEH]] $\sigma(M)$ of its [[intersection form|WK8GKRKAVD]] is divisible by $16$.

## Remarks

Vladimir Rokhlin proved the theorem in 1952.
One modern proof uses the Atiyah–Singer index theorem: the index of the Dirac operator of $M$ equals $-\sigma(M)/8$.
In dimension $4$ the spinor bundles are quaternionic, so the kernel and cokernel of the Dirac operator are quaternionic vector spaces and the index is even.
Hence $\sigma(M)/8$ is even.
The K3 surface is a spin $4$-manifold of signature $-16$, so $16$ cannot be replaced by a larger number.
Without smoothness the theorem fails: the [[$E_8$ manifold|BNDRN0G9BR]] is a closed topological $4$-manifold with even form and signature $8$.
Without the spin condition it fails: the complex projective plane has signature $1$.
Algebra alone gives less: an even unimodular form always has signature divisible by $8$.
For a simply connected $M$, being spin is the same as having an even intersection form.

## Explanations

### L1 human

The signature is a single integer that measures the crossings of surfaces in a four-dimensional space.
Rokhlin's theorem says that for smooth spaces of a special kind, called spin, this integer is always a multiple of $16$.
Algebra only predicts a multiple of $8$, and the gap of a factor $2$ is a purely smooth phenomenon.

### L2 human

The theorem is the first sign that smooth $4$-manifolds are more rigid than topological ones.
It shows that the $E_8$ manifold has no smooth structure.
It defines the Rokhlin invariant of an integral homology $3$-sphere: the signature of a smooth spin $4$-manifold it bounds, divided by $8$, modulo $2$.
The [[11/8 conjecture|90RA6TAT7H]] and Furuta's $10/8$ theorem refine it for spin $4$-manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/12CEWPMSG4.md" tag="12CEWPMSG4" kind="concept">
# Mackey machine

## Summary

The Mackey machine is a method to find the irreducible [[unitary representations|74BPYJNZTM]] of a [[locally compact group|AHV05ZBCYW]] $G$.
It builds them from a closed normal subgroup $N$ and from smaller subgroups.
In its basic form $G = N \rtimes H$ with $N$ abelian.
One lets $H$ act on the characters of $N$, picks one character in each orbit, and [[induces|6Q7AHAZ6Q4]] from its stabiliser.
When the orbits are well behaved, this gives every irreducible representation exactly once up to equivalence.
It explains the representations of the $ax + b$ group, of the Euclidean motion groups and of the Poincaré group.

## Explanations

### L1 human

Many groups are built from a simple abelian part, like translations, and a group that acts on it, like rotations.
The translations alone have one-dimensional representations, labelled by frequencies.
The rotations move these frequencies around, and frequencies in the same orbit give the same representation of the whole group.
So one picks one frequency per orbit, looks at the rotations that fix it, and combines the two pieces by induction.
For the motion group of the plane, the orbits are the circles of radius $r > 0$ and the origin, and each circle gives one infinite-dimensional irreducible representation.
</document>

<document source="https://tamnd.github.io/math2/node/12G9M694YG.md" tag="12G9M694YG" kind="concept">
# Mittag-Leffler function

## Summary

The Mittag-Leffler function $E_{\alpha, \beta}(z) = \sum_{k \ge 0} \frac{z^k}{\Gamma(\alpha k + \beta)}$ generalizes the [[exponential function|0J0FCWV32D]], which is the case $\alpha = \beta = 1$.
It is an [[entire function|0G4F0XB9X2]] for $\operatorname{Re} \alpha > 0$.
It plays the role in [[fractional calculus|T73XK15SA2]] that the exponential plays in ordinary calculus: it solves the basic linear fractional differential equations.

## Explanations

### L1 human

The exponential series is $\sum z^k / k!$, and $k! = \Gamma(k + 1)$.
If $k!$ is replaced by $\Gamma(\alpha k + 1)$, the result is the Mittag-Leffler function $E_\alpha$.
For $\alpha = 1$ it is $e^z$, and for $\alpha = 2$ it is $\cosh \sqrt{z}$.
For $0 < \alpha < 1$, the function $E_\alpha(-t^\alpha)$ decays more slowly than $e^{-t}$, like a power of $t$, which models relaxation in materials with memory.
</document>

<document source="https://tamnd.github.io/math2/node/12GTQ4WG8N.md" tag="12GTQ4WG8N" kind="definition">
# Geometric measure theory

## Definition

Let $n \ge 1$ and $0 \le m \le n$.
For $E \subseteq \R^n$, let $\mathcal{H}^m(E)$ be its $m$-dimensional [[Hausdorff measure|Q1TGZK8RN6]].
A set $E \subseteq \R^n$ is *countably $m$-rectifiable* if there are [[Lipschitz maps|0QNVQ4VD9T]] $f_j \colon \R^m \to \R^n$, $j = 1, 2, \dots$, with $\mathcal{H}^m\left( E \setminus \bigcup_j f_j(\R^m) \right) = 0$.
*Geometric measure theory* is the branch of analysis that studies:

1. the measure-theoretic structure of subsets of $\R^n$ and of Riemannian manifolds, measured by Hausdorff measures, in particular the [[rectifiable sets|7N0GFMW05R]] and their approximate tangent planes;
2. generalised surfaces built from them, such as [[currents|360H7MD9RD]] and [[varifolds|N3QA80M44M]];
3. variational problems for these objects, such as the minimisation of area, with the existence and regularity of their solutions.

## Remarks

The rectifiable sets are the sets on which the tools of calculus, such as tangent planes and the area formula, still work almost everywhere.

Example: the graph $E = \set{(x, \abs{x}) : x \in [-1, 1]}$ in $\R^2$ is $1$-rectifiable, since it is the image of a Lipschitz map of an interval, and $\mathcal{H}^1(E) = 2\sqrt{2}$.
Its tangent line exists at every point except the corner $(0, 0)$.

Non-example: the four-corner Cantor set in $\R^2$, obtained by keeping repeatedly the four corner squares of side $1/4$ of each square, has $0 < \mathcal{H}^1 < \infty$.
It is purely $1$-unrectifiable: it meets every Lipschitz image of $\R$ in a set of $\mathcal{H}^1$-measure $0$.

## Explanations

### L1 human

Geometric measure theory measures the size of rough sets and builds generalised surfaces from them, so that problems such as finding a least-area surface always have solutions.
</document>

<document source="https://tamnd.github.io/math2/node/12JQCGTQ8W.md" tag="12JQCGTQ8W" kind="concept">
# Transcendental extension

## Summary

A transcendental extension is a [[field extension|0EQWVHF3W5]] that is not [[algebraic|0PTBM7K0R2]].
So it contains at least one [[transcendental element|0NQMZ89X4A]], an element that is a root of no nonzero polynomial over the base field.
The size of such an extension is measured by its [[transcendence degree|D7TR0YWR9H]], the number of independent transcendental elements it needs.
Fields of rational functions and function fields of varieties are the main examples.

## Explanations

### L1 human

An algebraic extension adds roots of polynomials, such as $\sqrt 2$ added to $\Q$.
A transcendental extension adds something that solves no polynomial equation, such as $\pi$ added to $\Q$, or a free variable $t$.
The field $\Q(t)$ of rational functions in $t$ is the simplest example: no nonzero polynomial with rational coefficients vanishes at $t$.
</document>

<document source="https://tamnd.github.io/math2/node/12V4MHVV0N.md" tag="12V4MHVV0N" kind="example">
# Baker's map

## Example

Let $X = [0, 1) \times [0, 1)$ be the unit square with two-dimensional [[Lebesgue measure|2Z2QVWZ262]] $\lambda$.
The *baker's map* $B \colon X \to X$ is $B(x, y) = (2x, y/2)$ for $0 \le x < 1/2$.
For $1/2 \le x < 1$, it is $B(x, y) = (2x - 1, (y + 1)/2)$.
It stretches the square to twice its width and half its height.
Then it cuts the result in two and stacks the right half on top of the left half, as a baker folds dough.

It is a bijection: the left half $[0, 1/2) \times [0, 1)$ goes onto the bottom half $[0, 1) \times [0, 1/2)$, and the right half goes onto the top half.
On each half, $B$ is affine with Jacobian determinant $2 \cdot \frac{1}{2} = 1$, so $\lambda(B^{-1} E) = \lambda(E)$ for every Borel set $E$, and $B$ is a [[measure-preserving transformation|556WBDAHNP]].

Write $x = 0.x_1 x_2 x_3 \dots$ and $y = 0.y_1 y_2 y_3 \dots$ in binary, and code the point $(x, y)$ by the two-sided sequence $\dots y_2 y_1 . x_1 x_2 \dots$.
Then $B$ moves the binary point one step to the right: the digit $x_1$ leaves $x$ and becomes the first digit of the new $y$.
This coding is an isomorphism, up to null sets, from $B$ to the two-sided [[Bernoulli scheme|D33JK3AXRZ]] $B(1/2, 1/2)$.
Hence $B$ is invertible, [[mixing|S3RJQ9CX2P]], and has [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] $\log 2$.

## Explanations

### L1 human

Roll the dough out to double length, cut it in the middle, and put one piece on top of the other.
The area of the dough does not change, so the map preserves area.
Two raisins that start close together in the horizontal direction are pulled apart, doubling their distance at each step, while vertical distances shrink.
After many steps any small blob of dye is spread in thin layers across the whole square, which is what mixing means.
The baker's map is the invertible partner of the doubling map.
</document>

<document source="https://tamnd.github.io/math2/node/130FCDWRT1.md" tag="130FCDWRT1" kind="definition">
# Filtration

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]], and let $T$ be a totally ordered index set, such as $\N$, $\set{0, 1, \dots, N}$ or $[0, \infty)$.
A *filtration* is a family $(\mathcal{F}_t)_{t \in T}$ of [[sigma-algebras|0SV8K0TDGY]] on $\Omega$ such that:

1. $\mathcal{F}_t \subseteq \mathcal{F}$ for every $t \in T$;
2. $\mathcal{F}_s \subseteq \mathcal{F}_t$ whenever $s, t \in T$ and $s \le t$.

The quadruple $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in T}, P)$ is a *filtered probability space*.

## Remarks

Example: for a [[stochastic process|0QH5ENKZM1]] $(X_t)_{t \in T}$, the *natural filtration* is $\mathcal{F}_t^X = \sigma(X_s : s \le t)$, the smallest $\sigma$-algebra for which all $X_s$ with $s \le t$ are measurable.
If $s \le t$, every generator of $\mathcal{F}_s^X$ is a generator of $\mathcal{F}_t^X$, so clause 2 holds.

Non-example: let $X$ be a random variable that is not almost surely constant, so that $\sigma(X) \neq \set{\emptyset, \Omega}$.
On $T = \set{0, 1}$, the family $\mathcal{F}_0 = \sigma(X)$, $\mathcal{F}_1 = \set{\emptyset, \Omega}$ fails clause 2, since $\mathcal{F}_0 \not\subseteq \mathcal{F}_1$: information would be forgotten.

In continuous time one often assumes the *usual conditions*: $\mathcal{F}_0$ contains all $P$-null sets of the completion of $\mathcal{F}$, and the filtration is right-continuous, that is $\mathcal{F}_t = \bigcap_{s > t} \mathcal{F}_s$ for every $t$.
</document>

<document source="https://tamnd.github.io/math2/node/1313JFBBAD.md" tag="1313JFBBAD" kind="definition">
# Betti numbers of a module

## Definition

Let $(R, \mathfrak{m}, k)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] with [[residue field|NJ3APQ8BHH]] $k = R/\mathfrak{m}$.
Let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module.
For each $i \ge 0$, the module $\operatorname{Tor}^R_i(M, k)$ is annihilated by $\mathfrak{m}$, so it is a finite-dimensional $k$-vector space.
The $i$-th *Betti number* of $M$ is
$$\beta_i(M) = \dim_k \operatorname{Tor}^R_i(M, k).$$

In the graded case, $R = k[x_1, \ldots, x_n]$ is graded by degree, $\mathfrak{m} = (x_1, \ldots, x_n)$, and $M$ is a finitely generated graded $R$-module.
Then each $\operatorname{Tor}^R_i(M, k)$ is graded, and the *graded Betti numbers* are $\beta_{i,j}(M) = \dim_k \operatorname{Tor}^R_i(M, k)_j$.

## Remarks

Equivalently, $\beta_i(M)$ is the rank of the [[free module|0DZ9DKAKZR]] $F_i$ in a minimal free [[resolution|HK2AA1YSQ2]] $\cdots \to F_1 \to F_0 \to M \to 0$, that is, one whose differentials have all entries in $\mathfrak{m}$.

For $R = k[x, y]$ and $M = k = R/(x, y)$, the Koszul complex
$$0 \to R(-2) \to R(-1)^2 \to R \to k \to 0$$
is a minimal resolution, so $\beta_0 = 1$, $\beta_1 = 2$, $\beta_2 = 1$, and the nonzero graded Betti numbers are $\beta_{0,0} = 1$, $\beta_{1,1} = 2$, $\beta_{2,2} = 1$.

The ranks in a resolution that is not minimal are not the Betti numbers.
For $M = R/(x)$ over $R = k[x]$, the resolution $0 \to R \oplus R \to R \oplus R \to M \to 0$, with the second summand mapped identically, has ranks $2, 2$, while $\beta_0 = \beta_1 = 1$.

$\beta_0(M)$ is the least number of generators of $M$, and the largest $i$ with $\beta_i(M) \ne 0$ is the [[projective dimension|MR9QCH1PNA]] of $M$ when it is finite.
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# Geodesic curvature

## Definition

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]] with a smooth unit [[normal vector|0BJDQ2HBH1]] field $N$.
Let $\alpha \colon I \to S$ be a curve of class $C^2$ parametrised by [[arc length|00R1JDDV5X]] on an interval $I$.
The *geodesic curvature* of $\alpha$ at $s \in I$ is
$$\kappa_g(s) = \ip{\alpha''(s)}{N(\alpha(s)) \times \alpha'(s)},$$
where $\times$ is the [[cross product|0SEDGXG9S3]].
The *normal curvature* of $\alpha$ at $s$ is $\kappa_n(s) = \ip{\alpha''(s)}{N(\alpha(s))}$.

## Remarks

Since $\alpha'$, $N \times \alpha'$ and $N$ form an orthonormal basis and $\ip{\alpha''}{\alpha'} = 0$, one has $\alpha'' = \kappa_g \, N \times \alpha' + \kappa_n N$.
So $\kappa^2 = \kappa_g^2 + \kappa_n^2$, where $\kappa$ is the [[curvature|7JQS6VKFEZ]] of $\alpha$ as a space curve.
Example: on the unit sphere with the outer normal $N(p) = p$, a great circle $\alpha$ satisfies $\alpha'' = -\alpha$, which is normal to the sphere, so $\kappa_g = 0$.
A circle of latitude at polar angle $\varphi_0 \in (0, \pi)$ has $\kappa = 1 / \sin \varphi_0$ and $\abs{\kappa_g} = \abs{\cot \varphi_0}$.
Non-example: for a curve that is not of unit speed, the formula above is not the geodesic curvature; one must first reparametrise by arc length.
For a curve in the $xy$-plane with $N = (0, 0, 1)$, the geodesic curvature is the signed curvature of the plane curve.
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# Stokes flow

## Summary

Stokes flow is the flow of a viscous fluid in which viscous forces are much larger than inertial forces.
It is governed by the Stokes equations, the linear system obtained from the [[Navier–Stokes equations|18KXZHVF4M]] by dropping the nonlinear term and the time derivative.
Because the equations are linear, solutions can be added and many problems can be solved exactly.
Stokes flow models very small organisms, slow flows of very viscous fluids and flows in thin films.

## Explanations

### L1 human

When honey creeps down a spoon, or a bacterium swims in water, inertia hardly matters.
If the pushing stops, the fluid stops almost at once.
In this regime the pressure and the friction inside the fluid balance each other exactly.
A standard example is flow in a channel between two plates, where the speed profile is a parabola, fastest in the middle and zero at the walls.
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# Classical orthogonal polynomials

## Definition

Let $I = (a, b)$ be an open interval with $-\infty \le a < b \le \infty$, and let $w \colon I \to (0, \infty)$ be continuous with $\int_I \abs{x}^k w(x)\, dx < \infty$ for every $k \ge 0$.
A sequence $(p_n)_{n \ge 0}$ of real polynomials with [[degree|0P4JRMSWNH]] $\deg p_n = n$ is *orthogonal* for $w$ if $\int_I p_m(x)\, p_n(x)\, w(x)\, dx = 0$ for all $m \ne n$.
Such a sequence is a family of *classical orthogonal polynomials* if an affine change of variable $x \mapsto c x + d$ with $c \ne 0$ does the following.
It turns the pair $(I, w)$, up to a constant factor in $w$, into one of these:

1. (Jacobi) $I = (-1, 1)$ and $w(x) = (1 - x)^\alpha (1 + x)^\beta$ with $\alpha, \beta > -1$.
2. (Laguerre) $I = (0, \infty)$ and $w(x) = x^\alpha e^{-x}$ with $\alpha > -1$.
3. (Hermite) $I = \R$ and $w(x) = e^{-x^2}$.

## Remarks

The integral $\ip{f}{g} = \int_I f g\, w\, dx$ is an [[inner product|0NB0AWGJBW]] on real polynomials, and an orthogonal sequence for $w$ is determined by $w$ up to a nonzero constant factor in each $p_n$.
Example: the [[Chebyshev polynomials|M6EAR1JHPP]] $T_n$ are Jacobi with $\alpha = \beta = -\frac{1}{2}$; for instance $\int_{-1}^1 T_0 T_1 (1 - x^2)^{-1/2}\, dx = \int_{-1}^1 \frac{x}{\sqrt{1 - x^2}}\, dx = 0$, since the integrand is odd.
Example: the [[Hermite polynomials|MGACY98E9H]] are the Hermite case, and the weight $e^{-x^2/2}$ of the probabilists' version is the Hermite weight after the change $x \mapsto x / \sqrt{2}$.
A non-example: the orthogonal polynomials for $w(x) = e^{-x^4}$ on $\R$ exist, but they are not classical, since no affine change of variable turns $e^{-x^4}$ into one of the three weights.
The classical families are exactly the orthogonal families that are eigenfunctions of an operator $\sigma(x) y'' + \tau(x) y'$ with $\deg \sigma \le 2$ and $\deg \tau = 1$.
They are also exactly those whose derivatives $(p_{n+1}')_{n \ge 0}$ are again orthogonal, and those given by a Rodrigues formula.
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# Calabi flow

## Definition

Let $M$ be a compact complex manifold of complex dimension $n$ with a Kähler form $\omega_0$.
This is the form of a [[Hermitian metric|CK1DCPC3D3]] with $d\omega_0 = 0$.
For a smooth real function $\varphi$ on $M$ with $\omega_\varphi = \omega_0 + i \partial \bar\partial \varphi > 0$, let $S(\varphi)$ be the [[scalar curvature|ZHX2VYQDZ3]] of the Kähler metric $\omega_\varphi$.
Let $\underline{S}$ be the average of $S(\varphi)$ with respect to the volume form $\omega_\varphi^n$; it depends only on the Kähler class of $\omega_0$.
A smooth family $(\varphi_t)_{0 \le t < T}$ of such functions is a solution of the *Calabi flow* if
$$\frac{\partial \varphi_t}{\partial t} = S(\varphi_t) - \underline{S}.$$

## Remarks

Example: if $\omega_0$ has constant scalar curvature, such as a Kähler–Einstein metric or a flat metric on a complex torus, then $S(0) = \underline{S}$, and the constant family $\varphi_t = 0$ solves the flow.
Non-example: a family $\omega_t = (1 + t)\omega_0$ leaves the Kähler class of $\omega_0$, so it is not of the form $\omega_0 + i\partial\bar\partial\varphi_t$ and is not a Calabi flow.
In local coordinates the flow reads $\partial g_{j\bar k}/\partial t = \partial_j \partial_{\bar k} S$ for the metric coefficients $g_{j\bar k}$.
Along the flow the Calabi energy $\int_M (S - \underline{S})^2\, \omega_\varphi^n$ does not increase.
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# De Rham cohomology of varieties

## Summary

Let $X$ be a smooth variety over a field $k$.
Its algebraic de Rham cohomology is the hypercohomology of the complex of algebraic differential forms on $X$.
It replaces smooth differential forms by [[Kähler differentials|7HCFARHG7D]], so it makes sense over any field.
Over $\C$, Grothendieck showed that it agrees with the singular cohomology of $X(\C)$ with complex coefficients.
It is the algebraic form of [[de Rham cohomology|GX74C07T0G]] and a model for later cohomology theories such as [[crystalline cohomology|EVXCN06FW3]].

## Explanations

### L1 human

On a manifold, de Rham cohomology counts closed differential forms that are not derivatives.
On a variety, one can do the same with forms built from polynomials, such as $dx$ or $x \, dy$.
For the punctured line $\C \setminus \{0\}$, with functions $t^n$ for all integers $n$, every form $t^n \, dt$ with $n \ne -1$ is the derivative of $t^{n+1}/(n+1)$.
The form $dt/t$ is closed but not a derivative, and it spans the first cohomology.
This matches the one loop around the origin in the topology of the punctured plane.
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# Set-valued measure

## Summary

A set-valued measure assigns to each measurable set a nonempty subset of $\R^n$, and it is countably additive for the Minkowski sum of sets.
It generalises a [[vector measure|PS28BSAJKJ]], which assigns a single vector to each set.
Set-valued measures arise, for example, as integrals of set-valued functions over measurable sets.
They are studied in mathematical economics and in control theory.

## Explanations

### L1 human

An ordinary measure gives each set one number, its size.
A set-valued measure gives each set a whole region of possible values.
For two disjoint sets, the region of the union consists of all sums of a value from the first region and a value from the second.
For example, giving each set $A$ the interval $[0, \mu(A)]$ for a finite measure $\mu$ is a set-valued measure on the line.
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# Absolute geometry

## Definition

*Absolute geometry* is the theory whose axioms are the axioms of incidence, order, congruence and continuity of [[Hilbert's axioms|KM1FB8XZ2R]], without the axiom of parallels.
An *absolute plane* is a model of the plane part of these axioms.

## Remarks

Example: the Euclidean plane $\R^2$ is an absolute plane, since it satisfies all of Hilbert's axioms.
The hyperbolic plane is also an absolute plane: it satisfies all these axioms, but through a point off a line there pass several lines that miss it.
Non-example: the sphere with great circles as lines is not an absolute plane.
Two antipodal points lie on infinitely many great circles, so the incidence axiom "two points lie on exactly one line" fails.
With the continuity axioms, every absolute plane is isomorphic to the Euclidean plane or to the hyperbolic plane.
The name "neutral geometry" is common in textbooks.
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# Proof complexity

## Definition

Let $\Sigma$ be a finite alphabet, and let $\mathrm{TAUT} \subseteq \Sigma^*$ be the set of [[tautologies|0D1AHYPM8N]] of [[propositional logic|09K9MWD2QQ]], coded as strings.
A *propositional proof system*, in the sense of Cook and Reckhow, is a [[function|0XHDTE5YMZ]] $P \colon \Sigma^* \to \Sigma^*$ such that:

1. $P$ is computable in time polynomial in the length of its input;
2. the image of $P$ is exactly $\mathrm{TAUT}$.

A string $w$ with $P(w) = \varphi$ is a *$P$-proof* of $\varphi$.
The system $P$ is *polynomially bounded* if there is a polynomial $p$ such that every $\varphi \in \mathrm{TAUT}$ has a $P$-proof of length at most $p(\abs{\varphi})$.
*Proof complexity* is the study of the lengths of proofs in such systems, and of the relations between systems.

## Remarks

Example: the truth table system is a proof system.
Let $P$ send a string made of a formula $\varphi$ and its full truth table, with all rows true, to $\varphi$.
Let $P$ send every other string to the fixed tautology $p \vee \neg p$.
The check takes time polynomial in the length of the input, and the image of $P$ is $\mathrm{TAUT}$.
This system is not polynomially bounded, because the table of a formula with $n$ variables has $2^n$ rows.
Non-example: the map that sends each string to itself is not a proof system, because its image is all of $\Sigma^*$, not $\mathrm{TAUT}$.
Cook and Reckhow proved that a polynomially bounded proof system exists if and only if $\mathrm{NP} = \mathrm{coNP}$.
Haken proved in 1985 that resolution is not polynomially bounded, using the pigeonhole principle.
For stronger systems such as Frege systems, no superpolynomial lower bound is known.
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<document source="https://tamnd.github.io/math2/node/13RXPC2RJK.md" tag="13RXPC2RJK" kind="definition">
# Riemann domain over C^n

## Definition

A *Riemann domain over $\C^n$* is a pair $(X, \pi)$ where $X$ is a [[connected|0V032VWTHD]] [[Hausdorff space|0MXB4CKHFW]] and $\pi \colon X \to \C^n$ is a [[local homeomorphism|MDH71ZZ8XX]].
Let $W \subseteq X$ be open and $f \colon W \to \C$ a function.
Then $f$ is *holomorphic* if $f \circ (\pi|_V)^{-1}$ is a [[holomorphic function|507C34W0ZT]] on $\pi(V)$ for every open set $V \subseteq W$ on which $\pi$ is injective.
The Riemann domain is *schlicht* if $\pi$ is injective.

## Remarks

The charts $\pi|_V$ make $X$ a complex manifold of dimension $n$ on which $\pi$ is locally biholomorphic.
A schlicht Riemann domain is the same as a domain in $\C^n$, through the identification of $X$ with $\pi(X)$.

Example: $X = \C^2$ with $\pi(w, z_2) = (e^w, z_2)$.
The space $X$ is connected and Hausdorff, and $\pi$ is a local homeomorphism because $w \mapsto e^w$ is a local homeomorphism of $\C$ onto $\C \setminus \set{0}$.
It is not schlicht, since $\pi(w + 2\pi i, z_2) = \pi(w, z_2)$.

Non-example: $X = \C^2$ with $\pi(w, z_2) = (w^2, z_2)$.
Here $\pi$ is not injective on any neighbourhood of a point $(0, z_2)$, because $(w, z_2)$ and $(-w, z_2)$ have the same image; so $\pi$ is not a local homeomorphism there.
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# Unit-regular ring

## Definition

Let $R$ be a [[ring|0894J980RK]] with unit $1$.
The ring $R$ is *unit-regular* if for every $a \in R$ there is a [[unit|019NFGFPR5]] $u$ of $R$ such that
$$a u a = a.$$

## Remarks

A unit-regular ring is [[von Neumann regular|DEV24TSWQZ]].
Example: a division ring $D$; for $a \ne 0$ take $u = a^{-1}$, and for $a = 0$ take $u = 1$.
Example: the matrix ring $M_n(F)$ over a field $F$.
Write $A = PEQ$ with $P, Q$ invertible and $E = \operatorname{diag}(1, \ldots, 1, 0, \ldots, 0)$; then $u = Q^{-1} P^{-1}$ gives $AuA = PE^2Q = PEQ = A$.
A unit-regular ring is Dedekind-finite: if $ab = 1$ and $a = aua$, then $1 = ab = auab = au$, so $a = u^{-1}$ and $ba = 1$.
Non-example: let $V$ be a vector space with countable basis $e_1, e_2, \ldots$ over a field, and $R = \End(V)$.
The ring $R$ is von Neumann regular, but the shift $s(e_i) = e_{i+1}$ and $t$ with $t(e_1) = 0$, $t(e_{i+1}) = e_i$ satisfy $ts = 1 \ne st$.
So $R$ is not Dedekind-finite, hence not unit-regular.
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# Regular cardinal

## Summary

A regular cardinal is an infinite [[cardinal|0EBACQFAZW]] that cannot be reached by fewer smaller steps than itself: its [[cofinality|BAWHE1W6MH]] equals itself.
An infinite cardinal that is not regular is singular.
Regular cardinals are the natural setting for clubs, stationary sets and much of infinite combinatorics, and the [[precise definition|ZCE333FAWC]] is short.

## Explanations

### L1 human

The cardinal $\aleph_0$ is regular: a finite union of finite sets is finite, so finitely many finite steps never reach $\omega$.
The cardinal $\aleph_1$ is regular too, because a countable union of countable sets is countable.
But $\aleph_\omega$ is singular: the countably many smaller cardinals $\aleph_0, \aleph_1, \aleph_2, \ldots$ already climb up to it.
So regularity says a cardinal is not the limit of a short sequence of smaller cardinals.
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# Rate function

## Summary

A rate function is a lower semicontinuous function $I$ from a topological space to $[0, \infty]$.
In [[large deviations theory|Q702WCDQQ3]], it gives the exponential rate at which the probability of a rare event decays: roughly, $P(Z_n \approx x) \approx e^{-n I(x)}$.
It is called good when its sublevel sets are compact.
Examples are the Legendre transforms in [[Cramér's theorem|RZZK51CXMN]] and the [[Kullback–Leibler divergence|PQBW43X5RE]] in [[Sanov's theorem|QWEF2WAXJM]].

## Explanations

### L1 human

A rate function measures how costly it is for a random system to end up at a given state.
The typical state costs nothing, and states far from typical cost more.
The probability of a state falls like $e^{-n \cdot \text{cost}}$, where $n$ is the size of the system.
For the average of $n$ standard normal variables, the cost of the value $x$ is $x^2/2$.
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# Shearlet

## Definition

For $a > 0$ and $s \in \R$ let
$$A_a = \begin{pmatrix} a & 0 \\ 0 & \sqrt{a} \end{pmatrix}, \qquad S_s = \begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix}$$
be the parabolic scaling matrix and the shear matrix.
Let $\hat\psi$ denote the [[Fourier transform|DPVM34ADZV]] of $\psi \in L^2(\R^2)$, with variable $\xi = (\xi_1, \xi_2)$.
A function $\psi \in L^2(\R^2)$ is an *admissible shearlet* if
$$\int_{\R^2} \frac{\abs{\hat\psi(\xi)}^2}{\xi_1^2} \, d\xi < \infty.$$
The continuous shearlet system generated by $\psi$ consists of the functions
$$\psi_{a,s,t}(x) = a^{-3/4} \, \psi\left(A_a^{-1} S_s^{-1}(x - t)\right), \qquad a > 0, \ s \in \R, \ t \in \R^2.$$
The continuous shearlet transform of $f \in L^2(\R^2)$ is $\mathcal{SH}_\psi f(a, s, t) = \ip{f}{\psi_{a,s,t}}$.

## Remarks

The factor $a^{-3/4} = (\det A_a)^{-1/2}$ makes $\norm{\psi_{a,s,t}}_2 = \norm{\psi}_2$.
Discrete shearlet systems use scales $a = 2^{-j}$, shears $s = k \, 2^{-j/2}$ with $k \in \Z$, and translations on a lattice.

Example (classical shearlet): let $\hat\psi(\xi) = \hat\psi_1(\xi_1)\,\hat\psi_2(\xi_2/\xi_1)$, where $\hat\psi_1$ is smooth with support in $[-2, -1/2] \cup [1/2, 2]$ and $\hat\psi_2$ is smooth with support in $[-1, 1]$.
Then $\hat\psi$ is bounded and supported in the set where $1/2 \le \abs{\xi_1} \le 2$ and $\abs{\xi_2} \le \abs{\xi_1}$, which is bounded and stays away from $\xi_1 = 0$.
So the admissibility integral is finite.

Non-example: the Gaussian $\psi(x) = e^{-\abs{x}^2}$ is not admissible.
Its Fourier transform is a Gaussian with $\hat\psi(0) \ne 0$, so near $\xi = 0$ the integrand behaves like a positive constant divided by $\xi_1^2$, and the integral diverges.

For images that are smooth apart from smooth edges, suitable discrete shearlet frames give $N$-term approximations with squared error of order $N^{-2}(\log N)^3$, the same rate as [[curvelets|VEVPDJ8S7B]].

## Explanations

### L1 human

A shearlet is a generating function that is scaled more in one direction than the other, tilted by a shear and moved around the plane.
The admissibility condition asks that the function have essentially no energy at low horizontal frequencies, the analogue of a wavelet having zero mean.
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# Euler–Boole summation

## Summary

Euler–Boole summation expresses an alternating sum $\sum (-1)^j f(j + h)$ through the values of $f$ and its derivatives at the two ends of the range.
An integral remainder makes the formula exact.
The coefficients are values of the Euler polynomials, in the way that the [[Euler–Maclaurin formula|9P839VB8S7]] uses Bernoulli polynomials for ordinary sums.
It is the alternating counterpart of Euler–Maclaurin summation.
It gives exact formulas for alternating sums of powers and asymptotic expansions of alternating series.

## Explanations

### L1 human

The Euler–Maclaurin formula compares a sum $f(0) + f(1) + \cdots$ with an integral of $f$.
For an alternating sum $f(0) - f(1) + f(2) - \cdots$, the terms nearly cancel in pairs, and no integral is needed.
What remains is controlled by the ends of the sum: the values of $f$ and its derivatives there.
Euler–Boole summation writes this down exactly, with an error term that is small when high derivatives of $f$ are small.
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# Carathéodory's royal road

## Summary

Carathéodory's royal road is a method to prove that a curve minimises an integral functional of the [[calculus of variations|PQ25NHV0KS]].
One subtracts from the Lagrangian the total derivative of a function $S(t, x)$, which does not change the problem for curves with fixed endpoints.
If $S$ can be chosen so that the new Lagrangian is nonnegative and vanishes along the candidate curve, the candidate is a minimiser.
The function $S$ then solves the [[Hamilton–Jacobi equation|5TRXQKP7DY]], so the method links sufficient conditions, [[fields of extremals|K4DFT3H9BJ]] and Hamilton–Jacobi theory.

## Explanations

### L1 human

Adding the derivative of a function to the integrand changes the value of the integral only through the endpoints.
So one may look for such a change that makes the integrand nonnegative everywhere and zero along a chosen curve.
Then no curve can do better than that one.
For example, for $\int_0^1 \frac{1}{2} \dot x^2 \, dt$ with $x(0) = 0$ and $x(1) = 1$, subtracting the derivative of $x - t/2$ turns the integrand into $\frac{1}{2} (\dot x - 1)^2$.
This is nonnegative and vanishes only for $\dot x = 1$, so the straight line $x(t) = t$ is the minimiser.
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# Monge–Ampère equation

## Summary

The Monge–Ampère equation prescribes the determinant of the [[Hessian matrix|0693Y2METT]] of an unknown function: $\det D^2 u = f$.
It is the basic example of a [[fully nonlinear PDE|2D3N1ZJWBW]], since it is nonlinear in the highest derivatives.
It is elliptic when $u$ is convex and $f > 0$.
It appears in the problem of prescribing the Gauss curvature of a convex surface, in optimal transport and in complex geometry.

## Explanations

### L1 human

The Hessian of a function records how it curves in every direction, and its determinant multiplies these curvatures together.
The Monge–Ampère equation asks for a function whose total curvature, in this sense, is given at every point.
In optimal transport, the gradient of a convex solution moves one mass distribution onto another, and the determinant measures how much volume is stretched.
The function $u(x) = \abs{x}^2 / 2$ has Hessian the identity matrix, so it solves $\det D^2 u = 1$.
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# Laplace's method

## Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be twice continuously differentiable.
Assume that $f$ has a unique global [[maximum|04CPG9K1C4]] on $[a, b]$ at a point $x_0$ with $a < x_0 < b$, and that the [[second derivative|0Y9D2FJRQ4]] satisfies $f''(x_0) < 0$.
*Laplace's method* is the approximation of $I(M) = \int_a^b e^{M f(x)}\, dx$ for large $M$ by the integral of $e^{M (f(x_0) + \frac12 f''(x_0)(x - x_0)^2)}$ over $\R$.
Under these assumptions it gives the [[asymptotic equivalence|00SB5EA3VW]]
$$\int_a^b e^{M f(x)}\, dx \sim \sqrt{\frac{2\pi}{M \abs{f''(x_0)}}} \; e^{M f(x_0)} \qquad (M \to \infty).$$

## Remarks

Example: Stirling's formula.
Substituting $t = n s$ gives $n! = \int_0^\infty t^n e^{-t}\, dt = n^{n+1} \int_0^\infty e^{n(\ln s - s)}\, ds$.
The function $f(s) = \ln s - s$ has its maximum at $s = 1$, with $f(1) = -1$ and $f''(1) = -1$, and after a check that the tails are small the method gives $n! \sim \sqrt{2\pi n}\, (n/e)^n$.

Non-example: if the maximum is at an endpoint, for instance $f(x) = -x$ on $[0, 1]$, the hypothesis $a < x_0 < b$ fails.
Here $\int_0^1 e^{-Mx}\, dx = (1 - e^{-M})/M$, which decays like $1/M$ and not like $1/\sqrt{M}$.

The same idea applies to $\int_a^b g(x) e^{M f(x)}\, dx$ with $g$ continuous and $g(x_0) \ne 0$, where the right side gains the factor $g(x_0)$.
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# Sphere spectrum

## Description

The sphere spectrum $S$ is the [[spectrum|1GHF4EKBXS]] with $S_n = S^n$, the $n$-sphere, for $n \ge 0$, and with structure maps the canonical homeomorphisms $\Sigma S^n \cong S^{n+1}$.
It is the suspension spectrum $\Sigma^\infty S^0$ of the two-point space.
Its homotopy groups are the [[stable homotopy groups of spheres|HGZAYPQBXW]]:
$$\pi_k(S) = \operatorname{colim}_n \pi_{k+n}(S^n) = \pi_k^s.$$

## Remarks

One has $\pi_k(S) = 0$ for $k < 0$, $\pi_0(S) = \Z$, $\pi_1(S) = \Z/2$, $\pi_2(S) = \Z/2$ and $\pi_3(S) = \Z/24$.
For $k > 0$ every group $\pi_k(S)$ is finite, by Serre's finiteness theorem.
In the [[stable homotopy category|26GTNMAF7B]], $S$ is the unit for the smash product, and maps $S \to E$ of degree $k$ compute $\pi_k(E)$ for every spectrum $E$.
The multiplication $S^n \wedge S^m \cong S^{n+m}$ makes $S$ a commutative ring spectrum, and $\pi_*(S)$ is a graded commutative ring.

## Explanations

### L1 human

The sphere spectrum is the stable version of the two-point space $S^0$, or of the circle, the 2-sphere and all higher spheres at once.
It plays the role in stable homotopy theory that the integers play in algebra.
Its homotopy groups, the stable stems, are among the most studied and least understood objects in topology.
</document>

<document source="https://tamnd.github.io/math2/node/14DWEDXWCC.md" tag="14DWEDXWCC" kind="concept">
# Tarski's axioms

## Summary

Tarski's axioms are an axiom system for plane [[Euclidean geometry|KRZD96GSVC]] in first-order logic, due to Alfred Tarski.
They speak only of points and of two relations: [[betweenness|CE8GX6ADZS]] of three points and congruence of two segments.
Tarski proved that the resulting [[theory|92SWXST1ZM]] is [[complete|655K8Y7VS8]] and decidable: an algorithm decides, for each sentence, whether it is a theorem.

## Explanations

### L1 human

Tarski's axioms describe plane geometry with only points, no lines or circles as basic objects.
Two relations are enough: "$y$ is between $x$ and $z$" and "the segment $xy$ is as long as the segment $zw$".
A line, for example, is the set of points $y$ such that one of $x$, $y$, $z$ is between the other two, for two fixed points $x \ne z$.
Because the theory is decidable, every elementary question about the plane can be decided by a computer in principle.
</document>

<document source="https://tamnd.github.io/math2/node/14HX3QF45H.md" tag="14HX3QF45H" kind="definition">
# VC dimension

## Definition

Let $X$ be a set and let $\mathcal{F}$ be a [[family|0C21Q694AG]] of subsets of $X$.
A finite set $S \subseteq X$ is *shattered* by $\mathcal{F}$ if the set of [[intersections|0HBRW4E6G1]] $\set{A \cap S : A \in \mathcal{F}}$ is the whole [[power set|00E0YZCERH]] of $S$.
The *VC dimension* of $\mathcal{F}$ is the supremum of $|S|$ over all finite sets $S \subseteq X$ shattered by $\mathcal{F}$.
If $\mathcal{F}$ is nonempty, the empty set is shattered, so the VC dimension is a nonnegative integer or $\infty$.

## Remarks

Example: let $X = \R$ and let $\mathcal{F}$ be the family of closed intervals $[a, b]$ with $a \le b$.
The set $\set{0, 1}$ is shattered: the intersections $\emptyset$, $\set{0}$, $\set{1}$, $\set{0, 1}$ come from $[5, 6]$, $[0, 0]$, $[1, 1]$ and $[0, 1]$.
No set $\set{p < q < r}$ is shattered, because an interval that contains $p$ and $r$ contains $q$.
So the VC dimension is $2$.

Non-example: for the same family, the set $\set{0, 1, 2}$ is not shattered, since the pattern $\set{0, 2}$ fails the condition $A \cap S = \set{0, 2}$ for every interval $A$.

The family of all subsets of an infinite set has VC dimension $\infty$.
The half-planes in $\R^2$ have VC dimension $3$.
</document>

<document source="https://tamnd.github.io/math2/node/14KPTDH4ZN.md" tag="14KPTDH4ZN" kind="definition">
# Finitely based algebra

## Definition

Let $A$ be an algebra of a signature with only function symbols.
The algebra $A$ is *finitely based* if it has a finite [[equational basis|9Q008GMZSW]].
That is, there is a finite set $\Sigma$ of [[identities|PKD440G2XW]] satisfied by $A$ such that every identity of $A$ holds in all algebras satisfying $\Sigma$.
A [[variety|AJNZJD8442]] is *finitely based* if it is the class of all algebras satisfying some finite set of identities.

## Remarks

An algebra is finitely based exactly when the variety it generates is finitely based.
An algebra is finitely based exactly when its equational theory is the set of consequences of finitely many identities.
Every two-element algebra of finite signature is finitely based, by a theorem of Lyndon, while Lyndon also found a finite algebra that is not finitely based.

Example: the group $(\Z, +, -, 0)$ is finitely based.
It generates the variety of abelian groups, so the identities $x + (y + z) \approx (x + y) + z$, $x + 0 \approx x$, $x + (-x) \approx 0$ and $x + y \approx y + x$ form an equational basis.

Non-example: the six-element Brandt monoid $B_2^1$ is not finitely based as a semigroup, by a theorem of Perkins.
It consists of the four $2 \times 2$ matrix units, the zero matrix and the identity matrix, under multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/14RHHMGF2D.md" tag="14RHHMGF2D" kind="statement">
# Huisken's theorem on convex hypersurfaces

## Statement

Let $n \ge 2$ and let $M_0$ be a compact, connected, smooth hypersurface without boundary embedded in $\R^{n+1}$.
Assume that $M_0$ is uniformly convex, that is, all [[principal curvatures|FA8QE1SGP4]] of $M_0$ are positive at every point.
Then:

1. The [[mean curvature flow|2YYV01C7QQ]] starting from $M_0$ has a unique smooth solution $(M_t)_{0 \le t < T}$ on a maximal time interval with $T < \infty$.
2. Each $M_t$ is uniformly convex.
3. As $t \to T$, the hypersurfaces $M_t$ shrink to a single point $p \in \R^{n+1}$.
4. The rescaled hypersurfaces $(2n(T - t))^{-1/2}(M_t - p)$ converge smoothly to the unit sphere $S^n$ as $t \to T$.

## Remarks

The proof first shows that convexity is preserved by the maximum principle applied to the evolution of the second fundamental form.
The key estimate is a pinching estimate: the ratio of the smallest to the largest principal curvature tends to $1$ where the [[mean curvature|PA78NFEPCF]] blows up.
Together with bounds on the derivatives of the curvature, this forces the rescaled surfaces to become round.
For a round sphere of radius $r_0$, the flow gives spheres of radius $r(t) = \sqrt{r_0^2 - 2nt}$, so $T = r_0^2/(2n)$ and the rescaled surfaces are all the unit sphere.
Convexity is needed: a dumbbell, made of two large spheres joined by a thin neck, can pinch off at the neck before it shrinks to a point.
The case $n = 1$ of convex curves in the plane is the theorem of Gage and Hamilton, part of the [[Gage–Hamilton–Grayson theorem|FVGW1381TZ]].

## Explanations

### L1 human

Mean curvature flow moves each point of a surface inward at a speed given by how curved the surface is there.
Huisken's theorem says that a convex closed surface, such as an egg shape, stays convex under this flow.
It shrinks to a point in finite time.
Just before it disappears, it looks more and more like a round sphere.

### L2 human

Huisken proved this in 1984, for hypersurfaces of dimension at least two.
It is the model result for geometric flows that make a shape round.
It parallels Hamilton's theorem that the Ricci flow deforms a closed $3$-manifold with positive Ricci curvature into a round one.
The method combines the parabolic maximum principle with curvature pinching estimates.
Later work of Huisken and others studies what happens without convexity, where singularities such as neck pinches appear.
</document>

<document source="https://tamnd.github.io/math2/node/14SGZ5HA4Y.md" tag="14SGZ5HA4Y" kind="statement">
# Hales–Jewett theorem

## Statement

Let $k \ge 1$ and $n \ge 1$ be integers, and let $[k] = \set{1, \ldots, k}$.
A *variable word* of length $n$ is a word $w \in ([k] \cup \set{*})^n$ in which the symbol $*$ occurs at least once.
For $a \in [k]$, let $w(a) \in [k]^n$ be the word obtained from $w$ by replacing every $*$ by $a$.
The set $\set{w(1), \ldots, w(k)}$ is a *combinatorial line* in $[k]^n$.

Let $k \ge 1$ and $r \ge 1$ be integers.
Then there is an integer $N$ with the following property.
For every $n \ge N$ and every colouring of $[k]^n$ with $r$ colours, some combinatorial line in $[k]^n$ is monochromatic.

## Remarks

Example: for $k = 2$ and $r = 2$ one can take $N = 2$.
The square $[2]^2$ has the five lines $\set{11, 21}$, $\set{12, 22}$, $\set{11, 12}$, $\set{21, 22}$ and $\set{11, 22}$.
In a $2$-colouring, either $11$ and $22$ have the same colour, or the word $12$ has the colour of one of them and lies on a line with it.
For $n = 1$ the only line is $\set{1, 2}$, which is not monochromatic when $1$ and $2$ get different colours.
The original proof, by induction on $k$, gives very large bounds; Shelah found a proof with primitive recursive bounds.
The theorem implies [[van der Waerden's theorem|V96FRZ2ME1]].
Use the alphabet $\set{0, \ldots, k-1}$ and map a word $w$ to $w_1 + \cdots + w_n$.
A combinatorial line goes to an arithmetic progression of length $k$ whose common difference is the number of $*$ symbols.
The density version says that, for large $n$, every set containing a fixed positive fraction of $[k]^n$ contains a combinatorial line.
It was proved by Furstenberg and Katznelson, and later by an elementary method in the Polymath project.

## Explanations

### L1 human

Think of the game of tic-tac-toe played on a cube with $k$ cells along each side and $n$ dimensions.
The theorem says that, if the dimension is large enough, any colouring of the cells with $r$ colours has a full winning line in one colour.
In particular, a game of this kind in high enough dimension cannot end in a draw.

### L2 human

The Hales–Jewett theorem is the abstract core of many Ramsey statements about arithmetic structure.
It needs no arithmetic at all: only words and substitution.
From it follow [[van der Waerden's theorem|V96FRZ2ME1]] and its multidimensional form, the Gallai–Witt theorem.
Its density version similarly implies [[Szemerédi's theorem|B23B0QEMN5]].
</document>

<document source="https://tamnd.github.io/math2/node/14T6SJC0BC.md" tag="14T6SJC0BC" kind="definition">
# Mehler–Fock transform

## Definition

For real $\tau \ge 0$, let $P_{-1/2 + i\tau}$ be the conical [[Legendre function|69XMERM6KJ]] of the first kind on $[1, \infty)$.
Let $f \colon [1, \infty) \to \C$ be a function such that the integral below converges for every $\tau \ge 0$.
The *Mehler–Fock transform* of $f$ is the function
$$F(\tau) = \int_1^\infty f(x)\, P_{-1/2 + i\tau}(x)\, dx, \qquad \tau \ge 0.$$

## Remarks

Under suitable conditions on $f$, the inversion formula is
$$f(x) = \int_0^\infty \tau \tanh(\pi \tau)\, P_{-1/2 + i\tau}(x)\, F(\tau)\, d\tau, \qquad x > 1.$$
The function $P_{-1/2 + i\tau}(x)$ is real for real $\tau$ and $x \ge 1$, and it decays like $x^{-1/2}$ up to a logarithmic factor as $x \to \infty$.

Example: if $f$ is integrable and vanishes outside $[1, 2]$, then the integral is over a compact interval of a continuous kernel, so $F(\tau)$ is defined for every $\tau \ge 0$.

Non-example: $f(x) = x$ has no Mehler–Fock transform, since $x P_{-1/2 + i\tau}(x)$ grows like $x^{1/2}$ up to a logarithmic factor and is not integrable on $[1, \infty)$.

The substitution $x = \cosh t$ turns the transform into an integral over $t \in [0, \infty)$, the form used on the hyperbolic plane.
</document>

<document source="https://tamnd.github.io/math2/node/1513J40TKA.md" tag="1513J40TKA" kind="statement">
# Ryll-Nardzewski theorem

## Statement

Let $T$ be a [[complete theory|655K8Y7VS8]] in a countable signature $L$, and suppose that $T$ has infinite models.
The following are equivalent:

1. $T$ is [[$\aleph_0$-categorical|1JJCGNX240]].
2. For every $n \ge 1$, the set $S_n(T)$ of complete $n$-[[types|9V71Q3SD1G]] of $T$ is finite.
3. For every $n \ge 1$, there are only finitely many $L$-formulas in the variables $x_1, \dots, x_n$ up to equivalence modulo $T$.
4. For every $n \ge 1$, every complete $n$-type of $T$ is isolated.
5. $T$ has a countable model $M$ whose [[automorphism group|0K1SCPBNKJ]] has only finitely many [[orbits|0YQPZB5Q1G]] on $M^n$ for every $n \ge 1$.

## Remarks

If some $S_n(T)$ is infinite, then by compactness of $S_n(T)$ it has a non-isolated point.
The omitting types theorem gives a countable model omitting it, while another countable model realizes it, so $T$ is not $\aleph_0$-categorical.
Conversely, if all types are isolated, every countable model is atomic and also realizes every type, and a back-and-forth argument between two countable models gives an isomorphism.

Example: the theory of [[dense linear orders without endpoints|2W4S2H4VYA]] is $\aleph_0$-categorical.
In $(\Q, <)$, two $n$-tuples lie in the same orbit of the automorphism group exactly when they have the same order pattern, and there are finitely many patterns.

Non-example: the theory of $(\Z, <)$ is not $\aleph_0$-categorical.
The formulas "there are at least $k$ elements strictly between $x_1$ and $x_2$", for $k \in \N$, are pairwise inequivalent, so clause 3 fails for $n = 2$.

The theorem is named after Czesław Ryll-Nardzewski; it was also found independently by Erwin Engeler and Lars Svenonius.

## Explanations

### L1 human

A theory is $\aleph_0$-categorical when it has only one countable model up to isomorphism.
The theorem says that this happens exactly when the language can only say finitely many different things about any fixed number of elements.
For the rationals with their order, all one can say about two numbers is whether the first is smaller, equal or larger than the second.

### L2 human

The theorem turns a question about models into a question about counting types or formulas.
Clause 5 links model theory with permutation groups: the countable models of $\aleph_0$-categorical theories are exactly the countable structures with oligomorphic automorphism groups.
[[Fraïssé limits|HJVRBDQSGG]] of classes with finitely many structures of each finite size in a finite relational language, such as the Rado graph, are $\aleph_0$-categorical.
</document>

<document source="https://tamnd.github.io/math2/node/151RG1JCGB.md" tag="151RG1JCGB" kind="statement">
# Beta function in terms of the gamma function

## Statement

Let $x$ and $y$ be complex numbers with $\operatorname{Re} x > 0$ and $\operatorname{Re} y > 0$.
Let $B(x, y) = \int_0^1 t^{x-1} (1-t)^{y-1}\, dt$ be the [[beta function|8K9YQ1YGVS]] and let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
Then
$$B(x, y) = \frac{\Gamma(x)\, \Gamma(y)}{\Gamma(x + y)}.$$

## Remarks

Write $\Gamma(x)\Gamma(y)$ as the double integral $\int_0^\infty \int_0^\infty u^{x-1} v^{y-1} e^{-u-v}\, du\, dv$.
The change of variables $u = st$, $v = s(1-t)$ with $s > 0$ and $0 < t < 1$ has Jacobian $s$.
It turns the double integral into $\int_0^\infty s^{x+y-1} e^{-s}\, ds \cdot \int_0^1 t^{x-1}(1-t)^{y-1}\, dt = \Gamma(x+y)\, B(x, y)$.
Absolute convergence, which needs $\operatorname{Re} x > 0$ and $\operatorname{Re} y > 0$, justifies the change of order.

Example: $B(\frac12, \frac12) = \Gamma(\frac12)^2 / \Gamma(1) = (\sqrt{\pi})^2 = \pi$.
For positive integers $m, n$ the formula gives $B(m, n) = \frac{(m-1)!\,(n-1)!}{(m+n-1)!}$.

Without the condition on the real parts the integral defining $B$ diverges, for example at $x = 0$.
The right side still makes sense there, and it is used to extend $B$ to a meromorphic function of $x$ and $y$.

## Explanations

### L1 human

The beta function is an integral over the interval from $0$ to $1$, and the gamma function is an integral over the whole half line.
This theorem says that the beta function is a simple quotient of gamma values.
So every beta integral can be computed from values of the gamma function.
For whole numbers it becomes a formula with factorials.

### L2 human

The formula is the bridge between the two Euler integrals.
It shows at once that $B(x, y) = B(y, x)$ and gives the analytic continuation of $B$.
With $t = \sin^2 \theta$ it computes integrals $\int_0^{\pi/2} \sin^{a}\theta \cos^{b}\theta\, d\theta$, and with $x = y = \frac12$ it gives $\Gamma(\frac12) = \sqrt{\pi}$.
In probability it gives the normalising constant of the [[beta distribution|0EW824Z7G2]].
</document>

<document source="https://tamnd.github.io/math2/node/153N5DW55Z.md" tag="153N5DW55Z" kind="concept">
# Schatten class

## Summary

For $1 \le p < \infty$, the Schatten $p$-class consists of the compact operators on a Hilbert space whose sequence of singular values is $p$-summable.
These classes are the operator analogues of the sequence spaces $\ell^p$.
The case $p = 1$ is the trace class and the case $p = 2$ is the Hilbert–Schmidt class, the two most used ideals of operators.

## Explanations

### L1 human

A compact operator has a sequence of singular values that tends to zero, like the stretch factors of a matrix.
The Schatten classes sort compact operators by how fast these numbers decrease.
An operator lies in the $p$-class when the sum of the $p$-th powers of its singular values is finite.
For example, the diagonal operator with entries $1/n$ lies in the $p$-class exactly when $p > 1$, because $\sum 1/n^p$ converges exactly then.
</document>

<document source="https://tamnd.github.io/math2/node/155K06F4JF.md" tag="155K06F4JF" kind="definition">
# Newton polynomial

## Definition

Let $K$ be a [[field|04C5ZBG1WT]], let $n \ge 0$, and let $x_0, \dots, x_n \in K$ be pairwise distinct.
Let $f$ be a function defined at these points with values in $K$.
The *Newton polynomial* of $f$ at $x_0, \dots, x_n$ is
$$N(x) = \sum_{j=0}^{n} f[x_0, \dots, x_j] \prod_{k=0}^{j-1} (x - x_k),$$
where $f[x_0, \dots, x_j]$ are the [[divided differences|N5ZHV4HWDC]] of $f$ and the empty product for $j = 0$ is $1$.

## Remarks

$N$ has degree at most $n$ and $N(x_i) = f(x_i)$ for all $i$, so it is the [[interpolation polynomial|SS1YMS2NPC]] of the data, written in the Newton basis.
The first $j + 1$ terms form the interpolation polynomial at $x_0, \dots, x_j$.
It can be evaluated with nested multiplication, as in [[Horner's method|FYBQ3Z52Z7]].

Example: for $f(0) = 1$, $f(1) = 3$, $f(2) = 7$, the divided differences are $f[0] = 1$, $f[0, 1] = 2$, $f[1, 2] = 4$ and $f[0, 1, 2] = 1$.
So $N(x) = 1 + 2x + x(x - 1) = x^2 + x + 1$, and indeed $N(0) = 1$, $N(1) = 3$, $N(2) = 7$.

Non-example: $1 + 2x + 2x(x - 1)$ has the Newton shape but its last coefficient is not $f[0, 1, 2]$; its value at $2$ is $9 \ne 7$.

For equally spaced nodes $x_k = x_0 + kh$, the divided differences become forward differences divided by $k! \, h^k$, which gives the [[Newton–Gregory forward formula|Q4YAST023H]].
</document>

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# Nakayama's lemma

## Statement

Let $R$ be a [[commutative ring|0310REJJY8]] with [[Jacobson radical|XA0DCNH7WY]] $J(R)$.
Let $I \subseteq J(R)$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-[[module|0NZ8JA25WK]].
If $IM = M$, then $M = 0$.

## Remarks

Proof idea: let $m_1, \ldots, m_n$ generate $M$ with $n$ minimal, and suppose $n \ge 1$.
From $IM = M$, write $m_n = a_1 m_1 + \cdots + a_n m_n$ with $a_i \in I$.
Then $(1 - a_n) m_n$ lies in the submodule generated by $m_1, \ldots, m_{n-1}$.
Since $a_n \in J(R)$, the element $1 - a_n$ is a unit, so $m_n$ is not needed, contradicting minimality.
Another proof uses the [[determinant trick|A8HRT9H3XA]] to find $a \in I$ with $(1 + a) M = 0$.

Consequence: if $N \subseteq M$ is a [[submodule|05WDSN597C]] and $M = N + IM$, then $M = N$; apply the lemma to $M/N$.
For a [[local ring|MZ02BKPHJA]] $(R, \mathfrak m)$ with $I = \mathfrak m$, this says that elements of $M$ generate $M$ exactly when their images span the vector space $M/\mathfrak m M$ over $R/\mathfrak m$.

Both hypotheses are needed.
Without finite generation: over $R = \Z_{(p)}$ with $I = p\Z_{(p)} = J(R)$, the module $M = \Q$ satisfies $pM = M$ but $M \ne 0$.
Without $I \subseteq J(R)$: over $R = \Z$ with $I = 2\Z$, the module $M = \Z/3\Z$ satisfies $2M = M$, since $2$ is invertible modulo $3$, but $M \ne 0$.

## Explanations

### L1 human

Over a local ring, the maximal ideal consists of "small" elements.
Nakayama's lemma says that a finitely generated module cannot be made of small multiples of itself unless it is zero.
In practice it says that to find generators of a module, it is enough to find generators "modulo the maximal ideal", which is a question of linear algebra.

### L2 human

The lemma is the main tool for passing from the [[residue field|NJ3APQ8BHH]] back to the local ring.
It gives the equality between the minimal number of generators of a finitely generated module $M$ over a local ring and $\dim_k M/\mathfrak m M$.
With $M = \mathfrak m$, this identifies the [[embedding dimension|DP3KSCRKFP]] with the minimal number of generators of $\mathfrak m$.
It also shows that [[finitely generated projective modules over local rings are free|942FK83S09]], and it enters the proof of the [[Krull intersection theorem|1Q1AH2V41S]].
</document>

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# Piecewise linear manifold

## Summary

A piecewise linear manifold, or PL manifold, is a topological manifold together with an atlas whose changes of chart are piecewise linear.
Equivalently, it is a manifold given with a triangulation in which the neighbourhood of every vertex is a standard piecewise linear ball.
PL manifolds sit between topological and smooth manifolds.
They are the natural setting for combinatorial methods such as handle moves, regular neighbourhoods and simple homotopy theory.

## Explanations

### L1 human

A PL manifold is a manifold built from flat pieces, such as triangles or tetrahedra, glued along faces.
The surface of a cube is a PL $2$-sphere: it is made of flat squares, and its corners are no problem.
Maps between PL manifolds are allowed to bend, but only along flat creases, never in a smooth curve.
Every smooth manifold can be given such a flat-pieces structure, but some topological manifolds cannot.
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# Filter

## Summary

A filter on a set $X$ is a nonempty family of subsets of $X$ that does not contain $\emptyset$.
It contains every superset of its members and the intersection of any two of its members.
One thinks of the members of a filter as the "large" sets, or as the sets that contain "almost all" points in some sense.
Filters give a theory of [[convergence|70M1VAPFN0]] that works in every [[topological space|0CQHFPGJ45]], where sequences are not enough.
Maximal filters are the [[ultrafilters|J64Y3JC0RE]].

## Explanations

### L1 human

A filter is a way to say which sets are "big enough".
For a point $x$ of a topological space, the sets that contain a [[neighbourhood|0ZCY0Z80B6]] of $x$ form a filter: they are the sets that contain everything close enough to $x$.
On the natural numbers, the sets whose complement is finite form a filter: they contain all numbers from some point on.
Two big sets always overlap in a big set, and a set larger than a big set is big too.
</document>

<document source="https://tamnd.github.io/math2/node/15XEBGNGHC.md" tag="15XEBGNGHC" kind="concept">
# Proper map

## Summary

A proper map is a [[continuous map|0VGFHTM75Y]] between [[topological spaces|0CQHFPGJ45]] under which the preimage of every [[compact set|0R6EWFKRCF]] is compact.
It is a map that does not let points "escape to infinity".
Proper maps between locally compact Hausdorff spaces are [[closed|00FSJTXPJM]], and they are the maps that extend continuously to one-point compactifications.

## Explanations

### L1 human

Under a proper map, a bounded region of the target only receives a bounded region of the source.
So points that run off to infinity in the source must also run off to infinity in the target.
Every polynomial map $\R \to \R$ of degree at least $1$ is proper.
The map $\arctan \colon \R \to \R$ is not, since the whole line lands in the bounded interval $(-\pi/2, \pi/2)$.
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<document source="https://tamnd.github.io/math2/node/15Z6BT9XYR.md" tag="15Z6BT9XYR" kind="concept">
# Model predictive control

## Summary

Model predictive control is a feedback strategy that solves a finite-horizon [[optimal control|T3V6ZNAMDN]] problem at every sampling time.
From the measured state it predicts the future with a model and optimises the controls over a horizon of $N$ steps subject to constraints.
It applies only the first control and repeats at the next step.
This receding-horizon principle turns open-loop optimisation into a closed-loop controller that handles constraints on states and inputs directly.
It is standard in process industry, power electronics and autonomous driving.

## Explanations

### L1 human

A driver looks a little way down the road, plans the next few seconds, starts doing the first part of the plan, and then plans again with fresh information.
Model predictive control does the same with a mathematical model of the system and an optimiser.
Because it replans all the time, it corrects for disturbances and model errors.
Constraints such as a maximum speed or a limited motor force simply enter the optimisation problem.
The short horizon must be handled with care: if the plan ignores what happens after the horizon, the closed loop may fail to settle.
</document>

<document source="https://tamnd.github.io/math2/node/163A2WQQJX.md" tag="163A2WQQJX" kind="statement">
# Abel–Ruffini theorem

## Statement

Let $n \ge 5$ be an integer.

1. Let $t_1, \dots, t_n$ be indeterminates and $K = \Q(t_1, \dots, t_n)$. The general polynomial $x^n + t_1 x^{n-1} + \dots + t_{n-1} x + t_n \in K[x]$ is not [[solvable by radicals|J7YS0DA3HQ]] over $K$.
2. There is a polynomial of degree $n$ with rational coefficients that is not solvable by radicals over $\Q$.

## Remarks

By [[Galois' theorem on solvability|VHCWCFA3MX]], it suffices to find a Galois group that is not [[solvable|0P3KZK0FTV]].
The Galois group of the general polynomial of degree $n$ is the full [[symmetric group|0RJMNEN3BJ]] $S_n$, and [[$S_n$ is not solvable for $n \ge 5$|0J53NE674D]].
For clause 2, the polynomial $x^5 - 4x + 2$ has Galois group $S_5$ over $\Q$; see the [[quintic equation|S1WS18QEGJ]].
Multiplying it by $x^{n-5}$ gives a polynomial of degree $n$ with the same nonsolvable roots.
The bound $n \ge 5$ is necessary: every polynomial of degree at most $4$ is solvable by radicals, by the [[cubic formula|FB6J5WAT5V]] and the [[quartic formula|4CDFMKKZ01]].

## Explanations

### L1 human

For equations of degree $2$, $3$ and $4$ there are formulas that give the roots from the coefficients using arithmetic and roots.
The theorem says that no such formula can exist for degree $5$ or more.
Worse, some single equations of degree $5$ with whole-number coefficients have roots that cannot be written with radicals at all.

### L2 human

The theorem does not say that quintic equations have no solutions: by the fundamental theorem of algebra, each has five complex roots.
It says only that these roots are not always radical expressions in the coefficients.
Many quintics, such as $x^5 - 2$, are still solvable by radicals; Galois theory decides which ones.
The roots can be expressed with other functions, for instance with the [[Bring radical|BSDCDJR6RQ]] or with elliptic modular functions.
</document>

<document source="https://tamnd.github.io/math2/node/16B7TWCEX9.md" tag="16B7TWCEX9" kind="definition">
# Strongly minimal theory

## Definition

Let $L$ be a first-order language and let $T$ be a [[complete theory|655K8Y7VS8]] in $L$ that has infinite [[models|MVCH6N9SNW]].
Let $M$ be an infinite $L$-structure.
A subset of $M$ is [[definable|CY1TXHF6KQ]] in one variable if it has the form $\set{a \in M : M \models \varphi(a, \bar b)}$ for a formula $\varphi(x, \bar b)$ with parameters $\bar b$ in $M$.
The structure $M$ is *minimal* if every such subset is finite or cofinite in $M$.
The theory $T$ is *strongly minimal* if every model of $T$ is minimal.

## Remarks

Example: the theory of infinite sets in the empty language is strongly minimal.
By quantifier elimination, a formula $\varphi(x, \bar b)$ is equivalent to a Boolean combination of equations $x = b_i$, so it defines a finite or a cofinite set.
The theory of algebraically closed fields of a fixed characteristic is strongly minimal for a similar reason: a nonzero polynomial in one variable has finitely many roots.

Non-example: the theory of dense linear orders without endpoints is not strongly minimal.
In $(\Q, <)$ the formula $x > 0$ defines a set that is infinite and has an infinite complement.

A structure can be minimal while its theory is not strongly minimal, since the condition must hold in every model of the theory.
In a strongly minimal theory, algebraic closure satisfies the exchange property, so every model has a dimension.
If $L$ is countable, a strongly minimal theory is categorical in every uncountable cardinality.
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<document source="https://tamnd.github.io/math2/node/16J9DM8V57.md" tag="16J9DM8V57" kind="concept">
# Ideal quotient

## Summary

The ideal quotient $(I : J)$ of two ideals of a commutative ring is the set of elements that multiply $J$ into $I$.
It is an ideal that contains $I$.
It plays the role of division for ideals.
It is used in primary decomposition, in the study of [[associated primes|ERFZ6S8X0X]], and in computing the closure of a difference of two algebraic sets.

## Explanations

### L1 human

The ideal quotient asks: which elements $r$ make $rJ$ land inside $I$?
In $\Z$, the quotient $(12\Z : 4\Z)$ is $3\Z$, because $4r$ is a multiple of $12$ exactly when $r$ is a multiple of $3$.
So the quotient of ideals mirrors the division $12 / 4 = 3$.
In geometry, the ideal quotient corresponds to removing the zero set of $J$ from the zero set of $I$ and taking what is left, up to closure.
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<document source="https://tamnd.github.io/math2/node/16SRNY6AXK.md" tag="16SRNY6AXK" kind="definition">
# Lambert series

## Definition

Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers.
The *Lambert series* of $(a_n)$ is the [[series|0NM626FH6Q]]
$$L(q) = \sum_{n=1}^{\infty} a_n \frac{q^n}{1 - q^n},$$
considered as a function of the complex variable $q$.
Only the points $q$ with $\abs{q} < 1$ where the series converges are considered.

## Remarks

If $\abs{q} < 1$ and $\sum_n a_n q^n$ [[converges absolutely|0YBDSJARWN]], then the Lambert series converges absolutely and
$$L(q) = \sum_{m=1}^{\infty} \Big( \sum_{d \mid m} a_d \Big) q^m.$$
Check: $\frac{q^n}{1 - q^n} = \sum_{k \ge 1} q^{nk}$, and $\sum_{n, k} \abs{a_n} \abs{q}^{nk} \le \frac{1}{1 - \abs{q}} \sum_n \abs{a_n} \abs{q}^n < \infty$, so the double series may be regrouped by $m = nk$.
Example: for $a_n = 1$, the coefficient of $q^m$ is the [[number of divisors|0BKJT6WG96]] $d(m)$, so $\sum_n \frac{q^n}{1 - q^n} = \sum_m d(m) q^m$.
Example: for $a_n = \mu(n)$, the [[Möbius function|0C8QENHYZW]], the divisor sums vanish for $m > 1$, so $\sum_n \mu(n) \frac{q^n}{1 - q^n} = q$.
Example: for $a_n = \varphi(n)$, [[Euler's totient function|0VV04M41AN]], the [[divisor sum of the totient|09JXE6XMCC]] is $m$, so the Lambert series is $\sum_m m q^m = q/(1-q)^2$.
Non-example of convergence: for $a_n = n!$ and $0 < \abs{q} < 1$, the terms satisfy $\abs{a_n q^n / (1 - q^n)} \ge n! \abs{q}^n / 2$, which tends to $\infty$, so the series diverges.
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# Abelian theorems for integral transforms

## Statement

Let $f \colon [0, \infty) \to \C$ be locally integrable, and let $F(s) = \int_0^\infty f(t) e^{-st}\, dt$ be its [[Laplace transform|05XT5AVJBK]].
Let $c \in \C$.

1. (Behaviour at infinity.) If $\frac{1}{T} \int_0^T f(t)\, dt \to c$ as $T \to \infty$, then $F(s)$ converges for every real $s > 0$, and $s F(s) \to c$ as $s \to 0^+$.
In particular this holds when $f(t) \to c$ as $t \to \infty$.
2. (Behaviour at zero.) If $\int_0^\infty \abs{f(t)} e^{-s_0 t}\, dt < \infty$ for some real $s_0$, and $f(t) \to c$ as $t \to 0^+$, then $s F(s) \to c$ as $s \to +\infty$ through real values.

## Remarks

Both parts compare $f$ with the constant $c$, using $s \int_0^\infty e^{-st}\, dt = 1$ for $s > 0$.
For part 2, write $sF(s) - c = s\int_0^\infty (f(t) - c) e^{-st}\, dt$; the part of the integral over $[0, \delta]$ is small because $f$ is close to $c$ there, and the rest tends to $0$ by [[dominated convergence|BYNV84PF86]].
For part 1, integrate by parts with $U(T) = \int_0^T f$ to get $sF(s) = s^2 \int_0^\infty U(T) e^{-sT}\, dT$, and use $s^2 \int_0^\infty T e^{-sT}\, dT = 1$.
Example: for $f(t) = 1 + e^{-t}$, $F(s) = 1/s + 1/(s + 1)$, so $sF(s) \to 1 = \lim_{t \to \infty} f(t)$ as $s \to 0^+$, and $sF(s) \to 2 = f(0)$ as $s \to \infty$.
The converses are false without extra conditions; they are the [[Tauberian theorems|NV54MWKT6T]].

## Explanations

### L1 human

These results say that the Laplace transform remembers the limits of $f$.
The long-run value of $f$ shows up in $sF(s)$ for small $s$, and the starting value of $f$ shows up in $sF(s)$ for large $s$.
This is the integral analogue of Abel's theorem on power series, which is where the name "Abelian" comes from.

### L2 human

In engineering these are the final value theorem and the initial value theorem.
They read off the limits of a solution of a linear differential equation from its transform without inverting it.
Part 1 is the Laplace analogue of the fact that Cesàro summability implies Abel summability.
Analogous Abelian theorems hold for other transforms, such as the Stieltjes and Mellin transforms: an asymptotic behaviour of the function gives a matching asymptotic behaviour of the transform.
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<document source="https://tamnd.github.io/math2/node/16WR5EARVC.md" tag="16WR5EARVC" kind="concept">
# Bessel–Clifford function

## Summary

The Bessel–Clifford function $\mathcal{C}_n$ is an entire function of one variable.
It is the [[Bessel function of the first kind|CZCMRDWDYW]] without its power factor and its square root.
It satisfies $J_n(x) = \left( \frac{x}{2} \right)^n \mathcal{C}_n\left( -\frac{x^2}{4} \right)$.
Its power series has the simple coefficients $\frac{1}{k!\, \Gamma(n + k + 1)}$, which makes it convenient in computation and in algebraic arguments.

## Explanations

### L1 human

Bessel functions mix a power $x^n$ with a series in $x^2$.
The Bessel–Clifford function removes the power and replaces $x^2$ by a new variable, leaving a clean power series.
For $n = 0$ it is $\mathcal{C}_0(z) = \sum_{k \ge 0} \frac{z^k}{(k!)^2} = 1 + z + \frac{z^2}{4} + \cdots$.
From it one recovers $J_0(x) = \mathcal{C}_0\left( -\frac{x^2}{4} \right)$.
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<document source="https://tamnd.github.io/math2/node/1735VGRT2P.md" tag="1735VGRT2P" kind="concept">
# Tensor product of algebras

## Summary

The tensor product $A \otimes_R B$ of two [[algebras|DPBJ04JMH8]] over a commutative ring $R$ is the [[tensor product of modules|F0HQXQ0HCD]] with the multiplication $(a \otimes b)(a' \otimes b') = aa' \otimes bb'$.
For commutative algebras it is the coproduct.
A pair of algebra maps out of $A$ and $B$ to a commutative algebra is the same as one algebra map out of $A \otimes_R B$.
It describes base change, such as $\C \otimes_\R A$, and in geometry it gives the fibre product of affine schemes.

## Explanations

### L1 human

The tensor product of two algebras is the most general algebra built from them in which elements of the first commute with elements of the second.
Polynomial rings show the idea: $R[x] \otimes_R R[y]$ is $R[x, y]$.
Here $x^2 \otimes y$ corresponds to the monomial $x^2 y$.
Combining two algebras this way can also split them up: $\C \otimes_\R \C$ is isomorphic to $\C \times \C$.
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<document source="https://tamnd.github.io/math2/node/178GG6TXKQ.md" tag="178GG6TXKQ" kind="statement">
# Liénard's theorem

## Statement

Let $f, g \colon \R \to \R$ be [[continuously differentiable|0VW90BBSJ4]], and let $F(x) = \int_0^x f(s)\, ds$.
Assume:

1. $f$ is an [[even function|0FP31SX5EA]] and $g$ is an odd function;
2. $g(x) > 0$ for all $x > 0$;
3. $F$ has exactly one positive zero $a$, with $F(x) < 0$ for $0 < x < a$;
4. $F$ is increasing on $(a, \infty)$, and $F(x) \to \infty$ as $x \to \infty$.

Then the system $x' = y - F(x)$, $y' = -g(x)$, equivalent to $x'' + f(x) x' + g(x) = 0$, has exactly one [[limit cycle|0AAQ73WRFB]].
This limit cycle surrounds the origin, and it is stable.

## Remarks

The proof uses the energy $E(x, y) = \frac{1}{2} y^2 + G(x)$, where $G(x) = \int_0^x g(s)\, ds$.
Along solutions, $E' = -g(x) F(x)$.
By the symmetry $(x, y) \mapsto (-x, -y)$, it is enough to follow a solution from a point $(0, y_0)$ with $y_0 > 0$ to its next crossing of the $y$-axis.
Let $\Delta(y_0)$ be the change of $E$ on this half turn.
It is positive when the half turn stays in $0 \le x \le a$.
For the half turns that go beyond $x = a$, it decreases strictly in $y_0$ and tends to $-\infty$.
So $\Delta$ vanishes for exactly one $y_0$, and this gives the unique closed orbit.

For the van der Pol equation $x'' - \mu (1 - x^2) x' + x = 0$ with $\mu > 0$, one has $F(x) = \mu (x^3/3 - x)$ and $g(x) = x$.
Here $a = \sqrt{3}$, $F < 0$ on $(0, \sqrt{3})$, and $F' = \mu (x^2 - 1) > 0$ for $x > \sqrt{3}$.
So the van der Pol oscillator has exactly one limit cycle, and it is stable.

## Explanations

### L1 human

Liénard's theorem gives easy conditions under which an oscillator has one and only one self-sustained oscillation.
The friction must be negative for small motions and positive for large ones, with a single change of sign.
Then every motion except the rest state spirals onto the same closed loop.

### L2 human

Proving that a limit cycle exists is often done with the [[Poincaré–Bendixson theorem|0EW1X61ZT1]], but that theorem says nothing about how many there are.
Liénard's theorem gives existence and uniqueness at once for a whole family of equations.
Its proof is a careful estimate of how the energy changes on a half turn around the origin.
The van der Pol oscillator is the main application.
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<document source="https://tamnd.github.io/math2/node/17BH73YHVT.md" tag="17BH73YHVT" kind="definition">
# Concentration of measure

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and $\mu$ a Borel [[probability measure|059TJN0NZ1]] on $X$.
For a Borel set $A \subseteq X$ and $r > 0$, let $A_r = \set{x \in X : d(x, A) < r}$.
The *concentration function* of $(X, d, \mu)$ is
$$\alpha(r) = \sup \set{1 - \mu(A_r) : A \subseteq X \text{ Borel}, \ \mu(A) \ge 1/2}, \qquad r > 0.$$
A sequence $(X_k, d_k, \mu_k)$ of such spaces exhibits *concentration of measure*, and is called a *normal Lévy family*, if there are constants $C, c > 0$ with
$$\alpha_k(r) \le C e^{-c k r^2} \quad \text{for all } k \ge 1 \text{ and } r > 0.$$

## Remarks

Example: take the unit sphere $S^{k}$ of $\R^{k+1}$ with the geodesic metric and the normalized surface measure.
Lévy's isoperimetric inequality gives such a bound with absolute constants $C$ and $c$, so the spheres form a normal Lévy family.
The [[Gaussian measure|P0DV2TCH1N]] $\gamma_n$ on $\R^n$ concentrates with a bound that does not depend on the dimension: if $\gamma_n(A) \ge 1/2$, then $\gamma_n(A_r) \ge 1 - \frac{1}{2} e^{-r^2/2}$.
Non-example: the two-point space $\set{0, 1}$ with $d(0, 1) = 1$ and $\mu(\set{0}) = \mu(\set{1}) = 1/2$; for $A = \set{0}$ and $r \le 1$ one has $A_r = A$, so $\alpha(r) \ge 1/2$ for $r \le 1$.
An equivalent form: $\mu(\set{x : \abs{f(x) - M_f} \ge r}) \le 2 \alpha(r)$ for every $1$-Lipschitz function $f \colon X \to \R$ with median $M_f$.
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# Finitely additive measure

## Definition

Let $\mathcal{A}$ be an [[algebra of sets|PFMGZ7988V]] on a set $X$.
A *finitely additive measure* on $\mathcal{A}$ is a function $\mu \colon \mathcal{A} \to [0, \infty]$, with values in the [[extended real number line|0GSZ7B0CGR]], such that:

1. $\mu(\varnothing) = 0$.
2. $\mu(A \cup B) = \mu(A) + \mu(B)$ for all $A, B \in \mathcal{A}$ with $A \cap B = \varnothing$.

## Remarks

Let $\mathcal{A}$ be the algebra of subsets $A \subseteq \N$ such that $A$ or $\N \setminus A$ is finite.
Let $\nu(A) = 0$ for finite $A$ and $\nu(A) = 1$ otherwise.
Clause 1 holds, and clause 2 holds since two disjoint sets cannot both have finite complement.
This $\nu$ is not [[countably additive|D7KB9HFCGZ]], since $\nu(\N) = 1$ while $\nu(\set{n}) = 0$ for each $n$.
On the same algebra, the function with $\mu(\varnothing) = 0$ and $\mu(A) = 1$ for $A \ne \varnothing$ is not finitely additive, since clause 2 fails for $A = \set{1}$ and $B = \set{2}$.
By induction, clause 2 gives additivity over any finite disjoint union.
A finitely additive measure is monotone: $\mu(A) \le \mu(B)$ when $A \subseteq B$.
Other names are *content* and *charge*; the word content is also used for the same notion on a ring or a semiring of sets.
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<document source="https://tamnd.github.io/math2/node/17FVAN6HXX.md" tag="17FVAN6HXX" kind="definition">
# Closure operator

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
A *closure operator* on $P$ is a map $c \colon P \to P$ such that, for all $x, y \in P$:

1. $x \le c(x)$ (extensive);
2. if $x \le y$, then $c(x) \le c(y)$ (monotone);
3. $c(c(x)) = c(x)$ (idempotent).

An element $x$ with $c(x) = x$ is *closed*.

## Remarks

Let $X$ be a topological space and $P$ the set of subsets of $X$ ordered by inclusion.
The map sending $A$ to its [[closure|0DRKT1QJR7]] $\overline{A}$ is a closure operator: $A \subseteq \overline{A}$, $A \subseteq B$ gives $\overline{A} \subseteq \overline{B}$, and $\overline{\overline{A}} = \overline{A}$.
On $\R$, the map $c(x) = x + 1$ is extensive and monotone but not idempotent, since $c(c(0)) = 2 \ne 1 = c(0)$.
The three clauses together are equivalent to one condition: $x \le c(y)$ if and only if $c(x) \le c(y)$, for all $x, y \in P$.
The closed elements determine $c$: $c(x)$ is the least closed element above $x$.
If $(f, g)$ is a [[Galois connection|MBVJ4JQXC5]] with lower adjoint $f$, then $g \circ f$ is a closure operator.
The dual notion is an [[interior operator|BPXY5A22BB]].

## Explanations

### L1 human

A closure operator moves each element up to the nearest closed element.
Closed elements stay where they are.
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<document source="https://tamnd.github.io/math2/node/17NVR4XBHK.md" tag="17NVR4XBHK" kind="concept">
# Integral scheme

## Summary

An integral scheme is a nonempty [[scheme|5A35TYZD02]] whose ring of functions on each nonempty open set is an [[integral domain|0QVD7WBFGN]].
Equivalently, it is [[reduced|ZXSN8D4MCB]] and its underlying space is irreducible.
Integral schemes are the scheme version of varieties, and each has a function field: the local ring at its [[generic point|3A19C4YXGE]].

## Explanations

### L1 human

An integral scheme is "all in one piece" and has no nilpotent fuzz.
The line $\operatorname{Spec} k[x]$ and $\operatorname{Spec} \Z$ are integral.
The union of the two coordinate axes, $\operatorname{Spec} k[x, y]/(xy)$, is reduced but made of two pieces, so it is not integral: $x \cdot y = 0$ with $x \ne 0$ and $y \ne 0$.
The double point $\operatorname{Spec} k[x]/(x^2)$ is one piece but not reduced, so it is not integral either.
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<document source="https://tamnd.github.io/math2/node/17PWP8MB4X.md" tag="17PWP8MB4X" kind="definition">
# Vector measure

## Definition

Let $\mathcal{F}$ be an [[algebra of sets|PFMGZ7988V]] on a set $\Omega$, and let $B$ be a [[Banach space|06GZ956657]].
A *vector measure* is a map $F \colon \mathcal{F} \to B$ that is finitely additive: $F(E_1 \cup E_2) = F(E_1) + F(E_2)$ for all disjoint $E_1, E_2 \in \mathcal{F}$.
A vector measure $F$ is *countably additive* if, for every sequence $(E_n)$ of pairwise disjoint sets in $\mathcal{F}$ whose union $E$ lies in $\mathcal{F}$,
$$\lim_{N \to \infty} \Big\| F(E) - \sum_{n=1}^{N} F(E_n) \Big\| = 0.$$

## Remarks

Example: let $\mathcal{F}$ be the Lebesgue measurable subsets of $[0, 1]$, let $1 \le p < \infty$, and let $F(E) = \mathbf{1}_E \in L^p([0, 1])$.
Then $F$ is a countably additive vector measure.
Finite additivity holds because $\mathbf{1}_{E_1 \cup E_2} = \mathbf{1}_{E_1} + \mathbf{1}_{E_2}$ for disjoint sets.
For a disjoint sequence, $\norm{F(E) - \sum_{n \le N} F(E_n)}_p = \lambda\big( \bigcup_{n > N} E_n \big)^{1/p} \to 0$.

The same map with values in $L^\infty([0, 1])$ is a vector measure that is not countably additive.
For $E_n = [2^{-n}, 2^{-n+1})$, the remainder $\mathbf{1}_{(0, 2^{-N})}$ has $L^\infty$ norm $1$ for every $N$.

Non-example: $G(E) = \lambda(E)^2 \in \R$ is not a vector measure, since $G([0, 1]) = 1$ but $G([0, \tfrac12)) + G([\tfrac12, 1]) = \tfrac12$.

A real or complex measure that is finite is a countably additive vector measure with $B = \R$ or $B = \C$.
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# Polyharmonic function

## Summary

A polyharmonic function of order $m$ is a function whose $m$-fold iterated [[Laplacian|0E0B6RA8K0]] vanishes: $\Delta^m u = 0$.
For $m = 1$ these are the [[harmonic functions|0DEB0JMN0W]], and for $m = 2$ the solutions of the [[biharmonic equation|AS8A9F5SKE]].
Polyharmonic functions are the model solutions of higher-order elliptic equations.
They appear in elasticity and in the study of polyharmonic splines and interpolation.

## Explanations

### L1 human

Take the Laplacian of a function, then the Laplacian of the result, and so on, $m$ times.
If you end up with zero, the function is polyharmonic of order $m$.
For example, $\abs{x}^2$ on $\R^n$ is not harmonic, but its Laplacian is the constant $2n$, so it is polyharmonic of order $2$.
Every polynomial of degree less than $2m$ is polyharmonic of order $m$, because each Laplacian lowers the degree by $2$.
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<document source="https://tamnd.github.io/math2/node/17S8AM75AD.md" tag="17S8AM75AD" kind="example">
# Regular open algebra

## Example

Let $X$ be a [[topological space|0CQHFPGJ45]], and let $\operatorname{RO}(X)$ be the set of its [[regular open sets|833XAA2Q65]], the open sets $U$ with $U = \operatorname{int} \overline{U}$.
Order $\operatorname{RO}(X)$ by inclusion and define:

1. $U \wedge V = U \cap V$;
2. $U \vee V = \operatorname{int} \overline{U \cup V}$;
3. $\neg U = \operatorname{int}(X \setminus U)$;
4. $0 = \emptyset$ and $1 = X$.

Then $\operatorname{RO}(X)$ is a [[complete Boolean algebra|75BPV95AAS]], called the regular open algebra of $X$.
The supremum of a family $(U_i)$ is $\operatorname{int} \overline{\bigcup_i U_i}$, and the infimum is $\operatorname{int} \bigcap_i U_i$.

Check of the key points: the intersection of two regular open sets is regular open.
Also $\operatorname{int} \overline{W}$ is regular open for every open $W$, so the operations stay in $\operatorname{RO}(X)$.
For $U$ regular open, $\neg U$ is regular open, $U \cap \neg U = \emptyset$, and $\operatorname{int} \overline{U \cup \neg U} = X$, because $U \cup \neg U$ is dense in $X$.
For any family, $\operatorname{int} \overline{\bigcup_i U_i}$ is regular open, contains every $U_i$, and lies in every regular open $V$ that contains all $U_i$, since then $\overline{\bigcup_i U_i} \subseteq \overline{V}$.

## Remarks

The join is not the union: in $\R$, $(0, 1) \vee (1, 2) = (0, 2)$.
For $X = \R$, the algebra has no atoms, since every nonempty regular open set contains two disjoint open intervals.
Every Boolean algebra $B$ embeds densely into the regular open algebra of its [[Stone space|ZV38HA5PAQ]], and this gives the completion of $B$.
In Mathlib, the regular elements of a Heyting algebra are `Heyting.Regular`; for the open sets of $X$ they are the regular open sets.

## Explanations

### L1 human

The open sets of a space are almost a Boolean algebra, but the complement of an open set is not open.
The fix is to keep only the regular open sets, which have no cracks, and to fill in the cracks after each operation.
For example, on the line, $(0, 1)$ "or" $(1, 2)$ is $(0, 2)$: the missing point $1$ is filled in.
The result is a Boolean algebra in which every family of elements has a least upper bound.
</document>

<document source="https://tamnd.github.io/math2/node/17VG475BRV.md" tag="17VG475BRV" kind="example">
# Warsaw circle

## Example

Let $S = \set{(x, \sin(1/x)) : 0 < x \le 1} \cup (\set{0} \times [-1, 1])$ be the closed topologist's sine curve in the plane.
The *Warsaw circle* $W$ is the union of $S$ with an arc $A$ from $(0, -1)$ to $(1, \sin 1)$.
The arc meets $S$ only at these two end points.
For instance, $A$ can run from $(0, -1)$ down to $(0, -2)$, across to $(1, -2)$, and up to $(1, \sin 1)$ along straight segments.

The space $W$ is a [[continuum|NJQ196M1WR]].
It is closed and bounded in $\R^2$, hence compact.
It is connected, since $S$ is connected and $A$ is connected and meets $S$.

## Remarks

The space $W$ is [[path-connected|0WB6DH499T]], because the arc $A$ joins the vertical segment to the graph part, while $S$ alone is not path-connected.
It is not [[locally connected|FDVAHRJKF3]] at the points of the vertical segment.
All homotopy groups of $W$ are trivial, yet $W$ is not contractible.
So the map from $W$ to a point is a weak homotopy equivalence but not a homotopy equivalence.
In shape theory, $W$ has the shape of the circle, and its first Čech cohomology group is $\Z$, like that of a circle.

## Explanations

### L1 human

The Warsaw circle is a circle with a bad spot.
One stretch of it is replaced by the graph of $\sin(1/x)$, which wiggles faster and faster up to a vertical segment.
It looks like a circle from far away and it encloses a hole.
But no loop can go around it, because a path cannot run through the infinitely many wiggles in finite time.
It shows that homotopy groups can miss a hole that other tools can see.
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<document source="https://tamnd.github.io/math2/node/187NP5Z002.md" tag="187NP5Z002" kind="problem">
# Hilbert's twentieth problem

## Problem

Let $\Omega \subseteq \R^n$ be a bounded open set.
Consider a regular problem of the [[calculus of variations|PQ25NHV0KS]]: minimise $J(u) = \int_\Omega L(x, u, \nabla u) \, dx$ with given values of $u$ on $\partial\Omega$.
Here the Lagrangian $L$ is smooth and strictly convex in $\nabla u$.
Hilbert asked whether every such problem has a solution.
He allowed suitable assumptions on the [[boundary conditions|TM8VQJWGHZ]] and, if needed, a suitable extension of the notion of solution.

## Remarks

The problem is a programme rather than a single statement, and its answer depends on how "suitable" is made precise.
The modern answer is largely affirmative.
The [[direct method|TE379HBTY0]] gives minimisers in Sobolev spaces for Lagrangians that are convex in $\nabla u$ and satisfy growth conditions, which is the "suitably extended" notion of solution.
Regularity theory, which grew from [[Hilbert's nineteenth problem|QPMM2ZPWBW]], then shows that these generalised solutions are classical in many cases.
The same circle of ideas gives weak solutions of boundary value problems for linear and nonlinear elliptic equations.

## Explanations

### L1 human

Hilbert's twentieth problem asks whether every reasonable minimisation problem with boundary values has a solution, if one is willing to widen what counts as a solution.
The twentieth century answered this mostly in the affirmative, by working with weak solutions in Sobolev spaces and then proving that they are in fact smooth.
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<document source="https://tamnd.github.io/math2/node/18DPY9J8YP.md" tag="18DPY9J8YP" kind="definition">
# Kolmogorov complexity

## Definition

Let $\set{0,1}^*$ be the set of finite binary strings, and let $\abs{p}$ denote the length of a string $p$.
For a partial [[computable function|KZSB5E471K]] $\varphi \colon \set{0,1}^* \to \set{0,1}^*$ and a string $x$, let
$$C_\varphi(x) = \min \set{\abs{p} : \varphi(p) = x},$$
with $C_\varphi(x) = \infty$ if no such $p$ exists.
Fix a partial computable function $U$, given by a [[universal Turing machine|TC69PEHAX7]], with this property.
For every partial computable $\varphi$ there is a string $q_\varphi$ with $U(q_\varphi p) = \varphi(p)$ for all $p$, where $q_\varphi p$ is the concatenation.
The (plain) *Kolmogorov complexity* of $x$ is $C(x) = C_U(x)$.

## Remarks

Invariance theorem: for every partial computable $\varphi$ and every $x$, $C(x) \le C_\varphi(x) + \abs{q_\varphi}$.
So two choices of $U$ give complexities that differ by at most a constant.

Example: there is a constant $c$ with $C(x) \le \abs{x} + c$ for all $x$, by taking $\varphi$ the identity, which has $C_\varphi(x) = \abs{x}$.
For the string $0^n$ of $n$ zeros, $C(0^n) \le \log_2 n + c'$ for a constant $c'$, because a program can read $n$ in binary and print $n$ zeros.

Non-example: $C_{\mathrm{id}}(x) = \abs{x}$, the complexity for the identity function, is not the Kolmogorov complexity.
The identity is not universal, and $C_{\mathrm{id}}(0^n) = n$ exceeds $C(0^n)$ by more than any constant.

For each $n$ there is a string $x$ of length $n$ with $C(x) \ge n$, since there are $2^n$ such strings but only $2^n - 1$ strings of length less than $n$.
The function $C$ is not computable.
A variant, the prefix-free complexity $K$, uses a universal machine whose domain is a prefix-free set.
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<document source="https://tamnd.github.io/math2/node/18KPYRRPZY.md" tag="18KPYRRPZY" kind="statement">
# Spectral theorem for unbounded self-adjoint operators

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]], and let $A$ be a [[densely defined operator|BCJHNHKQFG]] on $H$ with domain $D(A)$.
Assume that $A$ is self-adjoint: $A = A^*$, with equal domains, where $A^*$ is the [[adjoint|86B7HF79PQ]] of $A$.
Then there is a unique [[projection-valued measure|ATGT25E5FF]] $E$ on the [[Borel sets|0WQXY32Q8D]] of $\R$ such that, with $E_{x,y}(B) = \ip{E(B) x}{y}$:

1. $D(A) = \set{x \in H : \int_{\R} \lambda^2 \, d E_{x,x}(\lambda) < \infty}$;
2. $\ip{A x}{y} = \int_{\R} \lambda \, d E_{x,y}(\lambda)$ for all $x \in D(A)$ and $y \in H$.

Moreover, $E(\R \setminus \sigma(A)) = 0$, where $\sigma(A) \subseteq \R$ is the [[spectrum|BMKATJ802A]] of $A$.

## Remarks

One proof uses the Cayley transform $U = (A - iI)(A + iI)^{-1}$, a unitary operator for which $1$ is not an eigenvalue.
The [[spectral theorem for bounded normal operators|X71XRPJR7D]] applied to $U$ gives a projection-valued measure on the unit circle.
It is carried over to $\R$ by the inverse of the map $\lambda \mapsto (\lambda - i)/(\lambda + i)$.
Equivalently, if $H$ is separable, $A$ is unitarily equivalent to the multiplication by a real measurable function $\varphi$ on some $L^2(\Omega, \mu)$, on the domain of all $f$ with $\varphi f \in L^2(\Omega, \mu)$.
Example: on $L^2(\R)$, the momentum operator $-i \, d/dx$ on the Sobolev space $H^1(\R)$ is self-adjoint; the Fourier transform turns it into multiplication by the variable $\xi$, with spectrum $\R$.
Self-adjointness, not mere symmetry, is needed.
The operator $-i \, d/dx$ on smooth functions with compact support in $(0, \infty)$ is symmetric, but it has no self-adjoint extension and no spectral resolution of this form.

## Explanations

### L1 human

Many important operators, such as derivatives, are unbounded: they can make vectors arbitrarily long.
The theorem says that a self-adjoint unbounded operator is still, in a precise sense, a real diagonal operator.
Its eigenvalues are replaced by a measure on the real line whose values are projections.

### L2 human

The theorem is the mathematical basis of quantum mechanics, where observables such as position, momentum and energy are unbounded self-adjoint operators.
It defines $f(A) = \int f(\lambda) \, dE(\lambda)$ for Borel functions $f$, in particular the unitary group $e^{itA}$ that gives the time evolution of a quantum system.
The probability that a measurement of $A$ in the unit state $x$ gives a value in $B$ is $\ip{E(B) x}{x}$.
Stone's theorem gives the converse correspondence between strongly continuous one-parameter unitary groups and self-adjoint operators.
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<document source="https://tamnd.github.io/math2/node/18KXZHVF4M.md" tag="18KXZHVF4M" kind="concept">
# Navier–Stokes equations

## Summary

The Navier–Stokes equations describe the motion of a viscous fluid, such as water or air.
They are Newton's second law for each small parcel of fluid, with forces from pressure, internal friction and external sources.
In the incompressible case they form a nonlinear system for the velocity field and the pressure.
Whether smooth solutions in three dimensions stay smooth for all time is one of the Millennium Prize Problems.

## Explanations

### L1 human

Think of a fluid as made of many tiny parcels, each with a velocity.
A parcel speeds up or slows down because of three effects.
Differences in pressure push it, friction with its neighbours drags it, and outside forces such as gravity act on it.
The Navier–Stokes equations write this balance as a formula, together with the rule that an incompressible fluid neither piles up nor thins out.
A simple solution is the shear flow $u = (y, 0, 0)$ with constant pressure, where layers of fluid slide past each other at speeds that grow with height.
The equations are used to design aircraft, predict weather and model blood flow.
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# Pseudofunctor

## Summary

A pseudofunctor is a map between [[bicategories|TAHV24ED74]] that preserves composition and identities up to specified invertible 2-morphisms, rather than on the nose.
These comparison 2-morphisms must be compatible with the associators and unitors.
Pseudofunctors into the 2-category of categories describe families of categories that vary with a parameter.
Examples are the categories of modules over varying rings, or of sheaves over varying spaces.
The [[Grothendieck construction|272T16NWZ9]] turns such a pseudofunctor into a fibred category.

## Explanations

### L1 human

A functor must send a composite $g \circ f$ to exactly $F(g) \circ F(f)$.
In many natural constructions this holds only up to a canonical isomorphism.
For example, extending scalars along $R \to S$ and then along $S \to T$ gives modules isomorphic, but not equal, to extending along $R \to T$ in one step.
A pseudofunctor keeps these isomorphisms as data and asks them to fit together consistently.
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<document source="https://tamnd.github.io/math2/node/18MMYZ7922.md" tag="18MMYZ7922" kind="concept">
# Jeffreys prior

## Summary

The Jeffreys prior is a [[prior distribution|00NBBSXRQS]] whose density is proportional to the square root of the determinant of the [[Fisher information|G1Z3WSTE8Q]].
Its defining property is invariance: it gives the same prior no matter how the parameter is written.
It is a standard choice of a non-informative prior in [[Bayesian inference|0JWRJX2RMP]], and it is often improper.

## Explanations

### L1 human

A flat prior on a parameter is not flat on a transformed parameter, such as its square or its logarithm.
So "flat" depends on how the parameter is written down.
The Jeffreys prior avoids this by measuring distances with the Fisher information, which measures how well data can tell nearby parameter values apart.
For a coin with chance $p$ of heads, it gives the density proportional to $1/\sqrt{p(1 - p)}$, which puts a little more weight near $0$ and $1$.
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<document source="https://tamnd.github.io/math2/node/18MQEGN13R.md" tag="18MQEGN13R" kind="concept">
# Isospectral manifolds

## Summary

Two closed Riemannian manifolds are isospectral if their Laplace–Beltrami operators have the same eigenvalues with the same multiplicities.
Isometric manifolds are isospectral, and the converse fails: Milnor found the first counterexample, a pair of flat tori of dimension $16$.
Isospectral pairs show the limits of the inverse problem of [[spectral geometry|QKAFRDPGE4]], the question whether one can [[hear the shape of a drum|X3VH8H1XV3]].
[[Sunada's theorem|7ZMWEDNCVQ]] gives a general method to construct them.

## Explanations

### L1 human

Two isospectral manifolds sound the same: every frequency of one is a frequency of the other, as often.
If two shapes are copies of each other, they certainly sound the same.
The surprise is that some shapes that are not copies of each other also sound the same.
So the list of frequencies does not always determine the shape, although it does determine the dimension, the volume and the total curvature.
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<document source="https://tamnd.github.io/math2/node/18Q1BBK6C0.md" tag="18Q1BBK6C0" kind="definition">
# Morrey–Campanato space

## Definition

Let $\Omega \subseteq \R^n$ be a bounded open set with diameter $d$, let $1 \le p < \infty$ and let $\lambda \ge 0$.
For $x_0 \in \Omega$ and $r > 0$ put $\Omega(x_0, r) = \Omega \cap B(x_0, r)$, where $B(x_0, r)$ is the open ball of radius $r$ about $x_0$.
For $f \in L^p(\Omega)$ let $f_{x_0, r}$ be the mean value of $f$ over $\Omega(x_0, r)$.

1. The *Morrey space* $L^{p, \lambda}(\Omega)$ is the set of $f$ in the [[Lebesgue space|SASP14MA1Z]] $L^p(\Omega)$ with
$$\sup_{x_0 \in \Omega,\ 0 < r < d} \ r^{-\lambda} \int_{\Omega(x_0, r)} \abs{f(x)}^p \, dx < \infty.$$
2. The *Campanato space* $\mathcal{L}^{p, \lambda}(\Omega)$ is the set of $f \in L^p(\Omega)$ with
$$[f]_{p, \lambda}^p = \sup_{x_0 \in \Omega,\ 0 < r < d} \ r^{-\lambda} \int_{\Omega(x_0, r)} \abs{f(x) - f_{x_0, r}}^p \, dx < \infty.$$

The norms are $\norm{f}_{L^p}$ plus the $p$-th root of the supremum.

## Remarks

Example: let $\Omega$ be the unit ball of $\R^n$, let $0 < a < n/p$ and let $f(x) = \abs{x}^{-a}$.
The function $\abs{x}^{-ap}$ is radially decreasing, so its integral over a ball of radius $r$ is largest for the ball centred at $0$.
That integral equals $c \, r^{n - ap}$ for a constant $c$.
Hence $f \in L^{p, \lambda}(\Omega)$ with $\lambda = n - ap$.

Non-example: if $\lambda > n$, then $f = 1$ is not in $L^{p, \lambda}(\Omega)$.
Indeed $r^{-\lambda} \abs{\Omega(x_0, r)}$ grows like $r^{n - \lambda}$ as $r \to 0$, and this tends to infinity.
In fact $L^{p, \lambda}(\Omega) = \set{0}$ for $\lambda > n$, by the Lebesgue differentiation theorem.

Further facts:

- $L^{p, 0}(\Omega) = \mathcal{L}^{p, 0}(\Omega) = L^p(\Omega)$, and $L^{p, n}(\Omega) = L^\infty(\Omega)$.
- $\mathcal{L}^{p, n}(\Omega)$ is a space of functions of [[bounded mean oscillation|QJYTXFT8Y4]].
- If $\Omega$ has Lipschitz boundary and $n < \lambda \le n + p$, then $\mathcal{L}^{p, \lambda}(\Omega)$ equals the space of [[Hölder continuous|012PBA0FMD]] functions with exponent $\alpha = (\lambda - n)/p$ (Campanato's theorem).
- If $\lambda > n + p$ and $\Omega$ is connected, then $\mathcal{L}^{p, \lambda}(\Omega)$ contains only the constant functions.
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# Resolvent set

## Definition

Let $\mathbb{K}$ be a field, let $A$ be an associative algebra over $\mathbb{K}$ with unit $1$, and let $a \in A$.
The *resolvent set* of $a$ is
$$\rho(a) = \set{\lambda \in \mathbb{K} : a - \lambda 1 \text{ is invertible in } A} = \mathbb{K} \setminus \sigma(a),$$
where invertible means being a [[unit|0XVEZDE9J8]] of $A$, and $\sigma(a)$ is the [[spectrum|EH0CZ19RKP]] of $a$.
For $\lambda \in \rho(a)$, the element $R(\lambda) = (a - \lambda 1)^{-1}$ is the *resolvent* of $a$ at $\lambda$.

Let $T$ be a closed linear operator with domain $D(T)$ in a complex [[Banach space|06GZ956657]] $X$.
The resolvent set of $T$ is the set of $\lambda \in \C$ such that $T - \lambda$ maps $D(T)$ bijectively onto $X$ with bounded inverse.

## Remarks

Some authors use $(\lambda 1 - a)^{-1}$ as the resolvent; this changes its sign only.
The resolvent satisfies the resolvent identity $R(\lambda) - R(\mu) = (\lambda - \mu) R(\lambda) R(\mu)$ for $\lambda, \mu \in \rho(a)$.
It follows from $R(\lambda) - R(\mu) = R(\lambda) \big( (a - \mu 1) - (a - \lambda 1) \big) R(\mu)$.

Example: in $C([0,1])$ with complex values, let $f(t) = t$.
Then $\rho(f) = \C \setminus [0,1]$, and $R(\lambda)(t) = 1/(t - \lambda)$.

Non-example: for the same $f$, the number $\lambda = 1/2$ is not in $\rho(f)$, because $f - 1/2$ vanishes at $t = 1/2$ and has no continuous inverse.
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<document source="https://tamnd.github.io/math2/node/1981Q0DGQW.md" tag="1981Q0DGQW" kind="definition">
# Holomorphic map

## Definition

Let $U \subseteq \C^n$ be an [[open set|02BH3MCBQB]] and let $F = (F_1, \dots, F_m) \colon U \to \C^m$ be a map with components $F_k \colon U \to \C$.
The map $F$ is *holomorphic* if each component $F_k$, $k = 1, \dots, m$, is a [[holomorphic function|507C34W0ZT]] on $U$.
If $V \subseteq \C^m$, a holomorphic map $U \to V$ is a holomorphic map $F \colon U \to \C^m$ with $F(U) \subseteq V$.

## Remarks

Equivalently, $F$ is differentiable at each point $a \in U$ in the real sense, and its derivative $DF(a) \colon \C^n \to \C^m$ is $\C$-linear.
This derivative is given by the complex [[Jacobian matrix|047NDTEPCV]] $\left( \frac{\partial F_k}{\partial z_j}(a) \right)_{k, j}$, an $m \times n$ matrix.
Example: $F(z_1, z_2) = (z_1 z_2, \ z_1 - z_2)$ is holomorphic from $\C^2$ to $\C^2$, since both components are polynomials.
Non-example: $G(z_1, z_2) = (z_1, \ \bar z_2)$ is not holomorphic, since its second component fails the Cauchy–Riemann equation $\partial G_2 / \partial \bar z_2 = 0$.
Compositions of holomorphic maps are holomorphic, by the chain rule.
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# Stable map

## Summary

A smooth map between [[smooth manifolds|F45VQ47GPV]] is stable if every map close enough to it is equivalent to it.
Here equivalent means equal after a change of coordinates in the source and in the target.
Small perturbations do not change its singularities in kind, only move them a little.
Stable maps are the maps whose singularities survive in nature and in experiments, which is why [[singularity theory|42V5XGZS4F]] aims to classify them.
Mather proved that stability is equivalent to a linear condition, [[infinitesimal stability|C3TFCVE2YK]], for proper maps.

## Explanations

### L1 human

Imagine a map drawn with a slightly shaky hand.
If the shaky copy can always be turned back into the original by smoothly bending the source and the target, the map is stable.
The height function on a circle is stable: tilt or wobble the circle a little and it still has one highest and one lowest point, of the same kind.
A constant function on a circle is not stable: a tiny wobble gives a function with peaks and valleys, which no change of coordinates can make constant again.
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<document source="https://tamnd.github.io/math2/node/199B14MWGC.md" tag="199B14MWGC" kind="definition">
# Fourier transform method for convolution equations

## Definition

Let $k \in L^1(\R)$, let $f \in L^2(\R)$, and let $\lambda \in \C$.
Write $\hat g(\xi) = \int_{-\infty}^{\infty} g(x) e^{-i x \xi} \, dx$ for the [[Fourier transform|DPVM34ADZV]] of $g$.
The *Fourier transform method* for the [[convolution integral equation|EY75YD988Z]]
$$\varphi(x) - \lambda \int_{-\infty}^{\infty} k(x - t) \, \varphi(t) \, dt = f(x), \quad x \in \R,$$
with unknown $\varphi \in L^2(\R)$ consists of the following steps.

1. Apply the Fourier transform to both sides; by the convolution theorem the equation becomes $(1 - \lambda \hat k(\xi)) \, \hat\varphi(\xi) = \hat f(\xi)$ for almost every $\xi \in \R$.
2. Solve for $\hat\varphi(\xi) = \hat f(\xi) / (1 - \lambda \hat k(\xi))$.
3. Recover $\varphi$ from $\hat\varphi$ with the inverse Fourier transform.

The function $\xi \mapsto 1 - \lambda \hat k(\xi)$ is the *symbol* of the equation.

## Remarks

If $1 - \lambda \hat k(\xi) \ne 0$ for every $\xi \in \R$, the method gives the unique solution in $L^2(\R)$.
Indeed $\hat k$ is continuous and tends to $0$ at infinity by the [[Riemann–Lebesgue lemma|WVKDC011KP]], so $\abs{1 - \lambda \hat k}$ is bounded below by a positive constant.
Then $\hat f / (1 - \lambda \hat k)$ lies in $L^2(\R)$, and the [[Plancherel theorem|HPZHRZ0502]] makes every step reversible.

Example: $k(s) = e^{-\abs{s}}$, $\lambda = \frac14$, $f(x) = \frac12 e^{-\abs{x}}$.
Here $\hat k(\xi) = \frac{2}{1 + \xi^2}$, the symbol $1 - \frac{1}{2(1 + \xi^2)}$ is at least $\frac12$, and step 2 gives $\hat\varphi(\xi) = \frac{1}{1/2 + \xi^2}$, so $\varphi(x) = \frac{1}{\sqrt 2} e^{-\abs{x}/\sqrt 2}$.

Non-example: for $k(s) = \frac12 e^{-\abs{s}}$ and $\lambda = 1$ the symbol $\frac{\xi^2}{1 + \xi^2}$ vanishes at $\xi = 0$.
Step 2 then gives a function $\hat f(\xi) (1 + \xi^2)/\xi^2$ that need not be in $L^2(\R)$, so the method does not produce a solution for every $f \in L^2(\R)$.

The method needs the integral to run over the whole group $\R$.
On a half-line one gets a [[Wiener–Hopf equation|QYY4NMDY5H]], which needs the [[Wiener–Hopf method|H4NXEY6QTZ]] instead.
For equations on $[0, \infty)$ of Volterra type, the [[Laplace transform method|EHGECREYB4]] plays the same role.
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# Univalent function

## Definition

Let $U \subseteq \C$ be an [[open set|01ZQQDCERP]].
A function $f \colon U \to \C$ is *univalent* on $U$ if it is [[holomorphic|0DG0Z72CQD]] on $U$ and [[injective|0PVG1CRAJN]] on $U$.

The class $S$ is the set of univalent functions $f$ on the [[unit disc|8G5ABXRK85]] $\mathbb{D}$ with $f(0) = 0$ and $f'(0) = 1$, so that $f(z) = z + \sum_{n \ge 2} a_n z^n$.

## Remarks

A univalent function has $f'(z) \ne 0$ for all $z \in U$, and its inverse $f^{-1} \colon f(U) \to U$ is holomorphic.
Example: $f(z) = e^z$ on the strip $\set{z : \abs{\operatorname{Im} z} < \pi}$; if $e^z = e^w$ then $z - w \in 2\pi i\Z$, and two points of the strip differ in imaginary part by less than $2\pi$, so $z = w$.
Non-example: $e^z$ on all of $\C$ is holomorphic but not injective, since $e^{0} = e^{2\pi i}$.
Non-example: $z \mapsto z^2$ on $\mathbb{D}$ is not univalent, though it is injective near every point except $0$.
The converse of the first remark fails: $f' \ne 0$ only gives local injectivity, as $e^z$ on $\C$ shows.
Some authors also call injective meromorphic functions univalent; then the class $\Sigma$ of univalent functions $g(z) = z + b_0 + b_1 z^{-1} + \cdots$ on $\set{\abs{z} > 1}$ is studied together with $S$.
The German term *schlicht* is used with the same meaning.
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# Interval exchange transformation

## Example

Let $d \ge 2$, let $\lambda_1, \dots, \lambda_d > 0$ be lengths with sum $\abs{\lambda} = \lambda_1 + \cdots + \lambda_d$, and let $\pi$ be a permutation of $\set{1, \dots, d}$.
Cut $I = [0, \abs{\lambda})$ into the intervals $I_j = [\lambda_1 + \cdots + \lambda_{j-1}, \lambda_1 + \cdots + \lambda_j)$, for $j = 1, \dots, d$.
The *interval exchange transformation* $T \colon I \to I$ translates each $I_j$ so that the intervals appear in the new order $I_{\pi^{-1}(1)}, I_{\pi^{-1}(2)}, \dots, I_{\pi^{-1}(d)}$ from left to right.
Explicitly, for $x \in I_j$,
$$T(x) = x - \sum_{i < j} \lambda_i + \sum_{\pi(i) < \pi(j)} \lambda_i.$$

Check that $T$ is a [[measure-preserving transformation|556WBDAHNP]] for [[Lebesgue measure|2Z2QVWZ262]].
Indeed, $T$ is a bijection of $I$, and it is a translation on each of the finitely many intervals $I_j$.
Translations preserve length, so for every Borel set $E \subseteq I$, $\abs{T^{-1}E} = \sum_j \abs{T^{-1}E \cap I_j} = \sum_j \abs{E \cap T(I_j)} = \abs{E}$.

Concrete case: $d = 2$ and $\pi$ the transposition, with $0 < \alpha < 1$ and lengths $\lambda_1 = 1 - \alpha$ and $\lambda_2 = \alpha$.
Then $T(x) = x + \alpha$ on $[0, 1 - \alpha)$ and $T(x) = x - (1 - \alpha)$ on $[1 - \alpha, 1)$, so $T(x) = x + \alpha \bmod 1$, the [[rotation of the circle|2Q1J7RZ4TK]].
Interval exchanges are thus a generalization of circle rotations.

For an irreducible permutation $\pi$ and Lebesgue almost every length vector, the interval exchange is [[uniquely ergodic|NZZE9KCAD8]].
Masur and Veech proved this independently in 1982.

## Explanations

### L1 human

Cut a stick into several pieces, shuffle the pieces into a new order without flipping them, and glue them back.
That rearrangement is an interval exchange.
Repeating it moves points around in a way that never changes the total length of any set.
With two pieces it is just a rotation of a circle; with more pieces it already leads to deep questions, linked to flat surfaces and billiards in polygons.
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# Meijer G-function

## Summary

The Meijer G-function $G^{m, n}_{p, q}$ is a very general special function defined by a [[Barnes integral|1PDGBGEHM2]] of a ratio of products of gamma functions.
Almost all classical special functions are particular G-functions, including the exponential function, the Bessel functions and every [[generalized hypergeometric function|VDBAW9PN88]] ${}_pF_q$ with $p \le q + 1$.
The class of G-functions is closed under many operations, such as multiplication by powers, differentiation, Laplace and Mellin transforms, and some integrals of products.
This makes it a tool for computing definite integrals in a systematic way, which is used in computer algebra systems.

## Explanations

### L1 human

The idea is to describe a function by its Mellin transform instead of by a series.
For the G-function, the Mellin transform is a ratio of products of gamma functions, and the function itself is recovered by a contour integral.
By choosing how many gamma factors go on top and on the bottom, one gets the exponential function, the Bessel functions, the hypergeometric functions and much more.
The simplest case is $G^{1, 0}_{0, 1}(- ; 0 \mid z) = e^{-z}$.
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# Random matrix

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]] and let $m, n \ge 1$ be integers.
Let $\mathbb{K}$ be $\R$ or $\C$, and give the space $\mathbb{K}^{m \times n}$ of $m \times n$ [[matrices|0VD9ZSWWJG]] the Borel $\sigma$-algebra of $\mathbb{K}^{mn}$.
A *random $m \times n$ matrix* is a measurable map $X \colon \Omega \to \mathbb{K}^{m \times n}$.

## Remarks

Equivalently, $X = (X_{ij})$ where each entry $X_{ij}$ is a [[random variable|0PNRKMRC55]] on the same space $(\Omega, \mathcal{F}, P)$.
The law of $X$ is the probability measure $P \circ X^{-1}$ on $\mathbb{K}^{m \times n}$.
Example: let $G$ be an $n \times n$ matrix whose entries are independent standard normal random variables, and let $H = (G + G^T)/\sqrt{2}$.
Each entry of $H$ is a sum of measurable functions, so $H$ is a random symmetric matrix; it is a Wigner matrix with Gaussian entries.
Non-example: $n^2$ random variables defined on different probability spaces do not form a random matrix, because the entries need one common probability space.
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# Lebesgue point

## Summary

Let $f$ be a locally integrable function on $\R^n$.
A Lebesgue point of $f$ is a point $x$ where the average of $\abs{f - f(x)}$ over small balls around $x$ tends to $0$.
At such a point, $f$ is well represented by its value $f(x)$ in an averaged sense.
By the Lebesgue differentiation theorem, almost every point is a Lebesgue point.
Points of continuity are Lebesgue points, but Lebesgue points need not be points of continuity.

## Explanations

### L1 human

Zoom into a function near a point and look at the average distance between $f$ and its value at the point.
If this average distance goes to zero as the window shrinks, the point is a Lebesgue point.
For the step function equal to $0$ on the negative numbers and $1$ elsewhere, the jump point $0$ is not a Lebesgue point.
On each small window around $0$ half of the values differ from $f(0) = 1$ by $1$.
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# Liouville's equation

## Definition

Let $\Omega \subseteq \R^2$ be an open set and let $K \in \R$ be a constant.
*Liouville's equation* on $\Omega$ is the [[semilinear|RCNY89A4NY]] equation
$$\Delta u + K e^{2u} = 0$$
for a twice continuously differentiable function $u \colon \Omega \to \R$, where $\Delta = \partial_x^2 + \partial_y^2$ is the [[Laplacian|0E0B6RA8K0]].

## Remarks

The metric $e^{2u}(dx^2 + dy^2)$ has [[Gaussian curvature|SQVQ5DQT6W]] $-e^{-2u} \Delta u$, so $u$ solves the equation exactly when this metric has constant curvature $K$.
Example: for $K = 1$, $u(x, y) = \log 2 - \log(1 + r^2)$ with $r^2 = x^2 + y^2$ is a solution on $\R^2$.
Indeed, $\Delta \log(1 + r^2) = 4/(1 + r^2)^2$, so $\Delta u = -4/(1 + r^2)^2 = -e^{2u}$.
Non-example: for $K = 1$, the function $u = 0$ is not a solution, since $\Delta u + e^{2u} = 1$.
Other normalizations are common: the substitution $u = v/2$ gives $\Delta v + 2K e^{v} = 0$, so the form $\Delta v + \lambda e^{v} = 0$ is equivalent.
Locally, the solutions for $K > 0$ are $u = \log\big( 2 \abs{f'} / (1 + K \abs{f}^2) \big)$ with $f$ a holomorphic function of $z = x + iy$ whose derivative has no zeros.
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# Entropy solution

## Summary

An entropy solution of a scalar conservation law $u_t + \operatorname{div} f(u) = 0$ is a [[weak solution|SJZX81A783]] that also satisfies a family of inequalities, the entropy conditions.
Weak solutions of conservation laws are in general not unique, because shocks can be drawn in many ways.
The entropy conditions select the physically relevant solution, the one obtained as the limit of solutions with small added viscosity.
In Kruzhkov's formulation, the entropy solution of the initial value problem exists and is unique for every bounded initial datum.

## Explanations

### L1 human

Solutions of conservation laws, such as models of traffic or gas flow, develop jumps called shocks even from smooth initial data.
Once jumps are allowed, the equation alone permits several solutions, some of them unphysical, like a shock that spontaneously forms from a smooth spreading wave.
The entropy condition is an extra rule saying that information may only flow into a shock, never out of it.
For Burgers' equation $u_t + (u^2/2)_x = 0$, a jump from $1$ down to $0$ is allowed, while a jump from $0$ up to $1$ must instead spread out into a fan.
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# Link

## Summary

A link is a finite collection of disjoint closed curves in three-dimensional space, considered up to deformation of the space.
Each curve is a component, and a link with one component is a [[knot|QYEAZTYGPT]].
Links show phenomena that single knots cannot, such as two components that cannot be pulled apart although each is unknotted.

## Explanations

### L1 human

Take several closed loops of string, possibly tangled together.
The [[Hopf link|JMWHKWD7EX]] is two rings linked like the links of a chain.
The [[Borromean rings|WX1Z2F3DCB]] are three rings that hold together although no two of them are linked.
Two links are the same if one can be moved to the other without cutting any string.
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# Convex join

## Summary

The convex join of two sets is the union of all [[line segments|VQ4Q2V45RP]] that start in the first set and end in the second.
When both sets are [[convex|02AS45T3H9]] and nonempty, the convex join is the [[convex hull|0FQSDC2EQ2]] of their union.
So it gives a concrete way to build the hull of two convex pieces from segments alone.

## Explanations

### L1 human

Take two shapes and join every point of the first to every point of the second by a straight segment.
The region covered by all these segments is the convex join.
For a point $p$ and a disc, the join is a cone shape: the disc together with all segments from $p$ to it.
For the two points $(0, 0)$ and $(1, 0)$ of the plane, the join is just the segment between them.
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# Leray–Schauder degree

## Definition

Let $X$ be a real [[Banach space|06GZ956657]] and let $\Omega \subseteq X$ be a [[bounded|09QSZVVM26]] open set.
Let $K \colon \overline{\Omega} \to X$ be a compact map, that is, continuous with $K(\overline{\Omega})$ relatively compact.
Let $y \in X$ with $y \notin (I - K)(\partial\Omega)$, where $I$ is the identity, and put $r = \inf_{u \in \partial\Omega} \norm{u - K(u) - y}$, which is positive.
Choose a continuous map $K_\varepsilon \colon \overline{\Omega} \to X$ with values in a finite-dimensional subspace $E \subseteq X$ that contains $y$, such that $\norm{K(u) - K_\varepsilon(u)} < r$ for all $u \in \overline{\Omega}$.
Let $\deg_B$ denote the Brouwer degree of continuous maps on closures of bounded open subsets of the finite-dimensional space $E$.
The *Leray–Schauder degree* of $I - K$ on $\Omega$ at $y$ is
$$\deg_{LS}(I - K, \Omega, y) = \deg_B\big((I - K_\varepsilon)|_{\overline{\Omega \cap E}}, \, \Omega \cap E, \, y\big).$$

## Remarks

Such $K_\varepsilon$ exist because $K(\overline{\Omega})$ is relatively compact, and the right-hand side does not depend on the choice of $K_\varepsilon$ and $E$; this is what makes the definition sound.
The degree has these properties.

1. Normalisation: $\deg_{LS}(I, \Omega, y) = 1$ if $y \in \Omega$.
2. Existence: if $\deg_{LS}(I - K, \Omega, y) \ne 0$, then $u - K(u) = y$ for some $u \in \Omega$.
3. Homotopy invariance: if $H \colon \overline{\Omega} \times [0, 1] \to X$ is compact and $u - H(u, t) \ne y$ for all $u \in \partial\Omega$ and $t \in [0, 1]$, then $\deg_{LS}(I - H(\cdot, t), \Omega, y)$ does not depend on $t$.

Example: let $\Omega$ be the open unit ball and let $K$ be compact with $\norm{K(u)} < 1$ on $\overline{\Omega}$.
The homotopy $H(u, t) = tK(u)$ has no solution of $u = tK(u)$ on the unit sphere, so $\deg_{LS}(I - K, \Omega, 0) = \deg_{LS}(I, \Omega, 0) = 1$, and $K$ has a fixed point in $\Omega$.

Non-example: if $X$ is infinite-dimensional, the map $K = -I$ on the closed unit ball is not compact, since that ball is not compact.
So this construction does not define a degree for $I - K = 2I$.

## Explanations

### L1 human

The Leray–Schauder degree counts solutions of $u - K(u) = y$ with signs, for compact $K$, by approximating $K$ by maps into finite-dimensional spaces and using the finite-dimensional degree there.
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# n-connected space

## Summary

A space is $n$-connected if it is nonempty, path-connected, and its [[homotopy groups|P976WYEFMB]] vanish in degrees $1$ to $n$.
It generalises path-connected ($0$-connected) and simply connected ($1$-connected) spaces.
Connectivity sets the range in which the Hurewicz, Freudenthal and Blakers–Massey theorems apply.

## Explanations

### L1 human

A space is $n$-connected if every sphere of dimension at most $n$ inside it can be shrunk to a point.
Being $0$-connected means any two points can be joined by a path.
Being $1$-connected means also that every loop can be contracted.
The $3$-sphere is $2$-connected: every loop and every small balloon on it can be shrunk.
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# Artin approximation theorem

## Statement

Let $k$ be a field and let $x = (x_1, \ldots, x_n)$ be variables.
Let $k[\![x]\!]$ be the [[ring of formal power series|3E8ZG3NJYZ]] in $x$.
Let $k\langle x \rangle \subseteq k[\![x]\!]$ be the ring of algebraic power series: those that are algebraic over the [[polynomial ring|0SDC934R52]] $k[x]$.
Let $y = (y_1, \ldots, y_N)$ be further variables and let $f_1, \ldots, f_m \in k[x, y]$ be polynomials.
Assume that $\hat{y} = (\hat{y}_1, \ldots, \hat{y}_N) \in k[\![x]\!]^N$ satisfies $f_j(x, \hat{y}) = 0$ for $j = 1, \ldots, m$.
Then for every integer $c \ge 1$ there is $\bar{y} \in k\langle x \rangle^N$ such that $f_j(x, \bar{y}) = 0$ for $j = 1, \ldots, m$ and $\bar{y}_i \equiv \hat{y}_i \pmod{(x_1, \ldots, x_n)^c}$ for $i = 1, \ldots, N$.

## Remarks

The ring $k\langle x \rangle$ is the [[Henselization|917PGHVZPM]] of the local ring of $k[x]$ at the ideal $(x_1, \ldots, x_n)$.
The theorem says that formal solutions of polynomial equations can be approximated, in the [[$(x)$-adic topology|2VF2TSJ3PY]], by algebraic solutions.
M. Artin proved it in 1969.
Popescu's theorem extends the result to every excellent Henselian local ring in place of $k\langle x \rangle$.

Example: the equation $y_1 - y_2 = 0$ has the formal solution $\hat{y}_1 = \hat{y}_2 = \sum_{i \ge 0} x^i / i!$ over $\Q$, which is not algebraic.
The truncations $\bar{y}_1 = \bar{y}_2 = \sum_{i < c} x^i / i!$ are polynomials, hence algebraic, and agree with $\hat{y}$ modulo $(x)^c$.
The theorem does not say that the formal solution itself is algebraic; it only gives algebraic solutions that agree with it to any given order.

## Explanations

### L1 human

Suppose a system of polynomial equations has a solution made of formal power series.
Such a solution may be wild: the series may not come from any algebraic equation.
The theorem says that there are tame solutions nearby, made of algebraic power series, which agree with the formal solution up to any order you choose.
So questions about formal solutions can often be moved to the much smaller world of algebraic functions.

### L2 human

In algebraic geometry, formal power series describe infinitesimal neighbourhoods, while algebraic power series describe étale neighbourhoods.
The theorem lets one pass from a formal statement about the completion of a local ring to a statement in the [[Henselization|917PGHVZPM]], and so to an étale neighbourhood.
This is used, for example, to compare the formal and the étale local structure of singularities.
It is also a key input in Artin's work on algebraic spaces and moduli problems.
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# Numerical stability

## Summary

An algorithm is numerically stable when the errors it makes, such as [[round-off errors|CPN6SD3BTV]], are not amplified beyond what the problem itself forces.
The strongest common form is backward stability: the computed answer is the exact answer for a slightly perturbed input, as in [[backward error analysis|CMXHRNWY35]].
A stable algorithm applied to a well-conditioned problem gives an accurate answer; for an ill-conditioned problem even a stable algorithm can give large errors.
For methods for differential equations, stability means instead that errors in the steps stay bounded, as checked by [[von Neumann stability analysis|2Z7NASDQB6]].

## Explanations

### L1 human

Two formulas can be equal in exact arithmetic but behave very differently on a computer.
For example, the roots of $x^2 - 10^8 x + 1 = 0$ are about $10^8$ and $10^{-8}$.
The textbook formula computes the small root as $(10^8 - \sqrt{10^{16} - 4})/2$, a difference of two almost equal numbers, and loses most of its digits.
The equivalent formula $2/(10^8 + \sqrt{10^{16} - 4})$ has no such subtraction and is accurate.
The second formula is a stable way to compute the small root and the first is not.
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# Equational theory

## Definition

Let $\sigma$ be a signature with only function symbols and let $X$ be a countably infinite set of variables.
For a class $\mathcal{K}$ of algebras of signature $\sigma$, the *equational theory* of $\mathcal{K}$ is the set
$$\operatorname{Id}(\mathcal{K}) = \set{s \approx t : s, t \text{ terms over } \sigma \text{ and } X, \ \mathcal{K} \models s \approx t}$$
of all [[identities|PKD440G2XW]] satisfied by $\mathcal{K}$.
A set $\Sigma$ of identities over $\sigma$ and $X$ is an *equational theory* if $\Sigma = \operatorname{Id}(\mathcal{K})$ for some class $\mathcal{K}$.

## Remarks

Every equational theory contains $t \approx t$ for every [[term|6D3WA7N2W4]] $t$, is closed under symmetry and transitivity, under applying an operation symbol to identities, and under substitution of terms for variables.
By Birkhoff's completeness theorem, these closure properties characterise equational theories.
For a set $\Sigma$ of identities, the equational theory of the class of all algebras satisfying $\Sigma$ is the set of consequences of $\Sigma$.

Example: for $\mathcal{K}$ the class of all commutative semigroups, $\operatorname{Id}(\mathcal{K})$ contains $x(yz) \approx (zy)x$, since both sides equal $xyz$ in every commutative semigroup.

Non-example: the set $\set{x y \approx y x}$ is not an equational theory.
Every set of the form $\operatorname{Id}(\mathcal{K})$ contains the identity $x \approx x$, which is not in this set.
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# Riesz projector

## Definition

Let $X$ be a complex Banach space with identity map $I$, and let $T \colon X \to X$ be a bounded operator with [[spectrum|BMKATJ802A]] $\sigma(T)$ and [[resolvent|J5YNNKSB9P]] $R(\lambda, T) = (T - \lambda I)^{-1}$.
Assume that $\sigma(T) = \sigma_1 \cup \sigma_2$, where $\sigma_1$ and $\sigma_2$ are disjoint closed sets.
Let $\Gamma$ be a finite union of positively oriented closed rectifiable curves in $\C \setminus \sigma(T)$.
Assume that $\Gamma$ has winding number $1$ around each point of $\sigma_1$ and $0$ around each point of $\sigma_2$.
The *Riesz projector* of $T$ for $\sigma_1$ is the operator
$$P_{\sigma_1} = -\frac{1}{2\pi i} \oint_\Gamma R(\lambda, T) \, d\lambda = \frac{1}{2\pi i} \oint_\Gamma (\lambda I - T)^{-1} \, d\lambda.$$

## Remarks

The integral does not depend on the choice of $\Gamma$, by the Cauchy theorem for analytic operator-valued functions.
The operator $P = P_{\sigma_1}$ is a [[projection|033DA1WE8Y]], $P^2 = P$, and it commutes with $T$.
So $X = PX \oplus (I - P)X$ with both summands closed and invariant under $T$, and the restrictions of $T$ have spectra $\sigma_1$ and $\sigma_2$.

Example: let $T = \operatorname{diag}(1, 2)$ on $\C^2$ and $\sigma_1 = \set{1}$, with $\Gamma$ the circle $\abs{\lambda - 1} = \tfrac12$.
Then $(\lambda I - T)^{-1} = \operatorname{diag}(1/(\lambda - 1), 1/(\lambda - 2))$, and [[Cauchy's integral formula|0DG10TVYPN]] gives $P_{\sigma_1} = \operatorname{diag}(1, 0)$.

Non-example: for the right shift on $\ell^2$, the spectrum is the closed unit disc, which is connected.
A subset such as the closed disc of radius $\tfrac12$ is not separated from the rest of the spectrum, so it has no Riesz projector.
Only $\sigma_1 = \emptyset$ and $\sigma_1 = \sigma(T)$ are allowed, giving $0$ and $I$.
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# Semiprimitive ring

## Summary

A semiprimitive ring is a ring whose [[Jacobson radical|XA0DCNH7WY]] is zero.
Equivalently, it is a subdirect product of [[primitive rings|8F6EHDVJ62]], or it has a faithful semisimple module.
Every semiprimitive ring is [[semiprime|5F0YMCV50R]], and a left Artinian ring is semiprimitive exactly when it is a [[semisimple ring|NY25HXA478]].
The ring $R/J(R)$ is always semiprimitive, which is why the Jacobson radical is the first thing one removes in the structure theory of rings.

## Explanations

### L1 human

The Jacobson radical is the part of a ring that every simple module ignores.
A semiprimitive ring has no such part: each nonzero element acts nontrivially on some simple module.
The integers are semiprimitive, because no nonzero integer lies in every ideal $p\Z$.
The ring $\Z/4\Z$ is not, because its only maximal ideal is $2\Z/4\Z \ne 0$.
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# Wagner's theorem

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]].
Then $G$ is [[planar|029JQEK15B]] if and only if neither the [[complete graph|03WKB2H474]] $K_5$ nor the [[complete bipartite graph|0V0GC7BQAN]] $K_{3,3}$ is a [[minor|VYQ32ZBPEE]] of $G$.

## Remarks

Deleting vertices or edges and contracting edges keep a plane drawing plane, so every minor of a planar graph is planar.
Since $K_5$ and $K_{3,3}$ are not planar, a planar graph has neither as a minor.
For the converse, if $G$ is not planar, [[Kuratowski's theorem|0J6G5NJC5R]] gives a subgraph that is a subdivision of $K_5$ or of $K_{3,3}$.
Contracting the subdivided paths of this subgraph back to single edges gives $K_5$ or $K_{3,3}$ as a minor.
For example, the [[Petersen graph|0NR95CTFCZ]] is not planar: contracting its five spokes gives $K_5$.
Klaus Wagner published the theorem in 1937.

## Explanations

### L1 human

A graph is planar when you can draw it without crossings.
Wagner's theorem says that the only obstacles are the five-vertex complete graph and the three-houses-three-utilities graph, found inside $G$ after deleting and shrinking parts.
So to prove a graph is not planar, it is enough to exhibit one of these two inside it.

### L2 human

Wagner's theorem is the minor version of Kuratowski's theorem.
It shows that the class of planar graphs is closed under minors and is described by exactly two forbidden minors.
The [[Robertson–Seymour theorem|WP3GDNWJJW]] says that every minor-closed class of graphs has such a finite description.
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# Congruence-permutable variety

## Summary

A congruence-permutable variety is a [[variety|AJNZJD8442]] in which any two congruences of any algebra permute under relational composition.
Groups, rings, modules and quasigroups form congruence-permutable varieties, while lattices and semilattices do not.
Congruence permutability makes the congruence lattices modular and gives strong control of direct and subdirect products.

## Explanations

### L1 human

A congruence of an algebra is a way to identify elements that respects the operations.
Two congruences permute when moving first by one and then by the other reaches the same pairs as doing it in the opposite order.
In a group, this says that $MN = NM$ for normal subgroups $M$ and $N$, which always holds.
A variety is congruence permutable when this happens in every one of its algebras.
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# Krull dimension

## Definition

Let $R$ be a nonzero [[commutative ring|0GYGHKX834]].
A chain of [[prime ideals|06HC6HK0Z8]] $P_0 \subsetneq P_1 \subsetneq \cdots \subsetneq P_n$ of $R$ has length $n$.
The *Krull dimension* $\dim R$ is the supremum of the lengths of all such chains, an element of $\set{0, 1, 2, \dots} \cup \set{\infty}$.

## Remarks

For the zero ring, which has no prime ideals, the dimension is set to $-1$ or $-\infty$ by convention.
Example: $\dim \Z = 1$; the chain $(0) \subsetneq (p)$ for a prime $p$ has length $1$, and no chain is longer, since every nonzero prime ideal of $\Z$ is maximal.
Non-example of a chain: $(0) \subsetneq (6) \subsetneq (3)$ does not show $\dim \Z \ge 2$, because $(6)$ is not a prime ideal.
For a field $k$, $\dim k[x_1, \dots, x_n] = n$, and the polynomial ring $k[x_1, x_2, \dots]$ in infinitely many variables has infinite dimension.
A [[Noetherian ring|0MVH4Y1CRZ]] can still have infinite dimension, by [[an example of Nagata|YEJVKCY78E]].

## Explanations

### L1 human

The Krull dimension of a ring is the length of the longest strictly increasing chain of prime ideals.
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# Euclidean distance matrix

## Summary

A Euclidean distance matrix is a square matrix whose entries are the squared distances between some points of a Euclidean space.
It records the whole shape of a finite point set up to isometry.
[[Schoenberg's criterion|4B0R70F4VE]] recognizes these matrices through a [[positive semidefinite|01MMVW8CKC]] condition, which links [[distance geometry|HPW5F0GQ4B]] with linear algebra.
They are used in multidimensional scaling, molecular conformation and sensor localization.

## Explanations

### L1 human

Take a few points, number them, and write the square of the distance from point $i$ to point $j$ in row $i$ and column $j$.
The table you get is a Euclidean distance matrix.
It is symmetric with zeros on the diagonal.
For the points $0$, $1$ and $3$ on a line, the table has rows $(0, 1, 9)$, $(1, 0, 4)$ and $(9, 4, 0)$.
The hard question is the reverse one: given such a table, do points exist that produce it?
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# Sumudu transform

## Summary

The Sumudu transform of a function $f$ on $[0, \infty)$ is $G(u) = \int_0^\infty f(ut)\, e^{-t}\, dt$.
It is the [[Laplace transform|0V49JEZW06]] in disguise: $G(u) = \frac{1}{u} F\big(\frac{1}{u}\big)$, where $F$ is the Laplace transform of $f$.
Its appeal is that it keeps units and scales: it sends $t^n$ to $n!\, u^n$ and constants to the same constants.
It is used, like the Laplace transform, to solve linear differential equations with constant coefficients.

## Explanations

### L1 human

The transform averages $f$ along the stretched line $t \mapsto f(ut)$, with exponentially decaying weights.
Because the weights $e^{-t}$ have total mass $1$, a constant function is sent to itself.
A power $t^n$ is sent to $n!\, u^n$, so a polynomial becomes a polynomial in $u$ of the same degree.
The variable $u$ plays the role of a time scale.
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<document source="https://tamnd.github.io/math2/node/1BK2B286W3.md" tag="1BK2B286W3" kind="definition">
# Integral of a Hopf algebra

## Definition

Let $K$ be a field and let $H$ be a [[Hopf algebra|7XFYP0N84H]] over $K$ with counit $\varepsilon$.
A *left integral* in $H$ is an element $\Lambda \in H$ such that $h \Lambda = \varepsilon(h) \Lambda$ for all $h \in H$.
A *right integral* in $H$ is an element $\Lambda \in H$ such that $\Lambda h = \varepsilon(h) \Lambda$ for all $h \in H$.

## Remarks

The left integrals form a subspace of $H$, often written $\int_H^l$.
Example: for a finite group $G$, the element $\Lambda = \sum_{g \in G} g$ of the [[group algebra|MNP2HBWB5Q]] $KG$ is a left and right integral.
The check: for $h \in G$, $h \Lambda = \sum_{g} hg = \Lambda = \varepsilon(h) \Lambda$, since $g \mapsto hg$ is a bijection of $G$; linearity gives all $h \in KG$.
Non-example: in $KG$ with $G \ne 1$, the element $1$ is not an integral, because $g \cdot 1 = g \ne 1 = \varepsilon(g) 1$ for $g \ne 1$.
In $K[x]$ with $\Delta(x) = x \otimes 1 + 1 \otimes x$ and $\varepsilon(x) = 0$, a left integral must satisfy $x \Lambda = 0$, so $0$ is the only integral.
Some authors call these elements integrals *in* $H$, and call integrals *on* $H$ the linear forms $\lambda \colon H \to K$ with $\sum h_{(1)} \lambda(h_{(2)}) = \lambda(h) 1$.
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<document source="https://tamnd.github.io/math2/node/1BK7TC6PW9.md" tag="1BK7TC6PW9" kind="definition">
# Exponential integral

## Definition

1. For a real number $x > 0$, the exponential integral $E_1$ is the [[improper integral|05KQ3F99BQ]]
$$E_1(x) = \int_x^\infty \frac{e^{-t}}{t}\, dt.$$
2. For a real number $x \ne 0$, the exponential integral $\operatorname{Ei}$ is
$$\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t}\, dt,$$
where for $x > 0$ the integral is a [[Cauchy principal value|0SYR10HD4R]] at $t = 0$.

## Remarks

The substitution $t \mapsto -t$ gives $E_1(x) = -\operatorname{Ei}(-x)$ for $x > 0$.
With the [[Euler–Mascheroni constant|00C5P9JZBK]] $\gamma$, the series
$$\operatorname{Ei}(x) = \gamma + \ln \abs{x} + \sum_{k=1}^\infty \frac{x^k}{k \cdot k!}, \qquad E_1(x) = -\gamma - \ln x - \sum_{k=1}^\infty \frac{(-x)^k}{k \cdot k!}$$
hold for $x \ne 0$ and $x > 0$ respectively.

Example: as $x \to \infty$, $E_1(x) \sim \frac{e^{-x}}{x}$, and $E_1(x) = \Gamma(0, x)$ is the upper [[incomplete gamma function|YB7ZV0HG7K]] with $s = 0$.

Non-example of a valid argument: $E_1(0)$ is not defined, since $\int_0^1 \frac{e^{-t}}{t}\, dt \ge e^{-1} \int_0^1 \frac{dt}{t} = \infty$.

$E_1$ extends to a holomorphic function on $\C \setminus (-\infty, 0]$.
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<document source="https://tamnd.github.io/math2/node/1BKVNEHX96.md" tag="1BKVNEHX96" kind="definition">
# Differential inclusion

## Definition

Let $I \subseteq \R$ be an interval and let $\Omega \subseteq \R^n$ be an open set.
Let $F$ be a [[multivalued function|03CPD1HGF3]] that assigns to each $(t, x) \in I \times \Omega$ a nonempty subset $F(t, x)$ of $\R^n$.
The *differential inclusion* given by $F$ is the relation
$$x'(t) \in F(t, x(t)).$$
A *solution* on a subinterval $J \subseteq I$ is an [[absolutely continuous|ZT6F4C2YHK]] function $x \colon J \to \Omega$ such that $x'(t) \in F(t, x(t))$ for almost every $t \in J$.

## Remarks

If $F(t, x) = \set{f(t, x)}$ for a function $f$, the solutions are the solutions of the [[ordinary differential equation|0NT64JV96A]] $x' = f(t, x)$ in the sense of Carathéodory.
Example: for $F(t, x) = [-1, 1]$ on $\R \times \R$, the function $x(t) = \abs{t}$ is a solution, since it is absolutely continuous and $x'(t) = \pm 1$ for every $t \ne 0$.
Non-example: $x(t) = 2t$ is not a solution of the same inclusion, because $x'(t) = 2 \notin [-1, 1]$ at every $t$.
An equation with a discontinuous right side, such as $x' = -\sgn x$, is often replaced by the inclusion $x' \in -\operatorname{Sgn}(x)$, where $\operatorname{Sgn}(0) = [-1, 1]$; then $x \equiv 0$ is a solution.
Local existence of solutions holds when the values $F(t, x)$ are nonempty, compact and convex and $F$ satisfies suitable [[upper hemicontinuity|XNMKX7H4FD]] and measurability conditions.
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<document source="https://tamnd.github.io/math2/node/1BXEJXQ2YT.md" tag="1BXEJXQ2YT" kind="statement">
# Banach–Alaoglu theorem

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a [[topological vector space|T32W553W95]] over $\mathbb{K}$ with [[continuous dual|4A82S5FVRT]] $E'$.
Let $U$ be a [[neighbourhood|0ZCY0Z80B6]] of $0$ in $E$.
Then the polar
$$U^\circ = \set{f \in E' : \abs{f(x)} \le 1 \text{ for all } x \in U}$$
is [[compact|0R6EWFKRCF]] in the weak-\* topology $\sigma(E', E)$, the [[weak topology|P6KHVSTPZV]] on $E'$.
In particular, if $E$ is a normed space, the closed unit ball $\set{f \in E' : \norm{f} \le 1}$ is weak-\* compact.

## Remarks

Since $U$ is absorbing, each $x \in E$ lies in $r_x U$ for some $r_x > 0$, so $\abs{f(x)} \le r_x$ for $f \in U^\circ$.
Hence $U^\circ$ sits inside the product $P = \prod_{x \in E} \set{c \in \mathbb{K} : \abs{c} \le r_x}$, which is compact by [[Tychonoff's theorem|0JTRSJKCFB]], and the weak-\* topology is the topology induced by the product topology.
The functions in $P$ that are linear and bounded by $1$ on $U$ form a closed subset of $P$.
Each such function is continuous, since it is bounded on a neighbourhood of $0$.
So this closed subset is $U^\circ$, and it is compact.
Example: for $E = c_0$, the dual is $\ell^1$, and the closed unit ball of $\ell^1$ is weak-\* compact, although it is not compact in norm.
The weak-\* topology is needed: in an infinite-dimensional normed space $E$, the closed unit ball of $E'$ is never compact in the norm topology.
The theorem gives compactness, not sequential compactness.
If $E$ is a separable normed space, the closed unit ball of $E'$ is weak-\* metrizable, and every bounded sequence in $E'$ has a weak-\* convergent subsequence.

## Explanations

### L1 human

In infinite dimension, closed bounded sets are usually not compact, which makes it hard to find limits.
The theorem rescues compactness by weakening the topology.
On the dual of a normed space, the unit ball becomes compact once convergence means convergence at each vector separately.
This is often enough to extract a limit from a bounded family of functionals.

### L2 human

The Banach–Alaoglu theorem is the standard source of compactness in functional analysis.
It is used to prove existence results, for instance minimizers in the calculus of variations, weak solutions of partial differential equations, and invariant means and measures.
With [[Goldstine's theorem|K4S9G3AQBM]], it characterizes reflexive Banach spaces by the weak compactness of their unit ball.
With the [[Krein–Milman theorem|RSRAY0Q68S]], it shows that the dual unit ball has many extreme points.
In duality theory, it says that polars of neighbourhoods of $0$ are weak-\* compact, which is a key step of the [[Mackey–Arens theorem|RKKWS7CDNF]].
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<document source="https://tamnd.github.io/math2/node/1C28287S75.md" tag="1C28287S75" kind="concept">
# Knot concordance

## Summary

Two oriented [[knots|QYEAZTYGPT]] are concordant if they cobound a smoothly embedded annulus in $S^3 \times [0, 1]$.
Concordance is an equivalence relation, coarser than isotopy.
The concordance classes form an abelian group under [[knot sum|AGZSZW7GYE]], the knot concordance group.
Its zero element is the class of [[slice knots|P68G28E4AD]].

## Explanations

### L1 human

Think of time as a fourth direction, and let a knot move through time.
During the motion it may change shape in ways that are not allowed in ordinary space, as long as the trace it sweeps out is a smooth tube without self-crossings.
Two knots are concordant if one can be turned into the other in this way.
Under this coarser notion of sameness, every knot has an inverse: its reversed mirror image.
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<document source="https://tamnd.github.io/math2/node/1C2TEAKJHJ.md" tag="1C2TEAKJHJ" kind="concept">
# Parry measure

## Summary

The Parry measure is a special shift-[[invariant measure|B232EZ1FMB]] on an irreducible [[subshift of finite type|624XDXX8FN]].
It is the unique invariant probability measure of maximal entropy.
So its [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] equals the [[topological entropy|BDZYBXBFDE]] $\log \lambda$, where $\lambda$ is the Perron eigenvalue of the transition matrix.
It is a Markov measure built from the left and right Perron eigenvectors.
It shows the [[variational principle for entropy|HHR8HP3R6G]] at work in the simplest case.

## Explanations

### L1 human

A subshift of finite type allows some words and forbids others.
Among all ways to pick an allowed sequence at random, the Parry measure is the one that is as unpredictable as possible.
It does not give each symbol equal weight; it weights them by how many long allowed words pass through them.
For the golden mean shift, where $11$ is forbidden, it gives the symbol $0$ probability $\varphi^2/(\varphi^2 + 1)$, with $\varphi$ the golden ratio.
See the [[definition|BE3E2AH0EV]].
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<document source="https://tamnd.github.io/math2/node/1C7TKGTB2F.md" tag="1C7TKGTB2F" kind="concept">
# Graded poset

## Summary

A graded poset is a [[partially ordered set|0QV2JS9RJ3]] with a rank function that goes up by exactly one along each covering step.
The elements then fall into levels, like the floors of a building.
Many posets of combinatorics are graded: subsets by size, divisors by the number of prime factors, partitions by their size.

## Explanations

### L1 human

Picture the Hasse diagram of a poset drawn so that every edge goes up exactly one level.
If this is possible, the poset is graded and the level of an element is its rank.
The subsets of $\set{1, 2, 3}$ are graded by their number of elements: $\varnothing$ has rank $0$ and $\set{1, 2, 3}$ has rank $3$.
A pentagon, where one side from bottom to top has two steps and the other has three, cannot be drawn this way.
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<document source="https://tamnd.github.io/math2/node/1CA6GZG0R6.md" tag="1CA6GZG0R6" kind="concept">
# Tate cohomology

## Summary

Tate [[cohomology|YXJXA0Q52R]] joins the cohomology and the [[homology|CFAG3VRXEA]] of a finite group $G$ into one sequence of groups $\hat H^n(G, M)$ indexed by all integers $n$.
It agrees with ordinary cohomology in positive degrees and with homology in degrees at most $-2$.
In the two middle degrees it is modified with the norm map.
A short exact sequence of modules gives a long exact sequence infinite in both directions.
Tate cohomology is a basic tool of class field theory and of [[Galois cohomology|9MPCZ3EAJM]], and for cyclic groups it is periodic of period $2$.

## Explanations

### L1 human

For a finite group, cohomology runs upward from degree $0$ and homology runs downward.
Tate noticed that the two can be glued into a single sequence that runs in both directions, after a small correction at the junction.
The correction uses the norm, the sum of all the group elements acting on a module.
Let a cyclic group of order $m$ act on $\Z$ with every element fixing every integer.
Then the groups alternate between $\Z/m\Z$ and $0$.
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<document source="https://tamnd.github.io/math2/node/1CBZA8MQJW.md" tag="1CBZA8MQJW" kind="concept">
# Ramanujan theta function

## Summary

Ramanujan's general theta function is $f(a, b) = \sum_{n \in \Z} a^{n(n+1)/2} b^{n(n-1)/2}$ for complex $a, b$ with $\abs{ab} < 1$.
It is a two-variable form of the Jacobi [[theta function|2F5CYQJVAF]], symmetric in $a$ and $b$.
By the [[Jacobi triple product|JFP2DGMNXT]] it is an infinite product.
Its special cases $\varphi(q)$, $\psi(q)$ and $f(-q)$ are the theta functions that Ramanujan used for partitions and modular equations.

## Explanations

### L1 human

The function $f(a, b)$ is a sum over all integers $n$, where positive $n$ mostly use powers of $a$ and negative $n$ mostly use powers of $b$.
Choosing $a$ and $b$ as powers of one variable $q$ gives famous series.
For example, $f(q, q) = \sum_{n \in \Z} q^{n^2} = 1 + 2q + 2q^4 + 2q^9 + \cdots$, where each square $n^2 > 0$ appears twice, once for $n$ and once for $-n$.
Its square counts the ways to write a number as a sum of two squares.
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<document source="https://tamnd.github.io/math2/node/1CCTH8X2FT.md" tag="1CCTH8X2FT" kind="concept">
# Parabolic partial differential equation

## Summary

A parabolic partial differential equation is an evolution equation of the form $\partial_t u + L u = f$, where $L$ is an elliptic operator in the space variables.
The model case is the [[heat equation|VP2HN86VAG]].
Parabolic equations describe diffusion: they smooth out the initial data, propagate information at infinite speed, and are well posed forward in time but not backward.
See the [[definition|WJGY7XG3QR]].

## Explanations

### L1 human

Drop some ink in still water: it spreads out, its sharp edges blur, and after a while the colour is nearly uniform.
A parabolic equation describes such a process.
The [[heat equation|VP2HN86VAG]] $\partial_t u = \partial_x^2 u$ says that the temperature rises where the graph of $u$ curves upward, that is, where the point is colder than its neighbours.
So hot spots cool down and cold spots warm up.
Running the process backward in time, from a blurred state to a sharp one, is unstable.
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<document source="https://tamnd.github.io/math2/node/1CF72TCVD2.md" tag="1CF72TCVD2" kind="definition">
# Graph of an operator

## Definition

Let $X$ and $Y$ be normed spaces, and let $T$ be an [[unbounded operator|KRJX5RA0PW]] from $X$ to $Y$ with domain $D(T)$.
The *graph* of $T$ is the set
$$\Gamma(T) = \set{(x, Tx) : x \in D(T)} \subseteq X \times Y.$$
The product $X \times Y$ carries the norm $\norm{(x, y)} = \norm{x} + \norm{y}$, or any equivalent norm.

## Remarks

It is the [[graph|0JBJYT69ZV]] of the function $T \colon D(T) \to Y$.
Since $T$ is linear, $\Gamma(T)$ is a [[linear subspace|00YC1D652K]] of $X \times Y$.

Example: for $X = Y = \R$ and $Tx = 3x$ with $D(T) = \R$, the graph is the line $\set{(x, 3x) : x \in \R}$.

Non-example: the subspace $\set{(0, y) : y \in \R}$ of $\R \times \R$ is not the graph of any operator, since it contains $(0, 1)$, while every graph contains only $(0, 0)$ above $x = 0$.

A linear subspace $G$ of $X \times Y$ is the graph of an operator if and only if $(0, y) \in G$ implies $y = 0$.
Then $D(T)$ is the projection of $G$ onto $X$, and $Tx$ is the unique $y$ with $(x, y) \in G$.
In this way $S \subseteq T$ holds exactly when $\Gamma(S) \subseteq \Gamma(T)$.
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<document source="https://tamnd.github.io/math2/node/1CPF7SWQ3S.md" tag="1CPF7SWQ3S" kind="concept">
# Wilson polynomials

## Summary

The Wilson polynomials $W_n(x^2; a, b, c, d)$ are orthogonal polynomials in the variable $x^2$ with four parameters, introduced by James Wilson in 1980.
They are given by terminating balanced ${}_4F_3$ hypergeometric series, and their weight on $(0, \infty)$ is a ratio of products of [[gamma functions|0H5SYBSED9]].
They are at the top of the [[Askey scheme|RR2S05C09B]].
Every other family in the scheme arises from them, or from the [[Racah polynomials|P6JYNDX5DR]], by specialisation or limits.
Their $q$-analogues are the [[Askey–Wilson polynomials|RSNZNG0WA1]].

## Explanations

### L1 human

The classical orthogonal polynomials, such as those of Jacobi, Laguerre and Hermite, come in a few families with few parameters.
The Wilson polynomials are one family with four parameters from which all the others can be reached.
Setting a parameter to a special value, or letting it go to infinity, moves down from the Wilson polynomials to the more familiar ones.
They are symmetric in their four parameters, which is not visible from the formula but is a deep identity.
</document>

<document source="https://tamnd.github.io/math2/node/1CPW6AG86Y.md" tag="1CPW6AG86Y" kind="definition">
# Homogeneous space

## Definition

Let $k$ be an algebraically closed field, and let $G$ be an [[algebraic group|GH81SYAPPN]] over $k$.
A *homogeneous space* for $G$ is a nonempty variety $X$ over $k$ with an [[action|RW85C34GQW]] of $G$ such that $G(k)$ acts [[transitively|02GK6HZYBT]] on $X(k)$.

## Remarks

Let $x \in X(k)$ and let $G_x$ be its [[stabilizer|0AYMP41THN]], a closed subgroup of $G$.
The orbit map $g \mapsto g \cdot x$ induces a bijective morphism $G / G_x \to X$.
In characteristic $0$ this morphism is an isomorphism, so $X \cong G / G_x$.
In characteristic $p$ it can fail to be an isomorphism, because the orbit map need not be separable.

Example: $\mathrm{GL}_{n+1}$ acts on $\mathbb{P}^n$ by $g \cdot [v] = [g v]$.
Any nonzero vector can be sent to any other by an invertible matrix, so the action is transitive and $\mathbb{P}^n$ is a homogeneous space.

Non-example: $\mathbb{G}_m$ acts on $\mathbb{A}^1$ by $t \cdot x = t x$.
There are two orbits, $\set{0}$ and $\mathbb{A}^1 \setminus \set{0}$, so the action is not transitive.

In other settings, such as Lie groups and topological groups, a homogeneous space is a manifold or a space with a transitive action of the group.
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<document source="https://tamnd.github.io/math2/node/1CRA2F3TY7.md" tag="1CRA2F3TY7" kind="definition">
# Kernel of an integral equation

## Definition

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ and $f \colon [a, b] \to \C$ be given [[functions|0EPAT51DXQ]], and let $\lambda \in \C$.
Consider an [[integral equation|R3SG1SN7JQ]] of one of the forms
$$\int_a^b K(x, s) \varphi(s) \, ds = f(x), \qquad \varphi(x) - \lambda \int_a^b K(x, s) \varphi(s) \, ds = f(x),$$
or the same forms with the upper limit $b$ replaced by $x$.
The function $K$ is the *kernel* of the equation.
The map $\varphi \mapsto \int_a^b K(\cdot, s) \varphi(s) \, ds$ is the *integral operator* with kernel $K$.

## Remarks

Example: in $\varphi(x) - \int_0^1 e^{x - s} \varphi(s) \, ds = x$ the kernel is $K(x, s) = e^{x - s}$ and $\lambda = 1$.

Non-example: in $\varphi(x) = x + \int_0^1 \varphi(s)^2 \, ds$ the unknown does not enter linearly under the integral, so the equation has no kernel in this sense.
For a nonlinear equation $\varphi(x) = f(x) + \int_a^b H(x, s, \varphi(s)) \, ds$, the function $H$ of three variables is sometimes also called the kernel.

For a Volterra equation, with upper limit $x$, only the values of $K$ with $s \le x$ matter, and one may set $K(x, s) = 0$ for $s > x$.
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<document source="https://tamnd.github.io/math2/node/1CS00CEAEH.md" tag="1CS00CEAEH" kind="definition">
# Algebraic curve

## Definition

Let $k$ be a field.
An *algebraic curve* over $k$ is an [[algebraic variety|HV6YBS0YDH]] over $k$ of [[dimension|36X2WTHYCB]] $1$.

## Remarks

Conventions differ on whether a curve must be irreducible, reduced, projective or smooth; each text states its choice.
For an integral variety $X$ over $k$, the dimension equals the transcendence degree of its function field over $k$.
So $X$ is a curve exactly when $k(X)$ has transcendence degree $1$.
Example: the plane curve $X \colon y^2 = x^3 - x$ in $\mathbb{A}^2_k$ is a curve.
Its function field $k(x)[y]/(y^2 - x^3 + x)$ is a finite extension of $k(x)$, so it has transcendence degree $1$.
Non-example: the affine plane $\mathbb{A}^2$ has dimension $2$, and a finite set of points has dimension $0$, so neither is a curve.
Let $k$ be algebraically closed.
Then each finitely generated field extension of $k$ of transcendence degree $1$ is the function field of exactly one smooth projective curve, up to isomorphism.
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<document source="https://tamnd.github.io/math2/node/1CW7Q8QH3N.md" tag="1CW7Q8QH3N" kind="statement">
# Heaviside expansion theorem

## Statement

Let $P$ and $Q$ be polynomials with complex coefficients with $\deg P < \deg Q$.
Suppose that $Q$ has $n = \deg Q$ distinct roots $a_1, \dots, a_n$.
Then $P/Q$ is the [[Laplace transform|0V49JEZW06]], for $\operatorname{Re} s > \max_k \operatorname{Re} a_k$, of the function
$$f(t) = \sum_{k=1}^n \frac{P(a_k)}{Q'(a_k)} \, e^{a_k t}, \qquad t \ge 0.$$

## Remarks

Since the roots are simple, the [[partial fraction decomposition|030VFEREKG]] of $P/Q$ is $\sum_k c_k / (s - a_k)$.
Multiplying by $s - a_k$ and letting $s \to a_k$ gives $c_k = P(a_k) / Q'(a_k)$, because $Q(s)/(s - a_k) \to Q'(a_k)$.
Each term $c_k/(s - a_k)$ is the Laplace transform of $c_k e^{a_k t}$ on $\operatorname{Re} s > \operatorname{Re} a_k$.

Example: $P = 1$, $Q(s) = s^2 + 1$, with roots $\pm i$ and $Q'(s) = 2s$.
The theorem gives $f(t) = \frac{e^{it}}{2i} + \frac{e^{-it}}{-2i} = \sin t$, the [[inverse Laplace transform|07WBTX1MHA]] of $1/(s^2 + 1)$.

In Heaviside's form, the response to a unit step is $\varphi(p) H$ with $\varphi = P/Q$, $\deg P \le \deg Q$ and $Q(0) \ne 0$.
If the roots $a_k$ of $Q$ are simple, the theorem applied to $P(s)/(s Q(s))$ gives, for $t \ge 0$,
$$\varphi(p) H = \frac{P(0)}{Q(0)} + \sum_k \frac{P(a_k)}{a_k Q'(a_k)} e^{a_k t}.$$

The hypothesis of simple roots is needed: $1/s^2$ is the transform of $t$, which is not a combination of exponentials.
For repeated roots, the residue formula gives terms $t^j e^{a_k t}$ instead.

## Explanations

### L1 human

A rational function of $s$ breaks into simple pieces $c/(s - a)$, and each piece is the transform of an exponential $c\,e^{at}$.
The theorem gives a direct recipe for the numbers $c$, without solving a system of equations.
So the inverse transform of a rational function with simple poles can be written down at once.

### L2 human

The coefficient $P(a_k)/Q'(a_k)$ is the residue of $P/Q$ at $a_k$, and the theorem is the special case of the residue form of the inverse Laplace transform, $f(t) = \sum_k \operatorname{Res}_{s = a_k} \left( e^{st} P(s)/Q(s) \right)$.
It is the main tool for solving linear constant-coefficient systems by the [[Heaviside operational calculus|QQ98C2AXZP]] or by the Laplace transform, because their transfer functions are rational.
The exponents $a_k$ are the natural frequencies of the system, and the theorem expresses the response as a superposition of its modes.
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<document source="https://tamnd.github.io/math2/node/1D541B576S.md" tag="1D541B576S" kind="conjecture">
# Park's conjecture

## Statement

Let $A$ be a finite algebra of finite signature, and let $V(A)$ be the [[variety|AJNZJD8442]] generated by $A$.
Suppose that $V(A)$ has a finite residual bound.
This means that there is a natural number $n$ such that every [[subdirectly irreducible algebra|DASF939439]] in $V(A)$ has fewer than $n$ elements.
Then $A$ is [[finitely based|YGYZAQ12MZ]].

## Remarks

Park stated the conjecture in his doctoral dissertation at UCLA in 1976.
It is open in general.
It holds when $V(A)$ is [[congruence-distributive|9GZ7F6ETC6]], by [[Baker's finite basis theorem|35G4HQG2FN]], which in that case needs no residual bound.
McKenzie proved it in 1987 when $V(A)$ is [[congruence-modular|QAQAS3JWXW]].
Willard proved it in 2000 when $V(A)$ is congruence meet-semidistributive.
Kearnes, Szendrei and Willard extended it to varieties with a difference term.
A variety with a finite residual bound is in particular [[residually small|TH2XQGGFWH]].

## Explanations

### L1 human

Every algebra in the variety of $A$ is built from "irreducible" pieces.
Park's conjecture says that if these pieces are all small, below a fixed size, then finitely many laws are enough to describe everything true in $A$.
All results so far agree with the conjecture, but nobody has proved it in general.
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# Function having a given Mellin transform

## Summary

A function $f$ on $(0, \infty)$ has Mellin transform $m$ at a point $s$ when the integral $\int_0^\infty x^{s-1} f(x) \, dx$ converges absolutely and its value is $m$.
The notion bundles the [[Mellin convergence|SGZZ4Y0D36]] of $f$ at $s$ with the value of the [[Mellin transform|N5PVGRJE1T]].
It is convenient in proofs, because a formula such as $\mathcal{M}f(s) = \Gamma(s)$ then carries its convergence claim with it.

## Explanations

### L1 human

Writing $\mathcal{M}f(s) = m$ quietly assumes that the integral makes sense.
Saying that $f$ has Mellin transform $m$ at $s$ makes this explicit: the integral converges absolutely, and it equals $m$.
For example, $e^{-x}$ has Mellin transform $\Gamma(2) = 1$ at $s = 2$, since $\int_0^\infty x e^{-x} \, dx = 1$.
The constant function $1$ has no Mellin transform at any $s$, so it has Mellin transform $m$ for no pair $(s, m)$.
</document>

<document source="https://tamnd.github.io/math2/node/1D99CSKX6N.md" tag="1D99CSKX6N" kind="statement">
# Dirichlet's unit theorem

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\mathcal{O}_K$.
Let $r_1$ be the number of field embeddings $K \to \R$ and $r_2$ the number of pairs of complex conjugate non-real embeddings $K \to \C$.
Let $\mu(K)$ be the set of [[roots of unity|0YV2CKP2AV]] in $K$.
Then $\mu(K)$ is a finite cyclic group, and the [[group of units|07C6ECNJET]] $\mathcal{O}_K^\times$ is finitely generated with
$$\mathcal{O}_K^\times \cong \mu(K) \times \Z^{r_1 + r_2 - 1}.$$

## Remarks

The proof uses the logarithmic embedding $u \mapsto \big(N_j \log \abs{\sigma_j(u)}\big)_j$ into $\R^{r_1 + r_2}$, where $\sigma_j$ runs over the real embeddings and one embedding from each complex pair, and $N_j$ is $1$ or $2$.
Its kernel on $\mathcal{O}_K^\times$ is the finite group $\mu(K)$, and its image lies in the hyperplane where the coordinates sum to $0$.
The hard part is that the image is a full [[lattice|WN9V7WSJKF]] in that hyperplane; this uses [[Minkowski's theorem|0K387S2PKQ]] to produce enough units.

A set of $r = r_1 + r_2 - 1$ units that generate $\mathcal{O}_K^\times$ modulo $\mu(K)$ is a system of *fundamental units*.

Example: for $K = \Q(\sqrt{2})$, $r_1 = 2$ and $r_2 = 0$, so the rank is $1$.
The units of $\Z[\sqrt{2}]$ are exactly $\pm (1 + \sqrt{2})^k$ with $k \in \Z$.
For an imaginary quadratic field, $r_1 = 0$ and $r_2 = 1$, so the rank is $0$ and every unit is a root of unity; for $\Q(i)$ the units are $\pm 1, \pm i$.

## Explanations

### L1 human

The units of a number field are the algebraic integers whose inverse is also an algebraic integer.
The theorem says that all of them are built from a finite list of basic units by multiplication, up to roots of unity.
The number of basic units is given by counting the real and complex embeddings of the field.

### L2 human

The theorem is one of the two finiteness results at the start of algebraic number theory, with the finiteness of the class number.
In terms of [[free abelian groups|0WJX8MGG37]], $\mathcal{O}_K^\times / \mu(K)$ is free of rank $r_1 + r_2 - 1$.
The covolume of the unit lattice is the [[regulator|RWK0HNF1NA]], which enters the [[class number formula|55XX5A7XZV]].
For real quadratic fields, the fundamental unit gives the solutions of Pell's equation.
</document>

<document source="https://tamnd.github.io/math2/node/1DBC9FYF7M.md" tag="1DBC9FYF7M" kind="concept">
# Martin kernel

## Summary

The Martin kernel of a domain is the [[Green's function|M36SD56MWQ]] normalised by its value at a fixed reference point: $M(x, y) = G(x, y) / G(x_0, y)$.
As the pole $y$ runs to the boundary, the functions $M(\cdot, y)$ accumulate at positive harmonic functions.
These limits are the points of the Martin boundary.
For a ball, the limits are the Poisson kernels, so the Martin kernel generalises the [[Poisson kernel|TCYSWCTYYM]] to arbitrary domains.

## Explanations

### L1 human

Put a unit charge at a point $y$ of a grounded domain and measure the potential at $x$.
Divide by the potential at a fixed point $x_0$, so that the value at $x_0$ is always $1$.
This ratio is the Martin kernel.
When $y$ approaches the boundary the potential itself tends to zero, but the ratio usually tends to a nonzero positive harmonic function.
For the unit ball these limits are the Poisson kernels, which describe how boundary values spread inside.
</document>

<document source="https://tamnd.github.io/math2/node/1DDYB6VVSR.md" tag="1DDYB6VVSR" kind="object">
# Real projective space

## Description

Let $n \ge 1$ be an integer.
The *real projective space* $\mathbb{RP}^n$ is the set of lines through the origin in $\R^{n+1}$.
As a topological space it is the [[quotient|0NQ1H2X495]] of the [[$n$-sphere|T62369RFH6]] $S^n$ by the relation that identifies each point $x$ with its antipode $-x$.
It is a compact [[topological manifold|QQQ9B1DK4P]] of dimension $n$.
For $i = 1, \ldots, n+1$, let $U_i$ be the image of $\set{x \in S^n : x_i \ne 0}$.
The map $U_i \to \R^n$ sending the class of $x$ to the $n$ ratios $x_j / x_i$ with $j \ne i$ is a [[homeomorphism|0QT00PB5G2]], and the $n + 1$ sets $U_i$ cover $\mathbb{RP}^n$.

## Remarks

$\mathbb{RP}^1$ is homeomorphic to the circle $S^1$.
$\mathbb{RP}^2$ is the [[real projective plane|MWDB7RACWQ]], a closed surface that cannot be embedded in $\R^3$.
$\mathbb{RP}^n$ is [[orientable|702SNFQG0W]] if and only if $n$ is odd.
For $n \ge 2$ the quotient map $S^n \to \mathbb{RP}^n$ is the universal covering, so the fundamental group of $\mathbb{RP}^n$ has order $2$.
$\mathbb{RP}^n$ is a CW complex with one cell in each dimension $0, 1, \ldots, n$.

## Explanations

### L1 human

Take a sphere and declare each point to be the same as the opposite point.
The space of these pairs of opposite points is the real projective space.
Equivalently, each point of it is a line through the centre of the sphere.
In dimension one this gives a circle again: half of the circle, with its two end points glued.
</document>

<document source="https://tamnd.github.io/math2/node/1DFG72YV3N.md" tag="1DFG72YV3N" kind="definition">
# Nonlinear Volterra equation

## Definition

Let $a < b$, let $\Delta = \set{(t, s) : a \le s \le t \le b}$, let $K \colon \Delta \times \R \to \R$ and $f \colon [a, b] \to \R$ be [[continuous|0X3WGX95FE]].
A *nonlinear Volterra equation of the second kind* for an unknown continuous function $\varphi$ is
$$\varphi(t) = f(t) + \int_a^t K\bigl(t, s, \varphi(s)\bigr)\, ds, \quad a \le t \le b.$$
A *nonlinear Volterra equation of the first kind* is $\int_a^t K(t, s, \varphi(s))\, ds = f(t)$.
A solution on a subinterval $[a, c]$ with $c \le b$ is a continuous function on $[a, c]$ that satisfies the equation there.

## Remarks

If $K(t, s, u) = k(t, s)\, u$, this is the linear [[Volterra integral equation|DND8CBEVXT]] of the second kind.
If $K$ satisfies a [[Lipschitz condition|0AW1E6YTJY]] $\abs{K(t, s, u) - K(t, s, v)} \le L \abs{u - v}$ on all of $\Delta \times \R$, the equation has exactly one continuous solution on $[a, b]$, and the successive approximations converge to it.
If the Lipschitz condition holds only for $u, v$ in bounded sets, there is still a unique solution on some interval $[a, c]$ with $c > a$.

Example: $\varphi(t) = 1 + \int_0^t \varphi(s)^2\, ds$ on $[0, 1)$, with $K(t, s, u) = u^2$.
The function $\varphi(t) = 1/(1 - t)$ solves it, since $1 + \int_0^t (1 - s)^{-2}\, ds = 1 + \frac{1}{1 - t} - 1 = \frac{1}{1 - t}$.
It is the integral form of $y' = y^2$, $y(0) = 1$, and the solution does not extend past $t = 1$.

Non-example: $\varphi(t) = f(t) + \int_a^b K(t, s, \varphi(s))\, ds$ has the fixed upper limit $b$; it is a Urysohn equation, not a Volterra one.
</document>

<document source="https://tamnd.github.io/math2/node/1DHZ81S0YE.md" tag="1DHZ81S0YE" kind="statement">
# Krylov–Bogolyubov theorem

## Statement

Let $X$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]], and let $T \colon X \to X$ be a [[continuous map|0X3WGX95FE]].
Then there is a Borel [[probability measure|059TJN0NZ1]] $\mu$ on $X$ that is [[$T$-invariant|B232EZ1FMB]], that is, $\mu(T^{-1}(A)) = \mu(A)$ for every Borel set $A \subseteq X$.

## Remarks

Fix $x \in X$ and let $\mu_n = \frac{1}{n} \sum_{k=0}^{n-1} \delta_{T^k(x)}$, where $\delta_y$ is the point mass at $y$.
Since $X$ is compact, the [[Borel probability measures|DYR1QFNGNJ]] on $X$ form a compact set for [[weak convergence|MBW9JCGZC1]], for instance by [[Prokhorov's theorem|D6FMEQYA76]].
So some subsequence $\mu_{n_j}$ converges weakly to a probability measure $\mu$.
For continuous $g \colon X \to \R$ one has $\int g \circ T \, d\mu_n - \int g \, d\mu_n = \frac{1}{n}\left(g(T^n(x)) - g(x)\right)$, which tends to $0$.
Passing to the limit gives $\int g \circ T \, d\mu = \int g \, d\mu$ for all continuous $g$, and this forces $T_* \mu = \mu$.
The same argument gives an invariant probability measure for a continuous flow on a compact metric space.

Compactness is needed: the translation $T(x) = x + 1$ on $\R$ has no invariant probability measure.
Indeed, invariance would give $\mu([k, k+1)) = \mu([0, 1))$ for every $k \in \Z$, and these countably many disjoint sets cover $\R$, so their measures cannot sum to $1$.

Example: for the rotation $x \mapsto x + \alpha \bmod 1$ of the circle, the theorem produces an invariant measure; when $\alpha$ is irrational, it is Lebesgue measure, which is the only invariant one.

## Explanations

### L1 human

Any continuous motion of a compact space has a way of spreading out mass that the motion leaves unchanged.
One finds it by following one orbit for a long time and recording how often it visits each region.
These visit frequencies settle down, at least along a subsequence, to an invariant measure.

### L2 human

The theorem makes ergodic theory available for every continuous map of a compact metric space: there is always at least one invariant probability measure to work with.
The set of invariant probability measures is a nonempty compact convex set, and its extreme points are the ergodic measures, so ergodic invariant measures exist as well.
When the invariant measure is unique, the map is called uniquely ergodic, and then Birkhoff averages of continuous functions converge uniformly.
The variational principle for topological entropy is a supremum over this set of invariant measures.
</document>

<document source="https://tamnd.github.io/math2/node/1DKCNS63DH.md" tag="1DKCNS63DH" kind="counterexample">
# Long line

## Counterexample

Let $\omega_1$ be the first uncountable [[ordinal|0YSCSX5YZC]].
The closed long ray is the set $L = \omega_1 \times [0, 1)$ with the [[lexicographic order|0B68KNPGCG]] and the [[order topology|RCWHDMWP1T]].
The long line is obtained by gluing two copies of $L$ along their least point $(0, 0)$, one of them with the reversed order.
The closed long ray $L$ is Hausdorff, [[connected|0V032VWTHD]] and [[sequentially compact|0FRMJH3CY7]], but it is not [[compact|0TVJZYB6SH]] and not [[metrizable|091GJ3FYD7]].

## Remarks

For each countable ordinal $\beta \ge 1$, the initial segment of $L$ below $(\beta, 0)$ is a countable stack of half-open intervals, and it is homeomorphic to $[0, 1)$.
So every point of $L$ other than $(0, 0)$ has a neighbourhood homeomorphic to an open interval of $\R$.
A sequence $(\alpha_n, t_n)$ in $L$ has a countable supremum $\gamma$ of the $\alpha_n$, because a countable union of countable sets is countable.
So the sequence lies in the set of points up to $(\gamma + 1, 0)$, which is homeomorphic to $[0, 1]$, and it has a convergent subsequence.
The open sets $\set{p \in L : p < (\alpha, 0)}$ for $\alpha < \omega_1$ cover $L$, but no finitely many of them do, because $\omega_1$ has no largest element.
A sequentially compact metrizable space is compact, so $L$ is not metrizable.
For the same reason the long line is a connected Hausdorff space, locally homeomorphic to $\R$, that is not second-countable and not metrizable.
So being Hausdorff and locally Euclidean does not make a space a [[topological manifold|QQQ9B1DK4P]] in the usual sense, which also asks for second countability.

## Explanations

### L1 human

The long line is made by laying uncountably many copies of the interval $[0, 1)$ end to end.
Near each of its points it looks just like the real line.
But it is so long that no sequence can climb to its end.
It shows that "looks like a line near every point" does not give a space that can be measured by a distance.
</document>

<document source="https://tamnd.github.io/math2/node/1DKFWGWHH5.md" tag="1DKFWGWHH5" kind="concept">
# Ind-completion

## Summary

The Ind-completion $\mathrm{Ind}(\mathcal{C})$ of a category $\mathcal{C}$ is the category of formal [[filtered|975PMY4FG8]] colimits of objects of $\mathcal{C}$.
Its objects are called ind-objects.
It is the free way to add filtered colimits to $\mathcal{C}$, and it can be realized inside the category of [[presheaves|AND2WHZC8P]] on $\mathcal{C}$.
For example, the Ind-completion of finite sets is the category of all sets.

## Explanations

### L1 human

Some objects are best seen as limits of growing chains of simpler objects.
An infinite set is the union of its finite subsets, and an infinite-dimensional vector space is the union of its finite-dimensional subspaces.
The Ind-completion adds such unions formally, even when the original category does not contain them.
So starting from finite-dimensional vector spaces, the Ind-completion gives all vector spaces.
</document>

<document source="https://tamnd.github.io/math2/node/1DRW7JP962.md" tag="1DRW7JP962" kind="definition">
# Weak topology

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $(X, Y, \langle \cdot, \cdot \rangle)$ be a [[dual pair|R68HNB5SVR]] over $\mathbb{K}$.
The *weak topology* $\sigma(X, Y)$ on $X$ is the [[topology generated by the family of seminorms|ZVFXQ7S6FY]] $p_y(x) = \abs{\langle x, y \rangle}$, $y \in Y$.
Let $E$ be a Hausdorff locally convex space with [[continuous dual|4A82S5FVRT]] $E'$, paired by $\langle x, f \rangle = f(x)$.
The *weak topology* of $E$ is $\sigma(E, E')$, and the *weak-\* topology* of $E'$ is $\sigma(E', E)$.

## Remarks

$\sigma(X, Y)$ is the coarsest topology on $X$ for which every map $x \mapsto \langle x, y \rangle$, $y \in Y$, is continuous; it is a Hausdorff locally convex topology.
A net $(x_i)$ converges to $x$ in $\sigma(X, Y)$ if and only if $\langle x_i, y \rangle \to \langle x, y \rangle$ for every $y \in Y$.
The continuous linear forms on $(X, \sigma(X, Y))$ are exactly the maps $x \mapsto \langle x, y \rangle$ with $y \in Y$.
Example: in $\ell^2$, the unit vectors $e_n$ converge to $0$ in $\sigma(\ell^2, \ell^2)$, since $\langle e_n, y \rangle = y_n \to 0$ for each $y \in \ell^2$.
Non-example: on $\ell^2$, the norm topology makes every map $x \mapsto \langle x, y \rangle$ continuous.
But it is not the coarsest such topology, so it is not the weak topology: the sequence $(e_n)$ converges weakly but not in norm.
</document>

<document source="https://tamnd.github.io/math2/node/1DSZV8QRS6.md" tag="1DSZV8QRS6" kind="definition">
# Translation surface

## Definition

A *translation surface* is a pair $(X, \omega)$ with the following data.
$X$ is a compact connected [[Riemann surface|5K0N8QMV6Q]].
$\omega$ is a holomorphic 1-form on $X$, an [[abelian differential|P014C13K5H]], that is not identically zero.
Two translation surfaces $(X, \omega)$ and $(X', \omega')$ are the same if there is a biholomorphic map $\phi \colon X \to X'$ with $\phi^* \omega' = \omega$.

## Remarks

Let $\Sigma$ be the finite set of zeros of $\omega$.
Near a point $p \notin \Sigma$, the function $z(q) = \int_p^q \omega$ is a local coordinate in which $\omega = dz$, and two such coordinates differ by a translation $z \mapsto z + c$.
So $X \setminus \Sigma$ gets a flat metric $\abs{\omega}$, and a zero of order $k$ is a cone point of angle $2\pi(k + 1)$.
If $X$ has [[genus|NJ787GZS1N]] $g$ and the zeros have orders $k_1, \dots, k_n$, then $k_1 + \cdots + k_n = 2g - 2$.

Equivalently, a translation surface is given by finitely many polygons in $\C$ whose sides are glued in pairs by translations.
Each side is glued to a parallel side of the same length with opposite outward normal.
The form $\omega$ is $dz$ on each polygon.

Example: the unit square with opposite sides glued is the [[torus|03NM7DZXXC]] $X = \C / (\Z + i\Z)$ with $\omega = dz$.
The form $dz$ is invariant under the translations by $\Z + i \Z$, so it descends to $X$, and it has no zeros, in line with $2g - 2 = 0$.

Example: the regular octagon with opposite sides glued has all eight vertices glued to one point, with total angle $8 \cdot \frac{3\pi}{4} = 6\pi$.
So $\omega$ has one zero of order $2$, and $2 = 2g - 2$ gives genus $g = 2$.

Non-example: $(X, 0)$ is not a translation surface, since the zero form gives no charts.
</document>

<document source="https://tamnd.github.io/math2/node/1DVRN1T2Z0.md" tag="1DVRN1T2Z0" kind="definition">
# Ehrenfeucht–Fraïssé game

## Definition

Let $L$ be a relational [[signature|WDVVEYD6AT]], that is, a signature with relation symbols and constant symbols but no function symbols.
Let $\mathcal{A}$ and $\mathcal{B}$ be $L$-[[structures|983GZBQYT4]] with domains $A$ and $B$, and let $n \ge 0$ be an integer.
The Ehrenfeucht–Fraïssé game $G_n(\mathcal{A}, \mathcal{B})$ is played by two players, Spoiler and Duplicator, in $n$ rounds.

1. In round $i$, Spoiler chooses one of the two structures and an element of it.
2. Duplicator then chooses an element of the other structure.
3. After round $i$, let $a_i \in A$ and $b_i \in B$ be the two elements chosen in that round.

Consider the correspondence $c^{\mathcal{A}} \mapsto c^{\mathcal{B}}$ for the constant symbols $c$, together with $a_i \mapsto b_i$ for $i = 1, \dots, n$.
Duplicator wins the play when it is a well-defined injective map that preserves and reflects each relation of $L$.
That is, it is an isomorphism between the [[substructures|676H7EGTKJ]] generated by the chosen elements and the constants.
Otherwise Spoiler wins.
Duplicator has a winning strategy when Duplicator has a rule for choosing moves that wins every play, whatever Spoiler does.

## Remarks

Example: let $\mathcal{A} = (\set{1, 2, 3}, <)$ and $\mathcal{B} = (\set{1, 2, 3, 4}, <)$.
Duplicator has a winning strategy in $G_2(\mathcal{A}, \mathcal{B})$.
In the first round, Duplicator answers the least element by the least, the greatest by the greatest, and an inner element by an inner element.
In the second round, an element below, equal to or above the first chosen element can then be matched on the other side.

Non-example: Duplicator has no winning strategy in $G_2((\Q, <), (\set{1, 2}, <))$.
Spoiler first picks $2$ in $\set{1, 2}$, and Duplicator answers some $q \in \Q$.
Spoiler then picks $q + 1$ in $\Q$, and no element of $\set{1, 2}$ lies above $2$.

Let $L$ be finite.
The theorem of Ehrenfeucht and Fraïssé says: Duplicator wins $G_n(\mathcal{A}, \mathcal{B})$ if and only if $\mathcal{A}$ and $\mathcal{B}$ satisfy the same sentences of quantifier depth at most $n$.
Hence $\mathcal{A} \equiv \mathcal{B}$ when Duplicator wins $G_n$ for every $n$.

## Explanations

### L1 human

The game turns the question "can first-order logic tell these two structures apart?" into a contest.
Each round of the game matches one quantifier of a sentence.
A winning strategy for Duplicator in $n$ rounds proves that no sentence with $n$ nested quantifiers separates the structures.
</document>

<document source="https://tamnd.github.io/math2/node/1DZMT4S9FB.md" tag="1DZMT4S9FB" kind="concept">
# Fiber product of schemes

## Summary

The fiber product $X \times_S Y$ of two [[morphisms of schemes|JXRKH6P1Z4]] $X \to S$ and $Y \to S$ is their [[pullback|Q05K04QJ71]] in the category of schemes.
It always exists, and for affine schemes it is the spectrum of a tensor product of rings.
Fiber products give products of varieties, fibers of morphisms, base change to another field, and the diagonal used to define separatedness.

## Explanations

### L1 human

For sets, the fiber product of $f \colon X \to S$ and $g \colon Y \to S$ is the set of pairs $(x, y)$ with $f(x) = g(y)$.
For schemes one cannot simply take pairs of points, because points of schemes carry more structure.
Instead the fiber product is defined by what maps into it look like.
In the affine case it is concrete: $\operatorname{Spec} B \times_{\operatorname{Spec} A} \operatorname{Spec} C = \operatorname{Spec}(B \otimes_A C)$.
For example, $\mathbb{A}^1_k \times_k \mathbb{A}^1_k = \operatorname{Spec}(k[x] \otimes_k k[y]) = \operatorname{Spec} k[x, y]$ is the plane.
</document>

<document source="https://tamnd.github.io/math2/node/1E8QNK0Q92.md" tag="1E8QNK0Q92" kind="statement">
# Bernstein–von Mises theorem

## Statement

Let $\set{P_\theta : \theta \in \Theta}$ be a [[parametric model|Q0DXKAQXE1]] with $\Theta \subseteq \R^k$ open, and let $X_1, X_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] with law $P_{\theta_0}$ for some $\theta_0 \in \Theta$.
Assume the following.

1. The model is differentiable in quadratic mean at $\theta_0$, with nonsingular [[Fisher information|G1Z3WSTE8Q]] matrix $I(\theta_0)$.
2. For every $\varepsilon > 0$ there are tests $\varphi_n = \varphi_n(X_1, \dots, X_n)$ with $E_{\theta_0}[\varphi_n] \to 0$ and $\sup_{\norm{\theta - \theta_0} \ge \varepsilon} E_\theta[1 - \varphi_n] \to 0$.
3. The [[prior|00NBBSXRQS]] has a density with respect to Lebesgue measure that is continuous and positive in a neighbourhood of $\theta_0$.

Let $\hat{\theta}_n$ be an efficient estimator, that is, $\sqrt{n}(\hat{\theta}_n - \theta_0) = \frac{1}{\sqrt{n}} \sum_{i=1}^{n} I(\theta_0)^{-1} \dot{\ell}_{\theta_0}(X_i) + o_P(1)$, where $\dot{\ell}_{\theta_0}$ is the score function.
Then the [[posterior distribution|0KJ7ZS8AJA]] $\Pi_n$ of $\theta$ given $X_1, \dots, X_n$ satisfies
$$\norm{\Pi_n - \mathcal{N}\left(\hat{\theta}_n, \tfrac{1}{n} I(\theta_0)^{-1}\right)}_{TV} \to 0$$
in $P_{\theta_0}$-[[probability|0EFDDXTVD6]], where $\norm{\cdot}_{TV}$ is the [[total variation distance|3SZESKX3N5]].

## Remarks

The proof expands the log-likelihood near $\theta_0$: by local asymptotic normality it is, up to $o_P(1)$, a quadratic function of $\sqrt{n}(\theta - \theta_0)$.
The tests of hypothesis 2 show that the posterior mass outside shrinking neighbourhoods of $\theta_0$ vanishes, and hypothesis 3 makes the prior nearly constant on these neighbourhoods.
So the posterior is close to the normalised exponential of a quadratic form, that is, to a [[multivariate normal|0G558GJQMW]] law.
Under the usual regularity conditions the maximum likelihood estimator is efficient and can serve as $\hat{\theta}_n$.
Example: for i.i.d. Bernoulli($p_0$) data with $0 < p_0 < 1$ and any beta prior, the posterior of $p$ is approximately $\mathcal{N}(\hat{p}_n, \hat{p}_n(1 - \hat{p}_n)/n)$ for large $n$, with $\hat{p}_n$ the proportion of successes.
The prior must charge every neighbourhood of $\theta_0$: a prior that vanishes near $\theta_0$ gives a posterior that stays away from $\theta_0$, and the conclusion fails.
The theorem can fail in nonparametric models.

## Explanations

### L1 human

With enough data, the prior stops mattering much.
The posterior then looks like a bell curve centred at the best classical estimate, with the spread that classical theory predicts.
So, in large samples, Bayesian credible intervals and frequentist confidence intervals nearly agree.

### L2 human

The theorem is the bridge between [[Bayesian inference|0JWRJX2RMP]] and frequentist asymptotics in regular parametric models.
It implies that an equal-tailed $95\%$ credible interval for a coordinate of $\theta$ has frequentist coverage tending to $95\%$.
Under additional integrability conditions, the posterior mean is also asymptotically efficient, like the maximum likelihood estimator.
Its hypotheses are the same local asymptotic normality conditions that drive the asymptotic theory of the maximum likelihood estimator.
</document>

<document source="https://tamnd.github.io/math2/node/1E8WCHSAE2.md" tag="1E8WCHSAE2" kind="concept">
# Horocycle

## Summary

A horocycle is a curve of the [[hyperbolic plane|GN4N94GQ0Z]] whose normal geodesics all run to the same point at infinity.
It is the limit of circles through a fixed point with a fixed tangent there, as the radius tends to infinity.
In the [[Poincaré disk model|85MM2EA0TK]] it is a Euclidean circle inside the disc tangent to the boundary circle, without the point of tangency.
Horocycles are not geodesics: they have constant geodesic curvature $1$ in curvature $-1$.

## Explanations

### L1 human

In Euclidean geometry, a circle of huge radius looks almost like a straight line.
In hyperbolic geometry this fails: as the radius grows, circles tend to a new kind of curve, the horocycle, which still bends.
All the radii of a horocycle, the geodesics that cross it at right angles, head toward one common point at infinity, its centre.
In the upper half-plane picture, the horizontal line $y = 1$ is a horocycle: the vertical half-lines cross it at right angles and all go up to the same point at infinity.
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# Malliavin calculus

## Summary

Malliavin calculus is a differential calculus for random variables that are functions of a [[Brownian motion|22735FGJV6]] path.
Its basic operator is the Malliavin derivative $D$, which measures how a random variable changes when the path is perturbed in directions of finite energy.
Its adjoint is the Skorokhod integral, which extends the [[Itô integral|JA95NH206E]] to integrands that are not adapted.
It is used to prove that laws of solutions of stochastic differential equations have smooth densities, and to compute sensitivities in finance.

## Explanations

### L1 human

A random variable built from a Brownian path is a function on the space of all paths, an infinite-dimensional space.
Malliavin calculus defines derivatives of such functions, much like partial derivatives in many variables.
For example, the value $B_T$ of the path at time $T$ has derivative $1$ at each time $t \le T$.
Indeed, a small push of the path at time $t$ moves $B_T$ by the same amount.
An integration by parts formula then turns questions about densities into questions about expectations.
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# Error function

## Definition

The *error function* is the function $\operatorname{erf} \colon \R \to \R$ given by the [[definite integral|04TMP0XGMJ]]
$$\operatorname{erf} x = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}\, dt,$$
where $e^{-t^2}$ uses the [[exponential function|0J0FCWV32D]].

## Remarks

Example: $\operatorname{erf} 0 = 0$, and the [[Gaussian integral|0DFT1GP1JP]] $\int_0^\infty e^{-t^2}\, dt = \frac{\sqrt{\pi}}{2}$ gives $\lim_{x \to \infty} \operatorname{erf} x = 1$.
The integrand is even, so $\operatorname{erf}(-x) = -\operatorname{erf} x$.
Its derivative is $\operatorname{erf}' x = \frac{2}{\sqrt{\pi}} e^{-x^2} > 0$, so $\operatorname{erf}$ increases strictly from $-1$ to $1$.

Non-example: $x \mapsto \int_0^x e^{-t^2}\, dt$ is not the error function; without the factor $\frac{2}{\sqrt{\pi}}$ its limit at $\infty$ is $\frac{\sqrt{\pi}}{2}$, not $1$.

The distribution function of the standard [[normal distribution|05AQE0QR0D]] is $\Phi(x) = \frac12\left(1 + \operatorname{erf}\frac{x}{\sqrt{2}}\right)$.
The same integral, along any path from $0$ to $z$, defines $\operatorname{erf} z$ as an entire function of $z \in \C$, with power series $\operatorname{erf} z = \frac{2}{\sqrt{\pi}} \sum_{n \ge 0} \frac{(-1)^n z^{2n+1}}{n!\,(2n+1)}$.
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# Busemann G-space

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]].
For points $x, y, z \in X$, say that $y$ lies *between* $x$ and $z$ when $y \ne x$, $y \ne z$ and $d(x, y) + d(y, z) = d(x, z)$.
The space $X$ is a *Busemann G-space* when it satisfies these four conditions.

1. Finite compactness: every bounded closed subset of $X$ is compact.
2. Menger convexity: for all distinct $x, z \in X$ there is a point between $x$ and $z$.
3. Local prolongability: each $p \in X$ has a radius $r_p > 0$ with this property. For distinct $x, y$ in the [[open ball|0DC3MPR7C9]] $B(p, r_p)$, some $z \in X$ has $y$ between $x$ and $z$.
4. Uniqueness of prolongation: if $y$ lies between $x$ and $z_1$ and between $x$ and $z_2$, and $d(y, z_1) = d(y, z_2)$, then $z_1 = z_2$.

## Remarks

Condition 1 says that $X$ is a [[proper metric space|0Z4V98VN5E]].
Example: the Euclidean plane $\R^2$.
It is proper, the midpoint lies between any two distinct points, and a segment from $x$ through $y$ extends along its line.
The extension of length $t$ beyond $y$ is unique, since equality in the triangle inequality forces $z$ onto the ray from $x$ through $y$.

Non-example: $\R^2$ with the taxicab distance $d_1$ fails condition 4.
Take $x = (0, 0)$, $y = (1, 0)$, $z_1 = (2, 0)$ and $z_2 = (\frac{3}{2}, \frac{1}{2})$.
Then $d_1(y, z_1) = d_1(y, z_2) = 1$ and $d_1(x, z_1) = d_1(x, z_2) = 2$, so $y$ lies between $x$ and both points, but $z_1 \ne z_2$.
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# Nevanlinna theory

## Summary

Nevanlinna theory is the quantitative study of how often a [[meromorphic function|0NSD1TPHR0]] on $\C$ takes each value.
For each value $a$ it uses a counting function, which counts the solutions of $f(z) = a$ in a disc.
It also uses a proximity function, which measures how close $f$ comes to $a$ on the boundary circle.
Its two main theorems compare these quantities with a single growth measure, the [[Nevanlinna characteristic|YT6Z0GWS0W]].
The theory refines [[Picard's little theorem|R2PHM118NG]] and has analogues in Diophantine approximation.

## Explanations

### L1 human

A nonconstant polynomial of degree $d$ takes every value exactly $d$ times.
An entire or meromorphic function can take a value infinitely often, or never, as $e^z$ never takes $0$.
Nevanlinna theory replaces the degree by a function $T(r, f)$ that grows with the radius $r$.
The [[first main theorem|QJWCFAK4AK]] says that each value is taken at most about $T(r, f)$ times, counted with a logarithmic weight.
The [[second main theorem|GWYE6745JK]] says that most values are taken about that often, and that at most two values can be strongly avoided.
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# Integral element

## Summary

An element $b$ of a [[commutative ring|0310REJJY8]] $B$ is integral over a subring $A$ if it is a root of a [[monic polynomial|0C3QBCR46M]] over $A$.
It extends to rings the notion of an [[algebraic element|0H628RSGWX]] of a field extension, with the extra demand that the leading coefficient is $1$.
Over $\Z$, the integral elements of $\C$ are the algebraic integers, the basic objects of algebraic number theory.
Integral elements are the starting point of the theory of [[integral extensions|B7X0KFY1S1]], integral closure and normal rings.

## Explanations

### L1 human

An element is integral over $A$ if it satisfies an equation $b^n + a_{n-1} b^{n-1} + \cdots + a_0 = 0$ with all $a_i$ in $A$.
The number $\sqrt{2}$ is integral over $\Z$, since it is a root of $x^2 - 2$.
The golden ratio $(1 + \sqrt{5})/2$ is also integral over $\Z$, since it is a root of $x^2 - x - 1$.
The number $1/2$ is not integral over $\Z$, although it is a root of $2x - 1$: the leading coefficient cannot be made $1$.
Integral elements behave like "integers" inside a larger ring.
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# Eckmann–Hilton duality

## Summary

Eckmann–Hilton duality is a correspondence in homotopy theory between pairs of dual notions, obtained by reversing the direction of maps.
Under it, the [[suspension|Q7RFNJH76Q]] corresponds to the [[loop space|2JFF5F3EGF]], [[cofibrations|W5WS5EY8ZM]] to fibrations, and the [[wedge sum|5JSWM1HBX2]] to the product.
It is a guiding principle, not a theorem: it suggests dual statements and dual proofs, but each dual statement must be checked on its own.

## Explanations

### L1 human

Many constructions in homotopy theory come in pairs that mirror each other.
One member of a pair is about maps going out of a space, and the other is about maps coming into a space.
For example, a homotopy can be seen as a map out of a cylinder $X \times [0, 1]$ or as a map into a space of paths.
Eckmann–Hilton duality is the habit of turning each result about one side into a guess about the other side.
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# Functional

## Summary

A functional is a map that takes an element of a vector space, usually a function, and returns a number.
Length of a curve, energy of a field and the value of an integral are functionals of the curve, the field or the integrand.
Functionals are the objects that the [[calculus of variations|PQ25NHV0KS]] minimises and maximises.
The linear ones, the [[linear forms|07YQ1C78G0]], are the elements of the [[dual space|07JBSZE0V4]] and are central in functional analysis.

## Explanations

### L1 human

An ordinary function eats a number and gives a number.
A functional eats a whole function and gives a single number.
Think of a curve drawn between two points: its length is one number, so "length" is a functional on the set of curves.
For example, $J(y) = \int_0^1 y(x)^2 \, dx$ sends $y(x) = x$ to $1/3$ and $y(x) = 1$ to $1$.
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# Hybrid system

## Summary

A hybrid system is a dynamical system whose state has a continuous part and a discrete part, called the mode.
The continuous part flows by [[ordinary differential equations|0NT64JV96A]], and the mode changes by jumps.
In each mode the continuous state follows its own [[vector field|0299VCW1TH]].
When it reaches a switching condition, the mode changes and the continuous state may be reset.
Hybrid systems model mechanical systems with impacts, digital controllers acting on physical plants, and switching circuits.

## Explanations

### L1 human

A thermostat is the classic picture.
The room temperature changes smoothly, but the heater is either on or off, and it switches when the temperature crosses a threshold.
A bouncing ball is another example: it falls smoothly under gravity, and at each impact its velocity jumps from downward to upward.
A hybrid system is any model that mixes such smooth motion with sudden switches.
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# Padé approximant

## Summary

A Padé approximant of a power series is a [[rational function|0WCZH4AFMT]] with prescribed degrees.
Its own power series agrees with the given one to as high an order as these degrees allow.
For numerator degree at most $m$ and denominator degree at most $n$, the agreement is up to the power $z^{m+n}$.
Padé approximants often approximate a function far beyond the disc of convergence of its Taylor series.
They can also show the location of poles.
They are closely linked to [[continued fractions|0BK6JBK58W]] and to orthogonal polynomials.

## Explanations

### L1 human

A [[Taylor polynomial|0HFSFVMWQB]] approximates a function by a polynomial, which cannot have poles and tends to infinity far away.
A Padé approximant uses a quotient of two polynomials instead, so it can imitate poles and limits at infinity.
The coefficients are chosen so that the quotient matches as many Taylor coefficients as possible.
For example, $(1 + z/2)/(1 - z/2)$ matches $e^z = 1 + z + z^2/2 + \cdots$ up to the term $z^2$.
The approximants for all pairs of degrees form a table, the Padé table.
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# Equivariant cohomology

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] and let $X$ be a space with a continuous left [[action|0WCY42EV9V]] of $G$.
Let $EG \to BG$ be a [[universal principal $G$-bundle|1HDH716Z24]]: $EG$ is a contractible space with a free right action of $G$, and $BG = EG / G$ is the [[classifying space|WSZ40H5E4T]].
The *Borel construction* (or homotopy quotient) is
$$X_G = EG \times_G X = (EG \times X) / \sim, \qquad (e g, x) \sim (e, g x) \text{ for } g \in G.$$
For a commutative ring $R$, the *equivariant cohomology* of $X$ is
$$H^\ast_G(X; R) = H^\ast(X_G; R),$$
the [[singular cohomology|6N7BC70CX0]] of the Borel construction.

## Remarks

Up to isomorphism, $H^\ast_G(X; R)$ does not depend on the choice of $EG$.
The projection $X_G \to BG$ makes $H^\ast_G(X; R)$ an algebra over $H^\ast(BG; R) = H^\ast_G(\mathrm{pt}; R)$.

Example: let $G = S^1$ act on a point.
One can take $EG = S^\infty$, the unit sphere in $\C^\infty$, so $X_G = BS^1 = \C P^\infty$ and $H^\ast_{S^1}(\mathrm{pt}; \Z) \cong \Z[u]$ with $u$ of degree $2$.

Non-example: the ordinary cohomology of the orbit space $X / G$ is not the equivariant cohomology in general.
For $S^1$ acting on a point, $X / G$ is a point with cohomology $\Z$ in degree $0$ only, unlike $\Z[u]$.
If $G$ is a compact Lie group acting freely on $X$, then the map $X_G \to X/G$ is a fibration with contractible fibre $EG$, and $H^\ast_G(X; R) \cong H^\ast(X/G; R)$.
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# Resistance form

## Definition

Let $X$ be a nonempty set and let $\ell(X)$ be the vector space of all functions $X \to \R$.
Let $\mathcal{F}$ be a linear subspace of $\ell(X)$ and let $\mathcal{E} \colon \mathcal{F} \times \mathcal{F} \to \R$ be a symmetric bilinear form, with $\mathcal{E}(u) = \mathcal{E}(u, u)$.
The pair $(\mathcal{E}, \mathcal{F})$ is a *resistance form* on $X$ if:

1. $\mathcal{F}$ contains the constant functions, $\mathcal{E}(u) \ge 0$ for all $u \in \mathcal{F}$, and $\mathcal{E}(u) = 0$ if and only if $u$ is constant.
2. Let $u \sim v$ mean that $u - v$ is constant. Then $\mathcal{E}$ induces an inner product on the quotient space $\mathcal{F} / {\sim}$, and $\mathcal{F} / {\sim}$ with this inner product is a [[Hilbert space|HM2HHE3J51]].
3. For all $x \ne y$ in $X$ there is $u \in \mathcal{F}$ with $u(x) \ne u(y)$.
4. For all $x, y \in X$,
$$R(x, y) = \sup \set{ \frac{(u(x) - u(y))^2}{\mathcal{E}(u)} : u \in \mathcal{F}, \ \mathcal{E}(u) > 0 } < \infty.$$
5. (Markov property.) For every $u \in \mathcal{F}$, the function $\bar u = \min(\max(u, 0), 1)$ lies in $\mathcal{F}$ and $\mathcal{E}(\bar u) \le \mathcal{E}(u)$.

The number $R(x, y)$ is the *effective resistance* between $x$ and $y$, and $R$ is a metric on $X$, the *resistance metric*.

## Remarks

Example: let $X$ be the vertex set of a finite connected graph, with a conductance $c_{xy} = c_{yx} > 0$ on each edge $\set{x, y}$, let $\mathcal{F} = \ell(X)$, and let $\mathcal{E}(u) = \sum_{\set{x, y}} c_{xy} (u(x) - u(y))^2$, the sum over edges.
Clause 1 holds because $\mathcal{E}(u) = 0$ forces $u$ to be equal at the two ends of every edge, hence constant on the connected graph.
Clauses 2 and 4 hold because $\mathcal{F} / {\sim}$ is finite-dimensional and $\mathcal{E}$ is positive definite on it.
Clause 3 holds with the indicator function of $\set{x}$.
Clause 5 holds because $t \mapsto \min(\max(t, 0), 1)$ does not increase distances, so no term $(u(x) - u(y))^2$ grows.
Here $R(x, y)$ is the [[effective resistance|XAE5YJ3H68]] of the electrical network.

Non-example: the same formula on a graph with two connected components fails clause 1.
The function equal to $0$ on one component and $1$ on the other has energy $0$ but is not constant.

On $X = [0, 1]$, the form $\mathcal{E}(u, v) = \int_0^1 u'(s) v'(s) \, ds$ on the absolutely continuous functions $u$ with $u' \in L^2$ is a resistance form with $R(x, y) = \abs{x - y}$.
On a domain of $\R^n$ with $n \ge 2$, the Dirichlet energy $\int \abs{\nabla u}^2$ does not give a resistance form.
There are smooth functions with arbitrarily small energy that equal $1$ at a given point $x$ and vanish at another point $y$, so clause 4 fails.
The definition and the main theory of resistance forms are due to Kigami.
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# hp-FEM

## Definition

Let $\Omega \subset \R^d$ be a bounded polygonal domain, and let $V$ be a [[Sobolev space|M4CEZ5K9HP]] on $\Omega$ such as $H^1_0(\Omega)$.
An *hp finite element space* is given by two data:

1. a mesh $\mathcal{T}$ of $\Omega$ into elements $K$;
2. a degree vector $p = (p_K)_{K \in \mathcal{T}}$ of positive integers.

The space $V_{\mathcal{T}, p}$ is the set of $v \in V$ whose restriction to each $K$ is a polynomial of degree at most $p_K$.
The *hp-FEM* is the [[Galerkin method|Y5B0PEBKE8]] on a sequence of such spaces.
In this sequence, both the mesh and the degree vector may change from one step to the next.

## Remarks

The $h$-version is the case of constant $p$ and finer meshes, and the $p$-version is the case of a fixed mesh and growing degrees.
Both are special cases of the [[finite element method|98GZ9DVHTA]] in this setting.

Example: on $\Omega = (0, 1)$ with a solution behaving like $x^\lambda$ near $0$, with $0 < \lambda < 1$, take a geometric mesh with nodes $0, \sigma^{n-1}, \ldots, \sigma, 1$ for some $0 < \sigma < 1$.
Take degrees that grow linearly away from $0$.
For such geometric meshes, the energy error decays exponentially in a root of the number of unknowns $N$.
In two dimensions the rate is $\exp(-b N^{1/3})$ for piecewise analytic data on polygons.

Non-example: let $V = H^1_0(\Omega)$ and take all functions that are polynomials of degree at most $p_K$ on each $K$, with no continuity across elements.
A function with a jump across an interior face is not in $H^1(\Omega)$, so this set fails the condition $v \in V$.
Such spaces are used by hp discontinuous Galerkin methods instead.
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# Grothendieck category

## Definition

A *Grothendieck category* is an [[abelian category|Y9CZ3MTNF1]] $\mathcal{A}$ such that:

1. $\mathcal{A}$ has all small [[colimits|BJS1CGTKQ5]];
2. filtered colimits are exact: for every small [[filtered category|975PMY4FG8]] $I$, the functor $\operatorname{colim} \colon \mathcal{A}^I \to \mathcal{A}$ is [[exact|S1G61TDW0K]];
3. $\mathcal{A}$ has a [[generator|CK08JW32WB]]: an object $G$ such that for any two distinct morphisms $f \ne g \colon X \to Y$ there is a morphism $h \colon G \to X$ with $f \circ h \ne g \circ h$.

## Remarks

Clauses 1 and 2 together form Grothendieck's axiom AB5.
Every Grothendieck category has enough [[injective objects|5QRFTBF4PG]], and it also has all small limits.

Example: for a ring $R$, the category of left [[$R$-modules|0NZ8JA25WK]] is a Grothendieck category.
It has all colimits, and a filtered colimit of short exact sequences of modules is exact.
The module $R$ is a generator: if $f \ne g \colon X \to Y$, pick $x$ with $f(x) \ne g(x)$ and let $h \colon R \to X$ be $r \mapsto r x$.

Non-example: the opposite of the category of [[abelian groups|082GHNPJ53]] is abelian, has all small colimits, and has the generator $\Q/\Z$, but it fails clause 2.
Filtered colimits in it are cofiltered limits of abelian groups, and these are not exact.
For a prime $p$, take the inverse system of short exact sequences $0 \to p^n \Z \to \Z \to \Z / p^n \Z \to 0$ for $n \ge 1$.
Its limit is $0 \to 0 \to \Z \to \Z_p \to 0$, where $\Z_p$ is the group of $p$-adic integers, and the map $\Z \to \Z_p$ is not surjective.
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# CAT(k) space

## Summary

A CAT($k$) space is a [[metric space|0VKD1TP7F3]] whose geodesic triangles are at least as thin as triangles in a model surface.
The model surface has constant curvature $k$.
It is the metric version of the condition "sectional curvature at most $k$".
The notion applies to spaces with no smooth structure, such as trees, polyhedral complexes and buildings.
CAT(0) spaces are central in geometric group theory.

## Explanations

### L1 human

Draw a triangle in the space whose sides are shortest paths.
Draw a triangle with the same side lengths on a flat sheet, on a sphere or on a saddle, according to $k$.
The space is CAT($k$) when the first triangle is never fatter than the model one: points on its sides are no farther apart than the matching points of the model.
A tree is CAT(0): every triangle in a tree is a tripod, as thin as possible.
A round sphere is not CAT(0), because its triangles bulge outward.
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# Gauss's hypergeometric theorem

## Statement

Let $a, b, c \in \C$ with $c \notin \set{0, -1, -2, \ldots}$ and $\operatorname{Re}(c - a - b) > 0$, and let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
Then the [[hypergeometric series|428M453CR2]] converges at $z = 1$, and
$${}_2F_1(a, b; c; 1) = \frac{\Gamma(c) \, \Gamma(c - a - b)}{\Gamma(c - a) \, \Gamma(c - b)}.$$

## Remarks

When moreover $\operatorname{Re} c > \operatorname{Re} b > 0$, put $z = 1$ in [[Euler's integral representation|812THX456R]]: the integral becomes the beta integral $B(b, c - a - b)$, and the gamma factors simplify to the right side.
Abel's theorem on power series identifies the value of the series at $1$ with the limit of the function as $z \to 1^-$.
The general case follows by analytic continuation in the parameters, or from a contiguous relation.

Example: for $a = b = 1$, $c = 3$ the series is $\sum_{n \ge 0} \frac{n! \, n!}{(n+2)!/2 \cdot n!} = \sum_{n \ge 0} \frac{2}{(n+1)(n+2)} = 2$, and the right side is $\frac{\Gamma(3) \Gamma(1)}{\Gamma(2) \Gamma(2)} = 2$.

With $a = -n$ for an integer $n \ge 0$, the theorem gives the [[Chu–Vandermonde identity|4YK46R1QDR]].

The condition $\operatorname{Re}(c - a - b) > 0$ is needed: for $a = b = 1$, $c = 2$ one has $c - a - b = 0$, and the series $\sum_n \frac{1}{n+1}$ diverges.

## Explanations

### L1 human

The hypergeometric series is an infinite sum depending on three parameters.
At the edge point $z = 1$ the sum can be computed in closed form, as a ratio of gamma function values.
This one formula evaluates a huge number of special series.

### L2 human

The theorem is the basic summation formula for ${}_2F_1$, and most closed-form evaluations of ${}_3F_2$ and higher series, such as those of Pfaff–Saalschütz and Dixon, are its relatives.
It also gives the connection coefficients between the solutions at $0$ and at $1$ of the hypergeometric equation, since these are values of ${}_2F_1$ at $1$.
The terminating case, Chu–Vandermonde, is a polynomial identity behind many binomial sums.
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# Tutte polynomial

## Definition

Let $M$ be a [[matroid|PX31ZNYS3Z]] on a finite ground set $E$ with [[rank function|SYRCTH853H]] $r$.
The *Tutte polynomial* of $M$ is
$$T_M(x, y) = \sum_{A \subseteq E} (x - 1)^{r(E) - r(A)} \, (y - 1)^{\abs{A} - r(A)}.$$
Let $G$ be a finite [[graph|00C0BR7D7H]], with loops and multiple edges allowed.
Its Tutte polynomial $T_G$ is the Tutte polynomial of its graphic matroid:
$$T_G(x, y) = \sum_{A \subseteq E(G)} (x - 1)^{k(A) - k(E)} \, (y - 1)^{k(A) + \abs{A} - \abs{V(G)}},$$
where $k(A)$ is the number of connected components of the graph $(V(G), A)$.

## Remarks

Let $e \in E$, and let $M \setminus e$ and $M / e$ be the [[deletion and contraction|4WCYBSH1P6]] of $e$.
If $e$ is neither a loop nor a coloop, then $T_M = T_{M \setminus e} + T_{M / e}$.
If $e$ is a coloop, then $T_M = x \, T_{M / e}$; if $e$ is a loop, then $T_M = y \, T_{M \setminus e}$.
For the dual matroid, $T_{M^*}(x, y) = T_M(y, x)$.
The value $T_M(1, 1)$ is the number of bases of $M$.
For a graph $G$, the [[chromatic polynomial|0B621RBVS9]] is $P_G(\lambda) = (-1)^{\abs{V(G)} - k(E)} \lambda^{k(E)} \, T_G(1 - \lambda, 0)$.

Example: for the triangle $K_3$, $\abs{V} = 3$ and $k(E) = 1$.
The empty set gives $(x-1)^2$, the $3$ single edges give $3(x-1)$, the $3$ pairs give $3$, and the full edge set gives $y - 1$.
The sum is $x^2 - 2x + 1 + 3x - 3 + 3 + y - 1 = x^2 + x + y$.
Then $T(1, 1) = 3$ spanning trees, and $(-1)^2 \lambda \, T(1 - \lambda, 0) = \lambda(\lambda - 1)(\lambda - 2)$ is the chromatic polynomial of $K_3$.

Non-example: the rank generating polynomial $\sum_{A} u^{r(E) - r(A)} v^{\abs{A} - r(A)}$ is not the Tutte polynomial itself; the Tutte polynomial is obtained from it by the substitution $u = x - 1$, $v = y - 1$.
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# Simpson's rule

## Definition

Let $a < b$, let $f \colon [a, b] \to \R$ be continuous, and let $m = (a + b)/2$.
*Simpson's rule* is the [[quadrature rule|ADSVMCY77A]]
$$S(f) = \frac{b - a}{6} \big( f(a) + 4 f(m) + f(b) \big).$$
For an even integer $N \ge 2$, let $h = (b - a)/N$ and $x_i = a + i h$.
The *composite Simpson's rule* is
$$S_N(f) = \frac{h}{3} \Big( f(x_0) + 4 f(x_1) + 2 f(x_2) + 4 f(x_3) + \cdots + 2 f(x_{N-2}) + 4 f(x_{N-1}) + f(x_N) \Big),$$
the sum of $S$ over the subintervals $[x_{2j}, x_{2j+2}]$.

## Remarks

$S$ is the closed [[Newton–Cotes formula|BQPQAY1FDJ]] with $n = 2$, and its [[degree of exactness|ZYT1JNT1Y0]] is $3$, one more than the degree of the interpolating parabola, by symmetry.
Example: for $f(x) = x^3$ on $[0, 1]$, $S(f) = \frac{1}{6}\big(0 + \frac{1}{2} + 1\big) = \frac{1}{4} = \int_0^1 x^3\, dx$.
Non-example of exactness: for $f(x) = x^4$ on $[0, 1]$, $S(f) = \frac{1}{6}\big(0 + \frac{1}{4} + 1\big) = \frac{5}{24}$, while the integral is $\frac{1}{5}$; so the degree is not $4$.
If $f$ has a continuous fourth derivative, then $\int_a^b f - S_N(f) = -\frac{(b - a) h^4}{180} f^{(4)}(\xi)$ for some $\xi \in [a, b]$.
For $x^4$ on $[0, 1]$ with $N = 2$ this gives $-\frac{1}{16 \cdot 180} \cdot 24 = -\frac{1}{120}$, matching $\frac{1}{5} - \frac{5}{24}$.
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# Herglotz–Riesz kernel

## Summary

The Herglotz–Riesz kernel of a disc is the holomorphic function of $z$ whose real part is the Poisson kernel.
Integrating it against the boundary values of a harmonic function $u$ gives a holomorphic function with real part $u$.
It is the kernel in the Schwarz integral formula and in the Herglotz representation of holomorphic functions with positive real part.

## Explanations

### L1 human

The Poisson kernel tells how much each boundary point of a disc counts when one rebuilds a harmonic function inside.
The Herglotz–Riesz kernel carries the same information, but as a holomorphic function.
For the unit disc and a boundary point $\zeta$ it is $(\zeta + z)/(\zeta - z)$.
At the centre $z = 0$ it equals $1$, and it blows up as $z$ approaches $\zeta$.
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# Polynomial function

## Summary

A real polynomial function is a function $\R \to \R$ given by a formula $a_0 + a_1 x + \dots + a_n x^n$ with real coefficients.
Polynomial functions are [[continuous|0C5TP6RYSV]] and infinitely differentiable, and their derivatives and antiderivatives are again polynomial functions.
They are the simplest functions beyond linear ones, and by the [[Weierstrass approximation theorem|0JQE130DHW]] they approximate every continuous function on a closed bounded interval uniformly.
The [[definition|MAM9H3SY3Y]] gives the precise form, and the algebraic side is the [[polynomial ring|0SDC934R52]].

## Explanations

### L1 human

A polynomial function is built from $x$ and constants using only addition and multiplication.
Its graph is smooth with no corners or jumps.
A polynomial function of degree $n \ge 1$ crosses each horizontal line at most $n$ times.
For example $f(x) = x^3 - 2x + 1$ has degree $3$ and $f(1) = 0$.
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# Arnold tongue

## Definition

For $\Omega \in \R$ and $0 \le K \le 1$, let $f_{\Omega, K}$ be the standard [[circle map|N2SVK69S53]] $x \mapsto x + \Omega - \frac{K}{2\pi} \sin(2\pi x)$ on $\R / \Z$, an orientation-preserving homeomorphism.
Let $F_{\Omega, K}(x) = x + \Omega - \frac{K}{2\pi} \sin(2\pi x)$ be its lift to $\R$, and let $\rho(F_{\Omega, K})$ be the translation number of this lift, as in the definition of the [[rotation number|DP4WWQBJJF]].
For a rational number $p/q$, the *Arnold tongue* of $p/q$ is the set
$$T_{p/q} = \set{(\Omega, K) \in \R \times [0, 1] : \rho(F_{\Omega, K}) = p/q}.$$

## Remarks

Example: the tongue $T_0$.
The translation number of $F_{\Omega, K}$ is $0$ exactly when $F_{\Omega, K}$ has a fixed point, that is, when $\Omega = \frac{K}{2\pi} \sin(2\pi x)$ for some $x$.
So $T_0 = \set{(\Omega, K) : \abs{\Omega} \le K / (2\pi)}$, a wedge with its tip at $(0, 0)$.
The point $(0.3, 0.5)$ is not in $T_0$, since $0.3 > 0.5/(2\pi) \approx 0.08$.

For $K = 0$ the map is the rotation by $\Omega$, so $T_{p/q}$ meets the line $K = 0$ only at $\Omega = p/q$.
For $0 < K \le 1$ the section of $T_{p/q}$ at height $K$ is a closed interval of positive length.
For fixed $K$, the function $\Omega \mapsto \rho(F_{\Omega, K})$ is continuous and nondecreasing; for $0 < K \le 1$ it is constant on each such interval and is a devil's staircase, like the [[Cantor function|NPBYYK63FE]].
Translation by $1$ in $\Omega$ maps $T_{p/q}$ onto $T_{p/q + 1}$.

## Explanations

### L1 human

Each rational number $p/q$ has its tongue: the set of settings of the circle map where the average turn per step is exactly $p/q$.
The example $T_0$ shows the typical shape: a wedge whose width grows with the strength $K$ of the kick.
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# Approximate point spectrum

## Definition

Let $X$ be a complex Banach space with identity map $I$, and let $T \colon X \to X$ be a bounded operator.
The *approximate point spectrum* of $T$ is the set $\sigma_{ap}(T)$ of $\lambda \in \C$ for which there is a sequence $(x_n)$ in $X$ with
$$\norm{x_n} = 1 \text{ for all } n \quad \text{and} \quad \norm{(T - \lambda I) x_n} \to 0.$$

## Remarks

Equivalently, $\lambda \in \sigma_{ap}(T)$ if and only if $T - \lambda I$ is not bounded below, that is, there is no $c > 0$ with $\norm{(T - \lambda I)x} \ge c \norm{x}$ for all $x$.
It satisfies $\sigma_p(T) \cup \sigma_c(T) \subseteq \sigma_{ap}(T) \subseteq \sigma(T)$, with the [[point spectrum|7FNVA0Q6YQ]], the [[continuous spectrum|YVH149HCAG]] and the [[spectrum|BMKATJ802A]].
It is closed, and it contains the boundary of $\sigma(T)$, so it is nonempty when $X \ne \set{0}$.

Example: for $D(x_n) = (x_n / n)$ on $\ell^2$, the unit vectors $e_n$ satisfy $\norm{D e_n} = 1/n \to 0$, so $0 \in \sigma_{ap}(D)$.

Non-example: for the right shift $S$ on $\ell^2$, $\norm{Sx} = \norm{x}$ for all $x$, so $S$ is bounded below and $0 \notin \sigma_{ap}(S)$, although $0 \in \sigma(S)$.
More generally $\norm{(S - \lambda I)x} \ge (1 - \abs{\lambda}) \norm{x}$, so $\sigma_{ap}(S)$ is the unit circle.

For a normal operator on a Hilbert space, $\sigma_{ap}(T) = \sigma(T)$.
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# Spectrum

## Summary

A spectrum, in algebraic topology, is a sequence of pointed spaces $E_n$ with maps from the suspension of each space to the next one.
Spectra are the objects of stable homotopy theory, much as spaces are the objects of unstable homotopy theory.
Every [[generalized cohomology theory|H28PSZR7XG]] is represented by a spectrum, and spectra have homotopy groups in all integer degrees, including negative ones.

## Explanations

### L1 human

A space can be suspended again and again, and a spectrum is what one gets by allowing the process to run in both directions.
It is a list of spaces $E_0, E_1, E_2, \ldots$ in which each space is glued to the next by a map from its suspension.
The simplest example takes $E_n = S^n$, the $n$-sphere, with the identity maps $\Sigma S^n = S^{n+1}$; this is the sphere spectrum.
Another example uses Eilenberg–MacLane spaces $K(G, n)$, and it gives ordinary cohomology with coefficients in $G$.
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# Multiresolution analysis

## Summary

A multiresolution analysis of $L^2(\R)$ is a nested chain of closed subspaces $V_j$, one for each scale $2^{-j}$.
The spaces are related by dilation by $2$, and $V_0$ is spanned by the integer translates of one scaling function.
Stéphane Mallat and Yves Meyer introduced it as the framework in which orthonormal [[wavelets|4PZ35C0GV3]] are built.
The wavelet spaces are the orthogonal complements of $V_j$ in $V_{j+1}$, and the [[discrete wavelet transform|P3H4MMC4ZH]] moves between the levels with two filters.

## Explanations

### L1 human

Think of a photograph seen at several resolutions.
The space $V_j$ holds all the pictures that can be drawn with pixels of size $2^{-j}$.
Each finer space contains the coarser ones, and as the pixels shrink one can approximate any picture.
The detail that one gains by going from $V_j$ to $V_{j+1}$ is carried by the wavelets of scale $2^{-j}$.
In the simplest case $V_j$ consists of the functions that are constant on each interval $[k 2^{-j}, (k+1) 2^{-j})$.
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# Harmonic analysis on groups

## Definition

*Harmonic analysis on groups* is the part of mathematics that studies a [[locally compact group|AHV05ZBCYW]] $G$, with a [[Haar measure|VY0HXD3GC4]], through the following questions:

1. how functions in spaces such as $L^1(G)$ and $L^2(G)$ decompose into components on which $G$ acts irreducibly;
2. how the [[unitary representations|74BPYJNZTM]] of $G$, in particular the regular representation on $L^2(G)$, decompose into irreducible ones;
3. how translation, convolution and invariant differential operators act on these components.

## Remarks

Example: for a [[locally compact abelian group|ZR09S9R3AW]] $G$, the irreducible unitary representations are the continuous characters.
The decomposition is then given by the [[Fourier transform|VHN5VW5Q92]] from functions on $G$ to functions on the dual group.
For $G = \mathbb{T}$ this is the theory of Fourier series.
For compact groups the decomposition is a direct sum over finite-dimensional irreducible representations, by the Peter–Weyl theorem.
Non-example: the study of the zeros of a single polynomial on $\R$ uses no group action, and it is not harmonic analysis on groups.
The terms abstract harmonic analysis and, for non-abelian $G$, non-commutative harmonic analysis are also used.
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# Spin(7)-manifold

## Summary

A Spin(7)-manifold is a Riemannian manifold of dimension $8$ whose holonomy group lies in the group $\operatorname{Spin}(7)$, viewed inside $SO(8)$.
The structure is encoded in a parallel $4$-form, the Cayley form.
Spin(7)-manifolds are Ricci-flat and form, with [[$G_2$ manifolds|P1TFW9QWAD]], the two exceptional cases of Berger's list.

## Explanations

### L1 human

The octonions form an eight-dimensional number system, and the group $\operatorname{Spin}(7)$ preserves a special $4$-form on them.
A Spin(7)-manifold is an eight-dimensional curved space in which such a form exists at every point and is unchanged by parallel transport.
The flat space $\R^8$ is the simplest example.
These spaces are cousins of the seven-dimensional $G_2$ manifolds: the product of a $G_2$ manifold with a circle is an example.
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# Lipschitz domain

## Summary

A Lipschitz domain is an open set whose boundary is locally the graph of a [[Lipschitz|0AW1E6YTJY]] function.
Near each boundary point the set lies on one side of the graph.
Such boundaries may have corners and edges, but no cusps or slits.
Lipschitz domains are the standard setting for boundary value problems with rough boundaries.
The divergence theorem, Sobolev traces and the theory of harmonic measure all work there.

## Explanations

### L1 human

Zoom in on any boundary point of the domain and turn your head suitably.
You see the domain lying above a graph whose slope is bounded.
A square is a Lipschitz domain: at a corner the boundary looks like the graph of $\abs{x}$ after a rotation.
A disc with a slit is not, because near the slit the domain lies on both sides of it.
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# Enriques–Kodaira classification

## Statement

Let $S$ be a smooth projective [[algebraic surface|KFTECAATKG]] over $\C$ that is minimal, meaning that it contains no smooth rational curve $E$ with $E^2 = -1$.
Let $\kappa(S)$ be its [[Kodaira dimension|ZW00Q0SAGR]] and $K_S$ its [[canonical divisor|W05VYY4HBR]].
Then exactly one of the following holds.

1. $\kappa(S) = -\infty$, and $S$ is the projective plane $\mathbb{P}^2$ or a [[ruled surface|5NK10NGY13]] $\mathbb{P}(\mathcal{E}) \to C$ over a smooth projective curve $C$.
2. $\kappa(S) = 0$, and $S$ is a [[K3 surface|Q3V566R6K2]], an [[Enriques surface|Y30JE2DC6Y]], an [[abelian surface|R7W03TXRS9]] or a bielliptic surface. In each case $12 K_S \sim 0$.
3. $\kappa(S) = 1$, and $S$ is an [[elliptic surface|VSC3F53DX4]].
4. $\kappa(S) = 2$, and $S$ is a [[surface of general type|SV4CK2XP8M]].

Moreover, every smooth projective surface is obtained from a minimal one by finitely many blow-ups of points, and if $\kappa(S) \ge 0$ this minimal model is unique.

## Remarks

The reduction to minimal surfaces uses [[Castelnuovo's contraction theorem|WJ5WRGG5DE]]: a smooth rational curve with self-intersection $-1$ can be blown down to a smooth point, which lowers the second Betti number.
For each value of $\kappa$, the finer classification uses the invariants $q = h^1(\mathcal{O}_S)$, $p_g = h^0(K_S)$, the plurigenera $P_n = h^0(nK_S)$, and the Chern numbers $K_S^2$ and $c_2(S)$.
[[Castelnuovo's rationality criterion|QXHMMQX1D7]] is the key step for $\kappa = -\infty$.
Kodaira extended the classification to all compact complex surfaces, which adds non-algebraic classes such as surfaces of class VII and Kodaira surfaces.
Bombieri and Mumford extended it to algebraically closed fields of positive characteristic, where new classes appear in characteristics $2$ and $3$.
Example: a smooth quartic surface in $\mathbb{P}^3$ has $K_S = 0$ by adjunction and lands in case 2 as a K3 surface.
A smooth quintic surface has $K_S$ ample and lands in case 4.

## Explanations

### L1 human

Every complex algebraic surface can be simplified, by collapsing certain special curves, to a minimal surface.
The classification sorts minimal surfaces into four families according to how fast the number of sections of powers of the canonical bundle grows.
The first three families are known in detail; the last one, the surfaces of general type, is by far the largest and is not classified in detail.

### L2 human

The classification is the two-dimensional model of the minimal model program.
In dimension one, curves split by genus into $g = 0$, $g = 1$ and $g \ge 2$, which matches $\kappa = -\infty, 0, 1$; for surfaces there is one more class.
The invariants that tell the classes apart are birational invariants, so they can be computed on any smooth model.
Results like the Bogomolov–Miyaoka–Yau inequality constrain the invariants of the last class.
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# Model theory

## Definition

Model theory is the part of mathematical logic whose objects are the following.

1. Formal languages, most often the language of [[first-order logic|086QKPF0A6]] over a signature.
2. [[Theories|92SWXST1ZM]], that is, sets of sentences of such a language.
3. [[Structures|983GZBQYT4]] for the signature, and the relation of truth between structures and sentences, in particular the [[models|MVCH6N9SNW]] of theories.

It studies the classes of models of theories and the sets definable in structures.

## Remarks

Example: the statement that every field elementarily equivalent to $\C$ is algebraically closed of characteristic $0$ is a result of model theory.
It concerns the class of models of the complete theory $\mathrm{Th}(\C)$.

Non-example: the computation of the homology of a sphere is not a result of model theory, because it concerns no formal language and no relation of truth.

Classical model theory works with first-order logic; other parts study [[infinitary logic|PQG7VAZ1R2]], finite structures or [[second-order logic|G6X7SZSNC5]].

## Explanations

### L1 human

Model theory is the study of mathematical structures through the sentences that are true in them.
It sits between logic and algebra: its theorems are about formulas, and its applications are often about fields, groups and orders.
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# Non-Archimedean absolute value

## Summary

A non-Archimedean absolute value on a field is an absolute value that satisfies the strong triangle inequality $\abs{x + y} \le \max(\abs{x}, \abs{y})$, see the [[definition|8XDYNK6TV0]].
It generalizes the usual [[absolute value|0FFXKTKKQC]] of real numbers in a different direction: small elements stay small under addition.
The [[p-adic absolute value|WGN82TCEF2]] is the main example, and the induced metric is an [[ultrametric|A6DPR620B5]].
These absolute values are the same as real-valued [[valuations|B85T5VKR3M]], up to a choice of base for the exponent.

## Explanations

### L1 human

The usual absolute value makes $1 + 1 + \cdots + 1$ grow without end.
A non-Archimedean absolute value never lets a sum be larger than its largest term.
For the $2$-adic absolute value, a number is small when it is divisible by a high power of $2$: $\abs{8}_2 = 1/8$ and $\abs{3}_2 = 1$.
Then $\abs{8 + 16}_2 = \abs{24}_2 = 1/8$, which is not more than the larger of $1/8$ and $1/16$.
In such a geometry every triangle is isosceles, and every point of a disc is a centre of it.
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<document source="https://tamnd.github.io/math2/node/1H650VRZVT.md" tag="1H650VRZVT" kind="definition">
# Algebraic approach to constraint satisfaction

## Definition

Let $\mathbb{A} = (A; R_1, \dots, R_k)$ be a relational structure with a finite nonempty universe $A$ and finitely many relations.
The *constraint satisfaction problem* $\operatorname{CSP}(\mathbb{A})$ is the following decision problem.
The input is a finite relational structure $\mathbb{X}$ with relations of the same arities.
The question is whether there is a homomorphism from $\mathbb{X}$ to $\mathbb{A}$, a map preserving each relation.
The *polymorphism clone* of $\mathbb{A}$ is the [[clone|PTKR703G6A]] $\operatorname{Pol}(\mathbb{A})$ of all operations on $A$ that are [[polymorphisms|A0K9EZRM7E]] of every $R_i$.
The *algebraic approach* is the study of the computational complexity of $\operatorname{CSP}(\mathbb{A})$ through the algebra $(A; \operatorname{Pol}(\mathbb{A}))$.

## Remarks

The approach rests on a reduction principle.
If $\mathbb{A}$ and $\mathbb{B}$ share the universe $A$ and $\operatorname{Pol}(\mathbb{A}) \subseteq \operatorname{Pol}(\mathbb{B})$, then $\operatorname{CSP}(\mathbb{B})$ reduces to $\operatorname{CSP}(\mathbb{A})$ in polynomial time.
The reason is that each relation of $\mathbb{B}$ is then primitive positive definable from $\mathbb{A}$, by the [[Pol–Inv Galois connection|BQ0AC9QX7E]].
Example: for the complete graph $K_2$, $\operatorname{CSP}(K_2)$ asks whether a graph is $2$-colourable, as a [[graph homomorphism|052JJ9YVNM]] to $K_2$.
The majority operation on $\set{0, 1}$ is a polymorphism of the edge relation $\set{(0, 1), (1, 0)}$, and the problem is in polynomial time.
Non-example of tractability: every polymorphism of $K_3$ has the form $\sigma(x_i)$ for a permutation $\sigma$ of $\set{0, 1, 2}$.
So $K_3$ has no Taylor polymorphism, and $3$-colouring is NP-complete.
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# Congruence subgroup

## Summary

A congruence subgroup of the [[modular group|VEKV1EM77B]] $\mathrm{SL}_2(\Z)$ is a subgroup defined by congruence conditions on the matrix entries modulo some integer $N$.
The main examples are $\Gamma(N)$, $\Gamma_1(N)$ and $\Gamma_0(N)$.
[[Modular forms|GWKN5VG78A]] of level $N$ are modular forms for these groups.
Most modular forms that arise in arithmetic, such as those attached to [[elliptic curves|427N6S6QQW]], have a congruence subgroup as their symmetry group.

## Explanations

### L1 human

Take the integer matrices of determinant $1$ and reduce their entries modulo $N$.
A congruence subgroup is a subgroup that can be described by conditions on these reductions.
For example, $\Gamma_0(N)$ is the set of matrices whose lower left entry is divisible by $N$.
Such a group is smaller than $\mathrm{SL}_2(\Z)$, so functions symmetric under it are easier to find.
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# Permutation pattern

## Summary

A permutation pattern is a short [[permutation|05RWFKVTJ7]] that may occur inside a longer one, see the [[definition|DF69BF6841]].
A permutation contains a pattern if some of its entries appear in the same relative order as the pattern, and avoids it otherwise.
The study of pattern avoidance counts the permutations that avoid given patterns.
Its central result is the [[Stanley–Wilf conjecture|KWJS2QYJ1K]], now a theorem, which bounds these counts by an exponential.

## Explanations

### L1 human

Write a permutation as a row of numbers, such as $3\,1\,4\,2$.
Pick some of the entries, keep their order, and look only at which is larger than which.
Picking $3, 1, 2$ gives the shape "high, low, middle", which is the pattern $312$.
The row $3\,1\,4\,2$ has no three entries that increase from left to right, so it avoids the pattern $123$.
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# Support function

## Summary

The support function of a nonempty set $A \subseteq \R^n$ assigns to each vector $x$ the [[supremum|0R3H5PANRQ]] of the [[dot products|0S52TNNTH9]] $x \cdot a$ over $a \in A$.
It is a [[sublinear function|K3356N0SAX]], and it determines a nonempty compact [[convex set|02AS45T3H9]] completely.
Support functions turn geometric operations into analytic ones: the support function of a Minkowski sum is the sum of the support functions.
They are a basic tool in the study of [[convex bodies|437FSG3419]].

## Explanations

### L1 human

Fix a direction $u$ of length $1$ and push a flat wall, perpendicular to $u$, towards the set from far away in direction $u$.
The value of the support function at $u$ is the signed distance from the origin to the wall at the moment it first touches the set.
For the unit disc centred at the origin, the wall always touches at distance $1$, so the support function is $h(u) = 1$ on unit vectors, and $h(x) = \abs{x}$ in general.
For the segment from $(-1, 0)$ to $(1, 0)$, one gets $h(x_1, x_2) = \abs{x_1}$.
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# Incidence algebra

## Summary

The incidence algebra of a locally finite [[partially ordered set|0QV2JS9RJ3]] is the algebra of functions on its intervals, multiplied by a convolution.
Its most important elements are the zeta function, which is $1$ on every interval, and its inverse, the Möbius function of the poset.
It gives a common setting for inclusion–exclusion, the number-theoretic [[Möbius inversion formula|07Q4TBTFVB]] and the calculus of finite differences, through [[Möbius inversion on posets|B8XJSJ517Q]].

## Explanations

### L1 human

Take a poset in which every interval $[x, y]$ has finitely many elements.
Attach a number to each pair $x \le y$, and multiply two such functions by summing over all the points $z$ between $x$ and $y$.
This is matrix multiplication in disguise.
For the chain $1 < 2 < \dots < n$ the incidence algebra is the algebra of upper triangular $n \times n$ matrices.
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# Regular local rings are UFDs

## Statement

Let $R$ be a [[regular local ring|63XXTF6B5M]].
Then $R$ is a [[unique factorization domain|0W4X9G6263]].

## Remarks

This is the Auslander–Buchsbaum theorem.
Proof idea: $R$ is a domain by [[regular local rings are domains|V3ZBMRKC08]]; one argues by induction on $\dim R$.
Pick $x \in \mathfrak m \setminus \mathfrak m^2$; then $x$ is prime, since $R/xR$ is regular, hence a domain.
By [[Nagata's criterion|GGEMJF449D]], it is enough to show that $R[1/x]$ is a UFD.
Its localizations at primes are regular of smaller dimension, hence UFDs by induction.
So each height one prime of $R[1/x]$ is locally principal.
Finite free resolutions, which exist by [[Serre's theorem|93RQRG9R1P]], then show that it is principal.

Example: the power series ring $k[\![x_1, \ldots, x_n]\!]$ over a field $k$ and the localizations $k[x_1, \ldots, x_n]_{\mathfrak p}$ of a polynomial ring at prime ideals are UFDs.

The regularity hypothesis matters.
For a field $k$, the local ring $R = k[x, y, z]_{(x, y, z)}/(xy - z^2)$ is a domain of dimension $2$ and embedding dimension $3$.
It is not a UFD, since $xy = z \cdot z$ are two factorizations of the same element into irreducible elements that are not associate.

## Explanations

### L1 human

At a smooth point of a variety, every function factors uniquely into irreducible pieces, just as integers factor uniquely into primes.
At a singular point this can fail: on the cone $xy = z^2$ the function $z^2$ factors in two different ways.

### L2 human

The proof sketched above relies on homological algebra, through Serre's theorem and finite free resolutions.
In geometry it means that on a smooth variety every hypersurface through a point is defined locally by one equation, so Weil divisors and Cartier divisors agree.
It is used with the [[divisor class group|WN8PK6RQZR]]: the class group of a regular local ring is trivial.
</document>

<document source="https://tamnd.github.io/math2/node/1HDH716Z24.md" tag="1HDH716Z24" kind="concept">
# Universal bundle

## Summary

A universal bundle for a [[topological group|92700QAB5Y]] $G$ is a [[principal $G$-bundle|8E37E74XVP]] $EG \to BG$ with a classifying property.
Every principal $G$-bundle over a CW complex is its pullback along a map, unique up to homotopy.
Its base $BG$ is the [[classifying space|WSZ40H5E4T]] of $G$.
A principal bundle with contractible total space has this property under mild conditions, which gives a practical way to recognize universal bundles.
Universal bundles turn the classification of bundles into the computation of homotopy classes of maps.

## Explanations

### L1 human

A universal bundle is a master bundle that contains every other bundle with the same group.
To get a bundle over a space $X$, one maps $X$ into the base of the master bundle and pulls the master bundle back.
Homotopic maps give the same bundle, so bundles over $X$ correspond to maps up to deformation.
For the group $\Z/2\Z$, the double cover $S^\infty \to \R P^\infty$ is a universal bundle, and it classifies the double covers of a CW complex.
</document>

<document source="https://tamnd.github.io/math2/node/1HGGRS57KT.md" tag="1HGGRS57KT" kind="definition">
# Boundary value problem

## Definition

Let $a < b$ and $n \ge 1$, and let $F \colon [a, b] \times \R^{n} \to \R$ be continuous.
Let $g \colon \R^{n} \times \R^{n} \to \R^n$ be a function.
A (two-point) *boundary value problem* on $[a, b]$ is the problem of finding a function $y \in C^n([a, b])$ such that:

1. $y$ is a [[solution|07VX34QXAV]] of the [[ordinary differential equation|0NT64JV96A]] $y^{(n)}(x) = F\left( x, y(x), \ldots, y^{(n-1)}(x) \right)$ for all $x \in [a, b]$, and
2. $y$ satisfies the boundary conditions $g\left( \mathbf{y}(a), \mathbf{y}(b) \right) = 0$, where $\mathbf{y}(x) = \left( y(x), y'(x), \ldots, y^{(n-1)}(x) \right)$.

The problem is *linear* if the equation is linear in $y, \ldots, y^{(n)}$ and $g$ is affine.
A linear problem is *homogeneous* if $y = 0$ satisfies the equation and the conditions.

## Remarks

If the conditions only prescribe $\mathbf{y}(a)$, that is, $g(u, v) = u - \mathbf{y}_0$ for a fixed $\mathbf{y}_0 \in \R^n$, the problem is an [[initial value problem|0FXDW992DS]], not a boundary value problem in the proper sense.
The same name is used for systems and for partial differential equations on a domain, with conditions on its boundary.

Example: $y'' = -y$ on $[0, \frac{\pi}{2}]$ with $y(0) = 0$ and $y(\frac{\pi}{2}) = 1$, that is, $g(u, v) = (u_1, v_1 - 1)$.
The function $y = \sin x$ solves the equation, $\sin 0 = 0$ and $\sin \frac{\pi}{2} = 1$; it is the only solution, since the general solution is $c_1 \cos x + c_2 \sin x$.

Non-example of solvability: $y'' = -y$ on $[0, \pi]$ with $y(0) = 0$ and $y(\pi) = 1$ is a boundary value problem without solution.
Every solution with $y(0) = 0$ is $c \sin x$, which vanishes at $\pi$.
</document>

<document source="https://tamnd.github.io/math2/node/1HJF40J5AK.md" tag="1HJF40J5AK" kind="concept">
# Hyperstructure

## Summary

An algebraic hyperstructure is a set with at least one hyperoperation, a binary operation whose value at two elements is a nonempty set of elements.
The main examples are hypergroups, introduced by Marty in 1934, which generalise [[groups|G5S1M9]] by replacing products with sets of products.
Hyperstructures are special [[multialgebras|3C930MVM0T]], and they arise naturally from quotients of groups by subgroups that are not normal.

## Explanations

### L1 human

In a group, two elements multiply to one element.
In a hypergroup, two elements multiply to a set of elements.
On the real line, let the product of $x$ and $y$ be every number between them; the product of $1$ and $3$ is then $[1, 3]$.
This product is associative in a natural sense.
Also every number $y$ lies in the product of $x$ and $y$, so the products of a fixed $x$ with all numbers cover the line.
These two facts make the line a hypergroup.
Hyperstructures study which facts about groups, rings and fields survive in this setting.
</document>

<document source="https://tamnd.github.io/math2/node/1HK0VYWTV7.md" tag="1HK0VYWTV7" kind="concept">
# Logical consequence

## Summary

A formula $\varphi$ is a logical consequence of a set of formulas $\Gamma$ when every interpretation making all of $\Gamma$ true makes $\varphi$ true.
This is written $\Gamma \models \varphi$.
It is the semantic notion of "follows from", in contrast with derivability $\Gamma \vdash \varphi$ in a proof system.
[[Soundness|4A84VHDTHV]] and [[Gödel's completeness theorem|WQ1ZE255SF]] say that the two notions agree in first-order logic.

## Explanations

### L1 human

A conclusion follows logically from premises when there is no way to make the premises true and the conclusion false.
For example, from "if it rains, the street is wet" and "it rains", it follows that "the street is wet".
This depends only on the form of the sentences, not on facts about weather.
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<document source="https://tamnd.github.io/math2/node/1HM7ZYMPD7.md" tag="1HM7ZYMPD7" kind="concept">
# Monoid ring

## Summary

The monoid ring $R[M]$ of a [[monoid|0N6R902FG1]] $M$ over a [[ring|0B1261B5ED]] $R$ is the ring of finite formal sums $\sum r_m m$, multiplied with the monoid law.
It contains $R$ and a copy of $M$ inside its multiplicative monoid.
Polynomial rings, Laurent polynomial rings and group rings are all monoid rings.
It is the free way to turn a monoid into a ring with coefficients in $R$.

## Explanations

### L1 human

Take the elements of a monoid as building blocks and form linear combinations of them with coefficients in a ring.
Multiply two combinations by expanding and multiplying the blocks with the monoid law.
For the monoid $\set{1, x, x^2, \dots}$ of powers of one letter, this gives the ordinary polynomials.
For example, $(1 + x)(2 + x^2) = 2 + 2x + x^2 + x^3$.
Changing the monoid changes the ring: the integers under addition give polynomials with negative exponents too.
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# Telegrapher's equations

## Summary

The telegrapher's equations are a pair of linear first-order equations for the voltage and the current along an electrical transmission line.
They were developed by Oliver Heaviside to describe signals on telegraph cables.
Eliminating the current gives one second-order equation for the voltage, the telegraph equation, which is a damped [[wave equation|5EHD2XBQDG]].
The equations show how resistance and leakage make a signal both decay and spread out.

## Explanations

### L1 human

A long cable has, per unit length, a resistance, an inductance, a capacitance between its wires, and a small leakage.
A voltage pulse sent along the cable travels like a wave, but it loses strength and changes shape.
The telegrapher's equations describe this with two equations linking how voltage and current change in space and in time.
Without resistance and leakage, they reduce to the wave equation with speed $1/\sqrt{LC}$, and a pulse travels without distortion.
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<document source="https://tamnd.github.io/math2/node/1HRMT3DSX2.md" tag="1HRMT3DSX2" kind="object">
# Octonions

## Description

The *octonions* $\mathbb{O}$ form an $8$-dimensional real algebra.
One construction is the Cayley–Dickson construction applied to the [[quaternions|0GE4DVQKR8]] $\mathbb{H}$.
An octonion is a pair $(a, b)$ of quaternions, with componentwise addition and the product
$$(a, b)(c, d) = (ac - \bar{d} b,\; da + b \bar{c}),$$
where $\bar{q}$ is the quaternion conjugate.
The conjugate of $(a, b)$ is $(\bar{a}, -b)$, and the norm is $N(a, b) = \abs{a}^2 + \abs{b}^2$.
The real numbers sit in $\mathbb{O}$ as the pairs $(r, 0)$ with $r \in \R$, and $(1, 0)$ is the unit.

## Remarks

$\mathbb{O}$ is not associative, so it is a [[nonassociative ring|YP5TQ6WFVN]] and not a ring in the usual sense.
Let $i, j, k$ be the quaternion units, identified with $(i, 0)$, $(j, 0)$, $(k, 0)$, and let $\ell = (0, 1)$.
The product formula gives $(a, 0)\,\ell = (0, a)$ and $(a, 0)(0, d) = (0, da)$.
Hence $(ij)\ell = k\ell = (0, k)$, while $i(j\ell) = (i, 0)(0, j) = (0, ji) = (0, -k)$, so $(ij)\ell \ne i(j\ell)$.
$\mathbb{O}$ is alternative: $x(xy) = (xx)y$ and $(yx)x = y(xx)$ for all $x, y$, and any two elements generate an associative subalgebra.
The norm is multiplicative, $N(xy) = N(x) N(y)$, so $\mathbb{O}$ is a normed division algebra.
By Hurwitz's theorem, $\R$, $\C$, $\mathbb{H}$ and $\mathbb{O}$ are the only normed division algebras over $\R$ with a unit.
The automorphism group of $\mathbb{O}$ is the compact exceptional Lie group $G_2$.

## Explanations

### L1 human

The real numbers, the complex numbers and the quaternions can each be doubled to get the next one: complex numbers are pairs of reals, quaternions are pairs of complex numbers.
Doubling the quaternions once more gives the octonions, eight real coordinates.
At each step something is lost: the complex numbers lose the ordering, the quaternions lose commutativity, and the octonions lose associativity.
Yet one can still divide by any nonzero octonion, and the length of a product is the product of the lengths.
After the octonions this stops: the next doubling has zero divisors.
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# Random matrix

## Summary

A random matrix is a matrix whose entries are [[random variables|0PNRKMRC55]] on a common probability space.
Random matrix theory studies the laws of its [[eigenvalues|0BQAA2BK3Q]] and singular values, mostly when the size of the matrix grows.
For a large symmetric matrix with independent centred entries of equal variance, the scaled eigenvalues follow the [[Wigner semicircle distribution|3QTT95DMFH]].
Random matrices are used in nuclear physics, multivariate statistics, number theory and wireless communication.

## Explanations

### L1 human

Fill a square table with numbers drawn at random, for example by coin tosses or from a normal distribution.
The result is a random matrix, and its eigenvalues are random numbers too.
For one small matrix the eigenvalues look erratic.
For a large symmetric matrix with independent entries of mean $0$ and equal variance, a histogram of the eigenvalues, after scaling, has an almost fixed shape: a half circle.
This shows that randomness at the level of entries can create order at the level of the spectrum.
</document>

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# Evans's theorem on word problems

## Statement

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of finite signature.
A finite [[partial algebra|H7RX5729ZB]] $P$ of the same signature *embeds in* $\mathcal{V}$ if there are $B \in \mathcal{V}$ and an injective map $e \colon P \to B$ with the following property.
Whenever $f(p_1, \ldots, p_n)$ is defined in $P$, one has $e(f(p_1, \ldots, p_n)) = f(e(p_1), \ldots, e(p_n))$ in $B$.
Then the following are equivalent:

1. The uniform [[word problem|7ZKBV18BFG]] for finitely presented algebras of $\mathcal{V}$ is solvable.
2. There is an algorithm that decides, for any finite partial algebra $P$ of the signature, whether $P$ embeds in $\mathcal{V}$.

## Remarks

From 1 to 2: take the [[presentation|43M1WFPYGQ]] whose generators are the elements of $P$ and whose relations are the entries of the partial operation tables.
Then $P$ embeds in $\mathcal{V}$ exactly when distinct elements of $P$ stay distinct in the presented algebra.
From 2 to 1: given relations $R$ and terms $s, t$, let $S$ be the finite set of subterms of $s$, $t$ and the terms in $R$.
Then $s \ne t$ in $\langle X \mid R \rangle_{\mathcal{V}}$ exactly when some quotient of $S$, as a partial algebra, satisfies $R$, separates $s$ and $t$, and embeds in $\mathcal{V}$.
There are finitely many quotients to test.
A useful sufficient condition: if $\mathcal{V}$ is finitely based and every finite partial algebra that embeds in $\mathcal{V}$ embeds in a finite algebra of $\mathcal{V}$, then condition 2 holds.
Indeed, one can search at the same time for two things.
One is a finite algebra of $\mathcal{V}$ in which $P$ embeds.
The other is an equational derivation that identifies two distinct elements of $P$ in the presented algebra.
Example: Boolean algebras are finitely based.
A finite partial Boolean algebra that embeds in some Boolean algebra embeds in the finite subalgebra it generates.
So finitely presented Boolean algebras have solvable word problem.

## Explanations

### L1 human

The word problem asks whether two expressions are equal in an algebra given by generators and relations.
The theorem says that this is the same question as deciding whether a small finite table of partial operations can be completed inside some algebra of the variety.
So word problems can be attacked by thinking about finite pieces of algebras.

### L2 human

The theorem, due to Trevor Evans, connects the word problem with embeddings of partial algebras.
Combined with the finite embeddability property, it explains why finitely based varieties such as Boolean algebras have solvable word problems.
It also explains why unsolvability results for semigroups and groups are hard: one must build finite partial algebras whose embeddability encodes a halting problem.
</document>

<document source="https://tamnd.github.io/math2/node/1HTW0F4ZZ7.md" tag="1HTW0F4ZZ7" kind="statement">
# Giroux correspondence

## Statement

Let $M$ be a closed oriented [[$3$-manifold|R20TB2KEQQ]].
An *open book decomposition* of $M$ is a pair $(B, \pi)$ of the following kind.
The *binding* $B \subset M$ is an oriented link, and $\pi \colon M \setminus B \to S^1$ is a fibration.
Each fiber $\pi^{-1}(\theta)$ is the interior of a compact surface $\Sigma_\theta$, a *page*, with oriented boundary $\partial \Sigma_\theta = B$.
A cooriented positive contact structure $\xi$ on $M$ is *supported* by $(B, \pi)$ if $\xi = \ker \alpha$ for some contact form $\alpha$ with two properties.
The form $\alpha$ is positive on the oriented binding $B$, and $d\alpha$ is a positive area form on each page.
Then:

1. Every open book decomposition of $M$ supports a [[contact structure|HBQ8BZ7XB5]], and any two contact structures supported by the same open book are isotopic.
2. Every positive contact structure on $M$ is supported by some open book decomposition.
3. Two open book decompositions support isotopic contact structures if and only if they become isotopic after finitely many positive stabilizations of each.

So supporting gives a bijection between two sets.
The first is the set of positive contact structures on $M$ up to isotopy.
The second is the set of open books of $M$ up to isotopy and positive stabilization.

## Remarks

Part 1, the existence, goes back to Thurston and Winkelnkemper; Giroux proved the uniqueness and parts 2 and 3.
The proof of part 2 builds a contact cell decomposition of $M$ and uses a ribbon of its Legendrian $1$-skeleton as a page.
A positive stabilization of an open book adds a $1$-handle to the page and composes the monodromy with a positive Dehn twist along a curve crossing the new handle once.

Example: the open book of $S^3$ whose binding is an unknot and whose pages are discs supports the standard tight contact structure of $S^3$.
The open book with annulus pages and monodromy a negative Dehn twist about the core supports an overtwisted contact structure on $S^3$.

## Explanations

### L1 human

An open book describes a closed three-dimensional space as a family of surfaces, the pages, that all share the same boundary, the binding.
They look like the pages of a book turning around its spine.
Giroux's theorem says that contact structures on such a space correspond exactly to open books, up to a simple move called positive stabilization.
This turns questions about twisting plane fields into questions about surfaces and their diffeomorphisms.

### L2 human

The correspondence lets one study contact $3$-manifolds through the monodromy of an open book, an element of the mapping class group of the page.
For example, a contact structure is Stein fillable exactly when it is supported by some open book whose monodromy is a product of positive Dehn twists.
This is due to work of Giroux, of Loi and Piergallini, and of Akbulut and Ozbagci.
It also gives a route from contact structures to Heegaard Floer invariants, through the contact invariant of Ozsváth and Szabó.
</document>

<document source="https://tamnd.github.io/math2/node/1J16RVRHYS.md" tag="1J16RVRHYS" kind="concept">
# Twelvefold way

## Summary

The twelvefold way is a table of twelve basic counting problems, see the [[definition|3BBE4XY4E9]].
Each problem counts maps from a finite set $N$ to a finite set $X$.
The maps may be arbitrary, injective or surjective.
They may be counted as they are, or up to permutations of $N$, of $X$, or of both.
The twelve answers bring together powers, [[falling factorials|0SJNCJCCQG]], [[binomial coefficients|0GBZM0MT0F]], [[Stirling numbers of the second kind|0JCQGPY8Z4]] and [[integer partitions|0JV30XN73Z]].

## Explanations

### L1 human

Think of $n$ balls that you put into $x$ boxes.
The balls may be labelled or identical, and the boxes may be labelled or identical.
You may also require at most one ball per box, or at least one ball per box.
Each choice gives one counting problem, and there are $2 \cdot 2 \cdot 3 = 12$ of them.
For example, $3$ labelled balls go into $2$ labelled boxes in $2^3 = 8$ ways, but $3$ identical balls go into $2$ labelled boxes in only $4$ ways.
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# Feedback linearization

## Summary

Feedback linearization transforms a nonlinear [[control system|ZPDTVZQN5K]] into a linear one by a change of state coordinates and a feedback law that cancels the nonlinearities.
After the transformation, the tools of linear control, such as eigenvalue placement, can be used to design a controller.
Exact conditions for it are given in terms of [[Lie brackets|68FGQEBD6J]] of the vector fields of the system.

## Explanations

### L1 human

Many machines obey nonlinear equations, which are hard to control directly.
If the input enters the equations in the right place, one can choose the input to cancel the nonlinear terms exactly.
What is left is a linear system, which is well understood.
For a pendulum $x_1' = x_2$, $x_2' = -\sin x_1 + u$, the choice $u = \sin x_1 + v$ gives $x_1' = x_2$, $x_2' = v$, a linear system in the new input $v$.
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# Positive-definite function on a group

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] with identity $e$.
A continuous function $\varphi \colon G \to \C$ is *positive-definite* if for every $n \ge 1$, all $g_1, \dots, g_n \in G$ and all $c_1, \dots, c_n \in \C$,
$$\sum_{i=1}^{n} \sum_{j=1}^{n} \overline{c_i} \, c_j \, \varphi(g_i^{-1} g_j) \ge 0.$$

## Remarks

Equivalently, each matrix $\big(\varphi(g_i^{-1}g_j)\big)_{i,j=1}^{n}$ is Hermitian and [[positive semidefinite|0KX66CZMBB]].
It follows that $\varphi(e) \ge 0$, $\varphi(g^{-1}) = \overline{\varphi(g)}$ and $\abs{\varphi(g)} \le \varphi(e)$ for all $g$.
For a discrete group, or when continuity is not part of the setting, the same condition defines positive-definite functions without the continuity requirement.

Example: let $\pi$ be a [[unitary representation|74BPYJNZTM]] of $G$ on $H$ and $v \in H$, and put $\varphi(g) = \ip{\pi(g)v}{v}$.
Then $\varphi$ is continuous, and
$$\sum_{i,j} \overline{c_i} c_j \ip{\pi(g_i^{-1}g_j)v}{v} = \sum_{i,j} \overline{c_i} c_j \ip{\pi(g_j)v}{\pi(g_i)v} = \Big\lVert \sum_j c_j \pi(g_j)v \Big\rVert^2 \ge 0.$$

Non-example: the constant function $\varphi \equiv -1$.
With $n = 1$ and $c_1 = 1$, the sum is $\varphi(e) = -1 < 0$.

Conversely, every continuous positive-definite function has the form $g \mapsto \ip{\pi(g)v}{v}$ for some unitary representation $\pi$ and a cyclic vector $v$, by a construction of Gelfand–Naimark–Segal type.
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# Mayer and Lagrange problems

## Problem

Let $U \subseteq \R^m$ be a nonempty set, and let $t_0 < t_1$ be fixed times.
Let $f \colon [t_0, t_1] \times \R^n \times U \to \R^n$ be continuous, and let $S \subseteq \R^n \times \R^n$ be a closed set of allowed endpoint pairs.
An [[admissible control|8M31QKB0GW]] is a measurable map $u \colon [t_0, t_1] \to U$ whose trajectory $x$ of the [[control system|ZPDTVZQN5K]] $x'(t) = f(t, x(t), u(t))$ is absolutely continuous on $[t_0, t_1]$ and satisfies $(x(t_0), x(t_1)) \in S$.

1. The *Mayer problem*: given a continuous function $\varphi \colon \R^n \times \R^n \to \R$, minimize $J(u) = \varphi(x(t_0), x(t_1))$ over all admissible controls $u$.
2. The *Lagrange problem*: given a continuous function $L \colon [t_0, t_1] \times \R^n \times U \to \R$, minimize the [[functional|1EQM0DJ4NC]] $J(u) = \int_{t_0}^{t_1} L(t, x(t), u(t)) \, dt$ over all admissible controls $u$.

## Remarks

The Mayer problem has only an endpoint cost, and the Lagrange problem has only a running cost.
Both are special cases of the Bolza problem, whose cost is the sum of the two.
They are equivalent to each other.
A Lagrange problem becomes a Mayer problem with the extra state $y' = L(t, x, u)$, $y(t_0) = 0$, and the endpoint cost $y(t_1)$.
A Mayer problem with $\varphi$ of class $C^1$ that depends only on $x(t_1)$, and a fixed initial state $x(t_0) = a$, becomes a Lagrange problem with $L(t, x, u) = \nabla \varphi(x) \cdot f(t, x, u)$, since $\varphi(x(t_1)) = \varphi(a) + \int_{t_0}^{t_1} \nabla\varphi(x(t)) \cdot x'(t) \, dt$.
The Mayer form is convenient for existence proofs and for geometric arguments with reachable sets.
The Lagrange form is the classical form of the calculus of variations.

## Explanations

### L1 human

In a Mayer problem one only cares about where the system ends up, for example the final altitude of a rocket.
In a Lagrange problem one only cares about the total cost along the way, for example the fuel spent.
A simple trick of bookkeeping, adding a counter for the cost so far, turns one kind of problem into the other.
</document>

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# Fundamental strip of a Mellin transform

## Statement

Let $f \colon (0, \infty) \to \C$ be integrable on every compact subinterval $[u, v] \subset (0, \infty)$.
Let $a, b \in \R$ with $b < a$, and assume that
$$f(x) = O(x^{-a}) \text{ as } x \to \infty, \qquad f(x) = O(x^{-b}) \text{ as } x \to 0^+.$$
Then $f$ is [[Mellin convergent|SGZZ4Y0D36]] at every $s \in \C$ with $b < \operatorname{Re} s < a$.
So the [[Mellin transform|N5PVGRJE1T]] $\mathcal{M}f(s)$ is defined on the open strip $b < \operatorname{Re} s < a$.

Here $O$ is [[big O notation|0TGV80E6A4]].

## Remarks

Let $\sigma = \operatorname{Re} s$.
Choose $0 < \delta < 1 < R$ with $\abs{f(x)} \le C x^{-b}$ on $(0, \delta]$ and $\abs{f(x)} \le C x^{-a}$ on $[R, \infty)$.
Then $x^{\sigma - 1} \abs{f(x)} \le C x^{\sigma - b - 1}$ on $(0, \delta]$, which is integrable since $\sigma - b > 0$.
Also $x^{\sigma - 1} \abs{f(x)} \le C x^{\sigma - a - 1}$ on $[R, \infty)$, which is integrable since $\sigma - a < 0$.
On $[\delta, R]$, $f$ is integrable and $x^{\sigma - 1}$ is bounded.

The largest open strip $\alpha < \operatorname{Re} s < \beta$ on which $f$ is Mellin convergent is called the fundamental strip and is written $\langle \alpha, \beta \rangle$.
Example: $f(x) = e^{-x}$ is $O(x^0)$ at $0$ and $O(x^{-a})$ at $\infty$ for every $a$, so its fundamental strip is $\langle 0, \infty \rangle$, where $\mathcal{M}f = \Gamma$.
Example: $f(x) = 1/(1 + x)$ is $O(1)$ at $0$ and $O(x^{-1})$ at $\infty$, giving the strip $\langle 0, 1 \rangle$.
The strict inequalities are needed: for this $f$ at $s = 1$, the bound $x^{\sigma - a - 1} = x^{-1}$ is not integrable at $\infty$, and indeed $\int_0^\infty dx/(1+x)$ diverges.

## Explanations

### L1 human

The Mellin integral has two dangerous ends, $0$ and $\infty$.
Assume that $f$ grows at most like $x^{-b}$ near $0$ and decays at least like $x^{-a}$ near $\infty$.
Then both ends are safe for every $s$ whose real part lies strictly between $b$ and $a$.
So the transform lives on a vertical strip whose edges are read off from the behaviour of $f$ at the two ends.

### L2 human

This is the first step in every computation with Mellin transforms: one finds the strip before anything else.
The edges of the strip are tied to the asymptotics of $f$.
Conversely, under suitable conditions, poles of the continued transform beyond the strip give terms of the asymptotic expansion of $f$ at $0$ or at $\infty$.
On the open strip the transform is also holomorphic, which is the companion result on differentiability of the Mellin transform.
Under the substitution $x = e^{-u}$, the statement becomes the familiar fact that a function with exponential decay at both ends of $\R$ has a two-sided Laplace transform on a strip.
</document>

<document source="https://tamnd.github.io/math2/node/1JCQTY728A.md" tag="1JCQTY728A" kind="definition">
# Semifield

## Definition

A *semifield* is a set $S$ with two binary operations $+$ and $\cdot$ such that:

1. $(S, +)$ is an [[abelian group|082GHNPJ53]], with identity element $0$.
2. Both distributive laws hold: $a(b + c) = ab + ac$ and $(a + b)c = ac + bc$ for all $a, b, c \in S$.
3. There is an element $1 \ne 0$ with $1 \cdot a = a \cdot 1 = a$ for all $a \in S$.
4. For all $a, b \in S$ with $a \ne 0$, each of the equations $a x = b$ and $y a = b$ has exactly one solution $x \in S$ and exactly one solution $y \in S$.

Multiplication is not required to be associative or commutative.

## Remarks

Clauses 3 and 4 say that $S \setminus \set{0}$ is a loop under multiplication.
A structure that satisfies clauses 1, 2 and 4 but not necessarily 3 is called a presemifield.
If $S$ is finite, clause 4 is equivalent to the absence of zero divisors: $ab = 0$ implies $a = 0$ or $b = 0$.
An associative semifield is a [[division ring|0EA5BTA0CX]], and a finite associative semifield is a finite [[field|04C5ZBG1WT]] by Wedderburn's little theorem.

Example: every field is a semifield, since it satisfies all four clauses and is moreover associative.
The real [[octonions|1HRMT3DSX2]] are a semifield in which multiplication is not associative.

Non-example: the ring $\Z$ satisfies clauses 1, 2 and 3 but fails clause 4, because $2x = 1$ has no solution $x \in \Z$.

In another usage, found for instance in tropical geometry, a semifield is a commutative semiring in which every nonzero element has a multiplicative inverse, such as $(\R \cup \set{-\infty}, \max, +)$.
</document>

<document source="https://tamnd.github.io/math2/node/1JD2FVGBM2.md" tag="1JD2FVGBM2" kind="definition">
# Arithmetical variety

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
The variety $\mathcal{V}$ is *arithmetical* if it is both a [[congruence-permutable variety|1B3N9T68YV]] and a [[congruence-distributive variety|9GZ7F6ETC6]].

## Remarks

Equivalently, $\mathcal{V}$ is arithmetical if and only if it has a Pixley term $p$, with $p(x, y, y) = x$, $p(x, x, y) = y$ and $p(x, y, x) = x$.
Example: the variety of Boolean algebras is arithmetical.
It is congruence distributive because Boolean algebras are lattices, and it is congruence permutable because the term $p(x, y, z) = (x \wedge \neg y) \vee (x \wedge z) \vee (\neg y \wedge z)$ is a Mal'cev term.
Non-example: the variety of groups is congruence permutable but not congruence distributive, so it is not arithmetical.
Non-example: the variety of lattices is congruence distributive but not congruence permutable.

## Explanations

### L1 human

A variety is arithmetical when the congruences of each of its algebras both permute and satisfy the distributive law, as the congruences of the integers do.
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<document source="https://tamnd.github.io/math2/node/1JFKTEQVRX.md" tag="1JFKTEQVRX" kind="statement">
# Cramér–Rao bound

## Statement

Let $\set{f(\cdot\,; \theta) : \theta \in \Theta}$ be a family of densities with respect to a fixed measure $\mu$, where $\Theta \subseteq \R$ is an open interval.
Assume the following regularity conditions.

1. The set $\set{x : f(x; \theta) > 0}$ does not depend on $\theta$.
2. For every such $x$, the map $\theta \mapsto f(x; \theta)$ is differentiable.
3. The [[Fisher information|G1Z3WSTE8Q]] $I(\theta)$ satisfies $0 < I(\theta) < \infty$ for every $\theta$.

Let $X$ have density $f(\cdot\,; \theta)$, and let $T = T(X)$ be a [[statistic|0Q5CG2NEQ8]] with $E_\theta[T^2] < \infty$ for every $\theta$.
Put $\psi(\theta) = E_\theta[T]$.
Assume that differentiation in $\theta$ may be taken under the integral sign in $\int f(x; \theta) \, \mu(dx) = 1$ and in $\int T(x) f(x; \theta) \, \mu(dx) = \psi(\theta)$.
Then $\psi$ is differentiable, and for every $\theta \in \Theta$,
$$\operatorname{Var}_\theta(T) \ge \frac{\psi'(\theta)^2}{I(\theta)}.$$
In particular, if $T$ is an [[unbiased|0BWGTN92FG]] [[estimator|0SSS6ERQ0S]] of $\theta$, then $\operatorname{Var}_\theta(T) \ge 1/I(\theta)$.

## Remarks

Let $s$ be the [[score|CP97GRVTQ0]].
Differentiating the two integrals gives $E_\theta[s] = 0$ and $E_\theta[T s] = \psi'(\theta)$, so $\operatorname{Cov}_\theta(T, s) = \psi'(\theta)$.
The [[Cauchy–Schwarz inequality|03PGHKAJG3]] then gives $\psi'(\theta)^2 \le \operatorname{Var}_\theta(T) \operatorname{Var}_\theta(s) = \operatorname{Var}_\theta(T) \, I(\theta)$.
For $n$ independent observations the information is $n I(\theta)$, so the bound becomes $\psi'(\theta)^2 / (n I(\theta))$.
In dimension $d$, the bound reads $\operatorname{Cov}_\theta(T) \succeq D \, I(\theta)^{-1} D^{\mathsf T}$, with $D$ the Jacobian matrix of $\psi$.

Example: for $n$ independent normal observations with mean $\theta$ and known variance $\sigma^2$, one has $I(\theta) = 1/\sigma^2$.
The sample mean is unbiased with variance $\sigma^2/n = 1/(n I(\theta))$, so it attains the bound.

The fixed support is necessary.
For $n$ independent uniform observations on $(0, \theta)$, the unbiased estimator $\frac{n+1}{n} \max_i X_i$ has variance $\theta^2 / (n(n+2))$.
This is smaller than $\theta^2/n$, the value of the formal bound with $E_\theta[s^2] = 1/\theta^2$.

## Explanations

### L1 human

No unbiased estimate can be more precise than a limit fixed by the model itself.
The limit is one over the Fisher information, so models whose data carry much information allow precise estimates.
An estimator that reaches the limit cannot be improved among unbiased estimators.

### L2 human

The bound is the benchmark for the efficiency of estimators: the ratio of the bound to the actual variance measures how much information an estimator wastes.
It is attained exactly only in special cases, essentially for exponential families and the natural statistic.
Asymptotically, under regularity conditions, the maximum likelihood estimator has variance close to $1/(n I(\theta))$ and so reaches the bound in the limit.
The regularity conditions matter: models with a moving support, such as the uniform family, can be estimated faster than the bound allows.
</document>

<document source="https://tamnd.github.io/math2/node/1JG3HFWX3T.md" tag="1JG3HFWX3T" kind="concept">
# Singularity of a smooth map

## Summary

A singularity, or singular point, of a smooth map between [[smooth manifolds|F45VQ47GPV]] is a point where the [[differential|6P9PXQDSEN]] has less than the largest possible rank.
At all other points the map is locally a linear projection or a linear inclusion, by the rank theorem.
So the singular points are the only places where the local shape of a map can be interesting.
Their study and classification is the subject of [[singularity theory|42V5XGZS4F]] of smooth maps.

## Explanations

### L1 human

Think of a smooth map as a way of projecting, folding or pressing one space onto another.
Where the map is regular, it looks like a flat projection, with nothing special happening nearby.
A singular point is a place where the map folds, pinches or crushes directions together.
For example, the map $x \mapsto x^2$ from the line to the line folds the line in half at $0$, and $0$ is its only singular point.
Shadows of curved objects show the same thing: the outline of the shadow is the image of the singular points of the projection.
</document>

<document source="https://tamnd.github.io/math2/node/1JJCGNX240.md" tag="1JJCGNX240" kind="concept">
# Categorical theory

## Summary

A theory is categorical in an infinite cardinal $\kappa$ when it has, up to isomorphism, exactly one model of size $\kappa$.
By the Löwenheim–Skolem theorems, a theory with an infinite model has models of many sizes.
So categoricity is always relative to a cardinal.
Categoricity gives completeness through the [[Łoś–Vaught test|ZR7Y5C235P]].
For countable theories, [[Morley's categoricity theorem|V7N84P60SJ]] says that categoricity in one uncountable cardinal implies it in all of them.
The study of categorical theories was the starting point of stability theory.

## Explanations

### L1 human

A theory is a list of axioms, and it usually has many different models.
A theory is categorical in a size when all its models of that size are copies of one another.
For example, any two countable dense linear orders without endpoints, such as the rationals and the rationals between $0$ and $1$, are isomorphic as orders.
So the theory of such orders is categorical in the countable size, but it is not categorical in larger sizes.
</document>

<document source="https://tamnd.github.io/math2/node/1JM66EN5R2.md" tag="1JM66EN5R2" kind="definition">
# Elliptic cohomology

## Definition

Let $E$ be a multiplicative [[generalized cohomology theory|H28PSZR7XG]] and let $R = E^0(\mathrm{pt})$, a [[commutative ring|0310REJJY8]].
An *elliptic cohomology theory* is a triple $(E, C, \phi)$ such that:

1. $E$ is even periodic: $E^i(\mathrm{pt}) = 0$ for all odd $i$, and $E^2(\mathrm{pt})$ contains a unit of the graded ring $E^*(\mathrm{pt})$.
2. $C$ is an [[elliptic curve|SJ2BVG9F4W]] over $R$.
3. $\phi$ is an isomorphism of formal groups over $R$ from $\operatorname{Spf} E^0(\C P^\infty)$ to the formal completion $\hat C$ of $C$ at its identity.

## Remarks

In coordinates, condition 3 says the following for a suitable complex orientation of $E$.
The [[formal group law|TNHP7XXYZT]] of $E$ is the expansion of the group law of $C$ in a local parameter at the identity.

Example: let $C$ be an elliptic curve over $R$ such that the map from $\operatorname{Spec} R$ to the moduli stack of elliptic curves is flat, for instance étale.
Choose a coordinate at the identity of $C$, so that its group law gives a formal group law $F$ over $R$, and let $u$ have cohomological degree $-2$.
Then the formal group law $u^{-1} F(u x, u y)$ over $R[u, u^{-1}]$ satisfies the hypothesis of the Landweber exact functor theorem.
So $E^*(X) = MU^*(X) \otimes_{MU^*} R[u, u^{-1}]$ is a cohomology theory, it is even periodic with the unit $u^{-1}$ in degree $2$, and its formal group is $\hat C$ by construction.
An earlier, closely related theory was built by Landweber, Ravenel and Stong from the Jacobi quartic.

Non-example: ordinary cohomology $H^*(-; \Z)$ fails condition 1, since $H^2(\mathrm{pt}; \Z) = 0$ contains no unit.

Complex K-theory is even periodic, but its formal group is the multiplicative one over $\Z$, and there is no elliptic curve over $\Z$ at all.
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<document source="https://tamnd.github.io/math2/node/1JNFB7XBDN.md" tag="1JNFB7XBDN" kind="definition">
# Collectively compact operator approximation

## Definition

Let $X$ be a [[Banach space|06GZ956657]] with closed unit ball $B$.
Let $\mathcal{K}$ be a set of bounded linear operators $X \to X$.
The set $\mathcal{K}$ is *collectively compact* if
$$\set{L\psi : L \in \mathcal{K}, \ \psi \in B}$$
is relatively compact in $X$.
This means that its closure is [[compact|0R6EWFKRCF]].
A sequence $(K_n)$ of bounded linear operators on $X$ is a *collectively compact approximation* of a bounded linear operator $K$ on $X$ if:

1. $K_n \psi \to K\psi$ for every $\psi \in X$, and
2. the set $\set{K_n : n \ge 1}$ is collectively compact.

## Remarks

Every operator in a collectively compact set is [[compact|M2NEP1K5AE]], and the limit $K$ in the definition is compact as well.
The main result assumes that $K_n$ is a collectively compact approximation of $K$ and that $I - K$ is injective.
Then for large $n$ the operators $I - K_n$ are invertible with uniformly bounded inverses.
The solutions of $\varphi_n - K_n\varphi_n = f$ then satisfy $\norm{\varphi_n - \varphi} \le C \norm{K_n\varphi - K\varphi}$.
The proof uses that $\norm{(K_n - K) K} \to 0$ and $\norm{(K_n - K) K_n} \to 0$, although $\norm{K_n - K}$ need not tend to $0$.

Example: let $X = C[0, 1]$, let $k$ be continuous on $[0, 1]^2$, and let $(K_n\psi)(x) = \frac{1}{n}\sum_{j=1}^{n} k(x, \frac{j}{n}) \psi(\frac{j}{n})$.
For $\psi \in B$ one has $\abs{K_n\psi(x)} \le \max \abs{k}$ and $\abs{K_n\psi(x) - K_n\psi(x')} \le \max_t \abs{k(x, t) - k(x', t)}$, so the family is uniformly bounded and equicontinuous, hence relatively compact by the [[Arzelà–Ascoli theorem|0381S8Z5P0]].
Riemann sums converge, so $K_n\psi \to K\psi$ uniformly with $(K\psi)(x) = \int_0^1 k(x, t) \psi(t) \, dt$.
In general $\norm{K_n - K} \not\to 0$ here: for $k = 1$, functions $\psi$ in $B$ that equal $-1$ at the nodes and have integral close to $1$ show $\norm{K_n - K} = 2$ for every $n$.

Non-example: on $X = \ell^2$, let $K_n$ be the projection onto the first $n$ coordinates.
Then $K_n \psi \to \psi$ for every $\psi$, but the set $\set{K_n e_n} = \set{e_n}$ has no convergent subsequence, so clause 2 fails; indeed the limit $I$ is not compact.
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<document source="https://tamnd.github.io/math2/node/1JPGWFMAGK.md" tag="1JPGWFMAGK" kind="definition">
# Hopf bifurcation

## Definition

Let $n \ge 2$, let $f \colon \R^n \times \R \to \R^n$ be [[smooth|0V33BNKZ9S]], and consider the family of systems $x' = f(x, \mu)$ with parameter $\mu$.
Assume $f(0, \mu) = 0$ for all $\mu$ near $0$, so that $0$ is an [[equilibrium point|0G4X36AE0X]] for each such $\mu$.
Let $A(\mu)$ be the [[Jacobian matrix|047NDTEPCV]] of $x \mapsto f(x, \mu)$ at $0$.
The family has a *Hopf bifurcation* at $\mu = 0$ if:

1. $A(\mu)$ has a pair of simple complex conjugate [[eigenvalues|0BQAA2BK3Q]] $\alpha(\mu) \pm i \beta(\mu)$ for $\mu$ near $0$, with $\alpha(0) = 0$ and $\beta(0) > 0$;
2. no other eigenvalue of $A(0)$ has zero real part;
3. $\alpha'(0) \ne 0$.

## Remarks

Under these conditions, a family of [[periodic orbits|0AA97XA70K]] branches off the equilibrium at $\mu = 0$, with periods close to $2\pi / \beta(0)$.
If moreover a certain coefficient $\ell_1$, the first Lyapunov coefficient, is nonzero, the orbits exist for $\mu$ on one side of $0$ only and have amplitude of order $\sqrt{\abs{\mu}}$.
The bifurcation is supercritical when $\ell_1 < 0$; in the plane the orbits are then stable limit cycles.
It is subcritical when $\ell_1 > 0$; the orbits are then unstable.

Example: $x' = \mu x - y - x(x^2 + y^2)$, $y' = x + \mu y - y(x^2 + y^2)$.
The Jacobian at $0$ is $\begin{pmatrix} \mu & -1 \\ 1 & \mu \end{pmatrix}$, with eigenvalues $\mu \pm i$.
So $\alpha(\mu) = \mu$, $\beta(\mu) = 1$, and all three clauses hold.
In polar coordinates the system is $r' = \mu r - r^3$, $\theta' = 1$, and the circle $r = \sqrt{\mu}$ is a stable limit cycle for $\mu > 0$.

Non-example: $x' = \mu^2 x - y$, $y' = x + \mu^2 y$.
The eigenvalues are $\mu^2 \pm i$, so $\alpha(\mu) = \mu^2$ and $\alpha'(0) = 0$; clause 3 fails.
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# Unramified morphism

## Summary

An unramified morphism of schemes is a morphism, locally of finite type, whose fibres are disjoint unions of spectra of finite separable field extensions.
It extends to schemes the notion of an unramified extension of number fields.
An unramified morphism that is also flat and locally of finite presentation is étale.

## Explanations

### L1 human

A map is unramified when no two sheets of it come together at a point.
For the squaring map $t \mapsto t^2$ on the line, over a field of characteristic not $2$, each nonzero value has two preimages.
The value $0$ has only one preimage, where the two sheets touch: the map is ramified there and unramified elsewhere.
In number theory, a prime $p$ ramifies in a number field when $p$ has a repeated prime ideal factor in the ring of integers.
This is the same phenomenon.
</document>

<document source="https://tamnd.github.io/math2/node/1JV7093B4C.md" tag="1JV7093B4C" kind="problem">
# Non-negative least squares

## Problem

Let $A \in \R^{m \times n}$ and $b \in \R^m$.
The *non-negative least squares problem* (NNLS) is to find
$$x^* \in \arg\min \set{ \norm{Ax - b}_2^2 : x \in \R^n, \ x \ge 0 },$$
where $x \ge 0$ means $x_i \ge 0$ for every $i$.

## Remarks

The problem always has a solution.
The image of the non-negative orthant under $A$ is a closed convex cone, and $Ax^*$ is the nearest point of that cone to $b$.
The objective is a [[convex function|07Q7CNM5WF]], so NNLS is a convex quadratic programming problem, and the solution is unique when $A$ has full column rank.
By the Karush–Kuhn–Tucker conditions, $x$ is a solution if and only if, with $w = A^T(b - Ax)$, one has $x \ge 0$, $w \le 0$ and $x_i w_i = 0$ for every $i$.
Example: for $A = I$ the solution is $x^*_i = \max(b_i, 0)$, which satisfies these conditions.
The classical algorithm is the active-set method of Lawson and Hanson, published in their 1974 book *Solving Least Squares Problems*.
It moves variables between a free set and a set fixed at zero.
At each step it solves an unconstrained least squares problem on the free set, and it stops after finitely many steps.
Projected gradient and coordinate descent methods are used for large problems.

## Explanations

### L1 human

Ordinary least squares finds the best fit of $Ax$ to $b$ with no restriction on $x$.
In many applications, such as concentrations, intensities or counts, the unknowns cannot be negative, and the unrestricted fit may give negative values.
NNLS asks for the best fit among non-negative $x$ only.
The answer is the closest point to $b$ in the cone of all non-negative combinations of the columns of $A$.
</document>

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# Fourier transform on Schwartz space

## Statement

Let $n \ge 1$ and let $\mathcal{S}(\R^n)$ be the [[Schwartz space|CG9JEATW3Q]] with its Fréchet topology.
Let $\mathcal{F} f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\, dx$ be the [[Fourier transform|DPVM34ADZV]].
Then $\mathcal{F}$ maps $\mathcal{S}(\R^n)$ onto $\mathcal{S}(\R^n)$, it is a linear bijection, and both $\mathcal{F}$ and its inverse are continuous.
The inverse is the [[inverse Fourier transform|JB8QMT8DCH]]
$$\mathcal{F}^{-1} g(x) = \int_{\R^n} g(\xi)\, e^{2\pi i x \cdot \xi}\, d\xi.$$

## Remarks

By the [[rules for differentiation and multiplication|HHNR31W3WD]], $\xi^\alpha \partial^\beta \hat f$ is, up to constants, the transform of $\partial^\alpha (x^\beta f)$.
That function is in $\mathcal{S}(\R^n) \subset L^1(\R^n)$, so its transform is bounded, which shows $\hat f \in \mathcal{S}(\R^n)$.
The same estimates, with a factor $(1 + \abs{x})^{-n-1}$ to control the integral, give continuity.
Bijectivity is the inversion formula $\mathcal{F}^{-1} \mathcal{F} f = f$, proved by pairing with a Gaussian of shrinking width.
Also $\mathcal{F}^4 f = f$, since $\mathcal{F}^2 f(x) = f(-x)$.

Example: the Gaussian $e^{-\pi \abs{x}^2}$ is a Schwartz function and is fixed by $\mathcal{F}$.

On $\mathcal{S}(\R^n)$ the transform also preserves the $L^2$ norm, $\int \abs{\hat f}^2 = \int \abs{f}^2$.
Since $\mathcal{S}(\R^n)$ is dense in $L^2(\R^n)$, this is how the transform is extended to a unitary map of $L^2(\R^n)$.
By duality it also extends to tempered distributions.
The space $L^1(\R^n)$ has no such property: the transform of an integrable function need not be integrable.

## Explanations

### L1 human

Schwartz functions are smooth and decay fast, and the Fourier transform turns smoothness into decay and decay into smoothness.
So it sends this space to itself, and nothing is lost: every Schwartz function is the transform of exactly one other.

### L2 human

This is the cleanest setting for Fourier analysis on $\R^n$: all formulas (inversion, Plancherel, the convolution theorem, differentiation rules) hold on $\mathcal{S}(\R^n)$ without any conditions.
Results on $L^p$ spaces are then obtained by density, and results on tempered distributions by duality.
</document>

<document source="https://tamnd.github.io/math2/node/1K31QJNYNF.md" tag="1K31QJNYNF" kind="definition">
# Handle decomposition

## Definition

Let $n \ge 1$ and $0 \le k \le n$, and let $D^j$ be the closed unit ball of $\R^j$.
An $n$-dimensional *$k$-handle* is $H = D^k \times D^{n-k}$; its *attaching region* is $\partial D^k \times D^{n-k}$.
Let $X$ be a smooth $n$-dimensional [[manifold with boundary|7DS6B2BFK8]], and let $\varphi \colon \partial D^k \times D^{n-k} \to \partial X$ be a smooth embedding onto a [[submanifold|VKFW27WXEZ]] of $\partial X$.
The result of *attaching a $k$-handle* to $X$ along $\varphi$ is $X \cup_\varphi H$.
It is the union of $X$ and $H$ with $y$ and $\varphi(y)$ identified, with corners smoothed.
Let $M$ be a compact smooth $n$-manifold, possibly with boundary.
A *handle decomposition* of $M$ is a finite chain $\emptyset = M_{-1} \subseteq M_0 \subseteq \cdots \subseteq M_r = M$ of smooth manifolds with boundary.
Each $M_i$ must be obtained from $M_{i-1}$ by attaching one handle, up to diffeomorphism.

## Remarks

The first step attaches a $0$-handle $D^0 \times D^n = D^n$ to the empty set, whose attaching region is empty.
A relative version starts from $\partial_0 M \times [0, 1]$ for a part $\partial_0 M$ of the boundary instead of the empty set; this is the form used for cobordisms.
Example: the sphere $S^n$ has a handle decomposition with one $0$-handle and one $n$-handle: the northern and southern hemispheres, glued along the equator $\partial D^n \times D^0 = S^{n-1}$.
Non-example: gluing a second disc $D^2$ to $D^2$ along a single arc of its boundary circle is not a handle attachment.
In dimension $2$ the attaching region of a $1$-handle is $\partial D^1 \times D^1$, two disjoint arcs.
That of a $2$-handle is the whole circle $\partial D^2 \times D^0$, and that of a $0$-handle is empty; none of them is a single arc.
If $f \colon M \to [a, b]$ is a Morse function with critical values in $(a, b)$, then each critical point of index $k$ gives one $k$-handle in a handle decomposition of $M$ relative to $f^{-1}(a)$.
</document>

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# Sheffer function

## Summary

A Sheffer function on a finite set $A$ is a single operation on $A$ that is [[functionally complete|TDZVCWYJ9F]] by itself.
Every operation on $A$ can then be written as a composition of copies of it.
On $\set{0, 1}$ the binary Sheffer functions are NAND and NOR.
Sheffer functions show that one well-chosen operation can carry the whole expressive power of a many-valued logic.

## Explanations

### L1 human

The NAND gate outputs $0$ only when both inputs are $1$.
Chips can be built from NAND gates alone, because NAND can imitate every other gate.
For example, $\neg x$ is $x \uparrow x$, and $x \wedge y$ is $(x \uparrow y) \uparrow (x \uparrow y)$.
An operation with this "one gate does everything" property is called a Sheffer function.
</document>

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# Transcritical bifurcation

## Definition

Let $f \colon \R \times \R \to \R$ be [[smooth|0V33BNKZ9S]], and consider the family of scalar equations $x' = f(x, \mu)$ with parameter $\mu$.
Assume $f(0, \mu) = 0$ for all $\mu$, so that $0$ is an [[equilibrium point|0G4X36AE0X]] for every $\mu$.
The family has a *transcritical bifurcation* at $(x, \mu) = (0, 0)$ if:

1. $f_x(0, 0) = 0$;
2. $f_{x\mu}(0, 0) \ne 0$;
3. $f_{xx}(0, 0) \ne 0$.

## Remarks

Under these conditions, besides $x = 0$, the equilibria near $(0, 0)$ form a curve $x = c(\mu)$ with $c(0) = 0$ and $c'(0) = -2 f_{x\mu}(0, 0) / f_{xx}(0, 0) \ne 0$.
So this second branch crosses the branch $x = 0$ at $\mu = 0$.
Since $f_x(0, \mu)$ changes sign at $\mu = 0$, the equilibrium $0$ changes stability there, and the other branch changes stability the opposite way.

Example: $f(x, \mu) = \mu x - x^2$.
Then $f(0, \mu) = 0$, $f_x(0, 0) = 0$, $f_{x\mu} = 1$ and $f_{xx} = -2$.
The second branch is $x = \mu$, with $c'(0) = 1 = -2 \cdot 1 / (-2)$.

Non-example: $f(x, \mu) = \mu x - x^3$.
Here $f_{xx}(0, 0) = 0$, so clause 3 fails; this family has a pitchfork bifurcation instead.
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# Kuranishi structure

## Definition

Let $X$ be a compact metrizable space.
A *Kuranishi neighbourhood* of a point $p \in X$ is a tuple $(V_p, E_p, \Gamma_p, s_p, \psi_p)$ where:

1. $V_p$ is a [[smooth manifold|F45VQ47GPV]] and $\Gamma_p$ is a finite group with a smooth [[action|0WCY42EV9V]] on $V_p$.
2. $E_p \to V_p$ is a finite-dimensional real [[vector bundle|P4DZM975BV]] with a $\Gamma_p$-action that lifts the action on $V_p$.
3. $s_p$ is a $\Gamma_p$-equivariant smooth [[section|WWX7H5W1RP]] of $E_p$, the Kuranishi map.
4. $\psi_p \colon s_p^{-1}(0) / \Gamma_p \to X$ is a [[homeomorphism|0QT00PB5G2]] onto an open neighbourhood of $p$, where $s_p^{-1}(0) / \Gamma_p$ is the [[quotient space|0BR3FY5Q8M]].

A *Kuranishi structure* on $X$ is a choice of a Kuranishi neighbourhood for every $p \in X$, together with coordinate changes, such that:

5. For each $q \in \psi_p(s_p^{-1}(0) / \Gamma_p)$ there is a coordinate change from the chart at $q$ to the chart at $p$.
It consists of a $\Gamma_q$-invariant open neighbourhood $V_{pq} \subseteq V_q$ of a lift of $q$ and an injective homomorphism $h_{pq} \colon \Gamma_q \to \Gamma_p$.
It also has an $h_{pq}$-equivariant smooth embedding $\phi_{pq} \colon V_{pq} \to V_p$ and an $h_{pq}$-equivariant bundle embedding $\hat{\phi}_{pq} \colon E_q|_{V_{pq}} \to E_p$ over $\phi_{pq}$.
These satisfy $\hat{\phi}_{pq} \circ s_q = s_p \circ \phi_{pq}$, and $\psi_p \circ \phi_{pq} = \psi_q$ on $s_q^{-1}(0) \cap V_{pq}$.
6. The coordinate changes are compatible.
Let $q$ lie in the image of the chart at $p$, and $r$ in the image of the chart at $q$.
Then on the common domain $\phi_{pq} \circ \phi_{qr}$ agrees with $\phi_{pr}$ up to the action of an element of $\Gamma_p$, and likewise for $\hat{\phi}$ and $h$.
7. The number $\dim V_p - \operatorname{rank} E_p$ is the same for all $p$; it is the *virtual dimension* of $X$.
8. Let $N_{pq}$ be the normal bundle of $\phi_{pq}(V_{pq})$ in $V_p$.
For each coordinate change, the derivative of $s_p$ induces an isomorphism from $N_{pq}$ onto $E_p / \hat{\phi}_{pq}(E_q)$ at the points of $\phi_{pq}(s_q^{-1}(0))$.

## Remarks

Example: let $V$ be a smooth manifold, $E \to V$ a vector bundle and $s$ a section whose zero set $X = s^{-1}(0)$ is compact.
Take for every $p$ the same chart $(V, E, \set{1}, s, \id)$, with identity coordinate changes.
All composites of coordinate changes are identities, the virtual dimension is $\dim V - \operatorname{rank} E$ everywhere, and the normal bundles in clause 8 are zero, as are the quotients $E / E$.
So this is a Kuranishi structure on $X$.
In particular a compact manifold $X$ has the Kuranishi structure with $V = X$ and $E = 0$.

Non-example: let $X$ be a single point.
One Kuranishi neighbourhood is $V_A = \R$, $E_A = \R \times \R$, $\Gamma_A$ trivial and $s_A(x) = x^2$, whose zero set is one point; another is $V_B$ a point with $E_B = 0$.
Both have virtual dimension $0$, so clause 7 holds.
But a coordinate change from $B$ to $A$ fails clause 8.
The normal space of the point $0$ in $\R$ is $\R$, the quotient $E_A / \hat\phi(E_B)$ is $\R$, and the derivative $s_A'(0) = 0$ is not an isomorphism.

Several variants of this definition are in use, due to Fukaya, Oh, Ohta and Ono, to Joyce, and to others.
They differ mainly in clauses 6 and 8 and in how the coordinate changes are organised.

Clause 8 is sometimes called the tangent bundle condition, and it is what makes orientations and virtual fundamental classes well defined.
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# Proximity space

## Definition

Let $X$ be a set.
A *proximity* on $X$ is a relation $\delta$ between subsets of $X$, written $A \mathrel{\delta} B$ and read "$A$ is near $B$", such that for all $A, B, C \subseteq X$:

1. If $A \mathrel{\delta} B$, then $B \mathrel{\delta} A$.
2. If $A \mathrel{\delta} B$, then $A \neq \emptyset$.
3. If $A \cap B \neq \emptyset$, then $A \mathrel{\delta} B$.
4. $A \mathrel{\delta} (B \cup C)$ if and only if $A \mathrel{\delta} B$ or $A \mathrel{\delta} C$.
5. If $A \mathrel{\delta} E$ or $B \mathrel{\delta} (X \setminus E)$ holds for every $E \subseteq X$, then $A \mathrel{\delta} B$.

A *proximity space* is a pair $(X, \delta)$ of a set $X$ and a proximity $\delta$ on $X$.

## Remarks

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $A \mathrel{\delta} B$ mean $\inf \set{d(a, b) : a \in A, b \in B} = 0$, with $\inf \emptyset = \infty$.
Clauses 1 to 4 follow directly.
For clause 5, suppose that $A$ is not near $B$, so the distance $r$ between them is positive.
Let $E = \set{x : d(x, A) \ge r/2}$.
Then $A$ is not near $E$, and each point of $X \setminus E$ is at distance more than $r/2$ from $B$, so $B$ is not near $X \setminus E$.
The relation that makes all pairs of subsets near fails clause 2, since it makes $\emptyset$ near $\emptyset$.
A proximity defines a topology: the closure of $A$ is the set of points $x$ with $\set{x} \mathrel{\delta} A$.
Many authors also ask that $\set{x} \mathrel{\delta} \set{y}$ imply $x = y$; such a proximity is *separated*.

## Explanations

### L1 human

A proximity is a rule that says which pairs of sets touch or almost touch.
The clauses say that nearness is symmetric, that the empty set is near nothing, and that sets that meet are near.
A set is near a union exactly when it is near one of the parts.
The last clause says that two sets that are not near can be kept apart by a set in between.
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<document source="https://tamnd.github.io/math2/node/1MDT9BF81H.md" tag="1MDT9BF81H" kind="concept">
# Cylindric algebra

## Summary

A cylindric algebra is a [[Boolean algebra|0XADJKM699]] with extra operations that model existential quantifiers and constants that model equality of variables.
Alfred Tarski and his collaborators introduced them to do for [[predicate logic|086QKPF0A6]] with equality what Boolean algebras do for propositional logic.
The cylindrification $c_\kappa$ plays the role of "there exists $x_\kappa$", and the diagonal element $d_{\kappa\lambda}$ plays the role of the formula $x_\kappa = x_\lambda$.

## Explanations

### L1 human

Think of the set of points of the plane that satisfy a formula $\varphi(x, y)$.
The formula "there exists $x$ such that $\varphi(x, y)$" describes the set swept out when that region is moved along the $x$-direction: a cylinder over the shadow of the region.
That is why the operation is called cylindrification.
The diagonal line $\set{(x, y) : x = y}$ is the set of the formula $x = y$.
A cylindric algebra keeps only the rules that these operations obey, so it can be studied with algebra.
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<document source="https://tamnd.github.io/math2/node/1MF0TVHDZD.md" tag="1MF0TVHDZD" kind="concept">
# Free Boolean algebra

## Summary

The free Boolean algebra on a set $S$ is the [[Boolean algebra|0XADJKM699]] generated by $S$ with no extra relations.
The only equalities in it are those that hold in every Boolean algebra.
Every map from $S$ into a Boolean algebra extends to exactly one homomorphism from it.
On $n$ generators it is the algebra of all [[Boolean functions|0X05GS525E]] of $n$ variables, with $2^{2^n}$ elements.
It is also the Lindenbaum–Tarski algebra of classical propositional logic.
Its propositional variables are the elements of $S$.

## Explanations

### L1 human

Take letters $x, y, z, \ldots$ and form all formulas with "and", "or" and "not".
Two formulas count as the same when they have the same truth table.
The resulting algebra is the free Boolean algebra on the letters.
With one letter $x$ it has four elements: false, $x$, not $x$, and true.
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<document source="https://tamnd.github.io/math2/node/1MGMQX43B8.md" tag="1MGMQX43B8" kind="concept">
# Hilbert system

## Summary

A Hilbert system is a proof system with many axiom schemas and very few rules of inference, often only modus ponens.
A proof is a finite list of formulas in which each formula is an axiom, a hypothesis, or follows from earlier ones by a rule.
Hilbert systems are simple to describe and to reason about, which makes them the standard tool for results about provability.
They are named after David Hilbert.

## Explanations

### L1 human

In a Hilbert system, almost all the work is done by the axioms.
The only move is modus ponens: from $\varphi$ and $\varphi \to \psi$, write down $\psi$.
A proof is then just a numbered list of lines, each justified by an axiom or by modus ponens.
For example, the formula $p \to p$ has a proof of five lines in the usual system.
Such proofs are easy to check but often hard to find, which is why [[natural deduction|6KT3BJZ7D1]] was invented.
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# Serre subcategory

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]].
A *Serre subcategory* of $\mathcal{A}$ is a nonempty [[full subcategory|V3BFGKBJF7]] $\mathcal{C}$ of $\mathcal{A}$ with the following property.
For every [[short exact sequence|KJKNJMGKZZ]] $0 \to A' \to A \to A'' \to 0$ in $\mathcal{A}$, the object $A$ lies in $\mathcal{C}$ if and only if $A'$ and $A''$ lie in $\mathcal{C}$.

## Remarks

Equivalently, $\mathcal{C}$ is nonempty and closed under subobjects, quotient objects and extensions.
It then contains a zero object and is closed under isomorphisms, and it is itself an abelian category with exact inclusion.
In a Grothendieck category, a Serre subcategory closed under arbitrary direct sums is called a *localizing subcategory*.

Example: the torsion abelian groups form a Serre subcategory of the category of abelian groups.
Subgroups and quotients of torsion groups are torsion.
If $0 \to A' \to A \to A'' \to 0$ is exact with $A'$ and $A''$ torsion and $a \in A$, then $na$ maps to $0$ in $A''$ for some $n \ge 1$, so $na \in A'$ and $mna = 0$ for some $m \ge 1$.

Non-example: the free abelian groups do not form a Serre subcategory.
The sequence $0 \to \Z \xrightarrow{2} \Z \to \Z/2\Z \to 0$ is exact with $\Z$ free, but the quotient $\Z/2\Z$ is not free.
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# Bernstein–Bézier form

## Definition

Let $a < b$ be real numbers and let $n \ge 0$ be an integer.
For $k = 0, \dots, n$ let
$$b_{k,n}^{[a,b]}(x) = \binom{n}{k} \left(\frac{x - a}{b - a}\right)^{k} \left(\frac{b - x}{b - a}\right)^{n-k}.$$
These $n + 1$ polynomials form a [[basis|0KCQ9Y3QKA]] of the space of real [[polynomials|1G0B73WPX8]] of degree at most $n$.
Let $p$ be a polynomial of degree at most $n$.
Its *Bernstein–Bézier form* on $[a, b]$ is its expansion
$$p(x) = \sum_{k=0}^{n} c_k\, b_{k,n}^{[a,b]}(x).$$
The uniquely determined numbers $c_0, \dots, c_n$ are its *Bézier coefficients* of degree $n$ on $[a, b]$.

## Remarks

On $[0, 1]$ the basis consists of the [[Bernstein basis polynomials|0C4VWBFY73]] $\binom{n}{k} x^k (1-x)^{n-k}$.
Since the basis functions are nonnegative on $[a, b]$ and sum to $1$, the values of $p$ on $[a, b]$ lie between $\min_k c_k$ and $\max_k c_k$, and $p(a) = c_0$, $p(b) = c_n$.
Example: on $[0, 1]$ with $n = 2$, $x = \tfrac{1}{2} \cdot 2x(1-x) + 1 \cdot x^2$, so $x$ has Bézier coefficients $0, 1/2, 1$.
Non-example: $p(x) = x^3$ has no Bernstein–Bézier form of degree $2$, since it is not a polynomial of degree at most $2$; one must take $n \ge 3$.
In several variables one uses barycentric coordinates $\lambda_0, \dots, \lambda_d$ of a simplex and the basis $\frac{n!}{\alpha_0! \cdots \alpha_d!} \lambda_0^{\alpha_0} \cdots \lambda_d^{\alpha_d}$ with $\abs{\alpha} = n$.
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# Airy function

## Summary

The Airy functions $\operatorname{Ai}$ and $\operatorname{Bi}$ are two standard independent solutions of the [[Airy equation|PNESAD6R5T]] $y'' = x y$.
$\operatorname{Ai}$ decays rapidly as $x \to +\infty$, and $\operatorname{Bi}$ grows; both oscillate as $x \to -\infty$.
They describe every simple turning point, where a wave changes from oscillating to decaying.
This makes them central in optics near caustics, in quantum mechanics near classical turning points, and in uniform asymptotics of other special functions.

## Explanations

### L1 human

The equation $y'' = x y$ acts like a spring on the left of $0$ and like exponential growth on the right of $0$.
The Airy function $\operatorname{Ai}$ is the solution that wiggles on the left and dies out quickly on the right.
Its graph looks like a sine wave whose waves get shorter and lower toward the left, then a single hump near $0$, then a fast decay.
The bright bands just inside a rainbow follow the square of $\operatorname{Ai}$.
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<document source="https://tamnd.github.io/math2/node/1N7TZCPBF0.md" tag="1N7TZCPBF0" kind="concept">
# Abstract Volterra equation

## Summary

An abstract Volterra equation is a Volterra integral equation whose unknown takes values in a [[Banach space|06GZ956657]].
Its memory term involves a linear operator, often unbounded.
Its standard form is $u(t) = f(t) + \int_0^t a(t - s) A u(s) \, ds$ with a scalar kernel $a$ and a [[closed operator|VM7B38PA87]] $A$.
It contains abstract first-order and second-order evolution equations as the cases $a = 1$ and $a(t) = t$.
Other kernels model heat conduction and viscoelasticity with memory, where $A$ is a differential operator in space.

## Explanations

### L1 human

Many physical systems change in time according to a rule that involves their whole history.
When the state at each time is a function of space, such as a temperature profile, the state is a point of an infinite-dimensional space.
The abstract Volterra equation writes the rule as an integral over the past.
A weight $a$ says how strongly the past is remembered, and an operator $A$ acts in space.
If $a = 1$, all of the past counts equally, and the equation is the integrated form of $u' = A u$.
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<document source="https://tamnd.github.io/math2/node/1NBACAEC2V.md" tag="1NBACAEC2V" kind="concept">
# Finite morphism of rings

## Summary

A ring homomorphism $f \colon A \to B$ is finite if $B$, viewed as an $A$-module through $f$, is [[finitely generated|05QY1VEBYD]].
Finite homomorphisms are exactly the [[integral|B7X0KFY1S1]] homomorphisms that are also of [[finite type|AC5TFFQT9M]].
They are closed under composition and base change.
In algebraic geometry they correspond to finite morphisms of affine schemes, such as the projection of a curve onto a line in general position.

## Explanations

### L1 human

A finite ring map makes the big ring a finite-dimensional object over the small ring, in the sense of modules.
The Gaussian integers $\Z[i]$ are finite over $\Z$: every element is $a \cdot 1 + b \cdot i$ with $a, b \in \Z$.
The polynomial ring $k[x]$ is not finite over the field $k$: the powers $1, x, x^2, \ldots$ are linearly independent, so no finite set spans it.
Finiteness is the ring version of a finite field extension.
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# Base change theorems

## Statement

Let $Y$ be a [[Noetherian scheme|ND6XEY68P4]] and let $f \colon X \to Y$ be a morphism.
For a sheaf $\mathcal{F}$ on $X$, $R^i f_* \mathcal{F}$ is the higher direct image, the sheaf on $Y$ associated to $V \mapsto H^i(f^{-1}(V), \mathcal{F})$, where $H^i$ is [[sheaf cohomology|PQ6Q143ARJ]].
For $y \in Y$ with [[residue field|G51E8MW7J1]] $k(y)$, let $X_y$ be the [[fiber|963QWX2E2Z]] and $\mathcal{F}_y$ the restriction of $\mathcal{F}$ to $X_y$.

1. (Flat base change.) Let $f$ be separated and of finite type, let $\mathcal{F}$ be [[quasi-coherent|KPKND0K7W6]], and let $u \colon Y' \to Y$ be a [[flat morphism|5M6KS8F47R]] with $Y'$ Noetherian. Let $v \colon X' = X \times_Y Y' \to X$ and $g \colon X' \to Y'$ be the projections. Then the natural maps $u^* R^i f_* \mathcal{F} \to R^i g_* (v^* \mathcal{F})$ are isomorphisms for all $i \ge 0$.

In the following, let $f$ be [[projective|Z5G47MQ99B]] and let $\mathcal{F}$ be a [[coherent sheaf|56RXEJ8Z6S]] on $X$ that is flat over $Y$.

2. (Semicontinuity.) For each $i \ge 0$, the function $y \mapsto \dim_{k(y)} H^i(X_y, \mathcal{F}_y)$ is upper semicontinuous on $Y$, and the Euler characteristic $y \mapsto \sum_i (-1)^i \dim_{k(y)} H^i(X_y, \mathcal{F}_y)$ is locally constant.
3. (Grauert.) If $Y$ is integral and $y \mapsto \dim_{k(y)} H^i(X_y, \mathcal{F}_y)$ is constant, then $R^i f_* \mathcal{F}$ is locally free and the natural map $R^i f_* \mathcal{F} \otimes k(y) \to H^i(X_y, \mathcal{F}_y)$ is an isomorphism for every $y \in Y$.
4. (Cohomology and base change.) Let $y \in Y$ and suppose that the natural map $\varphi^i(y) \colon R^i f_* \mathcal{F} \otimes k(y) \to H^i(X_y, \mathcal{F}_y)$ is surjective. Then $\varphi^i(y)$ is an isomorphism, and $\varphi^{i-1}(y)$ is surjective if and only if $R^i f_* \mathcal{F}$ is locally free in a neighbourhood of $y$.

## Remarks

The idea of the proof of parts 2 to 4: locally on an affine $Y = \operatorname{Spec} A$, there is a bounded complex $K^\bullet$ of finitely generated free $A$-modules with a special property.
For every $A$-algebra $B$, it computes the cohomology after base change: $H^i(X \times_Y \operatorname{Spec} B, \mathcal{F}_B) \cong H^i(K^\bullet \otimes_A B)$.
All statements then become facts about ranks of matrices of functions on $Y$.
Part 1 follows from the Čech complex of an affine cover, since flat base change commutes with taking cohomology of a complex.

Example: let $f \colon X \to Y$ be a flat projective family of curves over an integral base, with all fibers smooth of genus $g$.
Then $h^0(X_y, \mathcal{O}) = 1$ and $h^1(X_y, \mathcal{O}) = g$ for all $y$, so by part 3 $f_* \mathcal{O}_X$ and $R^1 f_* \mathcal{O}_X$ are locally free of ranks $1$ and $g$.

Flatness is needed in part 2.
Let $Y = \mathbb{A}^1_k$, and let $X$ be the disjoint union of $Y$ and the closed point $0$.
Let $f$ be the identity on $Y$ and the inclusion on the point, and let $\mathcal{F} = \mathcal{O}_X$.
Then $f$ is finite, hence projective, but $\mathcal{F}$ is not flat over $Y$.
The Euler characteristic of the fibers is $2$ at $0$ and $1$ elsewhere, so it is not locally constant.

## Explanations

### L1 human

Given a family of varieties, each with a sheaf on it, one wants to know how the cohomology of the fibers changes from one fiber to the next.
These theorems say that the dimensions can only jump up on special fibers, never down, and that their alternating sum stays constant in a flat family.
When the dimension does not jump, the cohomology groups of the fibers fit together into a vector bundle on the base.

### L2 human

The base change theorems are used to construct vector bundles on parameter spaces, such as the Hodge bundle $f_* \omega_{X/Y}$ on a moduli space of curves.
Semicontinuity says that a property like $H^1(X_y, \mathcal{F}_y) = 0$ is open in families, which is the starting point of deformation arguments.
Flat base change lets one compute higher direct images after a flat change of base, for example after passing to a completion or to an étale neighbourhood.
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# Homotopy limit

## Summary

A homotopy limit is the version of a [[limit|CHS5AFJGWE]] that respects weak equivalences.
Ordinary limits of diagrams of spaces can change drastically when the diagram is replaced by a weakly equivalent one; homotopy limits do not.
In a [[model category|SDTZVZ6B4Q]] the homotopy limit is the right derived functor of the limit functor.
Homotopy pullbacks, homotopy fixed points and the space of maps out of a homotopy colimit are typical instances.

## Explanations

### L1 human

The pullback of two maps to a space $Z$ consists of pairs of points with the same image.
In homotopy theory equality is too strict; one should allow a path in $Z$ between the two images.
The homotopy pullback of $X \to Z \leftarrow Y$ is therefore the space of triples $(x, \omega, y)$ with $\omega$ a path from the image of $x$ to the image of $y$.
For two points of the real line, the ordinary pullback is empty if the points differ, while the homotopy pullback is the space of paths between them, which is contractible.
Homotopy limits extend this idea from pullbacks to all diagrams.
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# Poincaré's theorem that the ball and polydisc are not biholomorphic

## Statement

Let $n \ge 2$.
Let $\mathbb{B}^n = \set{z \in \C^n : \abs{z_1}^2 + \cdots + \abs{z_n}^2 < 1}$ be the [[unit ball|PDQ9KW9FAP]] and $\Delta^n = \set{z \in \C^n : \abs{z_j} < 1 \text{ for all } j}$ the unit [[polydisc|DJBWP7D3M8]].
Then there is no [[biholomorphism|FX92PP3HZR]] from $\mathbb{B}^n$ onto $\Delta^n$.

## Remarks

One proof compares [[automorphism groups|0K1SCPBNKJ]].
The holomorphic automorphisms of $\mathbb{B}^n$ act transitively, so a biholomorphism $F \colon \Delta^n \to \mathbb{B}^n$ may be assumed to satisfy $F(0) = 0$.
Then conjugation by $F$ maps the automorphisms of $\Delta^n$ fixing $0$ isomorphically onto the automorphisms of $\mathbb{B}^n$ fixing $0$.
By [[Cartan's uniqueness theorem|EQHPSKC5SY]] both groups consist of linear maps.
For $\mathbb{B}^n$ it is the unitary group $U(n)$, which is connected and not abelian for $n \ge 2$.
For $\Delta^n$ it consists of the maps $z \mapsto (e^{i\theta_1} z_{\sigma(1)}, \dots, e^{i\theta_n} z_{\sigma(n)})$ with $\sigma$ a permutation, and its identity component is abelian.
So the two groups are not isomorphic as topological groups, a contradiction.
For $n = 1$ both sets are the unit disc, so the hypothesis $n \ge 2$ is needed.

## Explanations

### L1 human

In one complex variable, the Riemann mapping theorem says that all simply connected proper domains of the plane look the same to complex analysis.
In two or more variables this is false already for the two simplest domains: the round ball and the product of discs.
Both are bounded, convex and contractible, so the difference is not topological; it is a rigidity of holomorphic maps.

### L2 human

The theorem shows that there is no Riemann mapping theorem in several complex variables.
It opened the study of biholomorphic invariants of domains, such as automorphism groups, invariant metrics and the geometry of the boundary.
Another proof uses the boundary.
The boundary of $\Delta^n$ contains nonconstant holomorphic discs, while the sphere bounding $\mathbb{B}^n$ contains none.
A limiting argument applied to a would-be biholomorphism turns this difference into a contradiction.
Poincaré proved the result for $n = 2$ by computing automorphism groups.
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# Prenex normal form

## Definition

Let $\sigma$ be a signature.
A first-order [[formula|HYJ10AF6RT]] over $\sigma$ is in *prenex normal form* if it has the form
$$Q_1 x_1 \, Q_2 x_2 \cdots Q_n x_n \, \psi,$$
where $n \ge 0$ and each $Q_i$ is $\forall$ or $\exists$.
The $x_i$ are variables.
Here $\psi$ is a formula with no quantifiers.
The string $Q_1 x_1 \cdots Q_n x_n$ is the *prefix* and $\psi$ is the *matrix*.

## Remarks

The formula $\forall x \, \exists y \, (R(x, y) \lor \neg P(x))$ is in prenex normal form: its prefix is $\forall x \, \exists y$ and its matrix has no quantifiers.
The formula $\neg \forall x \, P(x)$ is not: the quantifier is inside a negation, so the formula is not a prefix followed by a quantifier-free matrix.
It is equivalent to $\exists x \, \neg P(x)$, which is in prenex normal form.
In classical first-order logic, every formula is logically equivalent to a formula in prenex normal form.
The proof moves quantifiers outwards with rules such as $(\forall x \, \varphi) \to \psi \equiv \exists x \, (\varphi \to \psi)$, when $x$ is not free in $\psi$.
This rule uses that structures have nonempty domains.
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# Compactness theorem

## Statement

Let $\sigma$ be a signature and let $\Gamma$ be a set of first-order [[sentences|DKFGQTRYZE]] over $\sigma$.
Then $\Gamma$ has a [[model|MVCH6N9SNW]] if and only if every finite subset of $\Gamma$ has a model.

## Remarks

One direction is clear, since a model of $\Gamma$ is a model of each subset.
For the other, if $\Gamma$ had no model, then by [[Gödel's completeness theorem|WQ1ZE255SF]] it would derive a contradiction.
That derivation uses only finitely many sentences of $\Gamma$, and this finite subset would have no model.
A second proof builds a model as an [[ultraproduct|AX9Q86GB8M]] of models of the finite subsets.
Application: let $\Gamma$ be the true sentences of arithmetic, plus $c > \underline{n}$ for a new constant $c$ and each numeral $\underline{n}$.
Each finite subset holds in $\N$ with $c$ a large number, so $\Gamma$ has a model, a [[non-standard model of arithmetic|1QWN5CV4CG]].
In Mathlib the theorem is `FirstOrder.Language.Theory.isSatisfiable_iff_isFinitelySatisfiable`.

## Explanations

### L1 human

If every finite part of a list of axioms can be made true, then the whole infinite list can be made true at once.
This sounds simple, but it has strong consequences.
For example, it gives models of arithmetic with numbers larger than every ordinary natural number.

### L2 human

Compactness is the main tool of [[model theory|HFTBG5PMXC]].
A typical use shows that some property cannot be expressed in first-order logic.
If a theory has finite models of every size, then it has an infinite model, so "the domain is finite" is not first-order.
The theorem is named after compactness in topology: it says that a certain space of complete theories, the Stone space, is compact.
By [[Lindström's theorem|FYZ3CJGZYV]], first-order logic is the strongest logic that has both compactness and the downward [[Löwenheim–Skolem theorem|484M0TZ1S7]].
</document>

<document source="https://tamnd.github.io/math2/node/1NTKG64F7N.md" tag="1NTKG64F7N" kind="concept">
# Radon–Nikodym derivative

## Summary

The Radon–Nikodym derivative $d\nu / d\mu$ is the density of one measure $\nu$ with respect to another measure $\mu$.
It exists when both measures are σ-finite and $\nu$ is absolutely continuous with respect to $\mu$, by the [[Radon–Nikodym theorem|1ZSZS5G1Q4]].
It turns integrals against $\nu$ into integrals against $\mu$.
Probability densities, likelihood ratios and conditional expectations are all Radon–Nikodym derivatives.

## Explanations

### L1 human

Think of $\mu$ as a way to measure size and $\nu$ as a way to measure mass.
The derivative $d\nu / d\mu$ at a point is the mass per unit size near that point.
For the standard normal distribution $\nu$ and Lebesgue measure $\mu$ on the line, the derivative is the bell curve $e^{-x^2/2}/\sqrt{2\pi}$.
The name comes from the rule $d\nu = \frac{d\nu}{d\mu} \, d\mu$, which looks like the chain rule of calculus.
</document>

<document source="https://tamnd.github.io/math2/node/1NWC3710WW.md" tag="1NWC3710WW" kind="concept">
# Initial topology

## Summary

Take a set and a family of maps from it into topological spaces.
The initial topology on the set is the coarsest topology that makes all these maps continuous.
The [[subspace topology|0WS9P12FQR]] and the [[product topology|0GBG8B49AA]] are both initial topologies.
It is characterised by a universal property: a map into the set is continuous exactly when its composites with all the given maps are continuous.
The dual notion is the [[final topology|6ERJRRZK32]].

## Explanations

### L1 human

Suppose you have a set and some maps from it into spaces you already understand.
You want the fewest open sets that still make every map continuous.
So you take the preimages of open sets under the maps, and only add what you must to get a topology.
For example, the projections $\R^2 \to \R$ onto the two coordinates give the usual topology of the plane.
Open rectangles are intersections of two such preimages.
</document>

<document source="https://tamnd.github.io/math2/node/1NX95TTMFT.md" tag="1NX95TTMFT" kind="definition">
# Semisimple Lie group

## Definition

Let $G$ be a connected real or complex [[Lie group|7JBRGCVJRB]], and let $\mathfrak{g}$ be its Lie algebra.
The group $G$ is a *semisimple Lie group* if $\mathfrak{g}$ is a semisimple Lie algebra, that is, the only solvable ideal of $\mathfrak{g}$ is $0$.

## Remarks

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic $0$.
The following are equivalent:

1. the only solvable ideal of $\mathfrak{g}$ is $0$;
2. $\mathfrak{g}$ is a direct sum of simple ideals;
3. the Killing form $B(x, y) = \tr(\operatorname{ad} x \operatorname{ad} y)$ is nondegenerate (Cartan's criterion).

Example: $G = SO(4)$.
Its Lie algebra $\mathfrak{so}(4)$ is isomorphic to $\mathfrak{so}(3) \oplus \mathfrak{so}(3)$, a direct sum of two simple ideals, so $G$ is semisimple.
It is not simple, since each summand is a proper nonzero ideal.

Non-example: $G = GL_n(\R)^+$, the invertible real $n \times n$ matrices with positive determinant, for $n \ge 1$.
Its Lie algebra $\mathfrak{gl}_n(\R)$ contains the ideal $\R I_n$ of scalar matrices, which is abelian, hence solvable, and nonzero.
So the condition fails.

Some authors do not require $G$ to be connected and ask only that $\mathfrak{g}$ be semisimple; some also ask for a finite centre.
</document>

<document source="https://tamnd.github.io/math2/node/1P47PCRVTZ.md" tag="1P47PCRVTZ" kind="concept">
# Levin-type transformations

## Summary

Levin-type transformations are nonlinear [[series acceleration|B1S2XJPGYD]] methods that use explicit estimates of the remainder of a series.
They assume that the error of the $n$-th partial sum is a known estimate $\omega_n$ times a slowly varying correction.
They eliminate this correction exactly when it is a polynomial in $1/(n + \beta)$.
Levin's $t$- and $u$-transformations take $\omega_n$ from the terms of the series.
They work well both for alternating series and for many series with terms of one sign.
They can also assign values to some divergent series.

## Explanations

### L1 human

For a series, the next term is often a good guess for the size of the remaining error.
A Levin-type transformation uses such a guess $\omega_n$ and fits the rest of the error by a short expansion in powers of $1/n$.
Solving for the sum then removes several orders of the error at once.
For the geometric series $\sum x^n$ with $\abs{x} < 1$, the error after $n$ terms is exactly a constant times the last term, and the simplest Levin transformation returns the exact sum $\frac{1}{1 - x}$.
</document>

<document source="https://tamnd.github.io/math2/node/1P4830BH6Z.md" tag="1P4830BH6Z" kind="concept">
# Comparability graph

## Summary

The comparability graph of a [[partially ordered set|0QV2JS9RJ3]] joins two elements by an edge when they are comparable.
A graph is a comparability graph when it arises in this way from some partial order.
Chains become cliques and antichains become independent sets, so order results such as [[Mirsky's theorem|5VPXR3RDF1]] turn into graph results.
Comparability graphs are [[perfect graphs|1YZ7NV161A]].

## Explanations

### L1 human

Start from a poset and forget which element is the smaller one in each related pair.
What remains is an ordinary [[graph|00C0BR7D7H]]: an edge just says "these two are related".
For the divisors of $6$, the graph has edges $1$–$2$, $1$–$3$, $1$–$6$, $2$–$6$ and $3$–$6$, but no edge $2$–$3$.
</document>

<document source="https://tamnd.github.io/math2/node/1P6TBEPHQH.md" tag="1P6TBEPHQH" kind="concept">
# Inversion of rational transforms by partial fractions

## Summary

A rational function $F(s) = P(s)/Q(s)$ with $\deg P < \deg Q$ is the [[Laplace transform|0V49JEZW06]] of a sum of terms $t^{k-1} e^{a t}$.
To find the [[inverse Laplace transform|07WBTX1MHA]], one splits $F$ by [[partial fraction decomposition|030VFEREKG]] into terms $c/(s - a)^k$ and inverts each term from the rule $\mathcal{L}\{t^{k-1} e^{a t}/(k-1)!\} = 1/(s - a)^k$.
This is the everyday method for the transforms that arise from linear differential equations with constant coefficients.

## Explanations

### L1 human

A complicated fraction in $s$ is hard to invert at once, but simple fractions such as $1/(s - a)$ are easy: they come from $e^{a t}$.
So one first breaks the fraction into simple pieces and then inverts piece by piece.
For example, $\frac{1}{s (s + 1)} = \frac{1}{s} - \frac{1}{s + 1}$, so it is the transform of $1 - e^{-t}$.
Each root $a$ of the denominator gives a term $e^{a t}$ in the answer, and repeated roots give extra powers of $t$.
</document>

<document source="https://tamnd.github.io/math2/node/1P8087YGYQ.md" tag="1P8087YGYQ" kind="definition">
# Linearly ordered group

## Definition

Let $G$ be a group and let $\le$ be a [[total order|0316AYH1SC]] on $G$.
The pair $(G, \le)$ is a *linearly ordered group*, or *totally ordered group*, if for all $a, b, c \in G$,
$$a \le b \implies ca \le cb \text{ and } ac \le bc.$$
Equivalently, it is an [[ordered group|A49BATWQEH]] whose order is total.

## Remarks

Example: $\Z^2$ under addition with the [[lexicographic order|0G77XFBV00]] is a linearly ordered group.
If $(a, b) \le (c, d)$, then either $a < c$, or $a = c$ and $b \le d$, and both cases survive adding $(x, y)$ to both sides.
Non-example: $\Z^2$ with the componentwise order is an ordered group, but $(1, 0)$ and $(0, 1)$ are not comparable, so the order is not total.
In terms of the positive cone $P = \set{g : e \le g}$, the order is total if and only if $G = P \cup P^{-1}$.
A linearly ordered group has no elements of finite order other than $e$: if $g > e$, then $g^n > e$ for all $n \ge 1$.
</document>

<document source="https://tamnd.github.io/math2/node/1P8D3MK79H.md" tag="1P8D3MK79H" kind="statement">
# Stone–Weierstrass theorem

## Statement

Let $X$ be a [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]].
Let $C(X, \R)$ be the space of continuous real functions on $X$ with the [[supremum norm|0PARCKE8WP]].
Let $A \subseteq C(X, \R)$ be a subalgebra with these two properties.

1. $A$ contains the constant functions.
2. $A$ separates points: for all $x \ne y$ in $X$ there is $f \in A$ with $f(x) \ne f(y)$.

Then $A$ is [[dense|0BSJSX7ME2]] in $C(X, \R)$.

## Remarks

The idea of the proof: the closure $\bar{A}$ is again an algebra.
By the [[Weierstrass approximation theorem|0JQE130DHW]], $\abs{t}$ is a uniform limit of polynomials on each interval, so $f \in \bar{A}$ implies $\abs{f} \in \bar{A}$.
Hence $\bar{A}$ is closed under $\max$ and $\min$.
A compactness argument then approximates any continuous function by finite maxima and minima of functions in $\bar{A}$.

Examples: polynomials in $n$ variables are dense in $C(K, \R)$ for each compact $K \subset \R^n$.
Trigonometric polynomials are dense in the continuous $2\pi$-periodic functions.

Both hypotheses are needed.
On $[-1, 1]$, the even polynomials form an algebra with constants, but it does not separate $1$ from $-1$, and $x$ is not a uniform limit of even functions.
The functions $f$ with $f(0) = 0$ separate points but miss the constants.

For complex functions, $A$ must also be closed under complex conjugation.
On the unit circle, the polynomials in $z$ separate points but are not dense: $\bar{z}$ is not a uniform limit of them.

## Explanations

### L1 human

You can approximate any continuous function on a compact space as closely as you like, provided your building blocks can tell points apart and include the constants.
This is a far-reaching form of Weierstrass's result that polynomials approximate continuous functions on an interval.

### L2 human

The theorem is the standard way to prove that a class of functions is dense in $C(X)$.
It gives density of polynomials, of trigonometric polynomials, and of sums of products $f(x) g(y)$ in $C(X \times Y)$.
In the theory of C*-algebras it is the main tool behind the commutative Gelfand–Naimark theorem.
</document>

<document source="https://tamnd.github.io/math2/node/1P8WX3EGHF.md" tag="1P8WX3EGHF" kind="concept">
# Region of convergence

## Summary

The region of convergence of a [[Z-transform|5FY0JQGHYZ]] is the set of complex numbers $z$ where its defining series converges absolutely.
It is an annulus centred at $0$, possibly empty, possibly with some boundary points.
It plays the part that the disc given by the [[radius of convergence|0N288P2TB5]] plays for a power series.
A Z-transform is only known together with its region of convergence: two different sequences can have the same formula for $X(z)$ on two different regions.
The position of the region with respect to the unit circle tells whether a system is causal and stable.

## Explanations

### L1 human

The Z-transform is an infinite sum, and an infinite sum only gives a number when it converges.
The region of convergence is the set of points where it does.
For the sequence $1, 2, 4, 8, \ldots$ the sum $\sum 2^n z^{-n}$ converges exactly when $\abs{z} > 2$, the outside of a circle of radius $2$.
For a sequence that is nonzero only for negative $n$, the region is instead the inside of a circle.
</document>

<document source="https://tamnd.github.io/math2/node/1P9ZR606JN.md" tag="1P9ZR606JN" kind="statement">
# Pettis measurability theorem

## Statement

Let $(X, \Sigma)$ be a measurable space and let $E$ be a Banach space over $\R$ or $\C$ with [[dual space|0M856Q1AMZ]] $E^*$.
Let $f \colon X \to E$ be a function.
Then $f$ is [[strongly measurable|JDM2M1FRY6]] if and only if both of the following hold.

1. The function $\varphi \circ f$ is [[measurable|YYNCJSR0PJ]] for every continuous linear functional $\varphi \in E^*$, that is, $f$ is *weakly measurable*.
2. The image $f(X)$ is a [[separable|0GKJ78948V]] subset of $E$.

## Remarks

The direct implication is easy: $\varphi \circ s_n \to \varphi \circ f$ pointwise, and $f(X)$ lies in the closure of the countably many values of the approximating simple functions $s_n$.
For the converse, one may replace $E$ by the closed linear span of $f(X)$, which is separable.
In a separable Banach space there is a sequence $(\varphi_k)$ in $E^*$ with $\norm{e} = \sup_k \abs{\varphi_k(e)}$ for all $e$.
So $x \mapsto \norm{f(x) - e}$ is measurable for each $e$, and one builds simple approximations by sending $x$ to the nearest point of a countable dense set among its first $n$ points.

When $E$ is separable, condition 2 is automatic, so weak and strong measurability coincide.
Without condition 2 the equivalence fails.
Take $X = [0, 1]$ with the Borel σ-algebra and the map $f(t) = \mathbf{1}_{\set{t}}$ into the non-separable Hilbert space $\ell^2([0, 1])$.
It is weakly measurable, since $\varphi \circ f$ vanishes outside a countable set for each $\varphi$.
But $f(X)$ is not separable, so $f$ is not strongly measurable.

There is also a version relative to a measure $\mu$.
Then $f$ is strongly $\mu$-measurable if and only if it is weakly $\mu$-measurable and takes values in a separable subset of $E$ outside a $\mu$-null set.

## Explanations

### L1 human

Testing measurability of a vector-valued function directly is hard.
The theorem says that it is enough to test each linear functional, one at a time.
This works as long as the values of the function stay inside a separable part of the space.
So questions about a function in a large space reduce to questions about ordinary real or complex functions.

### L2 human

The theorem is the main tool to check the hypotheses of the Bochner integral.
With Bochner's criterion, it shows when $f$ is Bochner integrable.
This holds as soon as $f$ is weakly measurable, essentially separably valued and $\int \norm{f} \, d\mu < \infty$.
For a separable space $E$, weak measurability alone suffices.
The theorem also explains why the Pettis integral, which only asks for weak measurability, covers more functions than the Bochner integral in non-separable spaces.
</document>

<document source="https://tamnd.github.io/math2/node/1PBQ6ARWND.md" tag="1PBQ6ARWND" kind="example">
# Solenoid

## Example

Let $p$ be a prime and let $\mathbb{T} = \set{z \in \C : \abs{z} = 1}$ be the [[circle group|04EAFVFTF7]].
The *$p$-adic solenoid* $\Sigma_p$ is the [[inverse limit|DZB378CNCK]] of the sequence
$$\cdots \to \mathbb{T} \xrightarrow{z \mapsto z^p} \mathbb{T} \xrightarrow{z \mapsto z^p} \mathbb{T}.$$
Its elements are sequences $(z_0, z_1, z_2, \dots)$ in $\mathbb{T}$ with $z_{k+1}^p = z_k$ for all $k$.
The group law is coordinatewise multiplication, and the topology is the one induced from the product $\mathbb{T}^{\N}$.

Check that it is a [[compact group|EVHCE5VR6Y]].
Each map $z \mapsto z^p$ is a continuous homomorphism, so the sequences with $z_{k+1}^p = z_k$ form a subgroup of $\mathbb{T}^{\N}$.
This subgroup is closed, since it is cut out by the equations $z_{k+1}^p = z_k$ between continuous functions.
The product $\mathbb{T}^{\N}$ is compact by Tychonoff's theorem, so $\Sigma_p$ is a compact Hausdorff abelian group.

## Remarks

The solenoid is connected, being an inverse limit of compact connected spaces.
It is not locally connected and not path-connected, so it is a compact connected group that is not a Lie group.
The projection $(z_k) \mapsto z_0$ onto $\mathbb{T}$ is onto, and its kernel is isomorphic to the group $\Z_p$ of [[p-adic integers|V6TJAFZ4MD]].
Its Pontryagin dual is the discrete group $\Z[1/p]$ of rationals whose denominators are powers of $p$.

## Explanations

### L1 human

Wrap a loop of string around a circle $p$ times, then wrap the result $p$ times again, and continue forever.
The limiting object is the solenoid.
Near any point it looks like a line segment times a Cantor set, so it is connected but full of gaps.
It is a standard example of a compact connected group that is not a Lie group.
</document>

<document source="https://tamnd.github.io/math2/node/1PDGBGEHM2.md" tag="1PDGBGEHM2" kind="concept">
# Barnes integral

## Summary

A Barnes integral, also called a Mellin–Barnes integral, is a kind of [[contour integral|06ZGPYTWK8]].
Its integrand is a quotient of products of values of the [[gamma function|0H5SYBSED9]], times a power $z^{-s}$.
The path runs from $-i\infty$ to $+i\infty$ and separates two families of poles of the integrand.
Many special functions, such as the [[hypergeometric function|463X5V31D9]], have such an integral as a representation.
Closing the path to the left or to the right and summing residues turns the integral into a series.
This gives convergent series and asymptotic expansions.

## Explanations

### L1 human

The gamma function has poles at $0, -1, -2, \dots$, and a Barnes integral is built so that these poles sit on known sides of a vertical path.
When one pushes the path far to one side, the [[residue theorem|032HKYS329]] collects one term of a series at each pole that the path crosses.
So a single integral can stand for a power series near $z = 0$ and for another expansion near $z = \infty$.
The smallest example is $\frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \Gamma(s)\, x^{-s}\, ds$ with $c > 0$ and $x > 0$.
The poles of $\Gamma(s)$ at $s = -n$ have residues $(-1)^n / n!$, and their sum gives $e^{-x}$.
</document>

<document source="https://tamnd.github.io/math2/node/1PFE27B2MG.md" tag="1PFE27B2MG" kind="statement">
# Birkhoff–Grothendieck theorem

## Statement

Let $k$ be a field and let $\mathcal{E}$ be a [[vector bundle|P4DZM975BV]] (a locally free sheaf) of rank $r \ge 1$ on the [[projective line|ZT9A6JK8QF]] $\mathbb{P}^1_k$.
Then there are integers $a_1 \ge a_2 \ge \dots \ge a_r$ such that
$$\mathcal{E} \cong \mathcal{O}(a_1) \oplus \mathcal{O}(a_2) \oplus \dots \oplus \mathcal{O}(a_r),$$
where $\mathcal{O}(a)$ is the $a$-th power of [[Serre's twisting sheaf|V93Y0XYPZD]].
The integers $a_1, \dots, a_r$ are uniquely determined by $\mathcal{E}$.

## Remarks

The idea of the proof: induct on $r$.
Choose the largest $a$ with $H^0(\mathcal{E}(-a)) \ne 0$; a section of $\mathcal{E}(-a)$ gives a subbundle $\mathcal{O}(a) \subseteq \mathcal{E}$ whose quotient is a vector bundle of rank $r - 1$, which splits by induction as $\bigoplus \mathcal{O}(b_i)$.
The maximality of $a$ forces $b_i \le a$, so the extension class lies in $\bigoplus H^1(\mathbb{P}^1, \mathcal{O}(a - b_i)) = 0$ and the sequence splits.
Uniqueness follows from the dimensions $h^0(\mathcal{E}(n))$ for all $n$.

Example: the tangent bundle of $\mathbb{P}^1_k$ is $\mathcal{O}(2)$.
The bundles $\mathcal{O}(1) \oplus \mathcal{O}(1)$ and $\mathcal{O} \oplus \mathcal{O}(2)$ both have rank $2$ and degree $2$, but they are not isomorphic.
Indeed, after twisting by $\mathcal{O}(-2)$, the first has no nonzero global sections and the second has a one-dimensional space of them.

The statement fails on other curves and in higher dimension.
On an elliptic curve there are indecomposable vector bundles of rank $2$, and the tangent bundle of $\mathbb{P}^2$ is not a direct sum of line bundles.

## Explanations

### L1 human

On the projective line, every vector bundle breaks into a sum of line bundles, and line bundles there are classified by one integer, the degree.
So a vector bundle on $\mathbb{P}^1$ is described by a finite list of integers.
This makes $\mathbb{P}^1$ the simplest possible base for vector bundles.

### L2 human

The integers $a_i$ are called the splitting type of $\mathcal{E}$.
The theorem is used to study vector bundles on $\mathbb{P}^n$ by restricting them to lines, where they split, and seeing how the splitting type varies.
In the analytic setting it says that every holomorphic vector bundle on the Riemann sphere is a sum of line bundles.
This is equivalent to a factorization theorem for matrices of functions on a circle.
</document>

<document source="https://tamnd.github.io/math2/node/1PJS951C8N.md" tag="1PJS951C8N" kind="definition">
# Mahler measure

## Definition

Let $f \in \C[x]$ be a nonzero polynomial.
By the [[fundamental theorem of algebra|0JP0W07Y9A]], write $f = a (x - \alpha_1) \cdots (x - \alpha_n)$ with $a \in \C \setminus \set{0}$ and [[roots|00EHX6FWC0]] $\alpha_1, \dots, \alpha_n \in \C$.
The Mahler measure of $f$ is
$$M(f) = \abs{a} \prod_{i=1}^{n} \max(1, \abs{\alpha_i}).$$

## Remarks

By Jensen's formula, $M(f) = \exp\left( \int_0^1 \log \abs{f(e^{2 \pi i \theta})} \, d\theta \right)$.
It is multiplicative: $M(f g) = M(f) M(g)$.
Example: $f = 2x - 1 = 2(x - \frac{1}{2})$ gives $M(f) = 2 \cdot \max(1, \frac{1}{2}) = 2$.
Non-example of a common confusion: $M(f)$ is not the largest absolute value of a coefficient; for $f = (x + 1)^2 = x^2 + 2x + 1$ that number is $2$, while $M(f) = 1$.
If $f \in \Z[x]$ is nonzero, then $M(f) \ge 1$.
By a theorem of Kronecker, a nonzero $f \in \Z[x]$ has $M(f) = 1$ if and only if $f$ is $\pm$ a product of a power of $x$ and [[cyclotomic polynomials|08KEYFAH66]].

## Explanations

### L1 human

The Mahler measure looks at where the roots are.
Roots inside the unit circle are ignored, and roots outside count with their size.
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<document source="https://tamnd.github.io/math2/node/1PKZW82H0C.md" tag="1PKZW82H0C" kind="definition">
# Efficiency of an estimator

## Definition

Let $\set{f(\cdot; \theta) : \theta \in \Theta}$ be a parametric family of densities with $\Theta \subseteq \R$ an open interval, regular enough for the [[Cramér–Rao bound|1JFKTEQVRX]] to hold.
Let $X = (X_1, \dots, X_n)$ be the data, and let $I_n(\theta)$ be the [[Fisher information|G1Z3WSTE8Q]] of $X$, assumed finite and positive.
Let $T$ be an unbiased [[estimator|0TTNRP5PFQ]] of $\theta$ (its [[bias|01BD2TFFP4]] is $0$) with finite positive [[variance|007QED3E0Y]].

1. The *efficiency* of $T$ at $\theta$ is $e(T, \theta) = \dfrac{1/I_n(\theta)}{\operatorname{Var}_\theta(T)}$.
2. $T$ is *efficient* if $e(T, \theta) = 1$ for every $\theta \in \Theta$.
3. For two unbiased estimators $T_1, T_2$ of $\theta$, the *relative efficiency* of $T_1$ with respect to $T_2$ is $\operatorname{Var}_\theta(T_2)/\operatorname{Var}_\theta(T_1)$.

## Remarks

By the Cramér–Rao bound, $0 < e(T, \theta) \le 1$.

Example: for i.i.d. $X_i \sim \mathcal{N}(\mu, \sigma^2)$ with $\sigma^2$ known, $I_n(\mu) = n/\sigma^2$ and the [[sample mean|0QNRBHWV35]] has variance $\sigma^2/n$, so $e(\bar{X}, \mu) = 1$ and $\bar{X}$ is efficient.

Non-example: in the same model, $T = X_1$ is unbiased with variance $\sigma^2$, so $e(T, \mu) = 1/n$, and $T$ is not efficient when $n \ge 2$.

The *asymptotic relative efficiency* compares two estimators $T_{1,n}, T_{2,n}$ with $\sqrt{n}(T_{j,n} - \theta) \xrightarrow{d} \mathcal{N}(0, v_j(\theta))$ by the ratio $v_2(\theta)/v_1(\theta)$.
For normal data, the asymptotic relative efficiency of the sample median with respect to the sample mean is $2/\pi \approx 0.64$.
</document>

<document source="https://tamnd.github.io/math2/node/1PQHRMQ4ZH.md" tag="1PQHRMQ4ZH" kind="definition">
# Non-measurable set

## Definition

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]].
A subset $A \subseteq X$ is *non-measurable* (with respect to $\Sigma$) if $A \notin \Sigma$.
Without further qualification, a non-measurable subset of $\R^n$ is a set that is not Lebesgue measurable.

## Remarks

Example: on $X = \set{1, 2}$ with $\Sigma = \set{\varnothing, X}$, the set $\set{1}$ is non-measurable.
Example: a Vitali set in $[0, 1]$ is not Lebesgue measurable.
Non-example: every interval, every open set and every closed subset of $\R^n$ is Lebesgue measurable, so none of them is non-measurable.
For Lebesgue measure, $A \subseteq \R^n$ is non-measurable if and only if there is a set $B$ with $\lambda^*(B) < \lambda^*(B \cap A) + \lambda^*(B \setminus A)$, where $\lambda^*$ is Lebesgue outer measure.
Every subset of $\R^n$ with positive outer measure contains a non-measurable subset.
Being non-measurable depends on $\Sigma$: a set that is not a Borel set may still be Lebesgue measurable.
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# Proof complexity

## Summary

Proof complexity is the part of [[proof theory|N4EDJNMTEF]] and of computational complexity that studies the size of formal proofs.
Its main objects are proof systems for [[propositional logic|09K9MWD2QQ]], and the question is how short a proof of a given [[tautology|05006AJ7DZ]] can be.
Cook and Reckhow showed in 1979 that a proof system with short proofs of all tautologies exists if and only if NP equals coNP.
So lower bounds on proof size for ever stronger systems are a route toward separating complexity classes.

## Explanations

### L1 human

A tautology is a formula that is true under every truth assignment.
One can check this with a truth table, but the table for $n$ variables has $2^n$ rows.
A proof can be much shorter, but it is not clear that a short proof always exists.
Proof complexity asks how long the shortest proof must be, in a given proof system.
For example, in the resolution system, every proof of the pigeonhole principle, "$n + 1$ pigeons do not fit in $n$ holes one per hole", has size exponential in $n$.
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# Krull intersection theorem

## Statement

Let $R$ be a commutative [[Noetherian ring|0670T59HGE]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module.
Then
$$\bigcap_{n \ge 0} I^n M = \set{x \in M : (1 + a) x = 0 \text{ for some } a \in I}.$$
In particular:

1. if $I$ is contained in the [[Jacobson radical|XA0DCNH7WY]] of $R$, then $\bigcap_{n} I^n M = 0$; this applies when $R$ is a [[local ring|MZ02BKPHJA]] and $I \neq R$;
2. if $R$ is an [[integral domain|0QVD7WBFGN]] and $I \neq R$, then $\bigcap_{n} I^n = 0$.

## Remarks

Let $N = \bigcap_n I^n M$.
By the [[Artin–Rees lemma|30H6B0G2FT]], $N = I^n M \cap N = I^{n-k}(I^k M \cap N) \subseteq I N$ for $n > k$, so $N = IN$.
As $N$ is finitely generated, the [[determinant trick|A8HRT9H3XA]] gives $a \in I$ with $(1 + a) N = 0$.
Conversely, if $(1 + a) x = 0$ then $x = -a x = (-a)^n x \in I^n M$ for every $n$.
Example: in $R = \Z/6\Z$ with $I = (2)$, one has $I^2 = I$, so the intersection is $I = \set{0, 2, 4}$, and indeed $3 = 1 + 2$ kills $2$ and $4$.
The Noetherian hypothesis is needed.
In the local ring of germs at $0$ of smooth functions $\R \to \R$, consider the germ of $e^{-1/x^2}$, extended by $0$ at $0$.
It is nonzero, but it lies in every power of the maximal ideal.

## Explanations

### L1 human

An element divisible by $p$, by $p^2$, by $p^3$ and so on forever must be $0$ in the integers.
The theorem says that the same holds in any Noetherian local ring and in any Noetherian domain: an element that lies in every power of a proper ideal is $0$.
In other words, the $I$-adic topology separates points.

### L2 human

Clause 1 shows that a Noetherian local ring embeds into its $\mathfrak m$-adic [[completion|KBN13NPYJQ]].
It also shows that two elements of a Noetherian local ring are equal if they agree modulo every power of $\mathfrak m$.
This is the basis of many arguments by successive approximation.
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# Topological dynamics

## Definition

A *topological dynamical system* is a pair $(X, T)$ where $X$ is a [[topological space|0CQHFPGJ45]] and $T \colon X \to X$ is a [[continuous map|0VGFHTM75Y]].
More generally, it is a triple $(G, X, \alpha)$ where $G$ is a [[topological group|92700QAB5Y]] and $\alpha \colon G \times X \to X$ is a continuous [[group action|0WCY42EV9V]].
*Topological dynamics* is the study of the properties of such systems that are preserved by topological conjugacy, that is, by homeomorphisms $h \colon X \to Y$ with $h \circ T = S \circ h$.

## Remarks

In most of the theory $X$ is a compact metric space, and $T$ is a homeomorphism or $G$ is $\Z$ or $\R$.
A map $T$ gives an action of the monoid $\N$ by $n \cdot x = T^n(x)$; a homeomorphism gives an action of $\Z$; a continuous flow gives an action of $\R$.
Example: the circle $X = \R / \Z$ with $T(x) = x + \alpha$ is a topological dynamical system, since $T$ is continuous.
Non-example: $X = [0, 1]$ with $T(x) = 0$ for $x < 1/2$ and $T(x) = 1$ for $x \ge 1/2$ is not one, because $T$ is not continuous at $1/2$.

## Explanations

### L1 human

A topological dynamical system is a space together with a continuous rule of motion.
Topological dynamics studies the features of the motion that survive a continuous change of coordinates.
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# Summability kernel

## Summary

A summability kernel is a sequence of periodic functions with mean $1$, uniformly bounded integrals of their absolute values, and mass concentrating at $0$.
Convolving a function with the terms of a summability kernel gives approximations that converge to it in $L^1$.
For a continuous function the convergence is uniform.
The [[Fejér kernel|DCCA2T9GKH]] and the [[Poisson kernel|TCYSWCTYYM]] are summability kernels, and they explain why [[Cesàro means|MA8X1G11DH]] and Abel means of Fourier series converge.
The [[Dirichlet kernel|3JY4BVE4DW]] is not one, which is why ordinary partial sums can fail to converge.

## Explanations

### L1 human

A summability kernel is a family of weights that become a sharper and sharper spike at $0$ while keeping total weight $1$.
Convolving a function with such a weight replaces each value by an average over a shrinking window.
So the averages approach the function itself.
For example, the Fejér kernels $F_N$ are nonnegative bumps of mean $1$ whose width near $0$ is about $1/N$.
They are the periodic case of an [[approximate identity|CDYXTW3PJF]].
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# Finite difference method

## Definition

Let a [[partial differential equation|GC5BDZ6QBC]] $\mathcal{L} u = f$ be given on a domain $\Omega \subseteq \R^d$, possibly with time, together with boundary or initial conditions.
The *finite difference method* consists of:

1. A grid: finitely many points $x_j$ of $\overline{\Omega}$, usually $x_j = jh$ with mesh size $h > 0$, and time levels $t_n = nk$ with time step $k > 0$.
2. Difference quotients: each [[partial derivative|09EGC6F8PT]] in $\mathcal{L}$ is replaced by a linear combination of grid values, such as $(u_{j+1} - u_j)/h$ for $\partial_x u$ or $(u_{j+1} - 2u_j + u_{j-1})/h^2$ for $\partial_x^2 u$.
3. The discrete problem: the resulting algebraic equations and the discretized side conditions are solved for the grid values $u_j \approx u(x_j)$.

## Remarks

The local truncation error is the residual left when the exact solution is put into the scheme.
The scheme is consistent of order $p$ if this error is $O(h^p)$; for example, Taylor expansion gives $\frac{u(x+h) - 2u(x) + u(x-h)}{h^2} = u''(x) + \frac{h^2}{12} u^{(4)}(\xi)$ for some $\xi$ in $(x - h, x + h)$ when $u \in C^4$.
Example: for the heat equation $u_t = u_{xx}$, the explicit scheme is $u_j^{n+1} = u_j^n + r (u_{j+1}^n - 2u_j^n + u_{j-1}^n)$ with $r = k/h^2$.
It is consistent of order $1$ in $k$ and $2$ in $h$.
It is stable in the maximum norm when $r \le \tfrac12$, since then $u_j^{n+1}$ is a convex combination of $u_{j-1}^n, u_j^n, u_{j+1}^n$.
Non-example: the finite element method also yields algebraic equations on a mesh, but it does not satisfy clause 2.
It imposes the equation in weak form against basis functions instead of replacing derivatives by difference quotients.
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# Jet

## Summary

The $k$-jet of a smooth map at a point records the value and all derivatives up to order $k$ there, and nothing else.
In coordinates, it is the [[Taylor polynomial|0ZZ4RHTR9X]] of degree $k$, but the notion makes sense on [[smooth manifolds|F45VQ47GPV]] without coordinates.
Jets turn differential equations into geometric objects: a differential equation of order $k$ becomes a subset of a space of $k$-jets.
They are the basic tool of the [[jet bundles|HC3VBBCJQP]] and of singularity theory.

## Explanations

### L1 human

Two curves through a point have the same $1$-jet if they pass through the point with the same velocity.
They have the same $2$-jet if, in addition, they have the same acceleration there, and so on.
For example, $\sin x$ and $x$ have the same $2$-jet at $0$, since both have value $0$, slope $1$ and second derivative $0$ there.
They have different $3$-jets, since the third derivative of $\sin$ at $0$ is $-1$ while that of $x$ is $0$.
A jet is a finite list of numbers that describes a map up to a chosen order of contact.
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# Dual graph

## Summary

The dual graph of a graph drawn in the plane has one vertex for each face of the drawing.
It has one edge for each edge of the drawing, joining the two faces on its two sides.
Duality exchanges faces and vertices, and cycles and minimal cuts.
It depends on the chosen [[embedding|CG91WNMG2T]], not only on the abstract graph, and the same idea works for graphs embedded in other surfaces.

## Explanations

### L1 human

Think of a map of countries.
Put a capital in each country, and join two capitals by a road crossing each stretch of border that the two countries share.
The road network is the dual graph of the map of borders.
For example, the dual of the graph of a cube, which has $6$ faces, is the graph of an octahedron, which has $6$ vertices.
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# Direct limit of rings

## Definition

Let $(I, \le)$ be a directed set: a nonempty preordered set in which any two elements have a common upper bound.
A *directed system of rings* over $I$ consists of [[rings|0B1261B5ED]] $R_i$ for $i \in I$ and [[ring homomorphisms|0VTH8KPCG2]] $f_{ij} \colon R_i \to R_j$ for $i \le j$, such that:

1. $f_{ii}$ is the identity of $R_i$ for every $i$.
2. $f_{jk} \circ f_{ij} = f_{ik}$ whenever $i \le j \le k$.

On the disjoint union $\bigsqcup_i R_i$, let $a \in R_i$ and $b \in R_j$ be [[equivalent|0G8VEZ3WTK]] if $f_{ik}(a) = f_{jk}(b)$ for some $k$ with $i \le k$ and $j \le k$.
The *direct limit* $\varinjlim R_i$ is the set of equivalence classes $[a]$.
For $a \in R_i$ and $b \in R_j$, choose $k \ge i, j$ and set $[a] + [b] = [f_{ik}(a) + f_{jk}(b)]$ and $[a][b] = [f_{ik}(a) f_{jk}(b)]$.

## Remarks

These operations do not depend on the choices, and they make $\varinjlim R_i$ a ring; the maps $\varphi_i \colon R_i \to \varinjlim R_i$, $a \mapsto [a]$, are ring homomorphisms with $\varphi_j \circ f_{ij} = \varphi_i$.
It is the [[direct limit|X9HEQ29Q49]] of the system in the category of rings: for every ring $S$ and homomorphisms $g_i \colon R_i \to S$ with $g_j \circ f_{ij} = g_i$, there is exactly one homomorphism $g \colon \varinjlim R_i \to S$ with $g \circ \varphi_i = g_i$.
Example: let $I = \N$, $R_n = k[x_1, \dots, x_n]$ and let $f_{mn}$ be the inclusion for $m \le n$.
The direct limit is the polynomial ring $k[x_1, x_2, \dots]$ in countably many variables, since each polynomial involves only finitely many variables.
This ring is not [[Noetherian|0670T59HGE]], although every $R_n$ is, as the chain of ideals $(x_1) \subset (x_1, x_2) \subset \cdots$ shows.
Non-example: if $I$ is not directed, say two incomparable indices $1, 2$ with rings $\Z$ and $\Z/2$ and no common upper bound.
Then the sum $[a] + [b]$ for $a \in R_1$ and $b \in R_2$ is not defined, and the construction does not give a ring.
A direct limit of fields is a field, and a direct limit of local rings along local homomorphisms is a local ring.
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# Rubin causal model

## Definition

Consider units $i = 1, \dots, n$ and a binary treatment.
In the *Rubin causal model* the data are described as follows.

1. Each unit $i$ has two potential outcomes $Y_i(1)$ and $Y_i(0)$: the outcome under treatment and under control.
2. Each unit receives a treatment $W_i \in \set{0, 1}$, and the observed outcome is $Y_i = W_i Y_i(1) + (1 - W_i) Y_i(0)$.
3. The potential outcomes of unit $i$ do not depend on the treatments of the other units.
   There is only one version of each treatment (the stable unit treatment value assumption, SUTVA).

The *unit causal effect* is $Y_i(1) - Y_i(0)$.
When the units are drawn from a population, the *average treatment effect* is $\tau = E[Y(1) - Y(0)]$, an [[expected value|0WQNRCNY3M]] over the population.

## Remarks

Since only one of $Y_i(1)$ and $Y_i(0)$ is observed, unit effects are never observed; this is called the fundamental problem of causal inference.
The assignment is *unconfounded* (ignorable) given covariates $X$ if $W$ is [[independent|0GQ938GDE0]] of $(Y(0), Y(1))$ conditionally on $X$.

Example: in a completely randomised experiment, randomization gives $W$ independent of $(Y(0), Y(1))$.
Then $E[Y \mid W = 1] = E[Y(1) \mid W = 1] = E[Y(1)]$, and likewise for $W = 0$, so $\tau = E[Y \mid W = 1] - E[Y \mid W = 0]$.

Non-example: in a vaccine trial in which unvaccinated people can be protected by their vaccinated neighbours, the outcome of a unit depends on the treatments of others.
Clause 3 fails, and the two potential outcomes per unit are not enough to describe the situation.
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# Non-standard model of arithmetic

## Example

Let $\mathsf{PA}$ be first-order [[Peano arithmetic|0QFT7Z76N5]] in the language with $0$, $S$, $+$, $\cdot$ and $<$.
A *non-standard model of arithmetic* is a [[model|MVCH6N9SNW]] of $\mathsf{PA}$ that is not isomorphic to the standard model $\N$ of the [[natural numbers|0MWKJ44V5A]].

Such a model exists.
Add a new constant symbol $c$, and let $T = \mathsf{PA} \cup \set{c > \underline{n} : n \in \N}$, where $\underline{n}$ is the term $S S \cdots S 0$ with $n$ copies of $S$.
Every finite subset of $T$ mentions only finitely many $n$, so it has a model: $\N$ with $c$ read as a large enough number.
By the [[compactness theorem|1NR3HAWG9D]], $T$ has a model $M$.
In $M$ the element $c^M$ is greater than the value of every numeral, so it is not of the form $\underline{n}^M$.
In a model isomorphic to $\N$ every element is the value of a numeral, so $M$ is a non-standard model of $\mathsf{PA}$.
By the downward Löwenheim–Skolem theorem, $M$ can be taken countable.

## Remarks

Every non-standard model $M$ starts with a copy of $\N$, the values of the numerals, followed by the non-standard elements.
The order type of every countable non-standard model of $\mathsf{PA}$ is $\omega + (\omega^* + \omega) \cdot \eta$, where $\omega^*$ is the reversed order of $\N$ and $\eta$ is the order type of $\Q$.
By [[Tennenbaum's theorem|W6D5ZJAY9D]], no countable non-standard model of $\mathsf{PA}$ has computable addition or multiplication.

## Explanations

### L1 human

The axioms of arithmetic, written in first-order logic, do not pin down the natural numbers.
There are other structures that satisfy every one of the axioms but contain "infinite" numbers larger than $0, 1, 2, \ldots$.
The compactness theorem builds one by asking for a number bigger than each ordinary number, one at a time.
Inside such a model, the infinite numbers look like ordinary numbers: each has a successor, a predecessor, and so on.
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# Log-rank test

## Definition

Let two groups give right-[[censored|MY8ZZP84NP]] lifetime data.
Let $t_1 < \dots < t_m$ be the distinct event times in the pooled data.
At $t_j$, let $n_j$ be the number at risk and $d_j$ the number of events in the pooled data.
Let $n_{1j}$ and $d_{1j}$ be the same numbers in group 1.
Set
$$E_j = d_j \frac{n_{1j}}{n_j}, \qquad V_j = d_j \frac{n_{1j}}{n_j} \left(1 - \frac{n_{1j}}{n_j}\right) \frac{n_j - d_j}{n_j - 1},$$
with $V_j = 0$ when $n_j = 1$.
The *log-rank statistic* is
$$Z = \frac{\sum_{j=1}^{m} (d_{1j} - E_j)}{\sqrt{\sum_{j=1}^{m} V_j}}.$$
The *log-rank test* of the null hypothesis that the two groups have the same [[hazard function|Y959NKQ3XW]] rejects at level $\alpha$ when $\abs{Z} > z_{1 - \alpha/2}$.
Here $z_{1 - \alpha/2}$ is the $(1 - \alpha/2)$-quantile of the standard [[normal distribution|05AQE0QR0D]].

## Remarks

Under the null hypothesis and non-informative censoring, $Z$ is approximately standard normal for large samples, and $Z^2$ is approximately [[chi-squared|0A9EB0KQBC]] with $1$ degree of freedom.
Given the numbers at risk, $E_j$ and $V_j$ are the mean and variance of $d_{1j}$ under the hypergeometric distribution.

Example: group 1 has events at times $1$ and $3$, group 2 at times $2$ and $4$, with no censoring.
The pairs $(n_j, n_{1j})$ are $(4, 2)$, $(3, 1)$, $(2, 1)$, $(1, 0)$, with $d_j = 1$ each time and $d_{1j} = 1, 0, 1, 0$.
So $\sum (d_{1j} - E_j) = 1/2 - 1/3 + 1/2 + 0 = 2/3$ and $\sum V_j = 1/4 + 2/9 + 1/4 = 13/18$, which gives $Z \approx 0.78$: no evidence of a difference.

Non-example: comparing the two Kaplan–Meier estimates at a single fixed time, say $t = 2$, is a different test.
It ignores the events at all other times, while the log-rank statistic combines them.
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# Laurent polynomial ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
The *Laurent polynomial ring* $R[x, x^{-1}]$ is the [[monoid ring|1HM7ZYMPD7]] $R[\Z]$ of the additive group $\Z$, with the element $n \in \Z$ written $x^n$.
Its elements are the finite sums $\sum_{i = m}^{n} a_i x^i$ with $m \le n$ integers and $a_i \in R$.
Addition is termwise, and multiplication is determined by $x^i x^j = x^{i + j}$ for $i, j \in \Z$ and by $a x^i = x^i a$ for $a \in R$.

## Remarks

Example: $f = x^{-1} + 2$ and $g = x - 2x^2$ are Laurent polynomials, and $fg = 1 - 2x + 2x - 4x^2 = 1 - 4x^2$.
Non-example: the series $\sum_{i \ge 0} x^i$ is not a Laurent polynomial, since it has infinitely many nonzero terms; it lies in the ring of [[Laurent series|0GXMPDNYX3]].
The ring contains the [[polynomial ring|0SDC934R52]] $R[x]$, and $x$ is a [[unit|0XVEZDE9J8]] with inverse $x^{-1}$.
For a commutative ring $R$, $R[x, x^{-1}]$ is the [[localization|4C1AE84N5N]] of $R[x]$ at the multiplicative set $\set{1, x, x^2, \dots}$.
If $R$ is an [[integral domain|0QVD7WBFGN]], the units of $R[x, x^{-1}]$ are the elements $u x^k$ with $u$ a unit of $R$ and $k \in \Z$.
The ring in several variables $R[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$ is the monoid ring $R[\Z^n]$.
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# Birkhoff min-max method

## Summary

The Birkhoff min-max method finds a [[closed geodesic|4MCCP8GG4J]] on a Riemannian sphere by sweeping the sphere with a one-parameter family of closed curves.
One takes, over all sweepouts, the length of the longest curve, and then the infimum of this number over all sweepouts.
George Birkhoff showed that this min-max value, the width, is positive and is the length of a closed geodesic.
The method is the prototype of min-max constructions in geometry, including [[Almgren–Pitts min-max theory|6KVJHG1KRZ]] for minimal hypersurfaces.

## Explanations

### L1 human

Slide a rubber band over a sphere from the south pole to the north pole, so that it passes over every point.
Somewhere along the way the band is longest.
Now try to choose the sweep so that this longest moment is as short as possible.
The best possible longest band cannot be shrunk further without the sweep getting stuck, and it sits on a closed geodesic.
On a round unit sphere, sweeping by circles of latitude gives longest band the equator, of length $2\pi$, and this is the width.
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# Riemannian metric

## Definition

Let $M$ be a smooth manifold with [[tangent bundle|WY6K3P4YCG]] $TM$.
A *Riemannian metric* on $M$ is a family $g = (g_p)_{p \in M}$ such that:

1. for each $p \in M$, $g_p$ is an [[inner product|0NB0AWGJBW]] on $T_pM$, that is, a symmetric positive definite bilinear form;
2. for all smooth vector fields $X, Y$ on $M$, the function $p \mapsto g_p(X_p, Y_p)$ is smooth.

## Remarks

Equivalently, $g$ is a smooth symmetric [[tensor field|WX302T93TW]] of type $(0, 2)$ that is positive definite at every point.
In a chart, $g = \sum_{i,j} g_{ij} \, dx^i \otimes dx^j$ where $(g_{ij}(p))$ is a symmetric positive definite matrix depending smoothly on $p$; one often writes $ds^2 = g_{ij} \, dx^i dx^j$.
Example: on the upper half-plane $\set{(x, y) : y > 0}$, $g = (dx^2 + dy^2)/y^2$ is a Riemannian metric; its matrix $y^{-2} I$ is symmetric and positive definite because $y > 0$, and its entries are smooth.
Non-example: on $\R^2$, $g = x^2 (dx^2 + dy^2)$ fails clause 1 on the line $x = 0$, where $g_p = 0$ is not positive definite.
The length of a piecewise smooth curve $\gamma : [a, b] \to M$ is $\int_a^b \sqrt{g(\gamma'(t), \gamma'(t))} \, dt$.
Every smooth manifold (Hausdorff and second countable) has a Riemannian metric: glue Euclidean metrics of charts with a [[partition of unity|NH739XXVXK]]; positive definiteness survives convex combinations.
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# Flux limiter

## Definition

Consider a scalar [[conservation law|8RA7AMWSAF]] $u_t + f(u)_x = 0$ and a conservative scheme $u_j^{n+1} = u_j^n - \lambda (F_{j+1/2} - F_{j-1/2})$ with $\lambda = \Delta t / \Delta x$.
Let $F^L_{j+1/2}$ be a low-order numerical flux, such as that of the [[upwind scheme|MTCRZC5QKT]].
Let $F^H_{j+1/2}$ be a high-order one, such as that of the [[Lax–Wendroff method|YHZ9BCNDRZ]].
For flow from left to right, the smoothness ratio at $j$ is
$$r_j = \frac{u_j^n - u_{j-1}^n}{u_{j+1}^n - u_j^n},$$
with a fixed convention when the denominator is zero.
A *flux limiter* is a function $\varphi \colon \R \to [0, \infty)$ with $\varphi(r) = 0$ for $r \le 0$, used to define the limited flux
$$F_{j+1/2} = F^L_{j+1/2} + \varphi(r_j) \left( F^H_{j+1/2} - F^L_{j+1/2} \right).$$

## Remarks

Take $u_t + a u_x = 0$ with $a > 0$ and $0 \le \nu \le 1$, with upwind and Lax–Wendroff fluxes.
Then Sweby's condition
$$0 \le \varphi(r) \le \min(2r, 2) \quad \text{for } r > 0$$
makes the scheme [[total variation diminishing|NX8RE9B72A]].
If moreover $\varphi(1) = 1$ and $\varphi$ is Lipschitz near $1$, the scheme is second-order accurate in smooth monotone regions.
Example: the minmod limiter $\varphi(r) = \max(0, \min(1, r))$ vanishes for $r \le 0$, satisfies $\varphi(r) \le \min(r, 1) \le \min(2r, 2)$ for $r > 0$, and $\varphi(1) = 1$.
Other standard limiters are superbee $\varphi(r) = \max(0, \min(2r, 1), \min(r, 2))$ and van Leer $\varphi(r) = (r + \abs{r}) / (1 + \abs{r})$.
Non-example: the constant function $\varphi \equiv 1$ is not a flux limiter, because $\varphi(r) \ne 0$ for $r \le 0$; it gives back the unlimited Lax–Wendroff scheme, which is not TVD.
At extrema $r_j \le 0$, so the scheme reduces to the low-order scheme there.

## Explanations

### L1 human

The limiter is a dial between $0$ (use only the safe low-order flux) and about $1$ or $2$ (use the high-order correction fully or more).
Its setting at each point depends on how smooth the solution looks there.
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<document source="https://tamnd.github.io/math2/node/1RHTVMH34G.md" tag="1RHTVMH34G" kind="concept">
# Weakly compact cardinal

## Summary

A weakly compact cardinal is an uncountable cardinal $\kappa$ for which the infinite Ramsey theorem holds at $\kappa$.
This means that every colouring of the pairs of elements of $\kappa$ with two colours has a homogeneous set of size $\kappa$.
Weakly compact cardinals are [[inaccessible|NW4DREQ5DQ]] and even [[Mahlo|CJFEPPP9KA]].
The name comes from an equivalent property: a compactness theorem for the infinitary logic $\mathcal{L}_{\kappa, \kappa}$ in the case of theories with at most $\kappa$ sentences.
Every [[measurable cardinal|AKBWT323RQ]] is weakly compact.

## Explanations

### L1 human

Ramsey's theorem says that if one colours each pair of natural numbers red or blue, there is an infinite set of numbers whose pairs all have the same colour.
A weakly compact cardinal is an uncountable size of infinity for which the same is true.
For the first uncountable cardinal this fails, so a weakly compact cardinal must be very large.
It is a large cardinal that combines combinatorics, trees and logic in one notion.
</document>

<document source="https://tamnd.github.io/math2/node/1RJJ33KKSS.md" tag="1RJJ33KKSS" kind="definition">
# Associated prime

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $M$ be an $R$-[[module|09VDS2W1E8]].
For $m \in M$, let $\operatorname{Ann}_R(m) = \set{r \in R : rm = 0}$ be the [[annihilator|HB4XDKJTW8]] of $m$.
A [[prime ideal|06HC6HK0Z8]] $P$ of $R$ is an *associated prime* of $M$ if $P = \operatorname{Ann}_R(m)$ for some $m \in M$.
The set of associated primes of $M$ is written $\operatorname{Ass}_R(M)$.

## Remarks

For an ideal $I$ of $R$, the associated primes of $I$ usually means $\operatorname{Ass}_R(R/I)$.
Example: for $M = \Z/12\Z$ over $\Z$, $\operatorname{Ann}(6) = (2)$ and $\operatorname{Ann}(4) = (3)$, so $(2)$ and $(3)$ are associated primes; in fact $\operatorname{Ass}(M) = \set{(2), (3)}$.
Non-example: $(0)$ is not an associated prime of $\Z/12\Z$, because $12$ kills every element, so no annihilator is $(0)$.
For $M = \Z$, the annihilator of every nonzero element is $(0)$, so $\operatorname{Ass}(\Z) = \set{(0)}$.
Equivalently, $P \in \operatorname{Ass}_R(M)$ when $M$ has a submodule isomorphic to $R/P$.
For rings that are not Noetherian, some authors use other variants, such as weakly associated primes.

## Explanations

### L1 human

An associated prime of a module is a prime ideal that is exactly the set of ring elements killing some one element of the module.
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<document source="https://tamnd.github.io/math2/node/1RP3NVGNA6.md" tag="1RP3NVGNA6" kind="statement">
# Szegő recurrence and Verblunsky coefficients

## Statement

Let $\mu$ be a probability measure on the unit circle $\partial\mathbb{D}$ with infinite support, and let $\Phi_n$ be its monic [[orthogonal polynomials|3F9G3DRQ62]].
For a polynomial $p$ of degree at most $n$, let $p^*(z) = z^n \, \overline{p(1/\bar z)}$ be its reversed polynomial of order $n$; so $\Phi_n^*$ has constant term $1$.

1. There are complex numbers $\alpha_0, \alpha_1, \alpha_2, \dots$ with $\abs{\alpha_n} < 1$ such that for every $n \ge 0$
$$\Phi_{n+1}(z) = z \, \Phi_n(z) - \overline{\alpha_n} \, \Phi_n^*(z).$$
They are given by $\alpha_n = -\overline{\Phi_{n+1}(0)}$ and are called the *Verblunsky coefficients* of $\mu$.
2. The norms satisfy $\norm{\Phi_{n+1}}^2 = (1 - \abs{\alpha_n}^2) \norm{\Phi_n}^2$, so $\norm{\Phi_n}^2 = \prod_{j=0}^{n-1} (1 - \abs{\alpha_j}^2)$.
3. (Verblunsky's theorem) Let $\mathbb{D}$ be the open unit disk. The map $\mu \mapsto (\alpha_n)_{n \ge 0}$ is a bijection from the probability measures on $\partial\mathbb{D}$ with infinite support onto the sequences in $\mathbb{D}$.

## Remarks

For part 1, the polynomial $\Phi_{n+1} - z\Phi_n$ has degree at most $n$.
It is orthogonal to $z, z^2, \dots, z^n$, because $\Phi_{n+1}$ is, and multiplication by $z$ is unitary on $L^2(\mu)$ and maps $\Phi_n \perp \set{1, \dots, z^{n-1}}$ onto $z\Phi_n \perp \set{z, \dots, z^n}$.
The polynomials of degree at most $n$ orthogonal to $z, \dots, z^n$ form a one-dimensional space spanned by $\Phi_n^*$, which gives the recurrence.
Taking norms, with $\norm{z\Phi_n} = \norm{\Phi_n^*} = \norm{\Phi_n}$, gives part 2, and $\norm{\Phi_{n+1}} > 0$ forces $\abs{\alpha_n} < 1$.
Part 3 builds $\mu$ from any sequence in $\mathbb{D}$ by running the recurrence and taking a weak limit of the measures $\abs{\varphi_n(e^{i\theta})}^{-2} d\theta/2\pi$.

Example: for $d\mu = d\theta/2\pi$, $\Phi_n(z) = z^n$ and $\Phi_n^* = 1$, so $\Phi_{n+1} = z\Phi_n$ and all $\alpha_n = 0$.
Conversely, by part 3, the uniform measure is the only one with all Verblunsky coefficients zero.

Infinite support is needed.
If $\mu$ is supported on $N$ points, then $L^2(\mu)$ has dimension $N$ and $\Phi_N$ has norm $0$; the recurrence ends with $\abs{\alpha_{N-1}} = 1$.

## Explanations

### L1 human

On the real line, orthogonal polynomials are built step by step by a rule with two numbers per step.
On the unit circle the rule is simpler: one complex number $\alpha_n$ per step, of modulus less than $1$.
Every measure on the circle gives such a sequence, and every such sequence comes from exactly one measure.
So the sequence of Verblunsky coefficients is a complete, free set of coordinates for measures on the circle.

### L2 human

The recurrence turns questions about the measure $\mu$ into questions about the sequence $(\alpha_n)$.
For example, [[Szegő's theorem|SZ8ARMDEMY]] says that $\sum \abs{\alpha_n}^2 < \infty$ exactly when $\log$ of the density of $\mu$ is integrable.
The $\alpha_n$ are the entries of the CMV matrix, the unitary analogue of the Jacobi matrix, which represents multiplication by $z$ on $L^2(\mu)$.
In prediction theory and signal processing the same numbers appear as reflection coefficients, computed by the Levinson algorithm from the moments of $\mu$.
</document>

<document source="https://tamnd.github.io/math2/node/1RVF7DWG8C.md" tag="1RVF7DWG8C" kind="problem">
# Post's problem

## Problem

Is there a [[computably enumerable set|TN9G24VABN]] $A \subseteq \N$ that is not computable and is not [[Turing equivalent|PA4PYQ1PZ0]] to the [[halting problem|Z7TN2B35B0]]?
In terms of [[Turing degrees|P5N2ND3VCG]]: is there a computably enumerable degree $\mathbf{a}$ with $\mathbf{0} < \mathbf{a} < \mathbf{0}'$?

## Remarks

Every computably enumerable set $A$ satisfies $A \le_T \emptyset'$, so the question asks whether the c.e. degrees contain anything besides $\mathbf{0}$ and $\mathbf{0}'$.
Emil Post posed the question in 1944.
The answer is yes.
Friedberg and Muchnik answered it independently, in 1957 and 1956, by constructing two c.e. sets that are not Turing reducible to each other; see the [[Friedberg–Muchnik theorem|RA358M9QCN]].
Their method of construction is the [[priority method|3BZFSC5QZ2]].
Post's own attempt was to find a structural property of c.e. sets, such as being simple, that forces the degree to lie strictly between $\mathbf{0}$ and $\mathbf{0}'$.

## Explanations

### L1 human

Among the sets that a computer can list, the easiest are the ones it can also decide, and the hardest is the set of halting programs.
Post asked whether anything lies in between.
It took more than ten years and a new technique to show that the answer is yes.
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<document source="https://tamnd.github.io/math2/node/1RY6DQJ1RV.md" tag="1RY6DQJ1RV" kind="concept">
# Symmetric spectrum

## Summary

A symmetric spectrum is a [[spectrum|1GHF4EKBXS]] in which the $n$-th space carries an action of the symmetric group on $n$ letters.
These actions are compatible with the structure maps.
This extra symmetry gives a smash product of spectra that is associative, commutative and unital on the nose, not only up to homotopy.
Symmetric spectra form a model for the [[stable homotopy category|26GTNMAF7B]] in which ring spectra with fully coherent multiplications are simply monoids.

## Explanations

### L1 human

The smash product $S^n \wedge S^m = S^{n+m}$ hides a choice: the coordinates of the first factor come before those of the second.
Swapping the factors permutes coordinates, which is a nontrivial map.
A symmetric spectrum keeps track of these permutations by letting the symmetric group act on each space.
With this bookkeeping the smash product of two symmetric spectra is defined directly, and it is strictly commutative.
The sphere spectrum, with the symmetric group acting on $S^n = S^1 \wedge \cdots \wedge S^1$ by permuting the factors, is the basic example.
</document>

<document source="https://tamnd.github.io/math2/node/1RZV6TRTPE.md" tag="1RZV6TRTPE" kind="definition">
# Structural rigidity

## Definition

Let $d \ge 1$ and let $G = (V, E)$ be a finite [[graph|00C0BR7D7H]].
A *framework* in $\R^d$ is a pair $(G, p)$ where $p \colon V \to \R^d$ is a map.
The framework $(G, p)$ is *rigid* if there is $\varepsilon > 0$ with the following property: every map $q \colon V \to \R^d$ such that

1. $\norm{q(u) - q(v)} = \norm{p(u) - p(v)}$ for every edge $uv \in E$, and
2. $\norm{q(v) - p(v)} < \varepsilon$ for every $v \in V$

also satisfies $\norm{q(u) - q(v)} = \norm{p(u) - p(v)}$ for all $u, v \in V$.
A framework that is not rigid is *flexible*.

## Remarks

Example: the triangle $K_3$ placed at three non-collinear points of $\R^2$ is rigid.
Clause 1 already fixes all three pairwise distances, so the conclusion holds for every $\varepsilon$.

Non-example: the $4$-cycle placed at the corners of the unit square in $\R^2$ is flexible.
For every $\varepsilon > 0$ there is a rhombus with sides $1$ close to the square, which satisfies clauses 1 and 2, but its diagonals have lengths different from $\sqrt{2}$.

Rigidity is equivalent to the absence of a nontrivial flex.
A flex is a continuous motion of the points, starting at $p$, that keeps all bar lengths; it is nontrivial if it changes some distance between two vertices.
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<document source="https://tamnd.github.io/math2/node/1S20A3R659.md" tag="1S20A3R659" kind="statement">
# Arakelyan's theorem

## Statement

Let $E \subseteq \C$ be a [[closed set|0G5AV9Q4RR]], and let $\hat\C = \C \cup \set{\infty}$ be the [[Riemann sphere|0CDMXGPSES]].
The following are equivalent.

1. Let $f \colon E \to \C$ be any function that is [[continuous|0P27W9RW0G]] on $E$ and [[holomorphic|0DG0Z72CQD]] on the [[interior|0C38S0KMJK]] of $E$, and let $\varepsilon > 0$ be arbitrary.
Then there is an [[entire function|0419794R0T]] $g$ with $\abs{f(z) - g(z)} < \varepsilon$ for all $z \in E$.
2. The set $\hat\C \setminus E$ is [[connected|0D939CMME7]] and locally connected.

## Remarks

For compact $E$, the set $\hat\C \setminus E$ is an open subset of the sphere, so it is automatically locally connected, and connectedness of $\hat\C \setminus E$ is the same as connectedness of $\C \setminus E$.
So for compact $E$ the theorem reduces to [[Mergelyan's theorem|5KKCSZJC29]], since entire functions can be approximated by polynomials on compact sets.
The new condition for unbounded $E$ is local connectedness of $\hat\C \setminus E$ at the point $\infty$.

Example: let $E = \R$.
Then $\hat\C \setminus \R$ is the union of the two open half-planes and the point $\infty$, which is connected and locally connected.
So every continuous function on $\R$ is a uniform limit on all of $\R$ of entire functions, a result first proved by Carleman.

Arakelyan also proved a version on an arbitrary open set $\Omega$ in place of $\C$, with $E$ closed in $\Omega$.
There the approximants are holomorphic on $\Omega$, and the one-point compactification of $\Omega$ replaces $\hat\C$.

## Explanations

### L1 human

Mergelyan's theorem tells when functions on a compact set can be approximated by polynomials.
Arakelyan's theorem does the same for closed sets that may go off to infinity, with entire functions instead of polynomials.
The set must have no holes, and in addition it must not wind around near infinity in a way that traps parts of its complement.
For the real line, the conclusion is that any continuous function can be uniformly approximated by entire functions on the whole line.

### L2 human

The theorem is the final answer to uniform holomorphic approximation on closed subsets of the plane.
It is used to construct entire functions with prescribed behaviour along unbounded sets, for example functions that tend to $0$ along every ray but are not identically $0$.
Its proof combines Mergelyan-type estimates with a careful exhaustion of $E$ by compact pieces, where local connectedness at $\infty$ controls how the pieces fit together.
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# t-structure

## Definition

Let $\mathcal{D}$ be a [[triangulated category|T2V7GJJ2EN]] with shift $X \mapsto X[1]$.
A *t-structure* on $\mathcal{D}$ is a pair $(\mathcal{D}^{\le 0}, \mathcal{D}^{\ge 0})$ of [[full subcategories|V3BFGKBJF7]], each closed under isomorphisms.
Write $\mathcal{D}^{\le n} = \mathcal{D}^{\le 0}[-n]$ and $\mathcal{D}^{\ge n} = \mathcal{D}^{\ge 0}[-n]$.
The pair must satisfy:

1. $\mathcal{D}^{\le -1} \subseteq \mathcal{D}^{\le 0}$ and $\mathcal{D}^{\ge 1} \subseteq \mathcal{D}^{\ge 0}$;
2. $\Hom(X, Y) = 0$ for every $X$ in $\mathcal{D}^{\le 0}$ and every $Y$ in $\mathcal{D}^{\ge 1}$;
3. every object $X$ fits into a [[distinguished triangle|ADKK56VZHW]] $A \to X \to B \to A[1]$ with $A$ in $\mathcal{D}^{\le 0}$ and $B$ in $\mathcal{D}^{\ge 1}$.

The *heart* of the t-structure is the full subcategory $\mathcal{D}^{\le 0} \cap \mathcal{D}^{\ge 0}$.

## Remarks

The heart is always an abelian category, and the triangle in clause 3 is unique up to a unique isomorphism; this gives truncation functors $\tau_{\le 0}$ and $\tau_{\ge 1}$.
Example: let $D(\mathcal{A})$ be the derived category of an abelian category, written with cochain complexes.
Its *standard* t-structure has $\mathcal{D}^{\le 0}$ the complexes with $H^i = 0$ for $i > 0$ and $\mathcal{D}^{\ge 0}$ those with $H^i = 0$ for $i < 0$.
The triangle of clause 3 is $\tau_{\le 0} X \to X \to \tau_{\ge 1} X \to$, built from the truncated complexes, and the heart is equivalent to $\mathcal{A}$.
Non-example: if $\mathcal{D} \ne 0$, taking $\mathcal{D}^{\le 0} = \mathcal{D}^{\ge 0} = \mathcal{D}$ fails clause 2, because for $X \ne 0$ the identity of $X$ is a nonzero element of $\Hom(X, X)$, while $X$ lies in both $\mathcal{D}^{\le 0}$ and $\mathcal{D}^{\ge 1} = \mathcal{D}$.
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<document source="https://tamnd.github.io/math2/node/1SAAZG46DG.md" tag="1SAAZG46DG" kind="concept">
# Scott domain

## Summary

A Scott domain is a directed-complete partial order that is algebraic and bounded complete.
Algebraic means that every element is the directed supremum of the [[compact elements|9F7TYES4KE]] below it.
Bounded complete means that every subset with an upper bound has a supremum.
They come from the work of Dana Scott on the semantics of programming languages and of the [[lambda calculus|BD756KR8FM]].
They form a cartesian closed category with Scott-continuous maps.

## Explanations

### L1 human

A Scott domain is a space of pieces of information in which every piece is a limit of finite pieces.
Two pieces that are consistent, meaning that some piece extends both, can always be combined into a least common extension.
The partial functions from $\N$ to $\N$ form a Scott domain, with the finite partial functions as compact elements.
Two partial functions that agree on their common domain combine by union.
Because maps between Scott domains again form a Scott domain, one can build domains of functions, of functions of functions, and so on.
</document>

<document source="https://tamnd.github.io/math2/node/1SCF8X47G1.md" tag="1SCF8X47G1" kind="statement">
# Robinson–Schensted correspondence

## Statement

Let $n \ge 1$ and let $S_n$ be the [[symmetric group|0515A6D2Y1]] of [[permutations|05RWFKVTJ7]] of $\set{1, \dots, n}$.
Row insertion defines a map $\sigma \mapsto (P(\sigma), Q(\sigma))$ from $S_n$ to pairs of standard [[Young tableaux|Q6TH9PH2R7]] of the same shape, a [[partition|0JV30XN73Z]] of $n$.

1. This map is a bijection onto the set of all pairs $(P, Q)$ of standard Young tableaux of the same shape with $n$ cells.
2. For every $\sigma \in S_n$, $P(\sigma^{-1}) = Q(\sigma)$ and $Q(\sigma^{-1}) = P(\sigma)$.
3. The length of the first row of the common shape equals the length of a longest increasing subsequence of $\sigma(1), \dots, \sigma(n)$.
4. The number of rows of the common shape equals the length of a longest decreasing subsequence of $\sigma(1), \dots, \sigma(n)$.

## Remarks

Row insertion of a number $x$ into a tableau works as follows.
If $x$ is larger than every entry of the first row, put $x$ at the end of that row.
Otherwise $x$ replaces the smallest entry of the row that is larger than $x$, and the replaced entry is inserted into the next row in the same way.
The tableau $P(\sigma)$ is obtained by inserting $\sigma(1), \dots, \sigma(n)$ in order into the empty tableau.
The tableau $Q(\sigma)$ records the step: it has entry $k$ in the cell that was created at step $k$.
Each step can be undone from the position of the new cell, which gives the inverse map and proves part 1.
Parts 3 and 4 are due to Schensted.

Example: for $\sigma = 2\,3\,1$ in one-line notation, inserting $2$ and $3$ gives the row $2\,3$.
Inserting $1$ bumps $2$ into the second row, so $P = \begin{smallmatrix} 1 & 3 \\ 2 & \end{smallmatrix}$ and $Q = \begin{smallmatrix} 1 & 2 \\ 3 & \end{smallmatrix}$.
The first row has length $2$, the length of the increasing subsequence $2\,3$.

## Explanations

### L1 human

The correspondence turns a permutation into two tableaux of the same shape, and back again without loss.
It is like a card-sorting game in which each new card pushes a bigger card down to the next row.
The shape you end with records how long the increasing and decreasing runs hidden in the permutation can be.

### L2 human

Counting both sides of part 1 gives $\sum_{\lambda \vdash n} (f^\lambda)^2 = n!$, where $f^\lambda$ is the number of standard tableaux of shape $\lambda$.
This matches the identity that the squares of the dimensions of the irreducible representations of $S_n$ add up to $\abs{S_n}$.
By part 2, involutions correspond to single tableaux, so the number of involutions in $S_n$ is $\sum_{\lambda \vdash n} f^\lambda$.
The [[Robinson–Schensted–Knuth correspondence|8JW8F4VVED]] extends the bijection to matrices with nonnegative integer entries and semistandard tableaux.
</document>

<document source="https://tamnd.github.io/math2/node/1SF8Y9M8YB.md" tag="1SF8Y9M8YB" kind="statement">
# Cap set problem

## Statement

Let $n \ge 1$ and let $\F_3^n$ be the [[vector space|000AYPA9AJ]] of dimension $n$ over the [[field|0ZXZ82FTX2]] $\F_3$ with three elements.
A *cap set* is a set $A \subseteq \F_3^n$ that contains no three distinct elements $x, y, z$ with $x + y + z = 0$, that is, no three distinct points on a line.
Let $m_n$ be the number of monomials $x_1^{e_1} \cdots x_n^{e_n}$ with each $e_i \in \set{0, 1, 2}$ and $e_1 + \cdots + e_n \le 2n/3$.
Then every cap set $A \subseteq \F_3^n$ satisfies $|A| \le 3 m_n$.
In particular $|A| = o(2.756^n)$ as $n \to \infty$, in the sense of [[asymptotic notation|0TGV80E6A4]].

## Remarks

This bound is due to Ellenberg and Gijswijt, following a method of Croot, Lev and Pach for $(\Z/4\Z)^n$.
The proof is an instance of the [[linear algebra method|DMYGZ3YZ6V]].
The function $T(x, y, z) = 1$ if $x + y + z = 0$ and $0$ otherwise, restricted to $A \times A \times A$, is diagonal with $|A|$ nonzero entries, so its slice rank is $|A|$.
Writing $T(x, y, z) = \prod_{i} \left(1 - (x_i + y_i + z_i)^2\right)$ and expanding, each monomial has degree at most $2n/3$ in one of the three variables, which gives slice rank at most $3 m_n$.
Earlier bounds, by Fourier methods, were of the form $3^n / n^{1 + c}$.

Example: in $\F_3^2$ the set $\set{(0,0), (0,1), (1,0), (1,1)}$ is a cap set of size $4$.
The sums of its four triples of distinct points are $(1,1)$, $(1,2)$, $(2,1)$ and $(2,2)$, none of which is $0$.
Cap sets of size larger than $2.2^n$ are known for large $n$, so the base of the exponential lies between $2.2$ and $2.756$.

## Explanations

### L1 human

In the card game SET, a cap set is a collection of cards that contains no SET.
The question is how large such a collection can be when cards have $n$ attributes instead of four.
The answer was long known to be a vanishing fraction of all $3^n$ cards, and the 2016 theorem showed it is exponentially smaller: at most about $2.756^n$.

### L2 human

The cap set problem is the analogue in $\F_3^n$ of [[Roth's theorem|E7H8NWXESF]] on three-term arithmetic progressions, since three distinct points $x, y, z$ with $x + y + z = 0$ form an arithmetic progression in $\F_3^n$.
The polynomial proof is very short and gives a power saving, which Fourier methods could not.
It introduced the slice rank of tensors and led to new bounds on related problems, such as sunflower-free families and matrix multiplication barriers.
In the integers, by contrast, there are sets without three-term progressions of size $N^{1 - o(1)}$, so the polynomial method does not transfer directly.
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<document source="https://tamnd.github.io/math2/node/1SH2WZVZ66.md" tag="1SH2WZVZ66" kind="statement">
# Huygens principle

## Statement

Let $n \ge 3$ be odd and let $g \in C^{m+1}(\R^n)$, $h \in C^{m}(\R^n)$ with $m = \frac{n+1}{2}$.
Let $u \in C^2(\R^n \times [0, \infty))$ be the solution of the [[wave equation|5EHD2XBQDG]]
$$u_{tt} = \Delta u \ \text{ on } \R^n \times (0, \infty), \qquad u(\cdot, 0) = g, \quad u_t(\cdot, 0) = h.$$
Let $x \in \R^n$ and $t > 0$.
Then $u(x, t)$ depends only on the values of $g$, $h$ and their derivatives of order at most $m$ on the sphere $\set{y : \abs{y - x} = t}$.
In particular, if $g$ and $h$ vanish outside a ball $B(0, R)$, then $u(x, t) = 0$ whenever $t > \abs{x} + R$ or $t < \abs{x} - R$.

## Remarks

For $n = 3$, Kirchhoff's formula makes this explicit:
$$u(x, t) = \frac{1}{4\pi t^2}\int_{\abs{y - x} = t} \big( t\, h(y) + g(y) + \nabla g(y) \cdot (y - x) \big)\, dS(y).$$
The proof in general odd dimension uses spherical means, which reduce the equation to a one-dimensional wave equation solved by d'Alembert's formula.
The hypothesis that $n$ is odd and at least $3$ is necessary.
For $n = 2$, Poisson's formula gives $u(x, t)$ as an integral over the whole disc $\abs{y - x} \le t$, and a pulse leaves a tail: $u(x, t)$ need not vanish for $t > \abs{x} + R$.
For $n = 1$, the integral term of d'Alembert's formula also depends on $h$ in the whole interval $[x - t, x + t]$.
This sharp form is also called the strong Huygens principle.

## Explanations

### L1 human

In three-dimensional space a short sound reaches you as a short sound: you hear it when the wave front arrives, and then silence.
The Huygens principle says this precisely: the solution at a point depends only on the data on a sphere around it, not inside the sphere.
In a flat, two-dimensional world this fails, and a short clap would be heard with a lingering echo.
The principle holds in every odd dimension from $3$ on and in no even dimension.

### L2 human

The statement follows from the explicit solution formulas for the wave equation obtained by spherical means and the method of descent.
In odd dimensions the formula contains only surface integrals over the sphere of radius $t$, while in even dimensions it contains integrals over the whole ball.
The weaker statement that $u(x, t)$ depends only on data in the closed ball of radius $t$ holds in all dimensions; it is finite speed of propagation.
The name refers to the physical Huygens principle of wave optics, where each point of a wave front acts as a source of secondary waves.
</document>

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# Locally path-connected space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *locally path-connected* if the following holds.
For every point $x \in X$ and every [[open set|02BH3MCBQB]] $U$ containing $x$, there is a [[path-connected|0WB6DH499T]] open set $V$ with $x \in V \subseteq U$.

## Remarks

Example: every open subset $W$ of $\R^n$ is locally path-connected.
For $x \in U \subseteq W$ with $U$ open, some open ball around $x$ lies in $U$, and a ball is path-connected by straight segments.

Non-example: the [[comb space|A2Q4RXD162]] is not locally path-connected.
Small open sets around $(0, 1)$ contain points of the teeth $\set{1/n} \times [0, 1]$, and these cannot be joined to $(0, 1)$ by a path inside the small set.

Every locally path-connected space is [[locally connected|FDVAHRJKF3]], because path-connected sets are connected.
The converse fails in general.
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<document source="https://tamnd.github.io/math2/node/1SYQ6QY95Y.md" tag="1SYQ6QY95Y" kind="problem">
# Kirkman's schoolgirl problem

## Problem

Fifteen schoolgirls go for a walk on each of seven days, in five rows of three.
Arrange the rows for the seven days so that no two girls walk in the same row on more than one day.

## Remarks

Each girl has $14$ companions and meets $2$ of them per day, so in a solution every two girls walk together exactly once.
So a solution is a [[Steiner triple system|7BPSC4T4M3]] $S(2, 3, 15)$ whose $35$ triples split into $7$ days of $5$ disjoint triples each.
A Steiner triple system with such a split into parallel classes is called a Kirkman triple system.
The problem has solutions; up to renaming the girls, there are seven different ones.
Thomas Kirkman posed the problem in 1850.
The general question, for $v$ girls with $v \equiv 3 \pmod 6$, was solved by Ray-Chaudhuri and Wilson in 1971: a Kirkman triple system of order $v$ exists for every such $v$.

## Explanations

### L1 human

The puzzle asks for a weekly walking plan in which every two of the fifteen girls walk side by side exactly once.
The numbers fit: there are $105$ pairs of girls, and each day the five rows of three cover $15$ pairs, so seven days cover $105$ pairs.
Finding an arrangement that really works is a nice exercise, and the puzzle started the theory of resolvable designs.
</document>

<document source="https://tamnd.github.io/math2/node/1SYY57DCWT.md" tag="1SYY57DCWT" kind="statement">
# Derivation of the conjugate gradient method

## Statement

Let $A$ be a real symmetric [[positive definite|0KX66CZMBB]] $n \times n$ matrix, let $b \in \R^n$, let $x^* = A^{-1} b$, and let $x_0 \in \R^n$.
Write $\norm{v}_A = \sqrt{v^T A v}$ and $r(x) = b - A x$.

1. (Conjugate directions) Let $p_0, \dots, p_{m-1}$ be nonzero vectors with $p_i^T A p_j = 0$ for $i \ne j$. Define $x_{k+1} = x_k + \alpha_k p_k$ with $\alpha_k = p_k^T r(x_k) / p_k^T A p_k$. Then for each $k \le m$, the residual $r(x_k)$ is orthogonal to $p_0, \dots, p_{k-1}$. Moreover $x_k$ minimizes $\norm{x - x^*}_A$ over $x_0 + \operatorname{span} \set{p_0, \dots, p_{k-1}}$. If $m = n$, then $x_n = x^*$.
2. (Short recurrence) Let $r_k = r(x_k)$, and suppose $r_0, \dots, r_k$ are nonzero. Let the directions arise by $A$-orthogonalizing the residuals. That is, $p_0 = r_0$, and $p_k$ is $r_k$ minus its $A$-orthogonal projection onto $\operatorname{span} \set{p_0, \dots, p_{k-1}}$. Then $\operatorname{span} \set{p_0, \dots, p_{k-1}} = \mathcal{K}_k(A, r_0)$, and $p_k = r_k + \beta_{k-1} p_{k-1}$ with $\beta_{k-1} = r_k^T r_k / r_{k-1}^T r_{k-1}$. Also $\alpha_k = r_k^T r_k / p_k^T A p_k$.

## Remarks

For clause 1, the function $f(x) = \frac12 \norm{x - x^*}_A^2$ restricted to $x_0 + \operatorname{span}\set{p_0, \dots, p_{k-1}}$ separates into one quadratic in each coordinate, by conjugacy.
The step $\alpha_k$ minimizes the quadratic in the $k$-th coordinate, and later steps do not change the earlier coordinates.
For clause 2, the residual $r_k$ is orthogonal to $\mathcal{K}_k(A, r_0)$ by clause 1.
Since $A p_j \in \mathcal{K}_{j+2}(A, r_0)$, the vector $r_k$ is $A$-orthogonal to $p_j$ for $j < k - 1$.
So only the last direction enters the [[Gram–Schmidt|0VT6P0GNQR]] step, and this gives the two-term recurrence.
Clause 2 with clause 1 is exactly the [[conjugate gradient method|AFXZGQB4EN]].
Example: with $p_i = e_i$ and $A$ diagonal, the directions are conjugate, and the method of clause 1 solves the $i$-th equation at step $i$.

## Explanations

### L1 human

If the search directions do not interfere with each other in the geometry defined by $A$, then minimizing along each direction once gives the minimum over all of them together.
Building such directions in general costs a lot of memory.
The miracle of conjugate gradients is that, when the directions come from the residuals, each new one only needs to be corrected against the previous one.

### L2 human

There is a second derivation from the [[Lanczos algorithm|7RRB4PCRJB]].
The [[Arnoldi iteration|CJ5EC820S7]] applied to a symmetric $A$ gives an orthonormal basis of $\mathcal{K}_k(A, r_0)$ in which $A$ becomes a tridiagonal matrix $T_k$.
The Galerkin condition $r_k \perp \mathcal{K}_k(A, r_0)$ asks to solve a system with $T_k$.
Updating an $L D L^T$ factorization of $T_k$ step by step gives the same recurrences.
This view explains the link between conjugate gradients and the Lanczos process for eigenvalues.
</document>

<document source="https://tamnd.github.io/math2/node/1T23294NKC.md" tag="1T23294NKC" kind="definition">
# Functional

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $V$ be a [[vector space|000AYPA9AJ]] over $\mathbb{K}$.
A *functional* on $V$ is a [[function|0EPAT51DXQ]] $J \colon V \to \mathbb{K}$.
More generally, a functional on a subset $A \subseteq V$ is a function $J \colon A \to \mathbb{K}$.
In the calculus of variations, $V$ is usually a space of real functions, such as $C^1([a, b])$, and $J$ is real-valued.

## Remarks

Example: on $V = C([0, 1])$, the map $J(y) = \int_0^1 y(x)^2 \, dx$ is a real functional.
It is not linear, since $J(2y) = 4 J(y)$ while a linear map would give $2 J(y)$.
A functional that is linear is a [[linear form|07YQ1C78G0]]; when $V$ is normed, the continuous linear functionals form the [[dual space|07JBSZE0V4]] $V^*$.
Non-example: the map $y \mapsto y'$ on $C^1([0, 1])$ takes values in $C([0, 1])$, not in $\mathbb{K}$, so it is an operator and not a functional.
Some authors restrict the word to linear functionals; in the calculus of variations it means any scalar-valued map as above.
</document>

<document source="https://tamnd.github.io/math2/node/1T7XKF6WKM.md" tag="1T7XKF6WKM" kind="definition">
# Affine morphism

## Definition

A [[morphism of schemes|JXRKH6P1Z4]] $f \colon X \to Y$ is *affine* if $Y$ has a cover by open [[affine|B3V3XJ2W8G]] subschemes $V_i$ with each $f^{-1}(V_i)$ affine.

## Remarks

If $f$ is affine, then $f^{-1}(V)$ is affine for every open affine subscheme $V$ of $Y$.
Example: every morphism $\operatorname{Spec} A \to \operatorname{Spec} B$ is affine; take the cover of $Y = \operatorname{Spec} B$ by itself, with preimage $\operatorname{Spec} A$.
Example: a closed immersion is affine, since over $\operatorname{Spec} B$ it is $\operatorname{Spec} B/I \to \operatorname{Spec} B$ for an ideal $I$.
Non-example: for a field $k$, the morphism $\mathbb{A}^2_k \setminus \set{0} \to \operatorname{Spec} k$ is not affine.
The only open affine of $\operatorname{Spec} k$ is the whole space, and its preimage $\mathbb{A}^2_k \setminus \set{0}$ is not an affine scheme.
Its ring of global functions is $k[x, y]$, whose spectrum also contains the origin.
</document>

<document source="https://tamnd.github.io/math2/node/1TC7W7BQT3.md" tag="1TC7W7BQT3" kind="concept">
# Slice category

## Summary

The slice category $\mathcal{C}/X$ of a [[category|A2DE66937D]] $\mathcal{C}$ over an object $X$ has the morphisms into $X$ as objects.
Its morphisms are the commuting triangles over $X$.
It is the [[comma category|PC6HG1JEGM]] $(1_{\mathcal{C}} \downarrow X)$.
Slice categories describe objects "over a base": families of sets indexed by $X$, spaces over a base space, and schemes over a base scheme.
The dual notion is the coslice category $X/\mathcal{C}$ of morphisms out of $X$, such as the category of pointed sets.

## Explanations

### L1 human

Fix an object $X$ and look at all arrows that end at $X$.
Each such arrow $A \to X$ is an object of the slice.
A map between two of them is a map of the sources that is compatible with the arrows to $X$.
For sets, a map $h \colon A \to I$ splits $A$ into the pieces $h^{-1}(i)$, one for each $i \in I$.
So a set over $I$ is a family of sets indexed by $I$, and maps over $I$ are maps sending each piece into the matching piece.
</document>

<document source="https://tamnd.github.io/math2/node/1TE3QF8EYC.md" tag="1TE3QF8EYC" kind="concept">
# Zeros and poles

## Summary

The zeros of a [[meromorphic function|0ZZ7ZYDR8W]] are the points where it takes the value $0$.
Its poles are the points where it tends to $\infty$.
See the [[definition|9DNDF7AX59]].
They are the two kinds of special points of a meromorphic function, and they are exchanged by $f \mapsto 1/f$.
Counting them, each with its [[order|23ASRP0MWR]], is the subject of the [[argument principle|0PN2BXRS8G]] and of [[Rouché's theorem|0D430QR02W]].

## Explanations

### L1 human

Picture the graph of $\abs{f}$ over the complex plane as a landscape.
The zeros are the points where the landscape touches the ground.
The poles are the points where it shoots up to infinity, like tent poles holding up a tent.
For a rational function such as $(z - 1)/(z^2 + 1)$, the zeros are the roots of the numerator and the poles are the roots of the denominator.
</document>

<document source="https://tamnd.github.io/math2/node/1TJ8TNH331.md" tag="1TJ8TNH331" kind="object">
# Fock space

## Description

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]].
For $n \ge 1$, let $H^{\otimes n}$ be the [[tensor product|EY0X2NKA3S]] of $n$ copies of $H$, and let $H^{\otimes 0} = \C$.
The *full Fock space* over $H$ is the [[Hilbert sum|WJ7PDMGQQW]]
$$\mathcal{F}(H) = \widehat{\bigoplus}_{n \ge 0} H^{\otimes n}.$$
Let $S_+$ and $S_-$ be the orthogonal projections of $H^{\otimes n}$ onto its symmetric and antisymmetric tensors.
The *bosonic Fock space* is $\mathcal{F}_+(H) = \widehat{\bigoplus}_{n \ge 0} S_+ H^{\otimes n}$, and the *fermionic Fock space* is $\mathcal{F}_-(H) = \widehat{\bigoplus}_{n \ge 0} S_- H^{\otimes n}$.
Each of them is a Hilbert space.
Indeed, each summand is a closed subspace of $H^{\otimes n}$, and a Hilbert sum of Hilbert spaces is a Hilbert space.

## Remarks

The summand of degree $n$ is the space of states with exactly $n$ particles; the vector $1 \in H^{\otimes 0}$ is the vacuum.
For $H = \C$, every symmetric power $S_+ \C^{\otimes n}$ has dimension $1$, so $\mathcal{F}_+(\C)$ is isomorphic to $\ell^2(\N)$.
For the same $H$, the antisymmetric part vanishes for $n \ge 2$, so $\mathcal{F}_-(\C) = \C \oplus \C$; this reflects the Pauli exclusion principle.
Creation and annihilation operators act on Fock space by raising and lowering the degree $n$.

## Explanations

### L1 human

Fock space is the state space for a quantum system in which the number of particles can change.
It stacks the spaces for $0, 1, 2, \ldots$ particles side by side.
For identical bosons one keeps only symmetric states, and for fermions only antisymmetric ones.
</document>

<document source="https://tamnd.github.io/math2/node/1TJXANASFE.md" tag="1TJXANASFE" kind="counterexample">
# Rossi's non-embeddable CR structure

## Counterexample

Let $S^3 = \set{(z_1, z_2) \in \C^2 : \abs{z_1}^2 + \abs{z_2}^2 = 1}$ be the unit [[sphere|P9CM4GQ2T0]].
Its standard [[CR structure|ZC78JKZX9S]] is spanned by the vector field
$$\bar L = z_2 \frac{\partial}{\partial \bar z_1} - z_1 \frac{\partial}{\partial \bar z_2},$$
which is tangent to $S^3$, and its conjugate is $L = \bar z_2 \frac{\partial}{\partial z_1} - \bar z_1 \frac{\partial}{\partial z_2}$.
For a real number $t$ with $0 < \abs{t} < 1$, let $\bar L_t = \bar L + t L$, and let $S^3_t$ be $S^3$ with the CR structure spanned by $\bar L_t$.
Then $S^3_t$ is a compact strongly pseudoconvex CR manifold of real dimension $3$.
Every [[CR function|JFKYYK2CEB]] $f$ on $S^3_t$, that is every $f$ with $\bar L_t f = 0$, is even: $f(-p) = f(p)$ for all $p \in S^3$.
So CR functions cannot separate the antipodal points $p$ and $-p$, and there is no CR embedding of $S^3_t$ into any $\C^N$.

## Remarks

The parity follows by expanding $f$ in spherical harmonics of bidegree $(a, b)$ in $(z, \bar z)$.
Both $L$ and $\bar L$ preserve the total degree $a + b$.
So the equation $\bar L_t f = 0$ splits into one equation for each total degree.
For odd total degree this equation has only the zero solution.
For $t = 0$ the structure is the standard one, and the coordinate functions $z_1, z_2$ embed it.
So arbitrarily small deformations destroy embeddability in dimension $3$.
This shows that the dimension hypothesis in [[Boutet de Monvel's embedding theorem|ZAH4SK5YX7]] cannot be dropped.

## Explanations

### L1 human

The round three-sphere sits in $\C^2$, and its CR structure is inherited from there.
Tilting this structure by a tiny amount gives a new abstract CR structure on the same sphere.
For the tilted structure, every function compatible with it takes the same value at opposite points.
So no collection of such functions can tell opposite points apart, and the tilted sphere cannot be placed inside any complex space as a hypersurface.
</document>

<document source="https://tamnd.github.io/math2/node/1TNS604ZST.md" tag="1TNS604ZST" kind="conjecture">
# Hochster's homological conjectures

## Statement

Let $(R, \mathfrak{m})$ be a commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] of dimension $d$.

1. Direct summand conjecture: let $R$ be a [[regular local ring|63XXTF6B5M]] and $R \subseteq S$ a ring extension with $S$ a finitely generated $R$-module. Then $R$ is a direct summand of $S$ as an $R$-module.
2. Monomial conjecture: if $x_1, \ldots, x_d$ is a [[system of parameters|95QTEPMDA4]] of $R$, then $(x_1 x_2 \cdots x_d)^t \notin (x_1^{t+1}, \ldots, x_d^{t+1})$ for every integer $t \ge 0$.
3. Big Cohen–Macaulay module conjecture: there is an $R$-module $B$ such that some system of parameters of $R$ is a [[regular sequence|69R5R2FCS5]] on $B$. The module $B$ need not be finitely generated.

## Remarks

These are three of the homological conjectures studied and organized by Hochster; there are others, such as the intersection conjectures.
Hochster proved all three when $R$ contains a field, using reduction to positive characteristic and the Frobenius map.
The monomial conjecture and the direct summand conjecture are known to be equivalent.
In mixed characteristic, André proved the direct summand conjecture and the existence of big Cohen–Macaulay algebras, using perfectoid methods; so the three statements are now theorems.
The [[direct summand conjecture|CQJ1FPKKHT]] has its own node.

## Explanations

### L1 human

These conjectures predict that some properties of polynomial rings over a field remain true for all Noetherian local rings.
For decades they were known only for rings that contain a field, and the case of rings like the $p$-adic integers resisted.
They were settled in that case only recently, with new tools from arithmetic geometry.
</document>

<document source="https://tamnd.github.io/math2/node/1TQ06R4Q66.md" tag="1TQ06R4Q66" kind="example">
# Unknot

## Example

The *unknot* is the round circle
$$U = \set{(x, y, 0) \in \R^3 : x^2 + y^2 = 1},$$
the image of the map $t \mapsto (\cos t, \sin t, 0)$, which is continuous and injective on the [[circle|060W5CBZP6]] $\R / 2\pi\Z$.
So $U$ is a [[knot|QYEAZTYGPT]], and it is tame since it is equivalent to a square, a closed polygon.
Any knot equivalent to $U$ is also called an unknot, or a trivial knot.

## Remarks

The unknot bounds the flat unit disk, a [[Seifert surface|K866ND3WF7]] of genus $0$, and a knot is trivial exactly when it bounds an embedded disk.
Its [[knot group|AQVHM3FP0D]] is $\Z$, its [[crossing number|SWJZ4QRF9W]] is $0$, and its Alexander and Jones polynomials are $1$.
A tame knot with knot group $\Z$ is the unknot.
Recognizing the unknot from a diagram is the [[unknotting problem|VXSD46NRX0]]; Wolfgang Haken showed that it is decidable.
Some diagrams of the unknot are complicated, and untangling them may first require more crossings.

## Explanations

### L1 human

The unknot is a plain loop of string with no knot in it, like a rubber band.
Any loop that can be moved in space, without cutting, to a round circle is also an unknot.
It is the starting point of knot theory: to show that a knot is really knotted is to show that it is not the unknot.
</document>

<document source="https://tamnd.github.io/math2/node/1TQ1JQXEWP.md" tag="1TQ1JQXEWP" kind="concept">
# Midpoint convex function

## Summary

A midpoint convex function, also called a Jensen-convex function, is a real function $f$ on an interval with $f\big(\tfrac{x + y}{2}\big) \le \tfrac{f(x) + f(y)}{2}$ for all $x, y$.
Every [[convex function|07Q7CNM5WF]] is midpoint convex, but the converse fails without some regularity.
Mild extra conditions close the gap: a midpoint convex function that is bounded above on a small interval, or that is measurable, is convex.
The notion is the starting point of the theory of convex functions as solutions of a [[functional inequality|YQB2B0EXY8]].

## Explanations

### L1 human

Convexity asks that every chord of the graph lies above the graph.
Midpoint convexity only asks this at the midpoint of each chord.
For nice functions this is the same thing, because midpoints of midpoints fill out the whole chord.
Strange functions built with the axiom of choice show that, in general, the midpoint condition alone is weaker.
</document>

<document source="https://tamnd.github.io/math2/node/1TQ2F9CFVE.md" tag="1TQ2F9CFVE" kind="statement">
# Minkowski bound

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] of degree $n$, with ring of integers $\mathcal{O}_K$ and [[discriminant|E0VZTNPJXA]] $d_K$.
Let $r_2$ be the number of pairs of complex conjugate embeddings $K \to \C$.
Put
$$M_K = \sqrt{\abs{d_K}} \left(\frac{4}{\pi}\right)^{r_2} \frac{n!}{n^n}.$$
Then every class of the [[ideal class group|WQ46BD3ARY]] of $K$ contains a nonzero ideal $\mathfrak{a} \subseteq \mathcal{O}_K$ whose [[norm|M54ADM0CVF]] satisfies $N(\mathfrak{a}) \le M_K$.

## Remarks

The proof embeds $\mathcal{O}_K$ and its ideals as [[lattices|WN9V7WSJKF]] in $\R^n$.
[[Minkowski's theorem|0K387S2PKQ]] on convex bodies then gives, in each nonzero ideal $\mathfrak{b}$, a nonzero element $\alpha$ with $\abs{N_{K/\Q}(\alpha)} \le M_K N(\mathfrak{b})$.
Applied to an ideal $\mathfrak{b}$ in the inverse class, the ideal $\alpha \mathfrak{b}^{-1}$ is integral, lies in the given class and has norm at most $M_K$.

Consequence: the class group is generated by the classes of the prime ideals of norm at most $M_K$.
Since there are only finitely many ideals of bounded norm, the class group is finite.

Example: for $K = \Q(\sqrt{-5})$ one has $n = 2$, $r_2 = 1$, $d_K = -20$, so $M_K = \sqrt{20} \cdot \frac{4}{\pi} \cdot \frac{2}{4} \approx 2.85$.
Only the prime ideal $\mathfrak{p} = (2, 1 + \sqrt{-5})$ of norm $2$ needs checking.
It is not principal, since $a^2 + 5b^2 = 2$ has no integer solution, and $\mathfrak{p}^2 = (2)$; so the class number is $2$.
For $K = \Q(i)$, $M_K = 2 \cdot \frac{4}{\pi} \cdot \frac{2}{4} \approx 1.27 < 2$, so every class contains the ideal $(1)$ and $\Z[i]$ is a principal ideal domain.

## Explanations

### L1 human

The class group of a number field measures how far its integers are from having unique factorisation.
The Minkowski bound is an explicit number such that only the prime ideals below it matter.
So the class group of a given field can be computed by checking finitely many small primes.

### L2 human

The bound is the standard tool for computing [[class numbers|KTKK9B1X41]] by hand for fields of small degree.
Since every class contains an integral ideal of norm at least $1$, it also gives $M_K \ge 1$, which yields a lower bound for $\abs{d_K}$; this is the [[Hermite–Minkowski theorem|KMVW45B0TT]].
The same geometry of numbers proves [[Dirichlet's unit theorem|1D99CSKX6N]].
</document>

<document source="https://tamnd.github.io/math2/node/1TZSV19D85.md" tag="1TZSV19D85" kind="definition">
# Associative algebra

## Definition

Let $K$ be a [[commutative ring|0GYGHKX834]].
Let $A$ be a [[ring|0894J980RK]] whose additive group carries a [[$K$-module|09VDS2W1E8]] structure.
Then $A$ is an *associative algebra over $K$*, or *$K$-algebra*, if
$$k(ab) = (ka)b = a(kb)$$
for all $k \in K$ and $a, b \in A$.

## Remarks

Equivalently, a $K$-algebra is a ring $A$ with a ring homomorphism $\eta \colon K \to A$ whose image lies in the center of $A$; the scalar product is then $ka = \eta(k)a$.
Example: for a field $F$, the matrix ring $M_n(F)$ is an $F$-algebra with the entrywise scalar product.
The check: $k(AB)$, $(kA)B$ and $A(kB)$ all have $(i,j)$ entry $k \sum_l a_{il} b_{lj}$, because $k$ commutes with the entries.
Non-example: the quaternions $\mathbb{H}$ with left multiplication by elements of $\C \subset \mathbb{H}$ are not a $\C$-algebra.
The condition $k(ab) = a(kb)$ fails for the scalar $i$ with $a = j$ and $b = 1$, since $i(j \cdot 1) = ij$ while $j(i \cdot 1) = ji = -ij \ne ij$.
They are an algebra over $\R$, because real numbers commute with all quaternions.
Every ring is a $\Z$-algebra in exactly one way.
Some authors do not require a unit element in $A$; this note assumes one.
</document>

<document source="https://tamnd.github.io/math2/node/1V11QM5C07.md" tag="1V11QM5C07" kind="statement">
# Clarkson's inequalities

## Statement

Let $(\Omega, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $f, g \in L^p(\mu)$, the [[Lp space|SASP14MA1Z]].
Write $\norm{\cdot}$ for the norm of $L^p(\mu)$.

1. If $2 \le p < \infty$, then
$$\Big\| \frac{f + g}{2} \Big\|^p + \Big\| \frac{f - g}{2} \Big\|^p \le \frac{1}{2} \big( \norm{f}^p + \norm{g}^p \big).$$
2. If $1 < p \le 2$ and $q = p/(p-1)$, then
$$\Big\| \frac{f + g}{2} \Big\|^q + \Big\| \frac{f - g}{2} \Big\|^q \le \Big( \frac{1}{2} \norm{f}^p + \frac{1}{2} \norm{g}^p \Big)^{q/p}.$$

## Remarks

The first inequality follows from the same inequality for numbers $a, b$, integrated over $\Omega$.
For numbers, with $\alpha = (a+b)/2$ and $\beta = (a-b)/2$, one has $\abs{\alpha}^p + \abs{\beta}^p \le (\abs{\alpha}^2 + \abs{\beta}^2)^{p/2}$ since $p \ge 2$.
The [[parallelogram law|026HPGYGS4]] gives $\abs{\alpha}^2 + \abs{\beta}^2 = (\abs{a}^2 + \abs{b}^2)/2$.
Convexity of $t \mapsto t^{p/2}$ then bounds $((\abs{a}^2 + \abs{b}^2)/2)^{p/2}$ by $(\abs{a}^p + \abs{b}^p)/2$.
The second inequality needs a finer argument.

For $p = 2$ both inequalities are the parallelogram law, with equality.
Clarkson proved them in 1936 to show that $L^p(\mu)$ is [[uniformly convex|BEAMC5011M]] for $1 < p < \infty$.
If $\norm{f} = \norm{g} = 1$ and $\norm{f - g} \ge \varepsilon$, then for $p \ge 2$ the first inequality gives $\norm{(f+g)/2}^p \le 1 - (\varepsilon/2)^p$.

## Explanations

### L1 human

In a space of functions with the $L^p$ norm, two unit vectors that are far apart have a midpoint that is shorter than $1$ by a definite amount.
Clarkson's inequalities make this precise with explicit numbers.
They generalise the parallelogram law of Euclidean geometry.

### L2 human

The inequalities are the standard route to uniform convexity of $L^p$ for $1 < p < \infty$.
By the Milman–Pettis theorem, uniform convexity then gives reflexivity of $L^p$.
Through uniform convexity, they can also be used to prove the duality of $L^p$ and $L^q$ without the Radon–Nikodym theorem.
The spaces $L^1$ and $L^\infty$ are not uniformly convex, and the inequalities have no analogue there.
</document>

<document source="https://tamnd.github.io/math2/node/1V2N8MFSHC.md" tag="1V2N8MFSHC" kind="concept">
# Riesz potential

## Summary

The Riesz potential of order $\alpha$, for $0 < \alpha < n$, is the operator $I_\alpha$ that convolves a function on $\R^n$ with a constant multiple of $\abs{x}^{\alpha - n}$.
It is the inverse of the [[fractional Laplacian|F8RSNQBMBE]] $(-\Delta)^{\alpha/2}$, so it integrates a function to the fractional order $\alpha$.
For $\alpha = 2$ and $n \ge 3$ it is, up to a constant factor, the [[Newtonian potential|DCMFQ01QGS]], which solves Poisson's equation.
Its main estimate is the [[Hardy–Littlewood–Sobolev inequality|HG65C9R1SP]], which underlies the Sobolev embedding theorem.

## Explanations

### L1 human

Taking a derivative makes a function rougher, and integrating makes it smoother.
The Riesz potential is a way to integrate "a fraction of a time" in several variables.
It averages the function with a weight $\abs{x - y}^{\alpha - n}$ that is large near $x$ and decays with distance.
The larger $\alpha$ is, the slower the decay and the smoother the result.
In $\R^3$ with $\alpha = 2$ the weight is $1/\abs{x - y}$, the gravitational potential of a point mass.
</document>

<document source="https://tamnd.github.io/math2/node/1V8H34F6N9.md" tag="1V8H34F6N9" kind="concept">
# Quantum enveloping algebra

## Summary

The quantum enveloping algebra $U_q(\mathfrak{g})$ is also called the quantized universal enveloping algebra or Drinfeld–Jimbo quantum group.
It is a one-parameter deformation of the [[universal enveloping algebra|SCZ8PRVY00]] of a semisimple Lie algebra $\mathfrak{g}$.
It is a [[Hopf algebra|7XFYP0N84H]] given by generators $E_i, F_i, K_i^{\pm 1}$ and relations read off from the Cartan matrix of $\mathfrak{g}$.
For $q$ not a root of unity its finite-dimensional representations closely mirror those of $\mathfrak{g}$.
It is the main example of a [[quantum group|AP2ZDXANKC]], and suitable versions of it are [[quasitriangular|NCRPWS9W5R]].

## Explanations

### L1 human

A Lie algebra is described by generators and relations, and its enveloping algebra turns these into an associative algebra.
The quantum enveloping algebra keeps the same generators but bends the relations by a number $q$.
For $\mathfrak{sl}_2$ the classical relation $[E, F] = H$ becomes $EF - FE = (K - K^{-1})/(q - q^{-1})$, where $K$ plays the role of $q^H$.
When $q$ tends to $1$, the right side tends formally to $H$, and the classical algebra comes back.
The deformation keeps a coproduct, so tensor products of representations still make sense, but they are no longer symmetric in the two factors.
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# Minimal clone

## Summary

A minimal clone on a set $A$ is a clone that lies just above the clone of [[projections|769B35T6WG]] in the [[lattice of clones|EX9NCDFYC7]].
It is generated by any one of its non-projection operations.
On a finite set, every clone other than the projection clone contains a minimal clone, and there are only finitely many minimal clones.
Minimal clones are the atoms of the clone lattice, dual to the maximal clones.

## Explanations

### L1 human

The smallest clone can only copy inputs.
A minimal clone adds the least possible amount of power to it.
For example, on $\set{0, 1}$ the clone generated by negation $\neg$ is minimal.
Every operation in it is a projection or a negated projection, and each negated projection generates $\neg$ back.
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# Functional data analysis

## Definition

Let $\mathcal T = [a, b]$ with $a < b$, and let $H = L^2(\mathcal T)$ be the [[Hilbert space|HM2HHE3J51]] of [[square-integrable|SASP14MA1Z]] real functions on $\mathcal T$, with inner product $\langle f, g \rangle = \int_{\mathcal T} f(t) g(t) \, dt$.
*Functional data analysis* is the statistical analysis of data $X_1, \dots, X_n$ that are realisations of a random element $X$ of $H$.
Such an $X$ is a [[stochastic process|0QH5ENKZM1]] $(X(t))_{t \in \mathcal T}$ with paths in $H$.
Its basic parameters, when $E\norm{X}^2 < \infty$, are:

1. the mean function $\mu(t) = E[X(t)]$, an [[expected value|0WQNRCNY3M]] for each $t$;
2. the [[covariance|0KBFAWYRTE]] function $C(s, t) = \operatorname{Cov}(X(s), X(t))$;
3. the covariance operator $\mathcal C \colon H \to H$, $(\mathcal C f)(t) = \int_{\mathcal T} C(s, t) f(s) \, ds$.

## Remarks

The operator $\mathcal C$ is compact, self-adjoint and positive semidefinite, so it has an orthonormal basis of eigenfunctions $\phi_k$ with eigenvalues $\lambda_1 \ge \lambda_2 \ge \dots \ge 0$.
Functional principal component analysis estimates these and writes $X = \mu + \sum_k \xi_k \phi_k$ with uncorrelated scores $\xi_k = \langle X - \mu, \phi_k \rangle$ of variance $\lambda_k$.
The sample versions are $\hat\mu = \frac1n \sum_i X_i$ and $\hat C(s, t) = \frac1n \sum_i (X_i(s) - \hat\mu(s))(X_i(t) - \hat\mu(t))$.

Example: let $X(t) = \xi \sqrt{2} \sin(\pi t)$ on $[0, 1]$, with $\xi$ a real random variable of mean $0$ and variance $1$.
Then $\mu = 0$ and $C(s, t) = 2 \sin(\pi s) \sin(\pi t)$, so $\mathcal C$ has the single nonzero eigenvalue $1$, with eigenfunction $\sqrt{2} \sin(\pi t)$ of norm $1$.

Non-example: a table of the heights of children measured once each, at a single age, is ordinary multivariate data, not functional data, since no observation is a function of $t$.
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# Commutant

## Summary

Let $S$ be a set of operators on a Hilbert space.
Its commutant $S'$ is the set of all bounded operators that commute with every element of $S$.
It is the operator version of the centralizer in an algebra.
The commutant of a self-adjoint set is always a [[von Neumann algebra|PSFWQVZ0FS]].
By the [[bicommutant theorem|BSA6GFT8SQ]], the von Neumann algebras are the self-adjoint unital algebras equal to their double commutant.

## Explanations

### L1 human

Given some operators, ask which operators can be applied before or after them with the same result.
These form the commutant.
For example, the only $n \times n$ matrices that commute with all $n \times n$ matrices are the multiples of the identity.
In the other direction, the matrices that commute with all diagonal matrices are exactly the diagonal matrices.
So a big set has a small commutant, and a small set has a big commutant.
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# Brouwer–Heyting–Kolmogorov interpretation

## Definition

Let $D$ be a domain of objects, and let $A$, $B$ and $P(x)$ be statements about $D$.
The *Brouwer–Heyting–Kolmogorov interpretation* says what a proof of a compound statement is, by these clauses.

1. A proof of $A \wedge B$ is a pair $(a, b)$, where $a$ is a proof of $A$ and $b$ is a proof of $B$.
2. A proof of $A \vee B$ is a pair $(i, c)$ with $i \in \set{0, 1}$. If $i = 0$, then $c$ is a proof of $A$. If $i = 1$, then $c$ is a proof of $B$.
3. A proof of $A \to B$ is a construction that transforms each proof of $A$ into a proof of $B$.
4. There is no proof of the absurdity $\bot$.
5. A proof of $\neg A$ is a proof of $A \to \bot$.
6. A proof of $\forall x\, P(x)$ is a construction that transforms each $d \in D$ into a proof of $P(d)$.
7. A proof of $\exists x\, P(x)$ is a pair $(d, p)$, where $d \in D$ and $p$ is a proof of $P(d)$.

These clauses give the meaning of [[conjunction|0M28ASWG8T]], [[disjunction|0H5GKDTB01]], [[implication|0ZNX6DQ3EJ]], [[negation|0XR7A7RSXB]], the [[universal quantifier|0B88PQTJ66]] and the [[existential quantifier|0JW2FHW7G3]] in [[intuitionistic logic|Q11N8X99FK]].

## Remarks

Example: $A \to \neg \neg A$ has a BHK proof.
Given a proof $a$ of $A$, the construction $f \mapsto f(a)$ sends each proof $f$ of $\neg A$ to a proof of $\bot$, so it is a proof of $\neg \neg A$.
Non-example: there is no known BHK proof of $\neg \neg A \to A$ for all $A$, because a proof of $\neg \neg A$ gives no construction of a proof of $A$.
The notion of "construction" is left informal.
Kleene's realizability reads it as a computable function, and the Curry–Howard correspondence reads it as a typed lambda term.
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# Conformal invariance of harmonic functions

## Statement

Let $U, V \subseteq \C$ be open sets and let $\varphi \colon U \to V$ be a [[holomorphic function|0NAHXR1FRH]].
Let $h \colon V \to \R$ be a [[harmonic function|0DEB0JMN0W]].
Then $h \circ \varphi$ is harmonic on $U$.
In particular, let $\varphi$ be a [[conformal map|0YYT36R4YD]] of $U$ onto $V$.
Then $h$ is harmonic on $V$ if and only if $h \circ \varphi$ is harmonic on $U$.

## Remarks

Proof idea: near a point $\varphi(z_0)$, [[harmonic conjugates exist locally|CVDGP7DKZ3]], so $h = \operatorname{Re} F$ for a holomorphic $F$ on a disc around $\varphi(z_0)$.
Then $h \circ \varphi = \operatorname{Re}(F \circ \varphi)$ near $z_0$, and $F \circ \varphi$ is holomorphic.
A direct computation gives $\Delta (h \circ \varphi) = \abs{\varphi'}^2 \, (\Delta h) \circ \varphi$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
Example: $h(w) = \log\abs{w}$ is harmonic on $\C \setminus \set{0}$, and with $\varphi(z) = z^2$ one gets $h(\varphi(z)) = 2 \log\abs{z}$, again harmonic.
Holomorphy of $\varphi$ is needed: for the smooth map $\varphi(x, y) = (x, y^2)$ and $h(s, t) = s^2 - t^2$, the function $h \circ \varphi = x^2 - y^4$ has Laplacian $2 - 12 y^2 \ne 0$.
The statement also holds for antiholomorphic $\varphi$, because $h \circ \bar\varphi$ equals $\tilde h \circ \varphi$ with $\tilde h(w) = h(\bar w)$, which is harmonic.

## Explanations

### L1 human

Harmonic functions in the plane survive a change of coordinates by a holomorphic map.
So one may move a problem about harmonic functions from a complicated region to a simple one, such as a disc, solve it there, and move the answer back.

### L2 human

This is the basis of conformal methods in planar potential theory.
With the Riemann mapping theorem, the Dirichlet problem on a simply connected domain reduces to the disc, where the Poisson integral solves it.
The same invariance makes harmonic measure and Green's functions conformally invariant.
It also shows that harmonicity makes sense on a Riemann surface, since the transition maps between charts are holomorphic.
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# Space form

## Summary

A space form is a complete connected [[Riemannian manifold|7CBT4SM83Y]] of constant [[sectional curvature|1X42JAA349]].
The three simply connected models are the round sphere, Euclidean space and hyperbolic space.
Every space form is a quotient of one of these models by a group of isometries.
So space forms are the simplest and most symmetric Riemannian manifolds.
They serve as the reference spaces in comparison geometry.

## Explanations

### L1 human

A space form is a space that curves by the same amount at every point and in every direction.
If the amount is positive, the space looks like a sphere; if it is zero, it looks flat.
If it is negative, it looks like the hyperbolic plane, a saddle everywhere.
A flat torus, made by gluing opposite sides of a square, is a space form of curvature zero.
An open disc in the plane is flat but has an edge one can reach in finite distance, so it is not complete and not a space form.
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# Representation theory of the symmetric group

## Summary

The representation theory of the [[symmetric group|0515A6D2Y1]] $S_n$ studies its [[representations|WYGRXCG2RN]] and characters.
Over the complex numbers, the irreducible representations of $S_n$ correspond to the [[partitions|0JV30XN73Z]] of $n$, and are realised by the [[Specht modules|NTFM7X5555]].
The theory is fully combinatorial: [[Young diagrams|096AHMB3P4]] and [[Young tableaux|Q6TH9PH2R7]] give dimensions and characters.
It connects with symmetric functions, the representations of general linear groups and many parts of combinatorics and physics.

## Explanations

### L1 human

The symmetric group consists of all shufflings of $n$ objects.
Its irreducible representations are labelled by the ways of writing $n$ as a sum of positive integers, drawn as left-justified rows of boxes.
For $n = 3$ there are three partitions, $3$, $2 + 1$ and $1 + 1 + 1$, matching the trivial, the two-dimensional and the sign representations of $S_3$.
The dimension of each one is the number of ways to fill its diagram with $1, \ldots, n$ increasing along rows and down columns.
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# Holmgren's uniqueness theorem

## Statement

Let $\Omega \subseteq \R^n$ be open, and let $P = \sum_{\abs{\alpha} \le m} a_\alpha(x) \, \partial^\alpha$ be a [[linear differential operator|6DZETZ5T9G]] of order $m$ whose coefficients $a_\alpha$ are [[real-analytic|081WKEDK6C]] on $\Omega$.
Let $S \subseteq \Omega$ be a real-analytic hypersurface and let $x_0 \in S$ be a point at which $S$ is [[noncharacteristic|CQJ2E2ARRF]] for $P$.
Let $U \subseteq \Omega$ be an open neighbourhood of $x_0$, and let $u \in C^m(U)$ satisfy $P u = 0$ in $U$ and $\partial^\alpha u = 0$ on $S \cap U$ for all $\abs{\alpha} \le m - 1$.
Then $u = 0$ in some neighbourhood of $x_0$.

## Remarks

Equivalently, two $C^m$ solutions of $P u = f$ with the same Cauchy data on $S$ coincide near $x_0$; here $f$ need not be analytic.
The proof is by duality: one solves the adjoint equation $P^{*} v = p$ for polynomials $p$, with zero Cauchy data on a nearby analytic hypersurface, using the [[Cauchy–Kovalevskaya theorem|97HMV9BT2W]].
Integration by parts then shows that $\int u \, p = 0$ on a lens-shaped region for every polynomial $p$, and the Weierstrass approximation theorem gives $u = 0$ there.
Example: for the wave equation $\partial_t^2 u - \partial_x^2 u = 0$, the line $t = 0$ is noncharacteristic, so a $C^2$ solution with $u(0, x) = \partial_t u(0, x) = 0$ vanishes near the line.
The analyticity of the coefficients cannot simply be dropped: there are linear operators with smooth coefficients for which the Cauchy problem has nonzero solutions with zero data.
Hörmander extended the theorem to distribution solutions and to $C^1$ hypersurfaces.

## Explanations

### L1 human

The Cauchy–Kovalevskaya theorem gives uniqueness only among analytic solutions.
Holmgren's theorem says that for linear equations with analytic coefficients there is no other, non-analytic, solution with the same starting data.
So the starting data on a good surface determine the solution near the surface.

### L2 human

The theorem is a unique continuation result across noncharacteristic hypersurfaces.
It applies to the Laplace, heat and wave equations, since their coefficients are constant.
For the wave equation it is a step towards domain of dependence arguments.
For the Laplace equation it shows that the Cauchy problem has at most one solution, even though it is not [[well posed|W09B9NZ9B5]].
The noncharacteristic hypothesis matters: for the heat operator the line $t = 0$ is characteristic, and [[Tychonoff's example|RN4J24CD83]] gives a nonzero solution of the heat equation with zero initial values.
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# CUSUM

## Summary

CUSUM, short for cumulative sum, is a sequential procedure for detecting a change in the distribution of a stream of observations.
It adds up evidence for a change and resets the sum to zero whenever it becomes negative.
It raises an alarm when the sum crosses a threshold.
E. S. Page introduced it in 1954, and it is the standard solution of the [[change detection|Z5H8ZZHVMC]] problem in quality control.

## Explanations

### L1 human

Picture a machine that fills bottles with $500$ ml on average, and suppose a fault would raise the average.
After each bottle, add how much it exceeds $500$ ml plus a small allowance, and subtract when it falls short, but never let the running total go below zero.
While the machine is fine, the total keeps hitting zero and stays small.
After a fault, the excesses pile up and the total soon crosses the alarm level.
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# Dynamic equation on time scales

## Definition

Let $\mathbb{T}$ be a time scale, that is, a nonempty closed subset of $\R$.
Let $\sigma$ be its forward jump operator, and let $\mathbb{T}^\kappa$ be as in the [[definition of the delta derivative|C4B1FA1H0J]].
For a function $x \colon \mathbb{T} \to \R^n$, write $x^\Delta$ for its delta derivative, taken componentwise.
Let $f \colon \mathbb{T} \times \R^n \to \R^n$ be a function.
The *first-order dynamic equation* on $\mathbb{T}$ with right side $f$ is
$$x^\Delta(t) = f(t, x(t)), \qquad t \in \mathbb{T}^\kappa.$$
A *solution* is a function $x \colon \mathbb{T} \to \R^n$ that is delta differentiable at every $t \in \mathbb{T}^\kappa$ and satisfies the equation there.
Dynamic equations of higher order are defined in the same way with iterated delta derivatives.

## Remarks

For $\mathbb{T} = \R$, the delta derivative is the ordinary derivative and the equation is the [[ordinary differential equation|0NT64JV96A]] $x' = f(t, x)$.
For $\mathbb{T} = \Z$, the equation reads $x(t + 1) = x(t) + f(t, x(t))$, a first-order [[recurrence relation|0FYX44GP92]].
Example: on $\mathbb{T} = h\Z$ with $h > 0$, the function $x(t) = (1 + h p)^{t/h}$ solves $x^\Delta = p x$, since $\frac{x(t + h) - x(t)}{h} = \frac{(1 + hp) - 1}{h} x(t) = p x(t)$.
Non-example: on $\mathbb{T} = \Z$, the function $x(t) = e^t$ does not solve $x^\Delta = x$, because $x(t + 1) - x(t) = (e - 1) e^t \ne e^t$.
At a point $t$ with $\sigma(t) > t$, the equation fixes $x(\sigma(t))$ from $x(t)$, so solutions of initial value problems advance across gaps by an explicit step.
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# Beltrami–Klein model

## Example

The *Beltrami–Klein model* of the [[hyperbolic plane|GN4N94GQ0Z]] is the open unit disc $D = \set{x \in \R^2 : \abs{x} < 1}$ with the following distance.
For distinct points $p, q \in D$, the line through $p$ and $q$ meets the unit circle in two points $a$ and $b$.
Label them so that the order along the line is $a, p, q, b$.
Then
$$d(p, q) = \frac{1}{2} \ln \frac{\abs{aq} \, \abs{pb}}{\abs{ap} \, \abs{qb}},$$
where $\abs{xy}$ is the Euclidean distance, and $d(p, p) = 0$.
This is the [[Cayley–Klein metric|J7ZE3EGRYX]] whose absolute is the unit circle, and it makes $D$ a [[metric space|0VKD1TP7F3]].
The lines of the model are the open chords of the disc.

Check on a diameter: take $p = (0, 0)$ and $q = (r, 0)$ with $0 < r < 1$.
Then $a = (-1, 0)$ and $b = (1, 0)$, so $\abs{aq} = 1 + r$, $\abs{pb} = 1$, $\abs{ap} = 1$ and $\abs{qb} = 1 - r$.
The formula gives $d(p, q) = \frac{1}{2} \ln \frac{1 + r}{1 - r} = \operatorname{artanh} r$, which tends to $\infty$ as $r \to 1$.
So the boundary circle lies at infinite distance, and the disc is a whole infinite plane.

## Remarks

Through a point $p$ not on a chord $\ell$ there pass infinitely many chords that do not meet $\ell$ inside the disc, so the parallel postulate fails.
The isometries of the model are the projective transformations of the plane that map the disc onto itself.
The model is not conformal: Euclidean angles between chords are in general not the hyperbolic angles.
The map $u \mapsto 2u / (1 + \abs{u}^2)$ is an isometry from the [[Poincaré disk model|85MM2EA0TK]] onto the Beltrami–Klein model.
Indeed, a point at Poincaré distance $t$ from the centre has Euclidean radius $\tanh(t/2)$ in the Poincaré disc and $\tanh t$ in the Klein disc, and $\tanh t = 2 \tanh(t/2) / (1 + \tanh^2(t/2))$.

## Explanations

### L1 human

The Beltrami–Klein model draws the hyperbolic plane inside a round disc, with straight chords as its lines.
Because lines look straight, problems about lines and their crossings become problems of ordinary straight-line drawing.
The price is that distances and angles are distorted: lengths blow up near the rim, and right angles do not look right.
For example, the segment from the centre to the point at Euclidean radius $0.9$ has hyperbolic length $\operatorname{artanh} 0.9 \approx 1.47$.
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# Queueing theory

## Definition

Let $A$ and $B$ be [[probability distributions|0DVDCMA2DK]] on $(0, \infty)$ and let $c \ge 1$ be an integer.
A *single-station queue* $A/B/c$, in Kendall's notation, is the following model:

1. customers arrive at times $0 < \tau_1 < \tau_2 < \cdots$ whose gaps $\tau_{n+1} - \tau_n$ and first time $\tau_1$ are [[independent and identically distributed|00MRN0FARV]] with law $A$;
2. customer $n$ needs a service time $\sigma_n$, where $\sigma_1, \sigma_2, \ldots$ are independent with law $B$ and independent of the arrivals;
3. there are $c$ servers and an unlimited waiting room, and customers are served in order of arrival by the first free server.

The queue length $Q(t)$ is the number of customers in the system at time $t$, waiting or in service.
Queueing theory studies such models and their networks.

## Remarks

The letter $M$ stands for an exponential law, which makes the model Markov, and $D$ for a deterministic law.
With mean gap $1/\lambda$ and mean service time $1/\mu$, the traffic intensity is $\rho = \lambda / (c \mu)$.
Example: the [[M/M/1 queue|GTFRPF5J2K]] takes $A$ exponential of rate $\lambda$, $B$ exponential of rate $\mu$ and $c = 1$; then $Q$ is a birth–death process.
Non-example: if customers who find the server busy leave at once, the model is a loss system, not an $A/B/c$ queue, since clause 3 asks for an unlimited waiting room.
For the $A/B/1$ queue with finite means, $\rho < 1$ ensures that the waiting times converge in law to a finite limit, and $\rho > 1$ makes them tend to infinity.
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# Pseudo-element

## Summary

A pseudo-element of an object $A$ in an [[abelian category|Y9CZ3MTNF1]] is an equivalence class of morphisms with codomain $A$.
Two such morphisms are identified when they agree after composing with suitable [[epimorphisms|RVXDSY04KG]].
Pseudo-elements behave enough like elements of a module to allow diagram chasing inside any abelian category.
With them, one proves diagram lemmas such as the [[snake lemma|Q8AYGD93CW]] without an embedding theorem.

## Explanations

### L1 human

In a category of modules one proves diagram lemmas by following elements around the diagram.
A general abelian category has objects without elements, so one uses morphisms $x \colon X \to A$ as "generalized elements" instead.
Two generalized elements are treated as the same if they cover each other after pulling back along surjections.
For example, in abelian groups the maps $\Z \to \Z$ given by $n \mapsto n$ and $n \mapsto -n$ give the same pseudo-element of $\Z$.
So pseudo-elements are coarser than elements, but they detect zero, monomorphisms, epimorphisms and exactness.
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# Cuthill–McKee algorithm

## Statement

Let $A$ be a [[symmetric|0VR671PHE5]] $n \times n$ matrix.
Its [[graph|00C0BR7D7H]] $G$ has vertices $1, \dots, n$ and an edge between $i \ne j$ when $a_{ij} \ne 0$; assume $G$ is connected.
The *bandwidth* of $A$ is $\max \set{\abs{i - j} : a_{ij} \ne 0}$.
The *Cuthill–McKee algorithm* orders the vertices as follows.
It picks a start vertex $v_1$.
For $k = 1, 2, \dots$ it appends to the list the neighbours of $v_k$ not yet listed, in order of increasing [[degree|0KS0KRDV4Y]].
Let $L_0 = \set{v_1}$ and, for $d \ge 1$, let $L_d$ be the set of vertices at distance $d$ from $v_1$ in $G$.
Then:

1. the algorithm lists every vertex exactly once, and each level $L_d$ forms a contiguous block of the list, in the order $L_0, L_1, L_2, \dots$;
2. if $P$ is the [[permutation matrix|08SP3DF8NB]] of this ordering, the bandwidth of $P A P^T$ is at most $\max_d \big(\abs{L_d} + \abs{L_{d+1}}\big) - 1$;
3. the reversed ordering, called *reverse Cuthill–McKee*, gives a matrix with the same bandwidth.

## Remarks

The list is built exactly as a breadth-first search from $v_1$ would build its queue, which lists the vertices by nondecreasing distance from $v_1$; this gives clause 1.
In a breadth-first level structure, every edge joins two vertices of the same level or of adjacent levels.
Two adjacent levels occupy $\abs{L_d} + \abs{L_{d+1}}$ consecutive positions, so the positions of the ends of an edge differ by at most $\abs{L_d} + \abs{L_{d+1}} - 1$; this gives clause 2.
Clause 3 holds because reversing the order maps the positions $p, q$ to $n + 1 - p, n + 1 - q$, which have the same difference.
So a start vertex with many small levels, such as a vertex far from the rest of the graph, gives a narrow [[band|Z58FRZCF8P]].
In practice one starts from a vertex of minimum degree or a pseudo-peripheral vertex.
Example: for the path graph $2 - 1 - 3$ (with $a_{12}, a_{13} \ne 0$), starting at $v_1 = 2$ gives the order $2, 1, 3$, and the reordered matrix is tridiagonal, with bandwidth $1$.
The reverse ordering usually produces less fill-in in a [[Cholesky factorization|0WKG8CYY6A]], which is why it is the version used in sparse solvers.

## Explanations

### L1 human

A [[sparse matrix|JGACZYEEY7]] is easier to factor when its nonzero entries sit close to the diagonal.
The Cuthill–McKee algorithm renumbers the unknowns so that this happens.
It starts at one unknown and numbers its neighbours, then their neighbours, and so on, layer by layer, like ripples spreading on a pond.

### L2 human

The algorithm is a heuristic: it does not find the smallest possible bandwidth, which is an NP-hard problem, but it is fast and usually good.
It runs in time roughly proportional to the number of nonzero entries, up to the sorting by degree.
Reverse Cuthill–McKee is a standard preprocessing step before banded or envelope Cholesky factorization of symmetric sparse matrices.
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# Functor category

## Summary

The functor category $[\mathcal{C}, \mathcal{D}]$, also written $\mathcal{D}^{\mathcal{C}}$, has the [[functors|28NQ0KRHD3]] from $\mathcal{C}$ to $\mathcal{D}$ as objects and the [[natural transformations|KYCPCC57FT]] between them as morphisms.
Many familiar categories are functor categories: sets with a group action, directed graphs, and [[presheaves|AND2WHZC8P]] on a topological space.
Its isomorphisms are the natural isomorphisms.
Functor categories into the category of sets are the setting of the [[Yoneda lemma|D6DF1ARPVD]].

## Explanations

### L1 human

Functors from one category to another can themselves be treated as objects, with natural transformations as the arrows between them.
For example, a group $G$ is a category with one object whose arrows are the elements of $G$.
A functor from it to sets is a set on which $G$ acts, and a natural transformation is a map that commutes with the action.
So the category of $G$-sets is a functor category.
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# Quasi-everywhere

## Summary

A property holds quasi-everywhere on a set if it holds at every point outside a set of [[capacity zero|W9N1TR1X8P]].
It plays the role in potential theory that "almost everywhere" plays in measure theory, with capacity in place of measure.
Since sets of capacity zero have Lebesgue measure zero, a property that holds quasi-everywhere also holds almost everywhere.
Many results, such as [[Frostman's theorem|TX0Z2VVKH3]] and the solvability of the Dirichlet problem at boundary points, hold quasi-everywhere but not everywhere.

## Explanations

### L1 human

"Almost everywhere" allows exceptions on a set of zero volume, which can still be quite large, like a whole line in the plane.
"Quasi-everywhere" allows only exceptions on a set of capacity zero, which is much thinner.
For example, the potential of the equilibrium charge on a conductor is constant at every point of the conductor except on a set of capacity zero.
So quasi-everywhere statements are sharper than almost-everywhere statements.
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# Jacobson ring

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]].
The ring $R$ is a *Jacobson ring* if every [[prime ideal|00MHHZ9SNE]] $\mathfrak{p}$ is the intersection of the [[maximal ideals|0PV74TR5FP]] that contain $\mathfrak{p}$.

## Remarks

Example: $\Z$ is a Jacobson ring.
A nonzero prime ideal $p\Z$ is maximal, and the prime $0$ equals $\bigcap_p p\Z$, because a nonzero integer has only finitely many prime divisors.

Non-example: the localization $\Z_{(p)}$ is not a Jacobson ring.
Its only maximal ideal is $p\Z_{(p)}$, so the intersection of the maximal ideals containing the prime $0$ is $p\Z_{(p)} \ne 0$.

Equivalent forms: $R$ is Jacobson if and only if every [[radical ideal|437YRCNYRJ]] is an intersection of maximal ideals.
It is also equivalent that for every ideal $I$, the [[nilradical|QQDN6EYQNJ]] of $R/I$ equals its [[Jacobson radical|XA0DCNH7WY]].
Some texts use the name Hilbert ring.
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# Kobayashi conjecture

## Statement

Let $n \ge 2$ and let $X \subset \mathbb{P}^{n+1}$ be a very general hypersurface of degree $d$ in complex [[projective space|ZT9A6JK8QF]].
Very general means that the defining polynomial lies outside a countable union of proper algebraic subsets of the space of polynomials of degree $d$.

1. If $d \ge 2n + 1$, then $X$ is [[Kobayashi hyperbolic|HRPAX8V7HA]].
2. If $d \ge 2n + 3$, then the complement $\mathbb{P}^{n+1} \setminus X$ is Kobayashi hyperbolic.

## Remarks

Kobayashi stated these problems in his 1970 book on hyperbolic manifolds.
The hypothesis $n \ge 2$ is needed in part 1 in this form.
For $n = 1$ the bound would be $d \ge 3$, but a smooth plane cubic is an elliptic curve, a complex torus, which is not hyperbolic.
For $n = 1$ the true statement is that smooth plane curves of degree at least $4$ are hyperbolic, since they have genus at least $3$.
Part 1 is known for all $d$ larger than a bound that depends on $n$; Yum-Tong Siu and Damian Brotbek gave proofs.
Later work lowered the bound, which is still far above $2n + 1$ for large $n$.
For $d \le 2n - 1$, every hypersurface of degree $d$ in $\mathbb{P}^{n+1}$ contains a line, so it is not hyperbolic.
Since $X$ is compact, part 1 asks equivalently that $X$ contains no entire curve, that is, that $X$ is [[Brody hyperbolic|Y7W36GJDJD]].

## Explanations

### L1 human

A complex manifold is hyperbolic when it contains no copy of the complex plane, in a strong sense.
The conjecture predicts that a typical hypersurface of large enough degree in projective space is hyperbolic, and so is the complement of such a hypersurface.
The degree needs to be large because hypersurfaces of small degree contain lines, which carry many copies of the plane.
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# Sectional curvature

## Summary

The sectional curvature of a [[Riemannian manifold|7CBT4SM83Y]] assigns a number to each $2$-dimensional plane in each tangent space.
It is the [[Gaussian curvature|SQVQ5DQT6W]] at $p$ of the surface swept out by the geodesics from $p$ tangent to that plane.
The sectional curvatures determine the full [[Riemann curvature tensor|JA29ETCM02]], and bounds on them are the hypotheses of the main comparison theorems of global Riemannian geometry.

## Explanations

### L1 human

In two dimensions, curvature at a point is a single number, the Gaussian curvature.
In higher dimensions a point has many surface directions, and the space can be curved differently in each.
Sectional curvature gives the Gaussian curvature for each choice of a $2$-plane of directions.
The round unit sphere has sectional curvature $1$ for every plane, flat space has $0$, and hyperbolic space has $-1$.
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# Balayage

## Definition

Let $n \ge 3$ and let $D \subset \R^n$ be a bounded domain whose boundary points are all [[regular|B7948M365K]].
For $x \in D$, let $\omega_x$ be the [[harmonic measure|CFD7BPM51W]] of $D$ at $x$, a Borel probability measure on $\partial D$.
Let $\mu$ be a finite positive [[Borel measure|DYR1QFNGNJ]] with compact support in $D$.
The *balayage* of $\mu$ onto $\partial D$ is the Borel measure $\mu'$ on $\partial D$ given by
$$\mu'(A) = \int_D \omega_x(A)\, d\mu(x) \qquad \text{for every Borel set } A \subseteq \partial D.$$

## Remarks

Write $U^\mu(y) = \int \abs{y - x}^{2-n}\, d\mu(x)$ for the [[Newtonian potential|DCMFQ01QGS]].
Then $\mu'(\partial D) = \mu(D)$ and $U^{\mu'} = U^\mu$ on $\R^n \setminus D$.
Indeed, for $y \notin \overline{D}$ the function $x \mapsto \abs{y - x}^{2-n}$ is harmonic on $D$ and continuous on $\overline{D}$, so it is the integral of its boundary values against $\omega_x$.
Inside $D$ one has $U^{\mu'} \le U^\mu$.
Example: let $D$ be the ball of radius $R$ about $0$ and $\mu = \delta_0$.
Then $\omega_0$ is the normalised surface measure on the sphere, so $\mu'$ is that measure.
Its potential is $\min(R^{2-n}, \abs{y}^{2-n})$, which equals $\abs{y}^{2-n} = U^{\delta_0}(y)$ for $\abs{y} \ge R$.
Non-example: $\mu$ itself is not a balayage of $\mu$ onto $\partial D$, since it is not carried by $\partial D$.
More generally, for any set $E \subseteq \R^n$ the balayage $\mu^E$ is the measure whose potential is the regularised [[reduced function|909B1WYQ4Q]] $\widehat{R}^E_{U^\mu}$.
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# Koszul complex

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $x = (x_1, \ldots, x_n)$ be a sequence of elements of $R$.
Let $e_1, \ldots, e_n$ be the standard basis of the [[free module|0DZ9DKAKZR]] $R^n$.
For $0 \le p \le n$, let $K_p = \Lambda^p R^n$ be its $p$-th [[exterior power|0MDMZ965VE]].
It is free with basis the elements $e_{i_1} \wedge \cdots \wedge e_{i_p}$ for $i_1 < \cdots < i_p$.
The *Koszul complex* $K_\bullet(x)$ is
$$0 \to K_n \xrightarrow{d_n} K_{n-1} \to \cdots \to K_1 \xrightarrow{d_1} K_0 \to 0,$$
with
$$d_p(e_{i_1} \wedge \cdots \wedge e_{i_p}) = \sum_{j=1}^{p} (-1)^{j+1} x_{i_j}\, e_{i_1} \wedge \cdots \wedge \widehat{e_{i_j}} \wedge \cdots \wedge e_{i_p},$$
where the hat marks an omitted factor.
For an [[$R$-module|0NZ8JA25WK]] $M$, the Koszul complex of $x$ on $M$ is $K_\bullet(x; M) = K_\bullet(x) \otimes_R M$.

## Remarks

One checks that $d_{p-1} \circ d_p = 0$, so $K_\bullet(x)$ is a complex, and $H_0(K_\bullet(x)) = R/(x_1, \ldots, x_n)$.
If $x_1, \ldots, x_n$ is a [[regular sequence|69R5R2FCS5]], then $H_p(K_\bullet(x)) = 0$ for $p > 0$, so the complex is a free resolution of $R/(x_1, \ldots, x_n)$.
Example: for $n = 2$, the complex is $0 \to R \to R^2 \to R \to 0$ with $e_1 \wedge e_2 \mapsto x_1 e_2 - x_2 e_1$ and $e_i \mapsto x_i$; the composite sends $e_1 \wedge e_2$ to $x_1 x_2 - x_2 x_1 = 0$.
Non-example of exactness: for $R = K[t]$ and $x = (t, t)$, which is not a regular sequence, $H_1$ is the set of $(a, -a)$ modulo the multiples of $(-t, t)$, which is isomorphic to $R/(t) \ne 0$.
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# Cartan–Thullen theorem

## Statement

Let $D \subseteq \C^n$ be a domain, that is, a nonempty connected open set.
The following are equivalent.

1. $D$ is a [[domain of holomorphy|4487CSC0MB]].
2. $D$ is [[holomorphically convex|D5J6CP5V2Z]]: for every compact set $K \subseteq D$, its [[holomorphically convex hull|CD6SMMZSW8]] $\hat K_D$ is compact.
3. There is a [[holomorphic function|507C34W0ZT]] $f$ on $D$ that cannot be continued beyond $D$. That is, there are no nonempty open sets $U_1 \subseteq D \cap U_2$, with $U_2$ connected and $U_2 \not\subseteq D$, and no holomorphic $g$ on $U_2$ with $g = f$ on $U_1$.

## Remarks

Clause 3 implies clause 1 directly from the definition.
For 1 implies 2, the key estimate concerns a domain of holomorphy $D$.
The distance from $\hat K_D$ to the boundary of $D$ equals the distance from $K$ to the boundary, measured with polydiscs.
It comes from the Cauchy estimates: the Taylor series at points of $\hat K_D$ of every holomorphic function on $D$ converge on polydiscs of a fixed radius.
For 2 implies 3, take a sequence of points of $D$ that accumulates at every boundary point.
Holomorphic convexity gives holomorphic functions that are large at chosen points of the sequence.
An infinite product or series of them gives a function unbounded near every boundary point.

Example: a convex domain is holomorphically convex, because $\abs{e^{\ell}} = e^{\operatorname{Re} \ell}$ for complex linear $\ell$ shows that $\hat K_D$ lies in the convex hull of $K$; so it is a domain of holomorphy.

## Explanations

### L1 human

There are three ways to say that a domain has no hidden extensions.
One can ask that not all functions extend across any boundary point, or that one single function refuses to extend anywhere, or a convexity condition defined with holomorphic functions.
The theorem says these three conditions are the same.

### L2 human

Holomorphic convexity is an intrinsic condition, stated without reference to points outside $D$.
This makes it the right condition to generalise to complex manifolds, where it leads to the notion of a Stein manifold.
It reduces the Levi problem to showing that pseudoconvex domains are holomorphically convex.
For $n = 1$ every domain satisfies all three conditions.
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# Degree of a projective variety

## Summary

Let $X \subseteq \mathbb{P}^n$ be a [[projective variety|A4H4CZ2V35]] of dimension $d$.
Its degree is the number of points in which $X$ meets a general linear subspace of dimension $n - d$.
Algebraically, it is $d!$ times the leading coefficient of the [[Hilbert polynomial|RMH4FBCFDA]] of $X$.
It extends the degree of a polynomial to varieties and is the basic quantity in [[Bézout's theorem|KWNVF194D8]].

## Explanations

### L1 human

A plane curve given by an equation of degree $e$ meets a general line in exactly $e$ points, so its degree is $e$.
For a surface in space one cuts with a general line instead, and for a curve in space with a general plane.
The twisted cubic, the image of $t \mapsto (t, t^2, t^3)$, meets a general plane $a + bx + cy + dz = 0$ where the cubic $a + bt + ct^2 + dt^3$ vanishes, that is, in $3$ points.
So its degree is $3$.
The degree depends on how the variety sits in projective space, not only on the variety itself.
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# Weak ordering

## Definition

Let $A$ be a set and let $<$ be a [[strict partial order|SWB1ZXFVKY]] on $A$.
Two elements $a, b \in A$ are *incomparable* when neither $a < b$ nor $b < a$.
The relation $<$ is a *strict weak ordering* when incomparability is transitive.
That is, for all $a, b, c \in A$: if $a, b$ are incomparable and $b, c$ are incomparable, then $a, c$ are incomparable.

## Remarks

Let $f \colon A \to \R$ be any function and set $a < b$ when $f(a) < f(b)$.
This is a strict weak ordering: $a$ and $b$ are incomparable exactly when $f(a) = f(b)$, and equality of values is transitive.
On $A = \set{a, b, c}$, the strict partial order whose only relation is $a < b$ is not a strict weak ordering.
Here $a$ and $c$ are incomparable and $c$ and $b$ are incomparable, but $a$ and $b$ are comparable.
In a strict weak ordering, incomparability is an [[equivalence relation|0SGTMBEWRF]], and $<$ induces a strict total order on its classes.
The non-strict form is a total preorder: a [[preorder|0RXS8R6K37]] $\lesssim$ in which $a \lesssim b$ or $b \lesssim a$ for all $a, b$.
Strict weak orderings and total preorders on $A$ correspond one to one, by $a \lesssim b$ if and only if not $b < a$.

## Explanations

### L1 human

A weak ordering is a ranking with ties.
Tied elements form groups, and the groups stand in a line.
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# Riemann zeta function

## Summary

The Riemann zeta function is given by the series $\zeta(s) = \sum_{n \ge 1} n^{-s}$ for complex $s$ with $\operatorname{Re} s > 1$.
It extends to a [[meromorphic function|0ZZ7ZYDR8W]] on $\C$ with a single pole at $s = 1$.
Through the [[Euler product formula|0C3PN4075R]] it encodes the [[prime numbers|01826KMG2H]], and the location of its zeros controls the error term in the [[prime number theorem|0EHBY5Y84N]].
The [[Riemann hypothesis|VCGF9YR36X]] about these zeros is one of the main open problems of mathematics.

## Explanations

### L1 human

Start with the sum $1 + 1/2^s + 1/3^s + \cdots$ of the reciprocals of the powers $n^s$.
For $s = 2$ the sum is $\pi^2/6$, and for $s = 1$ it is the harmonic series, which diverges.
Letting $s$ be a complex number turns this sum into a function with a rich structure.
Because each integer factors into primes in one way, the function carries information about all primes at once.
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# Algebraic set

## Definition

Let $k$ be a [[field|04C5ZBG1WT]], let $n \ge 0$ be an integer, and let $\mathbb{A}^n = k^n$ be the affine $n$-space over $k$.
Let $k[x_1, \dots, x_n]$ be the [[polynomial ring|0SDC934R52]] in $n$ variables over $k$.
For a subset $S \subseteq k[x_1, \dots, x_n]$, its zero set is
$$V(S) = \set{a \in k^n : f(a) = 0 \text{ for all } f \in S}.$$
A subset $X \subseteq k^n$ is an *algebraic set* (or affine algebraic set) if $X = V(S)$ for some subset $S \subseteq k[x_1, \dots, x_n]$.

## Remarks

Example: in $\R^2$, the unit circle is $V(x^2 + y^2 - 1)$, so it is an algebraic set.
Each point $a = (a_1, \dots, a_n)$ is algebraic, since $\set{a} = V(x_1 - a_1, \dots, x_n - a_n)$.

Non-example: the half-line $H = \set{(t, 0) : t \ge 0}$ in $\R^2$ is not an algebraic set.
If a polynomial $f$ vanishes on $H$, then $t \mapsto f(t, 0)$ is a polynomial in one variable with infinitely many roots, so it is zero.
Then $f$ also vanishes at $(-1, 0)$, so every $V(S)$ that contains $H$ contains the whole $x$-axis.

The zero set only depends on the [[ideal|096Z9MP2ME]] $I$ generated by $S$: $V(S) = V(I)$.
By [[Hilbert's basis theorem|0YKA9J3XEE]], $I$ is finitely generated, so every algebraic set is the zero set of finitely many polynomials.
The rules $V(I) \cup V(J) = V(IJ)$ and $\bigcap_i V(I_i) = V\left(\sum_i I_i\right)$ show that the algebraic sets are the closed sets of a topology on $k^n$.
This topology is the classical form of the [[Zariski topology|8SQK7RTB63]].
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# O-minimal theory

## Definition

Let $L$ be a first-order language that contains a binary relation symbol $<$.
Let $M$ be an $L$-[[structure|983GZBQYT4]] in which $<$ is interpreted as a [[total order|0FRXBVVR03]].
The structure $M$ is *o-minimal* if every subset of $M$ [[definable|CY1TXHF6KQ]] with parameters is a finite union of points and open [[intervals|0Q1D85B8G6]].
Here the endpoints of an interval $(a, b)$ lie in $M \cup \set{-\infty, +\infty}$.
An $L$-[[theory|92SWXST1ZM]] $T$ is *o-minimal* if every model of $T$ is o-minimal.

## Remarks

Example: the ordered field $(\R, +, \cdot, <, 0, 1)$ is o-minimal.
By the Tarski–Seidenberg theorem it has quantifier elimination, so a definable subset of $\R$ is a Boolean combination of sets $\set{x : p(x) > 0}$ and $\set{x : p(x) = 0}$ with $p$ a polynomial with real coefficients.
Each such set is a finite union of points and open intervals, because a nonzero polynomial has finitely many roots.

Non-example: the structure $(\Z, <, +)$ is not o-minimal.
The formula $\exists y \, (x = y + y)$ defines the set of even integers, which is not a finite union of points and intervals.

By a theorem of Knight, Pillay and Steinhorn, if a structure $M$ is o-minimal then its complete theory is o-minimal.
Dense linear orders without endpoints and real closed fields give o-minimal theories.
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# Saccheri–Legendre theorem

## Statement

Let $\Pi$ be an [[absolute plane|VKFPCSRNDH]] that satisfies the axiom of Archimedes.
Then the sum of the three [[angles|0GJAJC6TNZ]] of every [[triangle|0CTNFGYMM2]] in $\Pi$ is at most two right angles.

## Remarks

The proof starts from a triangle $ABC$ and builds a new triangle with the same angle sum, one of whose angles is at most half of the angle at $A$.
Repeating this $n$ times gives a triangle with the same angle sum and an angle at most $2^{-n}$ times the angle at $A$.
If the angle sum exceeded two right angles by some angle $\varepsilon$, the axiom of Archimedes would give a step where this small angle is less than $\varepsilon$.
Then the other two angles would sum to more than two right angles, which contradicts the [[exterior angle theorem|0NTSE7EREA]].
The axiom of Archimedes is needed: Max Dehn built non-Archimedean geometries in which the other axioms hold and some triangles have angle sum greater than two right angles.

## Explanations

### L1 human

Without the parallel postulate, one cannot prove that the angles of a triangle add up to $180^\circ$.
But one can still prove that they never add up to more than $180^\circ$.
In the Euclidean plane the sum is exactly $180^\circ$, and in the hyperbolic plane it is always less.

### L2 human

The theorem is one of the main results of absolute geometry.
With it, the parallel postulate is equivalent to the existence of a single triangle with angle sum $180^\circ$.
The defect of a triangle is two right angles minus its angle sum.
It is additive when a triangle is cut into two, and in the hyperbolic plane it is proportional to the area.
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# Poisson regression

## Definition

Let $x_1, \dots, x_n \in \R^p$ be known regressor vectors.
The *Poisson regression model* for responses $y_1, \dots, y_n \in \set{0, 1, 2, \dots}$ states three things.
The $y_i$ are independent, $y_i$ has the [[Poisson distribution|0XB9GQTN9G]] with mean $\mu_i > 0$, and
$$\log \mu_i = x_i^T \beta$$
for an unknown $\beta \in \R^p$.

## Remarks

It is the [[generalized linear model|K6C48KMRKC]] with Poisson responses and the log link, which is the canonical link.
The log-likelihood is $\ell(\beta) = \sum_i \big( y_i x_i^T \beta - e^{x_i^T \beta} - \log y_i! \big)$, a concave function.
The [[maximum likelihood|0FREPFKNS7]] estimate solves $\sum_i (y_i - \mu_i) x_i = 0$.

Example: with only an intercept, $x_i = 1$, the likelihood equation gives $\sum_i (y_i - e^{\beta}) = 0$, so $e^{\hat\beta} = \bar y$, the mean count.

When $y_i$ counts events over an exposure $t_i > 0$, one takes $\log \mu_i = \log t_i + x_i^T \beta$; the known term $\log t_i$ is an *offset*, and $e^{x_i^T \beta}$ is the rate per unit of exposure.

Non-example: if the counts have variance larger than their mean, the hypothesis that $y_i$ is Poisson fails (overdispersion).
The point estimates remain consistent when the mean is correctly specified, but the usual standard errors are too small.
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# Lebesgue's theorem on differentiability of monotone functions

## Statement

Let $a < b$ and let $f \colon [a, b] \to \R$ be a [[monotone function|0WFF6AZPBP]].
Then $f$ is [[differentiable|0F1WR9C83K]] at [[almost every|RARTMA0ZEC]] point of $[a, b]$.
That is, the set of points of $(a, b)$ where $f$ has no finite derivative is a [[null set|VQQM4Y550T]].

## Remarks

The standard proof uses the [[Dini derivatives|36DXKAD2RZ]] of $f$ and the [[Vitali covering lemma|FCA0J1X07K]].
For nondecreasing $f$, one shows that each set $\set{x : D^+ f(x) > q > p > D_- f(x)}$ with rationals $p < q$ is null.
One also shows that the set where $D^+ f(x) = \infty$ is null.
Together with the same argument for the other pairs of Dini derivatives, this gives a finite derivative almost everywhere.

For nondecreasing $f$, the derivative $f'$ is moreover Lebesgue integrable, and $\int_a^b f'(x) \, dx \le f(b) - f(a)$ for the [[Lebesgue integral|DWJF7ABPR1]].
The inequality can be strict: for the [[Cantor function|NPBYYK63FE]], the left side is $0$ and the right side is $1$.
Monotonicity cannot be replaced by continuity: the [[Weierstrass function|01A0MK1730]] is continuous and differentiable at no point.
The theorem extends to functions of bounded variation, which are differences of monotone functions by the [[Jordan decomposition|KDT3E0VE0K]].

## Explanations

### L1 human

A monotone function only goes up, or only goes down.
It can have jumps and corners, even infinitely many.
The theorem says that it is still differentiable at almost every point: the bad points take up no length at all.

### L2 human

The theorem is the starting point of the theory of differentiation of real functions in measure theory.
It leads to the differentiation of indefinite Lebesgue integrals and to the fundamental theorem of calculus for absolutely continuous functions.
[[Fubini's theorem on differentiation|XDEJCKRVQ4]] refines it for series of monotone functions.
In contrast with the Cantor function, absolutely continuous functions satisfy $\int_a^b f' = f(b) - f(a)$.
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# Hodge star operator

## Summary

The Hodge star operator is a linear map that sends $k$-forms to $(n-k)$-forms.
It is defined on an oriented $n$-dimensional inner product space and, pointwise, on an oriented [[Riemannian manifold|7CBT4SM83Y]].
It sends a form to its "orthogonal complement", using the metric and the orientation.
With it one defines the codifferential, the Hodge Laplacian and harmonic forms, and it underlies Hodge theory and Poincaré duality for de Rham cohomology.

## Explanations

### L1 human

In three-dimensional space, a small oriented plane piece and the arrow perpendicular to it carry the same information.
The Hodge star makes this exchange: it turns the plane spanned by $dx$ and $dy$ into the direction $dz$, and back.
In $n$ dimensions it exchanges $k$-dimensional pieces with $(n-k)$-dimensional ones.
For example, the cross product of two vectors in $\R^3$ is the Hodge star of their wedge product.
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# Analytic torsion

## Definition

Let $(M, g)$ be a closed oriented [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$.
Let $\rho \colon \pi_1(M) \to U(N)$ be a [[unitary representation|74BPYJNZTM]] of the [[fundamental group|S1XK3HXHFX]], and let $E_\rho$ be the associated flat Hermitian vector bundle.
Assume that $\rho$ is acyclic, that is, the cohomology $H^*(M; E_\rho)$ vanishes.
For $0 \le q \le n$ let $\Delta_q$ be the [[Hodge Laplacian|PWG3GKZG80]] on $E_\rho$-valued $q$-forms.
By acyclicity, all eigenvalues of $\Delta_q$ are positive.
Let $\zeta_q(s) = \sum_\lambda \lambda^{-s}$ be its spectral zeta function, summed over its eigenvalues with multiplicity.
The *analytic torsion* $T(M, \rho) > 0$ is defined by
$$\log T(M, \rho) = \frac12 \sum_{q = 0}^{n} (-1)^q \, q \, \zeta_q'(0).$$

## Remarks

Equivalently, $T(M, \rho) = \prod_q \big( \det_\zeta \Delta_q \big)^{(-1)^{q+1} q/2}$ in terms of [[zeta-regularized determinants|YT3ZQ2F7FG]].
Ray and Singer showed that $T(M, \rho)$ does not depend on the metric $g$ when $n$ is odd, and that $T(M, \rho) = 1$ when $n$ is even.
Without acyclicity, one defines the torsion as a metric on the determinant line of the cohomology, the Ray–Singer metric.
Conventions differ: some authors use the inverse of $T$, or omit the factor $\tfrac12$.

Example: let $M = \R / 2\pi\Z$ and let $\rho$ send the generator of $\pi_1(M) = \Z$ to $e^{2\pi i a}$ with $0 < a < 1$.
Sections of $E_\rho$ are functions with $f(\theta + 2\pi) = e^{2\pi i a} f(\theta)$, with eigenfunctions $e^{i(k + a)\theta}$, $k \in \Z$.
So $\Delta_0$ and $\Delta_1$ both have eigenvalues $(k + a)^2$, and $\zeta_1(s) = \zeta(2s, a) + \zeta(2s, 1 - a)$, with Hurwitz zeta functions.
From $\zeta'(0, a) = \log \Gamma(a) - \tfrac12 \log(2\pi)$ and $\Gamma(a)\Gamma(1 - a) = \pi / \sin(\pi a)$ one gets $\zeta_1'(0) = -2 \log(2 \sin \pi a)$, hence $T(M, \rho) = 2 \sin(\pi a) = \abs{1 - e^{2\pi i a}}$.

Non-example: for the trivial representation of $\pi_1(S^1)$, $H^0(S^1; \C) = \C$ is nonzero, so the representation is not acyclic, $\Delta_0$ has the eigenvalue $0$, and the formula above does not apply as stated.
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# Loewner's torus inequality

## Statement

Let $g$ be a [[Riemannian metric|Q61VFHCH2J]] on the torus $T^2$.
Let $\operatorname{sys}(g)$ be its [[systole|PZCJV40M4K]] and $\operatorname{area}(g)$ its area.
Then
$$\operatorname{sys}(g)^2 \le \frac{2}{\sqrt{3}} \operatorname{area}(g).$$
Let $\Lambda_h \subset \R^2$ be the hexagonal lattice spanned by $(1, 0)$ and $(\tfrac{1}{2}, \tfrac{\sqrt{3}}{2})$.
Equality holds exactly when $g$ is flat and isometric to a constant multiple of the metric of $\R^2 / \Lambda_h$.

## Remarks

By the uniformization theorem, $g$ is conformal to a flat metric $g_0$, so $g = f^2 g_0$ for a positive function $f$.
Averaging $f$ over the translations of the flat torus gives a flat metric.
By the Cauchy–Schwarz inequality its area is at most that of $g$, and by averaging lengths over translates of a closed geodesic its systole is at least that of $g$.
For a flat torus $\R^2 / \Lambda$, the claim is that the shortest nonzero vector $v$ of a lattice $\Lambda$ of covolume $A$ satisfies $\abs{v}^2 \le \tfrac{2}{\sqrt{3}} A$.
This follows by reduction of a basis of $\Lambda$, and the hexagonal lattice is the extreme case.

For the square torus $\R^2/\Z^2$ the two sides are $1$ and $2/\sqrt{3} \approx 1.155$.
For the hexagonal torus above, the systole is $1$ and the area is $\sqrt{3}/2$, so equality holds.

## Explanations

### L1 human

On any doughnut-shaped surface, take the shortest loop that goes around a hole.
The inequality says that the square of its length is at most about $1.155$ times the area.
So a surface whose short loops are all long must have a large area.
The best shape is the flat torus glued from a rhombus made of two equilateral triangles.

### L2 human

This is the first result of [[systolic geometry|PZCJV40M4K]], and the model for [[Gromov's systolic inequality|HB2K72PSZ6]] in higher dimensions.
The constant $2/\sqrt{3}$ is the Hermite constant $\gamma_2$ of the geometry of numbers.
The proof pattern, reduction to the most symmetric metrics in a conformal class by averaging, recurs in other optimal systolic inequalities.
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# Favard's theorem

## Statement

Let $(a_n)_{n \ge 0}$ be real numbers and let $(b_n)_{n \ge 1}$ be positive real numbers.
Define polynomials by $p_{-1} = 0$, $p_0 = 1$ and the [[recurrence|0FYX44GP92]]
$$p_{n+1}(x) = (x - a_n)\, p_n(x) - b_n\, p_{n-1}(x) \qquad (n \ge 0).$$
Then there is a positive [[Borel measure|DYR1QFNGNJ]] $\mu$ on $\R$ with infinite support and all moments finite such that
$$\int_\R p_m(x)\, p_n(x)\, d\mu(x) = 0 \quad \text{for } m \ne n.$$

## Remarks

The proof defines a linear functional $L$ on polynomials by $L(1) = 1$ and $L(p_n) = 0$ for $n \ge 1$, which is possible since $p_n$ is monic of degree $n$.
The recurrence gives $L(p_m p_n) = 0$ for $m \ne n$ and $L(p_n^2) = b_1 b_2 \cdots b_n > 0$, so $L$ is positive on nonzero squares of polynomials.
Then $L(f) \ge 0$ for every polynomial $f \ge 0$ on $\R$, since such $f$ is a sum of two squares, and the Hamburger moment problem gives a measure $\mu$ with $L(f) = \int f\, d\mu$.
The measure need not be unique.
It is unique when the moment problem is determinate, for instance when $(a_n)$ and $(b_n)$ are bounded, and then $\mu$ has compact support.
Example: $a_n = 0$ and $b_n = \frac{1}{4}$ give the monic Chebyshev polynomials of the second kind, and $\mu$ is the semicircle measure $\frac{2}{\pi} \sqrt{1 - x^2}\, dx$ on $[-1, 1]$.
The hypothesis $b_n > 0$ is needed.
For orthogonal polynomials of a positive measure with infinite support, $b_n = \frac{\int p_n^2\, d\mu}{\int p_{n-1}^2\, d\mu} > 0$, so a recurrence with some $b_n \le 0$ cannot come from such a measure.

## Explanations

### L1 human

Orthogonal polynomials always satisfy a three-term recurrence.
Favard's theorem says the converse: any polynomials built by such a recurrence, with positive coefficients in the right place, are orthogonal for some weight.
So one can produce orthogonal polynomials just by choosing two sequences of numbers.

### L2 human

The theorem identifies orthogonal polynomial sequences on the real line with pairs of sequences $(a_n)$, $(b_n)$ with $b_n > 0$, that is, with Jacobi matrices.
In operator language, $\mu$ is a spectral measure of the Jacobi operator with diagonal $a_n$ and off-diagonal $\sqrt{b_n}$.
It reverses the [[three-term recurrence|YRYWCP9TYV]], and its proof reduces to the Hamburger moment problem.
It is used to show that families defined by recurrences, such as many in the Askey scheme, are orthogonal before the weight is known.
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<document source="https://tamnd.github.io/math2/node/1YDR94WQMZ.md" tag="1YDR94WQMZ" kind="statement">
# Newton–Kantorovich theorem

## Statement

Let $X$ and $Y$ be [[Banach spaces|06GZ956657]], let $D \subseteq X$ be open, and let $F \colon D \to Y$ be [[Fréchet differentiable|06EH97V2A7]] with derivative $F'(x)$ at $x$.
Let $x_0 \in D$ be a point where $F'(x_0)$ is invertible with bounded inverse.
Assume that there are constants $\eta > 0$ and $L > 0$ such that:

1. $\norm{F'(x_0)^{-1} F(x_0)} \le \eta$;
2. $\norm{F'(x_0)^{-1} (F'(x) - F'(y))} \le L \norm{x - y}$ for all $x, y \in D$, a [[Lipschitz condition|0AW1E6YTJY]];
3. $h = L \eta \le \tfrac12$;
4. the closed ball $\bar B(x_0, t^*)$ lies in $D$, where $t^* = \dfrac{1 - \sqrt{1 - 2h}}{L}$.

Then the [[Newton iterates|07EXVNF9VP]] $x_{k+1} = x_k - F'(x_k)^{-1} F(x_k)$ are well defined, they lie in $\bar B(x_0, t^*)$, and they converge to a point $x^* \in \bar B(x_0, t^*)$ with $F(x^*) = 0$.

## Remarks

The proof compares the iterates with Newton's method for the real quadratic $p(t) = \tfrac{L}{2} t^2 - t + \eta$, started at $t_0 = 0$.
Those scalar iterates increase to the smallest root $t^*$ of $p$, and an induction shows $\norm{x_{k+1} - x_k} \le t_{k+1} - t_k$, so $(x_k)$ is a Cauchy sequence.
The quadratic $p$ has a real root exactly when $h \le \tfrac12$, which explains the condition on $h$.
If $h < \tfrac12$, the convergence is quadratic; if $h = \tfrac12$, it may be only linear.
Example: for $F(x) = x^2 - 2$ on $\R$ and $x_0 = 3/2$, one has $\eta = 1/12$ and $L = 2/3$ on $D = \R$, so $h = 1/18 \le 1/2$, and Newton's method converges to $\sqrt 2$.

## Explanations

### L1 human

Newton's method solves an equation by repeatedly replacing it with its linear approximation.
It can fail if the start is poor.
The Newton–Kantorovich theorem gives a test that uses only data at the starting point.
If the first Newton step is small compared with how fast the derivative can change, then the method converges.
The theorem also says in which ball the solution lies.

### L2 human

The theorem works in any Banach space, so it applies to nonlinear integral and differential equations, not only to systems in $\R^n$.
It is also an existence theorem: it proves that a solution exists near $x_0$ without knowing it in advance.
This is the basis of computer-assisted proofs, where $\eta$ and $L$ are bounded rigorously by interval arithmetic.
Compared with the Banach fixed point theorem, it needs a derivative but gives quadratic convergence when $h < \tfrac12$.
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# Local index theorem

## Statement

Let $M$ be a closed Riemannian manifold of even dimension $n$ with a [[spin structure|RSPZ9TCFQ8]] and spinor bundle $S = S^+ \oplus S^-$.
Let $W$ be a Hermitian vector bundle over $M$ with a unitary connection, and let $D_W$ be the [[Dirac operator|QMRAVX14BJ]] on $S \otimes W$.
Let $k_t(x, y)$ be the [[heat kernel|327NP6S7H5]] of $D_W^2$, and let $\operatorname{str}$ be the supertrace on $(S \otimes W)_x$, the trace on $S^+ \otimes W$ minus the trace on $S^- \otimes W$.
Then, uniformly in $x \in M$,
$$\lim_{t \to 0^+} \operatorname{str} k_t(x, x) \, \mathrm{vol}(x) = \Big[ \hat A(TM, \nabla) \operatorname{ch}(W, \nabla^W) \Big]_{n}(x).$$
Here $\hat A(TM, \nabla)$ and $\operatorname{ch}(W, \nabla^W)$ are the [[Chern–Weil|82ZPRD39J7]] forms of the Levi-Civita connection and of the connection on $W$.
They are normalized to represent the $\hat A$-class and the Chern character, and $[\cdot]_n$ is the component of degree $n$.

## Remarks

The [[McKean–Singer formula|F4WDYPQ9VP]] gives $\operatorname{ind} D_W^+ = \int_M \operatorname{str} k_t(x, x) \, \mathrm{vol}(x)$ for every $t > 0$.
Letting $t \to 0^+$ and using the theorem gives $\operatorname{ind} D_W^+ = \int_M \hat A(TM) \operatorname{ch}(W)$, which is the index theorem for twisted Dirac operators and contains [[the Â-genus formula|JR2M58RCW3]] for $W$ trivial.
The striking point is that the singular terms $t^{-n/2}, \ldots, t^{-1}$ of the asymptotic expansion of the supertrace cancel pointwise, not only after integration.
The cancellation was first proved by Patodi for the de Rham complex and by Gilkey with invariance theory.
Getzler gave a proof by a rescaling of the Clifford variables, and Bismut gave a probabilistic proof.
Example: for $n = 2$ and $W$ trivial, $\hat A$ has no component of degree $2$, so the limit is $0$, matching the vanishing index of the Dirac operator on a closed surface.

## Explanations

### L1 human

The index of a Dirac operator is a global count of solutions.
The local index theorem says that the heat kernel, a local object, already knows the curvature integrand that adds up to this count.
At every point, as time goes to zero, the heat flow reveals the characteristic form of the curvature.

### L2 human

The theorem is a sharpened form of the heat equation proof of the index theorem.
It gives the index density pointwise, which is what one needs on manifolds with boundary, for families, and for the [[Atiyah–Patodi–Singer theorem|GGFAH1M80J]].
Its proof by Getzler rescaling is now the standard route to the index theorem in courses on Dirac operators.
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# Lickorish–Wallace theorem

## Statement

Let $M$ be a closed connected orientable [[3-manifold|R20TB2KEQQ]].
Then there is a [[link|1A5WRCNTM3]] $L \subset S^3$ such that $M$ is homeomorphic to the result of integral [[Dehn surgery|ZA405B0Z1S]] on $L$.
Moreover, $L$ can be chosen so that each component is unknotted and each surgery coefficient is $+1$ or $-1$.

## Remarks

Wallace and Lickorish proved the theorem independently, by different methods.
Idea of Lickorish's proof: write $M$ as a [[Heegaard splitting|EDE9919B5R]] whose gluing map differs from that of a standard splitting of $S^3$ by a homeomorphism of the Heegaard surface.
By the [[Dehn–Lickorish theorem|BG3EH18KRE]], this homeomorphism is isotopic to a product of Dehn twists.
Each Dehn twist can be realised by a $\pm 1$ surgery on a copy of its curve pushed slightly off the surface into $S^3$, and such a copy is unknotted.

The theorem at work: $S^2 \times S^1$ is $0$-surgery on the unknot, and the lens space $L(p, 1)$ is integral surgery with coefficient $\pm p$ on the unknot, depending on the orientation convention.
The theorem implies that every closed orientable $3$-manifold bounds a compact orientable $4$-manifold, obtained from the $4$-ball by attaching $2$-handles along $L$.

## Explanations

### L1 human

Every closed orientable $3$-dimensional space can be built from the $3$-sphere by a finite number of drill-and-refill operations along circles.
The circles can even be taken unknotted, each refilled with a single twist.
So $3$-manifolds can be drawn as pictures of links with numbers attached.

### L2 human

The theorem is the starting point of the description of $3$-manifolds by framed links, and of [[Kirby calculus|AVQ9WN1MDV]], which says when two framed links give the same manifold.
Quantum invariants of $3$-manifolds, such as the Witten–Reshetikhin–Turaev invariants, are defined from link invariants through such surgery presentations.
Its $4$-dimensional reading, that every $3$-manifold bounds a $4$-manifold built with $2$-handles, connects it with the vanishing of the oriented cobordism group in dimension $3$.
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# Optimal truncation

## Summary

Optimal truncation is the rule of stopping a divergent [[asymptotic expansion|BJQTEPEZ9G]] just before its smallest term, for a fixed value of the variable.
For many series of factorial growth the error at this point is about the size of the smallest term.
This size is exponentially small in the large parameter.
So a divergent series can still give an approximation of very high accuracy.
The exponentially small error left after optimal truncation is the starting point of hyperasymptotics and of the theory of [[resurgent functions|37R4QJCARA]].

## Explanations

### L1 human

In a divergent asymptotic series the terms first get smaller and then start to grow again.
Adding terms helps while they shrink and hurts once they grow.
So the best one can do is to stop at the smallest term.
For example, for the series $1 - x + 2x^2 - 6x^3 + \cdots$ with terms $n!\, x^n$ and $x = 0.1$, the terms are smallest near $n = 10$, and stopping there gives an error of about $2 \cdot 10^{-4}$.
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# Perfect graph

## Summary

A perfect graph is a graph in which, for every induced subgraph, the [[chromatic number|0VHJRXNAAS]] equals the size of the largest [[clique|0BYX1D7C2E]].
For every graph the chromatic number is at least the [[clique number|04MQSQ1F5Q]], and perfect graphs are those where this lower bound is always exact.
Bipartite graphs, interval graphs and comparability graphs are perfect.
The [[strong perfect graph theorem|MFNNZQ484T]] characterizes perfect graphs by forbidden induced subgraphs.
On perfect graphs, colouring and finding a largest clique can be done in polynomial time.

## Explanations

### L1 human

To colour a graph you need at least as many colours as the largest group of mutually adjacent vertices.
In a perfect graph this is always enough, not just for the whole graph but for every piece of it you get by deleting vertices.
The $5$-cycle is the smallest graph that is not perfect: it has no triangle, so its largest clique has size $2$, but it needs $3$ colours.
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# Continuous additive functions are linear

## Statement

Let $f \colon \R \to \R$ be an [[additive function|KF719SZD5T]], that is, $f(x + y) = f(x) + f(y)$ for all $x, y \in \R$.
If $f$ is [[continuous|0X3WGX95FE]], then
$$f(x) = c x \quad \text{for all } x \in \R, \qquad \text{where } c = f(1).$$

## Remarks

Additivity gives $f(nx) = n f(x)$ for integers $n$, and then $f(q) = q f(1)$ for every rational $q = a/b$, since $b f(a/b) = f(a) = a f(1)$.
The rationals are [[dense|0BSJSX7ME2]] in $\R$, so every real $x$ is a limit of rationals $q_k$, and continuity gives $f(x) = \lim f(q_k) = \lim c q_k = c x$.
For example, if a continuous $f$ satisfies $f(x + y) = f(x) + f(y)$ and $f(1) = 2$, then $f(x) = 2x$, so $f(\pi) = 2\pi$.
Continuity cannot be dropped.
A basis of $\R$ as a vector space over $\Q$ exists by the axiom of choice, and with it one builds additive functions that are not of the form $cx$.
Weaker hypotheses suffice, such as continuity at a single point, monotonicity on an interval, or Lebesgue measurability.
The same argument shows that a continuous additive map between real normed spaces is $\R$-linear.

## Explanations

### L1 human

If a function turns sums into sums and has no jumps, it must be a straight line through the origin.
Additivity alone fixes the function on all fractions, and continuity fills in the remaining points.
This is why a rule like "the cost of two orders together is the sum of their costs", together with continuity, forces a fixed unit price.

### L2 human

This is the regular solution of [[Cauchy's functional equation|ZBMEVWN100]], and many other classical functional equations reduce to it.
For instance, a continuous $g \colon \R \to (0, \infty)$ with $g(x + y) = g(x) g(y)$ satisfies the hypothesis after taking logarithms, so $g(x) = a^x$ with $a = g(1)$.
In linear algebra terms, the theorem says that a continuous homomorphism of the group $(\R, +)$ to itself is automatically $\R$-linear.
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# Separably generated extension

## Summary

A separably generated extension is a field extension with a [[transcendence basis|E45PA267S4]] over which the whole extension is [[separable|0WCX7NB4N7]] algebraic.
Such a basis is a separating transcendence basis.
It extends the notion of a separable algebraic extension to transcendental extensions.
Every extension in characteristic $0$, and every finitely generated extension of a [[perfect field|VWBVKT794W]], is separably generated.

## Explanations

### L1 human

In characteristic $p$, adding a $p$-th root can give an inseparable extension, where an irreducible polynomial has a repeated root.
For example, over $\F_p(t)$ the polynomial $x^p - t$ is irreducible, but it equals $(x - t^{1/p})^p$ once $t^{1/p}$ is added.
A separably generated extension is one where one can choose free variables so that the rest is reached by separable steps only.
The function field $\F_p(x, y)$ with $y^2 = x^3 + 1$ and $p$ odd is separably generated over $\F_p$, with separating basis $\set{x}$.
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# Energy of a measure

## Definition

Let $n \ge 2$ and let $\mu$ be a finite positive [[Borel measure|DYR1QFNGNJ]] on $\R^n$ with compact [[support|FQTXV6BQZG]].
Let $k$ be the [[Newtonian kernel|6TJTWQ5HCR]] $k(x) = \abs{x}^{2-n}$ when $n \ge 3$, and the [[logarithmic kernel|VDZ5FKYC2T]] $k(x) = \log(1/\abs{x})$ when $n = 2$, with $k(0) = +\infty$.
The *energy* of $\mu$ is
$$I(\mu) = \iint k(x - y)\, d\mu(x)\, d\mu(y).$$
For $n \ge 3$ it lies in $[0, +\infty]$; for $n = 2$ it lies in $(-\infty, +\infty]$, because $k(x - y)$ is bounded below on the compact set $\operatorname{supp}\mu \times \operatorname{supp}\mu$.

## Remarks

Equivalently, $I(\mu) = \int U^\mu \, d\mu$, where $U^\mu(x) = \int k(x - y)\, d\mu(y)$ is the [[potential|JHKSF3XFY2]] of $\mu$.
Some authors include the normalising constant of the fundamental solution in $k$, which multiplies the energy by a constant.
The infimum of $I(\mu)$ over probability measures on a compact set $K$ determines the [[capacity|4YZVFGJVSZ]] of $K$.
For instance, when $n \ge 3$ and with this kernel, the Newtonian capacity of $K$ is the reciprocal of that infimum.

Example: let $\sigma$ be the uniform probability measure on the unit sphere in $\R^3$.
Its potential is $U^\sigma(x) = 1/\max(\abs{x}, 1)$, which equals $1$ on the sphere, so $I(\sigma) = 1$.
In the plane, the uniform probability measure on the unit circle has potential $\log(1/\max(\abs{z}, 1))$, which is $0$ on the circle, so its energy is $0$.

Non-example: the Dirac measure $\delta_0$ is a positive Borel measure with compact support, but $I(\delta_0) = k(0) = +\infty$, so it does not have finite energy.
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# Dedekind number

## Definition

Let $n \ge 0$ be an integer, and order $\set{0, 1}^n$ coordinatewise: $x \le y$ when $x_i \le y_i$ for every $i$.
A [[Boolean function|0X05GS525E]] $f \colon \set{0, 1}^n \to \set{0, 1}$ is *monotone* if $x \le y$ implies $f(x) \le f(y)$.
The *Dedekind number* $M(n)$ is the number of monotone Boolean functions $f \colon \set{0, 1}^n \to \set{0, 1}$.

## Remarks

Identify $\set{0, 1}^n$ with the subsets of $\set{1, \ldots, n}$.
A monotone $f$ is determined by the minimal sets $S$ with $f(S) = 1$, and these form an [[antichain|03HF498P1J]].
So $M(n)$ is also the number of antichains of subsets of $\set{1, \ldots, n}$, including the empty antichain.

Example: $M(2) = 6$.
The monotone functions of $x, y$ are $0$, $1$, $x$, $y$, $x \wedge y$ and $x \vee y$.
The matching antichains are $\emptyset$, $\set{\emptyset}$, $\set{\set{1}}$, $\set{\set{2}}$, $\set{\set{1, 2}}$ and $\set{\set{1}, \set{2}}$.

Non-example: the function $x \oplus y$, equal to $1$ exactly when $x \ne y$, is not monotone.
Indeed $(1, 0) \le (1, 1)$ but $f(1, 0) = 1 > 0 = f(1, 1)$.
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# Dedekind domain

## Definition

Let $R$ be an [[integral domain|0QVD7WBFGN]] that is not a field, and let $K$ be its [[field of fractions|0F2PTGX1A2]].
The ring $R$ is a *Dedekind domain* if:

1. $R$ is [[Noetherian|0670T59HGE]];
2. $R$ is integrally closed: every $x \in K$ that is a root of a [[monic polynomial|0C3QBCR46M]] with coefficients in $R$ lies in $R$;
3. every nonzero [[prime ideal|00MHHZ9SNE]] of $R$ is a [[maximal ideal|0PV74TR5FP]].

## Remarks

Example: $\Z$ is a Dedekind domain.
It is a principal ideal domain, hence Noetherian.
By the rational root test, a rational root of a monic integer polynomial is an integer.
Every nonzero prime ideal $(p)$ is maximal, since $\Z/(p)$ is a field.
More generally, the ring of integers of every [[number field|MWBDFS2KJ5]] is a Dedekind domain, for example $\Z[\sqrt{-5}]$, which is not a unique factorisation domain.
Non-example: $\Z[\sqrt{-3}]$ fails clause 2, since $\omega = (1 + \sqrt{-3})/2$ lies in its field of fractions, is a root of $x^2 - x + 1$, and is not in $\Z[\sqrt{-3}]$.
Another non-example: $k[x, y]$ for a field $k$ fails clause 3, since $(x)$ is a nonzero prime ideal contained in the larger ideal $(x, y)$.
For a Noetherian integral domain $R$ that is not a field, each of these conditions is equivalent to being Dedekind.
First: every nonzero proper ideal is a product of prime ideals, in a unique way up to order.
Second: every nonzero [[fractional ideal|YB4JTB4WJN]] of $R$ is invertible.
Some authors also count fields as Dedekind domains.
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<document source="https://tamnd.github.io/math2/node/1ZA98AGKGK.md" tag="1ZA98AGKGK" kind="concept">
# Regular chain

## Summary

A regular chain is a triangular set of polynomials, with one polynomial for each of some of the variables.
In it, the leading coefficient of each polynomial is not a zero divisor modulo the ideal described by the earlier polynomials.
Regular chains describe the solution sets of polynomial systems in a triangular form that can be solved one variable at a time.
Every polynomial system can be decomposed into finitely many regular chains, and this gives an alternative to [[Gröbner bases|C83KPT47R0]] in computer algebra.

## Explanations

### L1 human

A triangular system first constrains $x$, then $y$ in terms of $x$, then $z$ in terms of $x$ and $y$, and so on.
To solve it, one solves for $x$, substitutes, solves for $y$, and continues.
This fails if the coefficient of the highest power of $y$ vanishes for some allowed values of $x$.
A regular chain is a triangular system in which this never happens in a harmful way.
Each leading coefficient is invertible, or at least not a zero divisor, given the earlier equations.
For example, $x^2 - 1 = 0$, $xy - 1 = 0$ is a regular chain, because $x$ is nonzero whenever $x^2 = 1$.
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<document source="https://tamnd.github.io/math2/node/1ZAF1CC8M5.md" tag="1ZAF1CC8M5" kind="definition">
# Topological group structure of a vector space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a [[topological vector space|T32W553W95]] over $\mathbb{K}$.
The *[[topological group|92700QAB5Y]] structure* of $E$ is the additive group $(E, +)$ together with the topology of $E$.
It is a commutative topological group: the maps $(x, y) \mapsto x + y$ from $E \times E$ to $E$ and $x \mapsto -x$ from $E$ to $E$ are continuous.

## Remarks

The map $x \mapsto -x$ is continuous because it is scalar multiplication by $-1$.
For each $a \in E$ the translation $x \mapsto x + a$ is a homeomorphism of $E$, with inverse $x \mapsto x - a$.
So a set $U$ is a neighbourhood of $a$ if and only if $U - a$ is a neighbourhood of $0$.
Example: $\R^n$ with its Euclidean topology is a commutative topological group under addition, since addition and negation are continuous functions of the coordinates.
Non-example: give $\R$ the topology whose open sets are $\emptyset$, $\R$ and the intervals $(a, \infty)$.
This is not a topological group, because $x \mapsto -x$ is not continuous: the preimage of $(0, \infty)$ is $(-\infty, 0)$, which is not open.
The topological group structure gives a uniform structure on $E$; a net $(x_i)$ is Cauchy when $x_i - x_j$ tends to $0$.
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<document source="https://tamnd.github.io/math2/node/1ZB13N78XW.md" tag="1ZB13N78XW" kind="concept">
# Characteristic of a field

## Summary

The characteristic of a field is the least positive number of copies of $1$ that add up to $0$.
It is $0$ when no such number exists.
It is the [[characteristic of the field as a ring|0F6ES3BRYP]], and for a field it is always $0$ or a prime number.
It decides the [[prime subfield|0TNS65QB87]], and many results of field theory depend on it.
For example, the map $a \mapsto a^p$ is a ring homomorphism in characteristic $p$.

## Explanations

### L1 human

Start at $0$ and add $1$ again and again.
In the rational or real numbers you never come back to $0$, so the characteristic is $0$.
In the field $\F_7$ of integers modulo $7$, seven steps bring you back to $0$, so the characteristic is $7$.
A field can never have characteristic $6$, because $2 \cdot 3 = 0$ would make two nonzero elements multiply to $0$.
The [[definition|5ZXRZB41HE]] states this precisely.
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<document source="https://tamnd.github.io/math2/node/1ZBX4TKVZ9.md" tag="1ZBX4TKVZ9" kind="concept">
# Order (ring theory)

## Summary

An order is a [[subring|0GNCRPYXYV]] of a finite-dimensional algebra over a field.
It is a finitely generated module over a ring of integers such as $\Z$, and it spans the algebra over the field.
The basic examples are the subrings of an [[algebraic number field|MWBDFS2KJ5]] that are free of full rank over $\Z$, such as $\Z[\sqrt{-3}]$ in $\Q(\sqrt{-3})$.
Orders in noncommutative algebras include $M_n(\Z)$ in $M_n(\Q)$ and the integral group ring $\Z G$ in $\Q G$.
They are central in integral representation theory and in the arithmetic of quaternion algebras.
The largest orders, the [[maximal orders|E5DH1CSW1N]], generalize rings of integers.

## Explanations

### L1 human

The integers sit inside the rationals as a grid of points that still spans everything once fractions are allowed.
An order is the same idea for a bigger algebra: a subring that is a grid over $\Z$ and whose rational multiples fill the whole algebra.
The Gaussian integers $\Z[i]$ form an order in $\Q(i)$, with grid basis $1, i$.
The integer matrices form an order in the rational matrices.
Orders let one do number theory, with divisibility and ideals, inside algebras that are not fields.
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<document source="https://tamnd.github.io/math2/node/1ZBY4W4E81.md" tag="1ZBY4W4E81" kind="concept">
# Standard part function

## Summary

The standard part function sends each finite [[hyperreal number|421QMTFG9T]] $x$ to the unique [[real number|08J2M1JYAG]] $\operatorname{st}(x)$ that is infinitely close to it.
It is the bridge back from the hyperreals to the reals in [[nonstandard analysis|WM62GKCBWN]].
Derivatives, integrals and limits are defined as standard parts of hyperreal expressions.

## Explanations

### L1 human

A finite hyperreal number is a real number plus an infinitely small error.
The standard part throws away the error and keeps the real number.
For example, if $\varepsilon$ is a positive infinitesimal, then $\operatorname{st}(3 + \varepsilon) = 3$ and $\operatorname{st}(\varepsilon^2 - 5\varepsilon) = 0$.
An infinite hyperreal such as $1/\varepsilon$ has no standard part, because no real number is close to it.
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<document source="https://tamnd.github.io/math2/node/1ZC0ZW96YQ.md" tag="1ZC0ZW96YQ" kind="definition">
# Shelf

## Definition

A *shelf* is a [[magma|9689SFPVE9]] $(X, \triangleright)$ whose operation is [[left self-distributive|VEFE5WZMS3]]:
$$x \triangleright (y \triangleright z) = (x \triangleright y) \triangleright (x \triangleright z) \quad \text{for all } x, y, z \in X.$$

## Remarks

Equivalently, for every $x \in X$ the left translation $y \mapsto x \triangleright y$ is an endomorphism of $(X, \triangleright)$.
A shelf is a [[rack|X74VY86TSX]] when every left translation is a bijection.

Example: let $X = \set{1, 2}$ with $1 \triangleright y = 2$ and $2 \triangleright y = y$ for all $y$; this is the Laver table $A_1$.
If $x = 2$, both sides equal $y \triangleright z$.
If $x = 1$, the left side is $2$, and the right side is $2 \triangleright 2 = 2$.
So $X$ is a shelf; it is not a rack, since $y \mapsto 1 \triangleright y$ is not injective.

Non-example: $(\Z, +)$, since $x + (y + z) \ne (x + y) + (x + z)$ for $x \ne 0$.

Some authors use right self-distributivity instead, with $(z \triangleleft y) \triangleleft x = (z \triangleleft x) \triangleleft (y \triangleleft x)$; the two conventions are related by $x \triangleright y = y \triangleleft x$.
In Mathlib this structure is the class `Shelf`.
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<document source="https://tamnd.github.io/math2/node/1ZDAGGGRP4.md" tag="1ZDAGGGRP4" kind="definition">
# Dual cone

## Definition

Let $n \ge 1$ and let $C \subseteq \R^n$ be any subset.
The *dual cone* of $C$ is the set
$$C^* = \set{y \in \R^n : x \cdot y \ge 0 \text{ for all } x \in C},$$
where $x \cdot y$ is the [[dot product|0S52TNNTH9]].

## Remarks

The set $C^*$ is an intersection of closed half-spaces through $0$, so it is a closed [[convex cone|6VE4KA12VD]] that contains $0$.
If $C$ is a closed convex cone, then $(C^*)^* = C$.

Example: for $C = \R^n_{\ge 0}$, one has $C^* = \R^n_{\ge 0}$.
If $y$ has nonnegative coordinates, then $x \cdot y \ge 0$ for each $x \in C$; conversely, taking $x = e_i$ gives $y_i \ge 0$.
A cone with $C^* = C$ is called self-dual; the cone of positive semidefinite matrices with the trace inner product is another self-dual cone.

Non-example: the vector $y = (1, -1)$ is not in the dual of $\R^2_{\ge 0}$, since $(0, 1) \cdot y = -1 < 0$.

The *polar cone* is $C^\circ = -C^* = \set{y : x \cdot y \le 0 \text{ for all } x \in C}$.
In a general real inner product space, or with a pairing between a space and its dual, the same formula defines the dual cone.
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# Milnor number

## Summary

The Milnor number $\mu$ of a holomorphic germ $f$ at a critical point is the dimension of its local algebra, see the [[definition|HZCH4KNGY2]].
The local algebra is the quotient of the power series ring by the partial derivatives of $f$.
It is finite exactly when $f$ has an [[isolated singularity|VTGYEES3VS]].
It measures how many simple critical points are merged at the point: a small generic perturbation of $f$ has exactly $\mu$ nondegenerate critical points nearby.
Topologically, it is the number of spheres in the [[Milnor fibre|X5HMTTGB62]] of $f$.

## Explanations

### L1 human

A function like $z^2$ has a simple critical point at $0$, while $z^3$ has a worse one.
If you perturb $z^3$ to $z^3 - tz$ with small $t \ne 0$, the critical point splits into two simple ones, at $z = \pm \sqrt{t/3}$.
The Milnor number counts this: it is $1$ for $z^2$ and $2$ for $z^3$.
In general it is computed by pure algebra, as the dimension of a vector space of power series.
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<document source="https://tamnd.github.io/math2/node/1ZPMDE4ARK.md" tag="1ZPMDE4ARK" kind="concept">
# Split exact sequence

## Summary

A split exact sequence is a [[short exact sequence|KJKNJMGKZZ]] $0 \to A \to B \to C \to 0$ in which the middle object is the [[biproduct|BZEJKTQ76T]] $A \oplus C$, compatibly with the two maps.
These are the trivial extensions: $B$ is just $A$ and $C$ placed side by side.
By the [[splitting lemma|7ZGQKWYKQS]], a short exact sequence splits as soon as one of its two maps has a one-sided inverse.

## Explanations

### L1 human

Every short exact sequence builds $B$ from a piece $A$ and a quotient $C$.
In a split sequence the gluing is trivial, and $B$ is simply the direct sum of $A$ and $C$.
For example, $0 \to \Z \to \Z \oplus \Z/2\Z \to \Z/2\Z \to 0$ is split.
The sequence $0 \to \Z \xrightarrow{2} \Z \to \Z/2\Z \to 0$ is not, since $\Z$ is not $\Z \oplus \Z/2\Z$.
Over a field every short exact sequence of vector spaces splits, which is why linear algebra has no interesting extensions.
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<document source="https://tamnd.github.io/math2/node/1ZSTQGCCCQ.md" tag="1ZSTQGCCCQ" kind="definition">
# Sequential analysis

## Definition

Let $X_1, X_2, \dots$ be [[random variables|0DKRR22AEQ]] observed one at a time, and let $\mathcal{F}_n = \sigma(X_1, \dots, X_n)$ be the information after $n$ observations, with $\mathcal{F}_0$ trivial.
Let $D$ be a set of possible decisions, such as "accept $H_0$" and "reject $H_0$".
A *sequential procedure* is a pair $(N, \delta)$ such that:

1. $N$ is a [[stopping time|WND87FBAKC]] with respect to $(\mathcal{F}_n)_{n \ge 0}$, called the stopping rule.
2. $\delta$ is a terminal decision rule: on the event $\set{N = n}$, the decision $\delta$ is a function of $(X_1, \dots, X_n)$ with values in $D$.

*Sequential analysis* is the study of sequential procedures, in particular of their error probabilities and of the distribution of the sample size $N$.

## Remarks

Example: in a [[test|087P03NEND]] about the mean of $X_i \in \set{0, 1}$, the rule "stop at the first $n$ with $\abs{\sum_{i \le n} (2X_i - 1)} = 5$, and reject $H_0$ if the sum is positive" is a sequential procedure.
The event $\set{N = n}$ depends only on $X_1, \dots, X_n$, so $N$ is a stopping time.
Non-example: "stop at the $n$ for which $X_n$ is the largest of the first $100$ observations" is not a sequential procedure.
Deciding whether $N = n$ needs future observations, so clause 1 fails.
A procedure with $N = m$ constant is the usual fixed-sample procedure, so fixed-sample statistics is a special case.
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# Radon–Nikodym theorem

## Statement

Let $(X, \Sigma)$ be a measurable space and let $\mu$ and $\nu$ be [[σ-finite measures|WEBBT18D4X]] on $\Sigma$.
Assume that $\nu$ is [[absolutely continuous|P0P21KDV6E]] with respect to $\mu$, that is, $\mu(A) = 0$ implies $\nu(A) = 0$ for $A \in \Sigma$.
Then there is a [[measurable function|YYNCJSR0PJ]] $f \colon X \to [0, \infty)$ such that
$$\nu(A) = \int_A f \, d\mu \quad \text{for all } A \in \Sigma.$$
The function $f$ is unique up to equality $\mu$-almost everywhere.

## Remarks

A standard proof first treats finite measures.
Consider the measurable $g \ge 0$ with $\int_A g \, d\mu \le \nu(A)$ for all $A$, and take one that maximises $\int_X g \, d\mu$.
If the leftover measure $\nu - g \mu$ were not zero, a Hahn decomposition of $\nu - g\mu - \varepsilon \mu$ would allow $g$ to be increased, a contradiction.
Von Neumann gave another proof through the Riesz representation of a bounded linear functional on $L^2(\mu + \nu)$.
The σ-finite case follows by cutting $X$ into countably many pieces of finite measure.

Example: if $\nu$ is the standard normal distribution on $\R$ and $\mu$ is Lebesgue measure, then $f(x) = e^{-x^2/2} / \sqrt{2\pi}$.

The hypothesis of σ-finiteness of $\mu$ cannot be dropped.
Let $\mu$ be counting measure on the Borel sets of $[0, 1]$ and $\nu$ Lebesgue measure.
Then $\nu \ll \mu$, because only the empty set has $\mu$-measure $0$.
A density $f$ would satisfy $f(x) = \nu(\{x\}) = 0$ for every $x$, which gives $\nu = 0$, a contradiction.

The function $f$ is the [[measure with density|Y98C5T956Q]] description of $\nu$: $\nu = f \mu$.

## Explanations

### L1 human

If one measure never sees mass where another sees none, then the first is the second one weighted by a density function.
This is how a probability density arises from a distribution that has no point masses and no mass on sets of length zero.
The density plays the role of a derivative of one measure with respect to the other.

### L2 human

The theorem is the source of the Radon–Nikodym derivative $d\nu / d\mu$.
In probability it gives conditional expectation: the conditional expectation of an integrable variable given a sub-σ-algebra is a Radon–Nikodym derivative.
It also identifies the dual of $L^p(\mu)$ with $L^q(\mu)$ for $1 \le p < \infty$ and σ-finite $\mu$.
Together with the Lebesgue decomposition it describes every σ-finite measure relative to a fixed one.
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# Locally path-connected space

## Summary

A locally path-connected space is a [[topological space|0CQHFPGJ45]] in which every point has arbitrarily small [[path-connected|0WB6DH499T]] open neighbourhoods.
Every locally path-connected space is [[locally connected|FDVAHRJKF3]].
In a locally path-connected space, the path components and the connected components coincide.
So a connected, locally path-connected space is path-connected, a fact used throughout covering space theory.

## Explanations

### L1 human

Near any point of such a space, a small open piece can be chosen in which every two points are joined by a path inside the piece.
Open subsets of $\R^n$ have this property, because small balls are convex.
The [[comb space|A2Q4RXD162]] fails it: near a point on the left edge high up, nearby points on the teeth cannot be joined by short paths.
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<document source="https://tamnd.github.io/math2/node/20967BXZKA.md" tag="20967BXZKA" kind="concept">
# Clarke generalized gradient

## Summary

The Clarke generalized gradient is a set of generalized derivatives of a locally Lipschitz function at a point.
It extends both the gradient of a continuously differentiable function and the [[subdifferential|9C5WXYYQEK]] of a convex function.
It is always a nonempty, convex and weak-* compact set, and it contains $0$ at every local minimum and every local maximum.
Frank Clarke introduced it as the basis of nonsmooth analysis and of optimality conditions for problems without smoothness.
The [[definition|NCWYTR6PH0]] gives the precise construction.

## Explanations

### L1 human

A function like $\abs{x}$ has a corner, so at $0$ it has no derivative.
Near the corner its slopes are $-1$ on the left and $1$ on the right.
The Clarke generalized gradient collects all slopes that appear as limits near the point and fills in the gaps between them.
For $\abs{x}$ at $0$ this gives the interval $[-1, 1]$.
It gives the same interval for $-\abs{x}$, which has a peak at $0$, so it does not distinguish minima from maxima.
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# Berger's classification of holonomy groups

## Statement

Let $(M, g)$ be a simply connected Riemannian manifold of dimension $n$ with [[Levi-Civita connection|CAZYGE1T4E]] $\nabla$, and let $p \in M$.
Assume that the [[holonomy group|WHDZ3MK56B]] $\mathrm{Hol}_p(\nabla)$ acts irreducibly on $T_pM$, and that $(M, g)$ is not locally [[symmetric|PAA93R4WYD]], that is, $\nabla R \ne 0$ for its curvature tensor $R$.
Then $\mathrm{Hol}_p(\nabla)$, as a subgroup of $\mathrm{O}(T_pM) \cong \mathrm{O}(n)$, is conjugate to one of the following:

1. $\mathrm{SO}(n)$;
2. $\mathrm{U}(m)$, with $n = 2m \ge 4$;
3. $\mathrm{SU}(m)$, with $n = 2m \ge 4$;
4. $\mathrm{Sp}(k) \cdot \mathrm{Sp}(1)$, with $n = 4k \ge 8$;
5. $\mathrm{Sp}(k)$, with $n = 4k \ge 8$;
6. $G_2$, with $n = 7$;
7. $\mathrm{Spin}(7)$, with $n = 8$.

## Remarks

Berger's proof combines the Ambrose–Singer theorem with the Bianchi identity: the holonomy algebra must be spanned by curvature-type tensors.
Berger checked which irreducible subgroups of $\mathrm{SO}(n)$ admit enough of them while allowing $\nabla R \ne 0$.
Berger's original list also contained $\mathrm{Spin}(9)$ in dimension $16$; later work showed that such a metric is locally symmetric, so this case drops out.
All seven groups do occur; the existence of metrics with holonomy $G_2$ and $\mathrm{Spin}(7)$ was shown much later, first locally by Bryant and then on compact manifolds by Joyce.
The hypotheses are needed.
By the de Rham decomposition theorem, a reducible holonomy splits $M$ locally as a product.
A locally symmetric space can have the holonomy of its isotropy group, which is not in this list in general.
The geometric meaning: $\mathrm{U}(m)$ gives Kähler manifolds, $\mathrm{SU}(m)$ Ricci-flat Kähler (Calabi–Yau) manifolds, $\mathrm{Sp}(k)$ hyperkähler manifolds, and $\mathrm{Sp}(k)\cdot\mathrm{Sp}(1)$ quaternion-Kähler manifolds.

## Explanations

### L1 human

Every curved space has a holonomy group, which records how directions turn when carried around loops.
For a typical space this is the whole rotation group.
Berger's theorem says that, apart from products and highly symmetric spaces, only seven kinds of holonomy group can occur.
Each smaller group on the list comes with a special kind of geometry.

### L2 human

The theorem organises much of modern Riemannian geometry: each group on the list defines a class of manifolds with special holonomy and parallel tensors.
For example holonomy in $\mathrm{U}(m)$ means there is a parallel complex structure, which is the Kähler condition.
The groups $\mathrm{SU}(m)$, $\mathrm{Sp}(k)$, $G_2$ and $\mathrm{Spin}(7)$ force Ricci-flatness, and these are the geometries used in string theory compactifications.
Together with the de Rham decomposition and Cartan's classification of symmetric spaces, it describes the holonomy of every simply connected complete Riemannian manifold.
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# Self-similar measure

## Definition

Let $n \ge 1$ and $N \ge 2$.
For $i = 1, \dots, N$ let $S_i \colon \R^n \to \R^n$ be a similarity with ratio $r_i \in (0, 1)$, that is, $\abs{S_i(x) - S_i(y)} = r_i \abs{x - y}$ for all $x, y$.
Let $p_1, \dots, p_N > 0$ with $p_1 + \cdots + p_N = 1$.
A Borel [[probability measure|059TJN0NZ1]] $\mu$ on $\R^n$ is *self-similar* for $(S_i, p_i)$ if
$$\mu = \sum_{i=1}^{N} p_i \, (S_i)_* \mu,$$
where $(S_i)_* \mu = \mu \circ S_i^{-1}$ is the [[pushforward measure|3N66TM3TSQ]].

## Remarks

Each $S_i$ is a [[contraction|0QN69YA357]].
Hutchinson's theorem says that there is exactly one compactly supported such $\mu$.
It is the fixed point of a contraction on a space of measures, and its support is the attractor of $(S_i)$.
Example: on $\R$ let $S_1(x) = x/2$, $S_2(x) = x/2 + 1/2$ and $p_1 = p_2 = 1/2$, and let $\lambda$ be Lebesgue measure on $[0,1]$.
The measure $(S_1)_* \lambda$ is uniform on $[0, 1/2]$ with density $2$, and $(S_2)_* \lambda$ is uniform on $[1/2, 1]$ with density $2$.
Their average has density $1$ on $[0,1]$, so it equals $\lambda$, and $\lambda$ is self-similar.
Non-example: with the same maps and $p_1 = 1/3$, $p_2 = 2/3$, the average has density $2/3$ on $[0, 1/2]$, so $\lambda$ is not self-similar for these weights.
The Cantor measure is self-similar for $S_1(x) = x/3$, $S_2(x) = x/3 + 2/3$ and $p_1 = p_2 = 1/2$.
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# Artinian ring

## Summary

An Artinian ring is a ring in which every decreasing chain of ideals stops after finitely many steps.
For commutative rings, Artinian rings are exactly the [[Noetherian rings|0670T59HGE]] in which every prime ideal is maximal, by the [[Akizuki–Hopkins–Levitzki theorem|C7X657TF19]].
By the [[structure theorem for Artinian rings|HE7MP4HP6Q]], a commutative Artinian ring is a finite product of Artinian local rings.
Finite rings, fields and finite-dimensional algebras over a field are Artinian.

## Explanations

### L1 human

In an Artinian ring you cannot keep shrinking ideals forever: every decreasing chain becomes constant.
Such rings are small in a precise sense, like finite-dimensional algebras.
For example, $\Z/12\Z$ has only six ideals, so any decreasing chain of ideals must stop.
The ring $\Z$ is not Artinian, because $(2) \supset (4) \supset (8) \supset \cdots$ never stops.
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# Shell theorem

## Statement

Let $R > 0$ and $M > 0$.
Let $\mu$ be the measure on $\R^3$ of total mass $M$ spread uniformly over the sphere $S_R = \set{y : \abs{y} = R}$, that is, $\mu = \frac{M}{4\pi R^2} \, dS$ on $S_R$.
Let $U^\mu(x) = \int \frac{d\mu(y)}{\abs{x - y}}$ be its [[Newtonian potential|DCMFQ01QGS]].
Then
$$U^\mu(x) = \frac{M}{\abs{x}} \quad (\abs{x} \ge R), \qquad U^\mu(x) = \frac{M}{R} \quad (\abs{x} \le R).$$
Consequently the gravitational force of the shell on a point mass is $0$ inside the shell.
Outside the shell it equals the force of a point mass $M$ at the centre.

## Remarks

Proof idea: $U^\mu(x)$ is $M$ times the average of the function $y \mapsto 1/\abs{x - y}$ over the sphere $S_R$.
If $\abs{x} > R$, this function is harmonic on a neighbourhood of the closed ball of radius $R$, and the [[mean value property|0HYPR64JM2]] gives the value $1/\abs{x}$ at the centre.
If $\abs{x} < R$, fix $y_0 \in S_R$.
By rotation symmetry, the average of $1/\abs{x - y}$ over $y \in S_R$ equals the average of $1/\abs{x' - y_0}$ over $x'$ in the sphere of radius $\abs{x}$.
That function of $x'$ is harmonic on the ball of radius $R$, so the average is its value $1/\abs{y_0} = 1/R$ at the centre.
The value $\abs{x} = R$ follows by continuity of both formulas, or by [[Gauss's law|T2CYAYF0MT]] applied to spheres about the origin.
Since a ball with radial density is a union of concentric uniform shells, the potential of such a ball of mass $M$ is $M / \abs{x}$ outside the ball as well.

## Explanations

### L1 human

Newton proved that a hollow planet with a uniform crust pulls on outside objects as if all its mass sat at its centre.
Inside the hollow there is no pull at all: the near part of the crust pulls harder, but the far part is larger, and the two effects cancel exactly.
This is why one may treat the Sun and the planets as point masses.

### L2 human

The theorem is the first example of how potential theory turns symmetry into exact formulas.
It says that the potential of the shell is constant inside, so it is harmonic there, and that outside it is a multiple of the Newtonian kernel.
In $\R^n$ with $n \ge 3$ the same argument gives $M \abs{x}^{2-n}$ outside and $M R^{2-n}$ inside, and in the plane the logarithmic potential of a uniform circle behaves in the same way.
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# Locally compact group

## Definition

A *locally compact group* is a [[topological group|92700QAB5Y]] $G$ such that:

1. the topology of $G$ is [[Hausdorff|0MXB4CKHFW]];
2. the topology of $G$ is [[locally compact|0EQMWQ4WMJ]], that is, every point of $G$ has a compact neighbourhood.

## Remarks

Example: $(\R^n, +)$ is a locally compact group, since the closed ball of radius $1$ around $x$ is a compact neighbourhood of $x$ by the Heine–Borel theorem.
Every discrete group is locally compact, since $\set{x}$ is a compact neighbourhood of $x$.

Non-example: the additive group of an infinite-dimensional Hilbert space $H$ is a Hausdorff topological group but is not locally compact.
Suppose $0$ had a compact neighbourhood $K$.
Then $K$ contains a closed ball of some radius $r > 0$, and this ball is compact as a closed subset of $K$.
For an orthonormal sequence $(e_k)$, the points $r e_k$ lie in this ball.
They have no convergent subsequence, a contradiction, since $\norm{r e_j - r e_k} = r\sqrt{2}$ for $j \ne k$.

By homogeneity it is enough that the identity has a compact neighbourhood.
Some authors omit the Hausdorff condition; a locally compact group in that wider sense becomes Hausdorff after dividing by the closure of the identity.
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<document source="https://tamnd.github.io/math2/node/211PJYW8W6.md" tag="211PJYW8W6" kind="statement">
# Paley–Zygmund inequality

## Statement

Let $Z \ge 0$ be a real [[random variable|0PNRKMRC55]] with $0 < E[Z^2] < \infty$, and let $\theta \in [0, 1]$.
Then
$$P(Z > \theta \, E[Z]) \ge (1 - \theta)^2 \frac{E[Z]^2}{E[Z^2]}.$$

## Remarks

Proof: split $E[Z] = E[Z \mathbf{1}_{\set{Z \le \theta E[Z]}}] + E[Z \mathbf{1}_{\set{Z > \theta E[Z]}}]$.
The first term is at most $\theta E[Z]$, and by the [[Cauchy–Schwarz inequality|0RFWW56Y22]] the second is at most $\sqrt{E[Z^2] \, P(Z > \theta E[Z])}$.
Rearranging gives the bound.
With $\theta = 0$ it becomes the second moment method: $P(Z > 0) \ge E[Z]^2 / E[Z^2]$.
Example: if $Z$ counts the triangles in an [[Erdős–Rényi random graph|6E83YKANV3]] and $E[Z^2] / E[Z]^2 \to 1$, then $P(Z > 0) \to 1$, so a triangle exists with high probability.
The bound is a lower bound on the probability of being large, while [[Markov's inequality|05PWWHYBX5]] and [[Chebyshev's inequality|02HZ1BK3MW]] give upper bounds.
The hypothesis $Z \ge 0$ is needed: the constant $Z = -1$ has $E[Z]^2 / E[Z^2] = 1$, but with $\theta = 0$ the event $\set{Z > 0}$ is empty.

## Explanations

### L1 human

Take a nonnegative quantity whose second moment is not much larger than the square of its mean.
Then the quantity cannot be small most of the time.
It must exceed a fixed fraction of its mean with a probability bounded away from $0$.

### L2 human

The inequality is the basis of the second moment method in combinatorics and random graphs, where one shows that a count $Z$ is positive with high probability.
In analysis, Paley and Zygmund used it for random series, to show that a random trigonometric series is large with positive probability.
It complements the first moment method, which uses $P(Z > 0) \le E[Z]$ for integer-valued $Z \ge 0$.
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<document source="https://tamnd.github.io/math2/node/212MDDPN8A.md" tag="212MDDPN8A" kind="definition">
# Homotopy groups of spheres

## Definition

Let $n \ge 1$, and let $S^n = \set{x \in \R^{n+1} : \norm{x} = 1}$ be the unit $n$-sphere with base point $s_0 = (1, 0, \ldots, 0)$.
For $k \ge 1$, the group $\pi_k(S^n)$ is the [[homotopy group|P976WYEFMB]] $\pi_k(S^n, s_0)$.
The *homotopy groups of spheres* are the groups $\pi_k(S^n)$ for all $k, n \ge 1$.

## Remarks

Changing the base point gives isomorphic groups, because $S^n$ is path-connected; this is why it is usually omitted.
Example: $\pi_1(S^1) \cong \Z$, and the class of a loop is its winding number.
Example: $\pi_k(S^n) = 0$ for $1 \le k < n$.
Give $S^n$ the CW structure with one $0$-cell and one $n$-cell.
By cellular approximation, a based map $S^k \to S^n$ is homotopic to a map into the $k$-skeleton, which is a point.

Non-example: the groups are not the homology groups of spheres.
The homology $H_3(S^2)$ is zero, but $\pi_3(S^2) \cong \Z$, generated by the Hopf map.

Further values: $\pi_n(S^n) \cong \Z$ for all $n \ge 1$, $\pi_k(S^1) = 0$ for $k \ge 2$, and $\pi_{n+1}(S^n) \cong \Z/2$ for $n \ge 3$.
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<document source="https://tamnd.github.io/math2/node/212QCPWWA9.md" tag="212QCPWWA9" kind="definition">
# Crossing number

## Definition

Let $K$ be a tame [[knot|QYEAZTYGPT]] in $\R^3$.
A *diagram* of $K$ is a projection of $K$ to a plane that is injective except at finitely many transverse double points, called *crossings*.
At each crossing the diagram records which strand passes over.
The *crossing number* $c(K)$ is the least number of crossings of a diagram of a knot equivalent to $K$.

## Remarks

Since the minimum is taken over all knots equivalent to $K$, the number $c(K)$ is a [[knot invariant|GZQ5ZE3X6G]].
Example: $c(\text{unknot}) = 0$, since the round circle has a diagram with no crossings.
For the trefoil, $c = 3$: it has a diagram with $3$ crossings, and every diagram with at most $2$ crossings represents the unknot, which is not equivalent to the trefoil.
Non-example: a diagram of the unknot with one twist has $1$ crossing, but $1$ is not the crossing number of the unknot, because the minimum is $0$.
For a reduced alternating diagram, the number of crossings equals the crossing number; this was one of the Tait conjectures, proved with the Jones polynomial.
The same definition applies to links.
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<document source="https://tamnd.github.io/math2/node/218WAD58W3.md" tag="218WAD58W3" kind="definition">
# Grothendieck's Galois theory

## Definition

Let $k$ be a [[field|0QA8CHH8MG]] and let $k^s$ be a [[separable closure|JJ91A1V8NQ]] of $k$.
Let $G = \Gal(k^s/k)$ be the [[absolute Galois group|PGKWBJDTKX]] with its [[Krull topology|F3GTF5NT4M]].
A *finite étale $k$-algebra* is a $k$-algebra isomorphic to a finite product $L_1 \times \cdots \times L_r$.
Here each $L_i$ is a finite [[separable extension|0N019F6RE9]] of $k$.
For such an algebra $A$, let $F(A) = \Hom_k(A, k^s)$ be the set of $k$-algebra homomorphisms $A \to k^s$, with the [[action|0YSE802YCQ]] $g \cdot \varphi = g \circ \varphi$ of $G$.
Let $\mathcal{E}_k$ be the category of finite étale $k$-algebras, and let $\mathcal{S}_G$ be the category of finite sets with a continuous action of $G$.
*Grothendieck's Galois theory* for $k$ is the statement that $F \colon \mathcal{E}_k \to \mathcal{S}_G$ is an anti-equivalence of categories.

## Remarks

Under $F$, the dimension $\dim_k A$ equals the number of elements of $F(A)$.
A field $L$ goes to a transitive $G$-set, and a product of algebras goes to a disjoint union of $G$-sets.
For $k = \R$, the group $G$ has order $2$, and $F(\C)$ is a set of two points swapped by $G$, while $F(\R \times \R)$ is a set of two fixed points.
The algebra $k[x]/(x^2)$ is not finite étale, because it has a nonzero nilpotent element, and no product of fields has one.
The general form uses a Galois category with a fibre functor; for a connected scheme it defines the étale fundamental group, and for $\operatorname{Spec} k$ this group is $G$.
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<document source="https://tamnd.github.io/math2/node/21B8AHPD4P.md" tag="21B8AHPD4P" kind="concept">
# Devaney chaos

## Summary

Devaney chaos is a precise definition of a chaotic map, proposed by Robert Devaney in his textbook on chaotic dynamical systems.
A continuous map of a metric space is chaotic in this sense when three things hold.
It is [[topologically transitive|FR49Y9GQHR]], its periodic points are dense, and it has sensitive dependence on initial conditions.
It turns the informal ideas of [[chaos theory|DBKXCGDKTD]] into conditions that can be checked.
For infinite spaces the third condition follows from the first two.

## Explanations

### L1 human

A chaotic system mixes regular and irregular behaviour in a specific way.
It stirs every region into every other region, so it cannot be split into separate parts.
Regular, repeating orbits are found arbitrarily close to every point.
Yet any tiny change in the starting point can lead to a large difference later, which is the butterfly effect.
The doubling map $x \mapsto 2x \bmod 1$ of the circle has all three features.
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<document source="https://tamnd.github.io/math2/node/21BZEVVT7G.md" tag="21BZEVVT7G" kind="definition">
# Elementary substructure

## Definition

Let $L$ be a signature and let $\mathcal{M}$ and $\mathcal{N}$ be $L$-[[structures|983GZBQYT4]] with domains $M$ and $N$.
The structure $\mathcal{M}$ is an elementary substructure of $\mathcal{N}$, written $\mathcal{M} \preceq \mathcal{N}$, when the following hold.

1. $\mathcal{M}$ is a [[substructure|676H7EGTKJ]] of $\mathcal{N}$.
2. For every $L$-[[formula|HYJ10AF6RT]] $\varphi(x_1, \dots, x_n)$ and all $a_1, \dots, a_n \in M$, $\mathcal{M} \models \varphi(a_1, \dots, a_n)$ if and only if $\mathcal{N} \models \varphi(a_1, \dots, a_n)$.

Then $\mathcal{N}$ is an elementary extension of $\mathcal{M}$.

## Remarks

Example: $(\Q, <) \preceq (\R, <)$.
It follows from quantifier elimination for dense linear orders without endpoints: each formula is equivalent in these orders to a quantifier-free one.
A quantifier-free formula has the same truth value in a structure and in its substructures.

Non-example: let $P = \set{1, 2, 3, \dots}$.
Then $(P, <)$ is a substructure of $(\N, <)$, but not an elementary one.
For $\varphi(x) = \exists y \, (y < x)$, $(P, <) \models \neg \varphi(1)$ and $(\N, <) \models \varphi(1)$.

Taking $n = 0$ in clause 2 shows that $\mathcal{M} \preceq \mathcal{N}$ implies $\mathcal{M} \equiv \mathcal{N}$.
The non-example shows that the converse fails, since $(P, <)$ and $(\N, <)$ are isomorphic.
The [[Tarski–Vaught test|60SA99BER7]] gives a practical criterion.

## Explanations

### L1 human

In an elementary substructure, every statement about some of its elements is true there exactly when it is true in the bigger structure.
This is stronger than having the same sentences, because the statements may mention elements.
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<document source="https://tamnd.github.io/math2/node/21DD2334NR.md" tag="21DD2334NR" kind="concept">
# Tricomi equation

## Summary

The Tricomi equation, or Euler–Tricomi equation, is the linear second-order equation $y u_{xx} + u_{yy} = 0$ for a function $u(x, y)$ in the plane.
It is the model [[equation of mixed type|7WFKPAMQFG]]: it is [[elliptic|B4TTNHSTRX]] in the upper half-plane, [[hyperbolic|92C3S03AQZ]] in the lower half-plane, and degenerate on the $x$-axis.
It arises in the study of transonic flow, where the flow changes from subsonic to supersonic.

## Explanations

### L1 human

Some equations behave like the Laplace equation in one region and like the wave equation in another.
In the Tricomi equation the coefficient $y$ switches sign across the $x$-axis, and the behaviour switches with it.
Above the axis solutions are smooth and spread information in all directions, as for a steady temperature.
Below the axis information travels along curves, as for waves.
This models air flowing around a wing, where the flow is slower than sound in one region and faster in another.
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<document source="https://tamnd.github.io/math2/node/21FMF8VKXY.md" tag="21FMF8VKXY" kind="definition">
# Period-doubling cascade

## Definition

Let $I \subseteq \R$ be an open interval and let $f_\mu \colon X \to X$, $\mu \in I$, be a family of maps of a set $X \subseteq \R$ with $(\mu, x) \mapsto f_\mu(x)$ smooth.
A *period-doubling cascade* of the family is a strictly increasing sequence of parameters $\mu_1 < \mu_2 < \mu_3 < \cdots$ in $I$ with a limit $\mu_\infty \in I$, such that for each $k \ge 1$:

1. for $\mu$ slightly less than $\mu_k$, the family has an attracting [[periodic solution|3W358BSRN2]] of prime period $2^{k-1}$;
2. at $\mu = \mu_k$ this cycle undergoes a [[period-doubling bifurcation|NQDTTZ59X8]]: it loses stability, and an attracting cycle of prime period $2^k$ is born from it.

## Remarks

Example: the logistic family $f_r(x) = r x (1 - x)$ on $[0, 1]$ has such a cascade with $r_1 = 3$ and $r_2 = 1 + \sqrt{6} \approx 3.449$.
At $r_1 = 3$ the fixed point $1 - 1/r$ has multiplier $f_r'(1 - 1/r) = 2 - r = -1$, and a $2$-cycle appears.
The limit is $r_\infty \approx 3.5699$.

Non-example: the family $f_\mu(x) = \mu x$ on $\R$ has no period-doubling cascade.
Its only possible attracting cycle is the fixed point $0$ for $\abs{\mu} < 1$, and it never has an attracting cycle of period $2$, so clause 2 fails already for $k = 1$.

For typical smooth unimodal families with a quadratic maximum, such as the logistic family, the ratios $(\mu_k - \mu_{k-1})/(\mu_{k+1} - \mu_k)$ tend to the Feigenbaum constant $\delta \approx 4.6692$.
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<document source="https://tamnd.github.io/math2/node/21JJ523V95.md" tag="21JJ523V95" kind="statement">
# Zassenhaus lemma

## Statement

Let $G$ be a [[group|G5S1M9]].
Let $A$ and $B$ be [[subgroups|0X1JXT47GA]] of $G$, let $A^*$ be a [[normal subgroup|0CB0DX5BVV]] of $A$, and let $B^*$ be a normal subgroup of $B$.
Then:

1. $A^*(A \cap B^*)$ is a normal subgroup of $A^*(A \cap B)$;
2. $B^*(A^* \cap B)$ is a normal subgroup of $B^*(A \cap B)$;
3. there is an isomorphism of [[quotient groups|0KYX2N69J0]]
$$\frac{A^*(A \cap B)}{A^*(A \cap B^*)} \cong \frac{B^*(A \cap B)}{B^*(A^* \cap B)}.$$

## Remarks

Let $D = (A^* \cap B)(A \cap B^*)$.
It is a normal subgroup of $A \cap B$, and the [[second isomorphism theorem|0PPP2J2DJV]] shows that the left quotient is isomorphic to $(A \cap B)/D$.
The same argument with $A$ and $B$ exchanged shows that the right quotient is also isomorphic to $(A \cap B)/D$.
The lemma is also called the butterfly lemma, after the shape of the diagram of the subgroups involved.
For $A^* = \{e\}$ and $B^* = \{e\}$ the lemma says only that $A \cap B \cong A \cap B$; the content appears when $A^*$ and $B^*$ are large.

## Explanations

### L1 human

The lemma compares two pieces of a group, one cut out near $A$ and one cut out near $B$.
It says that the part of the first piece seen through $B$ is the same as the part of the second piece seen through $A$.
It is a technical tool, but it is exactly what makes two ways of cutting a group comparable.

### L2 human

The lemma is the key step in the proof of the [[Schreier refinement theorem|QQAWHC9X60]], which in turn proves the [[Jordan–Hölder theorem|0E088ZP533]].
It is a consequence of the second isomorphism theorem alone.
The same statement and proof hold for modules over a ring, with sums in place of products of subgroups.
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<document source="https://tamnd.github.io/math2/node/21JT983G60.md" tag="21JT983G60" kind="concept">
# Bayesian network

## Summary

A Bayesian network is a [[graphical model|Q3FQAD1BXG]] on a directed acyclic graph.
The joint distribution is the product of the conditional distributions of each variable given its parents in the graph.
The arrows can often be read as direct influences, which makes Bayesian networks a common language for reasoning under uncertainty and for [[causal inference|V3QJFN5800]].
Efficient algorithms compute conditional probabilities in networks with sparse graphs.

## Explanations

### L1 human

Rain makes the grass wet, and so does the sprinkler; rain also makes it less likely that the sprinkler is on.
Draw an arrow from each cause to each thing it directly affects, with no loops of arrows.
Then give, for each variable, its probabilities for every combination of its causes.
These small tables together fix the probability of every full scenario, as their product.
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<document source="https://tamnd.github.io/math2/node/21RJ8035KP.md" tag="21RJ8035KP" kind="statement">
# Ergodic decomposition

## Statement

Let $X$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]] and let $T \colon X \to X$ be a continuous map.
Let $M(X, T)$ be the set of $T$-[[invariant|B232EZ1FMB]] [[Borel|DYR1QFNGNJ]] [[probability measures|059TJN0NZ1]] on $X$, with the topology of [[weak convergence of measures|MBW9JCGZC1]].
Let $E(X, T) \subseteq M(X, T)$ be the subset of measures for which $T$ is [[ergodic|AZRPHS478B]].

1. $M(X, T)$ is a convex compact set, and $E(X, T)$ is exactly the set of its [[extreme points|3MSGFJM2NX]].
2. For every $\mu \in M(X, T)$ there is a unique Borel probability measure $\tau$ on $M(X, T)$ with $\tau(E(X, T)) = 1$ and
$$\mu(A) = \int_{E(X, T)} \nu(A) \, d\tau(\nu) \quad \text{for every Borel set } A \subseteq X.$$

## Remarks

For part 1, suppose $\mu$ is not ergodic, and let $A$ be an invariant set with $0 < \mu(A) < 1$.
Then $\mu$ is the convex combination $\mu(A) \mu_A + (1 - \mu(A)) \mu_{X \setminus A}$ of the normalized restrictions, which are distinct invariant measures, so $\mu$ is not extreme.
Conversely, if $\mu$ is ergodic and $\mu = t \nu_1 + (1-t) \nu_2$ with $0 < t < 1$, then $\nu_1$ is absolutely continuous with respect to $\mu$ with an invariant density, which ergodicity forces to be constant, so $\nu_1 = \mu$.
Part 2 follows from Choquet theory, since $M(X, T)$ is a compact convex metrizable set whose barycentric decompositions are unique.
The decomposition can also be built from the conditional measures of $\mu$ on the σ-algebra of invariant sets.
Example: for the identity map, every Borel probability measure is invariant, and the ergodic ones are the point masses $\delta_x$.
The decomposition of $\mu$ is $\mu = \int \delta_x \, d\mu(x)$, that is, $\tau$ is the image of $\mu$ under $x \mapsto \delta_x$.
Example: for the rotation of the circle by a quarter turn, Lebesgue measure is the average over $x \in [0, 1/4)$ of the uniform measures on the four-point orbits of $x$.

## Explanations

### L1 human

Every invariant measure can be split, in a unique way, into ergodic pieces.
Each piece is a measure that cannot be split further, and the original measure is a weighted average of the pieces.
So to understand all invariant measures, it is enough to understand the ergodic ones.

### L2 human

Many theorems are first proved for ergodic systems, where time averages are constant, and then extended to all systems by the ergodic decomposition.
For example, [[Birkhoff's ergodic theorem|J6NMBX3E55]] for a non-ergodic measure gives a limit that is constant on each ergodic component.
Two distinct ergodic measures are mutually singular.
By part 1, the unique invariant measure of a [[uniquely ergodic|NZZE9KCAD8]] map is ergodic.
That $M(X, T)$ is not empty is the [[Krylov–Bogolyubov theorem|1DHZ81S0YE]].
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<document source="https://tamnd.github.io/math2/node/21SXYJYC59.md" tag="21SXYJYC59" kind="concept">
# Convex combination

## Summary

A convex combination of finitely many points is a weighted sum of the points whose weights are nonnegative and add up to $1$.
It is the special kind of [[affine combination|0KJH6SRFZK]] in which no weight is negative.
The [[convex hull|0FQSDC2EQ2]] of a set is exactly the set of all convex combinations of its points.
A [[convex set|02AS45T3H9]] is a set closed under convex combinations.
So convex combinations are the basic operation of convex geometry.

## Explanations

### L1 human

Think of the weights as masses placed at the points, with total mass $1$.
The convex combination is the centre of mass of this system.
Because all masses are nonnegative, the centre of mass always lies inside the shape spanned by the points.
For example, with the points $0$ and $10$ on the real line and weights $0.3$ and $0.7$, the convex combination is $0.3 \cdot 0 + 0.7 \cdot 10 = 7$, which lies between $0$ and $10$.
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<document source="https://tamnd.github.io/math2/node/220PPH6N7Q.md" tag="220PPH6N7Q" kind="definition">
# Sweedler notation

## Definition

Let $K$ be a field and let $(C, \Delta, \varepsilon)$ be a [[coalgebra|A06RA8KB0T]] over $K$.
For $c \in C$, the element $\Delta(c) \in C \otimes C$ can be written as a finite sum $\sum_{i=1}^{r} a_i \otimes b_i$ with $a_i, b_i \in C$.
*Sweedler notation* writes this as
$$\Delta(c) = \sum c_{(1)} \otimes c_{(2)},$$
and an expression $\sum f(c_{(1)}, c_{(2)})$ means $\sum_{i} f(a_i, b_i)$ for a map $f$ that is $K$-bilinear in its two arguments.
By coassociativity, the element $(\Delta \otimes \id)\Delta(c) = (\id \otimes \Delta)\Delta(c)$ is written $\sum c_{(1)} \otimes c_{(2)} \otimes c_{(3)}$.
For a right [[comodule|7WEM2E6RTV]] $M$ with coaction $\rho$, one writes $\rho(m) = \sum m_{(0)} \otimes m_{(1)}$, with $m_{(0)} \in M$ and $m_{(1)} \in C$.

## Remarks

Because $f$ is bilinear, $\sum_i f(a_i, b_i)$ depends only on $\Delta(c)$ and not on the chosen sum; it equals $\tilde f(\Delta(c))$ for the linear map $\tilde f$ on $C \otimes C$ induced by $f$.
Example: the counit law reads $\sum \varepsilon(c_{(1)}) c_{(2)} = c = \sum c_{(1)} \varepsilon(c_{(2)})$.
Example: for a group-like element $g$, with $\Delta(g) = g \otimes g$, one may take $g_{(1)} = g_{(2)} = g$.
Non-example: $\sum c_{(1)} \otimes c_{(1)}$ is not a valid expression, because it is not bilinear in $(c_{(1)}, c_{(2)})$.
For $\Delta(x) = x \otimes 1 + 1 \otimes x$ in $K[x]$ it would give $x \otimes x + 1 \otimes 1$.
The same element written as $(x + 1) \otimes 1 + 1 \otimes x - 1 \otimes 1$ would give $(x+1) \otimes (x+1) + 1 \otimes 1 + 1 \otimes 1$, a different result.
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<document source="https://tamnd.github.io/math2/node/2230ZKE4W9.md" tag="2230ZKE4W9" kind="concept">
# Heegner point

## Summary

A Heegner point is a special point on the modular curve $X_0(N)$.
It corresponds to a pair of elliptic curves with [[complex multiplication|D3HEJTVY7G]] by the ring of integers of one imaginary [[quadratic field|7SV4WA92SM]] $K$.
Such points are defined over the [[Hilbert class field|WWCJ7CR2R3]] of $K$.
Through the [[modularity theorem|CS1NVJBZMH]], they give points on [[elliptic curves|427N6S6QQW]] over $\Q$.
The [[Gross–Zagier theorem|BX558DTV2D]] relates their heights to derivatives of $L$-functions.

## Explanations

### L1 human

Rational points of infinite order on an elliptic curve are hard to find, and there is in general no formula for them.
Heegner points are the main general construction that produces such points.
The idea is to start from very symmetric elliptic curves, those with complex multiplication, whose coordinates are algebraic numbers with known fields of definition.
Mapping these special points to a given elliptic curve and summing over their conjugates gives a point on that curve defined over the quadratic field $K$.
Heegner used such points in his solution of the class number one problem.
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<document source="https://tamnd.github.io/math2/node/22735FGJV6.md" tag="22735FGJV6" kind="concept">
# Wiener process

## Summary

The Wiener process, or standard Brownian motion, is the continuous-time [[stochastic process|0QH5ENKZM1]] with continuous paths and independent, normally distributed increments.
It is the scaling limit of random walks and the basic model of random motion.
It is a [[Gaussian process|22WEM6RTW0]], a [[Lévy process|8NT5CFG2ZR]], a [[martingale|0D0GBNDKDH]] and a Markov process at the same time.

## Explanations

### L1 human

Picture a pollen grain jiggling in water, or a stock price wobbling from second to second.
The Wiener process is the mathematical model of such motion: at every instant it moves by a small random amount, independent of the past.
Over a time span of length $t$ it moves by a normal amount with mean $0$ and variance $t$.
Its path is continuous but, almost surely, so rough that it has no derivative at any point.
If you take a fair coin-tossing walk with $n$ steps per unit of time and step size $1/\sqrt{n}$, you get a Wiener process as $n \to \infty$.
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<document source="https://tamnd.github.io/math2/node/227ERQ43PY.md" tag="227ERQ43PY" kind="statement">
# Jung's theorem

## Statement

Let $n \ge 1$, and let $S \subseteq \R^n$ be a bounded set with [[diameter|0KW3PN547K]] $d$.
Then $S$ is contained in a [[closed ball|0Q0H13G8ZB]] of radius
$$R = d \sqrt{\frac{n}{2(n+1)}}.$$

## Remarks

Take the smallest closed ball containing $S$; by a compactness argument its centre lies in the convex hull of the points of the closure of $S$ on its boundary sphere.
Since this centre is a convex combination of at most $n + 1$ such points, a computation with these points and the inequality $\abs{x_i - x_j} \le d$ gives the bound.
A classical alternative uses [[Helly's theorem|F1W3QRBNGJ]] to reduce to sets of $n + 1$ points.
The bound is sharp: the vertices of a regular $n$-simplex of edge $d$ have diameter $d$ and circumradius exactly $d \sqrt{n / (2(n+1))}$.
Example: for $n = 2$ the radius is $d / \sqrt{3}$, the circumradius of an equilateral triangle of side $d$.
For $n = 1$ the radius is $d/2$, as expected for a segment.

## Explanations

### L1 human

Any set of points whose pairwise distances are at most $d$ fits inside a ball of a fixed radius, a little larger than $d/2$.
In the plane the radius $d/\sqrt{3}$ suffices, and the equilateral triangle shows that nothing smaller always works.

### L2 human

The trivial bound is $R \le d$: a ball of radius $d$ centred at any point of $S$ contains $S$.
Jung's theorem gives the optimal constant, which grows from $1/2$ for $n = 1$ towards $1/\sqrt{2}$ as $n \to \infty$.
It is a typical result of convex geometry that relates two measures of size, the diameter and the circumradius.
It is often used together with the isodiametric inequality and in Borsuk-type partition problems.
</document>

<document source="https://tamnd.github.io/math2/node/22J8E7AB8V.md" tag="22J8E7AB8V" kind="definition">
# Hilbert–Samuel function

## Definition

Let $(R, \mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]], let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module, and let $I$ be an [[$\mathfrak m$-primary ideal|TMTPWVETQK]] of $R$.
The *Hilbert–Samuel function* of $M$ with respect to $I$ is the function $\chi_M^I \colon \N \to \N$ given by
$$\chi_M^I(n) = \ell(M/I^n M),$$
where $\ell$ denotes the [[length|CZSCNZH8VY]] of an $R$-module.

## Remarks

The length is finite because $M/I^n M$ is a finitely generated module over the Artinian ring $R/I^n$.
There is a polynomial $P_M^I$ with rational coefficients, the Hilbert–Samuel polynomial, such that $\chi_M^I(n) = P_M^I(n)$ for all large $n$.
If $M \neq 0$, its degree is the Krull dimension of $M$, so it does not depend on $I$.
Example: for $R = k[x, y]_{(x, y)}$, with $k$ a field, $M = R$ and $I = \mathfrak m$, the module $R/\mathfrak m^n$ has the $k$-basis of monomials $x^a y^b$ with $a + b < n$, so $\chi(n) = n(n+1)/2$.
Non-example: for the same ring and $I = (x)$, which is not $\mathfrak m$-primary, the module $R/(x)^n$ has infinite length, so the formula does not define a function to $\N$.
Some authors use $\ell(M/I^{n+1} M)$ instead, which shifts the variable by one.
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<document source="https://tamnd.github.io/math2/node/22K8CSQTYM.md" tag="22K8CSQTYM" kind="concept">
# Complete measure

## Summary

A measure is complete if every subset of a set of measure zero is measurable.
Then such subsets automatically have measure zero.
Lebesgue measure on the Lebesgue measurable sets is complete, while its restriction to the Borel sets is not.

## Explanations

### L1 human

If a set has size zero, then any part of it should also have size zero.
A complete measure makes sure that every such part is allowed to be measured at all.
Without completeness, a subset of a negligible set could be left out of the σ-algebra, which is awkward when one works with properties that hold almost everywhere.
Every measure can be completed by adding these subsets.
</document>

<document source="https://tamnd.github.io/math2/node/22M7FHJ3N8.md" tag="22M7FHJ3N8" kind="definition">
# Ideal sheaf

## Definition

Let $(X, \mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]].
An *ideal sheaf* on $X$ is a sub-$\mathcal{O}_X$-module $\mathcal{I} \subseteq \mathcal{O}_X$ in the sense of [[sheaves of modules|S33B2H3MYD]].
Equivalently, $\mathcal{I}$ is a subsheaf of $\mathcal{O}_X$ such that $\mathcal{I}(U)$ is an [[ideal|096Z9MP2ME]] of $\mathcal{O}_X(U)$ for every open set $U \subseteq X$.

## Remarks

On $X = \operatorname{Spec} A$, an ideal $I \subseteq A$ gives the ideal sheaf $\widetilde{I}$, with $\widetilde{I}(D(f)) = I_f \subseteq A_f$.
It is a sub-$\mathcal{O}_X$-module of $\widetilde{A} = \mathcal{O}_X$, since localization preserves ideals.
On $X = \mathbb{A}^1_k = \operatorname{Spec} k[t]$, the subsheaf of $\mathcal{O}_X$ of constant functions is not an ideal sheaf.
Its sections over $X$ form $k \subseteq k[t]$, which is not an ideal, since $t \cdot 1 = t$ is not constant.
For a closed subscheme $Z \subseteq X$ of a scheme, the kernel of $\mathcal{O}_X \to \mathcal{O}_Z$ is a quasi-coherent ideal sheaf $\mathcal{I}_Z$.
This gives a bijection between closed subschemes of $X$ and quasi-coherent ideal sheaves on $X$.
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<document source="https://tamnd.github.io/math2/node/22T0NVC56P.md" tag="22T0NVC56P" kind="concept">
# Euler–Maruyama method

## Summary

The Euler–Maruyama method is the simplest numerical scheme for stochastic differential equations.
It extends the [[Euler method|0JARTXX4XP]] for ordinary differential equations by adding, at each step, the diffusion coefficient times a Brownian increment.
Under Lipschitz and growth conditions on the coefficients, it converges with strong order $1/2$.
Under further smoothness conditions, it converges with weak order $1$.

## Explanations

### L1 human

To follow a random path driven by noise, one walks forward in small time steps.
At each step, one moves by the drift times the step length, as in the ordinary Euler method.
Then one adds a random kick, a normal random number with variance equal to the step length, scaled by the noise coefficient.
For example, for $dX = -X \, dt + dB$ with step $0.1$, the update is $Y_{n+1} = 0.9 \, Y_n + \sqrt{0.1} \, Z_n$ with $Z_n$ standard normal.
</document>

<document source="https://tamnd.github.io/math2/node/22WEM6RTW0.md" tag="22WEM6RTW0" kind="concept">
# Gaussian process

## Summary

A Gaussian process is a [[stochastic process|0QH5ENKZM1]] all of whose [[finite-dimensional distributions|Y0CN7V3B7E]] are [[multivariate normal|0XN3K7RSZY]].
Its law is determined by its mean function and its covariance function, and any positive semidefinite covariance function arises.
The [[Wiener process|22735FGJV6]], the [[Brownian bridge|3MCVMBAGD1]], the [[Ornstein–Uhlenbeck process|HZM2ASCDY4]] and [[fractional Brownian motion|VC6R7XSKRQ]] are Gaussian processes.
In statistics and machine learning, Gaussian processes serve as prior distributions on unknown functions.

## Explanations

### L1 human

A Gaussian process is a random function whose values at any finitely many points have a joint normal distribution.
One describes it by an average curve and by a rule saying how strongly the values at two points move together.
For example, take a standard normal number $Z$ and set $X_t = Z t$; this random straight line through the origin is a Gaussian process with mean $0$ and covariance $st$.
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<document source="https://tamnd.github.io/math2/node/230F5FXPMQ.md" tag="230F5FXPMQ" kind="statement">
# Weyl character formula for compact groups

## Statement

Let $G$ be a connected [[compact Lie group|9TFC6ADC1C]], and let $T$ be a [[maximal torus|3JVGZVVTB0]] with Lie algebra $\mathfrak{t}$.
Let $W$ be the [[Weyl group|EZGTEK3HQN]], acting on the dual space $\mathfrak{t}^*$.
Identify each continuous character of $T$ with the linear form $\mu \in \mathfrak{t}^*$ such that the character sends $\exp H$ to $e^{i\mu(H)}$ for $H \in \mathfrak{t}$.
Let $\Phi \subset \mathfrak{t}^*$ be the roots of $G$, choose a set $\Phi^+$ of positive roots, and let $\rho = \frac{1}{2}\sum_{\alpha \in \Phi^+} \alpha$.
For $w \in W$ let $\varepsilon(w) = \det(w) \in \set{1, -1}$.
Let $V$ be a finite-dimensional [[irreducible|FX3NEFYFFN]] continuous representation of $G$ with highest weight $\lambda$, and let $\chi_V$ be its [[character|RMWMYBY8TC]].
Then for every $H \in \mathfrak{t}$ with $\prod_{\alpha \in \Phi^+} \sin(\alpha(H)/2) \ne 0$,
$$\chi_V(\exp H) = \frac{\sum_{w \in W} \varepsilon(w)\, e^{i (w(\lambda + \rho))(H)}}{\sum_{w \in W} \varepsilon(w)\, e^{i (w\rho)(H)}}.$$

## Remarks

The denominator equals $\prod_{\alpha \in \Phi^+} \left(e^{i\alpha(H)/2} - e^{-i\alpha(H)/2}\right)$, by the Weyl denominator formula.
One proof multiplies $\chi_V$ by the denominator and notes that the product is $W$-alternating with integer coefficients.
The [[Weyl integration formula|QMH530N27M]] and $\int_G \abs{\chi_V}^2 \, dg = 1$ then show that only the term $\lambda + \rho$ and its $W$-images survive, each with coefficient $\pm 1$.
Example: for $G = SU(2)$ and $H = \operatorname{diag}(i\theta, -i\theta)$, there is one positive root with $\alpha(H) = 2\theta$, $\rho(H) = \theta$, and the representation of dimension $n + 1$ has $\lambda(H) = n\theta$.
The formula gives $\chi(\exp H) = \frac{e^{i(n+1)\theta} - e^{-i(n+1)\theta}}{e^{i\theta} - e^{-i\theta}} = \frac{\sin((n+1)\theta)}{\sin\theta}$, which is $e^{in\theta} + e^{i(n-2)\theta} + \cdots + e^{-in\theta}$.
Letting $H \to 0$ gives the Weyl dimension formula $\dim V = \prod_{\alpha \in \Phi^+} \frac{\ip{\lambda + \rho}{\alpha}}{\ip{\rho}{\alpha}}$ for a $W$-invariant inner product on $\mathfrak{t}^*$.

## Explanations

### L1 human

The character of a representation records the trace of each group element, and it determines the representation.
The Weyl formula gives this character in closed form, as a ratio of two alternating sums of exponentials.
Only the highest weight $\lambda$ of the representation enters, together with data of the group: its roots and its Weyl group.

### L2 human

By [[conjugacy of maximal tori|BE3XK5ERC1]], a character is determined by its values on $T$, so the formula gives it everywhere.
Together with the [[Peter–Weyl theorem|6VC9QE39XH]] it gives the theorem of the highest weight: irreducible representations of $G$ correspond to dominant integral weights.
The dimension formula and Kostant's multiplicity formula follow from it, and the [[Borel–Weil–Bott theorem|K10YADF5XV]] gives a geometric realisation of the same representations.
For $G = U(n)$ the formula expresses irreducible characters as Schur polynomials in the eigenvalues.
</document>

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# Quotient category

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
A *congruence* $R$ on $\mathcal{C}$ is a family of [[equivalence relations|0G8VEZ3WTK]] $\sim_{X,Y}$, one on each set $\Hom(X, Y)$ of [[morphisms|859F0TFVA3]], with this property.
If $f \sim f'$ for $f, f' \colon X \to Y$, then $g \circ f \circ h \sim g \circ f' \circ h$ for all $h \colon W \to X$ and $g \colon Y \to Z$.
The *quotient category* $\mathcal{C}/R$ has the same objects as $\mathcal{C}$.
Its morphisms $X \to Y$ are the equivalence classes $[f]$ of morphisms $f \colon X \to Y$, with composition $[g] \circ [f] = [g \circ f]$ and identities $[\id_X]$.

## Remarks

The compatibility condition makes the composition $[g] \circ [f] = [g \circ f]$ independent of the chosen representatives.
The quotient [[functor|28NQ0KRHD3]] $Q \colon \mathcal{C} \to \mathcal{C}/R$ is the identity on objects and sends $f$ to $[f]$.
It is universal: a functor $G \colon \mathcal{C} \to \mathcal{D}$ with $G(f) = G(f')$ whenever $f \sim f'$ factors uniquely as $G = \bar{G} \circ Q$.

Example: view the monoid $(\Z, +)$ as a category with one object, whose morphisms are the integers and whose composition is addition.
Let $m \sim n$ when $m - n$ is even.
If $m \sim m'$ then $(k + m + l) - (k + m' + l) = m - m'$ is even, so $\sim$ is a congruence, and the quotient is the one-object category of the group $\Z/2\Z$.

Non-example: on the same category let $m \sim n$ when $m = n$ or $\{m, n\} = \{0, 1\}$.
This is an equivalence relation, but $0 \sim 1$ while $0 + 1 = 1$ and $1 + 1 = 2$ are not related, so it is not compatible with composition and gives no quotient category.
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<document source="https://tamnd.github.io/math2/node/231TRBRZSY.md" tag="231TRBRZSY" kind="definition">
# Measure-preserving transformation

## Definition

Let $(X, \mathcal{A}, \mu)$ and $(Y, \mathcal{B}, \nu)$ be [[measure spaces|P15VQS5DJ9]].
A map $T \colon X \to Y$ is *measure-preserving* if two conditions hold.

1. $T$ is [[measurable|YYNCJSR0PJ]]: $T^{-1}(B) \in \mathcal{A}$ for every $B \in \mathcal{B}$.
2. $\mu(T^{-1}(B)) = \nu(B)$ for every $B \in \mathcal{B}$.

## Remarks

Condition 2 says that the [[pushforward measure|3N66TM3TSQ]] $T_* \mu$ equals $\nu$.
When $X = Y$ and $\mu = \nu$, one says that $T$ preserves $\mu$, and $(X, \mathcal{A}, \mu, T)$ is a measure-preserving system.
Example: the doubling map $T(x) = 2x \bmod 1$ on $[0,1)$ with Lebesgue measure $\lambda$.
For $0 \le a < b \le 1$ one has $T^{-1}([a, b)) = [a/2, b/2) \cup [(a+1)/2, (b+1)/2)$, of measure $b - a$.
The intervals form a $\pi$-system that generates the Borel sets, so condition 2 holds for all Borel sets.
Non-example: $T(x) = x^2$ on $[0,1]$ with Lebesgue measure is measurable but $\lambda(T^{-1}([0, 1/4])) = \lambda([0, 1/2]) = 1/2 \ne 1/4$.
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<document source="https://tamnd.github.io/math2/node/234F5DNAJH.md" tag="234F5DNAJH" kind="concept">
# Tricomi confluent hypergeometric function

## Summary

The Tricomi confluent hypergeometric function $U(a, b, z)$ is the solution of [[Kummer's equation|NT917KFR57]] that behaves like $z^{-a}$ as $z \to +\infty$.
It is a combination of two Kummer functions $M$.
It grows or decays only like a power of $z$ at infinity, while $M(a, b, z)$ in general grows like $e^z$.
Many functions of applied mathematics are special cases of $U$.
Examples are the complementary error function, the upper incomplete gamma function and the modified Bessel function $K_\nu$.

## Explanations

### L1 human

Kummer's equation has a two-dimensional space of solutions.
Kummer's function $M$ is the solution that is nice at $z = 0$.
Tricomi's function $U$ is the solution that is nice at $z = +\infty$: it behaves like a power of $z$ instead of growing exponentially.
For $\operatorname{Re} a > 0$ and $\operatorname{Re} z > 0$ it has the integral form $U(a, b, z) = \frac{1}{\Gamma(a)} \int_0^\infty e^{-zt}\, t^{a - 1} (1 + t)^{b - a - 1}\, dt$.
For $b = a + 1$ the factor $(1 + t)^{b - a - 1}$ is $1$, and the integral gives $U(a, a + 1, z) = z^{-a}$.
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<document source="https://tamnd.github.io/math2/node/2362AFNG8X.md" tag="2362AFNG8X" kind="definition">
# Formally real field

## Definition

Let $F$ be a [[field|0QA8CHH8MG]].
The field $F$ is *formally real* if $-1$ is not a sum of squares in $F$.
That is, there are no $n \ge 1$ and $a_1, \dots, a_n \in F$ with $a_1^2 + \cdots + a_n^2 = -1$.

## Remarks

Equivalently, for all $n \ge 1$ and $a_1, \dots, a_n \in F$, if $a_1^2 + \cdots + a_n^2 = 0$, then $a_1 = \cdots = a_n = 0$.
By the Artin–Schreier theorem on orderings, $F$ is formally real if and only if $F$ carries a structure of [[ordered field|010JCV9J4E]].
A formally real field has [[characteristic|1ZB13N78XW]] $0$, since in characteristic $p$ one has $-1 = 1^2 + \cdots + 1^2$ with $p - 1$ terms.
Example: $\Q$ is formally real, because a sum of rational squares is at least $0$ in the usual order.
Non-example: $\C$ is not formally real, because $-1 = i^2$; this is why [[$\C$ cannot be ordered|01CJRJKDAA]].
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<document source="https://tamnd.github.io/math2/node/237KR6FRCK.md" tag="237KR6FRCK" kind="concept">
# Lagrange interpolation

## Summary

Lagrange interpolation is an explicit formula for the polynomial of least degree through given points.
It writes the [[interpolation polynomial|SS1YMS2NPC]] as a combination of basis polynomials, each equal to $1$ at one node and $0$ at the others.
The formula works over every field and shows at once that the interpolation polynomial exists.
It is used in numerical analysis, in coding theory and in secret sharing.

## Explanations

### L1 human

Suppose you want a polynomial that takes chosen values at chosen points.
First build, for each point, a polynomial that is $1$ there and $0$ at all other points.
Then multiply each of them by the wanted value and add.
For the points $(0, 1)$, $(1, 3)$ and $(2, 7)$ this gives $x^2 + x + 1$.
The [[definition|QGQTYT6YZR]] gives the formula.
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<document source="https://tamnd.github.io/math2/node/23ASRP0MWR.md" tag="23ASRP0MWR" kind="concept">
# Order of a zero or pole

## Summary

The order of a [[meromorphic function|0ZZ7ZYDR8W]] at a point $a$ is the integer $n$ such that it behaves like $c (z - a)^n$ near $a$, with $c \neq 0$.
See the [[definition|K2S5TAW2BF]].
At a zero it is the [[order of the zero|0QX19C7K1C]].
At a pole it is minus the [[order of the pole|073MWACF1K]].
This single integer turns products of meromorphic functions into sums, like a logarithm.

## Explanations

### L1 human

Near $0$, the function $z^3$ vanishes like the third power of $z$, and $1/z^2$ blows up like the second power of $1/z$.
The order records both at once with a sign: $3$ for $z^3$ and $-2$ for $1/z^2$.
At ordinary points, where the function is neither $0$ nor infinite, the order is $0$.
When you multiply two functions, their orders add, just as exponents add when you multiply powers.
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<document source="https://tamnd.github.io/math2/node/23ATBZMJHR.md" tag="23ATBZMJHR" kind="concept">
# Rubin causal model

## Summary

The Rubin causal model, or potential outcomes framework, defines causal effects by comparing the outcomes that the same unit would have under different treatments.
Each unit has one potential outcome for each treatment, but only the one for the treatment actually received is observed.
Causal effects such as the average treatment effect are then parameters of the joint distribution of potential outcomes.
The framework goes back to Jerzy Neyman's work on experiments and was developed by Donald Rubin for observational studies.
It is a basic language of [[causal inference|V3QJFN5800]].

## Explanations

### L1 human

For a patient, think of two versions of the future: one in which she takes the drug and one in which she does not.
The effect of the drug for her is the difference between the two outcomes.
We can only ever see one of these futures, so individual effects are never observed directly.
Averages over many patients can still be learned, for example when a coin flip decides who gets the drug.
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# Nonexistence of infinite-dimensional Lebesgue measure

## Statement

Let $X$ be an infinite-dimensional separable real [[Banach space|06GZ956657]].
Let $\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$ such that these conditions hold:

1. $\mu$ is translation-invariant: $\mu(x + A) = \mu(A)$ for every Borel set $A$ and every $x \in X$.
2. $\mu$ is locally finite: every point of $X$ has an open neighbourhood of finite measure.

Then $\mu(A) = 0$ for every Borel set $A$.

## Remarks

By clause 1 and clause 2, some [[open ball|0DC3MPR7C9]] $B(0, r)$ with $r > 0$ has finite measure.
By [[Riesz's lemma|097C7KCAT5]], there are infinitely many points $x_1, x_2, \ldots$ with $\norm{x_i} = 3r/4$ and $\norm{x_i - x_j} \ge r/2$ for $i \ne j$.
The balls $B(x_i, r/4)$ are disjoint, lie in $B(0, r)$, and all have the same measure by translation invariance; so this measure is $0$.
By separability, $X$ is a countable union of balls of radius $r/4$, and so $\mu(X) = 0$.

The result is often stated in an equivalent way: a translation-invariant Borel measure that is not identically zero gives infinite measure to every nonempty open set.
In finite dimension the conclusion fails, because [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$ is translation-invariant and locally finite.
Gaussian measures on infinite-dimensional spaces avoid the theorem by giving up translation invariance.

## Explanations

### L1 human

In $\R^n$ volume does not change when a set is moved, and small balls have finite volume.
In infinite dimension no notion of volume can have both properties, except the one that gives volume zero to everything.
The reason is that a ball contains infinitely many disjoint smaller balls of a fixed size, which would all need the same positive volume.

### L2 human

The theorem explains why integration over spaces of functions, as in probability theory and quantum field theory, uses other measures.
The standard choices are Gaussian measures, such as Wiener measure on paths.
Their behaviour under translations is described by the Cameron–Martin theorem: only translations along a small subspace keep the measure class.
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<document source="https://tamnd.github.io/math2/node/23G7N7WZBH.md" tag="23G7N7WZBH" kind="definition">
# Primitive polynomial over a finite field

## Definition

Let $\F_q$ be a [[finite field|08RTDHX1SK]] and let $m \ge 1$.
A polynomial $f \in \F_q[x]$ of degree $m$ is *primitive* if it is [[irreducible|00X2A9G9W3]] and has a root $\alpha$ in $\F_{q^m}$ that is a [[primitive element|0YAA9CJQXF]].
That is, $\alpha$ has multiplicative [[order|0JPYSD5E0B]] $q^m - 1$.

## Remarks

Equivalently, $f$ is a scalar multiple of the [[minimal polynomial|0GK1WB9R98]] over $\F_q$ of a primitive element of $\F_{q^m}$.
Then all roots of $f$ are primitive elements, since they are the conjugates $\alpha^{q^i}$, which have the same order as $\alpha$.
Another standard form: an irreducible $f$ of degree $m$ with $f(0) \ne 0$ is primitive if and only if the least $e \ge 1$ such that $f$ divides $x^e - 1$ is $q^m - 1$.
There are $\varphi(q^m - 1)/m$ monic primitive polynomials of degree $m$ over $\F_q$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].
Example: over $\F_2$, the polynomial $x^3 + x + 1$ is irreducible, and a root $\alpha$ has order dividing $7$ but not $1$, so its order is $7 = 2^3 - 1$, and the polynomial is primitive.
Non-example: over $\F_2$, the polynomial $x^4 + x^3 + x^2 + x + 1$ is irreducible, but it divides $x^5 - 1$, so a root has order $5$, not $15$, and the polynomial is not primitive.
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<document source="https://tamnd.github.io/math2/node/23RK8EZ20P.md" tag="23RK8EZ20P" kind="statement">
# Fundamental theorem of Γ-convergence

## Statement

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]].
Let $F_n \colon X \to [-\infty, +\infty]$ be functions that [[Γ-converge|NW9MM370A5]] to $F \colon X \to [-\infty, +\infty]$.
Assume that the sequence is equi-coercive in the following sense: there is a nonempty [[compact set|0R6EWFKRCF]] $K \subseteq X$ with $\inf_X F_n = \inf_K F_n$ for every $n$.
Then:

1. $F$ attains its minimum on $X$, and
$$\min_X F = \lim_{n \to \infty} \inf_X F_n.$$
2. If $(x_n)$ is a sequence in $X$ with $\lim_n F_n(x_n) = \lim_n \inf_X F_n$, then every limit of a subsequence of $(x_n)$ is a minimum point of $F$.

## Remarks

For the proof, pick almost minimizers $x_n \in K$ of $F_n$ and a subsequence along which $\inf_X F_n$ tends to its limit inferior and $x_n$ converges to some $x \in K$.
The lower bound inequality gives $F(x) \le \liminf_n \inf_X F_n$.
For any $y \in X$, a recovery sequence $y_n \to y$ gives $\limsup_n \inf_X F_n \le \limsup_n F_n(y_n) \le F(y)$.
Together these show that $x$ minimizes $F$ and that the infima converge; part 2 follows from the same lower bound.

Example: for $F_n(x) = \sin(nx) + x^2$ on $\R$, the Γ-limit is $F(x) = x^2 - 1$, since adding a continuous real function commutes with Γ-convergence.
Since $F_n(x) \ge x^2 - 1 > 0 = F_n(0)$ for $\abs{x} > 1$, the infimum of $F_n$ over $\R$ equals its infimum over $K = [-1, 1]$.
The theorem gives $\min F_n \to -1$, and the minimizers of $F_n$ tend to $0$, the minimizer of $F$.

The coercivity hypothesis cannot be dropped.
Let $F_n(x) = -\max(0, 1 - \abs{x - n})$ on $\R$, a downward tent centred at $n$.
Its Γ-limit is $F = 0$, but $\inf F_n = -1$ for every $n$, so the infima do not converge to $\min F = 0$; the minimizers $x_n = n$ escape to infinity.

## Explanations

### L1 human

Suppose a hard minimization problem is close, in the sense of Γ-convergence, to a simpler limit problem.
The theorem says that the smallest values of the hard problems approach the smallest value of the limit problem.
It also says that their best points can only accumulate at best points of the limit problem.
The one extra condition is that the best points do not run away to infinity.

### L2 human

This theorem is the reason Γ-convergence is defined the way it is.
The lower bound inequality controls minimum values from below, and recovery sequences control them from above.
In applications one proves the two inequalities separately: the lower bound often by a lower semicontinuity argument, the upper bound by an explicit construction.
Equi-coercivity usually comes from a compactness theorem, such as the Rellich compactness theorem for Sobolev spaces.
Results such as the Modica–Mortola theorem on phase transitions are then read as statements about the behaviour of minimizers.
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# Thom's classification of elementary catastrophes

## Statement

Let $r \le 4$, and consider smooth families $V \colon \R^n \times \R^r \to \R$, $(x, u) \mapsto V_u(x)$, of functions of $n$ state variables depending on $r$ control parameters.
Give $C^\infty(\R^n \times \R^r, \R)$ the Whitney $C^\infty$ topology.
There is an open dense set of families $V$ with the following property at every point $(x_0, u_0)$.
View the germ of $V$ at $(x_0, u_0)$ as an unfolding of the [[germ|7SWSRGDT4A]] of $V_{u_0}$ at $x_0$.
This unfolding is equivalent to a nondegenerate quadratic form in some state variables plus one of the following:

1. a regular point: $x$;
2. a nondegenerate critical point: the function $0$, so that only the quadratic form remains;
3. the fold: $x^3 + a x$;
4. the cusp: $\pm(x^4 + a x^2 + b x)$;
5. the swallowtail: $x^5 + a x^3 + b x^2 + c x$;
6. the butterfly: $\pm(x^6 + a x^4 + b x^3 + c x^2 + d x)$;
7. the hyperbolic umbilic: $x^3 + y^3 + a xy + b x + c y$;
8. the elliptic umbilic: $x^3 - 3 x y^2 + a (x^2 + y^2) + b x + c y$;
9. the parabolic umbilic: $\pm(x^2 y + y^4 + a x^2 + b y^2 + c x + d y)$.

Here equivalence of unfoldings allows a change of control parameters and a change of state variables depending on the controls.
It also allows the addition of a smooth function of the controls.
The letters $a, b, c, d$ stand for the control parameters that are used.

## Remarks

Items 3 to 9 are the seven elementary catastrophes.
Each is the [[universal unfolding|KSAFWZVB8K]] of its organising centre, the germ obtained by setting the parameters to $0$.
The number of parameters is the [[codimension|B4FRQMWQKP]] of that germ.
It is $1$ for the fold, $2$ for the cusp, $3$ for the swallowtail and the hyperbolic and elliptic umbilics, and $4$ for the butterfly and the parabolic umbilic.
The idea of the proof has two parts.
By Thom's transversality theorem in jet space, germs of codimension greater than $r$ do not occur in a generic $r$-parameter family.
Up to sign and quadratic terms, the germs of codimension from $1$ to $4$ are exactly the seven organising centres.
These are $x^3$, $x^4$, $x^5$, $x^6$, $x^3 + y^3$, $x^3 - 3xy^2$ and $x^2 y + y^4$.
Each of them is finitely determined, and its universal unfolding is checked with the versality criterion.
The list of seven depends on the bound $r \le 4$: with five controls further catastrophes appear.
For $r \ge 6$, families of germs with a modulus, such as $x^3 + y^3 + z^3 + a xyz$, occur generically, and no finite list is possible.
Thom proposed the classification; complete proofs rest on the work of Mather on stability and finite determinacy.

## Explanations

### L1 human

Imagine a system that always settles in a minimum of some energy, while a few control knobs change the energy.
As the knobs turn, minima can merge and vanish, and the system then jumps.
Thom's theorem says that with at most four knobs, the ways this can happen are, after a change of coordinates, only seven.
They carry names that describe the shape of the set of knob settings where jumps occur: fold, cusp, swallowtail, butterfly and three umbilics.

### L2 human

The theorem combines the three main tools of local singularity theory: finite determinacy, the classification of germs of low codimension, and versal unfoldings.
Its organising centres are the real forms of the simple singularities $A_2$ to $A_5$, $D_4^{\pm}$ and $D_5$ from [[Arnold's classification|M5437FP73G]].
The [[fold catastrophe|E192Q8BQWJ]] and the [[cusp catastrophe|VQ7D5N3MV7]] are the two cases with one state variable and at most two controls, and they model sudden jumps and hysteresis.
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<document source="https://tamnd.github.io/math2/node/242TFD41KZ.md" tag="242TFD41KZ" kind="concept">
# Fredholm theory

## Summary

Fredholm theory is the theory of the [[Fredholm equation of the second kind|DEYYNVCV9J]] $\varphi - \lambda T_K \varphi = f$.
In its classical form it builds the solution from the [[Fredholm determinant|PPX0AK81QV]] and the [[Fredholm minors|HP5VJ9EBST]], entire functions of $\lambda$.
It proves [[Fredholm's theorem|XQ8DYKW4SH]]: the equation behaves like a square linear system.
Its abstract form is the [[spectral theory of compact operators|JEPDM5B5FC]] on Banach spaces, and it leads to the notion of a [[Fredholm operator|8HWGKWD6ZM]].
It was one of the sources of functional analysis.

## Explanations

### L1 human

A square system of linear equations either has exactly one solution for each right-hand side, or its homogeneous version has a nonzero solution.
Fredholm theory shows that integral equations of the second kind behave in the same way, although they involve infinitely many unknowns, one value of $\varphi$ for each point.
The trick is to replace the integral by a finite sum, solve the resulting linear system with determinants, and let the number of points grow.
For a kernel such as $K(x, y) = xy$, the integral equation reduces exactly to one linear equation in one unknown, and the theory can be checked by hand.
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<document source="https://tamnd.github.io/math2/node/246SBC2M2N.md" tag="246SBC2M2N" kind="statement">
# Cohn's irreducibility criterion

## Statement

Let $p \ge 10$ be a [[prime number|0SN0MAEKYX]] with decimal expansion $p = a_m 10^m + \dots + a_1 10 + a_0$, where $m \ge 1$, $a_i \in \set{0, 1, \dots, 9}$ and $a_m \ne 0$.
Then the polynomial $f = a_m x^m + \dots + a_1 x + a_0$ is [[irreducible|00X2A9G9W3]] in $\Z[x]$.

## Remarks

Proof idea: since the coefficients are digits, every complex root $\alpha$ of $f$ satisfies $\abs{10 - \alpha} > 1$.
So every factor $g \in \Z[x]$ of positive degree has $\abs{g(10)} > 1$.
A factorisation $f = g h$ into factors of positive degree would then write $p = f(10) = g(10) h(10)$ as a product of two integers of absolute value greater than $1$.
A constant factor is also excluded: it would divide all digits and $p$, but the digits are less than $p$.
Example: $113$ is prime, so $x^2 + x + 3$ is irreducible; indeed its discriminant $1 - 12 = -11$ is negative.
The theorem also holds in every base $b \ge 2$ in place of $10$.
The converse is false: $x^2 + 5$ is irreducible, but its value $105$ at $10$ is not prime.

## Explanations

### L1 human

Write a prime in decimal and read its digits as the coefficients of a polynomial.
That polynomial cannot be factored over the integers.
For example, the prime $1013$ gives $x^3 + x + 3$, which is irreducible.

### L2 human

The criterion links primality of a value with irreducibility of the polynomial.
The reverse direction, that an irreducible polynomial with suitable conditions takes infinitely many prime values, is the Bunyakovsky conjecture, which is open in every degree at least $2$.
The proof is a good example of locating roots in the complex plane to prove an algebraic fact, like [[Perron's criterion|5BCRQYVYR8]].
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<document source="https://tamnd.github.io/math2/node/249ZVY6WK3.md" tag="249ZVY6WK3" kind="definition">
# Serial ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
A right $R$-[[module|0NZ8JA25WK]] is *serial* if it is a direct sum of [[uniserial modules|Z1045A0V29]].
The ring $R$ is *right serial* if the right module $R_R$ is serial.
It is *left serial* if the left module ${}_R R$ is serial.
It is *serial* if it is both right serial and left serial.

## Remarks

Example: let $k$ be a field and $R = k[x]/(x^n)$ with $n \ge 1$.
Its ideals are the ideals $(x^i)/(x^n)$ for $0 \le i \le n$, and they form a chain.
So $R$ is uniserial over itself on both sides, and $R$ is serial.

Non-example: let $R = k[x, y]/(x, y)^2$.
This is a local ring, so $R$ has no idempotents other than $0$ and $1$, and $R$ is indecomposable as a module over itself.
If it were serial, it would therefore be uniserial.
But the ideals $(x)$ and $(y)$ of $R$ are not comparable, so $R$ is not serial.

Equivalently, $R$ is right serial when $R = e_1 R \oplus \cdots \oplus e_m R$ for some orthogonal idempotents $e_1, \dots, e_m$ with each $e_i R$ uniserial, since a direct summand of $R_R$ has the form $eR$ for an idempotent $e$.
A finite-dimensional serial algebra over a field is also called a [[Nakayama algebra|APRFMTZ10K]].
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<document source="https://tamnd.github.io/math2/node/24CEMRAGJ8.md" tag="24CEMRAGJ8" kind="definition">
# Schur-convex function

## Definition

Let $n \ge 1$.
For $x \in \R^n$, let $x_{[1]} \ge x_{[2]} \ge \dots \ge x_{[n]}$ be the entries of $x$ in decreasing order.
A vector $y \in \R^n$ *majorizes* $x \in \R^n$, written $x \prec y$, if
$$\sum_{i=1}^{k} x_{[i]} \le \sum_{i=1}^{k} y_{[i]} \quad \text{for } k = 1, \dots, n - 1, \qquad \sum_{i=1}^{n} x_{[i]} = \sum_{i=1}^{n} y_{[i]};$$
see [[majorization|5K7VT2VKZT]].
A function $f \colon \R^n \to \R$ is *Schur-convex* if $f(x) \le f(y)$ for all $x, y \in \R^n$ with $x \prec y$.
It is *Schur-concave* if $-f$ is Schur-convex.

## Remarks

A Schur-convex function is symmetric: if $y$ is obtained from $x$ by a [[permutation|0BQMWCNS8S]] of the entries, then $x \prec y$ and $y \prec x$, so $f(x) = f(y)$.
Every symmetric convex function is Schur-convex.
For a symmetric $f$ with continuous partial derivatives on $\R^n$, the Schur–Ostrowski criterion says that $f$ is Schur-convex if and only if $(x_i - x_j)\big(\partial_i f(x) - \partial_j f(x)\big) \ge 0$ for all $x$ and all $i, j$.

Example: $f(x) = \max_i x_i = x_{[1]}$.
If $x \prec y$, the case $k = 1$ of the definition gives $x_{[1]} \le y_{[1]}$, so $f(x) \le f(y)$.

Non-example: $f(x_1, x_2) = x_1$ on $\R^2$.
The vectors $(1, 0)$ and $(0, 1)$ majorize each other, but $f(1, 0) = 1 > 0 = f(0, 1)$.
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<document source="https://tamnd.github.io/math2/node/24CP8Q56CV.md" tag="24CP8Q56CV" kind="definition">
# Scalar curvature

## Definition

Let $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] with [[Ricci curvature|4GTNM0WHDH]] $\operatorname{Ric}$.
In local coordinates at $p$, let $g_{ij}$ and $R_{ij}$ be the components of $g$ and $\operatorname{Ric}$, and let $(g^{ij})$ be the inverse matrix of $(g_{ij})$.
The *scalar curvature* of $(M, g)$ is the function $S \colon M \to \R$ given by
$$S(p) = \sum_{i,j} g^{ij}(p) \, R_{ij}(p),$$
the [[trace|07PTPSZPME]] of $\operatorname{Ric}$ with respect to $g$.

## Remarks

The value does not depend on the coordinates; it is the trace of the linear map $A_p \colon T_pM \to T_pM$ with $g(A_p u, v) = \operatorname{Ric}(u, v)$.
In the Riemannian case, for an orthonormal basis $e_1, \ldots, e_n$ of $T_pM$, $S(p) = \sum_i \operatorname{Ric}(e_i, e_i) = \sum_{i \ne j} K(e_i, e_j)$.

Example: on the unit sphere $S^n$, $\operatorname{Ric} = (n - 1)g$, so $S = \sum_{i,j} g^{ij} (n - 1) g_{ij} = (n - 1) n$.

Non-example: the plain sum $\sum_i R_{ii}$, without the factor $g^{ij}$, depends on the coordinates and is not the scalar curvature.
On the sphere of radius $r$ in $\R^3$, $\operatorname{Ric} = r^{-2} g$; in coordinates at $p$ with $g_{ij}(p) = r^2 \delta_{ij}$, one gets $\sum_i R_{ii}(p) = 2$ for every $r$, while $S = 2 / r^2$.

For a surface, $S = 2K$, where $K$ is the [[Gaussian curvature|SQVQ5DQT6W]].
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<document source="https://tamnd.github.io/math2/node/24D1EGHP4G.md" tag="24D1EGHP4G" kind="statement">
# Expansion of iterated differences in shifts

## Statement

Let $M$ be an additive commutative monoid, let $G$ be an additive abelian group, let $h \in M$, and let $f \colon M \to G$ be a function.
Let $\Delta_h^n f$ be the [[$n$-th forward difference|E38801KVJ1]] of $f$ with step $h$.
Then for every $n \in \N$ and every $x \in M$,
$$\Delta_h^n f(x) = \sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} f(x + kh),$$
where $\binom{n}{k}$ is the [[binomial coefficient|092GEF38YD]].

## Remarks

Write $\Delta_h = E_h - I$, where $E_h$ is the [[shift operator|91QMF4R1CY]] and $I$ is the identity.
The operators $E_h$ and $I$ commute, so the [[binomial theorem|0EHWPWGHB2]] gives $(E_h - I)^n = \sum_k \binom{n}{k} (-1)^{n-k} E_h^k$, and $E_h^k f(x) = f(x + kh)$.
A direct proof is an induction on $n$ with Pascal's rule.
Example: for $n = 3$ the formula reads $\Delta_h^3 f(x) = f(x + 3h) - 3f(x + 2h) + 3f(x + h) - f(x)$.
With $f(x) = x^3$, $h = 1$ and $x = 0$ this gives $27 - 24 + 3 - 0 = 6 = 3!$.

## Explanations

### L1 human

The $n$-th difference of a function can be computed in one step from $n + 1$ equally spaced values.
The weights are the binomial coefficients with alternating signs, as in a row of Pascal's triangle.
For example, the second difference is $f(x + 2h) - 2f(x + h) + f(x)$.

### L2 human

The formula is the inverse of [[Newton's forward difference formula|DDHCMKHD2T]], which writes shifted values in terms of differences.
Together the two formulas say that the sequences $(f(x + kh))_k$ and $(\Delta_h^k f(x))_k$ are each other's binomial transform.
The second difference $f(x + h) - 2f(x) + f(x - h)$, divided by $h^2$, is the standard finite difference approximation of $f''(x)$.
Combined with the fact that [[differences of polynomials vanish beyond the degree|EFYGR426DF]], it gives identities.
An example is $\sum_{k=0}^n (-1)^k \binom{n}{k} k^m = 0$ for $0 \le m < n$.
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<document source="https://tamnd.github.io/math2/node/24GAZHVTGB.md" tag="24GAZHVTGB" kind="definition">
# Dirichlet L-function

## Definition

Let $q \ge 1$ be an integer and let $\chi$ be a [[Dirichlet character|7XB5VJKDPG]] modulo $q$.
For $s \in \C$ with $\operatorname{Re} s > 1$, the [[Dirichlet series|8MWTEF6462]]
$$L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}$$
converges absolutely and defines a holomorphic function of $s$.
The *Dirichlet L-function* of $\chi$ is the [[analytic continuation|0M3YQJ81NB]] of this function to a [[meromorphic function|0ZZ7ZYDR8W]] on $\C$, also written $L(s, \chi)$.
If $\chi$ is not principal, $L(s, \chi)$ is an [[entire function|0G4F0XB9X2]].
If $\chi = \chi_0$ is principal, $L(s, \chi_0)$ has a single pole, a simple pole at $s = 1$.

## Remarks

For $\operatorname{Re} s > 1$ there is an Euler product $L(s, \chi) = \prod_p (1 - \chi(p) p^{-s})^{-1}$ over the primes $p$.
For the principal character, $L(s, \chi_0) = \zeta(s) \prod_{p \mid q} (1 - p^{-s})$, where $\zeta$ is the [[Riemann zeta function|1XWBY4N0VT]].
Example: for the nonprincipal character $\chi$ modulo $4$, the series at $s = 1$ converges and $L(1, \chi) = 1 - \frac{1}{3} + \frac{1}{5} - \cdots = \frac{\pi}{4}$, by the [[Leibniz formula for pi|037FD5NBRT]].
Non-example: $\sum_{n} (-1)^{n+1} n^{-s}$ is a Dirichlet series with $2$-periodic coefficients, but it is not a Dirichlet L-function, since $f(n) = (-1)^{n+1}$ is not a Dirichlet character: $f(4) = -1 \ne 1 = f(2) f(2)$.
For a primitive character the completed L-function satisfies a functional equation relating $s$ and $1 - s$, as for the zeta function.
</document>

<document source="https://tamnd.github.io/math2/node/24RRYYGC7S.md" tag="24RRYYGC7S" kind="definition">
# Urysohn integral equation

## Definition

Let $\Omega$ be a measurable subset of $\R^m$, for example a closed interval $[a, b]$.
Let $K \colon \Omega \times \Omega \times \R \to \R$ and $f \colon \Omega \to \R$ be given functions.
A *Urysohn integral equation* for an unknown function $\varphi \colon \Omega \to \R$ is
$$\varphi(x) = f(x) + \int_\Omega K\bigl(x, t, \varphi(t)\bigr)\, dt, \quad x \in \Omega.$$
The function $K$ is the *Urysohn kernel*, and the map $\varphi \mapsto \int_\Omega K(x, t, \varphi(t))\, dt$ is the *Urysohn operator*.

## Remarks

In the classical setting $\Omega = [a, b]$, $K$ and $f$ are [[continuous|0X3WGX95FE]], and $\varphi$ is sought among continuous functions.
If $K(x, t, u) = k(x, t)\, u$, the equation is a linear Fredholm equation of the second kind.
If $K(x, t, u) = k(x, t)\, g(t, u)$, it is a Hammerstein equation.
Some authors also call $\int_\Omega K(x, t, \varphi(t))\, dt = f(x)$ a Urysohn equation of the first kind.

Example: on $[0, 1]$, take $f(x) = x$ and $K(x, t, u) = \sin\bigl(x(u - t)\bigr)$.
Then $\varphi(x) = x$ is a solution: the integrand is $\sin 0 = 0$, so the right side is $x$.

Non-example: $\varphi(x)^3 = f(x) + \int_\Omega k(x, t)\, \varphi(t)\, dt$ is a nonlinear integral equation, but not a Urysohn equation, since the unknown appears outside the integral through $\varphi(x)^3$ and not alone.
</document>

<document source="https://tamnd.github.io/math2/node/24YCKE2RPH.md" tag="24YCKE2RPH" kind="statement">
# Conjugate residual method

## Statement

Let $A$ be a real symmetric [[positive definite|0KX66CZMBB]] $n \times n$ matrix, let $b \in \R^n$, and let $x_0 \in \R^n$.
The *conjugate residual method* sets $r_0 = b - A x_0$ and $p_0 = r_0$, and for $k = 0, 1, \dots$ while $r_k \ne 0$ computes
$$\alpha_k = \frac{r_k^T A r_k}{(A p_k)^T (A p_k)}, \quad x_{k+1} = x_k + \alpha_k p_k, \quad r_{k+1} = r_k - \alpha_k A p_k,$$
$$\beta_k = \frac{r_{k+1}^T A r_{k+1}}{r_k^T A r_k}, \quad p_{k+1} = r_{k+1} + \beta_k p_k, \quad A p_{k+1} = A r_{k+1} + \beta_k A p_k.$$
In exact arithmetic the following hold.

1. $r_k = b - A x_k$, and $x_k$ minimizes $\norm{b - A x}_2$ over $x \in x_0 + \mathcal{K}_k(A, r_0)$, where $\mathcal{K}_k$ is the [[Krylov subspace|QB45PCGQPV]].
2. The residuals are $A$-conjugate, $r_i^T A r_j = 0$ for $i \ne j$, and the vectors $A p_i$ are mutually orthogonal.
3. Some $r_k$ with $k \le n$ is zero.

## Remarks

The method is the [[conjugate gradient method|AFXZGQB4EN]] with the inner product $\ip{u}{v} = u^T A v$ in place of $u^T v$.
Conjugate gradients in this inner product minimize $\norm{x - x^*}_{A^2}$, and $\norm{x - x^*}_{A^2} = \norm{A(x - x^*)}_2 = \norm{b - A x}_2$, which gives clause 1.
Clause 2 is the image of the orthogonality and conjugacy relations of conjugate gradients under the change of inner product.
The update of $A p_{k+1}$ saves a product with $A$, so each step needs one product, $A r_{k+1}$.
Example: if $r_0$ is an eigenvector of $A$ for $\lambda$, then $\alpha_0 = \lambda r_0^T r_0 / \lambda^2 r_0^T r_0 = 1 / \lambda$ and $r_1 = 0$.
For a symmetric indefinite $A$ the same formulas can be run, but $r_k^T A r_k$ can vanish and the method then breaks down.
The [[MINRES|3HRHSHTG9C]] method avoids this breakdown.

## Explanations

### L1 human

The conjugate residual method looks like conjugate gradients, but it chooses each step to make the residual, rather than the energy error, as small as possible.
So the size of the residual never grows from one step to the next.
It costs about the same as conjugate gradients, with one extra stored vector.

### L2 human

For symmetric positive definite systems, conjugate residuals and conjugate gradients minimize different norms over the same Krylov spaces.
In practice the residual norm of conjugate residuals is monotone, which is convenient for stopping tests.
For indefinite symmetric systems [[MINRES|3HRHSHTG9C]] is the robust choice, and for nonsymmetric systems the generalizations GCR and GMRES apply.
Like all Krylov [[iterative methods|2RPE77ACER]], it is normally combined with a preconditioner.
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<document source="https://tamnd.github.io/math2/node/256NP6JBEQ.md" tag="256NP6JBEQ" kind="concept">
# Sinai–Ruelle–Bowen measure

## Summary

A Sinai–Ruelle–Bowen measure, or SRB measure, is an [[invariant measure|B232EZ1FMB]] of a smooth dynamical system that is smooth along the expanding directions.
Such measures exist for many [[attractors|7FWX24X5MW]] that have zero volume, where no invariant measure can be absolutely continuous.
They are the natural measures to observe.
Take a uniformly hyperbolic attractor.
For almost every initial point of its basin, in the sense of volume, the time averages equal the SRB space averages.

## Explanations

### L1 human

Run a chaotic system on a computer from a random starting point and record how often the orbit visits each region.
The histogram usually settles down to the same limit for almost every start; that limit is the SRB measure.
On a strange attractor the measure lives on a set of zero volume, so it is not given by a density in space.
Along the stretched directions, however, it looks like length or area with a density.
For [[Arnold's cat map|BSCZN8Y1AN]] the SRB measure is plain area on the torus.
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<document source="https://tamnd.github.io/math2/node/259Y10BYJC.md" tag="259Y10BYJC" kind="concept">
# Delaunay triangulation

## Summary

A Delaunay triangulation of a finite point set in the plane is a special [[triangulation|V0B6C6TMWD]] of it.
The circumcircle of each of its triangles has no point of the set inside.
Among all triangulations of the set, it maximizes the smallest angle, so it avoids thin triangles.
It is the geometric dual of the [[Voronoi diagram|QDNEBBVZ84]], and it is widely used for meshes.

## Explanations

### L1 human

Take a triangulation of a set of points and look at the circle through the three corners of each triangle.
The triangulation is Delaunay when no other point of the set sits strictly inside any of these circles.
For four points at the corners of a square, all four lie on one circle, and both diagonals give a Delaunay triangulation.
If you push one corner slightly outward, only one diagonal remains Delaunay.
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<document source="https://tamnd.github.io/math2/node/25FRWWHFZV.md" tag="25FRWWHFZV" kind="concept">
# Choquet capacity

## Summary

A Choquet capacity is a monotone set function defined on all subsets of a space.
It is continuous along increasing sequences of sets and along decreasing sequences of compact sets.
Choquet introduced the notion to treat [[Newtonian capacity|4YZVFGJVSZ]] and outer measures in one framework.
Its main result, Choquet's capacitability theorem, says that analytic sets can be approximated from inside by compact sets for every Choquet capacity.
This theorem is a basic tool for measurability questions in probability and descriptive set theory.

## Explanations

### L1 human

A measure adds up sizes of disjoint pieces, but many useful notions of size do not add.
A Choquet capacity asks only for three softer things.
A bigger set is not smaller.
Growing sets have sizes that grow to the size of the limit.
Shrinking compact sets have sizes that shrink to the size of the limit.
The outer measure of a finite measure is an example, and so is the diameter of a set in a metric space.
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<document source="https://tamnd.github.io/math2/node/25G4AP9P4R.md" tag="25G4AP9P4R" kind="problem">
# Gelfand's problem on the index

## Problem

Let $M$ be a closed smooth manifold and let $P$ be an [[elliptic differential operator|MT9P2X85V8]] between vector bundles over $M$.
The [[analytic index|44GBFZ16SG]] $\operatorname{ind} P = \dim \ker P - \dim \operatorname{coker} P$ is unchanged under continuous deformations of $P$ through elliptic operators.
Find a formula for $\operatorname{ind} P$ in terms of topological invariants of $M$ and of the principal symbol of $P$.

## Remarks

Gelfand raised the question in an article on elliptic equations published in 1960.
Atiyah and Singer solved it with their index theorem, announced in 1963: the analytic index equals the [[topological index|QDCD33CS21]].
Earlier special cases were known, such as the Riemann–Roch theorem for curves and its generalization by Hirzebruch, and the Hirzebruch signature theorem.

## Explanations

### L1 human

The index of an elliptic operator is a whole number that does not change when the operator is deformed.
Numbers with this kind of stability are usually topological, so Gelfand asked for a topological formula for it.
The answer, the Atiyah–Singer index theorem, became one of the central results of twentieth-century mathematics.
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<document source="https://tamnd.github.io/math2/node/25HF1WPPKX.md" tag="25HF1WPPKX" kind="statement">
# Schur test

## Statement

Let $(X, \mu)$ and $(Y, \nu)$ be $\sigma$-finite [[measure spaces|P15VQS5DJ9]], and let $K \colon X \times Y \to \C$ be a [[measurable function|YYNCJSR0PJ]].
Assume that there are measurable functions $p \colon X \to (0, \infty)$ and $q \colon Y \to (0, \infty)$ and constants $\alpha, \beta > 0$ such that
$$\int_Y \abs{K(x, y)} \, q(y) \, d\nu(y) \le \alpha \, p(x) \quad \text{for almost every } x \in X,$$
$$\int_X \abs{K(x, y)} \, p(x) \, d\mu(x) \le \beta \, q(y) \quad \text{for almost every } y \in Y.$$
Then for every $f \in L^2(Y, \nu)$ the integral
$$(T f)(x) = \int_Y K(x, y) \, f(y) \, d\nu(y)$$
converges absolutely for almost every $x$.
Moreover, $T$ is a [[bounded linear operator|03TM135G0N]] from $L^2(Y, \nu)$ to $L^2(X, \mu)$ with operator norm $\norm{T} \le \sqrt{\alpha \beta}$.

## Remarks

Proof idea: write $\abs{K} \abs{f} = \big(\abs{K} q\big)^{1/2} \big(\abs{K} \abs{f}^2 / q\big)^{1/2}$ and apply the Cauchy–Schwarz inequality in $y$.
The first hypothesis gives $\abs{T f(x)}^2 \le \alpha \, p(x) \int_Y \abs{K(x, y)} \abs{f(y)}^2 q(y)^{-1} \, d\nu(y)$.
Integrating in $x$ and exchanging the integrals by Tonelli's theorem, the second hypothesis gives $\norm{T f}_2^2 \le \alpha \beta \norm{f}_2^2$.

With $p = q = 1$ the test says: if $\int_Y \abs{K(x, y)} \, d\nu(y) \le \alpha$ for almost every $x$ and $\int_X \abs{K(x, y)} \, d\mu(x) \le \beta$ for almost every $y$, then $\norm{T} \le \sqrt{\alpha \beta}$ on $L^2$.
For $\alpha = \beta$ this is the case $p = 2$ of [[Young's inequality for integral operators|48NS4ZCSR3]].
The freedom to choose the weights $p$ and $q$ is what makes the test useful for kernels such as $1/(x + y)$, which have no finite row or column integrals.

Example: let $X = Y = (0, \infty)$ with Lebesgue measure and $K(x, y) = 1/(x + y)$.
Take $p(x) = x^{-1/2}$ and $q(y) = y^{-1/2}$.
The substitution $y = x u$ gives $\int_0^\infty \frac{y^{-1/2}}{x + y} \, dy = x^{-1/2} \int_0^\infty \frac{u^{-1/2}}{1 + u} \, du = \pi x^{-1/2}$, and by symmetry the same holds for the second hypothesis.
So $\alpha = \beta = \pi$ and $\norm{T} \le \pi$, which is the continuous form of [[Hilbert's inequality|G4R4MWWP3C]].

## Explanations

### L1 human

An integral operator acts on a function by averaging it against a kernel $K(x, y)$.
The Schur test gives a simple way to see that such an operator does not blow up the size of functions, measured in $L^2$.
One only has to find one positive test function for the rows and one for the columns of the kernel, and check two integral inequalities.

### L2 human

The Schur test is the main elementary tool to prove that an integral operator is bounded on $L^2$ when its kernel is not square-integrable.
Such an operator is not [[Hilbert–Schmidt|W3BX66Y47G]].
It is used for kernels with a singularity or slow decay, such as homogeneous kernels on $(0, \infty)$ and kernels in the study of Bergman projections.
It is the continuous analogue of the bound of the norm of a matrix by the geometric mean of the largest row sum and the largest column sum.
For $L^p$ with $p \ne 2$, the same idea with Hölder's inequality in place of Cauchy–Schwarz gives a version with weights.
</document>

<document source="https://tamnd.github.io/math2/node/25RZTHSRS3.md" tag="25RZTHSRS3" kind="definition">
# Strictly convex set

## Definition

Let $V$ be a real [[topological vector space|T32W553W95]], such as $\R^n$, and let $C \subseteq V$.
The set $C$ is *strictly convex* if for all $x, y \in C$ with $x \ne y$, the [[open segment|CCZA91NE5X]] $(x, y)$ lies in the [[interior|0RZH6W98HK]] of $C$.

## Remarks

A strictly convex set is [[convex|02AS45T3H9]], since $(x, y) \subseteq \operatorname{int} C \subseteq C$.

Example: the closed unit disc $D = \set{p \in \R^2 : \abs{p} \le 1}$.
If $x \ne y$ in $D$ and $0 < t < 1$, then $\abs{(1 - t)x + t y} < 1$.
Indeed, equality would need $\abs{x} = \abs{y} = 1$ and equality in the triangle inequality.
For the Euclidean norm this means that $x$ and $y$ point in the same direction, which forces $x = y$.
So the open segment lies in the open disc, the interior of $D$.

Non-example: the closed square $[0, 1]^2$.
The points $(0, 0)$ and $(1, 0)$ lie in it, but the midpoint $(1/2, 0)$ is on the boundary, not in the interior.

Every open convex set is strictly convex, so the notion is of interest mainly for closed sets.
</document>

<document source="https://tamnd.github.io/math2/node/25V2MHPHZ7.md" tag="25V2MHPHZ7" kind="counterexample">
# Kolmogorov's everywhere divergent Fourier series

## Counterexample

Let $\mathbb{T} = \R / 2\pi\Z$ and let $L^1(\mathbb{T})$ be the [[space|SASP14MA1Z]] of integrable $2\pi$-periodic functions.
For $f \in L^1(\mathbb{T})$ let $S_N f(x) = \sum_{\abs{k} \le N} \hat f(k) e^{ikx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]], where $\hat f(k)$ are the [[Fourier coefficients|5NN2YZZR3V]].
Kolmogorov constructed a function $f \in L^1(\mathbb{T})$ such that the sequence $S_N f(x)$ diverges for every $x \in \R$.
So integrability alone gives no [[convergence of the Fourier series|V6Q24CHJG8]] at any point.

## Remarks

Kolmogorov first gave, in 1923, an integrable function whose Fourier series diverges almost everywhere, and in 1926 one whose Fourier series diverges everywhere.
The construction adds up a series of carefully chosen trigonometric polynomials.
Each has small $L^1$ norm, but some of its partial sums are very large on most of the circle.
The coefficients $\hat f(k)$ still tend to $0$, by the [[Riemann–Lebesgue lemma|WVKDC011KP]], so the divergence is not caused by large terms.
The example shows that [[Carleson's theorem|FBH6KG484Z]] and [[Hunt's extension|NAX51M6025]], which give convergence almost everywhere for $f \in L^p(\mathbb{T})$ with $1 < p < \infty$, have no analogue for $p = 1$.

## Explanations

### L1 human

A Fourier series tries to rebuild a periodic function from pure waves.
For square-integrable functions this works at almost every point.
Kolmogorov found an integrable function for which it fails at every single point: the partial sums never settle down anywhere.
So the right space for pointwise convergence questions is smaller than the space of all integrable functions.
</document>

<document source="https://tamnd.github.io/math2/node/25Y6SVM2D1.md" tag="25Y6SVM2D1" kind="definition">
# Wilson polynomials

## Definition

Let $a, b, c, d \in \C$ and let $n \ge 0$ be an integer.
Write $(u)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]].
The *Wilson polynomial* of degree $n$ is
$$W_n(x^2; a, b, c, d) = (a + b)_n (a + c)_n (a + d)_n \sum_{k=0}^{n} \frac{(-n)_k \, (n + a + b + c + d - 1)_k \, (a + ix)_k \, (a - ix)_k}{(a + b)_k \, (a + c)_k \, (a + d)_k \, k!},$$
that is, $(a + b)_n (a + c)_n (a + d)_n$ times the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_4F_3\left( -n, n + a + b + c + d - 1, a + ix, a - ix; a + b, a + c, a + d; 1 \right)$.

## Remarks

Since $(a + ix)_k (a - ix)_k = \prod_{j=0}^{k-1} \left( (a + j)^2 + x^2 \right)$, $W_n$ is a polynomial of degree $n$ in $x^2$.
Example: $W_1(x^2) = (abc + abd + acd + bcd) - (a + b + c + d) x^2$.
Check: $(a + b)(a + c)(a + d) - (a + b + c + d) a^2 = abc + abd + acd + bcd$, and the coefficient of $x^2$ is $-(a + b + c + d)$.
This example is symmetric in $a, b, c, d$; in fact every $W_n$ is.
Orthogonality: if $a, b, c, d$ have positive real parts and the nonreal ones come in complex conjugate pairs, then for $m \ne n$
$$\int_0^\infty W_m(x^2) W_n(x^2) \abs{\frac{\Gamma(a + ix) \Gamma(b + ix) \Gamma(c + ix) \Gamma(d + ix)}{\Gamma(2ix)}}^2 dx = 0,$$
where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
The positivity hypothesis matters: if, say, $a < 0$, the integral alone no longer gives orthogonality, and the orthogonality measure needs finitely many extra point masses.
Non-example: the polynomial $x$ is not a Wilson polynomial for any parameters, since every $W_n$ is a polynomial in $x^2$ and hence an even function of $x$.
</document>

<document source="https://tamnd.github.io/math2/node/263GGJFA21.md" tag="263GGJFA21" kind="example">
# Rotation group SO(3)

## Example

The *rotation group* is the [[special orthogonal group|0CGYNDCWRR]] in dimension $3$:
$$SO(3) = \set{R \in M_3(\R) : R^T R = I_3, \ \det R = 1}.$$
Each $R \in SO(3)$ is a rotation by some angle $\theta \in [0, \pi]$ about some line through the origin.
For example, the rotation by $\theta$ about the $z$-axis is
$$R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$
Check: its columns are orthonormal, so $R_z(\theta)^T R_z(\theta) = I_3$, and its determinant is $\cos^2\theta + \sin^2\theta = 1$.

$SO(3)$ is a [[compact Lie group|9TFC6ADC1C]]: it is a closed subgroup of $GL_3(\R)$, so a [[Lie group|7JBRGCVJRB]], and it is closed and bounded in $M_3(\R)$.

## Remarks

$SO(3)$ is connected and has dimension $3$.
Its Lie algebra $\mathfrak{so}(3)$ is the space of real skew-symmetric $3 \times 3$ matrices, which is isomorphic to $\R^3$ with the [[cross product|0N5N15HWZ7]] as bracket.
As a manifold, $SO(3)$ is diffeomorphic to the [[real projective space|1DDYB6VVSR]] $\R P^3$.
Its fundamental group has order $2$, so it is not simply connected.
Its universal cover is $SU(2)$, the group of unit quaternions, and the covering map has kernel $\set{\pm 1}$.

## Explanations

### L1 human

The rotation group is the set of all ways to turn a rigid object in space while keeping its centre fixed.
Doing one rotation after another is again a rotation, and every rotation can be undone.
Unlike rotations in the plane, rotations in space do not commute.
A quarter turn about one axis followed by a quarter turn about another axis gives a different result in the other order.
A loop of rotations that turns once by $360$ degrees cannot be shrunk to "doing nothing".
The loop that turns twice, by $720$ degrees, can be shrunk.
This is the meaning of the fundamental group of order $2$.
</document>

<document source="https://tamnd.github.io/math2/node/269C7M56G5.md" tag="269C7M56G5" kind="statement">
# Ball's existence theorem in nonlinear elasticity

## Statement

Let $\Omega \subseteq \R^3$ be a bounded open connected set with Lipschitz boundary, and let $\Gamma_0 \subseteq \partial \Omega$ have positive surface measure.
Let $\mathbb{M}^3_+$ be the set of $3 \times 3$ real matrices with positive [[determinant|00KDZRW4C0]], and let $W \colon \Omega \times \mathbb{M}^3_+ \to \R$ be a stored energy function such that:

1. ([[polyconvexity|8VYYWQ5CMB]]) $W(x, F) = G(x, F, \operatorname{cof} F, \det F)$ for a function $G \colon \Omega \times \R^9 \times \R^9 \times (0, \infty) \to \R$ that is convex in its last three arguments for almost every $x$, and is a Carathéodory function;
2. (behaviour under compression) $W(x, F) \to +\infty$ as $\det F \to 0^+$, for almost every $x$;
3. (coercivity) $W(x, F) \ge \alpha \big( \abs{F}^p + \abs{\operatorname{cof} F}^q + (\det F)^r \big) + \beta$ for constants $\alpha > 0$, $\beta \in \R$, $p \ge 2$, $q \ge p / (p - 1)$ and $r > 1$.

Here $\operatorname{cof} F$ is the [[cofactor|0XPKR18APA]] matrix.
Let $\varphi_0 \colon \Gamma_0 \to \R^3$ be given.
Let $\mathcal{A}$ be the set of $\varphi$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\Omega; \R^3)$ with $\varphi = \varphi_0$ on $\Gamma_0$ such that:
$\operatorname{cof} \nabla \varphi \in L^q$, $\det \nabla \varphi \in L^r$, and $\det \nabla \varphi > 0$ almost everywhere.
Let $I(\varphi) = \int_\Omega W(x, \nabla \varphi(x)) \, dx$.
If $\inf_{\mathcal{A}} I < +\infty$, then there is $\varphi \in \mathcal{A}$ with $I(\varphi) = \inf_{\mathcal{A}} I$.

## Remarks

The proof uses the [[direct method|TE379HBTY0]].
Coercivity bounds a minimising sequence $\varphi_k$, so along a subsequence $\varphi_k$, $\operatorname{cof} \nabla \varphi_k$ and $\det \nabla \varphi_k$ converge weakly in $L^p$, $L^q$ and $L^r$.
The key point is that the weak limits are $\operatorname{cof} \nabla \varphi$ and $\det \nabla \varphi$ of the limit map, because minors are null Lagrangians in divergence form.
Convexity of $G$ then gives lower semicontinuity, and condition 2 keeps $\det \nabla \varphi > 0$ almost everywhere.

The theorem is due to John Ball, in 1977, and it holds with dead loads added to $I$ as linear terms.
Convexity of $W$ in $F$ would be an unsuitable hypothesis, since the set $\mathbb{M}^3_+$ is not convex and $W$ must blow up as $\det F \to 0^+$.

Example: the energies $W(F) = a \abs{F}^2 + b \abs{\operatorname{cof} F}^2 + \Gamma(\det F)$, with $a, b > 0$ and $\Gamma$ convex on $(0, \infty)$ with $\Gamma(d) \to +\infty$ as $d \to 0^+$ and $\Gamma(d) \ge c \, d^r - C$ for some $c > 0$, $C$ and $r > 1$, satisfy the hypotheses with $p = q = 2$.

## Explanations

### L1 human

An elastic body, clamped on part of its boundary, settles into a shape of least stored energy.
Ball's theorem says that such a shape exists for a large class of realistic materials.
The materials may resist changes of length, of area and of volume, each in a convex way, and they resist being crushed to zero volume infinitely strongly.

### L2 human

Before this theorem, existence in nonlinear elasticity was known mainly near the reference configuration, by the implicit function theorem.
Ball's idea was to replace convexity, which is physically wrong, by polyconvexity, and to use the weak continuity of determinants and cofactors.
The minimiser is only a Sobolev map, and whether it satisfies the equilibrium equations, is injective, or is smooth are difficult further questions.
The theorem made polyconvexity a central notion of the calculus of variations, next to quasiconvexity and rank-one convexity.
</document>

<document source="https://tamnd.github.io/math2/node/26C8QTZSVY.md" tag="26C8QTZSVY" kind="definition">
# Low-discrepancy sequence

## Definition

Let $s \ge 1$ be an integer, and let $x_1, x_2, \ldots$ be a sequence of points of $[0, 1)^s$.
For $t = (t_1, \ldots, t_s) \in [0, 1]^s$ let $A_N(t)$ be the number of indices $i \le N$ with $x_i \in [0, t_1) \times \cdots \times [0, t_s)$.
The *star discrepancy* of the first $N$ points is
$$D_N^* = \sup_{t \in [0, 1]^s} \left| \frac{A_N(t)}{N} - t_1 t_2 \cdots t_s \right|.$$
The sequence is a *low-discrepancy sequence* if some constant $c$, independent of $N$, satisfies
$$D_N^* \le c\, \frac{(\log N)^s}{N} \qquad \text{for all } N \ge 2.$$

## Remarks

Example: the van der Corput sequence in base $2$, with $s = 1$, is $x_i = \sum_k a_k 2^{-k-1}$ when $i = \sum_k a_k 2^k$ in binary; its first terms are $\frac12, \frac14, \frac34, \frac18$.
It satisfies $D_N^* = O(\log N / N)$ in [[big O notation|0TGV80E6A4]], so it is a low-discrepancy sequence.
Non-example: the constant sequence $x_i = 0$ has $A_N(t) = N$ for every $t$ with all $t_j > 0$, so $D_N^* = 1$ for all $N$, and the bound fails.
A sequence has $D_N^* \to 0$ if and only if it is uniformly distributed in $[0, 1)^s$; for $s = 1$ this is [[equidistribution modulo one|XM6DH57BWY]].
So every low-discrepancy sequence is uniformly distributed, but the converse fails.
For independent uniform random points, $D_N^*$ is typically of order $N^{-1/2}$ up to logarithmic factors, much larger than $(\log N)^s / N$ for large $N$.
Some authors require the bound only for $N$ large, or use the extreme discrepancy over all boxes, not only those anchored at $0$; this changes only the constant.
</document>

<document source="https://tamnd.github.io/math2/node/26E0HYRDS8.md" tag="26E0HYRDS8" kind="concept">
# Dickey–Fuller test

## Summary

The Dickey–Fuller test is a [[statistical test|087P03NEND]] for an autoregressive time series.
Its null hypothesis is a unit root, which means random-walk behaviour, and its alternative is stationarity.
Its test statistic is an ordinary $t$-ratio.
Under the null hypothesis this ratio does not follow a Student or normal distribution, so special critical values are needed.
David Dickey and Wayne Fuller introduced it in 1979, and its augmented version is a standard first step in the analysis of economic time series.

## Explanations

### L1 human

Some series, like a stock price, wander without any pull back to a fixed level: a shock today moves them for good.
Other series fluctuate around a level and forget shocks over time.
The Dickey–Fuller test asks which kind a series is, by regressing each change on the previous level.
If high levels tend to be followed by falls, strongly enough by the test's standard, the series is pulled back and the random-walk hypothesis is rejected.
</document>

<document source="https://tamnd.github.io/math2/node/26GTNMAF7B.md" tag="26GTNMAF7B" kind="concept">
# Stable homotopy category

## Summary

The stable homotopy category has [[spectra|1GHF4EKBXS]] as objects and homotopy classes of maps of spectra as morphisms.
In it the stable weak equivalences are made invertible.
In it the suspension functor is an equivalence, every morphism set is an abelian group, and cofibre sequences are also fibre sequences.
It is a [[triangulated category|T2V7GJJ2EN]] with a symmetric monoidal smash product, and it is the setting of [[stable homotopy theory|545EV0W4CE]].

## Explanations

### L1 human

Spaces can be suspended, but not desuspended.
In the stable homotopy category every object can also be desuspended, so shifting up and down are inverse operations.
Maps between objects form abelian groups, much as linear maps between vector spaces do.
For two finite CW complexes, the maps between their suspension spectra are exactly the stable homotopy classes of maps between them.
For example, the maps from the sphere spectrum to itself form the group $\Z$, given by degree.
</document>

<document source="https://tamnd.github.io/math2/node/26RWFX4H2H.md" tag="26RWFX4H2H" kind="definition">
# Forgetful functor

## Definition

Let $\sigma$ be an algebraic signature and let $K$ be a class of $\sigma$-algebras.
Let $\mathcal{K}$ be the [[category|A2DE66937D]] whose objects are the members of $K$ and whose morphisms are the [[homomorphisms|KBXSZC6BV1]] between them.
The *forgetful functor* $U \colon \mathcal{K} \to \mathbf{Set}$ is the [[functor|28NQ0KRHD3]] that sends each algebra $A$ to its underlying set.
It sends each homomorphism $f \colon A \to B$ to the same map $f$ between the underlying sets.

More generally, let $\sigma' \subseteq \sigma$.
The functor that sends a $\sigma$-algebra to its reduct to the operations of $\sigma'$, and each homomorphism to itself, is also called forgetful.

## Remarks

$U$ is a functor because the identity homomorphism is the identity map and the composite of homomorphisms is their composite as maps.
It is faithful: two homomorphisms $A \to B$ with the same underlying map are equal.

Example: the forgetful functor $\mathbf{Ring} \to \mathbf{Ab}$ keeps the addition of a ring and drops its multiplication and unit.
A ring homomorphism is additive, so it is sent to a group homomorphism.

Non-example: consider the functor $\mathbf{Grp} \to \mathbf{Set}$ that sends a group $G$ to the set of its subgroups, and a homomorphism $f$ to $H \mapsto f(H)$.
It is a functor, but not a forgetful functor.
It does not send $G$ to its underlying set.

The term is also used informally for functors such as $\mathbf{Top} \to \mathbf{Set}$, which forget a topology rather than operations.
</document>

<document source="https://tamnd.github.io/math2/node/26XTJJT267.md" tag="26XTJJT267" kind="problem">
# Gauss circle problem

## Problem

For a real number $r > 0$, let $N(r)$ be the number of points $(x, y) \in \Z^2$ with $x^2 + y^2 \le r^2$, and let $E(r) = N(r) - \pi r^2$.
The problem is to determine the infimum $\theta$ of the real numbers $\alpha$ such that $E(r) = O(r^{\alpha})$ as $r \to \infty$, in [[big O notation|0TGV80E6A4]].

## Remarks

Gauss observed that $\abs{E(r)} \le 2\sqrt{2}\,\pi r$ for $r \ge 1$: the unit squares centred at the counted points lie between the discs of radius $r - \sqrt2/2$ and $r + \sqrt2/2$.
So $\theta \le 1$.
Sierpiński proved in 1906 that $E(r) = O(r^{2/3})$.
Hardy and Landau showed independently that $E(r)$ is not $O(r^{1/2})$, so $\theta \ge 1/2$.
It is conjectured that $\theta = 1/2$, that is, $E(r) = O(r^{1/2 + \varepsilon})$ for every $\varepsilon > 0$.
Martin Huxley proved $E(r) = O(r^{131/208 + \varepsilon})$ in 2003, and the problem remains open.
For example, $N(1) = 5$ and $N(2) = 13$, while $\pi \approx 3.14$ and $4\pi \approx 12.57$.

## Explanations

### L1 human

Draw a circle of radius $r$ around the origin on squared paper and count the grid points inside it.
The count is close to the area $\pi r^2$, since each grid point owns one unit square.
The question is how big the error can be.
The easy bound is about the length of the circle, but the true error appears to be much smaller, about the square root of that.
</document>

<document source="https://tamnd.github.io/math2/node/272KQSXNEA.md" tag="272KQSXNEA" kind="definition">
# Classifying space

## Definition

Let $G$ be a [[topological group|92700QAB5Y]].
A *classifying space* of $G$ is the base space $B$ of a [[universal principal $G$-bundle|1HDH716Z24]] $E \to B$.
It is written $BG$, and the total space is written $EG$.

## Remarks

Any two classifying spaces of $G$ are homotopy equivalent, so $BG$ is well defined up to homotopy.
For a CW complex $X$, isomorphism classes of [[principal $G$-bundles|8E37E74XVP]] over $X$ correspond to homotopy classes of maps $X \to BG$.
A principal $G$-bundle over a CW complex with [[contractible|R5WP8Z69Q8]] total space is universal, so its base is a $BG$.

Example: the circle group $S^1$ acts freely on the unit sphere $S^\infty$ of $\C^\infty$, with quotient $\C P^\infty$.
Since $S^\infty$ is contractible, $\C P^\infty$ is a classifying space of $S^1$.

Non-example: $\C P^1$ is not a classifying space of $S^1$.
The bundle $S^3 \to \C P^1$ is a principal $S^1$-bundle, but its total space $S^3$ is not contractible, and $\pi_3(\C P^1) \cong \Z$ while $\pi_3(\C P^\infty) = 0$.

For a discrete group $G$, a classifying space is the same as a [[classifying space of the group|WA796ZCNRX]], an Eilenberg–MacLane space $K(G, 1)$.
Other examples: $BO(n)$ and $BU(n)$ are infinite [[Grassmannians|CYPA882RB9]], and $B\Z = S^1$.
</document>

<document source="https://tamnd.github.io/math2/node/272T16NWZ9.md" tag="272T16NWZ9" kind="concept">
# Grothendieck construction

## Summary

A [[functor|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathbf{Cat}$ assigns a category to each object of $\mathcal{C}$.
The Grothendieck construction turns $F$ into a single [[category|A2DE66937D]] $\int F$ with a projection functor to $\mathcal{C}$.
It glues the categories $F(c)$ together along the functors $F(f)$.
It is the bridge between functors into $\mathbf{Cat}$ and categories fibred over $\mathcal{C}$.
It contains the category of elements of a set-valued functor as a special case.

## Explanations

### L1 human

Think of $\mathcal{C}$ as a base, and above each point $c$ of the base sits a whole category $F(c)$, called the fibre.
The Grothendieck construction puts all the fibres side by side and adds arrows that move from one fibre to another, following arrows of the base.
An object of $\int F$ is a pair: a base object $c$ and an object $x$ of the fibre over $c$.
For a small example, if $F$ is constant with value a category $\mathcal{D}$, then $\int F$ is the [[product category|AFNE9C4NH6]] $\mathcal{C} \times \mathcal{D}$.
If every fibre is a set with only identity arrows, the construction gives the category of elements of $F$.
</document>

<document source="https://tamnd.github.io/math2/node/273NZSYWZA.md" tag="273NZSYWZA" kind="definition">
# Subnormal operator

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $S \colon H \to H$ be a bounded linear operator.
The operator $S$ is *subnormal* when there are a Hilbert space $K \supseteq H$, with $H$ closed in $K$, and a bounded normal operator $N \colon K \to K$ such that

1. $N(H) \subseteq H$, that is, $H$ is an [[invariant subspace|0ACX5QT8TE]] of $N$;
2. $Nx = Sx$ for all $x \in H$.

Such an $N$ is a *normal extension* of $S$.

## Remarks

Example: the unilateral shift $S$ on $\ell^2(\N)$ is subnormal.
View $\ell^2(\N)$ as the sequences in $\ell^2(\Z)$ that vanish at negative indices, and let $N$ be the bilateral shift $(Nx)_n = x_{n-1}$, which is unitary, hence normal.
$N$ maps sequences vanishing at negative indices to sequences of the same kind, and it agrees with $S$ there.

Non-example: $S = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ on $\C^2$ is not subnormal.
A subnormal operator is hyponormal, that is $S^*S - SS^* \ge 0$, but here $S^*S - SS^* = \mathrm{diag}(-1, 1)$, which is not positive.

A normal extension $N$ can be chosen minimal: $K$ is the smallest closed subspace containing $H$ and invariant under $N$ and $N^*$.
The minimal normal extension is unique up to unitary equivalence.
</document>

<document source="https://tamnd.github.io/math2/node/275GQ9T7EB.md" tag="275GQ9T7EB" kind="definition">
# Exponential map

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and $p \in M$.
For $v$ in the [[tangent space|WY6K3P4YCG]] $T_pM$, let $\gamma_v$ be the maximal [[geodesic|NSYMQMHPDN]] with $\gamma_v(0) = p$ and $\gamma_v'(0) = v$.
Let $\mathcal{E}_p \subseteq T_pM$ be the set of $v$ such that $\gamma_v$ is defined on an interval containing $[0, 1]$.
The *exponential map* at $p$ is
$$\exp_p \colon \mathcal{E}_p \to M, \qquad \exp_p(v) = \gamma_v(1).$$

## Remarks

Example: in $\R^n$ with the Euclidean metric, $\gamma_v(t) = p + tv$, so $\mathcal{E}_p = \R^n$ and $\exp_p(v) = p + v$.

Non-example: in the punctured plane $M = \R^2 \setminus \set{0}$ with $p = (1, 0)$ and $v = (-2, 0)$, the geodesic $\gamma_v(t) = (1 - 2t, 0)$ hits the removed origin at $t = 1/2$.
So $v \notin \mathcal{E}_p$, and $\exp_p(v)$ is not defined.

The set $\mathcal{E}_p$ is open and star-shaped about $0$, and $\exp_p$ is smooth on it.
By rescaling, $\gamma_v(t) = \exp_p(tv)$.
The differential of $\exp_p$ at $0$ is the identity of $T_pM$.
So, by the inverse function theorem, $\exp_p$ maps a neighbourhood of $0$ diffeomorphically onto a neighbourhood of $p$.
</document>

<document source="https://tamnd.github.io/math2/node/275YT975A8.md" tag="275YT975A8" kind="concept">
# Complex geodesic

## Summary

A complex geodesic of a complex manifold $X$ is a holomorphic map from the unit disc into $X$ that preserves distances.
The disc carries its Poincaré distance, and $X$ carries its [[Kobayashi pseudodistance|DY8QQ5GTTX]].
Its image is a holomorphic disc along which the invariant geometry of $X$ is exactly that of the hyperbolic disc.
Edoardo Vesentini introduced the notion, first with the [[Carathéodory distance|SYNST4W3FR]].
By [[Lempert's theorem|2PANNSQ3G6]], in a bounded convex domain any two distinct points lie on a complex geodesic.

## Explanations

### L1 human

In the unit disc, holomorphic self-maps never increase hyperbolic distances.
A holomorphic disc in a larger manifold usually makes points look farther apart in the disc than they are in the manifold.
A complex geodesic is a disc for which no distance is lost: the distance of two points of the disc in the manifold equals their hyperbolic distance in the disc.
In the unit ball of $\C^n$, every slice of the ball by a complex line through the centre gives a complex geodesic.
</document>

<document source="https://tamnd.github.io/math2/node/2767J23XHN.md" tag="2767J23XHN" kind="concept">
# Jacobi sum

## Summary

A Jacobi sum is a sum $\sum_{a} \chi(a)\lambda(1 - a)$ over a [[finite field|0ZXZ82FTX2]], built from two multiplicative characters $\chi$ and $\lambda$.
It counts, in a weighted way, the solutions of $x + y = 1$.
When $\chi$, $\lambda$ and $\chi\lambda$ are nontrivial, it is a quotient of [[Gauss sums|EQ6A3ZN48K]].
Then its absolute value is the square root of the field size.
Jacobi sums are used to count points on curves such as $x^n + y^n = 1$ over finite fields and to prove [[cubic|EMVECB93EQ]] and [[quartic reciprocity|W0M20JWC9K]].

## Explanations

### L1 human

Gauss sums involve complex roots of unity of order $p$, which can be awkward.
Jacobi sums use only the values of multiplicative characters, so they live in a smaller ring of roots of unity.
For the Legendre symbol modulo $5$, the Jacobi sum is $\sum_a \left(\frac{a}{5}\right)\left(\frac{1-a}{5}\right) = -1$.
Writing a prime as a sum of two squares, or as $x^2 + 3y^2$, can be done by computing a suitable Jacobi sum.
</document>

<document source="https://tamnd.github.io/math2/node/279SXZFBJ2.md" tag="279SXZFBJ2" kind="definition">
# Left-invariant vector field

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] with identity $e$.
For $g \in G$, the *left translation* by $g$ is the diffeomorphism $L_g \colon G \to G$, $L_g(h) = gh$.
A [[vector field|0299VCW1TH]] $X$ on $G$ is *left-invariant* if
$$(dL_g)_h (X_h) = X_{gh} \quad \text{for all } g, h \in G,$$
where $(dL_g)_h$ is the [[differential|6P9PXQDSEN]] of $L_g$ at $h$.

## Remarks

Example: on $G = GL_n(\R)$, open in the space $M_n(\R)$ of matrices, fix $A \in M_n(\R)$ and put $X_g = gA$.
Left translation by $g$ is the linear map $h \mapsto gh$, so its differential is again $v \mapsto gv$, and $(dL_g)_h(X_h) = g h A = X_{gh}$.

Non-example: on the same group, the field $Y_g = Ag$ with $A \ne 0$ is not left-invariant in general.
For $n = 2$, $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $g = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$, one gets $(dL_g)_e(Y_e) = gA \ne Ag = Y_g$.
It is right-invariant instead.

The map $X \mapsto X_e$ is a linear isomorphism from the space of left-invariant vector fields onto the [[tangent space|E7A5DHFT01]] $T_eG$; the inverse sends $v$ to the field $g \mapsto (dL_g)_e(v)$.
The [[Lie bracket|68FGQEBD6J]] of two left-invariant vector fields is left-invariant.
So $T_eG$ inherits a bracket, and with it becomes the Lie algebra of $G$, written $\mathfrak{g}$ or $\operatorname{Lie}(G)$.
Every left-invariant vector field is complete: its integral curves are defined on all of $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/27BDDC590S.md" tag="27BDDC590S" kind="definition">
# Reflexive space

## Definition

Let $X$ be a [[normed vector space|03J5RAKB30]] over $\R$ or $\C$, and let $J \colon X \to X^{**}$ be the [[canonical embedding into the bidual|QKR3D65TTH]], $(Jx)(f) = f(x)$.
The space $X$ is *reflexive* if $J$ is surjective.

## Remarks

Since $J$ is a linear isometry, $X$ is reflexive exactly when $J$ is an isometric isomorphism of $X$ onto $X^{**}$.
A reflexive space is complete, because $X^{**}$ is complete.
Example: for $1 < p < \infty$ and $1/p + 1/q = 1$, the duals satisfy $(\ell^p)^* = \ell^q$ and $(\ell^q)^* = \ell^p$; under these identifications $J$ is the identity of $\ell^p$, so $\ell^p$ is reflexive.
Every [[Hilbert space|HM2HHE3J51]] is reflexive by the [[Riesz representation theorem|TTC40N49M6]].
Non-example: $c_0$ is not reflexive, since $J$ is the inclusion $c_0 \subset \ell^\infty$, which misses the constant sequence $(1, 1, \ldots)$.
The spaces $\ell^1$, $\ell^\infty$ and $C[0, 1]$ are not reflexive either.
The definition asks for the specific map $J$ to be onto: the [[James space|DYAWGXE2GV]] is isometrically isomorphic to its bidual by some other map, but it is not reflexive.
</document>

<document source="https://tamnd.github.io/math2/node/27C6J3YNZY.md" tag="27C6J3YNZY" kind="concept">
# Iterative root

## Summary

An iterative root of order $n$ of a map $f$ is a map $g$ whose $n$-th [[iterate|AYVTR2ES5B]] is $f$, that is, $g \circ g \circ \cdots \circ g = f$ with $n$ factors.
It is the analogue for composition of the $n$-th root of a number.
Iterative roots may fail to exist, and when they exist there are often many of them.
The search for them is one of the classical problems of iteration theory.

## Explanations

### L1 human

Think of $f$ as a step of a process, and ask for a smaller step $g$ such that $n$ small steps make one big step.
For $f(x) = x + 3$, the map $g(x) = x + 1$ is an iterative root of order $3$, since three shifts by $1$ make a shift by $3$.
For $f(x) = 8x$, the map $g(x) = 2x$ is an iterative root of order $3$.
Some maps have no nice root: no continuous map $g$ of the real line satisfies $g(g(x)) = -x$.
</document>

<document source="https://tamnd.github.io/math2/node/27K5X7S4XN.md" tag="27K5X7S4XN" kind="example">
# Somos sequence

## Example

For $k \ge 4$, the *Somos-$k$ sequence* is defined by $a_0 = a_1 = \cdots = a_{k-1} = 1$ and
$$a_n\, a_{n-k} = \sum_{j=1}^{\lfloor k/2 \rfloor} a_{n-j}\, a_{n-k+j} \qquad (n \ge k).$$
This is a [[nonlinear difference equation|7C7FBZGERM]]: each new term is a quotient of a sum of products by $a_{n-k}$.
For $k = 4$ the rule is $a_n = (a_{n-1} a_{n-3} + a_{n-2}^2)/a_{n-4}$.
The Somos-4 sequence begins $1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, \dots$.
Check: $a_4 = (1 + 1)/1 = 2$, $a_5 = (2 \cdot 1 + 1)/1 = 3$, $a_6 = (3 \cdot 1 + 4)/1 = 7$, $a_7 = (7 \cdot 2 + 9)/1 = 23$, $a_8 = (23 \cdot 3 + 49)/2 = 59$, $a_9 = (59 \cdot 7 + 529)/3 = 314$.
Although each step divides, the terms of Somos-$4$, $5$, $6$ and $7$ are all integers.
For Somos-$8$ this fails: a non-integer term appears.
Fomin and Zelevinsky showed more for $4 \le k \le 7$: with indeterminate initial values, each term is a Laurent polynomial in them with integer coefficients.
For Somos-4 and Somos-5, this follows from the [[Laurent phenomenon|6YYPNKK6WZ]] for cluster algebras.

## Explanations

### L1 human

Somos sequences are built by a rule that divides at every step, so one expects fractions to appear.
Surprisingly, for the first few members of the family, every division comes out exact.
The reason is a hidden algebraic structure: the terms are Laurent polynomials in the starting values, so setting the starting values to $1$ gives integers.
</document>

<document source="https://tamnd.github.io/math2/node/27QRARKWD5.md" tag="27QRARKWD5" kind="concept">
# Kleinian group

## Summary

A Kleinian group is a discrete subgroup of $\mathrm{PSL}_2(\C)$.
The group $\mathrm{PSL}_2(\C)$ acts on the Riemann sphere by Möbius transformations and on hyperbolic $3$-space by orientation-preserving isometries.
Quotients of hyperbolic $3$-space by torsion-free Kleinian groups are exactly the complete oriented [[hyperbolic 3-manifolds|GHVW1Z35BS]].
The theory links complex analysis, the geometry of $3$-manifolds, and fractal limit sets on the sphere.

## Explanations

### L1 human

Möbius transformations of the sphere, such as $z \mapsto (az + b)/(cz + d)$, move circles to circles.
A Kleinian group is a collection of these maps, closed under composition, with no maps arbitrarily close to the identity.
Repeating the maps of such a group often produces intricate fractal patterns of circles on the sphere.
The same maps act as rigid motions of three-dimensional hyperbolic space, and this is how they describe hyperbolic $3$-manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/27ZPTJP9PP.md" tag="27ZPTJP9PP" kind="definition">
# Free monoid

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]], called the alphabet.
A *word* over $A$ is a finite sequence $a_1 a_2 \cdots a_n$ with $n \ge 0$ and each $a_i \in A$; the word with $n = 0$ is the empty word $\varepsilon$.
The *free monoid* on $A$ is the set $A^*$ of all words over $A$ with the operation of concatenation,
$$(a_1 \cdots a_n)(b_1 \cdots b_m) = a_1 \cdots a_n b_1 \cdots b_m,$$
and with identity element $\varepsilon$.

## Remarks

Concatenation is associative and $\varepsilon$ is a two-sided identity, so $A^*$ is a [[monoid|0N6R902FG1]].
Universal property: for every monoid $M$ and every [[map|0XHDTE5YMZ]] $f \colon A \to M$, there is exactly one monoid homomorphism $\bar f \colon A^* \to M$ with $\bar f(a) = f(a)$ for $a \in A$, namely $\bar f(a_1 \cdots a_n) = f(a_1) \cdots f(a_n)$.
A monoid is called free if it is isomorphic to some $A^*$.
The free semigroup $A^+$ is $A^*$ without the empty word.

Example: for $A = \set{a}$, the map $a^n \mapsto n$ is an isomorphism from $A^*$ onto the monoid $(\N, +)$ with $0 \in \N$.

Non-example: the monoid $(\Z, +)$ is not free.
In a free monoid, $uv = \varepsilon$ forces $u = v = \varepsilon$, because lengths add; but in $\Z$, $1 + (-1) = 0$ with $1 \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/282MQP4X82.md" tag="282MQP4X82" kind="statement">
# Holomorphic inverse function theorem

## Statement

Let $U \subseteq \C^n$ be an open set, let $F = (F_1, \dots, F_n) \colon U \to \C^n$ be a [[holomorphic map|3MQTWKBBBT]], and let $a \in U$.
Suppose that the complex [[Jacobian determinant|0ZCSMYHTBP]] $\det \left( \frac{\partial F_k}{\partial z_j}(a) \right)_{k, j}$ is not zero.
Then there are open neighbourhoods $U_0 \subseteq U$ of $a$ and $V_0$ of $F(a)$ such that $F$ restricts to a [[biholomorphism|FX92PP3HZR]] from $U_0$ onto $V_0$.

## Remarks

Viewed as a map of $\R^{2n}$, $F$ has real Jacobian determinant $\abs{\det (\partial F_k / \partial z_j)}^2$, which is not zero at $a$.
The real [[inverse function theorem|0K9HSJFCMJ]] gives a $C^1$ local inverse $G$.
Its derivative at each point is the inverse of the $\C$-linear map $DF$, hence $\C$-linear, so $G$ is holomorphic.
Example: $F(z_1, z_2) = (e^{z_1}, \ z_2 e^{z_1})$ has complex Jacobian determinant $e^{2 z_1} \ne 0$ everywhere, so it is a local biholomorphism near every point, though it is not injective on $\C^2$, since $F(z_1 + 2\pi i, z_2) = F(z_1, z_2)$.
The hypothesis is needed: $F(z) = z^2$ on $\C$ has derivative $0$ at $0$, and it is not injective on any neighbourhood of $0$.

## Explanations

### L1 human

If the complex derivative of a holomorphic map is invertible at a point, the map can be undone near that point.
Moreover, the inverse is again holomorphic.
So holomorphic maps with invertible derivative are local changes of complex coordinates.

### L2 human

This theorem is the basic local tool for complex manifolds and holomorphic maps.
Together with the holomorphic implicit function theorem, it shows that the zero set of holomorphic functions with independent differentials is locally a complex submanifold.
A stronger fact holds in the complex case: an injective holomorphic map between open subsets of $\C^n$ automatically has nonzero Jacobian determinant, so it is a biholomorphism onto its image.
The real analogue fails, as $x \mapsto x^3$ shows.
</document>

<document source="https://tamnd.github.io/math2/node/284TYTVCC0.md" tag="284TYTVCC0" kind="definition">
# Kolmogorov backward equation

## Definition

Let $b \colon \R^n \to \R^n$ and $\sigma \colon \R^n \to \R^{n \times m}$ be functions, and let $a = \sigma \sigma^{T}$.
Consider the [[diffusion|9HDQ094ADQ]] given by the stochastic differential equation $dX_t = b(X_t) \, dt + \sigma(X_t) \, dB_t$, where $B$ is an $m$-dimensional [[Brownian motion|22735FGJV6]].
Its [[generator|VPA41PWRR1]] is the differential operator
$$\mathcal{L} g(x) = \sum_{i=1}^n b_i(x) \, \partial_i g(x) + \frac12 \sum_{i,j=1}^n a_{ij}(x) \, \partial_i \partial_j g(x),$$
with [[partial derivatives|09EGC6F8PT]] $\partial_i$ in the variable $x$.
The *Kolmogorov backward equation* of the diffusion is the equation
$$\partial_t u(t, x) = \mathcal{L} u(t, x), \quad t > 0, \ x \in \R^n,$$
for a function $u \colon [0, \infty) \times \R^n \to \R$, with an initial condition $u(0, x) = f(x)$.

## Remarks

Under regularity conditions on $b$, $\sigma$ and $f$, the function $u(t, x) = E[f(X_t) \mid X_0 = x]$ solves the backward equation.
If $X$ has transition densities $p(t, x, y)$, they satisfy $\partial_t p = \mathcal{L}_x p$, where $\mathcal{L}_x$ acts on the starting point $x$.
The forward, or Fokker–Planck, equation acts instead on the end point $y$.
Example: for the [[Ornstein–Uhlenbeck process|HZM2ASCDY4]] $dX = -X \, dt + dB$ in $\R$ and $f(y) = y$, one has $u(t, x) = x e^{-t}$.
Check: $\partial_t u = -x e^{-t}$ and $\mathcal{L} u = -x \, \partial_x u + \tfrac12 \partial_{xx} u = -x e^{-t}$.
Non-example: for the same process, $v(t, x) = x e^{t}$ has $\partial_t v = x e^{t}$ but $\mathcal{L} v = -x e^{t}$, so it does not solve the equation.
The equation follows from [[Itô's lemma|2J1RVQB0CM]] applied to $s \mapsto u(t - s, X_s)$.
</document>

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# Smooth morphism of rings

## Summary

A smooth morphism of rings is a map $R \to A$ of commutative rings that makes $A$ a [[finitely presented|764GZHTGV8]] $R$-algebra and is [[formally smooth|8KPDNVKX7B]].
It is the algebraic counterpart of a submersion of manifolds: the fibres are nonsingular and vary nicely.
A smooth algebra is flat over the base, and its module of [[Kähler differentials|7HCFARHG7D]] is finitely generated and projective.
Over a field, smooth algebras are the coordinate rings of nonsingular varieties, in a sense that is stable under extension of the field.

## Explanations

### L1 human

A polynomial ring $R[x_1, \ldots, x_n]$ is the model of a smooth algebra: it is like flat space over $R$.
A smooth algebra looks, near each point and to first order, like such a polynomial ring.
The circle $x^2 + y^2 = 1$ over the real numbers gives a smooth algebra $\R[x, y]/(x^2 + y^2 - 1)$.
The cusp $y^2 = x^3$ does not, because of its sharp point at the origin.
Smoothness is the property that lets one extend solutions step by step, as in Hensel's lemma.
</document>

<document source="https://tamnd.github.io/math2/node/287JS93ZDS.md" tag="287JS93ZDS" kind="definition">
# Young tableau

## Definition

Let $\lambda = (\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_\ell > 0)$ be a [[partition|0JV30XN73Z]] of $n$.
Let $D$ be its [[Young diagram|096AHMB3P4]], the set of cells $(i, j)$ with $1 \le i \le \ell$ and $1 \le j \le \lambda_i$, where $i$ is the row and $j$ the column.

1. A *Young tableau* of shape $\lambda$ is a map $T \colon D \to \set{1, 2, 3, \dots}$, called a filling of $D$.
2. $T$ is *semistandard* if $T(i, j) \le T(i, j+1)$ and $T(i, j) < T(i+1, j)$ whenever both cells lie in $D$. So entries weakly increase along rows and strictly increase down columns.
3. $T$ is *standard* if it is semistandard and $T$ is a bijection from $D$ onto $\set{1, \dots, n}$.

## Remarks

Example: for $\lambda = (2, 2)$, the filling with rows $1\,2$ and $3\,4$ is standard.
Each row increases ($1 < 2$, $3 < 4$), each column increases ($1 < 3$, $2 < 4$), and each of $1, 2, 3, 4$ appears once.
The filling with rows $1\,1$ and $2\,3$ is semistandard but not standard, because $1$ appears twice.

Non-example: the filling of shape $(2, 1)$ with first row $2\,1$ and second row $3$ is not semistandard, because the first row decreases.

Many authors say "tableau" for a semistandard tableau.
The *content* of a semistandard tableau $T$ is the sequence $(\mu_1, \mu_2, \dots)$, where $\mu_k$ is the number of entries equal to $k$.
The number $f^\lambda$ of standard tableaux of shape $\lambda$ is given by the [[hook length formula|VCMS1CP91N]].
</document>

<document source="https://tamnd.github.io/math2/node/28ADY9WTV3.md" tag="28ADY9WTV3" kind="definition">
# Maslov index

## Definition

Identify $\R^{2n}$ with $\C^n$ by $(x, y) \mapsto x + iy$, with the symplectic form $\omega = \sum_{j=1}^n dx_j \wedge dy_j$.
A linear subspace $\Lambda \subset \R^{2n}$ is *Lagrangian* if $\dim \Lambda = n$ and $\omega(u, v) = 0$ for all $u, v \in \Lambda$.
Let $\Lambda(n)$ be the set of Lagrangian subspaces, the *Lagrangian Grassmannian*.

Every $\Lambda \in \Lambda(n)$ is $\Lambda = U \R^n$ for some $U$ in the [[unitary group|0R9R0Y1HDK]] $U(n)$.
This $U$ is unique up to right multiplication by an element of the [[orthogonal group|0TKYAEJ8N7]] $O(n)$.
Hence the map $\Lambda \mapsto \det(U)^2 \in S^1 \subset \C$ is well defined on $\Lambda(n)$.
Let $\gamma \colon [0, 1] \to \Lambda(n)$ be a continuous loop, so $\gamma(0) = \gamma(1)$.
Its *Maslov index* is the [[winding number|0NQSV2EWDR]] about $0$ of the loop $t \mapsto \det(U(t))^2$, where $\gamma(t) = U(t) \R^n$.

## Remarks

Example: $n = 1$ and $\gamma(t) = e^{i\pi t} \R \subset \C$ for $t \in [0, 1]$, the line through $0$ turned by half a turn.
Here $U(t) = e^{i\pi t}$ and $\det(U(t))^2 = e^{2\pi i t}$ winds once around $0$, so the Maslov index is $1$.

Non-example: in $\R^4 = \C^2$, the plane $P = \C \times \set{0}$ has dimension $2$, but $\omega$ pairs the vectors $(1, 0)$ and $(i, 0)$ of $P$ to $1 \ne 0$.
So $P$ is not Lagrangian, and a loop of planes through $P$ is not a loop in $\Lambda(2)$ and has no Maslov index.

The Maslov index depends only on the homotopy class of the loop, and it gives an isomorphism of the [[fundamental group|S1XK3HXHFX]] $\pi_1(\Lambda(n))$ with $\Z$.
For a [[Lagrangian submanifold|H8ACPTMX08]] $L \subset \R^{2n}$, a loop in $L$ gives the loop of tangent spaces in $\Lambda(n)$, and its Maslov index defines a class in $H^1(L; \Z)$, the Maslov class.
</document>

<document source="https://tamnd.github.io/math2/node/28C5WWGX9J.md" tag="28C5WWGX9J" kind="statement">
# Minimal curvature property of natural cubic splines

## Statement

Let $k \ge 1$, let $a = x_0 < x_1 < \cdots < x_k = b$, and let $y_0, \dots, y_k \in \R$.
Let $s$ be the natural [[cubic spline|V42H0R3EKX]] with knots $x_0, \dots, x_k$ such that $s(x_i) = y_i$ for all $i$.
Let $g \colon [a, b] \to \R$ be twice continuously differentiable with $g(x_i) = y_i$ for all $i$.
Then
$$\int_a^b s''(x)^2 \, dx \le \int_a^b g''(x)^2 \, dx,$$
with equality only if $g = s$.

## Remarks

Let $h = g - s$, which vanishes at every knot.
[[Integration by parts|08Y1BBQTK9]] gives $\int_a^b s'' h'' \, dx = \bigl[s'' h'\bigr]_a^b - \int_a^b s''' h' \, dx$.
The boundary term is $0$ because $s''(a) = s''(b) = 0$.
On each $[x_{i-1}, x_i]$ the function $s'''$ is a constant $c_i$, so the last integral is $\sum_i c_i \bigl(h(x_i) - h(x_{i-1})\bigr) = 0$.
Hence $\int g''^2 = \int s''^2 + \int h''^2$, and equality forces $h'' = 0$, so $h$ is linear with two zeros and $h = 0$.

Example: for two data points ($k = 1$) the natural cubic spline is the straight line through them.
The statement says that every other $C^2$ curve through the two points has $\int g''^2 > 0$.

The end conditions matter.
For other cubic spline interpolants, such as the clamped one, the inequality holds only among functions $g$ that satisfy the same end conditions.

## Explanations

### L1 human

The integral of the squared second derivative measures how much a curve bends overall; for a thin elastic strip it is roughly its bending energy.
The theorem says that among all smooth curves through given points, the natural cubic spline bends the least.
This is why it imitates the draftsman's spline, a flexible strip held by pegs at the data points.

### L2 human

The theorem gives a variational characterisation of the natural cubic spline interpolant.
It minimises $\int_a^b g''^2$ among the $C^2$ functions $g$ that satisfy the interpolation conditions.
The same holds in the larger Sobolev space $H^2$.
The proof shows an orthogonality: $s''$ is orthogonal to $h''$ for every $h$ vanishing at the knots, and this orthogonality is what makes $s$ the minimiser.
Replacing the interpolation conditions by a penalised fit leads to [[smoothing splines|BNEXZF75QC]], and the same idea in two variables leads to [[thin plate splines|P9ZPH7EC8K]].
</document>

<document source="https://tamnd.github.io/math2/node/28DVNYNMZR.md" tag="28DVNYNMZR" kind="definition">
# Tautological line bundle

## Definition

Let $k$ be a field, $V$ a [[vector space|05EGHEQZAW]] over $k$ of dimension $n + 1$, and $\mathbb{P}(V)$ the [[projective space|ZT9A6JK8QF]] of lines in $V$.
The *tautological line bundle* on $\mathbb{P}(V)$ is the closed subvariety
$$L = \set{(\ell, v) \in \mathbb{P}(V) \times V : v \in \ell}$$
together with the projection $L \to \mathbb{P}(V)$.
Its fibre over a point $\ell$ is the line $\ell \subseteq V$ itself.
Its sheaf of sections is a [[vector bundle|P4DZM975BV]] of rank $1$, isomorphic to $\mathcal{O}_{\mathbb{P}(V)}(-1)$.

## Remarks

For $V = k^2$, a point of $\mathbb{P}^1$ is $\ell = [a : b]$, and the fibre of $L$ over it is $\set{t(a, b) : t \in k}$.
Over the open set $a \ne 0$, the map $(\ell, t(1, b/a)) \mapsto (\ell, t)$ trivializes $L$, so $L$ is locally a product with a line.
The projection $L \to \mathbb{P}(V)$ is a line bundle but not the trivial one when $n \ge 1$.
Indeed, $\mathcal{O}(-1)$ has no nonzero global sections, while a trivial bundle has a nowhere-zero section.
The dual of $L$ corresponds to [[Serre's twisting sheaf|V93Y0XYPZD]] $\mathcal{O}(1)$, whose global sections are the linear forms on $V$.
Some authors write $\mathbb{P}(V)$ for the space of hyperplanes of $V$, which changes the sign conventions.
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<document source="https://tamnd.github.io/math2/node/28E5Q5BM69.md" tag="28E5Q5BM69" kind="concept">
# Inner regular measure

## Summary

A measure on a topological space is inner regular when each measurable set is approximated from inside by compact subsets.
The approximation is in measure.
Inner regularity lets one reduce questions about general sets to questions about compact sets, where topology gives control.
It is one of the two halves of the definition of a Radon measure.
Some authors use closed sets instead of compact sets in the definition.

## Explanations

### L1 human

To measure a complicated set, fill it from inside with compact pieces, like filling a lake with larger and larger islands.
If the islands can capture as much of the measure as one likes, the measure is inner regular.
For length on the real line, the set of irrational numbers in $[0, 1]$ has length $1$.
It contains compact sets, built by removing small open intervals around the rationals, of length as close to $1$ as one likes.
</document>

<document source="https://tamnd.github.io/math2/node/28M8XD8EA9.md" tag="28M8XD8EA9" kind="concept">
# Words and profiles

## Summary

A word on a graph $G$ is any map $w:V(G)\to\mathcal C$ into a set of colors, proper or not. Its profile is the vector $\mathbf m(w)$ whose entry $m_X(w)=|w^{-1}(X)|$ counts the vertices that get color $X$.

Take $|\mathcal C|=\chi(G)$, the [[chromatic number|0VHJRXNAAS]] of $G$. Then each proper [[colouring|0NCPXH194Y]] with palette $\mathcal C$ uses every color, and a profile is feasible if some such coloring has it.
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<document source="https://tamnd.github.io/math2/node/28NQ0KRHD3.md" tag="28NQ0KRHD3" kind="concept">
# Functor

## Summary

A functor is a map between two [[categories|A2DE66937D]]: it sends objects to objects and [[morphisms|859F0TFVA3]] to morphisms, and it respects composition and identities.
Functors are how one field of mathematics is translated into another, as when the [[fundamental group|S1XK3HXHFX]] turns spaces and continuous maps into groups and homomorphisms.
Because functors preserve composition, they send isomorphisms to isomorphisms, which makes them a source of invariants.
Functors between two categories are compared by [[natural transformations|KYCPCC57FT]].

## Explanations

### L1 human

A functor takes every object of one category to an object of another, and every arrow to an arrow between the images.
It must send a chain of arrows to the chain of the images, and an identity arrow to an identity arrow.
A simple example forgets structure: send each group to its underlying set and each homomorphism to the same map seen as a function.
A deeper one sends each space to its fundamental group.
If two spaces are homeomorphic, their fundamental groups are isomorphic, so different groups prove that two spaces are different.
</document>

<document source="https://tamnd.github.io/math2/node/28PE4GRFD7.md" tag="28PE4GRFD7" kind="definition">
# Order type

## Definition

Let $(A, \le_A)$ and $(B, \le_B)$ be [[totally ordered sets|0316AYH1SC]].
They have the same *order type* when there is an [[order isomorphism|0Y74KN99A5]] from $A$ onto $B$.
The *order type* of $(A, \le_A)$, written $\operatorname{type}(A)$, is the class of all totally ordered sets that have the same order type as $A$.

## Remarks

The sets $\N$ and $2\N = \set{0, 2, 4, \dots}$ with their usual orders have the same order type, $\omega$, by the isomorphism $n \mapsto 2n$.
The sets $\N$ and $\Z$ do not: $\N$ has a least element, $\Z$ has none, and an order isomorphism maps a least element to a least element.
Having the same order type is an equivalence relation on totally ordered sets.
Each well-ordered set is order isomorphic to a unique [[ordinal number|0RDJK7EQYY]], which is used to represent its order type.
Common notation: $\omega$ for $\N$, $\omega^*$ for the reverse of $\N$, $\zeta$ for $\Z$, $\eta$ for $\Q$ and $\lambda$ for $\R$.
The same definition makes sense for [[partially ordered sets|011C29Q1S1]].

## Explanations

### L1 human

Two ordered sets have the same order type when one can be relabelled into the other without changing the order.
The order type is the shape of the order.
</document>

<document source="https://tamnd.github.io/math2/node/28VV812XWQ.md" tag="28VV812XWQ" kind="concept">
# Fractional Fourier transform

## Summary

The fractional Fourier transform is a family of unitary operators $\mathcal{F}_\alpha$ on $L^2(\R)$, indexed by an angle $\alpha$.
Here $\mathcal{F}_0$ is the identity and $\mathcal{F}_{\pi/2}$ is the [[Fourier transform|DPVM34ADZV]].
The family is a group: $\mathcal{F}_\alpha \mathcal{F}_\beta = \mathcal{F}_{\alpha + \beta}$, so the Fourier transform has a square root, a cube root, and so on.
In the time-frequency plane, $\mathcal{F}_\alpha$ acts as a rotation by the angle $\alpha$.
It is used in optics and signal processing to analyse chirps and other signals whose frequency changes in time.

## Explanations

### L1 human

Doing the Fourier transform four times gives back the original function, so it behaves like a rotation by a quarter turn.
The fractional Fourier transform fills in the rotations by all the other angles.
A signal is then seen from a direction between pure time and pure frequency.
For example, the Gaussian $e^{-x^2/2}$ is unchanged by every $\mathcal{F}_\alpha$, just as a round disc is unchanged by every rotation.
</document>

<document source="https://tamnd.github.io/math2/node/290G1NGAMR.md" tag="290G1NGAMR" kind="definition">
# Ind-completion

## Definition

Let $\mathcal{C}$ be a small category, and let $y \colon \mathcal{C} \to [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]$ be the [[Yoneda embedding|4DKB6AQ8J7]] into [[presheaves|AND2WHZC8P]].
A presheaf $P$ on $\mathcal{C}$ is an *ind-object* if there are a small [[filtered category|975PMY4FG8]] $\mathcal{I}$ and a functor $X \colon \mathcal{I} \to \mathcal{C}$ with $P \cong \operatorname{colim}_{i} y(X_i)$.
The *Ind-completion* $\mathrm{Ind}(\mathcal{C})$ is the full subcategory of $[\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]$ whose objects are the ind-objects.

## Remarks

So an ind-object is a filtered [[colimit|BJS1CGTKQ5]] of [[representable presheaves|KCCHH87H6E]].
In terms of the diagrams $X \colon \mathcal{I} \to \mathcal{C}$ and $Y \colon \mathcal{J} \to \mathcal{C}$, the morphisms are
$$\mathrm{Ind}(\mathcal{C})(X, Y) = \lim_{i} \operatorname{colim}_{j} \mathcal{C}(X_i, Y_j).$$
The functor $y$ gives a fully faithful functor $\mathcal{C} \to \mathrm{Ind}(\mathcal{C})$, and $\mathrm{Ind}(\mathcal{C})$ has all small filtered colimits.
It is universal: every functor from $\mathcal{C}$ to a category with filtered colimits extends, uniquely up to isomorphism, to a filtered-colimit-preserving functor on $\mathrm{Ind}(\mathcal{C})$.

Example: let $\mathcal{C}$ be a skeleton of the category of finite sets.
Every set is the filtered colimit of its finite subsets, and this gives an equivalence between $\mathrm{Ind}(\mathcal{C})$ and the category of sets.

Non-example: let $\mathcal{C} = \mathbf{1}$, the category with one object and one morphism.
A presheaf on $\mathbf{1}$ is a set, and the only representable is a one-point set.
A filtered category is nonempty and connected, so every filtered colimit of one-point sets along identities is a one-point set.
So a presheaf with two elements, or with none, is not an ind-object.
</document>

<document source="https://tamnd.github.io/math2/node/2941ECC4QQ.md" tag="2941ECC4QQ" kind="definition">
# Koopman operator

## Definition

Let $(X, \mathcal{B}, \mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \colon X \to X$ be a [[measure-preserving transformation|556WBDAHNP]].
The *Koopman operator* of $T$ is the [[composition operator|CFASPSF7BB]]
$$U_T \colon L^2(\mu) \to L^2(\mu), \qquad U_T f = f \circ T,$$
on the [[space|SASP14MA1Z]] $L^2(\mu)$ of square-integrable functions modulo equality almost everywhere.

## Remarks

The operator is well defined and isometric: since $\mu(T^{-1}A) = \mu(A)$ for all $A \in \mathcal{B}$, one gets $\int \abs{f \circ T}^2 \, d\mu = \int \abs{f}^2 \, d\mu$.
If $T$ is invertible with measurable measure-preserving inverse, then $U_T$ is a [[unitary operator|MM0EJ2GDE4]] with inverse $U_{T^{-1}}$.

Example: for the rotation $T(x) = x + \alpha \bmod 1$ with Lebesgue measure, the functions $e_k(x) = e^{2\pi i k x}$ satisfy $U_T e_k = e^{2\pi i k \alpha} e_k$.
They form an orthonormal basis of $L^2$, so $U_T$ is diagonal in this basis.

Non-example: if $T$ does not preserve $\mu$, the formula need not define a bounded operator on $L^2(\mu)$.
For $T(x) = x^2$ on $[0,1]$ with Lebesgue measure and $f(x) = x^{-1/3}$, the function $f$ is square-integrable but $f \circ T = x^{-2/3}$ is not.

The same formula defines an isometry of $L^p(\mu)$ for every $1 \le p \le \infty$.
The constants are always fixed by $U_T$, and $T$ is ergodic if and only if they are the only fixed vectors.
</document>

<document source="https://tamnd.github.io/math2/node/29AR3Z107J.md" tag="29AR3Z107J" kind="definition">
# Periodic tiling

## Definition

Let $\mathcal{T}$ be a [[tiling|GKJK293DXS]] of $\R^n$.
For $v \in \R^n$, let $\mathcal{T} + v = \set{T + v : T \in \mathcal{T}}$ be the image of $\mathcal{T}$ under the [[translation|087VJM5Q77]] $x \mapsto x + v$.
A vector $v$ is a *period* of $\mathcal{T}$ if $\mathcal{T} + v = \mathcal{T}$.
The tiling $\mathcal{T}$ is *periodic* if it has $n$ linearly independent periods.

## Remarks

Example: the tiling of $\R^2$ by the unit squares $[a, a+1] \times [b, b+1]$, $(a, b) \in \Z^2$, is periodic.
Translation by $(1, 0)$ maps $[a, a+1] \times [b, b+1]$ to $[a+1, a+2] \times [b, b+1]$, so $(1, 0)$ is a period, and in the same way $(0, 1)$ is a period; these two vectors are independent.

Non-example: shift the row of squares $[a, a+1] \times [0, 1]$ by $1/2$ to the right and leave all other rows as they are.
The vector $(1, 0)$ is still a period, but no vector with a nonzero second coordinate is one, because the shifted row is the only row with vertices at half-integers.
So the tiling has periods in only one direction and is not periodic.

A tiling with no nonzero period at all is called *non-periodic*.
The [[Penrose tilings|MBE83FTRBG]] are non-periodic.
</document>

<document source="https://tamnd.github.io/math2/node/29C7C41A52.md" tag="29C7C41A52" kind="concept">
# Regular summability method

## Summary

A [[summability method|8X25Q4XYMN]] is regular when it sums every convergent sequence to its ordinary limit.
Regularity is the basic requirement for a method to count as a generalization of convergence.
For matrix methods it is characterized by the [[Silverman–Toeplitz theorem|S4J7SRPXMZ]].

## Explanations

### L1 human

A new way of summing series should not change the answer for series that already have a sum.
Regular methods have this property: on convergent series they agree with the ordinary sum, and they only add new values for some divergent series.
Cesàro summation is regular, because the running averages of a convergent sequence tend to the same limit.
</document>

<document source="https://tamnd.github.io/math2/node/29T4NC87AR.md" tag="29T4NC87AR" kind="example">
# Ornstein–Zernike equation

## Example

Consider a homogeneous and isotropic classical fluid in $\R^3$ with number density $\rho > 0$.
Let $g(r)$ be its radial distribution function and $h(r) = g(r) - 1$ the total correlation function, where $r$ is the distance between two particles.
The Ornstein–Zernike equation defines the direct correlation function $c$ by
$$h(\abs{x - y}) = c(\abs{x - y}) + \rho \int_{\R^3} c(\abs{x - z}) \, h(\abs{z - y}) \, dz, \quad x, y \in \R^3.$$
For known $h$ it is a [[convolution integral equation|EY75YD988Z]] of the second kind for $c$ on $\R^3$, and for known $c$ one for $h$.

Check by the Fourier transform: the integral is the convolution $c * h$, so the equation becomes $\hat h(q) = \hat c(q) + \rho \, \hat c(q) \hat h(q)$.
Hence $\hat h = \frac{\hat c}{1 - \rho \hat c}$ and $\hat c = \frac{\hat h}{1 + \rho \hat h}$, wherever the denominators do not vanish.

## Remarks

The equation splits the correlation between two particles into a direct part $c$ and an indirect part carried through a chain of other particles.
By itself it has two unknown functions, so it needs a second relation, a closure, such as the [[Percus–Yevick approximation|K3E536PSRJ]] or the [[hypernetted-chain equation|MRJ2FHX6K1]].

## Explanations

### L1 human

In a liquid, the position of one molecule influences where its neighbours are likely to be.
The equation says that this influence is the sum of a direct effect and of effects passed on through a third molecule, then a fourth, and so on.
Because the indirect part is a convolution, it is simple in Fourier space.
</document>

<document source="https://tamnd.github.io/math2/node/2A5HZZRGFT.md" tag="2A5HZZRGFT" kind="statement">
# Malgrange preparation theorem

## Statement

Let $F(t, x)$ be a real-valued [[smooth function|0V33BNKZ9S]] on a neighbourhood of $(0, 0)$ in $\R \times \R^n$.
Let $k \ge 1$, and assume that
$$F(0, 0) = \partial_t F(0, 0) = \cdots = \partial_t^{k-1} F(0, 0) = 0, \qquad \partial_t^k F(0, 0) \ne 0.$$
Then there are smooth functions $q(t, x)$ near $(0, 0)$ with $q(0, 0) \ne 0$, and $a_0(x), \ldots, a_{k-1}(x)$ near $0 \in \R^n$ with $a_i(0) = 0$, such that near $(0, 0)$
$$q(t, x) \, F(t, x) = t^k + a_{k-1}(x) \, t^{k-1} + \cdots + a_1(x) \, t + a_0(x).$$

## Remarks

This is the smooth analogue of the [[Weierstrass preparation theorem|49JSD49P7X]] for holomorphic functions.
The hypothesis says that $t \mapsto F(t, 0)$ vanishes to order exactly $k$ at $0$.
It is necessary: since $q(0, 0) \ne 0$ and $a_i(0) = 0$, the function $F(t, 0)$ vanishes at $t = 0$ to the same order as $t^k$.
In the smooth case the proof is much harder than in the holomorphic case, because there is no Cauchy integral formula.
Unlike the holomorphic case, the factorisation need not be unique.
A closely related form is the division theorem: every smooth $G$ near $(0, 0)$ can be written as $G = Q F + \sum_{i < k} r_i(x) t^i$ with smooth $Q$ and $r_i$.
Mather gave a form for modules of [[germs|7SWSRGDT4A]].
Let $\varphi \colon (\R^n, 0) \to (\R^p, 0)$ be a smooth map germ, and let $A$ be a finitely generated module over the ring $\mathcal{E}_n$ of smooth germs on $\R^n$.
Let $\mathfrak{m}_p$ be the maximal ideal of the ring $\mathcal{E}_p$ of smooth germs on $\R^p$.
Then $A$ is finitely generated over $\mathcal{E}_p$ through $\varphi$ if and only if $A / \varphi^*(\mathfrak{m}_p) A$ is a finite-dimensional real vector space.
Example: for $F(t, x) = t^2 - x$ the theorem holds with $q = 1$, $a_1 = 0$ and $a_0(x) = -x$.

## Explanations

### L1 human

Near a point where a smooth function of $t$ vanishes to order $k$, the theorem says that the function behaves like a polynomial of degree $k$ in $t$.
The coefficients of this polynomial depend smoothly on the remaining variables.
So questions about the zeros of a smooth function reduce to questions about the roots of a polynomial.

### L2 human

The preparation theorem is the main algebraic tool of the local theory of smooth maps.
In Mather's module form it turns infinitesimal conditions into finite-dimensional linear algebra.
It is used to prove that [[infinitesimal stability implies stability|T6BK8Q8PK8]], and results on finite determinacy and versal unfoldings.
</document>

<document source="https://tamnd.github.io/math2/node/2A5Z21CE4P.md" tag="2A5Z21CE4P" kind="concept">
# Artin–Mazur zeta function

## Summary

The Artin–Mazur zeta function of a map packs the numbers of [[periodic points|T75CZ5H47N]] of all periods into one power series.
For a map $f$ with finitely many fixed points of each iterate it is $\exp \sum_{n \ge 1} N_n z^n / n$, where $N_n$ is the number of fixed points of $f^n$.
For many hyperbolic systems it is a rational function, and it then records all periodic orbit counts by finitely many numbers.
For the Frobenius map of a variety over a finite field it is the local zeta function of the variety.

## Explanations

### L1 human

Counting periodic points of a map gives a list of numbers $N_1, N_2, N_3, \dots$, one for each period.
The zeta function turns the whole list into a single function of one variable.
For the doubling map $z \mapsto z^2$ on the unit circle, $N_n = 2^n - 1$, and the zeta function turns out to be the simple fraction $(1 - z)/(1 - 2z)$.
The pole at $z = 1/2$ shows that the number of periodic points grows like $2^n$.
</document>

<document source="https://tamnd.github.io/math2/node/2A7CYKQ2XW.md" tag="2A7CYKQ2XW" kind="definition">
# Integrally closed domain

## Definition

Let $A$ be an [[integral domain|0QVD7WBFGN]] with [[field of fractions|0F2PTGX1A2]] $K$.
The domain $A$ is *integrally closed* if every element of $K$ that is [[integral|1EHN4KPSFS]] over $A$ lies in $A$.
Equivalently, $A$ is equal to its [[integral closure|G0V86SJBQW]] in $K$.

## Remarks

Example: $\Z$ is integrally closed.
If $p/q$ in lowest terms is a root of $x^n + a_{n-1} x^{n-1} + \cdots + a_0$ with $a_i \in \Z$, then $p^n = -q(a_{n-1} p^{n-1} + \cdots + a_0 q^{n-1})$, so $q$ divides $p^n$, which forces $q = \pm 1$.
The same argument shows that every [[unique factorisation domain|0W4X9G6263]] is integrally closed.

Non-example: $\Z[\sqrt{-3}]$ is not integrally closed.
The element $(1 + \sqrt{-3})/2$ of its field of fractions is a root of $x^2 - x + 1$, but it is not in $\Z[\sqrt{-3}]$.

An integrally closed domain is also called a normal domain.
A domain is integrally closed if and only if all its localizations at prime ideals are integrally closed.
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<document source="https://tamnd.github.io/math2/node/2AFNVJ54G7.md" tag="2AFNVJ54G7" kind="concept">
# Projection method for integral equations

## Summary

A projection method approximates the solution of an operator equation by an element of a finite-dimensional subspace.
The main case is an [[integral equation of the second kind|8KJCN49V64]].
One chooses subspaces $X_n$ and projections $P_n$ onto them, and asks that the equation holds after applying $P_n$.
This gives a finite linear system for the coefficients of the approximation.
The [[Galerkin method|Y5B0PEBKE8]] (orthogonal projection) and the collocation method (interpolation) are the two main cases.

## Explanations

### L1 human

An integral equation asks for a whole function, which is infinitely many numbers.
A projection method looks only among simple functions, for instance polynomials of degree at most $n$ or piecewise linear functions on a grid.
It cannot satisfy the equation exactly, so it asks that the error is invisible to the projection: zero at chosen points (collocation) or orthogonal to the chosen functions (Galerkin).
For example, for $\varphi(x) - \int_0^1 x t \, \varphi(t) \, dt = x$ on $[0, 1]$ with $X_1$ the multiples of $x$, the Galerkin method gives $\varphi = \frac{3}{2} x$, which here is the exact solution.
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<document source="https://tamnd.github.io/math2/node/2AHNVSH76M.md" tag="2AHNVSH76M" kind="statement">
# Osgood uniqueness criterion

## Statement

Let $D \subseteq \R \times \R^n$ be open and let $f : D \to \R^n$ be continuous.
Let $\omega : [0, \infty) \to [0, \infty)$ be continuous and non-decreasing, with $\omega(0) = 0$ and $\omega(r) > 0$ for $r > 0$, and with
$$\int_0^{\varepsilon} \frac{dr}{\omega(r)} = \infty \quad \text{for some } \varepsilon > 0.$$
Assume that $\norm{f(t, x) - f(t, y)} \le \omega(\norm{x - y})$ whenever $(t, x), (t, y) \in D$.
Let $I$ be an interval and let $\gamma_1, \gamma_2 : I \to \R^n$ be [[solutions|04SW7WECS8]] of $x' = f(t, x)$ whose graphs lie in $D$.
If $\gamma_1(t_0) = \gamma_2(t_0)$ for one $t_0 \in I$, then $\gamma_1 = \gamma_2$ on $I$.

## Remarks

Proof idea for $t \ge t_0$: the function $u = \norm{\gamma_1 - \gamma_2}$ satisfies $u(t) \le \delta + \int_{t_0}^t \omega(u(s))\, ds$ for every $\delta > 0$.
The [[Bihari–LaSalle inequality|9KMJSKNZ72]] with $c = \delta$ gives $\int_\delta^{u(t)} dr / \omega(r) \le t - t_0$ when $u(t) > \delta$.
If $u(t) > 0$, let $\delta \to 0$: the left side tends to $\infty$, a contradiction.
The Lipschitz case is $\omega(r) = K r$.
A weaker modulus that still works is $\omega(r) = r \ln(1/r)$ for $0 < r \le 1/e$, extended by the constant $1/e$ for $r > 1/e$; here $\int_0^{1/e} dr / (r \ln(1/r))$ diverges.
The divergence hypothesis is needed: $f(x) = 2\sqrt{\abs{x}}$ satisfies the condition with $\omega(r) = 2\sqrt{r}$, whose reciprocal is integrable at $0$, and $x' = 2\sqrt{\abs{x}}$, $x(0) = 0$ has the two solutions $x = 0$ and $x = t^2$ for $t \ge 0$.

## Explanations

### L1 human

Uniqueness of solutions usually comes from a Lipschitz condition on the right side.
Osgood's criterion shows that a slightly weaker modulus of continuity is enough.
What matters is whether a solution of $u' = \omega(u)$ can leave $0$ in finite time; when $\int_0 dr/\omega(r)$ diverges, it cannot.

### L2 human

The criterion extends [[uniqueness for Lipschitz vector fields|025PKW8AMR]], and it is sharp in the sense of the example $2\sqrt{\abs{x}}$.
Log-Lipschitz moduli such as $r \ln(1/r)$ appear in fluid mechanics, where velocity fields of the two-dimensional Euler equations with bounded vorticity have this regularity and still have unique trajectories.
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# Positive linear operator

## Definition

Let $X$ and $Y$ be real [[vector spaces|000AYPA9AJ]], each with a preorder $\le$, such as a [[partial order|0QV2JS9RJ3]], compatible with the vector operations.
Compatible means that $x \le y$ implies $x + z \le y + z$ and $tx \le ty$ for all vectors $z$ and all real $t \ge 0$.
A [[linear map|0NRYHJPQG2]] $L \colon X \to Y$ is a *positive linear operator* if $x \ge 0$ implies $L x \ge 0$ for every $x \in X$.

## Remarks

Equivalently, $L$ is monotone: $x \le y$ implies $L x \le L y$, since $L y - L x = L(y - x)$.

Example: on $C[0, 1]$ with the pointwise order $f \le g$ when $f(x) \le g(x)$ for all $x$, the Bernstein operator
$$B_n f(x) = \sum_{k=0}^{n} \binom{n}{k} x^k (1 - x)^{n-k} f(k/n)$$
is positive, because all weights $\binom{n}{k} x^k (1 - x)^{n-k}$ are nonnegative on $[0, 1]$.

Non-example: differentiation $D \colon C^1[0, 1] \to C[0, 1]$ is linear but not positive.
The function $f(x) = 1 - x$ satisfies $f \ge 0$, while $D f = -1 < 0$.

For a compact Hausdorff space $K$ and a positive linear operator $L \colon C(K) \to C(K)$ with the uniform norm, $L$ is bounded with $\norm{L} = \norm{L 1}$.
Indeed $-\norm{f} \cdot 1 \le f \le \norm{f} \cdot 1$ gives $\abs{L f} \le \norm{f} \, L 1$.
A positive linear functional is the case $Y = \R$; on $C(K)$ these are the integrals against finite positive measures.
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# Markov kernel

## Summary

A Markov kernel from a space $X$ to a space $Y$ assigns to each point of $X$ a [[probability measure|059TJN0NZ1]] on $Y$.
The assignment must be measurable in the point.
It generalises the [[transition matrix|0F462493EA]] of a Markov chain to arbitrary state spaces.
Kernels describe random transitions, conditional distributions and randomised maps.
They can be composed, and together with a starting measure they build measures on product spaces.

## Explanations

### L1 human

A Markov kernel is a rule that says: if you are at point $x$, here is the probability distribution of where you go next.
For a finite set of states it is just a table whose rows are probability vectors.
For example, a random walk on $\Z$ that moves to $x + 1$ or $x - 1$ with probability $\tfrac12$ each is given by the kernel $\kappa(x, \cdot) = \tfrac12 \delta_{x+1} + \tfrac12 \delta_{x-1}$.
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# Adjoint representation of a Lie group

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] with identity $e$ and Lie algebra $\mathfrak{g} = T_eG$, the [[tangent space|E7A5DHFT01]] at $e$.
For $g \in G$, let $C_g \colon G \to G$, $C_g(h) = g h g^{-1}$, be the [[inner automorphism|0RN00NTM3S]] given by $g$.
Since $C_g(e) = e$, its [[differential|6P9PXQDSEN]] at $e$ is a linear map $\operatorname{Ad}(g) = (dC_g)_e \colon \mathfrak{g} \to \mathfrak{g}$.
The *adjoint representation* of $G$ is the map
$$\operatorname{Ad} \colon G \to GL(\mathfrak{g}), \qquad g \mapsto \operatorname{Ad}(g).$$

## Remarks

From $C_{gh} = C_g \circ C_h$ and the chain rule, $\operatorname{Ad}(gh) = \operatorname{Ad}(g)\operatorname{Ad}(h)$, so $\operatorname{Ad}$ is a [[representation|WYGRXCG2RN]]; it is also smooth.

Example: for $G = GL_n(\R)$ and $\mathfrak{g} = M_n(\R)$, the map $C_g$ is the restriction of the linear map $X \mapsto g X g^{-1}$ of $M_n(\R)$.
So $\operatorname{Ad}(g)X = g X g^{-1}$.

Non-example: the map $g \mapsto (dL_g)_e$, with $L_g(h) = gh$, does not define a representation on $\mathfrak{g}$.
For $g \ne e$, $L_g$ does not fix $e$, so $(dL_g)_e$ sends $T_eG$ to $T_gG$, not to $\mathfrak{g}$.

The differential of $\operatorname{Ad}$ at $e$ is the adjoint representation $\operatorname{ad}$ of the Lie algebra $\mathfrak{g}$, given by $\operatorname{ad}(X)(Y) = [X, Y]$.
The two are related by $\operatorname{Ad}(\exp X) = e^{\operatorname{ad} X}$, and $g \exp(X) g^{-1} = \exp(\operatorname{Ad}(g)X)$.
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# Topological transformation group

## Summary

A topological transformation group is a [[topological group|92700QAB5Y]] acting on a topological space so that the action map is continuous.
It is the topological version of a [[group action|0WCY42EV9V]], and it is the basic object of topological dynamics.
Each group element then acts by a homeomorphism of the space.
Homogeneous spaces, flows and actions of Lie groups on manifolds are all examples.

## Explanations

### L1 human

A group can move the points of a space around: rotations move the points of a sphere, for example.
In a topological transformation group, the motion depends continuously both on the group element and on the point.
Rotating by a slightly different angle, or starting at a slightly different point, changes the result only slightly.
The rotation group acting on the sphere is the standard picture.
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# Shifted Legendre polynomials

## Definition

Let $P_n$ denote the [[Legendre polynomial|QFYGJVA996]] of degree $n \ge 0$.
The *shifted Legendre polynomial* of degree $n$ is the polynomial
$$\tilde{P}_n(x) = P_n(2x - 1).$$
Equivalently,
$$\tilde{P}_n(x) = (-1)^n \sum_{k=0}^{n} \binom{n}{k} \binom{n+k}{k} (-x)^k = \frac{1}{n!} \frac{d^n}{dx^n} \left( x^2 - x \right)^n.$$

## Remarks

The first ones are $\tilde{P}_0 = 1$, $\tilde{P}_1 = 2x - 1$ and $\tilde{P}_2 = 6x^2 - 6x + 1$.
Check for $\tilde{P}_2$: with $P_2(t) = (3t^2 - 1)/2$ and $t = 2x - 1$, one gets $(12x^2 - 12x + 2)/2 = 6x^2 - 6x + 1$.
They are orthogonal on $[0, 1]$: $\int_0^1 \tilde{P}_m(x) \tilde{P}_n(x) \, dx = \frac{\delta_{mn}}{2n + 1}$.
For instance $\int_0^1 (2x - 1) \, dx = 0$.
Non-example: the unshifted polynomials are not orthogonal on $[0, 1]$, since $\int_0^1 P_0(x) P_1(x) \, dx = \int_0^1 x \, dx = \frac12 \ne 0$.
Some authors, and some software libraries, use the reversed convention $P_n(1 - 2x) = (-1)^n \tilde{P}_n(x)$.
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# Admissibility condition for wavelets

## Summary

The admissibility condition asks that a function $\psi$ in $L^2(\R)$ have finite constant $C_\psi = \int \abs{\hat\psi(\xi)}^2 / \abs{\xi} \, d\xi$.
It makes the [[continuous wavelet transform|57FR21NDGV]] with respect to $\psi$ preserve energy up to the factor $C_\psi$.
So the transform can be inverted.
For integrable $\psi$ it forces $\int \psi = 0$: an admissible function must oscillate.

## Explanations

### L1 human

A wavelet transform measures a signal against stretched and shifted copies of one shape.
For this to lose no information, the shape must not contain a constant part, because very wide copies would then just measure the average of the signal again and again.
The admissibility condition makes this precise in terms of the Fourier transform near frequency zero.
The Mexican hat curve, which dips below zero on both sides, passes the test.
A plain bell curve, which is positive everywhere, fails it.
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# Permuting congruences

## Summary

Two [[congruences|DCSXXT4SER]] on an [[algebra|QJGPT2W0FT]] permute when their relational products in the two orders are equal.
Then their join in the [[congruence lattice|REE07CSXHR]] is simply their product, which makes it easy to compute.
An algebra whose congruences all permute is called congruence-permutable; groups, rings and modules are examples.
This property is the starting point of [[congruence-permutable varieties|1B3N9T68YV]] and of Mal'cev conditions.

## Explanations

### L1 human

Given two congruences, one can walk from $a$ to $c$ by first taking a step in one and then a step in the other.
The congruences permute when the order of the two steps does not matter for which pairs $(a, c)$ can be reached.
In a group, a step in the congruence of a normal subgroup $N$ is multiplication by an element of $N$, and two normal subgroups $N$ and $M$ satisfy $NM = MN$.
So all congruences of a group permute.
For a bare set without operations, many pairs of equivalence relations do not permute.
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# Unitary dual

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]].
Two [[unitary representations|74BPYJNZTM]] $\pi_1$ on $H_1$ and $\pi_2$ on $H_2$ are *unitarily equivalent* if there is a unitary operator $T \colon H_1 \to H_2$ with $T\pi_1(g) = \pi_2(g)T$ for all $g \in G$.
The *unitary dual* $\widehat{G}$ of $G$ is the set of unitary equivalence classes of [[irreducible|FX3NEFYFFN]] unitary representations of $G$.

## Remarks

Unitary equivalence is an equivalence relation, so $\widehat{G}$ is well defined.
It is usually given the Fell topology, which comes from the identification of $\widehat{G}$ with the spectrum of the group C\*-algebra $C^*(G)$.

Example: $G = \R$.
By Schur's lemma, every irreducible unitary representation of an abelian group is one-dimensional, so it is a continuous homomorphism $\R \to \set{z \in \C : \abs{z} = 1}$.
These are the maps $x \mapsto e^{ixy}$ with $y \in \R$, and different $y$ give inequivalent representations, so $\widehat{\R}$ is in bijection with $\R$.

Non-example: the representation of $\R$ on $L^2(\R)$ by translations $f \mapsto f(\cdot - t)$ does not define a point of $\widehat{\R}$, since it is not irreducible.
The closed subspace of functions whose Fourier transform vanishes on $(-\infty, 0)$ is invariant under all translations and is neither $0$ nor $L^2(\R)$.

For a locally compact abelian group, $\widehat{G}$ is the group of continuous characters, the Pontryagin dual.
For a compact group, every element of $\widehat{G}$ is finite-dimensional.
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# Universal coalgebra

## Summary

Universal coalgebra is the general study of coalgebras for an endofunctor of the category of sets.
Such a coalgebra is a set of states $X$ with a map $X \to F(X)$.
The map tells what can be observed of a state and which states come next.
It is the dual of [[universal algebra|W82R6N0GM9]]: algebras build elements from smaller ones, coalgebras unfold states into their behaviour.
Streams, automata, labelled transition systems and many other state-based systems are coalgebras, and the theory treats their behaviour and their bisimulations in one framework.
The general notion over any category is the [[F-coalgebra|6BS0300CMV]].

## Explanations

### L1 human

Think of a machine with hidden states that you can only watch from outside.
At each state it shows you something and then moves to a next state.
A coalgebra records exactly this: for each state, what you see and where you go.
For example, a stream of numbers is a coalgebra for $F(X) = \N \times X$: each state gives a head $n \in \N$ and a next state.
Two states are treated as equal when no experiment can tell them apart.
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# Factor congruence

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.
Let $\Delta_A = \set{(a, a) : a \in A}$ and $\nabla_A = A \times A$.
A [[congruence|DCSXXT4SER]] $\theta$ on $\mathbf{A}$ is a *factor congruence* when there is a congruence $\theta^*$ on $\mathbf{A}$ such that:

1. $\theta \cap \theta^* = \Delta_A$;
2. $\theta \circ \theta^* = \nabla_A$, where $(a, c) \in \theta \circ \theta^*$ means that $(a, b) \in \theta$ and $(b, c) \in \theta^*$ for some $b \in A$.

Then $\theta$ and $\theta^*$ are called a pair of complementary factor congruences.

## Remarks

For such a pair, the map $a \mapsto (a/\theta, a/\theta^*)$ is an isomorphism from $\mathbf{A}$ onto the [[direct product|WJ73XYBNE4]] $\mathbf{A}/\theta \times \mathbf{A}/\theta^*$.
Conversely, if $\mathbf{A} = \mathbf{B} \times \mathbf{C}$, the kernels of the two projections are complementary factor congruences.
The conditions say that $\theta$ and $\theta^*$ are complements in the [[congruence lattice|REE07CSXHR]] and that they permute.

Example: on the group $\Z/6\Z$, take $\theta$ to be congruence modulo $2$ and $\theta^*$ congruence modulo $3$.
An element congruent to $0$ modulo $2$ and modulo $3$ is $0$, so clause 1 holds.
For any $a, c$ there is $b$ with $b \equiv a \pmod 2$ and $b \equiv c \pmod 3$, by the Chinese remainder theorem, so clause 2 holds.

Non-example: on $\Z/4\Z$, congruence modulo $2$ is not a factor congruence.
The congruences of $\Z/4\Z$ are $\Delta$, congruence modulo $2$, and $\nabla$.
Only $\theta^* = \Delta$ satisfies clause 1, and then $\theta \circ \theta^* = \theta \ne \nabla$.

The congruences $\Delta_A$ and $\nabla_A$ are always factor congruences, complementary to each other.
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# Herbrand universe

## Example

Let $L$ be a first-order language with at least one constant symbol; if it has none, one constant is added.
The *Herbrand universe* $H(L)$ is the set of ground terms of $L$, the terms that contain no variables.
It is the smallest set that contains every constant symbol of $L$ and contains $f(t_1, \ldots, t_n)$ whenever $f$ is an $n$-ary function symbol and $t_1, \ldots, t_n \in H(L)$.

With the function symbols acting by $f(t_1, \ldots, t_n) \mapsto$ the term $f(t_1, \ldots, t_n)$, the set $H(L)$ is the [[term algebra|3YYKGWJJND]] on the empty set of variables.
So it is the [[absolutely free algebra|TTTEV3NWWM]] on no generators.
Check: every map from the empty set extends uniquely.
Indeed, a homomorphism to any algebra $A$ must send each ground term to its value in $A$.

For $L$ with one constant $0$ and one unary symbol $s$, the universe is $\set{0, s(0), s(s(0)), \ldots}$, a copy of $\N$.
For $L$ with one constant $a$ and one binary symbol $g$, it contains $a$, $g(a, a)$, $g(a, g(a, a))$, $g(g(a, a), a)$, and so on.
There is one element for each binary tree.

## Explanations

### L1 human

The Herbrand universe is the list of all objects that a language can name without variables.
Start from the constants and apply the function symbols again and again.
[[Herbrand's theorem|VZ21F10WRS]] reduces the unsatisfiability of a universal sentence to that of finitely many instances built from ground terms in this universe.
This is why it is central in automated theorem proving.
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# Grätzer–Schmidt theorem

## Statement

Let $L$ be an [[algebraic lattice|3J0BTXSCVN]].
Then there is an algebra $A$, a set with a family of finitary operations, with the following property.
The lattice $\operatorname{Con} A$ of congruences of $A$, ordered by inclusion, is isomorphic to $L$.

## Remarks

A congruence of an algebra $A$ is an equivalence relation on $A$ that is compatible with every operation of $A$.
For every algebra $A$, the lattice $\operatorname{Con} A$ is algebraic: its [[compact elements|9F7TYES4KE]] are the finitely generated congruences, and every congruence is the join of the congruences generated by single pairs.
So the theorem says that the algebraic lattices are exactly the congruence lattices of algebras, up to isomorphism.
The proof is an explicit construction: the operations of $A$ are chosen so that the finitely generated congruences of $A$ behave like the compact elements of $L$.
The algebra produced may have infinitely many operations; the representation of finite lattices as congruence lattices of finite algebras is a different and harder question.

## Explanations

### L1 human

Every algebra, such as a group, a ring or a lattice, has congruences, which are the ways to glue its elements together compatibly with the operations.
The congruences of one algebra form a lattice with a special property called algebraic.
The theorem says that this is the only restriction: every algebraic lattice arises in this way from some algebra.

### L2 human

In a course on universal algebra this theorem closes the description of congruence lattices.
It sits next to the [[Birkhoff–Frink theorem|DZGNHKE3X1]], which says the same thing for lattices of subalgebras.
For special classes of algebras the question changes: the congruence lattice of a lattice is always distributive, and which distributive algebraic lattices occur there is the congruence lattice problem.
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# Operational calculus

## Definition

Let $E$ be the space of locally integrable functions $f \colon [0, \infty) \to \C$ of exponential order, that is, with $\abs{f(t)} \le M e^{at}$ for some constants $M, a$ and almost all $t$.
The *operational calculus* based on the [[Laplace transform|0V49JEZW06]] is the correspondence $f \mapsto F = \mathcal{L} f$, $F(s) = \int_0^\infty e^{-st} f(t) \, dt$, used with the following rules.

1. $\mathcal{L}$ is linear and injective up to equality almost everywhere, so $f$ is recovered from $F$ by the [[inverse Laplace transform|07WBTX1MHA]].
2. If $f$ is continuous, $f'$ exists except at finitely many points on each bounded interval, and $f' \in E$, then $\mathcal{L}(f')(s) = s F(s) - f(0)$.
3. The [[convolution on the half-line|VDMCEPTTXB]] becomes a product: $\mathcal{L}(f * g) = \mathcal{L} f \cdot \mathcal{L} g$.

A linear problem for $f$ is solved by transforming it with these rules into an algebraic equation for $F$, solving for $F$, and recovering $f$.

## Remarks

Rule 2 is the [[Laplace transform of a derivative|0K090GVHVN]] and rule 3 is the [[convolution theorem|0JNMJEDD93]].
With them, a [[linear differential equation|0QQRP17Z7S]] with constant coefficients and given initial values becomes a linear equation for $F$.
Its coefficients are polynomials in $s$.
The name operational calculus also covers other frameworks with the same structure.
One is the formal calculus of Heaviside; another is the Mikusiński calculus, where $s$ is an element of a field and no integral transform is used.

Example: $y'' + y = 0$, $y(0) = 0$, $y'(0) = 1$.
Rule 2 twice gives $s^2 Y - 1 + Y = 0$, so $Y(s) = 1/(s^2 + 1)$, which is the transform of $\sin t$.
Check: $\sin'' + \sin = 0$, $\sin 0 = 0$, $\sin' 0 = 1$.

Non-example: $y' = t\, e^{t^2}$ fits the scheme only formally.
Its solutions are $e^{t^2}/2 + C$, which are not of exponential order, so they are outside $E$ and have no Laplace transform.
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# Homogeneous polynomial

## Summary

A homogeneous polynomial is a [[polynomial in several variables|07GNX8GCYP]] whose nonzero terms all have the same total degree.
Such polynomials are also called forms; the [[quadratic forms|0RBQS1JJCR]] are the homogeneous polynomials of degree $2$.
A homogeneous polynomial of degree $d$ satisfies $f(\lambda x) = \lambda^d f(x)$, so its zero set is a union of lines through the origin.
This is why homogeneous polynomials describe the subsets of projective space, and why they appear in the definition of [[quasi-algebraically closed fields|G7RM16N4PF]].

## Explanations

### L1 human

Take a polynomial and add the exponents in each term.
If every term gives the same total, the polynomial is homogeneous.
For example, $x^3 + 2xy^2 - 5y^3$ is homogeneous of degree $3$, since each term has total degree $3$.
The polynomial $x^2 + y$ is not homogeneous, because $x^2$ has degree $2$ and $y$ has degree $1$.
If you scale all variables by $2$, a homogeneous polynomial of degree $3$ is scaled by $2^3 = 8$.
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# Ordinal arithmetic

## Definition

Let $\alpha$ and $\beta$ be [[ordinal numbers|0RDJK7EQYY]].
In the clauses below, $\lambda$ is a nonzero limit ordinal, and $\sup$ is the [[supremum|0BTMQ04556]] of a set of ordinals, which is its union.
The sum, product and power of ordinals are defined by [[transfinite recursion|0ZHQMQ2C1N]] on $\beta$:

1. $\alpha + 0 = \alpha$, $\alpha + (\beta + 1) = (\alpha + \beta) + 1$, and $\alpha + \lambda = \sup_{\beta < \lambda} (\alpha + \beta)$.
2. $\alpha \cdot 0 = 0$, $\alpha \cdot (\beta + 1) = \alpha \cdot \beta + \alpha$, and $\alpha \cdot \lambda = \sup_{\beta < \lambda} \alpha \cdot \beta$.
3. $\alpha^0 = 1$, $\alpha^{\beta + 1} = \alpha^\beta \cdot \alpha$, and $\alpha^\lambda = \sup_{\beta < \lambda} \alpha^\beta$ when $\alpha > 0$, with $0^\lambda = 0$.

Here $\beta + 1 = \beta \cup \set{\beta}$ is the successor of $\beta$.

## Remarks

Example: $\omega + 1$ is the successor of $\omega$, and $\omega \cdot 2 = \omega + \omega$ by clauses 2 and 1.
Non-example of commutativity: $1 + \omega = \sup_{n < \omega} (1 + n) = \omega$, while $\omega + 1 \ne \omega$.
Likewise $2 \cdot \omega = \sup_n 2n = \omega$, while $\omega \cdot 2 = \omega + \omega > \omega$.
Equivalently, $\alpha + \beta$ is the order type of $\alpha$ followed by $\beta$.
And $\alpha \cdot \beta$ is the order type of the pairs $(b, a)$ with $b < \beta$ and $a < \alpha$, compared first by $b$ and then by $a$.
Addition and multiplication are associative, and $\alpha \cdot (\beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma$, but the other distributive law fails.
On finite ordinals the operations agree with those of the natural numbers.
Ordinal exponentiation differs from cardinal exponentiation: $2^\omega = \omega$ as ordinals, while $2^{\aleph_0} > \aleph_0$ as cardinals.
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# Semiperfect ring

## Definition

Let $R$ be a ring with identity and $J = J(R)$ its [[Jacobson radical|XA0DCNH7WY]].
$R$ is *semiperfect* if the following hold.

1. $R$ is [[semilocal|YH75M7XAM1]], that is, $R/J$ is a semisimple ring.
2. [[Idempotents lift|PJA5H9ZW4S]] modulo $J$: for every $x \in R$ with $x^2 - x \in J$ there is $e \in R$ with $e^2 = e$ and $e - x \in J$.

## Remarks

Example: a [[local ring|MZ02BKPHJA]] $R$ is semiperfect: $R/J$ is a field, hence semisimple, and its only idempotents $0$ and $1$ lift to $0$ and $1$.
Non-example: let $R = \set{a/b \in \Q : \gcd(b, 6) = 1}$.
Its maximal ideals are generated by $2$ and $3$, so $R/J \cong \mathbb{F}_2 \times \mathbb{F}_3$ and clause 1 holds.
Clause 2 fails: $R$ is a domain, so its only idempotents are $0$ and $1$, which cannot lift the idempotent $(1, 0)$ of $\mathbb{F}_2 \times \mathbb{F}_3$.
Equivalent forms: every finitely generated left $R$-module has a [[projective cover|WGYXR97NPK]]; or $1 = e_1 + \cdots + e_n$ for orthogonal idempotents $e_i$ with each $e_i R e_i$ local.
The definition is left-right symmetric.
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# Artin's induction theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]].
Then every complex [[character|RMWMYBY8TC]] of $G$ is a rational linear combination of [[induced|JJP4JFH3C6]] characters $\operatorname{Ind}_C^G \theta$.
Here $C$ runs over the [[cyclic|0KVGH0J5B3]] subgroups of $G$ and $\theta$ over the characters of $C$.

## Remarks

For the proof, let $\psi$ be a class function on $G$ orthogonal to all characters induced from cyclic subgroups.
By [[Frobenius reciprocity|PAP76H0ND9]], the restriction of $\psi$ to each cyclic subgroup is orthogonal to all its characters, so it is zero.
Every element of $G$ lies in a cyclic subgroup, so $\psi = 0$.
Hence the induced characters span the space of class functions over $\C$, and a rationality argument gives rational coefficients.
Rational coefficients are needed in general.
For example, let $V = \Z/2 \times \Z/2$ be the Klein four-group.
Its cyclic subgroups are proper, so every character induced from one of them has even degree, and so has every integer combination of such characters.
Hence the trivial character $1_V$, of degree $1$, is not an integer combination.
But $1_V = \frac{1}{2}\sum_{C} \operatorname{Ind}_C^V 1_C - \frac{1}{2}\operatorname{Ind}_{1}^{V} 1$, with $C$ running over the three subgroups of order $2$.
[[Brauer's theorem on induced characters|V8JVWE87TE]] reaches integer coefficients by using elementary subgroups instead of cyclic ones.

## Explanations

### L1 human

The theorem says that all characters of a finite group come from characters of its cyclic subgroups, the simplest subgroups there are, by induction and by taking rational combinations.
So, up to fractions, cyclic subgroups control the whole character theory.

### L2 human

Artin used this to show that a power of every Artin L-function is a quotient of products of abelian L-functions, so that some power of it is meromorphic.
Brauer's theorem later removed the power.
The theorem is a statement about the rational character ring: tensored with $\Q$, the induction maps from cyclic subgroups are jointly surjective.
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# Vietoris topology

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]], and let $\mathrm{CL}(X)$ be the set of nonempty [[closed|0ZCK8DQYN1]] subsets of $X$.
For an [[open|02BH3MCBQB]] set $U \subseteq X$, put
$$U^{+} = \set{A \in \mathrm{CL}(X) : A \subseteq U}, \qquad U^{-} = \set{A \in \mathrm{CL}(X) : A \cap U \ne \varnothing}.$$
The *Vietoris topology* on $\mathrm{CL}(X)$ is the topology generated by the [[subbasis|09S8DV34Y0]] of all sets $U^{+}$ and $U^{-}$ with $U$ open in $X$.

## Remarks

The same subbasis, intersected with a smaller family, defines the Vietoris topology on that family, for example on the nonempty [[compact|0R6EWFKRCF]] subsets $\mathrm{K}(X)$.
If $X$ is a metric space, the Vietoris topology on $\mathrm{K}(X)$ is the topology of the Hausdorff metric.
If $X$ is compact Hausdorff, then $\mathrm{CL}(X)$ with the Vietoris topology is compact Hausdorff.
If $X$ is $T_1$, the map $x \mapsto \set{x}$ is a [[topological embedding|00ZG9Y9TR4]] of $X$ into $\mathrm{CL}(X)$.

Example: let $X = [0, 1]$, let $A = [0, 1]$, and let $A_n = \set{k/n : 0 \le k \le n}$.
Each basic neighbourhood of $A$ has the form $U^+ \cap V_1^- \cap \cdots \cap V_m^-$ with $A \subseteq U$ and each $V_i$ open and nonempty.
For large $n$, the set $A_n$ meets each $V_i$ and lies in $U = X$, so $A_n \to A$.

Non-example: the sets $\set{0}$ do not converge to $[0, 1]$, though they lie in every $U^+$ with $[0, 1] \subseteq U$.
The open set $V = (1/2, 1]$ gives a neighbourhood $V^-$ of $[0, 1]$ that does not contain $\set{0}$.

The resulting space is called a hyperspace of $X$, and the topology is also called the finite topology or exponential topology.
</document>

<document source="https://tamnd.github.io/math2/node/2CHENV4DCC.md" tag="2CHENV4DCC" kind="definition">
# Well-formed formula

## Definition

Let $\sigma$ be a [[signature|WDVVEYD6AT]] and $V$ a set of variables.
The set of first-order *formulas* over $\sigma$ is the smallest set of strings such that:

1. if $t_1$ and $t_2$ are [[terms|6D3WA7N2W4]], then $t_1 = t_2$ is a formula;
2. if $R$ is a relation symbol of arity $n$ and $t_1, \dots, t_n$ are terms, then $R(t_1, \dots, t_n)$ is a formula;
3. if $\varphi$ and $\psi$ are formulas, then $\neg \varphi$, $(\varphi \land \psi)$, $(\varphi \lor \psi)$, $(\varphi \to \psi)$ and $(\varphi \leftrightarrow \psi)$ are formulas;
4. if $\varphi$ is a formula and $x \in V$, then $\forall x \, \varphi$ and $\exists x \, \varphi$ are formulas.

The formulas from clauses 1 and 2 are the *atomic formulas*.

## Remarks

In the signature of ordered sets, $\forall x \, \exists y \, (x < y)$ is a formula.
Here $x < y$ is atomic by clause 2, $\exists y \, (x < y)$ follows by clause 4, and then $\forall x$ by clause 4 again.
The string $\forall x \, (x \land y)$ is not a formula: $x$ and $y$ are terms, not formulas, so clause 3 does not apply to them.
In propositional logic the formulas are built from variables by clause 3 alone.
The [[connectives|QS0XVAN9BG]] used vary between books, since some are definable from others.
In Mathlib formulas with bounded variables are `FirstOrder.Language.BoundedFormula`.
</document>

<document source="https://tamnd.github.io/math2/node/2CKZETC3QA.md" tag="2CKZETC3QA" kind="definition">
# Discrete series representation

## Definition

Let $G$ be a [[unimodular|CXG6JZGMCK]] [[locally compact group|AHV05ZBCYW]] with a [[Haar measure|VY0HXD3GC4]] $dg$.
Let $\lambda$ be the left regular representation of $G$ on $L^2(G)$, given by $(\lambda(g) f)(x) = f(g^{-1} x)$.
Let $\pi$ be an [[irreducible|FX3NEFYFFN]] [[unitary representation|74BPYJNZTM]] of $G$ on a Hilbert space $H$.
Then $\pi$ is a *discrete series representation* if it is unitarily equivalent to a subrepresentation of $\lambda$.

## Remarks

For $u, v \in H$, the matrix coefficient of $\pi$ is the function $g \mapsto \ip{\pi(g) u}{v}$.
Equivalently, $\pi$ is in the discrete series if and only if some nonzero matrix coefficient lies in $L^2(G)$.
Then every matrix coefficient lies in $L^2(G)$.

Example: let $G$ be a compact group, with Haar measure of total mass $1$.
Each matrix coefficient of a unitary representation is continuous and bounded by $\norm{u} \norm{v}$, so it lies in $L^2(G)$.
So every irreducible unitary representation of $G$ is in the discrete series.

Non-example: the irreducible unitary representations of $G = \R$ are the characters $x \mapsto e^{ixt}$ with $t \in \R$.
Each matrix coefficient has constant absolute value, so it is not square-integrable on $\R$.
So $\R$ has no discrete series, and $L^2(\R)$ decomposes as a continuous direct integral by Fourier analysis.

For groups with non-compact center, such as $\mathrm{GL}_n(\R)$, one asks instead that the matrix coefficients be square-integrable modulo the center.
For connected semisimple $G$ with finite center and maximal compact subgroup $K$, discrete series exist if and only if $\rank G = \rank K$, by a theorem of Harish-Chandra.
For example, $\mathrm{SL}_2(\R)$ has discrete series, while $\mathrm{SL}_2(\C)$ has none.
</document>

<document source="https://tamnd.github.io/math2/node/2CM2MC5HQ6.md" tag="2CM2MC5HQ6" kind="definition">
# Cellular automaton

## Definition

Let $A$ be a finite set of states and let $d \ge 1$.
A configuration is an element $x = (x_u)_{u \in \Z^d}$ of $A^{\Z^d}$.
Let $N \subseteq \Z^d$ be a finite set, called the neighbourhood, and let $f \colon A^N \to A$ be a map, called the local rule.
The *cellular automaton* with neighbourhood $N$ and local rule $f$ is the map $F \colon A^{\Z^d} \to A^{\Z^d}$ given by
$$F(x)_u = f\big((x_{u + n})_{n \in N}\big) \quad \text{for all } u \in \Z^d.$$

## Remarks

Example: rule 90 has $d = 1$, $A = \set{0, 1}$, $N = \set{-1, 1}$ and $f(a, b) = a + b \bmod 2$.
So $F(x)_i = x_{i-1} + x_{i+1} \bmod 2$, and the value at $i$ depends only on the sites $i \pm 1$, as required.
Non-example: the map $F(x)_i = x_{2i}$ on $\set{0,1}^{\Z}$ is not a cellular automaton.
The value at $i$ reads the site $2i$, so no fixed finite set $N$ of offsets works for all $i$.
Equivalently, it does not commute with the shift, which the [[Curtis–Hedlund–Lyndon theorem|N35QM9G83G]] requires.
The dynamics is the iteration of $F$, with $F^t(x)$ the configuration at time $t$.
In one dimension with $A = \set{0,1}$ and $N = \set{-1, 0, 1}$, the $2^8 = 256$ local rules are the elementary cellular automata, numbered by Wolfram's convention.
</document>

<document source="https://tamnd.github.io/math2/node/2CNFVTWFRC.md" tag="2CNFVTWFRC" kind="definition">
# Geodesic

## Definition

Let $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] with [[Levi-Civita connection|CAZYGE1T4E]] $\nabla$, and let $I \subseteq \R$ be an interval.
A smooth curve $\gamma \colon I \to M$ is a *geodesic* if
$$D_t \gamma'(t) = 0 \quad \text{for all } t \in I,$$
where $D_t$ is the [[covariant derivative|5FZDQQHQKM]] along $\gamma$ induced by $\nabla$.

## Remarks

In local coordinates, with Christoffel symbols $\Gamma^k_{ij}$, the condition is the geodesic equation
$$\ddot\gamma^k + \sum_{i,j} \Gamma^k_{ij}(\gamma) \, \dot\gamma^i \dot\gamma^j = 0, \qquad k = 1, \ldots, n.$$

Example: in $\R^n$ with the Euclidean metric all $\Gamma^k_{ij}$ vanish, so the geodesics are the curves with $\ddot\gamma = 0$, that is $\gamma(t) = p + tv$.

Non-example: in $\R^2$, the curve $\gamma(t) = (t^2, 0)$ traces a straight line, but $\ddot\gamma = (2, 0) \ne 0$, so it is not a geodesic: a geodesic must have constant speed.

The quantity $g(\gamma', \gamma')$ is constant along a geodesic.
On a Riemannian manifold, every geodesic is locally length-minimizing: each $t \in I$ has a neighbourhood in $I$ on which $\gamma$ is a shortest curve between its endpoints.
For each $p \in M$ and $v \in T_pM$ there is a unique maximal geodesic $\gamma_v$ with $\gamma_v(0) = p$ and $\gamma_v'(0) = v$.
</document>

<document source="https://tamnd.github.io/math2/node/2CVH8BCHTF.md" tag="2CVH8BCHTF" kind="statement">
# Grauert's Oka principle for vector bundles

## Statement

Let $X$ be a [[Stein manifold|HMCD0JBFST]].

1. Two [[holomorphic vector bundles|TKA958Z4KZ]] on $X$ are holomorphically isomorphic if and only if they are isomorphic as topological complex [[vector bundles|P4DZM975BV]].
2. Every topological complex vector bundle on $X$ is topologically isomorphic to a holomorphic vector bundle.

## Remarks

Grauert proved the theorem in 1958, for holomorphic principal bundles over Stein spaces whose structure group is a complex Lie group.
Vector bundles of rank $r$ are the case of the group $GL_r(\C)$.
For line bundles it is the [[Oka principle|S1CJV1PA3D]] of 1939, which follows from the exponential sequence.
For higher rank the group $GL_r(\C)$ is not commutative, and the proof needs a new, nonlinear method.
Grauert deformed continuous transition functions to holomorphic ones by successive approximation over an exhaustion of $X$, using a Runge-type approximation for maps into complex Lie groups.
Example: on $\C^n$, which is contractible, every topological complex vector bundle is trivial, so every holomorphic vector bundle on $\C^n$ is holomorphically trivial.
The Stein hypothesis is needed.
On the projective line, the rank $2$ bundle $\mathcal{O}(1) \oplus \mathcal{O}(-1)$ and the trivial bundle of rank $2$ are topologically isomorphic, since both have first Chern class $0$.
They are not holomorphically isomorphic, since the first has a nonzero holomorphic section that vanishes at a point.

## Explanations

### L1 human

A vector bundle attaches a vector space to each point of a space.
Holomorphic vector bundles carry much more structure than continuous ones, and in general there are many more of them.
Grauert's theorem says that on a Stein manifold this extra structure brings nothing new: the holomorphic classification is the same as the topological one.

### L2 human

The theorem reduces the classification of holomorphic vector bundles on a Stein manifold to homotopy theory.
Bundles of rank $r$ correspond to homotopy classes of maps from $X$ to the classifying space of $GL_r(\C)$.
It is the starting point of modern Oka theory.
This theory asks for which target manifolds $Y$ the same principle holds for holomorphic maps from Stein manifolds to $Y$; these targets are the [[Oka manifolds|NXEAT9DMWD]].
</document>

<document source="https://tamnd.github.io/math2/node/2CX7RPPXA9.md" tag="2CX7RPPXA9" kind="example">
# Category of groups

## Example

The *category of groups* $\mathbf{Grp}$ has the [[groups|G5S1M9]] as objects and the [[group homomorphisms|0JYTG5K9Y4]] $f \colon G \to H$ as morphisms.
Composition is the composition of maps, and the identity morphism of $G$ is the identity map $\id_G$.
This is a [[category|A2DE66937D]].
If $f \colon G \to H$ and $g \colon H \to K$ are homomorphisms, then $g(f(xy)) = g(f(x)f(y)) = g(f(x))\,g(f(y))$, so $g \circ f$ is a homomorphism.
The identity map is a homomorphism, and composition of maps is associative, so both axioms of a category hold.

## Remarks

The trivial group is a [[zero object|AQET9YHH6F]]: from each group there is exactly one homomorphism to it, and from it exactly one homomorphism to each group.
The [[product|S5744QZV4N]] of $G$ and $H$ is the [[direct product|00HF4RJ5QN]] $G \times H$, and the [[coproduct|4FJ0S71P58]] is the [[free product|TM7QS7P7X9]] $G * H$.
The monomorphisms are the injective homomorphisms, and the epimorphisms are the surjective homomorphisms.
Forgetting the group structure gives a functor to the category of sets, whose left adjoint sends a set to the [[free group|020Z5SEN27]] on it.
$\mathbf{Grp}$ is not an [[abelian category|Y9CZ3MTNF1]]: for example, the coproduct $\Z * \Z$ differs from the product $\Z \times \Z$, since the first group is not abelian.

## Explanations

### L1 human

The category of groups collects all groups and all structure-preserving maps between them into one object of study.
Many group constructions become instances of general patterns there: the direct product is a categorical product and the free product is a coproduct.
For example, the map $\Z \to \Z/2\Z$ that sends an integer to its parity is a morphism of this category.
</document>

<document source="https://tamnd.github.io/math2/node/2CXH65EKY7.md" tag="2CXH65EKY7" kind="object">
# E8 lattice

## Description

The $E_8$ lattice is the [[lattice|WN9V7WSJKF]] in the [[Euclidean space|0XPR1E85W0]] $\R^8$ given by
$$E_8 = \set{x \in \Z^8 \cup (\Z + \tfrac12)^8 : x_1 + x_2 + \dots + x_8 \in 2\Z}.$$
So its points have all coordinates integers or all coordinates half-integers, with even coordinate sum.

## Remarks

It is closed under addition and negation, so it is a subgroup of $\R^8$, and it contains $8$ independent vectors.
It is even: $x \cdot x \in 2\Z$ for all $x \in E_8$.
It is unimodular: its covolume is $1$.
Up to isometry, it is the unique even unimodular lattice in $\R^8$.
Its shortest nonzero vectors have length $\sqrt 2$, and there are $240$ of them.
They are the $112$ vectors with two coordinates $\pm 1$ and six coordinates $0$.
The other $128$ are the vectors with all coordinates $\pm\tfrac12$ and an even number of minus signs.
These $240$ vectors form the root system of type $E_8$.
Balls of radius $\sqrt 2/2$ centred at the points of $E_8$ form the densest [[sphere packing|HF32CBN5MA]] in $\R^8$, with density $\pi^4/384$, as proved by Viazovska.
Each ball touches $240$ others, which is the [[kissing number|B408FZYXF9]] in dimension $8$.

## Explanations

### L1 human

The $E_8$ lattice is a grid of points in eight dimensions.
Take all points whose coordinates are either all whole numbers or all halves of odd numbers, and keep those whose coordinates add up to an even number.
This grid is extremely symmetric, and placing equal balls at its points gives the tightest possible packing of balls in eight dimensions.
</document>

<document source="https://tamnd.github.io/math2/node/2D3N1ZJWBW.md" tag="2D3N1ZJWBW" kind="concept">
# Fully nonlinear PDE

## Summary

A fully nonlinear PDE is a [[partial differential equation|GC5BDZ6QBC]] that depends nonlinearly on the highest-order derivatives of the unknown.
Examples are the [[Monge–Ampère equation|14AKAC0S0T]], the [[eikonal equation|HGMX1QGTAZ]] and the [[Hamilton–Jacobi equation|5TRXQKP7DY]].
Such equations often lack classical solutions, and notions such as [[viscosity solutions|H1ERJ3T75T]] were developed for them.
See the [[definition|MTQ3SGX56W]].

## Explanations

### L1 human

In a fully nonlinear equation the highest derivatives are squared, multiplied together or put inside other nonlinear functions.
For instance, $\abs{\nabla u} = 1$ asks for a function whose gradient always has length $1$.
The distance to a closed set has this property wherever it is differentiable away from the set, but it can have corners.
For example, the distance to the two points $\pm 1$ on the real line has a corner at $0$.
So even simple fully nonlinear equations force one to accept nonsmooth solutions.
</document>

<document source="https://tamnd.github.io/math2/node/2D4E6NY8Q4.md" tag="2D4E6NY8Q4" kind="definition">
# Kleene algebra

## Definition

Let $(K, +, \cdot, 0, 1)$ be an [[idempotent semiring|NA44T2NZE2]], with the order $a \le b \iff a + b = b$.
Let $^*$ be a unary operation on $K$.
The structure $(K, +, \cdot, ^*, 0, 1)$ is a *Kleene algebra* if for all $a, b, x \in K$:

1. $1 + a \cdot a^* \le a^*$;
2. $1 + a^* \cdot a \le a^*$;
3. if $b + a \cdot x \le x$, then $a^* \cdot b \le x$;
4. if $b + x \cdot a \le x$, then $b \cdot a^* \le x$.

## Remarks

Clauses 1 and 3 say that $a^* b$ is the least $x$ with $b + a x \le x$.
Example: let $\Sigma^*$ be the [[free monoid|27ZPTJP9PP]] on an alphabet $\Sigma$, and let $K$ be the set of languages $L \subseteq \Sigma^*$.
With union, concatenation, $0 = \emptyset$, $1 = \set{\varepsilon}$ and $L^* = \bigcup_{n \ge 0} L^n$, this is a Kleene algebra.
Indeed, $L^* B$ is the set of words $w_1 \cdots w_n u$ with $w_i \in L$ and $u \in B$, and each such word lies in any $X$ with $B \cup L X \subseteq X$, by induction on $n$.
Example: the [[binary relations|08EJC6Q25H]] on a set, with union, composition and reflexive transitive closure as $^*$, form a Kleene algebra.
Non-example: the Boolean semiring $\set{0, 1}$ with $a^* = a$ fails clause 1, since for $a = 0$ it says $1 + 0 \le 0$.
With $a^* = 1$ for all $a$, the Boolean semiring is a Kleene algebra.
In every Kleene algebra $a^* = 1 + a a^*$ and $a^* a^* = a^*$.
</document>

<document source="https://tamnd.github.io/math2/node/2D5SS6WM14.md" tag="2D5SS6WM14" kind="concept">
# Function field of a variety

## Summary

The function field $k(X)$ of an algebraic variety $X$ over $k$ is the field of rational functions on $X$.
A rational function is a function regular on some nonempty open subset.
Two of them are identified when they agree on a smaller open set.
For an affine variety it is the field of fractions of the coordinate ring.
It is a finitely generated field extension of $k$.
It is a birational invariant: two varieties have isomorphic function fields over $k$ exactly when they are birational.

## Explanations

### L1 human

A rational function is a fraction of polynomials, like $x/y$, which makes sense except where the denominator vanishes.
On a variety, such a fraction is defined on a large open piece, and we do not care which piece.
The function field collects all of these fractions, and it is a field because every nonzero fraction can be turned upside down.
The function field of the affine line is $k(t)$, the field of rational functions in one variable.
</document>

<document source="https://tamnd.github.io/math2/node/2D619WTPYC.md" tag="2D619WTPYC" kind="concept">
# Bergman space

## Summary

Let $\Omega$ be a domain in $\C$ and let $0 < p < \infty$.
The Bergman space $A^p(\Omega)$ is the space of [[holomorphic functions|0NAHXR1FRH]] $f$ on $\Omega$ such that $\abs{f}^p$ is integrable with respect to area.
Point evaluations are continuous on it, so for $p \ge 1$ it is a closed subspace of $L^p(\Omega)$ and a [[Banach space|06GZ956657]].
The space $A^2(\Omega)$ is a Hilbert space with a reproducing kernel, the [[Bergman kernel|8RWDHW0CXC]].
Bergman spaces differ from [[Hardy spaces|PNNNFJVF6W]] by measuring size over the whole domain instead of on circles near the boundary.

## Explanations

### L1 human

A function is in a Bergman space if the total area under $\abs{f}^p$ over the domain is finite.
This allows more growth near the boundary than a Hardy space does, because area integrals forgive spikes on thin regions.
On the unit disc, $1/(1 - z)$ lies in $A^1$ but not in $A^2$.
The square-integrable case $A^2$ has a rich geometry, because it carries an inner product.
</document>

<document source="https://tamnd.github.io/math2/node/2DC88BMQVX.md" tag="2DC88BMQVX" kind="example">
# Horocycle flow

## Example

Let $G = \mathrm{PSL}_2(\R)$, the quotient of the [[special linear group|0V9PMN0RBH]] $\mathrm{SL}_2(\R)$ by $\set{\pm I}$.
As in [[geodesic flow on a locally symmetric space|5TKBSHY6VC]], identify $G$ with the unit tangent bundle of the [[upper half-plane|X6XGGH5FPW]] $\mathbb{H}$.
Then the geodesic flow is $g \mapsto g a_t$, with $a_t$ the class of $\operatorname{diag}(e^{t/2}, e^{-t/2})$.
Let $h_s$ be the class of $\begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix}$.
The (stable) horocycle flow is $\psi_s(g) = g h_s$, for $s \in \R$.
For a [[lattice|BTYF5Y5PFY]] $\Gamma \subseteq G$, it descends to $\Gamma \backslash G$, because left and right multiplications commute.

Check of the name and of the link with the geodesic flow.
A matrix computation gives $a_{-t} h_s a_t = h_{e^{-t} s}$.
Hence $g h_s a_t = g a_t h_{e^{-t}s}$, so the geodesic orbits of $g$ and of $g h_s$ approach each other as $t \to +\infty$.
So the orbits of $\psi$ are the stable leaves of the geodesic flow.
They project to horocycles in $\mathbb{H}$: circles tangent to the boundary, and horizontal lines.

## Remarks

When $\Gamma$ is cocompact, Hedlund proved that the horocycle flow on $\Gamma \backslash G$ is minimal: every orbit is dense.
Furstenberg proved that it is then uniquely ergodic, with the Haar measure as the only invariant probability measure.
The relation $a_{-t} h_s a_t = h_{e^{-t}s}$ shows that the geodesic flow renormalises the horocycle flow.
The horocycle flow has zero entropy, in contrast with the geodesic flow, which is an Anosov flow.

## Explanations

### L1 human

On the hyperbolic plane, a horocycle is a curve that looks like a circle of infinite radius, touching the boundary at one point.
The horocycle flow slides a unit vector along such a curve, keeping it perpendicular to the curve.
Two vectors on the same horocycle, when pushed forward along geodesics, come closer and closer.
On a closed hyperbolic surface every horocycle orbit spreads evenly over the whole space, which makes this flow a model of rigid, slowly mixing motion.
</document>

<document source="https://tamnd.github.io/math2/node/2DDZXF5EBY.md" tag="2DDZXF5EBY" kind="concept">
# Mathieu function

## Summary

Mathieu functions are solutions of Mathieu's equation $y'' + (a - 2q \cos 2x) y = 0$.
For each $q$, only special values of $a$, the characteristic values, admit solutions of period $\pi$ or $2\pi$.
These periodic solutions $\operatorname{ce}_r(x, q)$ and $\operatorname{se}_r(x, q)$ are the Mathieu functions in the narrow sense.
They arise when the [[Helmholtz equation|MNXKTXJ35G]] is separated in elliptic coordinates, for instance for vibrations of an elliptic drum.
Mathieu's equation is the basic example of a [[Hill differential equation|YE7H8DGDPY]], studied by [[Floquet theory|BJVZXPB297]].

## Explanations

### L1 human

Mathieu's equation describes an oscillator whose stiffness is shaken periodically in time.
For $q = 0$ the shaking is off, and the equation $y'' + a y = 0$ has the periodic solutions $\cos rx$ and $\sin rx$ when $a = r^2$.
For $q \ne 0$ periodic solutions still exist, but only when $a$ is tuned to special values $a_r(q)$ or $b_r(q)$ that move with $q$.
The periodic solutions are deformed cosines $\operatorname{ce}_r$ and deformed sines $\operatorname{se}_r$.
For other values of $a$ the solutions are either bounded but not periodic, or they grow exponentially, which is the phenomenon of parametric resonance.
</document>

<document source="https://tamnd.github.io/math2/node/2DJT2AYY23.md" tag="2DJT2AYY23" kind="definition">
# Köthe–Toeplitz dual

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, let $\omega$ be the space of all sequences in $\mathbb{K}$, and let $X \subseteq \omega$ be a [[sequence space|XYTN19EPF2]].
The *$\alpha$-dual* of $X$ is
$$X^\alpha = \set{a \in \omega : \textstyle\sum_{n} \abs{a_n x_n} < \infty \text{ for every } x \in X}.$$
The *$\beta$-dual* of $X$ is
$$X^\beta = \set{a \in \omega : \textstyle\sum_{n} a_n x_n \text{ converges for every } x \in X}.$$
The $\alpha$-dual is called the Köthe–Toeplitz dual of $X$, and the $\beta$-dual is also called the generalized Köthe–Toeplitz dual.

## Remarks

Both duals are sequence spaces, and $X^\alpha \subseteq X^\beta$ because absolutely [[convergent series|0CVVKY7GY2]] converge.
If $X \subseteq Y$, then $Y^\alpha \subseteq X^\alpha$ and $Y^\beta \subseteq X^\beta$.

Example: $(\ell^1)^\alpha = (\ell^1)^\beta = \ell^\infty$, with $\ell^1$ and $\ell^\infty$ as in the [[spaces ℓ^p|08P6T2WH6R]].
If $a$ is bounded, then $\sum \abs{a_n x_n} \le \sup_n \abs{a_n} \sum \abs{x_n}$ for every $x \in \ell^1$.
If $a$ is unbounded, choose indices $n_1 < n_2 < \cdots$ with $\abs{a_{n_k}} \ge k^2$, and let $x_{n_k} = \overline{a_{n_k}} / \abs{a_{n_k}} \cdot k^{-2}$ and $x_n = 0$ otherwise.
Then $x \in \ell^1$, but the terms $a_{n_k} x_{n_k} = \abs{a_{n_k}} k^{-2} \ge 1$ do not tend to $0$, so $a \notin (\ell^1)^\beta$.

Non-example: $a = (n)_{n \ge 1}$ is not in $(\ell^1)^\beta$; for $x_n = 1/n^2$ the series $\sum a_n x_n = \sum 1/n$ diverges.

Other standard values: $\omega^\beta$ is the space $\varphi$ of sequences with finitely many nonzero terms, $\varphi^\beta = \omega$, and $(c_0)^\beta = (\ell^\infty)^\beta = \ell^1$.
</document>

<document source="https://tamnd.github.io/math2/node/2DSKCVDFAS.md" tag="2DSKCVDFAS" kind="definition">
# Montel space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a Hausdorff [[barrelled space|ZE6DXEJB8H]] over $\mathbb{K}$.
$E$ is a *Montel space* if every [[closed|0ZCK8DQYN1]] and [[bounded|3BAJ8ZTVPX]] subset of $E$ is [[compact|0R6EWFKRCF]].

## Remarks

A locally convex space in which closed bounded sets are compact, without the barrelled condition, is called *semi-Montel*.
Example: for an open set $\Omega \subseteq \C$, the space $H(\Omega)$ of holomorphic functions with the seminorms $p_K(f) = \sup_{z \in K} \abs{f(z)}$, $K \subseteq \Omega$ compact, is a Montel space.
It is a Fréchet space, hence barrelled.
A bounded set is a family that is uniformly bounded on each compact set, so [[Montel's theorem|ZVFVBVJMCB]] gives the compactness of closed bounded sets.
Non-example: an infinite-dimensional Banach space is barrelled, but its closed unit ball is closed and bounded and not compact, by Riesz's lemma.
Every Montel space is reflexive, and its strong dual is again a Montel space.
</document>

<document source="https://tamnd.github.io/math2/node/2DSN343YYW.md" tag="2DSN343YYW" kind="statement">
# Koksma–Hlawka inequality

## Statement

Let $s \ge 1$ and let $x_1, \ldots, x_N$ be points of $[0,1)^s$.
For $t = (t_1, \ldots, t_s) \in [0,1]^s$, let $A(t)$ be the number of indices $i$ with $x_i \in [0,t_1) \times \cdots \times [0,t_s)$.
The *star discrepancy* of the points is
$$D_N^* = \sup_{t \in [0,1]^s} \abs{\frac{A(t)}{N} - t_1 t_2 \cdots t_s}.$$
Let $f \colon [0,1]^s \to \R$ have finite variation $V(f)$ in the sense of Hardy and Krause.
Then
$$\abs{\frac{1}{N} \sum_{i=1}^{N} f(x_i) - \int_{[0,1]^s} f(u) \, du} \le V(f) \, D_N^*.$$

## Remarks

For $s = 1$ this is Koksma's inequality, and $V(f)$ is the ordinary [[total variation|8QA5DEWNYD]] of $f$ on $[0,1]$.
In that case the proof writes the error as a Riemann–Stieltjes integral $\int_0^1 \big(A(t)/N - t\big) \, df(t)$ by integration by parts, and bounds the integrand by $D_N^*$.
In higher dimension the same idea works with a multivariate integration by parts, which produces one term for each face of the cube.
The Hardy–Krause variation collects the variations of $f$ on all these faces.
The bound separates the two sources of error: $V(f)$ depends only on the integrand, and $D_N^*$ depends only on the points.
For a [[low-discrepancy sequence|XSPX1KW9AQ]] such as the [[Halton sequence|54MY2XHVGB]], $D_N^* = O\big((\log N)^s / N\big)$, so the [[quasi-Monte Carlo|ABSRB2KXA6]] error is $O\big((\log N)^s / N\big)$ for every $f$ of finite variation.
This beats the rate $N^{-1/2}$ of plain [[Monte Carlo integration|DAR3QJ5H7Y]] for large $N$.
The hypothesis of finite variation is needed: the indicator of a ball in $[0,1]^2$ has infinite Hardy–Krause variation, and the bound gives nothing for it.

## Explanations

### L1 human

To approximate the average of a function over a cube, one averages its values at $N$ sample points.
The inequality says the error is at most a product of two numbers.
One number measures how much the function wiggles.
The other measures how unevenly the points fill the cube.
So well-spread points give small errors for every function that does not wiggle too much.

### L2 human

The inequality is the basic error bound of quasi-Monte Carlo integration.
It turns the design of good integration rules into the purely geometric task of finding point sets with small star discrepancy.
Halton, Sobol and other digital sequences reach $D_N^* = O\big((\log N)^s / N\big)$.
The bound is sharp in a worst-case sense.
But $V(f)$ is often hard to compute, and it is infinite for simple functions with jumps along surfaces not aligned with the axes.
Variants replace $D_N^*$ and $V(f)$ by other pairs of dual quantities, for example $L^2$ discrepancy and a Sobolev-type norm.
</document>

<document source="https://tamnd.github.io/math2/node/2DT5DGTB8V.md" tag="2DT5DGTB8V" kind="definition">
# Bachmann's reflection geometry

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a set of involutions of $G$, that is, of elements of order $2$.
Assume that $S$ generates $G$ and is closed under conjugation by elements of $G$.
Call the elements of $S$ *lines*, written $a, b, c, g, h, j$.
For $\sigma, \tau \in G$, write $\sigma \mid \tau$ if $\sigma \tau$ is an involution.
Call the elements $ab$ with $a, b \in S$ and $a \mid b$ *points*, written $P, Q$.
For a point $P$ and a line $g$, $P \mid g$ means that $P$ lies on $g$.
For lines $a$ and $b$, $a \mid b$ means that they are perpendicular.
The pair $(G, S)$ is a *metric plane* if:

1. for any points $P, Q$ there is a line $g$ with $P \mid g$ and $Q \mid g$;
2. if $P \mid g$, $P \mid h$, $Q \mid g$ and $Q \mid h$, then $P = Q$ or $g = h$;
3. if $a, b, c \mid P$ for a point $P$, then $abc \in S$;
4. if $a, b, c \mid g$ for a line $g$, then $abc \in S$;
5. there are lines $g, h, j$ with $g \mid h$ such that none of $j \mid g$, $j \mid h$, $j \mid gh$ holds.

## Remarks

Clauses 3 and 4 are the three reflections theorems.
The product of the reflections in three lines through a common point, or in three lines perpendicular to a common line, is again a reflection.
Clause 5 excludes degenerate cases.

Example: let $G$ be the group of isometries of the Euclidean plane generated by the [[reflections|06BRDF3F0E]] in lines, and $S$ the set of these reflections.
Two reflections compose to an involution exactly when their lines are perpendicular, and then the product is the half-turn about the intersection point.
A half-turn about $P$ times the reflection in $g$ is an involution exactly when $P$ lies on $g$.
The product of reflections in three concurrent lines, or in three parallel lines, is a reflection, which gives clauses 3 and 4.

Non-example: let $G$ be cyclic of order $2$ and $S$ its non-identity element $s$.
Then $s s = 1$ is not an involution, so there are no points and clause 5 fails.

Metric planes give a group-theoretic form of plane [[absolute geometry|VKFPCSRNDH]].
The real hyperbolic plane and the real elliptic plane give further examples, with the same construction from their reflections.
</document>

<document source="https://tamnd.github.io/math2/node/2DTWB6F5TA.md" tag="2DTWB6F5TA" kind="example">
# Hilbert matrix

## Example

For $n \ge 1$, the Hilbert matrix $H_n$ is the $n \times n$ real matrix with entries
$$(H_n)_{ij} = \frac{1}{i + j - 1} \qquad (1 \le i, j \le n).$$
For $n = 3$ it is
$$H_3 = \begin{pmatrix} 1 & \tfrac12 & \tfrac13 \\ \tfrac12 & \tfrac13 & \tfrac14 \\ \tfrac13 & \tfrac14 & \tfrac15 \end{pmatrix}.$$

It is a [[Hankel matrix|PQ1D8FQB3E]]: the entry depends only on $i + j$, with $h_k = 1/(k-1)$.

It is a [[Cauchy matrix|95MTQMX59H]]: with $x_i = i$ and $y_j = 1 - j$, the entry is $1/(x_i - y_j)$, and all $x_i - y_j = i + j - 1$ are nonzero.

It is a [[positive definite matrix|0KX66CZMBB]].
Indeed, $(H_n)_{ij} = \int_0^1 t^{i-1} t^{j-1} \, dt$, so $H_n$ is the [[Gram matrix|00SPJSDT3Z]] of the functions $1, t, \dots, t^{n-1}$ for the inner product $\ip{f}{g} = \int_0^1 f(t) g(t) \, dt$.
For $c \in \R^n$ and $p(t) = \sum_i c_i t^{i-1}$, this gives $c^T H_n c = \int_0^1 p(t)^2 \, dt$, which is positive when $c \ne 0$, because then $p$ is a nonzero polynomial.

## Remarks

The inverse of $H_n$ has integer entries; for example $\det H_2 = 1/12$ and $H_2^{-1} = \begin{pmatrix} 4 & -6 \\ -6 & 12 \end{pmatrix}$.
The Hilbert matrix is a standard test case in numerical analysis, because it is badly conditioned: its condition number grows very fast with $n$.

## Explanations

### L1 human

The Hilbert matrix is a simple square table of fractions, $1$, $\tfrac12$, $\tfrac13$, and so on, arranged so that each anti-diagonal holds one fraction.
It comes from fitting a polynomial to a function by least squares on the interval from $0$ to $1$.
It looks harmless, but it is extremely close to a singular matrix, so computers lose many digits when they solve systems with it.
</document>

<document source="https://tamnd.github.io/math2/node/2DYR4C5Q5V.md" tag="2DYR4C5Q5V" kind="example">
# Index of the de Rham complex

## Example

Let $M$ be a closed oriented Riemannian manifold of dimension $n$, and let $\Omega^k(M)$ be the space of smooth real $k$-forms.
Let $d^*$ be the formal adjoint of the exterior derivative $d$.
The operator
$$D = d + d^* \colon \Omega^{\mathrm{even}}(M) \to \Omega^{\mathrm{odd}}(M)$$
is elliptic, since $D^* D$ is the Hodge Laplacian on even forms.
Its kernel is the space of harmonic even forms, and its cokernel is isomorphic to $\ker D^*$, the space of harmonic odd forms.
By the [[Hodge theorem|A127WFVYKW]], the harmonic $k$-forms form a space of dimension $b_k$, the $k$-th [[Betti number|ST71ZV3FES]].
So the [[analytic index|44GBFZ16SG]] is
$$\operatorname{ind} D = \sum_{k = 0}^{n} (-1)^k b_k = \chi(M),$$
the [[Euler characteristic|RNDDWDAFJ5]] of $M$.

## Remarks

For $n$ odd, $\chi(M) = 0$ by Poincaré duality, and the index is $0$.
For $n$ even, the [[topological index|QDCD33CS21]] of $D$ is the Euler number $\langle e(TM), [M] \rangle$.
So the Atiyah–Singer index theorem gives $\chi(M) = \langle e(TM), [M] \rangle$, and with the Chern–Weil form of the Euler class this is the [[Chern–Gauss–Bonnet theorem|GG6NYPKMHX]].
For the round sphere $S^2$: $b_0 = b_2 = 1$ and $b_1 = 0$, so the index is $2$.

## Explanations

### L1 human

The de Rham operator $d + d^*$ sends forms of even degree to forms of odd degree.
Its solutions are the harmonic forms, which by Hodge theory count the holes of the manifold in each dimension.
So its index is the alternating count of holes, the Euler characteristic.
This is the simplest case where the index theorem recovers a classical theorem, here Gauss–Bonnet.
</document>

<document source="https://tamnd.github.io/math2/node/2E1R3BDDMK.md" tag="2E1R3BDDMK" kind="definition">
# Relational structure

## Definition

Let $\sigma$ be a [[signature|WDVVEYD6AT]] with no function symbols and no constant symbols.
So each symbol $R \in \sigma$ is a relation symbol with an arity $n_R \ge 1$.
A *relational structure* of signature $\sigma$ is a [[structure|983GZBQYT4]] for $\sigma$.
Explicitly, it is a pair $(A, (R^A)_{R \in \sigma})$ where $A$ is a nonempty set and $R^A \subseteq A^{n_R}$ for each $R \in \sigma$.
Here $A^{n_R}$ is the [[Cartesian product|09CEDJYBKP]] of $n_R$ copies of $A$.

## Remarks

Example: a simple graph $(V, E)$, with $E \subseteq V^2$ the set of pairs $(u, v)$ such that $u$ and $v$ are adjacent, is a relational structure with one binary relation.

Non-example: a group $(G, \cdot)$ is not a relational structure, because its signature has a function symbol.
It can be turned into one by replacing $\cdot$ with its graph, the ternary relation $\set{(a, b, c) : a \cdot b = c}$.

Every structure can be encoded in this way as a relational structure.
Some authors allow an empty domain for relational structures.
</document>

<document source="https://tamnd.github.io/math2/node/2E24G3S4CK.md" tag="2E24G3S4CK" kind="definition">
# Homotopy

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], let $I = [0, 1]$, and let $f, g \colon X \to Y$ be [[continuous maps|0VGFHTM75Y]].
A *homotopy* from $f$ to $g$ is a continuous map $H \colon X \times I \to Y$, where $X \times I$ carries the [[product topology|0GBG8B49AA]], such that $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$ for all $x \in X$.

## Remarks

For each $t \in I$, the map $h_t = H(\cdot, t) \colon X \to Y$ is continuous, so a homotopy is a continuous family $(h_t)_{t \in I}$ of maps with $h_0 = f$ and $h_1 = g$.
Example: for $X = Y = \R$, $f = \id$ and $g = 0$, the map $H(x, t) = (1 - t)x$ is continuous, with $H(x, 0) = x$ and $H(x, 1) = 0$, so it is a homotopy from $f$ to $g$.
Non-example: for the same $f$ and $g$, the map $K(x, t) = x$ for $t < 1$ and $K(x, 1) = 0$ has the right ends but is not continuous at $(1, 1)$, so it is not a homotopy.
If $H$ is a homotopy from $f$ to $g$, then $(x, t) \mapsto H(x, 1 - t)$ is a homotopy from $g$ to $f$.
</document>

<document source="https://tamnd.github.io/math2/node/2E46CY6S3Z.md" tag="2E46CY6S3Z" kind="counterexample">
# Hodges' superefficient estimator

## Counterexample

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] $\mathcal{N}(\theta, 1)$, the [[normal distribution|07RXBVTVZB]] with unknown mean $\theta \in \R$, and let $\bar{X}_n$ be the [[sample mean|0QNRBHWV35]].
*Hodges' estimator* is
$$T_n = \begin{cases} \bar{X}_n & \text{if } \abs{\bar{X}_n} \ge n^{-1/4}, \\ 0 & \text{if } \abs{\bar{X}_n} < n^{-1/4}. \end{cases}$$
The [[Fisher information|G1Z3WSTE8Q]] of one observation is $1$, and $\sqrt{n}(\bar{X}_n - \theta) \sim \mathcal{N}(0, 1)$ exactly.

Check of the limit laws, with [[convergence in distribution|0JDAQHWA9N]]:

1. If $\theta \ne 0$, then $P_\theta(\abs{\bar{X}_n} < n^{-1/4}) \to 0$, because $\bar{X}_n \to \theta$ while $n^{-1/4} \to 0$. So $T_n = \bar{X}_n$ with probability tending to one, and $\sqrt{n}(T_n - \theta) \xrightarrow{d} \mathcal{N}(0, 1)$.
2. If $\theta = 0$, then $P_0(\abs{\bar{X}_n} < n^{-1/4}) = P(\abs{Z} < n^{1/4}) \to 1$ for $Z$ standard normal. So $T_n = 0$ with probability tending to one, and $\sqrt{n}\, T_n \xrightarrow{d} 0$.

So the asymptotic variance of $T_n$ is $1$ for $\theta \ne 0$ and $0$ for $\theta = 0$: at $\theta = 0$ it beats the bound $1/I(\theta) = 1$ that maximum likelihood attains.
This is *superefficiency*: the usual bound of [[efficiency|8VDNRZG3R1]] is beaten at one point.

## Remarks

The price appears near $0$.
Take $\theta_n = \frac{1}{2} n^{-1/4}$.
Then $\bar{X}_n - \theta_n$ is of order $n^{-1/2}$, so $P_{\theta_n}(T_n = 0) \to 1$, and the scaled [[mean squared error|0V4VSCBXP3]] satisfies $n\, E_{\theta_n}[(T_n - \theta_n)^2] \ge n \theta_n^2 P_{\theta_n}(T_n = 0) \approx \frac{1}{4}\sqrt{n} \to \infty$, while for $\bar{X}_n$ it equals $1$.

$T_n$ is not a regular estimator at $\theta = 0$: under $\theta_n = h/\sqrt{n}$, the law of $\sqrt{n}(T_n - \theta_n)$ tends to the point mass at $-h$, which depends on $h$.
This is why the [[Hájek–Le Cam convolution theorem|E5H2GFPK42]] assumes regularity.

## Explanations

### L1 human

Hodges' estimator rounds the sample mean to $0$ whenever it is small.
If the true mean really is $0$, this makes the estimate much more accurate than the sample mean.
But if the true mean is small and not $0$, the rounding causes large errors.
So beating the usual efficiency bound at one point costs much more at nearby points.
This shows that efficiency must be judged on small neighbourhoods of parameter values, not at one value at a time.
</document>

<document source="https://tamnd.github.io/math2/node/2E83D9ACDY.md" tag="2E83D9ACDY" kind="conjecture">
# Homotopy hypothesis

## Statement

Let $n$ be a natural number or $n = \infty$.
A *homotopy $n$-type* is a space $X$ whose [[homotopy groups|P976WYEFMB]] $\pi_k(X, x)$ vanish for all $k > n$ and all base points $x$.
For $n = \infty$ this is no condition.
Any good definition of weak $n$-[[groupoids|FH3C0T643F]] should satisfy the following.
Let $\Pi_n(X)$ be the fundamental $n$-groupoid of a space $X$.
Then $X \mapsto \Pi_n(X)$ induces an equivalence between the homotopy theory of homotopy $n$-types and that of weak $n$-groupoids.
For $n = \infty$ it says that weak $\infty$-groupoids and spaces up to weak homotopy equivalence are the same homotopy theory.

## Remarks

The hypothesis is a test that a definition of higher groupoid must pass, so its status depends on the chosen definition.
For $n = 0$ it says that sets correspond to spaces whose path components are weakly contractible, through $X \mapsto \pi_0(X)$.
For $n = 1$ it is classical: the [[fundamental groupoid|ZBHGXJF24E]] and the classifying space give an equivalence between groupoids up to equivalence and homotopy $1$-types.
If weak $\infty$-groupoids are modelled by [[Kan complexes|FDFYCM3YYC]], the hypothesis holds.
Quillen showed that [[geometric realization|TG6WHHR7CF]] and the [[singular simplicial set|RWBMXQW6S5]] functor form a [[Quillen equivalence|KZCXHTEYK7]].
It relates simplicial sets with the Kan–Quillen model structure and topological spaces with the Quillen model structure.
For other definitions of weak $\infty$-groupoids, in particular algebraic ones, the question is a subject of research.
Grothendieck stated the idea in his manuscript *Pursuing Stacks* (1983).

## Explanations

### L1 human

A space has points, paths between points, homotopies between paths, homotopies between homotopies, and so on.
Points and paths behave like objects and invertible arrows, and the higher homotopies like invertible arrows between arrows.
The homotopy hypothesis says that this structure loses nothing: the shape of a space, up to homotopy, is the same thing as an infinite groupoid.
For example, a circle is captured by one object with one invertible loop that generates a copy of $\Z$, and nothing above that.
</document>

<document source="https://tamnd.github.io/math2/node/2E8MQXDYNV.md" tag="2E8MQXDYNV" kind="statement">
# Martingale representation theorem

## Statement

Let $T > 0$ and let $(B_t)_{0 \le t \le T}$ be a standard [[Brownian motion|22735FGJV6]] on a probability space $(\Omega, \mathcal{F}, P)$.
Let $(\mathcal{F}_t)$ be the [[filtration|HNK1Y62G7V]] generated by $B$ and completed with the $P$-null sets.
Let $X$ be an $\mathcal{F}_T$-measurable random variable with $E[X^2] < \infty$.
Then there is a progressively measurable process $H$ with $E\left[\int_0^T H_t^2 \, dt\right] < \infty$ such that
$$X = E[X] + \int_0^T H_t \, dB_t,$$
where the integral is the [[Itô integral|JA95NH206E]].
The process $H$ is unique up to a set of measure zero for $dt \otimes dP$.

## Remarks

As a consequence, every square-integrable [[martingale|0D0GBNDKDH]] $M$ for $(\mathcal{F}_t)$ has the form $M_t = M_0 + \int_0^t H_s \, dB_s$, and so has a continuous version.
For the proof, the random variables $\mathcal{E}\left(\int_0^{\cdot} f \, dB\right)_T$ with deterministic $f$ are [[stochastic exponentials|3D88622PBQ]] with explicit representations.
Their linear span is dense in $L^2(\mathcal{F}_T)$, and the [[Itô isometry|5MDK7HSYRM]] passes the representation to limits.
Example: Itô's lemma gives $B_T^2 = T + \int_0^T 2 B_t \, dB_t$, so $H_t = 2 B_t$.
The filtration must be generated by $B$: if $\mathcal{F}_0$ contains a nonconstant variable $\xi$ independent of $B$ with $E[\xi] = 0$, then $\xi$ has no such representation, because $E\left[\int_0^T H \, dB \mid \mathcal{F}_0\right] = 0 \neq \xi$.

## Explanations

### L1 human

In a world where all randomness comes from one Brownian motion, every random payoff is a constant plus a stochastic integral against that motion.
In betting terms, any such payoff can be produced from its mean by a suitable trading strategy in the Brownian motion.

### L2 human

The theorem is the mathematical basis of hedging in complete markets, such as the Black–Scholes model.
Combined with the Girsanov theorem, it shows that each claim can be replicated by trading in the stock.
The Clark–Ocone formula of Malliavin calculus identifies the integrand $H$ for differentiable $X$.
The result also holds for a $d$-dimensional Brownian motion, with a vector-valued integrand.
</document>

<document source="https://tamnd.github.io/math2/node/2E9DMB7A6Q.md" tag="2E9DMB7A6Q" kind="concept">
# Quantum logic

## Summary

Quantum logic is the study of the propositions about a quantum system and of the logical operations on them.
In the approach of Birkhoff and von Neumann, a proposition is a closed subspace of the [[Hilbert space|HM2HHE3J51]] of the system.
These propositions form an [[orthomodular lattice|6FPKEFX0X4]].
When the Hilbert space has dimension at least $2$, this lattice is not distributive, so the distributive law of classical logic fails.
Abstract quantum logics are studied as orthomodular lattices or orthomodular partially ordered sets.

## Explanations

### L1 human

In classical physics, a statement such as "the position is in this interval" is true or false of each state.
Such statements combine like subsets, with "and", "or" and "not".
In quantum mechanics, a yes-no question corresponds to a closed subspace of a Hilbert space, the states for which the answer is surely yes.
"And" is intersection, "not" is the orthogonal complement, and "or" is the smallest closed subspace containing both.
With these operations "$p$ and ($q$ or $r$)" can differ from "($p$ and $q$) or ($p$ and $r$)".
This reflects the fact that some quantum questions cannot be answered at the same time.
</document>

<document source="https://tamnd.github.io/math2/node/2ECAM46NE2.md" tag="2ECAM46NE2" kind="concept">
# Permutation test

## Summary

A permutation test is a [[hypothesis test|087P03NEND]] that gets its null distribution by rearranging the data.
It recomputes the test statistic on all rearrangements that the null hypothesis allows.
In the two-sample case the null hypothesis says that the group labels do not matter, so every reassignment of the labels is equally likely.
The [[p-value|0GGKJGZF6D]] is the share of rearrangements whose statistic is at least as extreme as the observed one.
Under the null hypothesis the chance of a false rejection is at most the chosen level, with no assumption on the shape of the distribution.

## Explanations

### L1 human

Suppose a new method scores higher than an old one in a small trial.
If the method makes no difference, the labels "new" and "old" are arbitrary, and any other split of the same scores was just as likely.
So list all ways to split the scores into two groups of the same sizes, and compute the difference of means for each.
If the real difference is among the largest few percent of these, chance alone is a poor explanation.
</document>

<document source="https://tamnd.github.io/math2/node/2EE62A1QXA.md" tag="2EE62A1QXA" kind="definition">
# Spherical mean transform

## Definition

Let $n \ge 2$, let $S^{n-1} = \set{\theta \in \R^n : \abs{\theta} = 1}$ be the unit [[sphere|P9CM4GQ2T0]], let $\sigma$ be its surface measure, and let $\omega_{n-1} = \sigma(S^{n-1})$.
Let $f \colon \R^n \to \C$ be continuous.
The *spherical mean transform* of $f$ is the function on $\R^n \times [0, \infty)$ given by
$$(M f)(x, r) = \frac{1}{\omega_{n-1}} \int_{S^{n-1}} f(x + r\theta)\, d\sigma(\theta).$$

## Remarks

$M f(x, r)$ is the average of $f$ over the sphere of center $x$ and radius $r$, and $M f(x, 0) = f(x)$.
If $f$ is of class $C^2$, then $M f$ satisfies the Darboux equation
$$\Delta_x M f(x, r) = \left( \frac{\partial^2}{\partial r^2} + \frac{n - 1}{r} \frac{\partial}{\partial r} \right) M f(x, r), \qquad r > 0.$$

Example: $f(x) = \abs{x}^2$.
Since $\abs{x + r\theta}^2 = \abs{x}^2 + 2r\, x \cdot \theta + r^2$ and $\theta \mapsto x \cdot \theta$ has mean $0$ over the sphere, $M f(x, r) = \abs{x}^2 + r^2$.
Both sides of the Darboux equation equal $2n$.

Non-example: $f(y) = \abs{y}^{1-n}$ for $y \ne 0$ and $f(0) = 0$ is not continuous.
Its mean over the sphere of center $x \ne 0$ and radius $r = \abs{x}$, which passes through $0$, is $+\infty$.

The function $f$ may also be only locally integrable, and then $M f(x, r)$ is defined for almost every $(x, r)$.
</document>

<document source="https://tamnd.github.io/math2/node/2EEKCBNYS0.md" tag="2EEKCBNYS0" kind="definition">
# Integral equation of the second kind

## Definition

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ be a [[kernel|GSMBXT7FPK]], let $f \colon [a, b] \to \C$ be given, and let $\lambda \in \C$.
For each $x \in [a, b]$, let $I_x$ be either $[a, b]$ or $[a, x]$.
An *integral equation of the second kind* is a [[linear integral equation|AYZCKG34QK]] of the form
$$\varphi(x) = f(x) + \lambda \int_{I_x} K(x, s) \varphi(s) \, ds \quad \text{for all } x \in [a, b],$$
for the unknown function $\varphi \colon [a, b] \to \C$.

## Remarks

Example: $\varphi(x) = 1 + \lambda \int_0^1 \varphi(s) \, ds$ on $[0, 1]$, with $K = 1$.
Any solution is a constant $c$ with $c = 1 + \lambda c$, so for $\lambda \ne 1$ the unique solution is $\varphi = 1/(1 - \lambda)$.
For $\lambda = 1$ there is no solution.

Non-example: $\int_0^1 \varphi(s) \, ds = 1$ is not of the second kind, since $\varphi$ does not appear outside the integral; it is of the first kind.

In operator notation, $(I - \lambda K) \varphi = f$, where $I$ is the identity and $K$ is the integral operator.
Some authors write $\varphi(x) - \lambda \int K \varphi = f(x)$ or put $\lambda$ into the kernel.
</document>

<document source="https://tamnd.github.io/math2/node/2EM3EQEX2J.md" tag="2EM3EQEX2J" kind="definition">
# Catastrophe theory

## Definition

Let $V \colon \R^n \times \R^k \to \R$ be a [[smooth function|0V33BNKZ9S]], written $V(x, c)$, with state variables $x$ and control parameters $c$.
The *catastrophe manifold* is the set $M = \set{(x, c) : \nabla_x V(x, c) = 0}$ of [[critical points|0YTCMFB4HC]] of the functions $V(\cdot, c)$.
The *catastrophe map* is the projection $\chi \colon M \to \R^k$, $(x, c) \mapsto c$.
The *bifurcation set* is the set of $c$ for which $V(\cdot, c)$ has a degenerate critical point, one where the Hessian in $x$ is singular.
*Catastrophe theory* studies these sets and the local forms of such families up to smooth changes of coordinates.

## Remarks

Example: the cusp catastrophe $V(x, a, b) = x^4/4 + a x^2/2 + b x$ with $n = 1$ and $k = 2$.
The critical points solve $x^3 + a x + b = 0$, and they are degenerate when also $3x^2 + a = 0$.
Eliminating $x$ gives the bifurcation set $4a^3 + 27b^2 = 0$, a curve with a cusp at the origin.
Inside the cusp, where $4a^3 + 27b^2 < 0$, the function has two local minima, and outside it has one.
Non-example: $V(x, c) = x^2 + c x$ has the single critical point $x = -c/2$, which is nondegenerate for every $c$, so its bifurcation set is empty.
Away from the bifurcation set every $V(\cdot, c)$ is a [[Morse function|KPQ96640ZJ]] near its critical points, so the theory is about the degenerate points.
Thom's theorem lists the seven generic local forms for $k \le 4$: fold, cusp, swallowtail, butterfly, and the hyperbolic, elliptic and parabolic umbilics.
</document>

<document source="https://tamnd.github.io/math2/node/2ENTSSHEPZ.md" tag="2ENTSSHEPZ" kind="definition">
# Inaccessible cardinal

## Definition

Let $\kappa$ be a [[cardinal|0EBACQFAZW]].
The cardinal $\kappa$ is *(strongly) inaccessible* if:

1. $\kappa$ is [[uncountable|0W7FBYS1F1]].
2. $\kappa$ is [[regular|13ZV2F8JAR]]: $\kappa$ is not the union of fewer than $\kappa$ sets each of size less than $\kappa$.
3. $\kappa$ is a strong limit: $2^\lambda < \kappa$ for every cardinal $\lambda < \kappa$, where $2^\lambda$ is the cardinality of the [[power set|00E0YZCERH]] of a set of size $\lambda$.

The cardinal $\kappa$ is *weakly inaccessible* if it satisfies clauses 1 and 2 and is a limit cardinal: $\lambda^+ < \kappa$ for every cardinal $\lambda < \kappa$.

## Remarks

The cardinal $\aleph_0$ satisfies clauses 2 and 3, since a union of finitely many finite sets is finite and $2^n$ is finite; it fails only clause 1.
The cardinal $\aleph_1$ is regular but not inaccessible: it fails clause 3, since $2^{\aleph_0} \ge \aleph_1$.
Every inaccessible cardinal is weakly inaccessible, and under the generalized continuum hypothesis the two notions agree.
If $\kappa$ is inaccessible, then $V_\kappa$, the set of sets of rank less than $\kappa$, is a model of ZFC.
By Gödel's second incompleteness theorem, ZFC therefore cannot prove that an inaccessible cardinal exists, if ZFC is consistent.
In Mathlib the notion is `Cardinal.IsInaccessible`.
</document>

<document source="https://tamnd.github.io/math2/node/2EPRMNK1G2.md" tag="2EPRMNK1G2" kind="example">
# List of simple Lie groups

## Example

Every compact connected [[simple Lie group|AM7YJP8AR6]] is isomorphic to a quotient $\tilde{G}/C$.
Here $\tilde{G}$ is exactly one group of the following list, and $C$ is a subgroup of the centre of $\tilde{G}$.
Each $\tilde{G}$ is compact, connected and simply connected, and its centre is finite.

| Type | $\tilde{G}$ | Dimension | Centre |
|---|---|---|---|
| $A_n$, $n \ge 1$ | $SU(n+1)$ | $n(n+2)$ | $\Z/(n+1)$ |
| $B_n$, $n \ge 2$ | $\mathrm{Spin}(2n+1)$ | $n(2n+1)$ | $\Z/2$ |
| $C_n$, $n \ge 3$ | $\mathrm{Sp}(n)$ | $n(2n+1)$ | $\Z/2$ |
| $D_n$, $n \ge 4$ | $\mathrm{Spin}(2n)$ | $n(2n-1)$ | $\Z/4$ for $n$ odd, $\Z/2 \times \Z/2$ for $n$ even |
| $G_2$ | $G_2$ | $14$ | trivial |
| $F_4$ | $F_4$ | $52$ | trivial |
| $E_6$ | $E_6$ | $78$ | $\Z/3$ |
| $E_7$ | $E_7$ | $133$ | $\Z/2$ |
| $E_8$ | $E_8$ | $248$ | trivial |

Here $SU(n+1)$ is the [[special unitary group|CZCWFPNS4N]], $\mathrm{Spin}(m)$ is the [[spin group|JJSXXJQGEH]], and $\mathrm{Sp}(n)$ is the [[compact symplectic group|5NNPBFQA02]].
The groups [[G2|YX5FFSCR8P]] and [[E8|MKT1263M0A]] are the smallest and the largest exceptional groups.
The type is the type of the Dynkin diagram of the Lie algebra, and $n$ is the rank.
Check of one row: $SU(n+1)$ has Lie algebra of trace-zero skew-Hermitian $(n+1) \times (n+1)$ matrices, of dimension $(n+1)^2 - 1 = n(n+2)$.

## Remarks

The bounds on $n$ avoid repetitions and non-simple cases: $\mathrm{Sp}(1) \cong \mathrm{Spin}(3) \cong SU(2)$, $\mathrm{Sp}(2) \cong \mathrm{Spin}(5)$, $\mathrm{Spin}(6) \cong SU(4)$, and $\mathrm{Spin}(4) \cong SU(2) \times SU(2)$ is not simple.
The first four rows are the classical families, and the last five are the exceptional groups.
For example, $SO(3) = SU(2)/\set{\pm I}$ is the quotient for $A_1$ with $C$ the whole centre.
The same list of types classifies the complex simple Lie algebras.
The noncompact simple Lie groups are classified through the real forms of these complex Lie algebras, a classification due to Élie Cartan.

## Explanations

### L1 human

There are only a few kinds of compact simple Lie groups.
Four infinite families come from matrices that preserve lengths over the real numbers, the complex numbers or the quaternions.
Besides them there are exactly five exceptional groups, of dimensions $14$, $52$, $78$, $133$ and $248$.
Every compact simple Lie group is one of these, possibly divided by a finite group of central elements.
</document>

<document source="https://tamnd.github.io/math2/node/2ET068BZZH.md" tag="2ET068BZZH" kind="concept">
# Modified Bessel functions

## Summary

The modified Bessel functions $I_\nu$ and $K_\nu$ solve the modified Bessel equation $z^2 y'' + z y' - (z^2 + \nu^2) y = 0$.
They are the [[Bessel functions|CZCMRDWDYW]] evaluated on the imaginary axis, made real for real arguments.
On the positive real axis $I_\nu$ grows like $e^x$ and $K_\nu$ decays like $e^{-x}$.
They appear in heat flow, diffusion and electrostatics in cylindrical geometry.

## Explanations

### L1 human

Bessel's equation behaves like $y'' + y = 0$ for large $x$, so its solutions oscillate.
Changing one sign gives an equation that behaves like $y'' - y = 0$, whose solutions grow or decay like $e^{x}$ and $e^{-x}$.
The modified Bessel functions are the solutions of this changed equation.
$I_\nu$ is the growing one, finite at the origin, and $K_\nu$ is the decaying one, infinite at the origin.
For example, $I_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sinh x$ for $x > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/2F30ZHN1RT.md" tag="2F30ZHN1RT" kind="statement">
# Darboux's theorem on additive functions bounded on an interval

## Statement

Let $f \colon \R \to \R$ be an [[additive function|KF719SZD5T]], that is, $f(x + y) = f(x) + f(y)$ for all $x, y \in \R$.
Assume that there are real numbers $a < b$ and $M$ such that $f(x) \le M$ for all $x$ in the [[interval|0Q1D85B8G6]] $(a, b)$.
Then $f(x) = f(1)\, x$ for all $x \in \R$.

## Remarks

Set $c = f(1)$ and $g(x) = f(x) - cx$.
Then $g$ is additive and $g(q) = 0$ for every rational $q$, so $g(x + q) = g(x)$ for all real $x$ and rational $q$.
Every real $x$ has a rational $q$ with $x + q \in (a, b)$, and $g$ is bounded above on $(a, b)$, so $g$ is bounded above on all of $\R$.
If $g(x_0) \ne 0$ for some $x_0$, then $g(n x_0) = n g(x_0)$ or $g(-n x_0) = -n g(x_0)$ grows without bound as $n \to \infty$, a contradiction.
So $g = 0$ and $f(x) = cx$.
The same holds if $f$ is bounded below on an interval, since then $-f$ is bounded above.
Without any bound the conclusion fails: [[additive functions built from a Hamel basis|FFTNS66XG6]] are not linear.

## Explanations

### L1 human

An additive function is either a straight line through the origin or a wild function.
This theorem says that a wild additive function cannot stay below a fixed level even on a tiny interval.
So a single bound on a small piece of the line already forces the function to be linear.

### L2 human

This is the basic regularity criterion for [[Cauchy's functional equation|ZBMEVWN100]].
Continuity at a point, monotonicity on an interval and measurability all imply boundedness on some interval, so they all reduce to this result.
For measurable functions this reduction goes through the Steinhaus theorem; see [[measurable solutions of Cauchy's equation|TWANGP59RB]].
The same idea shows that the graph of a non-linear additive function is [[dense in the plane|MNAWWP9J1X]].
</document>

<document source="https://tamnd.github.io/math2/node/2F5C10ED1A.md" tag="2F5C10ED1A" kind="concept">
# Quantile regression

## Summary

Quantile regression models a conditional [[quantile|0X60VE98HN]] of a response, such as the conditional median, as a function of covariates.
Where ordinary [[linear regression|41FVC4P5JB]] fits the conditional mean by least squares, quantile regression minimises an asymmetric absolute loss, the check function.
Fitting several quantile levels shows how the whole conditional distribution changes with the covariates, not only its centre.
Koenker and Bassett introduced it in 1978, and the median case is robust to outliers in the response.

## Explanations

### L1 human

The usual regression line tells you the average weight of children at each age.
Quantile regression can tell you instead the weight below which $90\%$ of children of each age fall.
It does this by penalising points above the line nine times as much as points below it, so the line settles where $90\%$ of the points lie below.
With equal penalties on both sides, it gives the median line, which a few extreme points cannot drag far.
</document>

<document source="https://tamnd.github.io/math2/node/2F5CYQJVAF.md" tag="2F5CYQJVAF" kind="concept">
# Theta function

## Summary

The Jacobi theta function is $\vartheta(z, \tau) = \sum_{n \in \Z} e^{\pi i n^2 \tau + 2 \pi i n z}$ for $z \in \C$ and $\tau$ in the [[upper half-plane|X6XGGH5FPW]].
It is periodic in $z$ with period $1$ and quasi-periodic with period $\tau$.
Theta functions are the building blocks of elliptic functions and of functions on abelian varieties.
In the variable $\tau$, the theta series of quadratic forms are [[modular forms|GWKN5VG78A]] that count representations of numbers.

## Explanations

### L1 human

A theta function is a sum of exponentials whose exponents grow like squares, so it converges very fast.
Shifting $z$ by $1$ changes nothing, and shifting $z$ by $\tau$ multiplies the function by a simple exponential factor.
Setting $z = 0$ gives $\theta(\tau) = \sum_{n \in \Z} e^{\pi i n^2 \tau}$.
Raising it to the $k$-th power counts the ways to write numbers as sums of $k$ squares.
This is how [[Jacobi's four-square theorem|CG17WSS64W]] can be proved.
</document>

<document source="https://tamnd.github.io/math2/node/2F6AJ1DM11.md" tag="2F6AJ1DM11" kind="definition">
# Green's function for the Laplacian

## Definition

Let $\Omega$ be a bounded [[domain|0EWMDD6C14]] in $\C$ with [[boundary|05D5EGMFYV]] $\partial\Omega$, and let $w \in \Omega$.
A *Green's function of $\Omega$ with pole at $w$* is a function $g(\cdot, w) \colon \Omega \setminus \set{w} \to \R$ such that:

1. $g(\cdot, w)$ is [[harmonic|0DEB0JMN0W]] on $\Omega \setminus \set{w}$.
2. The function $z \mapsto g(z, w) + \log\abs{z - w}$ extends to a harmonic function on a neighbourhood of $w$.
3. $g(z, w) \to 0$ as $z \to \zeta$, for every $\zeta \in \partial\Omega$.

## Remarks

If such a function exists, it is unique, by the maximum principle applied to the difference of two of them.
It exists whenever the [[Dirichlet problem|8D56GQ4AG8]] on $\Omega$ is solvable for continuous boundary data: take $g(z, w) = h(z) - \log\abs{z - w}$, where $h$ is harmonic with boundary values $\log\abs{\zeta - w}$.
Example: for the unit disc, $g(z, w) = \log\abs{\dfrac{1 - \bar w z}{z - w}}$.
Here $\log\abs{1 - \bar w z}$ is harmonic on the disc because $1 - \bar w z \ne 0$ there, and on $\abs{z} = 1$ one has $\abs{1 - \bar w z} = \abs{\bar z - \bar w} = \abs{z - w}$, so $g = 0$ on the boundary.
Non-example: for $w \ne 0$, the function $z \mapsto -\log\abs{z - w}$ on the unit disc satisfies clauses 1 and 2, but at the boundary point $\zeta = -w/\abs{w}$ its limit is $-\log(1 + \abs{w}) \ne 0$, so clause 3 fails.
In the sense of distributions, $\Delta_z g(z, w) = -2\pi\delta_w$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
The Green's function is positive on $\Omega \setminus \set{w}$ and symmetric: $g(z, w) = g(w, z)$.
In $\R^n$ with $n \ge 3$, the analogous definition uses the singularity $\abs{x - y}^{2 - n}$ in place of $-\log\abs{z - w}$, up to normalizing constants.
</document>

<document source="https://tamnd.github.io/math2/node/2FE4P23NVC.md" tag="2FE4P23NVC" kind="statement">
# Von Neumann's theorem on self-adjoint extensions

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]], and let $A$ be a [[closed|VM7B38PA87]] [[symmetric operator|H971WGMV6C]] on $H$ with dense domain $D(A)$ and [[adjoint|86B7HF79PQ]] $A^*$.
Let $K_+ = \ker(A^* - iI)$ and $K_- = \ker(A^* + iI)$ be its [[deficiency subspaces|G4GMSNAX8X]], with deficiency indices $n_\pm = \dim K_\pm$.
Then:

1. $D(A^*) = D(A) \dotplus K_+ \dotplus K_-$, a direct sum of subspaces.
2. $A$ has a self-adjoint extension if and only if $n_+ = n_-$.
3. If $n_+ = n_-$, the self-adjoint extensions of $A$ are in bijection with the [[isometric isomorphisms|0ZXSF2YQV2]] $U \colon K_+ \to K_-$. The extension $A_U$ attached to $U$ has domain $D(A_U) = \set{f + k + U k : f \in D(A), \ k \in K_+}$, and $A_U(f + k + U k) = A f + i k - i U k$.

## Remarks

Every self-adjoint extension $B$ of $A$ satisfies $A \subseteq B = B^* \subseteq A^*$, so it is a restriction of $A^*$ to a subspace between $D(A)$ and $D(A^*)$.
On $D(A^*)$, the form $\ip{A^* x}{y} - \ip{x}{A^* y}$ measures the failure of symmetry, and it vanishes on $D(A)$.
Part 1 and a computation of this form on $K_+ \dotplus K_-$ show that the maximal subspaces on which it vanishes are exactly the graphs of the isometric isomorphisms $U \colon K_+ \to K_-$.
In particular, $A$ is self-adjoint if and only if $n_+ = n_- = 0$, and a symmetric operator is [[essentially self-adjoint|DT49GKR9AS]] if and only if its closure has $n_+ = n_- = 0$.
Example: for $A = -i \, d/dx$, closed from the smooth functions with compact support in $(0,1)$ in $L^2([0,1])$, the indices are $(1,1)$.
The isometric isomorphisms from $K_+$ to $K_-$ form a circle, and the self-adjoint extensions are the operators $-i \, d/dx$ with boundary conditions $u(1) = e^{i \theta} u(0)$, $\theta \in [0, 2\pi)$.
On $L^2([0, \infty))$, the same operator has indices $(1, 0)$, so it has no self-adjoint extension.
A symmetric operator that commutes with a conjugation, such as a real differential operator, has $n_+ = n_-$ and so always has self-adjoint extensions.

## Explanations

### L1 human

A symmetric operator can fail to be self-adjoint because its domain is too small.
Von Neumann's theorem says exactly when the domain can be enlarged to give a self-adjoint operator, and lists all the ways to do it.
The answer depends on two numbers, the deficiency indices.
An extension exists when they are equal, and the extensions are then labeled by unitary maps between two spaces.

### L2 human

In quantum mechanics, an observable must be self-adjoint, while a formula such as $-d^2/dx^2$ only gives a symmetric operator on nice functions.
The theorem turns the choice of a self-adjoint realization into linear algebra on the deficiency subspaces, and the choices correspond to boundary conditions.
For Sturm–Liouville operators, the Weyl limit point and limit circle alternative computes the deficiency indices from the behaviour of solutions at the endpoints.
Other constructions, such as the Friedrichs extension, pick out a distinguished self-adjoint extension of a semibounded symmetric operator.
</document>

<document source="https://tamnd.github.io/math2/node/2FQAKBVBX5.md" tag="2FQAKBVBX5" kind="concept">
# Conformal modulus

## Summary

The conformal modulus of a family of curves in the plane is the reciprocal of its [[extremal length|HX72KT9D0G]].
It is a conformal invariant.
It gives the modulus of a quadrilateral and of a ring domain, which classify these domains up to conformal equivalence.
Moduli are the basic quantities in the geometric definition of [[quasiconformal maps|SB8KG46NHP]], which distort them by a bounded factor.

## Explanations

### L1 human

Two annuli $\set{r < \abs{z} < R}$ and $\set{r' < \abs{z} < R'}$ are conformally equivalent exactly when $R/r = R'/r'$.
So one number, essentially $\log(R/r)$, tells which annulus one has, and this is the modulus of the ring.
The conformal modulus of a curve family generalises this: many curves that are short give a large modulus, few long curves give a small one.
For the curves crossing a rectangle of width $a$ and height $b$ from left to right, the modulus is $b/a$.
</document>

<document source="https://tamnd.github.io/math2/node/2FQVQCTV88.md" tag="2FQVQCTV88" kind="concept">
# Integral quadratic form

## Summary

An integral quadratic form is a [[quadratic form|0RBQS1JJCR]] $\sum_{i \le j} a_{ij} x_i x_j$ whose coefficients $a_{ij}$ are integers.
The main questions are arithmetic: which integers the form represents, in how many ways, and when two forms are equivalent under integral changes of variables.
Classical examples are sums of squares, such as $x_1^2 + x_2^2 + x_3^2 + x_4^2$.
Integral quadratic forms correspond to [[lattices|WN9V7WSJKF]] with an integer-valued quadratic function on them.

## Explanations

### L1 human

Take a quadratic polynomial such as $x^2 + xy + y^2$ or $x^2 + y^2 + z^2$ with whole number coefficients.
Plugging in whole numbers gives whole numbers, and one asks which numbers come out.
Lagrange proved that every nonnegative integer is a sum of four squares, so the form $x_1^2 + x_2^2 + x_3^2 + x_4^2$ takes every such value.
The form $x^2 + y^2 + z^2$ misses $7$, since $7$ is not a sum of three squares.
</document>

<document source="https://tamnd.github.io/math2/node/2FR1BWY3H6.md" tag="2FR1BWY3H6" kind="definition">
# dg category

## Definition

Let $k$ be a commutative ring.
A *dg category* over $k$ is a [[category|A2DE66937D]] $\mathcal{C}$ [[enriched|BPH9SJ30Y0]] over the monoidal category of cochain complexes of $k$-modules.
In detail, for objects $x, y$ of $\mathcal{C}$ there is a [[chain complex|WQ82C0M3QB]] $\Hom(x, y) = \bigoplus_{n \in \Z} \Hom^n(x, y)$ of $k$-modules with differential $d$ of degree $+1$, such that:

1. For objects $x, y, z$ there are $k$-bilinear composition maps $\Hom^p(y, z) \times \Hom^q(x, y) \to \Hom^{p+q}(x, z)$, written $(g, f) \mapsto g \circ f$.
2. Composition is associative.
3. Each object $x$ has an identity $1_x \in \Hom^0(x, x)$ with $d(1_x) = 0$, which is a unit for composition.
4. The Leibniz rule holds: $d(g \circ f) = d(g) \circ f + (-1)^{p} g \circ d(f)$ for $g \in \Hom^p(y, z)$ and $f \in \Hom^q(x, y)$.

Clauses 1 and 4 say that composition is a map of complexes $\Hom(y, z) \otimes_k \Hom(x, y) \to \Hom(x, z)$.

## Remarks

Example: the dg category $C_{dg}(k)$ has as objects the cochain complexes $X$ of $k$-modules, with $\Hom^n(X, Y) = \prod_{i} \Hom_k(X^i, Y^{i+n})$.
For $f$ of degree $n$ the differential is $d(f) = d_Y \circ f - (-1)^n f \circ d_X$.
One checks $d^2(f) = d_Y^2 \circ f - f \circ d_X^2 = 0$, because the two middle terms $\pm d_Y \circ f \circ d_X$ cancel, and the Leibniz rule follows from the same sign computation.
The closed morphisms of degree $0$ are the chain maps, and the exact ones are the null-homotopic chain maps.

Non-example: over $k = \Q$, use the same graded maps but drop the sign, so $d(f) = d_Y \circ f + f \circ d_X$.
Then $d^2(f) = 2\, d_Y \circ f \circ d_X$.
Let $C$ be the complex with $C^0 = C^1 = \Q$ and differential the identity, and let $f \in \Hom^{-1}(C, C)$ be the identity map $C^1 \to C^0$.
Then $d^2(f)$ is $2 \ne 0$ on $C^0$, so $\Hom(C, C)$ is not a complex.

A dg category with one object is the same as a differential graded algebra.
</document>

<document source="https://tamnd.github.io/math2/node/2G173YJC54.md" tag="2G173YJC54" kind="definition">
# Absolutely continuous measure

## Definition

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]] and let $\mu$ and $\nu$ be [[measures|PR11BZQQPG]] on it.
The measure $\nu$ is *absolutely continuous* with respect to $\mu$, written $\nu \ll \mu$, if for every $A \in \Sigma$,
$$\mu(A) = 0 \implies \nu(A) = 0.$$
For a [[signed|VG2P7R91Z7]] or complex measure $\nu$, one says $\nu \ll \mu$ if its [[total variation|EPBND51CR5]] satisfies $\abs{\nu} \ll \mu$.

## Remarks

Example: let $f \colon X \to [0, \infty]$ be measurable and $\nu(A) = \int_A f \, d\mu$.
If $\mu(A) = 0$, then $f \mathbf{1}_A = 0$ $\mu$-almost everywhere, so $\nu(A) = 0$; hence $\nu \ll \mu$.

Non-example: on $\R$ with the Borel sets, the Dirac measure $\delta_0$ is not absolutely continuous with respect to Lebesgue measure $\lambda$.
The set $A = \set{0}$ has $\lambda(A) = 0$ but $\delta_0(A) = 1$.

If $\nu$ is finite, then $\nu \ll \mu$ holds if and only if for every $\varepsilon > 0$ there is $\delta > 0$ with $\nu(A) < \varepsilon$ whenever $\mu(A) < \delta$.
If $\mu$ and $\nu$ are σ-finite, the [[Radon–Nikodym theorem|1ZSZS5G1Q4]] shows that $\nu \ll \mu$ if and only if $\nu$ has a density $f$ with respect to $\mu$.
</document>

<document source="https://tamnd.github.io/math2/node/2GD9V7SA0E.md" tag="2GD9V7SA0E" kind="concept">
# t-structure

## Summary

A t-structure on a [[triangulated category|T2V7GJJ2EN]] is a pair of full subcategories, of "non-positive" and "non-negative" objects.
They behave like the complexes with cohomology only in degrees $\le 0$ and $\ge 0$ in a [[derived category|5ST7NVJFSM]].
It gives truncation functors and cohomology functors with values in its heart, which is always an abelian category.
t-structures were introduced by Beilinson, Bernstein and Deligne to construct perverse sheaves as the heart of a non-standard t-structure.

## Explanations

### L1 human

In the derived category of an abelian category, a complex can be cut into its low degree part and its high degree part.
The original abelian category is recovered as the complexes with cohomology only in degree $0$.
A t-structure is an abstract way of making such a cut in any triangulated category.
Different t-structures on the same triangulated category can have quite different hearts.
This is how one category can hide several abelian categories, such as sheaves and perverse sheaves.
</document>

<document source="https://tamnd.github.io/math2/node/2GDFFH7Y7T.md" tag="2GDFFH7Y7T" kind="concept">
# Lagrange resolvent

## Summary

A Lagrange resolvent is a sum $\alpha + \zeta \sigma(\alpha) + \cdots + \zeta^{n-1} \sigma^{n-1}(\alpha)$ built from an element $\alpha$, an automorphism $\sigma$ of order $n$ and an $n$-th [[root of unity|0YV2CKP2AV]] $\zeta$.
The automorphism $\sigma$ multiplies it by $\zeta^{-1}$, so its $n$-th power is fixed by $\sigma$.
This turns a [[cyclic extension|VFE5B71E7Y]] into an extension obtained by adjoining an $n$-th root, which is the key step in solving equations by radicals.
Lagrange used such sums of the roots to explain the formulas for cubic and quartic equations.

## Explanations

### L1 human

Suppose the roots of an equation are permuted in a cycle by a symmetry.
Mix the roots with weights $1, \zeta, \zeta^2, \ldots$ that are powers of a root of unity.
Turning the cycle once only multiplies this mix by a root of unity, so a suitable power of the mix does not change at all.
That power is then a known quantity, and the mix is one of its $n$-th roots.
For a quadratic with roots $r_1, r_2$ the resolvent is $r_1 - r_2$, and its square $(r_1 - r_2)^2$ is the discriminant, which is why the quadratic formula has a square root.
</document>

<document source="https://tamnd.github.io/math2/node/2GF2675154.md" tag="2GF2675154" kind="definition">
# Reeb vector field

## Definition

Let $M$ be a smooth manifold of dimension $2n + 1$ and let $\alpha$ be a contact form on $M$.
So $\alpha \wedge (d\alpha)^n$ vanishes nowhere and $(M, \ker \alpha)$ is a [[contact manifold|HBQ8BZ7XB5]].
The *Reeb vector field* of $\alpha$ is the unique [[vector field|0299VCW1TH]] $R_\alpha$ on $M$ such that
$$\alpha(R_\alpha) = 1 \qquad \text{and} \qquad \iota_{R_\alpha} d\alpha = 0.$$

## Remarks

Existence and uniqueness: $d\alpha$ has rank $2n$ at each point, so its kernel is a line, and $\alpha$ is nonzero on this line because $\alpha \wedge (d\alpha)^n \ne 0$.
The flow of $R_\alpha$ preserves $\alpha$, by Cartan's formula $\mathcal{L}_{R_\alpha} \alpha = d(\alpha(R_\alpha)) + \iota_{R_\alpha} d\alpha = 0$.
The Reeb field depends on $\alpha$, not only on the contact structure: replacing $\alpha$ by $h\alpha$ changes it in general.

Example: on $\R^3$ with $\alpha = dz - y\,dx$, one has $d\alpha = dx \wedge dy$, and $R_\alpha = \partial_z$, since $\alpha(\partial_z) = 1$ and $\iota_{\partial_z}(dx \wedge dy) = 0$.
On $S^3 \subset \C^2$ with $\alpha = \sum_j (x_j\, dy_j - y_j\, dx_j)$, the Reeb field is $R_\alpha = \sum_j (x_j \partial_{y_j} - y_j \partial_{x_j})$, whose flow is $z \mapsto e^{it} z$.

Non-example: on $\R^3$ with $\alpha = dz - y\,dx$, the field $\partial_y$ satisfies $\iota_{\partial_y} d\alpha = -dx \ne 0$ and $\alpha(\partial_y) = 0$, so it is not the Reeb field.
</document>

<document source="https://tamnd.github.io/math2/node/2GFFGE6WK1.md" tag="2GFFGE6WK1" kind="definition">
# Mean curvature flow

## Definition

Let $M$ be a smooth $n$-dimensional [[manifold|F45VQ47GPV]] and let $T > 0$.
A *mean curvature flow* is a smooth map $F \colon M \times [0, T) \to \R^{n+1}$ such that:

1. for each $t \in [0, T)$, the map $F_t = F(\cdot, t)$ is an immersion of $M$ as a hypersurface;
2. $\partial_t F(p, t) = \vec{H}(p, t)$ for all $p \in M$ and $t \in [0, T)$.

Here $\vec{H}(p, t) = -H \nu$ is the mean curvature vector of $F_t$ at $p$.
The vector $\nu$ is a unit [[normal vector|0WQJJCCTC8]], and $H$ is the [[mean curvature|PA78NFEPCF]], the sum of the principal curvatures with respect to $\nu$.

## Remarks

Example: let $F(p, t) = R(t)\, p$ for $p$ in the unit sphere $S^n \subset \R^{n+1}$, with $R(t) = \sqrt{R_0^2 - 2nt}$.
The sphere of radius $R$ has mean curvature vector of length $n/R$ pointing to the centre, so the flow condition is $R'(t) = -n/R(t)$.
This holds since $\frac{d}{dt} R^2 = -2n$, and the flow exists until $T = R_0^2/(2n)$.
Non-example: $R(t) = R_0 + t$ moves the sphere outward, against the mean curvature vector, so it fails clause 2.
A plane or any minimal hypersurface, with $\vec H = 0$, is a stationary flow.
Sign conventions for $H$ and $\nu$ vary, but $\vec H$ always points toward the centre of a round sphere.
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# Phragmén–Lindelöf principle for subharmonic functions

## Statement

Let $S = \set{z \in \C : \abs{\operatorname{Im} z} < \pi/2}$ be the horizontal strip of width $\pi$, and let $u$ be [[subharmonic|Z1MHMGFNFW]] on $S$.
Assume that $\limsup_{z \to \zeta,\, z \in S} u(z) \le 0$ for every point $\zeta$ of the boundary of $S$.
Assume also that there are constants $A > 0$ and $\alpha < 1$ with $u(z) \le A \exp(\alpha \abs{\operatorname{Re} z})$ for all $z \in S$.
Then $u \le 0$ on $S$.

## Remarks

Proof idea: pick $\beta$ with $\max(\alpha, 0) < \beta < 1$ and put $w(z) = \operatorname{Re} \cosh(\beta z) = \cosh(\beta x) \cos(\beta y)$ for $z = x + iy$.
The function $w$ is [[harmonic|0DEB0JMN0W]], and on $\overline{S}$ it satisfies $w \ge \cos(\beta\pi/2) \cosh(\beta x) > 0$.
For $\varepsilon > 0$, the subharmonic function $u - \varepsilon w$ tends to $-\infty$ as $\abs{\operatorname{Re} z} \to \infty$, because $\beta > \alpha$.
The [[maximum principle for subharmonic functions|9XJ5FBQD7M]] on long rectangles gives $u \le \varepsilon w$, and letting $\varepsilon \to 0$ gives $u \le 0$.

The bound on $\alpha$ is sharp.
The function $u(z) = \operatorname{Re} e^{z} = e^{x}\cos y$ is harmonic on $S$, vanishes on the boundary lines $y = \pm\pi/2$, and satisfies $u \le e^{\abs{\operatorname{Re} z}}$, which is the case $\alpha = 1$.
But $u > 0$ on $S$.
For holomorphic $f$, applying the theorem to $\log\abs{f}$ gives the [[Phragmén–Lindelöf principle|FYR6V2NH6C]] for strips, and conformal maps carry it to sectors.

## Explanations

### L1 human

On a bounded region, a subharmonic function that is at most $0$ at the edge is at most $0$ inside.
On an unbounded strip this can fail, because the function may grow without limit along the strip.
The principle says it still holds as long as the function does not grow too fast: slower than $\exp(\abs{\operatorname{Re} z})$, by a definite margin.

### L2 human

This is the maximum principle for unbounded domains, with a growth condition replacing boundedness of the domain.
The threshold growth is the growth of the positive harmonic function $\operatorname{Re} e^{z}$, which vanishes on the boundary.
The proof compares $u$ with the harmonic function $\operatorname{Re}\cosh(\beta z)$, which grows more slowly than $\operatorname{Re} e^{z}$ but faster than $u$.
The subharmonic form is more flexible than the holomorphic one, since it applies to $\log\abs{f}$, $\abs{f}^p$ and to maxima of such functions.
</document>

<document source="https://tamnd.github.io/math2/node/2GRPCP5XK9.md" tag="2GRPCP5XK9" kind="concept">
# Symmetric decreasing rearrangement

## Summary

The symmetric decreasing rearrangement of a function $f$ on $\R^n$ is a radial function $f^*$, decreasing in $\abs{x}$.
Its superlevel sets are balls centred at the origin with the same volumes as the superlevel sets of $\abs{f}$.
It keeps every $L^p$ norm, does not increase the Dirichlet energy, and does not decrease certain double integrals.
So it is the main tool to show that balls and radial functions are optimal in variational problems.
Examples are the [[Faber–Krahn inequality|E7TY2TPXX7]] and the isoperimetric inequality.
It is the $n$-dimensional, symmetric version of the [[decreasing rearrangement|BWS042NC2P]] on $[0, \infty)$.

## Explanations

### L1 human

Think of the graph of $\abs{f}$ as a mountain landscape made of horizontal slices.
Replace each slice by a disc of the same area, centred over the origin, and stack the discs.
The result is a single round mountain with its peak at the origin and the same amount of material at each height.
For example, the indicator function of the interval $[1, 3]$ becomes the indicator function of $(-1, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/2GSRBDT96M.md" tag="2GSRBDT96M" kind="statement">
# Whitney embedding theorem

## Statement

Let $n \ge 1$ and let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, Hausdorff and second countable.
A *smooth embedding* is an [[immersion|6RW68JGBHC]] that is a homeomorphism onto its image.
Then there is a smooth embedding of $M$ into the [[Euclidean space|0XPR1E85W0]] $\R^{2n}$.
In particular $M$ is diffeomorphic to an [[embedded submanifold|VKFW27WXEZ]] of $\R^{2n}$.

## Remarks

There is also a weak version: every such $M$ embeds in $\R^{2n+1}$, and any smooth map $M \to \R^{2n+1}$ can be approximated by embeddings.
Its proof starts from an embedding into some large $\R^N$, built from charts and bump functions.
Then $N$ is lowered by generic linear projections.
Sard's theorem shows that almost every projection stays an injective immersion while $N > 2n + 1$.
To reach $2n$, one first takes an immersion into $\R^{2n}$ with only transverse double points.
Then one removes the double points in pairs; for $n \ge 3$ this step uses the Whitney trick, and $n = 1, 2$ are treated directly.
The bound $2n$ is sharp in general: for $n$ a power of $2$, the real projective space $\R P^n$ does not embed in $\R^{2n - 1}$.
Example: the circle ($n = 1$) embeds in $\R^2$, and the Klein bottle ($n = 2$) embeds in $\R^4$ but not in $\R^3$.

## Explanations

### L1 human

Every smooth manifold, however it was defined, can be realised as a smooth shape inside an ordinary Euclidean space.
Twice its dimension is always enough room.
So abstract manifolds are no more general than the curved surfaces and solids that sit inside some $\R^N$.

### L2 human

The theorem reconciles the intrinsic definition of a manifold by charts with the older picture of a manifold as a subset of $\R^N$.
It allows one to borrow structures from $\R^N$, such as a Riemannian metric induced from the dot product or a normal bundle and a tubular neighbourhood.
The weak version is a typical general position argument, a precursor of transversality theory.
The strong version introduced the Whitney trick, which later became the key step in the h-cobordism theorem.
</document>

<document source="https://tamnd.github.io/math2/node/2GY3131ZPS.md" tag="2GY3131ZPS" kind="concept">
# Projective determinacy

## Summary

Projective determinacy, written PD, says that every game of natural numbers whose payoff set is a projective set is [[determined|QXECDPAYTN]].
Unlike the [[axiom of determinacy|PEXTM3MJ5Z]], it is consistent with the axiom of choice, and it is often added to ZFC.
Donald A. Martin and John R. Steel proved that PD follows from the existence of infinitely many [[Woodin cardinals|CMC2C1D53W]].
Under PD, all projective sets of reals are Lebesgue measurable, have the [[Baire property|FZ7237VY65]] and have the [[perfect set property|H47ZGRR13V]].

## Explanations

### L1 human

Projective sets are the sets of reals one gets from Borel sets by taking continuous images and complements, again and again, finitely many times.
They include almost every set of reals that arises in ordinary analysis.
PD says that infinite games with such a payoff set always have a winning strategy for one player.
Many set theorists regard PD as the natural extension of ZFC for questions about projective sets, because it settles most of them.
</document>

<document source="https://tamnd.github.io/math2/node/2H0J4P4NQF.md" tag="2H0J4P4NQF" kind="definition">
# Local regression

## Definition

Let $(x_1, y_1), \dots, (x_n, y_n)$ be data with $x_i, y_i \in \R$.
Let $p \ge 0$ be an integer, the degree, and let $h > 0$ be a number, the bandwidth.
Let $K \colon \R \to [0, \infty)$ be a weight function, for example the tricube function $K(u) = (1 - \abs{u}^3)^3$ for $\abs{u} < 1$ and $K(u) = 0$ otherwise.
For a point $x \in \R$, let $\hat\beta(x) = (\hat\beta_0(x), \dots, \hat\beta_p(x))$ be a minimiser of
$$\sum_{i=1}^{n} K\Big(\frac{x_i - x}{h}\Big) \Big( y_i - \sum_{j=0}^{p} \beta_j (x_i - x)^j \Big)^2$$
over $\beta \in \R^{p+1}$.
The *local polynomial regression estimate* of degree $p$ at $x$ is $\hat m(x) = \hat\beta_0(x)$.

## Remarks

The minimisation is a [[weighted least squares|H5CGZHBR58]] problem with weights $w_i = K((x_i - x)/h)$.
It has a unique solution when at least $p + 1$ distinct points $x_i$ have $w_i > 0$.
In LOESS the bandwidth depends on $x$: $h(x)$ is the distance from $x$ to its $\lceil \alpha n \rceil$-th nearest point $x_i$, for a fixed fraction $\alpha \in (0, 1]$ called the span.
LOWESS adds rounds of reweighting that lower the weights of points with large residuals.

Example: take $p = 1$ and data that lie on a line, $y_i = a + b x_i$.
Then $\beta = (a + bx, b)$ makes every residual $y_i - \beta_0 - \beta_1 (x_i - x)$ zero, so it is a minimiser and $\hat m(x) = a + bx$; local linear regression reproduces lines exactly.

Non-example: ordinary least squares with a global polynomial uses the same weight for every point at every $x$.
Its weights do not depend on the distance $\abs{x_i - x}$, so it is not a local fit.

For $p = 0$ the estimate is the weighted average $\sum_i w_i y_i / \sum_i w_i$, the Nadaraya–Watson estimate of [[kernel regression|710VV73909]].
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# Garden of Eden theorem

## Statement

Let $A$ be a finite set, let $d \ge 1$, and let $F \colon A^{\Z^d} \to A^{\Z^d}$ be a [[cellular automaton|5QXNC6NTVN]].
Call two configurations $x, y$ asymptotic if they differ at only finitely many sites.
Call $F$ pre-injective if $F(x) \ne F(y)$ for all asymptotic configurations $x \ne y$.
Then $F$ is surjective if and only if $F$ is pre-injective.

## Remarks

A configuration outside the image of $F$ is called a Garden of Eden: it has no predecessor.
So the theorem says that a Garden of Eden exists exactly when two configurations that differ at finitely many sites have the same image.
The proof compares counts: in a large box of side $n$, the number of patterns grows like $\abs{A}^{n^d}$, while the boundary of the box has only about $n^{d-1}$ sites.
A missing pattern, or a pair of patterns with the same image, changes the exponential count by a factor that the boundary cannot compensate.
The theorem is due to Moore and Myhill.
Example: rule 90 in one dimension is surjective, so it is pre-injective.
It is not injective, since the all-$0$ and the all-$1$ configurations both map to the all-$0$ configuration, but these are not asymptotic.
Conway's Game of Life is not surjective: it has Garden of Eden configurations.
The theorem extends to cellular automata on amenable groups, and it fails for the free group on two generators.

## Explanations

### L1 human

A Garden of Eden is a picture that a cellular automaton can never produce, because nothing can lead to it.
The theorem says that such pictures exist exactly when the automaton loses information in a finite region.
Losing information means that two different finite changes of a picture give the same next picture.
So counting how much information is lost tells whether every picture can be reached.

### L2 human

Injective implies pre-injective, so every injective cellular automaton on $\Z^d$ is surjective.
This is a dynamical analogue of the fact that an injective map of a finite set to itself is onto.
The theorem is often used to show that a rule has Garden of Eden patterns by exhibiting two mutually erasable patterns.
Its extension to amenable groups by Ceccherini-Silberstein, Machì and Scarabotti links it to [[amenable group actions|ZCG4DJ7GH3]].
For the free group of rank two there are surjective cellular automata that are not pre-injective.
</document>

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# Chain recurrence

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
For $\varepsilon > 0$ and $x, y \in X$, an *$\varepsilon$-chain* from $x$ to $y$ is a finite sequence $x = x_0, x_1, \ldots, x_n = y$ with $n \ge 1$ and $d(f(x_{i-1}), x_i) < \varepsilon$ for $i = 1, \ldots, n$.
A point $x \in X$ is *chain recurrent* if for every $\varepsilon > 0$ there is an $\varepsilon$-chain from $x$ to $x$.
The *chain recurrent set* $\mathcal{R}(f)$ is the set of chain recurrent points.

## Remarks

For a continuous flow $\varphi_t$, an $(\varepsilon, T)$-chain from $x$ to $y$ is a sequence $x = x_0, \ldots, x_n = y$ with times $t_1, \ldots, t_n \ge T$.
It must satisfy $d(\varphi_{t_i}(x_{i-1}), x_i) < \varepsilon$ for each $i$.
A point $x$ is chain recurrent for the flow if such chains from $x$ to $x$ exist for all $\varepsilon > 0$ and $T > 0$.
The set $\mathcal{R}(f)$ is closed and contains the [[non-wandering set|EY2W34JZA4]] $\Omega(f)$.
Example: for the identity map, $x_0 = x_1 = x$ is an $\varepsilon$-chain from $x$ to $x$ for every $\varepsilon$, so every point is chain recurrent.
Non-example: let $f(x) = x/2$ on $[0, 1]$ and $x > 0$, $0 < \varepsilon < x/2$.
If $x_{i-1} \le x$, then $x_i < x_{i-1}/2 + \varepsilon < x$, so by induction every $x_i$ with $i \ge 1$ is less than $x$; no $\varepsilon$-chain returns to $x$.
For this map $\mathcal{R}(f) = \set{0}$.

## Explanations

### L1 human

An $\varepsilon$-chain is an orbit with small errors allowed at each step.
A point is chain recurrent if it can come back to itself along such chains, for every size of error.
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<document source="https://tamnd.github.io/math2/node/2HJGRABSPZ.md" tag="2HJGRABSPZ" kind="concept">
# Closure operator

## Summary

A closure operator on a [[partially ordered set|011C29Q1S1]] is a map $c$ that is extensive, monotone and idempotent.
It sends each element to the least "closed" element above it.
Examples on power sets are topological [[closure|0DRKT1QJR7]], the [[linear span|0HR573CHNM]] of vectors and the [[subgroup generated by|0RG9SPQSWM]] a set.
Closure operators on a power set correspond to [[Moore families|9ZVCJWYPB2]].

## Explanations

### L1 human

A closure operator completes an object to the smallest object of a special kind that contains it.
In a vector space, take a set of vectors and replace it by all their linear combinations.
Doing it again adds nothing new, a bigger set gives a bigger span, and the span always contains the original set.
These three features, growing, keeping order, and stopping after one step, define a closure operator.
</document>

<document source="https://tamnd.github.io/math2/node/2HKA02YDDM.md" tag="2HKA02YDDM" kind="definition">
# Nilpotency class

## Definition

Let $G$ be a [[nilpotent group|0Q7MMQHC1P]].
Let $\gamma_1(G) = G$ and $\gamma_{i+1}(G) = [\gamma_i(G), G]$ be its [[lower central series|0K98R7A4VC]].
The *nilpotency class* of $G$ is the least integer $c \ge 0$ such that $\gamma_{c+1}(G) = \{e\}$.

## Remarks

Equivalently, the class is the least $c$ such that the [[upper central series|0XW54DH28V]] satisfies $Z_c(G) = G$.
It is also the least length of a central series of $G$.
The trivial group has class $0$, and a nontrivial [[abelian group|082GHNPJ53]] has class $1$.
Example: the [[quaternion group|0NRJGW864D]] $Q_8$ has class $2$.
Indeed $\gamma_2(Q_8) = [Q_8, Q_8] = \{\pm 1\} \neq \{1\}$, and $\gamma_3(Q_8) = [\{\pm 1\}, Q_8] = \{1\}$ because $-1$ is central.
Non-example: the symmetric group $S_3$ has no class, because it is not nilpotent.
Its lower central series is $\gamma_1 = S_3$ and then $\gamma_i = A_3$ for all $i \ge 2$, so it never reaches $\{e\}$.

## Explanations

### L1 human

The class counts how many times you must take commutators with the whole group before everything becomes trivial.
</document>

<document source="https://tamnd.github.io/math2/node/2HNN80TMGH.md" tag="2HNN80TMGH" kind="definition">
# Wavelet packet decomposition

## Definition

Let $(h_n)$ and $(g_n)$ be the real low-pass and high-pass filters of an orthonormal wavelet, with $g_n = (-1)^n h_{1-n}$, as in the [[fast wavelet transform|PDVYQAW887]].
For a sequence $c = (c_m)_{m \in \Z}$ define the sequences $Hc$ and $Gc$ by
$$(Hc)_k = \sum_m h_{m-2k} \, c_m, \qquad (Gc)_k = \sum_m g_{m-2k} \, c_m.$$
Let $x$ be a sequence and let $J \ge 1$ be an integer.
The *wavelet packet decomposition* of $x$ to depth $J$ is the family of sequences $w_{j,p}$, for $0 \le j \le J$ and $0 \le p < 2^j$, defined by

1. $w_{0,0} = x$;
2. $w_{j+1,2p} = H w_{j,p}$ and $w_{j+1,2p+1} = G w_{j,p}$ for $0 \le j < J$.

The sequences $w_{j,p}$ are the nodes of a full binary tree, in which $w_{j,p}$ has children $w_{j+1,2p}$ and $w_{j+1,2p+1}$.

## Remarks

The [[discrete wavelet transform|P3H4MMC4ZH]] keeps only the nodes $w_{j,0}$ and $w_{j,1}$, that is, it splits only the low-pass branch again.

Each step $c \mapsto (Hc, Gc)$ is an orthogonal transform of $\ell^2(\Z)$, by the orthonormality of the filters.
So for every admissible subtree, in which each node is either a leaf or has both children, the leaves together determine $x$ and preserve its energy.
For a finite signal of length $N = 2^J$ with periodic extension, every level contains $N$ coefficients in total.

Example: with the Haar filters and $x = (1, 1, 1, 1)$, level $1$ gives $w_{1,0} = (\sqrt{2}, \sqrt{2})$ and $w_{1,1} = (0, 0)$.
Level $2$ gives $w_{2,0} = (2)$, $w_{2,1} = (0)$, $w_{2,2} = (0)$ and $w_{2,3} = (0)$.
The energy $4$ of $x$ is kept in each level.

Non-example: if only $H$ is applied at a node and $G$ is dropped, the leaves no longer determine the signal.
With the Haar filters, $(1, -1)$ and $(0, 0)$ both have $H$-output $(0)$.

## Explanations

### L1 human

At every node of the tree, the current list of numbers is split into a smooth half and a detail half.
Unlike the plain wavelet transform, the detail halves are split again too.
</document>

<document source="https://tamnd.github.io/math2/node/2HP4SA15DW.md" tag="2HP4SA15DW" kind="example">
# Riemann zeta function as a series

## Example

For a complex number $s$ with real part $\sigma = \operatorname{Re} s$, consider the series
$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \qquad n^{-s} = e^{-s \log n}.$$
It is the [[Dirichlet series|HN0AXS6VSG]] of the constant arithmetic function $a(n) = 1$.
Its terms satisfy $\abs{n^{-s}} = n^{-\sigma}$.
By the [[p-series test|0ZNSSK8ZK1]], $\sum_n n^{-\sigma}$ converges exactly when $\sigma > 1$.
So the series converges [[absolutely|0W3K3QPEWT]] for every $s$ with $\operatorname{Re} s > 1$.

Fix $\sigma_0 > 1$; on the half-plane $\operatorname{Re} s \ge \sigma_0$ the bound $\abs{n^{-s}} \le n^{-\sigma_0}$ holds.
So the [[Weierstrass M-test|0NX5NWAFC9]], in its form for complex-valued functions, shows that the series converges uniformly there.
On the whole half-plane $\operatorname{Re} s > 1$ the convergence is not uniform.
Indeed, for real $s \to 1^+$ the tail $\sum_{n > N} n^{-s}$ tends to $\sum_{n > N} n^{-1} = \infty$.
So for every $N$ the tail is unbounded on the half-plane.
At $s = 1$ the series is the [[harmonic series|0CXTMAY5PX]], which diverges.

## Remarks

The sum of this series on $\operatorname{Re} s > 1$ is the [[Riemann zeta function|1XWBY4N0VT]], which extends beyond the half-plane by analytic continuation.
At $s = 2$ the series gives $\zeta(2) = \pi^2/6$, the [[Basel problem|0DMY75D6FV]].

## Explanations

### L1 human

Add up $1$, then $1/2^s$, then $1/3^s$, and so on.
For $s = 2$ this is $1 + 1/4 + 1/9 + \cdots$, which adds up to $\pi^2/6$.
For $s = 1$ the terms shrink too slowly and the sum grows without bound.
The series works exactly when the real part of $s$ is larger than $1$, and it defines the zeta function there.
</document>

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# Poincaré asymptotic series

## Summary

A Poincaré asymptotic series of a function $f$ at infinity is a formal power series $\sum a_n z^{-n}$ with one property.
Every partial sum approximates $f$ with an error smaller than the last power used.
It is the [[asymptotic expansion|BJQTEPEZ9G]] for the [[asymptotic scale|XEGB8Y44GG]] of the powers $z^{-n}$, as defined by Poincaré in 1886.
The series may diverge for every $z$, and still give very accurate approximations.
The coefficients are unique when they exist, but different functions can have the same series.

## Explanations

### L1 human

Think of a series in powers of $1/z$ that is useful for large $z$ even though it may never converge.
The rule is: stop after any finite number of terms, and the error goes to zero faster than the last term you kept.
For example, $\frac{1}{z - 1} = \frac1z + \frac{1}{z^2} + \frac{1}{z^3} + \cdots$ is a Poincaré series at infinity that happens to converge.
The series $\sum_{n \ge 0} (-1)^n n!\, z^{-n-1}$ of the [[exponential integral|EV6ZDQ0NDY]] $e^z E_1(z)$ is one that diverges for every $z$.
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# Finite projective plane

## Description

A finite projective plane is a finite set of points with a family of subsets, called lines, with three properties.
Every two distinct points lie on exactly one line, and every two distinct lines meet in exactly one point.
There are four points with no three on a line.
Each such plane has an order: an integer $n \ge 2$ such that each line has $n + 1$ points and each point lies on $n + 1$ lines.
A plane of order $n$ has $n^2 + n + 1$ points and $n^2 + n + 1$ lines.

The standard construction is $PG(2, q)$ for a prime power $q$.
Let $\F_q$ be the [[finite field|08RTDHX1SK]] with $q$ elements.
The points are the $1$-dimensional [[subspaces|0EPTJA5QV4]] of $\F_q^3$ and the lines are its $2$-dimensional subspaces, with incidence given by inclusion.
Check: there are $(q^3 - 1)/(q - 1) = q^2 + q + 1$ points.
Two distinct $1$-dimensional subspaces span exactly one $2$-dimensional subspace.
Two distinct $2$-dimensional subspaces $U, W$ satisfy $\dim(U \cap W) = 2 + 2 - \dim(U + W) = 1$, so they meet in exactly one point.
The four points spanned by $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ and $(1,1,1)$ have no three on a line.
So $PG(2, q)$ is a projective plane of order $q$.

## Remarks

A projective plane of order $n$ is the same thing as a [[Steiner system|7BPSC4T4M3]] $S(2, n + 1, n^2 + n + 1)$, a symmetric [[block design|4JCSS2Z1BA]].
The plane of order $2$ is the [[Fano plane|QQ8GW1ASZM]].
A projective plane of order $n$ exists if and only if there are $n - 1$ [[mutually orthogonal Latin squares|H6TN6BADVY]] of order $n$.
Every prime power is the order of a plane.
The [[Bruck–Ryser–Chowla theorem|R43K0GTTDS]] rules out the orders $6$, $14$, $21$ and many more.
A long computer search by Lam and others, finished in 1989, showed that there is no plane of order $10$.
No plane of an order that is not a prime power is known.

## Explanations

### L1 human

A finite projective plane is a small geometry with finitely many points in which any two points lie on one line.
Unlike in the usual plane, any two lines meet.
There are no parallel lines.
The smallest one has $7$ points and $7$ lines, each line with $3$ points.
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# Infinite product measure

## Summary

Take a family of [[probability measures|059TJN0NZ1]].
Its infinite product measure is the probability measure on the product space under which the coordinates are independent with the given laws.
It is defined on the [[product σ-algebra|6HVCTJGB61]] and is determined by its values on cylinder sets.
It exists for every family of probability spaces, with an arbitrary index set.
It is the model for infinite sequences of independent trials, such as repeated coin tossing.

## Explanations

### L1 human

Imagine tossing a fair coin forever.
An outcome is an infinite sequence of heads and tails.
The infinite product measure says how likely a set of such sequences is: for example, the sequences that start with heads, tails, heads have probability $\tfrac18$.
Only probability measures are used, because a product of infinitely many total masses must make sense, and $1 \cdot 1 \cdot 1 \cdots = 1$.
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<document source="https://tamnd.github.io/math2/node/2HYK3KBQNA.md" tag="2HYK3KBQNA" kind="concept">
# Sheafification

## Summary

Sheafification turns a [[presheaf|AND2WHZC8P]] $P$ on a [[site|NYE0Z94MME]] into the closest [[sheaf|RT8A8FJ566]] $aP$.
It comes with a map $P \to aP$ through which every map from $P$ to a sheaf factors uniquely.
It is a functor that is [[left adjoint|TZHF6ZM92S]] to the inclusion of sheaves into presheaves, and it preserves finite limits.
It is how one builds sheaves such as cokernels, quotients and constant sheaves, which are not sheaves when computed objectwise.
On a small site it can be computed by applying the plus construction twice.

## Explanations

### L1 human

Some natural constructions give data that is defined locally but does not glue, so it is only a presheaf.
Sheafification repairs this: it forces local data that agree on overlaps to glue, and it identifies data that agree locally.
For example, on a space with two connected components, the constant presheaf with value $\Z$ has only constant functions, while its sheafification allows a different integer on each component.
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<document source="https://tamnd.github.io/math2/node/2J1RVQB0CM.md" tag="2J1RVQB0CM" kind="statement">
# Itô's lemma

## Statement

Let $(B_t)_{t \ge 0}$ be a [[Brownian motion|22735FGJV6]] for a filtration $(\mathcal{F}_t)$.
Let $X$ be an Itô process
$$X_t = X_0 + \int_0^t \mu_s \, ds + \int_0^t \sigma_s \, dB_s,$$
where the second integral is an [[Itô integral|JA95NH206E]].
The processes $\mu$ and $\sigma$ are progressively measurable with $\int_0^t (\abs{\mu_s} + \sigma_s^2) \, ds < \infty$ almost surely for every $t$.
Let $f \colon [0, \infty) \times \R \to \R$ be once continuously differentiable in $t$ and twice continuously differentiable in $x$.
Then, almost surely, for all $t \ge 0$,
$$f(t, X_t) = f(0, X_0) + \int_0^t \left( \partial_t f + \mu_s \, \partial_x f + \tfrac12 \sigma_s^2 \, \partial_{xx} f \right)(s, X_s) \, ds + \int_0^t \sigma_s \, \partial_x f(s, X_s) \, dB_s.$$
Here $\partial_t$, $\partial_x$ and $\partial_{xx}$ are [[partial derivatives|09EGC6F8PT]].

## Remarks

The proof expands $f$ by Taylor's formula to second order over a fine partition of $[0, t]$.
The sums of $(\Delta X)^2$ converge to $\int \sigma_s^2 \, ds$, the [[quadratic variation|X4DHMZD5JA]] of $X$, while higher-order terms vanish in the limit.
In differential form, $df = \partial_t f \, dt + \partial_x f \, dX + \tfrac12 \partial_{xx} f \, (dX)^2$ with the rules $(dB)^2 = dt$ and $dt \, dB = (dt)^2 = 0$.
Example: for $f(x) = x^2$ and $X = B$ with $B_0 = 0$, it gives $B_t^2 = 2 \int_0^t B_s \, dB_s + t$.
Example: for [[geometric Brownian motion|D48DR46B97]] $dS = \mu S \, dt + \sigma S \, dB$ with constants $\mu, \sigma$ and $S_0 > 0$, the function $\log$ gives $\log S_t = \log S_0 + (\mu - \tfrac12 \sigma^2) t + \sigma B_t$.
The $C^2$ hypothesis matters: for $f(x) = \abs{x}$ the formula needs an extra local time term, which is [[Tanaka's formula|GEM0ERJ5ZB]].

## Explanations

### L1 human

Itô's lemma is the chain rule for functions of a random process driven by Brownian motion.
Compared with the ordinary chain rule, it has one more term, half the second derivative times the squared noise intensity.
This term appears because a Brownian path moves by about $\sqrt{dt}$ in time $dt$, so squares of its moves do not vanish.

### L2 human

The lemma is the main computational tool of stochastic calculus.
It solves linear stochastic differential equations, such as those for geometric Brownian motion and the Ornstein–Uhlenbeck process.
It links expectations of diffusions to partial differential equations, as in the Kolmogorov backward equation and the Feynman–Kac formula.
The multidimensional and semimartingale versions have the same shape, with the quadratic covariations $d[X^i, X^j]$ in the second-order term.
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# Homology of a chain complex

## Summary

The homology of a [[chain complex|WQ82C0M3QB]] $(C_\bullet, d)$ is the sequence of objects $H_n(C) = \ker d_n / \im d_{n+1}$.
It measures the failure of the complex to be [[exact|JB7ZBFBA9C]]: $H_n(C) = 0$ exactly when the complex is exact at $C_n$.
Each [[chain map|HTK9WQZJ1G]] induces maps on homology, so homology is a functor.
The homology of the singular chain complex of a space is its [[singular homology|M0Q5CFJ7ZH]].
The same construction gives [[simplicial homology|C6J40QF82R]], group homology and derived functors.

## Explanations

### L1 human

In a chain complex, every element in the image of $d_{n+1}$ is killed by $d_n$.
The elements killed by $d_n$ are called cycles, and those in the image of $d_{n+1}$ are called boundaries.
Homology is "cycles modulo boundaries": it collects the cycles that are not explained by a boundary.
In topology, a loop around a hole is a cycle that is not the boundary of any surface, and it gives a nonzero homology class.
For the complex $0 \to \Z \xrightarrow{2} \Z \to 0$ in degrees $1$ and $0$, the homology is $0$ in degree $1$ and $\Z/2\Z$ in degree $0$.
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# Holomorphic automorphisms of C^n

## Summary

A holomorphic automorphism of $\C^n$ is a [[biholomorphism|FX92PP3HZR]] of $\C^n$ onto itself.
Under composition these maps form the group $\Aut(\C^n)$, an [[automorphism group|0K1SCPBNKJ]].
For $n = 1$ this group consists only of the affine maps $z \mapsto az + b$ with $a \ne 0$.
For $n \ge 2$ it is very large, and its study is a central topic of complex analysis in several variables.

## Explanations

### L1 human

An automorphism of $\C^n$ is a holomorphic way to rearrange all of complex space, with a holomorphic way back.
In one variable only stretching, rotating and shifting are possible.
In two variables one can also shear: the map $(z, w) \mapsto (z, w + g(z))$ is an automorphism for every entire function $g$, with inverse $(z, w) \mapsto (z, w - g(z))$.
Composing shears with linear maps already gives a huge, infinite-dimensional group.
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<document source="https://tamnd.github.io/math2/node/2J9GA17YAK.md" tag="2J9GA17YAK" kind="definition">
# Harmonic functions on Riemannian manifolds

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $U \subseteq M$ be an open set.
In local coordinates $x^1, \dots, x^n$, write $g_{ij}$ for the metric, $g^{ij}$ for its inverse matrix and $\abs{g} = \det(g_{ij})$.
The [[Laplace–Beltrami operator|F4NWZNTRMD]] of a function $u \in C^2(U)$ is
$$\Delta_g u = \frac{1}{\sqrt{\abs{g}}}\, \partial_i \Big( \sqrt{\abs{g}}\, g^{ij}\, \partial_j u \Big),$$
with summation over repeated indices; it equals $\operatorname{div}_g \operatorname{grad}_g u$ and does not depend on the coordinates.
A function $u \in C^2(U)$ is *harmonic* on $U$ if $\Delta_g u = 0$ on $U$.

## Remarks

For $M = \R^n$ with the Euclidean metric, $\Delta_g$ is the usual Laplacian, and the definition gives the classical [[harmonic functions|0DEB0JMN0W]].
Example: on the upper half-plane $\set{(x, y) : y > 0}$ with the hyperbolic metric $g = y^{-2}(dx^2 + dy^2)$, one has $\Delta_g = y^2(\partial_x^2 + \partial_y^2)$.
So the function $u(x, y) = x$ is harmonic, and so is the bounded function $\arctan(x/y)$.
Non-example: on the unit sphere $S^2 \subset \R^3$ with the round metric, the height function $z$ satisfies $\Delta_g z = -2z$, so it is not harmonic.
On a connected closed manifold every harmonic function is constant, since $\int_M \abs{\nabla u}^2 \, dV_g = -\int_M u\, \Delta_g u \, dV_g = 0$.
The definition depends on the metric: a function harmonic for one metric need not be harmonic for another.
In dimension $2$, however, harmonicity is unchanged under conformal changes of metric.
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# Stochastic geometry

## Summary

Stochastic geometry is the part of probability theory that studies random geometric structures in space.
Its main objects are [[point processes|DNDKBVHS3H]], random closed sets, random tessellations and random measures.
Typical models are the [[Poisson point process|0MTNR28K60]] and the [[Boolean model|NK3CZTVEJD]].
It supplies models and formulas for random patterns in materials science, wireless networks, biology and image analysis.

## Explanations

### L1 human

Stochastic geometry asks questions about shapes that are placed at random.
Think of raindrops on a pavement, trees in a forest, or cells seen under a microscope.
One models the centres as random points and the objects as random shapes around them.
For example, drop discs of radius $r$ at the points of a Poisson process of intensity $\lambda$ in the plane.
Then the fraction of the plane covered by the discs is $1 - e^{-\lambda \pi r^2}$.
Classical questions of geometric probability, such as [[Buffon's needle problem|06X5PF6YFP]], belong to the same circle of ideas.
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# Pro-p group

## Definition

Let $p$ be a prime number.
A *pro-$p$ group* is a [[profinite group|KA0YDB6QXE]] $G$ such that for every [[open|QYCFH68RD1]] [[normal subgroup|0CB0DX5BVV]] $N$ of $G$ the finite quotient $G/N$ is a [[p-group|06QYRTX9KQ]].

## Remarks

Equivalently, $G$ is isomorphic as a topological group to an inverse limit of finite $p$-groups.
Example: the [[p-adic integers|V6TJAFZ4MD]] $\Z_p = \varprojlim \Z/p^n\Z$.
Its open subgroups are the subgroups $p^n \Z_p$, and each quotient $\Z_p / p^n \Z_p \cong \Z/p^n\Z$ has order $p^n$.
Non-example: the profinite completion $\hat\Z$ of $\Z$ is profinite, but it has the open normal subgroup $6\hat\Z$ with quotient $\Z/6\Z$, which is not a $p$-group for any prime $p$.
Every finite $p$-group with the discrete topology is a pro-$p$ group.
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# Autoregressive model

## Summary

An autoregressive model describes each value of a [[time series|EA3MF4VTEH]] as a linear combination of a few past values plus a random shock.
The number of past values used is the order $p$, and the model is written AR($p$).
It is one of the simplest and most used models for dependent data, and a building block of ARMA and ARIMA models.

## Explanations

### L1 human

The model says that the present is a weighted echo of the recent past plus something new and unpredictable.
In the AR($1$) model $X_t = 0.8 X_{t-1} + \varepsilon_t$, each value keeps $80\%$ of the previous one and adds fresh noise.
A shock therefore fades away slowly, by a factor $0.8$ each step.
If the weight were $1$, shocks would never fade, and the series would wander like a random walk.
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# Hopf–Lax formula

## Statement

Let $H \colon \R^n \to \R$ be a smooth [[convex function|07Q7CNM5WF]] that is superlinear: $H(p)/\abs{p} \to \infty$ as $\abs{p} \to \infty$.
Let $L \colon \R^n \to \R$ be its [[Legendre transform|ZQYMWVT1TT]], $L(q) = \sup_{p \in \R^n} \big(p \cdot q - H(p)\big)$.
Let $g \colon \R^n \to \R$ be [[Lipschitz continuous|0AW1E6YTJY]].
For $x \in \R^n$ and $t > 0$ define
$$u(x, t) = \min_{y \in \R^n} \left( t\, L\!\left(\frac{x - y}{t}\right) + g(y) \right),$$
and set $u(x, 0) = g(x)$.
Then the minimum is attained, and $u$ is Lipschitz continuous on $\R^n \times [0, \infty)$.
Moreover, $u$ is differentiable at almost every point of $\R^n \times (0, \infty)$, and at each such point
$$\partial_t u + H(\nabla u) = 0.$$
So $u$ is a solution almost everywhere of the [[Hamilton–Jacobi equation|5TRXQKP7DY]] with initial value $g$.

## Remarks

The formula comes from the calculus of variations.
The value $u(x, t)$ is the least cost $\int_0^t L(\dot w(s))\, ds + g(w(0))$ over paths $w$ with $w(t) = x$.
By convexity of $L$, Jensen's inequality shows that straight paths are optimal, which gives the minimum over the starting point $y$.
The almost everywhere differentiability comes from the Lipschitz bound and Rademacher's theorem.
The equation itself follows from the semigroup property $u(x, t) = \min_y \big( (t - s) L((x - y)/(t - s)) + u(y, s) \big)$ for $0 \le s < t$.

Example: for $H(p) = \tfrac12 \abs{p}^2$ one has $L(q) = \tfrac12 \abs{q}^2$, and the formula reads $u(x, t) = \min_y \big( \abs{x - y}^2/(2t) + g(y) \big)$.
For $g(y) = \abs{y}$ in dimension $1$ this gives $u(x, t) = \abs{x} - t/2$ for $\abs{x} \ge t$ and $u(x, t) = x^2/(2t)$ for $\abs{x} \le t$.

Solutions almost everywhere are far from unique, so the formula alone does not settle the initial value problem.
Under the hypotheses above the Hopf–Lax function is also a [[viscosity solution|H1ERJ3T75T]], and this is the property that singles it out.

## Explanations

### L1 human

The Hopf–Lax formula writes down a solution of a nonlinear first-order equation with no differential equation solving at all: one only takes a minimum.
To get $u$ at a point $x$ and time $t$, one looks at all starting points $y$.
For each, one adds the start value $g(y)$ to the cost of travelling from $y$ to $x$ in time $t$, and keeps the cheapest.
The solution stays meaningful even after smooth solutions break down and corners appear.

### L2 human

The formula is the starting point of the theory of weak solutions of Hamilton–Jacobi equations in Evans' textbook.
Differentiating it in $x$ in dimension $1$ gives the Lax–Oleinik formula for the entropy solution of a scalar [[conservation law|8RA7AMWSAF]] with convex flux, since $v = \partial_x u$ solves $v_t + H(v)_x = 0$.
For $H(p) = \tfrac12 \abs{p}^2$ the formula is a Moreau envelope of $g$, and it is also the limit as $\nu \to 0$ of the solutions of the viscous equation obtained by the [[Cole–Hopf transformation|VBMQ9SJ3DH]].
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# Cauchy matrix

## Definition

Let $F$ be a [[field|04C5ZBG1WT]], and let $x_1, \dots, x_m$ and $y_1, \dots, y_n$ be elements of $F$ with $x_i \ne y_j$ for all $i, j$.
The *Cauchy matrix* of these elements is the $m \times n$ [[matrix|0VD9ZSWWJG]] $C$ with entries
$$C_{ij} = \frac{1}{x_i - y_j} \qquad (1 \le i \le m,\ 1 \le j \le n).$$

## Remarks

Example: with $x = (1, 2)$ and $y = (0, -1)$ the Cauchy matrix is $\begin{pmatrix} 1 & \tfrac12 \\ \tfrac12 & \tfrac13 \end{pmatrix}$; here $x_1 - y_2 = 2$ and $x_2 - y_2 = 3$.
Non-example: $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ is not a Cauchy matrix, because no entry $1/(x_i - y_j)$ can be $0$.
For $m = n$, the Cauchy determinant is
$$\det C = \frac{\prod_{1 \le i < j \le n} (x_j - x_i)(y_i - y_j)}{\prod_{i=1}^n \prod_{j=1}^n (x_i - y_j)}.$$
So a square Cauchy matrix is invertible exactly when the $x_i$ are distinct and the $y_j$ are distinct.
Every square submatrix of a Cauchy matrix is again a Cauchy matrix.
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# Countably generated structure

## Summary

A structure is countably generated when some [[countable|01TQB88H4C]] subset of it generates the whole structure under its operations.
It is the countable analogue of a [[finitely generated algebra|WR4WF1K2KW]].
In a countable signature, a structure is countably generated exactly when its universe is countable.

## Explanations

### L1 human

Start with a list of elements, one for each natural number, and close it up under the operations.
If this reaches every element, the structure is countably generated.
The rational numbers with addition, negation and $0$ are countably generated, by the list of all fractions $1/n$.
They are not finitely generated, so the countable notion is really weaker.
The real numbers with the same operations are not countably generated, since closing a countable set under these operations gives only a countable set.
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# Loop space

## Summary

The loop space $\Omega X$ of a [[pointed space|T9DG8DDE75]] $(X, x_0)$ is the space of all loops in $X$ that start and end at $x_0$.
Its path components are the elements of the [[fundamental group|S1XK3HXHFX]] of $X$, and in general $\pi_n(\Omega X) \cong \pi_{n+1}(X)$.
Concatenation of loops makes $\Omega X$ an H-space.
The loop space is dual to the [[suspension|Q7RFNJH76Q]] in the sense of [[Eckmann–Hilton duality|1EK3355WMC]].

## Explanations

### L1 human

Every point of the loop space is a whole loop in $X$ based at $x_0$.
Two points of the loop space are close when the two loops stay close at every time.
Moving continuously in the loop space means deforming a loop.
For example, the loop space of the circle falls into pieces, one for each integer winding number.
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# Weyl's law

## Statement

Let $d \ge 1$ and let $\Omega \subseteq \R^d$ be a nonempty bounded open set with Lebesgue measure $\abs{\Omega}$.
Let $0 < \lambda_1 \le \lambda_2 \le \cdots$ be the [[eigenvalues|0BQAA2BK3Q]] of $-\Delta$ on $\Omega$ with the [[Dirichlet boundary condition|EETVA6QYX4]], repeated according to multiplicity, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
For $\lambda > 0$ let $N(\lambda)$ be the number of indices $k$ with $\lambda_k \le \lambda$.
Let $\omega_d$ be the volume of the unit ball in $\R^d$.
Then
$$\lim_{\lambda \to \infty} \frac{N(\lambda)}{\lambda^{d/2}} = \frac{\omega_d \abs{\Omega}}{(2\pi)^d}.$$

## Remarks

For a cube the eigenvalues are explicit, and counting them amounts to counting lattice points in a ball of radius $\sqrt{\lambda}$, which gives the formula.
For a general set one fills $\Omega$ from inside and covers it from outside by small cubes.
Then one compares eigenvalues with Dirichlet and Neumann conditions on the cubes; this is Dirichlet–Neumann bracketing.
Example: for $\Omega = (0, L) \subseteq \R$, the eigenvalues are $(k\pi / L)^2$ for $k \ge 1$, so $N(\lambda) = \floor{L \sqrt{\lambda} / \pi}$.
This agrees with the formula, since $\omega_1 = 2$ and $\frac{2L}{2\pi} = \frac{L}{\pi}$.
Equivalently, $\lambda_k \sim \frac{(2\pi)^2}{(\omega_d \abs{\Omega})^{2/d}} \, k^{2/d}$ as $k \to \infty$.
Hermann Weyl proved the law in 1911.

## Explanations

### L1 human

Up to a constant factor, the eigenvalues of the Laplacian on a region are the squares of the pitches of a drum of that shape.
Weyl's law says that, for high pitches, the number of notes below a given pitch depends only on the dimension and the size of the drum, not on its shape.

### L2 human

Weyl's law shows that the volume of a domain can be read off from its spectrum.
So it gives a first positive answer to the question whether one can hear the shape of a drum.
The same leading term holds for the Neumann condition when the boundary is regular enough, and an analogous law holds on compact Riemannian manifolds.
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# Associativity equation

## Summary

The associativity equation is the [[functional equation|SRAN4EKWS0]] $F(F(x, y), z) = F(x, F(y, z))$ for an unknown binary operation $F$ on a set.
Its solutions are exactly the associative [[binary operations|0M418FE7BB]], so it turns the algebraic law of [[associativity|0K8F7PVDMX]] into a problem about unknown functions.
On a real interval, its continuous solutions that are strictly increasing in each variable are the operations $F(x, y) = g^{-1}(g(x) + g(y))$ with $g$ continuous and strictly monotone.
So, up to a change of scale, every such operation on an interval is ordinary addition.

## Explanations

### L1 human

Associativity says that brackets do not matter when one combines three things.
The equation asks which formulas $F(x, y)$ have this property.
Addition $x + y$ is the basic solution, and every regular solution on an interval is addition in disguise, after a change of scale $g$.
For example, $F(x, y) = x + y + xy$ on $(-1, \infty)$ is associative, because $1 + F(x, y) = (1 + x)(1 + y)$.
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# Simplex algorithm

## Definition

Let $A$ be a real $m \times n$ [[matrix|0VD9ZSWWJG]] of rank $m$, let $b \in \R^m$ and let $c \in \R^n$.
Consider the [[linear program|VNMXB0J2KA]]: maximize $c^T x$ subject to $A x = b$ and $x \ge 0$.
A *basis* is a set $B$ of $m$ column indices such that the submatrix $A_B$ is invertible; let $N$ be the other indices.
Its *basic solution* is the $x$ with $x_B = A_B^{-1} b$ and $x_N = 0$, and $B$ is *feasible* if $x_B \ge 0$.
The *simplex algorithm* starts from a feasible basis $B$ and repeats the following step.

1. Compute the reduced costs $\bar c_j = c_j - c_B^T A_B^{-1} A_j$ for $j \in N$, where $A_j$ is column $j$ of $A$.
If $\bar c_j \le 0$ for all $j \in N$, stop: the basic solution is optimal.
2. Otherwise choose an entering index $j \in N$ with $\bar c_j > 0$, and compute $d = A_B^{-1} A_j$.
If $d \le 0$, stop: the program is unbounded.
3. Otherwise choose a leaving index $i \in B$ with $d_i > 0$ that minimizes $x_i / d_i$ (the ratio test), and replace $B$ by $(B \setminus \set{i}) \cup \set{j}$.

## Remarks

Feasible basic solutions are exactly the vertices of the feasible region, and several bases can give the same vertex when it is degenerate.
One step moves along an edge to an adjacent vertex, or stays at the same vertex in a degenerate step, and it never decreases $c^T x$.
With an anti-cycling choice of $j$ and $i$, such as Bland's rule of taking the smallest eligible index, the algorithm stops after finitely many steps.

Example: maximize $x_1 + x_2$ subject to $x_1 + 2x_2 + s_1 = 4$, $3x_1 + x_2 + s_2 = 6$ and all variables $\ge 0$.
Start with $B = \set{s_1, s_2}$, the point $x_1 = x_2 = 0$; both reduced costs are $1$.
Let $x_1$ enter; then $d = (1, 3)$, the ratios are $4$ and $2$, so $s_2$ leaves, giving $x_1 = 2$, $s_1 = 2$ and value $2$.
Now $x_2$ has reduced cost $2/3$; it enters, the ratio test removes $s_1$, and the new point is $x_1 = 8/5$, $x_2 = 6/5$.
There the objective equals $14/5 - \frac25 s_1 - \frac15 s_2$, so all reduced costs are negative and $14/5$ is optimal.

Non-example: in the first step, letting $s_1$ leave instead of $s_2$ ignores the ratio test.
It gives $x_1 = 4$ and $s_2 = 6 - 12 = -6 < 0$, which is not a feasible basis.

A first feasible basis can be found by solving an auxiliary linear program, the phase one problem.
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# Cancellation property

## Definition

Let $\ast$ be a [[binary operation|0M418FE7BB]] on a set $S$ and let $a \in S$.

1. The element $a$ is *left cancellable* if $a \ast b = a \ast c$ implies $b = c$, for all $b, c \in S$.
2. The element $a$ is *right cancellable* if $b \ast a = c \ast a$ implies $b = c$, for all $b, c \in S$.
3. The element $a$ is *cancellable* if it is both left and right cancellable.

The pair $(S, \ast)$ is *left cancellative*, *right cancellative* or *cancellative* if every element of $S$ is left cancellable, right cancellable or cancellable.

## Remarks

Example: $(\N, +)$ is cancellative, because $a + b = a + c$ gives $b = c$ by comparing the numbers, and $+$ is commutative.

Non-example: $(\Z, \cdot)$ is not cancellative, because $0 \cdot 1 = 0 \cdot 2$ with $1 \ne 2$, so $0$ is not left cancellable.

An element $a$ is left cancellable if and only if the map $x \mapsto a \ast x$ is injective.
On a set with at least two elements, the operation $a \ast b = b$ is left cancellative but not right cancellative.
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# Holomorphic function of several variables

## Definition

Let $D$ be an [[open set|02BH3MCBQB]] in $\C^n$ and let $f \colon D \to \C$.
The function $f$ is *complex differentiable* at $a \in D$ if there is a $\C$-[[linear map|0NRYHJPQG2]] $L \colon \C^n \to \C$ such that
$$\lim_{h \to 0} \frac{\abs{f(a + h) - f(a) - L(h)}}{\abs{h}} = 0.$$
The function $f$ is *holomorphic* on $D$ if it is complex differentiable at every point of $D$.

## Remarks

The map $L$ is the [[Fréchet derivative|06EH97V2A7]] of $f$ at $a$, and it has the form $L(h) = \sum_{j=1}^n \frac{\partial f}{\partial z_j}(a) \, h_j$.
Example: $f(z_1, z_2) = z_1 z_2$ is holomorphic on $\C^2$, since $f(a + h) - f(a) = a_2 h_1 + a_1 h_2 + h_1 h_2$ and $\abs{h_1 h_2} \le \abs{h}^2$.
Non-example: $g(z_1, z_2) = \bar z_1$ is real differentiable with derivative $h \mapsto \bar h_1$, but this map is not $\C$-linear, since it sends $i h$ to $-i \bar h_1$.
Equivalent forms: $f$ is holomorphic on $D$ if and only if $f$ is continuous and holomorphic in each variable separately.
It is also equivalent that $f$ is, near each point $a$, the sum of a convergent [[power series in several variables|R3QTCPTJK7]] centred at $a$.
By [[Hartogs's theorem on separate holomorphicity|ECJG25SAJR]], the continuity condition in the first equivalent form may even be dropped.
The set of holomorphic functions on $D$ is written $\mathcal{O}(D)$.
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# Nilradical

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
The *nilradical* of $R$ is the set
$$N(R) = \set{r \in R : r^n = 0 \text{ for some integer } n \ge 1}$$
of [[nilpotent elements|0N9R5NN6TW]] of $R$.

## Remarks

$N(R)$ is the [[radical|437YRCNYRJ]] $\sqrt{(0)}$ of the zero ideal, so it is an ideal of $R$.
Example: in $\Z/12\Z$, $N = \set{0, 6}$; indeed $6^2 = 0$, and if $r^n \equiv 0 \pmod{12}$ then $2$ and $3$ divide $r$.
Non-example: $2 \notin N(\Z/12\Z)$, because $2^n$ is never divisible by $3$.
The quotient $R/N(R)$ has zero nilradical, that is, it is [[reduced|SZHY975FFX]].
$N(R)$ is the intersection of all [[prime ideals|06HC6HK0Z8]] of $R$; see [[the nilradical as an intersection of primes|3CZ8PXR9JY]].
Other notations are $\operatorname{nil}(R)$ and $\mathfrak{N}_R$.
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# Representation of a Lie group

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] and let $V$ be a finite-dimensional vector space over $\R$ or $\C$.
A *representation* of $G$ on $V$ is a continuous group homomorphism $\pi \colon G \to \mathrm{GL}(V)$, where $\mathrm{GL}(V)$ is the [[general linear group|0FCHE902D7]] of $V$.

Let $V$ be an infinite-dimensional complex Hilbert or Banach space.
A representation of $G$ on $V$ is a homomorphism $\pi$ from $G$ to the group of bounded invertible operators on $V$.
It must be strongly continuous: $g \mapsto \pi(g) v$ is continuous from $G$ to $V$ for each $v \in V$.

## Remarks

Example: $G = \mathrm{SO}(3)$, $V = \R^3$ and $\pi(g) = g$.
The map is the inclusion of a closed subgroup of $\mathrm{GL}_3(\R)$, so it is a continuous homomorphism.

Non-example: let $f \colon \R \to \R$ be an additive map that is not linear, which exists by choosing a basis of $\R$ over $\Q$.
Then $\pi(t) = e^{f(t)}$ is a homomorphism $\R \to \mathrm{GL}_1(\R)$, but it is not continuous, so it is not a representation of the Lie group $\R$.

A continuous homomorphism of Lie groups is automatically smooth.
So a finite-dimensional representation $\pi$ has a differential $d\pi$ at the identity, which is a representation of the Lie algebra of $G$.
If $G$ is connected, $d\pi$ determines $\pi$.
A unitary representation is one where each $\pi(g)$ preserves a fixed inner product on $V$.
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# Non-Euclidean geometry

## Definition

A plane geometry consists of points, [[lines|0MP9BCF6DP]], and relations of incidence, betweenness and congruence that satisfy a system of axioms.
A plane geometry is *non-Euclidean* if the [[parallel postulate|2PC3XY0PYZ]] fails in it.
It must satisfy the other axioms of [[Euclidean geometry|KRZD96GSVC]], with only the changes forced by the new parallel property.
The classical cases are the following.

1. Hyperbolic: for each line $\ell$ and point $P$ not on $\ell$, at least two lines through $P$ do not meet $\ell$.
2. Elliptic: any two lines meet.

## Remarks

In the hyperbolic case, all axioms of absolute geometry hold, and only the parallel axiom is replaced.
In the elliptic case, lines are closed curves, so the axioms of order must also be changed.
Both geometries have Riemannian models of constant [[Gaussian curvature|SQVQ5DQT6W]]: $-1$ for the hyperbolic plane and $+1$ for the elliptic plane.

Example: the [[Poincaré half-plane model|RHKSMPBSQ1]] is a hyperbolic plane.
Through $(1, 1)$, both the vertical line $x = 1$ and the semicircle centred at $(2, 0)$ of radius $\sqrt 2$ miss the line $x = 0$.

Non-example: in the Euclidean plane $\R^2$, the only line through $(1, 1)$ that misses $x = 0$ is $x = 1$.
So the parallel postulate holds there, and the Euclidean plane is not non-Euclidean.
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# Amalgamation property

## Definition

Let $L$ be a [[signature|WDVVEYD6AT]], and let $\mathcal{K}$ be a class of [[$L$-structures|983GZBQYT4]] closed under isomorphism.
An embedding is an injective map between $L$-structures that preserves and reflects all functions, relations and constants.
The class $\mathcal{K}$ has the *amalgamation property* if the following holds.
For all $A, B, C \in \mathcal{K}$ and all embeddings $f \colon A \to B$ and $g \colon A \to C$, there are $D \in \mathcal{K}$ and embeddings $f' \colon B \to D$ and $g' \colon C \to D$ with $f' \circ f = g' \circ g$.

## Remarks

The structure $D$ is called an amalgam of $B$ and $C$ over $A$.

Example: the class of finite [[graphs|00C0BR7D7H]] has the amalgamation property.
Given $f$ and $g$, take the disjoint union of $B$ and $C$, identify $f(a)$ with $g(a)$ for each $a \in A$, and add no new edges.
Two vertices of $B$ are adjacent in the result exactly when they are adjacent in $B$, and the same holds for $C$.
So the inclusions of $B$ and $C$ are embeddings, and they agree on $A$.

Non-example: the class of graphs with at most $2$ vertices fails the condition.
Let $A$ be one vertex $a$, let $B$ be an edge $\set{a, b}$, and let $C$ be two non-adjacent vertices $a, c$.
In any amalgam $D$, the images of $b$ and $c$ are distinct, since one is adjacent to the image of $a$ and the other is not.
So $D$ has at least $3$ vertices and is not in the class.

When $A$ may be the empty structure, the amalgamation property implies the joint embedding property: any two members of $\mathcal{K}$ embed into a common member.
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# Finite Hilbert transform

## Definition

Let $1 < p < \infty$ and let $f \in L^p(-1, 1)$, a function in the [[$L^p$ space|SASP14MA1Z]] of the interval.
The *finite Hilbert transform* of $f$ is the function $T f$ on $(-1, 1)$ given by
$$(T f)(x) = \frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{f(t)}{t - x}\, dt = \frac{1}{\pi} \lim_{\varepsilon \to 0^+} \int_{\set{t \in (-1, 1) : \abs{t - x} > \varepsilon}} \frac{f(t)}{t - x}\, dt,$$
the [[Cauchy principal value|0SYR10HD4R]] integral.
The limit exists for almost every $x \in (-1, 1)$.

## Remarks

If $\tilde f$ is $f$ on $(-1, 1)$ and $0$ elsewhere, then $T f = -H\tilde f$ on $(-1, 1)$, where $H$ is the [[Hilbert transform|NSCEBXM7ZM]] with kernel $\frac{1}{\pi (x - t)}$.
By the theorem of M. Riesz on $H$, the operator $T$ is bounded on $L^p(-1, 1)$ for $1 < p < \infty$.
Some authors use the kernel $\frac{1}{x - t}$ instead, which changes the sign of $T$.

Example: $f(t) = \sqrt{1 - t^2}$ gives $T f(x) = -x$.
Check: for $\abs{z} > 1$, $\int_{-1}^{1} \frac{\sqrt{1 - t^2}}{t - z}\, dt = \pi\bigl(\sqrt{z^2 - 1} - z\bigr)$ with the branch of the root that behaves like $z$ at infinity.
Its boundary values from above and below on $(-1, 1)$ are $\pi\bigl(\pm i\sqrt{1 - x^2} - x\bigr)$, and the principal value is their average $-\pi x$.

Non-example of injectivity: $g(t) = (1 - t^2)^{-1/2}$ lies in $L^p(-1, 1)$ for $1 < p < 2$ and $T g = 0$ on $(-1, 1)$.
So $T$ is not injective on these spaces.
For $p \ge 2$ the function $g$ is not in $L^p(-1, 1)$.
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# Hypergraph

## Definition

A hypergraph is a pair $H = (V, E)$ where $V$ is a [[set|0SJ3RX9TXA]] and $E$ is a [[family|05Q6G4EH2G]] of nonempty subsets of $V$.
The elements of $V$ are the vertices, and the members of $E$ are the hyperedges.
Let $k$ be a positive integer.
The hypergraph $H$ is $k$-uniform if each hyperedge has exactly $k$ elements.

## Remarks

Some authors allow repeated hyperedges or the empty hyperedge; others require $E$ to be a set.
A [[graph|0JD5C82QMB]] is the same thing as a $2$-uniform hypergraph without repeated edges.
Example: the [[Fano plane|QQ8GW1ASZM]], with the $7$ points as vertices and the $7$ lines as hyperedges, is a $3$-uniform hypergraph, since each line has $3$ points.
Non-example: $V = \set{1, 2, 3}$ with $E = \set{\set{1, 2}, \set{1, 2, 3}}$ is a hypergraph that is not $k$-uniform for any $k$, because its hyperedges have sizes $2$ and $3$.
Every hypergraph has an incidence matrix, with a row for each vertex and a column for each hyperedge.
</document>

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# Y transform

## Summary

The Y transform of order $\nu$ is an [[integral transform|YCK709QB06]] on $(0, \infty)$ with kernel $\sqrt{xy}\, Y_\nu(xy)$.
Here $Y_\nu$ is the [[Bessel function of the second kind|GRSNDJG363]].
It is the companion of the [[Hankel transform|2P9YWDGKC4]], which uses the Bessel function of the first kind $J_\nu$ in the same way.
Unlike the Hankel transform, it is not its own inverse.
It appears when boundary value problems in cylindrical coordinates need solutions that are singular on the axis.

## Explanations

### L1 human

Bessel functions come in two kinds: $J_\nu$ is finite at the origin, and $Y_\nu$ blows up there.
The Hankel transform uses $J_\nu$ as its family of test waves, and the Y transform uses $Y_\nu$.
For the order $\nu = 1/2$ the kernel is a cosine, so the Y transform is, up to a sign and a constant, the Fourier cosine transform.
For other orders it is a genuinely new transform with its own inversion theory.
</document>

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# Morse–Bott function

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] and let $f \colon M \to \R$ be a [[smooth function|0V33BNKZ9S]] with set of [[critical points|0YTCMFB4HC]] $\operatorname{Crit}(f) = \set{p \in M : df_p = 0}$.
The function $f$ is a *Morse–Bott function* if:

1. each connected component of $\operatorname{Crit}(f)$ is a closed submanifold of $M$; and
2. for each $p \in \operatorname{Crit}(f)$, the kernel of the Hessian $\operatorname{Hess}_p f$ on $T_p M$ equals $T_p C$, where $C$ is the component of $\operatorname{Crit}(f)$ through $p$.

## Remarks

The Hessian is defined at critical points as in the definition of a [[nondegenerate critical point|MGPPCDFQDG]].
Since $f$ is constant on each component $C$, the Hessian always vanishes on $T_p C$; the second condition says it is nondegenerate on the normal space $T_p M / T_p C$.
The *index* of a critical component is the number of negative directions of the Hessian on this normal space.
A [[Morse function|KPQ96640ZJ]] is a Morse–Bott function whose critical components are points.

Example: $f(x, y) = x^2$ on $\R^2$ has $\operatorname{Crit}(f) = \set{x = 0}$, a line.
At $(0, y)$ the Hessian matrix is $\operatorname{diag}(2, 0)$, whose kernel is spanned by $\partial_y$, the tangent direction of the line; so $f$ is Morse–Bott, with critical line of index $0$.

Non-example: $g(x, y) = x^4$ also has $\operatorname{Crit}(g) = \set{x = 0}$, a submanifold, but its Hessian at $(0, y)$ is $0$.
Its kernel is all of $\R^2$, not the tangent line, so the second condition fails.
Another non-example is $h(x, y) = x^2 y^2$, whose critical set is the union of the two axes, which is not a submanifold near the origin.
</document>

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# Inverse transform sampling

## Statement

Let $F \colon \R \to [0, 1]$ be a [[cumulative distribution function|0H8H7ZTHW2]], that is, non-decreasing, right-continuous, with limits $0$ at $-\infty$ and $1$ at $+\infty$.
Let $F^{-1} \colon (0, 1) \to \R$ be its generalized inverse, the [[quantile function|0X60VE98HN]]
$$F^{-1}(u) = \inf \set{x \in \R : F(x) \ge u}.$$
Let $U$ be a [[random variable|0PNRKMRC55]] with the [[uniform distribution|0AREFQ9Z65]] on $(0, 1)$.
Then the random variable $X = F^{-1}(U)$ has cumulative distribution function $F$.

## Remarks

The key fact is that for $u \in (0, 1)$ and $x \in \R$, $F^{-1}(u) \le x$ if and only if $u \le F(x)$; this uses right-continuity of $F$, which makes the infimum attained.
Hence $P(X \le x) = P(U \le F(x)) = F(x)$.
Example: the [[exponential distribution|0E99F51GVB]] with rate $\lambda > 0$ has $F(x) = 1 - e^{-\lambda x}$ for $x \ge 0$, so $F^{-1}(u) = -\ln(1 - u)/\lambda$; since $1 - U$ is also uniform, $-\ln(U)/\lambda$ is exponential too.
Example: for a discrete distribution with values $x_1 < x_2 < \cdots$ and probabilities $p_1, p_2, \ldots$, the method returns $x_k$ when $p_1 + \cdots + p_{k-1} < U \le p_1 + \cdots + p_k$.
The generalized inverse is needed: when $F$ has jumps or flat parts, the ordinary inverse does not exist.
A converse holds when $F$ is continuous: then $F(X)$ is uniform on $(0, 1)$, the probability integral transform.

## Explanations

### L1 human

A computer can produce uniform random numbers between $0$ and $1$.
To get random numbers with any other distribution, feed each uniform number into the inverse of the distribution function.
Values of $u$ near $0$ give values from the lower tail, values near $1$ give the upper tail, and the shape of the distribution comes out exactly right.
For the exponential distribution this means $-\ln(u)/\lambda$.

### L2 human

The method is exact and uses one uniform number per sample, so it is the first choice whenever $F^{-1}$ has a closed form or a fast accurate approximation.
For the normal distribution there is no closed form, and one uses numerical approximations of $F^{-1}$ or other methods such as the Box–Muller transform or the ziggurat algorithm.
Being monotone in $U$, it preserves the order of the uniform inputs, which is useful for antithetic variates, for quasi-Monte Carlo and for coupling arguments.
It also explains copulas: any multivariate distribution is obtained from its copula by applying the marginal quantile functions coordinatewise.
</document>

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# Matrix Wiener–Hopf factorization

## Statement

Let $n \ge 1$ and let $K$ be an $n \times n$ matrix of functions in $L^1(\R)$.
Write $\hat K(\xi) = \int_{\R} K(x)\, e^{i\xi x}\, dx$ entrywise, the [[Fourier transform|DPVM34ADZV]], and let $A(\xi) = I - \hat K(\xi)$.
Assume that $\det A(\xi) \ne 0$ for all $\xi \in \R$, so that $A(\xi)$ is an [[invertible matrix|0EB9N0S33C]] for every real $\xi$.
Let $r(\xi) = (\xi - i)/(\xi + i)$.
Then:

1. There are $n \times n$ matrices $K_+$ and $K_-$ of $L^1(\R)$ functions, with $K_+ = 0$ on $(-\infty, 0)$ and $K_- = 0$ on $(0, \infty)$. There are integers $\kappa_1 \ge \kappa_2 \ge \cdots \ge \kappa_n$ such that
$$A(\xi) = \bigl(I - \hat K_-(\xi)\bigr)\, \operatorname{diag}\bigl(r(\xi)^{\kappa_1}, \ldots, r(\xi)^{\kappa_n}\bigr)\, \bigl(I - \hat K_+(\xi)\bigr), \quad \xi \in \R.$$
Here $\det(I - \hat K_+)$ has no zeros in the closed upper half-plane and $\det(I - \hat K_-)$ has no zeros in the closed lower half-plane.
2. The integers $\kappa_1, \ldots, \kappa_n$, the *partial indices*, are uniquely determined by $A$, and their sum is the [[winding number|0X7SBS8E0F]] of $\xi \mapsto \det A(\xi)$ about $0$.
3. For $1 \le p < \infty$, the operator $\varphi \mapsto \varphi(x) - \int_0^\infty K(x - t)\, \varphi(t)\, dt$ on $L^p(0, \infty)^n$ is invertible if and only if $\kappa_1 = \cdots = \kappa_n = 0$.

## Remarks

The factors themselves are not unique, but the partial indices are.
For $n = 1$ this is the scalar [[Wiener–Hopf factorization|BYKP1V2AQN]], with the single index $\kappa_1$ equal to the winding number.
The proof cannot take logarithms, because matrices do not commute; one instead reduces to a matrix Riemann–Hilbert problem or works in the algebra of matrix functions with $L^1$ Fourier data.

The total index alone does not decide invertibility.
Example: $A = \operatorname{diag}\bigl(r, r^{-1}\bigr)$ has $\det A = 1$, so the winding number of the determinant is $0$.
Its partial indices are $1$ and $-1$, so the corresponding system of [[Wiener–Hopf equations|QYY4NMDY5H]] is not invertible.
The first equation has a range of codimension $1$ and the second a kernel of dimension $1$.
Each entry is admissible, since $r(\xi) = 1 - \frac{2i}{\xi + i}$ and $\frac{2i}{\xi + i}$ is the Fourier transform of the function equal to $2e^{-x}$ for $x > 0$ and to $0$ for $x \le 0$.
The inverse $r^{-1} = 1 + \frac{2i}{\xi - i}$ is admissible in the same way, with $\frac{2i}{\xi - i}$ the transform of a function supported on $(-\infty, 0]$.

## Explanations

### L1 human

A system of Wiener–Hopf equations has a matrix symbol instead of a scalar one.
The symbol can still be split into a forward-looking part, a backward-looking part and a simple diagonal middle part.
The middle part is described by a list of whole numbers, the partial indices, and these decide whether the system has exactly one solution.
Unlike the scalar case, the winding number of the determinant is not enough: the indices can cancel in the sum.

### L2 human

The partial indices are the matrix version of the index of a Fredholm operator, but they carry more information than the index.
They are unstable: a small perturbation of $A$ can change them, while their sum stays the same.
The same factorization governs the matrix [[Riemann–Hilbert problem|RY8YQ9JZF9]] on a closed contour, with the circle in place of the real line.
In applications, such as diffraction by a wedge or elasticity with mixed conditions, finding the factors explicitly is the main difficulty, and only special classes of matrices admit closed forms.
</document>

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# Unconditional convergence

## Summary

A series converges unconditionally when every reordering of its terms still converges.
For real or complex series this is the same as [[absolute convergence|0YBDSJARWN]], by the [[Riemann rearrangement theorem|0E7VBYGFF7]].
In infinite-dimensional Banach spaces the two notions differ, and unconditional convergence is the natural notion for sums without a preferred order.
The exact definition is in [[unconditional convergence|GA7Q813F50]].

## Explanations

### L1 human

Some infinite sums give different answers when you add the terms in a different order.
An unconditionally convergent series is one where the order never matters: every order converges, to the same sum.
For example $1 + \tfrac14 + \tfrac19 + \cdots$ converges in any order, while $1 - \tfrac12 + \tfrac13 - \cdots$ can be reordered to converge to any real number.
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# Weak mixing

## Summary

A [[measure-preserving transformation|556WBDAHNP]] is weakly mixing when the correlations $\mu(T^{-n}A \cap B) - \mu(A)\mu(B)$ tend to $0$ on average over $n$, in absolute value.
It sits strictly between [[mixing|S3RJQ9CX2P]] and [[ergodicity|AZRPHS478B]].
Weak mixing has a clean spectral meaning: the [[Koopman operator|37TRG5DBQC]] has no eigenfunctions other than the constants.
It describes a system with no rotation hidden inside it.

## Explanations

### L1 human

A weakly mixing system mixes sets together at almost all times, though it may fail to mix at some rare times.
The rare bad times have zero density among all times.
Rotations of a circle are not weakly mixing: they turn the space rigidly, and the function $e^{2\pi i x}$ just gets multiplied by a constant at each step.
Weak mixing says that no function on the space behaves in this rigid, periodic way.
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# Harish-Chandra module

## Summary

A Harish-Chandra module is the algebraic shadow of a representation of a real reductive group $G$.
It keeps the vectors that transform in a finite way under a maximal compact subgroup $K$.
On them it keeps the actions of $K$ and of the complexified Lie algebra $\mathfrak g$, and it forgets the topology.
Harish-Chandra showed that an [[admissible representation|SZV8Y806Y7]] is determined, up to a natural equivalence, by its Harish-Chandra module.
This turns questions about infinite-dimensional representations into questions of algebra.

## Explanations

### L1 human

A representation of a group like $\mathrm{SL}_2(\R)$ acts on a huge space of functions, where analysis is hard.
Inside it sit the vectors that only move in a finite-dimensional way under rotations, such as trigonometric polynomials on a circle.
On these vectors the group no longer acts, but its infinitesimal version, the Lie algebra, still does, together with the rotations.
The resulting algebraic object is the Harish-Chandra module.
For $\mathrm{SL}_2(\R)$ it can be drawn as a row of points, one per frequency $n$, with arrows that raise or lower $n$ by $2$.
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# Backward difference operator

## Definition

Let $G$ and $H$ be additive abelian groups, and let $h \in G$.
The *backward difference operator* with step $h$ sends a function $f \colon G \to H$ to the function $\nabla_h f \colon G \to H$ given by
$$\nabla_h f(x) = f(x) - f(x - h) \qquad (x \in G).$$
When $h = 1$, one writes $\nabla$ for $\nabla_1$.
For a sequence $u \colon \Z \to H$, this gives $(\nabla u)_n = u_n - u_{n-1}$.

## Remarks

It is a [[finite difference|7JSQZ126BP]] in the backward direction.
Example: for $f(x) = x^2$ on $\R$, $\nabla_h f(x) = x^2 - (x - h)^2 = 2hx - h^2$.
Non-example: $f(x + h) - f(x)$ is in general not $\nabla_h f(x)$; for $f(x) = x^2$ it equals $2hx + h^2$.
The two operators are related by $\nabla_h f(x) = \Delta_h f(x - h)$, where $\Delta_h$ is the [[forward difference operator|8E0EBA8X07]].
In operator form, $\nabla_h = I - E_{-h}$, with $E_{-h} f(x) = f(x - h)$ a [[shift operator|MBK6AE6734]].
</document>

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# Eberlein–Šmulian theorem

## Statement

Let $X$ be a [[Banach space|06GZ956657]] over $\R$ or $\C$, with the [[weak topology|P6KHVSTPZV]] $\sigma(X, X^*)$, and let $A \subseteq X$.
Then $A$ is [[compact|0R6EWFKRCF]] in the weak topology if and only if the following holds.
Every sequence in $A$ has a subsequence that converges weakly to a point of $A$.

## Remarks

The weak topology of an infinite-dimensional space is not metrizable, so compactness and sequential compactness need not agree a priori.
The theorem says they agree for the weak topology of a Banach space.
For "compact implies sequentially compact", a sequence lies in a separable closed subspace $Y$.
On weakly compact subsets of $Y$ the weak topology is metrizable, because countably many functionals separate the points of $Y$.
The converse direction is harder.
One proof first shows that $A$ is bounded, then uses the [[Banach–Alaoglu theorem|1BXEJXQ2YT]] in $X^{**}$.
It then shows that the weak* closure of the image $J(A)$ under the [[canonical embedding|QKR3D65TTH]] $J$ lies in $J(X)$.
The same equivalences hold for relative compactness: the weak closure of $A$ is weakly compact exactly when every sequence in $A$ has a weakly convergent subsequence with limit in $X$.

Example: the closed unit ball of a [[reflexive space|NN8RWPC8WG]] is weakly compact, so every bounded sequence in a reflexive space has a weakly convergent subsequence.
The analogue fails for the weak* topology: the unit ball of $(\ell^\infty)^*$ is weak*-compact by Banach–Alaoglu, but the evaluations $\delta_n(x) = x_n$ have no weak* convergent subsequence.

## Explanations

### L1 human

In general topology, compactness is defined with open covers, while in metric spaces one can test it with sequences.
The weak topology of a Banach space is not given by a metric, yet the theorem says that sequences still suffice to test compactness.
This makes weak compactness much easier to handle in practice.

### L2 human

The theorem is used in analysis and partial differential equations whenever one extracts weakly convergent subsequences from bounded sequences in reflexive spaces such as $L^p$ for $1 < p < \infty$ or Sobolev spaces.
It also combines with [[James's theorem|G1VWNQSAK5]] and the [[Krein–Smulian theorem|NYVHMQJP3E]] in the theory of weakly compact sets.
</document>

<document source="https://tamnd.github.io/math2/node/2N4RJCW5JK.md" tag="2N4RJCW5JK" kind="example">
# Space of convergent sequences c

## Example

Let $c$ be the set of real [[convergent sequences|0W4CN9E2W5]] $x = (x_n)_{n \ge 1}$, with the norm $\norm{x}_\infty = \sup_n \abs{x_n}$.
Then $c$ is a [[Banach space|06GZ956657]].

Check.
Every convergent sequence is [[bounded|0A8JNBGT2Z]], so $c$ is a linear subspace of the Banach space $\ell^\infty$ of bounded sequences.
It suffices to show that $c$ is closed in $\ell^\infty$.
Let $x^{(k)} \in c$ tend to $x$ in $\ell^\infty$.
For all $k$, $m$ and $n$,
$$\abs{x_m - x_n} \le 2 \norm{x - x^{(k)}}_\infty + \abs{x^{(k)}_m - x^{(k)}_n}.$$
Choose $k$ to make the first term small, then $m, n$ large to make the second term small.
So $x$ is a Cauchy sequence of real numbers, and it converges.

## Remarks

The sequences with limit $0$ form the closed subspace $c_0$ of codimension $1$ in $c$.
The map $x \mapsto (\ell, x_1 - \ell, x_2 - \ell, \dots)$, with $\ell = \lim_n x_n$, is a linear isomorphism of $c$ onto $c_0$.
But $c$ and $c_0$ are not isometric.
The constant sequence $(1, 1, \dots)$ is an extreme point of the unit ball of $c$, while the unit ball of $c_0$ has no extreme points.
The dual of $c$ is isometric to $\ell^1$.

## Explanations

### L1 human

The space $c$ collects all sequences of real numbers that settle down to a limit.
Two sequences are close when they are close at every index.
A uniform limit of sequences that settle down also settles down, so the space has no holes.
</document>

<document source="https://tamnd.github.io/math2/node/2N4Z69K7V6.md" tag="2N4Z69K7V6" kind="definition">
# Interval order

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]], with strict order $<$.
The poset $P$ is an *interval order* if there are real numbers $l_x \le r_x$ for each $x \in P$ such that, for all $x, y \in P$,
$$x < y \iff r_x < l_y.$$
The closed [[interval|0Q1D85B8G6]] $[l_x, r_x]$ is said to represent $x$.

## Remarks

Example: the poset on $\set{a, b, c}$ with $a < b$ as the only strict relation is an interval order.
The intervals $[0, 1]$, $[2, 3]$ and $[0.5, 2.5]$ for $a$, $b$, $c$ work: $1 < 2$ gives $a < b$, and the interval of $c$ meets both others.
Non-example: the poset $\mathbf{2} + \mathbf{2}$ with $a < b$, $c < d$ and no other strict relation is not an interval order.
Intervals would give $r_a < l_b$ and $r_c < l_d$, and also $l_d \le r_a$ and $l_b \le r_c$, so $r_a < l_b \le r_c < l_d \le r_a$, which is impossible.
Fishburn's theorem says that a finite poset is an interval order if and only if it has no subposet isomorphic to $\mathbf{2} + \mathbf{2}$.
Every total order is an interval order: on a finite total order, use the points $[i, i]$.
</document>

<document source="https://tamnd.github.io/math2/node/2N7CR03Q1X.md" tag="2N7CR03Q1X" kind="concept">
# Group von Neumann algebra

## Summary

The group von Neumann algebra $L(G)$ of a [[locally compact group|AHV05ZBCYW]] $G$ is the [[von Neumann algebra|PSFWQVZ0FS]] generated by the left translation operators on $L^2(G)$.
It is the weak closure of the group algebra acting on $L^2(G)$.
For discrete groups it is a main source of examples of factors.
When $G$ is a nontrivial discrete group in which every conjugacy class other than $\set{e}$ is infinite, $L(G)$ is a factor of type $\mathrm{II}_1$.

## Explanations

### L1 human

A group acts on functions on itself by shifting them, and each shift is a unitary operator on the Hilbert space $L^2(G)$.
Taking all these operators, their linear combinations and their limits in a weak sense gives the group von Neumann algebra.
For the integers, Fourier series turn shifting sequences into multiplying functions on the circle by $z^n$.
So $L(\Z)$ is the algebra of bounded measurable functions on the circle, acting by multiplication.
For noncommutative groups such as free groups, the algebra is much more mysterious, and its study is a large part of operator algebra theory.
</document>

<document source="https://tamnd.github.io/math2/node/2NHWCM0DK8.md" tag="2NHWCM0DK8" kind="concept">
# Relaxation oscillation

## Summary

A relaxation oscillation is a periodic motion that alternates long phases of slow change with short, abrupt jumps.
It arises in systems with two time scales, where one variable moves much faster than the other.
Examples include the [[van der Pol oscillator|WE89NFZ84Q]] with a large parameter, electronic circuits that charge and discharge a capacitor, and models of nerve impulses.
See the [[definition|JCMXX84BWT]].

## Explanations

### L1 human

Picture a bucket under a tap that tips over and empties when it is full, then slowly fills again.
The filling is slow and the tipping is sudden, and the cycle repeats with a fixed period.
The graph of such a motion against time looks like a series of plateaus and steep drops rather than a smooth sine wave.
The period is set mainly by the slow phases, since the jumps take almost no time.
</document>

<document source="https://tamnd.github.io/math2/node/2NVEYM6HN9.md" tag="2NVEYM6HN9" kind="definition">
# Jordan–Kinderlehrer–Otto scheme

## Definition

Let $\mathcal{P}_2(\R^n)$ be the set of Borel [[probability measures|059TJN0NZ1]] on $\R^n$ with finite second moment, with the [[Wasserstein distance|VEY29CS26X]] $W_2$.
Let $F \colon \mathcal{P}_2(\R^n) \to (-\infty, +\infty]$ be a functional, let $\tau > 0$ be a time step, and let $\rho_0 \in \mathcal{P}_2(\R^n)$ with $F(\rho_0) < \infty$.
The *Jordan–Kinderlehrer–Otto scheme* for $F$ with step $\tau$ is a sequence $(\rho_k^\tau)_{k \ge 0}$ in $\mathcal{P}_2(\R^n)$ with $\rho_0^\tau = \rho_0$.
For each $k \ge 0$, the measure $\rho_{k+1}^\tau$ is required to minimize the functional
$$\rho \mapsto F(\rho) + \frac{1}{2\tau} W_2(\rho, \rho_k^\tau)^2$$
over $\mathcal{P}_2(\R^n)$.
The associated discrete flow is the piecewise constant curve $\rho^\tau(t) = \rho_k^\tau$ for $t \in [k\tau, (k + 1)\tau)$.

## Remarks

Example: let $\beta > 0$, let $\Psi \colon \R^n \to [0, \infty)$ be smooth with $\abs{\nabla \Psi(x)} \le C(\Psi(x) + 1)$ for a constant $C$, and let
$$F(\rho) = \int \Psi \rho \, dx + \beta^{-1} \int \rho \log \rho \, dx$$
for $\rho$ with a density, and $F(\rho) = +\infty$ otherwise.
Jordan, Kinderlehrer and Otto proved that minimizers exist at each step, and that $\rho^\tau$ converges, as $\tau \to 0$, to the solution of the Fokker–Planck equation $\partial_t \rho = \operatorname{div}(\rho \nabla \Psi) + \beta^{-1} \Delta \rho$ with initial datum $\rho_0$.
With $\Psi = 0$ and $\beta = 1$ this gives the heat equation $\partial_t \rho = \Delta \rho$.

Non-example: replacing $W_2$ by the $L^2$ distance between densities gives a different scheme, the implicit Euler scheme for the $L^2$ gradient flow.
For the entropy $\int \rho \log \rho$ this flow is $\partial_t \rho = -(\log \rho + 1)$, which is not the heat equation.

The step is the Wasserstein analogue of the implicit Euler step $x_{k+1} \in \operatorname{argmin}_x \left( f(x) + \frac{1}{2\tau} \abs{x - x_k}^2 \right)$ for the gradient flow $\dot x = -\nabla f(x)$ in $\R^n$.
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# Poncelet–Steiner theorem

## Statement

Let $S$ be a set of points in the Euclidean plane, and suppose that one [[circle|060W5CBZP6]] and its centre are drawn in the plane.
Every point that can be constructed from $S$ with [[straightedge and compass|0DAECMZVRT]] can also be constructed from $S$ and the given circle with straightedge alone.

## Remarks

As for compass-only constructions, a circle counts as known when its centre and one of its points are known.
The proof shows how to find, with straightedge alone and the help of the fixed circle, the intersection points of a line and a circle, and of two circles.
Jean-Victor Poncelet stated the result in 1822, and Jakob Steiner proved it in 1833.
The centre is needed: with straightedge alone one cannot find the centre of a single given circle.
With straightedge alone and no circle, one obtains only points with coordinates in the field generated by the coordinates of $S$.
For example, $\sqrt 2$ cannot be constructed from $0$ and $1$ in this way.

## Explanations

### L1 human

A compass is needed only once.
If one circle and its centre are already on the page, a ruler alone can do every ruler-and-compass construction.
The single circle supplies all the square roots that the compass would otherwise give.

### L2 human

The theorem and the [[Mohr–Mascheroni theorem|G9VT2J3ZVK]] show that the constructible points depend on the algebraic operations available, not on the tools themselves.
Rational operations come from the straightedge, and square roots need some circle.
</document>

<document source="https://tamnd.github.io/math2/node/2P0V6X8GZ3.md" tag="2P0V6X8GZ3" kind="definition">
# Horn hypergeometric series

## Definition

Let $(A_{m, n})_{m, n \ge 0}$ be complex numbers with $A_{0, 0} = 1$.
Consider the [[formal power series|033WGN00BF]] $\sum_{m, n \ge 0} A_{m, n} x^m y^n$.
It is a *Horn hypergeometric series* if there are rational functions $f$ and $g$ of two variables such that for all $m, n \ge 0$:

1. $A_{m + 1, n} = f(m, n)\, A_{m, n}$;
2. $A_{m, n + 1} = g(m, n)\, A_{m, n}$.

Write $f = P / Q$ and $g = R / S$ with polynomials $P, Q, R, S$ in lowest terms.
The *order* of the series is the largest of the degrees of $P, Q, R, S$.

## Remarks

The two conditions mean that the ratios of neighbouring coefficients are rational in $(m, n)$, as for the Gauss [[hypergeometric series|428M453CR2]] in one variable.
Example: Appell's $F_1(a; b, b'; c; x, y)$ has $A_{m, n} = \frac{(a)_{m + n} (b)_m (b')_n}{(c)_{m + n}\, m!\, n!}$.
Then $f(m, n) = \frac{(a + m + n)(b + m)}{(c + m + n)(m + 1)}$ and $g(m, n) = \frac{(a + m + n)(b' + n)}{(c + m + n)(n + 1)}$, both rational, so $F_1$ is a Horn series of order $2$.
Non-example: $A_{m, n} = 2^{m^2}$ gives $A_{m + 1, n} / A_{m, n} = 2^{2m + 1}$, which is not a rational function of $m$, so the first clause fails.
Horn's list contains $34$ essentially distinct convergent series of order $2$: $14$ complete ones, among them the four Appell series, and $20$ confluent ones.
</document>

<document source="https://tamnd.github.io/math2/node/2P2184B6HE.md" tag="2P2184B6HE" kind="concept">
# Linear logic

## Summary

Linear logic, introduced by Jean-Yves Girard in 1987, is a logic in which hypotheses are resources that must be used exactly once.
It drops the structural rules of weakening and contraction, and allows them back only for formulas marked with the exponentials $!$ and $?$.
As a result, conjunction and disjunction split into multiplicative and additive versions.
It has applications in proof theory, the semantics of programming languages and the theory of concurrency.

## Explanations

### L1 human

In ordinary logic, a hypothesis can be used as often as you like, or not at all.
In linear logic, using a hypothesis consumes it, like spending a coin.
With one euro you can buy a coffee, and with one euro you can buy a tea, but one euro does not buy both.
Linear logic has a separate connective for each of these situations: "I can choose either" and "I have both at once".
</document>

<document source="https://tamnd.github.io/math2/node/2P3Y2FRCSZ.md" tag="2P3Y2FRCSZ" kind="concept">
# Free algebra

## Summary

The free algebra $k\langle X \rangle$ on a set $X$ over a [[commutative ring|0310REJJY8]] $k$ is the algebra of polynomials in noncommuting variables from $X$.
It is the free object among [[associative algebras|3CTW1P4HBP]] over $k$: any map from $X$ into an algebra extends uniquely to an algebra homomorphism.
So every algebra generated by a set $X$ is a quotient of $k\langle X \rangle$, and algebras are often given by generators and relations in this way.
It is the noncommutative analogue of the [[polynomial ring|0SDC934R52]] and is isomorphic to the [[tensor algebra|P4J5JZDW0B]] of the free module with basis $X$.

## Explanations

### L1 human

Take letters $x, y$ and form all words in them, such as $xy$, $yx$, $xxy$.
Unlike in ordinary polynomials, $xy$ and $yx$ are different words.
The free algebra consists of finite linear combinations of words, like $3xy - 2yx + 5$, multiplied by joining words end to end.
Every other algebra with two chosen elements is obtained from this one by imposing rules, for example $xy = yx$ gives back the polynomial ring $k[x, y]$.
</document>

<document source="https://tamnd.github.io/math2/node/2P4RDS0A6X.md" tag="2P4RDS0A6X" kind="concept">
# List of integral transforms

## Summary

Each [[integral transform|YCK709QB06]] has the form $(Tf)(u) = \int_{t_1}^{t_2} f(t)\, K(t, u)\, dt$, and it is fixed by its [[kernel|9M0NNNFJGM]] and its interval of integration.
The table below lists common transforms with one standard normalisation; books differ in constants and signs.

| Transform | Kernel $K(t, u)$ | Interval |
|---|---|---|
| [[Laplace transform|0V49JEZW06]] | $e^{-ut}$ | $(0, \infty)$ |
| [[Two-sided Laplace transform|6J2AGD0BYY]] | $e^{-ut}$ | $(-\infty, \infty)$ |
| [[Fourier transform|DPVM34ADZV]] | $e^{-2\pi i u t}$ | $(-\infty, \infty)$ |
| [[Mellin transform|N5PVGRJE1T]] | $t^{u - 1}$ | $(0, \infty)$ |
| [[Hankel transform|2P9YWDGKC4]] | $t\, J_\nu(ut)$ | $(0, \infty)$ |
| [[Stieltjes transform|RG1HFAYD98]] | $1/(u + t)$ | $(0, \infty)$ |
| [[Hilbert transform|NSCEBXM7ZM]] | $\frac{1}{\pi}\frac{1}{u - t}$ (principal value) | $(-\infty, \infty)$ |
| [[Weierstrass transform|5HYB6DDZH8]] | $\frac{1}{\sqrt{4\pi}} e^{-(u - t)^2/4}$ | $(-\infty, \infty)$ |
| [[Hartley transform|WEHYECYAVJ]] | $\frac{1}{\sqrt{2\pi}}(\cos ut + \sin ut)$ | $(-\infty, \infty)$ |

Here $J_\nu$ is the Bessel function of the first kind of order $\nu$.
Some transforms do not fit this one-line form.
The [[Radon transform|VJNZJ352PC]] integrates over lines, and the [[Abel transform|QMW1MWQH6J]] has a variable domain of integration.

## Explanations

### L1 human

All these transforms follow one recipe: multiply $f$ by a kernel and integrate.
What changes is the kernel and the range of integration.
Exponential kernels suit differential equations, power kernels $t^{u-1}$ suit problems with scaling, and Bessel kernels suit problems with circular symmetry.
For example, the Mellin transform of $e^{-t}$ is the gamma function $\Gamma(u)$.
</document>

<document source="https://tamnd.github.io/math2/node/2P90CV9QKH.md" tag="2P90CV9QKH" kind="concept">
# Fourier–Mukai transform

## Summary

A Fourier–Mukai transform is an exact functor between the [[derived categories of coherent sheaves|2VP2D51KB8]] of two varieties $X$ and $Y$.
It is given by an object on $X \times Y$ called the kernel.
Mukai showed that the Poincaré bundle gives an equivalence between the derived categories of an abelian variety and of its dual.
By a theorem of Orlov, all fully faithful exact functors between such derived categories of smooth projective varieties are of this form.

## Explanations

### L1 human

The classical Fourier transform turns a function $f(x)$ into a function of a new variable by integrating $f(x)$ against a kernel $e^{-2\pi i x y}$.
A Fourier–Mukai transform does the same with sheaves instead of functions: pull back, multiply by a kernel on the product, and push forward, which is the analogue of integration.
Different varieties can have equivalent derived categories, and Fourier–Mukai transforms are the tool that detects and produces such equivalences.
The simplest kernel is the structure sheaf of the diagonal, which gives the identity functor.
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# Hankel transform

## Summary

The Hankel transform of order $\nu$ of a function $f$ on $(0, \infty)$ is $\int_0^\infty f(r)\, J_\nu(kr)\, r\, dr$, where $J_\nu$ is the [[Bessel function of the first kind|CZCMRDWDYW]].
It is the radial part of the [[Fourier transform|DPVM34ADZV]].
The Fourier transform of a radial function in the plane is a Hankel transform of order $0$, and in higher dimensions other orders appear.
With this normalization it is its own inverse.
It is used to solve problems with circular or cylindrical symmetry, such as heat flow and vibrations of a drum.

## Explanations

### L1 human

The Fourier transform writes a function as a sum of plane waves.
For a function that only depends on the distance to a centre, circular waves are better suited, and their profiles are Bessel functions.
The Hankel transform measures how much of each circular wave of frequency $k$ the function contains.
For example, the Gaussian $e^{-r^2/2}$ has the Hankel transform $e^{-k^2/2}$ of order $0$, the same shape again.
</document>

<document source="https://tamnd.github.io/math2/node/2PANNSQ3G6.md" tag="2PANNSQ3G6" kind="statement">
# Lempert's theorem on convex domains

## Statement

Let $\Omega \subseteq \C^n$ be a bounded [[convex|02AS45T3H9]] domain.
Let $k_\Omega$ be its [[Kobayashi pseudodistance|DY8QQ5GTTX]] and $c_\Omega$ its [[Carathéodory pseudodistance|SYNST4W3FR]], both normalised to equal the Poincaré distance on the unit disc.
Then:

1. $c_\Omega(p, q) = k_\Omega(p, q)$ for all $p, q \in \Omega$;
2. for all $p \ne q$ in $\Omega$ there is a [[complex geodesic|275YT975A8]] $\varphi \colon \mathbb{D} \to \Omega$ whose image contains $p$ and $q$.

## Remarks

László Lempert proved the theorem in 1981.
The infinitesimal Kobayashi–Royden and Carathéodory–Reiffen metrics of $\Omega$ also agree.
Proof idea: an extremal disc through $p$ and $q$ exists by a normal families argument.
For smoothly bounded strongly convex domains, Lempert studied the boundary behaviour of extremal discs and constructed, for each one, a holomorphic left inverse $\Omega \to \mathbb{D}$.
This left inverse shows that the Carathéodory distance is at least the Kobayashi distance, and the general convex case follows by exhaustion.
Example: for the unit ball $B$ and $p = 0$, both distances equal $\tanh^{-1}\abs{q}$, and the complex geodesics through $0$ include the maps $\zeta \mapsto \zeta v$ for unit vectors $v$.
Convexity cannot simply be dropped: for an annulus $A \subset \C$, which is not simply connected, one has $c_A(p, q) < k_A(p, q)$ for $p \ne q$.

## Explanations

### L1 human

There are two natural ways to measure distances in a domain of $\C^n$ using holomorphic maps.
One uses discs mapped into the domain, the other uses functions from the domain to the disc.
In general the second way gives smaller numbers.
Lempert's theorem says that for convex domains both ways give the same answer, and that the distance between two points is realised by a single holomorphic disc through them.

### L2 human

The theorem makes the invariant geometry of convex domains computable: distances are realised by extremal discs, which solve a variational problem with a rich structure.
It is used to study biholomorphic maps of convex domains, boundary behaviour of holomorphic maps, and the [[pluricomplex Green function|3MQVKVGYYY]], which on convex domains is determined by the Kobayashi distance.
For general pseudoconvex domains the two distances can differ, so the theorem singles out convexity, and later work extended it to strongly linearly convex domains.
</document>

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# Group cohomology

## Definition

Let $G$ be a [[group|G5S1M9]].
A *$G$-module* is an [[abelian group|082GHNPJ53]] $M$ with an [[action|0WCY42EV9V]] of $G$ such that $g(m + m') = gm + gm'$ for all $g \in G$ and $m, m' \in M$.
Equivalently, it is a module over the [[group ring|MNP2HBWB5Q]] $\Z[G]$.
Let every element of $G$ act on $\Z$ as the identity.
For $n \ge 0$, the $n$-th *cohomology group* of $G$ with coefficients in $M$ is
$$H^n(G, M) = \operatorname{Ext}^n_{\Z[G]}(\Z, M),$$
where $\operatorname{Ext}$ is the [[Ext functor|4P7GBWJ4RR]].

## Remarks

With the [[bar resolution|8BGJ6J9PYF]], $H^n(G, M)$ is the cohomology of the complex of functions $f \colon G^n \to M$ with
$$(df)(g_1, \dots, g_{n+1}) = g_1 f(g_2, \dots, g_{n+1}) + \sum_{i=1}^{n} (-1)^i f(g_1, \dots, g_i g_{i+1}, \dots, g_{n+1}) + (-1)^{n+1} f(g_1, \dots, g_n).$$
So $H^0(G, M) = M^G$, the elements fixed by $G$.
And $H^1(G, M)$ is the group of crossed homomorphisms, the maps with $f(gh) = f(g) + g f(h)$, modulo those of the form $f(g) = gm - m$.
If every element of $G$ acts on $M$ as the identity, then $H^1(G, M) = \Hom(G, M)$.
Example: let $G = \Z/2\Z = \set{1, t}$ act on $M = \Z$ by $t m = -m$; this is a $G$-module, since $m \mapsto -m$ is additive.
Then $H^1(G, M) \cong \Z/2\Z$.
Indeed, a crossed homomorphism is determined by $f(t) \in \Z$, with $f(1) = 0$ and no further condition.
The principal ones have $f(t) = tm - m = -2m$.
Non-example: $\Z/2\Z$ acting on $\Z$ by $m \mapsto 1 - m$ is an action, but not by additive maps, since $0$ is not fixed, so $\Z$ is not a $G$-module for it.
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<document source="https://tamnd.github.io/math2/node/2PC3XY0PYZ.md" tag="2PC3XY0PYZ" kind="statement">
# Parallel postulate

## Statement

Let two [[lines|0MP9BCF6DP]] in a plane be cut by a third line, the transversal.
Suppose that the two interior [[angles|0GJAJC6TNZ]] on one side of the transversal have a sum less than two right angles.
Then the two lines, extended far enough, meet on that side of the transversal.

## Remarks

This is the fifth postulate of Book I of [[Euclid's Elements|83WHDAV68J]], an axiom and not a theorem.
Over the axioms of [[absolute geometry|VKFPCSRNDH]], it is equivalent to [[Playfair's axiom|PR7Q3QMEHZ]], and also to the statement that the angle sum of every triangle is two right angles.
It is independent of the other axioms: the Euclidean plane satisfies it, and the [[hyperbolic plane|GN4N94GQ0Z]] satisfies all axioms of absolute geometry but not this one.
Many attempts to prove it from the other postulates, such as those of Saccheri and Lambert, led in the end to the discovery of non-Euclidean geometry.

## Explanations

### L1 human

Take two lines and a third line that crosses both.
If the two lines lean toward each other on one side, they meet on that side.
This sounds obvious, but it cannot be proved from Euclid's other rules.

### L2 human

The postulate is the dividing line between Euclidean and hyperbolic geometry.
Results such as the angle sum $180^\circ$ of a triangle, the existence of rectangles and the theory of similar triangles depend on it.
Results of absolute geometry, such as the congruence criteria for triangles and the exterior angle theorem, do not.
In proofs one usually uses the equivalent Playfair's axiom, which is easier to apply.
</document>

<document source="https://tamnd.github.io/math2/node/2PCPEX85PX.md" tag="2PCPEX85PX" kind="definition">
# Temperley–Lieb algebra

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] with identity, let $\delta \in R$ and let $n \ge 1$.
The *Temperley–Lieb algebra* $TL_n(\delta)$ is the [[associative algebra|3CTW1P4HBP]] over $R$ generated by elements $e_1, \dots, e_{n-1}$, subject to the relations:

1. $e_i^2 = \delta e_i$ for $1 \le i \le n - 1$.
2. $e_i e_{i+1} e_i = e_i$ and $e_{i+1} e_i e_{i+1} = e_{i+1}$ for $1 \le i \le n - 2$.
3. $e_i e_j = e_j e_i$ for $\abs{i - j} > 1$.

## Remarks

The algebra $TL_n(\delta)$ is a free $R$-module of rank equal to the [[Catalan number|0YWZJEABB2]] $C_n = \frac{1}{n+1}\binom{2n}{n}$, with a basis given by non-crossing planar pairings of $2n$ points.
Example: $TL_3(\delta)$ has rank $C_3 = 5$, with basis $1, e_1, e_2, e_1 e_2, e_2 e_1$.
The check: every longer word reduces to one of these, for instance $e_1 e_2 e_1 = e_1$ and $e_1 e_2 e_2 = \delta e_1 e_2$.
Non-example: in the group algebra $R[S_3]$, the transpositions $s_1 = (1\,2)$ and $s_2 = (2\,3)$ satisfy $s_1 s_2 s_1 = (1\,3) \ne s_1$, so they do not satisfy relation 2.
For suitable $\delta$, $TL_n(\delta)$ is a quotient of the [[Hecke algebra|5J2P2AGY4N]] of the symmetric group $S_n$.
</document>

<document source="https://tamnd.github.io/math2/node/2PCVRG9NBN.md" tag="2PCVRG9NBN" kind="concept">
# Born series

## Summary

The Born series is the expansion of a scattered wave in powers of the scattering potential.
It is obtained by iterating the [[Lippmann–Schwinger equation|MW5R19P2X4]] $\psi = \phi + G_0 V \psi$, which gives $\psi = \phi + G_0 V \phi + G_0 V G_0 V \phi + \cdots$.
It is the [[Liouville–Neumann series|H1HWSWME6B]] of this integral equation.
So it converges when the operator $G_0 V$ is small, for example for weak potentials or at high energy.
Its first term beyond $\phi$ is the Born approximation, the standard first estimate of scattering amplitudes.

## Explanations

### L1 human

A wave meets an obstacle and is scattered once, then the scattered wave is scattered again, and so on.
The Born series adds up these contributions: no scattering, one scattering event, two events, and so on.
When the obstacle is weak, each extra event is much smaller than the last, so the first one or two terms already give a good answer.
</document>

<document source="https://tamnd.github.io/math2/node/2PEH9MPTAR.md" tag="2PEH9MPTAR" kind="statement">
# Uniqueness of adjoints

## Statement

Let $F \colon \mathcal{C} \to \mathcal{D}$ be a functor.

1. If $G$ and $G'$ from $\mathcal{D}$ to $\mathcal{C}$ are both [[right adjoint|TZHF6ZM92S]] to $F$, then there is a natural isomorphism $G \cong G'$.
2. Dually, if $F$ and $F'$ from $\mathcal{C}$ to $\mathcal{D}$ are both left adjoint to a functor $G$, then $F \cong F'$.

## Remarks

For each $d$, the adjunctions give bijections $\Hom(c, G d) \cong \Hom(F c, d) \cong \Hom(c, G' d)$, natural in $c$.
By the Yoneda lemma, these come from a unique isomorphism $G d \cong G' d$.
Naturality in $d$ makes these isomorphisms a [[natural isomorphism|KYCPCC57FT]].
The isomorphism is the unique one compatible with the two adjunctions.

Example: the free group on a set can be built from reduced words or as a quotient of a free monoid.
Both constructions are left adjoint to the forgetful functor, so they give isomorphic functors.
Likewise the abelianization $H \mapsto H/[H, H]$ is determined up to isomorphism by being left adjoint to the inclusion of abelian groups into groups.

## Explanations

### L1 human

An adjoint of a functor, when it exists, is unique up to isomorphism.
So it does not matter which construction one uses for a free object or a quotient, as long as it has the right universal property.
This is why one may speak of the left adjoint of a functor.

### L2 human

The proposition is a form of the uniqueness of objects with a universal property, applied object by object.
It lets one define a functor by asking that it be an adjoint, as for the tensor product, the abelianization or the Stone–Čech compactification.
It also shows that adjunctions compose up to isomorphism: if $F \dashv G$ and $F' \dashv G'$, then $F' F \dashv G G'$, and any other left adjoint of $G G'$ is isomorphic to $F' F$.
</document>

<document source="https://tamnd.github.io/math2/node/2PFQY2RZBC.md" tag="2PFQY2RZBC" kind="concept">
# Cofinal subset

## Summary

A subset $C$ of a preordered set $P$ is cofinal when every element of $P$ lies below some element of $C$.
A cofinal subset reaches arbitrarily far up in $P$.
For example, $C$ and $P$ have the same upper bounds, so a supremum of $P$ can be computed from $C$ alone.
The least cardinality of a cofinal subset is the [[cofinality|BAWHE1W6MH]] of $P$.

## Explanations

### L1 human

Think of $P$ as heights you can reach, and $C$ as a set of ladders.
The set $C$ is cofinal when, for any height, some ladder goes at least that high.
In the natural numbers, the even numbers are cofinal, and so are the powers of $2$.
A finite set of natural numbers is never cofinal in $\N$, because nothing in it is above its largest element plus one.
</document>

<document source="https://tamnd.github.io/math2/node/2PHQQ01C0F.md" tag="2PHQQ01C0F" kind="definition">
# Rectifiable set

## Definition

Let $n$ and $m$ be integers with $1 \le m \le n$, and let $\mathcal{H}^m$ be the $m$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] on $\R^n$.
A set $E \subseteq \R^n$ is *$m$-rectifiable* if there are [[Lipschitz maps|0TMZ21WKCN]] $f_i \colon \R^m \to \R^n$, for $i = 1, 2, \dots$, such that
$$\mathcal{H}^m\Big(E \setminus \bigcup_{i=1}^{\infty} f_i(\R^m)\Big) = 0.$$
A set $E \subseteq \R^n$ is *purely $m$-unrectifiable* if $\mathcal{H}^m(E \cap F) = 0$ for every $m$-rectifiable set $F \subseteq \R^n$.

## Remarks

This is the countable rectifiability of Federer and Mattila.
Some authors also require $\mathcal{H}^m(E) < \infty$, or use $C^1$ maps instead of Lipschitz maps; the second change gives the same class of sets.
Example: the unit circle $S \subseteq \R^2$ is $1$-rectifiable, since the Lipschitz map $f(t) = (\cos t, \sin t)$ satisfies $f(\R) = S$.
Non-example: the unit square $Q = [0,1]^2$ is not $1$-rectifiable.
Each set $f_i(\R)$ has $\sigma$-finite $\mathcal{H}^1$ measure, so it has Lebesgue measure zero in $\R^2$.
Hence $Q \setminus \bigcup_i f_i(\R)$ has positive area, and so its $\mathcal{H}^1$ measure is infinite, not zero.
The image of a [[rectifiable curve|0HATW2VWAZ]] is $1$-rectifiable.
</document>

<document source="https://tamnd.github.io/math2/node/2PJECK6ZGW.md" tag="2PJECK6ZGW" kind="statement">
# Bulirsch–Stoer algorithm

## Statement

Let $f \colon \R \times \R^d \to \R^d$ be infinitely differentiable, and consider the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$.
Fix a step $H > 0$, an even integer $n$, and put $h = H / n$.
Gragg's modified midpoint rule computes
$$z_0 = y_0, \qquad z_1 = z_0 + h f(t_0, z_0), \qquad z_{m+1} = z_{m-1} + 2 h f(t_0 + m h, z_m) \quad (1 \le m \le n - 1),$$
and sets $S(h) = z_n$.
Then $S(h)$ has an asymptotic expansion in even powers of $h$:
$$y(t_0 + H) - S(h) = e_1 h^2 + e_2 h^4 + \cdots + e_k h^{2k} + O(h^{2k+2})$$
for each $k \ge 1$, where the vectors $e_j$ do not depend on $h$.

The Bulirsch–Stoer algorithm computes $S(h_i)$ for $h_i = H / n_i$ with increasing even integers $n_1 < n_2 < \cdots$.
It extrapolates the values $T_{i,1} = S(h_i)$ to $h = 0$ by polynomial (or rational) extrapolation in $h^2$.
With polynomial extrapolation, the entry $T_{j,k}$ of the extrapolation table is a one-step method of order $2k$ in $H$.

## Remarks

The expansion in $h^2$ is due to Gragg, and it comes from the symmetry of the two-step midpoint rule.
For general $n$ the expansion of $z_n$ also contains terms with the factor $(-1)^n$.
Taking $n$ even makes these factors equal to $1$, so the coefficients $e_j$ are the same for all the $h_i$.
[[Richardson extrapolation|WMTZG71K2Y]] in the variable $h^2$ then removes one even power of $h$ in each column of the table.
For example, with $n_1 = 2$ and $n_2 = 4$, the value $T_{2,2} = S(h_2) + \frac{S(h_2) - S(h_1)}{(n_2/n_1)^2 - 1} = \frac{4 S(h_2) - S(h_1)}{3}$ has local error $O(H^5)$, so it is a method of order $4$.
Bulirsch and Stoer used rational extrapolation, and Gragg also proposed a final smoothing step; the order statement is the same.

## Explanations

### L1 human

The method takes one big step several times, each time cut into a different number of small midpoint steps.
The answers get better as the small steps shrink, and the way they improve is very regular.
One fits a curve through the answers and reads off its value at step size zero.
This gives very accurate results for smooth problems at a modest cost.

### L2 human

The algorithm is an extrapolation method: the order is raised by adding rows to the table, not by changing the base method.
The difference between the last two diagonal entries of the table estimates the error.
This estimate drives the choice of $H$ and of the number of rows, as in [[adaptive step size|6MREWBZR7R]] control.
It works best for smooth, non-stiff problems with tight tolerances; for stiff problems one uses an extrapolated linearly implicit method instead.
</document>

<document source="https://tamnd.github.io/math2/node/2PKHANZ8JS.md" tag="2PKHANZ8JS" kind="statement">
# Hahn decomposition theorem

## Statement

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]] and let $\nu$ be a [[signed measure|VG2P7R91Z7]] on it.
A set $A \in \Sigma$ is *positive* for $\nu$ if $\nu(E) \ge 0$ for every $E \in \Sigma$ with $E \subseteq A$, and *negative* if $\nu(E) \le 0$ for all such $E$.
Then there are disjoint sets $P, N \in \Sigma$ with $P \cup N = X$ such that $P$ is positive and $N$ is negative for $\nu$.
If $P', N'$ is another such pair, then $P \triangle P' = N \triangle N'$ is null for $\nu$: every measurable subset of it has $\nu$-measure $0$.

## Remarks

Assume $\nu$ does not take the value $+\infty$; otherwise apply the argument to $-\nu$.
Let $s$ be the supremum of $\nu(A)$ over positive sets $A$.
A maximising sequence of positive sets has a positive union $P$ with $\nu(P) = s$.
Suppose $N = X \setminus P$ contained a set of strictly positive measure.
Then a lemma gives a positive subset of $N$ of strictly positive measure, and adding it to $P$ contradicts maximality.

Example: let $\nu(E) = \int_E f \, d\mu$ with $f \in L^1(\mu)$ real.
Then $P = \set{f \ge 0}$ and $N = \set{f < 0}$ is a Hahn decomposition.
The set $\set{f = 0}$ can be put into either part, which shows that the decomposition is unique only up to null sets.

## Explanations

### L1 human

A signed measure has regions where it is positive and regions where it is negative.
The theorem says that one can split the whole space into two measurable pieces, one on which every part counts positively and one on which every part counts negatively.
The cut is not unique, but the possible cuts differ only by sets that do not matter for $\nu$.

### L2 human

The Hahn decomposition is the first step towards the [[Jordan decomposition|WAGJVK2NWE]] $\nu = \nu^+ - \nu^-$, which reduces questions about signed measures to questions about measures.
It is also used in the proof of the [[Radon–Nikodym theorem|1ZSZS5G1Q4]], applied to signed measures of the form $\nu - c \mu$.
</document>

<document source="https://tamnd.github.io/math2/node/2PMZ7GS0QY.md" tag="2PMZ7GS0QY" kind="definition">
# Transfinite diameter

## Definition

Let $K$ be a [[compact set|0R6EWFKRCF]] in $\C$.
For an integer $n \ge 2$, let
$$\delta_n(K) = \max \set{ \prod_{1 \le i < j \le n} \abs{z_i - z_j}^{2/(n(n-1))} : z_1, \dots, z_n \in K }.$$
The *transfinite diameter* of $K$ is $\delta(K) = \lim_{n \to \infty} \delta_n(K)$.

## Remarks

The maximum exists because $K^n$ is compact and the product is continuous.
The sequence $(\delta_n(K))$ is nonincreasing, so the limit exists.
To see this, drop one point from an optimal $(n+1)$-tuple in each of the $n + 1$ possible ways and multiply the resulting $n + 1$ inequalities.
The number $\delta_2(K)$ is the diameter of $K$.

Example: let $K$ be the unit circle and let $\omega = e^{2 \pi i / n}$.
The square of the Vandermonde determinant of $1, \omega, \dots, \omega^{n-1}$ has absolute value $n^n$, so these points give the product $n^{n/2}$.
By Hadamard's inequality no $n$ points of the closed unit disc give more, so $\delta_n(K) = n^{1/(n-1)}$, which tends to $1$.

Non-example: the limit of $\delta_n(K)$ is not the diameter in general.
For the unit circle $\delta_2 = 2$, while $\delta(K) = 1$.

A finite set $K$ has $\delta(K) = 0$, since for $n$ larger than the number of points two of the $z_i$ coincide.
</document>

<document source="https://tamnd.github.io/math2/node/2PSDHH34BJ.md" tag="2PSDHH34BJ" kind="statement">
# Ziggurat algorithm

## Statement

Let $f \colon [0, \infty) \to (0, \infty)$ be continuous, non-increasing and integrable.
Let $n \ge 2$, let $0 = x_0 < x_1 < \cdots < x_{n-1} = r$, and let $v > 0$ be such that:

1. $x_i \bigl(f(x_{i-1}) - f(x_i)\bigr) = v$ for $1 \le i \le n-1$;
2. $r f(r) + \int_r^\infty f(x)\, dx = v$.

Define the layers $L_i = [0, x_i] \times [f(x_i), f(x_{i-1})]$ for $1 \le i \le n-1$ and the base $L_n = \set{(x, y) : 0 \le x \le r,\ 0 \le y \le f(r)} \cup \set{(x, y) : x > r,\ 0 \le y \le f(x)}$.
The *ziggurat algorithm* repeats the following trial until it returns a value.
It chooses $i \in \set{1, \ldots, n}$ uniformly at random, draws a point $(X, Y)$ uniformly from $L_i$, and returns $X$ if $Y \le f(X)$.
Then the returned value has [[probability density|0ZY9DX8FBF]] $f / \int_0^\infty f$, and each trial returns a value with probability $\int_0^\infty f / (n v)$.

## Remarks

Each layer has area $v$ by clauses 1 and 2, and the layers overlap only in boundary segments of measure zero.
So a uniform choice of layer followed by a uniform point in it gives a point uniform on the union $Z = L_1 \cup \cdots \cup L_n$.
Since $f$ is non-increasing, $Z$ contains the region $\set{(x, y) : x \ge 0,\ 0 \le y \le f(x)}$ under the graph.
A uniform point of $Z$ conditioned to lie under the graph is uniform there, and its first coordinate has density proportional to $f$.
This is [[rejection sampling|R1QBNFTHFP]] with a uniform envelope on $Z$.
The speed comes from a shortcut: in layer $i \le n-1$, if $X < x_{i-1}$ then $Y \le f(x_{i-1}) \le f(X)$, so the point is accepted without evaluating $f$.
For large $n$, such as $n = 128$ or $256$, almost all trials end this way, after one random integer, one multiplication and one comparison.
A uniform point of the base $L_n$ is drawn as follows.
With probability $r f(r)/v$ it is a uniform point of the rectangle $[0, r] \times [0, f(r)]$.
Otherwise $X$ is drawn from the tail density proportional to $f$ on $(r, \infty)$, which needs a separate tail sampler, and $Y$ is uniform on $[0, f(X)]$.
For the standard [[normal distribution|05AQE0QR0D]], one samples $\abs{Z}$ with $f(x) = e^{-x^2/2}$ and attaches a random sign.
The numbers $x_i$, $r$ and $v$ are computed once, by solving the equations of clauses 1 and 2 numerically.

## Explanations

### L1 human

Picture the area under a decreasing density curve covered by a stack of horizontal slabs of equal area, like a stepped pyramid, the ziggurat.
Pick a slab at random and a random point in it.
Most of each slab lies completely under the curve, and a point there is accepted at once.
Only near the right edge of a slab, or in the bottom slab with its infinite tail, is extra work needed.
The result is an exact sampler that, most of the time, costs almost nothing.

### L2 human

The algorithm, due to Marsaglia and Tsang, is a form of rejection sampling with a piecewise constant envelope that fits the density tightly.
It is used for normal and exponential random numbers in many numerical libraries.
For symmetric densities such as the normal, it samples the right half and attaches a random sign; often the sign comes from a spare bit of the same random integer.
Its exactness depends on good uniform variates and an exact tail sampler; the tail of the normal is usually handled by a method of Marsaglia based on exponential variates.
</document>

<document source="https://tamnd.github.io/math2/node/2PVQY9PCT4.md" tag="2PVQY9PCT4" kind="statement">
# Elliptic regularity

## Statement

Let $U \subseteq \R^n$ be a bounded open set and let $m \ge 0$ be an integer.
Let
$$L u = -\sum_{i,j=1}^{n} \partial_j \big( a_{ij}(x) \, \partial_i u \big) + \sum_{i=1}^{n} b_i(x) \, \partial_i u + c(x) \, u,$$
where $a_{ij} = a_{ji}$, $b_i$ and $c$ are in $C^{m+1}(\overline{U})$.
Assume that $L$ is uniformly [[elliptic|XY8GEQSWPR]]: there is $\theta > 0$ with $\sum_{i,j} a_{ij}(x) \xi_i \xi_j \ge \theta \abs{\xi}^2$ for all $x \in U$ and $\xi \in \R^n$.
Let $f \in H^m(U)$, and let $u \in H^1(U)$ be a [[weak solution|SJZX81A783]] of $L u = f$ in $U$.
Here $H^k(U) = W^{k,2}(U)$ is the [[Sobolev space|M4CEZ5K9HP]] of functions on $U$ whose [[weak derivatives|C3EC148VZD]] up to order $k$ lie in $L^2(U)$.

1. Let $V$ be an open set whose closure is a compact subset of $U$. Then $u \in H^{m+2}(V)$, and
$$\norm{u}_{H^{m+2}(V)} \le C \big( \norm{f}_{H^m(U)} + \norm{u}_{L^2(U)} \big),$$
where $C$ depends only on $m$, $U$, $V$ and the coefficients of $L$.
2. If the coefficients of $L$ and $f$ are [[smooth|0V33BNKZ9S]] on $\overline{U}$, then $u$ agrees almost everywhere with a smooth function on $U$.

## Remarks

Proof idea for $m = 0$: multiply the weak form by a cut-off function and test it with difference quotients $D_k^h u$ of $u$ in the direction $e_k$.
Uniform ellipticity bounds $\norm{D_k^h \nabla u}_{L^2}$ on $V$ independently of $h$, and a uniform bound on difference quotients implies that the weak derivative $\partial_k \nabla u$ exists in $L^2(V)$.
The case of general $m$ follows by induction, since a derivative of $u$ solves an equation of the same form with a right-hand side in $H^{m-1}$.
Part 2 follows from part 1 for all $m$ together with the Sobolev embedding theorem, which puts $H^{k}$ inside $C^j$ for $k > j + n/2$.

Example: a weak solution $u \in H^1(U)$ of $\Delta u = 0$ is smooth; this is Weyl's lemma, and $u$ is then a classical harmonic function.

Ellipticity is needed.
For the wave operator $\partial_t^2 - \partial_x^2$ in the plane, $u(t, x) = \abs{x - t}$ is a weak solution of $\partial_t^2 u - \partial_x^2 u = 0$ that is not differentiable on the line $x = t$.

The regularity is local: without conditions on the boundary of $U$ and on the boundary values, $u$ need not be in $H^{m+2}(U)$ up to the boundary.
For the Dirichlet problem with $u \in H^1_0(U)$ and $\partial U$ of class $C^{m+2}$, the same conclusion holds on all of $U$.

## Explanations

### L1 human

Elliptic regularity says that solutions of elliptic equations are as smooth as the data allow, and in fact two derivatives smoother than the right-hand side.
One may first find a rough solution, defined only through integrals against test functions.
The theorem then shows that this rough solution is automatically smooth when the coefficients and the right-hand side are smooth.
For Laplace's equation, every weak solution is a genuine smooth harmonic function.

### L2 human

The standard way to solve a linear elliptic problem has two steps.
First one gets a weak solution in $H^1$ from the Lax–Milgram theorem or the Riesz representation theorem.
Then elliptic regularity upgrades it to $H^{m+2}$, and with Sobolev embeddings to a classical solution.
The same pattern holds in other function spaces: the Calderón–Zygmund estimates give the $W^{2,p}$ version for $1 < p < \infty$, and the Schauder estimates give the $C^{2,\alpha}$ version.
Hyperbolic operators have no such property, since their solutions propagate singularities of the data.
</document>

<document source="https://tamnd.github.io/math2/node/2PYANBGXFD.md" tag="2PYANBGXFD" kind="concept">
# Path algebra

## Summary

The path algebra $kQ$ of a [[quiver|RE9YFFS3TS]] $Q$ over a field $k$ is the vector space with basis the paths in $Q$.
Its multiplication is concatenation of paths.
For a quiver with finitely many vertices, its modules are the same as the [[representations of the quiver|TXVWFWP6VQ]].
Every finite-dimensional algebra over an algebraically closed field is Morita equivalent to a quotient of a path algebra by an ideal of relations.
This makes path algebras the basic examples of representation theory of finite-dimensional algebras.

## Explanations

### L1 human

A quiver is a set of dots joined by arrows.
A path is a walk along arrows, head to tail, and each dot also counts as a path of length zero.
To multiply two paths you join them if the end of the first is the start of the second, and otherwise the product is zero.
The quiver with one dot and one loop gives the polynomial ring $k[x]$, where $x^n$ is the walk around the loop $n$ times.
The quiver $1 \to 2$ with one arrow gives a three-dimensional algebra, isomorphic to the upper triangular $2 \times 2$ matrices.
</document>

<document source="https://tamnd.github.io/math2/node/2PYAZX5VD4.md" tag="2PYAZX5VD4" kind="statement">
# Steinitz's theorem

## Statement

Let $G$ be a finite simple [[graph|00C0BR7D7H]].
Then the following are equivalent.

1. $G$ is isomorphic to the graph of vertices and edges of a $3$-dimensional [[convex polytope|300K6CB1KH]].
2. $G$ is [[planar|0NC9R64TXP]] and [[3-vertex-connected|7FZB7YYBXA]].

## Remarks

One direction is easier.
The graph of a $3$-polytope is planar: project its boundary onto one face from a point just outside that face.
It is $3$-connected by Balinski's theorem.
For the converse, one proof reduces every $3$-connected planar graph to the complete graph $K_4$.
It uses moves that replace a triangle by a three-legged star and back, and shows that each move can be carried out on polytopes.
Another proof uses the [[circle packing theorem|GR1DDT1B6B]] to realize the graph with all edges tangent to the unit sphere.

Example: the cube graph, with $8$ vertices of degree $3$, is planar and $3$-connected, and the cube realizes it.
Without $3$-connectivity the conclusion fails: the cycle graph $C_4$ is planar but only $2$-connected.
No $3$-polytope has it as its graph, because every vertex of a $3$-polytope has degree at least $3$.

## Explanations

### L1 human

Which networks of dots and lines are the skeletons of convex solids in space?
Steinitz's theorem gives a complete answer: exactly those that can be drawn in the plane without crossings and that stay connected after removing any two dots.
So the combinatorics of three-dimensional polytopes is pure graph theory.

### L2 human

The theorem lets one count and list combinatorial types of $3$-polytopes by counting $3$-connected planar graphs.
It also shows that every combinatorial $3$-polytope can be realized with rational, even integer, coordinates, and with many extra properties, such as all edges tangent to a sphere.
In dimension $4$ and higher no such characterisation is known, and realizability of a given face lattice is a hard problem.
</document>

<document source="https://tamnd.github.io/math2/node/2Q008XWQDE.md" tag="2Q008XWQDE" kind="statement">
# Jordan–Brouwer separation theorem

## Statement

Let $n \ge 1$ and let $X \subseteq \R^{n+1}$ be a subset [[homeomorphic|0QT00PB5G2]] to the [[$n$-sphere|T62369RFH6]] $S^n$.
Then the complement $\R^{n+1} \setminus X$ has exactly two [[connected components|0V032VWTHD]], one bounded and one unbounded, and $X$ is the [[boundary|05D5EGMFYV]] of each of them.

## Remarks

The proof uses [[Alexander duality|KK05CSTM15]] in $S^{n+1} = \R^{n+1} \cup \set{\infty}$.
It gives $\tilde{H}_0(S^{n+1} \setminus X; \Z) \cong \tilde{H}^n(X; \Z) \cong \Z$, with Čech cohomology on the right.
So $S^{n+1} \setminus X$ has two path components, which are open, hence also its connected components.
Removing the point $\infty$ from the component that contains it does not disconnect it, since $n + 1 \ge 2$.
For $n = 1$ the theorem is the Jordan curve theorem in the plane.

Example: the unit sphere $S^n$ itself separates $\R^{n+1}$ into the open unit ball and the set $\set{x : \norm{x} > 1}$.

For $n \ge 2$ the theorem says nothing about the shape of the components.
For the Alexander horned sphere in $\R^3$, the unbounded component is not simply connected, unlike the outside of a round sphere.
The Schoenflies theorem, which says that the bounded component is an open disc, holds for $n = 1$ but needs extra hypotheses, such as local flatness, for $n \ge 2$.
The case $n = 0$ is excluded: two points on the line $\R$ leave three components.

## Explanations

### L1 human

A closed surface in space without self-intersections, however crumpled, has an inside and an outside.
To get from the inside to the outside you must cross the surface, and every point of the surface touches both sides.

### L2 human

The theorem is one of the first applications of homology to geometry.
It is used to prove [[invariance of domain|6X2D0JDFQW]], and through it the topological invariance of dimension.
The contrast with the Schoenflies problem shows that homology can detect the number of components but not their homeomorphism type.
</document>

<document source="https://tamnd.github.io/math2/node/2Q1J7RZ4TK.md" tag="2Q1J7RZ4TK" kind="example">
# Irrational rotation

## Example

Let $\alpha$ be an [[irrational number|0DZXAXEJQ9]] and let $\mathbb{T} = \R/\Z$ be the circle, identified with $[0, 1)$.
The *irrational rotation* by $\alpha$ is
$$R_\alpha \colon \mathbb{T} \to \mathbb{T}, \qquad R_\alpha(x) = x + \alpha \bmod 1.$$

It is a [[measure-preserving transformation|556WBDAHNP]] for [[Lebesgue measure|2Z2QVWZ262]] $\lambda$.
The preimage of an arc is an arc of the same length, and arcs generate the Borel sets.

It is [[ergodic|AZRPHS478B]]: let $f \in L^2(\lambda)$ with $f \circ R_\alpha = f$, with Fourier coefficients $c_k$.
Then $c_k e^{2\pi i k \alpha} = c_k$ for all $k \in \Z$.
Since $k \alpha \notin \Z$ for $k \ne 0$, one gets $c_k = 0$ for $k \ne 0$, so $f$ is constant almost everywhere.

It is even [[uniquely ergodic|NZZE9KCAD8]], and every orbit is dense in $\mathbb{T}$.
By [[Weyl's equidistribution theorem|NCCGNTBEYV]] every orbit $(x + n\alpha \bmod 1)_{n \ge 0}$ is equidistributed.

It is not [[weakly mixing|2MGZFRH5PX]]: $e^{2\pi i x}$ is a nonconstant eigenfunction of the Koopman operator, with eigenvalue $e^{2\pi i \alpha}$.
Its Kolmogorov–Sinai entropy is $0$.
If $\alpha = p/q$ is rational in lowest terms, then $R_\alpha^q$ is the identity and $R_\alpha$ is not ergodic.

## Explanations

### L1 human

Turn a wheel by a fixed angle again and again, where the angle is an irrational fraction of a full turn.
The marked point never comes back exactly to where it started, and over time it visits every part of the rim equally often.
This makes the irrational rotation the simplest ergodic system.
It is orderly rather than chaotic: nearby points stay at the same distance forever, and the system has zero entropy.
</document>

<document source="https://tamnd.github.io/math2/node/2QBWT7T02Y.md" tag="2QBWT7T02Y" kind="concept">
# Abel function

## Summary

An Abel function of a map $f$ is a solution $\alpha$ of the [[Abel equation|68SHB7Z721]] $\alpha(f(x)) = \alpha(x) + 1$.
It is a coordinate in which one step of $f$ becomes a shift by $1$.
When $\alpha$ is injective, it defines a whole family of iterates $f^t = \alpha^{-1} \circ (\alpha + t)$, one for each real $t$ in a suitable range.
Abel functions are the standard tool for fractional iteration, for example in the construction of iterates of the exponential function.

## Explanations

### L1 human

An Abel function is a clock that ticks once each time the map $f$ is applied.
For the map $f(x) = x + 3$ on the real line, the function $\alpha(x) = x/3$ is such a clock.
Reading the clock, one can run the map for any amount of time: running for time $t$ moves $x$ to $x + 3t$.
For more complicated maps the clock is harder to find, but the idea is the same.
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<document source="https://tamnd.github.io/math2/node/2QKRBPS8B9.md" tag="2QKRBPS8B9" kind="concept">
# Wavelet packet decomposition

## Summary

The wavelet packet decomposition generalizes the [[discrete wavelet transform|P3H4MMC4ZH]].
At every level it splits both the approximation and the detail coefficients again.
The result is a full binary tree of coefficient sequences, which splits the frequency axis into bands of equal width instead of octaves.
Each admissible subtree of this tree gives an orthonormal basis, so one signal has a very large library of bases to choose from.
A best-basis search picks the subtree that represents a given signal most economically, which is useful in compression and denoising.

## Explanations

### L1 human

The ordinary wavelet transform keeps refining only the low frequencies: it splits the low band in two, then the lower half in two, and so on.
Wavelet packets also refine the high bands.
Picture a tree where every node splits into a low and a high branch.
Cutting the tree at different places gives different ways to divide the signal into frequency pieces, all of which keep the full information.
A signal with sharp high-frequency tones, such as a whistle, is better served by a cut that refines the high bands, which the ordinary wavelet transform cannot do.
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<document source="https://tamnd.github.io/math2/node/2QSKZ59DB6.md" tag="2QSKZ59DB6" kind="definition">
# Étale morphism of rings

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $A$ be a commutative [[$R$-algebra|DPBJ04JMH8]].
Then $A$ is *étale* over $R$, and the structure map $R \to A$ is an *étale morphism*, if both conditions hold:

1. $A$ is a [[finitely presented|764GZHTGV8]] $R$-algebra.
2. $A$ is formally étale over $R$, in the following sense.

Let $C$ be a commutative $R$-algebra, let $J \subseteq C$ be an [[ideal|096Z9MP2ME]] with $J^2 = 0$, and let $u \colon A \to C/J$ be an $R$-algebra [[homomorphism|0VTH8KPCG2]].
Then there is exactly one $R$-algebra homomorphism $v \colon A \to C$ whose composite with $C \to C/J$ is $u$.

## Remarks

Condition 2 says that the lift required by [[formal smoothness|8KPDNVKX7B]] exists and is unique.
Equivalent forms: $A$ is [[smooth|286G0P1B69]] and [[unramified|GY3QA6NH0E]] over $R$; or $A$ is finitely presented, [[flat|73RYYEVDKA]] and unramified over $R$.

Example: let $f \in R[x]$ be monic and let $A = R[x]/(f)$; if the class of $f'$ is a unit in $A$, then $A$ is étale over $R$.
For instance, $R[x]/(x^2 - a)$ is étale when $2a$ is a unit of $R$, since $(2x)^2 = 4a$ in $A$.
Over a field $k$, a finite [[separable extension|0WCX7NB4N7]] $L$ of $k$ is étale, and the étale $k$-algebras are exactly the finite products of such fields.

Non-example: $A = k[x]$ over a field $k$ is smooth and finitely presented, but a map $A \to C/J$ has many lifts: $x$ may go to any preimage of $u(x)$.
So uniqueness in clause 2 fails; accordingly $\Omega_{A/k} = A \, dx \ne 0$, see [[Kähler differentials|7HCFARHG7D]].
</document>

<document source="https://tamnd.github.io/math2/node/2QV28WYQTQ.md" tag="2QV28WYQTQ" kind="concept">
# Least squares approximation

## Summary

Least squares approximation replaces a function by the element of a given finite-dimensional family that is closest in a mean square sense.
It is the [[best approximation|7N4988HN3P]] in an [[inner product space|0QVNCJXRY3]], so it is the [[orthogonal projection|0B8KZRT28M]] onto the family.
Unlike the uniform best approximation, it is found by solving a linear system, and an orthogonal basis makes it explicit.
It is the function version of the [[least squares|0BYTQ4X1HC]] method for linear systems.

## Explanations

### L1 human

Measure the error of an approximation by squaring the gap at each point and adding up, or integrating.
The least squares approximation is the choice with the smallest total squared error.
Large gaps count a lot, but a single bad point does not decide everything, as it does for the largest-error measure.
For example, the best constant approximation of $x^2$ on $[-1, 1]$ in this sense is its mean value $1/3$.
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<document source="https://tamnd.github.io/math2/node/2R3VH69BSA.md" tag="2R3VH69BSA" kind="definition">
# Clausen function

## Definition

The *Clausen function* is the function $\operatorname{Cl}_2 \colon \R \to \R$ given by
$$\operatorname{Cl}_2(\theta) = -\int_0^{\theta} \log \abs{2 \sin \tfrac{t}{2}} \, dt,$$
where $\log$ is the [[natural logarithm|04DXAFN2ES]].
The integral converges for every real $\theta$, because the logarithmic singularities of the integrand at the points of $2\pi\Z$ are integrable.

## Remarks

Equivalently, $\operatorname{Cl}_2$ is the sum of the uniformly convergent [[Fourier series|G12KXDHPPS]]
$$\operatorname{Cl}_2(\theta) = \sum_{k=1}^{\infty} \frac{\sin k\theta}{k^2},$$
and $\operatorname{Cl}_2(\theta) = \operatorname{Im} \operatorname{Li}_2(e^{i\theta})$, where $\operatorname{Li}_2$ is the [[dilogarithm|G0X0NWSDB7]].
The function is odd and $2\pi$-periodic, and it vanishes at every multiple of $\pi$.

Example: at $\theta = \pi/2$, $\sin(k\pi/2)$ is $0$ for even $k$ and $(-1)^{m}$ for $k = 2m + 1$, so $\operatorname{Cl}_2(\pi/2) = \sum_{m \ge 0} (-1)^m / (2m+1)^2 = G$, Catalan's constant.
Its maximum on $\R$ is $\operatorname{Cl}_2(\pi/3) \approx 1.01494$.

Non-example: the series $\sum_{k \ge 1} \cos(k\theta)/k^2$ is not the Clausen function; it is the real part of $\operatorname{Li}_2(e^{i\theta})$, which equals $\frac{\pi^2}{6} - \frac{\pi\theta}{2} + \frac{\theta^2}{4}$ for $0 \le \theta \le 2\pi$.

Higher Clausen functions $\operatorname{Cl}_n$ are defined from $\operatorname{Li}_n(e^{i\theta})$ in a similar way.
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# Bochner–Riesz means

## Definition

Let $n \ge 1$, let $\delta \ge 0$ and let $R > 0$.
Let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi} \, dx$ be the [[Fourier transform|DPVM34ADZV]] of a [[Schwartz function|CG9JEATW3Q]] $f$ on $\R^n$, and let $t_+ = \max(t, 0)$, with the convention $t_+^0 = \mathbf{1}_{t > 0}$.
The *Bochner–Riesz mean* of order $\delta$ and radius $R$ of $f$ is
$$S^\delta_R f(x) = \int_{\R^n} \Big(1 - \frac{\abs{\xi}^2}{R^2}\Big)_+^\delta \hat f(\xi) \, e^{2\pi i x \cdot \xi} \, d\xi.$$
The operator $S^\delta = S^\delta_1$ is the Bochner–Riesz operator of order $\delta$.

## Remarks

Example: for $\delta = 0$, $S^0_R f$ is the spherical partial sum, the inverse Fourier transform of $\hat f$ restricted to the open ball of radius $R$.
For $n = 1$ and $\delta = 1$, the multiplier is $(1 - \xi^2/R^2)_+$, which is a continuous function with compact support.

The multiplier of $S^\delta_R$ is the dilate of the multiplier of $S^\delta$ by $R$, and dilations do not change $L^p$ operator norms.
So $S^\delta_R$ is bounded on $L^p(\R^n)$ uniformly in $R$ if and only if $(1 - \abs{\xi}^2)_+^\delta$ is an [[$L^p$ multiplier|RAGR38Q40B]].
In that case, $S^\delta_R f \to f$ in $L^p(\R^n)$ as $R \to \infty$ for every $f \in L^p(\R^n)$, $1 \le p < \infty$.

Non-example of convergence: for $\delta = 0$, $n \ge 2$ and $p \ne 2$, the ball multiplier is not an $L^p$ multiplier, by [[Fefferman's ball multiplier theorem|4A5MJY4XPV]].

For $\delta > (n - 1)/2$, the kernel of $S^\delta$ is integrable, so $S^\delta$ is bounded on $L^p(\R^n)$ for all $1 \le p \le \infty$.
The value $(n - 1)/2$ is the critical index.
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<document source="https://tamnd.github.io/math2/node/2R8QQTP1CE.md" tag="2R8QQTP1CE" kind="concept">
# Lattice of equational theories

## Summary

The lattice of equational theories of a signature is the set of all equational theories of that signature, ordered by inclusion.
An equational theory is a set of identities closed under logical consequence.
This lattice is complete, and it is dually isomorphic to the lattice of all varieties of the signature, because larger theories have fewer models.
It can be described as the lattice of fully invariant congruences of a term algebra on countably many variables.

## Explanations

### L1 human

Collect all the identities that follow from some axioms; this is an equational theory.
Adding axioms gives a bigger theory and a smaller class of algebras that satisfy it.
The theories, ordered by inclusion, form a lattice that is the upside-down picture of the lattice of varieties.
For example, the theory of abelian groups contains the theory of groups, since it has the extra identity $xy \approx yx$.
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<document source="https://tamnd.github.io/math2/node/2RCTYXSJ67.md" tag="2RCTYXSJ67" kind="definition">
# Newtonian space

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] with a Borel [[measure|PR11BZQQPG]] $\mu$, and let $1 \le p < \infty$.
Let $u \colon X \to \R$ be a function, defined at every point.
For a Borel function $g \colon X \to [0, \infty]$ and a [[rectifiable curve|0HATW2VWAZ]] $\gamma \colon [a, b] \to X$, let $\int_\gamma g \, ds$ be the integral of $g$ along $\gamma$ with respect to arc length.
The function $g$ is an *upper gradient* of $u$ if
$$\abs{u(\gamma(a)) - u(\gamma(b))} \le \int_\gamma g \, ds$$
for every rectifiable curve $\gamma \colon [a, b] \to X$.
Let $\widetilde{N}^{1,p}(X)$ be the set of all $u$ in the [[Lp space|SASP14MA1Z]] $L^p(X, \mu)$ with an upper gradient in $L^p(X, \mu)$.
On it put
$$\norm{u}_{\widetilde{N}^{1,p}} = \left( \int_X \abs{u}^p \, d\mu + \inf_g \int_X g^p \, d\mu \right)^{1/p},$$
the infimum over all upper gradients $g$ of $u$.
The *Newtonian space* $N^{1,p}(X)$ is the quotient of $\widetilde{N}^{1,p}(X)$ by the relation $u \sim v$ if $\norm{u - v}_{\widetilde{N}^{1,p}} = 0$, with the induced norm.

## Remarks

Example: on $X = \R$ with Lebesgue measure, $u(x) = \max(0, 1 - \abs{x})$ lies in $\widetilde{N}^{1,p}(\R)$.
Indeed $u \in L^p$, and $g = \mathbf{1}_{[-1, 1]}$ is an upper gradient in $L^p$, since $\abs{u(s) - u(t)} \le \abs{\int_s^t g}$ for real $s, t$ and a curve from $\gamma(a)$ to $\gamma(b)$ covers at least the interval between them.

Non-example: on $X = [0, 1]$ with Lebesgue measure, $u = \mathbf{1}_{[1/2, 1]}$ is in $L^p$ but has no upper gradient in $L^p$.
An upper gradient $g$ would need $\int_{1/2 - \varepsilon}^{1/2 + \varepsilon} g \ge 1$ for every $\varepsilon > 0$, and these integrals tend to $0$ when $g$ is integrable.

The space $N^{1,p}(X)$ is a [[Banach space|06GZ956657]].
Its elements are classes of functions defined at every point, and two functions in one class agree outside a set of $p$-capacity zero, not only almost everywhere.
The definition is often given with $p$-weak upper gradients, which may fail the curve inequality on a family of curves of $p$-modulus zero.
This gives the same space and the same norm.
If $X$ contains no nonconstant rectifiable curves, for example a Cantor set, then $g = 0$ is an upper gradient of every function and $N^{1,p}(X) = L^p(X, \mu)$.
For an open set $\Omega \subset \R^n$ with Lebesgue measure and $1 < p < \infty$, $N^{1,p}(\Omega)$ agrees with the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\Omega)$, each class of $W^{1,p}$ being represented by suitable pointwise defined functions.
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<document source="https://tamnd.github.io/math2/node/2RD6T5KG59.md" tag="2RD6T5KG59" kind="concept">
# Projective subspace

## Summary

A projective subspace of a [[projective space|ZT9A6JK8QF]] $\mathbb{P}(V)$ is the set of points that come from a [[linear subspace|00YC1D652K]] of $V$.
Points, lines and planes of a projective space are its projective subspaces of dimension $0$, $1$ and $2$.
Equivalently, it is a set of points that contains the whole line through any two of its points.
The projective subspaces form a lattice that is isomorphic to the lattice of linear subspaces of $V$.

## Explanations

### L1 human

A projective space is made of lines through the origin of a vector space.
A plane through the origin contains many of these lines, and they form a projective line.
In the same way, every subspace through the origin gives a smaller projective space sitting inside the big one.
In the projective plane, the projective subspaces are the empty set, the points, the lines and the whole plane.
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<document source="https://tamnd.github.io/math2/node/2RDAA4PCK8.md" tag="2RDAA4PCK8" kind="concept">
# Prüfer domain

## Summary

A Prüfer domain is an [[integral domain|09S3PFBEPM]] in which every nonzero finitely generated ideal is [[invertible|E190TAV2S6]].
Equivalently, its [[localization|4C1AE84N5N]] at every prime ideal is a valuation ring.
Prüfer domains are the non-Noetherian analogue of Dedekind domains: a Noetherian Prüfer domain is exactly a Dedekind domain.
Every [[Bézout domain|HGEX40P0Z0]] is a Prüfer domain.

## Explanations

### L1 human

In a Dedekind domain, such as the ring of integers of a number field, every nonzero ideal can be "divided out": it has an inverse fractional ideal.
A Prüfer domain keeps this property for finitely generated ideals, without asking that every ideal be finitely generated.
Examples are $\Z$, every Dedekind domain, every valuation ring, and the ring of entire functions on $\C$.
The polynomial ring $\Z[x]$ is not Prüfer: its localization at the prime $(2, x)$ contains neither $2/x$ nor $x/2$, so it is not a valuation ring.
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# Ehrenfeucht–Fraïssé game

## Summary

The Ehrenfeucht–Fraïssé game is a two-player game played on two [[structures|983GZBQYT4]] of the same signature.
One player, Spoiler, tries to show that the structures differ, and the other, Duplicator, tries to show that they look alike.
For a finite relational signature, Duplicator wins the $n$-round game exactly when the structures satisfy the same sentences of quantifier depth at most $n$.
The game is a main tool to prove that a property is not first-order definable, in particular in finite model theory.

## Explanations

### L1 human

In each round, Spoiler points at an element of one of the two structures, and Duplicator answers with an element of the other.
After all rounds, Duplicator wins if the chosen elements relate to each other in the same way on both sides.
On the linear orders with $3$ and $4$ elements, Duplicator wins the game with $2$ rounds.
Spoiler wins with $3$ rounds, so some first-order sentence with three nested quantifiers tells the two orders apart.
</document>

<document source="https://tamnd.github.io/math2/node/2REN148QWD.md" tag="2REN148QWD" kind="definition">
# Repair radii and villainy

## Definition

**Repair radii and villainy** are defined for a finite graph $G$ as follows.

Use [[words and profiles|28M8XD8EA9]] with palette size $\chi(G)$. For a profile $\mathbf m$, let $\mathcal W_{\mathbf m}(G)$ be the set of words on $G$ with profile $\mathbf m$.

Let $\mathcal P_{\mathbf m}(G)$ be the set of [[proper colourings|0874138YP6]] of $G$ with profile $\mathbf m$. Let $\mathcal P(G)$ be the set of all proper colourings of $G$ with this palette.

For two words $w$ and $y$ on $G$, the Hamming distance $d_{\mathrm H}(w,y)$ is the number of vertices on which they differ.

1. The **repair radius** of $\mathbf m$ is $R_G(\mathbf m)=\max_{w\in\mathcal W_{\mathbf m}(G)}\min_{y\in\mathcal P_{\mathbf m}(G)} d_{\mathrm H}(w,y)$.

2. The **weak repair radius** of $\mathbf m$ is $R^{\mathrm w}_G(\mathbf m)=\max_{w\in\mathcal W_{\mathbf m}(G)}\min_{y\in\mathcal P(G)} d_{\mathrm H}(w,y)$.

3. The **villainy** of $G$ is $B(G)=\max R_G(\mathbf m)$, taken over all feasible profiles $\mathbf m$.

4. The **weak villainy** of $G$ is $B_{\mathrm w}(G)=\max R^{\mathrm w}_G(\mathbf m)$, taken over all feasible profiles $\mathbf m$.

Each inner quantity is the [[distance from a point to a set|07TXYYVYZR]] for the Hamming distance. Both $B(G)$ and $B_{\mathrm w}(G)$ are zero when $G$ is the empty graph.
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# Point spectrum

## Definition

Let $X$ be a complex Banach space with identity map $I$, and let $T \colon X \to X$ be a bounded operator.
The *point spectrum* of $T$ is the set
$$\sigma_p(T) = \set{\lambda \in \C : \ker(T - \lambda I) \ne \set{0}},$$
the set of [[eigenvalues|0BQAA2BK3Q]] of $T$.

## Remarks

Here $\ker$ is the [[kernel|0S23E6YE83]].
The point spectrum is a subset of the [[spectrum|BMKATJ802A]] $\sigma(T)$, and for $X = \C^n$ the two are equal.

Example: let $L(x_1, x_2, \ldots) = (x_2, x_3, \ldots)$ be the left shift on $\ell^2$.
For $\abs{\lambda} < 1$, the vector $x = (1, \lambda, \lambda^2, \ldots)$ is in $\ell^2$ and $Lx = \lambda x$, so $\lambda \in \sigma_p(L)$.
Conversely $Lx = \lambda x$ forces $x_{n+1} = \lambda x_n$, so $x = x_1 (1, \lambda, \lambda^2, \ldots)$, which is in $\ell^2$ with $x \ne 0$ only if $\abs{\lambda} < 1$; thus $\sigma_p(L)$ is the open unit disc.

Non-example: for the diagonal operator $D(x_n) = (x_n / n)$ on $\ell^2$, $0 \in \sigma(D)$ but $0 \notin \sigma_p(D)$, since $Dx = 0$ forces $x = 0$.

The point spectrum need not be closed, and it can be empty, as for the right shift on $\ell^2$.
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# Fredholm equation of the first kind

## Summary

A Fredholm equation of the first kind is an integral equation $\int_a^b K(x, y) \varphi(y) \, dy = f(x)$ with fixed limits $a$ and $b$.
The unknown function $\varphi$ appears only under the integral sign.
It is the [[Fredholm integral equation|526TCVMSHP]] form of an [[integral equation of the first kind|ZDS77Q0F06]].
Such equations model the recovery of a cause from a smoothed measurement, as in deblurring and tomography.
They are usually ill-posed: a solution may fail to exist, and small changes of $f$ can cause large changes of $\varphi$.

## Explanations

### L1 human

Think of $f$ as a blurred photograph and of $\varphi$ as the sharp picture that produced it.
The kernel $K(x, y)$ says how much the point $y$ of the sharp picture contributes to the point $x$ of the blurred one.
Solving the equation means undoing the blur, and blurring destroys fine detail, so this is hard.
For example, with $K(x, y) = 1$ on $[0, 1]$, the equation reads $\int_0^1 \varphi(y) \, dy = f(x)$.
It has a solution only when $f$ is constant, and then it has infinitely many.
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# Iterative method

## Summary

An iterative method solves a problem by producing a sequence of approximate solutions, each computed from the previous ones, that converges to a solution.
It contrasts with a direct method, which gives the answer after a fixed finite number of steps.
Iterative methods are the standard tool for nonlinear equations and for very large sparse linear systems, where direct elimination costs too much.

## Explanations

### L1 human

Start with a guess, improve it by a fixed rule, and repeat until the guess is good enough.
[[Newton's method|07EXVNF9VP]] for $\sqrt{2}$ is an example: from $x_0 = 1$ the rule $x_{k+1} = (x_k + 2 / x_k) / 2$ gives $1.5$, then $1.41\overline{6}$, then $1.41421\ldots$.
Each step is cheap, and one stops when the error is below a tolerance.
The questions are whether the sequence converges and how fast.
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<document source="https://tamnd.github.io/math2/node/2RXEM9BQFY.md" tag="2RXEM9BQFY" kind="definition">
# Pluriharmonic function

## Definition

Let $\Omega \subseteq \C^n$ be an open set.
A function $u \colon \Omega \to \R$ of class $C^2$ is *pluriharmonic* if
$$\frac{\partial^2 u}{\partial z_j \partial \bar z_k} = 0 \quad \text{on } \Omega \text{ for all } j, k \in \set{1, \dots, n},$$
where $\partial / \partial z_j = \frac{1}{2}(\partial / \partial x_j - i \, \partial / \partial y_j)$ and $\partial / \partial \bar z_k = \frac{1}{2}(\partial / \partial x_k + i \, \partial / \partial y_k)$ for $z_j = x_j + i y_j$.

## Remarks

Equivalently, for all $a \in \Omega$ and $b \in \C^n$, the function $\lambda \mapsto u(a + \lambda b)$ is [[harmonic|0DEB0JMN0W]] where it is defined.
Example: $u = \operatorname{Re} f$ for a holomorphic $f$ is pluriharmonic, because $\partial f / \partial \bar z_k = 0$ and $\partial \bar f / \partial z_j = 0$ give $\partial^2 (f + \bar f) / \partial z_j \partial \bar z_k = 0$.
Non-example: $u = \abs{z_1}^2$ has $\partial^2 u / \partial z_1 \partial \bar z_1 = 1 \ne 0$.
Conversely, on a ball, or more generally on a simply connected domain, every pluriharmonic function is the real part of a holomorphic function; on other domains this holds locally.
A pluriharmonic function is real analytic.
The condition can be written $\partial \bar\partial u = 0$.
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<document source="https://tamnd.github.io/math2/node/2S0MXDT4YH.md" tag="2S0MXDT4YH" kind="concept">
# Poisson bracket

## Summary

The Poisson bracket is an operation $\{f, g\}$ on pairs of smooth functions on a [[symplectic manifold|CAN490EPRT]], built from their [[Hamiltonian vector fields|C5EH3B4BEP]].
It is antisymmetric, satisfies the Jacobi identity and the Leibniz rule, so it makes the functions into a Lie algebra.
In mechanics it gives the time derivative of an observable along the motion, and its abstract properties lead to [[Poisson manifolds|A2KQ49Y4NF]] and to quantization.

## Explanations

### L1 human

In mechanics, a state is a point $(q, p)$ of positions and momenta, and an observable is a function of the state.
The Poisson bracket of two observables measures how one changes when the system flows along the energy defined by the other.
On the plane, $\{f, g\} = \frac{\partial f}{\partial q} \frac{\partial g}{\partial p} - \frac{\partial f}{\partial p} \frac{\partial g}{\partial q}$.
For example $\{q, p\} = 1$, which is the classical shadow of the quantum rule that position and momentum do not commute.
A quantity $g$ is conserved by the motion with energy $H$ exactly when $\{g, H\} = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/2S4K8FA8QY.md" tag="2S4K8FA8QY" kind="definition">
# Field of formal Laurent series

## Definition

Let $k$ be a [[field|0QA8CHH8MG]] and let $X$ be a symbol.
A *formal Laurent series* in $X$ over $k$ is a family $(a_n)_{n \in \Z}$ of elements of $k$ with the following property.
There is $N \in \Z$ with $a_n = 0$ for all $n < N$.
It is written $\sum_{n \ge N} a_n X^n$.
The set $k((X))$ of these series has the operations
$$\sum_n a_n X^n + \sum_n b_n X^n = \sum_n (a_n + b_n) X^n, \qquad \Big(\sum_n a_n X^n\Big)\Big(\sum_n b_n X^n\Big) = \sum_n \Big(\sum_{i + j = n} a_i b_j\Big) X^n.$$
With these operations $k((X))$ is a field, the *field of formal Laurent series* over $k$.

## Remarks

The inner sum $\sum_{i + j = n} a_i b_j$ is finite: if $a_i = 0$ for $i < N$ and $b_j = 0$ for $j < M$, only the pairs with $N \le i \le n - M$ contribute.
Example: $f = X - X^2$ has the inverse $g = \sum_{n \ge -1} X^n = X^{-1} + 1 + X + \cdots$; indeed $f g = (1 - X)(1 + X + X^2 + \cdots) = 1$.
Non-example: the family with $a_n = 1$ for all $n \in \Z$ is not a formal Laurent series, since no $N$ bounds its negative powers.
Its square would need the infinite sum $\sum_{i + j = 0} 1$.
The map $v(f) = \min \set{n : a_n \ne 0}$ for $f \ne 0$ is a [[discrete valuation|ZYK5F0GBK6]] on $k((X))$ with uniformizer $X$, and $k((X))$ is complete for it.
Its valuation ring is the ring $k[\![X]\!]$ of [[formal power series|072W67KFYK]], and $k((X))$ is the [[field of fractions|0VF11ZTTMD]] of $k[\![X]\!]$.
</document>

<document source="https://tamnd.github.io/math2/node/2S6XZCQWY7.md" tag="2S6XZCQWY7" kind="definition">
# Generalized Fourier series

## Definition

Let $V$ be an [[inner product space|0QVNCJXRY3]] with inner product $\ip{\cdot}{\cdot}$, linear in the first argument.
Let $(\varphi_k)_{k \ge 0}$ be a sequence of nonzero, pairwise [[orthogonal|0Y2X7P2WB7]] vectors of $V$.
For $f \in V$, the *generalized Fourier coefficients* of $f$ are the numbers
$$c_k = \frac{\ip{f}{\varphi_k}}{\ip{\varphi_k}{\varphi_k}}, \quad k \ge 0.$$
The formal series $\sum_{k \ge 0} c_k \varphi_k$ is the *generalized Fourier series* of $f$ with respect to $(\varphi_k)$.

## Remarks

If the $\varphi_k$ are orthonormal, $c_k = \ip{f}{\varphi_k}$.
The partial sum $\sum_{k \le N} c_k \varphi_k$ is the orthogonal projection of $f$ onto the span of $\varphi_0, \dots, \varphi_N$, the closest point of that span to $f$.
By [[Bessel's inequality|0RBT5PB74K]], $\sum_k \abs{c_k}^2 \norm{\varphi_k}^2 \le \norm{f}^2$.
In a [[Hilbert space|HM2HHE3J51]] the series converges to the projection of $f$ onto the closed span of the family.
It converges to $f$ for every $f$ exactly when the family is complete, and then [[Parseval's identity|047RFCNVSK]] holds.

Example: in $L^2(-1, 1)$, take the Legendre polynomials $\varphi_0 = 1$, $\varphi_1 = x$, $\varphi_2 = \frac12 (3x^2 - 1)$, and so on, and $f(x) = x^2$.
Then $\ip{f}{\varphi_0} = \frac23$, $\ip{\varphi_0}{\varphi_0} = 2$, $\ip{f}{\varphi_1} = 0$, $\ip{f}{\varphi_2} = \frac{4}{15}$ and $\ip{\varphi_2}{\varphi_2} = \frac25$.
So $c_0 = \frac13$, $c_1 = 0$, $c_2 = \frac23$, and indeed $x^2 = \frac13 + \frac23 \cdot \frac12(3x^2 - 1)$; all later coefficients vanish.

Non-example: the formula fails to reconstruct $f$ when the family is not orthogonal.
In $L^2(0, 1)$ with $\varphi_0 = 1$ and $\varphi_1 = x$, which are not orthogonal, the function $f = x$ gives $c_0 = \frac12$ and $c_1 = 1$, so $c_0 \varphi_0 + c_1 \varphi_1 = x + \frac12 \ne f$.
</document>

<document source="https://tamnd.github.io/math2/node/2SEF8QTYQF.md" tag="2SEF8QTYQF" kind="example">
# Chinese restaurant process

## Example

Fix a parameter $\theta > 0$.
Customers $1, 2, 3, \ldots$ enter a restaurant with infinitely many tables, one at a time.
Customer $1$ sits at a new table.
Suppose $n$ customers are seated, and the occupied tables hold $n_1, \ldots, n_k$ customers, so $n_1 + \cdots + n_k = n$.
Then customer $n + 1$ sits at the $j$-th occupied table with probability $n_j / (n + \theta)$, and at a new table with probability $\theta / (n + \theta)$.
These probabilities are nonnegative and sum to $(n + \theta)/(n + \theta) = 1$.
The seating of the first $n$ customers gives a random [[partition|08QMKTWF9T]] $\Pi_n$ of $\set{1, \ldots, n}$, whose blocks are the tables.

Multiplying the seating probabilities gives, for each partition $\pi$ of $\set{1, \ldots, n}$ with $k$ blocks of sizes $n_1, \ldots, n_k$,
$$P(\Pi_n = \pi) = \frac{\theta^k \, \Gamma(\theta)}{\Gamma(\theta + n)} \prod_{j=1}^{k} (n_j - 1)!,$$
where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
To check this, note that the denominators $\theta, \theta + 1, \ldots, \theta + n - 1$ multiply to $\Gamma(\theta + n)/\Gamma(\theta)$.
Each table contributes the factor $\theta$ when it is opened, and then the factors $1, 2, \ldots, n_j - 1$ from the later customers who join it.
The right side depends only on the block sizes, so the law of $\Pi_n$ is invariant under permutations of $\set{1, \ldots, n}$.
In this sense the random partition is [[exchangeable|4MR4YJS1ZD]].

For $n = 2$, customers $1$ and $2$ share a table with probability $1/(1 + \theta)$.
The number $K_n$ of occupied tables after $n$ customers has mean
$$E[K_n] = \sum_{i=0}^{n-1} \frac{\theta}{\theta + i},$$
which grows like $\theta \log n$.

## Remarks

Let $G$ follow a [[Dirichlet process|XT4SWG6J7N]] with concentration $\theta$ and a base measure without atoms.
Given $G$, draw $X_1, X_2, \ldots$ independently from $G$, and group the indices $i$ by equal values $X_i$.
The resulting random partition has the law of the Chinese restaurant process with parameter $\theta$.
This is why the process is used to sample from Dirichlet process mixture models.

## Explanations

### L1 human

Picture a restaurant where each new guest either joins a table or starts a new one.
A busy table attracts guests in proportion to how many already sit there, so the rich get richer.
The parameter $\theta$ measures how much guests like to start fresh tables.
With $\theta = 1$, the second guest joins the first with probability $1/2$.
The result is a random way of splitting the guests into groups.
</document>

<document source="https://tamnd.github.io/math2/node/2SFM73YET0.md" tag="2SFM73YET0" kind="statement">
# Classification of surfaces

## Statement

Let $S$ be a closed [[surface|VGJJVBZ1MQ]], that is, a compact connected surface without boundary.

1. $S$ is homeomorphic to exactly one surface in the following list.
   - The sphere $S^2$.
   - The [[connected sum|Y0K0K5NGWW]] of $g \ge 1$ copies of the [[torus|03NM7DZXXC]] $T^2$.
   - The connected sum of $k \ge 1$ copies of the [[real projective plane|MWDB7RACWQ]] $\R P^2$.
2. Two closed surfaces are homeomorphic if and only if both are [[orientable|0TD904MG06]] or both are non-orientable, and they have the same [[Euler characteristic|RNDDWDAFJ5]].

The Euler characteristic of the connected sum of $g$ tori is $2 - 2g$, and that of the connected sum of $k$ projective planes is $2 - k$.

## Remarks

Idea of the proof: by Radó's theorem every surface has a triangulation, so $S$ is obtained from a polygon by gluing its edges in pairs.
Cutting and regluing the polygon brings the gluing word to a normal form: $aa^{-1}$ for the sphere, $a_1 b_1 a_1^{-1} b_1^{-1} \cdots a_g b_g a_g^{-1} b_g^{-1}$ for $g$ tori, or $c_1 c_1 \cdots c_k c_k$ for $k$ projective planes.
The surfaces in the list are different because orientability and Euler characteristic are topological invariants that take different values on them.

The theorem at work: the [[Klein bottle|6J4C2P3ZHC]] is a closed non-orientable surface with Euler characteristic $0$, so it is homeomorphic to $\R P^2 \# \R P^2$.
Also $T^2 \# \R P^2$ is non-orientable with Euler characteristic $-1$, so it is homeomorphic to $\R P^2 \# \R P^2 \# \R P^2$.
Compactness is needed: the open disc and the open annulus are connected surfaces without boundary that are not in the list.

## Explanations

### L1 human

Every closed surface is a sphere with some number of handles, or a sphere with some number of crosscaps.
To tell two closed surfaces apart, two pieces of information suffice: whether they have a consistent inside and outside, and their Euler characteristic.
So the infinite variety of possible shapes reduces to a simple list.

### L2 human

The classification is the model of what one hopes for in manifold topology: a complete list with computable invariants.
The orientable number $g$ is the [[genus|NJ787GZS1N]] of the surface.
The proof through polygons and gluing words also computes the fundamental group of each surface.
Every closed surface other than $S^2$ and $\R P^2$ is aspherical, and every closed surface carries a metric of constant curvature, with sign given by the Euler characteristic.
In dimension $3$ no list this simple exists, and the analogue is the [[geometrization conjecture|B27HB43K86]].
</document>

<document source="https://tamnd.github.io/math2/node/2SM3JRJXA4.md" tag="2SM3JRJXA4" kind="concept">
# Regular category

## Summary

A regular category is a finitely complete category in which every morphism has a well-behaved image.
Precisely, coequalizers of kernel pairs exist and regular epimorphisms are stable under pullback.
In a regular category every morphism factors as a regular epimorphism followed by a monomorphism, and relations can be composed.
Sets, groups, modules, every abelian category and every topos are regular, and regular categories are the semantics of regular logic.

## Explanations

### L1 human

In sets, every function factors through its image: first onto the image, then into the target.
A regular category is one where this factorization exists for all morphisms and behaves well when one restricts to a part of the target.
This is what is needed to talk about images, to compose relations, and to interpret statements built from "and", "true" and "there exists".
The category of groups is regular, while the category of topological spaces is not.
</document>

<document source="https://tamnd.github.io/math2/node/2SM64J81YC.md" tag="2SM64J81YC" kind="statement">
# Holomorphic functions on compact complex manifolds are constant

## Statement

Let $X$ be a [[compact|0TVJZYB6SH]], [[connected|0V032VWTHD]] [[complex manifold|JSGWAN2RJV]], and let $f \colon X \to \C$ be a holomorphic function.
Then $f$ is constant.

## Remarks

Proof idea: since $X$ is compact, $\abs{f}$ attains its maximum at some point $p$.
In a connected chart around $p$, the [[maximum modulus principle in several variables|91DW8V0PET]] shows that $f$ equals $f(p)$ near $p$.
The set of points near which $f \equiv f(p)$ is open, and by the [[identity theorem|7GZ908QZ04]] it is also closed, so by connectedness it is all of $X$.

Both hypotheses are needed.
On the non-compact manifold $\C^n$, the coordinate function $z_1$ is holomorphic and not constant.
On a disjoint union of two compact complex manifolds, take the function equal to $0$ on one piece and $1$ on the other.
It is holomorphic and not constant; in general $f$ is only locally constant.

Consequence: a compact connected complex manifold of positive dimension is not a complex submanifold of any $\C^N$, since the coordinate functions restricted to it would be constant.
For example, [[complex projective space|7GJ4STCNNM]] does not embed holomorphically in $\C^N$.

## Explanations

### L1 human

On a compact complex manifold, such as the Riemann sphere or a torus, every holomorphic function is constant.
The reason is the maximum principle: the absolute value of a holomorphic function cannot have a peak inside its domain.
On a compact space there is no boundary to escape to.
This is a strong contrast with smooth functions, which exist in great supply on every manifold.

### L2 human

This result is the several-variable version of the fact that a compact Riemann surface has no nonconstant holomorphic functions, and it is a cousin of [[Liouville's theorem|0EHKVRF4J6]].
Because global holomorphic functions are useless on a compact complex manifold, one studies instead meromorphic functions and holomorphic sections of line bundles.
The opposite extreme is a [[Stein manifold|HMCD0JBFST]], which has enough holomorphic functions to embed in some $\C^N$.
</document>

<document source="https://tamnd.github.io/math2/node/2SRDN8EJG3.md" tag="2SRDN8EJG3" kind="concept">
# Lifting idempotents

## Summary

Lifting idempotents is the question of whether an [[idempotent|0N5NJZKKN3]] of a quotient ring $R/I$ comes from an idempotent of $R$.
When every idempotent of $R/I$ lifts, decompositions of $R/I$ into pieces can be pulled back to decompositions of $R$.
Idempotents always lift modulo a [[nil ideal|NZ1G9WRWFB]], and they lift modulo the Jacobson radical exactly in [[semiperfect rings|ZRC2S3ECFK]] among the semilocal rings.
The technique is central in the theory of finite-dimensional algebras, complete local rings and modular representation theory.

## Explanations

### L1 human

An idempotent is an element $e$ with $e^2 = e$, like a projection.
In a quotient ring, an element may be idempotent only "up to an error" in the ideal.
Lifting means correcting such an element to a true idempotent of the ring.
In $\Z/4\Z$, which is $\Z/8\Z$ modulo the nil ideal generated by $4$, every idempotent lifts.
In $\Z/6\Z$, which is $\Z$ modulo $6\Z$, the idempotent $3$ does not lift, since the only idempotents of $\Z$ are $0$ and $1$.
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<document source="https://tamnd.github.io/math2/node/2SRQCEB16J.md" tag="2SRQCEB16J" kind="statement">
# Ohsawa–Takegoshi extension theorem

## Statement

Let $n \ge 2$ and let $\Omega \subseteq \C^n$ be a bounded [[pseudoconvex|AHYZMSBBNH]] open set.
Let $H = \set{z \in \C^n : z_n = 0}$, and let $\psi$ be a [[plurisubharmonic function|FF2P7ER3QT]] on $\Omega$.
Then there is a constant $C > 0$, depending only on the diameter of $\Omega$, with the following property.
For every holomorphic function $f$ on $\Omega \cap H$ with
$$\int_{\Omega \cap H} \abs{f}^2 e^{-\psi}\, d\lambda_{n-1} < \infty,$$
there is a [[holomorphic function|507C34W0ZT]] $F$ on $\Omega$ with $F = f$ on $\Omega \cap H$ and
$$\int_\Omega \abs{F}^2 e^{-\psi}\, d\lambda_n \le C \int_{\Omega \cap H} \abs{f}^2 e^{-\psi}\, d\lambda_{n-1}.$$
Here $\lambda_k$ denotes Lebesgue measure on $\C^k$, and $\Omega \cap H$ is viewed as an open subset of $\C^{n-1}$.

## Remarks

The point is that $C$ does not depend on $\psi$ or on $f$.
The proof solves a $\bar\partial$-equation with $L^2$ estimates for a weight that is singular along $H$, using a twisted form of the Bochner–Kodaira–Morrey–Kohn identity.
One extends $f$ smoothly with a cutoff in $z_n$ and corrects the extension by a solution $u$ of $\bar\partial u = g$ whose weight $e^{-\psi}/\abs{z_n}^2$ forces $u$ to vanish on $H$.
The sharp value of the constant was later found by Błocki and by Guan and Zhou, and it yields the Suita conjecture on the Bergman kernel of plane domains.
Example: take $\psi = 0$ and let $\Omega$ be the unit ball.
Then every holomorphic $f$ on the unit ball of $\C^{n-1}$ with $\int \abs{f}^2 < \infty$ extends to a square integrable holomorphic function on the unit ball of $\C^n$.
For the ball this is easy, because the trivial extension $F(z', z_n) = f(z')$ is defined on the ball and satisfies $\int_\Omega \abs{F}^2 \le \pi \int_{\Omega \cap H} \abs{f}^2$.
For a general $\Omega$ the point $(z', 0)$ need not lie in $\Omega$ when $(z', z_n)$ does, so the trivial extension is not defined on $\Omega$, and the theorem is needed.

## Explanations

### L1 human

Take a holomorphic function on a slice of a domain cut by a hyperplane.
The theorem says it extends to a holomorphic function on the whole domain, and the extension is not much bigger than the original function.
The size is measured by weighted square integrals, and the bound holds uniformly for all plurisubharmonic weights.

### L2 human

The theorem refines Cartan's theorem B, which gives an extension from a closed submanifold of a Stein manifold but without control of its size.
Because the constant is uniform in the weight, the theorem can be applied to singular weights such as $\psi = c \log \abs{g}$.
This makes it a key tool in the theory of multiplier ideals: it is used in Demailly's approximation of plurisubharmonic functions and in Siu's proof of the invariance of plurigenera.
</document>

<document source="https://tamnd.github.io/math2/node/2T1AGV1K13.md" tag="2T1AGV1K13" kind="definition">
# Primitive ring

## Definition

Let $R$ be a [[ring|0894J980RK]] with identity.
$R$ is *left primitive* if there is a [[simple|G00C22AEDH]] left $R$-[[module|09VDS2W1E8]] $M$ that is [[faithful|TB40Y3SNSX]], that is, $rM = 0$ implies $r = 0$ for $r \in R$.
*Right primitive* rings are defined in the same way with right modules.

## Remarks

Example: let $V$ be a vector space over a field $k$ and $R = \End_k(V)$.
$V$ is a simple $R$-module, since any nonzero vector can be sent to any vector by a linear map.
It is faithful, since a linear map that kills every vector is $0$.
Non-example: $\Z$ is not primitive, since its simple modules are the groups $\Z/p\Z$, and $p$ kills $\Z/p\Z$; no simple module is faithful.
A commutative ring is primitive if and only if it is a field.
Every simple ring with identity is left and right primitive.
A left Artinian ring is left primitive if and only if it is isomorphic to $M_n(D)$ for a division ring $D$ and an integer $n \ge 1$.
There are left primitive rings that are not right primitive.
</document>

<document source="https://tamnd.github.io/math2/node/2T3218RXTD.md" tag="2T3218RXTD" kind="concept">
# Geometric group theory

## Summary

Geometric group theory studies [[finitely generated groups|082DWWXMRJ]] through the geometry of spaces on which they act, such as their [[Cayley graphs|0GYQK9NDJF]] with the [[word metric|VCHJVR9NGH]].
Its basic principle is to study groups up to [[quasi-isometry|6FKYSH729T]], which ignores small-scale details and distortions by bounded factors.
The [[Švarc–Milnor lemma|NTK1TVA50A]] links this view to group actions on metric spaces.
Central results include [[Gromov's theorem on groups of polynomial growth|RSBFHHPTWX]] and [[Stallings's theorem on ends of groups|GWZN3EKEWD]].
Central classes include [[hyperbolic groups|JWJ8YPR7EB]] and [[amenable groups|CEAT9BESKK]].
The field grew from combinatorial group theory and low-dimensional topology.
It took its modern form in the 1980s and 1990s, largely through the work of Gromov.

## Explanations

### L1 human

A group given by generators can be drawn as a graph: one dot for each element, and a line from $g$ to $gs$ for each generator $s$.
Seen from far away, this graph has a shape.
The Cayley graph of $\Z^2$ with its two standard generators looks like a grid and, from far away, like a plane.
The Cayley graph of a free group on two generators is an infinite tree, which looks nothing like a plane.
Geometric group theory reads algebraic properties of the group from such large-scale shapes, and the other way around.
</document>

<document source="https://tamnd.github.io/math2/node/2T33SA540E.md" tag="2T33SA540E" kind="statement">
# Monotone class theorem

## Statement

Let $X$ be a set.
A *monotone class* on $X$ is a family $\mathcal{M}$ of subsets of $X$ with these two properties.
If $A_1 \subseteq A_2 \subseteq \cdots$ are in $\mathcal{M}$, then $\bigcup_{n=1}^{\infty} A_n \in \mathcal{M}$.
If $A_1 \supseteq A_2 \supseteq \cdots$ are in $\mathcal{M}$, then $\bigcap_{n=1}^{\infty} A_n \in \mathcal{M}$.

Let $\mathcal{A}$ be an [[algebra of sets|PFMGZ7988V]] on $X$.
Then the smallest monotone class on $X$ that contains $\mathcal{A}$ is equal to the [[σ-algebra|0SV8K0TDGY]] $\sigma(\mathcal{A})$ generated by $\mathcal{A}$.

## Remarks

Every σ-algebra is a monotone class, so the smallest monotone class $m(\mathcal{A})$ is contained in $\sigma(\mathcal{A})$.
For the other inclusion, one shows that $m(\mathcal{A})$ is an algebra.
For fixed $A$, the sets $B$ with $A \cup B$, $A \setminus B$ and $B \setminus A$ in $m(\mathcal{A})$ form a monotone class; this gives closure first for $A \in \mathcal{A}$, then for $A \in m(\mathcal{A})$.
An algebra that is a monotone class is a σ-algebra, since a countable union is the increasing union of finite unions.

As an application, let $\mu$ and $\nu$ be finite measures on $(X, \sigma(\mathcal{A}))$ with $\mu(X) = \nu(X)$ that agree on $\mathcal{A}$.
The sets on which they agree form a monotone class, by continuity of measures, so $\mu = \nu$ on $\sigma(\mathcal{A})$.

The hypothesis that $\mathcal{A}$ is an algebra is needed.
On $X = \set{1, 2}$, the family $\set{\varnothing, \set{1}}$ is already a monotone class, but the σ-algebra it generates also contains $\set{2}$.

## Explanations

### L1 human

A monotone class is a family of sets that is stable under limits of growing sequences and of shrinking sequences.
The theorem says: start from an algebra of sets, and close it under such limits only.
Then you already get every set of the generated σ-algebra.
So a property that survives monotone limits and holds on the algebra holds on the whole σ-algebra.

### L2 human

The theorem is a tool to extend statements from simple sets to all measurable sets.
It is used to prove uniqueness of extended measures and to set up product measures and the Fubini–Tonelli theorem.
Dynkin's π-λ theorem is a variant in which the algebra is replaced by a π-system and the monotone class by a Dynkin system.
There is also a version for classes of functions, used in probability theory.
</document>

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# Finite element exterior calculus

## Summary

Finite element exterior calculus is a framework for designing and analysing finite element methods with the tools of [[differential forms|0E3G0GRPZ6]] and [[de Rham cohomology|GX74C07T0G]].
It was developed by Douglas Arnold, Richard Falk and Ragnar Winther.
Its central idea is that finite element spaces should form a subcomplex of the de Rham complex, with projections that commute with the [[exterior derivative|0ZHSTAG8ZY]].
Such spaces give stable [[mixed finite element methods|87VMQ892VW]] for the Hodge Laplacian and explain many earlier elements in one theory.
See the [[definition|GRBZT7SWK0]].

## Explanations

### L1 human

In three dimensions, the gradient, the curl and the divergence form a chain: the curl of a gradient is zero, and so is the divergence of a curl.
Good numerical methods for electromagnetism or fluid flow must respect this chain.
Finite element exterior calculus builds matching chains of finite element spaces, one space for each link.
For example, continuous piecewise linear functions, Nédélec edge elements, Raviart–Thomas face elements and piecewise constants form such a chain.
</document>

<document source="https://tamnd.github.io/math2/node/2TDSHJJYS4.md" tag="2TDSHJJYS4" kind="concept">
# Balanced set

## Summary

A subset of a real or complex vector space is balanced, or circled, if it is stable under scaling.
Here scaling means multiplication by any scalar of absolute value at most $1$.
In a [[topological vector space|T32W553W95]] the balanced neighbourhoods of $0$ form a neighbourhood base at $0$.
A set that is convex, balanced and absorbing gives a [[seminorm|FP5GVCN8GV]], through the [[Minkowski functional|PAX9SHDHDW]].

## Explanations

### L1 human

A balanced set is symmetric about the origin in a strong sense.
Over the reals, it contains $-x$ whenever it contains $x$, and it contains the whole segment from $-x$ to $x$.
Over the complex numbers, it also contains every rotation $e^{i \theta} x$.
A disc centred at $0$ is balanced.
The interval $[0, 1]$ is not, because it contains $1$ but not $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/2TE42ZWJ3E.md" tag="2TE42ZWJ3E" kind="concept">
# Design of experiments

## Summary

Design of experiments is the part of statistics that plans how data are collected when the experimenter controls the conditions.
Its aim is to answer the questions of interest validly and efficiently.
Its three classical principles, set out by Ronald Fisher, are [[randomization|734Q6Q51X7]], replication and local control by [[blocking|XCWKYS1DFA]].
Its main designs include completely randomized and block designs, Latin squares, [[factorial|EHJ3R589ZX]] and [[fractional factorial|PBK2XY4M0A]] designs, and response surface designs.
[[Optimal experimental design|PM5PQX6TVP]] chooses designs that make the estimates as precise as possible, and the data are usually analysed by [[analysis of variance|M9AQ5HKGP4]].

## Explanations

### L1 human

To learn whether a new fertiliser works, one could just look at farms that use it, but those farms may differ in many other ways.
In an experiment, the researcher decides which plots get which fertiliser, and that choice is the design.
A good design repeats each treatment several times, assigns treatments by chance, and compares treatments on similar plots.
Then a difference in yield can be credited to the fertiliser and not to something else.
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# Germ of a holomorphic function

## Definition

Let $a \in \C^n$.
Consider pairs $(U, f)$ where $U \subseteq \C^n$ is an open [[neighbourhood|0ZCY0Z80B6]] of $a$ and $f \colon U \to \C$ is a [[holomorphic function|507C34W0ZT]].
Two pairs $(U, f)$ and $(V, g)$ are equivalent if there is an open neighbourhood $W \subseteq U \cap V$ of $a$ with $f = g$ on $W$.
A *germ of a holomorphic function* at $a$ is an equivalence class for this relation.
The germ of $(U, f)$ is written $f_a$, and the set of all germs at $a$ is written $\mathcal{O}_{n,a}$, or $\mathcal{O}_a$.

## Remarks

This is the notion of [[germ|Y98MD3WVTG]] applied to holomorphic functions.
Sums and products of representatives on a common neighbourhood make $\mathcal{O}_{n,a}$ a commutative ring, and evaluation $f_a \mapsto f(a)$ is well defined.
Example: the pairs $(\C \setminus \set{1}, 1/(1 - z))$ and $(\set{\abs{z} < 1}, \sum_{k \ge 0} z^k)$ define the same germ at $0 \in \C$, since the two functions agree on the unit disc.
Non-example: the pairs $(\C, z)$ and $(\C, z + z^2)$ do not define the same germ at $0$, since $z^2 \ne 0$ at every point $z \ne 0$ near $0$.
By the identity theorem, two holomorphic functions on a connected open set containing $a$ have the same germ at $a$ only if they are equal.
Taking Taylor series at $a$ gives an isomorphism of $\mathcal{O}_{n,a}$ onto the ring of convergent power series in $z_1 - a_1, \dots, z_n - a_n$.
The germs at all points of an open set $D$ are the stalks of the sheaf $\mathcal{O}$ of holomorphic functions on $D$.
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# Mellin convergence

## Definition

Let $f \colon (0, \infty) \to \C$ be measurable and let $s \in \C$.
For $x > 0$, let $x^{s-1} = e^{(s-1)\log x}$ be the [[complex power|0RNGDMHT7M]].
The function $f$ is *Mellin convergent at $s$* if the function $x \mapsto x^{s-1} f(x)$ is [[integrable|X7GTF3D98F]] on $(0, \infty)$ with respect to Lebesgue measure, that is,
$$\int_0^\infty x^{\operatorname{Re} s - 1} \abs{f(x)} \, dx < \infty.$$

## Remarks

The two forms agree because $\abs{x^{s-1}} = x^{\operatorname{Re} s - 1}$.
So Mellin convergence at $s$ depends only on $\operatorname{Re} s$.
If $f$ is Mellin convergent at $\sigma_1$ and $\sigma_2$ with $\sigma_1 < \sigma_2$, then it is Mellin convergent at every $s$ with $\sigma_1 \le \operatorname{Re} s \le \sigma_2$, because $x^{\sigma - 1} \le x^{\sigma_1 - 1} + x^{\sigma_2 - 1}$ for $\sigma_1 \le \sigma \le \sigma_2$.
Example: $f(x) = 1/(1 + x)$ at $s = 1/2$.
Here $\int_0^\infty x^{-1/2}/(1 + x) \, dx = \pi$, by the substitution $x = t^2$, so $f$ is Mellin convergent at $1/2$; the same estimates show it is Mellin convergent exactly when $0 < \operatorname{Re} s < 1$.
Non-example: the same $f$ at $s = 1$.
The integral $\int_0^\infty dx/(1 + x)$ diverges at $\infty$, so $f$ is not Mellin convergent at $1$.
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# Generated subalgebra

## Summary

The subalgebra generated by a subset $X$ of an [[algebra|QJGPT2W0FT]] is the smallest [[subalgebra|676H7EGTKJ]] that contains $X$.
Its elements are the values of all terms in the operations applied to elements of $X$ and to the constants.
The [[subgroup generated by a subset|0RP7VWQ0YN]] and the [[ideal generated by a subset|0GSW76Q14N]] are special cases of this idea.
The map that sends $X$ to the subalgebra it generates is a [[closure operator|2HJGRABSPZ]], and an algebra generated by a finite set is [[finitely generated|WR4WF1K2KW]].

## Explanations

### L1 human

Start with some elements, then apply the operations to them again and again, and collect everything you get.
The result is the generated subalgebra: it contains the starting elements and is closed under the operations, and nothing smaller has both properties.
In the integers with addition, negation and $0$, starting from $4$ and $6$ you get $10$, $2 = 6 - 4$, and then all multiples of $2$.
So $\set{4, 6}$ generates the even integers.
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# Hankel contour integral for the gamma function

## Statement

Let $\mathcal{H}$ be a Hankel contour.
It is a path that comes from $-\infty$ below the negative real axis and goes once around $0$ counterclockwise.
Then it returns to $-\infty$ above the negative real axis.
For $t \notin (-\infty, 0]$ let $t^{-z} = e^{-z \operatorname{Log} t}$, with $\operatorname{Log}$ the principal [[complex logarithm|02QGGD5G87]].
On the two sides of the cut, $t^{-z}$ takes its limits from below and from above.
Then for every $z \in \C$,
$$\frac{1}{\Gamma(z)} = \frac{1}{2\pi i} \int_{\mathcal{H}} e^{t} \, t^{-z} \, dt,$$
where $1/\Gamma$ is the [[reciprocal gamma function|PY6K9AWR33]].

## Remarks

The integral converges for every $z$ because $e^t$ decays fast as $\operatorname{Re} t \to -\infty$, and it is an entire function of $z$.
For $\operatorname{Re} z < 1$, the path can be squeezed onto the cut, with $t = s e^{-i\pi}$ on the lower side and $t = s e^{i\pi}$ on the upper side, $s > 0$.
The small circle around $0$ contributes nothing in the limit.
The two sides give $\frac{1}{2\pi i}(e^{i\pi z} - e^{-i\pi z}) \int_0^\infty e^{-s} s^{-z}\, ds = \frac{\sin(\pi z)}{\pi}\,\Gamma(1 - z)$, which equals $1/\Gamma(z)$ by the [[reflection formula|WF7KQAQTS2]].
The identity theorem then extends the formula to all $z \in \C$.
At a positive integer $z = n$ the integrand $e^t t^{-n}$ is single-valued, and the formula reduces to the residue at $0$, which is $1/(n-1)!$.

## Explanations

### L1 human

Euler's integral defines the gamma function only for arguments with positive real part.
Hankel's idea is to integrate along a loop around the origin instead of along a half-line.
The resulting formula for $1/\Gamma$ works for every complex number at once.

### L2 human

The Hankel integral gives a direct proof that $1/\Gamma$ is entire, without first building the meromorphic continuation of $\Gamma$.
Similar loop integrals represent other functions with branch points, such as the Riemann zeta function, and the same technique gives their continuations.
It is also a starting point for asymptotic estimates of $1/\Gamma$ by the saddle point method.
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# Beurling's theorem

## Statement

Let $H^2$ be the [[Hardy space|PNNNFJVF6W]] of the unit disc $\mathbb{D}$, a [[Hilbert space|HM2HHE3J51]].
Let $S \colon H^2 \to H^2$ be the [[shift operator|MBK6AE6734]] $(S f)(z) = z f(z)$.
Let $M \neq \set{0}$ be a closed subspace of $H^2$.
Then $M$ is an [[invariant subspace|0EP59ESQ94]] of $S$, that is $S M \subseteq M$, if and only if
$$M = \theta H^2 = \set{\theta f : f \in H^2}$$
for some [[inner function|NN8XH5RJYB]] $\theta$.
The inner function $\theta$ is unique up to a constant factor of modulus $1$.

## Remarks

Proof idea: $S$ is an isometry and $\bigcap_n S^n M = \set{0}$, so $M \ominus S M$ is not zero.
Take $\theta$ of norm $1$ in $M \ominus S M$.
Then $\theta \perp S^n \theta$ for $n \ge 1$, which says that all Fourier coefficients of $\abs{\theta^*}^2$ vanish except the one of index $0$; hence $\abs{\theta^*} = 1$ almost everywhere and $\theta$ is inner.
A decomposition of $M$ into the orthogonal pieces $S^n (M \ominus S M)$ then shows $M = \theta H^2$.

Example: the functions in $H^2$ that vanish at a point $a \in \mathbb{D}$ form a closed invariant subspace; it equals $\theta H^2$ with $\theta(z) = \frac{z - a}{1 - \bar a z}$.

A consequence: $f \in H^2$ is cyclic for $S$, meaning that the polynomial multiples of $f$ are dense, if and only if $f$ is an [[outer function|KGJD4A4ATY]].

## Explanations

### L1 human

Multiplying by $z$ is a basic operator on the space $H^2$.
The theorem lists every closed subspace that this operator maps into itself.
Each one consists of all multiples of a single inner function.
So a question about operators becomes a question about functions, which complex analysis can answer.

### L2 human

Beurling's theorem was among the first complete descriptions of the invariant subspaces of a natural operator on an infinite-dimensional space.
Together with the [[inner-outer factorization|X6ZQPAQF76]], it shows that the invariant subspace generated by $f$ is $I H^2$, where $I$ is the inner factor of $f$.
Lax extended it to vector-valued Hardy spaces, which is the Beurling–Lax theorem; the model theory of contractions builds on it.
Its dual form describes the invariant subspaces of the backward shift as the model spaces $H^2 \ominus \theta H^2$.
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# Method of multiple scales

## Summary

The method of multiple scales builds approximations that stay valid over long times for problems with a small parameter.
It treats a fast time $t$ and a slow time $T = \varepsilon t$ as independent variables.
The slow dependence is chosen to remove secular terms, which would otherwise make the error grow.
It handles damping, slow modulation of amplitude and phase, and other effects that the [[Poincaré–Lindstedt method|YHJHPRKQ6D]] cannot capture.

## Explanations

### L1 human

A weakly damped oscillator swings quickly while its amplitude shrinks slowly.
The method describes the swing with a fast clock and the shrinking with a slow clock.
For $x'' + 2\varepsilon x' + x = 0$ it gives $x \approx e^{-\varepsilon t} (a \cos t + b \sin t)$.
A plain expansion in $\varepsilon$ would instead give $a \cos t$ plus a term growing like $\varepsilon t$, which is wrong once $t$ is of order $1/\varepsilon$.
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# Absorbing set

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $V$ be a [[vector space|05EGHEQZAW]] over $\mathbb{K}$.
A subset $A \subseteq V$ is *absorbing* if the following holds.
For every $x \in V$ there is $r > 0$ such that $x \in c A$ for every $c \in \mathbb{K}$ with [[absolute value|00S7C68D51]] $\abs{c} \ge r$.

## Remarks

An absorbing set $A$ contains $0$: for $x = 0$ there is $c \ne 0$ with $0 \in cA$, so $0 = c a$ for some $a \in A$, and then $a = 0$.
If $A$ is [[balanced|2TDSHJJYS4]], then $A$ is absorbing if and only if for every $x \in V$ there is $t > 0$ with $t x \in A$.
Example: the open unit ball $A = \set{x : \norm{x} < 1}$ of a normed space is absorbing; for $x \in V$ take $r = \norm{x} + 1$, and then $\norm{x / c} < 1$ whenever $\abs{c} \ge r$.
Non-example: in $\R^2$, the axis $A = \R \times \set{0}$ is not absorbing, because $(0, 1) \notin c A$ for every $c$.
In a [[topological vector space|T32W553W95]] every neighbourhood of $0$ is absorbing, since $c \mapsto c^{-1} x$ is continuous and tends to $0$.
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# Contravariant functor

## Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
A *contravariant functor* $F$ from $\mathcal{C}$ to $\mathcal{D}$ consists of:

1. for each object $A$ of $\mathcal{C}$, an object $F(A)$ of $\mathcal{D}$;
2. for each morphism $f \colon A \to B$ of $\mathcal{C}$, a morphism $F(f) \colon F(B) \to F(A)$ of $\mathcal{D}$;

such that:

1. $F(1_A) = 1_{F(A)}$ for every object $A$;
2. $F(g \circ f) = F(f) \circ F(g)$ for all $f \colon A \to B$ and $g \colon B \to C$.

## Remarks

A contravariant functor from $\mathcal{C}$ to $\mathcal{D}$ is the same as a [[functor|28NQ0KRHD3]] $\mathcal{C}^{\mathrm{op}} \to \mathcal{D}$ from the [[opposite category|DAMK8XE1FR]], via $f^{\mathrm{op}} \mapsto F(f)$.
Example: the [[power set|00E0YZCERH]] with preimages, $A \mapsto \mathcal{P}(A)$ and $f \mapsto f^{-1}$, is a contravariant functor from sets to sets.
Here $f^{-1} \colon \mathcal{P}(B) \to \mathcal{P}(A)$, and $(g \circ f)^{-1}(S) = f^{-1}(g^{-1}(S))$ gives clause 2.
Another example is the [[dual space|07JBSZE0V4]] functor $V \mapsto V^*$, $f \mapsto f^*$ on vector spaces over a field, where $(g \circ f)^* = f^* \circ g^*$.
Non-example: the power set with direct images, $f \mapsto (S \mapsto f(S))$, is not contravariant, because $f(\cdot)$ goes from $\mathcal{P}(A)$ to $\mathcal{P}(B)$, so clause 2 of the data fails; it is a covariant functor.
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# Matlis duality

## Statement

Let $(R, \mathfrak{m}, k)$ be a [[complete local ring|G9Q61B8A7C]] that is [[Noetherian|0670T59HGE]], with maximal ideal $\mathfrak{m}$ and [[residue field|NJ3APQ8BHH]] $k$.
Let $E = E_R(k)$ be the [[injective hull|QCJN05GN64]] of $k$, and for an $R$-module $M$ write $M^\vee = \Hom_R(M, E)$.
Then:

1. If $M$ is a [[Noetherian module|AK1ZRCXR32]], then $M^\vee$ is an [[Artinian module|YQNZTQGP1X]].
2. If $M$ is an Artinian module, then $M^\vee$ is a Noetherian module.
3. If $M$ is Noetherian or Artinian, then the natural map $M \to M^{\vee\vee}$ is an isomorphism.

Hence $M \mapsto M^\vee$ is an anti-equivalence between the category of Noetherian $R$-modules and the category of Artinian $R$-modules.

## Remarks

The key facts are that $E$ is an Artinian module, that $\Hom_R(k, E) \cong k$, and that the map $R \to \Hom_R(E, E)$ is an isomorphism when $R$ is complete.
For a Noetherian local ring that is not complete, $\Hom_R(E, E)$ is the completion $\hat{R}$, and the duality relates Artinian $R$-modules with finitely generated $\hat{R}$-modules.
For modules of finite length, the duality preserves length.
Example: for $R = k[\![x]\!]$ one has $E = k[x^{-1}]$, with $x$ acting by $x \cdot x^{-j} = x^{-j+1}$ and $x \cdot 1 = 0$; then $R^\vee = E$ and $E^\vee = R$.

## Explanations

### L1 human

Over a complete local ring there is one special injective module $E$.
Mapping into $E$ turns modules with no infinite increasing chains of submodules into modules with no infinite decreasing chains, and back.
Doing it twice returns the module you started with.

### L2 human

Matlis duality is the analogue, for modules over a complete local ring, of the duality $V \mapsto V^*$ for finite-dimensional vector spaces.
It is used to pass from local cohomology modules, which are Artinian, to finitely generated modules.
This is how [[Grothendieck's local duality|VA3DY2YX83]] is usually stated and applied.
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# Fredholm index

## Summary

The index of a Fredholm operator is the dimension of its kernel minus the dimension of its cokernel.
It is an integer that does not change under small perturbations, under compact perturbations, or along continuous paths of Fredholm operators.
The Atiyah–Singer index theorem computes it for elliptic differential operators in terms of topology.

## Explanations

### L1 human

The index counts how many more solutions a linear equation has than conditions it fails to meet.
The kernel counts independent solutions of $Tx = 0$, and the cokernel counts independent directions missed by $T$.
For a linear map from $\R^5$ to $\R^3$, the index is always $5 - 3 = 2$, whatever the map is.
The surprise is that in infinite dimensions the index is still stable under small changes, although the kernel and cokernel alone can jump.
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# Suspension

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]], and give $X \times [0, 1]$ the [[product topology|0GBG8B49AA]].
Let $\sim$ be the [[equivalence relation|0G8VEZ3WTK]] on $X \times [0, 1]$ generated by the following relations, for all $x, y \in X$:

1. $(x, 0) \sim (y, 0)$;
2. $(x, 1) \sim (y, 1)$.

The *suspension* of $X$ is the quotient
$$SX = (X \times [0, 1]) / {\sim},$$
with the [[quotient topology|0NQ1H2X495]].

## Remarks

The two classes $X \times \set{0}$ and $X \times \set{1}$ are the two cone points of $SX$.
A continuous map $f \colon X \to Y$ induces a continuous map $Sf \colon SX \to SY$ by $[x, t] \mapsto [f(x), t]$.

Example: let $X = S^0 = \set{-1, 1}$, a two-point discrete space.
Then $X \times [0, 1]$ is two segments, and the relation joins their lower ends and their upper ends.
The result is [[homeomorphic|0QT00PB5G2]] to the circle $S^1$.
More generally, $S(S^n)$ is homeomorphic to $S^{n+1}$ for every $n \ge 0$.

Non-example: collapsing only $X \times \set{1}$ gives the cone $CX$, not the suspension.
For $X = S^1$ the cone is a disc, which has a boundary circle, while $SS^1$ is a sphere.

For a pointed space $(X, x_0)$, the reduced suspension $\Sigma X$ also collapses the segment $\set{x_0} \times [0, 1]$.
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# I-adic topology

## Summary

Let $M$ be a module over a [[commutative ring|0310REJJY8]] $R$, and let $I$ be an [[ideal|096Z9MP2ME]] of $R$.
In the $I$-adic topology on $M$, the submodules $I^n M$ form a base of neighbourhoods of $0$.
Two elements are close when their difference lies in a high power of $I$.
It is the topology behind the [[completion of a ring|KBN13NPYJQ]], and the $p$-adic topology on $\Z$ is the main example.

## Explanations

### L1 human

In this topology an element is small when it is divisible by a high power of $I$.
For the integers and $I = (2)$, the number $2^{10}$ is very small and $1024 + 3$ is very close to $3$.
The powers $2, 4, 8, 16, \dots$ tend to $0$, which never happens for the usual distance on the real line.
The [[definition|MJDV3PFBRT]] makes this precise.
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# Derived category of coherent sheaves

## Summary

Let $X$ be a Noetherian scheme.
Its derived category of coherent sheaves $D^b(X)$ is the bounded [[derived category|5ST7NVJFSM]] of the abelian category of [[coherent sheaves|56RXEJ8Z6S]] on $X$.
It is a [[triangulated category|T2V7GJJ2EN]].
For $X$ smooth and projective over a field, the derived functors of pushforward, pullback, tensor product and Hom act on these categories.
It is the natural home of Serre and Grothendieck duality and of Fourier–Mukai transforms.
Two varieties with equivalent derived categories share many invariants, so $D^b(X)$ is studied as an invariant of $X$ in its own right.

## Explanations

### L1 human

Many constructions with sheaves, such as cohomology, do not produce a single sheaf but a whole complex of sheaves.
The derived category keeps these complexes and treats two complexes as the same when a map between them is an isomorphism on cohomology.
For a point, an object is just a list of finite-dimensional vector spaces, one in each degree.
For the projective line, every object is a direct sum of shifts of the line bundles $\mathcal{O}(n)$ and of shifts of torsion sheaves.
Some varieties that are not isomorphic have equivalent derived categories: by a theorem of Mukai, an abelian variety and its dual always do, and they need not be isomorphic.
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# Nichols algebra

## Definition

Let $k$ be a field and $V$ a [[vector space|05EGHEQZAW]] over $k$.
Let $c \colon V \otimes V \to V \otimes V$ be an invertible linear map that satisfies the braid relation $c_{12} c_{23} c_{12} = c_{23} c_{12} c_{23}$ on $V^{\otimes 3}$, the braid form of the [[Yang–Baxter equation|663BP8ZHS5]].
The pair $(V, c)$ is a *braided vector space*.
For $n \ge 2$ the [[braid group|SRS8WT4H8Y]] $B_n$ acts on $V^{\otimes n}$, the generator $\sigma_i$ acting as $c$ on the factors $i$ and $i+1$.
Let $s$ be a permutation in the [[symmetric group|0RJMNEN3BJ]] $S_n$, and take a reduced expression of $s$ in the simple transpositions.
Replace each transposition $(i\ i{+}1)$ by $\sigma_i$, and let $c_s$ be the action of the resulting element of $B_n$.
The *quantum symmetrizer* is $\Omega_n = \sum_{s \in S_n} c_s$.
The *Nichols algebra* of $(V, c)$ is
$$\mathfrak{B}(V) = T(V) \big/ \bigoplus_{n \ge 2} \ker \Omega_n,$$
where $T(V)$ is the [[tensor algebra|P4J5JZDW0B]] of $V$.

## Remarks

The lift $c_s$ does not depend on the reduced expression, by Matsumoto's theorem, and $\bigoplus_{n \ge 2} \ker \Omega_n$ is a graded two-sided ideal of $T(V)$.
Example: let $V = kx$ be one-dimensional with $c(x \otimes x) = q\, x \otimes x$, where $q \ne 1$ is a root of unity of order $N \ge 2$.
Each $c_s$ multiplies $x^{\otimes n}$ by $q^{\ell(s)}$, with $\ell(s)$ the length of $s$, so $\Omega_n(x^{\otimes n}) = (n)_q!\, x^{\otimes n}$, where $(m)_q = 1 + q + \cdots + q^{m-1}$.
This is zero exactly when $n \ge N$, so $\mathfrak{B}(V) = k[x]/(x^N)$.
Non-example: for $c$ the flip and $\dim V \ge 2$, the tensor algebra $T(V)$ itself is not the Nichols algebra, because $\Omega_2(u \otimes v - v \otimes u) = 0$ for $u, v \in V$, so $uv - vu$ is zero in $\mathfrak{B}(V)$.
If $\operatorname{char} k = 0$, then $\mathfrak{B}(V)$ is the symmetric algebra of $V$ for $c$ the flip, and the exterior algebra of $V$ for $c$ minus the flip.
Equivalently, $\mathfrak{B}(V)$ is the graded braided Hopf algebra $\bigoplus_{n \ge 0} R(n)$ with $R(0) = k$ and $R(1) = V$ that is generated by $V$ and whose primitive elements are exactly $V$.
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# Namioka--Phelps

## Statement

Let $X$ be a [[compact space|0QPHW703PV]]. Let $C(X)$ be the [[space of bounded continuous functions|D38YWGXRKJ]] on $X$, which is a [[Banach space|06GZ956657]] under the supremum norm.

A Banach space $E$ is *Asplund* when each closed separable subspace of $E$ has a separable [[continuous dual space|4A82S5FVRT]].

The space $X$ is *scattered* when each nonempty subset $A \subseteq X$ has a point that is [[isolated|01PKQYTTMT]] in $A$ for the subspace topology.

Then $C(X)$ is Asplund if and only if $X$ is scattered.
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# Noetherian topological space

## Summary

A Noetherian topological space is a [[topological space|0CQHFPGJ45]] in which every descending chain of closed subsets eventually stops.
It is the topological counterpart of a [[Noetherian ring|0670T59HGE]]: the spectrum of a Noetherian ring with the Zariski topology is a Noetherian space.
Every Noetherian space is [[compact|0TVJZYB6SH]], and so is each of its subspaces.
Every Noetherian space is a finite union of [[irreducible|38CDCK1RYB]] closed subsets, its irreducible components.

## Explanations

### L1 human

In a Noetherian space you cannot keep cutting closed sets smaller and smaller forever.
Any chain of shrinking closed sets must become constant after finitely many steps.
Finite spaces are Noetherian, and so is any set with the [[cofinite topology|0B906ZBTMZ]].
The real line is not: the closed sets $[n, \infty)$ for $n = 1, 2, 3, \dots$ shrink forever.
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# Auslander–Reiten theory

## Summary

Auslander–Reiten theory studies the category of finitely generated modules over an Artin algebra, such as a finite-dimensional algebra, through irreducible morphisms and [[almost split sequences|M5D95FN7G3]].
Its main tools are the Auslander–Reiten translation $\tau$ and the [[Auslander–Reiten quiver|125Z033VCQ]].
Maurice Auslander and Idun Reiten introduced it in the 1970s.
It is the standard method to describe module categories of algebras of finite representation type and of many tame algebras.

## Explanations

### L1 human

A module category has too many maps to look at all of them.
Auslander–Reiten theory keeps only the indecomposable modules and the maps that cannot be split into simpler maps, the irreducible ones.
These form a quiver, a map of the whole category.
The translation $\tau$ moves along this map, taking a module to the left end of the almost split sequence that ends at it.
For the path algebra of $1 \to 2$, the map has three modules and two irreducible maps between them.
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# Mac Lane's coherence theorem

## Statement

Let $(\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)$ be a [[monoidal category|M3MSQ3NKAE]].
Call a formal word any expression built from variables $x_1, x_2, \ldots$ and the unit $I$ with the binary operation $\otimes$.
Call a formal morphism between two words any composite of formal instances of $\alpha$, $\lambda$, $\rho$ and their inverses, tensored on both sides with identities.

1. Any two formal morphisms between the same two words are equal after every substitution of objects of $\mathcal{C}$ for the variables. In other words, every formal [[diagram|WV898P4X16]] built from $\alpha$, $\lambda$ and $\rho$ commutes.
2. $\mathcal{C}$ is equivalent, by a strong [[monoidal functor|PY430TXMG0]] that is an [[equivalence of categories|K7A04ZBD92]], to a strict monoidal category.

## Remarks

Proof idea for part 1: every word can be moved by a formal morphism to a normal form.
For example, the normal form can put all brackets to the right and remove all copies of $I$.
One shows that any two such moves to normal form are equal, using the [[pentagon identity|XM4NTS5BGG]] and the triangle identity to compare the possible first steps.
Then any formal morphism $u \to v$ equals the composite of the move $u \to$ normal form and the inverse of the move $v \to$ normal form.
Equivalently, the free monoidal category on a set of variables is a preorder.

The word "formal" is necessary.
A diagram of instances of $\alpha$, $\lambda$, $\rho$ on actual objects may fail to commute.
This can happen when the diagram only makes sense because two different words give equal objects, for example when $A \otimes B = C$ holds in $\mathcal{C}$.
Formal diagrams track each variable, so such accidental identifications are not allowed.

## Explanations

### L1 human

In a monoidal category, brackets and unit objects can be moved around using fixed isomorphisms.
The theorem says that it never matters which route of such moves one takes: all routes give the same isomorphism.
So one may safely write $A \otimes B \otimes C$ without brackets, as for an ordinary associative product.

### L2 human

Coherence is what makes calculations in monoidal categories practical: it justifies string diagrams and the habit of pretending that the tensor product is strictly associative.
It reduces infinitely many coherence conditions to the two axioms of the definition, the pentagon and the triangle.
For braided monoidal categories the analogous statement says that formal diagrams commute when they have the same underlying braid; for symmetric ones, when they have the same underlying permutation.
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# Dense linear order

## Example

A [[linear order|0FRXBVVR03]] $(D, <)$ is *dense* if for all $a < b$ in $D$ there is $c \in D$ with $a < c < b$.
The theory $\mathrm{DLO}$ of dense linear orders without endpoints is a theory in the signature $\set{<}$.
It says that $<$ is a dense linear order on a set with at least two elements.
It also says that there is no least and no greatest element.
The [[rational numbers|0NCJ5WXN6Z]] and the real numbers with their usual order are models.

$\mathrm{DLO}$ is $\aleph_0$-categorical: any two countable models are [[order isomorphic|0D3WA36K2Q]].
Check, by Cantor's back-and-forth argument: list the elements of two countable models $A$ and $B$, and build a chain of finite order-preserving partial maps.
At odd steps, send the next element of $A$ to an element of $B$ in the same position relative to the finite image.
This element exists because $B$ is dense and has no endpoints.
At even steps, do the same from $B$ back to $A$.
The union of the chain is an isomorphism from $A$ onto $B$.

## Remarks

Since $\mathrm{DLO}$ has no finite models, the [[Łoś–Vaught test|ZR7Y5C235P]] shows that it is complete; it is also decidable.
It has [[quantifier elimination|7FDDXV3JK7]].
It is not $\kappa$-categorical for any uncountable $\kappa$.
For $\kappa = 2^{\aleph_0}$, compare $\R$ with the order $\R + \Q$, a copy of $\R$ followed by a copy of $\Q$.
Both are dense without endpoints and have size $2^{\aleph_0}$.
They are not isomorphic.
In $\R$ every nonempty set with an upper bound has a least upper bound.
In $\R + \Q$ the copy of $\R$ is bounded above but has no least upper bound.

## Explanations

### L1 human

In a dense order, between any two elements there is a third, like the rational numbers on the number line.
Every countable dense order without a first or last element is a copy of the rational numbers.
This makes the theory of such orders one of the simplest examples in model theory.
It is complete, it has only one countable model, and every formula reduces to comparisons between the variables.
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# Completion of a normed space

## Summary

The completion of a normed space $X$ is a [[Banach space|06GZ956657]] that contains a copy of $X$ as a dense subspace, with the same norm.
It exists and is unique up to isometric isomorphism.
It lets one prove results for Banach spaces and apply them to incomplete spaces.
Many important spaces, such as $L^p$ spaces and Sobolev spaces, can be built as completions of spaces of nice functions.

## Explanations

### L1 human

A normed space may have holes: Cauchy sequences that do not converge to anything inside it.
The completion fills these holes and changes nothing else, much as the real numbers fill the gaps in the rationals.
For example, finitely supported sequences with the norm $\left(\sum_k \abs{x_k}^2\right)^{1/2}$ do not form a complete space.
Their completion is the space $\ell^2$ of all square-summable sequences.
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# Mathematical statistics

## Definition

Mathematical statistics is the study of [[statistical inference|FXCQWFY5RV]] with the methods of [[probability theory|0RWE7CPJ32]].
Its setting is a [[statistical model|JZ6R7J36K5]] $(S, \mathcal{P})$.
The data $x \in S$ are a value of a random element $X$ whose law $P$ is an unknown member of $\mathcal{P}$.
Its objects are [[statistics|036756NFD6]] $T = g(X)$ and the procedures built from them.
It judges each procedure by the laws that the procedure has under the members of $\mathcal{P}$.

## Remarks

Example: in the model of $n$ independent $\mathcal{N}(\mu, 1)$ observations with $\mu \in \R$, the sample mean $\bar{X}_n$ has law $\mathcal{N}(\mu, 1/n)$ for every $\mu$.
This law shows that $\bar{X}_n$ is unbiased for $\mu$ with variance $1/n$, a typical result of mathematical statistics.
Non-example: drawing a bar chart of one data set is descriptive statistics; it uses no model $\mathcal{P}$ and makes no claim about other possible data.
The main branches are point estimation, hypothesis testing, confidence sets and decision theory, each in a parametric or a nonparametric form.
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# Natural duality

## Definition

Let $M$ be a finite algebra and let $\mathcal{A} = \mathbf{ISP}(M)$ be the [[quasivariety|WRXEKT4NQS]] it generates.
An *alter ego* of $M$ is a structure $\underset{\sim}{M} = (M; G, H, R, \mathcal{T})$ on the same set, where $\mathcal{T}$ is the discrete [[topology|0CQHFPGJ45]] and:

1. $G$ is a set of operations $M^n \to M$ that are [[homomorphisms|KBXSZC6BV1]] from $M^n$ to $M$;
2. $H$ is a set of partial operations whose domains are [[subalgebras|7CHG6SD365]] of powers $M^n$ and which are homomorphisms on them;
3. $R$ is a set of relations $r \subseteq M^n$ that are subalgebras of $M^n$.

For $A \in \mathcal{A}$, the set $D(A)$ of homomorphisms $A \to M$ is a closed substructure of the power $\underset{\sim}{M}^A$ with the product topology.
Let $X$ be a closed substructure of a nonzero power of $\underset{\sim}{M}$.
The set $E(X)$ of continuous maps $X \to M$ that preserve $G$, $H$ and $R$ is a subalgebra of $M^X$.
Each $A$ has the evaluation homomorphism $e_A \colon A \to E(D(A))$, $e_A(a)(x) = x(a)$.
The alter ego $\underset{\sim}{M}$ *yields a natural duality* on $\mathcal{A}$ if $e_A$ is an isomorphism for every $A \in \mathcal{A}$.

## Remarks

The duality is *full* if, in addition, every closed substructure $X$ of a nonzero power of $\underset{\sim}{M}$ is isomorphic to $D(E(X))$ by the evaluation map.
Then $D$ and $E$ form a dual equivalence of categories.

Example: let $M$ be the two-element Boolean algebra and let $\underset{\sim}{M}$ be the two-element set with the discrete topology and $G = H = R = \emptyset$.
It yields a natural duality on Boolean algebras: this is Stone's representation of a Boolean algebra as the clopen subsets of its Stone space.
For the two-element bounded lattice, the alter ego with the order relation $\le$ on $\set{0, 1}$ yields Priestley duality.

Non-example: for the two-element bounded lattice $M$, the alter ego with no structure besides the topology does not yield a duality.
Take for $A$ the three-element chain $0 < a < 1$.
It has two homomorphisms to $M$, sending $a$ to $0$ or to $1$, so $D(A)$ is a two-point discrete space.
That space has four continuous maps to $M$, while $A$ has three elements, so $e_A$ is not onto.
The relation $\le$ is a subalgebra of $M^2$, and adding it cuts these maps down to the three order-preserving ones.
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# Ideal of compact operators

## Summary

The compact operators on a Hilbert space $H$ form a set $\mathcal{K}(H)$.
It is a norm-closed two-sided ideal of the algebra $\mathcal{B}(H)$ of bounded operators, closed under taking adjoints.
It is the norm closure of the [[finite-rank operators|Z8FABSYQTS]], and it is a C*-algebra in its own right.
For a separable infinite-dimensional $H$, it is the only closed two-sided ideal of $\mathcal{B}(H)$ other than $\set{0}$ and $\mathcal{B}(H)$, a theorem of Calkin.
The quotient $\mathcal{B}(H)/\mathcal{K}(H)$ is the [[Calkin algebra|6FE7JZ2VF1]], the setting of Fredholm theory.

## Explanations

### L1 human

Compact operators behave like small perturbations: adding or composing with a bounded operator keeps them compact.
That is what it means to be a two-sided ideal: a product $AK$ or $KA$ of a bounded operator and a compact one is again compact.
Working modulo this ideal ignores compact perturbations, which is the right point of view for questions such as whether an operator is invertible up to a finite-dimensional error.
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# Heegaard Floer homology

## Summary

Heegaard Floer homology is a family of invariants of closed oriented [[3-manifolds|R20TB2KEQQ]], introduced by Peter Ozsváth and Zoltán Szabó in the early 2000s.
It is a kind of [[Floer homology|NCKFPAGHXF]] built from a [[Heegaard splitting|EDE9919B5R]]: one counts holomorphic disks in a symmetric product of the Heegaard surface.
Its versions are written $\widehat{HF}$, $HF^+$, $HF^-$ and $HF^\infty$.
It has a relative version for knots, [[knot Floer homology|5P608WRETM]], and it gives invariants of smooth $4$-dimensional cobordisms.

## Explanations

### L1 human

A three-dimensional space can be cut along a surface into two simple pieces, and the way the pieces are glued is drawn as two families of curves on the surface.
Heegaard Floer homology turns this drawing into a chain complex: its generators come from the crossings of the two families, and its differential counts certain maps of disks.
The homology of this complex does not depend on the drawing, only on the space.
For the $3$-sphere, the simplest version $\widehat{HF}$ is a single copy of the coefficient field.
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# Prony's method

## Summary

Prony's method recovers a sum of exponentials $f(k) = \sum_{j=1}^{n} c_j z_j^k$ from equally spaced samples.
It uses the fact that such samples satisfy a [[linear recurrence|0NDVHBVBXB]] whose characteristic polynomial has the roots $z_j$.
One solves a [[Hankel|PQ1D8FQB3E]] linear system for the recurrence, finds the roots, and then solves a linear system for the amplitudes $c_j$.
The method underlies many techniques of signal processing and spectral estimation.
It is close to rational approximation, since the generating function $\sum_k f(k) w^k$ of such samples is rational.

## Explanations

### L1 human

A signal made of a few decaying or oscillating exponentials has a hidden regularity: each sample is a fixed combination of the previous few.
Prony's method first finds that combination from the data.
The roots of the associated polynomial reveal the rates and frequencies, and the remaining sizes come from a small linear system.
For example, the samples $2, 3, 5, 9$ of $2^k + 1$ satisfy $f(k+2) = 3 f(k+1) - 2 f(k)$, whose polynomial $z^2 - 3z + 2$ has roots $1$ and $2$.
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# F-test

## Definition

Let $d_1, d_2 \ge 1$ be integers and let $\alpha \in (0, 1)$.
An *F-test* is a [[hypothesis test|087P03NEND]] of a null hypothesis $H_0$ based on a [[statistic|036756NFD6]] $F \ge 0$ with the following property.
Under $H_0$, $F$ has the [[F-distribution|PBT8VRJADK]] with $d_1$ and $d_2$ degrees of freedom.
The test at level $\alpha$ rejects $H_0$ when $F > F_{1 - \alpha}(d_1, d_2)$, the $1 - \alpha$ quantile of that law.

## Remarks

Example: in one-way analysis of variance, there are $k$ groups and $N$ observations in total, with independent normal errors of common variance.
Let $\mathrm{SSB}$ be the sum of squares between groups and $\mathrm{SSW}$ the sum of squares within groups.
Under $H_0$ that all group means are equal, $F = \frac{\mathrm{SSB}/(k - 1)}{\mathrm{SSW}/(N - k)}$ has the F-distribution with $k - 1$ and $N - k$ degrees of freedom.
The check: $\mathrm{SSB}/\sigma^2$ and $\mathrm{SSW}/\sigma^2$ are independent chi-squared variables with $k - 1$ and $N - k$ degrees of freedom, by Cochran's theorem.
Non-example: for two non-normal samples, the variance ratio $S_1^2/S_2^2$ does not have an F-distribution under equal variances in general.
So comparing it with F quantiles is not an F-test in this sense, and its level can be far from $\alpha$.
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# Polylogarithm

## Definition

Let $s$ be a complex number.
For $z \in \C$ with $\abs{z} < 1$, the *polylogarithm of order $s$* is the [[power series|0D6KXBHKC0]]
$$\operatorname{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s},$$
together with its [[analytic continuation|0M3YQJ81NB]] in $z$ to $\C \setminus [1, \infty)$.

## Remarks

The series has radius of convergence $1$ for every $s$, because $\abs{k^{-s}}^{1/k} = k^{-\operatorname{Re} s/k} \to 1$.

Example: for $s = 1$ the series is $-\ln(1 - z)$, and for $s = 0$ and $s = -1$ it gives $\operatorname{Li}_0(z) = \frac{z}{1 - z}$ and $\operatorname{Li}_{-1}(z) = \frac{z}{(1 - z)^2}$.
For $\operatorname{Re} s > 1$ the series also converges at $z = 1$, and $\operatorname{Li}_s(1) = \zeta(s)$, the [[Riemann zeta function|1XWBY4N0VT]].

Non-example of a valid argument: for $s = 1$ the series diverges at $z = 1$ (it is the harmonic series), and $-\ln(1 - z)$ has a singularity there.

Termwise differentiation gives $z \frac{d}{dz} \operatorname{Li}_s(z) = \operatorname{Li}_{s-1}(z)$, so $\operatorname{Li}_{s+1}(z) = \int_0^z \frac{\operatorname{Li}_s(t)}{t}\, dt$.
For $s = 2$ this is the [[dilogarithm|G0X0NWSDB7]].
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# Z* theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]], and let $O(G)$ be the largest [[normal subgroup|0CB0DX5BVV]] of $G$ of odd order.
Let $T$ be a [[Sylow $2$-subgroup|0053CQE9Y2]] of $G$, and let $t \in T$ be an involution, that is, an element of [[order|0WE2NHZJ7B]] $2$.
Suppose that no element of $T$ other than $t$ is [[conjugate|0DXD4QSZ8A]] to $t$ in $G$.
Then $t \, O(G)$ lies in the [[centre|0DZ93493VN]] of the [[quotient group|0KYX2N69J0]] $G/O(G)$.

## Remarks

The theorem is due to George Glauberman, and its proof uses modular representation theory, in particular Brauer's theory of blocks.
The notation comes from $Z^*(G)$, the subgroup of $G$ containing $O(G)$ with $Z^*(G)/O(G) = Z(G/O(G))$; the conclusion says $t \in Z^*(G)$.

Example: $G = S_3$ and $t = (1\,2)$ in $T = \set{e, (1\,2)}$.
No other element of $T$ is conjugate to $t$, and $O(S_3) = A_3$.
Indeed $S_3 / A_3$ has order $2$, so it is abelian and $t A_3$ is central.

Corollary: if $G$ is a nonabelian [[simple group|0YMYMNQ0QZ]], then every involution $t$ in a Sylow $2$-subgroup $T$ is conjugate in $G$ to another element of $T$.
Indeed, $G$ has even order because it contains $t$, so $O(G) \ne G$; hence $O(G) = \set{e}$ by simplicity, and the centre of $G$ is trivial, so $t$ cannot be central.

## Explanations

### L1 human

An involution is an element that undoes itself, like a reflection.
Call an involution isolated if conjugation never moves it onto another element of its Sylow $2$-subgroup.
The Z* theorem says that an isolated involution becomes central after dividing out the largest normal subgroup of odd order.
So isolated involutions cannot exist in a nonabelian simple group.

### L2 human

The Z* theorem is a result about fusion of $2$-elements, like the [[focal subgroup theorem|H1BSG0J5G1]] but for single involutions.
It was an important tool in the classification of finite simple groups, where it rules out isolated involutions.
The proof by modular character theory is a striking case where representation theory is needed for a statement about conjugacy in a group.
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# Bifurcation diagram

## Summary

A bifurcation diagram is a picture of how the long-term behaviour of a one-parameter family of dynamical systems depends on the parameter.
The parameter runs along the horizontal axis, and above each value one plots the [[equilibria|0N8NGKDZDZ]], periodic orbits or [[attractor|7FWX24X5MW]] of the system.
Branches that split, merge or end mark the bifurcations studied by [[bifurcation theory|D9TW46P19Y]].
The best known example is the diagram of the [[logistic map|QN9X9BNKTX]], with its cascade of period doublings.

## Explanations

### L1 human

Run the system for many steps at one setting of the parameter and mark where it ends up.
Do this for many settings side by side, and you get a bifurcation diagram.
A single line means the system settles to one value; two lines mean it alternates between two values; a smeared band means chaos.
For the logistic map $x \mapsto r x (1 - x)$ the single line splits into two near $r = 3$, then into four, and so on, until chaos sets in.
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# Atomless Boolean algebra

## Definition

Let $B$ be a [[Boolean algebra|0XADJKM699]] with order $\le$.
An [[atom|TTVCTW6TNC]] of $B$ is an element $a \ne 0$ such that no $b \in B$ satisfies $0 < b < a$.
The algebra $B$ is *atomless* if it has no atoms, that is, for every $a \ne 0$ in $B$ there is $b \in B$ with $0 < b < a$.

## Remarks

The one-element algebra, with $0 = 1$, is atomless in a trivial way.
Every other atomless Boolean algebra is infinite, since in a finite algebra with $0 \ne 1$ a minimal element of $B \setminus \set{0}$ is an atom.

Example: let $B$ be the algebra of finite unions of half-open intervals $[s, t)$ with $0 \le s < t \le 1$, together with $\emptyset$, as subsets of $[0, 1)$.
A nonzero $a \in B$ contains some $[s, t)$ with $s < t$; then $b = [s, (s + t)/2)$ lies in $B$ and satisfies $\emptyset \ne b \subsetneq a$.

Non-example: the power set of a nonempty set $X$ is not atomless, since each singleton $\set{x}$ is an atom.
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# Laplace transform of the Dirac delta

## Example

Let $a \ge 0$ and let $\delta_a$ be the [[Dirac delta|0KPHENZ5S7]] at $a$, that is, the [[Dirac measure|099SXAG014]] that gives mass $1$ to the point $a$.
The delta is not a function, so its Laplace transform is defined as the [[Laplace–Stieltjes transform|KHRTBP37VD]] of the measure.
The point $0$ is included in the domain of integration:
$$\mathcal{L}\{\delta_a\}(s) = \int_{[0, \infty)} e^{-st}\, d\delta_a(t) = e^{-as} \quad (s \in \C).$$
Check: integrating a continuous function against $\delta_a$ evaluates it at $a$, and $t \mapsto e^{-st}$ is continuous.
In particular $\mathcal{L}\{\delta_0\} = 1$, with the convention that the lower limit is $0^-$.
This is consistent with the [[Laplace transform|05XT5AVJBK]] of the step $H(t - a)$, which is $e^{-as}/s$.
Indeed, the delta is the derivative of the step, and differentiation multiplies the transform by $s$ when the initial value is $0$.

## Explanations

### L1 human

The delta models an instant kick of total size $1$ at time $a$.
Its transform is $e^{-as}$, and for a kick at time $0$ it is just the constant $1$.
Since the transform of an ordinary function tends to $0$ as $s \to \infty$, the constant $1$ also shows that the delta is not an ordinary function.
In system theory, the response to the kick $\delta_0$ is the impulse response, and its transform is the transfer function.
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# Reductive group

## Summary

A reductive group is a connected [[linear algebraic group|GA81VVQHEP]] with no nontrivial connected normal unipotent subgroup.
Examples are $\mathrm{GL}_n$, $\mathrm{SL}_n$, the special orthogonal groups $\mathrm{SO}_n$, the symplectic groups, and the [[algebraic tori|66FQFAJNS4]].
Over an algebraically closed field, reductive groups are classified by root data.
In characteristic $0$, they are the connected linear groups whose representations are all semisimple.

## Explanations

### L1 human

Inside the group of invertible upper triangular matrices sits the subgroup of those with all diagonal entries $1$.
That subgroup is unipotent and normal, and it makes the group hard to analyze.
A reductive group is one with no such part: nothing unipotent hides in it as a normal subgroup.
This is the class of groups with a clean classification, much like compact Lie groups.
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# Errors-in-variables model

## Summary

An errors-in-variables model is a regression model in which the regressors are measured with error, not only the response.
In that case [[ordinary least squares|YM708KHV2W]] on the observed regressors is biased and not [[consistent|0AS800VAYX]].
In the simple linear case the slope is pulled towards zero, an effect called attenuation or regression dilution.
Consistent estimation needs extra information, such as the ratio of the error variances, repeated measurements, or [[instrumental variables|6ETK64HYK6]].
Deming regression and total least squares are methods for such models.

## Explanations

### L1 human

Suppose the true effect of a quantity $\xi$ on $y$ has slope $2$, but $\xi$ can only be measured with noise.
The noisy readings spread out more than the true values, and the extra spread is unrelated to $y$.
The fitted line therefore looks flatter than the true one.
If the measurement noise has the same variance as the true values, the least squares slope tends to $1$ instead of $2$.
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# Bialgebra

## Definition

Let $K$ be a field.
A *bialgebra* over $K$ is a vector space $B$ over $K$ with linear maps $m \colon B \otimes B \to B$, $\eta \colon K \to B$, $\Delta \colon B \to B \otimes B$ and $\varepsilon \colon B \to K$ such that:

1. $(B, m, \eta)$ is an [[associative algebra|3CTW1P4HBP]] with multiplication $m$ and unit $\eta(1)$.
2. $(B, \Delta, \varepsilon)$ is a [[coalgebra|A06RA8KB0T]].
3. $\Delta$ and $\varepsilon$ are algebra homomorphisms, where $B \otimes B$ has the product $(a \otimes b)(c \otimes d) = ac \otimes bd$.

## Remarks

Clause 3 means $\Delta(ab) = \Delta(a)\Delta(b)$, $\Delta(1) = 1 \otimes 1$, $\varepsilon(ab) = \varepsilon(a)\varepsilon(b)$ and $\varepsilon(1) = 1$.
It is equivalent to asking that $m$ and $\eta$ be coalgebra homomorphisms.
Example: the [[group algebra|MNP2HBWB5Q]] $KG$ of a group $G$, with $\Delta(g) = g \otimes g$ and $\varepsilon(g) = 1$ for $g \in G$.
The check of clause 3 on the basis: $\Delta(gh) = gh \otimes gh = (g \otimes g)(h \otimes h) = \Delta(g)\Delta(h)$ and $\varepsilon(gh) = 1 = \varepsilon(g)\varepsilon(h)$.
Non-example: on $K[x]$, the coalgebra with $\Delta(x^n) = \sum_{k=0}^{n} x^k \otimes x^{n-k}$ and $\varepsilon(x^n) = \delta_{n,0}$ fails clause 3.
Indeed $\Delta(x)^2 = (x \otimes 1 + 1 \otimes x)^2 = x^2 \otimes 1 + 2\, x \otimes x + 1 \otimes x^2$, while $\Delta(x^2) = x^2 \otimes 1 + x \otimes x + 1 \otimes x^2$, and $2 \ne 1$ in every field.
With $\Delta(x^n) = \sum_{k} \binom{n}{k} x^k \otimes x^{n-k}$ instead, $K[x]$ is a bialgebra.
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# Benamou–Brenier formula

## Statement

Let $\mathcal{P}_2(\R^n)$ be the set of Borel [[probability measures|059TJN0NZ1]] on $\R^n$ with finite second moment.
Let $W_2$ be the [[Wasserstein distance|VEY29CS26X]] of order $2$ on it.
Let $\mu_0, \mu_1 \in \mathcal{P}_2(\R^n)$.
Consider pairs $(\mu_t, v_t)_{t \in [0, 1]}$ such that:

1. $t \mapsto \mu_t$ is a weakly continuous curve in $\mathcal{P}_2(\R^n)$ from $\mu_0$ to $\mu_1$;
2. $(t, x) \mapsto v_t(x) \in \R^n$ is a Borel vector field with $\int_0^1 \int \abs{v_t}^2 \, d\mu_t \, dt < \infty$;
3. the continuity equation $\partial_t \mu_t + \operatorname{div}(v_t \mu_t) = 0$ holds in the sense of distributions on $(0, 1) \times \R^n$.

Then
$$W_2(\mu_0, \mu_1)^2 = \inf \int_0^1 \int_{\R^n} \abs{v_t(x)}^2 \, d\mu_t(x) \, dt,$$
where the infimum runs over all such pairs, and it is attained.

## Remarks

Jean-David Benamou and Yann Brenier introduced the formula in 2000, as a basis for numerical computation of optimal transport.
To see that the infimum is at most $W_2(\mu_0, \mu_1)^2$, take an optimal coupling $\pi$ and move mass along straight lines.
So $\mu_t$ is the image of $\pi$ under $(x, y) \mapsto (1 - t) x + t y$, and the velocity of each particle is $y - x$.
The energy is then $\int \abs{x - y}^2 \, d\pi = W_2(\mu_0, \mu_1)^2$.
For the other inequality, note that a curve with velocity field $v$ is the image of the flow of $v$.
Jensen's inequality then bounds the cost of the induced coupling by the kinetic energy.
Example: for $\mu_0 = \delta_a$ and $\mu_1 = \delta_b$, the curve $\mu_t = \delta_{(1 - t)a + tb}$ with constant velocity $b - a$ has energy $\abs{b - a}^2 = W_2(\delta_a, \delta_b)^2$.
In the substitution $m_t = v_t \mu_t$ the problem becomes convex in $(\mu, m)$, which is what makes it useful for computation.

## Explanations

### L1 human

The Wasserstein distance measures the cost of moving one distribution of mass onto another.
The Benamou–Brenier formula says that this cost is the least kinetic energy of a flow of mass that carries the first distribution to the second in unit time.
So the static problem of matching points becomes a dynamic problem of moving a fluid.

### L2 human

The formula shows that $(\mathcal{P}_2(\R^n), W_2)$ behaves like a Riemannian manifold, in Otto's formal calculus: tangent vectors are velocity fields, and the squared length of a tangent vector is $\int \abs{v}^2 \, d\mu$.
Optimal curves are the displacement interpolations, along which displacement convexity is measured.
The same structure underlies the interpretation of the heat and Fokker–Planck equations as gradient flows, as in the Jordan–Kinderlehrer–Otto scheme.
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# Knot group

## Definition

Let $K \subset \R^3$ be a [[knot|QYEAZTYGPT]], and let $x_0$ be a point of $\R^3 \setminus K$.
The *knot group* of $K$ is the [[fundamental group|S1XK3HXHFX]]
$$\pi_1(\R^3 \setminus K, x_0).$$

## Remarks

The complement $\R^3 \setminus K$ is path-connected, so the group does not depend on $x_0$ up to isomorphism.
If $K$ and $K'$ are equivalent, an ambient isotopy gives a homeomorphism of their complements, so their knot groups are isomorphic.
Removing a point at infinity does not change it: $\pi_1(S^3 \setminus K) \cong \pi_1(\R^3 \setminus K)$.
For a tame knot, the abelianization of the knot group is $\Z$.
Example: the complement of the [[unknot|1TQ06R4Q66]] deformation retracts onto the union of a $2$-sphere and a circle that meet in one point.
So its knot group is $\Z$, generated by a loop that goes once around the unknot.
For the [[trefoil knot|VYTYSVHN4A]], the knot group has the presentation $\langle x, y \mid xyx = yxy \rangle$; it is not abelian, so the trefoil is not equivalent to the unknot.
Non-example: the fundamental group of the knot $K$ itself is not the knot group.
It is $\Z$ for every knot, since $K$ is homeomorphic to a circle, so it distinguishes no knots.
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# Möbius function of a poset

## Summary

The Möbius function of a [[locally finite poset|TV1V4JDRDD]] assigns an integer $\mu(x, y)$ to each pair $x \le y$, by the recursion $\mu(x, x) = 1$ and $\sum_{x \le z \le y} \mu(x, z) = 0$ for $x < y$.
It is the inverse of the zeta function in the [[incidence algebra|1H9SFWHYEB]].
It drives [[Möbius inversion on posets|B8XJSJ517Q]], which contains inclusion–exclusion and the number-theoretic [[Möbius function|0FANE7P0DY]] as special cases.

## Explanations

### L1 human

Start at an element $x$ and give it weight $1$.
Then go up the poset, and give each element $y$ the weight that makes the total of all weights from $x$ to $y$ equal to zero.
For a chain $0 < 1 < 2 < \cdots$ this gives $1, -1, 0, 0, \ldots$.
For the subsets of a set, the weight of a set with $k$ more elements than the start is $(-1)^k$.
These weights are what you need to undo a sum over everything below.
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# Factor analysis

## Definition

Let $p > k \ge 1$ be integers.
The *factor analysis model* with $k$ factors for a random vector $X$ in $\R^p$ is
$$X = \mu + L F + \varepsilon,$$
where:

1. $\mu \in \R^p$ is a constant vector and $L$ is a constant $p \times k$ matrix, the matrix of *loadings*.
2. $F$ is a random vector in $\R^k$, the *common factors*, with $\mathbb{E}F = 0$ and covariance matrix $I_k$.
3. $\varepsilon$ is a random vector in $\R^p$, the *specific errors*, with $\mathbb{E}\varepsilon = 0$ and diagonal covariance matrix $\Psi$.
4. $F$ and $\varepsilon$ are uncorrelated.

The factors $F$ are not observed; only $X$ is.
*Factor analysis* is the estimation of $L$ and $\Psi$ from observations of $X$, and the interpretation of the factors.

## Remarks

The model implies that the [[covariance matrix|0V7G6WMFK0]] of $X$ is $\Sigma = L L^T + \Psi$.
Example: with $p = 3$, $k = 1$, $L = (1, 1, 1)^T$ and $\Psi = I_3$, the covariance matrix has diagonal entries $2$ and off-diagonal entries $1$.
Non-example: a model $X = \mu + L F + \varepsilon$ with $\Psi$ not diagonal fails clause 3; the errors would then carry shared variation that the factors are meant to explain.
The loadings are determined only up to rotation: if $Q$ is a $k \times k$ orthogonal matrix, then $L Q$ and $L$ give the same $\Sigma$.
For this reason, rotations such as varimax are chosen to make the loadings easier to interpret.
Under normality, $L$ and $\Psi$ are often estimated by maximum likelihood.
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# Series-parallel partial order

## Summary

A series-parallel partial order is a finite poset built from single points by two operations.
The series operation puts one poset entirely below another, as in the [[ordinal sum|6AJ7QQCTQQ]].
The parallel operation places two posets side by side, as in the [[disjoint union|TZWMEM7GKA]].
Many problems that are hard for general posets, such as counting linear extensions, become easy for this class.

## Explanations

### L1 human

Think of a project where some groups of tasks run one after another, and other groups run at the same time.
If the whole plan can be described this way, by nesting "then" and "at the same time", its order is series-parallel.
For example, "do $a$, then do $b$ and $c$ at the same time, then do $d$" gives the poset $a < b, c < d$ with $b$ and $c$ incomparable.
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<document source="https://tamnd.github.io/math2/node/2YDB5089FR.md" tag="2YDB5089FR" kind="concept">
# Todd class

## Summary

The Todd class is a characteristic class of vector bundles, built from the [[Chern classes|PAD37GKDPD]] by a fixed power series.
The Todd class of a smooth variety is the Todd class of its tangent bundle.
It is the correction term in the [[Hirzebruch–Riemann–Roch theorem|G3X02FKQCR]] and the [[Grothendieck–Riemann–Roch theorem|3HKREN3Y3F]], which compute Euler characteristics of sheaves from Chern classes.

## Explanations

### L1 human

The Riemann–Roch theorem expresses the dimension of a space of sections, an analytic quantity, through numbers such as degrees and the genus.
In higher dimensions the right expression needs a universal polynomial in the Chern classes of the tangent bundle, and this polynomial is the Todd class.
For a curve of genus $g$, the Todd class is $1 + \frac{1}{2} c_1$, and its degree part gives the term $1 - g$ in the Riemann–Roch formula $\chi(L) = \deg L + 1 - g$.
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<document source="https://tamnd.github.io/math2/node/2YHG0X6EA0.md" tag="2YHG0X6EA0" kind="definition">
# Disjoint union of posets

## Definition

Let $(P, \le_P)$ and $(Q, \le_Q)$ be [[partially ordered sets|0QV2JS9RJ3]] with $P \cap Q = \varnothing$.
The *disjoint union* $P + Q$ is the set $P \cup Q$ with the following relation $\le$.
For $x, y \in P \cup Q$, $x \le y$ holds if and only if either $x, y \in P$ and $x \le_P y$, or $x, y \in Q$ and $x \le_Q y$.

## Remarks

If $P$ and $Q$ are not disjoint, first replace them by disjoint copies, such as $P \times \set{0}$ and $Q \times \set{1}$.
The same construction works for any family of pairwise disjoint posets.
Example: if $P$ is the chain $a < b$ and $Q$ is the chain $c < d$, then $P + Q$ is the poset $\mathbf{2} + \mathbf{2}$ with exactly the strict relations $a < b$ and $c < d$.
Non-example: the poset with $a < b$, $c < d$ and also $a < d$ is not $P + Q$, since it relates an element of $P$ to an element of $Q$.
An antichain with $n$ elements is the disjoint union of $n$ one-element posets.
The width of $P + Q$ is the sum of the widths, and its height is the larger of the two heights.
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# Cartan formula

## Statement

Let $X$ be a space, let $p$ be a prime, and let $x, y \in H^*(X; \F_p)$ be homogeneous classes, with $\abs{x}$ the degree of $x$.
Write $\smile$ for the [[cup product|JX47MS1R1H]].

1. For $p = 2$, the [[Steenrod squares|VR7RHP41M0]] satisfy, for every $k \ge 0$,
$$Sq^k(x \smile y) = \sum_{i + j = k} Sq^i x \smile Sq^j y.$$
2. For $p$ odd, the [[Steenrod reduced powers|EZYHSE1E8P]] satisfy, for every $k \ge 0$,
$$P^k(x \smile y) = \sum_{i + j = k} P^i x \smile P^j y.$$
3. For $p$ odd, the [[Bockstein homomorphism|H7PCM776GR]] $\beta$ of $0 \to \Z/p \to \Z/p^2 \to \Z/p \to 0$ is a derivation:
$$\beta(x \smile y) = \beta x \smile y + (-1)^{\abs{x}} x \smile \beta y.$$

## Remarks

The same formulas hold for the cross product $H^*(X) \otimes H^*(Y) \to H^*(X \times Y)$, and the cup product case follows by pulling back along the diagonal $X \to X \times X$.
The cross product case is checked in the construction of the operations, which are built from a product-compatible map on cochains.

In terms of the total square $Sq = \sum_{i \ge 0} Sq^i$, part 1 says that $Sq$ is a ring homomorphism of $H^*(X; \F_2)$.

Example: let $x$ generate $H^1(\R P^\infty; \F_2)$, so $H^*(\R P^\infty; \F_2) = \F_2[x]$.
Since $Sq^0 x = x$ and $Sq^1 x = x^2$, one has $Sq(x) = x + x^2$.
Then $Sq(x^n) = (x + x^2)^n$, so $Sq^i(x^n) = \binom{n}{i} x^{n+i}$.

## Explanations

### L1 human

The Cartan formula says how a Steenrod operation acts on a product of two classes.
The answer looks like the product rule for derivatives, with a sum over all ways to split the operation between the two factors.
It turns many computations of Steenrod operations into simple algebra.

### L2 human

With the Cartan formula and the values on degree-one or degree-two generators, one computes all Steenrod operations on the cohomology of $\R P^n$, $\C P^n$ and products of these.
It also gives the Hopf algebra structure of the Steenrod algebra: the coproduct sends $Sq^k$ to $\sum_{i+j=k} Sq^i \otimes Sq^j$.
Milnor used this coproduct to describe the dual of the Steenrod algebra.
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<document source="https://tamnd.github.io/math2/node/2YPY6YTHX8.md" tag="2YPY6YTHX8" kind="definition">
# Relative homotopy group

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]], let $A \subseteq X$ be a subspace, let $x_0 \in A$, and let $n \ge 1$.
In the cube $I^n = [0, 1]^n$, let $I^{n-1}$ be the face where the last coordinate is $0$.
Let $J^{n-1}$ be the closure of $\partial I^n \setminus I^{n-1}$.
The *relative homotopy group* $\pi_n(X, A, x_0)$ is the set of [[homotopy|9YA6S6HD9V]] classes of maps $f \colon I^n \to X$ with $f(\partial I^n) \subseteq A$ and $f(J^{n-1}) = \set{x_0}$.
The homotopies must keep these two conditions at every time.
For $n \ge 2$, the sum $f + g$ is defined by the formula for absolute [[homotopy groups|P976WYEFMB]], putting $f$ and $g$ side by side in the first coordinate.

## Remarks

For $n \ge 2$ the sum makes $\pi_n(X, A, x_0)$ a group, and for $n \ge 3$ the group is abelian.
For $n = 1$ there is no group structure in general, and $\pi_1(X, A, x_0)$ is only a pointed set.

Example: when $A = \set{x_0}$, the conditions say that $f$ sends all of $\partial I^n$ to $x_0$, so $\pi_n(X, x_0, x_0) = \pi_n(X, x_0)$.

Example: for $n \ge 2$, $\pi_n(D^n, S^{n-1}, x_0) \cong \pi_{n-1}(S^{n-1}, x_0) \cong \Z$, by the long exact sequence of the pair and the contractibility of the disk $D^n$.

Non-example: a map $f$ with $f(\partial I^n) \subseteq A$ that does not send $J^{n-1}$ to $x_0$ is not a representative; the base point condition on $J^{n-1}$ is part of the definition.

Equivalently, $\pi_n(X, A, x_0)$ is the set of homotopy classes of maps of triples $(D^n, S^{n-1}, s_0) \to (X, A, x_0)$.
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# Milnor K-theory

## Summary

Milnor K-theory attaches to a field $K$ a sequence of abelian groups $K_n^M(K)$, see the [[definition|HEVWQ6B77W]].
They are generated by symbols $\set{a_1, \ldots, a_n}$ of nonzero elements.
The symbols are multilinear and satisfy the Steinberg relation $\set{a, 1 - a} = 0$.
In degrees $0$, $1$ and $2$ it agrees with the [[algebraic K-theory|5GVFZWMWC8]] groups $\Z$, $K^\times$ and $K_2(K)$.
The [[norm residue isomorphism theorem|5Q5VN56QH4]] identifies its reduction modulo $\ell$ with [[Galois cohomology|9MPCZ3EAJM]].
It also connects with quadratic forms through the Milnor conjecture.

## Explanations

### L1 human

Start with the nonzero elements of a field under multiplication.
Form "products" of $n$ of them, written $\set{a_1, \ldots, a_n}$, which behave like products in each slot.
Add one extra rule: a symbol vanishes when two of its entries add up to $1$, as in $\set{2, -1} = 0$ over $\Q$.
The groups that result are simple to define, yet they carry deep arithmetic information about the field.
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# Singular function

## Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function.
The function $f$ is *singular* if:

1. $f$ is [[continuous|0P27W9RW0G]] on $[a, b]$;
2. there is a set $N \subseteq [a, b]$ of Lebesgue measure $0$ such that for every $x \in [a, b] \setminus N$ the [[derivative|0F1WR9C83K]] $f'(x)$ exists and $f'(x) = 0$;
3. $f$ is not constant on $[a, b]$.

## Remarks

Example: the [[Cantor function|NPBYYK63FE]] $c$ on $[0, 1]$ is singular.
It is continuous, it is constant on each open interval removed in the construction of the Cantor set, so $c' = 0$ off the Cantor set, which has measure $0$, and $c(0) = 0 \ne 1 = c(1)$.

Non-example: the step function equal to $0$ on $[0, \frac12)$ and $1$ on $[\frac12, 1]$ satisfies clauses 2 and 3 but fails clause 1.

Conventions vary: some authors drop continuity, and some allow constants.
A singular function is never [[absolutely continuous|ZT6F4C2YHK]], since an absolutely continuous function with derivative $0$ almost everywhere is constant.
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# Mean curvature flow

## Summary

Mean curvature flow is the evolution of a hypersurface in which each point moves with velocity equal to the mean curvature vector.
It is the gradient flow of the area, so it decreases area as fast as possible.
The equation for the moving surface is a nonlinear, weakly parabolic system.
It plays for surfaces the role that the [[heat equation|VP2HN86VAG]] plays for functions.
It is a close relative of the [[inverse mean curvature flow|5XVEF5DKAG]] and of the [[Ricci flow|CR14D9YQTY]].

## Explanations

### L1 human

Imagine a soap film or a closed curve that tries to shorten itself.
Each point moves toward the inside of the bend, faster where the surface is more bent, as measured by its [[mean curvature|PA78NFEPCF]].
A round sphere of radius $R_0$ in $\R^{n+1}$ stays round and its radius shrinks as $\sqrt{R_0^2 - 2nt}$.
For curves in the plane the flow is called curve shortening flow, and every embedded closed curve becomes round before it shrinks to a point.
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# Convolution

## Summary

Convolution combines two functions $f$ and $h$ on a group into a new function $f * h$ by averaging translates of one function, weighted by the other.
On $\R^n$ it is $(f * h)(x) = \int f(y) h(x - y) \, dy$, and on a [[locally compact group|AHV05ZBCYW]] the integral is taken against a [[Haar measure|VY0HXD3GC4]].
It turns $L^1$ of the group into an algebra, it describes sums of independent random variables, and the Fourier transform turns it into pointwise multiplication.

## Explanations

### L1 human

To convolve a signal with a short bump, replace the value at each point by a weighted average of nearby values, with weights given by the bump.
The result is a smoothed version of the signal.
For sequences on the integers, convolving $(1, 1)$ with $(1, 2, 1)$ gives $(1, 3, 3, 1)$, exactly the coefficients of $(1 + x)(1 + 2x + x^2)$.
So convolution is the multiplication of polynomials and power series, written in terms of their coefficients.
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# Lebesgue measure

## Description

Let $n \ge 1$.
A box in $\R^n$ is a product $R = [a_1, b_1] \times \cdots \times [a_n, b_n]$ with $a_i \le b_i$, and its volume is $\operatorname{vol}(R) = (b_1 - a_1) \cdots (b_n - a_n)$.
Lebesgue [[outer measure|BJ314H17HC]] on [[Euclidean space|0XPR1E85W0]] $\R^n$ is
$$\lambda^*(A) = \inf \Big\{ \sum_{k=1}^{\infty} \operatorname{vol}(R_k) : R_1, R_2, \ldots \text{ boxes}, \ A \subseteq \bigcup_{k} R_k \Big\} \quad \text{for } A \subseteq \R^n.$$
A set $E \subseteq \R^n$ is Lebesgue measurable if $\lambda^*(A) = \lambda^*(A \cap E) + \lambda^*(A \setminus E)$ for all $A \subseteq \R^n$.
Lebesgue measure $\lambda$ (or $\lambda^n$) is $\lambda^*$ restricted to the Lebesgue measurable sets.

## Remarks

By [[Carathéodory's criterion|JDGQ135YK2]] the Lebesgue measurable sets form a $\sigma$-algebra and $\lambda$ is a complete [[measure|PR11BZQQPG]] on it.
Every [[Borel set|0WQXY32Q8D]] is Lebesgue measurable, and $\lambda(R) = \operatorname{vol}(R)$ for every box $R$; so $\lambda([a, b]) = b - a$ on $\R$.
Lebesgue measure is invariant under translations and rotations, and $\lambda(tA) = \abs{t}^n \lambda(A)$ for $t \in \R$.
Up to a constant factor, it is the only translation-invariant measure on the Borel sets of $\R^n$ that is finite on bounded sets.
Every countable set has measure $0$, and so does the Cantor set, which is uncountable.
Lebesgue measurable sets are regular: for every measurable $E$ and $\varepsilon > 0$ there are a closed set $F$ and an open set $U$ with $F \subseteq E \subseteq U$ and $\lambda(U \setminus F) < \varepsilon$.
Not every subset of $\R^n$ is Lebesgue measurable; a Vitali set is an example.

## Explanations

### L1 human

Lebesgue measure is length on the line, area in the plane and volume in space, extended to as many sets as possible.
To measure a set, cover it with countably many boxes and take the smallest possible total volume.
The interval $[0, 3]$ gets length $3$, a disk of radius $r$ gets area $\pi r^2$, and the set of rational numbers gets length $0$.
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# von Neumann stability analysis

## Statement

Let $\Delta x > 0$ and $\Delta t > 0$.
Consider a linear two-level [[finite difference|X7HYEBA5SA]] scheme on the grid $\Delta x \Z$:
$$\sum_{k=-K}^{K} b_k\, u_{j+k}^{n+1} = \sum_{k=-K}^{K} c_k\, u_{j+k}^n, \qquad j \in \Z,\ n \ge 0.$$
The unknown is scalar, and the complex coefficients $b_k$, $c_k$ do not depend on $j$ or $n$.
Assume that $\beta(\xi) = \sum_k b_k e^{i k \xi}$ has no zero for $\xi \in [-\pi, \pi]$.
The amplification factor of the scheme is
$$g(\xi) = \frac{\sum_k c_k e^{i k \xi}}{\sum_k b_k e^{i k \xi}}, \qquad \xi \in [-\pi, \pi].$$
The coefficients may depend on $\Delta t$ and $\Delta x$ through a relation $\Delta t = \Lambda(\Delta x)$ fixed in advance.
Use the norm $\norm{u}^2 = \Delta x \sum_j \abs{u_j}^2$.
The scheme is called stable if for each $T > 0$ there is $C_T$ with $\norm{u^n} \le C_T \norm{u^0}$ whenever $n \Delta t \le T$.
The scheme is stable if and only if there are constants $K_0$ and $\tau > 0$ such that
$$\abs{g(\xi)} \le 1 + K_0 \Delta t \qquad \text{for all } \xi \in [-\pi, \pi] \text{ and all } 0 < \Delta t \le \tau.$$
If $g$ does not depend on $\Delta t$, the condition is $\abs{g(\xi)} \le 1$ for all $\xi$.

## Remarks

The proof uses the [[discrete-time Fourier transform|RXFJ47VYGX]] $\hat u(\xi) = \Delta x \sum_j u_j e^{-i j \xi}$, which turns the scheme into $\hat u^{n+1}(\xi) = g(\xi)\, \hat u^n(\xi)$, so $\hat u^n = g^n \hat u^0$.
By Parseval's identity, $\norm{u^n}$ equals the $L^2$ norm of $g^n \hat u^0$ up to a constant factor, so stability is equivalent to a bound on $\sup_\xi \abs{g(\xi)}^n$ for $n \Delta t \le T$.
This bound holds exactly when $\abs{g} \le 1 + K_0 \Delta t$, because $(1 + K_0 \Delta t)^n \le e^{K_0 T}$.
Example: the [[upwind scheme|MTCRZC5QKT]] $u_j^{n+1} = u_j^n - \nu (u_j^n - u_{j-1}^n)$ for $u_t + a u_x = 0$, with $\nu = a \Delta t / \Delta x$, has $\abs{g(\xi)}^2 = 1 - 2 \nu (1 - \nu)(1 - \cos \xi)$; this is at most $1$ for all $\xi$ exactly when $0 \le \nu \le 1$, which is the [[Courant–Friedrichs–Lewy condition|FQVVNYV6N8]] for this scheme.
For systems of equations, the analogous condition on the spectral radius of the amplification matrix is necessary but not sufficient in general.
For variable coefficients or boundaries, the analysis is only a heuristic test.

## Explanations

### L1 human

Errors in a computed solution can be written as a sum of waves of different frequencies.
For a linear scheme with constant coefficients, each wave is multiplied by a fixed factor at every time step.
If some factor has size larger than one, that wave grows exponentially and ruins the computation; if all factors have size at most one, nothing blows up.
Von Neumann analysis computes these factors and checks this condition.

### L2 human

The analysis is the practical tool to test a new difference scheme: substitute $u_j^n = g^n e^{i j \xi}$ and solve for $g$.
Combined with consistency, stability gives convergence by the [[Lax equivalence theorem|D5M7TRD3MY]].
The term $K_0 \Delta t$ allows schemes for equations with lower-order terms, such as $u_t = u_{xx} + u$, whose exact solutions also grow.
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<document source="https://tamnd.github.io/math2/node/2ZB3NFZ94T.md" tag="2ZB3NFZ94T" kind="concept">
# Hermitian kernel

## Summary

A Hermitian kernel is a complex-valued [[kernel|GSMBXT7FPK]] with $K(x, s) = \overline{K(s, x)}$ for all $x$ and $s$.
It is the continuous analogue of a [[Hermitian matrix|0P17NGQQMQ]].
A square-integrable Hermitian kernel gives a [[self-adjoint operator|0NEGN5CRSC]] on $L^2$, so its characteristic values are real and its eigenfunctions can be chosen orthonormal.
The Hilbert–Schmidt theory of integral equations is built on such kernels.

## Explanations

### L1 human

Swapping the two variables of a Hermitian kernel gives its complex conjugate.
For a real kernel this simply means that the kernel is symmetric in its two variables.
It mirrors the rule $a_{ij} = \overline{a_{ji}}$ for the entries of a Hermitian matrix.
For example, $K(x, s) = e^{i (x - s)}$ is Hermitian, while $K(x, s) = i$ is not.
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<document source="https://tamnd.github.io/math2/node/2ZJ2G308QV.md" tag="2ZJ2G308QV" kind="concept">
# Smash product

## Summary

The smash product $A \# H$ is an algebra built from a [[Hopf algebra|7XFYP0N84H]] $H$ and an algebra $A$.
The Hopf algebra $H$ acts on $A$ compatibly with the multiplication of $A$.
It contains $A$ and $H$ as subalgebras, and the action of $H$ on $A$ becomes part of the multiplication.
For the group algebra of a group it is the [[skew group ring|PCKK4SFZEK]].
For the enveloping algebra of a Lie algebra acting by derivations it is a differential operator ring.
So it treats group actions and actions by derivations in one framework.

## Explanations

### L1 human

A group can act on an algebra by symmetries, and a Lie algebra can act on it by derivations.
A Hopf algebra is a common generalisation of both, and it can also act on an algebra.
The smash product puts the algebra and the Hopf algebra side by side, as $A \otimes H$.
When an element $h$ of $H$ moves past an element $b$ of $A$, it acts on $b$.
This works in the same way that a group element moves past a coefficient in a skew group ring.
</document>

<document source="https://tamnd.github.io/math2/node/2ZK4CE9MF8.md" tag="2ZK4CE9MF8" kind="statement">
# Wold's decomposition theorem

## Statement

Let $(X_t)_{t \in \Z}$ be a real weakly [[stationary process|DT0QXJE705]] with mean $0$.
Let $L^2$ be the [[Hilbert space|HM2HHE3J51]] of square-integrable random variables.
For $s \in \Z$, let $H_s$ be the closed linear span of $\set{X_r : r \le s}$ in $L^2$.
Let $P_s$ be the [[orthogonal projection|0B8KZRT28M]] onto $H_s$.
Assume that the one-step prediction error variance $\sigma^2 = E[(X_t - P_{t-1} X_t)^2]$ is positive.
Then
$$X_t = \sum_{j=0}^{\infty} \psi_j \varepsilon_{t-j} + V_t \qquad \text{for all } t \in \Z,$$
where:

1. $\varepsilon_t = X_t - P_{t-1} X_t$, and $(\varepsilon_t)$ is a white noise with variance $\sigma^2$: mean $0$, variance $\sigma^2$, pairwise uncorrelated.
2. $\psi_0 = 1$, $\sum_{j \ge 0} \psi_j^2 < \infty$, and $\psi_j = E[X_t \varepsilon_{t-j}] / \sigma^2$.
3. $E[\varepsilon_s V_t] = 0$ for all $s, t \in \Z$.
4. $V_t$ lies in $\bigcap_{s \in \Z} H_s$, and $(V_t)$ is deterministic: $V_t = P_{s} V_t$ for all $s$, so it is predicted without error from its own remote past.

The sequences $(\psi_j)$, $(\varepsilon_t)$ and $(V_t)$ are uniquely determined by these conditions.

## Remarks

The idea is to split $H_t$ orthogonally into the innovations $\varepsilon_t, \varepsilon_{t-1}, \dots$, which are orthogonal to each other, and the remote past $\bigcap_s H_s$.
The first sum is the projection of $X_t$ onto the closed span of the innovations, with coefficients given by inner products.
The remainder $V_t$ is the projection onto the remote past.
The process is called purely nondeterministic when $V_t = 0$; then $X_t$ is a [[moving-average model|PEB9DN0AHA]] of infinite order.
Example: a causal AR($1$) process $X_t = \phi X_{t-1} + \varepsilon_t$ with $\abs{\phi} < 1$ has $\psi_j = \phi^j$ and $V_t = 0$.
Example of a deterministic process: let $A$ and $B$ be uncorrelated with mean $0$ and variance $1$, let $0 < \lambda < \pi$, and let $X_t = A \cos(\lambda t) + B \sin(\lambda t)$.
This process is weakly stationary, and any two consecutive values determine $A$ and $B$, so $X_t$ lies in every $H_s$ and $\sigma^2 = 0$; the whole process is its own deterministic part.
The coefficients $\psi_j$ need not be absolutely summable, and the shocks $\varepsilon_t$ are only uncorrelated, not independent in general.

## Explanations

### L1 human

Any stationary series can be split into two parts that do not interact.
One part is fully predictable from the distant past, like a fixed periodic wave.
The other part is built from a sequence of unpredictable surprises, each passed on to the future with fading weights.
So, as far as linear prediction goes, every stationary series looks like a moving average of white noise plus a predictable part.

### L2 human

The theorem justifies the use of linear models, such as ARMA models, for stationary series: they are finite-parameter approximations of the infinite moving average in the decomposition.
The innovations $\varepsilon_t$ are the one-step prediction errors, which is why residual checks in model fitting test them for whiteness.
The proof is pure Hilbert space geometry: only means and covariances enter, so the result says nothing about nonlinear dependence.
It was proved by Herman Wold in his 1938 thesis.
</document>

<document source="https://tamnd.github.io/math2/node/2ZKWZ0A4MT.md" tag="2ZKWZ0A4MT" kind="statement">
# Schwartz kernel theorem

## Statement

Let $X \subseteq \R^m$ and $Y \subseteq \R^n$ be open sets.
Let $\mathcal{D}(X)$ be the space of test functions on $X$, and let $\mathcal{D}'(X)$ be the space of distributions on $X$.
For $\varphi \in \mathcal{D}(X)$ and $\psi \in \mathcal{D}(Y)$, let $(\varphi \otimes \psi)(x, y) = \varphi(x) \psi(y)$.

1. Each $K \in \mathcal{D}'(X \times Y)$ defines a linear map $A_K \colon \mathcal{D}(Y) \to \mathcal{D}'(X)$ by $\langle A_K \psi, \varphi \rangle = \langle K, \varphi \otimes \psi \rangle$. It is sequentially continuous: if $\psi_j \to 0$ in $\mathcal{D}(Y)$, then $\langle A_K \psi_j, \varphi \rangle \to 0$ for every $\varphi \in \mathcal{D}(X)$.
2. Conversely, for each linear map $A \colon \mathcal{D}(Y) \to \mathcal{D}'(X)$ that is sequentially continuous in this sense, there is exactly one $K \in \mathcal{D}'(X \times Y)$ with $A = A_K$.

The distribution $K$ is the *Schwartz kernel* of $A$.

## Remarks

The hard part is the existence of $K$ in part 2.
One proof expands test functions in Fourier series on large cubes and controls the coefficients.
Another uses that the spaces of test functions are [[nuclear spaces|XC6XPMC2FQ]], so that continuous bilinear forms on $\mathcal{D}(X) \times \mathcal{D}(Y)$ extend to $\mathcal{D}(X \times Y)$.

Example: the identity map of $\mathcal{D}(\R)$, viewed as a map into $\mathcal{D}'(\R)$, has kernel $\delta(x - y)$.
Indeed $\langle \delta(x - y), \varphi \otimes \psi \rangle = \int \varphi(x) \psi(x)\, dx$.
The operator $\psi \mapsto \psi'$ has kernel $\delta'(x - y)$, the derivative in $x$ of the [[Dirac delta|0KPHENZ5S7]] on the diagonal.

## Explanations

### L1 human

A matrix is the same as a linear map between spaces of columns.
The kernel theorem says that the same holds for very general linear maps on functions: each one is "integration against a kernel" $K(x, y)$.
The price is that $K$ may be a distribution rather than a function, like the delta function for the identity map.

### L2 human

The theorem lets one describe operators such as differential operators, pseudodifferential operators and Fourier integral operators through their kernels.
Properties of the operator then become properties of $K$, for example the singular support or the wave front set of $K$.
Function kernels are not enough: the identity operator of $L^2(\R)$ is not an integral operator with a function kernel, so one really needs distributions.
</document>

<document source="https://tamnd.github.io/math2/node/2ZM39HSFTQ.md" tag="2ZM39HSFTQ" kind="statement">
# Kirszbraun theorem

## Statement

Let $n, m \ge 1$, let $S \subseteq \R^n$ be any subset, and let $L \ge 0$.
Give $\R^n$ and $\R^m$ the Euclidean norm.
Let $f \colon S \to \R^m$ satisfy the [[Lipschitz condition|0AW1E6YTJY]] $\norm{f(x) - f(y)} \le L \norm{x - y}$ for all $x, y \in S$.
Then there is a map $F \colon \R^n \to \R^m$ with $F = f$ on $S$ and $\norm{F(x) - F(y)} \le L \norm{x - y}$ for all $x, y \in \R^n$.

## Remarks

The point is that the extension keeps the same Lipschitz constant $L$.
For $m = 1$ this follows from the [[McShane–Whitney extension|MV7ADC2DEH]].
Extending each coordinate separately for $m \ge 2$ only gives the constant $\sqrt{m} \, L$.
The theorem holds more generally for maps from a subset of one Hilbert space to another Hilbert space.
It can fail for norms that do not come from an [[inner product|0QVNCJXRY3]].
One proof first extends $f$ to one extra point, which reduces to a statement about intersections of balls in Euclidean space, and then uses Zorn's lemma or a compactness argument.

## Explanations

### L1 human

A Lipschitz map does not stretch distances by more than a factor $L$.
If such a map is given only on part of space, one can fill in the rest without stretching any more.
This is easy for real-valued maps, but for vector-valued maps it needs the geometry of Euclidean space.

### L2 human

The theorem is a basic tool in geometric measure theory, where Lipschitz maps are often defined on irregular sets.
It lets one assume that a Lipschitz map is defined on all of $\R^n$, and then apply results such as [[Rademacher's theorem|QEY26RT19R]].
The key geometric step is about balls.
Take two families of balls with the same radii, where the distances between centres in the second family are at most those in the first.
If the first family has a common point, then so does the second.
</document>

<document source="https://tamnd.github.io/math2/node/2ZTQX5M0TK.md" tag="2ZTQX5M0TK" kind="statement">
# Lyapunov's convexity theorem

## Statement

Let $(X, \Sigma)$ be a measurable space and let $\mu_1, \dots, \mu_n$ be finite real [[signed measures|VG2P7R91Z7]] on $\Sigma$.
Assume that each $\mu_i$ is [[non-atomic|SV96KN4WR9]], that is, its total variation $\abs{\mu_i}$ has no atoms.
Let $\mu \colon \Sigma \to \R^n$ be the [[vector measure|PS28BSAJKJ]] $\mu(E) = (\mu_1(E), \dots, \mu_n(E))$.
Then the range $\set{\mu(E) : E \in \Sigma}$ is a [[compact|0R6EWFKRCF]] [[convex|02AS45T3H9]] subset of $\R^n$.

## Remarks

A short proof considers the set $K$ of measurable $g \colon X \to [0, 1]$, compact in the weak-$*$ topology of $L^\infty(\abs{\mu_1} + \dots + \abs{\mu_n})$.
The map $g \mapsto \int g \, d\mu$ is linear and continuous, so its image is compact and convex, and it contains the range of $\mu$.
For a point $v$ of the image, the preimage of $v$ in $K$ is compact and convex and has an extreme point by the Krein–Milman theorem.
Non-atomicity forces such an extreme point to be an indicator function $\mathbf{1}_E$, so $v = \mu(E)$.

Example: for $n = 1$ and Lebesgue measure on $[0, 1]$, the range is $[0, 1]$, because $\lambda([0, t]) = t$.

Without non-atomicity the result fails: the Dirac measure $\delta_0$ has range $\{0, 1\}$, which is not convex.
The result also fails in infinite dimensions.
The vector measure $E \mapsto \mathbf{1}_E$ from the Borel sets of $[0, 1]$ to $L^2[0, 1]$ is non-atomic, but $\frac12 \mathbf{1}_{[0,1]}$ lies between $0$ and $\mathbf{1}_{[0,1]}$ and is not an indicator function.

## Explanations

### L1 human

Suppose several quantities are spread over a space without any point carrying a lump.
Then every proportion of these quantities can be cut out exactly by one piece of the space.
For instance, a cake with chocolate and cream spread smoothly can be cut so that one piece has exactly a third of the chocolate and exactly half of the cream.

### L2 human

The theorem is the basis of fair division results, such as the existence of exact divisions of a cake among several players.
In control theory it gives the bang-bang principle: the set of states reachable with measurable controls is already reached with controls taking only extreme values.
In economics it shows that non-atomic economies have convex aggregate behaviour.
</document>

<document source="https://tamnd.github.io/math2/node/300K6CB1KH.md" tag="300K6CB1KH" kind="concept">
# Convex polytope

## Summary

A convex polytope is the [[convex hull|0FQSDC2EQ2]] of finitely many points of $\R^n$.
Equivalently, it is a bounded set cut out by finitely many linear inequalities.
Convex polygons, convex polyhedra such as the cube and the regular tetrahedron, and their analogues in higher dimensions are convex polytopes.
They are the basic objects of polyhedral combinatorics and of linear programming, where the feasible region of a bounded problem is a convex polytope.

## Explanations

### L1 human

Hammer a few nails into a board and stretch a rubber band around them all: the region inside the band is a convex polygon.
A convex polytope is the same idea in any dimension: the smallest convex set containing a finite list of points.
It has flat sides, straight edges and finitely many corners.
For example, the unit square is the convex hull of the four points $(0, 0)$, $(1, 0)$, $(0, 1)$ and $(1, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/300V40FC1B.md" tag="300V40FC1B" kind="concept">
# Chromatic homotopy theory

## Summary

Chromatic homotopy theory studies the [[stable homotopy category|26GTNMAF7B]], one prime $p$ at a time, by splitting it into layers indexed by a height $n = 0, 1, 2, \ldots$.
The height comes from the height of formal groups, through [[Quillen's theorem|4E518M24MA]] and [[Brown–Peterson cohomology|S8S72AA9C8]].
Each layer is seen by a Morava K-theory $K(n)$ and is governed by $v_n$-periodic phenomena.
The [[nilpotence theorem|9N027RT5R6]] and the other [[Ravenel conjectures|B8XP2K7RS1]] proved by Devinatz, Hopkins, Ravenel and Smith are its foundations.

## Explanations

### L1 human

The stable homotopy groups of spheres look chaotic, but they contain families that repeat with fixed periods.
Chromatic homotopy theory sorts these families by a number $n$, like splitting white light into colours with a prism.
Height $0$ is rational information, which is simple.
Height $1$ is seen by K-theory and contains the image of the J-homomorphism, whose groups repeat with period $8$ at the prime $2$.
Higher heights contain more subtle periodic families, studied with Morava K-theories and Morava E-theories.
</document>

<document source="https://tamnd.github.io/math2/node/3017J7P6JD.md" tag="3017J7P6JD" kind="concept">
# Pseudogroup

## Summary

A pseudogroup on a [[topological space|0CQHFPGJ45]] is a collection of [[homeomorphisms|0QT00PB5G2]] between open subsets that is closed under composition, inversion, restriction and gluing.
It records the local symmetries of a space, which need not extend to global ones.
Pseudogroups are the transition rules of geometric structures: a smooth, complex or symplectic manifold is a space glued together by maps from the corresponding pseudogroup.
They also describe the transverse structure of foliations.

## Explanations

### L1 human

A group of symmetries acts on a whole space at once.
Many geometric questions only need symmetries that act on small pieces, such as a map from one disc to another.
A pseudogroup collects such local maps, with the rule that the collection is stable under the obvious operations: composing, inverting, shrinking the domain and patching pieces together.
For example, all smooth maps with smooth inverses between open subsets of $\R^n$ form a pseudogroup.
A smooth manifold is a space whose charts change by maps of this pseudogroup.
</document>

<document source="https://tamnd.github.io/math2/node/3039GWNHWS.md" tag="3039GWNHWS" kind="concept">
# Higher-order differences

## Summary

Higher-order differences are obtained by applying a difference operator several times.
The $n$-th forward difference $\Delta_h^n f$ is the discrete analogue of the $n$-th derivative.
They detect polynomial behaviour, appear in Newton's interpolation formula, and express linear difference equations in operator form.

## Explanations

### L1 human

Take a list of numbers, write down the gaps between neighbours, then the gaps between those gaps, and so on.
Each new row is a difference of higher order.
For the cubes $0, 1, 8, 27, 64, 125$ the rows are $1, 7, 19, 37, 61$, then $6, 12, 18, 24$, then $6, 6, 6$.
The third row is constant, which reflects that $x^3$ has degree $3$.
</document>

<document source="https://tamnd.github.io/math2/node/306PWGH7BF.md" tag="306PWGH7BF" kind="concept">
# Frame

## Summary

A frame is a complete lattice in which finite meets distribute over arbitrary joins.
The open sets of any topological space form a frame, and frames are the algebraic objects of [[pointless topology|CYEMGH98WN]].
As ordered sets, frames are exactly the complete [[Heyting algebras|NG943XNQWH]]; the word "frame" signals that the maps considered preserve finite meets and arbitrary joins.

## Explanations

### L1 human

Look at the open sets of a space, ordered by inclusion.
Any union of open sets is open, and the intersection of two open sets is open.
Moreover, intersecting with one open set $U$ commutes with taking unions: $U \cap \bigcup_i V_i = \bigcup_i (U \cap V_i)$.
A frame is any complete lattice with this one distributive law.
It captures what topology needs from open sets, without mentioning points.
</document>

<document source="https://tamnd.github.io/math2/node/30G1M35TK1.md" tag="30G1M35TK1" kind="concept">
# Time-scale calculus

## Summary

Time-scale calculus develops derivatives and integrals for functions defined on a time scale, that is, on any nonempty closed subset of $\R$.
On $\R$ its delta derivative is the ordinary [[derivative|0V1EJCT8G6]], and on $\Z$ it is the forward difference.
So it unifies differential equations and difference equations, and it covers mixed cases such as unions of intervals.

## Explanations

### L1 human

Some processes run in continuous time, some in steps, and some in both, like an insect population that grows in summer and is counted once each winter.
A time scale is the set of moments at which the process is observed.
At a moment followed by a gap, the derivative becomes a difference quotient across the gap; at a moment with no gap, it is the usual limit.
For example, on the time scale $\Z$ the derivative of $t^2$ is $(t + 1)^2 - t^2 = 2t + 1$, not $2t$.
</document>

<document source="https://tamnd.github.io/math2/node/30GF7TK8YP.md" tag="30GF7TK8YP" kind="definition">
# Meshfree methods

## Definition

Let $\Omega \subset \R^d$ be a domain and let $X = \set{x_1, \ldots, x_n} \subset \overline{\Omega}$ be a finite set of nodes.
A *meshfree method* for a differential equation on $\Omega$ approximates the solution by
$$u_h(x) = \sum_{i=1}^{n} \varphi_i(x) \, c_i,$$
where the functions $\varphi_i$ and the method satisfy two conditions:

1. each $\varphi_i$ is determined by the node set $X$ and fixed weight or kernel functions alone, with no elements or edges between nodes;
2. the coefficients $c_i$ are found by a [[Galerkin|Y5B0PEBKE8]] condition, by collocation at points of $\Omega$, or by a particle evolution law.

Usually each $\varphi_i$ has compact support, a ball of radius $\rho$ around $x_i$.

## Remarks

Common choices of $\varphi_i$ are [[moving least squares|T0JZN6WP5X]] shape functions and [[radial basis functions|ZBSXSTQM2J]] $\varphi_i(x) = \phi(\abs{x - x_i})$.
Moving least squares shape functions of order $m$ reproduce polynomials of degree at most $m$; since they reproduce constants, they form a [[partition of unity|NH739XXVXK]].
Essential boundary conditions need extra care, because these $\varphi_i$ usually do not satisfy $\varphi_i(x_j) = \delta_{ij}$.

Example: Gaussian radial basis function collocation for $-\Delta u = f$ uses $\varphi_i(x) = e^{-\varepsilon^2 \abs{x - x_i}^2}$ with a shape parameter $\varepsilon > 0$.
These functions depend only on the nodes and on $\varepsilon$, which satisfies clause 1.
The $c_i$ solve $-\Delta u_h(x_j) = f(x_j)$ at interior nodes and the boundary condition at boundary nodes, which is clause 2.

Non-example: the hat functions of linear finite elements on a triangulation are not meshfree.
Each hat function depends on which triangles contain its node, so clause 1 fails.
</document>

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# Artin–Rees lemma

## Statement

Let $R$ be a commutative [[Noetherian ring|0670T59HGE]] and let $I$ be an [[ideal|096Z9MP2ME]] of $R$.
Let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module, and let $N$ be a [[submodule|05WDSN597C]] of $M$.
Then there is an integer $k \ge 0$ such that
$$I^n M \cap N = I^{n-k} (I^k M \cap N) \quad \text{for all } n \ge k.$$

## Remarks

The [[Rees algebra|W8ZAH8TTHS]] $R[It]$ is Noetherian because $R$ is Noetherian and $I$ is finitely generated.
The module $\bigoplus_n I^n M$ is a finitely generated module over $R[It]$, so its graded submodule $\bigoplus_n (I^n M \cap N)$ is finitely generated too.
A finite set of homogeneous generators of degree at most $k$ gives the equality for $n \ge k$.
In words, the [[filtration|WHME2NGANG]] $(I^n M \cap N)_n$ of $N$ is a stable $I$-filtration.
Consequently the $I$-adic topology of $M$ induces the $I$-adic topology on $N$, since $I^n M \cap N \subseteq I^{n-k} N$.
Example: for $R = M = \Z$, $I = 2\Z$ and $N = 6\Z$, one has $2^n \Z \cap 6\Z = 3 \cdot 2^n \Z = 2^{n-1}(2\Z \cap 6\Z)$ for $n \ge 1$, so $k = 1$ works.

## Explanations

### L1 human

Take a big module and a smaller module inside it.
There are two ways to say that an element of the small module is tiny.
It can be a multiple of a high power of $I$ inside the big module, or inside the small module.
The lemma says that these two ways agree up to a fixed shift $k$.

### L2 human

The lemma is the main technical tool for completions of Noetherian rings.
It implies the [[Krull intersection theorem|1Q1AH2V41S]], and it shows that $I$-adic completion is exact on finitely generated modules, which gives [[flatness of completion|NPF42DJMBX]].
</document>

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# Toda bracket

## Summary

The Toda bracket is a secondary composition operation on homotopy classes of maps.
It takes three maps $f$, $g$, $h$ such that $g \circ f$ and $h \circ g$ are null-homotopic.
It produces a set of maps out of the suspension of the source of $f$.
Toda introduced it to compute the [[homotopy groups of spheres|YD6VAXXSBM]].
It remains a basic tool for finding relations in the [[stable homotopy groups of spheres|HGZAYPQBXW]].

## Explanations

### L1 human

If $g \circ f$ is homotopic to a constant map, there is a homotopy that witnesses this.
If also $h \circ g$ is null, there is a second witnessing homotopy.
Comparing the two witnesses for the triple composite $h \circ g \circ f$ gives a loop of maps, and this loop is a map out of a suspension.
The Toda bracket is the set of all maps that arise from such choices.
For example, in the stable homotopy groups of spheres the bracket $\langle 2, \eta, 2 \rangle$ is the single element $\eta^2$, where $\eta$ is the stable Hopf map.
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<document source="https://tamnd.github.io/math2/node/30R6VND1T7.md" tag="30R6VND1T7" kind="definition">
# Perfect map

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]].
A map $p \colon X \to Y$ is a *perfect map* if:

1. $p$ is [[continuous|0VGFHTM75Y]].
2. $p$ is [[closed|00FSJTXPJM]]: the image of every closed subset of $X$ is closed in $Y$.
3. $p$ is surjective.
4. For every $y \in Y$, the fibre $p^{-1}(y)$ is [[compact|0TVJZYB6SH]].

## Remarks

Some authors do not require clause 3, and some also require $X$ to be Hausdorff.
A perfect map is [[proper|15XEBGNGHC]]: the [[preimage|01AHG4G9RT]] of every compact set is compact.

Example: let $K$ be a compact space and $Y$ any space, and let $p \colon Y \times K \to Y$ be the projection.
It is continuous and surjective when $K \ne \emptyset$, each fibre $\set{y} \times K$ is homeomorphic to $K$, and the projection is closed because $K$ is compact, by the [[tube lemma|W8XEZR3VKQ]].

Non-example: the projection $p \colon \R^2 \to \R$, $p(x, y) = x$.
It is continuous and surjective, but the closed set $\set{(x, y) : xy = 1}$ has image $\R \setminus \set{0}$, which is not closed.
Also its fibres are lines, which are not compact.
</document>

<document source="https://tamnd.github.io/math2/node/30SKVHXFNQ.md" tag="30SKVHXFNQ" kind="definition">
# Restriction of a measure

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $A \in \Sigma$.
The *restriction* of $\mu$ to $A$ is the map $\mu|_A \colon \Sigma \to [0, \infty]$ given by
$$\mu|_A(B) = \mu(A \cap B) \quad \text{for } B \in \Sigma.$$

## Remarks

The map $\mu|_A$ is a [[measure|PR11BZQQPG]] on $(X, \Sigma)$, because intersecting a disjoint family with $A$ gives a disjoint family with the same union intersected with $A$.
Example: if $\lambda$ is Lebesgue measure on $\R$ and $A = [0, 1]$, then $\lambda|_A([0, 5]) = \lambda([0, 1]) = 1$.
For a measurable $f \colon X \to [0, \infty]$ one has $\int_X f \, d(\mu|_A) = \int_A f \, d\mu$.
Non-example: for a set $A \notin \Sigma$, the sets $A \cap B$ need not lie in $\Sigma$, so the formula need not define a measure on $\Sigma$.
A related view keeps only the points of $A$: the trace $\sigma$-algebra $\Sigma_A = \set{A \cap B : B \in \Sigma}$ on $A$, with the measure $\mu$ restricted to $\Sigma_A$.
Other notations are $\mu \llcorner A$ and $\mu(\,\cdot \cap A)$.
</document>

<document source="https://tamnd.github.io/math2/node/30TG40QA1J.md" tag="30TG40QA1J" kind="definition">
# Nevanlinna characteristic

## Definition

Let $f$ be a [[meromorphic function|0NSD1TPHR0]] on $\C$, not identically zero, and write $\log^+ x = \max(\log x, 0)$ for $x \ge 0$, with $\log^+ 0 = 0$.
For $t \ge 0$, let $n(t)$ be the number of [[poles|0WPQYW8MTE]] of $f$ in the disc $\abs{z} \le t$, counted with multiplicity.
For $r > 0$, let
$$m(r, f) = \frac{1}{2\pi} \int_0^{2\pi} \log^+ \abs{f(r e^{i\theta})} \, d\theta, \qquad N(r, f) = \int_0^r \frac{n(t) - n(0)}{t} \, dt + n(0) \log r.$$
The *Nevanlinna characteristic* of $f$ is the function
$$T(r, f) = m(r, f) + N(r, f), \qquad r > 0.$$

## Remarks

Example: for $f(z) = e^z$, $\log^+ \abs{f(re^{i\theta})} = \max(r\cos\theta, 0)$ and there are no poles, so $T(r, f) = \frac{1}{2\pi}\int_{-\pi/2}^{\pi/2} r \cos\theta \, d\theta = r/\pi$.
Non-example: the maximum modulus $\max_{\abs{z} = r} \abs{f(z)}$ is no growth measure for $f(z) = \tan z$.
It is infinite whenever the circle $\abs{z} = r$ passes through a pole $\pi/2 + k\pi$.
The characteristic avoids this by counting the poles in $N(r, f)$.
The function $T(r, f)$ is increasing and a convex function of $\log r$.
For an entire function $f$ and $M(r) = \max_{\abs{z} = r} \abs{f(z)}$, one has $T(r, f) \le \log^+ M(r) \le 3\, T(2r, f)$, so $T$ and $\log M$ define the same [[order|SHF8Z7GMFX]].
A meromorphic function is rational exactly when $T(r, f) = O(\log r)$, and a rational function of degree $d$ has $T(r, f) = d \log r + O(1)$.
The Ahlfors–Shimizu characteristic, defined from the spherical area of the image of the disc $\abs{z} < r$, differs from $T(r, f)$ by a bounded function.
</document>

<document source="https://tamnd.github.io/math2/node/30XVCMK916.md" tag="30XVCMK916" kind="statement">
# Stein's unbiased risk estimate

## Statement

Let $X$ be [[multivariate normal|0XN3K7RSZY]] on $\R^p$ with unknown mean $\theta$ and covariance matrix $\sigma^2 I_p$, where $\sigma^2 > 0$ is known.
Let $g = (g_1, \dots, g_p) \colon \R^p \to \R^p$ be [[weakly differentiable|W8W9AP79V5]], with $E_\theta \norm{g(X)}^2 < \infty$ and $E_\theta \abs{\partial_i g_i(X)} < \infty$ for $i = 1, \dots, p$.
Consider the estimator $\delta(X) = X + g(X)$ of $\theta$, and put
$$\mathrm{SURE}(X) = p\sigma^2 + \norm{g(X)}^2 + 2\sigma^2 \sum_{i=1}^{p} \partial_i g_i(X).$$
Then $\mathrm{SURE}(X)$ is an [[unbiased|0BWGTN92FG]] estimator of the [[risk|N26TSVRH2S]] of $\delta$ under squared error loss:
$$E_\theta\big[ \mathrm{SURE}(X) \big] = E_\theta \norm{\delta(X) - \theta}^2.$$

## Remarks

Expand $\norm{X - \theta + g(X)}^2 = \norm{X - \theta}^2 + \norm{g(X)}^2 + 2 \sum_i (X_i - \theta_i) g_i(X)$.
The first term has expectation $p\sigma^2$.
By the multivariate [[Stein's lemma|6PWC83R457]], $E_\theta[(X_i - \theta_i) g_i(X)] = \sigma^2 E_\theta[\partial_i g_i(X)]$, which gives the formula.

The point is that $\mathrm{SURE}(X)$ does not involve the unknown $\theta$.
So it can be computed from the data and minimized over a family of estimators, for example to choose a threshold or a smoothing parameter.

Example: the [[James–Stein estimator|709T196XCT]] with $\sigma = 1$ corresponds to $g(x) = -(p - 2) x / \norm{x}^2$.
Then $\mathrm{SURE}(X) = p - (p - 2)^2 / \norm{X}^2$, whose expectation is the risk of $\delta_{JS}$.
It is below $p$ for every $X$, which shows that $\delta_{JS}$ has risk below $p$ when $p \ge 3$.

## Explanations

### L1 human

Usually the error of an estimate cannot be computed, because the truth is unknown.
For normal data with known noise level, Stein found a formula built only from the data whose average equals the average squared error.
One can therefore compare estimators, or tune them, using the data alone.

### L2 human

The estimate is the standard tool to prove that shrinkage estimators beat the raw observation, as in Stein's example.
It is also used in practice to select tuning parameters, for instance the threshold in wavelet shrinkage, as an alternative to cross-validation.
The derivative term $\sum_i \partial_i g_i$ is the divergence of $g$.
The expected divergence of $\delta$, which is $p$ plus the expected divergence of $g$, is the effective number of degrees of freedom used by the estimator.
</document>

<document source="https://tamnd.github.io/math2/node/312GG23DN4.md" tag="312GG23DN4" kind="definition">
# Sylvester equation

## Definition

Let $\mathbb{K}$ be a [[field|04C5ZBG1WT]] and let $m, n \ge 1$.
Let $A$ be an $m \times m$ [[matrix|0VD9ZSWWJG]], $B$ an $n \times n$ matrix and $C$ an $m \times n$ matrix over $\mathbb{K}$.
The *Sylvester equation* for $A$, $B$, $C$ is the equation
$$AX + XB = C$$
for an unknown $m \times n$ matrix $X$ over $\mathbb{K}$.

## Remarks

The map $X \mapsto AX + XB$ is linear, and with the [[Kronecker product|0N7ENQRCXN]] the equation reads $(I_n \otimes A + B^T \otimes I_m) \operatorname{vec}(X) = \operatorname{vec}(C)$.
It has a unique solution for every $C$ if and only if $A$ and $-B$ have no common [[eigenvalue|0BQAA2BK3Q]] in an algebraic closure of $\mathbb{K}$.
Example: $A = \operatorname{diag}(1, 2)$, $B = 1$ and $C = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ give $x_1 + x_1 = 2$ and $2x_2 + x_2 = 3$, so $X = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$.
Non-example: the equation $AXB = C$ is not a Sylvester equation, since $X$ is multiplied on both sides in one term.
For $B = -A$, the equation $AX - XA = C$ never has a unique solution, because $A$ and $A$ share their eigenvalues.
</document>

<document source="https://tamnd.github.io/math2/node/314WH1WAP1.md" tag="314WH1WAP1" kind="definition">
# Stationary process

## Definition

Let $T$ be one of $\Z$, $\N$, $\R$ or $[0, \infty)$, so that $t + h \in T$ whenever $t, h \in T$.
Let $X = (X_t)_{t \in T}$ be a [[stochastic process|0QH5ENKZM1]] with values in a measurable space $S$.
The process $X$ is *stationary*, or *strictly stationary*, if for every $k \ge 1$, all $t_1, \dots, t_k \in T$ and every $h \in T$, the random vectors
$$(X_{t_1 + h}, \dots, X_{t_k + h}) \quad \text{and} \quad (X_{t_1}, \dots, X_{t_k})$$
have the same law.

## Remarks

Equivalently, all [[finite-dimensional distributions|Y0CN7V3B7E]] of $X$ are invariant under the time shift by $h$, for every $h \in T$.

A real process with $E[X_t^2] < \infty$ for all $t$ is *weakly stationary*, or *wide-sense stationary*, if $E[X_t] = m$ does not depend on $t$ and the [[covariance|0KBFAWYRTE]] $\operatorname{Cov}(X_{t+h}, X_t) = r(h)$ depends only on $h$.
A strictly stationary process with finite second moments is weakly stationary.
For [[Gaussian processes|22WEM6RTW0]] the two notions agree.

Example: an i.i.d. sequence $(X_n)_{n \in \Z}$ is stationary, since the law of $(X_{t_1 + h}, \dots, X_{t_k + h})$ is the product of $k$ copies of the common law, for distinct times, whatever $h$ is.

Non-example: a simple symmetric [[random walk|04VZE3FZXF]] $S_n$ with $S_0 = 0$ is not stationary.
Already for $k = 1$, the law of $S_0$ is the point mass at $0$, while $S_1$ takes the values $\pm 1$.
</document>

<document source="https://tamnd.github.io/math2/node/3173Y813V7.md" tag="3173Y813V7" kind="statement">
# Malgrange–Ehrenpreis theorem

## Statement

Let $P = \sum_{\abs{\alpha} \le m} c_\alpha \partial^\alpha$ be a [[linear differential operator|6DZETZ5T9G]] on $\R^n$ with constant complex coefficients $c_\alpha$, not all zero.
Then $P$ has a [[fundamental solution|K2NK9A8M5F]]: there is a distribution $E$ on $\R^n$ with
$$P E = \delta,$$
where $\delta$ is the Dirac distribution at the origin.
Consequently, for every [[test function|HS65DW0HXH]] $f$ on $\R^n$, the function $u = E * f$ is smooth and satisfies $P u = f$.

## Remarks

Formally, the Fourier transform turns $P E = \delta$ into multiplication of $\widehat{E}$ by a polynomial $p(\xi)$, so one wants $\widehat{E} = 1 / p$.
The difficulty is that $p$ may vanish on the real space.
One proof avoids its zeros by moving the Fourier integration into the complex domain.
Another uses the Hahn–Banach theorem together with an estimate $\norm{\varphi} \le C \norm{P^{*} \varphi}$ in suitable norms.
The theorem was proved independently by Bernard Malgrange and Leon Ehrenpreis.
Example: for $P = \Delta$ on $\R^3$, the function $E(x) = -1 / (4 \pi \abs{x})$ is a fundamental solution, and $u = E * f$ solves the Poisson equation $\Delta u = f$.
The hypothesis of constant coefficients is needed: Lewy's operator, which has polynomial coefficients, has no [[local solvability|P6E9RZNRBZ]] at all.
Constant-coefficient operators are even solvable in the large, by a further classical theorem.
That is, $P u = f$ has a smooth solution on $\R^n$ for every smooth $f$ on $\R^n$.

## Explanations

### L1 human

The theorem says that every linear differential equation with constant coefficients can be solved for a point source.
Once the response $E$ to a point source is known, the response to any source $f$ is obtained by adding up shifted copies of $E$, that is, by convolution.
So every such equation can be solved, at least for sources that vanish outside a bounded set.

### L2 human

The theorem is the starting point of the general theory of constant-coefficient operators.
For particular operators one writes the fundamental solution down explicitly, such as the Newtonian potential for the Laplacian or the [[heat kernel|KD404MF042]] for the heat operator.
The general theorem guarantees existence but gives little information on $E$; finer questions, such as whether $P$ is [[hypoelliptic|K9DSR21DQC]], depend on properties of the fundamental solution.
The theorem fails for variable coefficients, and the study of when local solvability holds then becomes a hard problem.
</document>

<document source="https://tamnd.github.io/math2/node/31CV5K5SAJ.md" tag="31CV5K5SAJ" kind="statement">
# Demailly's regularization of currents

## Statement

Let $\Omega \subseteq \C^n$ be a bounded [[pseudoconvex|AHYZMSBBNH]] domain and let $\varphi$ be a [[plurisubharmonic function|FF2P7ER3QT]] on $\Omega$, not identically $-\infty$.
For each integer $m \ge 1$, let $\mathcal{H}_m$ be the Hilbert space of holomorphic functions $f$ on $\Omega$ with $\int_\Omega \abs{f}^2 e^{-2m\varphi} \, dV < \infty$.
Let $(\sigma_{m,l})_{l \ge 1}$ be an orthonormal basis of $\mathcal{H}_m$, and let
$$\varphi_m = \frac{1}{2m} \log \sum_{l} \abs{\sigma_{m,l}}^2 .$$
Then there are constants $C_1, C_2 > 0$, independent of $m$, such that for all $z \in \Omega$ and all $r$ with $0 < r < \operatorname{dist}(z, \partial\Omega)$:

1. $\varphi(z) - \dfrac{C_1}{m} \le \varphi_m(z) \le \sup_{\abs{\zeta - z} < r} \varphi(\zeta) + \dfrac{1}{m} \log \dfrac{C_2}{r^n}$;
2. $\nu(\varphi, z) - \dfrac{n}{m} \le \nu(\varphi_m, z) \le \nu(\varphi, z)$, where $\nu$ denotes the [[Lelong number|WVB60RKC3Z]].

In particular $\varphi_m$ converges to $\varphi$ pointwise and in $L^1_{\mathrm{loc}}(\Omega)$ as $m \to \infty$.

## Remarks

Each $\varphi_m$ is plurisubharmonic and has analytic singularities: locally it is $\frac{1}{2m} \log \sum \abs{g_j}^2$ plus a bounded function, for finitely many holomorphic $g_j$, which generate the [[multiplier ideal|BKM0TQSCD9]] $\mathcal{I}(m\varphi)$.
Proof idea: $\varphi_m(z) = \frac{1}{m} \sup\set{\log\abs{f(z)} : f \in \mathcal{H}_m, \norm{f} \le 1}$, so it is the logarithm of a weighted Bergman kernel on the diagonal.
The upper bound follows from the mean value inequality for $\abs{f}^2$ on the ball of radius $r$ around $z$.
The lower bound follows from the [[Ohsawa–Takegoshi extension theorem|2SRQCEB16J]], which gives $f \in \mathcal{H}_m$ with $f(z) = e^{m\varphi(z)}$ and $\norm{f}$ bounded independently of $z$ and $m$.
The Lelong number inequalities follow from these bounds.
Example: for $\varphi = \log\abs{z}$ on the unit disc of $\C$, $\nu(\varphi, 0) = 1$, and the bounds force $1 - \frac{1}{m} \le \nu(\varphi_m, 0) \le 1$.
The method can be globalised with a partition of unity.
On a compact complex manifold, it approximates a closed [[positive current|BYJHP6A906]] of bidegree $(1, 1)$ by closed currents in the same cohomology class.
The approximating currents have analytic singularities, lose only a small amount of positivity, and have Lelong numbers converging to those of the current.

## Explanations

### L1 human

Plurisubharmonic functions can have very complicated singularities.
The theorem replaces such a function by a sequence of simpler ones, built from logarithms of sums of squares of holomorphic functions.
The simpler functions converge to the original one, and their singularities converge too, measured by Lelong numbers.

### L2 human

This is the main regularisation tool of analytic methods in algebraic geometry.
Functions with analytic singularities can be resolved by blowing up, so statements about them reduce to divisors with normal crossings.
The global version for closed positive $(1,1)$-currents is used to study pseudo-effective and nef line bundles.
It also gives a proof of Siu's theorem that the upper level sets of Lelong numbers are analytic.
It relies on the $L^2$ theory of $\bar\partial$, through the Ohsawa–Takegoshi theorem.
</document>

<document source="https://tamnd.github.io/math2/node/31DQH72ZJP.md" tag="31DQH72ZJP" kind="statement">
# Cauchy integral on Lipschitz curves

## Statement

Let $A \colon \R \to \R$ be a function that satisfies a [[Lipschitz condition|0AW1E6YTJY]], and let $\Gamma$ be its graph $\set{x + iA(x) : x \in \R}$ in $\C$.
For $\varepsilon > 0$ and $f$ continuous with compact support on $\R$, let
$$C_\varepsilon f(x) = \frac{1}{\pi i} \int_{\abs{x - y} > \varepsilon} \frac{f(y) \, (1 + iA'(y))}{(y - x) + i(A(y) - A(x))} \, dy, \quad x \in \R.$$
Then there is a constant $C$, depending only on $\norm{A'}_{L^\infty}$, such that
$$\norm{C_\varepsilon f}_{L^2(\R)} \le C \norm{f}_{L^2(\R)}$$
for all $\varepsilon > 0$ and all such $f$.
So the Cauchy integral on $\Gamma$ extends to a bounded operator on the [[Lp space|SASP14MA1Z]] $L^2(\R)$.

## Remarks

The operator $C_\varepsilon f$ is the truncated Cauchy integral $\frac{1}{\pi i}\int \frac{F(\zeta)}{\zeta - z} \, d\zeta$ over $\Gamma$, written in the parameter $x$, with $F(y + iA(y)) = f(y)$ and $z = x + iA(x)$.
A Lipschitz function is differentiable almost everywhere, with $A'$ bounded, so the formula makes sense.
Let $Hf(x) = \frac{1}{\pi} \lim_{\varepsilon \to 0} \int_{\abs{x - y} > \varepsilon} \frac{f(y)}{x - y} \, dy$ be the [[Hilbert transform|NSCEBXM7ZM]].
For $A = 0$ the curve is the real line and the operator is $iH$.
So the theorem generalizes the $L^2$ bound of the Hilbert transform.

The case of small $\norm{A'}_{L^\infty}$ is due to Calderón.
Expanding the kernel in powers of $A(y) - A(x)$ writes the operator as a series of Calderón commutators, and the series converges when $\norm{A'}_{L^\infty}$ is small.
The general case, for every Lipschitz function $A$, was proved by Coifman, McIntosh and Meyer.
Since the kernel is a Calderón–Zygmund kernel, the $L^2$ bound also gives boundedness on $L^p(\R)$ for $1 < p < \infty$.

## Explanations

### L1 human

The Cauchy integral is the basic way to build holomorphic functions from values on a curve.
On a smooth curve it behaves like the Hilbert transform on the line.
The theorem says that the same good behaviour holds on much rougher curves.
They may have corners, as long as they are graphs of Lipschitz functions, such as the graph of $\abs{x}$.

### L2 human

The theorem is a central example of a [[singular integral operator|HTMCEARGDD]] whose kernel is not of convolution type.
It is the starting point for the method of layer potentials on Lipschitz domains, which solves the Dirichlet problem for the Laplacian on domains with rough boundary.
</document>

<document source="https://tamnd.github.io/math2/node/31G7KHSYEC.md" tag="31G7KHSYEC" kind="concept">
# Lanczos sigma factor

## Summary

The Lanczos sigma factors are the weights $\sigma_k = \frac{\sin(\pi k/m)}{\pi k/m}$ by which one multiplies the terms of a truncated [[Fourier series|G12KXDHPPS]] of order $m$.
The weighted sum, called the sigma approximation, equals the average of the partial sum over a window of length $2\pi/m$ around each point.
This local averaging greatly reduces the overshoot near jumps known as the [[Gibbs phenomenon|VMF169AT67]].
The factors are named after Cornelius Lanczos.

## Explanations

### L1 human

A truncated Fourier series of a function with a jump overshoots near the jump and ripples around it.
Averaging the truncated series over a short window, exactly one period of the first frequency that was cut off, removes most of the ripple.
In terms of the coefficients, this averaging multiplies the term of frequency $k$ by $\frac{\sin(\pi k/m)}{\pi k/m}$.
For $m = 2$ the term of frequency $1$ is multiplied by $2/\pi \approx 0.64$.
</document>

<document source="https://tamnd.github.io/math2/node/31N0FH22ZT.md" tag="31N0FH22ZT" kind="statement">
# Jones index theorem

## Statement

Let $N \subseteq M$ be a [[subfactor|9RWS7TZHXH]] of type $\mathrm{II}_1$ factors with Jones index $[M:N] < \infty$.
Then
$$[M : N] \in \set{4 \cos^2(\pi / n) : n = 3, 4, 5, \dots} \cup [4, \infty).$$
Conversely, every number in this set is the index of a subfactor of the [[hyperfinite type II1 factor|NZ4NY8SK1C]] $R$.

## Remarks

The proof uses the basic construction: $M$ acts on $L^2(M)$, and adding the orthogonal projection $e_N$ onto $L^2(N)$ gives a larger factor $M_1 = \langle M, e_N \rangle$ with $[M_1 : M] = [M : N]$.
Iterating gives a tower $N \subseteq M \subseteq M_1 \subseteq M_2 \subseteq \cdots$ with projections $e_1, e_2, \dots$ that satisfy $e_i e_{i \pm 1} e_i = \lambda e_i$ with $\lambda = [M:N]^{-1}$ and $e_i e_j = e_j e_i$ for $\abs{i - j} \ge 2$.
The trace restricted to the algebras generated by $e_1, \dots, e_k$ must be positive, and this forces $\lambda^{-1}$ into the set above.
For the values below $4$, the construction of the converse uses these same projections inside $R$.

The first allowed values are $1$, $2$, $(3 + \sqrt 5)/2 \approx 2.618$ and $3$, for $n = 3, 4, 5, 6$.
So no subfactor has index strictly between $1$ and $2$, although the index need not be an integer.

## Explanations

### L1 human

The index of a subfactor measures how much bigger the large algebra is than the small one.
One might expect any number at least $1$ to be possible.
Jones found that below $4$ only a discrete list of values $4\cos^2(\pi/n)$ can occur, while every number from $4$ on can occur.

### L2 human

The relations among the projections $e_i$ are those of the Temperley–Lieb algebra.
Jones used representations of the braid group built from them to discover the Jones polynomial of knots.
Finer invariants of a subfactor, such as its principal graph and its standard invariant, lead to the classification of subfactors of small index.
For the fixed points $M^G$ of an outer action of a finite group $G$ on $M$, one has $[M : M^G] = \abs{G}$, so the index generalizes the index of a subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/31Q0RDPZ00.md" tag="31Q0RDPZ00" kind="concept">
# Clifford module

## Summary

A Clifford module is a module over a [[Clifford algebra|A38B1QFK20]].
It is a vector space on which the vectors of a quadratic space act as linear maps.
The square of each such map is a scalar fixed by the quadratic form: minus its value, or its value, depending on the sign convention.
Spinor spaces are the irreducible Clifford modules.
Bundles of Clifford modules over Riemannian manifolds carry Dirac operators, which makes Clifford modules the algebraic input of index theory.
Their classification shows a periodicity of period $8$ over the reals and $2$ over the complex numbers, closely related to Bott periodicity.

## Explanations

### L1 human

The complex numbers are the real plane with a rule saying that $i \cdot i = -1$.
A Clifford module generalises this: several directions $e_1, \ldots, e_n$ act on a space, each one squaring to $-1$, and different ones anticommuting.
For one direction, the plane $\C$ with $e_1$ acting as multiplication by $i$ is a Clifford module.
For two directions, the quaternions with $e_1$ and $e_2$ acting as left multiplication by $i$ and $j$ form one.
These modules are the spaces on which spinors live.
</document>

<document source="https://tamnd.github.io/math2/node/323GN6K117.md" tag="323GN6K117" kind="concept">
# Angle of parallelism

## Summary

Take a line of the [[hyperbolic plane|GN4N94GQ0Z]] and a point at distance $a$ from it.
The angle of parallelism $\Pi(a)$ is the angle between the perpendicular to the line and a limiting parallel through the point.
It depends only on the distance $a$, and in curvature $-1$ it is given by $\tan(\Pi(a)/2) = e^{-a}$.
It decreases from $\pi/2$ to $0$ as $a$ grows, which measures how far hyperbolic geometry departs from Euclid's [[parallel postulate|2PC3XY0PYZ]].
The notation $\Pi(a)$ goes back to Lobachevsky.

## Explanations

### L1 human

Stand at a point $P$ at distance $a$ from a line, and drop the perpendicular to it.
Turn a ray from the perpendicular toward the line: at first the ray meets the line, and past a certain angle it never does.
That critical angle is the angle of parallelism.
In Euclidean geometry it is always a right angle, but in hyperbolic geometry it is smaller, and the farther you stand, the smaller it gets.
For example, at distance $a = \ln 3 \approx 1.1$ one gets $\tan(\Pi/2) = 1/3$, so $\Pi \approx 37^\circ$.
</document>

<document source="https://tamnd.github.io/math2/node/3240RP14MT.md" tag="3240RP14MT" kind="concept">
# Lorentz space

## Summary

The Lorentz spaces $L^{p,q}(\mu)$ form a two-parameter family of function spaces that refines the scale of [[Lp spaces|SASP14MA1Z]].
They are defined with the [[decreasing rearrangement|BWS042NC2P]] of a function, so they only see the sizes of the level sets of the function.
The first index $p$ gives the main size and the second index $q$ gives a fine adjustment.
They contain $L^p = L^{p,p}$ and the [[weak Lp space|H4C888M5AZ]] $L^{p,\infty}$, and they appear in interpolation theory and in sharp forms of Sobolev inequalities.

## Explanations

### L1 human

To measure the size of a function, rearrange its values in decreasing order, like sorting a pile of sand by height.
The Lorentz norm then weighs this sorted profile in a way governed by two numbers $p$ and $q$.
With $q = p$ one gets the usual $L^p$ norm.
Letting $q$ grow makes the space larger, so for $p \ge 1$ one has $L^{p,1} \subseteq L^{p,p} \subseteq L^{p,\infty}$.
For example, $\abs{x}^{-1}$ on $\R$ is in $L^{1,\infty}$ but not in $L^{1,1} = L^1$.
</document>

<document source="https://tamnd.github.io/math2/node/327NP6S7H5.md" tag="327NP6S7H5" kind="concept">
# Heat kernel on a Riemannian manifold

## Summary

The heat kernel of a Riemannian manifold is the fundamental solution of the heat equation $\partial_t u + \Delta u = 0$, where $\Delta$ is the [[Laplace–Beltrami operator|F4NWZNTRMD]].
On a closed manifold it is the integral kernel of $e^{-t\Delta}$ and is the sum of $e^{-t\lambda_j} \phi_j(x) \phi_j(y)$ over an orthonormal basis of eigenfunctions.
Its behaviour for small time encodes local geometry, such as volume and curvature, and its trace encodes the spectrum.
It is the main tool of the heat equation approach to [[index theory|675MAXBQN6]] and [[spectral geometry|QKAFRDPGE4]].

## Explanations

### L1 human

Put a unit of heat at one point $y$ of a curved surface and let it spread.
The heat kernel $K(t, x, y)$ is the temperature at $x$ after time $t$.
For short times heat has not yet felt the shape of the surface, and $K$ looks like the Gaussian of flat space; the curvature appears in correction terms.
For long times on a closed surface, the heat spreads out evenly and $K$ tends to $1/\mathrm{area}$.
</document>

<document source="https://tamnd.github.io/math2/node/32C3D8FG9V.md" tag="32C3D8FG9V" kind="statement">
# De Giorgi–Nash–Moser theorem

## Statement

Let $\Omega \subseteq \R^n$ be open and let $0 < \lambda \le \Lambda$.
Let $a_{ij} \colon \Omega \to \R$, for $1 \le i, j \le n$, be measurable functions with $\abs{a_{ij}(x)} \le \Lambda$ and
$$\sum_{i,j=1}^n a_{ij}(x) \xi_i \xi_j \ge \lambda \abs{\xi}^2 \quad \text{for all } x \in \Omega \text{ and } \xi \in \R^n.$$
Let $u \in W^{1,2}_{\mathrm{loc}}(\Omega)$ be a [[weak solution|SJZX81A783]] of $\sum_{i,j} \partial_i (a_{ij} \partial_j u) = 0$ in $\Omega$, where $W^{1,2}_{\mathrm{loc}}$ is a local [[Sobolev space|M4CEZ5K9HP]].
There are constants $C > 0$ and $\alpha \in (0, 1)$, depending only on $n$, $\lambda$ and $\Lambda$, such that for every ball $B_{2r}(x_0)$ with closure in $\Omega$:

1. $u$ is essentially bounded on $B_r(x_0)$ and $\displaystyle \sup_{B_r(x_0)} \abs{u} \le C \Big( r^{-n} \int_{B_{2r}(x_0)} u^2 \, dx \Big)^{1/2}$.
2. After change on a null set, $u$ is continuous, and $\abs{u(x) - u(y)} \le C \big(\abs{x - y}/r\big)^{\alpha} \sup_{B_{2r}(x_0)} \abs{u}$ for all $x, y \in B_r(x_0)$.

## Remarks

The point is that the coefficients $a_{ij}$ are only measurable, so no smoothness of the equation is passed on to $u$; the [[Hölder continuity|012PBA0FMD]] comes from ellipticity alone.
De Giorgi and Nash proved the result independently; De Giorgi used truncations at many levels and an iteration on the measure of super-level sets.
Moser later gave a proof by iterating reverse Hölder inequalities for powers of $u$, and he obtained a Harnack inequality on the way.
The theorem solves Hilbert's nineteenth problem.
Let $F$ be smooth and uniformly convex, and let $u$ minimize $\int F(\nabla u)\,dx$ in $W^{1,2}$.
Then the derivatives $\partial_k u$ solve an equation of the form above, so they are Hölder continuous.
The Schauder estimates then make $u$ smooth.
The result fails for systems: De Giorgi gave an example of an elliptic system with bounded measurable coefficients with a discontinuous weak solution.

## Explanations

### L1 human

The theorem looks at the simplest kind of [[elliptic equation|B4TTNHSTRX]], but with very rough coefficients that may jump around.
Even so, every solution is continuous, and in fact Hölder continuous, with bounds that depend only on how elliptic the equation is.
This gives a floor of smoothness that rough data cannot break.

### L2 human

This is the key step of regularity theory for nonlinear elliptic equations in divergence form.
Differentiating a nonlinear equation gives a linear equation for the derivatives whose coefficients are only known to be bounded and elliptic; this theorem then makes the derivatives Hölder continuous.
After that, the Schauder estimates bootstrap the regularity to smoothness.
The parabolic version, also due to Nash and Moser, plays the same role for parabolic equations in divergence form.
The counterpart for equations in non-divergence form is the Krylov–Safonov theorem.
</document>

<document source="https://tamnd.github.io/math2/node/32CP806EHX.md" tag="32CP806EHX" kind="definition">
# Linear programming

## Definition

Let $A$ be a real $m \times n$ [[matrix|0VD9ZSWWJG]], let $b \in \R^m$ and let $c \in \R^n$.
For vectors $u, w \in \R^m$, write $u \le w$ when $u_i \le w_i$ for every $i$.
The *linear program* given by $(A, b, c)$ is the problem
$$\text{maximize } c^T x \quad \text{subject to} \quad A x \le b, \quad x \in \R^n.$$
A point $x$ with $A x \le b$ is a *feasible solution*, and the set of them is the *feasible region*.
A feasible solution $x^*$ is *optimal* if $c^T x^* \ge c^T x$ for every feasible solution $x$, and then $c^T x^*$ is the *optimal value*.
*Linear programming* is the study and solution of linear programs.

## Remarks

Example: maximize $x + y$ subject to $x \ge 0$, $y \ge 0$, $x + 2y \le 4$ and $3x + y \le 6$.
The feasible region is the quadrilateral with vertices $(0,0)$, $(2,0)$, $(8/5, 6/5)$ and $(0,2)$.
The objective takes the values $0$, $2$, $14/5$ and $2$ at these vertices, so the optimal value is $14/5$, attained at $(8/5, 6/5)$.

Non-example: maximizing $x + y$ subject to $x^2 + y^2 \le 1$ is not a linear program, because the constraint is not a linear inequality.

Every linear program is of exactly one of three kinds: infeasible, unbounded (the objective is not bounded above on the feasible region), or with an optimal solution.
Other standard forms, such as maximizing $c^T x$ subject to $A x = b$ and $x \ge 0$, are equivalent to this one after adding slack variables or splitting free variables.
The dual of the program above is to minimize $b^T y$ subject to $A^T y = c$ and $y \ge 0$.
If one of the two programs has an optimal solution, so does the other, with the same optimal value.
</document>

<document source="https://tamnd.github.io/math2/node/32PDB3WQ4D.md" tag="32PDB3WQ4D" kind="concept">
# Interpretation of varieties

## Summary

An interpretation of a [[variety|AJNZJD8442]] $\mathcal{V}$ in a variety $\mathcal{W}$ translates each basic operation of $\mathcal{V}$ into a term of $\mathcal{W}$.
The translation must turn every algebra of $\mathcal{W}$ into an algebra of $\mathcal{V}$ on the same set.
Interpretations order varieties by strength: if $\mathcal{V}$ is interpretable in $\mathcal{W}$, then $\mathcal{W}$ is at least as rich as $\mathcal{V}$.
Suppose $\mathcal{V}$ is interpretable in $\mathcal{W}$ and $\mathcal{V}$ has terms that satisfy some identities.
Then the translated terms of $\mathcal{W}$ satisfy the same identities, so such properties pass from $\mathcal{V}$ to $\mathcal{W}$.

## Explanations

### L1 human

Every ring can be seen as an abelian group by forgetting the multiplication.
More generally, one variety can sit inside another when its operations can be written as formulas in the operations of the other.
For example, in a Boolean ring the formula $x + y + xy$ behaves like the join of a Boolean algebra.
An interpretation records such a choice of formulas, one for each operation.
</document>

<document source="https://tamnd.github.io/math2/node/32SHBZX38H.md" tag="32SHBZX38H" kind="definition">
# Wigner D-matrix

## Definition

Let $j \in \set{0, \frac12, 1, \frac32, \dots}$, and let $V_j$ be the [[irreducible representation|FX3NEFYFFN]] of [[SU(2)|E5P8AG6CK7]] of dimension $2j + 1$, with an invariant inner product.
Let $J_x, J_y, J_z$ be the Hermitian operators on $V_j$ that represent the Lie algebra, with $[J_x, J_y] = i J_z$ and its cyclic permutations.
Let $\lvert j m \rangle$, for $m = j, j - 1, \dots, -j$, be the orthonormal basis of $V_j$ such that $J_z \lvert j m \rangle = m \lvert j m \rangle$ and $(J_x \pm i J_y) \lvert j m \rangle = \sqrt{j(j+1) - m(m \pm 1)}\, \lvert j, m \pm 1 \rangle$.
For Euler angles $\alpha, \beta, \gamma \in \R$, the *Wigner D-matrix* is the $(2j+1) \times (2j+1)$ matrix with entries
$$D^j_{m' m}(\alpha, \beta, \gamma) = \langle j m' \rvert\, e^{-i \alpha J_z} e^{-i \beta J_y} e^{-i \gamma J_z} \lvert j m \rangle,$$
where $m', m \in \set{j, j-1, \dots, -j}$.
The *small d-matrix* is $d^j_{m' m}(\beta) = D^j_{m' m}(0, \beta, 0)$.

## Remarks

Since $J_z$ is diagonal, $D^j_{m' m}(\alpha, \beta, \gamma) = e^{-i m' \alpha}\, d^j_{m' m}(\beta)\, e^{-i m \gamma}$.
The entries of $d^j(\beta)$ are real, and Wigner's formula is
$$d^j_{m' m}(\beta) = \sum_{s} \frac{(-1)^{m' - m + s} \sqrt{(j+m')!\,(j-m')!\,(j+m)!\,(j-m)!}}{(j+m-s)!\, s!\, (m'-m+s)!\, (j-m'-s)!} \left(\cos\tfrac{\beta}{2}\right)^{2j+m-m'-2s} \left(\sin\tfrac{\beta}{2}\right)^{m'-m+2s},$$
summed over the integers $s$ for which all factorials have nonnegative arguments.

Example: for $j = \frac12$, $J_y = \frac12 \sigma_y$ with the Pauli matrix $\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, and $\sigma_y^2 = I$.
So $e^{-i \beta J_y} = \cos\frac{\beta}{2}\, I - i \sin\frac{\beta}{2}\, \sigma_y$, which gives $d^{1/2}_{\frac12 \frac12} = \cos\frac{\beta}{2}$ and $d^{1/2}_{\frac12, -\frac12} = -\sin\frac{\beta}{2}$.
Wigner's formula with $m' = \frac12$, $m = -\frac12$ has only the term $s = 0$, and it also gives $-\sin\frac{\beta}{2}$.

Non-example: the matrix $\begin{pmatrix} \cos\frac{\beta}{2} & \sin\frac{\beta}{2} \\ \sin\frac{\beta}{2} & \cos\frac{\beta}{2} \end{pmatrix}$ is not a D-matrix, since it is not unitary for $\sin \beta \ne 0$, while every $D^j$ is unitary.

For integer $j = \ell$, $d^\ell_{00}(\beta) = P_\ell(\cos\beta)$, the [[Legendre polynomial|QFYGJVA996]].
Also $\overline{D^\ell_{m 0}(\alpha, \beta, \gamma)} = \sqrt{\frac{4\pi}{2\ell+1}}\, Y_\ell^m(\beta, \alpha)$, where $Y_\ell^m$ is a [[spherical harmonic|3F32Y21HP4]] with polar angle $\beta$ and azimuth $\alpha$.
The bra-ket notation $\langle u \rvert A \lvert v \rangle$ means the inner product of $u$ with $A v$.
</document>

<document source="https://tamnd.github.io/math2/node/32VAKG7CPS.md" tag="32VAKG7CPS" kind="definition">
# Minimal model program

## Definition

Let $X$ be a normal [[projective variety|A4H4CZ2V35]] over a field of characteristic $0$ with $\Q$-factorial terminal singularities.
Let $K_X$ be its canonical divisor, the divisor class of the [[canonical bundle|W05VYY4HBR]].
The variety $X$ is a *minimal model* if $K_X$ is [[nef|RZ66X0NAM5]].
The *minimal model program* applied to $X$ is the following procedure.

1. If $K_X$ is nef, stop: $X$ is a minimal model.
2. Otherwise, by the [[cone theorem|ZJSW0HP75H]] there is an extremal ray $R$ of the closed cone of curves of $X$ with $K_X \cdot R < 0$.
   It has a contraction $\varphi_R \colon X \to Z$ of exactly the curves whose classes lie in $R$.
3. If $\dim Z < \dim X$, stop: $\varphi_R$ is a *Mori fibre space*.
4. If $\varphi_R$ is [[birational|KA5890R9YM]] and its exceptional locus is a divisor, replace $X$ by $Z$ and go back to step 1.
5. If $\varphi_R$ is birational and its exceptional locus has codimension at least $2$, replace $X$ by the flip $X^+$ of $\varphi_R$.
   Then go back to step 1.

## Remarks

In steps 4 and 5 the new variety again has $\Q$-factorial terminal singularities, which is why the program allows these singularities.
The program terminates when there is no infinite sequence of flips.
Flips exist in all dimensions by a theorem of Birkar, Cascini, Hacon and McKernan, and termination of flips is known in low dimensions, including dimension $3$, but open in general.
Example: for a smooth projective surface, there are no small contractions, and step 4 contracts a $(-1)$-curve by [[Castelnuovo's contraction theorem|WJ5WRGG5DE]].
The surface $\mathbb{P}^2$ is not a minimal model, since $K_{\mathbb{P}^2} \cdot \ell = -3 < 0$ for a line $\ell$.
The program stops at step 3 for it, with $Z$ a point, so $\mathbb{P}^2$ is a Mori fibre space.
A smooth curve of genus $g \ge 1$ is a minimal model, since $\deg K = 2g - 2 \ge 0$.
</document>

<document source="https://tamnd.github.io/math2/node/32YJXG30MF.md" tag="32YJXG30MF" kind="concept">
# Hodge theory

## Summary

Hodge theory studies the cohomology of smooth manifolds through harmonic differential forms.
For complex algebraic varieties, it studies the extra structure that harmonic forms put on cohomology.
On a compact Riemannian manifold, the [[Hodge theorem|A127WFVYKW]] identifies [[de Rham cohomology|GX74C07T0G]] with the space of harmonic forms.
On a compact [[Kähler manifold|X60JG9AEQJ]], harmonic forms split by type, which gives the [[Hodge decomposition|HABCJ4Y0SR]] and a [[Hodge structure|ZSM083M97C]] on each cohomology group.
The theory gives strong topological restrictions on algebraic varieties and leads to deep open problems such as the [[Hodge conjecture|XAEEDWV3PM]].

## Explanations

### L1 human

A cohomology class on a manifold can be represented by many differential forms.
Hodge theory picks the best one, the harmonic form, which solves a Laplace equation and minimizes energy.
On complex manifolds with a compatible metric, harmonic forms can be sorted by how many $dz$ and $d\bar{z}$ factors they contain.
For a compact Riemann surface of genus $g$, this sorts the $2g$-dimensional first cohomology into $g$ holomorphic forms and $g$ antiholomorphic ones.
Such facts constrain the shape of algebraic varieties, for instance the odd Betti numbers of a smooth projective variety are even.
</document>

<document source="https://tamnd.github.io/math2/node/32ZHMZMACN.md" tag="32ZHMZMACN" kind="concept">
# Delta functor

## Summary

A delta functor, or $\delta$-functor, is a sequence of additive functors $T^0, T^1, T^2, \dots$ between abelian categories, together with connecting maps.
These turn each short exact sequence into a long exact sequence.
It is the axiomatic form of a cohomology theory.
The right [[derived functors|NPMC3K6J2F]] of a left exact functor form a universal $\delta$-functor, and universality is how one recognizes derived functors without resolutions.

## Explanations

### L1 human

Many cohomology theories share one feature: a short exact sequence gives a long exact sequence that runs through all degrees.
A delta functor keeps only this feature, as a list of functors with connecting maps between degrees.
It is universal if it is determined by its degree $0$ part.
For example, $\operatorname{Ext}^n(M, -)$ for $n \ge 0$ is a universal delta functor, determined by $\Hom(M, -)$.
</document>

<document source="https://tamnd.github.io/math2/node/337Z2AXRK2.md" tag="337Z2AXRK2" kind="definition">
# Laguerre plane

## Definition

Let $(\mathcal{P}, \mathcal{Z}, \in)$ be an [[incidence structure|FGKNAFNHA3]] of points and circles, where circles are subsets of $\mathcal{P}$.
Let $\parallel$ be an equivalence relation on $\mathcal{P}$, called parallelism.
The structure is a *Laguerre plane* if:

1. any three pairwise non-parallel points lie on exactly one circle;
2. for every circle $z$, every point $P \in z$ and every point $Q \notin z$ with $P \not\parallel Q$, there is exactly one circle $z'$ with $P, Q \in z'$ and $z \cap z' = \set{P}$;
3. for every circle $z$ and every point $P$, there is exactly one point $P' \in z$ with $P \parallel P'$;
4. every circle has at least three points, and there is a circle $z$ and a point not on $z$.

## Remarks

Example: the real parabola model has points $\R^2$, parallel classes the vertical lines $x = \text{const}$, and circles the graphs $y = ax^2 + bx + c$ with $a, b, c \in \R$.
Clause 3 holds because each circle is the graph of a function of $x$.
Clause 1 holds because through three points with distinct $x$-coordinates there is exactly one polynomial of degree at most $2$.
For clause 2, two such polynomials whose graphs touch at $P = (x_0, y_0)$ differ by $\lambda (x - x_0)^2$, and exactly one value of $\lambda$ makes the graph pass through $Q$.

Non-example: keep the same circles but let parallelism be equality of points.
Then three points on a vertical line are pairwise non-parallel and lie on no circle, so clause 1 fails.

The classical model is the set of plane sections of a circular cylinder not parallel to its axis, with the generators of the cylinder as parallel classes.
More generally, an oval cone in three-dimensional projective space with its vertex removed gives an ovoidal Laguerre plane.
</document>

<document source="https://tamnd.github.io/math2/node/3384ZV3PWM.md" tag="3384ZV3PWM" kind="concept">
# Poincaré's functional equation

## Summary

Poincaré's functional equation asks for a function $\Psi$ with $\Psi(\lambda z) = f(\Psi(z))$, where $f$ is a given holomorphic map with a [[fixed point|R6QVS6P21N]] at $0$ and $\lambda = f'(0)$.
It is the inverse form of [[Schröder's equation|VTJV7JVJEA]]: if $\Phi$ solves Schröder's equation and is invertible, then $\Psi = \Phi^{-1}$ solves Poincaré's equation.
For a repelling fixed point of an entire map, the solution extends to an entire function on all of $\C$, the Poincaré function.
Poincaré functions parametrise the dynamics of $f$ near the repelling point and are used to study Julia sets.

## Explanations

### L1 human

Schröder's equation looks for a coordinate in which the map $f$ becomes multiplication by $\lambda$.
Poincaré's equation looks at the same change of coordinates in the other direction: it maps the simple picture, multiplication by $\lambda$, onto the complicated one.
For $f(w) = w^2 + 2w$ and $\lambda = 2$, the function $\Psi(z) = e^z - 1$ works, since $f(e^z - 1) = e^{2z} - 1 = \Psi(2z)$.
For the logistic map $f(w) = 4w(1 - w)$ and $\lambda = 4$, the function $\Psi(z) = \sin^2 \sqrt{z}$ works, by the double-angle formula.
</document>

<document source="https://tamnd.github.io/math2/node/33C1KEK3SC.md" tag="33C1KEK3SC" kind="concept">
# Outer regular measure

## Summary

A measure on a topological space is outer regular when each measurable set is approximated from outside by open sets.
The measure of the set is the infimum of the measures of the open sets that contain it.
Lebesgue measure on $\R^n$ is outer regular, which is how it is often first constructed.
Outer regularity, together with inner regularity, is part of the usual definitions of a regular measure and of a Radon measure.

## Explanations

### L1 human

To measure a set, wrap it in open sets and shrink the wrapping.
If the wrappings can get as close to the set's measure as one likes, the measure is outer regular.
For length on the real line, a single point is covered by tiny open intervals of length as small as one likes, which matches its length $0$.
Counting measure fails: every open interval around a point contains infinitely many points.
</document>

<document source="https://tamnd.github.io/math2/node/33GY6P5A4T.md" tag="33GY6P5A4T" kind="concept">
# Casson invariant

## Summary

The Casson invariant is an integer-valued invariant of oriented integral [[homology 3-spheres|HK8NS02NNE]], introduced by Andrew Casson.
Informally it counts, with signs, the irreducible representations of the fundamental group into [[$SU(2)$|E5P8AG6CK7]], up to conjugation, divided by $2$.
It reduces modulo $2$ to the Rokhlin invariant, and it has a simple behaviour under [[Dehn surgery|ZA405B0Z1S]] on knots.
It was one of the first gauge-theoretic style invariants of $3$-manifolds and led to instanton Floer homology.

## Explanations

### L1 human

A homology $3$-sphere looks like the $3$-sphere to homology, but may be very different.
The Casson invariant is a whole number attached to each one, and it is $0$ for the $3$-sphere.
When it is not $0$, the fundamental group has a nontrivial representation into the group of unit quaternions, so the manifold cannot be the $3$-sphere.
For the Poincaré homology sphere it equals $\pm 1$, depending on the orientation.
</document>

<document source="https://tamnd.github.io/math2/node/33JWVAX32K.md" tag="33JWVAX32K" kind="definition">
# Quermassintegral

## Definition

Let $n \ge 1$, let $K$ be a nonempty compact convex subset of $\R^n$ (a [[convex body|437FSG3419]]), and let $B$ be the [[closed unit ball|0Q0H13G8ZB]] of $\R^n$.
For $j = 0, 1, \dots, n$, the $j$-th *quermassintegral* of $K$ is
$$W_j(K) = V(\underbrace{K, \dots, K}_{n - j}, \underbrace{B, \dots, B}_{j}),$$
where $V$ is the [[mixed volume|QB3Q5FBJ08]].

## Remarks

By the [[Steiner formula|F02XA3MYX8]], $\operatorname{vol}(K + tB) = \sum_{j=0}^n \binom{n}{j} W_j(K)\, t^j$ for $t \ge 0$.
Special values: $W_0(K) = \operatorname{vol}(K)$, $n W_1(K)$ is the surface area of $K$, and $W_n(K) = \kappa_n$, the volume of $B$.
Also $W_{n-1}(K) = \frac{\kappa_n}{2} w(K)$, where $w(K)$ is the mean width of $K$.
For $1 \le k \le n$, $W_{n-k}(K)$ equals $\kappa_n / \kappa_k$ times the average, over all $k$-dimensional linear subspaces $E$, of the $k$-dimensional volume of the orthogonal projection of $K$ onto $E$; this is Kubota's formula.

Example: $K = rB$, a ball of radius $r \ge 0$.
By multilinearity, $W_j(rB) = r^{n-j} V(B, \dots, B) = \kappa_n r^{n-j}$.
For $n = 2$ and $r = 1$ this gives $W_1 = \pi$, half the perimeter $2\pi$, as expected.

Non-example: the set $K = \set{0, 1} \subseteq \R$ is not convex, so its mixed volumes are not defined.
Indeed the length of $K + tB$ is $4t$ for $t \le \frac12$ and $1 + 2t$ for $t \ge \frac12$, not one polynomial in $t$.
</document>

<document source="https://tamnd.github.io/math2/node/33SJNF8V7Z.md" tag="33SJNF8V7Z" kind="concept">
# Mortar methods

## Summary

Mortar methods are finite element discretizations on a domain cut into subdomains whose meshes need not match across the interfaces.
Continuity across an interface is not imposed point by point but weakly.
The jump must be orthogonal to a space of multiplier functions, the mortar space.
They let each subdomain use its own mesh and polynomial degree while keeping optimal error estimates.
This makes them a standard tool of [[domain decomposition methods|AN054RHT1K]].

## Explanations

### L1 human

Imagine two brick walls built by different masons with bricks of different sizes, which must meet along a seam.
The bricks do not line up, so the mortar between them has to absorb the mismatch.
A mortar method does the same with meshes: two regions are meshed independently, and along their common edge the two sides are glued "on average" rather than at every node.
For example, a fine mesh around a corner singularity can meet a coarse mesh elsewhere without any transition layer.
</document>

<document source="https://tamnd.github.io/math2/node/340GV29CGE.md" tag="340GV29CGE" kind="definition">
# Transfer operator

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a map such that each $x \in X$ has finitely many preimages.
Let $g \colon X \to \C$ be a function, called the weight.
The *transfer operator* $\mathcal{L}_g$ acts on functions $\varphi \colon X \to \C$ by
$$(\mathcal{L}_g \varphi)(x) = \sum_{y \in f^{-1}(x)} g(y) \, \varphi(y),$$
where an empty sum is $0$.

## Remarks

In practice $X$ carries more structure, and $\mathcal{L}_g$ is studied on a suitable Banach space of functions, such as Hölder or bounded variation functions.

Example: let $f(x) = 2x \bmod 1$ on $[0,1)$ and $g = 1/2$.
Then $(\mathcal{L}_g \varphi)(x) = \frac{1}{2}\big(\varphi(x/2) + \varphi((x+1)/2)\big)$, and $\mathcal{L}_g 1 = 1$.
The substitutions $y = x/2$ and $y = (x+1)/2$ give, for integrable $\varphi$ and bounded $\psi$,
$$\int_0^1 (\mathcal{L}_g \varphi) \, \psi \, dx = \int_0^1 \varphi \, (\psi \circ f) \, dx.$$
So $\mathcal{L}_g$ is the adjoint of the [[Koopman operator|37TRG5DBQC]] $\psi \mapsto \psi \circ f$ for [[Lebesgue measure|2Z2QVWZ262]], and taking $\psi$ an indicator function shows that $\varphi \, dx$ is $f$-invariant exactly when $\mathcal{L}_g \varphi = \varphi$.

Non-example of this duality: with the weight $g = 1$ for the same map, $\mathcal{L}_1 1 = 2$, and $\int \mathcal{L}_1 \varphi \, dx = 2 \int \varphi \, dx$, so $\mathcal{L}_1$ is not the adjoint of a Koopman operator for Lebesgue measure.

For a piecewise smooth expanding map of an interval, the weight $g = 1/\abs{f'}$ gives the Perron–Frobenius operator, the adjoint of the Koopman operator for Lebesgue measure.
</document>

<document source="https://tamnd.github.io/math2/node/345DJTSPHM.md" tag="345DJTSPHM" kind="statement">
# Cauchy–Pompeiu formula

## Statement

Let $D \subset \C$ be a bounded open set whose boundary $\partial D$ consists of finitely many piecewise $C^1$ closed curves.
Orient $\partial D$ so that $D$ lies on the left.
Let $f$ be a [[continuously differentiable|06SKCHDQT5]] complex-valued function on an open set that contains the closure $\bar D$.
Then for each $z \in D$,
$$f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)}{\zeta - z} \, d\zeta - \frac{1}{\pi} \iint_D \frac{\partial f / \partial \bar \zeta \, (\zeta)}{\zeta - z} \, dA(\zeta),$$
where $\partial f / \partial \bar \zeta$ is the [[Wirtinger derivative|698R12MY4G]] and $dA$ is the area measure.

## Remarks

Proof idea: apply [[Green's theorem|0HR3N02317]], in the form $\oint_{\partial \Omega} g \, d\zeta = 2i \iint_\Omega \partial g / \partial \bar \zeta \, dA$, to $g(\zeta) = f(\zeta)/(\zeta - z)$ on $D$ minus a small disc around $z$.
Since $1/(\zeta - z)$ is holomorphic there, $\partial g / \partial \bar \zeta = (\partial f / \partial \bar \zeta)/(\zeta - z)$; the small circle contributes $2\pi i f(z)$ in the limit.
If $f$ is holomorphic, then $\partial f / \partial \bar \zeta = 0$ and the formula is [[Cauchy's integral formula|0DG10TVYPN]].
If $f$ has compact support in $\C$, the boundary term vanishes and $f(z) = \frac{1}{\pi} \iint \frac{\partial f / \partial \bar \zeta}{z - \zeta} \, dA(\zeta)$.
One deduces from it that, for $g$ of class $C^1$ with compact support, $u(z) = \frac{1}{\pi} \iint \frac{g(\zeta)}{z - \zeta} \, dA(\zeta)$ solves $\partial u / \partial \bar z = g$.

## Explanations

### L1 human

Cauchy's formula recovers a holomorphic function inside a region from its values on the boundary.
For a smooth function that is not holomorphic, the boundary values are not enough.
The Cauchy–Pompeiu formula adds an area integral that measures how far the function is from being holomorphic.

### L2 human

The formula is the basic tool for the inhomogeneous Cauchy–Riemann equation $\partial u / \partial \bar z = g$.
The kernel $\frac{1}{\pi z}$ is a fundamental solution of the operator $\partial / \partial \bar z$.
This is the starting point of the $\bar\partial$ method, of the theory of the Beltrami equation through the Cauchy and Beurling transforms, and of generalizations to several complex variables.
</document>

<document source="https://tamnd.github.io/math2/node/347JEZ7WTW.md" tag="347JEZ7WTW" kind="definition">
# Slowly varying function

## Definition

Let $L \colon (0, \infty) \to (0, \infty)$ be a [[measurable function|YYNCJSR0PJ]].
The function $L$ is *slowly varying* (at infinity) if for every real $a > 0$
$$\lim_{x \to \infty} \frac{L(ax)}{L(x)} = 1.$$

## Remarks

Example: $L(x) = \log(e + x)$ is slowly varying.
For fixed $a > 0$ and large $x$, $\log(e + ax) = \log x + \log a + o(1)$ and $\log(e + x) = \log x + o(1)$, so the ratio tends to $1$.
Every positive measurable function with a positive finite limit at infinity is slowly varying.

Non-example: $L(x) = x^\rho$ with $\rho \ne 0$ is not slowly varying, because $L(ax)/L(x) = a^\rho \ne 1$ for $a \ne 1$.

The uniform convergence theorem says that the limit is uniform for $a$ in each compact subset of $(0, \infty)$.
In the language of regular variation, slowly varying functions are the regularly varying functions of index $0$.
</document>

<document source="https://tamnd.github.io/math2/node/3482HSGW27.md" tag="3482HSGW27" kind="definition">
# Tension field

## Definition

Let $(M, g)$ and $(N, h)$ be [[Riemannian manifolds|7CBT4SM83Y]] of dimensions $m$ and $n$, and let $u \colon M \to N$ be a smooth map.
Regard $du$ as a section of the bundle $T^*M \otimes u^* TN$ over $M$, with the connection $\nabla$ induced by the [[Levi-Civita connections|CAZYGE1T4E]] of $g$ and $h$.
The *tension field* of $u$ is the section of $u^* TN$
$$\tau(u) = \tr_g \nabla du = \sum_{i=1}^m \big( \nabla_{e_i} du \big)(e_i),$$
where $(e_i)$ is a local orthonormal frame of $TM$.
In local coordinates $x^i$ on $M$ and $y^\gamma$ on $N$, its components are
$$\tau(u)^\gamma = \Delta_g u^\gamma + \sum_{i,j,\alpha,\beta} g^{ij} \, \Gamma^\gamma_{\alpha\beta}(u) \, \frac{\partial u^\alpha}{\partial x^i} \frac{\partial u^\beta}{\partial x^j}.$$
Here $\Delta_g$ is the [[Laplace–Beltrami operator|F4NWZNTRMD]] of $g$, with the sign for which $\Delta_g = \frac{d^2}{dx^2}$ on $\R$.
The $\Gamma^\gamma_{\alpha\beta}$ are the [[Christoffel symbols|FZBA1HA9ZM]] of $h$.

## Remarks

If $M$ is compact without boundary and $u_s$ is a smooth family of maps with $u_0 = u$ and variation field $V = \partial_s u_s |_{s=0}$, then the [[energy|E6HDE1DFXD]] satisfies
$$\frac{d}{ds}\Big|_{s=0} E(u_s) = -\int_M h\big(\tau(u), V\big) \, dV_g.$$
So $-\tau(u)$ is the gradient of $E$, and $u$ is a [[harmonic map|KZJYG9DS3M]] exactly when $\tau(u) = 0$.

Example: let $M = \R$ and $\gamma \colon \R \to N$ a curve; then $\tau(\gamma) = \nabla_{\dot\gamma} \dot\gamma$, the covariant acceleration, so $\tau(\gamma) = 0$ is the geodesic equation.
For $\gamma(t) = (\cos t, \sin t, 0)$ into the unit sphere $S^2$, a great circle, $\tau(\gamma) = 0$.

Non-example: the curve $\gamma(t) = (\cos t, \sin t)$ into $\R^2$ with the Euclidean metric has $\tau(\gamma) = \ddot\gamma = (-\cos t, -\sin t) \ne 0$, so it is not harmonic.
The same parametrized circle, seen as a map into the unit circle $S^1$, has tension $0$, so the tension field depends on the target.
For $N = \R^n$ with the Euclidean metric the Christoffel symbols vanish and $\tau(u) = (\Delta_g u^1, \dots, \Delta_g u^n)$.
</document>

<document source="https://tamnd.github.io/math2/node/34B1PW5EE2.md" tag="34B1PW5EE2" kind="statement">
# Crossing number inequality

## Statement

Let $G$ be a finite simple [[graph|00C0BR7D7H]] with $n$ vertices and $e$ edges, and let $\operatorname{cr}(G)$ be its [[crossing number|SWJZ4QRF9W]].
If $e \ge 4n$, then
$$\operatorname{cr}(G) \ge \frac{e^3}{64 n^2}.$$

## Remarks

A planar simple graph with $n \ge 3$ vertices has at most $3n - 6$ edges by the [[edge bound for planar graphs|0SM1REAVYJ]].
Removing one edge per crossing from a drawing gives a planar graph, so $\operatorname{cr}(G) \ge e - 3n$ for every simple graph.
Now keep each vertex of an optimal drawing independently with probability $p$, as in the [[probabilistic method|3B55HNS754]].
Taking expectations in the weak bound gives $p^4 \operatorname{cr}(G) \ge p^2 e - 3pn$, and the choice $p = 4n/e$ gives the result.
Example: the complete graph $K_n$ has $e = n(n-1)/2$ edges, which is at least $4n$ for $n \ge 9$, so $\operatorname{cr}(K_n) \ge c n^4$ for a constant $c > 0$.
Without a hypothesis like $e \ge 4n$ the bound fails: a graph with $e = n$ edges can be a cycle, which has crossing number $0$.

## Explanations

### L1 human

When a graph has many more edges than vertices, any drawing of it in the plane must have many crossings.
The inequality says how many: at least a constant times $e^3 / n^2$.
Doubling the number of edges, with the same vertices, forces about eight times as many crossings.

### L2 human

The inequality was proved by Ajtai, Chvátal, Newborn and Szemerédi and, independently, by Leighton, in the early 1980s.
Its main use in discrete geometry is through incidence bounds.
László Székely used it to give short proofs of the [[Szemerédi–Trotter theorem|NQBWFSC1E0]] on point-line incidences and of the $O(n^{4/3})$ bound in the [[unit distance problem|EPWB2JEXPE]].
One draws the points as vertices and the pieces of lines or circles as edges, and compares the forced crossings with the number of pairs of curves.
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<document source="https://tamnd.github.io/math2/node/34B7K77P31.md" tag="34B7K77P31" kind="statement">
# Donaldson's theorem

## Statement

Let $M$ be a closed, oriented, simply connected, [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]].
Let $Q_M$ be its [[intersection form|WK8GKRKAVD]] on $H_2(M; \Z) \cong \Z^n$.
If $Q_M$ is negative definite, then $Q_M$ is isomorphic over $\Z$ to the diagonal form $-I_n$, that is, $H_2(M; \Z)$ has a basis $e_1, \ldots, e_n$ with $Q_M(e_i, e_j) = -\delta_{ij}$.

## Remarks

Simon Donaldson proved the theorem in 1983 with gauge theory.
For a suitable Riemannian metric, the moduli space of anti-self-dual [[instantons|YWBTGC7MFA]] on the $SU(2)$-bundle with $c_2 = 1$ is a $5$-dimensional manifold with singular points.
Its singular points are the reducible connections, which correspond to the pairs $\pm e$ with $Q_M(e, e) = -1$, and near each of them it is a cone on $\C P^2$.
One end of the moduli space is a collar on $M$, so $M$ is cobordant to a disjoint union of copies of $\C P^2$ up to orientation.
Comparing signatures forces many vectors of square $-1$, and this makes the form diagonal.
Donaldson later removed the hypothesis that $M$ is simply connected.
Replacing the orientation of $M$ by its opposite gives the positive definite version.
Smoothness is essential.
By [[Freedman's classification|HSD6FKCYRC]], the negative of the $E_8$ form is the intersection form of a closed simply connected topological $4$-manifold.
This form is definite but not diagonalizable.

## Explanations

### L1 human

The intersection form records how surfaces cross inside a four-dimensional space.
Donaldson's theorem says that in a smooth such space, if all self-crossings have the same sign, the table is as simple as possible: a diagonal of $-1$'s.
Many tables allowed by algebra and by topology are forbidden by smoothness.

### L2 human

Combined with Freedman's theorem, it shows that most definite unimodular forms, such as $E_8$ and $E_8 \oplus E_8$, are realized by topological $4$-manifolds that admit no smooth structure.
It was the first major application of the [[Yang–Mills|JJWETW5AT8]] equations to topology and it started [[Donaldson theory|ZQW6M0S6M0]].
A short proof of the same result was later given with [[Seiberg–Witten invariants|PRDDMSTR4X]].
It also leads to the existence of [[exotic $\R^4$|X44PNC38E2]].
</document>

<document source="https://tamnd.github.io/math2/node/34JP0GZ3ED.md" tag="34JP0GZ3ED" kind="definition">
# Fermi–Dirac integral

## Definition

Let $j > -1$ be a real number and let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
The *complete Fermi–Dirac integral* of index $j$ is the function $F_j \colon \R \to \R$ given by
$$F_j(x) = \frac{1}{\Gamma(j+1)} \int_0^{\infty} \frac{t^j}{e^{t - x} + 1} \, dt.$$

## Remarks

The integral converges for $j > -1$: near $t = 0$ the integrand is bounded by $t^j$, and for large $t$ it decays like $t^j e^{x - t}$.

For $x < 0$, expanding $\frac{1}{e^{t-x} + 1} = \sum_{k \ge 1} (-1)^{k-1} e^{-k(t - x)}$ and integrating term by term gives
$$F_j(x) = \sum_{k=1}^{\infty} (-1)^{k-1} \frac{e^{kx}}{k^{j+1}} = -\operatorname{Li}_{j+1}(-e^x),$$
where $\operatorname{Li}_{j+1}$ is the [[polylogarithm|96CHDZKGQ0]], and the identity holds for all real $x$ by analytic continuation.
It also gives $F_j'(x) = F_{j-1}(x)$ for $j > 0$.

Example: for $j = 0$, $\int_0^\infty \frac{dt}{e^{t-x} + 1} = \log(1 + e^x)$, which matches $-\operatorname{Li}_1(-e^x) = \log(1 + e^x)$.

Non-example: for $j = -1$ the integrand behaves like $\frac{1}{t (e^{-x} + 1)}$ near $t = 0$, so the integral diverges; this is why $j > -1$ is required, and $F_{-1}$ is defined instead through the polylogarithm form.

Some authors omit the factor $1/\Gamma(j+1)$; the normalized version above is the one with $F_j(x) \sim e^x$ as $x \to -\infty$.
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<document source="https://tamnd.github.io/math2/node/34PT0GE8RA.md" tag="34PT0GE8RA" kind="statement">
# Remmert–Stein theorem

## Statement

Let $X$ be a [[complex manifold|JSGWAN2RJV]], let $E$ be an [[analytic subset|VMDTGTMN9P]] of $X$, and let $k \ge 1$ be an integer.
Let $A$ be an analytic subset of $X \setminus E$ of [[pure dimension|C2R5Q2Q717]] $k$.
Suppose that $\dim E < k$.
Then the closure $\overline{A}$ of $A$ in $X$ is an analytic subset of $X$, of pure dimension $k$.

## Remarks

The question is local, so one may assume that $X$ is a neighbourhood of a point of $E$ in $\C^n$.
The proof is by induction on the dimension of $E$ and uses projections.
After a generic linear change of coordinates, the projection of $\overline{A}$ to $\C^k$ is proper near the point.
Then $\overline{A}$ is described by the canonical defining polynomials of this branched covering.
Their coefficients are bounded holomorphic functions outside a thin set, so they extend by the Riemann extension theorem.
Example: $A = \set{z_2 = 0} \setminus \set{0}$ is an analytic subset of $\C^2 \setminus \set{0}$ of pure dimension $1$, and $E = \set{0}$ has dimension $0$, so $\overline{A}$, the $z_1$-axis, is analytic.
The hypothesis $\dim E < k$ is necessary.
Let $E = \set{z_1 = 0}$ in $\C^2$ and $A = \set{(z_1, e^{1/z_1}) : z_1 \ne 0}$, an analytic subset of $\C^2 \setminus E$ of pure dimension $1 = \dim E$.
Since $e^{1/z_1}$ takes every nonzero value, and tends to $0$ along the negative real axis, in every neighbourhood of $z_1 = 0$, the closure $\overline{A}$ is $A \cup E$.
It is not analytic: the components of an analytic curve other than $E$ would meet $E$ in a discrete set, but $A$ accumulates at every point of $E$.

## Explanations

### L1 human

Suppose an analytic set is known away from a smaller exceptional set.
The theorem says that if the exceptional set is of strictly smaller dimension, the analytic set extends across it: just add its limit points.
For instance, a curve defined outside a single point extends to a curve through that point.

### L2 human

The theorem is a basic extension result in complex analytic geometry, in the spirit of the Riemann extension theorem for functions.
It gives a short proof of Chow's theorem, by extending the cone over an analytic subset of projective space across the origin.
Bishop's theorem replaces the dimension condition on $E$ by a volume condition on $A$, and Shiffman's theorem by a Hausdorff measure condition on $E$.
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<document source="https://tamnd.github.io/math2/node/34SN01NWPS.md" tag="34SN01NWPS" kind="definition">
# Moore space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
For a family $\mathcal{G}$ of subsets of $X$ and a point $x \in X$, the *star* of $x$ in $\mathcal{G}$ is $\operatorname{st}(x, \mathcal{G}) = \bigcup \set{G \in \mathcal{G} : x \in G}$.
A *development* of $X$ is a sequence $\mathcal{G}_1, \mathcal{G}_2, \dots$ of [[open covers|049PXYHW1T]] of $X$ with the following property.
For every $x \in X$ and every open set $U$ with $x \in U$, there is $n$ with $\operatorname{st}(x, \mathcal{G}_n) \subseteq U$.
The space $X$ is *developable* if it has a development.
A *Moore space* is a developable space that is [[regular|03HMJASQBQ]] and [[Hausdorff|0S1FJ8VEX9]].

## Remarks

Every metric space $(X, d)$ is a Moore space.
Let $\mathcal{G}_n$ be the cover by all open balls of radius $1/n$.
Every ball of $\mathcal{G}_n$ that contains $x$ lies in the ball $B(x, 2/n)$, so $\operatorname{st}(x, \mathcal{G}_n) \subseteq B(x, 2/n)$, and this lies in $U$ for $n$ large.
A set $X$ with two points and the indiscrete topology is developable, with $\mathcal{G}_n = \set{X}$ for all $n$, because $X$ is the only open set containing a point.
It is not a Moore space, because it is not Hausdorff.
Some authors do not include regularity or the Hausdorff condition in the definition, so the word should be checked in each text.

## Explanations

### L1 human

A development is a sequence of open covers that get finer and finer around every point, like the covers by balls of radius $1/n$ in a metric space.
A Moore space is a well separated space with such a sequence.
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<document source="https://tamnd.github.io/math2/node/34VJ4T7ZSB.md" tag="34VJ4T7ZSB" kind="concept">
# Fourier–Bessel series

## Summary

A Fourier–Bessel series expands a function on an interval $[0, b]$ in the functions $J_\nu(j_{\nu,k} x / b)$.
Here $J_\nu$ is the Bessel function of the first kind and $j_{\nu,k}$ are its positive zeros.
These functions are orthogonal for the weight $x$, so the series is a [[generalized Fourier series|SNJC87TBHD]].
They are the eigenfunctions of a singular [[Sturm–Liouville problem|7WV6B2502K]] coming from [[Bessel's equation|EAV97QH9DF]].
Fourier–Bessel series solve problems with circular or cylindrical symmetry, such as the vibrations of a circular drum or heat flow in a cylinder.

## Explanations

### L1 human

A vibrating string has sine waves as its basic shapes, and any shape of the string is a sum of them.
A circular drum skin has different basic shapes: in the radial direction they are Bessel functions, scaled so that they vanish at the rim.
A Fourier–Bessel series writes a radial profile on the drum as a sum of these basic shapes.
For example, the constant profile $1$ on $[0, 1)$ is $\sum_k \frac{2}{j_k J_1(j_k)} J_0(j_k x)$, where $j_k$ are the zeros of $J_0$.
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<document source="https://tamnd.github.io/math2/node/351WM99D5W.md" tag="351WM99D5W" kind="definition">
# Free algebra in a variety

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] and let $X$ be a set.
A *free algebra of $\mathcal{V}$ on $X$* is an algebra $F \in \mathcal{V}$ together with a map $\eta \colon X \to F$ such that:

1. $F$ is [[generated|2TQD8NMG9P]] by $\eta(X)$.
2. For every $A \in \mathcal{V}$ and every map $f \colon X \to A$, there is a [[homomorphism|KBXSZC6BV1]] $h \colon F \to A$ with $h \circ \eta = f$.

It is written $F_{\mathcal{V}}(X)$, and $X$ is called a set of free generators.

## Remarks

Clause 1 makes the homomorphism $h$ in clause 2 unique, since two homomorphisms that agree on a generating set are equal.
A free algebra on $X$ is unique up to a unique isomorphism compatible with $\eta$.
If $\mathcal{V}$ contains an algebra with at least two elements, then $\eta$ is injective, and $X$ is usually regarded as a subset of $F$.
Example: in the variety of abelian groups, $F(X) = \Z^{(X)}$, the group of finitely supported functions $X \to \Z$, with $\eta(x)$ the indicator function of $x$.
A map $f \colon X \to A$ extends by $h(c) = \sum_{x} c(x) f(x)$, and this is a homomorphism.
Non-example: $\Z / 2\Z$ with $\eta(x) = 1$ is not free on $\set{x}$ in abelian groups, since the map $x \mapsto 1 \in \Z$ has no homomorphic extension $\Z / 2\Z \to \Z$.
When the signature has no constants and empty algebras are excluded, one takes $X$ nonempty.
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<document source="https://tamnd.github.io/math2/node/353YPJWBBR.md" tag="353YPJWBBR" kind="definition">
# Hermitian metric on a vector bundle

## Definition

Let $X$ be a smooth manifold and let $E \to X$ be a smooth complex [[vector bundle|P4DZM975BV]].
A *Hermitian metric* on $E$ is a family $h = (h_x)_{x \in X}$ such that:

1. For each $x \in X$, $h_x \colon E_x \times E_x \to \C$ is a [[Hermitian inner product|0NB0AWGJBW]] on the fibre $E_x$.
   So it is $\C$-linear in the first argument, $h_x(w, v) = \overline{h_x(v, w)}$, and $h_x(v, v) > 0$ for $v \ne 0$.
2. For any two smooth local [[sections|WWX7H5W1RP]] $s, t$ of $E$ over an open set $U$, the function $x \mapsto h_x(s(x), t(x))$ is smooth on $U$.

## Remarks

In a local frame $e_1, \dots, e_r$ of $E$ over $U$, a Hermitian metric is the same as a smooth map $H = (h(e_j, e_i))_{i,j}$ from $U$ to the positive definite Hermitian $r \times r$ matrices.
A vector bundle with a Hermitian metric is called a Hermitian vector bundle.
When $X$ is a complex manifold and $E$ is its holomorphic tangent bundle, a Hermitian metric on $E$ is a Hermitian metric on $X$ in the sense of [[Hermitian manifolds|CK1DCPC3D3]].

Example: on $\mathcal{O}(-1) \subset \mathbb{P}^n \times \C^{n+1}$, restrict the standard inner product $\sum_k v_k \bar w_k$ of $\C^{n+1}$ to each fibre.
Each restriction is a Hermitian inner product, and for a local holomorphic section such as $s(z) = (1, z_1, \dots, z_n)$ on $\set{z_0 \ne 0}$ one gets $h(s, s) = 1 + \abs{z_1}^2 + \cdots + \abs{z_n}^2$, which is smooth.

Non-example: on the trivial line bundle $\C \times \C \to \C$, the rule $h_z(v, w) = \abs{z}^2 v \bar w$ is smooth, but $h_0(v, v) = 0$ for all $v$, so clause 1 fails at $z = 0$.
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<document source="https://tamnd.github.io/math2/node/355XPVR564.md" tag="355XPVR564" kind="definition">
# Compactification

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A *compactification* of $X$ is a pair $(K, e)$ with the following properties.

1. $K$ is a [[compact space|0TVJZYB6SH]].
2. $e \colon X \to K$ is a [[topological embedding|00ZG9Y9TR4]].
3. The image $e(X)$ is [[dense|0BSJSX7ME2]] in $K$.

## Remarks

Example: $K = [0, 1]$ with the inclusion $e \colon (0, 1) \to [0, 1]$ is a compactification of $(0, 1)$.
The interval $[0, 1]$ is compact, the inclusion is an embedding for the subspace topology, and the closure of $(0, 1)$ is $[0, 1]$.

Non-example: $K = [0, 2]$ with the inclusion of $(0, 1)$ fails clause 3.
The closure of $(0, 1)$ in $[0, 2]$ is $[0, 1]$, not $[0, 2]$.

Many authors also require $K$ to be Hausdorff.
Two compactifications $(K, e)$ and $(K', e')$ are equivalent if there is a homeomorphism $h \colon K \to K'$ with $h \circ e = e'$.
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<document source="https://tamnd.github.io/math2/node/357FEAZ0Q4.md" tag="357FEAZ0Q4" kind="statement">
# Jacobi's formula

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, let $I \subseteq \R$ be an open interval, and let $A \colon I \to \mathbb{K}^{n \times n}$ be a [[differentiable|0K57YG3BB8]] map.
Then $t \mapsto \det A(t)$ is differentiable, and for all $t \in I$
$$\frac{d}{dt} \det A(t) = \tr\big(\operatorname{adj}(A(t))\, A'(t)\big),$$
where $\det$ is the [[determinant|00KDZRW4C0]], $\operatorname{adj}$ is the [[adjugate|07W9H8QF38]] and $\tr$ is the [[trace|0XC1RDBY7A]].
If $A(t)$ is invertible, this equals $\det A(t) \cdot \tr\big(A(t)^{-1} A'(t)\big)$.

## Remarks

The determinant is a polynomial in the entries $a_{ij}$.
By the [[Laplace expansion|0C08BQ3M05]] along row $i$, its partial derivative with respect to $a_{ij}$ is the [[cofactor|0XPKR18APA]] $C_{ij}$.
The chain rule gives $\frac{d}{dt}\det A = \sum_{i,j} C_{ij}\, a_{ij}'$.
This sum is $\tr(\operatorname{adj}(A) A')$, because $\operatorname{adj}(A)_{ji} = C_{ij}$.
The second form follows from $\operatorname{adj}(A) = \det(A) A^{-1}$.

A special case is $\frac{d}{dt}\det(I + tX)\big|_{t=0} = \tr X$: the derivative of the determinant at the identity is the trace.
For $A(t) = e^{tX}$, the formula gives $\frac{d}{dt}\det e^{tX} = \tr(X) \det e^{tX}$, so $\det e^{tX} = e^{t \tr X}$.

## Explanations

### L1 human

Jacobi's formula tells how fast the determinant of a matrix changes when the matrix changes.
Near the identity matrix, the rate of change of the determinant is the trace of the change.
For example, $\det\begin{pmatrix} 1 + t & 0 \\ 0 & 1 + 2t \end{pmatrix} = 1 + 3t + 2t^2$ has derivative $3$ at $t = 0$.
This is the trace of $\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$.

### L2 human

The formula says that the differential of $\det$ at $A$ is the linear map $X \mapsto \tr(\operatorname{adj}(A) X)$.
It gives Liouville's formula for the Wronskian of a linear system of differential equations.
It also gives the [[determinant of a matrix exponential|0A1QQN333A]], $\det e^A = e^{\tr A}$.
In Lie theory it says that the derivative of $\det \colon \mathrm{GL}_n(\mathbb{K}) \to \mathbb{K}^\times$ at the identity is the trace.
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<document source="https://tamnd.github.io/math2/node/35FFCR3GRH.md" tag="35FFCR3GRH" kind="definition">
# Minakshisundaram–Pleijel zeta function

## Definition

Let $(M, g)$ be a closed connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$.
Let $0 = \lambda_0 < \lambda_1 \le \lambda_2 \le \cdots$ be the [[spectrum of the Laplace–Beltrami operator|B8TYKV7AA1]], with an orthonormal basis of real eigenfunctions $\phi_0, \phi_1, \ldots$ of $L^2(M)$.
The *Minakshisundaram–Pleijel zeta function* is
$$Z(s, x, y) = \sum_{j \ge 1} \frac{\phi_j(x) \phi_j(y)}{\lambda_j^{s}}, \qquad x, y \in M,$$
and its trace on the diagonal is the *spectral zeta function*
$$Z(s) = \int_M Z(s, x, x) \, dv(x) = \sum_{j \ge 1} \lambda_j^{-s}.$$
Both series converge for $\operatorname{Re} s > n/2$.

## Remarks

By the [[Minakshisundaram–Pleijel expansion|45P2SEGGR1]] of the heat trace and the formula $\Gamma(s) Z(s) = \int_0^\infty t^{s-1} \big( \sum_{j \ge 0} e^{-t\lambda_j} - 1 \big) \, dt$, the function $Z(s)$ has a meromorphic [[continuation|0M3YQJ81NB]] to $\C$.
Its poles are simple and lie among the points $s = n/2 - k$, $k = 0, 1, 2, \ldots$, other than $0, -1, -2, \ldots$; in particular $Z$ is holomorphic at $s = 0$.
The residue at $s = n/2$ is $\operatorname{vol}(M) / \big( (4\pi)^{n/2} \Gamma(n/2) \big)$, which is equivalent to Weyl's law.

Example: for $M = \R / 2\pi\Z$ the nonzero eigenvalues are $k^2$, $k \ge 1$, each with multiplicity $2$.
So $Z(s) = 2 \sum_{k \ge 1} k^{-2s} = 2 \zeta(2s)$, with $\zeta$ the [[Riemann zeta function|1XWBY4N0VT]].
It has a single simple pole at $s = 1/2$ with residue $1$, and $Z(0) = 2\zeta(0) = -1$.

Non-example: the sum $\sum_{j \ge 0} \lambda_j^{-s}$ that includes $\lambda_0 = 0$ is not defined, since $0^{-s}$ has no meaning; this is why the zero eigenvalue is left out.
</document>

<document source="https://tamnd.github.io/math2/node/35G4HQG2FN.md" tag="35G4HQG2FN" kind="statement">
# Baker's finite basis theorem

## Statement

Let $A$ be a finite algebra of a finite signature with only function symbols.
Assume that the [[variety|AJNZJD8442]] generated by $A$ is [[congruence-distributive|9GZ7F6ETC6]].
Then $A$ is [[finitely based|YGYZAQ12MZ]].

## Remarks

Kirby Baker proved the theorem in 1977.
One ingredient is Jónsson's lemma, which shows that the subdirectly irreducible algebras in the variety generated by $A$ are quotients of subalgebras of $A$, so they are finite and few.
The other ingredient is the Jónsson terms of a congruence-distributive variety, which Baker used to find finitely many identities that axiomatise the variety.

Example: every finite lattice is finitely based, since lattices form a congruence-distributive variety.
The same holds for every finite Boolean algebra and every finite Heyting algebra, since these have lattice reducts and so generate congruence-distributive varieties.

The finite signature is needed.
Let $A$ be the two-element lattice $\set{0, 1}$ with infinitely many extra unary operations $f_1, f_2, \ldots$, all equal to the identity map.
It has a lattice reduct, so it generates a congruence-distributive variety, and it satisfies $f_i(x) \approx x$ for every $i$.
A finite set $\Sigma$ of identities of $A$ mentions only the $f_i$ with $i$ in some finite set $S$.
Change $A$ by letting $f_j$ be the constant map with value $0$ for every $j \notin S$.
The new algebra still satisfies $\Sigma$, but it fails $f_j(x) \approx x$ at $x = 1$, so $\Sigma$ is not an equational basis of $A$.

## Explanations

### L1 human

The theorem gives a large family of finite algebras whose laws follow from finitely many of them.
It applies whenever the congruences of the algebras in the variety behave like a distributive lattice, as they do for lattices and Boolean algebras.
So every finite lattice can be described by finitely many laws.

### L2 human

Baker's theorem is one of the main positive answers to Tarski's finite basis problem.
It extends McKenzie's earlier result that finite lattices are finitely based.
It was later generalised by McKenzie to congruence-modular varieties with a finite residual bound, and by Willard to congruence meet-semidistributive varieties with a finite residual bound.
These results are steps towards Park's conjecture, which predicts that a finite algebra of finite signature that generates a variety with a finite residual bound is finitely based.
</document>

<document source="https://tamnd.github.io/math2/node/35KB1QCD3Z.md" tag="35KB1QCD3Z" kind="statement">
# Abel–Jacobi theorem

## Statement

Let $X$ be a compact connected [[Riemann surface|5K0N8QMV6Q]] of genus $g \ge 1$, with [[Jacobian variety|ZQ84M9D9TC]] $J(X)$.
Fix $P_0 \in X$, and let $A$ be the [[Abel–Jacobi map|HE8F7KBF42]], extended additively to [[divisors|Z9DN23DAE2]] on $X$.

1. (Abel) A divisor $D$ of degree $0$ on $X$ is the divisor of a [[meromorphic function|0ZZ7ZYDR8W]] on $X$ if and only if $A(D) = 0$ in $J(X)$.
2. (Jacobi inversion) Every point of $J(X)$ equals $A(P_1 + \cdots + P_g) - g A(P_0)$ for some points $P_1, \ldots, P_g \in X$.

Let $\operatorname{Pic}^0(X)$ be the group of divisors of degree $0$ modulo divisors of meromorphic functions.
Then $A$ induces an isomorphism of groups $\operatorname{Pic}^0(X) \to J(X)$.

## Remarks

For the direct half of Abel's theorem, let $D = \operatorname{div}(f)$ and consider the fibres $f^{-1}(t)$ for $t \in \mathbb{P}^1$, counted with multiplicity.
The map $t \mapsto A(f^{-1}(t))$ is a holomorphic map from $\mathbb{P}^1$ to the torus $J(X)$, so it is constant, and $A(D) = A(f^{-1}(0)) - A(f^{-1}(\infty)) = 0$.
The converse builds $f$ as the exponential of an integral of a meromorphic differential with prescribed residues, whose periods are adjusted using $A(D) = 0$.
Jacobi inversion follows from a dimension count: the map from the $g$-fold symmetric product of $X$ to $J(X)$ has invertible differential at a general point, by Riemann–Roch.
So its image is a compact analytic subset of dimension $g$ in the connected torus $J(X)$ of dimension $g$, which must be all of $J(X)$.
Example: take $g = 1$ and $X = \C / \Lambda$ with base point $0$.
Abel's theorem says that $\sum n_i [z_i]$ with $\sum n_i = 0$ is the divisor of an elliptic function exactly when $\sum n_i z_i \in \Lambda$.

## Explanations

### L1 human

On a compact surface, a meromorphic function must have as many zeros as poles.
Abel's theorem gives the exact extra condition that a choice of zeros and poles must satisfy to come from a function: a certain sum of integrals must vanish modulo periods.
Jacobi inversion says that the target of these sums is completely filled out by sums of $g$ points.
So the group of divisor classes of degree zero is the same as the Jacobian torus.

### L2 human

The theorem identifies an algebraic object, the Picard group of degree $0$, with an analytic one, the torus of periods.
This identification is how one proves that the Jacobian is an algebraic variety and how one studies linear systems on curves through the geometry of theta divisors.
For an elliptic curve it gives the isomorphism between the curve with its group law and $\C / \Lambda$.
It is the starting point of Brill–Noether theory and of the Torelli theorem.
</document>

<document source="https://tamnd.github.io/math2/node/35KXRGMAHT.md" tag="35KXRGMAHT" kind="definition">
# Extended finite element method

## Definition

Let $\mathcal{T}_h$ be a mesh of a domain $\Omega \subset \R^d$ with nodes $x_i$, $i \in I$.
Let $N_i$ be standard [[finite element|98GZ9DVHTA]] shape functions with $\sum_{i \in I} N_i = 1$ on $\Omega$, a [[partition of unity|NH739XXVXK]].
Let $\psi \colon \Omega \to \R$ be an enrichment function that carries a known local feature of the solution.
Let $J \subseteq I$ be the set of nodes whose shape function support meets the region of that feature.
The *extended finite element method* is the [[Galerkin method|Y5B0PEBKE8]] in the space of functions
$$u_h(x) = \sum_{i \in I} N_i(x) \, u_i + \sum_{j \in J} N_j(x) \, \psi(x) \, a_j,$$
with unknown coefficients $u_i$ and $a_j$.

## Remarks

The extra functions $N_j \psi$ have small support, so the system matrix stays sparse.
If all $a_j$ equal one value $a$, then on a region where $\sum_{j \in J} N_j = 1$ the enrichment equals $a \psi$ exactly.
Common choices of $\psi$ are a [[Heaviside function|09C4JP7Y0M]] across a crack, and the singular functions of the crack tip, which behave like $\sqrt{r}$.
Often one uses the shifted form $\psi(x) - \psi(x_j)$, so that $u_h(x_i) = u_i$ at the nodes.

Example: on $\Omega = (0, 1)$ with nodes $x_i = ih$, let a crack sit at a point $c$ inside $(x_k, x_{k+1})$.
Take $\psi(x) = H(x - c)$, with $H$ the Heaviside step, and $J = \set{k, k+1}$.
Then $u_h$ can jump at $c$ by $N_k(c) a_k + N_{k+1}(c) a_{k+1}$, while the mesh does not have a node at $c$.

Non-example: the standard space with only the first sum is continuous on $\Omega$.
It has no enrichment, $J = \emptyset$, so it cannot jump at $c$ unless $c$ is a mesh node and the mesh is split there.
</document>

<document source="https://tamnd.github.io/math2/node/35NNSXJB1K.md" tag="35NNSXJB1K" kind="conjecture">
# Hadwiger conjecture

## Statement

Let $k \ge 1$ be an integer, and let $G$ be a finite graph with [[chromatic number|0C5M7AM6FH]] $\chi(G) \ge k$.
Then the [[complete graph|03WKB2H474]] $K_k$ is a [[minor|VYQ32ZBPEE]] of $G$.

## Remarks

Equivalently, every finite graph with no $K_k$ minor can be properly coloured with $k - 1$ colours.
Hugo Hadwiger stated the conjecture in 1943.
It is easy for $k \le 3$, and Hadwiger proved it for $k = 4$.
For $k = 5$ it is equivalent to the [[four colour theorem|0M406JC8J1]], by a structure theorem of Wagner on graphs without a $K_5$ minor.
For $k = 6$ Robertson, Seymour and Thomas proved it in 1993, by reduction to the four colour theorem.
The conjecture is open for every $k \ge 7$.

## Explanations

### L1 human

A minor of a graph is what you get by deleting vertices and edges and by shrinking edges to points.
Hadwiger's conjecture says that a graph needs many colours only for one reason: it contains a big complete graph in this loose sense.
The case of five colours already contains the four colour theorem, so the conjecture is a far-reaching generalization of it.
</document>

<document source="https://tamnd.github.io/math2/node/35XKRHEC3Q.md" tag="35XKRHEC3Q" kind="concept">
# Hyponormal operator

## Summary

A hyponormal operator is a bounded operator $T$ on a Hilbert space with $T^*T - TT^* \ge 0$, equivalently $\norm{T^*x} \le \norm{Tx}$ for all $x$.
The class contains the normal operators and the [[subnormal operators|81MZ023K1A]], and it is strictly larger than both.
Hyponormal operators share several properties of normal operators: for instance, the norm equals the spectral radius.

## Explanations

### L1 human

For a normal operator, $T$ and its adjoint stretch every vector by the same amount.
A hyponormal operator only requires that the adjoint stretches no vector more than $T$ does.
The unilateral shift on sequences is an example: it keeps every length, while its adjoint, the backward shift, shortens the vector $(1, 0, 0, \ldots)$ to $0$.
</document>

<document source="https://tamnd.github.io/math2/node/35XTXNEWQ9.md" tag="35XTXNEWQ9" kind="concept">
# Proper forcing axiom

## Summary

The proper forcing axiom, PFA, is a strong extension of [[Martin's axiom|V8KGT8MRFE]] at $\aleph_1$.
It replaces the countable chain condition by the much weaker condition of properness.
It says that for each proper forcing notion one can meet $\aleph_1$ dense sets with a single filter; the [[precise definition|KJT0AXWQXP]] is given separately.
Its consistency is known only from a large cardinal hypothesis, a supercompact cardinal.
It decides many questions of combinatorial set theory, and it implies that $2^{\aleph_0} = \aleph_2$.

## Explanations

### L1 human

A forcing axiom says that certain constructions of length $\omega_1$ can always be completed.
Martin's axiom allows this for a narrow class of construction methods, the ccc ones.
The proper forcing axiom allows it for a much wider class, the proper ones, which include all ccc methods and all countably closed ones.
With this freedom, many open-ended questions about sets of size $\aleph_1$ get definite answers.
</document>

<document source="https://tamnd.github.io/math2/node/35YQ94FNV1.md" tag="35YQ94FNV1" kind="statement">
# Lovász local lemma

## Statement

Let $A_1, \dots, A_n$ be [[events|031TPJVS74]] in a [[probability space|0NFPJVJJHX]].
Let $D$ be a [[graph|00C0BR7D7H]] on the vertex set $\set{1, \dots, n}$.
Suppose each $A_i$ is [[mutually independent|001B78R0DJ]] of the family of events $A_j$ with $j \ne i$ and $j$ not adjacent to $i$.

1. (General form.) Suppose there are real numbers $x_1, \dots, x_n \in [0, 1)$ with $\Pr(A_i) \le x_i \prod_{j \sim i} (1 - x_j)$ for each $i$, where $j \sim i$ means that $j$ is adjacent to $i$. Then $\Pr\big(\bigcap_i \overline{A_i}\big) \ge \prod_i (1 - x_i) > 0$.
2. (Symmetric form.) Suppose each vertex of $D$ has degree at most $d$, and $\Pr(A_i) \le p$ for each $i$, with $e \, p \, (d + 1) \le 1$. Then $\Pr\big(\bigcap_i \overline{A_i}\big) > 0$.

## Remarks

The proof shows by induction on $\abs{S}$ that $\Pr\big(A_i \mid \bigcap_{j \in S} \overline{A_j}\big) \le x_i$ for each set $S$ not containing $i$ for which the conditioning event has positive probability.
The symmetric form follows from the general form with $x_i = 1/(d + 1)$, using $(1 - 1/(d+1))^d \ge 1/e$.
For example, let $\varphi$ be a formula in conjunctive normal form in which each clause has exactly $k$ literals on distinct variables.
If each clause shares variables with at most $d$ other clauses and $e (d + 1) \le 2^k$, then $\varphi$ is satisfiable.
Indeed, under a uniformly random assignment each clause fails with probability $2^{-k}$.
Some limit on dependence is needed: if $A_1$ is an event of probability $1/2$ and $A_2$ is its complement, then no outcome avoids both.
Here $d = 1$ and $p = 1/2$, and $e p (d + 1) = e > 1$.

## Explanations

### L1 human

Suppose you want to avoid many bad events at once, and each bad event is unlikely.
If the events were independent, you could avoid all of them with positive probability.
The local lemma says that this still works when each event depends on only a few others.
This is often enough to prove that a good object exists.

### L2 human

The local lemma is one of the main tools of the [[probabilistic method|3B55HNS754]].
Unlike [[Boole's inequality|0EGSEE9J9R]], it can give positive probability even when the sum of the probabilities of the bad events is large.
Typical applications are colourings of hypergraphs, satisfiability of sparse formulas, and lower bounds for Ramsey numbers.
Moser and Tardos gave an efficient algorithm that finds an outcome avoiding all events, under the same hypotheses in the variable setting.
</document>

<document source="https://tamnd.github.io/math2/node/3601Z7V838.md" tag="3601Z7V838" kind="definition">
# Thue equation

## Definition

Let $n \ge 3$ be an integer and let
$$f(x, y) = a_n x^n + a_{n-1} x^{n-1} y + \dots + a_0 y^n$$
be a homogeneous polynomial of degree $n$ with integer coefficients $a_0, \dots, a_n$.
Assume that $f$ is [[irreducible|03FKX7Z5DC]] over $\Q$.
Let $m$ be a nonzero integer.
The *Thue equation* for $f$ and $m$ is the [[Diophantine equation|08B1RTW7VZ]] $f(x, y) = m$ in the unknowns $x, y \in \Z$.

## Remarks

Thue's theorem says that a Thue equation has only finitely many solutions $(x, y) \in \Z^2$.

Example: $x^3 - 2y^3 = 1$ is a Thue equation, since $x^3 - 2y^3$ is homogeneous of degree $3$ and irreducible over $\Q$, because $2$ is not the cube of a rational number.
It has the solutions $(1, 0)$ and $(-1, -1)$: indeed $(-1)^3 - 2(-1)^3 = 1$.

Non-example: $x^2 - 2y^2 = 1$ is not a Thue equation, because the degree is $2$.
It is a Pell equation, and it has infinitely many solutions, for example $(3, 2)$ and $(17, 12)$.
</document>

<document source="https://tamnd.github.io/math2/node/360H7MD9RD.md" tag="360H7MD9RD" kind="concept">
# Current

## Summary

An $m$-dimensional current is a continuous linear functional on the smooth compactly supported [[differential $m$-forms|0E3G0GRPZ6]] of an open set of $\R^n$.
Currents generalise oriented surfaces, in the way that distributions generalise functions.
They carry a boundary operator and a notion of mass, and limits of surfaces of bounded area and boundary length exist among them.
This makes currents a main tool for the Plateau problem and other minimisation problems in geometric measure theory.

## Explanations

### L1 human

An oriented surface $S$ can be known by what it does to forms: it sends a form $\omega$ to the number $\int_S \omega$.
A current is any rule of this kind that is linear and continuous, even when no surface lies behind it.
For example, a point $p$ gives the $0$-current that sends a function $f$ to $f(p)$.
A curve traced twice gives the current "twice the curve", and a curve with the opposite orientation gives its negative.
Stokes' theorem becomes the definition of the boundary: the boundary of a current is the current that sends $\omega$ to the value on $d\omega$.
</document>

<document source="https://tamnd.github.io/math2/node/361GJ6QWAR.md" tag="361GJ6QWAR" kind="statement">
# Albert–Brauer–Hasse–Noether theorem

## Statement

Let $K$ be a [[global field|634XKY8RZ0]] and let $A$ be a [[central simple algebra|5HRKJBAY2V]] over $K$.
For each place $v$ of $K$, let $K_v$ be the completion of $K$ at $v$, a [[local field|J85FZEQG38]] or $\R$ or $\C$.
If $K_v$ [[splits|NVD7R0T835]] $A$, that is, $A \otimes_K K_v$ is a matrix algebra over $K_v$, for every place $v$, then $A$ is a matrix algebra over $K$.
Equivalently, the natural map of [[Brauer groups|BEW48TA9WZ]] $\mathrm{Br}(K) \to \bigoplus_v \mathrm{Br}(K_v)$ is injective.

## Remarks

The proof uses class field theory: one reduces to cyclic algebras and then uses the Hasse norm theorem.
That theorem says that in a cyclic extension of global fields, an element that is a local norm everywhere is a global norm.
Example: the quaternion algebra $\left(\frac{-1, -1}{\Q}\right)$ splits over $\Q_p$ for every odd prime $p$, but not over $\Q_2$ or over $\R$.
So it is not split over $\Q$, in agreement with the theorem.
Class field theory gives more: the sequence $0 \to \mathrm{Br}(K) \to \bigoplus_v \mathrm{Br}(K_v) \to \Q/\Z \to 0$, with the sum of local invariants as the last map, is exact.
Because of this sum rule it is enough to check all places but one, and a quaternion algebra over $K$ is ramified at an even number of places.

## Explanations

### L1 human

Is a given central simple algebra over the rational numbers just a matrix algebra?
To decide, it is enough to check it over the real numbers and over each field of $p$-adic numbers.
These local questions are much easier.
The theorem is a local-global principle, like the Hasse–Minkowski theorem for quadratic forms.

### L2 human

The theorem is a cornerstone of the arithmetic of central simple algebras over number fields.
With the exact sequence above it gives the classification of central simple algebras over a global field by their local invariants.
Consequences are that every central simple algebra over a global field is cyclic and that its period equals its index.
For quaternion algebras it reduces to the Hasse–Minkowski theorem for the ternary quadratic form $a x^2 + b y^2 - a b z^2$.
</document>

<document source="https://tamnd.github.io/math2/node/361NAK68QK.md" tag="361NAK68QK" kind="object">
# Weierstrass elliptic function

## Description

Let $\omega_1, \omega_2 \in \C$ be linearly independent over $\R$, and let $\Lambda = \Z\omega_1 + \Z\omega_2$ be the lattice they span.
The *Weierstrass elliptic function* of $\Lambda$ is
$$\wp(z) = \frac{1}{z^2} + \sum_{\omega \in \Lambda \setminus \set{0}} \left( \frac{1}{(z - \omega)^2} - \frac{1}{\omega^2} \right), \qquad z \in \C \setminus \Lambda.$$
The series converges absolutely and uniformly on compact subsets of $\C \setminus \Lambda$, because $\sum_{\omega \ne 0} \abs{\omega}^{-3} < \infty$.

1. $\wp$ is a [[meromorphic function|0NSD1TPHR0]] on $\C$ whose [[poles|0WPQYW8MTE]] are exactly the points of $\Lambda$, each a pole of order $2$.
2. $\wp$ is even, and $\wp(z + \omega) = \wp(z)$ for all $\omega \in \Lambda$, so $\wp$ is an [[elliptic function|FF9ZPCZ59K]] for $\Lambda$.
3. $\wp$ satisfies the differential equation $(\wp')^2 = 4\wp^3 - g_2 \wp - g_3$, where $g_2 = 60 \sum_{\omega \ne 0} \omega^{-4}$ and $g_3 = 140 \sum_{\omega \ne 0} \omega^{-6}$.

## Remarks

Periodicity is checked on the derivative first: $\wp'(z) = -2 \sum_{\omega \in \Lambda} (z - \omega)^{-3}$ is visibly $\Lambda$-periodic.
So $\wp(z + \omega_1) - \wp(z)$ is a constant $c$; putting $z = -\omega_1/2$ and using that $\wp$ is even gives $c = 0$.
The differential equation follows because $(\wp')^2 - 4\wp^3 + g_2\wp + g_3$ is elliptic without poles, hence constant, and its Laurent expansion at $0$ shows the constant is $0$.
Every elliptic function for $\Lambda$ is a rational function of $\wp$ and $\wp'$.
The map $z \mapsto (\wp(z), \wp'(z))$ identifies the torus $\C / \Lambda$ minus one point with the affine cubic curve $y^2 = 4x^3 - g_2 x - g_3$.

## Explanations

### L1 human

The Weierstrass function is the simplest nonconstant function on the plane that repeats itself along two independent directions.
It is built by placing a double pole at each point of a lattice and adding up, with correction terms that make the sum converge.
Its square-derivative identity turns the torus into a cubic curve, which is the link between complex analysis and elliptic curves.
</document>

<document source="https://tamnd.github.io/math2/node/363770WNM2.md" tag="363770WNM2" kind="definition">
# Weyl group of a compact Lie group

## Definition

Let $G$ be a connected [[compact Lie group|9TFC6ADC1C]] and let $T$ be a [[maximal torus|3JVGZVVTB0]] of $G$.
Let $N_G(T) = \set{g \in G : gTg^{-1} = T}$ be the [[normaliser|0YGN3VAA0D]] of $T$.
The *Weyl group* of $G$ with respect to $T$ is the [[quotient group|0KYX2N69J0]]
$$W = W(G, T) = N_G(T)/T.$$
It acts on $T$ by $(nT) \cdot t = ntn^{-1}$ for $n \in N_G(T)$ and $t \in T$.

## Remarks

Example: for $G = SU(2)$ and $T = \set{\operatorname{diag}(z, \bar{z}) : \abs{z} = 1}$, the normaliser is $T \cup wT$ with $w = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.
Indeed $w \operatorname{diag}(z, \bar z) w^{-1} = \operatorname{diag}(\bar z, z)$, so $w \in N_G(T)$, and $W$ has order $2$, acting on $T$ by $z \mapsto \bar z$.
Non-example: for the abelian group $G = \mathbb{T}^n$, the maximal torus is $G$ itself, $N_G(T) = T$, and $W$ is trivial; so $W$ does not detect anything in the abelian case.
The group $W$ is finite, and it acts faithfully on $T$.
Up to isomorphism it does not depend on $T$, since all maximal tori are conjugate.
It is isomorphic to the Weyl group of the root system of $G$, the group generated by the reflections in the roots.
</document>

<document source="https://tamnd.github.io/math2/node/366D4ETGQH.md" tag="366D4ETGQH" kind="object">
# Conway groups

## Description

The Conway groups $\mathrm{Co}_1$, $\mathrm{Co}_2$ and $\mathrm{Co}_3$ are three [[sporadic groups|PZ49Z4CBFZ]], found by John Conway from the symmetries of the [[Leech lattice|XETV3657DG]].
Let $\mathrm{Co}_0$ be the [[automorphism group|0K1SCPBNKJ]] of the Leech lattice, that is, the group of linear isometries of $\R^{24}$ that map the lattice onto itself.
Its [[centre|0DZ93493VN]] is $\set{\pm 1}$, and its order is $8315553613086720000$.

1. $\mathrm{Co}_1 = \mathrm{Co}_0 / \set{\pm 1}$, of order $4157776806543360000 = 2^{21} \cdot 3^9 \cdot 5^4 \cdot 7^2 \cdot 11 \cdot 13 \cdot 23$.
2. $\mathrm{Co}_2$ is the stabiliser in $\mathrm{Co}_0$ of a lattice vector of norm $4$, of order $42305421312000 = 2^{18} \cdot 3^6 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23$.
3. $\mathrm{Co}_3$ is the stabiliser in $\mathrm{Co}_0$ of a lattice vector of norm $6$, of order $495766656000 = 2^{10} \cdot 3^7 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23$.

All three are [[simple groups|0YMYMNQ0QZ]].
Here the lattice is scaled so that its shortest nonzero vectors have norm $4$.

## Remarks

The group $\mathrm{Co}_0$ itself is not simple, because of its centre $\set{\pm 1}$.
Several other sporadic groups are subgroups of $\mathrm{Co}_0$, among them the [[Mathieu group|T1KR4KPX8P]] $M_{24}$, the Higman–Sims group and the McLaughlin group.
All three Conway groups are subquotients of the [[Monster group|C9VY20TAPT]].

## Explanations

### L1 human

The Leech lattice is an extremely symmetric arrangement of points in $24$-dimensional space.
Its symmetry group, divided by the symmetry $v \mapsto -v$, is the simple group $\mathrm{Co}_1$.
The groups $\mathrm{Co}_2$ and $\mathrm{Co}_3$ are the symmetries that keep one chosen lattice point fixed, for two kinds of points.
</document>

<document source="https://tamnd.github.io/math2/node/36DXKAD2RZ.md" tag="36DXKAD2RZ" kind="concept">
# Dini derivative

## Summary

The Dini derivatives of a real function at a point are the four one-sided upper and lower limits of its [[difference quotients|AZBDB5F4M4]].
Unlike the [[derivative|0F1WR9C83K]], they always exist, possibly as $+\infty$ or $-\infty$.
A function is differentiable at a point exactly when its four Dini derivatives there are equal and finite.
They are the main tool for proving differentiability almost everywhere, as in [[Lebesgue's theorem on monotone functions|1Y4N1DDQNR]].

## Explanations

### L1 human

At a corner or a wiggle, the slope of a function may fail to have a limit.
Dini derivatives still record what happens: the largest and smallest slopes seen from the right, and the largest and smallest seen from the left.
For $\abs{x}$ at $0$, the slopes from the right are all $1$ and the slopes from the left are all $-1$.
So the right Dini derivatives are $1$ and the left ones are $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/36J8PH2B0T.md" tag="36J8PH2B0T" kind="concept">
# Factorization system

## Summary

A factorization system on a category is a pair $(E, M)$ of classes of morphisms.
Every morphism factors as a member of $E$ followed by a member of $M$, in an essentially unique way.
The two classes determine each other through a unique [[lifting property|AZK4RD8XV0]].
The standard example in sets is (surjections, injections), which factors a function through its image.
A weaker variant, the weak factorization system, drops uniqueness and is a basic ingredient of model categories.

## Explanations

### L1 human

Every function factors as a surjection onto its image followed by the inclusion of the image.
A factorization system abstracts this: two kinds of maps such that every map is first one kind, then the other.
The factorization is unique up to a unique isomorphism of the middle object.
Groups have the same system with surjective and injective homomorphisms, and the middle object is the image subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/36KSZ7GA5S.md" tag="36KSZ7GA5S" kind="concept">
# Matrix Lie group

## Summary

A matrix Lie group is a closed subgroup of the [[general linear group|0FCHE902D7]] $GL_n(\C)$.
By the [[closed subgroup theorem|3NJGY8QSTE]], every matrix Lie group is a [[Lie group|7JBRGCVJRB]], and its Lie algebra can be computed with the [[matrix exponential|020M04FN1H]].
Most Lie groups met in practice, such as the orthogonal, unitary and symplectic groups, are matrix Lie groups.
So they are a common entry point to Lie theory.

## Explanations

### L1 human

A matrix Lie group is a set of invertible matrices that is closed under products and inverses.
It must also contain the limit of each convergent sequence of its elements, when that limit is invertible.
The rotation matrices of the plane form an example: products and inverses of rotations are rotations, and a limit of rotations is a rotation.
Working with matrices lets one do Lie theory with concrete calculations, using $e^A = I + A + A^2/2! + \cdots$ in place of abstract constructions.
Not every Lie group is of this kind, but the most important ones are.
</document>

<document source="https://tamnd.github.io/math2/node/36Q7XH9P7W.md" tag="36Q7XH9P7W" kind="example">
# Cuntz algebra

## Example

Let $n \ge 2$ be an integer.
The *Cuntz algebra* $\mathcal{O}_n$ is the universal C*-algebra generated by elements $s_1, \ldots, s_n$ subject to the relations
$$s_i^* s_i = 1 \quad (1 \le i \le n), \qquad \sum_{i=1}^{n} s_i s_i^* = 1.$$
So the $s_i$ are isometries, and the projections $s_i s_i^*$ onto their ranges add up to $1$.
The relations imply $s_i^* s_j = 0$ for $i \ne j$: the ranges are mutually orthogonal.

Concrete check for $n = 2$: on the Hilbert space $\ell^2(\N)$ with orthonormal basis $(e_k)_{k \ge 0}$, let $s_1 e_k = e_{2k}$ and $s_2 e_k = e_{2k+1}$.
Each $s_i$ maps an orthonormal basis to an orthonormal family, so it is an isometry.
The range of $s_1$ is spanned by the even basis vectors and that of $s_2$ by the odd ones, so $s_1 s_1^* + s_2 s_2^* = 1$.
The C\*-algebra generated by $s_1$ and $s_2$ is a copy of $\mathcal{O}_2$.

## Remarks

Cuntz showed that $\mathcal{O}_n$ is simple, so any $n$ isometries with these relations generate an isomorphic C\*-algebra.
It is also infinite: $s_1$ is an isometry with $s_1 s_1^* \ne 1$.
Its K-theory is $K_0(\mathcal{O}_n) \cong \Z / (n-1)\Z$ and $K_1(\mathcal{O}_n) = 0$, so the algebras $\mathcal{O}_n$ for different $n$ are not isomorphic.
Each $\mathcal{O}_n$ is separable and [[nuclear|D2M4VXJ7PA]].

## Explanations

### L1 human

In an infinite-dimensional space there is room to fit two full copies of the whole space side by side, as the even and odd positions of a sequence.
The Cuntz algebra is the algebra generated by such copying maps.
It is a basic example of a simple C\*-algebra that is very far from the matrix algebras.
</document>

<document source="https://tamnd.github.io/math2/node/36TXCRZ6GC.md" tag="36TXCRZ6GC" kind="concept">
# Choquet integral

## Summary

The Choquet integral integrates a non-negative function against a [[fuzzy measure|CRFQEGJDA9]], which need not be additive.
It adds up the sizes of the level sets $\set{f \ge t}$ over all levels $t$, as in the [[layer cake representation|H78RXJC1K1]].
For an ordinary measure it agrees with the Lebesgue integral, but in general it is not additive in the function.
It is used in decision theory, where it models attitudes towards uncertainty, and in the theory of capacities.

## Explanations

### L1 human

To integrate a non-negative function, cut its graph into thin horizontal slices.
At height $t$, the slice covers the set where the function is at least $t$.
The Choquet integral measures each such set with the given set function and adds up the slices over all heights.
With a non-additive set function, two functions can each have integral zero while their sum has a positive integral.
</document>

<document source="https://tamnd.github.io/math2/node/36WNFJTRRR.md" tag="36WNFJTRRR" kind="statement">
# Sacks–Uhlenbeck bubbling

## Statement

Let $(M, g)$ be a closed [[Riemann surface|5K0N8QMV6Q]] with a conformal metric, and let $(N, h)$ be a compact [[Riemannian manifold|7CBT4SM83Y]].
Let $u_k \colon M \to N$ be a sequence of [[harmonic maps|KZJYG9DS3M]] whose [[energies|E6HDE1DFXD]] $E(u_k)$ are bounded by a constant $C$.
Then there are a subsequence, still written $u_k$, a finite set $S \subset M$ and a harmonic map $u_\infty \colon M \to N$ with these properties.

1. The maps $u_k$ converge to $u_\infty$ in $C^\infty$ on compact subsets of $M \setminus S$.
2. For each $p \in S$ there are points $x_k \to p$ and radii $r_k \to 0$ with the following property.
The rescaled maps $v_k(y) = u_k(\exp_{x_k}(r_k y))$, defined on growing discs in $\R^2$, converge in $C^\infty$ on compact subsets of $\R^2$.
Their limit is a nonconstant harmonic map $v \colon \R^2 \to N$.
3. Each such limit $v$ has finite energy and extends to a nonconstant harmonic map $S^2 \to N$.

## Remarks

The key input is an $\varepsilon$-regularity estimate.
There is $\varepsilon > 0$ such that a harmonic map with energy less than $\varepsilon$ on a disc has all its derivatives bounded on the half disc.
Since the total energy is at most $C$, energy at least $\varepsilon$ can concentrate at no more than $C / \varepsilon$ points, and these points form $S$.
Near a point of $S$ one rescales the maps so that the concentrating energy stays visible, and the limit is a harmonic map on the plane.
Its extension to $S^2$ uses the removable singularity theorem for harmonic maps of finite energy on a punctured disc.
The energy of a map from a surface does not change under conformal changes of the domain, and this scale invariance is why bubbles can form.

If $N$ carries no nonconstant harmonic map $S^2 \to N$, then $S$ is empty and the convergence is smooth on all of $M$.
This happens, for example, when $N$ has nonpositive sectional curvature, since then every harmonic map from $S^2$ to $N$ is constant.
Sacks and Uhlenbeck used this analysis, for a perturbed energy, to show that every homotopy class of maps $M \to N$ contains a harmonic map when $\pi_2(N) = 0$.

## Explanations

### L1 human

Harmonic maps from a surface are like rubber sheets in equilibrium inside a curved space.
If the sheets in a sequence have bounded total stretching, most of each sheet settles down to a limit.
At a few points, however, a tiny piece of a sheet can wrap around a whole sphere inside the target and pinch off.
These pinched-off spheres are the bubbles.
The theorem says that this is the only thing that can go wrong.

### L2 human

The theorem is the prototype of compactness up to bubbling.
The same pattern reappears for pseudoholomorphic curves, for Yang–Mills connections in dimension four and for many other conformally invariant problems.
It explains why the existence theory for harmonic maps of surfaces depends on the second homotopy group of the target.
Compare with the [[Eells–Sampson theorem|AP4A0GX0V2]], where nonpositive curvature of the target rules out bubbles and the heat flow converges.
</document>

<document source="https://tamnd.github.io/math2/node/36X2WTHYCB.md" tag="36X2WTHYCB" kind="concept">
# Dimension of an algebraic variety

## Summary

The dimension of an algebraic variety is the length of the longest chain of nested irreducible closed subsets in it.
For an affine variety it equals the Krull dimension of the coordinate ring.
For any variety it equals the transcendence degree of the function field over the base field.
These different descriptions agree, which makes dimension one of the most robust invariants in algebraic geometry: points have dimension $0$, curves $1$, surfaces $2$.

## Explanations

### L1 human

Inside a surface, a chain of ever smaller irreducible pieces can go surface, then a curve on it, then a point on that curve.
That chain has two steps, and the surface has dimension $2$.
Another way to see it is to count how many coordinates can be chosen freely.
On the sphere $x^2 + y^2 + z^2 = 1$ over $\C$, two coordinates are free, and then the third has at most two possible values.
So the sphere is a surface, of dimension $2$.
</document>

<document source="https://tamnd.github.io/math2/node/36X3GET13B.md" tag="36X3GET13B" kind="object">
# Suzuki groups

## Description

The Suzuki groups $\mathrm{Sz}(q)$, also written ${}^2B_2(q)$, are an infinite family of finite [[groups of Lie type|NPEBABQ49M]].
The parameter is $q = 2^{2n+1}$ with $n \ge 0$.
The [[order|0B85S2V2DR]] of $\mathrm{Sz}(q)$ is
$$\abs{\mathrm{Sz}(q)} = q^2 (q^2 + 1)(q - 1).$$
For $n \ge 1$ the group $\mathrm{Sz}(q)$ is a non-abelian [[simple group|0YMYMNQ0QZ]]; the smallest one is $\mathrm{Sz}(8)$, of order $29120$.
The group $\mathrm{Sz}(2)$ has order $20$ and is not simple.

## Remarks

$\mathrm{Sz}(q)$ is the subgroup of the [[symplectic group|X12FS2JAP0]] $\mathrm{Sp}_4(\F_q)$ fixed by a certain automorphism.
This automorphism combines the exceptional graph automorphism of $\mathrm{Sp}_4$ in characteristic $2$ with a field automorphism, and it exists because $q$ is an odd power of $2$.
It acts doubly transitively on a set of $q^2 + 1$ points, an ovoid in the projective space of dimension $3$ over $\F_q$.
The Suzuki groups are the only non-abelian finite simple groups whose order is not divisible by $3$.
They were found by Michio Suzuki.
They are not related to the sporadic Suzuki group $\mathrm{Suz}$.

## Explanations

### L1 human

Most finite simple groups have an order divisible by $3$.
The Suzuki groups are the exception: an infinite family of simple groups whose sizes are not multiples of $3$.
They live inside $4 \times 4$ symplectic matrix groups over fields with $2^{2n+1}$ elements, and they exist because of a special symmetry that appears only in characteristic $2$.
</document>

<document source="https://tamnd.github.io/math2/node/36YNSTESMQ.md" tag="36YNSTESMQ" kind="concept">
# Selmer group

## Summary

The Selmer group of an isogeny of [[elliptic curves|427N6S6QQW]] over a [[number field|MWBDFS2KJ5]] is a group of [[Galois cohomology|9MPCZ3EAJM]] classes.
These classes satisfy local conditions at every place, and the group is finite and computable.
It contains a copy of the quotient $E'(K)/\phi(E(K))$, and the difference is the $\phi$-torsion of the [[Tate–Shafarevich group|BK9WBMF56G]].
Computing a Selmer group is the standard way to bound the rank of $E(K)$ from above.
It is the effective core of the proof of [[Mordell's theorem|BZHGKN6Q3K]].

## Explanations

### L1 human

To find all rational points of an elliptic curve, one would like to describe the points modulo doubles, $E(\Q)/2E(\Q)$.
This group sits inside a larger group of candidates that one can compute from local information, prime by prime.
That larger group is the $2$-Selmer group.
If it is as small as the known points allow, the rank is determined.
If it is larger, the extra candidates come from points not yet found or from elements of the Tate–Shafarevich group.
</document>

<document source="https://tamnd.github.io/math2/node/370PJ5BF70.md" tag="370PJ5BF70" kind="concept">
# Extremally disconnected space

## Summary

An extremally disconnected space is a [[topological space|0CQHFPGJ45]] in which the [[closure|0QEAJB91GZ]] of every open set is open.
Equivalently, any two disjoint open sets have disjoint closures.
Compact Hausdorff extremally disconnected spaces are exactly the Stone spaces of complete [[Boolean algebras|0XADJKM699]].
By [[Gleason's theorem|CPZKZ2AV0G]], they are also the projective objects among compact Hausdorff spaces.

## Explanations

### L1 human

In most familiar spaces, the closure of an open set adds a boundary that is not open.
For example, the closure of $(0, 1)$ in $\R$ is $[0, 1]$, and $[0, 1]$ is not open.
In an extremally disconnected space this never happens: closures of open sets stay open.
Discrete spaces and the Stone–Čech compactification $\beta \N$ are examples.
</document>

<document source="https://tamnd.github.io/math2/node/374WKANY3F.md" tag="374WKANY3F" kind="definition">
# Well-powered category

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
The category $\mathcal{C}$ is *well-powered* if for every object $X$ of $\mathcal{C}$ the [[subobjects|FS1FKAE89H]] of $X$ form a [[set|0BSJ7RQ9BW]], rather than a proper class.
Dually, $\mathcal{C}$ is *co-well-powered* if for every object $X$ the quotient objects of $X$, the equivalence classes of epimorphisms out of $X$, form a set.

## Remarks

Example: the category of sets is well-powered.
Each subobject of a set $X$ is represented by the inclusion of a subset.
So the subobjects of $X$ are in bijection with the [[power set|00E0YZCERH]] of $X$, which is a set.
The same holds for groups, rings and modules, where subobjects correspond to substructures.
In the category of topological spaces, a subobject of $X$ corresponds to a subset of $X$ with a topology finer than the subspace topology.
These pairs also form a set.

Non-example: let $\mathcal{C}$ have as objects all sets and one more object $X$.
Besides identities, let $\mathcal{C}$ have exactly one morphism $m_S \colon S \to X$ for each set $S$, and nothing else.
It is a category, since no two non-identity morphisms compose.
Each $m_S$ is a monomorphism, since there are no two different morphisms into $S$ from one object, and $m_S \equiv m_T$ only when $S = T$, since there is no morphism between distinct sets.
So the subobjects of $X$ form a proper class, and $\mathcal{C}$ is not well-powered.
</document>

<document source="https://tamnd.github.io/math2/node/375ZGE0V2W.md" tag="375ZGE0V2W" kind="statement">
# Marcinkiewicz multiplier theorem

## Statement

Let $m \colon \R \to \C$ be a bounded function, with $\abs{m(\xi)} \le A$ for all $\xi \in \R$.
For each integer $j$, let $I_j = [2^j, 2^{j+1}]$ and $I_j' = [-2^{j+1}, -2^j]$.
Assume that $m$ has bounded variation on each $I_j$ and each $I_j'$, with total variation at most $A$ on each of these intervals.
Then $m$ is an [[$L^p$ multiplier|RAGR38Q40B]] for every $1 < p < \infty$.
Let $\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$.
There is a constant $C_p$, depending only on $p$, with
$$\norm{(m \hat f)^\vee}_{L^p(\R)} \le C_p A \norm{f}_{L^p(\R)}$$
for every Schwartz function $f$.

## Remarks

The proof uses Littlewood–Paley theory.
The operators that cut the frequencies to the dyadic pieces $I_j$ and $I_j'$ split $f$ into pieces whose square function has $L^p$ norm comparable to that of $f$.
On each dyadic piece, a function of bounded variation is an average of indicators of intervals.
The multiplier of an interval is built from Hilbert transforms, which are bounded on $L^p$.
A vector-valued bound for these interval operators then gives the theorem.

Example: $m(\xi) = \abs{\xi}^{it}$ for a real $t \ne 0$ has modulus $1$, and on $[2^j, 2^{j+1}]$ its variation is $\int_{2^j}^{2^{j+1}} \abs{t}/\xi \, d\xi = \abs{t} \log 2$.
So the imaginary powers of $\abs{\xi}$ are $L^p$ multipliers for $1 < p < \infty$.

The conclusion fails at $p = 1$ and $p = \infty$: the multiplier $m = \mathbf{1}_{[0, \infty)}$ satisfies the hypotheses but gives an operator built from the Hilbert transform, which is unbounded on $L^1$.
There is also a version on $\R^n$, in which the condition is imposed on the dyadic rectangles, using mixed derivatives of $m$.

## Explanations

### L1 human

Cut the frequency line into blocks $[2^j, 2^{j+1}]$ that double in length, and the same on the negative side.
The theorem says that a bounded multiplier that does not wiggle much within each block preserves $L^p$ size for $1 < p < \infty$.
It may jump between blocks, and it may have infinitely many jumps in total.

### L2 human

This criterion needs no smoothness: only bounded variation on each dyadic block.
It is weaker in hypotheses than the [[Hörmander–Mikhlin multiplier theorem|F5V7CXW30E]] in one variable, where $\abs{m'(\xi)} \le A/\abs{\xi}$ implies a variation of at most $A \log 2$ on each block.
Its proof is a model use of Littlewood–Paley theory together with the $L^p$ boundedness of the Hilbert transform.
</document>

<document source="https://tamnd.github.io/math2/node/37CPGVHTAR.md" tag="37CPGVHTAR" kind="definition">
# Abelian variety

## Definition

Let $k$ be a field.
An *abelian variety* over $k$ is an [[algebraic group|GH81SYAPPN]] $A$ over $k$ such that:

1. $A$ is a geometrically integral [[variety|HV6YBS0YDH]] over $k$;
2. $A$ is complete, that is, the structure morphism $A \to \operatorname{Spec} k$ is [[proper|9537RKBZPP]].

Here the group structure is given by a multiplication morphism $m \colon A \times_k A \to A$, an inverse morphism $i \colon A \to A$ and a unit point $e \in A(k)$ that satisfy the group axioms.

## Remarks

Example: an elliptic curve $E$ over $k$, with its chord-and-tangent group law and the point at infinity as unit, is an abelian variety of dimension $1$.
It is a smooth projective curve, so it is geometrically integral and complete, and the group law is given by morphisms.

Non-example: the multiplicative group $\mathbb{G}_m = \operatorname{Spec} k[t, t^{-1}]$ is a geometrically integral algebraic group, but it is not complete.
It is an open subset of the projective line, and it is affine of positive dimension, so it is not proper over $k$.

The following three facts follow from the definition and are not part of it.
The group law of an abelian variety is commutative, so it is often written additively.
Every abelian variety is projective.
An abelian variety is smooth over $k$, since it is a geometrically reduced group variety.
</document>

<document source="https://tamnd.github.io/math2/node/37R4QJCARA.md" tag="37R4QJCARA" kind="concept">
# Resurgent function

## Summary

Resurgent functions are a class of analytic functions introduced by Jean Écalle to study divergent series.
A resurgent series is a formal series whose Borel transform converges near $0$.
This Borel transform must also continue analytically along every path that avoids a discrete set of singular points.
Many divergent series of analysis are resurgent, such as [[Stirling's series|K29QYWB6A5]] and the formal solutions of many analytic differential equations.
The singularities in the Borel plane link the divergent series to exponentially small terms.
This is the base of [[Borel summation|8DJPSYSHGJ]] beyond its simplest cases.

## Explanations

### L1 human

A divergent series with factorially growing coefficients can be tamed by dividing its $n$-th coefficient by $n!$.
The result is a convergent power series, a function on a small disc.
The series is resurgent when this function can be continued everywhere, except at some isolated points that can be walked around.
For example, $\sum_{n \ge 0} (-1)^n n! \, z^{-n-1}$ becomes $\sum_{n \ge 0} (-1)^n \zeta^n = \frac{1}{1 + \zeta}$, which continues to all of $\C$ except $\zeta = -1$.
The name says that the same information "resurges" at each singular point.
</document>

<document source="https://tamnd.github.io/math2/node/37TNNBQ4PK.md" tag="37TNNBQ4PK" kind="concept">
# Hasse–Weil zeta function

## Summary

The Hasse–Weil zeta function of a variety over a number field is an Euler product.
It packs together the numbers of points of the variety modulo all primes.
For a point it is the [[Riemann zeta function|1XWBY4N0VT]] or the [[Dedekind zeta function|KXS5DQDFYD]].
For an [[elliptic curve|427N6S6QQW]] it is built from the $L$-function of the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].
Its analytic continuation, functional equation and special values are the subject of deep conjectures, some of them proved through the [[modularity theorem|CS1NVJBZMH]].

## Explanations

### L1 human

Riemann's zeta function can be written as a product over all primes $p$ of the factors $(1 - p^{-s})^{-1}$.
The Hasse–Weil zeta function does the same for a system of polynomial equations with integer coefficients.
For each prime $p$, one counts the solutions modulo $p$ and in the finite fields of order $p^2, p^3, \ldots$, and turns these counts into a local factor.
The product of the local factors over all primes is the Hasse–Weil zeta function.
For an elliptic curve $E$ over $\Q$, it equals $\zeta(s)\zeta(s-1)/L(E, s)$, up to finitely many factors at the bad primes.
</document>

<document source="https://tamnd.github.io/math2/node/37TRG5DBQC.md" tag="37TRG5DBQC" kind="concept">
# Koopman operator

## Summary

The Koopman operator of a [[measure-preserving transformation|556WBDAHNP]] $T$ acts on functions by $f \mapsto f \circ T$.
It is the [[composition operator|CFASPSF7BB]] with symbol $T$ on the space $L^2$ of the invariant measure.
There it is an isometry, and a [[unitary operator|MM0EJ2GDE4]] when $T$ is invertible.
It turns a nonlinear dynamical system into a linear operator on a Hilbert space.
Ergodicity, weak mixing and mixing all become properties of this operator, which is the starting point of the [[spectral theory of dynamical systems|Z7GZ6SPFA5]].

## Explanations

### L1 human

Instead of following points, follow measurements.
A measurement is a function $f$ on the space, and after one step of the system the new measurement is $f \circ T$: measure where the point has moved to.
This rule is linear in $f$ even when $T$ is very nonlinear.
For a rotation $T(x) = x + \alpha$ of the circle, the function $e^{2\pi i x}$ becomes $e^{2\pi i \alpha} e^{2\pi i x}$, so it is an eigenfunction with eigenvalue $e^{2\pi i \alpha}$.
</document>

<document source="https://tamnd.github.io/math2/node/381RB634WS.md" tag="381RB634WS" kind="statement">
# Plateau's laws

## Statement

Consider a stable soap film or cluster of soap bubbles in equilibrium.

1. The film is made of smooth pieces of surface, and each piece has constant [[mean curvature|PA78NFEPCF]].
2. The pieces meet only in threes, along smooth curves, and any two of the three meet there at an angle of $120^\circ$.
3. These curves meet only in fours, at isolated points, and any two of the four meet there at an angle of $\arccos(-1/3) \approx 109.47^\circ$.

## Remarks

Joseph Plateau stated these laws from experiments with soap films, in his 1873 treatise on liquids under molecular forces.
The constant mean curvature in item 1 comes from the pressure difference across the film, by the Young–Laplace law.
A film spanning a wire frame, with the same pressure on both sides, is a [[minimal surface|41X0YW2R9N]].
The angle in item 2 comes from a balance of forces.
Three equal surface tensions pulling along a curve are in equilibrium only if they are at $120^\circ$ to each other.
The angle in item 3 is the angle between two segments joining the centre of a regular tetrahedron to two of its vertices.
Other junctions, such as four surfaces meeting along one curve, are unstable: a film with such a junction can lower its area by splitting it into two triple junctions.
Jean Taylor proved in 1976 that these are the only possible singularities for a mathematical model of soap films, the almost-minimizing sets of Almgren.

## Explanations

### L1 human

Soap films always join in the same few ways.
Three films meet along an edge, at equal angles of $120^\circ$, and four such edges meet at a corner, at the angle seen at the centre of a regular tetrahedron.
You can see this in any cluster of soap bubbles: four films never meet along one edge.
These rules come from the fact that a soap film always tries to reduce its area.
</document>

<document source="https://tamnd.github.io/math2/node/383CHJH46R.md" tag="383CHJH46R" kind="definition">
# Profinite completion

## Definition

Let $G$ be a [[group|G5S1M9]], and let $\mathcal{N}$ be the set of [[normal subgroups|0CB0DX5BVV]] of finite [[index|0DTXBQPA2S]] in $G$.
For $N \in \mathcal{N}$ give the finite [[quotient group|0KYX2N69J0]] $G/N$ the discrete topology.
The *profinite completion* of $G$ is
$$\hat G = \varprojlim_{N \in \mathcal{N}} G/N = \Big\{ (g_N N)_{N \in \mathcal{N}} \in \prod_{N \in \mathcal{N}} G/N : g_M N = g_N N \text{ whenever } M \subseteq N \Big\},$$
with the group law taken coordinatewise and the topology induced by the [[product topology|0GBG8B49AA]].
The map $\iota \colon G \to \hat G$, $\iota(g) = (gN)_{N \in \mathcal{N}}$, is the canonical homomorphism.

## Remarks

$\hat G$ is a [[profinite group|KA0YDB6QXE]], and $\iota(G)$ is dense in $\hat G$.
The kernel of $\iota$ is the intersection of all $N \in \mathcal{N}$, so $\iota$ is injective if and only if $G$ is [[residually finite|HEE922EHQ8]].
Example: for $G = \Z$, the finite-index subgroups are $n\Z$ with $n \ge 1$, and $\hat\Z = \varprojlim \Z/n\Z$; by the Chinese remainder theorem it is isomorphic to $\prod_p \Z_p$.
Here $\iota$ is injective, since $\bigcap_n n\Z = \set{0}$.
Non-example of injectivity: $(\Q, +)$ has no proper subgroup of finite index, so $\hat\Q$ is trivial and $\iota$ is not injective.
Universal property: every homomorphism from $G$ to a profinite group $P$ factors uniquely through $\iota$ by a continuous homomorphism $\hat G \to P$.
</document>

<document source="https://tamnd.github.io/math2/node/388FKFJAEJ.md" tag="388FKFJAEJ" kind="definition">
# Index of a singular integral operator

## Definition

Let $\Gamma$ be a smooth simple closed curve in $\C$, oriented counterclockwise.
Let $a$ and $b$ be [[Hölder continuous|012PBA0FMD]] functions on $\Gamma$ with $a(t)^2 - b(t)^2 \ne 0$ for all $t \in \Gamma$, and let $k$ be a Hölder continuous function on $\Gamma \times \Gamma$.
Consider the singular integral operator of a [[singular integral equation|E30EA6ETZ4]] of normal type
$$K \varphi(t) = a(t) \, \varphi(t) + \frac{b(t)}{\pi i} \, \mathrm{p.v.} \int_\Gamma \frac{\varphi(\tau)}{\tau - t} \, d\tau + \int_\Gamma k(t, \tau) \, \varphi(\tau) \, d\tau, \quad t \in \Gamma.$$
The *index* of $K$ is the integer
$$\kappa = \frac{1}{2\pi} \Big[ \arg \frac{a(t) - b(t)}{a(t) + b(t)} \Big]_\Gamma.$$
Here the bracket is the change of a continuous branch of the argument of $(a - b)/(a + b)$ after one run of $t$ around $\Gamma$.
In other words, $\kappa$ is the [[winding number|0NQSV2EWDR]] around $0$ of the closed curve $t \mapsto (a(t) - b(t))/(a(t) + b(t))$.

## Remarks

The condition $a^2 - b^2 \ne 0$ says that $a - b$ and $a + b$ do not vanish, so the quotient is a nonvanishing continuous function on $\Gamma$ and its winding number is defined.
Example: on the unit circle, $a(t) = (t + 1)/2$ and $b(t) = (t - 1)/2$ give $a + b = t$ and $a - b = 1$, so $a^2 - b^2 = t \ne 0$.
The quotient is $1/t = e^{-i\theta}$ for $t = e^{i\theta}$, whose argument decreases by $2\pi$, so $\kappa = -1$.
Example: for $a = 0$ and $b = 1$ the quotient is the constant $-1$, so $\kappa = 0$.
Non-example: $a = b = 1$ violates $a^2 - b^2 \ne 0$; the quotient is $0$ everywhere, and no index is defined.
The index does not depend on $k$, and it is additive: the index of a product of two such operators is the sum of their indices.
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# Hyperconnected space

## Summary

A hyperconnected space, also called an irreducible space, is a nonempty [[topological space|0CQHFPGJ45]] in which any two nonempty open sets meet.
Equivalently, it is not the union of two proper closed subsets.
Every hyperconnected space is [[connected|0V032VWTHD]], but the condition is much stronger.
Irreducible spaces are basic in algebraic geometry, where irreducible varieties and the spectra of integral domains have this property.

## Explanations

### L1 human

In a hyperconnected space, every nonempty open set is spread all over the space: it is dense.
This never happens in a Hausdorff space with two or more points, where two points have disjoint neighbourhoods.
An infinite set with the [[cofinite topology|0B906ZBTMZ]] is hyperconnected, since two nonempty open sets miss only finitely many points each.
Such coarse topologies appear naturally as Zariski topologies in algebraic geometry.
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<document source="https://tamnd.github.io/math2/node/38CEJPFESJ.md" tag="38CEJPFESJ" kind="definition">
# Minkowski functional

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, let $V$ be a [[vector space|05EGHEQZAW]] over $\mathbb{K}$, and let $K \subseteq V$ be an [[absorbing set|NM1BCK3ZX9]].
The *Minkowski functional* (or *gauge*) of $K$ is the map $p_K \colon V \to [0, \infty)$ given by
$$p_K(x) = \inf \set{t > 0 : x \in t K}.$$

## Remarks

Since $K$ is absorbing, the set $\set{t > 0 : x \in tK}$ is nonempty, so $p_K(x)$ is a finite number.
If $K$ is convex, then $p_K$ is sublinear: $p_K(x + y) \le p_K(x) + p_K(y)$ and $p_K(s x) = s \, p_K(x)$ for $s \ge 0$.
If $K$ is convex and [[balanced|2TDSHJJYS4]], then $p_K$ is a [[seminorm|FP5GVCN8GV]].
Example: let $\norm{\cdot}$ be a norm on $V$ and $K = \set{x : \norm{x} \le 1}$; then $x \in tK$ if and only if $\norm{x} \le t$, so $p_K(x) = \norm{x}$.
Non-example: in $\R^2$ the segment $K = [-1, 1] \times \set{0}$ is not absorbing, and for $x = (0, 1)$ the set $\set{t > 0 : x \in tK}$ is empty, so no finite value $p_K(x)$ is defined.
Some authors allow any set $K$ and set $p_K(x) = \infty$ when no $t$ exists.
In a topological vector space, the Minkowski functional of a convex open neighbourhood $K$ of $0$ is continuous, and $K = \set{x : p_K(x) < 1}$.
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<document source="https://tamnd.github.io/math2/node/38D9RPAXQ8.md" tag="38D9RPAXQ8" kind="statement">
# Bourget's hypothesis

## Statement

Let $n \ge 0$ and $m \ge 1$ be integers, and let $J_n$ and $J_{n+m}$ be [[Bessel functions of the first kind|CZCMRDWDYW]].
Then $J_n$ and $J_{n+m}$ have no common zero in $\C$ other than $z = 0$.

## Remarks

Bourget stated it as a conjecture in the nineteenth century, and Siegel proved it in 1929.
The proof is number-theoretic: a common nonzero zero would be a root of a polynomial with rational coefficients, hence algebraic.
But Siegel showed that $J_n(\alpha)$ is transcendental, so nonzero, for every nonzero algebraic $\alpha$.
Example: $J_0$ and $J_1$ have no common positive zero, which also follows from $J_0' = -J_1$ and the fact that the [[zeros|M324R146XH]] of $J_0$ are simple.
The cases $m = 1$ and $m = 2$ follow from the recurrence relations and the simplicity of nonzero zeros; the general case is the hard part.

## Explanations

### L1 human

Two Bessel functions of different whole-number orders never vanish at the same nonzero point.
In terms of a drum, two different families of vibration modes never share a frequency in this way.
It sounds like a simple fact about wiggly curves, but its proof needs deep results about transcendental numbers.

### L2 human

For $m = 1$, a common zero $z_0 \ne 0$ of $J_n$ and $J_{n+1}$ would give $J_n'(z_0) = \frac{n}{z_0} J_n(z_0) - J_{n+1}(z_0) = 0$, a double zero, which is impossible.
For $m = 2$ the relation $J_n + J_{n+2} = \frac{2(n+1)}{z} J_{n+1}$ reduces it to the case $m = 1$.
For larger $m$ the recurrence writes $J_{n+m}$ as a combination of $J_n$ and $J_{n+1}$ whose coefficients are polynomials in $1/z$ with rational coefficients (the Lommel polynomials).
So a common zero must be a root of one of these polynomials, and it is algebraic.
Siegel's transcendence theorem for $J_n$ at algebraic points, a starting point of the theory of $E$-functions, then settles it.
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<document source="https://tamnd.github.io/math2/node/38SETDT2HN.md" tag="38SETDT2HN" kind="concept">
# Wilcoxon signed-rank test

## Summary

The Wilcoxon signed-rank test is a [[hypothesis test|087P03NEND]] for the centre of a symmetric distribution, most often applied to the differences within matched pairs.
It ranks the absolute values of the observations and adds the ranks of the positive ones.
Unlike the [[sign test|NBK6G2VB7A]], it uses how far each observation lies from the hypothesised centre.
So it is usually more powerful, but it assumes that the distribution is symmetric.
It is the rank-based counterpart of the one-sample $t$-test.

## Explanations

### L1 human

Suppose you measure each patient before and after a treatment and take the differences.
If the treatment does nothing, positive and negative differences of each size should be equally likely.
Sort the differences by size, ignoring the sign, and number them $1, 2, 3, \dots$ from smallest to largest.
Add up the numbers that belong to positive differences.
A total near the maximum means that the large differences are mostly positive, which is evidence of an effect.
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<document source="https://tamnd.github.io/math2/node/390Z8QV5ZG.md" tag="390Z8QV5ZG" kind="definition">
# Sporadic group

## Definition

Let $G$ be a finite [[simple group|0YMYMNQ0QZ]].
The group $G$ is *sporadic* when it is not isomorphic to any of the following:

1. a [[cyclic group|0KVGH0J5B3]] of prime order;
2. an [[alternating group|0Z5905Q73D]] $A_n$ with $n \ge 5$;
3. a simple [[group of Lie type|NPEBABQ49M]].

## Remarks

By the [[classification of finite simple groups|06J6P03MG7]], there are exactly $26$ sporadic groups up to isomorphism.
The [[Tits group|WC16TBY3FN]] ${}^2F_4(2)'$ is usually counted with the groups of Lie type in clause 3; some authors count it as a $27$th sporadic group.
Example: the [[Mathieu group|T1KR4KPX8P]] $M_{11}$ is sporadic.
It is simple and not cyclic, its order $7920$ is not of the form $n!/2$, and no simple group of Lie type has order $7920$.
Non-example: $A_5$ is simple but not sporadic, since it is an alternating group, so clause 2 fails.
The sporadic groups include the five Mathieu groups, the four [[Janko groups|PSEC8D7CX0]], the three [[Conway groups|366D4ETGQH]] and the three [[Fischer groups|AYJ1CM1XT8]].
The others are the Higman–Sims, McLaughlin, Held, Rudvalis, Suzuki, O'Nan, Harada–Norton, Lyons and Thompson groups, the [[Baby Monster|TNPQ9AWVBC]] and the [[Monster|C9VY20TAPT]].

## Explanations

### L1 human

A sporadic group is a finite simple group that does not belong to the regular infinite families.
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<document source="https://tamnd.github.io/math2/node/393PNMXBFJ.md" tag="393PNMXBFJ" kind="definition">
# Quasiconvex function

## Definition

Let $V$ be a real [[vector space|05EGHEQZAW]], let $S \subseteq V$ be a [[convex set|0K06X9QXDX]], and let $f \colon S \to \R$ be a function.
The function $f$ is *quasiconvex* if
$$f(t x + (1 - t) y) \le \max\set{f(x), f(y)}$$
for all $x, y \in S$ and all $t \in [0, 1]$.

## Remarks

Equivalently, for each $a \in \R$ the sublevel set $\set{x \in S : f(x) \le a}$ is convex; compare [[level sets|0Y5Q8Q9WJ7]].
The function $f$ is *quasiconcave* if $-f$ is quasiconvex.
A [[convex function|0M1GJSWZDA]] is quasiconvex, because $t f(x) + (1 - t) f(y) \le \max\set{f(x), f(y)}$.

Example: $f(x) = \sqrt{\abs{x}}$ on $\R$.
Its sublevel sets are $\emptyset$ for $a < 0$ and $[-a^2, a^2]$ for $a \ge 0$, all convex.
It is not convex, since $f(1/2) = \sqrt{1/2} > 1/2 = \frac{1}{2} f(0) + \frac{1}{2} f(1)$.

Non-example: $f(x) = -x^2$ on $\R$.
With $x = -1$, $y = 1$ and $t = 1/2$, one gets $f(0) = 0 > -1 = \max\set{f(-1), f(1)}$.
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<document source="https://tamnd.github.io/math2/node/39HWNYW5EF.md" tag="39HWNYW5EF" kind="definition">
# Symbolic method

## Definition

A *combinatorial class* is a set $\mathcal{A}$ with a size function $\abs{\cdot} \colon \mathcal{A} \to \N$ such that each set $\mathcal{A}_n = \set{a \in \mathcal{A} : \abs{a} = n}$ is finite.
Its ordinary [[generating function|0PHAC833JF]] is $A(z) = \sum_{n \ge 0} \abs{\mathcal{A}_n} z^n$.
A construction $\Phi$ builds a class from given classes $\mathcal{B}, \mathcal{C}, \ldots$.
It is *admissible* if the counts of $\Phi(\mathcal{B}, \mathcal{C}, \ldots)$ depend only on the counts of $\mathcal{B}, \mathcal{C}, \ldots$.
Let $\mathcal{E}$ be the neutral class, with one object of size $0$, and $\mathcal{Z}$ the atomic class, with one object of size $1$.
The *symbolic method* has two steps:

1. describe the class by an equation built from $\mathcal{E}$, $\mathcal{Z}$ and admissible constructions;
2. replace each construction by its rule on generating functions, which gives an equation for $A(z)$.

## Remarks

For unlabelled classes the basic rules are as follows.
The disjoint union $\mathcal{B} + \mathcal{C}$ gives $B(z) + C(z)$.
The Cartesian product $\mathcal{B} \times \mathcal{C}$, with sizes added, gives $B(z) C(z)$.
The sequence construction $\operatorname{SEQ}(\mathcal{B})$, for $\mathcal{B}_0 = \emptyset$, gives $1 / (1 - B(z))$.
The multiset construction $\operatorname{MSET}(\mathcal{B})$, for $\mathcal{B}_0 = \emptyset$, gives $\exp\big(\sum_{k \ge 1} B(z^k) / k\big)$.
For labelled classes one uses [[exponential generating functions|0JHXPVFKT0]].
Then the labelled product gives $B(z) C(z)$, and, for $\mathcal{B}_0 = \emptyset$, sets give $\exp(B(z))$ and cycles give $\log \frac{1}{1 - B(z)}$.

Example: binary trees satisfy $\mathcal{B} = \mathcal{E} + \mathcal{Z} \times \mathcal{B} \times \mathcal{B}$, with size the number of internal nodes.
The rules give $B(z) = 1 + z B(z)^2$.
Example: permutations are labelled sets of cycles, so their exponential generating function is $\exp\big(\log \frac{1}{1 - z}\big) = \frac{1}{1 - z}$, which gives $n!$.

Non-example: the operation "keep the objects of $\mathcal{B}$ that are trees of height at most $2$" is not admissible.
Its counts depend on the inner structure of the objects of $\mathcal{B}$, and not only on the numbers $\abs{\mathcal{B}_n}$.
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# Strong maximal function

## Summary

The strong maximal function of a function $f$ on $\R^n$ is a supremum of averages of $\abs{f}$.
At a point $x$, it is taken over all rectangles that contain $x$ and have sides parallel to the axes.
Unlike the [[Hardy–Littlewood maximal function|VX1A56P5KQ]], it allows rectangles of any shape, long and thin in some directions.
It is bounded on $L^p(\R^n)$ for $1 < p \le \infty$ but is not of [[weak type|9534R6VMY0]] $(1, 1)$ when $n \ge 2$.
It controls multiparameter averages and is the basic maximal operator of product, or multiparameter, harmonic analysis.

## Explanations

### L1 human

The usual maximal function looks at balls around a point, which have the same size in every direction.
The strong maximal function also looks at boxes that can be very long in one direction and very short in another.
Because there are many more boxes than balls, it can be much larger than the usual maximal function.
Still, it is bounded on $L^p$ for $p > 1$, because in each coordinate direction it is no worse than a one-dimensional maximal function.
For example, take the indicator of the unit square in the plane and the point $(a, 1/2)$ with $a > 1$.
The rectangle $[0, a] \times [0, 1]$ gives the average $1/a$.
For large $a$, balls centred at the point only give averages of order $1/a^2$.
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<document source="https://tamnd.github.io/math2/node/39PCDRDB01.md" tag="39PCDRDB01" kind="concept">
# Delay differential equation

## Summary

In a delay differential equation, the rate of change at time $t$ depends on the state at earlier times, such as $t - \tau$.
To solve it, one must give the whole history of the solution on an initial interval, not just one initial value.
So the natural state space is a space of functions, and the equation is infinite-dimensional even for one unknown.
Delay equations model systems with reaction times, such as feedback control, population dynamics and epidemics.

## Explanations

### L1 human

A shower whose temperature responds to the tap only after a few seconds is a classic picture of delay.
You react to the water you feel now, which reflects the tap setting of a moment ago, and so you tend to overshoot.
The equation $x'(t) = -x(t - 1)$ says that the state decreases at a rate given by its value one time unit earlier.
Unlike $x' = -x$, which only decays, this equation has oscillating solutions.
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# Generic point

## Summary

A generic point of a closed set $Z$ in a topological space is a point whose closure is all of $Z$.
In a [[scheme|5A35TYZD02]], every irreducible closed subset has exactly one generic point.
Generic points let one speak of a property holding "at a general point" of a variety.
Functions at the generic point form the function field.

## Explanations

### L1 human

In classical geometry, every point of a curve is a single spot on it.
A scheme adds one more point for the whole curve, which lies "everywhere on it at once".
In $\operatorname{Spec} \C[x, y]$, the prime ideal $(y - x^2)$ is such a point: its closure contains all the points $(x - a, y - a^2)$ of the parabola.
Similarly, $(0)$ is the generic point of $\operatorname{Spec} \Z$, and its closure is the whole space.
A statement true at the generic point is often true on a dense open set.
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<document source="https://tamnd.github.io/math2/node/3A1D37TVF6.md" tag="3A1D37TVF6" kind="concept">
# Small cancellation theory

## Summary

Small cancellation theory studies groups given by [[presentations|0JGEQXQ4AC]] whose relators have only short common pieces.
Under such conditions, such as $C'(1/6)$, a nontrivial word equal to the identity must contain a long part of some relator.
This gives Dehn's algorithm for the [[word problem|PBGTTAVMMT]], and finitely presented $C'(1/6)$ groups are [[hyperbolic|JWJ8YPR7EB]].
The theory is also a tool to build groups with unusual properties, as in the construction of [[Tarski monster groups|TE19BGFFF1]].

## Explanations

### L1 human

A presentation lists generators and relators, words that are declared equal to $1$.
If two relators share a long piece, multiplying them can cancel a lot and produce surprising consequences.
Small cancellation conditions forbid this: any piece shared by two relators is a small part of each.
Then the relators behave almost independently, and one can tell when a word equals $1$ by a simple greedy rule.
For example, the fundamental group of a closed orientable surface of genus $2$, $\langle a, b, c, d \mid [a, b][c, d] \rangle$, satisfies the condition $C'(1/6)$.
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# Higher category theory

## Summary

Higher category theory studies structures with morphisms between objects, 2-morphisms between morphisms, 3-morphisms between 2-morphisms, and so on.
Its basic notions are $n$-categories and $\infty$-categories.
They come in strict forms such as [[2-categories|CQACXG10JZ]] and in weak forms such as [[bicategories|TAHV24ED74]], where composition is associative only up to higher morphisms.
The theory of [[(∞,1)-categories|YFY1FWA1MY]], modelled for example by [[quasi-categories|9328WEVJEK]], is its most developed part.
It is now a standard tool in homotopy theory and algebraic geometry.
A guiding idea is the [[homotopy hypothesis|2E83D9ACDY]], which relates $\infty$-groupoids to topological spaces.

## Explanations

### L1 human

In an ordinary category, two arrows between the same objects are either equal or not.
In many examples, though, there are ways to compare arrows, such as homotopies between continuous maps or natural transformations between functors.
Higher category theory treats these comparisons as arrows of a higher level, and keeps going to comparisons of comparisons.
For example, the paths in a space, the homotopies between paths, the homotopies between homotopies, and so on, form an $\infty$-groupoid.
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# A translation of the integers is not a homomorphism

## Example

Let $G = (\mathbb{Z}, +)$ be the [[integers|0C4JS9RF5M]] with addition. Define $t : G \to G$ by $t(u) = u + 1$.
A [[homomorphism|0HV9Y7WG5R]] sends the identity to the identity. But $t(0) = 1$, and $1$ is not the identity $0$.
We can also compute directly: $t(0 + 0) = 1$, but $t(0) + t(0) = 2$.
Thus $t$ is not a homomorphism.

## Remarks

$t$ is a bijection, so a bijection between groups need not be an isomorphism.
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# Henselian field

## Summary

A Henselian field is a [[valued field|B85T5VKR3M]] in which the conclusion of [[Hensel's lemma|010QVB71T2]] holds, see the [[definition|Z3KPTM2YWX]].
In it, a simple root of a polynomial modulo the maximal ideal lifts to a true root.
Its [[valuation ring|QSX12D8RQM]] is a [[Henselian ring|RNXPV3KC5J]].
Complete fields for a discrete valuation, such as the [[p-adic numbers|BT6T702C28]] and [[fields of formal Laurent series|YZWYF7DXTW]], are Henselian.
In a Henselian field the valuation extends in only one way to each algebraic extension, which makes their extension theory clean.

## Explanations

### L1 human

Over the $7$-adic numbers, one can solve $x^2 = 2$ by first solving it modulo $7$ and then correcting the answer step by step.
Modulo $7$, the number $3$ works, since $9 - 2 = 7$.
A field is Henselian when this "solve approximately, then refine" method always works for simple roots.
The rationals with the $7$-adic valuation are not Henselian: the approximate root $3$ exists, but $\sqrt 2$ is not rational.
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# Regulator of a number field

## Definition

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\mathcal{O}_K$.
Let $\sigma_1, \ldots, \sigma_{r_1}$ be the real embeddings of $K$, and let $\sigma_{r_1 + 1}, \ldots, \sigma_{r_1 + r_2}$ be one embedding from each pair of complex conjugate non-real embeddings.
Put $N_j = 1$ for $j \le r_1$ and $N_j = 2$ for $j > r_1$, and let $r = r_1 + r_2 - 1$.
Let $u_1, \ldots, u_r$ be a system of fundamental units, as in [[Dirichlet's unit theorem|1D99CSKX6N]].
If $r \ge 1$, the *regulator* of $K$ is
$$R_K = \abs{\det\big(N_j \log \abs{\sigma_j(u_i)}\big)_{1 \le i, j \le r}}.$$
This is the absolute value of the [[determinant|00KDZRW4C0]] of an $r \times r$ matrix.
The matrix keeps the first $r$ of the $r + 1$ columns $j = 1, \ldots, r_1 + r_2$.
If $r = 0$, the regulator is $R_K = 1$.

## Remarks

The value does not depend on the choices.
Each row of the full $r \times (r+1)$ matrix sums to $0$, because $\prod_j \abs{\sigma_j(u)}^{N_j} = \abs{N_{K/\Q}(u)} = 1$ for a unit $u$; so deleting another column changes only the sign of the determinant.
Another system of fundamental units differs by a matrix in $\mathrm{GL}_r(\Z)$, which has determinant $\pm 1$.
Here $\log$ is the [[natural logarithm|04DXAFN2ES]].

Example: for $K = \Q(\sqrt{2})$, $r = 1$ and $u_1 = 1 + \sqrt{2}$, so $R_K = \abs{\log(1 + \sqrt{2})} \approx 0.881$.

Non-example: in $\Q(\sqrt{2})$, the unit $3 + 2\sqrt{2} = (1 + \sqrt{2})^2$ is not fundamental, since it does not generate the units modulo $\pm 1$.
Using it in the formula gives $2 \log(1 + \sqrt{2})$, which is twice the regulator.

For $\Q$ and for imaginary quadratic fields, $r = 0$ and $R_K = 1$.
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# p-adic Lie group

## Summary

A $p$-adic Lie group is a [[topological group|92700QAB5Y]] that is also an analytic manifold over the field $\Q_p$ of [[p-adic numbers|BT6T702C28]], with analytic group operations.
It is the $p$-adic analogue of a real [[Lie group|7JBRGCVJRB]].
Such groups are totally disconnected, so their structure is studied through compact open subgroups, which are [[pro-p groups|C13D6W3MM9]] up to finite index.
They appear as Galois groups in number theory and as the groups $G(\Q_p)$ of points of algebraic groups.

## Explanations

### L1 human

A real Lie group looks near each point like a piece of $\R^n$, and its multiplication is given by convergent power series.
A $p$-adic Lie group is the same idea with $\R$ replaced by the $p$-adic numbers $\Q_p$.
The additive group $\Z_p$ of $p$-adic integers is the simplest example: it is an open piece of $\Q_p$, and addition is a polynomial.
The matrix group $GL_n(\Q_p)$ is another example, since products and inverses of matrices are given by rational functions of the entries.
Unlike real Lie groups, these groups fall apart into tiny open and closed pieces, because $\Q_p$ itself is totally disconnected.
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<document source="https://tamnd.github.io/math2/node/3ANCZ518BN.md" tag="3ANCZ518BN" kind="definition">
# Quadratic polynomial dynamics

## Definition

For $c \in \C$ let $f_c \colon \C \to \C$ be the polynomial $f_c(z) = z^2 + c$.
The family $(f_c)_{c \in \C}$ is the *quadratic family*.
*Quadratic polynomial dynamics* is the [[complex dynamics|BQQZ0ZCGQF]] of this family.
It studies the [[iterates|AYVTR2ES5B]] $f_c^n$ for each $c$, and how they change with $c$.
The *filled Julia set* of $f_c$ is $K_c = \set{z \in \C : (f_c^n(z))_{n \ge 0} \text{ is bounded}}$.
Its boundary $\partial K_c$ is the [[Julia set|5M4S7P04GR]] $J_c$ of $f_c$.

## Remarks

Every [[polynomial|1G0B73WPX8]] $p(z) = a z^2 + b z + d$ with $a \ne 0$ is conjugate to exactly one $f_c$ by an affine map $z \mapsto \alpha z + \beta$.
This is why one parameter suffices.
The only critical point of $f_c$ in $\C$ is $0$.
The set $K_c$ is connected if and only if the orbit of $0$ is bounded, and otherwise $K_c$ is a Cantor set.
Example: for $c = 0$, $f_0^n(z) = z^{2^n}$ is bounded exactly when $\abs{z} \le 1$, so $K_0$ is the closed unit disc and $J_0$ is the unit circle.
Non-example: $z \mapsto z^3 + c$ is not in the quadratic family, since its degree is $3$.
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<document source="https://tamnd.github.io/math2/node/3AQFY4PAC2.md" tag="3AQFY4PAC2" kind="definition">
# Triangular matrix

## Definition

Let $R$ be a [[ring|0B1261B5ED]] and let $A = (a_{ij})$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $A$ is *upper triangular* when $a_{ij} = 0$ for all $i > j$.
It is *lower triangular* when $a_{ij} = 0$ for all $i < j$.
It is *triangular* when it is upper triangular or lower triangular.

## Remarks

Example: $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix}$ is triangular, because its entries below the diagonal, in positions $(2,1)$, $(3,1)$, $(3,2)$, are zero.
Non-example: $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is not triangular, because $a_{21} = 3 \ne 0$ and $a_{12} = 2 \ne 0$.
A matrix is both upper and lower triangular exactly when it is diagonal.
A triangular matrix is *strictly* triangular when its diagonal entries are also zero, and *unit* triangular when they are all $1$.
See also [[upper triangular matrix|0N4XY7Z3NN]] and [[lower triangular matrix|00J3KZP8Y0]].
</document>

<document source="https://tamnd.github.io/math2/node/3AX549FFFM.md" tag="3AX549FFFM" kind="concept">
# Maximal chain

## Summary

A maximal chain in a [[partially ordered set|011C29Q1S1]] is a [[chain|0GP9ASVQ1X]] that is not contained in any strictly larger chain.
The [[Hausdorff maximal principle|0KM4KVR9MV]] says that every chain lies in a maximal chain, and this statement is equivalent to the axiom of choice.
In finite posets, maximal chains are used to define the height and the grading of a poset.

## Explanations

### L1 human

A chain is a set of elements that stand on one line in the order.
A maximal chain is a chain that cannot be extended: no other element can be added while keeping all elements comparable.
In the subsets of $\set{1, 2}$, the chain $\emptyset \subset \set{1} \subset \set{1, 2}$ is maximal.
The chain $\emptyset \subset \set{1, 2}$ is not maximal, because $\set{1}$ can be inserted.
</document>

<document source="https://tamnd.github.io/math2/node/3B0HB3DHW1.md" tag="3B0HB3DHW1" kind="problem">
# Stefan problem

## Problem

The one-dimensional one-phase Stefan problem describes a solid at its melting temperature $0$ that occupies the half-line $x > 0$ and melts when heat enters at $x = 0$.
Let $a > 0$ be the thermal diffusivity of the liquid, $k > 0$ its thermal conductivity and $L > 0$ the latent heat per unit volume.
Let $g \colon [0, \infty) \to [0, \infty)$ be a given boundary temperature.
Find a function $s \colon [0, \infty) \to [0, \infty)$ with $s(0) = 0$, the position of the interface, and a temperature $u(x, t)$ defined for $0 \le x \le s(t)$, such that:

1. $\partial_t u = a \, \partial_x^2 u$ for $0 < x < s(t)$ and $t > 0$;
2. $u(0, t) = g(t)$ for $t > 0$;
3. $u(s(t), t) = 0$ for $t > 0$;
4. $L \, s'(t) = -k \, \partial_x u(s(t), t)$ for $t > 0$.

The unknown interface $s$ is the free boundary.
Condition 4, the Stefan condition, says that the heat flux arriving at the interface is used to melt the solid.

## Remarks

The problem is nonlinear, although the [[heat equation|VP2HN86VAG]] in clause 1 is linear, because the domain where it holds depends on the solution.
For a constant boundary temperature $g = T > 0$ there is an explicit similarity solution, due to Neumann.
In units where $a = k = L = 1$, it is $s(t) = 2 \lambda \sqrt{t}$ and $u(x, t) = T \left(1 - \operatorname{erf}\left(x / (2 \sqrt{t})\right) / \operatorname{erf}(\lambda)\right)$.
Here $\lambda > 0$ is the unique root of $\lambda e^{\lambda^2} \operatorname{erf}(\lambda) = T / \sqrt{\pi}$, which comes from inserting $u$ and $s$ into the Stefan condition.
The interface thus moves like $\sqrt{t}$, as is typical for diffusion.
Two-phase versions, in which the temperature of the solid also evolves, and versions in several space dimensions are studied as well.
Lamé and Clapeyron had studied an equivalent problem in 1831.

## Explanations

### L1 human

A block of ice is heated from one side.
Inside the water the heat spreads by diffusion, and at the boundary between water and ice the heat is used up to melt more ice.
The position of this boundary is not known in advance: it is part of the answer.
The Stefan problem asks for both the temperature in the water and the motion of the melting front.
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<document source="https://tamnd.github.io/math2/node/3B1743XS7S.md" tag="3B1743XS7S" kind="concept">
# Betti numbers of a module

## Summary

The Betti numbers of a finitely generated module over a local ring are the ranks of the free modules in its minimal free resolution.
The same holds for graded modules over a polynomial ring.
They are computed as the dimensions of the [[Tor|DDQRERAQCV]] modules against the residue field.
They are the basic numerical invariants of syzygies, and they refine the Hilbert function and the [[projective dimension|MR9QCH1PNA]].

## Explanations

### L1 human

To describe a module, one picks generators, then relations among them, then relations among the relations, and so on.
If one picks as few as possible at each step, the counts do not depend on the choices.
These counts are the Betti numbers.
For the residue field $k$ of $k[x, y]$, one needs $1$ generator, $2$ relations $x$ and $y$, and $1$ relation among them, so the Betti numbers are $1, 2, 1$.
The name comes from the Betti numbers of topology, which count holes in a space.
</document>

<document source="https://tamnd.github.io/math2/node/3B451RD8AJ.md" tag="3B451RD8AJ" kind="statement">
# Borel–Harish-Chandra theorem

## Statement

Let $G \subseteq \mathrm{GL}_n$ be a [[linear algebraic group|GA81VVQHEP]] defined over $\Q$, and let $G^\circ$ be its identity component.
A *$\Q$-character* of $G^\circ$ is a homomorphism of algebraic groups $G^\circ \to \mathrm{GL}_1$ defined over $\Q$.
Let $G(\Z) = G(\R) \cap \mathrm{GL}_n(\Z)$, an [[arithmetic group|4XWG6S7R4Z]].
Then $G(\Z)$ is a [[lattice|NS2057MFWP]] in $G(\R)$ if and only if $G^\circ$ has no nontrivial $\Q$-character.

In particular, if $G$ is semisimple, then $G(\Z)$ is a lattice in $G(\R)$.

## Remarks

The proof uses reduction theory.
One builds an explicit Siegel set $\mathfrak{S}$ in $G(\R)$, of finite Haar measure, such that $G(\R) = \mathfrak{S} \, F \, G(\Z)$ for a finite set $F$.
For $\mathrm{GL}_n$ and $\mathrm{SL}_n$, Siegel sets come from Minkowski's reduction of quadratic forms; the general case reduces to these.

Example: $G = \mathrm{SL}_n$ has no nontrivial characters, so $\mathrm{SL}_n(\Z)$ is a lattice in $\mathrm{SL}_n(\R)$.

The condition on characters is necessary.
For $G = \mathrm{GL}_1$, the identity map is a $\Q$-character.
Here $G(\Z) = \set{\pm 1}$, and $\R^\times / \set{\pm 1}$ has infinite Haar measure $\int_0^\infty dx / x$.
So $G(\Z)$ is not a lattice in $G(\R) = \R^\times$.

A companion result, proved by Borel and Harish-Chandra and by Mostow and Tamagawa, says when $G(\R)/G(\Z)$ is compact.
For reductive $G$, this happens exactly when $G$ has no nontrivial unipotent elements in $G(\Q)$ and no nontrivial $\Q$-characters.

## Explanations

### L1 human

Take a group of matrices defined by polynomial equations with rational coefficients, like the matrices of determinant $1$.
Its integer points form a discrete subgroup of its real points.
The theorem says when this discrete subgroup is large enough that the quotient has finite volume.
The answer is: unless the group has a homomorphism to the multiplicative group, defined over $\Q$, that would stretch the volume to infinity.

### L2 human

This theorem is the source of most lattices in semisimple Lie groups.
Groups like $\mathrm{SL}_n(\Z)$, $\mathrm{Sp}_{2n}(\Z)$ and integral orthogonal groups of indefinite forms are lattices because of it.
The [[Margulis arithmeticity theorem|FEJK846N5X]] gives a converse in higher rank: every irreducible lattice arises from this construction.
For $\mathrm{SL}_2$ it recovers the fact that the modular surface $\mathrm{SL}_2(\Z) \backslash \mathbb{H}$ has finite area.
</document>

<document source="https://tamnd.github.io/math2/node/3B55HNS754.md" tag="3B55HNS754" kind="concept">
# Probabilistic method

## Summary

The probabilistic method proves that an object with a given property exists.
It chooses an object at random and shows that it has the property with positive probability.
It is a nonconstructive method: it usually gives no explicit example.
Erdős made it a main tool of combinatorics, starting with lower bounds for [[Ramsey numbers|0K9QVPJYJF]].
Its tools include [[linearity of expectation|0P7PTC8DDE]] and [[Boole's inequality|0EGSEE9J9R]].
More refined tools are the [[Lovász local lemma|35YQ94FNV1]] and concentration inequalities.

## Explanations

### L1 human

To show that something exists, it is enough to show that a random choice finds it with a chance above zero.
For example, colour the edges of a complete graph red or blue by fair coin flips.
If the expected number of single-coloured large cliques is below $1$, some colouring has none.
So a good colouring exists, even though the argument does not tell you which one it is.
</document>

<document source="https://tamnd.github.io/math2/node/3BAJ8ZTVPX.md" tag="3BAJ8ZTVPX" kind="concept">
# von Neumann bounded set

## Summary

A subset of a [[topological vector space|T32W553W95]] is von Neumann bounded if a large enough scaling shrinks it into any given neighbourhood of $0$.
It replaces the notion of a [[bounded set|09QSZVVM26]] of a metric space.
That notion does not suit topological vector spaces, since a metric may make the whole space bounded.
In a normed space the von Neumann bounded sets are the sets of finite norm diameter.
Bounded sets are used to define the [[strong dual space|RG52DZ1A8Y]], [[bornological spaces|4R6SQD60FN]] and [[Montel spaces|NWMYW2WK9S]].

## Explanations

### L1 human

Pick any small neighbourhood of the origin, however tiny.
A set is bounded if, after you divide it by a large enough number, it fits inside that neighbourhood.
In $\R^2$ this is the usual idea: a set is bounded if it fits inside some disc.
A straight line through the origin is not bounded, because no rescaling of it fits in a small disc.
In a space of functions with seminorms $p_n$, a set is bounded when each $p_n$ is bounded on it, possibly with a different bound for each $n$.
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<document source="https://tamnd.github.io/math2/node/3BBE4XY4E9.md" tag="3BBE4XY4E9" kind="definition">
# Twelvefold way

## Definition

Let $N$ and $X$ be finite sets with $\abs{N} = n$ and $\abs{X} = x$.
Consider the [[functions|0XHDTE5YMZ]] $f \colon N \to X$ in one of three classes: all functions, [[injective functions|0PVG1CRAJN]], or [[surjective functions|0XKH881BZ7]].
Count them in one of four ways:

1. as functions;
2. up to permutations of $N$, so $f$ and $f \circ \sigma$ count as the same for every permutation $\sigma$ of $N$;
3. up to permutations of $X$, so $f$ and $\tau \circ f$ count as the same for every permutation $\tau$ of $X$;
4. up to permutations of both, so $f$ and $\tau \circ f \circ \sigma$ count as the same for all such $\sigma$ and $\tau$.

The *twelvefold way* is the list of these $3 \cdot 4 = 12$ counting problems with their answers.

## Remarks

Write $x^{\underline{n}}$ for the [[falling factorial|0N5M924N3P]], $S(n, k)$ for the [[Stirling numbers of the second kind|0JCQGPY8Z4]], and $p_k(m)$ for the number of [[partitions|0JV30XN73Z]] of $m$ into exactly $k$ parts.
Write $[n \le x]$ for $1$ if $n \le x$ and $0$ otherwise.
The answers, with $n$ balls $N$ and $x$ boxes $X$, are as follows.

| count | all functions | injective | surjective |
|---|---|---|---|
| as functions | $x^n$ | $x^{\underline{n}}$ | $x!\, S(n, x)$ |
| up to permutations of $N$ | $\binom{x + n - 1}{n}$ | $\binom{x}{n}$ | $\binom{n - 1}{x - 1}$ |
| up to permutations of $X$ | $\sum_{k=0}^{x} S(n, k)$ | $[n \le x]$ | $S(n, x)$ |
| up to both | $\sum_{k=0}^{x} p_k(n)$ | $[n \le x]$ | $p_x(n)$ |

The entry $\binom{n-1}{x-1}$ needs $n \ge 1$; for $n = x = 0$ there is exactly one surjection, the empty map.

Example: take $n = 3$ and $x = 2$, counted up to permutations of $N$.
A class of functions is then fixed by how many balls go into each box, so the classes are $(3, 0)$, $(2, 1)$, $(1, 2)$ and $(0, 3)$.
This gives $4 = \binom{4}{3}$, as in the table.

Non-example: counting functions up to a permutation of $N$ that is not arbitrary, for instance only up to cyclic rotations of $N$, is not one of the twelve problems.
It leads to necklace counting, which is handled by the [[Pólya enumeration theorem|MPYQ39P01F]].

The second row is the theory of [[combinations with repetition|0VQQCVG8WA]].
The first entry of the first row is the [[number of functions between finite sets|08QQH9F7G0]], and the second is the [[number of injections|0X0N833FF8]].
</document>

<document source="https://tamnd.github.io/math2/node/3BBSXXAZKE.md" tag="3BBSXXAZKE" kind="definition">
# Anger function

## Definition

Let $\nu \in \C$ and $z \in \C$.
The *Anger function* of order $\nu$ is
$$\mathbf{J}_\nu(z) = \frac{1}{\pi} \int_0^\pi \cos(\nu\tau - z \sin\tau)\, d\tau.$$
The *Weber function* of order $\nu$ is
$$\mathbf{E}_\nu(z) = \frac{1}{\pi} \int_0^\pi \sin(\nu\tau - z \sin\tau)\, d\tau.$$

## Remarks

Both are entire functions of $z$ and of $\nu$, since the integrands are entire and the interval is compact.
If $n$ is an integer, then $\mathbf{J}_n = J_n$, the [[Bessel function of the first kind|CZCMRDWDYW]], by Bessel's integral.
Example: at $z = 0$, $\mathbf{J}_\nu(0) = \frac{1}{\pi} \int_0^\pi \cos(\nu\tau)\, d\tau = \frac{\sin(\nu\pi)}{\nu\pi}$ for $\nu \ne 0$.
A non-example: for $\nu = \frac{1}{2}$, $\mathbf{J}_{1/2}(0) = \frac{2}{\pi} \ne 0$, while $J_{1/2}(0) = 0$, so $\mathbf{J}_{1/2}$ is not the Bessel function $J_{1/2}$.
The Anger function solves $z^2 w'' + z w' + (z^2 - \nu^2)\, w = \frac{(z - \nu)\sin(\nu\pi)}{\pi}$, so it solves Bessel's equation exactly when $\nu$ is an integer.
The Weber function $\mathbf{E}_\nu$ here is different from the Bessel function of the second kind, which is also sometimes called a Weber function.
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<document source="https://tamnd.github.io/math2/node/3BCYVS9E17.md" tag="3BCYVS9E17" kind="concept">
# Discriminant of a field extension

## Summary

The discriminant of a basis $b_1, \ldots, b_n$ of a finite extension $L/K$ is the determinant of the matrix $\left(\mathrm{Tr}_{L/K}(b_i b_j)\right)$, the Gram matrix of the [[trace form|YWX2R53ED6]].
Changing the basis multiplies it by the square of a nonzero element of $K$, so its class modulo squares depends only on the extension.
It is nonzero exactly when $L/K$ is separable.
For a power basis $1, \alpha, \ldots, \alpha^{n-1}$ it equals the [[discriminant of the minimal polynomial|0VVK64Q4JK]] of $\alpha$.
For an integral basis of a number field it gives the discriminant of the number field.

## Explanations

### L1 human

Pick a basis of the extension and pair basis elements with the trace form.
The determinant of the resulting square table is the discriminant.
It is nonzero exactly when the trace pairing is nondegenerate.
For $\Q(i)$ with basis $1, i$, the table is $\begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix}$, so the discriminant is $-4$.
For the basis $1, \alpha$ of $\Q(\alpha)$ with $\alpha^2 = d$, it is $4d$, the discriminant $b^2 - 4ac$ of $x^2 - d$.
</document>

<document source="https://tamnd.github.io/math2/node/3BJTDXB9KK.md" tag="3BJTDXB9KK" kind="definition">
# Orthogonal polynomials on the unit circle

## Definition

Let $\partial\mathbb{D} = \set{z \in \C : \abs{z} = 1}$ and let $\mu$ be a Borel [[probability measure|0VGMEBVW1M]] on $\partial\mathbb{D}$ whose [[support|FQTXV6BQZG]] is an infinite set.
Give the polynomials the inner product $\ip{p}{q} = \int_{\partial\mathbb{D}} p(z) \overline{q(z)} \, d\mu(z)$ of the [[Lebesgue space|SASP14MA1Z]] $L^2(\partial\mathbb{D}, \mu)$.
For $n \ge 0$, the *monic orthogonal polynomial* $\Phi_n$ of degree $n$ is the unique polynomial of the form $\Phi_n(z) = z^n + \text{(terms of lower degree)}$ such that
$$\ip{\Phi_n}{z^k} = 0 \quad \text{for } k = 0, 1, \dots, n - 1.$$
The *orthonormal polynomials* are $\varphi_n = \Phi_n / \norm{\Phi_n}$.

## Remarks

The condition on the support makes $\ip{p}{p} > 0$ for every nonzero polynomial $p$, since a nonzero polynomial has finitely many zeros.
Then $\Phi_n$ is $z^n$ minus its orthogonal projection onto the span of $1, z, \dots, z^{n-1}$, which shows existence and uniqueness.
Such measures are called *nontrivial*.

Example: let $\mu$ be the normalized arc length $d\theta / 2\pi$, with $z = e^{i\theta}$.
Then $\ip{z^n}{z^k} = \frac{1}{2\pi} \int_0^{2\pi} e^{i(n - k)\theta} \, d\theta = 0$ for $k \ne n$, so $\Phi_n(z) = z^n$ and $\norm{\Phi_n} = 1$.

Non-example: let $\mu = \delta_1$, the point mass at $1$.
Its support is finite, and the polynomial $z - 1$ has norm $0$.
The orthogonality conditions for $n = 1$ give $\ip{z + c}{1} = 1 + c = 0$, so $\Phi_1 = z - 1$, which is zero in $L^2(\mu)$; the construction breaks down and does not give an orthonormal sequence.
</document>

<document source="https://tamnd.github.io/math2/node/3BTA41B7M1.md" tag="3BTA41B7M1" kind="definition">
# Walsh–Hadamard transform

## Definition

Let $m \ge 0$ and $N = 2^m$.
Let $H_0 = (1)$, and for $m \ge 1$ let $H_m = H_1 \otimes H_{m-1}$, where $\otimes$ is the [[Kronecker product|0N7ENQRCXN]] and
$$H_1 = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.$$
The *Walsh–Hadamard transform* of a vector $x \in \C^N$ is the vector $H_m x \in \C^N$.
Index rows and columns by $0, \dots, N-1$.
Then $(H_m)_{jk} = (-1)^{j \cdot k}$, where $j \cdot k$ counts the positions where the binary expansions of $j$ and $k$ both have the digit $1$.

## Remarks

$H_m$ is a symmetric [[Hadamard matrix|J0N4J606X0]] of order $N$, so $H_m^2 = N I$ and the inverse transform is $x = N^{-1} H_m (H_m x)$.
Many authors use the normalized matrix $N^{-1/2} H_m$, which is orthogonal and its own inverse.
Writing $x$ as two halves $x = (x', x'')$, one has $H_m x = (H_{m-1}(x' + x''), H_{m-1}(x' - x''))$; iterating this gives the fast transform with $m N$ additions and subtractions.

Example: for $m = 2$,
$$H_2 = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{pmatrix},$$
and $x = (1, 0, 0, 0)$ goes to $(1, 1, 1, 1)$, while $x = (1, 1, 1, 1)$ goes to $(4, 0, 0, 0)$.
Check of one entry: $(H_2)_{3,3} = (-1)^{2} = 1$, since $3 = 11_2$ shares two digits $1$ with itself.

Non-example: the discrete Fourier transform matrix $(e^{-2\pi i jk/4})_{j,k}$ of order $4$ has entries $\pm i$, so it is not $H_2$, although its rows are also orthogonal.
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<document source="https://tamnd.github.io/math2/node/3BTG6VVCGZ.md" tag="3BTG6VVCGZ" kind="concept">
# Loss function

## Summary

A loss function assigns a number to each pair of a true state of nature and an action.
The number measures the cost of taking that action in that state.
In statistics the state is an unknown parameter and the action is often an estimate.
Loss functions are the starting point of [[statistical decision theory|CPFSBBHXT5]]: averaging the loss over the data gives the risk, by which procedures are compared.

## Explanations

### L1 human

A loss function says how bad a guess is when the truth is known.
Guessing $7$ when the truth is $5$ costs $(7 - 5)^2 = 4$ under squared error loss, and $\abs{7 - 5} = 2$ under absolute error loss.
Different losses express different priorities: squared error punishes large mistakes heavily, while a $0$-$1$ loss only cares whether the answer is right.
The choice of loss changes which estimator is best.
</document>

<document source="https://tamnd.github.io/math2/node/3BYHP9E9VF.md" tag="3BYHP9E9VF" kind="definition">
# Elliptic partial differential equation

## Definition

Let $P = \sum_{\abs{\alpha} \le m} a_\alpha(x) \, \partial^\alpha$ be a [[linear differential operator|6DZETZ5T9G]] of order $m$ on an open set $\Omega \subseteq \R^n$, with [[principal symbol|F31T650DKY]] $\sigma_m(P)$.
The operator $P$ is *elliptic at* $x \in \Omega$ if $\sigma_m(P)(x, \xi) \ne 0$ for every $\xi \in \R^n \setminus \set{0}$.
It is *elliptic in* $\Omega$ if it is elliptic at every point of $\Omega$.
The equation $P u = f$ is then an *elliptic partial differential equation*.

For a second-order operator $\sum_{i,j} a_{ij}(x) \partial_i \partial_j + \sum_i b_i(x) \partial_i + c(x)$ with real symmetric $(a_{ij})$, this means that the matrix $(a_{ij}(x))$ is positive definite or negative definite at each $x$.
It is *uniformly elliptic* in $\Omega$ if there is a constant $\theta > 0$ with
$$\sum_{i,j=1}^{n} a_{ij}(x) \, \xi_i \xi_j \ge \theta \abs{\xi}^2 \quad \text{for all } x \in \Omega, \ \xi \in \R^n.$$

## Remarks

Example: the [[Laplacian|0E0B6RA8K0]] $\Delta = \sum_i \partial_i^2$ has principal symbol $\abs{\xi}^2$, which is nonzero for $\xi \ne 0$, so it is elliptic; it is uniformly elliptic with $\theta = 1$.
Example: the [[Cauchy–Riemann operator|PPSSTHTEDE]] $\partial_x + i \partial_y$ on $\R^2$ has symbol $\xi_1 + i \xi_2$, which vanishes only at $\xi = 0$, so it is elliptic of order $1$.
Non-example: the wave operator $\partial_t^2 - \partial_x^2$ has symbol $\tau^2 - \xi^2$, which vanishes at $(\tau, \xi) = (1, 1)$, so it is not elliptic.
If all coefficients are real and $n \ge 2$, an elliptic operator has even order $m$.
Indeed $\sigma_m(x, -\xi) = (-1)^m \sigma_m(x, \xi)$, and a real continuous function that does not vanish on the connected unit sphere cannot change sign.
For nonlinear second-order equations $F(D^2 u, D u, u, x) = 0$, ellipticity is a monotonicity condition of $F$ in the matrix variable $D^2 u$.

## Explanations

### L1 human

An operator is elliptic when its top-order part does not vanish in any direction, which for second-order equations means that the coefficient matrix is definite.
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<document source="https://tamnd.github.io/math2/node/3BZAPDBPC8.md" tag="3BZAPDBPC8" kind="definition">
# Urysohn space

## Definition

A [[topological space|0CQHFPGJ45]] $X$ is an *Urysohn space* when the following holds.
For all $x, y \in X$ with $x \ne y$, there are [[neighbourhoods|0ZCY0Z80B6]] $U$ of $x$ and $V$ of $y$ such that $\overline{U} \cap \overline{V} = \varnothing$.

## Remarks

Equivalently, any two distinct points have disjoint [[closed|0ZCK8DQYN1]] neighbourhoods.
Every Urysohn space is [[Hausdorff|0S1FJ8VEX9]], since the interiors of $\overline{U}$ and $\overline{V}$ are disjoint open sets around $x$ and $y$.
Every [[regular|03HMJASQBQ]] Hausdorff space is Urysohn.

Example: $\R$ is Urysohn.
For $x < y$, put $r = (y - x)/3$; the closures $[x - r, x + r]$ and $[y - r, y + r]$ of the open intervals of radius $r$ are disjoint.

Non-example: an infinite set with the [[cofinite topology|0B906ZBTMZ]] is not Urysohn.
Any two nonempty open sets in it meet, so even the Hausdorff condition fails.

Notation: $T_{2\frac12}$.
A space in which distinct points are separated by a continuous function $f \colon X \to [0, 1]$ is called completely Hausdorff; some authors call that an Urysohn space.
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# Priority method

## Summary

The priority method is a technique to build [[computably enumerable sets|TN9G24VABN]] with prescribed properties.
The desired property is split into infinitely many requirements, ordered by priority, and the set is built in stages.
When requirements conflict, the one of higher priority wins, and a requirement that loses is injured and starts again.
It was introduced to solve [[Post's problem|1RVF7DWG8C]] and is the main tool for constructions in the theory of [[Turing degrees|P5N2ND3VCG]].

## Explanations

### L1 human

Imagine a list of infinitely many people who each want something from a shared building project.
Their wishes may conflict, so each has a rank: the first person always gets their way, the second gets their way unless the first objects, and so on.
A person whose work is undone just starts over.
If each person is disturbed only finitely often, in the end everyone is satisfied.
In the [[Friedberg–Muchnik theorem|RA358M9QCN]], the wishes say "set $A$ is not computed from $B$ by the $e$-th program" and the same with $A$ and $B$ swapped.
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# Lagrange's theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $H$ be a subgroup of $G$.
Then the order of $H$ divides the order of $G$.

## Remarks

The quotient $|G| / |H|$ is the number of left cosets of $H$ in $G$.

## Explanations

### L1 agent

The left cosets of a subgroup cut the group into pieces of the same size.
Each piece has as many elements as the subgroup.
Thus the size of the group is a whole number of copies of the size of the subgroup.
For example, a group with $12$ elements can have subgroups of order $1, 2, 3, 4, 6$ or $12$, and of no other order.

### L2 agent

Lagrange's theorem gives a strong limit on the subgroups of a finite group. The order of each subgroup is a divisor of the order of the group.

The idea of the proof is to count cosets:

1. The left cosets $aH$ of $H$ partition $G$.
2. Each left coset has exactly $|H|$ elements, because $h \mapsto ah$ is a bijection from $H$ to $aH$.
3. If there are $k$ cosets, then $|G| = k\,|H|$.

The number $k$ is the index $[G : H]$. Thus $|G| = [G : H] \cdot |H|$.

A worked example with $G = \mathbb{Z}/12\mathbb{Z}$ and $H = \{0, 4, 8\}$:

1. $H$ is a subgroup of order $3$.
2. The cosets are $H$, $1 + H = \{1, 5, 9\}$, $2 + H = \{2, 6, 10\}$ and $3 + H = \{3, 7, 11\}$.
3. There are $4$ cosets of $3$ elements each, and $4 \cdot 3 = 12$.

Two standard consequences are that the order of each element divides $|G|$, and that a group of prime order is cyclic.

A common error is to believe the converse. A divisor $d$ of $|G|$ need not be the order of a subgroup. The alternating group $A_4$ has order $12$, but it has no subgroup of order $6$. The converse is true for some classes of groups, for example for cyclic groups and for groups of prime power order.

The theorem needs $G$ to be finite. For an infinite group the index can still be finite, as for $2\mathbb{Z}$ in $\mathbb{Z}$, but the orders are not numbers.
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# Feigenbaum–Cvitanović functional equation

## Definition

The *Feigenbaum–Cvitanović functional equation* is the [[functional equation|SRAN4EKWS0]]
$$g(x) = -\alpha \, g\big(g(x/\alpha)\big) \quad \text{for all } x \in [-1, 1],$$
in which the unknowns are a real number $\alpha > 1$ and a [[function|0EPAT51DXQ]] $g \colon [-1, 1] \to [-1, 1]$ such that:

1. $g$ is even, that is, $g(-x) = g(x)$ for all $x \in [-1, 1]$.
2. $g(0) = 1$.
3. $g$ is twice differentiable at $0$ with $g''(0) < 0$, so that $0$ is a quadratic maximum.

## Remarks

Setting $x = 0$ in the equation gives $1 = -\alpha \, g(1)$, so any solution has $\alpha = -1/g(1)$.
Lanford proved, with computer help, that there is a real-analytic solution, and its constant is the [[Feigenbaum constant|F1EHT2CPHC]] $\alpha = 2.502907875\ldots$.
For that solution $g(1) = -1/\alpha \approx -0.3995$.

Non-example: a constant function $g = c$ does not solve the equation together with clause 2, since the equation gives $c = -\alpha c$, so $c = 0$, while clause 2 asks $g(0) = 1$.

The right side is $T(g)$ for the renormalization, or period-doubling, operator $T(g)(x) = -\alpha \, g(g(x/\alpha))$.
So the solution is a fixed point of $T$.
The other Feigenbaum constant $\delta \approx 4.6692$ is the only eigenvalue of modulus greater than $1$ of the derivative of $T$ at this fixed point.
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# Automorphic form

## Summary

An automorphic form is a function on a [[Lie group|7JBRGCVJRB]] $G$ that is invariant under a discrete subgroup $\Gamma$.
It must also satisfy strong finiteness and growth conditions.
[[Modular forms|GWKN5VG78A]] and [[Maass wave forms|3R959YSNYP]] give the basic examples, for the group $\mathrm{SL}_2(\R)$ and $\Gamma = \mathrm{SL}_2(\Z)$ or a congruence subgroup.
Automorphic forms carry deep arithmetic information, and they are one side of the correspondences predicted by the [[Langlands program|J9T2M084W5]].

## Explanations

### L1 human

A periodic function on the line repeats under the group of integer shifts.
An automorphic form is the same idea for a much richer group of symmetries.
An example is the group of real $2 \times 2$ matrices of determinant one, with its subgroup of integer matrices.
Because so many symmetries are imposed, only special functions survive, and the space of such functions with given extra conditions is finite-dimensional.
For example, the discriminant function $\Delta(z) = q \prod_{n \ge 1} (1 - q^n)^{24}$ with $q = e^{2\pi i z}$ is a modular form, and so gives an automorphic form on $\mathrm{SL}_2(\R)$.
Its coefficients encode facts about primes, which is why number theorists care about these functions.
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# Thomson problem

## Problem

Let $N \ge 2$ be an integer and let $S^2 \subset \R^3$ be the unit sphere.
Find the minimum of the energy
$$E(x_1, \dots, x_N) = \sum_{1 \le i < j \le N} \frac{1}{\abs{x_i - x_j}}$$
over all $N$-tuples of distinct points $x_1, \dots, x_N \in S^2$, where $\abs{x_i - x_j}$ is the [[Euclidean distance|0NPPGHQNXQ]].
Find also the configurations that attain it.

## Remarks

The energy is the electrostatic potential energy of $N$ equal point charges that repel by Coulomb's law and are confined to the sphere.
Thomson posed the question in 1904 in connection with his model of the atom.
By compactness a minimizer exists for each $N$.
Rigorous solutions are known for $N = 2, 3, 4, 5, 6$ and $12$.
They are two antipodal points, an equilateral triangle on a great circle, the regular tetrahedron, the triangular bipyramid, the regular octahedron and the regular icosahedron.
The case $N = 5$ was proved by Richard Schwartz with computer assistance.
For larger $N$ only numerical candidates are known.
The minimizers need not maximize the minimal distance, so the Thomson problem differs from the [[Tammes problem|9WD2TZ4ZB5]] in general.

## Explanations

### L1 human

Put $N$ electrons on the surface of a ball and let them push each other apart.
They settle in a pattern of lowest energy, and the problem asks which pattern this is.
For a few small numbers the answer is a regular shape, but in general it is unknown.
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# Multialgebra

## Summary

A multialgebra is a set with multivalued operations.
Each operation sends a tuple of elements to a nonempty set of elements, not to a single element.
It generalises the [[algebraic structure|QJGPT2W0FT]] of universal algebra, where each value is one element.
Multialgebras include the [[hyperstructures|1HJF40J5AK]] such as hypergroups, and they appear when one passes to quotients, to sets of cosets, or to relations instead of functions.

## Explanations

### L1 human

An ordinary operation, like addition, gives exactly one answer.
A multivalued operation may give several possible answers at once.
For example, on the real numbers let the "combination" of $x$ and $y$ be every number between them.
Then $1$ combined with $3$ is the whole interval $[1, 3]$.
A multialgebra is a set with one or more operations of this kind, of any number of arguments.
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# Deformation of complex structure

## Definition

Let $X$ be a compact [[complex manifold|JSGWAN2RJV]].
A *complex analytic family* of compact complex manifolds is a [[holomorphic map|3MQTWKBBBT]] $\varpi \colon \mathcal{X} \to B$ between connected complex manifolds such that:

1. $\varpi$ is [[proper|15XEBGNGHC]];
2. $\varpi$ is a submersion: its differential is surjective at every point of $\mathcal{X}$.

Then each fibre $X_t = \varpi^{-1}(t)$ is a compact complex manifold.
A *deformation* of $X$ is such a family together with a point $0 \in B$ and a [[biholomorphism|FX92PP3HZR]] $X \to X_0$.

## Remarks

Two deformations of $X$ over the same pointed base are isomorphic if there is a biholomorphism of the total spaces over $B$ compatible with the identifications of $X_0$ with $X$.
By Ehresmann's theorem, conditions 1 and 2 imply that $\varpi$ is a smooth fiber bundle, so all fibres are diffeomorphic to $X$.
The fibres need not be biholomorphic to each other.

Example: let $\mathbb{H}$ be the upper half-plane and let $\Z^2$ act on $\C \times \mathbb{H}$ by $(m, n) \cdot (z, \tau) = (z + m + n\tau, \tau)$.
The quotient $\mathcal{X}$ is a complex manifold, and the projection $\mathcal{X} \to \mathbb{H}$ is a proper submersion with fibre $\C / (\Z + \tau\Z)$ over $\tau$.
So it is a deformation of the elliptic curve $\C/(\Z + i\Z)$ over $(\mathbb{H}, i)$.

Non-example: remove one point from the total space $\mathcal{X}$ of the previous family.
The restricted projection is still a holomorphic submersion, but it is not proper, since one fibre is now a punctured torus, which is not compact.
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# Milnor fibration theorem

## Statement

Let $f \colon \C^{n+1} \to \C$ be a nonconstant polynomial with $f(0) = 0$, where $n \ge 0$.
For $\varepsilon > 0$ let $S_\varepsilon$ be the [[sphere|P9CM4GQ2T0]] of radius $\varepsilon$ about $0$ in $\C^{n+1}$, and let $K = f^{-1}(0) \cap S_\varepsilon$.

1. For all sufficiently small $\varepsilon > 0$, the map
$$\phi = \frac{f}{\abs{f}} \colon S_\varepsilon \setminus K \to S^1$$
is a smooth locally trivial [[fibration|CX4M8K61AB]].
2. Suppose moreover that $0$ is an isolated [[critical point|0YTCMFB4HC]] of $f$, with Milnor number
$$\mu = \dim_\C \C\{z_0, \ldots, z_n\} \big/ \left( \frac{\partial f}{\partial z_0}, \ldots, \frac{\partial f}{\partial z_n} \right).$$
Then each fibre of $\phi$ has the homotopy type of a [[wedge|5JSWM1HBX2]] of $\mu$ spheres of dimension $n$.

## Remarks

The fibration is the Milnor fibration, its fibre is the Milnor fibre, and $K$ is the link of the singularity.
In the isolated case $K$ is a smooth manifold of real dimension $2n - 1$, and Milnor showed that it is $(n-2)$-connected.
Example: for $f = z_0^2 + z_1^3$ with $n = 1$, the link $K$ in $S^3$ is the trefoil knot, and $\mu = 1 \cdot 2 = 2$.
The Milnor fibre is a punctured torus, which has the homotopy type of a wedge of two circles.
For $f = z_0^2 + z_1^2$ the link is the Hopf link and the fibre is an annulus, homotopic to one circle, in agreement with $\mu = 1$.
The theorem holds equally for germs of holomorphic functions at $0$.
Milnor proved it in his book *Singular points of complex hypersurfaces*; the proof uses a curve selection lemma from real algebraic geometry.

## Explanations

### L1 human

Take a complex polynomial with a singular point at the origin and look at it on a tiny sphere around that point.
The zero set cuts the sphere in a knot or link, such as the trefoil for $z_0^2 + z_1^3$.
The rest of the sphere is swept out by a circle's worth of pieces, one for each argument of $f$.
All of them have the same shape, and all are bounded by the link.
The theorem says that these pieces are as simple as possible up to homotopy: each is a bouquet of spheres, as many as the Milnor number.

### L2 human

The Milnor fibration is the basic topological structure attached to an isolated hypersurface singularity.
The monodromy of the fibration acts on the middle homology of the fibre, a free abelian group of rank $\mu$.
Its eigenvalues and the Seifert form carry much of the topology of the singularity.
For $n = 1$ the link is an algebraic knot or link in $S^3$, and the fibre is a Seifert surface, so plane curve singularities give fibred links.
The Milnor number appearing in the theorem is also the dimension of the local algebra used in [[finite determinacy|EGPP69XA4F]] and the classification of [[simple critical points|M5437FP73G]].
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# Internal language

## Definition

Let $\mathcal{E}$ be a [[topos|6837RS7N82]] with [[subobject classifier|NKP5TPQ79X]] $\top \colon 1 \to \Omega$, and write $PX$ for the [[exponential object|RHWD8DY41R]] $\Omega^X$.
The *internal language* (Mitchell–Bénabou language) of $\mathcal{E}$ is the typed language described as follows.

1. The types are the objects of $\mathcal{E}$; the type of a [[product|S5744QZV4N]] $X \times Y$ is the product type, and $PX$ is the power type of $X$.
2. There are variables of each type, and each morphism $f \colon X \to Y$ is a function symbol from type $X$ to type $Y$.
3. Terms are built from variables by function symbols and by pairing $\langle \sigma, \tau \rangle$.
For terms $\sigma, \tau$ of the same type, $\sigma = \tau$ is a term of type $\Omega$.
For $\sigma$ of type $X$ and $\tau$ of type $PX$, the membership $\sigma \in \tau$ is a term of type $\Omega$.
For a term $\varphi$ of type $\Omega$, the comprehension $\set{x \in X \mid \varphi}$ is a term of type $PX$.
4. A *formula* is a term of type $\Omega$.

A term $\tau$ of type $Y$ whose free variables are $x_1 : X_1, \dots, x_n : X_n$ is interpreted as a morphism $X_1 \times \cdots \times X_n \to Y$.
Variables are interpreted by projections, function symbols by composition, $\sigma = \tau$ by the characteristic morphism of the diagonal $Y \to Y \times Y$, and $\sigma \in \tau$ by the evaluation morphism $X \times PX \to \Omega$.
A formula $\varphi$ is *valid* in $\mathcal{E}$ when its interpretation equals $\top$ composed with the unique morphism $X_1 \times \cdots \times X_n \to 1$.

## Remarks

The connectives $\wedge, \vee, \Rightarrow, \neg$ and the quantifiers $\forall, \exists$ are definable from equality, membership and comprehension; on subobjects they act as the Heyting algebra operations and as the adjoints to [[pullback|Q05K04QJ71]].
The valid formulas are closed under the rules of higher-order intuitionistic logic, but not in general under the law of excluded middle.
Example: in $\mathbf{Set}$, a formula is valid exactly when it is true for all values of its variables in the usual sense.
Example: a morphism $f \colon X \to Y$ is a monomorphism exactly when the formula $\forall x, x' \in X\, (f(x) = f(x') \Rightarrow x = x')$ is valid.
Non-example: in the topos $\mathrm{Sh}(\R)$ of sheaves on the real line, the formula $\forall p \in \Omega\, (p \vee \neg p)$ is not valid, so excluded middle is not part of the internal logic.
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# Erdős–Mordell inequality

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane and let $P$ be a point inside $ABC$ or on its boundary.
Let $p$, $q$, $r$ be the distances from $P$ to the lines $BC$, $CA$, $AB$.
Then
$$PA + PB + PC \ge 2(p + q + r).$$
Equality holds if and only if $ABC$ is equilateral and $P$ is its centre.

## Remarks

Write $a = BC$, $b = CA$ and $c = AB$.
One proof reflects $P$ in the bisector of the angle at $A$ and compares areas; this gives $a \cdot PA \ge b r + c q$.
Adding the three such bounds and using $t + 1/t \ge 2$ for $t > 0$ gives the inequality.
Example: let $ABC$ be equilateral with side $2$ and let $P$ be its centre.
Each distance to a vertex is $2/\sqrt 3$ and each distance to a side is $1/\sqrt 3$, so both sides equal $2\sqrt 3$.
The hypothesis that $P$ lies in the closed triangle is needed.
For some points far outside the triangle, the distances to the side lines are large compared to the distances to the vertices, and the inequality fails.
Erdős posed the inequality as a problem, and Mordell and Barrow gave the first proof.

## Explanations

### L1 human

Stand at a point inside a triangle.
Add up your distances to the three corners, then add up your distances to the three sides.
The first sum is always at least twice the second.
Only the centre of an equilateral triangle makes the two sides equal.

### L2 human

The inequality is a classic of olympiad geometry and a model for many inequalities that compare vertex distances with side distances.
Its proofs use [[Euclidean distance|0NPPGHQNXQ]] and area arguments, or the law of cosines combined with the inequality between arithmetic and geometric means.
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# Fourier transform of radial functions

## Statement

Let $n \ge 2$ and let $f \in L^1(\R^n)$ be radial, that is, $f(x) = f_0(\abs{x})$ for a measurable function $f_0$ on $(0, \infty)$ with $\int_0^\infty \abs{f_0(r)}\, r^{n-1}\, dr < \infty$.
Let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\, dx$ be its [[Fourier transform|DPVM34ADZV]], and let $J_\nu$ denote the Bessel function of the first kind of order $\nu$.
Then $\hat f$ is radial, and for every $\xi \ne 0$, with $\rho = \abs{\xi}$,
$$\hat f(\xi) = 2\pi\, \rho^{1 - n/2} \int_0^\infty f_0(r)\, J_{n/2 - 1}(2\pi r \rho)\, r^{n/2}\, dr.$$

## Remarks

The transform commutes with [[rotations|0HFREYRSZP]]: if $R$ is an orthogonal matrix, then the transform of $f \circ R$ is $\hat f \circ R$.
So the transform of a radial function is radial.
To compute it, write the integral in polar coordinates $x = r \omega$ and use the formula $\int_{S^{n-1}} e^{-2\pi i r \rho\, \omega \cdot e_1}\, d\sigma(\omega) = 2\pi (r\rho)^{1 - n/2} J_{n/2-1}(2\pi r \rho)$ for the surface measure $\sigma$ on the unit sphere.
The integral converges absolutely, because $z^{-\nu} J_\nu(z)$ is bounded on $(0, \infty)$ for $\nu \ge -\frac12$ and the hypothesis on $f_0$ holds.

Example: for $n = 3$, $J_{1/2}(z) = \sqrt{2/(\pi z)}\, \sin z$, and the formula becomes
$$\hat f(\xi) = \frac{2}{\rho} \int_0^\infty f_0(r)\, r \sin(2\pi r \rho)\, dr.$$
For $n = 1$ the same formula, with $J_{-1/2}(z) = \sqrt{2/(\pi z)} \cos z$, reduces to the cosine transform of an even function.

Up to the factors of $\rho$ and $r$, the one-dimensional integral is a Hankel transform of order $n/2 - 1$.

## Explanations

### L1 human

A radial function looks the same in every direction, like the ripples around a stone thrown into water.
Its Fourier transform is radial too, so it is a function of one variable.
The formula computes it by a single integral over the radius, in which a Bessel function replaces the sine and cosine of the one-dimensional case.

### L2 human

This reduction is the main tool for radial problems on $\R^n$.
The transforms of the indicator of a ball, of the surface measure on a sphere and of the Bochner–Riesz multipliers are all computed this way.
The decay of Bessel functions, $J_\nu(z) = O(z^{-1/2})$ as $z \to \infty$, has a direct consequence.
The Fourier transform of the surface measure on a sphere decays like $\abs{\xi}^{-(n-1)/2}$, a fact at the heart of restriction theory.
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# Associative algebra

## Summary

An associative algebra over a commutative ring $K$ is a [[ring|0B1261B5ED]] whose elements can also be multiplied by scalars from $K$.
The scalar product is compatible with the ring product.
Matrix algebras, polynomial algebras, group algebras and endomorphism algebras of vector spaces are the standard examples.
The notion puts ring theory and linear algebra in one frame, so that dimension and bases are available when $K$ is a field.
A [[commutative algebra over a ring|DPBJ04JMH8]] is the special case where the ring product is commutative.

## Explanations

### L1 human

An algebra is a vector space, or more generally a module, in which vectors can also be multiplied with each other.
The multiplication is associative and works well with scaling: scaling a product is the same as scaling one factor.
The $n \times n$ real matrices form an algebra over $\R$: they can be added, multiplied, and scaled by real numbers.
The complex numbers are a $2$-dimensional algebra over $\R$, with basis $1, i$ and the rule $i^2 = -1$.
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# Equilibrium of a difference equation

## Definition

Let $X$ be a set and let $k \ge 1$.
Let $F \colon X^k \to X$ be a function, and consider the autonomous [[recurrence|07TZV87Y0T]]
$$x_{n+1} = F(x_n, x_{n-1}, \dots, x_{n-k+1}) \qquad (n \ge k - 1).$$
A point $x^* \in X$ is an *equilibrium* (or *equilibrium point*) of this difference equation if $F(x^*, x^*, \dots, x^*) = x^*$.
For $k = 1$, this says that $x^*$ is a [[fixed point|W46BP9W06H]] of $F$.

## Remarks

The point $x^*$ is an equilibrium if and only if the constant sequence $x_n = x^*$ is a solution.
Example: for the logistic recurrence $x_{n+1} = r x_n (1 - x_n)$ on $\R$ with $r \ne 0$, the equation $r x (1 - x) = x$ gives the equilibria $0$ and $1 - 1/r$.
For $r = 2$, check: $2 \cdot \tfrac12 \cdot \tfrac12 = \tfrac12$.
Non-example: $x_{n+1} = x_n + 1$ on $\R$ has no equilibrium, since $x + 1 \ne x$ for every $x$.
For a non-autonomous equation $x_{n+1} = F(n, x_n)$, an equilibrium is a point $x^*$ with $F(n, x^*) = x^*$ for all $n$.
Other names are fixed point, steady state and stationary solution.
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# Strassen algorithm

## Summary

The Strassen algorithm multiplies two $n \times n$ matrices with $O(n^{\log_2 7})$ arithmetic operations, where $\log_2 7 \approx 2.807$.
It rests on a way to multiply $2 \times 2$ matrices with $7$ multiplications instead of $8$, applied recursively to blocks.
Published by Volker Strassen in 1969, it was the first method faster than the usual $O(n^3)$ [[matrix multiplication|032Q9FB8MT]].
It started the search for the [[matrix multiplication exponent|X0G02CGQ5J]].

## Explanations

### L1 human

The schoolbook product of two $2 \times 2$ matrices uses $8$ multiplications of entries.
Strassen found clever sums and differences of entries so that $7$ products are enough, at the cost of more additions.
Because the entries can themselves be blocks, the trick can be applied again inside each block.
Saving one multiplication out of eight at every level adds up: for large matrices the total work grows like $n^{2.81}$ instead of $n^3$.
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# Hilbert–Schmidt expansion theorem

## Statement

Let $a < b$ and let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]] that is [[Hermitian|2ZB3NFZ94T]]: $K(y, x) = \overline{K(x, y)}$ for all $x, y$.
Let $(\varphi_n)_{n \in N}$ be an orthonormal family of [[eigenfunctions|4NSACSWRJZ]] of $T_K \varphi = \int_a^b K(\cdot, y) \varphi(y) \, dy$, with real nonzero eigenvalues $\mu_n$.
Assume that, for each nonzero eigenvalue, the $\varphi_n$ with that eigenvalue form a basis of its eigenspace.
These eigenfunctions are continuous, since $\varphi_n = \mu_n^{-1} T_K \varphi_n$.
Let $h \in L^2[a, b]$ and $f = T_K h$, that is,
$$f(x) = \int_a^b K(x, y) \, h(y) \, dy.$$
Then for every $x \in [a, b]$,
$$f(x) = \sum_{n \in N} \ip{f}{\varphi_n} \varphi_n(x) = \sum_{n \in N} \mu_n \ip{h}{\varphi_n} \varphi_n(x),$$
where $\ip{u}{v} = \int_a^b u \overline{v}$, and the series converges absolutely and [[uniformly|08VZ57PT26]] on $[a, b]$.

## Remarks

For fixed $x$, the numbers $\mu_n \overline{\varphi_n(x)}$ are the Fourier coefficients of the function $y \mapsto \overline{K(x, y)}$, so [[Bessel's inequality|0RBT5PB74K]] gives $\sum_n \mu_n^2 \abs{\varphi_n(x)}^2 \le \int_a^b \abs{K(x, y)}^2 \, dy \le C$ for a constant $C$ independent of $x$.
The Cauchy–Schwarz inequality then bounds the tail $\sum_{n \ge m} \abs{\mu_n \ip{h}{\varphi_n} \varphi_n(x)}$ by $\sqrt{C} \left(\sum_{n \ge m} \abs{\ip{h}{\varphi_n}}^2\right)^{1/2}$, which tends to $0$ uniformly in $x$.
By the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]], the series converges to $f$ in $L^2$; the uniform limit is continuous and so is $f$, hence they agree everywhere.

Example: for $K(x, y) = \min(x, y)$ on $[0, 1]$, the eigenfunctions are $\sqrt{2} \sin\left((n - \tfrac12) \pi x\right)$.
Every function $f = T_K h$ with $h \in L^2[0, 1]$ is continuous with $f(0) = 0$, and its expansion in these sines converges uniformly on $[0, 1]$.

The hypothesis $f = T_K h$ matters.
For a general function $f \in L^2[a, b]$, the eigenfunction series converges only in $L^2$.
Its sum is the orthogonal projection of $f$ onto the closed span of the $\varphi_n$, which may differ from $f$.

## Explanations

### L1 human

The eigenfunctions of a symmetric kernel are like the axes of an ellipse.
Any function produced by the integral operator can be written as a sum of these eigenfunctions.
The theorem says that this sum converges in the strongest usual sense, uniformly at every point.

### L2 human

This is the classical form of the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]], adapted to continuous kernels.
It is used to solve the Fredholm equation of the second kind with a symmetric kernel: the solution of $\varphi - \lambda T_K \varphi = f$, for $\lambda$ not a characteristic value, is $\varphi = f + \lambda \sum_n \frac{\mu_n \ip{f}{\varphi_n}}{1 - \lambda \mu_n} \varphi_n$.
Through Green's functions it gives the uniform convergence of eigenfunction expansions for Sturm–Liouville problems.
</document>

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# Nilradical is the intersection of the prime ideals

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]].
Then the [[nilradical|QQDN6EYQNJ]] of $R$ is the intersection of all [[prime ideals|06HC6HK0Z8]] of $R$:
$$N(R) = \bigcap_{P \text{ prime}} P.$$

## Remarks

One inclusion is direct: if $r^n = 0$, then $r^n$ lies in every prime ideal $P$, so $r \in P$.
For the other inclusion, let $f$ be an element that is not [[nilpotent|0N9R5NN6TW]].
Then the set $S = \set{1, f, f^2, \ldots}$ does not contain $0$, and [[Zorn's lemma|04RXTQA5ZA]] gives an ideal $P$ that is maximal among the ideals disjoint from $S$.
Such an ideal is prime, and $f \notin P$.
Example: in $\Z/12\Z$ the prime ideals are $2\Z/12\Z$ and $3\Z/12\Z$, and their intersection $6\Z/12\Z = \set{0, 6}$ is the set of nilpotent elements.
If $R$ is the zero ring, it has no prime ideals, the empty intersection is $R$, and every element of $R$ is nilpotent, so the statement still holds.
Applied to $R/I$, the statement shows that the [[radical|437YRCNYRJ]] of an ideal $I$ is the intersection of the prime ideals that contain $I$.

## Explanations

### L1 human

An element of a commutative ring is nilpotent exactly when it lies in every prime ideal.
Being nilpotent is an algebraic property: some power is zero.
Lying in every prime ideal is a geometric property: the element, seen as a function on the [[spectrum|FB45BHHA2J]], vanishes at every point.
The theorem says that these two properties agree.

### L2 human

The theorem is the starting point of the link between ideals and closed subsets of the spectrum.
It gives $\sqrt{I} = \bigcap_{P \supseteq I} P$, so two ideals have the same zero locus in $\operatorname{Spec} R$ exactly when they have the same radical.
The proof is a standard use of Zorn's lemma, in the same style as the [[existence of maximal ideals|02HJYT53F7]].
Over a polynomial ring over an algebraically closed field, [[Hilbert's Nullstellensatz|D85VZWS9QK]] sharpens it: maximal ideals alone already suffice.
Compare the [[Jacobson radical|XA0DCNH7WY]], which is the intersection of the maximal ideals only and contains the nilradical.
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# Weierstrass zeta and sigma functions

## Definition

Let $\omega_1, \omega_2 \in \C$ be linearly independent over $\R$, and let $\Lambda = \Z\omega_1 + \Z\omega_2$ be the [[lattice|WN9V7WSJKF]] they span.
The *Weierstrass sigma function* of $\Lambda$ is the [[infinite product|JYV1Y8XX16]]
$$\sigma(z) = z \prod_{\omega \in \Lambda \setminus \set{0}} \left(1 - \frac{z}{\omega}\right) \exp\!\left(\frac{z}{\omega} + \frac{z^2}{2\omega^2}\right), \qquad z \in \C.$$
The *Weierstrass zeta function* of $\Lambda$ is the meromorphic function
$$\zeta(z) = \frac{\sigma'(z)}{\sigma(z)} = \frac{1}{z} + \sum_{\omega \in \Lambda \setminus \set{0}} \left(\frac{1}{z - \omega} + \frac{1}{\omega} + \frac{z}{\omega^2}\right), \qquad z \in \C \setminus \Lambda.$$

## Remarks

The product converges absolutely and locally uniformly, because each factor is $1 + O(\abs{z/\omega}^3)$ and $\sum_{\omega \ne 0}\abs{\omega}^{-3} < \infty$.
So $\sigma$ is an [[entire function|0G4F0XB9X2]], odd, with simple zeros exactly at the points of $\Lambda$.
Also $\zeta'(z) = -\wp(z)$, where $\wp$ is the [[Weierstrass elliptic function|361NAK68QK]] of $\Lambda$.
Since $\wp$ is periodic, $\eta_j = \zeta(z + \omega_j) - \zeta(z)$ is a constant, the *quasi-period*, for $j = 1, 2$.
Then $\sigma(z + \omega_j) = -\exp\big(\eta_j(z + \omega_j/2)\big)\,\sigma(z)$.
If $\operatorname{Im}(\omega_2/\omega_1) > 0$, the Legendre relation $\eta_1\omega_2 - \eta_2\omega_1 = 2\pi i$ holds; it follows by integrating $\zeta$ around a fundamental parallelogram, where $\zeta$ has one simple pole of residue $1$.
Example: replace $\Lambda$ by the one-dimensional group $\Z\pi$ in the same formulas.
Since $\sum_{n \ne 0} 1/(n\pi)^2 = 1/3$, they give $\zeta(z) = \cot z + z/3$ and $\sigma(z) = e^{z^2/6}\sin z$.
This is the limit of the lattice case when $\omega_1 = \pi$ is fixed and $\omega_2 \to i\infty$.
Non-example: $\zeta$ is not an elliptic function.
A nonconstant elliptic function cannot have only one simple pole in a period parallelogram, and indeed the Legendre relation shows that $\eta_1, \eta_2$ are not both $0$.
</document>

<document source="https://tamnd.github.io/math2/node/3D1J3AXS69.md" tag="3D1J3AXS69" kind="definition">
# Rigidity of frameworks

## Definition

Let $d \ge 1$ and let $G = (V, E)$ be a finite [[graph|0JD5C82QMB]].
A *framework* in $\R^d$ is a pair $(G, p)$ with $p \colon V \to \R^d$.
Write $\norm{\cdot}$ for the [[Euclidean distance|0C95TG69SG]].
The framework $(G, p)$ is *rigid* if there is $\varepsilon > 0$ with the following property.
Every $q \colon V \to \R^d$ that satisfies both conditions below also satisfies $\norm{q(u) - q(v)} = \norm{p(u) - p(v)}$ for all $u, v \in V$.

1. $\norm{q(v) - p(v)} < \varepsilon$ for every $v \in V$.
2. $\norm{q(u) - q(v)} = \norm{p(u) - p(v)}$ for every edge $uv \in E$.

A framework that is not rigid is called *flexible*.

## Remarks

So a framework is rigid when every nearby placement with the same bar lengths is congruent to it.
Example: a triangle $K_3$ placed at three non-collinear points of $\R^2$ is rigid.
Its three bar lengths are all the pairwise distances, so condition 2 already gives the conclusion.

Non-example: the cycle on four vertices placed at the corners of a unit square in $\R^2$ is flexible.
For small $t > 0$, the rhombus with corners $(0, 0)$, $(1, 0)$, $(1 + \sin t, \cos t)$ and $(\sin t, \cos t)$ keeps all four bar lengths equal to $1$.
Its diagonal from $(0, 0)$ to $(1 + \sin t, \cos t)$ has length different from $\sqrt{2}$.

An equivalent form: every continuous motion of the vertices that keeps all bar lengths is the restriction of a family of isometries of $\R^d$.
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# Gelfand–Mazur theorem

## Statement

Let $A$ be a complex [[Banach algebra|9KYWN94QDN]] with unit $1 \ne 0$.
Suppose every nonzero element of $A$ is invertible, so that $A$ is a [[division ring|0WZXK05ME7]].
Then every $a \in A$ equals $\lambda 1$ for a unique $\lambda \in \C$, and the map $\C \to A$, $\lambda \mapsto \lambda 1$, is an isomorphism of algebras.

## Remarks

Proof idea: the spectrum $\set{\lambda \in \C : a - \lambda 1 \text{ is not invertible}}$ of each $a \in A$ is nonempty.
Suppose it were empty, and let $\varphi$ be a continuous linear functional on $A$.
Then $\lambda \mapsto \varphi((a - \lambda 1)^{-1})$ would be a bounded entire function that tends to $0$, hence zero by Liouville's theorem.
As this holds for every $\varphi$, the Hahn–Banach theorem would give $(a - \lambda 1)^{-1} = 0$, a contradiction.
So some $a - \lambda 1$ is not invertible; in a division algebra this forces $a - \lambda 1 = 0$.
The complex scalars are needed.
The quaternions $\mathbb{H}$ with the Euclidean norm on $\R^4$ form a real Banach algebra in which every nonzero element is invertible, but $\mathbb{H}$ is not $\R$ or $\C$.
Consequence: a field that contains $\C$ and has an absolute value extending the usual one on $\C$, complete for it, is $\C$ itself.

## Explanations

### L1 human

A Banach algebra is a space where one can add, multiply, scale by complex numbers and measure size, and where limits exist.
The theorem says that if such an algebra lets you divide by every nonzero element, it is nothing more than the complex numbers.
So there is no room for a bigger "complex number system" with a good notion of size.

### L2 human

The theorem is a basic step in the theory of commutative Banach algebras.
If $A$ is a commutative unital complex Banach algebra and $\mathfrak{m}$ a maximal ideal, then $\mathfrak{m}$ is closed and $A/\mathfrak{m}$ is a Banach division algebra, hence $\C$.
This gives the characters $A \to \C$ that underlie the Gelfand transform.
Some proofs of Ostrowski's theorem, that a field complete for an Archimedean absolute value is $\R$ or $\C$, use the Gelfand–Mazur theorem.
</document>

<document source="https://tamnd.github.io/math2/node/3D88622PBQ.md" tag="3D88622PBQ" kind="concept">
# Doléans-Dade exponential

## Summary

The Doléans-Dade exponential, or stochastic exponential, $\mathcal{E}(X)$ of a [[semimartingale|4EJB6E2AZK]] $X$ is the solution of the equation $dZ = Z \, dX$ with $Z_0 = 1$.
It plays the role of the exponential function in [[Itô calculus|JA95NH206E]].
For a continuous semimartingale with $X_0 = 0$ it equals $\exp(X - \tfrac12 [X])$.
It is the density process in the [[Girsanov theorem|Z76HD1BZ29]].

## Explanations

### L1 human

In ordinary calculus, the solution of $dz = z \, dx$ with $z(0) = 1$ is $z = e^{x}$.
For a random driver such as Brownian motion, $e^{X}$ grows a little too fast, because Itô's lemma adds a drift.
The stochastic exponential removes this drift by the factor $e^{-[X]/2}$.
For $X = \sigma B$ with a constant $\sigma$, it is $\exp(\sigma B_t - \tfrac12 \sigma^2 t)$, a positive martingale with mean $1$.
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<document source="https://tamnd.github.io/math2/node/3DBAMXA80B.md" tag="3DBAMXA80B" kind="concept">
# Bilateral Z-transform

## Summary

The bilateral Z-transform of a sequence $(x_n)_{n \in \Z}$ sums over all integers: $X(z) = \sum_{n \in \Z} x_n z^{-n}$.
It is the form of the [[Z-transform|5FY0JQGHYZ]] for signals that extend into the past as well as the future.
Its set of convergence is an annulus centred at $0$, possibly empty.
The same formula for $X(z)$ can come from different sequences on different annuli.
It is the discrete analogue of the two-sided Laplace transform.

## Explanations

### L1 human

The bilateral transform looks at a sequence in both directions of time.
Terms with $n \ge 0$ need $\abs{z}$ large to converge, and terms with $n < 0$ need $\abs{z}$ small.
So the transform lives on a ring between two circles.
For example, $x_n = 2^{-\abs{n}}$ has transform defined for $1/2 < \abs{z} < 2$.
</document>

<document source="https://tamnd.github.io/math2/node/3DGHC142FT.md" tag="3DGHC142FT" kind="statement">
# L2 convergence of Fourier series

## Statement

Let $\mathbb{T} = \R / 2\pi\Z$ and let $f \in L^2(\mathbb{T})$, the [[Lebesgue space|SASP14MA1Z]] of square integrable functions with the norm $\norm{g}_2 = \left( \frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{g(x)}^2 \, dx \right)^{1/2}$.
Let $\hat f(n)$ be the [[Fourier coefficients|5NN2YZZR3V]] of $f$ and let $S_N f(x) = \sum_{n=-N}^{N} \hat f(n) e^{inx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]].
Then
$$\lim_{N \to \infty} \norm{f - S_N f}_2 = 0 \qquad \text{and} \qquad \frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{f(x)}^2 \, dx = \sum_{n=-\infty}^{\infty} \abs{\hat f(n)}^2.$$

## Remarks

Both claims follow from the [[Fourier basis theorem|NXQX1KFMDB]], which says that the monomials $e_n$ form an orthonormal basis of $L^2(\mathbb{T})$.
$S_N f$ is the orthogonal projection of $f$ onto the span of $e_{-N}, \dots, e_N$, so $\norm{f - S_N f}_2^2 = \norm{f}_2^2 - \sum_{\abs{n} \le N} \abs{\hat f(n)}^2$.
Completeness of the basis makes the right side tend to $0$; the second formula is Parseval's identity for this basis.
Example: for $f(x) = x$ on $(-\pi, \pi)$, $\hat f(0) = 0$ and $\abs{\hat f(n)} = 1/\abs{n}$ for $n \ne 0$, and Parseval gives $\frac{\pi^2}{3} = 2 \sum_{n \ge 1} \frac{1}{n^2}$, which is the value $\frac{\pi^2}{6}$ of the [[Basel problem|0DMY75D6FV]].
Convergence in $L^2$ does not give convergence at each point; pointwise convergence almost everywhere for $L^2$ functions is the much deeper [[Carleson's theorem|FBH6KG484Z]].
The converse direction is the Riesz–Fischer theorem: every sequence $(c_n) \in \ell^2(\Z)$ is the sequence of Fourier coefficients of some $f \in L^2(\mathbb{T})$.

## Explanations

### L1 human

The finite sums of the Fourier series of a square integrable function get closer and closer to the function, when distance is measured by the average of the squared error.
The energy of the function, the average of $\abs{f}^2$, equals the sum of the energies of its frequency components.

### L2 human

This is the simplest and most robust convergence theorem for Fourier series.
It needs no smoothness, only square integrability, and it holds for every function in $L^2(\mathbb{T})$.
The analogous statement in $L^p(\mathbb{T})$ for $1 < p < \infty$ is true but harder; it rests on the boundedness of the conjugate function, a theorem of M. Riesz.
For $p = 1$ and in $C(\mathbb{T})$ with the supremum norm the analogous statement is false, because the Lebesgue constants of the Dirichlet kernel are unbounded.
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<document source="https://tamnd.github.io/math2/node/3DHNP5NEWH.md" tag="3DHNP5NEWH" kind="concept">
# Short-time Fourier transform

## Summary

The short-time Fourier transform of a function $f$ on $\R$ is a [[Fourier transform|DPVM34ADZV]] taken through a window.
One multiplies $f$ by a window $g$ shifted to a point $x$, and transforms the product.
It is a function of position $x$ and frequency $\omega$, and so it shows which frequencies are present near each time.
It is the basic tool of [[time–frequency analysis|GQFPVXEWK1]]; its squared modulus is the spectrogram.
For a window in $L^2$ it is an isometry up to the factor $\norm{g}_2$, so no information is lost.

## Explanations

### L1 human

Listen to a piece of music through a small moving window in time.
At each position of the window, ask which pitches are sounding: this is a Fourier transform of the windowed piece.
Collecting the answers for all positions gives a two-dimensional picture, time against pitch.
A narrow window shows well when notes start but blurs their pitch, and a wide window does the opposite.
</document>

<document source="https://tamnd.github.io/math2/node/3DJ46V21KH.md" tag="3DJ46V21KH" kind="example">
# Joukowsky transform

## Example

The Joukowsky transform is the map $J(z) = z + \dfrac{1}{z}$ on $\C \setminus \set{0}$.
It is [[holomorphic|0DG0Z72CQD]] with $J'(z) = 1 - z^{-2}$, which vanishes only at $z = \pm 1$.
So $J$ is a [[conformal map|0VHNKP1N3K]] near each point other than $0$ and $\pm 1$.

Circles go to ellipses.
For $z = r e^{i\theta}$ with $r > 1$,
$$J(z) = \left(r + \frac{1}{r}\right) \cos \theta + i \left(r - \frac{1}{r}\right) \sin \theta,$$
so the circle $\abs{z} = r$ goes onto the [[ellipse|0FF6CAQT2Y]] with semi-axes $r + 1/r$ and $r - 1/r$ and foci $\pm 2$.
For $r = 1$ one gets $J(e^{i\theta}) = 2 \cos \theta$, so the unit circle is folded onto the segment $[-2, 2]$.

The exterior of the unit disc is mapped bijectively onto $\C \setminus [-2, 2]$.
Check of injectivity: $J(z) - J(w) = (z - w)(1 - \frac{1}{zw})$, and $zw \neq 1$ when $\abs{z} > 1$ and $\abs{w} > 1$.
Check of surjectivity: for $c \in \C$, the two roots of $z^2 - cz + 1 = 0$ have product $1$.
If neither root has modulus greater than $1$, both have modulus $1$, and then $c = 2\cos\theta$ lies in $[-2, 2]$.

## Remarks

Circles that pass through $-1$ or $1$ and contain the other point go to curves with one sharp edge, the Joukowsky airfoils.
This is why the map is used to model the flow of air around a wing.
Some authors use $\frac{1}{2}(z + 1/z)$ instead, which only rescales the picture by $\frac{1}{2}$.

## Explanations

### L1 human

The Joukowsky map adds a number and its reciprocal.
Far from the origin $1/z$ is tiny, so the map is almost the identity there.
Near the unit circle the two terms are of the same size, and the circle is squashed flat onto a segment.
Circles a bit larger than the unit circle become ellipses, and suitably shifted circles become wing shapes.
</document>

<document source="https://tamnd.github.io/math2/node/3DJMFT0SQM.md" tag="3DJMFT0SQM" kind="definition">
# Similarity solution

## Definition

Let $n \ge 1$ and consider a [[partial differential equation|GC5BDZ6QBC]] for a function $u \colon \R^n \times (0, \infty) \to \R$.
Let $\alpha, \beta \in \R$, and for $\lambda > 0$ put $u_\lambda(x, t) = \lambda^{\alpha} u(\lambda^{\beta} x, \lambda t)$.
Assume that the equation is invariant under these scalings: if $u$ is a solution, then so is $u_\lambda$ for every $\lambda > 0$.
A solution $u$ is a *similarity solution*, or *self-similar solution*, with exponents $(\alpha, \beta)$ if $u_\lambda = u$ for every $\lambda > 0$.

## Remarks

Taking $\lambda = 1/t$ shows that $u$ is self-similar exactly when
$$u(x, t) = t^{-\alpha}\, v\!\left( \frac{x}{t^{\beta}} \right)$$
for a function $v$ on $\R^n$, namely $v = u(\cdot, 1)$.
Substituting this form turns the equation into an equation for $v$ in the variable $\eta = x / t^\beta$.

Example: the [[heat equation|VP2HN86VAG]] $u_t = \Delta u$ is invariant for $\beta = 1/2$ and every $\alpha$.
The heat kernel $\Phi(x, t) = (4\pi t)^{-n/2} e^{-\abs{x}^2 / (4t)}$ is self-similar with $\alpha = n/2$, $\beta = 1/2$.
Check: $\lambda^{n/2} \Phi(\lambda^{1/2} x, \lambda t) = \lambda^{n/2} (4\pi\lambda t)^{-n/2} e^{-\lambda \abs{x}^2/(4\lambda t)} = \Phi(x, t)$.

Non-example: $u(x, t) = e^{-t} \sin x$ solves $u_t = u_{xx}$ for $n = 1$, but it is not self-similar.
A self-similar solution would have zero set $\sqrt{t}\, Z$ at time $t$, with $Z$ the zero set of $v$.
But the zero set of $u(\cdot, t)$ is $\pi \Z$ for every $t$.
At $t = 1$ this forces $Z = \pi \Z$, and then at $t = 4$ the zero set would be $2 \pi \Z \ne \pi \Z$.

For a scalar conservation law $u_t + F(u)_x = 0$, the scalings with $\alpha = 0$, $\beta = 1$ are symmetries.
The corresponding similarity solutions $u = v(x/t)$ include the [[rarefaction waves|MFN4Q5SH8H]] and the solutions of the [[Riemann problem|GJWKX22VVE]].

## Explanations

### L1 human

A similarity solution looks the same at every time after the right change of scale.
Its whole history is encoded in the profile $v$ at time $1$.
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<document source="https://tamnd.github.io/math2/node/3DK4V3A10K.md" tag="3DK4V3A10K" kind="example">
# Hough transform

## Example

The Hough transform detects straight lines in a finite set of points $p_1, \dots, p_N$ of the plane, such as the edge pixels of an image.
Each line is written as $x \cos\theta + y \sin\theta = \rho$ with $\theta \in [0, \pi)$ and $\rho \in \R$.
A point $p_i = (x_i, y_i)$ lies on this line exactly when $\rho = x_i \cos\theta + y_i \sin\theta$.
So each point gives a sinusoidal curve in the $(\theta, \rho)$ plane, and the curves of collinear points meet at the parameters of their common line.
The transform is the count
$$H(\theta, \rho) = \#\set{i : x_i \cos\theta + y_i \sin\theta = \rho},$$
computed in practice on a grid of cells, and lines are found as peaks of $H$.

Check on three points: $(1, 0)$, $(0, 1)$ and $(2, -1)$ lie on the line $x + y = 1$.
For $\theta = \pi/4$ and $\rho = 1/\sqrt{2}$, each point satisfies $x \cos\theta + y \sin\theta = (x + y)/\sqrt{2} = 1/\sqrt{2}$.
So $H(\pi/4, 1/\sqrt{2}) = 3$.

The [[Radon transform|VJNZJ352PC]] integrates a function along each line, and $H$ counts the given points on each line.
So the Hough transform is a discrete analogue of the Radon transform, applied to an edge image.

## Explanations

### L1 human

Every point votes for all the lines that pass through it.
A line that passes through many points collects many votes.
After the vote, the lines with the most votes are the lines in the picture.
The method still works when some points are missing or extra noise points are present, which makes it popular in computer vision.
</document>

<document source="https://tamnd.github.io/math2/node/3DRCZF6J9J.md" tag="3DRCZF6J9J" kind="concept">
# Oscillatory solution of a difference equation

## Summary

A solution of a difference equation is oscillatory if it keeps changing sign, however far one goes along the sequence.
A solution that is eventually of one sign is called nonoscillatory.
Oscillation theory looks for conditions on the coefficients that force all solutions, or some solutions, to oscillate.
It is the discrete counterpart of the [[oscillation theory|ZH2S50NSZ6]] of differential equations, such as Sturm theory.

## Explanations

### L1 human

Watch the signs of the terms of a sequence.
If they settle down to always plus or always minus, the sequence does not oscillate.
If a sign change keeps happening again and again, it oscillates.
The solution $x_n = (-1)^n$ of $x_{n+1} = -x_n$ oscillates, while the solution $x_n = 2^n$ of $x_{n+1} = 2 x_n$ does not.
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<document source="https://tamnd.github.io/math2/node/3DRNRZEW9D.md" tag="3DRNRZEW9D" kind="counterexample">
# Knaster–Kuratowski fan

## Counterexample

Let $C$ be the [[Cantor set|0BXNDZFQME]] on the segment $[0, 1] \times \set{0}$ of the plane, and let $p = (1/2, 1/2)$.
Let $E \subseteq C$ be the countable set of end points of the open intervals removed in the construction of $C$.
For $c \in C$, let $L(c)$ be the closed segment from $(c, 0)$ to $p$.
Let $X_c$ be the set of points of $L(c)$ whose second coordinate is rational if $c \in E$, and irrational if $c \in C \setminus E$.
The *Knaster–Kuratowski fan* is $F = \bigcup_{c \in C} X_c \cup \set{p}$, with the topology of a subspace of the plane.
Then $F$ is [[connected|0V032VWTHD]], but $F \setminus \set{p}$ is [[totally disconnected|0RJSBR9GW1]].
So removing one point from a connected space can leave no connected subset with more than one point.

## Remarks

For two points $c < c'$ of $C$, pick $t$ in a removed interval between them.
The line through $p$ and $(t, 0)$ misses $F \setminus \set{p}$, and the two open half-planes it bounds split $F \setminus \set{p}$ into two open pieces.
So a connected subset of $F \setminus \set{p}$ lies in a single set $X_c$.
Each $X_c$ is homeomorphic to the rational or to the irrational points of an interval, which is totally disconnected.
The proof that $F$ is connected is harder; it uses the [[Baire category theorem|0A7KGAYHGV]] on the Cantor set.
The point $p$ is called a dispersion point of $F$.
The space is also called Cantor's leaky tent or Cantor's teepee.

## Explanations

### L1 human

Picture a tent whose poles are the segments from the points of the Cantor set up to a common top.
Each pole is full of holes, keeping only points at rational heights or only at irrational heights.
The whole tent hangs together in one piece, but if one removes the top point, it falls apart into dust: no two points stay in one connected piece.
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<document source="https://tamnd.github.io/math2/node/3DXGJTEQZH.md" tag="3DXGJTEQZH" kind="concept">
# Dirac operator

## Summary

A Dirac operator is a first-order differential operator $D$ whose square $D^2$ is a Laplace-type operator.
It is a formal square root of the [[Laplacian|0E0B6RA8K0]], made possible by coefficients in a [[Clifford algebra|A38B1QFK20]].
Dirac introduced such an operator to write a first-order relativistic equation for the electron, the [[Dirac equation|XNZX7VXXR2]].
In geometry, Dirac operators on [[spinor bundles|PYYQQKAZ0C]] are elliptic, and their indices are computed by the Atiyah–Singer index theorem.

## Explanations

### L1 human

The Laplacian $\partial_1^2 + \cdots + \partial_n^2$ is a sum of squares, and one may ask for a first-order operator whose square is it.
With ordinary numbers this is impossible beyond one variable, because the cross terms $2\partial_j\partial_k$ do not cancel.
Dirac's idea was to use matrices $\gamma_j$ whose products anticommute, so the cross terms cancel.
In one variable it is easy: $D = -i\, d/dx$ satisfies $D^2 = -d^2/dx^2$.
In the plane, the Cauchy–Riemann operators of complex analysis assemble into a Dirac operator.
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<document source="https://tamnd.github.io/math2/node/3E740YTT9V.md" tag="3E740YTT9V" kind="definition">
# Fejér kernel

## Definition

For an integer $n \ge 0$, the Dirichlet kernel is the function $D_n(x) = \sum_{k=-n}^{n} e^{ikx}$ for real $x$.
For an integer $N \ge 1$, the *Fejér kernel* $F_N$ is the [[Cesàro mean|0P0DJHBGBB]] of $D_0, \dots, D_{N-1}$:
$$F_N(x) = \frac{1}{N} \sum_{n=0}^{N-1} D_n(x) = \sum_{k=-(N-1)}^{N-1} \left(1 - \frac{\abs{k}}{N}\right) e^{ikx}.$$

## Remarks

For $x$ not a multiple of $2\pi$, the closed form is
$$F_N(x) = \frac{1}{N} \left( \frac{\sin(Nx/2)}{\sin(x/2)} \right)^2,$$
and $F_N(2\pi m) = N$ for integers $m$.
So $F_N \ge 0$, $F_N$ is $2\pi$-periodic, and $\frac{1}{2\pi} \int_{-\pi}^{\pi} F_N(x) \, dx = 1$, because only the term $k = 0$ has a nonzero integral.
For $\delta \le \abs{x} \le \pi$ one has $F_N(x) \le \frac{1}{N \sin^2(\delta/2)}$, which tends to $0$.

Example: $F_1 = D_0 = 1$, and $F_2(x) = \frac12 (1 + (1 + 2\cos x)) = 1 + \cos x$, which agrees with $\frac12 \left(\frac{\sin x}{\sin(x/2)}\right)^2 = 2\cos^2(x/2)$.

Non-example: the Dirichlet kernel $D_1(x) = 1 + 2\cos x$ is not a Fejér kernel, and it takes the negative value $-1$ at $x = \pi$.

For a $2\pi$-periodic integrable function $f$ with [[Fourier series|G12KXDHPPS]] partial sums $S_n f$, one has $\frac{1}{N} \sum_{n=0}^{N-1} S_n f(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} F_N(x - t) f(t) \, dt$.
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<document source="https://tamnd.github.io/math2/node/3E8ZG3NJYZ.md" tag="3E8ZG3NJYZ" kind="concept">
# Multivariate formal power series

## Summary

A formal power series in $X_1, \dots, X_n$ over a [[commutative ring|0310REJJY8]] $R$ is a possibly infinite sum $\sum a_\alpha X^\alpha$ over all monomials $X^\alpha$.
No convergence condition is imposed.
These series form the ring $R[\![X_1, \dots, X_n]\!]$, which extends the one-variable [[formal power series|033WGN00BF]] ring and contains the [[polynomial ring|07GNX8GCYP]] $R[X_1, \dots, X_n]$.
Over a field, this ring is the basic example of a [[complete local ring|G9Q61B8A7C]] and of a regular local ring of dimension $n$.

## Explanations

### L1 human

A polynomial in several variables is a finite sum of monomials such as $3x^2 y$.
A formal power series in several variables allows infinitely many monomials, like $1 + x + y + x^2 + xy + y^2 + \cdots$.
You add and multiply such series as you would polynomials, and every coefficient of a product only needs finitely many terms to compute.
The [[definition|KCBFS516DE]] makes this precise.
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<document source="https://tamnd.github.io/math2/node/3EC1CQKMK2.md" tag="3EC1CQKMK2" kind="definition">
# Regular measure

## Definition

Let $X$ be a topological space and let $\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.
The measure $\mu$ is *regular* if both of the following hold.

1. $\mu$ is [[inner regular|28E5Q5BM69]]: $\mu(A) = \sup \set{\mu(K) : K \subseteq A, \ K \text{ compact and Borel}}$ for every Borel set $A$.
2. $\mu$ is [[outer regular|33C1KEK3SC]]: $\mu(A) = \inf \set{\mu(U) : U \supseteq A, \ U \text{ open}}$ for every Borel set $A$.

## Remarks

Example: Lebesgue measure on the Borel sets of $\R^n$ is regular.
Outer regularity follows from its construction by covers with open boxes.
For inner regularity, a Borel set $A$ contains a closed set $F$ with $\lambda(A \setminus F)$ arbitrarily small when $\lambda(A) < \infty$.
The sets $F \cap [-k, k]^n$ are compact, and their measures tend to $\lambda(F)$.
The case $\lambda(A) = \infty$ follows by cutting $A$ into bounded pieces.

Non-example: counting measure on $\R$ satisfies clause 1 but not clause 2.
Finite sets are compact, which gives clause 1; the set $\{0\}$ has measure $1$, but every open set around it has measure $\infty$.

Conventions vary.
Many texts use closed sets instead of compact sets in clause 1.
On a metric space, every finite Borel measure is regular in that closed-set sense.
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<document source="https://tamnd.github.io/math2/node/3EG3HMEGWC.md" tag="3EG3HMEGWC" kind="definition">
# Formal concept analysis

## Definition

A *formal context* is a triple $(G, M, I)$ of a set $G$ of objects, a set $M$ of attributes and a [[binary relation|05AVARPXA2]] $I \subseteq G \times M$.
One reads $g \mathrel{I} m$ as "object $g$ has attribute $m$".
For $A \subseteq G$ and $B \subseteq M$, put
$$A' = \set{m \in M : g \mathrel{I} m \text{ for all } g \in A}, \qquad B' = \set{g \in G : g \mathrel{I} m \text{ for all } m \in B}.$$
A *formal concept* of $(G, M, I)$ is a pair $(A, B)$ with $A \subseteq G$, $B \subseteq M$, $A' = B$ and $B' = A$.
The set $A$ is its extent and $B$ is its intent.
Formal concepts are ordered by $(A_1, B_1) \le (A_2, B_2)$ if and only if $A_1 \subseteq A_2$, which holds if and only if $B_2 \subseteq B_1$.
The ordered set of all formal concepts is the *concept lattice* $\mathfrak{B}(G, M, I)$.
*Formal concept analysis* is the study of formal contexts through their concept lattices.

## Remarks

The two maps $A \mapsto A'$ and $B \mapsto B'$ between the [[power sets|00E0YZCERH]] of $G$ and $M$ form an antitone [[Galois connection|Q5JHEPRYQ6]].
The extents are the closed sets of the closure operator $A \mapsto A''$.
The concept lattice is a [[complete lattice|D7A0ZGZT5S]]: the meet of concepts $(A_i, B_i)$ is $\left(\bigcap_i A_i, \left(\bigcup_i B_i\right)''\right)$.
Conversely, every complete lattice $L$ is isomorphic to the concept lattice of the context $(L, L, \le)$.
Example: for $G = \set{1, 2}$, $M = \set{a, b}$ and $I = \set{(1, a), (2, a), (2, b)}$, the concepts are $(\set{1, 2}, \set{a})$ and $(\set{2}, \set{a, b})$; for instance $\set{1, 2}' = \set{a}$ and $\set{a}' = \set{1, 2}$.
Non-example: in the same context, $(\set{1}, \set{a})$ is not a concept, because $\set{a}' = \set{1, 2} \ne \set{1}$.
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<document source="https://tamnd.github.io/math2/node/3EH4C3M5GV.md" tag="3EH4C3M5GV" kind="concept">
# Finitely presented module

## Summary

A finitely presented module is a module given by finitely many generators and finitely many relations among them.
It is the cokernel of a linear map between two finitely generated [[free modules|0DZ9DKAKZR]], so it is described by a finite matrix.
Every finitely presented module is [[finitely generated|05QY1VEBYD]], and over a [[Noetherian ring|0670T59HGE]] the two notions agree.
Many statements about modules, such as the behaviour of $\Hom$ under localization, need finite presentation rather than finite generation.

## Explanations

### L1 human

A module is finitely presented when you can describe it completely with a finite list of generators and a finite list of rules.
For example, $\Z/6\Z$ has one generator $e$ and one rule $6e = 0$.
A presentation is like a finite recipe for building the module from a free module.
Over familiar rings, such as $\Z$ or polynomial rings in finitely many variables over a field, every finitely generated module has such a recipe.
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<document source="https://tamnd.github.io/math2/node/3EH4F5H7FT.md" tag="3EH4F5H7FT" kind="concept">
# Algebraic approach to constraint satisfaction

## Summary

The algebraic approach studies the constraint satisfaction problem of a fixed finite structure through the [[polymorphisms|A0K9EZRM7E]] of that structure.
Its basic fact is that the complexity of the problem depends only on the clone of polymorphisms.
Polymorphisms satisfying good identities, such as [[Taylor|QEAMRNRJJD]], [[Siggers|JYW1H3J8SX]] or cyclic identities, give efficient algorithms.
The approach led to the proof of the [[CSP dichotomy theorem|ZA4YXJ9T1P]]: every such problem is either in polynomial time or NP-complete.

## Explanations

### L1 human

Many puzzles ask for values of variables that satisfy given constraints, for example colouring a map so that neighbours get different colours.
If the allowed constraints are fixed, one can ask how hard the puzzle is.
The algebraic approach looks for operations that combine several solutions into a new solution.
For example, for $2$-colouring a graph, the majority of three valid colourings is again a valid colouring, and $2$-colouring is easy.
For $3$-colouring no such good operation exists, and the problem is NP-complete.
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<document source="https://tamnd.github.io/math2/node/3ENQE05H1H.md" tag="3ENQE05H1H" kind="example">
# Lakes of Wada

## Example

The *Lakes of Wada* are three disjoint, connected, open subsets of the [[plane|0V3XAMF3C1]] that all have the same [[boundary|05D5EGMFYV]].
They can be built from a closed square of land with three lakes of water in it.
At step $n$, each lake in turn is extended by a thin canal of open water that winds through the land.
After the step, every point of the remaining land is within distance $2^{-n}$ of each lake.
The canals never meet each other, and the land left after each step is a nonempty compact set, so some land remains at the end.
After all steps, each lake is the union of its stages, so it is [[open|02BH3MCBQB]] and [[connected|0V032VWTHD]].
Every point of the remaining land is a limit of points of each lake, and the lakes are open and disjoint.
So the remaining land is the boundary of each of the three lakes.

## Remarks

The construction works the same way for any finite number of lakes.
In contrast, a simple closed curve in the plane is the common boundary of exactly two regions, by the Jordan curve theorem.
So the common boundary of the Lakes of Wada is not a simple closed curve; it is a continuum of a much wilder kind.
The example is credited to Takeo Wada; an earlier construction of the same kind is due to L. E. J. Brouwer.

## Explanations

### L1 human

Imagine an island with three lakes, and keep digging canals so that the water of each lake comes ever closer to every spot of dry land.
In the end, every point of the shore touches all three lakes at once.
It sounds impossible, because on a map two countries usually meet along a line and three meet only at points.
The example shows that boundaries in the plane can be far wilder than drawn lines.
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# Baskakov operator

## Definition

Let $f \colon [0, \infty) \to \R$ satisfy $\abs{f(t)} \le C (1 + t)^m$ for some constants $C \ge 0$, $m \ge 0$ and all $t \ge 0$.
For an integer $n \ge 1$, the *Baskakov operator* $V_n$ maps $f$ to the function
$$V_n(f)(x) = \sum_{k=0}^{\infty} \binom{n + k - 1}{k} \frac{x^k}{(1 + x)^{n + k}} \, f\left(\frac{k}{n}\right), \qquad x \ge 0,$$
where $\binom{n + k - 1}{k}$ is a [[binomial coefficient|092GEF38YD]].

## Remarks

With $y = x/(1 + x)$, the weights are $\binom{n + k - 1}{k} y^k (1 - y)^n$, the negative binomial probabilities, so they are nonnegative and sum to $1$.
Hence $V_n$ is a [[positive linear operator|AQDXWH4812]] with $V_n(1) = 1$.

Example: the mean of these probabilities is $n y / (1 - y) = n x$, so $V_n(t)(x) = x$.
Their variance is $n x (1 + x)$, which gives $V_n(t^2)(x) = x^2 + \frac{x(1 + x)}{n}$.
By Korovkin's theorem, $V_n f \to f$ uniformly on each interval $[0, b]$ for every bounded continuous $f$.

Non-example: $f(t) = e^{t}$ is not in the domain: the terms grow like $k^{n-1} (x/(1 + x))^k e^{k/n}$, which diverges when $\frac{x}{1 + x} e^{1/n} \ge 1$.

Baskakov's general construction takes functions $\varphi_n$ and sets $L_n(f)(x) = \sum_{k \ge 0} \frac{(-x)^k \varphi_n^{(k)}(x)}{k!} f(k/n)$.
The choice $\varphi_n(x) = (1 + x)^{-n}$ gives $V_n$, $\varphi_n(x) = e^{-nx}$ gives the Szász–Mirakyan operator, and $\varphi_n(x) = (1 - x)^n$ gives the Bernstein polynomials on $[0, 1]$.
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# Vector fields on spheres

## Statement

Let $n \ge 1$ and write $n = 2^{4a + b} m$ with $m$ odd, $a \ge 0$ and $0 \le b \le 3$.
Let $\rho(n) = 2^b + 8a$, the Radon–Hurwitz number of $n$.
Then the largest number of continuous tangent vector fields on the [[sphere|T62369RFH6]] $S^{n-1}$ that are linearly independent at every point is $\rho(n) - 1$.

## Remarks

The lower bound is a construction from Radon and Hurwitz: $\rho(n) - 1$ linear maps $A_i$ of $\R^n$ such that the fields $x \mapsto A_i x$ on $S^{n-1}$ are pointwise independent.
These maps come from modules over Clifford algebras.
The upper bound was proved by J. F. Adams in 1962 using topological K-theory and Adams operations on real projective spaces.

Example: for $n$ odd, $\rho(n) = 1$, so an even-dimensional sphere $S^{n-1}$ has no nowhere-zero vector field; this is the [[hairy ball theorem|Z49RAKVCWG]].
For $n = 4$, $\rho(4) = 4$, and $S^3$ has $3$ independent fields, given by multiplication by the quaternions $i$, $j$ and $k$.
For $n = 16 = 2^{4}$, $a = 1$ and $b = 0$, so $\rho(16) = 9$ and $S^{15}$ has $8$ independent fields but not $9$.

As a consequence, the [[tangent bundle|WY6K3P4YCG]] of $S^{n-1}$ is trivial, that is, $S^{n-1}$ is parallelizable, exactly when $\rho(n) = n$, which happens for $n = 1, 2, 4, 8$: the spheres $S^0$, $S^1$, $S^3$ and $S^7$.

## Explanations

### L1 human

A vector field on a sphere attaches an arrow tangent to the sphere at each point, varying continuously.
One can ask how many such fields one can choose so that at every point the arrows point in genuinely different directions.
The answer depends on how often $2$ divides $n$, in a precise pattern with period $8$.
For the ordinary sphere in space the answer is zero: every field vanishes somewhere.

### L2 human

The problem is equivalent to asking for sections of the Stiefel bundle $V_{k+1}(\R^n) \to S^{n-1}$.
Adams' proof of the upper bound works with real projective spaces and computes their real K-theory $\widetilde{KO}(\R P^m)$ together with the Adams operations on it.
The theorem is closely related to the Hopf invariant one problem and to the classification of normed division algebras over $\R$, which also singles out the dimensions $1, 2, 4, 8$.
</document>

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# Ending lamination theorem

## Statement

Let $M$ be a [[hyperbolic 3-manifold|GHVW1Z35BS]] with finitely generated [[fundamental group|S1XK3HXHFX]].
Each end of $M$ carries an *end invariant*.
For a geometrically finite end, it is a point of a Teichmüller space.
For a degenerate end, it is an ending lamination on a surface.
Then $M$ is determined up to isometry by its topological type together with its end invariants.

## Remarks

Thurston conjectured the theorem.
Brock, Canary and Minsky proved it, first for manifolds homeomorphic to $S \times \R$ for a surface $S$, building on earlier work of Minsky.
The general case uses the [[tameness theorem|DE1Z7R9T05]], which makes the ends of $M$ products of a surface and a half-line, so that end invariants make sense.
The proof builds a combinatorial model manifold from the end invariants, using the curve complex, and shows that $M$ is bilipschitz to this model.
For example, a quasi-Fuchsian manifold $S \times \R$ has two geometrically finite ends.
The theorem then recovers Bers simultaneous uniformization: the manifold is determined by a pair of points in the [[Teichmüller space|ZZGC6BZNJ5]] of $S$.

## Explanations

### L1 human

A hyperbolic $3$-manifold with finitely generated fundamental group may have ends that open out to infinity.
Each end leaves a trace at infinity, either a shape of surface or a pattern of curves called a lamination.
The theorem says that the topology and these traces at infinity pin down the whole geometry.

### L2 human

The theorem completes the classification of hyperbolic $3$-manifolds with finitely generated fundamental group.
Together with tameness, it was used to prove the density theorem: every finitely generated Kleinian group is a limit of geometrically finite ones.
It also makes rigorous Thurston's picture of doubly degenerate surface groups, such as the fiber group of a fibered hyperbolic $3$-manifold.
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# Universal property of free algebras

## Statement

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of signature $\sigma$, and let $X$ be a nonempty set of variables.
Let $T(X)$ be the [[term algebra|3YYKGWJJND]] of $\sigma$ over $X$.
Let $\theta$ be the set of pairs $(s, t)$ of terms such that the [[identity|PKD440G2XW]] $s \approx t$ holds in every algebra of $\mathcal{V}$.
Then:

1. $\theta$ is a congruence of $T(X)$, and the [[quotient algebra|N5KE7FH6CX]] $F = T(X) / \theta$ belongs to $\mathcal{V}$.
2. With $\eta(x) = x / \theta$, the algebra $F$ is a [[free algebra|GTS4DRRMZR]] of $\mathcal{V}$ on $X$. So for every $A \in \mathcal{V}$ and every map $f \colon X \to A$, there is exactly one homomorphism $h \colon F \to A$ with $h \circ \eta = f$.
3. For terms $s, t$ over $X$, the elements $s / \theta$ and $t / \theta$ are equal if and only if $\mathcal{V}$ satisfies $s \approx t$.

## Remarks

The map $f$ extends to a homomorphism $T(X) \to A$ by evaluating terms, and its kernel contains $\theta$ because $A$ satisfies every identity of $\mathcal{V}$; this gives $h$.
To see $F \in \mathcal{V}$, choose, for each pair $(s, t) \notin \theta$, an algebra of $\mathcal{V}$ and an assignment that separate $s$ and $t$.
The induced maps embed $F$ in the product of these algebras, and $\mathcal{V}$ is [[closed under subalgebras and products|5083DHSK82]].
Example: for $\mathcal{V}$ the variety of commutative rings with identity, $F$ is the polynomial ring $\Z[X]$, and the unique homomorphism is evaluation of polynomials.
The variable set must be nonempty only when $\sigma$ has no constants and empty algebras are not allowed; otherwise $X = \emptyset$ is fine and gives the initial algebra of $\mathcal{V}$.

## Explanations

### L1 human

Every variety has free algebras on every set of generators.
They are built from formal expressions in the generators, where two expressions are identified exactly when the laws of the variety prove them equal.
Any assignment of the generators to an algebra of the variety extends in one and only one way to a structure-preserving map.

### L2 human

The theorem gives free groups, free modules, polynomial rings and free lattices by one construction.
Part 3 says that the free algebra on countably many generators encodes all identities of $\mathcal{V}$, which links free algebras with equational logic.
Every algebra $A \in \mathcal{V}$ is a quotient of $F_{\mathcal{V}}(A)$, by extending the identity map of $A$; presentations by generators and relations come from this.
In categorical terms, the construction is a left adjoint to the forgetful functor from $\mathcal{V}$ to sets.
</document>

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# Spherical harmonics

## Summary

Spherical harmonics are the restrictions to the unit sphere of [[homogeneous polynomials|2C3PAQ7H50]] that are [[harmonic|0DEB0JMN0W]].
Those of degree $\ell$ form a finite-dimensional space, and these spaces give an orthogonal decomposition of the square-integrable functions on the sphere.
They are the eigenfunctions of the [[Laplace–Beltrami operator|F4NWZNTRMD]] of the sphere, and they play for the sphere the role that $e^{in\theta}$ plays for the circle.
On the sphere in $\R^3$ they are built from the [[associated Legendre polynomials|VPJCSVRY6S]].

## Explanations

### L1 human

Any reasonable function on a circle can be written as a sum of sine and cosine waves of increasing frequency.
Spherical harmonics are the analogue for a sphere: basic wave patterns from which every function on the sphere can be built.
Degree $0$ is the constant pattern.
Degree $1$ is a pattern that is high at one pole and low at the other, like the function $z$.
Higher degrees have more and more lobes.
They describe the shapes of atomic orbitals, the gravity field of the Earth and the temperature map of the cosmic microwave background.
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# Regular boundary point

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]] with [[boundary|05D5EGMFYV]] $\partial\Omega$.
For a [[continuous function|0X3WGX95FE]] $f \colon \partial\Omega \to \R$, let $H_f$ be its [[Perron–Wiener–Brelot solution|R12KQGVGN6]].
A point $\xi \in \partial\Omega$ is a *regular boundary point* of $\Omega$ if
$$\lim_{x \to \xi, \, x \in \Omega} H_f(x) = f(\xi)$$
for every continuous function $f$ on $\partial\Omega$.
A boundary point that is not regular is *irregular*.

## Remarks

Example: if $\Omega$ is the open unit ball, every boundary point is regular, because $H_f$ is the Poisson integral of $f$, which tends to $f(\xi)$ at each $\xi$.
Non-example: for the punctured ball $\Omega = \set{0 < \abs{x} < 1}$, the point $0$ is irregular: for $f = 0$ on the sphere and $f(0) = 1$, one has $H_f = 0$, which does not tend to $1$.
Equivalently, $\xi$ is regular if and only if there is a [[barrier|ED0SZS1VED]] at $\xi$.
The Dirichlet problem on $\Omega$ has a solution continuous on $\overline{\Omega}$ for every continuous $f$ if and only if every point of $\partial\Omega$ is regular; the solution is then $H_f$.
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# Whitney stratification

## Summary

A Whitney stratification of a singular space is a partition into smooth manifolds, the strata, see the [[definition|F19BQTN832]].
The strata fit together in a controlled way along their boundaries.
The control is given by Whitney's conditions (a) and (b) on limits of tangent planes and secant lines.
Hassler Whitney proved that every complex [[analytic set|3RWWZYPAFV]] has such a stratification, and the result was later extended to semialgebraic and subanalytic sets.
By the Thom–Mather isotopy theorem, a Whitney stratified space is topologically locally trivial along each stratum.
This makes these stratifications the standard setting for [[perverse sheaves|5WHXRZDZFK]] and intersection homology.

## Explanations

### L1 human

A singular space, such as a cone or a surface with a crease, can be cut into smooth pieces of different dimensions: the smooth part, the crease, the tip.
Whitney's conditions ask that the large pieces approach the small ones nicely, without spiralling or twisting in a way that changes the local picture.
Then the space looks the same near every point of a given piece.
For the cone $x^2 + y^2 = z^2$, the pieces are the two halves of the cone without its tip, and the tip.
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# Nuclear operator

## Definition

Let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\R$ or $\C$, and let $X'$ be the [[continuous dual|4A82S5FVRT]] of $X$.
A [[bounded linear map|03TM135G0N]] $T \colon X \to Y$ is *nuclear* if there are sequences $(f_n)$ in $X'$ and $(y_n)$ in $Y$ such that
$$\sum_{n=1}^{\infty} \norm{f_n} \norm{y_n} < \infty \quad \text{and} \quad T x = \sum_{n=1}^{\infty} f_n(x) \, y_n \quad \text{for all } x \in X.$$
The *nuclear norm* $\nu(T)$ is the infimum of $\sum_n \norm{f_n} \norm{y_n}$ over all such representations.

## Remarks

Every [[finite-rank operator|Z8FABSYQTS]] is nuclear, with a finite sum.
Example: on $\ell^2$, let $T x = \sum_n d_n x_n e_n$ with $\sum_n \abs{d_n} < \infty$; take $f_n(x) = d_n x_n$ and $y_n = e_n$, so $\norm{f_n} \norm{y_n} = \abs{d_n}$ is summable and $T$ is nuclear.
Non-example: the identity of an infinite-dimensional Banach space is not nuclear, because it is not compact.
Every nuclear operator is a norm limit of the partial sums, which are of finite rank, so it is [[compact|M2NEP1K5AE]].
When $X$ and $Y$ are Hilbert spaces, the nuclear operators are exactly the [[trace class operators|Z2CP530WDE]], and $\nu(T)$ is the trace norm.
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# Orthogonal polynomials on the unit circle

## Summary

Orthogonal polynomials on the unit circle are the polynomials obtained by Gram–Schmidt orthogonalization of $1, z, z^2, \dots$ in $L^2$ of a probability measure on the unit circle.
They are the circle analogue of [[orthogonal polynomials|8TXASARQN1]] on the real line.
Instead of a three-term recurrence they satisfy the Szegő recurrence, governed by a sequence of numbers in the unit disk, the Verblunsky coefficients.
The theory, started by Gábor Szegő, is linked to [[Toeplitz matrices|GKY2WSHE26]], to prediction theory of stationary processes, and to spectral theory of CMV matrices.

## Explanations

### L1 human

On the unit circle, the powers $z^n = e^{in\theta}$ are orthogonal for the uniform measure, which is the basis of Fourier series.
If the circle carries a different weight, the powers are no longer orthogonal.
Straightening them one at a time with respect to the new weight gives a new sequence of polynomials, one of each degree.
These polynomials record the weight in a form well suited to computation and to asymptotic questions.
</document>

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# T(1) theorem

## Statement

Let $n \ge 1$, let $\mathcal{D}$ be the space of smooth functions with compact support on $\R^n$, and let $\mathcal{D}'$ be its dual, the space of distributions.
Let $T \colon \mathcal{D} \to \mathcal{D}'$ be a continuous linear operator with transpose $T^t$, defined by $\langle T^t f, g \rangle = \langle Tg, f \rangle$ for $f, g \in \mathcal{D}$.
Assume that $T$ is associated with a [[Calderón–Zygmund kernel|78N6AXP9VV]] $K$: for $f, g \in \mathcal{D}$ with disjoint supports,
$$\langle Tf, g \rangle = \int\!\!\int K(x, y) f(y) g(x) \, dy \, dx.$$
Then $T$ extends to a bounded operator on $L^2(\R^n)$ if and only if the following three conditions hold:

1. $T(1)$ lies in the space $\mathrm{BMO}$ of functions of [[bounded mean oscillation|QJYTXFT8Y4]];
2. $T^t(1)$ lies in $\mathrm{BMO}$;
3. $T$ has the weak boundedness property.

Here $T(1)$ and $T^t(1)$ are defined as elements of $\mathrm{BMO}$ modulo constants, using the kernel bounds.
The weak boundedness property means that there are constants $C$ and $N$ with the following property.
Let $B$ be a ball of radius $R$, and let $\phi, \psi \in \mathcal{D}$ be supported in $B$ with $\norm{\partial^\alpha\phi}_\infty, \norm{\partial^\alpha\psi}_\infty \le R^{-\abs{\alpha}}$ for $\abs{\alpha} \le N$.
Then $\abs{\langle T\phi, \psi \rangle} \le C R^n$.

## Remarks

Guy David and Jean-Lin Journé proved the theorem in 1984.
The three conditions are necessary.
An $L^2$-bounded operator with a Calderón–Zygmund kernel maps $L^\infty$ into $\mathrm{BMO}$, and so does its transpose.
The weak boundedness property follows from the $L^2$ bound and the Cauchy–Schwarz inequality.
For sufficiency, one subtracts paraproducts with symbols $T(1)$ and $T^t(1)$, which are $L^2$ bounded, and reduces to the case $T(1) = T^t(1) = 0$.
That case is handled by an almost orthogonality argument, such as the Cotlar–Stein lemma, applied to a Littlewood–Paley decomposition of $T$.

Example: for an antisymmetric kernel, $K(x, y) = -K(y, x)$, with $T$ defined as a principal value, the weak boundedness property holds automatically.
Then the theorem reduces to the single condition $T(1) \in \mathrm{BMO}$.
A model case is the Hilbert transform, for which $T(1) = 0$.

Without conditions 1 and 2 the conclusion fails: the kernel bounds alone do not give $L^2$ boundedness, since they carry no information on the cancellation of $T$.

## Explanations

### L1 human

For a singular integral operator, the hard part is to show that it is bounded on $L^2$; the rest of the theory then gives $L^p$ bounds.
The $T(1)$ theorem reduces this hard question to testing the operator on the constant function $1$.
If $T$ and its transpose send $1$ to functions of bounded mean oscillation, and $T$ is mild on bump functions, then $T$ is bounded on $L^2$.

### L2 human

The theorem turns $L^2$ bounds for non-convolution operators, which cannot be checked with the Fourier transform, into simple tests.
It gives a short proof of the $L^2$ boundedness of the Cauchy integral on Lipschitz curves and of the Calderón commutators.
Its output, an $L^2$-bounded operator with a Calderón–Zygmund kernel, is a [[Calderón–Zygmund operator|3SCXCW4AH3]], hence bounded on $L^p$ for $1 < p < \infty$.
There are versions with $T(b)$ in place of $T(1)$, for suitable accretive functions $b$.
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# Quaternionic analysis

## Definition

Let $\mathbb{H}$ be the [[quaternions|0GE4DVQKR8]], the real [[division ring|0EA5BTA0CX]] with basis $1, i, j, k$ and $i^2 = j^2 = k^2 = ijk = -1$.
Write $q = x_0 + x_1 i + x_2 j + x_3 k$ with $x_0, \ldots, x_3 \in \R$, and identify $\mathbb{H}$ with $\R^4$.
Let $U \subset \mathbb{H}$ be open.
A function $f \colon U \to \mathbb{H}$ with continuous first [[partial derivatives|09EGC6F8PT]] is *left regular*, or *Fueter regular*, if
$$\frac{\partial f}{\partial x_0} + i \frac{\partial f}{\partial x_1} + j \frac{\partial f}{\partial x_2} + k \frac{\partial f}{\partial x_3} = 0 \quad \text{on } U.$$
*Quaternionic analysis* is the study of functions $U \to \mathbb{H}$ by analogy with complex analysis, and above all of regular functions.

## Remarks

Example: $f(q) = x_1 - i x_0$ is left regular, since $\partial_{x_0} f + i \partial_{x_1} f = -i + i = 0$ and the other two derivatives vanish.

Non-example: $f(q) = q$ is not left regular.
The sum is $1 + i \cdot i + j \cdot j + k \cdot k = 1 - 3 = -2 \ne 0$.
So, unlike in complex analysis, the identity function and polynomials in $q$ are in general not regular.

Write $\bar\partial = \partial_{x_0} + i \partial_{x_1} + j \partial_{x_2} + k \partial_{x_3}$ and $\partial = \partial_{x_0} - i \partial_{x_1} - j \partial_{x_2} - k \partial_{x_3}$.
Then $\partial \bar\partial$ is the [[Laplacian|0E0B6RA8K0]] of $\R^4$, so each component of a regular function of class $C^2$ is harmonic.
Right regular functions are defined with the units multiplied on the right of the derivatives.
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<document source="https://tamnd.github.io/math2/node/3FTH87V9GD.md" tag="3FTH87V9GD" kind="definition">
# Mikusiński field of operators

## Definition

Let $\mathcal{C}$ be the set of continuous functions $[0, \infty) \to \C$, with pointwise addition and the [[convolution on the half-line|FCKX8KFVGN]] $(f * g)(t) = \int_0^t f(\tau) g(t - \tau) \, d\tau$ as multiplication.
This is a commutative ring without unit and [[without zero divisors|E9QNNPQD2P]].
The *Mikusiński field of operators* $\mathcal{M}$ is its [[field of fractions|0F2PTGX1A2]]:

1. its elements are the classes $f/g$ of pairs $(f, g) \in \mathcal{C} \times (\mathcal{C} \setminus \set{0})$, where $(f, g)$ and $(f_1, g_1)$ are identified when $f * g_1 = f_1 * g$;
2. addition is $f/g + f_1/g_1 = (f * g_1 + f_1 * g)/(g * g_1)$;
3. multiplication is $(f/g)(f_1/g_1) = (f * f_1)/(g * g_1)$.

The elements of $\mathcal{M}$ are called *operators*.

## Remarks

The construction works without a unit in $\mathcal{C}$.
The identification is an equivalence relation because $\mathcal{C}$ has no zero divisors, and $\mathcal{M}$ is a field with unit $g/g$ for any $g \ne 0$.
Each $f \in \mathcal{C}$ is identified with $(f * g)/g$, which does not depend on $g$, and this embeds $\mathcal{C}$ as a subring.
Let $l \in \mathcal{C}$ be the constant function $1$, so that $l * f$ is the integral $\int_0^t f$.
A complex number $\alpha$ is identified with $(\alpha l)/l$; then the product of $\alpha$ and $f$ in $\mathcal{M}$ is the ordinary multiple $\alpha f$.
The operator $s = l/(l * l) = 1/l$ satisfies $s f = f' + f(0)$ for every continuously differentiable $f$, because $l * f' = f - f(0) l$.

Example: the function $t$ equals $l * l$, so its image in $\mathcal{M}$ is $(l * l * l)/l$, and $s \cdot t = \frac{l}{l * l} \cdot \frac{l * l * l}{l} = \frac{l * l * l * l}{l * l * l} = l$.
So $s t = l$, the constant function $1$, in line with $s f = f' + f(0)$ for $f(t) = t$.

Non-example: the pair $(l, 0)$ does not define an element of $\mathcal{M}$, since clause 1 requires a nonzero denominator.
Allowing the denominator $0$ would make every pair $(f, 0)$ equivalent to every $(f_1, 0)$, since $f * 0 = f_1 * 0 = 0$, and the division rules would fail.
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<document source="https://tamnd.github.io/math2/node/3FW4J270FV.md" tag="3FW4J270FV" kind="concept">
# Periodic boundary conditions

## Summary

Periodic boundary conditions ask a solution on an interval $[a, b]$ to take the same value and the same derivatives at both ends.
They turn a [[boundary value problem|TM8VQJWGHZ]] on an interval into a problem on a circle.
They appear in the study of equations with periodic coefficients and in Fourier series.
The eigenfunctions of $-y''$ with these conditions are the sines and cosines.

## Explanations

### L1 human

Imagine bending the interval $[a, b]$ into a ring so that its two ends meet.
A function that fits smoothly around the ring must have the same value and the same slope at both ends.
That is what periodic boundary conditions demand.
For example, on $[0, 2\pi]$ the function $\cos x$ satisfies them, since $\cos 0 = \cos 2\pi$ and $-\sin 0 = -\sin 2\pi$.
The function $x$ does not, since $0 \ne 2\pi$.
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<document source="https://tamnd.github.io/math2/node/3FX0JBH9S1.md" tag="3FX0JBH9S1" kind="definition">
# Vector bundle

## Definition

Let $X$ be a [[scheme|5A35TYZD02]] and $n \ge 0$ an integer.
A *vector bundle of rank $n$* on $X$ is a [[sheaf of $\mathcal{O}_X$-modules|S33B2H3MYD]] $\mathcal{E}$ with the following property.
Every point of $X$ has an open neighbourhood $U$ such that $\mathcal{E}|_U \cong \mathcal{O}_U^{\oplus n}$ as $\mathcal{O}_U$-modules.
It is also called a *locally free sheaf of rank $n$*.

## Remarks

The free sheaf $\mathcal{O}_X^{\oplus n}$ is a vector bundle of rank $n$, with $U = X$ for every point.
On $\mathbb{A}^1_k = \operatorname{Spec} k[t]$ over a field $k$, the skyscraper sheaf $\widetilde{M}$ with $M = k[t]/(t)$ is not a vector bundle.
Its stalk at the origin is $k$, which is killed by $t$ and so is not a free module over the local ring $k[t]_{(t)}$; its stalks elsewhere are $0$.
A vector bundle is a [[quasi-coherent sheaf|KPKND0K7W6]], and it is coherent when $X$ is locally Noetherian.
To $\mathcal{E}$ corresponds the geometric vector bundle $\mathbb{V}(\mathcal{E}^\vee) = \operatorname{Spec}_X \operatorname{Sym}(\mathcal{E}^\vee)$, whose sheaf of sections is $\mathcal{E}$.
Hartshorne writes $\mathbb{V}(\mathcal{E}) = \operatorname{Spec}_X \operatorname{Sym}(\mathcal{E})$, whose sheaf of sections is $\mathcal{E}^\vee$, so the placement of the dual differs between sources.
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# Continuous geometry

## Definition

Let $L$ be a lattice with join $\vee$ and meet $\wedge$.
The lattice $L$ is a *continuous geometry* if:

1. $L$ is a [[modular lattice|QX7WZ92QAK]];
2. $L$ is a [[complete lattice|D7A0ZGZT5S]], with least element $0$ and greatest element $1$;
3. $L$ is [[complemented|ZT47BK9Y6K]];
4. for every [[chain|0S8FE7ZMFM]] $C \subseteq L$ and every $b \in L$,
$$b \wedge \bigvee_{c \in C} c = \bigvee_{c \in C} (b \wedge c) \quad\text{and}\quad b \vee \bigwedge_{c \in C} c = \bigwedge_{c \in C} (b \vee c);$$
5. $L$ is irreducible: the only elements of $L$ that have exactly one complement are $0$ and $1$.

## Remarks

Some authors omit the irreducibility clause 5 and call the irreducible ones continuous geometries in the strict sense.

Example: the lattice of subspaces of $D^n$ for a division ring $D$ and $n \ge 2$.
It is modular and complemented; it is complete and clause 4 holds because every chain in it is finite.
It is irreducible: a subspace $U \ne 0, D^n$ has more than one complement.
Indeed, a complement $W$ can be changed by adding a nonzero vector of $U$ to one basis vector of $W$.

Non-example: the lattice of all subsets of a set $X$ with at least two elements.
It satisfies clauses 1 to 4, but every subset has exactly one complement, so clause 5 fails.

Von Neumann proved that a continuous geometry has a dimension function $d \colon L \to [0, 1]$ with $d(0) = 0$, $d(1) = 1$ and $d(a \vee b) + d(a \wedge b) = d(a) + d(b)$.
Moreover $d(a) = d(b)$ if and only if $a$ and $b$ have a common complement.
Its range is either $\set{0, 1/n, \dots, 1}$ for some $n \ge 1$ or the whole interval $[0, 1]$.
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<document source="https://tamnd.github.io/math2/node/3G1CJ6HJMK.md" tag="3G1CJ6HJMK" kind="statement">
# Cheng–Yau gradient estimate

## Statement

Let $(M, g)$ be a complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$, and let $K \ge 0$ be a constant such that the [[Ricci curvature|4GTNM0WHDH]] satisfies $\operatorname{Ric} \ge -(n - 1) K g$.
Let $x_0 \in M$, let $R > 0$, and let $B(x_0, r)$ denote the geodesic ball of radius $r$ about $x_0$.
There is a constant $C_n$, depending only on $n$, with the following property.
If $u$ is a positive [[harmonic function|PK2KNMWNW3]] on $B(x_0, 2R)$, then
$$\frac{\abs{\nabla u}}{u} \le C_n \left( \frac{1}{R} + \sqrt{K} \right) \quad \text{on } B(x_0, R),$$
where $\nabla u$ is the [[gradient|0ABK9V9Y64]] of $u$.

## Remarks

The proof applies the Bochner formula to $w = \abs{\nabla \log u}^2$.
With the lower bound on Ricci curvature, it gives a lower bound for $\Delta w$ in which $w^2$ appears with a positive coefficient, up to terms in $\nabla w$ and $K w$.
One multiplies $w$ by a cutoff function of the distance to $x_0$ and applies the maximum principle at an interior maximum.
The Laplacian comparison theorem controls the Laplacian of the distance function in this step.
Example: if $\operatorname{Ric} \ge 0$, take $K = 0$ and let $R \to \infty$.
Then $\abs{\nabla u} = 0$ for every positive harmonic function $u$ on $M$, which is Yau's Liouville theorem: positive harmonic functions on a complete manifold with nonnegative Ricci curvature are constant.
Integrating the estimate along geodesics gives a Harnack inequality: $\sup_{B(x_0, R)} u \le e^{C_n (2 + 2\sqrt{K} R)} \inf_{B(x_0, R)} u$.
The term $\sqrt{K}$ cannot be dropped: on the hyperbolic upper half-plane, where $\operatorname{Ric} = -g$, the positive harmonic function $u = y$ has $\abs{\nabla u}/u = 1$ everywhere.

## Explanations

### L1 human

A positive harmonic function cannot change too fast compared with its own size.
The theorem gives a precise limit on the relative rate of change.
It is at most a constant over the radius of the ball, plus a term that depends on how negative the curvature can be.
On a space with no negative curvature and infinite size, the limit tends to $0$, so positive harmonic functions must be constant.
This explains why such spaces resemble Euclidean space from the point of view of harmonic functions.

### L2 human

The estimate is a geometric form of the [[gradient estimates for harmonic functions|N76R7JWMNJ]] in $\R^n$ and of [[Harnack's inequality|YNH832D50W]].
Its proof, through the Bochner formula and a maximum principle, became a model for many later estimates, such as the Li–Yau estimate for positive solutions of the heat equation.
It gives Liouville theorems, Harnack inequalities and control of the growth of harmonic functions in terms of curvature.
</document>

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# Upper nilradical

## Summary

The upper nilradical of a ring is its largest [[nil ideal|NZ1G9WRWFB]], the sum of all two-sided ideals that consist of nilpotent elements.
It contains the [[prime radical|WF117EJ2HF]] and the [[Levitzki radical|9DD8M9R868]].
For a [[commutative ring|0310REJJY8]] all three coincide with the [[nilradical|QQDN6EYQNJ]].
Whether the upper nilradical also contains every nil one-sided ideal is the open [[Köthe conjecture|RT2D9JEVRR]].

## Explanations

### L1 human

An ideal is nil when each of its elements has some power equal to zero.
The upper nilradical gathers all such ideals into one.
In a commutative ring this is just the set of nilpotent elements.
In a noncommutative ring a nilpotent element need not lie in any nil ideal: in $M_2(\Q)$ the matrix unit $e_{12}$ is nilpotent, but the only nil ideal is $0$.
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<document source="https://tamnd.github.io/math2/node/3G7WAP9DJJ.md" tag="3G7WAP9DJJ" kind="definition">
# Toeplitz matrix

## Definition

Let $R$ be a [[ring|0B1261B5ED]] and let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$.
The matrix $A$ is a *Toeplitz matrix* when $a_{ij} = a_{i+1, j+1}$ for all $1 \le i < m$ and $1 \le j < n$.
Equivalently, there are elements $t_k \in R$, for $1 - n \le k \le m - 1$, with $a_{ij} = t_{i-j}$ for all $i, j$.

## Remarks

Example: $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 1 & 2 \\ 5 & 4 & 1 \end{pmatrix}$ is Toeplitz, with $t_0 = 1$, $t_{-1} = 2$, $t_{-2} = 3$, $t_1 = 4$, $t_2 = 5$.
Non-example: $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is not Toeplitz, because $a_{11} = 1 \ne 4 = a_{22}$.
For $m = n$, the circulant matrices are the Toeplitz matrices with $t_k = t_{k-n}$ for $1 \le k \le n-1$.
If $J$ is the $m \times m$ matrix with ones on the anti-diagonal and zeros elsewhere, then $A$ is Toeplitz exactly when $JA$ is a [[Hankel matrix|PQ1D8FQB3E]].
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# Fréchet space

## Summary

A Fréchet space is a [[locally convex space|B6GY2KJMWW]] that is also an [[F-space|86BYEV9AX8]]: its topology comes from a complete translation-invariant metric.
Equivalently, it is a complete Hausdorff space whose topology is generated by a countable family of seminorms.
Fréchet spaces generalize Banach spaces and include the main spaces of smooth and holomorphic functions.
The Hahn–Banach theorem, the open mapping theorem and the uniform boundedness principle all hold for them.

## Explanations

### L1 human

A Banach space is complete for one norm.
A Fréchet space is complete for a whole sequence of seminorms used together.
The space of smooth functions on $\R$ is a typical case: one controls $f$, $f'$, $f''$, and so on, each on bounded intervals.
No single norm captures all this control, but countably many seminorms do.
Convergence means convergence for each seminorm separately.
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<document source="https://tamnd.github.io/math2/node/3GHGFTMFQR.md" tag="3GHGFTMFQR" kind="concept">
# Faddeeva function

## Summary

The Faddeeva function is the scaled complex [[complementary error function|TZ3CT1YA9B]] $w(z) = e^{-z^2} \operatorname{erfc}(-iz)$.
It is an [[entire function|0G4F0XB9X2]], and in the upper half-plane it is a Cauchy-type integral of the Gaussian.
On the real line its real part is $e^{-x^2}$ and its imaginary part is a multiple of the [[Dawson function|EQ4QB1QW60]].
It is used to compute the Voigt line profile in spectroscopy and complex error functions in numerical work.

## Explanations

### L1 human

The error function of a complex number can be enormous, because $e^{-z^2}$ is huge for some complex $z$.
The Faddeeva function multiplies by a factor that cancels this growth, so it stays of moderate size in the upper half-plane.
On the real axis it packs two real functions into one complex value.
Its real part is the Gaussian bell curve, and its imaginary part is the Dawson function up to a factor.
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<document source="https://tamnd.github.io/math2/node/3GNKSCQSCD.md" tag="3GNKSCQSCD" kind="statement">
# Hurwitz's automorphisms theorem

## Statement

Let $X$ be a compact connected [[Riemann surface|5K0N8QMV6Q]] of [[genus|NJ787GZS1N]] $g \ge 2$.
Then the [[automorphism group|0K1SCPBNKJ]] $\Aut(X)$ of biholomorphic self-maps of $X$ is finite, and its order satisfies
$$\abs{\Aut(X)} \le 84 (g - 1).$$

## Remarks

Finiteness is a separate step; a standard proof uses the finite set of Weierstrass points of $X$, which every automorphism permutes.
For the bound, let $G = \Aut(X)$ and apply the [[Riemann–Hurwitz formula|42A5ZT6AWN]] to the quotient map $X \to X / G$, of degree $\abs{G}$.
Here $X/G$ has genus $h$, and the branch values have ramification indices $m_1, \ldots, m_k$.
It gives $2g - 2 = \abs{G} \left( 2h - 2 + \sum_{i=1}^{k} (1 - 1/m_i) \right)$.
The bracket is positive, and its smallest positive value is $1/42$, reached for $h = 0$ and $(m_1, m_2, m_3) = (2, 3, 7)$; this gives $\abs{G} \le 84(g - 1)$.
The hypothesis $g \ge 2$ is needed: a torus $\C / \Lambda$ of genus $1$ has infinitely many automorphisms, the translations, and so does the Riemann sphere.
Example: the Klein quartic $x^3 y + y^3 z + z^3 x = 0$ has genus $3$ and automorphism group of order $168 = 84 \cdot 2$, so the bound is attained.
The bound is not attained for every genus; for $g = 2$ the largest automorphism group has order $48$.

## Explanations

### L1 human

A surface with at least two holes can have only finitely many symmetries that respect its complex structure.
Hurwitz's theorem gives an explicit ceiling: a surface of genus $g$ has at most $84(g - 1)$ such symmetries.
The number $84$ comes from the triangle with angles $\pi/2$, $\pi/3$ and $\pi/7$, the smallest triangle that tiles the hyperbolic plane in this way.

### L2 human

The theorem is a typical application of the Riemann–Hurwitz formula to a quotient by a group action.
The surfaces that reach the bound are called Hurwitz surfaces, and their automorphism groups are the finite quotients of the $(2, 3, 7)$ triangle group, called Hurwitz groups.
The same bound holds for smooth projective curves over an algebraically closed field of characteristic $0$, but in positive characteristic it can fail because of wild ramification.
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<document source="https://tamnd.github.io/math2/node/3GSQJXG19A.md" tag="3GSQJXG19A" kind="definition">
# Schwarzian derivative

## Definition

Let $U \subseteq \C$ be a [[domain|097KNJFJ7V]], and let $f \colon U \to \C$ be [[holomorphic|0DG0Z72CQD]] with $f'(z) \ne 0$ for all $z \in U$.
The *Schwarzian derivative* of $f$ is the holomorphic function $S(f) \colon U \to \C$ given by
$$S(f) = \left(\frac{f''}{f'}\right)' - \frac{1}{2}\left(\frac{f''}{f'}\right)^2 = \frac{f'''}{f'} - \frac{3}{2}\left(\frac{f''}{f'}\right)^2.$$

## Remarks

Other notations are $\set{f, z}$ and $S_f$.
The two expressions agree, since $(f''/f')' = f'''/f' - (f''/f')^2$.
Example: for $f(z) = e^z$ on $\C$, $f'' / f' = 1$, so $S(f) = 0 - \frac{1}{2} = -\frac{1}{2}$.
For a [[Möbius transformation|0B346DDAM5]] $T(z) = (az + b)/(cz + d)$ with $ad - bc \ne 0$, one gets $S(T) = 0$; conversely $S(f) = 0$ on a domain only for restrictions of Möbius maps.
Non-example: $f(z) = z^2$ on $\C$ has $f'(0) = 0$, so it is excluded; indeed $S(f)(z) = -\frac{3}{2z^2}$ on $\C \setminus \set{0}$ has a pole at $0$.
The chain rule is $S(f \circ g) = (S(f) \circ g)\,(g')^2 + S(g)$, so $S(T \circ f) = S(f)$ for every Möbius $T$.
The definition extends to locally injective meromorphic functions, with $S(f)$ holomorphic, since $S(1/f) = S(f)$.
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# Minimal surface equation

## Summary

The minimal surface equation for a function $u$ of $n$ variables says that its graph is a [[minimal surface|41X0YW2R9N]].
That is, the graph has zero mean curvature.
It reads $\operatorname{div}\big(\nabla u / \sqrt{1 + \abs{\nabla u}^2}\big) = 0$.
It is the Euler–Lagrange equation of the area of the graph, $\int \sqrt{1 + \abs{\nabla u}^2} \, dx$.
It is a model [[quasilinear|7X9N9H3RK6]] elliptic equation, and its study led to Bernstein's problem and to deep results on regularity.

## Explanations

### L1 human

Imagine a soap film over a wire frame, seen as the graph of a height function $u(x, y)$.
The film settles where its area cannot be lowered by small changes, and this condition is a differential equation for the height.
For gentle slopes, the equation is close to Laplace's equation $u_{xx} + u_{yy} = 0$.
Planes are the simplest solutions, and Scherk's surface $u(x, y) = \log(\cos y / \cos x)$ is a famous curved one.
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<document source="https://tamnd.github.io/math2/node/3GZ99NCX22.md" tag="3GZ99NCX22" kind="definition">
# Full and faithful functors

## Definition

Let $F \colon \mathcal{C} \to \mathcal{D}$ be a [[functor|28NQ0KRHD3]].
For objects $A$, $B$ of $\mathcal{C}$, let $F_{A, B} \colon \Hom_{\mathcal{C}}(A, B) \to \Hom_{\mathcal{D}}(F(A), F(B))$ be the map $f \mapsto F(f)$.

1. $F$ is *faithful* if $F_{A, B}$ is [[injective|09NWETVTNY]] for all objects $A$, $B$.
2. $F$ is *full* if $F_{A, B}$ is [[surjective|01RVCMSN5N]] for all objects $A$, $B$.
3. $F$ is *fully faithful* if $F_{A, B}$ is bijective for all objects $A$, $B$.

## Remarks

A faithful functor may still identify two morphisms with different domains or codomains, and it may send different objects to the same object.
Example: the forgetful functor from [[groups|G5S1M9]] to sets is faithful, since two [[homomorphisms|0JYTG5K9Y4]] with the same underlying map are equal.
It is not full: the constant map $\Z \to \Z$ with value $1$ is a function between the underlying sets but not a homomorphism.
The inclusion of a [[full subcategory|V3BFGKBJF7]] is fully faithful.
Non-example: the functor from groups to the category with one object and one morphism is not faithful.
The two homomorphisms $\Z \to \Z$ given by $x \mapsto 0$ and $x \mapsto x$ are sent to the same morphism.
A fully faithful functor reflects isomorphisms: if $F(f)$ is an isomorphism, then so is $f$.
</document>

<document source="https://tamnd.github.io/math2/node/3H1YKSPH19.md" tag="3H1YKSPH19" kind="definition">
# Standard probability space

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]] whose measure is [[complete|22K8CSQTYM]].
A *model space* is built from a number $a \in [0,1]$ and a finite or countable family of numbers $p_k > 0$ with $a + \sum_k p_k = 1$.
It is the disjoint union of $[0, a]$ with points $x_k$.
Lebesgue measurable subsets of $[0,a]$ get their [[Lebesgue measure|2Z2QVWZ262]], and each point $x_k$ gets mass $p_k$.
Two complete probability spaces are *isomorphic mod $0$* if a [[null set|VQQM4Y550T]] can be removed from each so that the following holds.
There is a bijection between the remaining sets that is measurable in both directions and preserves the measure.
The space $(\Omega, \mathcal{F}, P)$ is *standard* if it is isomorphic mod $0$ to a model space.

## Remarks

Example: the unit square $[0,1]^2$ with completed Lebesgue measure is standard with $a = 1$ and no atoms.
Remove the null set of points with a dyadic rational coordinate, and interleave the binary digits of the two coordinates.
This gives a bijection mod $0$ with $[0,1]$ that sends dyadic squares to dyadic intervals of the same measure, so it preserves the measure.
Non-example: let $Z \subseteq [0,1]$ have Lebesgue inner measure $0$ and outer measure $1$, and let $\Omega = [0,1] \setminus Z$ with $P(A \cap \Omega) = \lambda(A)$ for Lebesgue measurable $A$.
This "perforated interval" is a complete probability space that is known not to be standard.
Rokhlin's Lebesgue spaces are the same spaces, defined through a countable separating family of sets and a completeness condition.
</document>

<document source="https://tamnd.github.io/math2/node/3H2W1MQFGM.md" tag="3H2W1MQFGM" kind="definition">
# Absolutely free algebra

## Definition

Let $\sigma$ be a signature of function symbols and let $X$ be a set.
An *absolutely free algebra* of signature $\sigma$ on $X$ is an algebra $F$ of signature $\sigma$ with a map $\eta \colon X \to F$ such that:

1. $F$ is [[generated|2TQD8NMG9P]] by $\eta(X)$.
2. For every algebra $A$ of signature $\sigma$ and every map $f \colon X \to A$, there is a [[homomorphism|KBXSZC6BV1]] $h \colon F \to A$ with $h \circ \eta = f$.

## Remarks

This is a free algebra in the variety of all algebras of signature $\sigma$, the variety defined by the empty set of identities.
The [[term algebra|3YYKGWJJND]] $T(X)$, whose elements are terms over $X$, is an absolutely free algebra on $X$.
Every absolutely free algebra on $X$ is isomorphic to it.
Let $F$ be generated by $\eta(X)$.
Then $F$ is absolutely free on $X$ exactly when $\eta$ and each basic operation are injective, and the images of different operations are disjoint and do not meet $\eta(X)$.
Example: for one unary symbol $s$ and $X = \set{x}$, the algebra $\N$ with $s(n) = n + 1$ and $\eta(x) = 0$ is absolutely free; the extension of $f$ sends $n$ to $s^n(f(x))$.
Non-example: $\Z$ with $s(n) = n + 1$ and $\eta(x) = 0$ is not absolutely free on $\set{x}$, since it is not generated by $0$.
</document>

<document source="https://tamnd.github.io/math2/node/3H5SDW6JTX.md" tag="3H5SDW6JTX" kind="definition">
# Co-Heyting algebra

## Definition

Let $(L, \vee, \wedge, 0, 1)$ be a [[bounded lattice|72V1Z82M5F]] with order $\le$.
A *co-Heyting algebra* structure on $L$ is a binary operation $\setminus$ on $L$ such that for all $a, b, c \in L$,
$$a \setminus b \le c \iff a \le b \vee c.$$
So $a \setminus b$ is the least element $c$ with $a \le b \vee c$.
A *co-Heyting algebra* is a bounded lattice with such an operation.

## Remarks

The operation is unique when it exists, since it is a least element.
A bounded lattice is a co-Heyting algebra if and only if its order dual is a [[Heyting algebra|NG943XNQWH]].
The element $\neg a = 1 \setminus a$ is called the supplement of $a$; it is the least $c$ with $a \vee c = 1$.

Example: let $X$ be a topological space and $L$ its family of closed sets, ordered by inclusion.
For closed $A, B$, put $A \setminus B = \overline{A - B}$, the closure of the set difference.
If $A \subseteq B \cup C$ with $C$ closed, then $A - B \subseteq C$, so $\overline{A - B} \subseteq C$.
If $\overline{A - B} \subseteq C$, then $A \subseteq B \cup (A - B) \subseteq B \cup C$.
So the defining equivalence holds.

Non-example: the diamond lattice $M_3$, with atoms $p, q, r$, is bounded but has no such operation.
Every co-Heyting algebra is distributive, and $M_3$ is not, since $p \vee (q \wedge r) = p$ while $(p \vee q) \wedge (p \vee r) = 1$.
Directly: the elements $c$ with $1 \le p \vee c$ are $q$, $r$ and $1$, and they have no least one.

In Mathlib the notion is `CoheytingAlgebra`, with difference `\` and supplement `￢`.
</document>

<document source="https://tamnd.github.io/math2/node/3H6SQA6JWM.md" tag="3H6SQA6JWM" kind="statement">
# Removable singularities for harmonic functions

## Statement

Let $n \ge 2$, let $U \subseteq \R^n$ be an open set, let $a \in U$, and let $u$ be [[harmonic|0DEB0JMN0W]] on $U \setminus \set{a}$.
Assume that
$$\lim_{x \to a} \frac{u(x)}{\log(1/\abs{x - a})} = 0 \quad (n = 2), \qquad \lim_{x \to a} \abs{x - a}^{n-2} \, u(x) = 0 \quad (n \ge 3).$$
Then $u$ extends to a harmonic function on $U$.
In particular this holds when $u$ is bounded on a punctured neighbourhood of $a$.

## Remarks

Proof idea for $n \ge 3$ and $a = 0$: take a closed ball $\overline{B}(0, r) \subseteq U$.
Let $h$ be harmonic on $B(0, r)$ and continuous on its closure, with $h = u$ on the sphere, given by the [[Poisson integral formula|PCZBE7EX01]].
For $\varepsilon > 0$, the functions $u - h \mp \varepsilon(\abs{x}^{2-n} - r^{2-n})$ are harmonic on the punctured ball and have the right sign near $0$ and on the outer sphere.
The [[maximum principle|T62HHSRHP5]] on annuli gives $\abs{u - h} \le \varepsilon(\abs{x}^{2-n} - r^{2-n})$, and letting $\varepsilon \to 0$ gives $u = h$ off $0$.
In the plane the same argument works with $\log(r/\abs{x})$.
The growth condition is sharp: the [[fundamental solution of the Laplacian|BSJ8X2XRN9]], $\abs{x}^{2-n}$ for $n \ge 3$ or $\log(1/\abs{x})$ for $n = 2$, is harmonic off $0$ and has no harmonic extension.
Example: $u(x, y) = \frac{x}{x^2 + y^2}$ is harmonic on $\R^2 \setminus \set{0}$, and on the positive $x$-axis it equals $1/x$, which grows faster than $\log(1/x)$.
So the hypothesis fails, and indeed $u$ is unbounded near $0$ and has no harmonic extension.

## Explanations

### L1 human

An isolated bad point of a harmonic function can be repaired, as long as the function does not blow up too fast there.
Bounded is certainly fine: a bounded harmonic function on a punctured ball is really harmonic on the whole ball once one fills in the right value at the centre.

### L2 human

This is the harmonic counterpart of Riemann's theorem on removable singularities of holomorphic functions.
The precise threshold is the growth of the fundamental solution, which is the only kind of singularity a positive harmonic function can have at an isolated point, by Bôcher's theorem.
For larger exceptional sets, bounded harmonic functions extend across closed sets of zero capacity, which shows the role of capacity in describing negligible sets.
</document>

<document source="https://tamnd.github.io/math2/node/3HAV1FH6V9.md" tag="3HAV1FH6V9" kind="definition">
# Hypergeometric function of a matrix argument

## Definition

Let $m \ge 1$ be an integer, let $\alpha > 0$, and let $X$ be a complex $m \times m$ [[symmetric matrix|0VR671PHE5]] with [[eigenvalues|0BQAA2BK3Q]] $x_1, \dots, x_m$.
Write $\kappa \vdash k$ when $\kappa = (\kappa_1 \ge \kappa_2 \ge \dots)$ is an [[integer partition|0JV30XN73Z]] of $k$.
Let $C_\kappa^{(\alpha)}(X)$ be the Jack polynomial of $\kappa$ in $x_1, \dots, x_m$, normalised so that $\sum_{\kappa \vdash k} C_\kappa^{(\alpha)}(X) = (\operatorname{tr} X)^k$.
It is $0$ when $\kappa$ has more than $m$ nonzero parts.
For a complex number $a$, the *generalized Pochhammer symbol* is
$$(a)_\kappa^{(\alpha)} = \prod_{i \ge 1} \prod_{j = 1}^{\kappa_i} \left( a - \frac{i - 1}{\alpha} + j - 1 \right).$$
Let $a_1, \dots, a_p$ and $b_1, \dots, b_q$ be complex numbers such that $(b_l)_\kappa^{(\alpha)} \ne 0$ for every $l$ and every $\kappa$ with at most $m$ parts.
The *hypergeometric function of matrix argument* is the series
$${}_pF_q^{(\alpha)}(a_1, \dots, a_p; b_1, \dots, b_q; X) = \sum_{k = 0}^{\infty} \sum_{\kappa \vdash k} \frac{(a_1)_\kappa^{(\alpha)} \cdots (a_p)_\kappa^{(\alpha)}}{(b_1)_\kappa^{(\alpha)} \cdots (b_q)_\kappa^{(\alpha)}} \frac{C_\kappa^{(\alpha)}(X)}{k!},$$
wherever it converges.

## Remarks

For $m = 1$ there is one partition $\kappa = (k)$ of each $k$, $C_{(k)}^{(\alpha)}(x) = x^k$ and $(a)_{(k)}^{(\alpha)} = (a)_k$, so the series is the scalar [[generalized hypergeometric function|VDBAW9PN88]].
The value depends only on the eigenvalues of $X$.
For $\alpha = 2$ the polynomials $C_\kappa^{(2)}$ are the zonal polynomials of real symmetric matrices; for $\alpha = 1$ they are multiples of [[Schur polynomials|43VFMJCDYJ]], suited to complex Hermitian matrices.
Example: ${}_0F_0^{(\alpha)}(X) = \sum_k \frac{1}{k!} \sum_{\kappa \vdash k} C_\kappa^{(\alpha)}(X) = \sum_k \frac{(\operatorname{tr} X)^k}{k!} = e^{\operatorname{tr} X}$.
Non-example: for $m = 2$ and $\alpha = 1$, the choice $b_1 = 1$ is excluded, since $(1)_{(1, 1)}^{(1)} = 1 \cdot (1 - 1) = 0$ appears in a denominator, although $b_1 = 1$ is allowed in the scalar case.
</document>

<document source="https://tamnd.github.io/math2/node/3HD30YG3DG.md" tag="3HD30YG3DG" kind="statement">
# Löb's theorem

## Statement

Let $T$ be a [[theory|92SWXST1ZM]] in the language of arithmetic that contains the [[Peano axioms|0QFT7Z76N5]] and whose set of axioms is [[computably enumerable|TN9G24VABN]].
Let $\mathrm{Prov}_T(x)$ be the standard provability predicate of $T$ for a [[Gödel numbering|MFJXPAF0R0]].
For every [[sentence|DKFGQTRYZE]] $\varphi$, if $T \vdash \mathrm{Prov}_T(\ulcorner \varphi \urcorner) \to \varphi$, then $T \vdash \varphi$.

## Remarks

Proof idea: by the [[diagonal lemma|3PRW76471J]] there is $\psi$ with $T \vdash \psi \leftrightarrow (\mathrm{Prov}_T(\ulcorner \psi \urcorner) \to \varphi)$.
The derivability conditions of $\mathrm{Prov}_T$, which $T$ proves, give $T \vdash \mathrm{Prov}_T(\ulcorner \psi \urcorner) \to \mathrm{Prov}_T(\ulcorner \varphi \urcorner)$.
With the hypothesis this yields $T \vdash \mathrm{Prov}_T(\ulcorner \psi \urcorner) \to \varphi$, hence $T \vdash \psi$, hence $T \vdash \mathrm{Prov}_T(\ulcorner \psi \urcorner)$, and finally $T \vdash \varphi$.

Example: take $\varphi$ to be $0 = S0$.
If $T$ proved $\mathrm{Con}(T)$, that is, $\neg \mathrm{Prov}_T(\ulcorner 0 = S0 \urcorner)$, then $T$ would prove $\mathrm{Prov}_T(\ulcorner 0 = S0 \urcorner) \to 0 = S0$, and so $T \vdash 0 = S0$.
So a consistent $T$ does not prove its own consistency, which is the second of [[Gödel's incompleteness theorems|X9WYNYE29C]].
Another consequence: the sentence that says "I am provable" is provable, which answers a question of Henkin.

## Explanations

### L1 human

A theory can never prove "if I prove $\varphi$, then $\varphi$ is true" unless it already proves $\varphi$ itself.
So a theory cannot vouch for its own soundness, even one sentence at a time.

### L2 human

Löb's theorem generalizes the second incompleteness theorem, as the example shows.
Its formalized version, $\square(\square p \to p) \to \square p$, is the characteristic axiom of the modal [[provability logic|WJK0YA4GP2]] $\mathsf{GL}$, read with $\square$ as "it is provable in $T$ that".
</document>

<document source="https://tamnd.github.io/math2/node/3HDRK3QYSD.md" tag="3HDRK3QYSD" kind="example">
# Weeks manifold

## Example

Take the complement of the [[Whitehead link|TPGBASPKE0]] in $S^3$.
The Weeks manifold $W$ is the closed orientable $3$-manifold obtained from it by [[Dehn filling|ZA405B0Z1S]] with coefficients $(5, 1)$ and $(5, 2)$ on the two components.
It is a closed [[hyperbolic 3-manifold|GHVW1Z35BS]].
Its hyperbolic volume is approximately $0.9427$.
Its first homology group is $H_1(W; \Z) \cong \Z/5 \oplus \Z/5$.

The hyperbolic structure can be checked by computer, for example with the program SnapPea of Jeffrey Weeks.

## Remarks

Gabai, Meyerhoff and Milley proved in 2009 that $W$ has the smallest volume among all closed orientable hyperbolic $3$-manifolds.
By [[Mostow rigidity|4BEWKR0EG7]], its volume is a topological invariant, so this is a statement about topology.

## Explanations

### L1 human

The Weeks manifold is the smallest closed orientable hyperbolic $3$-dimensional space, measured by volume.
It is made by closing up the two tube-shaped ends of the complement of a simple link in two particular ways.
Its volume, about $0.94$, is a lower bound for the volume of every other closed orientable hyperbolic $3$-manifold.
</document>

<document source="https://tamnd.github.io/math2/node/3HFP9ZQ580.md" tag="3HFP9ZQ580" kind="statement">
# Cramér–Wold theorem

## Statement

Let $k \ge 1$, and let $X, X_1, X_2, \dots$ be random vectors in $\R^k$.
Then $X_n$ [[converges in distribution|0SRYJGXKR4]] to $X$ if and only if, for every $t \in \R^k$, the real random variables $t \cdot X_n$ converge in distribution to $t \cdot X$.
Here $t \cdot x$ is the [[dot product|0S52TNNTH9]] of $t$ and $x$.

## Remarks

Proof idea: the [[characteristic function|0V5B6D6BQH]] of $X_n$ at $t$ is $E[e^{i t \cdot X_n}]$, which is the characteristic function of $t \cdot X_n$ at $1$.
So the condition gives pointwise convergence of the characteristic functions of $X_n$, and [[Lévy's continuity theorem|0FPWF7TTMB]] concludes.
The converse holds because $x \mapsto t \cdot x$ is continuous.
A special case: the law of a random vector $X$ is determined by the laws of all its one-dimensional projections $t \cdot X$.
Convergence of the coordinates alone is not enough.
If $Z$ is standard normal, then $(Z, Z)$ and $(Z, -Z)$ have the same marginals but different laws.
So the constant sequence $(Z, Z)$ has coordinates that converge in law to those of $(Z, -Z)$, but it does not converge in law to $(Z, -Z)$.

## Explanations

### L1 human

To check that random vectors settle down to a limit law, it is enough to look at every straight-line projection of them.
This reduces questions about many dimensions to questions about one dimension.

### L2 human

The standard use is the multivariate [[central limit theorem|0EPZC4Z1FC]]: for i.i.d. random vectors with finite covariance matrix, each projection $t \cdot X_i$ obeys the one-dimensional theorem, and the Cramér–Wold theorem lifts this to a [[multivariate normal|0XN3K7RSZY]] limit.
It is also called the Cramér–Wold device.
</document>

<document source="https://tamnd.github.io/math2/node/3HGJCQBWVY.md" tag="3HGJCQBWVY" kind="statement">
# Leapfrog integration

## Statement

Let $V \colon \R^d \to \R$ be three times continuously differentiable, put $a = -\nabla V$, and consider the equation $x'' = a(x)$ with $x(0) = x_0$ and $x'(0) = v_0$.
Fix a step size $h > 0$.
The leapfrog method, in its kick-drift-kick form, maps $(x_n, v_n)$ to $(x_{n+1}, v_{n+1})$ by
$$v_{n+1/2} = v_n + \tfrac{h}{2} a(x_n), \qquad x_{n+1} = x_n + h\, v_{n+1/2}, \qquad v_{n+1} = v_{n+1/2} + \tfrac{h}{2} a(x_{n+1}).$$
Let $\Phi_h \colon \R^{2d} \to \R^{2d}$ denote this map.

1. $\Phi_h$ is symplectic: it preserves the symplectic form $\sum_{i=1}^d dx_i \wedge dv_i$.
2. $\Phi_h$ is time-reversible: $\Phi_{-h}(\Phi_h(x, v)) = (x, v)$ for all $(x, v)$.
3. The method has order $2$: on a bounded time interval $[0, T]$, the error $\max_{nh \le T} \abs{(x_n, v_n) - (x(nh), x'(nh))}$ is $O(h^2)$ as $h \to 0$.

## Remarks

The equation is the [[Hamiltonian system|0EDKGA2FAK]] with $H(x, v) = \frac12 \abs{v}^2 + V(x)$.
Each of the three substeps is the exact flow, for time $h/2$, $h$ or $h/2$, of the Hamiltonian $V(x)$ or $\frac12\abs{v}^2$.
Exact Hamiltonian flows are symplectic, and so is their composition, which gives clause 1.
The composition is symmetric, which gives clause 2, and a symmetric method has even order, which together with consistency gives clause 3.
So leapfrog is a [[symplectic integrator|B4C25SXFRE]], and its energy error stays bounded over very long times instead of growing.
Example: for $a(x) = -x$ in one dimension, $\Phi_h$ is a linear map with determinant $1$ and trace $2 - h^2$.
For $0 < h < 2$ its eigenvalues are distinct and lie on the unit circle, so all orbits stay bounded.

## Explanations

### L1 human

Positions and velocities are updated in turn, as if they leapt over each other.
The velocity takes half a kick from the force, the position drifts for a full step, and the velocity takes the second half kick.
The method costs one force evaluation per step, and it keeps the energy of a planet or a molecule from slowly creeping away.

### L2 human

Leapfrog is the Störmer–Verlet method written in one-step form, and it is the standard integrator of molecular dynamics and of $N$-body simulations.
It is the simplest example of a splitting method: it is the Strang composition of the kinetic and potential flows.
Higher-order symplectic methods are often built by composing leapfrog steps with well-chosen step sizes.
</document>

<document source="https://tamnd.github.io/math2/node/3HKREN3Y3F.md" tag="3HKREN3Y3F" kind="statement">
# Grothendieck–Riemann–Roch theorem

## Statement

Let $k$ be a field and let $f \colon X \to Y$ be a [[projective morphism|Z5G47MQ99B]] of [[smooth|EXKGG9X6J0]] quasi-projective varieties over $k$.
Let $K_0(X)$ be the [[Grothendieck group|5FDMY7PDYY]] of [[coherent sheaves|56RXEJ8Z6S]] on $X$, and let $f_! \colon K_0(X) \to K_0(Y)$ be the map $[\mathcal{F}] \mapsto \sum_{i \ge 0} (-1)^i [R^i f_* \mathcal{F}]$.
Let $A^*(X)_\Q$ be the rational [[Chow ring|EBEF6PEK4N]], with proper pushforward $f_* \colon A^*(X)_\Q \to A^*(Y)_\Q$.
Let $\operatorname{ch} \colon K_0(X) \to A^*(X)_\Q$ be the Chern character, built from the [[Chern classes|PAD37GKDPD]], and let $\operatorname{td}$ be the [[Todd class|2YDB5089FR]] of a tangent bundle.
Then for every $x \in K_0(X)$,
$$\operatorname{ch}(f_! x) \, \operatorname{td}(T_Y) = f_*\big(\operatorname{ch}(x) \, \operatorname{td}(T_X)\big) \quad \text{in } A^*(Y)_\Q.$$

## Remarks

The idea of the proof: factor $f$ as a closed immersion $X \to \mathbb{P}^N_Y$ followed by the projection $\mathbb{P}^N_Y \to Y$.
For the projection, both sides are computed from the explicit structure of $K_0$ and of the Chow ring of a projective bundle.
For a closed immersion, one deforms to the normal bundle and reduces to the zero section of a vector bundle, where the Koszul complex gives the formula.

Example: for $Y = \operatorname{Spec} k$, $A^*(Y)_\Q = \Q$, $\operatorname{td}(T_Y) = 1$ and $\operatorname{ch}(f_! [\mathcal{F}]) = \chi(X, \mathcal{F})$.
So the theorem becomes the Hirzebruch–Riemann–Roch formula $\chi(X, \mathcal{F}) = \int_X \operatorname{ch}(\mathcal{F}) \operatorname{td}(T_X)$.

Without the Todd classes the formula is false: the Chern character alone does not commute with pushforward, as already the case $f \colon \mathbb{P}^1_k \to \operatorname{Spec} k$ with $\mathcal{F} = \mathcal{O}$ shows, where $\chi = 1$ but $f_* \operatorname{ch}(\mathcal{O}) = 0$.

## Explanations

### L1 human

Grothendieck–Riemann–Roch is a version of Riemann–Roch for families.
It does not compute one number, the Euler characteristic of a sheaf on one variety.
Instead it computes how the cohomology of a sheaf varies over a base, as a class on the base.
The correction factor is the Todd class, which measures the geometry of the varieties involved.

### L2 human

The theorem says that the Chern character, corrected by Todd classes, is compatible with pushforward: a commutative square between K-theory and the Chow ring.
It is used to compute Chern classes of Hodge bundles and other natural bundles on moduli spaces, such as Mumford's computation on the moduli space of curves.
The case of a point recovers the Hirzebruch–Riemann–Roch theorem, and the topological analogue is the Atiyah–Singer family index theorem.
</document>

<document source="https://tamnd.github.io/math2/node/3HRHSHTG9C.md" tag="3HRHSHTG9C" kind="statement">
# Minimal residual method

## Statement

Let $A$ be a real [[symmetric|0VR671PHE5]] [[invertible|0EB9N0S33C]] $n \times n$ matrix, possibly indefinite, let $b \in \R^n$, and let $x_0 \in \R^n$ with $r_0 = b - A x_0$.
For $k \ge 1$, the $k$-th iterate of the *minimal residual method* (MINRES) is the vector $x_k \in x_0 + \mathcal{K}_k(A, r_0)$ that minimizes $\norm{b - A x}_2$ there.
Here $\mathcal{K}_k$ is the [[Krylov subspace|QB45PCGQPV]].

1. The iterates can be computed by recurrences of fixed length. Each step costs one product with $A$ and a number of vector operations that does not depend on $k$.
2. Let $r_k = b - A x_k$, and let $P_k$ be the set of polynomials $p$ of degree at most $k$ with $p(0) = 1$. Then $$\norm{r_k}_2 = \min_{p \in P_k} \norm{p(A) r_0}_2 \le \min_{p \in P_k} \max_{\lambda} \abs{p(\lambda)} \, \norm{r_0}_2,$$ where $\lambda$ runs over the eigenvalues of $A$.
3. In exact arithmetic some $r_k$ with $k \le n$ is zero.

## Remarks

The [[Lanczos algorithm|7RRB4PCRJB]] gives an orthonormal basis $Q_{k+1}$ of $\mathcal{K}_{k+1}(A, r_0)$ with $A Q_k = Q_{k+1} \tilde{T}_k$, where $\tilde{T}_k$ is $(k+1) \times k$ and [[tridiagonal|DARX1KZRHZ]].
The minimization reduces to a small least squares problem with $\tilde{T}_k$.
A QR factorization of $\tilde{T}_k$ by Givens rotations can be updated with each new column, and this yields the short recurrences of clause 1.
Clause 2 uses that $A$ is orthogonally diagonalizable, so $\norm{p(A)}_2 = \max_\lambda \abs{p(\lambda)}$.
Example: if $A$ has only the [[eigenvalues|0BQAA2BK3Q]] $1$ and $-1$, then $p(t) = 1 - t^2$ gives $r_2 = 0$.
For a nonsymmetric $A$ the Lanczos recurrence no longer produces an orthonormal basis, and one needs [[GMRES|A3FR6PNEPB]] with its long recurrences.

## Explanations

### L1 human

MINRES solves symmetric systems that need not be positive definite, such as saddle point problems.
At each step it picks, among the vectors it can reach with the matrix, the one that makes the residual as small as possible.
Thanks to symmetry, it only needs to remember a few vectors.

### L2 human

MINRES is mathematically [[GMRES|A3FR6PNEPB]] specialized to symmetric matrices, where the Arnoldi process becomes the Lanczos process.
It fills the gap left by the [[conjugate gradient method|AFXZGQB4EN]], which can break down when $A$ is indefinite.
For symmetric positive definite $A$ both apply: conjugate gradients minimize the energy norm of the error, MINRES the Euclidean norm of the residual.
Like other Krylov [[iterative methods|2RPE77ACER]], it is used with a preconditioner, which must be symmetric positive definite to keep the symmetry.
The method is due to Paige and Saunders.
</document>

<document source="https://tamnd.github.io/math2/node/3J0BTXSCVN.md" tag="3J0BTXSCVN" kind="concept">
# Algebraic lattice

## Summary

An algebraic lattice is a complete lattice in which every element is the supremum of the [[compact elements|9F7TYES4KE]] below it.
Lattices of subgroups, of ideals, of subspaces and of congruences of algebras are algebraic, which explains the name.
The [[Birkhoff–Frink theorem|DZGNHKE3X1]] shows that the algebraic lattices are exactly the lattices of subalgebras of algebras.
Algebraic lattices are also called compactly generated lattices.

## Explanations

### L1 human

In the lattice of subgroups of a group, every subgroup is the union of its finitely generated subgroups.
An algebraic lattice is a complete lattice with the same feature: every element is built from "finite" elements below it.
The power set of a set is algebraic, since every set is the union of its finite subsets.
The lattice of closed subsets of the interval $[0, 1]$ is not algebraic.
Its only compact element is the empty set, so no nonempty closed set is a join of compact elements.
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# Mollifier

## Definition

Let $n \ge 1$.
A *mollifier* on $\R^n$ is a function $\varphi \colon \R^n \to \R$ such that:

1. $\varphi$ is a [[bump function|HS65DW0HXH]], that is, [[smooth|0V33BNKZ9S]] with compact support;
2. the support of $\varphi$ lies in the closed unit ball $\set{x : \abs{x} \le 1}$;
3. $\varphi(x) \ge 0$ for all $x$;
4. $\int_{\R^n} \varphi(x) \, dx = 1$.

For $\varepsilon > 0$ the rescaled function is $\varphi_\varepsilon(x) = \varepsilon^{-n} \varphi(x / \varepsilon)$.

## Remarks

Each $\varphi_\varepsilon$ is again smooth and nonnegative, has integral $1$ by the change of variables $x = \varepsilon y$, and has support in the ball of radius $\varepsilon$.
Some authors drop the sign condition 3 or replace condition 2 by compact support anywhere; the approximation properties remain.

Example: the standard mollifier is
$$\varphi(x) = \begin{cases} c \, \exp\left( \dfrac{-1}{1 - \abs{x}^2} \right), & \abs{x} < 1, \\ 0, & \abs{x} \ge 1, \end{cases}$$
with $c > 0$ chosen so that the integral is $1$.
It is positive inside the ball and vanishes outside, so conditions 2 and 3 hold.
It is smooth because $t \mapsto e^{-1/t}$ for $t > 0$, extended by $0$ for $t \le 0$, is smooth, and $\varphi(x) = c \, e^{-1/t}$ with $t = 1 - \abs{x}^2$.

Non-example: the Gaussian $\varphi(x) = \pi^{-n/2} e^{-\abs{x}^2}$ is smooth, positive and has integral $1$.
Its support is all of $\R^n$, so it fails condition 1 and condition 2.
It still gives an approximate identity, but convolution with it does not preserve compact support.
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# Polygamma function

## Summary

The polygamma function of order $m$ is the $m$-th derivative of the [[digamma function|HWRRXB0YXS]], so the $(m+1)$-th derivative of $\log \Gamma$.
For $m \ge 1$ it is a series of inverse powers, closely tied to the [[Hurwitz zeta function|G4SW7HJG88]].
Its values at $1$ are, up to sign and a factorial, values $\zeta(m+1)$ of the [[Riemann zeta function|1XWBY4N0VT]].
The case $m = 1$ is the [[trigamma function|5VGVPENA0F]].

## Explanations

### L1 human

Start from the logarithm of the gamma function and differentiate it again and again.
The first derivative is the digamma function, the second is the trigamma function, and the later ones are the polygamma functions.
From order one on, each of them is a simple sum such as $\sum_{k \ge 0} \frac{1}{(z+k)^2}$.
For example, the trigamma function at $1$ is $1 + \frac14 + \frac19 + \cdots = \frac{\pi^2}{6}$.
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# Derivation

## Summary

A derivation is a linear map on an algebra that satisfies the product rule $D(ab) = D(a) b + a D(b)$, see the [[definition|R1F0HCCE6F]].
It is the algebraic form of the [[derivative|0V1EJCT8G6]], and it makes sense without limits.
Derivations are basic in [[differential algebra|GA5PTQGC6G]], algebraic geometry and Lie theory.
The derivations of a Lie algebra obey the same rule for the Lie bracket.
The commutator of two derivations is again a derivation, so the derivations of an algebra form a Lie algebra.

## Explanations

### L1 human

The derivative of a product obeys the product rule: $(fg)' = f'g + fg'$.
A derivation is any map that adds up linearly and obeys this rule.
For polynomials, $d/dX$ is a derivation: it sends $X^3$ to $3X^2$ without using limits.
Another example on polynomials in $X$ and $Y$ is $X \, \partial/\partial Y$, which sends $Y^2$ to $2XY$.
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# Complex cobordism

## Definition

For $n \ge 0$, let $BU(n)$ be the [[classifying space|WSZ40H5E4T]] of the unitary group $U(n)$.
Let $\gamma_n \to BU(n)$ be the [[universal|1HDH716Z24]] complex vector bundle of rank $n$, and let $MU(n)$ be its [[Thom space|5MNYT3AFPH]].
The bundle $\gamma_n \oplus \C$ is classified by a map $BU(n) \to BU(n+1)$ covered by a bundle map $\gamma_n \oplus \C \to \gamma_{n+1}$, which induces a map $\Sigma^2 MU(n) \to MU(n+1)$ of Thom spaces.
The *complex cobordism spectrum* $MU$ is the [[spectrum|1GHF4EKBXS]] with
$$MU_{2n} = MU(n), \qquad MU_{2n+1} = \Sigma MU(n),$$
whose structure maps are the identity $\Sigma MU_{2n} = MU_{2n+1}$ and the map $\Sigma MU_{2n+1} = \Sigma^2 MU(n) \to MU(n+1) = MU_{2n+2}$.
*Complex cobordism* is the [[generalized cohomology theory|H28PSZR7XG]] represented by $MU$: for a pointed CW complex $X$ and $k \in \Z$,
$$\widetilde{MU}^k(X) = \operatorname{colim}_n \, [\Sigma^n X, MU_{n+k}]_*.$$

## Remarks

$MU$ is a [[Thom spectrum|6H8AGEN6NY]], and it is a commutative ring spectrum, with multiplication induced by the Whitney sum maps $BU(n) \times BU(m) \to BU(n+m)$.
By the Pontryagin–Thom construction, $\pi_n(MU)$ is the [[cobordism|T8BVGPPSZP]] group of closed smooth $n$-manifolds with a complex structure on the stable normal bundle.
Milnor and Novikov showed that $\pi_*(MU) \cong \Z[x_1, x_2, \ldots]$ with $\deg x_i = 2i$; in particular $\pi_n(MU) = 0$ for odd $n$.

Example: $\pi_2(MU) \cong \Z$, generated by the class of $\C P^1$ with its complex structure.

Non-example: $\R P^2$ is a closed smooth surface, but it is not orientable, so it has no stably complex structure and gives no class in $\pi_2(MU)$.
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# Oscillator representation

## Definition

Let $n \ge 1$ and let $W = \R^{2n}$ with the standard [[symplectic form|PQ7JJY0VPH]] $\omega$.
Let $\mathbb H = W \times \R$ be the [[Heisenberg group|ZZYCV5EX9J]] with product $(w, s)(w', s') = (w + w', s + s' + \tfrac12 \omega(w, w'))$.
Let $\rho$ be the Schrödinger representation of $\mathbb H$ on $L^2(\R^n)$ with central character $(0, s) \mapsto e^{is}$.
Let $\mathrm{Sp}(W) = \mathrm{Sp}_{2n}(\R)$ be the [[symplectic group|X12FS2JAP0]], which acts on $\mathbb H$ by automorphisms $g \cdot (w, s) = (gw, s)$.
Let $p \colon \mathrm{Mp}(W) \to \mathrm{Sp}(W)$ be the [[metaplectic group|8B0JM2YDE2]], the connected double cover.
The *oscillator representation* is the unitary representation $\omega_\psi$ of $\mathrm{Mp}(W)$ on $L^2(\R^n)$ such that
$$\omega_\psi(\tilde g) \, \rho(w, s) \, \omega_\psi(\tilde g)^{-1} = \rho(p(\tilde g) w, s)$$
for all $\tilde g \in \mathrm{Mp}(W)$ and $(w, s) \in \mathbb H$.

## Remarks

Such a representation exists and is unique.
For each $g \in \mathrm{Sp}(W)$, the representation $(w, s) \mapsto \rho(gw, s)$ has the same central character as $\rho$.
So by the [[Stone–von Neumann theorem|Z877XDBMQ3]] there is a unitary $T_g$ with $T_g \rho(w, s) T_g^{-1} = \rho(gw, s)$, unique up to a scalar of modulus $1$.
The map $g \mapsto T_g$ is a projective representation of $\mathrm{Sp}(W)$, and the scalars can be chosen so that it lifts to a genuine representation of $\mathrm{Mp}(W)$.
Two lifts differ by a character of $\mathrm{Mp}(W)$, which is trivial since $\mathrm{Mp}(W)$ is connected with semisimple Lie algebra, hence equal to its commutator subgroup.

Example: for $n = 1$, the dilation $g = \operatorname{diag}(a, a^{-1})$ with $a > 0$ lifts to the operator $\psi(x) \mapsto a^{-1/2} \psi(x / a)$, up to sign.
It is unitary on $L^2(\R)$.
Conjugation by it turns the position operator $x$ into $x / a$ and the momentum operator $-i \, d/dx$ into $a \cdot (-i \, d/dx)$, which is the action of a diagonal symplectic map on position and momentum.

Non-example: the operators $T_g$ cannot be normalised to give a representation of $\mathrm{Sp}(W)$ itself.
Only the double cover carries a genuine representation, so the condition on $\mathrm{Mp}(W)$ in the definition cannot be replaced by one on $\mathrm{Sp}(W)$.

The oscillator representation splits into two irreducible pieces, on the even and on the odd functions.
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# Berkovich space

## Definition

Let $K$ be a field that is complete for a [[non-Archimedean absolute value|1H5F1BP6XS]] $\abs{\cdot}$.
Let $A$ be a commutative [[Banach algebra|9KYWN94QDN]] over $K$ with unit, with norm $\norm{\cdot}$.
A *bounded multiplicative seminorm* on $A$ is a map $x \colon A \to [0, \infty)$, written $f \mapsto \abs{f(x)}$, such that:

1. $\abs{1(x)} = 1$ and $\abs{(f + g)(x)} \le \abs{f(x)} + \abs{g(x)}$ for all $f, g \in A$;
2. $\abs{(fg)(x)} = \abs{f(x)} \, \abs{g(x)}$ for all $f, g \in A$;
3. $\abs{f(x)} \le \norm{f}$ for all $f \in A$.

The *Berkovich spectrum* $\mathcal{M}(A)$ is the set of bounded multiplicative [[seminorms|FP5GVCN8GV]] on $A$.
It carries the weakest topology for which the map $x \mapsto \abs{f(x)}$ is continuous for every $f \in A$.
A $K$-affinoid algebra is a quotient of a [[Tate algebra|B1TEQ98JZD]] or of one of its versions with other radii.
For such $A$, $\mathcal{M}(A)$ is a *$K$-affinoid space*.
A *Berkovich analytic space* over $K$ is a space glued from $K$-affinoid spaces along affinoid domains.

## Remarks

Berkovich proved that $\mathcal{M}(A)$ is a nonempty compact Hausdorff space when $A \ne 0$.

Example: let $A = K\langle T \rangle$ be the Tate algebra of power series $\sum_n a_n T^n$ with $a_n \to 0$, with the Gauss norm $\norm{\sum a_n T^n} = \max_n \abs{a_n}$.
Take $a \in K$ with $\abs{a} \le 1$ and a real number $0 \le r \le 1$.
Expand $f \in A$ as $f = \sum_n c_n (T - a)^n$ and set $\abs{f(\zeta_{a,r})} = \max_n \abs{c_n} r^n$.
This is a point $\zeta_{a,r}$ of $\mathcal{M}(A)$, the point of the disc of centre $a$ and radius $r$.
Clause 3 holds because $\abs{c_n} \le \norm{f}$ when $\abs{a} \le 1$, and clause 2 is a form of Gauss's lemma.
For $a = 0$ and $r = 1$ it is the Gauss norm itself.
For $r = 0$ it is $f \mapsto \abs{f(a)}$, evaluation at $a$, a classical point.

Non-example: the supremum over the two-point set $\set{0, 1}$, namely $f \mapsto \max(\abs{f(0)}, \abs{f(1)})$, satisfies clauses 1 and 3 but not clause 2.
For $f = T$ and $g = T - 1$ both values are $1$, while $fg$ vanishes at both points and has value $0$.

Let $X = \operatorname{Spec} B$ be an affine variety of finite type over $K$.
Its analytification $X^{\mathrm{an}}$ is the set of multiplicative seminorms on $B$ that extend $\abs{\cdot}$ on $K$, with the same kind of topology.
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# Filter limit

## Definition

Let $S$ be a set and let $\mathcal{F}$ be a [[filter|15PF73BKY6]] on $S$.
Let $X$ be a [[topological space|0CQHFPGJ45]], let $f \colon S \to X$ be a map, and let $x \in X$.
The map $f$ *tends to $x$ along $\mathcal{F}$*, and $x$ is a *limit of $f$ along $\mathcal{F}$*, if $f^{-1}(U) \in \mathcal{F}$ for every [[neighbourhood|0ZCY0Z80B6]] $U$ of $x$.

## Remarks

Equivalently, the image filter $f(\mathcal{F}) = \set{B \subseteq X : f^{-1}(B) \in \mathcal{F}}$ [[converges|70M1VAPFN0]] to $x$.

Example: on $S = \N$ let $\mathcal{F}$ be the filter of sets that contain $\set{n : n \ge N}$ for some $N$; in Mathlib this is `Filter.atTop`.
A sequence $f(n) = x_n$ tends to $x$ along $\mathcal{F}$ exactly when for every neighbourhood $U$ of $x$ there is $N$ with $x_n \in U$ for all $n \ge N$.
This is the usual [[convergence of a sequence|0W4CN9E2W5]].

Non-example: the sequence $f(n) = (-1)^n$ in $\R$ has no limit along this filter.
For $x \ne 1$, some neighbourhood of $x$ misses $1$, so its preimage misses all even numbers and contains no tail.
For $x = 1$, the preimage of $(0, 2)$ is the set of even numbers, which contains no tail either.

Example: for $a \in \R$, the sets $V \setminus \set{a}$ with $V$ a neighbourhood of $a$ generate a filter on $\R \setminus \set{a}$, and the limit along it is the usual limit $\lim_{t \to a} f(t)$.

If $X$ is [[Hausdorff|0S1FJ8VEX9]] and $\emptyset \notin \mathcal{F}$, then $f$ has at most one limit along $\mathcal{F}$.
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# Honeycomb theorem

## Statement

Let $\Gamma \subseteq \R^2$ be a locally finite graph made of smooth curves, such that $\R^2 \setminus \Gamma$ has infinitely many bounded connected components, all of area $1$.
Let $C$ be the union of these bounded components.
For $r > 0$ let $B_r$ be the disc of radius $r$ about the origin.
Then
$$\limsup_{r \to \infty} \frac{\operatorname{perimeter}(C \cap B_r)}{\operatorname{area}(C \cap B_r)} \ge \sqrt[4]{12},$$
and the regular hexagonal tiling by hexagons of area $1$ attains equality.

## Remarks

Thomas Hales proved the theorem in 1999.
The proof reduces to a single inequality, the hexagonal isoperimetric inequality.
It bounds the perimeter of each cell from below, in terms of its area and of how far the cell is from a hexagon.
Curving an edge outward helps one cell and hurts its neighbour, and the inequality controls this trade.
Before Hales, László Fejes Tóth had proved the result under the hypothesis that all cells are convex.
Check of the constant: a [[regular hexagon|01KJJ9A400]] of side $s$ has area $\frac{3\sqrt{3}}{2} s^2$, so area $1$ means $s^2 = \frac{2}{3\sqrt{3}}$.
Each edge is shared by two cells, so the length per cell is $3s$, and $(3s)^2 = \frac{6}{\sqrt{3}} = 2\sqrt{3} = \sqrt{12}$, so $3s = \sqrt[4]{12} \approx 1.861$.
For comparison, a [[tiling|GKJK293DXS]] by unit squares has length $2$ per cell, which is larger.

## Explanations

### L1 human

Bees build their combs from hexagonal cells.
The theorem says that hexagons are the best choice.
Among all ways to divide the plane into regions of equal area, the hexagonal grid uses the least total length of walls.
This was believed since antiquity and was proved only at the end of the twentieth century.

### L2 human

The theorem is the two-dimensional case of the problem of partitions of least interface area.
The three-dimensional analogue, the Kelvin problem, is open.
Allowing curved edges is what makes the problem hard, since a cell can gain by bending its walls into its neighbours.
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# Maximal torus

## Summary

A maximal torus of a compact Lie group is a subgroup isomorphic to a torus $\mathbb{T}^k$ that is not contained in any larger such subgroup.
In a compact connected Lie group all maximal tori are conjugate, and every element lies in one of them.
Much of the theory of compact Lie groups is read off from a maximal torus and the action of the Weyl group on it.

## Explanations

### L1 human

A torus inside a group is a family of commuting rotations, like several independent clock hands.
A maximal torus is as large as such a family can be.
In the group $U(n)$ of unitary matrices, the diagonal matrices with entries on the unit circle form a maximal torus: there are $n$ independent angles.
Every unitary matrix can be diagonalised by a unitary change of basis, so every element of $U(n)$ lies in a conjugate of this torus.
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# Trace of an integral operator

## Summary

The trace of an integral operator with a continuous kernel $K(s, t)$ on $[a, b]$ is the integral of the kernel along the diagonal, $\int_a^b K(s, s) \, ds$.
It is the continuous analogue of the sum of the diagonal entries of a matrix.
In Fredholm theory, minus the trace is the coefficient of $\lambda$ in the Fredholm determinant $D(\lambda) = 1 - \lambda \int_a^b K(s, s) \, ds + \cdots$.
For a [[trace class operator|Z2CP530WDE]] with continuous kernel, it equals the operator trace and the sum of the [[eigenvalues|0BQAA2BK3Q]].

## Explanations

### L1 human

Think of the kernel $K(s, t)$ as a matrix with a continuum of rows $s$ and columns $t$.
The diagonal of this matrix is the set of points with $s = t$.
Summing the diagonal entries becomes integrating $K(s, s)$ over $s$.
For example, the kernel $K(s, t) = st$ on $[0, 1]$ has trace $\int_0^1 s^2 \, ds = 1/3$.
This operator sends $x$ to $s \mapsto s \int_0^1 t x(t) \, dt$, and its only nonzero eigenvalue is also $1/3$, with eigenfunction $s$.
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# Dirichlet kernel

## Summary

The Dirichlet kernel $D_N(x) = \sum_{n=-N}^{N} e^{inx}$ is the [[trigonometric polynomial|J3NJQ5FZJ5]] whose [[convolution|2Z29JN3VES]] with a periodic function gives the $N$-th partial sum of its [[Fourier series|G12KXDHPPS]].
In closed form $D_N(x) = \frac{\sin((N + \frac{1}{2}) x)}{\sin(x/2)}$.
Its integral over a period is constant, but the integral of its absolute value grows like $\log N$.
This growth is the source of most convergence problems of Fourier series, including the existence of continuous functions with divergent Fourier series.

## Explanations

### L1 human

To get the $N$-th partial Fourier sum of a function at a point, one averages the function against a fixed weight centred at that point, the Dirichlet kernel.
The kernel has a tall peak of height $2N + 1$ at $0$ and oscillates with slowly shrinking waves elsewhere, taking negative values.
Because of these negative lobes, the averaging is not a true local average, and it can be fooled by functions that oscillate in step with the kernel.
The kernel of [[Fejér|DCCA2T9GKH]], an average of Dirichlet kernels, has no negative values and behaves much better.
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# Square root of a positive operator

## Summary

Every [[positive operator|77020CY5ZW]] $A$ on a Hilbert space has exactly one positive operator $B$ with $B^2 = A$, written $A^{1/2}$ or $\sqrt{A}$.
It extends the [[square root of a positive semidefinite matrix|0BJVDCF38Y]] to infinite dimensions.
The square root defines the absolute value $\abs{T} = (T^*T)^{1/2}$ of an operator, which enters the polar decomposition and the definition of singular values.

## Explanations

### L1 human

A positive operator behaves like a nonnegative number, so it should have a square root that is again nonnegative.
For a diagonal operator with nonnegative entries $d_1, d_2, \ldots$, the square root is the diagonal operator with entries $\sqrt{d_1}, \sqrt{d_2}, \ldots$.
An operator can have many square roots, for instance $I$ and $-I$ both square to $I$, but only one of them is positive.
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# Robinson arithmetic

## Summary

Robinson arithmetic, often written $\mathsf{Q}$, is a first-order theory of the natural numbers with zero, successor, addition and multiplication.
It has seven axioms and no induction.
It is much weaker than the [[Peano axioms|0QFT7Z76N5]], yet strong enough to represent every computable function.
For this reason the incompleteness theorems and many undecidability results already hold for $\mathsf{Q}$ and every consistent theory that extends it.

## Explanations

### L1 human

Robinson arithmetic keeps only the recursive rules for addition and multiplication and a few facts about successor.
It can check any particular computation, such as $2 + 2 = 4$.
But without induction it cannot prove general laws such as $x + y = y + x$.
Its value is that it is a small, finite list of axioms that still captures all of computation.
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# Moving frame

## Summary

A moving frame is a basis of the tangent space that varies smoothly from point to point.
It may live on a manifold or along a curve or surface.
Élie Cartan developed the method of moving frames.
One expresses the derivatives of the frame in terms of the frame itself, and the coefficients become the invariants of the geometry.
The Frenet frame of a space curve, whose derivative gives curvature and torsion, is the classical example.
In modern terms a moving frame is a local section of the [[frame bundle|F4HWTRNXAP]].

## Explanations

### L1 human

Instead of using one fixed set of axes for the whole space, attach a small set of axes to each point and let it move with the point.
For a curve in space, one axis can point forward, one toward the centre of bending, and one perpendicular to both.
How these axes turn as one moves along the curve tells everything about its shape.
In the plane without the origin, the polar frame, with one arrow pointing away from the origin and one turning around it, is a moving frame.
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# Carleman matrix

## Summary

The Carleman matrix of a [[formal power series|033WGN00BF]] $f$ is the infinite matrix whose row $j$ lists the coefficients of the $j$-th power $f(x)^j$.
It turns composition of power series into [[matrix multiplication|032Q9FB8MT]].
If $g$ has no constant term, the matrix of $f \circ g$ is the product of the matrices of $f$ and $g$.
So, when $f$ has no constant term, the [[iterates|AYVTR2ES5B]] of $f$ correspond to powers of a matrix.
This gives a linear-algebra approach to iteration and to [[fractional iteration|5M3THA1S6E]].

## Explanations

### L1 human

Composing power series by hand is messy, while multiplying matrices is routine.
The Carleman matrix records a power series together with all its powers in one table.
With these tables, substituting one series into another becomes a matrix product.
For $f(x) = ax$ the table is diagonal with entries $1, a, a^2, \ldots$, and composing $ax$ with $bx$ multiplies the diagonals.
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# Truncation error

## Definition

Let $A$ be a real number defined as the [[limit|05M5SWSZZD]] of a [[real sequence|0CTV783HYT]] $(A_n)$, or of a family $A(h)$ as $h \to 0$.
Assume that each $A_n$ or $A(h)$ is computable by finitely many arithmetic operations.
The *truncation error* of the approximation $A_n$ is $A - A_n$, and that of $A(h)$ is $A - A(h)$.
Here $A_n$ and $A(h)$ are evaluated in exact arithmetic.

## Remarks

Example: let $f$ have a continuous second derivative and take the forward difference $A(h) = (f(x+h) - f(x))/h$, with limit $A = f'(x)$.
Taylor's theorem gives $A - A(h) = -\tfrac12 f''(\xi) h$ for some $\xi$ between $x$ and $x + h$, so the truncation error is of order $h$.
Example: for $A_n = \sum_{k=0}^{n} 1/k!$ and $A = e$, the truncation error is $\sum_{k > n} 1/k!$, which lies between $1/(n+1)!$ and $2/(n+1)!$ for $n \ge 1$.
Non-example: the difference between $1/3$ and its stored value $0.333$ in three-digit decimal arithmetic is a round-off error, since no limit process is cut off.
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# Minimum principle for harmonic functions

## Statement

Let $U \subseteq \R^n$ be a domain, that is, a connected open set, and let $u \colon U \to \R$ be [[harmonic|0DEB0JMN0W]].

1. (Strong minimum principle.) If there is $a \in U$ with $u(a) = \inf_U u$, then $u$ is constant on $U$.
2. (Weak minimum principle.) If $U$ is bounded and $u$ extends continuously to the closure $\overline{U}$, then $\min_{\overline{U}} u = \min_{\partial U} u$.

## Remarks

Proof: $-u$ is harmonic, and the [[maximum principle|T62HHSRHP5]] for $-u$ is exactly this statement for $u$.
Consequence: if $U$ is bounded, $u$ is continuous on $\overline{U}$ and $u \ge 0$ on $\partial U$, then $u \ge 0$ on $U$; if moreover $u$ is not constant, then $u > 0$ on $U$.
Example: on the unit disc, $u(x, y) = 1 + x$ is $\ge 0$ on the boundary circle, with value $0$ only at $(-1, 0)$, and it is positive inside.
Boundedness is needed in part 2: on the half-plane $\set{x > 0}$, $u = -x$ is $0$ on the boundary and negative inside.
For subharmonic functions only the maximum principle holds, and for superharmonic functions only the minimum principle.

## Explanations

### L1 human

A harmonic function cannot have a dip inside its domain.
Its smallest values on a bounded region are reached on the edge.
For a steady temperature distribution, the coldest point is on the boundary.

### L2 human

Together with the maximum principle, it says that the values of a harmonic function inside a bounded domain lie between its smallest and largest boundary values.
The positivity form, nonnegative boundary values give nonnegative solutions, is the comparison principle used everywhere in potential theory, for example to show that Green's functions are positive.
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# Gelfand's formula

## Statement

Let $n \ge 1$, let $A$ be an $n \times n$ complex matrix, and let $\norm{\cdot}$ be a [[matrix norm|54T6Y6DSY4]] on $\C^{n \times n}$.
Then the [[spectral radius|9V7YQA85PC]] of $A$ is the [[limit|05M5SWSZZD]]
$$\rho(A) = \lim_{k \to \infty} \norm{A^k}^{1/k}.$$

## Remarks

For a submultiplicative norm, $\rho(A)^k = \rho(A^k) \le \norm{A^k}$, which gives $\rho(A) \le \norm{A^k}^{1/k}$ for every $k$.
For $\varepsilon > 0$, the matrix $A / (\rho(A) + \varepsilon)$ has spectral radius less than $1$, so its powers tend to $0$ by the Jordan normal form; this gives $\norm{A^k} \le (\rho(A) + \varepsilon)^k$ for large $k$.
Any two norms are equivalent, and $c^{1/k} \to 1$ for $c > 0$, so the limit does not depend on the norm.
Example: $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ has $A^k = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$, and with the maximum row sum norm $\norm{A^k}^{1/k} = (1 + k)^{1/k} \to 1 = \rho(A)$, although $\norm{A} = 2$.
The same formula holds for elements of a complex Banach algebra; see the [[spectral radius formula|KX2XY7S3X6]].

## Explanations

### L1 human

The spectral radius is the largest absolute value of an eigenvalue.
A norm of $A$ can overestimate it, but the norms of the powers $A^k$ grow like $\rho(A)^k$.
Taking the $k$-th root and letting $k$ grow recovers $\rho(A)$ exactly, whatever norm is used.

### L2 human

The formula explains why $A^k \to 0$ if and only if $\rho(A) < 1$, which is the convergence criterion for iterative methods such as Jacobi and Gauss–Seidel.
It also shows that $\rho(A) \le \norm{A}$ for every submultiplicative norm, and that for every $\varepsilon > 0$ there is an operator norm with $\norm{A} \le \rho(A) + \varepsilon$.
In the Banach algebra setting, the same statement is proved with the resolvent and complex analysis instead of the Jordan form.
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<document source="https://tamnd.github.io/math2/node/3KY1RMEDMJ.md" tag="3KY1RMEDMJ" kind="statement">
# Birkhoff–von Neumann theorem

## Statement

Let $n \ge 1$ and let $A$ be an $n \times n$ [[doubly stochastic matrix|JJA90EB2PS]].
So $A$ has real nonnegative entries, and its rows and columns each sum to $1$.
Then $A$ is a [[convex combination|21SXYJYC59]] of $n \times n$ [[permutation matrices|0Q5SM60HTH]].
That is, there are permutation matrices $P_1, \dots, P_k$ and real numbers $\theta_1, \dots, \theta_k \ge 0$ with $\theta_1 + \cdots + \theta_k = 1$ such that
$$A = \theta_1 P_1 + \cdots + \theta_k P_k.$$

## Remarks

The proof is by induction on the number of nonzero entries of $A$.
Form the bipartite graph whose two sides are the rows and the columns, with an edge for each nonzero entry.
Any $r$ rows carry total mass $r$, and this mass lies in the columns joined to them, which carry at most $1$ each.
So [[Hall's marriage theorem|08WCB90WND]] gives a perfect matching.
It is a permutation matrix $P$ supported on nonzero entries of $A$.
If $\theta$ is the least entry of $A$ on this support and $A \ne P$, then $(A - \theta P)/(1 - \theta)$ is doubly stochastic with fewer nonzero entries.
For example,
$$\begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix} = \frac12 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \frac12 \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.$$
Equivalently, the set of doubly stochastic matrices, the [[Birkhoff polytope|ADEXY830S8]], is the [[convex hull|0TCHAS04WK]] of the $n!$ permutation matrices, and these are its vertices.

## Explanations

### L1 human

A doubly stochastic matrix spreads one unit from each row over the columns so that every column also receives one unit.
The theorem says that every such spreading is a mixture of perfect one-to-one assignments.
So a fractional assignment can always be written as a weighted average of whole assignments.

### L2 human

The theorem is the matrix form of Hall's and Kőnig's theorems on bipartite matchings.
It shows that the linear program for the assignment problem has a permutation matrix among its optimal solutions.
It is also used in the theory of majorization and in proofs of inequalities for permanents and eigenvalues.
</document>

<document source="https://tamnd.github.io/math2/node/3KZW2930V2.md" tag="3KZW2930V2" kind="concept">
# Ridge regression

## Summary

Ridge regression estimates the coefficients of a linear model by least squares with a penalty on the squared Euclidean norm of the coefficients.
The penalty shrinks the coefficients towards zero.
It makes the estimate unique and stable even when the regressors are nearly collinear or more numerous than the observations.
It is a [[shrinkage estimator|GPBSS1V3SS]]: it accepts some bias in exchange for a smaller variance.
In numerical analysis the same idea is called Tikhonov regularization.

## Explanations

### L1 human

When two regressors are almost the same, [[ordinary least squares|YM708KHV2W]] can give huge coefficients of opposite signs that nearly cancel.
Ridge regression adds a cost for large coefficients, so such wild solutions are no longer worth it.
A tuning parameter $\lambda$ sets the strength of the cost: $\lambda = 0$ is ordinary least squares and a very large $\lambda$ pushes all coefficients to zero.
With a single regressor and no intercept, the ridge slope is $\sum_i x_i y_i / (\sum_i x_i^2 + \lambda)$, a shrunk version of the least squares slope.
Unlike the [[lasso|KZ226116HW]], ridge regression does not set coefficients exactly to zero.
</document>

<document source="https://tamnd.github.io/math2/node/3M724K3B50.md" tag="3M724K3B50" kind="concept">
# Inverse semigroup

## Summary

An inverse semigroup is a [[semigroup|0XMEM406JM]] in which each element $a$ has exactly one element $a^{-1}$ with $a a^{-1} a = a$ and $a^{-1} a a^{-1} = a^{-1}$.
The model example is the semigroup of partial bijections of a set, just as the model group is a group of permutations.
By the Wagner–Preston theorem, every inverse semigroup embeds in a semigroup of partial bijections.
Inverse semigroups describe partial symmetries, and they appear in the theory of tilings and of operator algebras.

## Explanations

### L1 human

A group collects symmetries that move a whole object onto itself.
Many structures have only partial symmetries, which match one part of the object with another part.
Partial symmetries can be composed where they fit and reversed, and they form an inverse semigroup.
For example, the map that sends $1$ to $2$ and is undefined elsewhere is a partial bijection of $\set{1, 2, 3}$, and its inverse sends $2$ to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/3M8Y76E4AG.md" tag="3M8Y76E4AG" kind="concept">
# Jónsson terms

## Summary

Jónsson terms are a finite chain of ternary terms of a [[variety|AJNZJD8442]] that connect the first projection to the third by identities.
By Jónsson's theorem, a variety has Jónsson terms if and only if it is a [[congruence-distributive variety|9GZ7F6ETC6]].
They give a Mal'cev condition for congruence distributivity, so this property of all algebras of a variety reduces to the existence of finitely many terms.

## Explanations

### L1 human

Jónsson terms are a list of formulas with three inputs that step from the formula $x$ to the formula $z$.
Each formula returns $x$ when the first and the last input agree.
Neighbouring formulas agree when two particular inputs are equal.
For lattices, one middle step suffices: the majority formula, which returns the value that appears at least twice.
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<document source="https://tamnd.github.io/math2/node/3MCVMBAGD1.md" tag="3MCVMBAGD1" kind="concept">
# Brownian bridge

## Summary

A Brownian bridge is a Brownian motion on $[0, 1]$ forced to return to $0$ at time $1$.
It is a continuous centred [[Gaussian process|22WEM6RTW0]] with covariance $\min(s, t) - st$.
It is the limit of the uniform [[empirical process|XKFG1676CG]], which makes it central in goodness-of-fit tests such as the Kolmogorov–Smirnov test.

## Explanations

### L1 human

Imagine a Brownian path that starts at $0$, wanders at random, and is tied down so that it must be back at $0$ at time $1$.
This pinned random path is a Brownian bridge.
A simple recipe: take a Brownian path $W$ and subtract the straight line from $(0, 0)$ to $(1, W_1)$.
Near the two ends it is almost fixed, and in the middle, at $t = 1/2$, it varies most, with variance $1/4$.
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<document source="https://tamnd.github.io/math2/node/3MCZ97SW6V.md" tag="3MCZ97SW6V" kind="concept">
# Einstein field equations

## Summary

The Einstein field equations are the equations of general relativity.
They relate the curvature of a four-dimensional [[Lorentzian manifold|BY4QYYS788]], the spacetime, to the energy and momentum of matter in it.
For the metric they form a system of nonlinear second-order partial differential equations, which is hyperbolic after a suitable choice of coordinates.
Their solutions include the flat Minkowski space, the Schwarzschild black hole and the expanding cosmological models.

## Explanations

### L1 human

Gravity in Einstein's theory is not a force but the curvature of spacetime.
Matter and energy tell spacetime how to curve, and the curvature tells matter how to move.
The field equations are the precise form of the first half of this slogan.
Empty, flat spacetime with no matter is the simplest solution, and the field around a single star or black hole is the next.
</document>

<document source="https://tamnd.github.io/math2/node/3MDPTW005V.md" tag="3MDPTW005V" kind="definition">
# Linear congruential generator

## Definition

Let $m \ge 2$ be an integer, the modulus, and let $a$, $c$ and $X_0$ be integers with $0 < a < m$, $0 \le c < m$ and $0 \le X_0 < m$.
The *linear congruential generator* with multiplier $a$, increment $c$ and seed $X_0$ is a [[pseudorandom number generator|C5FG6ARWVV]].
Its state space is $\set{0, \ldots, m-1}$ and its states are
$$X_{n+1} = (a X_n + c) \bmod m, \qquad n \ge 0,$$
where $\bmod$ is the [[modulo operation|03BR7SAX0Q]].
Its outputs are the numbers $u_n = X_n / m \in [0, 1)$.
If $c = 0$ it is called a *multiplicative* congruential generator, or Lehmer generator.

## Remarks

Example: $m = 8$, $a = 5$, $c = 1$, $X_0 = 0$.
The states are $0, 1, 6, 7, 4, 5, 2, 3, 0, \ldots$, for instance $5 \cdot 7 + 1 = 36 \equiv 4 \pmod 8$; all $8$ residues appear, so the period is $m$.
Non-example: $X_{n+1} = X_n^2 \bmod m$ is not a linear congruential generator, because the map $x \mapsto x^2$ is not of the form $x \mapsto ax + c$.
The map $x \mapsto ax + c$ is an affine map of the ring of [[integers modulo $m$|03VFZ2WETT]], so the period is at most $m$.
The Hull–Dobell theorem treats $c \ne 0$.
The period equals $m$ for every seed if and only if three conditions hold.
They are: $c$ and $m$ are coprime, every prime factor of $m$ divides $a - 1$, and $4$ divides $a - 1$ if $4$ divides $m$.
In the example, $\gcd(1, 8) = 1$ and $a - 1 = 4$, so the conditions hold.
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<document source="https://tamnd.github.io/math2/node/3MGF59XBD3.md" tag="3MGF59XBD3" kind="definition">
# Complete intersection

## Definition

Let $k$ be an algebraically closed field.
Let $X \subseteq \mathbb{P}^n$ be a [[projective variety|A4H4CZ2V35]] of [[dimension|36X2WTHYCB]] $d$, and let $r = n - d$ be its codimension.
$X$ is a *complete intersection* if its [[homogeneous ideal|6ZBNGS0P98]] $I(X) \subseteq k[x_0, \dots, x_n]$ is generated by $r$ [[homogeneous polynomials|2C3PAQ7H50]].
In the same way, an [[affine variety|JERVQT4C12]] $X \subseteq k^n$ of dimension $d$ is a complete intersection if its [[ideal|HH6VEVM6Z0]] $I(X)$ is generated by $n - d$ polynomials.

## Remarks

By [[Krull's principal ideal theorem|SJZW2HN1SC]], every irreducible component of a nonempty set $V(f_1, \dots, f_s) \subseteq k^n$ has dimension at least $n - s$.
So $n - d$ is the least number of generators that $I(X)$ can have.
If $X \subseteq \mathbb{P}^n$ and $I(X)$ is generated by $r = n - d$ homogeneous polynomials $f_1, \dots, f_r$, then $f_1, \dots, f_r$ is a [[regular sequence|69R5R2FCS5]] in $k[x_0, \dots, x_n]$.
In both the affine and the projective case, the local rings of a complete intersection are [[complete intersection rings|EPM2HKGMZS]].
A weaker notion is a set-theoretic complete intersection: $X = V(f_1, \dots, f_r)$ as sets, without asking that the $f_i$ generate $I(X)$.

Example: if $f$ is an irreducible homogeneous polynomial of positive degree, the hypersurface $V(f) \subseteq \mathbb{P}^n$ has dimension $n - 1$ and ideal $(f)$, so it is a complete intersection.
A linear subspace defined by $r$ independent linear forms is a complete intersection of codimension $r$.

Non-example: the twisted cubic $C \subseteq \mathbb{P}^3$, the image of the [[Veronese embedding|DQA60Z3FD7]] $[s : t] \mapsto [s^3 : s^2 t : s t^2 : t^3]$, has dimension $1$ and codimension $2$.
Its ideal contains no nonzero linear form, because $s^3, s^2 t, s t^2, t^3$ are linearly independent.
Its quadrics form the $3$-dimensional space spanned by $xz - y^2$, $yw - z^2$ and $xw - yz$.
Indeed, this space is the kernel of the surjective map from the $10$ quadrics in $x, y, z, w$ to the $7$ forms of degree $6$ in $s, t$.
A set of homogeneous generators must therefore contain at least $3$ quadrics, so $C$ is not a complete intersection.
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<document source="https://tamnd.github.io/math2/node/3MHF6QBW4Q.md" tag="3MHF6QBW4Q" kind="statement">
# Number of points of a finite projective plane

## Statement

Let $\Pi$ be a finite [[projective plane|DANFNN8C4Y]] of [[order|K2YVMTGTBB]] $n$.
Then every point of $\Pi$ lies on exactly $n + 1$ lines, and $\Pi$ has exactly $n^2 + n + 1$ points and exactly $n^2 + n + 1$ lines.

## Remarks

Fix a point $P$.
It lies on $n + 1$ lines, and each of them has $n$ points other than $P$.
Every other point $Q$ lies on exactly one line through $P$, namely $PQ$, so the number of points is $1 + (n + 1) n = n^2 + n + 1$.
The count of lines follows by the same argument in the dual plane, which is again a projective plane of order $n$.

Example: the [[Fano plane|QQ8GW1ASZM]] has order $2$ and $4 + 2 + 1 = 7$ points.
The plane $PG(2, q)$ has $(q^3 - 1)/(q - 1) = q^2 + q + 1$ points, as it should.
In Mathlib the count of points is `Configuration.ProjectivePlane.card_points`.

## Explanations

### L1 human

Once one knows how many points lie on a line of a finite projective plane, one knows the size of the whole plane.
With $n + 1$ points on each line, there are $n^2 + n + 1$ points and the same number of lines.
For order $2$ this gives the $7$ points of the Fano plane, and for order $3$ it gives $13$.

### L2 human

The proof is a standard double count from a single point, and the same method gives the counts of flags and of triangles.
It shows that a projective plane of order $n$ is a symmetric block design with parameters $(n^2 + n + 1, n + 1, 1)$, a [[Steiner system|7BPSC4T4M3]] $S(2, n + 1, n^2 + n + 1)$.
This turns existence questions for planes into questions about designs, which is the setting of the Bruck–Ryser–Chowla theorem.
</document>

<document source="https://tamnd.github.io/math2/node/3MHP17HY7N.md" tag="3MHP17HY7N" kind="definition">
# Uniform convergence of series

## Definition

Let $S$ be a set, and let $f_n \colon S \to \C$ be functions for $n \ge 0$.
Let $s_N = \sum_{n=0}^{N} f_n$ be the partial sums of the [[series of functions|0MRE6XRE5V]] $\sum_{n=0}^{\infty} f_n$, and let $f \colon S \to \C$ be a function.
The series *converges uniformly* to $f$ on $S$ if the sequence $(s_N)$ converges [[uniformly|08FPABDQ4C]] to $f$ on $S$.
That is, for every $\varepsilon > 0$ there is an integer $N_0$ such that $\abs{s_N(x) - f(x)} < \varepsilon$ for all $N \ge N_0$ and all $x \in S$.

## Remarks

Equivalently, the series converges at each point of $S$ and $\sup_{x \in S} \abs{\sum_{n > N} f_n(x)} \to 0$ as $N \to \infty$.
A Cauchy form also holds: the series converges uniformly on $S$ if and only if for every $\varepsilon > 0$ there is $N_0$ with $\abs{\sum_{n=M+1}^{N} f_n(x)} < \varepsilon$ for all $N > M \ge N_0$ and all $x \in S$.
For a subset of a larger domain, this is [[uniform convergence on a set|08HJRTG5N7]] of the partial sums.

Example: let $0 < r < 1$ and $f_n(x) = x^n$ on $S = [0, r]$, with $f(x) = 1/(1 - x)$.
The tail is $\sum_{n > N} x^n = x^{N+1}/(1 - x) \le r^{N+1}/(1 - r)$ for all $x \in S$, and this bound tends to $0$.
So the series converges uniformly on $[0, r]$.

Non-example: the same series on $S = [0, 1)$ converges at each point to $1/(1 - x)$, but not uniformly.
For each $N$, the tail $x^{N+1}/(1 - x)$ tends to $\infty$ as $x \to 1^-$, so its supremum over $S$ is infinite.

The [[Weierstrass M-test|0NX5NWAFC9]] is the usual way to prove uniform convergence.

## Explanations

### L1 human

At each point $x$, the series is an ordinary series of numbers, and its partial sums approach $f(x)$.
The series converges uniformly when one number of terms is enough to make the error small at every point at once.
For $\sum x^n$, this works on $[0, r]$ with $r < 1$, but not on $[0, 1)$, where points near $1$ need more and more terms.
</document>

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# Deformation quantization

## Summary

Deformation quantization equips the smooth functions on a [[Poisson manifold|A2KQ49Y4NF]] with a noncommutative star product.
The star product is a formal power series in a parameter $\hbar$.
The commutator of the star product reproduces the [[Poisson bracket|2S0MXDT4YH]] at first order in $\hbar$.
The Moyal product on $\R^{2n}$ is the first example.
[[Kontsevich's formality theorem|XXA9D45E0N]] shows that every Poisson manifold admits a star product.

## Explanations

### L1 human

In classical mechanics, observables are functions on phase space, and they commute: $fg = gf$.
In quantum mechanics, observables are operators, and position and momentum do not commute.
Deformation quantization keeps the functions but changes their product, adding correction terms in powers of Planck's constant $\hbar$.
At $\hbar = 0$ the old commutative product comes back.
For position $x$ and momentum $p$ the new product satisfies $x * p - p * x = \hbar$, the formal version of the canonical commutation relation.
</document>

<document source="https://tamnd.github.io/math2/node/3MKZST79MK.md" tag="3MKZST79MK" kind="statement">
# Ultrafilter lemma

## Statement

Let $B$ be a [[Boolean algebra|0XADJKM699]], and let $F$ be a proper filter of $B$.
Then there is an [[ultrafilter|CQM84EW10G]] $U$ of $B$ with $F \subseteq U$.

## Remarks

Proof idea: consider the proper filters that contain $F$, ordered by inclusion.
This family is nonempty, since it contains $F$.
Every nonempty chain of them has an upper bound, its union, which is still a proper filter.
By [[Zorn's lemma|04RXTQA5ZA]] there is a maximal one, and it is an ultrafilter.
Applied to the power set of a set $X$, the lemma says that every filter on $X$ lies in an [[ultrafilter|J64Y3JC0RE]] on $X$.
For an infinite $X$ and the filter of cofinite sets, this gives non-principal ultrafilters.
Applied to the filter generated by an element $a \ne 0$, it gives an ultrafilter containing $a$, which is the key step of [[Stone's representation theorem|S4QJ395PA5]].
The lemma is not provable in ZF alone, but it is strictly weaker than the [[axiom of choice|0HTGB13FE2]] over ZF; it is equivalent to the Boolean prime ideal theorem.

## Explanations

### L1 human

A filter is a partial, consistent way of saying which elements are true.
The lemma says that such a partial decision can always be completed to a full decision about every element, without contradiction.
Its proof uses a form of the axiom of choice, because the completion is built by infinitely many free choices.

### L2 human

The ultrafilter lemma is the form of choice that much of algebra and topology needs.
It proves the compactness theorem of propositional logic, Tychonoff's theorem for compact Hausdorff spaces, and the representation of Boolean algebras as fields of sets.
For a countable Boolean algebra no choice is needed: one lists the elements and decides them one at a time.
</document>

<document source="https://tamnd.github.io/math2/node/3MP41FQHZ5.md" tag="3MP41FQHZ5" kind="statement">
# Gabriel–Popescu theorem

## Statement

Let $\mathcal{C}$ be a [[Grothendieck category|A7W468BQSM]] and let $U$ be a [[generator|CK08JW32WB]] of $\mathcal{C}$.
Let $R = \End(U)$ be the endomorphism [[ring|0B1261B5ED]] of $U$, so that $\Hom(U, X)$ is a right $R$-module for each object $X$, with $R$ acting by composition.
Let $G = \Hom(U, -) \colon \mathcal{C} \to \mathrm{Mod}\text{-}R$ be the [[hom functor|VJXS8AK67E]] into the [[category of right $R$-modules|JBSSXP4FFY]].
Then:

1. $G$ is [[full and faithful|GRG8WEBSH5]];
2. $G$ has a [[left adjoint|TZHF6ZM92S]] $T \colon \mathrm{Mod}\text{-}R \to \mathcal{C}$;
3. $T$ is an [[exact functor|S1G61TDW0K]].

## Remarks

The functor $T$ is the colimit-preserving functor with $T(R) = U$, often written $T(M) = M \otimes_R U$.
The hard part of the proof is the exactness of $T$; it uses that filtered colimits are exact in a Grothendieck category and that $U$ generates.
As a consequence, $\mathcal{C}$ is equivalent to the quotient of $\mathrm{Mod}\text{-}R$ by the [[Serre subcategory|CSD6W0JV0P]] of modules $M$ with $T(M) = 0$.
That subcategory is closed under arbitrary direct sums, so it is a localizing subcategory.
Example: for $\mathcal{C} = \mathrm{Mod}\text{-}S$ and $U = S$, one gets $R \cong S$, the functor $G$ is an equivalence, and the localizing subcategory is $0$.
Example: the category of abelian sheaves on a topological space is a Grothendieck category, so the theorem presents it as a quotient of a module category.

## Explanations

### L1 human

A Grothendieck category is an abelian category with good infinite colimits and a generating object, like a category of modules or of sheaves.
The theorem says that every such category sits inside a category of modules in a very controlled way.
It is obtained from the modules by declaring a certain class of modules to be zero.

### L2 human

The theorem is an analogue of the [[Freyd–Mitchell embedding theorem|8V296XR2M3]] for large categories with a generator.
The embedding $G$ is left exact but in general not exact; in exchange, it has an exact left adjoint, so $\mathcal{C}$ is a reflective subcategory of $\mathrm{Mod}\text{-}R$ with exact reflector.
This makes it a basic tool for transferring module-theoretic results to Grothendieck categories.
It is also the starting point of the description of Grothendieck categories as localizations of module categories.
</document>

<document source="https://tamnd.github.io/math2/node/3MQTWKBBBT.md" tag="3MQTWKBBBT" kind="concept">
# Holomorphic map

## Summary

A holomorphic map from an open subset of $\C^n$ to $\C^m$ is a map whose $m$ components are [[holomorphic functions of several variables|507C34W0ZT]].
Holomorphic maps are the morphisms of [[several complex variables|BH57D2FBNR]] and of complex manifolds.
Their derivative at each point is a complex linear map, given by the complex Jacobian matrix $(\partial F_k / \partial z_j)$.
For $n = m = 1$ they are the holomorphic functions of one variable, and [[holomorphic maps between Riemann surfaces|T19P7J53A1]] are the same notion for complex curves.

## Explanations

### L1 human

A holomorphic function returns one complex number; a holomorphic map returns several, each depending holomorphically on the inputs.
Near each point it looks like a complex linear map, up to a small error.
For example, $F(z_1, z_2) = (z_1 + z_2^2, \ e^{z_1})$ is a holomorphic map from $\C^2$ to $\C^2$.
Compositions of holomorphic maps are again holomorphic.
</document>

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# Pluricomplex Green function

## Summary

Let $\Omega \subseteq \C^n$ be a domain and $w \in \Omega$.
The pluricomplex Green function $g_\Omega(\cdot, w)$ is the largest negative [[plurisubharmonic function|FF2P7ER3QT]] on $\Omega$ with a logarithmic pole at $w$.
It was introduced by Maciej Klimek as the analogue of the classical [[Green's function|M36SD56MWQ]] of a planar domain.
On bounded hyperconvex domains it is continuous and solves a homogeneous [[complex Monge–Ampère|MXRC891TQS]] equation off the pole, with a Dirac mass at the pole.
On bounded convex domains it is expressed through the [[Kobayashi distance|DY8QQ5GTTX]], by a theorem of Lempert.
The version with pole at infinity for a compact set is the [[Siciak extremal function|QBJ9R06GYE]].

## Explanations

### L1 human

In the plane, the Green function of a domain with pole at $w$ is harmonic away from $w$.
It tends to $-\infty$ like $\log\abs{z - w}$ at the pole and tends to $0$ at the boundary.
In several complex variables, harmonic is replaced by plurisubharmonic, and one takes the largest such function that is negative and has the logarithmic pole.
For the unit ball and the pole at the origin, the pluricomplex Green function is simply $\log\abs{z}$.
It measures how far a point is from the pole in a way that holomorphic maps cannot increase.
</document>

<document source="https://tamnd.github.io/math2/node/3MSGFJM2NX.md" tag="3MSGFJM2NX" kind="concept">
# Extreme point

## Summary

An extreme point of a convex set is a point of the set that lies strictly inside no segment between two other points of it.
Extreme points are the corners of a convex set.
The [[Krein–Milman theorem|RSRAY0Q68S]] says that a compact convex set in a locally convex space is rebuilt from its extreme points by taking closed convex hulls.
They also matter in linear optimization, where a linear function on a compact convex set in $\R^n$ reaches its maximum at an extreme point.

## Explanations

### L1 human

Think of a convex shape, such as a filled polygon or a disc.
A point is extreme if you cannot place it in the middle of a segment that stays in the shape and has other endpoints.
For a filled square the extreme points are the four corners.
For a filled disc every point of the boundary circle is extreme, and no inner point is.
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# Jordan–Kinderlehrer–Otto scheme

## Summary

The Jordan–Kinderlehrer–Otto scheme, or JKO scheme, is a time discretization of a gradient flow in the space of probability measures with the [[Wasserstein distance|VEY29CS26X]].
At each step it minimizes the energy plus a penalty for moving far, in Wasserstein distance, from the previous measure.
Richard Jordan, David Kinderlehrer and Felix Otto introduced it in 1998 and proved that it converges to solutions of the [[Fokker–Planck equation|FVQWR7NS00]].
It showed that the [[heat equation|VP2HN86VAG]] is the gradient flow of the entropy for the Wasserstein distance, and started the theory of Wasserstein gradient flows.

## Explanations

### L1 human

Imagine a drop of ink spreading in water.
The JKO scheme describes this spreading in small time steps.
At each step, the ink chooses a new distribution that lowers its energy, an entropy term that favours spreading out.
It also pays a cost for how far its mass has to travel.
When the steps become very short, the result is the heat equation, so diffusion is a way of decreasing entropy as fast as possible for the cost of moving mass.
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<document source="https://tamnd.github.io/math2/node/3MXR03096V.md" tag="3MXR03096V" kind="statement">
# Ratner's theorems

## Statement

Let $G$ be a connected [[Lie group|7JBRGCVJRB]], let $\Gamma$ be a [[lattice|BTYF5Y5PFY]] in $G$, and let $X = G/\Gamma$.
An element $u \in G$ is *unipotent* if $\mathrm{Ad}(u)$ is a unipotent linear map of the Lie algebra of $G$.
Let $H$ be a connected closed subgroup of $G$ generated by one-parameter subgroups of unipotent elements.
A probability measure $\mu$ on $X$ is *homogeneous* if there are a closed subgroup $L$ of $G$ and $x \in X$ with the following properties.
The orbit $Lx$ is closed, and $\mu$ is the $L$-invariant [[probability measure|059TJN0NZ1]] supported on $Lx$.
Then:

1. (Measure classification) every $H$-invariant and ergodic Borel probability measure on $X$ is homogeneous;
2. (Orbit closures) for every $x \in X$, the [[closure|0QEAJB91GZ]] of $Hx$ equals $Lx$ for a closed subgroup $L \supseteq H$ such that $Lx$ carries an $L$-invariant probability measure;
3. (Equidistribution) let $(u_t)_{t \in \R}$ be a one-parameter subgroup of unipotent elements, and let $x \in X$.
   Then there is a homogeneous measure $\mu_x$ such that, for every bounded continuous $f$ on $X$,
$$\lim_{T \to \infty} \frac{1}{T} \int_0^T f(u_t x) \, dt = \int_X f \, d\mu_x.$$

## Remarks

Marina Ratner proved these theorems around 1990 and 1991.
The proof of the measure classification uses the polynomial divergence of nearby unipotent orbits: two such orbits drift apart slowly and mainly along the centraliser of the flow.
Orbit closures and equidistribution then follow from the measure classification together with the non-divergence results of Margulis and Dani.

Example: $G = \mathrm{SL}_2(\R)$, $\Gamma = \mathrm{SL}_2(\Z)$, and $H = \set{\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} : t \in \R}$.
Then $X$ is the unit tangent bundle of the modular surface, and $H$ acts by the horocycle flow.
Each $H$-orbit is either a periodic closed horocycle or dense in $X$, a fact known before Ratner; here $L = H$ or $L = G$.

The unipotent hypothesis is necessary.
For the diagonal subgroup of $\mathrm{SL}_2(\R)$, acting by the geodesic flow, there are orbits whose closures are fractal sets, not orbits of any subgroup.

## Explanations

### L1 human

Imagine a flow on a finite-volume space built from a Lie group, such as the motion of horocycles on a hyperbolic surface.
For unipotent flows, Ratner proved that the behaviour is always very orderly.
Each orbit either closes up or fills out a nice smooth piece of the space evenly.
There are no strange fractal-shaped orbit closures, which do occur for other flows like the geodesic flow.

### L2 human

These theorems are central tools of homogeneous dynamics.
Clause 2 is [[Raghunathan's conjecture|ME4ZSRAT5Z]].
They have applications in number theory, such as the Oppenheim conjecture on values of indefinite quadratic forms, which Margulis had proved earlier by related methods.
They also give counting results for integer points on homogeneous varieties.
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# Ergodic hypothesis

## Definition

Let $H \colon \R^{2n} \to \R$ be a smooth Hamiltonian and let $\varphi_t$ be the flow of the [[Hamiltonian system|0EDKGA2FAK]] of $H$ on the [[phase space|CPEC2NW2KF]] $\R^{2n}$.
Let $E$ be a regular value of $H$ such that the energy surface $\Sigma_E = H^{-1}(E)$ is compact.
The flow preserves $\Sigma_E$ and a finite measure $\mu_E$ on it, the Liouville (microcanonical) measure $\mu_E = d\sigma / \norm{\nabla H}$, where $\sigma$ is the surface measure.
Normalise $\mu_E$ to a probability measure.
The system satisfies the *ergodic hypothesis* on $\Sigma_E$ if for every $g \in L^1(\Sigma_E, \mu_E)$ and for $\mu_E$-almost every $x \in \Sigma_E$,
$$\lim_{T \to \infty} \frac{1}{T} \int_0^T g(\varphi_t(x)) \, dt = \int_{\Sigma_E} g \, d\mu_E .$$

## Remarks

By Birkhoff's ergodic theorem, this holds exactly when the flow on $\Sigma_E$ is [[ergodic|AZRPHS478B]] for $\mu_E$: every measurable invariant subset of $\Sigma_E$ has measure $0$ or $1$.
Example: for $H(q, p) = \tfrac12 (p^2 + q^2)$ on $\R^2$ and $E > 0$, the energy surface is a circle, the flow is a rotation at constant speed, and $\mu_E$ is the normalised arc length.
Every orbit runs once around the circle in time $2\pi$, so time averages equal the circle average and the hypothesis holds.
Non-example: for two uncoupled oscillators, $H = \tfrac12(p_1^2 + q_1^2) + \tfrac12(p_2^2 + q_2^2)$, each energy $h_1 = \tfrac12(p_1^2 + q_1^2)$ is conserved.
The set where $h_1 < E/2$ is then an invariant subset of $\Sigma_E$ of measure strictly between $0$ and $1$, so the hypothesis fails.

## Explanations

### L1 human

The definition makes precise the slogan "time average equals space average".
One fixes the total energy, follows a typical state forever, and compares the long-run average of a measured quantity with its average over all states of that energy.
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# von Mises–Fisher distribution

## Definition

Let $p \ge 2$ and let $S^{p-1} \subseteq \R^p$ be the unit [[sphere|T62369RFH6]] with its surface measure $\sigma$.
Let $\mu \in S^{p-1}$ and $\kappa > 0$.
Let $I_\nu$ be the modified Bessel function of the first kind of order $\nu$.
The *von Mises–Fisher distribution* $\operatorname{vMF}(\mu, \kappa)$ is the [[distribution|0DVDCMA2DK]] on $S^{p-1}$ with this [[density|0ZY9DX8FBF]] with respect to $\sigma$:
$$f(x) = C_p(\kappa) \exp(\kappa \mu^T x), \qquad C_p(\kappa) = \frac{\kappa^{p/2 - 1}}{(2\pi)^{p/2} I_{p/2 - 1}(\kappa)}.$$
For $\kappa = 0$ it is defined as the uniform distribution on $S^{p-1}$.
The vector $\mu$ is the *mean direction* and $\kappa$ is the *concentration parameter*.

## Remarks

Example: for $p = 3$, $I_{1/2}(\kappa) = \sqrt{2/(\pi \kappa)} \sinh \kappa$, so $C_3(\kappa) = \kappa / (4 \pi \sinh \kappa)$.
Check: in spherical coordinates with $\mu$ as north pole, $\int_{S^2} e^{\kappa \cos \theta} \, d\sigma = 2\pi \int_0^\pi e^{\kappa \cos \theta} \sin \theta \, d\theta = 4 \pi \sinh(\kappa)/\kappa$.
Non-example: a density proportional to $\exp(\kappa (\mu^T x)^2)$ with $\kappa > 0$ is not von Mises–Fisher.
It takes the same value at $x$ and $-x$, so its maximum is attained at both $\mu$ and $-\mu$, while a von Mises–Fisher density has a single maximum.
The density is largest at $x = \mu$ and smallest at $x = -\mu$, and it depends on $x$ only through the angle to $\mu$.
For $p = 2$, writing $x = (\cos \theta, \sin \theta)$ gives the von Mises density $e^{\kappa \cos(\theta - \theta_0)} / (2 \pi I_0(\kappa))$.
The maximum likelihood estimate of $\mu$ from a sample is its mean direction.
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# Milnor–Thurston kneading theory

## Statement

Let $f \colon [0, 1] \to [0, 1]$ be a [[continuous|0X3WGX95FE]] unimodal map.
So there is a turning point $c \in (0, 1)$ such that $f$ is strictly increasing on $[0, c]$ and strictly decreasing on $[c, 1]$.
Assume that $f^n(c) \neq c$ for all $n \ge 1$.
For $x \in [0, 1]$ let $\varepsilon(x) = 1$ if $x < c$, and $\varepsilon(x) = -1$ if $x > c$.
The *kneading determinant* of $f$ is the [[formal power series|033WGN00BF]]
$$D(t) = 1 + \sum_{n=1}^{\infty} \varepsilon(f(c)) \, \varepsilon(f^2(c)) \cdots \varepsilon(f^n(c)) \, t^n.$$
Let $\ell(f^n)$ be the number of maximal intervals on which $f^n$ is monotone (the lap number of $f^n$).
Then:

1. The limit $s = \lim_{n \to \infty} \ell(f^n)^{1/n}$ exists, $1 \le s \le 2$, and the [[topological entropy|BDZYBXBFDE]] of $f$ is $h(f) = \log s$.
2. The power series $D(t)$ converges for $\abs{t} < 1$ and has no zero in the disc $\abs{t} < 1/s$.
3. If $s > 1$, then $D(1/s) = 0$, so $1/s$ is the smallest positive zero of $D$.
4. If $s > 1$, then $f$ is [[semiconjugate|AEXJNRBASV]] to a piecewise linear unimodal map $g$ whose slopes are $s$ and $-s$. That is, there is a continuous nondecreasing surjection $h \colon [0, 1] \to [0, 1]$ with $h \circ f = g \circ h$.

## Remarks

The coefficients of $D(t)$ are $\pm 1$, which gives convergence for $\abs{t} < 1$.
The sign sequence $\varepsilon(f(c)), \varepsilon(f^2(c)), \dots$, the kneading sequence, records on which side of the turning point the orbit of the critical value falls.
Milnor and Thurston show that it determines the combinatorics of all orbits: the lap numbers, the numbers of periodic points, and the entropy can all be read off from it.

Example: the full [[tent map|D0A22MEB1F]] $T(x) = 1 - \abs{1 - 2x}$ with $c = 1/2$.
Here $T(c) = 1 > c$ and $T^n(c) = 0 < c$ for $n \ge 2$, so the signs are $-1, 1, 1, \dots$ and
$$D(t) = 1 - t - t^2 - t^3 - \cdots = \frac{1 - 2t}{1 - t}.$$
Its only zero is $t = 1/2$, and indeed $\ell(T^n) = 2^n$, so $s = 2$ and $h(T) = \log 2$.

## Explanations

### L1 human

A unimodal map folds an interval once, like kneading dough.
Everything about its orbits is encoded in one list of signs: where the image of the fold point lands, left or right of the fold, after each step.
Milnor and Thurston turned this list into a power series and showed that its first zero gives the entropy, the exponential rate at which the map creates complexity.

### L2 human

The theorem reduces the topological dynamics of unimodal maps, such as the logistic family, to symbolic data.
Part 4 says that, up to collapsing some intervals, every unimodal map with positive entropy is a tent-like map of constant slope, so the entropy is visible as a slope.
Kneading theory was a main tool in the proof that topological entropy of the logistic family $x \mapsto r x (1 - x)$ is a nondecreasing function of $r$.
It also explains the order in which periodic orbits appear in such families, which refines [[Sharkovskii's theorem|YBCZ9VY7SX]].
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# Pushforward measure

## Summary

The pushforward measure moves a [[measure|PR11BZQQPG]] along a [[measurable function|YYNCJSR0PJ]] from one space to another.
The new measure gives a set the old measure of its preimage.
In probability theory, the distribution of a random variable is the pushforward of the probability measure along the variable.
It also gives the abstract change of variables formula for integrals.

## Explanations

### L1 human

Think of the measure as sand spread over a space $X$, and of the map $f$ as a rule that moves each grain to a point of $Y$.
The pushforward is the new pile of sand on $Y$.
To weigh a region $B$ of $Y$, weigh all the grains of $X$ that land in $B$.
For example, let $X = \set{1, 2, 3, 4, 5, 6}$ with mass $1/6$ at each point, and let $f(x) = 0$ for even $x$ and $f(x) = 1$ for odd $x$.
The pushforward puts mass $1/2$ at $0$ and mass $1/2$ at $1$.
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# Forcing

## Definition

A *forcing notion* is a [[partial order|0QV2JS9RJ3]] $(P, \le)$ with a greatest element $1$.
Its elements are called *conditions*, and $q \le p$ is read "$q$ is stronger than $p$".

Let $M$ be a countable transitive model of [[ZFC|0HE5WDQ9FJ]] with $P \in M$.
A *$P$-name* is a set $\tau$ of pairs $(\sigma, p)$, where $\sigma$ is a $P$-name and $p \in P$; this is a definition by recursion on rank.
For a filter $G \subseteq P$, the *value* of a name is defined by recursion:
$$\tau^G = \set{\sigma^G : (\sigma, p) \in \tau \text{ for some } p \in G}.$$
For a [[generic filter|WEDP73Y7GK]] $G$ over $M$, the *forcing extension* is $M[G] = \set{\tau^G : \tau \in M \text{ is a } P\text{-name}}$.
Let $p \in P$, let $\varphi(x_1, \ldots, x_n)$ be a formula, and let $\tau_1, \ldots, \tau_n \in M$ be names.
The condition $p$ *forces* $\varphi(\tau_1, \ldots, \tau_n)$, written $p \Vdash \varphi(\tau_1, \ldots, \tau_n)$, if $M[G] \models \varphi(\tau_1^G, \ldots, \tau_n^G)$ for every generic $G$ over $M$ with $p \in G$.

## Remarks

The basic theorem of forcing says that $M[G]$ is a countable transitive model of ZFC, that $M \subseteq M[G]$ and $G \in M[G]$, and that $M$ and $M[G]$ have the same ordinals.
The forcing theorem says that a statement holds in $M[G]$ if and only if some $p \in G$ forces it.
It also says that the relation $\Vdash$ restricted to each formula is definable in $M$.
Example: *Cohen forcing* is the set of finite partial functions from $\omega$ to $\set{0, 1}$, ordered by reverse inclusion, with $1 = \emptyset$.
For a generic $G$, the union $\bigcup G$ is a function from $\omega$ to $\set{0, 1}$ that is not in $M$.
A non-example of a forcing notion is a partial order without a greatest element; this is a minor point, since one can add a new top element.
Some authors use preorders instead of partial orders, or write $p \le q$ for "$p$ is weaker", so the notation must be checked in each source.
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# Ill-posedness of real Laplace inversion

## Definition

Let $L^2(0, \infty)$ be the space of square-[[integrable|DWJF7ABPR1]] functions on $(0, \infty)$ with the norm $\norm{f}_2 = \left( \int_0^\infty \abs{f(t)}^2 \, dt \right)^{1/2}$.
For $f \in L^2(0, \infty)$, its [[Laplace transform|05XT5AVJBK]] $\mathcal{L} f(s) = \int_0^\infty f(t) e^{-s t} \, dt$ is defined for real $s > 0$, and $\mathcal{L}$ is a bounded linear operator from $L^2(0, \infty)$ to $L^2(0, \infty)$.
The *real Laplace inversion problem* is: given $F \in L^2(0, \infty)$, find $f \in L^2(0, \infty)$ with $\mathcal{L} f = F$ on $(0, \infty)$.
*Ill-posedness* of this problem means that it is not [[well posed|FZ64956KX2]] in the sense of Hadamard.
Here the solution is unique when it exists, but the inverse map $\mathcal{L} f \mapsto f$ is not continuous.
That is, there are $f_n$ with $\norm{f_n}_2$ bounded away from $0$ and $\norm{\mathcal{L} f_n}_2 \to 0$.

## Remarks

Uniqueness holds by [[Lerch's uniqueness theorem|V3YTF5WJHY]].
Instability, with the check: let $f_n(t) = e^{-t} \sin(n t)$.
Then $\norm{f_n}_2^2 = \int_0^\infty e^{-2t} \sin^2(n t) \, dt = \frac{n^2}{4(1 + n^2)} \ge \frac{1}{8}$ for $n \ge 1$.
Its transform is $\mathcal{L} f_n(s) = \frac{n}{(s + 1)^2 + n^2}$, and $\norm{\mathcal{L} f_n}_2^2 \le \int_{-\infty}^{\infty} \frac{n^2}{(u^2 + n^2)^2} \, du = \frac{\pi}{2 n} \to 0$.
So no estimate $\norm{f}_2 \le C \norm{\mathcal{L} f}_2$ can hold.
Existence also fails for most data: the range of $\mathcal{L}$ consists of restrictions of holomorphic functions, so a generic $F \in L^2(0, \infty)$ is not a transform at all.
Contrast: inversion from values of $F$ on a vertical line in the complex plane, by the Bromwich integral, is a Fourier inversion and is stable in $L^2$.
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# Complemented subspace

## Definition

Let $X$ be a [[Banach space|06GZ956657]] over $\R$ or $\C$, and let $M$ be a [[closed|0ZCK8DQYN1]] [[linear subspace|00YC1D652K]] of $X$.
The subspace $M$ is *complemented* in $X$ if there is a closed linear subspace $N$ of $X$ such that $M \cap N = \set{0}$ and $M + N = X$.
In other words, $M$ and $N$ are [[complementary subspaces|0M83V17KHX]].

## Remarks

Equivalently, $M$ is complemented if and only if there is a continuous linear map $P \colon X \to X$ with $P^2 = P$ and $P(X) = M$, a bounded [[projection|033DA1WE8Y]] onto $M$.
Given such $P$, take $N = \ker P$; given $N$, the algebraic projection onto $M$ along $N$ has closed graph, so it is continuous by the [[closed graph theorem|GXM8BX8YM7]].
Example: in a [[Hilbert space|HM2HHE3J51]] every closed subspace $M$ is complemented by its orthogonal complement $M^\perp$, which is closed with $M \cap M^\perp = \set{0}$ and $M + M^\perp$ equal to the whole space.
Every finite-dimensional subspace of a Banach space is complemented: extend a dual basis by the [[Hahn–Banach theorem|QB5AE3VYZZ]] to build the projection.
Non-example: $c_0$ is a closed subspace of $\ell^\infty$ that is not complemented; every closed subspace $N$ with $c_0 \cap N = \set{0}$ has $c_0 + N \ne \ell^\infty$.
A theorem of Lindenstrauss and Tzafriri says that a Banach space in which every closed subspace is complemented is isomorphic to a Hilbert space.
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# Closed subgroup theorem

## Statement

Let $G$ be a [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{g}$ and [[exponential map|H3QX1ZFC8K]] $\exp$.
Let $H$ be a [[subgroup|0X1JXT47GA]] of $G$ that is a [[closed subset|0ZCK8DQYN1]] of $G$.
Then $H$ is an embedded submanifold of $G$, and with this smooth structure $H$ is an embedded [[Lie subgroup|NM8XCPS1Q3]] of $G$.
Its Lie algebra is
$$\mathfrak{h} = \set{X \in \mathfrak{g} : \exp(tX) \in H \text{ for all } t \in \R}.$$

## Remarks

The proof first shows that $\mathfrak{h}$ is a linear subspace, using the limits $\exp(X + Y) = \lim_n (\exp(X/n)\exp(Y/n))^n$ and closedness of $H$.
Then one chooses a complement $\mathfrak{m}$ of $\mathfrak{h}$ in $\mathfrak{g}$ and shows that $(Y, X) \mapsto \exp(Y)\exp(X)$ on $\mathfrak{m} \times \mathfrak{h}$ is a chart near $e$ in which $H$ is the slice $Y = 0$.
The key point is that $H$ has no elements of the form $\exp(Y)$ with small nonzero $Y \in \mathfrak{m}$; otherwise a limit argument gives a nonzero element of $\mathfrak{h} \cap \mathfrak{m}$.
Example: $SO(n) = \set{A \in GL_n(\R) : A^T A = I, \det A = 1}$ is closed, so it is a Lie group; its Lie algebra is the space of skew-symmetric matrices.
The closedness hypothesis is necessary: the image of $t \mapsto (e^{it}, e^{i\sqrt2 t})$ in the torus $S^1 \times S^1$ is a subgroup that is dense and not closed, and it is not an embedded submanifold.

## Explanations

### L1 human

To check that a group of matrices is a Lie group, one would normally need to build charts and check smoothness.
This theorem removes the work: any subgroup of a Lie group that is closed as a set is automatically a smooth submanifold and a Lie group.
So all the usual groups cut out by equations, such as orthogonal, unitary and symplectic groups, are Lie groups at once.

### L2 human

The theorem is often stated for matrix groups and attributed to von Neumann in that case and to Élie Cartan in general.
A typical use: the stabiliser $H$ of a point under a smooth action is closed, hence a Lie subgroup.
Then the quotient $G / H$ is a manifold, a homogeneous space.
The formula for $\mathfrak{h}$ is the practical way to compute Lie algebras: for $SL_n(\R)$ it gives the matrices of trace $0$, because $\det e^{tX} = e^{t \tr X}$.
It also gives Cartan's theorem that a continuous homomorphism of Lie groups is smooth, by applying it to the graph.
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# Brouwer's characterization of the Cantor space

## Statement

Let $X$ be a [[topological space|0CQHFPGJ45]].
Then $X$ is [[homeomorphic|0QT00PB5G2]] to the [[Cantor space|W9VYJRBF9N]] $\set{0, 1}^{\N}$ if and only if $X$ is nonempty, [[compact|0TVJZYB6SH]], [[metrizable|091GJ3FYD7]], [[totally disconnected|0RJSBR9GW1]] and has no isolated points.

## Remarks

A compact metrizable totally disconnected space has a base of clopen sets.
With this and the absence of isolated points, one splits $X$ into two nonempty clopen pieces, each piece again into two, and so on, with diameters tending to $0$.
Sending a point to the sequence of left or right choices that leads to it gives a homeomorphism onto $\set{0, 1}^{\N}$.
For example, the product $\set{0,1}^{\N} \times \set{0,1}^{\N}$ satisfies the conditions, so it is homeomorphic to the Cantor space.
No condition can be dropped.
The rationals $\Q$ are metrizable, totally disconnected and without isolated points, but not compact.
The space $\set{0, 1}$ is compact, metrizable and totally disconnected, but its points are isolated.
The interval $[0, 1]$ satisfies all conditions except total disconnectedness.
A product of uncountably many copies of $\set{0, 1}$ satisfies all conditions except metrizability.

## Explanations

### L1 human

The theorem says that the Cantor space is the only compact metric space that is pure dust and has no lonely points.
So many sets built in quite different ways, such as the middle-thirds Cantor set or products of Cantor sets, are all the same space in disguise.

### L2 human

The theorem is used to recognise Cantor spaces in analysis, dynamics and descriptive set theory, for instance the perfect compact subsets of the line that contain no interval.
It combines with the fact that every nonempty compact metric space is a continuous image of the Cantor space.
A similar characterization holds for the [[Baire space of sequences|P2WS16HKD7]], by the Alexandrov–Urysohn theorem.
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# Additive combinatorics

## Summary

Additive combinatorics studies finite subsets of abelian groups, such as sets of integers, and how they behave under addition.
Typical questions ask how large a [[sumset|XNBN31HQWB]] $A + B$ must be, and what a set with a small sumset looks like.
Others ask when a set contains additive patterns such as arithmetic progressions.
It combines combinatorics, Fourier analysis, ergodic theory and number theory.
Its results include [[Szemerédi's theorem|B23B0QEMN5]], [[Freiman's theorem|KEPFYE6WYZ]] and the [[Green–Tao theorem|H8BAHWSEMC]].

## Explanations

### L1 human

Take a finite set of numbers and add every pair of them, allowing a number to be added to itself.
Additive combinatorics asks how many different sums you get, and what that number tells you about the set.
For example, $\set{1, 2, 3, 4}$ gives only the sums $2, \dots, 8$, seven values, because it is very structured.
A random-looking set like $\set{1, 2, 5, 11}$ gives ten different sums.
The field also asks when a large set must contain evenly spaced numbers.
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# Eisenstein series

## Definition

Let $k \ge 4$ be an even integer and let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]].
The *Eisenstein series of weight $k$* is the function $G_k \colon \mathbb{H} \to \C$ given by
$$G_k(\tau) = \sum_{(m, n) \in \Z^2 \setminus \set{(0, 0)}} \frac{1}{(m\tau + n)^k}.$$
The *normalized Eisenstein series* is $E_k = G_k / (2\zeta(k))$, where $\zeta$ is the [[Riemann zeta function|1XWBY4N0VT]].

## Remarks

For $k \ge 3$ the series converges absolutely and locally uniformly on $\mathbb{H}$.
So $G_k$ is holomorphic, and it is a [[modular form|GWKN5VG78A]] of weight $k$ for $\mathrm{SL}_2(\Z)$.
With $q = e^{2 \pi i \tau}$, $\sigma_{k-1}(n) = \sum_{d \mid n} d^{k-1}$ (a [[divisor function|04J4Z73J5B]]) and $B_k$ the $k$-th [[Bernoulli number|0KC00K7KY4]]:
$$E_k(\tau) = 1 - \frac{2k}{B_k} \sum_{n \ge 1} \sigma_{k-1}(n) q^n.$$
Example: $B_4 = -1/30$ gives $E_4 = 1 + 240 \sum_{n \ge 1} \sigma_3(n) q^n$, and $B_6 = 1/42$ gives $E_6 = 1 - 504 \sum_{n \ge 1} \sigma_5(n) q^n$.
Non-example: for odd $k$, the terms for $(m, n)$ and $(-m, -n)$ cancel, so the sum is $0$; this is why $k$ is even.
For $k = 2$ the double series does not converge absolutely, and the function $E_2$ obtained by summing in a fixed order is not a modular form.
</document>

<document source="https://tamnd.github.io/math2/node/3P246RHXMF.md" tag="3P246RHXMF" kind="statement">
# Thue's theorem

## Statement

Every packing of equal circles in the plane $\R^2$ has upper [[packing density|4P5PSWADJD]] at most
$$\frac{\pi}{\sqrt{12}} = \frac{\pi}{2\sqrt 3} \approx 0.9069.$$
The hexagonal [[circle packing|S360JBEYKD]], in which the centres form a triangular lattice and each circle touches six others, has density exactly $\pi/\sqrt{12}$.

## Remarks

For the hexagonal packing with radius $1$, the centres form a lattice whose fundamental parallelogram has area $2\sqrt 3$.
Each cell carries one disc of area $\pi$, and this gives the density.
For the upper bound, one standard proof joins the centres of a saturated packing into a Delaunay triangulation.
Each triangle has sides at least $2$.
One shows that the discs cover at most the fraction $\pi/\sqrt{12}$ of each triangle, with equality for the equilateral triangle of side $2$.
Lagrange proved the bound for lattice packings in 1773.
Axel Thue gave proofs for general packings in 1890 and 1910, and László Fejes Tóth gave the first proof regarded as complete in 1940.

## Explanations

### L1 human

Pack equal coins on a big table as tightly as possible.
The best you can do is the honeycomb pattern, where each coin touches six others, and it covers about $90.7\%$ of the table.
The theorem says that no clever irregular arrangement can beat it.

### L2 human

Thue's theorem is the two-dimensional case of the sphere packing problem; its three-dimensional case is the [[Kepler conjecture|AFMBQT06ZV]].
The local argument works in the plane because the densest local arrangement, six circles around one, fits together into a global tiling.
In three dimensions this fails, which is one reason why the Kepler conjecture is so much harder.
</document>

<document source="https://tamnd.github.io/math2/node/3P5R3WQPT6.md" tag="3P5R3WQPT6" kind="definition">
# Perspective projection

## Definition

Let $E$ be Euclidean 3-space, let $O \in E$ be a point, the centre, and let $\pi$ be a [[plane|0V3XAMF3C1]] not containing $O$, the picture plane.
Let $P \ne O$ be a point such that the [[line|0MP9BCF6DP]] $OP$ is not parallel to $\pi$.
The *perspective projection* from $O$ onto $\pi$ sends $P$ to the point where $OP$ meets $\pi$.
In coordinates with $O = (0, 0, 0)$ and $\pi = \set{z = f}$ for some $f > 0$, it is
$$(x, y, z) \mapsto \left( \frac{f x}{z}, \frac{f y}{z} \right), \qquad z \ne 0.$$

## Remarks

Example: the two rails $(1, -1, t)$ and $(-1, -1, t)$, $t > 0$, are parallel lines in the direction $(0, 0, 1)$.
Their images $(f/t, -f/t)$ and $(-f/t, -f/t)$ lie on the lines $y = -x$ and $y = x$, which meet at $(0, 0)$, and both images tend to $(0, 0)$ as $t \to \infty$.
So $(0, 0)$ is the vanishing point of the direction $(0, 0, 1)$; in general the vanishing point of a direction $v$ with $v_3 \ne 0$ is the image of $v$ itself, $(f v_1 / v_3, f v_2 / v_3)$.
Non-example: the orthographic projection $(x, y, z) \mapsto (x, y)$ is not a perspective projection, since its projecting lines are parallel and do not pass through one centre.
It maps parallel lines to parallel lines.
The points of a line not through $O$ that lie in the domain map to points of one line, and lines parallel to $\pi$ keep their direction.
In [[homogeneous coordinates|FS9FRJ17D2]] the map is linear: $[x : y : z : 1] \mapsto [f x : f y : z]$, which is how graphics systems implement it.
</document>

<document source="https://tamnd.github.io/math2/node/3PGXH0T1K6.md" tag="3PGXH0T1K6" kind="statement">
# Lyusternik's theorem on tangent spaces

## Statement

Let $X$ and $Y$ be real [[Banach spaces|06GZ956657]], let $U \subseteq X$ be open, and let $g \colon U \to Y$ be continuously [[Fréchet differentiable|06EH97V2A7]].
Let $x_0 \in U$ with $g(x_0) = 0$, and suppose that the derivative $Dg(x_0) \colon X \to Y$ is surjective.
Let $M = \set{x \in U : g(x) = 0}$.
Then for every $h$ in the [[kernel|0S23E6YE83]] of $Dg(x_0)$ there are $\varepsilon > 0$ and a map $r \colon (-\varepsilon, \varepsilon) \to X$ with
$$\lim_{t \to 0} \frac{\norm{r(t)}}{\abs{t}} = 0 \quad \text{and} \quad x_0 + t h + r(t) \in M \text{ for all } \abs{t} < \varepsilon.$$
Consequently, the set of tangent vectors to $M$ at $x_0$ is exactly $\ker Dg(x_0)$.

## Remarks

The converse inclusion is easy: if $x_0 + th + r(t) \in M$ with $r(t) = o(t)$, then differentiating $g(x_0 + th + r(t)) = 0$ at $t = 0$ gives $Dg(x_0) h = 0$.
The main inclusion is proved by a Newton-type iteration that corrects $x_0 + th$ back onto $M$.
The [[open mapping theorem|03YEM7JEDN]] gives a constant $C$ such that each $v \in Y$ equals $Dg(x_0) w$ for some $w$ with $\norm{w} \le C \norm{v}$, and this makes the corrections converge.
When $\ker Dg(x_0)$ has a closed complement, for example when $X$ is a Hilbert space, the result also follows from the implicit function theorem.

Surjectivity cannot be dropped.
For $g \colon \R^2 \to \R$, $g(x_1, x_2) = x_1^2 + x_2^2$, one has $M = \set{0}$ and $Dg(0) = 0$, so $\ker Dg(0) = \R^2$, while the only tangent vector to the single point $M$ is $0$.

## Explanations

### L1 human

The zero set of a nice map looks, near a regular point, like a flat space: the kernel of the derivative.
The theorem says that each direction in this kernel is really followed by a curve inside the zero set, up to an error that is small compared with the step.
It works in infinite dimensions, where no coordinates are available.

### L2 human

This is the key step in the proof of [[Lagrange multipliers on Banach spaces|Q7QK3PF806]].
In the calculus of variations, $X$ is a function space and $g$ encodes constraints such as a fixed integral, so the theorem describes the admissible variations of a constrained problem.
The condition that $Dg(x_0)$ is onto plays the role of the constraint qualification.
</document>

<document source="https://tamnd.github.io/math2/node/3PNJYFPJ3P.md" tag="3PNJYFPJ3P" kind="definition">
# Homogeneous ideal

## Definition

Let $R = \bigoplus_{n \ge 0} R_n$ be a [[graded ring|8KB4SQGC0C]].
An [[ideal|096Z9MP2ME]] $I$ of $R$ is *homogeneous* if for every $f \in I$ with homogeneous components $f_n \in R_n$, each $f_n$ lies in $I$.

## Remarks

Equivalent conditions: $I = \bigoplus_{n \ge 0} (I \cap R_n)$; and $I$ is [[generated|0GSW76Q14N]] by homogeneous elements.
Example: in $k[x, y, z]$, with $k$ a field, the ideal $(x^2 - yz)$ is homogeneous, since it is generated by the homogeneous element $x^2 - yz$.
Non-example: in $k[x]$ the ideal $(x - 1)$ is not homogeneous; it contains $x - 1$, whose component $-1$ of degree $0$ is a unit and does not lie in the proper ideal $(x - 1)$.
If $I$ is homogeneous, then $R/I$ is graded by the images of the $R_n$.
The ideal $R_+ = \bigoplus_{n \ge 1} R_n$ is homogeneous; it is called the irrelevant ideal.
</document>

<document source="https://tamnd.github.io/math2/node/3PR407YKHN.md" tag="3PR407YKHN" kind="definition">
# Spectral radius

## Definition

Let $A$ be an $n \times n$ complex [[square matrix|0JXZKX526R]], and let $\sigma(A) \subset \C$ be its [[spectrum|0XPX5NC67E]], the set of its [[eigenvalues|0BQAA2BK3Q]].
The *spectral radius* of $A$ is
$$\rho(A) = \max \set{\abs{\lambda} : \lambda \in \sigma(A)}.$$

## Remarks

Example: $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ has characteristic polynomial $x^2 + 1$, so $\sigma(A) = \set{i, -i}$ and $\rho(A) = 1$.

The spectral radius is not a norm on matrices.
For $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, the only eigenvalue is $0$, so $\rho(N) = 0$ although $N \ne 0$.

For every [[operator norm|06EKW1AKXF]] $\norm{\cdot}$, $\rho(A) \le \norm{A}$, and [[Gelfand's formula|3KX0KQ7V05]] gives $\rho(A) = \lim_{k \to \infty} \norm{A^k}^{1/k}$.
The powers $A^k$ tend to $0$ if and only if $\rho(A) < 1$.
For a [[normal matrix|0MR0BYF0K2]], $\rho(A)$ equals the operator norm induced by the Euclidean norm.
The same definition applies to a linear map of a finite-dimensional complex vector space.

## Explanations

### L1 human

The spectral radius is the size of the largest eigenvalue.
It says how fast repeated application of the matrix makes vectors grow or shrink in the long run.
</document>

<document source="https://tamnd.github.io/math2/node/3PRW76471J.md" tag="3PRW76471J" kind="statement">
# Diagonal lemma

## Statement

Let $T$ be a [[theory|92SWXST1ZM]] in the language of arithmetic that contains [[Robinson arithmetic|3K1AKGXF8A]] $\mathsf{Q}$.
Fix a [[Gödel numbering|MFJXPAF0R0]], and let $\ulcorner \psi \urcorner$ be the numeral of the code of $\psi$.
For every [[formula|HYJ10AF6RT]] $\varphi(x)$ with exactly one free variable $x$, there is a [[sentence|DKFGQTRYZE]] $\psi$ such that
$$T \vdash \psi \leftrightarrow \varphi(\ulcorner \psi \urcorner).$$

## Remarks

Proof idea: the diagonal function $d$, which sends the code of a formula $\theta(x)$ to the code of $\theta(\ulcorner \theta \urcorner)$, is computable.
So it is represented in $\mathsf{Q}$ by a formula $D(x, y)$.
Let $\theta(x)$ be $\exists y\, (D(x, y) \wedge \varphi(y))$ and let $\psi$ be $\theta(\ulcorner \theta \urcorner)$.
Then $\psi$ says that $\varphi$ holds of $d(\ulcorner \theta \urcorner)$, which is the code of $\psi$, and $\mathsf{Q}$ proves this equivalence.

Example: with $\varphi(x) = \neg \mathrm{Prov}_T(x)$, the lemma gives a sentence that says of itself "I am not provable in $T$", the Gödel sentence of [[Gödel's incompleteness theorems|X9WYNYE29C]].
No consistency assumption on $T$ is needed.

## Explanations

### L1 human

The lemma says that arithmetic can build sentences that talk about themselves.
For any property of sentences that arithmetic can express, there is a sentence that says "I have this property".
This self-reference is the engine behind the incompleteness theorems.

### L2 human

The diagonal lemma is the common core of [[Gödel's incompleteness theorems|X9WYNYE29C]], [[Tarski's undefinability theorem|YRV6SYK2BY]] and [[Löb's theorem|3HD30YG3DG]].
Each of them applies the lemma to a well-chosen formula: unprovability, falsity, or a provability implication.
It is an arithmetic version of [[Kleene's recursion theorem|F3Z30M606H]], which lets a program refer to its own code.
</document>

<document source="https://tamnd.github.io/math2/node/3PRZ02APFA.md" tag="3PRZ02APFA" kind="concept">
# Positive-definite function on a group

## Summary

A positive-definite function on a group $G$ is a complex function $\varphi$ on $G$ with the following property.
For all $g_1, \dots, g_n \in G$, the matrix $\big(\varphi(g_i^{-1}g_j)\big)_{i,j}$ is [[positive semidefinite|0KX66CZMBB]].
The diagonal matrix coefficients $g \mapsto \ip{\pi(g)v}{v}$ of [[unitary representations|74BPYJNZTM]] are positive-definite, and conversely every continuous positive-definite function arises in this way.
This correspondence links functions on a group with its representations, and it underlies Bochner's theorem and the Gelfand–Raikov theorem.

## Explanations

### L1 human

On the real line, the function $\cos x$ is positive-definite, while $-1$ is not.
The idea is that $\varphi(g^{-1}h)$ behaves like an inner product between the elements $g$ and $h$, so sums $\sum \bar c_i c_j \varphi(g_i^{-1}g_j)$ behave like squared lengths and cannot be negative.
Indeed $\cos(y - x) = \cos x \cos y + \sin x \sin y$ is the dot product of the unit vectors at angles $x$ and $y$.
Every positive-definite function is, in this sense, a hidden inner product, and the hidden space is a unitary representation of the group.
</document>

<document source="https://tamnd.github.io/math2/node/3PS1JB7V95.md" tag="3PS1JB7V95" kind="definition">
# Heteroscedasticity

## Definition

Let $\varepsilon_1, \dots, \varepsilon_n$ be [[random variables|0PNRKMRC55]] with finite [[variances|0477R9PYAE]].
They are *homoscedastic* if $\operatorname{Var}(\varepsilon_1) = \cdots = \operatorname{Var}(\varepsilon_n)$, and *heteroscedastic* otherwise.

Now let $y_i = f(x_i) + \varepsilon_i$ be a regression model with random regressors $x_i$.
Its errors are *homoscedastic* if the [[conditional variance|0VGPX25NXQ]] $\operatorname{Var}(\varepsilon_i \mid x_i)$ is a constant $\sigma^2$, not depending on $x_i$.
They are *heteroscedastic* otherwise.

## Remarks

Example: let $x_i = i$ and $\varepsilon_i \sim N(0, \sigma^2 x_i^2)$ for $i = 1, \dots, n$ with $n \ge 2$ and $\sigma > 0$.
Then $\operatorname{Var}(\varepsilon_1) = \sigma^2 \ne 4 \sigma^2 = \operatorname{Var}(\varepsilon_2)$, so the errors are heteroscedastic.

Non-example: if $\varepsilon_i = x_i u_i$ with $u_i \sim N(0, 1)$ independent of $x_i$ and $x_i \in \set{-1, 1}$, then $\operatorname{Var}(\varepsilon_i \mid x_i) = x_i^2 = 1$ for all $i$, so the errors are homoscedastic, although they depend on $x_i$.

In [[linear regression|41FVC4P5JB]] with heteroscedastic, uncorrelated errors, the covariance of the least squares estimator is $(X^T X)^{-1} X^T \Omega X (X^T X)^{-1}$ with $\Omega = \operatorname{diag}(\operatorname{Var}(\varepsilon_i))$.
The heteroscedasticity-consistent estimator of White replaces $\Omega$ by $\operatorname{diag}(e_i^2)$, where $e_i$ are the residuals.
The spelling *heteroskedastic* is also used.
</document>

<document source="https://tamnd.github.io/math2/node/3PZ3TMW86C.md" tag="3PZ3TMW86C" kind="definition">
# Boolean model

## Definition

Let $d \ge 1$ and $\lambda > 0$.
Let $\Phi = \set{x_1, x_2, \ldots}$ be a homogeneous [[Poisson point process|05TG0SE4HT]] in $\R^d$ with intensity $\lambda$; its points are the *germs*.
Let $K_1, K_2, \ldots$ be [[independent and identically distributed|00MRN0FARV]] random nonempty [[compact sets|00R0X1WGPX]] in $\R^d$, independent of $\Phi$; they are the *grains*.
Let $K$ denote a random set with their common law.
Let $V_d$ be Lebesgue measure on $\R^d$, and let $B$ be the [[closed unit ball|05DSV40EB9]] centred at $0$.
Assume $E[V_d(K + B)] < \infty$, where $K + B = \set{k + b : k \in K, b \in B}$.
The *Boolean model* with these germs and grains is the random set
$$\Xi = \bigcup_{i \ge 1} (x_i + K_i).$$

## Remarks

Under the integrability assumption, only finitely many grains $x_i + K_i$ meet each bounded set almost surely, so $\Xi$ is a random closed set.
Its law is invariant under translations of $\R^d$.

Example: let $d = 2$ and let every grain be the closed disc of radius $r > 0$ centred at $0$.
Then $V_2(K + B) = \pi (r + 1)^2$ is finite, so the assumption holds.
A point $y$ is uncovered exactly when $\Phi$ has no point in the disc of radius $r$ around $y$, which has probability $e^{-\lambda \pi r^2}$.
So the covered fraction of the plane is $p = 1 - e^{-\lambda \pi r^2}$.
In general, $P(y \in \Xi) = 1 - e^{-\lambda E[V_d(K)]}$ for every $y \in \R^d$.

Non-example: put the same discs at the points of the lattice $\Z^2$ instead of a Poisson process.
The union is a union of discs, but the germs do not form a Poisson point process, so it is not a Boolean model.

The Boolean model is also called the Poisson germ-grain model.
</document>

<document source="https://tamnd.github.io/math2/node/3Q6CFPQZDF.md" tag="3Q6CFPQZDF" kind="concept">
# Thue equation

## Summary

A Thue equation is a [[Diophantine equation|08B1RTW7VZ]] $f(x, y) = m$ with $m$ a nonzero integer.
Here $f$ is an irreducible homogeneous polynomial in two variables, with integer coefficients and degree at least $3$.
Axel Thue proved in 1909 that such an equation has only finitely many integer solutions.
His proof came from a sharpening of Liouville's bound on rational approximation of algebraic numbers, the line that led to the [[Thue–Siegel–Roth theorem|R9JZA3G2NX]].
Later, [[Baker's theorem|WMBJSNG57F]] gave explicit bounds on the solutions.

## Explanations

### L1 human

Take an equation such as $x^3 - 2y^3 = 1$ and look for whole number solutions.
If $(x, y)$ is a large solution, then $x/y$ is an extremely good rational approximation of $\sqrt[3]{2}$.
Thue showed that algebraic numbers like $\sqrt[3]{2}$ cannot have too many such good approximations.
So there are only finitely many solutions.
For degree $2$ this fails: the Pell equation $x^2 - 2y^2 = 1$ has infinitely many solutions.
</document>

<document source="https://tamnd.github.io/math2/node/3Q99FQH4QX.md" tag="3Q99FQH4QX" kind="definition">
# Quasitriangular Hopf algebra

## Definition

Let $k$ be a field and let $H$ be a [[Hopf algebra|7XFYP0N84H]] over $k$ with coproduct $\Delta$.
Let $\tau \colon H \otimes H \to H \otimes H$ be the flip $a \otimes b \mapsto b \otimes a$, and let $\Delta^{\mathrm{op}} = \tau \circ \Delta$.
For $R = \sum_i a_i \otimes b_i$ in the [[tensor product|0BF2DE2SYK]] $H \otimes H$, put $R_{12} = \sum_i a_i \otimes b_i \otimes 1$, $R_{13} = \sum_i a_i \otimes 1 \otimes b_i$ and $R_{23} = \sum_i 1 \otimes a_i \otimes b_i$.
The pair $(H, R)$ is a *quasitriangular Hopf algebra* if $R$ is invertible in $H \otimes H$ and:

1. $R \, \Delta(x) \, R^{-1} = \Delta^{\mathrm{op}}(x)$ for all $x \in H$.
2. $(\Delta \otimes \id)(R) = R_{13} R_{23}$.
3. $(\id \otimes \Delta)(R) = R_{13} R_{12}$.

The element $R$ is the *universal R-matrix*.

## Remarks

Example: let $H = k[G]$ be the group algebra of a group $G$, with $\Delta(g) = g \otimes g$, and let $R = 1 \otimes 1$.
Then $\Delta^{\mathrm{op}} = \Delta$, so clause 1 holds, and both sides of clauses 2 and 3 equal $1 \otimes 1 \otimes 1$.
Non-example: let $G$ be a finite non-abelian group and let $H = k^G$ be the dual Hopf algebra of functions on $G$.
Since $k^G$ is commutative, $R \, \Delta(x) \, R^{-1} = \Delta(x)$ for every invertible $R$, so clause 1 would force $\Delta = \Delta^{\mathrm{op}}$.
This fails because $k^G$ is not cocommutative, so no $R$ makes $k^G$ quasitriangular.
The axioms imply $(\varepsilon \otimes \id)(R) = 1 = (\id \otimes \varepsilon)(R)$ for the counit $\varepsilon$.
They also imply the Yang–Baxter equation $R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$.
A *triangular* Hopf algebra is a quasitriangular one with $R_{21} R = 1$, where $R_{21} = \tau(R)$.
</document>

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# Nonparametric statistics

## Definition

Let $\mathcal{P}$ be a [[statistical model|K3AMMSY8DD]], a set of probability distributions on a sample space.
The model $\mathcal{P}$ is *nonparametric* if it is not a [[parametric model|MSC0MR8VMH]].
That is, it cannot be indexed in a natural way by a parameter in a subset of $\R^k$ for a fixed $k$.
*Nonparametric statistics* is the study of inference in nonparametric models.
It also covers *distribution-free* procedures: tests or intervals whose law under the null hypothesis is the same for all distributions in a nonparametric class.

## Remarks

"Natural" means that the indexing map has the regularity required of a parametric model.
Any set of the size of $\R$ is in bijection with $\R$, so a bare bijection does not count.
Example: the set of all distributions on $\R$ with a continuous distribution function is nonparametric.
Example: the sign test of $H_0 \colon$ the median is $0$ is distribution-free.
Under $H_0$, the number of positive observations is binomial with parameters $n$ and $1/2$, for every continuous law with median $0$.
Non-example: the family $\set{\mathcal{N}(\mu, \sigma^2) : \mu \in \R, \sigma > 0}$ is parametric, with $k = 2$.
Semiparametric models mix both kinds, with a finite-dimensional parameter of interest and an infinite-dimensional nuisance part.
</document>

<document source="https://tamnd.github.io/math2/node/3QDDSSH4FB.md" tag="3QDDSSH4FB" kind="statement">
# Convergence of the Neumann series for Volterra kernels

## Statement

Let $a < b$, let $\Delta = \set{(t, s) : a \le s \le t \le b}$, and let $K \colon \Delta \to \C$ be continuous with $M = \max_\Delta \abs{K}$.
Define the [[iterated kernels|9QNBFERMHD]] on $\Delta$ by $K_1 = K$ and
$$K_{n+1}(t, s) = \int_s^t K(t, u) \, K_n(u, s) \, du, \quad n \ge 1.$$
Then the following hold.

1. Each $K_n$ is continuous on $\Delta$ and $\abs{K_n(t, s)} \le M^n \dfrac{(t - s)^{n - 1}}{(n - 1)!}$ for all $(t, s) \in \Delta$.
2. For every $\lambda \in \C$, the series $R(t, s; \lambda) = \sum_{n \ge 1} \lambda^{n - 1} K_n(t, s)$ converges absolutely and uniformly on $\Delta$, and uniformly for $\lambda$ in bounded sets.
3. The sum $R$ is continuous on $\Delta$, entire in $\lambda$, and $\abs{R(t, s; \lambda)} \le M e^{\abs{\lambda} M (t - s)}$.
4. For every continuous $f \colon [a, b] \to \C$, the [[Liouville–Neumann series|H1HWSWME6B]] $x = f + \sum_{n \ge 1} \lambda^n \int_a^t K_n(t, s) f(s) \, ds$ converges uniformly on $[a, b]$, and its sum
$$x(t) = f(t) + \lambda \int_a^t R(t, s; \lambda) \, f(s) \, ds$$
solves the [[Volterra equation of the second kind|ATPSAHYG7C]] $x(t) = f(t) + \lambda \int_a^t K(t, s) \, x(s) \, ds$.

## Remarks

The bound in item 1 is proved by induction: $\abs{K_{n+1}(t, s)} \le \int_s^t M \cdot M^n \frac{(u - s)^{n - 1}}{(n - 1)!} \, du = M^{n+1} \frac{(t - s)^n}{n!}$.
The factorial makes the series of item 2 converge like the exponential series, by the Weierstrass M-test, whatever the size of $\lambda$.
The function $R$ is the [[resolvent kernel|ARTJV1XCRX]] of $K$.
Example: for $K = 1$ on $[0, b]$, $K_n(t, s) = (t - s)^{n-1}/(n-1)!$ and $R(t, s; \lambda) = e^{\lambda (t - s)}$.
Then $x = 1 + \lambda \int_0^t x$ has the solution $x(t) = 1 + \lambda \int_0^t e^{\lambda(t - s)} \, ds = e^{\lambda t}$.
For a Fredholm kernel on $[a, b] \times [a, b]$ the same series converges only for $\abs{\lambda}$ small; the Volterra structure, with integration over $[s, t]$ only, is what produces the factorial.

## Explanations

### L1 human

The Neumann series solves an equation by substituting it into itself again and again.
In general, this only works if the kernel is small.
For a Volterra kernel, each substitution integrates over a shrinking triangle of past times, and the size of the $n$-th term drops like $1/(n - 1)!$.
So the iteration always converges, as the exponential series does.

### L2 human

The theorem shows that the Volterra integral operator has spectral radius $0$: it is quasinilpotent.
Consequently $I - \lambda V$ is invertible for every $\lambda$, which gives the [[unique solvability of Volterra equations of the second kind|WG4GVC9HR8]].
The same estimate, applied to a nonlinearity that is Lipschitz in $x$ uniformly on $[a, b]$, gives a global solution of the Picard–Lindelöf type on all of $[a, b]$.
The bound in item 3 is of [[Grönwall|07EJHQYR9A]] type, and it controls how solutions depend on $f$.
</document>

<document source="https://tamnd.github.io/math2/node/3QG8F9ER3S.md" tag="3QG8F9ER3S" kind="definition">
# Functional equation

## Definition

Let $X$ and $Y$ be sets, let $k \ge 1$, and let $D \subseteq X^k$.
Let $f \colon X \to Y$ be an unknown [[function|0EPAT51DXQ]].
Let $E_1(f)$ and $E_2(f)$ be expressions built from $f$, from given functions and from variables $x_1, \dots, x_k$ by finitely many substitutions and compositions.
A *functional equation* in $f$ is a condition
$$E_1(f)(x_1, \dots, x_k) = E_2(f)(x_1, \dots, x_k) \quad \text{for all } (x_1, \dots, x_k) \in D.$$
In particular the expressions contain no derivatives, integrals or limits of $f$.
A *solution* is a function $f \colon X \to Y$ that satisfies the condition.
Systems of such conditions and several unknown functions are allowed in the same way.

## Remarks

Example: Jensen's equation $f\!\left(\frac{x + y}{2}\right) = \frac{f(x) + f(y)}{2}$ on $D = \R^2$ is a functional equation with $X = Y = \R$.
The function $f(x) = 3x + 1$ solves it, since both sides equal $\frac{3(x + y)}{2} + 1$.

Non-example: $f'(x) = f(x)$ for all $x \in \R$ is not a functional equation in this sense, because it involves the derivative of $f$; it is a differential equation.

Difference equations such as $f(n + 2) = f(n + 1) + f(n)$ on $\Z$ are functional equations in one variable.
Iterative equations, such as $f(f(x)) = g(x)$ for a given $g$, are another important class.
</document>

<document source="https://tamnd.github.io/math2/node/3QQ2PBGY3N.md" tag="3QQ2PBGY3N" kind="concept">
# Sullivan minimal model

## Summary

A Sullivan minimal model is a free graded-commutative [[differential graded algebra|GQVAXW5X09]] over $\Q$ with a decomposable differential.
It models the rational homotopy type of a space.
Every simply connected space of finite type has a minimal model, unique up to isomorphism.
Its generators in degree $n$ form the dual of the rational homotopy group $\pi_n(X) \otimes \Q$, and its cohomology is the rational cohomology of $X$.
So the minimal model is the main computational tool of [[rational homotopy theory|8G9QPMBT7R]].

## Explanations

### L1 human

The minimal model describes a space by a list of generators, one batch for each degree.
A rule $d$ says how each generator is "killed" by products of earlier ones.
The generators count the rational homotopy groups, and the rule records how they interact.
For the sphere $S^2$, one needs a generator $x$ in degree $2$ for the class of the sphere and a generator $y$ in degree $3$ with $dy = x^2$.
The second generator kills the square $x^2$, which must vanish in the cohomology of $S^2$, and it matches the rational homotopy group $\pi_3(S^2) \otimes \Q \cong \Q$ given by the Hopf map.
</document>

<document source="https://tamnd.github.io/math2/node/3QTT95DMFH.md" tag="3QTT95DMFH" kind="statement">
# Wigner semicircle law

## Statement

Let $(Y_{ij})_{1 \le i < j}$ be [[independent and identically distributed|00MRN0FARV]] real random variables with mean $0$ and variance $1$.
Let $(Y_{ii})_{i \ge 1}$ be independent and identically distributed real random variables with mean $0$ and finite variance, independent of the $Y_{ij}$ with $i < j$.
For $n \ge 1$, let $W_n$ be the real symmetric $n \times n$ matrix with entries $W_n(i, j) = W_n(j, i) = Y_{ij} / \sqrt n$ for $i \le j$.
Let $\lambda_1, \ldots, \lambda_n$ be the [[eigenvalues|0BQAA2BK3Q]] of $W_n$, and let
$$\mu_n = \frac1n \sum_{k=1}^n \delta_{\lambda_k}$$
be its empirical spectral measure.
Then, almost surely, $\mu_n$ [[converges weakly|0SRYJGXKR4]] as $n \to \infty$ to the semicircle distribution, the probability measure on $\R$ with density
$$\rho(x) = \frac{1}{2\pi} \sqrt{4 - x^2} \quad \text{for } \abs{x} \le 2, \qquad \rho(x) = 0 \quad \text{for } \abs{x} > 2.$$

## Remarks

The moment method is the classical proof.
One shows that $\mathbb{E} \int x^k \, d\mu_n = \frac1n \mathbb{E} \tr W_n^k$ tends to the $k$-th moment of the semicircle law, which is $0$ for odd $k$ and the Catalan number $\frac{1}{m+1}\binom{2m}{m}$ for $k = 2m$.
Expand the trace as a sum over closed paths.
In the limit only paths that trace a tree and cross each edge exactly twice contribute, and Catalan numbers count them.
A variance bound and a truncation of the entries then give almost sure convergence.
The same result holds for complex [[Hermitian|0EKZCX4K3P]] matrices with independent and identically distributed entries above the diagonal, of mean $0$ and $\mathbb{E}\abs{Y_{ij}}^2 = 1$.

## Explanations

### L1 human

Fill a large symmetric matrix with independent random numbers and scale it by $1/\sqrt n$.
Its eigenvalues then spread over the interval $[-2, 2]$, and their histogram takes the shape of a half-disc.
The shape does not depend on the law of the entries, only on their mean and variance.

### L2 human

It is the law of large numbers of random matrix theory, the analogue for spectra of the law of large numbers for averages.
Wigner introduced such matrices as models for the energy levels of heavy atomic nuclei.
Finer questions, such as the fluctuations of the largest eigenvalue, lead to the [[Tracy–Widom distribution|89ZY5YRETE]].
For sample covariance matrices the analogue is the [[Marchenko–Pastur law|DCK63WTY5E]].
</document>

<document source="https://tamnd.github.io/math2/node/3QWJB8GHDE.md" tag="3QWJB8GHDE" kind="concept">
# Saturation of approximation methods

## Summary

Many linear approximation methods cannot converge faster than a certain rate, except for a small class of trivial functions.
This phenomenon is called saturation, and that rate is the saturation order of the method.
The functions approximated with exactly this rate form the saturation class.
Saturation explains why smoothness beyond a certain level does not improve methods such as Bernstein polynomials or Fejér means.

## Explanations

### L1 human

Some approximation methods hit a speed limit.
Bernstein polynomials of a function with a nonzero second derivative have error of order $1/n$ and not smaller, however smooth the function is.
Only the straight lines, which the method reproduces exactly, do better.
Knowing the speed limit of a method tells when it is worth switching to a better one.
</document>

<document source="https://tamnd.github.io/math2/node/3QWVVM56JF.md" tag="3QWVVM56JF" kind="statement">
# Bendixson–Dulac theorem

## Statement

Let $D \subseteq \R^2$ be a [[simply connected domain|0QW64ARSHB]] and let $f = (P, Q) \colon D \to \R^2$ be continuously differentiable.
Let $\varphi \colon D \to \R$ be continuously differentiable.
Assume that the [[divergence|0MWV6ZTK6S]]
$$\nabla \cdot (\varphi f) = \frac{\partial (\varphi P)}{\partial x} + \frac{\partial (\varphi Q)}{\partial y}$$
satisfies $\nabla \cdot (\varphi f) \ge 0$ on $D$ and does not vanish identically on any nonempty open subset of $D$.
Then the [[autonomous system|07BNRFQ0PJ]] $x' = P(x, y)$, $y' = Q(x, y)$ has no [[periodic orbit|0AA97XA70K]] contained in $D$.
The same holds when $\nabla \cdot (\varphi f) \le 0$ instead.

## Remarks

Suppose a periodic orbit $\Gamma \subseteq D$ exists, and let $R$ be the region inside $\Gamma$; since $D$ is simply connected, $R \subseteq D$.
By [[Green's theorem|0HR3N02317]], $\iint_R \nabla \cdot (\varphi f) \, dx \, dy$ equals the flux $\oint_\Gamma \varphi \, (P \, dy - Q \, dx)$.
The flux is $0$, because along $\Gamma$ one has $dx = P \, dt$ and $dy = Q \, dt$; but the double integral is positive, since $R$ is a nonempty open set.
The case $\varphi = 1$ is Bendixson's criterion, and a function $\varphi$ that works is called a Dulac function.
Example: for $x' = y$, $y' = -x - y + x^2$ one has $\nabla \cdot f = -1$, so the system has no periodic orbit in $\R^2$.
Simple connectivity is needed: the system $x' = x(1 - x^2 - y^2) - y$, $y' = y(1 - x^2 - y^2) + x$ has $\nabla \cdot f = 2 - 4(x^2 + y^2) < 0$ on the annulus $0.8 < \sqrt{x^2 + y^2} < 2$, yet the unit circle in this annulus is a periodic orbit.

## Explanations

### L1 human

A periodic orbit in the plane encloses a region, and the flow neither gains nor loses area across the orbit.
If the flow, after reweighting by a function $\varphi$, always expands area or always shrinks it, such a closed loop cannot exist.
So the theorem is a simple test that rules out cycles.

### L2 human

Bendixson–Dulac is the standard negative criterion for periodic orbits in the plane, and it complements the [[Poincaré–Bendixson theorem|0EW1X61ZT1]], which proves their existence.
Finding a good Dulac function is an art; for Lotka–Volterra type systems, $\varphi = 1 / (x y)$ is a common choice.
On a domain with one hole, the same argument shows that there is at most one periodic orbit that goes around the hole.
</document>

<document source="https://tamnd.github.io/math2/node/3R064KFF68.md" tag="3R064KFF68" kind="statement">
# Poincaré conjecture

## Statement

Let $M$ be a closed connected [[3-manifold|R20TB2KEQQ]] whose [[fundamental group|S1XK3HXHFX]] is trivial.
Then $M$ is [[homeomorphic|0QT00PB5G2]] to the [[3-sphere|T62369RFH6]] $S^3$.

## Remarks

Poincaré asked this question in 1904.
Perelman proved it in 2002 and 2003, following Hamilton's program of [[Ricci flow|CR14D9YQTY]].
One runs the Ricci flow from any Riemannian metric on $M$ and cuts out regions where the curvature blows up (surgery).
For simply connected $M$ the flow with surgery becomes extinct in finite time, and this shows that $M$ is a sphere.
It is also a special case of the [[geometrization conjecture|B27HB43K86]].
Both hypotheses matter: $\R^3$ is simply connected but not closed, and the [[Poincaré homology sphere|NHERCK7C43]] is closed with the homology of $S^3$ but has nontrivial fundamental group.
The analogous statement in dimensions $n \ge 5$ is due to Smale and in dimension $4$ to Freedman, in their topological versions.

## Explanations

### L1 human

A closed $3$-dimensional space in which every loop can be shrunk to a point must be the $3$-sphere.
It was the most famous open problem of topology for a century.
It was one of the Clay Millennium Prize problems.

### L2 human

The statement is a recognition theorem: a single algebraic condition identifies $S^3$ among all closed $3$-manifolds.
In practice it is now used as part of geometrization, which gives every closed $3$-manifold with finite fundamental group the form $S^3 / \Gamma$ for a finite group $\Gamma$ of isometries.
In the smooth category, it also follows that every smooth homotopy $3$-sphere is diffeomorphic to $S^3$, because topological $3$-manifolds have unique smooth structures.
</document>

<document source="https://tamnd.github.io/math2/node/3R2VHPXXWS.md" tag="3R2VHPXXWS" kind="example">
# Desargues configuration

## Example

The *Desargues configuration* is a [[configuration|Q2QFXPP9EE]] $(10_3)$ in the projective plane.
Take two triangles $a_1 a_2 a_3$ and $b_1 b_2 b_3$ in perspective from a point $O$, so that $O$, $a_i$, $b_i$ are collinear for $i = 1, 2, 3$.
Let $c_{ij}$ be the intersection of the sides $a_i a_j$ and $b_i b_j$.
By [[Desargues's theorem|5Z19Y02MME]], the three points $c_{12}$, $c_{13}$, $c_{23}$ lie on a line, the axis.

Its $10$ points are $O$, the six vertices and the three points $c_{ij}$.
Its $10$ lines are the three lines $O a_i b_i$, the six sides of the triangles and the axis.
Check: each line has $3$ of the points, for instance the side $a_1 a_2$ contains $a_1$, $a_2$, $c_{12}$.
Each point lies on $3$ lines.
For instance, $O$ lies on the lines $O a_i b_i$, the vertex $a_1$ on $O a_1 b_1$, $a_1 a_2$, $a_1 a_3$, and $c_{12}$ on $a_1 a_2$, $b_1 b_2$ and the axis.

A combinatorial model: the points are the $2$-element subsets of $\set{1, 2, 3, 4, 5}$ and the lines are the $3$-element subsets, with incidence by inclusion.
Each $3$-set contains three $2$-sets, and each $2$-set lies in three $3$-sets.
Its incidence graph is the Desargues graph.

## Explanations

### L1 human

Draw two triangles whose corresponding corners are lined up with a common point.
Extend the corresponding sides until they cross; the three crossing points then lie on one line.
The whole drawing has $10$ points and $10$ lines, with $3$ points on each line and $3$ lines through each point.
The pattern is perfectly balanced, and any of its $10$ points can play the role of the center.
</document>

<document source="https://tamnd.github.io/math2/node/3R2Y906QAQ.md" tag="3R2Y906QAQ" kind="concept">
# Transfer homomorphism

## Summary

The transfer is a [[group homomorphism|0JYTG5K9Y4]] from a group $G$ to the [[abelianisation|0AQZV0905N]] of a subgroup $H$ of finite [[index|0DTXBQPA2S]].
It is built from the way $G$ permutes the [[left cosets|03DEH43F4H]] of $H$.
The transfer gives normal subgroups and abelian quotients of $G$ from information about $H$, usually a [[Sylow subgroup|0053CQE9Y2]].
It proves results such as Burnside's normal $p$-complement theorem and the [[focal subgroup theorem|H1BSG0J5G1]].

## Explanations

### L1 human

Let $H$ be a subgroup of $G$ with $n$ cosets.
An element $g$ of $G$ shuffles the cosets, and each step of the shuffle leaves a small correction term in $H$.
The transfer multiplies these $n$ correction terms together, in a version of $H$ where the order of multiplication does not matter.
When $G$ is abelian, the result is simply $g^n$.
For the integers with $H$ the multiples of $n$, the transfer sends $1$ to $n$.
</document>

<document source="https://tamnd.github.io/math2/node/3R959YSNYP.md" tag="3R959YSNYP" kind="concept">
# Maass wave form

## Summary

A Maass wave form is a smooth function on the [[upper half-plane|X6XGGH5FPW]] that is invariant under a [[congruence subgroup|1H6R856P3C]] of the [[modular group|VEKV1EM77B]].
It is also an eigenfunction of the hyperbolic [[Laplace–Beltrami operator|F4NWZNTRMD]].
It is the non-holomorphic counterpart of a [[modular form|GWKN5VG78A]].
Maass forms are basic examples of [[automorphic forms|3C73HND6XA]], and their eigenvalues are the subject of spectral theory and trace formulas.

## Explanations

### L1 human

Think of the upper half-plane as a drum whose shape is folded up by the modular group.
A Maass form is a standing wave on this drum: a smooth wave that repeats under the symmetries and has a pure "frequency", its Laplace eigenvalue.
Holomorphic modular forms are rigid objects built from complex analysis; Maass forms are the wave-like analogues that need no complex differentiability.
A simple example is the real-analytic Eisenstein series $E(z, s)$, an average of the function $y^s$ over the modular group.
Square-integrable Maass forms for $\mathrm{SL}_2(\Z)$ exist, but no one knows a closed formula for any of them.
</document>

<document source="https://tamnd.github.io/math2/node/3RCAY97W1E.md" tag="3RCAY97W1E" kind="concept">
# Graph of an operator

## Summary

The graph of an operator $T$ from $X$ to $Y$ is the set of pairs $(x, Tx)$ with $x$ in the domain of $T$.
It is a linear subspace of $X \times Y$, and it determines the operator completely, domain included.
Topological properties of the graph, such as being closed, replace boundedness in the theory of unbounded operators.

## Explanations

### L1 human

For a function of one real variable, the graph is the curve of points $(x, f(x))$ in the plane.
The graph of an operator is the same picture in the product space $X \times Y$.
For example, the graph of $x \mapsto 2x$ on $\R$ is the line $y = 2x$ in $\R^2$.
Because operators are linear, their graphs are always flat subspaces through the origin.
</document>

<document source="https://tamnd.github.io/math2/node/3RDB8B5GFA.md" tag="3RDB8B5GFA" kind="statement">
# Hurwitz's theorem (number theory)

## Statement

Let $\xi$ be an [[irrational number|0DZXAXEJQ9]].
Then there are infinitely many [[rational numbers|0NCJ5WXN6Z]] $p/q$, with $p, q \in \Z$ and $q \ge 1$, such that
$$\abs{\xi - \frac{p}{q}} < \frac{1}{\sqrt{5}\, q^2}.$$
The constant $\sqrt{5}$ is best possible.
That is, for each real $A > \sqrt{5}$ there is an irrational $\xi$ with only finitely many rationals $p/q$ that satisfy $\abs{\xi - p/q} < 1/(A q^2)$.

## Remarks

The proof uses the [[continued fraction expansion|04VA28F3HB]] of $\xi$.
Among any three consecutive convergents $p_n/q_n$ of $\xi$, at least one satisfies the inequality, and there are infinitely many convergents.
For the second part, take the [[golden ratio|0SC8T7548D]] $\varphi = (1 + \sqrt{5})/2$, whose continued fraction is $[1; 1, 1, 1, \dots]$.
Its convergents are quotients of consecutive Fibonacci numbers, and $q^2 \abs{\varphi - p/q}$ tends to $1/\sqrt{5}$ along them.
The theorem improves the constant $1$ of [[Dirichlet's approximation theorem|04JAY9DGB7]].
If one removes $\varphi$ and all numbers equivalent to it under the action of $\mathrm{GL}_2(\Z)$, the best constant becomes $\sqrt{8}$.

## Explanations

### L1 human

Each irrational number can be approached by fractions $p/q$ with an error much smaller than $1/q^2$.
Hurwitz's theorem gives the exact best factor: the error can be made smaller than $1/(\sqrt{5} q^2)$ infinitely often.
The golden ratio is the hardest number to approximate, and it shows that $\sqrt{5}$ cannot be improved.

### L2 human

The theorem belongs to the theory of the Lagrange spectrum, which records the best constants for each irrational number.
Continued fractions are the main tool: the convergents give the best approximations.
Numbers with bounded partial quotients, such as quadratic irrationals, are badly approximable, and the golden ratio is the extreme case.
</document>

<document source="https://tamnd.github.io/math2/node/3RGBA24QVA.md" tag="3RGBA24QVA" kind="concept">
# Galois category

## Summary

A Galois category is a [[category|A2DE66937D]] with a [[functor|28NQ0KRHD3]] to finite sets, called a fibre functor.
The pair satisfies axioms modelled on finite covering spaces and on finite separable field extensions.
The main theorem says that such a category is equivalent to the category of finite sets with a continuous action of a [[profinite group|KA0YDB6QXE]].
This group is the automorphism group of the fibre functor.
It is the abstract frame of [[Grothendieck's Galois theory|CX7BDQ47ZK]].
With finite étale covers of a connected scheme it defines the [[étale fundamental group|KJ11DAXMR0]].

## Explanations

### L1 human

Classical Galois theory and the theory of covering spaces share one pattern: a group acts on a fibre, and the objects correspond to sets with an action of that group.
A Galois category isolates the properties needed for this pattern.
Its objects behave like finite covers, and the fibre functor sends each one to the finite set of points above a base point.
For example, finite sets with an action of a finite group $G$, with the functor that forgets the action, form a Galois category, and the group it recovers is $G$.
</document>

<document source="https://tamnd.github.io/math2/node/3RJ1K36A9A.md" tag="3RJ1K36A9A" kind="statement">
# Michael selection theorem

## Statement

Let $X$ be a [[paracompact|8R29G6NG9Q]] [[Hausdorff space|0S1FJ8VEX9]] and let $Y$ be a [[Banach space|06GZ956657]].
Let $F$ be a [[multivalued function|03CPD1HGF3]] from $X$ to $Y$ such that:

1. $F$ is [[lower hemicontinuous|XNMKX7H4FD]].
2. For every $x \in X$, the set $F(x)$ is nonempty, [[convex|02AS45T3H9]] and [[closed|0ZCK8DQYN1]] in $Y$.

Then there is a [[continuous|0VGFHTM75Y]] map $f \colon X \to Y$ with $f(x) \in F(x)$ for every $x \in X$.

## Remarks

Such a map $f$ is a *continuous selection* of $F$.
The proof first builds, for each $\varepsilon > 0$, a continuous map that lies within distance $\varepsilon$ of $F(x)$ at every $x$.
It glues local choices with a partition of unity, which exists because $X$ is paracompact.
Convexity keeps the convex combinations close to $F(x)$.
Repeating this with $\varepsilon = 2^{-n}$, each step close to the previous one, gives a uniformly Cauchy sequence of continuous maps.
Its limit exists since $Y$ is complete, it is continuous, and it lies in $F(x)$ since $F(x)$ is closed.

Example: let $T \colon Y \to Z$ be a continuous linear surjection between Banach spaces and set $F(z) = T^{-1}(z)$.
The values are closed affine subspaces, and the open mapping theorem makes $F$ lower hemicontinuous.
So $T$ has a continuous, possibly nonlinear, right inverse.

Lower hemicontinuity cannot be replaced by upper hemicontinuity.
On $X = \R$, let $F(x) = \set{-1}$ for $x < 0$, $F(0) = [-1, 1]$ and $F(x) = \set{1}$ for $x > 0$.
This $F$ is upper hemicontinuous with nonempty closed convex values, but a selection must jump from $-1$ to $1$ at $0$, so no continuous selection exists.

## Explanations

### L1 human

Suppose that at every point of a space you are given a set of allowed values, and the sets change in a reasonable way.
The theorem says that you can choose one allowed value at each point so that the choice varies continuously.
This needs the allowed sets to be convex and closed, the space of values to be a complete normed space, and the sets not to shrink suddenly.

### L2 human

The theorem is the main tool to turn set-valued information into continuous functions.
It gives continuous right inverses of linear surjections between Banach spaces, and it is used in the theory of [[absolute neighborhood retracts|DK3MT3HYX5]] and in fixed-point and control theory.
Paracompactness enters only through partitions of unity, in the same way as in the proof of the [[Dugundji extension theorem|RBDJBK63HX]].
Compare with the fixed-point theorems for upper hemicontinuous maps, such as Kakutani's, where the opposite kind of hemicontinuity is the right one.
</document>

<document source="https://tamnd.github.io/math2/node/3RQ2PT36KQ.md" tag="3RQ2PT36KQ" kind="definition">
# Daniell integral

## Definition

Let $X$ be a set and let $H$ be a set of functions $X \to \R$ with the following property.
$H$ is a [[vector space|05EGHEQZAW]] under pointwise operations, and $\max(h, 0) \in H$ for every $h \in H$.
A *Daniell integral*, or elementary integral, on $H$ is a map $I \colon H \to \R$ such that:

1. $I$ is linear.
2. $I$ is positive: $I(h) \ge 0$ for every $h \in H$ with $h \ge 0$.
3. If $h_1 \ge h_2 \ge \cdots$ in $H$ and $h_n \to 0$ [[pointwise|056FSKG32J]] on $X$, then $I(h_n) \to 0$.

## Remarks

Example: let $H$ be the continuous functions $\R \to \R$ with compact support, and $I(h) = \int h(x) \, dx$, the Riemann integral.
Linearity and positivity are clear.
For clause 3, all $h_n$ vanish outside a compact interval $K$ that contains the support of $h_1$.
By Dini's theorem $h_n \to 0$ uniformly on $K$, so $0 \le I(h_n) \le \max h_n \cdot \operatorname{length}(K) \to 0$.

Non-example: let $H$ be the convergent real sequences on $X = \N$, and $I(h) = \lim_{n \to \infty} h(n)$.
Clauses 1 and 2 hold.
Clause 3 fails: $h_n = \mathbf{1}_{\set{n, n+1, \dots}}$ decreases to $0$ pointwise, but $I(h_n) = 1$ for all $n$.

Daniell's construction extends $I$ to a class of integrable functions on which the monotone and dominated convergence theorems hold.
</document>

<document source="https://tamnd.github.io/math2/node/3RQEK50CP0.md" tag="3RQEK50CP0" kind="object">
# 120-cell

## Description

The *120-cell* is the convex [[regular polytope|F60PD912D5]] in $\R^4$ with [[Schläfli symbol|MAE6BF4DFK]] $\set{5, 3, 3}$.
Its cells are $120$ regular dodecahedra, three around each edge, and four cells meet at each vertex.
It has $600$ vertices, $1200$ edges, $720$ pentagonal faces and $120$ cells.

Check of the counts: each dodecahedron has $12$ pentagons, and each pentagon lies in exactly $2$ cells, so there are $120 \cdot 12 / 2 = 720$ faces.
Each pentagon has $5$ edges, and each edge lies in $3$ pentagons, so there are $720 \cdot 5 / 3 = 1200$ edges.
The vertex figure is a regular tetrahedron $\set{3,3}$, so each vertex lies on $4$ edges, and there are $1200 \cdot 2 / 4 = 600$ vertices.
Then $600 - 1200 + 720 - 120 = 0$, as Euler's relation for convex $4$-polytopes requires.

## Remarks

The 120-cell is the dual of the [[600-cell|8MNQDK5X5W]].
Its symmetry group is the reflection group $H_4$, of order $14400$.
It is the four-dimensional analogue of the regular dodecahedron $\set{5, 3}$.
Three dodecahedra fit around an edge with a small angular gap, since the dihedral angle of the dodecahedron is about $116.57^\circ$.

## Explanations

### L1 human

The 120-cell is a four-dimensional solid made of $120$ regular dodecahedra.
It has $600$ corners, and it is the largest and most complicated of the six regular solids of four-dimensional space.
It plays the role in four dimensions that the dodecahedron plays in three.
</document>

<document source="https://tamnd.github.io/math2/node/3RT74FQYG1.md" tag="3RT74FQYG1" kind="statement">
# Transfer principle

## Statement

Let ${}^*\R$ be the field of [[hyperreal numbers|421QMTFG9T]] built as an ultrapower of the [[real numbers|08J2M1JYAG]].
For each $k$-ary function $f$ on $\R$, let ${}^*f$ be its extension to ${}^*\R$, defined termwise on sequences.
Extend relations on $\R$ in the same way.
Let $\varphi$ be a sentence of [[first-order logic|086QKPF0A6]] built from real constants, functions and relations on $\R$.
Let ${}^*\varphi$ be the sentence obtained by replacing each function $f$ by ${}^*f$ and each relation $R$ by ${}^*R$.
Then $\varphi$ is true in $\R$ if and only if ${}^*\varphi$ is true in ${}^*\R$.

## Remarks

The principle is a special case of [[Łoś's theorem|AGYBSTD7BG]].
That theorem says that a first-order formula holds at a tuple of classes exactly when it holds termwise on a set of indices in the ultrafilter.

Example: the sentence $\forall x \, (x > 0 \to \exists y \, (y \cdot y = x))$ is true in $\R$.
By transfer, every positive hyperreal has a square root in ${}^*\R$.

The restriction to first-order sentences, where quantifiers range over elements and not over arbitrary subsets, is necessary.
Every nonempty subset of $\R$ that is bounded above has a supremum, but in ${}^*\R$ the set of infinitesimals is bounded above by $1$ and has no supremum.
Indeed, a supremum $s$ would be positive; if $s$ were infinitesimal then $2s$ would be a larger infinitesimal, and otherwise $s/2$ would be a smaller upper bound.

## Explanations

### L1 human

The hyperreals contain new numbers, infinitely small and infinitely large ones.
The transfer principle says that despite this, the hyperreals obey exactly the same elementary rules as the reals.
Take any statement that uses only "for all numbers", "there is a number" and the usual functions and relations.
It is true for the reals exactly when it is true for the hyperreals.

### L2 human

Transfer is what makes nonstandard proofs valid.
One proves a statement in ${}^*\R$ using infinitesimals, checks that it is the star version of a first-order statement, and transfers it back to $\R$.
For instance, a real function $f$ is continuous at a real $a$ exactly when ${}^*f(x) - f(a)$ is infinitesimal for every $x$ with $x - a$ infinitesimal.
Transfer is the key step in proving this equivalence.
Statements about all subsets of $\R$ transfer only to statements about internal subsets of ${}^*\R$, such as ${}^*A$ for $A \subseteq \R$.
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# Analytic set

## Summary

An analytic set is a subset of a [[Polish space|584GXXY73M]] that is a continuous image of the [[Baire space|P2WS16HKD7]].
Equivalently, it is the projection of a closed set or of a Borel set in a product.
Every [[Borel set|0WQXY32Q8D]] is analytic, but some analytic sets are not Borel.
[[Suslin's theorem|9418TGREZX]] says that a set is Borel exactly when both it and its complement are analytic.
Analytic sets are Lebesgue measurable, and an uncountable analytic set contains a perfect set.

## Explanations

### L1 human

The projection of a set in the plane onto the $x$-axis is its shadow.
The shadow of a closed set is simple, but the shadow of a complicated Borel set in the plane need not be Borel.
Analytic sets are exactly such shadows.
They are still well behaved: they can be measured, and they cannot be uncountable without containing a copy of the Cantor set.
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# Leavitt path algebra

## Definition

Let $K$ be a [[field|04C5ZBG1WT]] and let $E = (E^0, E^1, s, r)$ be a [[directed graph|0KFYGSHHF5]], with vertex set $E^0$, edge set $E^1$, and source and range maps $s, r \colon E^1 \to E^0$.
For each edge $e$ take a new symbol $e^*$, called a ghost edge.
The *Leavitt path algebra* $L_K(E)$ is the quotient of the [[free algebra|2P3Y2FRCSZ]] over $K$ on the set $E^0 \cup E^1 \cup \set{e^* : e \in E^1}$ by the [[two-sided ideal|0CD87SFM2N]] of these relations:

1. $v w = \delta_{v, w} v$ for all $v, w \in E^0$.
2. $s(e) e = e = e \, r(e)$ for all $e \in E^1$.
3. $r(e) e^* = e^* = e^* s(e)$ for all $e \in E^1$.
4. $e^* f = \delta_{e, f} \, r(e)$ for all $e, f \in E^1$.
5. $v = \sum_{e \in E^1, \, s(e) = v} e e^*$ for every vertex $v$ that is the source of at least one and at most finitely many edges.

Here $\delta_{a, b}$ is $1$ if $a = b$ and $0$ otherwise.

## Remarks

Relations 4 and 5 are the Cuntz–Krieger relations.
The algebra has an identity, $\sum_{v \in E^0} v$, exactly when $E^0$ is finite.
Relations 1 and 2 alone define the [[path algebra|2PYANBGXFD]] of $E$, which maps into $L_K(E)$.
Example: let $E$ have one vertex $v$ and one loop $e$.
Relation 4 gives $e^* e = v$ and relation 5 gives $e e^* = v$, with $v$ the identity, so $L_K(E)$ is the [[Laurent polynomial ring|QMY5JTZZB8]] $K[x, x^{-1}]$ with $x = e$ and $x^{-1} = e^*$.
Example: if $E$ is $u \to v$ with one edge $e$, then $L_K(E)$ is isomorphic to the [[matrix ring|0FHVPRAE60]] $M_2(K)$, via $u \mapsto E_{11}$, $v \mapsto E_{22}$, $e \mapsto E_{12}$ and $e^* \mapsto E_{21}$.
Example: one vertex with $n \ge 2$ loops gives the Leavitt algebra $L(1, n)$, in which the free module of rank $1$ is isomorphic to the free module of rank $n$.
Non-example: in the path algebra of the one-loop graph, the element $e$ has no inverse, since every product of $e$ with a combination of paths has no constant term.
Relation 4 needs the ghost edge $e^*$.
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# Markov partition

## Definition

Let $M$ be a compact Riemannian manifold with distance $d$, and let $f \colon M \to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]].
Let $\Lambda \subseteq M$ be a [[hyperbolic set|MPFD24TYW2]] of $f$.
For $\varepsilon > 0$ and $x \in M$, let $W^s_\varepsilon(x) = \set{y \in M : d(f^n x, f^n y) \le \varepsilon \text{ for all } n \ge 0}$ and $W^u_\varepsilon(x) = \set{y \in M : d(f^{-n} x, f^{-n} y) \le \varepsilon \text{ for all } n \ge 0}$.
Assume that $\Lambda$ has a local product structure, with constants $\varepsilon > 0$ and $\delta > 0$.
This means that for all $x, y \in \Lambda$ with $d(x, y) < \delta$, the set $W^s_\varepsilon(x) \cap W^u_\varepsilon(y)$ is a single point $[x, y]$ of $\Lambda$.
A *rectangle* is a nonempty set $R \subseteq \Lambda$ of diameter less than $\delta$ such that $[x, y] \in R$ for all $x, y \in R$.
It is *proper* if $R$ is the closure of its interior, taken relative to $\Lambda$.
For $x \in R$ put $W^s(x, R) = W^s_\varepsilon(x) \cap R$ and $W^u(x, R) = W^u_\varepsilon(x) \cap R$.
A *Markov partition* of $\Lambda$ is a finite family $R_1, \ldots, R_k$ of proper rectangles such that:

1. $\Lambda = R_1 \cup \cdots \cup R_k$.
2. The interiors of $R_i$ and $R_j$ are disjoint for $i \ne j$.
3. If $x$ lies in the interior of $R_i$ and $f(x)$ lies in the interior of $R_j$, then $f(W^s(x, R_i)) \subseteq W^s(f(x), R_j)$ and $f(W^u(x, R_i)) \supseteq W^u(f(x), R_j)$.

## Remarks

Clause 3 says that $f$ maps stable slices into stable slices and stretches unstable slices across the whole rectangle.
Then the transition matrix $T_{ij} = 1$ when $f(\operatorname{int} R_i) \cap \operatorname{int} R_j \ne \emptyset$, and $T_{ij} = 0$ otherwise, defines a [[subshift of finite type|624XDXX8FN]] $\Sigma_T$.
The map $\pi(x) = \bigcap_{n \in \Z} f^{-n}(R_{x_n})$ is a continuous surjection $\Sigma_T \to \Lambda$ with $\pi \circ \sigma = f \circ \pi$.
Example: let $\Lambda$ be the invariant Cantor set of the [[Smale horseshoe|N63Z6EA6YS]], identified with $\set{0,1}^{\Z}$ so that $f$ becomes the shift.
Fix $m \ge 1$, and for each word $w$ of length $2m + 1$ let $R_w$ be the set of points whose coordinates $-m, \ldots, m$ spell $w$.
For large $m$ these sets are small.
The point $[x, y]$ takes its coordinates of index $n \ge 0$ from $x$ and of index $n < 0$ from $y$, so each $R_w$ is a rectangle.
It is open and closed, so it is proper.
Applying the shift moves the agreeing future of a stable slice to the future of $f(x)$, which gives clause 3.
Non-example: removing one set $R_w$ from this family leaves a family that violates clause 1.
Sinai constructed Markov partitions for Anosov diffeomorphisms, and Bowen for the basic sets of Axiom A diffeomorphisms.
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# Local property

## Summary

In commutative algebra, a property of [[modules|0NZ8JA25WK]] or rings is local when it holds globally exactly when it holds after [[localization at every prime ideal|X6JD4AG973]].
Local properties let one prove a statement by checking it over [[local rings|MZ02BKPHJA]], where tools such as [[Nakayama's lemma|15JHBYR6KH]] are available.
Being zero, being [[flat|73RYYEVDKA]], and the exactness of a sequence are local; being Noetherian or being an [[integral domain|0QVD7WBFGN]] is not.

## Explanations

### L1 human

A property is local if you can test it one point at a time.
For a ring $R$, the "points" are its [[prime ideals|00MHHZ9SNE]], and testing at a point $\mathfrak p$ means passing to the localization $M_{\mathfrak p}$.
For example, a module is zero exactly when all its localizations are zero.
So to show that a module vanishes, it is enough to show it near every point.
The [[definition|FD3T25A1DX]] states the condition.
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# Nagata ring

## Summary

A Nagata ring is a [[Noetherian ring|0670T59HGE]] in which [[integral closures|G0V86SJBQW]] are well behaved.
For every prime ideal $\mathfrak{p}$, the integral closure of $R/\mathfrak{p}$ in any finite extension of its fraction field is a finite module.
The notion is named after Masayoshi Nagata, who gave examples of Noetherian domains whose integral closures are not finite.
Every [[excellent ring|YF7YFQ58QJ]] is a Nagata ring, and every finitely generated algebra over a Nagata ring is again a Nagata ring.
The finiteness of integral closures is what makes normalization a finite operation in algebraic geometry.

## Explanations

### L1 human

Starting from a domain, one often wants to add all elements of the fraction field that are roots of monic polynomials, to make the ring "normal".
For the integers inside a number field this adds only finitely many new generators, and the result is a finite module over $\Z$.
For some Noetherian domains, though, infinitely many new generators are needed.
Nagata rings are the rings where this never happens, not even after passing to quotients by primes or to finite field extensions.
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# Dissipative operator

## Summary

A dissipative operator is a linear operator $A$ on a Banach space such that $\norm{(\lambda - A)x} \ge \lambda \norm{x}$ for every $\lambda > 0$.
On a Hilbert space this means $\operatorname{Re}\ip{Ax}{x} \le 0$ for every $x$ in the domain.
The densely defined maximal dissipative operators are exactly the generators of semigroups that do not increase norms, by the [[Lumer–Phillips theorem|Z2N8MPD8DH]].
The name comes from physics: the evolution $u' = Au$ loses energy and never gains it.

## Explanations

### L1 human

Think of $u' = Au$ as a law of motion and of $\norm{u}^2$ as an energy.
On a Hilbert space, the derivative of $\norm{u(t)}^2$ is $2 \operatorname{Re}\ip{Au}{u}$, so dissipativity says the energy can only go down.
The heat equation is the main example: heat spreads out and the total squared temperature decreases.
In one dimension, $\ip{f''}{f} = -\int \abs{f'}^2 \, dx \le 0$ for smooth $f$ that vanish at the ends, so the second derivative is dissipative.
The operator $-A$ of a dissipative $A$ is called accretive.
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# Calderón–Zygmund operator

## Summary

A Calderón–Zygmund operator is a bounded linear operator on $L^2(\R^n)$ that is given away from the diagonal by integration against a [[Calderón–Zygmund kernel|78N6AXP9VV]].
This class contains the Hilbert transform, the [[Riesz transforms|FRNDFJ9WQ5]], the operators with smooth multipliers of Hörmander–Mikhlin type, and many operators that are not translation invariant.
Every Calderón–Zygmund operator is bounded on $L^p(\R^n)$ for $1 < p < \infty$ and of weak type $(1, 1)$.
This makes it the standard class of [[singular integral operators|HTMCEARGDD]].

## Explanations

### L1 human

A Calderón–Zygmund operator averages a function against a kernel that blows up near the diagonal at the borderline rate.
The definition asks for two things: the operator is already known to be bounded in the energy norm $L^2$, and its kernel has the right size and smoothness.
The theory then shows that the operator is bounded in every $L^p$ norm with $1 < p < \infty$.
The Hilbert transform $f \mapsto \frac{1}{\pi}\,\mathrm{p.v.}\int \frac{f(y)}{x - y} \, dy$ is the basic example.
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# Toeplitz algebra

## Description

Let $\ell^2(\N)$ be the Hilbert space of square-summable sequences, with orthonormal basis $e_0, e_1, e_2, \ldots$.
The unilateral shift is the operator $S$ with $S e_n = e_{n+1}$ for all $n$.
The *Toeplitz algebra* $\mathcal{T}$ is the C*-subalgebra of the [[algebra of bounded operators|BMDG4JQPEH]] $B(\ell^2(\N))$ generated by $S$.
It is the norm closure of the set of polynomials in $S$ and $S^*$.
It is a C*-algebra by construction.
Identify $\ell^2(\N)$ with the [[Hardy space|PNNNFJVF6W]] $H^2$ of the unit circle.
Then $\mathcal{T}$ is the C*-algebra generated by the [[Toeplitz operators|GQ0X3R3WM2]] $T_f$ with continuous symbol $f$.

Check that $\mathcal{T}$ contains the [[compact operators|M2NEP1K5AE]]: $S^* S = 1$, and $P = 1 - S S^*$ is the projection onto $\C e_0$.
Then $S^m P (S^*)^n$ is the rank-one operator sending $e_n$ to $e_m$ and the other basis vectors to $0$.
These matrix units lie in $\mathcal{T}$, and the norm closure of their span is the [[ideal of compact operators|2WAZF1RV2Y]] $K$.

## Remarks

The quotient $\mathcal{T} / K$ is isomorphic to the algebra $C(\mathbb{T})$ of continuous functions on the unit circle, by a map that sends the class of $T_f$ to $f$.
This gives the exact sequence $0 \to K \to \mathcal{T} \to C(\mathbb{T}) \to 0$, often called the Toeplitz extension.
Coburn's theorem states that the C*-algebra generated by any isometry that is not unitary is isomorphic to $\mathcal{T}$, by an isomorphism that sends the isometry to $S$.
The index of the Fredholm operator $T_f$, for a continuous nowhere-vanishing symbol $f$, is minus the winding number of $f$ around $0$.

## Explanations

### L1 human

The shift operator moves each entry of a sequence one place to the right and puts a zero in front.
It almost has an inverse, the backward shift, but shifting back and then forward loses the first entry.
The Toeplitz algebra is everything one can build from these two operators with sums, products and limits.
Modulo the compact operators, the shift becomes multiplication by $z$ on the circle, so the algebra links operators with functions on the circle and with winding numbers.
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# Variational capacity

## Definition

Let $n \ge 1$, let $1 < p < \infty$, let $\Omega \subseteq \R^n$ be an open set, and let $K \subseteq \Omega$ be a [[compact set|0R6EWFKRCF]].
Let $C_c^\infty(\Omega)$ be the set of smooth real functions with compact support in $\Omega$.
The *variational capacity* of $K$ relative to $\Omega$ is
$$\operatorname{cap}_p(K, \Omega) = \inf \set{ \int_\Omega \abs{\nabla u}^p \, dx : u \in C_c^\infty(\Omega), \ u \ge 1 \text{ on } K }.$$

## Remarks

For an open set $U \subseteq \Omega$ one sets $\operatorname{cap}_p(U, \Omega) = \sup \set{ \operatorname{cap}_p(K, \Omega) : K \subseteq U \text{ compact} }$.
For an arbitrary set $E \subseteq \Omega$ one sets $\operatorname{cap}_p(E, \Omega) = \inf \set{ \operatorname{cap}_p(U, \Omega) : E \subseteq U \subseteq \Omega, \ U \text{ open} }$.
The infimum does not change if the admissible functions are taken in the [[Sobolev space|M4CEZ5K9HP]] $W_0^{1,p}(\Omega)$ and continuous, with $u \ge 1$ on $K$.

Example: let $0 < r < R$, $p \ne n$, $\alpha = (p - n)/(p - 1)$, and let $\sigma_{n-1}$ be the area of the unit sphere in $\R^n$.
The radial function $u$ with $u = 1$ for $\abs{x} \le r$, $u = (\abs{x}^\alpha - R^\alpha)/(r^\alpha - R^\alpha)$ for $r \le \abs{x} \le R$, and $u = 0$ outside, solves the $p$-Laplace equation on the shell.
Its gradient has size $\abs{\alpha} \abs{x}^{\alpha - 1} / \abs{r^\alpha - R^\alpha}$, and integration in polar coordinates gives
$$\operatorname{cap}_p(\bar B_r, B_R) = \sigma_{n-1} \abs{\alpha}^{p-1} \abs{r^\alpha - R^\alpha}^{1-p}.$$
For $p = 2$ and $n = 3$ this is $4 \pi r R / (R - r)$.

Non-example of an admissible function: a function $u \in C_c^\infty(\Omega)$ with $u = 1/2$ on $K$ fails the condition $u \ge 1$ on $K$ and does not enter the infimum.

A related notion is the Sobolev $p$-capacity, which uses $\int_{\R^n} (\abs{u}^p + \abs{\nabla u}^p) \, dx$ and no reference set $\Omega$; for subsets of a bounded open set $\Omega$ both capacities have the same null sets.
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# Radial basis function

## Definition

Let $d \ge 1$ and let $\norm{\cdot}$ be the Euclidean norm on [[Euclidean space|0XPR1E85W0]] $\R^d$.
A *radial basis function* is a [[function|0EPAT51DXQ]] $\Phi \colon \R^d \to \R$ of the form
$$\Phi(x) = \varphi(\norm{x}),$$
for some function $\varphi \colon [0, \infty) \to \R$.
For a centre $c \in \R^d$, the translate $x \mapsto \varphi(\norm{x - c})$ depends only on the [[Euclidean distance|0NPPGHQNXQ]] from $x$ to $c$.

## Remarks

Common choices of $\varphi$, with a shape parameter $\varepsilon > 0$:

1. Gaussian: $\varphi(r) = e^{-(\varepsilon r)^2}$;
2. multiquadric: $\varphi(r) = \sqrt{1 + (\varepsilon r)^2}$;
3. inverse multiquadric: $\varphi(r) = 1 / \sqrt{1 + (\varepsilon r)^2}$;
4. polyharmonic splines: $\varphi(r) = r^k$ for odd $k$ and $\varphi(r) = r^k \log r$ for even $k$, with $\varphi(0) = 0$.

Example: $\Phi(x) = e^{-\norm{x}^2}$ on $\R^2$ is radial: $\Phi(x_1, x_2) = e^{-(x_1^2 + x_2^2)}$ only depends on $\norm{x}$.
For instance $\Phi(1, 0) = \Phi(0, 1) = e^{-1}$.

Non-example: $\Psi(x_1, x_2) = e^{-(x_1^2 + 2 x_2^2)}$ is not radial: $\norm{(1, 0)} = \norm{(0, 1)} = 1$, but $\Psi(1, 0) = e^{-1} \ne e^{-2} = \Psi(0, 1)$.
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# Exponential of a bounded operator

## Definition

Let $X$ be a [[Banach space|06GZ956657]] over $\R$ or $\C$, and let $T \colon X \to X$ be a [[continuous linear map|03TM135G0N]].
Write $T^0 = I$, the identity map of $X$.
The *exponential* of $T$ is the bounded operator
$$e^T = \exp(T) = \sum_{n=0}^{\infty} \frac{T^n}{n!},$$
where the series converges in [[operator norm|06EKW1AKXF]].

## Remarks

The series converges because $\norm{T^n / n!} \le \norm{T}^n / n!$, and the numbers $\norm{T}^n / n!$ have the finite sum $e^{\norm{T}}$; see [[absolutely convergent series in a Banach space converge|082MGNWNA8]].
This also gives $\norm{e^T} \le e^{\norm{T}}$.

Example: let $D$ be the diagonal operator $D(x_n) = (\lambda_n x_n)$ on $\ell^2$, with $(\lambda_n)$ a bounded sequence of complex numbers.
Then $D^k(x_n) = (\lambda_n^k x_n)$, so summing coordinatewise gives $e^D(x_n) = (e^{\lambda_n} x_n)$.

For $X = \C^n$, the definition gives the [[matrix exponential|020M04FN1H]].
If $ST = TS$, then $e^{S + T} = e^S e^T$; in particular $e^T$ is [[invertible|CPAD5W1CGW]] with inverse $e^{-T}$.
The identity $e^{S+T} = e^S e^T$ can fail when $S$ and $T$ do not commute, already for $2 \times 2$ matrices.
The map $t \mapsto e^{tT}$ from $\R$ to bounded operators is differentiable in operator norm, with derivative $T e^{tT}$.
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# Von Neumann universe

## Description

The von Neumann universe $V$ is the [[class|Y48FA69563]] of all sets, arranged in stages indexed by the [[ordinal numbers|0YSCSX5YZC]].
The stages $V_\alpha$ are defined by [[transfinite recursion|0ZHQMQ2C1N]] on the ordinal $\alpha$:

1. $V_0 = \varnothing$.
2. $V_{\alpha + 1} = \mathcal{P}(V_\alpha)$, the [[power set|0BC7KX51Z8]] of $V_\alpha$.
3. $V_\lambda = \bigcup_{\beta < \lambda} V_\beta$ for each limit ordinal $\lambda$.

Then $V = \bigcup_{\alpha} V_\alpha$, the union over all ordinals.
The rank of a set $x$ in $V$ is the least ordinal $\alpha$ such that $x \subseteq V_\alpha$.

## Remarks

Each $V_\alpha$ is a transitive set, and $V_\beta \subseteq V_\alpha$ when $\beta \le \alpha$.
The first stages are $V_1 = \{\varnothing\}$ and $V_2 = \{\varnothing, \{\varnothing\}\}$, and in general $\abs{V_{n+1}} = 2^{\abs{V_n}}$ for finite $n$.
The stage $V_\omega$ is the set of hereditarily finite sets, and it is countable.
Over the other axioms of ZF, the [[axiom of regularity|NFQ9FVCP31]] is equivalent to the statement that every set lies in some $V_\alpha$.
If $\kappa$ is an [[inaccessible cardinal|NW4DREQ5DQ]], then $V_\kappa$ is a model of ZFC.

## Explanations

### L1 human

Start with nothing, the empty set.
At each step, form all sets whose elements were already built; at a limit step, collect everything built so far.
The von Neumann universe is everything that ever appears in this process, and in ZF this is every set.
For example, $\varnothing$ is an element of $V_1$, $\{\varnothing\}$ is an element of $V_2$, and $\{\{\varnothing\}\}$ is an element of $V_3$.
The picture is often drawn as a V-shaped cone that widens as the stages go up.
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# Total variation distance between probability measures

## Summary

The total variation distance between two probability measures is the largest difference between the probabilities they give to the same event.
It is half the [[total variation|EPBND51CR5]] norm of their difference.
It is the natural distance for questions of distinguishing two distributions: it controls the best possible success probability of a test between them.

## Explanations

### L1 human

Two distributions are close in total variation if no event at all has very different chances under them.
For two coins with heads probabilities $p$ and $q$, the distance is $\abs{p - q}$.
For densities, it is half the area between the two density curves.
A distance of $0$ means the distributions are equal, and a distance of $1$ means they live on disjoint sets.
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# Wave front set

## Summary

The wave front set of a distribution records where the distribution fails to be smooth, and in which frequency directions it fails.
It is a closed subset of $\Omega \times (\R^n \setminus \set{0})$ that is conic in the second variable.
Its projection to $\Omega$ is the singular support, the set of points near which the distribution is not a smooth function.
It is the basic object of [[microlocal analysis|5B6WEQX7XK]] and the language of [[propagation of singularities|P2QJ7NPW3Y]].
See the [[definition|PF3G0J5JD6]].

## Explanations

### L1 human

A function with a jump along a line is rough at the points of the line, but only in one direction: across the line.
Along the line it looks perfectly regular.
The wave front set keeps track of this: at each bad point it lists the directions in which the function oscillates without control.
One finds these directions by cutting the function off near the point and checking in which directions the [[Fourier transform|DPVM34ADZV]] of the piece decays slowly.
For the jump function $H(x_1)$ in the plane, the bad points form the line $x_1 = 0$, and the bad directions there are $(\pm 1, 0)$, the two normal directions.
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# Metric outer measure

## Summary

A metric outer measure is an [[outer measure|BJ314H17HC]] on a [[metric space|0VKD1TP7F3]] with one extra property.
It is additive on any two sets lying at positive distance from each other.
This condition links the outer measure with the geometry of the space.
Its main use is that for a metric outer measure every Borel set is measurable in the sense of Carathéodory.
Lebesgue outer measure and the [[Hausdorff outer measures|Q1TGZK8RN6]] are metric outer measures.

## Explanations

### L1 human

If two sets are separated by a gap, their total size should be the sum of their sizes.
A metric outer measure is an outer measure that respects this rule.
For example, in the plane, two disks that do not touch have total area equal to the sum of their areas.
This simple rule is enough to make all open and closed sets measurable.
</document>

<document source="https://tamnd.github.io/math2/node/3T9TT5SSGD.md" tag="3T9TT5SSGD" kind="conjecture">
# Uniform boundedness conjecture for preperiodic points

## Statement

Let $N \ge 1$, $d \ge 2$ and $D \ge 1$ be integers.
There is a constant $B = B(N, d, D)$ with the following property.
Let $K$ be a [[number field|MWBDFS2KJ5]] with $[K : \Q] \le D$.
Let $\phi \colon \mathbb{P}^N \to \mathbb{P}^N$ be a morphism of degree $d$ defined over $K$.
Then the number of preperiodic points of $\phi$ in $\mathbb{P}^N(K)$ is at most $B$.

## Remarks

Here a point $P$ of [[projective space|ZT9A6JK8QF]] is preperiodic for $\phi$ when its forward orbit $\set{\phi^n(P) : n \ge 0}$ is finite, as in [[arithmetic dynamics|BE4X0QG8XW]].
For each single $\phi$, the set of preperiodic points in $\mathbb{P}^N(K)$ is finite by Northcott's theorem on heights.
The conjecture asks for a bound that depends only on $N$, $d$ and $D$.

The conjecture is open, even for the maps $\phi(z) = z^2 + c$ with $c \in \Q$ acting on $\mathbb{P}^1(\Q)$.

The case $N = 1$, $d = 4$ implies the uniform boundedness of torsion of elliptic curves, proved by Merel in 1996.
Indeed, let $E$ be an elliptic curve over $K$.
The $x$-coordinates of its torsion points are preperiodic points of the degree $4$ Lattès map that describes doubling on $E$.
At most two points of $E$ share an $x$-coordinate.

## Explanations

### L1 human

Fix a kind of polynomial or rational map, like $z \mapsto z^2 + c$, and a kind of number system, like the fractions.
For each single map, only finitely many fractions eventually repeat under iteration.
The conjecture says more: there is one bound on how many such fractions there can be, the same for all maps of that degree.
It is a dynamical version of famous finiteness results about elliptic curves.
</document>

<document source="https://tamnd.github.io/math2/node/3TERJ5C8QK.md" tag="3TERJ5C8QK" kind="statement">
# Characterization of Shannon entropy

## Statement

For each integer $n \ge 2$, let $\Delta_n$ be the set of probability vectors $(p_1, \dots, p_n)$ with $p_i \ge 0$ and $p_1 + \cdots + p_n = 1$.
Let $H$ be a function defined on $\bigcup_{n \ge 2} \Delta_n$ with real values.
Assume:

1. The function $p \mapsto H(p, 1 - p)$ is [[continuous|0X3WGX95FE]] on $[0, 1]$.
2. $H$ is symmetric: $H(p_1, \dots, p_n)$ does not change when the $p_i$ are permuted.
3. $H(\frac{1}{2}, \frac{1}{2}) = 1$.
4. For all $n \ge 2$, all $(p_1, \dots, p_n) \in \Delta_n$ with $p_n > 0$, and all $q_1, q_2 \ge 0$ with $q_1 + q_2 = p_n$,
$$H(p_1, \dots, p_{n-1}, q_1, q_2) = H(p_1, \dots, p_n) + p_n \, H\left(\frac{q_1}{p_n}, \frac{q_2}{p_n}\right).$$

Then for every $n \ge 2$ and every $(p_1, \dots, p_n) \in \Delta_n$,
$$H(p_1, \dots, p_n) = -\sum_{i=1}^{n} p_i \log_2 p_i,$$
with the convention $0 \log_2 0 = 0$.
Conversely, this function has properties 1 to 4.

## Remarks

The formula is the Shannon entropy of a finite [[probability distribution|0DVDCMA2DK]], measured in bits.
This set of axioms is due to Faddeev; Shannon and Khinchin gave earlier characterizations with stronger hypotheses.
The proof first studies $F(n) = H(\frac{1}{n}, \dots, \frac{1}{n})$: property 4 gives $F(mn) = F(m) + F(n)$, and the remaining properties force $F(n) = \log_2 n$.
Grouping equal masses then gives the formula for rational probabilities, and continuity extends it to all probability vectors.
The normalization 3 only fixes the unit; without it, $H$ is a constant multiple $c$ of the Shannon entropy.

## Explanations

### L1 human

Entropy measures how uncertain the outcome of a random experiment is.
This theorem says that once one asks for a few natural rules, there is only one possible formula.
The first rule says that small changes of the probabilities give small changes of the uncertainty.
The other rules say that the order of the outcomes does not matter, that a fair coin carries one bit, and that uncertainty can be computed in stages.
The last rule says that splitting one outcome into two adds the uncertainty of the split, weighted by the probability of that outcome.

### L2 human

The theorem is a solution of a [[functional equation|SRAN4EKWS0]]: setting $f(p) = H(p, 1 - p)$, the properties reduce to the fundamental equation of information $f(x) + (1 - x) f\left(\frac{y}{1 - x}\right) = f(y) + (1 - y) f\left(\frac{x}{1 - y}\right)$ for $x, y \in [0, 1)$ with $x + y \le 1$.
It explains why the logarithm, and no other function, appears in the entropy formula.
Dropping or changing the recursivity axiom leads to other entropies, such as the Rényi entropies.
Similar axiomatic characterizations exist for relative entropy and mutual information.
</document>

<document source="https://tamnd.github.io/math2/node/3TGTNRBKW0.md" tag="3TGTNRBKW0" kind="definition">
# Hopf module

## Definition

Let $K$ be a field and let $H$ be a [[Hopf algebra|7XFYP0N84H]] over $K$ with comultiplication $\Delta$.
A *right Hopf module* over $H$ is a vector space $M$ over $K$ such that:

1. $M$ is a right $H$-[[module|0NZ8JA25WK]], with action $m \otimes h \mapsto m \cdot h$.
2. $M$ is a right $H$-[[comodule|7WEM2E6RTV]], with coaction $\rho \colon M \to M \otimes H$.
3. For all $m \in M$ and $h \in H$, in [[Sweedler notation|NC26CE0Y8B]],
$$\rho(m \cdot h) = \sum m_{(0)} \cdot h_{(1)} \otimes m_{(1)} h_{(2)}.$$

The *coinvariants* of $M$ are the elements of $M^{\mathrm{co}H} = \set{m \in M : \rho(m) = m \otimes 1}$.

## Remarks

Clause 3 says that $\rho$ is a module map when $M \otimes H$ carries the action $(m \otimes x) \cdot h = \sum m \cdot h_{(1)} \otimes x h_{(2)}$.
Example: for a vector space $V$, the space $M = V \otimes H$ with $(v \otimes x) \cdot h = v \otimes xh$ and $\rho(v \otimes x) = \sum v \otimes x_{(1)} \otimes x_{(2)}$.
Clause 3 holds because $\Delta(xh) = \Delta(x)\Delta(h)$.
Non-example: $H = KG$ for a group $G \ne 1$, acting on itself by multiplication, with the coaction $\rho(x) = x \otimes 1$.
This is a module and a comodule, but for $g \ne 1$ in $G$, $\rho(1 \cdot g) = g \otimes 1$ while the right side of clause 3 is $g \otimes g$.
The fundamental theorem of Hopf modules states that the map $M^{\mathrm{co}H} \otimes H \to M$, $m \otimes h \mapsto m \cdot h$, is an isomorphism of Hopf modules.
</document>

<document source="https://tamnd.github.io/math2/node/3THE3Q96KK.md" tag="3THE3Q96KK" kind="statement">
# Brianchon's theorem

## Statement

Work in the real [[projective plane|DANFNN8C4Y]] $\mathbb{P}^2(\R)$, and let $\mathcal{C}$ be a nondegenerate [[conic|00HJCMC2CD]].
Let $t_1, \dots, t_6$ be six distinct lines [[tangent|01R1HW3Y42]] to $\mathcal{C}$.
Put $A = t_1 \cap t_2$, $B = t_2 \cap t_3$, $C = t_3 \cap t_4$, $D = t_4 \cap t_5$, $E = t_5 \cap t_6$ and $F = t_6 \cap t_1$.
Then the three lines $AD$, $BE$ and $CF$ pass through a common point.

## Remarks

So $ABCDEF$ is a hexagon whose sides touch the conic, and the statement says that its three main diagonals are concurrent.
No three tangent lines of a nondegenerate real conic pass through one point, which makes the six vertices distinct and the three diagonals well defined.

The theorem is the dual of [[Pascal's theorem|94AKWRHND0]].
The tangent lines of a nondegenerate conic form a nondegenerate conic of the dual plane, and the dual of the statement "three points are collinear" is "three lines are concurrent".
So the principle of [[duality|DCMP1A0138]] applied to Pascal's theorem gives Brianchon's theorem.

Example: for a regular hexagon circumscribed about a circle, the three main diagonals pass through the centre of the circle.

## Explanations

### L1 human

Draw six lines that just touch an ellipse, so that they enclose a hexagon around it.
Join opposite corners of the hexagon by three diagonals.
These three diagonals always meet at a single point.

### L2 human

Being the dual of Pascal's theorem, it is a classic illustration of projective [[duality|DCMP1A0138]].
It is used to construct tangent lines of a conic from five given tangent lines with a straightedge.
The same argument works over any field of characteristic different from $2$.
In characteristic $2$ all tangent lines of a nondegenerate conic pass through one point, so the hypotheses of the statement cannot be met there.
</document>

<document source="https://tamnd.github.io/math2/node/3TNVVAA1AG.md" tag="3TNVVAA1AG" kind="definition">
# Compact element

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
An element $c \in P$ is *compact* if for every [[directed subset|D9RY4667K6]] $D \subseteq P$ that has a [[supremum|0R3H5PANRQ]] $\bigvee D$ in $P$ with $c \le \bigvee D$, there is $d \in D$ with $c \le d$.

## Remarks

In a [[complete lattice|D7A0ZGZT5S]] $L$, an element $c$ is compact if and only if for every $S \subseteq L$ with $c \le \bigvee S$ there is a finite $F \subseteq S$ with $c \le \bigvee F$.
Compact elements are also called finite elements, and in domain theory $c$ is compact exactly when $c \ll c$ for the way-below relation.
Example: in the power set of a set $X$, a finite set $c$ is compact.
If $c \subseteq \bigcup_{d \in D} d$ for a directed family $D$, each of the finitely many points of $c$ lies in some member of $D$.
Directedness then gives one member containing all of them.
Non-example: in the power set of $\N$, the set $\N$ is not compact.
It is the union of the directed family of sets $\set{0, \ldots, n}$ and lies in none of them.
The least element, when it exists, is always compact.
</document>

<document source="https://tamnd.github.io/math2/node/3TVXB7X266.md" tag="3TVXB7X266" kind="statement">
# Commutant lifting theorem

## Statement

Let $H$ be a [[Hilbert space|HM2HHE3J51]] and let $T \colon H \to H$ be a contraction, that is, a bounded linear operator with $\norm{T} \le 1$.
Let $V \colon K \to K$ be a minimal isometric dilation of $T$.
This means that $K$ is a Hilbert space containing $H$ as a closed subspace and $V$ is an [[isometry|02HPP5VCZH]].
It also means that $T^n = P_H V^n|_H$ for all $n \ge 0$, and that $K$ is the closed span of $\bigcup_{n \ge 0} V^n H$.
Here $P_H$ is the [[orthogonal projection|0B8KZRT28M]] of $K$ onto $H$.
Let $A \colon H \to H$ be a bounded linear operator with $AT = TA$.
Then there is a bounded linear operator $B \colon K \to K$ such that
$$BV = VB, \qquad \norm{B} = \norm{A}, \qquad P_H B = A P_H.$$

## Remarks

The condition $P_H B = A P_H$ says that $B^*$ maps $H$ into $H$ and $B^*|_H = A^*$; so $B$ lifts $A$ to the dilation space without increasing the norm.
The proof constructs $B$ step by step on the pieces of $K$, with a one-step extension lemma for operator matrices that keeps the norm.
The theorem is due to Sz.-Nagy and Foias, and Sarason proved an earlier special case.

Example: take $T$ to be the compression of the shift $S$ on the [[Hardy space|PNNNFJVF6W]] $H^2$ to a model space $H^2 \ominus \theta H^2$, with $\theta$ an [[inner function|NN8XH5RJYB]].
Then the lifts of operators commuting with $T$ commute with $S$, so they are multiplications by bounded analytic functions.
This gives Sarason's theorem: each operator commuting with $T$ is the compression of multiplication by some $\varphi \in H^\infty$ with the same norm.

## Explanations

### L1 human

A contraction can be seen as a piece of an isometry acting on a larger space.
The theorem says that any operator that commutes with the contraction can be extended to the larger space so that it commutes with the isometry, with no loss in norm.

### L2 human

Commutant lifting gives a unified operator-theoretic approach to interpolation problems.
The Nevanlinna–Pick and Carathéodory–Fejér problems ask for a bounded analytic function on the disc with prescribed values or Taylor coefficients and least supremum norm.
Both reduce to lifting an operator that commutes with a compressed shift.
The theorem builds on [[Sz.-Nagy's dilation theorem|YB7Q3EK0ND]], which provides the minimal isometric dilation.
</document>

<document source="https://tamnd.github.io/math2/node/3TWMBFQJJ8.md" tag="3TWMBFQJJ8" kind="concept">
# Closable operator

## Summary

A closable operator is an unbounded operator that has a closed extension.
Many operators are first defined on a convenient small domain, such as smooth functions, and are not closed there.
If they are closable, they have a smallest closed extension, the closure, which is the operator one actually studies.

## Explanations

### L1 human

Sometimes a formula is written down only for nice inputs, and the resulting operator is not closed.
One then tries to enlarge the domain by taking limits, keeping the rule that limits of inputs go with limits of outputs.
This works when the same input limit can never come with two different output limits.
For example, differentiation on polynomials, in the space $C[0, 1]$ with the supremum norm, is closable, and its closure is differentiation on continuously differentiable functions.
</document>

<document source="https://tamnd.github.io/math2/node/3TXY35FR30.md" tag="3TXY35FR30" kind="definition">
# n-term approximation

## Definition

Let $X$ be a real or complex [[normed vector space|03J5RAKB30]] with norm $\norm{\cdot}$, and let $\mathcal{D} \subseteq X$ be a set, called the *dictionary*.
Let $n \ge 1$ be an integer.
The set of *$n$-term approximants* is
$$\Sigma_n(\mathcal{D}) = \set{\textstyle\sum_{k=1}^{n} c_k g_k : c_k \text{ scalars}, \ g_k \in \mathcal{D}} .$$
The *error of best $n$-term approximation* of $f \in X$ is
$$\sigma_n(f, \mathcal{D}) = \inf\set{\norm{f - g} : g \in \Sigma_n(\mathcal{D})} .$$
*$n$-term approximation* is the approximation of elements $f \in X$ by elements of $\Sigma_n(\mathcal{D})$, measured by $\sigma_n(f, \mathcal{D})$.

## Remarks

This is the [[error of best approximation|YDN0T4MEX0]] from the set $\Sigma_n(\mathcal{D})$, which is a union of subspaces and not a subspace.
Indeed $\Sigma_n + \Sigma_n \subseteq \Sigma_{2n}$, but in general $\Sigma_n + \Sigma_n \ne \Sigma_n$.
Example: let $X$ be a Hilbert space with orthonormal basis $\mathcal{D} = (e_j)$ and $f = \sum_j a_j e_j$.
Keeping the $n$ coefficients $a_j$ of largest absolute value gives a best approximant, and $\sigma_n(f, \mathcal{D})^2$ is the sum of $\abs{a_j}^2$ over the remaining indices.
Check: for $g = \sum_{j \in S} c_j e_j$ with $\abs{S} \le n$, $\norm{f - g}^2 = \sum_{j \in S} \abs{a_j - c_j}^2 + \sum_{j \notin S} \abs{a_j}^2$, which is smallest when $c_j = a_j$ and $S$ carries the largest $\abs{a_j}$.
Non-example: in $\R^3$ with the standard basis, $3e_1 + e_2 + 2e_3$ is not in $\Sigma_2$, since it has three nonzero coordinates.
When $\mathcal{D}$ is redundant, the best approximant may be hard to compute, and greedy algorithms are used instead.
</document>

<document source="https://tamnd.github.io/math2/node/3TY9Y91AMZ.md" tag="3TY9Y91AMZ" kind="definition">
# K-function

## Definition

Let $\Gamma$ be the [[gamma function|0H5SYBSED9]] and $\ln$ the [[natural logarithm|04DXAFN2ES]].
For a real number $x > 0$, the *K-function* is
$$K(x) = (2\pi)^{-(x-1)/2} \exp\left(\binom{x}{2} + \int_0^{x-1} \ln \Gamma(t + 1)\, dt\right),$$
where $\binom{x}{2} = \frac{x(x-1)}{2}$.

## Remarks

Raabe's formula $\int_a^{a+1} \ln \Gamma(u)\, du = \frac12 \ln 2\pi + a \ln a - a$ for $a > 0$ gives
$$K(x + 1) = x^x K(x) \qquad (x > 0).$$

Example: $\int_0^1 \ln \Gamma(t+1)\, dt = \int_0^1 (\ln t + \ln \Gamma(t))\, dt = -1 + \frac12 \ln 2\pi$.
Hence $K(2) = (2\pi)^{-1/2} \exp(1 - 1 + \frac12 \ln 2\pi) = 1$.
The recurrence then gives $K(3) = 2^2 = 4$, $K(4) = 4 \cdot 3^3 = 108$, and in general $K(n + 1) = \prod_{k=1}^n k^k$.

Non-example: the function $x \mapsto x^x$ does not satisfy the recurrence; for instance $3^3 = 27 \ne 2^2 \cdot 2^2 = 16$.

The formula above is stated for $x > 0$, where $t + 1 > 0$ on the path of integration.
The same formula, with the principal branch of $\ln \Gamma$, defines $K$ as a holomorphic function on $\C \setminus (-\infty, 0]$.
</document>

<document source="https://tamnd.github.io/math2/node/3V9GYYXM1H.md" tag="3V9GYYXM1H" kind="example">
# Moulton plane

## Example

The points of the *Moulton plane* are the points of $\R^2$.
Its lines are of three kinds:

1. the vertical lines $x = c$;
2. the lines $y = m x + b$ with slope $m \ge 0$;
3. for each slope $m < 0$ and each $b \in \R$, the bent line made of the points $(x, m x + b)$ with $x \le 0$ and the points $(x, 2 m x + b)$ with $x \ge 0$.

So lines of negative slope are bent where they cross the $y$-axis, and their slope doubles on the right.
With these lines it is an [[affine plane|EC7XT2ZM5S]]: any two points lie on exactly one line.
For a fixed $m < 0$, the bent lines with parameter $m$ are pairwise parallel, as are the lines of each other slope.
Some authors use another factor than $2$; any positive factor other than $1$ gives a plane with the same properties.

The Moulton plane does not satisfy the affine form of [[Desargues's theorem|5Z19Y02MME]].
Let $O = (-10, -10)$, $A = (-8, -6)$, $B = (-6, 2)$, $C = (-2, -2)$, and $A' = (-7, -4)$, $B' = (-4, 8)$, $C' = (2, 2)$.
Check:

- $A' = O + \tfrac{3}{2}(A - O)$, and likewise for $B'$ and $C'$.
- The lines $OA$, $OB$, $OC$ have slopes $2$, $3$, $1$, so they are ordinary lines; they contain $A'$, $B'$, $C'$.
- $AB$ and $A'B'$ both have slope $4$, and $AC$ and $A'C'$ both have slope $2/3$; these are ordinary distinct lines, so $AB \parallel A'B'$ and $AC \parallel A'C'$.
- $B$ and $C$ lie in $x < 0$ on $y = -x - 4$, so $BC$ is the bent line with $m = -1$, $b = -4$.
- The line through $B'$ and $C'$ is bent with $8 = -4m + b$ and $2 = 4m + b$, so $m = -3/4$ and $b = 5$.

Since $-3/4 \ne -1$, the line $B'C'$ is not parallel to $BC$.
In the Euclidean plane the same data would force $BC \parallel B'C'$.

## Remarks

Add a line at infinity, with one point for each class of parallel lines, to get a [[projective plane|DANFNN8C4Y]].
In it the triangles $ABC$ and $A'B'C'$ are in perspective from $O$, the points $AB \cap A'B'$ and $AC \cap A'C'$ lie on the line at infinity, but $BC \cap B'C'$ is a finite point.
So the three points are not collinear, and the projective completion is a [[non-Desarguesian plane|HYE5FT4M34]].

## Explanations

### L1 human

The Moulton plane is the ordinary plane with a strange rule for lines: lines going downhill bend as they cross the vertical axis, like light entering water.
Uphill and vertical lines stay straight.
Through two points there is still exactly one line, and parallels behave as usual.
But Desargues's theorem, which always works in the ordinary plane, fails here, which shows that this theorem is not a consequence of the incidence axioms alone.
</document>

<document source="https://tamnd.github.io/math2/node/3VB65PEZQS.md" tag="3VB65PEZQS" kind="statement">
# Hermite–Hadamard inequality

## Statement

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a [[convex function|0VV0HSW1ZS]].
Then $f$ is Riemann integrable on $[a, b]$ and
$$f\Big( \frac{a + b}{2} \Big) \le \frac{1}{b - a} \int_a^b f(x) \, dx \le \frac{f(a) + f(b)}{2}.$$

## Remarks

Proof idea: a convex function on $[a, b]$ is bounded and is continuous on $(a, b)$, so it is integrable.
For the left inequality, convexity gives $f\big(\frac{a+b}{2}\big) \le \frac{1}{2}\big(f(x) + f(a + b - x)\big)$; integrate over $x \in [a, b]$.
For the right inequality, the graph lies below the chord: $f(x) \le \frac{b - x}{b - a} f(a) + \frac{x - a}{b - a} f(b)$; integrate this bound, whose mean value is $\frac{f(a) + f(b)}{2}$.

Example: $f(x) = x^2$ on $[0, 1]$ gives $\frac{1}{4} \le \frac{1}{3} \le \frac{1}{2}$.

For a concave function both inequalities are reversed.

## Explanations

### L1 human

The average value of a convex function over an interval lies between two simple numbers.
It is at least the value at the midpoint, and at most the average of the two end values.
Geometrically, the graph lies above a supporting line at the midpoint and below the chord joining the ends.

### L2 human

The inequality gives error bounds for the midpoint rule and the trapezoidal rule: for a convex integrand the midpoint rule underestimates the integral and the trapezoidal rule overestimates it.
It is a two-sided refinement of [[Jensen's inequality|0RWMRTPC87]] for the uniform distribution on $[a, b]$.
</document>

<document source="https://tamnd.github.io/math2/node/3VBRS94V1G.md" tag="3VBRS94V1G" kind="concept">
# Mild solution

## Summary

A mild solution of a linear evolution equation $u' = A u + f$ in a [[Banach space|06GZ956657]] is a continuous function.
It is given by the variation of constants formula $u(t) = T(t) u_0 + \int_0^t T(t - s) f(s) \, ds$, where $T(t)$ is the [[C0-semigroup|EVF180QJXY]] generated by $A$.
It needs no differentiability in time and no assumption that $u_0$ lies in the domain of $A$.
Every classical solution is a mild solution, so mild solutions extend the notion of solution to rough data.
Many semilinear parabolic and hyperbolic equations are solved by finding a mild solution as a fixed point of this integral formula.

## Explanations

### L1 human

For a scalar equation $u' = a u + f(t)$, the solution is $u(t) = e^{at} u_0 + \int_0^t e^{a(t - s)} f(s) \, ds$.
A partial differential equation such as the heat equation can be seen as such an equation, with $a$ replaced by an operator and $e^{ta}$ by a semigroup of operators.
The same formula then makes sense even when the initial state is too rough to be differentiated.
The function it defines is called the mild solution.
</document>

<document source="https://tamnd.github.io/math2/node/3VCD0T7QNG.md" tag="3VCD0T7QNG" kind="definition">
# Toeplitz operator

## Definition

Let $\mathbb{T}$ be the unit circle with normalised Lebesgue measure, and let $L^2 = L^2(\mathbb{T})$ ([[Lp space|SASP14MA1Z]]).
Let $H^2 \subseteq L^2$ be the [[Hardy space|PNNNFJVF6W]], the closed subspace of functions whose Fourier coefficients of negative index vanish.
Let $P \colon L^2 \to H^2$ be the [[orthogonal projection|0Q8HZTJC7G]].
For $\varphi \in L^\infty(\mathbb{T})$, the *Toeplitz operator* with symbol $\varphi$ is the operator $T_\varphi \colon H^2 \to H^2$ given by
$$T_\varphi f = P(\varphi f).$$

## Remarks

$T_\varphi$ is a bounded linear operator, and in fact $\norm{T_\varphi} = \norm{\varphi}_\infty$.
With respect to the orthonormal basis $e_n(t) = e^{int}$, $n \ge 0$, its matrix entries are $\ip{T_\varphi e_k}{e_j} = \hat\varphi(j - k)$, so the matrix is a [[Toeplitz matrix|GKY2WSHE26]].
If $\varphi$ is continuous and has no zeros on $\mathbb{T}$, then $T_\varphi$ is a [[Fredholm operator|8HWGKWD6ZM]] whose index is minus the winding number of $\varphi$ around $0$.

Example: $\varphi(e^{it}) = e^{it}$.
For $f \in H^2$ the product $\varphi f$ has no negative Fourier coefficients, so $P(\varphi f) = \varphi f$ and $T_\varphi$ is the [[shift operator|MBK6AE6734]] $f \mapsto z f$.

Non-example: for $\varphi(e^{it}) = e^{-it}$, plain multiplication by $\varphi$ does not map $H^2$ into itself, since it sends $1$ to $e^{-it} \notin H^2$.
So the projection $P$ in the definition cannot be omitted; the Toeplitz operator $T_\varphi$ is the backward shift.
</document>

<document source="https://tamnd.github.io/math2/node/3VJC5Q9MYA.md" tag="3VJC5Q9MYA" kind="concept">
# Leray–Schauder degree

## Summary

The Leray–Schauder degree is an integer that counts, with signs, the solutions of $u - K(u) = y$ in a bounded open set of a Banach space.
Here $K$ is a compact map.
It extends the Brouwer degree of maps between finite-dimensional spaces to this infinite-dimensional class.
It does not change under compact deformations that keep solutions away from the boundary, and a nonzero degree guarantees a solution.
This makes it a main tool to prove existence for nonlinear elliptic equations by continuation from a simpler problem.

## Explanations

### L1 human

In finite dimensions, the degree of a map counts how many times it covers a point, with orientation signs.
The degree survives deformations, so one can compute it on an easy map and transfer the answer to a hard one.
In infinite dimensions this fails for general continuous maps, but it works for maps of the form identity minus a compact map.
Such maps are close to finite-dimensional ones, because a compact map can be approximated by maps with values in finite-dimensional spaces.
For example, $u \mapsto u - K(u)$ on a ball, with $K$ small and compact, has degree $1$ at $0$, so the equation $u = K(u)$ has a solution in the ball.
</document>

<document source="https://tamnd.github.io/math2/node/3VMJGQCD7S.md" tag="3VMJGQCD7S" kind="statement">
# Hartman's uniqueness theorem for harmonic maps

## Statement

Let $M$ be a compact [[Riemannian manifold|7CBT4SM83Y]] without boundary.
Let $N$ be a compact Riemannian manifold whose [[sectional curvature|1X42JAA349]] is nonpositive everywhere.

1. Let $u_0, u_1 \colon M \to N$ be [[harmonic maps|KZJYG9DS3M]] that are [[homotopic|9YA6S6HD9V]].
Then there is a smooth homotopy $u_t$, $t \in [0, 1]$, from $u_0$ to $u_1$ such that each $u_t$ is harmonic and has the same [[energy|E6HDE1DFXD]] as $u_0$.
Moreover, for each $x \in M$ the curve $t \mapsto u_t(x)$ is a [[geodesic|NSYMQMHPDN]] traversed at constant speed, and this speed does not depend on $x$.
2. Suppose that the sectional curvature of $N$ is negative everywhere.
Let $u \colon M \to N$ be a harmonic map whose image is neither a point nor a closed geodesic.
Then $u$ is the only harmonic map in its homotopy class.

## Remarks

Lift to the universal cover of $N$, which is a Hadamard manifold, and join $u_0(x)$ to $u_1(x)$ by the unique geodesic in the right homotopy class.
In nonpositive curvature the squared distance between the two lifted maps is a subharmonic function on $M$, so it is constant on the compact manifold $M$.
Following the geodesics gives the homotopy $u_t$, and the second variation of the energy shows that every $u_t$ is again harmonic.
With strictly negative curvature the homotopy must be constant, unless all the geodesics $t \mapsto u_t(x)$ run along one closed geodesic, which forces the image of $u$ into that geodesic.

The curvature hypothesis is needed.
For $M = N = S^2$ with the round metric, every rotation is a harmonic map homotopic to the identity.
So harmonic maps in a homotopy class are far from unique.
Uniqueness also fails in the flat case: on a flat torus $N$, composing a harmonic map with a translation gives a family of homotopic harmonic maps, as part 1 allows.

## Explanations

### L1 human

When the target space curves like a saddle everywhere, harmonic maps are rigid.
Two harmonic maps that can be deformed into each other differ only by sliding along geodesics in a uniform way.
When the curvature is strictly negative, even this sliding is impossible except in degenerate cases, so the harmonic map is unique.

### L2 human

The theorem complements the [[Eells–Sampson theorem|AP4A0GX0V2]], which gives existence of a harmonic map in each homotopy class under the same curvature hypothesis.
Together they say that, for targets of negative curvature, a homotopy class of maps usually has a canonical harmonic representative.
This is used in rigidity theory, for example for maps between locally symmetric spaces and in the study of Teichmüller space.
</document>

<document source="https://tamnd.github.io/math2/node/3VQ2V5HH3R.md" tag="3VQ2V5HH3R" kind="concept">
# Sphere bundle

## Summary

A sphere bundle is a [[fiber bundle|CX4M8K61AB]] whose fibers are spheres of a fixed dimension.
The main source of examples is the unit sphere bundle of a real vector bundle with an inner product.
Sphere bundles carry the [[Euler class|BJX6QS0ANT]] and the [[Gysin sequence|QJSPWM9F9P]], and the [[Hopf fibration|M5AKNYC4QQ]] is a circle bundle over the $2$-sphere.
Milnor's exotic $7$-spheres were found as total spaces of $S^3$-bundles over $S^4$.

## Explanations

### L1 human

Over each point of a base space, place a sphere, and let these spheres vary continuously.
The torus is a circle bundle over the circle: above each point of one circle sits another circle.
The Klein bottle is also a circle bundle over the circle, but with a reflection as one goes around.
For a surface, the unit tangent vectors at each point form a circle, and together they make a circle bundle over the surface.
</document>

<document source="https://tamnd.github.io/math2/node/3VRZ80X45P.md" tag="3VRZ80X45P" kind="statement">
# Rückert Nullstellensatz

## Statement

Let $a \in \C^n$, let $\mathcal{O}_{n,a}$ be the ring of [[germs of holomorphic functions|7PJ8KVZZCQ]] at $a$, and let $I \subseteq \mathcal{O}_{n,a}$ be an ideal.
Let $V(I)$ be the [[germ of the analytic set|Q9B74BYZ8C]] at $a$ defined by $I$, and let $\mathcal{I}(V(I))$ be the ideal of germs vanishing on $V(I)$.
Then
$$\mathcal{I}(V(I)) = \sqrt{I},$$
the [[radical|437YRCNYRJ]] of $I$.
In other words, if a germ $f \in \mathcal{O}_{n,a}$ vanishes on $V(I)$, then $f^m \in I$ for some integer $m \ge 1$.

## Remarks

The inclusion $\sqrt{I} \subseteq \mathcal{I}(V(I))$ is clear: if $f^m \in I$, then $f^m$, and hence $f$, vanishes on $V(I)$.
For the other inclusion, write $\sqrt{I}$ as a finite intersection of prime ideals, which is possible because $\mathcal{O}_{n,a}$ is Noetherian, and reduce to the case of a prime ideal $\mathfrak{p}$.
For $\mathfrak{p}$ prime, the local parametrization of the irreducible germ $V(\mathfrak{p})$ shows that a germ vanishing on $V(\mathfrak{p})$ lies in $\mathfrak{p}$.
Example: for $I = (z_1^2) \subseteq \mathcal{O}_{2,0}$, the germ $V(I)$ is the line $\set{z_1 = 0}$, and $\mathcal{I}(V(I)) = (z_1) = \sqrt{I}$; here $z_1^2 \in I$.
As a consequence, $I \mapsto V(I)$ and $A_a \mapsto \mathcal{I}(A_a)$ are mutually inverse bijections between radical ideals of $\mathcal{O}_{n,a}$ and germs of analytic sets at $a$, and prime ideals correspond to irreducible germs.
As for Hilbert's Nullstellensatz, the base field matters: the statement fails for real analytic germs, since $x^2 + y^2$ vanishes only at $0 \in \R^2$ but $x \notin \sqrt{(x^2 + y^2)}$.

## Explanations

### L1 human

The theorem says that the germs vanishing on the common zeros of some germs are exactly those of which some power can be built from them.
It is the local, holomorphic version of Hilbert's Nullstellensatz for polynomials.
It means that the shape of an analytic set near a point and the algebra of the ideal of germs determine each other.

### L2 human

The Nullstellensatz is the dictionary of local analytic geometry.
With it, the decomposition of an analytic germ into irreducible components corresponds to the primary decomposition of its ideal.
The dimension of a germ $A_a$ equals the Krull dimension of $\mathcal{O}_{n,a}/\mathcal{I}(A_a)$.
It is also an ingredient in Cartan's coherence theorem for the ideal sheaf of an analytic set.
</document>

<document source="https://tamnd.github.io/math2/node/3VX06XRVZG.md" tag="3VX06XRVZG" kind="statement">
# Schoenflies theorem

## Statement

1. Let $C \subset \R^2$ be a [[Jordan curve|687RGCHBTP]]. Then there is a [[homeomorphism|0QT00PB5G2]] $h \colon \R^2 \to \R^2$ with $h(C) = S^1$, the unit circle. In particular, the closure of the bounded component of $\R^2 \setminus C$ is homeomorphic to the [[closed disk|0Q0H13G8ZB]] $D^2$.
2. (Generalized Schoenflies theorem.) Let $n \ge 2$ and let $e \colon S^{n-1} \times [-1, 1] \to S^n$ be a [[topological embedding|00ZG9Y9TR4]], where $S^n$ is the [[$n$-sphere|T62369RFH6]]. Let $\Sigma = e(S^{n-1} \times \set{0})$. Then $S^n \setminus \Sigma$ has two components, and the closure of each is homeomorphic to the closed ball $D^n$.

## Remarks

The hypothesis in part 2 says that the sphere $\Sigma$ is bicollared.
For a codimension-one sphere this is equivalent to being locally flat.
It cannot be dropped for $n = 3$: the Alexander horned sphere is an embedded $2$-sphere one of whose complementary regions is not simply connected, so its closure is not a ball.
In the plane no such hypothesis is needed, because every Jordan curve is tame.
One proof of part 1 uses the Riemann mapping theorem and Carathéodory's extension theorem for the interior; part 2 was proved by Morton Brown in 1960.
For smooth embedded spheres in $\R^4$, whether the closed inside is always a smooth ball is open.

## Explanations

### L1 human

The Jordan curve theorem says a loop in the plane has an inside and an outside.
The Schoenflies theorem says more: the inside, with the loop added, is just a deformed disk.
One can straighten the whole plane so that the loop becomes a round circle.

### L2 human

In higher dimensions, the naive analogue fails: there are wild embeddings of $S^2$ in $\R^3$ whose outside is not the outside of a ball.
The generalized theorem shows that all such wildness lives at points where the sphere is not locally flat.
It is a basic tool in topological manifold theory, for example for the annulus theorem and for showing that the connected sum is well defined.
</document>

<document source="https://tamnd.github.io/math2/node/3W0FP5DCEV.md" tag="3W0FP5DCEV" kind="definition">
# Projective cover

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity and let $M$ be a left $R$-[[module|0NZ8JA25WK]].
A [[submodule|05WDSN597C]] $K$ of a module $P$ is *superfluous* if for every submodule $L$ of $P$, $L + K = P$ implies $L = P$.
A *projective cover* of $M$ is a surjective $R$-linear map $p \colon P \to M$ such that:

1. $P$ is a [[projective module|BFFA03GE3V]].
2. The kernel $\ker p$ is superfluous in $P$.

## Remarks

If $p \colon P \to M$ and $p' \colon P' \to M$ are projective covers, there is an isomorphism $f \colon P \to P'$ with $p' \circ f = p$.
Example: let $R = \Z/4$ and $p \colon R \to \Z/2$ the reduction map.
The module $R$ is free, hence projective, and $\ker p = 2R$.
The submodules of $R$ are $0$, $2R$ and $R$, and $0 + 2R$ and $2R + 2R$ are not $R$, so $2R$ is superfluous and $p$ is a projective cover.
Non-example: over $R = \Z$, the reduction map $\Z \to \Z/2$ has kernel $2\Z$, and $2\Z + 3\Z = \Z$ with $3\Z \ne \Z$, so the kernel is not superfluous.
In fact the $\Z$-module $\Z/2$ has no projective cover at all.
Over a local ring $R$ with maximal ideal $\mathfrak m$, the map $R \to R / \mathfrak m$ is a projective cover, by Nakayama's lemma.
Bass proved two characterizations.
A ring is semiperfect if and only if every finitely generated left module has a projective cover.
A ring is left perfect if and only if every left module has a projective cover.
</document>

<document source="https://tamnd.github.io/math2/node/3W358BSRN2.md" tag="3W358BSRN2" kind="concept">
# Periodic solution of a difference equation

## Summary

A periodic solution of a difference equation is a solution that is a [[periodic sequence|Q3B6K9HYRG]].
For a first-order autonomous equation $x_{n+1} = f(x_n)$, it is the orbit of a [[periodic point|T75CZ5H47N]] of $f$.
Equilibria are the periodic solutions of period $1$.
Periodic solutions, their periods and their stability describe much of the long-run behaviour of nonlinear recursions.
Their birth is studied through bifurcations such as period doubling.
It is the discrete counterpart of a [[periodic solution|FHNDX22WMK]] of a differential equation.

## Explanations

### L1 human

A periodic solution is a sequence produced by the recursion that comes back to the same state after a fixed number of steps.
From then on it repeats the same cycle forever.
For $x_{n+1} = 1 - x_n$ with $x_0 = 0$, the solution is $0, 1, 0, 1, \dots$, a cycle of period $2$.
In the logistic map $x_{n+1} = r x_n (1 - x_n)$ with $r$ a little above $3$, solutions settle near a cycle of period $2$.
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<document source="https://tamnd.github.io/math2/node/3W5JGDZAN0.md" tag="3W5JGDZAN0" kind="statement">
# Tarski–Seidenberg theorem

## Statement

Let $R$ be a [[real closed field|DDY3Q3PJ6W]], for example $\R$.
A subset of $R^m$ is *semialgebraic* if it is a finite union of sets of the following form.
$$\set{x \in R^m : f(x) = 0, \ g_1(x) > 0, \dots, g_k(x) > 0}$$
Here $k \ge 0$, and $f, g_1, \dots, g_k$ are [[polynomials|07GNX8GCYP]] in $m$ variables with coefficients in $R$.
Let $n \ge 1$, let $\pi \colon R^{n+1} \to R^n$ be the projection $\pi(x_1, \dots, x_{n+1}) = (x_1, \dots, x_n)$, and let $S \subseteq R^{n+1}$ be semialgebraic.
Then $\pi(S)$ is a semialgebraic subset of $R^n$.

## Remarks

The theorem is the geometric form of [[quantifier elimination for real closed fields|FSJ9NGRRPA]].
The set $\pi(S)$ is defined by the formula $\exists x_{n+1} \, \varphi(x_1, \dots, x_{n+1})$, where $\varphi$ is a quantifier-free formula that defines $S$.
Quantifier elimination replaces it by a quantifier-free formula with parameters in $R$, and the set defined by such a formula is semialgebraic.

Example: let $S = \set{(a, x) \in \R^2 : x^2 - a = 0}$, which is semialgebraic.
Its projection to the first coordinate is $\set{a : a \ge 0}$, which is semialgebraic as the union of $\set{a : a = 0}$ and $\set{a : a > 0}$.

The analogous statement fails for algebraic sets.
The projection of the algebraic set $S$ above is $[0, \infty)$, which is not the zero set of any family of polynomials in one variable.
This is why inequalities are part of the definition.

## Explanations

### L1 human

Semialgebraic sets are the sets cut out by polynomial equations and inequalities, such as discs, half-planes and parabolic regions.
The theorem says that the shadow of such a set on a lower-dimensional space is again of this kind.
So the class of semialgebraic sets is stable under the operations that geometric constructions use.

### L2 human

By induction on $n$, the image of a semialgebraic set under any polynomial map is semialgebraic, and so are closures, interiors and sets defined with quantifiers over $R$.
This makes semialgebraic sets the natural setting of real algebraic geometry, in contrast with real algebraic sets, which lack this closure property.
It also shows that every subset of $R$ definable with $+$, $\cdot$ and $<$ is semialgebraic, hence a finite union of points and open intervals.
This is the starting point of o-minimality.
Algorithms for quantifier elimination over $\R$, such as cylindrical algebraic decomposition, make the projection effective.
</document>

<document source="https://tamnd.github.io/math2/node/3WDZY1J002.md" tag="3WDZY1J002" kind="concept">
# Krohn–Rhodes theory

## Summary

Krohn–Rhodes theory studies finite [[semigroups|0XMEM406JM]] and finite automata by decomposing them into simple building blocks.
Its central result is the Krohn–Rhodes theorem.
It says that every finite semigroup divides an iterated [[wreath product|W1X984PKJ9]] of finite [[simple groups|0YMYMNQ0QZ]] and copies of the flip-flop, a monoid of resets.
The simple groups in such a decomposition can be chosen among the simple groups that divide the semigroup.
So every finite automaton can be built in a cascade from reversible parts, which are groups, and memory parts, which are flip-flops.
The least number of group levels needed is the complexity of the semigroup, a central object of study in the theory.

## Explanations

### L1 human

A finite automaton is a machine with finitely many states that reads input one symbol at a time.
Krohn–Rhodes theory says that any such machine can be simulated by a chain of very simple machines, each feeding its state to the next ones.
The simple machines are of two kinds: those that permute their states reversibly, like the hands of a clock, and one-bit memories that can be set or reset.
This is a kind of prime factorization for machines.
</document>

<document source="https://tamnd.github.io/math2/node/3WJFWAGKXR.md" tag="3WJFWAGKXR" kind="statement">
# Polar sets and Brownian hitting

## Statement

Let $d \ge 2$ and let $K \subset \R^d$ be a compact set.
For $x \in \R^d$, let $(B_t)$ be a [[Brownian motion|22735FGJV6]] in $\R^d$ started at $x$, with law $P^x$, and let $T_K = \inf\set{t > 0 : B_t \in K}$.
Then $K$ is a [[polar set|C95QJ7YH5T]] if and only if
$$P^x(T_K < \infty) = 0 \quad \text{for every } x \in \R^d.$$

## Remarks

Proof idea: the hitting probability $h(x) = P^x(T_K < \infty)$ is harmonic off $K$ and is the equilibrium potential of $K$, up to regularization.
For $d \ge 3$ it is the potential of the equilibrium measure of $K$.
It vanishes identically exactly when the Newtonian capacity of $K$ is $0$.
For $d = 2$ a similar argument, with logarithmic capacity and Brownian motion killed on leaving a large disc, applies.
Polar sets are exactly the sets of capacity zero, which gives the statement.
Example: a single point is polar for $d \ge 2$, so planar Brownian motion started away from $0$ never hits $0$.
A countable union of polar sets is polar, so planar Brownian motion almost surely avoids all points with rational coordinates, except possibly its starting point at time $0$.
A line segment in $\R^2$ is not polar, so Brownian motion hits it with positive probability.
In dimension $1$ the picture is different: Brownian motion hits every point almost surely, and only the empty set is polar.

## Explanations

### L1 human

Some sets are so thin that a random path will never touch them, even though the path wanders everywhere.
In the plane, a single point is such a set.
Polar sets in potential theory are exactly these sets that Brownian motion cannot see.

### L2 human

This is the probabilistic description of the exceptional sets of potential theory.
Statements that hold "except on a polar set", such as the boundary behaviour of Perron solutions, become statements that hold almost surely for Brownian paths.
It explains why polar sets are removable for bounded harmonic functions: Brownian motion never feels them.
In the language of Hunt processes, a set is polar for a process when the process almost surely never enters it, and this definition generalizes the classical one.
</document>

<document source="https://tamnd.github.io/math2/node/3WZAKQ3EJY.md" tag="3WZAKQ3EJY" kind="definition">
# Perspectivity

## Definition

Let $\ell$ and $m$ be two distinct lines of a [[projective plane|DANFNN8C4Y]], and let $O$ be a point that lies on neither of them.
The *perspectivity* from $\ell$ to $m$ with centre $O$ sends each point $X$ of $\ell$ to the common point of the lines $OX$ and $m$.

## Remarks

The map is a bijection from the points of $\ell$ to the points of $m$; its inverse is the perspectivity from $m$ to $\ell$ with the same centre.
It fixes the common point of $\ell$ and $m$.
A composition of finitely many perspectivities is a *projectivity*.
Dually, let $A$ and $B$ be points and $a$ a line through neither.
The perspectivity with axis $a$ sends each line $x$ through $A$ to the line through $B$ and the point $x \cap a$.

Example: in the real affine plane, completed to $\mathbb{P}^2(\R)$, let $\ell$ be $y = 1$, let $m$ be $y = 2$, and let $O = (0, 0)$.
The perspectivity sends $(t, 1)$ to $(2t, 2)$.
Check: the line through $O$ and $(t, 1)$ is $\set{(st, s)}$, and it meets $y = 2$ at $s = 2$.
The common point of $\ell$ and $m$ is their point at infinity, which is fixed.

Non-example: if the centre $O$ lies on $\ell$, the rule fails at $X = O$, since there is no line $OX$ for $X = O$.
</document>

<document source="https://tamnd.github.io/math2/node/3X4YMR25ST.md" tag="3X4YMR25ST" kind="definition">
# Type and cotype

## Definition

Let $X$ be a real or complex [[Banach space|06GZ956657]].
For vectors $x_1, \dots, x_n$ in $X$, write
$$R(x_1, \dots, x_n) = \Big( \frac{1}{2^n} \sum_{\varepsilon \in \set{-1, 1}^n} \Big\| \sum_{i=1}^n \varepsilon_i x_i \Big\|^2 \Big)^{1/2},$$
the quadratic mean of the norm over all choices of signs.

1. Let $1 \le p \le 2$. The space $X$ has *type* $p$ if there is a constant $C$ with $R(x_1, \dots, x_n) \le C \big( \sum_{i=1}^n \norm{x_i}^p \big)^{1/p}$ for all $n$ and all $x_1, \dots, x_n \in X$.
2. Let $2 \le q < \infty$. The space $X$ has *cotype* $q$ if there is a constant $C$ with $\big( \sum_{i=1}^n \norm{x_i}^q \big)^{1/q} \le C \, R(x_1, \dots, x_n)$ for all $n$ and all $x_1, \dots, x_n \in X$.

## Remarks

Example: a Hilbert space $H$ has type $2$ and cotype $2$ with $C = 1$.
Expanding the squares, the cross terms $\pm 2 \operatorname{Re} \ip{x_i}{x_j}$ cancel in the average over signs.
So $R(x_1, \dots, x_n)^2 = \sum_i \norm{x_i}^2$ exactly.

Non-example: $\ell^1$ has no type $p > 1$.
For the unit vectors $e_1, \dots, e_n$, every sign sum has norm $n$, so $R = n$.
The right side of the type inequality is $C n^{1/p}$, and $n \le C n^{1/p}$ fails for large $n$.

Every Banach space has type $1$, by the triangle inequality.
Type $p$ implies type $p'$ for $p' < p$, and cotype $q$ implies cotype $q'$ for $q' > q$.
By the Kahane inequality, the quadratic mean in $R$ can be replaced by the plain mean of the norm, with other constants.
For $1 \le p < \infty$, the space $L^p$ has type $\min(p, 2)$ and cotype $\max(p, 2)$.
</document>

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# Ten martini problem

## Problem

For real numbers $\lambda, \alpha, \theta$, let $H_{\lambda,\alpha,\theta}$ be the [[almost Mathieu operator|D08SKTH2T2]] on $\ell^2(\Z)$.
A Cantor set in $\R$ is a nonempty compact set with no isolated points and with empty interior.
Is the [[spectrum|BMKATJ802A]] of $H_{\lambda,\alpha,\theta}$ a Cantor set for every irrational $\alpha$, every $\lambda \ne 0$ and every $\theta$?

## Remarks

The answer is yes.
Avila and Jitomirskaya proved it in a paper titled "The Ten Martini Problem", published in the Annals of Mathematics in 2009.
Before that, Bellissard and Simon showed that the spectrum is a Cantor set for a generic set of pairs $(\lambda, \alpha)$.
The name comes from Mark Kac, who offered ten martinis for a solution; Barry Simon gave the problem this name in his list of problems on almost periodic Schrödinger operators.

## Explanations

### L1 human

A Cantor set is like the [[middle-thirds Cantor set|0BXNDZFQME]]: a closed set full of gaps at every scale, with no intervals inside it and no isolated points.
The problem asks whether the possible energies of an electron in a quasi-periodic chain always form such a set.
Its solution shows that quasi-periodicity, however weak, breaks the spectrum into infinitely many pieces at every scale.
</document>

<document source="https://tamnd.github.io/math2/node/3X6C5SC7AF.md" tag="3X6C5SC7AF" kind="statement">
# Durand–Kerner method

## Statement

Let $p$ be a [[monic polynomial|0BDX4MBKT4]] of degree $n \ge 2$ with [[complex|09FTAET5Z4]] coefficients, whose roots $r_1, \dots, r_n$ are pairwise distinct.
The *Durand–Kerner method*, also called the Weierstrass method, updates $n$ approximations $z_1^{(k)}, \dots, z_n^{(k)}$ simultaneously by
$$z_i^{(k+1)} = z_i^{(k)} - \frac{p\big(z_i^{(k)}\big)}{\prod_{j \ne i} \big(z_i^{(k)} - z_j^{(k)}\big)}, \qquad i = 1, \dots, n.$$
Then there is $\delta > 0$ such that, if $\abs{z_i^{(0)} - r_i} < \delta$ for all $i$, the iterates are defined and $z_i^{(k)} \to r_i$ for each $i$, with quadratic [[convergence|YWZQC5E54B]].

## Remarks

The update is [[Newton's method|07EXVNF9VP]] applied to the system of $n$ equations in $n$ unknowns that says that $\prod_i (x - z_i) = p(x)$ as polynomials.
Equivalently, it is Newton's method for the coefficients of the factorization, and the Jacobian of this system is invertible exactly when the $z_i$ are distinct.
This explains the local quadratic convergence at simple roots.
The [[fundamental theorem of algebra|0JP0W07Y9A]] guarantees that $p$ has $n$ complex roots counted with multiplicity.
A usual start takes $z_i^{(0)} = w^{i-1}$ for a complex number $w$ that is neither real nor a root of unity, for example $w = 0.4 + 0.9 \sqrt{-1}$.
With such starts the method usually converges in practice, but the theorem above is only local.
At multiple roots the convergence is only linear.

## Explanations

### L1 human

Instead of hunting one root at a time, the Durand–Kerner method moves $n$ guesses at once, one for each root.
Each guess is pushed by a Newton-like step, in which the other guesses play the role of the other roots.
The guesses spread out and settle on the $n$ roots together.

### L2 human

The method avoids deflation, which can lose accuracy when roots are removed one by one.
Each step costs $O(n^2)$ operations.
The Aberth method is a refinement with cubic local convergence.
</document>

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# Doubly stochastic matrix

## Definition

Let $n \ge 1$ and let $A = (a_{ij})$ be a real $n \times n$ [[matrix|0JXZKX526R]].
The matrix $A$ is *doubly stochastic* if:

1. $a_{ij} \ge 0$ for all $i$ and $j$.
2. $\sum_{j=1}^{n} a_{ij} = 1$ for each row index $i$.
3. $\sum_{i=1}^{n} a_{ij} = 1$ for each column index $j$.

## Remarks

Clause 1 says that $A$ is a [[nonnegative matrix|JPQCCFCY8X]].
With $e = (1, \ldots, 1)^T$, clauses 2 and 3 say $Ae = e$ and $e^T A = e^T$.
Example: $A = \frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$; all entries are $\frac{1}{2} \ge 0$, and each row and each column sums to $1$.
Every [[permutation matrix|08SP3DF8NB]] is doubly stochastic.
Non-example: $\begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}$ satisfies clauses 1 and 2, but its columns sum to $2$ and $0$, so clause 3 fails.
The set of doubly stochastic $n \times n$ matrices is the [[Birkhoff polytope|ADEXY830S8]].
</document>

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# Subadditive function

## Summary

A subadditive function is a function $f$ with $f(x + y) \le f(x) + f(y)$ for all $x, y$ in its domain.
The triangle inequality says that every [[norm|0V2WP9B1MF]] is subadditive, and every [[sublinear function|K3356N0SAX]] is subadditive.
For subadditive sequences, [[Fekete's lemma|8A4GGX1MEY]] shows that $f(n)/n$ converges to its infimum, which may be $-\infty$.
This makes subadditivity a basic tool for growth rates in analysis, combinatorics and ergodic theory.

## Explanations

### L1 human

A function is subadditive when the whole costs at most the sum of its parts.
Shipping costs are a picture: sending two parcels together usually costs no more than sending them apart.
The square root is subadditive on $[0, \infty)$: for example $\sqrt{9 + 16} = 5 \le 3 + 4$.
The absolute value on $\R$ is subadditive by the triangle inequality.
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<document source="https://tamnd.github.io/math2/node/3XB4PTTK68.md" tag="3XB4PTTK68" kind="definition">
# Minimal surface equation

## Definition

Let $n \ge 1$ and let $\Omega \subseteq \R^n$ be an open set.
Consider a twice continuously differentiable function $u \colon \Omega \to \R$.
The *minimal surface equation* on $\Omega$ is the [[quasilinear|7X9N9H3RK6]] equation
$$\operatorname{div} \left( \frac{\nabla u}{\sqrt{1 + \abs{\nabla u}^2}} \right) = 0.$$
Here $\nabla u$ is the [[gradient|0ABK9V9Y64]] and $\operatorname{div}$ is the [[divergence|0MWV6ZTK6S]].

## Remarks

Expanding the divergence gives the equivalent form
$$(1 + \abs{\nabla u}^2) \Delta u - \sum_{i,j=1}^n \partial_i u \, \partial_j u \, \partial_i \partial_j u = 0.$$
For $n = 2$ this is $(1 + u_y^2) u_{xx} - 2 u_x u_y u_{xy} + (1 + u_x^2) u_{yy} = 0$.
A function solves it exactly when the mean curvature of its graph in $\R^{n+1}$ is zero, that is, when the graph is a [[minimal surface|41X0YW2R9N]].
Example: Scherk's function $u(x, y) = \log(\cos y) - \log(\cos x)$ on the square $\abs{x}, \abs{y} < \pi/2$ is a solution.
Here $u_x = \tan x$, $u_y = -\tan y$, $u_{xx} = \sec^2 x$, $u_{yy} = -\sec^2 y$ and $u_{xy} = 0$, so the left side is $\sec^2 y \sec^2 x - \sec^2 x \sec^2 y = 0$.
Non-example: $u(x, y) = x^2 + y^2$ is not a solution, since at the origin $\nabla u = 0$ and the left side equals $\Delta u = 4$.
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<document source="https://tamnd.github.io/math2/node/3XC1C1NC0M.md" tag="3XC1C1NC0M" kind="statement">
# Hardy's uncertainty principle

## Statement

Let $f \colon \R \to \C$ be measurable, and let $a, b > 0$ and $C \ge 0$ be constants.
Suppose that $\abs{f(x)} \le C e^{-\pi a x^2}$ for all $x \in \R$, so that $f$ is integrable.
Let $\hat f(\xi) = \int_\R f(x) e^{-2\pi i x \xi}\, dx$ be the [[Fourier transform|DPVM34ADZV]] of $f$.
Suppose that $\abs{\hat f(\xi)} \le C e^{-\pi b \xi^2}$ for all $\xi \in \R$.

1. If $ab > 1$, then $f = 0$.
2. If $ab = 1$, then $f(x) = c\, e^{-\pi a x^2}$ for some constant $c \in \C$.

## Remarks

By the [[Fourier transform of the Gaussian|ZNEHMR2CCP]], $e^{-\pi a x^2}$ has transform $a^{-1/2} e^{-\pi \xi^2 / a}$, so case 2 cannot be improved.
If $ab < 1$, then $e^{-\pi c x^2}$ satisfies both bounds, with a suitable constant $C$, for every $c$ with $a \le c \le 1/b$, so there are nonzero functions of this kind.

The proof uses complex analysis.
The decay of $f$ makes $\hat f$ extend to an entire function with $\abs{\hat f(\xi + i\eta)} \le C' e^{\pi \eta^2 / a}$.
In case 2, the entire function $g(z) = e^{\pi z^2 / a} \hat f(z)$ is bounded on the real and on the imaginary axis.
The [[Phragmén–Lindelöf principle|FYR6V2NH6C]], applied in each quadrant, shows that $g$ is bounded, hence constant by Liouville's theorem, and Fourier inversion gives $f$.
Case 1 reduces to case 2 with $b$ replaced by $1/a$, and then the bound $\abs{\hat f(\xi)} \le C e^{-\pi b \xi^2}$ with $b > 1/a$ forces the constant to be $0$.

The result is due to G. H. Hardy.
There are versions on $\R^n$, and versions with $L^p$ bounds in place of pointwise bounds.

## Explanations

### L1 human

The bell curve $e^{-\pi x^2}$ and its transform decay at the same Gaussian rate.
Hardy's theorem says that no nonzero function can do better on both sides at once.
If both $f$ and $\hat f$ decay faster than this matched pair of Gaussians, the function must be zero.

### L2 human

Unlike the Heisenberg inequality, which measures spread with variances, Hardy's theorem measures concentration through pointwise decay.
It is a sharp qualitative statement, and the borderline case singles out the Gaussians, just as the equality case of Heisenberg's inequality does.
Its proof is a model for the use of Phragmén–Lindelöf arguments in Fourier analysis.
It has been extended to Lie groups and to the Schrödinger equation, where it gives unique continuation results.
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<document source="https://tamnd.github.io/math2/node/3XJYJHMPKY.md" tag="3XJYJHMPKY" kind="definition">
# Zernike polynomials

## Definition

Let $n$ and $m$ be integers with $0 \le m \le n$ and $n - m$ even.
The *radial polynomial* $R_n^m$ on $[0, 1]$ is
$$R_n^m(\rho) = \sum_{k=0}^{(n - m)/2} \frac{(-1)^k \, (n - k)!}{k! \, \left( \frac{n + m}{2} - k \right)! \, \left( \frac{n - m}{2} - k \right)!} \, \rho^{n - 2k}.$$
The *Zernike polynomials* are the functions on the closed unit disk, written in polar coordinates $(\rho, \varphi)$ with $0 \le \rho \le 1$, given by
$$Z_n^m(\rho, \varphi) = R_n^m(\rho) \cos(m\varphi),$$
and, for $m > 0$,
$$Z_n^{-m}(\rho, \varphi) = R_n^m(\rho) \sin(m\varphi).$$

## Remarks

Each $Z_n^{\pm m}$ is a [[polynomial in the two variables|07GNX8GCYP]] $x = \rho \cos \varphi$ and $y = \rho \sin \varphi$, of degree $n$.
Examples: $Z_0^0 = 1$, $Z_1^1 = \rho \cos \varphi = x$ and $Z_2^0 = 2\rho^2 - 1 = 2x^2 + 2y^2 - 1$.
Check for $R_2^0$: the term $k = 0$ is $\frac{2!}{0! \, 1! \, 1!} \rho^2 = 2\rho^2$ and the term $k = 1$ is $-\frac{1!}{1! \, 0! \, 0!} = -1$.
Orthogonality: $\int_0^1 R_n^m(\rho) R_{n'}^m(\rho) \, \rho \, d\rho = \frac{\delta_{n n'}}{2(n + 1)}$; for example $\int_0^1 (2\rho^2 - 1) \rho \, d\rho = \frac12 - \frac12 = 0$.
The normalisation is $R_n^m(1) = 1$.
The radial parts are Jacobi polynomials: $R_n^m(\rho) = (-1)^{(n - m)/2} \rho^m P_{(n - m)/2}^{(m, 0)}(1 - 2\rho^2)$.
Non-example: $n = 1$, $m = 0$ is excluded because $n - m$ is odd.
The natural candidate $\rho = \sqrt{x^2 + y^2}$ is rotation invariant but is not a polynomial in $x$ and $y$.
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<document source="https://tamnd.github.io/math2/node/3XNJ04AMCW.md" tag="3XNJ04AMCW" kind="definition">
# Lerch zeta function

## Definition

Let $\lambda \in \R$, let $\alpha$ be a real number with $0 < \alpha \le 1$, and let $s \in \C$.
The *Lerch zeta function* is the generalized [[Dirichlet series|8MWTEF6462]]
$$L(\lambda, \alpha, s) = \sum_{n=0}^{\infty} \frac{e^{2\pi i \lambda n}}{(n + \alpha)^s}.$$
It converges for $\operatorname{Re} s > 1$, and also for $\operatorname{Re} s > 0$ when $\lambda \notin \Z$.

## Remarks

The *Lerch transcendent* is $\Phi(z, s, a) = \sum_{n \ge 0} z^n (n + a)^{-s}$, defined for $\abs{z} < 1$ and $a \notin \set{0, -1, -2, \ldots}$, and for $\abs{z} = 1$ when $\operatorname{Re} s > 1$.
The two are related by $L(\lambda, \alpha, s) = \Phi(e^{2\pi i \lambda}, s, \alpha)$.

Example: for $\lambda \in \Z$ every phase factor is $1$, so $L(\lambda, \alpha, s) = \zeta(s, \alpha)$ is the [[Hurwitz zeta function|G4SW7HJG88]], and $L(0, 1, s) = \zeta(s)$.
With $a = 1$ the transcendent gives the [[polylogarithm|96CHDZKGQ0]]: $z \, \Phi(z, s, 1) = \sum_{n \ge 0} z^{n+1} / (n+1)^s = \operatorname{Li}_s(z)$.

Non-example: $\lambda = 0$, $\alpha = 1$, $s = 1$ gives the harmonic series $\sum 1/(n+1)$, which diverges; the condition $\operatorname{Re} s > 1$ is needed when $\lambda \in \Z$.
For $\lambda = \frac12$ and the same $\alpha, s$, the series is $1 - \frac12 + \frac13 - \cdots = \log 2$, which converges.

As a function of $s$, $L(\lambda, \alpha, s)$ extends to a meromorphic function on $\C$; for $\lambda \notin \Z$ it is entire.
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# Covering space

## Summary

A covering space of a space $X$ is a space $E$ with a continuous map $p \colon E \to X$ of a special kind.
Over a small neighbourhood of each point of $X$, the map looks like a stack of disjoint copies of that neighbourhood.
Such a map $p$ is a covering map.
Covering spaces have the path lifting and homotopy lifting properties, and for nice spaces they are classified by subgroups of the [[fundamental group|S1XK3HXHFX]].
They give the computation of $\pi_1(S^1)$ and link topology with group theory.

## Explanations

### L1 human

Picture an infinite spiral staircase above a circle, with each step projecting straight down onto the floor.
Near each point of the circle, the part of the staircase above it is a pile of separate small pieces, one on each turn.
That is the covering $\R \to S^1$, $t \mapsto e^{2\pi i t}$.
Another example is the map $z \mapsto z^2$ from the circle to itself, which wraps the circle twice around itself.
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<document source="https://tamnd.github.io/math2/node/3YF7P33WRM.md" tag="3YF7P33WRM" kind="definition">
# Steiner symmetrization

## Definition

Let $n \ge 2$, let $K \subseteq \R^n$ be a [[convex body|437FSG3419]], and let $u \in \R^n$ be a unit vector.
Let $u^\perp = \set{x \in \R^n : x \cdot u = 0}$.
For $x \in u^\perp$, the set $K \cap \set{x + t u : t \in \R}$ is empty or a closed segment; let $\ell(x)$ be its length when it is nonempty.
The *Steiner symmetrization* of $K$ in the direction $u$ is the set
$$S_u K = \set{x + t u : x \in u^\perp, \ K \cap (x + \R u) \ne \emptyset, \ \abs{t} \le \tfrac{1}{2} \ell(x)}.$$

## Remarks

The set $S_u K$ is again a convex body, it is symmetric under the reflection in $u^\perp$, and it has the same volume as $K$.
Its surface area and its diameter are at most those of $K$.
Example: let $K$ be the triangle with vertices $(0, 0)$, $(2, 0)$, $(0, 2)$ in $\R^2$ and $u = (0, 1)$.
The vertical chord over $x \in [0, 2]$ has length $2 - x$.
So $S_u K = \set{(x, t) : 0 \le x \le 2, \ \abs{t} \le (2 - x)/2}$, the triangle with vertices $(0, 1)$, $(0, -1)$, $(2, 0)$, of area $2$ like $K$.
Non-example: the set $\set{x + t u : 0 \le t \le \ell(x)}$ has the same chord lengths as $S_u K$, but its chords are not centred on $u^\perp$, so it is not the Steiner symmetrization.
Some authors write $\mathrm{st}_u K$ for $S_u K$; the operation is also defined for compact sets, with $\ell(x)$ the one-dimensional measure of the section.
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<document source="https://tamnd.github.io/math2/node/3YYKGWJJND.md" tag="3YYKGWJJND" kind="concept">
# Term algebra

## Summary

The term algebra over a set of variables is the algebra whose elements are the [[terms|6D3WA7N2W4]] themselves.
Its operations build bigger terms from smaller ones.
It is the absolutely free algebra on the variables.
Every map from the variables into an algebra of the same signature extends to a unique homomorphism, given by [[term evaluation|DH30WTA26T]].
All free algebras of varieties are quotients of term algebras, and the syntax of equational logic lives in it.

## Explanations

### L1 human

In a term algebra, nothing is simplified.
With one binary symbol $\cdot$ and variables $x, y$, the elements are expressions such as $x$, $x \cdot y$, $(x \cdot y) \cdot x$ and $x \cdot (y \cdot x)$.
Two expressions are equal only if they are written the same way, so $(x \cdot y) \cdot x$ and $x \cdot (y \cdot x)$ are different elements.
The operation simply puts two expressions side by side with the symbol between them.
</document>

<document source="https://tamnd.github.io/math2/node/3Z7YMK2WDS.md" tag="3Z7YMK2WDS" kind="definition">
# Excellent ring

## Definition

Let $R$ be a [[Noetherian ring|0670T59HGE]].
A Noetherian ring $A$ is *regular* if $A_{\mathfrak{q}}$ is a [[regular local ring|63XXTF6B5M]] for every prime ideal $\mathfrak{q}$ of $A$.
An algebra $A$ over a field $\kappa$ is *geometrically regular* if $A \otimes_\kappa L$ is regular for every finite field extension $L$ of $\kappa$.
The ring $R$ is *excellent* if:

1. $R$ is [[universally catenary|5WAFPTH09V]].
2. $R$ is a G-ring: let $\mathfrak{p}$ be a prime ideal of $R$ and let $\widehat{R_{\mathfrak{p}}}$ be the [[completion|KBN13NPYJQ]] of the [[localization|X6JD4AG973]] $R_{\mathfrak{p}}$.
Then for every prime ideal $\mathfrak{q}$ of $R_{\mathfrak{p}}$ with residue field $\kappa(\mathfrak{q})$, the ring $\widehat{R_{\mathfrak{p}}} \otimes_{R_{\mathfrak{p}}} \kappa(\mathfrak{q})$ is geometrically regular over $\kappa(\mathfrak{q})$.
3. $R$ is J-2: let $S$ be a [[finitely generated|AC5TFFQT9M]] $R$-algebra.
Then the set of prime ideals $\mathfrak{q}$ of $S$ with $S_{\mathfrak{q}}$ regular is open in the [[spectrum|FB45BHHA2J]] of $S$ with the [[Zariski topology|8SQK7RTB63]].

## Remarks

Example: every field $k$ is excellent.
It is universally catenary because finitely generated $k$-algebras are catenary.
Clause 2 holds because the only prime of $k$ is $0$, the ring $k$ is its own completion, and $k \otimes_k k = k$ is geometrically regular over $k$.
The regular locus of a finitely generated $k$-algebra is open, which is clause 3.
More generally, $\Z$, every complete Noetherian local ring, and every Dedekind domain of characteristic $0$ are excellent.
Localizations and finitely generated algebras of excellent rings are excellent.

Non-example: there are Noetherian local domains whose completion has nonzero nilpotent elements.
Such a ring fails clause 2, because if a reduced Noetherian local ring is a G-ring, then its completion is reduced.

Some authors call a ring that satisfies only clauses 2 and 3 quasi-excellent.
</document>

<document source="https://tamnd.github.io/math2/node/3Z9DCWYXQZ.md" tag="3Z9DCWYXQZ" kind="concept">
# Picard group

## Summary

The Picard group of a ringed space $X$ is the group of isomorphism classes of invertible sheaves on $X$.
Its group law is the tensor product.
It is a basic invariant of a scheme, and it measures the line bundles available for maps to projective space.
For the spectrum of a Dedekind domain it is the ideal class group.
For projective space $\mathbb{P}^n$ over a field, with $n \ge 1$, it is $\Z$, generated by $\mathcal{O}(1)$.

## Explanations

### L1 human

Line bundles can be multiplied by the tensor product, and each line bundle has an inverse, its dual.
So the line bundles on a space, up to isomorphism, form a group: the Picard group.
On affine space every line bundle is trivial, so the Picard group is trivial.
On projective space every line bundle is a power of one basic bundle, so the Picard group is the integers.
</document>

<document source="https://tamnd.github.io/math2/node/3ZCJRT8FPT.md" tag="3ZCJRT8FPT" kind="definition">
# Hyperkähler manifold

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] with [[Levi-Civita connection|CAZYGE1T4E]] $\nabla$.
The manifold $M$ is *hyperkähler* if it carries three [[almost complex structures|9SYWRGVTRS]] $I$, $J$, $K$ such that:

1. $g(IX, IY) = g(JX, JY) = g(KX, KY) = g(X, Y)$ for all tangent vectors $X, Y$;
2. $\nabla I = \nabla J = \nabla K = 0$;
3. $IJ = K$.

## Remarks

From $I^2 = J^2 = -\id$ and $IJ = K$ with $K^2 = -\id$ one gets the quaternion relations $IJ = -JI = K$.
Clauses 1 and 2 say that $(M, g)$ is [[Kähler|CK1DCPC3D3]] for each of $I$, $J$, $K$, and in fact for $aI + bJ + cK$ whenever $a^2 + b^2 + c^2 = 1$.
Equivalently, the [[holonomy|WHDZ3MK56B]] group of $g$ is contained in $Sp(n)$, where $\dim M = 4n$.

Example: $\mathbb{H}^n = \R^{4n}$ with the flat metric and $I, J, K$ given by left multiplication by the quaternions $i, j, k$ on each coordinate.
These maps are orthogonal, have constant coefficients so they are parallel, and satisfy $ij = k$.

Non-example: $\C P^2$ with the Fubini–Study metric is Kähler of real dimension $4$, but it is not hyperkähler.
A hyperkähler metric is Ricci-flat, while the Fubini–Study metric has positive Ricci curvature.
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<document source="https://tamnd.github.io/math2/node/3ZEW7MC05Z.md" tag="3ZEW7MC05Z" kind="definition">
# Method of multiple scales

## Definition

Consider an [[ordinary differential equation|0NT64JV96A]] $F(t, x, x', x'', \varepsilon) = 0$ for $x(t)$, depending on a small parameter $\varepsilon > 0$.
The *method of multiple scales* (two-timing, in its simplest form) proceeds as follows.

1. Introduce the slow time $T = \varepsilon t$ and seek $x(t) = X(t, T)$, where $X$ is a function of two independent variables.
2. Replace $\frac{d}{dt}$ by $\partial_t + \varepsilon \partial_T$, and expand $X = X_0(t, T) + \varepsilon X_1(t, T) + \cdots$.
3. Collect powers of $\varepsilon$; this gives equations in $t$ for $X_0, X_1, \ldots$ in which $T$ is a parameter.
4. Choose the dependence of $X_0, X_1, \ldots$ on $T$ so that each $X_k$ stays bounded in $t$.
   This means that the equations have no resonant forcing that produces secular terms.

The result is an [[asymptotic expansion|Q26QN6GYZX]] intended to be valid for $0 \le t \le C / \varepsilon$.
More slow times $\varepsilon^2 t, \varepsilon^3 t, \ldots$ can be added to reach longer time intervals.

## Remarks

Example: $x'' + 2\varepsilon x' + x = 0$.
Order $0$ gives $\partial_t^2 X_0 + X_0 = 0$, so $X_0 = A(T) \cos t + B(T) \sin t$.
Order $1$ gives $\partial_t^2 X_1 + X_1 = -2 \partial_t \partial_T X_0 - 2 \partial_t X_0 = 2(A' + A) \sin t - 2(B' + B) \cos t$.
Bounded $X_1$ requires $A' = -A$ and $B' = -B$, so $X_0 = e^{-T} (a \cos t + b \sin t)$.
Every exact solution has the form $e^{-\varepsilon t} \bigl(c_1 \cos(\sqrt{1 - \varepsilon^2}\, t) + c_2 \sin(\sqrt{1 - \varepsilon^2}\, t)\bigr)$.
Its frequency differs from $1$ by $O(\varepsilon^2)$, so the approximation has error $O(\varepsilon)$ for $0 \le t \le C / \varepsilon$.

Non-example: the regular expansion $x = x_0(t) + \varepsilon x_1(t)$ with one time gives $x_1$ containing $t \cos t$ and $t \sin t$ terms, so the error is not small when $t \ge 1/\varepsilon$.

## Explanations

### L1 human

Use a fast clock and a slow clock at the same time, and let the slow clock govern how the fast motion changes.
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# Wiener–Khinchin theorem

## Statement

Let $(X_t)_{t \in \R}$ be a complex-valued weakly [[stationary process|DT0QXJE705]]: $E[\abs{X_t}^2] < \infty$, $E[X_t] = m$ for all $t$, and the autocovariance $r(\tau) = E[(X_{t+\tau} - m)\overline{(X_t - m)}]$ does not depend on $t$.
Assume that $r$ is [[continuous|0X3WGX95FE]] at $0$.
Then there is a unique finite Borel measure $F$ on $\R$, the *spectral measure* of $X$, such that
$$r(\tau) = \int_{\R} e^{i \tau \omega} \, F(d\omega) \quad \text{for all } \tau \in \R.$$
If moreover $r$ is integrable on $\R$, then $F$ has the continuous density
$$S(\omega) = \frac{1}{2\pi} \int_{\R} r(\tau) \, e^{-i \tau \omega} \, d\tau,$$
the *power spectral density* of $X$, and $S(\omega) \ge 0$ for all $\omega$.

## Remarks

The function $r$ is positive definite: for all $\tau_1, \dots, \tau_k \in \R$ and $c_1, \dots, c_k \in \C$, $\sum_{j, l} c_j \overline{c_l} \, r(\tau_j - \tau_l) = E\bigl[\abs{\sum_j c_j (X_{\tau_j} - m)}^2\bigr] \ge 0$.
A positive definite function that is continuous at $0$ is continuous everywhere, and [[Bochner's theorem|ZRHAG4NXHC]] then gives the measure $F$.
The density formula is Fourier inversion, see the [[Fourier transform|DPVM34ADZV]].

For a weakly stationary sequence $(X_n)_{n \in \Z}$, the analogue is Herglotz's theorem, with a finite measure on $[-\pi, \pi)$ and no continuity assumption.

Example: if $r(\tau) = e^{-\abs{\tau}}$, as for a suitably normalised stationary [[Ornstein–Uhlenbeck process|HZM2ASCDY4]], then $\int_{\R} e^{-\abs{\tau}} e^{-i\tau\omega} \, d\tau = \dfrac{2}{1 + \omega^2}$, so $S(\omega) = \dfrac{1}{\pi(1 + \omega^2)}$.

## Explanations

### L1 human

A stationary random signal can be analysed in two ways: by how its values at two times are correlated, or by how much of its power lies at each frequency.
The Wiener–Khinchin theorem says that these two descriptions carry the same information: the power spectrum is the Fourier transform of the autocovariance.
This is why engineers can measure the spectrum of noise from its correlations.

### L2 human

The theorem is the basis of spectral analysis of time series.
It is used to estimate the spectrum from data and to classify noise; for example, white noise has a flat spectrum.
It is also used to design filters, since a linear time-invariant filter multiplies the spectral density by the squared modulus of its transfer function.
The spectral measure also leads to the spectral representation of the process itself as a stochastic integral against a process with orthogonal increments.
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# Periodogram

## Summary

The periodogram of a finite [[time series|EA3MF4VTEH]] measures how much of the variation of the series is due to oscillations at each frequency.
It is the squared modulus of the discrete Fourier transform of the data, suitably scaled.
It is the basic estimate of the spectral density of a stationary process and the starting point of spectral analysis.
By itself, however, it is a noisy estimate.

## Explanations

### L1 human

A series that rises and falls every $12$ months has a strong rhythm at the frequency of one cycle per year.
The periodogram scans through all frequencies and asks how well a sine wave of that frequency fits the data.
Frequencies with a good fit show up as peaks.
For example, for monthly sales with a December peak, the periodogram has a peak at the frequency $2\pi/12$ radians per month.
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<document source="https://tamnd.github.io/math2/node/3ZZSXKM16Z.md" tag="3ZZSXKM16Z" kind="statement">
# Kolmogorov–Arnold–Moser theorem

## Statement

Let $n \ge 2$, let $D \subseteq \R^n$ be a bounded open set and let $\mathbb{T}^n = \R^n / 2\pi \Z^n$.
Use [[action-angle coordinates|M3T81YNSKC]] $(I, \theta)$ on $D \times \mathbb{T}^n$, with symplectic form $\sum_k dI_k \wedge d\theta_k$.
Consider the [[Hamiltonian system|0EDKGA2FAK]] with Hamiltonian
$$H_\varepsilon(I, \theta) = H_0(I) + \varepsilon H_1(I, \theta),$$
where $H_0$ and $H_1$ are real-analytic on a neighbourhood of $\overline{D} \times \mathbb{T}^n$.
Assume that $H_0$ is nondegenerate: $\det \left( \partial^2 H_0 / \partial I^2 \right) \ne 0$ on $\overline{D}$.
Then there is $\varepsilon_0 > 0$ such that for $0 \le \varepsilon < \varepsilon_0$:

1. the phase space contains a family of invariant $n$-[[tori|03NM7DZXXC]] of $H_\varepsilon$. Each is close to an invariant torus $\set{I = I_0}$ of $H_0$ with Diophantine frequency vector $\omega = \nabla H_0(I_0)$. This means that $\abs{\langle k, \omega \rangle} \ge \gamma \abs{k}^{-\tau}$ for all $k \in \Z^n \setminus \set{0}$, for some constants $\gamma > 0$ and $\tau > n - 1$;
2. on each of these tori the flow of $H_\varepsilon$ is conjugate to the linear flow $\theta \mapsto \theta + t \omega$;
3. the Lebesgue measure of the complement of the union of these tori in $D \times \mathbb{T}^n$ tends to $0$ as $\varepsilon \to 0$.

## Remarks

The proof is a Newton-type iteration.
At each step a canonical change of coordinates removes the angle dependence of the perturbation to a higher order.
Each step needs to divide by $\langle k, \omega \rangle$, the small divisors, and the Diophantine condition controls them.
The quadratic convergence of the Newton scheme beats the loss caused by the small divisors.

Kolmogorov announced the result in 1954, Arnold proved it for analytic Hamiltonians, and Moser proved a version for twist maps of finite smoothness.

Tori with resonant frequencies, where $\langle k, \omega \rangle = 0$ for some $k \ne 0$, are in general destroyed by the perturbation.
Some nondegeneracy condition on $H_0$ is needed, because the proof uses the map $I \mapsto \nabla H_0(I)$ to choose tori with Diophantine frequencies; other versions use the isoenergetic nondegeneracy condition instead.

## Explanations

### L1 human

An [[integrable system|N0N7CMRCMH]] has all its motions wound around invariant tori.
The theorem says that a small push does not destroy this order completely.
Most of the tori survive, slightly deformed, and the motions on them stay quasi-periodic forever.
Only the tori whose frequencies are close to rational relations break up.
This explains why, for example, a planetary system that is close to an integrable one can be stable for most initial conditions.

### L2 human

KAM theory answers a question of Poincaré on the persistence of quasi-periodic motion in nearly integrable systems.
Its key ingredients are the Diophantine condition on frequencies and a rapidly converging iteration.
For two degrees of freedom the surviving $2$-tori separate each $3$-dimensional energy level, which gives stability of the actions for all time.
In three or more degrees of freedom the tori do not separate the energy levels, and [[Arnold diffusion|ZVY046R4XB]] becomes possible; the [[Nekhoroshev estimates|PYEN6SFS9P]] bound its speed.
The same ideas apply to area-preserving twist maps such as the [[standard map|8TEV7CSX2H]].
</document>

<document source="https://tamnd.github.io/math2/node/405TSJS5PA.md" tag="405TSJS5PA" kind="definition">
# Bloch space

## Definition

Let $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ be the [[unit disc|8G5ABXRK85]].
The *Bloch space* $\mathcal{B}$ is the set of [[holomorphic functions|0NAHXR1FRH]] $f \colon \mathbb{D} \to \C$ such that
$$\sup_{z \in \mathbb{D}} (1 - \abs{z}^2) \abs{f'(z)} < \infty.$$
Its norm is $\norm{f}_{\mathcal{B}} = \abs{f(0)} + \sup_{z \in \mathbb{D}} (1 - \abs{z}^2) \abs{f'(z)}$.

## Remarks

With this norm, $\mathcal{B}$ is a [[Banach space|06GZ956657]].
Example: let $f(z) = \log(1 - z)$ with the principal branch, so $f'(z) = -1/(1 - z)$.
Since $\abs{1 - z} \ge 1 - \abs{z}$, one has $(1 - \abs{z}^2)\abs{f'(z)} \le (1 - \abs{z})(1 + \abs{z})/(1 - \abs{z}) < 2$, so $f \in \mathcal{B}$, although $f$ is unbounded.
Non-example: $g(z) = 1/(1 - z)$ has $g'(z) = 1/(1 - z)^2$, and for real $z = r \in (0, 1)$ one gets $(1 - r^2)/(1 - r)^2 = (1 + r)/(1 - r) \to \infty$, so $g \notin \mathcal{B}$.
Every bounded holomorphic function is in $\mathcal{B}$: if $\abs{f} \le 1$, the [[Schwarz–Pick theorem|V1QYA9RW73]] gives $(1 - \abs{z}^2)\abs{f'(z)} \le 1$.
The closure of the polynomials in $\mathcal{B}$ is the little Bloch space $\mathcal{B}_0$, the set of $f$ with $(1 - \abs{z}^2)\abs{f'(z)} \to 0$ as $\abs{z} \to 1$.
</document>

<document source="https://tamnd.github.io/math2/node/4085B8A528.md" tag="4085B8A528" kind="statement">
# Ahlswede–Daykin inequality

## Statement

Let $L$ be a finite [[distributive lattice|0HM0HQCK6N]] with join $\vee$ and meet $\wedge$.
Let $f_1, f_2, f_3, f_4 \colon L \to [0, \infty)$ be functions such that
$$f_1(x) \, f_2(y) \le f_3(x \vee y) \, f_4(x \wedge y) \quad \text{for all } x, y \in L.$$
For $A, B \subseteq L$, let $A \vee B = \set{a \vee b : a \in A, b \in B}$, let $A \wedge B = \set{a \wedge b : a \in A, b \in B}$, and let $f(A) = \sum_{a \in A} f(a)$.
Then
$$f_1(A) \, f_2(B) \le f_3(A \vee B) \, f_4(A \wedge B) \quad \text{for all } A, B \subseteq L.$$

## Remarks

It is also called the four functions theorem.
Every finite distributive lattice embeds in the lattice of subsets of a finite set, so it suffices to treat the power set of $\set{1, \ldots, n}$.
The proof is by induction on $n$; the case $n = 1$ is a direct check of an inequality between four products of nonnegative numbers.

Example: with all four functions equal to $1$, the hypothesis holds and the conclusion is $|A| \, |B| \le |A \vee B| \, |A \wedge B|$ for any subsets $A, B$ of $L$.
Taking $f_1 = f_2 = f_3 = f_4 = \mu$ for a function $\mu$ with $\mu(x)\mu(y) \le \mu(x \vee y)\mu(x \wedge y)$ gives the [[FKG inequality|DCXJGS6E6D]] and, as a special case, the [[Harris inequality|W1YH3YD3S4]].

## Explanations

### L1 human

The theorem turns an inequality between single elements into the same inequality between whole sets of elements.
Suppose the weights of any $x$ and $y$ are controlled by the weights of their join and meet.
Then the total weights of two sets are controlled by the total weights of all their joins and meets.

### L2 human

The four functions theorem is the common source of the correlation inequalities of combinatorics.
It contains the [[FKG inequality|DCXJGS6E6D]], the [[Harris inequality|W1YH3YD3S4]] and Kleitman's lemma on monotone families of sets.
These are used for random graphs, percolation and the statistical mechanics of spin systems, where increasing events tend to occur together.
</document>

<document source="https://tamnd.github.io/math2/node/40ATJEVR32.md" tag="40ATJEVR32" kind="concept">
# Bootstrap

## Summary

The bootstrap estimates the [[sampling distribution|656Y8FA3RY]] of a [[statistic|0Q5CG2NEQ8]] by resampling the observed data with replacement.
It treats the [[empirical distribution function|A05WR6TEK5]] of the sample as if it were the unknown true distribution.
It then recomputes the statistic on many simulated samples drawn from it.
From these replicates one gets standard errors, bias estimates and confidence intervals without explicit formulas.
Bradley Efron introduced it in 1979, and it is one of the most used tools of modern statistics.

## Explanations

### L1 human

You have one sample and want to know how much your estimate would vary if you could repeat the study.
You cannot repeat it, so you pretend that the sample is the whole population.
Draw a new sample of the same size from your data, with replacement, so that some points appear twice and some not at all.
Compute the estimate again, and repeat this a few thousand times.
The spread of these estimates approximates the spread you would see over real repetitions.
</document>

<document source="https://tamnd.github.io/math2/node/40F8KNF8Q1.md" tag="40F8KNF8Q1" kind="definition">
# Auslander–Reiten theory

## Definition

Let $k$ be a field and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $k$.
Let $\operatorname{mod} A$ be the category of finitely generated left $A$-[[modules|0NZ8JA25WK]].

1. A morphism $f \colon M \to N$ between [[indecomposable|E2P29GCD9T]] modules is *irreducible* if it is neither a split monomorphism nor a split epimorphism, and the following holds.
In every factorization $f = g h$, either $h$ is a split monomorphism or $g$ is a split epimorphism.
2. For $M$ in $\operatorname{mod} A$ with minimal [[projective|BFFA03GE3V]] presentation $P_1 \xrightarrow{p} P_0 \to M \to 0$, the *transpose* $\operatorname{Tr} M$ is the cokernel of $\Hom_A(p, A) \colon \Hom_A(P_0, A) \to \Hom_A(P_1, A)$, a right $A$-module.
3. The *Auslander–Reiten translation* is $\tau M = D \operatorname{Tr} M$, where $D = \Hom_k(\cdot, k)$.

*Auslander–Reiten theory* is the study of $\operatorname{mod} A$ through irreducible morphisms, the translation $\tau$, [[almost split sequences|M5D95FN7G3]] and the [[Auslander–Reiten quiver|125Z033VCQ]].

## Remarks

The basic existence theorem: if $M$ is indecomposable and not projective, there is an almost split sequence $0 \to \tau M \to E \to M \to 0$.
Dually, for $N$ indecomposable and not injective there is an almost split sequence starting at $N$, and it ends at $\tau^{-1} N = \operatorname{Tr} D N$.
Example: for the path algebra of $1 \to 2$, with simple modules $S_1, S_2$ and the projective cover $P_1$ of $S_1$, one has $\tau S_1 = S_2$.
The almost split sequence is $0 \to S_2 \to P_1 \to S_1 \to 0$.
The translation $\tau$ sends projective modules to $0$.
The theory works more generally for Artin algebras over a commutative Artinian ring $R$, with $D = \Hom_R(\cdot, E)$ for the injective envelope $E$ of $R/\operatorname{rad} R$.
</document>

<document source="https://tamnd.github.io/math2/node/40SV748R64.md" tag="40SV748R64" kind="concept">
# Hilbert dimension

## Summary

The Hilbert dimension of a [[Hilbert space|HM2HHE3J51]] is the cardinality of any of its orthonormal bases, that is of any maximal [[orthonormal family|0RW9TNNT87]].
All orthonormal bases of a given Hilbert space have the same cardinality, so this number is well defined.
Two Hilbert spaces over the same field are isometrically isomorphic exactly when they have the same Hilbert dimension.
A Hilbert space is separable exactly when its Hilbert dimension is finite or countable.

## Explanations

### L1 human

The Hilbert dimension counts the perpendicular directions of unit length needed to build every vector of the space as a convergent series.
For a finite-dimensional space it is the usual dimension.
For $\ell^2$ it is countably infinite, with the unit vectors $e_1, e_2, \dots$ as basis.
This count differs from the algebraic dimension, which only allows finite sums and is uncountable for $\ell^2$.
</document>

<document source="https://tamnd.github.io/math2/node/40TPSVZGJB.md" tag="40TPSVZGJB" kind="definition">
# Schwarz–Christoffel mapping

## Definition

Let $P$ be the bounded open region enclosed by a simple polygon with $n \ge 3$ vertices $w_1, \ldots, w_n$, listed in the positive direction.
Let $\pi\alpha_k$ be the interior angle of $P$ at $w_k$, with $0 < \alpha_k < 2$; then $\sum_{k=1}^{n} (\alpha_k - 1) = -2$.
Let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]].
A *Schwarz–Christoffel mapping* for $P$ is a [[conformal map|0VHNKP1N3K]] $f$ of $\mathbb{H}$ onto $P$.

The *Schwarz–Christoffel formula* gives the form of every such $f$:
$$f(z) = A + C \int_{z_0}^{z} \prod_{k=1}^{n} (\zeta - x_k)^{\alpha_k - 1}\, d\zeta, \qquad z \in \mathbb{H},$$
with real *prevertices* $x_1 < x_2 < \cdots < x_n$, a point $z_0 \in \mathbb{H}$ and constants $A, C \in \C$.
The powers are the branches that are holomorphic on $\mathbb{H}$.
Here $f$ extends continuously to the real axis with $f(x_k) = w_k$.
If one prevertex is placed at $\infty$, its factor is omitted from the product.

## Remarks

Such an $f$ exists by the [[Riemann mapping theorem|02WHMEG5ZR]] composed with a Möbius map from the disc to $\mathbb{H}$, and it extends to the boundary by [[Carathéodory's theorem|AZ6H4ZKK55]].
On each interval $(x_k, x_{k+1})$ the argument of $f'$ is constant, so the interval is mapped onto a straight segment; at $x_k$ the argument jumps by $\pi(1 - \alpha_k)$, the exterior angle.
Three of the prevertices can be chosen freely, because $\mathbb{H}$ has a three-parameter group of conformal automorphisms; the others are determined by $P$ and usually found numerically.
Example: for the right isosceles triangle, one may take prevertices $0, 1, \infty$ with angles $\pi/2, \pi/4, \pi/4$, giving $f(z) = \int_0^z \zeta^{-1/2} (\zeta - 1)^{-3/4}\, d\zeta$ up to the constants.
The same formula, with vertices at infinity allowed, describes maps onto unbounded polygonal regions.
For example, $\arcsin$ maps $\mathbb{H}$ onto the half-strip $\set{w : \abs{\operatorname{Re} w} < \pi/2,\ \operatorname{Im} w > 0}$.
Non-example: a conformal map of $\mathbb{H}$ onto the unit disc is not a Schwarz–Christoffel mapping.
The disc is not bounded by a polygon, and the image of the real axis has no straight pieces.
</document>

<document source="https://tamnd.github.io/math2/node/41009J4M5D.md" tag="41009J4M5D" kind="concept">
# Quasi-compact morphism

## Summary

A quasi-compact morphism of schemes is one for which the preimage of every open affine subscheme is quasi-compact.
Quasi-compact means that every open cover has a finite subcover; no Hausdorff condition is asked.
It is a finiteness condition on a [[morphism of schemes|JXRKH6P1Z4]].
It is part of the definition of morphisms of finite type and of proper morphisms.

## Explanations

### L1 human

Spaces in algebraic geometry are rarely Hausdorff, so "compact" is replaced by "quasi-compact": every open cover has a finite subcover.
Every affine scheme is quasi-compact.
A morphism is quasi-compact when, over each affine piece of the target, the source can be covered by finitely many affine pieces.
Mapping infinitely many separate points onto a single point fails this test.
</document>

<document source="https://tamnd.github.io/math2/node/410BCE4PAM.md" tag="410BCE4PAM" kind="definition">
# Mapping class group

## Definition

Let $S$ be a connected oriented [[surface|VGJJVBZ1MQ]], possibly with boundary $\partial S$.
Let $\mathrm{Homeo}^+(S, \partial S)$ be the group, under composition, of orientation-preserving [[homeomorphisms|0QT00PB5G2]] $f \colon S \to S$ with $f(x) = x$ for all $x \in \partial S$.
Two such homeomorphisms $f$ and $g$ are *isotopic* if there is a continuous map $H \colon S \times [0, 1] \to S$ such that:

1. each $H_t = H(\cdot, t)$ lies in $\mathrm{Homeo}^+(S, \partial S)$;
2. $H_0 = f$ and $H_1 = g$.

The *mapping class group* of $S$ is the quotient group
$$\mathrm{Mod}(S) = \mathrm{Homeo}^+(S, \partial S) / \mathrm{Homeo}_0(S, \partial S),$$
where $\mathrm{Homeo}_0(S, \partial S)$ is the normal subgroup of homeomorphisms isotopic to the identity.
Its elements, the isotopy classes, are *mapping classes*.

## Remarks

For a surface with marked points or punctures, the homeomorphisms may permute the marked points, and the isotopies must keep them fixed.
Some authors also allow orientation-reversing homeomorphisms; this gives the extended mapping class group, which contains $\mathrm{Mod}(S)$ with index $2$ when $S$ admits an orientation-reversing homeomorphism fixing $\partial S$.

Example: for the torus $T^2 = \R^2 / \Z^2$, the map $\mathrm{Mod}(T^2) \to SL(2, \Z)$ that records the action on the first homology group $H_1(T^2; \Z) \cong \Z^2$ is an isomorphism.
A matrix $A \in SL(2, \Z)$ maps $\Z^2$ onto itself, so it induces a homeomorphism of $T^2$, and since $\det A = 1$ it preserves orientation.

Non-example: let $A = S^1 \times [0, 1]$ be the annulus, with points written $(\theta, t)$, and let $T(\theta, t) = (\theta + 2\pi t, t)$, a [[Dehn twist|W890NRHGS4]].
The maps $H_s(\theta, t) = (\theta + 2\pi s t, t)$ for $s \in [0, 1]$ deform the identity into $T$, but for $0 < s < 1$ they rotate the boundary circle $S^1 \times \set{1}$, so they fail clause 1.
In fact $T$ is not isotopic to the identity in the sense above, and $\mathrm{Mod}(A) \cong \Z$, generated by the class of $T$.
For the closed disk, the Alexander lemma gives $\mathrm{Mod}(D^2) = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/410CS90H3R.md" tag="410CS90H3R" kind="definition">
# Calculus of variations

## Definition

Let $\Omega \subseteq \R^n$ be a bounded open set and let $L \colon \Omega \times \R \times \R^n \to \R$ be a continuous function, the *Lagrangian*, written $L(x, z, p)$.
Let $\mathcal{A}$ be a class of functions $u \colon \Omega \to \R$ with prescribed values $u = g$ on $\partial\Omega$, for which the integral below makes sense.
For example, $\mathcal{A}$ may lie in $C^1(\overline{\Omega})$ or in a [[Sobolev space|M4CEZ5K9HP]] $W^{1,q}(\Omega)$.
The *integral functional* with Lagrangian $L$ is
$$J(u) = \int_\Omega L\big(x, u(x), \nabla u(x)\big) \, dx, \qquad u \in \mathcal{A}.$$
A *variational problem* asks for a minimiser of $J$ on $\mathcal{A}$, or more generally for its critical points.
A minimiser is a $u_0 \in \mathcal{A}$ with $J(u_0) \le J(u)$ for all $u \in \mathcal{A}$.
The *calculus of variations* is the study of such problems: existence of minimisers, necessary and sufficient conditions for a minimum, and properties of minimisers.

## Remarks

The same words are used for functionals with higher derivatives, for vector-valued $u$, and for constraints such as fixed volume or fixed $L^2$ norm.

Example: for $n = 2$, the Dirichlet energy $L(x, z, p) = \frac{1}{2}\abs{p}^2$ gives $J(u) = \frac{1}{2} \int_\Omega \abs{\nabla u}^2 \, dx$.
Its minimisers with boundary values $g$ are the harmonic functions with these boundary values, which is Dirichlet's principle.

Example: the area of the graph of $u$ has Lagrangian $L(x, z, p) = \sqrt{1 + \abs{p}^2}$, and its critical points solve the [[minimal surface equation|3GV2E2YTZX]].

Non-example: minimising $u \mapsto \max_{\overline\Omega} \abs{u}$ is an optimisation over functions.
But this functional is not the integral of a Lagrangian, so it is not a variational problem in this sense.

## Explanations

### L1 human

A variational problem asks for the function that makes an integral quantity, such as energy or area, as small as possible.
The calculus of variations is the theory of these problems.
</document>

<document source="https://tamnd.github.io/math2/node/4135XQ4WHK.md" tag="4135XQ4WHK" kind="statement">
# Minimum degree algorithm

## Statement

Let $A$ be a real symmetric [[positive definite|0KX66CZMBB]] $n \times n$ [[sparse matrix|JGACZYEEY7]].
Let $G$ be its [[graph|00C0BR7D7H]]: the vertices are $1, \dots, n$, and $i \ne j$ are adjacent when $a_{ij} \ne 0$.
To *eliminate* a vertex $v$ of a graph means to join every two neighbours of $v$ by an edge and then to delete $v$.
For an ordering $v_1, \dots, v_n$ of the vertices, let $G_0 = G$ and let $G_k$ be the graph obtained from $G_{k-1}$ by eliminating $v_k$.

1. Let $P$ be the permutation matrix of the ordering, and let $L$ be the [[Cholesky factor|0WKG8CYY6A]] of $P A P^T$. If no numerical cancellation occurs, then for $i > j$ the entry $l_{ij}$ is nonzero exactly when $v_i$ and $v_j$ are adjacent in $G_{j-1}$.
2. The *minimum degree algorithm* builds the ordering greedily: for $k = 1, \dots, n$, it takes for $v_k$ a vertex of least [[degree|0KS0KRDV4Y]] in $G_{k-1}$.

## Remarks

Clause 1 holds because one step of symmetric [[Gaussian elimination|0JVJ2WCXZH]] with pivot $v$ subtracts a multiple of the row of $v$ from the row of each neighbour of $v$.
That update creates a nonzero at every pair of neighbours of $v$, which is exactly the set of edges added when $v$ is eliminated.
The new nonzeros are called *fill-in*.
A vertex of degree $d$ creates at most $d(d-1)/2$ new edges when it is eliminated, so the greedy rule tries to keep the fill small.
Example: take the star graph with centre $1$ and leaves $2, \dots, n$.
Eliminating the centre first fills the whole matrix.
Eliminating the leaves first, each of degree $1$, creates no fill at all.
The rule is a heuristic and need not give an ordering of least fill; finding an ordering of least fill is NP-hard.
Practical codes, such as approximate minimum degree, use quotient graphs and degree bounds to run fast.

## Explanations

### L1 human

Solving a large sparse linear system by elimination can turn many zeros into nonzeros, which costs memory and time.
The order in which the unknowns are eliminated decides how much of this fill appears.
The minimum degree algorithm always eliminates next the unknown that is linked to the fewest others.
It is a simple greedy rule that works well in practice.

### L2 human

The graph model turns a question about matrices into a question about graphs, and it predicts the sparsity of the Cholesky factor before any arithmetic.
Minimum degree is one of the standard fill-reducing orderings, together with [[nested dissection|9DTJ87P4R1]] and bandwidth-reducing orderings such as [[Cuthill–McKee|1X0E3JHCH9]].
It works well on irregular problems, while nested dissection has better guarantees on grid-like and planar problems.
</document>

<document source="https://tamnd.github.io/math2/node/415776K9Z3.md" tag="415776K9Z3" kind="definition">
# Perfect set property

## Definition

Let $X$ be a [[Polish space|584GXXY73M]] and let $A \subseteq X$.
The set $A$ has the *perfect set property* if $A$ is [[countable|01TQB88H4C]] or $A$ contains a nonempty [[perfect set|628242KN0T]].
Here a perfect set is a closed subset of $X$ with no isolated points.
A class $\Gamma$ of subsets of Polish spaces has the perfect set property if every set in $\Gamma$ has it.

## Remarks

Example: the Cantor set $C \subseteq \R$ has the property, because it is nonempty, closed and has no isolated points, so it contains the perfect set $C$ itself.

Non-example: under the axiom of choice there is a Bernstein set $B \subseteq \R$, a set such that both $B$ and $\R \setminus B$ meet every uncountable closed set.
Then $B$ is uncountable, but $B$ contains no nonempty perfect set $P$, because $P$ is an uncountable closed set and so meets $\R \setminus B$.

Every closed set has the property by the Cantor–Bendixson theorem, and so does every Borel set and every analytic set.
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<document source="https://tamnd.github.io/math2/node/417ZRDCHY6.md" tag="417ZRDCHY6" kind="concept">
# Quotient of a topological group

## Summary

The quotient of a [[topological group|92700QAB5Y]] $G$ by a subgroup $H$ is the set $G/H$ of left cosets with the quotient topology.
The projection $G \to G/H$ is continuous and open.
The quotient is Hausdorff exactly when $H$ is closed, and when $H$ is normal the quotient is again a topological group.
Quotients give homogeneous spaces such as spheres, which are quotients of rotation groups.

## Explanations

### L1 human

To form the quotient, glue together all elements of $G$ that differ by an element of $H$.
The result is a new space, where a set is open when the set of all elements that land in it is open in $G$.
For example, gluing the real line by the integers wraps it into a circle: $\R / \Z$ is the circle group.
Each point of the circle stands for a whole family $x + \Z$ of real numbers.
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<document source="https://tamnd.github.io/math2/node/41D1KJWM78.md" tag="41D1KJWM78" kind="definition">
# Lp multiplier

## Definition

Let $n \ge 1$ and $1 \le p \le \infty$.
Let $\mathcal{S}(\R^n)$ be the [[Schwartz space|CG9JEATW3Q]], let $\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$, and let $g^\vee$ be the inverse Fourier transform of $g$.
Let $m \colon \R^n \to \C$ be a bounded measurable function, and let $T_m f = (m \hat f)^\vee$ for $f \in \mathcal{S}(\R^n)$.
The function $m$ is an *$L^p$ multiplier* if there is a constant $C$ such that
$$\norm{T_m f}_{L^p(\R^n)} \le C \norm{f}_{L^p(\R^n)} \quad \text{for all } f \in \mathcal{S}(\R^n).$$
The smallest such $C$ is the multiplier norm of $m$, and $\mathcal{M}_p$ denotes the space of $L^p$ multipliers.

## Remarks

Example: for $h \in \R^n$, let $m(\xi) = e^{2\pi i \xi \cdot h}$, with the convention $\hat f(\xi) = \int f(x) e^{-2\pi i x \cdot \xi} \, dx$.
Then $T_m f(x) = f(x + h)$, so $\norm{T_m f}_{L^p} = \norm{f}_{L^p}$ and $m \in \mathcal{M}_p$ with norm $1$ for every $p$.

Non-example: in $\R^1$, $m(\xi) = \xi^2$ is not bounded, so it is not an $L^p$ multiplier.
Indeed, for $p = 2$ the Plancherel theorem gives $\norm{T_m f}_{L^2} = \norm{\xi^2 \hat f}_{L^2}$, and its ratio to $\norm{f}_{L^2} = \norm{\hat f}_{L^2}$ is unbounded when $\hat f$ is a fixed bump moved far out.

By the Plancherel theorem, $\mathcal{M}_2 = L^\infty(\R^n)$ with equal norms.
By duality, $\mathcal{M}_p = \mathcal{M}_{p'}$ when $1/p + 1/p' = 1$.
The space $\mathcal{M}_1 = \mathcal{M}_\infty$ consists of the Fourier transforms of finite Borel measures.
By the [[Riesz–Thorin theorem|AGRKRG1TMJ]], $\mathcal{M}_p \subseteq \mathcal{M}_q$ when $q$ lies between $p$ and $2$.
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# Linear regression

## Summary

Linear regression models a response variable as a linear function of explanatory variables plus a random error.
The model is linear in the unknown coefficients, not necessarily in the explanatory variables themselves.
Its coefficients are most often estimated by [[ordinary least squares|YM708KHV2W]].
It is one of the most widely used statistical models, and the base of the [[general linear model|A06SGFCSCE]] and its extensions.

## Explanations

### L1 human

Plot the heights of children against their ages, and the points lie roughly along a line.
Linear regression describes the height as an intercept plus a slope times the age, plus a random deviation for each child.
The slope says how much height increases per year on average.
With several explanatory variables, such as age and parents' heights, the line becomes a plane or a higher-dimensional flat surface.
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<document source="https://tamnd.github.io/math2/node/41S2EZ0KR3.md" tag="41S2EZ0KR3" kind="concept">
# Einstein manifold

## Summary

An Einstein manifold is a [[Riemannian manifold|7CBT4SM83Y]] whose [[Ricci curvature|4GTNM0WHDH]] is a constant multiple of the metric.
The condition is the vacuum Einstein equation of general relativity, with a cosmological constant, written for a Riemannian metric.
Einstein metrics are natural candidates for a "best" metric on a manifold, and their existence is a central question of Riemannian geometry.

## Explanations

### L1 human

The Ricci curvature measures how the volume of small balls differs from the flat case, direction by direction.
On an Einstein manifold this effect is the same in every direction and at every point.
The round sphere is the basic example: it looks the same everywhere and in every direction.
Flat space is also Einstein, with constant zero.
A product of two round spheres of different dimensions and the same radius is not Einstein, because the two factors bend volumes by different amounts.
</document>

<document source="https://tamnd.github.io/math2/node/41X0YW2R9N.md" tag="41X0YW2R9N" kind="concept">
# Minimal surface

## Summary

A minimal surface is a surface in space whose mean curvature is zero at every point.
Equivalently, small pieces of it are critical points of area: no small deformation that fixes the boundary changes the area to first order.
Soap films spanning a wire frame are physical models of minimal surfaces.
The subject links differential geometry with complex analysis, the calculus of variations and partial differential equations.

## Explanations

### L1 human

Dip a wire loop into soapy water and pull it out: the film that forms tries to have as little area as possible.
Its shape is a minimal surface.
At each point it bends upward in one direction exactly as much as it bends downward in the perpendicular direction, so the two bendings cancel.
A flat disc in a flat loop is the simplest example; two parallel rings give a waisted tube called a catenoid.
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<document source="https://tamnd.github.io/math2/node/421QMTFG9T.md" tag="421QMTFG9T" kind="concept">
# Hyperreal number

## Summary

The hyperreal numbers form an [[ordered field|0P3VCBDEE2]] ${}^*\R$ that contains the [[real numbers|08J2M1JYAG]] together with infinitesimal and infinite numbers.
The usual construction is an [[ultrapower|AX9Q86GB8M]] of $\R$: sequences of reals modulo agreement on a set in a nonprincipal [[ultrafilter|J64Y3JC0RE]] on $\N$.
By [[Łoś's theorem|AGYBSTD7BG]], ${}^*\R$ satisfies the same first-order statements as $\R$, which makes it the basic number system of [[nonstandard analysis|WM62GKCBWN]].
It is not [[Archimedean|0XH79S06J3]]: an infinite hyperreal is larger than every natural number.

## Explanations

### L1 human

Think of a hyperreal number as a sequence of real numbers, where two sequences count as equal if they agree "almost everywhere" in a precise sense.
The constant sequence $(3, 3, 3, \dots)$ is the real number $3$.
The sequence $(1, 1/2, 1/3, \dots)$ is positive but eventually below any positive real, so it is an infinitesimal.
The sequence $(1, 2, 3, \dots)$ is larger than every real number, so it is infinite.
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<document source="https://tamnd.github.io/math2/node/427N6S6QQW.md" tag="427N6S6QQW" kind="concept">
# Elliptic curve

## Summary

An elliptic [[curve|XJM2MT4S3Y]] over a [[field|04C5ZBG1WT]] $K$ is a smooth projective curve of [[genus|6W8TDV9VVN]] one over $K$ together with a chosen point defined over $K$.
Away from characteristics $2$ and $3$, it is given by an equation $y^2 = x^3 + ax + b$ with $4a^3 + 27b^2 \ne 0$.
Its points form an abelian group under the [[chord-and-tangent law|P7V1BQJPSS]], which makes it the simplest [[abelian variety|R7W03TXRS9]].
Elliptic curves are central in number theory, for example in the proof of Fermat's Last Theorem, and in cryptography.

## Explanations

### L1 human

Over the real numbers, an elliptic curve looks like a smooth curve in the plane, in one or two pieces, symmetric about the $x$-axis.
Its key feature is that one can "add" two points on it: draw the line through them, take the third intersection point, and reflect it in the $x$-axis.
For example, $y^2 = x^3 - x$ is an elliptic curve, since $4 \cdot (-1)^3 + 27 \cdot 0^2 = -4 \ne 0$.
The curve $y^2 = x^3$ is not one, because it has a sharp cusp at the origin.
Despite the name, an elliptic curve is not an ellipse; the name comes from elliptic integrals.
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<document source="https://tamnd.github.io/math2/node/428M453CR2.md" tag="428M453CR2" kind="concept">
# Hypergeometric series

## Summary

The Gauss hypergeometric series is the [[power series|0MZB06ST2B]] $\sum_{n \ge 0} \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}$, written ${}_2F_1(a, b; c; z)$, where $(x)_n$ is the rising factorial.
The ratio of consecutive coefficients is a rational function of $n$, of degree $2$ over degree $2$.
Many elementary functions are special cases, and its sum on the unit disc is the starting point of the [[hypergeometric function|463X5V31D9]].

## Explanations

### L1 human

The geometric series $1 + z + z^2 + \cdots$ has the same ratio $z$ between all consecutive terms.
In the hypergeometric series the ratio of the term with $z^{n+1}$ to the term with $z^n$ is $\frac{(n+a)(n+b)}{(n+c)(n+1)} z$, which changes with $n$ but tends to $z$.
So the series converges for $\abs{z} < 1$, just like the geometric series.
With $a = 1$ and $b = c$ it is exactly the geometric series $1/(1 - z)$.
With $a = b = 1$ and $c = 2$ it gives $-\log(1 - z)/z$.
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<document source="https://tamnd.github.io/math2/node/428R02EYWS.md" tag="428R02EYWS" kind="statement">
# Cartan–Hadamard theorem

## Statement

Let $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ whose [[sectional curvature|1X42JAA349]] satisfies $K \le 0$ everywhere.
Let $p \in M$.
Then the [[exponential map|Y7Z0AGY1A7]] $\exp_p \colon T_p M \to M$ is a [[covering map|3XZRRE165N]].
In particular, the [[universal cover|Y1YPE6PM52]] of $M$ is diffeomorphic to $\R^n$, and if $M$ is simply connected, then $M$ is diffeomorphic to $\R^n$.

## Remarks

The idea of the proof: if $K \le 0$, then for a [[Jacobi field|9AHMM41PJ3]] $J$ with $J(0) = 0$ the function $\abs{J(t)}^2$ is convex.
So a nonzero such $J$ never vanishes again, and there are no [[conjugate points|ZPXE5VSZ35]].
Hence $\exp_p$ is a local diffeomorphism.
Pull the metric back to $T_p M$; the lines through $0$ are geodesics defined for all time, so the pulled-back metric is complete by the [[Hopf–Rinow theorem|QCCD2JNFBT]].
A local isometry from a complete connected Riemannian manifold onto a connected one is a covering map.

Example: for a flat torus $\R^n / \Z^n$, the map $\exp_p$ is the quotient map $\R^n \to \R^n / \Z^n$ followed by a translation.
For a compact hyperbolic surface, $\exp_p$ is the universal covering by the hyperbolic plane.

A consequence is that all higher homotopy groups of $M$ vanish.
The sign condition is needed: on the round sphere $S^n$, the map $\exp_p$ sends the whole sphere of radius $\pi$ in $T_p S^n$ to the antipode of $p$.
So it is not a covering.
Completeness is needed too: the punctured plane $\R^2 \setminus \set{0}$ is flat, but $\exp_p$ is not defined on all of $T_p M$.

## Explanations

### L1 human

On a space that curves like a saddle everywhere, geodesics that start at one point spread apart and never meet again.
So straight lines from one point unroll the space onto a flat plane, perhaps wrapping it several times around.
In particular, such a space with no holes is just a stretched copy of $\R^n$.

### L2 human

The theorem shows that curvature restricts topology: a compact manifold with $K \le 0$ is aspherical, its universal cover is $\R^n$, and its topology is governed by its fundamental group.
For example, no sphere $S^n$ with $n \ge 2$ carries a metric of nonpositive curvature.
It is the starting point of the study of manifolds of negative curvature.
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<document source="https://tamnd.github.io/math2/node/429JJ5ZNBY.md" tag="429JJ5ZNBY" kind="example">
# Baumslag–Solitar group

## Example

Let $m$ and $n$ be nonzero integers.
The *Baumslag–Solitar group* $BS(m, n)$ is the group with the [[presentation|0JGEQXQ4AC]]
$$BS(m, n) = \langle a, b \mid b a^m b^{-1} = a^n \rangle.$$
It is a [[one-relator group|68AFPHR4GN]], since the presentation has the single relator $b a^m b^{-1} a^{-n}$.
It is also the [[HNN extension|YEYERRY37K]] of the infinite cyclic group $\langle a \rangle$ that conjugates $\langle a^m \rangle$ onto $\langle a^n \rangle$ by $a^m \mapsto a^n$, with stable letter $b$.

## Remarks

Small cases: $BS(1, 1) \cong \Z^2$, and $BS(1, -1)$ is the fundamental group of the Klein bottle.
The group $BS(1, n)$ is isomorphic to the semidirect product $\Z[1/n] \rtimes \Z$, where the generator of $\Z$ acts by multiplication by $n$.
The isomorphism sends $a$ to $1 \in \Z[1/n]$ and $b$ to the generator of $\Z$.
In particular $BS(1, n)$ is solvable.

$BS(m, n)$ is [[residually finite|HEE922EHQ8]] if and only if $\abs{m} = 1$, $\abs{n} = 1$ or $\abs{m} = \abs{n}$.

$BS(2, 3) = \langle a, b \mid b a^2 b^{-1} = a^3 \rangle$ is not [[Hopfian|PHEN4E1JPT]].
Let $\varphi$ be the endomorphism with $\varphi(a) = a^2$ and $\varphi(b) = b$.
It is well defined, since $\varphi(b a^2 b^{-1}) = b a^4 b^{-1} = (b a^2 b^{-1})^2 = a^6 = \varphi(a^3)$.
It is surjective, since $a = a^3 a^{-2} = b a^2 b^{-1} a^{-2} = \varphi(b a b^{-1} a^{-1})$.
It is not injective: the element $[b a b^{-1}, a]$ is not $1$ by Britton's lemma, but $\varphi$ sends it to $[b a^2 b^{-1}, a^2] = [a^3, a^2] = 1$.

## Explanations

### L1 human

A Baumslag–Solitar group has two generators and one rule: conjugating $a^m$ by $b$ gives $a^n$.
For $m = n = 1$ the rule says that $a$ and $b$ commute, and the group is $\Z^2$.
For $m = 1$ and $n = 2$, think of $a$ as the shift $x \mapsto x + 1$ and $b$ as the doubling $x \mapsto 2x$ of the real line.
Then $b a b^{-1}$ is $x \mapsto x + 2$, which is $a^2$.
Other values of $m$ and $n$ give groups with surprising properties, and they serve as standard test cases in group theory.
</document>

<document source="https://tamnd.github.io/math2/node/42A2M9S2C1.md" tag="42A2M9S2C1" kind="definition">
# Type theory

## Definition

A *type theory* is a formal system with these parts.

1. A collection of *types*, built from base types by type constructors.
2. A collection of *terms*, built from variables and constants by term constructors.
3. *Judgments*, in particular the typing judgment $\Gamma \vdash t : A$, which says that in the context $\Gamma = (x_1 : A_1, \dots, x_n : A_n)$ the term $t$ has type $A$.
4. *Inference rules*, which say which judgments can be derived from which, and a notion of computation or equality on terms.

Every well-formed term has a type, and a term can only be used where its type allows.

## Remarks

Example: Church's simply typed lambda calculus is a type theory.
Its types are built from base types by the arrow $A \to B$.
Its rules include: if $\Gamma, x : A \vdash t : B$, then $\Gamma \vdash \lambda x.\, t : A \to B$; and if $\Gamma \vdash f : A \to B$ and $\Gamma \vdash a : A$, then $\Gamma \vdash f\, a : B$.
Non-example: the untyped lambda calculus is not a type theory, because its terms carry no types.
In it, the self-application $\lambda x.\, x\, x$ is a legal term, while in the simply typed calculus it has no type.
In *dependent* type theories, types may depend on terms, as in the type $\mathrm{Vec}(n)$ of lists of length $n$.
Unlike a [[set|0BSJ7RQ9BW]], a term has its type as part of its form, and "$t$ has type $A$" is a judgment, not a proposition that can be true or false.
This is how type theory avoids [[Russell's paradox|0HESMMSYEG]].
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<document source="https://tamnd.github.io/math2/node/42A5ZT6AWN.md" tag="42A5ZT6AWN" kind="statement">
# Riemann–Hurwitz formula

## Statement

Let $X$ and $Y$ be compact connected [[Riemann surfaces|5K0N8QMV6Q]] of [[genus|NJ787GZS1N]] $g_X$ and $g_Y$.
Let $f \colon X \to Y$ be a nonconstant holomorphic map of degree $n$.
For $P \in X$, let $e_P \ge 1$ be the ramification index of $f$ at $P$, so that near $P$ the map looks like $z \mapsto z^{e_P}$ in local coordinates.
Then $e_P = 1$ for all but finitely many $P$, and
$$2 g_X - 2 = n\,(2 g_Y - 2) + \sum_{P \in X} (e_P - 1).$$

## Remarks

Triangulate $Y$ so that every branch value is a vertex, and lift the triangulation to $X$ along the [[branched covering|YARENZRXTD]] $f$.
Each edge and face has exactly $n$ preimages, and a vertex $y$ has $n - \sum_{f(P) = y} (e_P - 1)$ preimages.
Comparing [[Euler characteristics|RNDDWDAFJ5]] with $\chi = 2 - 2g$ gives the formula.
The same formula holds for a finite separable morphism of smooth projective curves over an algebraically closed field when every $e_P$ is prime to the characteristic.
With wild ramification the term $e_P - 1$ must be replaced by a larger number.
Example: the map $z \mapsto z^n$ on the Riemann sphere has degree $n$ and ramifies only at $0$ and $\infty$ with $e = n$; the formula reads $-2 = n(-2) + 2(n - 1)$, which is true.
Example: a double cover $X \to \mathbb{P}^1$ branched at $2g + 2$ points has $2 g_X - 2 = -4 + 2g + 2$, so $g_X = g$; this is how the genus of a hyperelliptic curve is found.

## Explanations

### L1 human

A map between surfaces that wraps one surface $n$ times around another almost everywhere, except at finitely many branch points, links the numbers of holes of the two surfaces.
Without branch points, the Euler characteristic simply multiplies by $n$.
Each branch point, where several sheets come together, makes a correction.
The formula records this exactly.

### L2 human

The formula is the main tool to compute genera of curves given as covers, such as hyperelliptic curves, Fermat curves and modular curves.
It shows at once that there is no nonconstant map from a curve of genus $g$ to a curve of larger genus.
It also shows that an unramified cover of a genus-$1$ curve has genus $1$.
Applied to the quotient map by a group of automorphisms, it gives [[Hurwitz's automorphisms theorem|3GNKSCQSCD]].
It is the one-dimensional case of the comparison of canonical divisors $K_X = f^* K_Y + R$, where $R$ is the ramification divisor.
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<document source="https://tamnd.github.io/math2/node/42AGEAAQ8H.md" tag="42AGEAAQ8H" kind="definition">
# Neural network approximation

## Definition

Let $\sigma \colon \R \to \R$ be a function, called the *activation function*, and let $d, n \ge 1$ be integers.
A *shallow neural network* with $n$ neurons is a function $\R^d \to \R$ of the form
$$x \mapsto \sum_{k=1}^{n} c_k \, \sigma(w_k \cdot x + b_k), \qquad c_k, b_k \in \R, \ w_k \in \R^d .$$
Let $\Sigma_n(\sigma)$ be the set of these functions.
Let $K \subseteq \R^d$ be a [[compact set|0TVJZYB6SH]] and let $f \colon K \to \R$ be [[continuous|0X3WGX95FE]].
*Neural network approximation* of $f$ is its approximation by elements of $\Sigma_n(\sigma)$, measured by the error
$$E_n(f) = \inf\set{\sup_{x \in K} \abs{f(x) - g(x)} : g \in \Sigma_n(\sigma)} .$$

## Remarks

Other [[norms|03J5RAKB30]], such as $L^p$ norms on $K$, are also used.
A *deep* network with $L$ layers is a composition $T_L \circ \sigma \circ T_{L-1} \circ \cdots \circ \sigma \circ T_1$ of affine maps $T_j \colon \R^{d_{j-1}} \to \R^{d_j}$, with $d_0 = d$ and $d_L = 1$, where $\sigma$ acts on each coordinate; the shallow case is $L = 2$.
The set $\Sigma_n(\sigma)$ is in general not a linear space, since the sum of two networks with $n$ neurons may need $2n$ neurons.
For continuous $\sigma$, $E_n(f) \to 0$ as $n \to \infty$ for all $K$ and $f$ if and only if $\sigma$ is not a polynomial.
Example: for the ReLU $\sigma(t) = \max(t, 0)$, $d = 1$, $K = [-1, 1]$ and $f(x) = \abs{x}$, one has $f = \sigma(x) + \sigma(-x) \in \Sigma_2(\sigma)$, so $E_2(f) = 0$.
Non-example: for $\sigma(t) = t$, every element of $\Sigma_n(\sigma)$ is an affine function, so on $K = [-1, 1]$ the function $x^2$ has $E_n(x^2) = 1/2$ for every $n$ and is not approximated.
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<document source="https://tamnd.github.io/math2/node/42DDDZHP7F.md" tag="42DDDZHP7F" kind="statement">
# Kahane–Katznelson theorem

## Statement

Let $E \subset \R$ be a set of Lebesgue measure zero, that is, a [[null set|VQQM4Y550T]].
Then there is a [[continuous|0X3WGX95FE]] $2\pi$-periodic function $f \colon \R \to \C$ whose [[Fourier series|G12KXDHPPS]] diverges at every point of $E$.

## Remarks

Kahane and Katznelson proved this in 1966.
Together with [[Carleson's theorem|FBH6KG484Z]] it describes the sets of divergence exactly.
A set lies in the divergence set of the Fourier series of some continuous function if and only if it has measure zero.
The proof builds $f$ as a uniformly convergent sum of trigonometric polynomials, using covers of $E$ by intervals of small total length.
For a single point $E = \set{0}$ the result is the classical [[example of du Bois-Reymond|FPDWWD12FE]].
For example, $E$ can be the set of rational multiples of $\pi$, or the Cantor set.
Then some continuous function has a Fourier series diverging at each point of $E$.

## Explanations

### L1 human

Carleson's theorem says that the Fourier series of a continuous function can fail to converge only on a set of total length zero.
This theorem says that every set of length zero really occurs: one can choose a continuous function whose Fourier series fails to converge at each point of that set.
So nothing better than "almost everywhere" can be true.

### L2 human

The result shows that Carleson's theorem is sharp for continuous functions.
It contrasts with smooth or Hölder continuous functions, whose Fourier series converge everywhere by the [[Dini test|F5PHA35M50]].
So the divergence sets of continuous functions are as large as Carleson's theorem allows.
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<document source="https://tamnd.github.io/math2/node/42NZSQ65DQ.md" tag="42NZSQ65DQ" kind="concept">
# Response surface methodology

## Summary

Response surface methodology is a set of statistical techniques for learning how a response depends on several input factors.
Its goal is to find the factor settings that make the response best.
It fits low-order polynomial models to data from designed experiments and moves through the factor space in a sequence of experiments.
George Box and K. B. Wilson introduced it in 1951, and it is widely used to improve industrial processes.

## Explanations

### L1 human

Think of the response, such as the yield of a chemical process, as an unknown hilly landscape over the settings of temperature and pressure.
A few runs near the current settings show which way is uphill, and a flat plane fitted to them gives the direction to walk.
Near the top the landscape bends, so a quadratic surface is fitted instead and its peak is computed.
For example, if the fitted surface near the top is $y = 80 - (x_1 - 1)^2 - 2 (x_2 + 0.5)^2$, the best settings are $x_1 = 1$ and $x_2 = -0.5$.
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<document source="https://tamnd.github.io/math2/node/42V5XGZS4F.md" tag="42V5XGZS4F" kind="concept">
# Singularity theory

## Summary

Singularity theory studies the points where a variety or a map fails to be smooth, and classifies them up to local changes of coordinates.
Its objects are germs: the behaviour of a function, a map or a variety near one point.
Main tools are invariants such as the [[Milnor number|1ZE51P7GRQ]], and resolutions by [[blowing up|C4KGYAFPN7]].
It connects algebraic geometry with differential topology, through [[Morse functions|KPQ96640ZJ]] and catastrophe theory.

## Explanations

### L1 human

Under a microscope, most points of a curve or surface look like a flat piece of space.
A few may not, such as the tip of a cone or the crossing point of a figure eight.
Singularity theory asks what such special points can look like, and when two of them are the same up to a smooth change of coordinates.
For functions, the singular points are the [[critical points|0YTCMFB4HC]].
Over the complex numbers, near a nondegenerate critical point, a function is $x_1^2 + \cdots + x_n^2$ after a change of coordinates, so all of them look alike.
Degenerate ones, such as $x^3$ or $x^2 + y^3$, give a rich classification.
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<document source="https://tamnd.github.io/math2/node/42ZCE14AD2.md" tag="42ZCE14AD2" kind="concept">
# Reductive Lie group

## Summary

A reductive Lie group is a [[Lie group|7JBRGCVJRB]] whose Lie algebra is a reductive Lie algebra.
Such a Lie algebra is the direct sum of its centre and a semisimple Lie algebra.
The class contains the semisimple groups, the tori, and the general linear groups $GL_n(\R)$ and $GL_n(\C)$, as well as every compact Lie group.
It is the natural setting for much of representation theory and harmonic analysis.
One reason is that Levi factors of parabolic subgroups of reductive groups are again reductive, while those of semisimple groups need not be semisimple.
Authors differ on the extra conditions they impose, such as finitely many components or conditions on the adjoint action.

## Explanations

### L1 human

A reductive Lie group is a group that is semisimple up to a commutative part that sits in the centre.
The basic example is $GL_n(\R)$: every invertible matrix is a scalar times a matrix of determinant $\pm 1$, and the scalars commute with everything.
The Lie algebra of $GL_n(\R)$ splits as the scalar matrices plus the trace-zero matrices.
The group of upper triangular invertible matrices is not reductive: its strictly upper triangular part is a non-central normal piece that cannot be split off.
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<document source="https://tamnd.github.io/math2/node/437FSG3419.md" tag="437FSG3419" kind="concept">
# Convex body

## Summary

A convex body is a [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]] in $\R^n$ with nonempty [[interior|0RZH6W98HK]].
Convex bodies are the main objects of convex geometry: balls, cubes, simplices and all convex polytopes of full dimension are examples.
Volume, surface area, mixed volumes and the Brunn–Minkowski theory are all developed for convex bodies.

## Explanations

### L1 human

A convex body is a solid, bounded, convex shape that includes its boundary.
It must have some thickness in every direction, so a flat disc lying in space does not count as a convex body in $\R^3$.
A solid ball, a solid cube and a solid tetrahedron are convex bodies in space.
A filled triangle is a convex body in the plane.
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<document source="https://tamnd.github.io/math2/node/437YRCNYRJ.md" tag="437YRCNYRJ" kind="concept">
# Radical of an ideal

## Summary

The radical of an [[ideal|0MYSA6MYNA]] $I$ of a commutative ring is the set of elements some power of which lies in $I$.
It is again an ideal, and it contains $I$.
The radical is the intersection of the [[prime ideals|06HC6HK0Z8]] that contain $I$.
It is central in the link between ideals and geometry given by [[Hilbert's Nullstellensatz|D85VZWS9QK]].

## Explanations

### L1 human

The radical forgets multiplicities.
In $\Z$, the ideal $12\Z$ records the number $12 = 2^2 \cdot 3$, and its radical is $6\Z$, which only records the primes $2$ and $3$.
An integer $r$ has a power divisible by $12$ exactly when $r$ is divisible by $2$ and by $3$.
For polynomials, the ideals $(x^2)$ and $(x)$ have the same zero set, and the radical of $(x^2)$ is $(x)$.
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<document source="https://tamnd.github.io/math2/node/43DJVJYRBN.md" tag="43DJVJYRBN" kind="concept">
# Complete intersection

## Summary

A complete intersection is a variety in affine or projective space whose ideal can be generated by as many equations as its codimension.
Each equation cuts the dimension down by exactly one, so the variety is defined by the smallest possible number of equations.
Complete intersections are among the simplest varieties to study.
Their degree is the product of the degrees of the equations, and their coordinate rings have good algebraic properties.

## Explanations

### L1 human

In three-dimensional space, one equation usually defines a surface and two equations usually define a curve.
A curve in space is a complete intersection when two equations are enough to describe it exactly.
A line, such as $x = y = 0$, is a complete intersection.
The twisted cubic curve, the image of $t \mapsto (t, t^2, t^3)$ completed in projective space, is a curve in $\mathbb{P}^3$ whose ideal needs three equations, so it is not a complete intersection.
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<document source="https://tamnd.github.io/math2/node/43DXNAFZ9T.md" tag="43DXNAFZ9T" kind="statement">
# Interlacing of zeros of orthogonal polynomials

## Statement

Let $\mu$ be a positive Borel measure on $\R$ with finite moments of all orders and infinite [[support|FQTXV6BQZG]].
Let $(p_n)_{n \ge 0}$ be a sequence of real [[orthogonal polynomials|8TXASARQN1]] for $\mu$, and let $I$ be the smallest closed interval that contains the support of $\mu$.

1. For every $n \ge 1$, the polynomial $p_n$ has $n$ distinct real zeros, and all of them lie in the interior of $I$.
2. Let $x_{n,1} < \cdots < x_{n,n}$ be the zeros of $p_n$. Then the zeros of $p_n$ and $p_{n+1}$ strictly interlace:
$$x_{n+1,k} < x_{n,k} < x_{n+1,k+1} \quad \text{for } k = 1, \dots, n.$$

## Remarks

For part 1, let $y_1, \dots, y_m$ be the zeros of odd multiplicity of $p_n$ in the interior of $I$, the points where $p_n$ changes sign.
Suppose $m < n$.
Then $p_n(x) \prod_j (x - y_j)$ does not change sign on $I$ and is not zero $\mu$-almost everywhere, so its integral against $\mu$ is not zero.
But $\prod_j (x - y_j)$ has degree $m < n$, and $p_n$ is orthogonal to all polynomials of lower degree, a contradiction.
Part 2 follows from the three-term recurrence through the Christoffel–Darboux formula.
If the leading coefficients are positive, it gives $p_{n+1}'(x) p_n(x) - p_{n+1}(x) p_n'(x) > 0$ for all real $x$.
Evaluating at consecutive zeros of $p_{n+1}$ shows that $p_n$ changes sign between them.

Example: for the Legendre polynomials, $p_1 = x$ has the zero $0$, and $p_2 = \frac12(3x^2 - 1)$ has the zeros $\pm 1/\sqrt{3}$.
Indeed $-1/\sqrt{3} < 0 < 1/\sqrt{3}$, and all lie in $(-1, 1)$.

## Explanations

### L1 human

The $n$-th orthogonal polynomial crosses the axis exactly $n$ times, inside the region where the measure lives.
Moreover, the crossings of consecutive polynomials alternate: between two crossings of one there is exactly one crossing of the other.
This regular pattern is a fingerprint of orthogonality.

### L2 human

The zeros of $p_n$ are the eigenvalues of the $n \times n$ truncated Jacobi matrix, the symmetric [[tridiagonal matrix|DARX1KZRHZ]] built from the recurrence coefficients.
In this language, interlacing is the [[Cauchy interlacing theorem|8QQPVNR80F]] for a matrix and its principal submatrix.
The zeros of $p_n$ are the nodes of Gaussian quadrature for $\mu$, and the fact that they are real, simple and inside $I$ is what makes that quadrature work.
As $n \to \infty$, the counting measures of the zeros, divided by $n$, describe the asymptotic distribution of the zeros, a central topic of the theory.
</document>

<document source="https://tamnd.github.io/math2/node/43HZJ7DRT0.md" tag="43HZJ7DRT0" kind="concept">
# Godunov's scheme

## Summary

Godunov's scheme is a finite volume method for hyperbolic systems of [[conservation laws|8RA7AMWSAF]].
It stores cell averages, solves the [[Riemann problem|GJWKX22VVE]] exactly at each cell interface, and averages the result again.
It is conservative, first-order accurate, and captures shocks without spurious oscillations.
Its idea of using Riemann solvers is the basis of many modern high-resolution schemes.
See the [[definition|FBJ4JCP5SR]].

## Explanations

### L1 human

Divide the line into cells and keep only the average of the solution in each cell.
At each cell boundary two constant values meet, and the exact evolution of such a jump is known.
Let these jumps evolve for one short time step, then compute the new average in each cell.
For the equation $u_t + a u_x = 0$ with $a > 0$, this gives the upwind scheme, which takes information from the left neighbour.
</document>

<document source="https://tamnd.github.io/math2/node/43J9XSE6M8.md" tag="43J9XSE6M8" kind="definition">
# Barnes integral

## Definition

Let $a_1, \dots, a_m$, $b_1, \dots, b_n$, $c_1, \dots, c_p$ and $d_1, \dots, d_q$ be complex numbers, and let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
Assume that $a_j + b_k$ is not in $\set{0, -1, -2, \dots}$ for all $j$ and $k$.
Then the poles $s = -a_j - \ell$ of the factors $\Gamma(a_j + s)$ are disjoint from the poles $s = b_k + \ell$ of the factors $\Gamma(b_k - s)$, where $\ell \in \N$.
Let $L$ be a path from $-i\infty$ to $+i\infty$.
Assume that all poles of the first family lie on its left and all poles of the second family on its right.
Fix $z \ne 0$ and a branch of $\log z$, and put $z^{-s} = e^{-s \log z}$.
A *Barnes integral* is a [[contour integral|06ZGPYTWK8]] of the form
$$\frac{1}{2\pi i} \int_L \frac{\prod_{j=1}^{m} \Gamma(a_j + s) \prod_{k=1}^{n} \Gamma(b_k - s)}{\prod_{j=1}^{p} \Gamma(c_j + s) \prod_{k=1}^{q} \Gamma(d_k - s)}\, z^{-s}\, ds,$$
for those $z$ for which it converges.

## Remarks

Example: take $m = 1$, $a_1 = 0$, no other factors, $x > 0$, and $L$ the line $\operatorname{Re} s = c$ with $c > 0$.
All poles $s = 0, -1, -2, \dots$ of $\Gamma(s)$ lie to the left of $L$, so $L$ is admissible.
Shifting the line to the left and summing the residues $(-1)^\ell x^\ell / \ell!$ at $s = -\ell$ gives
$$\frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \Gamma(s)\, x^{-s}\, ds = e^{-x}.$$
Non-example: the same integrand on the line $\operatorname{Re} s = -\tfrac12$ is not a Barnes integral in this sense, since the pole $s = 0$ lies on the wrong side.
Its value is $e^{-x} - 1$, because the residue $1$ at $s = 0$ is lost.
Barnes' representation of the [[hypergeometric function|463X5V31D9]] reads, for $\abs{\arg(-z)} < \pi$ and $a, b$ not in $\set{0, -1, -2, \dots}$,
$$\frac{\Gamma(a)\Gamma(b)}{\Gamma(c)}\, {}_2F_1(a, b; c; z) = \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{\Gamma(a + s)\Gamma(b + s)\Gamma(-s)}{\Gamma(c + s)} (-z)^s\, ds,$$
with the path separating the poles of $\Gamma(a + s)\Gamma(b + s)$ from those of $\Gamma(-s)$.
The Meijer G-function and the Fox H-function are defined by integrals of this type.
</document>

<document source="https://tamnd.github.io/math2/node/43M1WFPYGQ.md" tag="43M1WFPYGQ" kind="concept">
# Presentation of an algebra

## Summary

A presentation of an algebra in a [[variety|AJNZJD8442]] describes it by generators and relations.
The relations are equations between terms in the generators.
The presented algebra is the quotient of the free algebra on the generators by the smallest congruence that makes the relations true.
Every algebra of the variety has a presentation, and finitely presented algebras are those with finitely many generators and relations.
[[Presentations of groups|0JGEQXQ4AC]] are the case of the variety of groups.

## Explanations

### L1 human

To describe an algebra, name some elements that generate it and list the equations they satisfy.
The algebra is then the "most general" one in which these equations hold, and nothing more.
For example, in commutative rings with identity, one generator $x$ and the relation $x^2 = -1$ present the Gaussian integers $\Z[i]$.
In monoids, generators $a, b$ with relation $ab = ba$ present the monoid $\N^2$ under addition.
</document>

<document source="https://tamnd.github.io/math2/node/43MP9NWS6H.md" tag="43MP9NWS6H" kind="concept">
# Quotient category

## Summary

A quotient category is obtained from a [[category|A2DE66937D]] by declaring some parallel morphisms equal.
The identification must be an [[equivalence relation|0G8VEZ3WTK]] on each set of morphisms that is compatible with composition, called a congruence.
Quotient categories are how one builds homotopy categories, where homotopic maps become equal.

## Explanations

### L1 human

Keep all the objects, but glue some arrows together.
For this to make sense, gluing $f$ to $f'$ must also glue $g \circ f$ to $g \circ f'$, so that composition still works on the glued arrows.
The main example is the homotopy category of topological spaces, where two continuous maps become the same arrow when one can be deformed into the other.
For a small example, a monoid is a category with one object, and a quotient of that category is a quotient monoid.
Gluing integers of the same parity turns $\Z$ under addition into $\Z/2\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/43NRBE72C6.md" tag="43NRBE72C6" kind="concept">
# Nilmanifold

## Summary

A nilmanifold is a homogeneous space $N/H$ of a connected [[nilpotent Lie group|DK87YN458D]] $N$ by a closed subgroup $H$.
The most studied ones are the compact quotients $N/\Gamma$ of a simply connected nilpotent Lie group by a lattice $\Gamma$.
The torus is the abelian case, and the Heisenberg manifold is the first nonabelian one.
Compact nilmanifolds give many examples in geometry, such as symplectic manifolds that are not Kähler.

## Explanations

### L1 human

The torus is the plane $\R^2$ folded up by the integer translations.
A nilmanifold does the same with a group that is almost commutative, a nilpotent group.
The main example uses upper triangular $3 \times 3$ matrices with ones on the diagonal and real entries above it.
Folding them up by the matrices with integer entries gives a compact space of dimension $3$, the Heisenberg manifold.
It looks like a torus bundle that twists as one goes around a circle.
</document>

<document source="https://tamnd.github.io/math2/node/43R0MPN4T1.md" tag="43R0MPN4T1" kind="definition">
# Lagrangian singularity

## Definition

Let $Q$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.
Let $T^*Q$ be its cotangent bundle with the canonical [[symplectic form|CAN490EPRT]] $\omega = \sum_{i=1}^n dp_i \wedge dq_i$ in local coordinates $(q, p)$, and let $\pi \colon T^*Q \to Q$ be the bundle projection.
Let $L \subset T^*Q$ be a [[Lagrangian submanifold|H8ACPTMX08]].
The *Lagrangian map* of $L$ is the restriction $\pi|_L \colon L \to Q$, a smooth map between manifolds of the same dimension $n$.
A point $x \in L$ is a *Lagrangian singularity* if the [[differential|6P9PXQDSEN]] $d(\pi|_L)_x \colon T_x L \to T_{\pi(x)} Q$ is not invertible, that is, if its rank is less than $n$.

## Remarks

Example: $Q = \R$, $T^*Q = \R^2$ with coordinates $(q, p)$, and $L = \set{(p^2, p) : p \in \R}$.
Every curve in a symplectic surface is Lagrangian, so $L$ is Lagrangian.
In the parameter $p$ the Lagrangian map is $p \mapsto p^2$, whose derivative $2p$ vanishes at $p = 0$.
So $(0, 0)$ is a Lagrangian singularity, of fold type $A_2$.

Non-example: the zero section $L_0 = \set{p = 0}$, or more generally the graph $\set{(q, dS(q))}$ of the differential of a smooth function $S$ on $Q$.
The projection restricted to such a graph is a diffeomorphism onto $Q$, so its differential is invertible at every point and there is no Lagrangian singularity.

Lagrangian singularities are studied up to Lagrangian equivalence: symplectomorphisms of $T^*Q$ that map fibres of $\pi$ to fibres.
Locally every Lagrangian submanifold is given by a generating family $F(q, y)$ with $y \in \R^m$, as $L = \set{(q, \partial_q F) : \partial_y F = 0}$.
Its Lagrangian singularities then lie over the points $q$ where $F(q, \cdot)$ has a degenerate critical point, which links the theory to the [[bifurcation set|JV45S0KQE6]] of catastrophe theory.
</document>

<document source="https://tamnd.github.io/math2/node/43VFMJCDYJ.md" tag="43VFMJCDYJ" kind="concept">
# Schur polynomial

## Summary

The Schur polynomial $s_\lambda(x_1, \ldots, x_n)$ is a [[symmetric polynomial|0VHSHB9B6J]] attached to an [[integer partition|0JV30XN73Z]] $\lambda$, see the [[definition|DA03KQ311J]].
It is a quotient of two alternating determinants, and it is also a sum over semistandard [[Young tableaux|Q6TH9PH2R7]] of shape $\lambda$.
For partitions $\lambda$ of $d$ with at most $n$ parts, the $s_\lambda$ form a basis of the degree-$d$ symmetric polynomials in $n$ variables.
They are the characters of the irreducible polynomial representations of $GL_n(\C)$, and their products are described by the [[Littlewood–Richardson rule|Q490XBH1Q2]].

## Explanations

### L1 human

Draw the [[Young diagram|096AHMB3P4]] of a partition, and fill its boxes with numbers from $1$ to $n$.
The numbers must weakly increase along rows and strictly increase down columns.
Each filling gives a monomial, with $x_i$ raised to the number of times $i$ appears.
The sum of these monomials is the Schur polynomial.
For one row of two boxes and $n = 2$ the fillings are $11$, $12$ and $22$, so $s_{(2)} = x_1^2 + x_1 x_2 + x_2^2$.
</document>

<document source="https://tamnd.github.io/math2/node/441JWPAV3W.md" tag="441JWPAV3W" kind="definition">
# Stochastic partial differential equation

## Definition

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, let $D \subseteq \R^d$ be open and let $T > 0$.
Let $L$ be a linear differential operator in the variable $x \in D$, and let $F$ and $G$ be functions.
Let $\xi$ be a random noise on $[0, T] \times D$, for example space-time white noise, the formal time derivative of a cylindrical [[Wiener process|22735FGJV6]] $W$.
A *stochastic partial differential equation* is an equation for a random field $u(t, x, \omega)$ of the form
$$\partial_t u = L u + F(u) + G(u) \, \xi, \qquad t \in (0, T], \ x \in D,$$
with initial and boundary conditions.
In Itô form it reads $du = (L u + F(u)) \, dt + G(u) \, dW$.
A solution is a [[stochastic process|0QH5ENKZM1]], adapted to the filtration of $W$, that satisfies the equation in a [[weak|SJZX81A783]] or [[mild|3VBRS94V1G]] sense.
Classical derivatives are not used, because $\xi$ is too irregular.

## Remarks

Example: the stochastic heat equation $\partial_t u = \partial_x^2 u + \xi$ on $(0, 1)$ with $u = 0$ at $x = 0, 1$ and $u(0, \cdot) = 0$, where $\xi$ is space-time white noise.
Its mild solution is $u(t, x) = \int_0^t \int_0^1 p_{t-s}(x, y) \, W(ds, dy)$, with $p$ the Dirichlet heat kernel of $(0, 1)$.
This is a well-defined Gaussian random field, continuous in $(t, x)$.
Non-example: the stochastic differential equation $dX = -X \, dt + dB$, with $B$ a Brownian motion, is driven by noise.
But it involves no derivatives in a space variable, so it is not a partial differential equation.
The same stochastic heat equation in dimension $d \ge 2$ still has a mild solution, but it is a random distribution rather than a function.
</document>

<document source="https://tamnd.github.io/math2/node/441PZ3T7F0.md" tag="441PZ3T7F0" kind="concept">
# Credible interval

## Summary

A credible interval is an interval of parameter values that has a given probability, such as $95\%$, under the [[posterior distribution|0KJ7ZS8AJA]].
It is the Bayesian counterpart of a [[confidence interval|0S6T9KQNC0]].
Its probability statement is about the parameter given the observed data, which is how confidence intervals are often, wrongly, read.

## Explanations

### L1 human

After seeing the data, a Bayesian has a probability distribution for the unknown quantity.
A $95\%$ credible interval is a range that holds $95\%$ of this probability.
So one may say "given the data and the prior, the parameter lies in this range with probability $95\%$".
For example, if the posterior of a mean is normal with mean $10$ and standard deviation $1$, then $[8.04, 11.96]$ is a $95\%$ credible interval.
</document>

<document source="https://tamnd.github.io/math2/node/443ZWDX88P.md" tag="443ZWDX88P" kind="concept">
# Survival analysis

## Summary

Survival analysis is the part of statistics that studies the time until an event happens.
Examples of events are death, relapse of a disease or failure of a machine.
Its central objects are the [[survival function|QTM1HZM53Q]] and the [[hazard function|Y959NKQ3XW]] of the event time.
Its typical difficulty is [[censoring|MY8ZZP84NP]].
For some subjects the event is not observed during the study, and only a lower bound for their time is known.
Standard tools include the [[Kaplan–Meier estimator|B7AJ8RWZRC]], the [[log-rank test|F6CT1YXZ0G]] and the [[proportional hazards model|N029V3JX60]].

## Explanations

### L1 human

A clinical trial follows patients for five years and records when each one has a relapse.
Some patients have no relapse before the trial ends, and some move away; for them we only know that the relapse came later than their last visit.
Survival analysis uses these incomplete records correctly instead of throwing them away.
It answers questions such as "what fraction of patients is still free of relapse after two years?" and "does the new drug delay relapse?".
In engineering the same methods are called reliability analysis.
</document>

<document source="https://tamnd.github.io/math2/node/447CBEA2Z1.md" tag="447CBEA2Z1" kind="definition">
# Nörlund mean

## Definition

Let $(p_k)_{k \ge 0}$ be a sequence of real numbers with $p_0 > 0$ and $p_k \ge 0$ for all $k$, and put $P_n = p_0 + p_1 + \cdots + p_n$.
Let $(s_k)_{k \ge 0}$ be a [[sequence|0QE8STGJH2]] of complex numbers.
The *Nörlund means* of $(s_k)$ with respect to $(p_k)$ are
$$t_n = \frac{p_n s_0 + p_{n-1} s_1 + \cdots + p_0 s_n}{P_n} \quad (n \ge 0).$$
The sequence $(s_k)$ is $(N, p)$ summable to $s$ if $t_n \to s$.
A series is $(N, p)$ summable to $s$ if the sequence of its [[partial sums|0NB643K60Q]] is.

## Remarks

Example: for $p_k = 1$ for all $k$, one has $P_n = n + 1$ and $t_n = \frac{s_0 + \cdots + s_n}{n+1}$, the [[Cesàro mean|0P0DJHBGBB]].

The method is regular, that is, it sends every convergent sequence to its limit, if and only if $p_n / P_n \to 0$.

Non-example of regularity: for $p_k = 2^k$ one has $P_n = 2^{n+1} - 1$ and $p_n / P_n \to \frac12$.
For the sequence $s = (1, 0, 0, \dots)$, which converges to $0$, one gets $t_n = \frac{2^n}{2^{n+1} - 1} \to \frac12$.
So this Nörlund method is not regular.
The condition $p_0 > 0$ is needed so that every $P_n$ is positive and the means are defined.
</document>

<document source="https://tamnd.github.io/math2/node/4487CSC0MB.md" tag="4487CSC0MB" kind="concept">
# Domain of holomorphy

## Summary

A domain of holomorphy is an open set in $\C^n$ that is the natural domain of its [[holomorphic functions|507C34W0ZT]].
No part of its boundary can be crossed by all of these functions at once.
In one variable every open set has this property, but for $n \ge 2$ many open sets do not, by the Hartogs phenomenon.
Domains of holomorphy are characterised by holomorphic convexity and, by the solution of the Levi problem, by pseudoconvexity.
They are the domains in $\C^n$ on which the global theory of holomorphic functions works best.

## Explanations

### L1 human

Some regions in $\C^n$ have hidden extensions: every holomorphic function on them automatically extends further.
A ball with a small closed ball removed from the middle is such a region in $\C^2$, since every function fills in the hole.
A domain of holomorphy has no such hidden extension: for each boundary point there is a function that really breaks down there.
The open unit ball is a domain of holomorphy; near a boundary point $p$, the function $1/(\ip{z}{p} - 1)$ blows up, where $\ip{z}{p} = \sum_j z_j \bar p_j$.
</document>

<document source="https://tamnd.github.io/math2/node/44GBFZ16SG.md" tag="44GBFZ16SG" kind="concept">
# Analytic index

## Summary

The analytic index of an elliptic operator $P$ on a closed manifold is $\dim \ker P - \dim \operatorname{coker} P$.
Both dimensions are finite because elliptic operators on closed manifolds are [[Fredholm operators|8HWGKWD6ZM]] between suitable Sobolev spaces.
So the analytic index is a [[Fredholm index|2V9Q4KHY1T]].
The Atiyah–Singer index theorem computes it as the [[topological index|QDCD33CS21]].

## Explanations

### L1 human

The kernel of $P$ counts the independent solutions of $Pu = 0$.
The cokernel counts the independent conditions that $f$ must satisfy for $Pu = f$ to be solvable.
The analytic index is the difference of these two counts.
Take the Cauchy–Riemann operator $\bar\partial$ on functions on a closed Riemann surface of genus $g$.
Its kernel is the constants and its cokernel has dimension $g$, so the index is $1 - g$.
</document>

<document source="https://tamnd.github.io/math2/node/44HTG620GB.md" tag="44HTG620GB" kind="definition">
# Latin square

## Definition

Let $n$ be a positive integer and let $S$ be a [[finite set|0BBTH3YSZA]] with $n$ elements, the symbols.
A Latin square of order $n$ on $S$ is an $n \times n$ array $L = (L_{ij})$ with entries in $S$ such that:

1. In each row, each element of $S$ occurs exactly once.
2. In each column, each element of $S$ occurs exactly once.

## Remarks

Example: for $i, j \in \set{0, 1, \dots, n - 1}$, put $L_{ij} = (i + j) \bmod n$.
In row $i$ the map $j \mapsto i + j \bmod n$ is a bijection of $\set{0, \dots, n-1}$, so each symbol occurs once; the same holds for each column.
This square is the addition table of the [[integers modulo n|01N3AYAA6N]].
More generally, the Cayley table of each finite [[group|G5S1M9]] is a Latin square.
Non-example: the array with rows $1\,2$ and $1\,2$ is not a Latin square, because the symbol $1$ occurs twice in the first column.
A Latin square is the same thing as the multiplication table of a finite quasigroup.
Two Latin squares can be combined into a pair of [[orthogonal Latin squares|H6TN6BADVY]].
</document>

<document source="https://tamnd.github.io/math2/node/44KMEHYN1R.md" tag="44KMEHYN1R" kind="counterexample">
# Sorgenfrey plane

## Counterexample

The Sorgenfrey plane is the product $\R_\ell \times \R_\ell$ of two copies of the [[Sorgenfrey line|Y4T3GVK8GV]] with the [[product topology|0GBG8B49AA]].
Each factor is normal, but the Sorgenfrey plane is not [[normal|0NSR9V8TED]].

## Remarks

The basic open sets are the half-open rectangles $[a, b) \times [c, d)$.
The anti-diagonal $D = \set{(x, -x) : x \in \R}$ is [[closed|0ZCK8DQYN1]], since it is closed in the coarser Euclidean topology.
It is discrete: the rectangle $[x, x + 1) \times [-x, -x + 1)$ meets $D$ only in $(x, -x)$.
The plane is [[separable|0Z2P41SVDP]], because $\Q \times \Q$ is dense.
By Jones' lemma, a separable normal space has no closed discrete subset $D$ with $2^{\abs{D}} > 2^{\aleph_0}$.
Here $\abs{D} = 2^{\aleph_0}$, so the plane is not normal.
Concretely, the points of $D$ with rational first coordinate and those with irrational first coordinate form two disjoint closed sets that no disjoint open sets separate.
The plane also shows that a product of two [[Lindelöf|SW0SN1NEJE]] spaces need not be Lindelöf.

## Explanations

### L1 human

The Sorgenfrey line is a well-behaved space.
Multiplying it by itself gives a plane in which two disjoint closed sets cannot be pulled apart by open sets.
So normality is a property that does not survive products.
The trouble sits on the line $y = -x$, where the half-open rectangles touch the line in only one point each.
</document>

<document source="https://tamnd.github.io/math2/node/44RN97ZXDZ.md" tag="44RN97ZXDZ" kind="statement">
# Scheffé's lemma

## Statement

Let $(X, \Sigma, \mu)$ be a measure space.
Let $f_1, f_2, \dots$ and $f$ be real or complex [[integrable functions|X7GTF3D98F]] on $X$ with $f_n \to f$ [[almost everywhere|RARTMA0ZEC]].
Then
$$\lim_{n \to \infty} \int_X \abs{f_n - f} \, d\mu = 0 \quad \text{if and only if} \quad \lim_{n \to \infty} \int_X \abs{f_n} \, d\mu = \int_X \abs{f} \, d\mu.$$

## Remarks

Proof idea: the direct implication follows from $\abs{\int \abs{f_n} - \int \abs{f}} \le \int \abs{f_n - f}$.
For the converse, the functions $g_n = \abs{f_n} + \abs{f} - \abs{f_n - f}$ are non-negative and tend to $2\abs{f}$ almost everywhere.
[[Fatou's lemma|S95TR54TCG]] gives $2 \int \abs{f} \le 2 \int \abs{f} - \limsup_n \int \abs{f_n - f}$, and $\int \abs{f}$ is finite.

Example: let $p_n$ and $p$ be probability densities on $\R$ with $p_n \to p$ almost everywhere.
Then $\int p_n = \int p = 1$, so $\int \abs{p_n - p} \to 0$.
The distributions with densities $p_n$ then converge to the one with density $p$ in total variation.

The convergence of the norms is needed.
For $f_n = n \mathbf{1}_{(0, 1/n)}$ on $\R$, $f_n \to 0$ everywhere but $\int \abs{f_n - 0} = 1$.

## Explanations

### L1 human

Suppose functions converge at almost every point, and their total sizes converge to the total size of the limit.
Then the area between their graphs and the graph of the limit goes to zero.
No mass can escape, because the total mass is kept.

### L2 human

Scheffé's lemma is a quick criterion for convergence in $L^1$.
Its main use is in probability and statistics: pointwise convergence of densities implies convergence of the distributions in total variation, hence in distribution.
It is close in spirit to the Vitali convergence theorem, but it holds on any measure space and its proof is a direct application of Fatou's lemma.
</document>

<document source="https://tamnd.github.io/math2/node/44WEBQSKEY.md" tag="44WEBQSKEY" kind="example">
# Pseudosphere

## Example

For $t > 0$ and $\theta \in \R$, let
$$x(t, \theta) = \left(\operatorname{sech} t \cos\theta, \; \operatorname{sech} t \sin\theta, \; t - \tanh t\right).$$
This is the [[surface of revolution|NE09EDK19V]] of the tractrix $t \mapsto (\operatorname{sech} t, \, t - \tanh t)$ about the $z$-axis, and its image is called the pseudosphere.
Its [[Gaussian curvature|SQVQ5DQT6W]] is $-1$ at every point.

Check.
Write $r(t) = \operatorname{sech} t$ and $z(t) = t - \tanh t$.
Then $r'(t) = -\operatorname{sech} t \tanh t$ and $z'(t) = \tanh^2 t$, so $r'^2 + z'^2 = \tanh^2 t \,(\operatorname{sech}^2 t + \tanh^2 t) = \tanh^2 t$.
For $t > 0$ the speed is $\tanh t > 0$, and the arc length from $t = 0$ is $s = \ln\cosh t$.
So in terms of arc length, $r = 1/\cosh t = e^{-s}$.
For a surface of revolution with profile parametrized by arc length, $K = -r''(s)/r(s)$, and here $K = -e^{-s}/e^{-s} = -1$.

## Remarks

The surface is not complete: as $t \to 0$ it ends in a sharp circular edge of radius $1$, where the parametrization stops being regular.
This agrees with [[Hilbert's theorem|XM772FD0BS]], which says that no complete surface of constant curvature $-1$ sits smoothly and isometrically in $\R^3$.
Pieces of the pseudosphere are locally isometric to pieces of the hyperbolic plane, so they were used as early models of [[hyperbolic geometry|GN4N94GQ0Z]].

## Explanations

### L1 human

The pseudosphere looks like a trumpet: a long thin horn that widens to a sharp rim.
A sphere has the same positive curvature everywhere; the pseudosphere has the same negative curvature everywhere.
At each point it bends like a saddle, curving up in one direction and down in the other.
</document>

<document source="https://tamnd.github.io/math2/node/4503R8BVF8.md" tag="4503R8BVF8" kind="concept">
# M-estimator

## Summary

An M-estimator is an estimator defined as the minimizer of a sum of a loss function over the observations.
Least squares, least absolute deviations and maximum likelihood are all M-estimators.
The class was introduced in robust statistics, where the loss is chosen to limit the influence of outliers.
A single general theory gives consistency and asymptotic normality for the whole class.

## Explanations

### L1 human

Choose a way to measure how badly a candidate value $\theta$ fits one observation, for example the squared distance $(x - \theta)^2$.
Add up these penalties over all observations and take the $\theta$ with the smallest total.
With squared distance you get the sample mean; with absolute distance $\abs{x - \theta}$ you get the sample median.
Other penalties, which grow more slowly for far-away points, give estimates that are less disturbed by outliers.
</document>

<document source="https://tamnd.github.io/math2/node/456E0RCQR1.md" tag="456E0RCQR1" kind="concept">
# Digital topology

## Summary

Digital topology studies notions such as connectedness, boundaries and curves for sets of points of the integer lattice $\Z^2$ or $\Z^3$.
Such sets model the pixels or voxels of digital images.
Its basic tools are adjacency relations between lattice points and the paths and connected components they define.
One also uses genuine topologies on $\Z^n$, such as the Khalimsky topology.
These notions underlie image processing tasks such as counting objects, tracing borders and thinning shapes.

## Explanations

### L1 human

A black-and-white picture on a screen is a set of black squares on a grid.
To say that a shape in the picture is "in one piece" or "has a hole", one must decide when two squares touch.
They may touch only along an edge, or also at a corner.
For example, two black squares that meet only at a corner form one piece under the corner rule but two pieces under the edge rule.
Digital topology makes such choices precise and checks that familiar facts, such as "a closed curve separates the inside from the outside", still hold.
</document>

<document source="https://tamnd.github.io/math2/node/457T9T7Y75.md" tag="457T9T7Y75" kind="concept">
# Day convolution

## Summary

Day convolution is a tensor product on the category of set-valued functors on a small [[monoidal category|M3MSQ3NKAE]].
It is built from the tensor product of the source category.
It makes the [[functor category|1X0WKFQX8K]] into a closed monoidal category.
The Yoneda embedding becomes a strong monoidal functor.
So Day convolution is the universal way to extend a monoidal structure from a category to its category of set-valued functors.
It generalizes the convolution of functions on a group and the Cauchy product of power series.

## Explanations

### L1 human

Think of a functor $F$ from a monoidal category to sets as a family of sets $F(a)$, one for each object $a$.
To combine two such families at an object $c$, one collects all pairs of elements $x \in F(a)$, $y \in G(b)$ together with a way to map $a \otimes b$ to $c$.
For example, if the objects are the natural numbers with $+$ and only identity arrows, a functor is a graded set.
The Day convolution of two graded sets has in degree $n$ the pairs of elements whose degrees add up to $n$, exactly like multiplying power series.
</document>

<document source="https://tamnd.github.io/math2/node/45P0QAMAFW.md" tag="45P0QAMAFW" kind="definition">
# Wright omega function

## Definition

Let $W_k$, for $k \in \Z$, be the branches of the [[Lambert W function|FW2JKAN14B]] over $\C$.
For $z \in \C$, let $K(z) = \ceil{\frac{\operatorname{Im} z - \pi}{2\pi}}$ be the unwinding number.
The *Wright omega function* is
$$\omega(z) = W_{K(z)}(e^z).$$

## Remarks

Example: for real $x$, $\operatorname{Im} x = 0$ and $K(x) = \ceil{-\tfrac12} = 0$, so $\omega(x) = W_0(e^x)$.
It is the unique real $y > 0$ with $y + \ln y = x$, since $y e^y = e^x$ is the same equation and $y \mapsto y + \ln y$ is a strictly increasing bijection from $(0, \infty)$ onto $\R$.
For instance $\omega(1) = 1$, and $\omega(0) = W_0(1) \approx 0.5671$, the omega constant.

Non-example: $W_0(e^z)$ is not $\omega(z)$ in general.
For $z = 4\pi i$, $K(z) = \ceil{3/2} = 2$, so $\omega(4\pi i) = W_2(1)$, which is not real, while $W_0(e^{4\pi i}) = W_0(1)$ is real.

The function $\omega$ is single-valued.
With the principal branch of the logarithm, $\omega(z) + \ln \omega(z) = z$ holds for every $z$ except possibly on the two half-lines $\set{t \pm i\pi : t \le -1}$.
</document>

<document source="https://tamnd.github.io/math2/node/45P2SEGGR1.md" tag="45P2SEGGR1" kind="statement">
# Minakshisundaram–Pleijel asymptotic expansion

## Statement

Let $(M, g)$ be a closed [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$, and let $K(t, x, y)$ be its [[heat kernel|327NP6S7H5]].
Then there are smooth functions $a_0, a_1, a_2, \ldots$ on $M$ such that for every integer $N \ge 0$
$$K(t, x, x) = (4\pi t)^{-n/2} \left( \sum_{k = 0}^{N} a_k(x) \, t^k + O(t^{N+1}) \right) \quad \text{as } t \to 0^+,$$
uniformly in $x \in M$.
Each $a_k(x)$ is a universal polynomial in the curvature tensor and its covariant derivatives at $x$.
The first two are $a_0 = 1$ and $a_1 = \tfrac16 S$, where $S$ is the [[scalar curvature|ZHX2VYQDZ3]].
Integrating over $M$, the heat trace satisfies
$$\sum_{j \ge 0} e^{-t\lambda_j} \sim (4\pi t)^{-n/2} \sum_{k \ge 0} t^k \int_M a_k \, dv,$$
where $\lambda_0 \le \lambda_1 \le \cdots$ is the [[spectrum of the Laplace–Beltrami operator|B8TYKV7AA1]].

## Remarks

Idea of the proof: build an approximate heat kernel near the diagonal as the Euclidean Gaussian in normal coordinates times a power series $\sum u_k(x, y) t^k$.
The coefficients $u_k$ solve transport equations along geodesics, and $a_k(x) = u_k(x, x)$.
A Duhamel argument shows that the true heat kernel differs from the truncated approximation by $O(t^{N + 1 - n/2})$.
Consequences: the spectrum determines the dimension $n$, the volume $\int_M a_0 = \operatorname{vol}(M)$ and the total scalar curvature $\int_M S$.
The next integrated coefficient is $\int_M a_2 = \frac{1}{360} \int_M \left( 2\abs{\mathrm{Rm}}^2 - 2\abs{\mathrm{Ric}}^2 + 5 S^2 \right) dv$.
Example: on a flat torus $\R^n / \Gamma$, $K(t, x, x) = (4\pi t)^{-n/2} \sum_{\gamma \in \Gamma} e^{-\abs{\gamma}^2 / 4t}$, so $a_0 = 1$ and $a_k = 0$ for $k \ge 1$, with an exponentially small error.

## Explanations

### L1 human

For a very short time, heat spreading from a point only feels a tiny neighbourhood, which looks almost flat.
So the heat kernel at the point is close to the flat-space value $(4\pi t)^{-n/2}$.
The corrections come in powers of $t$, and each coefficient is computed from the curvature at the point.

### L2 human

Minakshisundaram and Pleijel used the expansion to continue the spectral zeta function of the Laplacian meromorphically.
It shows that the spectrum determines the volume and the total scalar curvature.
This answers some spectral rigidity questions.
For example, a closed surface with the spectrum of the round sphere is a round sphere.
The same construction works for Laplace-type operators on vector bundles, and the coefficients there are the Seeley–DeWitt coefficients.
Combined with the McKean–Singer formula it leads to the heat equation proof of the index theorem.
</document>

<document source="https://tamnd.github.io/math2/node/45V0W838BQ.md" tag="45V0W838BQ" kind="concept">
# Renewal theory

## Summary

Renewal theory studies processes that start afresh at random times, with independent and identically distributed waiting times between two restarts.
Its central object is the renewal process, which counts the restarts up to time $t$.
Its main results, such as the renewal theorems, describe the long-run rate of restarts.
It applies to the replacement of machine parts, to queues and to returns of a Markov chain to a state.

## Explanations

### L1 human

A light bulb burns out at a random time and is replaced at once by a new bulb of the same kind.
Each replacement is a renewal: the system starts over as if new.
Renewal theory asks how many bulbs are used by time $t$.
If a bulb lasts $\mu$ hours on average, then in the long run one uses about $t / \mu$ bulbs in $t$ hours.
The Poisson process is the renewal process with exponential waiting times.
</document>

<document source="https://tamnd.github.io/math2/node/45VSJ0RG2S.md" tag="45VSJ0RG2S" kind="definition">
# Hodge star operator

## Definition

Let $V$ be a real vector space of dimension $n$ with an [[inner product|0NB0AWGJBW]] $\ip{\cdot}{\cdot}$ and an [[orientation|0QJJAQ6ZYZ]].
Let $\operatorname{vol} = e_1 \wedge \dots \wedge e_n$ for any positively oriented orthonormal basis $e_1, \dots, e_n$.
On the [[exterior power|0MDMZ965VE]] $\Lambda^k V$ use the inner product with $\ip{v_1 \wedge \dots \wedge v_k}{w_1 \wedge \dots \wedge w_k} = \det\bigl(\ip{v_i}{w_j}\bigr)$.
The *Hodge star operator* is the unique linear map $\star \colon \Lambda^k V \to \Lambda^{n-k} V$ such that
$$\alpha \wedge \star\beta = \ip{\alpha}{\beta} \operatorname{vol} \quad \text{for all } \alpha, \beta \in \Lambda^k V.$$
On an oriented [[Riemannian manifold|7CBT4SM83Y]] $(M, g)$, the Hodge star is applied pointwise to [[differential forms|0E3G0GRPZ6]], with $V = T_p^* M$ and the inner product induced by $g$.

## Remarks

Example: in $\R^3$ with the standard orientation, $\star dx = dy \wedge dz$, $\star dy = dz \wedge dx$ and $\star dz = dx \wedge dy$.
Check for $dx$: $dx \wedge \star dx = dx \wedge dy \wedge dz = \ip{dx}{dx} \operatorname{vol}$, and $dy \wedge \star dx = 0 = \ip{dy}{dx} \operatorname{vol}$, and the same for $dz$.

Non-example: in $\R^2$ the map with $dx \mapsto -dy$ and $dy \mapsto dx$ is not the Hodge star, since $dx \wedge (-dy) = -\operatorname{vol}$, while $\ip{dx}{dx} \operatorname{vol} = \operatorname{vol}$.
The Hodge star of $\R^2$ is $\star dx = dy$, $\star dy = -dx$, a rotation by a right angle.

On orthonormal bases, $\star(e_{i_1} \wedge \dots \wedge e_{i_k}) = \pm e_{j_1} \wedge \dots \wedge e_{j_{n-k}}$, where the $j$'s are the complementary indices and the sign is that of the permutation $(i_1, \dots, i_k, j_1, \dots, j_{n-k})$.
It satisfies $\star 1 = \operatorname{vol}$ and $\star\star = (-1)^{k(n-k)}$ on $\Lambda^k V$, and it is an isometry.
Reversing the orientation changes the sign of $\star$.
</document>

<document source="https://tamnd.github.io/math2/node/4628Z96WVZ.md" tag="4628Z96WVZ" kind="conjecture">
# Union-closed sets conjecture

## Statement

Let $\mathcal{F}$ be a finite [[family|0C21Q694AG]] of finite sets that contains at least one nonempty set.
Suppose that $\mathcal{F}$ is union-closed: $A \cup B \in \mathcal{F}$ for all $A, B \in \mathcal{F}$, where $\cup$ is the [[union|0X1GSWNBN1]].
Then there is an element $x$ that belongs to at least $|\mathcal{F}| / 2$ of the sets in $\mathcal{F}$.

## Remarks

The conjecture is attributed to Péter Frankl and is also called Frankl's conjecture.
It is open.
In 2022 Justin Gilmer proved that some element lies in at least $0.01 \, |\mathcal{F}|$ of the sets, the first bound by a positive constant fraction.
Several authors then improved the constant to about $0.38$.
The conjecture is known when $\mathcal{F}$ contains a set with one or two elements: such an element, or one of the two elements, works.

Example: $\mathcal{F} = \set{\emptyset, \set{1}, \set{2}, \set{1, 2}}$ is union-closed, and the element $1$ lies in $2 = |\mathcal{F}|/2$ of its sets, so the bound $1/2$ cannot be improved.
The nonempty hypothesis is needed: $\mathcal{F} = \set{\emptyset}$ is union-closed, but no element lies in any of its sets.

## Explanations

### L1 human

Take a collection of sets in which the union of any two sets of the collection is again in the collection.
The conjecture says that some single element appears in at least half of the sets.
It is very easy to state, yet it resisted all attempts for decades, and only recently was a constant fraction proved.
</document>

<document source="https://tamnd.github.io/math2/node/462AEY7MWF.md" tag="462AEY7MWF" kind="statement">
# Regula falsi

## Statement

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a [[continuous function|0X3WGX95FE]] with $f(a) f(b) < 0$.
The *regula falsi* (method of false position) sets $a_0 = a$, $b_0 = b$, and for $k \ge 0$ lets
$$c_k = \frac{a_k f(b_k) - b_k f(a_k)}{f(b_k) - f(a_k)},$$
the zero of the chord through $(a_k, f(a_k))$ and $(b_k, f(b_k))$.
If $f(c_k) = 0$ it stops; if $f(a_k) f(c_k) < 0$ it sets $(a_{k+1}, b_{k+1}) = (a_k, c_k)$; otherwise it sets $(a_{k+1}, b_{k+1}) = (c_k, b_k)$.

1. As long as the method runs, $a_k < c_k < b_k$ and $f(a_k) f(b_k) < 0$, so each $[a_k, b_k]$ contains a zero of $f$.
2. Assume moreover that $f$ is [[twice continuously differentiable|0VW90BBSJ4]] on $[a, b]$ with [[second derivative|0Q4QJWEHW3]] $f'' > 0$, and $f(a) < 0 < f(b)$. Then $f$ has exactly one zero $r$ in $(a, b)$, $b_k = b$ for all $k$, the sequence $(c_k)$ increases strictly to $r$, and
$$\lim_{k \to \infty} \frac{r - c_{k+1}}{r - c_k} = 1 - \frac{f'(r)(b - r)}{f(b)} \in (0, 1).$$

## Remarks

Clause 1: the chord has values of opposite signs at $a_k$ and $b_k$, so its zero $c_k$ lies strictly between them.
The update keeps a sign change, and the [[intermediate value theorem|00KAFBR32E]] gives a zero in each bracket.
Clause 2: by convexity the chord lies above the graph, so $f(c_k) < 0$ and $c_k$ always replaces the left endpoint.
Then $c_{k+1} = g(c_k)$ with $g(x) = x - f(x)(b - x)/(f(b) - f(x))$.
The increasing bounded sequence $(c_k)$ has a limit, which is a fixed point of $g$, hence $r$, and the error ratio tends to $g'(r)$.
So under the hypotheses of clause 2 the method converges only linearly, see [[rate of convergence|YWZQC5E54B]], and the bracket width $b - c_k$ does not tend to $0$.
Example: for $f(x) = x^2 - 2$ on $[1, 2]$, $c_0 = 4/3$, $c_1 = 7/5$, $c_2 = 24/17 \approx 1.41176$, increasing to $\sqrt{2}$ with $b = 2$ fixed, and the limit ratio is $1 - 2\sqrt{2}(2 - \sqrt{2})/2 = 3 - 2\sqrt{2} \approx 0.17$.
The other cases of fixed sign of $f''$ and of $f(a)$ follow by replacing $f(x)$ by $-f(x)$ or by $f(-x)$; then the other endpoint stays fixed.

## Explanations

### L1 human

Like bisection, regula falsi keeps an interval on which the function changes sign.
Instead of cutting the interval in the middle, it draws the straight line between the two end values and cuts where that line crosses zero.
This is often a much better guess than the midpoint.
But when the graph bends the same way throughout, one endpoint never moves and progress can become slow.

### L2 human

Regula falsi is a bracketing [[root-finding algorithm|TR4CNS5PMJ]] like the bisection method, and it uses the same chord formula as the secant method.
The secant method uses the two most recent points instead of the two bracketing ones; it loses the guarantee of a bracket but converges superlinearly near a simple zero.
The fixed-endpoint problem of clause 2 led to modified versions such as the Illinois algorithm.
It halves the stored function value at an endpoint that stays fixed twice in a row.
</document>

<document source="https://tamnd.github.io/math2/node/463X5V31D9.md" tag="463X5V31D9" kind="concept">
# Hypergeometric function

## Summary

The Gauss hypergeometric function ${}_2F_1(a, b; c; z)$ is the sum of the [[hypergeometric series|428M453CR2]] on the unit disc.
It is extended by [[analytic continuation|0M3YQJ81NB]] to the cut plane $\C \setminus [1, \infty)$.
It solves the [[hypergeometric differential equation|W8VE25R5A7]], to which every second-order linear equation with three regular singular points can be reduced.
Most classical special functions, such as Legendre functions, complete elliptic integrals, Jacobi polynomials and many elementary functions, are special cases or limits of it.

## Explanations

### L1 human

The hypergeometric function is a single formula with three parameters $a, b, c$ that produces a large family of familiar functions.
For different choices of the parameters it gives powers $(1-z)^{-a}$, logarithms, inverse sines, and orthogonal polynomials.
The power series only converges for $\abs{z} < 1$, but the function can be extended to all complex $z$ except a cut along $[1, \infty)$.
For example, ${}_2F_1(\tfrac12, \tfrac12; \tfrac32; z^2) = \frac{\arcsin z}{z}$ for $0 < \abs{z} < 1$.
Because so many functions are special cases, a single identity for ${}_2F_1$ often gives many classical identities at once.
</document>

<document source="https://tamnd.github.io/math2/node/46436KE7NT.md" tag="46436KE7NT" kind="concept">
# Excessive function

## Summary

An excessive function of a Markov process is a nonnegative function that decreases on average along the paths of the process.
It is also recovered as the limit of its averages at small times.
Excessive functions are the probabilistic counterpart of nonnegative superharmonic functions.
For Brownian motion the two classes coincide.
For general Markov processes, excessive functions play the role of superharmonic functions in an abstract potential theory.

## Explanations

### L1 human

Imagine a quantity attached to each position, and a particle moving at random.
The function is excessive if its expected value at the particle's later position never exceeds its value at the starting position.
Constants are excessive, and so is the expected total time that the particle will spend in a given region.
For Brownian motion in space, $1/\abs{x}$ is excessive, matching the fact that it is superharmonic.
</document>

<document source="https://tamnd.github.io/math2/node/464NN2K4FZ.md" tag="464NN2K4FZ" kind="concept">
# Equilibrium measure

## Summary

The equilibrium measure of a compact set $K$ is the way a unit charge distributes itself on $K$ when it minimizes its energy.
In the plane the energy is the logarithmic energy, and in $\R^n$ with $n \ge 3$ it is the Newtonian energy.
When $K$ has positive [[capacity|DBQXRTC0VN]], the minimizing measure exists and is unique.
Its potential is constant on most of $K$, and this constant determines the [[capacity of the set|4YZVFGJVSZ]].

## Explanations

### L1 human

Put one unit of electric charge on a metal body $K$ and let it move freely.
Like charges repel, so the charge spreads out until the electrostatic energy is as small as possible.
The final distribution is the equilibrium measure, and the body is then at one constant voltage.
On the unit circle the equilibrium measure is the uniform measure, with total mass $1$ spread evenly along the circle.
On the closed unit disc the charge also sits on the boundary circle and leaves the interior empty.
</document>

<document source="https://tamnd.github.io/math2/node/464TFVZ9JA.md" tag="464TFVZ9JA" kind="definition">
# Block system

## Definition

Let a group $G$ [[act|0WCY42EV9V]] on a set $\Omega$.
A *block* of $G$ is a nonempty subset $B \subseteq \Omega$ such that, for every $g \in G$, either $gB = B$ or $gB \cap B = \emptyset$.
A *block system* of $G$ is a partition $\mathcal{P}$ of $\Omega$ such that $gB \in \mathcal{P}$ for every $g \in G$ and every part $B \in \mathcal{P}$.

## Remarks

Each part of a block system is a block.
Conversely, if $G$ is [[transitive|02GK6HZYBT]] and $B$ is a block, then $\set{gB : g \in G}$ is a block system.
For transitive $G$ all parts of a block system have the same size, so for finite $\Omega$ this size divides $\abs{\Omega}$.
The partition into singletons and the partition $\set{\Omega}$ are the *trivial* block systems.
A transitive action whose only block systems are trivial is [[primitive|Y10S0751KG]].

Example: the eight symmetries of a square acting on the corners $1, 2, 3, 4$ in cyclic order, and $\mathcal{P} = \set{\set{1, 3}, \set{2, 4}}$.
Each symmetry maps a pair of opposite corners to a pair of opposite corners, so it maps each part of $\mathcal{P}$ to a part of $\mathcal{P}$.

Non-example: for the same action, $\mathcal{P}' = \set{\set{1, 2}, \set{3, 4}}$.
The quarter turn $(1\,2\,3\,4)$ maps $\set{1, 2}$ to $\set{2, 3}$, which is not a part of $\mathcal{P}'$.

If $G$ is transitive and $\omega \in \Omega$, the blocks containing $\omega$ correspond to the subgroups $H$ with $G_\omega \le H \le G$, where $G_\omega$ is the [[stabiliser|0AYMP41THN]]: $H$ gives the block $H\omega$.
The [[orbits|0YQPZB5Q1G]] of a [[normal subgroup|0CB0DX5BVV]] of $G$ form a block system.
</document>

<document source="https://tamnd.github.io/math2/node/4670W2WNFX.md" tag="4670W2WNFX" kind="definition">
# Series in Banach spaces

## Definition

Let $X$ be a [[Banach space|0BDTTM24DP]] over $\R$ or $\C$ with norm $\norm{\cdot}$, and let $(x_n)_{n \ge 0}$ be a sequence in $X$.
The [[partial sums|0NB643K60Q]] of the series $\sum_n x_n$ are $S_N = \sum_{n=0}^{N} x_n$.
The series *converges* to $s \in X$, written $\sum_{n \ge 0} x_n = s$, if $\norm{S_N - s} \to 0$ as $N \to \infty$.
The series *converges absolutely* if the real series $\sum_{n \ge 0} \norm{x_n}$ [[converges|0CVVKY7GY2]].

## Remarks

[[An absolutely convergent series in a Banach space converges|082MGNWNA8]].
More generally, if $\norm{x_n} \le b_n$ for all $n$ and $\sum b_n$ converges, then $\sum x_n$ converges; in Mathlib this is `Summable.of_norm_bounded`.
A normed space in which every absolutely convergent series converges is complete, so this property characterizes Banach spaces.

Example: in $\ell^2$, see [[spaces ℓ^p|08P6T2WH6R]], let $x_n = e_n / n$ for $n \ge 1$, where $e_n$ is the $n$-th unit vector.
Then $\norm{S_N - s}_2 = \left( \sum_{n > N} 1/n^2 \right)^{1/2} \to 0$ for $s = (1, 1/2, 1/3, \dots)$, so the series converges.

Non-example: the same series does not converge absolutely, because $\sum_n \norm{x_n}_2 = \sum_n 1/n$ diverges.
This series even converges after every rearrangement.
In $\R$ this cannot happen: a real series that converges after every rearrangement converges absolutely.
</document>

<document source="https://tamnd.github.io/math2/node/468EN8KVPJ.md" tag="468EN8KVPJ" kind="definition">
# Injective algebra

## Definition

Fix a signature and let $\mathcal{K}$ be a class of algebras of this signature.
An algebra $A \in \mathcal{K}$ is *injective* in $\mathcal{K}$ if the following holds.
For all $B, C \in \mathcal{K}$, every [[embedding|SCRB74SWQN]] $g \colon B \to C$ and every [[homomorphism|KBXSZC6BV1]] $f \colon B \to A$, there is a homomorphism $h \colon C \to A$ with $h \circ g = f$.

## Remarks

When $\mathcal{K}$ is a variety, the embeddings are exactly the monomorphisms of the category of algebras of $\mathcal{K}$.
So in a variety, $A$ is injective in this sense exactly when it is an injective object of that category.

Example: $\Q$ is injective in the variety of abelian groups.
An abelian group is injective there exactly when it is divisible, and $\Q$ is divisible since $x = n \cdot (x/n)$ for every $x \in \Q$ and $n \ge 1$.
In the variety of modules over a ring, the injective algebras are the [[injective modules|85H7PRHDPZ]].

Non-example: $\Z$ is not injective in the variety of abelian groups.
Let $g \colon \Z \to \Q$ be the inclusion and $f = \id_{\Z}$.
A homomorphism $h \colon \Q \to \Z$ with $h(1) = 1$ would give $2 h(1/2) = 1$ in $\Z$, which is impossible.

In many varieties, such as groups, only the trivial algebra is injective.
</document>

<document source="https://tamnd.github.io/math2/node/4695YA7F87.md" tag="4695YA7F87" kind="definition">
# Collineation

## Definition

Let $S$ and $S'$ be [[projective geometries|N32Z68F0D3]], with point sets $P$ and $P'$.
A *collineation* from $S$ to $S'$ is a bijection $f \colon P \to P'$ with the following property.
For all points $a, b, c$ of $P$, the points $a, b, c$ are [[collinear|0AFNNTRTJE]] exactly when $f(a), f(b), f(c)$ are collinear.
A collineation from $S$ to itself is a collineation of $S$.

## Remarks

Equivalently, $f$ is a bijection on points such that the image of every line is a line and every line of $S'$ is such an image.
The same definition applies to affine spaces and to other incidence structures whose lines are sets of points.

Example: every [[projective transformation|P1MKTF7MWQ]] $[v] \mapsto [Tv]$ of $\mathbb{P}(V)$ is a collineation.
Check: the line through $[u]$ and $[w]$ is $\mathbb{P}(W)$ for the plane $W$ spanned by $u, w$.
The map sends it onto $\mathbb{P}(T W)$, again a line, and the inverse map comes from $T^{-1}$.

Non-example: in the Fano plane $\mathbb{P}^2(\F_2)$, whose lines are the sets $\set{a, b, a + b}$ of nonzero vectors of $\F_2^3$, swap $e_1$ and $e_2$ and fix all other points.
The line $\set{e_1, e_3, e_1 + e_3}$ goes to $\set{e_2, e_3, e_1 + e_3}$, which is not a line, because $e_2 + e_3 \ne e_1 + e_3$.

The collineations of $S$ form a group under composition, the collineation group of $S$.
</document>

<document source="https://tamnd.github.io/math2/node/46ESFR9MSE.md" tag="46ESFR9MSE" kind="definition">
# Ample line bundle

## Definition

Let $X$ be a [[Noetherian scheme|ND6XEY68P4]] and let $\mathcal{L}$ be an [[invertible sheaf|7JCF2TE3V5]] on $X$.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
It is *globally generated* if for every $x \in X$ the images of $H^0(X, \mathcal{F})$ generate the stalk $\mathcal{F}_x$ as an $\mathcal{O}_{X,x}$-module.
The sheaf $\mathcal{L}$ is *ample* if the following holds for every [[coherent sheaf|56RXEJ8Z6S]] $\mathcal{F}$ on $X$.
There is an integer $n_0$ such that $\mathcal{F} \otimes \mathcal{L}^{\otimes n}$ is globally generated for all $n \ge n_0$.

## Remarks

If $X$ is of finite type over a Noetherian ring $A$, then $\mathcal{L}$ is ample if and only if $\mathcal{L}^{\otimes m}$ is very ample over $\operatorname{Spec} A$ for some $m \ge 1$.
Here very ample means $\mathcal{L}^{\otimes m} \cong i^*\mathcal{O}(1)$ for an immersion $i \colon X \to \mathbb{P}^N_A$.

Example: on $X = \mathbb{P}^n_k$ the [[twisting sheaf|V93Y0XYPZD]] $\mathcal{O}(1)$ is ample.
By a theorem of Serre, every coherent $\mathcal{F}$ on $\mathbb{P}^n_k$ has $\mathcal{F}(n)$ globally generated for $n$ large.

Non-example: on $\mathbb{P}^1_k$, $\mathcal{L} = \mathcal{O}(-1)$ is not ample.
Take $\mathcal{F} = \mathcal{O}_X$: then $\mathcal{L}^{\otimes n} = \mathcal{O}(-n)$ has no nonzero global sections for $n \ge 1$, so it is not globally generated.

Ampleness is preserved by positive tensor powers, and $\mathcal{L}$ is ample if and only if $\mathcal{L}^{\otimes m}$ is ample for one $m \ge 1$.
</document>

<document source="https://tamnd.github.io/math2/node/46JCQ6KDA4.md" tag="46JCQ6KDA4" kind="concept">
# Formalism

## Summary

Formalism is the view that mathematics is about formal systems: strings of symbols and the rules that transform them.
On this view, a theorem is a string derived from the axioms by the rules.
It need not describe any abstract objects.
David Hilbert is the name most linked with formalism, through [[Hilbert's program|P1RW6JWX73]].
It is one of the classical answers in the [[foundations of mathematics|EFFRJFW67Y]].

## Explanations

### L1 human

Think of mathematics as a game like chess, played with symbols instead of pieces.
The axioms are the starting position and the rules of inference are the legal moves.
For example, in a formal system for arithmetic the string $0 + 0 = 0$ is a theorem because a sequence of legal moves leads to it.
A strict formalist says that this is all there is to it, and that the question "is it really true?" does not arise.
</document>

<document source="https://tamnd.github.io/math2/node/46MFCQQ9VW.md" tag="46MFCQQ9VW" kind="object">
# Sinogram

## Description

Let $f$ be a function on $\R^2$ with [[Radon transform|VJNZJ352PC]] $R f$.
Parametrize directions by an angle, $\theta(\varphi) = (\cos\varphi, \sin\varphi)$.
The *sinogram* of $f$ is the function $(\varphi, s) \mapsto R f(\theta(\varphi), s)$ on $[0, \pi) \times \R$.
It is drawn as an image with the angle $\varphi$ on one axis and the offset $s$ on the other.
The range $[0, \pi)$ suffices, because $R f(-\theta, -s) = R f(\theta, s)$.

Check of the name: let $f$ be concentrated near a single point $p = (\abs{p}\cos\alpha, \abs{p}\sin\alpha)$, a narrow bump of total mass $1$.
A line $x \cdot \theta(\varphi) = s$ meets the bump only when $s$ is close to $p \cdot \theta(\varphi) = \abs{p} \cos(\varphi - \alpha)$.
So the sinogram is concentrated near the curve $s = \abs{p} \cos(\varphi - \alpha)$, a sinusoid in $\varphi$.

Second check: take the indicator of a disk of radius $a$ centered at $0$.
Its sinogram is $2\sqrt{a^2 - s^2}$ for $\abs{s} \le a$ and $0$ otherwise, independent of $\varphi$: a band of constant profile.

## Explanations

### L1 human

A CT scanner measures, for each angle of the X-ray source, a profile of absorption values along the detector.
Stacking these profiles row by row, one row per angle, gives an image called the sinogram.
Each small dense spot in the body traces a sine wave across this image, because its shadow moves back and forth on the detector as the source turns.
A whole body gives a superposition of many such sine waves, and reconstruction means turning the sinogram back into the picture of the body.
</document>

<document source="https://tamnd.github.io/math2/node/46MSX1FXZT.md" tag="46MSX1FXZT" kind="concept">
# Convenient vector space

## Summary

A convenient vector space is a [[locally convex space|B6GY2KJMWW]] that is complete in a weak sense: every Mackey–Cauchy sequence converges.
These spaces are the setting of the convenient calculus of Frölicher, Kriegl and Michor.
In it, a map is smooth when it sends smooth curves to smooth curves.
The category of convenient spaces and smooth maps is cartesian closed.

This makes spaces of smooth maps, and manifolds of mappings, easy to handle without Banach-space estimates.

## Explanations

### L1 human

Calculus beyond Banach spaces has a problem: there are many inequivalent notions of a derivative.
The convenient approach tests smoothness only along curves: a map is smooth if every smooth path is sent to a smooth path.
For this to work, the space must contain enough limits, at least those that integrating a smooth curve produces.
Convenient vector spaces are the spaces with exactly this much completeness.
Every Banach space and every [[Fréchet space|3GAA614673]] is convenient.
</document>

<document source="https://tamnd.github.io/math2/node/46NHWJHW44.md" tag="46NHWJHW44" kind="concept">
# Logarithmic potential

## Summary

The logarithmic potential of a finite measure $\mu$ with compact support in the plane is $p_\mu(z) = \int \log(1/\abs{z - w}) \, d\mu(w)$.
It is the [[potential of the measure|JHKSF3XFY2]] $\mu$ with respect to the [[logarithmic kernel|VDZ5FKYC2T]].
It is harmonic off the support of $\mu$ and superharmonic on the whole plane.
Logarithmic potentials connect potential theory with complex analysis.
For a polynomial $p$ with zeros $z_1, \dots, z_d$, $\log(1/\abs{p})$ is, up to a constant, the logarithmic potential of the point masses at the zeros.

## Explanations

### L1 human

In the plane a point charge creates a potential that changes like the logarithm of the distance.
Spread charge along a curve or over a region, and add up the logarithmic contributions: this is the logarithmic potential.
For one unit charge at the origin it is $\log(1/\abs{z})$.
For a monic polynomial $p(z) = (z - z_1)\cdots(z - z_d)$, the potential of unit charges at the roots is $-\log\abs{p(z)}$.
</document>

<document source="https://tamnd.github.io/math2/node/46WGVM5VS6.md" tag="46WGVM5VS6" kind="statement">
# Jordan–von Neumann theorem

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $(V, \norm{\cdot})$ be a [[normed vector space|03J5RAKB30]] over $\mathbb{K}$.
Then the following are equivalent.

1. There is an inner product $\ip{\cdot}{\cdot}$ on $V$ whose [[induced norm|0MFJJV97RJ]] is $\norm{\cdot}$, that is, $\norm{x}^2 = \ip{x}{x}$ for all $x \in V$.
2. The norm satisfies the parallelogram law: $\norm{x + y}^2 + \norm{x - y}^2 = 2 \norm{x}^2 + 2 \norm{y}^2$ for all $x, y \in V$.

When these hold, the inner product is unique and is given by the polarization identity.
For $\mathbb{K} = \R$ it is
$$\ip{x}{y} = \frac{1}{4} \left( \norm{x + y}^2 - \norm{x - y}^2 \right),$$
and for $\mathbb{K} = \C$, with $\ip{\cdot}{\cdot}$ linear in the first argument, it is
$$\ip{x}{y} = \frac{1}{4} \sum_{k=0}^{3} i^k \norm{x + i^k y}^2.$$

## Remarks

The direction from 1 to 2 is the [[parallelogram law|026HPGYGS4]] in an [[inner product space|0QVNCJXRY3]].
For the converse in the real case, define $\ip{x}{y}$ by the polarization formula.
Symmetry and $\ip{x}{x} = \norm{x}^2$ are immediate.
The parallelogram law gives $\ip{x + x'}{y} = \ip{x}{y} + \ip{x'}{y}$, so $t \mapsto \ip{tx}{y}$ is an [[additive function|KF719SZD5T]] of $t \in \R$.
It is continuous, because the norm is continuous, so it equals $t \ip{x}{y}$.
The norm $\abs{x_1} + \abs{x_2}$ on $\R^2$ fails the parallelogram law for $x = (1, 0)$, $y = (0, 1)$, since $4 + 4 \ne 2 + 2$, so it comes from no inner product.

## Explanations

### L1 human

Some ways of measuring length come from an inner product, which also measures angles, and some do not.
This theorem gives a simple test: a length comes from an inner product exactly when it obeys the parallelogram law.
So one identity about four sides and two diagonals decides whether the space has a full Euclidean geometry.

### L2 human

The theorem shows that among the spaces $L^p([0, 1])$ with $1 \le p \le \infty$, only $L^2([0, 1])$ has a norm that comes from an inner product.
For $p \ne 2$ the parallelogram law fails for the indicator functions of $[0, 1/2]$ and $(1/2, 1]$.
It is a characterization of Hilbert spaces among Banach spaces, because completeness is not affected.
The proof is a functional equation argument: the parallelogram law is the [[quadratic functional equation|7MZGZFQ347]] for $f(x) = \norm{x}^2$, and continuity turns additivity into linearity.
The law only involves two vectors at a time, so a norm is an inner product norm as soon as its restriction to every two-dimensional subspace is.
</document>

<document source="https://tamnd.github.io/math2/node/46Z1TVS2K6.md" tag="46Z1TVS2K6" kind="conjecture">
# Navier–Stokes existence and smoothness

## Statement

Let $\nu > 0$.
Let $u_0 \colon \R^3 \to \R^3$ be a [[smooth|0V33BNKZ9S]] vector field with [[divergence|0MWV6ZTK6S]] $\nabla \cdot u_0 = 0$.
Assume that for every multi-index $\alpha$ and every $K > 0$ there is a constant $C_{\alpha K}$ with $\abs{\partial^\alpha u_0(x)} \le C_{\alpha K} (1 + \abs{x})^{-K}$ for all $x \in \R^3$.
Then there exist smooth functions $u \colon \R^3 \times [0, \infty) \to \R^3$ and $p \colon \R^3 \times [0, \infty) \to \R$ such that:

1. $u$ and $p$ satisfy the [[Navier–Stokes equations|18KXZHVF4M]] $\partial_t u + (u \cdot \nabla) u = \nu \Delta u - \nabla p$ and $\nabla \cdot u = 0$ on $\R^3 \times [0, \infty)$, with no external force.
2. $u(x, 0) = u_0(x)$ for all $x \in \R^3$.
3. There is a constant $E$ with $\int_{\R^3} \abs{u(x, t)}^2 \, dx \le E$ for all $t \ge 0$.

## Remarks

The problem is open and is one of the seven Millennium Prize Problems of the Clay Mathematics Institute.
The official problem asks for a proof of this statement or of its analogue for periodic data on the torus, or for a counterexample showing a breakdown of smooth solutions.
Charles Fefferman wrote the official problem description.
Smooth solutions are known to exist for a short time for such data, and for all time when the data are small in a suitable norm.
In two space dimensions the corresponding statement is true.
Jean Leray proved that global weak solutions with finite energy exist, see [[existence of Leray weak solutions|DHHR5PZZ7E]].
Caffarelli, Kohn and Nirenberg proved that the singular set of a suitable weak solution has one-dimensional parabolic Hausdorff measure zero.

## Explanations

### L1 human

The Navier–Stokes equations describe how water and air flow.
The question is whether a smooth, calm initial flow in all of space can develop an infinitely sharp feature in finite time.
An example of such a feature would be an infinite speed at a point.
Nobody knows, even though the equations are used every day in engineering.
A proof either way earns a prize of one million dollars.
</document>

<document source="https://tamnd.github.io/math2/node/47096P24SB.md" tag="47096P24SB" kind="concept">
# Morse–Bott function

## Summary

A Morse–Bott function is a smooth function whose critical points form submanifolds, and which is nondegenerate in the directions normal to these submanifolds.
It generalizes a [[Morse function|KPQ96640ZJ]], whose critical submanifolds are points.
Raoul Bott introduced the notion for functions with symmetry, whose critical points come in families.
An example is the energy on spaces of paths in a symmetric space.

## Explanations

### L1 human

Think of a ring-shaped doughnut lying flat on a table, and take the height as the function.
The lowest points form a whole circle, and so do the highest points, so the critical points are not isolated.
Still, in the direction across each circle the height curves up or down in a clean way.
Such a function is Morse–Bott: degenerate only along its critical circles and nondegenerate across them.
A simpler example is $f(x, y) = x^2$ on the plane, whose critical points form the line $x = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/4724KRNHAT.md" tag="4724KRNHAT" kind="concept">
# Palais–Smale compactness condition

## Summary

The Palais–Smale condition is a compactness hypothesis on a continuously differentiable functional on a Banach space.
It asks that every sequence along which the functional stays bounded and its derivative tends to zero has a convergent subsequence.
It replaces the compactness of closed bounded sets, which fails in infinite dimensions, in the search for [[critical points|0YTCMFB4HC]].
It is the main hypothesis of the [[mountain pass theorem|EPXNCSP2MN]] and of many other minimax results.

## Explanations

### L1 human

To find a lowest point or a mountain pass of a landscape, one usually walks along points where the ground gets flatter and flatter.
In infinite dimensions such a walk can run away to infinity or spread out without ever reaching a point.
The Palais–Smale condition forbids this: every such walk with bounded heights must have a part that settles at an actual point.
For example, $f(x) = e^x$ on $\R$ fails it, since its slope tends to $0$ as $x \to -\infty$ while no flat point exists.
</document>

<document source="https://tamnd.github.io/math2/node/476ZQPMZHP.md" tag="476ZQPMZHP" kind="statement">
# Clenshaw algorithm

## Statement

Let $n \ge 1$, let $x$ be a fixed point, and let $\phi_0(x), \ldots, \phi_n(x)$ be numbers that satisfy the three-term [[recurrence|0FYX44GP92]]
$$\phi_{k+1}(x) = \alpha_k(x)\, \phi_k(x) + \beta_k(x)\, \phi_{k-1}(x), \qquad 1 \le k \le n-1,$$
for given coefficients $\alpha_k(x)$ and $\beta_k(x)$.
Let $a_0, \ldots, a_n$ be numbers and $S(x) = \sum_{k=0}^{n} a_k \phi_k(x)$.
Set $b_{n+1} = b_{n+2} = 0$ and, for $k = n, n-1, \ldots, 1$,
$$b_k = a_k + \alpha_k(x)\, b_{k+1} + \beta_{k+1}(x)\, b_{k+2},$$
where $\alpha_n(x)$, $\beta_n(x)$ and $\beta_{n+1}(x)$ may be given any values in these formulas, since they multiply $b_{n+1} = 0$ or $b_{n+2} = 0$.
Then
$$S(x) = a_0\, \phi_0(x) + b_1\, \phi_1(x) + \beta_1(x)\, b_2\, \phi_0(x).$$

## Remarks

Write $a_k = b_k - \alpha_k b_{k+1} - \beta_{k+1} b_{k+2}$ for $1 \le k \le n$ and substitute in $\sum_{k \ge 1} a_k \phi_k$.
After collecting terms, the coefficient of $b_j$ for $j \ge 3$ is $\phi_j - \alpha_{j-1}\phi_{j-1} - \beta_{j-1}\phi_{j-2} = 0$, and the coefficient of $b_2$ is $\phi_2 - \alpha_1 \phi_1 = \beta_1 \phi_0$; only the terms in the formula remain.
For the [[Chebyshev polynomials|M6EAR1JHPP]] $T_k$ one has $\phi_0 = 1$, $\phi_1 = x$, $\alpha_k = 2x$ and $\beta_k = -1$, so $\sum_{k=0}^n a_k T_k(x) = a_0 + x b_1 - b_2$.
Check with $n = 2$: $b_2 = a_2$, $b_1 = a_1 + 2x a_2$, and $a_0 + x(a_1 + 2x a_2) - a_2 = a_0 + a_1 x + a_2(2x^2 - 1)$, which is $a_0 T_0 + a_1 T_1 + a_2 T_2$.
For the monomials $\phi_k = x^k$, with $\alpha_k = x$ and $\beta_k = 0$, the algorithm is [[Horner's method|FYBQ3Z52Z7]].
The cost is $O(n)$ operations, and the values $\phi_k(x)$ for $k \ge 2$ are never formed.

## Explanations

### L1 human

Many useful functions are sums of special polynomials, such as Chebyshev or Legendre polynomials, each defined by a recurrence from the two before.
Computing every polynomial value and then adding is possible, but Clenshaw found a shorter route.
It runs a similar recurrence backwards, starting from the last coefficient, and reads off the sum at the end.
It plays for these families the role that Horner's rule plays for ordinary polynomials.

### L2 human

The algorithm is the standard way to evaluate a truncated Chebyshev series, for instance in function libraries and in Chebyshev spectral methods.
It works for any family defined by a three-term recurrence, so for all classical orthogonal polynomials and also for some special functions such as Bessel functions.
The backward recurrence is the transpose of the forward one, which explains why only $b_1$ and $b_2$ appear in the result.
Its rounding error behaviour is good for Chebyshev series on $[-1, 1]$, though near the endpoints $\pm 1$ the errors can be amplified.
</document>

<document source="https://tamnd.github.io/math2/node/47CVS9RJ9D.md" tag="47CVS9RJ9D" kind="problem">
# Levi problem

## Problem

Let $n \ge 2$ and let $\Omega \subseteq \C^n$ be a domain.
Every [[domain of holomorphy|4487CSC0MB]] is [[pseudoconvex|AHYZMSBBNH]].
Is every pseudoconvex domain $\Omega \subseteq \C^n$ a domain of holomorphy?

## Remarks

The answer is yes.
Kiyoshi Oka proved it for $n = 2$ in 1942.
For all $n$ it was proved in the early 1950s, independently by Oka, by Hans-Joachim Bremermann and by François Norguet.
Hans Grauert solved the analogue for complex manifolds in 1958: a complex manifold with a smooth strictly plurisubharmonic exhaustion function is a Stein manifold.
A modern proof solves the equation $\bar\partial u = f$ with Hörmander's $L^2$ estimates for weights built from a plurisubharmonic exhaustion function.
The problem has no content for $n = 1$, where every domain is both pseudoconvex and a domain of holomorphy.

## Explanations

### L1 human

In several complex variables, not every domain is the natural home of some holomorphic function: on some domains, every holomorphic function extends to a larger domain.
The domains that are natural homes are the domains of holomorphy.
Levi found a geometric condition that they must satisfy, a complex form of convexity of the boundary called pseudoconvexity.
The Levi problem asks whether this geometric condition is enough.
It is, and the proof was one of the main achievements of complex analysis in the twentieth century.
</document>

<document source="https://tamnd.github.io/math2/node/47DYWZTQMC.md" tag="47DYWZTQMC" kind="definition">
# External ray

## Definition

Let $P$ be a monic [[polynomial|1G0B73WPX8]] of degree $d \ge 2$ whose filled Julia set $K$ is connected.
Let $\overline{\mathbb{D}}$ be the closed unit disc.
By [[Böttcher's theorem|YBBDYHBADD]] there is a unique [[conformal map|0YYT36R4YD]] $\varphi \colon \C \setminus K \to \C \setminus \overline{\mathbb{D}}$ with $\varphi(z)/z \to 1$ as $z \to \infty$.
It satisfies $\varphi(P(z)) = \varphi(z)^d$.
For $\theta \in \R/\Z$, the *external ray of angle* $\theta$ is
$$R_\theta = \varphi^{-1}\big(\set{r e^{2\pi i \theta} : r > 1}\big).$$
The ray $R_\theta$ *lands* at $z_0 \in \partial K$ if $\varphi^{-1}(r e^{2\pi i \theta}) \to z_0$ as $r \to 1^+$.

## Remarks

The relation $\varphi \circ P = \varphi^d$ gives $P(R_\theta) = R_{d\theta}$.
Example: for $P(z) = z^2$, $K$ is the closed unit disc and $\varphi$ is the identity.
So $R_\theta$ is the half-line $\set{r e^{2\pi i \theta} : r > 1}$, and it lands at $e^{2\pi i \theta}$.
Non-example: for $P(z) = z^2$, the spiral $\set{r e^{2\pi i (\theta + \log r)} : r > 1}$ is not an external ray, since $\varphi$ maps it to a curve whose angle varies.
Douady and Hubbard showed that every ray with a rational angle lands, at a periodic or preperiodic point of the [[Julia set|5M4S7P04GR]].
For the [[quadratic family|WDZFV2ACXR]], external rays of the Mandelbrot set are defined in the same way, with the conformal map from its complement onto $\C \setminus \overline{\mathbb{D}}$.
</document>

<document source="https://tamnd.github.io/math2/node/47KD8V5A7G.md" tag="47KD8V5A7G" kind="definition">
# Volterra series

## Definition

Let $h_0 \in \R$ and, for each $n \ge 1$, let $h_n \colon [0, \infty)^n \to \R$ be an integrable function, the *$n$-th Volterra kernel*.
Assume that $\sum_{n \ge 1} \norm{h_n}_{L^1} r^n < \infty$ for some $r > 0$.
For a bounded measurable input $x \colon \R \to \R$ with $\sup \abs{x} \le r$, the *Volterra series* with these kernels gives the output
$$y(t) = h_0 + \sum_{n=1}^{\infty} \int_0^\infty \!\!\cdots\! \int_0^\infty h_n(\tau_1, \ldots, \tau_n)\, x(t - \tau_1) \cdots x(t - \tau_n)\, d\tau_1 \cdots d\tau_n, \quad t \in \R.$$
The $n$-th term is bounded by $\norm{h_n}_{L^1}\, r^n$, so the series converges absolutely and uniformly in $t$.

## Remarks

The operator $x \mapsto y$ is causal, since $y(t)$ depends only on $x(s)$ for $s \le t$, and time-invariant, since shifting $x$ in time shifts $y$ by the same amount.
The kernels can be taken symmetric in their variables without changing $y$.
If only $h_1$ is nonzero, $y = h_0 + h_1 * x$ is a [[convolution|2Z29JN3VES]], the output of a linear system with impulse response $h_1$.
Some authors also allow kernels with delta functions, or integrals over all of $\R^n$ for noncausal systems.

Example: let $z(t) = \int_0^\infty e^{-\tau} x(t - \tau)\, d\tau$ and $y = z^2$.
Then $y(t) = \int_0^\infty \int_0^\infty e^{-\tau_1 - \tau_2}\, x(t - \tau_1)\, x(t - \tau_2)\, d\tau_1\, d\tau_2$ by Fubini's theorem.
So $y$ is a Volterra series with $h_2(\tau_1, \tau_2) = e^{-\tau_1 - \tau_2}$, of $L^1$ norm $1$, and all other kernels $0$.

Non-example: the system $y(t) = t\, x(t)$ is not given by any Volterra series, because it is not time-invariant.
The input $x = 1$ gives $y(t) = t$, which is not constant, while a Volterra series maps a constant input to a constant output.
</document>

<document source="https://tamnd.github.io/math2/node/47NFRVB564.md" tag="47NFRVB564" kind="concept">
# Separation of variables

## Summary

Separation of variables is a method for solving differential equations by splitting the dependence on different variables.
For an ordinary differential equation $y' = g(x)\, h(y)$, one moves the $y$ terms to one side and the $x$ terms to the other and integrates.
Such equations are called [[separable equations|074E9A8ZJK]].
For a linear partial differential equation, one looks for solutions that are products of functions of one variable each.
This turns the equation into several ordinary differential equations.
It is one of the oldest and most used solution methods.
See the [[definition|PR2DV03M4H]].

## Explanations

### L1 human

The idea is to put everything about one variable on one side of the equation and everything about the other variable on the other side.
For $y' = x y$, one writes $dy / y = x\, dx$, integrates both sides to get $\ln\abs{y} = x^2/2 + C$, and finds $y = A e^{x^2/2}$.
For the heat equation $u_t = u_{xx}$, one guesses $u(x, t) = X(x)\, T(t)$, and the equation splits into one equation for $X$ and one for $T$.
Sums of such product solutions then give many more solutions, which leads to Fourier series.
</document>

<document source="https://tamnd.github.io/math2/node/47NSS1R24Y.md" tag="47NSS1R24Y" kind="definition">
# Monotone operator

## Definition

Let $X$ be a real [[Banach space|06GZ956657]] with [[dual space|07JBSZE0V4]] $X^*$, and write $\ip{\ell}{u} = \ell(u)$ for $\ell \in X^*$ and $u \in X$.
Let $D \subseteq X$ and let $T \colon D \to X^*$ be a map, not necessarily linear.
The map $T$ is a *monotone operator* if
$$\ip{T(u) - T(v)}{u - v} \ge 0 \quad \text{for all } u, v \in D.$$
It is *strictly monotone* if equality holds only for $u = v$.

## Remarks

A set-valued map $T$ from $X$ to subsets of $X^*$ is monotone if $\ip{u^* - v^*}{u - v} \ge 0$ whenever $u^* \in T(u)$ and $v^* \in T(v)$.
It is *maximal monotone* if its graph is not properly contained in the graph of another monotone map.
For a real Hilbert space one identifies $X^*$ with $X$.
A linear operator $T$ is then monotone if and only if $\ip{Tu}{u} \ge 0$ for all $u$, that is, if and only if $-T$ is [[dissipative|3S7HJ5GQRF]].
The subdifferential of a proper convex lower semicontinuous function is maximal monotone.

Example: let $f \colon \R^n \to \R$ be a differentiable [[convex function|07Q7CNM5WF]].
Adding the inequalities $f(v) \ge f(u) + \nabla f(u) \cdot (v - u)$ and $f(u) \ge f(v) + \nabla f(v) \cdot (u - v)$ gives $(\nabla f(u) - \nabla f(v)) \cdot (u - v) \ge 0$, so $\nabla f$ is monotone.

Non-example: $T(u) = -u$ on $\R$ gives $(T(1) - T(0))(1 - 0) = -1 < 0$.

On $X = \R$, a map $T \colon \R \to \R$ is monotone exactly when it is a nondecreasing [[monotone function|0A7KV2WB7A]].
</document>

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# Classical orthogonal polynomials

## Summary

The classical orthogonal polynomials are the three families of Jacobi, Laguerre and [[Hermite polynomials|MGACY98E9H]].
They include special cases such as the Gegenbauer, Legendre and [[Chebyshev polynomials|M6EAR1JHPP]].
Each family is orthogonal for a weight on an interval.
The weights are of beta type on $(-1, 1)$, of gamma type on $(0, \infty)$, and Gaussian on $\R$.
They are the only orthogonal polynomial families that are eigenfunctions of a second-order differential operator, and they have Rodrigues formulas and explicit recurrences.
They are the basic tools of approximation, Gaussian quadrature, and separation of variables in mathematical physics.

## Explanations

### L1 human

Two polynomials are orthogonal for a weight when the integral of their product times the weight is zero, like perpendicular vectors.
For a given weight, the Gram–Schmidt process applied to $1, x, x^2, \dots$ gives one orthogonal polynomial of each degree.
For three kinds of weight this produces families with especially nice formulas, and these are the classical ones.
For example, the weight $e^{-x^2}$ on the whole line gives the Hermite polynomials $1$, $2x$, $4x^2 - 2$, and so on.
</document>

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# Second-order logic

## Definition

Let $L$ be a signature.
The formulas of second-order logic over $L$ are built like the formulas of [[first-order logic|086QKPF0A6]] over $L$, with these additions.

1. For each $n \ge 1$ there are relation variables $X, Y, \dots$ of arity $n$, and $X(t_1, \dots, t_n)$ is an atomic formula for terms $t_1, \dots, t_n$.
2. If $\varphi$ is a formula and $X$ is a relation variable, then $\forall X \, \varphi$ and $\exists X \, \varphi$ are formulas.

In the standard semantics, a formula is evaluated in an $L$-[[structure|983GZBQYT4]] $\mathcal{M}$ with domain $M$ as in first-order logic.
An $n$-ary relation variable ranges over the whole [[power set|00E0YZCERH]] of $M^n$.

## Remarks

Function variables may be added in the same way; they can be expressed by relation variables.
Monadic second-order logic allows only relation variables of arity $1$, that is, variables for subsets of $M$.

Example: the sentence $\forall X \, \big( (X(0) \land \forall x \, (X(x) \to X(S(x)))) \to \forall x \, X(x) \big)$ is a second-order sentence over the signature $\set{0, S}$.
It says that every subset of the domain that contains $0$ and is closed under $S$ is the whole domain.

Non-example: a formula with a quantifier over sets of subsets of $M$ is not second-order, because no clause provides such variables; it belongs to third-order logic.

With the standard semantics, second-order logic has neither a complete proof system nor the compactness property.
With Henkin semantics, in which the relation variables range over a chosen family of relations, both properties return.

## Explanations

### L1 human

Second-order logic lets formulas speak about all subsets and all relations of the domain, not only about its elements.
This makes it much more expressive than first-order logic.
For example, finiteness of the domain, or the completeness of the real line, can be written as a single second-order sentence.
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# Dedekind number

## Summary

The Dedekind number $M(n)$ is the number of monotone [[Boolean functions|0X05GS525E]] of $n$ variables.
It also counts the [[antichains|03HF498P1J]] of subsets of an $n$-element set.
For $n \ge 1$, $M(n) - 2$ is the number of elements of the [[free distributive lattice|T5GK6DH8BR]] on $n$ generators.
The first values are $M(0) = 2$, $M(1) = 3$, $M(2) = 6$, $M(3) = 20$, $M(4) = 168$ and $M(5) = 7581$.
The numbers grow very fast, and no closed formula for them is known; finding one is [[Dedekind's problem|RM64J2CP1M]].

## Explanations

### L1 human

A monotone Boolean function is a yes/no rule on $n$ switches.
In such a rule, turning a switch on can never change the answer from yes to no.
The Dedekind number counts these rules.
For two switches $x$ and $y$ there are six: always no, always yes, $x$, $y$, "$x$ and $y$", and "$x$ or $y$".
The rule "exactly one of $x$, $y$" is not monotone.
With $x$ on, turning on $y$ changes the answer from yes to no.
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# Abscissa of absolute convergence

## Definition

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$ with $T > 0$.
The *abscissa of absolute convergence* of the [[Laplace transform|05XT5AVJBK]] of $f$ is
$$\sigma_a = \inf \set{ \sigma \in \R : \int_0^\infty \abs{f(t)} e^{-\sigma t}\, dt < \infty } \in [-\infty, +\infty],$$
with $\inf \emptyset = +\infty$.

## Remarks

For $\operatorname{Re} s > \sigma_a$ the Laplace integral converges as a [[Lebesgue integral|XVEWR93EMV]], and for $\operatorname{Re} s < \sigma_a$ it does not converge absolutely.
Always $\sigma_c \le \sigma_a$, where $\sigma_c$ is the [[abscissa of convergence|FTHFSJ30WJ]].
Example: for $f(t) = e^{at}$ with $a \in \R$, $\abs{f} = f$, so $\sigma_a = \sigma_c = a$.
Example with $\sigma_c < \sigma_a$: let $f(t) = e^t \sin(e^t)$.
The substitution $u = e^t$ turns $\int_0^\infty f(t) e^{-\sigma t}\, dt$ into $\int_1^\infty \sin(u)\, u^{-\sigma}\, du$.
This converges for $\sigma > 0$ by the Dirichlet test and diverges for $\sigma \le 0$, so $\sigma_c = 0$.
It converges absolutely exactly for $\sigma > 1$, since $\int_1^\infty \abs{\sin u}\, u^{-\sigma}\, du < \infty$ if and only if $\sigma > 1$; so $\sigma_a = 1$.
For a function of exponential order $a$ that is integrable on bounded intervals, $\sigma_a \le a$.
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# Löwenheim–Skolem theorem

## Statement

Let $\sigma$ be a [[signature|WDVVEYD6AT]], and let $\kappa = \max(\abs{\sigma}, \aleph_0)$.
Let $M$ be an infinite $\sigma$-[[structure|983GZBQYT4]], and let $\lambda$ be an infinite [[cardinal|0EBACQFAZW]].

1. Downward: if $\kappa \le \lambda \le \abs{M}$ and $A \subseteq M$ has $\abs{A} \le \lambda$, then $M$ has an [[elementary substructure|8WZF9SBP39]] $N$ with $A \subseteq N$ and $\abs{N} = \lambda$.
2. Upward: if $\lambda \ge \max(\abs{M}, \kappa)$, then $M$ has an elementary extension of cardinality $\lambda$.

## Remarks

The downward part closes $A$ under Skolem functions, which pick witnesses for existential formulas, and uses the [[Tarski–Vaught test|60SA99BER7]].
The upward part adds $\lambda$ new constants that must be pairwise distinct, uses the [[compactness theorem|1NR3HAWG9D]], and then applies the downward part.
For example, the field $\R$ has a countable elementary substructure: the real algebraic numbers form one, by quantifier elimination for real closed fields, the [[Tarski–Seidenberg theorem|3W5JGDZAN0]].
The hypothesis that $M$ is infinite is needed: a finite structure has no elementary extension of a different size.
As a consequence, a countable first-order theory with an infinite model has models of every infinite cardinality.
So no first-order theory with an infinite model determines its models up to isomorphism; [[Skolem's paradox|68QZWDXAQ4]] is one effect of this.
In Mathlib the downward part is `FirstOrder.Language.exists_elementarySubstructure_card_eq`.

## Explanations

### L1 human

First-order logic cannot control the size of an infinite structure.
If some first-order axioms have an infinite model, then they have models of every infinite size, small and large, at least as large as the language.
For example, there are countable structures that satisfy exactly the same first-order sentences as the real numbers.

### L2 human

The theorem is a basic tool of [[model theory|HFTBG5PMXC]].
It shows that categoricity in all infinite cardinalities is impossible, so model theory studies categoricity in one cardinality, as in [[Morley's categoricity theorem|V7N84P60SJ]].
The [[Łoś–Vaught test|ZR7Y5C235P]] combines it with categoricity in one cardinality to prove that a theory is complete.
Second-order logic has no such theorem, since the second-order Peano axioms determine $\N$ up to isomorphism.
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# Near-field

## Definition

A *near-field* is a set $N$ with two binary operations $+$ and $\cdot$ such that:

1. $(N, +)$ is an [[abelian group|082GHNPJ53]], with identity element $0$.
2. $N \setminus \set{0}$ is closed under $\cdot$ and is a [[group|G5S1M9]] under $\cdot$.
3. $a \cdot 0 = 0 \cdot a = 0$ for all $a \in N$.
4. $(a + b) \cdot c = a \cdot c + b \cdot c$ for all $a, b, c \in N$.

## Remarks

Clauses 2 and 3 make multiplication associative on all of $N$, with an identity $1$.
Some authors use the left distributive law $a(b + c) = ab + ac$ in clause 4 instead; the two versions are mirror images of each other.
A near-field that also satisfies the other distributive law is a [[division ring|0EA5BTA0CX]].
The commutativity of $+$ in clause 1 is often omitted, because it follows from the other clauses.

Example: let $F = \F_9$ be the [[finite field|0ZXZ82FTX2]] with $9$ elements, and let $\phi(a) = a^3$ be its [[Frobenius automorphism|0T3WVTD675]].
Define $a \circ b = a b$ if $b$ is a square in $F$ (including $b = 0$) and $a \circ b = \phi(a) b$ otherwise.
Clause 4 holds because $\phi$ is additive.
Clause 2 holds because $\phi$ preserves squares.
So $b \circ c$ is a nonzero square exactly when $b$ and $c$ are both squares or both non-squares, and this gives associativity.
The element $1$ is the identity.
A nonzero $b$ has the inverse $c$ with $\phi^{k}(c) = b^{-1}$, where $k = 0$ if $b$ is a square and $k = 1$ otherwise.
Write $\F_9 = \F_3(i)$ with $i^2 = -1$; then $i$ is a square and $1 + i$ is not, and $i \circ (1 + i) = 1 - i$ while $(1 + i) \circ i = i - 1$.
So $(F, +, \circ)$ is a near-field with non-commutative multiplication; since every finite division ring is commutative, it is not a division ring.

Non-example: $\Z$ with its usual operations satisfies clauses 1, 3 and 4 but fails clause 2, because $2$ has no multiplicative inverse in $\Z$.
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# Interpretation of varieties

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of signature $\sigma$ and let $\mathcal{W}$ be a variety of signature $\tau$.
An *interpretation* of $\mathcal{V}$ in $\mathcal{W}$ is a map $f \mapsto t_f$ with the following properties.
It assigns to each $n$-ary operation symbol $f$ of $\sigma$ a $\tau$-term $t_f(x_1, \dots, x_n)$.
For $A$ in $\mathcal{W}$, let $A^{*}$ be the algebra on the universe of $A$ where each $f$ acts as the [[term operation|N9M2N27END]] $t_f^{A}$.
Then $A^{*}$ belongs to $\mathcal{V}$ for every $A$ in $\mathcal{W}$.
The variety $\mathcal{V}$ is *interpretable* in $\mathcal{W}$ if an interpretation of $\mathcal{V}$ in $\mathcal{W}$ exists.

## Remarks

Example: let $\mathcal{V}$ be the variety of Boolean algebras, with operations $\vee, \wedge, \neg, 0, 1$, and let $\mathcal{W}$ be the variety of Boolean rings.
The assignment $x \vee y \mapsto x + y + xy$, $x \wedge y \mapsto xy$, $\neg x \mapsto 1 + x$, $0 \mapsto 0$, $1 \mapsto 1$ is an interpretation.
In every Boolean ring these term operations satisfy the Boolean algebra identities.
Non-example: let $\mathcal{V}$ be the variety of semigroups and $\mathcal{W}$ the variety of groups, and assign to the semigroup product the group term $x y^{-1}$.
This is not an interpretation, since in the group $(\Z, +)$ the operation $a - b$ is not associative: $(1 - 1) - 1 = -1$ but $1 - (1 - 1) = 1$.
It suffices to check the defining identities of $\mathcal{V}$ on the free algebras of $\mathcal{W}$.

## Explanations

### L1 human

An interpretation writes each operation of one variety as a formula in the operations of another, in a way that keeps all the laws true.
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# q-convex manifold

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of complex dimension $n$, and let $q$ be an integer with $1 \le q \le n$.
Let $\varphi \colon X \to \R$ be a [[smooth|0V33BNKZ9S]] function.
It is *strongly $q$-convex* at a point $x$ if its [[Levi form|80MTNXWW80]] at $x$, in local holomorphic coordinates, has at least $n - q + 1$ positive eigenvalues.
The manifold $X$ is *$q$-convex* if there are a smooth function $\varphi \colon X \to \R$ and a [[compact set|0R6EWFKRCF]] $K \subseteq X$ such that:

1. $\varphi$ is an exhaustion: the set $\set{x \in X : \varphi(x) < c}$ is relatively compact in $X$ for every $c \in \R$.
2. $\varphi$ is strongly $q$-convex at every point of $X \setminus K$.

The manifold $X$ is *$q$-complete* if this holds with $K = \emptyset$.

## Remarks

This is the convention of Andreotti and Grauert; some authors shift the index by one.
The number of positive eigenvalues of the Levi form does not depend on the choice of local coordinates.
Example: $\C^n$ is $1$-complete with $\varphi(z) = \abs{z}^2$, whose Levi form $\abs{w}^2$ has $n$ positive eigenvalues.
Non-example: a compact complex manifold of positive dimension is $q$-convex with $K = X$, but it is not $1$-complete.
Indeed a strongly $1$-convex function is strictly plurisubharmonic, so it has no maximum, while every continuous function on a compact manifold has one.
The $1$-complete manifolds are exactly the Stein manifolds, and the $1$-convex manifolds are also called strongly pseudoconvex manifolds.
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# Sumset

## Definition

Let $G$ be an [[abelian group|082GHNPJ53]] written additively, and let $A$ and $B$ be subsets of $G$.
The *sumset* of $A$ and $B$ is
$$A + B = \set{a + b : a \in A, \ b \in B}.$$
The *difference set* is $A - B = \set{a - b : a \in A, \ b \in B}$.
For an integer $k \ge 1$, the iterated sumset $kA$ is $A + \dots + A$ with $k$ summands.

## Remarks

In $\Z$, take $A = \set{0, 1, 2}$ and $B = \set{0, 5}$.
Then $A + B = \set{0, 1, 2, 5, 6, 7}$, since each sum $a + b$ lies in this set and each element of it is such a sum.
The set $\set{a + a : a \in A} = \set{0, 2, 4}$ is not the sumset $A + A$, because it misses sums of two different elements such as $0 + 1$.
The iterated sumset $2A = A + A$ is different from the dilate $2 \cdot A = \set{2a : a \in A}$.
For finite nonempty $A, B \subseteq \Z$ one has $\abs{A + B} \ge \abs{A} + \abs{B} - 1$, with equality for arithmetic progressions of the same common difference.
For subsets of $\R^n$ the sumset is called the Minkowski sum.
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# Lopatinskii–Shapiro condition

## Summary

The Lopatinskii–Shapiro condition, also called the complementing condition, is an algebraic condition on boundary operators.
It is what a boundary value problem for an [[elliptic operator|XY8GEQSWPR]] needs to be well behaved.
It is checked at each boundary point by freezing coefficients and looking at an ordinary differential equation in the normal direction.
When it holds, the problem has a priori estimates in Sobolev spaces and is a Fredholm problem.
It plays for boundary conditions the role that ellipticity plays for the operator itself.

## Explanations

### L1 human

An elliptic equation inside a domain needs the right number and the right kind of conditions on the boundary.
To test them, zoom in on one boundary point until the boundary looks flat and the coefficients look constant.
Then look for solutions that oscillate along the boundary and decay into the interior.
The condition says that the boundary conditions kill all such solutions.
For the Laplacian, prescribing the values $u$ on the boundary passes this test, and so does prescribing the normal derivative.
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# Young's inequality for integral operators

## Statement

Let $(X, \mu)$ and $(Y, \nu)$ be $\sigma$-finite [[measure spaces|P15VQS5DJ9]], let $1 \le p \le \infty$ and $C > 0$.
Let $K \colon X \times Y \to \C$ be a [[measurable function|YYNCJSR0PJ]] such that
$$\int_Y \abs{K(x, y)} \, d\nu(y) \le C \quad \text{for almost every } x \in X,$$
$$\int_X \abs{K(x, y)} \, d\mu(x) \le C \quad \text{for almost every } y \in Y.$$
Let $f \in L^p(Y, \nu)$.
Then the integral
$$(T f)(x) = \int_Y K(x, y) \, f(y) \, d\nu(y)$$
converges absolutely for almost every $x \in X$, the function $T f$ lies in $L^p(X, \mu)$, and
$$\norm{T f}_p \le C \norm{f}_p.$$

## Remarks

Proof idea for $1 < p < \infty$: let $p'$ be the conjugate exponent and write $\abs{K} = \abs{K}^{1/p'} \abs{K}^{1/p}$.
Hölder's inequality in $y$ and the first hypothesis give $\abs{T f(x)}^p \le C^{p/p'} \int_Y \abs{K(x, y)} \abs{f(y)}^p \, d\nu(y)$.
Integrating in $x$, Tonelli's theorem and the second hypothesis give $\norm{T f}_p^p \le C^{p/p'} \cdot C \norm{f}_p^p = C^p \norm{f}_p^p$.
The case $p = \infty$ uses only the first hypothesis, and the case $p = 1$ uses only the second.

Example: for $X = Y = \R^n$ with Lebesgue measure and $K(x, y) = g(x - y)$ with $g \in L^1(\R^n)$, both hypotheses hold with $C = \norm{g}_1$.
The result is $\norm{g * f}_p \le \norm{g}_1 \norm{f}_p$, the case $q = 1$ of [[Young's convolution inequality|JXDKJCBXJ8]].

Both hypotheses are needed for the full range of $p$.
On $X = Y = (0, \infty)$, the Hardy kernel $K(x, y) = 1/x$ for $y < x$ and $K(x, y) = 0$ otherwise has $\int_0^\infty K(x, y) \, dy = 1$ for every $x$, but $\int_0^\infty K(x, y) \, dx = \infty$ for every $y$.
The operator $T f(x) = \frac{1}{x} \int_0^x f(y) \, dy$ is bounded on $L^p$ for $1 < p \le \infty$ by [[Hardy's inequality|S7A8YFGG8X]], but not on $L^1$: for $f$ the indicator function of $(0, 1]$, $T f(x) = 1/x$ for $x > 1$, which is not integrable.

## Explanations

### L1 human

An integral operator replaces a function by weighted averages of its values, with weights $K(x, y)$.
Suppose that every row and every column of the weights has total mass at most $C$.
Then the operator enlarges the $L^p$ size of a function by at most the factor $C$, for every $p$.
This is the continuous analogue of the fact that a matrix whose absolute row sums and column sums are at most $C$ has norm at most $C$ on every $\ell^p$.

### L2 human

This result is sometimes called the generalized Young inequality, since it contains the case $\norm{g * f}_p \le \norm{g}_1 \norm{f}_p$ of [[Young's convolution inequality|JXDKJCBXJ8]].
It is the quickest way to show that an integral operator with an integrable kernel acts boundedly on all $L^p$ spaces at once.
For $p = 2$ it is the [[Schur test|25HF1WPPKX]] with constant weights; the Schur test allows other weights and so applies to more kernels.
In the theory of [[integral equations|R3SG1SN7JQ]] it gives the bounds that the contraction principle needs for second-kind equations in $L^p$.
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# Hermitian form

## Summary

A Hermitian form on a complex vector space is a [[sesquilinear form|054YEQ0Y8M]] $h$ with $h(v, u) = \overline{h(u, v)}$ for all vectors $u, v$.
It is the complex analogue of a [[symmetric bilinear form|0AXZNE5XNH]].
The value $h(v, v)$ is always real, so Hermitian forms can be positive, negative or indefinite.
Complex [[inner products|0NB0AWGJBW]] are exactly the positive definite Hermitian forms.

## Explanations

### L1 human

Over the real numbers, symmetric bilinear forms such as the dot product measure squared lengths.
Over the complex numbers, one has to conjugate one argument to keep $h(v, v)$ real; this gives a Hermitian form.
On $\C^n$, the Hermitian forms are exactly $h(x, y) = y^* M x$ with $M$ a [[Hermitian matrix|0P17NGQQMQ]].
For example, $h(x, y) = x_1 \overline{y_1} - x_2 \overline{y_2}$ on $\C^2$ is Hermitian, with $h(v, v) = \abs{v_1}^2 - \abs{v_2}^2$.
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# Homotopy category

## Summary

The homotopy category of a [[model category|SDTZVZ6B4Q]] is obtained by making all weak equivalences invertible.
It is the setting in which homotopy theory compares objects.
Two objects are isomorphic in it when they are connected by a chain of weak equivalences.
For topological spaces it can be described by CW complexes and homotopy classes of maps.
Quillen's theorem identifies its morphisms with homotopy classes of maps between suitably replaced objects.

## Explanations

### L1 human

In homotopy theory, two maps are treated as equal when one can be deformed into the other.
The homotopy category records only this information: objects are spaces, and an arrow is a deformation class of maps.
In it, a disc and a point become isomorphic, since a disc can be shrunk to a point.
The circle and a point stay different, because the circle has a loop that cannot be shrunk.
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# Representable functor

## Definition

Let $\mathcal{C}$ be a locally small [[category|A2DE66937D]] and let $\mathbf{Set}$ be the category of [[sets|0BSJ7RQ9BW]].
A [[functor|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathbf{Set}$ is *representable* if there are an object $A$ of $\mathcal{C}$ and a natural isomorphism $\Hom(A, -) \cong F$.
A natural isomorphism is a [[natural transformation|KYCPCC57FT]] whose components are all bijections.
A functor $G \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ is *representable* if there are an object $A$ and a natural isomorphism $\Hom(-, A) \cong G$.
Here $\Hom(A, -)$ and $\Hom(-, A)$ are the [[hom functors|VJXS8AK67E]], and $A$ is called a *representing object*.

## Remarks

A representation $\Phi \colon \Hom(A, -) \cong F$ is determined by the element $u = \Phi_A(\id_A) \in F(A)$, called a universal element.
The pair $(A, u)$ represents $F$ exactly when for every object $X$ and every $x \in F(X)$ there is a unique $f \colon A \to X$ with $F(f)(u) = x$.

Example: the contravariant [[power set|00E0YZCERH]] functor $P \colon \mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}$ sends $X$ to its set of subsets and $f \colon X \to Y$ to the preimage map $S \mapsto f^{-1}(S)$.
It is represented by $A = \{0, 1\}$: a function $\chi \colon X \to \{0, 1\}$ corresponds to the subset $\chi^{-1}(1)$, and this bijection commutes with preimages.

Non-example: the constant functor $F \colon \mathbf{Set} \to \mathbf{Set}$ with value the empty set is not representable.
For every set $A$ the set $\Hom(A, A)$ contains $\id_A$, so it is not in bijection with $F(A) = \emptyset$.
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# Jensen's formula

## Statement

Let $r > 0$, and let $f$ be a [[meromorphic function|0NSD1TPHR0]] on an open set that contains the closed disc $\abs{z} \le r$.
Assume that $f(0) \ne 0$, that $0$ is not a pole of $f$, and that $f$ has no zeros and no poles on the circle $\abs{z} = r$.
Let $a_1, \dots, a_m$ be the [[zeros|0CEGEASY49]] and $b_1, \dots, b_n$ the [[poles|0WPQYW8MTE]] of $f$ in the disc $\abs{z} < r$, each repeated according to its multiplicity.
Then
$$\log \abs{f(0)} = \frac{1}{2\pi} \int_0^{2\pi} \log \abs{f(r e^{i\theta})} \, d\theta - \sum_{j=1}^{m} \log \frac{r}{\abs{a_j}} + \sum_{k=1}^{n} \log \frac{r}{\abs{b_k}}.$$

## Remarks

Divide $f$ by the [[Blaschke factors|9N6R5ADJXT]] $r(z - a_j)/(r^2 - \overline{a_j} z)$ and multiply by the analogous factors for the poles.
The quotient $g$ has no zeros or poles on the closed disc, and $\abs{g} = \abs{f}$ on the circle $\abs{z} = r$.
Then $\log \abs{g}$ is [[harmonic|0W5NJQ8MKY]], and its mean value property gives the formula, since each factor has modulus $\abs{a_j}/r$ or $\abs{b_k}/r$ at $0$.
For example, take $f(z) = z - a$ with $0 < \abs{a} < 1$ and $r = 1$.
The integral is $0$, because $\abs{e^{i\theta} - a} = \abs{1 - \overline{a} e^{i\theta}}$ and $\log \abs{1 - \overline{a} z}$ is harmonic on the closed unit disc with value $0$ at $0$; both sides then equal $\log \abs{a}$.
The formula remains true when $f$ has zeros or poles on the circle, since the integral still converges.

## Explanations

### L1 human

Jensen's formula relates the size of a function on a circle to its zeros inside the circle.
For a function without zeros or poles, $\log \abs{f}$ is harmonic and its value at the centre is its average on the circle.
Each zero inside the disc pulls the value at the centre down, and each pole pushes it up, by an exact amount.
So a function that is not large on the circle cannot have many zeros inside.

### L2 human

For a holomorphic $f$, the formula gives $\sum_j \log(r/\abs{a_j}) \le \log M(r) - \log \abs{f(0)}$, where $M(r)$ is the maximum of $\abs{f}$ on $\abs{z} = r$.
This bounds the number of zeros in terms of the growth, which is the first step of the [[Hadamard factorization theorem|KYSR7NHNFQ]].
The sum can be written as $\int_0^r n(t)/t \, dt$, where $n(t)$ counts the zeros in $\abs{z} \le t$.
In this form the formula is the starting point of [[Nevanlinna theory|1EHH63VNAD]] and of its [[first main theorem|QJWCFAK4AK]].
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# Rational difference equation

## Definition

Let $k \ge 0$ be an integer, and let $\alpha, \beta_0, \dots, \beta_k, A, B_0, \dots, B_k$ be real numbers.
A *rational difference equation* is a [[nonlinear difference equation|7C7FBZGERM]] of the form
$$x_{n+1} = \frac{\alpha + \sum_{i=0}^{k} \beta_i\, x_{n-i}}{A + \sum_{i=0}^{k} B_i\, x_{n-i}} \qquad (n \ge 0),$$
for real sequences $(x_n)_{n \ge -k}$.
The initial values $x_{-k}, \dots, x_0$ are taken such that the denominator is nonzero for every $n \ge 0$.

## Remarks

Example: $x_{n+1} = 1/(1 + x_n)$ is a rational difference equation with $k = 0$, $\alpha = 1$, $\beta_0 = 0$, $A = 1$, $B_0 = 1$.
For $x_0 = 1$ all denominators are at least $1$, and $x_n = F_{n+1}/F_{n+2}$ with $F$ the [[Fibonacci numbers|0GA9K57A39]].
Non-example: $x_{n+1} = 1/(1 + x_n^2)$ is not of this form, since its denominator is not affine in $x_n$.
If all $B_i = 0$ and $A \ne 0$, the equation is linear; the interesting cases have some $B_i \ne 0$.
The case $k = 0$, $x_{n+1} = (a x_n + b)/(c x_n + d)$ with $c \ne 0$ and $ad - bc \ne 0$, is the [[Riccati difference equation|D4Q8VM7826]].
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<document source="https://tamnd.github.io/math2/node/498VVT12YJ.md" tag="498VVT12YJ" kind="definition">
# Functional inequality

## Definition

Let $D$ be a set, let $\mathcal{F}$ be a set of [[functions|0EPAT51DXQ]] $f \colon D \to \R$, and let $n \ge 1$.
Let $S \subseteq D^n$ be a set.
Let $E_1$ and $E_2$ be rules that assign a real number to each $f \in \mathcal{F}$ and each $(x_1, \ldots, x_n) \in S$.
The rules are built from $f$, the variables and fixed given functions by composition and arithmetic operations.
The *functional inequality* given by $E_1$ and $E_2$ is the condition
$$E_1(f; x_1, \ldots, x_n) \le E_2(f; x_1, \ldots, x_n) \quad \text{for all } (x_1, \ldots, x_n) \in S,$$
and a *solution* is a function $f \in \mathcal{F}$ that satisfies it.

## Remarks

Example: the subadditivity inequality $f(x + y) \le f(x) + f(y)$ on $D = [0, \infty)$, with $S = D^2$.
The function $f(x) = \sqrt{x}$ is a solution: both sides are nonnegative, and squaring gives $x + y \le x + y + 2\sqrt{xy}$.

Non-example: $f(x) = x^2$ is not a solution of the same inequality, since $f(1 + 1) = 4 > 2 = f(1) + f(1)$.

As with a [[functional equation|SRAN4EKWS0]], the class $\mathcal{F}$ matters: one may ask for all solutions, or only for continuous, measurable or bounded ones.
A system that mixes equations and inequalities is also common, for example a function that is additive and bounded above.
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<document source="https://tamnd.github.io/math2/node/49JSD49P7X.md" tag="49JSD49P7X" kind="statement">
# Weierstrass preparation theorem

## Statement

Let $(A, \mathfrak m)$ be a [[complete Noetherian local ring|G9Q61B8A7C]], and let $f = \sum_{i \ge 0} a_i X^i$ be an element of the [[formal power series|033WGN00BF]] ring $A[\![X]\!]$.
Suppose that some coefficient $a_i$ does not lie in $\mathfrak m$, and let $d$ be the least such index.
Then there are a unique [[unit|0XVEZDE9J8]] $u$ of $A[\![X]\!]$ and a unique [[monic polynomial|0C3QBCR46M]]
$$g = X^d + b_{d-1} X^{d-1} + \cdots + b_1 X + b_0, \qquad b_0, \dots, b_{d-1} \in \mathfrak m,$$
such that $f = u g$.

## Remarks

The proof goes through the Weierstrass division theorem.
Under the same hypothesis, every $h \in A[\![X]\!]$ can be written uniquely as $h = q f + r$ with $q \in A[\![X]\!]$ and $r$ a polynomial of degree less than $d$.
Dividing $X^d$ by $f$ gives $X^d = q f + r$, and one checks that $q$ is a unit and $g = X^d - r$ has its lower coefficients in $\mathfrak m$.
The division itself is built by successive approximation modulo powers of $\mathfrak m$, which converges because $A$ is complete.
The hypothesis on the coefficients is needed.
For $A = k[\![Y]\!]$ and $f = Y$, every coefficient lies in $\mathfrak m = (Y)$, and $f$ is not of the form $u g$.
Indeed, modulo $\mathfrak m$ the product $u g$ becomes $\bar u X^d \neq 0$, while $f$ becomes $0$.
Applied to $A = k[\![X_1, \dots, X_{n-1}]\!]$, the theorem says that a power series in $n$ variables which is nonzero modulo $(X_1, \dots, X_{n-1})$ is, up to a unit, a polynomial in $X_n$.
The original form of the theorem is analytic.
Let $f$ be a [[holomorphic function|0NAHXR1FRH]] near $0 \in \C^n$ such that $z_n \mapsto f(0, \dots, 0, z_n)$ has a zero of finite order $d$ at $0$.
Then the same factorization holds with convergent series.
Here $g$ is a monic polynomial in $z_n$ whose lower coefficients are holomorphic in $z_1, \dots, z_{n-1}$ and vanish at $0$.

## Explanations

### L1 human

A power series can have infinitely many terms, which makes it hard to handle.
The theorem says that, near the origin, such a series behaves like an ordinary polynomial in one of its variables, after multiplication by a series that never vanishes there.
So questions about zeros of power series reduce to questions about zeros of polynomials.

### L2 human

The theorem is the main tool to prove, by induction on $n$, that $k[\![X_1, \dots, X_n]\!]$ over a field $k$ is a Noetherian unique factorization domain.
The same holds for the ring of convergent power series in $n$ complex variables.
In number theory, for $A = \Z_p$, it shows that a power series in $\Z_p[\![X]\!]$ not divisible by $p$ is a unit times a polynomial, which is used in Iwasawa theory.
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<document source="https://tamnd.github.io/math2/node/49MX22EDS7.md" tag="49MX22EDS7" kind="definition">
# Catastrophic cancellation

## Definition

Let $x, y$ be real numbers with $x \ne y$, and let $\tilde x = x(1 + \delta_1)$ and $\tilde y = y(1 + \delta_2)$ be approximations of them with relative errors $\delta_1, \delta_2$, for example [[round-off errors|CPN6SD3BTV]].
Then $\tilde x - \tilde y = (x - y)(1 + \delta)$ with
$$\delta = \frac{x \delta_1 - y \delta_2}{x - y}, \qquad \abs{\delta} \le \frac{\abs{x} + \abs{y}}{\abs{x - y}} \max(\abs{\delta_1}, \abs{\delta_2}).$$
*Catastrophic cancellation* is the case $\abs{x - y} \ll \abs{x} + \abs{y}$.
Then the relative error $\delta$ of the computed difference can be much larger than the relative errors of the inputs.

## Remarks

Example: $x = 1.2341$, $y = 1.2259$ rounded to three significant digits give $\tilde x = \tilde y = 1.23$, so $\tilde x - \tilde y = 0$ while $x - y = 0.0082$; the relative error is $\abs{\delta} = 1$, although $\abs{\delta_1}, \abs{\delta_2} < 0.004$.
Here the factor $(\abs{x} + \abs{y})/\abs{x - y}$ is about $300$.
Non-example: for $x = 1.2341$ and $y = -1.2259$ the factor is $1$, so the relative error of $\tilde x - \tilde y$ is at most the larger input error; nothing cancels.
The factor $(\abs{x} + \abs{y})/\abs{x - y}$ is the relative [[condition number|AB3S6MHJ8X]] of subtraction.
By the Sterbenz lemma, if $x$ and $y$ are floating-point numbers with $y/2 \le x \le 2y$, then $x - y$ is a floating-point number, so the subtraction itself adds no error.
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<document source="https://tamnd.github.io/math2/node/49SVWS83ZX.md" tag="49SVWS83ZX" kind="definition">
# Sectional curvature

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] with curvature $R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$ (see [[Riemann curvature tensor|JA29ETCM02]]).
Let $p \in M$ and let $\sigma \subseteq T_pM$ be a $2$-dimensional linear subspace, spanned by $u, v$.
The *sectional curvature* of $\sigma$ is
$$K(\sigma) = K(u, v) = \frac{g\bigl(R(u, v)v, u\bigr)}{g(u, u) \, g(v, v) - g(u, v)^2}.$$

## Remarks

The value does not depend on the choice of basis $u, v$ of $\sigma$, because numerator and denominator both change by the square of the determinant of the change of basis.

Example: on the unit sphere $S^n$, $R(X, Y)Z = g(Y, Z)X - g(X, Z)Y$, so for orthonormal $u, v$ the numerator is $g(g(v, v)u - g(u, v)v, u) = 1$ and $K(\sigma) = 1$ for every plane.

Non-example: $K(u, v)$ is not defined when $u$ and $v$ are linearly dependent, since then they span no plane and the denominator is $0$.

For a surface, the only plane at $p$ is $T_pM$, and $K$ is the [[Gaussian curvature|SQVQ5DQT6W]].
A manifold has constant sectional curvature $k$ if and only if $R(X, Y)Z = k\bigl(g(Y, Z)X - g(X, Z)Y\bigr)$.
With the opposite sign convention for $R$, the numerator is written $g(R(u, v)u, v)$.
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<document source="https://tamnd.github.io/math2/node/49VHC2VWNF.md" tag="49VHC2VWNF" kind="definition">
# Hopf algebra

## Definition

Let $K$ be a field.
A *Hopf algebra* over $K$ is a [[bialgebra|Q9HHVN606G]] $(H, m, \eta, \Delta, \varepsilon)$ over $K$ together with a linear map $S \colon H \to H$ such that
$$m \circ (S \otimes \id_H) \circ \Delta = \eta \circ \varepsilon = m \circ (\id_H \otimes S) \circ \Delta.$$
The map $S$ is the [[antipode|PG3ECG5D0G]] of $H$.

## Remarks

Example: the [[group algebra|MNP2HBWB5Q]] $KG$ of a group $G$, with $\Delta(g) = g \otimes g$, $\varepsilon(g) = 1$ and $S(g) = g^{-1}$.
The check on a basis element $g$: $m(S \otimes \id)\Delta(g) = g^{-1} g = 1 = \varepsilon(g) 1$, and likewise $g g^{-1} = 1$.
Example: the universal enveloping algebra of a Lie algebra $\mathfrak{g}$, with $\Delta(x) = x \otimes 1 + 1 \otimes x$, $\varepsilon(x) = 0$ and $S(x) = -x$ for $x \in \mathfrak{g}$.
Non-example: the monoid algebra $K[x]$ of $(\N, +)$, with $\Delta(x) = x \otimes x$ and $\varepsilon(x) = 1$, is a bialgebra but not a Hopf algebra.
An antipode would need $S(x) x = \varepsilon(x) 1 = 1$, but $x$ has no inverse in $K[x]$.
The antipode of a Hopf algebra is unique, because it is the inverse of $\id_H$ in the convolution algebra $\End_K(H)$.
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# Perfect ring

## Summary

A left perfect ring is a ring $R$ such that $R/J(R)$ is semisimple and the [[Jacobson radical|XA0DCNH7WY]] $J(R)$ is left T-nilpotent.
By [[Bass's theorem P|D2M8BSW720]], these are exactly the rings over which every left module has a [[projective cover|WGYXR97NPK]].
Left perfect rings are [[semiperfect|ZRC2S3ECFK]], and [[semiprimary rings|XVP517QRH5]], in particular left Artinian rings, are left and right perfect.
Left perfect and right perfect are different conditions.

## Explanations

### L1 human

In a semiperfect ring the finitely generated modules have projective covers, a "best" projective module mapping onto them.
A perfect ring asks this of all modules, however large.
This needs a strong smallness condition on the Jacobson radical, called T-nilpotence.
It says that for every infinite sequence of elements of the radical, some initial product is already zero.
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<document source="https://tamnd.github.io/math2/node/4A5MJY4XPV.md" tag="4A5MJY4XPV" kind="statement">
# Fefferman's ball multiplier theorem

## Statement

Let $n \ge 2$, let $B = \set{\xi \in \R^n : \abs{\xi} \le 1}$ be the closed unit ball, and let $\mathbf{1}_B$ be its indicator function.
Then $\mathbf{1}_B$ is an [[$L^p$ multiplier|RAGR38Q40B]] on $\R^n$ if and only if $p = 2$.

## Remarks

For $p = 2$, the indicator is bounded, so the Plancherel theorem gives the bound.
For $p \ne 2$, Charles Fefferman's 1971 proof uses [[Kakeya sets|6PEN7AKC7M]] of small measure in the plane.
By a lemma of Yves Meyer, a bound for the ball multiplier would give a vector-valued bound for the multipliers of many half-planes with different normal directions.
A Kakeya construction gives many thin rectangles whose union has small area while their translates in the direction of their long sides are disjoint, and this contradicts that vector-valued bound.
By duality, it suffices to disprove the case $p > 2$.

The theorem contrasts with dimension one, where the indicator of an interval is an $L^p$ multiplier for all $1 < p < \infty$, since it is a combination of modulated Hilbert transforms.
It also contrasts with cubes: the indicator of a cube is a product of indicators of intervals, so it is an $L^p$ multiplier for $1 < p < \infty$.

A consequence is that the spherical partial sums $(\mathbf{1}_{RB} \hat f)^\vee$ of the Fourier integral of $f \in L^p(\R^n)$ need not converge to $f$ in $L^p$ when $p \ne 2$ and $n \ge 2$.

## Explanations

### L1 human

Take a function in several variables and keep only its frequencies inside a ball, throwing the rest away.
For the energy norm $L^2$ this is harmless.
The theorem says that for every other $L^p$ norm, this sharp round cut can make functions arbitrarily large compared to what they were.
The curved boundary of the ball is the problem: cutting along a flat face, as with a cube, is fine.

### L2 human

This result explains why one studies the smoothed [[Bochner–Riesz means|AFS2DCWCD3]] instead of sharp spherical partial sums in higher dimensions.
Its proof brought the Kakeya problem into Fourier analysis, a link that runs through the [[restriction conjecture|9CE11G48JY]] as well.
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<document source="https://tamnd.github.io/math2/node/4A7H5AG0NF.md" tag="4A7H5AG0NF" kind="definition">
# Homotopic maps

## Definition

Let $X$ and $Y$ be topological spaces and let $f, g \colon X \to Y$ be [[continuous maps|0VGFHTM75Y]].
The maps $f$ and $g$ are *homotopic*, written $f \simeq g$, if there is a [[homotopy|9YA6S6HD9V]] from $f$ to $g$.

## Remarks

The relation $\simeq$ is an [[equivalence relation|0G8VEZ3WTK]] on the set of continuous maps $X \to Y$.
Reflexivity uses the constant homotopy $H(x, t) = f(x)$, symmetry reverses time, and transitivity runs two homotopies one after the other, each at double speed.
The equivalence class of $f$ is its *homotopy class* $[f]$, and the set of classes is written $[X, Y]$.
Homotopy is compatible with composition: if $f \simeq f'$ and $g \simeq g'$, then $g \circ f \simeq g' \circ f'$.
Example: the maps $f(x) = x$ and $g(x) = 0$ from $[0, 1]$ to $\R$ are homotopic via $H(x, t) = (1 - t)x$.
Non-example: the identity of the [[circle|060W5CBZP6]] $S^1$ and a constant map $S^1 \to S^1$ are not homotopic, because they induce different maps on the [[fundamental group|S1XK3HXHFX]].
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<document source="https://tamnd.github.io/math2/node/4A7VAXPG4F.md" tag="4A7VAXPG4F" kind="concept">
# Pluriharmonic function

## Summary

A pluriharmonic function is a real function on an open subset of $\C^n$ whose restriction to every complex line is [[harmonic|0DEB0JMN0W]].
Equivalently, all its mixed derivatives $\partial^2 u / \partial z_j \partial \bar z_k$ vanish.
Locally, pluriharmonic functions are exactly the real parts of [[holomorphic functions|507C34W0ZT]].
They are the functions $u$ such that both $u$ and $-u$ are [[plurisubharmonic|FF2P7ER3QT]].

## Explanations

### L1 human

In one complex variable, harmonic functions are the real parts of holomorphic functions.
In several variables, the real parts of holomorphic functions satisfy more equations than the Laplace equation: they are harmonic on each complex line.
These functions are called pluriharmonic.
For example, $u(z_1, z_2) = \operatorname{Re}(z_1 z_2)$ is pluriharmonic on $\C^2$.
The function $\abs{z_1}^2 - \abs{z_2}^2$ is harmonic on $\R^4$ but not pluriharmonic, since on the line $z_2 = 0$ it is $\abs{z_1}^2$, which is not harmonic.
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<document source="https://tamnd.github.io/math2/node/4A82S5FVRT.md" tag="4A82S5FVRT" kind="concept">
# Continuous dual space

## Summary

The continuous dual space of a [[topological vector space|T32W553W95]] $E$ is the vector space $E'$ of all continuous [[linear forms|07YQ1C78G0]] on $E$.
It is a subspace of the algebraic [[dual space|07JBSZE0V4]], usually much smaller in infinite dimension.
For a normed space it is a Banach space with the dual norm.
Duality theory studies $E$ through $E'$ and the topologies on both, such as the [[weak topology|P6KHVSTPZV]] and the [[strong dual|RG52DZ1A8Y]] topology.

## Explanations

### L1 human

A linear form assigns a number to each vector, in a linear way.
In infinite dimension most linear forms are wild, so one keeps only the continuous ones.
Small changes of the vector should give small changes of the number.
For example, on continuous functions on $[0, 1]$, the map $f \mapsto \int_0^1 f$ and the map $f \mapsto f(1/2)$ are continuous linear forms for uniform convergence.
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<document source="https://tamnd.github.io/math2/node/4A84VHDTHV.md" tag="4A84VHDTHV" kind="concept">
# Soundness

## Summary

A proof system is sound when everything it proves is true.
More precisely, whenever $\Gamma \vdash \varphi$ in the system, also $\Gamma \models \varphi$ as a [[logical consequence|1HK0VYWTV7]].
Soundness is the converse of completeness, and the standard proof systems for propositional and first-order logic have both.
A sound system can never prove a false statement from true premises.

## Explanations

### L1 human

A proof system is sound if its rules never lead from true premises to a false conclusion.
For example, modus ponens is a safe rule: if $p$ and "if $p$ then $q$" are true, then $q$ is true.
A rule such as "from $q$ and if $p$ then $q$, infer $p$" is unsafe, because $q$ may be true while $p$ is false.
In philosophy, an argument is also called sound when it is valid and all its premises are true.
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<document source="https://tamnd.github.io/math2/node/4ACJ5TPHJX.md" tag="4ACJ5TPHJX" kind="definition">
# Knot sum

## Definition

Let $K_1$ and $K_2$ be oriented tame [[knots|QYEAZTYGPT]] in $S^3$.
Choose disjoint closed $3$-balls $B_1, B_2 \subset S^3$ with $K_i \subset B_i$.
Choose a band $b \cong [0,1] \times [0,1]$ in $S^3$ that meets $B_1 \cup B_2$ only in $\set{0, 1} \times [0,1]$.
Require $\set{i-1} \times [0,1] \subset K_i$ for $i = 1, 2$.
Choose the band so that the orientations of $K_1$ and $K_2$ agree with an orientation of the knot below.
The *knot sum* $K_1 \# K_2$ is the oriented knot
$$\big( (K_1 \cup K_2) \setminus (\set{0,1} \times (0,1)) \big) \cup \big( [0,1] \times \set{0, 1} \big).$$

## Remarks

The isotopy class of $K_1 \# K_2$ does not depend on the balls or the band.
Example: if $K_2$ is the [[unknot|1TQ06R4Q66]], the band slides into a trivial arc and $K_1 \# K_2$ is isotopic to $K_1$.
Non-example: if the band is allowed to enter $B_1$ away from $K_1$, it may wind around $K_1$.
Gluing along such a band can give a knot different from $K_1 \# K_2$, so the condition on the band matters.
The sum is commutative and associative on isotopy classes of oriented knots.
The genus of a [[Seifert surface|K866ND3WF7]] is additive, $g(K_1 \# K_2) = g(K_1) + g(K_2)$, so a nontrivial knot has no inverse.
In a diagram, the sum is drawn by placing the two diagrams side by side and replacing two facing arcs by two parallel arcs.
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<document source="https://tamnd.github.io/math2/node/4AF7AF7690.md" tag="4AF7AF7690" kind="problem">
# Conjugacy problem

## Problem

Let $G$ be a [[finitely generated group|082DWWXMRJ]] with a finite generating set $S$.
A word is a finite sequence of elements of $S$ and their inverses.
The *conjugacy problem* for $G$ asks for an algorithm that takes two words $u$ and $v$ and decides whether they represent [[conjugate|0DXD4QSZ8A]] elements of $G$.
That is, the algorithm decides whether $v = g u g^{-1}$ in $G$ for some $g \in G$.

## Remarks

Whether the problem is solvable does not depend on the finite generating set.
A word $w$ represents the identity if and only if it is conjugate to the empty word.
So an algorithm for the conjugacy problem solves the [[word problem|PBGTTAVMMT]], and every group with unsolvable word problem has unsolvable conjugacy problem.
The converse fails: there are finitely presented groups with solvable word problem and unsolvable conjugacy problem.

The problem is solvable in many classes of groups.
In a [[free group|0KTW26C80J]], two elements are conjugate if and only if their cyclically reduced forms are cyclic permutations of each other.
It is also solvable for finitely generated abelian groups, where it is the word problem, for [[hyperbolic groups|JWJ8YPR7EB]], and for [[braid groups|BNR4H5GCTP]].
It is one of the three decision problems, with the word problem and the [[isomorphism problem|MP3F49QT3C]], that Max Dehn posed for groups given by presentations.

## Explanations

### L1 human

Two elements of a group are conjugate when one becomes the other after a change of point of view, $v = g u g^{-1}$.
The conjugacy problem asks whether a computer can always decide this for a group given by generators.
In a free group the answer is easy: cancel what you can, also around the ends of the word, and compare the results up to rotation.
For example, in the free group on $a$ and $b$, the word $b a b^{-1}$ reduces cyclically to $a$, so it is conjugate to $a$ and not to $b$.
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<document source="https://tamnd.github.io/math2/node/4AG8ZWK2BB.md" tag="4AG8ZWK2BB" kind="definition">
# Ringed space

## Definition

A *ringed space* is a pair $(X, \mathcal O_X)$ where $X$ is a [[topological space|0CQHFPGJ45]] and $\mathcal O_X$ is a [[sheaf|TB6TM6R4MD]] of [[commutative rings|0310REJJY8]] on $X$.
The sheaf $\mathcal O_X$ is called the *structure sheaf*.
A *morphism of ringed spaces* $(f, f^\#) \colon (X, \mathcal O_X) \to (Y, \mathcal O_Y)$ consists of two parts.
The first is a [[continuous map|0VGFHTM75Y]] $f \colon X \to Y$.
The second is a ring homomorphism $f^\#_V \colon \mathcal O_Y(V) \to \mathcal O_X(f^{-1}(V))$ for each open set $V \subseteq Y$, compatible with restriction to smaller open sets.

## Remarks

Example: let $X$ be a topological space and let $\mathcal O_X(U)$ be the ring of continuous functions $U \to \R$; then $(X, \mathcal O_X)$ is a ringed space.
The [[stalk|JTAKAPTT8H]] at each point $x$ is a [[local ring|MZ02BKPHJA]], whose maximal ideal consists of the germs that vanish at $x$.

Example: an [[algebraic variety|HV6YBS0YDH]] $X$ over $k$, with $\mathcal O_X(U)$ the ring of [[regular functions|4M94FZH9Y8]] on $U$, is a ringed space.

Example: any topological space $X$ with the sheaf of locally constant functions to a commutative ring $A$ is a ringed space.

Non-example: let $\mathcal B(U)$ be the ring of bounded continuous functions on an open set $U \subseteq \R$.
Then $(\R, \mathcal B)$ is not a ringed space, because $\mathcal B$ is not a sheaf: the sections $x \mapsto x$ on the intervals $(-m, m)$ do not glue.

A ringed space whose stalks are all local rings is a locally ringed space; schemes and varieties are of this kind.
</document>

<document source="https://tamnd.github.io/math2/node/4AHBMJYCX1.md" tag="4AHBMJYCX1" kind="statement">
# Hamiltonian Monte Carlo

## Statement

Let $U \colon \R^d \to \R$ be continuously differentiable with $Z = \int_{\R^d} e^{-U(q)} \, dq < \infty$, and let $\pi$ be the [[probability density|0ZY9DX8FBF]] $\pi(q) = e^{-U(q)} / Z$.
Let $H(q,p) = U(q) + \tfrac{1}{2} \abs{p}^2$ for $q, p \in \R^d$.
Fix a step size $\varepsilon > 0$ and a number of steps $L \ge 1$.
The leapfrog step $\Psi(q,p) = (q', p')$ is
$$p_{1/2} = p - \tfrac{\varepsilon}{2} \nabla U(q), \qquad q' = q + \varepsilon \, p_{1/2}, \qquad p' = p_{1/2} - \tfrac{\varepsilon}{2} \nabla U(q'),$$
where $\nabla U$ is the [[gradient|0ABK9V9Y64]] of $U$, and $\Phi = \Psi^{L}$ is $L$ leapfrog steps.
One *Hamiltonian Monte Carlo* step from $q \in \R^d$ is as follows.

1. Draw $p$ from the standard [[normal distribution|0XN3K7RSZY]] on $\R^d$.
2. Compute $(q^*, p^*) = \Phi(q, p)$.
3. With probability $\min\big(1, e^{H(q,p) - H(q^*, p^*)}\big)$ move to $q^*$; otherwise stay at $q$.

Then $\pi$ is a [[stationary distribution|0KRH5157WP]] of this Markov transition: if $q$ has density $\pi$, the new state also has density $\pi$.

## Remarks

The proof works on the joint density proportional to $e^{-H(q,p)}$ on $\R^{2d}$, whose $q$-marginal is $\pi$ and whose $p$-marginal is standard normal.
Let $R(q,p) = (q,-p)$; then $F = R \circ \Phi$ is an [[involution|GP56NBN9RY]], since each leapfrog step is reversed by flipping the momentum, and $F$ preserves volume, since each of the three substeps is a shear.
A Metropolis step with a deterministic, volume-preserving, involutive proposal $F$ and acceptance probability $\min(1, e^{H - H \circ F})$ satisfies [[detailed balance|CTGN6CK1ZS]] for $e^{-H}$, and $H \circ R = H$.
Resampling $p$ in step 1 also leaves $e^{-H}$ invariant, so the composition does, and hence its $q$-marginal $\pi$ is invariant.
The leapfrog map is a [[symplectic integrator|B4C25SXFRE]] for the [[Hamiltonian system|0EDKGA2FAK]] $\dot q = p$, $\dot p = -\nabla U(q)$.
The exact flow would keep $H$ constant and always accept, and the accept step corrects the discretisation error.
Example: for $U(q) = \tfrac{1}{2} q^2$ on $\R$, the target is standard normal, and with small $\varepsilon$ the trajectories follow circles in the $(q,p)$ plane, so the chain moves far in each step.
Invariance alone does not give convergence.
For this $U$, if $\varepsilon L$ is half a period of the exact flow, the chain nearly alternates between $q$ and $-q$.
For this reason $\varepsilon$ or $L$ is often randomised in practice.
More generally $p$ may be drawn from a normal law with a positive definite mass matrix $M$, with $\tfrac{1}{2} p^T M^{-1} p$ in $H$ and $q' = q + \varepsilon M^{-1} p_{1/2}$.

## Explanations

### L1 human

Hamiltonian Monte Carlo is a way to draw random samples from a probability distribution on a high-dimensional space.
It imagines a ball rolling on a landscape whose height is $U$, so that low places are likely places.
At each step the ball gets a random kick, rolls for a while under simulated physics, and its end point is the next proposal.
A final accept-or-reject check makes the method exact despite the errors of the simulation.
The result says that, in the long run, the samples follow the intended distribution.

### L2 human

HMC is a [[Markov chain Monte Carlo|ZGNBR2SPZ0]] method that uses the gradient of the log-density to make long, directed moves.
It is a special case of the [[Metropolis–Hastings algorithm|MKHF449E22]] on the extended space of positions and momenta.
The two properties of the leapfrog map that make the acceptance rule simple are reversibility and volume preservation; energy conservation is only needed for a high acceptance rate.
Compared with random-walk proposals, HMC scales much better with dimension for smooth targets, which is why it is the default sampler in several statistics packages.
Choosing $\varepsilon$ and $L$ is the main practical difficulty, and adaptive schemes such as the no-U-turn sampler address it.
</document>

<document source="https://tamnd.github.io/math2/node/4AKCQERM89.md" tag="4AKCQERM89" kind="concept">
# Truncation error

## Summary

Truncation error is the error made by replacing an infinite or exact mathematical process by a finite approximation, even when all arithmetic is exact.
Examples are cutting a [[Taylor series|0Z49YH6ZWY]] after finitely many terms, replacing a derivative by a difference quotient, or an integral by a finite sum.
It usually shrinks as the number of terms grows or the step size $h$ goes to $0$, often like a power $h^p$.
For ODE methods, the error made in one step is the [[local truncation error|T220HF0562]].
It is the counterpart of [[round-off error|CPN6SD3BTV]], which comes from finite-precision arithmetic.

## Explanations

### L1 human

Many quantities are defined by a process that never ends, such as an infinite sum or a limit.
A computer must stop after finitely many steps, and what is left out is the truncation error.
For example, $e^x = 1 + x + x^2/2 + \cdots$, and stopping after $1 + x$ gives an error of about $x^2/2$ for small $x$.
Taking more terms or a smaller step makes the truncation error smaller, but it may make the round-off error larger, so one has to balance the two.
</document>

<document source="https://tamnd.github.io/math2/node/4AVF0RN7HV.md" tag="4AVF0RN7HV" kind="concept">
# Graph enumeration

## Summary

Graph enumeration is the part of combinatorics that counts [[graphs|00C0BR7D7H]] with a given property, either as labelled graphs or up to [[isomorphism|0F7QKGJS6B]].
There are $2^{\binom{n}{2}}$ labelled graphs on $n$ given vertices, and [[Cayley's formula|0M2SSGSYPE]] counts the labelled trees.
Unlabelled counts are harder, and tools such as the [[Pólya enumeration theorem|MPYQ39P01F]] and generating functions are used for them.
It is a central part of [[enumerative combinatorics|EX382NQTGP]].

## Explanations

### L1 human

Graph enumeration asks "how many graphs of this kind are there?".
There are two ways to count.
In labelled counting the vertices have names, so two drawings with different names count as different graphs.
In unlabelled counting only the shape matters.
For three vertices there are $8$ labelled graphs but only $4$ shapes: no edge, one edge, a path with two edges, and a triangle.
</document>

<document source="https://tamnd.github.io/math2/node/4AWC5Q4FHE.md" tag="4AWC5Q4FHE" kind="definition">
# Superharmonic function

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]].
For $a \in \R^n$ and $r > 0$, let $B(a, r)$ be the open ball and $S(a, r)$ the sphere of centre $a$ and radius $r$.
Let $\sigma_{a,r}$ be the surface measure on $S(a, r)$ normalised to total mass $1$.
A function $u \colon \Omega \to (-\infty, +\infty]$ is *superharmonic* if:

1. $u$ is [[lower semicontinuous|RN4G8SGP6P]].
2. $u$ is not identically $+\infty$ on any connected component of $\Omega$.
3. For every $a \in \Omega$ and $r > 0$ with $\overline{B(a, r)} \subseteq \Omega$,
$$u(a) \ge \int_{S(a, r)} u \, d\sigma_{a,r}.$$

## Remarks

A function $u$ is superharmonic if and only if $-u$ is [[subharmonic|Z1MHMGFNFW]] and not identically $-\infty$ on any component.
It suffices to require clause 3 for all small enough $r$ at each point.
A function $u$ of class $C^2$ is superharmonic if and only if its [[Laplacian|0E0B6RA8K0]] satisfies $\Delta u \le 0$.
A function is [[harmonic|0DEB0JMN0W]] if and only if it is both superharmonic and subharmonic.

Example: for $n \ge 3$ let $u(x) = \abs{x}^{2-n}$ on $\R^n$, with $u(0) = +\infty$.
It is lower semicontinuous and finite off $0$.
Its mean over $S(a, r)$ equals $\max(\abs{a}, r)^{2-n}$, which is at most $\abs{a}^{2-n} = u(a)$, and at $a = 0$ the value $+\infty$ dominates.
So $u$ is superharmonic.
In the plane, $u(x) = -\log\abs{x}$ is superharmonic in the same way.

Non-example: $u(x) = \abs{x}^2$ is continuous on $\R^n$, but its mean over $S(a, r)$ is $\abs{a}^2 + r^2 > u(a)$, so clause 3 fails.
</document>

<document source="https://tamnd.github.io/math2/node/4AXC8NESM4.md" tag="4AXC8NESM4" kind="definition">
# Proper holomorphic map

## Definition

Let $\Omega_1 \subset \C^n$ and $\Omega_2 \subset \C^m$ be domains, that is, nonempty connected open sets.
A [[holomorphic map|3MQTWKBBBT]] $F \colon \Omega_1 \to \Omega_2$ is *proper* if $F^{-1}(K)$ is [[compact|0TVJZYB6SH]] for every compact set $K \subset \Omega_2$.
The same condition defines proper holomorphic maps between complex manifolds.

## Remarks

Equivalent form: $F$ is proper if and only if it has the following property.
For every sequence $(z_k)$ in $\Omega_1$ that eventually leaves every compact subset of $\Omega_1$, the sequence $(F(z_k))$ eventually leaves every compact subset of $\Omega_2$.
For bounded domains this says that $\operatorname{dist}(F(z), \partial\Omega_2) \to 0$ as $\operatorname{dist}(z, \partial\Omega_1) \to 0$.

Example: let $\mathbb{D}$ be the unit disc and $F(z) = z^k$ with $k \ge 1$.
A compact $K \subset \mathbb{D}$ lies in a closed disc $\set{\abs{w} \le r}$ with $r < 1$, and $F^{-1}(K)$ is a closed subset of $\set{\abs{z} \le r^{1/k}}$, hence compact.
The map $(z_1, z_2) \mapsto (z_1, z_2^2)$ of the unit polydisc in $\C^2$ to itself is proper for the same reason.

Non-example: the inclusion $F \colon \set{\abs{z} < 1/2} \to \mathbb{D}$ is holomorphic but not proper.
For the compact set $K = \set{\abs{w} \le 1/2}$, the preimage $F^{-1}(K) = \set{\abs{z} < 1/2}$ is not compact.

A proper holomorphic map between domains of the same dimension $n$ is surjective, its fibres are finite, and it is a finite branched covering.
A [[biholomorphism|FX92PP3HZR]] is a proper holomorphic map.
The proper holomorphic self-maps of $\mathbb{D}$ are exactly the finite [[Blaschke products|9N6R5ADJXT]].
</document>

<document source="https://tamnd.github.io/math2/node/4AY3RMCTA3.md" tag="4AY3RMCTA3" kind="definition">
# Goldie ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
For a subset $X \subseteq R$, the *right annihilator* of $X$ is $r(X) = \set{a \in R : x a = 0 \text{ for all } x \in X}$, a right ideal of $R$.
The ring $R$ is *right Goldie* if:

1. $R$ has finite right uniform dimension: there is no infinite family of nonzero right ideals $I_1, I_2, \ldots$ of $R$ whose sum is direct;
2. $R$ satisfies the [[ascending chain condition|0GZ6Y1H4RN]] on right annihilators: every increasing chain $r(X_1) \subseteq r(X_2) \subseteq \cdots$ is eventually constant.

Left Goldie rings are defined in the same way with left ideals and left [[annihilators|HB4XDKJTW8]].
A ring is *Goldie* if it is both right and left Goldie.

## Remarks

Example: a right [[Noetherian ring|0670T59HGE]] $R$ is right Goldie.
An infinite direct sum $I_1 \oplus I_2 \oplus \cdots$ of nonzero right ideals would give a strictly increasing chain $I_1 \subset I_1 \oplus I_2 \subset \cdots$, which is impossible.
Right annihilators are right ideals, so they satisfy the ascending chain condition.

Example: every commutative integral domain is Goldie, since two nonzero ideals $I, J$ always meet in the nonzero ideal $IJ$, and the only annihilators are $0$ and $R$.

Non-example: let $k$ be a field and $R = \prod_{n \ge 1} k$.
The ideals $I_n = k e_n$, where $e_n$ has $1$ in coordinate $n$ and $0$ elsewhere, are nonzero and their sum is direct, so clause 1 fails.

Clause 1 can be stated with [[uniform modules|YXEVE39Q31]]: $R_R$ has finite uniform dimension exactly when it contains an essential submodule that is a finite direct sum of uniform right ideals.
</document>

<document source="https://tamnd.github.io/math2/node/4B0R70F4VE.md" tag="4B0R70F4VE" kind="statement">
# Schoenberg's embedding criterion

## Statement

Let $n \ge 2$ and let $D = (d_{ij})$ be a real symmetric $n \times n$ matrix with $d_{ii} = 0$ for all $i$.
Let $G$ be the $(n - 1) \times (n - 1)$ matrix with entries
$$G_{ij} = \frac{1}{2} \left( d_{1i} + d_{1j} - d_{ij} \right), \qquad 2 \le i, j \le n.$$
Then the following hold.

1. $D$ is a [[Euclidean distance matrix|1BCJ7A73KS]] if and only if $G$ is [[positive semidefinite|01MMVW8CKC]].
2. In that case, points of $\R^k$ realize $D$ if and only if $k$ is at least the [[rank|0ZC9YTXZRY]] of $G$.

## Remarks

If $x_1, \ldots, x_n$ realize $D$, translate so that $x_1 = 0$.
The polarization identity $\ip{x_i}{x_j} = \frac{1}{2}(\norm{x_i}^2 + \norm{x_j}^2 - \norm{x_i - x_j}^2)$ shows that $G$ is the [[Gram matrix|00SPJSDT3Z]] of $x_2, \ldots, x_n$, hence positive semidefinite.
Conversely, a positive semidefinite $G$ of rank $k$ factors as $G = X^T X$ with $X$ a $k \times (n - 1)$ matrix.
Its columns, together with $x_1 = 0$, realize $D$ in $\R^k$.

Example: the matrix with rows $(0, 1, 9)$, $(1, 0, 1)$ and $(9, 1, 0)$ gives $G = \begin{pmatrix} 1 & 4.5 \\ 4.5 & 9 \end{pmatrix}$.
Its determinant is $9 - 20.25 < 0$, so $G$ is not positive semidefinite and $D$ is not a Euclidean distance matrix.

An equivalent form uses the centring matrix $J = I - \frac{1}{n} \mathbf{1}\mathbf{1}^T$: $D$ is a Euclidean distance matrix if and only if $-\frac{1}{2} J D J$ is positive semidefinite.

## Explanations

### L1 human

A table of squared distances comes from real points exactly when a related matrix has no negative eigenvalues.
This turns a geometric question into a check that a computer does quickly.
The number of nonzero eigenvalues tells you how many dimensions the points need.

### L2 human

The criterion is the basis of classical multidimensional scaling.
One forms $-\frac{1}{2} J D J$, takes its eigendecomposition, and reads off coordinates from the top eigenvectors.
It is the matrix side of [[distance geometry|HPW5F0GQ4B]], and it is closely related to the [[Cayley–Menger determinant|KGC5CYDF53]].
The criterion is named after Isaac Schoenberg.
</document>

<document source="https://tamnd.github.io/math2/node/4B66270PGS.md" tag="4B66270PGS" kind="definition">
# Essential extension

## Definition

Fix a signature, let $B$ be an algebra and let $A$ be a [[subalgebra|7CHG6SD365]] of $B$.
The algebra $B$ is an *essential extension* of $A$ if every [[congruence|DCSXXT4SER]] $\theta$ of $B$ with $\theta \cap (A \times A) = \Delta_A$ is the trivial congruence $\Delta_B$.
Here $\Delta_X = \set{(x, x) : x \in X}$.

## Remarks

Equivalently, every [[homomorphism|KBXSZC6BV1]] $\varphi$ from $B$ whose restriction to $A$ is injective is itself injective.
An embedding $e \colon A \to B$ is called essential when $B$ is an essential extension of $e(A)$.
For modules over a ring, congruences correspond to submodules, and the condition says that every nonzero submodule of $B$ meets $A$ in a nonzero element.

Example: $\Q$ is an essential extension of $\Z$ as abelian groups.
Congruences of $\Q$ correspond to subgroups $N$, and $\theta \cap (\Z \times \Z) = \Delta_{\Z}$ means $N \cap \Z = \set{0}$.
If $N$ contains $p/q \ne 0$, then it contains the nonzero integer $p$, so $N \cap \Z = \set{0}$ forces $N = \set{0}$.

Non-example: $\Z \times \Z$ is not an essential extension of $\Z \times \set{0}$ as abelian groups.
The subgroup $N = \set{0} \times \Z$ meets $\Z \times \set{0}$ only in $0$, but $N \ne \set{0}$.
</document>

<document source="https://tamnd.github.io/math2/node/4B7M4A1MH0.md" tag="4B7M4A1MH0" kind="concept">
# Partial difference equation

## Summary

A partial difference equation is an equation for an unknown function of two or more integer variables.
It relates the values of the function at a point and at finitely many shifted points.
It is to an ordinary difference equation what a [[partial differential equation|GC5BDZ6QBC]] is to an ordinary differential equation.
Such equations arise from discretizing partial differential equations on a grid, from counting problems in combinatorics, and from lattice models in physics.

## Explanations

### L1 human

Fill the squares of an infinite grid with numbers by a local rule.
Each value is tied to the values at a few neighbouring squares.
Pascal's triangle is the classic case: each binomial coefficient is the sum of the two above it, $\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}$.
A computer that simulates heat flow on a grid also solves a partial difference equation at every time step.
</document>

<document source="https://tamnd.github.io/math2/node/4BBW1NZ1CC.md" tag="4BBW1NZ1CC" kind="definition">
# Jacobi polynomials

## Definition

Let $\alpha, \beta > -1$ be real numbers and let $n \ge 0$ be an integer.
Write $(a)_k = a (a + 1) \cdots (a + k - 1)$ for the [[Pochhammer symbol|YZHM3HSM0F]], with $(a)_0 = 1$.
The *Jacobi polynomial* of degree $n$ with parameters $\alpha, \beta$ is
$$P_n^{(\alpha, \beta)}(x) = \sum_{k=0}^{n} \frac{(n + \alpha + \beta + 1)_k \, (\alpha + k + 1)_{n-k}}{k! \, (n - k)!} \left( \frac{x - 1}{2} \right)^k.$$

## Remarks

Equivalently, $P_n^{(\alpha, \beta)}(x) = \frac{(\alpha + 1)_n}{n!} \, {}_2F_1\left( -n, n + \alpha + \beta + 1; \alpha + 1; \frac{1 - x}{2} \right)$.
Rodrigues' formula: $(1 - x)^\alpha (1 + x)^\beta P_n^{(\alpha, \beta)}(x) = \frac{(-1)^n}{2^n n!} \frac{d^n}{dx^n} \left( (1 - x)^{\alpha + n} (1 + x)^{\beta + n} \right)$.
They are orthogonal: $\int_{-1}^{1} P_m^{(\alpha, \beta)}(x) P_n^{(\alpha, \beta)}(x) (1 - x)^\alpha (1 + x)^\beta \, dx = 0$ for $m \ne n$.
Example: $P_1^{(\alpha, \beta)}(x) = (\alpha + 1) + (\alpha + \beta + 2) \frac{x - 1}{2}$; for $\alpha = \beta = 0$ this is $x$, the Legendre polynomial $P_1$.
The value at $1$ is $P_n^{(\alpha, \beta)}(1) = \frac{(\alpha + 1)_n}{n!}$, since only the term $k = 0$ survives.
Non-example: for $\alpha = -1$ the weight $(1 - x)^{-1}$ is not integrable near $x = 1$, so the clause $\alpha > -1$ fails and the orthogonality integrals diverge.
</document>

<document source="https://tamnd.github.io/math2/node/4BEWKR0EG7.md" tag="4BEWKR0EG7" kind="statement">
# Mostow rigidity theorem

## Statement

Let $n \ge 3$, and let $M$ and $N$ be closed connected hyperbolic $n$-manifolds.
That is, they are compact connected [[Riemannian manifolds|7CBT4SM83Y]] without boundary, complete, with constant [[sectional curvature|1X42JAA349]] $-1$.
Let $\phi \colon \pi_1(M) \to \pi_1(N)$ be an [[isomorphism|08MG7H8JW5]] of [[fundamental groups|S1XK3HXHFX]].
Then there is a unique [[isometry|02HPP5VCZH]] $f \colon M \to N$ that induces $\phi$, up to the choice of base points.

## Remarks

Here is the idea of Mostow's proof.
A homotopy equivalence $M \to N$ inducing $\phi$ lifts to a quasi-isometry of the universal covers, both equal to hyperbolic space $\mathbb{H}^n$.
This quasi-isometry extends to a quasiconformal homeomorphism of the sphere at infinity $S^{n-1}$, equivariant for the two actions of $\pi_1$.
Ergodicity of the action of $\pi_1(M)$ on $S^{n-1}$ forces this map to be conformal, so it comes from an isometry of $\mathbb{H}^n$.

Consequence: the volume of a closed hyperbolic manifold of dimension at least $3$ depends only on its fundamental group.
So the volume is a topological invariant.

The hypothesis $n \ge 3$ is necessary.
For a closed orientable surface of genus $g \ge 2$, the hyperbolic metrics, up to isometries isotopic to the identity, form the Teichmüller space of real dimension $6g - 6$.
Its quotient by the mapping class group still has positive dimension, so two hyperbolic surfaces of genus $g$ need not be isometric.

The theorem extends to complete hyperbolic manifolds of finite volume, by work of Marden for $n = 3$ and of Prasad in general.

## Explanations

### L1 human

A hyperbolic manifold is a space that looks everywhere like the hyperbolic plane or its higher-dimensional analogues.
The fundamental group records how loops in the space can be deformed.
In dimension $3$ and higher, Mostow proved that this purely topological information determines the whole geometry.
So the shape and the size of such a manifold cannot be changed without changing its topology.

### L2 human

In the language of Lie groups, the theorem says the following.
Two cocompact lattices in $\mathrm{SO}(n, 1)$, $n \ge 3$, that are isomorphic as abstract groups are conjugate in the isometry group of $\mathbb{H}^n$.
Mostow extended it to irreducible cocompact lattices in semisimple Lie groups not locally isomorphic to $\mathrm{SL}_2(\R)$.
In higher rank, the [[Margulis superrigidity theorem|DSKV65NHZZ]] gives a much stronger form.
By contrast, surfaces have flexible moduli, which leads to Teichmüller theory.
</document>

<document source="https://tamnd.github.io/math2/node/4BT3Y8TCDN.md" tag="4BT3Y8TCDN" kind="definition">
# Schrödinger equation

## Definition

Let $n \ge 1$, let $\hbar > 0$ and $m > 0$ be constants, and let $V \colon \R^n \to \R$ be a measurable function, the *potential*.
Let $\psi(x, t)$, for $x \in \R^n$ and $t \in \R$, be an unknown complex-valued function, and let $\Delta$ be the [[Laplacian|0E0B6RA8K0]] in $x$.
The *Schrödinger equation* with potential $V$ is the [[partial differential equation|GC5BDZ6QBC]]
$$i\hbar \, \frac{\partial \psi}{\partial t}(x, t) = -\frac{\hbar^2}{2m} \Delta \psi(x, t) + V(x) \psi(x, t).$$
With $H = -\frac{\hbar^2}{2m}\Delta + V$, the [[Schrödinger operator|89AA6XJY5K]], it reads $i\hbar\, \partial_t \psi = H\psi$.
The *time-independent* Schrödinger equation is the eigenvalue equation $H\varphi = E\varphi$; if it holds, then $\psi(x, t) = e^{-iEt/\hbar}\varphi(x)$ solves the time-dependent equation.

## Remarks

In mathematics one often sets $\hbar = 1$ and $2m = 1$, giving $i\partial_t u = -\Delta u + Vu$.

When $H$ is self-adjoint on $L^2(\R^n)$, Stone's theorem gives the solution $\psi(t) = e^{-itH/\hbar}\psi(0)$ for every initial value in $L^2(\R^n)$, and $e^{-itH/\hbar}$ is unitary, so $\norm{\psi(t)}_{L^2} = \norm{\psi(0)}_{L^2}$.

Example: with $V = 0$, $\hbar = 1$, $2m = 1$ and $k \in \R^n$, the plane wave $u(x, t) = e^{i(k \cdot x - \abs{k}^2 t)}$ is a solution.
Indeed $i\partial_t u = i \cdot (-i\abs{k}^2) u = \abs{k}^2 u$ and $-\Delta u = \abs{k}^2 u$.

Non-example: the heat equation $\partial_t u = \Delta u$ differs only by the factor $i$, but it is not a Schrödinger equation.
Its solutions decay in $L^2$ instead of keeping their norm: $u = e^{i k\cdot x - \abs{k}^2 t}$ solves it, and its modulus $e^{-\abs{k}^2 t}$ decreases.

## Explanations

### L1 human

The Schrödinger equation says that the time derivative of the wave function, times $i\hbar$, equals the energy operator applied to the wave function.
The factor $i$ makes solutions oscillate and spread rather than decay, and it keeps the total probability equal to one.
</document>

<document source="https://tamnd.github.io/math2/node/4BVWH0BN28.md" tag="4BVWH0BN28" kind="concept">
# Relative homology

## Summary

Relative homology assigns to a space $X$ and a subspace $A$ the groups $H_n(X, A)$.
They measure the homology of $X$ while ignoring everything that happens inside $A$.
It is built from the singular chains of $X$ modulo the chains of $A$.
Relative groups fit into the long exact sequence of the pair and are the setting of the excision theorem.
This makes them the main tool for computing [[singular homology|M0Q5CFJ7ZH]].

## Explanations

### L1 human

Think of $A$ as a region that is painted over and treated as if it were a single point.
A relative cycle is a chain whose boundary lies in $A$, so it may have loose ends as long as they end inside $A$.
For the interval $X = [0, 1]$ with $A = \set{0, 1}$, the path from $0$ to $1$ is a relative cycle, because its boundary consists of the two end points.
It represents a generator of $H_1([0,1], \set{0,1}) \cong \Z$, just as if the two end points were glued into a circle.
</document>

<document source="https://tamnd.github.io/math2/node/4BW0Y31347.md" tag="4BW0Y31347" kind="definition">
# Horn

## Definition

For $n \ge 0$ let $[n] = \set{0, 1, \dots, n}$, and let $\Delta^n$ be the standard $n$-simplex [[simplicial set|WW81ARB8K9]], whose $m$-simplices are the order-preserving maps $[m] \to [n]$.
Let $n \ge 1$ and $0 \le k \le n$.
The *horn* $\Lambda^n_k$ is the simplicial subset of $\Delta^n$ whose $m$-simplices are the order-preserving maps $\alpha \colon [m] \to [n]$ such that
$$\alpha([m]) \cup \set{k} \ne [n].$$
The horn is *inner* if $0 < k < n$ and *outer* if $k = 0$ or $k = n$.

## Remarks

Equivalently, $\Lambda^n_k$ is the union of the faces $d^i \Delta^{n-1} \subseteq \Delta^n$ for $i \ne k$, where $d^i$ is the face that omits the vertex $i$.
Its geometric realization is the boundary of the $n$-simplex with the interior of the face opposite the vertex $k$ removed.

Example: $\Lambda^2_1$ consists of the edges $\set{0, 1}$ and $\set{1, 2}$ and their vertices.
The edge $\set{0, 2}$ is the map $[1] \to [2]$ with image $\set{0, 2}$, and $\set{0, 2} \cup \set{1} = [2]$, so it is not in $\Lambda^2_1$.
The edge $\set{0, 1}$ has image $\set{0, 1}$, and $\set{0, 1} \cup \set{1} \ne [2]$, so it is in $\Lambda^2_1$.

Non-example: the boundary $\partial \Delta^2$ contains all three edges, so it is not a horn.
The edge $\set{0, 2}$ fails the condition for $k = 1$, the edge $\set{1, 2}$ for $k = 0$, and the edge $\set{0, 1}$ for $k = 2$.

A map $\Lambda^n_k \to X$ to a simplicial set $X$ is the same as a family of $n$ simplices $x_i \in X_{n-1}$, $i \ne k$, that agree on their common faces.
</document>

<document source="https://tamnd.github.io/math2/node/4BY024AVR8.md" tag="4BY024AVR8" kind="statement">
# Solution of equations with degenerate kernels

## Statement

Let $a < b$, let $m \ge 1$, and let $a_1, \dots, a_m, b_1, \dots, b_m \in L^2[a, b]$.
Let $K(x, y) = \sum_{i=1}^{m} a_i(x) b_i(y)$ be the corresponding [[degenerate kernel|6Y1RY5767R]], let $\lambda \in \C$ and $f \in L^2[a, b]$.
Let $A$ be the $m \times m$ matrix and $F \in \C^m$ the vector with entries
$$A_{ij} = \int_a^b b_i(y) \, a_j(y) \, dy, \qquad F_i = \int_a^b b_i(y) \, f(y) \, dy.$$
Consider the [[Fredholm equation of the second kind|DEYYNVCV9J]]
$$\varphi(x) - \lambda \int_a^b K(x, y) \varphi(y) \, dy = f(x).$$
Its solutions $\varphi \in L^2[a, b]$ are exactly the functions $\varphi = f + \lambda \sum_{i=1}^{m} c_i a_i$, where $c \in \C^m$ solves $(I_m - \lambda A) c = F$.
Distinct solutions $c$ give distinct functions $\varphi$.
In particular, if $\det(I_m - \lambda A) \ne 0$, the integral equation has exactly one solution for every $f$.

## Remarks

If $\varphi$ is a solution, put $c_i = \int_a^b b_i(y) \varphi(y) \, dy$.
The equation says $\varphi = f + \lambda \sum_i c_i a_i$, and multiplying this by $b_i$ and integrating gives $c_i = F_i + \lambda \sum_j A_{ij} c_j$.
Conversely, if $c$ solves the system and $\varphi = f + \lambda \sum_i c_i a_i$, the same computation shows $\int_a^b b_i \varphi = c_i$, hence $\lambda \int_a^b K(\cdot, y) \varphi(y) \, dy = \lambda \sum_i c_i a_i = \varphi - f$.
This also recovers $c$ from $\varphi$, which gives the injectivity.

Example: on $[0, 1]$ with $K(x, y) = xy$, so $m = 1$ and $a_1(x) = b_1(x) = x$, one gets $A = (1/3)$.
For $\lambda = 1$ and $f(x) = x$, $F = 1/3$, so $c = (1/3)/(2/3) = 1/2$ and $\varphi(x) = 3x/2$.
For $\lambda = 3$ the system is $0 \cdot c = F$, solvable only when $F = \int_0^1 y f(y) \, dy = 0$.

The polynomial $\det(I_m - \lambda A)$ is the [[determinant|00KDZRW4C0]] that governs the problem; it equals the Fredholm determinant of $K$ when $K$ is continuous.

## Explanations

### L1 human

When the kernel is a finite sum of products of a function of $x$ and a function of $y$, the integral equation hides only finitely many unknowns.
These unknowns are the numbers $\int b_i \varphi$, and they satisfy an ordinary square linear system.
Solve the system, and the formula gives the solution of the integral equation.

### L2 human

Degenerate kernels are the base case of Fredholm theory.
Every square-integrable kernel can be approximated by degenerate kernels, and the general theorems are proved by reducing to this case plus a small remainder that a Neumann series handles.
The same reduction is a practical numerical method: expand the kernel in a few separable terms and solve a small linear system.
</document>

<document source="https://tamnd.github.io/math2/node/4C1AE84N5N.md" tag="4C1AE84N5N" kind="concept">
# Localization

## Summary

The localization $S^{-1}R$ of a [[commutative ring|0310REJJY8]] $R$ at a [[multiplicative set|AN9SKEZ3VY]] $S$ is the ring of fractions $a/s$ with $a \in R$ and $s \in S$.
It is the most general ring in which the elements of $S$ become [[units|0XVEZDE9J8]].
It generalizes the construction of the [[field of fractions|0F2PTGX1A2]] of an [[integral domain|0QVD7WBFGN]].
Localization lets one study a ring "near" a [[prime ideal|00MHHZ9SNE]], which is the algebraic form of looking at a space near a point.

## Explanations

### L1 human

Localization means: allow some new denominators.
Starting from the integers and allowing every nonzero denominator gives the rational numbers.
Allowing only the powers of $2$ gives the fractions $a / 2^k$, such as $3/8$, but not $1/3$.
In general one picks a set $S$ of allowed denominators that contains $1$ and is closed under products.
The [[definition|KET2GSPZFG]] explains how to compare two fractions when the ring has zero divisors.
</document>

<document source="https://tamnd.github.io/math2/node/4C3BANJ8AH.md" tag="4C3BANJ8AH" kind="concept">
# Second variation

## Summary

The second variation of a [[functional|1EQM0DJ4NC]] at a function $y_0$ is the second derivative of $\varepsilon \mapsto J(y_0 + \varepsilon h)$ at $\varepsilon = 0$.
It plays the role of the second derivative test of ordinary calculus.
At a minimiser the [[first variation|5B8NFYB2XQ]] vanishes and the second variation is nonnegative in every direction $h$.
For integral functionals it is a quadratic functional in $h$, and its study leads to the Legendre condition, the Jacobi equation and conjugate points.

## Explanations

### L1 human

At the bottom of a valley the ground is flat, and it curves upward in every direction.
The first variation measures the slope of a functional, and the second variation measures its curvature.
For $J(y) = \int_0^1 y'(x)^2 \, dx$ one gets $J(y_0 + \varepsilon h) = J(y_0) + 2\varepsilon \int_0^1 y_0' h' \, dx + \varepsilon^2 \int_0^1 h'^2 \, dx$.
So the second variation is $2\int_0^1 h'(x)^2 \, dx$, which is never negative.
This fits the fact that straight lines minimise $J$ among functions with fixed end values.
The same idea in Riemannian geometry gives the [[second variation of arc length|ZB91TH6FKV]].
</document>

<document source="https://tamnd.github.io/math2/node/4C3RRDA4WZ.md" tag="4C3RRDA4WZ" kind="definition">
# Regular semigroup

## Definition

Let $S$ be a [[semigroup|0XMEM406JM]].
An element $a \in S$ is *regular* if there is $x \in S$ with $axa = a$.
The semigroup $S$ is *regular* if every element of $S$ is regular.

## Remarks

If $axa = a$, then $y = xax$ satisfies $aya = a$ and $yay = y$; such a $y$ is called an inverse of $a$.
So $S$ is regular if and only if every element has at least one inverse in this sense.
If $axa = a$, then $ax$ and $xa$ are [[idempotents|0N5NJZKKN3]].
A semigroup is regular if and only if every $\mathcal{R}$-class, or equivalently every $\mathcal{L}$-class, of [[Green's relations|S7WJW2YES2]] contains an idempotent.

Example: the monoid $T_X$ of all maps $X \to X$ under composition, $(fg)(x) = f(g(x))$.
Given $f$, choose $g$ such that $g(y)$ is a preimage of $y$ under $f$ for each $y$ in the image of $f$, and $g(y)$ is arbitrary otherwise.
Then $f(g(f(x))) = f(x)$ for all $x$, so $fgf = f$.

Non-example: the positive integers under addition.
For $a = 1$, the equation $1 + x + 1 = 1$ has no positive solution $x$, so $1$ is not regular.
</document>

<document source="https://tamnd.github.io/math2/node/4C5RG7P4VT.md" tag="4C5RG7P4VT" kind="definition">
# Pointless topology

## Definition

A *[[frame|306PWGH7BF]]* is a complete lattice $L$ such that $a \wedge \bigvee_{i \in I} b_i = \bigvee_{i \in I} (a \wedge b_i)$ for every $a \in L$ and every family $(b_i)_{i \in I}$ in $L$.
A *frame homomorphism* is a map between frames that preserves finite meets, including the top element, and arbitrary joins, including the bottom element.
The category of *[[locales|G2QNFT210W]]* is the opposite of the category of frames and frame homomorphisms.
*Pointless topology* is the study of locales.
Each topological space $X$ gives the locale $\Omega(X)$, the frame of [[open sets|02BH3MCBQB]] of $X$ ordered by inclusion.
Each [[continuous map|0X3WGX95FE]] $f \colon X \to Y$ gives the frame homomorphism $f^{-1} \colon \Omega(Y) \to \Omega(X)$, hence a locale map $\Omega(X) \to \Omega(Y)$.

## Remarks

Example: for $X = \R$, the open sets form a frame, because $U \cap \bigcup_i V_i = \bigcup_i (U \cap V_i)$ for open sets $U, V_i$.
Non-example: the lattice of subspaces of $\R^2$ is complete but not a frame; for three distinct lines $a, b, c$ through $0$, $a \wedge (b \vee c) = a$ while $(a \wedge b) \vee (a \wedge c) = 0$.
A point of a locale $L$ is a frame homomorphism $L \to \set{0, 1}$; a locale is spatial if it is isomorphic to $\Omega(X)$ for some space $X$.
The functor $\Omega$ restricts to an equivalence between [[sober spaces|4N4DNDR22Z]] and spatial locales.
Every frame is a complete [[Heyting algebra|NG943XNQWH]]; Mathlib calls frames `Order.Frame`.
</document>

<document source="https://tamnd.github.io/math2/node/4CDFMKKZ01.md" tag="4CDFMKKZ01" kind="statement">
# Quartic formula

## Statement

Let $a, b, c, d, e \in \C$ with $a \ne 0$.
Put
$$p = \frac{8ac - 3b^2}{8a^2}, \qquad q = \frac{b^3 - 4abc + 8a^2 d}{8a^3}, \qquad r = \frac{16ab^2c - 3b^4 - 64a^2bd + 256a^3e}{256a^4}.$$
Then $x$ is a [[root|00EHX6FWC0]] of $ax^4 + bx^3 + cx^2 + dx + e$ if and only if $y = x + \frac{b}{4a}$ is a root of $y^4 + py^2 + qy + r$.
Assume $q \ne 0$.
Let $m$ be a root of the resolvent cubic
$$8m^3 + 8pm^2 + (2p^2 - 8r)m - q^2,$$
and let $s$ be a square root of $2m$.
Then $m \ne 0$, $s \ne 0$, and the four roots of $y^4 + py^2 + qy + r$, counted with multiplicity, are
$$\frac{s \pm \sqrt{-2p - 2m - 2q/s}}{2} \qquad \text{and} \qquad \frac{-s \pm \sqrt{-2p - 2m + 2q/s}}{2},$$
where $\sqrt{z}$ denotes either square root of $z$.

## Remarks

Since $q \ne 0$, the constant term $-q^2$ of the resolvent cubic is nonzero, so $m \ne 0$.
The resolvent cubic says exactly that $(y^2 + \frac{p}{2} + m)^2 - (y^4 + py^2 + qy + r) = 2my^2 - qy + (m^2 + pm + \frac{p^2}{4} - r)$ is the square $(sy - \frac{q}{2s})^2$.
So the quartic factors as $(y^2 + sy + \frac{p}{2} + m - \frac{q}{2s})(y^2 - sy + \frac{p}{2} + m + \frac{q}{2s})$, and the quadratic formula gives the four roots.
The resolvent cubic itself is solved by the [[cubic formula|FB6J5WAT5V]], so all roots are radical expressions in the coefficients.
If $q = 0$, the equation $y^4 + py^2 + r = 0$ is a quadratic in $y^2$, and $y^2 = \frac{-p \pm \sqrt{p^2 - 4r}}{2}$.
Example: for $y^4 - 2y^2 + 8y - 3$, the value $m = 2$ is a root of $8m^3 - 16m^2 + 32m - 64$, so $s = 2$ and the factors are $y^2 + 2y - 1$ and $y^2 - 2y + 3$.

## Explanations

### L1 human

The quartic formula gives the roots of any equation of degree $4$ using arithmetic, square roots and cube roots.
The trick is to add a well-chosen quantity $m$ to both sides so that both become perfect squares.
Finding $m$ needs a cubic equation, which the cubic formula solves.

### L2 human

This method is due to Ferrari, and it shows that every quartic is [[solvable by radicals|J7YS0DA3HQ]].
In Galois terms, $S_4$ is solvable, with the normal subgroup $V_4$ of order $4$ and quotient $S_3$.
If $y_1, y_2, y_3, y_4$ are the roots, each root $m$ of the resolvent cubic has the form $\frac{1}{2}(y_i + y_j)^2$, and $(y_1 + y_2)^2$ is fixed by $V_4$.
This is why a cubic appears.
For degree $5$ no such reduction exists, by the [[Abel–Ruffini theorem|163A2WQQJX]].
</document>

<document source="https://tamnd.github.io/math2/node/4CDFWST967.md" tag="4CDFWST967" kind="statement">
# Ramanujan–Petersson conjecture

## Statement

Let $N \ge 1$ and $k \ge 2$ be integers.
Let $f = \sum_{n \ge 1} a_n q^n$ be a [[cusp form|A50Y012RBP]] of weight $k$ for the [[congruence subgroup|1H6R856P3C]] $\Gamma_0(N)$, with $q = e^{2 \pi i \tau}$.
Assume that $a_1 = 1$ and that $f$ is an eigenvector of the [[Hecke operator|S4SV3QZ8B1]] $T_p$ for every [[prime|01826KMG2H]] $p$ not dividing $N$.
Then for every prime $p$ not dividing $N$,
$$\abs{a_p} \le 2 p^{(k-1)/2}.$$
In particular, the [[Ramanujan tau function|6DNZAGCSQS]] satisfies $\abs{\tau(p)} \le 2 p^{11/2}$ for every prime $p$.

## Remarks

Ramanujan conjectured the bound for $\tau(p)$ in 1916, and Petersson extended the conjecture to other modular forms.
Deligne proved it in 1974 as a consequence of his proof of the Weil conjectures.
Earlier work of Eichler, Shimura and Deligne had identified $a_p$ with the trace of Frobenius on an étale cohomology group, whose eigenvalues have absolute value $p^{(k-1)/2}$ by the Weil conjectures.
Example: $\tau(2) = -24$ and $2 \cdot 2^{11/2} = 2^{13/2} \approx 90.5$; $\tau(3) = 252$ and $2 \cdot 3^{11/2} \approx 841.8$.
The normalization $a_1 = 1$ and the eigenform hypothesis are needed: the cusp form $100\Delta$ has $a_2 = -2400$, which breaks the bound.
The analogous statement for Maass forms is still open.

## Explanations

### L1 human

The coefficients of a Hecke eigenform look random, but they cannot be too large.
At a prime $p$, the coefficient is at most about $2 p^{(k-1)/2}$, roughly the square root of the trivial size $p^{k-1}$.
For Ramanujan's function this says $\abs{\tau(p)} \le 2 p^{11/2}$.
It took almost sixty years, and a revolution in algebraic geometry, to prove it.

### L2 human

For a normalized eigenform, the Euler factor at $p \nmid N$ is $1 - a_p p^{-s} + p^{k-1-2s}$.
Writing it as $(1 - \alpha_p p^{-s})(1 - \beta_p p^{-s})$, the bound is equivalent to $\abs{\alpha_p} = \abs{\beta_p} = p^{(k-1)/2}$.
So the theorem is an analogue of the Riemann hypothesis for the local factors of the L-function of $f$.
Together with multiplicativity, it gives $\abs{a_n} \le d(n)\, n^{(k-1)/2}$ for $n$ prime to $N$, where $d(n)$ is the number of divisors of $n$.
The [[Sato–Tate conjecture|TRGBJYPBXH]] describes how the normalized values $a_p / (2 p^{(k-1)/2}) \in [-1, 1]$ are distributed.
</document>

<document source="https://tamnd.github.io/math2/node/4CGTN9NRR9.md" tag="4CGTN9NRR9" kind="definition">
# Teichmüller space

## Definition

Let $S$ be a closed, connected, oriented topological surface of [[genus|NJ787GZS1N]] $g \ge 0$.
A *marked Riemann surface* of type $S$ is a pair $(X, f)$, where $X$ is a [[Riemann surface|5K0N8QMV6Q]] and $f \colon S \to X$ is an orientation-preserving [[homeomorphism|05GZ11PWRY]].
Two marked Riemann surfaces $(X, f)$ and $(Y, h)$ are *equivalent* if there is a [[biholomorphism|8AJEMCZ29B]] $\varphi \colon X \to Y$ such that $\varphi \circ f$ is isotopic to $h$.
The *Teichmüller space* $T(S)$ is the set of equivalence classes of marked Riemann surfaces of type $S$.
It depends up to natural bijection only on $g$ and is written $T_g$.

## Remarks

Riemann surfaces carry their orientation from the complex structure, which is why the marking is required to preserve orientation.
$T_g$ carries a natural complete metric, the Teichmüller metric, defined from the least dilatation of [[quasiconformal maps|SB8KG46NHP]] between marked surfaces.
For $g \ge 2$, $T_g$ is a complex manifold of dimension $3g - 3$, homeomorphic to $\R^{6g - 6}$; $T_0$ is a point.
The mapping class group of $S$ acts on $T(S)$ by changing the marking, and the quotient is the moduli space $M_g$.

Example: $g = 1$.
For $\tau$ in the upper half-plane let $X_\tau = \C / (\Z + \tau \Z)$.
Mark it by the homeomorphism $f_\tau$ from the standard torus $\R^2/\Z^2$ induced by the real linear map sending $(1, 0)$ to $1$ and $(0, 1)$ to $\tau$.
Every marked torus is equivalent to exactly one such $(X_\tau, f_\tau)$, so $T_1$ is identified with the upper half-plane.

Non-example: a pair $(X, f)$ with $f$ orientation-reversing is not a marked Riemann surface, so it is not a point of $T(S)$.
</document>

<document source="https://tamnd.github.io/math2/node/4CK2GSRAGD.md" tag="4CK2GSRAGD" kind="concept">
# Reproducing kernel Hilbert space

## Summary

A reproducing kernel Hilbert space is a [[Hilbert space|HM2HHE3J51]] of functions on a set $X$.
In it, evaluation at each point is a continuous linear functional.
By the [[Riesz representation theorem|TTC40N49M6]], evaluation at $x$ is the inner product with a function $k_x$, and $K(x, y) = k_y(x)$ is the reproducing kernel.
The kernel is a positive-definite kernel, and by the Moore–Aronszajn theorem each positive-definite kernel comes from exactly one such space.
These spaces appear in complex analysis, as Hardy and Bergman spaces, and in statistics and machine learning, through kernel methods.

## Explanations

### L1 human

In such a space, functions that are close in norm are also close at every single point.
The value of a function at a point can be read off by taking the inner product with one special function for that point.
A simple example: the vectors of $\C^n$, seen as functions on $\set{1, \dots, n}$, form such a space.
The special function for the point $i$ is the unit vector $e_i$.
</document>

<document source="https://tamnd.github.io/math2/node/4CMPPTSJNP.md" tag="4CMPPTSJNP" kind="statement">
# Galois group of a prime degree polynomial with two nonreal roots

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]] and let $f \in \Q[x]$ be an [[irreducible|00X2A9G9W3]] polynomial of degree $p$.
Assume that $f$ has exactly two nonreal roots in $\C$.
Then the [[Galois group|0JZY3R908H]] of $f$ over $\Q$, acting on the $p$ roots of $f$, is the full [[symmetric group|0RJMNEN3BJ]] $S_p$.

## Remarks

Let $G$ be the Galois group, seen as a group of permutations of the $p$ roots.
Since $f$ is irreducible, adjoining one root gives a field of degree $p$, so $p$ divides $\abs{G}$.
By [[Cauchy's theorem|0JM7PA288B]], $G$ contains an element of order $p$, which is a $p$-cycle.
[[Complex conjugation|0BJV24C4SY]] restricts to an element of $G$ that swaps the two nonreal roots and fixes the others, so $G$ contains a [[transposition|02DTRYKGE8]].
For $p$ prime, a $p$-cycle and any transposition generate $S_p$.
Example: $x^5 - 4x + 2$ has Galois group $S_5$; see the [[quintic equation|S1WS18QEGJ]].
The prime degree matters: in degree $4$, the group generated by $(1\,2\,3\,4)$ and $(1\,3)$ is a dihedral group of order $8$, not $S_4$.

## Explanations

### L1 human

The theorem gives an easy test for when the symmetries of the roots of a polynomial include every permutation.
Count the real roots: if the degree is prime and exactly two roots are not real, the group is as big as possible.
This is the usual way to produce explicit equations that cannot be solved by radicals.

### L2 human

Combined with [[Galois' theorem on solvability|VHCWCFA3MX]] and the fact that [[$S_p$ is not solvable for $p \ge 5$|0J53NE674D]], the theorem proves the second part of the [[Abel–Ruffini theorem|163A2WQQJX]].
The number of real roots can be found with calculus or with [[Sturm's theorem|R44CV0KD9T]].
The proof uses only that $\Q \subseteq \R$; it works for any subfield of $\R$ in place of $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/4CMRM3975W.md" tag="4CMRM3975W" kind="concept">
# Growth rate of a group

## Summary

The growth function of a [[finitely generated group|082DWWXMRJ]] counts how many elements can be written as words of length at most $n$ in the generators.
Its rate of increase, up to a natural equivalence, does not depend on the generating set, and it is a [[quasi-isometry|6FKYSH729T]] invariant.
Groups split into groups of polynomial growth, of exponential growth, and of intermediate growth.
[[Gromov's theorem|RSBFHHPTWX]] identifies the groups of polynomial growth as the virtually nilpotent groups.

## Explanations

### L1 human

Start at the identity of the group and take $n$ steps, each step a multiplication by a generator or its inverse.
Count how many different elements you can reach.
In $\Z$ with generator $1$, after at most $n$ steps you reach the $2n + 1$ integers from $-n$ to $n$, which grows like a line.
In $\Z^2$ you reach a diamond of about $2n^2$ points, which grows like an area.
In a free group on two generators, the number of reachable elements roughly triples with each step, which is exponential growth.
</document>

<document source="https://tamnd.github.io/math2/node/4CRH8NNFEF.md" tag="4CRH8NNFEF" kind="concept">
# Near-unanimity term

## Summary

A near-unanimity term is a term of arity at least $3$ that returns $x$ whenever all arguments but at most one equal $x$.
Having such a term is a [[Mal'cev condition|6R0CA4JWWY]] on a [[variety|AJNZJD8442]].
A variety with a near-unanimity term is congruence-distributive.
For a finite algebra, such a term makes the clone of term operations determined by a finite set of relations, as in the [[Baker–Pixley theorem|97SRYBF41G]].

## Explanations

### L1 human

Think of a vote among $n$ voters where at most one voter disagrees with the rest.
A near-unanimity term is an operation that always follows the large group in this situation.
In a lattice, the term $(x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$ is such a term with three arguments.
For example, with $x = z$ it gives $(x \wedge y) \vee (y \wedge x) \vee x = x$.
When many voters disagree, the term may return anything; only the nearly unanimous inputs are fixed.
</document>

<document source="https://tamnd.github.io/math2/node/4CSMQFTGD8.md" tag="4CSMQFTGD8" kind="concept">
# Tate module

## Summary

Fix a prime $\ell$.
The $\ell$-adic Tate module of an abelian group collects its elements of $\ell$-power order into one module over the [[$\ell$-adic integers|V6TJAFZ4MD]].
Let $A$ be an [[abelian variety|R7W03TXRS9]] of dimension $g$ over a field $k$ whose characteristic is not $\ell$.
Then the Tate module of $A$ is free of rank $2g$.
The [[absolute Galois group|PGKWBJDTKX]] of $k$ acts on it, which gives an $\ell$-adic [[Galois representation|JQEYEQJY2G]].
These representations carry much of the arithmetic of $A$.

## Explanations

### L1 human

On an elliptic curve over $\C$ there are $\ell^{2n}$ points that give zero when added to themselves $\ell^n$ times.
These torsion points form a tower as $n$ grows, each level mapping onto the one below by multiplication by $\ell$.
The Tate module is the limit of this tower.
It behaves like a lattice $\Z_\ell^2$, an $\ell$-adic version of the lattice that defines the curve as a quotient of $\C$.
Over a number field, the Galois group permutes the torsion points, and so it acts on the Tate module.
</document>

<document source="https://tamnd.github.io/math2/node/4CZMWQVC30.md" tag="4CZMWQVC30" kind="statement">
# g-theorem

## Statement

Let $d \ge 1$ and let $(h_0, h_1, \dots, h_d)$ be a sequence of integers.
Let $g_0 = h_0$ and $g_i = h_i - h_{i-1}$ for $1 \le i \le \floor{d/2}$.
Then $(h_0, \dots, h_d)$ is the [[h-vector|DCK64GFPD7]] of the boundary complex of a simplicial convex polytope of dimension $d$ if and only if the following hold:

1. $h_i = h_{d-i}$ for $0 \le i \le d$.
2. $g_0 = 1$.
3. $0 \le g_{i+1} \le g_i^{\langle i \rangle}$ for $1 \le i \le \floor{d/2} - 1$, and $g_1 \ge 0$.

Here, for integers $a \ge 0$ and $i \ge 1$, write $a = \binom{n_i}{i} + \binom{n_{i-1}}{i-1} + \dots + \binom{n_j}{j}$ with $n_i > n_{i-1} > \dots > n_j \ge j \ge 1$ (this expansion exists and is unique), and set $a^{\langle i \rangle} = \binom{n_i + 1}{i + 1} + \binom{n_{i-1} + 1}{i} + \dots + \binom{n_j + 1}{j + 1}$, with $0^{\langle i \rangle} = 0$.

## Remarks

Condition 1 is the [[Dehn–Sommerville equations|4WKHF5BCTA]].
Conditions 2 and 3 say that $(g_0, \dots, g_{\floor{d/2}})$ is an M-sequence, the Hilbert function of a graded quotient of a polynomial ring generated in degree $1$; in particular $h_0 \le h_1 \le \dots \le h_{\floor{d/2}}$.
The conditions were conjectured by McMullen.
Billera and Lee proved sufficiency by constructing polytopes, and Stanley proved necessity with the hard Lefschetz theorem for toric varieties.

Example: the boundary of the $4$-dimensional cross-polytope has $h = (1, 4, 6, 4, 1)$ and $g = (1, 3, 2)$.
Here $3 = \binom{3}{1}$, so $3^{\langle 1 \rangle} = \binom{4}{2} = 6 \ge 2$, and all conditions hold.

The sequence $(1, 1, 3, 1, 1)$ is symmetric, but $g = (1, 0, 2)$ and $0^{\langle 1 \rangle} = 0 < 2$, so it is not the h-vector of a simplicial $4$-polytope.

## Explanations

### L1 human

The theorem lists exactly which face counts a polytope with simplex faces can have.
Besides the mirror symmetry of Dehn–Sommerville, the h-vector must rise up to its middle, and it cannot rise too fast.
"Too fast" is measured by a precise rule from commutative algebra due to Macaulay.

### L2 human

The g-theorem is a model of how commutative algebra and algebraic geometry solve combinatorial problems.
The h-vector of a simplicial polytope is the Hilbert function of an Artinian reduction of its Stanley–Reisner ring.
The hard Lefschetz property gives an injective multiplication map, and this forces the g-vector to be an M-sequence.
The same characterization for all simplicial spheres was a long-standing conjecture; Adiprasito announced a proof in 2018.
</document>

<document source="https://tamnd.github.io/math2/node/4D073CCYGV.md" tag="4D073CCYGV" kind="concept">
# Rational map

## Summary

A rational map from a variety $X$ to a variety $Y$ is a morphism defined on some nonempty open subset of $X$.
Two such morphisms give the same rational map when they agree on a common open subset.
It is written $X \dashrightarrow Y$, because it may fail to be defined at some points.
Rational maps are the maps of birational geometry, which studies varieties up to changes on lower-dimensional subsets.

## Explanations

### L1 human

Some natural maps are given by formulas with denominators, so they are undefined where a denominator vanishes.
The projection of the plane from the origin onto the line of directions, $(x, y) \mapsto [x : y]$, is defined everywhere except at the origin.
A rational map keeps such a formula and simply forgets the bad points.
Because varieties are irreducible, the open set where the map is defined is dense, so the formula determines the map.
</document>

<document source="https://tamnd.github.io/math2/node/4D090WJ84H.md" tag="4D090WJ84H" kind="statement">
# Carathéodory's existence theorem

## Statement

Let $t_0 \in \R$, $x_0 \in \R^n$ and $a, b > 0$, and let $R = [t_0 - a, t_0 + a] \times \set{x \in \R^n : \norm{x - x_0} \le b}$.
Let $f : R \to \R^n$ satisfy these three conditions.

1. For each $x$, the map $t \mapsto f(t, x)$ is Lebesgue measurable.
2. For almost every $t$, the map $x \mapsto f(t, x)$ is continuous.
3. There is a Lebesgue integrable function $m : [t_0 - a, t_0 + a] \to [0, \infty)$ with $\norm{f(t, x)} \le m(t)$ for all $(t, x) \in R$.

Then there are $\beta \in (0, a]$ and an absolutely continuous function $x : [t_0 - \beta, t_0 + \beta] \to \R^n$ with $\norm{x(t) - x_0} \le b$ for all $t$, with $x(t_0) = x_0$, and with $x'(t) = f(t, x(t))$ for almost every $t$.

## Remarks

Such a function $x$ is called a solution in the sense of Carathéodory; it is the same as a continuous solution of the integral equation $x(t) = x_0 + \int_{t_0}^{t} f(s, x(s))\, ds$.
One can take any $\beta$ with $\int_{t_0 - \beta}^{t_0 + \beta} m(s)\, ds \le b$.
The proof follows that of the [[Peano existence theorem|098YP831QY]]: one builds approximate solutions with delayed arguments.
The bound by $m$ makes them equicontinuous, and the dominated convergence theorem allows passage to the limit.
Example: let $f(t, x) = 0$ for $t < 0$ and $f(t, x) = 1$ for $t \ge 0$, with $x(0) = 0$.
The function $x(t) = \max(t, 0)$ is a Carathéodory solution.
There is no solution that is differentiable at every point, because a derivative cannot have a jump discontinuity, by Darboux's theorem.
So the theorem covers equations with discontinuous time dependence, where the classical theory gives nothing.

## Explanations

### L1 human

Classical existence theorems ask the right side of $x' = f(t, x)$ to be continuous.
In applications the forcing is often switched on and off, so $f$ jumps in time.
Carathéodory's theorem still gives a solution, if one only asks the equation to hold at almost every time.
The solution is then a continuous curve with corners where the forcing jumps.

### L2 human

The theorem replaces continuity in $t$ by measurability, and boundedness of $f$ by an integrable bound.
Solutions are absolutely continuous functions, which is the natural class for the fundamental theorem of calculus of the Lebesgue integral.
This framework underlies control theory, where controls are measurable functions of time.
As for Peano's theorem, there is no uniqueness in general; a Lipschitz condition in $x$ with an integrable Lipschitz constant gives uniqueness, as in the [[Picard–Lindelöf theorem|0P6KRD5B7A]].
</document>

<document source="https://tamnd.github.io/math2/node/4D1E2YYJNZ.md" tag="4D1E2YYJNZ" kind="example">
# Bilateral Laplace transform of a probability law

## Example

Let $\mu$ be a [[probability measure|059TJN0NZ1]] on $\R$, for example the law of a real random variable $X$.
Its [[two-sided Laplace transform|6J2AGD0BYY]] is
$$\mathcal{B}\mu(s) = \int_{\R} e^{-sx} \, d\mu(x) = E\left[e^{-sX}\right],$$
for the complex $s$ at which the integral converges absolutely.
Since $\abs{e^{-sx}} = e^{-x \operatorname{Re} s}$, convergence depends only on $\operatorname{Re} s$, and it always holds on the line $\operatorname{Re} s = 0$, where the integrand has modulus $1$.
For real $t$, the [[moment-generating function|0WQVXP7PDB]] is $M_X(t) = \mathcal{B}\mu(-t)$.

Take $\mu$ the standard [[normal distribution|05AQE0QR0D]], with density $(2\pi)^{-1/2} e^{-x^2/2}$.
Completing the square gives $-sx - x^2/2 = -(x + s)^2/2 + s^2/2$.
So
$$\mathcal{B}\mu(s) = e^{s^2/2} \cdot \frac{1}{\sqrt{2\pi}} \int_{\R} e^{-(x+s)^2/2} \, dx = e^{s^2/2}$$
for every $s \in \C$, where the last integral equals $\sqrt{2\pi}$ by a shift of the line of integration.
Here the strip of convergence is the whole plane.

Take $\mu$ the [[exponential distribution|0E99F51GVB]] with rate $\lambda > 0$, with density $\lambda e^{-\lambda x}$ on $[0, \infty)$.
Then $\int_0^\infty e^{-sx} \lambda e^{-\lambda x} \, dx = \lambda / (\lambda + s)$ when $\operatorname{Re} s > -\lambda$, and the integral diverges when $\operatorname{Re} s \le -\lambda$.
Here the strip of convergence is the half-plane $\operatorname{Re} s > -\lambda$.

## Explanations

### L1 human

A probability law can be turned into a function of a complex variable by averaging the exponential $e^{-sX}$.
This is the moment-generating function read with the opposite sign and allowed to take complex arguments.
For the bell curve the answer is the very simple function $e^{s^2/2}$.
For the exponential law the answer $\lambda / (\lambda + s)$ only makes sense to the right of the line $\operatorname{Re} s = -\lambda$.
The reason is that the tail of this law decays only like $e^{-\lambda x}$.
</document>

<document source="https://tamnd.github.io/math2/node/4D8CDGZNHE.md" tag="4D8CDGZNHE" kind="definition">
# Partial isometry

## Definition

Let $H$ and $K$ be [[Hilbert spaces|HM2HHE3J51]] and let $V \colon H \to K$ be a bounded linear operator.
The operator $V$ is a *partial isometry* when
$$\norm{Vx} = \norm{x} \quad \text{for all } x \in (\ker V)^\perp,$$
where $\ker V$ is the [[kernel|02YFME3RGW]] of $V$ and $(\ker V)^\perp$ is its [[orthogonal complement|079D68S3AW]].
The space $(\ker V)^\perp$ is the *initial space* of $V$, and the range of $V$ is its *final space*.

## Remarks

Example: the backward shift $V(x_0, x_1, x_2, \ldots) = (x_1, x_2, \ldots)$ on $\ell^2(\N)$ has kernel spanned by $(1, 0, 0, \ldots)$.
A vector orthogonal to the kernel has $x_0 = 0$, and then $\norm{Vx}^2 = \sum_{n \ge 1} \abs{x_n}^2 = \norm{x}^2$; so $V$ is a partial isometry.

Non-example: $T = 2I$ on $\C$ has kernel $\set{0}$, so its initial space is $\C$, but $\norm{T 1} = 2 \ne 1$.

Equivalent forms: $V$ is a partial isometry if and only if $V^*V$ is an orthogonal projection.
Then $V^*V$ is the projection onto the initial space and $VV^*$ is the projection onto the final space.
Another equivalent condition is $VV^*V = V$.
The final space of a partial isometry is closed.
</document>

<document source="https://tamnd.github.io/math2/node/4D9HP5TS6W.md" tag="4D9HP5TS6W" kind="example">
# Minkowski question-mark function

## Example

Every irrational $x \in (0, 1)$ has a unique infinite [[continued fraction|0BK6JBK58W]] expansion $x = [0; a_1, a_2, a_3, \dots]$ with positive integers $a_k$.
The question-mark function $? \colon [0, 1] \to [0, 1]$ is defined at such $x$ by
$$?(x) = 2 \sum_{k=1}^{\infty} (-1)^{k+1} \, 2^{-(a_1 + a_2 + \cdots + a_k)}.$$
For a rational $x = [0; a_1, \dots, a_n]$ the same formula with a finite sum is used, and $?(0) = 0$.
The value does not depend on which of the two finite expansions of a rational is chosen.
For example, $\frac12 = [0; 2] = [0; 1, 1]$ gives $2 \cdot 2^{-2} = \frac12$ and $2(2^{-1} - 2^{-2}) = \frac12$.

Some values: $?(1) = 1$, $?(\frac13) = 2 \cdot 2^{-3} = \frac14$, and for $x = [0; 1, 1, 1, \dots] = (\sqrt5 - 1)/2$ the geometric series gives $?(x) = 2 \cdot \frac{1/2}{1 + 1/2} = \frac23$.

The function $?$ is continuous and strictly increasing, so it is a [[homeomorphism|05GZ11PWRY]] of $[0, 1]$ onto itself.
Its derivative exists and equals $0$ at almost every point of $[0, 1]$.
So $?$ is a [[singular function|P5AWRD2793]].
Unlike the [[Cantor function|NPBYYK63FE]], it is strictly increasing, so it is constant on no interval.

The function maps the rationals of $[0, 1]$ onto the dyadic rationals $m/2^n$ of $[0, 1]$.
It maps the quadratic irrationals of $(0, 1)$ onto the rationals that are not dyadic.

## Explanations

### L1 human

The question-mark function reads the continued fraction of a number and turns it into a binary expansion: each partial quotient becomes a run of equal binary digits.
The first run has length $a_1 - 1$, and the $k$-th run has length $a_k$ for $k \ge 2$.
It is a continuous, strictly increasing function that still has slope zero almost everywhere.
This shows that a function can climb steadily from $0$ to $1$ while being flat in the sense of derivatives almost everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/4DD3GM2K4T.md" tag="4DD3GM2K4T" kind="definition">
# Local cohomology

## Definition

Let $R$ be a commutative [[Noetherian ring|0670T59HGE]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be an [[$R$-module|0NZ8JA25WK]].
The *$I$-torsion functor* is
$$\Gamma_I(M) = \set{m \in M : I^n m = 0 \text{ for some } n \ge 1}.$$
It is a left exact functor from $R$-modules to $R$-modules.
For $i \ge 0$, the *$i$-th local cohomology module* $H^i_I(M)$ is the $i$-th right derived functor of $\Gamma_I$ at $M$.
Explicitly, choose an injective resolution $0 \to M \to E^0 \to E^1 \to \cdots$ by [[injective modules|85H7PRHDPZ]].
Then $H^i_I(M)$ is the $i$-th cohomology of the complex $0 \to \Gamma_I(E^0) \to \Gamma_I(E^1) \to \cdots$.

## Remarks

One has $H^0_I(M) = \Gamma_I(M)$, and a short exact sequence of modules gives a long exact sequence of local cohomology.
Equivalently, $H^i_I(M) \cong \varinjlim_n \operatorname{Ext}^i_R(R/I^n, M)$, a direct limit of [[Ext modules|4P7GBWJ4RR]].
If $I = (x_1, \ldots, x_n)$, then $H^i_I(M)$ is also the cohomology of the Čech complex $0 \to M \to \bigoplus_i M_{x_i} \to \bigoplus_{i<j} M_{x_i x_j} \to \cdots \to M_{x_1 \cdots x_n} \to 0$.
Example: for $R = K[x]$ and $I = (x)$, the Čech complex of $R$ is $0 \to R \to R_x \to 0$, so $H^0_I(R) = 0$ and $H^1_I(R) = K[x, x^{-1}]/K[x]$.
The functor $\Gamma_I$ is not right exact: for $0 \to K[x] \xrightarrow{x} K[x] \to K \to 0$ with $I = (x)$ one gets $0 \to 0 \to K$, and the last map is not onto.
If $(R, \mathfrak{m})$ is local and $M \ne 0$ is finitely generated, then $H^i_{\mathfrak{m}}(M) = 0$ for $i < \operatorname{depth} M$ and $i > \dim M$, and it is nonzero for $i = \operatorname{depth} M$ and $i = \dim M$.
</document>

<document source="https://tamnd.github.io/math2/node/4DEGC6JADR.md" tag="4DEGC6JADR" kind="statement">
# Center manifold theorem

## Statement

Let $U$ be an open subset of $\R^n$, let $k$ be an integer with $1 \le k < \infty$, and let $f \colon U \to \R^n$ be of class $C^k$.
Let $x_0 \in U$ be an [[equilibrium point|0G4X36AE0X]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.
Let $E^c$ be the sum of the generalised eigenspaces of the [[Jacobian matrix|047NDTEPCV]] $Df(x_0)$ for the eigenvalues with real part $0$.
Then $x_0$ has a [[local center manifold|F35WJP8VHG]] $W^c$ of class $C^k$.
That is, $W^c$ is an embedded $C^k$ submanifold of a neighbourhood of $x_0$ that contains $x_0$.
Its tangent space at $x_0$ is $E^c$, and it is locally [[invariant|WWZTW11664]] under the [[flow|02903SSA48]] of $f$.

## Remarks

The usual proof first cuts $f$ off outside a small ball, so that the nonlinear part is globally small in $C^1$.
For the modified equation, a global center manifold is the graph of a map $E^c \to E^s \oplus E^u$.
This map is found as a fixed point of a contraction on a space of functions with controlled growth; this is the Lyapunov–Perron method.
The center manifold need not be unique.
For $x' = x^2$, $y' = -y$, every curve made of $y = c e^{1/x}$ for $x < 0$ and $y = 0$ for $x \ge 0$ is a center manifold of the origin.
The finiteness of $k$ matters.
For a $C^\infty$ system there is a $C^k$ center manifold for each finite $k$, but in general no $C^\infty$ one, and the neighbourhood may shrink as $k$ grows.
If no eigenvalue of $Df(x_0)$ has positive real part, the stability of $x_0$ is the same as for the restriction of the equation to $W^c$.

## Explanations

### L1 human

At a rest point, the linear part of the equation splits the directions into contracting, expanding and neutral ones.
In the contracting and expanding directions the linear part already tells what happens.
The theorem says that the neutral directions bend into a curved invariant surface through the rest point.
All the delicate behaviour happens on this surface, which has fewer dimensions than the whole space.

### L2 human

The theorem is the main tool for studying non-hyperbolic equilibria, where the [[Hartman–Grobman theorem|0S8WX8A4V9]] gives no information.
One computes a Taylor expansion of the center manifold as a graph over $E^c$ by matching coefficients in the invariance equation.
Then one studies the reduced equation on $E^c$, which in a bifurcation is often one- or two-dimensional.
This reduction underlies the normal forms of the saddle-node and Hopf bifurcations.
It complements the [[stable manifold theorem|WF22KK7C3R]], which treats the contracting and expanding directions.
</document>

<document source="https://tamnd.github.io/math2/node/4DJ4V5VBSH.md" tag="4DJ4V5VBSH" kind="concept">
# Normed algebra

## Summary

A normed algebra is an algebra over the real or complex numbers with a norm that is compatible with the product.
The norm of a product is at most the product of the norms.
This inequality makes multiplication continuous and lets power series of elements converge.
A complete normed algebra is a Banach algebra.
Algebras of bounded operators and of bounded continuous functions are the main examples.

## Explanations

### L1 human

A normed algebra is a space where one can add, scale and multiply elements, and also measure their size.
The rule $\norm{xy} \le \norm{x} \norm{y}$ says that multiplying cannot blow sizes up more than the sizes themselves allow.
For square matrices with the operator norm, this rule says that applying two maps in a row stretches vectors by at most the product of the two stretch factors.
</document>

<document source="https://tamnd.github.io/math2/node/4DKB6AQ8J7.md" tag="4DKB6AQ8J7" kind="concept">
# Yoneda embedding

## Summary

The Yoneda embedding of a locally small [[category|A2DE66937D]] $\mathcal{C}$ is a [[functor|28NQ0KRHD3]] into presheaves.
It sends each object $A$ to its [[hom functor|VJXS8AK67E]] $\Hom(-, A)$, a [[presheaf|AND2WHZC8P]] on $\mathcal{C}$.
By the [[Yoneda lemma|D6DF1ARPVD]] it is fully faithful, so $\mathcal{C}$ is equivalent to a full subcategory of a category of presheaves.
This lets one study objects of any category through set-valued functors, where many constructions are easy.

## Explanations

### L1 human

Replace each object $A$ by the record of all arrows that end at $A$, one set for each possible starting object.
The Yoneda embedding says that this record loses nothing.
Two objects with matching records in a natural way are isomorphic, and arrows between objects are the same as natural maps between their records.
For a group $G$ seen as a one-object category, the record of the single object is $G$ itself with $G$ acting by multiplication, and this gives Cayley's theorem.
</document>

<document source="https://tamnd.github.io/math2/node/4DQGSSWREK.md" tag="4DQGSSWREK" kind="definition">
# General recursive function

## Definition

A partial function $f \colon \N^k \rightharpoonup \N$ is a function from a subset of $\N^k$ to $\N$, where $\N$ is the set of [[natural numbers|0MWKJ44V5A]].
For a partial function $g \colon \N^{k+1} \rightharpoonup \N$, the *minimisation* $\mu g \colon \N^k \rightharpoonup \N$ is defined as follows.
The value $\mu g(\bar x)$ is the least $y$ such that $g(y, \bar x) = 0$ and $g(z, \bar x)$ is defined for all $z < y$.
If no such $y$ exists, then $\mu g(\bar x)$ is undefined.

The class of *partial recursive functions*, or *$\mu$-recursive functions*, is the smallest class of partial functions $\N^k \rightharpoonup \N$ with these properties.

1. It contains the zero function, the successor function and the projections, as for [[primitive recursive functions|5NVA2JRHPV]].
2. It is closed under composition and primitive recursion, where a value is defined only when all values used to compute it are defined.
3. It is closed under minimisation.

A *general recursive function* is a partial recursive function that is total, that is, defined at every point of $\N^k$.

## Remarks

Example: let $g(y, x) = x \mathbin{\dot-} y \cdot y$, where $a \mathbin{\dot-} b = \max(a - b, 0)$.
The function $g$ is primitive recursive, and $g(y, x) = 0$ exactly when $y^2 \ge x$.
So $\mu g(x) = \ceil{\sqrt x}$, which is defined for all $x$, and $\mu g$ is general recursive.

Non-example: let $g(y, x) = y + 1$, which is never $0$.
Then $\mu g$ is the empty function, which is partial recursive but not total, so it is not general recursive.

Terminology varies: some authors use "general recursive" for all partial recursive functions.
Every primitive recursive function is general recursive, and the [[Ackermann function|N470JF7ZJX]] shows that the converse fails.
</document>

<document source="https://tamnd.github.io/math2/node/4DRPG6VR70.md" tag="4DRPG6VR70" kind="definition">
# Net of a polyhedron

## Definition

Let $P \subseteq \R^3$ be a three-dimensional [[convex polytope|300K6CB1KH]], and let $G$ be its edge graph, whose vertices and edges are those of $P$.
Let $T$ be a [[spanning tree|0FQ5B8S1ZC]] of $G$.
Cutting the surface of $P$ along the edges of $T$ leaves a surface that is connected and can be laid flat in the plane.
The *edge unfolding* of $P$ along $T$ is a placement of the faces of $P$ in $\R^2$ such that:

1. each face is mapped into the plane by an [[isometry|02HPP5VCZH]];
2. two faces that share an edge not in $T$ have images that share the image of that edge and lie on opposite sides of it.

The unfolding is a *net* of $P$ if, in addition, the interiors of the images of distinct faces are pairwise disjoint.

## Remarks

Example: let the unit cube have top face $ABCD$ and bottom face $EFGH$, with $A$ above $E$, $B$ above $F$, $C$ above $G$ and $D$ above $H$.
Let $T$ consist of the four vertical edges $AE$, $BF$, $CG$, $DH$ and the three top edges $BC$, $CD$, $DA$.
These $7$ edges join all $8$ vertices without a cycle, so $T$ is a spanning tree.
The unfolding along $T$ puts the bottom square in the middle, the four side squares around it, and the top square attached to the side $ABFE$ along $AB$.
This is the familiar cross of six squares, and the squares do not overlap, so it is a net.
Up to symmetry the cube has $11$ distinct nets, and none of its edge unfoldings overlaps.

Non-example: cutting the cube only along the four edges of its top square does not use a spanning tree, since these edges form a cycle and miss the bottom vertices.
The top face falls off, so the surface is not unfolded as one piece.

For a convex polytope, the cut edges of any unfolding of the surface into one flat piece along edges form a spanning tree, so the definition loses no unfoldings.
Some edge unfoldings of some convex polytopes overlap, which is why a net requires the extra disjointness condition.
</document>

<document source="https://tamnd.github.io/math2/node/4DSVM5Q17H.md" tag="4DSVM5Q17H" kind="concept">
# Operator system

## Summary

An operator system is a subspace of a unital C*-algebra that contains the unit and is closed under taking adjoints.
Such a subspace has enough positive elements to make sense of positive and [[completely positive maps|CGY299QMC3]] on it.
Operator systems are the setting of the Arveson extension theorem and of much of the theory of completely positive maps.
A closed operator system is in particular an [[operator space|VB1J9AA4WX]].

## Explanations

### L1 human

Inside an algebra of operators, an operator system keeps just enough structure to talk about positivity.
It must contain the identity, and with each element it must contain the adjoint.
It need not be closed under multiplication.
For example, the functions $a + b z + c \bar z$ on the unit circle, with $a, b, c$ complex, form an operator system in the continuous functions on the circle.
</document>

<document source="https://tamnd.github.io/math2/node/4DV7PEPZ7R.md" tag="4DV7PEPZ7R" kind="definition">
# Jordan algebra

## Definition

Let $F$ be a [[field|0QA8CHH8MG]] of characteristic different from $2$.
A *Jordan algebra* over $F$ is a [[vector space|05EGHEQZAW]] $J$ over $F$ with a bilinear product $J \times J \to J$, $(x, y) \mapsto xy$, such that for all $x, y \in J$:

1. $xy = yx$;
2. $(xy)(xx) = x(y(xx))$.

Clause 2 is the *Jordan identity*.
The product need not be associative, so $J$ is a [[nonassociative ring|YP5TQ6WFVN]] with an $F$-vector space structure.

## Remarks

Example: for an associative $F$-algebra $A$, put $x \circ y = \frac{1}{2}(xy + yx)$.
Clause 1 is clear, and clause 2 follows by expanding: with $x^2 = xx$ in $A$, one has $x \circ x = x^2$, and both $(x \circ y) \circ x^2$ and $x \circ (y \circ x^2)$ equal $\frac{1}{4}(xyx^2 + yx^3 + x^3y + x^2yx)$.
The resulting Jordan algebra is written $A^+$; the Hermitian $n \times n$ complex matrices form a real Jordan subalgebra of $M_n(\C)^+$.
Non-example: $\R^3$ with the cross product satisfies clause 2, since $x \times x = 0$ makes both sides $0$, but fails clause 1, because $x \times y = -y \times x$.
A Jordan algebra is power-associative: the powers of a single element do not depend on the bracketing.
</document>

<document source="https://tamnd.github.io/math2/node/4DVJQJV0DG.md" tag="4DVJQJV0DG" kind="concept">
# Net

## Summary

A net in a space $X$ is a family of points of $X$ indexed by a [[directed set|D9RY4667K6]].
It generalizes a sequence, which is indexed by $\N$.
Convergence of nets is defined just as for sequences: eventually the net stays in every neighbourhood of the limit.
In every [[topological space|0CQHFPGJ45]], nets describe closures, continuity, compactness and the Hausdorff property, which sequences do only in special spaces.
Nets are an alternative to the [[convergence of filters|70M1VAPFN0]], and the two languages are equivalent.

## Explanations

### L1 human

A sequence moves forward along $1, 2, 3, \dots$.
A net moves forward along any set of indices where any two indices have a common later index.
For example, the finite subsets of a set, ordered by inclusion, form such an index set; this is how infinite sums over arbitrary sets are defined.
A net converges to a point when, from some index on, all its points stay as close to the point as one wants.
</document>

<document source="https://tamnd.github.io/math2/node/4DWC4XFS34.md" tag="4DWC4XFS34" kind="example">
# Lagrange's theorem in S3

## Example

The [[symmetric group|0515A6D2Y1]] $S_3$ has order $6$.
Its [[subgroups|0X1JXT47GA]] are $\{e\}$, the three subgroups $\{e, (1\,2)\}$, $\{e, (1\,3)\}$ and $\{e, (2\,3)\}$, the subgroup $A_3$ of order $3$, and $S_3$.
Their orders are $1$, $2$, $3$ and $6$. Each of these numbers divides $6$, as [[Lagrange's theorem|3C53TD]] states.

## Remarks

No subgroup has order $4$ or $5$, because these numbers do not divide $6$.
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<document source="https://tamnd.github.io/math2/node/4E4RXKV9W3.md" tag="4E4RXKV9W3" kind="statement">
# van Dantzig's theorem

## Statement

Let $G$ be a Hausdorff [[locally compact group|AHV05ZBCYW]] that is [[totally disconnected|AW3J9AQSA0]].
Then every [[neighbourhood|0ZCY0Z80B6]] of the identity $e$ contains a [[compact|0R6EWFKRCF]] [[open subgroup|QYCFH68RD1]] of $G$.

## Remarks

The proof has two steps.
A locally compact Hausdorff totally disconnected space has a basis of compact open sets, so the given neighbourhood contains a compact open set $V \ni e$.
By compactness of $V$ there is a symmetric open neighbourhood $W$ of $e$ with $V W \subseteq V$.
Then all products of elements of $W$ lie in $V$, so the subgroup generated by $W$ is open and lies in $V$.
An open subgroup is closed, so this subgroup is compact.

Example: in the $p$-adic numbers $\Q_p$, the subgroups $p^k \Z_p$ with $k \ge 0$ are compact and open, and they shrink to $0$.
In $\mathrm{GL}_n(\Q_p)$, the congruence subgroups $I + p^k \mathrm{M}_n(\Z_p)$ with $k \ge 1$ play the same role.

Local compactness is necessary.
The group $\Q$ with the topology of $\R$ is totally disconnected and not locally compact.
An open subgroup of it contains an interval around $0$, so it is all of $\Q$, which is not compact.

## Explanations

### L1 human

In a totally disconnected group, points cannot be joined by any connected piece.
Van Dantzig's theorem says that, when the group is also locally compact, there are arbitrarily small compact subgroups that are also open.
They play the role of small balls around the identity.
For example, in the $p$-adic numbers the balls $p^k \Z_p$ are themselves subgroups.

### L2 human

The theorem is the starting point of the structure theory of totally disconnected locally compact groups.
It shows that such a group is a union of cosets of a compact open subgroup $U$, with a discrete coset space $G/U$.
A totally disconnected compact group is profinite, so these groups are locally profinite.
Compact open subgroups are used to define [[Cayley–Abels graphs|QWG66JE1JE]] and the [[Hecke algebras|6H3WCCHZB5]] of $p$-adic groups.
</document>

<document source="https://tamnd.github.io/math2/node/4E518M24MA.md" tag="4E518M24MA" kind="statement">
# Quillen's theorem on complex cobordism

## Statement

Let $L$ be the Lazard ring, the commutative ring generated by elements $a_{ij}$, $i, j \ge 0$, subject to the relations that make $F_L(x, y) = \sum a_{ij} x^i y^j$ a [[formal group law|TNHP7XXYZT]].
Give $a_{ij}$ the degree $2(i + j - 1)$.
Let $MU$ be the [[complex cobordism|5CZHFNY9F3]] spectrum with its standard complex orientation, and let $F_{MU}$ be its formal group law over $MU_* = \pi_* MU$, in homological grading.
Let $\theta \colon L \to MU_*$ be the unique ring homomorphism that carries $F_L$ to $F_{MU}$.
Then $\theta$ is an isomorphism of graded rings.

## Remarks

Both rings are polynomial rings $\Z[x_1, x_2, \ldots]$ with one generator in each degree $2i$: this is Lazard's theorem for $L$ and the theorem of Milnor and Novikov for $MU_*$.
So, degree by degree, $\theta$ is a map between free abelian groups of the same finite rank, and it suffices to show that it is surjective.
Quillen proved surjectivity in 1969 with his geometric description of cobordism classes and with cohomology operations in cobordism.
Rationally, the logarithm of $F_{MU}$ is Mishchenko's series $\sum_{n \ge 0} \frac{[\C P^n]}{n + 1} x^{n + 1}$.

At work: a ring map $MU_* \to R$ is the same as a formal group law over $R$.
For instance, the additive law over $\Z$ corresponds to the map that kills all positive-degree classes, and the multiplicative law $x + y - xy$ over $\Z$ corresponds to the Todd genus.

## Explanations

### L1 human

Complex cobordism classifies manifolds with a complex structure on their stable normal bundle up to cobordism.
Formal group laws are an algebraic notion, power series that act like addition.
Quillen's theorem says that the ring of complex cobordism classes of manifolds is exactly the ring that classifies formal group laws.
It is a surprising bridge between geometry and algebra.

### L2 human

The theorem turns questions about $MU$ into questions about formal groups.
Since $MU_*MU = MU_*[b_1, b_2, \ldots]$, the pair $(MU_*, MU_*MU)$ is the Hopf algebroid that classifies formal group laws and their strict isomorphisms.
This identifies the $E_2$-page of the [[Adams–Novikov spectral sequence|D34QHZVPES]] with cohomology over the moduli of formal groups.
Localizing at a prime $p$ leads to Brown–Peterson cohomology and $p$-typical laws, and the height filtration of formal groups organizes chromatic homotopy theory.
</document>

<document source="https://tamnd.github.io/math2/node/4E7ZAWAJ7S.md" tag="4E7ZAWAJ7S" kind="definition">
# Topological index

## Definition

Let $M$ be a closed smooth manifold, with cotangent bundle $\pi \colon T^*M \to M$.
Let $P \colon C^\infty(M; E) \to C^\infty(M; F)$ be an [[elliptic operator|MT9P2X85V8]] between complex [[vector bundles|P4DZM975BV]].
Its [[principal symbol|S8GC34D5ZG]] $\sigma(P)$ is a bundle map $\pi^* E \to \pi^* F$ that is invertible outside the zero section.
So it defines a class $[\sigma(P)]$ in the compactly supported K-theory $K(T^*M)$.
Choose a smooth embedding $i \colon M \to \R^N$.
It induces a Gysin map $i_! \colon K(T^*M) \to K(T^*\R^N) = K(\R^{2N})$, and Bott periodicity identifies $K(\R^{2N})$ with $\Z$.
The *topological index* of $P$ is the integer
$$\operatorname{t-ind} P = i_! [\sigma(P)] \in \Z.$$
It does not depend on the choice of the embedding.

## Remarks

In cohomology, the topological index is the integral over $T^*M$ of $\operatorname{ch}[\sigma(P)] \cdot \pi^* \operatorname{Td}(TM \otimes \C)$, up to a sign fixed by orientation conventions.
Here $\operatorname{ch}$ is the Chern character and $\operatorname{Td}$ is the [[Todd class|2YDB5089FR]].
Let $M$ be a closed oriented Riemannian manifold of even dimension and $P = d + d^*$ from even to odd forms.
Then the topological index is the Euler number $\langle e(TM), [M] \rangle$, which is the Euler characteristic of $M$.

Example: let $P \colon C^\infty(M; E) \to C^\infty(M; F)$ be the operator of order $0$ given by a bundle isomorphism $\phi \colon E \to F$.
Its symbol is $\pi^*\phi$, which is invertible on all of $T^*M$, so $[\sigma(P)] = 0$ in $K(T^*M)$ and $\operatorname{t-ind} P = 0$.
This agrees with the analytic index, since $P$ is bijective.

Non-example: for a non-elliptic operator, the principal symbol is not invertible at some nonzero covector.
Then $\sigma(P)$ does not define a class in $K(T^*M)$, and there is no topological index.
</document>

<document source="https://tamnd.github.io/math2/node/4EBHAWJJ8C.md" tag="4EBHAWJJ8C" kind="concept">
# Chain homotopy

## Summary

A chain homotopy between two [[chain maps|HTK9WQZJ1G]] $f, g \colon C_\bullet \to D_\bullet$ is a family of maps $h_n \colon C_n \to D_{n+1}$ with $f - g = dh + hd$.
Chain homotopic maps induce the same maps on [[homology|2J4JJJYP22]].
The notion is the algebraic shadow of a [[homotopy|4A7H5AG0NF]] between continuous maps, and it is used to define the [[homotopy category of chain complexes|9Y5Y92R4D5]].

## Explanations

### L1 human

Two chain maps can look different in each degree yet have the same effect on homology.
A chain homotopy is a certificate of this: a family of maps one degree up whose "boundary" is the difference $f - g$.
In topology, a homotopy between two maps of spaces sweeps out a cylinder, and taking chains of that cylinder gives exactly such maps $h_n$.
For example, on the complex $0 \to \Z \xrightarrow{1} \Z \to 0$ the identity is chain homotopic to zero, which matches the fact that this complex has zero homology.
</document>

<document source="https://tamnd.github.io/math2/node/4ECWKRYM8P.md" tag="4ECWKRYM8P" kind="definition">
# Dirac equation

## Definition

Write points of $\R^4$ as $x = (x^0, x^1, x^2, x^3)$ with $x^0 = t$, and let $\eta = \mathrm{diag}(1, -1, -1, -1)$.
Let $\gamma^0, \gamma^1, \gamma^2, \gamma^3$ be complex $4 \times 4$ matrices with
$$\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu\nu} I \quad \text{for all } \mu, \nu \in \set{0, 1, 2, 3}.$$
Let $m \ge 0$ be a constant, the mass.
The *Dirac equation* is the [[partial differential equation|GC5BDZ6QBC]]
$$\Big( i \sum_{\mu=0}^{3} \gamma^\mu \partial_\mu - m \Big) \psi = 0$$
for an unknown function $\psi \colon \R^4 \to \C^4$, where $\partial_\mu$ is the [[partial derivative|09EGC6F8PT]] in $x^\mu$.
Units are chosen with $\hbar = c = 1$.

## Remarks

The matrices $\gamma^\mu$ generate a representation of the [[Clifford algebra|A38B1QFK20]] of $(\R^4, \eta)$.
A standard choice is the Dirac representation, in which $\gamma^0 = \mathrm{diag}(1, 1, -1, -1)$.

Example: in the Dirac representation, $\psi(x) = e^{-imt} (1, 0, 0, 0)$ is a solution.
Indeed $\psi$ depends only on $t$, and $i \gamma^0 \partial_t \psi = i \gamma^0 (-im) \psi = m \gamma^0 \psi = m \psi$, since $(1, 0, 0, 0)$ is an eigenvector of $\gamma^0$ for the eigenvalue $1$.

Non-example: for $m > 0$, $\psi(x) = e^{imt} (1, 0, 0, 0)$ is not a solution, because $i \gamma^0 \partial_t \psi - m \psi = -2m \psi \ne 0$.

Applying the operator $i \gamma^\nu \partial_\nu + m$ to the equation and using the anticommutation relations gives $(\partial_t^2 - \Delta + m^2) \psi = 0$, the Klein–Gordon equation for each component.
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<document source="https://tamnd.github.io/math2/node/4EH60YPD9Q.md" tag="4EH60YPD9Q" kind="statement">
# Dini–Lipschitz criterion

## Statement

Let $f \colon \R \to \C$ be a [[continuous|0X3WGX95FE]] $2\pi$-periodic function with [[modulus of continuity|AXCJBSQZWF]]
$$\omega(\delta) = \sup \set{\abs{f(x) - f(y)} : x, y \in \R, \ \abs{x - y} \le \delta}.$$
Suppose that
$$\lim_{\delta \to 0^+} \omega(\delta) \log \frac{1}{\delta} = 0.$$
Then the partial sums of the [[Fourier series|G12KXDHPPS]] of $f$ converge to $f$ [[uniformly|08VZ57PT26]] on $\R$.

## Remarks

Let $E_N(f)$ be the uniform distance from $f$ to the trigonometric polynomials of degree at most $N$.
Since $S_N T = T$ for such a polynomial $T$, one gets $\sup \abs{f - S_N f} \le (1 + L_N) E_N(f)$, where $L_N$ is the [[Lebesgue constant|RWKQ5EP4YF]].
Jackson's theorem gives $E_N(f) \le C \omega(1/N)$ with an absolute constant $C$, and $L_N = O(\log N)$, so the error is $O(\omega(1/N) \log N)$, which tends to $0$ by hypothesis.
Example: every Hölder continuous $f$, with $\omega(\delta) \le C \delta^\alpha$ for some $\alpha > 0$, satisfies the hypothesis; so does $f$ with $\omega(\delta) \le C / (\log(1/\delta))^{2}$ for small $\delta$.

## Explanations

### L1 human

A continuous function can have a badly behaved Fourier series.
The criterion says that a little more regularity than continuity is enough to rule this out.
The required regularity just beats the logarithmic growth of the Lebesgue constants.

### L2 human

The proof is a model of a general principle: if a linear approximation process has norms $L_N$, then it converges for all functions whose best approximation error is $o(1/L_N)$.
Here the norms grow like $\log N$, and Jackson's theorem converts best approximation error into modulus of continuity.
The criterion contains the uniform convergence of Fourier series of Hölder and Lipschitz continuous functions.
</document>

<document source="https://tamnd.github.io/math2/node/4EJB6E2AZK.md" tag="4EJB6E2AZK" kind="concept">
# Semimartingale

## Summary

A semimartingale is a process that is the sum of a [[local martingale|XC2YPC2BSW]] and a process of finite variation.
Semimartingales are the largest natural class of integrators for stochastic integrals.
Brownian motion, Lévy processes and solutions of stochastic differential equations are semimartingales.

## Explanations

### L1 human

A semimartingale is "noise plus drift".
The noise part is a fair game, like Brownian motion, and the drift part changes in a tame way, like an ordinary function of bounded variation.
For example, a share price modelled as $X_t = W_t + 0.05\, t$ is a fair-game noise $W_t$ plus a steady drift.
Itô's formula and stochastic integration work for exactly this kind of process.
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<document source="https://tamnd.github.io/math2/node/4EJED0YZBB.md" tag="4EJED0YZBB" kind="concept">
# C-infinity ring

## Summary

A $C^\infty$-ring is a set on which every [[smooth function|0V33BNKZ9S]] $f \colon \R^n \to \R$ acts as an $n$-ary operation, compatibly with composition.
The basic example is the ring $C^\infty(M)$ of smooth functions on a [[smooth manifold|F45VQ47GPV]], where the operations are composition with $f$.
Every $C^\infty$-ring is in particular a commutative $\R$-algebra, but it also knows how to apply $\exp$, $\sin$ and all other smooth functions.
$C^\infty$-rings play for smooth manifolds the role that commutative rings play in algebraic geometry, and they give models of synthetic differential geometry.

## Explanations

### L1 human

In a ring one can add and multiply, so one can plug elements into polynomials.
In a $C^\infty$-ring one can plug elements into any smooth function, such as $e^x$ or $\sin(x + y)$.
For functions on a manifold this is natural: if $a$ and $b$ are smooth functions, so is $\sin(a + b)$.
Rings of germs and rings such as $\R[\varepsilon]/(\varepsilon^2)$, with $\varepsilon^2 = 0$, are also $C^\infty$-rings, and they let one treat infinitesimal points algebraically.
</document>

<document source="https://tamnd.github.io/math2/node/4F0N5J5QT2.md" tag="4F0N5J5QT2" kind="statement">
# Wald–Wolfowitz optimality of the sequential probability ratio test

## Statement

Let $X_1, X_2, \dots$ be [[independent and identically distributed|00MRN0FARV]] observations with density $f$, and test $H_0 \colon f = f_0$ against $H_1 \colon f = f_1$, where $f_0$ and $f_1$ are distinct densities.
Let $S$ be the [[sequential probability ratio test|SXR93CAP1R]] with boundaries $0 < A < 1 < B$.
Let $N$ be its sample size, and let $\alpha = P_0(S \text{ rejects } H_0)$ and $\beta = P_1(S \text{ accepts } H_0)$ be its [[error probabilities|03N8C8ZR4B]].
Let $S'$ be any [[sequential procedure|9CWYNB9003]] for the same hypotheses with sample size $N'$ and error probabilities $\alpha' \le \alpha$ and $\beta' \le \beta$.
Then
$$E_0[N] \le E_0[N'] \quad \text{and} \quad E_1[N] \le E_1[N'],$$
where $E_i$ is the [[expectation|0WQNRCNY3M]] when $f = f_i$.

## Remarks

The proof is Bayesian.
Fix a prior probability of $H_1$, a cost per observation and losses for wrong decisions.
The Bayes sequential procedure is then an optimal stopping rule, and it has the form of an SPRT on the posterior odds.
One shows that every SPRT with $A < 1 < B$ is Bayes for suitable priors and costs.
Comparing the Bayes risks of $S$ and $S'$ over a range of priors and costs for which $S$ is Bayes then gives both inequalities.
The theorem was proved by Wald and Wolfowitz in 1948.
The optimality holds only at the two simple hypotheses.
For a composite family of distributions, the expected sample size of an SPRT can be large at parameter values between $f_0$ and $f_1$.

## Explanations

### L1 human

The SPRT stops as soon as the evidence is clear enough.
This theorem says that no other method with the same or smaller chances of error can do better on average, whichever of the two hypotheses is true.
So the SPRT is the most economical test for choosing between two fully specified explanations of the data.

### L2 human

Fixed-sample tests, such as those from the [[Neyman–Pearson lemma|0EE19CFEEF]], optimize power for a given sample size.
The SPRT optimizes the opposite quantity: the expected [[stopping time|WND87FBAKC]] for given error probabilities.
In practice the SPRT often needs far fewer observations on average than the best fixed-sample test with the same error rates.
The theorem is the starting point of optimal stopping methods in sequential statistics.
</document>

<document source="https://tamnd.github.io/math2/node/4F1ABY39W1.md" tag="4F1ABY39W1" kind="definition">
# Weakly compact cardinal

## Definition

Let $\kappa$ be a [[cardinal|0EBACQFAZW]], and write $[\kappa]^2$ for the set of two-element subsets of $\kappa$.
The cardinal $\kappa$ is *weakly compact* if:

1. $\kappa$ is [[uncountable|0W7FBYS1F1]].
2. For every function $c \colon [\kappa]^2 \to \set{0, 1}$ there is a set $H \subseteq \kappa$ with $\abs{H} = \kappa$ such that $c$ is constant on $[H]^2$.

Clause 2 is written $\kappa \to (\kappa)^2_2$.

## Remarks

The cardinal $\aleph_0$ satisfies clause 2 by Ramsey's theorem, and it fails only clause 1.
The cardinal $\aleph_1$ fails clause 2: Sierpiński's colouring compares the order of $\omega_1$ with the order of an injective family of real numbers, and it has no uncountable homogeneous set.
Equivalent form: $\kappa$ is weakly compact if and only if $\kappa$ is [[inaccessible|NW4DREQ5DQ]] and has the tree property.
The tree property says that every tree of height $\kappa$ whose levels have size less than $\kappa$ has a branch of length $\kappa$.
Another equivalent form: $\kappa$ is inaccessible, and a compactness theorem holds for the infinitary logic $\mathcal{L}_{\kappa, \kappa}$.
It says that a set $\Sigma$ of at most $\kappa$ sentences has a model whenever every subset of $\Sigma$ of size less than $\kappa$ has one.
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<document source="https://tamnd.github.io/math2/node/4F3KWXYXBA.md" tag="4F3KWXYXBA" kind="concept">
# Homotopy fiber

## Summary

The homotopy fiber of a map $f \colon X \to Y$ over $y_0$ is the space of pairs of a point $x \in X$ and a path from $f(x)$ to $y_0$.
It replaces the ordinary fiber $f^{-1}(y_0)$ by a version that depends only on the homotopy class of $f$.
For a [[Serre fibration|PB9JXQ2KC3]] the two fibers have the same homotopy groups.
So homotopy fibers extend the [[long exact sequence of homotopy groups|E0RV5D1GQ0]] to arbitrary maps.
The homotopy fiber of the inclusion of a point is the [[loop space|2JFF5F3EGF]].

## Explanations

### L1 human

The ordinary fiber of $f$ over $y_0$ collects the points that land exactly on $y_0$.
The homotopy fiber is more forgiving: a point $x$ counts if it lands anywhere, as long as one also says how to walk from $f(x)$ to $y_0$.
So its points are pairs, a point $x$ and a path from $f(x)$ to $y_0$.
If $X$ is a single point mapped to $y_0$, the pairs are just the loops at $y_0$.
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<document source="https://tamnd.github.io/math2/node/4F812GHFZK.md" tag="4F812GHFZK" kind="concept">
# Resonance

## Summary

Resonance is the strong response of a linear system when it is forced at one of its own natural frequencies.
For a linear equation with constant coefficients and forcing $e^{\lambda t}$, it happens when $\lambda$ is a root of the characteristic polynomial.
Then the response is not a multiple of $e^{\lambda t}$ but picks up a power of $t$, and an undamped oscillator has unbounded solutions.
It explains the collapse of structures under periodic loads and the tuning of circuits and instruments.
See the [[definition|NTB5JQGPGC]] and the worked [[forced oscillator|0X4NT20S3G]].

## Explanations

### L1 human

A child on a swing goes higher and higher when pushed in time with the swing.
The pushes add up because they match the swing's own rhythm.
For the undamped oscillator $x'' + x = \cos t$, the forcing has the natural frequency $1$.
A solution is $x(t) = \frac{1}{2} t \sin t$, whose swings grow without bound.
With forcing $\cos(2t)$ instead, every solution stays bounded.
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<document source="https://tamnd.github.io/math2/node/4FBC8218YE.md" tag="4FBC8218YE" kind="definition">
# Sparse grid

## Definition

Let $d \ge 1$ and let $X^1 \subseteq X^2 \subseteq X^3 \subseteq \cdots$ be a nested sequence of finite subsets of an interval $I$.
For an integer $q \ge d$, the *sparse grid* of level $q$ in dimension $d$ is the set
$$H(q, d) = \bigcup_{\substack{\mathbf{i} \in \N^d,\ i_1, \dots, i_d \ge 1 \\ i_1 + \cdots + i_d \le q}} X^{i_1} \times X^{i_2} \times \cdots \times X^{i_d} \subseteq I^d.$$
Let $U^i$ be a [[quadrature rule|ADSVMCY77A]] on $I$ with nodes $X^i$, let $U^0 = 0$, and let $\Delta^i = U^i - U^{i-1}$.
The Smolyak rule is
$$A(q, d) = \sum_{i_1 + \cdots + i_d \le q} \Delta^{i_1} \otimes \cdots \otimes \Delta^{i_d}.$$
It is a rule on $I^d$ whose nodes lie in $H(q, d)$.

## Remarks

The full tensor grid $X^{q - d + 1} \times \cdots \times X^{q - d + 1}$ is the union of the products with every $i_j \le q - d + 1$; the sparse grid replaces this box condition on $\mathbf{i}$ by the simplex condition $i_1 + \cdots + i_d \le q$.
Write $\ell = q - d + 1$ for the finest level in each direction.
When $X^i$ has about $2^i$ points and $d$ is fixed, the sparse grid has $O(2^\ell \ell^{d-1})$ points, while the full grid $X^\ell \times \cdots \times X^\ell$ has about $2^{\ell d}$ points.

Example: take $I = [-1, 1]$, $X^1 = \set{0}$ and $X^2 = \set{-1, 0, 1}$, and $q = d + 1$.
The allowed $\mathbf{i}$ are $(1, \dots, 1)$ and those with one entry $2$ and the others $1$.
So $H(d + 1, d)$ consists of the origin and the $2d$ points $\pm e_j$, which makes $2d + 1$ points.

Non-example: the full grid $\set{-1, 0, 1}^d$ is not $H(d + 1, d)$ for $d \ge 2$, since it contains $(1, 1, 0, \dots, 0)$, which would need $\mathbf{i}$ with two entries equal to $2$ and so $i_1 + \cdots + i_d = d + 2 > q$.
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# Coproduct

## Summary

The coproduct of a family of objects in a [[category|A2DE66937D]] is an object with an injection from each member of the family.
Any family of morphisms out of the members factors uniquely through it.
It is the notion dual to the [[product|S5744QZV4N]], and it is the [[colimit|BJS1CGTKQ5]] of a diagram with no non-identity arrows.
It is the disjoint union for sets and topological spaces, the direct sum for abelian groups and modules, and the free product for groups.

## Explanations

### L1 human

A map out of a coproduct is the same thing as a list of maps, one out of each piece.
For sets, the coproduct of $X$ and $Y$ is their disjoint union.
A function on it is given by saying what it does on $X$ and what it does on $Y$, with no further constraint.
For abelian groups the answer is different: the coproduct of $A$ and $B$ is the [[direct sum|0S7W6EG32V]] $A \oplus B$.
This works because homomorphisms out of $A$ and out of $B$ can be added together.
In a partially ordered set seen as a category, the coproduct of two elements is their least upper bound.
</document>

<document source="https://tamnd.github.io/math2/node/4FK55CYMVZ.md" tag="4FK55CYMVZ" kind="definition">
# Pringsheim convergence

## Definition

Let $(a_{m,n})_{m, n \ge 1}$ be a [[double sequence|B70G8FYPG0]] of real or complex numbers, and let $L$ be a number.
The double sequence *converges to $L$ in the sense of Pringsheim* if the following holds.
For every $\varepsilon > 0$ there is an integer $N$ such that $\abs{a_{m,n} - L} < \varepsilon$ for all $m \ge N$ and $n \ge N$.
Then $L$ is unique; it is called the *double limit* and is written $\lim_{m, n \to \infty} a_{m,n} = L$.

## Remarks

Example: $a_{m,n} = \frac{1}{m} + \frac{1}{n}$ converges to $0$, because $m, n \ge N$ gives $\abs{a_{m,n}} \le \frac{2}{N}$, which is less than $\varepsilon$ once $N > 2/\varepsilon$.
Non-example: $a_{m,n} = \frac{m}{m+n}$ has no double limit, because $a_{N,N} = \frac12$ while $a_{N,n} \to 0$ as $n \to \infty$, so no single $L$ works for $\varepsilon = \frac14$.
A Pringsheim convergent double sequence need not be bounded: put $a_{1,n} = n$ and $a_{m,n} = 0$ for $m \ge 2$; the double limit is $0$, yet the first row diverges.
So the double limit may exist while an iterated limit $\lim_m \lim_n a_{m,n}$ does not; the [[Moore–Osgood theorem|QN2210XCST]] gives conditions under which they agree.

## Explanations

### L1 human

Both indices must be large at the same time.
It is not enough to send one index to infinity first and then the other.
</document>

<document source="https://tamnd.github.io/math2/node/4G25S7PJ22.md" tag="4G25S7PJ22" kind="concept">
# Monomial ideal

## Summary

A monomial ideal is an ideal of a polynomial ring over a field that is generated by monomials.
Membership in a monomial ideal is a combinatorial question about exponent vectors, so many invariants of these ideals can be computed by counting.
Every ideal can be deformed to a monomial ideal, its ideal of leading terms with respect to a [[monomial order|E874NARWPV]].
This is the bridge from general ideals to combinatorics through [[Gröbner bases|C83KPT47R0]].

## Explanations

### L1 human

A monomial is a product of variables such as $x^2 y$.
A monomial ideal is what one gets by taking some monomials and all their multiples, then all sums of those.
Picture the exponents $(a, b)$ of the monomials $x^a y^b$ as points of a grid: a monomial ideal is a staircase region that is closed under moving up and to the right.
For example, $(x^2, xy, y^3)$ contains every monomial except $1, x, y, y^2$.
A polynomial lies in the ideal exactly when each of its terms lies in the staircase.
</document>

<document source="https://tamnd.github.io/math2/node/4G53E7BRXF.md" tag="4G53E7BRXF" kind="statement">
# Doob's martingale inequality

## Statement

Let $n \ge 0$, let $(X_k)_{0 \le k \le n}$ be a [[submartingale|EQ055QMACW]] with respect to a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_k)_{0 \le k \le n}$, and let $\lambda > 0$.
Then
$$\lambda\, P\Bigl(\max_{0 \le k \le n} X_k \ge \lambda\Bigr) \le E\Bigl[X_n \mathbf{1}\bigl\{\max_{0 \le k \le n} X_k \ge \lambda\bigr\}\Bigr] \le E[X_n^+],$$
where $X_n^+ = \max(X_n, 0)$.

## Remarks

Let $\tau$ be the first $k \le n$ with $X_k \ge \lambda$, and split the event $A = \{\max_k X_k \ge \lambda\}$ into the disjoint events $\{\tau = k\}$.
Since $\{\tau = k\} \in \mathcal{F}_k$, the submartingale property gives $E[X_n \mathbf{1}\{\tau = k\}] \ge E[X_k \mathbf{1}\{\tau = k\}] \ge \lambda P(\tau = k)$.
Summing over $k$ gives the first inequality, and $X_n \mathbf{1}_A \le X_n^+$ gives the second.
If $(M_k)$ is a [[martingale|0CDVK9RQBH]], then $\abs{M_k}$ is a submartingale, so $P(\max_{k \le n} \abs{M_k} \ge \lambda) \le E[\abs{M_n}]/\lambda$.
For $M_k$ a sum of independent centred square-integrable variables, applying this to the submartingale $M_k^2$ gives [[Kolmogorov's inequality|D9B4HE4B4Q]].
The inequality is stronger than [[Markov's inequality|05PWWHYBX5]] for $X_n$ alone: it controls the whole path up to time $n$ by the last value.
For a non-negative submartingale and $p > 1$, it implies Doob's $L^p$ inequality $E[\max_{k \le n} X_k^p] \le \bigl(\tfrac{p}{p-1}\bigr)^p E[X_n^p]$.

## Explanations

### L1 human

For a process that tends to increase, the chance that it ever climbs above a level $\lambda$ before time $n$ is controlled by its final value alone.
The bound is the same as the Markov bound for the single variable $X_n$.
So looking at the maximum over the whole path costs nothing extra.

### L2 human

The inequality is the basic maximal inequality of martingale theory.
It gives Kolmogorov's inequality, which is a step in the strong law of large numbers and the three-series theorem.
Its $L^p$ form is used to prove $L^p$ convergence of martingales and to define the norm on spaces of continuous martingales.
In continuous time it holds for right-continuous submartingales, with the maximum replaced by the supremum over $[0, t]$.
</document>

<document source="https://tamnd.github.io/math2/node/4G65M0ZWAZ.md" tag="4G65M0ZWAZ" kind="definition">
# Brun sieve

## Definition

Let $\mathcal{A}$ be a finite set of integers, let $\mathcal{P}$ be a set of primes and let $z \ge 2$ be real.
Let $P(z)$ be the product of the primes $p \in \mathcal{P}$ with $p < z$.
For $d \ge 1$ let $\mathcal{A}_d = \set{a \in \mathcal{A} : d \mid a}$.
The sifting function is
$$S(\mathcal{A}, \mathcal{P}, z) = \#\set{a \in \mathcal{A} : \gcd(a, P(z)) = 1}.$$
Let $\mu$ be the [[Möbius function|0FANE7P0DY]] and $\omega(d)$ the [[number of distinct prime factors|08RQ62VAVA]] of $d$.
The *Brun sieve* in its simplest form, Brun's pure sieve, uses the truncated sums
$$S_r = \sum_{\substack{d \mid P(z) \\ \omega(d) \le r}} \mu(d)\, \#\mathcal{A}_d,$$
together with the inequalities, valid for every integer $r \ge 0$,
$$S_{2r+1} \le S(\mathcal{A}, \mathcal{P}, z) \le S_{2r}.$$

## Remarks

Example: let $\mathcal{A} = \set{1, \ldots, 10}$, let $\mathcal{P}$ be all primes and $z = 4$, so $P(z) = 6$.
Then $S(\mathcal{A}, \mathcal{P}, 4) = \#\set{1, 5, 7} = 3$.
Here $S_0 = 10$ and $S_1 = 10 - 5 - 3 = 2$, so $S_1 \le 3 \le S_0$; and $S_2 = 2 + \#\mathcal{A}_6 = 3$.
The inequalities are the Bonferroni inequalities: for $m \ge 1$, the partial sums $\sum_{j \le k} (-1)^j \binom{m}{j}$ are $\ge 0$ for even $k$ and $\le 0$ for odd $k$.
Non-example: the full Legendre sum $S_r$ with $r \ge \omega(P(z))$ is exact, but it is not a truncation.
It has $2^{\omega(P(z))}$ terms, and the errors in approximating each $\#\mathcal{A}_d$ add up.
Brun's later, more refined sieve replaces the condition $\omega(d) \le r$ by conditions on the sizes of the prime factors of $d$.
</document>

<document source="https://tamnd.github.io/math2/node/4G6CFK4T7F.md" tag="4G6CFK4T7F" kind="definition">
# Homotopy fiber

## Definition

Let $X$ be a topological space, let $(Y, y_0)$ be a [[pointed space|T9DG8DDE75]], and let $f \colon X \to Y$ be a [[continuous map|0X3WGX95FE]].
Let $I = [0, 1]$ and let $Y^I$ be the space of [[paths|03NAMV1DHJ]] $\gamma \colon I \to Y$ with the compact-open topology.
The *homotopy fiber* of $f$ over $y_0$ is
$$F_f = \set{(x, \gamma) \in X \times Y^I : \gamma(0) = f(x), \ \gamma(1) = y_0},$$
with the [[subspace topology|0WS9P12FQR]] of the [[product|0GBG8B49AA]] $X \times Y^I$.

## Remarks

The map $F_f \to X$, $(x, \gamma) \mapsto x$, is a fibration, and its fiber over $x$ is the space of paths from $f(x)$ to $y_0$.
The ordinary fiber $f^{-1}(y_0)$ embeds in $F_f$ by $x \mapsto (x, c_{y_0})$, where $c_{y_0}$ is the constant path.
If $f$ is a [[Serre fibration|PB9JXQ2KC3]], this embedding is a weak homotopy equivalence.

Example: if $X = \set{x_0}$ and $f(x_0) = y_0$, then $F_f$ is the space of paths from $y_0$ to $y_0$, which is the [[loop space|2JFF5F3EGF]] $\Omega Y$.

Non-example: the ordinary fiber need not be a homotopy fiber.
Let $X = \set{x_0}$, $Y = \R$, $f(x_0) = 0$ and $y_0 = 1$.
Then $f^{-1}(1)$ is empty, while $F_f$ is the space of paths in $\R$ from $0$ to $1$, which is nonempty and contractible.

The notations $\operatorname{hofib}(f)$ and $\operatorname{hofib}_{y_0}(f)$ are also used.
</document>

<document source="https://tamnd.github.io/math2/node/4G7BGMGGK2.md" tag="4G7BGMGGK2" kind="definition">
# Smale's degree for proper Fredholm maps

## Definition

Let $M$ and $N$ be connected $C^2$ [[Banach manifolds|9K15AMGBV9]], with $M$ separable.
Let $f \colon M \to N$ be a $C^2$ [[Fredholm map|GFXQ6HEM65]] of [[Fredholm index|2V9Q4KHY1T]] $0$ that is [[proper|15XEBGNGHC]].
A point $y \in N$ is a *regular value* of $f$ if $df_x$ is surjective for every $x \in f^{-1}(y)$.
For a regular value $y$, the fibre $f^{-1}(y)$ is finite.
The *Smale degree* of $f$ is
$$\deg_2 f = \# f^{-1}(y) \bmod 2 \in \Z/2,$$
for any regular value $y$ of $f$.

## Remarks

The definition makes sense because regular values exist and the parity does not depend on $y$.
Regular values form a residual subset of $N$ by the [[Sard–Smale theorem|Z0NPFYDVP2]].
At a point of $f^{-1}(y)$ the derivative is a surjective Fredholm operator of index $0$, hence an isomorphism, so $f^{-1}(y)$ is discrete; properness makes it compact, hence finite.
The degree is also unchanged under a $C^2$ homotopy $H \colon M \times [0, 1] \to N$ that is proper and is Fredholm of index $0$ at each time.

Example: $f \colon \R \to \R$, $f(x) = x^3$, is proper, and every smooth map between manifolds of the same finite dimension is Fredholm of index $0$.
The value $y = 1$ is regular with the single preimage $x = 1$, so $\deg_2 f = 1$.

Non-example: $g \colon \R \to \R$, $g(x) = e^x$, is Fredholm of index $0$ but not proper.
The regular value $1$ has one preimage and the regular value $-1$ has none, so the parity depends on the point, and no degree is defined.

In finite dimensions, for compact connected manifolds of the same dimension, $\deg_2 f$ is the mod $2$ reduction of the [[degree of a continuous map|TMFG2P912R]].
For compact perturbations of the identity on a Banach space, the [[Leray–Schauder degree|3VJC5Q9MYA]] gives an integer-valued theory.
</document>

<document source="https://tamnd.github.io/math2/node/4GG1HGDEZC.md" tag="4GG1HGDEZC" kind="statement">
# XYZ inequality

## Statement

Let $(P, \le)$ be a finite [[partially ordered set|0QV2JS9RJ3]] and let $x, y, z \in P$.
Choose a [[linear extension|W8DZ5KGGHN]] $\prec$ of $P$ uniformly at random among all linear extensions, and write $\Pr$ for the resulting probability.
Then
$$\Pr(x \prec y \text{ and } x \prec z) \ge \Pr(x \prec y) \, \Pr(x \prec z).$$

## Remarks

Shepp proved the inequality using the [[FKG inequality|DCXJGS6E6D]] on a suitable distributive lattice of integer labellings of $P$.
It says that the events "$x$ comes before $y$" and "$x$ comes before $z$" are positively correlated.
This matches intuition: if $x$ comes early relative to $y$, it tends to come early in general.

Example: let $P$ be the antichain on $\set{x, y, z}$.
All $6$ orderings are linear extensions, so $\Pr(x \prec y) = \Pr(x \prec z) = 1/2$, and $x$ comes first in $2$ of the $6$ orderings, so the left side is $1/3 \ge 1/4$.

## Explanations

### L1 human

Put the elements of a poset in a random order that respects the poset.
Knowing that $x$ comes before $y$ makes it more likely, or at least not less likely, that $x$ also comes before $z$.
The XYZ inequality states this in numbers.

### L2 human

The inequality belongs to the family of correlation inequalities for linear extensions, studied in connection with sorting and with the 1/3–2/3 conjecture.
Its proof is a model use of the FKG inequality.
One looks at order-preserving labellings of $P$ by a large chain, which form a distributive lattice.
One applies FKG there and then lets the chain grow.
</document>

<document source="https://tamnd.github.io/math2/node/4GG7VRW0DY.md" tag="4GG7VRW0DY" kind="definition">
# Chern class

## Definition

Let $X$ be a smooth [[quasi-projective variety|EQSHJPE7H8]] over a field, with [[Chow ring|4HV4WTAX9P]] $CH^\bullet(X)$ built from its [[Chow groups|EBEF6PEK4N]].
For an [[invertible sheaf|7JCF2TE3V5]] $L$ on $X$, let $c_1(L) \in CH^1(X)$ be the class of a [[Cartier divisor|FHV05ZK99F]] $D$ with $L \cong \mathcal{O}_X(D)$.
Let $E$ be a [[vector bundle|P4DZM975BV]] of rank $r$ on $X$.
Let $\pi \colon \mathbb{P}(E) \to X$ be the associated projective bundle, $\mathbb{P}(E) = \operatorname{Proj} \operatorname{Sym} E$, and let $\xi = c_1(\mathcal{O}_{\mathbb{P}(E)}(1)) \in CH^1(\mathbb{P}(E))$, where $\mathcal{O}_{\mathbb{P}(E)}(1)$ is the [[twisting sheaf|V93Y0XYPZD]].
Then $CH^\bullet(\mathbb{P}(E))$ is a free $CH^\bullet(X)$-module, through $\pi^*$, with basis $1, \xi, \ldots, \xi^{r-1}$.
The *Chern classes* $c_i(E) \in CH^i(X)$, for $0 \le i \le r$, are the unique classes with $c_0(E) = 1$ and
$$\sum_{i=0}^{r} (-1)^i \, \pi^* c_i(E) \, \xi^{r-i} = 0.$$
One sets $c_i(E) = 0$ for $i > r$, and the *total Chern class* is $c(E) = \sum_i c_i(E)$.

## Remarks

This is Grothendieck's definition.
The Chern classes satisfy the Whitney formula $c(E) = c(E') \, c(E'')$ for every exact sequence $0 \to E' \to E \to E'' \to 0$, and they are compatible with pullback.
Example: for a line bundle $L$, $\mathbb{P}(L) = X$ and $\mathcal{O}_{\mathbb{P}(L)}(1) = L$, so $\xi = c_1(L)$, and the relation $\xi - c_1(L) = 0$ agrees with the definition of $c_1(L)$ for invertible sheaves.
On $\mathbb{P}^n$ with hyperplane class $h$, the Euler sequence gives $c(T_{\mathbb{P}^n}) = (1 + h)^{n+1}$.
Non-example: the skyscraper sheaf $k(P)$ at a point $P$ of a curve is coherent but not locally free.
So it is not a vector bundle, and the definition does not apply to it directly.
Over $\C$, the cycle class map sends $c_i(E)$ to the topological Chern class $c_i(E) \in H^{2i}(X(\C), \Z)$.
</document>

<document source="https://tamnd.github.io/math2/node/4GM75FC4PP.md" tag="4GM75FC4PP" kind="definition">
# Maass wave form

## Definition

Let $\mathbb{H} = \set{z = x + iy : y > 0}$ be the [[upper half-plane|X6XGGH5FPW]].
Let $\Gamma$ be a [[congruence subgroup|1H6R856P3C]] of $\mathrm{SL}_2(\Z)$, acting on $\mathbb{H}$ by $\gamma z = \frac{az + b}{cz + d}$ for $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.
Let $\Delta = -y^2 \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right)$ be the hyperbolic [[Laplace operator|F4NWZNTRMD]].
A *Maass wave form* for $\Gamma$ is a [[smooth function|0V33BNKZ9S]] $f \colon \mathbb{H} \to \C$ such that:

1. $f(\gamma z) = f(z)$ for all $\gamma \in \Gamma$ and $z \in \mathbb{H}$.
2. There is $\lambda \in \C$ with $\Delta f = \lambda f$.
3. $f$ has moderate growth at every cusp: for every $\sigma \in \mathrm{SL}_2(\Z)$ there is $N > 0$ with $f(\sigma z) = O(y^N)$ as $y \to \infty$, uniformly in $x$.

The number $\lambda$ is the [[eigenvalue|0BQAA2BK3Q]] of $f$.
A *Maass cusp form* is a Maass form whose constant term $\int_0^1 f(\sigma(x + iy)) \, dx$ vanishes at every cusp, that is, for every $\sigma \in \mathrm{SL}_2(\Z)$.

## Remarks

Example: for $\Gamma = \mathrm{SL}_2(\Z)$ and $\mathrm{Re}(s) > 1$, the real-analytic Eisenstein series
$$E(z, s) = \frac{1}{2} \sum_{\substack{(c, d) \in \Z^2 \\ \gcd(c, d) = 1}} \frac{y^s}{\abs{cz + d}^{2s}}$$
is a Maass form with eigenvalue $s(1 - s)$.
Each term is $(\mathrm{Im}\, \gamma z)^s$ for a matrix $\gamma$ with bottom row $(c, d)$, so the sum is $\Gamma$-invariant, and $\Delta y^s = -y^2 s(s-1) y^{s-2} = s(1-s) y^s$ term by term.
It is not a cusp form, since its constant term is $y^s + \varphi(s) y^{1-s}$ for some function $\varphi$.

Non-example: $f(z) = y^s$ satisfies clauses 2 and 3 and is invariant under $z \mapsto z + 1$, but it is not invariant under $z \mapsto -1/z$, so clause 1 fails.

A cusp form decays rapidly at the cusps and is square-integrable on $\Gamma \backslash \mathbb{H}$; the eigenvalue of a nonzero Maass cusp form is then real and positive.
One often writes $\lambda = \frac{1}{4} + r^2$ or $\lambda = s(1 - s)$.
Some authors also allow a weight or a character in clause 1.
</document>

<document source="https://tamnd.github.io/math2/node/4GNPFMT7DZ.md" tag="4GNPFMT7DZ" kind="example">
# Lamplighter group

## Example

Let $L = \bigoplus_{i \in \Z} \Z/2\Z$ be the group of finitely supported functions $f \colon \Z \to \Z/2\Z$, with pointwise addition.
Let $\Z$ act on $L$ by shifts: $(n \cdot f)(i) = f(i - n)$.
The *lamplighter group* is the [[semidirect product|0H3WZXCZNF]]
$$\Z/2\Z \wr \Z = L \rtimes \Z,$$
the restricted [[wreath product|W1X984PKJ9]] of $\Z/2\Z$ by $\Z$.
An element is a pair $(f, n)$.
Here $f$ says which lamps on the integer line are lit, finitely many of them, and $n$ is the position of the lamplighter.

It is generated by $t = (0, 1)$, which moves the lamplighter one step, and $a = (\delta_0, 0)$, which switches the lamp at $0$.
So it is a [[finitely generated group|082DWWXMRJ]].
It is an extension of the abelian group $L$ by the abelian group $\Z$, so it is [[solvable|0FXJ7YFMMX]] of derived length $2$.
Solvable groups are amenable, so it is [[amenable|CEAT9BESKK]].

## Remarks

The group has exponential [[growth|4CMRM3975W]].
Consider the $2^n$ words $a^{\varepsilon_1} t \, a^{\varepsilon_2} t \cdots a^{\varepsilon_n} t$ with $\varepsilon_i \in \set{0, 1}$, of length at most $2n$.
They give distinct elements, because the lamps lit along the way, at $n$ consecutive positions, record the sequence $(\varepsilon_1, \ldots, \varepsilon_n)$.
So it is a solvable group that is not virtually nilpotent.
It has the infinite presentation $\langle a, t \mid a^2, \ [a, t^k a t^{-k}] \text{ for } k \in \Z \rangle$, and it is not finitely presented.

## Explanations

### L1 human

Picture an infinite street of lamps, one at each integer, and a lamplighter walking along it.
A state is a finite set of lit lamps and a position of the lamplighter.
The two moves are: walk one step, or switch the lamp where you stand.
Combining moves is the group operation.
This simple group is a standard source of examples and counterexamples in [[geometric group theory|2T3218RXTD]].
</document>

<document source="https://tamnd.github.io/math2/node/4GPAX3B1MK.md" tag="4GPAX3B1MK" kind="definition">
# Radon transform

## Definition

Let $n \ge 2$, and let $S^{n-1}$ be the unit [[sphere|P9CM4GQ2T0]] in $\R^n$.
For $\theta \in S^{n-1}$ and $s \in \R$, the set $\set{x \in \R^n : x \cdot \theta = s}$ is a hyperplane, and every hyperplane is of this form.
Let $\theta^\perp = \set{y \in \R^n : y \cdot \theta = 0}$, with its $(n-1)$-dimensional Lebesgue measure $dy$.
Let $f \colon \R^n \to \C$ be continuous and [[integrable|X7GTF3D98F]] on every hyperplane.
The *Radon transform* of $f$ is the function on $S^{n-1} \times \R$ given by
$$(R f)(\theta, s) = \int_{\theta^\perp} f(s\theta + y)\, dy.$$

## Remarks

$R f(\theta, s)$ is the integral of $f$ over the hyperplane $x \cdot \theta = s$.
Since $(-\theta, -s)$ gives the same hyperplane, $R f(-\theta, -s) = R f(\theta, s)$.
The transform is also defined for $f \in L^1(\R^n)$, at almost every $(\theta, s)$, by Fubini's theorem.
One often writes $R_\theta f(s) = R f(\theta, s)$.

Example: $f(x) = e^{-\abs{x}^2}$.
For $y \in \theta^\perp$, $\abs{s\theta + y}^2 = s^2 + \abs{y}^2$, so $R f(\theta, s) = e^{-s^2} \int_{\R^{n-1}} e^{-\abs{y}^2}\, dy = \pi^{(n-1)/2} e^{-s^2}$.

Example: in $\R^2$, the indicator of the unit disk has $R f(\theta, s) = 2\sqrt{1 - s^2}$ for $\abs{s} \le 1$ and $0$ otherwise.

Non-example: $f = 1$ is not integrable on any hyperplane, so it has no Radon transform.

For $n = 2$ the Radon transform coincides with the [[X-ray transform|V9G9J2C10P]] up to the parametrization of lines.
</document>

<document source="https://tamnd.github.io/math2/node/4GRWMXWJNZ.md" tag="4GRWMXWJNZ" kind="definition">
# Semisimple module

## Definition

Let $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].
The module $M$ is *semisimple* if there is a family $(S_i)_{i \in I}$ of [[simple|G00C22AEDH]] [[submodules|0RGZ10DFPG]] of $M$ such that $M = \bigoplus_{i \in I} S_i$.

## Remarks

The empty family is allowed, so the zero module is semisimple.
Equivalently, $M$ is a sum of simple submodules, or every submodule of $M$ is a direct summand.
Example: the $\Z$-module $\Z/6\Z$ is semisimple.
The check: $\Z/6\Z = \set{0, 3} \oplus \set{0, 2, 4}$, and these subgroups have prime orders $2$ and $3$, so they are simple.
Non-example: the $\Z$-module $\Z/4\Z$ is not semisimple.
Its only simple submodule is $\set{0, 2}$, so every sum of simple submodules lies inside $\set{0, 2} \ne \Z/4\Z$.
Every vector space over a field, or over a division ring, is semisimple, as the direct sum of the lines spanned by the vectors of a basis.
</document>

<document source="https://tamnd.github.io/math2/node/4GTNM0WHDH.md" tag="4GTNM0WHDH" kind="concept">
# Ricci curvature

## Summary

The Ricci curvature of a [[Riemannian manifold|7CBT4SM83Y]] is a trace of the [[Riemann curvature tensor|JA29ETCM02]].
It is a symmetric bilinear form on each tangent space.
Let $v$ be a unit vector and $e_2, \ldots, e_n$ an orthonormal basis of $v^\perp$.
Then $\operatorname{Ric}(v, v)$ is the sum of the [[sectional curvatures|1X42JAA349]] of the planes spanned by $v$ and $e_i$.
It controls how the volume of small balls and cones of geodesics differs from the Euclidean value.
It appears in Einstein's equations of general relativity, in the Ricci flow, and in theorems such as [[Myers's theorem|FPMJEX3DSJ]] and the [[Bishop–Gromov inequality|82F5Q4F1S6]].

## Explanations

### L1 human

Ricci curvature is an average of curvatures in all the planes containing a given direction.
Positive Ricci curvature means that geodesics leaving a point in nearby directions tend to come together, so volumes grow more slowly than in flat space.
Negative Ricci curvature means they spread apart and volumes grow faster.
On the unit sphere of dimension $n$, the Ricci curvature is $(n - 1)$ times the metric.
</document>

<document source="https://tamnd.github.io/math2/node/4H1E4KW1JX.md" tag="4H1E4KW1JX" kind="definition">
# Hamilton–Jacobi equation

## Definition

Let $n \ge 1$ and let $H \colon \R^n \times \R^n \to \R$ be a continuous function, called the *Hamiltonian*.
Let $T \in (0, \infty]$.
The *Hamilton–Jacobi equation* with Hamiltonian $H$ is the [[partial differential equation|GC5BDZ6QBC]]
$$\partial_t u + H(\nabla u, x) = 0$$
for $u \colon \R^n \times (0, T) \to \R$.
Here $\nabla u$ is the [[gradient|0ABK9V9Y64]] of $u$ in $x$.
The *stationary* Hamilton–Jacobi equation is $H(\nabla u, x) = 0$ for $u$ defined on an open subset of $\R^n$.

## Remarks

Often $H$ does not depend on $x$, and the equation reads $u_t + H(\nabla u) = 0$, usually with an initial condition $u(x, 0) = g(x)$.
In mechanics the Hamiltonian also depends on time, and the equation is $\partial_t S + H(\nabla S, x, t) = 0$ for Hamilton's principal function $S$.
The [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] of optimal control is a Hamilton–Jacobi equation whose Hamiltonian is an infimum or a supremum over controls.

Example: $H(p, x) = \tfrac12 \abs{p}^2$ and $u(x, t) = \dfrac{\abs{x}^2}{2(t + 1)}$ for $t > 0$.
Check: $\partial_t u = -\dfrac{\abs{x}^2}{2(t + 1)^2}$ and $\nabla u = \dfrac{x}{t + 1}$, so $\tfrac12 \abs{\nabla u}^2 = \dfrac{\abs{x}^2}{2(t + 1)^2}$ and the sum is $0$.

Non-example: the heat equation $u_t - \Delta u = 0$ is not a Hamilton–Jacobi equation, since it involves second derivatives of $u$.
Adding a term $-\varepsilon \Delta u$ to a Hamilton–Jacobi equation gives a viscous approximation, whose limits as $\varepsilon \to 0$ lead to viscosity solutions.
</document>

<document source="https://tamnd.github.io/math2/node/4H31EM5T0C.md" tag="4H31EM5T0C" kind="problem">
# Erdős distinct distances problem

## Problem

Let $n \ge 2$ be an integer.
For a set $S$ of $n$ points in the plane $\R^2$, consider the [[Euclidean distances|0NPPGHQNXQ]] $\norm{p - q}$ with $p, q \in S$ and $p \ne q$.
Let $g(n)$ be the least number of distinct values of these distances, over all such sets $S$.
Find the order of growth of $g(n)$ as $n \to \infty$.
Erdős conjectured that the square grid is essentially optimal.
A weaker form of his conjecture says that $g(n) \ge n^{1 - \varepsilon}$ for every $\varepsilon > 0$ and all large enough $n$.

## Remarks

Erdős posed the problem in 1946 and proved the lower bound $g(n) = \Omega(\sqrt{n})$.
A $\sqrt{n} \times \sqrt{n}$ piece of the integer lattice determines only $O(n / \sqrt{\log n})$ distinct distances, so $g(n) = O(n / \sqrt{\log n})$.
Guth and Katz proved the lower bound $g(n) = \Omega(n / \log n)$.
This settles the weaker conjecture $g(n) \ge n^{1 - \varepsilon}$.
The exact order of $g(n)$ is still open: the known bounds differ by a factor of order $\sqrt{\log n}$.
The analogous questions in higher dimensions are also studied.

## Explanations

### L1 human

Put $n$ dots in the plane and list all the distances between pairs of dots.
How few different values can this list contain?
If the dots are spread at random, almost all distances differ.
A square grid does much better, because many pairs of dots have the same distance.
The problem asks whether one can do much better than a grid, and the answer is essentially no.
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<document source="https://tamnd.github.io/math2/node/4H35VQ16DC.md" tag="4H35VQ16DC" kind="concept">
# Realizability

## Summary

Realizability is a way to read intuitionistic statements as statements about programs.
A realizer of a formula is a program, often the code of a [[computable function|KZSB5E471K]], that carries the computational content of a proof of it.
Kleene introduced number realizability for [[Heyting arithmetic|CK8ZQRN183]] in 1945, as a precise form of the [[Brouwer–Heyting–Kolmogorov interpretation|7B53ZGXXEQ]].
Realizability shows that intuitionistic proofs of existence yield algorithms, and it gives models in which classical principles fail.

## Explanations

### L1 human

In constructive mathematics, to prove "for every $n$ there is an $m$" one must in effect give a way to find $m$ from $n$.
Realizability makes "a way to find" precise: it is a program.
A program realizes a statement when it provides all the data the statement asks for.
For example, a program that sends $n$ to $2n$, together with a trivial check that $2n = n + n$, realizes "every number has a double".
Every theorem of Heyting arithmetic has a realizer, so its existence theorems come with algorithms.
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<document source="https://tamnd.github.io/math2/node/4H3AHHDSVS.md" tag="4H3AHHDSVS" kind="definition">
# Poisson manifold

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] and let $C^\infty(M)$ be the algebra of smooth real functions on $M$.
A *Poisson bracket* on $M$ is a map $\{\cdot, \cdot\} \colon C^\infty(M) \times C^\infty(M) \to C^\infty(M)$ such that for all $f, g, h \in C^\infty(M)$:

1. it is bilinear over $\R$;
2. it is antisymmetric: $\{f, g\} = -\{g, f\}$;
3. it satisfies the Jacobi identity: $\{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0$;
4. it satisfies the Leibniz rule: $\{f, gh\} = \{f, g\}\, h + g\, \{f, h\}$.

A *Poisson manifold* is a smooth manifold with a Poisson bracket.

## Remarks

Clauses 1 to 3 say that $(C^\infty(M), \{\cdot, \cdot\})$ is a Lie algebra over $\R$.
By clause 4, there is a unique bivector field $\pi$, a section of $\Lambda^2 TM$, with $\{f, g\} = \pi(df, dg)$; the Jacobi identity is then the condition $[\pi, \pi] = 0$ for the Schouten bracket.

Example: every [[symplectic manifold|CAN490EPRT]] with its [[Poisson bracket|2S0MXDT4YH]] is a Poisson manifold.
Another example is the dual $\mathfrak{g}^*$ of a finite-dimensional Lie algebra $\mathfrak{g}$, with $\{f, g\}(\xi) = \langle \xi, [df_\xi, dg_\xi] \rangle$, where $df_\xi \in (\mathfrak{g}^*)^* = \mathfrak{g}$.
On $\mathfrak{g}^*$ for $\mathfrak{g} = \mathfrak{so}(3)$ with suitable coordinates, $\{x, y\} = z$, $\{y, z\} = x$ and $\{z, x\} = y$.

Non-example: on $\R^3$, define a bracket by clauses 1, 2, 4 and $\{x, y\} = 1$, $\{z, x\} = x$, $\{y, z\} = 0$.
Then $\{x, \{y, z\}\} + \{y, \{z, x\}\} + \{z, \{x, y\}\} = 0 + \{y, x\} + \{z, 1\} = -1$, so the Jacobi identity fails.
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<document source="https://tamnd.github.io/math2/node/4H7DF303WF.md" tag="4H7DF303WF" kind="definition">
# Determinacy

## Definition

Let $A$ be a nonempty set and let $X \subseteq A^{\N}$ be a set of infinite [[sequences|0G9EKM0E15]] of elements of $A$.
The game $G(A, X)$ is played by two players, I and II.
Player I chooses $a_0 \in A$, then player II chooses $a_1 \in A$, then player I chooses $a_2 \in A$, and so on forever.
Player I wins the play $x = (a_0, a_1, a_2, \dots)$ if $x \in X$, and player II wins it otherwise.

1. A *strategy* for player I is a map $\sigma$ from the finite sequences of elements of $A$ of even length to $A$.
A strategy for player II is a map $\tau$ from the finite sequences of odd length to $A$.
2. A play *follows* $\sigma$ if $a_{2n} = \sigma(a_0, \dots, a_{2n-1})$ for all $n \ge 0$, and similarly for $\tau$ and the odd positions.
3. A strategy for a player is *winning* if that player wins every play that follows it.

The game $G(A, X)$, and the set $X$, is *determined* if player I or player II has a winning strategy.

## Remarks

Example: let $A = \N$ and $X = \set{x \in \N^{\N} : x_0 = 0}$.
The strategy of player I that plays $0$ at the first move, and anything later, is winning, so $X$ is determined.

Non-example: under the axiom of choice there is a set $X \subseteq \set{0, 1}^{\N}$ for which neither player has a winning strategy.
It is built by a diagonal argument against a well-ordering of all strategies.

At most one player can have a winning strategy: two winning strategies $\sigma$ and $\tau$ would produce a single play that both players win.
The case $A = \N$ is the main one; then $A^{\N}$ is the [[Baire space|P2WS16HKD7]] with its product topology.
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<document source="https://tamnd.github.io/math2/node/4HH0RPA7E6.md" tag="4HH0RPA7E6" kind="definition">
# Linear recurrence with polynomial coefficients

## Definition

Let $K$ be a field and let $r \ge 0$ be an integer.
Let $p_0, \dots, p_r \in K[n]$ be polynomials in the [[polynomial ring|0SDC934R52]] over $K$, with $p_r \ne 0$.
The *linear recurrence with polynomial coefficients* $p_0, \dots, p_r$ is the [[recurrence relation|07TZV87Y0T]]
$$p_r(n)\, u_{n+r} + p_{r-1}(n)\, u_{n+r-1} + \cdots + p_0(n)\, u_n = 0 \qquad (n \in \N)$$
for a sequence $u \colon \N \to K$.
Its order is $r$, when $p_0 \ne 0$.

## Remarks

Example: the [[Catalan numbers|0YWZJEABB2]] $C_n$ satisfy $(n + 2)\, C_{n+1} - (4n + 2)\, C_n = 0$.
Check for $n = 0, 1$: $2 C_1 = 2 = 2 C_0$ and $3 C_2 = 6 = 6 C_1$, with $C_0 = C_1 = 1$ and $C_2 = 2$.
Non-example: $u_{n+1} - 2^n u_n = 0$ fails the definition, since the coefficient $2^n$ is not a polynomial in $n$.
When all $p_i$ are constant, the recurrence has constant coefficients.
If $p_r$ has a root $n_0 \in \N$, the equation for $n = n_0$ does not determine $u_{n_0 + r}$, so the initial values may not fix the solution.
Other names are P-recursive or holonomic recurrence; an inhomogeneous version allows a polynomial right side.
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<document source="https://tamnd.github.io/math2/node/4HH8C9V8JQ.md" tag="4HH8C9V8JQ" kind="statement">
# Fredholm alternative

## Statement

Let $F$ be a [[field|04C5ZBG1WT]], let $n \ge 1$ and let $A$ be an $n \times n$ matrix over $F$, with [[transpose|09DXWYRD43]] $A^{T}$.
Then exactly one of the following two cases holds.

1. For every $b \in F^n$, the [[system|0GT4WWGWH2]] $Ax = b$ has exactly one solution $x \in F^n$.
2. The [[homogeneous system|04KGXJG1A2]] $Ax = 0$ has a solution $x \ne 0$.

In case 2, for $b \in F^n$, the system $Ax = b$ has a solution if and only if $y^{T} b = 0$ for every $y \in F^n$ with $A^{T} y = 0$.

## Remarks

The first part follows from the [[rank–nullity theorem|0W4FSW1TVB]]: the map $x \mapsto Ax$ on $F^n$ is injective if and only if it is surjective.
For the solvability condition, the [[column space|03AJKXH1AV]] of $A$ and the solution space of $A^{T} y = 0$ have dimensions $r$ and $n - r$, where $r$ is the [[rank|0ZC9YTXZRY]] of $A$.
Every column $a$ of $A$ satisfies $y^{T} a = 0$ for such $y$.
So the column space lies in the space of all $b$ with $y^{T} b = 0$ for all such $y$.
Both spaces have dimension $r$, so they are equal.
Example: for $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, case 2 holds with $x = (1, -1)$.
The solutions of $A^{T} y = 0$ are the multiples of $y = (1, -1)$, so $Ax = b$ is solvable exactly when $b_1 = b_2$.
The name comes from Fredholm's theory of integral equations, where the same alternative holds for compact perturbations of the identity on Banach spaces.

## Explanations

### L1 human

A square linear system behaves in one of two ways.
Either it is perfect: every right-hand side has one and only one solution.
Or the homogeneous system has a nonzero solution, and then some right-hand sides have no solution at all.
In the second case, a simple test tells which right-hand sides work: $y^{T} b$ must be $0$ for every solution $y$ of the transposed homogeneous system $A^{T} y = 0$.

### L2 human

The theorem says that for square systems, uniqueness and existence go together.
To show that $Ax = b$ is always solvable, it is enough to show that $Ax = 0$ has only the zero solution.
The solvability condition in case 2 is the statement that the column space of $A$ is the annihilator of the null space of $A^{T}$.
It is the finite-dimensional model for the Fredholm alternative for integral equations and for elliptic boundary value problems.
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<document source="https://tamnd.github.io/math2/node/4HHDPZ4BDY.md" tag="4HHDPZ4BDY" kind="concept">
# Covering relation

## Summary

In a [[partially ordered set|011C29Q1S1]], an element $y$ covers an element $x$ when $x < y$ and nothing lies strictly between them.
The covering relation records the immediate steps of the order.
For a finite poset it determines the whole order, and it is exactly what a [[Hasse diagram|00EYM7MDKS]] draws.

## Explanations

### L1 human

The element $y$ covers $x$ when $y$ is the next step up from $x$, with no element in between.
In the integers, $n + 1$ covers $n$.
In the rational numbers nothing covers anything, because between two rationals there is always a third.
For the subsets of $\set{1, 2, 3}$ ordered by inclusion, $\set{1, 2}$ covers $\set{1}$, but $\set{1, 2, 3}$ does not cover $\set{1}$.
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<document source="https://tamnd.github.io/math2/node/4HMBNA4RMN.md" tag="4HMBNA4RMN" kind="definition">
# Function having a given Mellin transform

## Definition

Let $f \colon (0, \infty) \to \C$ be measurable, let $s \in \C$ and let $m \in \C$.
The function $f$ *has Mellin transform $m$ at $s$* if both conditions hold:

1. $f$ is [[Mellin convergent|SGZZ4Y0D36]] at $s$, that is, $x \mapsto x^{s-1} f(x)$ is integrable on $(0, \infty)$.
2. The [[Mellin transform|N5PVGRJE1T]] of $f$ at $s$ equals $m$: $\int_0^\infty x^{s-1} f(x) \, dx = m$.

## Remarks

Example: $f = \mathbf{1}_{(0, 1]}$, the indicator function of $(0, 1]$, has Mellin transform $1/s$ at every $s$ with $\operatorname{Re} s > 0$.
Indeed $\int_0^1 x^{\operatorname{Re} s - 1} \, dx = 1/\operatorname{Re} s < \infty$, and $\int_0^1 x^{s-1} \, dx = 1/s$.
Non-example: the same $f$ at $s = 0$ fails clause 1, because $\int_0^1 x^{-1} \, dx$ diverges.
So $f$ does not have Mellin transform $m$ at $0$ for any $m$, although the function $1/s$ has a pole there and no value.
In Mathlib this predicate is `HasMellin f s m`.
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<document source="https://tamnd.github.io/math2/node/4HMR4KQ01T.md" tag="4HMR4KQ01T" kind="statement">
# Max-flow min-cut theorem

## Statement

Let $D = (V, E)$ be a finite [[directed graph|0R5HTV7REV]] with distinct vertices $s$ (the source) and $t$ (the sink), and let $c \colon E \to [0, \infty)$ be a capacity function.
A flow is a function $f \colon E \to \R$ such that:

1. $0 \le f(e) \le c(e)$ for every edge $e$.
2. For every vertex $w \ne s, t$, the total of $f$ on edges entering $w$ equals the total of $f$ on edges leaving $w$.

The value of $f$ is the total of $f$ on edges leaving $s$ minus the total of $f$ on edges entering $s$.
An $s$–$t$ cut is a partition $V = S \cup T$ with $s \in S$ and $t \in T$.
Its capacity is the sum of $c(e)$ over the edges $e$ from a vertex of $S$ to a vertex of $T$.
Then the maximum value of a flow exists and equals the minimum capacity of an $s$–$t$ cut.

## Remarks

Let $f$ be a flow and $(S, T)$ a cut.
The value of $f$ equals the flow from $S$ to $T$ minus the flow back, so it is at most the capacity of the cut.
For the reverse inequality, take a flow of maximum value and let $S$ be the set of vertices reachable from $s$ by an augmenting path.
Then $t \notin S$, every edge out of $S$ is full and every edge into $S$ carries no flow, so the cut $(S, V \setminus S)$ has capacity equal to the value.
The Ford–Fulkerson method pushes flow along augmenting paths, one at a time.
When all capacities are integers, it stops after finitely many steps with a maximum flow.
If all capacities are integers, there is a maximum flow with integer values on all edges.
This integrality gives [[Menger's theorem|E6GYGZDBS7]] and [[Kőnig's theorem|0JP0ZF57PN]] as special cases.
The theorem is also an instance of linear programming duality.

## Explanations

### L1 human

Think of pipes from a source to a sink, each with a maximum rate.
The most water you can send through equals the smallest total rate of a set of pipes whose removal cuts the sink off from the source.
The bottleneck of the network decides the throughput.

### L2 human

The theorem is the central result about flow networks.
Its proof gives an algorithm and a certificate: a maximum flow and a minimum cut prove each other optimal.
Many combinatorial min-max theorems, such as Menger's theorem, Kőnig's theorem and [[Hall's marriage theorem|08WCB90WND]], follow by building a suitable network with capacities $0$ and $1$.
</document>

<document source="https://tamnd.github.io/math2/node/4HP429BZQ5.md" tag="4HP429BZQ5" kind="definition">
# WZ theory

## Definition

Let $F(n, k)$ and $G(n, k)$ be functions defined for integers $n \ge 0$ and $k$.
The pair $(F, G)$ is a *WZ pair* if, for all $n$ and $k$,
$$F(n + 1, k) - F(n, k) = G(n, k + 1) - G(n, k).$$
The *WZ method* proves an identity $\sum_k f(n, k) = r(n)$, with $r(n) \ne 0$, in three steps:

1. set $F(n, k) = f(n, k) / r(n)$, so the identity reads $\sum_k F(n, k) = 1$;
2. give a rational function $R(n, k)$, the *WZ certificate*, such that $(F, G)$ with $G = R F$ is a WZ pair;
3. check that $\sum_k F(0, k) = 1$ and that $G(n, k) \to 0$ as $k \to \pm\infty$ for each $n$.

## Remarks

Why it works: summing the WZ equation over all $k$ makes the right side telescope to $0$.
So $\sum_k F(n + 1, k) = \sum_k F(n, k)$ for all $n$, and the sum stays equal to its value $1$ at $n = 0$.
When $F$ is a hypergeometric term, the WZ equation divided by $F(n, k)$ becomes an identity of rational functions, which is checked by algebra.

Example: the identity $\sum_k \binom{n}{k} = 2^n$ gives $F(n, k) = \binom{n}{k} / 2^n$.
The certificate is $R(n, k) = -\frac{k}{2(n - k + 1)}$, so $G(n, k) = -\binom{n}{k-1} / 2^{n+1}$.
Both sides of the WZ equation equal $\big(\binom{n}{k-1} - \binom{n}{k}\big) / 2^{n+1}$, by Pascal's rule.
Also $G(n, k) = 0$ outside $1 \le k \le n + 1$, and $\sum_k F(0, k) = 1$.

Non-example: for $F(n, k) = \binom{n}{k} / 2^n$, the choice $G = 0$ does not give a WZ pair.
The equation would require $F(n + 1, k) = F(n, k)$, which fails for $n = 0$ and $k = 1$, since $1/2 \ne 0$.

The [[Wilf–Zeilberger pair|WC0QNABPZ4]] node treats the pairs themselves in more detail.
</document>

<document source="https://tamnd.github.io/math2/node/4HQBRDQ3A9.md" tag="4HQBRDQ3A9" kind="concept">
# Bifurcation from a characteristic value

## Summary

Consider a nonlinear equation $\varphi = \lambda\bigl(K\varphi + R(\varphi)\bigr)$ with a compact linear operator $K$ and a term $R$ that is small compared with $\varphi$ near $0$.
Then $\varphi = 0$ is a solution for every $\lambda$.
A value $\lambda_0$ is a bifurcation point if nonzero solutions branch off the trivial solution at $\lambda = \lambda_0$.
Such a point must be a [[characteristic value|DC7HMHVREP]] of $K$, that is, a value where the linearized equation $\varphi = \lambda K\varphi$ has a nonzero solution.
This is the classical setting of [[bifurcation theory|D9TW46P19Y]] for nonlinear integral equations, with results such as Krasnosel'skii's theorem on characteristic values of odd multiplicity.

## Explanations

### L1 human

A straight elastic rod pushed at its ends stays straight under a small load.
At a critical load it can suddenly bow sideways, and new bent shapes appear next to the straight one.
The critical loads are exactly the values where the linear approximation of the problem has a nonzero solution.
In the scalar model $\varphi = \lambda(\varphi - \varphi^3)$, nonzero solutions $\varphi = \pm\sqrt{1 - 1/\lambda}$ appear when $\lambda$ passes $1$, the characteristic value of the linear part.
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<document source="https://tamnd.github.io/math2/node/4HS3C12472.md" tag="4HS3C12472" kind="concept">
# Feller process

## Summary

A Feller process is a Markov process whose transition semigroup acts nicely on continuous functions that vanish at infinity.
The semigroup maps such functions to such functions and is continuous in time at $0$.
These two conditions give a version with right-continuous paths that have left limits, the strong Markov property, and an [[infinitesimal generator|VPA41PWRR1]].
Brownian motion, [[Lévy processes|8NT5CFG2ZR]] and many [[diffusions|9HDQ094ADQ]] are Feller processes.

## Explanations

### L1 human

A Markov process forgets its past: only the present position matters for the future.
For a Feller process, the future also depends continuously on the starting point.
If two particles start very close, their laws at a later time are close.
Also, after a very short time the process has barely moved in law.
Brownian motion on the line is the main example.
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<document source="https://tamnd.github.io/math2/node/4HS60JABC5.md" tag="4HS60JABC5" kind="concept">
# Fukaya category

## Summary

The Fukaya [[category|A2DE66937D]] of a symplectic manifold is an algebraic structure whose objects are [[Lagrangian submanifolds|H8ACPTMX08]] and whose morphisms are Lagrangian [[Floer|NCKFPAGHXF]] cochain complexes.
Its composition laws count [[pseudoholomorphic|N469E47V75]] polygons with boundary on the Lagrangians.
Composition is associative only up to homotopy, so it is an $A_\infty$-category rather than an ordinary category.
It was introduced by Kenji Fukaya and is one side of [[homological mirror symmetry|B5N9WJH5F0]].

## Explanations

### L1 human

Lagrangian submanifolds are the most important subspaces of a symplectic manifold, for example curves on a surface.
Two of them, if in general position, meet in finitely many points, and these points span the space of arrows from one to the other.
To compose arrows, one counts holomorphic triangles whose three sides lie on three Lagrangians and whose corners are the given intersection points.
On a torus, for example, closed curves of different slopes play the role of the Lagrangians, and the triangles are flat triangles in the plane that covers the torus.
</document>

<document source="https://tamnd.github.io/math2/node/4HV4WTAX9P.md" tag="4HV4WTAX9P" kind="concept">
# Intersection theory

## Summary

Intersection theory studies how subvarieties of an algebraic variety meet, and assigns numbers and classes to their intersections.
Its central object is the Chow ring of a smooth variety, made of the [[Chow groups|EBEF6PEK4N]].
Its product sends two cycle classes to the class of their intersection, counted with [[intersection multiplicity|GPQF9GK4T9]].
It gives a rigorous form to classical enumerative geometry, including [[Bézout's theorem|KWNVF194D8]] and [[Schubert calculus|FWGHMADJR4]].
It is also the home of [[Chern classes|PAD37GKDPD]] and of the [[Hirzebruch–Riemann–Roch theorem|G3X02FKQCR]].

## Explanations

### L1 human

Two curves in the plane usually meet in finitely many points, and one wants to know how many.
The answer is stable only if one counts with multiplicity, works over the complex numbers, and includes points at infinity.
Intersection theory does this in any dimension, and it allows one to move cycles to general position without changing the answer.
On the projective plane, a curve of degree $d$ and a curve of degree $e$ have intersection number $de$.
The reason is that each is equivalent to a multiple of a line, and two lines meet in one point.
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<document source="https://tamnd.github.io/math2/node/4HZX2EN319.md" tag="4HZX2EN319" kind="statement">
# Legendre condition

## Statement

Let $a < b$, and let $L \colon [a, b] \times \R^n \times \R^n \to \R$, written $L(x, y, p)$, be of class $C^2$.
Let $J(y) = \int_a^b L(x, y(x), y'(x)) \, dx$ be the [[integral functional|JRQZ803Q5Y]] on $C^1([a, b]; \R^n)$.
Let $y_0 \in C^1([a, b]; \R^n)$ be a [[weak local minimiser|WMPCR723MJ]] of $J$ among functions with the same values at $a$ and $b$.
Then for every $x \in [a, b]$ the $n \times n$ matrix $L_{pp}(x, y_0(x), y_0'(x))$ of second derivatives in $p$ is [[positive semidefinite|01MMVW8CKC]].
For $n = 1$ this reads $L_{pp}(x, y_0(x), y_0'(x)) \ge 0$.

## Remarks

The [[second variation|4C3BANJ8AH]] at a minimiser is nonnegative: $\int_a^b \big( h'^T L_{pp} h' + 2 h^T L_{yp} h' + h^T L_{yy} h \big) \, dx \ge 0$ for all $h \in C^1$ with $h(a) = h(b) = 0$.
If $L_{pp}$ had a negative direction $v$ at some interior point $x_0$, take $h$ to be a small zigzag in direction $v$ near $x_0$, with slopes $\pm 1$ and tiny height.
Then the term with $h'^2$ dominates the others, because $h$ itself is small, and the second variation becomes negative.
The endpoints follow by continuity.

Example: for $J(y) = \int_0^1 (1 - y'^2) \, dx$ one has $L_{pp} = -2 < 0$ everywhere, so $J$ has no weak local minimiser with fixed end values.

For a maximiser the matrix is negative semidefinite.
The condition is only necessary: the extremal $y_0 = 0$ of $\int_0^{2\pi} (y'^2 - y^2) \, dx$ satisfies $L_{pp} = 2 > 0$ but is not a minimiser.
The *strengthened Legendre condition* asks that $L_{pp}$ be positive definite; together with the Jacobi condition it gives a sufficient condition for a weak minimum.

## Explanations

### L1 human

At a minimum of an ordinary function, the second derivative cannot be negative.
The Legendre condition is the analogue for curves that minimise an integral.
It says that the integrand must curve upward as a function of the slope, along the minimising curve.

### L2 human

The Legendre condition is the second necessary condition of the classical theory, after the Euler–Lagrange equation.
It only involves the highest-order part of the second variation, the coefficient of $h'^2$.
The full sign of the second variation is controlled by the Jacobi equation and conjugate points.
In several variables its analogue is the Legendre–Hadamard condition, and in optimal control it becomes the Legendre–Clebsch condition on the Hamiltonian.
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# Random group

## Summary

A random group is a group given by a [[presentation|0JGEQXQ4AC]] whose relators are chosen at random.
The random model has a size parameter that tends to infinity.
One then asks which properties hold with probability tending to $1$.
Gromov's density model has a sharp threshold at density $\frac{1}{2}$.
Below it, random groups are infinite and [[hyperbolic|JWJ8YPR7EB]]; above it, they are trivial or of order $2$.
Random groups show what a typical finitely presented group looks like, and they provide examples of groups with unusual combinations of properties.

## Explanations

### L1 human

Choose a few letters as generators and write down many random words in them as relations.
The group you get is a random group.
With few relations compared to the number of possible words, the group is almost surely infinite and looks like a tree at large scales.
With too many relations, they collapse everything, and the group is almost surely trivial or has two elements.
</document>

<document source="https://tamnd.github.io/math2/node/4J4EW7FXWW.md" tag="4J4EW7FXWW" kind="statement">
# Mazur–Ulam theorem

## Statement

Let $X$ and $Y$ be [[normed vector spaces|03J5RAKB30]] over $\R$.
Let $f \colon X \to Y$ be a surjective [[isometry|02HPP5VCZH]], that is, $f$ is onto and $\norm{f(x) - f(y)} = \norm{x - y}$ for all $x, y \in X$.
Then $f$ is [[affine|0JF0TVYGWT]]: the map $x \mapsto f(x) - f(0)$ is linear.

## Remarks

The heart of the proof is that $f$ preserves [[midpoints|0S706JBZBG]]: $f\left(\frac{x + y}{2}\right) = \frac{f(x) + f(y)}{2}$.
One proof describes the midpoint by distances only: it is fixed by every surjective isometry of $X$ that fixes $x$ and $y$.
The argument uses the point reflection $z \mapsto x + y - z$, which swaps $x$ and $y$.
Midpoint preservation and continuity then give real linearity of $f - f(0)$.

Surjectivity is needed.
The map $f \colon \R \to \R^2$, $f(t) = (t, \sin t)$, with the norm $\norm{(a, b)} = \max(\abs{a}, \abs{b})$ on $\R^2$, is an isometry because $\abs{\sin t - \sin s} \le \abs{t - s}$, but it is not affine.
The theorem is about real structure only: complex conjugation on $\C$ is a surjective isometry that is real-linear but not complex-linear.

## Explanations

### L1 human

If a map between normed spaces keeps all distances and covers the whole target, then it does not bend anything.
It is a linear map followed by a shift.
So the metric of a normed space already knows its linear structure.

### L2 human

The theorem shows that two real normed spaces are isometric as metric spaces exactly when they are isometrically isomorphic as normed spaces.
So isometric classification of Banach spaces can ignore nonlinear isometries.
For a Euclidean space it recovers the fact that every distance-preserving map of $\R^n$ onto itself is an orthogonal map followed by a translation.
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# Hilbert symbol

## Definition

Let $K$ be a [[local field|J85FZEQG38]] whose characteristic is not $2$.
For $a, b \in K^\times$, the *Hilbert symbol* $(a, b)_K$ is
$$(a, b)_K = \begin{cases} 1 & \text{if } z^2 = a x^2 + b y^2 \text{ has a solution } (x, y, z) \in K^3 \setminus \set{(0, 0, 0)}, \\ -1 & \text{otherwise.} \end{cases}$$

## Remarks

The value only depends on the classes of $a$ and $b$ modulo squares.
Equivalently, $(a, b)_K = 1$ exactly when $a$ is a norm from the field $K(\sqrt{b})$.
The symbol is symmetric and bimultiplicative, and $(a, 1 - a)_K = 1$ for $a \neq 1$.
Example: $(-1, -1)_\R = -1$, because $z^2 = -x^2 - y^2$ forces $x = y = z = 0$ over $\R$.
Example: $(2, 7)_\R = 1$, because $(x, y, z) = (1, 1, 3)$ solves $z^2 = 2x^2 + 7y^2$.
Non-example: the solution $(0, 0, 0)$ must be excluded; it solves every such equation, so allowing it would make every symbol equal to $1$.
For $K = \Q_p$ with $p$ odd, write $a = p^\alpha u$ and $b = p^\beta v$ with units $u, v \in \Z_p^\times$.
Then $(a, b)_p = (-1)^{\alpha \beta (p - 1)/2} \left(\frac{u}{p}\right)^\beta \left(\frac{v}{p}\right)^\alpha$, with [[Legendre symbols|0JTC045NNZ]] of the residues of $u$ and $v$.
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# Kahan summation algorithm

## Statement

Let $x_1, \ldots, x_n$ be numbers in a binary [[floating-point|RTKRSC3KPH]] system with rounding to nearest and unit roundoff $u$ (see [[machine epsilon|JA3QXVF2B8]]).
Assume that no overflow or underflow occurs.
The *Kahan summation algorithm* computes $\hat s$ as follows, in floating-point operations:
start with $s = 0$ and $c = 0$; for $i = 1, \ldots, n$ set $y = x_i - c$, $t = s + y$, $c = (t - s) - y$, $s = t$; finally $\hat s = s$.
Then
$$\hat s = \sum_{i=1}^{n} (1 + \mu_i) x_i \quad\text{with}\quad \abs{\mu_i} \le 2u + O(n u^2),$$
and hence
$$\Bigl| \hat s - \sum_{i=1}^{n} x_i \Bigr| \le \bigl( 2u + O(n u^2) \bigr) \sum_{i=1}^{n} \abs{x_i}.$$

## Remarks

The variable $c$ holds the low-order part lost when $y$ is added to the large running sum $s$.
The difference $t - s$ recovers what was actually added, and subtracting $y$ gives minus the [[round-off error|CPN6SD3BTV]] of the addition.
This correction is fed back into the next term.
The proof tracks the rounding errors of the four operations through the loop; the key point is that, in binary arithmetic, $c$ captures the rounding error of $s + y$ almost exactly.
For comparison, plain recursive summation only gives the bound $\frac{(n-1)u}{1 - (n-1)u} \sum \abs{x_i}$, which grows linearly in $n$.
The result is a [[backward error|CMXHRNWY35]] bound: $\hat s$ is the exact sum of the slightly perturbed terms $(1 + \mu_i) x_i$.
The relative error $\abs{\hat s - s}/\abs{s}$ can still be large when $\sum \abs{x_i} \gg \abs{\sum x_i}$, that is, when the sum is ill-conditioned.
Optimizing compilers that treat floating-point addition as associative can simplify $c$ to $0$ and destroy the method.

## Explanations

### L1 human

Adding many numbers on a computer loses a little accuracy at each addition, and these losses can add up.
Kahan's trick keeps a second variable that remembers the small part lost at each step and adds it back in the next step.
With this, the error no longer grows with the number of terms: it stays about twice the error of a single rounding, as long as $n u$ is small.

### L2 human

The bound should be compared with the bound $\gamma_{n-1} \sum \abs{x_i}$, $\gamma_k = ku/(1 - ku)$, for recursive summation and $\gamma_{\lceil \log_2 n \rceil} \sum \abs{x_i}$ for [[pairwise summation|SVSVMWA9NQ]].
Compensated summation costs four floating-point operations per term instead of one, but removes the dependence on $n$ from the leading term.
It is the simplest example of an error-free transformation: the pair $(t, -c)$ represents $s + y$ exactly or nearly exactly.
The same idea underlies compensated algorithms for dot products and polynomial evaluation.
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# Block design

## Summary

A block design is a finite set of points together with a family of subsets of the same size, the blocks.
It is balanced: every set of $t$ points lies in the same number of blocks.
The most studied case is $t = 2$, the balanced incomplete block design.
Block designs are a central kind of [[combinatorial design|PAJTMQ604P]].
They appear in the planning of experiments, in finite geometry and in coding theory.
The node [[block design|TGZATGBHTH]] gives the precise definition.

## Explanations

### L1 human

Think of a tournament organiser who must split players into groups of the same size so that every two players meet equally often.
The players are the points and the groups are the blocks.
A small example uses $7$ points and $7$ blocks of size $3$, in which every two points share exactly one block.
This example is the [[Fano plane|QQ8GW1ASZM]].
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# Determinantal variety

## Definition

Let $k$ be a field, and let $m, n \ge 1$ and $0 \le r < \min(m, n)$ be integers.
Identify the space of $m \times n$ matrices over $k$ with the affine space $\mathbb{A}^{mn}$, with coordinates $x_{ij}$.
The *determinantal variety* $M_r$ is the set of $m \times n$ matrices of [[rank|0ZC9YTXZRY]] at most $r$.
It is the [[algebraic set|VHFMQQYSTN]] defined by the [[determinantal ideal|SXD7X0WQH1]] $I_{r+1}$, generated by all $(r + 1) \times (r + 1)$ [[minors|0BT9BYWVRY]] of the matrix $(x_{ij})$.

## Remarks

A matrix has rank at most $r$ if and only if all its $(r + 1) \times (r + 1)$ minors vanish, so the two descriptions agree.
For $k$ algebraically closed, $M_r$ is irreducible of dimension $r(m + n - r)$, and the ideal $I_{r+1}$ is prime.
Since $M_r$ is defined by homogeneous equations, it also defines a projective variety in $\mathbb{P}^{mn - 1}$.

Example: $m = n = 2$ and $r = 1$.
$M_1$ is the quadric cone $x_{11} x_{22} - x_{12} x_{21} = 0$ in $\mathbb{A}^4$, of dimension $3 = 1 \cdot (2 + 2 - 1)$.

Non-example: the set of $2 \times 2$ matrices of rank exactly $1$ is not a determinantal variety.
It is not closed, since its closure contains the zero matrix, which has rank $0$.
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# Siggers term

## Definition

Let $\mathcal{V}$ be a [[variety of algebras|AJNZJD8442]].
A *Siggers term* for $\mathcal{V}$ is a $6$-ary [[term|N9M2N27END]] $s(x_1, \dots, x_6)$ such that $\mathcal{V}$ satisfies the [[identities|PKD440G2XW]]
$$s(x, x, x, x, x, x) \approx x \quad \text{and} \quad s(x, y, x, z, y, z) \approx s(y, x, z, x, z, y).$$
The first identity says that $s$ is [[idempotent|9FS1JFZSX4]].

## Remarks

Example: in the variety of semilattices, $s = x_1 \wedge \cdots \wedge x_6$ is a Siggers term.
Check: it is idempotent, and both sides of the second identity equal $x \wedge y \wedge z$, by commutativity, associativity and idempotence.
Non-example: in the variety of sets with no operations, the only terms are variables, and no variable $x_i$ is a Siggers term.
The two sides of the second identity differ in every place, so for $s = x_i$ it would read $x \approx y$, $x \approx z$ or $y \approx z$.
Every Siggers term is a [[Taylor term|QEAMRNRJJD]]: substituting $z = x$, $y = x$ or $x = y$ in the second identity gives, for each place, an identity in two variables that differs in that place.
A $4$-ary variant, an idempotent term with $s(a, r, e, a) \approx s(r, a, r, e)$, is also used; a finite algebra has a Taylor term exactly when it has a term of this kind.
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# Discrete valuation ring

## Definition

A *discrete valuation ring* is a [[principal ideal domain|0CMD0BCH8T]] $R$ that has exactly one nonzero [[prime ideal|00MHHZ9SNE]].

## Remarks

Equivalently, $R$ is an [[integral domain|0QVD7WBFGN]] with [[field of fractions|0F2PTGX1A2]] $K$ such that there is a surjective [[discrete valuation|A2KKZAV29W]] $v \colon K^{\times} \to \Z$ with $R = \set{x \in K^{\times} : v(x) \ge 0} \cup \set{0}$.
A generator $\pi$ of the maximal ideal is a uniformizer; every nonzero $x \in K$ is $x = u \pi^{v(x)}$ with $u$ a unit of $R$.
The nonzero ideals of $R$ are exactly $(\pi^n)$ for $n \ge 0$.

Example: for a prime $p$, the ring $\Z_{(p)} = \set{a/b : a, b \in \Z,\ p \nmid b}$ is a discrete valuation ring.
Every nonzero element is $p^n u$ with $u$ a unit and $n \ge 0$, so every nonzero ideal is $(p^n)$, which is principal.
The only nonzero prime among these is $(p)$.
The ring $k[\![t]\!]$ of formal power series over a field $k$ and the ring $\Z_p$ of $p$-adic integers are other examples.

Non-example: $\Z$ is a principal ideal domain, but it has infinitely many nonzero prime ideals $(2), (3), (5), \ldots$, so it is not a discrete valuation ring.
A field is not a discrete valuation ring either: it has no nonzero prime ideal.
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# Eilenberg–Steenrod axioms

## Definition

A pair of spaces $(X, A)$ is a topological space $X$ with a subspace $A$.
A map of pairs $f \colon (X, A) \to (Y, B)$ is a continuous map $f \colon X \to Y$ with $f(A) \subseteq B$.
Write $X$ for the pair $(X, \emptyset)$.
An *(ordinary) homology theory* consists of [[functors|28NQ0KRHD3]] $h_n$, for $n \in \Z$, from pairs of spaces to [[abelian groups|082GHNPJ53]], and [[natural transformations|KYCPCC57FT]] $\partial \colon h_n(X, A) \to h_{n-1}(A)$.
The *Eilenberg–Steenrod axioms* are the following conditions:

1. (Homotopy) If $f, g \colon (X, A) \to (Y, B)$ are [[homotopic|F2F143KQ2M]] through maps of pairs, then $f_* = g_*$.
2. (Exactness) For each pair, the sequence $\cdots \to h_n(A) \to h_n(X) \to h_n(X, A) \xrightarrow{\partial} h_{n-1}(A) \to \cdots$, with the unnamed maps induced by inclusions, is [[exact|JB7ZBFBA9C]].
3. (Excision) If $Z \subseteq A$ and the [[closure|0QEAJB91GZ]] of $Z$ lies in the [[interior|0RZH6W98HK]] of $A$, then the inclusion $(X \setminus Z, A \setminus Z) \to (X, A)$ induces isomorphisms $h_n(X \setminus Z, A \setminus Z) \to h_n(X, A)$.
4. (Dimension) For a one-point space $P$, $h_n(P) = 0$ for all $n \ne 0$.
5. (Additivity) If $X$ is the disjoint union of spaces $X_\alpha$, the inclusions induce an isomorphism $\bigoplus_\alpha h_n(X_\alpha) \to h_n(X)$.

The group $h_0(P)$ is the *coefficient group* of the theory.

## Remarks

The additivity axiom was added later by Milnor; it is automatic for finite disjoint unions.
A *cohomology theory* is defined dually by contravariant functors $h^n$ and natural maps $\delta \colon h^n(A) \to h^{n+1}(X, A)$, with products in place of direct sums in the additivity axiom.

Example: [[singular homology|M0Q5CFJ7ZH]] with coefficients in an abelian group $G$ satisfies all five axioms, and its coefficient group is $H_0(P; G) = G$.

Non-example: complex topological K-theory, written as a cohomology theory $K^n$, satisfies the first three axioms and additivity for finite unions, but $K^{-2}(P) \cong \Z$, so it fails the dimension axiom.
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# Legendre polynomials

## Definition

The *Legendre polynomials* are the real polynomials $P_n$, for $n \ge 0$, determined by the following conditions:

1. $P_n$ has [[degree|0P4JRMSWNH]] exactly $n$.
2. $\int_{-1}^{1} P_m(x)\, P_n(x)\, dx = 0$ for all $m \ne n$.
3. $P_n(1) = 1$.

## Remarks

Conditions 1 and 2 fix $P_n$ up to a constant factor, and condition 3 fixes the factor.
Example: $P_2(x) = \frac{3x^2 - 1}{2}$; it has degree $2$, $P_2(1) = 1$, and $\int_{-1}^1 P_2\, dx = \frac{1}{2}\left( 2 - 2 \right) = 0$ and $\int_{-1}^1 x P_2\, dx = 0$ since $x P_2$ is odd.
A non-example: $x^2$ has degree $2$ and value $1$ at $1$, but $\int_{-1}^1 x^2 \cdot 1\, dx = \frac{2}{3} \ne 0$, so it fails condition 2 against $P_0 = 1$.
They are given by $P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n$, and they satisfy $(n + 1) P_{n+1}(x) = (2n + 1)\, x P_n(x) - n P_{n-1}(x)$ and $\int_{-1}^1 P_n(x)^2\, dx = \frac{2}{2n + 1}$.
Their generating function is $\frac{1}{\sqrt{1 - 2xt + t^2}} = \sum_{n \ge 0} P_n(x)\, t^n$ for $\abs{t}$ small.
They are the [[classical orthogonal polynomials|47WJ34NC9K]] of Jacobi type with $\alpha = \beta = 0$, and $P_n$ solves $(1 - x^2) y'' - 2x y' + n(n+1) y = 0$.
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# Character sum

## Definition

Let $q \ge 1$ be an integer, and let $\chi$ be a [[Dirichlet character|7XB5VJKDPG]] modulo $q$.
Let $M$ be an integer and $N \ge 1$.
The *character sum* of $\chi$ over the interval $(M, M + N]$ is
$$S_\chi(M, N) = \sum_{n = M + 1}^{M + N} \chi(n).$$
More generally, a character sum is a sum $\sum_{n \in A} \chi(f(n))$ over a finite set $A$ of integers, where $f$ is a polynomial with integer coefficients.

## Remarks

Example: let $\chi$ be the Legendre symbol modulo $5$, so $\chi(1) = \chi(4) = 1$, $\chi(2) = \chi(3) = -1$ and $\chi(5) = 0$.
Then $S_\chi(0, 3) = 1 - 1 - 1 = -1$ and $S_\chi(0, 5) = 0$.
Non-example: the sum $\sum_{n \le N} \mu(n)$ of the [[Möbius function|0FANE7P0DY]] is not a character sum, since $\mu$ is not a Dirichlet character: $\mu(4) = 0$ but $\mu(2)\mu(2) = 1$, so $\mu$ is not completely multiplicative.
Orthogonality gives the complete sum: $S_\chi(M, q) = 0$ if $\chi$ is not principal, and $S_\chi(M, q) = \varphi(q)$ if $\chi$ is principal, where $\varphi$ is [[Euler's totient function|0DAMHCT17R]].
Thus $\abs{S_\chi(M, N)} \le \varphi(q)$ for every nonprincipal $\chi$.
The Pólya–Vinogradov inequality improves this to $\abs{S_\chi(M, N)} \le C \sqrt{q} \log q$ for nonprincipal $\chi$, with an absolute constant $C$.
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# Markov random field

## Definition

Let $G = (V, E)$ be a finite undirected [[graph|00C0BR7D7H]] and let $X = (X_v)_{v \in V}$ be [[random variables|0PNRKMRC55]].
For $A \subseteq V$ write $X_A = (X_v)_{v \in A}$.
A set $S \subseteq V$ *separates* $A$ from $B$ if every path from a vertex of $A$ to a vertex of $B$ passes through $S$.
Then $X$ is a *Markov random field* with respect to $G$ if it has the global Markov property.
This means: whenever $A, B, S \subseteq V$ are disjoint and $S$ separates $A$ from $B$, $X_A$ and $X_B$ are [[conditionally independent|0THJ7416DX]] given $X_S$.

## Remarks

Two weaker properties are also used.
In the local Markov property, each $X_v$ is conditionally independent of all non-neighbours given its neighbours $X_{N(v)}$.
In the pairwise Markov property, $X_u \perp X_v \mid X_{V \setminus \set{u, v}}$ for all non-adjacent $u \ne v$.
Global implies local implies pairwise, and all three are equivalent when the joint density is strictly positive.
Example: on the path $1 - 2 - 3$, a Markov chain $X_1, X_2, X_3$ is a Markov random field, since $\set{2}$ separates $1$ from $3$ and $X_1 \perp X_3 \mid X_2$.
Non-example: let $X_1, X_2$ be independent fair coins and $X_3 = X_1$.
Then $X_1$ and $X_3$ are dependent given $X_2$, so $X$ is not a Markov random field on this path.
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# Kolmogorov quotient

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
Two points $x, y \in X$ are topologically indistinguishable when every [[open set|02BH3MCBQB]] of $X$ that contains one of them also contains the other.
This is an [[equivalence relation|0G8VEZ3WTK]] $\equiv$ on $X$.
The *Kolmogorov quotient* of $X$ is the quotient set $\mathrm{KQ}(X) = X / {\equiv}$ with the [[quotient topology|0NQ1H2X495]].

## Remarks

Equivalently, $x \equiv y$ if and only if $\overline{\set{x}} = \overline{\set{y}}$.
The space $\mathrm{KQ}(X)$ is a [[T0 space|07SG12TQH3]].
The quotient map $q \colon X \to \mathrm{KQ}(X)$ is open and closed, and $U \mapsto q(U)$ is a bijection from the open sets of $X$ to the open sets of $\mathrm{KQ}(X)$.
If $X$ is already $T_0$, then $q$ is a homeomorphism.

Example: let $(X, d)$ be a [[pseudometric space|0A3HJX3KTF]] with its topology.
Here $x \equiv y$ if and only if $d(x, y) = 0$, because a point with $d(x, y) > 0$ has a ball around $x$ that misses $y$.
So $\mathrm{KQ}(X)$ is the metric space obtained by identifying points at distance $0$.

Non-example: the identity map of a two-point set $\set{a, b}$ with the trivial topology is not a Kolmogorov quotient map.
Its target is not $T_0$, since $a$ and $b$ have the same neighbourhoods; the Kolmogorov quotient of this space is a single point.
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# Spherical Bessel functions

## Summary

The spherical Bessel functions $j_n$ and $y_n$ are the Bessel functions of half-integer order $n + \frac{1}{2}$, rescaled by $\sqrt{\pi / (2x)}$.
They solve the radial part of the Helmholtz equation in spherical coordinates.
They are elementary: each is a combination of $\sin x$, $\cos x$ and powers of $x$.
They describe waves scattered by a sphere and the radial motion of a free quantum particle.

## Explanations

### L1 human

When a wave spreads out from a point in space, its strength along each ray is described by a spherical Bessel function.
The simplest one is $j_0(x) = \frac{\sin x}{x}$, a sine wave whose size falls off like $\frac{1}{x}$.
The next one is $j_1(x) = \frac{\sin x}{x^2} - \frac{\cos x}{x}$.
Higher ones come from a simple recurrence, so they can all be written with sines, cosines and powers.
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# Turán's inequalities

## Statement

Let $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$, and let $n \ge 1$.
Then for every $x$ with $-1 < x < 1$,
$$P_n(x)^2 - P_{n-1}(x) P_{n+1}(x) > 0.$$

## Remarks

At $x = \pm 1$ the left side is $0$, since $P_k(1) = 1$ and $P_k(-1) = (-1)^k$; so the strict inequality needs $\abs{x} < 1$.
Check for $n = 1$: $x^2 - \frac{3x^2 - 1}{2} = \frac{1 - x^2}{2} > 0$ for $\abs{x} < 1$.
The known proofs combine the three-term recurrence of the $P_n$ with their differential equation, and show that the left side cannot vanish inside $(-1, 1)$.
The left side is called the Turán determinant, since it is $\det \begin{pmatrix} P_n & P_{n+1} \\ P_{n-1} & P_n \end{pmatrix}$.
Analogues hold for other families: for the [[Chebyshev polynomials|M6EAR1JHPP]], $T_n(x)^2 - T_{n-1}(x) T_{n+1}(x) = 1 - x^2$.
This follows from $\cos^2(n\theta) - \cos((n - 1)\theta) \cos((n + 1)\theta) = \sin^2 \theta$ with $x = \cos \theta$.
For the [[Hermite polynomials|MGACY98E9H]] one has $H_n(x)^2 - H_{n-1}(x) H_{n+1}(x) > 0$ for all real $x$.

## Explanations

### L1 human

Fix a point $x$ strictly between $-1$ and $1$, and look at the sequence of numbers $P_0(x), P_1(x), P_2(x), \dots$.
Turán's inequality says that each term squared is larger than the product of its two neighbours.
For a sequence of positive numbers this means the sequence is log-concave, a kind of smoothness without sudden jumps.

### L2 human

Turán found the inequality for Legendre polynomials, and Szegő published several proofs of it in 1948.
Similar Turán-type inequalities hold for ultraspherical, Laguerre and Hermite polynomials, and for Bessel functions in the order variable.
Such inequalities for the Taylor coefficients of entire functions are related to the Laguerre–Pólya class.
This link is the reason they appear in work on the Riemann hypothesis through the Jensen polynomials of the Riemann $\xi$ function.
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# Beltrami equation

## Definition

Let $U$ be an [[open set|02BH3MCBQB]] in $\C$.
Let $\mu \colon U \to \C$ be a measurable function with $\norm{\mu}_\infty \le k$ for some constant $k < 1$.
The *Beltrami equation* with *Beltrami coefficient* $\mu$ is
$$f_{\bar z} = \mu f_z,$$
where $f_z$ and $f_{\bar z}$ are the [[Wirtinger derivatives|WMXDVF36ZT]] of $f$.
A *solution* is a function $f \in W^{1,2}_{\mathrm{loc}}(U)$, that is, with locally square-integrable weak first partial derivatives, that satisfies the equation almost everywhere on $U$.

## Remarks

Example: for $\mu = 0$ the equation is $f_{\bar z} = 0$, which is the Cauchy–Riemann system, and its solutions are the [[holomorphic functions|0NAHXR1FRH]] on $U$.
Example: for a constant $c$ with $\abs{c} < 1$, the function $f(z) = z + c \bar z$ has $f_z = 1$ and $f_{\bar z} = c$, so it solves the equation with $\mu = c$.
Non-example: the coefficient $\mu = 1$ fails the condition $\norm{\mu}_\infty < 1$.
Then the equation reads $f_{\bar z} - f_z = i f_y = 0$, and every function of $x$ alone solves it, so the equation is degenerate and its solutions are never homeomorphisms.
A homeomorphic solution is a [[quasiconformal mapping|SB8KG46NHP]] whose complex dilatation is $\mu$.
</document>

<document source="https://tamnd.github.io/math2/node/4K6F157663.md" tag="4K6F157663" kind="definition">
# Integral extension

## Definition

Let $B$ be a commutative ring and let $A$ be a [[subring|0GNCRPYXYV]] of $B$.
The ring $B$ is *integral over $A$*, and $A \subseteq B$ is an *integral extension*, if every element of $B$ is [[integral|1EHN4KPSFS]] over $A$.
A [[ring homomorphism|0GGE0Z0JQ1]] $f \colon A \to B$ of commutative rings is *integral* if $B$ is integral over the subring $f(A)$.

## Remarks

Example: $\Z \subseteq \Z[i]$ is integral.
The element $a + bi$ with $a, b \in \Z$ is a root of the monic polynomial $x^2 - 2a x + a^2 + b^2$ with integer coefficients.

Non-example: $\Z \subseteq \Q$ is not integral, because $1/2$ is not integral over $\Z$.

An extension is integral and of finite type if and only if it is [[finite|1NBACAEC2V]].
Integrality is [[transitive|ZXZMQM818D]], and it is kept by quotients and localizations.
If $B$ is integral over $A$, then $B/J$ is integral over $A/(J \cap A)$ for every ideal $J$ of $B$.
Also $S^{-1}B$ is integral over $S^{-1}A$ for every multiplicative set $S$ of $A$.
</document>

<document source="https://tamnd.github.io/math2/node/4K7TJE5KM4.md" tag="4K7TJE5KM4" kind="definition">
# Fresnel integrals

## Definition

The *Fresnel integrals* are the functions $S, C \colon \R \to \R$ given by the [[definite integrals|04TMP0XGMJ]]
$$S(x) = \int_0^x \sin(t^2)\, dt, \qquad C(x) = \int_0^x \cos(t^2)\, dt,$$
where $\sin$ and $\cos$ are the [[sine and cosine|08WFZP6P89]].

## Remarks

Example: $S$ and $C$ are odd, since the integrands are even, and $S(0) = C(0) = 0$.
The curve $t \mapsto (C(t), S(t))$ has velocity $(\cos t^2, \sin t^2)$ of length $1$, so $t$ is its arc length, and its direction angle $t^2$ gives curvature $2t$.
This curve is the Euler spiral.

Non-example: $x \mapsto \int_0^x \sin(t)^2\, dt = \frac{x}{2} - \frac{\sin 2x}{4}$ is not a Fresnel integral; the square must be inside the sine.

As $x \to \infty$, $S(x) \to \sqrt{\pi/8}$ and $C(x) \to \sqrt{\pi/8}$, as one sees by rotating the contour of the [[Gaussian integral|0DFT1GP1JP]].
Many references use the normalisation $\int_0^x \sin\left(\frac{\pi t^2}{2}\right) dt$ and $\int_0^x \cos\left(\frac{\pi t^2}{2}\right) dt$ instead; then both limits are $\frac12$.
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<document source="https://tamnd.github.io/math2/node/4K9RYSNPG3.md" tag="4K9RYSNPG3" kind="definition">
# Open subgroup

## Definition

Let $G$ be a [[topological group|92700QAB5Y]].
An *open subgroup* of $G$ is a [[subgroup|0X1JXT47GA]] $H$ of $G$ that is an [[open set|02BH3MCBQB]] of $G$.

## Remarks

Example: in $\R \times \Z$ with the product topology, where $\Z$ is discrete, the subgroup $\R \times \set{0}$ is open, because $\set{0}$ is open in $\Z$.
Non-example: $\Z$ is a subgroup of $(\R, +)$ but it is not open, since no interval around $0$ lies in $\Z$.
A subgroup $H$ is open if and only if it is a [[neighbourhood|0ZCY0Z80B6]] of the identity $e$.
Indeed, if $U \subseteq H$ is a neighbourhood of $e$, then $hU \subseteq H$ is a neighbourhood of each $h \in H$.
</document>

<document source="https://tamnd.github.io/math2/node/4KAWK0FD4V.md" tag="4KAWK0FD4V" kind="definition">
# Isolated hypersurface singularity

## Definition

Let $n \ge 0$, let $U$ be an open neighbourhood of $0$ in $\C^{n+1}$ and let $f \colon U \to \C$ be a [[holomorphic function|507C34W0ZT]] with $f(0) = 0$.
The [[germ|VZXF7RNFKS]] $f$ has an *isolated singularity* at $0$ if:

1. $\frac{\partial f}{\partial z_j}(0) = 0$ for $j = 0, \dots, n$, so $0$ is a [[critical point|0YTCMFB4HC]] of $f$;
2. there is a neighbourhood $U' \subseteq U$ of $0$ in which $0$ is the only point where all the partial derivatives $\partial f / \partial z_j$ vanish.

The germ at $0$ of the hypersurface $V = f^{-1}(0)$ is then called an *isolated hypersurface singularity*.

## Remarks

Example: $f(z) = z_0^2 + z_1^2 + \cdots + z_n^2$.
Its gradient is $(2z_0, \dots, 2z_n)$, which vanishes only at $0$, so both clauses hold.

Non-example: $f(z_0, z_1, z_2) = z_0 z_1$ on $\C^3$.
Its gradient is $(z_1, z_0, 0)$, which vanishes on the whole $z_2$-axis, so clause 2 fails.

Let $J$ be the ideal of the ring $\C\{z_0, \dots, z_n\}$ of convergent power series generated by the partial derivatives of $f$.
By the Rückert Nullstellensatz, clause 2 holds if and only if $\C\{z_0, \dots, z_n\} / J$ is finite-dimensional over $\C$.
Its dimension is the Milnor number of $f$.
If $f$ has no repeated factor, the points near $0$ where $V$ is singular are the points of $V$ where the gradient of $f$ vanishes.
</document>

<document source="https://tamnd.github.io/math2/node/4KCBFQS9WR.md" tag="4KCBFQS9WR" kind="concept">
# Daniell integral

## Summary

The Daniell integral is an approach to integration that starts from a positive linear functional on a space of elementary functions.
It does not start from a measure.
The functional must satisfy a continuity condition along decreasing sequences.
It then extends to a large class of integrable functions, and the measure of a set appears afterwards as the integral of its indicator function.
This approach gives the Lebesgue integral and integrals on locally compact spaces in one framework.

## Explanations

### L1 human

The usual route defines lengths or areas first and integrals afterwards.
Daniell's route goes the other way: it starts with an integral that one already knows on easy functions, such as continuous functions or step functions.
The only demand is that the integral behaves well when a sequence of functions shrinks down to zero.
For example, the Riemann integral of continuous functions with compact support on the real line is such a starting point, and its extension is the Lebesgue integral.
</document>

<document source="https://tamnd.github.io/math2/node/4KCECXYW81.md" tag="4KCECXYW81" kind="statement">
# Hom-set characterization of adjunctions

## Statement

Let $F \colon \mathcal{C} \to \mathcal{D}$ and $G \colon \mathcal{D} \to \mathcal{C}$ be functors.

1. Let $\varphi$ be an [[adjunction|TZHF6ZM92S]] between $F$ and $G$, with [[unit and counit|MYGWT305AB]] $\eta$ and $\varepsilon$.
Then $\eta$ and $\varepsilon$ are [[natural transformations|KYCPCC57FT]] and satisfy $\varepsilon_{F c} \circ F(\eta_c) = \id_{F c}$ and $G(\varepsilon_d) \circ \eta_{G d} = \id_{G d}$.
2. Conversely, let $\eta \colon \mathrm{Id}_{\mathcal{C}} \Rightarrow G F$ and $\varepsilon \colon F G \Rightarrow \mathrm{Id}_{\mathcal{D}}$ be natural transformations that satisfy these two triangle identities.
Then $\varphi(g) = G(g) \circ \eta_c$, for $g \colon F c \to d$, is an adjunction.
Its inverse is $f \mapsto \varepsilon_d \circ F(f)$ for $f \colon c \to G d$.
3. These two constructions are inverse to each other.

## Remarks

For part 1, naturality of $\varphi$ gives $\varphi(g) = G(g) \circ \eta_c$ and $\varphi^{-1}(f) = \varepsilon_d \circ F(f)$.
Applying these to identities gives the triangle identities.
For part 2, the triangle identities are exactly what is needed for the two maps to be inverse to each other, and naturality is a direct check.

Example: for sets, $F = - \times Y$ and $G = (-)^Y$, the unit sends $x \in X$ to the map $y \mapsto (x, y)$.
The counit is evaluation $Z^Y \times Y \to Z$.
The triangle identities then say that currying and evaluating undo each other.

## Explanations

### L1 human

There are two ways to describe an adjunction.
One matches maps $F c \to d$ with maps $c \to G d$.
The other gives two families of maps, the unit and the counit, which satisfy two simple equations.
The theorem says that the two descriptions carry the same information.

### L2 human

The hom-set form is useful for recognizing adjunctions, since it is often a familiar bijection such as currying or the universal property of free objects.
The unit and counit form is better for computation, because it is a finite set of equations between natural transformations.
It also makes sense in any 2-category, so it is the form used to define adjunctions in general.
A third equivalent form says that each $\eta_c$ is an initial object of the comma category $c \downarrow G$.
</document>

<document source="https://tamnd.github.io/math2/node/4KFC9BE02Y.md" tag="4KFC9BE02Y" kind="example">
# 1 + 2 + 3 + 4 + ⋯

## Example

The series of the positive integers is
$$\sum_{n=1}^{\infty} n = 1 + 2 + 3 + 4 + \cdots.$$
Its $N$-th partial sum is $s_N = N(N+1)/2$, which tends to $+\infty$.
So it is a [[divergent series|0JWMJSTBSG]].

## Remarks

It is not summable by [[Cesàro summation|N8ZGG7EW6J]], since the averages of $s_N$ also tend to $+\infty$.
It is not [[Abel summable|EZXPY4A19C]] either: for $0 < x < 1$, $\sum_{n \ge 1} n x^n = x/(1-x)^2$, which tends to $+\infty$ as $x \to 1^-$.
[[Zeta function regularization|VJCJYXGTDY]] assigns it the value $-1/12$.
For $\operatorname{Re} s > 1$, $\sum_n n^{-s} = \zeta(s)$, where $\zeta$ is the [[Riemann zeta function|1XWBY4N0VT]], and the series of the example is formally the case $s = -1$; the analytic continuation of $\zeta$ has $\zeta(-1) = -1/12$.
[[Ramanujan summation|E3SRYCE5VN]] also gives $-1/12$.
The value $-1/12$ is not a sum in the usual sense: the partial sums are all positive and grow without bound.

## Explanations

### L1 human

Adding $1 + 2 + 3 + \cdots$ gives totals that grow without limit, so the series has no ordinary sum.
The famous value $-1/12$ comes from a different procedure: one extends the function $\sum n^{-s}$ beyond the region where the series converges and evaluates it at $s = -1$.
This value appears in physics and in number theory, but it should not be read as the result of ordinary addition.
</document>

<document source="https://tamnd.github.io/math2/node/4KN39NQZ02.md" tag="4KN39NQZ02" kind="definition">
# Spectrum of an element

## Definition

Let $\mathbb{K}$ be a field, let $A$ be an [[associative algebra|3CTW1P4HBP]] over $\mathbb{K}$ with unit $1$, and let $a \in A$.
The *spectrum* of $a$ is the set
$$\sigma(a) = \set{\lambda \in \mathbb{K} : a - \lambda 1 \text{ is not invertible in } A}.$$
Here invertible means that $a - \lambda 1$ is a [[unit|0XVEZDE9J8]] of $A$.
If $A$ has no unit, the spectrum of $a$ is its spectrum in the [[unitization|7SB03M850M]] $A^+$.

## Remarks

The spectrum depends on the algebra: if $a$ lies in a subalgebra $B$ of $A$ that contains $1$, then $\sigma_A(a) \subseteq \sigma_B(a)$, and the inclusion can be strict.
In a non-unital algebra, $0$ always lies in the spectrum, because no $(0, a)$ is invertible in $A^+$.

Example: for $A = M_n(\C)$, $a - \lambda 1$ is invertible exactly when $\det(a - \lambda 1) \ne 0$.
So $\sigma(a)$ is the set of eigenvalues of $a$, the [[spectrum of the matrix|0XPX5NC67E]].

Example: for $A = C([0,1])$ with complex values and $f \in A$, $\sigma(f) = f([0,1])$.
If $\lambda \notin f([0,1])$, then $1/(f - \lambda)$ is continuous and inverts $f - \lambda$; if $\lambda = f(t_0)$, then $f - \lambda$ vanishes at $t_0$ and has no inverse.

For bounded operators on $\ell^2$, the right shift $S$ has no eigenvalues, but $0 \in \sigma(S)$ because $S$ is not surjective.
So in infinite dimension the spectrum can contain points that are not eigenvalues.
</document>

<document source="https://tamnd.github.io/math2/node/4KPNX1SZYR.md" tag="4KPNX1SZYR" kind="statement">
# Barbalat's lemma

## Statement

Let $f \colon [0, \infty) \to \R$ be differentiable.
Assume that the limit $\lim_{t \to \infty} f(t)$ exists and is finite.
Assume that $f'$ is [[uniformly continuous|0M4JJT780X]] on $[0, \infty)$.
Then $\lim_{t \to \infty} f'(t) = 0$.

## Remarks

Suppose $f'(t) \not\to 0$.
Then there are $\varepsilon > 0$ and times $t_n \to \infty$ with $\abs{f'(t_n)} \ge \varepsilon$.
Uniform continuity gives $\delta > 0$, independent of $n$, with $\abs{f'(t)} \ge \varepsilon / 2$ and constant sign on $[t_n, t_n + \delta]$.
Then $\abs{f(t_n + \delta) - f(t_n)} \ge \varepsilon \delta / 2$ for all $n$, which contradicts the Cauchy criterion for the limit of $f$.

The hypothesis on $f'$ is necessary.
For $f(t) = \sin(t^2) / (t + 1)$, the limit of $f$ is $0$, but $f'(t) = 2t \cos(t^2) / (t + 1) - \sin(t^2) / (t + 1)^2$ does not tend to $0$.
Here $f'$ is not uniformly continuous.

An equivalent form: if $g \colon [0, \infty) \to \R$ is uniformly continuous and $\int_0^\infty g(t) \, dt$ converges, then $g(t) \to 0$.
In stability theory, it is applied to $f(t) = V(t, x(t))$ for a [[Lyapunov function|01REYCJ371]] $V$ along a solution $x$.
If $V$ is bounded below, $\dot V \le 0$ and $\dot V$ is uniformly continuous in $t$, then $\dot V(t, x(t)) \to 0$.

## Explanations

### L1 human

If a quantity settles down to a limit and its rate of change cannot jump suddenly, then the rate of change must die out.
Without the second condition, the quantity could settle while still making ever faster tiny wiggles.

### L2 human

For time-varying systems, LaSalle's invariance principle does not apply directly.
Barbalat's lemma replaces it: from a Lyapunov function with $\dot V \le 0$ one gets $\dot V \to 0$ along solutions, which often forces some components of the state to tend to $0$.
This is a standard step in adaptive control.
A common sufficient condition for uniform continuity of $\dot V$ is that $\ddot V$ is bounded.
</document>

<document source="https://tamnd.github.io/math2/node/4KTBRP27P6.md" tag="4KTBRP27P6" kind="statement">
# Siegel's theorem on integral points

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\mathcal{O}_K$.
Let $C$ be a smooth projective geometrically irreducible [[curve|XJM2MT4S3Y]] over $K$ of [[genus|6W8TDV9VVN]] $g \ge 1$.
Let $f$ be a nonconstant rational function on $C$ defined over $K$.
Then the set $\set{P \in C(K) : P \text{ is not a pole of } f \text{ and } f(P) \in \mathcal{O}_K}$ is finite.

## Remarks

Equivalently, an affine curve of genus at least $1$ over $K$, embedded in affine space, has only finitely many points with coordinates in $\mathcal{O}_K$.
For genus $0$ the same conclusion holds when $f$ has at least three distinct poles over $\overline{K}$.
The hypothesis $g \ge 1$ is necessary: the Pell equation $x^2 - 2y^2 = 1$ defines a curve of genus $0$ with infinitely many integral points, here $f = x$ has only two poles.
Example: for each integer $k \ne 0$, the Mordell equation $y^2 = x^3 + k$ defines an elliptic curve, so it has only finitely many solutions with $x, y \in \Z$.
The proof combines the weak Mordell–Weil theorem for the Jacobian with a Diophantine approximation theorem of Thue–Siegel type, now usually Roth's theorem; it is not effective in general.
Siegel proved the theorem in 1929.

## Explanations

### L1 human

A curve of genus at least one can have infinitely many rational points, but only finitely many of them have integer coordinates.
For instance, $y^2 = x^3 + k$ has only finitely many integer solutions for each fixed nonzero $k$.

### L2 human

The theorem applies to [[elliptic curves|427N6S6QQW]], where rational points may be infinite in number by [[Mordell's theorem|BZHGKN6Q3K]] but integral points are finite.
Baker's method in transcendence theory later gave effective bounds in many cases, such as Mordell's equation and [[Thue equations|3Q6CFPQZDF]].
For genus at least $2$, [[Faltings's theorem|H0DB07WM20]] gives the stronger statement that even rational points are finite.
</document>

<document source="https://tamnd.github.io/math2/node/4KX81Q85DT.md" tag="4KX81Q85DT" kind="definition">
# Seminorm

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $V$ be a [[vector space|05EGHEQZAW]] over $\mathbb{K}$.
A *seminorm* on $V$ is a map $p \colon V \to \R$ such that:

1. $p(x + y) \le p(x) + p(y)$ for all $x, y \in V$.
2. $p(\lambda x) = \abs{\lambda} \, p(x)$ for all $\lambda \in \mathbb{K}$ and $x \in V$, where $\abs{\lambda}$ is the [[absolute value|00S7C68D51]].

## Remarks

The two conditions imply $p(0) = 0$ and $p(x) \ge 0$ for all $x$.
Indeed $p(0) = p(0 \cdot 0) = 0 \cdot p(0) = 0$, and then $0 = p(x - x) \le p(x) + p(-x) = 2 p(x)$.
A seminorm is a [[norm|08NYZ2FS6B]] exactly when $p(x) = 0$ implies $x = 0$.
Example: on the space $C([0, 1])$ of continuous real functions, $p(f) = \sup_{0 \le t \le 1/2} \abs{f(t)}$ is a seminorm; both conditions follow from the same inequalities for the absolute value at each point.
It is not a norm, since it vanishes on a nonzero function supported in $(1/2, 1]$.
Non-example: on $\R$, $p(x) = x^2$ is not a seminorm, since $p(2 \cdot 1) = 4 \ne 2 = \abs{2} \, p(1)$.
The kernel $\set{x : p(x) = 0}$ is a linear subspace, and $p$ induces a norm on the quotient of $V$ by it.
</document>

<document source="https://tamnd.github.io/math2/node/4KXHGXZQY8.md" tag="4KXHGXZQY8" kind="counterexample">
# Weierstrass's counterexample to the Dirichlet principle

## Counterexample

Consider the [[integral functional|JRQZ803Q5Y]]
$$J(u) = \int_{-1}^{1} x^2 \, u'(x)^2 \, dx$$
on the set $\mathcal{A}$ of functions $u \in C^1([-1, 1])$ with $u(-1) = -1$ and $u(1) = 1$.
Then $\inf_{\mathcal{A}} J = 0$, but no $u \in \mathcal{A}$ has $J(u) = 0$, so $J$ has no minimiser on $\mathcal{A}$.

Check that the infimum is $0$.
For $\varepsilon > 0$, put $u_\varepsilon(x) = \arctan(x / \varepsilon) / \arctan(1 / \varepsilon)$, which lies in $\mathcal{A}$.
With $c_\varepsilon = \arctan(1 / \varepsilon)$, one has $u_\varepsilon'(x) = \varepsilon / \big(c_\varepsilon (\varepsilon^2 + x^2)\big)$.
Since $x^2 \le \varepsilon^2 + x^2$,
$$J(u_\varepsilon) \le \frac{1}{c_\varepsilon^2} \int_{-1}^{1} \frac{\varepsilon^2}{\varepsilon^2 + x^2} \, dx = \frac{2 \varepsilon}{c_\varepsilon},$$
which tends to $0$ as $\varepsilon \to 0$, because $c_\varepsilon \to \pi / 2$.

Check that $0$ is not attained.
If $J(u) = 0$, then $x u'(x) = 0$ for all $x$, by continuity of the integrand.
So $u' = 0$ for $x \ne 0$, hence also at $0$, and $u$ is constant.
This contradicts $u(-1) = -1 \ne 1 = u(1)$.

## Remarks

Riemann had used the Dirichlet principle, the claim that a nonnegative energy with given boundary values attains its infimum.
Weierstrass's example shows that this claim can fail, even for a nonnegative functional with a smooth integrand.
The minimising functions $u_\varepsilon$ tend to the sign function, which jumps at $0$ and does not belong to the admissible class.
The integrand $x^2 \xi^2$ is convex in $\xi$, but its coefficient vanishes at $x = 0$.
So $J$ gives no bound on $u'$ near $0$, and minimising sequences can develop a jump there.
This is the failure of the compactness step of the [[direct method|TE379HBTY0]].
Hilbert later justified the Dirichlet principle for the Dirichlet energy by the direct method.

## Explanations

### L1 human

The problem asks for a curve from height $-1$ to height $1$ that makes a weighted energy as small as possible.
The weight $x^2$ is zero at the middle, so steep slopes there are almost free.
Curves that jump more and more steeply at $0$ get energies closer and closer to $0$.
But a curve with energy exactly $0$ would have to be flat everywhere, and then it could not climb from $-1$ to $1$.
So the infimum is never reached: the best candidates converge to a jump, which is not an allowed curve.
</document>

<document source="https://tamnd.github.io/math2/node/4KZ458T2QA.md" tag="4KZ458T2QA" kind="concept">
# Variational integrator

## Summary

A variational integrator is a numerical method for Lagrangian mechanics obtained by discretising Hamilton's principle of stationary action instead of the equations of motion.
One approximates the action over a single time step by a discrete Lagrangian.
Asking the sum of these to be stationary gives the discrete Euler–Lagrange equations.
These methods are automatically symplectic and keep the momenta that come from symmetries, by a discrete Noether theorem, so they are [[geometric integrators|VQ2BTYFST4]].

## Explanations

### L1 human

In mechanics, the true motion makes a quantity called the action stationary.
A variational integrator copies this principle on a time grid: it looks for a list of positions that makes a discrete action stationary.
Because it inherits the principle and not only the equations, it inherits the structure that comes with it, such as conservation of angular momentum for a rotation-invariant system.
The Störmer–Verlet method used in molecular dynamics is the simplest example.
It comes from the discrete action with kinetic energy $\frac12((q_{k+1} - q_k)/h)^2$ and the potential averaged at the two ends of each step.
</document>

<document source="https://tamnd.github.io/math2/node/4M1SYSPGA2.md" tag="4M1SYSPGA2" kind="definition">
# Rational map

## Definition

Let $X$ and $Y$ be [[algebraic varieties|HV6YBS0YDH]] over an algebraically closed field $k$.
Consider pairs $(U, \varphi_U)$ where $U \subseteq X$ is a nonempty [[open subset|02BH3MCBQB]] and $\varphi_U \colon U \to Y$ is a [[morphism|CDS1VWB7A3]].
Two pairs $(U, \varphi_U)$ and $(V, \varphi_V)$ are equivalent if $\varphi_U = \varphi_V$ on $U \cap V$.
A *rational map* $\varphi \colon X \dashrightarrow Y$ is an equivalence class of such pairs.
It is *dominant* if for some, hence every, pair $(U, \varphi_U)$ in the class, the image $\varphi_U(U)$ is [[dense|0BSJSX7ME2]] in $Y$.

## Remarks

This relation is an [[equivalence relation|0G8VEZ3WTK]].
The key fact is that two morphisms $U \to Y$ that agree on a nonempty open subset of $U$ are equal.
The reason is that the set where they agree is closed, while a nonempty open subset of the irreducible space $U$ is dense.
The union of the domains of all pairs in the class is the largest open set on which $\varphi$ is defined, its domain of definition.

Example: $\varphi \colon \mathbb{P}^2 \dashrightarrow \mathbb{P}^1$, $[x : y : z] \mapsto [x : y]$, is a rational map.
It is a morphism on $U = \mathbb{P}^2 \setminus \set{[0 : 0 : 1]}$, and it cannot be extended to a morphism on all of $\mathbb{P}^2$.
It is dominant, since it is surjective onto $\mathbb{P}^1$.

Non-example: on $X = k$, the assignment $x \mapsto 1/x$ for $x \ne 0$ and $0 \mapsto 0$ is a function $X \to k$ but not a morphism, so it is not a pair in the definition.
Its restriction to $k \setminus \set{0}$ is a morphism and defines a rational map $k \dashrightarrow k$, whose domain of definition is $k \setminus \set{0}$.

Rational maps can only be composed when the first one is dominant; otherwise the image may lie outside the domain of the second.
A rational map $X \dashrightarrow \mathbb{A}^1$ is the same as a rational function on $X$.
</document>

<document source="https://tamnd.github.io/math2/node/4M5JXYVQ2X.md" tag="4M5JXYVQ2X" kind="concept">
# Compactly generated space

## Summary

A compactly generated space, or k-space, is a [[topological space|0CQHFPGJ45]] whose topology is determined by its compact pieces.
A set is closed as soon as its trace on every compact piece is closed.
Metric spaces and [[locally compact|0FE2SB8WVJ]] Hausdorff spaces are compactly generated.
Algebraic topology often works in the category of compactly generated spaces, because there products and function spaces behave well.

## Explanations

### L1 human

To check whether a set is closed, one may look at how it meets every compact part of the space.
In a compactly generated space, this local test is enough.
For example, in the plane a set is closed exactly when its intersection with every closed disc is closed.
So the topology of the plane is rebuilt from its compact discs.
</document>

<document source="https://tamnd.github.io/math2/node/4M94FZH9Y8.md" tag="4M94FZH9Y8" kind="concept">
# Regular function

## Summary

A regular function on a variety is a function to the base field that is locally a quotient of two polynomials.
The denominator must not vanish where the quotient is used.
Regular functions are the algebraic analogue of smooth or holomorphic functions.
On an affine variety the regular functions are exactly the polynomial functions, so they form the coordinate ring.
They are used to define morphisms of varieties and the structure sheaf of a variety.

## Explanations

### L1 human

On the affine line, a regular function on the whole line is just a polynomial.
On the line with the origin removed, the function $1/x$ is also regular, because its denominator never vanishes there.
In general a function is regular when near each point it can be written as a fraction of polynomials whose bottom part is nonzero at that point.
On a projective variety, the only regular functions on the whole variety are the constants.
</document>

<document source="https://tamnd.github.io/math2/node/4MBR0W0S6N.md" tag="4MBR0W0S6N" kind="statement">
# Artin–Tate lemma

## Statement

Let $A \subseteq B \subseteq C$ be commutative rings with $A$ a [[Noetherian ring|0670T59HGE]].
Suppose that $C$ is a [[finitely generated|AC5TFFQT9M]] $A$-algebra and that $C$ is a [[finitely generated|05QY1VEBYD]] $B$-module.
Then $B$ is a finitely generated $A$-algebra.

## Remarks

Let $x_1, \ldots, x_m$ generate $C$ as an $A$-algebra and $y_1, \ldots, y_n$ generate $C$ as a $B$-module.
Write $x_i = \sum_j b_{ij} y_j$ and $y_i y_j = \sum_l b_{ijl} y_l$ with $b_{ij}, b_{ijl} \in B$.
Let $B_0$ be the $A$-algebra generated by all $b_{ij}$ and $b_{ijl}$; it is Noetherian by [[Hilbert's basis theorem|0YKA9J3XEE]], and $C$ is a finitely generated $B_0$-module.
Then $B$ is a $B_0$-submodule of $C$, hence finitely generated as a module over the Noetherian ring $B_0$, so $B$ is of finite type over $A$.

Example: let $A = k$ be a field, $C = k[x]$ and $B = k[x^2]$; then $C$ is generated by $1$ and $x$ as a $B$-module, and indeed $B$ is generated by $x^2$ as a $k$-algebra.

## Explanations

### L1 human

If a big ring is finitely generated over a small one, and is a finite module over a middle ring, then the middle ring is finitely generated too.
Finite generation passes down to an intermediate ring as long as the top is finite over it.

### L2 human

The lemma gives a short proof of [[Zariski's lemma|YBJ2VQHFDQ]].
Suppose a field $K$ is a finitely generated $k$-algebra but not finite over $k$, and choose a transcendence basis $t_1, \ldots, t_r$ with $r \ge 1$.
Then $K$ is finite over $k(t_1, \ldots, t_r)$, and the lemma would make $k(t_1, \ldots, t_r)$ a finitely generated $k$-algebra, which is impossible.
The same idea shows that rings of invariants of finite groups acting on finitely generated algebras over a field are finitely generated.
</document>

<document source="https://tamnd.github.io/math2/node/4MBYAVB0PY.md" tag="4MBYAVB0PY" kind="concept">
# Commuting probability

## Summary

The commuting probability of a finite group is the probability that two elements chosen independently and uniformly at random commute.
It equals $1$ exactly for abelian groups, and it equals the number of conjugacy classes divided by the order of the group.
A non-abelian finite group has commuting probability at most $5/8$.
So the value measures how far a group is from being abelian.

## Explanations

### L1 human

Pick two elements of a finite group at random, with repetition allowed, and ask whether $xy = yx$.
In an abelian group the answer is always yes.
In the symmetric group $S_3$, of the $36$ ordered pairs exactly $18$ commute, so the probability is $1/2$.
A group cannot be "almost abelian" in this sense: once some pair fails to commute, at least $3/8$ of all pairs fail.
</document>

<document source="https://tamnd.github.io/math2/node/4MCCP8GG4J.md" tag="4MCCP8GG4J" kind="concept">
# Closed geodesic

## Summary

A closed geodesic on a [[Riemannian manifold|7CBT4SM83Y]] is a [[geodesic|NSYMQMHPDN]] that closes up smoothly, returning to its starting point with its starting velocity.
Equivalently, it is a nonconstant critical point of the energy on the space of closed loops.
Closed geodesics are the periodic orbits of the geodesic flow.
Their existence and their number are central questions of the variational theory of geodesics and of Hamiltonian dynamics.

## Explanations

### L1 human

A closed geodesic is a loop on a curved surface that goes straight ahead at every point and comes back to where it started, pointing the same way.
A rubber band stretched around a surface and allowed to settle often sits on one.
On a round sphere the great circles are closed geodesics.
Glue opposite sides of a square to get a flat torus.
On it, a straight line of rational slope closes up into a closed geodesic, while a line of irrational slope never closes.
</document>

<document source="https://tamnd.github.io/math2/node/4MR4YJS1ZD.md" tag="4MR4YJS1ZD" kind="concept">
# Exchangeable random variables

## Summary

A finite or infinite sequence of [[random variables|0PNRKMRC55]] is exchangeable when its [[joint distribution|0DR81R1F2B]] does not change if finitely many of the variables are permuted.
Every sequence of [[independent and identically distributed|00MRN0FARV]] random variables is exchangeable, but exchangeable variables need not be independent.
Exchangeability is the basic symmetry assumption of Bayesian statistics.
By [[de Finetti's theorem|W19129PVKQ]], an infinite exchangeable sequence is a mixture of i.i.d. sequences.

## Explanations

### L1 human

Exchangeable means that the order of the observations carries no information.
Draw balls one by one, without replacement, from an urn with $3$ red and $2$ blue balls.
The chance that the first ball is red and the second is blue equals the chance that the first is blue and the second is red: both are $\frac{3}{5} \cdot \frac{2}{4} = \frac{2}{5} \cdot \frac{3}{4} = \frac{3}{10}$.
The colours are exchangeable, yet they are not independent, since a red first ball makes a red second ball less likely.
</document>

<document source="https://tamnd.github.io/math2/node/4MWPJBF1Z7.md" tag="4MWPJBF1Z7" kind="definition">
# Wavelet

## Definition

Let $\psi \in L^2(\R)$ (see [[Lp space|SASP14MA1Z]]).
For $j, k \in \Z$ put
$$\psi_{j,k}(x) = 2^{j/2} \, \psi(2^j x - k).$$
The function $\psi$ is an *orthonormal wavelet* if the family $(\psi_{j,k})_{j, k \in \Z}$ is an [[orthonormal basis|0S9GGTR0EX]] of $L^2(\R)$.

## Remarks

The factor $2^{j/2}$ makes $\norm{\psi_{j,k}}_2 = \norm{\psi}_2$.
Then every $f \in L^2(\R)$ has the expansion $f = \sum_{j,k} \ip{f}{\psi_{j,k}} \psi_{j,k}$, convergent in $L^2(\R)$.
If $\psi$ is also integrable, then $\int_\R \psi(x) \, dx = 0$.
Weaker notions are also called wavelets: $\psi$ such that $(\psi_{j,k})$ is a Riesz basis or a frame, and the admissible functions of the continuous wavelet transform.

Example: the [[Haar wavelet|JW5PA9XA7F]] $\psi = 1_{[0, 1/2)} - 1_{[1/2, 1)}$ is an orthonormal wavelet.
For instance, $\psi_{0,0}$ and $\psi_{1,0}$ are orthogonal, because $\psi_{1,0}$ lives on $[0, \tfrac12)$, where $\psi_{0,0} = 1$, and has integral $0$.

Non-example: the Gaussian $\psi(x) = e^{-\pi x^2}$ is not an orthonormal wavelet.
Its translates $\psi_{0,0}$ and $\psi_{0,1}$ are both positive, so $\ip{\psi_{0,0}}{\psi_{0,1}} > 0$ and they are not orthogonal.
</document>

<document source="https://tamnd.github.io/math2/node/4N1WKH2ZQV.md" tag="4N1WKH2ZQV" kind="proof">
# Proof that subgroups of index two are normal

## Proof

1. Let $H$ be a subgroup of a group $G$ with [[index|04QEBF8S49]] $2$.
2. There are two [[left cosets|03DEH43F4H]]. One is $H$, so the other is $G \setminus H$.
3. In the same way, there are two right cosets, $H$ and $G \setminus H$.
4. Let $g \in G$. If $g \in H$, then $gH = H = Hg$.
5. If $g \notin H$, then $gH \ne H$ and $Hg \ne H$. Thus $gH = G \setminus H = Hg$.
6. So $gH = Hg$ for all $g \in G$, and [[$H$ is normal|01E7SWYJX3]].

## Idea

With only two cosets, one of them is the subgroup and the other is its complement. This is true on the left and on the right, so the left and right cosets agree.
</document>

<document source="https://tamnd.github.io/math2/node/4N4DNDR22Z.md" tag="4N4DNDR22Z" kind="concept">
# Sober space

## Summary

A sober space is a topological space in which every irreducible closed set is the closure of exactly one point.
That point is called the generic point of the set.
Every sober space is $T_0$, and every Hausdorff space is sober.
Spectra of commutative rings with the Zariski topology are sober.
A sober space can be recovered, up to homeomorphism, from its lattice of open sets.

## Explanations

### L1 human

Some closed sets cannot be split into two smaller closed pieces; they are called irreducible.
A sober space has, for each such set, one special point whose closure is the whole set.
In algebraic geometry, the irreducible curve $y = x^2$ in the plane has such a point in the spectrum of the polynomial ring: the prime ideal $(y - x^2)$.
In a Hausdorff space, the only irreducible closed sets are single points, so each is its own generic point.
</document>

<document source="https://tamnd.github.io/math2/node/4N53T1BZ9S.md" tag="4N53T1BZ9S" kind="statement">
# Mehta integral

## Statement

Let $n \ge 1$ be an integer and let $\gamma \in \C$ with $\operatorname{Re} \gamma > -\frac{1}{n}$.
Let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
Then
$$\frac{1}{(2\pi)^{n/2}} \int_{\R^n} \prod_{1 \le i < j \le n} \abs{x_i - x_j}^{2\gamma}\, e^{-(x_1^2 + \cdots + x_n^2)/2} \, dx_1 \cdots dx_n = \prod_{j=1}^{n} \frac{\Gamma(1 + j\gamma)}{\Gamma(1 + \gamma)}.$$

## Remarks

The formula follows from the [[Selberg integral|S8VTVSJ6EX]] by a limit.
Put $\alpha = \beta = \lambda$ and $t_i = \frac12 + \frac{x_i}{2\sqrt{2\lambda}}$; then $(t_i(1 - t_i))^{\lambda - 1}$ tends, after rescaling, to $e^{-x_i^2/2}$ as $\lambda \to \infty$, and Stirling's formula evaluates the limit of the right side.
For $\gamma = 0$ both sides equal $1$, by the [[Gaussian integral|0DFT1GP1JP]].
Example: $n = 2$ and $\gamma = 1$.
Then $x_1 - x_2$ is normal with mean $0$ and variance $2$ under the Gaussian weight, so the left side is $\mathbb{E}[(x_1 - x_2)^2] = 2$, and the right side is $\frac{\Gamma(2)}{\Gamma(2)} \cdot \frac{\Gamma(3)}{\Gamma(2)} = 2$.
The bound $\operatorname{Re} \gamma > -\frac1n$ is needed: for smaller $\gamma$ the singularity where all $x_i$ coincide makes the integral diverge.

## Explanations

### L1 human

Take $n$ independent standard normal numbers and look at the product of all their pairwise distances, raised to the power $2\gamma$.
The Mehta integral gives the average of this quantity as a short product of gamma values.

### L2 human

For $\gamma = \frac12, 1, 2$ the integrand is, up to the Gaussian factor, the joint density of the eigenvalues of the Gaussian orthogonal, unitary and symplectic ensembles of [[random matrices|1HS4RT7HK5]].
So the formula gives the normalizing constants of these densities.
It is the Gaussian member of a family of integrals over root systems, the Macdonald–Mehta integrals, of which this is the case of type $A_{n-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/4N5FGYKAFW.md" tag="4N5FGYKAFW" kind="definition">
# Whitney C-infinity topology

## Definition

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] and let $C^\infty(M, N)$ be the set of smooth maps from $M$ to $N$.
For an integer $k \ge 0$, let $J^k(M, N)$ be the manifold of [[$k$-jets|1QE0Z0756D]] of maps from $M$ to $N$, the [[jet bundle|HC3VBBCJQP]] of the trivial bundle $M \times N \to M$.
Each $f \in C^\infty(M, N)$ has the jet map $j^k f \colon M \to J^k(M, N)$, $p \mapsto j^k_p f$.
For an open set $U \subseteq J^k(M, N)$, put
$$\mathcal{M}(U) = \set{ f \in C^\infty(M, N) : j^k f(M) \subseteq U }.$$
The sets $\mathcal{M}(U)$, with $U$ open in $J^k(M, N)$, form a base of a [[topology|0CQHFPGJ45]] on $C^\infty(M, N)$, the *Whitney $C^k$ topology*.
The *Whitney $C^\infty$ topology* is the topology whose base is the union over all $k \ge 0$ of these bases.

## Remarks

With the Whitney $C^\infty$ topology, $C^\infty(M, N)$ is a [[Baire space|0ZH5AJ2ED7]].
If $M$ is compact, it coincides with the weak $C^\infty$ topology, the topology of uniform convergence of all derivatives, which is built from the [[compact-open topology|RG15F0TWPW]] on jets.
If $M$ is not compact, it is much finer.

Example: for $M = N = \R$, let $\varepsilon \colon \R \to (0, \infty)$ be continuous and let $f \in C^\infty(\R, \R)$.
The set $U = \set{ (x, y) \in \R \times \R : \abs{y - f(x)} < \varepsilon(x) }$ is open in $J^0(\R, \R) = \R \times \R$.
So $\mathcal{M}(U) = \set{ g : \abs{g(x) - f(x)} < \varepsilon(x) \text{ for all } x }$ is an open neighbourhood of $f$.

Non-example: the constant maps $f_n = 1/n$ do not converge to $0$ in the Whitney $C^\infty$ topology on $C^\infty(\R, \R)$, although they converge uniformly with all derivatives.
Take $f = 0$ and $\varepsilon(x) = 1/(1 + x^2)$ above.
For each $n$, $1/n > \varepsilon(x)$ for large $\abs{x}$, so no $f_n$ lies in this neighbourhood of $0$.
</document>

<document source="https://tamnd.github.io/math2/node/4NCB2ZY08G.md" tag="4NCB2ZY08G" kind="counterexample">
# Lewy's example

## Counterexample

On $\R^3$ with coordinates $(x, y, t)$, consider the first-order linear operator
$$L = \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} - 2i(x + iy) \frac{\partial}{\partial t}.$$
Its coefficients are polynomials, so they are smooth and even real-analytic.
Hans Lewy proved that there is a function $f \in C^\infty(\R^3)$ with the following property.
The [[partial differential equation|GC5BDZ6QBC]] $Lu = f$ has no solution $u$ of class $C^1$ on any nonempty open subset of $\R^3$.
So a linear partial differential equation with smooth coefficients and a smooth right-hand side need not be locally solvable.

## Remarks

The key step concerns right-hand sides that depend only on $t$.
If $\psi$ is a real-valued $C^1$ function of $t$ and $Lu = \psi'(t)$ has a $C^1$ solution near the origin, then $\psi$ is real-analytic near $t = 0$.
Choosing $\psi$ smooth but not real-analytic gives an equation with no solution near the origin; a Baire category argument then produces one $f$ that works at every point.
The operator $L$ is, up to a constant factor, the [[tangential Cauchy–Riemann operator|X7GP9BECD8]] of the hypersurface $\set{(z, w) \in \C^2 : \operatorname{Im} w = \abs{z}^2}$, written in the coordinates $z = x + iy$ and $t = \operatorname{Re} w$.
This contrasts with the Cauchy–Kovalevskaya theorem, which gives local solutions when the coefficients and the data are all real-analytic.
Hörmander later explained the phenomenon by a necessary condition for local solvability involving the commutator of $L$ and its complex conjugate $\bar L$.

## Explanations

### L1 human

Before this example, it was widely expected that a linear differential equation with nice coefficients always has solutions at least on small regions.
Lewy wrote down a simple equation in three real variables, with polynomial coefficients, that has no solution at all near any point for a suitable smooth right-hand side.
The obstruction comes from complex analysis: solutions would force certain smooth functions to be real-analytic, which they need not be.
</document>

<document source="https://tamnd.github.io/math2/node/4NCYA7GME1.md" tag="4NCYA7GME1" kind="concept">
# Signature operator

## Summary

The signature operator of a closed oriented Riemannian manifold of dimension $4k$ is the operator $d + d^*$ on complex [[differential forms|0E3G0GRPZ6]].
It is restricted to the forms that are even under a natural involution built from the [[Hodge star|1Y4WX4XQWJ]].
It is an [[elliptic operator|MT9P2X85V8]], and its index equals the signature of the manifold, the signature of the [[intersection form|WK8GKRKAVD]] on middle cohomology.
Applied to it, the Atiyah–Singer index theorem gives the [[Hirzebruch signature theorem|TEG1ZX1AGJ]].
It is one of the classical operators of index theory, together with the de Rham, Dolbeault and spin Dirac operators.

## Explanations

### L1 human

The signature of a manifold of dimension $4k$ is a whole number that counts how the middle-dimensional cycles intersect, positive pairs minus negative pairs.
The signature operator is a differential operator built from the metric whose index, the number of solutions minus the number of obstructions, recovers this number.
In this way a purely topological count is expressed through analysis.
For the complex projective plane the signature is $1$, and so is the index of its signature operator.
</document>

<document source="https://tamnd.github.io/math2/node/4NFHBBNZBK.md" tag="4NFHBBNZBK" kind="definition">
# Integro-differential equation

## Definition

Let $I = [t_0, t_0 + T]$ with $T > 0$, let $n \ge 1$, and let $f \colon I \times \R^n \to \R^n$ and $K \colon I \times I \times \R^n \to \R^n$ be continuous.
A (Volterra type) *integro-differential equation* is an equation
$$x'(t) = f(t, x(t)) + \int_{t_0}^{t} K(t, s, x(s)) \, ds, \quad t \in I,$$
for an unknown continuously differentiable $x \colon I \to \R^n$.
More generally, an integro-differential equation is any equation with the unknown both under a derivative and under an [[integral|0W29TBTY12]].
When the integral is over a fixed interval, the equation is of Fredholm type.

## Remarks

Example: $x'(t) = -\int_0^t x(s) \, ds$, $x(0) = 1$.
Differentiating gives $x'' = -x$ with $x(0) = 1$ and $x'(0) = 0$, so $x = \cos t$.
Check: $-\int_0^t \cos s \, ds = -\sin t = x'(t)$.

Non-example: $x'(t) = -x(t)$ contains no integral of the unknown, so it is an [[ordinary differential equation|0NT64JV96A]].
Also $x(t) = 1 + \int_0^t x(s) \, ds$ contains no derivative; it is a [[Volterra integral equation|DND8CBEVXT]].

Integrating the Volterra form from $t_0$ to $t$ turns the problem with $x(t_0) = x_0$ into a Volterra integral equation for $x$.
For a convolution kernel $K(t, s, x) = k(t - s) x$, the Laplace transform turns the linear equation into an algebraic one.

## Explanations

### L1 human

The rate of change is given partly by the present value and partly by a weighted sum of all past values.
</document>

<document source="https://tamnd.github.io/math2/node/4NKP73KVVW.md" tag="4NKP73KVVW" kind="concept">
# Area-minimizing current

## Summary

An area-minimizing current is an [[integral current|EG4AJVEFT2]] whose mass is not larger than that of any other integral current with the same boundary.
It is the solution concept of the Plateau problem in geometric measure theory, and it generalises oriented [[minimal surfaces|41X0YW2R9N]] that minimise area.
Such currents exist for every boundary that bounds some integral current.
Their regularity is a central topic: they are smooth away from a singular set of small dimension.

## Explanations

### L1 human

Take a closed wire loop and ask for the surface of least area spanning it, counting area with orientation and multiplicity.
An area-minimizing current is such a least-area surface in a generalised sense, so that one exists even when no smooth one does.
For a round circle in space, the flat disc is area-minimizing; a hemisphere with the same rim is not, because it has twice the area.
In high dimensions minimisers can have singular points, for example cones over products of spheres.
</document>

<document source="https://tamnd.github.io/math2/node/4NMTSX0XDS.md" tag="4NMTSX0XDS" kind="concept">
# Group ring

## Summary

The group ring $R[G]$ of a [[group|G5S1M9]] $G$ over a [[ring|0B1261B5ED]] $R$ is the ring of finite formal sums $\sum_g a_g g$ with coefficients in $R$.
Such sums are multiplied using the group law.
It turns questions about $G$ and its actions into questions about modules over one ring.
When $R$ is a field, it is the [[group algebra|MNP2HBWB5Q]], the basic object of representation theory.
It is the special case of the [[monoid ring|1HM7ZYMPD7]] in which the monoid is a group.

## Explanations

### L1 human

Think of the elements of $G$ as basis vectors, and allow coefficients from $R$.
Multiply two formal sums by expanding and multiplying group elements in $G$.
For the integers $\Z$ and the cyclic group $\set{1, t}$ of order $2$, the element $1 - t$ satisfies $(1 - t)^2 = 1 - 2t + t^2 = 2 - 2t = 2(1 - t)$.
The integral group ring $\Z[G]$ is important in topology and in the study of group extensions.
</document>

<document source="https://tamnd.github.io/math2/node/4NSACSWRJZ.md" tag="4NSACSWRJZ" kind="concept">
# Eigenfunction of an integral operator

## Summary

An eigenfunction of an integral operator $T_K$ is a nonzero function $\varphi$ with $T_K \varphi = \mu \varphi$ for some number $\mu$.
It is an [[eigenvector|0N6960W0BV]] of $T_K$ in a space of functions.
For symmetric kernels the eigenfunctions can be chosen to form an orthonormal system, and expanding in them diagonalizes the operator, as in the [[Hilbert–Schmidt theorem|A1G25R99PY]].

## Explanations

### L1 human

Applying the integral operator to an eigenfunction does not change its shape; it only multiplies it by a number.
So eigenfunctions are the natural modes of the operator, like the pure tones of a vibrating string.
For the kernel $K(x, y) = \min(x, y)$ on $[0, 1]$, the function $\sin(\pi x / 2)$ is an eigenfunction with eigenvalue $4 / \pi^2$.
</document>

<document source="https://tamnd.github.io/math2/node/4P3S0DRZXB.md" tag="4P3S0DRZXB" kind="concept">
# Nerve of a cover

## Summary

The nerve of a cover of a space is the [[simplicial complex|V8GZ0W1ZZ5]] that records which members of the cover meet.
It has one vertex for each nonempty member, and a set of members spans a simplex when they have a common point.
By the nerve theorem, a good cover has a nerve with the same homotopy type as the space.
This turns topology into combinatorics, and nerves are also the basis of Čech cohomology.

## Explanations

### L1 human

Cover a space with overlapping patches.
Draw a dot for each patch, an edge between two dots when the patches overlap, a triangle when three patches share a point, and so on.
The resulting shape is the nerve.
Cover a circle by three arcs, each overlapping the next, with no point in all three.
The nerve is the boundary of a triangle, which is again a circle.
</document>

<document source="https://tamnd.github.io/math2/node/4P4Z4ZGXCY.md" tag="4P4Z4ZGXCY" kind="concept">
# Borel hierarchy

## Summary

The Borel hierarchy sorts the Borel sets of a [[Polish space|584GXXY73M]] into levels $\Sigma^0_\alpha$, $\Pi^0_\alpha$ and $\Delta^0_\alpha$, indexed by the countable [[ordinals|0YSCSX5YZC]] $\alpha \ge 1$.
It starts with the open and the closed sets, and each new level allows countable unions of complements of earlier sets.
Every [[Borel set|0WQXY32Q8D]] appears at some countable level, and in an uncountable Polish space every level contains new sets.
The level of a set measures how complicated it is, and this is a basic tool of [[descriptive set theory|TF8HMVXKYC]].

## Explanations

### L1 human

Open and closed sets are the simplest sets.
Countable unions of closed sets, and countable intersections of open sets, are one step more complex.
Repeating this, one climbs a ladder of complexity with one rung for each countable ordinal.
For example, the set of rational numbers is a countable union of points, so it sits on the second rung, but it is not a countable intersection of open sets.
</document>

<document source="https://tamnd.github.io/math2/node/4P5PSWADJD.md" tag="4P5PSWADJD" kind="concept">
# Packing density

## Summary

The packing density of a packing is the fraction of space covered by its bodies.
It is measured inside large balls and passed to the limit, since space is infinite.
The packing density of a body is the largest density of a packing of its congruent copies.
For equal balls this is the optimal [[sphere packing|HF32CBN5MA]] density, known exactly only in dimensions $1$, $2$, $3$, $8$ and $24$.

## Explanations

### L1 human

Fill a huge box with equal coins laid flat, as tightly as you can, and compute the share of the box that the coins cover.
Making the box larger and larger, this share settles near one number, the density of the arrangement.
Coins in a square grid cover $\pi/4 \approx 78.5\%$ of the plane.
Coins in a hexagonal pattern cover $\pi/\sqrt{12} \approx 90.7\%$, and no arrangement does better.
</document>

<document source="https://tamnd.github.io/math2/node/4P7GBWJ4RR.md" tag="4P7GBWJ4RR" kind="concept">
# Ext functor

## Summary

The Ext functors $\operatorname{Ext}^n_R(M, N)$ are the right derived functors of $\Hom_R(-, N)$.
They measure how far $\Hom$ is from being exact.
The group $\operatorname{Ext}^1_R(M, N)$ classifies the extensions of $M$ by $N$, and the vanishing of Ext detects [[projective modules|BFFA03GE3V]], injective modules and the [[projective dimension|MR9QCH1PNA]].

## Explanations

### L1 human

Applying $\Hom(-, N)$ to a short exact sequence loses exactness on the right.
The Ext groups record exactly what is lost, in a long sequence of correction terms.
To compute them, replace $M$ by a [[projective resolution|HK2AA1YSQ2]], apply $\Hom(-, N)$, and take cohomology.
For example, $\operatorname{Ext}^1_\Z(\Z/n\Z, \Z) \cong \Z/n\Z$, which reflects the extension $0 \to \Z \xrightarrow{n} \Z \to \Z/n\Z \to 0$ that does not split.
</document>

<document source="https://tamnd.github.io/math2/node/4P7KPA3Y7R.md" tag="4P7KPA3Y7R" kind="statement">
# Smoothing by convolution

## Statement

Let $n \ge 1$ and let $k \ge 0$ be an integer or $k = \infty$.
Let $f \colon \R^n \to \C$ be locally integrable, that is, integrable on every compact set.
Let $g \colon \R^n \to \C$ be a $C^k$ function with compact support.
Then the [[convolution|2Z29JN3VES]] $f * g$ is a $C^k$ function on $\R^n$, and $\partial^\alpha (f * g) = f * \partial^\alpha g$ for every multi-index $\alpha$ with $\abs{\alpha} \le k$.
In particular, if $g$ is a [[smooth function|0V33BNKZ9S]] with compact support, then $f * g$ is smooth.

## Remarks

For $k = 0$, continuity of $f * g$ follows from the uniform continuity of $g$ and the integrability of $f$ on compact sets.
The case $k \ge 1$ follows by induction from the [[derivative of a convolution|9FN0PFZZXZ]], applied to the $C^{k-1}$ functions $\partial_j g$.
The case $k = \infty$ is the conjunction of all finite cases.

Example: $f = \mathbf{1}_{[0, 1]}$ is discontinuous on $\R$.
If $0 < \varepsilon < 1/2$ and $\varphi$ is a smooth function with support in $[-\varepsilon, \varepsilon]$ and integral $1$, then $f * \varphi$ is smooth, equals $1$ on $[\varepsilon, 1 - \varepsilon]$ and $0$ outside $[-\varepsilon, 1 + \varepsilon]$.

The smoothness comes only from $g$: if $g$ is merely continuous, $f * g$ need not be differentiable.
For $f = g = \mathbf{1}_{[0, 1]}$, neither factor is continuous, yet $f * g$ is the continuous tent function, which is not differentiable at $0$, $1$, $2$.

## Explanations

### L1 human

Averaging a rough function against a smooth bump produces a smooth function.
The result is at least as smooth as the bump, however rough the original function was.
This is how blurring removes sharp edges from a picture.

### L2 human

With a [[mollifier|CQMK4YPE1J]] $\varphi_\varepsilon$, the functions $f * \varphi_\varepsilon$ are smooth and tend to $f$ in $L^p$ as $\varepsilon \to 0$ when $f \in L^p$, $1 \le p < \infty$.
This is the standard way to approximate integrable or continuous functions by smooth ones.
It is used to prove density theorems, to define weak derivatives through smooth approximations, and to regularize distributions.
</document>

<document source="https://tamnd.github.io/math2/node/4PBC71YCTK.md" tag="4PBC71YCTK" kind="definition">
# Sequential probability ratio test

## Definition

Let $X_1, X_2, \dots$ be [[independent and identically distributed|00MRN0FARV]] random variables with a density $f$ with respect to a $\sigma$-finite measure.
Let $f_0 \ne f_1$ be two densities, and consider the [[hypotheses|087P03NEND]] $H_0 \colon f = f_0$ and $H_1 \colon f = f_1$.
The ratio of the [[likelihoods|0VW9E4QXR6]] after $n$ observations is
$$\Lambda_n = \prod_{i=1}^{n} \frac{f_1(X_i)}{f_0(X_i)}, \qquad \Lambda_0 = 1.$$
Let $0 < A < 1 < B$ be constants.
The *sequential probability ratio test* with boundaries $A$ and $B$ is the [[sequential procedure|9CWYNB9003]] such that:

1. It stops at the [[stopping time|WND87FBAKC]] $N = \min \set{n \ge 1 : \Lambda_n \le A \text{ or } \Lambda_n \ge B}$.
2. It rejects $H_0$ if $\Lambda_N \ge B$ and accepts $H_0$ if $\Lambda_N \le A$.

## Remarks

Example: for coin tosses $X_i \in \set{0, 1}$, let $f_0$ be the Bernoulli law with parameter $1/2$ and $f_1$ the one with parameter $2/3$.
Then each head multiplies $\Lambda_n$ by $4/3$ and each tail by $2/3$.
With $A = 1/4$ and $B = 4$, the test stops the first time the running product leaves the interval $(1/4, 4)$.

Non-example: comparing $\Lambda_{100}$ with a single threshold after exactly $100$ tosses is a fixed-sample likelihood ratio test, not an SPRT, because the stopping rule of clause 1 is not used.

Since $f_0 \ne f_1$, the time $N$ is finite with probability one under both hypotheses and has finite expectation.
In logarithms, $\log \Lambda_n$ is a [[random walk|04VZE3FZXF]] with the two absorbing barriers $\log A < 0 < \log B$.
Let $\alpha$ and $\beta$ be the error probabilities of [[type I and type II|03N8C8ZR4B]].
Wald's inequalities are $B \le (1 - \beta)/\alpha$ and $A \ge \beta/(1 - \alpha)$, and in practice the boundaries $B = (1 - \beta)/\alpha$ and $A = \beta/(1 - \alpha)$ are used as approximations.
</document>

<document source="https://tamnd.github.io/math2/node/4PDKJ8N1S9.md" tag="4PDKJ8N1S9" kind="statement">
# Strichartz estimate

## Statement

Let $n \ge 1$.
For $f \in L^2(\R^n)$ and $t \in \R$, let $e^{it\Delta} f$ be the solution at time $t$ of the free [[Schrödinger equation|EPZR6H2WJN]] $i u_t + \Delta u = 0$ with $u(0) = f$.
It is given by the [[Fourier transform|DPVM34ADZV]] as $\widehat{e^{it\Delta} f}(\xi) = e^{-it\abs{\xi}^2} \hat f(\xi)$.
Call a pair $(q, r)$ admissible if $2 \le q \le \infty$, $2 \le r \le \infty$,
$$\frac{2}{q} + \frac{n}{r} = \frac{n}{2},$$
and $(q, r, n) \ne (2, \infty, 2)$.
Let $(q, r)$ be admissible, and let $L^r$ denote the [[Lebesgue space|SASP14MA1Z]].
Then there is a constant $C = C(n, q, r)$ with
$$\Big( \int_{\R} \norm{e^{it\Delta} f}_{L^r(\R^n)}^q \, dt \Big)^{1/q} \le C \norm{f}_{L^2(\R^n)} \quad \text{for all } f \in L^2(\R^n).$$
For $q = \infty$ the left side means $\sup_{t} \norm{e^{it\Delta} f}_{L^r(\R^n)}$.

## Remarks

The proof combines two facts.
The first is the dispersive estimate $\norm{e^{it\Delta} f}_{L^\infty} \le (4\pi\abs{t})^{-n/2} \norm{f}_{L^1}$, which follows from the explicit kernel of $e^{it\Delta}$; interpolation with the conservation of the $L^2$ norm gives the $L^{r'} \to L^r$ bounds.
The second is the $TT^*$ argument, which reduces the estimate to a bound for a time convolution handled by the Hardy–Littlewood–Sobolev inequality when $q > 2$.
The endpoint $q = 2$, $r = \frac{2n}{n - 2}$ for $n \ge 3$ was proved by Keel and Tao in 1998.
The relation $\frac{2}{q} + \frac{n}{r} = \frac{n}{2}$ is forced by scaling: if $f_\lambda(x) = f(\lambda x)$, then $e^{it\Delta} f_\lambda (x) = (e^{i\lambda^2 t\Delta} f)(\lambda x)$, and comparing powers of $\lambda$ on both sides gives the relation.
Example: for $n = 1$ the pair $(q, r) = (4, \infty)$ is admissible, and for $n = 2$ the pair $(4, 4)$ is admissible, the case first proved by Strichartz.

## Explanations

### L1 human

Waves of the Schrödinger equation spread out, so the solution cannot stay concentrated for long.
The Strichartz estimate measures this: although the total mass $\norm{u(t)}_{L^2}$ never changes, the solution is small in other norms for most times.
Averaged in time, the solution is more integrable in space than the initial data alone would suggest.
This extra integrability is what makes nonlinear equations tractable.

### L2 human

Strichartz estimates are the basic tool for the local and global well-posedness of nonlinear [[dispersive equations|AYMJ2NFR4W]], such as the nonlinear Schrödinger equation $i u_t + \Delta u = \pm\abs{u}^{p-1} u$.
One writes the equation in Duhamel form and runs a fixed point argument in the spaces $L^q_t L^r_x$, using the inhomogeneous version of the estimate.
Similar estimates hold for the wave and Klein–Gordon equations, with different admissibility conditions and a loss of derivatives.
Abstract versions, due to Keel and Tao, need only an energy bound and a dispersive decay estimate for the evolution.
</document>

<document source="https://tamnd.github.io/math2/node/4PK2T949SQ.md" tag="4PK2T949SQ" kind="counterexample">
# Zaremba's punctured ball

## Counterexample

Let $n \ge 2$ and let $\Omega = \set{x \in \R^n : 0 < \abs{x} < 1}$ be the unit [[open ball|0YE5868HZB]] with its centre removed.
Its boundary is the unit sphere together with the point $0$.
Let $f = 0$ on the unit sphere and $f(0) = 1$; this $f$ is continuous on $\partial\Omega$, since $0$ is an isolated point of $\partial\Omega$.
The [[Perron–Wiener–Brelot solution|R12KQGVGN6]] of $f$ is $H_f = 0$, so $0$ is not a [[regular boundary point|B7948M365K]].
Check: the constant $0$ is in the lower class, so $H_f \ge 0$.
For $\varepsilon > 0$, let $u_\varepsilon(x) = \varepsilon (\abs{x}^{2-n} - 1)$ if $n \ge 3$ and $u_\varepsilon(x) = \varepsilon \log (1/\abs{x})$ if $n = 2$.
This function is harmonic, hence [[superharmonic|X4V2S56YQQ]], on $\Omega$.
It is $0$ on the unit sphere and tends to $+\infty$ at $0$, so it is in the upper class and $H_f \le u_\varepsilon$.
Letting $\varepsilon \to 0$ gives $H_f \le 0$, so $H_f = 0$, which does not tend to $f(0) = 1$.

## Remarks

So the [[Dirichlet problem|8D56GQ4AG8]] on the punctured ball has no solution for this continuous boundary function.
A classical solution $u$ would be bounded and harmonic near $0$, so $0$ would be a removable singularity.
The extended function on the ball would vanish by the maximum principle, which contradicts $u(0) = 1$.
The point $0$ is too small to be seen by harmonic functions: it has capacity zero.

## Explanations

### L1 human

Take a ball, keep the surface at temperature $0$, and try to hold the single centre point at temperature $1$.
A single point cannot hold heat in equilibrium, so the temperature in the ball is $0$ everywhere.
This shows that the Dirichlet problem can fail for a very simple open set.
</document>

<document source="https://tamnd.github.io/math2/node/4PPT08K2M1.md" tag="4PPT08K2M1" kind="statement">
# Importance sampling

## Statement

Let $p$ and $g$ be [[probability density functions|0ZY9DX8FBF]] on $\R^d$, and let $h \colon \R^d \to \R$ be measurable with $\int \abs{h}\, p < \infty$ and $q = \int h(x) p(x)\, dx$.
Assume that $g(x) > 0$ whenever $h(x) p(x) \ne 0$.
Let $Y_1, \ldots, Y_n$ be independent with density $g$, and let
$$\hat q_n = \frac{1}{n} \sum_{i=1}^{n} h(Y_i)\, \frac{p(Y_i)}{g(Y_i)},$$
with the convention that a term with $g(Y_i) = 0$ is $0$.
Then:

1. $\mathbb E \hat q_n = q$;
2. $\operatorname{Var} \hat q_n = \frac{1}{n} \left( \int \frac{h(x)^2 p(x)^2}{g(x)}\, dx - q^2 \right)$, where the integral runs over $\set{g > 0}$ and may be $+\infty$;
3. among all such $g$, the variance is smallest for $g^* = \abs{h} p / \int \abs{h} p$ (when $\int \abs{h} p > 0$), and its minimum is $\frac1n \bigl( (\int \abs{h} p)^2 - q^2 \bigr)$; in particular it is $0$ if $h \ge 0$.

## Remarks

Clause 1: $\mathbb E[h(Y) p(Y)/g(Y)] = \int_{g > 0} h p = q$, since $h p = 0$ where $g = 0$.
Clause 2 is the same computation for the second moment, divided by $n$ for independent samples.
Clause 3 follows from the [[Cauchy–Schwarz inequality|0RFWW56Y22]]: $\left( \int \abs{h} p \right)^2 = \left( \int \frac{\abs{h} p}{\sqrt g} \sqrt g \right)^2 \le \int \frac{h^2 p^2}{g}$, with equality when $g$ is proportional to $\abs{h} p$.
The optimal $g^*$ needs $\int \abs{h} p$, which is essentially the unknown; in practice one picks $g$ close in shape to $\abs{h} p$.
Example: let $X$ be standard normal and $q = P(X > 4) \approx 3.17 \times 10^{-5}$.
Plain sampling sees almost no hits.
Sampling from the normal density with mean $4$ puts half the samples in the event and reduces the relative error by orders of magnitude.
A badly chosen $g$ with lighter tails than $\abs{h} p$ can make the integral in clause 2 infinite, so the estimate converges very slowly and erratically.

## Explanations

### L1 human

Importance sampling estimates an average by sampling more often where the integrand matters most, and correcting for this bias with weights.
Each sample is weighted by the ratio $p/g$ of the target density to the sampling density, so the estimate stays exactly right on average.
If the sampling density follows the shape of the integrand, the weighted values are nearly constant and the estimate becomes very precise.
It is especially useful for rare events, which plain sampling almost never hits.

### L2 human

This is a [[variance reduction|A01GEGWKVB]] technique for the [[Monte Carlo method|HWP11G3HTN]], and also a change of measure: $g$ defines a new probability measure and $p/g$ is the Radon–Nikodym derivative.
When $p$ is known only up to a constant, one uses the self-normalised estimator $\sum w_i h(Y_i) / \sum w_i$ with $w_i = p(Y_i)/g(Y_i)$, which is biased but consistent.
The effective sample size $(\sum w_i)^2 / \sum w_i^2$ is a common diagnostic of weight degeneracy.
In rare-event simulation, exponential tilting of $p$ is a standard way to build $g$, and in sequential Monte Carlo the idea is applied step by step.
</document>

<document source="https://tamnd.github.io/math2/node/4PVP1C3JZG.md" tag="4PVP1C3JZG" kind="concept">
# Multidimensional transform

## Summary

A multidimensional transform is an [[integral transform|YCK709QB06]], or a discrete analogue of one, applied to a function of several variables.
The main examples are the multidimensional [[Fourier transform|DPVM34ADZV]], the [[multidimensional Laplace transform|QT3M720TCV]], the [[multidimensional Mellin transform|92JY94ZJD8]] and, for sequences indexed by $\Z^n$, the [[multidimensional Z-transform|A7NF067M7N]].
When the kernel is a product of one-variable kernels, the transform is the composition of one-variable transforms in each variable.
Such transforms are used for partial differential equations, for images and other multidimensional signals, and in probability for joint distributions.

## Explanations

### L1 human

An image is a function of two variables, the position across and the position down.
A multidimensional transform analyzes such a function in all its variables at once.
Often it is enough to transform in the first variable, then in the second, and so on.
For example, the two-dimensional Fourier transform of a picture is obtained by transforming each row and then each column.
</document>

<document source="https://tamnd.github.io/math2/node/4PZ35C0GV3.md" tag="4PZ35C0GV3" kind="concept">
# Wavelet

## Summary

A wavelet is a function $\psi$ on $\R$ whose dilations by powers of $2$ and integer translations form an [[orthonormal basis|0S9GGTR0EX]] of $L^2(\R)$.
Each basis function is localized near a position and at a scale, so a wavelet expansion describes a function at all levels of detail.
In a wider sense, any oscillating function with integral zero that is used in a [[continuous wavelet transform|57FR21NDGV]] is called a wavelet.
Wavelets are used in signal and image compression, numerical analysis and harmonic analysis.

## Explanations

### L1 human

A wavelet is a small wave: it wiggles up and down and then dies out.
Stretching it gives slow wiggles for coarse features, and squeezing it gives fast wiggles for fine detail.
Sliding these copies along the line, one can match the features of a signal wherever they occur.
The simplest wavelet is the Haar wavelet, which is $+1$ on $[0, \tfrac12)$, $-1$ on $[\tfrac12, 1)$ and $0$ elsewhere.
</document>

<document source="https://tamnd.github.io/math2/node/4Q1CK359NB.md" tag="4Q1CK359NB" kind="definition">
# S-finite measure

## Definition

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]].
A [[measure|PR11BZQQPG]] $\mu$ on $(X, \Sigma)$ is *s-finite* if there are [[finite measures|63Z1Q4FS23]] $\mu_1, \mu_2, \ldots$ on $(X, \Sigma)$ such that $\mu(A) = \sum_{n=1}^{\infty} \mu_n(A)$ for every $A \in \Sigma$.

## Remarks

Let $\lambda$ be Lebesgue measure on the Borel sets of $\R$.
Let $\mu(A) = \sum_{n=1}^{\infty} \lambda(A \cap [0, 1])$, which is $0$ when $\lambda(A \cap [0, 1]) = 0$ and $\infty$ otherwise.
It is s-finite, since each term is a finite measure.
It is not [[σ-finite|WEBBT18D4X]]: a set of finite $\mu$-measure meets $[0, 1]$ in a $\lambda$-null set, and countably many such sets cannot cover $[0, 1]$.
Every σ-finite measure is s-finite: write $X$ as a disjoint union of sets $B_n$ of finite measure, and let $\mu_n(A) = \mu(A \cap B_n)$.
Counting measure on the Borel sets of $\R$ is not s-finite.
A finite measure gives positive mass to at most countably many points.
So countably many finite measures give positive mass to at most countably many points in total.
Counting measure gives mass $1$ to every point of the uncountable set $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/4Q3WY1RX91.md" tag="4Q3WY1RX91" kind="concept">
# Weak homotopy equivalence

## Summary

A weak homotopy equivalence is a continuous map that induces a bijection on path components and isomorphisms on all [[homotopy groups|P976WYEFMB]], at every base point.
Every [[homotopy equivalence|XZG6MTSWCN]] is a weak homotopy equivalence, and between CW complexes the converse holds by the [[Whitehead theorem|6P1E2FJPSD]].
Every space is weakly equivalent to a CW complex, so weak equivalences are the natural notion of sameness in homotopy theory.
They are the model for weak equivalences in abstract homotopy theory, such as in model categories.

## Explanations

### L1 human

Homotopy groups only see a space through the spheres that map into it.
A weak homotopy equivalence is a map that looks like an equivalence to every sphere.
For nice spaces this is the same as a true equivalence up to deformation.
For strange spaces, such as the Warsaw circle, a map to a point can fool every sphere and still not be an equivalence.
</document>

<document source="https://tamnd.github.io/math2/node/4QAQ2TRAY0.md" tag="4QAQ2TRAY0" kind="definition">
# Percolation theory

## Definition

Let $G = (V, E)$ be a [[graph|00C0BR7D7H]] with countably many vertices, each of finite degree, and let $p \in [0, 1]$.
*Bernoulli bond percolation* on $G$ with parameter $p$ is the product [[probability measure|059TJN0NZ1]] $P_p$ on $\Omega = \set{0, 1}^E$.
Under $P_p$ the coordinates $\omega(e)$, $e \in E$, are independent and $P_p(\omega(e) = 1) = p$.
An edge $e$ is *open* if $\omega(e) = 1$ and *closed* otherwise.
The *open cluster* $C(x)$ of a vertex $x$ is the vertex set of the [[connected component|0JG9W3KA6B]] of $x$ in the subgraph $(V, \set{e : \omega(e) = 1})$.
For a fixed vertex $o$, the *percolation probability* is $\theta(p) = P_p(\abs{C(o)} = \infty)$.
Site percolation is defined in the same way, with open and closed vertices in place of edges.
Percolation theory is the study of these random clusters.

## Remarks

Example: on the line graph $\Z$, a cluster is infinite only if all edges on one side of $o$ up to infinity are open.
This has probability $\lim_n p^n = 0$ for $p < 1$, so $\theta(p) = 0$ for $p < 1$ and $\theta(1) = 1$.
Non-example: a random subgraph in which two edges at the same vertex are never both open is not Bernoulli percolation, since its edge states are not independent.
The function $\theta$ is nondecreasing in $p$, by coupling all $P_p$ with one family of uniform random variables on $[0, 1]$.
</document>

<document source="https://tamnd.github.io/math2/node/4QKRPRCSXH.md" tag="4QKRPRCSXH" kind="concept">
# Axiom of constructibility

## Summary

The axiom of constructibility, written $V = L$, says that every set is constructible, that is, every set lies in the [[constructible universe|6TF2JBTPPJ]] $L$.
It is consistent with [[ZF|0HE5WDQ9FJ]] when ZF is consistent, and it decides many questions that ZFC leaves open.
It implies the [[axiom of choice|0HTGB13FE2]] and the generalized [[continuum hypothesis|0MN2MER9WN]].
It is incompatible with large cardinals such as [[measurable cardinals|AKBWT323RQ]], so most set theorists do not accept it as true.

## Explanations

### L1 human

The constructible universe is the thinnest possible world of sets that can be built from definable pieces.
The axiom of constructibility says that there are no other sets at all.
It is like saying that every set has a description at some stage of one fixed construction.
In such a world many hard questions get a definite answer; for example, there are exactly $\aleph_1$ real numbers.
</document>

<document source="https://tamnd.github.io/math2/node/4QQQP1Q886.md" tag="4QQQP1Q886" kind="concept">
# Factor analysis

## Summary

Factor analysis models many observed variables as linear combinations of a few unobserved common factors plus a noise term specific to each variable.
It is a [[latent variable model|C63RVH96TF]] that explains the correlations between the observed variables through the shared factors.
The model implies that the [[covariance matrix|0V7G6WMFK0]] is a low-rank matrix plus a diagonal matrix.
It is widely used in psychology and the social sciences.

## Explanations

### L1 human

Students who do well in algebra tend to do well in geometry, and good readers tend to be good writers.
Factor analysis tries to explain such patterns by a few hidden abilities, for example a mathematical ability and a verbal ability.
Each test score is then a weighted sum of the hidden abilities plus a part special to that test.
The weights, called loadings, show which tests depend on which abilities.
</document>

<document source="https://tamnd.github.io/math2/node/4R6SQD60FN.md" tag="4R6SQD60FN" kind="concept">
# Bornological space

## Summary

A bornological space is a [[locally convex space|B6GY2KJMWW]] whose topology is determined by its [[bounded sets|3BAJ8ZTVPX]].
In such a space, every convex balanced set that absorbs all bounded sets is a neighbourhood of $0$.
The main consequence concerns a linear map from a bornological space to a locally convex space.
Such a map is continuous as soon as it maps bounded sets to bounded sets.
Every metrizable locally convex space, in particular every normed space, is bornological.

## Explanations

### L1 human

For linear maps between normed spaces, continuity is the same as sending bounded sets to bounded sets.
Bornological spaces are the locally convex spaces where this convenient test still works.
They are the spaces whose topology can be read off from the knowledge of which sets are bounded.
A Banach space with its weak topology is a standard example that is not bornological when it is infinite-dimensional.
</document>

<document source="https://tamnd.github.io/math2/node/4R6XZSDH3E.md" tag="4R6XZSDH3E" kind="definition">
# Weak and strong extremum

## Definition

Let $a < b$, let $\mathcal{A} \subseteq C^1([a, b])$ be a class of [[admissible functions|G7T008BEWY]], and let $J \colon \mathcal{A} \to \R$ be a [[functional|1EQM0DJ4NC]].
For $y \in C^1([a, b])$ use the [[norms|0V2WP9B1MF]]
$$\norm{y}_0 = \max_{[a, b]} \abs{y}, \qquad \norm{y}_1 = \max_{[a, b]} \abs{y} + \max_{[a, b]} \abs{y'}.$$
Let $y_0 \in \mathcal{A}$.

1. $y_0$ is a *strong local minimum* of $J$ if there is $\varepsilon > 0$ with $J(y_0) \le J(y)$ for all $y \in \mathcal{A}$ with $\norm{y - y_0}_0 < \varepsilon$.
2. $y_0$ is a *weak local minimum* of $J$ if there is $\varepsilon > 0$ with $J(y_0) \le J(y)$ for all $y \in \mathcal{A}$ with $\norm{y - y_0}_1 < \varepsilon$.

Strong and weak local maxima are defined with $J(y_0) \ge J(y)$.
A strong (weak) local extremum is a strong (weak) local minimum or maximum.

## Remarks

Since $\norm{y}_0 \le \norm{y}_1$, every strong local minimum is a weak local minimum.

Example: let $J(y) = \int_0^1 y'^2 \, dx$ on $\mathcal{A} = \set{y \in C^1([0, 1]) : y(0) = 0, \ y(1) = 1}$, and $y_0(x) = x$.
For $y \in \mathcal{A}$ write $y' = 1 + \eta$; then $\int_0^1 \eta \, dx = 0$, so $J(y) = 1 + \int_0^1 \eta^2 \, dx \ge J(y_0)$.
So $y_0$ is a global, hence strong, minimum.

Non-example: let $J(y) = \int_0^1 y'^3 \, dx$ on the same class, and $y_0(x) = x$.
With $y' = 1 + \eta$ one gets $J(y) = 1 + \int_0^1 \eta^2 (3 + \eta) \, dx$, which is at least $1$ when $\abs{\eta} < 3$; so $y_0$ is a weak local minimum.
It is not a strong local minimum.
Fix $h > 0$, let $N$ be large and $\delta = h/N$, and let $y$ have slope $-N$ on $[0, \delta]$ and slope $(1 + h)/(1 - \delta)$ on $[\delta, 1]$, with corners rounded off.
Then $\norm{y - y_0}_0 \le h + \delta$, while $J(y)$ is close to $-N^2 h + (1 + h)^3/(1 - \delta)^2$, which tends to $-\infty$ as $N \to \infty$.
</document>

<document source="https://tamnd.github.io/math2/node/4R7HW6X4JY.md" tag="4R7HW6X4JY" kind="definition">
# Exponential map of a Lie group

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] with identity $e$ and Lie algebra $\mathfrak{g} = T_eG$, the [[tangent space|E7A5DHFT01]] at $e$.
For $X \in \mathfrak{g}$, let $\gamma_X \colon \R \to G$ be the unique [[one-parameter subgroup|YHA64AP1HN]] with $\gamma_X'(0) = X$.
The *exponential map* of $G$ is
$$\exp \colon \mathfrak{g} \to G, \qquad \exp(X) = \gamma_X(1).$$

## Remarks

Equivalently, $\gamma_X$ is the integral curve through $e$ of the [[left-invariant vector field|SNA3DVPK0C]] with value $X$ at $e$.
By rescaling, $\exp(tX) = \gamma_X(t)$ for all $t \in \R$.

Example: for $G = GL_n(\R)$, $\mathfrak{g} = M_n(\R)$ and $\gamma_A(t) = e^{tA}$ is a one-parameter subgroup with $\gamma_A'(0) = A$.
So $\exp(A) = e^A$ is the [[matrix exponential|020M04FN1H]].

Non-example: in $G = SL_2(\R)$, the matrix $\begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix}$ is not of the form $\exp(X)$ for any $X$ in the Lie algebra $\mathfrak{sl}_2(\R)$.
So $\exp$ need not be surjective, even for a connected group.

The map $\exp$ is smooth and its differential at $0$ is the identity of $\mathfrak{g}$.
By the inverse function theorem, it restricts to a diffeomorphism from a neighbourhood of $0$ in $\mathfrak{g}$ onto a neighbourhood of $e$ in $G$.
For a compact connected Lie group, $\exp$ is surjective.
</document>

<document source="https://tamnd.github.io/math2/node/4R8XEAV3P6.md" tag="4R8XEAV3P6" kind="definition">
# Laguerre polynomials

## Definition

Let $\alpha > -1$ be a real number and let $n \ge 0$ be an integer.
The *generalized Laguerre polynomial* of degree $n$ and parameter $\alpha$ is
$$L_n^{(\alpha)}(x) = \sum_{k=0}^{n} (-1)^k \frac{\Gamma(n + \alpha + 1)}{(n - k)! \, \Gamma(k + \alpha + 1)} \frac{x^k}{k!},$$
where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
The *Laguerre polynomial* of degree $n$ is $L_n = L_n^{(0)}$.

## Remarks

For $\alpha = 0$ the coefficient is a binomial coefficient: $L_n(x) = \sum_{k=0}^{n} \binom{n}{k} \frac{(-x)^k}{k!}$.
Example: $L_1^{(\alpha)}(x) = 1 + \alpha - x$ and $L_2(x) = 1 - 2x + \frac{x^2}{2}$.
Rodrigues' formula: $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} e^x}{n!} \frac{d^n}{dx^n} \left( e^{-x} x^{n + \alpha} \right)$.
Orthogonality: $\int_0^\infty L_m^{(\alpha)}(x) L_n^{(\alpha)}(x) \, x^\alpha e^{-x} \, dx = \frac{\Gamma(n + \alpha + 1)}{n!} \delta_{mn}$.
The polynomial $L_n^{(\alpha)}$ solves the [[Laguerre equation|FTGTM3JB1F]] $x y'' + (\alpha + 1 - x) y' + n y = 0$.
Check for $n = 1$: $y = 1 + \alpha - x$ gives $0 - (\alpha + 1 - x) + (1 + \alpha - x) = 0$.
Non-example: the clause $\alpha > -1$ is needed for orthogonality, because for $\alpha \le -1$ the weight $x^\alpha e^{-x}$ is not integrable near $0$.
</document>

<document source="https://tamnd.github.io/math2/node/4RB8834SEJ.md" tag="4RB8834SEJ" kind="concept">
# F-algebra

## Summary

For an endofunctor $F$ of a category $\mathcal{C}$, an $F$-algebra is an object $A$ with a morphism $F A \to A$.
For a suitable functor on sets, $F$-algebras are exactly the algebras of a given signature, so the notion generalizes universal algebra without equations.
Inductive data types are [[initial F-algebras|R76EC56V9R]], and the dual notion, [[F-coalgebra|6BS0300CMV]], models state-based systems.
Unlike an algebra over a monad, an $F$-algebra need not satisfy any unit or associativity law.

## Explanations

### L1 human

A structure map $F A \to A$ says how to combine data of the "shape" $F$ into a single element of $A$.
For $F X = 1 \sqcup X$, an $F$-algebra is a set with a chosen element and a function from the set to itself.
The natural numbers with $0$ and $n \mapsto n + 1$ are such an algebra.
For $F X = 1 \sqcup X \times X$, an $F$-algebra is a set with a constant and a binary operation, which is the signature of monoids.
</document>

<document source="https://tamnd.github.io/math2/node/4RDG2SJ70D.md" tag="4RDG2SJ70D" kind="definition">
# Hankel functions

## Definition

Let $\nu \in \C$, and let $J_\nu$ and $Y_\nu$ be the [[Bessel functions of the first kind|CZCMRDWDYW]] and of the [[second kind|GRSNDJG363]] on $\C \setminus (-\infty, 0]$.
The *Hankel functions* of order $\nu$ are
$$H^{(1)}_\nu(z) = J_\nu(z) + i\, Y_\nu(z), \qquad H^{(2)}_\nu(z) = J_\nu(z) - i\, Y_\nu(z).$$

## Remarks

Both functions solve Bessel's equation $z^2 y'' + z y' + (z^2 - \nu^2) y = 0$, since they are linear combinations of $J_\nu$ and $Y_\nu$.
They are independent: their Wronskian is $W(H^{(1)}_\nu, H^{(2)}_\nu)(z) = -\frac{4i}{\pi z}$.
Example: for $\nu = \frac{1}{2}$ and $x > 0$, $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$ and $Y_{1/2}(x) = -\sqrt{\frac{2}{\pi x}} \cos x$, so $H^{(1)}_{1/2}(x) = -i \sqrt{\frac{2}{\pi x}}\, e^{ix}$.
For real $\nu$ and $x > 0$, $H^{(2)}_\nu(x)$ is the complex conjugate of $H^{(1)}_\nu(x)$.
A non-example: $J_\nu + Y_\nu$ also solves Bessel's equation, but it is not a Hankel function, because the coefficient of $Y_\nu$ is not $\pm i$.
They are also called Bessel functions of the third kind.
</document>

<document source="https://tamnd.github.io/math2/node/4READX002Z.md" tag="4READX002Z" kind="statement">
# Stein factorization

## Statement

Let $Y$ be a locally [[Noetherian scheme|ND6XEY68P4]] and let $f \colon X \to Y$ be a [[proper morphism|9537RKBZPP]].
Then $f = g \circ f'$ for [[morphisms|JXRKH6P1Z4]] $f' \colon X \to Y'$ and $g \colon Y' \to Y$ with the following properties.

1. $g$ is a [[finite morphism|N9C4X7XKE4]].
2. $f'$ is proper, and the map $\mathcal{O}_{Y'} \to f'_* \mathcal{O}_X$ is an isomorphism.
3. Every fibre of $f'$ is nonempty and connected.

One may take $Y'$ to be the relative spectrum over $Y$ of the sheaf of $\mathcal{O}_Y$-algebras $f_* \mathcal{O}_X$.

## Remarks

The sheaf $f_* \mathcal{O}_X$ is coherent because $f$ is proper, so $g$ is finite.
Clause 3 follows from clause 2 by Zariski's connectedness theorem, which in turn rests on the theorem on formal functions.
For example, if $f$ is finite, then $f$ is affine and $X$ is the relative spectrum of $f_* \mathcal{O}_X$.
So $Y' = X$, $f'$ is the identity and $g = f$.
At the other extreme, for $X$ proper and connected over $k$ with $\mathcal{O}(X) = k$, the map $X \to \operatorname{Spec} k$ already has $Y' = Y$.
The properness hypothesis is necessary: for a non-proper $f$, the sheaf $f_* \mathcal{O}_X$ need not be coherent.
For $\mathbb{A}^1_k \to \operatorname{Spec} k$, the algebra $f_* \mathcal{O}_X = k[t]$ is not finite over $k$.

## Explanations

### L1 human

Every proper map splits into two simpler maps.
The first has connected fibres, so it only collapses connected pieces.
The second is finite, so each point of the target has finitely many preimages.
One can picture it as first squashing each connected part of each fibre to a point, and then mapping the result finitely onto the target.

### L2 human

The Stein factorization reduces many questions about proper morphisms to two cases: finite morphisms and morphisms with connected fibres.
It is used in the study of contractions in birational geometry.
The contractions of the minimal model program are by definition proper morphisms $f$ with $f_* \mathcal{O}_X = \mathcal{O}_Y$, exactly the maps $f'$ of the factorization.
The name comes from the analogous result of Karl Stein for proper holomorphic maps of complex spaces.
</document>

<document source="https://tamnd.github.io/math2/node/4RF117CH63.md" tag="4RF117CH63" kind="example">
# Steiner quasigroup

## Example

Let $S$ be a set with a Steiner triple system on it, that is, a [[Steiner system|7BPSC4T4M3]] $S(2, 3, n)$.
Its blocks are $3$-element subsets of $S$, and every two distinct points lie in exactly one block.
The *Steiner quasigroup* of this system is $S$ with the binary operation $\cdot$ given by:

1. $x \cdot x = x$ for every $x \in S$;
2. $x \cdot y = z$ for $x \ne y$, where $\set{x, y, z}$ is the unique block that contains $x$ and $y$.

This operation is [[commutative|XAC78GN2YJ]] and [[idempotent|9FS1JFZSX4]], and it satisfies $x \cdot (x \cdot y) = y$ for all $x, y$.
For $x \ne y$ the last law holds because $x \cdot y = z$ and $x \cdot z = y$ come from the same block.

Check that $S$ is a quasigroup, that is, every equation $a \cdot y = b$ and $y \cdot a = b$ has exactly one solution $y$.
The law $x \cdot (x \cdot y) = y$ says that the map $y \mapsto a \cdot y$ is its own inverse, so it is a bijection.
Hence the equation $a \cdot y = b$ has the unique solution $y = a \cdot b$.
By commutativity the equation $y \cdot a = b$ has the same unique solution.

## Remarks

The smallest case is $S = \set{0, 1, 2}$ with the single block $\set{0, 1, 2}$.
There the operation is $x \cdot y = -x - y \bmod 3$.
For instance $1 \cdot 2 = -3 \equiv 0$, which is the third point of the block, and $1 \cdot 1 = -2 \equiv 1$.
Conversely, a commutative idempotent quasigroup that satisfies $x \cdot (x \cdot y) = y$ comes from a Steiner triple system, whose blocks are the sets $\set{x, y, x \cdot y}$ with $x \ne y$.
The multiplication table of a Steiner quasigroup is a symmetric [[Latin square|CDVRMVED2V]] with entry $x$ in row $x$ and column $x$.

## Explanations

### L1 human

A Steiner triple system groups points into triangles so that any two points share exactly one triangle.
The Steiner quasigroup turns this into a multiplication: the product of two points is the third corner of their triangle.
The product of a point with itself is the point.
Multiplying twice by the same point brings you back, which makes every equation $a \cdot y = b$ solvable in exactly one way.
On the three points $0, 1, 2$ the product of two different points is the remaining one.
</document>

<document source="https://tamnd.github.io/math2/node/4RG1RB1MXH.md" tag="4RG1RB1MXH" kind="definition">
# Biproduct

## Definition

Let $\mathcal{C}$ be a category with [[zero morphisms|YF9F9EG6M3]], and let $X_1, X_2$ be objects of $\mathcal{C}$.
A *biproduct* of $X_1$ and $X_2$ is an object $X$ with morphisms $p_k \colon X \to X_k$ and $i_k \colon X_k \to X$ for $k = 1, 2$ such that:

1. $p_k \circ i_k = \id_{X_k}$ for $k = 1, 2$, and $p_l \circ i_k = 0$ for $k \ne l$;
2. $(X, p_1, p_2)$ is a [[product|S5744QZV4N]] of $X_1$ and $X_2$;
3. $(X, i_1, i_2)$ is a [[coproduct|4FJ0S71P58]] of $X_1$ and $X_2$.

The biproduct is written $X_1 \oplus X_2$; biproducts of finite families are defined in the same way.

## Remarks

In a [[preadditive category|SC6FD5F4J0]], an object $X$ with maps $p_k$, $i_k$ satisfying clause 1 and $i_1 \circ p_1 + i_2 \circ p_2 = \id_X$ is a biproduct, and every biproduct has this form.
In particular, in a preadditive category every binary product, and every binary coproduct, is a biproduct.

Example: for [[abelian groups|082GHNPJ53]] $A$ and $B$, the direct sum $A \oplus B$ with $p_1(a, b) = a$, $p_2(a, b) = b$, $i_1(a) = (a, 0)$, $i_2(b) = (0, b)$ is a biproduct.
Clause 1 is immediate, and $i_1 p_1 + i_2 p_2$ sends $(a, b)$ to $(a, 0) + (0, b) = (a, b)$.

Non-example: in the category of groups, with the trivial group as zero object, the product $\Z \times \Z$ with its inclusions satisfies clauses 1 and 2 but not clause 3.
If it were a coproduct, the maps $\Z \to F_2$ onto the two generators $a, b$ of the free group $F_2$ would extend to one homomorphism $\Z \times \Z \to F_2$.
Its image would be abelian and contain $a$ and $b$, but $ab \ne ba$ in $F_2$.
The coproduct in groups is the [[free product|TM7QS7P7X9]] instead.
</document>

<document source="https://tamnd.github.io/math2/node/4RH1D40WG1.md" tag="4RH1D40WG1" kind="concept">
# Buchberger's algorithm

## Summary

Buchberger's algorithm turns a finite set of generators of an ideal of a polynomial ring into a [[Gröbner basis|C83KPT47R0]] of the same ideal.
It repeatedly forms S-polynomials of pairs of elements, divides them by the current set with the [[multivariate division algorithm|T8EN3CAEQF]], and adds the nonzero remainders.
It was introduced by Bruno Buchberger, who also named Gröbner bases after his advisor Wolfgang Gröbner.
It plays the role for polynomial systems that Gaussian elimination plays for linear systems and the Euclidean algorithm plays for polynomials in one variable.

## Explanations

### L1 human

Two polynomials can combine so that their leading terms cancel, and what is left may have a new leading term that the generators do not see.
The S-polynomial of two generators is the simplest such combination.
Buchberger's algorithm computes all these combinations, reduces them by the current generators, and adds whatever does not reduce to zero.
Each addition makes the ideal of leading terms bigger, and this cannot go on forever, so the algorithm stops.
At the end, all S-polynomials reduce to zero, and this is exactly the test for a Gröbner basis.
</document>

<document source="https://tamnd.github.io/math2/node/4RJSN0QARD.md" tag="4RJSN0QARD" kind="concept">
# Power of a test

## Summary

The power of a [[hypothesis test|087P03NEND]] at a parameter value is the probability that the test rejects the null hypothesis.
Here the data are assumed to come from that parameter value.
Under the alternative it equals one minus the probability of a [[type II error|03N8C8ZR4B]].
Power is the main criterion to compare tests of the same level, and it guides the choice of a sample size.

## Explanations

### L1 human

Power measures how good a test is at detecting an effect that is really there.
A test with power $0.8$ finds a true effect of the given size in $80\%$ of repeated experiments.
For a sensible test, such as a one-sided z-test, the power grows with the sample size and with the size of the effect.
For example, a test for a biased coin with $10$ tosses rarely detects a probability of heads of $0.55$, while $10000$ tosses detect it almost always.
</document>

<document source="https://tamnd.github.io/math2/node/4RK8T21D27.md" tag="4RK8T21D27" kind="definition">
# Fiber bundle

## Definition

Let $E$, $B$ and $F$ be [[topological spaces|0CQHFPGJ45]], and let $\pi \colon E \to B$ be a [[continuous|0X3WGX95FE]] surjection.
The quadruple $(E, B, \pi, F)$ is a *fiber bundle* if the following holds.
Every point $b \in B$ has an open neighbourhood $U$ and a [[homeomorphism|0QT00PB5G2]]
$$\varphi \colon \pi^{-1}(U) \to U \times F$$
such that $\operatorname{pr}_1 \circ \varphi = \pi$ on $\pi^{-1}(U)$, where $\operatorname{pr}_1 \colon U \times F \to U$ is the first projection.
Such a pair $(U, \varphi)$ is a *local trivialization*.
The spaces $E$, $B$ and $F$ are the total space, the base and the fiber, and $\pi$ is the projection.

## Remarks

For any spaces $B$ and $F$, the product $B \times F$ with $\pi = \operatorname{pr}_1$ is a fiber bundle, the trivial bundle; one trivialization over $U = B$ suffices.
The Möbius band is a fiber bundle over the circle with fiber $[0, 1]$ that is not trivial.

Non-example: let $E = \set{(x, y) \in \R^2 : xy = 0}$ and $\pi(x, y) = x$, so $\pi \colon E \to \R$.
The preimage of $0$ is a line, while the preimage of each $x \neq 0$ is a single point.
A local trivialization around $0$ would make all preimages near $0$ homeomorphic to the same $F$, which is impossible.

Each fiber $\pi^{-1}(b)$ is homeomorphic to $F$.
A *smooth fiber bundle* uses smooth manifolds, a smooth submersion $\pi$ and diffeomorphisms $\varphi$.
The notation $F \to E \to B$ is common.

## Explanations

### L1 human

A fiber bundle is a map onto a base space that looks like the projection of a product over small open sets of the base.
</document>

<document source="https://tamnd.github.io/math2/node/4RPERPNBK1.md" tag="4RPERPNBK1" kind="definition">
# Eckmann–Hilton duality

## Definition

*Eckmann–Hilton duality* is an informal correspondence between notions of homotopy theory of [[pointed spaces|T9DG8DDE75]].
It reverses the direction of all maps and exchanges the members of each of the following pairs:

1. maps out of the cylinder $X \times [0, 1]$, and maps into the path space $Y^{[0, 1]}$, as two descriptions of a homotopy;
2. the reduced [[suspension|Q7RFNJH76Q]] $\Sigma X$, and the [[loop space|2JFF5F3EGF]] $\Omega Y$;
3. [[cofibrations|W5WS5EY8ZM]], defined by extending homotopies, and fibrations, defined by lifting homotopies;
4. the [[wedge sum|5JSWM1HBX2]] $X \vee Y$, and the product $X \times Y$;
5. mapping cones and cofibre sequences, and homotopy fibres and fibre sequences;
6. the homotopy groups $\pi_n(Y) = [S^n, Y]_*$, and the cohomology groups $H^n(X; G) \cong [X, K(G, n)]$ of a CW complex $X$, with $K(G, n)$ an Eilenberg–MacLane space.

## Remarks

The pair in clause 2 is linked by the natural bijection $[\Sigma X, Y]_* \cong [X, \Omega Y]_*$.
Example: the homotopy extension property of a cofibration corresponds under clause 3 to the homotopy lifting property of a fibration.
Likewise the cofibre sequence of a map corresponds to its fibre sequence.
The correspondence is not a theorem: the dual of a true statement need not be true, and each dual statement needs its own proof.
The duality was named after Beno Eckmann and Peter Hilton.
</document>

<document source="https://tamnd.github.io/math2/node/4RPGSCQVR9.md" tag="4RPGSCQVR9" kind="definition">
# Minkowski content

## Definition

Let $A \subseteq \R^n$, and let $\mathcal{L}^n$ be [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$.
For $r > 0$, let $A_r = \set{x \in \R^n : \abs{x - a} < r \text{ for some } a \in A}$.
Let $0 \le m \le n$ be an integer, and let $\alpha(k)$ be the volume of the unit ball of $\R^k$, with $\alpha(0) = 1$.
The *upper* and *lower $m$-dimensional Minkowski contents* of $A$ are
$$\mathcal{M}^{*m}(A) = \limsup_{r \to 0^+} \frac{\mathcal{L}^n(A_r)}{\alpha(n - m) \, r^{n - m}}, \qquad \mathcal{M}^m_*(A) = \liminf_{r \to 0^+} \frac{\mathcal{L}^n(A_r)}{\alpha(n - m) \, r^{n - m}}.$$
If they are equal, their common value $\mathcal{M}^m(A)$ is the *$m$-dimensional Minkowski content* of $A$.

## Remarks

The [[limit superior|0T3RZQXDYD]] and limit inferior always exist in $[0, \infty]$.
The same formulas with a real exponent $s$ in place of $m$ are also used, with a suitable constant.

Example: let $A \subseteq \R^2$ be a segment of length $L$ and $m = 1$.
Then $A_r$ is a rectangle of area $2 r L$ with two half-discs of radius $r$, so $\mathcal{L}^2(A_r) = 2 r L + \pi r^2$.
Since $\alpha(1) = 2$, the ratio is $L + \pi r / 2 \to L$, so $\mathcal{M}^1(A) = L$.

Minkowski content does not ignore countable sets.
For $A = \Q \cap [0, 1]$ in $\R$ with $m = 1$, one has $A_r = (-r, 1 + r)$, so $\mathcal{M}^1(A) = 1$, although $A$ has Lebesgue measure $0$.
Thus Minkowski content is not countably additive.

For a closed set $A \subseteq \R^n$ that is the image of a bounded subset of $\R^m$ under a Lipschitz map, $\mathcal{M}^m(A)$ exists and equals its normalized $m$-dimensional [[Hausdorff measure|Q1TGZK8RN6]].

## Explanations

### L1 human

Thicken the set a little, measure the volume of the thickening, and divide by the volume one would expect from an $m$-dimensional set.
The limit of this ratio is the $m$-dimensional size of the set.
</document>

<document source="https://tamnd.github.io/math2/node/4RXXE3BB6W.md" tag="4RXXE3BB6W" kind="statement">
# Harish-Chandra embedding

## Statement

Let $G$ be a connected noncompact simple real Lie group, contained in its simply connected [[complexification|DRYYK2WX0M]] $G_\C$.
Let $K \subset G$ be a maximal compact subgroup such that $X = G/K$ is a [[Hermitian symmetric space|NT4797Z9H8]].
Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the Cartan decomposition, and let $\mathfrak{p}_\C = \mathfrak{p}^+ \oplus \mathfrak{p}^-$ be the splitting into the $\pm i$ eigenspaces of the complex structure.
Put $P^\pm = \exp \mathfrak{p}^\pm$, and let $K_\C$ be the complexification of $K$ in $G_\C$.
Then:

1. Every $g \in G$ can be written uniquely as $g = p^+ k p^-$ with $p^\pm \in P^\pm$ and $k \in K_\C$.
2. The map $\zeta \colon X \to \mathfrak{p}^+$, $gK \mapsto \log p^+$, is a holomorphic embedding onto a bounded domain in the complex vector space $\mathfrak{p}^+ \cong \C^n$, where $n = \dim_\C X$.

## Remarks

The image is a [[bounded symmetric domain|Y7Q2SNKG0C]], the Harish-Chandra realization of $X$, and it is convex and circled.
The map extends to the Borel embedding of $X$ as an open subset of the compact dual $G_\C / K_\C P^-$.
In this picture $\zeta$ is the restriction of an affine chart $P^+ \cong \mathfrak{p}^+$ of the compact dual.
The same holds for semisimple $G$ by taking products.

Example: for $G = SU(1, 1)$ and $K = U(1)$ the Harish-Chandra realization of $G/K$ is the unit disc.
For $G = Sp(2n, \R)$ and $K = U(n)$, the space $G/K$ is the [[Siegel upper half-space|5PTYE97PF6]] $\mathbb{H}_n$, and the realization is the domain $\set{W : W^T = W,\ I - W^* W > 0}$, reached by the Cayley transform $Z \mapsto (Z - iI)(Z + iI)^{-1}$.

## Explanations

### L1 human

A Hermitian symmetric space of noncompact type is first defined abstractly, as a quotient of a Lie group.
The theorem says that it can always be drawn as a bounded region in $\C^n$.
So abstract symmetric spaces and concrete bounded symmetric domains are the same objects.

### L2 human

Together with the fact that bounded symmetric domains with the Bergman metric are Hermitian symmetric, this gives an equivalence between the two notions.
The embedding is the tool behind holomorphic discrete series representations and the analysis of automorphic forms on $G/K$.
It also explains why $X$ sits as an open subset inside its compact dual, as the disc sits inside the Riemann sphere.
</document>

<document source="https://tamnd.github.io/math2/node/4S1HC0N5E8.md" tag="4S1HC0N5E8" kind="definition">
# Perfect set

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A subset $P \subseteq X$ is *perfect* if $P$ is [[closed|0ZCK8DQYN1]] in $X$ and $P$ has no [[isolated points|0A1YXSKB48]].
Equivalently, $P = P'$, where $P'$ is the [[derived set|8ZAPAPYJFG]] of $P$.

## Remarks

The two forms agree because $P$ is closed if and only if $P' \subseteq P$, and $P$ has no isolated points if and only if $P \subseteq P'$.

Example: $[0, 1]$ is closed in $\R$, and every $x \in [0, 1]$ is a limit of other points of $[0, 1]$, so $[0, 1]$ is perfect.
The [[Cantor set|0BXNDZFQME]] is also perfect.

Non-example: $P = \set{0} \cup \set{1/n : n \ge 1}$ is closed in $\R$, but $1$ is an isolated point of $P$, so $P$ is not perfect.

Non-example: $\Q$ has no isolated points, but it is not closed in $\R$, so it is not perfect.

Every nonempty perfect subset of $\R$ is uncountable.
</document>

<document source="https://tamnd.github.io/math2/node/4S4Z7SQ47G.md" tag="4S4Z7SQ47G" kind="concept">
# Conformal geometric algebra

## Summary

Conformal geometric algebra is the [[geometric algebra|N8H460W3E2]] of $\R^{n+1,1}$, used as a model of Euclidean space $\R^n$.
In it, points, spheres and planes are all represented by vectors.
A point $x \in \R^n$ is embedded as a null vector $X$, and the inner product of two embedded points is $-\tfrac{1}{2}$ times their squared distance.
Spheres and planes are the vectors orthogonal to the points that lie on them.
All Möbius transformations of $\R^n \cup \set{\infty}$ act by conjugation with elements of the algebra.
It is used in computer graphics and robotics, where rigid motions, intersections and tangencies become algebraic operations.

## Explanations

### L1 human

The trick is to add two extra dimensions to space, one that squares to $+1$ and one that squares to $-1$.
In this bigger space a point becomes a vector of length zero, and the distance between two points is read from their inner product.
A sphere is then also a single vector, and asking whether a point lies on it is just asking whether an inner product is zero.
Rotations, translations, scalings and inversions in spheres all become the same kind of algebraic move.
</document>

<document source="https://tamnd.github.io/math2/node/4S50G4PS26.md" tag="4S50G4PS26" kind="statement">
# Oppenheim conjecture

## Statement

Let $n \ge 3$ and let $Q$ be a real [[quadratic form|0RBQS1JJCR]] in $n$ variables.
Assume that $Q$ is nondegenerate, that $Q$ is [[indefinite|01RBSSGMY5]], and that $Q$ is not a real multiple of a form with rational coefficients.
Then the set of values $Q(\Z^n) = \set{Q(x) : x \in \Z^n}$ is [[dense|0BSJSX7ME2]] in $\R$.
In particular, for every $\varepsilon > 0$ there is $x \in \Z^n \setminus \set{0}$ with $0 < \abs{Q(x)} < \varepsilon$.

## Remarks

Margulis proved the statement in 1987, so the name "conjecture" is historical.
Following an idea of Raghunathan, it reduces to the case $n = 3$ and to the orbits of the group $\mathrm{SO}(Q)$ on the space $\mathrm{SL}_3(\R)/\mathrm{SL}_3(\Z)$ of unimodular lattices in $\R^3$.
The heart of the proof is Margulis's theorem that a relatively compact orbit of $\mathrm{SO}(2,1)$ in this space is compact, while the irrationality of $Q$ rules out a compact orbit.
[[Ratner's theorems|3MXR03096V]] later gave a general picture of such orbit closures, as predicted by [[Raghunathan's conjecture|ME4ZSRAT5Z]].
An example: $Q(x, y, z) = x^2 + y^2 - \sqrt{2}\, z^2$ satisfies the hypotheses, so its values at integer points are dense in $\R$.
Each hypothesis is needed.
If $Q$ is definite, the values on $\Z^n$ form a discrete set.
If $Q$ is a multiple $c Q_0$ of a rational form $Q_0$ with denominators dividing $d$, its values lie in the discrete set $(c/d)\Z$.
For $n = 2$ the form $x^2 - \varphi^2 y^2 = (x - \varphi y)(x + \varphi y)$, with $\varphi$ the golden ratio, is irrational and indefinite.
But $\abs{x - \varphi y} \ge c/\abs{y}$ for a constant $c > 0$ and all integers $x$ and $y \ne 0$.
This shows that the nonzero values of the form at integer points stay away from $0$.

## Explanations

### L1 human

A quadratic form is a polynomial like $x^2 + y^2 - \sqrt{2}\, z^2$.
Feed it only whole numbers and look at the outputs.
The theorem says that, with at least three variables, mixed signs and an irrational coefficient that cannot be scaled away, the outputs come arbitrarily close to every real number.
This is a statement about integers that was proved with the dynamics of group actions.

### L2 human

The theorem was a turning point for the use of homogeneous dynamics in number theory.
The key move is to replace integer points by the lattice $\Z^3$.
Then one moves this lattice by elements of the orthogonal group of $Q$, which do not change the values of $Q$.
Then density of values becomes a statement about the closure of an orbit of $\mathrm{SO}(2,1)$ in the space of lattices.
This viewpoint led to Ratner's classification of orbit closures for unipotent flows and to quantitative versions counting integer points with $a < Q(x) < b$.
</document>

<document source="https://tamnd.github.io/math2/node/4S6DXRPZ4Z.md" tag="4S6DXRPZ4Z" kind="definition">
# Essential submodule

## Definition

Let $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].
A [[submodule|0RGZ10DFPG]] $N$ of $M$ is *essential* in $M$ if $N \cap L \ne 0$ for every nonzero submodule $L$ of $M$.
In that case $M$ is called an *essential extension* of $N$.

## Remarks

Equivalently, for every nonzero $m \in M$ there is $r \in R$ with $rm \in N$ and $rm \ne 0$.
Notation: $N \subseteq_e M$ or $N \trianglelefteq_e M$, depending on the author.
Example: for $n \ne 0$, the submodule $n\Z$ is essential in the $\Z$-module $\Z$.
The check: a nonzero submodule $a\Z$ of $\Z$ contains $na \ne 0$, which lies in $n\Z$.
Non-example: $\Z \oplus 0$ is not essential in $\Z \oplus \Z$, since the nonzero submodule $0 \oplus \Z$ meets it only in $0$.
A nonzero direct summand $N \ne M$ is never essential, since its complement meets it in $0$.
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<document source="https://tamnd.github.io/math2/node/4SBZ4EMPBZ.md" tag="4SBZ4EMPBZ" kind="statement">
# Maschke's theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $F$ be a field whose [[characteristic|0F6ES3BRYP]] does not divide the order $\abs{G}$.
Let $V$ be a [[representation|WYGRXCG2RN]] of $G$ over $F$, that is, a module over the [[group algebra|MNP2HBWB5Q]] $F[G]$.
Then every $G$-invariant subspace $W$ of $V$ has a $G$-invariant complement $W'$ with $V = W \oplus W'$.
Equivalently, the group algebra $F[G]$ is a [[semisimple ring|NY25HXA478]].

## Remarks

Take any linear projection $p \colon V \to W$ onto $W$ and average it over the group:
$$\tilde{p}(v) = \frac{1}{\abs{G}} \sum_{g \in G} g \, p(g^{-1} v).$$
The division by $\abs{G}$ is allowed by the hypothesis on the characteristic, $\tilde{p}$ is $G$-equivariant and is still a projection onto $W$, and $W' = \ker \tilde{p}$ is the required complement.
For example, every finite-dimensional complex representation of a finite group is a direct sum of irreducible representations.
The hypothesis is necessary: let $G = \set{1, g}$ have order $2$ and $F = \F_2$, acting on $F^2$ by swapping coordinates.
The line spanned by $(1, 1)$ is invariant, but it is the only invariant line, so it has no invariant complement.

## Explanations

### L1 human

Let a finite group act linearly on a space, and suppose one can divide by the size of the group.
Then an invariant piece can always be split off cleanly from the rest.
So each representation breaks into irreducible building blocks.
This is the reason why representations of finite groups over the complex numbers are so well behaved.

### L2 human

By the Artin–Wedderburn theorem, Maschke's theorem gives $\C[G] \cong M_{n_1}(\C) \times \cdots \times M_{n_k}(\C)$, which leads to $\abs{G} = \sum n_i^2$ and to character theory.
The converse also holds: if the characteristic of $F$ divides $\abs{G}$, then $F[G]$ is not semisimple.
Indeed, $N = \sum_{g \in G} g$ spans a nonzero two-sided ideal with $N^2 = \abs{G} N = 0$.
A semisimple ring has no such nilpotent ideal.
That case is the subject of modular representation theory.
The averaging argument also works for compact groups with the Haar integral in place of the sum.
</document>

<document source="https://tamnd.github.io/math2/node/4SCS55K5MA.md" tag="4SCS55K5MA" kind="example">
# The integers modulo the even integers

## Example

Let $G = (\mathbb{Z}, +)$ be the [[integers|0C4JS9RF5M]] with addition, and let $N = 2\mathbb{Z}$ be the even integers. $N$ is normal, because $G$ is abelian.
$N$ has two cosets: $N$, the even integers, and $1 + N$, the odd integers.
The sum of two odd integers is even, so $(1 + N) + (1 + N) = N$.
Thus the [[quotient group|0YYWCNTPWR]] $\mathbb{Z}/2\mathbb{Z}$ has two elements, with identity $N$. It is the [[integers modulo $2$|03VFZ2WETT]].
</document>

<document source="https://tamnd.github.io/math2/node/4SH2ZB984E.md" tag="4SH2ZB984E" kind="concept">
# Global attractor

## Summary

The global attractor of a dissipative dynamical system on a metric space is a compact invariant set.
It attracts every bounded set of initial data.
It contains all the long-time dynamics: all equilibria, periodic orbits and bounded complete trajectories.
Many dissipative PDEs have a global attractor of finite fractal dimension, although the phase space is infinite-dimensional.
Examples are reaction–diffusion equations and the two-dimensional [[Navier–Stokes equations|18KXZHVF4M]].

## Explanations

### L1 human

Start a dissipative system from any state and wait.
Friction or diffusion wears down most of the details, and the state comes closer and closer to a certain set, the global attractor.
For $u' = u - u^3$ on the real line, every solution tends to $-1$, $0$ or $1$.
The global attractor is the interval $[-1, 1]$, made of the three equilibria and the orbits joining them.
For a dissipative PDE, the attractor is often a finite-dimensional object living inside an infinite-dimensional space.
</document>

<document source="https://tamnd.github.io/math2/node/4SNJWQBAHF.md" tag="4SNJWQBAHF" kind="definition">
# Almost convergence

## Definition

Let $x = (x_n)_{n \ge 1}$ be a [[bounded sequence|0A8JNBGT2Z]] of real numbers and let $L$ be a real number.
The sequence $x$ is *almost convergent* to $L$ if $\varphi(x) = L$ for every [[Banach limit|SYF2ND6J97]] $\varphi$.

## Remarks

Example: every [[convergent sequence|0W4CN9E2W5]] with limit $L$ is almost convergent to $L$, because every Banach limit agrees with the ordinary limit on convergent sequences.

Example: $x = (1, 0, 1, 0, \dots)$ is almost convergent to $1/2$.
Let $S$ be the shift and $\mathbf{1} = (1, 1, \dots)$.
Then $x + Sx = \mathbf{1}$, so for every Banach limit $\varphi$ one has $\varphi(x) + \varphi(Sx) = 1$.
Since $\varphi(Sx) = \varphi(x)$, this gives $\varphi(x) = 1/2$.

Non-example: let $x_n = 1$ if $4^k \le n < 2 \cdot 4^k$ for some integer $k \ge 0$, and $x_n = 0$ otherwise.
This sequence has longer and longer blocks of ones and of zeros, and different Banach limits give it different values.
This follows from Lorentz's characterization, since averages over long blocks of consecutive terms can be $0$ or $1$.

The limit $L$ is unique when it exists, because Banach limits exist.
</document>

<document source="https://tamnd.github.io/math2/node/4SPC17FPXE.md" tag="4SPC17FPXE" kind="concept">
# Jones polynomial

## Summary

The Jones polynomial is an invariant of oriented [[links|1A5WRCNTM3]] with values in Laurent polynomials in $t^{1/2}$.
Vaughan Jones discovered it in 1984 through operator algebras.
It can be computed from any diagram by a skein relation or by the [[Kauffman bracket|JYKVYS4N32]].
Unlike the [[Alexander polynomial|JKA25KDPQP]], it distinguishes many knots from their mirror images, such as the two trefoils.

## Explanations

### L1 human

The Jones polynomial turns a knot drawing into a polynomial, and different drawings of the same knot give the same answer.
You compute it by undoing crossings one at a time with a fixed rule, the skein relation.
The plain circle gets the value $1$.
One trefoil gets $t + t^3 - t^4$ and its mirror image gets $t^{-1} + t^{-3} - t^{-4}$.
So the left-handed and right-handed trefoils are different knots.
It is still unknown whether a nontrivial knot can have Jones polynomial $1$.
</document>

<document source="https://tamnd.github.io/math2/node/4SQHHRPKMV.md" tag="4SQHHRPKMV" kind="concept">
# Recurrent point

## Summary

A recurrent point of a continuous map is a point whose orbit returns arbitrarily close to it again and again.
Formally, the point lies in its own $\omega$-[[limit set|YF4XH2BEG5]].
Recurrence is the topological form of the idea that a system comes back near its initial state.
Every point of a [[minimal set|CRGZZKQ7Z9]] is recurrent, so on a nonempty compact metric space every continuous map has recurrent points.

## Explanations

### L1 human

Follow a point under repeated application of a map.
The point is recurrent if, however small a window you draw around it, the orbit enters that window at infinitely many later times.
A periodic point is recurrent, since it comes back exactly.
For a rotation of the circle by an irrational angle, every point is recurrent without ever coming back exactly.
For the map $x \mapsto x/2$ on the real line, the point $1$ is not recurrent: its orbit $1, 1/2, 1/4, \ldots$ runs off to $0$.
</document>

<document source="https://tamnd.github.io/math2/node/4SQYNP0B4Y.md" tag="4SQYNP0B4Y" kind="concept">
# Nilpotency class

## Summary

The nilpotency class of a nilpotent group is the length of its shortest central series.
It measures how far the group is from being abelian: the trivial group has class $0$ and a nontrivial abelian group has class $1$.
It equals the number of steps that the lower central series, or the upper central series, needs to reach its end.

## Explanations

### L1 human

In an abelian group, all elements commute.
In a nilpotent group, the failure to commute dies out after a fixed number of steps: commutators of commutators of commutators, and so on, become trivial.
The class is that number of steps.
For example, in the quaternion group $Q_8$ every commutator is $\pm 1$, and $\pm 1$ commutes with everything, so $Q_8$ has class $2$.
</document>

<document source="https://tamnd.github.io/math2/node/4SSJ589G6T.md" tag="4SSJ589G6T" kind="statement">
# Portmanteau theorem

## Statement

Let $X$ be a [[metric space|0VKD1TP7F3]] and let $\mu, \mu_1, \mu_2, \dots$ be Borel [[probability measures|059TJN0NZ1]] on $X$.
For a set $A \subseteq X$, let $\partial A$ be its boundary.
The following conditions are equivalent.

1. $\mu_n$ [[converges weakly|MBW9JCGZC1]] to $\mu$: $\int f \, d\mu_n \to \int f \, d\mu$ for every bounded continuous $f \colon X \to \R$.
2. $\int f \, d\mu_n \to \int f \, d\mu$ for every bounded [[Lipschitz|0AW1E6YTJY]] function $f \colon X \to \R$.
3. $\limsup_{n \to \infty} \mu_n(F) \le \mu(F)$ for every [[closed set|0ZCK8DQYN1]] $F \subseteq X$.
4. $\liminf_{n \to \infty} \mu_n(G) \ge \mu(G)$ for every [[open set|02BH3MCBQB]] $G \subseteq X$.
5. $\mu_n(A) \to \mu(A)$ for every Borel set $A$ with $\mu(\partial A) = 0$.

## Remarks

Condition 1 implies 2 at once.
For 2 implies 3, the Lipschitz functions $f_k(x) = \max(0, 1 - k \, d(x, F))$ decrease to the indicator of $F$ as $k \to \infty$.
Conditions 3 and 4 are equivalent by taking complements.
Conditions 3 and 4 applied to the closure and the interior of $A$ give 5.
For 5 implies 1, write $\int f \, d\mu$ as an integral of $\mu(\set{f > t})$ over $t$; for all but countably many $t$ the set $\set{f > t}$ has a $\mu$-null boundary.

Example: $\delta_{1/n}$ converges weakly to $\delta_0$ on $\R$.
For the open set $G = (0, 1)$, one has $\delta_{1/n}(G) = 1$ for $n \ge 2$ and $\delta_0(G) = 0$, so the inequality in 4 can be strict.
For $A = \set{0}$, the boundary is $\set{0}$, which has $\delta_0$-measure $1$.
So condition 5 makes no claim, and indeed $\delta_{1/n}(A) = 0$ does not tend to $\delta_0(A) = 1$.

## Explanations

### L1 human

The theorem lists several equivalent ways to say that measures converge weakly.
In the limit, a closed set can gain mass and an open set can lose mass, but not the other way round.
Sets whose boundary carries no limit mass have converging measures.

### L2 human

One uses whichever form is easiest to check.
Condition 2 is handy because Lipschitz functions are easy to control, while condition 5 gives convergence of probabilities of concrete events.
For real random variables, condition 5 applied to half-lines gives convergence of distribution functions at continuity points.
The theorem is a standard tool together with Prokhorov's theorem and the continuous mapping theorem.
</document>

<document source="https://tamnd.github.io/math2/node/4ST57PN8K7.md" tag="4ST57PN8K7" kind="definition">
# Spheroidal wave function

## Definition

Let $m$ be a nonnegative integer and let $c \ge 0$.
The *prolate spheroidal wave equation* is the [[linear differential equation|0QQRP17Z7S]]
$$(1 - x^2) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + \left( \lambda - c^2 x^2 - \frac{m^2}{1 - x^2} \right) y = 0, \qquad -1 < x < 1,$$
with a real parameter $\lambda$.
Call $\lambda$ an eigenvalue if the equation has a nonzero solution that is bounded on $(-1, 1)$.
The eigenvalues form an increasing sequence $\lambda_{m, m}(c) < \lambda_{m, m + 1}(c) < \lambda_{m, m + 2}(c) < \cdots$.
For each eigenvalue, the bounded solution is unique up to a constant factor.
The *angular prolate spheroidal wave function* $S_{mn}(c, x)$, for $n \ge m$, is that bounded solution for $\lambda = \lambda_{mn}(c)$, with a chosen normalisation.
The *oblate* functions are defined in the same way with $c^2$ replaced by $-c^2$.

## Remarks

The eigenvalue problem is a singular [[Sturm–Liouville|7WV6B2502K]] problem on $(-1, 1)$, which explains the discrete increasing sequence of eigenvalues.
Example: for $c = 0$ the equation is the associated Legendre equation, $\lambda_{mn}(0) = n(n + 1)$, and $S_{mn}(0, x)$ is a multiple of the [[associated Legendre polynomial|VPJCSVRY6S]] $P_n^m(x)$.
For $c = 0$, $m = 0$ and $n = 0$: $y = 1$ is bounded and satisfies $-2x \cdot 0 + (0 - 0) \cdot 1 = 0$ with $\lambda = 0$.
Non-example: for $c = 0$, $m = 0$ and $\lambda = 0$, the function $y = \frac12 \log \frac{1 + x}{1 - x}$ also solves the equation, but it is unbounded near $x = \pm 1$, so it is not a spheroidal wave function.
The radial spheroidal wave functions solve the same equation on $x > 1$ and describe the dependence on the other coordinate.
Normalisations differ between authors; a common one makes $S_{mn}(c, x)$ tend to $P_n^m(x)$ as $c \to 0$.
</document>

<document source="https://tamnd.github.io/math2/node/4T1MNB8Z11.md" tag="4T1MNB8Z11" kind="definition">
# Quadrature rule

## Definition

Let $a < b$ be real numbers and let $n \ge 1$.
A *quadrature rule* on $[a, b]$ with $n$ nodes consists of distinct nodes $x_1, \dots, x_n \in [a, b]$ and real weights $w_1, \dots, w_n$.
It assigns to each [[continuous function|0X3WGX95FE]] $f \colon [a, b] \to \R$ the number
$$Q(f) = \sum_{i=1}^{n} w_i\, f(x_i),$$
which is used as an approximation of the [[definite integral|021WEM680W]] $I(f) = \int_a^b f(x)\, dx$.

## Remarks

A quadrature rule is a [[numerical integration|7HYF6YG0RS]] method, and $Q$ is a linear functional on the continuous functions.
More generally, a rule for a weight function $\omega \ge 0$ approximates $\int_a^b f(x)\, \omega(x)\, dx$ by the same kind of sum.
Example: the midpoint rule has $n = 1$, $x_1 = (a + b)/2$, $w_1 = b - a$; for $f = 1$ it gives $b - a = I(1)$.
Non-example: $f \mapsto (b - a) \max_{[a, b]} f$ is not a quadrature rule, because it is not of the form $\sum w_i f(x_i)$ with fixed nodes; it is not even linear.
A rule is called interpolatory if $Q(f)$ is the integral of the polynomial that interpolates $f$ at the nodes.
A rule with nonnegative weights is stable, since $\abs{Q(f)} \le \big(\sum_i w_i\big) \max \abs{f}$.
</document>

<document source="https://tamnd.github.io/math2/node/4T72ZP5XHF.md" tag="4T72ZP5XHF" kind="example">
# Mumford–Shah functional

## Example

Let $\Omega \subseteq \R^2$ be a bounded open set, the image domain.
Let $g \in L^\infty(\Omega)$ be a grey-level image, and let $\alpha, \beta > 0$ be parameters.
For a relatively [[closed set|0ZCK8DQYN1]] $K \subseteq \Omega$ and a function $u \in C^1(\Omega \setminus K)$, the *Mumford–Shah functional* is
$$E(u, K) = \int_{\Omega \setminus K} \abs{\nabla u}^2 \, dx + \alpha \int_{\Omega \setminus K} (u - g)^2 \, dx + \beta \, \mathcal{H}^1(K).$$
Here $\mathcal{H}^1$ is the one-dimensional [[Hausdorff measure|Q1TGZK8RN6]], the length of $K$.
A minimizing pair $(u, K)$ gives a smoothed image $u$ and a set of edges $K$.

Check on a simple image: let $D$ be a closed disc of radius $r$ in $\Omega$, and let $g = \mathbf{1}_D$.
The pair $u = g$, $K = \partial D$ has no gradient term and no fidelity term, so $E(u, K) = 2 \pi \beta r$.
The pair $u = 0$, $K = \emptyset$ has energy $\alpha \pi r^2$.
So for $\alpha > 2\beta/r$ the first pair is cheaper than the second, and for $\alpha < 2\beta/r$ the second is cheaper.
Large weights on fidelity favour keeping the edge, and small weights favour erasing it.

## Remarks

The functional was proposed by David Mumford and Jayant Shah for image segmentation.
It is a free discontinuity problem: the unknown set $K$ is where $u$ may jump.
Ennio De Giorgi, Michele Carriero and Antonio Leaci proved the existence of minimizers.
Their method first solves a weak form, where $u$ is a special function of bounded variation and $K$ is its jump set.
Luigi Ambrosio and Vincenzo Maria Tortorelli approximated it, in the sense of [[Γ-convergence|NW9MM370A5]], by elliptic functionals of two smooth functions, which is the basis of many numerical methods.

## Explanations

### L1 human

An image has smooth regions separated by sharp edges.
The Mumford–Shah functional looks for a clean image $u$ that stays close to the given image $g$.
The clean image must be smooth away from a set of edges, and the total length of the edges must be small.
Each wish has a weight, and the best compromise gives both a cleaned image and a drawing of its contours.
For a white disc on a black background, the answer depends on the weights: either the circle is kept as an edge, or the disc is wiped out.
</document>

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# Sobol sequence

## Description

A *Sobol sequence* is a sequence of points in $[0,1)^s$ built from binary digits.
For each coordinate $j = 1, \ldots, s$ one fixes direction numbers $v_{j,1}, v_{j,2}, \ldots \in (0,1)$, constructed as follows.

1. Choose a [[primitive polynomial over $\F_2$|HQ7KMZYPSQ]] $x^{d} + c_1 x^{d-1} + \cdots + c_{d-1} x + 1$ of degree $d = d_j$, with different polynomials for different $j$.
2. Choose odd integers $m_1, \ldots, m_d$ with $m_k < 2^k$.
3. For $k > d$ set $m_k = 2 c_1 m_{k-1} \oplus 2^2 c_2 m_{k-2} \oplus \cdots \oplus 2^{d-1} c_{d-1} m_{k-d+1} \oplus 2^d m_{k-d} \oplus m_{k-d}$, where $\oplus$ is the bitwise exclusive or of binary integers.
4. Set $v_{j,k} = m_k / 2^k$.

Write $n \ge 0$ in binary as $n = \sum_{k \ge 1} \beta_k 2^{k-1}$ with $\beta_k \in \set{0,1}$.
The $j$-th coordinate of the $n$-th point is
$$x_{n,j} = \beta_1 v_{j,1} \oplus \beta_2 v_{j,2} \oplus \cdots,$$
where $\oplus$ now acts digit by digit on binary expansions.
The first coordinate usually takes $v_{1,k} = 2^{-k}$, which gives the van der Corput sequence in base $2$.

## Remarks

Take $s = 2$, the van der Corput sequence in the first coordinate, and the polynomial $x + 1$ with $m_1 = 1$ in the second.
The recurrence gives $m_k = 2 m_{k-1} \oplus m_{k-1}$, so $m_1, m_2, m_3 = 1, 3, 5$ and $v_{2,k} = 1/2, 3/4, 5/8$.
The points for $n = 0, 1, 2, 3$ are $(0,0)$, $(1/2, 1/2)$, $(1/4, 3/4)$, $(3/4, 1/4)$; for $n = 3$ the second coordinate is $0.10_2 \oplus 0.11_2 = 0.01_2 = 1/4$.
Each of the four squares of side $1/2$ contains exactly one of these points.
So does each vertical strip of width $1/4$ and each horizontal strip of height $1/4$.
In general Sobol sequences are $(t,s)$-sequences in base $2$ in the sense of Niederreiter, and their star discrepancy is $O\big((\log N)^s / N\big)$.
So they are [[low-discrepancy sequences|XSPX1KW9AQ]], widely used in [[quasi-Monte Carlo methods|ABSRB2KXA6]].
The quality of the points in practice depends on the choice of the initial numbers $m_k$.
Because the construction uses only bit operations, the points are very fast to generate, especially in Gray code order.

## Explanations

### L1 human

A Sobol sequence is a list of points that fills a square or a cube very evenly.
Each coordinate is made by combining a few fixed binary fractions, chosen by the binary digits of the point's index.
The construction makes the first $2^m$ points, for each $m$, spread out evenly over grids of small boxes.
For example, the first four points in the plane put one point in each quarter of the unit square.
These points serve as sample points for computing averages of functions with high accuracy.
</document>

<document source="https://tamnd.github.io/math2/node/4TCFNSTXAD.md" tag="4TCFNSTXAD" kind="definition">
# Nef line bundle

## Definition

Let $X$ be a scheme that is [[proper|9537RKBZPP]] over a field $k$, and let $\mathcal{L}$ be an [[invertible sheaf|7JCF2TE3V5]] on $X$.
For a closed integral [[curve|XJM2MT4S3Y]] $C \subseteq X$, proper over $k$, let $\deg(\mathcal{L}|_C)$ be the degree of the restriction of $\mathcal{L}$ to $C$, also written $(\mathcal{L} \cdot C)$.
The sheaf $\mathcal{L}$ is *nef* if $(\mathcal{L} \cdot C) \ge 0$ for every closed integral curve $C \subseteq X$.

## Remarks

A Cartier divisor $D$ is nef when $\mathcal{O}_X(D)$ is nef, that is, $D \cdot C \ge 0$ for all curves $C$.

Example: every [[ample|RRK8P6TXG2]] $\mathcal{L}$ is nef.
Some power $\mathcal{L}^{\otimes m}$ is very ample, so $m \, (\mathcal{L} \cdot C)$ is the degree of $C$ in a projective embedding, which is positive.
The trivial bundle $\mathcal{O}_X$ is nef, with degree $0$ on every curve, and it is not ample when $\dim X \ge 1$ and $X$ is projective.

Non-example: on $\mathbb{P}^1_k$, $\mathcal{O}(-1)$ is not nef, since its degree on the curve $C = \mathbb{P}^1_k$ itself is $-1$.

On a projective scheme, Kleiman's criterion says that $\mathcal{L}$ is nef if and only if $(\mathcal{L}^{\dim V} \cdot V) \ge 0$ for every closed subvariety $V$.
It also says that the nef cone is the closure of the ample cone.
</document>

<document source="https://tamnd.github.io/math2/node/4THTYJ07RT.md" tag="4THTYJ07RT" kind="definition">
# Limit point compact space

## Definition

A [[topological space|0CQHFPGJ45]] $X$ is *limit point compact* when every infinite subset $A \subseteq X$ has a [[limit point|0E3X7TXS67]] in $X$.
Here $x \in X$ is a limit point of $A$ when every neighbourhood of $x$ contains a point of $A$ other than $x$.

## Remarks

Every [[compact|0TVJZYB6SH]] space is limit point compact, and so is every [[countably compact|S3N3CHZV89]] space.
For [[T1 spaces|0NW6CH38YR]], limit point compactness is equivalent to countable compactness.
For [[metric spaces|0VKD1TP7F3]], it is equivalent to compactness and to sequential compactness.
Other names in use are weakly countably compact and Fréchet compact.

Example: $[0, 1]$ is limit point compact.
An infinite subset contains a sequence of distinct points, which has a convergent subsequence by the Bolzano–Weierstrass theorem, and its limit is a limit point of the set.

Non-example: $\R$ is not limit point compact.
The infinite set $\Z$ has no limit point, since each $x \in \R$ has a neighbourhood $(x - 1/2, x + 1/2)$ that contains at most one integer.

Without the T1 condition, limit point compact need not imply countably compact.
Let $X = \N \times \set{0, 1}$, where $\N$ is discrete and $\set{0, 1}$ has the trivial topology.
Each nonempty subset has a limit point, since $(n, 1 - i)$ is a limit point of any set that contains $(n, i)$.
But the countable open cover by the sets $\set{n} \times \set{0, 1}$ has no finite subcover.
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<document source="https://tamnd.github.io/math2/node/4TKVXC9N46.md" tag="4TKVXC9N46" kind="definition">
# Kardar–Parisi–Zhang equation

## Definition

Let $\nu > 0$, $\lambda \in \R$ and $D > 0$ be constants.
Let $\xi$ be a Gaussian space-time white noise on $\R \times [0, \infty)$, that is, a centred Gaussian random distribution with formal covariance $\mathbb{E}[\xi(x, t)\xi(y, s)] = \delta(x - y)\delta(t - s)$.
The *Kardar–Parisi–Zhang equation* is the [[stochastic partial differential equation|PRCQG82QTK]] for a random function $h(x, t)$:
$$\partial_t h = \nu\, \partial_x^2 h + \frac{\lambda}{2} \big(\partial_x h\big)^2 + \sqrt{D}\, \xi.$$

## Remarks

The equation is formal: a solution $h(\cdot, t)$ is only Hölder continuous of every order less than $\frac{1}{2}$ in $x$, so $(\partial_x h)^2$ is not defined.
For $\lambda \ne 0$, a standard meaning is the Cole–Hopf solution $h = \frac{2\nu}{\lambda}\log Z$, where $Z > 0$ solves the stochastic heat equation $\partial_t Z = \nu\, \partial_x^2 Z + \frac{\lambda\sqrt{D}}{2\nu} Z \xi$ in the Itô sense; see [[Cole–Hopf transformation|VBMQ9SJ3DH]].
Hairer's theory of regularity structures gives a solution theory that agrees with the Cole–Hopf solution, up to an infinite renormalization constant.
Example: with $\lambda = 0$ the equation is linear, the additive stochastic heat equation, and its solution is a Gaussian process given by a stochastic convolution with the heat kernel.
Non-example: the deterministic equation $h_t = \nu h_{xx} + \frac{\lambda}{2}(h_x)^2$ has no noise term, so it is not the KPZ equation; it is a viscous Hamilton–Jacobi equation.
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# Loewner order

## Definition

Let $n \ge 1$ and let $A$ and $B$ be $n \times n$ [[Hermitian matrices|0P17NGQQMQ]].
Then $A \succeq B$ in the *Loewner order* if $A - B$ is [[positive semidefinite|01MMVW8CKC]].
Also $A \succ B$ if $A - B$ is [[positive definite|0KX66CZMBB]].

## Remarks

The relation $\succeq$ is a [[partial order|0QV2JS9RJ3]].
It is transitive because sums of positive semidefinite matrices are positive semidefinite.
It is antisymmetric because a matrix that is both positive semidefinite and negative semidefinite is $0$.
Example: $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \succeq I$, because the difference $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ has eigenvalues $2$ and $0$.
Non-example: $\operatorname{diag}(1, 0) \not\succeq \operatorname{diag}(0, 1)$, because the difference $\operatorname{diag}(1, -1)$ has the eigenvalue $-1$; the converse fails too, so the order is not total.
The order does not respect squares: for $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $A \succeq B$, but $A^2 - B^2 = \begin{pmatrix} 4 & 3 \\ 3 & 2 \end{pmatrix}$ has determinant $-1$.
On the other hand, $A \succeq B \succeq 0$ implies $A^{1/2} \succeq B^{1/2}$.
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<document source="https://tamnd.github.io/math2/node/4TWY53TBKK.md" tag="4TWY53TBKK" kind="concept">
# Coplanarity

## Summary

A set of points is coplanar when some plane contains all of them.
Any three points are coplanar, so the notion first says something for four or more points.
It is the next step after [[collinearity|0AFNNTRTJE]], and in space it is tested by a vanishing determinant.

## Explanations

### L1 human

Points are coplanar if a flat sheet can pass through all of them.
Three points always fit on a sheet, like a three-legged stool that never wobbles.
Four points may not: the corners of a tetrahedron are not coplanar.
The four corners of a square are coplanar, and so are any points of the floor of a room.
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<document source="https://tamnd.github.io/math2/node/4TXG13F5N0.md" tag="4TXG13F5N0" kind="statement">
# Feldman–Hájek theorem

## Statement

Let $H$ be a real separable [[Hilbert space|HM2HHE3J51]].
Let $\mu = N(a, Q)$ and $\nu = N(b, R)$ be [[Gaussian measures|P0DV2TCH1N]] on $H$ with means $a, b \in H$ and covariance operators $Q, R$.
Here $Q$ and $R$ are positive [[self-adjoint|0NEGN5CRSC]] [[trace class|Z2CP530WDE]] operators on $H$.

1. Either $\mu$ and $\nu$ are equivalent, that is, each is [[absolutely continuous|P0P21KDV6E]] with respect to the other, or they are [[mutually singular|EBGECZNCV5]].
2. They are equivalent if and only if the following three conditions hold:
   (a) $Q^{1/2}(H) = R^{1/2}(H)$; call this space $H_0$.
   (b) $a - b \in H_0$.
   (c) The operator $(Q^{-1/2} R^{1/2})(Q^{-1/2} R^{1/2})^* - I$ is a [[Hilbert–Schmidt operator|W3BX66Y47G]] on the closure of $H_0$.

## Remarks

Here $Q^{-1/2}$ is the inverse of $Q^{1/2}$ on its image, and $Q^{1/2}(H)$ is the Cameron–Martin space of $\mu$.
When $Q = R$, the theorem reduces to the Cameron–Martin theorem: $N(a, Q)$ and $N(b, Q)$ are equivalent exactly when $a - b$ lies in the Cameron–Martin space.
The proof compares the two measures along an orthonormal basis and applies Kakutani's theorem on infinite products of measures, which also has an all-or-nothing form.

Example: on $\R^n$, two Gaussian measures with invertible covariance matrices are always equivalent, since both have everywhere positive densities.
Condition (c) is automatic in finite dimensions.

Example: on an infinite-dimensional $H$ with $Q$ injective, $N(0, Q)$ and $N(0, c Q)$ with $c > 0$, $c \ne 1$, are mutually singular.
Indeed the operator in (c) is $(c - 1) I$, which is not Hilbert–Schmidt.

## Explanations

### L1 human

In finite dimensions, two non-degenerate Gaussian laws have the same null sets.
In infinite dimensions this is false, and two Gaussian measures either have exactly the same null sets or live on disjoint sets.
The theorem says precisely when the first case happens.

### L2 human

The dichotomy is a basic fact for statistics of Gaussian processes.
If two models give singular laws, one can tell them apart without error from a single infinite-dimensional observation.
For example, the variance parameter of Brownian motion can be recovered exactly from one path, which matches the scaling example above.
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<document source="https://tamnd.github.io/math2/node/4V09M20SJB.md" tag="4V09M20SJB" kind="definition">
# Limit set

## Definition

Let $U$ be an open subset of $\R^n$, let $f \colon U \to \R^n$ be locally Lipschitz, and let $\varphi$ be the [[flow|02903SSA48]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.
Let $x \in U$ be such that the solution through $x$ is defined for all $t \ge 0$.
The $\omega$-limit set of $x$ is the set $\omega(x)$ of points $y \in U$ for which there is a sequence $t_k \to +\infty$ with $\varphi_{t_k}(x) \to y$.
Let the solution through $x$ be defined for all $t \le 0$.
The $\alpha$-limit set $\alpha(x)$ is the set of points $y \in U$ with $\varphi_{t_k}(x) \to y$ for some sequence $t_k \to -\infty$.

## Remarks

Equivalently, $\omega(x) = \bigcap_{T \ge 0} \overline{\set{\varphi_t(x) : t \ge T}} \cap U$, where the bar denotes the closure in $\R^n$.
The set $\omega(x)$ is closed in $U$ and [[invariant|WWZTW11664]].
If the forward orbit of $x$ lies in a compact subset of $U$, then $\omega(x)$ is nonempty, compact and connected, and $\varphi_t(x)$ tends to $\omega(x)$ as $t \to \infty$.
Example: in polar coordinates, let $r' = r(1 - r^2)$ and $\theta' = 1$ in the plane.
For $x \ne 0$ the radius tends to $1$ while the angle keeps turning.
So every point of the unit circle is a limit of $\varphi_{t_k}(x)$ for suitable $t_k \to \infty$, and $\omega(x)$ is the unit circle.
For $0 < \abs{x} < 1$, the radius tends to $0$ as $t \to -\infty$, so $\alpha(x) = \set{0}$.
Non-example: for $x' = 1$ on $\R$, the solution $\varphi_t(x) = x + t$ has no convergent sequence $\varphi_{t_k}(x)$ with $t_k \to \infty$, so $\omega(x)$ is empty.
This is possible because the forward orbit is not bounded.

## Explanations

### L1 human

The $\omega$-limit set collects all points that the forward motion comes close to again and again.
The $\alpha$-limit set does the same for the motion run backwards in time.
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# Fractional calculus

## Definition

Let $a \in \R$ and let $f$ be a locally integrable real function on $[a, \infty)$.
For a real number $\alpha > 0$, the fractional integral of order $\alpha$ of $f$ is the [[Riemann–Liouville integral|JTKCZ44NN3]]
$$I_a^\alpha f(x) = \frac{1}{\Gamma(\alpha)} \int_a^x (x - t)^{\alpha - 1} f(t) \, dt, \qquad x > a,$$
where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
For $\alpha > 0$ let $n = \ceil{\alpha}$.
The Riemann–Liouville fractional derivative of order $\alpha$ is
$$D_a^\alpha f(x) = \frac{d^n}{dx^n} \, I_a^{n - \alpha} f(x),$$
whenever the right side exists, with $I_a^0 f = f$.
*Fractional calculus* is the study of these operators and of their variants, for real or complex orders $\alpha$.

## Remarks

For a positive integer $\alpha = n$, $I_a^n f$ is the $n$-fold iterated integral of $f$ from $a$, and $D_a^n f$ is the ordinary [[higher derivative|04TKTF8KDK]] $f^{(n)}$.
Example: with $a = 0$ and $f(t) = 1$, $I_0^{1/2} f(x) = \frac{1}{\Gamma(1/2)} \int_0^x (x - t)^{-1/2} \, dt = \frac{2\sqrt{x}}{\sqrt{\pi}}$.
Non-example: for $\alpha = 0$ the integral formula is not defined, since $\Gamma(0)$ is not defined; the operator $I_a^0$ is set to be the identity by convention.
The fractional integrals satisfy $I_a^\alpha I_a^\beta = I_a^{\alpha + \beta}$ for $\alpha, \beta > 0$.
Other fractional derivatives, such as the [[Caputo derivative|6VMGQYF8RW]] and the [[Grünwald–Letnikov derivative|M9Z8VM4K1Z]], agree with $D_a^\alpha$ on suitable classes of functions.

## Explanations

### L1 human

The fractional integral replaces "integrate $n$ times" by a single weighted integral whose weight makes sense for any positive order.
A fractional derivative of order $\alpha$ is then obtained by integrating a little, to order $n - \alpha$, and differentiating a whole number $n$ of times.
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# Field of rational functions

## Summary

The field of rational functions $K(x)$ over a field $K$ consists of the quotients $p/q$ of polynomials with $q \ne 0$.
It is the [[field of fractions|0F2PTGX1A2]] of the [[polynomial ring|0SDC934R52]] $K[x]$.
It is the simplest field that contains $K$ and an element $x$ that is not a root of any nonzero polynomial over $K$.
It is the basic example of a [[purely transcendental extension|G5PVJMM37Z]] and of a [[function field|XKEXQDRTGH]].

## Explanations

### L1 human

Polynomials can be added, subtracted and multiplied, but not always divided.
Allowing fractions of polynomials, as with fractions of integers, gives a field.
For example, $\frac{1}{x}$ and $\frac{x^2 + 1}{x - 1}$ lie in $\Q(x)$.
Two fractions are equal when cross multiplication gives the same polynomial, as for $\frac{x}{x^2} = \frac{1}{x}$.
The [[definition|JX6CHG7HWV]] makes this precise.
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<document source="https://tamnd.github.io/math2/node/4V9V0AHJCS.md" tag="4V9V0AHJCS" kind="definition">
# Characteristic value of a kernel

## Definition

Let $a < b$ and let $K \in L^2([a, b] \times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]].
A number $\lambda \in \C$ is a *characteristic value* of $K$ if the [[homogeneous equation|W6CYS54QXS]]
$$\varphi(x) = \lambda \int_a^b K(x, y) \, \varphi(y) \, dy$$
has a solution $\varphi \in L^2[a, b]$ that is not $0$ almost everywhere.

## Remarks

A characteristic value is never $0$, since for $\lambda = 0$ the equation forces $\varphi = 0$.
So $\lambda$ is a characteristic value exactly when $1/\lambda$ is an eigenvalue of the integral operator $T_K$, and the nonzero solutions $\varphi$ are the [[eigenfunctions|4NSACSWRJZ]] for $1/\lambda$.

Example: for $K(x, y) = xy$ on $[0, 1]$, the number $\lambda = 3$ is a characteristic value with solution $\varphi(x) = x$, since $3 \int_0^1 x y \cdot y \, dy = 3 \cdot x/3 = x$.

Non-example: for the same kernel, $\lambda = 1$ is not a characteristic value.
Every solution has the form $\varphi(x) = c x$ with $c = \int_0^1 y \varphi(y) \, dy = c/3$, so $c = 0$ and $\varphi = 0$.

Many classical texts call $\lambda$ an eigenvalue of the kernel; the name characteristic value avoids confusion with the eigenvalue $1/\lambda$ of the operator.
A kernel may have no characteristic value at all: the Volterra kernel, equal to $1$ for $y \le x$ and $0$ otherwise, is an example.
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<document source="https://tamnd.github.io/math2/node/4VD8XXQ4P3.md" tag="4VD8XXQ4P3" kind="concept">
# Dirichlet eigenvalue

## Summary

A Dirichlet eigenvalue of a bounded domain $\Omega$ is a number $\lambda$ such that $-\Delta u = \lambda u$ has a nonzero solution vanishing on the boundary.
These numbers form an increasing sequence of positive numbers that tends to infinity.
They are the squared frequencies of a vibrating membrane fixed along its edge.
They also control the decay of heat in $\Omega$ with zero boundary temperature.
How they depend on the shape of $\Omega$ is the subject of spectral geometry, with results such as the [[Faber–Krahn inequality|E7TY2TPXX7]].

## Explanations

### L1 human

Fix the edge of a drum and strike it.
The drum can vibrate only in certain patterns, and each pattern has its own pitch.
The Dirichlet eigenvalues are the squares of these pitches, up to a physical constant.
For the interval $(0, \pi)$, the patterns are $\sin(kx)$ and the eigenvalues are $1, 4, 9, 16, \dots$.
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<document source="https://tamnd.github.io/math2/node/4VDFK4ZF0V.md" tag="4VDFK4ZF0V" kind="definition">
# Rack and quandle

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]] with a [[binary operation|0M418FE7BB]] $\triangleleft$.
The pair $(X, \triangleleft)$ is a *rack* if:

1. $a \triangleleft (b \triangleleft c) = (a \triangleleft b) \triangleleft (a \triangleleft c)$ for all $a, b, c \in X$.
2. For all $a, b \in X$ there is exactly one $c \in X$ with $a \triangleleft c = b$.

A rack is a *quandle* if moreover:

3. $a \triangleleft a = a$ for all $a \in X$.

## Remarks

Clause 2 says that for each $a$ the map $x \mapsto a \triangleleft x$ is a [[bijection|0GG9TVW90M]] of $X$, and clause 1 says that this bijection is an automorphism of $(X, \triangleleft)$.

Example: the conjugation quandle of a [[group|G5S1M9]] $G$, with $a \triangleleft b = a b a^{-1}$, the [[conjugate|0DXD4QSZ8A]] of $b$ by $a$.
Clause 1: both sides equal $a b c b^{-1} a^{-1}$, since $(aba^{-1})(aca^{-1})(aba^{-1})^{-1} = abcb^{-1}a^{-1}$.
Clause 2: $c = a^{-1} b a$ is the only solution.
Clause 3: $a a a^{-1} = a$.

Example: the dihedral quandle on $\Z/n\Z$, with $a \triangleleft b = 2a - b$.
Both sides of clause 1 equal $2a - 2b + c$, clause 2 has the solution $c = 2a - b$, and $2a - a = a$.

Non-example: a rack that is not a quandle.
Let $\sigma$ be a permutation of $X$ other than the identity, and let $a \triangleleft b = \sigma(b)$.
Both sides of clause 1 equal $\sigma(\sigma(c))$, and clause 2 holds since $\sigma$ is a bijection.
But $a \triangleleft a = \sigma(a) \ne a$ for each $a$ moved by $\sigma$, so clause 3 fails.

Some authors write the operation on the other side, with $(c \triangleright b) \triangleright a = (c \triangleright a) \triangleright (b \triangleright a)$.
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<document source="https://tamnd.github.io/math2/node/4VE006XQ33.md" tag="4VE006XQ33" kind="definition">
# Grünwald–Letnikov derivative

## Definition

Let $\alpha > 0$ be real and let $k \ge 0$ be an integer.
The generalised [[binomial coefficient|0GBZM0MT0F]] is $\binom{\alpha}{k} = \frac{\alpha (\alpha - 1) \cdots (\alpha - k + 1)}{k!}$, with $\binom{\alpha}{0} = 1$.
Let $a < x$ be real numbers and let $f \colon [a, x] \to \R$.
The *Grünwald–Letnikov derivative* of order $\alpha$ of $f$ at $x$, with lower terminal $a$, is the [[limit|0PKRYB8ATM]]
$${}^{GL}D_a^\alpha f(x) = \lim_{h \to 0^+} \frac{1}{h^\alpha} \sum_{k=0}^{\floor{(x - a)/h}} (-1)^k \binom{\alpha}{k} f(x - kh),$$
when this limit exists.

## Remarks

Example: for $\alpha = 1$, $\binom{1}{0} = \binom{1}{1} = 1$ and $\binom{1}{k} = 0$ for $k \ge 2$.
The sum is $f(x) - f(x - h)$ once $h \le x - a$, so ${}^{GL}D_a^1 f(x)$ is the left derivative of $f$ at $x$, and it equals $f'(x)$ when $f$ is differentiable at $x$.
More generally, for an integer $\alpha = n$ the sum is the $n$-th backward difference, and the limit is $f^{(n)}(x)$ when $f$ is $n$ times differentiable at $x$.
Non-example: for $f(t) = t^{1/3}$ (the real cube root), $a = -1$, $x = 0$ and $\alpha = 1$, the quotient is $(0 - (-h)^{1/3})/h = h^{-2/3}$.
It tends to $\infty$, so the limit does not exist in $\R$.
The definition only sees values to the left of $x$: for $f(t) = \abs{t}$ at $x = 0$ it gives the one-sided value $-1$, although $f'(0)$ does not exist.
For $n - 1 < \alpha < n$ and $f$ of class $C^n$ on $[a, x]$, the Grünwald–Letnikov derivative equals the Riemann–Liouville derivative of [[fractional calculus|T73XK15SA2]].

## Explanations

### L1 human

The definition takes the backward difference quotients that give whole-order derivatives and lets the order be any positive number.
The weights $(-1)^k \binom{\alpha}{k}$ come from expanding $(1 - z)^\alpha$ as a power series.
Truncating the limit at a small step $h$ gives a practical numerical approximation.
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# Birational geometry

## Definition

Let $k$ be a field and let $X$ and $Y$ be integral [[algebraic varieties|HV6YBS0YDH]] over $k$.
The varieties $X$ and $Y$ are *birationally equivalent* if there is a [[birational map|KA5890R9YM]] $X \dashrightarrow Y$, that is, a [[rational map|4D073CCYGV]] with a rational inverse.
A *birational invariant* is a property or a quantity attached to varieties that takes the same value on birationally equivalent varieties.
*Birational geometry* is the study of varieties up to birational equivalence and of their birational invariants.

## Remarks

Two integral varieties over $k$ are birationally equivalent exactly when their [[function fields|2D5SS6WM14]] are isomorphic as extensions of $k$.
Example: $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$ are birationally equivalent.
The map $[x : y : z] \mapsto ([x : z], [y : z])$ is defined where $z \ne 0$, and its inverse $([x : 1], [y : 1]) \mapsto [x : y : 1]$ is defined on the dense open set where both second coordinates are nonzero.
Both function fields are $k(s, t)$.
Non-example: $\mathbb{P}^1$ and a smooth plane cubic curve $E$ over $\C$ are not birationally equivalent.
For smooth projective curves a birational map extends to an isomorphism, and $\mathbb{P}^1$ has genus $0$ while $E$ has genus $1$.
The dimension, the geometric genus and the Kodaira dimension of smooth projective varieties are birational invariants.
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<document source="https://tamnd.github.io/math2/node/4VMSSYB5DN.md" tag="4VMSSYB5DN" kind="statement">
# Nevanlinna–Pick interpolation

## Statement

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].
Let $n \ge 1$, let $z_1, \ldots, z_n \in \mathbb{D}$ be distinct, and let $w_1, \ldots, w_n \in \mathbb{D}$.
Let $P$ be the [[Pick matrix|TSPTN464CM]] with entries $P_{ij} = \frac{1 - w_i \overline{w_j}}{1 - z_i \overline{z_j}}$.

1. There is a [[holomorphic function|0DG0Z72CQD]] $f \colon \mathbb{D} \to \mathbb{D}$ with $f(z_i) = w_i$ for $i = 1, \ldots, n$ if and only if $P$ is [[positive semidefinite|0880F9EQCP]].
2. Let $P$ be positive semidefinite and singular, of rank $m$.
Then such an $f$ is unique, and it is a constant of modulus $1$ times a [[Blaschke product|9N6R5ADJXT]] of degree $m$.

## Remarks

Proof idea for necessity: if $\abs{f} \le 1$, the multiplication operator by $f$ on the [[Hardy space|PNNNFJVF6W]] $H^2$ has norm at most $1$.
Its adjoint sends the reproducing kernel $k_{z_i}(z) = \frac{1}{1 - \overline{z_i} z}$ to $\overline{w_i} k_{z_i}$, and the condition $\norm{M_f^* g} \le \norm{g}$ on combinations of these kernels is exactly $P \ge 0$.
Sufficiency can be proved by Pick's original induction, which reduces $n$ points to $n - 1$ points with a Möbius change of variables, or by the commutant lifting theorem.

Example: for $n = 2$ with $z_1 = w_1 = 0$, $z_2 = a \ne 0$ and $w_2 = b$, the matrix is positive semidefinite exactly when $\abs{b} \le \abs{a}$.
This is the [[Schwarz lemma|0MB48XDBY3]]; when $\abs{b} = \abs{a}$, the matrix has rank $1$ and the unique solution is $f(z) = (b/a) z$.

## Explanations

### L1 human

You choose some points in the disc and where you want them to go.
The question is whether a holomorphic map of the disc into itself can do this.
The theorem gives a test: build the Pick matrix and check that it has no negative eigenvalue.
When the test passes only just, the map is forced and is a finite Blaschke product.

### L2 human

The theorem generalises the Schwarz–Pick lemma from one pair of points to any finite set.
Its operator form says that multipliers of $H^2$ of norm at most $1$ are characterised by positivity of a kernel.
This viewpoint later led to the theory of complete Pick spaces.
The same method treats interpolation in the upper half-plane and boundary interpolation problems, and it has applications in control theory.
For infinitely many points the condition becomes positivity of an infinite Pick matrix, and the interpolating sequences are characterised by Carleson.
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# Markov partition

## Summary

A Markov partition of a hyperbolic system is a cover of the space by finitely many pieces, called rectangles.
The map moves these rectangles in a very regular way.
Each rectangle is a product of a stable and an unstable piece, and the map stretches the unstable pieces across whole rectangles.
Recording which rectangle a point visits at each time turns the system into a [[subshift of finite type|624XDXX8FN]].
In this way Markov partitions make [[symbolic dynamics|EWFWTSVY0B]] a tool for [[Anosov diffeomorphisms|RCQH8P6DPQ]] and for the basic sets of [[Axiom A|Z26Z4XZFVA]] systems.

## Explanations

### L1 human

Cut the space into a few boxes, and name each box by a letter.
Follow a point under the map and write down the letter of the box it is in at each time.
For a Markov partition the boxes are chosen so well that the allowed strings of letters are exactly those that avoid a finite list of forbidden pairs.
So the complicated motion is described by a simple rule on sequences of letters.
For the [[Smale horseshoe|N63Z6EA6YS]], two boxes already suffice, and every sequence of two letters occurs.
See the [[definition|3S543TYT8C]].
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<document source="https://tamnd.github.io/math2/node/4VSZ22TX4B.md" tag="4VSZ22TX4B" kind="example">
# Particular point topology

## Example

Let $X$ be a set and let $p \in X$.
The *particular point topology* on $X$ declares a set $U \subseteq X$ [[open|02BH3MCBQB]] if $U = \emptyset$ or $p \in U$.
This is a topology: $\emptyset$ and $X$ are open.
A union of sets that contain $p$ contains $p$, and so does a finite intersection of them.
So $X$ is a [[topological space|0CQHFPGJ45]].

- $X$ is a [[T0 space|07SG12TQH3]]: let $x \ne y$, and name them so that $x \ne p$.
  Then $\set{p, y}$ is open, contains $y$, and does not contain $x$.
- If $X$ has at least two points, $X$ is not a [[T1 space|0NW6CH38YR]]: every open set that contains a point $x \ne p$ also contains $p$.
- $X$ is [[connected|0V032VWTHD]]: any two nonempty open sets meet at $p$, so $X$ is not a union of two disjoint nonempty open sets.
- The set $\set{p}$ is [[dense|0BSJSX7ME2]], since every nonempty open set contains $p$.
- $X$ is [[compact|0TVJZYB6SH]] if and only if $X$ is finite.
  Indeed, the open sets $\set{p, x}$ for $x \in X$ cover $X$, and a finite subcover exists only when $X$ is finite.

## Explanations

### L1 human

In this topology one point $p$ is special: a nonempty set is open exactly when it contains $p$.
So $p$ is "close" to every other point, and every small open set around any point must reach $p$.
This makes the space connected and lets the single point $p$ be dense.
It is a standard source of counterexamples in general topology.
</document>

<document source="https://tamnd.github.io/math2/node/4W0EG94RZ5.md" tag="4W0EG94RZ5" kind="statement">
# Fredholm series for the resolvent

## Statement

Let $a < b$ and let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]].
Let $D(\lambda)$ be its [[Fredholm determinant|PPX0AK81QV]] and $D(x, y; \lambda)$ its first [[Fredholm minor|HP5VJ9EBST]].
Let $\lambda \in \C$ with $D(\lambda) \ne 0$, and put
$$R(x, y; \lambda) = \frac{D(x, y; \lambda)}{D(\lambda)}, \qquad x, y \in [a, b].$$
Then:

1. $R(\cdot, \cdot; \lambda)$ is continuous on $[a, b]^2$ and satisfies
$$R(x, y; \lambda) - K(x, y) = \lambda \int_a^b K(x, t) R(t, y; \lambda) \, dt = \lambda \int_a^b R(x, t; \lambda) K(t, y) \, dt;$$
2. for every continuous $f \colon [a, b] \to \C$, the [[Fredholm equation of the second kind|DEYYNVCV9J]] $\varphi(x) - \lambda \int_a^b K(x, y) \varphi(y) \, dy = f(x)$ has exactly one continuous solution, namely
$$\varphi(x) = f(x) + \lambda \int_a^b R(x, y; \lambda) f(y) \, dy;$$
3. $\lambda$ is not a [[characteristic value|DC7HMHVREP]] of $K$.

So $R$ is the [[resolvent kernel|ARTJV1XCRX]] of $K$, and it is a meromorphic function of $\lambda$ whose poles lie among the zeros of $D$.

## Remarks

The proof rests on two identities between entire functions of $\lambda$.
They come from expanding the determinants in the series along the first row or the first column:
$$D(x, y; \lambda) = K(x, y) D(\lambda) + \lambda \int_a^b K(x, t) D(t, y; \lambda) \, dt,$$
and the same with $K$ and $D$ exchanged in the integral.
Dividing by $D(\lambda)$ gives part 1, and part 1 shows by direct substitution that the formula in part 2 solves the equation.
Uniqueness comes from the same identities applied to a solution of the homogeneous equation, which must then be $0$.

Example: for $K(x, y) = xy$ on $[0, 1]$, $D(\lambda) = 1 - \lambda/3$ and $D(x, y; \lambda) = xy$.
So $R(x, y; \lambda) = \frac{3xy}{3 - \lambda}$ for $\lambda \ne 3$.
For $|\lambda| < 3$ this agrees with the sum $\sum_{n \ge 0} \lambda^n xy / 3^n$ of the [[Liouville–Neumann series|H1HWSWME6B]].

The Neumann series converges only for small $\abs{\lambda}$, while Fredholm's formula is valid for every $\lambda$ with $D(\lambda) \ne 0$.

## Explanations

### L1 human

This is Cramer's rule for integral equations.
The solution of a linear system is a ratio of determinants, and here the solution kernel is a ratio of two everywhere convergent power series in $\lambda$.
The formula works as long as the denominator does not vanish.

### L2 human

The result turns the [[Fredholm equation of the second kind|DEYYNVCV9J]] into an explicit computation whenever $D(\lambda) \ne 0$.
It extends the [[resolvent kernel|ARTJV1XCRX]] beyond the disc where the Neumann series converges, as a meromorphic function of $\lambda$.
At a zero $\lambda_0$ of $D$, the homogeneous equation has nonzero solutions, and these can be read off from the first Fredholm minor that does not vanish identically at $\lambda_0$.
In operator language, $R$ is the kernel of $\lambda^{-1}((I - \lambda T_K)^{-1} - I)$, where $T_K$ is the integral operator with kernel $K$.
</document>

<document source="https://tamnd.github.io/math2/node/4W27TKGQWR.md" tag="4W27TKGQWR" kind="concept">
# Buekenhout geometry

## Summary

A Buekenhout geometry, or diagram geometry, is a set of objects of several types with a symmetric incidence relation.
Every flag of pairwise incident objects extends to a chamber, which contains one object of each type.
It generalizes [[projective spaces|ZT9A6JK8QF]], [[generalized polygons|PCT2HY9AAZ]] and Tits buildings.
Francis Buekenhout introduced a diagram that records, for each pair of types, which rank $2$ geometries appear.
These diagrams are used to describe and classify geometries, including geometries for sporadic simple groups.

## Explanations

### L1 human

A projective space has points, lines, planes and so on, and a point lies on a line, a line lies in a plane, and so forth.
A Buekenhout geometry keeps only this pattern: objects of different types and a relation saying when two objects are incident.
Each pair of types gives a small two-type geometry, such as a projective plane or a generalized quadrangle.
A diagram with one node per type and labelled edges records these pairs, much like a Dynkin diagram.
</document>

<document source="https://tamnd.github.io/math2/node/4W31MKE01V.md" tag="4W31MKE01V" kind="concept">
# Approximate derivative

## Summary

The approximate derivative of a real function at a point is a limit of its difference quotients.
In this limit one may ignore a set that is very thin near the point.
The ignored set must have density $0$ at the point, in the sense of [[Lebesgue's density theorem|WSVXYSNHNX]].
Approximate derivatives exist in many cases where ordinary derivatives do not, and they are the derivatives used in the [[Khinchin integral|Q1TV4F608D]].

## Explanations

### L1 human

An ordinary derivative looks at all points near $x$.
An approximate derivative is allowed to skip a set of points, as long as that set takes up a smaller and smaller share of each small interval around $x$.
For example, change the zero function to the value $1$ at the points $1, \frac12, \frac13, \dots$.
The new function has no derivative at $0$, but its approximate derivative at $0$ is $0$, since the changed points form a countable set, which is invisible to measure.
</document>

<document source="https://tamnd.github.io/math2/node/4W40NHWPQT.md" tag="4W40NHWPQT" kind="example">
# Log-normal distribution

## Example

Let $\mu \in \R$ and $\sigma > 0$, and let $Y$ be [[normal|05AQE0QR0D]] with mean $\mu$ and variance $\sigma^2$.
The law of $X = e^Y$ is the *log-normal distribution* with parameters $\mu$ and $\sigma^2$; so $X > 0$ and $\ln X$ is normal.
Its [[density|0ZY9DX8FBF]] is
$$f(x) = \frac{1}{x \sigma \sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu)^2}{2 \sigma^2}\right) \qquad (x > 0),$$
and $f(x) = 0$ for $x \le 0$.
Check: for $0 < a < b$, $P(a < X < b) = P(\ln a < Y < \ln b)$, and the substitution $y = \ln x$, $dy = dx / x$, turns the normal density of $Y$ into $f$.
The [[moments|01XAHPADBA]] are $E[X^n] = E[e^{nY}] = e^{n\mu + n^2 \sigma^2 / 2}$ for every real $n$, by the moment-generating function of the normal law.
So $E[X] = e^{\mu + \sigma^2/2}$ and $\operatorname{Var}(X) = (e^{\sigma^2} - 1) e^{2\mu + \sigma^2}$.
The median is $e^{\mu}$, which is smaller than the mean.

## Remarks

All moments are finite, but the [[moment-generating function|0WQVXP7PDB]] $E[e^{tX}]$ is infinite for every $t > 0$.
The log-normal law is a standard example of a distribution that is not determined by its moments: other distributions have exactly the same moments.
If $X_1$ and $X_2$ are independent and log-normal, then $X_1 X_2$ is log-normal, because $\ln X_1 + \ln X_2$ is normal.

## Explanations

### L1 human

A log-normal quantity is one whose logarithm follows a bell curve.
It arises when many independent factors multiply together, just as the normal law arises when many independent terms add.
Its graph is skewed to the right: it is always positive, has a long tail of large values, and its mean lies above its median.
Sizes of particles, incomes and stock prices in the Black–Scholes model are often modelled with it.
</document>

<document source="https://tamnd.github.io/math2/node/4W9NM803BB.md" tag="4W9NM803BB" kind="concept">
# Pi-system

## Summary

A π-system on a set is a nonempty family of subsets that is closed under intersections of two members.
It is the kind of generating family used in the uniqueness theorems of measure theory.
Two probability measures that agree on a π-system agree on the σ-algebra it generates.

## Explanations

### L1 human

A π-system is a family of sets in which the overlap of any two members is again a member.
The rays $(-\infty, a]$ on the real line are the key example: the overlap of two rays is the shorter one.
These rays generate all Borel sets.
So a probability distribution on the line is fixed by its values on the rays, which is why a distribution function determines the distribution.
</document>

<document source="https://tamnd.github.io/math2/node/4WCJ49Z90S.md" tag="4WCJ49Z90S" kind="statement">
# Lichnerowicz formula

## Statement

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with a [[spin structure|RSPZ9TCFQ8]], and let $S$ be its [[spinor bundle|PYYQQKAZ0C]] with spin connection $\nabla$.
Let $D$ be the [[Dirac operator|QMRAVX14BJ]] of $M$, defined with Clifford multiplication satisfying $v \cdot v = -g(v, v)$.
Let $\nabla^* \nabla = -\sum_{i=1}^n \bigl(\nabla_{e_i} \nabla_{e_i} - \nabla_{\nabla_{e_i} e_i}\bigr)$ be the rough Laplacian, for a local orthonormal frame $e_1, \ldots, e_n$, and let $R$ be the [[scalar curvature|ZHX2VYQDZ3]] of $g$.
Then
$$D^2 \psi = \nabla^* \nabla \psi + \frac{R}{4} \, \psi$$
for every smooth section $\psi$ of $S$.

## Remarks

Work at a point with a frame satisfying $\nabla e_i = 0$ there.
Then $D^2 = \sum_{i,j} e_i e_j \nabla_{e_i} \nabla_{e_j}$, the terms with $i = j$ give $\nabla^* \nabla$, and the terms with $i \ne j$ give $\frac{1}{2} \sum_{i,j} e_i e_j R^S(e_i, e_j)$, where $R^S$ is the curvature of $S$.
The curvature of the spin connection is the Riemann tensor acting through Clifford multiplication, and the first Bianchi identity reduces this sum to $\frac{R}{4}$.

Example: on flat $\R^n$, $R = 0$ and the formula says that $D^2 = -\sum_j \partial_j^2$ on spinor fields, the defining property of the flat Dirac operator.
On the round sphere $S^n$ of radius $1$, $R = n(n-1) > 0$, so $D^2 \ge \frac{n(n-1)}{4}$ as an operator, which bounds the eigenvalues of $D$ away from $0$.

## Explanations

### L1 human

The square of the Dirac operator is a Laplacian plus one extra term, and that term is a quarter of the scalar curvature.
So the Dirac operator feels the curvature of the manifold only through one number at each point.
When that number is positive, it pushes the square of the operator up and leaves no room for spinors killed by it.

### L2 human

The formula is a special case of the Bochner–Weitzenböck formula for the Dirac operator of a [[Dirac bundle|K0GVMM3YPF]], like the [[Weitzenböck identity|BE9GTWMN0B]] for forms.
Integrating it on a closed manifold gives $\norm{D\psi}^2 = \norm{\nabla\psi}^2 + \frac{1}{4} \int_M R \abs{\psi}^2$, which is the source of the Lichnerowicz vanishing theorem.
For a spinor bundle twisted by a bundle $E$ with connection, an extra term appears: Clifford multiplication by the curvature of $E$.
</document>

<document source="https://tamnd.github.io/math2/node/4WCYBSH1P6.md" tag="4WCYBSH1P6" kind="concept">
# Matroid minor

## Summary

A minor of a [[matroid|PX31ZNYS3Z]] is a matroid obtained from it by deleting some elements and contracting others.
Deletion and contraction generalize the deletion and contraction of edges of a graph, and they are exchanged by [[matroid duality|6QF83CFK2G]].
Many classes of matroids are closed under minors and are characterized by a list of excluded minors.
This parallels the role of [[graph minors|VYQ32ZBPEE]] for classes of graphs.

## Explanations

### L1 human

Deleting an element just throws it away together with every independent set that uses it.
Contracting an element that is not a loop means treating it as already chosen: a set counts as independent when it stays independent after adding that element.
For a graph, deletion removes an edge and contraction shrinks an edge to a point.
A minor is anything you can reach by doing these two operations several times.
</document>

<document source="https://tamnd.github.io/math2/node/4WKHF5BCTA.md" tag="4WKHF5BCTA" kind="statement">
# Dehn–Sommerville equations

## Statement

Let $P$ be a simplicial convex polytope of dimension $d \ge 1$, that is, a polytope all of whose proper faces are simplices.
Let $(h_0, h_1, \dots, h_d)$ be the [[h-vector|DCK64GFPD7]] of the boundary complex of $P$.
Then
$$h_i = h_{d-i} \quad \text{for } 0 \le i \le d.$$

## Remarks

One proof uses a shelling of the boundary complex, an order of the facets in which each new facet meets the earlier ones in a union of its own facets.
For each facet, record the size of its smallest new face; then $h_i$ is the number of facets for which this size is $i$.
The reverse of a line shelling is again a shelling, and reversing the order turns size $i$ into size $d - i$, which gives the symmetry.

The equation $h_0 = h_d$ is equivalent to Euler's relation $f_0 - f_1 + f_2 - \dots + (-1)^{d-1} f_{d-1} = 1 - (-1)^d$ for the boundary.
For $d = 3$ the equations give $f_1 = 3 f_0 - 6$ and $f_2 = 2 f_0 - 4$, so the number of vertices determines the whole f-vector of a simplicial $3$-polytope.

Example: the octahedron has $h = (1, 3, 3, 1)$, which is symmetric.

The hypothesis that the complex is the boundary of a polytope (more generally, a sphere) is necessary.
The full triangle, viewed as a simplicial complex of dimension $2$, has $f = (1, 3, 3, 1)$ and $h = (1, 0, 0, 0)$, which is not symmetric.

## Explanations

### L1 human

The face numbers of a polytope with triangular faces are not free: they satisfy linear equations.
In the h-vector form the equations are a simple mirror symmetry.
For example, a convex polyhedron in space made of triangles with $v$ vertices always has $3v - 6$ edges and $2v - 4$ triangles.

### L2 human

The equations hold more generally for simplicial spheres and for Eulerian complexes.
They show that about half of the entries of the f-vector of a simplicial $d$-polytope determine the rest.
The remaining question, which h-vectors occur at all, is answered by the [[g-theorem|4CZMWQVC30]].
</document>

<document source="https://tamnd.github.io/math2/node/4WSR1VXJNW.md" tag="4WSR1VXJNW" kind="definition">
# Tate algebra

## Definition

Let $K$ be a field that is complete with respect to a nontrivial [[non-archimedean absolute value|1H5F1BP6XS]] $\abs{\cdot}$, and let $n \ge 0$.
For a multi-index $\nu = (\nu_1, \dots, \nu_n) \in \N^n$ write $X^\nu = X_1^{\nu_1} \cdots X_n^{\nu_n}$ and $\abs{\nu} = \nu_1 + \cdots + \nu_n$.
The *Tate algebra* in $n$ variables over $K$ is the subring
$$T_n = K\langle X_1, \dots, X_n \rangle = \Big\{ \sum_{\nu \in \N^n} a_\nu X^\nu \in K[\![X_1, \dots, X_n]\!] : \abs{a_\nu} \to 0 \text{ as } \abs{\nu} \to \infty \Big\}$$
of the [[formal power series|033WGN00BF]] ring.
Its *Gauss norm* is $\norm{\sum a_\nu X^\nu} = \max_\nu \abs{a_\nu}$.

## Remarks

The maximum exists because $\abs{a_\nu} \to 0$.
$T_n$ is closed under multiplication, and the Gauss norm is multiplicative: $\norm{fg} = \norm{f} \norm{g}$.
With this norm $T_n$ is a [[Banach algebra|9KYWN94QDN]] over $K$, and the polynomial ring $K[X_1, \dots, X_n]$ is dense in it.
Let $\bar K$ be an algebraic closure of $K$ with the extended absolute value.
A formal power series lies in $T_n$ exactly when it converges at every point $x \in \bar K^n$ with all $\abs{x_i} \le 1$.
$T_n$ is Noetherian, regular and a unique factorization domain, and it satisfies a Noether normalization lemma.
The residue field of each maximal ideal of $T_n$ is a finite extension of $K$.

Example: over the [[completion|W168DS966V]] $\Q_p$ of $\Q$ for the $p$-adic absolute value, $f = \sum_{k \ge 0} p^k X^k$ lies in $T_1$, because $\abs{p^k}_p = p^{-k} \to 0$; its Gauss norm is $\abs{1}_p = 1$.

Non-example: $\sum_{k \ge 0} X^k$ has all coefficients of absolute value $1$, which do not tend to $0$, so it is not in $T_1$; it diverges at $X = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/4WX0HAK9X9.md" tag="4WX0HAK9X9" kind="definition">
# Graded poset

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
For $x, y \in P$, say that $y$ covers $x$ when $x < y$ and no $z \in P$ satisfies $x < z < y$; this is the [[covering relation|4HHDPZ4BDY]].
A *rank function* on $P$ is a map $\rho \colon P \to \Z$ such that:

1. if $x < y$, then $\rho(x) < \rho(y)$;
2. if $y$ covers $x$, then $\rho(y) = \rho(x) + 1$.

A *graded poset* is a partially ordered set together with a rank function.

## Remarks

Example: the power set of a finite set, ordered by inclusion, is graded by $\rho(A) = \abs{A}$.
A subset $B$ covers $A$ exactly when $B = A \cup \set{b}$ with $b \notin A$, so the rank goes up by one.
Example: the positive divisors of an integer $n \ge 1$, ordered by divisibility, are graded by the number of prime factors counted with multiplicity.
Non-example: the [[pentagon lattice|DHC8TZXGYS]] with elements $0 < a < b < 1$ and $0 < c < 1$ has no rank function.
The covers $0, a, b, 1$ would give $\rho(1) = \rho(0) + 3$, while the covers $0, c, 1$ would give $\rho(1) = \rho(0) + 2$.
Some authors also require $\rho$ to take values in $\N$, or require all maximal chains to have the same length; the conventions should be checked in each source.
</document>

<document source="https://tamnd.github.io/math2/node/4WY3A9YGDR.md" tag="4WY3A9YGDR" kind="concept">
# Kuranishi structure

## Summary

A Kuranishi structure on a compact space describes it locally as a quotient by a finite group.
The thing divided is the zero set of a smooth section of a vector bundle.
The bundle has finite rank and lives over a finite-dimensional manifold.

The local descriptions, called Kuranishi charts, are glued by coordinate changes that embed one chart into another.
The notion was introduced by Fukaya and Ono to handle moduli spaces of pseudoholomorphic curves, which are usually singular and of the wrong dimension.
A Kuranishi structure with an orientation has a [[virtual fundamental class|5VXYQ4BXRB]], which is used to define [[Gromov–Witten invariants|XVJA0JD79P]] and Floer homology in general symplectic manifolds.

## Explanations

### L1 human

Many moduli spaces are cut out by equations that, near each solution, reduce to finitely many equations in finitely many unknowns.
A Kuranishi chart records exactly this: a smooth space of unknowns, a finite list of equations, and a finite symmetry group.
The solutions may form a singular set, but the chart still knows the expected dimension, the number of unknowns minus the number of equations.
For example, the zero set of $s(x, y) = x y$ on $\R^2$, a pair of crossing lines, is described by one chart with $V = \R^2$, $E = \R$ and expected dimension $1$.
</document>

<document source="https://tamnd.github.io/math2/node/4X0A0CQ0B0.md" tag="4X0A0CQ0B0" kind="definition">
# Harmonic at a point

## Definition

Let $U \subseteq \R^n$ be an [[open set|02BH3MCBQB]], let $f \colon U \to \R$ and let $a \in U$.
The function $f$ is *harmonic at $a$* if there is an open set $V$ with $a \in V \subseteq U$ with two properties.

1. The function $f$ is twice continuously differentiable on $V$.
2. Its [[Laplacian|0E0B6RA8K0]] satisfies $\Delta f = 0$ at every point of $V$.

## Remarks

Example: $f(x, y) = \log(x^2 + y^2)$ on $U = \R^2 \setminus \set{0}$ is harmonic at $a = (1, 0)$.
Indeed, on $V = U$ one computes $\partial_x^2 f + \partial_y^2 f = \frac{2(y^2 - x^2)}{(x^2 + y^2)^2} + \frac{2(x^2 - y^2)}{(x^2 + y^2)^2} = 0$.
Non-example: $f(x, y) = x^3$ on $\R^2$ is not harmonic at $(0, 0)$.
Its Laplacian $6x$ vanishes at $(0, 0)$, but it is nonzero at points $(t, 0)$ with $t \ne 0$ arbitrarily close, so no neighbourhood $V$ works.
A function is [[harmonic|0DEB0JMN0W]] on $U$ if and only if it is harmonic at every point of $U$.
In Mathlib the condition reads: $f$ is $C^2$ at $a$ and $\Delta f = 0$ on a neighbourhood of $a$.
This is the same condition, since a function that is $C^2$ at a point is $C^2$ on a neighbourhood of it.
</document>

<document source="https://tamnd.github.io/math2/node/4X6WC3E7P8.md" tag="4X6WC3E7P8" kind="statement">
# Céa's lemma

## Statement

Let $V$ be a real [[Hilbert space|HM2HHE3J51]] with norm $\norm{\cdot}$.
Let $a \colon V \times V \to \R$ be a [[bilinear form|0SQZMXB6GP]] that is bounded, $\abs{a(u, v)} \le M \norm{u} \norm{v}$ for all $u, v \in V$, and [[coercive|86KRHS6X22]], $a(v, v) \ge \alpha \norm{v}^2$ for all $v \in V$, with constants $M \ge 0$ and $\alpha > 0$.
Let $f$ be an element of the [[continuous dual space|4A82S5FVRT]] of $V$, and let $u \in V$ satisfy $a(u, v) = f(v)$ for all $v \in V$.
Let $V_h$ be a closed subspace of $V$, and let $u_h \in V_h$ satisfy $a(u_h, v) = f(v)$ for all $v \in V_h$.
Then
$$\norm{u - u_h} \le \frac{M}{\alpha} \, \inf_{v \in V_h} \norm{u - v}.$$

## Remarks

Both $u$ and $u_h$ exist and are unique by the [[Lax–Milgram theorem|85KK79FCN5]], applied on $V$ and on $V_h$.
Subtracting the two equations gives Galerkin orthogonality: $a(u - u_h, w) = 0$ for all $w \in V_h$.
For $v \in V_h$, take $w = v - u_h$: then $\alpha \norm{u - u_h}^2 \le a(u - u_h, u - u_h) = a(u - u_h, u - v) \le M \norm{u - u_h} \norm{u - v}$.
Dividing by $\alpha \norm{u - u_h}$ and taking the infimum over $v$ gives the claim.
If $a$ is also symmetric, then $u_h$ is the $a$-orthogonal projection of $u$ onto $V_h$, and the constant improves to $\sqrt{M / \alpha}$.

Example: for $-u'' = f$ on $(0, 1)$ with $u(0) = u(1) = 0$, take $V = H^1_0(0, 1)$, $a(u, v) = \int_0^1 u' v' \, dx$, and $V_h$ the continuous piecewise linear functions on a mesh of width $h$.
Choosing $v$ to be the piecewise linear interpolant of $u$ in the infimum gives $\norm{u - u_h}_{H^1} \le C h \norm{u''}_{L^2}$ when $u \in H^2(0, 1)$.

Coercivity is needed.
Take an indefinite form, such as $a(u, v) = \int_0^1 (u' v' - k^2 u v) \, dx$ with $k$ large.
Then a discrete solution may fail to exist on a given $V_h$.
When it exists, it need not be close to the best approximation unless $h$ is small enough.

## Explanations

### L1 human

The finite element method looks for an approximate solution inside a smaller space of simple functions.
Céa's lemma says that this approximate solution is almost as good as the best possible approximation from that space.
The loss is at most a fixed factor $M / \alpha$, which depends on the problem but not on the space.

### L2 human

The lemma splits error analysis of [[Galerkin methods|Y5B0PEBKE8]] into two independent questions.
One is stability, measured by $M / \alpha$, which comes from the continuous problem.
The other is approximation: how well functions in $V_h$ approximate the exact solution, which is a question of interpolation theory and is answered with the [[Bramble–Hilbert lemma|K4P0R83G1Q]].
For problems that are not coercive, such as saddle-point problems, the role of $\alpha$ is taken by an inf-sup constant, as in the [[Ladyzhenskaya–Babuška–Brezzi condition|SC5FV3GHV5]].
</document>

<document source="https://tamnd.github.io/math2/node/4XEGZK0H36.md" tag="4XEGZK0H36" kind="definition">
# Stieltjes function

## Definition

A *Stieltjes function* is a function $F \colon \R \to \R$ such that:

1. $F$ is non-decreasing, that is, $F(x) \le F(y)$ whenever $x \le y$; see [[monotone function|0A7KV2WB7A]].
2. $F$ is right-continuous, that is, $F(x) = \lim_{y \to x^+} F(y)$ for every $x \in \R$.

## Remarks

Example: the floor function $F(x) = \floor{x}$ is a Stieltjes function.
It is non-decreasing, and on each interval $[n, n + 1)$ with $n \in \Z$ it is constant, so it is right-continuous at every point.
Its [[Lebesgue–Stieltjes measure|QV75XJBFCS]] puts mass $1$ at each integer.

Non-example: the ceiling function $F(x) = \ceil{x}$ is non-decreasing but is not right-continuous at the integers.
At $x = 0$ one has $F(0) = 0$, while $F(y) = 1$ for all $y \in (0, 1)$, so the limit from the right is $1 \ne F(0)$.

Every non-decreasing function has one-sided limits at every point, so right-continuity only fixes the value at the countably many jumps.
Two Stieltjes functions give the same measure exactly when they differ by a constant.
Some authors write "distribution function" for the same notion.
</document>

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# Nonexpansive map

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|0VKD1TP7F3]].
A map $f \colon X \to Y$ is *nonexpansive* if
$$d_Y(f(x), f(x')) \le d_X(x, x') \quad \text{for all } x, x' \in X.$$

## Remarks

Equivalently, $f$ is [[Lipschitz|0AW1E6YTJY]] with Lipschitz constant at most $1$.
In operator theory, $X = Y = C$ is often a subset of a Banach space with the norm distance, and the condition reads $\norm{f(x) - f(x')} \le \norm{x - x'}$.

Example: let $H$ be a real Hilbert space, $C \subseteq H$ a nonempty closed convex set, and $P_C(x)$ the nearest point of $C$ to $x$.
The variational inequality $\ip{x - P_C x}{c - P_C x} \le 0$ for $c \in C$ gives $\norm{P_C x - P_C y}^2 \le \ip{x - y}{P_C x - P_C y} \le \norm{x - y}\norm{P_C x - P_C y}$, so $P_C$ is nonexpansive.

Non-example: $f(x) = 2x$ on $\R$ gives $\abs{f(1) - f(0)} = 2 > 1 = \abs{1 - 0}$.

Every [[contraction mapping|0QN69YA357]] is nonexpansive, but not conversely.
The Browder–Göhde–Kirk theorem gives fixed points: a nonexpansive map of a nonempty bounded closed convex subset of a uniformly convex Banach space into itself has a fixed point.
</document>

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# Weak operator topology

## Summary

The weak operator topology is a topology on the bounded operators on a Hilbert space.
In it, a net of operators converges when all its matrix coefficients $\langle T x, y \rangle$ converge.
It is much coarser than the norm topology, so more sets are compact and more nets converge.
It is the topology used to define [[von Neumann algebras|PSFWQVZ0FS]], and its closed unit ball is compact.

## Explanations

### L1 human

An operator on a Hilbert space can be tested by feeding it a vector $x$ and reading off one component $\langle Tx, y \rangle$ of the output.
In the weak operator topology, operators are close when these numbers are close for finitely many chosen $x$ and $y$.
This is a very forgiving notion of closeness.
For example, the right shift $S$ on sequences moves each entry one step to the right, so the powers $S^n$ push every vector away to infinity.
Every fixed component of $S^n x$ then tends to $0$, so $S^n \to 0$ in this topology, although each $S^n$ preserves lengths.
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# Cayley–Abels graph

## Definition

Let $G$ be a [[totally disconnected|AW3J9AQSA0]] [[locally compact group|AHV05ZBCYW]] that is compactly generated.
A *Cayley–Abels graph* for $G$ is a graph $\Gamma$ with an action of $G$ by graph automorphisms such that:

1. $\Gamma$ is [[connected|0KWFFMDF8T]];
2. $\Gamma$ is locally finite: every vertex has finite [[degree|0KS0KRDV4Y]];
3. $G$ acts [[transitively|02GK6HZYBT]] on the vertices of $\Gamma$;
4. the stabiliser in $G$ of each vertex is a compact [[open subgroup|QYCFH68RD1]] of $G$.

## Remarks

Example: let $G$ be a finitely generated group with the discrete topology, and let $S$ be a finite symmetric generating set.
The [[Cayley graph|0GYQK9NDJF]] of $(G, S)$, with $G$ acting by left multiplication, is connected because $S$ generates $G$.
Each vertex has degree at most $\abs{S}$, the action is transitive, and stabilisers are trivial, hence compact and open.

Example: $G = \mathrm{Aut}(T_d)$, the automorphism group of the regular tree $T_d$ of degree $d \ge 3$, acting on $T_d$.
Vertex stabilisers are compact and open in the topology of pointwise convergence.

Non-example: $\Z$ acting on its Cayley graph with respect to the infinite generating set $\Z$.
The graph is complete, so every vertex has infinite degree, and clause 2 fails.

Construction: choose a compact open subgroup $U$ and a finite symmetric set $A$ with $G$ generated by $U \cup A$.
Take the cosets $gU$ as vertices, and join $gU$ and $hU$ when $g^{-1} h \in U A U$.
This is a Cayley–Abels graph, since $UAU$ is a finite union of cosets of $U$.
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<document source="https://tamnd.github.io/math2/node/4XR88RPJ4K.md" tag="4XR88RPJ4K" kind="definition">
# Bohr compactification

## Definition

Let $G$ be a [[topological group|92700QAB5Y]].
A *Bohr compactification* of $G$ is a pair $(bG, b)$ of a compact Hausdorff group $bG$ and a [[continuous homomorphism|9J5PQMG3HJ]] $b \colon G \to bG$ with this property.
For every compact Hausdorff group $K$ and every continuous homomorphism $f \colon G \to K$, there is exactly one continuous homomorphism $\tilde{f} \colon bG \to K$ with $f = \tilde{f} \circ b$.

## Remarks

The Bohr compactification exists for every topological group, and it is unique up to a unique isomorphism compatible with the maps from $G$.
Example: if $G$ is a [[compact|EVHCE5VR6Y]] Hausdorff group, then $(G, \id_G)$ is a Bohr compactification, since each $f$ factors as $f = f \circ \id_G$ and the factor must be $f$.
Non-example: $(\mathbb{T}, x \mapsto e^{2\pi i x})$ is not a Bohr compactification of $\R$.
The continuous homomorphism $x \mapsto e^{2\pi i \sqrt{2} x}$ from $\R$ to $\mathbb{T}$ does not factor through $x \mapsto e^{2\pi i x}$.
A factor $\tilde f$ would give $e^{2\pi i \sqrt{2}} = \tilde f(e^{2\pi i}) = \tilde f(1) = 1$, which is false.
If $G$ is a locally compact abelian group, $bG$ is the [[Pontryagin dual|FMW57WNPDB]] of the group $\widehat{G}$ with the discrete topology, and $b(x)$ is evaluation at $x$.
The map $b$ need not be injective, and its image is dense in $bG$.
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# Finite model theory

## Summary

Finite model theory is the part of [[model theory|HFTBG5PMXC]] that studies logics on finite [[structures|983GZBQYT4]].
Many classical tools fail there; for example, the [[compactness theorem|1NR3HAWG9D]] does not hold when only finite structures are allowed.
Instead the field uses combinatorial tools such as [[Ehrenfeucht–Fraïssé games|2RE7YAZK8R]] and locality arguments.
It is closely tied to computer science, through databases and descriptive complexity, where [[Fagin's theorem|BG8Q0BYSQ3]] matches a logic with a complexity class.

## Explanations

### L1 human

A finite structure is, for example, a finite [[graph|00C0BR7D7H]] or a finite database table.
Finite model theory asks which properties of such objects can be written as sentences of a given logic.
A typical result: "the graph is connected" cannot be written as a single first-order sentence about finite graphs.
The proof uses a game between two players that compares two graphs move by move.
Questions of this kind relate the power of a logic to the computing time needed to check a property.
</document>

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# Poincaré–Hopf theorem

## Statement

Let $M$ be a compact [[smooth manifold|F45VQ47GPV]] of dimension $n$ without boundary.
Let $v$ be a continuous [[vector field|0299VCW1TH]] on $M$ with only isolated zeros.
For a zero $x$ of $v$, choose a chart around $x$ and a small sphere $S$ around $x$ in it that encloses no other zero.
The *index* $\operatorname{ind}_x(v)$ is the [[degree|TMFG2P912R]] of the map $S \to S^{n-1}$, $u \mapsto v(u) / \norm{v(u)}$.
Then $v$ has finitely many zeros, and
$$\sum_{v(x) = 0} \operatorname{ind}_x(v) = \chi(M),$$
the [[Euler characteristic|RNDDWDAFJ5]] of $M$.

## Remarks

The index does not depend on the chart.
One proof first shows that the sum of indices does not depend on $v$.
For this it compares $v$ with a field on a tubular neighbourhood of $M$ in $\R^N$ and uses the degree of the Gauss map.
Then one computes the sum for the gradient of a Morse function, whose zeros are the critical points.
The index at a critical point of Morse index $\lambda$ is $(-1)^\lambda$, and the sum of these signs is $\chi(M)$ by Morse theory.
For a manifold with boundary the same formula holds if $v$ points outward along the boundary.

Example: on the sphere $S^2$, the gradient of the height function vanishes only at the two poles, a source and a sink, each of index $+1$, and $1 + 1 = 2 = \chi(S^2)$.
On the torus, the field tangent to one family of circles has no zero, and $\chi(T^2) = 0$.

Compactness is needed: the constant field on $\R^n$ has no zero, but $\chi(\R^n) = 1$.

## Explanations

### L1 human

Count the points where a flow on a closed surface stops, giving each one a sign or multiplicity according to how the flow turns around it.
The total does not depend on the flow; it is the Euler characteristic, a number that only depends on the shape of the surface.
On a sphere the total is $2$, so every flow on a sphere must stop somewhere.

### L2 human

The theorem links analysis, through vector fields, with topology, through the Euler characteristic.
It gives the [[hairy ball theorem|Z49RAKVCWG]] at once.
It shows that a closed manifold has a nowhere vanishing vector field only if $\chi(M) = 0$, and Hopf proved the converse for connected closed manifolds.
It can be derived from the Lefschetz fixed-point theorem applied to the flow of $v$.
It is closely related to the Gauss–Bonnet theorem and to the Euler class of the tangent bundle.
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# Arithmetic group

## Summary

An arithmetic group is a group of integer matrices cut out by polynomial equations, such as $\mathrm{SL}_n(\Z)$.
A group that agrees with such a group up to finite index is also arithmetic.
Arithmetic groups link number theory with the geometry of [[Lie groups|7JBRGCVJRB]].
For a semisimple group, a theorem of Borel and Harish-Chandra says that the arithmetic subgroups are [[lattices|BTYF5Y5PFY]] in the real Lie group.
The [[Margulis arithmeticity theorem|FEJK846N5X]] shows that in higher rank all irreducible lattices arise this way.

## Explanations

### L1 human

Take a group of real matrices defined by equations, such as the matrices of determinant $1$.
Keep only the matrices whose entries are integers.
The result, such as $\mathrm{SL}_2(\Z)$, is an arithmetic group.
It sits inside the big continuous group like the integer points $\Z^2$ sit inside the plane $\R^2$.
The [[modular group|VEKV1EM77B]] $\mathrm{SL}_2(\Z)$, which acts on the upper half plane, is the most famous example.
</document>

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# Darboux integral

## Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a bounded function.
For a [[partition|069P20KYQT]] $P$ given by $a = x_0 < x_1 < \cdots < x_n = b$, let $M_i = \sup_{[x_{i-1}, x_i]} f$ and $m_i = \inf_{[x_{i-1}, x_i]} f$, and form the [[upper and lower Darboux sums|0QRQVH0CRW]]
$$U(f, P) = \sum_{i=1}^{n} M_i (x_i - x_{i-1}), \qquad L(f, P) = \sum_{i=1}^{n} m_i (x_i - x_{i-1}).$$
The [[upper and lower Darboux integrals|09FQT057T7]] are $\overline{I}(f) = \inf_P U(f, P)$ and $\underline{I}(f) = \sup_P L(f, P)$.
The function $f$ is *Darboux integrable* on $[a, b]$ if $\overline{I}(f) = \underline{I}(f)$.
Its *Darboux integral* $\int_a^b f(x)\, dx$ is then this common value.

## Remarks

Example: $f(x) = x$ on $[0, 1]$.
For the partition $P_n$ into $n$ equal parts, $U(f, P_n) - L(f, P_n) = \sum_{i=1}^{n} \frac1n \cdot \frac1n = \frac1n$, so the two Darboux integrals agree, and the integral is $\lim_n U(f, P_n) = \lim_n \frac{n+1}{2n} = \frac12$.

Non-example: the [[Dirichlet function|03Z4JC4YRJ]], equal to $1$ on rationals and $0$ on irrationals in $[0, 1]$.
Every subinterval contains both kinds of numbers, so $U(f, P) = 1$ and $L(f, P) = 0$ for every $P$, and the Darboux integrals $1$ and $0$ differ.

Riemann's criterion: $f$ is Darboux integrable if and only if for every $\varepsilon > 0$ there is a partition $P$ with $U(f, P) - L(f, P) < \varepsilon$.
A bounded function is Darboux integrable exactly when it is [[Riemann integrable|0N1JQXZXHC]], with the same integral.
</document>

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# Association scheme

## Definition

Let $X$ be a nonempty [[finite set|070R982G4B]] and let $d \ge 1$.
An *association scheme* with $d$ classes on $X$ is a [[partition|08QMKTWF9T]] of $X \times X$ into nonempty relations $R_0, R_1, \dots, R_d$ such that:

1. $R_0 = \set{(x, x) : x \in X}$.
2. For each $i$, the relation $R_i^T = \set{(y, x) : (x, y) \in R_i}$ is equal to $R_{i'}$ for some $i'$.
3. For all $i, j, k$ there is an integer $p^k_{ij}$ such that for every $(x, y) \in R_k$ the number of $z \in X$ with $(x, z) \in R_i$ and $(z, y) \in R_j$ equals $p^k_{ij}$.

The scheme is *symmetric* if $R_i^T = R_i$ for every $i$, and *commutative* if $p^k_{ij} = p^k_{ji}$ for all $i, j, k$.

## Remarks

Many authors, following Bose and Mesner, require the scheme to be symmetric in the definition.
In terms of the [[adjacency matrices|08B9CX109G]] $A_i$ of the relations, clause 3 says $A_i A_j = \sum_k p^k_{ij} A_k$.
So the $A_i$ span an algebra of matrices, the Bose–Mesner algebra, which is commutative exactly when the scheme is commutative.

Example: on any finite set $X$ with at least $2$ points, let $R_0$ be the diagonal and $R_1$ the set of pairs of distinct points.
Clauses 1 and 2 hold, and clause 3 holds with $p^0_{11} = \abs{X} - 1$ and $p^1_{11} = \abs{X} - 2$, while every other $p^k_{ij}$ is $0$ or $1$.

Non-example: let $X = \set{a, b, c}$ with $R_0$ the diagonal, $R_1$ the edges of the path $a - b - c$ in both directions, and $R_2 = \set{(a, c), (c, a)}$.
Clause 3 fails for $i = j = 1$, $k = 0$: for $(a, a)$ there is one $z$ with $(a, z), (z, a) \in R_1$, but for $(b, b)$ there are two.

A symmetric scheme with $2$ classes is the same as a [[strongly regular graph|C22WGRP3T8]] that is neither complete nor edgeless, with $R_1$ as its edge relation.
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# Approximately finite-dimensional C*-algebra

## Summary

An approximately finite-dimensional (AF) C\*-algebra is a C*-algebra that is the closure of an increasing sequence of finite-dimensional C\*-subalgebras.
Finite-dimensional C\*-algebras are finite direct sums of matrix algebras, so an AF algebra is built from matrices by a limit process.
Bratteli described AF algebras by diagrams, and Elliott classified them by their ordered K_0 groups.
Examples are the compact operators, the UHF algebras such as the CAR algebra, and $C(X)$ for the Cantor set $X$.

## Explanations

### L1 human

Start with a matrix algebra, put it inside a bigger matrix algebra, then a bigger one, and so on, and take the closure of the union.
The resulting algebra is usually infinite-dimensional, but every element is a limit of matrices of finite size.
For example, $M_2 \subset M_4 \subset M_8 \subset \cdots$, with each matrix $a$ sent to the block matrix $\operatorname{diag}(a, a)$, gives the CAR algebra used for fermions in physics.
</document>

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# Descriptive geometry

## Summary

Descriptive geometry represents three-dimensional objects by two-dimensional drawings, usually several orthogonal projections, and solves spatial problems by constructions on these drawings.
Its classical form is due to Gaspard Monge.
There a point of space is given by its projections onto a horizontal and a vertical plane, drawn together on one sheet.
From such drawings one can construct true lengths, true angles, intersections of surfaces and shadows.
It was the theoretical basis of technical drawing in engineering and architecture.

## Explanations

### L1 human

An engineer's drawing of a machine part usually shows a top view and a front view, side by side, carefully lined up.
Each view alone loses one dimension, but together they fix the object completely.
Descriptive geometry is the set of rules for making such views and for reading answers from them.
Examples are the true length of a slanted rod or the line where two pipes meet.
It does by careful drawing what analytic geometry does with coordinates.
</document>

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# Iwasawa decomposition of a Lie group

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center and Lie algebra $\mathfrak{g}$.
Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the Cartan decomposition for a Cartan involution, as in the [[Cartan decomposition|A5JB9JQKR0]].
Let $K$ be the connected subgroup with Lie algebra $\mathfrak{k}$.
Let $\mathfrak{a} \subseteq \mathfrak{p}$ be a maximal abelian subspace.
For a linear form $\lambda \ne 0$ on $\mathfrak{a}$, put $\mathfrak{g}_\lambda = \set{X \in \mathfrak{g} : [H, X] = \lambda(H) X \text{ for all } H \in \mathfrak{a}}$, and let $\Sigma$ be the set of $\lambda$ with $\mathfrak{g}_\lambda \ne 0$, the restricted roots.
Fix $H_0 \in \mathfrak{a}$ with $\lambda(H_0) \ne 0$ for all $\lambda \in \Sigma$, put $\Sigma^+ = \set{\lambda \in \Sigma : \lambda(H_0) > 0}$ and $\mathfrak{n} = \bigoplus_{\lambda \in \Sigma^+} \mathfrak{g}_\lambda$.
Let $A$ and $N$ be the connected subgroups of $G$ with Lie algebras $\mathfrak{a}$ and $\mathfrak{n}$.
Then $A$ is abelian, $N$ is [[nilpotent|DK87YN458D]], both are closed and simply connected, and the map
$$K \times A \times N \to G, \quad (k, a, n) \mapsto kan,$$
is a diffeomorphism.

## Remarks

On the Lie algebra level, $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$ as vector spaces; this follows from the restricted root space decomposition and the fact that $\theta$ maps $\mathfrak{g}_\lambda$ to $\mathfrak{g}_{-\lambda}$.
One then shows that $AN$ is a closed simply connected solvable subgroup, and that $K \times AN \to G$ is a bijective local diffeomorphism.

Example: $G = \mathrm{SL}_n(\R)$ with $K = \mathrm{SO}(n)$.
Here $A$ is the group of diagonal matrices with positive entries and determinant $1$, and $N$ is the group of upper triangular matrices with $1$ on the diagonal.
The decomposition $g = kan$ is the Gram–Schmidt process applied to the columns of $g$, also known as the QR decomposition.

## Explanations

### L1 human

Every element of a semisimple Lie group is, in exactly one way, a rotation times a scaling times a shear.
For matrices, the scaling is a positive diagonal matrix and the shear is an upper triangular matrix with ones on the diagonal.
This is what Gram–Schmidt orthogonalisation does to the columns of an invertible matrix.

### L2 human

The Iwasawa decomposition shows that $G/K$ is diffeomorphic to the solvable group $AN$.
So the symmetric space $G/K$ is a solvable Lie group with a left-invariant metric.
The subgroup $MAN$, with $M$ the centralizer of $\mathfrak{a}$ in $K$, is the minimal parabolic subgroup.
Inducing to $G$ the representations of $MAN$ that are trivial on $N$ gives the principal series representations.
The dimension of $\mathfrak{a}$ is the real rank of $G$.
</document>

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# Chu–Vandermonde identity

## Statement

Let $n \ge 0$ be an integer.

1. For all $x, y \in \C$,
$$\binom{x + y}{n} = \sum_{k=0}^{n} \binom{x}{k} \binom{y}{n - k},$$
where $\binom{x}{k} = \frac{x(x-1)\cdots(x-k+1)}{k!}$ is the [[binomial coefficient|0GBZM0MT0F]].
2. Equivalently, for all $b, c \in \C$ with $c \notin \set{0, -1, -2, \ldots}$,
$${}_2F_1(-n, b; c; 1) = \frac{(c - b)_n}{(c)_n},$$
where ${}_2F_1$ is the terminating [[hypergeometric series|428M453CR2]] and $(x)_n$ is the [[Pochhammer symbol|YZHM3HSM0F]].

## Remarks

Both sides of part 1 are polynomials in $x$ and $y$.
For natural numbers $x = m$ and $y = p$ the identity is [[Vandermonde's identity|01JADB9VW5]], proved by counting $n$-element subsets of a union of an $m$-set and a $p$-set.
A polynomial in two variables that vanishes on $\N \times \N$ is zero, so part 1 holds for all $x, y$.
Part 2 is also the case $a = -n$ of [[Gauss's hypergeometric theorem|1FC0T0GCCX]], since $\frac{\Gamma(c) \Gamma(c + n - b)}{\Gamma(c + n) \Gamma(c - b)} = \frac{(c - b)_n}{(c)_n}$.

Example: with $x = y = \frac12$ and $n = 2$, the left side is $\binom{1}{2} = 0$, and the right side is $\binom{1/2}{2} + \binom{1/2}{1}^2 + \binom{1/2}{2} = -\frac18 + \frac14 - \frac18 = 0$.

The name records that the identity was known to Zhu Shijie long before Vandermonde.

## Explanations

### L1 human

Vandermonde's identity counts the ways to choose $n$ people from two groups: choose $k$ from the first and $n - k$ from the second.
The Chu–Vandermonde identity says that the same formula stays true when the group sizes are replaced by arbitrary numbers, even fractions or complex numbers.
This works because both sides are polynomials that agree at all whole numbers.

### L2 human

In hypergeometric language it is the summation of a terminating ${}_2F_1$ at $z = 1$.
It is the simplest case of a whole hierarchy of terminating summations, followed by the Pfaff–Saalschütz formula for ${}_3F_2$ and Dougall's formula.
Its $q$-analogue is the [[q-Vandermonde identity|MBRSCQRGGH]].
</document>

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# Universal property of the free magma

## Statement

Let $X$ be a set, let $M(X)$ be the [[free magma|ZZF8XPBG3Z]] on $X$, and let $\iota \colon X \to M(X)$ be the inclusion of the generators.
Let $N$ be a [[magma|9689SFPVE9]] and let $f \colon X \to N$ be a map.
Then there is exactly one magma [[homomorphism|KBXSZC6BV1]] $h \colon M(X) \to N$ such that $h \circ \iota = f$.

## Remarks

Existence: define $h$ by recursion on the construction of elements, with $h(x) = f(x)$ for $x \in X$ and $h(u \cdot v) = h(u) \cdot h(v)$.
This recursion is valid because each element of $M(X)$ is either a generator or a product $u \cdot v$ for exactly one pair $(u, v)$.
Uniqueness: two homomorphisms that agree on $X$ agree on all elements, by induction on the construction.
Example: with $N = (\Z, +)$ and $f(x) = 1$ for all $x$, the homomorphism $h$ sends each element to its number of leaves.
With $N = (\N, \max(a, b) + 1)$ and $f(x) = 0$, the homomorphism $h$ sends each element to the height of its binary tree.
In Mathlib this is `FreeMagma.lift`, an equivalence between maps $X \to N$ and magma homomorphisms $M(X) \to N$.

## Explanations

### L1 human

To map the free magma into another magma, it is enough to say where each letter goes.
The image of every bracketed product is then forced: it is the same product, bracketed the same way, of the images of the letters.
Since the free magma has no rules, nothing can go wrong.

### L2 human

The proposition says that the free magma is the free algebra of the variety of all magmas.
The free semigroup on $X$ has the same kind of universal property, and it is the quotient of $M(X)$ by the congruence generated by associativity.
Recursive definitions on binary trees, such as size and height, are instances of this proposition.
</document>

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# Starred transform

## Summary

The starred transform is the [[Laplace transform|0V49JEZW06]] of the impulse-sampled version of a continuous-time signal.
For a sampling period $T$ it equals $X^*(s) = \sum_{n \ge 0} x(nT) e^{-nTs}$.
It is the [[Z-transform|5FY0JQGHYZ]] of the samples with $z = e^{sT}$, and so it links the continuous-time analysis of a system with its discrete-time model.
It is used in the theory of sampled-data control systems.

## Explanations

### L1 human

A digital controller sees a continuous signal only at the instants $0, T, 2T, \ldots$.
One models this by a train of spikes at those instants, each spike weighted by the value of the signal there.
The starred transform is the Laplace transform of that spike train.
For $x(t) = 1$ it is $\sum_{n \ge 0} e^{-nTs} = 1/(1 - e^{-Ts})$ for $\operatorname{Re} s > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/4YSCK1DF26.md" tag="4YSCK1DF26" kind="concept">
# Subvariety

## Summary

A subvariety of a [[variety|AJNZJD8442]] $\mathcal{V}$ is a variety of the same signature that is contained in $\mathcal{V}$.
It is obtained from $\mathcal{V}$ by adding more identities.
Abelian groups form a subvariety of groups, and distributive lattices form a subvariety of lattices.
The subvarieties of a variety form a lattice, which organises the classification of varieties.

## Explanations

### L1 human

A variety is described by equations; a subvariety obeys those equations and some extra ones.
Adding the equation $xy = yx$ to the group axioms gives the abelian groups.
Adding $x^2 = e$ as well gives the abelian groups in which every element is its own inverse, the vector spaces over the field with two elements.
Each extra equation can only shrink the class.
</document>

<document source="https://tamnd.github.io/math2/node/4YZVFGJVSZ.md" tag="4YZVFGJVSZ" kind="concept">
# Capacity of a set

## Summary

The Newtonian capacity of a compact set $K \subseteq \R^n$, $n \ge 3$, measures how much electric charge $K$ can hold at unit potential.
It is the least Dirichlet energy of a function that is at least $1$ on $K$ and decays at infinity.
Capacity is the right notion of size in [[potential theory|9PM8Q2E30Q]]: sets of capacity zero are the negligible sets for harmonic functions and for Sobolev functions.
In the plane its role is played by [[logarithmic capacity|DBQXRTC0VN]].

## Explanations

### L1 human

Imagine $K$ as a piece of metal held at voltage $1$, with voltage $0$ far away.
The capacity is the amount of charge that $K$ then carries, and it depends on the shape of $K$ and not only on its volume.
A ball of radius $r$ in $\R^3$ has capacity $4 \pi r$ in the normalisation used here.
A single point and a line segment in $\R^3$ have capacity zero, while a flat disc has positive capacity.
So capacity sees thin sets in a different way from volume.
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<document source="https://tamnd.github.io/math2/node/4Z1YXT957Z.md" tag="4Z1YXT957Z" kind="object">
# Modular lambda function

## Description

Let $\mathbb{H} = \set{\tau \in \C : \operatorname{Im}\tau > 0}$ be the [[upper half-plane|X6XGGH5FPW]], and for $\tau \in \mathbb{H}$ let $q = e^{\pi i \tau}$.
Let $\theta_2$ and $\theta_3$ be the [[theta functions|2F5CYQJVAF]]
$$\theta_2(\tau) = \sum_{n \in \Z} q^{(n + 1/2)^2}, \qquad \theta_3(\tau) = \sum_{n \in \Z} q^{n^2}.$$
The *modular lambda function* is
$$\lambda(\tau) = \frac{\theta_2(\tau)^4}{\theta_3(\tau)^4}, \qquad \tau \in \mathbb{H}.$$

1. $\lambda$ is holomorphic on $\mathbb{H}$ and never takes the values $0$ and $1$.
2. $\lambda(\tau + 1) = \dfrac{\lambda(\tau)}{\lambda(\tau) - 1}$ and $\lambda(-1/\tau) = 1 - \lambda(\tau)$.
3. $\lambda$ is invariant under the principal [[congruence subgroup|1H6R856P3C]] $\Gamma(2)$ of level $2$.
4. $\lambda(\tau) = 16q + O(q^2)$ as $\operatorname{Im}\tau \to \infty$.

## Remarks

The transformation rules in point 2 follow from those of the theta functions together with Jacobi's identity $\theta_3^4 = \theta_2^4 + \theta_4^4$, where $\theta_4(\tau) = \sum_{n} (-1)^n q^{n^2}$.
For example, $\tau \mapsto \tau + 1$ changes $\theta_2^4$ into $-\theta_2^4$ and exchanges $\theta_3$ and $\theta_4$, which gives $-\theta_2^4/\theta_4^4 = \lambda/(\lambda - 1)$.
The map $\lambda \colon \mathbb{H} \to \C \setminus \set{0, 1}$ is a holomorphic covering map.
An entire function that omits $0$ and $1$ lifts through this covering to an entire function with values in $\mathbb{H}$.
Composing with a Möbius map from $\mathbb{H}$ onto the unit disc gives a bounded entire function, which is therefore constant; this is the classical proof of [[Picard's little theorem|R2PHM118NG]].
The [[j-invariant|FEC2XH822R]] is a rational function of $\lambda$:
$$j = 256\,\frac{(1 - \lambda + \lambda^2)^3}{\lambda^2 (1 - \lambda)^2}.$$

## Explanations

### L1 human

The lambda function is a holomorphic function on the upper half-plane with many symmetries.
It never takes the values $0$ and $1$, and every other complex value is taken infinitely often.
This makes it the tool that shows a nonconstant entire function can miss at most one complex value.
</document>

<document source="https://tamnd.github.io/math2/node/4Z6EED4616.md" tag="4Z6EED4616" kind="definition">
# Ore localization

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity and let $S \subseteq R$ be a [[multiplicative set|AN9SKEZ3VY]].
A *right ring of fractions* of $R$ with respect to $S$ is a ring $Q$ together with a [[ring homomorphism|0VTH8KPCG2]] $\varphi \colon R \to Q$ such that:

1. $\varphi(s)$ is a [[unit|0XVEZDE9J8]] of $Q$ for every $s \in S$;
2. every element of $Q$ has the form $\varphi(a) \varphi(s)^{-1}$ with $a \in R$ and $s \in S$;
3. the kernel of $\varphi$ is $\set{a \in R : a t = 0 \text{ for some } t \in S}$.

When it exists, it is unique up to a unique isomorphism compatible with $\varphi$, and it is written $R S^{-1}$.
It is called the *Ore localization* of $R$ at $S$.

## Remarks

Ore's theorem: $R S^{-1}$ exists if and only if $S$ is a right denominator set.
This means that $S$ satisfies the right [[Ore condition|TJP4ARC2X7]] and is right reversible: if $s a = 0$ with $s \in S$, then $a t = 0$ for some $t \in S$.
In that case one may build $R S^{-1}$ as the set of classes of pairs $(a, s) \in R \times S$, where $(a, s) \sim (b, t)$ when there are $c, d \in R$ with $s c = t d \in S$ and $a c = b d$.
It then has the universal property: each ring homomorphism $f \colon R \to B$ with $f(S)$ consisting of units factors uniquely through $\varphi$.

Example: if $R$ is commutative, every multiplicative set is a right denominator set, and $R S^{-1}$ is the usual [[localization|4C1AE84N5N]] $S^{-1} R$.
Clause 3 matches the kernel description there.

Non-example: for the free algebra $R = k\langle x, y \rangle$ over a field $k$ and $S = R \setminus \set{0}$, no right ring of fractions exists.
Suppose $\varphi$ satisfied clauses 1 to 3; then $\varphi$ would be injective by clause 3.
By clause 2, $\varphi(y)^{-1} \varphi(x)$ would equal $\varphi(b) \varphi(t)^{-1}$, giving $x t = y b$ with $t \ne 0$, which is impossible.

The left ring of fractions $S^{-1} R$ is defined in the same way with elements $\varphi(s)^{-1} \varphi(a)$.
When both exist, they are isomorphic as rings over $R$.
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<document source="https://tamnd.github.io/math2/node/4Z73AFN7WT.md" tag="4Z73AFN7WT" kind="concept">
# Monotone likelihood ratio

## Summary

Consider a family of densities indexed by a real parameter.
It has a monotone likelihood ratio in a statistic $T$ when larger parameters make larger values of $T$ relatively more likely.
The ratio of the densities at two parameter values is then a nondecreasing function of $T$.
This property makes one-sided testing simple: tests that reject for large $T$ are optimal.
One-parameter exponential families with an increasing natural parameter have this property.

## Explanations

### L1 human

Think of $T$ as a signal that points towards larger parameter values.
In a family with a monotone likelihood ratio, the larger $T$ is, the more the data favour a larger parameter over a smaller one, for every pair of parameters.
For coin tosses with head probability $p$, the number of heads $T$ is such a signal.
Seeing more heads always favours a larger $p$ over a smaller one.
</document>

<document source="https://tamnd.github.io/math2/node/4Z75CMNQ1P.md" tag="4Z75CMNQ1P" kind="definition">
# Newtonian potential

## Definition

Let $n \ge 3$ and let $\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\R^n$ with finite total mass and compact [[support|FQTXV6BQZG]].
The *Newtonian potential* of $\mu$ is the function $U^\mu \colon \R^n \to [0, +\infty]$,
$$U^\mu(x) = \int \frac{d\mu(y)}{\abs{x - y}^{n-2}}.$$
It is the [[potential|JHKSF3XFY2]] of $\mu$ with respect to the [[Newtonian kernel|6TJTWQ5HCR]].

## Remarks

Example: for $n = 3$ and $\mu = m \, \delta_a$ a point mass, $U^\mu(x) = m / \abs{x - a}$.
Non-example: in $\R^3$ the function $u(x) = \abs{x}^2$ is not the Newtonian potential of any such $\mu$, since a Newtonian potential tends to $0$ at infinity.
The function $U^\mu$ is lower semicontinuous, [[harmonic|0DEB0JMN0W]] on $\R^n \setminus \operatorname{supp} \mu$, and $U^\mu(x) \abs{x}^{n-2} \to \mu(\R^n)$ as $\abs{x} \to \infty$.
If $\mu = f \, dx$ with $f \ge 0$ twice continuously differentiable with compact support, then $U^\mu$ is twice continuously differentiable and $\Delta U^\mu = -(n-2)\sigma_n f$, where $\sigma_n$ is the area of the unit sphere.
Equivalently, $U^\mu = (n-2)\sigma_n \, \Phi * \mu$, where $\Phi$ is the [[fundamental solution of the Laplacian|BSJ8X2XRN9]].
Some texts call $\Phi * f$ itself the Newtonian potential of $f$, and use the same name in the plane with the logarithmic kernel.
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<document source="https://tamnd.github.io/math2/node/4ZTTPDXCQT.md" tag="4ZTTPDXCQT" kind="definition">
# Affine differential geometry

## Definition

Let $n \ge 1$.
The *equi-affine group* of $\R^{n+1}$ is the group of maps $x \mapsto Ax + b$ with $A$ in the [[general linear group|0FCHE902D7]] $GL(n+1, \R)$, $\det A = 1$ and $b \in \R^{n+1}$.
*Affine differential geometry* is the study of the properties of smooth curves and hypersurfaces in $\R^{n+1}$ that are invariant under the equi-affine group.

## Remarks

For a [[regular curve|06JMR2XWXW]] $\gamma \colon I \to \R^2$ with $\det(\gamma', \gamma'') > 0$, the *affine arc length* is
$$\sigma(t) = \int_{t_0}^{t} \det\big(\gamma'(u), \gamma''(u)\big)^{1/3} \, du.$$
It is invariant: if $\Phi(x) = Ax + b$ with $\det A = 1$, then $\det(A\gamma', A\gamma'') = \det(\gamma', \gamma'')$.
For the parabola $\gamma(t) = (t, t^2/2)$, one has $\gamma' = (1, t)$ and $\gamma'' = (0, 1)$, so the determinant is $1$ and the affine arc length from $0$ is $t$.

Euclidean arc length is not an affine invariant.
The shear $(x, y) \mapsto (x + y, y)$ has determinant $1$, and it maps the segment from $(0,0)$ to $(0,1)$, of length $1$, to the segment from $(0,0)$ to $(1,1)$, of length $\sqrt{2}$.

For a nondegenerate hypersurface, the basic invariants are the affine normal vector field and the induced Blaschke metric.
Some authors use the full affine group, with any invertible $A$; then volume is no longer preserved and the invariants change.

## Explanations

### L1 human

Affine differential geometry replaces the rigid motions of Euclidean geometry by the larger group of area- or volume-preserving affine maps.
It asks which features of a curved shape survive all of these maps.
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<document source="https://tamnd.github.io/math2/node/4ZY2D6C8A6.md" tag="4ZY2D6C8A6" kind="statement">
# Benedicks' theorem

## Statement

Let $n \ge 1$ and let $f \in L^1(\R^n)$.
Let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\, dx$ be its [[Fourier transform|DPVM34ADZV]].
Suppose that the sets $\set{x \in \R^n : f(x) \ne 0}$ and $\set{\xi \in \R^n : \hat f(\xi) \ne 0}$ both have finite [[Lebesgue measure|2Z2QVWZ262]].
Then $f = 0$ almost everywhere.

## Remarks

The theorem strengthens the [[qualitative uncertainty principle|S1FB2K14NV]], which assumes bounded sets instead of sets of finite measure.
With finite measure only, the analytic continuation argument is no longer available.
The proof periodizes instead.
After a dilation one may assume that the set where $f \ne 0$ has measure less than $1$.
For almost every $\xi$, the $\Z^n$-periodization of $x \mapsto f(x) e^{-2\pi i x \cdot \xi}$ has the numbers $\hat f(\xi + k)$, $k \in \Z^n$, as Fourier coefficients, as in the [[Poisson summation formula|GARGJYSGFG]], and only finitely many of them are nonzero.
So this periodization is a trigonometric polynomial, and it vanishes on a set of positive measure in the torus.
A trigonometric polynomial that vanishes on a set of positive measure is zero, so $\hat f(\xi + k) = 0$ for all $k$ and almost every $\xi$, and $f = 0$.

The theorem is due to M. Benedicks.
A version for $L^2(\R^n)$ was proved independently by W. O. Amrein and A. M. Berthier.

Example: the Gaussian $e^{-\pi \abs{x}^2}$ is nowhere zero, as is its transform, so both sets have infinite measure, consistent with the theorem.

## Explanations

### L1 human

A nonzero function cannot live on a set of finite total size while its spectrum also lives on a set of finite total size.
The sets may be scattered and unbounded; only their total size matters.

### L2 human

This is the measure-theoretic form of the principle that $f$ and $\hat f$ cannot both be concentrated.
A quantitative version, due to Nazarov in dimension one and extended to $\R^n$, bounds $\norm{f}_2$ by a constant times the $L^2$ norms of $f$ and $\hat f$ outside such sets.
A pair of sets with this property is called an annihilating pair, and the theorem says that any two sets of finite measure form one.
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<document source="https://tamnd.github.io/math2/node/507C34W0ZT.md" tag="507C34W0ZT" kind="concept">
# Holomorphic function of several variables

## Summary

A holomorphic function of several variables is a function on an open subset of $\C^n$ that is complex differentiable at every point.
It extends the notion of a [[holomorphic function|0NAHXR1FRH]] of one variable.
Such a function is holomorphic in each variable separately and infinitely differentiable.
It is locally the sum of a convergent [[power series in several variables|R3QTCPTJK7]].
These functions are the main objects of the theory of [[several complex variables|BH57D2FBNR]].

## Explanations

### L1 human

A function of one complex variable is holomorphic when it has a complex derivative.
For several variables, one asks that near each point the function is well approximated by a complex linear function.
Polynomials in $z_1, \dots, z_n$, such as $z_1^3 - 2 z_1 z_2$, are holomorphic on all of $\C^n$.
The function $1/(1 - z_1 - z_2)$ is holomorphic where $z_1 + z_2 \ne 1$.
A function that involves $\bar z_1$, such as $\abs{z_1}^2$, is not holomorphic.
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<document source="https://tamnd.github.io/math2/node/5083DHSK82.md" tag="5083DHSK82" kind="statement">
# Birkhoff's HSP theorem

## Statement

Let $K$ be a class of algebras of a fixed signature $\sigma$.
Then $K$ is a [[variety|AJNZJD8442]] if and only if $K$ is closed under the [[class operators|S7MJVSP79N]] $H$, $S$ and $P$.
Here a variety is the class of all models of some set of [[identities|PKD440G2XW]].

## Remarks

One direction is a direct check: identities are preserved by homomorphic images, by subalgebras and by direct products.
For the converse, let $\Sigma$ be the set of identities that hold in every member of $K$, and let $A$ satisfy $\Sigma$.
The free algebra of $HSP(K)$ on a set of the size of $A$ is a subalgebra of a product of members of $K$.
Since $A$ satisfies $\Sigma$, the algebra $A$ is a homomorphic image of it.
So $A \in HSP(K) = K$.
Example: groups form a variety, and the theorem confirms that subgroups, quotient groups and products of groups are groups.
Closure under products cannot be dropped.
The class of groups with at most two elements is closed under $H$ and $S$, but it does not contain $\Z / 2\Z \times \Z / 2\Z$.

## Explanations

### L1 human

There are two ways to describe a class of algebras.
One is by the equations its members obey.
The other is by the constructions it is closed under.
Birkhoff's theorem says that being described by equations is the same as being closed under taking images, subalgebras and products.

### L2 human

The theorem lets one decide whether a class is a variety without writing identities.
To show that a class is not equationally definable, one finds a product, subalgebra or image that leaves the class.
For example, torsion-free abelian groups are not a variety, because their quotient $\Z / 2\Z$ has torsion.
The proof introduces free algebras of $HSP(K)$, which are the main tool of the theory.
A finer form, Tarski's formula $V(K) = HSP(K)$, describes the variety generated by any class.
</document>

<document source="https://tamnd.github.io/math2/node/509ZWA0R26.md" tag="509ZWA0R26" kind="statement">
# Paris–Harrington theorem

## Statement

Call a finite set $Y \subseteq \N$ *large* if $\abs{Y} \ge \min Y$.
The *strengthened finite Ramsey theorem* $\mathrm{PH}$ is the following statement.
For all positive integers $n$, $k$, $m$ there is an integer $N$ with this property.
For every colouring of the $n$-element subsets of $\set{1, 2, \ldots, N}$ with $k$ colours, there is a large set $Y \subseteq \set{1, \ldots, N}$ with $\abs{Y} \ge m$.
Moreover, all $n$-element subsets of $Y$ have the same colour.

1. $\mathrm{PH}$ is true; it follows from the infinite Ramsey theorem.
2. $\mathrm{PH}$, written as a sentence of first-order arithmetic, is not provable in [[Peano arithmetic|0QFT7Z76N5]] $\mathsf{PA}$.

## Remarks

Part 1 is a compactness argument.
If $\mathrm{PH}$ failed for some $n, k, m$, the bad colourings for larger and larger $N$ would fit together by König's lemma.
They would give a colouring of the $n$-element subsets of $\N$ whose restriction to each $\set{1, \ldots, N}$ is bad.
The infinite Ramsey theorem gives an infinite homogeneous set $H$ for it.
An initial segment of $H$ with $\max(m, \min H)$ elements is then large and homogeneous, a contradiction.
Part 2 is proved with models.
In a non-standard model of $\mathsf{PA}$, $\mathrm{PH}$ gives initial segments that are themselves models of $\mathsf{PA}$.
From this, $\mathsf{PA}$ plus $\mathrm{PH}$ proves the consistency of $\mathsf{PA}$.
The function that sends $(n, k, m)$ to the least $N$ grows faster than every function that $\mathsf{PA}$ proves total.

Without the word "large", the statement is the ordinary finite Ramsey theorem, which is provable in $\mathsf{PA}$.

## Explanations

### L1 human

The theorem gives a natural statement about colouring finite sets that is true but cannot be proved from the usual axioms of arithmetic.
It differs from the finite Ramsey theorem by one small demand: the same-coloured set must have at least as many elements as its smallest member.
This small change makes the required numbers grow so fast that arithmetic cannot prove they exist.

### L2 human

Paris and Harrington proved the theorem in 1977.
It was one of the first natural examples from combinatorics of a true statement independent of $\mathsf{PA}$, after the artificial sentences of [[Gödel's incompleteness theorems|X9WYNYE29C]].
[[Goodstein's theorem|XQMNJF241J]] is another example of the same kind.
</document>

<document source="https://tamnd.github.io/math2/node/50D3KZHQ7B.md" tag="50D3KZHQ7B" kind="definition">
# Partial algebra

## Definition

Let $\Sigma$ be a signature, that is, a set of function symbols, each with an [[arity|MSXX2P86CY]] $n \ge 0$.
A *partial $\Sigma$-algebra* is a [[set|0SJ3RX9TXA]] $A$ with the following data for each symbol $f \in \Sigma$ of arity $n$.
It has a subset $\operatorname{dom}(f^A)$ of the [[Cartesian power|09CEDJYBKP]] $A^n$ and a map $f^A \colon \operatorname{dom}(f^A) \to A$.
The map $f^A$ is a *partial operation* on $A$, and $f^A(a_1, \dots, a_n)$ is defined exactly when $(a_1, \dots, a_n) \in \operatorname{dom}(f^A)$.

## Remarks

When $\operatorname{dom}(f^A) = A^n$ for every $f$, the partial algebra is an ordinary (total) algebra.

Example: the natural numbers $\N$ with subtraction, where $m - n$ is defined exactly when $m \ge n$.
The domain is $\set{(m, n) \in \N^2 : m \ge n}$, and on it $m - n \in \N$, so subtraction is a partial operation on $\N$.
It is not total, since $(0, 1)$ is not in its domain.

Non-example: subtraction on all of $\N^2$ with values in $\Z$ is not a partial operation on $\N$, since its value $0 - 1 = -1$ does not lie in the carrier $\N$.

A field $K$ with the partial operation $x \mapsto x^{-1}$ defined on $K \setminus \set{0}$ is a partial algebra for the signature $\set{+, \cdot, -, ^{-1}, 0, 1}$.
Homomorphisms, subalgebras and congruences admit several inequivalent versions for partial algebras, for example weak and strong homomorphisms, so the version in use must be stated.
</document>

<document source="https://tamnd.github.io/math2/node/5115KV924G.md" tag="5115KV924G" kind="definition">
# Linear difference equation

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $k \ge 1$ be an integer.
Let $a_0, a_1, \dots, a_k$ and $b$ be sequences in $K$, indexed by $n \in \N$, with $a_k(n) \ne 0$ for all $n$.
The *linear difference equation* with these coefficients is the equation
$$a_k(n)\, u_{n+k} + a_{k-1}(n)\, u_{n+k-1} + \dots + a_0(n)\, u_n = b(n) \qquad (n \ge 0)$$
for an unknown sequence $(u_n)_{n \ge 0}$ in $K$.
It is *homogeneous* if $b(n) = 0$ for all $n$, and *inhomogeneous* otherwise.
It has *constant coefficients* if each $a_i$ is a constant sequence.

## Remarks

Since $a_k(n) \ne 0$, the equation can be solved for $u_{n+k}$, so it is a [[recurrence relation|07TZV87Y0T]] of order at most $k$.
Example: $u_{n+1} - (n+1) u_n = 0$ is a homogeneous linear difference equation with $k = 1$, $a_1 = 1$ and $a_0(n) = -(n+1)$; with $u_0 = 1$ its solution is $u_n = n!$.
Non-example: $u_{n+1} = u_n^2$ is not linear, because the unknown appears squared; the sequences $u = (1, 1, 1, \dots)$ and $2u = (2, 2, 2, \dots)$ show that multiples of solutions need not be solutions.
In terms of the [[shift operator|91QMF4R1CY]] $E$, the equation reads $L u = b$ with $L = a_k E^k + \dots + a_0$, a linear operator on sequences.
The constant coefficient case is the [[linear recurrence with constant coefficients|081THG13JZ]].
</document>

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# Volterra integro-differential equation

## Summary

A Volterra integro-differential equation is an [[integro-differential equation|YRKW2KX2CW]] in which the integral runs only over the past.
The integral goes from a fixed initial time up to the current time.
So the rate of change at time $t$ depends on the present state and on the history up to $t$, not on the future.
Together with an initial condition it is equivalent to a [[Volterra integral equation|DND8CBEVXT]], and under Lipschitz conditions it has a unique solution.
Such equations model materials with memory, such as viscoelastic bodies, and populations whose growth depends on past sizes.

## Explanations

### L1 human

Imagine a spring made of a material that remembers how it was stretched before.
Its force now depends on its current stretch and on a weighted sum of all past stretches.
Newton's law then gives an equation with both a derivative and an integral over the past: a Volterra integro-differential equation.
A small example is $x'(t) = \int_0^t x(s) \, ds$ with $x(0) = 1$, whose solution is $x(t) = \cosh t$.
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<document source="https://tamnd.github.io/math2/node/5152FTFE13.md" tag="5152FTFE13" kind="definition">
# Thom space

## Definition

Let $\pi \colon E \to B$ be a real vector bundle of rank $n$ with a continuous [[inner product|0NB0AWGJBW]] on its fibers.
Let $D(E) = \set{v \in E : \abs{v} \le 1}$ be the disc bundle and $S(E) = \set{v \in E : \abs{v} = 1}$ the unit [[sphere bundle|3VQ2V5HH3R]].
The *Thom space* of $E$ is the [[quotient space|0BR3FY5Q8M]]
$$\operatorname{Th}(E) = D(E) / S(E),$$
in which $S(E)$ is collapsed to one point, the base point.

## Remarks

Up to homeomorphism, $\operatorname{Th}(E)$ does not depend on the inner product.
If $B$ is compact Hausdorff, $\operatorname{Th}(E)$ is homeomorphic to the [[one-point compactification|0GWN3CMRK8]] of $E$.
The notations $T(E)$, $B^E$ and $M(E)$ are also used.

Example: if $B$ is a point, then $E = \R^n$, $D(E) = D^n$ and $S(E) = S^{n-1}$, so $\operatorname{Th}(E) = D^n / S^{n-1} \cong S^n$.
More generally, for the trivial bundle $B \times \R^n$ over a compact space $B$, the Thom space is $(B \times D^n)/(B \times S^{n-1})$, the $n$-fold reduced suspension of $B$ with a disjoint base point added.
The Thom space of the tautological line bundle over $\R P^m$ is $\R P^{m+1}$.

Non-example: the disc bundle $D(E)$ without the collapse is not the Thom space; it deformation retracts onto the zero section, so it is homotopy equivalent to $B$.
For $B$ a point and $n \ge 1$ this gives a disc, which is contractible, while $\operatorname{Th}(E) = S^n$ is not.
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<document source="https://tamnd.github.io/math2/node/51G4QJZ345.md" tag="51G4QJZ345" kind="concept">
# Loewner differential equation

## Summary

The Loewner differential equation describes a family of conformal maps that change continuously in time.
Each map sends the complement of a growing set onto a fixed reference domain.
The whole family is driven by a single real or circle-valued function of time, the driving function.
Charles Loewner introduced it to prove the case $n = 3$ of the [[Bieberbach conjecture|ZMRFSYVWQT]].
Louis de Branges used it in the proof of the full conjecture.
With a Brownian motion as driving function it defines the [[Schramm–Loewner evolution|AQ75M2JTB2]].

## Explanations

### L1 human

Picture a crack growing into the upper half-plane from a point on the real axis.
At each time, there is a conformal map that opens the cracked half-plane back into the whole half-plane.
These maps change smoothly in time, and the Loewner equation says how.
The speed of the map at a point depends only on the value of the map there and on where the tip of the crack is sent.
For a straight vertical crack of height $2\sqrt{t}$ at $0$, the map is $z \mapsto \sqrt{z^2 + 4t}$.
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<document source="https://tamnd.github.io/math2/node/51GRFQ4WY6.md" tag="51GRFQ4WY6" kind="concept">
# Rademacher functions

## Summary

The Rademacher functions $r_1, r_2, \dots$ on $[0, 1)$ are square waves that record the binary digits of a point.
The value $r_k(t)$ is $1$ or $-1$ according as the $k$-th binary digit of $t$ is $0$ or $1$.
They form an orthonormal system in $L^2([0, 1))$ which is not complete.
For [[Lebesgue measure|2Z2QVWZ262]] they are independent random signs, so they model coin tossing inside analysis.
Their finite products are the [[Walsh functions|F12A5QKW8Z]].

## Explanations

### L1 human

Split $[0, 1)$ into halves and put $+1$ on the left half and $-1$ on the right: that is $r_1$.
Split into quarters and alternate $+1, -1, +1, -1$: that is $r_2$.
Each next function flips twice as fast.
If one picks $t$ at random, the values $r_1(t), r_2(t), \dots$ behave like fair coin tosses, one for each binary digit of $t$.
</document>

<document source="https://tamnd.github.io/math2/node/51PH7Z03TJ.md" tag="51PH7Z03TJ" kind="concept">
# Legendre integral transform

## Summary

The Legendre integral transform sends a function $f$ on $[-1, 1]$ to the sequence of numbers $\int_{-1}^{1} P_n(x) f(x)\, dx$, where $P_n$ is the $n$-th [[Legendre polynomial|QFYGJVA996]].
These numbers are, up to the factors $\frac{2n+1}{2}$, the coefficients of the [[Fourier–Legendre series|RK2G9HN63E]] of $f$, so the series is the inverse transform.
It plays for the Legendre differential operator the role that Fourier coefficients play for $d^2/dx^2$.
It turns Legendre's differential equation into an algebraic equation.
It should not be confused with the Legendre transformation of convex analysis.

## Explanations

### L1 human

The Legendre polynomials $1, x, \frac{3x^2 - 1}{2}, \dots$ are like a set of perpendicular directions for functions on $[-1, 1]$.
The transform measures how much of each direction a function contains.
For example, the function $f(x) = x$ is exactly $P_1$, so its transform is $\frac{2}{3}$ at $n = 1$ and $0$ at every other $n$.
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<document source="https://tamnd.github.io/math2/node/51YVV85GWB.md" tag="51YVV85GWB" kind="concept">
# Birational geometry

## Summary

Birational geometry is the part of algebraic geometry that studies varieties up to [[birational equivalence|KA5890R9YM]].
Two varieties are birationally equivalent when they become isomorphic after removing lower-dimensional closed subsets, or equivalently when they have isomorphic [[function fields|2D5SS6WM14]].
Its main questions are which varieties are [[rational|XV4K9GREJ9]], and how to choose a simplest variety in each birational class.
The [[minimal model program|Q8CTDCZH01]] is its central project in higher dimensions.

## Explanations

### L1 human

In birational geometry one is allowed to cut out a curve or a point and glue in something else, as long as the bulk of the variety stays the same.
The plane $\mathbb{P}^2$ and the quadric surface $\mathbb{P}^1 \times \mathbb{P}^1$ are not isomorphic, but they agree on dense open subsets, so they are birational.
[[Blowing up|C4KGYAFPN7]] a point is the basic move: it replaces a point by a line of directions.
Numbers that do not change under such moves, such as the geometric genus or the Kodaira dimension, are the main tools to tell birational classes apart.
</document>

<document source="https://tamnd.github.io/math2/node/5252P3NHT7.md" tag="5252P3NHT7" kind="concept">
# Selberg sieve

## Summary

The Selberg sieve is an upper bound sieve of Atle Selberg.
It replaces the indicator of "no prime factor below $z$" by the square of a sum of real weights.
It then chooses the weights to minimise the main term, which is a quadratic form.
It gives sharp upper bounds with a simple remainder term, for example the Brun–Titchmarsh bound for primes in arithmetic progressions.
It is also called the $\Lambda^2$ sieve.

## Explanations

### L1 human

A sieve wants to count the numbers in a list that are not divisible by any small prime.
Selberg's trick is to use the fact that a square is never negative.
He builds, for each number, a square of a sum of weights that equals $1$ when the number survives and is $\ge 0$ otherwise.
Summing these squares gives an upper bound for the count, and he then picks the weights that make this bound as small as possible.
The method gives only upper bounds, but these are often close to the truth.
</document>

<document source="https://tamnd.github.io/math2/node/526TCVMSHP.md" tag="526TCVMSHP" kind="concept">
# Fredholm integral equation

## Summary

A Fredholm integral equation is a [[linear integral equation|AYZCKG34QK]] whose integral runs over a fixed set, such as $[a, b]$.
The upper limit is not the variable $x$.
It comes in a [[first kind|2RMWAVR1HV]], $\int_a^b K(x, s) \varphi(s) \, ds = f(x)$, and a [[second kind|DEYYNVCV9J]], $\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds$.
For the second kind with a continuous or square-integrable kernel, [[Fredholm theory|242TFD41KZ]] gives a complete picture of solvability.
This picture is much as for square systems of linear equations.
[[Boundary value problems|TM8VQJWGHZ]] for differential equations lead to Fredholm equations through [[Green's functions|SH6JY9RN2E]].

## Explanations

### L1 human

In a Fredholm equation, the value of the unknown at each point depends on its values on the whole interval.
This is unlike a [[Volterra integral equation|DND8CBEVXT]], where only the values up to the current point matter.
A Fredholm equation behaves like a very large square system of linear equations.
For example, $\varphi(x) = x + \int_0^1 x s \, \varphi(s) \, ds$ is a Fredholm equation of the second kind on $[0, 1]$, with solution $\varphi(x) = 3x/2$.
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<document source="https://tamnd.github.io/math2/node/5276MDSDQV.md" tag="5276MDSDQV" kind="concept">
# Unisolvent point set

## Summary

Let $V$ be a finite-dimensional space of functions.
A set of points is unisolvent for $V$ when each choice of values at the points is taken by exactly one function in $V$.
It is the condition under which an interpolation problem has a unique answer.
In one variable, any $n + 1$ distinct points are unisolvent for the polynomials of degree at most $n$, which is the basis of [[polynomial interpolation|SS1YMS2NPC]].
In several variables the position of the points matters, and choosing unisolvent sets is a central problem of [[multivariate interpolation|BM703GV34G]].

## Explanations

### L1 human

A space of functions with $N$ free parameters needs $N$ conditions to fix one function.
Values at $N$ points give $N$ conditions, but they only work when the points are in a good position.
The affine functions $a + bx + cy$ on the plane have three parameters.
Three points that form a triangle fix such a function, so they are unisolvent.
Three points on one line are not: the function $x - y$ is zero at $(0, 0)$, $(1, 1)$ and $(2, 2)$, and so it can be added to any solution.
The [[definition|681HPCS6G6]] makes this precise.
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<document source="https://tamnd.github.io/math2/node/528G11SDWR.md" tag="528G11SDWR" kind="concept">
# Schauder basis

## Summary

A Schauder basis of a [[Banach space|06GZ956657]] is a sequence $(e_n)$ such that every vector is the sum of a unique norm-convergent series $\sum \alpha_n e_n$.
It replaces the finite linear combinations of a [[basis|0KCQ9Y3QKA]] by infinite series, and so it fits infinite-dimensional spaces.
The unit vectors form a Schauder basis of $\ell^p$ for $1 \le p < \infty$ and of $c_0$.
A Banach space with a Schauder basis is separable, but Enflo showed that not every separable Banach space has a Schauder basis.

## Explanations

### L1 human

In $\R^n$ every vector is a combination of the $n$ unit vectors.
In a space of infinite sequences one would like to write every vector as an infinite combination $\alpha_1 e_1 + \alpha_2 e_2 + \cdots$ of basic vectors.
A Schauder basis is a sequence of vectors for which this works, with the coefficients determined uniquely and the series converging in the norm of the space.
For example, a sequence $x$ in $\ell^2$ is the sum of the series $x_1 e_1 + x_2 e_2 + \cdots$, where $e_k$ has $1$ in place $k$ and $0$ elsewhere.
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<document source="https://tamnd.github.io/math2/node/529D1C4W9H.md" tag="529D1C4W9H" kind="concept">
# Koenigs function

## Summary

The Koenigs function of a holomorphic map $f$ with an attracting, not superattracting, [[fixed point|R6QVS6P21N]] is the normalised solution of [[Schröder's equation|VTJV7JVJEA]] at that point.
It is the limit of the rescaled iterates $f^n(z) / \lambda^n$, where $\lambda = f'(0)$.
In its coordinate the map $f$ becomes multiplication by $\lambda$; this is the content of the [[Koenigs linearization theorem|G02NBNN6CJ]].
It is a basic tool in complex dynamics and in the study of composition operators.

## Explanations

### L1 human

If a map pulls points towards $0$ and shrinks distances near $0$ by the factor $\lambda$, then $n$ steps shrink by about $\lambda^n$.
Dividing the $n$-th iterate by $\lambda^n$ undoes this shrinking, and in the limit one gets a new coordinate.
In that coordinate the map is exactly multiplication by $\lambda$.
For $f(z) = z / (2 - z)$ the factor is $\lambda = 1/2$, and the Koenigs function is $z / (1 - z)$.
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<document source="https://tamnd.github.io/math2/node/52GCNZ1MZ8.md" tag="52GCNZ1MZ8" kind="definition">
# Hermitian symmetric space

## Definition

Let $(M, g)$ be a connected [[Hermitian manifold|CK1DCPC3D3]].
So $M$ is a complex manifold with integrable [[almost complex structure|9SYWRGVTRS]] $J$.
The Riemannian metric $g$ satisfies $g(JX, JY) = g(X, Y)$ for all tangent vectors $X, Y$.
The manifold $M$ is a *Hermitian symmetric space* if each point $p \in M$ has an [[isometry|R9YWQS15AA]] $s_p \colon M \to M$ with three properties.

1. $s_p$ is holomorphic.
2. $s_p \circ s_p = \id$.
3. $p$ is an isolated fixed point of $s_p$.

## Remarks

Conditions 2 and 3 force $(ds_p)_p = -\id$, so $(M, g)$ is a Riemannian [[symmetric space|PAA93R4WYD]] and $s_p$ is its geodesic symmetry.
The metric $g$ is then Kähler.

Example: the unit disc $D = \set{z \in \C : \abs{z} < 1}$ with the metric $\frac{4\,\abs{dz}^2}{(1 - \abs{z}^2)^2}$.
At $0$ take $s_0(z) = -z$: it is holomorphic, an isometry, an involution, and $0$ is its only fixed point.
The holomorphic isometries $z \mapsto e^{i t} \frac{z - a}{1 - \bar a z}$ act transitively, and conjugating $s_0$ by them gives $s_p$ at every point $p$.

Non-example: the round sphere $S^4$ is a Riemannian symmetric space, but it has no almost complex structure at all, so it is not a Hermitian symmetric space.

Each simply connected Hermitian symmetric space is a product of a flat part $\C^k$ and irreducible ones of compact type, such as $\C P^n$, and of noncompact type, such as $D$.
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<document source="https://tamnd.github.io/math2/node/52N6S8WXM9.md" tag="52N6S8WXM9" kind="definition">
# Brownian bridge

## Definition

A *Brownian bridge* is a real stochastic process $(X_t)_{0 \le t \le 1}$ such that:

1. $(X_t)$ is a [[Gaussian process|22WEM6RTW0]] with $E[X_t] = 0$ for all $t \in [0, 1]$.
2. Its [[covariance|0KBFAWYRTE]] is $E[X_s X_t] = \min(s, t) - st$ for all $s, t \in [0, 1]$.
3. Almost surely, the path $t \mapsto X_t$ is continuous.

## Remarks

If $W$ is a [[Wiener process|552AWX3QXN]], then $X_t = W_t - t W_1$ is a Brownian bridge.
Check: $X$ is a linear image of the Gaussian process $W$, hence Gaussian and centred, with continuous paths, and $E[X_s X_t] = \min(s, t) - st - st + st = \min(s, t) - st$.
The Wiener process $W$ restricted to $[0, 1]$ satisfies clauses 1 and 3 but fails clause 2, since $E[W_s W_t] = \min(s, t)$; for instance $\operatorname{Var}(W_1) = 1$, while clause 2 forces $X_1 = 0$.
Clause 2 gives $X_0 = X_1 = 0$ almost surely and $\operatorname{Var}(X_t) = t(1 - t)$.
The law of a Brownian bridge is the conditional law of $(W_t)_{0 \le t \le 1}$ given $W_1 = 0$, in a sense made precise by a limit.
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<document source="https://tamnd.github.io/math2/node/52QT0EK3GJ.md" tag="52QT0EK3GJ" kind="definition">
# Analytic set

## Definition

Let $X$ be a [[Polish space|584GXXY73M]], and let $\mathcal{N} = \N^\N$ be the [[Baire space|P2WS16HKD7]].
A subset $A \subseteq X$ is *analytic* if $A = \emptyset$ or there is a [[continuous map|0VGFHTM75Y]] $f \colon \mathcal{N} \to X$ with $f(\mathcal{N}) = A$.
The class of analytic subsets of $X$ is written $\Sigma^1_1$.

## Remarks

Every closed subset $F$ of $X$ is analytic: if $F$ is nonempty, it is a Polish space, and every nonempty Polish space is a continuous image of $\mathcal{N}$.
More generally every Borel subset of $X$ is analytic.
Non-example by counting: in $X = \R$ there are only $2^{\aleph_0}$ analytic sets.
Indeed, each nonempty one is the image of a continuous map on $\mathcal{N}$, and such a map is determined by its values on a countable dense set.
Since $\R$ has $2^{2^{\aleph_0}}$ subsets, some subset of $\R$ is not analytic.
The following are equivalent forms of the definition.
$A$ is the projection to $X$ of a closed subset of $X \times \mathcal{N}$.
$A$ is the projection to $X$ of a Borel subset of $X \times Y$ for some Polish space $Y$.
$A$ is empty or a continuous image of a Polish space.
Complements of analytic sets are called coanalytic, and they form the class $\Pi^1_1$.
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<document source="https://tamnd.github.io/math2/node/52VS07H6GK.md" tag="52VS07H6GK" kind="concept">
# Hermite functions

## Summary

The Hermite functions $\psi_n$ are the [[Hermite polynomials|MGACY98E9H]] multiplied by the Gaussian $e^{-x^2/2}$ and normalised.
They form an [[orthonormal basis|0S9GGTR0EX]] of the [[Hilbert space|HM2HHE3J51]] $L^2(\R)$.
Each $\psi_n$ is an eigenfunction of the [[Fourier transform|DPVM34ADZV]] and of the quantum harmonic oscillator.

## Explanations

### L1 human

A Hermite polynomial grows without bound, so it cannot describe a wave that dies out far away.
Multiplying by the bell curve $e^{-x^2/2}$ makes it decay fast in both directions.
The first Hermite function is a bell curve itself, $\psi_0(x) = \pi^{-1/4} e^{-x^2/2}$.
The next ones are bell curves with more and more wiggles, and in quantum mechanics they are the states of a particle on a spring.
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<document source="https://tamnd.github.io/math2/node/52XCYMK0WW.md" tag="52XCYMK0WW" kind="definition">
# Intuitionistic logic

## Definition

The formulas of intuitionistic propositional logic are built from propositional variables and $\bot$ with $\land$, $\lor$ and $\to$.
The negation $\neg\varphi$ abbreviates $\varphi \to \bot$.
Intuitionistic propositional logic is the set of formulas derivable in the [[Hilbert system|1MGMQX43B8]] with the following axiom schemes, for all formulas $\varphi$, $\psi$, $\chi$.

1. $\varphi \to (\psi \to \varphi)$.
2. $(\varphi \to (\psi \to \chi)) \to ((\varphi \to \psi) \to (\varphi \to \chi))$.
3. $\varphi \land \psi \to \varphi$ and $\varphi \land \psi \to \psi$.
4. $\varphi \to (\psi \to \varphi \land \psi)$.
5. $\varphi \to \varphi \lor \psi$ and $\psi \to \varphi \lor \psi$.
6. $(\varphi \to \chi) \to ((\psi \to \chi) \to (\varphi \lor \psi \to \chi))$.
7. $\bot \to \varphi$.

The only rule is [[modus ponens|0Y0MQYST0M]]: from $\varphi$ and $\varphi \to \psi$, derive $\psi$.

## Remarks

Adding the scheme $\varphi \lor \neg\varphi$, the [[law of excluded middle|0RYYK8PG4S]], gives classical [[propositional logic|09K9MWD2QQ]].
Intuitionistic first-order logic adds quantifiers with the usual axioms and rules, again without excluded middle.

Example: $\varphi \to \neg\neg\varphi$, that is, $\varphi \to ((\varphi \to \bot) \to \bot)$, is derivable.
By the deduction theorem it suffices to derive $\bot$ from $\varphi$ and $\varphi \to \bot$, which is one step of modus ponens.

Non-example: $p \lor \neg p$ is not derivable for a variable $p$.
In the Kripke model with worlds $u \le v$ and $p$ true only at $v$, neither $p$ nor $\neg p$ holds at $u$, and derivable formulas hold at every world.

Likewise $\neg\neg p \to p$ is not derivable, while $\neg\neg\neg p \to \neg p$ is.

## Explanations

### L1 human

Intuitionistic logic keeps every rule of ordinary logic except the principle that each statement is true or false.
It reads a formula as a task: to prove $A \to B$ is to give a method that turns any proof of $A$ into a proof of $B$.
</document>

<document source="https://tamnd.github.io/math2/node/53750WSJ6K.md" tag="53750WSJ6K" kind="statement">
# Stirling's formula for the gamma function

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], let $\operatorname{Log}$ be the principal [[complex logarithm|02QGGD5G87]], and for $z \in \C \setminus (-\infty, 0]$ let $z^{z - 1/2} = e^{(z - 1/2)\operatorname{Log} z}$.
Fix $\delta$ with $0 < \delta < \pi$.
Then, as $\abs{z} \to \infty$ with $\abs{\arg z} \le \pi - \delta$,
$$\Gamma(z) = \sqrt{2\pi}\; z^{z - 1/2} \, e^{-z} \left(1 + O\!\left(\frac{1}{\abs{z}}\right)\right),$$
where the constant in $O$ depends only on $\delta$.
In particular $\Gamma(z)$ is [[asymptotically equivalent|0QR92XBV6H]] to $\sqrt{2\pi}\, z^{z - 1/2} e^{-z}$ in this sector.

## Remarks

The proof starts from [[Binet's formula|AGBWPB994F]], which writes $\operatorname{Log}\Gamma(z) - (z - \tfrac12)\operatorname{Log} z + z - \tfrac12\ln(2\pi)$ as an integral that is $O(1/\abs{z})$ for $\operatorname{Re} z > 0$; rotating the line of integration extends this bound to the whole sector.
Another route is Laplace's method applied to Euler's integral $\Gamma(x + 1) = \int_0^\infty e^{x \ln t - t}\,dt$, whose integrand peaks at $t = x$.
For real $x \to \infty$ it gives $\Gamma(x + 1) \sim \sqrt{2\pi x}\,(x/e)^x$; at integers this is [[Stirling's formula|0KC0F51T6A]] for $n!$.
The sector condition is needed: on the negative real axis $\Gamma$ has poles, so no such estimate holds near it.
The full asymptotic series for $\operatorname{Log}\Gamma(z)$ is [[Stirling's series|K29QYWB6A5]].

## Explanations

### L1 human

Stirling's formula tells how big $\Gamma(z)$ is when $z$ is large.
It extends the classical estimate $n! \approx \sqrt{2\pi n}\,(n/e)^n$ from whole numbers to real and complex arguments.

### L2 human

The complex form gives the decay of $\Gamma$ on vertical lines: for fixed real $x$, $\abs{\Gamma(x + iy)} \sim \sqrt{2\pi}\,\abs{y}^{x - 1/2} e^{-\pi\abs{y}/2}$ as $\abs{y} \to \infty$.
This decay is what makes Mellin–Barnes integrals and the inverse Mellin transform converge, and it is used to estimate the Riemann zeta function through its functional equation.
Dividing two instances gives $\Gamma(x + a)/\Gamma(x) \sim x^a$ as $x \to \infty$.
</document>

<document source="https://tamnd.github.io/math2/node/537J79VYQR.md" tag="537J79VYQR" kind="problem">
# First Cousin problem

## Problem

Let $X$ be a [[complex manifold|JSGWAN2RJV]] and let $(U_i)_{i \in I}$ be an open cover of $X$.
For each $i$, let $f_i$ be a [[meromorphic function|W6FTSF57T1]] on $U_i$, such that $f_i - f_j$ is [[holomorphic|507C34W0ZT]] on $U_i \cap U_j$ for all $i, j$.
Find a meromorphic function $f$ on $X$ such that $f - f_i$ is holomorphic on $U_i$ for every $i$.

## Remarks

The data $(U_i, f_i)$ are called Cousin I data; they prescribe the principal parts of $f$.
The differences $g_{ij} = f_i - f_j$ form a Čech $1$-cocycle with values in the sheaf $\mathcal{O}$ of holomorphic functions.
The problem is solvable for these data if and only if this cocycle is a coboundary, that is, $g_{ij} = h_j - h_i$ for holomorphic $h_i$ on $U_i$.
Then $f = f_i + h_i$ on $U_i$ is a solution.
So if $H^1(X, \mathcal{O}) = 0$, every first Cousin problem on $X$ is solvable.
By Cartan's theorem B this holds on every Stein manifold, in particular on every domain of holomorphy in $\C^n$.
For $n = 1$ and $X$ a domain in $\C$, the problem is solved by the Mittag-Leffler theorem.
It can fail on other domains: on $X = \C^2 \setminus \set{0}$, with $U_1 = \set{z_1 \ne 0}$, $U_2 = \set{z_2 \ne 0}$, $f_1 = 1/(z_1 z_2)$ and $f_2 = 0$, there is no solution.
Pierre Cousin studied the problem in 1895 for product domains in $\C^n$.

## Explanations

### L1 human

In one variable, the Mittag-Leffler theorem builds a meromorphic function with prescribed poles and principal parts.
The first Cousin problem asks the same in several variables.
The local singular behaviour is given on pieces of a manifold, with the pieces agreeing up to holomorphic terms.
One wants one global meromorphic function with exactly this behaviour.
Whether this is possible depends on the shape of the manifold.
On Stein manifolds, for example on $\C^n$ and on convex domains, it always is.
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<document source="https://tamnd.github.io/math2/node/53H8EVRC1J.md" tag="53H8EVRC1J" kind="definition">
# Lax–Friedrichs method

## Definition

Let $F \colon \R^m \to \R^m$ be continuous, and consider the [[conservation law|8RA7AMWSAF]] $u_t + F(u)_x = 0$ for $u \colon \R \times [0, \infty) \to \R^m$.
Fix $\Delta x > 0$ and $\Delta t > 0$, and put $\lambda = \Delta t / \Delta x$.
Let $u_j^n$ denote the approximation of $u$ at $x_j = j \Delta x$ and $t_n = n \Delta t$.
The *Lax–Friedrichs method* is the explicit scheme
$$u_j^{n+1} = \frac{u_{j+1}^n + u_{j-1}^n}{2} - \frac{\lambda}{2} \left( F(u_{j+1}^n) - F(u_{j-1}^n) \right), \qquad j \in \Z,\ n \ge 0.$$

## Remarks

It is in conservation form $u_j^{n+1} = u_j^n - \lambda \left( G_{j+1/2} - G_{j-1/2} \right)$ with the numerical flux
$$G_{j+1/2} = \frac{F(u_j^n) + F(u_{j+1}^n)}{2} - \frac{u_{j+1}^n - u_j^n}{2 \lambda}.$$
Its truncation error is of first order in $\Delta x$ when $\lambda$ is fixed.

Example: for $u_t + a u_x = 0$ with $a \in \R$, the scheme reads $u_j^{n+1} = \frac{1 - a\lambda}{2} u_{j+1}^n + \frac{1 + a\lambda}{2} u_{j-1}^n$.
If $\abs{a} \lambda \le 1$, both weights lie in $[0, 1]$ and sum to $1$.
So $\max_j \abs{u_j^{n+1}} \le \max_j \abs{u_j^n}$, and the scheme is stable in the maximum norm.

Non-example: the forward-time centred-space scheme $u_j^{n+1} = u_j^n - \frac{\lambda}{2}(F(u_{j+1}^n) - F(u_{j-1}^n))$ differs only in the first term.
For $u_t + a u_x = 0$ with $a \ne 0$, it multiplies the Fourier mode $e^{i j \xi}$ by $1 - i a \lambda \sin \xi$, of modulus greater than $1$ when $\sin \xi \ne 0$.
So it is unstable for every $\lambda > 0$.

## Explanations

### L1 human

The update averages the two neighbours and then corrects by the flux difference across them.
The averaging is what makes the method work, at the price of some blurring of sharp fronts.
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<document source="https://tamnd.github.io/math2/node/53HPW59SJC.md" tag="53HPW59SJC" kind="definition">
# Shimura variety

## Definition

Let $\mathbb{S} = \operatorname{Res}_{\C/\R} \mathbb{G}_m$ be the Deligne torus, so that $\mathbb{S}(\R) = \C^\times$.
Let $G$ be a connected [[reductive group|2XS84PWE3S]] over $\Q$ and let $X$ be a $G(\R)$-conjugacy class of homomorphisms $h \colon \mathbb{S} \to G_\R$.
The pair $(G, X)$ is a *Shimura datum* if for every $h \in X$:

1. The [[Hodge structure|ZSM083M97C]] on the Lie algebra of $G_\R$ defined by $\operatorname{Ad} \circ h$ has only the types $(-1, 1)$, $(0, 0)$ and $(1, -1)$.
2. Conjugation by $h(i)$ is a Cartan involution of the adjoint group $G^{\mathrm{ad}}_\R$.
3. $G^{\mathrm{ad}}$ has no factor defined over $\Q$ on which the projection of $h$ is trivial.

Let $\mathbb{A}_f$ be the finite [[adeles|SEH68BA1QQ]] of $\Q$ and let $K \subseteq G(\mathbb{A}_f)$ be a compact open subgroup.
The *Shimura variety* of level $K$ is
$$\operatorname{Sh}_K(G, X) = G(\Q) \backslash \big( X \times G(\mathbb{A}_f) / K \big),$$
with $G(\Q)$ acting diagonally and $K$ acting on the right of $G(\mathbb{A}_f)$.

## Remarks

The conditions make $X$ a finite union of Hermitian [[symmetric spaces|PAA93R4WYD]], and $\operatorname{Sh}_K(G, X)$ is a finite disjoint union of quotients $\Gamma \backslash X^+$ by arithmetic subgroups $\Gamma$.
By the theorem of Baily and Borel, it is a [[quasi-projective variety|EQSHJPE7H8]] over $\C$, and it is smooth when $K$ is small enough.
By work of Shimura, Deligne and others it has a canonical model over a number field, the reflex field.

Example: let $G = \mathrm{GL}_2$ and let $X$ be the conjugacy class of $h(a + bi) = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}$.
Then $X$ is identified with $\C \setminus \R$, the union of the upper and lower half-planes, and the Shimura varieties $\operatorname{Sh}_K(\mathrm{GL}_2, X)$ are finite unions of [[modular curves|T5W0CMS0DM]].
For $G = \mathrm{GSp}_{2g}$ one obtains the Siegel modular varieties, which parametrize principally polarized abelian varieties of dimension $g$ with level structure.

Non-example: $G = \mathrm{GL}_3$ admits no Shimura datum, because the symmetric space of $\mathrm{SL}_3(\R)$ has no invariant complex structure, so clauses 1 and 2 cannot both hold.
</document>

<document source="https://tamnd.github.io/math2/node/53K0FDVSHE.md" tag="53K0FDVSHE" kind="definition">
# Inertial manifold

## Definition

Let $H$ be a real [[Hilbert space|HM2HHE3J51]] and let $(S(t))_{t \ge 0}$ be a [[semigroup|0XMEM406JM]] of continuous maps $S(t) \colon H \to H$ with $S(0) = \id$ and $S(t + s) = S(t) S(s)$.
An *inertial manifold* for $S$ is a subset $\mathcal{M} \subseteq H$ such that:

1. $\mathcal{M}$ is a finite-dimensional Lipschitz manifold: there are an orthogonal projection $P$ of finite rank and a [[Lipschitz|0AW1E6YTJY]] map $\Phi \colon PH \to (I - P)H$ with $\mathcal{M} = \set{p + \Phi(p) : p \in PH}$;
2. $\mathcal{M}$ is positively invariant: $S(t) \mathcal{M} \subseteq \mathcal{M}$ for all $t \ge 0$;
3. $\mathcal{M}$ attracts exponentially: there is $\gamma > 0$ and, for each $u_0 \in H$, a constant $C(u_0)$ with $\operatorname{dist}(S(t) u_0, \mathcal{M}) \le C(u_0) e^{-\gamma t}$ for all $t \ge 0$.

## Remarks

If the semigroup has a [[global attractor|4SH2ZB984E]] $\mathcal{A}$, then $\mathcal{A} \subseteq \mathcal{M}$.
The dynamics on $\mathcal{M}$ are given by an ordinary differential equation for $p = P u$ in the finite-dimensional space $PH$, the inertial form.
Some authors ask in clause 3 for exponential tracking: each solution approaches some solution that lies on $\mathcal{M}$.

Example: let $H = \R^2$ and $S(t)(x, y) = (e^{-t} x, e^{-10 t} y)$.
Take $P(x, y) = (x, 0)$ and $\Phi = 0$, so $\mathcal{M}$ is the $x$-axis.
It is invariant, and $\operatorname{dist}(S(t)(x, y), \mathcal{M}) = e^{-10 t} \abs{y}$, so all clauses hold with $\gamma = 10$.

Non-example: for the same semigroup, the parabola $\set{(x, x^2)}$ is the graph of a function, but it is not invariant.
The point $(1, 1)$ goes to $(e^{-t}, e^{-10 t})$, which is not on the parabola for $t > 0$, so clause 2 fails.

Consider a dissipative equation $u' + A u = F(u)$ with $A$ self-adjoint with compact inverse and $F$ globally Lipschitz.
An inertial manifold exists when the eigenvalues of $A$ have large enough gaps compared with the Lipschitz constant of $F$.
</document>

<document source="https://tamnd.github.io/math2/node/53MWMX2HWT.md" tag="53MWMX2HWT" kind="statement">
# Bochner's classification of orthogonal polynomial eigenfunctions

## Statement

Let $\sigma$ and $\tau$ be real polynomials, not both zero, with $\deg \sigma \le 2$ and $\deg \tau \le 1$, and let $\mathcal{D} = \sigma(x) \frac{d^2}{dx^2} + \tau(x) \frac{d}{dx}$.
Let $(p_n)_{n \ge 0}$ be real polynomials with $\deg p_n = n$, each an eigenfunction of $\mathcal{D}$: $\mathcal{D} p_n = \lambda_n p_n$ for some real [[eigenvalue|0BQAA2BK3Q]] $\lambda_n$.
Assume that $(p_n)$ is orthogonal for a weight.
That is, there are an open interval $I$ and a continuous $w \colon I \to (0, \infty)$ with all moments finite such that $\int_I p_m p_n\, w\, dx = 0$ for $m \ne n$.
Then, up to an affine change of variable and nonzero constant factors, $(p_n)$ is one of the [[classical orthogonal polynomial|47WJ34NC9K]] families: Jacobi, Laguerre or Hermite.

## Remarks

Every operator $\mathcal{D}$ of this form maps polynomials of degree at most $n$ to themselves.
In fact $\mathcal{D} x^n = \left( n(n-1)\sigma_2 + n \tau_1 \right) x^n + \text{lower terms}$, where $\sigma_2$ and $\tau_1$ are the coefficients of $x^2$ in $\sigma$ and of $x$ in $\tau$.
The proof sorts $\sigma$ by its degree and its roots, up to affine changes.
Orthogonality for a positive weight forces $w$ to solve the Pearson equation $(\sigma w)' = \tau w$, and only the Jacobi, Laguerre and Hermite cases give a positive weight with finite moments.
Example: $\sigma = 1$ and $\tau = -2x$ give $\mathcal{D} = \frac{d^2}{dx^2} - 2x \frac{d}{dx}$, with eigenfunctions the [[Hermite polynomials|MGACY98E9H]] and $\lambda_n = -2n$; the check for $H_2 = 4x^2 - 2$ is $8 - 2x \cdot 8x = -16x^2 + 8 = -4 H_2$.
The orthogonality hypothesis is necessary: $\sigma = x^2$ and $\tau = 2x + 2$ have the Bessel polynomials as eigenfunctions, which are not orthogonal for any positive weight on a real interval.
Without the orthogonality hypothesis, Bochner's list also contains the Bessel polynomials and the powers $x^n$, which are eigenfunctions of $x^2 \frac{d^2}{dx^2}$.

## Explanations

### L1 human

Many famous polynomial families solve a second-order differential equation, one polynomial for each degree.
Bochner asked which families can do this, and the answer is a short list.
If one also asks for orthogonality with a positive weight, only the Jacobi, Laguerre and Hermite families remain.

### L2 human

This is the reason the classical families are singled out among all orthogonal polynomials.
The operator $\mathcal{D}$ becomes symmetric for the inner product $\int f g\, w\, dx$ when $(\sigma w)' = \tau w$, the Pearson equation for the weight, and $\sigma w$ vanishes fast enough at the ends of $I$.
Then $\mathcal{D}$ is a [[Sturm–Liouville operator|7WV6B2502K]], and the $p_n$ are its eigenfunctions, orthogonal because they belong to different eigenvalues.
Discrete analogues with difference operators lead to the Hahn, Meixner, Krawtchouk and Charlier polynomials, and $q$-analogues to the Askey scheme.
</document>

<document source="https://tamnd.github.io/math2/node/53MZ2DGDTQ.md" tag="53MZ2DGDTQ" kind="statement">
# Weierstrass product for the gamma function

## Statement

Let $1/\Gamma$ be the [[reciprocal gamma function|PY6K9AWR33]] and let $\gamma$ be the [[Euler–Mascheroni constant|00C5P9JZBK]].
For every $z \in \C$,
$$\frac{1}{\Gamma(z)} = z \, e^{\gamma z} \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) e^{-z/n}.$$
The [[infinite product|RZ0Z2BTMT4]] converges absolutely and uniformly on every compact subset of $\C$.

## Remarks

Each factor is $1 + O(z^2/n^2)$, and $\sum 1/n^2 < \infty$ gives the convergence.
The partial product up to $n$ equals $\frac{z(z+1)\cdots(z+n)}{n!}\, e^{z(\gamma - H_n)}$, where $H_n$ is the [[harmonic number|0NANDX7V02]].
Since $H_n - \ln n \to \gamma$, this is the reciprocal of the quotient in the [[Euler–Gauss limit formula|SKRP9NDXR4]] times a factor tending to $1$, which proves the identity.
The product exhibits the simple zeros of $1/\Gamma$ at $0, -1, -2, \ldots$; it is the [[Weierstrass factorization|11VC51YHQJ]] of this entire function of order $1$.
Multiplying the products for $1/\Gamma(z)$ and $1/\Gamma(-z)$ and using [[Euler's sine product|DQEHD56649]] gives $\Gamma(z)\Gamma(-z) = -\pi/(z\sin \pi z)$, a form of the reflection formula.

## Explanations

### L1 human

The reciprocal of the gamma function can be written as an infinite product, much like a polynomial is a product of factors for its roots.
Here the roots are $0, -1, -2, \ldots$, and the exponential factors are there to make the infinite product converge.

### L2 human

Taking the logarithmic derivative of the product gives the series $\psi(z) = -\gamma - \frac1z + \sum_{n \ge 1}\left(\frac1n - \frac{1}{z+n}\right)$ for the [[digamma function|HWRRXB0YXS]].
Differentiating once more gives $(\ln\Gamma)''(x) = \sum_{n \ge 0} (x+n)^{-2} > 0$ on $(0, \infty)$, so $\ln \Gamma$ is convex there.
The product also yields the Taylor expansion of $\ln\Gamma(1 + z)$ around $0$, whose coefficients are values of the [[Riemann zeta function|1XWBY4N0VT]] at the integers $k \ge 2$.
</document>

<document source="https://tamnd.github.io/math2/node/53TSQ57J0H.md" tag="53TSQ57J0H" kind="definition">
# Kleisli category

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $(T, \eta, \mu)$ be a [[monad|FF6XXA6D1A]] on $\mathcal{C}$, with unit $\eta \colon \mathrm{Id} \to T$ and multiplication $\mu \colon T T \to T$.
The *Kleisli category* $\mathcal{C}_T$ is the category given as follows.

1. Its objects are the objects of $\mathcal{C}$.
2. A morphism $X \to Y$ in $\mathcal{C}_T$ is a morphism $f \colon X \to T Y$ in $\mathcal{C}$.
3. The composite of $f \colon X \to T Y$ and $g \colon Y \to T Z$ is $g \circ_T f = \mu_Z \circ T g \circ f \colon X \to T Z$.
4. The identity on $X$ is $\eta_X \colon X \to T X$.

## Remarks

The monad laws $\mu \circ T \eta = \mathrm{id} = \mu \circ \eta T$ and $\mu \circ T \mu = \mu \circ \mu T$ give the identity laws and the associativity of $\circ_T$.
The functor $\mathcal{C} \to \mathcal{C}_T$, $f \mapsto \eta \circ f$, has a right adjoint $\mathcal{C}_T \to \mathcal{C}$ that sends $X$ to $T X$ and $f$ to $\mu \circ T f$, and this adjunction induces $T$.

Example: let $T$ be the monad $X \mapsto X \sqcup \set{\ast}$ on $\mathbf{Set}$, with $\eta_X$ the inclusion and $\mu_X$ sending both added points to $\ast$.
A Kleisli morphism $X \to Y \sqcup \set{\ast}$ is a partial function from $X$ to $Y$, undefined where the value is $\ast$, and $\circ_T$ is composition of partial functions.
So $\mathbf{Set}_T$ is isomorphic to the category of sets and partial functions.

Non-example: composing $f \colon X \to T Y$ and $g \colon Y \to T Z$ by $T g \circ f$ alone gives a morphism $X \to T T Z$, not $X \to T Z$.
Without the multiplication $\mu$, clause 3 does not produce a morphism of the right type.

For the monad of a variety on $\mathbf{Set}$, the Kleisli category is equivalent to the category of free algebras of the variety and the homomorphisms between them.
</document>

<document source="https://tamnd.github.io/math2/node/541S6DXQAR.md" tag="541S6DXQAR" kind="definition">
# Orthocentric system

## Definition

A set $S$ of points of the Euclidean plane is an *orthocentric system* if both conditions hold.

1. $S$ has exactly four points.
2. Some three points of $S$ form a [[triangle|0JDB8YJBSR]] whose [[orthocentre|0QMYQ1X03H]] is the fourth point.

## Remarks

Condition 1 forces the triangle in condition 2 to have no right angle.
For a right triangle the orthocentre is the vertex of the right angle, so only three distinct points would remain.

Example: let $A = (0, 2)$, $B = (-1, 0)$, $C = (1, 0)$ and $H = (0, \frac{1}{2})$.
The altitude from $A$ is the line $x = 0$.
The altitude from $B$ is perpendicular to $C - A = (1, -2)$, so it is the line $(x + 1) - 2y = 0$, which meets $x = 0$ at $(0, \frac{1}{2})$.
So $H$ is the orthocentre of $ABC$, and $\set{A, B, C, H}$ is an orthocentric system.

Non-example: $(0, 0)$, $(1, 0)$, $(0, 1)$ and $(1, 1)$.
The orthocentre of each three of them is a vertex of that triangle, since each such triangle has a right angle, so condition 2 fails.

If $\set{A, B, C, H}$ is an orthocentric system, then each point is the orthocentre of the other three.
Indeed, $H$ is the orthocentre of $ABC$ when $AH \perp BC$, $BH \perp CA$ and $CH \perp AB$.
The same three relations say that $A$ is the orthocentre of $HBC$, and similarly for $B$ and $C$.
</document>

<document source="https://tamnd.github.io/math2/node/54387XJ9QW.md" tag="54387XJ9QW" kind="statement">
# Consistency of the maximum likelihood estimator

## Statement

Let $\set{f(\cdot; \theta) : \theta \in \Theta}$ be a family of probability densities with respect to a measure $\nu$, and let $X_1, X_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] with density $f(\cdot; \theta_0)$ for some $\theta_0 \in \Theta$.
Assume:

1. ([[identifiability|605M5NMFD5]]) if $\theta \ne \theta_0$, then $f(\cdot; \theta) \ne f(\cdot; \theta_0)$ on a set of positive $\nu$-measure;
2. $\Theta$ is a [[compact|00R0X1WGPX]] subset of $\R^k$;
3. for $\nu$-almost every $x$, the map $\theta \mapsto \log f(x; \theta)$ is continuous on $\Theta$;
4. $E_{\theta_0}\big[\sup_{\theta \in \Theta} \abs{\log f(X_1; \theta)}\big] < \infty$.

Let $\hat{\theta}_n$ be a measurable maximizer over $\Theta$ of the log-[[likelihood|05FSTXM92W]] $\ell_n(\theta) = \sum_{i=1}^n \log f(X_i; \theta)$.
Then $\hat{\theta}_n \to \theta_0$ in [[probability|0EFDDXTVD6]]: the [[maximum likelihood estimator|0FREPFKNS7]] is [[consistent|0XJVF81QF6]].

## Remarks

Idea of proof: let $L(\theta) = E_{\theta_0}[\log f(X_1; \theta)]$.
By [[Jensen's inequality|0RWMRTPC87]] for the strictly concave logarithm, and by identifiability, $L(\theta) < L(\theta_0)$ for every $\theta \ne \theta_0$.
Conditions 2 to 4 give a uniform [[law of large numbers|0BFDHF2H7N]]: $\sup_{\theta} \abs{\ell_n(\theta)/n - L(\theta)} \to 0$ in probability.
Since $L$ is continuous on the compact set $\Theta$, it stays below $L(\theta_0)$ by a fixed margin outside each neighbourhood of $\theta_0$.
So the maximizer of $\ell_n/n$ must eventually lie inside that neighbourhood.

Example: for i.i.d. $\mathcal{N}(\mu, 1)$ data with $\mu \in [-K, K]$, the conditions hold and $\hat{\mu}_n$ is the sample mean clipped to $[-K, K]$, which tends to $\mu_0$.

Identifiability is necessary: if $f(\cdot; \theta_1) = f(\cdot; \theta_0)$ for some $\theta_1 \ne \theta_0$, the data cannot distinguish $\theta_0$ from $\theta_1$, and no estimator converges to $\theta_0$ under both.
Compactness can be replaced by other conditions, such as concavity of $\theta \mapsto \log f(x; \theta)$, but some condition of this kind is needed.

## Explanations

### L1 human

With more and more data, the parameter value that makes the data most likely gets closer and closer to the true value.
The reason is that, on average, the true parameter gives the data a higher log-likelihood than any other parameter.
The conditions make sure that this average behaviour is reached uniformly over all parameter values.

### L2 human

This is Wald's approach to consistency, and the same argument proves consistency of general M-estimators and extremum estimators.
Consistency is the first step in proving the asymptotic normality of the maximum likelihood estimator, which needs a consistent estimator to linearize the likelihood equations around $\theta_0$.
The limit $L(\theta_0) - L(\theta)$ is the Kullback–Leibler divergence of $f(\cdot; \theta)$ from $f(\cdot; \theta_0)$, so maximum likelihood can be seen as minimizing an empirical version of this divergence.
</document>

<document source="https://tamnd.github.io/math2/node/545EV0W4CE.md" tag="545EV0W4CE" kind="concept">
# Stable homotopy theory

## Summary

Stable homotopy theory is the part of homotopy theory that studies the phenomena that do not change under iterated [[suspension|Q7RFNJH76Q]].
By the [[Freudenthal suspension theorem|7DM2KSS2DB]], the homotopy groups of $\Sigma^k X$ stabilize as $k$ grows.
The stable values are often more regular than the unstable ones.
Its natural objects are [[spectra|1GHF4EKBXS]], which represent generalized cohomology theories, and its main computational problem is the [[stable homotopy groups of spheres|HGZAYPQBXW]].

## Explanations

### L1 human

Suspending a space makes it one dimension bigger, and suspending a map does the same to the map.
After enough suspensions, the set of homotopy classes of maps between two finite complexes stops changing.
Stable homotopy theory studies these stable answers.
For example, $\pi_{n+1}(S^n)$ is $\Z$ for $n = 2$ but $\Z/2$ for all $n \ge 3$, so the stable value is $\Z/2$.
In the stable world, maps can be added and fibre sequences behave like exact sequences, so many tools of algebra apply.
</document>

<document source="https://tamnd.github.io/math2/node/54BD10ZZDN.md" tag="54BD10ZZDN" kind="definition">
# Saturation of approximation methods

## Definition

Let $X$ be a [[Banach space|06GZ956657]] and let $L_n \colon X \to X$, $n \ge 1$, be [[continuous linear maps|03TM135G0N]].
Let $\varphi(n) > 0$ with $\varphi(n) \to 0$.
The sequence $(L_n)$ is *saturated* with *saturation order* $\varphi(n)$ if both conditions hold.

1. If $f \in X$ and $\norm{L_n f - f} = o(\varphi(n))$, then $L_n f = f$ for all $n$.
2. Some $f \in X$ with $L_n f \ne f$ for some $n$ satisfies $\norm{L_n f - f} = O(\varphi(n))$.

The *saturation class* is the set of $f \in X$ with $\norm{L_n f - f} = O(\varphi(n))$, in [[big O notation|0TGV80E6A4]].

## Remarks

Example: the Bernstein operators $B_n$ on $C[0, 1]$ with the uniform norm are saturated with order $1/n$.
For $f(x) = x^2$ one has $B_n f(x) = x^2 + \frac{x(1 - x)}{n}$, so $\norm{B_n f - f} = \frac{1}{4n}$, which is $O(1/n)$ but not $o(1/n)$.
The only $f$ with $\norm{B_n f - f} = o(1/n)$ are the linear functions, which $B_n$ reproduces.

Non-example: the Fourier partial sum operators $S_n$ on $L^2$ of the circle are not saturated for any order.
A trigonometric polynomial $f$ of degree $k \ge 1$ has $S_n f = f$ for $n \ge k$, so $\norm{S_n f - f} = o(\varphi(n))$, but $S_0 f \ne f$, so clause 1 fails.

The Fejér means of Fourier series are saturated with order $1/n$ in $C$ of the circle, and only constants are approximated with error $o(1/n)$.
Saturation theorems typically identify the saturation class with a smoothness class, such as functions with Lipschitz first derivative for Bernstein operators.
</document>

<document source="https://tamnd.github.io/math2/node/54KHZGC0CV.md" tag="54KHZGC0CV" kind="definition">
# Compact group

## Definition

A *compact group* is a [[topological group|92700QAB5Y]] $G$ whose topology is Hausdorff and [[compact|0TVJZYB6SH]].

## Remarks

Example: the unitary group $U(n) = \set{A \in M_n(\C) : A^* A = I}$ with the topology from $M_n(\C) \cong \C^{n^2}$ is a compact group.
It is Hausdorff as a subspace of $\C^{n^2}$.
It is closed, as the zero set of the continuous map $A \mapsto A^*A - I$.
It is bounded, since each column of a unitary matrix has norm $1$, so it is compact by the Heine–Borel theorem.
Non-example: the group $\R$ with addition and the usual topology is a Hausdorff topological group.
It is not compact, since the open cover by the intervals $(-n, n)$ has no finite subcover.
Every finite group with the discrete topology is a compact group.
Some authors do not require the Hausdorff condition; others include it in the definition of a topological group.
</document>

<document source="https://tamnd.github.io/math2/node/54MY2XHVGB.md" tag="54MY2XHVGB" kind="object">
# Halton sequence

## Description

Let $b \ge 2$ be an integer.
Each integer $n \ge 0$ has a unique base-$b$ expansion $n = \sum_{k \ge 0} a_k b^k$ with digits $a_k \in \set{0, \ldots, b-1}$, almost all zero.
The *radical inverse* in base $b$ is
$$\varphi_b(n) = \sum_{k \ge 0} a_k \, b^{-k-1} \in [0,1).$$
It mirrors the digits of $n$ at the radix point.
The sequence $(\varphi_b(n))_{n \ge 0}$ is the van der Corput sequence in base $b$.
Let $s \ge 1$ and let $b_1, \ldots, b_s \ge 2$ be pairwise coprime integers.
The *Halton sequence* in these bases is the sequence of points
$$x_n = \big(\varphi_{b_1}(n), \ldots, \varphi_{b_s}(n)\big) \in [0,1)^s, \qquad n = 0, 1, 2, \ldots$$
The usual choice is $b_1, \ldots, b_s$ the first $s$ [[prime numbers|01826KMG2H]].

## Remarks

In bases $2$ and $3$, the points for $n = 1, 2, 3, 4$ are $(1/2, 1/3)$, $(1/4, 2/3)$, $(3/4, 1/9)$, $(1/8, 4/9)$.
For example $3 = 10_3$ in base $3$, so $\varphi_3(3) = 0.01_3 = 1/9$.
For pairwise coprime bases, the star discrepancy of the first $N$ points is $O\big((\log N)^s / N\big)$, with a constant depending on $s$ and the bases.
So the Halton sequence is a [[low-discrepancy sequence|XSPX1KW9AQ]], and it is a standard point set for [[quasi-Monte Carlo methods|ABSRB2KXA6]].
The coprimality condition matters: with $b_1 = b_2 = 2$, all points lie on the diagonal of the unit square, which they do not fill.
In high dimension the coordinates with large bases are correlated for moderate $N$, and scrambled variants are used in practice.

## Explanations

### L1 human

Write the counting numbers $1, 2, 3, \ldots$ in base $2$ and read each one backwards after a binary point.
This gives $1/2, 1/4, 3/4, 1/8, 5/8, \ldots$, and each new point falls into one of the largest gaps left by the earlier ones.
Do the same in base $3$, base $5$, and so on, and use the results as the coordinates of a point.
The points fill a square or cube far more evenly than random points do.
This makes them good sample points for computing averages.
</document>

<document source="https://tamnd.github.io/math2/node/54QMKJF64A.md" tag="54QMKJF64A" kind="statement">
# Monodromy theorem

## Statement

Let $D \subseteq \C$ be an open set, let $a, b \in D$, and let $f$ be a [[holomorphic function|0DG0Z72CQD]] on a disc around $a$.
Let $\gamma_0$ and $\gamma_1$ be [[paths|0JDE56HWQ8]] in $D$ from $a$ to $b$.
Let $H \colon [0, 1] \times [0, 1] \to D$ be a homotopy from $\gamma_0$ to $\gamma_1$ with fixed end points, and write $\gamma_s = H(s, \cdot)$.
Assume that $f$ has an [[analytic continuation|0M3YQJ81NB]] along every path $\gamma_s$, $0 \le s \le 1$.
Then the continuations of $f$ along $\gamma_0$ and along $\gamma_1$ agree near $b$.

## Remarks

The idea of the proof is that the continuation along $\gamma_s$ depends continuously on $s$.
For $s'$ close to $s$, the two paths stay close, and the continuations along them use overlapping discs, so they give the same germ at $b$.
Compactness of $[0, 1]$ then passes from $s = 0$ to $s = 1$.
An important consequence concerns a [[simply connected domain|075C629XTH]] $D$.
If $f$ can be continued along every path in $D$ that starts at $a$, then there is a holomorphic function $F$ on $D$ that equals $f$ near $a$.
The homotopy hypothesis is needed.
On $D = \C \setminus \set{0}$, a branch of the [[complex logarithm|0X2FY491PF]] near $a = 1$ continues along every path, but continuation along the unit circle returns to $1$ with the value increased by $2 \pi i$.
The unit circle is not homotopic to a constant path in $D$, so there is no contradiction.

## Explanations

### L1 human

Analytic continuation extends a function step by step along a path.
In general, two paths to the same point can give two different values, as for the logarithm going around $0$.
The monodromy theorem says this cannot happen when one path can be deformed into the other without leaving the region where continuation works.
So on a region without holes, continuation produces a single-valued function.

### L2 human

The theorem is the standard way to construct functions on simply connected domains, such as logarithms and square roots of zero-free holomorphic functions.
It is the analytic form of a topological fact: a covering space over a simply connected space is trivial.
Seen in this way, the continuations of a germ form a Riemann surface spread over the plane.
The theorem says that this surface has only one sheet over a simply connected region.
The monodromy group measures the failure over regions that are not simply connected.
</document>

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# Matrix norm

## Summary

A matrix norm is a [[norm|0V2WP9B1MF]] on a space of matrices, often required to be submultiplicative: $\norm{AB} \le \norm{A} \norm{B}$.
It measures the size of a matrix and so the size of errors and perturbations in matrix computations.
The main examples are the [[operator norms|06EKW1AKXF]] induced by vector norms and the [[Frobenius norm|JT8Q204E4R]].
Every submultiplicative matrix norm bounds the [[spectral radius|9V7YQA85PC]] from above.

## Explanations

### L1 human

A matrix norm gives one nonnegative number that says how big a matrix is.
One way is to ask how much the matrix can stretch a vector of length $1$.
Another way is to treat the matrix as a long list of numbers and take its usual length.
For the diagonal matrix with entries $3$ and $4$, the largest stretch is $4$, and the length of the list $(3, 0, 0, 4)$ is $5$.
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<document source="https://tamnd.github.io/math2/node/550P8PFMVC.md" tag="550P8PFMVC" kind="concept">
# Geodesic curvature

## Summary

The geodesic curvature of a curve on a surface measures how much the curve bends within the surface.
The acceleration of a unit-speed curve on a surface splits into a part normal to the surface and a part tangent to it.
The geodesic curvature is the size of the tangent part, with a sign.
Curves with geodesic curvature zero are the geodesics, the straightest curves of the surface.
It is intrinsic: it depends only on the first fundamental form.

## Explanations

### L1 human

Walk on a large ball along a path.
Some of the bending of your path is forced on you by the ball itself; the rest comes from how you steer.
Geodesic curvature measures only the steering.
Walking along the equator you never steer, so its geodesic curvature is zero, while a small circle around the north pole needs constant steering.
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<document source="https://tamnd.github.io/math2/node/552AWX3QXN.md" tag="552AWX3QXN" kind="definition">
# Wiener process

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space.
A *Wiener process*, or standard Brownian motion, is a real [[stochastic process|0QH5ENKZM1]] $(W_t)_{t \ge 0}$ such that:

1. $W_0 = 0$ almost surely.
2. For all $0 \le t_0 < t_1 < \dots < t_n$, the increments $W_{t_1} - W_{t_0}, \dots, W_{t_n} - W_{t_{n-1}}$ are [[independent|0GBJACNAP6]].
3. For all $0 \le s < t$, the increment $W_t - W_s$ has the [[normal distribution|05AQE0QR0D]] with mean $0$ and variance $t - s$.
4. Almost surely, the path $t \mapsto W_t$ is continuous.

## Remarks

Such a process exists; one construction takes the uniform limit of random series $\sum_k \xi_k \phi_k(t)$, with independent standard normal $\xi_k$ and suitable functions $\phi_k$ such as the Schauder functions.
The process $X_t = 2 W_t$ satisfies clauses 1, 2 and 4 but fails clause 3, since $X_t - X_s$ has variance $4(t - s)$.
Equivalently, $W$ is a centred [[Gaussian process|22WEM6RTW0]] with $E[W_s W_t] = \min(s, t)$ and continuous paths.
A process $W_t + \mu t$, or $\sigma W_t$, is called a Brownian motion with drift $\mu$ or with variance parameter $\sigma^2$.
A $d$-dimensional Wiener process is a vector of $d$ independent real Wiener processes.
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# The special linear group is the kernel of the determinant

## Example

Let $n \ge 1$. The [[determinant|0C3CFHDN6R]] is a homomorphism $\det : \mathrm{GL}_n(\mathbb{R}) \to \mathbb{R}^\times$, because $\det(AB) = \det(A)\det(B)$.
Here $\mathrm{GL}_n(\mathbb{R})$ is the [[general linear group|0FCHE902D7]], and $\mathbb{R}^\times$ is the group of nonzero real numbers under multiplication.
The [[kernel|0Z424HK8CQ]] of $\det$ is the set of matrices with determinant $1$. This is the [[special linear group|0V9PMN0RBH]] $\mathrm{SL}_n(\mathbb{R})$.
Thus $\mathrm{SL}_n(\mathbb{R})$ is a [[normal subgroup|0C9ZKYSPYR]] of $\mathrm{GL}_n(\mathbb{R})$.

## Remarks

For $n \ge 2$, the group $\mathrm{GL}_n(\mathbb{R})$ is not abelian. Thus the normality does not come from commutativity.
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# Measure-preserving transformation

## Summary

A measure-preserving transformation is a measurable map between [[measure spaces|P15VQS5DJ9]] under which the preimage of every measurable set has the same measure as the set.
Equivalently, the [[pushforward|3N66TM3TSQ]] of the first measure is the second one.
A measure-preserving map of a space to itself is the basic object of ergodic theory.
It models a system that evolves in time without creating or destroying mass, as in Hamiltonian mechanics.

## Explanations

### L1 human

Think of a fluid in a container that is stirred, where no fluid is added or lost.
After one step of stirring, the fluid that ends up in a region came from a region of the same volume.
A rotation of a circle preserves length, so it is measure-preserving.
The doubling map $x \mapsto 2x \bmod 1$ on $[0,1)$ is also measure-preserving, although it stretches: each interval has two preimage intervals of half the length.
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<document source="https://tamnd.github.io/math2/node/558XADG64M.md" tag="558XADG64M" kind="concept">
# Numerical methods for ordinary differential equations

## Summary

Numerical methods for ordinary differential equations compute approximate values of the solution of an [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$, at a finite set of times.
Most [[ordinary differential equations|0NT64JV96A]] have no solution in closed form, so these methods are the main way to solve them in science and engineering.
The two big families are one-step methods, such as the [[Euler method|0JARTXX4XP]] and the [[Runge–Kutta methods|7DBPJKYC6C]], and [[linear multistep methods|BX4D85A79P]].
Their theory studies consistency, measured by the [[local truncation error|T220HF0562]], together with stability and convergence.
It pays special attention to [[stiff equations|RR5FCG92PN]] and to [[adaptive step size|6MREWBZR7R]] control.

## Explanations

### L1 human

A differential equation tells how fast a quantity changes, given its current value.
A numerical method starts from the known initial value and takes small time steps, using the rate of change to predict the next value each time.
For $y' = y$, $y(0) = 1$, steps of length $0.1$ with the rule "new value equals old value times $1.1$" give $1.1^{10} \approx 2.594$ at time $1$, close to the true value $e \approx 2.718$.
Better methods sample the rate of change more cleverly and get much closer with the same number of steps.
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# Godunov's theorem

## Statement

Let $a \in \R$, and consider the [[transport equation|PVD7VZJZ9B]] $u_t + a u_x = 0$.
Fix $\Delta x, \Delta t > 0$ and put $\nu = a \Delta t / \Delta x$.
Consider a linear explicit scheme
$$u_j^{n+1} = \sum_{k=-K}^{K} c_k\, u_{j+k}^n, \qquad j \in \Z,$$
with real coefficients $c_k$ that depend only on $\nu$.
Call the scheme monotone if $c_k \ge 0$ for all $k$, and of order at least $2$ if
$$\sum_k c_k\, k^m = (-\nu)^m \qquad \text{for } m = 0, 1, 2.$$
If the scheme is monotone and of order at least $2$, then $\nu$ is an integer and the scheme is the exact shift $u_j^{n+1} = u_{j - \nu}^n$.
In particular, for $\nu \notin \Z$, a monotone linear scheme is at most first-order accurate.

## Remarks

The conditions for $m = 0, 1, 2$ say that the scheme is exact on polynomials of degree at most $2$, which is the same as a local truncation error $O(\Delta x^3)$ for smooth solutions.
Proof: the numbers $c_k \ge 0$ sum to $1$, so they are the weights of a probability distribution on the integers $k$, with mean $-\nu$ and second moment $\nu^2$.
So its variance is $0$, and all the weight sits on the single point $k = -\nu$, which must then be an integer.
For a linear scheme, monotone is the same as monotonicity preserving: an increasing step $u_i = 1$ for $i > J$ and $0$ otherwise gives $u_{j+1}^{n+1} - u_j^{n+1} = c_{J-j}$.
Examples: the [[upwind scheme|MTCRZC5QKT]] with $0 \le \nu \le 1$ is monotone and of first order.
The [[Lax–Wendroff method|YHZ9BCNDRZ]] is of second order, has the negative coefficient $c_1 = -\nu(1 - \nu)/2$ for $0 < \nu < 1$, and oscillates near jumps.

## Explanations

### L1 human

When a computer simulates a moving sharp front, simple schemes either smear the front or create wiggles next to it.
Godunov's theorem says that, for linear schemes, this is unavoidable: a linear scheme that never creates new wiggles is at best first-order accurate, and so it smears.
To have both sharp fronts and no wiggles, one must use schemes whose coefficients depend on the solution itself.

### L2 human

Godunov proved this in 1959, and it is the reason why modern high-resolution schemes are nonlinear.
[[Total variation diminishing|NX8RE9B72A]] schemes, flux limiters, MUSCL and ENO/WENO schemes switch to a first-order monotone flux near discontinuities and use a higher-order flux in smooth regions.
The theorem is a statement about linear schemes; it does not limit nonlinear ones, which can be second order away from extrema.
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# Rational surface

## Summary

A rational surface is an [[algebraic surface|KFTECAATKG]] that is birational to the projective plane.
It is the two-dimensional case of a [[rational variety|XV4K9GREJ9]].
Rational surfaces include the projective plane, the quadric $\mathbb{P}^1 \times \mathbb{P}^1$, the Hirzebruch surfaces, the blow-ups of the plane in points, and all smooth cubic surfaces.
They are the surfaces whose points can be parametrized, outside a lower-dimensional set, by two free parameters.

## Explanations

### L1 human

A surface is rational when its points can be described by two independent parameters through rational functions, with an inverse that is also given by rational functions.
For instance, the sphere $x^2 + y^2 + z^2 = 1$ can be parametrized by stereographic projection from a point, so it is rational.
Two rational surfaces can look very different, but they agree with the plane after removing some curves.
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# Matroid rank

## Definition

Let $M = (E, \mathcal{I})$ be a [[matroid|PX31ZNYS3Z]].
The *rank function* of $M$ is the map $r \colon \mathcal{P}(E) \to \set{0, 1, 2, \dots}$ given by
$$r(A) = \max \set{\abs{I} : I \in \mathcal{I}, \ I \subseteq A}.$$
The *rank* of $M$ is $r(E)$.

## Remarks

The maximum exists because $\emptyset \subseteq A$ is independent.
All maximal independent subsets of $A$ have $r(A)$ elements, and $r(E)$ is the size of every [[basis|QTSQR7130G]] of $M$.
A set $A$ is independent if and only if $r(A) = \abs{A}$.

The rank function satisfies, for all $A, B \subseteq E$:

1. $0 \le r(A) \le \abs{A}$.
2. If $A \subseteq B$, then $r(A) \le r(B)$.
3. $r(A \cup B) + r(A \cap B) \le r(A) + r(B)$ (submodularity).

Conversely, every function with these three properties is the rank function of exactly one matroid.

Example: for the matroid of the columns of a matrix, $r(A)$ is the [[rank|0ZC9YTXZRY]] of the submatrix formed by the columns in $A$.
For the graphic matroid of a graph with vertex set $V$, $r(A) = \abs{V} - c(A)$, where $c(A)$ is the number of connected components of the graph $(V, A)$.

Non-example: in the uniform matroid $U_{2,3}$, whose independent sets are the subsets of $\set{1, 2, 3}$ with at most $2$ elements, the size function $A \mapsto \abs{A}$ is not the rank function, because $r(\set{1, 2, 3}) = 2 \ne 3$.
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# Numerical diffusion

## Summary

Numerical diffusion is the smearing of a computed solution caused by the discretisation itself.
It acts like an extra diffusion term that the original equation does not have.
It is most visible in advection problems: a sharp front that should move without change of shape spreads out more and more.
It is measured by the coefficient of the second-derivative term in the modified equation of the scheme.
It is typical of first-order schemes such as the [[upwind scheme|MTCRZC5QKT]].
Some numerical diffusion is often needed for stability, but too much of it destroys the resolution of fine features.

## Explanations

### L1 human

Imagine a square pulse of dye carried by a steady current.
The exact pulse keeps its shape, but a simple computer simulation shows its edges blurring, as if the dye were also spreading by diffusion.
This blur is not physics; it is an error of the method, and it behaves like a diffusion whose strength shrinks with the grid size.
For the upwind scheme, the extra diffusion coefficient is $\frac{a \Delta x}{2}(1 - \nu)$, where $a$ is the speed and $\nu = a \Delta t / \Delta x$.
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# p-adic absolute value

## Definition

Let $p$ be a [[prime number|01826KMG2H]].
For a nonzero rational number $x$, let $v_p(x)$ be its [[p-adic valuation|03HDF50VN4]], the integer $k$ with $x = p^k a / b$ for integers $a, b$ not divisible by $p$.
The *$p$-adic absolute value* is the map $\abs{\cdot}_p \colon \Q \to \R$ given by
$$\abs{x}_p = p^{-v_p(x)} \text{ for } x \neq 0, \qquad \abs{0}_p = 0.$$

## Remarks

It is an [[absolute value|0FFXKTKKQC]] on $\Q$ in the sense of fields:

1. $\abs{x}_p = 0$ exactly when $x = 0$;
2. $\abs{xy}_p = \abs{x}_p \abs{y}_p$;
3. $\abs{x + y}_p \le \max(\abs{x}_p, \abs{y}_p)$, which is stronger than the triangle inequality.

Example: $\abs{12}_2 = 2^{-2} = 1/4$ and $\abs{3/8}_2 = 2^{3} = 8$.
Non-example: the map $x \mapsto p^{v_p(x)}$, without the minus sign, is not an absolute value.
Indeed, it gives $1$ to both $p^2 - 1$ and $1$, but $p^2$ to their sum $p^2$, so the triangle inequality fails.
Product formula: for every nonzero rational $x$, $\abs{x} \prod_p \abs{x}_p = 1$, where $\abs{x}$ is the usual absolute value.
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# Finite measure

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
The [[measure|PR11BZQQPG]] $\mu$ is *finite* if $\mu(X) < \infty$.

## Remarks

Lebesgue measure on the Borel sets of $[0, 1]$ is finite, since $\mu([0, 1]) = 1$.
Lebesgue measure on the Borel sets of $\R$ is not finite, since $\mu(\R) = \infty$.
If $\mu$ is finite, then $\mu(A) \le \mu(X) < \infty$ for every $A \in \Sigma$, by monotonicity.
Every [[probability measure|059TJN0NZ1]] is finite.
If $0 < \mu(X) < \infty$, then $\mu / \mu(X)$ is a probability measure.
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# Inverse distance weighting

## Definition

Let $(X, \delta)$ be a [[metric space|0VKD1TP7F3]], usually $\R^d$ with the [[Euclidean distance|0NPPGHQNXQ]].
Let $x_1, \dots, x_N \in X$ be pairwise distinct, let $u_1, \dots, u_N \in \R$, and let $p > 0$.
For $x \in X$ and $i = 1, \dots, N$ with $x \ne x_i$, put $w_i(x) = \delta(x, x_i)^{-p}$.
The *inverse distance weighting interpolant* of the data is the function $u \colon X \to \R$ given as follows.
If $x$ is not one of the $x_i$, then
$$u(x) = \frac{\sum_{i=1}^{N} w_i(x) \, u_i}{\sum_{i=1}^{N} w_i(x)}.$$
At the data points, $u(x_i) = u_i$ for $i = 1, \dots, N$.

## Remarks

The interpolant reproduces the data by definition, and it is continuous at each $x_i$, because $w_i(x) \to \infty$ as $x \to x_i$ while the other weights stay bounded.
Each value $u(x)$ is a convex combination of the $u_i$, so $\min_i u_i \le u(x) \le \max_i u_i$.
This scheme is also called Shepard's method.
It is the case of constant fits with weight $r^{-p}$ in [[moving least squares|T0JZN6WP5X]].

Example: on $\R$ with $p = 2$, data $u_1 = 10$ at $x_1 = 0$ and $u_2 = 40$ at $x_2 = 3$, at $x = 1$ the weights are $1$ and $1/4$.
So $u(1) = \frac{10 + 40/4}{1 + 1/4} = 16$, which lies between $10$ and $40$.

Non-example: the plain average $\sum_i u_i / N$ with equal weights does not depend on the distances and does not reproduce the data; for the same data it gives $25 \ne 10$ at $x_1$.
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# Fodor's lemma

## Statement

Let $\kappa$ be an uncountable [[regular cardinal|ZCE333FAWC]].
Let $S \subseteq \kappa$ be [[stationary|6FDYJVG1CK]] with $0 \notin S$.
Let $f \colon S \to \kappa$ be a [[function|0XHDTE5YMZ]] that is regressive, that is, $f(\alpha) < \alpha$ for each $\alpha \in S$.
Then there is $\gamma < \kappa$ such that the set $\set{\alpha \in S : f(\alpha) = \gamma}$ is stationary in $\kappa$.

## Remarks

The proof is by contradiction.
If each fibre $f^{-1}(\gamma)$ is nonstationary, pick a [[club set|GWS5QXH7WW]] $C_\gamma$ disjoint from it.
The diagonal intersection $D = \set{\alpha < \kappa : \alpha \in \bigcap_{\gamma < \alpha} C_\gamma}$ is club, so it meets $S$ in some ordinal $\alpha$.
Then $\gamma = f(\alpha) < \alpha$ gives $\alpha \in C_\gamma$, which contradicts $f(\alpha) = \gamma$.
Example: on the limit ordinals below $\omega_1$, any function with $f(\alpha) < \alpha$ is constant on a stationary set.
Stationarity is necessary: on the successor ordinals, $f(\beta + 1) = \beta$ is regressive and injective, so all its fibres have one element.
The lemma is also called the pressing-down lemma.

## Explanations

### L1 human

Suppose that to each point of a large set of countable ordinals you assign a smaller ordinal.
Fodor's lemma says that you must then assign the same value to a large set of points.
You cannot keep pushing every point down to a different place.
"Large" here means stationary, the right notion of size for these sets.

### L2 human

Fodor's lemma is the basic tool for working with stationary sets.
It is the infinite analogue of the pigeonhole principle for regressive functions, and it relies on closure of the club filter under diagonal intersections.
It is used in Solovay's splitting theorem for stationary sets and in Silver's theorem on singular cardinals.
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# Well-founded order

## Summary

A well-founded order is a partial order in which every nonempty subset has a minimal element.
Equivalently, assuming the axiom of dependent choice, there is no infinite strictly decreasing sequence.
It is the order case of a [[well-founded relation|0DQPNJNPYK]].
It is exactly what is needed for proofs by well-founded induction and definitions by well-founded recursion.
A [[well-order|01CGSJ0GNQ]] is a well-founded order that is also total.

## Explanations

### L1 human

In a well-founded order you cannot keep stepping down forever: every descent stops after finitely many steps.
The natural numbers with the usual order are the basic example.
The order need not be total: the positive integers ordered by divisibility are well-founded.
Indeed, a sequence in which each term is a proper divisor of the previous one must stop.
The integers with the usual order are not well-founded, since $0 > -1 > -2 > \cdots$ never stops.
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# Modular curve

## Definition

Let $\mathbb{H} = \set{\tau \in \C : \operatorname{Im} \tau > 0}$ be the [[upper half-plane|X6XGGH5FPW]], on which $\mathrm{SL}_2(\Z)$ acts by $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \tau = \frac{a\tau + b}{c\tau + d}$.
Let $\Gamma \subseteq \mathrm{SL}_2(\Z)$ be a [[congruence subgroup|1H6R856P3C]], that is, a subgroup that contains $\Gamma(N) = \set{\gamma \in \mathrm{SL}_2(\Z) : \gamma \equiv I \pmod N}$ for some $N \ge 1$.
The *open modular curve* of $\Gamma$ is the quotient $Y(\Gamma) = \Gamma \backslash \mathbb{H}$.
The *modular curve* of $\Gamma$ is
$$X(\Gamma) = \Gamma \backslash \big( \mathbb{H} \cup \mathbb{P}^1(\Q) \big),$$
where $\mathbb{P}^1(\Q) = \Q \cup \set{\infty}$ is acted on by the same formula; the finitely many points of $X(\Gamma) \setminus Y(\Gamma)$ are the cusps.

## Remarks

With the standard charts at elliptic points and at cusps, $X(\Gamma)$ is a compact [[Riemann surface|5K0N8QMV6Q]], hence a projective algebraic curve over $\C$.
The standard families are $X(N) = X(\Gamma(N))$, $X_0(N)$ for $\Gamma_0(N) = \set{\gamma : c \equiv 0 \pmod N}$ and $X_1(N)$ for $\Gamma_1(N) = \set{\gamma : c \equiv 0,\ a \equiv d \equiv 1 \pmod N}$.
The curves $X_0(N)$ and $X_1(N)$ have models over $\Q$.
The open curve $Y_0(N)$ parametrizes pairs $(E, C)$ of an elliptic curve over $\C$ and a cyclic subgroup $C$ of order $N$, up to isomorphism.

Example: for $\Gamma = \mathrm{SL}_2(\Z)$, the [[modular group|VEKV1EM77B]] up to sign, the $j$-invariant identifies $Y(1)$ with $\C$ and $X(1)$ with $\mathbb{P}^1(\C)$, of genus $0$.
The curve $X_0(11)$ has genus $1$ and is an elliptic curve over $\Q$.

Non-example: a subgroup of finite index in $\mathrm{SL}_2(\Z)$ that contains no $\Gamma(N)$ is not a congruence subgroup.
Its quotient is still a compact Riemann surface, but it is not a modular curve in the sense above.
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# Class number formula

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]].
Let $r_1$ be the number of real embeddings of $K$ and $r_2$ the number of pairs of complex conjugate non-real embeddings.
Let $h_K$ be the [[class number|KTKK9B1X41]] of $K$, $R_K$ its [[regulator|RWK0HNF1NA]], $w_K$ the number of [[roots of unity|0YV2CKP2AV]] in $K$ and $d_K$ its [[discriminant|E0VZTNPJXA]].
Then the [[Dedekind zeta function|KXS5DQDFYD]] $\zeta_K(s)$ has a simple pole at $s = 1$, and
$$\lim_{s \to 1^+} (s - 1)\, \zeta_K(s) = \frac{2^{r_1} (2\pi)^{r_2}\, h_K\, R_K}{w_K \sqrt{\abs{d_K}}}.$$

## Remarks

The proof counts ideals of norm at most $x$ in each ideal class.
This count reduces to counting lattice points in a region of $\R^{r_1} \times \C^{r_2}$ cut out using the units.
So its leading term is a volume; the volume brings in $R_K$ and $\sqrt{\abs{d_K}}$, and the classes bring in $h_K$.
Example: for $K = \Q$ one has $r_1 = 1$, $r_2 = 0$, $h = R = 1$, $w = 2$, $d = 1$, and the formula gives residue $1$, as for the Riemann zeta function.
For an imaginary quadratic field of discriminant $d < 0$, the formula becomes $h_K = \frac{w_K \sqrt{\abs{d}}}{2\pi} L(1, \chi_d)$, where $\chi_d$ is the Kronecker character of $K$ and $L$ its [[Dirichlet L-function|9D2Z543EJQ]].
For $K = \Q(i)$: $w = 4$, $d = -4$ and $L(1, \chi_{-4}) = 1 - \frac13 + \frac15 - \dots = \frac{\pi}{4}$, so $h_K = 1$.

## Explanations

### L1 human

The class number measures how far unique factorisation fails in a number field.
The formula says that this number, together with other basic invariants of the field, can be read off from the behaviour of an analytic function near the point $s = 1$.
So a purely algebraic quantity is computed by analysis.

### L2 human

For an abelian field $K$, the function $\zeta_K(s)$ is the product of $\zeta(s)$ and Dirichlet $L$-functions, so the residue becomes a product of values $L(1, \chi)$.
This gives Dirichlet's class number formulas for quadratic fields, and the fact that $L(1, \chi) \ne 0$ for real characters, which is the key step in [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]].
The units enter through the regulator, which is defined by means of [[Dirichlet's unit theorem|1D99CSKX6N]].
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# Hochster–Roberts theorem

## Statement

Let $k$ be a [[field|04C5ZBG1WT]] and let $G$ be a linearly reductive [[linear algebraic group|GA81VVQHEP]] over $k$.
Let $V$ be a finite-dimensional rational representation of $G$.
Let $k[V]$ be the [[polynomial ring|0SDC934R52]] of polynomial functions on $V$, with the induced action of $G$.
Then the [[ring of invariants|6VTR5EBNYC]] $k[V]^G$ is a [[Cohen–Macaulay ring|N1QJQEGBFC]].

## Remarks

A group is linearly reductive when every rational representation is a direct sum of irreducible ones.
In characteristic $0$ this holds exactly for the [[reductive groups|2XS84PWE3S]]; in characteristic $p > 0$ it holds for tori and for finite groups of order prime to $p$, for example.
Proof idea: linear reductivity gives the Reynolds operator, a $k[V]^G$-linear projection $k[V] \to k[V]^G$, so $k[V]^G$ is a direct summand of the regular ring $k[V]$.
Hochster and Roberts showed that such summands are Cohen–Macaulay here, first in characteristic $p$ with the Frobenius map, and then in characteristic $0$ by reduction to characteristic $p$.

Example: let $G = \set{1, g}$ act on $k[x, y]$ by $g \cdot x = -x$, $g \cdot y = -y$, where the characteristic of $k$ is not $2$.
Then $k[x, y]^G = k[x^2, xy, y^2]$ is a free module over the polynomial ring $k[x^2, y^2]$ with basis $1, xy$, which is the Cohen–Macaulay property in graded form.

Linear reductivity cannot be dropped: for a finite group whose order is divisible by the characteristic of $k$, the ring of invariants of a linear action need not be Cohen–Macaulay.

## Explanations

### L1 human

Invariant rings of nice groups are well behaved.
The theorem says that one can choose finitely many algebraically independent homogeneous invariants such that the whole invariant ring is a free module over the polynomial ring they generate.
So every invariant has a unique expression of a fixed simple form.

### L2 human

The Cohen–Macaulay property makes the Hilbert series of $k[V]^G$ a rational function with a numerator with nonnegative coefficients, and it controls the free resolution of the invariant ring.
The proof is an early major use of reduction to positive characteristic, the method that later gave [[tight closure|KJ9MMP99G6]].
The result fits with the finite generation of rings of invariants of reductive groups, the positive side of [[Hilbert's fourteenth problem|M5N08HD75V]].
Boutot later proved the stronger fact that, in characteristic $0$, these invariant rings have rational singularities.
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<document source="https://tamnd.github.io/math2/node/563ANQ3MY7.md" tag="563ANQ3MY7" kind="concept">
# Wright omega function

## Summary

The Wright omega function $\omega$ is a single-valued function built from the branches of the [[Lambert W function|FW2JKAN14B]].
It is $\omega(z) = W_{K(z)}(e^z)$, where the unwinding number $K(z)$ picks the branch.
It solves the equation $\omega + \ln \omega = z$ for all $z$ off two horizontal half-lines.
On the real line it is the unique real solution of this equation.
It is convenient in computer algebra because it avoids the choice of a branch.

## Explanations

### L1 human

The Lambert W function has many branches, and it is easy to pick the wrong one.
The Wright omega function packages the right branch for the input $e^z$ into one function of $z$.
On the real line it is simple: $\omega(x)$ is the positive number $y$ with $y + \ln y = x$.
For example, $\omega(1) = 1$ because $1 + \ln 1 = 1$.
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<document source="https://tamnd.github.io/math2/node/564VVAQTNR.md" tag="564VVAQTNR" kind="concept">
# Coset

## Summary

In mathematics, specifically group theory, a [[subgroup|0X1JXT47GA]] $H$ of a group $G$ may be used to decompose the underlying [[set|0BSJ7RQ9BW]] of $G$ into [[disjoint|0W4RPA8QH3]], equal-size [[subsets|0BJR9QR08E]] called **cosets**. There are *left cosets* and *right cosets*. Cosets (both left and right) have the same number of elements ([[cardinality|0G0ASKNA6Z]]) as does $H$. Furthermore, $H$ itself is both a left coset and a right coset. The number of left cosets of $H$ in $G$ is equal to the number of right cosets of $H$ in $G$. This common value is called the [[index|0DTXBQPA2S]] of $H$ in $G$ and is usually denoted by $[G : H]$.
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<document source="https://tamnd.github.io/math2/node/565B0Y76XE.md" tag="565B0Y76XE" kind="definition">
# Differential of a smooth map

## Definition

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], let $F \colon M \to N$ be a [[smooth|0V33BNKZ9S]] map, and let $p \in M$.
Tangent vectors are taken as derivations, as in the [[tangent bundle|WY6K3P4YCG]].
The *differential* of $F$ at $p$ is the linear map
$$dF_p \colon T_pM \to T_{F(p)}N, \qquad (dF_p\, v)(f) = v(f \circ F)$$
for $v \in T_pM$ and $f \in C^\infty(N)$.
The maps $dF_p$ for all $p$ together form the smooth map $dF \colon TM \to TN$, $(p, v) \mapsto (F(p), dF_p v)$.

## Remarks

In charts $(x^1, \ldots, x^m)$ near $p$ and $(y^1, \ldots, y^n)$ near $F(p)$, with $F^j = y^j \circ F$,
$$dF_p\left(\left.\frac{\partial}{\partial x^i}\right|_p\right) = \sum_{j=1}^{n} \frac{\partial F^j}{\partial x^i}(p) \left.\frac{\partial}{\partial y^j}\right|_{F(p)},$$
so the matrix of $dF_p$ is the [[Jacobian matrix|047NDTEPCV]] of $F$ in these coordinates.

Example: for $F \colon \R \to \R^2$, $F(t) = (\cos t, \sin t)$, one gets $dF_t(d/dt) = -\sin t \, \partial_x + \cos t \, \partial_y$.
Check: for $f \in C^\infty(\R^2)$, $(d/dt)(f(\cos t, \sin t)) = -\sin t \, f_x + \cos t \, f_y$ by the chain rule.

The formula $v \mapsto v(f \circ F)$ needs $F$ to be smooth: if $F$ is only continuous, $f \circ F$ need not be smooth and $v(f \circ F)$ is not defined.

The chain rule reads $d(G \circ F)_p = dG_{F(p)} \circ dF_p$.
Other notations are $F_*$, $DF$, $TF$ and $F'$.

## Explanations

### L1 human

The differential of a smooth map at a point is the linear map that sends each velocity vector at the point to the velocity vector of the image curve.
</document>

<document source="https://tamnd.github.io/math2/node/56J7DSVAZH.md" tag="56J7DSVAZH" kind="concept">
# Indecomposable continuum

## Summary

An indecomposable continuum is a [[continuum|NJQ196M1WR]] that is not the union of two proper subcontinua.
Familiar continua such as arcs, circles and discs are decomposable, so the indecomposable ones are a source of strange examples.
They appear in plane topology, as common boundaries of domains, and in dynamics, as attractors.

## Explanations

### L1 human

The interval $[0, 1]$ is the union of the two smaller pieces $[0, 1/2]$ and $[1/2, 1]$, each a continuum.
An indecomposable continuum cannot be cut up like this: any two smaller continua inside it leave something out.
Pictures of them look like infinitely many strands wound back and forth, such as the Knaster bucket handle.
The [[pseudo-arc|VGCTDCGS6V]] is an extreme case, where every subcontinuum is again indecomposable.
</document>

<document source="https://tamnd.github.io/math2/node/56MJSVTHG0.md" tag="56MJSVTHG0" kind="definition">
# Measure space

## Definition

A *measure space* is a triple $(X, \Sigma, \mu)$, where $(X, \Sigma)$ is a [[measurable space|N4KDT83V3S]] and $\mu$ is a [[measure|PR11BZQQPG]] on $(X, \Sigma)$.

## Remarks

Let $X = \set{1, \ldots, 6}$, let $\Sigma$ be the power set of $X$, and let $\mu(A) = \abs{A} / 6$.
Then $\Sigma$ is a σ-algebra and $\mu$ is a measure, since the number of elements is additive over disjoint sets, so $(X, \Sigma, \mu)$ is a measure space.
On $\N$, let $\mathcal{A}$ be the family of finite sets and of sets with finite complement, and let $\mu(A) = \abs{A}$ for finite $A$ and $\mu(A) = \infty$ otherwise.
Then $(\N, \mathcal{A}, \mu)$ is not a measure space, since $\mathcal{A}$ is not a σ-algebra: it does not contain the set of even numbers.
A measure space with $\mu(X) = 1$ is a [[probability space|0NFPJVJJHX]].
</document>

<document source="https://tamnd.github.io/math2/node/56NVR0SS61.md" tag="56NVR0SS61" kind="concept">
# Empirical risk minimization

## Summary

Empirical risk minimization is the principle of choosing, from a class of prediction rules, one that has the smallest average [[loss|3BTG6VVCGZ]] on the training data.
The average loss on the data, the empirical risk, stands in for the true risk, which depends on an unknown distribution.
[[Ordinary least squares|YM708KHV2W]] and [[maximum likelihood estimation|0FREPFKNS7]] are instances.
Whether it works is a question of [[statistical learning theory|5MR9ZSC3CX]].
It succeeds when the empirical risk is uniformly close to the true risk over the class.
In binary classification this holds for classes of finite [[VC dimension|KCFXR9K3GD]].

## Explanations

### L1 human

You cannot test a rule on all future data, so you test it on the data you have and pick the rule that does best.
For example, to fit a line through points, choose the line with the smallest average squared vertical distance to the points.
This is a good idea when the family of rules is not too rich.
If the family can fit anything, the winner may just memorise the data and predict badly on new points.
Adding a penalty for complexity, as in [[ridge regression|3KZW2930V2]], is the usual fix.
</document>

<document source="https://tamnd.github.io/math2/node/56QC25W5AT.md" tag="56QC25W5AT" kind="definition">
# Empirical Bayes method

## Definition

Let $\theta_1, \dots, \theta_N$ be i.i.d. with a [[prior|00NBBSXRQS]] distribution $G$ on a parameter space $\Theta$, and given them let $X_1, \dots, X_N$ be independent with $X_i \sim f(\cdot \mid \theta_i)$, for a known [[parametric model|Q0DXKAQXE1]] $f$.
The prior $G$ is unknown, either fully or as $G = G_\eta$ with an unknown hyperparameter $\eta$.
An *empirical Bayes method* proceeds in two steps.

1. It estimates $G$ by some $\hat{G}$ from $X_1, \dots, X_N$, using that the $X_i$ are i.i.d. with the [[marginal|0AXP2Z5N9H]] density $m_G(x) = \int_\Theta f(x \mid \theta) \, G(d\theta)$.
2. It makes inference on each $\theta_i$ with the [[posterior|0KJ7ZS8AJA]] of $\theta_i$ given $X_i$ computed under the prior $\hat{G}$.

## Remarks

In the parametric case a common choice is to take $\hat{\eta}$ maximising the marginal likelihood $\prod_{i} m_{G_\eta}(X_i)$.
Example: let $X_i \mid \theta_i \sim \mathcal{N}(\theta_i, 1)$ and $\theta_i \sim \mathcal{N}(0, \tau^2)$ with $\tau$ unknown.
The posterior mean is $(1 - B) X_i$ with $B = 1/(1 + \tau^2)$, and marginally $X_i \sim \mathcal{N}(0, 1 + \tau^2)$.
Since $\sum_i X_i^2 / (1 + \tau^2)$ has the chi-squared law with $N$ degrees of freedom, $(N - 2)/\sum_i X_i^2$ is an unbiased estimator of $B$ for $N \ge 3$; plugging it in gives the James–Stein estimator.
Example: for Poisson data, $f(x \mid \theta) = e^{-\theta}\theta^x/x!$, the posterior mean is $E[\theta \mid X = x] = (x + 1)\, m_G(x + 1)/m_G(x)$, and Robbins' method replaces $m_G$ by the observed frequencies of the counts.
Non-example: a full Bayesian analysis puts a fixed hyperprior on $\eta$ and integrates over it.
This is a hierarchical Bayes method, not an empirical Bayes method, since step 1 does not estimate $G$.
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# Coherent sheaf

## Summary

A coherent sheaf on a Noetherian scheme is a quasi-coherent sheaf that is locally given by finitely generated modules.
Coherent sheaves are the algebraic analogue of finite-dimensional data: on a proper scheme over a field, their cohomology groups are finite-dimensional.
They include vector bundles, ideal sheaves of closed subschemes, and structure sheaves of closed subschemes.

## Explanations

### L1 human

A coherent sheaf is a quasi-coherent sheaf that is finite in each small region.
On an affine piece it is described by a finitely generated module, so by finitely many generators and relations.
Vector bundles are coherent, but coherent sheaves can also do things vector bundles cannot.
For example, the sheaf that has the value $k$ at the origin of a line and $0$ elsewhere is coherent, with a jump in rank at one point.
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<document source="https://tamnd.github.io/math2/node/56V4WT0BT2.md" tag="56V4WT0BT2" kind="definition">
# Modification of a complex space

## Definition

Let $X$ and $Y$ be [[reduced complex spaces|831N2Q4GCB]].
A holomorphic map $f \colon X \to Y$ is a *modification* if:

1. $f$ is [[proper|15XEBGNGHC]] and surjective;
2. there is a [[nowhere dense|0VQYCCQVJ5]] [[analytic subset|GB7M582M5Y]] $B \subseteq Y$ such that $f^{-1}(B)$ is nowhere dense in $X$;
3. $f$ restricts to a biholomorphic map $X \setminus f^{-1}(B) \to Y \setminus B$.

## Remarks

Example: the [[blow-up|C4KGYAFPN7]] $f \colon X \to \C^2$ of $\C^2$ at the origin is a modification with $B = \set{0}$.
It is proper and surjective, $f^{-1}(0)$ is a projective line, which is nowhere dense in the $2$-dimensional manifold $X$, and $f$ is biholomorphic off it.

Non-example: let $X$ be the disjoint union of $\C$ and a point $p$, and let $f \colon X \to \C$ be the identity on $\C$ and send $p$ to $0$.
Then $f$ is proper and surjective, and with $B = \set{0}$ clause 3 holds.
But $f^{-1}(B)$ contains the isolated point $p$, so it is not nowhere dense and clause 2 fails.
No other choice of $B$ helps, since $f$ is not injective over $0$, so $p$ must lie over $B$.

The set $f^{-1}(B)$ is the exceptional set.
Some authors call these maps proper modifications.
</document>

<document source="https://tamnd.github.io/math2/node/56YF8W9NXP.md" tag="56YF8W9NXP" kind="statement">
# h-cobordism theorem

## Statement

Let $n \ge 5$.
Let $W$ be a compact smooth $(n+1)$-manifold whose boundary is a disjoint union $\partial W = M_0 \sqcup M_1$ of two closed $n$-manifolds.
Assume that $W$ is an *h-cobordism*: both inclusions $M_0 \hookrightarrow W$ and $M_1 \hookrightarrow W$ are [[homotopy equivalences|XZG6MTSWCN]].
Assume also that $M_0$ is connected and [[simply connected|S1XK3HXHFX]].
Then $W$ is [[diffeomorphic|0KZZ70ZHXP]] to $M_0 \times [0, 1]$ by a diffeomorphism that is the identity on $M_0$.
In particular $M_0$ and $M_1$ are diffeomorphic.

## Remarks

The theorem is due to Smale.
Idea of the proof: a Morse function on $W$ gives a [[handle decomposition|9V11N8VVTP]] of $W$ on $M_0$.
The homotopy equivalence condition makes the handle chain complex acyclic, and handles are traded and cancelled in pairs until none remain.
Cancelling a pair needs the Whitney trick to remove extra intersection points of spheres in a level set.
The Whitney trick needs simple connectivity and level sets of dimension at least $5$.
A cobordism with no handles is a product.

The theorem at work: let $\Sigma$ be a smooth closed $m$-manifold homotopy equivalent to $S^m$ with $m \ge 6$.
Removing two disjoint open balls from $\Sigma$ gives an h-cobordism between two copies of $S^{m-1}$, which is therefore a product.
It follows that $\Sigma$ is homeomorphic to $S^m$, the [[generalized Poincaré conjecture|8KDV43WZ3Z]] in these dimensions.

Both extra hypotheses are needed.
Without simple connectivity there are h-cobordisms that are not products; the obstruction is the Whitehead torsion, see the [[s-cobordism theorem|E1E5ABP89W]].
For $n = 4$ the smooth statement fails: Donaldson found smooth $5$-dimensional h-cobordisms between simply connected $4$-manifolds that are not products.

## Explanations

### L1 human

Two manifolds are h-cobordant when they form the two ends of a bigger manifold that each end fills up, up to homotopy.
The theorem says that in high dimensions, when there are no loops to worry about, such a bigger manifold is just a cylinder.
So the two ends are the same manifold.
In this way the theorem proves the Poincaré conjecture in dimensions at least six.

### L2 human

The h-cobordism theorem converts a homotopy-theoretic condition into a diffeomorphism.
It is the base of the classification of high-dimensional manifolds through [[surgery theory|MHHN8386TG]]: surgery produces cobordisms, and the theorem shows that suitable cobordisms are products.
Its proof is the model application of Morse theory and handle cancellation.
The [[s-cobordism theorem|E1E5ABP89W]] extends it to manifolds with a nontrivial fundamental group.
In dimension $4$ the topological version holds by Freedman's work, but the smooth version fails.
</document>

<document source="https://tamnd.github.io/math2/node/572QKEYYKX.md" tag="572QKEYYKX" kind="statement">
# Lieb–Thirring inequality

## Statement

Let $n \ge 1$ be an integer and let $\gamma$ be a real number with
$$\gamma \ge \tfrac{1}{2} \text{ if } n = 1, \qquad \gamma > 0 \text{ if } n = 2, \qquad \gamma \ge 0 \text{ if } n \ge 3.$$
Then there is a constant $L_{\gamma,n}$, depending only on $\gamma$ and $n$, with the following property.
Let $V \colon \R^n \to \R$ be measurable with positive part $V_+ = \max(V, 0)$ locally integrable and negative part $V_- = \max(-V, 0)$ in the Lebesgue space $L^{\gamma + n/2}(\R^n)$.
Let $E_1 \le E_2 \le \cdots < 0$ be the negative [[eigenvalues|0BQAA2BK3Q]], counted with multiplicity, of the [[Schrödinger operator|89AA6XJY5K]] $-\Delta + V$ on $L^2(\R^n)$, defined through its quadratic form.
Then
$$\sum_j \abs{E_j}^\gamma \le L_{\gamma,n}\int_{\R^n} V_-(x)^{\gamma + n/2} \, dx,$$
where for $\gamma = 0$ the left side is the number of negative eigenvalues.

## Remarks

The right side is, up to a constant, the semiclassical value: counting phase-space volume gives $\sum_j \abs{E_j}^\gamma \approx L^{\mathrm{cl}}_{\gamma,n}\int V_-^{\gamma + n/2} \, dx$ for large potentials.
Lieb and Thirring proved the inequality for $\gamma > 1/2$ when $n = 1$ and $\gamma > 0$ when $n \ge 2$.
The case $\gamma = 0$, $n \ge 3$ is the Cwikel–Lieb–Rozenblum bound, and the endpoint $\gamma = 1/2$, $n = 1$ is due to Weidl.
The restrictions on $\gamma$ are needed.
For $n = 1, 2$, every nonzero $V \le 0$ produces a negative eigenvalue, so no bound with $\gamma = 0$ is possible there.
For $n = 1$ the bound also fails for $0 < \gamma < 1/2$.

For $\gamma = 1$, a duality argument turns the inequality into a lower bound on kinetic energy.
For an $N$-fermion wave function with one-particle density $\rho$, the kinetic energy is at least $K_n\int \rho(x)^{1 + 2/n} \, dx$ for a constant $K_n > 0$.

## Explanations

### L1 human

The negative eigenvalues of $-\Delta + V$ are the bound states of a quantum particle in a potential well $V$.
The inequality says that a shallow or small well cannot hold many deep bound states.
The total, measured by the sum of powers of the binding energies, is controlled by a single integral of the well depth.

### L2 human

Lieb and Thirring found the inequality to give a short proof of the stability of matter.
This says that the energy of $N$ electrons and $N$ nuclei is at least $-CN$ for a constant $C$ independent of $N$.
The case $\gamma = 1$ is the key, through the kinetic energy bound, which resembles the Thomas–Fermi model.
The sharp constants are an active topic; for $n = 1$ and $\gamma = 3/2$ the sharp constant equals the semiclassical one.
The inequality is a quantitative version of the fact that $-\Delta + V$ has finitely many negative eigenvalues when $V_-$ is small at infinity.
</document>

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# Similarity solution

## Summary

A similarity solution, or self-similar solution, of a partial differential equation is a solution that is unchanged by a scaling of its variables.
Its graph at one time is a rescaled copy of its graph at any other time.
Such a solution has the form $u(x, t) = t^{-\alpha} v(x / t^{\beta})$, so finding it reduces to an equation in fewer variables.
Many fundamental solutions, [[rarefaction waves|MFN4Q5SH8H]] and blow-up profiles are of this kind.
See the [[definition|3DJMFT0SQM]].

## Explanations

### L1 human

Drop ink into still water: the blot grows and fades, but its shape stays the same, only stretched.
A similarity solution is a solution with this property.
If one measures distance in units that grow like a power of time, the picture stops changing.
For the heat equation the blot has width proportional to $\sqrt{t}$ and height proportional to $t^{-1/2}$ in one dimension.
The variable $x / \sqrt{t}$ is then the only one that matters, and the PDE becomes an ordinary differential equation.
</document>

<document source="https://tamnd.github.io/math2/node/579PT4D349.md" tag="579PT4D349" kind="definition">
# Analytic geometry

## Definition

Let $E$ be a [[Euclidean space|0XPR1E85W0]] of dimension $n$, and fix a [[Cartesian coordinate system|0RWKYX8339]] on $E$.
It is a bijection $P \mapsto x(P) \in \R^n$ given by an origin and $n$ mutually perpendicular unit axes.
*Analytic geometry* is the study of figures in $E$ through their coordinates:

1. a figure $F \subseteq E$ is described by equations and inequalities, as the set of points $P$ whose coordinates satisfy them;
2. the [[distance|0NPPGHQNXQ]] between $P$ and $Q$ is computed as $\abs{x(P) - x(Q)} = \sqrt{\sum_{i=1}^n (x_i(P) - x_i(Q))^2}$, and angles through the dot product of coordinate vectors;
3. geometric statements are proved by algebraic computation on these equations.

## Remarks

Example: in the plane, the line $y = x$ and the circle $x^2 + y^2 = 2$ meet where $2x^2 = 2$, that is at $(1, 1)$ and $(-1, -1)$.
The distance between these points is $\sqrt{2^2 + 2^2} = 2\sqrt{2}$, the diameter of the circle, as it should be for a line through the centre.
Non-example: with oblique axes at an angle of $60^\circ$ and unit lengths on both axes, the formula of clause 2 fails.
The points with coordinates $(1, 0)$ and $(0, 1)$ are then at distance $1$, since with the origin they form an equilateral triangle, and not $\sqrt{2}$.
Affine notions such as lines, midpoints and parallelism can be studied with any affine coordinates; metric notions need Cartesian ones.
In the plane, a line is $\set{(x, y) : ax + by = c}$ with $(a, b) \ne (0, 0)$, and the circle with centre $(a, b)$ and radius $r$ is $\set{(x, y) : (x - a)^2 + (y - b)^2 = r^2}$.
</document>

<document source="https://tamnd.github.io/math2/node/57BXX32WF0.md" tag="57BXX32WF0" kind="statement">
# Giraud's theorem

## Statement

Let $\mathcal{E}$ be a locally small category.
Then $\mathcal{E}$ is a [[Grothendieck topos|S3YVE24AQK]] if and only if it has the following properties:

1. $\mathcal{E}$ has all finite limits.
2. $\mathcal{E}$ has all small [[coproducts|4FJ0S71P58]], and they are disjoint and stable under [[pullback|Q05K04QJ71]].
3. Every [[epimorphism|RVXDSY04KG]] in $\mathcal{E}$ is a [[coequalizer|DVSTWGVCSZ]].
4. Every equivalence relation $R \rightrightarrows E$ in $\mathcal{E}$ has a coequalizer $E \to Q$ and is the [[kernel pair|E70DQGM8X5]] of $E \to Q$.
5. Every coequalizer in $\mathcal{E}$ is stable under pullback.
6. $\mathcal{E}$ has a small set of [[generators|CK08JW32WB]].

## Remarks

Coproducts are disjoint when the pullback of two different coproduct inclusions is an initial object.
A construction is stable under pullback when pulling it back along any morphism gives a construction of the same kind.
One direction is a check in the category of sheaves, where limits are computed objectwise and colimits are sheafifications of objectwise colimits.
For the converse, take a small full subcategory $\mathcal{C}$ of generators.
Give it the topology $J$ whose covering sieves are the families that are jointly epimorphic in $\mathcal{E}$.
Then one shows that $E \mapsto \Hom(-, E)|_{\mathcal{C}}$ is an equivalence of $\mathcal{E}$ with $\mathrm{Sh}(\mathcal{C}, J)$.
Example: $\mathbf{Set}$ has all six properties, with the one-point set as generator.
Example: the category of finite sets satisfies conditions 1, 3, 4, 5 and 6.
It fails condition 2, since it has no infinite coproducts, and it is not a Grothendieck topos.

## Explanations

### L1 human

Giraud's theorem tells when a category is "sheaves on some site" without naming a site.
The conditions say that the category has limits and well-behaved colimits, like the category of sets, and that a small set of objects is enough to probe all objects.
It makes the notion of Grothendieck topos intrinsic.

### L2 human

The theorem is used to show that categories built in other ways are toposes, for example categories of sheaves glued from local pieces or categories of equivariant sheaves.
It also shows that a Grothendieck topos determines a site only up to choice: any small generating full subcategory with its canonical topology is a site for it.
Compared with the elementary definition of a [[topos|6837RS7N82]], the extra conditions are the small coproducts and the small set of generators.
</document>

<document source="https://tamnd.github.io/math2/node/57DPMSN7H8.md" tag="57DPMSN7H8" kind="definition">
# Effective resistance

## Definition

Let $(G, c)$ be a finite [[electrical network|T7KRE7J3B4]] with vertex set $V$, and let $a, b \in V$ with $a \ne b$.
Let $v \colon V \to \R$ be the unique function with $v(a) = 1$ and $v(b) = 0$ that is [[harmonic|WRY0M7SW59]] on $V \setminus \set{a, b}$.
This is the solution of a [[discrete Dirichlet problem|YKETR75ZYW]] with boundary $\set{a, b}$.
Let $I = \sum_{y} c(a, y)\big(1 - v(y)\big)$ be the net current that leaves $a$.
The *effective resistance* between $a$ and $b$ is $R_{\mathrm{eff}}(a, b) = 1/I$.
The number $C_{\mathrm{eff}}(a, b) = I$ is the *effective conductance*.

## Remarks

The current $I$ is positive: $0 \le v \le 1$ by the maximum principle, and $v(y) < 1$ for some neighbour $y$ of $a$ since $G$ is connected.
Scaling $v$ shows that $R_{\mathrm{eff}}(a, b)$ is the voltage difference between $a$ and $b$ when a current of strength $1$ enters at $a$ and leaves at $b$.
Example: for the path $a - m - b$ with unit conductances, $v(m) = \tfrac12$, so $I = \tfrac12$ and $R_{\mathrm{eff}}(a, b) = 2$, the sum of the two resistances in series.
If an edge of conductance $1$ is added between $a$ and $b$, the current becomes $\tfrac12 + 1$, so $R_{\mathrm{eff}}(a, b) = \tfrac23$, as for resistors in parallel.
Non-example: if $a$ and $b$ lie in different components of a disconnected graph, no current can flow between them.
Then the effective resistance is not a finite number, which is why connectedness is part of the definition of a network.
With $L$ the weighted [[graph Laplacian|TN6M6DVZRP]] and $L^{+}$ its Moore–Penrose pseudoinverse, $R_{\mathrm{eff}}(a, b) = (e_a - e_b)^{T} L^{+} (e_a - e_b)$, where $e_x$ is the indicator vector of $x$.
The function $(a, b) \mapsto R_{\mathrm{eff}}(a, b)$, extended by $0$ on the diagonal, is a metric on $V$, the resistance distance.
</document>

<document source="https://tamnd.github.io/math2/node/57FR21NDGV.md" tag="57FR21NDGV" kind="concept">
# Continuous wavelet transform

## Summary

The continuous wavelet transform compares a function $f$ with all rescaled and shifted copies $\abs{a}^{-1/2} \psi((t - b)/a)$ of one analyzing function $\psi$.
It gives a function of scale $a$ and position $b$, which shows where $f$ has features of each size.
When $\psi$ satisfies the [[admissibility condition|2BBB8NY5ZA]], the transform preserves energy up to a constant, and $f$ can be rebuilt from it.
Unlike the [[short-time Fourier transform|3DHNP5NEWH]], its time resolution adapts to the scale.

## Explanations

### L1 human

Pick a small wave shape, like the Mexican hat curve.
Stretch it to many widths and slide it to every position.
At each width and position, measure how well the signal matches the shape.
Wide shapes pick out slow trends and narrow shapes pick out sharp details, such as a jump in the signal.
The resulting map of matches, scale against position, is the continuous wavelet transform.
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<document source="https://tamnd.github.io/math2/node/57PTMS2ADY.md" tag="57PTMS2ADY" kind="definition">
# Ordinal analysis

## Definition

Let $T$ be a recursively axiomatized theory that contains elementary arithmetic.
An *ordinal analysis* of $T$ is the determination of its [[proof-theoretic ordinal|C19Z3B6V4P]] $\abs{T}$.
It consists of three parts.

1. A computable notation system: a computable well-order $\prec$ on a set of codes in $\N$, whose order type is an ordinal $\alpha$.
2. A lower bound: for each $\beta < \alpha$, a proof in $T$ of [[transfinite induction|0ZHQMQ2C1N]] along the initial segment of $\prec$ of order type $\beta$.
3. An upper bound: a proof that $T$ does not prove transfinite induction along $\prec$ itself. More precisely, each computable well-order that $T$ proves well-founded has order type below $\alpha$.

Then $\abs{T} = \alpha$.

## Remarks

Example: Gentzen's analysis of [[Peano arithmetic|0QFT7Z76N5]] uses the Cantor normal forms below [[epsilon nought|KT50SMH3EC]] as notation system, and gives $\abs{\mathrm{PA}} = \varepsilon_0$.
The upper bound comes from a cut-elimination argument for PA, together with transfinite induction up to $\varepsilon_0$.
Non-example: a proof that $T$ is consistent, by itself, is not an ordinal analysis, because it gives no ordinal.
The upper bound is usually proved in a weak base theory together with transfinite induction along $\prec$.
This also gives a proof of the consistency of $T$ in that base theory.
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<document source="https://tamnd.github.io/math2/node/57TCAKSBQT.md" tag="57TCAKSBQT" kind="definition">
# Fuchsian group

## Definition

Let $\mathrm{PSL}_2(\R) = \mathrm{SL}_2(\R) / \set{\pm I}$, where [[SL(2,R)|Q0GBCB9RA4]] has the topology of a subset of $\R^4$ and $\mathrm{PSL}_2(\R)$ has the quotient topology.
A *Fuchsian group* is a [[discrete subgroup|5M7D1BSHVZ]] of $\mathrm{PSL}_2(\R)$.

## Remarks

The group $\mathrm{PSL}_2(\R)$ acts on the [[upper half-plane|X6XGGH5FPW]] $\mathbb H$ by [[Möbius transformations|0SG74WBFA2]] $z \mapsto (az + b)/(cz + d)$, and this is the group of orientation-preserving isometries of the hyperbolic metric.
A subgroup $\Gamma \subseteq \mathrm{PSL}_2(\R)$ is Fuchsian if and only if its action on $\mathbb H$ is [[properly discontinuous|J5X5ED0B2Y]].

Example: $\mathrm{PSL}_2(\Z)$, the image of the integer matrices of determinant $1$, is Fuchsian.
Two distinct integer matrices differ by at least $1$ in some entry, so a small neighbourhood of $\pm I$ contains no other element of the group.

Non-example: the group generated by $z \mapsto z + 1$ and $z \mapsto z + \sqrt2$ consists of the translations $z \mapsto z + a + b\sqrt2$ with $a, b \in \Z$.
The translation lengths $a + b\sqrt2$ come arbitrarily close to $0$ without being $0$, so the group is not discrete, and it is not Fuchsian.

Some authors call a group Fuchsian when it is a discrete group of Möbius transformations that preserves a disc or half-plane in the Riemann sphere.
Such a group is conjugate to a group as above.
Since $\mathrm{PSL}_2(\R) \subset \mathrm{PSL}_2(\C)$, every Fuchsian group is a Kleinian group.
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<document source="https://tamnd.github.io/math2/node/581AGFCCRX.md" tag="581AGFCCRX" kind="concept">
# Robin boundary condition

## Summary

A Robin boundary condition prescribes a linear combination of the value of the unknown function and its outward normal derivative on the boundary.
It interpolates between the [[Dirichlet condition|EETVA6QYX4]] and the [[Neumann condition|GS3KT2H645]].
It models heat exchange with the surroundings, where the flux through the wall is proportional to the temperature difference.
The separated boundary conditions of [[Sturm–Liouville theory|7WV6B2502K]] are of this type.
See the [[definition|Y19WM1EMS7]].

## Explanations

### L1 human

Think of a hot rod whose ends lose heat to the air.
The heat flowing out of an end is proportional to how much hotter the end is than the air.
This links the slope of the temperature at the end to its value there.
For an end at $x = 1$ with outside temperature $0$, the condition reads $u'(1) = -h\, u(1)$ for a constant $h > 0$.
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<document source="https://tamnd.github.io/math2/node/584GXXY73M.md" tag="584GXXY73M" kind="concept">
# Polish space

## Summary

A Polish space is a [[separable|0Z2P41SVDP]] [[topological space|0CQHFPGJ45]] whose topology comes from a [[complete metric|0BY52JW6NK]].
Examples are the real line, $\R^n$, the [[Cantor set|0BXNDZFQME]], the [[Baire space|P2WS16HKD7]] and every separable Banach space.
Polish spaces are the natural setting of [[descriptive set theory|TF8HMVXKYC]] and of much of probability theory.
The name honours the Polish topologists and logicians, such as Sierpiński and Kuratowski, who first studied these spaces in depth.

## Explanations

### L1 human

A Polish space is a space where limits of Cauchy sequences exist and where a countable set of points comes close to every point.
The real line is the model case: the rationals are dense, and Cauchy sequences converge.
The topology matters, not the chosen metric: the open interval $(0, 1)$ is Polish, even though its usual metric is not complete, because it is homeomorphic to the real line.
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<document source="https://tamnd.github.io/math2/node/5858M5PGN6.md" tag="5858M5PGN6" kind="statement">
# Dolbeault–Grothendieck lemma

## Statement

Let $\Delta = \set{z \in \C^n : \abs{z_j} < r_j \text{ for } j = 1, \dots, n}$ be a [[polydisc|DJBWP7D3M8]] with radii $0 < r_j \le \infty$.
Let $p \ge 0$ and $q \ge 1$ be integers, and let $\alpha$ be a smooth [[(p,q)-form|DC81TCR98Y]] on $\Delta$ with $\bar\partial \alpha = 0$, where $\bar\partial$ is the [[dbar operator|M1VKCW3RFW]].
Then there is a smooth $(p,q-1)$-form $\beta$ on $\Delta$ with $\bar\partial \beta = \alpha$.

## Remarks

The key case is $n = 1$, $q = 1$: for a smooth function $g$ with compact support in $\C$, the function
$$u(z) = \frac{1}{2\pi i} \int_\C \frac{g(w)}{w - z}\, dw \wedge d\bar w$$
is smooth and satisfies $\partial u / \partial \bar z = g$.
In general one first proves the result on a slightly smaller polydisc, by induction on the largest index $k$ such that some $d\bar z_k$ occurs in $\alpha$.
The coefficients involving $d\bar z_k$ are holomorphic in $z_{k+1}, \dots, z_n$, and solving the one-variable equation in $z_k$ with a cutoff removes $d\bar z_k$ without destroying this holomorphy.
An exhaustion of $\Delta$ by smaller polydiscs, together with Runge approximation to correct the successive solutions, gives the result on $\Delta$ itself.
Example: on $\C$, the $(0,1)$-form $\bar z \, d\bar z$ is $\bar\partial$ of $\bar z^2/2$.
The condition $q \ge 1$ is necessary to make the statement meaningful; for $q = 0$ a $\bar\partial$-closed $(p,0)$-form is a holomorphic $p$-form and need not vanish.
The conclusion fails on general open sets: on $\C^2 \setminus \set{0}$ the Dolbeault group $H^{0,1}$ is not zero.

## Explanations

### L1 human

The lemma says that on a polydisc every form with $\bar\partial \alpha = 0$ is of the form $\bar\partial \beta$.
It is the complex analogue of the Poincaré lemma, which says that closed forms on a ball are exact.
In particular, on a small polydisc around any point of a complex manifold, the equation $\bar\partial \beta = \alpha$ can always be solved.

### L2 human

The lemma is the local input to the Dolbeault isomorphism.
It says that the sequence of sheaves $0 \to \Omega^p \to \mathcal{A}^{p,0} \to \mathcal{A}^{p,1} \to \cdots$ is exact, where $\Omega^p$ is the sheaf of holomorphic $p$-forms and $\mathcal{A}^{p,q}$ is the sheaf of smooth $(p,q)$-forms.
So every Dolbeault cohomology class is locally trivial, and the global groups measure only the topology and the complex structure of the manifold as a whole.
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<document source="https://tamnd.github.io/math2/node/58895A7T4H.md" tag="58895A7T4H" kind="concept">
# Trajectory optimization

## Summary

Trajectory optimization is the computation of a state trajectory and an open-loop control that minimise a cost for a [[control system|ZPDTVZQN5K]], subject to constraints.
It is the numerical side of [[optimal control|T3V6ZNAMDN]]: one wants one good trajectory from a given initial state, not a feedback law for every state.
Indirect methods solve the necessary conditions, as in [[shooting|RDTESNZTKQ]], while direct methods discretise the problem into a nonlinear program, as in [[direct collocation|T58W34D60G]].
It is used to plan rocket ascents, satellite transfers, walking robots and vehicle manoeuvres.

## Explanations

### L1 human

Suppose a robot arm must move from one pose to another in two seconds while using as little energy as possible and not hitting an obstacle.
Trajectory optimization finds the whole motion in advance: the positions over time and the motor commands that produce them.
A small example is the double integrator $x'' = u$ that must go from rest at $0$ to rest at $1$ in unit time.
Minimising $\int_0^1 u^2 \, dt$ gives the smooth motion $x(t) = 3t^2 - 2t^3$ with $u(t) = 6 - 12t$.
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<document source="https://tamnd.github.io/math2/node/589GJMH5EF.md" tag="589GJMH5EF" kind="statement">
# Griffiths transversality

## Statement

Let $\varpi \colon \mathcal{X} \to B$ be a complex analytic [[family|VKQSM3BCE4]] of compact [[Kähler manifolds|X60JG9AEQJ]] over a simply connected base $B$.
Fix $k$, let $H = H^k(X_0, \C)$, and let $\mathcal{P}^p \colon B \to \operatorname{Gr}(f^p, H)$, $t \mapsto F^p H^k(X_t)$, be the components of its [[period map|YGNXQVTWP3]].
Then:

1. Each $\mathcal{P}^p$ is holomorphic.
2. For each $t \in B$ and each tangent vector $v \in T_t B$, the differential $d\mathcal{P}^p_t(v)$, viewed in $T_{F^p}\operatorname{Gr}(f^p, H) = \Hom(F^p, H/F^p)$, has image in $F^{p-1}/F^p$, where $F^j = F^j H^k(X_t)$.

## Remarks

Equivalently, if $s(t)$ is a holomorphic family of classes with $s(t) \in F^p H^k(X_t)$, then the derivative of $s$ in any direction, taken in the fixed space $H$, lies in $F^{p-1} H^k(X_t)$.
The idea of the proof: represent $s(t)$ by closed forms of type $(r, k-r)$ with $r \ge p$ on $X_t$.
To first order, a change of complex structure replaces $dz_j$ by $dz_j + \sum_k \varphi_{jk} d\bar z_k$, so it lowers the holomorphic degree of a form by at most one.
Hence the derivative has components only in types $(r, k-r)$ with $r \ge p - 1$.

Example: for a family of curves and $k = 1$, $F^0 = H$, so condition 2 holds automatically.
For Calabi–Yau threefolds and $k = 3$, condition 2 says that the derivative of the line $F^3 = H^{3,0}$ lies in $F^2$, which is a genuine restriction.

## Explanations

### L1 human

The Hodge pieces of the cohomology move holomorphically as the complex structure varies.
Moreover, each step of the Hodge filtration can only move "one step down" to first order.
This restricts which curves in the period domain can come from families of manifolds.

### L2 human

Transversality is the key constraint of variations of Hodge structure; it is built into their definition.
Period domains are usually not Hermitian symmetric, and the transversality condition defines a horizontal distribution on them.
Curvature estimates along horizontal directions give the Griffiths–Schmid results on period maps, for instance that horizontal holomorphic maps from a disc are distance decreasing for a suitable metric.
The derivative of $\mathcal{P}^p$ is computed by the [[Kodaira–Spencer map|EWZ1G10MAY]] followed by cup product and contraction.
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<document source="https://tamnd.github.io/math2/node/58AM5FK67J.md" tag="58AM5FK67J" kind="concept">
# Neville's algorithm

## Summary

Neville's algorithm computes the value at one point $x$ of the [[interpolation polynomial|SS1YMS2NPC]] through given data, without computing its coefficients.
It builds a triangular table.
Each entry is the value at $x$ of the interpolation polynomial through a block of consecutive nodes.
It is computed from two entries for shorter blocks.
It costs $O(n^2)$ operations for $n + 1$ nodes.
It is often used for extrapolation, for example to the limit $h \to 0$ in Richardson-type methods.

## Explanations

### L1 human

Through each single point there is a constant, through each pair a line, through each triple a parabola, and so on.
Neville's idea is to get the value of the curve through a block of points by blending two shorter curves.
One goes through the block without its last point, the other through the block without its first point.
The blend is a weighted average whose weights depend on how far $x$ is from the two end nodes.
For the points $(0, 1)$ and $(1, 3)$, the line has value $1 \cdot (1 - x) + 3 \cdot x$, and at $x = 2$ this gives $5$.
The [[definition|KVZ7D84541]] gives the recursion.
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<document source="https://tamnd.github.io/math2/node/58MV5BK731.md" tag="58MV5BK731" kind="definition">
# Parallelohedron

## Definition

Let $d \ge 1$ and let $P \subset \R^d$ be a $d$-dimensional [[convex polytope|300K6CB1KH]].
For $t \in \R^d$ write $P + t = \set{x + t : x \in P}$ for the [[translate|087VJM5Q77]] of $P$ by $t$.
The polytope $P$ is a *parallelohedron* if there is a set $T \subset \R^d$ such that:

1. the translates $P + t$ with $t \in T$ cover $\R^d$;
2. for distinct $s, t \in T$, the interiors of $P + s$ and $P + t$ are disjoint;
3. the tiling is face to face: the intersection of any two of the translates is empty or a face of both.

## Remarks

Example: the square $P = [0, 1]^2$ with $T = \Z^2$.
The squares $[a, a + 1] \times [b, b + 1]$ cover the plane, two distinct ones have disjoint interiors, and two of them meet in a common edge, a common vertex, or not at all.
Non-example: an equilateral triangle.
All its translates point the same way, and an upward triangle cannot be surrounded by upward triangles without gaps, so clause 1 or clause 2 fails for every $T$.
The Venkov–McMullen theorem characterizes the convex polytopes that tile $\R^d$ by translations.
They are those that are centrally symmetric, with centrally symmetric facets, and whose belts of facets parallel to a $(d-2)$-face have $4$ or $6$ facets.
Such a polytope then admits a face-to-face tiling in which $T$ is a lattice.
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<document source="https://tamnd.github.io/math2/node/58MWCEYTZD.md" tag="58MWCEYTZD" kind="definition">
# Expansive homeomorphism

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \colon X \to X$ be a [[homeomorphism|0QT00PB5G2]].
The map $f$ is *expansive* if there is a constant $c > 0$ such that for all $x, y \in X$ with $x \neq y$ there is an integer $n \in \Z$ with $d(f^n(x), f^n(y)) > c$.
Such a $c$ is an *expansivity constant* of $f$.

## Remarks

On a compact metric space, expansiveness does not depend on the choice of a metric that gives the topology, although the constant does.
Example: the two-sided full [[shift|A1XE59ZHNM]] $\sigma$ on $\set{0, 1}^{\Z}$ with $d(x, y) = 2^{-\min\set{\abs{i} : x_i \neq y_i}}$ and $c = 1/2$.
If $x \neq y$, pick $i$ with $x_i \neq y_i$; then $\sigma^i(x)$ and $\sigma^i(y)$ differ at coordinate $0$, so their distance is $1 > 1/2$.
Non-example: an [[isometry|02HPP5VCZH]] such as the rotation $x \mapsto x + \alpha \bmod 1$ of the circle.
For any $c > 0$, two distinct points at distance less than $c$ stay at that distance for all $n \in \Z$.
An expansive homeomorphism of a compact metric space has finite [[topological entropy|BDZYBXBFDE]].

## Explanations

### L1 human

A homeomorphism is expansive when any two distinct points are pushed at least a fixed distance apart at some time, past or future.
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<document source="https://tamnd.github.io/math2/node/58NHGZDSRE.md" tag="58NHGZDSRE" kind="concept">
# Order statistic

## Summary

The $k$-th order statistic of a sample is its $k$-th smallest value.
The minimum, the maximum, the median and the sample quantiles are order statistics or simple functions of them.
They are the basis of many nonparametric and robust methods, and their distributions have explicit formulas for i.i.d. samples.

## Explanations

### L1 human

Write the numbers of a sample in increasing order.
The first number in the sorted list is the first order statistic, the second is the second, and so on.
For the sample $5, 1, 3$, the sorted list is $1, 3, 5$, so the order statistics are $1$, $3$ and $5$.
The middle one, $3$, is the sample median.
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<document source="https://tamnd.github.io/math2/node/58P0EGAFVV.md" tag="58P0EGAFVV" kind="statement">
# Artin–Schreier theorem

## Statement

Let $F$ be a [[field|0QA8CHH8MG]] and let $C$ be an [[algebraic closure|0MCXX29HWY]] of $F$.
Suppose that the [[degree|0GS74FGFEK]] $[C : F]$ is finite and greater than $1$.
Then $[C : F] = 2$, the field $F$ has [[characteristic|1ZB13N78XW]] $0$, $F$ is [[real closed|778KFSM09P]], and $C = F(i)$ with $i^2 = -1$.

## Remarks

The proof uses Galois theory on the finite extension $C/F$.
One first shows that $C/F$ is separable, so that $C/F$ is a finite Galois extension.
In characteristic $p$ such an extension of degree greater than $1$ cannot exist, by an argument with the roots of $x^p - x - a$.
In characteristic $0$, one shows that $i \notin F$, that $C = F(i)$, and that $-1$ is not a sum of squares in $F$, by studying cyclic subextensions of prime degree.
Example: $F = \R$ and $C = \C$, where $[\C : \R] = 2$, as the [[fundamental theorem of algebra|0JP0W07Y9A]] says.
Without the finiteness hypothesis nothing of this kind holds: for $F = \Q$, the degree $[\overline{\Q} : \Q]$ is infinite.

## Explanations

### L1 human

The theorem says that the pair $\R \subset \C$ is the only kind of example where a field is a finite step away from being algebraically closed.
If finitely many new elements make a field algebraically closed, then one square root of $-1$ is enough, and the field behaves like the real numbers.

### L2 human

In terms of Galois groups: a field whose absolute Galois group is finite and nontrivial has absolute Galois group of order $2$, and it is real closed.
So the only nontrivial finite subgroups of absolute Galois groups have order $2$, and they come from orderings of the field.
The theorem is part of the theory of real closed fields developed by Artin and Schreier.
That theory also shows that a field can be ordered if and only if it is [[formally real|X32TSWJSYT]].
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# Almost periodic sequence

## Definition

Let $(x_n)_{n \in \Z}$ be a two-sided [[sequence|0G9EKM0E15]] of complex numbers.
For $\varepsilon > 0$, an integer $p$ is an *$\varepsilon$-almost period* of the sequence if
$$\abs{x_{n+p} - x_n} < \varepsilon \quad \text{for all } n \in \Z.$$
The sequence is *almost periodic* if the following holds for every $\varepsilon > 0$.
There is an integer $L \ge 1$ such that every set of $L$ consecutive integers contains an $\varepsilon$-almost period.

## Remarks

Example: a [[periodic sequence|Q3B6K9HYRG]] with period $p$ is almost periodic.
Every multiple of $p$ is an $\varepsilon$-almost period for every $\varepsilon > 0$, and every set of $p$ consecutive integers contains a multiple of $p$, so $L = p$ works.

Non-example: let $x_n = 0$ for $n < 0$ and $x_n = 1$ for $n \ge 0$.
For $p > 0$ take $n = -1$, and for $p < 0$ take $n = 0$: in both cases $\abs{x_{n+p} - x_n} = 1$.
So for $\varepsilon = 1/2$ the only almost period is $0$, and no $L$ works.

Equivalent forms: a bounded sequence is almost periodic if and only if the set of its translates $(x_{n+p})_{n \in \Z}$, $p \in \Z$, is relatively compact for the supremum norm.
It is also equivalent that the sequence is a uniform limit on $\Z$ of trigonometric polynomials $\sum_{j=1}^{m} c_j e^{i \lambda_j n}$ with $c_j \in \C$ and $\lambda_j \in \R$.
An almost periodic sequence is bounded.
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<document source="https://tamnd.github.io/math2/node/58WSWYBCWA.md" tag="58WSWYBCWA" kind="example">
# Complex projective plane

## Example

The *complex projective plane* $\mathbb{CP}^2$ is the set of 1-dimensional complex linear subspaces of $\C^3$, the [[projective space|ZT9A6JK8QF]] of dimension $2$ over $\C$.
A point is written in [[homogeneous coordinates|FS9FRJ17D2]] $[z_0 : z_1 : z_2]$ with $(z_0, z_1, z_2) \ne 0$, up to multiplication by a nonzero complex number.
A line is the set $\set{[z_0 : z_1 : z_2] : a_0 z_0 + a_1 z_1 + a_2 z_2 = 0}$ for some $(a_0, a_1, a_2) \in \C^3 \setminus \set{0}$.

Check that it is a [[projective plane|DANFNN8C4Y]]: the arguments for the real case work over any field.
Two distinct points span a unique 2-dimensional subspace of $\C^3$, so they lie on a unique line.
Two distinct 2-dimensional subspaces of $\C^3$ meet in a 1-dimensional subspace, so two distinct lines meet in a unique point.
The points $[1 : 0 : 0]$, $[0 : 1 : 0]$, $[0 : 0 : 1]$, $[1 : 1 : 1]$ have no three on a line.

## Remarks

Each line is a copy of $\mathbb{CP}^1$, which is the [[Riemann sphere|0J78KBPFZB]].
So, as a real manifold, $\mathbb{CP}^2$ is a compact simply connected manifold of real dimension $4$ that contains many 2-spheres meeting in single points.
It is a compact complex manifold of complex dimension $2$, and with the Fubini–Study metric it is a Kähler manifold.
Its integral cohomology is $\Z$ in degrees $0$, $2$, $4$ and zero otherwise, so its [[Euler characteristic|RNDDWDAFJ5]] is $3$.
Plane algebraic curves over $\C$ live naturally in $\mathbb{CP}^2$, and there Bézout's theorem holds without exceptions.
Two curves of degrees $d$ and $e$ with no common component meet in exactly $de$ points, counted with multiplicity.

## Explanations

### L1 human

The complex projective plane is built like the real one, but with complex numbers as coordinates.
Each point is a line through the origin in $\C^3$, and parallel lines of $\C^2$ are made to meet at infinity.
Every line in it is a sphere, so the plane is a four-dimensional space woven out of spheres, any two of which cross at exactly one point.
It is the natural home of plane algebraic curves: there, a line and a circle always meet in exactly two points, counted correctly.
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# Woodin cardinal

## Definition

Let $\kappa$ be a [[cardinal|0EBACQFAZW]].
For an ordinal $\gamma$, let $V_\gamma$ be the set of sets of rank less than $\gamma$ in the [[von Neumann universe|3SZADQXVZJ]].
The cardinal $\kappa$ is a *Woodin cardinal* if for every function $f \colon \kappa \to \kappa$ there is an ordinal $\alpha < \kappa$ such that:

1. $f(\beta) < \alpha$ for every $\beta < \alpha$.
2. There is an [[elementary embedding|CGH8C8EA9N]] $j \colon V \to M$, with $M$ a transitive class, whose critical point is $\alpha$.
3. $V_{j(f)(\alpha)} \subseteq M$.

## Remarks

If $\kappa$ is a Woodin cardinal, then $\kappa$ is Mahlo, and there are $\kappa$ measurable cardinals below $\kappa$.
Every supercompact cardinal is a Woodin cardinal.
A non-example: a measurable cardinal $\kappa$ that is the least measurable cardinal is not Woodin, since a Woodin cardinal has measurable cardinals below it.
Martin and Steel proved that if there are $n$ Woodin cardinals with a measurable cardinal above them all, then every $\Pi^1_{n+1}$ set of reals is determined.
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# Potential operator of a Markov process

## Summary

The potential operator of a Markov process integrates the transition semigroup over all times.
Applied to the indicator of a set, it gives the expected total time that the process spends in that set.
It is the probabilistic analogue of the Newtonian potential, and formally it inverts minus the generator of the process.
Its finiteness distinguishes transient processes from recurrent ones.

## Explanations

### L1 human

Start a random particle at a point and ask how long, in total over its whole life, it will spend inside a given region.
The potential operator answers this question, on average, for every starting point and every region.
For Brownian motion in three-dimensional space the answer is finite for bounded regions, because the particle drifts away for good.
In the plane the particle keeps coming back, and the expected time in any disc is infinite.
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<document source="https://tamnd.github.io/math2/node/59CF2D7BRB.md" tag="59CF2D7BRB" kind="definition">
# Continuous kernel

## Definition

Let $a < b$.
A *continuous kernel* on $[a, b]$ is a [[kernel|GSMBXT7FPK]] $K \colon [a, b] \times [a, b] \to \C$ that is a [[continuous function|0X3WGX95FE]] of the pair $(x, s)$ on the square $[a, b] \times [a, b]$.

## Remarks

Example: $K(x, s) = \min(x, s)$ on $[0, 1]$.
On the closed triangle $s \le x$ it equals $s$, and on the closed triangle $x \le s$ it equals $x$.
Both formulas are continuous and agree on the diagonal, so $K$ is continuous on the square.

Non-example: $K(x, s) = \abs{x - s}^{-1/2}$ for $x \ne s$, with any value on the diagonal, is not continuous on $[0, 1]^2$, since it is unbounded near the diagonal.
It is a weakly singular kernel.

A continuous kernel is bounded, $\abs{K(x, s)} \le M$, and uniformly continuous on the compact square.
So $x \mapsto \int_a^b K(x, s) \varphi(s) \, ds$ is continuous for every continuous $\varphi$, and it is bounded by $M (b - a) \max \abs{\varphi}$.
The same definition is used with $[a, b]$ replaced by a compact subset of $\R^m$.
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<document source="https://tamnd.github.io/math2/node/59GFP54D78.md" tag="59GFP54D78" kind="definition">
# Dynamical system

## Definition

Let $\mathbb{T}$ be an additive [[monoid|0N6R902FG1]], the *time*, usually $\N$, $\Z$, $[0, \infty)$ or $\R$.
Let $X$ be a [[set|0BSJ7RQ9BW]], the *phase space* or *state space*.
A *dynamical system* with time $\mathbb{T}$ on $X$ is a [[map|0EPAT51DXQ]] $\Phi \colon \mathbb{T} \times X \to X$ such that:

1. $\Phi(0, x) = x$ for every $x \in X$;
2. $\Phi(s, \Phi(t, x)) = \Phi(s + t, x)$ for all $s, t \in \mathbb{T}$ and $x \in X$.

The map $\Phi^t = \Phi(t, \cdot) \colon X \to X$ is the *time-$t$ map*, and the set $\set{\Phi(t, x) : t \in \mathbb{T}}$ is the *orbit* of $x$.

## Remarks

Clauses 1 and 2 say that $t \mapsto \Phi^t$ is a monoid homomorphism from $\mathbb{T}$ to the maps $X \to X$ under composition, that is, an action of $\mathbb{T}$ on $X$.
In practice $X$ carries more structure, and $\Phi$ is required to respect it.
Examples are continuous maps on a topological space, smooth maps on a manifold, or measurable maps on a measure space.
With $\mathbb{T} = \R$ the system is a flow, with $\mathbb{T} = [0, \infty)$ a semiflow, and with $\mathbb{T} = \N$ or $\Z$ a discrete-time system.

Example: $\mathbb{T} = \R$, $X = \R$ and $\Phi(t, x) = e^{-t} x$, the solutions of $x' = -x$.
Clause 1 holds since $e^{0} x = x$, and clause 2 holds since $e^{-s}(e^{-t} x) = e^{-(s + t)} x$.

Non-example: $\Phi(t, x) = x + t^2$ on $\R$ with $\mathbb{T} = \R$.
Clause 1 holds, but $\Phi(1, \Phi(1, 0)) = 2$ while $\Phi(2, 0) = 4$, so clause 2 fails.
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<document source="https://tamnd.github.io/math2/node/59PG49KH6V.md" tag="59PG49KH6V" kind="definition">
# Esakia space

## Definition

Let $(X, \tau, \le)$ be a [[Priestley space|ZGBV8V3823]].
So $(X, \tau)$ is a compact topological space and $\le$ is a [[partial order|0QV2JS9RJ3]] on $X$.
Also, for all $x, y \in X$ with $x \not\le y$, there is a [[clopen|DQ4E55T3BW]] [[up-set|ETCE287E2R]] $U$ with $x \in U$ and $y \notin U$.
For $S \subseteq X$, write ${\downarrow} S = \set{x \in X : x \le s \text{ for some } s \in S}$.
The space $(X, \tau, \le)$ is an *Esakia space* if ${\downarrow} C$ is clopen for every clopen set $C \subseteq X$.

## Remarks

In a Priestley space, ${\downarrow} C$ is always closed when $C$ is closed; the extra condition is that it is also open when $C$ is clopen.
For clopen up-sets $U$ and $V$ of an Esakia space, the set $X \setminus {\downarrow}(U \setminus V)$ is a clopen up-set, and it is the Heyting implication $U \to V$.

Example: a finite poset $(P, \le)$ with the discrete topology.
Every subset is clopen, so the separation condition holds with $U = {\uparrow} x$, and every down-set is clopen.

Non-example: let $X = \set{a_1, a_2, \ldots} \cup \set{\infty, b}$, where $a_n \to \infty$ as in the one-point compactification of $\N$, and $b$ is an isolated point.
Order $X$ by $\infty \le b$ and no other strict relations.
This is a Priestley space, but $\set{b}$ is clopen and ${\downarrow}\set{b} = \set{\infty, b}$ is not open, since every neighbourhood of $\infty$ contains some $a_n$.
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# Kac's theorem

## Statement

Let $k$ be an algebraically closed field and let $Q$ be a finite [[quiver|RE9YFFS3TS]] without loops.
Let $B_{ij}$ be the number of edges between $i$ and $j$ in the underlying graph of $Q$, and let $C = 2I - B$.
Let $\Delta_+$ be the set of positive roots of the Kac–Moody algebra with generalized Cartan matrix $C$.

1. The dimension vectors of the finite-dimensional [[indecomposable|E2P29GCD9T]] [[representations|TXVWFWP6VQ]] of $Q$ over $k$ are exactly the elements of $\Delta_+$.
In particular, this set does not depend on the orientation of $Q$.
2. If $\alpha \in \Delta_+$ is a real root, there is exactly one indecomposable representation of dimension vector $\alpha$, up to isomorphism.
3. If $\alpha \in \Delta_+$ is an imaginary root, there are infinitely many pairwise non-isomorphic indecomposable representations of dimension vector $\alpha$.

## Remarks

Victor Kac proved the theorem by counting representations over finite fields and passing to algebraically closed fields.
Over a finite field with $q$ elements, count the isomorphism classes of absolutely indecomposable representations of dimension vector $\alpha$.
This number is a polynomial in $q$ that does not depend on the orientation.
Example: for the Kronecker quiver, the root system is of affine type $\tilde{A}_1$, and $(1, 1)$ is an imaginary root.
The representations $k \rightrightarrows k$ given by the pairs of scalars $(1, \lambda)$ are indecomposable and pairwise non-isomorphic, as part 3 predicts.
For Dynkin quivers all roots are real and finite in number, and the theorem reduces to [[Gabriel's theorem|FC78F2E87E]].

## Explanations

### L1 human

Every quiver, not only the Dynkin ones, has a root system attached to it, built from its graph.
Kac's theorem says that the possible sizes of the indecomposable representations are exactly the positive roots.
Real roots give a single representation each, and imaginary roots give infinite families.
So the root system controls the shape of the classification, even when a full list is out of reach.

### L2 human

The theorem generalizes [[Gabriel's theorem|FC78F2E87E]] from Dynkin quivers to all quivers without loops.
Real roots are the images of simple roots under the Weyl group, and they satisfy $q(\alpha) = 1$ for the Tits form $q$.
The counting polynomials in the remarks are the Kac polynomials, studied in later work on quiver varieties.
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# Littlewood polynomial

## Summary

A Littlewood polynomial is a polynomial all of whose coefficients are $+1$ or $-1$.
On the unit circle such a polynomial of degree $n$ has mean square $n + 1$, so its typical size there is $\sqrt{n+1}$.
Littlewood asked how close to constant its absolute value on the circle can be.
This question about flat polynomials links harmonic analysis, combinatorics and signal design.
The [[Rudin–Shapiro polynomials|PSAX8W3ZYV]] are a classical family that is flat from above.

## Explanations

### L1 human

Write a polynomial using only the coefficients $+1$ and $-1$, for example $1 + z - z^2$.
On the unit circle $\abs{z} = 1$, the sum of squares of the coefficients controls the average of $\abs{p(z)}^2$.
The question is whether the signs can be chosen so that $\abs{p(z)}$ is nearly the same everywhere on the circle.
Such flat sign sequences are useful in signal processing, where they spread energy evenly over all frequencies.
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# Scheme

## Summary

A scheme is a [[locally ringed space|B7FPT8JTXR]] that looks, near each point, like an [[affine scheme|B3V3XJ2W8G]].
Schemes were introduced by Alexander Grothendieck as the basic objects of algebraic geometry.
They include algebraic varieties over any field, and also objects such as $\operatorname{Spec} \Z$ that come from number theory.
They also allow nilpotent functions, which record multiplicities and infinitesimal information.

## Explanations

### L1 human

A manifold is built by gluing open pieces of $\R^n$; a scheme is built by gluing affine schemes, the spectra of rings.
Each piece carries its own ring of functions, and the pieces agree where they overlap.
The projective line is the main small example.
It is glued from two copies of the affine line, with coordinates $t$ and $s$, along the rule $s = 1/t$ where both are nonzero.
The result is not an affine scheme, but every point has an affine neighbourhood.
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# Discrete-time dynamical system

## Summary

A discrete-time dynamical system is a [[dynamical system|BBZP1R872N]] in which time advances in whole steps.
It is given by a single map $f$ of a space to itself, and the state after $n$ steps is the [[iterate|AYVTR2ES5B]] $f^n(x)$.
Such systems arise from sampling a continuous motion at regular times, from population models with generations, and from numerical methods.
Many basic phenomena of dynamics, such as chaos and period doubling, are already visible in maps of an interval.

## Explanations

### L1 human

Imagine pressing a button that changes the state of a system by a fixed rule, and pressing it again and again.
The sequence of states $x, f(x), f(f(x)), \dots$ is the orbit of the starting state.
For example, take the rule $f(x) = x^2$ on the real numbers and start at $x = 1/2$.
The orbit is $1/2, 1/4, 1/16, 1/256, \dots$, and it rushes to $0$.
Starting at $x = 2$ instead gives $2, 4, 16, 256, \dots$, which runs off to infinity.
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# Extension and contraction of ideals

## Summary

A [[ring homomorphism|0GGE0Z0JQ1]] $f \colon A \to B$ of commutative rings moves ideals in both directions.
An ideal of $A$ extends to the ideal of $B$ generated by its image, and an ideal of $B$ contracts to its preimage in $A$.
Contraction keeps [[prime ideals|06HC6HK0Z8]] prime, while extension in general does not.
These two operations describe how ideals behave under localization, under integral extensions and in number fields, where they describe how primes split.

## Explanations

### L1 human

Think of the inclusion $\Z \subseteq \Z[i]$ of the integers in the [[Gaussian integers|0B268GVX95]].
The ideal $5\Z$ extends to the ideal $5\Z[i]$, and since $5 = (2 + i)(2 - i)$ this ideal is no longer prime.
Going the other way, the prime ideal $(2 + i)$ of $\Z[i]$ contracts to $(2+i) \cap \Z = 5\Z$.
Extension pushes ideals up to the bigger ring, and contraction pulls them back down.
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# Killing vector field

## Summary

A Killing vector field on a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] is a vector field along which the metric does not change.
That is, the [[Lie derivative|PJWK0PKN6Q]] of the metric along it is zero.
Its local flow consists of [[isometries|R9YWQS15AA]], so Killing fields are the infinitesimal symmetries of the metric.
They form a Lie algebra, which is the Lie algebra of the isometry group when the manifold is complete.
In physics they give conserved quantities along geodesics.

## Explanations

### L1 human

Imagine sliding every point of a space a little along a vector field.
If the slide never stretches or shrinks any distance, the vector field is a Killing field.
On the round sphere, the field that turns everything around the north–south axis is a Killing field, since a rotation keeps distances.
In the plane, the constant field $(1, 0)$ is a Killing field, because translating the plane keeps distances, while the field $(x, y)$ is not, because its flow is a scaling.
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# Adic space

## Definition

A *Huber ring* is a [[topological ring|TSF8RZDD5B]] $A$ with an open subring $A_0$ and a finitely generated ideal $I$ of $A_0$.
The topology of $A_0$ must be the [[$I$-adic topology|2VF2TSJ3PY]].
An element $a$ of $A$ is *power-bounded* if the set $\set{a^n : n \ge 1}$ is bounded, and $A^{\circ}$ denotes the subring of power-bounded elements.
A *Huber pair* $(A, A^+)$ is a Huber ring $A$ with an open subring $A^+ \subseteq A^{\circ}$ that is [[integrally closed|G0V86SJBQW]] in $A$.

A *valuation* on $A$ is a map $v \colon A \to \Gamma \cup \set{0}$, where $\Gamma$ is a totally ordered abelian group written multiplicatively, with $v(0) = 0$, $v(1) = 1$, $v(ab) = v(a) v(b)$ and $v(a + b) \le \max(v(a), v(b))$.
It is *continuous* if $\set{a \in A : v(a) < \gamma}$ is open for every $\gamma$ in the value group of $v$.
The *adic spectrum* $\operatorname{Spa}(A, A^+)$ is the set of equivalence classes of continuous [[valuations|B85T5VKR3M]] $v$ on $A$ with $v(f) \le 1$ for all $f \in A^+$.
Its topology is generated by the rational subsets
$$U\!\left(\tfrac{f_1, \dots, f_n}{g}\right) = \set{v : v(f_i) \le v(g) \ne 0 \text{ for } i = 1, \dots, n},$$
where $f_1, \dots, f_n$ generate an open ideal of $A$.
It carries a structure presheaf $\mathcal{O}$ of complete topological rings, built from completions of the localizations $A[1/g]$.

An *adic space* is a triple $(X, \mathcal{O}_X, (v_x)_{x \in X})$ such that:

1. $X$ is a topological space and $\mathcal{O}_X$ is a [[sheaf|TB6TM6R4MD]] of complete topological rings on $X$;
2. for each $x \in X$, $v_x$ is an equivalence class of valuations on the stalk $\mathcal{O}_{X,x}$;
3. every point has an open neighbourhood isomorphic, with these structures, to $\operatorname{Spa}(A, A^+)$ for a Huber pair whose structure presheaf is a sheaf.

## Remarks

Example: $\operatorname{Spa}(\Q_p, \Z_p)$ is a single point.
The $p$-adic valuation is continuous and bounded by $1$ on $\Z_p$.
The trivial valuation, with $v(a) = 1$ for $a \ne 0$, is not continuous, since $\set{a : v(a) < 1} = \set{0}$ is not open in $\Q_p$.
Other valuations of $\Q_p$ bounded on $\Z_p$ are equivalent to one of these two.

Non-example of a Huber ring: $\R$ with its usual topology.
An open subring contains an interval $(-\varepsilon, \varepsilon)$, which generates $\R$ as an additive group, so the only open subring is $\R$.
Its topology is neither discrete nor indiscrete, so it is not the $I$-adic topology for any ideal $I$ of $\R$.

The structure presheaf of $\operatorname{Spa}(A, A^+)$ need not be a sheaf in general; this is why clause 3 asks for it.
Huber proved that it is a sheaf when, for example, $A$ is a strongly Noetherian Tate ring, which covers the affinoid algebras of rigid geometry.
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# Microlocal analysis

## Summary

Microlocal analysis studies functions, distributions and linear operators on $\R^n$ or on a manifold by localizing both in position $x$ and in frequency direction $\xi$.
Its natural home is phase space, the cotangent bundle, whose points are pairs $(x, \xi)$.
Its main tools are the [[wave front set|3T5Q9WVQX8]], [[pseudo-differential operators|FF27N0BWTP]] and [[Fourier integral operators|EVS4JPEE45]].
It explains where and in which directions solutions of linear [[partial differential equations|GC5BDZ6QBC]] are singular, and how these singularities move.

## Explanations

### L1 human

A function can be smooth in some places and rough in others.
Microlocal analysis also records the direction in which it is rough.
Picture a function on the plane that jumps across the line $x_1 = 0$: it is singular only on that line, and only in the directions $\pm(1, 0)$ normal to it.
The [[Fourier transform|DPVM34ADZV]] of the function, cut off near a point of the line, decays fast in every direction except those two.
This finer picture shows, for example, that singularities of waves travel along rays, as in [[propagation of singularities|P2QJ7NPW3Y]].
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# First variation

## Summary

The first variation of a [[functional|1EQM0DJ4NC]] $J$ at $y$ in the direction $\eta$ is the rate of change of $J(y + t\eta)$ at $t = 0$.
It is the analogue for functionals of the [[directional derivative|06KWGHDN2J]].
At a minimiser it vanishes in every admissible direction, and for [[integral functionals|JRQZ803Q5Y]] this condition leads to the [[Euler–Lagrange equation|PS0VV149ET]].

## Explanations

### L1 human

Push a function $y$ a little in the direction of another function $\eta$, and watch how the value of the functional changes.
The first variation is the speed of that change at the start.
At the bottom of a valley every direction is flat to first order, so at a minimiser the first variation is zero.
For $J(y) = \int_0^1 y^2 \, dx$, the first variation at $y$ in the direction $\eta$ is $2 \int_0^1 y \eta \, dx$.
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# Hyperfunction

## Definition

Let $U \subseteq \R$ be an open set.
Let $V \subseteq \C$ be an open set with $V \cap \R = U$; then $U$ is a closed subset of $V$.
For an open set $W \subseteq \C$, let $\mathcal{O}(W)$ be the space of [[holomorphic functions|0NAHXR1FRH]] on $W$.
The space of *hyperfunctions* on $U$ is the quotient vector space
$$\mathcal{B}(U) = \mathcal{O}(V \setminus U) / \mathcal{O}(V),$$
where $\mathcal{O}(V)$ is embedded in $\mathcal{O}(V \setminus U)$ by restriction.
The class of $F \in \mathcal{O}(V \setminus U)$ is written $[F]$.

## Remarks

Up to canonical isomorphism, $\mathcal{B}(U)$ does not depend on the choice of $V$.
Write $F_+$ and $F_-$ for the restrictions of $F$ to the parts of $V$ above and below the real axis.
One thinks of $[F]$ as $F_+(x + i0) - F_-(x - i0)$.
When both boundary values exist as distributions, this difference is a distribution, and every distribution on $U$ arises this way.

Example: on $U = \R$ with $V = \C$, let $F(z) = -1/(2\pi i z)$ on $\C \setminus \R$.
The boundary values give $F_+(x + i0) - F_-(x - i0) = -\frac{1}{2\pi i}\left(\frac{1}{x + i0} - \frac{1}{x - i0}\right)$, and this is the [[Dirac delta|0KPHENZ5S7]].
So $\delta = [F]$.
A function holomorphic on all of $V$ gives the zero hyperfunction.

Non-example: for $U = (0,1)$, the set $V = \C$ does not qualify, because $V \cap \R = \R \neq U$.
With that choice, $\mathcal{O}(\C \setminus (0,1)) / \mathcal{O}(\C)$ would also see singularities at the end points $0$ and $1$, which lie outside $U$.

In several variables the definition uses relative sheaf cohomology of the sheaf of holomorphic functions instead of a quotient.
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# Perron's irreducibility criterion

## Statement

Let $n \ge 1$ and let $f = x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ be a [[monic polynomial|0BDX4MBKT4]] with integer coefficients and $a_0 \ne 0$.
If
$$\abs{a_{n-1}} > 1 + \abs{a_{n-2}} + \dots + \abs{a_1} + \abs{a_0},$$
then $f$ is [[irreducible|00X2A9G9W3]] in $\Z[x]$ and in $\Q[x]$.

## Remarks

Proof idea: by [[Rouché's theorem|0D430QR02W]], comparing $f$ with $a_{n-1} x^{n-1}$ on the unit circle, $f$ has exactly $n - 1$ roots in the open unit disc and one root outside it.
If $f = g h$ with monic $g, h \in \Z[x]$ of positive degree, one factor has all its roots in the open unit disc.
Its constant term is then a nonzero integer of absolute value less than $1$, which is impossible.
Irreducibility over $\Q$ follows by [[Gauss's lemma|0QFV5Y1Z5X]].
Example: $x^3 + 5x^2 + 2x + 1$ is irreducible, since $5 > 1 + 2 + 1$.
The hypothesis $a_0 \ne 0$ is necessary: $x^2 + 3x = x(x + 3)$ satisfies the inequality $3 > 1 + 0$.
A second form of the criterion allows equality in the inequality, provided that $f(1) \ne 0$ and $f(-1) \ne 0$.

## Explanations

### L1 human

If the second coefficient of a monic integer polynomial is larger than all the other coefficients together, plus one, the polynomial cannot be factored.
The check needs only additions of absolute values.

### L2 human

The criterion is a typical analytic proof of irreducibility.
One locates the complex roots, then uses that the constant term of an integer factor is a nonzero integer.
It complements algebraic tests such as [[Eisenstein's criterion|02BPKD7TMD]] and [[reduction modulo a prime|0KWWEXW0RN]].
For example, it shows at once that $x^n + 3x^{n-1} + 1$ is irreducible for every $n \ge 2$.
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# Proper group action

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] and let $X$ be a topological space.
Let $G$ act continuously on $X$, that is, let a [[group action|0WCY42EV9V]] $G \times X \to X$, $(g, x) \mapsto g \cdot x$, be continuous.
The action is *proper* if the map
$$\theta \colon G \times X \to X \times X, \qquad (g, x) \mapsto (g \cdot x, x)$$
is a [[proper map|15XEBGNGHC]], that is, continuous, closed, and with compact fibres.

## Remarks

When $G$ and $X$ are Hausdorff and [[locally compact|0FE2SB8WVJ]], $\theta$ is proper if and only if the preimage of every [[compact|0TVJZYB6SH]] set is compact.
Then the action is proper if and only if, for every compact $K \subseteq X$, the set $\set{g \in G : (g \cdot K) \cap K \ne \emptyset}$ is compact.

Example: let $G = \Z$ act on $X = \R$ by $n \cdot x = x + n$.
For compact $K \subseteq [-R, R]$, the condition $(n + K) \cap K \ne \emptyset$ forces $\abs{n} \le 2R$, so the set of such $n$ is finite, hence compact in $\Z$.
So the action is proper.

Non-example: let $G = (0, \infty)$ under multiplication act on $X = \R$ by $t \cdot x = tx$.
For $K = [-1, 1]$, the point $0$ lies in $t \cdot K \cap K$ for every $t$, so the set of such $t$ is all of $(0, \infty)$, which is not compact.
So the action is not proper.

Every continuous action of a compact Hausdorff group on a Hausdorff space is proper.
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# Opposite category

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
The *opposite category* $\mathcal{C}^{\mathrm{op}}$ is the category defined as follows:

1. its objects are the objects of $\mathcal{C}$;
2. for objects $A$, $B$, its [[morphisms|859F0TFVA3]] are $\Hom_{\mathcal{C}^{\mathrm{op}}}(A, B) = \Hom_{\mathcal{C}}(B, A)$; a morphism $f \colon B \to A$ of $\mathcal{C}$ is written $f^{\mathrm{op}} \colon A \to B$ when seen in $\mathcal{C}^{\mathrm{op}}$;
3. the composite is $g^{\mathrm{op}} \circ f^{\mathrm{op}} = (f \circ g)^{\mathrm{op}}$, and the identity of $A$ is $1_A^{\mathrm{op}}$.

## Remarks

The category axioms for $\mathcal{C}^{\mathrm{op}}$ follow from those of $\mathcal{C}$ read backwards, and $(\mathcal{C}^{\mathrm{op}})^{\mathrm{op}} = \mathcal{C}$.
Example: let $(P, \le)$ be a [[preorder|0VABYWJWH9]], seen as a category with one morphism $a \to b$ when $a \le b$.
Its opposite has one morphism $a \to b$ when $b \le a$, so it is the preorder $(P, \ge)$.
Non-example: the rule "keep the arrows as they are and only reverse the order of composition" does not give a category in general.
For $f \colon A \to B$ and $g \colon B \to C$, the expression $f \circ g$ has no meaning in $\mathcal{C}$ unless $C = A$.
A morphism $f$ is a monomorphism in $\mathcal{C}$ exactly when $f^{\mathrm{op}}$ is an epimorphism in $\mathcal{C}^{\mathrm{op}}$.
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# Parseval formula for the Mellin transform

## Statement

Let $\sigma \in \R$ and let $f, g \colon (0, \infty) \to \C$ be measurable.
Write $\mathcal{M}$ for the [[Mellin transform|N5PVGRJE1T]].

1. Assume that $x^{\sigma - 1} f(x)$ is [[integrable|X7GTF3D98F]] on $(0, \infty)$ and that $\int_0^\infty x^{2\sigma - 1} \abs{f(x)}^2 \, dx < \infty$. Then
$$\int_0^\infty x^{2\sigma - 1} \abs{f(x)}^2 \, dx = \frac{1}{2\pi} \int_{-\infty}^{\infty} \abs{\mathcal{M}f(\sigma + i\tau)}^2 \, d\tau.$$
2. Assume moreover that $x^{-\sigma} g(x)$ is integrable on $(0, \infty)$ and that $\int_0^\infty x^{1 - 2\sigma} \abs{g(x)}^2 \, dx < \infty$. Then
$$\int_0^\infty f(x) g(x) \, dx = \frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{M}f(\sigma + i\tau) \, \mathcal{M}g(1 - \sigma - i\tau) \, d\tau.$$

## Remarks

The integrability hypotheses say that $f$ is [[Mellin convergent|SGZZ4Y0D36]] at $\sigma$ and $g$ at $1 - \sigma$, so the transforms on the right are defined pointwise.
Put $h_f(u) = e^{\sigma u} f(e^{u})$ and $h_g(u) = e^{(1 - \sigma) u} g(e^{u})$.
The substitution $x = e^{u}$ turns the hypotheses into $h_f, h_g \in L^1(\R) \cap L^2(\R)$ and the left sides into $\int \abs{h_f}^2 \, du$ and $\int h_f h_g \, du$.
It also turns $\mathcal{M}f(\sigma + i\tau)$ and $\mathcal{M}g(1 - \sigma - i\tau)$ into Fourier transforms of $h_f$ and $h_g$ at $\mp \tau / 2\pi$.
The [[Plancherel theorem|HPZHRZ0502]] for the Fourier transform on $\R$ then gives both formulas.
Example: $f(x) = e^{-x}$ and $\sigma = 1/2$ give $\int_0^\infty e^{-2x} \, dx = \frac{1}{2} = \frac{1}{2\pi} \int_{\R} \abs{\Gamma(\tfrac12 + i\tau)}^2 \, d\tau$; since $\abs{\Gamma(\tfrac12 + i\tau)}^2 = \pi / \cosh(\pi \tau)$, the right side is $\frac12 \int_{\R} \frac{d\tau}{\cosh \pi \tau} = \frac12$.
By density, part 1 extends $\mathcal{M}$ to a unitary map, up to the factor $2\pi$, from $L^2((0, \infty), x^{2\sigma - 1} dx)$ onto $L^2(\R)$.

## Explanations

### L1 human

The total energy of a function, measured with the weight $x^{2\sigma - 1}$, equals the total energy of its Mellin transform along the line $\operatorname{Re} s = \sigma$.
The second formula is the same fact for products of two functions.
It is the Mellin version of the Plancherel and Parseval identities for Fourier series and integrals.

### L2 human

The formula is used to evaluate integrals of products: one knows $\mathcal{M}f$ and $\mathcal{M}g$, and computes $\int_0^\infty f g \, dx$ by a single integral on a vertical line, often by residues.
Part 2 can also be read as the convolution rule for the Mellin convolution of $f$ and $x \mapsto g(1/x)/x$, combined with Mellin inversion at the point $x = 1$.
On the critical line $\sigma = 1/2$ the weight disappears, and part 1 says that $f \mapsto \mathcal{M}f(\tfrac12 + i\tau) / \sqrt{2\pi}$ preserves the norm of $L^2(0, \infty)$.
</document>

<document source="https://tamnd.github.io/math2/node/5BW26GPS2A.md" tag="5BW26GPS2A" kind="concept">
# FBI transform

## Summary

The FBI transform, named after Fourier, Bros and Iagolnitzer, is a phase-space transform.
It looks at a function near a point $x$ at frequency $\xi$ through a Gaussian window of width about $\abs{\xi}^{-1/2}$.
It is a tool of microlocal analysis.
Smoothness and real analyticity of a function near a point, in given directions, can be read off from the decay of its FBI transform.
In particular, it characterizes the analytic [[wave front set|3T5Q9WVQX8]] through exponential decay.
It is a close relative of the [[Bargmann transform|QPH28QH036]] and of the short-time Fourier transform with a Gaussian window.

## Explanations

### L1 human

To see whether a function is smooth near a point, one can zoom in there and look at its high-frequency content.
The FBI transform does this with a Gaussian magnifying glass placed at $x$ and tuned to frequency $\xi$.
The higher the frequency, the narrower the glass.
If the result dies off very fast as the frequency grows, the function is real analytic near $x$ in that direction of frequency.
</document>

<document source="https://tamnd.github.io/math2/node/5BW600RVX5.md" tag="5BW600RVX5" kind="definition">
# Hartogs domain

## Definition

Let $n \ge 2$ and write a point of $\C^n$ as $(z', z_n)$ with $z' \in \C^{n-1}$ and $z_n \in \C$.
A *Hartogs domain* is a [[connected|0SPE8WNWFV]] [[open set|02BH3MCBQB]] $D \subseteq \C^n$ such that $(z', e^{i\theta} z_n) \in D$ for all $(z', z_n) \in D$ and all $\theta \in \R$.
It is a *complete Hartogs domain* if moreover $(z', \lambda z_n) \in D$ for all $(z', z_n) \in D$ and all $\lambda \in \C$ with $\abs{\lambda} \le 1$.

## Remarks

Example: $D = \set{(z_1, z_2) \in \C^2 : \abs{z_1} < 1, \ \abs{z_2} < 1 - \abs{z_1}^2}$.
It is open and connected, and the condition on $z_2$ involves only $\abs{z_2}$.
This number does not change under $z_2 \mapsto e^{i\theta} z_2$, and it does not increase under $z_2 \mapsto \lambda z_2$ with $\abs{\lambda} \le 1$.
So $D$ is a complete Hartogs domain.

Non-example: $D = \set{(z_1, z_2) \in \C^2 : \abs{z_1 - z_2} < 1}$.
The point $(1, 1/2)$ lies in $D$, but its rotation $(1, -1/2)$ does not, since $\abs{1 + 1/2} > 1$.

The [[Hartogs figure|GHJC53RFBT]] is a Hartogs domain that is not complete.
A [[Reinhardt domain|WA0SE56FVH]] is invariant under rotation of each coordinate separately, so it is a Hartogs domain in particular.
</document>

<document source="https://tamnd.github.io/math2/node/5BXMSPDPRJ.md" tag="5BXMSPDPRJ" kind="definition">
# Elliptic function

## Definition

Let $\omega_1, \omega_2 \in \C$ be linearly independent over $\R$, and let $\Lambda = \set{m\omega_1 + n\omega_2 : m, n \in \Z}$ be the lattice they span.
An *elliptic function* for $\Lambda$ is a [[meromorphic function|0NSD1TPHR0]] $f$ on $\C$ such that
$$f(z + \omega_1) = f(z) \quad \text{and} \quad f(z + \omega_2) = f(z)$$
for all $z \in \C$ that are not poles of $f$.
A function is *elliptic* if it is elliptic with respect to some such lattice.

## Remarks

Then $f(z + \omega) = f(z)$ for every $\omega \in \Lambda$, so $f$ is [[periodic|089KWVYD69]] with each lattice vector as a period.
A *fundamental parallelogram* is a set $\set{a + s\omega_1 + t\omega_2 : 0 \le s, t < 1}$ with $a \in \C$; an elliptic function is determined by its values there.
Example: the [[Weierstrass elliptic function|361NAK68QK]] $\wp$ of $\Lambda$ is elliptic with respect to $\Lambda$, and so is its derivative $\wp'$.
Constant functions are elliptic for every lattice.
Non-example: $e^z$ is meromorphic and periodic, but its periods are exactly the numbers $2\pi i k$ with $k \in \Z$.
They all lie on one line through $0$, so there is no second period independent over $\R$.
Equivalently, the elliptic functions for $\Lambda$ are the meromorphic functions on the compact Riemann surface $\C / \Lambda$.
</document>

<document source="https://tamnd.github.io/math2/node/5C0N2ENRF2.md" tag="5C0N2ENRF2" kind="concept">
# Neutral differential equation

## Summary

A neutral differential equation is a [[delay differential equation|39PCDRDB01]] in which the derivative of the unknown also appears with a delay.
A typical form is $x'(t) = f(t, x(t), x(t - \tau), x'(t - \tau))$.
Such equations do not smooth their initial data over time, unlike retarded delay equations.
They arise in models of transmission lines, coupled oscillators and some population models.

## Explanations

### L1 human

In an ordinary delay equation, the rate of change depends on past states.
In a neutral equation, it also depends on past rates of change.
For $x'(t) = \frac{1}{2} x'(t - 1) - x(t)$, a kink in the history is copied forward forever, only shrinking by the factor $\frac{1}{2}$ each time unit.
So the solution never becomes smoother than its initial history.
</document>

<document source="https://tamnd.github.io/math2/node/5C3411G4C0.md" tag="5C3411G4C0" kind="definition">
# Stochastic Volterra equation

## Definition

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, P)$ be a probability space with a [[filtration|130FCDWRT1]], and let $W$ be a one-dimensional [[Wiener process|22735FGJV6]] with respect to this filtration.
Let $\tau > 0$ and $\Delta = \set{(t, s) : 0 \le s \le t \le \tau}$.
Let $a, b \colon \Delta \times \R \to \R$ be measurable functions, and let $X_0$ be a continuous [[adapted process|YRNFE016FF]] on $[0, \tau]$.
The *stochastic Volterra equation* with coefficients $a$, $b$ and free term $X_0$ is
$$X(t) = X_0(t) + \int_0^t a(t, s, X(s)) \, ds + \int_0^t b(t, s, X(s)) \, dW(s), \qquad 0 \le t \le \tau,$$
where the last integral is an Itô integral, see [[Itô calculus|JA95NH206E]].
A *solution* is a continuous adapted process $X$ on $[0, \tau]$ such that, for every $t \in [0, \tau]$:

1. almost surely $\int_0^t \abs{a(t, s, X(s))} \, ds < \infty$ and $\int_0^t b(t, s, X(s))^2 \, ds < \infty$;
2. the equation holds almost surely.

## Remarks

If $a(t, s, x) = \tilde a(s, x)$ and $b(t, s, x) = \tilde b(s, x)$ do not depend on $t$ and $X_0(t) = \xi$ is constant in time, the equation is the integral form of the [[stochastic differential equation|B219G9SHJA]] $dX = \tilde a(t, X) \, dt + \tilde b(t, X) \, dW$, $X(0) = \xi$.
If $b = 0$, the equation is a deterministic [[Volterra integral equation|DND8CBEVXT]] for each path.
Systems with values in $\R^n$ and several Wiener processes are defined in the same way.

Example: $a = 0$, $b(t, s, x) = x$ and $X_0 = 1$.
The equation is $X(t) = 1 + \int_0^t X(s) \, dW(s)$, and $X(t) = \exp(W(t) - t/2)$ is a solution.
Indeed, Itô's formula applied to $f(t, w) = e^{w - t/2}$ gives $dX = X \, dW + \big(-\tfrac{1}{2} + \tfrac{1}{2}\big) X \, dt = X \, dW$.

Example with memory: $a = 0$, $X_0 = 0$ and $b(t, s, x) = (t - s)^{H - 1/2}$ with $H > 0$.
Clause 1 holds since $\int_0^t (t - s)^{2H - 1} \, ds = t^{2H} / (2H) < \infty$, and the solution is the process $X(t) = \int_0^t (t - s)^{H - 1/2} \, dW(s)$, up to a choice of a continuous version.

Non-example: for $b(t, s, x) = (t - s)^{-1/2}$, the case $H = 0$, the integral $\int_0^t (t - s)^{-1} \, ds$ diverges, so clause 1 fails for every $t > 0$ and the equation has no solution.

## Explanations

### L1 human

It is a Volterra equation in which part of the input arrives through random kicks $dW$, whose effect may depend on the time elapsed since they happened.
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<document source="https://tamnd.github.io/math2/node/5C91SFVTY7.md" tag="5C91SFVTY7" kind="concept">
# Pixley term

## Summary

A Pixley term of a [[variety|AJNZJD8442]] is a ternary term $p$ such that $p(x, y, y) = x$, $p(x, x, y) = y$ and $p(x, y, x) = x$ hold.
By Pixley's theorem, a variety has a Pixley term if and only if it is an [[arithmetical variety|GW78A05TNA]].
A Pixley term is a [[Mal'cev term|Y203PPTS61]] that also yields a [[majority term|9879ZMW5AA]].

## Explanations

### L1 human

A Pixley term is a three-input formula that behaves like the Mal'cev term of groups, and in addition returns $x$ on inputs $x, y, x$.
For Boolean algebras it can be written with "and", "or" and "not".
Its existence makes the congruences of every algebra in the variety both permute and satisfy the distributive law.
</document>

<document source="https://tamnd.github.io/math2/node/5CET1KWWWH.md" tag="5CET1KWWWH" kind="concept">
# Continuous geometry

## Summary

A continuous geometry is a complete, complemented, [[modular lattice|QX7WZ92QAK]] in which meets and joins are continuous along chains, and which is irreducible.
John von Neumann introduced continuous geometries as analogues of projective geometries.
In them the dimension can take every value in $[0, 1]$, not only integer values.
The lattice of projections of a type II$_1$ factor is the motivating example.
Every continuous geometry carries a dimension function, and von Neumann related these lattices to [[von Neumann regular rings|DXX7GNY3HE]].

## Explanations

### L1 human

In a projective geometry, subspaces have dimensions $0, 1, 2, \dots, n$, and after rescaling by $n$ these are the numbers $0, 1/n, \dots, 1$.
Von Neumann asked for a geometry whose subspaces can have any real dimension between $0$ and $1$.
Such a "geometry" has no points or lines, only subspaces, so it is described by the lattice of its subspaces.
The axioms ask that this lattice behaves like a subspace lattice and that limits of increasing or decreasing families behave well.
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<document source="https://tamnd.github.io/math2/node/5CHFRPQ03W.md" tag="5CHFRPQ03W" kind="statement">
# Hardy inequality in Sobolev spaces

## Statement

Let $n \ge 2$ and $1 \le p < n$.
Then every continuously differentiable function $u \colon \R^n \to \R$ with compact support satisfies
$$\int_{\R^n} \frac{\abs{u(x)}^p}{\abs{x}^p} \, dx \le \Big( \frac{p}{n - p} \Big)^p \int_{\R^n} \abs{\nabla u(x)}^p \, dx.$$

## Remarks

For $p = 2$ and $n \ge 3$ there is a short proof.
Let $a = (n - 2)/2$ and expand $0 \le \int \abs{\nabla u + a u \, x / \abs{x}^2}^2 \, dx$.
The cross term is $a \int \nabla (u^2) \cdot x / \abs{x}^2 \, dx = -a (n - 2) \int u^2 / \abs{x}^2 \, dx$ by integration by parts, since $\operatorname{div}(x / \abs{x}^2) = (n - 2) / \abs{x}^2$.
Collecting terms gives $\frac{(n-2)^2}{4} \int u^2 / \abs{x}^2 \, dx \le \int \abs{\nabla u}^2 \, dx$, which is the statement.

The constant $\big( \frac{p}{n - p} \big)^p$ is the best possible, but there is no nonzero function for which equality holds.
By density, the inequality extends to every $u$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\R^n)$.
It is the $n$-dimensional analogue of the one-dimensional [[Hardy's inequality|S7A8YFGG8X]].

The hypothesis $p < n$ is needed.
For $p = n$, take $u$ smooth with compact support and $u(0) \ne 0$: then $\abs{u}^n / \abs{x}^n$ is not integrable near $0$, while the right side is finite.

## Explanations

### L1 human

The weight $1/\abs{x}^p$ is large near the origin, so the left side punishes functions that are large near $0$.
The inequality says that the gradient energy alone pays for this, as long as the dimension is larger than $p$.
In space with $p = 2$, it says that $\int u^2 / \abs{x}^2 \le 4 \int \abs{\nabla u}^2$.

### L2 human

Hardy's inequality controls singular potentials by kinetic energy.
In quantum mechanics it shows that the operator $-\Delta - c / \abs{x}^2$ on $\R^n$ is bounded below when $c \le (n-2)^2 / 4$.
In partial differential equations it is used to treat lower-order terms with singular coefficients and to study equations with critical singular weights.
</document>

<document source="https://tamnd.github.io/math2/node/5CN7KY8MWP.md" tag="5CN7KY8MWP" kind="statement">
# Karhunen–Loève theorem

## Statement

Let $a < b$ and let $(X_t)_{t \in [a, b]}$ be a real [[stochastic process|0QH5ENKZM1]] such that the map $(t, \omega) \mapsto X_t(\omega)$ is jointly measurable, $E[X_t] = 0$ and $E[X_t^2] < \infty$ for every $t$.
Assume that the [[covariance|0KBFAWYRTE]] function $K(s, t) = E[X_s X_t]$ is continuous on $[a, b]^2$.
Let $T_K$ be the [[integral operator|H8P36XFV8K]] on $L^2[a, b]$ given by $(T_K f)(s) = \int_a^b K(s, t) f(t) \, dt$.
Let $(\lambda_k)_{k \ge 1}$ be its non-zero [[eigenvalues|0BQAA2BK3Q]], listed with multiplicity, and let $(e_k)$ be corresponding [[orthonormal|0S9GGTR0EX]] eigenfunctions.
Then:

1. each $\lambda_k$ is positive and each $e_k$ is continuous;
2. the random variables $Z_k = \int_a^b X_t \, e_k(t) \, dt$ satisfy $E[Z_k] = 0$ and $E[Z_j Z_k] = \lambda_k$ if $j = k$ and $0$ otherwise;
3. $X_t = \sum_{k} Z_k e_k(t)$, where the series converges in mean square uniformly in $t$:
$$\lim_{n \to \infty} \sup_{t \in [a, b]} E\left[\left(X_t - \sum_{k=1}^{n} Z_k e_k(t)\right)^2\right] = 0.$$

If $X$ is moreover a Gaussian process, the $Z_k$ are independent normal random variables.

## Remarks

The proof rests on [[Mercer's theorem|PKGADWXPYH]]: $K$ is continuous, symmetric and positive semidefinite, so $K(s, t) = \sum_k \lambda_k e_k(s) e_k(t)$ with uniform convergence on $[a, b]^2$.
Fubini's theorem gives the moments of the $Z_k$ and $E[X_t Z_k] = \lambda_k e_k(t)$.
Expanding the square then gives
$$E\left[\left(X_t - \sum_{k \le n} Z_k e_k(t)\right)^2\right] = K(t, t) - \sum_{k \le n} \lambda_k e_k(t)^2,$$
which tends to $0$ uniformly in $t$ by Mercer's theorem.

Example: for the standard [[Wiener process|22735FGJV6]] on $[0, 1]$, $K(s, t) = \min(s, t)$.
Solving the eigenvalue equation gives $e_k(t) = \sqrt{2} \sin\left((k - \tfrac{1}{2})\pi t\right)$ and $\lambda_k = \dfrac{1}{(k - \frac{1}{2})^2 \pi^2}$ for $k \ge 1$.
So $W_t = \sum_{k \ge 1} \sqrt{2} \, \xi_k \dfrac{\sin\left((k - \frac{1}{2})\pi t\right)}{(k - \frac{1}{2})\pi}$ with i.i.d. standard normal $\xi_k$.

## Explanations

### L1 human

A random curve can be written as a sum of fixed shapes multiplied by random coefficients.
The Karhunen–Loève theorem gives the best such shapes for a given process: the eigenfunctions of its covariance.
With this choice, the random coefficients are uncorrelated, and the first few terms capture as much of the variance as possible.

### L2 human

The expansion is the continuous analogue of principal component analysis, which diagonalises a covariance matrix.
Truncating it after $n$ terms gives the best approximation by $n$ terms for the mean square error integrated over $[a, b]$, among all expansions in an orthonormal basis of $L^2[a, b]$.
It is used to simulate Gaussian processes, to reduce dimension in signal processing and uncertainty quantification, and to compute laws of quadratic functionals such as $\int_0^1 W_t^2 \, dt = \sum_k \lambda_k \xi_k^2$.
</document>

<document source="https://tamnd.github.io/math2/node/5CSCTKRTB2.md" tag="5CSCTKRTB2" kind="definition">
# Regular variation

## Definition

Let $f \colon (0, \infty) \to (0, \infty)$ be a [[measurable function|YYNCJSR0PJ]].
The function $f$ is *regularly varying* (at infinity) if for every real $a > 0$ the limit
$$g(a) = \lim_{x \to \infty} \frac{f(ax)}{f(x)}$$
exists and satisfies $0 < g(a) < \infty$.

## Remarks

For such $f$ there is a real number $\rho$ with $g(a) = a^\rho$ for all $a > 0$; this is Karamata's characterization theorem.
The number $\rho$ is the *index* of $f$.
Equivalently, $f(x) = x^\rho L(x)$ where $L$ is a [[slowly varying function|EHJ7D5CRZS]].
The regularly varying functions of index $0$ are exactly the slowly varying functions.

Example: $f(x) = x^2 \log(e + x)$ is regularly varying of index $2$.
Indeed $f(ax)/f(x) = a^2 \cdot \log(e + ax)/\log(e + x)$, and the second factor tends to $1$.

Non-example: $f(x) = e^x$ is not regularly varying.
For $a = 2$ the ratio $f(2x)/f(x) = e^x$ tends to $\infty$, so the limit is not finite.
</document>

<document source="https://tamnd.github.io/math2/node/5CTHC6612N.md" tag="5CTHC6612N" kind="statement">
# Atiyah–Guillemin–Sternberg convexity theorem

## Statement

Let $(M, \omega)$ be a compact connected [[symplectic manifold|CAN490EPRT]].
Let a torus $T = (S^1)^k$ act on $M$ with a [[moment map|ZARM34FZ6Z]] $\mu \colon M \to \mathfrak{t}^*$, where $\mathfrak{t}^* \cong \R^k$ is the dual of the Lie algebra of $T$.
Let $M^T$ be the set of points fixed by every element of $T$.
Then:

1. $\mu(M^T)$ is a finite set.
2. $\mu(M)$ is the [[convex hull|0FQSDC2EQ2]] of $\mu(M^T)$, so it is a [[convex polytope|300K6CB1KH]].
3. Every fiber $\mu^{-1}(a)$, $a \in \mathfrak{t}^*$, is connected or empty.

## Remarks

The theorem was proved independently by Atiyah and by Guillemin and Sternberg.
The fixed set $M^T$ has finitely many connected components, and $\mu$ is constant on each of them, which gives part 1.
The heart of the proof is that each component $\mu^\xi = \langle \mu, \xi \rangle$ is a Morse–Bott function whose critical submanifolds have even index and coindex.
Such functions have connected level sets, and an induction on $k$ gives convexity.

Example: $T^n$ acts on $\mathbb{CP}^n$ by $(t_1, \ldots, t_n) \cdot [z_0 : z_1 : \cdots : z_n] = [z_0 : t_1 z_1 : \cdots : t_n z_n]$.
A moment map is $\mu([z]) = -\frac{1}{2} \left( \abs{z_1}^2, \ldots, \abs{z_n}^2 \right) / \abs{z}^2$, for a suitable normalization of the Fubini–Study form, with image a simplex whose $n + 1$ vertices are the images of the $n + 1$ coordinate points.

Compactness is needed.
For the circle acting on $\C$ by rotation with $\mu(z) = -\frac{1}{2}\abs{z}^2$, the image is the half-line $(-\infty, 0]$, which is not the hull of the single fixed value $0$.

## Explanations

### L1 human

A torus symmetry of a compact symplectic manifold has a moment map with values in a vector space.
The theorem says that the image is always a convex polytope, and its corners are the images of the points that the torus does not move.
So a complicated manifold casts a very simple convex shadow.

### L2 human

The theorem is the start of the link between symplectic geometry and convex geometry.
For toric manifolds the polytope determines the manifold completely, by [[Delzant's theorem|WBKCGFMT0P]].
The [[Duistermaat–Heckman formula|EWJK7NY6ZB]] describes how the symplectic volume is spread over the polytope.
Kirwan extended the convexity statement to actions of compact nonabelian groups, where one intersects the image with a positive Weyl chamber.
</document>

<document source="https://tamnd.github.io/math2/node/5CVYTHWY96.md" tag="5CVYTHWY96" kind="definition">
# Algebraic independence

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $S \subseteq L$.
The set $S$ is *algebraically independent over $K$* if, for every $n \ge 1$, all distinct $s_1, \dots, s_n \in S$ and every nonzero polynomial $f \in K[x_1, \dots, x_n]$, one has $f(s_1, \dots, s_n) \ne 0$.
Otherwise $S$ is *algebraically dependent over $K$*.

## Remarks

Equivalently, the evaluation map from the [[polynomial ring|0Q9EZS90GY]] $K[x_s : s \in S]$ to $L$ that sends $x_s$ to $s$ is injective.
A one-element set $\set{s}$ is algebraically independent if and only if $s$ is [[transcendental|00SW8B91E7]] over $K$.
Example: in $K(x, y)$, the set $\set{x, y}$ is independent, since a polynomial $f(x, y)$ that is zero as an element of $K(x, y)$ is the zero polynomial.
Example: $\set{\pi}$ is independent over $\Q$, because $\pi$ is a [[transcendental number|079EGSD32Y]].
Non-example: $\set{\pi, \pi^2}$ is dependent over $\Q$, since $f(x_1, x_2) = x_2 - x_1^2$ is nonzero and vanishes at $(\pi, \pi^2)$.
Algebraic independence implies [[linear independence|0EVHJFTG9Z]] over $K$, by taking $f$ linear; the converse fails, as $\set{1, \pi}$ shows.
</document>

<document source="https://tamnd.github.io/math2/node/5CYEQY4SWR.md" tag="5CYEQY4SWR" kind="concept">
# Torsor

## Summary

A torsor, or principal homogeneous space, for a group $G$ is a space with a [[group action|0WCY42EV9V]] of $G$ that is free and transitive.
It looks exactly like $G$ except that no point is singled out as the identity.
Let $G$ be an [[algebraic group|GH81SYAPPN]] over a field $k$.
A $G$-torsor may have no point over $k$ even though it becomes isomorphic to $G$ over a larger field.
Such torsors are classified by the [[Galois cohomology|9MPCZ3EAJM]] set $H^1(k, G)$, at least when $G$ is smooth.

## Explanations

### L1 human

The points of a line form a torsor for the group of translations.
Two points differ by exactly one translation, but there is no natural origin, so one cannot add points, only move them.
Similarly, the musical notes form a torsor under shifts by intervals, and calendar dates form a torsor under shifts by days.
A torsor is a group that has forgotten which element is the identity.
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# Complex cobordism

## Summary

Complex cobordism $MU^*$ is the [[generalized cohomology theory|H28PSZR7XG]] represented by the Thom spectrum $MU$ of the universal complex vector bundles.
Its coefficient ring is the cobordism ring of stably complex manifolds, a polynomial ring over $\Z$ with one generator in each positive even degree.
By a theorem of Quillen this ring carries the universal formal group law.
This link makes complex cobordism the central theory of modern stable homotopy theory.

## Explanations

### L1 human

On a point, complex cobordism is built from closed manifolds with a complex structure on their stable normal bundle.
Two such manifolds give the same class when together they bound such a manifold.
The theory extends this count of manifolds to a cohomology theory defined for all spaces.
It is the most refined of the complex-oriented cohomology theories, in the sense that it maps to each of them.
For example, the coefficient group $\pi_2(MU)$ is $\Z$, generated by the Riemann sphere $\C P^1$.
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# Differential dynamic programming

## Summary

Differential dynamic programming is an iterative method for discrete-time [[optimal control|T3V6ZNAMDN]] problems.
Around the current trajectory it builds second-order models of the cost-to-go and solves them backward in time, as in dynamic programming.
This gives a correction of the control together with a local linear feedback, which is applied in a forward simulation to get the next trajectory.
Near a solution it converges quadratically, like [[Newton's method|07EXVNF9VP]], and it is widely used in robotics.

## Explanations

### L1 human

Dynamic programming would compute the best decision for every possible state, which is far too much work for a robot with many joints.
Differential dynamic programming only looks near the current guess of the motion.
There it replaces the costs by quadratic approximations, which can be solved exactly, and so learns how to improve each step and how to react to small deviations.
For a linear system with quadratic costs, the approximations are exact and one iteration gives the optimal controller.
For nonlinear problems, a few iterations usually suffice.
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# Pettis integral

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
Let $E$ be a [[Banach space|0BDTTM24DP]] over $\R$ or $\C$ with [[dual space|0M856Q1AMZ]] $E^*$.
Let $f \colon X \to E$ be a function such that $\varphi \circ f$ is [[integrable|X7GTF3D98F]] for every $\varphi \in E^*$.
The function $f$ is *Pettis integrable* if for every $A \in \Sigma$ there is a vector $x_A \in E$ with
$$\varphi(x_A) = \int_A \varphi \circ f \, d\mu \quad \text{for every } \varphi \in E^*,$$
where the right side is a [[Lebesgue integral|DWJF7ABPR1]].
The vector $x_A$ is unique, and it is the *Pettis integral* $\int_A f \, d\mu$.

## Remarks

The vector $x_A$ is unique because the continuous linear functionals separate the points of $E$, by the Hahn–Banach theorem.

Example: let $E = \R^n$ and let each component $f_k$ of $f$ be integrable.
Then $x_A = (\int_A f_1 \, d\mu, \dots, \int_A f_n \, d\mu)$ works, because every $\varphi \in E^*$ is a linear combination of the coordinates.

Non-example: let $X = \N$ with the counting measure, $E = c_0$, the space of sequences tending to $0$, and $f(n) = e_n$, the $n$-th unit vector.
For each $\varphi \in E^* = \ell^1$, the sum $\sum_n \abs{\varphi(e_n)}$ is finite, so $\varphi \circ f$ is integrable.
But for $A = \N$ the only candidate is the constant sequence $(1, 1, \dots)$, which is not in $c_0$, so $f$ is not Pettis integrable.

Every Bochner integrable function is Pettis integrable, with the same integral.
In infinite-dimensional spaces the converse fails.
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# Minkowski–Weyl theorem

## Statement

Let $n \ge 1$, and let $P \subseteq \R^n$ be nonempty.
The following are equivalent.

1. There are a real $m \times n$ matrix $A$ and $b \in \R^m$, for some $m \ge 0$, with $P = \set{x \in \R^n : A x \le b}$, where $\le$ is taken coordinatewise.
2. There are finite sets $V, W \subseteq \R^n$ with $V \ne \emptyset$ and $P = \mathrm{conv}(V) + \mathrm{cone}(W)$.

Here $\mathrm{conv}(V)$ is the [[convex hull|0FQSDC2EQ2]] of $V$, and $\mathrm{cone}(W)$ is the [[convex cone|6VE4KA12VD]] of all nonnegative linear combinations of $W$, with $\mathrm{cone}(\emptyset) = \set{0}$.
In particular, $P$ is a [[convex polytope|300K6CB1KH]] if and only if it is a bounded set of the form in 1.

## Remarks

For 2 implies 1, write $P$ as the projection to $\R^n$ of the set of all $(x, \lambda, \mu)$ with $x = \sum_v \lambda_v v + \sum_w \mu_w w$, $\lambda, \mu \ge 0$ and $\sum_v \lambda_v = 1$.
Fourier–Motzkin elimination shows that the projection of a set given by finitely many linear inequalities is again such a set.
For 1 implies 2, one can apply the first direction to a suitable polar cone.
The polytope case follows, because a bounded set $\mathrm{conv}(V) + \mathrm{cone}(W)$ must have $\mathrm{cone}(W) = \set{0}$.
Example: the square $[0, 1]^2$ is $\set{x : 0 \le x_1 \le 1, \ 0 \le x_2 \le 1}$, given by four inequalities, and also $\mathrm{conv}\set{(0, 0), (1, 0), (0, 1), (1, 1)}$.
The quadrant $\set{x : x_1 \ge 0, x_2 \ge 0}$ is $\mathrm{conv}\set{0} + \mathrm{cone}\set{(1, 0), (0, 1)}$.
Finiteness is needed: the closed unit disc is an intersection of infinitely many half-planes and the convex hull of infinitely many points, but of neither form above.

## Explanations

### L1 human

There are two ways to describe a convex polytope: by its corners, or by the flat walls that bound it.
The theorem says that both ways describe the same shapes.
Any finite list of points spans a shape that can be cut out by finitely many walls.
Any bounded region cut out by finitely many walls is spanned by finitely many corners.

### L2 human

The two descriptions are the [[V-representation and H-representation|W8P2G5R02N]] of a polytope.
Linear programming uses both: the feasible region is given by inequalities, while the simplex method moves between vertices.
Changing from one description to the other is the vertex and facet enumeration problem.
The sizes can differ a lot: the cube $[0, 1]^n$ needs $2n$ inequalities but has $2^n$ vertices.
The theorem is a finite-dimensional cousin of Farkas' lemma, and the two can be derived from each other.
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# Strongly measurable function

## Definition

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]] and let $E$ be a [[Banach space|0BDTTM24DP]] over $\R$ or $\C$.
A function $s \colon X \to E$ is a *simple measurable function* if it takes finitely many values and $s^{-1}(\set{e}) \in \Sigma$ for each $e \in E$.
This extends the notion of a [[simple function|7WMBX7M4P6]] to values in $E$.
A function $f \colon X \to E$ is *strongly measurable* if there is a sequence $(s_n)$ of simple measurable functions $s_n \colon X \to E$ with
$$\lim_{n \to \infty} \norm{s_n(x) - f(x)} = 0 \quad \text{for every } x \in X,$$
that is, $s_n \to f$ [[pointwise|056FSKG32J]] on $X$.

## Remarks

Example: let $X = [0, 1]$ with the Borel σ-algebra and let $f \colon X \to E$ be continuous.
The function $s_n$ that equals $f(k/n)$ on $[k/n, (k+1)/n)$ for $0 \le k < n$ and $f(1)$ at $1$ is simple and measurable, and $s_n(x) \to f(x)$ by continuity, so $f$ is strongly measurable.

Non-example: let $X = [0, 1]$ with the σ-algebra $\set{\varnothing, X}$ and $f(x) = x$ with values in $E = \R$.
A simple measurable function is then constant, so a pointwise limit of such functions is constant, and $f$ is not strongly measurable.

When $E = \R$ or $E = \C$, a function is strongly measurable if and only if it is measurable.
For a measure $\mu$ on $(X, \Sigma)$, a function is called *strongly $\mu$-measurable* or *Bochner measurable* when it agrees $\mu$-almost everywhere with a strongly measurable function.
The image of a strongly measurable function is a separable subset of $E$, because it lies in the closure of the countable set of values of the $s_n$.
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# Ring with involution

## Definition

Let $R$ be a [[ring|0894J980RK]] with unit $1$.
An *involution* on $R$ is a map $R \to R$, $a \mapsto a^*$, such that for all $a, b \in R$:

1. $(a + b)^* = a^* + b^*$.
2. $(ab)^* = b^* a^*$.
3. $1^* = 1$.
4. $(a^*)^* = a$.

A *ring with involution*, or *$*$-ring*, is a ring together with an involution.

## Remarks

Clause 3 follows from the others: $1^* a = (a^* \cdot 1)^* = a$ and likewise $a 1^* = a$ for all $a$, so $1^*$ is a unit element and equals $1$.
An element $a$ is *self-adjoint* or *Hermitian* if $a^* = a$.
Example: $R = M_n(\C)$ with $A^* = \overline{A}^{\,T}$, the [[conjugate transpose|0EX045E2BP]].
Clauses 1, 3 and 4 are clear, and clause 2 holds because $(AB)^T = B^T A^T$ and conjugation is multiplicative.
Example: the transpose on $M_n(F)$ for a field $F$, and $g \mapsto g^{-1}$ extended linearly on a group ring $\Z[G]$.
Non-example: the identity map on $M_2(\R)$ satisfies clauses 1, 3 and 4 but fails clause 2, since $AB \ne BA$ for some matrices $A, B$.
If $R$ is an algebra over a commutative $*$-ring $K$, a *$*$-algebra* also asks $(\lambda a)^* = \lambda^* a^*$ for $\lambda \in K$.
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# Brown–McCoy radical

## Definition

Let $R$ be a [[ring|0894J980RK]] with identity.
A *maximal ideal* of $R$ is a [[two-sided ideal|0DDA6CY7GJ]] $M \ne R$ such that no two-sided ideal $I$ satisfies $M \subsetneq I \subsetneq R$.
The *Brown–McCoy radical* of $R$ is the intersection of all maximal ideals of $R$.

## Remarks

Equivalently, it is the intersection of the ideals $M$ such that $R/M$ is a simple ring.
Example: for $R = \Z$ the maximal ideals are $p\Z$ for primes $p$, and $\bigcap_p p\Z = 0$, so the Brown–McCoy radical of $\Z$ is $0$.
Non-example of a maximal ideal: in $\Z$ the ideal $4\Z$ is not maximal, since $4\Z \subsetneq 2\Z \subsetneq \Z$.
Every maximal ideal is a [[primitive ideal|M0JDQR0V5S]], because a simple ring with identity has a faithful simple module; hence the [[Jacobson radical|XA0DCNH7WY]] is contained in the Brown–McCoy radical.
The inclusion can be strict.
For the ring $R$ of linear maps of a vector space of countably infinite dimension, $J(R) = 0$, while the only maximal ideal is the ideal of maps of finite rank.
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# Determinant line bundle

## Summary

The determinant line bundle of a continuous family of [[Fredholm operators|8HWGKWD6ZM]] is a complex line bundle over the parameter space.
Its fibre at a parameter $y$ is built from the top exterior powers of the kernel and the cokernel of the operator $T_y$.
The kernels and cokernels jump in dimension, but these lines fit together into a locally trivial line bundle.
Quillen used it for families of Cauchy–Riemann operators and gave it a metric through [[zeta-regularized determinants|YT3ZQ2F7FG]].
It is basic in the study of anomalies in physics.

## Explanations

### L1 human

For an invertible matrix one has a number, the determinant.
For a family of operators that are sometimes not invertible, one cannot have a single number, but one can have a line at each parameter where a determinant would live.
When the operator is invertible, the line has a canonical nonzero element, the determinant itself.
The twisting of this family of lines over the parameter space is a topological invariant of the family.
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# Seiberg–Witten invariants

## Definition

Let $M$ be a closed, oriented, [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]] with $b_2^+(M) \ge 2$, where $b_2^+(M)$ is the number of positive eigenvalues of its [[intersection form|WK8GKRKAVD]].
Fix a [[Riemannian metric|Q61VFHCH2J]] $g$ on $M$ and an orientation of $H^1(M; \R) \oplus H^2_+(M; \R)$.
Let $\mathfrak{s}$ be a spin$^c$ structure on $M$, a generalization of a [[spin structure|RSPZ9TCFQ8]], with spinor bundles $W^\pm$ and determinant line bundle $L$.
For a $U(1)$-connection $A$ on $L$, let $D_A \colon \Gamma(W^+) \to \Gamma(W^-)$ be the associated [[Dirac operator|3DXGJTEQZH]].

1. Let $F_A^+$ be the self-dual part of the curvature of $A$.
Let $\sigma(\psi)$ be the trace-free part of the endomorphism $\psi \otimes \psi^*$ of $W^+$, viewed as a self-dual $2$-form by Clifford multiplication.
Let $\eta$ be a generic self-dual $2$-form.
The *Seiberg–Witten equations* for a pair $(A, \psi)$ with $\psi \in \Gamma(W^+)$ are
$$D_A \psi = 0, \qquad F_A^+ = \sigma(\psi) + i\eta.$$
2. The gauge group $\mathrm{Map}(M, S^1)$ acts by $u \cdot (A, \psi) = (A - 2u^{-1} du, u \psi)$, and the *moduli space* $\mathcal{M}(\mathfrak{s})$ is the set of solutions modulo this action.
3. Let $c_1(L)$ be the first [[Chern class|PAD37GKDPD]] of $L$, $\chi(M)$ the Euler characteristic and $\sigma(M)$ the signature.
The moduli space is a compact smooth oriented manifold of dimension
$$d(\mathfrak{s}) = \frac{c_1(L)^2 - 2\chi(M) - 3\sigma(M)}{4}.$$

The *Seiberg–Witten invariant* $SW_M(\mathfrak{s}) \in \Z$ is defined as follows.
If $d(\mathfrak{s}) = 0$, it is the signed number of points of $\mathcal{M}(\mathfrak{s})$.
If $d(\mathfrak{s}) > 0$ is even, it is the integral over $\mathcal{M}(\mathfrak{s})$ of a power of a natural degree-$2$ class.
In the other cases it is $0$.

## Remarks

When $b_2^+(M) \ge 2$, the number $SW_M(\mathfrak{s})$ does not depend on $g$ or $\eta$, so it is an invariant of $M$ with its orientation data.
For example, a closed symplectic $4$-manifold with $b_2^+ \ge 2$ has $SW = \pm 1$ on its canonical spin$^c$ structure, by a theorem of Clifford Taubes.
In contrast, a connected sum $M_1 \# M_2$ with $b_2^+(M_i) \ge 1$ for both summands has all Seiberg–Witten invariants zero.
When $b_2^+(M) = 1$ the count can jump as $g$ and $\eta$ vary, so the definition then depends on a chamber.
Normalizations of $\sigma(\psi)$ and of the gauge action differ by constants between authors.
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# Cross-polytope

## Example

For $d \ge 1$, let $e_1, \dots, e_d$ be the standard basis of $\R^d$.
The $d$-dimensional *cross-polytope* is
$$X_d = \operatorname{conv}\set{\pm e_1, \dots, \pm e_d} = \set{x \in \R^d : \abs{x_1} + \cdots + \abs{x_d} \le 1}.$$
It is a [[convex polytope|300K6CB1KH]] with the $2d$ [[vertices|R7V3S9V891]] $\pm e_i$.

A set of vertices spans a proper face exactly when it contains no pair $e_i, -e_i$.
Indeed, for signs $\epsilon_i \in \set{1, -1}$ the vector $c = (\epsilon_1, \dots, \epsilon_d)$ gives $c \cdot x \le 1$ on $X_d$, with equality on the hull of $\epsilon_1 e_1, \dots, \epsilon_d e_d$.
So the [[facets|DPDXN2FJC4]] are the $2^d$ simplices $\operatorname{conv}\set{\epsilon_1 e_1, \dots, \epsilon_d e_d}$, and $X_d$ is a [[simplicial polytope|YVJ15PGKDM]].
Choosing $k + 1$ of the $d$ axes and a sign for each gives the number of $k$-faces:
$$f_k(X_d) = 2^{k+1} \binom{d}{k+1}, \qquad 0 \le k \le d-1.$$

## Remarks

For $d = 2$ the cross-polytope is a square turned by $45$ degrees, and for $d = 3$ it is the regular octahedron.
It is the [[polar polytope|6DXF1T506B]] of the [[hypercube|P9PYM71GMM]] $[-1,1]^d$: a vector $y$ satisfies $x \cdot y \le 1$ for all $x \in [-1,1]^d$ exactly when $\abs{y_1} + \cdots + \abs{y_d} \le 1$.
Its unit ball description shows that it is the unit ball of the $\ell^1$ norm.

## Explanations

### L1 human

Put a point at distance one on each side of the origin along every axis, and wrap a skin around these points.
In the plane you get a diamond, in space an octahedron.
The cross-polytope is the same shape in any dimension.
Its faces are all triangles or their higher versions, simplices, which makes it the dual of the cube.
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# Representation theory of SL2(R)

## Example

Let $G = \mathrm{SL}_2(\R)$, the [[group|Q0GBCB9RA4]] of real $2 \times 2$ matrices of determinant $1$.
It is a connected simple Lie group that is not compact.
So each nontrivial [[irreducible|FX3NEFYFFN]] [[unitary representation|74BPYJNZTM]] of $G$ is infinite-dimensional.
Indeed, a nontrivial unitary representation on $\C^n$ has a differential that embeds $\mathfrak{sl}_2(\R)$ into $\mathfrak{u}(n)$.
Each subalgebra of $\mathfrak{u}(n)$ has an inner product for which every $\operatorname{ad} X$ is skew-symmetric.
The algebra $\mathfrak{sl}_2(\R)$ has none, since $\operatorname{ad} X$ has real nonzero eigenvalues for $X = \operatorname{diag}(1, -1)$.

Principal series.
Fix $\varepsilon \in \set{0, 1}$ and $\nu \in i\R$.
Let $H_{\varepsilon, \nu}$ be the space of functions $f$ on $\R^2 \setminus \set{0}$ with $f(\lambda x) = \sgn(\lambda)^{\varepsilon} \abs{\lambda}^{-1-\nu} f(x)$ for $\lambda \ne 0$.
Its norm is the norm of $L^2$ on the unit circle.
The action is $(\pi(g) f)(x) = f(g^{-1} x)$.
Here $\abs{f}^2$ is homogeneous of degree $-2$.
For such functions the integral over the circle is unchanged by linear maps of determinant $1$, so $\pi$ is unitary.

Discrete series.
For an integer $k \ge 2$, let $D_k^+$ be the space of holomorphic functions $f$ on the upper half plane $\set{z = x + iy : y > 0}$ with $\int \abs{f(z)}^2 y^{k-2} \, dx \, dy < \infty$.
For $g^{-1} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the action is $(\pi(g) f)(z) = (cz + d)^{-k} f\big(\frac{az + b}{cz + d}\big)$.
This is an irreducible unitary representation whose matrix coefficients are square integrable on $G$.

## Remarks

Bargmann classified the irreducible unitary representations of $G$ in 1947.
Up to equivalence they are the following.

1. The trivial representation.
2. The irreducible principal series.
3. The complementary series, a family with a real parameter in an interval.
4. The discrete series $D_k^{\pm}$ for $k \ge 2$.
5. The two limits of discrete series $D_1^{\pm}$, the two pieces of the reducible principal series with $\varepsilon = 1$ and $\nu = 0$.

Here $D_k^-$ is the complex conjugate of $D_k^+$, realized on antiholomorphic functions.

The Casimir element of $\mathfrak{sl}_2(\R)$ acts on each irreducible representation by a scalar, and this scalar is a main invariant in the classification.
These representations govern the spectral theory of the Laplacian on hyperbolic surfaces $\Gamma \backslash \mathbb{H}$, and holomorphic cusp forms of weight $k$ give copies of the discrete series $D_k^{\pm}$ in $L^2(\Gamma \backslash G)$.

## Explanations

### L1 human

The group $\mathrm{SL}_2(\R)$ moves the points of the hyperbolic plane.
Unlike a compact group, it has no nontrivial length-preserving action on a finite-dimensional space.
So its interesting representations live on spaces of functions, such as functions on the circle or holomorphic functions on the upper half plane.
Bargmann found all of them, and they come in a few families.
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# Harmonic analysis on groups

## Summary

Harmonic analysis on groups studies functions on a locally compact group by writing them as superpositions of basic oscillations attached to the group.
For abelian groups the oscillations are the characters and the main tool is the Fourier transform on the dual group.
For non-abelian groups the characters are replaced by irreducible unitary representations.
The subject contains Fourier series, the Fourier transform on $\R^n$, and the representation theory of compact and Lie groups.

## Explanations

### L1 human

A musical sound can be split into pure tones; harmonic analysis does the same for functions.
Which "pure tones" are available depends on the symmetry of the space: on a circle they are the waves $e^{in\theta}$, on the line they are $e^{i\xi x}$.
On a sphere, where rotations act, the pure tones are spherical harmonics, and they come from representations of the rotation group.
Harmonic analysis on groups is the general study of this splitting for any group with a reasonable topology.
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# Brieskorn manifold

## Example

Let $n \ge 1$ and let $a_0, \dots, a_n \ge 2$ be integers.
The *Brieskorn manifold* is
$$\Sigma(a_0, \dots, a_n) = \set{z \in \C^{n+1} : z_0^{a_0} + z_1^{a_1} + \cdots + z_n^{a_n} = 0} \cap S^{2n+1},$$
where $S^{2n+1}$ is the unit sphere in $\C^{n+1}$.
It is the [[link|C831AN3RFM]] at $0$ of the hypersurface $f^{-1}(0)$, where $f = z_0^{a_0} + \cdots + z_n^{a_n}$.

Check: the partial derivatives of $f$ are $a_j z_j^{a_j - 1}$, which vanish together only at $z = 0$.
So $f$ has an [[isolated singularity|VTGYEES3VS]] at $0$ and $f^{-1}(0)$ is smooth away from $0$.
Since $f$ is weighted homogeneous, the sphere of radius $1$ gives the same link as any small sphere.
So $\Sigma(a_0, \dots, a_n)$ is a smooth closed manifold of dimension $2n - 1$.
The [[Milnor number|1ZE51P7GRQ]] of $f$ is $(a_0 - 1)(a_1 - 1) \cdots (a_n - 1)$, because the monomials $z_0^{b_0} \cdots z_n^{b_n}$ with $0 \le b_j \le a_j - 2$ form a basis of the local algebra.

## Remarks

$\Sigma(2, 3, 5)$ is the [[Poincaré homology sphere|NHERCK7C43]], a $3$-manifold with the homology of $S^3$ but a nontrivial fundamental group of order $120$.
For $n = 1$, $\Sigma(p, q)$ is the torus link of type $(p, q)$ in $S^3$, a knot when $p$ and $q$ are coprime.
Brieskorn showed that the $7$-manifolds $\Sigma(6k - 1, 3, 2, 2, 2)$ for $k = 1, \dots, 28$ are homeomorphic to $S^7$ and realize the $28$ oriented diffeomorphism classes of [[exotic spheres|6WYEMXSHD9]] of dimension $7$.

## Explanations

### L1 human

Brieskorn manifolds come from the simplest polynomials one can write: sums of powers of the variables.
Cut the zero set of such a polynomial with the unit sphere, and the result is a smooth manifold.
Changing the exponents changes the manifold in a computable way.
Some choices give spheres that are equal to the usual sphere as topological spaces but not as smooth manifolds.
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# Wave equation

## Summary

The wave equation $u_{tt} = c^2 \Delta u$ is the basic [[hyperbolic partial differential equation|92C3S03AQZ]].
It describes vibrating strings and membranes, sound, and light in vacuum.
Its solutions travel at the finite speed $c$.
In one space dimension every solution is a sum of a wave moving left and a wave moving right.
It is the model for the theory of hyperbolic equations, just as the [[heat equation|VP2HN86VAG]] is for parabolic ones.

## Explanations

### L1 human

Pluck a guitar string: the displacement $u(x, t)$ of the string obeys the wave equation.
The acceleration $u_{tt}$ of a piece of string is proportional to its curvature $u_{xx}$, because tension pulls a bent piece back toward straight.
A bump on the string splits into two bumps that move apart at speed $c$ without changing shape.
For example, $u(x, t) = \sin(x - ct)$ is a wave that moves to the right at speed $c$.
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# Uniformization theorem

## Statement

Let $X$ be a simply connected [[Riemann surface|5K0N8QMV6Q]], that is, its [[fundamental group|S1XK3HXHFX]] is trivial.
Then $X$ is [[biholomorphic|8AJEMCZ29B]] to exactly one of the following three Riemann surfaces:

1. the [[Riemann sphere|0CDMXGPSES]] $\hat\C$;
2. the [[complex plane|0CTE7A31XQ]] $\C$;
3. the open [[unit disc|8G5ABXRK85]] $\mathbb{D}$.

## Remarks

The three surfaces are pairwise not biholomorphic: $\hat\C$ is compact and the others are not, and a holomorphic map $\C \to \mathbb{D}$ is constant by Liouville's theorem.
Applied to the [[universal cover|Y1YPE6PM52]], the theorem shows that every Riemann surface $X$ is biholomorphic to a quotient $\tilde X / \Gamma$.
Here $\tilde X$ is one of the three surfaces and $\Gamma$ is a group of automorphisms of $\tilde X$ acting freely and properly discontinuously.
Only $\hat\C$ itself is covered by $\hat\C$, and only $\C$, $\C \setminus \set{0}$ and the tori $\C / \Lambda$ are covered by $\C$.
Every other Riemann surface is a quotient $\mathbb{D} / \Gamma$ with $\Gamma$ a [[Fuchsian group|VNJWSCVP6V]].
For simply connected open subsets of $\C$ the theorem contains the [[Riemann mapping theorem|02WHMEG5ZR]].

The theorem was proved in 1907, independently by Koebe and by Poincaré.
One proof constructs a harmonic Green's function or a harmonic dipole function on $X$ and uses it to build a holomorphic injection into $\hat\C$.

Example: $\C \setminus \set{0}$ is not simply connected; its universal cover is $\exp \colon \C \to \C \setminus \set{0}$, and $\C \setminus \set{0} \cong \C / 2\pi i \Z$.

## Explanations

### L1 human

There are infinitely many Riemann surfaces, but only three that have no loops that cannot be shrunk: the sphere, the plane and the disc.
Every other Riemann surface is obtained from one of these three by rolling it up, that is, by identifying points that differ by a group of symmetries.
So the study of all Riemann surfaces reduces to the study of these three and their symmetry groups.

### L2 human

The theorem gives every Riemann surface a metric of constant curvature $+1$, $0$ or $-1$ compatible with its complex structure.
Compact surfaces of genus $0$ are spherical, genus $1$ flat, and genus at least $2$ hyperbolic, a fact that links complex analysis to hyperbolic geometry.
It is the starting point of Teichmüller theory, because a complex structure on a surface of genus $g \ge 2$ becomes a Fuchsian group up to conjugation.
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<document source="https://tamnd.github.io/math2/node/5ENHTF01Z8.md" tag="5ENHTF01Z8" kind="concept">
# Saddle point of a Lagrangian

## Summary

A saddle point of a Lagrangian is a pair made of a point and a vector of multipliers.
At this pair the Lagrangian of a constrained optimization problem is minimal in the point and maximal in the multipliers.
Whenever such a pair exists, the point solves the constrained problem and the multipliers satisfy complementary slackness.
For convex problems that satisfy a constraint qualification, saddle points exist and they are the basis of Lagrangian duality.
The [[definition|QQMQGXS0XE]] gives the precise conditions.

## Explanations

### L1 human

A constrained problem asks for the smallest value of $f$ among points that obey some constraints.
The Lagrangian adds to $f$ a penalty for each constraint, with a price $\lambda_i \ge 0$ per unit of violation.
At a saddle point, the chooser of the point cannot gain by moving and the chooser of the prices cannot gain by changing them.
For example, to minimize $x^2$ subject to $x \ge 1$, the point $x = 1$ with price $\lambda = 2$ is a saddle point.
Unlike the [[saddle point|0JW6MDHAZD]] of a surface in calculus, it is defined by inequalities, not by derivatives.
</document>

<document source="https://tamnd.github.io/math2/node/5EZRN3812E.md" tag="5EZRN3812E" kind="statement">
# Zeros of holomorphic functions are not isolated

## Statement

Let $n \ge 2$, let $D \subseteq \C^n$ be an open set, and let $f$ be a [[holomorphic function|507C34W0ZT]] on $D$.
If $p \in D$ and $f(p) = 0$, then every neighbourhood of $p$ contains a point $q \ne p$ with $f(q) = 0$.

## Remarks

Suppose instead that $p$ is the only zero of $f$ in a small open ball $B$ around $p$.
Then $1/f$ is holomorphic on $B \setminus \set{p}$, which is connected.
By [[Hartogs's extension theorem|FSGES670AZ]] with $K = \set{p}$, it extends to a holomorphic function $g$ on $B$.
Then $g f = 1$ on $B \setminus \set{p}$, hence at $p$ by continuity, which contradicts $f(p) = 0$.

Example: $f(z_1, z_2) = z_1^2 + z_2^2$ vanishes at the origin, and its zeros near the origin form the two complex lines $z_1 = i z_2$ and $z_1 = -i z_2$.

The hypothesis $n \ge 2$ is necessary: $f(z) = z$ on $\C$ has the isolated zero $0$.

## Explanations

### L1 human

In one complex variable, a nonzero holomorphic function has isolated zeros, like the roots of a polynomial.
In two or more variables this is never the case: a zero always lies on a whole curve, surface or larger set of zeros.
So one cannot hope to describe zero sets by listing points.

### L2 human

The zero set of a holomorphic function that is not identically zero is an analytic hypersurface, of complex dimension $n - 1$ at each of its points.
This proposition is the first step toward that fact.
Divisors in several variables are made of hypersurfaces instead.
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<document source="https://tamnd.github.io/math2/node/5F0YMCV50R.md" tag="5F0YMCV50R" kind="concept">
# Semiprime ring

## Summary

A semiprime ring is a ring in which $aRa = 0$ forces $a = 0$.
Equivalently, it has no nonzero nilpotent two-sided ideals, or its [[prime radical|WF117EJ2HF]] is zero.
Semiprime rings are the subdirect products of [[prime rings|Z9TJSZZYG1]], and a commutative ring is semiprime exactly when it is [[reduced|SZHY975FFX]].
They form the natural setting of Goldie's theorem and of much of the theory of polynomial identity rings.

## Explanations

### L1 human

A reduced commutative ring has no nonzero nilpotent elements.
In noncommutative rings this is too strong, since matrix rings have nilpotent matrices.
A semiprime ring instead has no nonzero ideal whose elements all multiply to zero in long enough products.
For example $M_2(\Q)$ is semiprime, while the ring of upper triangular $2 \times 2$ matrices is not, because its strictly upper triangular matrices form an ideal with square zero.
</document>

<document source="https://tamnd.github.io/math2/node/5F76XA1ABC.md" tag="5F76XA1ABC" kind="statement">
# Brent's method

## Statement

Let $f \colon [a, b] \to \R$ be continuous with $f(a) f(b) < 0$, and let $\varepsilon > 0$ be a tolerance.
*Brent's method* keeps at each step a bracket, an interval whose endpoints have values of opposite signs.
At each step it proposes a new point by [[inverse quadratic interpolation|NW1XQ95R5P]] or by the [[secant method|9ZC177PVPC]].
It falls back to a [[bisection|7VMYC7MB3Q]] step when the proposed point fails tests on its position or on the decrease of the bracket.
Then:

1. Every bracket contains a zero of $f$.
2. The method stops after finitely many evaluations of $f$ with a bracket of length at most $\varepsilon$, up to the rounding allowance in its tests.
3. If bisection needs $N$ steps to reach the tolerance, Brent's method needs at most about $N^2$ evaluations of $f$.

## Remarks

Claim 1 follows from the [[intermediate value theorem|00KAFBR32E]], since the method only keeps brackets with a sign change.
Claims 2 and 3 come from the safeguards: the tests force a bisection step whenever the interpolation steps have not shrunk the bracket enough over the last few steps.
The bound in claim 3 is a worst case.
For a smooth function with a simple zero, the method ends up taking interpolation steps only, and then it converges superlinearly.
So it combines the reliability of bisection with the speed of the interpolation methods.
It refines an earlier method of Dekker, which could take a very large number of tiny steps.
It is the standard root finder for one equation in many numerical libraries, for example as `fzero` in MATLAB.

## Explanations

### L1 human

Bisection is slow but cannot fail; interpolation is fast but can go wrong.
Brent's method tries a fast interpolation step first and checks that it behaves.
If it does not, the method takes a safe bisection step instead, so the root stays trapped in a shrinking interval.

### L2 human

The main design point is the bracket: the method never gives up the sign change, so convergence is guaranteed for every continuous $f$.
For well-behaved $f$ the interpolation steps dominate and the speed is close to that of inverse quadratic interpolation.
To use it, one needs a starting interval where $f$ changes sign.
</document>

<document source="https://tamnd.github.io/math2/node/5F9MP8VCJ4.md" tag="5F9MP8VCJ4" kind="definition">
# Real Hardy space H1

## Definition

Let $n \ge 1$ and let $L^1(\R^n)$ be the [[Lp space|SASP14MA1Z]] of integrable functions for Lebesgue measure.
For $j = 1, \dots, n$ let $R_j$ be the $j$-th [[Riesz transform|FRNDFJ9WQ5]], which maps each $f \in L^1(\R^n)$ to a [[tempered distribution|AES2T5JPFP]] $R_j f$.
The *real Hardy space* $H^1(\R^n)$ is the set of $f \in L^1(\R^n)$ such that each $R_j f$, $j = 1, \dots, n$, is given by a function in $L^1(\R^n)$.
It is a Banach space with the norm
$$\norm{f}_{H^1} = \norm{f}_{L^1} + \sum_{j=1}^n \norm{R_j f}_{L^1}.$$

## Remarks

For $n = 1$ the only Riesz transform is the [[Hilbert transform|NSCEBXM7ZM]] $H$, so $H^1(\R) = \set{f \in L^1(\R) : Hf \in L^1(\R)}$.
Equivalent definitions use maximal functions or atoms.
Fix a Schwartz function $\varphi$ with $\int \varphi = 1$ and let $\varphi_t(x) = t^{-n}\varphi(x/t)$.
Then $f \in H^1(\R^n)$ exactly when $f$ is a tempered distribution with $\sup_{t > 0} \abs{f * \varphi_t} \in L^1(\R^n)$, and the two norms are comparable.
Every $f \in H^1(\R^n)$ has $\int f = 0$.

Example: $n = 1$ and $a = \mathbf{1}_{[0, 1]} - \mathbf{1}_{[-1, 0]}$.
For $\abs{x} > 2$, using $\int a = 0$, $Ha(x) = \frac{1}{\pi} \int a(y) \left( \frac{1}{x - y} - \frac{1}{x} \right) dy$, which is bounded by $C/x^2$.
Near the points $-1$, $0$ and $1$, $Ha$ has only logarithmic singularities, which are integrable.
So $Ha \in L^1(\R)$ and $a \in H^1(\R)$.

Non-example: $f = \mathbf{1}_{[0, 1]}$ on $\R$ is integrable but not in $H^1(\R)$.
Its Hilbert transform is $Hf(x) = \frac{1}{\pi} \log \abs{\frac{x}{x - 1}}$, which behaves like $\frac{1}{\pi x}$ as $\abs{x} \to \infty$, so it is not integrable.
This agrees with the fact that $\int f = 1 \ne 0$.
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<document source="https://tamnd.github.io/math2/node/5FC2T66AQD.md" tag="5FC2T66AQD" kind="definition">
# Totally unimodular matrix

## Definition

Let $A$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] with entries in $\Z$.
The matrix $A$ is *totally unimodular* when every [[minor|0BT9BYWVRY]] of $A$, that is, the determinant of every square submatrix, is $0$, $1$ or $-1$.

## Remarks

Example: $\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix}$ is totally unimodular.
Its $1 \times 1$ minors are entries in $\{0, 1\}$, and its $2 \times 2$ minors are $1$, $1$ and $1$.
Non-example: $\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}$ is not totally unimodular, because its determinant is $2$.
A totally unimodular matrix has all entries in $\{0, 1, -1\}$, since the entries are its $1 \times 1$ minors.
A square totally unimodular matrix need not be [[unimodular|R6RBQHW191]], because its determinant may be $0$.
The incidence matrix of a [[bipartite graph|0ZVQKTRKVQ]] and the incidence matrix of a [[directed graph|0KFYGSHHF5]] are totally unimodular.
By a theorem of Hoffman and Kruskal, an integer matrix $A$ is totally unimodular exactly when the polyhedron $\set{x \in \R^n : Ax \le b,\ x \ge 0}$ has integral vertices for every integral vector $b$.
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<document source="https://tamnd.github.io/math2/node/5FC6BMM1NA.md" tag="5FC6BMM1NA" kind="concept">
# Classification of second-order PDEs

## Summary

Linear second-order partial differential equations are sorted into elliptic, parabolic and hyperbolic equations.
The type is given by the signs of the eigenvalues of the matrix of second-order coefficients.
The three types behave very differently: elliptic equations describe equilibria, parabolic equations describe diffusion, and hyperbolic equations describe waves.
The model cases are the [[Laplace equation|S8RG6N1PRT]], the [[heat equation|VP2HN86VAG]] and the [[wave equation|5EHD2XBQDG]].
See the [[definition|J53A2A04P7]].

## Explanations

### L1 human

In two variables the second-order part $A u_{xx} + 2B u_{xy} + C u_{yy}$ looks like the quadratic form $A x^2 + 2B xy + C y^2$.
Such a form describes an ellipse, a parabola or a hyperbola, depending on the sign of $AC - B^2$, and the names of the PDE types come from this.
For $u_{xx} + u_{yy}$ one has $AC - B^2 = 1 > 0$, an ellipse, and the equation is elliptic.
For $u_{xx} - u_{yy}$ one has $AC - B^2 = -1 < 0$, a hyperbola, and the equation is hyperbolic.
The type tells which extra data, such as boundary values or initial values, give a good problem.
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<document source="https://tamnd.github.io/math2/node/5FD8V6MQNX.md" tag="5FD8V6MQNX" kind="statement">
# Cartan's theorem A

## Statement

Let $X$ be a [[Stein space|Z4AMJS5KA6]] and let $\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.
Then $\mathcal{F}$ is generated by its global sections.
That is, for every $x \in X$, the germs at $x$ of the elements of $\Gamma(X, \mathcal{F})$ generate the stalk $\mathcal{F}_x$ as a module over $\mathcal{O}_{X,x}$.

## Remarks

Henri Cartan proved theorems A and B in his seminar of 1951–52, building on the work of Oka.
In the usual proof, $X$ is exhausted by compact holomorphically convex sets.
On each of them, $\mathcal{F}$ is generated by finitely many sections, by a local argument.
Cartan's theorem B on these pieces and a Runge-type approximation theorem then pass to the limit.
Example: let $\mathcal{I}$ be the ideal sheaf of a closed complex submanifold $Y$ of a Stein manifold $X$.
Theorem A says that near each point, $Y$ is cut out by global holomorphic functions on $X$ that vanish on $Y$.
For $X = \C^n$ and $\mathcal{F} = \mathcal{O}$ the theorem is trivial, because the constant function $1$ generates every stalk.
The Stein hypothesis is needed: on the projective line $\mathbb{P}^1$, the sheaf $\mathcal{O}(-1)$ is coherent but has no nonzero global sections, so it is not generated by them.

## Explanations

### L1 human

A coherent sheaf describes local data, such as the holomorphic functions that vanish on a given analytic set near each point.
Theorem A says that on a Stein space all this local data comes from global data.
Everything one sees near a point is produced by objects defined on the whole space.
It is the complex-analytic counterpart of the fact that every quasi-coherent sheaf on an affine scheme is generated by its global sections.

### L2 human

Theorem A is usually proved together with [[Cartan's theorem B|R7VKRV8AGQ]], and each is used in the proof of the other on exhausting compact sets.
A typical consequence: global holomorphic functions give local coordinates at every point of a Stein manifold.
Another one: every analytic subset of a Stein space is the common zero set of global holomorphic functions.
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<document source="https://tamnd.github.io/math2/node/5FDMY7PDYY.md" tag="5FDMY7PDYY" kind="concept">
# Grothendieck group

## Summary

The Grothendieck group of a commutative [[monoid|0N6R902FG1]] is the [[abelian group|082GHNPJ53]] obtained by adding formal differences, in the universal way.
It turns the natural numbers into the integers.
Applied to isomorphism classes of finitely generated [[projective modules|BFFA03GE3V]] under direct sum, it gives the group $K_0$ of a ring, the start of [[algebraic K-theory|5GVFZWMWC8]].
The same idea applies to vector bundles and to abelian categories.

## Explanations

### L1 human

Some structures can be added but not subtracted, such as natural numbers, or vector spaces under direct sum.
The Grothendieck group adds negatives in the most economical way.
Its elements are formal differences $a - b$, with $a - b = c - d$ when $a + d$ and $c + b$ become equal after adding a common term.
From the natural numbers one gets the integers.
From finite-dimensional vector spaces over a field one also gets the integers, through the dimension.
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<document source="https://tamnd.github.io/math2/node/5FG1BKZXKS.md" tag="5FG1BKZXKS" kind="concept">
# Haagerup property

## Summary

A locally compact group has the Haagerup property when it has a unitary representation with almost invariant vectors whose matrix coefficients vanish at infinity.
Equivalently, it acts on a Hilbert space by affine isometries in a proper way; for this reason Gromov called such groups a-T-menable.
The property is a weak form of [[amenability|CEAT9BESKK]] and is opposite to [[property (T)|GTFZ277047]]: a group with both properties is compact.
Haagerup showed that free groups have it, and it is used in the Baum–Connes conjecture and in the theory of von Neumann algebras.

## Explanations

### L1 human

Imagine a group acting on an infinite-dimensional space by rigid motions, so that points far out in the group move a given point far away.
Groups with such an action have the Haagerup property.
For the integers, translating along a line already works: the element $n$ moves the origin to distance $\abs{n}$.
Free groups also have such actions, built from the tree on which they act.
Groups with property (T), such as $\mathrm{SL}_3(\Z)$, are rigid in the opposite way: every such action of them has a fixed point.
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<document source="https://tamnd.github.io/math2/node/5FKKCM9ESE.md" tag="5FKKCM9ESE" kind="definition">
# Quasi-projective variety

## Definition

Let $k$ be an algebraically closed field and let $\mathbb{P}^n$ be the [[projective space|ZT9A6JK8QF]] over $k$ with the Zariski topology.
A *quasi-projective variety* is a nonempty [[open subset|02BH3MCBQB]] $X$ of a [[projective variety|A4H4CZ2V35]] $Y \subseteq \mathbb{P}^n$, with the [[subspace topology|0WS9P12FQR]].

## Remarks

Equivalently, $X$ is an [[irreducible|SY2MWC434Z]] subset of $\mathbb{P}^n$ of the form $Y \setminus Z$, where $Y$ and $Z$ are projective algebraic sets.
A nonempty open subset of an irreducible space is irreducible and dense, so $X$ is irreducible and its closure is $Y$.

Example: every [[affine variety|JERVQT4C12]] $X \subseteq k^n$ is quasi-projective.
Identify $k^n$ with the chart $U_0 = \set{x_0 \ne 0}$ of $\mathbb{P}^n$; then $X$ is open in its closure $\overline{X} \subseteq \mathbb{P}^n$, which is a projective variety.
Every projective variety is quasi-projective, taking $X = Y$.

Example: $k^2 \setminus \set{(0, 0)}$ is quasi-projective, as an open subset of $\mathbb{P}^2$; it is neither affine nor projective.

Non-example: $V(xy) \setminus \set{[0 : 0 : 1]} \subseteq \mathbb{P}^2$ is open in a projective algebraic set but is not a quasi-projective variety, because it is the union of two proper closed subsets, the two punctured lines.
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<document source="https://tamnd.github.io/math2/node/5FM1TC4FG4.md" tag="5FM1TC4FG4" kind="definition">
# Spectral density estimation

## Definition

Let $(X_t)_{t \in \Z}$ be a real weakly [[stationary process|DT0QXJE705]] with autocovariance $\gamma$ satisfying $\sum_{h} \abs{\gamma(h)} < \infty$.
Its *spectral density* is
$$f(\omega) = \frac{1}{2\pi} \sum_{h \in \Z} \gamma(h) e^{-i h \omega}, \qquad \omega \in [-\pi, \pi],$$
so that $\gamma(h) = \int_{-\pi}^{\pi} e^{i h \omega} f(\omega) \, d\omega$.
*Spectral density estimation* is the construction of estimators $\hat f_n(\omega)$ of $f(\omega)$ from observations $X_1, \dots, X_n$.
Such an estimator is a function of the data that should be close to $f$ for large $n$.

## Remarks

Nonparametric estimators smooth the [[periodogram|3ZYM0HXA3E]] $I_n$.
Let $\hat\gamma$ be the sample autocovariance, defined with the [[autocorrelation function|YCZHYY8VE5]].
Let $w$ be a weight function with $w(0) = 1$, and let $r = r_n$ be a truncation point.
A lag-window estimator has the form
$$\hat f_n(\omega) = \frac{1}{2\pi} \sum_{\abs{h} \le r} w(h / r) \, \hat\gamma(h) e^{-i h \omega}.$$
Under standard conditions, such as $r_n \to \infty$ and $r_n / n \to 0$, it is consistent.
Averaging periodograms of several segments of the data, as in the methods of Bartlett and Welch, is another common choice.

Parametric estimators fit an [[ARMA model|B1HNX0YT0Z]] and use its spectral density $\frac{\sigma^2}{2\pi} \abs{\theta(e^{-i\omega})}^2 / \abs{\phi(e^{-i\omega})}^2$.

Example: for a white noise with variance $\sigma^2$, the spectral density is the constant $\sigma^2 / (2\pi)$, and the averaged estimate $\frac{1}{2\pi} \cdot \frac{1}{m} \sum_j I_n(\omega_j)$ over $m$ Fourier frequencies near $\omega$ estimates it.

The rescaled periodogram $I_n(\omega) / (2\pi)$ is the simplest estimator, but it is not consistent.
Under standard conditions, for $0 < \abs{\omega} < \pi$, it is asymptotically unbiased while its variance tends to $f(\omega)^2$ rather than to $0$.
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# Convolution integral equation

## Definition

Let $k \in L^1(\R)$, let $f \colon \R \to \C$ be a given function, and let $\lambda \in \C$.
A *convolution integral equation of the second kind* on $\R$ is the equation
$$\varphi(x) - \lambda \int_{-\infty}^{\infty} k(x - t) \, \varphi(t) \, dt = f(x), \quad x \in \R,$$
for an unknown function $\varphi$.
A *convolution integral equation of the first kind* is the equation
$$\int_{-\infty}^{\infty} k(x - t) \, \varphi(t) \, dt = f(x), \quad x \in \R.$$
In both cases the kernel $(x, t) \mapsto k(x - t)$ is a [[difference kernel|B5AB451TB7]], and the integral is the [[convolution|2Z29JN3VES]] $(k * \varphi)(x)$.
The same names are used on $\R^n$, and on an interval such as $[0, \infty)$ when the integral runs only over that interval.

## Remarks

Example: take $k(s) = e^{-\abs{s}}$, $\lambda = \frac{1}{4}$ and $f(x) = \frac{1}{2} e^{-\abs{x}}$.
The Fourier transform of $e^{-a\abs{s}}$ is $\frac{2a}{a^2 + \xi^2}$ for $a > 0$, so the transformed equation reads $\hat\varphi(\xi) \left(1 - \frac{1}{2(1 + \xi^2)}\right) = \frac{1}{1 + \xi^2}$.
Hence $\hat\varphi(\xi) = \frac{2}{1 + 2\xi^2} = \frac{1}{1/2 + \xi^2}$, and $\varphi(x) = \frac{1}{\sqrt{2}} e^{-\abs{x}/\sqrt{2}}$.

Non-example: the equation $\varphi(x) - \int_{-\infty}^{\infty} e^{-x^2 - t^2} \varphi(t) \, dt = f(x)$ is a linear integral equation, but its kernel $e^{-x^2 - t^2}$ is not a function of $x - t$ alone, so it is not of convolution type.

The equation on the half-line $\varphi(x) - \int_0^\infty k(x - t) \varphi(t) \, dt = f(x)$, $x > 0$, is the [[Wiener–Hopf equation|QYY4NMDY5H]].
The Volterra form $\varphi(x) - \int_0^x k(x - t) \varphi(t) \, dt = f(x)$ is the [[Volterra equation of convolution type|SNFYTC1TTA]].
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# Hilbert polynomial

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $S = k[x_1, \dots, x_r]$ be the [[polynomial ring|07GNX8GCYP]] with each $x_i$ of degree $1$.
Let $M = \bigoplus_{n \in \Z} M_n$ be a [[finitely generated|05QY1VEBYD]] [[graded|8KB4SQGC0C]] $S$-module.
The Hilbert function of $M$ is $H_M(n) = \dim_k M_n$.
The *Hilbert polynomial* of $M$ is the unique polynomial $P_M \in \Q[t]$ such that $H_M(n) = P_M(n)$ for all sufficiently large integers $n$.

## Remarks

Such a polynomial exists by the [[Hilbert–Serre theorem|HP56DJECMT]], and it is unique because two polynomials that agree at infinitely many integers are equal.
Example: for $M = k[x, y, z]/(f)$ with $f$ homogeneous of degree $d$, the exact sequence $0 \to S(-d) \to S \to M \to 0$ gives $H_M(n) = \binom{n+2}{2} - \binom{n-d+2}{2}$ for $n \ge d$, so $P_M(t) = dt - d(d-3)/2$.
Non-example: the Hilbert function itself need not be a polynomial on all of $\N$; for $M = k = S/(x_1, \dots, x_r)$ one has $H_M(0) = 1$ and $H_M(n) = 0$ for $n \ge 1$, so $P_M = 0$ but $H_M(0) \neq P_M(0)$.
For $M = S$, the Hilbert polynomial is $\binom{t + r - 1}{r - 1}$, of degree $r - 1$.
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# Z-transform

## Summary

The Z-transform turns a sequence $(x_n)$ of complex numbers into the function $X(z) = \sum_n x_n z^{-n}$ of a complex variable $z$, defined where the series converges.
It is the discrete analogue of the [[Laplace transform|0V49JEZW06]].
It turns shifts of sequences into multiplication by powers of $z$ and convolution into products.
So it is the main tool for linear difference equations and for discrete-time signals and control systems.

## Explanations

### L1 human

A sequence is a list of numbers $x_0, x_1, x_2, \ldots$, such as the samples of a signal.
The Z-transform packs the whole list into one function, using $z^{-n}$ as a label for position $n$.
Delaying the list by one step then just multiplies the function by $z^{-1}$.
For the constant list $1, 1, 1, \ldots$ the transform is $\sum_{n \ge 0} z^{-n} = z/(z - 1)$ for $\abs{z} > 1$.
</document>

<document source="https://tamnd.github.io/math2/node/5FZDQQHQKM.md" tag="5FZDQQHQKM" kind="concept">
# Covariant derivative

## Summary

A covariant derivative differentiates a vector field in the direction of a tangent vector, and the result is again a tangent vector.
It applies more generally to sections of a vector bundle and to tensor fields.
It is the operator $\nabla_X$ that a [[connection|5ZYRRVE8BW]] provides.
Unlike partial derivatives of components, it does not depend on the choice of coordinates.
It also differentiates vector fields along curves, which leads to parallel transport and geodesics.

## Explanations

### L1 human

The partial derivatives of the components of a vector field change when you change coordinates in a non-linear way.
The covariant derivative corrects them by a term that accounts for how the coordinate axes themselves bend.
For a car driving on a sphere, the covariant derivative of its velocity is the part of the acceleration that is tangent to the sphere.
A curve with zero covariant acceleration is as straight as possible on the surface; on a sphere these are the great circles.
</document>

<document source="https://tamnd.github.io/math2/node/5GCANV7337.md" tag="5GCANV7337" kind="statement">
# Integral extensions of fields

## Statement

Let $A \subseteq B$ be [[integral domains|0QVD7WBFGN]] with $B$ [[integral over|B7X0KFY1S1]] $A$.
Then $A$ is a [[field|04C5ZBG1WT]] if and only if $B$ is a field.

## Remarks

Suppose $A$ is a field and let $b \in B$ be nonzero.
Take a monic relation $b^n + a_{n-1} b^{n-1} + \cdots + a_0 = 0$ of least degree; since $B$ is a domain, $a_0 \ne 0$.
Then $b^{-1} = -a_0^{-1}(b^{n-1} + a_{n-1} b^{n-2} + \cdots + a_1)$ lies in $B$.
Conversely, suppose $B$ is a field and let $a \in A$ be nonzero.
Then $a^{-1} \in B$ is integral over $A$, say $a^{-m} + c_{m-1} a^{-m+1} + \cdots + c_0 = 0$, and multiplying by $a^{m-1}$ gives $a^{-1} = -(c_{m-1} + c_{m-2} a + \cdots + c_0 a^{m-1}) \in A$.

Example: $\Q \subseteq \Q(\sqrt{2})$ is integral and both are fields; $\Z \subseteq \Z[i]$ is integral and neither is a field.

The hypothesis that $B$ is a domain is needed for "$A$ field implies $B$ field": the ring $k[x]/(x^2)$ is integral over the field $k$, but it is not a field.

## Explanations

### L1 human

In an integral extension of domains, the small ring is a field exactly when the big ring is.
So integral extensions cannot turn a non-field into a field, nor the reverse.

### L2 human

This is the step that links maximal ideals across an integral extension: for a prime $\mathfrak{q}$ of $B$, apply it to $A/(\mathfrak{q} \cap A) \subseteq B/\mathfrak{q}$.
It gives that $\mathfrak{q}$ is maximal if and only if $\mathfrak{q} \cap A$ is maximal, and it is used in the proofs of [[lying over|YZ97GKYY3P]], [[incomparability|Q2XBHJ96GV]] and [[Zariski's lemma|YBJ2VQHFDQ]].
</document>

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# Hypergeometric series

## Definition

Let $a, b, c \in \C$ with $c \notin \set{0, -1, -2, \ldots}$.
For $x \in \C$ and an integer $n \ge 0$, let $(x)_n = x(x+1)\cdots(x+n-1)$ be the [[Pochhammer symbol|YZHM3HSM0F]], with $(x)_0 = 1$.
The *hypergeometric series* is the [[power series|0MZB06ST2B]]
$${}_2F_1(a, b; c; z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \, \frac{z^n}{n!}.$$

## Remarks

If $a$ or $b$ is a non-positive integer $-m$, then $(a)_n$ or $(b)_n$ vanishes for $n > m$ and the series is a polynomial of degree at most $m$.
Otherwise the ratio test gives [[radius of convergence|0N288P2TB5]] $1$.
On the circle $\abs{z} = 1$ the series converges absolutely when $\operatorname{Re}(c - a - b) > 0$.

Example: for $a = b = 1$, $c = 2$, the coefficient is $\frac{n! \, n!}{(n+1)! \, n!} = \frac{1}{n+1}$, so ${}_2F_1(1, 1; 2; z) = \sum_n \frac{z^n}{n+1} = -\frac{\log(1 - z)}{z}$ for $0 < \abs{z} < 1$.
Also ${}_2F_1(a, b; b; z) = \sum_n \frac{(a)_n}{n!} z^n = (1 - z)^{-a}$ by the binomial series.

Non-example: $c = 0$ is excluded, because $(0)_n = 0$ for every $n \ge 1$ and the coefficients are undefined.
For $c = -m$ a negative integer the series is still meaningful when $a = -k$ with $0 \le k \le m$ and the sum is cut at $n = k$, but this needs a separate convention.

The notation $F(a, b; c; z)$ is also common.
</document>

<document source="https://tamnd.github.io/math2/node/5GDFYDF2R2.md" tag="5GDFYDF2R2" kind="statement">
# Cartan–Iwasawa–Malcev theorem

## Statement

Let $G$ be a connected [[Lie group|7JBRGCVJRB]].
Then:

1. $G$ has a [[maximal compact subgroup|T2EB8R9HEV]], and each compact subgroup of $G$ is contained in a maximal compact subgroup;
2. any two maximal compact subgroups of $G$ are conjugate in $G$;
3. each maximal compact subgroup $K$ is connected, and $G$ is homeomorphic to $K \times \R^n$ for some $n \ge 0$.

## Remarks

For semisimple groups with finite center, the result follows from the [[Cartan decomposition|A5JB9JQKR0]] $G = K \exp(\mathfrak{p})$ and from Cartan's fixed point theorem.
That theorem says that a compact group acting by isometries on the nonpositively curved space $G/K$ has a fixed point.
So each compact subgroup lies in the stabilizer of a point, a conjugate of $K$.
The general case is due to Malcev and Iwasawa.

Example: $G = \mathrm{GL}_n^+(\R)$, the invertible real matrices of positive determinant.
A maximal compact subgroup is $\mathrm{SO}(n)$, and the polar decomposition gives a homeomorphism of $G$ with $\mathrm{SO}(n) \times \R^{n(n+1)/2}$, the second factor being the symmetric matrices.

The conclusions extend to Lie groups with finitely many connected components, except that $K$ then has one component for each component of $G$.
For groups with infinitely many components, maximal compact subgroups may fail to exist.

## Explanations

### L1 human

Every connected Lie group contains a largest bounded subgroup, unique up to conjugation.
The whole group is this bounded subgroup times a flat space with no holes.
So all the topology of the group, for example its loops and its holes, lives in the compact part.

### L2 human

The theorem reduces the homotopy type of a connected Lie group to that of a compact Lie group.
For example $\mathrm{SL}_2(\R)$ has the homotopy type of a circle, and $\mathrm{SL}_n(\C)$ that of $\mathrm{SU}(n)$, which is simply connected.
It also says that the classifying space of $G$ is that of $K$, which is why characteristic classes of $G$-bundles only depend on $K$.
Uniqueness up to conjugacy makes $K$ an invariant of $G$, used for example to define the symmetric space $G/K$.
</document>

<document source="https://tamnd.github.io/math2/node/5GN0RYRVBA.md" tag="5GN0RYRVBA" kind="concept">
# Totally positive matrix

## Summary

A totally positive matrix is a real matrix all of whose [[minors|0BT9BYWVRY]] are positive.
When the minors are only required to be nonnegative, the matrix is called totally nonnegative.
Totally positive matrices have strong spectral properties: a square one has only positive, distinct eigenvalues.
They occur in approximation theory, in probability, in combinatorics and in the study of oscillating mechanical systems.

## Explanations

### L1 human

Every entry of a matrix is a $1 \times 1$ minor, and determinants of larger square pieces are larger minors.
A totally positive matrix is one where every such determinant, of every size and position, is positive.
For example, $\begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$ has positive entries and determinant $1$, so it is totally positive.
The condition is much stronger than having positive entries: $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$ has positive entries but determinant $-3$.
</document>

<document source="https://tamnd.github.io/math2/node/5GNSQRHSSC.md" tag="5GNSQRHSSC" kind="example">
# Involution as solution of Babbage's equation

## Example

Babbage's functional equation asks for maps $f \colon X \to X$ with $f(f(x)) = x$ for all $x \in X$.
Its solutions are exactly the [[involutions|GP56NBN9RY]] of $X$, and also exactly the [[iterative roots|27C6J3YNZY]] of order $2$ of the identity.

Take $X = \R \setminus \set{-1}$ and $f(x) = \frac{1 - x}{1 + x}$.
First, $f$ maps $X$ into $X$: the equation $\frac{1 - x}{1 + x} = -1$ would give $1 - x = -1 - x$, which is impossible.
Second, for $x \in X$,
$$f(f(x)) = \frac{1 - \frac{1-x}{1+x}}{1 + \frac{1-x}{1+x}} = \frac{(1 + x) - (1 - x)}{(1 + x) + (1 - x)} = \frac{2x}{2} = x.$$
So $f$ is an involution and solves Babbage's equation.
Other solutions on suitable sets are $x \mapsto c - x$, $x \mapsto 1/x$ and the identity.

## Remarks

On an interval $I \subseteq \R$, every [[continuous|0X3WGX95FE]] solution other than the identity is strictly decreasing.
Indeed, a continuous involution is injective, hence strictly monotone.
If it is increasing and $f(x) > x$ for some $x$, then $x = f(f(x)) > f(x) > x$, a contradiction; the case $f(x) < x$ is the same.
So an increasing continuous solution is the identity.
The more general equation $f^n(x) = x$ asks for the iterative roots of order $n$ of the identity; for example the rotation $z \mapsto e^{2\pi i/n} z$ of $\C$ is one.

## Explanations

### L1 human

Babbage's equation asks for a rule that, applied twice, brings every point back to where it started.
Mirror reflections, the change of sign $x \mapsto -x$ and the reciprocal $x \mapsto 1/x$ are such rules.
The map $x \mapsto \frac{1 - x}{1 + x}$ is a less obvious one, and a short computation checks it.
</document>

<document source="https://tamnd.github.io/math2/node/5GVFZWMWC8.md" tag="5GVFZWMWC8" kind="concept">
# Algebraic K-theory

## Summary

Algebraic K-theory attaches to a [[ring|0B1261B5ED]] $R$ a sequence of abelian groups $K_n(R)$.
The group $K_0(R)$ is the [[Grothendieck group|5FDMY7PDYY]] of finitely generated [[projective modules|BFFA03GE3V]], and $K_1(R)$ is the abelianization of the infinite general linear group.
Daniel Quillen defined the higher groups by homotopy theory, for instance through the plus construction.
The groups encode deep arithmetic information, such as class groups, units and special values of zeta functions.

## Explanations

### L1 human

Algebraic K-theory measures a ring through its modules and its invertible matrices.
The group $K_0$ counts projective modules, with the rule that a direct sum becomes a sum.
For a field, projective modules are vector spaces, and $K_0$ just records the dimension, so it is $\Z$.
The group $K_1$ remembers invertible matrices up to elementary row operations; for a field this is the determinant, so $K_1$ is the group of nonzero elements.
The higher groups are harder to compute, and they connect algebra with topology and number theory.
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<document source="https://tamnd.github.io/math2/node/5GW2T2AM1A.md" tag="5GW2T2AM1A" kind="concept">
# Lattice ideal

## Summary

An ideal of a [[lattice|WH4SFAACAS]] is a nonempty subset that is closed downward and closed under joins.
The typical example is the principal ideal $\downarrow a = \set{x : x \le a}$ of all elements below a fixed element $a$.
Lattice ideals are the order-theoretic analogue of [[ideals|096Z9MP2ME]] of a ring, and the dual notion is the [[filter|Z15FXM9WM1]].
Prime ideals of lattices are the key to representation theorems for distributive lattices and Boolean algebras.

## Explanations

### L1 human

An ideal of a lattice is a collection of "small" elements.
Anything below a small element is small, and the join of two small elements is still small.
In the subsets of the natural numbers, the finite subsets form an ideal: a subset of a finite set is finite, and the union of two finite sets is finite.
In the positive integers ordered by divisibility, the divisors of $12$ form an ideal.
</document>

<document source="https://tamnd.github.io/math2/node/5GX76EM5HS.md" tag="5GX76EM5HS" kind="statement">
# Strichartz estimates in Lorentz space

## Statement

Let $d \ge 1$.
For $f \in L^2(\R^d)$, let $e^{it\Delta} f$ be the solution at time $t$ of the free [[Schrödinger equation|EPZR6H2WJN]] with data $f$.
It is given by the [[Fourier transform|DPVM34ADZV]] as $\widehat{e^{it\Delta} f}(\xi) = e^{-it\abs{\xi}^2} \hat f(\xi)$.

Call a pair $(q, r)$ admissible if $2 \le q \le \infty$, $2 \le r < \infty$, $(q, r, d) \ne (2, \infty, 2)$ and
$$\frac{2}{q} + \frac{d}{r} = \frac{d}{2}.$$
For $1 \le s \le \infty$, let $s'$ be the conjugate exponent, with $\frac{1}{s} + \frac{1}{s'} = 1$.
For an interval $I \subseteq \R$, let $\norm{u}_{L^q_t L^{r,2}_x(I \times \R^d)}$ be the $L^q$ norm over $t \in I$ of the function $t \mapsto \norm{u(t)}_{L^{r,2}(\R^d)}$.
Here $L^{r,2}$ is the [[Lorentz space|B7X9M15Q92]] with Lebesgue measure, and $L^q$ is a [[Lebesgue space|SASP14MA1Z]].

Then the following two estimates hold, where the implied constants depend only on $d$ and the exponents.

1. Let $(q, r)$ be admissible. Then for all $f \in L^2(\R^d)$,
$$\norm{e^{it\Delta} f}_{L^q_t L^{r,2}_x(\R \times \R^d)} \lesssim \norm{f}_{L^2(\R^d)}.$$

2. Let $(q_1, r_1)$ and $(q_2, r_2)$ be admissible. Let $I \subseteq \R$ be an interval and let $t_0 \in I$. Then for all $F \in L^{q_2'}_t L^{r_2',2}_x(I \times \R^d)$,
$$\Big\| \int_{t_0}^t e^{i(t-\tau)\Delta} F(\tau) \, d\tau \Big\|_{L^{q_1}_t L^{r_1,2}_x(I \times \R^d)} \lesssim \norm{F}_{L^{q_2'}_t L^{r_2',2}_x(I \times \R^d)}.$$

## Remarks

The classical [[Strichartz estimate|4PDKJ8N1S9]] is the same bound with $L^r$ in place of $L^{r,2}$.
Since $L^{r,2}$ is contained in $L^{r,r} = L^r$ for $r \ge 2$, the first estimate is stronger than the classical one.
The paper allows $r = \infty$ in its admissible pairs. This node requires $r < \infty$, since the linked Lorentz space needs a finite first index.
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<document source="https://tamnd.github.io/math2/node/5H16H7FY5D.md" tag="5H16H7FY5D" kind="concept">
# Extensive category

## Summary

An extensive category is a category with finite [[coproducts|4FJ0S71P58]] that behave like disjoint unions of sets.
Coproducts are disjoint and stable under pullback.
So an object over $A + B$ splits uniquely into a part over $A$ and a part over $B$.
Sets, topological spaces, small categories and every topos are extensive, while categories of modules are not.
Extensivity is the categorical form of the idea that a space can be cut into disjoint pieces.

## Explanations

### L1 human

If a set $X$ maps into a disjoint union $A \sqcup B$, then $X$ breaks into the part that lands in $A$ and the part that lands in $B$.
These two parts do not overlap, and their union is $X$.
An extensive category is one where every object over a sum breaks up in this way.
In vector spaces this fails: the diagonal line in $\R \oplus \R$ meets both axes only at zero, yet it is not the sum of those two meetings.
</document>

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# Minkowski's second theorem

## Statement

Let $n \ge 1$ and let $\Lambda \subset \R^n$ be a [[lattice|WN9V7WSJKF]] with covolume $\det \Lambda$.
Let $K \subset \R^n$ be a [[convex|02AS45T3H9]] [[compact|0R6EWFKRCF]] set with nonempty interior that is symmetric about the origin, that is, $-x \in K$ for all $x \in K$.
For $k = 1, \dots, n$, the $k$-th successive minimum $\lambda_k$ is the infimum of the real numbers $\lambda > 0$ such that $\lambda K$ contains $k$ [[linearly independent|0CRMGPAH83]] vectors of $\Lambda$.
Then $0 < \lambda_1 \le \lambda_2 \le \dots \le \lambda_n < \infty$ and
$$\frac{2^n}{n!} \det \Lambda \le \lambda_1 \lambda_2 \cdots \lambda_n \operatorname{vol}(K) \le 2^n \det \Lambda.$$

## Remarks

The lower bound is elementary: $K$ contains the cross-polytope spanned by $\pm v_i/\lambda_i$ for suitable independent lattice vectors $v_i$, and this cross-polytope has volume at least $2^n \det\Lambda / (n! \, \lambda_1 \cdots \lambda_n)$.
The upper bound is the deep part; Minkowski's proof compares $K$ with a body built from the successive minima and uses a packing argument.
Since $\lambda_1^n \le \lambda_1 \cdots \lambda_n$, the upper bound implies $\lambda_1^n \operatorname{vol}(K) \le 2^n \det \Lambda$; so if $\operatorname{vol}(K) \ge 2^n \det\Lambda$, then $\lambda_1 \le 1$ and $K$ contains a nonzero lattice point, which is [[Minkowski's theorem|0K387S2PKQ]].
Both bounds are sharp for $\Lambda = \Z^n$: for the cube $K = [-1, 1]^n$ all $\lambda_k = 1$ and $\operatorname{vol}(K) = 2^n$, and for the cross-polytope $\set{x : \abs{x_1} + \dots + \abs{x_n} \le 1}$ all $\lambda_k = 1$ and $\operatorname{vol}(K) = 2^n/n!$.

## Explanations

### L1 human

Blow up a symmetric convex body around the origin until it contains one lattice direction, then two independent ones, and so on.
The scaling factors needed are the successive minima.
The theorem says that their product, times the volume of the body, is about the volume of a lattice cell, up to factors that depend only on the dimension.

### L2 human

Minkowski's first theorem controls only the first minimum; the second theorem controls all of them at once.
This is what one needs to find a whole basis of short vectors.
It is used, for instance, in the reduction theory of quadratic forms and in bounds for units and ideals in number fields.
It is also the starting point of transference theorems that relate a lattice to its dual.
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# Wiener algebra

## Summary

The Wiener algebra $A(\mathbb{T})$ is the set of continuous periodic functions with an [[absolutely convergent Fourier series|HHBGB5R5S2]].
Its norm is the sum of the absolute values of the Fourier coefficients.
With pointwise multiplication it is a commutative [[Banach algebra|9KYWN94QDN]], isometrically isomorphic to the sequence algebra $\ell^1(\Z)$ with convolution.
It is the setting of [[Wiener's lemma|NC768TWZ3D]] and a basic example in the Gelfand theory of commutative Banach algebras.

## Explanations

### L1 human

Take all periodic functions whose Fourier coefficients have a finite sum of absolute values.
Adding or multiplying two such functions gives another one, so they form an algebra.
Measuring a function by the total size of its coefficients makes this algebra complete.
The functions $\cos x$ and $\sum_{n \in \Z} 2^{-\abs{n}} e^{inx}$ belong to it, while the square wave does not, because it is not even continuous.
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# Lasker–Noether theorem

## Statement

Let $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $I \ne R$ be an [[ideal|0MYSA6MYNA]] of $R$.

1. $I$ has a minimal [[primary decomposition|M2VPVJH33K]] $I = Q_1 \cap \cdots \cap Q_n$.
2. In every minimal primary decomposition of $I$, the set of prime ideals $\set{\sqrt{Q_1}, \dots, \sqrt{Q_n}}$ is the same: it is the set of [[associated primes|1RJJ33KKSS]] of $R/I$.
3. If $\sqrt{Q_i}$ is minimal in this set, then the component $Q_i$ is the same in every minimal primary decomposition of $I$.

## Remarks

For existence, call an ideal irreducible if it is not the intersection of two strictly larger ideals.
By the [[Noetherian|0670T59HGE]] property, every proper ideal is a finite intersection of [[irreducible ideals|7PDWB550C9]], and in a Noetherian ring every irreducible ideal is primary.
Removing redundant terms and joining terms with the same radical gives a minimal decomposition.
For example, in $k[x, y]$ the ideal $(x^2, xy)$ has the minimal decompositions $(x) \cap (x^2, y)$ and $(x) \cap (x^2, xy, y^2)$.
In both, the primes are $(x)$ and $(x, y)$, and the component $(x)$ at the minimal prime is the same, while the [[embedded|RVKPM94MDF]] component changes.
Without the Noetherian hypothesis, existence can fail.
In the product ring $R = \prod_{i \in \N} \F_2$, every prime ideal is maximal and there are infinitely many, such as the kernels of the projections.
If $0 = Q_1 \cap \cdots \cap Q_n$ with $Q_i$ primary, then every prime contains some $\sqrt{Q_i}$, so $R$ would have at most $n$ prime ideals.

## Explanations

### L1 human

In a Noetherian ring every ideal can be written as an intersection of finitely many primary ideals.
The pieces are not always unique, but the primes behind them are.
This extends the unique factorisation of integers into prime powers.

### L2 human

The primes in the theorem are the associated primes, and the minimal ones among them are the minimal primes over $I$.
For an ideal of a polynomial ring, they describe the irreducible components of the zero set and the embedded points.
The same theorem holds for submodules of finitely generated modules over a Noetherian ring.
The uniqueness of the isolated components comes from localization: $Q_i$ is the preimage of $I R_P$ in $R$, for $P = \sqrt{Q_i}$ minimal.
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<document source="https://tamnd.github.io/math2/node/5HDG31YEPW.md" tag="5HDG31YEPW" kind="definition">
# Riemannian manifold

## Definition

A *Riemannian manifold* is a pair $(M, g)$ where $M$ is a [[smooth manifold|F45VQ47GPV]] and $g$ is a [[Riemannian metric|Q61VFHCH2J]] on $M$.
An isometry between Riemannian manifolds $(M, g)$ and $(N, h)$ is a diffeomorphism $\varphi : M \to N$ with $\varphi^* h = g$.

## Remarks

Example: the unit sphere $S^n \subset \R^{n+1}$ with $g_p(u, v) = u \cdot v$ for $u, v \in T_pS^n \subset \R^{n+1}$ is a Riemannian manifold.
Indeed $g_p$ is the restriction of the dot product, so it is symmetric and positive definite, and it depends smoothly on $p$.
More generally, every immersed submanifold of $\R^N$ is a Riemannian manifold with the induced metric, and by the Nash embedding theorem every Riemannian manifold arises this way.
Non-example: $\R^2$ with $g = dx^2 - dy^2$ is not a Riemannian manifold, because $g(\partial_y, \partial_y) = -1 < 0$; it is a pseudo-Riemannian manifold.
A connected Riemannian manifold is a metric space, with distance the infimum of the lengths of piecewise smooth curves joining two points, and this metric induces the manifold topology.
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# Hyperconnected space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *hyperconnected*, or *irreducible*, if the following two conditions hold.

1. $X$ is nonempty.
2. For all nonempty [[open sets|02BH3MCBQB]] $U, V \subseteq X$, the intersection $U \cap V$ is nonempty.

## Remarks

Equivalent forms: every nonempty open subset of $X$ is [[dense|0BSJSX7ME2]] in $X$.
Also equivalent: $X$ is nonempty and is not the union of two closed subsets different from $X$.

Example: an infinite set $X$ with the [[cofinite topology|0B906ZBTMZ]].
Nonempty open sets have finite complements, so the union of the complements of $U$ and $V$ is finite and cannot be all of $X$.

Non-example: a [[Hausdorff space|0S1FJ8VEX9]] with at least two points fails clause 2.
Two distinct points have disjoint open neighbourhoods.
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# Central simple algebra

## Summary

A central simple algebra over a field $K$ is a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $K$ that is a [[simple ring|62M0NC3E2T]].
Its [[center|MVT64B3B6H]] must be exactly $K$.
By Wedderburn's theorem each one is a matrix algebra $M_n(D)$ over a central division algebra $D$.
They are stable under tensor product over $K$, and their classes up to matrix size form the [[Brauer group|BEW48TA9WZ]] of $K$.
They are central in algebraic number theory, in Galois cohomology and in the arithmetic of quadratic forms.

## Explanations

### L1 human

The basic example is the algebra of $n \times n$ matrices over a field $K$.
It is simple, since its only two-sided ideals are $0$ and itself, and its center consists of the scalar matrices, which form a copy of $K$.
Central simple algebras are the algebras that look like matrix algebras after the field is enlarged.
For example, Hamilton's [[quaternions|0GE4DVQKR8]] are central simple over $\R$, and over $\C$ they become the $2 \times 2$ complex matrices.
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<document source="https://tamnd.github.io/math2/node/5HYB6DDZH8.md" tag="5HYB6DDZH8" kind="concept">
# Weierstrass transform

## Summary

The Weierstrass transform of a function $f$ on the real line is its average against a Gaussian weight of variance $2$ centred at each point.
It is the solution at time $t = 1$ of the [[heat equation|VP2HN86VAG]] $u_t = u_{xx}$ with initial value $f$.
It smooths every function it is applied to, and on the Fourier side it multiplies by a Gaussian.
Weierstrass used this smoothing in his proof of the approximation theorem that bears his name.

## Explanations

### L1 human

Imagine $f$ as a temperature profile along a long thin rod.
Let the heat flow for one unit of time: sharp peaks spread out and the profile becomes smooth.
The new profile is the Weierstrass transform of $f$.
Each new value is a weighted mean of old values, with the largest weights on nearby points.
For example, the transform of $x^2$ is $x^2 + 2$, because averaging adds the variance $2$ of the Gaussian weight.
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<document source="https://tamnd.github.io/math2/node/5J0D97M548.md" tag="5J0D97M548" kind="statement">
# Cartan's coherence theorem for ideal sheaves

## Statement

Let $\Omega \subseteq \C^n$ be open and let $A \subseteq \Omega$ be an [[analytic subset|GB7M582M5Y]].
For each open $U \subseteq \Omega$ let $\mathcal{I}_A(U)$ be the set of [[holomorphic functions|507C34W0ZT]] on $U$ that vanish at every point of $A \cap U$.
Then the [[ideal sheaf|N3PGYJJQ7W]] $\mathcal{I}_A$ is a [[coherent analytic sheaf|DSJET02WPR]] on $\Omega$.

## Remarks

The question is local, and one may assume that the germ of $A$ at a point is irreducible.
The [[local parametrization theorem|K18PKZY5RS]] presents $A$ as a branched covering of a domain in $\C^d$, and from this one writes down explicit holomorphic functions that generate $\mathcal{I}_A$ near the point.
Coherence then follows from the [[Oka coherence theorem|MSX3VKM9Z5]], since $\mathcal{I}_A$ is a subsheaf of finite type of the coherent sheaf $\mathcal{O}_\Omega$.

Example: for $A = \set{z_1 = z_2 = 0} \subseteq \C^3$, the sheaf $\mathcal{I}_A$ is generated by $z_1$ and $z_2$ at every point.
For the cusp $A = \set{z_1^2 = z_2^3}$, it is generated by $z_1^2 - z_2^3$.
In contrast, the sheaf generated by $z_1^2$ has the same zero set $\set{z_1 = 0}$ but is strictly smaller than the ideal sheaf of that set.

## Explanations

### L1 human

An analytic set is the common zero set of some holomorphic equations.
The theorem says that near each point, finitely many equations produce every holomorphic function that vanishes on the set.
Moreover, the same equations keep doing so at all nearby points.

### L2 human

The theorem gives every analytic set $A$ a canonical structure of [[reduced complex space|831N2Q4GCB]], with structure sheaf $\mathcal{O}_\Omega / \mathcal{I}_A$ restricted to $A$.
By the Rückert Nullstellensatz, the nilradical of a local model $\mathcal{O}_D / \mathcal{I}$ is $\mathcal{I}_Z / \mathcal{I}$, where $Z$ is the zero set of $\mathcal{I}$.
So the theorem shows that the nilradical of the structure sheaf of any complex space is coherent, and every complex space has a reduction.
Together with Oka's theorem it is one of the two pillars of the theory of coherent analytic sheaves.
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# Macdonald constant term conjecture

## Statement

Let $R$ be a reduced finite root system of rank $l$ in a Euclidean space $V$, with Weyl group $W$.
Let $d_1, \dots, d_l$ be the degrees of the basic polynomial invariants of $W$.
For $\alpha \in R$, let $e^\alpha$ be the corresponding element of the group algebra $\Q[P]$ of the weight lattice $P$ of $R$.
Let $k \ge 0$ be an integer.
Then the constant term (the coefficient of $e^0$) of
$$\prod_{\alpha \in R} \left( 1 - e^\alpha \right)^k$$
equals
$$\prod_{i=1}^{l} \binom{k\, d_i}{k}.$$

## Remarks

Macdonald stated this conjecture in 1982, together with a $q$-version: the constant term of $\prod_{\alpha \in R_+} (e^\alpha; q)_k\, (q e^{-\alpha}; q)_k$ is $\prod_{i=1}^{l} \binom{k d_i}{k}_q$, with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]] and the [[Gaussian binomial coefficients|YS09Z1YNEK]].
Cherednik proved both versions for all reduced root systems in 1995, with double affine Hecke algebras.
Example: $R = A_1 = \set{\alpha, -\alpha}$, with $l = 1$ and $d_1 = 2$.
Writing $x = e^\alpha$, the product is $(1 - x)^k (1 - x^{-1})^k = (-1)^k x^{-k} (1 - x)^{2k}$, whose constant term is $(-1)^k (-1)^k \binom{2k}{k} = \binom{2k}{k}$, as predicted.
For $R = A_{n-1}$ the degrees are $2, 3, \dots, n$, and the formula is the equal-parameter case of the [[Dyson conjecture|NDYA85381W]], since $\prod_{j=1}^{n} \binom{jk}{k} = \frac{(nk)!}{(k!)^n}$.

## Explanations

### L1 human

Every root system, such as the one behind the symmetries of a square or a cube, comes with a reflection group and a list of numbers called degrees.
Take the product of the factors $1 - e^\alpha$ over all roots, raise it to the power $k$, and expand.
The constant term is a product of binomial coefficients built from the degrees.

### L2 human

The constant term of a Laurent polynomial is its average over a torus.
So the statement evaluates a Selberg-type integral over the maximal torus of a [[compact Lie group|9TFC6ADC1C]] whose [[Weyl group|EZGTEK3HQN]] is $W$.
The degrees enter because $\prod_i d_i = \abs{W}$ and $\sum_i (d_i - 1)$ is the number of positive roots.
The analogous Gaussian integrals are the Macdonald–Mehta integrals, of which the [[Mehta integral|4N53T1BZ9S]] is the type $A$ case.
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# Nonlinear Schrödinger equation

## Definition

Let $d \ge 1$ be an integer, let $p > 1$ be a real number, and let $\mu \in \set{1, -1}$.
The *nonlinear Schrödinger equation* with power nonlinearity is the [[partial differential equation|GC5BDZ6QBC]]
$$i \partial_t u + \Delta u = \mu \abs{u}^{p-1} u$$
for an unknown function $u \colon \R \times \R^d \to \C$, $(t, x) \mapsto u(t, x)$.
Here $\Delta$ is the [[Laplacian|0E0B6RA8K0]] in the variable $x$.
The equation is *defocusing* if $\mu = 1$ and *focusing* if $\mu = -1$.
The case $p = 3$ is the *cubic* nonlinear Schrödinger equation.

## Remarks

Example: for $A \in \C$ and $k \in \R^d$, the plane wave $u(t,x) = A e^{i(k \cdot x - \omega t)}$ is a solution when $\omega = \abs{k}^2 + \mu \abs{A}^{p-1}$.
Indeed $i \partial_t u = \omega u$, $\Delta u = -\abs{k}^2 u$ and $\abs{u}^{p-1} u = \abs{A}^{p-1} u$.

Non-example: the linear [[Schrödinger equation|EPZR6H2WJN]] $i \partial_t u + \Delta u = 0$ corresponds to $\mu = 0$, which the definition excludes.

Smooth solutions that decay at infinity conserve the mass $\int \abs{u}^2 \, dx$ and the energy
$$E(u) = \int_{\R^d} \left( \tfrac{1}{2} \abs{\nabla u}^2 + \tfrac{\mu}{p+1} \abs{u}^{p+1} \right) dx.$$
Some authors write the equation as $i \partial_t u = -\Delta u + \mu \abs{u}^{p-1} u$, which is the same equation.
</document>

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# Hecke algebra

## Summary

The name Hecke algebra covers several related algebras; the most common is the Iwahori–Hecke algebra of a [[Coxeter group|G6FND7X69J]].
The Iwahori–Hecke algebra is a deformation of the group algebra of the Coxeter group, depending on a parameter $q$.
At $q = 1$ it is the group algebra.
Let $G$ be a finite group of Lie type over a field with $q$ elements.
Iwahori showed that the Hecke algebra is the endomorphism algebra of the permutation module of $G$ on the cosets of a Borel subgroup.
Other Hecke algebras are built from double cosets of a subgroup, as for the [[Hecke operators|S4SV3QZ8B1]] of number theory, or from locally compact groups.

## Explanations

### L1 human

A Coxeter group, such as the [[symmetric group|0515A6D2Y1]], is generated by reflections $s$ with $s^2 = 1$ and some braid relations.
The Hecke algebra keeps the braid relations but replaces $s^2 = 1$ by a rule with a parameter $q$, namely $T_s^2 = (q - 1) T_s + q$.
When $q = 1$ this is again $T_s^2 = 1$, so the Hecke algebra is a one-parameter family of algebras through the group algebra.
For the group with two elements, the Hecke algebra is spanned by $1$ and $T$, with $T^2 = (q-1)T + q$.
These algebras appear in the representation theory of finite and $p$-adic groups, in knot invariants and in the combinatorics of Kazhdan–Lusztig polynomials.
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# Induced representation of a locally compact group

## Definition

Let $G$ be a second countable [[locally compact group|AHV05ZBCYW]], and let $H$ be a closed subgroup of $G$.
Assume that the coset space $G/H$ carries a nonzero $G$-invariant Radon measure $\nu$.
Let $\sigma$ be a [[unitary representation|74BPYJNZTM]] of $H$ on a [[Hilbert space|HM2HHE3J51]] $K$.
Let $\mathcal{H}$ be the space of measurable functions $f \colon G \to K$ such that:

1. $f(gh) = \sigma(h)^{-1} f(g)$ for all $g \in G$ and $h \in H$;
2. $\int_{G/H} \norm{f(g)}^2 \, d\nu(gH) < \infty$.

Functions that agree almost everywhere are identified.
The norm $\norm{f(g)}$ depends only on the [[coset|03DEH43F4H]] $gH$, because $\sigma$ is unitary, so clause 2 makes sense.
The *induced representation* $\operatorname{Ind}_H^G \sigma$ is the representation of $G$ on $\mathcal{H}$ given by $(\lambda(x) f)(g) = f(x^{-1} g)$.

## Remarks

Example: take $H = \set{e}$ and $\sigma$ trivial on $K = \C$.
Clause 1 is empty, $G/H = G$, and $\nu$ is a left [[Haar measure|VY0HXD3GC4]].
So $\operatorname{Ind}_{\set{e}}^G 1$ is the left regular representation of $G$ on $L^2(G)$.

Example: take $G = \R$, $H = \Z$, and $\sigma(n) = e^{2 \pi i \theta n}$ for a fixed $\theta \in \R$.
Clause 1 asks that $f(x + n) = e^{-2 \pi i \theta n} f(x)$, and clause 2 asks that $\int_0^1 \abs{f(x)}^2 \, dx < \infty$.

Non-example: in the second example, the constant function $f = 1$ fails clause 1 when $\theta \notin \Z$.

When $G/H$ has no invariant measure, one uses a quasi-invariant measure instead.
The action then includes the square root of a Radon–Nikodym factor, so that $\lambda(x)$ stays unitary.
For finite index and finite groups, the result agrees with the [[induced representation|JJP4JFH3C6]] of finite group theory.
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# Spectral graph theory

## Summary

Spectral graph theory studies a graph through the [[eigenvalues|0BQAA2BK3Q]] and eigenvectors of matrices attached to it, mainly the [[adjacency matrix|08B9CX109G]] and the [[Laplacian matrix|TN6M6DVZRP]].
These eigenvalues are invariants of the graph up to isomorphism, and many of them have combinatorial meaning.
For example, a finite graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric about $0$.
Its number of connected components equals the multiplicity of the Laplacian eigenvalue $0$.
The [[Cheeger constant|Y7S6W12N64]] is controlled by the second Laplacian eigenvalue, which links the subject to [[expander graphs|DXJXVFV79H]].

## Explanations

### L1 human

A graph can be turned into a matrix, and a matrix has eigenvalues.
Spectral graph theory asks what these numbers tell us about the graph.
They detect, for example, whether the graph is connected, whether it is bipartite, and how hard it is to cut it into two large pieces.
They do not tell everything: the star with four leaves and the $4$-cycle together with an isolated vertex have the same adjacency eigenvalues $2, 0, 0, 0, -2$, but they are different graphs.
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<document source="https://tamnd.github.io/math2/node/5JHFB384BF.md" tag="5JHFB384BF" kind="concept">
# de Boor's algorithm

## Summary

De Boor's algorithm evaluates a spline written in the [[B-spline|62AFX32JGT]] basis at a point.
It replaces the coefficients of the B-splines that are active at the point by repeated [[convex combinations|21SXYJYC59]], until a single value remains.
It needs no explicit B-spline values, is numerically stable, and generalizes [[de Casteljau's algorithm|HT4Q9K3P2Z]] for Bézier curves.

## Explanations

### L1 human

A spline curve is controlled by a list of control points, but only a few of them affect a given parameter value.
De Boor's algorithm takes those few points and repeatedly replaces each pair of neighbours by a point on the segment between them.
After as many rounds as the degree, one point is left, and that point lies on the curve.
For a quadratic spline, three control points are blended into two, and the two into one.
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<document source="https://tamnd.github.io/math2/node/5JQG4S5JVH.md" tag="5JQG4S5JVH" kind="definition">
# Glimm scheme

## Definition

Let $u_t + F(u)_x = 0$ be a strictly [[hyperbolic system of conservation laws|XN081NCPF5]], with $F \colon \Omega \to \R^m$ smooth on an open set $\Omega \subseteq \R^m$.
Assume that every [[Riemann problem|GJWKX22VVE]] with states in $\Omega$ close enough to each other has an entropy solution.
Let $\Lambda$ be an upper bound for the absolute values of the eigenvalues of $DF(u)$ on $\Omega$.
Fix a space step $\Delta x > 0$ and a time step $\Delta t > 0$ with $\Lambda \Delta t \le \Delta x / 2$.
Set $x_j = j \Delta x$ for $j \in \Z$ and $t_n = n \Delta t$ for $n \ge 0$.
Fix a sequence $(\theta_n)_{n \ge 1}$ in $(0, 1)$.

The *Glimm scheme* defines an approximate solution $u^\Delta$ of the initial value problem with data $u_0$ as follows.

1. At $t = 0$, set $u^\Delta(x, 0) = u_0(x_j)$ for $x_{j-1} < x \le x_j$.
2. If $u^\Delta(\cdot, t_n)$ is constant on each cell $(x_{j-1}, x_j]$, then on $t_n \le t < t_{n+1}$ let $u^\Delta$ be the exact solution of the Riemann problems at the points $x_j$.
3. At $t = t_{n+1}$, set $u^\Delta(x, t_{n+1}) = u^\Delta(x_{j-1} + \theta_{n+1} \Delta x,\, t_{n+1} - 0)$ for $x_{j-1} < x \le x_j$.

When the $\theta_n$ are chosen independently and uniformly at random, it is also called the *random choice method*.

## Remarks

The bound $\Lambda \Delta t \le \Delta x/2$ is a [[Courant–Friedrichs–Lewy condition|FQVVNYV6N8]].
It ensures that waves from neighbouring points $x_j$ do not meet before time $t_{n+1}$.
So in step 2 the solution on the strip is exactly the union of the local Riemann solutions.

Glimm proved that, if the total variation of $u_0$ is small enough, the total variation of $u^\Delta(\cdot, t)$ stays bounded uniformly in $\Delta x$ and $t$.
For almost every random sequence $(\theta_n)$, as $\Delta x \to 0$ with $\Delta t / \Delta x$ fixed, a subsequence of the approximations converges to a weak solution.
T.-P. Liu later showed that any equidistributed sequence $(\theta_n)$ gives convergence.

Example: for the linear equation $u_t + a u_x = 0$ with $0 < a$ and $a \Delta t \le \Delta x/2$, the Riemann solution at $x_j$ moves the jump to $x_j + a(t - t_n)$.
At $t_{n+1}$, a cell takes its old value if $\theta_{n+1} \Delta x > a \Delta t$, and the value of its left neighbour otherwise.

Non-example: step 3 with the cell average instead of a sampled value is not the Glimm scheme.
It is [[Godunov's scheme|43HZJ7DRT0]], which smears shocks over several cells, while sampling keeps them sharp.

## Explanations

### L1 human

The scheme replaces the data by steps, solves each jump exactly, and then resamples.
Sampling at random points means that a jump moves by a whole cell or not at all in each step.
On average it moves at the right speed, so shocks stay sharp and travel correctly.
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<document source="https://tamnd.github.io/math2/node/5JQRYAKHSB.md" tag="5JQRYAKHSB" kind="definition">
# Accelerated failure time model

## Definition

Let $T$ be an event time with values in $(0, \infty)$ and let $x \in \R^p$ be a vector of covariates.
An *accelerated failure time model* is the following [[statistical model|K3AMMSY8DD]].
The conditional [[survival function|QTM1HZM53Q]] of $T$ given $x$ is
$$S(t \mid x) = S_0\!\left(t \, e^{-\beta^T x}\right), \qquad t \ge 0,$$
where $\beta \in \R^p$ is a parameter and $S_0$ is a baseline survival function.

## Remarks

Equivalently, $T = e^{\beta^T x} T_0$, where $T_0$ has survival function $S_0$, so that $\log T = \beta^T x + \log T_0$.
Parametric versions fix the law of $\log T_0$, for example normal (log-normal model), logistic (log-logistic model) or extreme value (Weibull model).
If $S_0$ has [[hazard function|Y959NKQ3XW]] $h_0$, then $h(t \mid x) = e^{-\beta^T x} h_0(t e^{-\beta^T x})$.

Example: if $T_0$ is exponential with rate $1$, then $S(t \mid x) = \exp(-t e^{-\beta^T x})$, so $T$ given $x$ is exponential with rate $e^{-\beta^T x}$.
Its median $e^{\beta^T x} \log 2$ is the baseline median times $e^{\beta^T x}$, as the model requires.

Non-example: the family $S(t \mid x) = \exp(-t^{k(x)})$, with a shape $k(x) > 0$ that varies with $x$, is not an accelerated failure time model.
If two members with shapes $k_1 \ne k_2$ were related by a change of time scale, then $\exp(-t^{k_2}) = \exp(-(ct)^{k_1})$ for all $t > 0$ and some $c > 0$.
This is impossible, since $t^{k_2} / t^{k_1}$ is not constant in $t$.
</document>

<document source="https://tamnd.github.io/math2/node/5JSWM1HBX2.md" tag="5JSWM1HBX2" kind="concept">
# Wedge sum

## Summary

The wedge sum of a family of [[pointed spaces|T9DG8DDE75]] is their disjoint union with all base points glued to one point.
It is the coproduct in the category of pointed spaces.
Wedges of circles and of spheres are the basic building blocks of CW complexes and appear throughout homotopy theory.

## Explanations

### L1 human

Take several spaces, each with a marked point, and glue them together at the marked points only.
The result looks like a bouquet of flowers tied at one spot.
For example, the wedge of two circles is the figure eight.
Its fundamental group is the free group on two generators.
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# Borel subgroup

## Summary

A Borel subgroup of a [[linear algebraic group|GA81VVQHEP]] is a maximal connected [[solvable|0FXJ7YFMMX]] closed subgroup.
In $\mathrm{GL}_n$ the upper triangular matrices form a Borel subgroup.
Over an algebraically closed field all Borel subgroups are conjugate, and the quotient by one of them is a projective variety, the [[flag variety|NHRWCQQKTH]].
Borel subgroups are the starting point of the structure theory of [[reductive groups|2XS84PWE3S]].

## Explanations

### L1 human

Solvable groups are the easy groups, built up from commutative pieces.
A Borel subgroup is a largest easy piece of a linear algebraic group that is still connected.
For invertible matrices, the upper triangular ones are such a piece.
Every other Borel subgroup of $\mathrm{GL}_n$ is the stabilizer of some full flag of subspaces, a conjugate of the triangular group.
</document>

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# Sheaf of modules

## Definition

Let $(X, \mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]].
A *sheaf of $\mathcal{O}_X$-modules* is a [[sheaf|TB6TM6R4MD]] $\mathcal{F}$ of [[abelian groups|082GHNPJ53]] on $X$ with the following data and condition.

1. For each open set $U \subseteq X$, $\mathcal{F}(U)$ is a [[module|0NZ8JA25WK]] over the ring $\mathcal{O}_X(U)$.
2. For open sets $V \subseteq U$, all $a \in \mathcal{O}_X(U)$ and all $s \in \mathcal{F}(U)$, one has $(a s)|_V = (a|_V)(s|_V)$.

A morphism of $\mathcal{O}_X$-modules is a morphism of sheaves $\varphi \colon \mathcal{F} \to \mathcal{G}$ such that each $\varphi_U$ is $\mathcal{O}_X(U)$-linear.

## Remarks

The structure sheaf $\mathcal{O}_X$ is a sheaf of $\mathcal{O}_X$-modules, with each $\mathcal{O}_X(U)$ acting on itself by multiplication.
Condition 2 holds because restriction maps of $\mathcal{O}_X$ are ring homomorphisms.
On $X = \operatorname{Spec} \Z$, the constant sheaf $\underline{\Z}$ admits no structure of $\mathcal{O}_X$-module.
Its sections over $D(2)$ form the group $\Z$, and clause 1 would need a module structure over $\mathcal{O}_X(D(2)) = \Z[1/2]$.
This is impossible, since multiplication by $2$ is not invertible on $\Z$.
For each point $x \in X$, the stalk $\mathcal{F}_x$ is a module over the local ring $\mathcal{O}_{X,x}$.
The $\mathcal{O}_X$-modules form an abelian category, with tensor products $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ and sheaves of homomorphisms.
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# Finite Hilbert transform

## Summary

The finite Hilbert transform sends a function $f$ on the interval $(-1, 1)$ to the [[Cauchy principal value|0SYR10HD4R]] integral $\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{f(t)}{t - x}\, dt$.
Up to sign, it is the [[Hilbert transform|NSCEBXM7ZM]] of $f$ extended by zero, looked at only on the interval.
It is the singular part of most singular integral equations on an arc, such as the [[airfoil equation|W6Z6CWRE25]].
Unlike the Hilbert transform on the line, it is not invertible on $L^p(-1, 1)$ for any $1 < p < \infty$.
For $1 < p < 2$ it even sends a nonzero function to $0$.

## Explanations

### L1 human

Take a function on a segment and average it against $1/(t - x)$, which is large and changes sign at $t = x$.
The two sides of the singularity nearly cancel, and the principal value keeps what is left.
For example, the transform of $\sqrt{1 - t^2}$ is $-x$, and the transform of $1/\sqrt{1 - t^2}$ is $0$.
So the transform kills a nonzero function, which is why equations with it need an extra condition to have one solution.
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# Riemann surface

## Summary

A Riemann surface is a connected space that looks locally like an open piece of the complex plane, with changes of coordinates that are holomorphic.
It is a [[complex manifold|JSGWAN2RJV]] of complex dimension one.
On a Riemann surface one can speak of [[holomorphic|0DG0Z72CQD]] and [[meromorphic|0NSD1TPHR0]] functions.
This makes it the natural home of multivalued functions such as $\sqrt{z}$ and $\log z$.
Compact Riemann surfaces are the same objects as smooth projective complex algebraic curves.

## Explanations

### L1 human

Imagine a surface built by gluing pieces of the complex plane, where the gluing maps are holomorphic.
Because the gluing respects holomorphy, you can tell whether a function on the surface is holomorphic by looking at it in any piece.
The plane itself, the [[Riemann sphere|0CDMXGPSES]] and the torus $\C/(\Z + i\Z)$ are examples.
The graph of $w^2 = z$ is a Riemann surface on which $\sqrt{z}$ becomes a single-valued function, namely $w$.
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# Majorization

## Summary

Majorization compares two vectors of real numbers with the same sum by how spread out their entries are.
A vector $x$ majorizes $y$ when, for each $k$, the sum of the $k$ largest entries of $x$ is at least that of $y$.
Equivalently, $y$ is obtained from $x$ by a [[doubly stochastic matrix|JJA90EB2PS]], that is, by averaging.
It organises many inequalities, including the [[Schur–Horn theorem|Y553MSYDAH]] on diagonals and eigenvalues of Hermitian matrices.

## Explanations

### L1 human

Think of a fixed amount of money shared among $n$ people.
One sharing majorizes another when it is less equal: the richest person, the two richest people, and so on, always hold at least as much.
Moving an amount from a richer person to a poorer one, no larger than the gap between them, gives a sharing that is majorized by the old one.
For example, $(1, 0, 0)$ majorizes $(\frac{1}{2}, \frac{1}{2}, 0)$, which majorizes $(\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$.
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# Duhamel's integral

## Statement

Let $m > 0$ and $c, k \in \R$, and let $F \colon [0, \infty) \to \R$ be continuous.
Let $h$ be the solution of $m h'' + c h' + k h = 0$ with $h(0) = 0$ and $h'(0) = 1/m$.
Then the solution of the [[initial value problem|0FXDW992DS]]
$$m x'' + c x' + k x = F(t), \qquad x(0) = 0, \quad x'(0) = 0,$$
is the convolution
$$x(t) = \int_0^t F(\tau) \, h(t - \tau) \, d\tau, \qquad t \ge 0.$$

## Remarks

Differentiate under the integral sign: $x'(t) = h(0) F(t) + \int_0^t F(\tau) h'(t - \tau) \, d\tau = \int_0^t F(\tau) h'(t - \tau) \, d\tau$, and $x''(t) = h'(0) F(t) + \int_0^t F(\tau) h''(t - \tau) \, d\tau$.
Then $m x'' + c x' + k x = m h'(0) F(t) = F(t)$, and $x(0) = x'(0) = 0$ because the integrals run over an interval of length $0$.
The function $h$ is the unit impulse response: the motion caused by a unit impulse at time $0$, which gives the mass the velocity $1/m$.

Example: for an underdamped oscillator, with $\omega_n = \sqrt{k/m}$, $\zeta = c / (2\sqrt{km})$ and $0 \le \zeta < 1$, put $\omega_d = \omega_n \sqrt{1 - \zeta^2}$.
Then
$$h(t) = \frac{1}{m \omega_d} e^{-\zeta \omega_n t} \sin(\omega_d t),$$
and the response to $F$ is $x(t) = \frac{1}{m \omega_d} \int_0^t F(\tau) e^{-\zeta \omega_n (t - \tau)} \sin(\omega_d (t - \tau)) \, d\tau$.
Nonzero initial data are handled by adding the free motion with those data.

## Explanations

### L1 human

A general force can be cut into many short kicks.
Each kick sets the system moving in the same way, only scaled and delayed.
Because the system is linear, the total motion is the sum of all these delayed responses, and in the limit the sum becomes an integral.

### L2 human

Duhamel's integral is the [[convolution on the half-line|FCKX8KFVGN]] of the load with the impulse response, so the Laplace transform turns it into the product $X(s) = H(s) F(s)$ with $H(s) = 1/(m s^2 + c s + k)$.
The same idea works for any [[linear differential equation|0QQRP17Z7S]] with constant coefficients, and in the form of [[Duhamel's principle|Q28DM6ZCMF]] for linear evolution equations.
It is a special case of [[variation of parameters|0Y6TH5PQVS]].
In structural dynamics it is used to compute responses to shocks and other non-periodic loads.
</document>

<document source="https://tamnd.github.io/math2/node/5KERJ85BXJ.md" tag="5KERJ85BXJ" kind="definition">
# Stable homotopy theory

## Definition

Stable homotopy theory is the study of homotopy invariants that are unchanged by suspension, such as the following groups.
Let $X$ and $Y$ be [[pointed|T9DG8DDE75]] [[CW complexes|5W95QFHXPR]], with $X$ finite, and let $\Sigma$ be the reduced [[suspension|2VD7GR64JJ]].
Let $[A, B]_*$ be the set of pointed homotopy classes of pointed maps $A \to B$.
The group of *stable maps* from $X$ to $Y$ is the colimit
$$\set{X, Y} = \operatorname{colim}_{k} \, [\Sigma^k X, \Sigma^k Y]_*,$$
taken along the maps $[\Sigma^k X, \Sigma^k Y]_* \to [\Sigma^{k+1} X, \Sigma^{k+1} Y]_*$ that send $f$ to $\Sigma f$.
For $n \ge 0$, the $n$-th *stable homotopy group* of $Y$ is
$$\pi_n^s(Y) = \operatorname{colim}_k \, \pi_{n+k}(\Sigma^k Y) = \set{S^n, Y}.$$

## Remarks

For $k \ge 2$ the set $[\Sigma^k X, \Sigma^k Y]_*$ is an abelian group, so the colimit is an abelian group.
Let $Y$ be $(c-1)$-connected and let $X$ have dimension $d$.
The [[Freudenthal suspension theorem|7DM2KSS2DB]] shows that the maps in the colimit are bijective once $d + k < 2(c + k) - 1$, that is, once $k > d - 2c + 1$.
So the colimit is reached at a finite stage.

Example: $\pi_0^s(S^0) = \Z$, since $\pi_k(S^k) = \Z$ for all $k \ge 1$ and suspension preserves degree.
Also $\pi_1^s(S^0) = \Z/2$, generated by the stable class of the Hopf map $\eta$.

Non-example: the group $\pi_3(S^2) \cong \Z$, generated by the Hopf map, is not a stable group.
Its suspension lands in $\pi_4(S^3) \cong \Z/2$, and the stable value $\pi_1^s(S^0)$ is $\Z/2$.

Today stable homotopy theory is usually set in the stable homotopy category of spectra, where $\set{X, Y}$ becomes a group of morphisms between suspension spectra.
</document>

<document source="https://tamnd.github.io/math2/node/5KGKGTW2XY.md" tag="5KGKGTW2XY" kind="definition">
# Satellite knot

## Definition

Let $V = S^1 \times D^2$ be the solid torus, and let $P \subset \operatorname{int} V$ be a tame [[knot|QYEAZTYGPT]], the *pattern*, such that:

1. $P$ is not contained in any $3$-ball in $V$;
2. $P$ is not isotopic in $V$ to the core circle $S^1 \times \set{0}$.

Let $C \subset S^3$ be a knot that is not the [[unknot|1TQ06R4Q66]], the *companion*.
Let $h \colon V \to S^3$ be an embedding onto a closed tubular neighbourhood of $C$.
A knot in $S^3$ is a *satellite knot* if it is isotopic to $h(P)$ for some such $P$, $C$ and $h$.

## Remarks

Equivalently, a knot $K$ is a satellite knot if $S^3 \setminus K$ contains an incompressible torus that is not parallel to the boundary of a neighbourhood of $K$.
Example: let $C$ be a [[trefoil knot|VYTYSVHN4A]] and let $P$ be the $(2, 1)$-curve on a smaller concentric torus in $V$.
Then $h(P)$ is a cable of the trefoil and a satellite knot.
Non-example: if $P$ is the core circle, clause 2 fails and $h(P)$ is just $C$.
Every [[knot sum|AGZSZW7GYE]] $K_1 \# K_2$ of nontrivial knots is a satellite with companion $K_1$.
Whitehead doubles are satellites whose pattern has winding number $0$ in $V$.
</document>

<document source="https://tamnd.github.io/math2/node/5KKCSZJC29.md" tag="5KKCSZJC29" kind="statement">
# Mergelyan's theorem

## Statement

Let $K \subseteq \C$ be a [[compact set|00R0X1WGPX]] such that $\C \setminus K$ is [[connected|0D939CMME7]].
Let $f \colon K \to \C$ be [[continuous|0P27W9RW0G]] on $K$ and [[holomorphic|0DG0Z72CQD]] on the [[interior|0C38S0KMJK]] of $K$.
Then for every $\varepsilon > 0$ there is a polynomial $p$ such that
$$\abs{f(z) - p(z)} < \varepsilon \quad \text{for all } z \in K.$$

## Remarks

The conditions on $f$ are necessary, since a [[uniform limit|08HJRTG5N7]] of polynomials on $K$ is continuous on $K$ and holomorphic on its interior.
The condition on $K$ is also necessary.
On the unit circle, $f(z) = 1/z$ is continuous and the interior is empty, yet $f$ is not a uniform limit of polynomials.
Indeed $\oint p(z) \, dz = 0$ for polynomials while $\oint z^{-1} \, dz = 2\pi i$.

Two special cases were known before.
If $f$ is holomorphic on a neighbourhood of $K$, the result is [[Runge's theorem|6DG4CTGXJD]].
If $K$ is the closed unit disc, it suffices to approximate $f$ by $f(rz)$ with $r < 1$ and then truncate its Taylor series.
The general proof is much harder; it uses estimates on Cauchy integrals that are uniform over the shape of $K$.

Example: let $K = [0, 1]$, a segment with empty interior and connected complement.
The theorem says every continuous function on $[0, 1]$ is a uniform limit of complex polynomials, which recovers the Weierstrass approximation theorem.

## Explanations

### L1 human

If a compact set in the plane has no holes, then every function that is continuous on it and holomorphic inside it can be approximated by polynomials.
The approximation is uniform: the error is small at every point of the set at once.
Without holes there is no "hidden" Cauchy integral that polynomials cannot imitate.
This is the best possible result of its kind.

### L2 human

Mergelyan's theorem completes the theory of polynomial approximation on compact sets with connected complement.
It contains the Weierstrass approximation theorem and its complex analogue for sets with empty interior.
For sets with holes, the question of approximation by rational functions is harder and is described by analytic capacity (Vitushkin's theory).
On unbounded closed sets the corresponding statement is [[Arakelyan's theorem|1S20A3R659]].
</document>

<document source="https://tamnd.github.io/math2/node/5KMSW7139A.md" tag="5KMSW7139A" kind="definition">
# Idele class group

## Definition

Let $K$ be a [[number field|MWBDFS2KJ5]].
For each [[place|BT5V3TSNY6]] $v$ of $K$, let $K_v$ be the completion of $K$ at $v$.
For each finite place $v$, let $\mathcal{O}_v$ be the ring of integers of $K_v$ and $\mathcal{O}_v^\times$ its group of units.
The *idele group* $I_K$ is the set of families $x = (x_v)_v$ such that:

1. $x_v \in K_v^\times$ for every place $v$;
2. $x_v \in \mathcal{O}_v^\times$ for all but finitely many finite places $v$.

It is a group under componentwise multiplication.
Its topology has as a basis the sets $\prod_{v \in S} U_v \times \prod_{v \notin S} \mathcal{O}_v^\times$.
Here $S$ is a finite set of places containing all infinite places, and each $U_v$ is open in $K_v^\times$.
The map $a \mapsto (a, a, \ldots)$ embeds $K^\times$ in $I_K$ as a discrete subgroup.
The *idele class group* of $K$ is the [[quotient group|0KYX2N69J0]]
$$C_K = I_K / K^\times,$$
with the quotient topology.

## Remarks

Example: for $K = \Q$, the family with $x_\infty = 2$, $x_3 = 1/3$ and $x_p = 1$ for all other $p$ is an idele, because only finitely many entries are not units.
Non-example: the family with $x_p = p$ for every prime $p$ and $x_\infty = 1$ is not an idele, because $x_p \notin \Z_p^\times$ for infinitely many $p$.
The ideles are the units of the [[adele ring|SEH68BA1QQ]] $\mathbb{A}_K$, but the topology above is finer than the subspace topology from $\mathbb{A}_K$.
The idele norm $\abs{x} = \prod_v \abs{x_v}_v$ equals $1$ on $K^\times$ by the product formula, so it is defined on $C_K$.
Its kernel $C_K^1$ is compact.
The [[ideal class group|WQ46BD3ARY]] of $K$ is a quotient of $C_K$.
The same construction works for every [[global field|634XKY8RZ0]].
</document>

<document source="https://tamnd.github.io/math2/node/5KNAXNHRY6.md" tag="5KNAXNHRY6" kind="concept">
# Bromwich integral

## Summary

The Bromwich integral is the complex line integral $\frac{1}{2 \pi i} \int_{\gamma - i \infty}^{\gamma + i \infty} e^{s t} F(s) \, ds$ along a vertical line $\operatorname{Re} s = \gamma$ in the half-plane where a [[Laplace transform|0V49JEZW06]] $F$ is defined.
It is the standard integral formula for the [[inverse Laplace transform|07WBTX1MHA]]: under mild conditions it returns the function $f$ with $\mathcal{L}\{f\} = F$.
It also opens the way to computing inverse transforms by [[contour integration|06ZGPYTWK8]] and residues.

## Explanations

### L1 human

The Laplace transform packs a function of time into a function of a complex variable $s$.
The Bromwich integral unpacks it: it adds up the exponentials $e^{s t}$, weighted by $F(s)$, over a vertical line in the complex plane.
For example, for $F(s) = 1/s$ and any line $\operatorname{Re} s = \gamma > 0$, the integral gives $1$ for $t > 0$ and $0$ for $t < 0$, which is the step function whose transform is $1/s$.
The line must lie to the right of all singularities of $F$.
</document>

<document source="https://tamnd.github.io/math2/node/5M0HYY3RYV.md" tag="5M0HYY3RYV" kind="conjecture">
# Brauer's height zero conjecture

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime.
Let $B$ be a $p$-block of $G$, that is, a block of the [[group algebra|MNP2HBWB5Q]] of $G$ over an algebraically closed field of characteristic $p$.
Let $D$ be a defect group of $B$.
Write $\abs{G}_p = p^a$ and $\abs{D} = p^d$.
Each complex irreducible [[character|RMWMYBY8TC]] $\chi$ in $B$ has degree $\chi(1)$ with $p$-part $\chi(1)_p = p^{a - d + h}$ for an integer $h \ge 0$, the height of $\chi$.
Then every irreducible character in $B$ has height $0$ if and only if $D$ is abelian.

## Remarks

Richard Brauer posed the conjecture in 1955.
For the principal block, a defect group is a [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$.
The direction "if $D$ is abelian, then all heights are zero" was proved by Kessar and Malle in 2013, using the [[classification of finite simple groups|06J6P03MG7]].
The converse direction was proved by Malle, Navarro, Schaeffer Fry and Tiep, so the conjecture is now a theorem.
For $p$-solvable groups the conjecture was proved earlier by Gluck and Wolf.

## Explanations

### L1 human

A block is a piece of the representation theory of a finite group that belongs together when one works modulo a prime $p$.
Each block has a defect group, a $p$-subgroup that measures how complicated the block is.
The conjecture says that a simple numerical pattern in the character degrees of the block, all heights being zero, happens exactly when the defect group is commutative.
So local information, a small subgroup, is read off from global information, the character degrees.
</document>

<document source="https://tamnd.github.io/math2/node/5M223XDM43.md" tag="5M223XDM43" kind="concept">
# Essential extension

## Summary

An essential extension of an algebra $A$ is a larger algebra $B$ containing $A$ that cannot collapse anything without collapsing part of $A$.
Every nontrivial [[congruence|DCSXXT4SER]] of $B$ already identifies two distinct elements of $A$.
Essential extensions are the building blocks of injective hulls: in good cases, the injective hull of $A$ is a maximal essential extension of $A$.

## Explanations

### L1 human

Think of $A$ as sitting inside $B$ and holding it together.
If you glue together some elements of $B$, you are forced to glue together some elements of $A$ as well.
For example, the integers sit essentially inside the rationals as additive groups.
Any nonzero subgroup of $\Q$ contains some fraction $p/q \ne 0$, and then it contains the nonzero integer $p = q \cdot (p/q)$.
</document>

<document source="https://tamnd.github.io/math2/node/5M3THA1S6E.md" tag="5M3THA1S6E" kind="concept">
# Fractional iteration

## Summary

Fractional iteration extends the [[iterates|AYVTR2ES5B]] $f^n$ of a map $f$ from whole numbers $n$ to real numbers $t \ge 0$.
One asks for maps $f^t$ with $f^0 = \id$, $f^1 = f$ and $f^s \circ f^t = f^{s+t}$, so that $f$ becomes the time-one map of a flow.
Then $f^{1/n}$ is an [[iterative root|27C6J3YNZY]] of order $n$ of $f$.
The main tools are [[Schröder's equation|VTJV7JVJEA]] and the [[Abel equation|68SHB7Z721]], which turn iteration into multiplication or into addition.

## Explanations

### L1 human

Applying $f$ twice is easy to picture, but what does applying $f$ one and a half times mean?
Fractional iteration gives an answer: a family of maps $f^t$, one for each time $t$, that fit together like the positions of a moving object.
For $f(x) = 2x$, the natural choice is $f^t(x) = 2^t x$.
For $f(x) = x + c$, it is $f^t(x) = x + tc$.
For most maps there is no single natural choice, and one adds conditions such as continuity or analyticity in $t$.
</document>

<document source="https://tamnd.github.io/math2/node/5M4S7P04GR.md" tag="5M4S7P04GR" kind="concept">
# Julia set

## Summary

The Julia set of a [[rational function|0XDK9CGYKJ]] is the set of points near which the iterates of the function behave chaotically.
It is the complement of the [[Fatou set|C0N0XGC6GK]], where the iterates form a [[normal family|JTVWRXXAKM]].
For a map of degree at least $2$ it is a nonempty compact set.
The map sends it onto itself, and it is often a fractal.
Julia sets are the central objects of [[complex dynamics|BQQZ0ZCGQF]].

## Explanations

### L1 human

Iterate a rule such as $z \mapsto z^2 + c$ and colour each starting point by what its orbit does.
In some regions nearby starting points keep behaving alike; this is the Fatou set.
The Julia set is the border where tiny changes of the starting point can change the long-run behaviour completely.
For $z \mapsto z^2$ the Julia set is the unit circle: inside it orbits go to $0$, outside they go to infinity.
For most values of $c$ the border is no longer a smooth curve but an intricate fractal.
</document>

<document source="https://tamnd.github.io/math2/node/5M6KS8F47R.md" tag="5M6KS8F47R" kind="concept">
# Flat morphism

## Summary

A flat morphism of schemes is a morphism whose local rings are flat modules over the local rings of the target.
Flatness is the algebraic expression of a family whose fibres vary continuously.
In a flat family of projective schemes over a connected base, numerical invariants such as the Hilbert polynomial stay constant from fibre to fibre.

## Explanations

### L1 human

A flat morphism is a family of spaces with no sudden jumps.
Open immersions and the projection $\mathbb{A}^2 \to \mathbb{A}^1$ are flat.
The blow-up of the plane at a point is not flat: all its fibres are points except one, which is a whole line.
Flatness forbids such a jump in the fibre dimension, and also subtler jumps, such as a fibre suddenly gaining an embedded point.
</document>

<document source="https://tamnd.github.io/math2/node/5M7D1BSHVZ.md" tag="5M7D1BSHVZ" kind="concept">
# Discrete subgroup

## Summary

A discrete subgroup of a topological group is a subgroup whose elements are isolated from each other.
In a Hausdorff topological group, discrete subgroups are closed.
A discrete subgroup of a Lie group $G$ acts properly discontinuously on each quotient $G/K$ with $K$ compact.
The resulting quotient spaces are central objects of geometry and number theory.
Lattices, Fuchsian groups, Kleinian groups and crystallographic groups are all discrete subgroups.

## Explanations

### L1 human

Inside the real line, the integers form a subgroup in which each element has some room around it with no other element of the subgroup.
That is what discrete means.
The numbers $a + b\sqrt2$ with $a, b$ integers also form a subgroup, but they crowd together everywhere, so this subgroup is not discrete.
Discrete subgroups behave like crystals or grids sitting inside a continuous group.
</document>

<document source="https://tamnd.github.io/math2/node/5MDK7HSYRM.md" tag="5MDK7HSYRM" kind="statement">
# Itô isometry

## Statement

Let $(B_t)_{t \ge 0}$ be a [[Brownian motion|22735FGJV6]] for a filtration $(\mathcal{F}_t)$, and let $T > 0$.
Let $H$ be a progressively measurable process with $E\left[\int_0^T H_t^2 \, dt\right] < \infty$.
Then the [[Itô integral|JA95NH206E]] satisfies
$$E\left[\left(\int_0^T H_t \, dB_t\right)^2\right] = E\left[\int_0^T H_t^2 \, dt\right],$$
where $E$ is the [[expected value|0DCT0QZVQ1]].

## Remarks

For a simple process $H = \sum_i \xi_i \mathbf{1}_{(t_i, t_{i+1}]}$, expand the square of $\sum_i \xi_i \Delta B_i$.
A cross term $E[\xi_i \xi_j \Delta B_i \Delta B_j]$ with $i < j$ vanishes, because $\Delta B_j$ has mean zero and is independent of $\mathcal{F}_{t_j}$.
A diagonal term equals $E[\xi_i^2] (t_{i+1} - t_i)$, because $\Delta B_i$ is independent of $\mathcal{F}_{t_i}$ with variance $t_{i+1} - t_i$.
The general case follows by approximation, and this is how the integral is defined.
Example: for $H_t = B_t$ with $B_0 = 0$, the right side is $\int_0^T t \, dt = T^2/2$, and indeed $\int_0^T B_t \, dB_t = \tfrac12 (B_T^2 - T)$ has variance $\tfrac14 (3T^2 - 2T^2 + T^2) = T^2/2$.

## Explanations

### L1 human

The Itô integral of $H$ is a random number with mean zero.
Its variance equals the expected total of $H^2$ over time.
So the size of the integral can be computed by an ordinary integral, without the random noise.

### L2 human

The isometry is the key step in the construction of the Itô integral: it extends the integral from simple processes to all square-integrable adapted ones.
Its polarised form gives $E\left[\int H \, dB \int K \, dB\right] = E\left[\int H K \, dt\right]$.
For a general continuous martingale $M$ in place of $B$, the same identity holds with $dt$ replaced by $d\langle M \rangle_t$.
</document>

<document source="https://tamnd.github.io/math2/node/5MDSVRCDZW.md" tag="5MDSVRCDZW" kind="definition">
# Naive set theory

## Definition

*Naive set theory*, in its formal reading, is the [[set theory|JRBRNZT680]] in first-order logic with the relation symbol $\in$ and these two axioms.

1. Extensionality: two [[sets|0SJ3RX9TXA]] with the same elements are equal, as in the [[axiom of extensionality|0TB16T2C97]].
2. Unrestricted comprehension: for each formula $\varphi(x)$, possibly with parameters, in which $y$ is not free, $\exists y\, \forall x\, (x \in y \leftrightarrow \varphi(x))$.

In a wider sense, naive set theory is the informal use of sets in ordinary language, without a fixed list of axioms.

## Remarks

Example: the comprehension instance for $\varphi(x) = (x \ne x)$ gives a set with no elements, the empty set, and extensionality makes it unique.
Non-example: the formal theory is inconsistent, so it has no model.
Indeed, take $\varphi(x) = (x \notin x)$; comprehension gives $y$ with $\forall x\, (x \in y \leftrightarrow x \notin x)$, and $x = y$ gives $y \in y \leftrightarrow y \notin y$.
This is [[Russell's paradox|0HESMMSYEG]].
[[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] replaces unrestricted comprehension by the separation scheme $\exists y\, \forall x\, (x \in y \leftrightarrow x \in a \wedge \varphi(x))$, which only forms subsets of a given set $a$.
</document>

<document source="https://tamnd.github.io/math2/node/5MEKEETGDA.md" tag="5MEKEETGDA" kind="concept">
# Weak dimension

## Summary

The weak dimension, or flat dimension, of a [[module|0NZ8JA25WK]] is the shortest length of a resolution of it by [[flat modules|73RYYEVDKA]].
The weak global dimension of a [[ring|0B1261B5ED]] is the supremum of the weak dimensions of its modules.
It is measured by the vanishing of the [[Tor functors|DDQRERAQCV]] and is the same for left and right modules.
It is at most the [[global dimension|GWA0T1WKCN]] and often easier to compute; rings of weak global dimension $0$ are exactly the von Neumann regular rings.

## Explanations

### L1 human

Flat modules are the modules that behave well under tensor products, and every free module is flat.
A general module can be approximated by flat modules in a chain, called a flat resolution.
The weak dimension counts how many steps such a chain needs at least.
Over the integers, a module like $\Z/2$ needs one step: $0 \to \Z \to \Z \to \Z/2 \to 0$, so its weak dimension is $1$.
A ring of small weak dimension has a simple homological theory.
</document>

<document source="https://tamnd.github.io/math2/node/5MN8MX0JPA.md" tag="5MN8MX0JPA" kind="definition">
# Heegaard splitting

## Definition

Let $\Gamma \subset \R^3$ be a connected finite graph with Euler characteristic $1 - g$.
A *handlebody of genus $g$* is a $3$-manifold homeomorphic to a closed regular neighbourhood of $\Gamma$ in $\R^3$.
Its boundary is a closed orientable surface of [[genus|NJ787GZS1N]] $g$.

Let $M$ be a closed connected orientable [[3-manifold|R20TB2KEQQ]].
A *Heegaard splitting* of $M$ is a pair $(V, W)$ of subsets of $M$ such that:

1. $M = V \cup W$;
2. $V$ and $W$ are [[handlebodies|AP1KY4EHMA]] of the same genus $g$;
3. $V \cap W = \partial V = \partial W$.

The closed [[surface|VGJJVBZ1MQ]] $\Sigma = V \cap W$ is the *Heegaard surface*, and $g$ is the genus of the splitting.

## Remarks

Every closed connected orientable $3$-manifold has a Heegaard splitting.
Take a triangulation and let $V$ be a regular neighbourhood of its $1$-skeleton.
The closure $W$ of the complement is a regular neighbourhood of the dual $1$-skeleton.
The *Heegaard genus* of $M$ is the least genus of a Heegaard splitting.

Example: $S^3 = \set{(z, w) \in \C^2 : \abs{z}^2 + \abs{w}^2 = 1}$ is the union of the solid tori $V = \set{\abs{z} \le \abs{w}}$ and $W = \set{\abs{z} \ge \abs{w}}$, which meet in the torus $\abs{z} = \abs{w} = 1/\sqrt{2}$; this is a splitting of genus $1$.
Lens spaces also have splittings of genus $1$.

Non-example: let $N$ be a closed tubular neighbourhood of a trefoil knot in $S^3$ and $E$ the closure of its complement.
Then $S^3 = N \cup E$ with $N \cap E = \partial N = \partial E$ a torus, and $N$ is a solid torus, but $E$ is not a handlebody.
Indeed the fundamental group of $E$ is $\langle x, y \mid x^2 = y^3 \rangle$, which is not free, while handlebodies have free fundamental groups.
So clause 2 fails.

## Explanations

### L1 human

The definition splits a closed $3$-dimensional space into two thickened graphs that share their whole boundary surface.
All the topology of the space is then in how the two boundary surfaces are glued.
</document>

<document source="https://tamnd.github.io/math2/node/5MNYT3AFPH.md" tag="5MNYT3AFPH" kind="concept">
# Thom space

## Summary

The Thom space of a real vector bundle $E \to B$ is its unit disc bundle with the unit [[sphere bundle|3VQ2V5HH3R]] collapsed to a point.
For compact $B$ it is the [[one-point compactification|0GWN3CMRK8]] of the total space $E$.
The [[Thom isomorphism|N685RXHDZX]] identifies its reduced cohomology with the cohomology of $B$ shifted by the rank.
Thom spaces of universal bundles form the [[Thom spectra|6H8AGEN6NY]] that compute cobordism groups.

## Explanations

### L1 human

Over each point of the base there is a disc, the vectors of length at most one.
The Thom space glues the boundary circles or spheres of all these discs to one common point.
So each disc becomes a sphere, and all these spheres touch at the collapsed point.
For a bundle over a single point, the Thom space is one disc with its boundary collapsed, which is a sphere of the same dimension.
</document>

<document source="https://tamnd.github.io/math2/node/5MP7DM0V5V.md" tag="5MP7DM0V5V" kind="counterexample">
# Failure of Kalman's conjecture

## Counterexample

Kalman's conjecture concerns systems of the form
$$\dot x = P x + q\, f(e), \qquad e = r^{T} x.$$
Here $x \in \R^n$, $P$ is a constant real $n \times n$ matrix, and $q, r \in \R^n$ are constant vectors.
The function $f \colon \R \to \R$ is differentiable with $f(0) = 0$.
Assume that $k_1 < f'(e) < k_2$ for all $e$, and that for every $k \in (k_1, k_2)$ the [[linear system|037ZNY7NAG]] $\dot x = (P + k\, q r^{T}) x$ is [[asymptotically stable|0NB689782W]].
The conjecture claims that then the [[equilibrium|0N8NGKDZDZ]] $x = 0$ attracts every solution.

The claim is false in general.
For $n \le 3$ it is true.
But for some $n > 3$ there are $P$, $q$, $r$ and $f$ that satisfy all the hypotheses, while the system has a non-constant [[periodic orbit|0AA97XA70K]].
A solution on that orbit does not tend to $0$, so $0$ is not a global attractor.

## Remarks

Kalman stated the problem in 1957.
In the counterexamples constructed by Leonov, Kuznetsov and coauthors, the periodic orbit is stable and coexists with the stable equilibrium at $0$.
Its basin of attraction does not touch a small neighbourhood of the equilibrium, so it is a hidden oscillation: simulation from initial points near $0$ does not reveal it.
Once a counterexample exists in dimension $n$, adding decoupled coordinates with $\dot w = -w$ gives one in every larger dimension.
For the discrete-time analogue the conjecture holds only for $n = 1$.

## Explanations

### L1 human

Replace a nonlinear part of a feedback loop by straight lines whose slopes lie in the range of the slopes of the nonlinearity.
If every such linear system is stable, it is tempting to think that the nonlinear system is stable as well.
Kalman's conjecture says so, and it is true in low dimension.
In higher dimensions it can fail: the nonlinear system can keep oscillating forever, although every linear comparison system is stable.
</document>

<document source="https://tamnd.github.io/math2/node/5MR9ZSC3CX.md" tag="5MR9ZSC3CX" kind="concept">
# Statistical learning theory

## Summary

Statistical learning theory is the mathematical theory of learning a prediction rule from examples.
It models the data as independent draws from an unknown distribution and measures a rule by its expected [[loss|3BTG6VVCGZ]] on new data, its risk.
Its central results bound the gap between the error on the training data and the true risk.
The bounds depend on the size of the class of rules, measured for example by its [[VC dimension|KCFXR9K3GD]].
It was developed by Vladimir Vapnik and Alexey Chervonenkis among others, and it underlies [[empirical risk minimization|56NVR0SS61]] and support vector machines.

## Explanations

### L1 human

A spam filter is trained on a few thousand labelled emails and must then work on emails it has never seen.
Statistical learning theory asks when doing well on the training emails guarantees doing well on new ones.
The answer depends on how many examples there are and on how flexible the family of possible filters is.
A very flexible family can memorise the training data and still fail on new data; this is overfitting.
The theory gives precise bounds that balance these two effects.
</document>

<document source="https://tamnd.github.io/math2/node/5N1825CM10.md" tag="5N1825CM10" kind="definition">
# Energy of a map between Riemannian manifolds

## Definition

Let $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]] of dimension $m$, possibly with boundary, and let $(N, h)$ be a Riemannian manifold of dimension $n$.
Let $u \colon M \to N$ be a smooth map.
The *energy density* of $u$ is the function $e(u) = \frac{1}{2} \abs{du}^2$ on $M$.
Here $\abs{du}^2$ is the squared norm of $du_x \colon T_x M \to T_{u(x)} N$ for the metrics $g$ and $h$.
In local coordinates $x^i$ on $M$ and $y^\alpha$ on $N$ it is
$$\abs{du}^2 = \sum_{i,j,\alpha,\beta} g^{ij}(x) \, h_{\alpha\beta}(u(x)) \, \frac{\partial u^\alpha}{\partial x^i} \frac{\partial u^\beta}{\partial x^j}.$$
The *energy* of $u$ is
$$E(u) = \int_M e(u) \, dV_g = \frac{1}{2} \int_M \abs{du}^2 \, dV_g,$$
where $dV_g$ is the Riemannian volume measure of $g$.

## Remarks

Equivalently, $\abs{du}^2 = \sum_i \abs{du(e_i)}_h^2$ for an orthonormal basis $(e_i)$ of $T_x M$, which is the trace with respect to $g$ of the pull-back metric $u^* h$.
The same formula defines $E(u)$ for maps of Sobolev class $W^{1,2}$, and for non-compact $M$ when the integral converges.
For $N = \R$ it is the Dirichlet energy $\frac12 \int_M \abs{\nabla u}^2 \, dV_g$, and for $M$ an interval it is the [[energy of a path|Q12AB1MJ64]].
When $\dim M = 2$, the energy is invariant under conformal changes of the metric $g$.
Its critical points are the [[harmonic maps|KZJYG9DS3M]].

Example: for the identity map of a compact $(M, g)$ of dimension $m$, $du(e_i) = e_i$, so $\abs{du}^2 = m$ and $E(\id) = \frac{m}{2} \operatorname{Vol}(M, g)$.

Non-example: the formula gives no finite energy for the identity map of $\R$ with the Euclidean metric, since $\R$ is not compact.
There $e = \frac{1}{2}$ everywhere, and the integral over $\R$ diverges.
</document>

<document source="https://tamnd.github.io/math2/node/5N2CJZ47D4.md" tag="5N2CJZ47D4" kind="concept">
# Left ideal

## Summary

A left ideal of a [[ring|0B1261B5ED]] $R$ is an additive subgroup that is closed under multiplication on the left by every element of $R$.
Left ideals are exactly the submodules of $R$ viewed as a left module over itself.
In a commutative ring left ideals, right ideals and [[two-sided ideals|0CD87SFM2N]] all coincide with [[ideals|096Z9MP2ME]].
In a noncommutative ring the three notions differ, and left ideals carry much of the module theory of $R$.

## Explanations

### L1 human

A left ideal is a part of a ring that absorbs multiplication from the left: multiply one of its elements on the left by anything, and the product stays inside.
Multiplication on the right may lead out of it.
In the ring of $2 \times 2$ matrices, the matrices whose second column is zero form a left ideal.
Multiplying such a matrix on the left by any matrix keeps the second column zero.
Multiplying it on the right can fill the second column, so it is not a right ideal.
</document>

<document source="https://tamnd.github.io/math2/node/5N3P64YHQR.md" tag="5N3P64YHQR" kind="example">
# Pareto distribution

## Example

Let $x_m > 0$ and $\alpha > 0$.
A random variable $X$ has the *Pareto distribution* with scale $x_m$ and shape $\alpha$ if
$$P(X > x) = \left(\frac{x_m}{x}\right)^{\alpha} \quad \text{for } x \ge x_m, \qquad P(X > x) = 1 \quad \text{for } x < x_m.$$
This is a valid law: the [[distribution function|0H8H7ZTHW2]] $1 - (x_m/x)^\alpha$ is continuous and increasing from $0$ at $x_m$ to the limit $1$ as $x \to \infty$.
Its [[density|0ZY9DX8FBF]] is $f(x) = \alpha x_m^{\alpha} / x^{\alpha + 1}$ for $x \ge x_m$ and $0$ otherwise.
For $k > 0$, $E[X^k] = \int_{x_m}^\infty \alpha x_m^\alpha x^{k - \alpha - 1} \, dx$ is finite if and only if $k < \alpha$, and then equals $\frac{\alpha x_m^k}{\alpha - k}$.
So the mean is $\frac{\alpha x_m}{\alpha - 1}$ for $\alpha > 1$ and infinite for $\alpha \le 1$, and the variance is $\frac{\alpha x_m^2}{(\alpha - 1)^2 (\alpha - 2)}$ for $\alpha > 2$.

## Remarks

The variable $\ln(X / x_m)$ is [[exponential|0E99F51GVB]] with rate $\alpha$, since $P(\ln(X/x_m) > s) = P(X > x_m e^s) = e^{-\alpha s}$ for $s \ge 0$.
The law is scale free in the following sense: for $y \ge x_m$, given $X > y$ the variable $X$ is Pareto with scale $y$ and the same shape $\alpha$.
It is the basic example of a heavy-tailed law, with a tail that decays like a power of $x$; only the [[moments|01XAHPADBA]] of order less than $\alpha$ are finite.

## Explanations

### L1 human

A Pareto variable is at least some minimum value, and the chance that it exceeds $x$ falls like a power $x^{-\alpha}$.
Doubling the threshold always divides the chance of exceeding it by the same factor $2^\alpha$, at every scale.
For small $\alpha$, rare huge values dominate, and the mean can even be infinite.
Vilfredo Pareto used such laws to describe the distribution of wealth, and they also model city sizes and insurance losses.
</document>

<document source="https://tamnd.github.io/math2/node/5N5DXQMVP7.md" tag="5N5DXQMVP7" kind="concept">
# Riesz potential operator

## Summary

The Riesz potential operator $I_\alpha$ of order $\alpha$, with $0 < \alpha < n$, is the integral operator on $\R^n$ with kernel a constant times $\abs{x - y}^{\alpha - n}$.
It is the inverse of the fractional power $(-\Delta)^{\alpha/2}$ of the [[Laplacian|0E0B6RA8K0]], so it acts as integration of order $\alpha$ in $n$ variables.
For $n = 3$ and $\alpha = 2$ it is the Newtonian potential of physics.
Its mapping properties between [[Lp spaces|SASP14MA1Z]], given by the [[Hardy–Littlewood–Sobolev inequality|HG65C9R1SP]], underlie the Sobolev embedding theorems.

## Explanations

### L1 human

The gravitational potential of a mass distribution in space is obtained by adding up contributions $1/\abs{x - y}$ from each piece of mass at $y$.
The Riesz potential replaces the power $1/\abs{x - y}$ by $\abs{x - y}^{\alpha - n}$ in $n$ dimensions.
Small $\alpha$ gives a sharply peaked kernel that smooths a function only a little, and larger $\alpha$ smooths more.
In one dimension with $0 < \alpha < 1$, it is a two-sided version of the [[Riemann–Liouville integral|JTKCZ44NN3]] of order $\alpha$.
</document>

<document source="https://tamnd.github.io/math2/node/5N5SDY3K53.md" tag="5N5SDY3K53" kind="statement">
# Poincaré classification of circle homeomorphisms

## Statement

Let $f$ be an orientation-preserving [[homeomorphism|0QT00PB5G2]] of the circle $S^1 = \R / \Z$, with [[rotation number|DP4WWQBJJF]] $\rho = \rho(f)$.

1. Let $\rho = p/q$ with $p, q$ coprime integers and $q \ge 1$. Then $f$ has a [[periodic point|T75CZ5H47N]], and every periodic point of $f$ has least period $q$. The omega-limit set of every point is a periodic orbit.
2. Let $\rho$ be irrational. Then $f$ has no periodic point. There is a continuous surjective map $h \colon S^1 \to S^1$ of degree one, weakly preserving the cyclic order, with $h \circ f = R_\rho \circ h$. Here $R_\rho(x) = x + \rho$.
3. If $\rho$ is irrational and $f$ has a [[dense|0BSJSX7ME2]] orbit, then $h$ is a homeomorphism, so $f$ is [[topologically conjugate|HNXXAPQKG3]] to the rotation $R_\rho$.

## Remarks

The converse parts also hold: $f$ has a periodic point if and only if $\rho$ is rational.
For part 1, one shows that $\rho(F) = p/q$ for a lift $F$ forces the equation $F^q(x) = x + p$ to have a solution, which gives a periodic point.
Since $f$ preserves the cyclic order, a periodic orbit of $f$ is arranged on the circle like an orbit of the rotation by $p/q$.
This forces its least period to be $q$.
Between two consecutive fixed points of $f^q$ the map $f^q$ moves every point in the same direction.
So every orbit of $f^q$ is monotone and converges to a fixed point of $f^q$.
For part 2, $h$ is built by sending the orbit of a point $x$ to the orbit of $0$ under $R_\rho$.
This works because $f$ orders an orbit on the circle in the same way as $R_\rho$ does, and $h$ then extends by monotonicity.
When $\rho$ is irrational and $f$ has no dense orbit, the omega-limit set of every point is the same [[Cantor set|0BXNDZFQME]], and $f$ is only [[semiconjugate|AEXJNRBASV]] to $R_\rho$.
Example: for the rotation $R_{2/5}$ every point has period $5$, and for an irrational $\alpha$ the map $R_\alpha$ is an [[irrational rotation|2Q1J7RZ4TK]] with $h = \id$.

## Explanations

### L1 human

A homeomorphism of the circle moves points around while keeping their order.
Poincaré showed that the long-run average turn, the rotation number, tells what kind of motion is possible.
If it is a fraction $p/q$, there are points that come back after exactly $q$ steps, and all other points drift towards such cycles.
If it is irrational, nothing ever comes back exactly, and the map looks like a rigid irrational rotation, possibly after collapsing some intervals.

### L2 human

The theorem reduces the topological study of circle homeomorphisms to one number plus a little extra structure.
In the irrational case the remaining question is whether the semiconjugacy $h$ is a conjugacy, that is, whether $f$ has wandering intervals.
Denjoy's theorem answers this for smooth enough diffeomorphisms: a $C^2$ diffeomorphism with irrational rotation number is conjugate to the rotation.
In the rational case the periodic orbits all have the same period and the same combinatorics as the rotation by $p/q$.
This is the starting point of the study of Arnold tongues.
</document>

<document source="https://tamnd.github.io/math2/node/5NAGGDQQTM.md" tag="5NAGGDQQTM" kind="definition">
# Residue field of a point

## Definition

Let $X$ be a [[scheme|5A35TYZD02]] and let $x \in X$.
The [[stalk|JTAKAPTT8H]] $\mathcal{O}_{X,x}$ is a [[local ring|MZ02BKPHJA]]; let $\mathfrak{m}_x$ be its maximal ideal.
The *residue field* of $X$ at $x$ is the field
$$\kappa(x) = \mathcal{O}_{X,x} / \mathfrak{m}_x.$$
For an open set $U \ni x$ and $s \in \mathcal{O}_X(U)$, the *value* $s(x)$ is the image of $s$ in $\kappa(x)$.

## Remarks

Example: for $X = \operatorname{Spec} A$ and a prime ideal $\mathfrak{p}$ of $A$, the residue field is $\kappa(\mathfrak{p}) = A_{\mathfrak p} / \mathfrak{p} A_{\mathfrak p}$.
It is isomorphic to the field of fractions of $A / \mathfrak{p}$.
For $A = \Z$ this gives $\kappa((p)) = \F_p$ for a prime $p$ and $\kappa((0)) = \Q$.
For $A = \R[x]$ and $\mathfrak{p} = (x^2 + 1)$ it gives $\R[x]/(x^2+1) \cong \C$, so this point has residue field larger than $\R$.
Non-example: the stalk $\mathcal{O}_{X,x}$ itself is in general not a field; for $\Z$ at $(p)$ it is $\Z_{(p)}$, which has the non-unit $p \ne 0$.
For a field $K$, morphisms $\operatorname{Spec} K \to X$ correspond to pairs of a point $x \in X$ and a field embedding $\kappa(x) \to K$.
</document>

<document source="https://tamnd.github.io/math2/node/5NB63N9W3P.md" tag="5NB63N9W3P" kind="definition">
# Absolute continuity

## Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function.
The function $f$ is *absolutely continuous* on $[a, b]$ if for every $\varepsilon > 0$ there is $\delta > 0$ with the following property.
For every finite family of pairwise disjoint open [[intervals|0TK8Y3J5RW]] $(a_1, b_1), \dots, (a_n, b_n)$ contained in $[a, b]$,
$$\sum_{k=1}^{n} (b_k - a_k) < \delta \quad \text{implies} \quad \sum_{k=1}^{n} \abs{f(b_k) - f(a_k)} < \varepsilon.$$

## Remarks

Example: a [[Lipschitz function|0QNVQ4VD9T]] with constant $L > 0$ is absolutely continuous.
Given $\varepsilon$, take $\delta = \varepsilon / L$; then $\sum_k \abs{f(b_k) - f(a_k)} \le L \sum_k (b_k - a_k) < \varepsilon$.

Non-example: the [[Cantor function|NPBYYK63FE]] $c$ on $[0, 1]$ is continuous but not absolutely continuous.
For each $\delta > 0$, the Cantor set is covered by finitely many disjoint open intervals of total length less than $\delta$.
The changes of $c$ across these intervals add up to $c(1) - c(0) = 1$.

Equivalent form: $f$ is absolutely continuous on $[a, b]$ if and only if there is a Lebesgue integrable $g$ on $[a, b]$ with $f(x) = f(a) + \int_a^x g(t)\, dt$ for all $x$.
Then $f' = g$ almost everywhere.
With $n = 1$ only, the definition reduces to uniform continuity.
</document>

<document source="https://tamnd.github.io/math2/node/5NEJPXA6N4.md" tag="5NEJPXA6N4" kind="statement">
# Congruence criteria for triangles

## Statement

Let $ABC$ and $A'B'C'$ be [[triangles|0CTNFGYMM2]] in the Euclidean plane.
Write $XY$ for the [[distance|0NPPGHQNXQ]] between points $X$ and $Y$, and $\angle A$ for the [[angle|0GJAJC6TNZ]] $\angle BAC$, and so on.
The triangles are [[congruent|093EDENVA1]] under $A \mapsto A'$, $B \mapsto B'$, $C \mapsto C'$ in each of these cases.

1. Side-side-side: $AB = A'B'$, $BC = B'C'$ and $CA = C'A'$.
2. Side-angle-side: $AB = A'B'$, $AC = A'C'$ and $\angle A = \angle A'$.
3. Angle-side-angle: $\angle A = \angle A'$, $AB = A'B'$ and $\angle B = \angle B'$.
4. Angle-angle-side: $\angle A = \angle A'$, $\angle B = \angle B'$ and $BC = B'C'$.

## Remarks

The [[side-side-side criterion|0BRBYRK6PA]] has its own node.
For side-angle-side, the [[law of cosines|060ZVZBV3E]] gives $BC^2 = AB^2 + AC^2 - 2 \, AB \cdot AC \cos \angle A$, so $BC = B'C'$, and case 1 applies.
Angle-angle-side reduces to angle-side-angle, because the [[angle sum|05MDKQBEAF]] gives $\angle C = \angle C'$.
For angle-side-angle, the law of sines then fixes the two other sides.

Two other patterns are not criteria.
Side-side-angle fails: $AB = 2$, $BC = \sqrt{2}$ and $\angle A = 30^\circ$ fit two triangles.
By the law of sines $\sin \angle C = \frac{1}{\sqrt{2}}$, so $\angle C = 45^\circ$ or $\angle C = 135^\circ$.
Angle-angle-angle fails: triangles with equal angles are [[similar|0YF6QXXWB5]], but they can have different sizes.

## Explanations

### L1 human

These rules say how little you need to know about a triangle to pin it down.
Three sides are enough, and so are two sides with the angle between them, or one side with two angles.
Each rule says that two triangles that agree in these pieces are copies of each other.

### L2 human

In Euclid's Elements these are early propositions of Book I, and in Hilbert's axioms a form of side-angle-side is taken as an axiom.
The criteria are the main tool of synthetic proofs: one shows two segments or angles are equal by fitting them into congruent triangles.
Over the reals, each criterion follows from the law of cosines or the law of sines.
</document>

<document source="https://tamnd.github.io/math2/node/5NH4HVMEXA.md" tag="5NH4HVMEXA" kind="definition">
# Polygon triangulation

## Definition

Let $P \subset \R^2$ be a simple polygon, that is, a closed region bounded by a closed polygonal chain with $n \ge 3$ vertices that does not cross itself.
A *triangulation* of $P$ is a finite set $T$ of [[triangles|0CTNFGYMM2]] such that:

1. the union of the triangles of $T$ is $P$;
2. the vertices of each triangle of $T$ are vertices of $P$;
3. any two distinct triangles of $T$ meet in the empty set, in one common vertex, or in one common side.

Equivalently, a triangulation is given by a maximal set of diagonals of $P$ that pairwise meet only at endpoints.
A diagonal is a [[line segment|VQ4Q2V45RP]] joining two vertices of $P$ that lies in $P$ and meets the boundary of $P$ only at its endpoints.

## Remarks

Example: let $P$ be the square with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$.
The diagonal from $(0,0)$ to $(1,1)$ splits it into two triangles.
Their union is $P$, their vertices are vertices of $P$, and they meet in a common side, so all three clauses hold.

Non-example: split the same square into four triangles by joining the centre $(1/2, 1/2)$ to the four corners.
Clauses 1 and 3 hold, but clause 2 fails, since the centre is not a vertex of $P$.

Every simple polygon with $n$ vertices has a triangulation, and every triangulation has $n - 2$ triangles and $n - 3$ diagonals.
A convex polygon with $n$ vertices has exactly $C_{n-2}$ triangulations, where $C_m = \frac{1}{m+1}\binom{2m}{m}$ is the $m$-th Catalan number.
</document>

<document source="https://tamnd.github.io/math2/node/5NH9BXWAKR.md" tag="5NH9BXWAKR" kind="problem">
# Matrix Riemann–Hilbert problem

## Problem

Let $\Gamma \subset \C$ be a smooth simple closed curve, oriented counterclockwise, with interior domain $D^+$ and exterior domain $D^-$.
Let $n \ge 1$ and let $G \colon \Gamma \to \mathrm{GL}_n(\C)$ be a [[Hölder continuous|012PBA0FMD]] map into the [[invertible|0EB9N0S33C]] complex $n \times n$ matrices.
Let $g \colon \Gamma \to \C^n$ be Hölder continuous.
Find a function $\Phi$ with values in $\C^n$ that is [[holomorphic|0NAHXR1FRH]] on $D^+$ and on $D^-$.
It may have at most a pole of prescribed order at $\infty$.
It must have continuous boundary values $\Phi^+$ from $D^+$ and $\Phi^-$ from $D^-$.
These must satisfy
$$\Phi^+(t) = G(t)\, \Phi^-(t) + g(t), \quad t \in \Gamma.$$
The *homogeneous* problem is the case $g = 0$.

## Remarks

This is the vector form of the [[Riemann–Hilbert problem|RY8YQ9JZF9]]; for $n = 1$ it is the scalar problem, which is solved in closed form with logarithms and Cauchy integrals.
For $n \ge 2$ the scalar method fails, because the values $G(t)$ need not commute.

The general theory rests on a factorization $G = G_+\, \Lambda\, G_-$ on $\Gamma$, where $G_+$ extends holomorphically and invertibly to $D^+$, $G_-$ does the same on $D^-$ including $\infty$, and $\Lambda(t) = \operatorname{diag}(t^{\kappa_1}, \ldots, t^{\kappa_n})$ when $0 \in D^+$.
Such a factorization exists for every Hölder continuous invertible $G$, and the integers $\kappa_1 \ge \cdots \ge \kappa_n$, the partial indices, are unique.
Their sum is the [[winding number|0X7SBS8E0F]] of $\det G$ about $0$.
The number of solutions of the homogeneous problem and the solvability conditions of the inhomogeneous problem are expressed by the partial indices, not by their sum alone.
The same structure appears for matrix symbols on the real line in the [[matrix Wiener–Hopf factorization|2M672PP7QR]].
No general method computes the partial indices or the factors of a given $G$ in closed form.

Example: if $G$ is diagonal, the problem splits into $n$ scalar problems, and the partial indices are the winding numbers of the diagonal entries.
If $G$ is triangular, the problem is solved by a chain of scalar problems.

The problem is linked to Hilbert's twenty-first problem, which asks for a linear system of differential equations with regular singular points and prescribed monodromy.
Plemelj gave a positive answer for regular singular systems in 1908 through this problem.
Bolibruch showed in 1989 that the version asking for Fuchsian systems has a negative answer in general.
Matrix Riemann–Hilbert problems on contours with prescribed jumps are also the basis of the nonlinear steepest descent method for integrable systems and random matrices.

## Explanations

### L1 human

One looks for a vector of functions that is holomorphic on both sides of a closed curve.
Across the curve the vector jumps: the values from inside equal a given matrix times the values from outside.
For a single function this is solved exactly, but with matrices the order of multiplication matters, and the problem becomes much harder.
Its answer is controlled by a list of integers, the partial indices, which are hidden inside the jump matrix.
</document>

<document source="https://tamnd.github.io/math2/node/5NK10NGY13.md" tag="5NK10NGY13" kind="concept">
# Ruled surface

## Summary

In algebraic geometry, a ruled surface is a smooth projective [[algebraic surface|KFTECAATKG]] that maps onto a curve with every fibre a projective line.
So it is a family of lines parametrized by a curve, the algebraic analogue of a surface swept out by a moving line.
Ruled surfaces, together with the projective plane, are the minimal surfaces of Kodaira dimension $-\infty$.
They play a basic role in the classification of surfaces.

## Explanations

### L1 human

Imagine a curve, and attach to each of its points a copy of the projective line, varying smoothly.
The result is a surface ruled by these lines.
The simplest example is the product of a curve with a line.
The Hirzebruch surfaces are ruled surfaces over the projective line that are twisted versions of $\mathbb{P}^1 \times \mathbb{P}^1$.
</document>

<document source="https://tamnd.github.io/math2/node/5NN2YZZR3V.md" tag="5NN2YZZR3V" kind="concept">
# Fourier coefficient

## Summary

The $n$-th Fourier coefficient $\hat f(n)$ of a $2\pi$-periodic integrable function $f$ measures how much of the wave $e^{inx}$ is present in $f$.
It is the average of $f(x) e^{-inx}$ over one period, and it is the inner product of $f$ with the [[Fourier monomial|TZYSW0975T]] $e_n$.
The sequence $(\hat f(n))_{n \in \Z}$ determines $f$ up to a null set, and the [[Fourier series|G12KXDHPPS]] of $f$ is $\sum_n \hat f(n) e^{inx}$.
The size of the coefficients for large $\abs{n}$ reflects the smoothness of $f$.

## Explanations

### L1 human

To find how much of the frequency $n$ is in a signal, multiply the signal by the wave $e^{-inx}$ and average over one period.
The other frequencies average out to zero, and only the part with frequency $n$ survives.
For example, for $f(x) = 3 + 5 \cos 2x$ the coefficient $\hat f(0)$ is $3$, the coefficients $\hat f(2)$ and $\hat f(-2)$ are $\frac{5}{2}$, and all the others are $0$.
</document>

<document source="https://tamnd.github.io/math2/node/5NNPBFQA02.md" tag="5NNPBFQA02" kind="example">
# Compact symplectic group

## Example

Let $n \ge 1$ and let $\mathbb{H}$ be the [[quaternions|0GE4DVQKR8]].
For a quaternionic matrix $A \in M_n(\mathbb{H})$, let $A^*$ be its quaternionic conjugate transpose.
The *compact symplectic group* is
$$\mathrm{Sp}(n) = \set{A \in M_n(\mathbb{H}) : A^* A = I_n},$$
the group of $\mathbb{H}$-linear maps of $\mathbb{H}^n$ that preserve the quaternionic inner product $\ip{x}{y} = \sum_k \bar{x}_k y_k$.
It is also called the quaternionic unitary group.

For $n = 1$, the condition reads $\bar{q} q = 1$, so $\mathrm{Sp}(1)$ is the group of unit quaternions, which is isomorphic to [[SU(2)|E5P8AG6CK7]].

Writing $\mathbb{H}^n = \C^{2n}$ turns $\mathrm{Sp}(n)$ into a group of complex $2n \times 2n$ matrices, and then
$$\mathrm{Sp}(n) = \mathrm{Sp}(2n, \C) \cap U(2n),$$
the intersection of the complex [[symplectic group|X12FS2JAP0]] with the [[unitary group|0R9R0Y1HDK]].
This is a closed and bounded subgroup of $GL_{2n}(\C)$, so $\mathrm{Sp}(n)$ is a [[compact Lie group|9TFC6ADC1C]].

## Remarks

$\mathrm{Sp}(n)$ is connected and simply connected, and its real dimension is $n(2n + 1)$.
Its Lie algebra is simple of type $C_n$.
It is a maximal compact subgroup of $\mathrm{Sp}(2n, \C)$.
The maximal compact subgroup of the real group $\mathrm{Sp}(2n, \R)$ is different: it is isomorphic to $U(n)$.
The notation $\mathrm{Sp}(n)$ for this group clashes with the notation $\mathrm{Sp}(2n)$ for the noncompact symplectic groups, and some authors write $\mathrm{Sp}(2n)$ or $USp(2n)$ for it.

## Explanations

### L1 human

The compact symplectic group plays for the quaternions the role that the orthogonal group plays for the real numbers and the unitary group plays for the complex numbers.
It collects the quaternionic matrices that keep the length of every vector.
For $n = 1$ it is the set of quaternions of length $1$, which is a $3$-sphere.
Together with the special orthogonal and special unitary groups it gives the infinite families of compact simple Lie groups.
</document>

<document source="https://tamnd.github.io/math2/node/5NPVX29Y0C.md" tag="5NPVX29Y0C" kind="concept">
# Rounding

## Summary

Rounding replaces a number by a nearby number from a fixed set, such as the integers, the numbers with two decimals, or the [[floating-point numbers|RTKRSC3KPH]].
Rounding to nearest picks the closest element and needs a rule for ties, such as rounding half to even.
Directed rounding always goes down, up or toward zero; the [[floor function|08GR54CKEA]] is rounding down to an integer.
In a computer every arithmetic result is rounded, and the difference is the [[round-off error|CPN6SD3BTV]].

## Explanations

### L1 human

Rounding makes a number shorter and simpler, at the price of a small error.
A price of $3.47$ euros rounded to whole euros is $3$; rounded to one decimal it is $3.5$.
The number $2.5$ is exactly halfway between $2$ and $3$, so one needs a tie rule: "half up" gives $3$, "half to even" gives $2$.
Rounding half to even avoids a systematic upward drift when many rounded numbers are added.
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<document source="https://tamnd.github.io/math2/node/5NVA2JRHPV.md" tag="5NVA2JRHPV" kind="concept">
# Primitive recursive function

## Summary

A primitive recursive function is a function on the [[natural numbers|0MWKJ44V5A]] built from zero, successor and projections by composition and primitive recursion.
Primitive recursion defines $f(n + 1)$ from $f(n)$, like a for-loop whose number of steps is fixed in advance.
Every primitive recursive function is total and [[computable|KZSB5E471K]], and most functions met in ordinary number theory are primitive recursive.
Not every total computable function is primitive recursive; the [[Ackermann function|N470JF7ZJX]] is the classical example.

## Explanations

### L1 human

Start with the simplest functions: the constant $0$, adding one, and picking one argument out of a list.
Then allow two ways of building new functions: plugging functions into each other, and defining a value at $n + 1$ from the value at $n$.
Addition comes from adding one repeatedly, multiplication from adding repeatedly, and powers from multiplying repeatedly.
Every function obtained this way can be computed by a program whose loops have a bound known before the loop starts.
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<document source="https://tamnd.github.io/math2/node/5P5EMYHXE5.md" tag="5P5EMYHXE5" kind="concept">
# Hierarchical matrix

## Summary

A hierarchical matrix, or $\mathcal{H}$-matrix, is a matrix split into a hierarchy of blocks.
Each block that couples well-separated groups of indices has low [[rank|0ZC9YTXZRY]].
Such matrices are stored and multiplied with a vector in about $n \log n$ operations instead of $n^2$.
Approximate inverses and factorizations can be kept in the same format.
This gives fast solvers for the dense matrices of integral equations and for finite element matrices.

## Explanations

### L1 human

Think of $n$ points in space and a matrix whose entry $(i, j)$ measures the interaction of point $i$ with point $j$, such as $1 / \abs{x_i - x_j}$.
Two clusters of points that are far apart interact smoothly, so the block of the matrix linking them is close to a matrix of small rank.
Such a block can be stored with a few vectors instead of all its entries.
Only the blocks of nearby points are stored in full, and these are small.
A hierarchical matrix applies this idea at all scales at once.
</document>

<document source="https://tamnd.github.io/math2/node/5P608WRETM.md" tag="5P608WRETM" kind="concept">
# Knot Floer homology

## Summary

Knot Floer homology is an invariant of [[knots|QYEAZTYGPT]] in $3$-manifolds built from [[Heegaard Floer homology|2WJJBN2T5P]].
It was introduced by Ozsváth and Szabó, and independently by Rasmussen.
For a knot in $S^3$ it is a bigraded vector space whose graded Euler characteristic is the [[Alexander polynomial|JKA25KDPQP]].
It detects the Seifert genus of a knot and whether the knot is fibered.

## Explanations

### L1 human

Knot Floer homology describes a knot by a finite list of vector spaces, arranged in a grid by two gradings.
Adding up their dimensions with suitable signs recovers the Alexander polynomial.
But the list knows more: the highest row that is not zero tells you the least genus of a surface bounded by the knot.
It comes from counting certain holomorphic disks, a technique from symplectic geometry.
</document>

<document source="https://tamnd.github.io/math2/node/5P6NB8P1HG.md" tag="5P6NB8P1HG" kind="statement">
# Representer theorem

## Statement

Let $\mathcal X$ be a nonempty set and let $k \colon \mathcal X \times \mathcal X \to \R$ be a positive-definite kernel with [[reproducing kernel Hilbert space|4CK2GSRAGD]] $H$ and norm $\norm{\cdot}_H$.
Let $x_1, \dots, x_n \in \mathcal X$, let $c \colon \R^n \to \R \cup \set{\infty}$ be any function, and let $g \colon [0, \infty) \to \R$ be strictly increasing.
Then every minimiser $f \in H$ of
$$c(f(x_1), \dots, f(x_n)) + g(\norm{f}_H)$$
has the form
$$f = \sum_{i=1}^{n} \alpha_i \, k(\cdot, x_i)$$
for some real numbers $\alpha_1, \dots, \alpha_n$.

## Remarks

Proof idea: let $V$ be the span of $k(\cdot, x_1), \dots, k(\cdot, x_n)$ in the [[Hilbert space|HM2HHE3J51]] $H$, and write $f = f_V + f_\perp$ with $f_V \in V$ and $f_\perp \perp V$.
By the reproducing property $f(x_i) = \langle f, k(\cdot, x_i) \rangle_H = f_V(x_i)$, so the first term only depends on $f_V$.
Since $\norm{f}_H^2 = \norm{f_V}_H^2 + \norm{f_\perp}_H^2$ and $g$ is strictly increasing, any $f$ with $f_\perp \ne 0$ is beaten by $f_V$.
If $g$ is only nondecreasing, the same argument shows that, if a minimiser exists, some minimiser has this form.

Example: kernel ridge regression minimises $\frac1n \sum_{i=1}^{n} (y_i - f(x_i))^2 + \lambda \norm{f}_H^2$ with $\lambda > 0$.
By the theorem $f = \sum_i \alpha_i k(\cdot, x_i)$, and with the Gram matrix $K_{ij} = k(x_i, x_j)$ the problem becomes a quadratic problem in $\alpha \in \R^n$, solved by $\alpha = (K + n \lambda I)^{-1} y$.

Without the penalty $g(\norm{f}_H)$ the conclusion fails: if $H$ is infinite-dimensional, adding to a minimiser any nonzero $f_\perp \perp V$ gives another minimiser of $c(f(x_1), \dots, f(x_n))$ that is not in $V$.

## Explanations

### L1 human

Searching for the best function in an infinite-dimensional space looks hopeless.
The representer theorem says that, for this kind of penalised fitting problem, the best function is a combination of $n$ simple bumps, one centred at each data point.
So the problem reduces to finding $n$ numbers.

### L2 human

The theorem is what makes kernel methods computable: support vector machines, kernel ridge regression and smoothing splines all reduce to finite-dimensional problems in $\alpha$ involving only the Gram matrix $K$.
It is a form of regularized [[empirical risk minimization|56NVR0SS61]] in a [[reproducing kernel Hilbert space|4CK2GSRAGD]], where the penalty $\norm{f}_H$ controls smoothness.
The "kernel trick" follows: the solution can be evaluated at a new point $x$ as $\sum_i \alpha_i k(x, x_i)$ without ever working with elements of $H$ explicitly.
Versions of the theorem allow an additional unpenalised finite-dimensional part, such as a constant term.
</document>

<document source="https://tamnd.github.io/math2/node/5PFJBYDD2H.md" tag="5PFJBYDD2H" kind="concept">
# Discrete Painlevé equations

## Summary

Discrete Painlevé equations are integrable nonlinear second-order difference equations that play the role of the [[Painlevé equations|AZ17JCMRKB]] for recurrences.
They are non-autonomous, and in a suitable continuum limit they tend to one of the six Painlevé differential equations.
Their integrability shows up as singularity confinement, as a Lax pair of linear problems, and as a description by birational maps of rational surfaces.
They arose in the theory of orthogonal polynomials and random matrix models, where recurrence coefficients satisfy them.

## Explanations

### L1 human

The Painlevé equations are special nonlinear differential equations whose solutions behave unusually well.
Discrete Painlevé equations are recurrences with the same good behaviour: each new term is a rational function of the two previous ones and of the step number.
A typical one is $x_{n+1} + x_n + x_{n-1} = (\alpha n + \beta)/x_n + \gamma$, with constants $\alpha, \beta, \gamma$.
If a term happens to be $0$, the next terms blow up for a moment, but the recursion recovers after a few steps instead of losing information.
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<document source="https://tamnd.github.io/math2/node/5PGSQYB5Y1.md" tag="5PGSQYB5Y1" kind="concept">
# Logicism

## Summary

Logicism is the view that mathematics, or at least arithmetic, is a part of logic.
Its claim is that mathematical notions can be defined in purely logical terms and that mathematical truths can be proved from logical laws alone.
Gottlob Frege, and later Bertrand Russell and Alfred North Whitehead in Principia Mathematica, tried to carry out this program.
It is one of the classical answers in the [[foundations of mathematics|EFFRJFW67Y]].

## Explanations

### L1 human

A logicist says that numbers are not new things beyond logic.
For example, the number $2$ can be described as what all two-element collections have in common.
And "two-element" can be said in logic: "there are $x$ and $y$ with $x \ne y$, and nothing else".
Then a fact such as $1 + 1 = 2$ would be a truth of logic.
Frege's system turned out to be inconsistent because of Russell's paradox.
Later systems avoided the paradox, but they needed axioms, such as an axiom of infinity, that are hard to call purely logical.
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<document source="https://tamnd.github.io/math2/node/5PJF1NDCTT.md" tag="5PJF1NDCTT" kind="statement">
# Fukushima correspondence between Dirichlet forms and processes

## Statement

Let $X$ be a [[locally compact|0FE2SB8WVJ]] separable metric space and let $m$ be a positive [[Radon measure|HZNEQQBCW4]] on $X$ with full support.
Let $(\mathcal{E}, \mathcal{F})$ be a regular [[Dirichlet form|MH20Q2N1AH]] on $L^2(X, m)$, and let $(T_t)_{t \ge 0}$ be the strongly continuous symmetric Markovian [[contraction semigroup|XPWE94BV6N]] on $L^2(X, m)$ associated with it.
Then there is an $m$-symmetric [[Hunt process|Z7T669BQ9B]] $M$ on $X$ whose transition function $(p_t)_{t \ge 0}$ satisfies, for every $t > 0$ and every bounded Borel function $f \in L^2(X, m)$:
$$p_t f = T_t f \quad m\text{-almost everywhere}.$$
Such a process is unique up to equivalence.
That is, two $m$-symmetric Hunt processes with this property have the same transition functions outside a set of zero capacity for $\mathcal{E}$.
This exceptional set can be chosen so that neither process enters it.

## Remarks

The process is built from the semigroup, but the semigroup only acts on classes of functions defined almost everywhere.
The difficulty is to choose good pointwise versions, and this is done with the capacity of $\mathcal{E}$.
Every element of $\mathcal{F}$ has a quasi-continuous version, which is continuous outside open sets of small capacity.
Regularity of the form provides enough continuous functions in $\mathcal{F}$ to separate points and to build the process on $X$ itself.
Moreover $p_t f$ is a quasi-continuous version of $T_t f$.
Example: the form $\tfrac12 \int_{\R^n} \nabla u \cdot \nabla v \, dx$ on $W^{1,2}(\R^n)$ corresponds to Brownian motion on $\R^n$.
Ma and Röckner extended the correspondence to quasi-regular Dirichlet forms, which include forms on infinite-dimensional spaces.

## Explanations

### L1 human

A Dirichlet form is an energy for functions, and a Hunt process is a random motion that may jump or be killed.
The theorem says that every regular energy of this kind comes from a random motion.
Heat flow defined by the energy is the same as averaging over the random paths of the process.
So questions about the energy can be answered with random paths, and conversely.

### L2 human

The theorem is a central result of the monograph of Fukushima, Oshima and Takeda on Dirichlet forms.
Together with the [[Beurling–Deny formula|EK58S2872F]], it gives a meaning to the three parts of the form.
The local part, the jumping measure and the killing measure describe the continuous motion, the jumps and the killing of the process.
A regular Dirichlet form has the local property exactly when the process is a diffusion, that is, has continuous paths up to its lifetime.
A nearly Borel set has zero capacity if and only if the process, started from $m$-almost every point, never hits it, which gives a probabilistic meaning to capacity.
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# Pseudoholomorphic curve

## Definition

Let $(M, J)$ be an [[almost complex manifold|9SYWRGVTRS]], so $J$ is a smooth endomorphism of $TM$ with $J^2 = -\id$.
Let $(\Sigma, j)$ be a [[Riemann surface|5K0N8QMV6Q]], with $j$ its complex structure on $T\Sigma$.
A *pseudoholomorphic curve* (or *$J$-holomorphic curve*) is a smooth map $u \colon \Sigma \to M$ such that
$$du \circ j = J \circ du.$$

## Remarks

In a local coordinate $z = s + it$ on $\Sigma$, the condition reads $\partial_s u + J(u) \, \partial_t u = 0$, a nonlinear Cauchy–Riemann equation.

Example: let $M = \C^n$ with $J$ the multiplication by $i$, and $\Sigma$ an open subset of $\C$.
Then $u$ is pseudoholomorphic exactly when each component is a holomorphic function, by the Cauchy–Riemann equations.
So $u(z) = (z, z^2)$ is a pseudoholomorphic curve in $\C^2$.

Non-example: $u(z) = \bar z$ from $\C$ to $\C$.
With $z = s + it$ and target coordinate $x + iy$, one has $du(j \partial_s) = du(\partial_t) = -\partial_y$, while $J\,du(\partial_s) = J \partial_x = \partial_y$.
So $du \circ j = -J \circ du \ne J \circ du$; the map is antiholomorphic.

If $J$ is tamed by a symplectic form $\omega$, meaning $\omega(v, Jv) > 0$ for $v \ne 0$, then the energy $\int_\Sigma u^* \omega$ of a nonconstant pseudoholomorphic curve is positive.
When $\Sigma$ is closed and $\omega$ is closed, this energy depends only on the homology class of $u$.
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# Tropical semiring

## Example

Let $T = \R \cup \set{+\infty}$.
For $a, b \in T$, define
$$a \oplus b = \min(a, b), \qquad a \otimes b = a + b,$$
with the conventions $\min(a, +\infty) = a$ and $a + (+\infty) = +\infty$.
Then $(T, \oplus, \otimes)$ is a commutative [[semiring|D31S3GR51T]], the *tropical semiring* in its min-plus form.

The check of the semiring axioms:

1. $\oplus$ is associative and commutative, and $+\infty$ is its neutral element, since $\min(a, +\infty) = a$.
2. $\otimes$ is associative and commutative, and $0$ is its neutral element, since $a + 0 = a$.
3. $\otimes$ distributes over $\oplus$: $a + \min(b, c) = \min(a + b, a + c)$, because adding $a$ preserves the order.
4. $+\infty$ is absorbing for $\otimes$: $a + (+\infty) = +\infty$.

For instance, $3 \oplus 5 = 3$ and $3 \otimes 5 = 8$, and $2 \otimes (3 \oplus 7) = 5 = (2 \otimes 3) \oplus (2 \otimes 7)$.

## Remarks

The semiring is [[idempotent|JCSAC9Y1CV]]: $a \oplus a = \min(a, a) = a$.
It is not a ring, since $3$ has no additive inverse: $\min(3, b) = +\infty$ has no solution $b$ in $T$.
Every element $a \ne +\infty$ has the inverse $-a$ for $\otimes$, so the tropical semiring is a [[semifield|AHH97EJ6VH]].
The map $x \mapsto -x$ is an isomorphism onto the max-plus semiring $(\R \cup \set{-\infty}, \max, +)$, the base of [[max-plus algebra|64G0VSC5F4]].
Some authors call that max-plus form the tropical semiring instead.

## Explanations

### L1 human

In tropical arithmetic, "plus" means "take the smaller" and "times" means "add".
So $3 \oplus 5 = 3$ and $3 \otimes 5 = 8$.
These rules still obey the usual laws of reordering, regrouping and distributing.
They describe shortest paths: along a route the lengths add, and between two routes one takes the shorter.
</document>

<document source="https://tamnd.github.io/math2/node/5PP333WZEK.md" tag="5PP333WZEK" kind="definition">
# Soundness

## Definition

Let $\vdash$ be the derivability relation of a proof system, such as a [[Hilbert system|1MGMQX43B8]].
Let $\models$ be [[logical consequence|1HK0VYWTV7]] for the same language.
The proof system is *sound* if $\Gamma \vdash \varphi$ implies $\Gamma \models \varphi$, for every set of formulas $\Gamma$ and every formula $\varphi$.
It is *weakly sound* if $\vdash \varphi$ implies $\models \varphi$ for every formula $\varphi$.

## Remarks

The standard Hilbert system for propositional logic is sound.
Each axiom is a [[tautology|05006AJ7DZ]], checked by a truth table, and modus ponens preserves truth under every assignment.
Induction on the length of a derivation then gives soundness.
Adding the rule "from $\psi$ and $\varphi \to \psi$, infer $\varphi$" gives an unsound system.
With it, $\set{p \to q, \; q} \vdash p$, but the assignment $p = \mathsf{F}$, $q = \mathsf{T}$ shows $\set{p \to q, \; q} \not\models p$.
The converse property, that $\Gamma \models \varphi$ implies $\Gamma \vdash \varphi$, is completeness.
For first-order logic it is [[Gödel's completeness theorem|WQ1ZE255SF]].
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# Eulerian number

## Summary

The Eulerian number $A(n, k)$ counts the [[permutations|05RWFKVTJ7]] of $\set{1, \ldots, n}$ with exactly $k$ descents, see the [[definition|B2EMR1MA1H]].
For each $n \ge 1$ the numbers $A(n, 0), \ldots, A(n, n-1)$ add up to $n!$.
They are the coefficients of the Eulerian polynomials, which appear in the sums $\sum_{m \ge 0} m^n t^m$.
They should not be confused with the Euler numbers or with Eulerian paths in graphs.

## Explanations

### L1 human

Write a permutation as a row of numbers and mark each place where the next number is smaller.
These places are the descents.
For $n = 3$, the row $1\,2\,3$ has no descent, the row $3\,2\,1$ has two, and the other four rows have exactly one.
So the Eulerian numbers for $n = 3$ are $1, 4, 1$.
</document>

<document source="https://tamnd.github.io/math2/node/5PTYE97PF6.md" tag="5PTYE97PF6" kind="object">
# Siegel upper half-space

## Description

Let $g \ge 1$ be an integer.
The *Siegel upper half-space* of degree $g$ is
$$\mathbb{H}_g = \set{Z \in M_{g \times g}(\C) : Z^T = Z,\ \operatorname{Im} Z \text{ is positive definite}}.$$
It is an open subset of the space of complex symmetric $g \times g$ matrices, so it is a complex manifold of dimension $g(g+1)/2$.
For $g = 1$ it is the [[upper half-plane|X6XGGH5FPW]].

The real [[symplectic group|X12FS2JAP0]] $Sp(2g, \R)$, the group of real $2g \times 2g$ matrices $M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$ with $M^T J M = J$ for $J = \begin{pmatrix} 0 & I_g \\ -I_g & 0 \end{pmatrix}$, acts on $\mathbb{H}_g$ by
$$M \cdot Z = (AZ + B)(CZ + D)^{-1}.$$
For $Z = X + iY \in \mathbb{H}_g$, the matrix $\begin{pmatrix} Y^{1/2} & X Y^{-1/2} \\ 0 & Y^{-1/2} \end{pmatrix}$ is symplectic and sends $iI_g$ to $Z$, so the action is transitive.
The stabilizer of $iI_g$ is the subgroup with $A = D$ and $C = -B$, which is isomorphic to the unitary group $U(g)$ via $A + iB$.
Hence $\mathbb{H}_g \cong Sp(2g, \R)/U(g)$.

## Remarks

The Siegel upper half-space is a [[Hermitian symmetric space|NT4797Z9H8]] of noncompact type, the type $III_g$ in Cartan's list.
The Cayley transform $Z \mapsto (Z - iI)(Z + iI)^{-1}$ maps it biholomorphically onto the bounded domain $\set{W : W^T = W,\ I - W^* W > 0}$.
The quotient $Sp(2g, \Z) \backslash \mathbb{H}_g$ is the moduli space of principally polarized abelian varieties of dimension $g$: the point $Z$ corresponds to the complex torus $\C^g / (\Z^g + Z \Z^g)$.
Siegel modular forms are functions on $\mathbb{H}_g$ with a transformation law under $Sp(2g, \Z)$.

## Explanations

### L1 human

The upper half-plane is the set of complex numbers with positive imaginary part.
The Siegel upper half-space replaces the number by a symmetric matrix and "positive" by "positive definite".
Just as the upper half-plane parametrizes elliptic curves, the Siegel space parametrizes higher-dimensional complex tori with a polarization, through their period matrices.
</document>

<document source="https://tamnd.github.io/math2/node/5PVK5PVZBA.md" tag="5PVK5PVZBA" kind="definition">
# Barycentric subdivision

## Definition

Let $K$ be a geometric simplicial complex in $\R^N$, so each simplex $\sigma$ of $K$ is the [[convex hull|0FQSDC2EQ2]] of affinely independent vertices $v_0, \dots, v_n$.
The *barycentre* of $\sigma$ is the point
$$b(\sigma) = \frac{1}{n + 1} \sum_{i=0}^{n} v_i.$$
The *barycentric subdivision* $\operatorname{sd} K$ is the geometric simplicial complex with the following simplices.
For each chain $\sigma_0 \subsetneq \sigma_1 \subsetneq \cdots \subsetneq \sigma_k$ of simplices of $K$, it has the convex hull of $\set{b(\sigma_0), \dots, b(\sigma_k)}$.

## Remarks

The union of the simplices of $\operatorname{sd} K$ equals the union of the simplices of $K$, and each simplex of $\operatorname{sd} K$ lies in a simplex of $K$.
If every simplex of $K$ has diameter at most $d$ and $K$ has dimension at most $n$, then every simplex of $\operatorname{sd} K$ has diameter at most $\frac{n}{n + 1} d$.

For an abstract [[simplicial complex|V8GZ0W1ZZ5]] $K$, the vertices of $\operatorname{sd} K$ are the nonempty faces of $K$.
The faces of $\operatorname{sd} K$ are the finite chains of nonempty faces under inclusion.

Example: let $K$ be a $2$-simplex with vertices $v_0, v_1, v_2$ and all its faces.
The maximal chains are $\set{v_i} \subsetneq \set{v_i, v_j} \subsetneq \set{v_0, v_1, v_2}$ for the $6$ ordered pairs $i \ne j$, so $\operatorname{sd} K$ has $6$ triangles, $7$ vertices and $12$ edges.

Non-example: cutting the triangle into $3$ triangles by joining its centroid to the three corners gives a subdivision, but not the barycentric one.
The barycentres of the edges are not vertices of it, though each edge is a simplex of $K$.
</document>

<document source="https://tamnd.github.io/math2/node/5PVSQXCTCT.md" tag="5PVSQXCTCT" kind="concept">
# Fourier transform of tempered distributions

## Summary

The Fourier transform of a tempered distribution $T$ is again a tempered distribution.
It acts on a Schwartz function $\varphi$ as $T$ acts on the Fourier transform of $\varphi$.
It extends the Fourier transform from integrable and square-integrable functions to all tempered distributions.
It is a linear bijection of the space of tempered distributions onto itself.
It turns derivatives into multiplication by polynomials, which makes it a main tool for linear partial differential equations.

## Explanations

### L1 human

Many useful objects, such as constant functions, polynomials and the Dirac delta, have no Fourier transform as an integral.
The trick is to move the transform onto smooth, fast-decaying test functions, where the integral always makes sense.
In this way the delta at $0$ has Fourier transform the constant function $1$, and the constant $1$ has Fourier transform the delta.
This matches the idea that a spike contains all frequencies equally.
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# Frobenius theorem

## Statement

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.
Let $D \subset TM$ be a smooth subbundle of rank $k$ of the [[tangent bundle|WY6K3P4YCG]], called a distribution.
Call $D$ *involutive* if for all smooth vector fields $X, Y$ with values in $D$, the [[Lie bracket|68FGQEBD6J]] $[X, Y]$ also has values in $D$.
Call $D$ *integrable* if every point of $M$ lies in an integral manifold of $D$.
An integral manifold is an immersed submanifold $N$ with $T_qN = D_q$ for all $q \in N$.
Then the following are equivalent.

1. $D$ is involutive.
2. $D$ is integrable.
3. Every point of $M$ has a chart with coordinates $x^1, \dots, x^n$ in which $D$ is spanned by $\partial_1, \dots, \partial_k$.

## Remarks

Condition 3 implies condition 2, since the slices $x^{k+1} = c_{k+1}, \dots, x^n = c_n$ are integral manifolds.
Condition 2 implies condition 1, since vector fields tangent to a submanifold have a bracket tangent to it.
The main step is that condition 1 implies condition 3.
One picks local vector fields spanning $D$ that pairwise commute, which involutivity allows after a change of basis, and then uses their commuting flows as coordinates.
Example: on $\R^3$ the distribution spanned by $X = \partial_x$ and $Y = \partial_y + x \, \partial_z$ is the kernel of $dz - x \, dy$.
Here $[X, Y] = \partial_z$, which is not in $D$, so no surface is tangent to $D$; this is the standard contact structure.
In the global form of the theorem, an involutive distribution splits $M$ into maximal connected integral manifolds, the leaves of a foliation.

## Explanations

### L1 human

At every point of a space, choose a plane of allowed directions.
Can the space be sliced into surfaces that are tangent to these planes everywhere, like the pages of a book?
The Frobenius theorem says this happens exactly when moving back and forth along allowed directions never produces a forbidden direction, a condition tested by the Lie bracket.

### L2 human

The theorem turns a question about solving a system of first-order PDEs into an algebraic check on brackets.
For $k = 1$ every line field is involutive, and the theorem reduces to the existence of integral curves of a vector field.
A dual form uses differential forms: $D = \ker \theta^1 \cap \dots \cap \ker \theta^{n-k}$ is involutive if and only if each $d\theta^i$ lies in the ideal generated by the $\theta^j$.
The theorem is used to integrate Lie subalgebras to Lie subgroups, to show that flat connections are locally trivial, and as the foundation of the theory of foliations.
</document>

<document source="https://tamnd.github.io/math2/node/5Q53XWV0QN.md" tag="5Q53XWV0QN" kind="concept">
# De Morgan algebra

## Summary

A De Morgan algebra is a bounded [[distributive lattice|0HM0HQCK6N]] with a negation $\sim$ that is an involution and satisfies [[De Morgan's laws|09PKJKA6J0]].
Unlike the complement of a [[Boolean algebra|0XADJKM699]], this negation need not satisfy $a \wedge {\sim} a = 0$ or $a \vee {\sim} a = 1$.
De Morgan algebras give the algebraic semantics of several many-valued and relevance logics.
Every Boolean algebra is a De Morgan algebra.

## Explanations

### L1 human

Take the numbers from $0$ to $1$ as degrees of truth, with "and" as the minimum, "or" as the maximum, and "not $x$" as $1 - x$.
This negation turns "and" into "or" and back, and doing it twice gives the start again.
But the statement "$x$ and not $x$" need not be fully false: for $x = 1/2$ it has truth degree $1/2$.
A De Morgan algebra keeps these rules of negation and drops the laws $x \wedge {\sim} x = 0$ and $x \vee {\sim} x = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/5Q5VN56QH4.md" tag="5Q5VN56QH4" kind="statement">
# Norm residue isomorphism theorem

## Statement

Let $K$ be a field, let $\ell$ be a [[prime number|0SN0MAEKYX]] different from the [[characteristic|1ZB13N78XW]] of $K$, and let $n \ge 0$.
Let $K^s$ be a [[separable closure|JJ91A1V8NQ]] of $K$, and let $\mu_\ell \subseteq K^s$ be the group of $\ell$-th roots of unity.
Let $\mu_\ell^{\otimes n}$ be its $n$-fold tensor power over $\Z$, with $\mu_\ell^{\otimes 0} = \Z/\ell\Z$.
Let $\delta \colon K^\times \to H^1(K, \mu_\ell)$ be the Kummer map.
Then the norm residue map
$$K_n^M(K)/\ell \longrightarrow H^n(K, \mu_\ell^{\otimes n}), \qquad \set{a_1, \ldots, a_n} \longmapsto \delta(a_1) \cup \cdots \cup \delta(a_n),$$
from [[Milnor K-theory|2YXDQ7Q445]] modulo $\ell$ to [[Galois cohomology|EESNTXFWHZ]], is well defined and is an isomorphism of groups.

## Remarks

The map is well defined because the cup product $\delta(a) \cup \delta(1 - a)$ vanishes for $a \ne 0, 1$.
For $n = 0$ the statement is trivial, and for $n = 1$ it is Kummer theory with [[Hilbert's Theorem 90|YFF8B5A0QG]]: $K^\times/(K^\times)^\ell \cong H^1(K, \mu_\ell)$.
For $n = 2$ it is the Merkurjev–Suslin theorem, which identifies $K_2(K)/\ell$ with the $\ell$-torsion of the [[Brauer group|BEW48TA9WZ]] when $K$ contains the $\ell$-th roots of unity.
The case $\ell = 2$ is the Milnor conjecture, proved by Voevodsky; the general case, formerly the Bloch–Kato conjecture, was completed by Voevodsky with essential contributions of Rost.
The proof uses motivic cohomology and is far beyond elementary methods.
The theorem also holds with $\ell$ replaced by any positive integer $m$ invertible in $K$, with $\mu_m^{\otimes n}$ in place of $\mu_\ell^{\otimes n}$.

## Explanations

### L1 human

Milnor K-groups are defined by simple generators and relations, while Galois cohomology groups are defined through the symmetries of field extensions.
The theorem says that, modulo a prime $\ell$, these two very different constructions give the same groups.
So questions about Galois cohomology can be answered with explicit symbols, and the other way round.

### L2 human

In degree $2$, suppose $K$ contains the $\ell$-th roots of unity.
Then every central simple algebra of exponent $\ell$ is Brauer equivalent to a tensor product of cyclic algebras.
For $\ell = 2$ it yields, with further work, the description of the graded Witt ring of quadratic forms by Galois cohomology, also called the Milnor conjecture on quadratic forms.
</document>

<document source="https://tamnd.github.io/math2/node/5Q9BNSZ5XZ.md" tag="5Q9BNSZ5XZ" kind="definition">
# Epigraph

## Definition

Let $X$ be a set and let $f \colon X \to [-\infty, +\infty]$ be a function with values in the [[extended real line|0GSZ7B0CGR]].
The *epigraph* of $f$ is the subset of the [[Cartesian product|09CEDJYBKP]] $X \times \R$ given by
$$\operatorname{epi} f = \set{(x, r) \in X \times \R : f(x) \le r}.$$

## Remarks

Only real numbers $r$ are used, so points $x$ with $f(x) = +\infty$ contribute nothing, and points with $f(x) = -\infty$ contribute the whole line $\set{x} \times \R$.
The *strict epigraph* uses $f(x) < r$ instead.
If $X$ is a real vector space, $f$ is convex if and only if $\operatorname{epi} f$ is a convex set.
If $X$ is a topological space, $f$ is lower semicontinuous if and only if $\operatorname{epi} f$ is closed in $X \times \R$.
The function is recovered as $f(x) = \inf \set{r \in \R : (x, r) \in \operatorname{epi} f}$, with $\inf \emptyset = +\infty$.

Example: for $f(x) = x^2$ on $\R$, the point $(1, 2)$ lies in $\operatorname{epi} f$, since $f(1) = 1 \le 2$.

Non-example: the point $(2, 1)$ does not lie in $\operatorname{epi} f$, since $f(2) = 4 > 1$.
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# Sphere theorem for 3-manifolds

## Statement

Let $M$ be an [[orientable|702SNFQG0W]] [[3-manifold|R20TB2KEQQ]] whose second [[homotopy group|P976WYEFMB]] $\pi_2(M)$ is not trivial.
Then there is an embedding $S^2 \to M$ of the $2$-sphere that is not homotopic to a constant map.

## Remarks

Papakyriakopoulos proved the theorem in 1957 with the tower construction also used for [[Dehn's lemma|FH5WSPD79F]].
For example, in $M = S^2 \times S^1$ the sphere $S^2 \times \set{1}$ is embedded and generates $\pi_2(M) \cong \Z$.
A main consequence: if $M$ is orientable and irreducible, meaning that every embedded $2$-sphere bounds a $3$-ball, then $\pi_2(M) = 0$.
If moreover $\pi_1(M)$ is infinite, then $M$ is aspherical, that is, its universal cover is contractible.
Orientability cannot be dropped: in the non-orientable case one must also allow embedded two-sided projective planes, as in Epstein's version of the theorem.

## Explanations

### L1 human

The second homotopy group records spheres in a space that cannot be shrunk to a point.
The sphere theorem says that, in an orientable $3$-manifold, if there is any such sphere, then there is one that does not cross itself.
So a nontrivial $\pi_2$ is always witnessed by an embedded sphere.

### L2 human

The theorem explains why irreducible orientable $3$-manifolds have trivial $\pi_2$.
With infinite fundamental group, they are then Eilenberg–MacLane spaces, determined up to homotopy by $\pi_1$.
This is the starting point for the study of $3$-manifolds through their fundamental groups.
</document>

<document source="https://tamnd.github.io/math2/node/5QJYCNRYMQ.md" tag="5QJYCNRYMQ" kind="concept">
# Bilinear interpolation

## Summary

Bilinear interpolation estimates a function of two variables inside a rectangle from its values at the four corners.
It uses the unique function of the form $a + bx + cy + dxy$ that matches the four values.
It is the two-dimensional analogue of linear interpolation and a basic tool for resampling images and gridded data.

## Explanations

### L1 human

Suppose the temperature is known at the four corners of a rectangular field.
To estimate it at a point inside, first interpolate linearly along the bottom edge and along the top edge, and then interpolate linearly between those two results.
Doing it in the other order, first along the vertical edges, gives the same answer.
For example, on the unit square with corner values $0$, $0$, $0$ and $1$ (the last at $(1, 1)$), the estimate at $(x, y)$ is $xy$, so the center gets $1/4$.
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<document source="https://tamnd.github.io/math2/node/5QRFTBF4PG.md" tag="5QRFTBF4PG" kind="concept">
# Injective object

## Summary

An injective object of a [[category|A2DE66937D]] is an object $I$ such that maps into $I$ extend along every [[monomorphism|Y2NP1JFR86]].
In a category of modules, the injective objects are exactly the [[injective modules|85H7PRHDPZ]].
Among abelian groups they are the [[divisible groups|ZXVBEPMTCN]], such as $\Q$ and $\Q/\Z$.
Injective objects are used to build injective resolutions, from which right [[derived functors|NPMC3K6J2F]] such as sheaf cohomology are computed.
The dual notion is the [[projective object|YXMXAGVC1Q]].

## Explanations

### L1 human

Suppose $A$ sits inside a larger object $B$, and a map is given on $A$ with values in $I$.
The object $I$ is injective if such a map can always be extended to all of $B$.
The rational numbers have this property among abelian groups, because one can always divide in $\Q$ to find a value for a new element.
The integers do not: the map $2\Z \to \Z$ sending $2$ to $1$ cannot be extended to $\Z$, since its value at $1$ would be an integer whose double is $1$.
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# Cellular automaton

## Summary

A cellular automaton is a grid of cells, each in one of finitely many states.
All cells update at the same time by one local rule.
The new state of a cell depends only on the old states of the cells in a fixed finite neighbourhood.
As a dynamical system it is a continuous map of the configuration space that commutes with the shifts, by the [[Curtis–Hedlund–Lyndon theorem|N35QM9G83G]].
Very simple rules can produce complex behaviour, so cellular automata serve as models in physics and biology and as a model of computation.

## Explanations

### L1 human

Imagine a row of light bulbs, each on or off.
At each tick of a clock, every bulb looks at itself and its two neighbours and decides whether to be on or off by a fixed table.
All bulbs follow the same table and switch at the same moment.
In rule 90, a bulb is on at the next tick exactly when one of its two neighbours, but not both, is on now.
Started from a single lit bulb, this rule draws a Sierpinski triangle over time.
Conway's Game of Life is a famous two-dimensional example.
See the [[definition|2CM2MC5HQ6]].
</document>

<document source="https://tamnd.github.io/math2/node/5R6QZ388BB.md" tag="5R6QZ388BB" kind="concept">
# Finsler manifold

## Summary

A Finsler manifold is a [[smooth manifold|F45VQ47GPV]] with a smoothly varying, strongly convex [[norm|0V2WP9B1MF]]-like function on each tangent space.
It generalises a [[Riemannian manifold|7CBT4SM83Y]], where the length of a tangent vector comes from an inner product.
Lengths of curves, distances and geodesics are defined as in Riemannian geometry.
But the unit ball in each tangent space may be any smooth strongly convex body, not only an ellipsoid.

## Explanations

### L1 human

In Riemannian geometry, the set of tangent vectors of length one at a point is always an ellipse or an ellipsoid.
Finsler geometry allows this unit set to be any smooth convex shape around the origin.
Then the cost of moving can depend on the direction, as when walking with or against a wind.
The length need not even be symmetric: going one way along a path can cost less than coming back.
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<document source="https://tamnd.github.io/math2/node/5R76HPDB8W.md" tag="5R76HPDB8W" kind="definition">
# Monoidal category

## Definition

A *monoidal category* is a [[category|A2DE66937D]] $\mathcal{C}$ together with the following data:

1. a [[functor|28NQ0KRHD3]] $\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ on the [[product category|AFNE9C4NH6]], called the tensor product;
2. an object $I$, called the unit object;
3. a natural isomorphism $\alpha$ with components $\alpha_{A,B,C} \colon (A \otimes B) \otimes C \to A \otimes (B \otimes C)$, called the associator;
4. natural isomorphisms $\lambda$ and $\rho$ with components $\lambda_A \colon I \otimes A \to A$ and $\rho_A \colon A \otimes I \to A$, called the left and right unitors.

Here natural means natural as a [[natural transformation|KYCPCC57FT]] in each object variable.
These data must satisfy two axioms for all objects $A, B, C, D$:

1. pentagon: $(\id_A \otimes \alpha_{B,C,D}) \circ \alpha_{A, B \otimes C, D} \circ (\alpha_{A,B,C} \otimes \id_D) = \alpha_{A,B,C \otimes D} \circ \alpha_{A \otimes B, C, D}$ as morphisms $((A \otimes B) \otimes C) \otimes D \to A \otimes (B \otimes (C \otimes D))$;
2. triangle: $(\id_A \otimes \lambda_B) \circ \alpha_{A,I,B} = \rho_A \otimes \id_B$ as morphisms $(A \otimes I) \otimes B \to A \otimes B$.

## Remarks

Example: $\mathbf{Set}$ with the Cartesian product $A \times B$, a one-point set $I = \set{*}$, $\alpha((a,b),c) = (a,(b,c))$, $\lambda(*,a) = a$ and $\rho(a,*) = a$.
Both sides of the pentagon send $(((a,b),c),d)$ to $(a,(b,(c,d)))$, and both sides of the triangle send $((a,*),b)$ to $(a,b)$.
Non-example: on the category of $\R$-vector spaces with $\otimes$ the usual tensor product, take $\alpha$ to be $-1$ times the usual associator and the usual unitors.
For nonzero spaces the pentagon then fails: its left side contains three copies of $\alpha$ and its right side two, so the two sides differ by the sign $-1$.
A monoidal category is *strict* when $\alpha$, $\lambda$ and $\rho$ are identities.
</document>

<document source="https://tamnd.github.io/math2/node/5R9ZRK8YB5.md" tag="5R9ZRK8YB5" kind="definition">
# MacCormack method

## Definition

Let $f \colon \R^m \to \R^m$ be smooth, and consider the [[conservation law|8RA7AMWSAF]] $u_t + f(u)_x = 0$.
Fix $\Delta x > 0$ and $\Delta t > 0$, put $\lambda = \Delta t / \Delta x$, and let $u_j^n$ approximate $u(j \Delta x, n \Delta t)$.
The *MacCormack method* computes $u^{n+1}$ from $u^n$ in two steps.

1. Predictor, with a forward difference: $u_j^{*} = u_j^n - \lambda \left( f(u_{j+1}^n) - f(u_j^n) \right)$.
2. Corrector, with a backward difference: $u_j^{n+1} = \frac12 \left( u_j^n + u_j^{*} \right) - \frac{\lambda}{2} \left( f(u_j^{*}) - f(u_{j-1}^{*}) \right)$.

## Remarks

Example: for $f(u) = a u$ and $\nu = a \lambda$, the predictor gives $u_j^* - u_{j-1}^* = (u_j^n - u_{j-1}^n) - \nu (u_{j+1}^n - 2u_j^n + u_{j-1}^n)$.
Putting this into the corrector gives
$$u_j^{n+1} = u_j^n - \frac{\nu}{2} (u_{j+1}^n - u_{j-1}^n) + \frac{\nu^2}{2} (u_{j+1}^n - 2u_j^n + u_{j-1}^n),$$
which is the [[Lax–Wendroff method|YHZ9BCNDRZ]]; so the method is second-order accurate and stable for $\abs{\nu} \le 1$ in this case.
Non-example: using a forward difference in the corrector as well gives, for $f(u) = a u$, the scheme $u_j^{n+1} = u_j^n - \nu (u_{j+1}^n - u_j^n) + \frac{\nu^2}{2}(u_{j+2}^n - 2u_{j+1}^n + u_j^n)$, whose Taylor expansion leaves the error term $-\frac{a \Delta x}{2} u_{xx}$, so it is only first order.
The order of the one-sided differences may be reversed (backward in the predictor, forward in the corrector); alternating the two orders from step to step is common.

## Explanations

### L1 human

Predict with differences pointing forward, correct with differences pointing back, and average.
For linear problems this is exactly the Lax–Wendroff scheme.
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<document source="https://tamnd.github.io/math2/node/5RD6ZFKPDS.md" tag="5RD6ZFKPDS" kind="definition">
# Translation equation

## Definition

Let $X$ be a set.
The *translation equation* on $X$ is the [[functional equation|SRAN4EKWS0]]
$$F(F(x, s), t) = F(x, s + t) \quad \text{for all } x \in X \text{ and } s, t \in \R,$$
for an unknown [[function|0EPAT51DXQ]] $F \colon X \times \R \to X$.
A solution that also satisfies $F(x, 0) = x$ for all $x \in X$ is called a *flow* or *one-parameter group of transformations* of $X$.

## Remarks

Example: $F(x, t) = x e^t$ on $X = \R$ is a solution, since $F(F(x, s), t) = x e^s e^t = x e^{s + t}$.
Non-example: $F(x, t) = x + t^2$ is not a solution, since $F(F(0, 1), 1) = 2$ while $F(0, 2) = 4$.
For a flow, each map $f^t = F(\cdot, t)$ is a bijection of $X$ with inverse $f^{-t}$, and $f^s \circ f^t = f^{s + t}$.
If $\alpha \colon X \to \R$ is a bijection, then $F(x, t) = \alpha^{-1}(\alpha(x) + t)$ is a flow, and this is how [[Abel functions|2QBWT7T02Y]] produce fractional iterates.
The same equation is studied with $\R$ replaced by $[0, \infty)$, $\Z$, $\C$ or any group or semigroup.
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# Invariant subspace problem

## Problem

Let $H$ be a complex, separable, infinite-dimensional [[Hilbert space|HM2HHE3J51]] and let $T \colon H \to H$ be a bounded linear operator.
Does $T$ always have a nontrivial closed [[invariant subspace|0ACX5QT8TE]], that is, a closed linear subspace $M$ with $M \ne \set{0}$, $M \ne H$ and $T(M) \subseteq M$?

## Remarks

The problem is open.
The hypotheses rule out easy cases.
In a finite-dimensional complex space of dimension at least $2$, an eigenvector spans an invariant line.
In a non-separable space, the closed span of $x, Tx, T^2 x, \ldots$ for any $x \ne 0$ is a nontrivial invariant subspace.

Positive answers are known for many classes of operators.
Compact operators have nontrivial invariant subspaces (Aronszajn and Smith, 1954), and so do operators that commute with a nonzero compact operator, by [[Lomonosov's theorem|JEBP2FYZFM]].
Normal operators have them by the spectral theorem, and subnormal operators have them by a theorem of Scott Brown.
Bernstein and Robinson settled the case of polynomially compact operators in 1966, a question that Halmos had posed for operators whose square is compact.

For general Banach spaces the answer is negative.
Per Enflo constructed a bounded operator on a separable Banach space with no nontrivial closed invariant subspace, and Charles Read gave such an operator on $\ell^1$.

## Explanations

### L1 human

An invariant subspace of an operator is a smaller space that the operator never leaves.
In finite dimensions over the complex numbers, the line through an eigenvector is one.
The problem asks whether every bounded operator on an infinite-dimensional separable Hilbert space has such a proper closed piece.
Despite many partial results, nobody knows the answer.
</document>

<document source="https://tamnd.github.io/math2/node/5RFPT46DXX.md" tag="5RFPT46DXX" kind="statement">
# Oates–Powell theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]].
Then $G$ is [[finitely based|YGYZAQ12MZ]]: there is a finite set of [[identities|PKD440G2XW]] satisfied by $G$ from which every identity of $G$ follows.

## Remarks

Oates and Powell proved that the variety generated by a finite group is a Cross variety.
This means that it is finitely based, it is locally finite, and it contains only finitely many critical groups.
A finite group is critical if it does not lie in the variety generated by its proper subgroups and proper quotients.
The proof is a long induction that controls the finite groups in the variety through these critical groups.

Example: the symmetric group $S_3$ is finitely based, although its equational theory is not as simple as that of abelian groups.

The group axioms matter: a finite semigroup need not be finitely based.
Perkins proved that the six-element Brandt monoid $B_2^1$ is not finitely based as a semigroup.
The analogous statement for finite rings holds, by theorems of Kruse and of L'vov.

## Explanations

### L1 human

The laws satisfied by a finite group can always be summarised by a finite list.
For example, the laws of a finite abelian group of exponent $n$ follow from the group axioms, commutativity and $x^n \approx e$.
The theorem says that every finite group, abelian or not, has such a finite list of laws.

### L2 human

The theorem answers the finite basis question for groups, a special case of Tarski's finite basis problem for finite algebras.
It is the group case of a family of results: finite rings and finite lattices are also finitely based, while finite semigroups and finite groupoids may fail to be.
The notion of a Cross variety from the proof is a main tool in the theory of varieties of groups.
The result is stated with the group signature of multiplication, inverse and identity, so the identities are group words.
</document>

<document source="https://tamnd.github.io/math2/node/5RN835HTBD.md" tag="5RN835HTBD" kind="statement">
# Takens's theorem

## Statement

Let $M$ be a compact [[smooth manifold|F45VQ47GPV]] of dimension $m$.
For a $C^2$ [[diffeomorphism|0KZZ70ZHXP]] $\phi \colon M \to M$ and a $C^2$ function $y \colon M \to \R$, define the delay map
$$\Phi_{(\phi, y)} \colon M \to \R^{2m+1}, \qquad \Phi_{(\phi, y)}(x) = \big( y(x), y(\phi(x)), \ldots, y(\phi^{2m}(x)) \big).$$
An embedding here is an injective [[immersion|6RW68JGBHC]] that is a homeomorphism onto its image.
Give the space of all pairs $(\phi, y)$ the $C^2$ topology.
Then the set of pairs for which $\Phi_{(\phi, y)}$ is an embedding contains an open dense subset.

## Remarks

The proof follows the proof of the [[Whitney embedding theorem|2GSRBDT96M]].
For generic $\phi$ the periodic points of low period are finite in number and have distinct eigenvalues.
A perturbation of $y$ near them makes the delay map an immersion and injective there.
Away from these points a transversality argument shows that a generic perturbation of $y$ removes self-intersections, since $2m + 1 > 2m$.

The point of the theorem is that $2m + 1$ consecutive measurements of one scalar quantity along an orbit determine the state.
In applications $M$ is often an attractor of a flow, $\phi$ is the time-$\tau$ map, and $y$ is a measured quantity.
The image of the attractor under $\Phi$ is a faithful copy that can be reconstructed from a time series.
Sauer, Yorke and Casdagli extended the result to compact sets of box-counting dimension $d$, with $2m + 1$ replaced by any integer larger than $2d$, in a probabilistic sense of genericity.

Genericity is needed.
If $y$ is constant, then $\Phi_{(\phi, y)}$ is constant and is not injective.
If $\phi = \id$, then $\Phi_{(\phi, y)}(x) = (y(x), \ldots, y(x))$, which cannot be injective on a manifold of dimension $m \ge 1$.

## Explanations

### L1 human

Suppose a complicated system, like the weather, moves on some hidden space of states, and you can measure only one number, such as the temperature, at regular times.
The theorem says that, in typical situations, a list of enough consecutive measurements is as good as knowing the full state.
So the shape of the hidden dynamics can be rebuilt from a single time series.

### L2 human

Takens's theorem is the basis of delay-coordinate reconstruction in time series analysis.
One plots vectors $(y_n, y_{n+1}, \ldots, y_{n + k})$ built from data and studies the resulting cloud of points.
Since the delay map is an embedding, invariants that survive smooth changes of coordinates, such as [[Lyapunov exponents|NJ1FJSNK6A]], dimensions and entropies, can be estimated from the reconstruction.
The theorem is a dynamical analogue of the Whitney embedding theorem, which embeds an $m$-manifold in $\R^{2m}$ without any dynamics.
</document>

<document source="https://tamnd.github.io/math2/node/5RQGCEQYK5.md" tag="5RQGCEQYK5" kind="statement">
# Wallace–Bolyai–Gerwien theorem

## Statement

Let $P$ and $Q$ be two simple [[polygons|8VEFWD3EJB]] in the Euclidean plane with the same area.
Then $P$ can be cut into finitely many polygonal pieces that can be moved by [[isometries|0J1N3WTKM4]] to form $Q$, with disjoint interiors.

## Remarks

The proof cuts a polygon into triangles.
Each triangle can be cut and reassembled into a rectangle, and each rectangle into a rectangle with one side of length $1$.
Stacking these rectangles gives a rectangle of width $1$ and height equal to the area.
So $P$ and $Q$ can both be cut into the same rectangle, and the two dissections combine, since the relation "can be cut into" is transitive.
The analogue in dimension $3$ is false: this is the negative answer to [[Hilbert's third problem|81035XQ5HE]].

## Explanations

### L1 human

Take any two polygons with the same area, for example a square and a triangle.
You can always cut the first into finitely many pieces and reassemble them, like a jigsaw puzzle, into the second.
So in the plane, area is the only thing that matters for cutting and rearranging.

### L2 human

The theorem justifies the old method of finding areas by cutting and rearranging.
An example is the proof that a parallelogram has the area of a rectangle with the same base and height.
It shows that in the plane, equal area is the same as scissors congruence.
In space there is a second invariant, the [[Dehn invariant|S0TGRX3ARS]], and equal volume is not enough.
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# Homogenization

## Summary

Homogenization is the study of partial differential equations whose coefficients oscillate on a small scale $\varepsilon$, in the limit $\varepsilon \to 0$.
The solutions converge to the solution of a *homogenized* equation with effective, slowly varying or constant coefficients.
For periodic media the effective coefficients are computed from a cell problem on one period.
It explains, for example, the effective conductivity of a composite material made of a fine mixture of two materials.

## Explanations

### L1 human

Look at a material made of many thin alternating layers of copper and plastic.
At the scale of the whole piece, heat flows as if through one uniform material with some average conductivity.
Homogenization computes this average, which is in general not the plain arithmetic mean.
Across the layers, heat must cross both materials in series, and the effective conductivity is the harmonic mean, as for resistors in series.
Along the layers, heat flows through both in parallel, and the effective conductivity is the arithmetic mean.
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# Frame bundle

## Definition

Let $M$ be a smooth manifold of dimension $n$ with [[tangent bundle|WY6K3P4YCG]] $TM$.
A *frame* at $p \in M$ is an ordered [[basis|0KCQ9Y3QKA]] $e = (e_1, \ldots, e_n)$ of the tangent space $T_pM$.
The *frame bundle* of $M$ is the set
$$F(M) = \set{(p, e) : p \in M, \ e \text{ a frame at } p}.$$
It comes with the projection $\pi(p, e) = p$ and the right action of the [[general linear group|0FCHE902D7]] $GL(n, \R)$ given by
$$(p, e) \cdot g = (p, e g), \qquad (e g)_j = \sum_{i=1}^{n} g_{ij} \, e_i.$$
For a chart $(U, x)$ of $M$ and a frame $e$ at $p \in U$, let $A$ be the matrix with $e_j = \sum_i A_{ij} \, \partial/\partial x^i|_p$.
$F(M)$ carries the smooth structure for which each map $(p, e) \mapsto (p, A)$ is a diffeomorphism $\pi^{-1}(U) \to U \times GL(n, \R)$.
With this structure $F(M)$ is a [[principal|8E37E74XVP]] $GL(n, \R)$-bundle over $M$.

## Remarks

For $M = \R^n$, a frame at $p$ is the same as an invertible matrix, so $F(\R^n) = \R^n \times GL(n, \R)$.
Check of the action: $(e g) h = e (g h)$, because $((eg)h)_k = \sum_j h_{jk} \sum_i g_{ij} e_i = \sum_i (gh)_{ik} e_i$.

A tuple $(e_1, \ldots, e_n)$ of tangent vectors at $p$ that is linearly dependent, such as $e_1 = e_2$, is not a frame and is not a point of $F(M)$.

If $M$ is a Riemannian manifold, the orthonormal frames form the orthonormal frame bundle, a principal $O(n)$-bundle.
The same construction applies to any vector bundle $E$ of rank $k$ and gives a principal $GL(k, \R)$-bundle.

## Explanations

### L1 human

The frame bundle of a manifold is the set of all pairs made of a point and an ordered basis of the tangent space at that point.
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# Bicategory

## Definition

A *bicategory* $\mathcal{B}$ consists of:

1. a class of objects;
2. for each pair of objects $a, b$, a [[category|A2DE66937D]] $\mathcal{B}(a, b)$ of 1-morphisms $f \colon a \to b$ and 2-morphisms $\theta \colon f \Rightarrow g$;
3. for each triple $a, b, c$, a [[functor|28NQ0KRHD3]] $\circ \colon \mathcal{B}(b, c) \times \mathcal{B}(a, b) \to \mathcal{B}(a, c)$, called horizontal composition;
4. for each object $a$, an identity 1-morphism $1_a \in \mathcal{B}(a, a)$;
5. invertible 2-morphisms $\alpha_{h,g,f} \colon (h \circ g) \circ f \Rightarrow h \circ (g \circ f)$, $\lambda_f \colon 1_b \circ f \Rightarrow f$ and $\rho_f \colon f \circ 1_a \Rightarrow f$, [[natural|KYCPCC57FT]] in $f$, $g$, $h$.

These data satisfy, for all composable $f \colon a \to b$, $g \colon b \to c$, $h \colon c \to d$, $k \colon d \to e$:

1. pentagon: $(1_k \circ \alpha_{h,g,f}) \cdot \alpha_{k, h \circ g, f} \cdot (\alpha_{k,h,g} \circ 1_f) = \alpha_{k,h,g \circ f} \cdot \alpha_{k \circ h, g, f}$;
2. triangle: $(1_g \circ \lambda_f) \cdot \alpha_{g, 1_b, f} = \rho_g \circ 1_f$.

Here $\cdot$ is vertical composition in the hom-categories, and $1_f$ also denotes the identity 2-morphism of $f$.

## Remarks

Example: the bicategory $\mathbf{Bim}$ has rings as objects, $S$-$R$-bimodules as 1-morphisms $R \to S$, and bimodule maps as 2-morphisms.
Its horizontal composition is $N \circ M = N \otimes_S M$, its identity is $1_R = R$, and $\alpha$, $\lambda$, $\rho$ are the usual isomorphisms of tensor products.
The pentagon and triangle hold because, on elementary tensors, both sides of each identity give the same rebracketed tensor.
Non-example: in the same data, replace each identity $1_R$ by the zero bimodule; then $0 \otimes_R M = 0$, which is not isomorphic to a nonzero $M$, so no unitor $\lambda_M$ exists.
A one-object bicategory is the same as a [[monoidal category|M3MSQ3NKAE]], and a bicategory whose $\alpha$, $\lambda$, $\rho$ are identities is a [[2-category|CQACXG10JZ]].
</document>

<document source="https://tamnd.github.io/math2/node/5S30FAJNGM.md" tag="5S30FAJNGM" kind="definition">
# Levi form

## Definition

Let $U \subseteq \C^n$ be an [[open set|02BH3MCBQB]] and let $\rho \colon U \to \R$ be a function of class $C^2$ (see [[smooth function|0V33BNKZ9S]]).
The *Levi form* of $\rho$ at a point $p \in U$ is the [[Hermitian form|48P4A1CX85]]
$$L_\rho(p; w) = \sum_{j,k=1}^{n} \frac{\partial^2 \rho}{\partial z_j \partial \bar z_k}(p)\, w_j \bar w_k, \qquad w \in \C^n.$$
It is also called the complex Hessian of $\rho$.

Now let $\Omega \subseteq \C^n$ be a domain with $C^2$ boundary.
A *defining function* of $\Omega$ is a $C^2$ function $\rho$ on a neighbourhood $U$ of $\partial\Omega$ with $\Omega \cap U = \set{z \in U : \rho(z) < 0}$ and $d\rho \ne 0$ on $\partial\Omega$.
For $p \in \partial\Omega$, the complex tangent space of $\partial\Omega$ at $p$ is
$$H_p = \set{w \in \C^n : \sum_{j=1}^{n} \frac{\partial \rho}{\partial z_j}(p)\, w_j = 0}.$$
The *Levi form of $\partial\Omega$ at $p$* is the restriction of $L_\rho(p; \cdot)$ to $H_p$.

## Remarks

Example: for the unit ball, $\rho(z) = \abs{z_1}^2 + \cdots + \abs{z_n}^2 - 1$ has $\partial^2 \rho / \partial z_j \partial \bar z_k = \delta_{jk}$, so $L_\rho(p; w) = \abs{w}^2 > 0$ for every $w \ne 0$.
Non-example of positivity: for the bidisc, near a boundary point $p = (1, b)$ with $\abs{b} < 1$, a defining function is $\rho(z) = \abs{z_1}^2 - 1$.
Here $H_p = \set{w : w_1 = 0}$, and $L_\rho(p; (0, w_2)) = 0$, so the Levi form vanishes on $H_p$.
If $\rho$ is replaced by another defining function $h \rho$ with $h > 0$, the Levi form of $\partial\Omega$ at $p$ is multiplied by $h(p)$.
So its signature does not depend on the choice.
The signature is also unchanged under holomorphic changes of coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/5S7ER95XAC.md" tag="5S7ER95XAC" kind="concept">
# Symmetric monoidal category

## Summary

A symmetric monoidal category is a [[braided monoidal category|VT8SFWT0XW]] whose braiding, applied twice, gives the identity.
It is the categorical form of a commutative operation: $A \otimes B$ and $B \otimes A$ are identified by a swap that undoes itself.
Sets with the Cartesian product, vector spaces with the [[tensor product|0BF2DE2SYK]], and modules over a commutative ring are symmetric monoidal.
Symmetric monoidal categories are the setting of commutative monoid objects, of topological quantum field theories and of linear logic.

## Explanations

### L1 human

In a symmetric monoidal category, swapping two objects placed side by side and then swapping them back changes nothing.
So only the order of the objects matters, not how they were moved past each other.
For finite sets with the product, the swap $(a, b) \mapsto (b, a)$ is the symmetry, and doing it twice gives back $(a, b)$.
This is the familiar commutativity, made into a structure rather than an equation.
</document>

<document source="https://tamnd.github.io/math2/node/5S7FCJMCD7.md" tag="5S7FCJMCD7" kind="definition">
# Tensor product of Hilbert spaces

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $H_1$ and $H_2$ be [[Hilbert spaces|HM2HHE3J51]] over $\mathbb{K}$.
Let $H_1 \otimes H_2$ be their algebraic [[tensor product|0BF2DE2SYK]] over $\mathbb{K}$.
There is exactly one inner product on $H_1 \otimes H_2$ such that
$$\ip{x_1 \otimes x_2}{y_1 \otimes y_2} = \ip{x_1}{y_1} \ip{x_2}{y_2}$$
for all $x_1, y_1 \in H_1$ and $x_2, y_2 \in H_2$.
The *tensor product of Hilbert spaces* $H_1 \mathbin{\widehat{\otimes}} H_2$ is the completion of $H_1 \otimes H_2$ for this inner product.

## Remarks

If $(e_i)_{i \in I}$ and $(f_j)_{j \in J}$ are [[orthonormal bases|0S9GGTR0EX]] of $H_1$ and $H_2$, then $(e_i \otimes f_j)_{(i, j) \in I \times J}$ is an orthonormal basis of $H_1 \mathbin{\widehat{\otimes}} H_2$.

Example: $\ell^2(I) \mathbin{\widehat{\otimes}} \ell^2(J)$ is unitarily equivalent to $\ell^2(I \times J)$, by $e_i \otimes f_j \mapsto e_{(i, j)}$.
Both sides have orthonormal bases indexed by $I \times J$, and the map sends one onto the other.
Similarly, for $\sigma$-finite measure spaces, $L^2(X) \mathbin{\widehat{\otimes}} L^2(Y)$ is $L^2(X \times Y)$, with $f \otimes g$ sent to $(x, y) \mapsto f(x) g(y)$.

Non-example: if $H_1$ and $H_2$ are infinite-dimensional, the algebraic tensor product $H_1 \otimes H_2$ with the inner product above is not complete.
For instance, $\sum_{n \ge 1} 2^{-n} e_n \otimes f_n$ converges in the completion but is not a finite sum of elementary tensors.
In finite dimension no completion is needed, and $H_1 \mathbin{\widehat{\otimes}} H_2 = H_1 \otimes H_2$.

Other completions of $H_1 \otimes H_2$, for instance the projective tensor product, are Banach spaces with different norms.
</document>

<document source="https://tamnd.github.io/math2/node/5SB8EKSQRN.md" tag="5SB8EKSQRN" kind="concept">
# Hyperelliptic curve

## Summary

A hyperelliptic curve is a smooth projective [[algebraic curve|XJM2MT4S3Y]] of genus at least $2$ that is a double cover of the projective line.
Away from characteristic $2$, it has an affine model $y^2 = f(x)$ with $f$ a squarefree polynomial.
Hyperelliptic curves are the simplest curves of higher genus.
They are the exceptions in many results, such as [[Clifford's theorem|RSW29AVKGM]] and the embedding by the canonical divisor.

## Explanations

### L1 human

Take a polynomial $f(x)$ with distinct roots and look at all solutions of $y^2 = f(x)$.
For most values of $x$ there are two values of $y$, one the negative of the other, and they come together where $f(x) = 0$.
So the curve is two copies of the line glued along cuts between the roots.
For $y^2 = x^5 - x$ there are six branch points, counting the point at infinity, and the curve has genus $2$.
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# Milnor conjecture on slice genus of torus knots

## Statement

Let $p, q \ge 2$ be coprime integers and let $T(p, q)$ be the [[torus knot|NS60VV786W]] of type $(p, q)$.
For a knot $K \subset S^3 = \partial B^4$, consider the compact oriented surfaces smoothly and properly embedded in $B^4$ with boundary $K$.
The *slice genus* $g_4(K)$ is the least genus of such a surface.
Then
$$g_4(T(p, q)) = u(T(p, q)) = \frac{(p-1)(q-1)}{2},$$
where $u$ is the [[unknotting number|PW9D9QKAD4]].

## Remarks

For any knot, $g_4(K) \le u(K)$: each crossing change can be realized by adding a handle to a surface in $B^4$.
An explicit unknotting shows $u(T(p, q)) \le (p-1)(q-1)/2$.
The hard part is the lower bound $g_4(T(p, q)) \ge (p-1)(q-1)/2$.
Kronheimer and Mrowka proved it in 1993 using gauge theory.
Rasmussen later gave a combinatorial proof with his invariant from [[Khovanov homology|XB24W24B20]].
Example: for the [[trefoil knot|VYTYSVHN4A]] $T(2, 3)$, the formula gives $g_4 = u = 1$.
Milnor asked the question in his study of singular points of complex plane curves, where torus knots appear as links of singularities.

## Explanations

### L1 human

A torus knot is wound on a doughnut, going $p$ times one way and $q$ times the other.
The theorem gives the exact number of crossing changes needed to untie it: $(p-1)(q-1)/2$.
It also says that no surface in the four-dimensional ball bounded by it can have fewer handles than this.
Both quantities are easy to bound from one side and very hard to bound from the other.

### L2 human

The upper bounds are elementary, but lower bounds on the slice genus need deep tools.
Kronheimer and Mrowka obtained the bound from gauge theory, through lower bounds on the genus of embedded surfaces in $4$-manifolds.
Rasmussen's $s$-invariant, the Ozsváth–Szabó $\tau$-invariant from [[knot Floer homology|5P608WRETM]], and the signature give bounds of the form $g_4(K) \ge \abs{s(K)}/2$ and $g_4(K) \ge \abs{\tau(K)}$.
For torus knots, the $s$- and $\tau$-invariants are large enough to give the exact value.
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<document source="https://tamnd.github.io/math2/node/5SE8EMSDNY.md" tag="5SE8EMSDNY" kind="concept">
# Atomic lattice

## Summary

A lattice with a least element $0$ is atomic if every nonzero element lies above at least one [[atom|TTVCTW6TNC]].
Every finite lattice is atomic, and so is the lattice of all subsets of a set.
The notion separates lattices that are built from minimal pieces from atomless ones such as the interval $[0, 1]$.
It is weaker than being [[atomistic|99DVVV7BH2]], where every element is a join of atoms.

## Explanations

### L1 human

In an atomic lattice, whatever nonzero element one picks, one can always find a smallest possible nonzero piece inside it.
In the subsets of a set, a nonempty set contains a one-element subset.
In the positive integers ordered by divisibility, every number $n > 1$ has a prime divisor.
In the interval $[0, 1]$ there is no smallest positive number, so it is not atomic.
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<document source="https://tamnd.github.io/math2/node/5SE9H1SPBN.md" tag="5SE9H1SPBN" kind="concept">
# L-curve

## Summary

The L-curve is a log-log plot of the size of the regularized solution against the size of the residual.
Each point corresponds to one value of the regularization parameter.
For many discrete ill-posed problems the curve is L-shaped.
The parameter at the corner of the L balances fitting the data against amplifying the noise, and choosing it is the L-curve criterion for [[regularization|WTY54BFC46]].

## Explanations

### L1 human

In Tikhonov regularization a parameter controls how much the solution is smoothed.
Small values fit the data closely but let noise blow up the solution.
Large values keep the solution small but fit the data poorly.
Plotting the two errors against each other on logarithmic axes gives an L-shaped curve with a steep part and a flat part.
The corner is a natural compromise, where neither error can be reduced much without a large increase of the other.
</document>

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# Saccheri quadrilateral

## Summary

A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to a common base.
Giovanni Gerolamo Saccheri studied it in 1733 in an attempt to prove the [[parallel postulate|2PC3XY0PYZ]] from the other axioms.
Its two top angles are equal, and they are right in [[Euclidean geometry|KRZD96GSVC]] and acute in [[hyperbolic geometry|GN4N94GQ0Z]].
So it measures how far an [[absolute plane|VKFPCSRNDH]] is from being Euclidean.

## Explanations

### L1 human

Draw a segment as a base, raise two posts of equal length at right angles at its ends, and join the tops of the posts.
On a flat sheet you get a rectangle, so the top angles are right angles.
Saccheri asked whether the top angles could be smaller than right angles without a contradiction.
They can: this happens in hyperbolic geometry, although Saccheri himself believed he had ruled it out.
</document>

<document source="https://tamnd.github.io/math2/node/5SMK5KEBQE.md" tag="5SMK5KEBQE" kind="statement">
# May recognition principle

## Statement

Let $\mathcal{O}$ be an [[$E_\infty$ operad|T78YG90VVT]], and let $X$ be an [[$\mathcal{O}$-algebra|ES9YATDWA3]] in spaces, with the homotopy type of a CW complex.
Suppose $X$ is grouplike: the monoid $\pi_0(X)$, with the product induced by any binary operation of $\mathcal{O}$, is a group.
Then $X$ is weakly equivalent to an [[infinite loop space|A7VB2NSGJ2]].
More precisely, there is a connective $\Omega$-spectrum $E$ with $E_0$ [[weakly equivalent|4Q3WY1RX91]] to $X$, and this equivalence is compatible with the $\mathcal{O}$-structures.

## Remarks

May proved the principle in 1972 by building the spectrum from iterated bar constructions.
The key input is his approximation theorem.
It says that, for a connected space $Y$, the free algebra on $Y$ over the little $n$-cubes operad is weakly equivalent to $\Omega^n \Sigma^n Y$.
Boardman and Vogt gave another proof with their theory of homotopy-everything spaces.
The finite version holds too: a grouplike algebra over the little $n$-cubes operad is weakly equivalent to an $n$-fold [[loop space|2JFF5F3EGF]].
If $X$ is not grouplike, the zeroth space of the spectrum is instead the group completion of $X$.

At work: the disjoint union $\coprod_{n \ge 0} B\Sigma_n$ is an algebra over the Barratt–Eccles operad, but $\pi_0 = \N$ is not a group.
Its group completion is $\Omega^\infty \Sigma^\infty S^0 = QS^0$, which is the [[Barratt–Priddy–Quillen theorem|ZD7BNADK4H]].

## Explanations

### L1 human

Infinite loop spaces carry a multiplication that is commutative and associative up to all higher homotopies.
The recognition principle says that this property characterizes them.
Any space with such a coherent multiplication, in which every element has an inverse up to homotopy, is an infinite loop space.
So to show that a space comes from a cohomology theory, it is enough to find the multiplicative structure.

### L2 human

The principle is an "infinite loop space machine": it turns $E_\infty$ spaces into spectra.
It applies to classifying spaces of symmetric monoidal categories, which is how algebraic K-theory spectra are built.
It also shows that spaces such as $BO$, $BU$ and $BTop$ are infinite loop spaces, which gives new cohomology theories and operations.
Different machines, by May, Segal and Boardman–Vogt, give equivalent spectra by the uniqueness theorem of May and Thomason.
</document>

<document source="https://tamnd.github.io/math2/node/5SNCPXVH1J.md" tag="5SNCPXVH1J" kind="definition">
# Quasi-arithmetic mean

## Definition

Let $I$ be a real [[interval|0Q1D85B8G6]] and let $\varphi \colon I \to \R$ be a [[continuous|0X3WGX95FE]] and [[strictly monotone|011EJQWWJS]] function.
Let $n \ge 1$ and let $x_1, \dots, x_n \in I$.
The *quasi-arithmetic mean* of $x_1, \dots, x_n$ generated by $\varphi$ is
$$M_\varphi(x_1, \dots, x_n) = \varphi^{-1}\left( \frac{1}{n} \sum_{i=1}^{n} \varphi(x_i) \right),$$
where $\varphi^{-1}$ is the [[inverse function|0J2WAR83WP]] of $\varphi$ on the interval $\varphi(I)$.

## Remarks

The value is well defined: the average of the $\varphi(x_i)$ lies between their minimum and maximum, hence in the interval $\varphi(I)$.
Example: on $I = (0, \infty)$ with $\varphi = \ln$, $M_\varphi(x_1, \dots, x_n) = \exp\big(\frac{1}{n} \sum \ln x_i\big) = (x_1 \cdots x_n)^{1/n}$, the geometric mean.
Non-example: $\varphi(x) = x^2$ on $I = \R$ is not strictly monotone, and the formula breaks down.
For the numbers $-1$ and $1$ the value $\varphi^{-1}(1)$ is not defined, since both $-1$ and $1$ map to $1$.
Other standard cases are $\varphi(x) = x$ (arithmetic mean), $\varphi(x) = 1/x$ (harmonic mean) and $\varphi(x) = x^p$ on $(0, \infty)$ with $p \ne 0$ (power mean).
Two functions $\varphi$ and $\psi$ generate the same mean exactly when $\psi = a \varphi + b$ with real $a \ne 0$ and $b$.
The weighted version replaces $\frac{1}{n}$ by weights $w_i \ge 0$ with $\sum_i w_i = 1$.
The mean is also called the Kolmogorov mean or the generalized $f$-mean.
</document>

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# CW complex

## Definition

Let $X$ be a [[Hausdorff space|0MXB4CKHFW]].
For $n \ge 0$ let $D^n$ be the closed unit [[ball|05DSV40EB9]] in $\R^n$, with interior $\mathring{D}^n$ and boundary $S^{n-1}$.
A *CW structure* on $X$ is a partition of $X$ into subsets $e_\alpha$, called *cells*, with extra data.
Each cell has a dimension $n_\alpha \ge 0$ and a [[continuous map|0SY9ECQ19Q]] $\Phi_\alpha \colon D^{n_\alpha} \to X$, its *characteristic map*.
These data satisfy:

1. $\Phi_\alpha$ restricts to a [[homeomorphism|05GZ11PWRY]] from $\mathring{D}^{n_\alpha}$ onto $e_\alpha$.
2. $\Phi_\alpha(S^{n_\alpha - 1})$ is contained in the union of finitely many cells $e_\beta$ with $n_\beta < n_\alpha$ (closure finiteness).
3. A subset $A \subseteq X$ is closed if and only if $\Phi_\alpha^{-1}(A)$ is closed in $D^{n_\alpha}$ for every $\alpha$ (weak topology).

A *CW complex* is a Hausdorff space with a CW structure.
The cell $e_\alpha$ is an $n_\alpha$-cell, and the union $X^n$ of all cells of dimension at most $n$ is the $n$-skeleton.

## Remarks

Equivalently, a CW complex is built inductively.
The skeleton $X^0$ is a discrete space, and $X^n$ is obtained from $X^{n-1}$ by attaching $n$-cells along maps $S^{n-1} \to X^{n-1}$.
The union $X = \bigcup_n X^n$ has the topology in which a set is closed when its intersection with each $X^n$ is closed.

Example: the sphere $S^n$ with $n \ge 1$ has a CW structure with one $0$-cell $\set{p}$ and one $n$-cell $S^n \setminus \set{p}$.
The characteristic map $D^n \to S^n$ collapses $S^{n-1}$ to $p$ and is a homeomorphism on the interior, so clause 1 holds.
Clause 2 holds since $\Phi(S^{n-1}) = \set{p}$, and clause 3 holds since $S^n$ is a quotient of $D^n \sqcup \set{p}$.

Non-example: the closed disk $D^2$ with one $0$-cell at each point of the boundary circle and one $2$-cell, the interior, fails clause 2.
The boundary of the $2$-cell meets infinitely many $0$-cells.

A CW complex is finite if it has finitely many cells; a finite CW complex is compact.
</document>

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# Ribbon Hopf algebra

## Definition

Let $(H, R)$ be a [[quasitriangular Hopf algebra|NCRPWS9W5R]] over a field $k$, with coproduct $\Delta$, counit $\varepsilon$ and antipode $S$.
Write $R = \sum_i a_i \otimes b_i$, let $R_{21} = \sum_i b_i \otimes a_i$, and let $u = \sum_i S(b_i)\, a_i$ be the Drinfeld element.
A *ribbon element* of $H$ is a central invertible element $v \in H$ such that:

1. $v^2 = u\, S(u)$.
2. $S(v) = v$.
3. $\varepsilon(v) = 1$.
4. $\Delta(v) = (R_{21} R)^{-1} (v \otimes v)$.

A *ribbon Hopf algebra* is a quasitriangular Hopf algebra together with a ribbon element.

## Remarks

Example: let $H = k[G]$ be the group algebra of a group $G$ with $R = 1 \otimes 1$.
Then $u = S(1) \cdot 1 = 1$, and $v = 1$ is central and invertible.
Clauses 1 to 3 read $1 = 1$, and clause 4 reads $1 \otimes 1 = (1 \otimes 1)^{-1}(1 \otimes 1)$.
Non-example: in the same algebra with $\operatorname{char} k \ne 2$, the element $v = -1$ satisfies clauses 1 and 2 but fails clause 3, since $\varepsilon(-1) = -1$.
The Drinfeld element $u$ is invertible and satisfies $S^2(x) = u x u^{-1}$ for all $x \in H$.
Some authors use $v^{-1}$ in place of $v$, which changes clause 4 to $\Delta(v) = (v \otimes v) R_{21} R$.
</document>

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# Derived category

## Summary

The derived category $D(\mathcal{A})$ of an [[abelian category|Y9CZ3MTNF1]] $\mathcal{A}$ is obtained from the category of [[chain complexes|WQ82C0M3QB]] by declaring every [[quasi-isomorphism|C9JEC2WQKV]] to be an isomorphism.
In it, an object and any of its resolutions become isomorphic, so derived functors become ordinary functors between derived categories.
It is a [[triangulated category|T2V7GJJ2EN]], and $\operatorname{Ext}^n(X, Y)$ is the group of morphisms from $X$ to $Y[n]$.
Derived categories are the standard setting for duality theorems, for sheaf theory and for modern algebraic geometry.

## Explanations

### L1 human

In homological algebra one replaces an object by a resolution, and many different resolutions carry the same information.
The derived category builds this freedom into the category itself: two complexes count as the same object when a map between them induces isomorphisms on homology.
For example, the complex $\Z \xrightarrow{2} \Z$ and the group $\Z/2\Z$ become isomorphic.
This makes the machinery of derived functors much simpler to state and compose.
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# Stable homotopy groups of spheres

## Definition

Let $k \ge 0$ be an integer.
For $n \ge 1$, let $E \colon \pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1})$ be the suspension homomorphism, which sends the class of $f$ to the class of its [[suspension|Q7RFNJH76Q]] $\Sigma f$.
The *$k$-th stable homotopy group of spheres* is the colimit
$$\pi_k^s = \operatorname{colim}_{n} \pi_{n+k}(S^n)$$
of the sequence of [[homotopy groups of spheres|YD6VAXXSBM]] $\pi_{k+1}(S^1) \to \pi_{k+2}(S^2) \to \pi_{k+3}(S^3) \to \cdots$ along the maps $E$.

## Remarks

By the [[Freudenthal suspension theorem|7DM2KSS2DB]], $E \colon \pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1})$ is an isomorphism for $n \ge k + 2$.
So $\pi_k^s \cong \pi_{n+k}(S^n)$ for every $n \ge k + 2$.

Example: $\pi_0^s \cong \Z$, because $\pi_n(S^n) \cong \Z$ for all $n$ and $E$ preserves the degree.

Example: $\pi_1^s \cong \Z/2$, generated by the suspensions of the Hopf map $S^3 \to S^2$; the group $\pi_3(S^2) \cong \Z$ itself is not yet in the stable range, since $n = 2 < k + 2 = 3$.

The next values are $\pi_2^s \cong \Z/2$, $\pi_3^s \cong \Z/24$, $\pi_4^s = \pi_5^s = 0$, $\pi_6^s \cong \Z/2$ and $\pi_7^s \cong \Z/240$.
Composition of maps makes $\pi_*^s = \bigoplus_k \pi_k^s$ a graded commutative ring.
For $k \ge 1$, the group $\pi_k^s$ is finite.
</document>

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# Polish group

## Definition

A *Polish group* is a [[topological group|92700QAB5Y]] $G$ whose underlying topological space is a [[Polish space|584GXXY73M]].
That is, the topology of $G$ is separable and is induced by some complete metric.

## Remarks

Example: the [[symmetric group|0515A6D2Y1]] $S_\infty$ of all bijections $\N \to \N$, with the topology of pointwise convergence.
It is a $G_\delta$ subset of the Polish space $\N^\N$, hence Polish, and composition and inversion are continuous for this topology.
Non-example: $(\Q, +)$ with the topology from $\R$.
It is separable and metrizable, but it is countable without isolated points.
So it is a countable union of nowhere dense singletons and is not a [[Baire space|0ZH5AJ2ED7]]; hence no complete metric induces its topology.
The metric need not be invariant: a Polish group always has a compatible left-invariant metric, but this metric may fail to be complete.
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# Vandermonde matrix

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]], let $m, n \ge 1$, and let $x_1, \dots, x_m \in R$.
The *Vandermonde matrix* of $x_1, \dots, x_m$ with $n$ columns is the $m \times n$ [[matrix|0VD9ZSWWJG]] $V$ with entries
$$V_{ij} = x_i^{\,j-1} \qquad (1 \le i \le m,\ 1 \le j \le n),$$
where $x_i^0 = 1$.

## Remarks

Example: for $x_1 = 1$, $x_2 = 2$, $x_3 = 3$ and $n = 3$, the matrix is $\begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{pmatrix}$; row $i$ is $(1, x_i, x_i^2)$.
Non-example: $\begin{pmatrix} 2 & 4 \\ 3 & 9 \end{pmatrix}$ is not a Vandermonde matrix, because a Vandermonde matrix has first column $1, \dots, 1$.
It is obtained from the Vandermonde matrix $\begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}$ of $2, 3$ by multiplying row $i$ by $x_i$.
In the square case $m = n$, the [[Vandermonde determinant|0P84B554AS]] is $\det V = \prod_{1 \le i < j \le n} (x_j - x_i)$.
For a polynomial $p(t) = c_0 + c_1 t + \cdots + c_{n-1} t^{n-1}$, the vector $V c$ lists the values $p(x_1), \dots, p(x_m)$.
Some authors use the transpose of $V$ instead.
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# Final functor

## Definition

Let $F \colon \mathcal{C} \to \mathcal{D}$ be a [[functor|28NQ0KRHD3]].
For an object $d$ of $\mathcal{D}$, let $d \downarrow F$ be the [[comma category|PC6HG1JEGM]] whose objects are pairs $(c, f)$ with $c$ an object of $\mathcal{C}$ and $f \colon d \to F c$.
A morphism $(c, f) \to (c', f')$ in it is a morphism $h \colon c \to c'$ with $F(h) \circ f = f'$.
$F$ is *final* if for every object $d$ of $\mathcal{D}$ the category $d \downarrow F$ is nonempty and connected.

## Remarks

Here a category is connected when any two objects are joined by a finite zigzag of morphisms.
If $F$ is final, then for every functor $G \colon \mathcal{D} \to \mathcal{E}$ the [[colimit|BJS1CGTKQ5]] of $G \circ F$ exists if and only if the colimit of $G$ exists.
The canonical morphism $\operatorname{colim} (G \circ F) \to \operatorname{colim} G$ is then an isomorphism.
For small $\mathcal{C}$ and $\mathcal{D}$ this property for all $G$ is equivalent to finality.

Example: let $\mathcal{D}$ have a terminal object $t$, and let $F$ be the functor from the one-object category picking out $t$.
For each $d$ the category $d \downarrow F$ has one object, the unique arrow $d \to t$, and only its identity, so it is nonempty and connected.
Hence $\operatorname{colim} G = G(t)$ for every $G$.

Non-example: let $\mathcal{D}$ be the poset $\set{0 < 1}$ and let $F$ pick out $0$.
There is no arrow $1 \to 0$, so $1 \downarrow F$ is empty and $F$ is not final.

The inclusion of a cofinal subset of a directed set is a final functor.
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<document source="https://tamnd.github.io/math2/node/5TBC06NNS1.md" tag="5TBC06NNS1" kind="concept">
# Completely distributive lattice

## Summary

A completely distributive lattice is a complete lattice in which arbitrary meets distribute over arbitrary joins, in the strongest possible form.
It is the infinite version of a [[distributive lattice|0HM0HQCK6N]].
Power sets and the closed interval $[0, 1]$ are completely distributive, and every completely distributive lattice is a [[frame|306PWGH7BF]].

## Explanations

### L1 human

In a distributive lattice, one can expand $a \wedge (b \vee c)$ into $(a \wedge b) \vee (a \wedge c)$, like multiplying out brackets.
Complete distributivity allows this expansion for infinitely many factors, each of which is an infinite join.
The result is a join over all ways of choosing one term from each factor, just as $(a_1 + a_2)(b_1 + b_2)$ is the sum of the four products $a_i b_j$.
In the lattice of subsets of a set, with unions and intersections, this expansion always works.
</document>

<document source="https://tamnd.github.io/math2/node/5TKBSHY6VC.md" tag="5TKBSHY6VC" kind="example">
# Geodesic flow on a locally symmetric space

## Example

Let $G = \mathrm{PSL}_2(\R)$, acting on the upper half-plane $\mathbb{H} = \set{z \in \C : \operatorname{Im} z > 0}$ by Möbius transformations $z \mapsto (az + b)/(cz + d)$.
These maps are the orientation-preserving isometries of the hyperbolic metric $\abs{dz}/\operatorname{Im} z$.
The action of $G$ on the unit [[tangent bundle|DQB1BVQJWD]] $T^1\mathbb{H}$ is free and transitive.
Let $v_0$ be the upward unit vector at $i$; then $g \mapsto g \cdot v_0$ identifies $G$ with $T^1\mathbb{H}$.
For $t \in \R$ let $a_t$ be the class of $\begin{pmatrix} e^{t/2} & 0 \\ 0 & e^{-t/2} \end{pmatrix}$.
Then $a_t \cdot i = e^t i$, and the path $t \mapsto e^t i$ is the unit-speed [[geodesic|NSYMQMHPDN]] through $v_0$, since the hyperbolic distance from $i$ to $e^t i$ is $t$ for $t \ge 0$.
As $g$ is an isometry, $g a_t \cdot v_0$ is the unit vector reached after time $t$ along the geodesic starting at $g \cdot v_0$.
So the geodesic flow on $T^1\mathbb{H}$ is the right multiplication $g \mapsto g a_t$.

Now let $\Gamma$ be a [[lattice|BTYF5Y5PFY]] in $G$, for example the [[modular group|VEKV1EM77B]] or a torsion-free cocompact lattice.
Left multiplication by $\Gamma$ commutes with right multiplication by $a_t$.
So the flow descends to $\Gamma \backslash G$, which is the unit tangent bundle of the locally symmetric surface $\Gamma \backslash \mathbb{H}$ when $\Gamma$ is torsion-free.
It is an action of the Lie group $\R$ on $\Gamma \backslash G$.
The same recipe works for a quotient $\Gamma \backslash G / K$ of a Riemannian symmetric space $G/K$.
There geodesics are orbits of one-parameter subgroups, and the flow becomes right multiplication on a homogeneous space of $G$.

## Explanations

### L1 human

On a curved surface, a geodesic flow moves every direction arrow forward along its straightest path at unit speed.
On a hyperbolic surface built from the group $\mathrm{PSL}_2(\R)$, this complicated-looking motion is just multiplication on the right by a diagonal matrix.
That turns a question in geometry into a question about groups.
Facts about group actions, such as their mixing behaviour, then say how geodesics wander over the surface.
</document>

<document source="https://tamnd.github.io/math2/node/5TNA2HBG7Y.md" tag="5TNA2HBG7Y" kind="statement">
# Whitman's condition

## Statement

Let $X$ be a set and let $\mathrm{FL}(X)$ be the [[free lattice|WP20S9NZ15]] on $X$, with [[join and meet|J2T75ATRV5]] $\vee$ and $\wedge$.
Then $\mathrm{FL}(X)$ satisfies *Whitman's condition* (W): for all $a, b, c, d \in \mathrm{FL}(X)$, if
$$a \wedge b \le c \vee d,$$
then $a \le c \vee d$, or $b \le c \vee d$, or $a \wedge b \le c$, or $a \wedge b \le d$.

## Remarks

The converse implication is automatic in any lattice, since $a \wedge b \le a$, $a \wedge b \le b$, $c \le c \vee d$ and $d \le c \vee d$.
So (W) says that an inequality between a meet and a join holds in $\mathrm{FL}(X)$ only for one of these four trivial reasons.
Proof idea: Whitman described the order of $\mathrm{FL}(X)$ on lattice terms by recursive rules, the solution of the [[word problem for free lattices|Z86VGP5RMG]].
He showed that these rules define a lattice with the universal property.
The rule for comparing a meet with a join is exactly (W).
Every [[sublattice|PV90EW00XK]] of a lattice satisfying (W) satisfies (W), since joins and meets in a sublattice are computed in the big lattice.
Example of failure: in the lattice of subsets of $\set{1, 2, 3, 4}$, take $a = \set{1, 2, 3}$, $b = \set{1, 2, 4}$, $c = \set{1}$ and $d = \set{2}$.
Then $a \cap b = \set{1, 2} = c \cup d$, but $a \not\subseteq \set{1, 2}$, $b \not\subseteq \set{1, 2}$, $\set{1, 2} \not\subseteq c$ and $\set{1, 2} \not\subseteq d$.
So this lattice is not a sublattice of any free lattice.

## Explanations

### L1 human

In a free lattice nothing holds by accident.
If a meet $a \wedge b$ lies below a join $c \vee d$, then there is an obvious reason.
Either one of $a$ or $b$ lies below $c \vee d$, or $a \wedge b$ lies below one of $c$ or $d$.
Lattices of sets do not behave this way, since two sets can overlap in exactly the union of two smaller sets.

### L2 human

Whitman's condition is the key structural property of free lattices.
Together with the rules for generators, it gives an algorithm that decides the order of $\mathrm{FL}(X)$.
It is also the main test for deciding which finite lattices embed in free lattices, because it passes to sublattices.
Since the subsets of a four-element set fail (W), a free lattice cannot contain this Boolean lattice as a sublattice.
</document>

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# Gegenbauer polynomials

## Summary

The Gegenbauer polynomials $C_n^{(\lambda)}$, also called ultraspherical polynomials, are orthogonal on $[-1, 1]$ for the weight $(1 - x^2)^{\lambda - 1/2}$.
Up to constant factors they are the [[Jacobi polynomials|TRD7DSF2KR]] with $\alpha = \beta = \lambda - \frac12$.
They generalize the [[Legendre polynomials|QFYGJVA996]] ($\lambda = \frac12$) and give the [[zonal spherical harmonics|Q5D54R3RGE]] on spheres of every dimension at least $2$.

## Explanations

### L1 human

The Legendre polynomials are tied to the ordinary sphere in space.
On a sphere in a space of another dimension, the same role is played by a related family, the Gegenbauer polynomials.
They all come from one formula: expand $(1 - 2xt + t^2)^{-\lambda}$ in powers of $t$, and the coefficients are the $C_n^{(\lambda)}(x)$.
For $\lambda = \frac12$ this is the generating function of the Legendre polynomials.
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<document source="https://tamnd.github.io/math2/node/5TRXQKP7DY.md" tag="5TRXQKP7DY" kind="concept">
# Hamilton–Jacobi equation

## Summary

A Hamilton–Jacobi equation is a first-order nonlinear [[partial differential equation|GC5BDZ6QBC]] of the form $u_t + H(\nabla u, x) = 0$, where $H$ is a given function called the Hamiltonian.
It comes from classical mechanics, where its solutions generate the motions of a [[Hamiltonian system|0EDKGA2FAK]].
It also appears in optimal control, geometric optics and front propagation.
Classical solutions usually stop existing after some time, and the right notion of global solution is the [[viscosity solution|H1ERJ3T75T]].
See the [[definition|4H1E4KW1JX]].

## Explanations

### L1 human

Imagine many travellers who all want to reach a goal as cheaply as possible.
Let $u(x, t)$ be the least cost of a journey that ends at the point $x$ at time $t$.
The Hamilton–Jacobi equation is the rule that this least cost must obey when one moves $x$ or $t$ a little.
For $H(p) = \abs{p}^2/2$ the equation reads $u_t + \tfrac12 \abs{\nabla u}^2 = 0$, and for example $u(x, t) = \abs{x}^2 / (2(t + 1))$ is a solution for $t > -1$.
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<document source="https://tamnd.github.io/math2/node/5V22013VEQ.md" tag="5V22013VEQ" kind="definition">
# Modulus of smoothness

## Definition

Let $a < b$, let $k \ge 1$ be an integer, and let $1 \le p \le \infty$.
Let $f \in L^p[a, b]$, an [[Lp space|SASP14MA1Z]] element; for $p = \infty$ let $f$ be [[continuous|0X3WGX95FE]] on $[a, b]$.
For $h > 0$, the $k$-th forward difference of $f$ with step $h$ is
$$\Delta_h^k f(x) = \sum_{i=0}^{k} (-1)^{k-i} \binom{k}{i} f(x + i h), \qquad a \le x \le b - k h.$$
For $0 < t \le (b - a)/k$, the *modulus of smoothness of order $k$* of $f$ is
$$\omega_k(f, t)_p = \sup_{0 < h \le t} \norm{\Delta_h^k f}_{L^p[a, \, b - k h]}.$$

## Remarks

Example: for $f(x) = x$ on $[0, 1]$ and $p = \infty$, one has $\Delta_h^1 f = h$ and $\Delta_h^2 f = 0$, so $\omega_1(f, t)_\infty = t$ and $\omega_2(f, t)_\infty = 0$.

Non-example: the norm of the difference at a single step is not a modulus of smoothness, because it need not grow with $t$.
For $f(x) = \sin(2 \pi x)$ on $[0, 2]$ and $p = \infty$, one has $\Delta_1^1 f = 0$ but $\norm{\Delta_{1/2}^1 f}_\infty = 2$; the supremum over $0 < h \le t$ is what makes $\omega_k(f, \cdot)_p$ nondecreasing.

For $k = 1$ and $p = \infty$, $\omega_1(f, t)_\infty$ is the [[modulus of continuity|AXCJBSQZWF]] of $f$.
One has $\omega_k(f, t)_p \to 0$ as $t \to 0$, and $\omega_{k+1}(f, t)_p \le 2 \omega_k(f, t)_p$.
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# Numerical continuation

## Summary

Numerical continuation is a family of methods that trace the solutions of a system of equations $F(x, \lambda) = 0$ as a parameter $\lambda$ changes.
Starting from one known solution, it computes a chain of nearby solutions along a curve.
Each new point usually comes from a predictor step and a [[Newton's method|07EXVNF9VP]] corrector step.
It is the main tool for studying how the solutions of nonlinear models depend on parameters, and for finding bifurcations.
It also gives a way to solve a hard system by deforming it from an easy one.

## Explanations

### L1 human

Imagine a curve of solutions drawn in space, and you stand on one point of it.
You take a small step along the tangent of the curve, which lands you near the curve, and then you walk back onto the curve with a few Newton steps.
Repeating this, you follow the whole curve.
For example, the solutions of $x^2 + \lambda^2 = 1$ form a circle, and continuation from $(x, \lambda) = (1, 0)$ walks around this circle, even through the points where $\lambda$ turns back.
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# Grandi's series

## Example

*Grandi's series* is the series
$$\sum_{n=0}^{\infty} (-1)^n = 1 - 1 + 1 - 1 + \cdots.$$
Its partial sums are $s_N = \sum_{n=0}^{N} (-1)^n$, which equal $1$ for even $N$ and $0$ for odd $N$.
So the partial sums have no limit, and the series is a [[divergent series|0JWMJSTBSG]].
Its terms do not tend to $0$, which already rules out convergence.

## Remarks

Several summation methods assign it the value $1/2$.
[[Cesàro summation|N8ZGG7EW6J]]: the averages $(s_0 + \cdots + s_N)/(N+1)$ equal $1/2$ for odd $N$ and $(N/2 + 1)/(N+1)$ for even $N$, so they tend to $1/2$.
[[Abel summation|EZXPY4A19C]]: for $0 < x < 1$, $\sum_{n \ge 0} (-1)^n x^n = 1/(1 + x)$, which tends to $1/2$ as $x \to 1^-$.
[[Borel summation|8DJPSYSHGJ]] also gives $1/2$, since it sums the geometric series $\sum z^n$ to $1/(1 - z)$ at $z = -1$.
The value $1/2$ is forced for any linear and stable method that sums the series: if $S = 1 - 1 + 1 - \cdots$, stability and linearity give $S = 1 - S$.

## Explanations

### L1 human

Add $1$, subtract $1$, add $1$, subtract $1$, forever.
The running total jumps between $1$ and $0$ and never settles, so the series has no sum in the ordinary sense.
On average the running total is $1/2$, and this is the value that the usual summation methods give it.
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# Siegel zero

## Summary

A Siegel zero, or Landau–Siegel zero, is a hypothetical real zero of a [[Dirichlet L-function|9D2Z543EJQ]] of a real character that lies very close to $1$.
The classical zero-free region for Dirichlet L-functions excludes every zero near $s = 1$ except possibly one such zero.
Whether Siegel zeros exist is open; the [[generalized Riemann hypothesis|DNKA4WBKDS]] rules them out.
Their possible existence is the source of ineffective constants in results such as the [[Siegel–Walfisz theorem|XEPW251A97]].

## Explanations

### L1 human

Dirichlet L-functions are cousins of the Riemann zeta function, one for each Dirichlet character.
Their zeros control how primes are spread over residue classes.
One knows that no zero comes very close to the point $s = 1$, with one possible exception: a single real zero for a real character.
That possible exception is called a Siegel zero.
Nobody has found one, and most experts believe they do not exist, but nobody can rule them out either.
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# Algebraic combinatorics

## Definition

*Algebraic combinatorics* is the branch of mathematics with two complementary aims:

1. to solve problems of [[combinatorics|0JKW7W7XFR]] with tools of abstract algebra, such as groups, [[group actions|0WCY42EV9V]], rings, linear algebra and representation theory;
2. to describe algebraic objects, such as bases, structure constants, characters and dimensions, by combinatorial rules that count explicit finite structures.

## Remarks

Example of aim 1: the number of [[orbits|0YQPZB5Q1G]] of a group acting on colourings is computed by [[Burnside's lemma|0Z47M1B2VS]] and the [[Pólya enumeration theorem|MPYQ39P01F]].
Example of aim 2: the product of two [[Schur polynomials|43VFMJCDYJ]] is a sum of Schur polynomials.
The [[Littlewood–Richardson rule|Q490XBH1Q2]] gives the coefficients as numbers of certain tableaux.

Non-example: a proof that a graph with a property exists, by showing that a random graph has it with positive probability, meets neither aim.
It belongs to the [[probabilistic method|3B55HNS754]].

The main objects include [[symmetric functions|PAJEHWXW6M]], [[Young tableaux|Q6TH9PH2R7]], [[association schemes|C338GKB4VJ]] and [[incidence algebras|1H9SFWHYEB]] of posets.
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# Trigamma function

## Summary

The trigamma function $\psi_1$ is the derivative of the [[digamma function|HWRRXB0YXS]], the second derivative of $\log \Gamma$.
It is the [[polygamma function|3J3QGKPMZQ]] of order $1$ and equals $\sum_{k \ge 0} (z+k)^{-2}$.
It is positive on the positive real axis, which expresses the log-convexity of the [[gamma function|0H5SYBSED9]].

## Explanations

### L1 human

The trigamma function measures how fast the slope of $\log \Gamma$ changes, that is, the bending of the graph of $\log \Gamma$.
It is a sum of squares of reciprocals: $\frac{1}{z^2} + \frac{1}{(z+1)^2} + \frac{1}{(z+2)^2} + \cdots$.
For $z = 1$ this is the famous Basel sum $1 + \frac14 + \frac19 + \cdots = \frac{\pi^2}{6}$.
Since every term is positive for $z > 0$, the graph of $\log \Gamma$ always bends upward there.
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# Indefinite sum

## Definition

Let $f \colon \R \to \R$ be a function, and let $\Delta$ be the [[forward difference operator|TXTR68V4R1]] with step $1$.
An *indefinite sum* of $f$ is a function $F \colon \R \to \R$ with $\Delta F = f$, that is,
$$F(x + 1) - F(x) = f(x) \quad \text{for all } x \in \R.$$
Another name is *antidifference*.
The same definition applies to functions $f \colon \Z \to \R$, with $F \colon \Z \to \R$.

## Remarks

Common notations are $\Delta^{-1} f$ and $\sum_x f(x)$.
Example: $F(x) = 2^x$ is an indefinite sum of $f(x) = 2^x$, since $2^{x+1} - 2^x = 2^x$.
Non-example: $F(x) = x^2/2$ is not an indefinite sum of $f(x) = x$, since $F(x + 1) - F(x) = x + \tfrac{1}{2}$.
Indefinite sums are not unique.
On $\Z$, two indefinite sums of $f$ differ by a constant.
On $\R$, two indefinite sums of $f$ differ by a function $C$ with $C(x + 1) = C(x)$ for all $x$, such as $\sin(2\pi x)$.
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# Kodaira dimension

## Definition

Let $X$ be a smooth projective [[variety|HV6YBS0YDH]] of [[dimension|36X2WTHYCB]] $n$ over a field $k$, and let $\omega_X$ be its [[canonical bundle|W05VYY4HBR]].
For $m \ge 1$, the *$m$-th plurigenus* is $P_m(X) = \dim_k H^0(X, \omega_X^{\otimes m})$.
The *Kodaira dimension* $\kappa(X)$ is defined as follows:

1. If $P_m(X) = 0$ for all $m \ge 1$, then $\kappa(X) = -\infty$.
2. Otherwise, $\kappa(X)$ is the least integer $\kappa \ge 0$ such that $P_m(X) / m^{\kappa}$ is bounded as $m \to \infty$.

## Remarks

Always $\kappa(X) \in \set{-\infty, 0, 1, \dots, n}$.
Equivalently, when some $P_m(X) > 0$, $\kappa(X)$ is the largest dimension of the image of the rational map $X \dashrightarrow \mathbb{P}(H^0(X, \omega_X^{\otimes m}))$ over all $m$ with $P_m(X) > 0$.
The variety $X$ is of *general type* if $\kappa(X) = n$.

Example: for $X = \mathbb{P}^n_k$ with $n \ge 1$, $\omega_X \cong \mathcal{O}(-n-1)$, so $\omega_X^{\otimes m} \cong \mathcal{O}(-m(n+1))$ has no nonzero sections and $\kappa(X) = -\infty$.
For a smooth projective curve of genus $g \ge 2$, $\deg \omega_X^{\otimes m} = m(2g-2)$ and Riemann–Roch gives $P_m = (2m-1)(g-1)$ for $m \ge 2$, which grows like $m^1$, so $\kappa(X) = 1$.

Non-example of clause 2 with $\kappa = 0$: the curve of genus $2$ has $P_m / m^0 = P_m$ unbounded, so its Kodaira dimension is not $0$.
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# High indices theorem

## Statement

Let $(n_k)_{k \ge 1}$ be a strictly increasing sequence of positive integers, and assume there is a real $q > 1$ with $n_{k+1} / n_k \ge q$ for all $k$.
Let $(a_k)_{k \ge 1}$ be complex numbers such that the power series
$$f(x) = \sum_{k \ge 1} a_k x^{n_k}$$
converges for $0 \le x < 1$, and assume $f(x) \to s$ as $x \to 1^-$.
Then the series $\sum_{k \ge 1} a_k$ [[converges|0CVVKY7GY2]] and its sum is $s$.

## Remarks

In the language of [[Abel summation|EZXPY4A19C]], the theorem says that a series with Hadamard gaps that is Abel summable is convergent.
So the gap condition acts as a Tauberian condition, and no condition on the size of the coefficients $a_k$ is needed.
The proof is considerably harder than that of [[Tauber's theorem|8SK3GQ4S0M]].
The gap hypothesis cannot be dropped: $\sum_{n \ge 0} (-1)^n x^n = 1/(1 + x) \to 1/2$ as $x \to 1^-$, but $\sum (-1)^n$ diverges.
As a consequence, $f(x) = \sum_{k \ge 0} (-1)^k x^{2^k}$ has no limit as $x \to 1^-$, because $\sum_k (-1)^k$ diverges.

## Explanations

### L1 human

A power series with very sparse exponents, such as $x^2, x^4, x^8, \dots$, cannot hide divergence behind smoothing.
If its value approaches a limit as $x$ goes up to $1$, then the series of coefficients already converges to that limit.
For ordinary series this is false, so the theorem shows how strong a constraint sparse exponents are.

### L2 human

[[Abel's theorem|0S4EMDQS2F]] goes from convergence to Abel summability, and Tauberian theorems go back under extra hypotheses.
Usual Tauberian conditions bound the terms, as in $n a_n = O(1)$.
Here the extra hypothesis is only on where the nonzero terms sit.
The same gap condition appears in the [[Hadamard gap theorem|DA8M8VZZHC]], where it forces the unit circle to be a natural boundary.
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# Mirsky's theorem

## Statement

Let $(P, \le)$ be a finite [[partially ordered set|0QV2JS9RJ3]].
Let $h$ be the largest number of elements of a [[chain|0S8FE7ZMFM]] in $P$, that is, the [[height|MPR6MH4YSQ]] of $P$.
Then $P$ is the union of $h$ pairwise disjoint [[antichains|03HF498P1J]], and $P$ is not the union of fewer than $h$ antichains.

## Remarks

A chain meets an antichain in at most one element, so a chain with $h$ elements needs $h$ different antichains to cover it.
For the other half, let $f(x)$ be the largest number of elements of a chain whose greatest element is $x$.
If $x < y$, then a chain ending at $x$ extends by $y$, so $f(x) < f(y)$.
Thus each level set $\set{x \in P : f(x) = k}$ is an antichain, and the values of $f$ are $1, \ldots, h$.
Example: in the divisors of $12$ ordered by divisibility, the chain $1, 2, 4, 12$ has $4$ elements, and the levels $\set{1}$, $\set{2, 3}$, $\set{4, 6}$, $\set{12}$ are $4$ antichains that cover the set.
The same proof works for an infinite partially ordered set of finite height.
The theorem is the dual of [[Dilworth's theorem|GT224XDA4E]], which exchanges the roles of chains and antichains, but its proof is much easier.

## Explanations

### L1 human

Think of the elements as tasks, where $x < y$ means that $x$ must be done before $y$.
An antichain is a set of tasks that can be done in the same round.
The theorem says that the least number of rounds equals the length of the longest sequence of tasks that must be done one after the other.

### L2 human

The proof gives an explicit partition: put each element into the level given by the longest chain that ends at it.
This is the layering used in scheduling and in drawing directed acyclic graphs.
In the language of graphs, the comparability graph of a poset has chromatic number equal to its clique number.
The same holds for each induced subgraph, so comparability graphs are perfect.
Together with Dilworth's theorem it gives the [[Erdős–Szekeres theorem|0SNJM8HSD3]] on monotone subsequences.
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# Homotopy continuation for polynomial systems

## Summary

Homotopy continuation for polynomial systems computes all isolated solutions of a system of polynomial equations in several complex variables.
It deforms a start system, whose solutions are known, into the target system, and it follows each start solution along the deformation by [[numerical continuation|5V3GJ84CPQ]].
It is the basic algorithm of numerical algebraic geometry, and it is used for systems from kinematics, chemistry and other applications.

## Explanations

### L1 human

Solving many polynomial equations at once is hard, but some systems are easy, such as $x^2 = 1$, $y^3 = 1$.
Homotopy continuation slides the easy system into the hard one step by step.
Each solution of the easy system moves along a path as the equations change, and where the paths end are the solutions of the hard system.
For a system of two equations of degrees $2$ and $3$, one follows $2 \cdot 3 = 6$ paths, and these endpoints include every isolated solution.
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# Riemann extension theorem in several variables

## Statement

Let $D \subseteq \C^n$ be a connected [[open set|02BH3MCBQB]].
Let $A \subseteq D$ be a closed subset with the following property.
Each point $p \in D$ has a connected open neighbourhood $U \subseteq D$ and a [[holomorphic function|507C34W0ZT]] $h \not\equiv 0$ on $U$ with $A \cap U \subseteq \set{z \in U : h(z) = 0}$.

1. Let $f$ be holomorphic on $D \setminus A$, and let each point of $A$ have a neighbourhood $V$ with $f$ bounded on $V \setminus A$. Then $f$ extends uniquely to a holomorphic function on $D$.
2. Let $A$ be moreover a complex analytic subset of $D$ of dimension at most $n - 2$. Then every holomorphic function on $D \setminus A$ extends uniquely to a holomorphic function on $D$.

## Remarks

Under the hypothesis, $D \setminus A$ is dense in $D$ and connected, which gives uniqueness by the identity theorem.
For part 1, take a point of $A$ and choose coordinates and a small polydisc around it.
They can be chosen so that $h$ has no zeros on the part of the boundary where $\abs{z_n}$ equals the radius.
The Cauchy integral in $z_n$ over that circle defines a holomorphic function $F$.
On each line where $z'$ is fixed, $f$ is a bounded holomorphic function of $z_n$ off finitely many points, so by the one-variable [[removable singularity|09R8GC85RB]] theorem $F = f$ there.
Part 2 is proved with Hartogs figures.
Near a point of a set of codimension at least $2$, one finds Hartogs figures in $D \setminus A$ whose filled-in polydiscs cover a neighbourhood of the point.

Example: every holomorphic function on $\C^2 \setminus \set{0}$ extends to $\C^2$, by part 2 with $A = \set{0}$.

The boundedness in part 1 cannot be dropped when $A$ has codimension $1$.
The function $1/z_1$ is holomorphic on $\C^2 \setminus \set{z_1 = 0}$ but unbounded near that line, and it has no holomorphic extension.

## Explanations

### L1 human

A holomorphic function that stays bounded near a thin bad set, such as a complex curve or a point, can be extended across that set.
If the bad set is very thin, of complex codimension at least two, no boundedness is needed at all.
This generalises Riemann's theorem on removable singularities from one variable.

### L2 human

Part 1 is used constantly when holomorphic functions are defined only off an analytic set, for example off the singular points of an analytic variety.
Part 2 is a form of the Hartogs phenomenon: a holomorphic function cannot have a singular set that is an analytic set of codimension two or more.
Part 1 is also the reason why a continuous function on $D$ that is holomorphic off such a set $A$ is holomorphic on all of $D$.
</document>

<document source="https://tamnd.github.io/math2/node/5VWTS78ZPJ.md" tag="5VWTS78ZPJ" kind="definition">
# Flat norm

## Definition

Let $0 \le m \le n$, and for a [[current|360H7MD9RD]] $R$ on $\R^n$ let $\mathbf{M}(R) \in [0, \infty]$ be its mass.
The *flat norm* of an $m$-current $T$ on $\R^n$ is
$$\mathbf{F}(T) = \inf \set{ \mathbf{M}(R) + \mathbf{M}(S) : R \text{ an } m\text{-current}, \ S \text{ an } (m+1)\text{-current}, \ T = R + \partial S },$$
with $\inf \emptyset = \infty$.
The *flat distance* between $m$-currents $T_1$ and $T_2$ is $\mathbf{F}(T_1 - T_2)$.
For [[integral currents|EG4AJVEFT2]], the *integral flat norm* is defined in the same way with $R$ and $S$ required to be integral currents.

## Remarks

Taking $S = 0$ shows $\mathbf{F}(T) \le \mathbf{M}(T)$.
If $\mathbf{F}(T_j - T) \to 0$, then $T_j(\omega) \to T(\omega)$ for every smooth compactly supported $m$-form $\omega$, so flat convergence implies weak convergence.
When $\mathbf{F}(T) < \infty$, there is a dual description of $\mathbf{F}(T)$ as the supremum of $T(\omega)$ over the forms $\omega$ such that $\omega$ and $d\omega$ have comass at most $1$ everywhere.

Example: in $\R$, let $a < b$ and $T = \delta_b - \delta_a$, a $0$-current.
Then $T = \partial [\![a, b]\!]$, so $\mathbf{F}(T) \le b - a$, and $\mathbf{F}(T) \le \mathbf{M}(T) = 2$.
Testing $T$ on a function $\varphi$ with $\abs{\varphi} \le 1$ and $\abs{\varphi'} \le 1$, chosen with $\varphi(b) - \varphi(a) = \min(b - a, 2)$, shows that $\mathbf{F}(T) = \min(b - a, 2)$.
So $\delta_b - \delta_a \to 0$ in the flat norm as $b \to a$, while its mass stays $2$.

Non-example: the mass $\mathbf{M}$ is a norm on currents of finite mass, but it is not the flat norm, as the example shows.

## Explanations

### L1 human

The flat norm of a current is the cheapest way to write it as a piece plus a boundary, where each part costs its mass.
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# Virtual fundamental class

## Summary

A virtual fundamental class is a homology class that plays the role of the [[fundamental class|ZEA0PEM3B2]].
It is attached to a moduli space that need not be a manifold of the expected dimension.
It lives in the expected, or virtual, dimension: the number of unknowns minus the number of equations that cut out the space.
It is built by perturbing the equations to transverse ones.
In algebraic geometry it is built from a perfect obstruction theory, as done by Li and Tian and by Behrend and Fantechi.
Integrating cohomology classes against it defines invariants such as [[Gromov–Witten invariants|XVJA0JD79P]] and Donaldson–Thomas invariants.

## Explanations

### L1 human

When a space is cut out by equations, one expects its dimension to be the number of unknowns minus the number of equations.
Sometimes the equations are degenerate and the actual solution space is bigger, or singular.
The virtual fundamental class is what the solution space would contribute if the equations were in general position.
For example, the zero vector field on the sphere vanishes everywhere, so its zero set is the whole sphere.

A generic vector field on the sphere vanishes at points whose signed count is $2$.
The virtual class is these points counted with signs, which is $2$ times the class of a point.
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# Dedekind-finite ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with $1$.
The ring $R$ is *Dedekind-finite* if, for all $a, b \in R$, the equation $ab = 1$ implies $ba = 1$.

## Remarks

Example: for a field $F$ and $n \ge 1$, the [[matrix ring|0FHVPRAE60]] $M_n(F)$ is Dedekind-finite.
Indeed, if $AB = I$, then $\det A \det B = 1$, so $A$ is invertible, and $B = A^{-1}(AB) = A^{-1}$, hence $BA = I$.

Non-example: let $V$ be the vector space of sequences $(x_1, x_2, \dots)$ over a field and $R = \End(V)$.
Let $a(x_1, x_2, \dots) = (x_2, x_3, \dots)$ and $b(x_1, x_2, \dots) = (0, x_1, x_2, \dots)$.
Then $ab = 1$, but $ba(x_1, x_2, \dots) = (0, x_2, x_3, \dots)$, so $ba \ne 1$.

Other names are *directly finite* and *von Neumann finite*.
Equivalently, $R$ is not isomorphic as a left $R$-module to $R \oplus N$ for any nonzero module $N$.
Every commutative ring, every finite ring, and every left Noetherian ring is Dedekind-finite.
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# Eigenfunction of an integral operator

## Definition

Let $a < b$, let $K \in L^2([a, b] \times [a, b])$, and let $T_K$ be the [[Hilbert–Schmidt integral operator|CB5G212Q03]] $(T_K \varphi)(x) = \int_a^b K(x, y) \varphi(y) \, dy$ on $L^2[a, b]$.
An *eigenfunction* of $T_K$ is a function $\varphi \in L^2[a, b]$, not $0$ almost everywhere, such that
$$\int_a^b K(x, y) \, \varphi(y) \, dy = \mu \, \varphi(x) \quad \text{for almost every } x \in [a, b]$$
for some $\mu \in \C$.
The number $\mu$ is the [[eigenvalue|0BQAA2BK3Q]] of $\varphi$.

## Remarks

So an eigenfunction is an [[eigenvector|0N6960W0BV]] of $T_K$ in the vector space $L^2[a, b]$.
When $\mu \ne 0$, the eigenfunctions for $\mu$ are the nonzero solutions of the homogeneous equation $\varphi = \lambda T_K \varphi$ with $\lambda = 1/\mu$.

Example: for $K(x, y) = \min(x, y)$ on $[0, 1]$ and $\varphi(x) = \sin(\pi x / 2)$, one has $\int_0^1 \min(x, y) \varphi(y) \, dy = \frac{4}{\pi^2} \varphi(x)$.
To check, call the left side $g(x)$; then $g(0) = 0$, $g'(1) = 0$ and $g'' = -\varphi$, and $\frac{4}{\pi^2} \varphi$ satisfies the same three conditions, which determine $g$.

Non-example: for the same kernel, $\varphi = 1$ is not an eigenfunction, since $\int_0^1 \min(x, y) \, dy = x - x^2/2$ is not a constant multiple of $1$.

The same definition is used with $L^2[a, b]$ replaced by $C[a, b]$ for continuous kernels, or by the $L^2$ space of a measure space.
For continuous kernels and $\mu \ne 0$, every eigenfunction in $L^2$ agrees almost everywhere with a continuous one.
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# Stratonovich integral

## Definition

Let $X$ and $Y$ be continuous real [[semimartingales|4EJB6E2AZK]] on a filtered probability space, and let $t \ge 0$.
The *Stratonovich integral* of $Y$ with respect to $X$ is
$$\int_0^t Y_s \circ dX_s = \int_0^t Y_s \, dX_s + \frac12 [Y, X]_t.$$
Here $\int_0^t Y_s \, dX_s$ is the [[Itô integral|JA95NH206E]].
The process $[Y, X]$ is the [[quadratic covariation|X4DHMZD5JA]] of $Y$ and $X$.

## Remarks

Equivalently, it is the limit in probability of the sums $\sum_i \tfrac12 (Y_{t_i} + Y_{t_{i+1}})(X_{t_{i+1}} - X_{t_i})$ over partitions $0 = t_0 < \cdots < t_n = t$ whose mesh tends to $0$.
Example: let $B$ be a [[Brownian motion|22735FGJV6]] with $B_0 = 0$.
Then $\int_0^t B_s \, dB_s = \tfrac12 (B_t^2 - t)$ and $[B, B]_t = t$, so $\int_0^t B_s \circ dB_s = \tfrac12 B_t^2$.
Non-example: the left-point sums $\sum_i B_{t_i} (B_{t_{i+1}} - B_{t_i})$ converge to $\tfrac12 (B_t^2 - t)$, not to the Stratonovich integral, because they omit the covariation term.
For $f$ three times continuously differentiable, the chain rule takes the classical form $f(X_t) = f(X_0) + \int_0^t f'(X_s) \circ dX_s$.
The notation $\circ \, dX$, with a small circle, marks the Stratonovich integral.
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# Free spectrum of a variety

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]], and for each natural number $n$ let $F_{\mathcal{V}}(n)$ be the [[free algebra|GTS4DRRMZR]] of $\mathcal{V}$ on a set of $n$ generators.
The *free spectrum* of $\mathcal{V}$ is the function
$$f_{\mathcal{V}}(n) = \abs{F_{\mathcal{V}}(n)}, \qquad n \in \N,$$
with values in $\N \cup \set{\infty}$.

## Remarks

The values are all finite exactly when $\mathcal{V}$ is a [[locally finite variety|8QPW65PZVW]].
When $\mathcal{V}$ is generated by a finite algebra $A$, the free algebra $F_{\mathcal{V}}(n)$ embeds in $A^{A^n}$, so $f_{\mathcal{V}}(n) \le \abs{A}^{\abs{A}^n}$.
Example: for Boolean algebras, $f(n) = 2^{2^n}$, since $F(n)$ is the algebra of all Boolean functions $\set{0, 1}^n \to \set{0, 1}$.
Example: for semilattices, $f(n) = 2^n - 1$, since $F(n)$ is the set of nonempty subsets of the generators under union.
Non-example: the variety of all groups has no finite spectrum value for $n \ge 1$, since the free group on one generator is $\Z$.
Some authors start the spectrum at $n = 1$, or use the related function $\log_2 f_{\mathcal{V}}(n)$.
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# CW complex

## Summary

A CW complex is a space built by gluing closed balls of increasing dimension, called cells.
Each ball is glued along its boundary sphere to the cells already present.
The class of CW complexes is flexible enough to contain spheres, projective spaces, Grassmannians, all [[simplicial complexes|V8GZ0W1ZZ5]] and, up to homotopy equivalence, all manifolds.
It is rigid enough for homotopy theory to work well.
Weak homotopy equivalences between CW complexes are homotopy equivalences, and homology can be computed cell by cell.
The letters C and W stand for closure-finite and weak topology, the two conditions of J. H. C. Whitehead's definition.

## Explanations

### L1 human

Start with a set of points, the $0$-cells.
Glue intervals to them by their ends, giving a graph.
Then glue disks along their boundary circles to the graph, then solid balls along their boundary spheres, and so on.
The sphere $S^2$ is a CW complex with one point and one disk whose whole boundary circle is glued to that point.
The torus is a CW complex with one point, two loops and one square glued along the word $aba^{-1}b^{-1}$.
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# E-infinity operad

## Definition

An *operad* $\mathcal{O}$ in [[topological spaces|0CNQXVBHCX]] consists of spaces $\mathcal{O}(n)$ for $n \ge 0$ and the following data.
There is a right [[action|0YSE802YCQ]] of the [[symmetric group|0RJMNEN3BJ]] $\Sigma_n$ on each $\mathcal{O}(n)$ and an identity element $1 \in \mathcal{O}(1)$.
There are continuous composition maps
$$\mathcal{O}(k) \times \mathcal{O}(j_1) \times \cdots \times \mathcal{O}(j_k) \to \mathcal{O}(j_1 + \cdots + j_k),$$
which are associative, unital and equivariant in the standard sense.
An operad $\mathcal{O}$ is an *$E_\infty$ operad* if, for every $n \ge 0$:

1. the space $\mathcal{O}(n)$ is [[contractible|R5WP8Z69Q8]], and
2. the action of $\Sigma_n$ on $\mathcal{O}(n)$ is free.

## Remarks

Often one also asks the action to be nice enough that $\mathcal{O}(n) \to \mathcal{O}(n)/\Sigma_n$ is a principal $\Sigma_n$-bundle, so that $\mathcal{O}(n)/\Sigma_n$ is a model of $B\Sigma_n$.

Example: the linear isometries operad, with $\mathcal{L}(n)$ the space of linear isometric embeddings $(\R^\infty)^n \to \R^\infty$ and composition by composing maps.
Each $\mathcal{L}(n)$ is contractible, and $\Sigma_n$ acts by permuting the $n$ copies of $\R^\infty$.
The action is free: an isometric embedding sends the $n$ summands to mutually orthogonal subspaces, so it is not invariant under a nontrivial permutation.

Example: the Barratt–Eccles operad has $\mathcal{O}(n) = E\Sigma_n$, which is contractible with a free $\Sigma_n$-action.

Non-example: the commutative operad, with $\mathcal{O}(n)$ a single point for every $n$, satisfies clause 1 but not clause 2 when $n \ge 2$, since $\Sigma_n$ fixes the point.
Its algebras are strictly commutative monoids, which are too restrictive in homotopy theory.
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# Universally catenary ring

## Summary

A universally catenary ring is a [[Noetherian ring|0670T59HGE]] whose finitely generated algebras are all [[catenary|B2CYR292J6]].
Being catenary is not preserved when one adjoins variables, so this stronger condition is the one that behaves well under the usual constructions.
Fields, $\Z$, Cohen–Macaulay rings and [[excellent rings|YF7YFQ58QJ]] are universally catenary.

## Explanations

### L1 human

A catenary ring is one where all maximal chains of prime ideals between two primes have the same length.
A universally catenary ring keeps this property after any finite number of polynomial variables are added and after any quotient is taken.
For example, every finitely generated algebra over a field, such as $k[x, y]/(xy)$, is catenary because the field $k$ is universally catenary.
The [[definition|PYABBA1QQE]] makes this precise.
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# Semitopological group

## Summary

A semitopological group is a [[group|G5S1M9]] with a topology in which multiplication is continuous in each variable separately.
Equivalently, every left translation and every right translation is continuous, and then each translation is a homeomorphism.
It weakens the notion of a [[topological group|92700QAB5Y]], which asks for joint continuity of multiplication and continuity of inversion.
Ellis's theorem says that a locally compact Hausdorff semitopological group is already a topological group.

## Explanations

### L1 human

In a semitopological group you may move one factor at a time and the product changes continuously.
Moving both factors at once may cause a jump.
A simple example is the group of integers under addition, with the topology in which the open sets are the empty set and the sets with finite complement.
Shifting by a fixed integer keeps such sets open, but addition of two varying integers is not continuous in this topology.
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# Weierstrass equation

## Definition

Let $K$ be a [[field|04C5ZBG1WT]].
A *Weierstrass equation* over $K$ is an equation
$$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$$
with $a_1, a_2, a_3, a_4, a_6 \in K$.
Its associated quantities are
$$b_2 = a_1^2 + 4a_2, \quad b_4 = 2a_4 + a_1 a_3, \quad b_6 = a_3^2 + 4a_6,$$
$$b_8 = a_1^2 a_6 + 4a_2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 - a_4^2,$$
and its *discriminant* is
$$\Delta = -b_2^2 b_8 - 8 b_4^3 - 27 b_6^2 + 9 b_2 b_4 b_6.$$

## Remarks

The projective closure of the curve in the [[projective plane|ZT9A6JK8QF]] adds one point $O = [0 : 1 : 0]$.
This closure is smooth exactly when $\Delta \ne 0$, and then it is an [[elliptic curve|427N6S6QQW]] with origin $O$.
If the characteristic of $K$ is not $2$ or $3$, a linear change of variables brings the equation to the short form $y^2 = x^3 + Ax + B$, for which $\Delta = -16(4A^3 + 27B^2)$.
Example: $y^2 = x^3 - x$ has $A = -1$ and $B = 0$, so $\Delta = -16 \cdot (-4) = 64 \ne 0$, and it defines an elliptic curve over $\Q$.
Non-example: $y^2 = x^3$ is a Weierstrass equation with $\Delta = 0$; the curve has a cusp at $(0, 0)$, where both partial derivatives of $y^2 - x^3$ vanish, so it is not an elliptic curve.
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# Stein interpolation theorem

## Statement

Let $(X, \mu)$ and $(Y, \nu)$ be $\sigma$-finite measure spaces, and let $S = \set{z \in \C : 0 \le \operatorname{Re} z \le 1}$.
Call a simple function admissible if its support has finite measure.
For each $z \in S$ let $T_z$ be a linear operator from the admissible simple functions on $X$ to the measurable functions on $Y$.
Assume that $(T_z)$ is an admissible analytic family, which means the following.
Let $f$ on $X$ and $g$ on $Y$ be admissible simple functions.
It is required that $T_z(f) g$ is integrable for each $z$, and that $\Phi(z) = \int_Y T_z(f) \, g \, d\nu$ is continuous on $S$ and [[holomorphic|0NAHXR1FRH]] in the interior of $S$.
Moreover $\log \abs{\Phi(z)} \le C e^{a \abs{\operatorname{Im} z}}$ on $S$ for some constants $C$ and $a < \pi$.

Let $1 \le p_0, p_1, q_0, q_1 \le \infty$.
Let $M_0, M_1 \colon \R \to (0, \infty)$ satisfy $\log M_j(y) \le C' e^{b \abs{y}}$ for some constants $C'$ and $b < \pi$.
Assume that for all admissible simple $f$ on $X$ and all $y \in \R$,
$$\norm{T_{iy} f}_{L^{q_0}} \le M_0(y) \norm{f}_{L^{p_0}}, \qquad \norm{T_{1 + iy} f}_{L^{q_1}} \le M_1(y) \norm{f}_{L^{p_1}}.$$
Let $0 < \theta < 1$ and define $p$ and $q$ by
$$\frac{1}{p} = \frac{1 - \theta}{p_0} + \frac{\theta}{p_1}, \qquad \frac{1}{q} = \frac{1 - \theta}{q_0} + \frac{\theta}{q_1}.$$
Then $\norm{T_\theta f}_{L^q} \le M_\theta \norm{f}_{L^p}$ for all admissible simple $f$ on $X$, where
$$\log M_\theta = \frac{\sin \pi\theta}{2} \int_{\R} \left( \frac{\log M_0(y)}{\cosh \pi y - \cos \pi\theta} + \frac{\log M_1(y)}{\cosh \pi y + \cos \pi\theta} \right) dy.$$

## Remarks

The proof follows the proof of the [[Riesz–Thorin theorem|AGRKRG1TMJ]].
For fixed simple $f$ and $g$ one builds families $f_z$, $g_z$ of simple functions whose absolute values are powers of $\abs{f}$ and $\abs{g}$ depending on $z$.
Then one studies the holomorphic function $z \mapsto \int_Y T_z(f_z) g_z \, d\nu$ on the strip.
The hypotheses bound it on the two boundary lines.
A form of the [[Hadamard three-lines theorem|WHS58R43N0]] for functions of admissible growth, proved with the Poisson kernel of the strip, then bounds it at $z = \theta$.
The two kernels in the formula for $M_\theta$ are the Poisson kernels of the strip at the point $\theta$.

If $T_z = T$ does not depend on $z$ and $M_0$, $M_1$ are constants, the formula gives $M_\theta = M_0^{1 - \theta} M_1^{\theta}$, which is the Riesz–Thorin theorem.
A typical use is the family of fractional integration or Bochner–Riesz type operators $T_z$, where the exponent of a kernel depends analytically on $z$.
The bound $a < \pi$ in the growth condition cannot be relaxed to $a = \pi$.
The function $h(z) = \exp(i e^{-i\pi z})$ has modulus $1$ on both boundary lines of $S$, but $\abs{h(\tfrac12 + iy)} = \exp(e^{\pi y})$ is unbounded.

## Explanations

### L1 human

Suppose a family of operators depends on a complex parameter in a smooth, holomorphic way.
If the operators are bounded between two pairs of $L^p$ spaces on the two edges of a strip, then the operator in the middle is bounded between the intermediate spaces.
This is a version of Riesz–Thorin interpolation in which the operator itself is allowed to move.

### L2 human

The Riesz–Thorin theorem interpolates one fixed operator.
Many problems in harmonic analysis come instead with a natural family, for example operators whose kernel contains a power $\abs{x}^{z}$ or a factor $(1 - \abs{\xi}^2)^{z}_+$.
At one end of the strip the family is easy to bound in $L^2$ by Plancherel's theorem.
At the other end it is bounded in $L^\infty$ by a pointwise estimate of the kernel.
Stein's theorem gives the bounds in between, as in results on Bochner–Riesz means and on Fourier restriction.
It belongs to the complex method of interpolation, while the [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]] belongs to the real method.
</document>

<document source="https://tamnd.github.io/math2/node/5WHXRZDZFK.md" tag="5WHXRZDZFK" kind="concept">
# Perverse sheaf

## Summary

A perverse sheaf on a complex algebraic variety or analytic space is an object of the [[derived category|5ST7NVJFSM]] of constructible sheaves, see the [[definition|JYW9312EE6]].
It satisfies dimension conditions on its cohomology sheaves and on those of its dual.
Despite the name, a perverse sheaf is a complex of sheaves, not a single sheaf.
Perverse sheaves form an abelian category, the heart of the perverse [[t-structure|2GD9V7SA0E]], introduced by Beilinson, Bernstein, Deligne and Gabber.
They include the intersection cohomology complexes of Goresky and MacPherson, and by the [[Riemann–Hilbert correspondence|KMAEWSVGRS]] they correspond to regular holonomic D-modules.

## Explanations

### L1 human

On a smooth space, the constant sheaf, which assigns to each open set the locally constant functions on it, behaves well with respect to Poincaré duality.
On a singular space it does not.
The fix is to use complexes of sheaves with cohomology in several degrees, with rules about how large the support of each degree may be.
These rules are chosen so that the class of objects is preserved by duality.
On a smooth variety of dimension $d$, the constant sheaf placed in degree $-d$ is perverse, and so is the sheaf of a single point placed in degree $0$.
</document>

<document source="https://tamnd.github.io/math2/node/5WR304YRST.md" tag="5WR304YRST" kind="concept">
# Brjuno number

## Summary

A Brjuno number is an [[irrational number|0DZXAXEJQ9]] whose [[continued fraction|04P2RJ4WPS]] convergents have denominators that do not grow too fast.
The condition is that the series $\sum_n \log(q_{n+1}) / q_n$ converges.
Brjuno numbers are exactly the rotation numbers for which every holomorphic germ with that rotation at a fixed point can be linearized.
They are central to small divisor problems and to the existence of [[Siegel discs|KNDYA9T8ZK]].

## Explanations

### L1 human

Some irrational numbers are very close to fractions with small denominators, and others stay away from all such fractions.
A Brjuno number is one that is not approximated too well by fractions, measured in a precise, fairly generous way.
The golden mean is a Brjuno number, and in fact almost every real number is one.
The condition decides when a map that turns points near a fixed point by such an angle can be straightened into an exact rotation.
</document>

<document source="https://tamnd.github.io/math2/node/5WR98YG0AX.md" tag="5WR98YG0AX" kind="definition">
# Virtual fundamental class

## Definition

Let $V$ be an [[oriented|913WEQERBE]] [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $E \to V$ be an oriented real [[vector bundle|P4DZM975BV]] of rank $r$.
Let $s$ be a smooth [[section|WWX7H5W1RP]] of $E$ whose zero set $X = s^{-1}(0)$ is compact.
Let $U$ be an open neighbourhood of $X$ with compact closure in $V$.
Choose a smooth section $s'$ of $E$ with these properties:

1. $s'$ agrees with $s$ outside a compact subset of $U$.
2. $s'$ is [[transverse|A2PDZ8GA9C]] to the zero section.
3. No section $(1 - t) s + t s'$ with $t \in [0, 1]$ has a zero on $\overline{U} \setminus U$.

Then $Z = s'^{-1}(0) \cap U$ is a compact oriented submanifold of dimension $d = n - r$.
The *virtual fundamental class* of $X$ in $U$ is the image of the [[fundamental class|ZEA0PEM3B2]] of $Z$ in [[singular homology|M0Q5CFJ7ZH]]:
$$[X]^{\mathrm{vir}} = [Z] \in H_d(U; \Q).$$
The number $d$ is the *virtual dimension*.

## Remarks

The class does not depend on the choice of $s'$.
Two choices are joined by a generic homotopy whose zero set is an oriented cobordism between the two zero sets inside $U$.
The classes for smaller neighbourhoods map to the classes for larger ones.
If $s$ itself is transverse to the zero section, then $X$ is a compact oriented manifold of dimension $d$ and $[X]^{\mathrm{vir}}$ is its fundamental class.
For a general Kuranishi structure the same construction is done chart by chart with multivalued perturbations, which is why rational coefficients are used.

Example: let $V = S^2$, $E = T S^2$ and $s = 0$, so $X = S^2$ and $d = 0$.
A generic vector field on $S^2$ has isolated zeros whose signed count is $2$, the value of the [[Euler class|BJX6QS0ANT]] of $T S^2$ on the fundamental class of $S^2$.
So $[X]^{\mathrm{vir}} = 2 \, [\mathrm{pt}] \in H_0(S^2; \Q)$, although $X$ has dimension $2$.


Non-example: for $V = \R$, $E = \R \times \R$ and $s = 0$, the zero set $X = \R$ is not compact.
The perturbations $s' = \varepsilon$ and $s' = \varepsilon x$ have $0$ and $1$ zeros, so no well-defined class arises; this is why compactness is required.
</document>

<document source="https://tamnd.github.io/math2/node/5WVZSM42XR.md" tag="5WVZSM42XR" kind="statement">
# Domination principle

## Statement

Let $n \ge 3$ and let $\mu$ and $\nu$ be positive [[Radon measures|HZNEQQBCW4]] on $\R^n$ with compact support.
For a positive measure $\lambda$, let $U^\lambda(x) = \int \abs{x - y}^{2-n} \, d\lambda(y)$ be its [[Newtonian potential|DCMFQ01QGS]].
Assume that $\mu$ has finite [[energy|SFSRQCM3EG]] $\int U^\mu \, d\mu < \infty$, and that $U^\mu(x) \le U^\nu(x)$ for $\mu$-almost every $x$.
Then $U^\mu(x) \le U^\nu(x)$ for every $x \in \R^n$.

## Remarks

Proof idea: standard proofs combine balayage with the positive definiteness of the Newtonian energy.
Balayage sweeps a measure onto a set without changing its potential there.
The finite energy of $\mu$ is what makes these energy arguments possible.

Example: let $\nu = \delta_0$, a unit point mass at the origin of $\R^3$.
Let $\mu$ be the uniform measure of mass $1$ on the sphere of radius $r$ centred at $0$.
On the sphere, $U^\mu = 1/r = U^\nu$, so the hypothesis holds.
The theorem gives $U^\mu \le U^\nu$ everywhere, and indeed $U^\mu(x) = \min(1/r, 1/\abs{x}) \le 1/\abs{x} = U^\nu(x)$.

The finite energy of $\mu$ is needed.
For $\mu = \delta_0$ and $\nu = \frac12 \delta_0$, both potentials are $+\infty$ at $0$, so $U^\mu \le U^\nu$ holds $\mu$-almost everywhere, but $U^\mu(x) = 1/\abs{x} > U^\nu(x)$ for $x \ne 0$.
Take for $\nu$ a uniform measure on a large sphere, whose potential equals a constant $M$ inside the sphere and is at most $M$ outside.
This gives back the maximum principle for potentials of finite energy with support inside the sphere.

## Explanations

### L1 human

The potential of a charge $\mu$ of finite energy is controlled everywhere as soon as it is controlled on the charge itself.
If the potential of $\nu$ is at least as large as that of $\mu$ wherever $\mu$ sits, then it is at least as large everywhere.
So comparing two potentials only needs a check on the support of one of them.

### L2 human

The domination principle is one of the basic properties of the Newtonian kernel, and it is taken as an axiom in abstract potential theory.
It is used in the theory of balayage and in comparison arguments for equilibrium potentials.
It holds for Riesz kernels $\abs{x - y}^{\alpha - n}$ with $0 < \alpha \le 2$ and $\alpha < n$.
In the plane with the logarithmic kernel, an extra condition on the total masses is needed.
</document>

<document source="https://tamnd.github.io/math2/node/5WXY2T8JG0.md" tag="5WXY2T8JG0" kind="definition">
# Stokes flow

## Definition

Let $\Omega \subseteq \R^3$ be an open set, let $\nu > 0$ be a constant, and let $f \colon \Omega \to \R^3$ be a given force.
Consider a [[vector field|0299VCW1TH]] $u \colon \Omega \to \R^3$, the velocity, and a function $p \colon \Omega \to \R$, the pressure.
The pair $(u, p)$ is a *Stokes flow* in $\Omega$ if it satisfies the *Stokes equations*
$$- \nu \Delta u + \nabla p = f, \qquad \nabla \cdot u = 0 \quad \text{in } \Omega.$$
Here $\Delta$ is the [[Laplacian|0E0B6RA8K0]] acting on each component, $\nabla p$ is the [[gradient|0ABK9V9Y64]] and $\nabla \cdot u$ is the [[divergence|0MWV6ZTK6S]].

## Remarks

Example: in the channel $\Omega = \set{(x, y, z) : -1 < y < 1}$ with $f = 0$, the Poiseuille flow $u = (1 - y^2, 0, 0)$ with $p = -2\nu x$ is a Stokes flow.
Indeed $\Delta u = (-2, 0, 0)$, so $-\nu \Delta u + \nabla p = (2\nu, 0, 0) + (-2\nu, 0, 0) = 0$, and $\nabla \cdot u = 0$.
It also vanishes on the walls $y = \pm 1$.

Non-example: the same $u$ with constant pressure is not a Stokes flow when $f = 0$, because then $-\nu \Delta u + \nabla p = (2\nu, 0, 0) \ne 0$.

The Stokes equations arise from the [[Navier–Stokes equations|18KXZHVF4M]] when the Reynolds number is small, so that the term $(u \cdot \nabla) u$ can be neglected, and the flow is steady.
The time-dependent version $\partial_t u - \nu \Delta u + \nabla p = f$ is also called the Stokes equations.
</document>

<document source="https://tamnd.github.io/math2/node/5WYZPX1ZVW.md" tag="5WYZPX1ZVW" kind="statement">
# Bilinear series of a symmetric kernel

## Statement

Let $a < b$ and let $K \in L^2([a, b] \times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]] that is [[Hermitian|2ZB3NFZ94T]]: $K(y, x) = \overline{K(x, y)}$ for almost all $x, y$.
Let $(\varphi_n)_{n \in N}$ be an orthonormal family of [[eigenfunctions|4NSACSWRJZ]] of the integral operator $T_K$, with nonzero eigenvalues $\mu_n$.
Assume that, for each nonzero eigenvalue, the $\varphi_n$ with that eigenvalue form a basis of its eigenspace.
Let $\lambda_n = 1/\mu_n$ be the corresponding [[characteristic values|DC7HMHVREP]].
Then
$$K(x, y) = \sum_{n \in N} \mu_n \, \varphi_n(x) \overline{\varphi_n(y)} = \sum_{n \in N} \frac{\varphi_n(x) \overline{\varphi_n(y)}}{\lambda_n},$$
where the series converges in $L^2([a, b] \times [a, b])$, and
$$\sum_{n \in N} \mu_n^2 = \int_a^b \int_a^b \abs{K(x, y)}^2 \, dx \, dy.$$

## Remarks

The functions $\varphi_n(x) \overline{\varphi_m(y)}$ form an orthonormal family in $L^2([a, b]^2)$.
The coefficient of $K$ on $\varphi_n(x) \overline{\varphi_m(y)}$ is $\int \int K(x, y) \varphi_m(y) \overline{\varphi_n(x)} \, dy \, dx = \mu_m \ip{\varphi_m}{\varphi_n}$, which is $\mu_n$ if $m = n$ and $0$ otherwise.
So the series is the orthogonal projection of $K$ onto the span of this family.
The difference between $K$ and the series is a kernel with operator $T_K - \sum_n \mu_n \ip{\cdot}{\varphi_n} \varphi_n$.
This operator is $0$ by the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]], so the difference is $0$ almost everywhere.
The identity for $\sum \mu_n^2$ is then [[Parseval's identity|047RFCNVSK]].

Example: for $K(x, y) = \min(x, y)$ on $[0, 1]$,
$$\min(x, y) = \sum_{n \ge 1} \frac{2 \sin\left((n - \tfrac12)\pi x\right) \sin\left((n - \tfrac12)\pi y\right)}{(n - \tfrac12)^2 \pi^2},$$
and the identity gives $\sum_{n \ge 1} \frac{1}{(n - \frac12)^4 \pi^4} = \int_0^1 \int_0^1 \min(x, y)^2 = \frac{1}{6}$.

In general the convergence is only in $L^2$.
Under the extra hypotheses of [[Mercer's theorem|PKGADWXPYH]], continuity and positivity of the kernel, the series converges absolutely and uniformly.

## Explanations

### L1 human

A symmetric kernel can be rebuilt from its eigenfunctions and eigenvalues.
Each eigenfunction contributes one simple product term, weighted by its eigenvalue.
This is the integral form of writing a symmetric matrix as a sum of rank-one pieces along its eigenvectors.

### L2 human

The bilinear series diagonalizes the kernel itself, not only the operator.
It gives the estimate $\sum \mu_n^2 < \infty$, so the characteristic values grow fast enough that $\sum 1/\lambda_n^2$ converges.
For positive continuous kernels it is upgraded to uniform convergence by Mercer's theorem, which also gives the trace formula $\sum \mu_n = \int K(x, x) \, dx$.
</document>

<document source="https://tamnd.github.io/math2/node/5X1YMSR204.md" tag="5X1YMSR204" kind="concept">
# Maximum and minimum

## Summary

The maximum of a real function on a set is its largest value, and the minimum is its smallest value.
A point where the maximum or the minimum is attained is a global extremum; this is stronger than a [[local extremum|0WFBH1Q3WC]].
A function need not attain a maximum, but by the [[extreme value theorem|0WS1A8E2P9]] a continuous function on a closed bounded interval does.
Finding maxima and minima is the aim of optimization, and derivatives are the main tool.

## Explanations

### L1 human

The maximum of a function is the highest point of its graph, and the minimum is the lowest point.
On $[-1, 2]$, the function $x^2$ has its maximum $4$ at $x = 2$ and its minimum $0$ at $x = 0$.
Some functions have no highest point: on the open interval $(0, 1)$, the function $x$ gets close to $1$ but never reaches it.
</document>

<document source="https://tamnd.github.io/math2/node/5X6E97EMFR.md" tag="5X6E97EMFR" kind="concept">
# Topological dynamics

## Summary

Topological dynamics is the part of the theory of [[dynamical systems|BBZP1R872N]] that studies continuous maps and group actions on topological spaces.
It uses only the topology, not a measure or a smooth structure.
Its questions concern the long-term behaviour of orbits: recurrence, density of orbits, minimal sets, mixing and the growth of complexity.
Typical objects are a compact metric space $X$ with a [[homeomorphism|0QT00PB5G2]] or a continuous map $T \colon X \to X$.
It sits beside ergodic theory, which studies the same questions with a measure in place of a topology.

## Explanations

### L1 human

Imagine a space of states and a rule that moves each state to the next one, where nearby states move to nearby states.
Topological dynamics asks qualitative questions about this motion without measuring anything.
Does a point come back near where it started?
Does some orbit pass near every point?
For a rotation of a circle by an irrational angle, every orbit comes back arbitrarily close to its start and passes near every point of the circle.
</document>

<document source="https://tamnd.github.io/math2/node/5X6G3Q5CSQ.md" tag="5X6G3Q5CSQ" kind="definition">
# Lie supergroup

## Definition

A *Lie supergroup* is a real [[supermanifold|Z0SB9HQBAJ]] $G$ with three morphisms of supermanifolds
$$m \colon G \times G \to G, \qquad i \colon G \to G, \qquad e \colon \R^{0|0} \to G,$$
called multiplication, inverse and unit.
These must make the following diagrams of morphisms commute:

1. associativity: $m \circ (m \times \id) = m \circ (\id \times m)$ as morphisms $G \times G \times G \to G$;
2. unit: $m \circ (e \times \id) = \id = m \circ (\id \times e)$, after identifying $\R^{0|0} \times G$ and $G \times \R^{0|0}$ with $G$;
3. inverse: $m \circ (i \times \id) \circ \Delta = e \circ \pi = m \circ (\id \times i) \circ \Delta$, where $\Delta \colon G \to G \times G$ is the diagonal and $\pi \colon G \to \R^{0|0}$ is the unique morphism to the point.

## Remarks

Example: on $\R^{1|1}$, with even coordinate $t$ and odd coordinate $\theta$, define $m$ by $(t, \theta)(t', \theta') = (t + t' + \theta\theta', \theta + \theta')$, inverse $(t, \theta) \mapsto (-t, -\theta)$ and unit $(0, 0)$.
Associativity holds since both bracketings give $(t + t' + t'' + \theta\theta' + \theta\theta'' + \theta'\theta'', \theta + \theta' + \theta'')$.
The inverse works since $(t, \theta)(-t, -\theta) = (-\theta\theta, 0) = (0, 0)$, as $\theta^2 = 0$.

Non-example: on $\R^{1|1}$ the law $(t, \theta)(t', \theta') = (tt', \theta + \theta')$ is associative with unit $(1, 0)$.
But it has no inverse morphism: on underlying points it is multiplication of real numbers, and $0$ has no inverse.

The underlying manifold of a Lie supergroup is an ordinary [[Lie group|7JBRGCVJRB]].
The left-invariant vector fields on $G$ form a Lie superalgebra, the Lie superalgebra of $G$.
Ordinary Lie groups are the Lie supergroups with no odd coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/5X7NA4TTK8.md" tag="5X7NA4TTK8" kind="definition">
# Clifford algebra

## Definition

Let $F$ be a field, let $V$ be a [[vector space|000AYPA9AJ]] over $F$, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
Let $T(V)$ be the [[tensor algebra|P4J5JZDW0B]] of $V$.
Let $I_Q$ be the [[two-sided ideal|0CD87SFM2N]] of $T(V)$ generated by the elements $v \otimes v - Q(v) \cdot 1$ with $v \in V$.
The *Clifford algebra* of $(V, Q)$ is the [[quotient ring|0076HWHGHQ]]
$$\mathrm{Cl}(V, Q) = T(V) / I_Q.$$

## Remarks

Universal property: for every associative unital $F$-algebra $A$ and every linear map $f \colon V \to A$ with $f(v)^2 = Q(v) \cdot 1_A$ for all $v \in V$, there is exactly one unital algebra homomorphism $\mathrm{Cl}(V, Q) \to A$ that extends $f$.
If $\dim V = n$, then $\dim \mathrm{Cl}(V, Q) = 2^n$, with basis the products $e_{i_1} \cdots e_{i_k}$, $i_1 < \cdots < i_k$, for a basis $e_1, \ldots, e_n$ of $V$.
Example: $V = \R^2$ with $Q(x, y) = -(x^2 + y^2)$.
The relation $(e_1 + e_2)^2 = Q(e_1 + e_2) = -2$ with $e_1^2 = e_2^2 = -1$ gives $e_1 e_2 = -e_2 e_1$, and then $(e_1 e_2)^2 = -1$; so $\mathrm{Cl}(V, Q)$ is the quaternion algebra with $i = e_1$, $j = e_2$, $k = e_1 e_2$.
For $Q = 0$, the Clifford algebra is the [[exterior algebra|0MDMZ965VE]] of $V$.
Non-example: the tensor algebra $T(V)$ for $V \ne 0$ is not a Clifford algebra of $(V, Q)$, because $v \otimes v$ is not a scalar there.
Some authors use the opposite sign convention $v^2 = -Q(v)$.
</document>

<document source="https://tamnd.github.io/math2/node/5X8PK7H6W3.md" tag="5X8PK7H6W3" kind="concept">
# Fourier–Stieltjes transform

## Summary

The Fourier–Stieltjes transform extends the [[Fourier transform|DPVM34ADZV]] from integrable functions to finite [[complex measures|NY6TYHRGMS]] on $\R^n$.
It sends a measure $\mu$ to the function $\hat\mu(\xi) = \int e^{-2\pi i x \cdot \xi}\, d\mu(x)$, which is bounded and uniformly continuous.
Up to a change of variable it is the [[characteristic function|0V5B6D6BQH]] of a probability distribution.
[[Bochner's theorem|ZRHAG4NXHC]] describes the transforms of positive finite measures as the continuous positive-definite functions.
See the [[definition|JKCFHKT13F]].

## Explanations

### L1 human

Some objects that one wants to analyse by frequency are not functions but masses placed at points, such as a unit mass at the origin.
The Fourier–Stieltjes transform handles them by integrating the waves $e^{-2\pi i x \xi}$ against the mass distribution.
A unit mass at $0$ has the constant transform $1$: it contains every frequency equally.
A unit mass at $a$ has transform $e^{-2\pi i a \xi}$, a pure oscillation.
The name refers to the Stieltjes integral $\int e^{-2\pi i x \xi}\, dF(x)$ against a function $F$ of bounded variation, which is how the transform was first written on the line.
</document>

<document source="https://tamnd.github.io/math2/node/5XSY1Y133X.md" tag="5XSY1Y133X" kind="definition">
# Kobayashi metric

## Definition

Let $\mathbb{D} = \set{\zeta \in \C : \abs{\zeta} < 1}$ and let $\rho$ be the Poincaré distance on $\mathbb{D}$,
$$\rho(a, b) = \tanh^{-1} \abs{\frac{a - b}{1 - \bar a b}} .$$
Let $X$ be a connected [[complex manifold|JSGWAN2RJV]] and $p, q \in X$.
A *holomorphic chain* from $p$ to $q$ consists of points $p = p_0, p_1, \dots, p_k = q$ of $X$, points $a_1, b_1, \dots, a_k, b_k$ of $\mathbb{D}$, and [[holomorphic maps|3MQTWKBBBT]] $f_1, \dots, f_k \colon \mathbb{D} \to X$ with $f_j(a_j) = p_{j-1}$ and $f_j(b_j) = p_j$.
The *Kobayashi pseudodistance* of $X$ is
$$k_X(p, q) = \inf \sum_{j=1}^k \rho(a_j, b_j),$$
the infimum over all holomorphic chains from $p$ to $q$.

## Remarks

Holomorphic chains exist because $X$ is connected.
The function $k_X$ is a [[pseudodistance|0A3HJX3KTF]]: it is symmetric, satisfies the triangle inequality and vanishes on the diagonal, but it may vanish for distinct points.
Some authors use $2\rho$, the distance of curvature $-1$, which doubles $k_X$.
The infinitesimal form is the Kobayashi–Royden pseudometric $F_X(p, v) = \inf\set{1/R : \text{there is } f \colon \mathbb{D} \to X \text{ holomorphic with } f(0) = p,\ f'(0) = R v}$ for $v$ tangent at $p$; by a theorem of Royden, $k_X$ is the integrated form of $F_X$.
Distance-decreasing property: if $g \colon X \to Y$ is holomorphic, then $k_Y(g(p), g(q)) \le k_X(p, q)$, because $g$ maps holomorphic chains to holomorphic chains.
Example: $k_{\mathbb{D}} = \rho$; the inequality $k_{\mathbb{D}} \le \rho$ comes from the chain with $f = \id$, and $k_{\mathbb{D}} \ge \rho$ comes from the [[Schwarz–Pick theorem|V1QYA9RW73]].
Non-example of a distance: $k_\C \equiv 0$; for $p \ne q$ in $\C$ and $R > 1$, the map $f(\zeta) = p + R \zeta (q - p)$ sends $0$ to $p$ and $1/R$ to $q$, so $k_\C(p, q) \le \rho(0, 1/R) \to 0$ as $R \to \infty$.
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<document source="https://tamnd.github.io/math2/node/5XVEF5DKAG.md" tag="5XVEF5DKAG" kind="concept">
# Inverse mean curvature flow

## Summary

Inverse [[mean curvature flow|2YYV01C7QQ]] is a geometric evolution of hypersurfaces.
Each point moves outward along the unit normal, with speed equal to the reciprocal of the [[mean curvature|PA78NFEPCF]].
It needs a hypersurface with positive mean curvature.
Under this flow the area of the hypersurface grows exponentially in time.
Huisken and Ilmanen used a weak form of this flow to prove the Riemannian Penrose inequality for a connected horizon.

## Explanations

### L1 human

Think of a closed surface that expands outward.
Where the surface is strongly curved it moves slowly, and where it is nearly flat it moves fast.
This is the opposite of mean curvature flow, where curved parts move fastest and the surface shrinks.
A round sphere of radius $r$ in $\R^3$ has mean curvature $2/r$ (with $H$ the sum of the principal curvatures), so it expands with speed $r/2$.
Its radius is then $r_0 e^{t/2}$ at time $t$, and its area is $e^t$ times the initial area.
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# Oscillation of a function

## Summary

The oscillation of a function at a point measures the size of the spread of its values near that point.
It is $0$ exactly at the points where the function is [[continuous|0EEE2CZB2A]], so it measures how badly continuity fails.
It is a basic tool in the study of the set of discontinuities, for example in [[Lebesgue's criterion for Riemann integrability|0565EGK6XP]].

## Explanations

### L1 human

Look at the values of a function on a small window around a point, and measure the gap between the largest and the smallest of them.
Shrink the window and see what this gap tends to.
That limit is the oscillation at the point.
For the sign function, the values near $0$ are $-1$, $0$ and $1$, so the oscillation at $0$ is $2$.
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# Ordinary least squares

## Definition

Consider the [[linear regression|41FVC4P5JB]] model $y = X \beta + \varepsilon$ with an $n \times p$ design matrix $X$ and response $y \in \R^n$.
The *ordinary least squares* [[estimator|0SSS6ERQ0S]] of $\beta$ is a vector $\hat\beta \in \R^p$ that minimises the residual sum of squares
$$S(b) = \abs{y - X b}^2 = \sum_{i=1}^n (y_i - x_i^T b)^2, \qquad b \in \R^p.$$
If $X$ has [[rank|0ZC9YTXZRY]] $p$, then $X^T X$ is [[invertible|0EB9N0S33C]] and the minimiser is unique:
$$\hat\beta = (X^T X)^{-1} X^T y.$$

## Remarks

The minimisers are exactly the solutions of the normal equations $X^T X b = X^T y$.
The fitted vector $\hat y = X \hat\beta = H y$, with $H = X (X^T X)^{-1} X^T$, is the [[orthogonal projection|0B8KZRT28M]] of $y$ onto the column space of $X$.
So the residual vector $y - \hat y$ is orthogonal to every column of $X$.

Example: fit $y = b_0 + b_1 t$ to the points $(0, 0)$, $(1, 1)$, $(2, 3)$.
Here $\bar t = 1$ and $\bar y = 4/3$, so $b_1 = \sum (t_i - \bar t)(y_i - \bar y) / \sum (t_i - \bar t)^2 = 3/2$ and $b_0 = \bar y - b_1 \bar t = -1/6$.
Check: the residuals are $1/6$, $-1/3$, $1/6$, which sum to $0$ and are orthogonal to $(0, 1, 2)$.
Non-example: if two columns of $X$ are equal, $X$ has rank less than $p$ and $X^T X$ is singular.
Then $S$ has infinitely many minimisers, so the formula above does not apply and $\beta$ is not identified.
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# Noetherian scheme

## Definition

A [[scheme|5A35TYZD02]] $X$ is *locally Noetherian* if it can be covered by open [[affine schemes|B3V3XJ2W8G]] $\operatorname{Spec} A_i$ in which every $A_i$ is a [[Noetherian ring|0670T59HGE]].
It is *Noetherian* if it can be covered by finitely many such open sets.

## Remarks

A scheme is locally Noetherian if and only if $A$ is Noetherian for every open affine subscheme $\operatorname{Spec} A$.
So $\operatorname{Spec} A$ is Noetherian if and only if $A$ is a Noetherian ring.
The underlying space of a Noetherian scheme is a [[Noetherian topological space|2VQX2E8EAH]].
Example: projective space $\mathbb{P}^n_k$ over a field $k$ is Noetherian, being covered by $n+1$ copies of $\operatorname{Spec} k[x_1, \ldots, x_n]$, which are Noetherian by the Hilbert basis theorem.
Non-example: $\operatorname{Spec} k[x_1, x_2, \ldots]$ is not Noetherian, since the ideal $(x_1, x_2, \ldots)$ is not finitely generated.
Non-example: a disjoint union of infinitely many copies of $\operatorname{Spec} k$ is locally Noetherian but not Noetherian, since it is not quasi-compact.
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# Krylov–Safonov estimate

## Statement

Let $\Omega \subseteq \R^n$ be open and let $0 < \lambda \le \Lambda$.
Let $a_{ij} = a_{ji} \colon \Omega \to \R$ be measurable functions with
$$\lambda \abs{\xi}^2 \le \sum_{i,j=1}^n a_{ij}(x) \xi_i \xi_j \le \Lambda \abs{\xi}^2 \quad \text{for all } x \in \Omega \text{ and } \xi \in \R^n.$$
Let $u \in W^{2,n}_{\mathrm{loc}}(\Omega)$, a local [[Sobolev space|M4CEZ5K9HP]], satisfy $\sum_{i,j} a_{ij} \partial_i \partial_j u = 0$ almost everywhere in $\Omega$.
There are constants $C > 0$ and $\alpha \in (0,1)$, depending only on $n$, $\lambda$ and $\Lambda$, such that for every ball $B_{2r}(x_0)$ with closure in $\Omega$:

1. (Harnack inequality) If $u \ge 0$ in $B_{2r}(x_0)$, then $\sup_{B_r(x_0)} u \le C \inf_{B_r(x_0)} u$.
2. (Hölder estimate) $\abs{u(x) - u(y)} \le C \big(\abs{x-y}/r\big)^{\alpha} \sup_{B_{2r}(x_0)} \abs{u}$ for all $x, y \in B_r(x_0)$.

## Remarks

The equation is an [[elliptic equation|B4TTNHSTRX]] in non-divergence form and the coefficients are only measurable, so the energy methods of De Giorgi, Nash and Moser do not apply.
The proof uses the Aleksandrov–Bakelman–Pucci maximum principle to show that a supersolution which is large on a fixed fraction of a ball is bounded below on a smaller ball.
A covering argument with dyadic cubes in the style of Calderón and Zygmund then turns this into a decay of the measure of level sets, which gives the Harnack inequality.
The [[Hölder continuity|012PBA0FMD]] estimate follows from the Harnack inequality applied to $\sup u - u$ and $u - \inf u$ on shrinking balls.
For $a_{ij} = \delta_{ij}$ the first part is the classical [[Harnack inequality|YNH832D50W]] for harmonic functions.

## Explanations

### L1 human

A non-negative solution of an elliptic equation cannot be huge at one point and tiny at a nearby point: its largest and smallest values on a ball are comparable.
This stays true even when the coefficients of the equation are very rough.
From this one gets that solutions are Hölder continuous.

### L2 human

This is the non-divergence analogue of the De Giorgi–Nash–Moser theorem.
It is the starting point of the regularity theory of fully nonlinear elliptic equations $F(D^2 u) = 0$: differences of solutions satisfy linear equations with measurable coefficients, so they are Hölder continuous.
With concavity of $F$ this leads to the Evans–Krylov theorem on $C^{2,\alpha}$ regularity.
Caffarelli extended the theory to viscosity solutions.
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# Closed monoidal category

## Definition

Let $(\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)$ be a [[monoidal category|M3MSQ3NKAE]].
$\mathcal{C}$ is *closed* (more precisely, right closed) if for every object $B$ the functor $- \otimes B \colon \mathcal{C} \to \mathcal{C}$ has a right [[adjoint|TZHF6ZM92S]], written $[B, -] \colon \mathcal{C} \to \mathcal{C}$.
Equivalently, for all objects $A, B, C$ there is a bijection
$$\Hom_{\mathcal{C}}(A \otimes B, C) \cong \Hom_{\mathcal{C}}(A, [B, C]),$$
natural in $A$ and $C$.
The object $[B, C]$ is called the internal hom.

## Remarks

If also every functor $B \otimes -$ has a right adjoint, $\mathcal{C}$ is called biclosed; for a symmetric monoidal category the two notions agree.
Example: $\mathbf{Vect}_k$ with $[B, C]$ the space of linear maps $B \to C$; a linear map $f \colon A \otimes B \to C$ corresponds to $a \mapsto (b \mapsto f(a \otimes b))$.
Non-example: $\mathbf{Set}$ with the disjoint union $\sqcup$ as tensor product and the empty set as unit is monoidal but not closed.
A left adjoint preserves the initial object, but $\emptyset \sqcup B = B$ is not empty when $B$ is nonempty, so $- \sqcup B$ has no right adjoint.
A [[Cartesian closed category|BFBNQ36QS8]] is a closed monoidal category whose tensor product is the categorical product.
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# String diagram

## Summary

A string diagram is a picture of a morphism in a [[monoidal category|M3MSQ3NKAE]]: wires stand for objects and boxes stand for morphisms.
Composition is drawn by stacking boxes and the monoidal product by placing them side by side.
A theorem of Joyal and Street says that diagrams that deform into each other in the plane, with wires kept upward, denote the same morphism.
So many computations with coherence isomorphisms become moves of pictures.

## Explanations

### L1 human

Draw each object as a vertical line and each morphism as a box with lines entering at the bottom and leaving at the top.
Doing one process after another stacks the boxes; doing two processes in parallel puts them next to each other.
The equation $(g \otimes \id) \circ (\id \otimes f) = (\id \otimes f) \circ (g \otimes \id)$ becomes obvious: one may slide box $g$ above or below box $f$ on its own wire.
String diagrams are used in quantum computing, in linear algebra with tensors, and in the study of knots.
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# Hille–Yosida theorem

## Statement

Let $X$ be a real or complex [[Banach space|0BDTTM24DP]].
Let $A \colon D(A) \to X$ be a linear operator with domain $D(A) \subseteq X$.
Then $A$ is the [[generator|60HSWZBTK9]] of a [[strongly continuous|EVF180QJXY]] [[contraction semigroup|XPWE94BV6N]] $(T(t))_{t \ge 0}$ on $X$ if and only if the following two conditions hold.

1. $A$ is a [[closed operator|VM7B38PA87]] and is [[densely defined|BCJHNHKQFG]].
2. Every real $\lambda > 0$ lies in the [[resolvent set|J5YNNKSB9P]] of $A$, and $\norm{(\lambda I - A)^{-1}} \le \dfrac{1}{\lambda}$.

## Remarks

The necessity comes from the resolvent formula $(\lambda I - A)^{-1} x = \int_0^\infty e^{-\lambda t} T(t) x \, dt$, whose norm is at most $\norm{x} / \lambda$ when $\norm{T(t)} \le 1$.
For the converse, one forms the Yosida approximations $A_\lambda = \lambda A (\lambda I - A)^{-1} = \lambda^2 (\lambda I - A)^{-1} - \lambda I$.
They are bounded operators, the exponentials $e^{t A_\lambda}$ are contractions, and they converge strongly as $\lambda \to \infty$ to a semigroup with generator $A$.
The general form concerns semigroups with $\norm{T(t)} \le M e^{\omega t}$ for constants $M \ge 1$ and $\omega \in \R$.
The operator $A$ generates such a semigroup if and only if it is closed and densely defined, $(\omega, \infty)$ lies in its resolvent set, and $\norm{(\lambda I - A)^{-n}} \le M (\lambda - \omega)^{-n}$ for all $\lambda > \omega$ and $n \ge 1$.
Example: on $X = \C$, the operator $A x = a x$ with $\operatorname{Re} a \le 0$ satisfies $\abs{(\lambda - a)^{-1}} \le 1/\lambda$ for $\lambda > 0$, and it generates the contractions $T(t) = e^{a t}$.
If $a > 0$ is real, then $\lambda = a$ is not in the resolvent set, and indeed $e^{a t} > 1$ for $t > 0$, so no contraction semigroup has this generator.
The [[Lumer–Phillips theorem|Z2N8MPD8DH]] replaces condition 2 by dissipativity and a range condition, which are often easier to check.

## Explanations

### L1 human

Many evolution equations have the form $u'(t) = A u(t)$, where $A$ is an unbounded operator such as a second derivative.
One wants a solution map $T(t)$ that sends the initial state to the state at time $t$.
The theorem says exactly which operators $A$ give such a family of maps that never increases the norm.
The test only uses the inverse of $\lambda I - A$, which is often easy to study.

### L2 human

The theorem turns an abstract Cauchy problem $u' = A u$, $u(0) = u_0$, in a [[Banach space|0BDTTM24DP]] into a question about the resolvent of $A$.
For a [[densely defined|BCJHNHKQFG]] closed operator, the resolvent bound is checked by solving the stationary equation $\lambda u - A u = f$ with an a priori estimate $\lambda \norm{u} \le \norm{f}$.
This is how one shows that the heat equation and many parabolic equations are well posed.
The semigroup is then the abstract replacement for the matrix exponential $e^{tA}$ of a system of linear ordinary differential equations.
The [[Lumer–Phillips theorem|Z2N8MPD8DH]] is the form used most often in Hilbert spaces.
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# Rayleigh–Ritz method

## Summary

The Rayleigh–Ritz method approximates the [[eigenvalues|0BQAA2BK3Q]] of a self-adjoint operator by restricting its [[Rayleigh quotient|06WHP8SE98]] to a finite-dimensional space of trial vectors.
This reduces the problem to a small matrix eigenvalue problem.
By the [[min-max theorem|G11NE3AT6E]], each approximate eigenvalue is an upper bound for the corresponding true one, and the bounds improve as the trial space grows.
The method underlies the finite element computation of vibration modes and many iterative eigenvalue solvers.

## Explanations

### L1 human

The lowest frequency of a vibrating string or drum is the smallest value of an energy ratio over all possible shapes.
If you only try a few shapes, you get a number that is too large, but often close.
The Rayleigh–Ritz method tries all combinations of a few chosen shapes at once and solves a small matrix problem for the best ones.
For a string of length $1$ fixed at both ends, the single trial shape $x(1 - x)$ gives $10$, close to the true value $\pi^2 \approx 9.87$.
</document>

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# Classical Wiener space

## Description

Let $T > 0$ and $n \ge 1$.
The classical Wiener space is the set
$$C_0([0, T]; \R^n) = \set{w \colon [0, T] \to \R^n : w \text{ continuous}, \ w(0) = 0},$$
with the norm $\norm{w} = \sup_{0 \le t \le T} \abs{w(t)}$ and the Wiener measure $\mu$.
The Wiener measure is the unique Borel probability measure on this space under which the coordinate process $W_t(w) = w(t)$ is an $n$-dimensional [[Wiener process|552AWX3QXN]] on $[0, T]$.

Check that $C_0([0, T]; \R^n)$ is a [[Banach space|06GZ956657]]:

1. It is a vector space, and $\norm{\cdot}$ is a norm on it, since a continuous function on a compact interval is bounded.
2. A Cauchy sequence for $\norm{\cdot}$ converges uniformly to a function $w$; a uniform limit of continuous functions is continuous, and $w(0) = 0$.

It is also separable: piecewise linear paths with rational vertices are dense.

## Remarks

The Wiener measure is the law of Brownian motion, seen as a random element of the path space.
The Cameron–Martin space is the subspace of absolutely continuous paths $h$ with $h(0) = 0$ and $\int_0^T \abs{h'(t)}^2 \, dt < \infty$.
Translations of $\mu$ by elements of the Cameron–Martin space give measures equivalent to $\mu$, but the Cameron–Martin space itself has $\mu$-measure zero.
The space is the standard setting of [[Malliavin calculus|1EC7VPHFAY]].

## Explanations

### L1 human

The classical Wiener space is the set of all possible paths of a Brownian particle that starts at the origin.
Each point of the space is a whole continuous path, and two paths are close when they stay close at every time.
The Wiener measure tells how likely each set of paths is, so that "pick a random path" means "run a Brownian motion".
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# Desargues's theorem

## Statement

Let $k$ be a field, or more generally a division ring, and work in the [[projective plane|DANFNN8C4Y]] $\mathbb{P}^2(k)$.
Let $A, B, C$ and $A', B', C'$ be two triangles, that is, two triples of non-[[collinear|0AFNNTRTJE]] points.
Assume $A \ne A'$, $B \ne B'$, $C \ne C'$, and that the corresponding sides $AB$ and $A'B'$, $BC$ and $B'C'$, $CA$ and $C'A'$ are distinct lines.
Then the lines $AA'$, $BB'$, $CC'$ pass through a common point if and only if the three points
$$AB \cap A'B', \quad BC \cap B'C', \quad CA \cap C'A'$$
are collinear.

## Remarks

The two triangles are then said to be in perspective from a point and in perspective from a line.
For two triangles in different planes of a [[projective space|ZT9A6JK8QF]] of dimension $3$, the proof is short: the three intersection points lie on both planes, hence on their common line.
The plane case follows by viewing the plane figure as a projection of a figure in space, or by a computation with [[homogeneous coordinates|FS9FRJ17D2]].
The converse direction is the dual statement, so it follows by the principle of [[duality|DCMP1A0138]].

In the Euclidean plane, if two triangles are homothetic, as $A' = 2A$, $B' = 2B$, $C' = 2C$ with centre the origin, the corresponding sides are parallel, so they meet on the line at infinity.

The theorem does not follow from the axioms of a projective plane alone.
It fails in the projective completion of the [[Moulton plane|3V9GYYXM1H]] and in other [[non-Desarguesian planes|HYE5FT4M34]].
It holds in a projective plane exactly when the plane is isomorphic to $\mathbb{P}^2(D)$ for some division ring $D$.

## Explanations

### L1 human

Draw two triangles so that the lines through matching corners all meet at one point, as if one triangle were the shadow of the other cast by a lamp.
Then extend matching sides until they cross: the three crossing points always lie on one straight line.
The statement only talks about points and lines, with no lengths or angles.

### L2 human

Desargues's theorem is the dividing line between planes that come from algebra and planes that do not.
A projective plane in which it holds can be coordinatized by a division ring.
It holds automatically in every projective space of dimension at least $3$, so the [[Veblen–Young theorem|NPNNXWMVCC]] needs no extra axiom there.
[[Pappus's hexagon theorem|B6C3J2GJWP]] implies it, by Hessenberg's theorem, and Pappus holds exactly when the coordinate division ring is commutative.
The little Desargues theorem, where the centre lies on the axis, characterizes [[Moufang planes|7F4HSNJJH8]].
</document>

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# Fibration and cofibration

## Definition

Let $\mathcal{M}$ be a [[model category|SDTZVZ6B4Q]] with [[weak equivalences|E2870TEDG7]] $W$.
A *fibration* is a morphism in the class $\mathrm{Fib}$ of the model structure, and a *cofibration* is a morphism in the class $\mathrm{Cof}$.
A *trivial* (or *acyclic*) fibration is a fibration that is also in $W$, and a trivial cofibration is a cofibration that is also in $W$.
An object $X$ is *fibrant* if the morphism $X \to \ast$ to the terminal object is a fibration.
It is *cofibrant* if the morphism $\emptyset \to X$ from the initial object is a cofibration.

## Remarks

The axioms of a model category imply characterizations by the [[lifting property|AZK4RD8XV0]].
A morphism is a cofibration if and only if it has the left lifting property with respect to all trivial fibrations.
A morphism is a fibration if and only if it has the right lifting property with respect to all trivial cofibrations.
Dually, the trivial cofibrations are the maps with the left lifting property with respect to fibrations.
The trivial fibrations are the maps with the right lifting property with respect to cofibrations.
So, given $W$, the fibrations determine the cofibrations and conversely.

Example: in the Quillen model structure on topological spaces, the fibrations are the [[Serre fibrations|PB9JXQ2KC3]], and every space is fibrant.
In the Strøm model structure on topological spaces, the fibrations are the [[Hurewicz fibrations|Y773B7EQA7]] and the cofibrations are the closed [[cofibrations|W5WS5EY8ZM]] in the sense of the homotopy extension property.
In the projective model structure on non-negatively graded chain complexes of modules over a ring, the fibrations are the maps that are surjective in each positive degree.

Non-example: in the Quillen model structure, the inclusion $p \colon \set{0} \to [0, 1]$ is not a fibration.
The path $t \mapsto t$ in $[0, 1]$ starts at the image $0$ of $p$, but it has no lift to $\set{0}$, because it leaves the image of $p$.
So $p$ fails the right lifting property against the trivial cofibration $\set{0} \to [0, 1]$.
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# Gauss–Legendre quadrature

## Definition

Let $n \ge 1$ and let $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$.
Let $x_1, \dots, x_n$ be the zeros of $P_n$; they are distinct and lie in $(-1, 1)$.
The $n$-point *Gauss–Legendre quadrature* rule is
$$\int_{-1}^{1} f(x)\,dx \approx \sum_{i=1}^{n} w_i f(x_i), \qquad w_i = \frac{2}{(1 - x_i^2)\, P_n'(x_i)^2}.$$
It is the $n$-point [[Gaussian quadrature|FP37BTP93K]] rule for the weight $w = 1$ on $(-1, 1)$.

## Remarks

The rule is exact for every polynomial of degree at most $2n - 1$.
For an interval $[a, b]$ one uses $\int_a^b g(t)\,dt = \frac{b - a}{2} \int_{-1}^{1} g\big(\frac{b - a}{2} x + \frac{a + b}{2}\big)\,dx$.

Example: $n = 2$.
Here $P_2(x) = \frac12 (3x^2 - 1)$, with zeros $\pm 1/\sqrt{3}$, and $P_2'(x) = 3x$.
The weights are $2 / \big((2/3) \cdot 3\big) = 1$.
For $f(x) = x^2$ the rule gives $2/3$, and for $f(x) = x^3$ it gives $0$, both exact.

Non-example: for $n = 2$ and $f(x) = x^4$ the rule gives $2/9$ while the integral is $2/5$.
This is expected, since $4 > 2n - 1 = 3$.
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# Bayes factor

## Definition

Let $M_0$ and $M_1$ be two models for data $x$.
For $i = 0, 1$, model $M_i$ has a parameter $\theta_i \in \Theta_i$, a density or mass function $f_i(x \mid \theta_i)$ and a proper [[prior|00NBBSXRQS]] $\pi_i$ on $\Theta_i$.
The *marginal likelihood* of $M_i$ is
$$m_i(x) = \int_{\Theta_i} f_i(x \mid \theta_i) \, \pi_i(d\theta_i).$$
If $m_0(x) > 0$, the *Bayes factor* of $M_1$ against $M_0$ is
$$B_{10}(x) = \frac{m_1(x)}{m_0(x)}.$$

## Remarks

By [[Bayes' theorem|0NM6E9VW04]], with prior model probabilities $P(M_0), P(M_1) > 0$, the posterior odds are
$$\frac{P(M_1 \mid x)}{P(M_0 \mid x)} = B_{10}(x) \cdot \frac{P(M_1)}{P(M_0)}.$$
If both models are simple, that is each $\Theta_i$ is a single point, then $B_{10}$ is the [[likelihood|05FSTXM92W]] ratio.
Example: let $M_0$ say that a coin is fair and $M_1$ that its chance $p$ of heads is uniform on $[0, 1]$, and observe two heads in two tosses.
Then $m_0 = 1/4$ and $m_1 = \int_0^1 p^2 \, dp = 1/3$, so $B_{10} = 4/3$.
Non-example: with an improper prior $\pi_1$, the integral $m_1(x)$ is defined only up to an arbitrary constant factor, so the ratio is not a well-defined Bayes factor.
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# Levinson's theorem on asymptotic integration

## Statement

Let $t_0 \in \R$ and let $n \ge 1$.
Let $\Lambda(t) = \operatorname{diag}(\lambda_1(t), \dots, \lambda_n(t))$ be a continuous [[diagonal matrix|044JNC7ZAA]] function on $[t_0, \infty)$ with complex entries.
Let $R(t)$ be a continuous complex $n \times n$ matrix function on $[t_0, \infty)$ with
$$\int_{t_0}^{\infty} \norm{R(t)} \, dt < \infty.$$
Assume the dichotomy condition: there is a constant $K$ such that for each pair $i \ne j$ one of the following holds, where $D_{ij}(s, t) = \int_s^t \operatorname{Re}\bigl(\lambda_i(u) - \lambda_j(u)\bigr) \, du$:

1. $D_{ij}(t_0, t) \to \infty$ as $t \to \infty$, and $D_{ij}(s, t) \ge -K$ for all $t \ge s \ge t_0$;
2. $D_{ij}(s, t) \le K$ for all $t \ge s \ge t_0$.

Then the [[linear system|037ZNY7NAG]]
$$x' = \bigl(\Lambda(t) + R(t)\bigr) x$$
has a [[fundamental matrix|0H1GN6P9R5]] of the form
$$X(t) = \bigl(I + E(t)\bigr) \exp\Bigl(\int_{t_0}^{t} \Lambda(s) \, ds\Bigr), \qquad E(t) \to 0 \text{ as } t \to \infty.$$
Equivalently, for each $k$ there is a solution $x_k(t) = \bigl(e_k + o(1)\bigr) \exp\bigl(\int_{t_0}^t \lambda_k(s) \, ds\bigr)$, where $e_k$ is the $k$-th standard basis vector.

## Remarks

The proof writes the $k$-th solution as $x_k = \exp\bigl(\int \lambda_k\bigr) z$ and turns the equation for $z$ into an integral equation.
The dichotomy condition splits the indices into those whose integrals are taken from $t_0$ to $t$ and those taken from $t$ to $\infty$.
With this split the integral operator is a contraction on a far half-line $[T, \infty)$, because $\int_T^\infty \norm{R}$ is small.
Its fixed point gives $z(t) \to e_k$.

Example: if $\Lambda$ is a constant diagonal matrix, then each $D_{ij}(s, t)$ is $(t - s)$ times a constant, so the dichotomy condition holds automatically.
Hence for a constant diagonal $\Lambda$ with distinct or equal eigenvalues and an integrable perturbation $R$, each solution of $x' = (\Lambda + R) x$ behaves like $e^{\lambda_k t}$ times a vector close to $e_k$.

The integrability of $R$ cannot be dropped.
For $n = 1$, $\Lambda = 0$ and $R(t) = 1/t$ on $[1, \infty)$, the solutions are $c t$, which do not tend to a constant multiple of $e^0 = 1$.

## Explanations

### L1 human

Many linear systems look like a simple diagonal system plus a small disturbance.
For a diagonal system the solutions are explicit exponentials, one for each coordinate.
Levinson's theorem says that if the disturbance is small in total, measured by the integral of its size, then the true solutions still look like these exponentials for large times.
A mild condition on the growth rates is needed, so that the exponentials can be told apart.

### L2 human

The theorem is the basic tool of asymptotic integration of linear systems.
One first brings a system into the form diagonal part plus integrable part, often by a change of variables that diagonalises the leading term.
Then the theorem gives the leading behaviour of a fundamental matrix.
The WKB, or Liouville–Green, approximation for $y'' = Q(t) y$ can be justified in this way after a suitable substitution.
Hartman and Wintner proved a related result in which $R$ is only square integrable, at the cost of a stronger dichotomy condition and a correction term in the exponent.
</document>

<document source="https://tamnd.github.io/math2/node/5ZJDCNSGHJ.md" tag="5ZJDCNSGHJ" kind="definition">
# Singular cohomology

## Definition

Let $X$ be a topological space, let $G$ be an [[abelian group|082GHNPJ53]], and let $C_\bullet(X)$ be the singular [[chain complex|WQ82C0M3QB]] of $X$ with integer coefficients.
The group of *singular $n$-cochains* is $C^n(X; G) = \Hom(C_n(X), G)$, that is, the group of all functions from singular $n$-simplices of $X$ to $G$.
The *coboundary* $\delta \colon C^n(X; G) \to C^{n+1}(X; G)$ is $\delta \varphi = \varphi \circ \partial_{n+1}$.
Then $\delta \circ \delta = 0$.
The *singular cohomology* group $H^n(X; G)$ is $\ker(\delta \colon C^n \to C^{n+1}) / \im(\delta \colon C^{n-1} \to C^n)$.
For a subspace $A \subseteq X$, the relative group $H^n(X, A; G)$ is defined in the same way from $\Hom(C_n(X, A), G)$.
These are the cochains that vanish on all simplices in $A$.

## Remarks

A continuous map $f \colon X \to Y$ induces $f^* \colon H^n(Y; G) \to H^n(X; G)$, so $H^n(-; G)$ is a contravariant functor; compare the [[Hom functor|VJXS8AK67E]].
Evaluation $\langle \varphi, c \rangle = \varphi(c)$ of a cochain on a chain gives the *Kronecker pairing* $H^n(X; G) \times H_n(X) \to G$.
The [[universal coefficient theorem|GTA8ZFGZ3N]] gives a split exact sequence $0 \to \operatorname{Ext}(H_{n-1}(X), G) \to H^n(X; G) \to \Hom(H_n(X), G) \to 0$, with [[Ext|4P7GBWJ4RR]] the Ext functor and $H_n$ [[singular homology|M0Q5CFJ7ZH]].
When $G = \Z$ one writes $H^n(X)$.

Example: a $0$-cochain $\varphi$ is a function from points of $X$ to $G$, and $(\delta \varphi)(\sigma) = \varphi(\sigma(1)) - \varphi(\sigma(0))$ for a path $\sigma$.
So $\varphi$ is a cocycle exactly when it is constant on each path component.
Hence $H^0(X; G)$ is the group of functions from the set of path components of $X$ to $G$.

Non-example: on $X = [0, 1]$ the $0$-cochain $\varphi(x) = x$ with $G = \R$ is not a cocycle, because $\delta\varphi$ takes the value $1$ on the path $t \mapsto t$.
</document>

<document source="https://tamnd.github.io/math2/node/5ZKBBTBXVB.md" tag="5ZKBBTBXVB" kind="definition">
# Log-gamma function

## Definition

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], holomorphic and without zeros on $\C \setminus \set{0, -1, -2, \ldots}$, and let $D = \C \setminus (-\infty, 0]$.
The *log-gamma function* is the unique [[holomorphic function|0NAHXR1FRH]] $\operatorname{Log}\Gamma \colon D \to \C$ such that:

1. $e^{\operatorname{Log}\Gamma(z)} = \Gamma(z)$ for every $z \in D$;
2. $\operatorname{Log}\Gamma(x)$ is real for every real $x > 0$.

On $(0, \infty)$ it equals the real function $x \mapsto \ln\Gamma(x)$, with $\ln$ the [[natural logarithm|04DXAFN2ES]].

## Remarks

Existence: $D$ is simply connected and $\Gamma$ has no zeros on $D$, so $\Gamma$ has a holomorphic logarithm on $D$.
This logarithm is unique up to adding $2\pi i k$, and condition 2 fixes $k$.
Example: $\operatorname{Log}\Gamma(3) = \ln 2$, which is real, and $e^{\ln 2} = 2 = \Gamma(3)$.
Non-example: the composition $z \mapsto \operatorname{Log}(\Gamma(z))$ with the principal logarithm satisfies condition 1 but is not continuous on $D$.
The reason is that the argument of $\Gamma(x + iy)$ grows without bound as $y \to \infty$, so $\Gamma(z)$ crosses the negative real axis.
The functional equation becomes $\operatorname{Log}\Gamma(z + 1) = \operatorname{Log}\Gamma(z) + \operatorname{Log} z$ on $D$.
Its derivative is the digamma function $\psi = \Gamma'/\Gamma$.

## Explanations

### L1 human

The log-gamma function is the logarithm of the gamma function, chosen so that it is smooth and real on the positive axis.
</document>

<document source="https://tamnd.github.io/math2/node/5ZKP6E4JCE.md" tag="5ZKP6E4JCE" kind="definition">
# Gröbner basis

## Definition

Let $k$ be a [[field|04C5ZBG1WT]], let $S = k[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]], and fix a [[monomial order|E874NARWPV]] on $S$.
For a nonzero $f \in S$, let $\operatorname{LT}(f)$ be its leading term.
Let $I$ be an [[ideal|096Z9MP2ME]] of $S$, and let $\operatorname{LT}(I)$ be the ideal generated by the leading terms $\operatorname{LT}(f)$ of all nonzero $f \in I$.
A finite subset $G = \set{g_1, \ldots, g_t}$ of $I \setminus \set{0}$ is a *Gröbner basis* of $I$ if
$$(\operatorname{LT}(g_1), \ldots, \operatorname{LT}(g_t)) = \operatorname{LT}(I).$$

## Remarks

$\operatorname{LT}(I)$ is a [[monomial ideal|4G25S7PJ22]].
Equivalent forms: for every nonzero $f \in I$, some $\operatorname{LT}(g_i)$ divides $\operatorname{LT}(f)$.
Or: for every $f \in S$, the remainder of $f$ under the [[multivariate division algorithm|T8EN3CAEQF]] by $G$ is zero if and only if $f \in I$.
A Gröbner basis of $I$ generates $I$, and every ideal has a Gröbner basis; this gives another proof of [[Hilbert's basis theorem|0YKA9J3XEE]].
A Gröbner basis is *reduced* if each $\operatorname{LT}(g_i)$ has coefficient $1$ and no monomial of $g_i$ is divisible by $\operatorname{LT}(g_j)$ for $j \ne i$.
For a given monomial order, each ideal has a unique reduced Gröbner basis.
[[Buchberger's algorithm|4RH1D40WG1]] computes a Gröbner basis from any finite set of generators.

Example: $G = \set{x, y}$ is a Gröbner basis of $I = (x, y)$ in $k[x, y]$, for any monomial order.
Every nonzero $f \in I$ has no constant term, so each of its monomials, in particular $\operatorname{LT}(f)$, is divisible by $x$ or by $y$.

Non-example: for the lexicographic order with $x > y$, the set $G = \set{x + y, \; x}$ generates the same ideal $I$, and both leading terms equal $x$.
But $y = x - (x + y) \in I$ and $\operatorname{LT}(y) = y \notin (x)$, so $G$ is not a Gröbner basis of $I$.
</document>

<document source="https://tamnd.github.io/math2/node/5ZN4VXR3X5.md" tag="5ZN4VXR3X5" kind="definition">
# Perron integral

## Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function.
For $U \colon [a, b] \to \R$ and $x \in [a, b]$, the lower and upper derivatives of $U$ at $x$ are
$$\underline{D} U(x) = \liminf_{y \to x} \frac{U(y) - U(x)}{y - x}, \qquad \overline{D} U(x) = \limsup_{y \to x} \frac{U(y) - U(x)}{y - x},$$
where $y$ ranges over $[a, b] \setminus \set{x}$.
A *major function* of $f$ is a function $U$ with $\underline{D} U(x) \ne -\infty$ and $\underline{D} U(x) \ge f(x)$ for every $x \in [a, b]$.
A *minor function* of $f$ is a function $V$ with $\overline{D} V(x) \ne +\infty$ and $\overline{D} V(x) \le f(x)$ for every $x \in [a, b]$.
Let $\mathcal{U}$ be the set of major functions and $\mathcal{V}$ the set of minor functions of $f$.
The function $f$ is *Perron integrable* on $[a, b]$ if both sets are nonempty and
$$\inf_{U \in \mathcal{U}} \bigl(U(b) - U(a)\bigr) = \sup_{V \in \mathcal{V}} \bigl(V(b) - V(a)\bigr) \in \R.$$
The *Perron integral* of $f$ is this value.

## Remarks

For any major function $U$ and minor function $V$, the difference $U - V$ is nondecreasing, so $V(b) - V(a) \le U(b) - U(a)$ always holds.

Example: if $F$ is differentiable at every point of $[a, b]$ and $f = F'$, then $F$ is both a major and a minor function of $f$.
So $f$ is Perron integrable with integral $F(b) - F(a)$, even when $f$ is not Lebesgue integrable.

Non-example: $f(x) = 1/x$ for $0 < x \le 1$ and $f(0) = 0$ is not Perron integrable on $[0, 1]$.
A nonnegative Perron integrable function is Lebesgue integrable, and $\int_0^1 dx / x = \infty$.

A function is Perron integrable if and only if it is [[Henstock–Kurzweil integrable|FCGHXDGVV9]], and the two integrals agree.
</document>

<document source="https://tamnd.github.io/math2/node/5ZNN348M6X.md" tag="5ZNN348M6X" kind="statement">
# Crainic–Fernandes integrability theorem

## Statement

Let $A \to M$ be a Lie algebroid with anchor $\rho \colon A \to TM$.
For $x \in M$, let $\mathfrak{g}_x = \ker(\rho_x)$ be the isotropy Lie algebra at $x$, with centre $Z(\mathfrak{g}_x)$.
Let $\mathcal{N}_x(A) \subseteq Z(\mathfrak{g}_x)$ be the monodromy group at $x$.
It is the set of $v \in Z(\mathfrak{g}_x)$ such that the constant $A$-path with value $v$ is $A$-homotopic to the zero $A$-path at $x$.
Fix a continuous [[norm|08NYZ2FS6B]] on the fibres of $A$, and let $r(x)$ be the distance from $0$ to $\mathcal{N}_x(A) \setminus \set{0}$, with $r(x) = \infty$ when $\mathcal{N}_x(A) = \set{0}$.
Then $A$ is isomorphic to the Lie algebroid of some [[Lie groupoid|HNVW0E732J]] if and only if, for every $x \in M$,

1. $\mathcal{N}_x(A)$ is a discrete subset of $A_x$, and
2. $\liminf_{y \to x} r(y) > 0$.

## Remarks

Crainic and Fernandes build a candidate groupoid $\mathcal{G}(A)$, the Weinstein groupoid, as the quotient of the space of $A$-paths by $A$-homotopy.
It is always a topological groupoid, and $A$ is integrable exactly when $\mathcal{G}(A)$ is smooth; it is then the unique integration whose source fibres are simply connected.
The monodromy groups measure the failure of smoothness, and the two conditions say that they stay uniformly discrete as the point moves.
The condition does not depend on the chosen norm.

Example: when $M$ is a point, $A$ is a finite-dimensional Lie algebra and the monodromy groups are trivial.
So every Lie algebra integrates to a Lie group, which is Lie's third theorem.
Unlike Lie algebras, Lie algebroids need not be integrable, and the theorem identifies the obstruction.

## Explanations

### L1 human

Every Lie group has a Lie algebra, and every finite-dimensional Lie algebra comes from a Lie group.
Lie algebroids play the role of Lie algebras for Lie groupoids, but the second half of this correspondence can fail for them.
The theorem gives an exact test for when a Lie algebroid comes from a Lie groupoid.
The test checks that certain discrete groups attached to each point do not shrink towards zero as the point moves.

### L2 human

The theorem settles the integrability problem for Lie algebroids, which asks for the analogue of Lie's third theorem.
It applies to Poisson manifolds through their cotangent Lie algebroids, where integrability means the existence of a symplectic groupoid.
The monodromy group at $x$ is the image of a homomorphism defined on the second homotopy group of the orbit through $x$.
So integrability is automatic when every orbit has trivial second homotopy group.
</document>

<document source="https://tamnd.github.io/math2/node/5ZXRZB41HE.md" tag="5ZXRZB41HE" kind="definition">
# Characteristic of a field

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] with unit element $1$.
For a positive integer $n$, write $n \cdot 1 = 1 + 1 + \dots + 1$, with $n$ terms.
If $n \cdot 1 = 0$ for some positive integer $n$, the characteristic $\operatorname{char} K$ is the least such $n$.
Otherwise $\operatorname{char} K = 0$.

## Remarks

This is the [[characteristic of a ring|0P7CH1CND5]] applied to $K$.
Since a field is an integral domain, the [[characteristic is zero or a prime|00758QM6JJ]].
Example: in $\F_5$, $5 \cdot 1 = 0$ and $n \cdot 1 \ne 0$ for $1 \le n \le 4$, so $\operatorname{char} \F_5 = 5$.
Non-example of positive characteristic: in $\Q$, $n \cdot 1 = n \ne 0$ for every $n \ge 1$, so no positive $n$ works and $\operatorname{char} \Q = 0$.
The [[prime subfield|0VPFBAY7MW]] of $K$ is isomorphic to $\Q$ if $\operatorname{char} K = 0$ and to $\F_p$ if $\operatorname{char} K = p$.
If $\operatorname{char} K = p > 0$, then $(a + b)^p = a^p + b^p$ for all $a, b \in K$, which gives the [[Frobenius endomorphism|0GHDDVAH8P]].

## Explanations

### L1 human

The characteristic counts how many ones you must add to reach zero.
In the usual number fields you never reach zero, and the characteristic is $0$.
In a finite field you always reach zero, after a prime number of steps.
</document>

<document source="https://tamnd.github.io/math2/node/5ZYRRVE8BW.md" tag="5ZYRRVE8BW" kind="concept">
# Connection on a vector bundle

## Summary

A connection on a vector bundle is a rule for differentiating [[sections|WWX7H5W1RP]] of the bundle in the direction of tangent vectors.
It is needed because the fibres over different points are different vector spaces.
So there is no built-in way to compare a section at nearby points.
A connection gives covariant derivatives, parallel transport, curvature and holonomy.
Connections on the tangent bundle are the affine connections, such as the Levi-Civita connection.
Connections on other bundles are the gauge fields of physics.

## Explanations

### L1 human

A section of a vector bundle puts a vector in a separate vector space above each point.
To say whether such a section is "constant" you must compare vectors in different spaces, and nothing tells you how.
A connection is the extra data that says how to differentiate the section, and so how to compare nearby fibres.
On the trivial bundle $\R^n \times \R^r$ one can just differentiate each component, and this is the simplest connection.
On a twisted bundle, such as the tangent bundle of a sphere, there is no such obvious choice.
</document>

<document source="https://tamnd.github.io/math2/node/605M5NMFD5.md" tag="605M5NMFD5" kind="concept">
# Identifiability

## Summary

A [[statistical model|K3AMMSY8DD]] is identifiable when different parameter values give different laws of the data.
Then the law of the data determines the parameter, and it makes sense to estimate the parameter.
Without identifiability no amount of data can separate parameters that give the same law, and consistent estimation of them is impossible.

## Explanations

### L1 human

Imagine that you see only the sum of two unknown numbers plus some noise.
You can learn the sum, but you can never learn the two numbers apart, since $1 + 2$ and $2 + 1$ give the same data.
Such a model is not identifiable.
In an identifiable model each parameter value leaves its own fingerprint on the distribution of the data.
</document>

<document source="https://tamnd.github.io/math2/node/6068BQ4XZF.md" tag="6068BQ4XZF" kind="definition">
# Absorbing element

## Definition

Let $\ast$ be a [[binary operation|0M418FE7BB]] on a set $S$ and let $z \in S$.

1. The element $z$ is *left absorbing* if $z \ast a = z$ for all $a \in S$.
2. The element $z$ is *right absorbing* if $a \ast z = z$ for all $a \in S$.
3. The element $z$ is *absorbing*, or a *zero element*, if it is both left and right absorbing.

## Remarks

Example: in $(\Z, \cdot)$ the integer $0$ is absorbing, because $0 \cdot a = a \cdot 0 = 0$ for all $a \in \Z$.

Non-example: in $(\Z, +)$ the integer $0$ is not absorbing, because $0 + 1 = 1 \ne 0$; it is an identity element instead.

If $z$ is left absorbing and $z'$ is right absorbing, then $z = z \ast z' = z'$.
So an operation has at most one absorbing element.
On a set with at least two elements, the operation $a \ast b = a$ makes every element left absorbing and no element right absorbing.
</document>

<document source="https://tamnd.github.io/math2/node/6080X3FAD4.md" tag="6080X3FAD4" kind="concept">
# Regular surface

## Summary

A regular surface is a subset of $\R^3$ that looks, near each of its points, like a smoothly bent piece of the plane.
Near each point it is the image of a smooth injective map from an open set of the plane, with linearly independent partial derivatives.
Regular surfaces are the setting of the classical differential geometry of surfaces: tangent planes, fundamental forms and curvatures are defined on them.

## Explanations

### L1 human

Think of the skin of a ball, or the surface of a doughnut.
If you zoom in at any point, you see something that is almost a flat sheet, bent but without folds, creases or sharp tips.
A regular surface is a shape in space with this property at every point.
The tip of a cone is a sharp point, so a cone with its tip is not a regular surface, but the sphere is.
</document>

<document source="https://tamnd.github.io/math2/node/60A5CC7S3A.md" tag="60A5CC7S3A" kind="statement">
# Pullback pasting lemma

## Statement

Let $\mathcal{C}$ be a [[category|A2DE66937D]], and consider a [[commutative diagram|WV898P4X16]] of two squares side by side:
$$\begin{array}{ccccc} A & \xrightarrow{a} & B & \xrightarrow{b} & C \\ \downarrow{\scriptstyle u} & & \downarrow{\scriptstyle v} & & \downarrow{\scriptstyle w} \\ D & \xrightarrow{d} & E & \xrightarrow{e} & F \end{array}$$
so that $v \circ a = d \circ u$ and $w \circ b = e \circ v$.
Assume that the right square is a [[pullback|Q05K04QJ71]].
The outer rectangle has top side $b \circ a$, bottom side $e \circ d$ and vertical sides $u$ and $w$.
Then the left square is a pullback if and only if the outer rectangle is a pullback.

## Remarks

Both directions are checked with the universal property.
A cone over the outer rectangle gives, through the right pullback, a cone over the left square, and back again.
The uniqueness parts match because the right square is a pullback.
In the category of sets, a pullback along an inclusion is a preimage.
The lemma then says that the preimage under $e \circ d$ is the preimage under $d$ of the preimage under $e$.

The hypothesis on the right square is needed.
Let $A$, $C$, $D$, $E$ and $F$ be one-point sets, let $B$ have two points, and let $a$ pick one of them.
The outer rectangle is then a pullback, but the left square is not, because the pullback of $B \to E \leftarrow D$ has two points while $A$ has one.

## Explanations

### L1 human

Pullbacks can be stacked.
If you glue a pullback square to a second square, the big rectangle is a pullback exactly when the second square is.
It means that pulling back in two steps is the same as pulling back once along the composite map.

### L2 human

The lemma is the main tool for computing with pullbacks in diagrams.
It shows, for example, that pullback along $g \circ f$ is the same as pullback along $g$ followed by pullback along $f$, up to isomorphism.
In algebraic geometry it gives the transitivity of base change, $(X \times_S S') \times_{S'} S'' \cong X \times_S S''$.
The dual statement holds for pushouts: if the left square is a pushout, then the right square is a pushout if and only if the outer rectangle is.
</document>

<document source="https://tamnd.github.io/math2/node/60FNZ35H6R.md" tag="60FNZ35H6R" kind="concept">
# Strict partial order

## Summary

A strict partial order is a binary relation $<$ that is irreflexive and transitive.
It is the "strictly less than" version of a [[partial order|011C29Q1S1]].
Strict partial orders on a set correspond one to one with partial orders on the same set.
So either one can serve as the basic notion.

## Explanations

### L1 human

A strict partial order says when one element is strictly below another.
No element is strictly below itself, and if $a$ is below $b$ and $b$ is below $c$, then $a$ is below $c$.
Some pairs may be incomparable.
For example, "is a proper subset of" on the subsets of $\set{1, 2}$ puts $\emptyset$ below $\set{1}$, but $\set{1}$ and $\set{2}$ are incomparable.
</document>

<document source="https://tamnd.github.io/math2/node/60G5ABJQ75.md" tag="60G5ABJQ75" kind="example">
# Cantor group

## Example

The *Cantor group* is the product $C = (\Z/2\Z)^{\N}$ of countably many copies of the group of order $2$.
Its elements are sequences $(a_0, a_1, a_2, \dots)$ with $a_k \in \set{0, 1}$, added coordinatewise modulo $2$.
Each factor has the discrete topology, and $C$ has the [[product topology|0GBG8B49AA]].

Check that it is a [[profinite group|KA0YDB6QXE]].
Addition and negation are continuous, since in each coordinate they depend on only one coordinate of the inputs.
The space $C$ is compact by Tychonoff's theorem and Hausdorff as a product of Hausdorff spaces.
It is totally disconnected: two different sequences differ in some coordinate $k$.
The sets where $a_k = 0$ and where $a_k = 1$ are open, closed and separate them.
A compact Hausdorff totally disconnected group is profinite.
Equivalently, $C$ is the inverse limit of the finite groups $(\Z/2\Z)^n$ under the maps that forget the last coordinate.

## Remarks

As a topological space, $C$ is homeomorphic to the [[Cantor set|0BXNDZFQME]], via $(a_k) \mapsto \sum_k 2 a_k 3^{-k-1}$.
The normalized [[Haar measure|VY0HXD3GC4]] of $C$ is the product of the uniform measures on $\set{0, 1}$, the law of an infinite sequence of fair coin tosses.
Every neighbourhood of $0$ contains a subgroup of the form $\set{a : a_0 = \cdots = a_n = 0}$, so $C$ fails the [[no small subgroups|HPMAJV2RDY]] condition and is far from a Lie group.

## Explanations

### L1 human

Think of an infinite sequence of light switches, each on or off.
Adding two such sequences means flipping the switches that are on in the second one.
Two sequences are close when they agree on a long first stretch of switches.
This gives a group whose space of points is the Cantor set, the classical dust of points obtained by removing middle thirds.
</document>

<document source="https://tamnd.github.io/math2/node/60HSWZBTK9.md" tag="60HSWZBTK9" kind="concept">
# Generator of a semigroup

## Summary

The generator of a [[strongly continuous semigroup|EVF180QJXY]] $(T(t))_{t \ge 0}$ of operators is the derivative of $T(t)$ at $t = 0$.
It is usually an unbounded, closed, densely defined operator $A$.
The semigroup solves the abstract Cauchy problem $u'(t) = Au(t)$, $u(0) = x$, through $u(t) = T(t)x$, and one writes $T(t) = e^{tA}$.
The generator determines the semigroup uniquely, so questions about evolution equations become questions about one operator.

## Explanations

### L1 human

For a number $a$, the function $t \mapsto e^{ta}$ has derivative $a$ at $t = 0$.
A semigroup is a family of operators that behaves like $e^{ta}$, and its generator plays the role of $a$.
For the heat equation, the semigroup moves an initial temperature forward in time, and the generator is the Laplacian.
For the shift $(T(t)f)(s) = f(s + t)$, the generator is the derivative $d/ds$.
The generator is often a differential operator, which is why it is defined only on a dense set of nice vectors.
</document>

<document source="https://tamnd.github.io/math2/node/60MQBQRACM.md" tag="60MQBQRACM" kind="definition">
# Closable operator

## Definition

Let $X$ and $Y$ be normed spaces, and let $T$ be an [[unbounded operator|KRJX5RA0PW]] from $X$ to $Y$.
The operator $T$ is *closable* if it has an extension $S \supseteq T$ that is a [[closed operator|VM7B38PA87]].

## Remarks

The following are equivalent.

1. The operator $T$ is closable.
2. The [[closure|0QEAJB91GZ]] of the [[graph|3RCAY97W1E]] $\Gamma(T)$ in $X \times Y$ is the graph of an operator.
3. For every sequence $(x_n)$ in $D(T)$ with $x_n \to 0$ and $T x_n \to y$, one has $y = 0$.

Example: on $C[0, 1]$ with the supremum norm, let $D(T)$ be the space of polynomials and $Tp = p'$.
If $p_n \to 0$ and $p_n' \to g$ uniformly, then $p_n(x) - p_n(0) = \int_0^x p_n'$ gives $0 = \int_0^x g$ for all $x$, so $g = 0$; hence $T$ is closable.

Non-example: on $L^2(0, 1)$, let $D(T) = C[0, 1]$ and $Tf = f(0) \cdot \mathbf{1}$, where $\mathbf{1}$ is the constant function $1$.
The functions $f_n(x) = \max(0, 1 - n x)$ satisfy $f_n \to 0$ in $L^2(0, 1)$ but $T f_n = \mathbf{1}$ for all $n$, so $T$ is not closable.
</document>

<document source="https://tamnd.github.io/math2/node/60P2XWY5QX.md" tag="60P2XWY5QX" kind="concept">
# Turing jump

## Summary

The Turing jump of a set $A$ of natural numbers is the halting problem for [[oracle machines|Z5CCVBME2V]] with oracle $A$.
It produces from $A$ a set $A'$ that is strictly harder than $A$ in the sense of [[Turing reducibility|PA4PYQ1PZ0]].
Iterating the jump starting from the empty set gives the sets $\emptyset', \emptyset'', \ldots$, which line up with the levels of the [[arithmetical hierarchy|7CVJCQKY41]].

## Explanations

### L1 human

Give a computer access to a set $A$ as an oracle.
Even then it cannot decide which of its own programs halt; that question is the jump $A'$.
So each set leads to a harder one, and there is no hardest set.
For the empty set, which gives the computer no help at all, the jump is the ordinary [[halting problem|Z7TN2B35B0]].
</document>

<document source="https://tamnd.github.io/math2/node/60SA99BER7.md" tag="60SA99BER7" kind="statement">
# Tarski–Vaught test

## Statement

Let $L$ be a signature and let $\mathcal{M}$ be a [[substructure|676H7EGTKJ]] of an $L$-[[structure|983GZBQYT4]] $\mathcal{N}$, with domains $M \subseteq N$.
Then $\mathcal{M}$ is an [[elementary substructure|8WZF9SBP39]] of $\mathcal{N}$ if and only if the following holds.
For every $L$-[[formula|HYJ10AF6RT]] $\varphi(x_1, \dots, x_n, y)$ and all $a_1, \dots, a_n \in M$, if $\mathcal{N} \models \exists y \, \varphi(a_1, \dots, a_n, y)$, then there is $b \in M$ with $\mathcal{N} \models \varphi(a_1, \dots, a_n, b)$.

## Remarks

The condition mentions truth only in $\mathcal{N}$, which makes it easier to check than the definition.
The forward direction is direct: if $\mathcal{N} \models \exists y \, \varphi(\bar a, y)$, then $\mathcal{M} \models \exists y \, \varphi(\bar a, y)$, so some $b \in M$ has $\mathcal{M} \models \varphi(\bar a, b)$, hence $\mathcal{N} \models \varphi(\bar a, b)$.
For the converse, one proves by induction on $\varphi$ that $\mathcal{M} \models \varphi(\bar a)$ if and only if $\mathcal{N} \models \varphi(\bar a)$.
Atomic formulas agree because $\mathcal{M}$ is a substructure, the connectives are immediate, and the condition handles the existential quantifier.

Example of use: in the proof of the downward [[Löwenheim–Skolem theorem|484M0TZ1S7]], one closes a subset of $N$ under witnesses for all existential formulas.
The test then shows that the result is an elementary substructure.

## Explanations

### L1 human

The test says: a substructure is elementary when every object that the big structure can describe with parameters from the small one has a witness already in the small one.
The positive integers inside $(\N, <)$ fail the test: the formula $y < 1$ has the witness $0$ in $\N$ but none among the positive integers.

### L2 human

The Tarski–Vaught test is the standard tool to build elementary substructures.
It underlies the downward Löwenheim–Skolem theorem and the construction of Skolem hulls.
Combined with automorphisms of $\mathcal{N}$ that fix finitely many elements of $M$, it also proves results such as $(\Q, <) \preceq (\R, <)$: if $\mathcal{N} \models \exists y \, \varphi(\bar a, y)$, an automorphism fixing $\bar a$ moves a witness into $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/60TBPSWD32.md" tag="60TBPSWD32" kind="example">
# Symm's integral equation

## Example

Let $\Gamma \subset \R^2$ be a closed $C^2$ curve, and let $f$ be a continuous function on $\Gamma$.
Symm's integral equation asks for a density $\psi$ on $\Gamma$ with
$$-\frac{1}{2\pi} \int_\Gamma \ln \abs{x - y} \, \psi(y) \, ds(y) = f(x), \quad x \in \Gamma.$$
The left side is the [[single layer potential|9T6ZEV4125]] of $\psi$ in the plane, a [[logarithmic potential|46NHWJHW44]], restricted to $\Gamma$.
The unknown appears only under the integral, so it is an [[integral equation of the first kind|ZDS77Q0F06]], with the [[weakly singular kernel|94TNBAAJYX]] $-\frac{1}{2\pi}\ln\abs{x - y}$.
Solving it gives a single layer potential that solves the interior Dirichlet problem for the Laplace equation with boundary values $f$.

Check on the circle $\Gamma = \set{x : \abs{x} = a}$, with $y = a e^{i\theta}$ and $x = a e^{i\phi}$:
by the mean value property, $\frac{1}{2\pi} \int_0^{2\pi} \ln \abs{x - a e^{i\theta}} \, d\theta = \ln a$ for $\abs{x} = a$, and $\ln \abs{e^{i\phi} - e^{i\theta}} = -\sum_{n \ge 1} \frac{\cos n(\phi - \theta)}{n}$.
Hence the operator maps $\psi = 1$ to the constant $-a \ln a$, and $\psi(\theta) = e^{in\theta}$ with $n \ne 0$ to $\frac{a}{2\abs{n}} e^{in\phi}$.
So on a circle of radius $a \ne 1$ the equation is uniquely solvable for every trigonometric polynomial $f$.
For $a = 1$ the constant density $\psi = 1$ solves the homogeneous equation.

## Remarks

The circle example shows the general picture: the homogeneous equation has only the zero solution unless $\Gamma$ has logarithmic capacity $1$, and this exceptional case can be avoided by rescaling.
The eigenvalues $\frac{a}{2\abs{n}}$ tend to $0$, so the solution depends unstably on $f$ in the supremum norm, as for every first-kind equation with a compact operator.
Because of its simple structure, the equation is a standard test case for boundary element and collocation methods.

## Explanations

### L1 human

Put an unknown charge density on a closed curve in the plane, and ask that its potential takes prescribed values on the curve.
That is Symm's equation.
On a circle everything can be computed with Fourier series: each wave pattern of the charge is just multiplied by a number.
</document>

<document source="https://tamnd.github.io/math2/node/60TDCPY4Z4.md" tag="60TDCPY4Z4" kind="concept">
# Poincaré map

## Summary

The Poincaré map, or first return map, belongs to a [[periodic orbit|0AA97XA70K]] and a cross-section of it.
It follows nearby solutions once around the orbit until they hit the cross-section again.
It replaces a continuous-time flow in dimension $n$ by a discrete map in dimension $n - 1$.
The periodic orbit becomes a fixed point of the map, and the stability of the orbit is the stability of this fixed point.
The idea is due to Henri Poincaré and is a basic tool in the study of periodic orbits, bifurcations and chaos.

## Explanations

### L1 human

Put a small screen across a closed loop traced by the system.
Start a solution on the screen and wait until it crosses the screen again; the new crossing point is the image of the old one.
If the screen points come closer and closer to the loop, the loop attracts nearby motions.
For motions that spiral onto the unit circle, each crossing of the positive $x$-axis lies closer to the point $(1, 0)$ than the one before.
</document>

<document source="https://tamnd.github.io/math2/node/60TNVD82Q5.md" tag="60TNVD82Q5" kind="concept">
# Harmonic at a point

## Summary

A function on an open subset of $\R^n$ is harmonic at a point when it is harmonic on some neighbourhood of that point.
This local notion lets one say where a function is harmonic without fixing a domain in advance.
A function is a [[harmonic function|0DEB0JMN0W]] on an open set exactly when it is harmonic at each point of the set.

## Explanations

### L1 human

Being harmonic at a point is not a condition on the value at that single point.
It asks that the function solves Laplace's equation in a small ball around the point.
For example, $\log(x^2 + y^2)$ is harmonic at every point of the plane except the origin, where it is not even defined.
The function $x^3$ in the plane has Laplacian $6x$, which vanishes on the line $x = 0$ but on no ball, so it is harmonic at no point.
</document>

<document source="https://tamnd.github.io/math2/node/60VZDSFKNF.md" tag="60VZDSFKNF" kind="definition">
# Projective line over a ring

## Definition

Let $R$ be an associative [[ring|0B1261B5ED]] with identity, and let $U$ be its [[group of units|07C6ECNJET]].
A pair $(a, b) \in R^2$ is *admissible* if there are $c, d \in R$ such that the matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ lies in the [[general linear group|0FCHE902D7]] $\mathrm{GL}(2, R)$.
The *projective line over $R$* is the set
$$\mathrm{P}(R) = \set{U(a, b) : (a, b) \in R^2 \text{ admissible}},$$
where $U(a, b) = \set{(ua, ub) : u \in U}$.

## Remarks

The group $\mathrm{GL}(2, R)$ acts on $\mathrm{P}(R)$ by $U(a, b) \mapsto U\big((a, b) M\big)$.
If $R$ is commutative, $(a, b)$ is admissible if and only if $aR + bR = R$.
Indeed, if $ax + by = 1$ then $\begin{pmatrix} a & b \\ -y & x \end{pmatrix}$ has determinant $1$; conversely the first row of an invertible matrix generates $R$, by expanding its determinant.

Example: if $R$ is a field $F$, every $(a, b) \ne (0, 0)$ is admissible, and $\mathrm{P}(F)$ is the usual projective line $\set{U(x, 1) : x \in F} \cup \set{U(1, 0)}$.

Example: for $R = \Z$, admissible means $\gcd(a, b) = 1$ and $U = \set{1, -1}$, so $\mathrm{P}(\Z)$ is in bijection with $\Q \cup \set{\infty}$ by $U(a, b) \mapsto a / b$.

Non-example: in $\Z$, the pair $(2, 0)$ is not admissible, because $2\Z + 0\Z = 2\Z \ne \Z$.
</document>

<document source="https://tamnd.github.io/math2/node/60XZ6DNKKW.md" tag="60XZ6DNKKW" kind="statement">
# Levinson recursion

## Statement

Let $T = (t_{i-j})_{1 \le i, j \le N}$ be an $N \times N$ [[Toeplitz matrix|GKY2WSHE26]] over $\mathbb{K}$, where $\mathbb{K}$ is $\R$ or $\C$, and let $y \in \mathbb{K}^N$.
For $n \le N$, let $T^n$ be the leading $n \times n$ submatrix of $T$, and assume that every $T^n$ is [[invertible|0EB9N0S33C]].
Let $f^n, b^n, x^n \in \mathbb{K}^n$ solve $T^n f^n = e_1$, $T^n b^n = e_n$ and $T^n x^n = (y_1, \dots, y_n)^T$, where $e_1, e_n$ are the first and last unit vectors.
Put
$$\varepsilon_f^n = \sum_{j=1}^n t_{n+1-j} f^n_j, \qquad \varepsilon_b^n = \sum_{j=1}^n t_{-j} b^n_j, \qquad \varepsilon_x^n = \sum_{j=1}^n t_{n+1-j} x^n_j.$$
Then $1 - \varepsilon_f^n \varepsilon_b^n \ne 0$, and
$$f^{n+1} = \frac{1}{1 - \varepsilon_f^n \varepsilon_b^n} \left( \begin{pmatrix} f^n \\ 0 \end{pmatrix} - \varepsilon_f^n \begin{pmatrix} 0 \\ b^n \end{pmatrix} \right), \qquad b^{n+1} = \frac{1}{1 - \varepsilon_f^n \varepsilon_b^n} \left( \begin{pmatrix} 0 \\ b^n \end{pmatrix} - \varepsilon_b^n \begin{pmatrix} f^n \\ 0 \end{pmatrix} \right),$$
$$x^{n+1} = \begin{pmatrix} x^n \\ 0 \end{pmatrix} + (y_{n+1} - \varepsilon_x^n) \, b^{n+1}.$$
Starting from $f^1 = b^1 = 1/t_0$ and $x^1 = y_1 / t_0$, the *Levinson recursion* computes the solution $x^N$ of $T x = y$ in $O(N^2)$ arithmetic operations.

## Remarks

The key fact is that both the leading and the trailing $n \times n$ blocks of $T^{n+1}$ equal $T^n$.
So $T^{n+1} (f^n, 0)^T = (1, 0, \dots, 0, \varepsilon_f^n)^T$ and $T^{n+1} (0, b^n)^T = (\varepsilon_b^n, 0, \dots, 0, 1)^T$, and the stated combinations give $e_1$ and $e_{n+1}$.
Similarly $T^{n+1} (x^n, 0)^T = (y_1, \dots, y_n, \varepsilon_x^n)^T$, and adding $(y_{n+1} - \varepsilon_x^n) b^{n+1}$ fixes the last entry.
If $1 - \varepsilon_f^n \varepsilon_b^n$ were $0$, the nonzero vector $(f^n, 0)^T - \varepsilon_f^n (0, b^n)^T$ would lie in the kernel of $T^{n+1}$.
This vector is nonzero because its last entry is $-\varepsilon_f^n b^n_n$, with $\varepsilon_f^n \ne 0$ and $b^n_n = \det T^{n-1} / \det T^n \ne 0$ by Cramer's rule.
Each step costs $O(n)$ operations, which gives the total $O(N^2)$, against $O(N^3)$ for Gaussian elimination.
The hypothesis on all leading [[minors|0BT9BYWVRY]] is needed: for $T = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ the matrix is invertible but $t_0 = 0$, and the recursion cannot start.
For symmetric positive definite $T$ every leading submatrix is invertible, and the special case of the [[Yule–Walker equations|GA46NC0CJZ]] is the Levinson–Durbin recursion.

## Explanations

### L1 human

A Toeplitz matrix is constant along each diagonal, so it is described by only $2N - 1$ numbers.
The Levinson recursion uses this structure to solve a Toeplitz system by growing the solution one size at a time.
Each growth step reuses the previous answer and costs only a few vector operations.

### L2 human

Toeplitz systems arise from stationary time series, in linear prediction, and in signal processing, where $T$ is a matrix of autocovariances.
The recursion solves them in $O(N^2)$ time and $O(N)$ memory, which made it a standard tool in speech coding.
It can be unstable when some leading submatrix is nearly singular, even if $T$ itself is well conditioned; superfast $O(N \log^2 N)$ algorithms also exist.
</document>

<document source="https://tamnd.github.io/math2/node/610F4NS6GD.md" tag="610F4NS6GD" kind="definition">
# Graph enumeration

## Definition

Let $\mathcal{P}$ be a class of finite simple [[graphs|00C0BR7D7H]] that is closed under [[isomorphism|0F7QKGJS6B]].
For an integer $n \ge 1$:

1. The labelled count $\ell_{\mathcal{P}}(n)$ is the number of graphs in $\mathcal{P}$ with vertex set $\set{1, \ldots, n}$.
2. The unlabelled count $u_{\mathcal{P}}(n)$ is the number of isomorphism classes of graphs in $\mathcal{P}$ with $n$ vertices.

*Graph enumeration* is the study of the functions $\ell_{\mathcal{P}}$ and $u_{\mathcal{P}}$: exact formulas, generating functions and asymptotic behaviour as $n \to \infty$.

## Remarks

Example: for the class of all graphs, $\ell(n) = 2^{\binom{n}{2}}$, since each of the $\binom{n}{2}$ pairs is an edge or not.
For $n = 3$ this gives $8$, while $u(3) = 4$, one class for each number of edges $0, 1, 2, 3$.

Non-example: the class of graphs in which $1$ and $2$ are adjacent is not closed under isomorphism, so it has no unlabelled count.

Always $u_{\mathcal{P}}(n) \le \ell_{\mathcal{P}}(n) \le n! \, u_{\mathcal{P}}(n)$, because an isomorphism class contains at most $n!$ graphs on $\set{1, \ldots, n}$.
For all graphs, $u(n)$ is asymptotic to $2^{\binom{n}{2}} / n!$, because almost all graphs have no nontrivial automorphism.
</document>

<document source="https://tamnd.github.io/math2/node/6155D300EB.md" tag="6155D300EB" kind="statement">
# Diaconescu's theorem

## Statement

Let $\mathcal{E}$ be an elementary [[topos|6837RS7N82]].
Suppose that every [[epimorphism|RVXDSY04KG]] in $\mathcal{E}$ has a section, that is, $\mathcal{E}$ satisfies the [[axiom of choice|0HTGB13FE2]].
Then $\mathcal{E}$ is Boolean: every [[subobject|FS1FKAE89H]] of every object of $\mathcal{E}$ has a complement.
Equivalently, the internal logic of $\mathcal{E}$ satisfies the [[law of excluded middle|0RYYK8PG4S]].

## Remarks

The same argument works in constructive set theory, where it is often given in the form of Goodman and Myhill.
Let $P$ be a proposition, and let $U = \set{x \in \set{0,1} : x = 0 \lor P}$ and $V = \set{x \in \set{0,1} : x = 1 \lor P}$.
Both sets are inhabited, so a choice function on $\set{U, V}$ gives $u \in U$ and $v \in V$.
Equality on $\set{0,1}$ is decidable, so either $u \ne v$ or $u = v$.
If $u \ne v$, then $P$ is false, because $P$ would give $U = V$ and hence $u = v$; if $u = v$, then either $u = 1 \in U$ or $v = 0 \in V$, and in both cases $P$ holds.
So $P \lor \lnot P$ holds.

The converse fails.
For a nontrivial group $G$, the topos of $G$-sets is Boolean.
But the epimorphism from $G$, acting on itself, to the one-point $G$-set has no section, since $G$ has no fixed point.
In the topos $\mathbf{Set}$, both choice and excluded middle hold, in line with the theorem.

## Explanations

### L1 human

The theorem says that the axiom of choice is not a neutral addition to constructive mathematics.
Once one can always choose an element from each nonempty set in a family, every statement becomes either true or false.
So a constructive setting that wants to avoid excluded middle must also avoid full choice.

### L2 human

In a [[topos|6837RS7N82]] the logic is intuitionistic in general: the [[subobject classifier|NKP5TPQ79X]] is an internal [[Heyting algebra|NG943XNQWH]], not a Boolean algebra.
Diaconescu's theorem shows that choice forces this algebra to be Boolean.
This is why constructive set theories and type theories use weaker forms such as countable choice or dependent choice.
</document>

<document source="https://tamnd.github.io/math2/node/617CH1WRQN.md" tag="617CH1WRQN" kind="definition">
# Deformation quantization

## Definition

Let $M$ be a [[Poisson manifold|A2KQ49Y4NF]] with [[Poisson bracket|2S0MXDT4YH]] $\set{\cdot, \cdot}$, and let $C^\infty(M)$ be the algebra of real [[smooth functions|0V33BNKZ9S]] on $M$.
Let $C^\infty(M)[\![\hbar]\!]$ be the space of [[formal power series|033WGN00BF]] in $\hbar$ with coefficients in $C^\infty(M)$.
A *star product* on $M$ is an $\R[\![\hbar]\!]$-bilinear associative product $*$ on $C^\infty(M)[\![\hbar]\!]$ such that:

1. $f * g = \sum_{n \ge 0} \hbar^n B_n(f, g)$ for $f, g \in C^\infty(M)$, where each $B_n$ is a bidifferential operator;
2. $B_0(f, g) = fg$;
3. $B_1(f, g) - B_1(g, f) = \set{f, g}$;
4. $1 * f = f * 1 = f$.

A *deformation quantization* of $M$ is a choice of star product on $M$.
Two star products are equivalent when they are equivalent as [[formal deformations|QP92G4ZGJE]] by a series of differential operators.

## Remarks

Clause 3 says $f * g - g * f = \hbar \set{f, g} + O(\hbar^2)$.
Many authors use complex coefficients and ask for $i \hbar \set{f, g}$ instead.
Example: the Moyal product on $\R^2$ with coordinates $x, p$ and $\set{f, g} = \partial_x f \, \partial_p g - \partial_p f \, \partial_x g$ is
$$f * g = \sum_{n \ge 0} \frac{1}{n!} \left(\frac{\hbar}{2}\right)^n \sum_{j=0}^{n} (-1)^j \binom{n}{j} (\partial_x^{n-j} \partial_p^{j} f)(\partial_x^{j} \partial_p^{n-j} g).$$
Its term $B_1(f, g) = \frac{1}{2}\set{f, g}$ is antisymmetric, so clause 3 holds, and $x * p - p * x = \hbar$.
Non-example: the undeformed product $f * g = fg$ satisfies clauses 1, 2 and 4, but fails clause 3 whenever the bracket is nonzero.
</document>

<document source="https://tamnd.github.io/math2/node/61B8B11Y0Y.md" tag="61B8B11Y0Y" kind="statement">
# Ryll-Nardzewski fixed-point theorem

## Statement

Let $E$ be a real [[normed vector space|03J5RAKB30]] and let $K \subseteq E$ be a nonempty [[convex set|02AS45T3H9]] that is compact in the [[weak topology|P6KHVSTPZV]] of $E$.
Let $S$ be a [[semigroup|0XMEM406JM]], under composition, of maps $s \colon K \to K$ such that each $s \in S$ is [[affine|0JF0TVYGWT]] and an [[isometry|02HPP5VCZH]] for the norm.
Then the maps in $S$ have a common fixed point: there is $x \in K$ with $s(x) = x$ for all $s \in S$.

## Remarks

The isometry condition can be weakened to the condition that $S$ is noncontracting: for $x \ne y$ in $K$, the vector $0$ is not in the norm closure of $\set{s(x) - s(y) : s \in S}$.
In this form the theorem holds for locally convex spaces and weakly continuous affine maps.
By weak compactness of $K$, it suffices to treat finitely generated semigroups.
For these, one takes a fixed point of a suitable average of the generators.
The noncontracting condition and a geometric lemma on weakly compact convex sets show that it is fixed by each generator.

The isometry hypothesis cannot be dropped.
On $K = [0, 1]$, the constant maps $s_0(x) = 0$ and $s_1(x) = 1$ are affine and generate the semigroup $\set{s_0, s_1}$, but they have no common fixed point.

Example: let $G$ be the symmetric group on $\set{1, \ldots, n}$, acting on the simplex $K = \set{x \in \R^n : x_i \ge 0, \sum x_i = 1}$ by permuting coordinates.
These maps are affine isometries for the Euclidean norm, and the common fixed point is the barycentre $(1/n, \ldots, 1/n)$.

## Explanations

### L1 human

If a family of symmetries moves a compact convex set into itself without distorting distances, then some point is left in place by all of them at once.
In finite dimensions one can often take a centre of mass, but in infinite dimensions there is no obvious centre, and the theorem supplies one.

### L2 human

A classical application, in the locally convex form, is the existence of Haar measure on a compact group.
The group acts by affine maps on a compact convex set of means on the continuous functions.
A common fixed point is an invariant mean, which gives the Haar integral.
The theorem should be compared with the Markov–Kakutani theorem, which asks for commuting continuous affine maps instead of isometries.
</document>

<document source="https://tamnd.github.io/math2/node/61CDKWSXV4.md" tag="61CDKWSXV4" kind="concept">
# Discrete calculus

## Summary

Discrete calculus studies change and accumulation for functions on discrete sets, such as the integers or the vertices and cells of a grid.
Difference quotients take the place of derivatives, and finite sums take the place of integrals.
Its results mirror those of ordinary calculus, often with exact formulas instead of limits, and it includes the [[calculus of finite differences|FT9JQA58KJ]].
It is used in difference equations, numerical analysis, combinatorics, and in discrete models of geometry and physics.

## Explanations

### L1 human

Ordinary calculus lets the step size shrink to zero; discrete calculus keeps a fixed step.
A table of daily prices has no derivative, but it has daily changes, and adding the changes gives the total change.
For example, the daily changes of the values $1, 3, 7, 15$ are $2, 4, 8$, and their sum $14$ is the change from $1$ to $15$.
This is the discrete form of the fundamental theorem of calculus.
</document>

<document source="https://tamnd.github.io/math2/node/61F4HEY84K.md" tag="61F4HEY84K" kind="definition">
# Schreier–Sims algorithm

## Definition

Let $\Omega = \set{1, \dots, n}$ and let $G \le \operatorname{Sym}(\Omega)$ be the [[permutation group|ZDJQRKPWGA]] [[generated|0RP7VWQ0YN]] by a finite set $S$ of permutations.
A *base* of $G$ is a sequence $(b_1, \dots, b_k)$ of points of $\Omega$ such that only the identity of $G$ fixes every $b_i$.
It gives the *stabiliser chain* $G = G^{(1)} \ge G^{(2)} \ge \dots \ge G^{(k+1)} = \set{e}$, where $G^{(i+1)}$ is the pointwise [[stabiliser|0AYMP41THN]] of $b_1, \dots, b_i$ in $G$.
A *strong generating set* for this base is a subset $T \subseteq G$ such that $T \cap G^{(i)}$ generates $G^{(i)}$ for each $i = 1, \dots, k$.

The *Schreier–Sims algorithm* computes, from $S$, a base of $G$ and a strong generating set for it.
It builds the chain from the top, and it repeats these steps until no new generator appears:

1. For each level $i$, compute the [[orbit|0YQPZB5Q1G]] $\Delta_i = G^{(i)} b_i$ and, for each $\delta \in \Delta_i$, an element $u_\delta \in G^{(i)}$ with $u_\delta b_i = \delta$.
2. By Schreier's lemma, $G^{(i+1)}$ is generated by the *Schreier generators* $u_{s\delta}^{-1} \, s \, u_\delta$, with $s$ in the current generators of $G^{(i)}$ and $\delta \in \Delta_i$.
3. Sift each Schreier generator through the levels below $i$, that is, divide it by the coset representatives $u_\delta$ found there. If the remainder is not the identity, add it to the generators of the level where sifting stopped, with a new base point if needed.

## Remarks

When the algorithm ends, $\abs{G} = \abs{\Delta_1} \abs{\Delta_2} \cdots \abs{\Delta_k}$, because $\abs{\Delta_i}$ is the [[index|0DTXBQPA2S]] $[G^{(i)} : G^{(i+1)}]$ by the orbit-stabiliser theorem.
A permutation $\sigma$ lies in $G$ exactly when sifting $\sigma$ through the whole chain ends at the identity.

Example: $G = S_3$ generated by $S = \set{(1\,2\,3), (1\,2)}$, with base $(1, 2)$.
The orbit of $1$ is $\Delta_1 = \set{1, 2, 3}$.
The stabiliser of $1$ is $G^{(2)} = \set{e, (2\,3)}$, and the orbit of $2$ under it is $\Delta_2 = \set{2, 3}$.
So $\set{(1\,2\,3), (1\,2), (2\,3)}$ is a strong generating set, and $\abs{G} = 3 \cdot 2 = 6$.

Non-example: the sequence $(1)$ is not a base of $S_3$, because the element $(2\,3) \ne e$ fixes $1$.
Also, $S = \set{(1\,2\,3), (1\,2)}$ is not a strong generating set for the base $(1, 2)$: no element of $S$ fixes $1$, so $S \cap G^{(2)}$ generates only $\set{e}$.
</document>

<document source="https://tamnd.github.io/math2/node/61THBNS8WE.md" tag="61THBNS8WE" kind="definition">
# Provability logic

## Definition

The provability logic $\mathrm{GL}$ is the smallest normal [[modal logic|C3RZY20ETK]] that contains Löb's axiom
$$\square(\square p \to p) \to \square p.$$
Its intended reading interprets $\square$ as provability in [[Peano arithmetic|0QFT7Z76N5]].
An arithmetical realization $*$ maps each propositional variable to a sentence of arithmetic and commutes with the connectives.
It sets $(\square\varphi)^* = \mathrm{Prov}(\ulcorner \varphi^* \urcorner)$, where $\mathrm{Prov}$ is the provability predicate of Peano arithmetic and $\ulcorner \cdot \urcorner$ is a [[Gödel numbering|MFJXPAF0R0]].

## Remarks

Solovay's theorem (1976) says: a modal formula $\varphi$ is in $\mathrm{GL}$ if and only if Peano arithmetic proves $\varphi^*$ for every arithmetical realization $*$.

Example: $\square p \to \square\square p$ is in $\mathrm{GL}$.
Its realizations are instances of the fact that Peano arithmetic proves "if $\sigma$ is provable, then it is provable that $\sigma$ is provable".

Non-example: $\square p \to p$ is not in $\mathrm{GL}$.
Under a realization with $p^* = (0 = 1)$, it becomes the consistency statement of Peano arithmetic, which Peano arithmetic does not prove by Gödel's second incompleteness theorem.

On Kripke frames, $\mathrm{GL}$ is the logic of the finite transitive irreflexive frames.
The name "provability logic" also covers the study of related systems for other theories and other provability notions.

## Explanations

### L1 human

The system $\mathrm{GL}$ is named after Gödel and Löb.
Its one extra axiom is the modal form of [[Löb's theorem|3HD30YG3DG]].
It captures every general law of provability that Peano arithmetic can prove about itself.
</document>

<document source="https://tamnd.github.io/math2/node/61VG0XSDM6.md" tag="61VG0XSDM6" kind="problem">
# Neumann problem

## Problem

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a bounded connected [[open set|02BH3MCBQB]] whose [[boundary|05D5EGMFYV]] $\partial\Omega$ is a $C^1$ hypersurface, with outward unit normal $\nu$.
Let $g \colon \partial\Omega \to \R$ be a [[continuous function|0X3WGX95FE]].
Find a function $u$ that is twice continuously differentiable on $\Omega$, continuously differentiable on $\overline{\Omega}$, [[harmonic|0DEB0JMN0W]] on $\Omega$, and satisfies the [[Neumann boundary condition|GS3KT2H645]]
$$\frac{\partial u}{\partial \nu} = g \quad \text{on } \partial\Omega.$$

## Remarks

A necessary condition is $\int_{\partial\Omega} g \, dS = 0$: by the [[divergence theorem|0JX86PSSA9]], $\int_{\partial\Omega} \partial u / \partial \nu \, dS = \int_\Omega \Delta u \, dx = 0$.
A solution is unique up to an additive constant: the difference $w$ of two solutions satisfies $\int_\Omega \abs{\nabla w}^2 \, dx = \int_{\partial\Omega} w \, \partial w / \partial \nu \, dS = 0$, so $w$ is constant on the connected set $\Omega$.
For bounded domains with sufficiently smooth boundary, for example $C^2$, a solution exists whenever the condition $\int_{\partial\Omega} g \, dS = 0$ holds; one proof uses layer potentials and Fredholm theory.
Example: on the unit disc with $g(\theta) = \cos\theta$, the condition holds and $u(x, y) = x$ is a solution, since $\partial u / \partial \nu = x = \cos\theta$ on the unit circle.
The problem is the companion of the [[Dirichlet problem|8D56GQ4AG8]], which prescribes $u$ instead of its normal derivative.

## Explanations

### L1 human

Imagine a metal body in thermal equilibrium, where you control how much heat flows in or out through each piece of its surface.
The Neumann problem asks for the temperature inside.
In equilibrium the total inflow must be zero, and the temperature is then determined up to adding a constant.
</document>

<document source="https://tamnd.github.io/math2/node/61WT6CE08W.md" tag="61WT6CE08W" kind="statement">
# Schwarz integral formula

## Statement

Let $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$.
Let $f$ be a [[holomorphic function|0NAHXR1FRH]] on $\mathbb{D}$ that extends continuously to the closed disc $\overline{\mathbb{D}}$, and let $u = \operatorname{Re} f$.
Then for every $z \in \mathbb{D}$,
$$f(z) = \frac{1}{2\pi}\int_0^{2\pi} \frac{e^{it} + z}{e^{it} - z}\, u(e^{it})\, dt + i \operatorname{Im} f(0).$$

## Remarks

Call the integral $S(z)$.
The kernel $(e^{it} + z)/(e^{it} - z)$ is holomorphic in $z$, so $S$ is holomorphic on $\mathbb{D}$, and its real part is the Poisson integral of $u$ with the [[Poisson kernel|W19567AVYG]].
Since $u$ is [[harmonic|0DEB0JMN0W]] in $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$, the Poisson integral of $u$ equals $u$.
So $f - S$ is holomorphic with real part $0$, and hence it is a purely imaginary constant.
At $z = 0$ the integral is the mean of $u$ over the circle, which is the real number $u(0)$, so the constant is $i \operatorname{Im} f(0)$.
Example: for $f(z) = z$ one has $u(e^{it}) = \cos t$, and the formula gives $\frac{1}{2\pi}\int_0^{2\pi} \frac{e^{it} + z}{e^{it} - z}\cos t \, dt = z$.

## Explanations

### L1 human

A holomorphic function on the disc is determined, up to an imaginary constant, by its real part on the boundary circle.
The Schwarz formula writes the function explicitly as an integral of that real part.
So the real part on the boundary carries all the information about the function inside.

### L2 human

The formula is the holomorphic counterpart of the Poisson integral: taking real parts of both sides gives the Poisson representation of $u$.
It solves the problem of finding a holomorphic function with prescribed real part on the boundary, which is a simple Riemann–Hilbert type problem.
It is used to prove the [[Herglotz representation theorem|9ZBAX4ZPX0]] and the Borel–Carathéodory theorem, and it has versions for the upper half-plane.
</document>

<document source="https://tamnd.github.io/math2/node/61Z8MZ2M28.md" tag="61Z8MZ2M28" kind="definition">
# Hecke algebra of a locally compact group

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] with a left [[Haar measure|VY0HXD3GC4]] $\mu$, and let $K$ be a [[compact|0R6EWFKRCF]] [[open subgroup|QYCFH68RD1]] of $G$.
The *Hecke algebra* $\mathcal{H}(G, K)$ is the set of functions $f \colon G \to \C$ such that:

1. $f(k_1 g k_2) = f(g)$ for all $g \in G$ and $k_1, k_2 \in K$;
2. $f$ has compact support.

Its product is [[convolution|2Z29JN3VES]]:
$$(f_1 * f_2)(x) = \int_G f_1(y) f_2(y^{-1} x) \, d\mu(y).$$

## Remarks

Each double coset $KgK$ is compact and open, and a compact set meets only finitely many of them.
So the indicator functions $\mathbf{1}_{KgK}$ form a basis of $\mathcal{H}(G, K)$ as a vector space.
The function $\mu(K)^{-1} \mathbf{1}_K$ is the unit element.

Example: $G$ discrete and $K = \set{e}$, with $\mu$ the counting measure.
Both clauses say only that $f$ is finitely supported, and $\mathbf{1}_{\set{a}} * \mathbf{1}_{\set{b}} = \mathbf{1}_{\set{ab}}$.
So $\mathcal{H}(G, \set{e})$ is the group algebra $\C[G]$.

Example: $G = \mathrm{GL}_2(\Q_p)$ and $K = \mathrm{GL}_2(\Z_p)$.
Then $\mathcal{H}(G, K)$ is commutative.
It is generated by the indicators of $K \operatorname{diag}(p, 1) K$ and $K \operatorname{diag}(p, p) K$ and the inverse of the latter.

Non-example: for $G = \Q_p$ and $K = \Z_p$, the indicator of $p \Z_p$ fails clause 1, since it is not constant on the coset $\Z_p$.

When $G$ is totally disconnected, the union of the $\mathcal{H}(G, K)$ over all compact open $K$ is the full Hecke algebra $\mathcal{H}(G)$ of locally constant compactly supported functions.
</document>

<document source="https://tamnd.github.io/math2/node/624XDXX8FN.md" tag="624XDXX8FN" kind="concept">
# Subshift of finite type

## Summary

A subshift of finite type is a [[shift space|A1XE59ZHNM]] described by a finite list of forbidden words.
After a recoding of the symbols, every subshift of finite type becomes the set of bi-infinite walks on a finite [[directed graph|0KFYGSHHF5]].
These spaces are the basic models of [[symbolic dynamics|EWFWTSVY0B]].
Through [[Markov partitions|4VNHVDHTNN]] they also code hyperbolic smooth systems, so questions about such systems become questions about matrices.

## Explanations

### L1 human

Think of an infinite sequence of symbols, written in both directions.
A subshift of finite type allows every such sequence except those that contain some word from a fixed finite blacklist.
For example, take the symbols $0$ and $1$ and forbid the word $11$.
Then the allowed sequences are those in which two $1$s never stand next to each other.
Whether a sequence is allowed can always be checked by looking through a window of fixed width.
See the [[definition|M7RPDE39FM]].
</document>

<document source="https://tamnd.github.io/math2/node/628242KN0T.md" tag="628242KN0T" kind="concept">
# Perfect set

## Summary

A perfect set in a [[topological space|0CQHFPGJ45]] is a [[closed set|0ZCK8DQYN1]] without [[isolated points|0A1YXSKB48]].
Equivalently, it equals its own [[derived set|8ZAPAPYJFG]].
In the real line, every nonempty perfect set is uncountable, and the [[Cantor set|0BXNDZFQME]] is the standard example.

## Explanations

### L1 human

A perfect set contains all the points it accumulates at, and none of its points stands alone.
Every point of it has other points of the set crowding around it.
The closed interval $[0, 1]$ is perfect.
The Cantor set, obtained by removing middle thirds again and again, is perfect too, although it contains no interval at all.
</document>

<document source="https://tamnd.github.io/math2/node/62AFX32JGT.md" tag="62AFX32JGT" kind="concept">
# B-spline

## Summary

A B-spline is a [[spline|84TJ8KQ926]] of given degree on a given knot sequence that is nonnegative and nonzero only on a few consecutive knot intervals.
The B-splines of degree $p$ on a knot sequence form a basis of the corresponding spline space.
So every such spline is a unique combination of them.
They are computed by a stable recursion, they sum to $1$, and each one is local.
This makes them the standard representation of splines in numerical work and in computer-aided design.
Curves built from them, including [[NURBS|Y8B2N9ERPE]], are evaluated with [[de Boor's algorithm|5JHFB384BF]].

## Explanations

### L1 human

A B-spline is a small smooth bump made of polynomial pieces.
Of degree $0$ it is a box, equal to $1$ on one interval and $0$ elsewhere.
Of degree $1$ it is a tent, rising from $0$ to $1$ and back down over two intervals.
Higher degrees give smoother and wider bumps.
Any spline is a weighted sum of such bumps, and changing one weight only changes the curve nearby.
The [[definition|8R4CVVV8WT]] gives the recursion.
</document>

<document source="https://tamnd.github.io/math2/node/62FGRXS27K.md" tag="62FGRXS27K" kind="concept">
# Strictly convex set

## Summary

A strictly convex set is a [[convex set|02AS45T3H9]] in which the open segment between any two distinct points lies in the [[interior|0RZH6W98HK]] of the set.
Its boundary contains no flat piece: no nontrivial segment lies on the boundary.
A normed space is a [[strictly convex space|GDTKW9KPH8]] exactly when its closed unit ball is a strictly convex set.
Strict convexity gives uniqueness, for example of the point of a set where a nonzero linear function is largest.

## Explanations

### L1 human

A disc is strictly convex: join two of its points and every point strictly between them is well inside the disc.
A square is convex but not strictly convex.
Two points on the same side give a segment that runs along the edge and never enters the inside.
So strict convexity means that the boundary bulges outward everywhere and has no straight parts.
</document>

<document source="https://tamnd.github.io/math2/node/62M0NC3E2T.md" tag="62M0NC3E2T" kind="concept">
# Simple ring

## Summary

A simple ring is a nonzero ring whose only [[two-sided ideals|0CD87SFM2N]] are $0$ and the whole ring.
It cannot be broken into smaller rings by passing to quotients.
Fields, [[division rings|0EA5BTA0CX]] and full [[matrix rings|0FHVPRAE60]] over them are simple.
By Wedderburn's theorem, the simple Artinian rings are exactly the matrix rings over division rings.

## Explanations

### L1 human

For a commutative ring, having no ideals except $0$ and itself means being a field.
For noncommutative rings, one asks this only of two-sided ideals, and many more rings qualify.
The ring of $2 \times 2$ real matrices is simple, although it has zero divisors and one-sided ideals.
Simple rings play the role among rings that simple groups play among groups.
</document>

<document source="https://tamnd.github.io/math2/node/62NS7PV1M9.md" tag="62NS7PV1M9" kind="statement">
# Perron–Frobenius theorem

## Statement

Let $n \ge 1$, let $A$ be a real $n \times n$ [[nonnegative matrix|JPQCCFCY8X]], and let $r = \rho(A)$ be its [[spectral radius|9V7YQA85PC]].

1. The number $r$ is an [[eigenvalue|0BQAA2BK3Q]] of $A$, and it has an [[eigenvector|0N6960W0BV]] $v \ne 0$ with all entries nonnegative.
2. If $n \ge 2$ and $A$ is [[irreducible|7Q60BB01YG]], then $r > 0$ and $r$ is a simple root of the [[characteristic polynomial|0VSSPAFM2C]] of $A$.
3. If $n \ge 2$ and $A$ is irreducible, then $r$ has an eigenvector $v$ with all entries positive.
4. If $n \ge 2$ and $A$ is irreducible, then every nonnegative eigenvector of $A$ is a positive multiple of $v$.
5. If $n \ge 2$, $A$ is irreducible, and $h$ is the number of eigenvalues of $A$ of absolute value $r$, then these eigenvalues are $r e^{2 \pi i k / h}$ for $k = 0, 1, \ldots, h - 1$.
6. If all entries of $A$ are positive, then the conclusions of 2, 3 and 4 hold, and $\abs{\lambda} < r$ for every eigenvalue $\lambda \ne r$ of $A$.

## Remarks

One proof of part 6 looks at the map $x \mapsto Ax / \sum_i (Ax)_i$ on the simplex of nonnegative vectors with entry sum $1$; Brouwer's fixed point theorem gives a positive eigenvector.
For an irreducible matrix, another proof uses the Collatz–Wielandt formula $r = \max_{x \ge 0,\, x \ne 0} \min_{x_i > 0} (Ax)_i / x_i$.
Part 1 follows from part 6 by continuity, applied to $A + \varepsilon J$ where $J$ has all entries $1$ and $\varepsilon \to 0$.
Example: $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is irreducible with $r = 1$, eigenvector $(1, 1)$, and eigenvalues $1$ and $-1$, so $h = 2$; the strict inequality of part 6 fails because $A$ has zero entries.
Without irreducibility, simplicity can fail: the identity matrix $I_2$ has $r = 1$ as a double eigenvalue.
For the transition matrix of a finite irreducible [[Markov chain|0G48PYM96Q]], the theorem gives a unique stationary distribution.

## Explanations

### L1 human

A matrix with nonnegative entries has a special eigenvalue: its spectral radius, the largest absolute value of an eigenvalue, is itself an eigenvalue, a real number $r \ge 0$.
Its eigenvector can be chosen with nonnegative entries, so it makes sense as a list of sizes, shares or probabilities.
When all entries are positive, this eigenvalue strictly beats all the others, and repeated multiplication by $A$ pushes every positive vector towards the direction of that eigenvector.
This is why rankings such as PageRank and long-run behaviour of population models are computed from it.

### L2 human

The theorem has three layers: nonnegative matrices give a nonnegative eigenvector for $\rho(A)$; irreducible ones give simplicity and positivity; positive ones give a spectral gap.
The number $h$ in part 5 is the period of the directed graph of $A$, and $A$ is called primitive when $h = 1$.
For a primitive matrix, $A^k / r^k$ converges to the rank one matrix $v w^T / (w^T v)$, where $w$ is the positive left eigenvector.
The theorem underlies the convergence of finite Markov chains, the [[M-matrix|DNKXHMGQY1]] theory and power iteration for the dominant eigenvalue.
</document>

<document source="https://tamnd.github.io/math2/node/62PG3KVVVK.md" tag="62PG3KVVVK" kind="definition">
# Homotopy extension property

## Definition

Let $X$ be a topological space, let $A \subseteq X$ carry the [[subspace topology|0WS9P12FQR]], let $Y$ be a topological space, and let $I = [0, 1]$.
Let $f \colon X \to Y$ be continuous, and let $h \colon A \times I \to Y$ be a [[homotopy|9YA6S6HD9V]] with $h(a, 0) = f(a)$ for all $a \in A$.
An *extension* of $h$ along $f$ is a homotopy $H \colon X \times I \to Y$ such that:

1. $H(x, 0) = f(x)$ for all $x \in X$;
2. $H(a, t) = h(a, t)$ for all $a \in A$ and $t \in I$.

The pair $(X, A)$ has the *homotopy extension property with respect to $Y$* if every such $h$ has an extension along every such $f$.
The pair $(X, A)$ has the *homotopy extension property* if it has it with respect to every topological space $Y$.

## Remarks

If $A$ is closed in $X$, then $(X, A)$ has the homotopy extension property if and only if $X \times \set{0} \cup A \times I$ is a retract of $X \times I$.
Example: $(D^n, S^{n-1})$ has it, because the radial projection from the point $(0, \ldots, 0, 2)$ of $\R^{n+1}$ gives a retraction of $D^n \times I$ onto $D^n \times \set{0} \cup S^{n-1} \times I$.
Non-example: let $X = \set{0} \cup \set{1/n : n \ge 1}$ and $A = \set{0}$.
A retraction $r$ of $X \times I$ onto $X \times \set{0} \cup A \times I$ would fix $(0, 1)$, and would send each connected set $\set{1/n} \times I$ to $(1/n, 0)$.
Then $r(1/n, 1) = (1/n, 0) \to (0, 0) \neq r(0, 1)$, so $r$ is not continuous and the pair fails the property.
Every CW pair has the homotopy extension property.
</document>

<document source="https://tamnd.github.io/math2/node/62SNWC5E45.md" tag="62SNWC5E45" kind="problem">
# n-body problem

## Problem

Let $n \ge 2$, let $G > 0$ be the gravitational constant, and let $m_1, \ldots, m_n > 0$ be masses.
Let the positions $x_1(t), \ldots, x_n(t) \in \R^3$ satisfy Newton's equations
$$m_i \, x_i'' = \sum_{j \ne i} \frac{G m_i m_j (x_j - x_i)}{\norm{x_j - x_i}^3}, \qquad i = 1, \ldots, n.$$
Given the initial positions, with $x_i(0) \ne x_j(0)$ for $i \ne j$, and the initial velocities, describe the solution for all time.

## Remarks

The equations are a system of [[ordinary differential equations|0NT64JV96A]] of order two.
They form a [[Hamiltonian system|0EDKGA2FAK]] with Hamiltonian
$$H = \sum_i \frac{\norm{p_i}^2}{2 m_i} - \sum_{i < j} \frac{G m_i m_j}{\norm{x_i - x_j}}, \qquad p_i = m_i x_i'.$$
There are ten classical [[first integrals|0262JP0NV9]].
They are the energy, the three components of the total momentum, the three components of the angular momentum, and three from the uniform motion of the centre of mass.

For $n = 2$ the problem reduces to the Kepler problem and is solved completely: the relative motion follows a conic section.
For $n \ge 3$ there is no general solution by elementary formulas, and the [[three-body problem|JE1Z920QKT]] already shows chaotic behaviour.
A solution can stop existing in finite time at a singularity.
Collisions are one kind of singularity.
Xia proved in 1992 that for $n = 5$ there are also singularities without collision, in which bodies escape to infinity in finite time.

## Explanations

### L1 human

Put several stars or planets in space, give them initial positions and speeds, and let gravity act.
The $n$-body problem asks how they move from then on.
With two bodies the answer is the ellipses, parabolas and hyperbolas of Kepler.
With three or more bodies the motion can be extremely complicated, and much of the theory of dynamical systems grew out of attempts to understand it.
</document>

<document source="https://tamnd.github.io/math2/node/62TMXRHZ76.md" tag="62TMXRHZ76" kind="statement">
# Pickands–Balkema–De Haan theorem

## Statement

Let $F$ be a [[cumulative distribution function|0MXMS3AE13]] on $\R$ and let $x_F = \sup\set{x : F(x) < 1} \le \infty$ be its right endpoint.
For $u < x_F$, let $F_u$ be the excess distribution over the threshold $u$:
$$F_u(y) = \frac{F(u + y) - F(u)}{1 - F(u)}, \qquad 0 \le y < x_F - u,$$
the law of $X - u$ given $X > u$ when $X$ has distribution function $F$.
For $\xi \in \R$ and $\beta > 0$, let $G_{\xi, \beta}$ be the generalised Pareto distribution function
$$G_{\xi, \beta}(y) = 1 - \Big(1 + \frac{\xi y}{\beta}\Big)^{-1/\xi} \ (\xi \ne 0), \qquad G_{0, \beta}(y) = 1 - e^{-y/\beta},$$
for $y \ge 0$ if $\xi \ge 0$ and for $0 \le y \le -\beta/\xi$ if $\xi < 0$.
Say that $F$ is in the maximum domain of attraction of $H_\xi(x) = \exp\big(-(1 + \xi x)^{-1/\xi}\big)$ (with $H_0(x) = \exp(-e^{-x})$) if, for [[i.i.d.|0AJN7WHBJZ]] $X_1, X_2, \dots$ with distribution function $F$, there are $a_n > 0$ and $b_n \in \R$ such that $(\max_{i \le n} X_i - b_n)/a_n$ [[converges in distribution|0JDAQHWA9N]] to $H_\xi$.

Then the following holds for each $\xi \in \R$.
The function $F$ is in the maximum domain of attraction of $H_\xi$ if and only if there is a positive measurable function $\beta(u)$ with
$$\lim_{u \uparrow x_F} \ \sup_{0 \le y < x_F - u} \abs{F_u(y) - G_{\xi, \beta(u)}(y)} = 0.$$

## Remarks

Idea of the proof.
Membership in the domain of attraction of $H_\xi$ is equivalent to the existence of a positive function $a(u)$ with $\frac{1 - F(u + a(u) y)}{1 - F(u)} \to (1 + \xi y)^{-1/\xi}$ as $u \uparrow x_F$, for each admissible $y$.
The left side is $1 - F_u(a(u) y)$, so this is pointwise convergence of rescaled excess laws to $G_{\xi, 1}$, and $\beta(u) = a(u)$ works.
Uniformity in $y$ follows because the functions are monotone and the limit is continuous.

Examples.
For the exponential law $F(x) = 1 - e^{-x}$, $F_u(y) = 1 - e^{-y}$ for every $u$, so $\xi = 0$ and $\beta(u) = 1$ give equality.
For the [[Pareto law|5N3P64YHQR]] $1 - F(x) = x^{-\alpha}$ for $x \ge 1$, $F_u(y) = 1 - (1 + y/u)^{-\alpha} = G_{1/\alpha, \, u/\alpha}(y)$ for $u \ge 1$.
For the uniform law on $[0, 1]$, $F_u(y) = y/(1 - u) = G_{-1, \, 1 - u}(y)$.

The hypothesis is not automatic: some laws lie in no domain of attraction, for example the Poisson and geometric laws.
Their excesses over high thresholds have no generalised Pareto limit.

## Explanations

### L1 human

Pick a high threshold, such as a flood level, and look only at how far the observations exceed it.
The theorem says that, for most continuous distributions met in practice, these excesses follow approximately a generalised Pareto law once the threshold is high enough.
This family has just two parameters: a shape $\xi$, which says how heavy the tail is, and a scale.
So one can model rare extreme events without knowing the full distribution.

### L2 human

The theorem is the threshold counterpart of the [[Fisher–Tippett–Gnedenko theorem|Y3Z99461HJ]], which describes the limit laws of maxima.
The same shape parameter $\xi$ appears in both.
One has $\xi > 0$ for heavy, Pareto-like tails, $\xi = 0$ for light tails such as the normal and exponential laws, and $\xi < 0$ for laws with a finite right endpoint.
In practice it justifies the peaks-over-threshold method of extreme value statistics.
One fits $G_{\xi, \beta}$ to the excesses over a high threshold by maximum likelihood and uses the fit to estimate high quantiles and tail probabilities.
</document>

<document source="https://tamnd.github.io/math2/node/62WCKX1EZT.md" tag="62WCKX1EZT" kind="definition">
# Locale

## Definition

The category of *locales* is the opposite of the category of [[frames|306PWGH7BF]] and frame homomorphisms.
So a *locale* $X$ is given by a frame $\mathcal{O}(X)$, and a *locale map* $f \colon X \to Y$ is a frame homomorphism $f^* \colon \mathcal{O}(Y) \to \mathcal{O}(X)$.

## Remarks

Example: each [[topological space|0CQHFPGJ45]] $X$ gives the locale with frame $\Omega(X)$ of [[open sets|02BH3MCBQB]].
Each [[continuous map|0X3WGX95FE]] $f \colon X \to Y$ gives the locale map with $f^* = f^{-1} \colon \Omega(Y) \to \Omega(X)$, since preimages preserve unions and finite intersections.
A point of a locale $X$ is a locale map from the one-point locale, that is, a frame homomorphism $\mathcal{O}(X) \to \set{0, 1}$.
A locale is spatial if it comes from a topological space; the functor $\Omega$ is an equivalence between [[sober spaces|4N4DNDR22Z]] and spatial locales.
Non-example of spatiality: a nontrivial complete Boolean algebra without atoms is a frame with no frame homomorphisms to $\set{0, 1}$.
As a locale it has no points, so it is not spatial.
Locales are the subject of [[pointless topology|CYEMGH98WN]].
</document>

<document source="https://tamnd.github.io/math2/node/634XKY8RZ0.md" tag="634XKY8RZ0" kind="concept">
# Global field

## Summary

A global field is either a [[number field|MWBDFS2KJ5]] or a global function field.
A number field is a finite extension of $\Q$, and a global function field is a finite extension of $\F_q(t)$ for a finite field $\F_q$.
The two kinds share much of their arithmetic: rings of integers, class groups, the product formula, adeles and [[class field theory|BEKCEYHCHX]].
Their completions at places are the [[local fields|J85FZEQG38]].

## Explanations

### L1 human

The rational numbers $\Q$ and the rational functions $\F_p(t)$ over a finite field behave alike in many ways.
In both, there is a notion of prime, unique factorization in a suitable ring, and many absolute values.
A global field is a finite extension of one of these two basic fields.
For example, $\Q(i)$, the numbers $a + bi$ with $a, b$ rational, is a global field of the first kind.
Studying both kinds side by side often helps: a hard question about numbers can have an easier twin about functions.
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<document source="https://tamnd.github.io/math2/node/63AMHDV2HX.md" tag="63AMHDV2HX" kind="definition">
# Principal series representation

## Definition

Let $G$ be a [[real reductive Lie group|42ZCE14AD2]] with [[maximal compact subgroup|T2EB8R9HEV]] $K$.
Let $P$ be a minimal [[parabolic subgroup|Z2FEBW5MQM]] of $G$ with [[Langlands decomposition|BM3TWJ0D4P]] $P = MAN$, and let $\mathfrak a$ be the Lie algebra of $A$.
Let $\rho \in \mathfrak a^*$ be half the sum of the roots of $\mathfrak a$ in the Lie algebra of $N$, counted with multiplicity.
For $a \in A$ and $\lambda \in \mathfrak a^*_\C$, write $a^\lambda = e^{\lambda(\log a)}$.
Let $\sigma$ be an [[irreducible|FX3NEFYFFN]] unitary representation of $M$ on $V_\sigma$, and let $\nu \in \mathfrak a^*_\C$.
The *principal series representation* $I(\sigma, \nu) = \mathrm{Ind}_P^G(\sigma \otimes e^\nu \otimes 1)$ acts on the space of measurable functions $f \colon G \to V_\sigma$ such that
$$f(x m a n) = a^{-(\nu + \rho)} \sigma(m)^{-1} f(x) \qquad (x \in G, \ m \in M, \ a \in A, \ n \in N)$$
and $\int_K \norm{f(k)}^2 \, dk < \infty$, by left translation $(I(\sigma, \nu)(g) f)(x) = f(g^{-1} x)$.

## Remarks

The shift by $\rho$ makes $I(\sigma, \nu)$ unitary when $\nu \in i \mathfrak a^*$; these are the unitary principal series.
Every function in the space is determined by its restriction to $K$, because $G = KP$ by the Iwasawa decomposition.

Example: for $G = \mathrm{SL}_2(\R)$, let $P$ be the upper triangular matrices, so $M = \set{\pm 1}$, $A$ is the positive diagonal matrices and $N$ the unipotent upper triangular ones.
Here $\rho(\log \operatorname{diag}(t, t^{-1})) = \log t$, so $\operatorname{diag}(t, t^{-1})^\rho = t$.
The restriction to $K = \mathrm{SO}(2)$ identifies the space with the functions on the circle that are even or odd according to $\sigma(-1) = \pm 1$.
So each character $e^{in\theta}$ of $K$ with $(-1)^n = \sigma(-1)$ occurs once, and $I(\sigma, \nu)$ is admissible.

Non-example: inducing the trivial representation from the compact subgroup $K$ instead of from $P$ gives the representation of $G$ on $L^2(G/K)$.
It is not a principal series representation: $K$ is not parabolic, and the representation is not even admissible.

Some authors call representations induced from any parabolic subgroup, with discrete series or tempered data on $M$, generalised principal series.
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<document source="https://tamnd.github.io/math2/node/63D4AXYCT4.md" tag="63D4AXYCT4" kind="concept">
# Soliton

## Summary

A soliton is a localised wave that moves at constant speed without changing its shape.
It keeps its shape and speed after it collides with other solitons; the only trace of the collision is a shift in position.
Solitons arise when nonlinear steepening and dispersion balance, in equations such as the [[Korteweg–De Vries equation|KTT75QC72Z]], the [[nonlinear Schrödinger equation|GEJHRXC176]] and the [[sine-Gordon equation|802QSF9NM3]].
They are central in the theory of integrable equations and in models of water waves and of light pulses in optical fibres.

## Explanations

### L1 human

A soliton is a single hump of water, or of light, that travels far without spreading out or breaking.
Usually a wave packet spreads because its parts travel at different speeds, and a tall wave steepens and breaks because its top moves faster.
In a soliton these two effects cancel.
For the KdV equation $u_t + 6 u u_x + u_{xxx} = 0$ the soliton is $u = \frac{c}{2} \operatorname{sech}^2\big(\frac{\sqrt{c}}{2}(x - ct)\big)$: taller solitons are narrower and faster.
When a tall fast soliton overtakes a small slow one, both come out of the collision unchanged.
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<document source="https://tamnd.github.io/math2/node/63FAKMYZ2E.md" tag="63FAKMYZ2E" kind="concept">
# Square root of a matrix

## Summary

A square root of a square matrix $A$ is a matrix $B$ with $B^2 = A$.
A matrix can have no square root, finitely many, or infinitely many.
A [[positive semidefinite matrix|01MMVW8CKC]] has exactly one positive semidefinite square root; see [[square root of a positive semidefinite matrix|0BJVDCF38Y]].
This square root is used in statistics, polar decompositions and quantum mechanics.

## Explanations

### L1 human

For numbers, a square root of $a$ is a number $b$ with $b^2 = a$.
For matrices the idea is the same, with matrix multiplication.
The diagonal matrix with entries $4$ and $9$ has the square root with entries $2$ and $3$, and also those with entries $\pm 2$ and $\pm 3$.
The identity matrix of size $2$ even has infinitely many square roots, such as every reflection of the plane.
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<document source="https://tamnd.github.io/math2/node/63FS7T8R18.md" tag="63FS7T8R18" kind="definition">
# Zeta function regularization

## Definition

Let $(\lambda_n)_{n \ge 1}$ be a [[sequence|0QE8STGJH2]] of real numbers such that:

1. $0 < \lambda_1 \le \lambda_2 \le \cdots$ and $\lambda_n \to \infty$;
2. the series $\zeta_\lambda(s) = \sum_{n \ge 1} \lambda_n^{-s}$ converges for all complex $s$ with $\operatorname{Re} s > \sigma_0$, for some real $\sigma_0$;
3. $\zeta_\lambda$ has a meromorphic [[analytic continuation|0M3YQJ81NB]] to a domain that contains $s = -1$ and $s = 0$.

If the continuation is holomorphic at $s = -1$, the *zeta-regularized sum* of $\sum \lambda_n$ is $\zeta_\lambda(-1)$.
If it is holomorphic at $s = 0$, the *zeta-regularized product* of $\prod \lambda_n$ is $\exp(-\zeta_\lambda'(0))$.

## Remarks

Here $\lambda_n^{-s} = e^{-s \ln \lambda_n}$, which needs $\lambda_n > 0$.
The function $\zeta_\lambda$ is a generalized [[Dirichlet series|8MWTEF6462]].

Example: for $\lambda_n = 2^n$ one has $\zeta_\lambda(s) = \sum_{n \ge 1} 2^{-ns} = \frac{1}{2^s - 1}$ for $\operatorname{Re} s > 0$, and the right side is meromorphic on $\C$.
So the regularized sum of $2 + 4 + 8 + \cdots$ is $\frac{1}{2^{-1} - 1} = -2$.
For $\lambda_n = n$, $\zeta_\lambda$ is the Riemann zeta function, so the regularized sum of $1 + 2 + 3 + \cdots$ is $\zeta(-1) = -\frac{1}{12}$, and the regularized product $\prod n$ is $e^{-\zeta'(0)} = \sqrt{2\pi}$.

Non-example: the sequence $\lambda_n = (-1)^n n$ fails clause 1, since it has negative terms and $\lambda_n^{-s}$ is not defined by a real logarithm.
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# Distributive property

## Summary

The distributive property links two [[binary operations|0M418FE7BB]] on one set: one of them can be applied term by term over the other.
The basic case is $a \cdot (b + c) = a \cdot b + a \cdot c$ for numbers, the [[distributive law|0KBEMAJAYA]].
Distributivity of multiplication over addition is an axiom of rings and fields, and distributivity of meet over join defines [[distributive lattices|0HM0HQCK6N]].

## Explanations

### L1 human

To compute $3 \cdot 12$, one can split $12$ as $10 + 2$ and compute $3 \cdot 10 + 3 \cdot 2 = 36$.
That this always works is the distributive property of multiplication over addition.
For sets, intersection distributes over union: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.
Addition does not distribute over multiplication: $1 + 2 \cdot 3 = 7$ but $(1 + 2) \cdot (1 + 3) = 12$.
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<document source="https://tamnd.github.io/math2/node/63TX60RTRQ.md" tag="63TX60RTRQ" kind="concept">
# Modular function

## Summary

The modular function of a [[locally compact group|AHV05ZBCYW]] measures how far a left [[Haar measure|VY0HXD3GC4]] is from being right-invariant.
It is a continuous homomorphism $\Delta$ from the group to the multiplicative group of positive reals.
It appears in every formula that moves a Haar integral across a right translation or an inversion.
It is identically $1$ exactly for unimodular groups.

## Explanations

### L1 human

A left Haar measure does not change when a set is moved on the left, but it may grow or shrink when the set is moved on the right.
The growth factor depends only on the element used to move, and that factor is the modular function.
For the group of maps $x \mapsto ax + b$ with $a > 0$, moving a set on the right by such a map multiplies its left Haar measure by $1/a$.
For commutative groups left and right are the same, so the factor is always $1$.
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<document source="https://tamnd.github.io/math2/node/63XHDVA5FS.md" tag="63XHDVA5FS" kind="concept">
# Kolmogorov epsilon-entropy

## Summary

Let $K$ be a [[totally bounded set|020Q06TYYH]] in a [[metric space|0VKD1TP7F3]].
Its Kolmogorov $\varepsilon$-entropy is the binary logarithm of the least number of pieces of diameter at most $2\varepsilon$ needed to cover $K$.
It measures how many bits are needed to describe an element of $K$ to accuracy $\varepsilon$.
Kolmogorov and Tikhomirov studied its growth as $\varepsilon \to 0$ for classes of smooth and analytic functions.
It gives lower bounds for every method of approximation or coding of the class, linear or not.

## Explanations

### L1 human

Suppose you must store any element of a set $K$, but only up to an error $\varepsilon$.
Cut $K$ into pieces so small that every piece has diameter at most $2\varepsilon$, and store only the label of the piece.
The fewest pieces that work is the covering number, and its logarithm in base $2$ is the number of bits you need.
For example, the interval $[0, 1]$ needs $\ceil{1/(2\varepsilon)}$ pieces, so its $\varepsilon$-entropy is about $\log_2(1/\varepsilon)$; a set of functions needs far more.
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<document source="https://tamnd.github.io/math2/node/63XXTF6B5M.md" tag="63XXTF6B5M" kind="concept">
# Regular local ring

## Summary

A regular local ring is a Noetherian [[local ring|MZ02BKPHJA]] whose maximal ideal can be generated by as many elements as its [[Krull dimension|MVGBS3J1P7]].
Equivalently, its [[embedding dimension|DP3KSCRKFP]] equals its dimension.
Regular local rings are the algebraic form of smooth points of varieties.
They are [[integral domains|V3ZBMRKC08]] and even [[unique factorization domains|1HD3EK5392]], and by [[Serre's theorem|93RQRG9R1P]] they are the Noetherian local rings of finite global dimension.

## Explanations

### L1 human

At a smooth point of a surface, two coordinates describe everything nearby, and the surface has dimension two.
At a singular point, like the tip of a cone, you need more coordinates than the dimension.
A regular local ring is the local ring of a point where the number of needed coordinates equals the dimension.
Examples are fields, the ring $\Z_{(p)}$, and power series rings $k[\![x_1, \ldots, x_n]\!]$.
The [[definition|KTQC33ZJTN]] gives the condition.
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<document source="https://tamnd.github.io/math2/node/63Z1Q4FS23.md" tag="63Z1Q4FS23" kind="concept">
# Finite measure

## Summary

A finite measure is a measure that gives the whole space a finite value.
Every measurable set then has finite measure.
Probability measures are the finite measures of total mass $1$, and every nonzero finite measure is a multiple of a probability measure.

## Explanations

### L1 human

A finite measure is a way of measuring in which the total amount is finite.
Length on the interval $[0, 1]$ is a finite measure, with total $1$.
Length on the whole real line is not, since the line is infinitely long.
Dividing a nonzero finite measure by its total gives a probability measure.
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<document source="https://tamnd.github.io/math2/node/63ZT8D3DXD.md" tag="63ZT8D3DXD" kind="problem">
# Yamabe problem

## Problem

Let $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]] without boundary, of dimension $n \ge 3$.
Is there a smooth positive function $u$ on $M$ such that the metric $\tilde g = u^{4/(n-2)} g$ has constant [[scalar curvature|ZHX2VYQDZ3]]?

## Remarks

The scalar curvature $\tilde R$ of $\tilde g$ is related to the scalar curvature $R$ of $g$ by
$$-\frac{4(n-1)}{n-2} \Delta_g u + R u = \tilde R \, u^{(n+2)/(n-2)},$$
where $\Delta_g$ is the Laplace–Beltrami operator of $g$, the analogue of the [[Laplacian|0E0B6RA8K0]].
So the problem asks for a positive solution of this semilinear elliptic equation with $\tilde R$ constant.
The exponent $(n+2)/(n-2)$ is the critical Sobolev exponent, and this is why the direct variational method does not apply at once.
The answer is yes.
Yamabe proposed a proof that Neil Trudinger found to contain a gap.
Trudinger, Thierry Aubin and Richard Schoen completed the proof, with Schoen settling the remaining cases with the positive mass theorem.
In dimension $2$ the analogous statement is a consequence of the uniformization theorem.

## Explanations

### L1 human

On a curved space one can stretch lengths by a factor that varies from point to point, without changing angles.
The Yamabe problem asks whether such a stretching can always make the scalar curvature, an average of the curvature at each point, the same everywhere.
It is a higher-dimensional version of the fact that every surface can be given a metric of constant curvature, and the answer is again yes.
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<document source="https://tamnd.github.io/math2/node/640A1MTM4Z.md" tag="640A1MTM4Z" kind="definition">
# Paraproduct

## Definition

Let $n \ge 1$, and let $\chi$, $\psi$ and $\Delta_j$, $j \in \Z$, be as in the [[Littlewood–Paley decomposition|D1PAT0K446]] on $\R^n$.
For $m \in \Z$ and a bounded function $g$, let $S_m g$ be defined by $\widehat{S_m g}(\xi) = \chi(2^{-m}\xi) \hat g(\xi)$, where $\hat{\ }$ is the [[Fourier transform|DPVM34ADZV]] of tempered distributions.
For $f$ in the [[Lp space|SASP14MA1Z]] $L^2(\R^n)$ and $g \in L^\infty(\R^n)$, the *paraproduct* of $f$ by $g$ is
$$\Pi_g f = \sum_{j \in \Z} (S_{j-3} g) \, (\Delta_j f),$$
where the series converges in $L^2(\R^n)$.

## Remarks

The function $S_m g$ is the convolution of $g$ with $2^{mn} \check\chi(2^m \cdot)$, a smooth average of $g$ at scale $2^{-m}$, so $\abs{S_m g} \le C \norm{g}_\infty$.
The Fourier transform of the $j$-th term is supported in $\set{2^{j-2} \le \abs{\xi} \le 2^{j+2}}$, because $\Delta_j f$ has frequencies between $2^{j-1}$ and $2^{j+1}$ and $S_{j-3} g$ has frequencies below $2^{j-2}$.
So the terms are almost orthogonal, which gives the convergence and the bound $\norm{\Pi_g f}_2 \le C \norm{g}_\infty \norm{f}_2$.
The shift $3$ is a convention; other authors use a different shift or a different cut-off.

Grouping the terms of $fg = \sum_{j, k} \Delta_j f \, \Delta_k g$ according to $k \le j - 3$, $j \le k - 3$ or $\abs{j - k} \le 2$ gives Bony's decomposition
$$fg = \Pi_g f + \Pi_f g + \sum_{\abs{j - k} \le 2} \Delta_j f \, \Delta_k g$$
for $f$ and $g$ both in $L^2(\R^n) \cap L^\infty(\R^n)$, since $\sum_{k \le m} \Delta_k g = S_m g$.

Example: if $g = c$ is a constant, then $\hat g$ is a multiple of the point mass at $0$ and $\chi(0) = 1$, so $S_m g = c$ for every $m$.
Hence $\Pi_g f = c \sum_j \Delta_j f = c f$.

Non-example: the term $\Delta_j f \, \Delta_j g$, a product of two pieces at the same frequency, is not part of a paraproduct.
Its Fourier transform can be supported near $0$, for instance when $\widehat{\Delta_j g}$ is the reflection of $\widehat{\Delta_j f}$, so high frequencies combine into low ones; such terms go into the remainder.
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# Monogenic function

## Summary

A monogenic function is a function on an open subset of $\R^n$ with values in a [[Clifford algebra|A38B1QFK20]] that is annihilated by the [[Dirac operator|3DXGJTEQZH]].
It is the higher-dimensional analogue of a holomorphic function and the main object of [[Clifford analysis|DKVSAZQWF7]].
Monogenic functions satisfy a Cauchy integral formula, have local power series expansions, and their components are harmonic.

## Explanations

### L1 human

A holomorphic function solves the Cauchy–Riemann equations, which can be packed into one equation $\partial f / \partial \bar z = 0$.
In $\R^n$ one packs the derivatives into one operator $\sum_j e_j \partial_{x_j}$ with anticommuting units $e_j$.
A monogenic function is one sent to $0$ by this operator.
For example, a constant function is monogenic, and so is $x \mapsto x / \abs{x}^n$ away from $0$, where $x = \sum_j x_j e_j$; this is the Cauchy kernel up to a constant.
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<document source="https://tamnd.github.io/math2/node/64F9VMG27D.md" tag="64F9VMG27D" kind="definition">
# Invariant mean

## Definition

Let $G$ be a [[group|G5S1M9]] and let $\ell^\infty(G)$ be the real vector space of bounded functions $f \colon G \to \R$.
For $g \in G$ and $f \in \ell^\infty(G)$, let $g \cdot f$ be the function $x \mapsto f(g^{-1} x)$.
A *left-invariant mean* on $G$ is a [[linear form|07YQ1C78G0]] $m \colon \ell^\infty(G) \to \R$ with these three properties.

1. Positivity: $m(f) \ge 0$ for every $f \ge 0$.
2. Normalization: $m(\mathbf{1}) = 1$, where $\mathbf{1}$ is the constant function $1$.
3. Invariance: $m(g \cdot f) = m(f)$ for every $g \in G$ and $f \in \ell^\infty(G)$.

## Remarks

Right-invariant and two-sided invariant means are defined in the same way.
A group has a left-invariant mean if and only if it is [[amenable|CEAT9BESKK]].
A left-invariant mean $m$ gives a left-invariant [[finitely additive measure|KFWQ24D9M6]] $\nu(A) = m(\mathbf{1}_A)$ on all subsets of $G$, with $\nu(G) = 1$, and every such $\nu$ arises in this way.

Example: for a finite group $G$, the average $m(f) = \frac{1}{\abs{G}} \sum_{x \in G} f(x)$ is a left-invariant mean.
Positivity and normalization are clear, and invariance holds because $x \mapsto g^{-1} x$ is a bijection of $G$.

Non-example: on $G = \Z$, the evaluation $m(f) = f(0)$ is positive and normalized but not invariant.
For $f = \mathbf{1}_{\set{0}}$ and $g = 1$, the function $g \cdot f$ is $\mathbf{1}_{\set{1}}$, so $m(g \cdot f) = 0 \ne 1 = m(f)$.

On $\Z$, invariant means exist by the Hahn–Banach theorem, for example Banach limits, but none can be given by an explicit formula.
For a locally compact group, one uses $L^\infty(G)$ with respect to Haar measure in place of $\ell^\infty(G)$.

## Explanations

### L1 human

An invariant mean is an averaging rule for all bounded functions on a group that ignores translations.
It always exists on finite groups and abelian groups, and never on free groups of rank two.
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# Max-plus algebra

## Summary

Max-plus algebra is linear algebra in which addition is replaced by taking the maximum and multiplication is replaced by ordinary addition.
These operations make $\R \cup \set{-\infty}$ a semiring, the max-plus semiring.
It is one form of the tropical semiring.
Matrices over it describe longest paths in weighted graphs.
They also model the timing of systems that wait for several events, such as train schedules.

## Explanations

### L1 human

In max-plus arithmetic, "$a$ plus $b$" means $\max(a, b)$ and "$a$ times $b$" means $a + b$.
So $3 \oplus 5 = 5$ and $3 \otimes 5 = 8$.
A matrix product then picks the best route: entry $(i, j)$ of $A \otimes B$ is $\max_k (a_{ik} + b_{kj})$.
If $a_{ik}$ is the time to go from station $i$ to station $k$, then powers of $A$ give the longest travel times along chains of steps.
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# Khovanov homology

## Definition

Let $D$ be a diagram of an oriented tame [[link|1A5WRCNTM3]] $L$, with crossings ordered $1, \ldots, n$, of which $n_+$ are positive and $n_-$ are negative.
Let $A = \Z[x]/(x^2)$, graded by $\deg 1 = 1$ and $\deg x = -1$.
For $v \in \set{0,1}^n$, let $D_v$ be the diagram where crossing $i$ gets the $0$-smoothing (A-smoothing of the [[Kauffman bracket|JYKVYS4N32]]) if $v_i = 0$ and the $1$-smoothing otherwise.
Let $k_v$ be the number of circles of $D_v$ and $r(v) = v_1 + \cdots + v_n$.
Put $C_v = A^{\otimes k_v}$, with quantum grading shifted up by $r(v) + n_+ - 2n_-$.
The chain group in homological degree $i$ is $C^i(D) = \bigoplus_{r(v) = i + n_-} C_v$.
For an edge $v \to v'$ that changes one $0$ to a $1$, the map $C_v \to C_{v'}$ is:

1. $m \colon A \otimes A \to A$ if two circles merge, with $m(1 \otimes 1) = 1$, $m(1 \otimes x) = m(x \otimes 1) = x$, $m(x \otimes x) = 0$;
2. $\Delta \colon A \to A \otimes A$ if one circle splits, with $\Delta(1) = 1 \otimes x + x \otimes 1$ and $\Delta(x) = x \otimes x$;

and the identity on the other circles.
The differential $d \colon C^i(D) \to C^{i+1}(D)$ is the sum of these edge maps, each with a sign $(-1)^{s}$.
Here $s$ is the number of $1$s in $v$ before the changed position.
The *Khovanov homology* of $L$ is $Kh^{i,j}(L) = H^i(C(D))$ in quantum degree $j$.

## Remarks

The map $d$ satisfies $d^2 = 0$ and preserves the quantum grading.
Up to isomorphism, $Kh^{i,j}(L)$ does not depend on the diagram $D$.
Example: the round diagram of the [[unknot|1TQ06R4Q66]] has $n = 0$, so the complex is $A$ in degree $0$ and $Kh$ is $\Z$ in bidegrees $(0, 1)$ and $(0, -1)$.
Its graded Euler characteristic $q + q^{-1}$ is the unnormalized Jones polynomial of the unknot.
In general $\sum_{i,j} (-1)^i q^j \rank Kh^{i,j}(L) = (q + q^{-1}) V_L(t)$ at $t^{1/2} = -q$, where $V_L$ is the [[Jones polynomial|4SPC17FPXE]].
Non-example: the homology of the complex $\bigoplus_v C_v$ with the signs omitted is not defined, because without signs the faces of the cube commute and $d^2 \ne 0$.
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# Schur–Zassenhaus theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$ such that $\abs{N}$ and the [[index|0DTXBQPA2S]] $[G : N]$ are [[coprime|0VZ48N083F]].

1. $N$ has a complement in $G$: a subgroup $H$ with $NH = G$ and $N \cap H = \set{e}$. Hence $G$ is a [[semidirect product|0H3WZXCZNF]] $N \rtimes H$.
2. Any two complements of $N$ in $G$ are [[conjugate|0H11ZPX4ND]] in $G$.

## Remarks

In other words, every [[extension|MF4N3VGGFV]] of a group of order $m$ by a group of order $n$ with $\gcd(m, n) = 1$ splits.
When $N$ is abelian, the proof is cohomological.
The obstruction to a complement lies in $H^2(G/N, N)$, and the obstruction to conjugacy lies in $H^1(G/N, N)$.
Both cohomology groups are zero, because they are killed by both $\abs{N}$ and $[G : N]$.
The general case of part 1 reduces to the abelian case by induction, using a Sylow subgroup of $N$ and the Frattini argument.
For part 2 in general one needs $N$ or $G/N$ to be [[solvable|0FXJ7YFMMX]]; one of them has odd order, so this follows from the Feit–Thompson theorem.

Example: $G = S_3$ and $N = A_3$, with $\abs{N} = 3$ and $[G : N] = 2$.
Each subgroup of order $2$, such as $\set{e, (1\,2)}$, is a complement, and the three of them are conjugate.

The coprimality is necessary.
In $G = \Z/4\Z$ with $N = \set{0, 2}$, both $\abs{N}$ and $[G : N]$ are $2$, and $N$ has no complement, because the only subgroup of order $2$ is $N$ itself.

## Explanations

### L1 human

Suppose a finite group has a normal subgroup $N$, and the size of $N$ shares no prime factor with the number of its cosets.
Then the group splits cleanly: there is a second subgroup that meets $N$ only in the identity and fills up the rest.
So the group is built from $N$ and the quotient in the simplest possible way, as a semidirect product.

### L2 human

A normal subgroup $N$ with $\gcd(\abs{N}, [G : N]) = 1$ is a normal [[Hall subgroup|KPWW9X14V2]], so the theorem says that normal Hall subgroups have complements, unique up to conjugacy.
It is a key step in the proof of Hall's theorem on solvable groups and in many arguments with coprime actions.
Without coprimality, extensions may fail to split, and then one needs cohomology to describe them.
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# Validated numerics

## Definition

Let $y \in \R^n$ be a quantity defined by a mathematical problem, such as a zero of a function or the value of an integral.
A *validated* (or *verified*) computation of $y$ is an algorithm, run in floating-point arithmetic, with the following property.
It either reports failure or outputs a set $Y \subseteq \R^n$, usually a box of [[intervals|QPKKTE1ZY0]], such that $y \in Y$ is guaranteed.
The guarantee must account for every [[round-off error|CPN6SD3BTV]] and every [[truncation error|4AKCQERM89]] of the computation.
*Validated numerics* is the study and use of such algorithms.

## Remarks

Example: the interval Newton test.
Let $f(x) = x^2 - 2$ and $X = [1.4, 1.5]$; with midpoint $m = 1.45$ and the enclosure $[2.8, 3.0]$ of $f'$ on $X$, the interval Newton step is $N(X) = m - f(m)/[2.8, 3.0] = 1.45 - 0.1025/[2.8, 3.0] \subseteq [1.4133, 1.4159]$.
Since $N(X) \subseteq X$ and $0 \notin [2.8, 3.0]$, the interval Newton theorem shows that $f$ has exactly one zero in $X$, and it lies in $N(X)$.
Done with outward rounding, this proves $1.4133 \le \sqrt{2} \le 1.4159$.
Non-example: running Newton's method in floating point until two iterates agree to $15$ digits gives an excellent approximation, but no guarantee, so it is not a validated computation.
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# Quasi-identity

## Definition

Let $\sigma$ be an algebraic signature and let $X$ be a set of variables.
A *quasi-identity* of type $\sigma$ is an expression
$$s_1 \approx t_1 \wedge \cdots \wedge s_n \approx t_n \rightarrow s \approx t,$$
where $n \ge 0$ and $s_1, t_1, \ldots, s_n, t_n, s, t$ are [[terms|3YYKGWJJND]] of type $\sigma$ over $X$.
A $\sigma$-algebra $A$ *satisfies* this quasi-identity if every assignment $h \colon X \to A$ with $h(s_i) = h(t_i)$ for all $i \le n$ also has $h(s) = h(t)$.
Here $h(u)$ denotes the value of the term $u$ in $A$ under $h$.

## Remarks

A quasi-identity with $n = 0$ is an [[identity|PKD440G2XW]] $s \approx t$.

Example: the right cancellation law $x \cdot z \approx y \cdot z \rightarrow x \approx y$ is satisfied by every group.
Indeed, if $a z = b z$ in a group, then multiplying on the right by $z^{-1}$ gives $a = b$.

Non-example: the monoid $(\Z, \cdot, 1)$ does not satisfy this quasi-identity.
The assignment $x \mapsto 1$, $y \mapsto 2$, $z \mapsto 0$ satisfies the hypothesis $1 \cdot 0 = 2 \cdot 0$ but not the conclusion $1 = 2$.

In first-order terms, a quasi-identity stands for the universal sentence $\forall \bar x \, (s_1 = t_1 \wedge \cdots \wedge s_n = t_n \rightarrow s = t)$.
Another example is $x^2 \approx 1 \rightarrow x \approx 1$, which a group satisfies exactly when it has no element of order $2$.
This property cannot be expressed by identities, since the class of such groups is not closed under quotients: the additive group $\Z$ has it, and its quotient $\Z / 2\Z$ does not.
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<document source="https://tamnd.github.io/math2/node/64W1MJFRNT.md" tag="64W1MJFRNT" kind="definition">
# Codimension of a singularity

## Definition

Let $\mathcal{E}_n$ be the ring of [[germs|7SWSRGDT4A]] at $0$ of smooth functions $\R^n \to \R$, and let $\mathfrak{m}$ be its maximal [[ideal|096Z9MP2ME]] of germs vanishing at $0$.
Let $f \in \mathcal{E}_n$ have a [[critical point|0YTCMFB4HC]] at $0$, and let $J(f) \subseteq \mathfrak{m}$ be the ideal generated by $\partial f / \partial x_1, \ldots, \partial f / \partial x_n$.
The *codimension* of $f$ is
$$\operatorname{codim}(f) = \dim_\R \mathfrak{m} / J(f),$$
which may be infinite.

## Remarks

Example: for $f = x^4$ on $\R$, $J(f) = (x^3)$ and $\mathfrak{m} / J(f)$ has basis the classes of $x$ and $x^2$, so $\operatorname{codim}(f) = 2$.
For a nondegenerate critical point $J(f) = \mathfrak{m}$, so the codimension is $0$.
Non-example: the definition needs a critical point at $0$; for $f = x$ one has $J(f) = \mathcal{E}_1 \not\subseteq \mathfrak{m}$, and a regular point is not given a codimension in this sense.
When it is finite, $\operatorname{codim}(f) = \dim_\R \mathcal{E}_n / J(f) - 1$.
For a holomorphic germ, the same quotient of the ring of convergent power series has dimension the Milnor number $\mu(f)$.
It equals the least number of parameters of a [[versal unfolding|KSAFWZVB8K]] of $f$, not counting an additive constant.
Conventions vary between authors; some count the additive constant as well and use $\dim_\R \mathcal{E}_n / J(f)$.
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# Complete theory

## Summary

A complete theory is a satisfiable [[theory|92SWXST1ZM]] that decides every sentence of its language: it implies the sentence or its negation.
All models of a complete theory are [[elementarily equivalent|P3B1V182N4]].
The theory of a single structure is always complete, and proving that a list of axioms is complete is a central task of model theory.

## Explanations

### L1 human

A complete theory leaves no first-order question open.
The axioms of groups are not complete: they do not decide whether $x \cdot y = y \cdot x$ holds for all $x$ and $y$.
The axioms of dense linear orders without endpoints are complete: every first-order question about such an order has the same answer in $(\Q, <)$ and in $(\R, <)$.
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<document source="https://tamnd.github.io/math2/node/656Y8FA3RY.md" tag="656Y8FA3RY" kind="concept">
# Sampling distribution

## Summary

The sampling distribution of a [[statistic|0Q5CG2NEQ8]] is its probability distribution when the data are random.
It describes how the statistic would vary if the sampling were repeated many times.
Standard errors, confidence intervals and p-values of [[frequentist inference|YDSNHKM3EG]] are all computed from sampling distributions.

## Explanations

### L1 human

Take a sample of $50$ people and compute their average height.
Another sample of $50$ people would give a slightly different average.
The sampling distribution is the pattern of all these averages over all possible samples.
For averages it is centred at the true mean, and its spread shrinks like $1/\sqrt{n}$ as the sample size $n$ grows.
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# Inverse semigroup

## Definition

Let $S$ be a [[semigroup|0XMEM406JM]].
The semigroup $S$ is an *inverse semigroup* if for every $a \in S$ there is exactly one $b \in S$ with
$$aba = a \quad \text{and} \quad bab = b.$$
This $b$ is written $a^{-1}$.

## Remarks

Equivalently, $S$ is [[regular|XGJKF6T7FZ]] and any two [[idempotents|0N5NJZKKN3]] of $S$ commute.
In an inverse semigroup, $(a^{-1})^{-1} = a$ and $(ab)^{-1} = b^{-1} a^{-1}$.

Example: the symmetric inverse monoid $I_X$ on a set $X$, the set of all bijections $f \colon A \to B$ between subsets $A, B \subseteq X$.
The product $fg$ is the composite $f \circ g$, defined on the points $x$ with $x$ in the domain of $g$ and $g(x)$ in the domain of $f$.
For $f \colon A \to B$, the inverse bijection $f^{-1} \colon B \to A$ satisfies $f f^{-1} f = f$ and $f^{-1} f f^{-1} = f^{-1}$, and it is the only element that does.
Every group is also an inverse semigroup.

Non-example: the left zero semigroup $\set{e, f}$ with $xy = x$ for all $x, y$.
Here $exe = e$ and $xex = x$ for both $x = e$ and $x = f$, so $e$ has two such partners and uniqueness fails.
Accordingly, the idempotents $e$ and $f$ do not commute: $ef = e$ but $fe = f$.
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# Duality in projective geometry

## Definition

Let $\Pi = (P, L, I)$ be a [[projective plane|DANFNN8C4Y]], seen as an [[incidence structure|FGKNAFNHA3]].
Its *dual plane* is $\Pi^{*} = (L, P, I^{\top})$, whose points are the lines of $\Pi$, whose lines are the points of $\Pi$, and where $(\ell, p) \in I^{\top}$ exactly when $(p, \ell) \in I$.
The *dual* of a statement about points, lines and incidence is the statement obtained by exchanging "point" with "line" and "lies on" with "passes through".
A *duality* (or *correlation*) of $\Pi$ is an isomorphism of incidence structures from $\Pi$ to $\Pi^{*}$.

## Remarks

The axioms of a projective plane are self-dual, so $\Pi^{*}$ is again a projective plane.
Hence the *principle of duality*: if a statement holds in every projective plane, so does its dual.

Example: $\mathbb{P}^2(k)$ over a field $k$ has a duality sending the point $[a : b : c]$ to the line $a x_0 + b x_1 + c x_2 = 0$ and the line $a x_0 + b x_1 + c x_2 = 0$ to the point $[a : b : c]$.
Check: the point $[x_0 : x_1 : x_2]$ lies on the line with coefficients $(a, b, c)$ exactly when $a x_0 + b x_1 + c x_2 = 0$, a condition symmetric in the two triples, so incidence is preserved in both directions.

Non-example: the dual of an [[affine plane|EC7XT2ZM5S]] is not an affine plane.
Two parallel lines become two points with no line through both, which fails the first axiom of an affine plane.

Let $\mathbb{P}(V)$ be a projective space of finite dimension $n$.
The map $W \mapsto W^{0}$, sending a subspace to its annihilator in the [[dual space|07JBSZE0V4]], reverses inclusion and exchanges projective dimensions $d$ and $n - 1 - d$.
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<document source="https://tamnd.github.io/math2/node/65H4W6TYKF.md" tag="65H4W6TYKF" kind="definition">
# Butcher tableau

## Definition

Let $s \ge 1$ be an integer.
An *$s$-stage Butcher tableau* is an array
$$\begin{array}{c|c} c & A \\ \hline & b^{T} \end{array} \;=\; \begin{array}{c|ccc} c_1 & a_{11} & \cdots & a_{1s} \\ \vdots & \vdots & & \vdots \\ c_s & a_{s1} & \cdots & a_{ss} \\ \hline & b_1 & \cdots & b_s \end{array}$$
with real entries: a vector $c \in \R^s$, a matrix $A = (a_{ij}) \in \R^{s \times s}$ and a vector $b \in \R^s$.
It specifies the $s$-stage [[Runge–Kutta method|7DBPJKYC6C]] for $y' = f(t, y)$ with step $h$:
$$K_i = f\Big(t_n + c_i h, \; y_n + h \sum_{j=1}^{s} a_{ij} K_j\Big), \quad i = 1, \ldots, s, \qquad y_{n+1} = y_n + h \sum_{i=1}^{s} b_i K_i.$$

## Remarks

The method is explicit when $a_{ij} = 0$ for $j \ge i$, so the tableau is strictly lower triangular and blank entries are read as $0$.
Usually $c_i = \sum_j a_{ij}$ for each $i$.
An embedded pair adds a second row $\hat b^{T}$ of weights that gives a method of another order from the same stages.
Example: the classical fourth-order method has $c = (0, \tfrac12, \tfrac12, 1)^T$, $a_{21} = a_{32} = \tfrac12$, $a_{43} = 1$, all other $a_{ij} = 0$, and $b = (\tfrac16, \tfrac13, \tfrac13, \tfrac16)^T$.
Check of two order conditions: $\sum b_i = 1$ and $\sum b_i c_i = \frac16 + \frac16 + \frac16 = \frac12$.
Non-example: an array with $c \in \R^2$, $A \in \R^{2 \times 2}$ and $b \in \R^3$ is not a Butcher tableau, since the number of weights must equal the number of stages.

## Explanations

### L1 human

A Butcher tableau is a small table of numbers that encodes a Runge–Kutta method completely.
The left column says when in the step each stage is evaluated.
The square block says how earlier stages enter it, and the bottom row says how to average the stages.
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# Normed field

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
An *absolute value* on $K$ is a map $\abs{\cdot} \colon K \to \R$ such that:

1. $\abs{x} \ge 0$ for all $x \in K$, and $\abs{x} = 0$ if and only if $x = 0$.
2. $\abs{xy} = \abs{x} \abs{y}$ for all $x, y \in K$.
3. $\abs{x + y} \le \abs{x} + \abs{y}$ for all $x, y \in K$.

A *normed field* is a field $K$ together with an absolute value on $K$.
It is a metric space with the distance $d(x, y) = \abs{x - y}$.

## Remarks

Example: $\C$ with the modulus $\abs{a + bi} = \sqrt{a^2 + b^2}$ is a normed field; clause 2 is $\abs{zw} = \abs{z}\abs{w}$, and clause 3 is the triangle inequality in the plane.
Example: $\Q$ with the [[p-adic absolute value|WGN82TCEF2]] for a prime $p$ is a normed field.
Example: every field with the trivial absolute value, $\abs{x} = 1$ for $x \ne 0$, is a normed field.
Non-example: on $\R$, the map $x \mapsto x^2$ satisfies clauses 1 and 2 but fails clause 3, since $(1 + 1)^2 = 4 > 2 = 1^2 + 1^2$.
Clauses 1 and 2 give $\abs{1} = 1$ and $\abs{-x} = \abs{x}$.
An absolute value that satisfies the stronger inequality $\abs{x + y} \le \max(\abs{x}, \abs{y})$ is a [[non-Archimedean absolute value|1H5F1BP6XS]].
A normed field is a [[topological field|D7YQW9RG24]] for the topology of its metric.
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# Gelfand–Levitan theory

## Definition

Let $q \colon [0, \infty) \to \R$ be continuous, let $h \in \R$, and consider the [[Schrödinger operator|89AA6XJY5K]] $-y'' + q y$ on $[0, \infty)$ with the boundary condition $y'(0) = h y(0)$.
For $\lambda \in \C$, let $\varphi(x, \lambda)$ be the solution of $-y'' + q y = \lambda y$ with $\varphi(0, \lambda) = 1$ and $\varphi'(0, \lambda) = h$.
Let $\rho$ be the spectral function of the operator.
This is the nondecreasing function for which $f \mapsto \int_0^\infty f(x) \varphi(x, \lambda) \, dx$ extends to a unitary map from $L^2(0, \infty)$ onto $L^2(\R, d\rho)$.
Let $\rho_0(\lambda) = \frac{2}{\pi}\sqrt{\lambda}$ for $\lambda \ge 0$ and $\rho_0(\lambda) = 0$ for $\lambda < 0$; this is the spectral function for $q = 0$ and $h = 0$.
Set
$$F(x, t) = \int_{-\infty}^{\infty} \cos(\sqrt{\lambda}\, x) \cos(\sqrt{\lambda}\, t) \, d(\rho - \rho_0)(\lambda),$$
where the integral converges in a suitable regularised sense.
The Gelfand–Levitan equation is the linear integral equation for an unknown function $K(x, t)$:
$$K(x, t) + F(x, t) + \int_0^x K(x, s) F(s, t) \, ds = 0, \qquad 0 \le t < x.$$
Gelfand–Levitan theory is the method that recovers $q$ and $h$ from $\rho$ by solving this equation and setting
$$q(x) = 2 \frac{d}{dx} K(x, x), \qquad h = K(0, 0).$$

## Remarks

The solution $K$ is the transformation kernel with $\varphi(x, \lambda) = \cos(\sqrt{\lambda}\, x) + \int_0^x K(x, t) \cos(\sqrt{\lambda}\, t) \, dt$.
Example: for $q = 0$ and $h = 0$, one has $\rho = \rho_0$, so $F = 0$, the equation gives $K = 0$, and the formulas return $q = 0$ and $h = 0$.
Non-example: an arbitrary nondecreasing function $\rho$ need not be a spectral function of such an operator; Gelfand and Levitan gave conditions on $\rho$ that characterise spectral functions.
For problems on a finite interval $[0, \pi]$ the same method works with $\rho$ a step function whose jumps sit at the eigenvalues.

## Explanations

### L1 human

The Gelfand–Levitan equation turns spectral data into a potential.
One measures how the given spectral function differs from the one of the free equation, solves a linear integral equation, and reads $q$ off the diagonal of the solution.
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<document source="https://tamnd.github.io/math2/node/65PV4VFP9Y.md" tag="65PV4VFP9Y" kind="conjecture">
# Schanuel's conjecture

## Statement

Let $z_1, \dots, z_n$ be complex numbers that are [[linearly independent|0CRMGPAH83]] over $\Q$.
Then the field $\Q(z_1, \dots, z_n, e^{z_1}, \dots, e^{z_n})$ has transcendence degree at least $n$ over $\Q$.
Here $e^{z}$ is the [[complex exponential|07GJV36896]].
The transcendence degree of a field $K \supseteq \Q$ is the largest number of elements of $K$ that are algebraically independent over $\Q$.

## Remarks

The conjecture is open.
The case $n = 1$ is known: if $z \ne 0$, then $z$ or $e^z$ is [[transcendental|0VTQXTCR2G]], which is the Hermite–Lindemann theorem.
If $z_1, \dots, z_n$ are algebraic, the conjecture gives the [[Lindemann–Weierstrass theorem|0JNCBFMPXR]], which is proved.
With $z_1 = 1$ and $z_2 = i\pi$, it would imply that $e$ and $\pi$ are algebraically independent; it is not even known whether $e + \pi$ is irrational.
It would also imply the [[Gelfond–Schneider theorem|RN58Z2NWAC]] and [[Baker's theorem|WMBJSNG57F]], and even that $\Q$-linearly independent logarithms of algebraic numbers are algebraically independent.

## Explanations

### L1 human

Take numbers $z_1, \dots, z_n$ with no rational linear relation among them, together with their exponentials.
The conjecture says that among these $2n$ numbers, at least $n$ are transcendental and free of any polynomial relation.
It is a single statement that would contain most known transcendence results and many famous open ones.
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# Discrete-time Fourier transform

## Definition

Let $x = (x_n)_{n \in \Z}$ be an absolutely summable sequence of complex numbers, that is, $x \in \ell^1(\Z)$, a case of the [[$L^p$ spaces|SASP14MA1Z]].
The *discrete-time Fourier transform* of $x$ is the function $X \colon \R \to \C$ given by
$$X(\omega) = \sum_{n \in \Z} x_n\, e^{-i \omega n}, \qquad \omega \in \R,$$
where $e^{-i\omega n}$ is the [[complex exponential|07GJV36896]].

## Remarks

The series converges absolutely and uniformly in $\omega$, so $X$ is continuous and $2\pi$-periodic, with $\abs{X(\omega)} \le \sum_n \abs{x_n}$.
The sequence is recovered by
$$x_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\omega)\, e^{i \omega n}\, d\omega,$$
so $x_{-n}$ is the $n$-th coefficient of the [[Fourier series|G12KXDHPPS]] of $X$.

Example: for $\abs{a} < 1$, let $x_n = a^n$ for $n \ge 0$ and $x_n = 0$ for $n < 0$.
Then $\sum_n \abs{x_n} = 1/(1 - \abs{a}) < \infty$, and the geometric series gives $X(\omega) = \sum_{n \ge 0} (a e^{-i\omega})^n = \frac{1}{1 - a e^{-i\omega}}$.

Non-example: the constant sequence $x_n = 1$ is not absolutely summable, and $\sum_n e^{-i\omega n}$ diverges for every $\omega$ because its terms do not tend to $0$.
As a distribution its transform is a Dirac comb of period $2\pi$, scaled by $2\pi$.

By the Riesz–Fischer and Parseval theorems the transform extends to an isometry, up to the factor $2\pi$, from $\ell^2(\Z)$ onto $L^2$ of a period, with the series converging in $L^2$.
Some authors use the frequency $f = \omega / (2\pi)$, which gives period $1$.
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# Adjunction space

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], let $A$ be a [[closed|0ZCK8DQYN1]] subset of $Y$, and let $f \colon A \to X$ be a [[continuous map|0VGFHTM75Y]].
Let $\sim$ be the smallest [[equivalence relation|0G8VEZ3WTK]] on the [[disjoint union|ZXJKSB30WF]] $X \sqcup Y$ such that $a \sim f(a)$ for every $a \in A$.
The *adjunction space* is the quotient set
$$X \cup_f Y = (X \sqcup Y) / {\sim},$$
with the [[quotient topology|0NQ1H2X495]].
The map $f$ is called the attaching map.

## Remarks

The composite $X \to X \sqcup Y \to X \cup_f Y$ is a closed [[topological embedding|00ZG9Y9TR4]], and $Y \setminus A$ maps homeomorphically onto an open subset.
Many authors only ask that $A$ be a subspace; the closedness of $A$ is needed for these two facts.

Example: let $Y = D^2$ be the closed unit disc, $A = S^1$ its boundary circle, and $X = S^1$ with $f(z) = z^2$.
Then $f$ is continuous and $A$ is closed in $Y$, so $X \cup_f D^2$ is an adjunction space.
It is homeomorphic to the real projective plane.

Example: if $X$ is a one-point space, then $X \cup_f Y$ is the quotient $Y / A$ that collapses $A$ to a point.

Non-example: let $Y = [0, 1]$, let $A = (0, 1]$, let $X$ be a one-point space, and let $f$ be the constant map.
The clause that $A$ is closed fails, and the quotient is a two-point space whose open sets are $\varnothing$, $\set{[A]}$ and the whole space.
There the image of $X$ is the open point $[A]$, which is not closed.
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# Yang–Baxter equation

## Summary

The Yang–Baxter equation is the identity $R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$ for a linear map $R$ of $V \otimes V$.
The indices say on which two of three tensor factors $R$ acts.
It first appeared in exactly solvable models of statistical mechanics and in quantum scattering in one dimension.
Composing with the flip turns a solution into a solution of the braid relation.
So an invertible solution gives representations of the [[braid group|BNR4H5GCTP]].
The universal R-matrix of a [[quasitriangular Hopf algebra|NCRPWS9W5R]] satisfies it, and [[quantum groups|AP2ZDXANKC]] are a rich source of solutions.

## Explanations

### L1 human

Picture three particles on a line that pass each other, and let $R$ describe how two of them interact when they cross.
The three particles can be brought from one order to the reverse order in two different sequences of pairwise crossings.
The Yang–Baxter equation says that both sequences give the same result.
In pictures of braids, it says that a strand can slide across the crossing of two other strands.
The plain swap of tensor factors is the simplest solution.
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# Cardinal characteristics of the continuum

## Summary

Cardinal characteristics of the continuum are cardinals of a special kind.
Each is the least size of a family of reals, or of sets of reals, with some combinatorial property.
Each of them lies between $\aleph_1$ and $\mathfrak{c} = 2^{\aleph_0}$, the [[cardinality of the continuum|0WC28W9DNM]].
Under the [[continuum hypothesis|0MN2MER9WN]] they all equal $\aleph_1$, but without it they can take different values.
The provable inequalities between them form a rich theory.
Standard examples, such as the bounding and dominating numbers, are [[defined|R8S00S3Z94]] separately.

## Explanations

### L1 human

How many meagre sets do you need to cover the real line?
Countably many are never enough, by the Baire category theorem, and continuum many points, each one a meagre set, are always enough.
So the answer is some cardinal between $\aleph_1$ and $2^{\aleph_0}$.
Questions of this type define the cardinal characteristics.
If the continuum hypothesis holds there is nothing in between, but otherwise these numbers can differ, and comparing them is a large part of modern set theory.
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# Central polynomial

## Summary

A central polynomial for the matrix ring $M_n(k)$ is a noncommutative polynomial whose values on $n \times n$ matrices are always scalar matrices, but not always zero.
Their existence for every $n$ was proved independently by Formanek and by Razmyslov.
Central polynomials produce many central elements in [[polynomial identity rings|C0EDT81EGH]], and they are a main tool in the structure theory of these rings.

## Explanations

### L1 human

The values of most expressions in matrices, such as $xy - yx$, are ordinary matrices that are not multiples of the identity.
A central polynomial is a special expression whose value is always a multiple of the identity matrix, whatever matrices one substitutes, and is sometimes nonzero.
For $2 \times 2$ matrices, $(xy - yx)^2$ is such an expression.
For example, with $x = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ one gets $xy - yx = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, whose square is the identity matrix.
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# Euclidean distance matrix

## Definition

Let $n \ge 1$.
Let $D = (d_{ij})$ be a real $n \times n$ matrix.
It is a *Euclidean distance matrix* if some integer $k \ge 1$ and points $x_1, \ldots, x_n \in \R^k$ satisfy
$$d_{ij} = \norm{x_i - x_j}^2 \quad \text{for all } i, j,$$
where $\norm{x_i - x_j}$ is the [[Euclidean distance|0C95TG69SG]] of $x_i$ and $x_j$.
The points $x_1, \ldots, x_n$ are then said to realize $D$.

## Remarks

Every Euclidean distance matrix is symmetric, has zero diagonal and has nonnegative entries.

Example: the points $0, 1, 3 \in \R$ give
$$D = \begin{pmatrix} 0 & 1 & 9 \\ 1 & 0 & 4 \\ 9 & 4 & 0 \end{pmatrix},$$
since $1^2 = 1$, $3^2 = 9$ and $2^2 = 4$.

Non-example: the matrix with rows $(0, 1, 9)$, $(1, 0, 1)$ and $(9, 1, 0)$ is symmetric with zero diagonal, but it is not a Euclidean distance matrix.
Points realizing it would have distances $1$, $1$ and $3$, and $3 > 1 + 1$ violates the triangle inequality.

Some authors use the distances themselves instead of their squares.
</document>

<document source="https://tamnd.github.io/math2/node/66FQFAJNS4.md" tag="66FQFAJNS4" kind="concept">
# Algebraic torus

## Summary

An algebraic torus is a [[linear algebraic group|GA81VVQHEP]] that becomes a power of the multiplicative group $\mathbb{G}_m$ over an algebraic closure of the base field.
Over an algebraically closed field it is a group of invertible diagonal matrices.
Maximal tori control the structure of [[reductive groups|2XS84PWE3S]] through their characters and roots.
Tori also give [[toric varieties|YWB981MEF2]].

## Explanations

### L1 human

The simplest torus is the set of nonzero numbers under multiplication.
A torus of dimension $n$ is the set of $n$-tuples of nonzero numbers, multiplied coordinatewise, or equivalently the invertible diagonal $n \times n$ matrices.
Over the real numbers there are also twisted tori, such as the circle $x^2 + y^2 = 1$ with the multiplication of complex numbers $x + iy$.
Over $\C$ the circle becomes the same as $\C^{*}$ through $(x, y) \mapsto x + iy$.
</document>

<document source="https://tamnd.github.io/math2/node/66G7TEZ3JW.md" tag="66G7TEZ3JW" kind="definition">
# Energy functional on paths

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $a < b$.
For a continuous, piecewise smooth path $\omega \colon [a, b] \to M$, the *energy* of $\omega$ is
$$E(\omega) = \frac{1}{2} \int_a^b g\big(\dot\omega(t), \dot\omega(t)\big) \, dt.$$
For $p, q \in M$, the *energy functional on paths* from $p$ to $q$ is the function $E$ on the set of such paths with $\omega(a) = p$ and $\omega(b) = q$.

## Remarks

Some authors omit the factor $\frac{1}{2}$.
With the [[length|0TH78S7GBZ]] $L(\omega) = \int_a^b \abs{\dot\omega} \, dt$, the Cauchy–Schwarz inequality gives $L(\omega)^2 \le 2 (b - a) E(\omega)$, with equality if and only if $\omega$ has constant speed.
The critical points of $E$ among paths with fixed ends are the geodesics from $p$ to $q$ parametrized on $[a, b]$.
A minimizer of $E$ is a length-minimizing geodesic with constant speed.
To apply Morse theory one extends $E$ to paths of Sobolev class $H^1$, which form a Hilbert manifold.

Example: in $\R^n$ with the Euclidean metric, the path $\omega(t) = p + t(q - p)$ on $[0, 1]$ has $E(\omega) = \frac{1}{2} \abs{q - p}^2$, and $L(\omega)^2 = 2 E(\omega)$, so it has constant speed.

Non-example: the reparametrization $\sigma(t) = p + t^2 (q - p)$ of the same segment has $E(\sigma) = \frac{2}{3} \abs{q - p}^2 > \frac{1}{2} \abs{q - p}^2$.
It has the same length but is not a critical point of $E$, since its speed is not constant.
</document>

<document source="https://tamnd.github.io/math2/node/66NMTCQNQV.md" tag="66NMTCQNQV" kind="definition">
# Many-valued logic

## Definition

Fix a finite set $C$ of connectives, each with an arity.
A logical matrix for $C$ is a triple $(V, D, \set{f_c}_{c \in C})$ with these parts.

1. A set $V$ of truth values with at least three elements.
2. A nonempty proper subset $D \subseteq V$ of designated values.
3. For each $n$-ary connective $c$, a [[function|0EPAT51DXQ]] $f_c \colon V^n \to V$, its truth table.

A valuation assigns a value in $V$ to each propositional variable and extends to all formulas through the functions $f_c$.
A formula is a tautology of the matrix when it takes a value in $D$ under every valuation.
The many-valued logic of the matrix is the set of its tautologies.

## Remarks

With $V = \set{0, 1}$, $D = \set{1}$ and the usual tables, the tautologies are those of classical [[propositional logic|09K9MWD2QQ]]; clause 1 excludes this two-valued case.

Example: Łukasiewicz's three-valued logic $\mathrm{L}_3$ has $V = \set{0, \tfrac12, 1}$ and $D = \set{1}$.
Its tables are $\neg x = 1 - x$, $x \to y = \min(1, 1 - x + y)$, $x \lor y = \max(x, y)$ and $x \land y = \min(x, y)$.
The formula $p \to p$ is a tautology, since $\min(1, 1 - x + x) = 1$ for every $x$.

Non-example: $p \lor \neg p$ is a classical [[tautology|05006AJ7DZ]] but not a tautology of $\mathrm{L}_3$.
The valuation $p \mapsto \tfrac12$ gives $\max(\tfrac12, \tfrac12) = \tfrac12 \notin D$.

Infinite-valued logics use, for example, $V = [0, 1]$ with the same tables.

## Explanations

### L1 human

A many-valued logic replaces the two truth values by a larger set and gives a table for each connective on that set.
A formula is a law when it always lands among the designated values.
Some classical laws survive, and others, such as excluded middle in Łukasiewicz logic, do not.
</document>

<document source="https://tamnd.github.io/math2/node/672HNBV803.md" tag="672HNBV803" kind="concept">
# Reflexive operator algebra

## Summary

A reflexive operator algebra is an algebra of operators on a Hilbert space that is determined by its [[invariant subspaces|0ACX5QT8TE]].
It contains every operator that leaves all of them invariant.
Reflexivity links the algebraic structure of operator algebras with the lattice of their invariant subspaces.
Von Neumann algebras and nest algebras are reflexive, and the question of which algebras are reflexive is close to the [[invariant subspace problem|5RF055GK4N]].

## Explanations

### L1 human

Each set of operators has a family of subspaces that it maps into themselves.
Starting from these subspaces, one can collect all operators that respect them, and this collection contains the original algebra.
The algebra is reflexive when nothing new appears.
For example, the upper triangular $2 \times 2$ matrices are reflexive, while the matrices $\begin{pmatrix} a & b \\ 0 & a \end{pmatrix}$ are not, since they have the same invariant subspaces as all upper triangular matrices.
</document>

<document source="https://tamnd.github.io/math2/node/672SS33S5H.md" tag="672SS33S5H" kind="definition">
# Convex integrand

## Definition

Let $\Omega \subseteq \R^n$ be open and let $m \ge 1$.
Let $L \colon \Omega \times \R^m \times \R^{m \times n} \to \R$, written $L(x, u, \xi)$, be the integrand of an [[integral functional|JRQZ803Q5Y]] $J(u) = \int_\Omega L(x, u, \nabla u) \, dx$.
The integrand $L$ is *convex* (in the gradient variable) if for every $x \in \Omega$ and $u \in \R^m$ the map $\xi \mapsto L(x, u, \xi)$ is a [[convex function|07GR3TNW4H]] on $\R^{m \times n}$.

## Remarks

Example: $L(x, u, \xi) = \frac{1}{2} \abs{\xi}^2 - f(x) u$ is convex in $\xi$ for any function $f$.
Indeed $\xi \mapsto \frac{1}{2} \abs{\xi}^2$ is convex, and the term $-f(x) u$ does not depend on $\xi$.

Non-example: $L(\xi) = (\xi^2 - 1)^2$ with $n = m = 1$ is not convex.
Here $L(-1) = L(1) = 0$, but $L(0) = 1 > \frac{1}{2} L(-1) + \frac{1}{2} L(1)$.

A $C^2$ integrand is convex in $\xi$ exactly when its Hessian $\big(\partial^2 L / \partial \xi^i_\alpha \partial \xi^j_\beta\big)$ is positive semidefinite everywhere.
Some authors use the term for integrands that are jointly convex in $(u, \xi)$, which is a stronger condition.
For $m \ge 2$ and $n \ge 2$, convexity is stronger than polyconvexity, quasiconvexity and rank-one convexity.
</document>

<document source="https://tamnd.github.io/math2/node/675MAXBQN6.md" tag="675MAXBQN6" kind="concept">
# Index theory

## Summary

Index theory studies the [[Fredholm index|2V9Q4KHY1T]] of [[elliptic operators|XY8GEQSWPR]] on manifolds.
Its central result is the Atiyah–Singer index theorem.
It says that the [[analytic index|44GBFZ16SG]] of an elliptic operator on a closed manifold equals its [[topological index|QDCD33CS21]].
The topological index is an integer built from characteristic classes.
The theory links analysis, topology and geometry.
Classical results such as the [[Chern–Gauss–Bonnet theorem|GG6NYPKMHX]], the [[Hirzebruch signature theorem|TEG1ZX1AGJ]] and the [[Hirzebruch–Riemann–Roch theorem|G3X02FKQCR]] are special cases.

## Explanations

### L1 human

An elliptic equation $Pu = f$ on a closed manifold has finitely many independent solutions of $Pu = 0$ and finitely many independent obstructions to solving it.
The index is the first number minus the second.
Each number alone can jump when the operator changes a little, but the difference stays fixed.
Index theory explains this stability: the index depends only on the topology of the manifold and of the operator.
For example, on a closed surface of genus $g$ the operator $d + d^*$ from even to odd differential forms has index $2 - 2g$, the Euler characteristic.
</document>

<document source="https://tamnd.github.io/math2/node/676H7EGTKJ.md" tag="676H7EGTKJ" kind="concept">
# Substructure

## Summary

A substructure of a [[structure|983GZBQYT4]] is a subset of its domain that is closed under the functions and contains the constants.
Each symbol is interpreted on it by restriction.
It is the common form of subgroups, subrings and induced subgraphs.
Which subsets are substructures depends on the signature, not only on the structure.

## Explanations

### L1 human

Take a structure and keep only some of its elements.
If applying the operations to kept elements always gives kept elements, and the constants are kept, the kept part is again a structure of the same kind.
For example, the even integers with addition, negation and $0$ form a substructure of $(\Z, +, -, 0)$.
In a graph seen as a structure with one edge relation, any nonempty set of vertices with the edges between them is a substructure.
</document>

<document source="https://tamnd.github.io/math2/node/676K167XS8.md" tag="676K167XS8" kind="concept">
# Cyclic homology

## Summary

Cyclic homology is a homology theory for [[associative algebras|3CTW1P4HBP]] introduced by Alain Connes and, independently, by Boris Tsygan.
It refines [[Hochschild homology|S5H153HFWC]] by taking into account the cyclic rotation of tensor factors.
For smooth commutative algebras in characteristic $0$ it is computed from de Rham cohomology.
So it plays the role of de Rham theory in noncommutative geometry.
It is also the target of trace maps from [[algebraic K-theory|5GVFZWMWC8]].

## Explanations

### L1 human

Hochschild homology is built from strings $a_0 \otimes a_1 \otimes \cdots \otimes a_n$ of elements of an algebra.
Cyclic homology treats such a string as written around a circle, so that rotating it gives the same object up to a sign.
This extra symmetry makes cyclic homology closer to topology and to de Rham cohomology.
For the ground field $k$ itself, cyclic homology is $k$ in every even degree and $0$ in odd degrees.
Connes used it to define index-type invariants for noncommutative spaces.
</document>

<document source="https://tamnd.github.io/math2/node/67A4D4FAYJ.md" tag="67A4D4FAYJ" kind="statement">
# Iwasawa main conjecture

## Statement

Let $p$ be an odd prime.
Let $K_n = \Q(\zeta_{p^{n+1}})$ for $n \ge 0$ be the [[cyclotomic fields|NN77VYSGXR]] of $p$-power roots of unity, with union $K_\infty$.
Let $A_n$ be the $p$-part of the [[ideal class group|WQ46BD3ARY]] of $K_n$, and let $X = \varprojlim_n A_n$, the inverse limit under the norm maps.
Let $\Lambda = \Z_p[\![T]\!]$ be the Iwasawa algebra over the [[p-adic integers|V6TJAFZ4MD]], acting on $X$ through $\Gal(K_\infty/K_0)$ as in [[Iwasawa theory|Y2SPPSQ5D4]].
Let $\Delta = \Gal(K_0/\Q) \cong (\Z/p\Z)^\times$, and let $\omega \colon \Delta \to \Z_p^\times$ be the Teichmüller character.
For an integer $i$, let $X^{(i)}$ be the part of $X$ on which $\Delta$ acts by $\omega^i$.
Let $i$ be odd with $i \not\equiv 1 \pmod{p-1}$.
Then the $\Lambda$-module $X^{(i)}$ is finitely generated and torsion.
Its characteristic ideal is generated by the element of $\Lambda$ that interpolates the Kubota–Leopoldt $p$-adic $L$-function $L_p(s, \omega^{1-i})$.
Here the variable $T$ has its standard normalisation.

## Remarks

The $p$-adic $L$-function $L_p(s, \omega^{1-i})$ interpolates values at negative integers of [[Dirichlet L-functions|9D2Z543EJQ]].
These are attached to powers of $\omega$ viewed as [[Dirichlet characters|7XB5VJKDPG]], and the values are given by generalised [[Bernoulli numbers|0KC00K7KY4]].
So the statement links an algebraic object, built from class groups, to an analytic one, built from $L$-values.
Mazur and Wiles proved it using the geometry of modular curves, and Wiles later proved the analogue for totally real fields.
Rubin gave another proof using Kolyvagin's Euler systems of cyclotomic units.

Example of its content: by the structure theory of $\Lambda$-modules, $X^{(i)}$ is finite exactly when its characteristic ideal is all of $\Lambda$.
So $X^{(i)}$ is finite exactly when the interpolating power series has a constant term that is a $p$-adic unit.
When $p$ is a [[regular prime|B4VK33PCXZ]], $p$ does not divide the class number of $K_0$, and then all the $A_n$ are trivial.
Correspondingly, all the relevant power series are units of $\Lambda$.

## Explanations

### L1 human

The theorem says that two very different ways of measuring the same arithmetic agree.
One way counts ideal classes in the tower of fields generated by $p$-power roots of unity.
The other way uses $p$-adic versions of the special values of $L$-functions.
The agreement is exact, not only up to some error.

### L2 human

The result is a refinement of Kummer's criterion and of the Herbrand–Ribet theorem, which relate the $p$-divisibility of Bernoulli numbers to the class group of $\Q(\zeta_p)$.
Here the tower of all $p$-power cyclotomic fields is treated at once, with modules over $\Lambda$ in place of finite groups.
It is a model for many later main conjectures, for example for elliptic curves, where the class group is replaced by a Selmer group.
</document>

<document source="https://tamnd.github.io/math2/node/67RJ0S8F74.md" tag="67RJ0S8F74" kind="statement">
# Andreotti–Grauert theorem

## Statement

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of complex dimension $n$, let $1 \le q \le n$, and let $\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.

1. If $X$ is [[$q$-convex|VBHR4R5W7F]], then the [[sheaf cohomology|PQ6Q143ARJ]] group $H^r(X, \mathcal{F})$ is a finite-dimensional complex vector space for every $r \ge q$.
2. If $X$ is $q$-complete, then $H^r(X, \mathcal{F}) = 0$ for every $r \ge q$.

## Remarks

Here $q$-convexity is taken in the convention of Andreotti and Grauert: there is a smooth exhaustion function whose Levi form has at least $n - q + 1$ positive eigenvalues outside a compact set.
Andreotti and Grauert proved the theorem in 1962, for complex spaces as well as for manifolds.
The proof extends Grauert's bumping method.
The cohomology on a sublevel set $\set{\varphi < c}$ is compared with the cohomology on a slightly larger one.
Local vanishing results near strongly $q$-convex boundary points control the restriction maps in degrees $r \ge q$.
The $1$-complete manifolds are the Stein manifolds.
For them, part 2 says that $H^r(X, \mathcal{F}) = 0$ for $r \ge 1$, which is Cartan's theorem B.
For $q = 1$, part 1 gives Grauert's finiteness theorem for strongly pseudoconvex manifolds.
On $\C^2 \setminus \set{0}$, which is $2$-complete but not $1$-complete, the group $H^1(X, \mathcal{O})$ is infinite-dimensional, so the bound $r \ge q$ cannot be lowered.

## Explanations

### L1 human

The cohomology groups of a coherent sheaf measure the obstructions to solving holomorphic problems on a complex manifold, such as finding a function with prescribed local behaviour.
The theorem says that convexity of the manifold kills these obstructions in high degrees.
The more convex the manifold is, the more degrees are controlled: a Stein manifold has no obstructions at all in positive degree.

### L2 human

The theorem puts Cartan's theorem B and Grauert's finiteness theorem into one framework, indexed by $q$.
Applied to the sheaf of holomorphic $p$-forms and combined with the Dolbeault isomorphism, part 2 says that on a $q$-complete manifold every $\bar\partial$-closed smooth $(p, r)$-form with $r \ge q$ is $\bar\partial$-exact.
</document>

<document source="https://tamnd.github.io/math2/node/681HPCS6G6.md" tag="681HPCS6G6" kind="definition">
# Unisolvent point set

## Definition

Let $\Omega$ be a set and let $V$ be a [[vector space|000AYPA9AJ]] of functions $\Omega \to \R$ of finite [[dimension|03GSZ8RTZJ]] $N$.
Points $x_1, \dots, x_N \in \Omega$ are *unisolvent* for $V$ if for every $(y_1, \dots, y_N) \in \R^N$ there is exactly one $f \in V$ with $f(x_i) = y_i$ for $i = 1, \dots, N$.

## Remarks

Equivalently, the only $f \in V$ with $f(x_1) = \cdots = f(x_N) = 0$ is $f = 0$.
Equivalently, for a basis $\varphi_1, \dots, \varphi_N$ of $V$, the $N \times N$ matrix $(\varphi_j(x_i))$ is invertible.
The usual case is $V$ equal to the [[polynomials in $d$ variables|07GNX8GCYP]] of total degree at most $n$, where $N = \binom{n + d}{d}$.

Example: in one variable, the points $0, 1, 2$ are unisolvent for the polynomials of degree at most $2$.
A polynomial of degree at most $2$ with three roots is $0$, so the equivalent condition holds.

Non-example: for $V = \set{a + bx + cy : a, b, c \in \R}$ on $\R^2$, the points $(0, 0)$, $(1, 1)$, $(2, 2)$ are not unisolvent.
The nonzero function $x - y \in V$ vanishes at all three points.
</document>

<document source="https://tamnd.github.io/math2/node/681NY7506V.md" tag="681NY7506V" kind="definition">
# Global attractor

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $(S(t))_{t \ge 0}$ be a semigroup of continuous maps $S(t) \colon X \to X$, with $S(0) = \id$ and $S(t + s) = S(t) S(s)$.
For subsets $A, B \subseteq X$, let $\operatorname{dist}(B, A) = \sup_{b \in B} \inf_{a \in A} d(a, b)$.
A *global attractor* of the semigroup is a set $\mathcal{A} \subseteq X$ such that:

1. $\mathcal{A}$ is [[compact|00R0X1WGPX]];
2. $\mathcal{A}$ is [[invariant|WWZTW11664]]: $S(t) \mathcal{A} = \mathcal{A}$ for every $t \ge 0$;
3. $\mathcal{A}$ attracts bounded sets: $\operatorname{dist}(S(t) B, \mathcal{A}) \to 0$ as $t \to \infty$ for every bounded set $B \subseteq X$.

## Remarks

A global attractor, when it exists, is unique: it is the largest compact invariant set and the smallest closed set that attracts bounded sets.

Example: let $X = \R$ and let $S(t)$ be the flow of $u' = u - u^3$.
Every solution with $u(0) > 0$ tends to $1$, every solution with $u(0) < 0$ tends to $-1$, and $0$ is fixed.
The set $\mathcal{A} = [-1, 1]$ is compact, invariant since the flow maps $[-1, 1]$ onto itself, and attracts every bounded interval.

Non-example: for the same flow, the set $\set{-1, 1}$ is compact and invariant, but it does not attract the bounded set $\set{0}$, since $S(t) 0 = 0$ for all $t$.
So clause 3 fails.

A standard sufficient condition is that the semigroup has a bounded absorbing set and is asymptotically compact.
</document>

<document source="https://tamnd.github.io/math2/node/6837RS7N82.md" tag="6837RS7N82" kind="concept">
# Topos

## Summary

An (elementary) topos is a [[category|A2DE66937D]] with finite limits, [[exponential objects|RHWD8DY41R]] and a [[subobject classifier|NKP5TPQ79X]].
The category of sets is a topos, and so is the category of [[sheaves|RT8A8FJ566]] on any [[site|NYE0Z94MME]].
A topos behaves like a universe of sets in which one can do much of mathematics, but its internal logic is in general [[intuitionistic|Q11N8X99FK]].
Toposes connect geometry, where they arise as generalized spaces, with logic, where they serve as models of higher-order theories.

## Explanations

### L1 human

The category of sets has products, function sets and a set $\set{0, 1}$ of truth values that classifies subsets.
A topos is any category with these three features.
Sets on which a fixed group acts form a topos, as do sets that vary continuously over a space (sheaves).
In such a world most constructions with sets still work.
But the law of excluded middle may fail, because a statement can be true on part of a space and false elsewhere.
</document>

<document source="https://tamnd.github.io/math2/node/685ZV73JJ8.md" tag="685ZV73JJ8" kind="concept">
# Projective tensor product

## Summary

Let $X$ and $Y$ be Banach spaces.
Their projective tensor product is the completion of the algebraic [[tensor product|0BF2DE2SYK]] $X \otimes Y$ for the largest reasonable norm on it.
It turns bounded bilinear maps on $X \times Y$ into bounded linear maps, with the same norm.
Its dual is the space of bounded linear maps from $X$ to the dual of $Y$.
With the [[injective tensor product|RQXC5D3M1Y]] it is one of the two extreme ways to make $X \otimes Y$ a Banach space, as studied by Grothendieck.

## Explanations

### L1 human

A tensor $u$ in $X \otimes Y$ can be written in many ways as a sum of simple products $x_i \otimes y_i$.
Each way has a cost, the sum of the products of the lengths $\norm{x_i} \, \norm{y_i}$.
The projective norm of $u$ is the cheapest cost over all ways to write it.
For example, in $\ell^1 \otimes \ell^1$ this norm turns a tensor into a double sequence $(a_{mn})$ with norm $\sum_{m,n} \abs{a_{mn}}$.
</document>

<document source="https://tamnd.github.io/math2/node/68650624FF.md" tag="68650624FF" kind="counterexample">
# Telescope conjecture

## Counterexample

Fix a prime $p$ and $n \ge 1$.
Let $X$ be a finite $p$-local [[spectrum|1GHF4EKBXS]] of type $n$.
Let $v \colon \Sigma^d X \to X$ be a $v_n$ self-map, as given by the periodicity theorem among the [[Ravenel conjectures|B8XP2K7RS1]].
The telescope of $v$ is the homotopy colimit
$$v^{-1} X = \operatorname{hocolim}\left( X \xrightarrow{v} \Sigma^{-d} X \xrightarrow{v} \Sigma^{-2d} X \to \cdots \right).$$
The telescope conjecture asserted that the natural map $v^{-1} X \to L_{K(n)} X$ is an equivalence, where $L_{K(n)}$ is Bousfield localization at the $n$-th Morava K-theory.
The conjecture is false for every prime $p$ and every height $n \ge 2$.
In 2023 Burklund, Hahn, Levy and Schlank showed that for such $p$ and $n$ the map $v^{-1} X \to L_{K(n)} X$ is not an equivalence.

## Remarks

The telescope $v^{-1} X$ does not depend on the choice of $v$, and its Bousfield class depends only on $n$; the localization at it is written $L_{T(n)}$.
So the conjecture says $L_{T(n)} = L_{K(n)}$.
At height $1$ the conjecture is true: Mahowald proved it for $p = 2$ and Miller for $p$ odd.
The counterexample of Burklund, Hahn, Levy and Schlank uses algebraic K-theory to build $T(n)$-local spectra that are not $K(n)$-local.
It shows that the homotopy of telescopes is genuinely larger than what $K(n)$-local computations see.

## Explanations

### L1 human

A finite spectrum of height $n$ has a periodic self-map, and inverting it gives a "telescope".
Ravenel conjectured that the telescope is the same as the localization at Morava K-theory, which is much easier to compute.
This is true at heights $0$ and $1$.
At every height from $2$ on it is false, so the telescope holds information that Morava K-theory does not see.
</document>

<document source="https://tamnd.github.io/math2/node/687RGCHBTP.md" tag="687RGCHBTP" kind="statement">
# Jordan curve theorem

## Statement

Let $C \subset \R^2$ be a Jordan curve, that is, the image of a [[topological embedding|00ZG9Y9TR4]] $\gamma \colon S^1 \to \R^2$ of the [[circle|060W5CBZP6]] into the [[plane|0XPR1E85W0]].
Then $\R^2 \setminus C$ has exactly two [[connected components|0V032VWTHD]].
One component is bounded and the other is unbounded.
The curve $C$ is the boundary of each component.

## Remarks

For a polygon the theorem is elementary: the parity of the number of crossings of a ray with the polygon tells the two sides apart.
For a general curve, one proof approximates by polygons; another computes with homology, where Alexander duality gives $\tilde H_0(\R^2 \setminus C) \cong \Z$, so there are two components.
The theorem is hard because a Jordan curve can be very irregular, for example nowhere differentiable, or of positive area as an Osgood curve.
The bounded component is called the interior of $C$ and the unbounded one the exterior.
The same holds on the sphere $S^2$, without the words bounded and unbounded.

## Explanations

### L1 human

A closed loop drawn in the plane without crossing itself cuts the plane into an inside and an outside.
To go from a point inside to a point outside, a path must cross the loop.
This seems obvious, but the loop can be extremely wiggly, so the proof is subtle.

### L2 human

The theorem is the first case of the Jordan–Brouwer separation theorem: an embedded $(n-1)$-sphere in $\R^n$ separates it into two components.
Its strengthening in the plane, the Schoenflies theorem, says that the inside is a topological disk.
In higher dimensions this strengthening fails without extra hypotheses, as the Alexander horned sphere shows.
In complex analysis, the theorem gives meaning to "the inside of a contour" in Cauchy's integral theorem.
</document>

<document source="https://tamnd.github.io/math2/node/68AFPHR4GN.md" tag="68AFPHR4GN" kind="concept">
# One-relator group

## Summary

A one-relator group is a [[group|G5S1M9]] with a [[presentation|0JGEQXQ4AC]] that has a single defining relator.
Examples are $\Z^2$, the fundamental groups of closed surfaces, and the [[Baumslag–Solitar groups|429JJ5ZNBY]].
They are the natural next class after [[free groups|0KTW26C80J]], and they have a rich general theory.
Magnus proved the [[Freiheitssatz|B2TRERFWEH]] for them and showed that their [[word problem|PBGTTAVMMT]] is solvable.

## Explanations

### L1 human

Take a free group, where no rule holds between the generators, and impose exactly one rule.
For example, $\langle a, b \mid a b a^{-1} b^{-1} \rangle$ imposes $ab = ba$, and the result is the group $\Z^2$ of pairs of integers.
One rule is little, so these groups stay close to free groups.
Still they include many important groups, such as the fundamental groups of closed surfaces.
</document>

<document source="https://tamnd.github.io/math2/node/68E0F8N94E.md" tag="68E0F8N94E" kind="concept">
# Localization of a module

## Summary

Let $M$ be a [[module|0NZ8JA25WK]] over a [[commutative ring|0310REJJY8]] $R$, and let $S$ be a [[multiplicative set|AN9SKEZ3VY]].
The localization of $M$ at $S$ is the module $S^{-1}M$ of fractions $m/s$ with $m \in M$ and $s \in S$.
It is a module over the [[localization|4C1AE84N5N]] $S^{-1}R$.
Localization of modules preserves [[exact sequences|JB7ZBFBA9C]], so it is a basic tool to reduce statements about modules to the local case.

## Explanations

### L1 human

A module is like a vector space whose scalars come from a ring.
Localizing the module means allowing the same new denominators as in the ring.
Elements killed by some allowed denominator become zero.
For example, over $\Z$ the module $\Z/2\Z$ disappears once you may divide by $2$, because $1 = 2 \cdot (1/2)$ and $2 \cdot 1 = 0$ there.
The [[definition|S357810Z25]] gives the construction.
</document>

<document source="https://tamnd.github.io/math2/node/68FGQEBD6J.md" tag="68FGQEBD6J" kind="concept">
# Lie bracket of vector fields

## Summary

The Lie bracket of two vector fields $X$ and $Y$ on a smooth manifold is a third vector field $[X, Y]$.
It measures how much the flows of $X$ and $Y$ fail to commute.
It is zero exactly when, for short times, moving along $X$ then $Y$ gives the same result as moving along $Y$ then $X$.
With this bracket the vector fields on a manifold form a Lie algebra.
It appears in the Frobenius theorem, in the definition of curvature and torsion, and in control theory.

## Explanations

### L1 human

Think of a car: it can drive forward and it can turn the wheels, but it cannot slide sideways.
By alternating small forward and turning moves, you can still end up shifted sideways, as in parallel parking.
The sideways motion comes from Lie brackets of the motions you are allowed.
On the plane, the fields $\partial_x$ and $x\,\partial_y$ have bracket $\partial_y$, a direction neither field gives at the origin.
</document>

<document source="https://tamnd.github.io/math2/node/68MA0RM5P6.md" tag="68MA0RM5P6" kind="concept">
# Moore space

## Summary

A Moore space is a regular Hausdorff space with a development, that is, a sequence of open covers that shrink to each point.
Every [[metrizable space|091GJ3FYD7]] is a Moore space, and Moore spaces share many properties of metric spaces.
They are the setting of the [[normal Moore space problem|Z6T00E3JQH]], a central question of set-theoretic topology.

## Explanations

### L1 human

In a metric space, the balls of radius $1, 1/2, 1/3, \dots$ around a point get smaller and close in on the point.
A Moore space keeps this picture without a distance.
It has a sequence of covers by open sets, and the sets of the $n$-th cover that contain a point close in on it as $n$ grows.
The plane with the usual topology is an example, with the cover by all open discs of radius $1/n$ as the $n$-th cover.
The [[Moore plane|SAR7PBYF07]] is a Moore space that is not metrizable.
</document>

<document source="https://tamnd.github.io/math2/node/68P2VKSEP5.md" tag="68P2VKSEP5" kind="concept">
# q-gamma function

## Summary

The $q$-gamma function $\Gamma_q$ is a [[q-analog|GFSDMJHHVA]] of the [[gamma function|0H5SYBSED9]].
It is a quotient of two [[q-Pochhammer symbols|Y1X8ZZQ7VM]] and satisfies $\Gamma_q(x + 1) = [x]_q\, \Gamma_q(x)$, where $[x]_q = \frac{1 - q^x}{1 - q}$ is the $q$-number.
As $q \to 1^-$ it tends to $\Gamma(x)$.
It appears throughout the theory of basic hypergeometric series.

## Explanations

### L1 human

In $q$-analogs, a number $n$ is replaced by $[n]_q = 1 + q + \cdots + q^{n-1}$, which becomes $n$ again when $q = 1$.
The $q$-factorial is then $[1]_q [2]_q \cdots [n]_q$.
The $q$-gamma function fills in this $q$-factorial between the whole numbers, as the gamma function does for the factorial.
For example, $\Gamma_q(3) = [2]_q = 1 + q$, which is $2 = 2!$ when $q = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/68PFSGP39B.md" tag="68PFSGP39B" kind="concept">
# Retract

## Summary

A retract of an [[algebra|QJGPT2W0FT]] is a [[subalgebra|7CHG6SD365]] onto which the whole algebra can be mapped by a [[homomorphism|KBXSZC6BV1]] that fixes the subalgebra pointwise.
It is the universal-algebra form of a [[section and retraction|DXJH3DGXBC]] in a category.
Retracts are used to describe projective algebras: in a variety, the projective algebras are the retracts of free algebras.

## Explanations

### L1 human

Think of a subalgebra as a piece inside a bigger algebra.
It is a retract when the whole algebra can be folded onto the piece, respecting the operations, without moving the points of the piece.
In the group $\Z \times \Z$, the subgroup $\Z \times \set{0}$ is a retract: the map $(m, n) \mapsto (m, 0)$ folds everything onto it.
The subgroup $2\Z$ of $\Z$ is not a retract, since no homomorphism $\Z \to 2\Z$ fixes $2$.
</document>

<document source="https://tamnd.github.io/math2/node/68QZWDXAQ4.md" tag="68QZWDXAQ4" kind="example">
# Skolem's paradox

## Example

Assume that [[ZFC|0HE5WDQ9FJ]], written in first-order logic, is consistent.
Its signature has one binary relation symbol $\in$, so it is countable.
By the [[Löwenheim–Skolem theorem|484M0TZ1S7]], ZFC has a [[countable|01TQB88H4C]] [[model|MVCH6N9SNW]] $M$.
By [[Cantor's theorem|0WTEMGYV2C]], ZFC proves that the power set of $\omega$ is [[uncountable|0W7FBYS1F1]].
So $M$ contains an element $p$ that $M$ believes is uncountable: no element of $M$ is a bijection between $\omega^M$ and $p$.
But $M$ is countable, so the set of elements of $p$ in $M$ is countable when seen from outside.
There is no contradiction.
A bijection between $\omega^M$ and the members of $p$ exists outside $M$, but it is not an element of $M$.

## Remarks

The example shows that "uncountable" is not an absolute notion: it depends on which functions the model contains.
The same reasoning shows that no countable first-order theory with an infinite model can pin down the size of its models.
Thoralf Skolem pointed out this situation in the 1920s.

## Explanations

### L1 human

Set theory proves that some sets are uncountable, yet it has a model with only countably many objects.
This sounds impossible, but it is not.
Inside the small model, the counting functions that would list an uncountable set are simply missing.
From outside, one can list the set, but the model cannot see this list.
</document>

<document source="https://tamnd.github.io/math2/node/68RX8E3NQP.md" tag="68RX8E3NQP" kind="definition">
# Null set

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
A set $N \subseteq X$ is a *null set*, or a $\mu$-null set, if $N \in \Sigma$ and $\mu(N) = 0$.

## Remarks

For Lebesgue measure $\lambda$ on $\R$, every [[countable set|01TQB88H4C]] $\set{x_1, x_2, \ldots}$ is null.
Each point has measure $0$, since $\set{x} \subseteq [x - \varepsilon, x + \varepsilon]$ for every $\varepsilon > 0$, and countable additivity gives total $0$.
The [[Cantor set|0BXNDZFQME]] is also a Lebesgue null set, though it is uncountable.
The interval $[0, 1]$ is not a null set, since $\lambda([0, 1]) = 1$.
A countable union of null sets is null, by countable subadditivity.
Some authors call a set null when it is merely contained in a measurable set of measure $0$; the two notions agree when the measure is complete.
</document>

<document source="https://tamnd.github.io/math2/node/68SHB7Z721.md" tag="68SHB7Z721" kind="concept">
# Abel equation

## Summary

The Abel equation is the [[iterative functional equation|BCGBQ3HK26]] $\alpha(f(x)) = \alpha(x) + 1$, where $f$ is a given map and $\alpha$ is unknown.
A solution conjugates the map $f$ to the translation $t \mapsto t + 1$.
An injective solution gives a natural way to define non-integer iterates of $f$, by $f^t(x) = \alpha^{-1}(\alpha(x) + t)$.
It is closely related to [[Schröder's equation|VTJV7JVJEA]]: if $\psi$ solves $\psi(f(x)) = s \psi(x)$ with $s > 0$, $s \ne 1$ and $\psi > 0$, then $\alpha = \log_s \psi$ solves the Abel equation.

## Explanations

### L1 human

The Abel equation asks for a counter that goes up by exactly one each time the map $f$ is applied.
For the doubling map $f(x) = 2x$ on the positive numbers, $\alpha(x) = \log_2 x$ is such a counter.
In the new coordinate $\alpha$, applying $f$ becomes simply adding one.
Then half an application of $f$ is adding one half, which in this example gives $f^{1/2}(x) = \sqrt{2}\, x$.
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<document source="https://tamnd.github.io/math2/node/68SJQEC11F.md" tag="68SJQEC11F" kind="definition">
# Automorphic form

## Definition

Let $G$ be a connected semisimple [[Lie group|7JBRGCVJRB]] with finite center that is a closed [[subgroup|0X1JXT47GA]] of $\mathrm{GL}_n(\R)$.
Let $K$ be a maximal compact subgroup of $G$, and let $\mathfrak{g}$ be the complexified Lie algebra of $G$.
Let $Z(\mathfrak{g})$ be the center of its universal enveloping algebra.
It acts on smooth functions on $G$ as left-invariant differential operators.
Let $\Gamma$ be a discrete subgroup of $G$, in the main case an arithmetic subgroup.
For $g \in G$ let $\norm{g} = \max_{i,j} \max(\abs{g_{ij}}, \abs{(g^{-1})_{ij}})$.
An *automorphic form* for $\Gamma$ is a [[smooth function|0V33BNKZ9S]] $f \colon G \to \C$ such that:

1. ($\Gamma$-invariance) $f(\gamma g) = f(g)$ for all $\gamma \in \Gamma$ and $g \in G$.
2. ($K$-finiteness) The functions $g \mapsto f(gk)$ for $k \in K$ span a finite-dimensional vector space.
3. ($Z(\mathfrak{g})$-finiteness) There is an ideal $J$ of finite codimension in $Z(\mathfrak{g})$ with $Df = 0$ for all $D \in J$.
4. (Moderate growth) There are constants $C, N > 0$ with $\abs{f(g)} \le C \norm{g}^N$ for all $g \in G$.

## Remarks

Example: let $G = \mathrm{SL}_2(\R)$, $K = \mathrm{SO}(2)$, $\Gamma = \mathrm{SL}_2(\Z)$, and let $h$ be a [[modular form|GWKN5VG78A]] of weight $k$.
For $g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ put $f(g) = h(g \cdot i)\,(ci + d)^{-k}$, where $g \cdot i = \frac{ai + b}{ci + d}$.
The modularity of $h$ gives clause 1.
Right translation by the rotation $k_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ multiplies $f$ by $e^{-ik\theta}$, since $k_\theta \cdot i = i$ and $i \sin\theta + \cos\theta = e^{i\theta}$, which gives clause 2.
Holomorphy of $h$ makes the Casimir operator act on $f$ by a scalar, which gives clause 3.
The bound of $h$ at the cusp gives clause 4.

Non-example: on the same $G$, the function $f(g) = (\mathrm{Im}\, g \cdot i)^s$ satisfies clauses 2, 3 and 4, but it is not invariant under $\gamma = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \in \Gamma$, so clause 1 fails.

This is the definition of Harish-Chandra.
A classical variant works on a symmetric space $G/K$ and allows a factor of automorphy $j(\gamma, z)$ in clause 1, as for modular forms of weight $k$.
In the adelic setting, $G$ is a reductive group over a number field $F$ and automorphic forms are functions on $G(F) \backslash G(\mathbb{A}_F)$.
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# Gauss–Codazzi equations

## Statement

Let $M \subseteq \R^3$ be a [[regular surface|6080X3FAD4]] with a smooth unit normal field $N$ and [[shape operator|DQCD11X1VM]] $S$.
Let $I\!I(X, Y) = \ip{S X}{Y}$ be its [[second fundamental form|9KHQR16P1B]].
Let $\nabla$ be the [[Levi-Civita connection|CAZYGE1T4E]] of the induced metric on $M$, with [[Riemann curvature tensor|JA29ETCM02]] $R(X, Y) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$.
Then for all tangent vector fields $X, Y, Z, W$ on $M$:

1. (Gauss equation) $\ip{R(X, Y) Z}{W} = I\!I(Y, Z) \, I\!I(X, W) - I\!I(X, Z) \, I\!I(Y, W)$.
2. (Codazzi–Mainardi equation) $(\nabla_X S) Y = (\nabla_Y S) X$, where $(\nabla_X S) Y = \nabla_X (S Y) - S(\nabla_X Y)$.

## Remarks

Let $D$ be the flat connection of $\R^3$, given by directional derivatives.
The Gauss formula $D_X Y = \nabla_X Y + I\!I(X, Y) N$ and the Weingarten formula $D_X N = -S X$ split $D$ into tangent and normal parts.
Since $D$ has zero curvature, the tangent part of $D_X D_Y Z - D_Y D_X Z - D_{[X, Y]} Z = 0$ gives the Gauss equation, and the normal part gives the Codazzi–Mainardi equation.
Taking $X, Y$ orthonormal and $Z = Y$, $W = X$ in the Gauss equation gives $\ip{R(X, Y) Y}{X} = \det S = K$, the [[Gaussian curvature|SQVQ5DQT6W]].
Example: for the sphere of radius $r$ with the inner normal, $S = \frac{1}{r} \id$, so $(\nabla_X S) Y = \frac{1}{r} (\nabla_X Y - \nabla_X Y) = 0$ and both sides of the Codazzi equation vanish, while the Gauss equation gives sectional curvature $1/r^2$.

## Explanations

### L1 human

A surface in space has two kinds of data: how lengths are measured on it, and how it bends in space.
These cannot be chosen freely.
The Gauss–Codazzi equations are the compatibility conditions between them.

### L2 human

The Gauss equation contains the Theorema Egregium: the left side depends only on the metric, so $K = \det S$ does too.
Conversely, Bonnet's theorem says that on a simply connected domain, a metric and a symmetric form satisfying these equations come from a surface in $\R^3$, unique up to rigid motion.
This is the surface analogue of the fundamental theorem of space curves.
The same equations hold for hypersurfaces of $\R^{n+1}$, and with extra terms for submanifolds of a general Riemannian manifold.
</document>

<document source="https://tamnd.github.io/math2/node/68WXFBJEA6.md" tag="68WXFBJEA6" kind="definition">
# Direct product of algebras

## Definition

Let $I$ be a set and let $(\mathbf{A}_i)_{i \in I}$ be a family of [[algebras|QJGPT2W0FT]] of the same signature, with universes $A_i$.
The *direct product* $\prod_{i \in I} \mathbf{A}_i$ has as universe the [[Cartesian product|09CEDJYBKP]] $\prod_{i \in I} A_i$, the set of functions $a$ on $I$ with $a(i) \in A_i$ for each $i$.
Each $n$-ary fundamental operation $f$ is defined coordinatewise:
$$f(a_1, \dots, a_n)(i) = f^{\mathbf{A}_i}(a_1(i), \dots, a_n(i)) \quad \text{for all } i \in I.$$

## Remarks

For each $j \in I$ the projection $\pi_j \colon a \mapsto a(j)$ is a homomorphism onto $\mathbf{A}_j$ when the product is nonempty.
For two factors one writes $\mathbf{A}_1 \times \mathbf{A}_2$, with elements written as pairs.
The product of the empty family is a one-element algebra.

Example: the group $\Z/2\Z \times \Z/3\Z$ has six elements and is cyclic, generated by $(1, 1)$.
Coordinatewise, $k \cdot (1, 1) = (k \bmod 2, k \bmod 3)$, which is $(0, 0)$ first for $k = 6$.

Non-example: the group $\Z/4\Z$ is not isomorphic to the direct product $\Z/2\Z \times \Z/2\Z$, although both have four elements.
In the product, $x + x = 0$ for every $x$, since this holds in each coordinate, while $1 + 1 = 2 \ne 0$ in $\Z/4\Z$.

A direct product of algebras is the product in the category of algebras of the given signature, by the universal property of the projections.
</document>

<document source="https://tamnd.github.io/math2/node/69460Z6DMW.md" tag="69460Z6DMW" kind="definition">
# Dynkin system

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]].
A *Dynkin system* on $X$ is a family $\mathcal{D}$ of subsets of $X$ such that:

1. $X \in \mathcal{D}$.
2. If $A, B \in \mathcal{D}$ and $A \subseteq B$, then $B \setminus A \in \mathcal{D}$.
3. If $A_1 \subseteq A_2 \subseteq \cdots$ is an increasing sequence of members of $\mathcal{D}$, then the [[union|07KVB4JSC7]] $\bigcup_{n=1}^{\infty} A_n$ is in $\mathcal{D}$.

## Remarks

On $X = \set{1, 2, 3, 4}$, the family of subsets with an even number of elements is a Dynkin system.
Clause 1 holds since $X$ has $4$ elements.
Clause 2 holds since $B \setminus A$ has $\abs{B} - \abs{A}$ elements when $A \subseteq B$.
Clause 3 holds since an increasing sequence in a finite family is eventually constant.
On $X = \set{1, 2}$, the family $\set{\varnothing, \set{1}, X}$ is not a Dynkin system, since clause 2 fails: $X \setminus \set{1} = \set{2}$ is missing.
An equivalent definition: $X \in \mathcal{D}$, $\mathcal{D}$ is closed under [[complements|0B1FSS1VZW]], and $\mathcal{D}$ is closed under countable unions of pairwise disjoint members.
Every [[σ-algebra|04WSW8T3SY]] is a Dynkin system.
A Dynkin system that is also a [[π-system|4W9NM803BB]] is a σ-algebra.
Dynkin systems are also called λ-systems.
</document>

<document source="https://tamnd.github.io/math2/node/698R12MY4G.md" tag="698R12MY4G" kind="definition">
# Wirtinger derivatives

## Definition

Let $U \subseteq \C$ be an open set and write $z = x + iy$ with $x, y$ real.
Let $f \colon U \to \C$ be a function whose [[partial derivatives|05AXG3QRKR]] $\partial f / \partial x$ and $\partial f / \partial y$ exist at a point $z \in U$.
The *Wirtinger derivatives* of $f$ at $z$ are
$$\frac{\partial f}{\partial z} = \frac{1}{2} \left( \frac{\partial f}{\partial x} - i \frac{\partial f}{\partial y} \right), \qquad \frac{\partial f}{\partial \bar z} = \frac{1}{2} \left( \frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y} \right).$$

## Remarks

Example: for $f(z) = \abs{z}^2 = x^2 + y^2$, one gets $\partial f / \partial z = \frac{1}{2}(2x - 2iy) = \bar z$ and $\partial f / \partial \bar z = \frac{1}{2}(2x + 2iy) = z$.
Non-example of holomorphy: since $\partial f / \partial \bar z = z \neq 0$ for $z \neq 0$, the function $\abs{z}^2$ is not [[holomorphic|0DG0Z72CQD]] on any open set.
For $f$ of class $C^1$ on $U$, the equation $\partial f / \partial \bar z = 0$ on $U$ is equivalent to the [[Cauchy–Riemann equations|0T71SSYZ39]], so it holds exactly when $f$ is holomorphic on $U$; then $\partial f / \partial z = f'$.
The operators obey the rules $\partial z / \partial z = 1$, $\partial \bar z / \partial z = 0$, $\partial z / \partial \bar z = 0$ and $\partial \bar z / \partial \bar z = 1$, together with linearity and the product rule.
The Laplacian is $\Delta f = 4 \, \dfrac{\partial^2 f}{\partial z \, \partial \bar z}$ for $f$ of class $C^2$.
The notations $\partial_z f$ and $\partial_{\bar z} f$ are also common.
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<document source="https://tamnd.github.io/math2/node/69HK75YAD1.md" tag="69HK75YAD1" kind="concept">
# Mori domain

## Summary

A Mori domain is an [[integral domain|09S3PFBEPM]] that satisfies the ascending chain condition on divisorial ideals.
Every Noetherian domain and every [[Krull domain|CB9WJCN9BM]] is a Mori domain.
A Mori domain is a Krull domain exactly when it is completely integrally closed.
So Mori domains relate to Krull domains as Noetherian domains relate to Noetherian integrally closed domains.

## Explanations

### L1 human

Divisorial ideals are the ideals that are determined by which principal ideals contain them.
In a Mori domain one cannot build an infinite strictly increasing chain of such ideals.
Principal ideals are divisorial, so in particular there is no infinite strictly increasing chain of principal ideals, and every nonzero nonunit factors into irreducibles.
The ring of all algebraic integers is not a Mori domain: the principal ideals generated by $2^{1/2}, 2^{1/4}, 2^{1/8}, \ldots$ form a strictly increasing chain.
</document>

<document source="https://tamnd.github.io/math2/node/69R5R2FCS5.md" tag="69R5R2FCS5" kind="concept">
# Regular sequence

## Summary

A regular sequence is a list of ring elements in which each element is a non-zero-divisor modulo the earlier ones.
It is the algebraic version of a set of independent equations, like the coordinates $x_1, \ldots, x_n$ of a polynomial ring.
Regular sequences define the [[depth|M2RADBFW99]] of a module, and the [[Koszul complex|YZY11W08SD]] of a regular sequence is a free resolution.

## Explanations

### L1 human

Cut a space by one equation, then cut the result by a second equation, and so on.
The sequence of equations is regular when each new equation genuinely cuts down what is left, and never vanishes on a whole piece of it.
In $K[x, y]$ the sequence $x, y$ is regular: $x$ is a non-zero-divisor in $K[x,y]$, and $y$ is a non-zero-divisor in $K[x,y]/(x) \cong K[y]$.
The sequence $x, xy$ is not regular, because $xy$ is already zero modulo $x$.
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<document source="https://tamnd.github.io/math2/node/69XMERM6KJ.md" tag="69XMERM6KJ" kind="concept">
# Legendre functions

## Summary

Legendre functions are the solutions of the general Legendre differential equation, which has two parameters, a degree $\nu$ and an order $\mu$.
The solutions of the first kind, $P_\nu^\mu$, and of the second kind, $Q_\nu^\mu$, are the standard basis of solutions.
When $\mu = 0$ and $\nu$ is a nonnegative integer, the first kind gives the [[Legendre polynomials|QFYGJVA996]].
They arise when the Laplace equation or the Helmholtz equation is separated in spherical coordinates.

## Explanations

### L1 human

Legendre polynomials solve [[Legendre's equation|W6VE30V10Y]] only for special integer values of a parameter.
If the parameter is any number, the equation still has solutions, but they are no longer polynomials.
These solutions are the Legendre functions.
For example, $Q_0(x) = \frac12 \ln \frac{1 + x}{1 - x}$ on $(-1, 1)$ solves the equation with parameter $0$, and it blows up at both ends of the interval.
</document>

<document source="https://tamnd.github.io/math2/node/6AJ7QQCTQQ.md" tag="6AJ7QQCTQQ" kind="concept">
# Ordinal sum of posets

## Summary

The ordinal sum of two partially ordered sets places the second one entirely above the first.
Inside each part the order is unchanged, and every element of the first part is below every element of the second.
It is also called the linear sum or series composition, and it extends the addition of [[ordinal numbers|0YSCSX5YZC]].

## Explanations

### L1 human

Stack two posets on top of each other, like two floors of a building.
Everything on the lower floor is below everything on the upper floor.
Stacking a single point on top of a two-element antichain gives a "V" shape turned upside down, with the point as the top.
The order of stacking matters: a point below the antichain gives a different poset.
</document>

<document source="https://tamnd.github.io/math2/node/6ANP035KG8.md" tag="6ANP035KG8" kind="definition">
# Self-injective ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
The ring $R$ is *left self-injective* if $R$, viewed as a left [[module|0NZ8JA25WK]] over itself by left multiplication, is an [[injective module|85H7PRHDPZ]].
It is *right self-injective* if $R$ is injective as a right module over itself, and *self-injective* if it is both.

## Remarks

Example: $R = \Z/n\Z$ for $n \ge 1$ is self-injective.
By Baer's criterion it suffices to extend maps $f$ from an ideal $d\Z/n\Z$, with $d \mid n$, to $R$.
The image $a = f(d)$ satisfies $(n/d) a = 0$, so $a = db$ for some $b \in R$, and $x \mapsto bx$ extends $f$.
Non-example: $\Z$ is not self-injective.
The map $2\Z \to \Z$, $2m \mapsto m$, has no extension $g \colon \Z \to \Z$, since $2 g(1) = 1$ has no solution in $\Z$.
Left and right self-injectivity are different conditions in general.
A ring is [[quasi-Frobenius|KP1PP7A6JR]] if and only if it is left Noetherian and left self-injective.
</document>

<document source="https://tamnd.github.io/math2/node/6AT64EQEJQ.md" tag="6AT64EQEJQ" kind="definition">
# Free product of algebras

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras and let $(A_i)_{i \in I}$ be a family of algebras in $\mathcal{V}$.
A *free product* of the family in $\mathcal{V}$ is an algebra $A$ in $\mathcal{V}$ with [[homomorphisms|KBXSZC6BV1]] $\iota_i \colon A_i \to A$ such that the following holds.
For every algebra $B$ in $\mathcal{V}$ and all homomorphisms $f_i \colon A_i \to B$, there is a unique homomorphism $f \colon A \to B$ with $f \circ \iota_i = f_i$ for all $i \in I$.

## Remarks

The free product exists for every family in a variety, and it is unique up to a unique isomorphism compatible with the maps $\iota_i$.
It is the coproduct of the family in the category of algebras of $\mathcal{V}$.
Some authors require in addition that every $\iota_i$ be injective.

Example: in the variety of abelian groups, $A_1 \times A_2$ with $\iota_1(a) = (a, 0)$ and $\iota_2(b) = (0, b)$ is a free product.
Given $f_1, f_2$ into an abelian group $B$, the map $f(a, b) = f_1(a) + f_2(b)$ is a homomorphism because $B$ is abelian, it satisfies $f \circ \iota_k = f_k$, and it is the only such map since $(a, b) = \iota_1(a) + \iota_2(b)$.

Non-example: in the variety of all groups, $\Z \times \Z$ with the two coordinate inclusions is not a free product of $\Z$ and $\Z$.
Take $f_1, f_2 \colon \Z \to S_3$ sending $1$ to the transpositions $(1\,2)$ and $(2\,3)$.
A homomorphism $f$ with $f \circ \iota_k = f_k$ would send the commuting elements $(1, 0)$ and $(0, 1)$ to these two transpositions, which do not commute.

In the variety of groups, the free product of a family is the group of reduced words in the factors.
</document>

<document source="https://tamnd.github.io/math2/node/6AW49BZ8TM.md" tag="6AW49BZ8TM" kind="concept">
# Koszul duality

## Summary

Koszul duality pairs a [[Koszul algebra|FTKFCCZ8YA]] $A$ with its quadratic dual $A^!$, which is again Koszul and computes the Ext algebra of the trivial $A$-module.
The basic pair is the symmetric algebra of a vector space and the exterior algebra of its dual.
Under suitable finiteness conditions it gives an equivalence between [[derived categories|5ST7NVJFSM]] of graded modules over $A$ and $A^!$.
The same pattern appears in representation theory, algebraic topology and the theory of operads.

## Explanations

### L1 human

Koszul duality is a way to trade one algebra for another, simpler or more familiar one, without losing homological information.
The polynomial ring $k[x_1, \dots, x_n]$, where the variables commute, is paired with the exterior algebra, where the variables anticommute.
The relations of one algebra are, roughly, the orthogonal complement of the relations of the other.
Problems about modules over one side can be translated into problems about modules over the other side.
For instance, the Bernstein–Gelfand–Gelfand correspondence relates sheaves on projective space with modules over an exterior algebra.
</document>

<document source="https://tamnd.github.io/math2/node/6AZ4ZBN1M8.md" tag="6AZ4ZBN1M8" kind="concept">
# Confluent hypergeometric function

## Summary

A confluent hypergeometric function is a solution of [[Kummer's equation|NT917KFR57]] $z w'' + (b - z) w' - a w = 0$.
The main one is Kummer's function $M(a, b, z) = {}_1F_1(a; b; z)$, the [[generalized hypergeometric series|VDBAW9PN88]] with one upper and one lower parameter.
It is an [[entire function|0G4F0XB9X2]] of $z$.
Its special cases include the exponential function, the [[Laguerre polynomials|WAMRHWW82R]], the [[error function|7Y56QCZSZJ]] and the [[incomplete gamma function|YB7ZV0HG7K]].

## Explanations

### L1 human

Kummer's function is a power series in which each coefficient is obtained from the previous one by a simple rational factor.
The coefficient of $z^n$ is $\frac{a (a + 1) \cdots (a + n - 1)}{b (b + 1) \cdots (b + n - 1)} \cdot \frac{1}{n!}$.
When $a = b$ the two products cancel and the series is $\sum z^n / n! = e^z$.
So Kummer's function is a two-parameter deformation of the exponential function.
It appears whenever a physical problem leads to a second order equation with one regular and one irregular singular point, as for the hydrogen atom.
</document>

<document source="https://tamnd.github.io/math2/node/6B05T6XHY0.md" tag="6B05T6XHY0" kind="definition">
# Simple random sample

## Definition

Let $U = \set{1, \dots, N}$ be a finite population of $N$ units, and let $1 \le n \le N$.
Let $S \subseteq U$ be a random subset, defined on a [[probability space|0NFPJVJJHX]].
Then $S$ is a *simple random sample* of size $n$ from $U$ if for every subset $s \subseteq U$ with $\abs{s} = n$,
$$P(S = s) = \binom{N}{n}^{-1}.$$
Here $\binom{N}{n}$ is the [[binomial coefficient|0GBZM0MT0F]], the number of [[combinations|098PHYBTV9]] of $n$ units out of $N$.

## Remarks

Equivalently, one draws $n$ units one after the other, each uniformly at random among the units not yet drawn.
Each unit is in $S$ with probability $n/N$.
For values $y_1, \dots, y_N$ attached to the units, the sample mean $\bar y_S = \frac{1}{n}\sum_{i \in S} y_i$ is unbiased for the population mean $\bar Y$.
Its variance is $\big(1 - \frac{n}{N}\big) \frac{S_y^2}{n}$, where $S_y^2 = \frac{1}{N-1} \sum_{i=1}^{N} (y_i - \bar Y)^2$.
Sampling *with replacement*, where $n$ independent uniform draws are made, is a different design.

Example: $N = 4$ and $n = 2$, with $S$ uniform on the $\binom{4}{2} = 6$ pairs.
Each pair has probability $1/6$, as required, and unit $1$ lies in $3$ of the pairs, so it is chosen with probability $1/2 = n/N$.

Non-example: with $N = 4$, choose $r \in \set{1, 2}$ uniformly and take $S = \set{r, r + 2}$.
Every unit is chosen with probability $1/2$, but the pair $\set{1, 2}$ has probability $0$ instead of $1/6$.
So equal inclusion probabilities are not enough for a simple random sample.
</document>

<document source="https://tamnd.github.io/math2/node/6B1PSZ42MC.md" tag="6B1PSZ42MC" kind="concept">
# Directly indecomposable algebra

## Summary

A directly indecomposable algebra is a nontrivial [[algebra|QJGPT2W0FT]] that is not isomorphic to a [[direct product|WJ73XYBNE4]] of two nontrivial algebras.
Every finite algebra is a direct product of finitely many directly indecomposable ones.
The notion is weaker than [[subdirect irreducibility|DASF939439]], which forbids also the more flexible subdirect decompositions.

## Explanations

### L1 human

Some algebras split into two independent parts, like $\Z/6\Z$, which behaves as $\Z/2\Z$ and $\Z/3\Z$ side by side.
A directly indecomposable algebra admits no such splitting.
Any algebra with a prime number of elements is directly indecomposable, because a product of two nontrivial algebras has a composite number of elements.
The ring $\Z$ is directly indecomposable too, even though it is not subdirectly irreducible.
</document>

<document source="https://tamnd.github.io/math2/node/6BB0KD6752.md" tag="6BB0KD6752" kind="concept">
# Birkhoff's axioms

## Summary

Birkhoff's axioms are four postulates for plane [[Euclidean geometry|KRZD96GSVC]] that George David Birkhoff gave in 1932.
They take the [[real numbers|08J2M1JYAG]] as known and measure distances with a ruler and angles with a protractor.
Because the real numbers do much of the work, the list is very short, and it became the basis of many school geometry courses.

## Explanations

### L1 human

Hilbert's axioms build geometry from scratch, which makes the list long.
Birkhoff instead allows a ruler and a protractor: every line is a copy of the number line, and every angle has a size in degrees.
With these tools and one axiom about similar triangles, all of plane geometry follows.
For example, the angle sum of a triangle is a short consequence of the similarity axiom.
</document>

<document source="https://tamnd.github.io/math2/node/6BJB149MDK.md" tag="6BJB149MDK" kind="statement">
# Oka's solution of the Levi problem

## Statement

Let $\Omega \subseteq \C^n$ be a domain.
The following are equivalent:

1. $\Omega$ is a [[domain of holomorphy|4487CSC0MB]].
2. $\Omega$ is [[pseudoconvex|AHYZMSBBNH]].
3. $\Omega$ has a continuous [[plurisubharmonic exhaustion function|FGSKR6ZS3Q]].

## Remarks

The implication from 1 to 3 is the easy one.
Let $d(z)$ be the distance from $z$ to $\partial\Omega$; on a domain of holomorphy $\Omega \ne \C^n$, the function $-\log d$ is [[plurisubharmonic|FF2P7ER3QT]].
Then $\max(-\log d(z), \abs{z}^2)$ is a continuous plurisubharmonic exhaustion function.
The hard implication, from 3 to 1, is the solution of the [[Levi problem|47CVS9RJ9D]].
For each boundary point one needs a holomorphic function on $\Omega$ that cannot be continued past it.
Oka built such functions with his own methods; his proof for general $n$ used his theory of coherent ideal sheaves.
Hörmander later obtained them by solving $\bar\partial u = f$ with $L^2$ estimates for weights $e^{-k\varphi}$, where $\varphi$ is the exhaustion function.
Example: any convex domain $\Omega$ is pseudoconvex, so it is a domain of holomorphy.
For convex domains this can also be seen directly with the functions $1/\ell$.
Here $\ell$ is an affine complex-linear function that vanishes at a given boundary point and nowhere on $\Omega$.
The same equivalence holds for unramified Riemann domains over $\C^n$.

## Explanations

### L1 human

A domain of holomorphy is a domain that is the natural home of some holomorphic function, one that cannot be pushed past any boundary point.
Pseudoconvexity is a convexity condition on the shape of the domain.
The theorem says that the two notions are the same.
So an analytic question, whether every holomorphic function extends, becomes a geometric question about the boundary.

### L2 human

The theorem completes the [[Cartan–Thullen theorem|1XCBCB3MD7]], which says that domains of holomorphy are the holomorphically convex domains.
Together they give four equivalent conditions on a domain in $\C^n$: domain of holomorphy, holomorphically convex, pseudoconvex, and Stein.
For domains with $C^2$ boundary, pseudoconvexity can be checked with the [[Levi form|80MTNXWW80]], so this gives a computable test.
Grauert's theorem extends the result from domains to complex manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/6BQ71KQ6QT.md" tag="6BQ71KQ6QT" kind="definition">
# Kripke semantics

## Definition

A Kripke frame is a pair $(W, R)$ of a nonempty set $W$ of worlds and a [[binary relation|05AVARPXA2]] $R$ on $W$, the accessibility relation.
A Kripke model is a triple $\mathfrak{M} = (W, R, V)$ where $V$ maps each propositional variable to a subset of $W$, an element of the [[power set|00E0YZCERH]].
Truth of a [[modal formula|C3RZY20ETK]] at a world $w \in W$ is defined by induction.

1. $\mathfrak{M}, w \models p$ if and only if $w \in V(p)$, and $\mathfrak{M}, w \not\models \bot$.
2. The connectives $\neg$, $\land$, $\lor$ and $\to$ are evaluated at $w$ as in classical logic.
3. $\mathfrak{M}, w \models \square\varphi$ if and only if $\mathfrak{M}, v \models \varphi$ for every $v \in W$ with $w R v$.

A formula is valid on the frame $(W, R)$ when it is true at every world of every model $(W, R, V)$.

## Remarks

Then $\mathfrak{M}, w \models \lozenge\varphi$ if and only if some $v$ with $w R v$ has $\mathfrak{M}, v \models \varphi$.

Example: on any frame, the axiom $\square(p \to q) \to (\square p \to \square q)$ is valid.
At a world $w$ where $\square(p \to q)$ and $\square p$ hold, every $v$ with $w R v$ satisfies $p \to q$ and $p$, hence $q$.

Non-example: $\square p \to p$ is not valid on the frame $(\set{w}, \emptyset)$.
With $V(p) = \emptyset$, the formula $\square p$ holds at $w$ because no world is accessible, while $p$ fails at $w$.

The formulas valid on all frames form the logic $\mathrm{K}$.
The formulas valid on all reflexive frames form $\mathrm{T}$, and those valid on all reflexive and transitive frames form $\mathrm{S4}$.

For intuitionistic logic, the relation is a preorder, $V(p)$ is closed upward, and $\varphi \to \psi$ holds at $w$ when $\psi$ holds at every $v \ge w$ where $\varphi$ holds.

## Explanations

### L1 human

Kripke semantics reads "necessarily" as "in every world I can see".
Different conditions on the arrows between worlds give different laws of necessity.
For instance, if every world sees itself, then what is necessary is true.
</document>

<document source="https://tamnd.github.io/math2/node/6BQJ50NEJ0.md" tag="6BQJ50NEJ0" kind="concept">
# Building

## Summary

A building is a [[simplicial complex|V8GZ0W1ZZ5]] glued together from copies of a Coxeter complex, called apartments, under strong compatibility rules.
Jacques Tits introduced buildings to give a geometric picture of reductive algebraic groups and of the [[groups of Lie type|NPEBABQ49M]].
Each group with a BN-pair acts on a building, and the [[Bruhat decomposition|T91E86D0RH]] becomes a statement about the relative position of two chambers.
Buildings of $p$-adic groups are spaces of nonpositive curvature, used much like symmetric spaces for real Lie groups.

## Explanations

### L1 human

Think of a building as a large structure made of identical flat floors, the apartments.
Any two rooms of the structure lie together on at least one floor.
Two floors that share two rooms can be laid on top of each other without moving those rooms.
The simplest infinite example is a [[tree|062D9KXJB1]] in which every vertex has at least three neighbours.
Its apartments are the infinite straight lines through the tree.
</document>

<document source="https://tamnd.github.io/math2/node/6BS0300CMV.md" tag="6BS0300CMV" kind="concept">
# F-coalgebra

## Summary

For an endofunctor $F$ of a category, an $F$-coalgebra is an object $A$ with a morphism $A \to F(A)$.
It is the dual of an $F$-algebra.
Coalgebras model state-based systems such as automata, streams and transition systems.
The map describes what one can observe of a state and where it goes next.
Terminal coalgebras give infinite data such as streams, and the uniqueness of maps into them gives proofs by coinduction.

## Explanations

### L1 human

Think of $A$ as a set of states of a machine.
The coalgebra map tells, for each state, what the machine shows and what state comes next.
For example, with $F(X) = \N \times X$ on sets, take $A = \N$ and send $n$ to $(n, n + 1)$.
Starting at $0$, the machine shows $0, 1, 2, \dots$ forever, which is an infinite stream.
</document>

<document source="https://tamnd.github.io/math2/node/6C1S8HD38Z.md" tag="6C1S8HD38Z" kind="example">
# Shannon wavelet

## Example

Write $\operatorname{sinc}(t) = \sin(\pi t)/(\pi t)$ for $t \ne 0$ and $\operatorname{sinc}(0) = 1$.
Use the [[Fourier transform|DPVM34ADZV]] $\hat f(\omega) = \int_{\R} f(t) e^{-i\omega t} \, dt$.
The real *Shannon wavelet* is
$$\psi(t) = 2\operatorname{sinc}(2t) - \operatorname{sinc}(t) = \operatorname{sinc}\left(\frac{t}{2}\right) \cos\left(\frac{3\pi t}{2}\right).$$
Its Fourier transform is the indicator function of the set $\set{\omega : \pi < \abs{\omega} \le 2\pi}$, up to a set of measure zero.

Check.
The transform of $\operatorname{sinc}$ is the indicator of $[-\pi, \pi]$.
The transform of $2\operatorname{sinc}(2t)$ is the indicator of $[-2\pi, 2\pi]$.
Their difference is the indicator of the two bands.
The second formula follows from $\sin(2\pi t) - \sin(\pi t) = 2\cos(3\pi t/2)\sin(\pi t/2)$.

## Remarks

The functions $\psi_{j,k}(t) = 2^{j/2}\psi(2^j t - k)$, $j, k \in \Z$, form an orthonormal basis of $L^2(\R)$.
The reason is that the two bands, shifted by multiples of $2\pi$, tile the line, and their dyadic dilates tile $\R \setminus \set{0}$.
The scaling function of the underlying [[multiresolution analysis|1GHQMBR7DE]] is $\varphi = \operatorname{sinc}$, and its approximation spaces are spaces of band-limited functions.
The Shannon wavelet is perfectly localized in frequency but poorly localized in time: $\abs{\psi(t)}$ decays only like $1/\abs{t}$, so $\psi$ is not integrable.
The [[Meyer wavelet|Q5RN2FNX6Z]] smooths the sharp edges of $\hat\psi$ to gain fast decay in time.

## Explanations

### L1 human

The Shannon wavelet is an ideal band-pass filter turned into a function.
It keeps exactly the frequencies between $\pi$ and $2\pi$ in absolute value and nothing else.
Its stretched copies split the frequency line into octaves with no overlap.
The price for these sharp cuts is a function that ripples on for a long time before it fades.
</document>

<document source="https://tamnd.github.io/math2/node/6C2MCVB8C7.md" tag="6C2MCVB8C7" kind="concept">
# Jordan algebra

## Summary

A Jordan algebra is a commutative [[nonassociative|YP5TQ6WFVN]] algebra whose product satisfies the Jordan identity $(xy)(xx) = x(y(xx))$.
The model is the symmetrized product $x \circ y = \frac{1}{2}(xy + yx)$ on an associative algebra, for example on Hermitian matrices.
Algebras obtained this way are special; the others are exceptional, and the Albert algebra of $3 \times 3$ Hermitian octonion matrices is the basic exceptional one.
Jordan algebras were introduced by Pascual Jordan to formalize observables in quantum mechanics, and they are linked to Lie algebras, symmetric spaces and projective geometry.

## Explanations

### L1 human

The product of two Hermitian matrices need not be Hermitian, but the average $\frac{1}{2}(AB + BA)$ is.
This symmetrized product is commutative but not associative.
It still satisfies a weak form of associativity, the Jordan identity.
A Jordan algebra is any algebra with a commutative product and this identity.
</document>

<document source="https://tamnd.github.io/math2/node/6C67F22FTD.md" tag="6C67F22FTD" kind="statement">
# Frisch–Waugh–Lovell theorem

## Statement

Let $y \in \R^n$, let $X_1$ be a real $n \times p_1$ matrix and let $X_2$ be a real $n \times p_2$ matrix.
Assume that the block matrix $X = [X_1 \; X_2]$ has [[rank|0ZC9YTXZRY]] $p_1 + p_2$.
Let $\hat\beta_1, \hat\beta_2$ be the [[ordinary least squares|YM708KHV2W]] coefficients of the regression of $y$ on $X$, and let $e = y - X_1 \hat\beta_1 - X_2 \hat\beta_2$ be its residual vector.
Let $M_1 = I_n - X_1 (X_1^T X_1)^{-1} X_1^T$, the [[orthogonal projection|0B8KZRT28M]] onto the orthogonal complement of the column space of $X_1$.
Then:

1. $\hat\beta_2 = (X_2^T M_1 X_2)^{-1} X_2^T M_1 y$, that is, $\hat\beta_2$ is the least squares coefficient of the regression of $M_1 y$ on $M_1 X_2$.
2. The residual vector of that regression equals $e$.

## Remarks

Multiply $y = X_1 \hat\beta_1 + X_2 \hat\beta_2 + e$ by $M_1$.
Since $M_1 X_1 = 0$ and $X_1^T e = 0$, this gives $M_1 y = M_1 X_2 \hat\beta_2 + e$.
Since $X_2^T e = 0$ and $M_1 e = e$, the vector $e$ is orthogonal to the columns of $M_1 X_2$.
So the last equation is the least squares decomposition of $M_1 y$ on $M_1 X_2$.
The matrix $X_2^T M_1 X_2$ is invertible because $M_1$ is symmetric and [[idempotent|0KCE8CG86P]] and $M_1 X_2$ has rank $p_2$ by the rank hypothesis.

Example: take $X_1$ the column of ones.
Then $M_1$ subtracts the mean.
The theorem says that the slopes of a regression with intercept equal the slopes of the regression of the centred response on the centred regressors.

## Explanations

### L1 human

In a regression with many variables, the coefficient of one group of variables can be found in two steps.
First remove from the response and from that group everything that the other variables explain.
Then regress what is left of the response on what is left of the group.
The answer is the same coefficient as in the full regression.

### L2 human

The theorem explains the phrase "the effect of $X_2$ holding $X_1$ fixed": a coefficient in a multiple regression is the slope after partialling out the other regressors.
It justifies centring data before a fit with intercept.
In econometrics it justifies removing fixed effects or trends before estimating the coefficients of interest.
It is a purely algebraic fact about projections, so it holds whatever the distribution of the errors.
</document>

<document source="https://tamnd.github.io/math2/node/6CHHCTGQ18.md" tag="6CHHCTGQ18" kind="concept">
# False discovery rate and the Benjamini–Hochberg procedure

## Summary

The false discovery rate of a multiple testing procedure is the expected proportion of false rejections among all rejections.
Controlling it is less strict than controlling the [[family-wise error rate|GQCY1N8JZT]], so it allows more discoveries when many hypotheses are tested.
The Benjamini–Hochberg procedure is a simple step-up rule on the ordered [[p-values|0GGKJGZF6D]] that controls the false discovery rate under independence.

## Explanations

### L1 human

In a study with thousands of tests, such as gene screens, a few false alarms are acceptable if most discoveries are real.
The false discovery rate is the average fraction of false alarms in the list of discoveries.
The Benjamini–Hochberg procedure sorts the p-values and compares the $i$-th smallest with $i q/m$.
It rejects all hypotheses up to the last p-value below its threshold.
With $q = 0.1$, on average at most about one discovery in ten is false.
</document>

<document source="https://tamnd.github.io/math2/node/6CR9A0EWPN.md" tag="6CR9A0EWPN" kind="concept">
# Cancellation property

## Summary

An element $a$ has the cancellation property for a [[binary operation|0M418FE7BB]] when it can be removed from both sides of an equation $a \ast b = a \ast c$ or $b \ast a = c \ast a$.
A structure in which every element is cancellable is called cancellative.
Every [[group|G5S1M9]] is cancellative by the [[cancellation laws|0VSNX4P2Q0]], and the nonzero elements of an [[integral domain|0QVD7WBFGN]] are cancellable for multiplication.
Cancellative commutative [[semigroups|0XMEM406JM]] are exactly those that embed in a group.

## Explanations

### L1 human

Cancelling means: from $a + b = a + c$, conclude $b = c$.
This works for addition of natural numbers even though there is no subtraction inside the natural numbers.
It fails for multiplication by $0$: $0 \cdot 1 = 0 \cdot 2$, yet $1 \ne 2$.
So $0$ is not cancellable for multiplication, while every nonzero integer is.
</document>

<document source="https://tamnd.github.io/math2/node/6CW61T5BB2.md" tag="6CW61T5BB2" kind="definition">
# Rogers polynomials

## Definition

Let $q, \beta \in \C$ with $\abs{q} < 1$ and $\abs{\beta} < 1$, and let $x = \cos\theta$.
The *Rogers polynomials* $C_n(x; \beta \mid q)$, $n \ge 0$, are the coefficients of the [[power series|0MZB06ST2B]] in $t$
$$\sum_{n=0}^{\infty} C_n(x; \beta \mid q)\, t^n = \frac{(\beta t e^{i\theta}; q)_\infty\, (\beta t e^{-i\theta}; q)_\infty}{(t e^{i\theta}; q)_\infty\, (t e^{-i\theta}; q)_\infty}, \qquad \abs{t} < 1,$$
where $(\cdot\,; q)_\infty$ is the infinite [[q-Pochhammer symbol|Y1X8ZZQ7VM]].

## Remarks

Expanding each factor by the $q$-binomial theorem gives the explicit form
$$C_n(x; \beta \mid q) = \sum_{k=0}^{n} \frac{(\beta; q)_k\, (\beta; q)_{n-k}}{(q; q)_k\, (q; q)_{n-k}}\, e^{i(n - 2k)\theta}.$$
The terms for $k$ and $n - k$ pair to $2\cos((n - 2k)\theta)$ times the same coefficient.
So $C_n$ is a polynomial in $x$ of degree $n$, with leading coefficient $2^n \frac{(\beta; q)_n}{(q; q)_n} \ne 0$.

Example: $\beta = q$.
Then $\frac{(q; q)_k (q; q)_{n-k}}{(q; q)_k (q; q)_{n-k}} = 1$, so $C_n(x; q \mid q) = \sum_{k=0}^{n} e^{i(n-2k)\theta} = \frac{\sin((n+1)\theta)}{\sin\theta}$, the Chebyshev polynomial $U_n(x)$ of the [[second kind|7CATGEN6KH]].

Non-example: $\beta = 1$ fails the condition $\abs{\beta} < 1$.
There the right side of the generating function is $1$, so all coefficients of $t^n$ with $n \ge 1$ vanish, and there is no polynomial of degree $n$.

For $\beta = 0$, $C_n(x; 0 \mid q) = \frac{H_n(x \mid q)}{(q; q)_n}$, with the [[continuous q-Hermite polynomials|XFKVZJ55Q5]].
For $0 < q < 1$ and $-1 < \beta < 1$, the $C_n$ are orthogonal on $[-1, 1]$ for the weight $\frac{(e^{2i\theta}, e^{-2i\theta}; q)_\infty}{(\beta e^{2i\theta}, \beta e^{-2i\theta}; q)_\infty}$ with respect to $d\theta$, $x = \cos\theta$.
As $q \to 1$, $C_n(x; q^\lambda \mid q) \to C_n^{(\lambda)}(x)$, the [[Gegenbauer polynomial|5TRFA6F79B]].
</document>

<document source="https://tamnd.github.io/math2/node/6CXPGB13WD.md" tag="6CXPGB13WD" kind="concept">
# Mapping cone

## Summary

The mapping cone of a [[chain map|HTK9WQZJ1G]] $f \colon C_\bullet \to D_\bullet$ is a [[chain complex|WQ82C0M3QB]] $\mathrm{cone}(f)$.
It is built from $D$ and a shifted copy of $C$, with a differential that uses $f$.
It measures the failure of $f$ to be a [[quasi-isomorphism|C9JEC2WQKV]]: $f$ is one exactly when the cone has zero homology.
The cone fits into a short exact sequence of complexes, and in the [[homotopy category|9Y5Y92R4D5]] and the [[derived category|5ST7NVJFSM]] it gives the [[distinguished triangles|ADKK56VZHW]].

## Explanations

### L1 human

The cone glues the source of $f$, moved up one degree, onto the target of $f$.
In topology, the mapping cone of a map of spaces $X \to Y$ is formed by attaching the cone on $X$ to $Y$, and its chains are this algebraic cone.
The homology of the cone sits in a long exact sequence between the homologies of $C$ and $D$, with the maps $H_n(f)$ as connecting maps.
So the cone is acyclic exactly when $f$ induces isomorphisms on homology.
For example, the cone of multiplication by $2$ on $\Z$ in degree $0$ is $0 \to \Z \xrightarrow{2} \Z \to 0$, whose homology $\Z/2\Z$ is the cokernel of $f$.
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# Almost everywhere

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]], and let $P(x)$ be a statement about the points $x \in X$.
The statement $P$ holds *$\mu$-almost everywhere* if there is a [[null set|VQQM4Y550T]] $N \in \Sigma$ such that $P(x)$ is true for every $x \in X \setminus N$.

## Remarks

Let $\lambda$ be Lebesgue measure on $\R$ and let $f(x) = 1$ for rational $x$ and $f(x) = 0$ otherwise.
Then $f = 0$ holds $\lambda$-almost everywhere: take $N = \Q$, a countable set, so $\lambda(\Q) = 0$.
The statement $x \ne 0 \Rightarrow x < 0$ does not hold $\lambda$-almost everywhere, since it fails on $(0, \infty)$, and every measurable set containing $(0, \infty)$ has measure $\infty$.
The set where $P$ fails is only required to lie inside a null set; it need not be measurable itself.
The usual abbreviation is a.e., or $\mu$-a.e.
On a probability space, almost everywhere is called [[almost surely|0ST21PGDAP]].
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# Rough path

## Definition

Let $T > 0$, let $d \ge 1$, and let $\alpha \in (1/3, 1/2]$.
For a path $X \colon [0, T] \to \R^d$ write $X_{s,t} = X_t - X_s$.
An *$\alpha$-Hölder rough path* on $[0, T]$ with values in $\R^d$ is a pair $(X, \mathbb{X})$ of a path $X \colon [0, T] \to \R^d$ and a map $\mathbb{X} \colon [0, T]^2 \to \R^d \otimes \R^d$ such that:

1. (Chen's relation) $\mathbb{X}_{s,t} - \mathbb{X}_{s,u} - \mathbb{X}_{u,t} = X_{s,u} \otimes X_{u,t}$ for all $s, u, t \in [0, T]$;
2. $\sup_{s \ne t} \abs{X_{s,t}} / \abs{t - s}^{\alpha} < \infty$;
3. $\sup_{s \ne t} \abs{\mathbb{X}_{s,t}} / \abs{t - s}^{2\alpha} < \infty$.

Here $\otimes$ is the [[tensor product|0BF2DE2SYK]], and $\R^d \otimes \R^d$ is identified with the $d \times d$ matrices.

## Remarks

The map $\mathbb{X}_{s,t}$ stands for the iterated integral $\int_s^t X_{s,r} \otimes dX_r$.
Clause 2 says that $X$ is [[Hölder continuous|012PBA0FMD]] with exponent $\alpha$.
Example: for a continuously differentiable path $X$, set $\mathbb{X}_{s,t} = \int_s^t X_{s,r} \otimes \dot{X}_r \, dr$.
Splitting $\int_s^t$ at $u$ and writing $X_{s,r} = X_{s,u} + X_{u,r}$ for $r \ge u$ gives Chen's relation, and clauses 2 and 3 hold with exponents $1$ and $2$.
Example: for a [[Brownian motion|22735FGJV6]] $B$, the [[Itô integrals|JA95NH206E]] $\mathbb{B}_{s,t} = \int_s^t B_{s,r} \otimes dB_r$ give a rough path almost surely, for each $\alpha \in (1/3, 1/2)$.
Non-example: a smooth path $X$ with $\mathbb{X} = 0$ fails Chen's relation when $X_{s,u} \otimes X_{u,t} \ne 0$.
The second level is not determined by $X$: if $g \colon [0, T] \to \R^d \otimes \R^d$ is $2\alpha$-Hölder, then $\mathbb{X}_{s,t} + g_t - g_s$ also satisfies the three clauses.
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# Right-angled Artin group

## Summary

A right-angled Artin group is a group given by a finite simple graph, with one generator for each vertex.
Two generators commute whenever their vertices are joined by an edge, and there are no other relations.
These groups interpolate between [[free groups|0KTW26C80J]], for graphs with no edges, and free abelian groups $\Z^n$, for complete graphs.
They are the [[Artin groups|SAWBDBHE41]] in which every relation is a commutation.
They are [[CAT(0) groups|XB9QTY77ZA]], and they play a central role in the study of 3-manifolds and of special cube complexes.

## Explanations

### L1 human

Choose some letters, and decide for each pair whether the two letters may be swapped when they stand next to each other in a word.
Draw a graph with one vertex per letter and an edge for each swappable pair.
The right-angled Artin group of this graph is the group of words up to these swaps and cancellations of a letter with its inverse.
If no letters commute you get a free group; if all of them commute you get $\Z^n$.
For a path $a - b - c$, the letter $b$ commutes with everything, and the group is $F_2 \times \Z$.
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# Term

## Summary

A term is an expression of a first-order language that names an object.
Terms are built from variables and constant symbols by applying function symbols of the [[signature|WDVVEYD6AT]].
In a [[structure|983GZBQYT4]], once the variables have values, each term has a value, which is an element of the structure.
Terms play the role of nouns in logic, while formulas play the role of statements.

## Explanations

### L1 human

In the language of arithmetic, $x + 1$ and $(x \cdot y) + 0$ are terms.
They stand for numbers once you say what $x$ and $y$ are.
For example, with $x = 3$ the term $x + 1$ has value $4$.
An expression such as $x < y$ is not a term: it does not name a number but makes a claim.
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<document source="https://tamnd.github.io/math2/node/6D4EW91AJY.md" tag="6D4EW91AJY" kind="statement">
# Capacitability of Borel sets

## Statement

Let $X$ be a [[Polish space|584GXXY73M]] and let $I$ be a [[Choquet capacity|25FRWWHFZV]] on $X$.
Let $A \subseteq X$ be a Borel set, that is, a member of the [[Borel sigma-algebra|0WQXY32Q8D]] of $X$.
Then $A$ is capacitable for $I$:
$$I(A) = \sup \set{ I(K) : K \subseteq A, \ K \text{ compact} }.$$

## Remarks

The statement is a special case of Choquet's capacitability theorem, which gives the same conclusion for every [[analytic set|3RWWZYPAFV]] of $X$; every Borel subset of a Polish space is analytic.
Proof idea: an analytic set is the image of the space $\N^\N$ under a continuous map.
One uses the continuity of $I$ along increasing sequences to choose, coordinate by coordinate, a closed piece of $\N^\N$ whose image still has capacity close to $I(A)$.
The image of this piece is contained in a decreasing intersection of compact sets, and continuity along decreasing sequences of [[compact sets|0R6EWFKRCF]] finishes the proof.

Example: in $\R^n$ with $n \ge 3$, the outer Newtonian capacity is a Choquet capacity.
So for each Borel set $A$, inner and outer capacity agree: the capacity of $A$ can be computed from compact subsets or from open supersets with the same result.

The hypothesis on the set is needed in general.
Under the axiom of choice, some subsets of $\R$ have positive Lebesgue outer measure but contain no compact set of positive measure.
Lebesgue outer measure restricted to $[0, 1]$ is a Choquet capacity, so such sets are not capacitable.

## Explanations

### L1 human

A capacity measures size, but unlike a measure it is not additive, so it is not clear that the size of a set can be found from its compact pieces.
Choquet's theorem says that this works for every reasonable set: the size of a Borel set is the supremum of the sizes of the compact sets inside it.
This makes capacities usable in the same way as measures.

### L2 human

In potential theory the theorem shows that every Borel set has a well-defined capacity, given equally by compact subsets and by open supersets.
So statements such as "a Borel set of capacity zero is polar" do not depend on which of the two definitions is used.
In probability the same theorem, applied to the capacity given by the probability that a process hits a set, gives the measurability of hitting times of Borel sets.
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# Zak transform

## Definition

Let $f$ be a function in the [[Schwartz space|CG9JEATW3Q]] $\mathcal{S}(\R)$.
The *Zak transform* of $f$ is the function $Zf \colon \R^2 \to \C$,
$$Zf(x, \omega) = \sum_{k \in \Z} f(x + k) \, e^{-2\pi i k \omega}.$$
For general $f \in L^2(\R)$ (see [[Lp space|SASP14MA1Z]]), $Zf$ is defined by continuous extension, with the series converging in $L^2([0, 1)^2)$.

## Remarks

For fixed $x$, $Zf(x, \cdot)$ is a [[Fourier series|G12KXDHPPS]] with coefficients $f(x + k)$.
The map $Z$ extends to a unitary operator from $L^2(\R)$ onto $L^2([0, 1)^2)$.
The values are quasi-periodic: $Zf(x, \omega + 1) = Zf(x, \omega)$ and $Zf(x + 1, \omega) = e^{2\pi i \omega} Zf(x, \omega)$.
The second rule follows by putting $j = k + 1$ in the series.
Some authors include a lattice parameter $a > 0$ and use $f(x + ak)$.

Example: let $f = 1_{[0, 1)}$.
For $x \in [0, 1)$ only the term $k = 0$ is nonzero, so $Zf(x, \omega) = 1$ on $[0, 1)^2$.
Then $\norm{Zf}^2_{L^2([0,1)^2)} = 1 = \norm{f}_2^2$, as unitarity requires.

Non-example: the map $f \mapsto \sum_{k \in \Z} f(x + k)$ without phase factors is the periodization of $f$.
It loses information, for instance it sends $1_{[0, 1/2)} - 1_{[1, 3/2)}$ to $0$.
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<document source="https://tamnd.github.io/math2/node/6D73FFD9DC.md" tag="6D73FFD9DC" kind="statement">
# Runge–Kutta–Fehlberg method

## Statement

Consider the explicit six-stage [[Butcher tableau|RH8BD7PW5F]] with two weight rows
$$\begin{array}{c|cccccc}
0 \\
\frac14 & \frac14 \\
\frac38 & \frac{3}{32} & \frac{9}{32} \\
\frac{12}{13} & \frac{1932}{2197} & -\frac{7200}{2197} & \frac{7296}{2197} \\
1 & \frac{439}{216} & -8 & \frac{3680}{513} & -\frac{845}{4104} \\
\frac12 & -\frac{8}{27} & 2 & -\frac{3544}{2565} & \frac{1859}{4104} & -\frac{11}{40} \\
\hline
b^{T} & \frac{25}{216} & 0 & \frac{1408}{2565} & \frac{2197}{4104} & -\frac15 & 0 \\
\hat b^{T} & \frac{16}{135} & 0 & \frac{6656}{12825} & \frac{28561}{56430} & -\frac{9}{50} & \frac{2}{55}
\end{array}$$
Then:

1. the [[Runge–Kutta method|7DBPJKYC6C]] with weights $b$ has order $4$, and the one with weights $\hat b$ has order $5$;
2. both use the same six stages $K_1, \ldots, K_6$, so a step gives $y_{n+1} = y_n + h \sum_i b_i K_i$ and $\hat y_{n+1} = y_n + h \sum_i \hat b_i K_i$ for six evaluations of $f$;
3. the difference $\hat y_{n+1} - y_{n+1}$ equals the [[local truncation error|T220HF0562]] of the fourth-order step up to a term $O(h^6)$, for $f$ smooth enough.

## Remarks

Clause 1 is checked against Butcher's order conditions.
There are $8$ conditions for the trees with at most $4$ vertices, for $b$, and $17$ for the trees with at most $5$ vertices, for $\hat b$.
For instance $\sum b_i = \sum \hat b_i = 1$ and $\sum \hat b_i c_i^4 = \frac15$.
Clause 3 follows from clause 1: the local errors of the two steps are $O(h^5)$ and $O(h^6)$, so their difference is the first one up to $O(h^6)$.
The estimate drives [[adaptive step size|6MREWBZR7R]] control: a step is accepted when the estimate is below the tolerance, and the next step is scaled by about $(\mathrm{tol}/\mathrm{err})^{1/5}$.
Erwin Fehlberg proposed such embedded pairs in 1969; in his original use the fourth-order value is propagated, while many codes propagate the fifth-order value.

## Explanations

### L1 human

To choose good step sizes, a solver must know how large its error is.
The Fehlberg method computes two answers of different accuracy from the same six slope evaluations.
Their difference estimates the error at almost no extra cost, and the step is then made longer or shorter.

### L2 human

This is the classic example of an embedded Runge–Kutta pair, often called RKF45.
It was largely replaced in modern software by the [[Dormand–Prince method|CDMXAH4A7G]], whose coefficients are tuned to make the error of the fifth-order solution small and which reuses the last stage.
The idea of embedding, two weight vectors on one set of stages, is the basis of nearly all explicit adaptive ODE solvers.
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# Regular sequence

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].
A sequence $x_1, \ldots, x_n$ of elements of $R$ is an *$M$-regular sequence* if:

1. for each $i = 1, \ldots, n$, the element $x_i$ is a non-zero-divisor on $M / (x_1, \ldots, x_{i-1}) M$, that is, $x_i m = 0$ in this quotient implies $m = 0$ in it;
2. $M / (x_1, \ldots, x_n) M \ne 0$.

A *regular sequence* of $R$ is an $R$-regular sequence.

## Remarks

Example: in $R = K[x_1, \ldots, x_n]$ with $K$ a field, the variables form a regular sequence, since $R/(x_1, \ldots, x_{i-1}) \cong K[x_i, \ldots, x_n]$ is a domain in which $x_i \ne 0$, and $R/(x_1, \ldots, x_n) \cong K \ne 0$.
Non-example: in $K[x, y]$ the sequence $x, xy$ fails clause 1, because $xy$ is zero in $K[x,y]/(x)$ and so kills the nonzero element $1$.
A sequence that satisfies only clause 1 is called *weakly regular*.
The order matters in general: in $K[x, y, z]$, the sequence $x, y(1-x), z(1-x)$ is regular but $y(1-x), z(1-x), x$ is not.
If $R$ is a Noetherian local ring, $M$ is finitely generated and all $x_i$ lie in the maximal ideal, then every permutation of an $M$-regular sequence is $M$-regular.
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# Homoclinic orbit

## Summary

A homoclinic orbit is a [[trajectory|07TZ2Q4C1M]] that leaves an [[equilibrium point|0N8NGKDZDZ]] and returns to the same equilibrium.
It lies in both the stable and the unstable manifold of the equilibrium.
Homoclinic orbits separate regions of different behaviour in the phase plane, and they appear in global bifurcations.
In higher dimensions their transverse versions are a source of chaos.

## Explanations

### L1 human

Picture a loop in the phase plane that starts at a saddle point, swings around, and comes back to the same saddle.
A state on this loop takes infinitely long to leave the saddle and infinitely long to come back.
For the equation $x'' = x - x^2$, the solution $x(t) = \frac{3}{2} \operatorname{sech}^2(t/2)$ starts near $0$, rises to $\frac{3}{2}$ and falls back to $0$.
Its path in the $(x, x')$ plane is a homoclinic loop at the origin.
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# Runge's theorem

## Statement

Let $K \subseteq \C$ be a [[compact set|00R0X1WGPX]].
Let $f$ be a [[holomorphic function|0DG0Z72CQD]] on an open set containing $K$.
Let $A \subseteq \C \setminus K$ be a set that contains at least one point of each bounded connected component of $\C \setminus K$.
Then for every $\varepsilon > 0$ there is a [[rational function|0WCZH4AFMT]] $r$ whose poles all lie in $A$ such that
$$\abs{f(z) - r(z)} < \varepsilon \quad \text{for all } z \in K.$$
In particular, if $\C \setminus K$ is [[connected|0D939CMME7]], then $f$ is a [[uniform limit|08HJRTG5N7]] on $K$ of polynomials.

## Remarks

Proof idea: by the Cauchy integral formula over a cycle around $K$, $f$ is a uniform limit on $K$ of finite sums of terms $c/(z - a)$ with $a$ off $K$.
Each pole $a$ can then be moved to a point of $A$ in the same component of $\C \setminus K$, by expanding in power series.
For the unbounded component, the pole is moved to $\infty$ instead.
A pole at $\infty$ means a polynomial.

The hypothesis on $A$ is necessary.
Let $K$ be the unit circle and $f(z) = 1/z$; the bounded component of $\C \setminus K$ is the open disc.
For every polynomial $p$ one has $\oint_K p(z) \, dz = 0$, while $\oint_K z^{-1} \, dz = 2\pi i$.
If $\abs{p - 1/z} < \varepsilon$ on $K$, then $2\pi = \abs{\oint_K (z^{-1} - p(z)) \, dz} \le 2\pi \varepsilon$, which fails for $\varepsilon < 1$.
With $A = \set{0}$ the rational function $r(z) = 1/z$ itself has its only pole in $A$ and approximates $f$ exactly.

## Explanations

### L1 human

A holomorphic function near a compact set can be approximated, uniformly on the set, by rational functions.
The poles of the rational functions are only needed inside the holes of the set, one chosen point per hole being enough.
If the set has no holes, polynomials are enough.
The circle example shows that holes really matter.

### L2 human

Runge's theorem is the starting point of approximation theory in the complex plane.
It is used to build holomorphic functions with prescribed properties, for example sequences of polynomials that converge pointwise on $\C$ to a discontinuous limit.
It gives the existence of solutions of the $\bar\partial$-equation and the Mittag-Leffler theorem on general open sets.
[[Mergelyan's theorem|5KKCSZJC29]] strengthens the polynomial case to functions that are only continuous on $K$ and holomorphic in its interior.
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# Penny graph

## Summary

A penny graph is the contact graph of a set of equal coins lying flat on a table without overlapping.
Each coin is a vertex, and two coins are joined when they touch.
Penny graphs are [[planar graphs|0NC9R64TXP]] with maximum degree at most $6$, and they are a test case for questions about unit distances and [[circle packings|S360JBEYKD]].

## Explanations

### L1 human

Lay some identical coins on a table so that none overlap.
Draw a dot at the centre of each coin and a line between the centres of two coins that touch.
The drawing is a penny graph.
Six coins around a central coin, all touching it and their neighbours, give a hexagon with its centre, a penny graph with $7$ vertices and $12$ edges.
No coin can touch more than six others, since a seventh would not fit around it.
</document>

<document source="https://tamnd.github.io/math2/node/6DHQTFAJ03.md" tag="6DHQTFAJ03" kind="statement">
# Five-term relation for the dilogarithm

## Statement

Let $\operatorname{Li}_2$ be the [[dilogarithm|G0X0NWSDB7]], and for $0 < x < 1$ let
$$L(x) = \operatorname{Li}_2(x) + \tfrac12 \log x \, \log(1 - x)$$
be the Rogers dilogarithm, where $\log$ is the [[natural logarithm|04DXAFN2ES]].
Let $D$ be the [[Bloch–Wigner function|Q8K5MWEXB9]].

1. For all real $x, y$ with $0 < x < 1$ and $0 < y < 1$,
$$L(x) + L(y) = L(xy) + L\left(\frac{x(1-y)}{1-xy}\right) + L\left(\frac{y(1-x)}{1-xy}\right).$$
2. For all complex $x, y$ such that $x$, $y$, $1 - xy$, $\frac{1-x}{1-xy}$ and $\frac{1-y}{1-xy}$ all lie in $\C \setminus \set{0, 1}$,
$$D(x) + D(1 - xy) + D(y) + D\left(\frac{1-x}{1-xy}\right) + D\left(\frac{1-y}{1-xy}\right) = 0.$$

## Remarks

For the real form, fix $y$ and differentiate both sides in $x$.
Using $\frac{d}{dx}\operatorname{Li}_2(x) = -\frac{\log(1-x)}{x}$, the derivatives of the two sides agree, and both sides tend to $L(y)$ as $x \to 0^+$, since $L(x) \to 0$.
The complex form follows in the same way from the differential of $D$, which is an explicit real-analytic $1$-form on $\C \setminus \set{0, 1}$.

Example: with $y = x$, the third argument is $\frac{x(1-x)}{1-x^2} = \frac{x}{1+x}$, so the relation gives
$$2L(x) = L(x^2) + 2L\left(\frac{x}{1+x}\right), \qquad 0 < x < 1.$$

The five arguments in the complex form are cross-ratios of five points on the Riemann sphere taken four at a time.
Geometrically, $D(z)$ is the hyperbolic volume of the ideal tetrahedron with vertices $0, 1, \infty, z$.
The relation says that two ways of cutting an ideal polyhedron with five vertices into ideal tetrahedra give the same volume.

The correction term $\frac12 \log x \log(1-x)$ is needed: the plain dilogarithm $\operatorname{Li}_2$ satisfies a five-term relation only with extra products of logarithms.

## Explanations

### L1 human

The dilogarithm is a function defined by a power series, much like the logarithm.
The logarithm turns products into sums; the dilogarithm has a more complicated rule that ties together its values at five related points.
This five-term rule is the basic functional equation of the dilogarithm, and many other identities for it can be derived from it.

### L2 human

The relation is usually stated for the Rogers dilogarithm on $(0, 1)$ or for the Bloch–Wigner function on $\C \setminus \set{0, 1}$, because these versions have no elementary correction terms.
It is the defining relation of the Bloch group in algebraic K-theory, and it explains why sums of values $D(z_i)$ compute volumes of hyperbolic $3$-manifolds.
Its quantum analogue is the pentagon identity of the [[quantum dilogarithm|6HMT17DVX7]].
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# Concrete category of algebras

## Definition

A [[concrete category|FYBRP16KAN]] is a pair $(\mathcal{C}, U)$ of a category $\mathcal{C}$ and a faithful functor $U \colon \mathcal{C} \to \mathbf{Set}$.
Faithful means: for all morphisms $f, g \colon A \to B$, $U(f) = U(g)$ implies $f = g$.
Let $\sigma$ be an algebraic signature and $K$ a class of $\sigma$-algebras closed under isomorphism.
Let $\mathcal{K}$ be the category with the members of $K$ as objects and the [[homomorphisms|KBXSZC6BV1]] between them as morphisms.
The *concrete category of algebras* of $K$ is the pair $(\mathcal{K}, U)$, where $U$ is the [[forgetful functor|WV3BPYCHVG]].

## Remarks

The functor $U$ is faithful because a homomorphism is determined by its underlying map.

Example: the category $\mathbf{Grp}$ of groups with its forgetful functor to $\mathbf{Set}$ is a concrete category of algebras.

Non-example: the category $\mathbf{Grp}$ with the functor that sends $G$ to the underlying set of its abelianization $G / [G, G]$ is not a concrete category, since this functor is not faithful.
The two homomorphisms $\Z \to S_3$ that send $1$ to the transpositions $(1\,2)$ and $(1\,3)$ are different, but both induce the map $\Z \to \Z / 2\Z$ that sends $1$ to the nontrivial element.

Not every category admits a faithful functor to $\mathbf{Set}$: by a theorem of Freyd, the homotopy category of topological spaces does not.
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# Local asymptotic normality

## Summary

A sequence of statistical models is locally asymptotically normal (LAN) at a parameter value if its log-likelihood ratios have a special form at scale $1/\sqrt{n}$.
At that scale they look like those of a single observation from a normal location model.
The notion is due to Lucien Le Cam.
Smooth i.i.d. parametric models are LAN, with the Fisher information as the covariance.
LAN is the setting of the asymptotic optimality theory of estimators and tests: the convolution theorem and the local asymptotic minimax theorem.

## Explanations

### L1 human

With $n$ observations, parameter values that differ by much more than $1/\sqrt{n}$ are easy to tell apart, and values that differ by much less are impossible to tell apart.
The interesting scale is exactly $1/\sqrt{n}$.
LAN says that at this scale, every smooth model behaves like the simplest model there is: one normal observation whose mean is the unknown local parameter.
So asymptotic questions about complicated models reduce to easy questions about the normal distribution.
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# Ramanujan tau function

## Summary

The Ramanujan tau function $\tau(n)$ gives the Fourier coefficients of the [[cusp form|A50Y012RBP]] $\Delta = q \prod_{n \ge 1} (1 - q^n)^{24}$ of weight $12$.
Ramanujan studied it in 1916 and conjectured that it is multiplicative, which Mordell proved in 1917.
Its properties come from the theory of [[Hecke operators|S4SV3QZ8B1]].
The bound $\abs{\tau(p)} \le 2 p^{11/2}$ for primes $p$ is the [[Ramanujan–Petersson conjecture|4CDFWST967]], proved by Deligne.

## Explanations

### L1 human

Multiply out the infinite product $q(1 - q)^{24}(1 - q^2)^{24}(1 - q^3)^{24} \cdots$ and read off the coefficient of $q^n$; that is $\tau(n)$.
The first values are $\tau(1) = 1$, $\tau(2) = -24$, $\tau(3) = 252$, $\tau(4) = -1472$ and $\tau(5) = 4830$.
These numbers look random, yet $\tau(6) = \tau(2)\tau(3)$, and in general $\tau(mn) = \tau(m)\tau(n)$ when $m$ and $n$ share no factor.
They also satisfy the congruence $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$.
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<document source="https://tamnd.github.io/math2/node/6DSWPHSWQW.md" tag="6DSWPHSWQW" kind="concept">
# Commutator theory

## Summary

Commutator theory extends the commutator of normal subgroups of a group to congruences of arbitrary algebras.
For two [[congruences|DCSXXT4SER]] $\alpha$ and $\beta$ of an algebra it defines a congruence $[\alpha, \beta]$ by a condition on term operations, the term condition.
For groups it gives the usual commutator subgroup $[M, N]$ of normal subgroups, and for rings it gives the ideal $MN + NM$.
The theory works best in [[congruence-modular varieties|QAQAS3JWXW]], where it yields notions of abelian, solvable and nilpotent algebras that behave as in group theory.

## Explanations

### L1 human

In a group, the commutator $x y x^{-1} y^{-1}$ measures how far two elements are from commuting.
Commutator theory finds the same measure in any kind of algebra, using congruences in place of normal subgroups.
An algebra whose commutator of the largest congruence with itself is trivial is called abelian; for groups these are exactly the abelian groups.
With this one notion, ideas such as solvable and nilpotent groups make sense for rings, loops and many other algebras.
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<document source="https://tamnd.github.io/math2/node/6DXF1T506B.md" tag="6DXF1T506B" kind="concept">
# Polar polytope

## Summary

Let $P$ be a [[convex polytope|300K6CB1KH]] with the origin in its interior.
Its polar polytope is the set $P^\circ$ of vectors $y$ with $x \cdot y \le 1$ for all $x \in P$.
It is again a convex polytope, and taking the polar twice gives back $P$.
Polarity reverses the order of faces: vertices of $P$ correspond to [[facets|DPDXN2FJC4]] of $P^\circ$ and facets of $P$ to vertices of $P^\circ$.
It turns [[simple polytopes|7E6FXTGH2Y]] into [[simplicial polytopes|YVJ15PGKDM]] and back, and it is the geometric form of the dual polyhedron.

## Explanations

### L1 human

Polarity swaps corners and walls.
The polar of a cube centred at the origin is an octahedron.
The $6$ faces of the cube become the $6$ corners of the octahedron, and the $8$ corners of the cube become its $8$ faces.
The polar of a regular dodecahedron is an icosahedron.
A large polytope has a small polar, and the other way round.
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<document source="https://tamnd.github.io/math2/node/6DYAGW68AV.md" tag="6DYAGW68AV" kind="concept">
# Semi-local ring

## Summary

A semi-local ring is a [[commutative ring|0310REJJY8]] with only finitely many [[maximal ideals|0PV74TR5FP]].
It generalizes a [[local ring|MZ02BKPHJA]], which has exactly one.
Semi-local rings arise when one localizes at the complement of a finite union of prime ideals.
They also appear in the study of integral extensions of local rings.

## Explanations

### L1 human

A local ring looks at one point; a semi-local ring looks at finitely many points at once.
For example, the fractions $a/b$ with $b$ prime to both $2$ and $3$ form a ring in which only the primes $2$ and $3$ remain non-invertible.
This ring has exactly two maximal ideals, generated by $2$ and by $3$.
The [[definition|Q256RTTNTN]] gives the condition.
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# Linear PDE

## Summary

A linear PDE is a [[partial differential equation|GC5BDZ6QBC]] in which the unknown and its derivatives appear only linearly.
Its coefficients depend on the variables alone.
For a linear homogeneous equation, sums and multiples of solutions are again solutions.
This superposition principle makes linear PDEs the best understood class, and it is the setting of Fourier methods, fundamental solutions and distribution theory.
See the [[definition|B54GEJWQAV]].

## Explanations

### L1 human

In a linear PDE each term is a known function times one derivative of the unknown $u$, plus possibly a known source term.
No products such as $u \cdot \partial_x u$ and no powers such as $u^2$ occur.
The [[heat equation|VP2HN86VAG]] $\partial_t u = \partial_x^2 u$ is linear: if $u$ and $v$ solve it, so does $2u + 3v$.
The equation $\partial_t u + u \, \partial_x u = 0$ is not linear, and the sum of two of its solutions is usually not a solution.
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<document source="https://tamnd.github.io/math2/node/6E83YKANV3.md" tag="6E83YKANV3" kind="concept">
# Erdős–Rényi model

## Summary

The Erdős–Rényi model is the simplest model of a [[random graph|TNT3DG0HRJ]].
In the form $G(n, p)$, each of the $\binom{n}{2}$ possible edges on $n$ labelled vertices is present independently with probability $p$.
In the form $G(n, M)$, a graph with exactly $M$ edges is chosen uniformly at random.
The model shows sharp thresholds, such as the sudden appearance of a giant component, and it is a basic tool of the [[probabilistic method|3B55HNS754]].

## Explanations

### L1 human

Take $n$ dots and, for every pair of dots, flip a biased coin that shows heads with probability $p$.
Draw a line between the two dots exactly when the coin shows heads.
The picture you get is a random graph from $G(n, p)$.
For example, with $n = 3$ and $p = 1/2$, each of the $8$ graphs on the vertices $1, 2, 3$ appears with probability $1/8$.
When $p$ grows, the graph changes quickly from a few scattered edges to one large connected piece.
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<document source="https://tamnd.github.io/math2/node/6E8KRJKXHS.md" tag="6E8KRJKXHS" kind="statement">
# Weierstrass–Erdmann corner conditions

## Statement

Let $a < b$ and let $L \colon [a, b] \times \R^n \times \R^n \to \R$, written $L(x, y, p)$, be of class $C^1$.
Let $J(y) = \int_a^b L(x, y, y') \, dx$ on piecewise $C^1$ functions $y \colon [a, b] \to \R^n$ with fixed values at $a$ and $b$.
Let $y_0$ be a piecewise $C^1$ [[strong local minimiser|WMPCR723MJ]] of $J$ in this class.
Let $c \in (a, b)$ be a corner of $y_0$, with one-sided derivatives $p_- = y_0'(c^-)$ and $p_+ = y_0'(c^+)$.
Then
$$L_p(c, y_0(c), p_-) = L_p(c, y_0(c), p_+),$$
$$L(c, y_0(c), p_-) - p_- \cdot L_p(c, y_0(c), p_-) = L(c, y_0(c), p_+) - p_+ \cdot L_p(c, y_0(c), p_+).$$

## Remarks

The first condition comes from variations $y_0 + \varepsilon h$ with $h$ supported near $c$.
The du Bois-Reymond form of the [[Euler–Lagrange equation|PS0VV149ET]] says that $L_p(x, y_0, y_0')$ equals a continuous function of $x$, so it has no jump at $c$.
The second condition comes from moving the corner point sideways, which is a variation of the independent variable $x$.

Example: let $J(y) = \int_0^2 (y'^2 - 1)^2 \, dx$ with $y(0) = y(2) = 0$.
The zigzag $y_0(x) = \min(x, 2 - x)$ has $J(y_0) = 0$, so it is a minimiser, with a corner at $1$ where $p_- = 1$ and $p_+ = -1$.
Here $L_p = 4p(p^2 - 1)$ and $L - p L_p = (p^2 - 1)^2 - 4p^2(p^2 - 1)$, and both vanish at $p = \pm 1$, as the theorem requires.

If $L_{pp} > 0$ everywhere, the map $p \mapsto L_p(x, y, p)$ is injective when $n = 1$, and the first condition forces $p_- = p_+$.
So for such Lagrangians minimisers have no corners.

## Explanations

### L1 human

Sometimes the best curve has a sharp bend, a corner.
The corner conditions say which bends are allowed.
Two quantities built from the integrand, often called the momentum and the energy, must take the same value just before and just after the corner.

### L2 human

The Euler–Lagrange equation holds on each smooth piece of a broken extremal, but it says nothing about how the pieces join.
The Weierstrass–Erdmann conditions supply the missing junction conditions, two of them, which match the freedom in the location and the slopes at a corner.
In Hamiltonian terms they say that the momentum $L_p$ and the Hamiltonian $p \cdot L_p - L$ are continuous along the extremal.
They explain why problems with Lagrangians that are not convex in $p$ often have broken minimisers.
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# Rearrangement inequality

## Statement

Let $x_1 \le x_2 \le \dots \le x_n$ and $y_1 \le y_2 \le \dots \le y_n$ be real numbers, and let $\sigma$ be a [[permutation|0BQMWCNS8S]] of $\set{1, \dots, n}$.
Then
$$\sum_{i=1}^{n} x_i y_{n+1-i} \le \sum_{i=1}^{n} x_i y_{\sigma(i)} \le \sum_{i=1}^{n} x_i y_i.$$

## Remarks

Proof idea: suppose $i < j$ but $y_{\sigma(i)} > y_{\sigma(j)}$.
Swapping the two values changes the sum by $(x_j - x_i)(y_{\sigma(i)} - y_{\sigma(j)}) \ge 0$.
Repeated swaps sort $\sigma$ into the identity without decreasing the sum, which gives the upper bound.
The lower bound follows by applying the upper bound to $-y_n \le \dots \le -y_1$.

Example: with $x = (1, 2, 3)$ and $y = (4, 5, 6)$, the sums range from $1 \cdot 6 + 2 \cdot 5 + 3 \cdot 4 = 28$ to $1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 = 32$.

Mathlib states a form as `Monovary.sum_smul_comp_perm_le_sum_smul`.

## Explanations

### L1 human

You have $n$ boxes of banknotes and may take some number of notes from each box, with the counts given in advance.
To get the most money, take the most notes from the box with the largest bills.
To get the least, take the most notes from the box with the smallest bills.
The rearrangement inequality says exactly this: matching large with large maximises a sum of products.

### L2 human

The rearrangement inequality gives quick proofs of several classical inequalities.
Averaging it over the cyclic shifts of $y$ gives [[Chebyshev's sum inequality|S8X2N1DW8P]].
Applied to suitable sequences it also gives the [[AM–GM inequality|0DRF0F3XC9]] and the [[Cauchy–Schwarz inequality|0RFWW56Y22]] in special cases.
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<document source="https://tamnd.github.io/math2/node/6EEAEDWXZ2.md" tag="6EEAEDWXZ2" kind="concept">
# Discrete Poisson equation

## Summary

The discrete Poisson equation $\Delta u = f$ asks for a function on a grid or graph whose [[discrete Laplacian|ZN30Y1ZDB6]] is a given function $f$.
It is the finite difference version of Poisson's equation $\Delta u = f$, and with $f = 0$ it becomes the discrete form of [[Laplace's equation|S8RG6N1PRT]].
With boundary values prescribed, it becomes a large sparse linear system.
Solving this system fast is a central task of numerical analysis, done with iterative methods, multigrid or the fast Fourier transform.

## Explanations

### L1 human

Lay a square grid over a metal plate and keep the edges at fixed temperatures.
Inside, each grid value should equal the average of its four neighbours, corrected by a heat source term.
Writing this rule at every inner point gives one linear equation per point.
Solving all of them at once gives the temperature at every grid point.
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<document source="https://tamnd.github.io/math2/node/6EEF2E8XPF.md" tag="6EEF2E8XPF" kind="concept">
# Poisson boundary

## Summary

The Poisson boundary of a [[random walk on a group|WAS9EANSJJ]] is a measure space that records where the walk goes at infinity.
Every bounded harmonic function for the walk is the Poisson integral of a unique bounded function on this boundary.
This mirrors the Poisson integral formula for bounded harmonic functions on the disc.
The boundary is trivial exactly when all bounded harmonic functions are constant, which is the [[Liouville property|Y1EGR8DF84]].
It was introduced by Furstenberg and developed by Kaimanovich and Vershik, who related its triviality to the entropy of the walk.

## Explanations

### L1 human

Let a random walker move on a group by multiplying by random elements, again and again.
Some information about the walk's position survives forever, for example on a free group the infinite word that the walk spells out in the limit.
The Poisson boundary collects exactly this surviving information, together with the probabilities of each outcome.
On $\Z^d$ nothing survives: the boundary is a single point, and every bounded harmonic function is constant.
On a free group with two generators, the boundary is the set of infinite reduced words, and it carries many bounded harmonic functions.
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# Chow's theorem

## Statement

Let $n \ge 0$ and let $A$ be a closed [[analytic subset|VMDTGTMN9P]] of the [[complex projective space|7GJ4STCNNM]] $\mathbb{P}^n(\C)$.
Then $A$ is algebraic: there are finitely many homogeneous polynomials $P_1, \dots, P_m \in \C[z_0, \dots, z_n]$ such that
$$A = \set{[z_0 : \cdots : z_n] \in \mathbb{P}^n(\C) : P_1(z) = \cdots = P_m(z) = 0}.$$

## Remarks

A short proof uses the Remmert–Stein theorem.
Let $p \colon \C^{n+1} \setminus \set{0} \to \mathbb{P}^n(\C)$ be the projection, and let $C = p^{-1}(A)$, the cone over $A$; it is analytic in $\C^{n+1} \setminus \set{0}$, and each irreducible component has dimension at least $1$.
By the Remmert–Stein theorem, $C \cup \set{0}$ is an analytic subset of $\C^{n+1}$.
Near $0$ it is the zero set of holomorphic functions $f_1, \dots, f_r$.
Since $C$ is invariant under $z \mapsto \lambda z$, the homogeneous parts of the Taylor series of the $f_j$ also vanish on $C$.
By Hilbert's basis theorem, finitely many of these homogeneous polynomials define $C \cup \set{0}$.
Example: the image of $\mathbb{P}^1(\C)$ under $[s : t] \mapsto [s^3 : s^2 t : s t^2 : t^3]$ is a closed analytic subset of $\mathbb{P}^3(\C)$, and indeed it is the twisted cubic, defined by $z_0 z_2 = z_1^2$, $z_1 z_3 = z_2^2$ and $z_0 z_3 = z_1 z_2$.
Compactness of projective space is essential: in $\C$, the analytic subset $\Z$ is not the zero set of polynomials, since a nonzero polynomial has finitely many zeros.

## Explanations

### L1 human

In projective space, any set cut out locally by holomorphic equations is cut out globally by polynomial equations.
So every compact complex submanifold of projective space, however it is presented, is an algebraic variety.
This is a striking rigidity: holomorphic functions are much more flexible than polynomials, but the compactness of projective space forces algebraicity.

### L2 human

Chow's theorem is the first instance of the principle that compact analytic objects in projective space are algebraic, which Serre's GAGA theorems extend to coherent sheaves and their cohomology.
Combined with the Kodaira embedding theorem, it shows that every compact Kähler manifold with an integral Kähler class is a projective algebraic variety.
It also implies that holomorphic maps between projective varieties are algebraic, by applying the theorem to their graphs.
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<document source="https://tamnd.github.io/math2/node/6ERBSBVF14.md" tag="6ERBSBVF14" kind="concept">
# Essentially surjective functor

## Summary

A [[functor|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathcal{D}$ is essentially surjective when every object of $\mathcal{D}$ is isomorphic to $F(A)$ for some object $A$ of $\mathcal{C}$.
It is surjectivity on objects up to isomorphism, which is the right notion in category theory, where isomorphic objects are interchangeable.
A [[fully faithful|GRG8WEBSH5]] and essentially surjective functor is an [[equivalence of categories|K7A04ZBD92]].

## Explanations

### L1 human

A functor might not hit every object of its target, but it can still reach every object up to isomorphism.
For example, the vector spaces $\R^n$ for $n = 0, 1, 2, \dots$ are only a few of all finite-dimensional real vector spaces.
Yet each finite-dimensional real vector space is isomorphic to one of them.
So including the spaces $\R^n$ into all finite-dimensional spaces is essentially surjective, though far from surjective on objects.
</document>

<document source="https://tamnd.github.io/math2/node/6ERJRRZK32.md" tag="6ERJRRZK32" kind="concept">
# Final topology

## Summary

Take a set and a family of maps from topological spaces into it.
The final topology on the set is the finest topology that makes all these maps continuous.
The [[quotient topology|0NQ1H2X495]] and the [[disjoint union topology|ZXJKSB30WF]] are both final topologies.
A map out of the set is continuous for it exactly when its composites with all the given maps are continuous.
The dual notion is the [[initial topology|1NWC3710WW]].

## Explanations

### L1 human

Suppose some spaces you understand map into a set.
You want as many open sets as possible while keeping every map continuous.
So you declare a set open exactly when each map pulls it back to an open set.
For example, the map from $[0, 1]$ onto a circle that glues $0$ to $1$ gives the circle its usual topology this way.
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<document source="https://tamnd.github.io/math2/node/6ETEBA8827.md" tag="6ETEBA8827" kind="concept">
# Kähler–Einstein metric

## Summary

A Kähler–Einstein metric is a Kähler metric whose Ricci curvature is a constant multiple of the metric.
It is both a Kähler metric and an [[Einstein metric|41S2EZ0KR3]], so it is a canonical choice of metric on a complex manifold.
On a compact manifold its existence forces the first Chern class to be negative, zero or positive, and the three cases behave very differently.
The negative case was settled by Aubin and by Yau, and the zero case by Yau; the positive case leads to the [[Yau–Tian–Donaldson conjecture|DHYSPSYSCB]].

## Explanations

### L1 human

On a complex manifold there are many ways to measure lengths that respect the complex structure.
A Kähler–Einstein metric is one whose curvature, in the Ricci sense, is the same in every direction at every point.
The round sphere, seen as the complex projective line, is an example with positive curvature.
A flat torus is an example with zero curvature, and the hyperbolic disc is one with negative curvature.
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<document source="https://tamnd.github.io/math2/node/6ETK64HYK6.md" tag="6ETK64HYK6" kind="concept">
# Instrumental variables estimation

## Summary

Instrumental variables estimation recovers the coefficients of a linear model when some regressors are correlated with the error term, a situation called endogeneity.
Endogeneity arises from omitted variables, simultaneous equations or [[measurement error|2XX0A85WSF]], and it makes [[ordinary least squares|YM708KHV2W]] inconsistent.
An instrument is a variable that is correlated with the endogenous regressor but uncorrelated with the error.
It isolates the part of the regressor's variation that is free of the confounding.
Two-stage least squares is the standard instrumental variables estimator, and it is a special case of the [[generalized method of moments|ZPVMCCDYZ5]].

## Explanations

### L1 human

Suppose one wants the effect of years of schooling on wages, but ability raises both, and ability is not observed.
Least squares then mixes the effect of schooling with the effect of ability.
An instrument is something that shifts schooling but has no direct link to wages or to ability.
An example is a rule that changes the age at which pupils may leave school.
Comparing how wages and schooling both move with the instrument gives the effect of schooling alone.
With one regressor and one instrument $z$, the estimate is the ratio $\operatorname{Cov}(z, y) / \operatorname{Cov}(z, x)$ of sample covariances.
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# Algebraic torus

## Definition

Let $k$ be a field with algebraic closure $\bar{k}$.
Let $\mathbb{G}_m = \operatorname{Spec} k[t, t^{-1}]$ be the multiplicative group.
An *algebraic torus* over $k$ is an [[algebraic group|GH81SYAPPN]] $T$ over $k$ such that $T \times_k \bar{k} \cong \mathbb{G}_{m, \bar{k}}^n$ as algebraic groups for some $n \ge 0$.
The torus is *split* if $T \cong \mathbb{G}_m^n$ already over $k$.

## Remarks

An algebraic torus is a [[linear algebraic group|GA81VVQHEP]], and $n$ is its dimension.
One can replace $\bar{k}$ by a separable closure of $k$ in the definition.

Example: over $k = \R$, let $T$ be the circle $x^2 + y^2 = 1$ with $(x, y)(x', y') = (x x' - y y', x y' + x' y)$.
Over $\C$, the map $(x, y) \mapsto x + iy$ is an isomorphism $T_{\C} \cong \mathbb{G}_{m, \C}$, so $T$ is a torus of dimension $1$.
It is not split, because $T(\R)$ is compact while $\mathbb{G}_m(\R) = \R^{*}$ is not.

Non-example: the additive group $\mathbb{G}_a = \operatorname{Spec} k[x]$ is not a torus.
The ring $\bar{k}[x]$ has only constant units, while $\bar{k}[t, t^{-1}]$ has the unit $t$, so $\mathbb{G}_{a, \bar{k}} \not\cong \mathbb{G}_{m, \bar{k}}$.
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# Variety as a reflective subcategory

## Statement

Let $\sigma$ be an algebraic signature, and let $\mathbf{Alg}(\sigma)$ be the category of all $\sigma$-algebras and their [[homomorphisms|KBXSZC6BV1]].
Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of $\sigma$-algebras, viewed as a [[full subcategory|V3BFGKBJF7]].
For a $\sigma$-algebra $A$, let $\theta_A$ be the intersection of all [[congruences|DCSXXT4SER]] $\theta$ of $A$ with $A / \theta \in \mathcal{V}$, and let $q_A \colon A \to A / \theta_A$ be the [[quotient map|N5KE7FH6CX]].
Then $A / \theta_A \in \mathcal{V}$, and for every $B \in \mathcal{V}$ and every homomorphism $f \colon A \to B$ there is a unique homomorphism $\bar f \colon A / \theta_A \to B$ with $\bar f \circ q_A = f$.
Hence $\mathcal{V}$ is a [[reflective subcategory|FC7VB4WSG0]] of $\mathbf{Alg}(\sigma)$, with reflection $q_A$, and its reflector $A \mapsto A / \theta_A$ is left adjoint to the inclusion.

## Remarks

The algebra $A / \theta_A$ embeds into the product of the algebras $A / \theta$ over all such $\theta$, so it lies in $\mathcal{V}$ by closure under subalgebras and products.
For $f \colon A \to B$ with $B \in \mathcal{V}$, the image $f(A)$ is a subalgebra of $B$, so $A / \ker f \cong f(A) \in \mathcal{V}$.
Thus $\ker f \supseteq \theta_A$, and $f$ factors through $q_A$; the factorization is unique because $q_A$ is surjective.

Only closure under isomorphic copies, subalgebras and products is used, so the same holds for every quasivariety and every [[prevariety|EEAJ7AXBCT]].
Example: for the variety of abelian groups inside groups, $\theta_G$ is the congruence of the commutator subgroup $[G, G]$, and the reflection is the abelianization $G \to G / [G, G]$.
Applied to the absolutely free term algebra on a set $X$, the reflection gives the [[free algebra|GTS4DRRMZR]] of $\mathcal{V}$ on $X$.

## Explanations

### L1 human

Any algebra can be pushed into a given variety in a best possible way: impose just the equations of the variety, and nothing more.
For groups and the variety of abelian groups, this means forcing all elements to commute, which gives the abelianization.
Every homomorphism into an algebra of the variety factors through this best approximation in exactly one way.

### L2 human

The reflection $A \mapsto A / \theta_A$ is a universal construction, and $\theta_A$ is the congruence generated by the pairs $(s(\bar a), t(\bar a))$ for identities $s \approx t$ of $\mathcal{V}$ and tuples $\bar a$ of $A$.
Since the reflector is a left adjoint, it preserves colimits: the free product in $\mathcal{V}$ of two members is the reflection of their free product in $\mathbf{Alg}(\sigma)$.
Composing with the free functor of $\mathbf{Alg}(\sigma)$ recovers the [[free functor|N353JF8RBK]] of $\mathcal{V}$.
Reflective subcategories are closed under limits, which matches the closure of varieties under products and under equalizers, which are subalgebras.
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# Lévy–Steinitz theorem

## Statement

Let $d \ge 1$ and let $(x_n)_{n \ge 0}$ be a sequence in the [[Euclidean space|0XPR1E85W0]] $\R^d$.
Let $S$ be the set of all $s \in \R^d$ such that some [[rearrangement|0HRJ2Y74T3]] $\sum_n x_{\sigma(n)}$ converges to $s$.
Let $\Gamma$ be the set of [[linear functionals|0M856Q1AMZ]] $f \colon \R^d \to \R$ such that $\sum_n \abs{f(x_n)} < \infty$.
If $S$ is not empty, then
$$S = s_0 + \set{v \in \R^d : f(v) = 0 \text{ for all } f \in \Gamma}$$
for any $s_0 \in S$.
In particular, $S$ is either empty or an [[affine subspace|0FBXFBZBH8]] of $\R^d$.

## Remarks

For $d = 1$ this is the [[Riemann rearrangement theorem|0E7VBYGFF7]].
If the series converges [[absolutely|0W3K3QPEWT]], then $\Gamma$ contains all functionals and $S$ is a single point.
If it converges conditionally, then $\Gamma = \set{0}$ and $S = \R$.
Example in $\R^2$: take $x_n = ((-1)^n/(n+1), 1/2^n)$.
The second coordinate converges absolutely, the first only conditionally, so $S$ is a horizontal line $\set{(t, 2) : t \in \R}$.
The proof is harder than in dimension one.
Its core is a lemma of Steinitz on finitely many vectors in $\R^d$ with sum zero.
They can be ordered so that every partial sum has norm at most $C_d$ times the largest norm of the vectors, where $C_d$ depends only on $d$.
In infinite-dimensional Banach spaces the conclusion can fail: the set of sums need not be an affine subspace.

## Explanations

### L1 human

Reordering a convergent series in the plane or in space can change its sum.
The theorem says that the possible sums always form a point, a line, a plane, or a higher flat piece of space.
The directions in which the sum cannot move are the directions in which the series converges absolutely.

### L2 human

The theorem extends Riemann's rearrangement theorem from $\R$ to $\R^d$.
The subspace of possible shifts is cut out by the functionals that make the series absolutely convergent.
It explains why unconditional convergence and absolute convergence coincide in $\R^d$: the set $S$ is a single point exactly when every functional is in $\Gamma$.
</document>

<document source="https://tamnd.github.io/math2/node/6EZ400WM94.md" tag="6EZ400WM94" kind="statement">
# Fuchs' theorem

## Statement

Let $R > 0$ and $z_0 \in \C$, and let $D = \set{z : 0 < \abs{z - z_0} < R}$.
Let $p, q$ be functions such that $P(z) = (z - z_0)\, p(z)$ and $Q(z) = (z - z_0)^2 q(z)$ are analytic on $\abs{z - z_0} < R$.
Let $r_1, r_2$ be the roots of the indicial equation $r (r - 1) + P(z_0)\, r + Q(z_0) = 0$, labelled so that $\operatorname{Re} r_1 \ge \operatorname{Re} r_2$.
Fix a branch of $\log (z - z_0)$ on $D$ minus a ray from $z_0$, and put $(z - z_0)^r = e^{r \log (z - z_0)}$.
Then the equation $y'' + p(z) y' + q(z) y = 0$ has two linearly independent solutions $y_1, y_2$ there, of the following form, where all the [[power series|0MZB06ST2B]] converge on $\abs{z - z_0} < R$.

1. $y_1 = (z - z_0)^{r_1} \sum_{k \ge 0} a_k (z - z_0)^k$ with $a_0 = 1$.
2. If $r_1 - r_2 \notin \Z$, then $y_2 = (z - z_0)^{r_2} \sum_{k \ge 0} b_k (z - z_0)^k$ with $b_0 = 1$.
3. If $r_1 = r_2$, then $y_2 = y_1 \log (z - z_0) + (z - z_0)^{r_1} \sum_{k \ge 1} b_k (z - z_0)^k$.
4. If $r_1 - r_2$ is a positive integer, then $y_2 = c\, y_1 \log (z - z_0) + (z - z_0)^{r_2} \sum_{k \ge 0} b_k (z - z_0)^k$ with $b_0 = 1$ and a constant $c$, which may be $0$.

## Remarks

The coefficients come from the recurrence of the [[Frobenius method|W8ZWXC2HQS]].
For $r = r_1$, the factor $F(r_1 + k)$ in front of $a_k$ never vanishes for $k \ge 1$, because $r_1$ has the largest real part.
Convergence on the whole disc follows by comparing the recurrence with a majorant series built from Cauchy estimates for the coefficients of $P$ and $Q$.
The second solution comes from reduction of order, which is where the [[logarithm|02QGGD5G87]] enters.

Example: for the Bessel equation of order $0$, $z^2 y'' + z y' + z^2 y = 0$, the indicial equation is $r^2 = 0$, so case 3 applies.
The solutions are $J_0(z) = \sum_{m \ge 0} \frac{(-1)^m}{(m!)^2} \left( \frac{z}{2} \right)^{2m}$ and a second solution with a term $J_0(z) \log z$.

If $z_0$ is an ordinary point, then $P(z_0) = Q(z_0) = 0$, the exponents are $0$ and $1$, and the theorem gives two analytic solutions, with [[radius of convergence|0N288P2TB5]] at least $R$.
The hypothesis that $z_0$ is at worst a regular singular point is needed: at the irregular point $0$ of $z^2 y'' + y' = 0$, the solution $\int e^{1/z} dz$ has no such form.
For real equations, the same holds on $0 < \abs{x - x_0} < R$ with $(x - x_0)^r$ replaced by $\abs{x - x_0}^r$.

## Explanations

### L1 human

The theorem says that the Frobenius method always works at a regular singular point.
There is always one solution of the form $z^r$ times a convergent power series.
The second solution has the same form, unless the two exponents differ by a whole number; then a logarithm may appear.
The series converge as far as the coefficients of the equation stay nice.

### L2 human

The theorem is the basic local result for linear equations in the complex domain.
It explains the behaviour of Bessel, Legendre, Laguerre and hypergeometric functions near their singular points.
Going around $z_0$ once multiplies $(z - z_0)^r$ by $e^{2 \pi i r}$ and adds $2 \pi i$ to the logarithm.
So the theorem computes the local [[monodromy|C8HD62AB6Q]] of the equation from the exponents.
At an irregular singular point, solutions are described instead by asymptotic expansions in sectors, which leads to the [[Stokes phenomenon|TTGF8KCA8C]].
</document>

<document source="https://tamnd.github.io/math2/node/6F4JHFDM19.md" tag="6F4JHFDM19" kind="definition">
# Simplicial polytope

## Definition

A $k$-simplex is the [[convex hull|0FQSDC2EQ2]] of $k + 1$ affinely independent points.
Let $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d \ge 1$.
$P$ is *simplicial* if every [[facet|DPDXN2FJC4]] of $P$ is a $(d-1)$-simplex.

## Remarks

Equivalently, every proper face of $P$ is a simplex, since each face of a simplex is a simplex.
Equivalently again, every facet of $P$ has exactly $d$ vertices.

Example: the octahedron $\operatorname{conv}\set{\pm e_1, \pm e_2, \pm e_3}$ in $\R^3$ is simplicial.
Its facets are the eight triangles $\operatorname{conv}\set{\epsilon_1 e_1, \epsilon_2 e_2, \epsilon_3 e_3}$ with $\epsilon_i \in \set{1, -1}$, and each is the hull of $3$ affinely independent points.

Non-example: the cube $[0,1]^3$ is not simplicial, because its facets are squares with $4 > 3$ vertices.

Every polygon is simplicial, since its facets are segments.
The convex hull of finitely many points in general position in $\R^d$, with at least $d+1$ of them, is simplicial.
</document>

<document source="https://tamnd.github.io/math2/node/6F5B8KPEGH.md" tag="6F5B8KPEGH" kind="statement">
# Carlson's theorem

## Statement

Let $f$ be a continuous function on the closed half-plane $\operatorname{Re} z \ge 0$ that is [[holomorphic|0DG0Z72CQD]] on the open half-plane $\operatorname{Re} z > 0$.
Assume that:

1. there are constants $C, \tau > 0$ such that $\abs{f(z)} \le C e^{\tau \abs{z}}$ for $\operatorname{Re} z \ge 0$;
2. there are constants $C' > 0$ and $c < \pi$ such that $\abs{f(iy)} \le C' e^{c \abs{y}}$ for all real $y$;
3. $f(n) = 0$ for every integer $n \ge 0$.

Then $f$ is identically zero.

## Remarks

Condition 1 says that $f$ has [[exponential type|9P8D8YAHQS]] in the half-plane.
The proof compares $f$ with $\sin(\pi z)$, which vanishes at the integers, and uses the Phragmén–Lindelöf principle to bound $f(z)/\sin(\pi z)$.
The bound $c < \pi$ in condition 2 is sharp: $f(z) = \sin(\pi z)$ satisfies conditions 1 and 3, and $\abs{\sin(\pi i y)} = \abs{\sinh(\pi y)}$, which grows like $e^{\pi \abs{y}}/2$.
Applied to a difference, the theorem shows that two functions of this kind that agree at all nonnegative integers are equal.

## Explanations

### L1 human

A function on the integers can be extended to the complex plane in many ways, for example by adding $\sin(\pi z)$.
Carlson's theorem says that the extension becomes unique once its growth is limited.
The function may grow exponentially, but on the imaginary axis more slowly than $\sin(\pi z)$.

### L2 human

The theorem justifies many formulas that are first proved for integer values and then used for complex values.
For example, two expressions in $z$ that agree for $z = 0, 1, 2, \dots$ and satisfy the growth conditions agree everywhere.
It is used for interpolation of sequences and for questions of uniqueness of Newton series.
Like the [[Paley–Wiener theorem|THY12GPZ8C]], it shows how the exponential type of an entire function controls its behaviour.
</document>

<document source="https://tamnd.github.io/math2/node/6FDYJVG1CK.md" tag="6FDYJVG1CK" kind="definition">
# Stationary set

## Definition

Let $\kappa$ be an uncountable [[regular cardinal|ZCE333FAWC]], and let $S \subseteq \kappa$.
The set $S$ is stationary in $\kappa$ if $S \cap C \ne \varnothing$ for each [[club set|GWS5QXH7WW]] $C$ in $\kappa$.

## Remarks

Example: each club set $D$ in $\kappa$ is stationary.
Check: for each club $C$, the intersection $C \cap D$ is club, hence unbounded, hence nonempty.
Example: the set $\set{\alpha < \omega_2 : \operatorname{cf}(\alpha) = \omega}$ is stationary in $\omega_2$.
Check: each club $C$ in $\omega_2$ contains the supremum of its first $\omega$ elements, and this supremum has cofinality $\omega$.
Non-example: a bounded set $S \subseteq \kappa$, with $S \subseteq \beta$ for some $\beta < \kappa$, misses the club set $\set{\alpha : \beta \le \alpha < \kappa}$.
The sets that are not stationary form a $\kappa$-complete ideal, the nonstationary ideal; it is closed under diagonal unions.
The same definition is used for a limit ordinal of uncountable cofinality in place of $\kappa$.
By a theorem of Solovay, each stationary subset of $\kappa$ is the union of $\kappa$ pairwise disjoint stationary sets.
</document>

<document source="https://tamnd.github.io/math2/node/6FE7JZ2VF1.md" tag="6FE7JZ2VF1" kind="object">
# Calkin algebra

## Description

Let $H$ be an infinite-dimensional separable complex [[Hilbert space|HM2HHE3J51]].
Let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$, and let $K(H)$ be the [[ideal of compact operators|2WAZF1RV2Y]] in it.
The *Calkin algebra* of $H$ is the [[quotient|9240K052CS]] $Q(H) = B(H) / K(H)$.
It carries the quotient norm and the induced product.
Its involution is $\pi(T)^* = \pi(T^*)$, where $\pi \colon B(H) \to Q(H)$ is the quotient map.

## Remarks

Since $K(H)$ is a closed two-sided ideal of $B(H)$ that is closed under adjoints, $Q(H)$ is a unital C*-algebra.
Calkin showed that for separable $H$, $K(H)$ is the only closed two-sided ideal of $B(H)$ other than $\set{0}$ and $B(H)$, so $Q(H)$ is a simple C*-algebra.
An operator $T \in B(H)$ is Fredholm exactly when $\pi(T)$ is invertible in $Q(H)$.
The spectrum of $\pi(T)$ in $Q(H)$ is the essential spectrum of $T$.
The algebra $Q(H)$ is infinite-dimensional and not separable as a normed space.
The construction makes sense for any Hilbert space, but for non-separable $H$ the ideal $K(H)$ is no longer the only nontrivial closed ideal.

## Explanations

### L1 human

The Calkin algebra is what remains of the bounded operators when compact operators are treated as zero.
Two operators are the same element of it when they differ by a compact operator, a small error in a precise sense.
In this algebra, the right shift on sequences becomes invertible, since it has an inverse up to a rank-one error.
It is the natural home of the Fredholm index and of the essential spectrum.
</document>

<document source="https://tamnd.github.io/math2/node/6FFVXT5KMH.md" tag="6FFVXT5KMH" kind="concept">
# Arc in a finite plane

## Summary

A $k$-arc in a finite [[projective plane|DANFNN8C4Y]] is a set of $k$ points with no three on a common line.
In a plane of order $n$ a $k$-arc has at most $n + 2$ points, and at most $n + 1$ points when $n$ is odd.
Arcs of maximal size, such as conics, are central objects of finite geometry and are linked to maximum distance separable codes.

## Explanations

### L1 human

Pick points in a finite plane so that no line passes through three of them.
Such a set is an arc, a discrete version of a curve that bends everywhere.
In the plane over the field with $q$ elements, the points $(1 : t : t^2)$ for $t$ in the field, together with $(0 : 0 : 1)$, form an arc with $q + 1$ points.
It is the finite analogue of a parabola together with its point at infinity.
</document>

<document source="https://tamnd.github.io/math2/node/6FKJ6SXP3G.md" tag="6FKJ6SXP3G" kind="concept">
# Catastrophe theory

## Summary

Catastrophe theory is the part of [[singularity theory|42V5XGZS4F]] that studies how the [[critical points|0YTCMFB4HC]] of a smooth family of functions change as the parameters vary.
Small smooth changes of the parameters can make a minimum vanish, so that the state jumps suddenly; such a jump is a catastrophe.
René Thom showed that for families with at most four parameters only seven elementary catastrophes occur generically, such as the fold and the cusp.
The theory was popularised by Christopher Zeeman as a model of sudden changes in physical and other systems.

## Explanations

### L1 human

Think of a ball resting in a valley of a landscape that slowly changes shape.
Most of the time the ball moves smoothly as the valley shifts.
But if the valley flattens out and disappears, the ball suddenly rolls to another valley far away.
Catastrophe theory describes the few typical ways this can happen.
In the cusp catastrophe, for example, two valleys exist for some settings and the jump between them depends on the direction from which you approach.
</document>

<document source="https://tamnd.github.io/math2/node/6FKYSH729T.md" tag="6FKYSH729T" kind="concept">
# Quasi-isometry

## Summary

A quasi-isometry is a map between [[metric spaces|0VKD1TP7F3]] that distorts distances by at most a fixed factor and a fixed additive error.
In addition, its image comes within a bounded distance of every point.
It ignores all small-scale structure and keeps only the large-scale shape of a space.
Two spaces joined by a quasi-isometry are called quasi-isometric, and this is an equivalence relation.
It is the basic notion of sameness in [[geometric group theory|2T3218RXTD]], since all [[word metrics|VCHJVR9NGH]] of a finitely generated group are quasi-isometric to one another.

## Explanations

### L1 human

Look at a space from very far away, so far that bounded errors become invisible.
Two spaces are quasi-isometric if they look the same from there.
The integers $\Z$ and the real line $\R$ are quasi-isometric: the inclusion $\Z \to \R$ keeps distances exactly, and every real number is within $1/2$ of an integer.
Likewise the grid $\Z^2$ looks like the plane $\R^2$.
A bounded space, such as a disc, is quasi-isometric to a single point.
</document>

<document source="https://tamnd.github.io/math2/node/6FPKEFX0X4.md" tag="6FPKEFX0X4" kind="concept">
# Orthomodular lattice

## Summary

An orthomodular lattice is an [[orthocomplemented lattice|J7KN14DYV3]] that satisfies the orthomodular law: if $a \le b$, then $b = a \vee (a^\perp \wedge b)$.
The lattice of closed subspaces of any [[Hilbert space|HM2HHE3J51]] is orthomodular, even when it is not [[modular|QX7WZ92QAK]].
Every [[Boolean algebra|0XADJKM699]] and every modular ortholattice is orthomodular.
Orthomodular lattices are the standard algebraic model of the propositions of [[quantum logic|2E9DMB7A6Q]].

## Explanations

### L1 human

If a subspace $A$ sits inside a closed subspace $B$ of a Hilbert space, then $B$ splits as $A$ plus the part of $B$ that is perpendicular to $A$.
The orthomodular law asks for this splitting in an abstract ortholattice.
It is weaker than distributivity and weaker than modularity, and it still holds for the closed subspaces of an infinite-dimensional Hilbert space.
For example, in $\R^3$, a plane $B$ containing a line $A$ is the join of $A$ with the line in $B$ perpendicular to $A$.
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<document source="https://tamnd.github.io/math2/node/6FY0KHMK5A.md" tag="6FY0KHMK5A" kind="problem">
# Isoperimetric problem of the calculus of variations

## Problem

Let $a < b$ and $A, B, \ell \in \R$, and let $L, G \colon [a, b] \times \R \times \R \to \R$, written $L(x, z, p)$ and $G(x, z, p)$, be of class $C^2$.
Set $J(y) = \int_a^b L(x, y, y') \, dx$ and $K(y) = \int_a^b G(x, y, y') \, dx$, two [[integral functionals|JRQZ803Q5Y]].
The *isoperimetric problem* asks for a minimiser of $J$ among the [[admissible functions|G7T008BEWY]] $y \in C^1([a, b])$ with $y(a) = A$, $y(b) = B$ and the integral constraint $K(y) = \ell$.

## Remarks

The name comes from the classical problem of the largest area enclosed by a curve of given length, the case where $K$ is arc length.
[[Dido's problem|FABJSZ9D3S]] is a version of it.

The basic necessary condition is Euler's multiplier rule.
Let $y_0 \in C^2([a, b])$ be a minimiser, and suppose that $y_0$ does not solve the [[Euler–Lagrange equation|PS0VV149ET]] of $K$.
Then there is $\lambda \in \R$ such that $y_0$ solves the Euler–Lagrange equation of $L - \lambda G$:
$$\frac{d}{dx} \big( L_p - \lambda G_p \big) = L_z - \lambda G_z \quad \text{on } [a, b].$$
It follows from [[Lagrange multipliers on Banach spaces|Q7QK3PF806]] with constraint space $\R$.
If $y_0$ solves the Euler–Lagrange equation of $K$, the rule must be replaced by one with multipliers $(\lambda_0, \lambda) \ne (0, 0)$ for $\lambda_0 L - \lambda G$, where $\lambda_0 = 0$ is possible.

The [[catenary problem|8REBFQD6E0]] is an isoperimetric problem: it minimises potential energy at fixed length.

## Explanations

### L1 human

Many problems ask for the best curve when some total amount is fixed: the length of a fence, the mass of a chain, the area under a graph.
The fixed amount is written as a second integral, and the problem is to minimise the first integral while the second keeps its value.
As for constrained problems in calculus, the answer is found by adding a multiple of the constraint to the functional, with an unknown multiplier.
</document>

<document source="https://tamnd.github.io/math2/node/6G0079XRKK.md" tag="6G0079XRKK" kind="definition">
# Domain of holomorphy

## Definition

Let $D \subseteq \C^n$ be an [[open set|02BH3MCBQB]].
Then $D$ is a *domain of holomorphy* if there are no nonempty open sets $U_1, U_2 \subseteq \C^n$ such that:

1. $U_2$ is [[connected|0SPE8WNWFV]] and $U_2 \not\subseteq D$;
2. $U_1 \subseteq D \cap U_2$;
3. for every [[holomorphic function|507C34W0ZT]] $f$ on $D$ there is a holomorphic function $g$ on $U_2$ with $f = g$ on $U_1$.

## Remarks

Example: every convex open set $D \subsetneq \C^n$ is a domain of holomorphy.
For a boundary point $p$ there is a complex linear function $\ell$ with $\operatorname{Re} \ell(z) < \operatorname{Re} \ell(p)$ for all $z \in D$, and $f_p(z) = 1/(\ell(z) - \ell(p))$ is holomorphic on $D$ and unbounded near $p$.
Suppose $U_1, U_2$ satisfied the three clauses, and let $W$ be a component of $D \cap U_2$ that meets $U_1$.
Since $U_2$ is connected and not contained in $D$, it contains a boundary point $p$ of $W$, and $p$ lies on the boundary of $D$.
The extension $g$ of $f_p$ equals $f_p$ on $W$ by the identity theorem, and $g$ is continuous at $p$.
So $f_p$ is bounded near $p$ on $W$, which is a contradiction.

Non-example: $D = \set{z \in \C^2 : 0 < \abs{z} < 1}$.
Take $U_2 = \set{\abs{z} < 1}$ and $U_1 = D$.
Clauses 1 and 2 hold, and clause 3 holds by [[Hartogs's extension theorem|FSGES670AZ]] with $K = \set{0}$.

Every open subset of $\C$ is a domain of holomorphy.
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<document source="https://tamnd.github.io/math2/node/6G03V0BVZT.md" tag="6G03V0BVZT" kind="concept">
# Co-Heyting algebra

## Summary

A co-Heyting algebra is a bounded lattice with a difference operation $a \setminus b$, the least element $c$ such that $a \le b \vee c$.
It is the order dual of a [[Heyting algebra|NG943XNQWH]]: turning a co-Heyting algebra upside down gives a Heyting algebra, and conversely.
Every co-Heyting algebra is a [[distributive lattice|0HM0HQCK6N]].
The closed sets of a topological space form a co-Heyting algebra, and co-Heyting algebras serve as algebraic models of paraconsistent logics.

## Explanations

### L1 human

In a Heyting algebra one asks for the largest element that, together with $a$, stays below $b$.
In a co-Heyting algebra one asks instead for the smallest piece $c$ that one must add to $b$ in order to cover $a$.
The picture is the family of closed sets of the plane, where $A \setminus B$ is the closure of the ordinary set difference.
For example, on the real line, if $A = [0, 2]$ and $B = [1, 2]$, then the difference is $[0, 1]$, not $[0, 1)$, since it must be closed.
</document>

<document source="https://tamnd.github.io/math2/node/6G73DJH9T6.md" tag="6G73DJH9T6" kind="statement">
# Preissmann's theorem

## Statement

Let $(M, g)$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] with negative [[sectional curvature|1X42JAA349]].
Then every nontrivial abelian subgroup of the [[fundamental group|S1XK3HXHFX]] $\pi_1(M)$ is infinite cyclic.

## Remarks

The idea of the proof: by the [[Cartan–Hadamard theorem|428R02EYWS]], the universal cover $\tilde M$ is diffeomorphic to $\R^n$ and $\pi_1(M)$ acts on it by deck isometries.
Each nontrivial element has infinite order and, by compactness, moves points along a unique invariant [[geodesic|NSYMQMHPDN]], its axis.
Negative curvature forces two commuting elements to have the same axis.
So an abelian subgroup acts by translations on one geodesic line, and a discrete group of translations of a line is infinite cyclic.

Example: let $M_1$ and $M_2$ be compact manifolds whose fundamental groups contain elements of infinite order, such as $M_1 = M_2 = S^1$.
Then $\pi_1(M_1 \times M_2)$ contains $\Z^2$, so $M_1 \times M_2$ carries no metric of negative sectional curvature.

Negative curvature is needed: the flat torus $\R^2 / \Z^2$ has curvature $0$ and fundamental group $\Z^2$.

## Explanations

### L1 human

In a negatively curved compact space, two loops that commute must go "in the same direction": both are powers of a single loop.
So the fundamental group of such a space contains no copy of $\Z^2$, the group of loops on a torus.

### L2 human

Preissmann's theorem is a group-theoretic obstruction to negative curvature.
It rules out, for example, every product of two compact manifolds whose fundamental groups have elements of infinite order.
It is an early instance of the link between negative curvature and the algebra of the fundamental group that later led to Gromov's hyperbolic groups.
</document>

<document source="https://tamnd.github.io/math2/node/6GACD52XKX.md" tag="6GACD52XKX" kind="definition">
# Generic point

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $Z \subseteq X$ be a [[closed set|0ZCK8DQYN1]].
A point $\eta \in Z$ is a *generic point* of $Z$ if the [[closure|0QEAJB91GZ]] of $\set{\eta}$ is $Z$.
A generic point of $X$ is a generic point of the closed set $X$.

## Remarks

A closed set with a generic point is irreducible: it is not the union of two smaller closed sets.
In a [[scheme|5A35TYZD02]], every irreducible closed subset has a unique generic point, so the underlying space of a scheme is a [[sober space|4N4DNDR22Z]].
Example: in $X = \operatorname{Spec} \Z$, the closure of $\set{(0)}$ is $V(0) = X$, because $(0)$ is contained in every prime ideal.
So $(0)$ is the generic point of $X$.
Non-example: the point $(p)$, for a prime $p$, is not a generic point of $X$, since $\set{(p)}$ is already closed.
Non-example: $X$ itself has no generic point when $X = \operatorname{Spec} k[x, y]/(xy)$ for a field $k$, since $X$ is the union of the two lines $V(x)$ and $V(y)$.
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<document source="https://tamnd.github.io/math2/node/6GATRFMJYY.md" tag="6GATRFMJYY" kind="statement">
# Basu's theorem

## Statement

Let $\set{P_\theta : \theta \in \Theta}$ be a family of probability measures for the data $X$.
Let $T = T(X)$ be a [[sufficient|QNMR5F525C]] statistic that is boundedly [[complete|PF4RHTZWXH]], and let $V = V(X)$ be an [[ancillary|NR6MGAB5V0]] statistic.
Then, for every $\theta \in \Theta$, the statistics $T$ and $V$ are [[independent|0XCE070V8B]] under $P_\theta$.

## Remarks

Proof idea: fix a measurable set $B$ and let $q = P_\theta(V \in B)$, which does not depend on $\theta$ since $V$ is ancillary.
By sufficiency, the [[conditional probability|0YAMJRK04E]] $k(T) = P_\theta(V \in B \mid T)$ can be chosen free of $\theta$.
Then $k(T) - q$ is bounded and $E_\theta[k(T) - q] = 0$ for all $\theta$, so bounded completeness gives $k(T) = q$ almost surely; that is, $V \in B$ is independent of $T$.
Example: for $X_1, \dots, X_n$ i.i.d. $\mathcal{N}(\mu, \sigma^2)$ with $\sigma^2$ known and $\mu \in \R$, $\bar{X}$ is complete and sufficient for $\mu$, and the sample variance $S^2$ is ancillary; so $\bar{X}$ and $S^2$ are independent.
Since $\sigma^2$ was arbitrary, this proves the independence of $\bar{X}$ and $S^2$ for every normal sample.
Completeness cannot be dropped: for $n \ge 2$ the whole sample $T = (X_1, \dots, X_n)$ is sufficient, and $V = X_1 - X_2$ is ancillary, but $V$ is a non-constant function of $T$, so they are not independent.

## Explanations

### L1 human

A complete sufficient statistic holds all the information about the parameter, and an ancillary statistic holds none.
Basu's theorem says that two such statistics do not even influence each other: they are independent.
This gives quick proofs of independence results that are otherwise hard to compute.

### L2 human

The best-known use is the independence of the sample mean and the sample variance of a normal sample, which underlies the $t$-distribution of $\sqrt{n}(\bar{X} - \mu)/S$.
More generally, in an exponential family of full rank the natural sufficient statistic is complete, so it is independent of every ancillary statistic.
The theorem also allows one to compute $E_\theta[V \mid T]$ as $E_\theta[V]$ for an integrable ancillary $V$, which simplifies Rao–Blackwell computations.
</document>

<document source="https://tamnd.github.io/math2/node/6GD25S53ND.md" tag="6GD25S53ND" kind="concept">
# Group algebra of a locally compact group

## Summary

The group algebra of a [[locally compact group|AHV05ZBCYW]] $G$ is the space $L^1(G)$ of integrable functions on $G$.
Its product is [[convolution|2Z29JN3VES]], and it has a natural involution.
It is a Banach $*$-algebra, and it plays for $G$ the role that the finite [[group algebra|MNP2HBWB5Q]] plays for a finite group.
Unitary representations of $G$ correspond to nondegenerate $*$-representations of $L^1(G)$, so the representation theory of $G$ becomes the representation theory of one algebra.

## Explanations

### L1 human

For a finite group one can add and multiply formal combinations of group elements.
For a group like the real line there are too many elements for finite sums, so one uses functions instead, with integrals in place of sums.
Multiplying two functions in this algebra means convolving them, and the point mass at a group element would play the role of that element.
For the integers, the group algebra consists of absolutely summable sequences, multiplied like the coefficients of power series in $z$ and $z^{-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/6GD30FGYDZ.md" tag="6GD30FGYDZ" kind="statement">
# Cantor normal form

## Statement

Let $\alpha > 0$ be an [[ordinal number|0RDJK7EQYY]], with [[ordinal arithmetic|KAEDCQ8RM6]].
Then there are an integer $k \ge 1$, ordinals $\beta_1 > \beta_2 > \cdots > \beta_k$ and positive integers $c_1, \ldots, c_k$ such that
$$\alpha = \omega^{\beta_1} \cdot c_1 + \omega^{\beta_2} \cdot c_2 + \cdots + \omega^{\beta_k} \cdot c_k.$$
The integer $k$, the ordinals $\beta_i$ and the integers $c_i$ are unique.

## Remarks

The proof is by induction on $\alpha$.
Let $\beta_1$ be the largest ordinal with $\omega^{\beta_1} \le \alpha$.
Division with remainder gives $\alpha = \omega^{\beta_1} \cdot c_1 + \rho$ with $0 < c_1 < \omega$ and $\rho < \omega^{\beta_1}$, and the induction hypothesis applies to $\rho$.
Example: $\omega^2 + \omega \cdot 3 + 5$ is already in normal form, with exponents $2 > 1 > 0$.
The normal form has $\beta_1 \le \alpha$, and equality can occur: for $\varepsilon_0$, the least ordinal with $\omega^{\varepsilon_0} = \varepsilon_0$, the normal form is $\omega^{\varepsilon_0}$.
For ordinals below [[epsilon nought|KT50SMH3EC]], the exponents are smaller than the ordinal, and repeating the normal form on them gives a finite notation.
The same holds with any base $b \ge 2$ in place of $\omega$, with coefficients $0 < c_i < b$.

## Explanations

### L1 human

Every ordinal can be written as a polynomial in $\omega$, with decreasing exponents and positive whole coefficients.
This is like writing a whole number in base ten, but with base $\omega$.
For instance, $\omega^2 \cdot 2 + 7$ means two copies of $\omega^2$ followed by seven more points.
The expression is unique, so it can be used to name and compare ordinals.

### L2 human

The Cantor normal form makes ordinal arithmetic computable on notations.
Two ordinals in normal form are compared by their leading terms, as polynomials are compared by degree.
Iterating the form on exponents gives the standard notation system for ordinals below $\varepsilon_0$, which appears in Gentzen's consistency proof for Peano arithmetic and in [[Goodstein's theorem|XQMNJF241J]].
Beyond $\varepsilon_0$ one needs stronger notations, such as those of the [[Veblen function|JQFAE9S2F4]].
</document>

<document source="https://tamnd.github.io/math2/node/6GG8AS9XX4.md" tag="6GG8AS9XX4" kind="definition">
# Legendre integral transform

## Definition

Let $f \colon [-1, 1] \to \C$ be Lebesgue integrable, and let $P_n$ denote the [[Legendre polynomial|QFYGJVA996]] of degree $n$.
The *Legendre transform* of $f$ is the sequence $(\tilde f(n))_{n \ge 0}$ given by
$$\tilde f(n) = \int_{-1}^{1} P_n(x)\, f(x)\, dx, \qquad n = 0, 1, 2, \dots$$

## Remarks

The integrals converge because $\abs{P_n(x)} \le 1$ on $[-1, 1]$.
Let $f \in L^2([-1, 1])$, the [[$L^2$ space|SASP14MA1Z]].
Then the inverse transform is the [[Fourier–Legendre series|RK2G9HN63E]]
$$f(x) = \sum_{n=0}^{\infty} \frac{2n + 1}{2}\, \tilde f(n)\, P_n(x),$$
with convergence in $L^2([-1, 1])$; this follows from $\int_{-1}^{1} P_m P_n\, dx = \frac{2}{2n+1} \delta_{mn}$ and the completeness of the Legendre polynomials.
If $f$ is twice continuously differentiable on $[-1, 1]$, integration by parts gives $\int_{-1}^{1} P_n(x)\, \frac{d}{dx}\big[(1 - x^2) f'(x)\big]\, dx = -n(n+1)\, \tilde f(n)$, the rule that makes the transform useful for Legendre's equation.

Example: $f(x) = x^2$.
Since $x^2 = \frac{1}{3} P_0(x) + \frac{2}{3} P_2(x)$, orthogonality gives $\tilde f(0) = \frac{1}{3} \cdot 2 = \frac{2}{3}$, $\tilde f(2) = \frac{2}{3} \cdot \frac{2}{5} = \frac{4}{15}$, and $\tilde f(n) = 0$ otherwise.
Directly, $\tilde f(0) = \int_{-1}^{1} x^2\, dx = \frac{2}{3}$.

Non-example: $f(x) = \frac{1}{1 - x}$ is not integrable on $[-1, 1]$, and $\tilde f(0) = \int_{-1}^{1} \frac{dx}{1 - x}$ diverges.
</document>

<document source="https://tamnd.github.io/math2/node/6GSTSR4V09.md" tag="6GSTSR4V09" kind="example">
# Category of rings

## Example

The *category of rings* $\mathbf{Ring}$ has the [[rings|0B1261B5ED]] with identity as objects and the [[ring homomorphisms|0VTH8KPCG2]] that send $1$ to $1$ as morphisms.
Composition is the composition of maps, and the identity of $R$ is $\id_R$.
This is a [[category|A2DE66937D]].
A composite of two maps that preserve sums, products and $1$ again preserves them, and identity maps preserve them, so composites and identities are morphisms.
Composition of maps is associative.

## Remarks

The ring $\Z$ is an [[initial object|EK85VSSRMR]]: for each ring $R$ the only homomorphism $\Z \to R$ sends $n$ to $n \cdot 1_R$.
The zero ring is a terminal object, since every ring has exactly one map to it.
The zero ring is not initial, because a homomorphism from it must send $1 = 0$ to $1_R = 0_R$, which fails when $R$ is nonzero; so $\mathbf{Ring}$ has no zero object.
The [[product|S5744QZV4N]] is the direct product $R \times S$ with componentwise operations.
The commutative rings form a full subcategory $\mathbf{CRing}$, in which the [[coproduct|4FJ0S71P58]] of $R$ and $S$ is the tensor product $R \otimes_{\Z} S$.
The inclusion $\Z \to \Q$ is an epimorphism of $\mathbf{Ring}$ that is not surjective: two homomorphisms out of $\Q$ that agree on $\Z$ agree on all fractions $m/n$.

## Explanations

### L1 human

The category of rings collects all rings and the maps between them that respect addition, multiplication and the unit.
The integers sit at the start of this category, since every ring receives exactly one map from $\Z$.
For example, reduction modulo $5$ is a morphism $\Z \to \Z/5\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/6GVPDQMBN7.md" tag="6GVPDQMBN7" kind="definition">
# QR algorithm

## Definition

Let $A$ be an $n \times n$ real [[matrix|0VD9ZSWWJG]].
The (unshifted) *QR algorithm* builds a sequence of matrices $A_0, A_1, A_2, \ldots$ as follows.

1. Set $A_0 = A$.
2. For $k \ge 0$, compute a [[QR decomposition|0ERWG4NXQ9]] $A_k = Q_k R_k$, with $Q_k$ an [[orthogonal matrix|0W02QESPJQ]] and $R_k$ [[upper triangular|0N4XY7Z3NN]].
3. Set $A_{k+1} = R_k Q_k$.

The same definition applies to complex matrices with unitary $Q_k$.

## Remarks

Since $R_k = Q_k^T A_k$, one has $A_{k+1} = Q_k^T A_k Q_k$, so every $A_k$ is [[similar|0E89HQATM8]] to $A$ and has the same [[eigenvalues|0BQAA2BK3Q]].
Suppose the eigenvalues of $A$ have pairwise distinct absolute values, and a mild condition on the eigenvectors holds.
Then the entries of $A_k$ below the diagonal tend to $0$, and the diagonal entries tend to the eigenvalues.
Example: for $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$, $Q_0 = \tfrac{1}{\sqrt5}\begin{pmatrix} 2 & -1 \\ 1 & 2 \end{pmatrix}$ and $R_0 = \tfrac{1}{\sqrt5}\begin{pmatrix} 5 & 4 \\ 0 & 3 \end{pmatrix}$.
So $A_1 = R_0 Q_0 = \tfrac15 \begin{pmatrix} 14 & 3 \\ 3 & 6 \end{pmatrix}$; the trace stays $4$ and the off-diagonal entry drops from $1$ to $0.6$.
Non-example of convergence: for the rotation $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, which is orthogonal, one may take $Q_0 = A$, $R_0 = I$, so $A_1 = A$ and the sequence is constant; its eigenvalues $\pm i$ have equal absolute value.
Practical versions first reduce $A$ to [[Hessenberg form|YCEEWJTX56]], use shifts $A_k - \sigma_k I = Q_k R_k$, $A_{k+1} = R_k Q_k + \sigma_k I$, and split off converged eigenvalues.
The algorithm was found independently by John G. F. Francis and Vera Kublanovskaya around 1960.
</document>

<document source="https://tamnd.github.io/math2/node/6GW1FHF9MB.md" tag="6GW1FHF9MB" kind="definition">
# Schubert calculus

## Definition

Let $1 \le d < n$, and let $\mathrm{Gr}(d, n)$ be the complex [[Grassmannian|CYPA882RB9]].
For each partition $\lambda$ in the $d \times (n - d)$ rectangle, let $\sigma_\lambda \in H^{2 \abs{\lambda}}(\mathrm{Gr}(d, n); \Z)$ be the class of the [[Schubert variety|Z014Q8KPDD]] $X_\lambda$.
These classes form a $\Z$-basis of the [[cohomology ring|RDT4785M4D]] $H^{*}(\mathrm{Gr}(d, n); \Z)$.
*Schubert calculus* is the computation of the structure constants $c_{\lambda \mu}^{\nu}$ in
$$\sigma_\lambda \cdot \sigma_\mu = \sum_\nu c_{\lambda \mu}^{\nu} \, \sigma_\nu,$$
and their use to count intersection points of Schubert varieties in general position.
The same name is used for the analogous computations in the [[Chow ring|EBEF6PEK4N]] and for flag varieties.

## Remarks

For Grassmannians the constants $c_{\lambda \mu}^{\nu}$ are the Littlewood–Richardson coefficients, and they are nonnegative.
The Pieri rule computes the product with $\sigma_k$, for $k = 1, \ldots, n - d$.
It says that $\sigma_k \sigma_\lambda$ is the sum of $\sigma_\nu$ over the $\nu$ obtained from $\lambda$ by adding $k$ boxes, no two in the same column.
Only the $\nu$ that fit in the $d \times (n - d)$ rectangle are counted.

Example: in $\mathrm{Gr}(2, 4)$, the Pieri rule gives $\sigma_1^2 = \sigma_2 + \sigma_{1,1}$, $\sigma_1 \sigma_2 = \sigma_1 \sigma_{1,1} = \sigma_{2,1}$ and $\sigma_1 \sigma_{2,1} = \sigma_{2,2}$.
So $\sigma_1^4 = 2 \sigma_{2,2}$, and $\sigma_{2,2}$ is the class of a point.
Hence $2$ lines in $\mathbb{P}^3$ meet four general lines.

Non-example: the count of lines meeting four lines in a common plane is not given by $\sigma_1^4$.
The four Schubert varieties are not in general position, and they meet in a positive-dimensional set.
</document>

<document source="https://tamnd.github.io/math2/node/6GWMGFGFJJ.md" tag="6GWMGFGFJJ" kind="statement">
# Hahn–Banach separation theorem

## Statement

Let $X$ be a topological vector space over $\R$, and let $A, B \subseteq X$ be nonempty disjoint [[convex sets|02AS45T3H9]].

1. If $A$ is open, then there are a continuous linear functional $f \colon X \to \R$ and a real number $t$ such that $f(a) < t \le f(b)$ for all $a \in A$ and $b \in B$.
2. If $X$ is a [[locally convex space|B6GY2KJMWW]], $A$ is [[compact|0R6EWFKRCF]] and $B$ is [[closed|0ZCK8DQYN1]], then the sets can be strictly separated.
   There are a continuous linear functional $f \colon X \to \R$ and real numbers $s < t$ such that $f(a) < s$ and $f(b) > t$ for all $a \in A$ and $b \in B$.

## Remarks

For part 1, the set $C = A - B + x_0$, with $x_0 \in B - A$ fixed, is an open convex neighbourhood of $0$ that misses $x_0$.
Its [[Minkowski functional|PAX9SHDHDW]] $p$ is sublinear, and the [[Hahn–Banach theorem|QB5AE3VYZZ]] extends the functional $\lambda x_0 \mapsto \lambda$ to a linear $f \le p$; then $f < 1 = f(x_0)$ on $C$, which gives the separation.
Part 2 reduces to part 1: compactness of $A$ gives a convex open neighbourhood $U$ of $0$ with $(A + U) \cap B = \emptyset$.
For a complex space, apply the theorem to the underlying real space and take the real part of a complex functional.
The hypotheses matter: in $\R^2$ the closed convex sets $A = \set{(x, y) : y \le 0}$ and $B = \set{(x, y) : x > 0,\ y \ge 1/x}$ are disjoint, but no line separates them strictly, so part 2 needs compactness of one set.

## Explanations

### L1 human

Two convex sets that do not meet can be put on the two sides of a flat wall.
In the plane the wall is a line and in space it is a plane.
In an infinite-dimensional space it is the set where a continuous linear functional takes one value.
If one set is compact and the other closed, there is even a gap between the sets and the wall.

### L2 human

This is the geometric form of the [[Hahn–Banach theorem|QB5AE3VYZZ]].
It shows that a closed convex set in a locally convex space is the intersection of the closed half-spaces that contain it.
Hence closed convex sets are closed for the [[weak topology|P6KHVSTPZV]], which is the key step in [[Mazur's lemma|7ZR88SCSRF]].
It is also the main tool in the proofs of the [[Krein–Milman theorem|RSRAY0Q68S]], the [[Goldstine theorem|K4S9G3AQBM]] and the [[bipolar theorem|FX5PWKCMBQ]].
</document>

<document source="https://tamnd.github.io/math2/node/6GWRCGBGWC.md" tag="6GWRCGBGWC" kind="definition">
# Dolbeault cohomology

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] and let $p, q \ge 0$ be integers.
Let $\mathcal{A}^{p,q}(X)$ be the complex vector space of smooth [[(p,q)-forms|DC81TCR98Y]] on $X$, with $\mathcal{A}^{p,-1}(X) = 0$.
Let $\bar\partial \colon \mathcal{A}^{p,q}(X) \to \mathcal{A}^{p,q+1}(X)$ be the [[dbar operator|M1VKCW3RFW]], which satisfies $\bar\partial \circ \bar\partial = 0$.
The *Dolbeault cohomology group* of bidegree $(p,q)$ is the quotient vector space
$$H^{p,q}_{\bar\partial}(X) = \frac{\ker\big(\bar\partial \colon \mathcal{A}^{p,q}(X) \to \mathcal{A}^{p,q+1}(X)\big)}{\im\big(\bar\partial \colon \mathcal{A}^{p,q-1}(X) \to \mathcal{A}^{p,q}(X)\big)}.$$

## Remarks

The class of a $\bar\partial$-closed form $\alpha$ is written $[\alpha]$; two closed forms define the same class when they differ by $\bar\partial \beta$.
Example: on $X = \C$, every smooth $(0,1)$-form $g\, d\bar z$ with compact support is $\bar\partial$ of the function $u(z) = \frac{1}{2\pi i} \int_\C \frac{g(w)}{w - z}\, dw \wedge d\bar w$, so it gives the zero class in $H^{0,1}_{\bar\partial}(\C)$; in fact $H^{0,1}_{\bar\partial}(\C) = 0$.
Example: $H^{p,0}_{\bar\partial}(X)$ is the space of holomorphic $p$-forms on $X$, since there are no $(p,-1)$-forms to divide by.
A nonzero class: let $X = \C/\Lambda$ with $\Lambda = \Z + i\Z$.
The form $d\bar z$ is $\bar\partial$-closed on $X$, but it is not $\bar\partial u$ for a smooth function $u$ on $X$.
Indeed, if it were, the lift $\tilde u$ of $u$ to $\C$ would satisfy $\partial \tilde u / \partial \bar z = 1$, so $h = \tilde u - \bar z$ would be entire.
Then $h(z + 1) = h(z) - 1$ and $h(z + i) = h(z) + i$, so $h'$ is a doubly periodic entire function; it is bounded, hence a constant $a$ by Liouville's theorem.
The two relations give $a = -1$ and $a = 1$, a contradiction.
So $[d\bar z] \ne 0$ in $H^{0,1}_{\bar\partial}(X)$.
The groups are also written $H^{p,q}(X)$ when no confusion with other cohomologies arises.
</document>

<document source="https://tamnd.github.io/math2/node/6GWZN8FXC6.md" tag="6GWZN8FXC6" kind="definition">
# Harmonic majorant

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]], and let $u \colon \Omega \to [-\infty, \infty)$ be a function, usually [[subharmonic|Z1MHMGFNFW]].
A *harmonic majorant* of $u$ is a [[harmonic function|0DEB0JMN0W]] $h$ on $\Omega$ such that $u(x) \le h(x)$ for every $x \in \Omega$.

## Remarks

Symmetrically, a harmonic minorant of $v$ is a harmonic $h$ with $h \le v$, so $h$ is a harmonic minorant of $v$ exactly when $-h$ is a harmonic majorant of $-v$.
For a holomorphic $f$ on the unit disc and $0 < p < \infty$, $f$ lies in the [[Hardy space|PNNNFJVF6W]] $H^p$ if and only if the subharmonic function $\abs{f}^p$ has a harmonic majorant.

Example: on the unit disc $\mathbb{D} \subseteq \C$, let $u(z) = \abs{z}^2$.
The constant $h = 1$ is harmonic and $\abs{z}^2 < 1$ on $\mathbb{D}$, so $h$ is a harmonic majorant of $u$.

Non-example: on all of $\C$, $u(z) = \abs{z}^2$ has no harmonic majorant.
A harmonic $h$ with $h \ge \abs{z}^2 \ge 0$ would be a harmonic function bounded below on $\C$, hence constant by [[Liouville's theorem for harmonic functions|QK1KWYZNRD]].
No constant is at least $\abs{z}^2$ for all $z$.
</document>

<document source="https://tamnd.github.io/math2/node/6GY3X4Y8X5.md" tag="6GY3X4Y8X5" kind="concept">
# Inverse scattering transform

## Summary

The inverse scattering transform is a method to solve the initial value problem of some nonlinear evolution equations, such as the [[Korteweg–De Vries equation|KTT75QC72Z]].
The solution at time $t$ is the potential of a linear spectral problem, such as a [[Schrödinger operator|89AA6XJY5K]] in one variable.
Its scattering data evolve in time by simple explicit formulas, and the potential is then recovered from the data.
It plays for these nonlinear equations the role that the Fourier transform plays for linear equations with constant coefficients.

## Explanations

### L1 human

The method works in three steps.
First, one turns the initial profile of the wave into spectral data: its bound states and how it reflects incoming waves.
Second, one moves these data forward in time, which is easy because each piece evolves on its own.
Third, one rebuilds the profile from the new data by solving a linear integral equation.
The bound states give the [[solitons|63D4AXYCT4]], which is why a localised initial wave splits into a train of solitons and a small spreading tail.
</document>

<document source="https://tamnd.github.io/math2/node/6GY4GAMY9Z.md" tag="6GY4GAMY9Z" kind="definition">
# Riemann's differential equation

## Definition

Let $a, b, c \in \C$ be distinct, and let $\alpha, \alpha', \beta, \beta', \gamma, \gamma' \in \C$ satisfy
$$\alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1.$$
*Riemann's differential equation* with these data is the [[ordinary differential equation|0NT64JV96A]]
$$\begin{aligned} w'' &+ \left[\frac{1 - \alpha - \alpha'}{z - a} + \frac{1 - \beta - \beta'}{z - b} + \frac{1 - \gamma - \gamma'}{z - c}\right] w' \\ &+ \left[\frac{\alpha\alpha'(a - b)(a - c)}{z - a} + \frac{\beta\beta'(b - c)(b - a)}{z - b} + \frac{\gamma\gamma'(c - a)(c - b)}{z - c}\right] \frac{w}{(z - a)(z - b)(z - c)} = 0. \end{aligned}$$

## Remarks

Its singular points are $a$, $b$, $c$, all [[regular singular points|H96HXG6CA0]], and $\infty$ is an ordinary point.
The exponents are $\alpha, \alpha'$ at $a$, $\beta, \beta'$ at $b$ and $\gamma, \gamma'$ at $c$.

Check at $z = a$: multiplying by $(z - a)^2$ and letting $z \to a$, the indicial equation is $r(r - 1) + (1 - \alpha - \alpha') r + \alpha\alpha' = (r - \alpha)(r - \alpha') = 0$, so the exponents at $a$ are $\alpha$ and $\alpha'$.

Example: take the singular points $0$, $\infty$, $1$, using the limiting form in which the factors with $b$ are dropped as $b \to \infty$.
With exponents $0, 1 - C$ at $0$, $A, B$ at $\infty$ and $0, C - A - B$ at $1$, the equation becomes the [[hypergeometric differential equation|W8VE25R5A7]] with parameters $A, B, C$.

Non-example: exponents with sum different from $1$ are not allowed, because the equation then acquires a fourth singular point at $\infty$.
The condition on the sum is Fuchs's relation for three singular points.

The solutions are written with the [[Riemann P-symbol|WWW053WXP2]].
</document>

<document source="https://tamnd.github.io/math2/node/6GYQNB2862.md" tag="6GYQNB2862" kind="statement">
# Littlewood–Paley inequality

## Statement

Let $n \ge 1$ and $1 < p < \infty$, and let $Sf$ be the [[Littlewood–Paley square function|H4E1PW67Z1]] on $\R^n$, built from a fixed cut-off $\chi$.
Then there are constants $0 < c \le C$, depending only on $n$, $p$ and $\chi$, such that
$$c \norm{f}_{L^p(\R^n)} \le \norm{Sf}_{L^p(\R^n)} \le C \norm{f}_{L^p(\R^n)}$$
for every $f$ in the [[Lp space|SASP14MA1Z]] $L^p(\R^n)$.

## Remarks

For $p = 2$ the inequality follows from Plancherel's theorem.
For the upper bound, view $f \mapsto (\Delta_j f)_{j \in \Z}$ as a single operator from $L^p(\R^n)$ to $L^p(\R^n; \ell^2)$.
Its kernel $(\check\psi_j)_j$ satisfies the size and smoothness estimates of a vector-valued [[Calderón–Zygmund operator|3SCXCW4AH3]], so the $L^2$ bound extends to $1 < p < \infty$.
The lower bound follows by duality.
Let $\tilde\Delta_j = \Delta_{j-1} + \Delta_j + \Delta_{j+1}$; its multiplier equals $1$ on the support of the multiplier of $\Delta_j$, so $\int f \bar g = \sum_j \int \Delta_j f \, \overline{\tilde\Delta_j g}$ for nice $f$ and $g$.
The Cauchy–Schwarz inequality in $j$ and [[Hölder's inequality|JQ9WXC729D]] give $\abs{\int f \bar g} \le \norm{Sf}_p \norm{\tilde S g}_{p'}$, and the upper bound for the square function $\tilde S$ of the $\tilde\Delta_j$ in $L^{p'}$ finishes the proof.

The inequality fails for $p = 1$ and $p = \infty$.
At $p = 1$ the upper bound fails already for an integrable $f$ with $\int f \neq 0$, and the correct substitute replaces $L^1$ by the real Hardy space $H^1$.

## Explanations

### L1 human

In $L^2$, a function split into frequency octaves has a norm that is exactly computable from the norms of the octaves.
The Littlewood–Paley inequality says that in $L^p$, for $1 < p < \infty$, the norm of $f$ can still be read off, up to constants, from the square function.
The square function combines the sizes of the octave pieces at each point.
So the octave pieces of $f$ do not cancel each other too much in $L^p$.

### L2 human

A typical consequence: for any choice of signs $\varepsilon_j \in \set{-1, 1}$, the operator $f \mapsto \sum_j \varepsilon_j \Delta_j f$ is bounded on $L^p$ with a bound independent of the signs.
This is the first step towards the Marcinkiewicz and Hörmander–Mikhlin multiplier theorems, where a multiplier is cut into dyadic pieces and each piece is handled separately.
The inequality is also the starting point for the characterization of Sobolev spaces $W^{k,p}$, Besov spaces and Triebel–Lizorkin spaces through the sizes of the $\Delta_j f$.
On the torus, the analogous statement with sharp dyadic blocks of the Fourier series is the original theorem of Littlewood and Paley.
</document>

<document source="https://tamnd.github.io/math2/node/6GZ8DXT7TN.md" tag="6GZ8DXT7TN" kind="example">
# Prandtl's lifting-line equation

## Example

Consider a wing of span $2s$ along the axis $-s \le y \le s$, in a uniform flow of speed $V$.
Let $c(y)$ be the chord, $\alpha(y)$ the geometric angle of attack and $\alpha_0(y)$ the angle of zero lift of the section at $y$.
Assume the sections have the thin airfoil lift slope $2\pi$.
*Prandtl's lifting-line equation* for the circulation $\Gamma(y)$ is
$$\Gamma(y) = \pi V c(y) \bigl(\alpha(y) - \alpha_0(y)\bigr) - \frac{c(y)}{4}\, \mathrm{PV}\!\int_{-s}^{s} \frac{\Gamma'(\eta)}{y - \eta}\, d\eta, \quad -s < y < s,$$
with $\Gamma(\pm s) = 0$, where $\mathrm{PV}$ is the [[Cauchy principal value|0SYR10HD4R]].
The integral term is $\pi V c(y)\, \alpha_i(y)$, where the induced angle is $\alpha_i(y) = \frac{1}{4\pi V}\, \mathrm{PV}\!\int_{-s}^{s} \frac{\Gamma'(\eta)}{y - \eta}\, d\eta$.
It is a singular integro-differential equation: the derivative of the unknown appears under a Cauchy kernel, so it is close to a [[singular integral equation|E30EA6ETZ4]].

Check with the elliptic distribution $\Gamma(y) = \Gamma_0 \sqrt{1 - y^2/s^2}$.
Substituting $y = s x$ and $\eta = s t$ gives
$$\alpha_i = \frac{\Gamma_0}{4\pi V s}\, \mathrm{PV}\!\int_{-1}^{1} \frac{t}{\sqrt{1 - t^2}\,(t - x)}\, dt = \frac{\Gamma_0}{4 V s},$$
since $\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{t\, dt}{\sqrt{1 - t^2}\,(t - x)} = 1$ for $-1 < x < 1$.
So the induced angle is constant along the span.
For an untwisted wing with $\alpha$ constant, $\alpha_0 = 0$ and elliptic chord $c(y) = c_0 \sqrt{1 - y^2/s^2}$, the equation reduces to $\Gamma_0 = \pi V c_0 \bigl(\alpha - \Gamma_0/(4 V s)\bigr)$.
It gives $\Gamma_0 = \pi V c_0 \alpha / \bigl(1 + \pi c_0/(4 s)\bigr)$.

## Remarks

The integral term is a [[finite Hilbert transform|5JYBXH9BP6]] of $\Gamma'$ after scaling to $[-1, 1]$.
A standard way to solve the equation in general expands $\Gamma$ in the Fourier sine series $\Gamma = \sum_n A_n \sin(n\theta)$ with $y = -s \cos\theta$.
The Cauchy integral acts diagonally on this basis, and the equation becomes a linear system for the coefficients $A_n$.

## Explanations

### L1 human

A wing of finite length leaves trailing vortices at its tips, and these push the air near the wing downwards.
This downwash lowers the effective angle of attack, so the lift at each point depends on the lift everywhere else along the span.
Prandtl's equation writes this coupling as one equation for the circulation along the span.
For an elliptic wing the downwash is the same everywhere, and the equation can be solved by hand.
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<document source="https://tamnd.github.io/math2/node/6H3WCCHZB5.md" tag="6H3WCCHZB5" kind="concept">
# Hecke algebra of a locally compact group

## Summary

Let $K$ be a compact open subgroup of a [[locally compact group|AHV05ZBCYW]] $G$.
The Hecke algebra of $(G, K)$ is the algebra of compactly supported $K$-bi-invariant functions on $G$ under [[convolution|2Z29JN3VES]].
It has a basis given by the indicator functions of the double cosets $K g K$.
The $K$-fixed vectors of a smooth representation of $G$ form a module over it.
This makes it a main tool for representations of $p$-adic groups.
For $G = \mathrm{GL}_n(\Q_p)$ and $K = \mathrm{GL}_n(\Z_p)$ it is commutative, and it is closely related to the [[Hecke operators|S4SV3QZ8B1]] on modular forms.

## Explanations

### L1 human

Pick a compact open subgroup $K$, like $\mathrm{GL}_2(\Z_p)$ inside $\mathrm{GL}_2(\Q_p)$.
The group is then cut into double cosets $KgK$, which are like coarse blocks of the group.
The Hecke algebra records how these blocks multiply: the product of two blocks spreads over several blocks with integer multiplicities.
For a discrete group with $K$ trivial, the blocks are single elements and one gets the ordinary group algebra.
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<document source="https://tamnd.github.io/math2/node/6H8AGEN6NY.md" tag="6H8AGEN6NY" kind="concept">
# Thom spectrum

## Summary

A Thom spectrum is a [[spectrum|1GHF4EKBXS]] whose spaces are the [[Thom spaces|5MNYT3AFPH]] of a compatible sequence of vector bundles.
The basic examples are $MO$, $MSO$ and $MU$, built from the universal bundles over the classifying spaces $BO(n)$, $BSO(n)$ and $BU(n)$.
By the Pontryagin–Thom construction, the homotopy groups of these spectra are the [[cobordism|T8BVGPPSZP]] groups of manifolds with the matching structure, as in [[Thom's cobordism theorem|HM23BV561Z]].
So Thom spectra turn geometric classification problems into problems of stable homotopy theory.

## Explanations

### L1 human

The Thom space of a vector bundle is obtained by adding one point at infinity to every fibre and then gluing all these points together.
If one adds a trivial line to a bundle, its Thom space becomes the suspension of the old Thom space.
A Thom spectrum strings together Thom spaces of bigger and bigger bundles with these suspension maps.
For example, $MO$ uses the universal $n$-plane bundle over $BO(n)$ for every $n$.
Its zeroth homotopy group is $\Z/2$, which counts finite sets of points up to cobordism: a set is a boundary exactly when it has an even number of points.
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<document source="https://tamnd.github.io/math2/node/6HAZ0AKKAR.md" tag="6HAZ0AKKAR" kind="definition">
# Frame

## Definition

Let $L$ be a [[complete lattice|D7A0ZGZT5S]].
It is a *frame* if for every $a \in L$ and every family $(b_i)_{i \in I}$ of elements of $L$,
$$a \wedge \bigvee_{i \in I} b_i = \bigvee_{i \in I} (a \wedge b_i).$$
A *frame homomorphism* between frames is a map that preserves finite meets, including the top element, and arbitrary joins, including the bottom element.

## Remarks

Example: for a [[topological space|0CQHFPGJ45]] $X$, the [[open sets|02BH3MCBQB]] of $X$ ordered by inclusion form a frame $\Omega(X)$; joins are unions, finite meets are intersections, and $U \cap \bigcup_i V_i = \bigcup_i (U \cap V_i)$.
Non-example: the lattice of subspaces of $\R^2$ is complete but not a frame; for three distinct lines $a, b, c$ through $0$, $a \wedge (b \vee c) = a$ while $(a \wedge b) \vee (a \wedge c) = \set{0}$.
A complete lattice is a frame if and only if it is a [[Heyting algebra|NG943XNQWH]], with implication $a \to b = \bigvee \set{c : a \wedge c \le b}$.
Every frame is a [[distributive lattice|0HM0HQCK6N]].
As ordered sets, frames, locales and complete Heyting algebras are the same objects; the three names differ in the maps one uses between them.
</document>

<document source="https://tamnd.github.io/math2/node/6HMT17DVX7.md" tag="6HMT17DVX7" kind="concept">
# Quantum dilogarithm

## Summary

The quantum dilogarithm is a [[q-analog|GFSDMJHHVA]] of the [[dilogarithm|G0X0NWSDB7]].
In its basic form it is the infinite product $(x; q)_\infty = \prod_{n \ge 0} (1 - x q^n)$, whose logarithm is a deformed dilogarithm series.
It satisfies a pentagon identity for $q$-commuting variables, a quantum version of the five-term relation.
It appears in quantum groups, quantum Teichmüller theory, cluster algebras and knot invariants.

## Explanations

### L1 human

The ordinary dilogarithm is the series $\sum_{n \ge 1} x^n / n^2$.
Replace one factor $n$ in the denominator by the "$q$-number" $1 - q^n$, and you get the series $\sum_{n \ge 1} x^n / \big(n (1 - q^n)\big)$.
Its exponential is a simple product, $\prod_{n \ge 0} (1 - x q^n)^{-1}$.
As $q$ tends to $1$, the deformed series times $1 - q$ tends back to the dilogarithm, so the classical function is a limit of the quantum one.
For example, with $q = 0$ the product $(x; q)_\infty$ is just $1 - x$, since every factor after the first equals $1$.
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<document source="https://tamnd.github.io/math2/node/6HSJBJ8TTQ.md" tag="6HSJBJ8TTQ" kind="concept">
# Hilbert space adjoint

## Summary

The adjoint of a bounded operator $T \colon H \to K$ between Hilbert spaces is the bounded operator $T^* \colon K \to H$ characterised by $\ip{Tx}{y} = \ip{x}{T^* y}$.
It extends the [[adjoint of a linear map|0R9NMSM1ZP]] between finite-dimensional inner product spaces, and the conjugate transpose of matrices.
The adjoint is used to define self-adjoint, normal, unitary and positive operators, and it makes the bounded operators on a Hilbert space a C*-algebra.

## Explanations

### L1 human

The adjoint lets one move an operator from one side of an inner product to the other.
For a complex matrix, it is the conjugate transpose.
For the right shift on square-summable sequences, which moves every entry one place to the right, the adjoint is the left shift, which deletes the first entry.
Operators equal to their own adjoint play the role of real numbers among operators.
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<document source="https://tamnd.github.io/math2/node/6HVAGKSQWB.md" tag="6HVAGKSQWB" kind="definition">
# Bump function

## Definition

Let $n \ge 1$.
A *bump function* on $\R^n$ is a function $\varphi \colon \R^n \to \R$ such that:

1. $\varphi$ is [[smooth|0BE3QDH73A]], that is, of class $C^\infty$;
2. the support $\overline{\set{x \in \R^n : \varphi(x) \ne 0}}$ of $\varphi$ is [[compact|00R0X1WGPX]].

The set of bump functions on $\R^n$ is often written $C_c^\infty(\R^n)$.

## Remarks

Example: let $\psi(x) = e^{-1/(1 - x^2)}$ for $\abs{x} < 1$ and $\psi(x) = 0$ for $\abs{x} \ge 1$.
Let $g(t) = e^{-1/t}$ for $t > 0$ and $g(t) = 0$ for $t \le 0$, which is [[smooth|09RTYYHXKA]].
Then $\psi(x) = g(1 - x^2)$ is a composition of smooth functions, so it is smooth.
Its support is $[-1, 1]$, which is compact.

Non-example: the Gaussian $e^{-x^2}$ is smooth, but it is nonzero at every point, so its support $\R$ is not compact.
The tent function $\max(0, 1 - \abs{x})$ has compact support $[-1, 1]$, but it is not differentiable at $0$ and $\pm 1$.

A bump function that is not identically zero is not [[analytic|0JD5ZNW0F4]] on $\R^n$, since an analytic function that vanishes on a nonempty open set of $\R^n$ is zero.
For each compact set $K$ and open set $U \supseteq K$ in $\R^n$, some bump function equals $1$ on $K$, takes values in $[0, 1]$, and has support in $U$.
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<document source="https://tamnd.github.io/math2/node/6HVCTJGB61.md" tag="6HVCTJGB61" kind="concept">
# Product sigma-algebra

## Summary

The product σ-algebra on a Cartesian product of [[measurable spaces|N4KDT83V3S]] is the smallest [[σ-algebra|0SV8K0TDGY]] that makes every coordinate projection measurable.
For two factors it is generated by the measurable rectangles $A \times B$.
It is the σ-algebra on which [[product measures|77PV0ZAVXT]] live and on which Fubini's theorem is stated.
For infinite products it is generated by the cylinder sets, which constrain only finitely many coordinates.

## Explanations

### L1 human

To measure sets in a product $X \times Y$, start from the rectangles $A \times B$ built from measurable pieces of each factor.
The product σ-algebra contains these rectangles and everything one can build from them with countable unions, countable intersections and complements.
For the real line with its Borel sets, this produces exactly the Borel sets of the plane.
For infinitely many coin tosses, it contains the events that depend on finitely many tosses, and countable combinations of them, such as "heads appears infinitely often".
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<document source="https://tamnd.github.io/math2/node/6HVDTYAWX1.md" tag="6HVDTYAWX1" kind="definition">
# Complete set of orthogonal idempotents

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $n \ge 1$ be an integer.
A *complete set of orthogonal idempotents* of $R$ is a family $e_1, \dots, e_n$ of elements of $R$ such that:

1. each $e_i$ is an [[idempotent|0X4JKZXJ86]], that is, $e_i^2 = e_i$;
2. $e_i e_j = 0$ for all $i \ne j$;
3. $e_1 + \cdots + e_n = 1$.

## Remarks

Many authors also require each $e_i$ to be nonzero.
Example: in $M_n(K)$ over a field $K$, the diagonal matrix units $E_{11}, \dots, E_{nn}$ form a complete set of orthogonal idempotents.
The check: $E_{ii} E_{jj}$ equals $E_{ii}$ if $i = j$ and $0$ otherwise, and $E_{11} + \cdots + E_{nn} = I_n$.
Non-example: in $M_2(K)$, the family consisting of $E_{11}$ alone satisfies clauses 1 and 2 but not clause 3, since $E_{11} \ne I_2$.
For any idempotent $e$, the pair $e, 1 - e$ is a complete set of orthogonal idempotents.
A complete set of orthogonal idempotents gives a decomposition $R = Re_1 \oplus \cdots \oplus Re_n$ of $R$ as a left module over itself.
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<document source="https://tamnd.github.io/math2/node/6HVZ30TS5F.md" tag="6HVZ30TS5F" kind="definition">
# Circle map

## Definition

Let $\Omega \in \R$ and $K \ge 0$.
Let $F_{\Omega, K} \colon \R \to \R$ be the map
$$F_{\Omega, K}(x) = x + \Omega - \frac{K}{2\pi} \sin(2\pi x).$$
Since $F_{\Omega, K}(x + 1) = F_{\Omega, K}(x) + 1$, it induces a map $f_{\Omega, K}$ of the [[circle|060W5CBZP6]] $\R / \Z$, given by $f_{\Omega, K}(x + \Z) = F_{\Omega, K}(x) + \Z$.
The map $f_{\Omega, K}$ is the *(standard) circle map* with parameters $\Omega$ and $K$, and $F_{\Omega, K}$ is its lift.

## Remarks

Example: for $K = 0$ the circle map is the rotation $x \mapsto x + \Omega$; for irrational $\Omega$ it is an [[irrational rotation|2Q1J7RZ4TK]].

The derivative is $F'_{\Omega, K}(x) = 1 - K \cos(2\pi x)$.
For $0 \le K < 1$ it is positive everywhere, so $f_{\Omega, K}$ is a [[diffeomorphism|0KZZ70ZHXP]] of the circle.
For $K = 1$ the derivative vanishes at $x = 0$, so $f_{\Omega, 1}$ is a [[homeomorphism|0QT00PB5G2]] with a [[critical point|0YTCMFB4HC]], but not a diffeomorphism.

Non-example: for $K > 1$ the derivative takes negative values, so $F_{\Omega, K}$ is not monotone and $f_{\Omega, K}$ is not invertible.
The map is still defined, but it is not a homeomorphism.

The parameter $\Omega$ is often taken in $[0, 1)$, since $f_{\Omega + 1, K} = f_{\Omega, K}$.
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<document source="https://tamnd.github.io/math2/node/6HWFBXYPZP.md" tag="6HWFBXYPZP" kind="concept">
# Schur complement

## Summary

The Schur complement of an invertible block $A$ in a [[block matrix|0ASDFEQS6P]] $\begin{pmatrix} A & B \\ C & D \end{pmatrix}$ is the matrix $D - C A^{-1} B$.
It is what remains of the block $D$ after block Gaussian elimination removes the block $C$.
It gives the [[determinant via the Schur complement|0KA7A4ENVJ]], formulas for the inverse of a block matrix, and tests for positive definiteness.

## Explanations

### L1 human

Think of a system of two vector equations in two unknown vectors $x$ and $y$.
If you solve the first equation for $x$ and put the result into the second, the matrix in front of $y$ is the Schur complement.
For numbers, take $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $a \ne 0$.
The Schur complement of $a$ is the number $d - cb/a$.
Then the determinant $ad - bc$ equals $a \cdot (d - cb/a)$.
</document>

<document source="https://tamnd.github.io/math2/node/6HYGN2WAQQ.md" tag="6HYGN2WAQQ" kind="definition">
# Blocking set

## Definition

Let $\Pi$ be a [[projective plane|DANFNN8C4Y]].
A *blocking set* of $\Pi$ is a set $B$ of points such that every line of $\Pi$ contains a point of $B$.
A blocking set is *nontrivial* if it contains no line of $\Pi$.

## Remarks

Some authors include "contains no line" in the definition of a blocking set.

Bruen's bound: a nontrivial blocking set in a plane of [[order|K2YVMTGTBB]] $n$ has at least $n + \sqrt{n} + 1$ points, with equality exactly for the point set of a Baer subplane.

Example: let $q$ be a prime power and view the [[plane $\mathrm{PG}(2, q)$|CSMB49BDPT]] as a subplane of $\mathrm{PG}(2, q^2)$.
Its $q^2 + q + 1$ points form a nontrivial blocking set of $\mathrm{PG}(2, q^2)$.
Check: each point of the subplane lies on $q^2 + 1$ lines of the large plane, of which $q + 1$ are lines of the subplane.
A line of the large plane through two points of the subplane is a line of the subplane, so the other $q^2 - q$ lines meet the subplane in one point.
This gives $(q^2 + q + 1)(q^2 - q) = q^4 - q$ such lines, and with the $q^2 + q + 1$ lines of the subplane that is $q^4 + q^2 + 1$, all lines of $\mathrm{PG}(2, q^2)$.
No line of the large plane lies inside the subplane, since it has $q^2 + 1 > q + 1$ points.

Non-example: the point set of a line $\ell$ meets every line, but it contains $\ell$, so it is a trivial blocking set.
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<document source="https://tamnd.github.io/math2/node/6J1X63Z7QZ.md" tag="6J1X63Z7QZ" kind="concept">
# Bergman projection

## Summary

The Bergman projection of a domain $\Omega \subseteq \C^n$ is the [[orthogonal projection|0B8KZRT28M]] of $L^2(\Omega)$ onto its closed subspace of holomorphic functions, the [[Bergman space|2D619WTPYC]].
It is an integral operator whose kernel is the [[Bergman kernel|8RWDHW0CXC]].
Its regularity, for example whether it maps smooth functions on the closure to smooth functions, governs the boundary behaviour of biholomorphic maps.
It is closely tied to the $\bar\partial$-Neumann problem, through Kohn's formula $P = I - \bar\partial^* N \bar\partial$.

## Explanations

### L1 human

Among all square-integrable functions on a domain, the holomorphic ones form a closed subspace.
The Bergman projection sends each function to the holomorphic function closest to it in the $L^2$ sense.
It does so by integrating the function against the Bergman kernel.
For example, on the unit disc it sends $\bar z$ to $0$ and $z^k$ to itself.
</document>

<document source="https://tamnd.github.io/math2/node/6J2AGD0BYY.md" tag="6J2AGD0BYY" kind="concept">
# Two-sided Laplace transform

## Summary

The two-sided, or bilateral, Laplace transform of a function $f$ on the whole real line is $\int_{-\infty}^{\infty} f(t) e^{-s t} \, dt$.
It treats past and future times alike, unlike the one-sided [[Laplace transform|0V49JEZW06]], which only sees $t \ge 0$.
It converges in a vertical strip of the complex plane, and on the imaginary axis it becomes the [[Fourier transform|DPVM34ADZV]].
In probability it gives the moment generating function, with $s$ replaced by $-s$.

## Explanations

### L1 human

The ordinary Laplace transform starts the clock at time $0$; the two-sided transform looks at all times, negative and positive.
Because of this, it needs $f$ to be small at both ends, and it usually exists only for $s$ in a band between two vertical lines.
For example, $e^{-\abs{t}}$ has two-sided transform $2/(1 - s^2)$ for $-1 < \operatorname{Re} s < 1$.
</document>

<document source="https://tamnd.github.io/math2/node/6J4C2P3ZHC.md" tag="6J4C2P3ZHC" kind="example">
# Klein bottle

## Example

Let $Q = [0, 1] \times [0, 1]$ be the unit square.
The *Klein bottle* $K$ is the quotient of $Q$ by the identifications
$$(0, y) \sim (1, y) \quad \text{and} \quad (x, 0) \sim (1 - x, 1) \qquad \text{for } x, y \in [0, 1].$$
Then $K$ is a closed [[surface|VGJJVBZ1MQ]].
Check: every point has a disc neighbourhood, made of pieces of $Q$ glued along the identified edges.
Also $K$ is compact and connected as a quotient of $Q$.
The four corners become one vertex, the edges give two edges, and $Q$ gives one face, so the Euler characteristic is $\chi(K) = 1 - 2 + 1 = 0$.
The strip $\set{(x, y) : \abs{x - 1/2} \le 1/4}$ with its top and bottom edges glued by $(x, 0) \sim (1 - x, 1)$ is a [[Möbius strip|04AJE0M271]] in $K$, so $K$ is not orientable.

## Remarks

By the [[classification of surfaces|2SFM73YET0]], $K$ is homeomorphic to the connected sum $\R P^2 \# \R P^2$.
Its fundamental group has the presentation $\langle a, b \mid a b a^{-1} b \rangle$.
Gluing two Möbius strips along their boundary circles also gives $K$.
The Klein bottle cannot be embedded in $\R^3$, since a closed surface embedded in $\R^3$ bounds a region and is therefore orientable.
It has immersions in $\R^3$ and embeddings in $\R^4$.

## Explanations

### L1 human

Take a rubber sheet, glue two opposite sides to make a tube, then glue the two ends of the tube with a flip.
In ordinary space the flip forces the tube to pass through its own wall, which gives the familiar glass bottle picture.
The result has no inside and no outside: a walker on it can come back mirror-reversed.
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<document source="https://tamnd.github.io/math2/node/6J921PEDN2.md" tag="6J921PEDN2" kind="concept">
# Period-doubling cascade

## Summary

A period-doubling cascade is an infinite sequence of [[period-doubling bifurcations|NQDTTZ59X8]] in a [[family of maps|EE9X3NZW81]].
In it, stable cycles of periods $1, 2, 4, 8, \dots$ appear one after another.
The parameter values of the bifurcations increase to a finite limit, beyond which the dynamics can be chaotic.
For the logistic map $x \mapsto r x (1 - x)$ the cascade starts at $r = 3$ and accumulates near $r \approx 3.5699$.
For typical smooth unimodal maps, the gaps between successive bifurcation values shrink at a universal geometric rate.
This rate is the [[Feigenbaum constant|F1EHT2CPHC]] $\delta \approx 4.6692$.

## Explanations

### L1 human

Turn up the growth rate in a simple population model.
First the population settles to one value, then it alternates between two values, then it cycles through four, then eight, and so on.
Each doubling happens sooner than the last, and the doublings pile up at a finite setting.
Past that point the population can jump around in a way that never repeats.
This road from order to chaos is seen in many models and in experiments with fluids and electronic circuits.
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<document source="https://tamnd.github.io/math2/node/6J9FXT6HAV.md" tag="6J9FXT6HAV" kind="concept">
# Limit-point and limit-circle

## Summary

For a singular [[Sturm–Liouville|7WV6B2502K]] equation, each singular endpoint is either in the limit-point case or in the limit-circle case.
In the limit-circle case every solution is square integrable near the endpoint; in the limit-point case some solution is not.
This dichotomy, found by Hermann Weyl, tells whether a boundary condition is needed at that endpoint to get a self-adjoint operator.
At a limit-point end no boundary condition is needed, and at a limit-circle end one must be imposed.

## Explanations

### L1 human

Near a bad endpoint, such as $+\infty$, solutions of the equation may be small enough to be square integrable or may be too big.
If all of them are small, the endpoint behaves almost like an ordinary endpoint and one must choose a boundary condition there.
If some of them are too big, the endpoint chooses for itself.
For $-y'' = 0$ on $(0, \infty)$ the solutions $1$ and $x$ are not square integrable near $\infty$, so $\infty$ is a limit-point end.
The name comes from Weyl's picture of a shrinking family of circles in the complex plane, which tend either to a point or to a circle.
</document>

<document source="https://tamnd.github.io/math2/node/6JB30BXDA2.md" tag="6JB30BXDA2" kind="concept">
# Local cohomology

## Summary

Let $M$ be a module over a commutative Noetherian ring and let $I$ be an ideal.
The local cohomology modules $H^i_I(M)$ are the right derived functors of the functor of sections supported on $I$.
It was introduced by Grothendieck.
Over a Noetherian local ring, local cohomology at the maximal ideal detects the [[depth|M2RADBFW99]] and the [[Krull dimension|MVGBS3J1P7]] of a finitely generated module.
It is the subject of [[Grothendieck's local duality|VA3DY2YX83]].

## Explanations

### L1 human

Take the elements of a module that are killed by some power of an ideal $I$.
Geometrically, these are the sections that live only near the closed set cut out by $I$.
Taking these elements is not an exact operation, and local cohomology records the correction terms.
For $R = K[x]$ and $I = (x)$, no nonzero polynomial is killed by a power of $x$, so $H^0_I(R) = 0$.
The first local cohomology is $H^1_I(R) = K[x, x^{-1}]/K[x]$, spanned by the classes of $x^{-1}, x^{-2}, \ldots$.
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<document source="https://tamnd.github.io/math2/node/6JBE71PZWA.md" tag="6JBE71PZWA" kind="definition">
# Semiconjugacy and factor map

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], and let $f \colon X \to X$ and $g \colon Y \to Y$ be [[continuous maps|0X3WGX95FE]].
A *semiconjugacy* from $f$ to $g$ is a continuous [[surjective|01RVCMSN5N]] map $h \colon X \to Y$ such that
$$h \circ f = g \circ h,$$
where $\circ$ is [[composition|0THBK9V825]].
Then $h$ is called a *factor map*, the system $(Y, g)$ is a *factor* of $(X, f)$, and $(X, f)$ is an *extension* of $(Y, g)$.

## Remarks

Some authors use the word semiconjugacy without the surjectivity condition and say factor map only when $h$ is onto.
If $h$ is moreover a homeomorphism, it is a topological conjugacy.
The relation gives $h \circ f^n = g^n \circ h$, so $h$ maps the orbit of $x$ onto the orbit of $h(x)$, and periodic points of $f$ to periodic points of $g$.

Example: let $\Sigma = \set{0, 1}^{\N}$ with the product topology, let $\sigma$ be the shift $(\sigma x)_i = x_{i+1}$, and let $g(y) = 2y \bmod 1$ on the circle $\R / \Z$.
The map $h(x) = \sum_{i \ge 0} x_i 2^{-i-1} \bmod 1$ is continuous and onto, since every number in $[0, 1]$ has a binary expansion.
One has $h(\sigma x) = \sum_{i \ge 0} x_{i+1} 2^{-i-1} = 2 h(x) - x_0$, which equals $2 h(x)$ modulo $1$, so $h \circ \sigma = g \circ h$.
This $h$ is not injective, since $(1, 0, 0, \dots)$ and $(0, 1, 1, \dots)$ have the same image.

Non-example: the inclusion $h \colon \set{0} \to \R$ satisfies $h \circ f = g \circ h$ for $f = \id$ and $g(y) = y/2$, but it is not surjective, so it is not a factor map.
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<document source="https://tamnd.github.io/math2/node/6JGM1K39FB.md" tag="6JGM1K39FB" kind="concept">
# Hamilton–Jacobi theory of the calculus of variations

## Summary

Hamilton–Jacobi theory links a problem of the [[calculus of variations|PQ25NHV0KS]] with a single first-order partial differential equation, the [[Hamilton–Jacobi equation|5TRXQKP7DY]].
A solution $S$ of this equation, such as the action along extremals, generates a whole family of extremals, a [[field of extremals|K4DFT3H9BJ]].
Conversely, a suitable family of solutions, a complete integral, gives all solutions of Hamilton's canonical equations by Jacobi's theorem.
The theory gives sufficient conditions for minima and is the classical ancestor of dynamic programming.

## Explanations

### L1 human

In a variational problem, one looks for the curve that makes some cost as small as possible, such as the travel time of light.
Instead of following one curve, Hamilton–Jacobi theory looks at the least cost to reach every point, like wave fronts spreading out from a source.
The curves of least cost cross these fronts in a definite direction, as light rays cross the wave fronts.
So knowing the fronts, which solve one partial differential equation, gives all the best curves at once.
</document>

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# Representation theory of the symmetric group

## Definition

Let $n \ge 1$ be an integer, let $S_n$ be the [[symmetric group|0515A6D2Y1]] on $\set{1, \ldots, n}$, and let $F$ be a field.
A *representation of the symmetric group* $S_n$ over $F$ is a [[group representation|WYGRXCG2RN]] $\rho \colon S_n \to GL(V)$ with $V$ a vector space over $F$.
The *representation theory of the symmetric group* is the study of these representations, of their decomposition into [[irreducible representations|FX3NEFYFFN]], and of their characters.

## Remarks

Example: the sign map $\sgn \colon S_n \to \set{\pm 1} \subseteq GL_1(\C)$ is a representation of degree $1$, because $\sgn(\sigma\tau) = \sgn(\sigma)\sgn(\tau)$.
Non-example: let $\rho \colon S_3 \to GL_1(\C)$ send $1$ to $1$, each transposition to $-1$ and each $3$-cycle to $\omega = e^{2\pi i/3}$.
It is not a representation: the product of the transpositions $(1\,2)$ and $(2\,3)$ is a $3$-cycle, while $(-1)(-1) = 1 \ne \omega$.
Over $\C$, the number of irreducible representations of $S_n$ is the number of [[conjugacy classes|0DEBXAJ5SA]] of $S_n$.
Conjugacy classes correspond to cycle types, so this is the number $p(n)$ of [[partitions|0JV30XN73Z]] of $n$.
The Specht modules $S^\lambda$, for $\lambda$ a partition of $n$, are a complete list of irreducible complex representations, without repetition.
Over a field of characteristic $p$ with $p \le n$, the theory is the modular representation theory of $S_n$, and is much less understood.
</document>

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# Superalgebra

## Summary

A superalgebra is an algebra graded by $\Z/2\Z$: it splits as an even part and an odd part, and the product adds parities.
It is the algebraic setting for supersymmetry, where even elements behave like ordinary commuting quantities and odd elements like anticommuting ones.
The [[exterior algebra|0MDMZ965VE]] and [[Clifford algebras|A38B1QFK20]] are basic examples, and Lie superalgebras are the corresponding nonassociative notion.
Many constructions, such as tensor products and commutators, acquire a sign $(-1)^{\abs{a}\abs{b}}$ when two odd elements are moved past each other.

## Explanations

### L1 human

Sort the elements of an algebra into two kinds, even and odd, like even and odd numbers.
The product follows the rule for adding parities: even times even and odd times odd are even, and even times odd is odd.
In the exterior algebra of the plane, with basis $1, e_1, e_2, e_1 e_2$, the even part is spanned by $1$ and $e_1 e_2$ and the odd part by $e_1$ and $e_2$.
There $e_1 e_2 = -e_2 e_1$: two odd elements anticommute, which is the typical "super" sign rule.
</document>

<document source="https://tamnd.github.io/math2/node/6K8GPMHWBB.md" tag="6K8GPMHWBB" kind="statement">
# Evans potential theorem

## Statement

Let $K$ be a [[compact set|0R6EWFKRCF]] in $\C$ of logarithmic [[capacity zero|W9N1TR1X8P]], or equivalently a compact [[polar set|C95QJ7YH5T]].
Then there is a Borel [[probability measure|059TJN0NZ1]] $\mu$ on $K$ whose [[logarithmic potential|46NHWJHW44]]
$$U^\mu(z) = \int_K \log \frac{1}{\abs{z - w}} \, d\mu(w)$$
satisfies $U^\mu(z) = +\infty$ for every $z \in K$.

## Remarks

The potential $U^\mu$ is harmonic on $\C \setminus K$ and superharmonic on $\C$, so $u = -U^\mu$ is a subharmonic function equal to $-\infty$ exactly on $K$.
Such a potential is called an Evans potential of $K$.

Proof idea: let $K_k = \set{z : \operatorname{dist}(z, K) \le 1/k}$.
Since $K$ has capacity zero, the capacities of the compact sets $K_k$ tend to $0$, so their Robin constants $V(K_k)$ tend to $+\infty$.
By Frostman's theorem, the potential of the equilibrium measure of $K_k$ equals $V(K_k)$ on $K$.
One combines these equilibrium measures with suitable weights and passes to a limit measure carried by $K$.
The delicate step is to keep the potential infinite at every point of $K$ in the limit.

Example: for $K = \set{0}$, the measure $\mu = \delta_0$ has $U^\mu(z) = \log(1/\abs{z})$, which is $+\infty$ exactly at $0$.

The theorem is false without polarity.
If $K$ is compact and not polar, then for every probability measure $\mu$ on $K$ the set where $U^\mu = +\infty$ is polar, so it cannot contain $K$.
The analogous statement in $\R^n$, $n \ge 3$, holds with the Newtonian potential $\int \abs{x - y}^{2-n} \, d\mu(y)$.

## Explanations

### L1 human

A polar set is one where some subharmonic function equals $-\infty$.
The theorem says that for a compact polar set one can choose this function very concretely: as minus the potential of a charge placed on the set itself.
It also says that the infinite values can be made to occur exactly on the set and nowhere else.

### L2 human

Evans potentials are a key tool in the study of exceptional sets.
They are used to show that compact polar sets are removable for bounded harmonic functions and to build harmonic functions with prescribed singular behaviour.
They also show that the definition of polar sets by subharmonic functions agrees with the definition by capacity for compact sets.
</document>

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# Chattering control

## Summary

A chattering control is a control that switches between values infinitely many times in a bounded time interval.
Its switching times accumulate at some point.
Chattering controls can be optimal, as in [[Fuller's phenomenon|RSZS9SFFZ7]], and they also appear as approximations of [[relaxed controls|SFMD8RKRSE]].
They show that the [[bang–bang|GK1Z3Q1WRZ]] structure of an optimal control may involve infinitely many switches.

## Explanations

### L1 human

Imagine a heater that must be fully on or fully off, and that should bring a room to an exact temperature as well as possible.
The best strategy may switch on and off more and more often as the target gets close, with shorter and shorter periods.
The switching times pile up at one moment, like the points $1/2, 3/4, 7/8, \ldots$ piling up at $1$.
A control with such infinitely many switches before a finite time is called chattering.
</document>

<document source="https://tamnd.github.io/math2/node/6KAVKR1HC8.md" tag="6KAVKR1HC8" kind="definition">
# Stochastic optimal control

## Definition

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] that carries a $d$-dimensional [[Wiener process|22735FGJV6]] $W$.
Let $T > 0$ and let $U \subseteq \R^m$ be a nonempty closed set, the control set.
Let $b \colon \R^n \times U \to \R^n$ and $\sigma \colon \R^n \times U \to \R^{n \times d}$ be continuous.
Assume that they are [[Lipschitz|0QNVQ4VD9T]] in $x$ uniformly in $u$, and that $\abs{b(x, u)} + \abs{\sigma(x, u)} \le K (1 + \abs{x} + \abs{u})$ for a constant $K$.
Let $f \colon \R^n \times U \to \R$ and $g \colon \R^n \to \R$ be continuous with $\abs{f(x, u)} \le K (1 + \abs{x}^2 + \abs{u}^2)$ and $\abs{g(x)} \le K (1 + \abs{x}^2)$.

1. An *admissible control* is a progressively measurable process $u = (u_t)_{0 \le t \le T}$ with values in $U$ and $E \int_0^T \abs{u_t}^2 \, dt < \infty$.
2. For an admissible control $u$ and $x \in \R^n$, the *state* is the solution $X^u$ of the [[stochastic differential equation|B219G9SHJA]] $dX_t = b(X_t, u_t) \, dt + \sigma(X_t, u_t) \, dW_t$ with $X_0 = x$.
3. The *cost* of $u$ is $J(x; u) = E\left[ \int_0^T f(X^u_t, u_t) \, dt + g(X^u_T) \right]$, an [[expected value|0WQNRCNY3M]].

The *stochastic optimal control problem* is to find an admissible control $u^*$ with $J(x; u^*) = \inf_u J(x; u)$, the infimum over all admissible controls.
Such a $u^*$ is an *optimal control*, and $V(x) = \inf_u J(x; u)$ is the *value*.

## Remarks

Under these hypotheses the state equation has a unique strong solution and the cost is finite.
Starting the problem at time $t$ in state $x$ gives the value function $V(t, x)$.
When $V$ is smooth enough, it solves the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]]
$$V_t + \min_{u \in U} \left\{ b(x, u) \cdot \nabla_x V + \tfrac{1}{2} \tr\left( \sigma \sigma^T(x, u) D_x^2 V \right) + f(x, u) \right\} = 0, \qquad V(T, x) = g(x).$$
With $\sigma = 0$ the problem is a deterministic optimal control problem.

Example: $n = m = d = 1$, $U = \R$, $b(x, u) = u$, $\sigma = 1$, $f(x, u) = u^2 / 2$ and $g(x) = x^2 / 2$.
The function $V(t, x) = \dfrac{x^2}{2 (1 + T - t)} + \dfrac{1}{2} \ln(1 + T - t)$ satisfies $V(T, x) = x^2 / 2$.
The minimum over $u$ of $u V_x + u^2 / 2$ is $-\frac{1}{2} V_x^2$, attained at $u = -V_x$.
With $s = 1 + T - t$, one has $V_t = \frac{x^2}{2 s^2} - \frac{1}{2 s}$, $\frac{1}{2} V_x^2 = \frac{x^2}{2 s^2}$ and $\frac{1}{2} V_{xx} = \frac{1}{2 s}$, so the equation holds.
The optimal control is the feedback $u_t = -X_t / (1 + T - t)$.

Non-example: the process $u_t = W_T$ for all $t$ is not admissible, because it is not adapted: it uses the future of the noise.

## Explanations

### L1 human

The controller picks its action at each time using only what it has seen so far.
The noise makes the cost random, so one minimises its average.
</document>

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# Bargmann transform

## Definition

Let $\mathcal{F}^2$ be the space of [[entire functions|0G4F0XB9X2]] $F$ with
$$\norm{F}^2 = \frac{1}{\pi} \int_\C \abs{F(z)}^2 e^{-\abs{z}^2}\, dA(z) < \infty,$$
where $dA$ is Lebesgue measure on $\C = \R^2$.
It is a [[Hilbert space|HM2HHE3J51]], the Segal–Bargmann space.
The *Bargmann transform* of $f \in L^2(\R)$, the [[$L^2$ space|SASP14MA1Z]], is the function
$$Bf(z) = \pi^{-1/4} \int_\R \exp\Big( -\frac{z^2}{2} + \sqrt{2}\, z x - \frac{x^2}{2} \Big) f(x)\, dx, \qquad z \in \C.$$

## Remarks

For fixed $z$ the kernel is a Gaussian in $x$, so it lies in $L^2(\R)$, and the integral converges by the Cauchy–Schwarz inequality.
The function $Bf$ is entire, and $B$ is a [[unitary operator|MM0EJ2GDE4]] from $L^2(\R)$ onto $\mathcal{F}^2$.
The monomials $z^n / \sqrt{n!}$ form an orthonormal basis of $\mathcal{F}^2$, and $B$ maps the normalized [[Hermite function|52VS07H6GK]] $\psi_n$ to $z^n / \sqrt{n!}$.
Conventions differ: some authors use the weight $e^{-\pi \abs{z}^2}$ or $e^{-\abs{z}^2/\hbar}$ and rescale the kernel accordingly.

Example: $f = \psi_0 = \pi^{-1/4} e^{-x^2/2}$.
Then $Bf(z) = \pi^{-1/2} e^{-z^2/2} \int_\R e^{-x^2 + \sqrt{2} z x}\, dx = \pi^{-1/2} e^{-z^2/2} \cdot \sqrt{\pi}\, e^{z^2/2} = 1$.
The constant $1$ has norm $\frac{1}{\pi} \int_\C e^{-\abs{z}^2}\, dA = 1$, equal to $\norm{\psi_0}_{L^2} = 1$.

Non-example: the function $F(z) = e^{z^2}$ is entire but not in $\mathcal{F}^2$.
On the real axis $\abs{F(x)}^2 e^{-x^2} = e^{x^2}$, and the integral of $\abs{F}^2 e^{-\abs{z}^2}$ over a neighbourhood of the real axis diverges, so $F$ is not a Bargmann transform.
</document>

<document source="https://tamnd.github.io/math2/node/6KFS240R42.md" tag="6KFS240R42" kind="definition">
# Sheffer function

## Definition

Let $A$ be a finite set with at least two elements.
An [[operation|EJ8YJVM9BW]] $f$ on $A$ is a *Sheffer function* if the set $\set{f}$ is [[functionally complete|TDZVCWYJ9F]].
That is, the [[clone|PTKR703G6A]] generated by $f$ contains every finitary operation on $A$.

## Remarks

Example: NAND, $x \uparrow y = \neg(x \wedge y)$ on $\set{0, 1}$, is a Sheffer function.
Check: $\neg x = x \uparrow x$ and $x \wedge y = \neg(x \uparrow y)$, and $\set{\wedge, \neg}$ is functionally complete.
Non-example: $\wedge$ on $\set{0, 1}$ is not a Sheffer function.
Every operation generated by $\wedge$ maps $(0, \dots, 0)$ to $0$, so $\neg$ is not generated.
NOR, $x \downarrow y = \neg(x \vee y)$, is the only other binary Sheffer function on $\set{0, 1}$.
A Sheffer function must have arity at least $2$, since a unary operation generates only essentially unary operations.
</document>

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# Charlier polynomials

## Summary

The Charlier polynomials $C_n(x; a)$ are the orthogonal polynomials of the [[Poisson distribution|0XB9GQTN9G]] with mean $a > 0$.
They are orthogonal for a weight on the nonnegative integers rather than on an interval.
They belong to the [[Askey scheme|RR2S05C09B]] as a limit of the [[Krawtchouk|CJGKW80KBR]] and [[Meixner polynomials|CEC3DESCFT]].

## Explanations

### L1 human

Count random events, such as calls arriving at a desk in an hour, with an average of $a$ events.
The count follows the Poisson distribution, which puts weight $e^{-a} a^x / x!$ on each whole number $x$.
The Charlier polynomials are the polynomials in $x$ that are perpendicular to each other when one averages with this weight.
The first nonconstant one is $1 - x/a$, whose average is $1 - a/a = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/6KQ6E3YHKR.md" tag="6KQ6E3YHKR" kind="definition">
# Stone space

## Definition

A *Stone space* is a [[topological space|0CQHFPGJ45]] $X$ such that:

1. $X$ is [[compact|0TVJZYB6SH]].
2. $X$ is [[Hausdorff|0S1FJ8VEX9]].
3. $X$ is [[totally disconnected|0RJSBR9GW1]]: its only connected subsets with at least one point are the singletons.

## Remarks

For a compact Hausdorff space, clause 3 is equivalent to having a basis of [[clopen sets|DQ4E55T3BW]], that is, to being [[zero-dimensional|JJ7D0CYECZ]].

Example: the Cantor space $\set{0, 1}^{\N}$ with the product topology.
It is compact by Tychonoff's theorem and Hausdorff as a product of Hausdorff spaces.
Two different sequences differ at some index $n$, and the set of sequences with a fixed $n$-th term is clopen, so no connected set contains both.

Non-example: the interval $[0, 1]$ is compact and Hausdorff but connected, so clause 3 fails.
The space $\Q$ is Hausdorff and totally disconnected but not compact, so clause 1 fails.
</document>

<document source="https://tamnd.github.io/math2/node/6KQCNP7P01.md" tag="6KQCNP7P01" kind="definition">
# Vanishing mean oscillation

## Definition

Let $n \ge 1$ and let $f$ be a function of [[bounded mean oscillation|QJYTXFT8Y4]] on $\R^n$.
For a cube $Q$ with sides parallel to the axes, let $\abs{Q}$ be its [[Lebesgue measure|2Z2QVWZ262]] and $f_Q = \frac{1}{\abs{Q}} \int_Q f$.
For $\delta > 0$ let
$$\omega_f(\delta) = \sup_{\abs{Q} \le \delta} \frac{1}{\abs{Q}} \int_Q \abs{f(x) - f_Q} \, dx.$$
The function $f$ has *vanishing mean oscillation* if $\omega_f(\delta) \to 0$ as $\delta \to 0$.
The space of such functions is denoted $\mathrm{VMO}(\R^n)$.

## Remarks

VMO is a closed subspace of BMO, and Sarason showed that it is the closure in BMO of the uniformly continuous functions in BMO.
Some authors use a smaller space on $\R^n$, also called VMO: the closure in BMO of the continuous functions with compact support.
The dual space of this smaller space is the real Hardy space $H^1$.

Example: every uniformly continuous $f$ in BMO is in VMO.
If $\abs{f(x) - f(y)} \le \varepsilon$ whenever $\abs{x - y} \le \eta$, then on a cube $Q$ of diameter at most $\eta$ one has $\abs{f(x) - f_Q} \le \varepsilon$ for $x \in Q$, so $\omega_f(\delta) \le \varepsilon$ for $\delta$ small.

Example with a singularity: if $n \ge 2$ and $\nabla f \in L^n(\R^n)$, then the [[Poincaré inequality|JS2CCR3RNS]] on cubes gives $\frac{1}{\abs{Q}} \int_Q \abs{f - f_Q} \le C \left( \int_Q \abs{\nabla f}^n \right)^{1/n}$, which tends to $0$ uniformly as $\abs{Q} \to 0$.
So such $f$ are in VMO, although they can be unbounded.

Non-example: $f(x) = \log \abs{x}$ on $\R$ is in BMO but not in VMO.
For $r > 0$, $f(r x) = f(x) + \log r$, so the mean oscillation of $f$ on $[-r, r]$ equals its mean oscillation on $[-1, 1]$, which is a fixed positive number.
Hence $\omega_f(\delta)$ does not tend to $0$.
</document>

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# Successive over-relaxation

## Statement

Let $A$ be a real $n \times n$ matrix with nonzero diagonal entries, and let $b \in \R^n$.
Write $A = D + L + U$, with $D$ diagonal, $L$ strictly lower triangular and $U$ strictly upper triangular.
Let $\omega \in \R$ with $\omega \ne 0$.
The method of *successive over-relaxation* (SOR) with parameter $\omega$ starts from any $x^{(0)} \in \R^n$ and sets
$$x^{(k+1)} = (D + \omega L)^{-1} \left( \omega b - \left( \omega U + (\omega - 1) D \right) x^{(k)} \right).$$
Let $G_\omega = (D + \omega L)^{-1} \left( (1 - \omega) D - \omega U \right)$ be its iteration matrix.

1. (Kahan) The [[spectral radius|9V7YQA85PC]] satisfies $\rho(G_\omega) \ge \abs{\omega - 1}$. So if SOR converges for every $b$ and every $x^{(0)}$, then $0 < \omega < 2$.
2. (Ostrowski–Reich) If $A$ is symmetric and [[positive definite|0KX66CZMBB]] and $0 < \omega < 2$, then $\rho(G_\omega) < 1$. Then for every $b$ and every $x^{(0)}$ the iterates converge to $A^{-1} b$.

## Remarks

Componentwise, SOR computes the [[Gauss–Seidel|G0EDYEDG4S]] value $\tilde{x}_i$ for the $i$-th unknown and takes $x^{(k+1)}_i = (1 - \omega) x^{(k)}_i + \omega \tilde{x}_i$.
For $\omega = 1$ it is the Gauss–Seidel method.
Kahan's bound comes from the determinant: $\det G_\omega = \det(D)^{-1} \det((1 - \omega) D) = (1 - \omega)^n$, and this is the product of the eigenvalues.
The Ostrowski–Reich theorem follows because the energy norm of the error decreases strictly at each step when $A$ is positive definite and $0 < \omega < 2$.
Assume moreover that $A$ is consistently ordered, for example tridiagonal.
Assume also that the [[Jacobi|B8FVHNXAN1]] iteration matrix has real eigenvalues and spectral radius $\mu < 1$.
Then Young's theory gives the best parameter $\omega_{\mathrm{opt}} = 2 / (1 + \sqrt{1 - \mu^2})$, with $\rho(G_{\omega_{\mathrm{opt}}}) = \omega_{\mathrm{opt}} - 1$.
For the model Poisson problem with mesh width $h$ this improves the convergence factor from $1 - O(h^2)$ to $1 - O(h)$.

## Explanations

### L1 human

SOR takes the step that Gauss–Seidel suggests and stretches it by a factor $\omega$.
When the iterates creep slowly towards the solution, going a bit further than suggested, with $\omega$ between $1$ and $2$, saves many steps.
The parameter must stay between $0$ and $2$; outside that range it cannot converge for every starting guess.

### L2 human

SOR was for decades the main [[iterative method|2RPE77ACER]] for large elliptic problems, because a good $\omega$ makes it far faster than Jacobi or Gauss–Seidel.
The best $\omega$ depends on the spectrum of the Jacobi matrix, which is known exactly for model problems and estimated otherwise.
Symmetric SOR runs a forward and a backward sweep and gives a symmetric preconditioner for the conjugate gradient method.
</document>

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# Natural deduction

## Summary

Natural deduction is a proof system whose rules mimic the way mathematicians reason.
Each connective has introduction rules, which say how to prove a formula with it, and elimination rules, which say how to use such a formula.
Assumptions can be made for a while and then discharged, as in "suppose $\varphi$; then ...; hence $\varphi \to \psi$".
It was introduced by Gerhard Gentzen, and independently by Stanisław Jaśkowski, and it is a basic system of [[proof theory|N4EDJNMTEF]].

## Explanations

### L1 human

To prove "if $A$ then $B$", you usually assume $A$ and show $B$.
Natural deduction turns this habit into a formal rule, called implication introduction.
Likewise, to prove "$A$ and $B$" you prove $A$ and prove $B$, and from "$A$ and $B$" you may conclude $A$.
Proofs are drawn as trees, with the assumptions at the leaves and the conclusion at the root.
Compared with a [[Hilbert system|1MGMQX43B8]], proofs are shorter and closer to ordinary arguments.
</document>

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# Almgren–Pitts min-max theory

## Summary

Almgren–Pitts min-max theory constructs closed [[minimal hypersurfaces|41X0YW2R9N]] in a closed Riemannian manifold by a min-max over families of hypersurfaces that sweep out the manifold.
It is the higher-dimensional analogue of the [[Birkhoff min-max method|1R5X9HW6D3]] for closed geodesics.
Frederick Almgren set up the topology of spaces of cycles, and Jon Pitts proved the regularity of the min-max hypersurface in low dimensions.
Fernando Marques and André Neves developed the theory further and used it to prove the Willmore conjecture.
It is also the main tool behind [[Yau's conjecture on infinitely many minimal surfaces|VAP45DFXW3]].

## Explanations

### L1 human

Fill a closed curved space with a family of soap films that starts as nothing, grows to fill the whole space, and shrinks back to nothing.
At some moment the film in the family has the largest area.
Choose the family so that this largest area is as small as possible.
The film at the critical moment cannot be pushed down further, and it settles into a minimal hypersurface, a film whose mean curvature is zero.
For a round $3$-sphere, sweeping by parallel $2$-spheres gives the equatorial $2$-sphere as the min-max surface.
</document>

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# Bass–Serre theory

## Summary

Bass–Serre theory describes the groups that act on [[trees|062D9KXJB1]] by building them from simpler groups.
Its central object is a [[graph of groups|DC1CQMD3WK]]: a graph with a group at each vertex and each edge.
The structure theorem says that a group acting on a tree without inversions is the fundamental group of the quotient graph of groups.
The vertex and edge groups of this graph of groups are stabilisers.
Conversely, the fundamental group of a graph of groups acts on a tree, the Bass–Serre tree.
It unifies [[free products|TM7QS7P7X9]], [[amalgamated free products|M848AHBRCM]] and [[HNN extensions|YEYERRY37K]].
It gives short proofs of subgroup theorems such as the [[Nielsen–Schreier theorem|W97X7WX5G8]] and the [[Kurosh subgroup theorem|RY8K5RQJSR]].

## Explanations

### L1 human

A tree is a graph with no cycles, so there is exactly one path between any two points.
When a group moves a tree around by symmetries, the quotient picture is a graph, and each vertex and edge of it remembers the group of symmetries fixing it.
Bass–Serre theory says one can rebuild the whole group from this decorated graph.
For instance, $\mathrm{SL}_2(\Z)$ acts on a tree with quotient a single edge.
This shows that it is an amalgam of a cyclic group of order $4$ and a cyclic group of order $6$ over a common subgroup of order $2$.
A subgroup of the group still acts on the same tree, which is why the theory proves so much about subgroups.
</document>

<document source="https://tamnd.github.io/math2/node/6M852B4RA8.md" tag="6M852B4RA8" kind="concept">
# Submartingale

## Summary

A submartingale is an adapted, integrable [[stochastic process|0QH5ENKZM1]] whose [[conditional expectation|0TTDGTEFMN]] of a future value, given the present information, is at least the present value.
It models the fortune in a game that is favourable or fair to the player.
Every [[martingale|0D0GBNDKDH]] is a submartingale, and convex functions of martingales give many more examples.
Doob's inequalities and convergence theorems are proved for submartingales.

## Explanations

### L1 human

A submartingale tends to go up, or at least not down, on average.
Given everything known today, the best forecast of tomorrow's value is at least today's value.
The value can still go down on some days; only the average trend is upward.
If $S_n$ is a fair coin-tossing walk, then $S_n^2$ is a submartingale: squaring adds $1$ on average at each step.
</document>

<document source="https://tamnd.github.io/math2/node/6MB2G4T913.md" tag="6MB2G4T913" kind="definition">
# Homogeneity equation

## Definition

Let $n \ge 1$ be an integer and let $k$ be a real number.
The *homogeneity equation* of degree $k$ is the [[functional equation|SRAN4EKWS0]]
$$f(tx) = t^k f(x) \quad \text{for all } t > 0 \text{ and } x \in \R^n \setminus \set{0},$$
for an unknown [[function|0EPAT51DXQ]] $f \colon \R^n \setminus \set{0} \to \R$.
A solution is called a *positively homogeneous function of degree $k$*.

## Remarks

Example: $f(x, y) = x^2 + xy$ is a solution with $k = 2$, since $f(tx, ty) = t^2 x^2 + t^2 xy = t^2 f(x, y)$.
Non-example: $f(x) = x + 1$ on $\R \setminus \set{0}$ is not a solution for any $k$, since $f(2) = 3$, $f(1) = 2$ would need $2^k = 3/2$, while $f(4) = 5$, $f(2) = 3$ would need $2^k = 5/3$.
The general solution is $f(x) = \norm{x}^k g(x / \norm{x})$, where $g$ is an arbitrary function on the unit sphere and $\norm{\cdot}$ is the Euclidean norm.
For differentiable $f$, Euler's homogeneous function theorem says that $f$ is a solution exactly when $\sum_i x_i \, \partial f / \partial x_i = k f$.
Other versions require the identity for all nonzero real $t$ or for $t$ in a field, which gives a stronger notion of homogeneity.
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# Jacobson topology

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $\operatorname{Prim}(R)$ be the set of its [[primitive ideals|M0JDQR0V5S]].
For a subset $S \subseteq \operatorname{Prim}(R)$, its *kernel* is $k(S) = \bigcap_{P \in S} P$, with $k(\varnothing) = R$.
For a [[two-sided ideal|0DDA6CY7GJ]] $I$ of $R$, its *hull* is $h(I) = \set{P \in \operatorname{Prim}(R) : P \supseteq I}$.
The *Jacobson topology*, or *hull-kernel topology*, on $\operatorname{Prim}(R)$ is the [[topology|0CNQXVBHCX]] whose closure operator is
$$\overline{S} = h(k(S)).$$
Equivalently, its closed sets are exactly the hulls $h(I)$ of the two-sided ideals $I$ of $R$.

## Remarks

The map $S \mapsto h(k(S))$ satisfies the Kuratowski closure axioms, so it does define a topology.
The space $\operatorname{Prim}(R)$ is $T_0$: two distinct primitive ideals $P \ne Q$ have different closures $h(P) \ne h(Q)$.
Example: for $R = \Z$ the primitive ideals are the ideals $p\Z$ with $p$ prime.
The kernel of an infinite set of them is $0$, so its closure is all of $\operatorname{Prim}(\Z)$; the closed sets are the finite sets and the whole space.
Non-example of a closed set: $\set{p\Z : p \text{ an odd prime}}$ is not closed, because its closure contains $2\Z$.
For a commutative ring $R$, the primitive ideals are the maximal ideals and the Jacobson topology is the subspace topology from the Zariski topology on the prime spectrum.
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<document source="https://tamnd.github.io/math2/node/6MFS8FEYHD.md" tag="6MFS8FEYHD" kind="definition">
# Triangulation of a point set

## Definition

Let $P$ be a finite set of points in $\R^2$, not all [[collinear|0AFNNTRTJE]], and let $\operatorname{conv}(P)$ be its [[convex hull|0FQSDC2EQ2]].
A *triangulation* of $P$ is a finite set $T$ of closed non-degenerate [[triangles|0CTNFGYMM2]] such that:

1. the vertices of every triangle of $T$ belong to $P$;
2. any two distinct triangles of $T$ are either disjoint or meet in one common vertex or in one common edge;
3. the union of the triangles of $T$ is $\operatorname{conv}(P)$;
4. every point of $P$ is a vertex of some triangle of $T$.

## Remarks

Example: let $P = \set{(0,0), (1,0), (1,1), (0,1)}$.
The two triangles with vertices $(0,0), (1,0), (1,1)$ and $(0,0), (1,1), (0,1)$ form a triangulation: they meet along the diagonal, and their union is the unit square.

Non-example: add the centre $(1/2, 1/2)$ to $P$ and keep the same two triangles.
Clauses 1 to 3 still hold, but the centre is not a vertex of either triangle, so clause 4 fails.

If $P$ has $n$ points, of which $h$ lie on the boundary of $\operatorname{conv}(P)$, then every triangulation of $P$ has $2n - h - 2$ triangles and $3n - h - 3$ edges.
Some authors drop clause 4 and allow points of $P$ that are not used as vertices.
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# Butcher group

## Description

Let $\mathcal{T}$ be the set of finite non-planar [[rooted trees|0Y0JGC2K9T]].
They are written recursively: $\bullet$ is the tree with one vertex, and $[t_1, \ldots, t_k]$ is the tree whose root is joined to the roots of $t_1, \ldots, t_k$.
Let $H$ be the commutative polynomial algebra over $\R$ generated by $\mathcal{T}$, whose monomials are rooted forests.
Give $H$ the coproduct $\Delta(t) = t \otimes 1 + 1 \otimes t + \sum_{c} P^c(t) \otimes R^c(t)$, where $c$ runs over the admissible cuts of $t$.
Here $R^c(t)$ is the part that keeps the root and $P^c(t)$ is the forest cut off.
Then $H$ is the Connes–Kreimer [[Hopf algebra|7XFYP0N84H]] of rooted trees.

The *Butcher group* $G$ is the set of algebra homomorphisms $\varphi \colon H \to \R$.
Such a $\varphi$ is the same as a map $\mathcal{T} \to \R$, extended multiplicatively to forests.
The product of $\varphi, \psi \in G$ is the convolution
$$(\varphi \psi)(t) = (\varphi \otimes \psi)\big(\Delta(t)\big).$$
It is a [[group|G5S1M9]]: the product is associative because $\Delta$ is coassociative, and the identity is the counit, which is $0$ on every tree.
The inverse of $\varphi$ is $\varphi \circ S$, where $S$ is the antipode of $H$.

## Remarks

Each [[Runge–Kutta method|7DBPJKYC6C]] with [[Butcher tableau|RH8BD7PW5F]] $(A, b, c)$ gives an element $\Phi$ of $G$ through its elementary weights: $\Phi(t) = \sum_i b_i \Phi_i(t)$, where $\Phi_i(\bullet) = 1$ and $\Phi_i([t_1, \ldots, t_k]) = \prod_{m=1}^{k} \sum_j a_{ij} \Phi_j(t_m)$.
The exact flow of the differential equation corresponds to $e(t) = 1/\gamma(t)$, where $\gamma(\bullet) = 1$ and $\gamma([t_1, \ldots, t_k]) = \abs{t} \, \gamma(t_1) \cdots \gamma(t_k)$, with $\abs{t}$ the number of vertices.
A method has order $p$ if and only if $\Phi(t) = 1/\gamma(t)$ for every tree with at most $p$ vertices; these are Butcher's order conditions.
Check for small trees: $\Phi(\bullet) = \sum_i b_i$ must equal $1$, and $\Phi([\bullet]) = \sum_i b_i c_i$ with $c_i = \sum_j a_{ij}$ must equal $\frac{1}{\gamma([\bullet])} = \frac12$.
With a suitable ordering convention, applying one method after another corresponds to the product in $G$, which explains the name: the group encodes composition of methods.
John C. Butcher introduced this group in 1972 in his algebraic theory of integration methods.

## Explanations

### L1 human

Every Runge–Kutta method can be described by a list of numbers, one for each rooted tree.
The trees keep track of how the derivatives of the right-hand side combine in the Taylor expansion of the solution.
Doing two steps with two methods in a row gives a new list, and this operation turns the set of all such lists into a group.
A method is accurate to order $p$ when its numbers agree with those of the exact solution on all trees with at most $p$ vertices.

### L2 human

The Butcher group organises the theory of B-series, the formal series indexed by rooted trees that describe Runge–Kutta methods and many other integrators.
Its identification with the character group of the Connes–Kreimer Hopf algebra links numerical analysis with the renormalization of quantum field theory, where the same Hopf algebra appears.
Effective order, the composition of methods and the backward error analysis of geometric integrators are all computations in this group.
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# Homotopy type

## Definition

Two topological spaces $X$ and $Y$ are *homotopy equivalent*, or *have the same homotopy type*, written $X \simeq Y$, if there is a [[homotopy equivalence|XZG6MTSWCN]] $f \colon X \to Y$.
The *homotopy type* of $X$ is the class of all spaces homotopy equivalent to $X$.

## Remarks

The relation $\simeq$ is reflexive, symmetric and transitive, so it behaves as an [[equivalence relation|0G8VEZ3WTK]] on topological spaces.
Symmetry holds because a homotopy inverse is itself a homotopy equivalence.
Example: $\R^n$ has the homotopy type of a point; the maps $\R^n \to \set{0}$ and $\set{0} \to \R^n$ are mutually inverse up to the homotopy $H(x, t) = (1 - t) x$.
Non-example: $S^1$ and a point do not have the same homotopy type, since their [[fundamental groups|S1XK3HXHFX]] differ.
Homeomorphic spaces have the same homotopy type, but the converse fails, as $\R$ and a point show.
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# Adaptive step size

## Summary

Adaptive step size control lets a [[numerical method for an ordinary differential equation|558XADG64M]] change its step length as it goes.
At each step the method estimates its [[local truncation error|T220HF0562]] and compares it with a tolerance given by the user.
Steps whose error estimate is too large are rejected and repeated with a smaller step.
The next step size is predicted from the current estimate.
In this way the method takes long steps where the solution is smooth and short steps where it changes fast.
It reaches a target accuracy with far fewer steps than a fixed step size.

## Explanations

### L1 human

Think of driving a car along a winding road: on straight parts you go fast, and before sharp bends you slow down.
An adaptive solver does the same with its time step.
For an orbit of a comet, it takes tiny steps when the comet sweeps close to the sun and very large steps far away, where little happens.
The error estimate plays the role of the driver's eyes.
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# Spectral decomposition theorem

## Statement

Let $M$ be a compact smooth manifold and let $f \colon M \to M$ be a $C^1$ diffeomorphism that satisfies [[Axiom A|S31RTTJPDC]], with [[non-wandering set|EY2W34JZA4]] $\Omega(f)$.
Then:

1. There are pairwise disjoint closed sets $\Lambda_1, \ldots, \Lambda_k$ with $\Omega(f) = \Lambda_1 \cup \cdots \cup \Lambda_k$ such that $f(\Lambda_i) = \Lambda_i$ and $f|_{\Lambda_i}$ is [[topologically transitive|FR49Y9GQHR]] for each $i$.
2. Each $\Lambda_i$ is a union of pairwise disjoint closed sets $X_{i,1}, \ldots, X_{i,n_i}$ such that $f(X_{i,j}) = X_{i,j+1}$ for $j < n_i$, $f(X_{i,n_i}) = X_{i,1}$, and $f^{n_i}|_{X_{i,j}}$ is [[topologically mixing|B763QH5521]] for each $j$.

## Remarks

The sets $\Lambda_i$ are called basic sets; the decomposition into them is unique.
The idea of the proof is to group periodic points whose stable and unstable manifolds meet transversally in both directions.
The local product structure of the hyperbolic set makes the closures of these groups open and closed in $\Omega(f)$.
Compactness then makes them finite in number.
Part 1 is due to Smale, and part 2 to Bowen.
Example: for the north-south map of the circle, the basic sets are the two hyperbolic fixed points, with $n_i = 1$.
For an [[Anosov diffeomorphism|RCQH8P6DPQ]] of the torus, $\Omega(f)$ is the whole torus and there is one basic set.
Without hyperbolicity the conclusion fails: for the identity of the circle every point is non-wandering, and $\Omega(f)$ is connected but $f$ is not transitive on it.

## Explanations

### L1 human

The long-term dynamics of an Axiom A system breaks into finitely many separate blocks.
On each block the motion is irreducible: some orbit comes close to every point of it.
Each block in turn is made of a few pieces that the map visits in a cycle, and on each piece a power of the map mixes completely.
So the global behaviour is a finite combination of well-understood chaotic pieces.

### L2 human

The theorem is the starting point of the theory of Axiom A systems.
Orbits outside $\Omega(f)$ go from one basic set to another.
When there are no cycles among the basic sets, this gives a partial order on them, as in Conley's theory of [[chain recurrence|6MX5R482KJ]].
Each basic set admits a [[Markov partition|4VNHVDHTNN]], so the dynamics on it is a factor of a [[subshift of finite type|624XDXX8FN]].
This leads to equilibrium states, [[Sinai–Ruelle–Bowen measures|256NP6JBEQ]] and the counting of periodic orbits.
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# Chain recurrence

## Summary

A point is chain recurrent when it can return to itself along a path that follows the dynamics up to arbitrarily small jumps.
Such paths are called $\varepsilon$-chains or $\varepsilon$-pseudo-orbits.
The chain recurrent set is the weakest of the usual notions of recurrence: it contains the [[non-wandering set|EY2W34JZA4]].
By [[Conley's fundamental theorem|D07MBM0S3V]], outside the chain recurrent set the dynamics is gradient-like.

## Explanations

### L1 human

Imagine following the orbit of a point, but allowing yourself a tiny error at each step, as a computer with rounding would.
The point is chain recurrent if, for every allowed error size, you can get back to the starting point this way.
A recurrent point is chain recurrent: its orbit comes back close to it, and one small final jump closes the loop.
For the map $x \mapsto x/2$ on $[0, 1]$, a point $x > 0$ is not chain recurrent: with small errors you still cannot climb back from $x/2$ to $x$.
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<document source="https://tamnd.github.io/math2/node/6N268PHKQA.md" tag="6N268PHKQA" kind="concept">
# Factor congruence

## Summary

A factor congruence of an [[algebra|QJGPT2W0FT]] is a [[congruence|DCSXXT4SER]] that comes from splitting the algebra as a [[direct product|WJ73XYBNE4]] of two factors.
It is the kernel of the projection onto one factor.
Factor congruences describe all the ways an algebra decomposes into a product of two algebras.
An algebra is [[directly indecomposable|6B1PSZ42MC]] when its only factor congruences are the trivial ones.

## Explanations

### L1 human

If an algebra is a product $\mathbf{B} \times \mathbf{C}$, two pairs $(b, c)$ and $(b', c')$ can be compared by their first coordinates only.
The relation "same first coordinate" is a congruence, and it is a factor congruence.
Factor congruences let one detect such a product splitting from inside the algebra, without knowing $\mathbf{B}$ and $\mathbf{C}$ in advance.
For example, $\Z/6\Z$ splits as $\Z/2\Z \times \Z/3\Z$, and congruence modulo $2$ is the matching factor congruence.
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# Witt vectors

## Summary

Fix a prime $p$ and a [[commutative ring|0310REJJY8]] $A$.
The ring $W(A)$ of $p$-typical Witt vectors consists of infinite sequences of elements of $A$ with a special addition and multiplication.
These operations are built so that certain polynomial functions of the entries, the ghost components, turn them into ordinary addition and multiplication.
The construction lifts rings of characteristic $p$ to characteristic $0$: the Witt vectors of the [[finite field|0ZXZ82FTX2]] $\F_p$ form the ring $\Z_p$ of [[p-adic integers|V6TJAFZ4MD]].
Witt vectors are a basic tool in arithmetic geometry and in the theory of [[perfect fields|VWBVKT794W]] of characteristic $p$.

## Explanations

### L1 human

A $p$-adic integer can be written as a digit expansion $a_0 + a_1 p + a_2 p^2 + \cdots$ with digits in $\set{0, \ldots, p-1}$.
Adding two such expansions needs carries, and the carry rule does not come from the arithmetic of the digits modulo $p$ alone.
Witt vectors make the carry rule into universal polynomials, so the same construction works with "digits" taken from any ring.
For example, in $W(\F_2)$ the sum of $(1, 0, 0, \ldots)$ with itself is $(0, 1, 0, \ldots)$, just as $1 + 1 = 0 + 1 \cdot 2$ in base $2$.
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<document source="https://tamnd.github.io/math2/node/6N7BC70CX0.md" tag="6N7BC70CX0" kind="concept">
# Singular cohomology

## Summary

Singular cohomology assigns to a topological space $X$ and an abelian group $G$ the groups $H^n(X; G)$.
They are built from functions on singular simplices with values in $G$.
It is the dual of [[singular homology|M0Q5CFJ7ZH]], and it turns continuous maps into homomorphisms in the opposite direction.
Its extra structure, the cup product, makes $H^\bullet(X; R)$ a ring, which distinguishes spaces that have the same homology groups.

## Explanations

### L1 human

Homology builds things out of simplices; cohomology measures them.
A cochain is a rule that gives a number to every simplex, like a test that is applied to each piece of the space.
A cocycle is a test that gives zero on every boundary, so its value on a cycle depends only on the homology class.
For example, on a circle the rule "count how many times a loop winds around" is a $1$-cocycle, and it generates $H^1(S^1; \Z) \cong \Z$.
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# Incompressible surface

## Summary

An incompressible surface is a [[surface|VGJJVBZ1MQ]] inside a [[3-manifold|R20TB2KEQQ]] that cannot be simplified by cutting it along a disk in the manifold.
For two-sided surfaces other than spheres and disks, this means that the inclusion of the surface induces an injective map on fundamental groups.
Incompressible surfaces are the main tool to cut $3$-manifolds into simpler pieces, as in the theory of Haken manifolds and the JSJ decomposition.

## Explanations

### L1 human

Imagine a surface sitting inside a $3$-dimensional space.
Suppose some loop on the surface bounds a disk in the surrounding space, but not on the surface.
Then one can cut the surface along that loop and glue in two copies of the disk, making the surface simpler.
A surface is incompressible when no such move is possible.
The boundary torus of a solid doughnut is compressible: the loop around the tube bounds a disk inside the doughnut.
A torus $T^2 \times \set{1/2}$ inside the thickened torus $T^2 \times [0, 1]$ is incompressible.
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<document source="https://tamnd.github.io/math2/node/6NN1HP54GB.md" tag="6NN1HP54GB" kind="example">
# Catenoid

## Example

For $u \in \R$ and $v \in \R$, let
$$x(u, v) = (\cosh v \cos u, \; \cosh v \sin u, \; v).$$
Its image, the catenoid, is the [[surface of revolution|NE09EDK19V]] obtained by rotating the catenary $r = \cosh z$ about the $z$-axis.
It is a [[minimal surface|41X0YW2R9N]].

Check.
The partial derivatives are $x_u = (-\cosh v \sin u, \cosh v \cos u, 0)$ and $x_v = (\sinh v \cos u, \sinh v \sin u, 1)$.
So $E = \cosh^2 v$, $F = 0$ and $G = \sinh^2 v + 1 = \cosh^2 v$, and the parametrization is conformal.
For a conformal parametrization, $x_{uu} + x_{vv} = 2 E H N$, where $H$ is the mean curvature and $N$ the unit normal.
Here $x_{uu} = (-\cosh v \cos u, -\cosh v \sin u, 0)$ and $x_{vv} = (\cosh v \cos u, \cosh v \sin u, 0)$, so $x_{uu} + x_{vv} = 0$ and $H = 0$.

## Remarks

The principal curvatures are $\pm 1/\cosh^2 v$, and the Gaussian curvature is $K = -1/\cosh^4 v$.
Apart from pieces of planes, the catenoids are the only minimal surfaces of revolution.
The catenoid is locally isometric to the [[helicoid|EQ5JSYGWG2]], and one can be bent into the other through a family of minimal surfaces.

## Explanations

### L1 human

Dip two parallel circular rings into soapy water and pull them a little apart.
The soap film between them forms a tube with a narrow waist: that is a catenoid.
Its profile is the catenary, the curve of a hanging chain.
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# Whitehead theorem

## Statement

Let $X$ and $Y$ be connected [[CW complexes|5W95QFHXPR]], let $x_0 \in X$, and let $f \colon X \to Y$ be a continuous map.
Assume that $f_* \colon \pi_n(X, x_0) \to \pi_n(Y, f(x_0))$ is an isomorphism of [[homotopy groups|P976WYEFMB]] for every $n \ge 1$.
Then $f$ is a [[homotopy equivalence|XZG6MTSWCN]].

## Remarks

Replace $Y$ by the mapping cylinder of $f$, so that $f$ becomes the inclusion of a subcomplex.
The relative homotopy groups of the pair then vanish, and one deforms $Y$ onto $X$ cell by cell, using compression of maps of disks.
In other words, a [[weak homotopy equivalence|4Q3WY1RX91]] between CW complexes is a homotopy equivalence.
The CW hypothesis is needed: the map from the [[Warsaw circle|17VG475BRV]] to a point induces isomorphisms on all homotopy groups, but it is not a homotopy equivalence.
The isomorphisms must be induced by one map: $\R P^2 \times S^3$ and $S^2 \times \R P^3$ have isomorphic homotopy groups, but they are not homotopy equivalent, as their homology differs.

## Explanations

### L1 human

Homotopy groups record how spheres of each dimension sit in a space.
For spaces built from cells, a map that matches all these records perfectly is as good as an equivalence: it can be inverted up to deformation.
So for such spaces, homotopy groups detect sameness of shape, provided a map realises the match.

### L2 human

The theorem justifies working with CW complexes in homotopy theory, where weak and genuine homotopy equivalences coincide.
Combined with the [[Hurewicz theorem|ZA11N5XPZM]], it gives a homology version.
A map between simply connected CW complexes that induces isomorphisms on all integral homology groups is a homotopy equivalence.
For example, a simply connected CW complex with the integral homology of a point is contractible.
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# Discontinuous Galerkin method

## Summary

A discontinuous Galerkin method is a [[Galerkin method|Y5B0PEBKE8]] whose trial and test functions are polynomials on each element of a mesh.
No continuity is required between elements.
Elements are coupled only through numerical fluxes on their common faces, as in a [[finite volume method|HGKN8E09RS]].
It combines the high order of [[finite element methods|98GZ9DVHTA]] with the local conservation and upwinding of finite volume schemes.
It was introduced by Reed and Hill for the neutron transport equation, and is widely used for [[conservation laws|8RA7AMWSAF]].
See the [[definition|J332CQ8SPB]].

## Explanations

### L1 human

Cut the domain into pieces and, on each piece, approximate the solution by its own polynomial.
The polynomials of neighbouring pieces need not agree at the boundary, so the approximation may jump there.
The pieces talk to each other only through a flux at their common boundary, chosen so that what leaves one piece enters the next.
With polynomials of degree zero, each piece holds one number, and the method is a finite volume scheme.
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# Amenable Banach algebra

## Summary

An amenable [[Banach algebra|9KYWN94QDN]] is a Banach algebra for which every bounded [[derivation|3J89REV9WQ]] into every dual bimodule is inner.
Barry Johnson introduced the notion.
He proved that for a locally compact group $G$, the group algebra $L^1(G)$ is amenable exactly when $G$ is an [[amenable group|CEAT9BESKK]].
For C*-algebras, amenability is equivalent to being [[nuclear|D2M4VXJ7PA]].
It is one of the main finiteness conditions in the cohomology of Banach algebras.

## Explanations

### L1 human

A derivation is a linear map that obeys the product rule $D(ab) = a D(b) + D(a) b$.
Some derivations come from a simple formula, $D(a) = a f - f a$ for a fixed $f$; these are called inner.
An algebra is amenable if, in a whole class of settings, every derivation is of this simple kind.
For example, the complex numbers $\C$ form an amenable Banach algebra, and so does every algebra $C(X)$ of continuous functions on a compact Hausdorff space.
The name comes from amenable groups, whose group algebras are exactly the amenable ones.
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# Differential of a smooth map

## Summary

The differential of a smooth map between manifolds is its best linear approximation at a point.
It sends tangent vectors at a point of the source to tangent vectors at the image point in the target.
In local coordinates it is given by the Jacobian matrix of partial derivatives.
It extends the derivative of calculus to curved spaces, and it obeys the chain rule.

## Explanations

### L1 human

Imagine a smooth map that wraps a line around a circle.
If a point moves along the line with some velocity, its image moves along the circle with some velocity.
The differential is the rule that turns the first velocity into the second.
For $t \mapsto (\cos t, \sin t)$, the velocity $1$ at time $t$ becomes the tangent arrow $(-\sin t, \cos t)$.
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# Torsion submodule

## Summary

Let $M$ be a module over a commutative ring $R$.
Its torsion submodule is the set of elements of $M$ that are killed by some non-zero-divisor of $R$.
It is a [[submodule|05WDSN597C]] of $M$, and it measures how far $M$ is from being torsion-free.
For $R = \Z$ it is the [[torsion subgroup|0DBWXQRXQJ]] of an [[abelian group|082GHNPJ53]].
Over a [[principal ideal domain|0CMD0BCH8T]], a finitely generated module splits as its torsion submodule plus a free module.

## Explanations

### L1 human

Think of the elements of a module that "vanish after scaling" by a harmless scalar.
A harmless scalar is one that is not a zero divisor, such as a nonzero integer.
These elements form the torsion part, and dividing them out leaves a torsion-free module.
For example, in the abelian group $\Z \oplus \Z/6\Z$, the element $(0, 1)$ is torsion because $6 \cdot (0, 1) = 0$.
The element $(1, 0)$ is not torsion, because $n \cdot (1, 0) = (n, 0) \ne 0$ for every $n \ne 0$.
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# Collatz conjecture

## Statement

Let $T$ be the map on the positive integers given by $T(n) = n/2$ if $n$ is even and $T(n) = 3n + 1$ if $n$ is odd.
For every positive integer $n$, there is an integer $k \ge 0$ with $T^k(n) = 1$, where $T^k$ is the $k$-fold iterate of $T$.

## Remarks

The conjecture is open.
It has been checked by computer for all starting values up to very large bounds.
Riho Terras showed in 1976 that the set of $n$ having some iterate $T^k(n) < n$ has [[natural density|RVG9Z8NYA6]] $1$.
Terence Tao showed in 2019 that almost all orbits, in the sense of logarithmic density, reach values below $f(n)$ for any function $f$ that tends to infinity.
For example, starting from $6$ the [[sequence|0G9EKM0E15]] is $6, 3, 10, 5, 16, 8, 4, 2, 1$.
Starting from $27$ it takes $111$ steps to reach $1$ and climbs as high as $9232$ on the way.
Once the sequence reaches $1$, it cycles through $1, 4, 2, 1$; the conjecture says that this is the only cycle and that no orbit grows without bound.

## Explanations

### L1 human

Pick a positive whole number.
If it is even, halve it; if it is odd, triple it and add one.
Repeat.
The conjecture says that you always end up at $1$, whatever number you start from.
The rule is simple enough for a child, yet nobody has found a proof, and many think current methods are not strong enough.
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# Lie bracket of vector fields

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]], and let $C^\infty(M)$ be the algebra of [[smooth functions|0V33BNKZ9S]] on $M$.
A smooth vector field $X$ on $M$ is a smooth [[section|WWX7H5W1RP]] of the [[tangent bundle|WY6K3P4YCG]]; it acts on $f \in C^\infty(M)$ by $(Xf)(p) = X_p(f)$.
For smooth vector fields $X$ and $Y$ on $M$, the *Lie bracket* $[X, Y]$ is the vector field given by
$$[X, Y] f = X(Y f) - Y(X f), \qquad f \in C^\infty(M).$$

## Remarks

In local coordinates, with $X = \sum_i X^i \partial_i$ and $Y = \sum_j Y^j \partial_j$,
$$[X, Y] = \sum_{j} \sum_{i} \left( X^i \frac{\partial Y^j}{\partial x^i} - Y^i \frac{\partial X^j}{\partial x^i} \right) \partial_j.$$
Example on $\R^2$: for $X = \partial_x$ and $Y = x \, \partial_y$, one gets $[X, Y] f = \partial_x(x f_y) - x f_{yx} = f_y$, so $[X, Y] = \partial_y$.

The product $f \mapsto X(Y f)$ alone is not a vector field.
For $X = Y = \partial_x$ on $\R$, it is $f \mapsto f''$, and $(fg)'' = f'' g + 2 f' g' + f g''$, so the product rule fails.
In the bracket the second-order terms cancel, and the product rule holds.

The bracket is bilinear and antisymmetric, and it satisfies the Jacobi identity $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$.
So the smooth vector fields on $M$ form a Lie algebra.
The bracket $[X, Y]$ equals the [[Lie derivative|PJWK0PKN6Q]] $\mathcal{L}_X Y$.

## Explanations

### L1 human

The Lie bracket of two vector fields is the commutator $XY - YX$ of the two derivations they define, which is again a vector field.
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# Kakeya set

## Description

Let $n \ge 2$.
A *Kakeya set*, or *Besicovitch set*, is a [[compact set|0R6EWFKRCF]] $K \subseteq \R^n$ that contains a line segment of length $1$ in every direction.
That is, for each unit vector $e \in \R^n$ there is a point $a \in \R^n$ with $\set{a + t e : 0 \le t \le 1} \subseteq K$.
The closed ball of radius $1/2$ about the origin is a Kakeya set.
For a unit vector $e$, the segment from $-e/2$ to $e/2$ lies in the ball.
Besicovitch showed that there are Kakeya sets in $\R^2$ with [[Lebesgue measure|2Z2QVWZ262]] zero.
If $K \subseteq \R^2$ is such a set, then $K \times [0,1]^{n-2}$ is a Kakeya set in $\R^n$ of measure zero.
Indeed, let $e = (e_1, e_2, e')$ be a unit vector.
Take a segment of $K$ in the direction of $(e_1, e_2)$ with length $\abs{(e_1, e_2)}$.
The last $n - 2$ coordinates then move by $te'$ for $0 \le t \le 1$, starting from a suitable corner of the cube.

## Remarks

Although a Kakeya set may have measure zero, Davies proved in 1971 that every Kakeya set in $\R^2$ has [[Hausdorff dimension|SSNFTHZ6YJ]] $2$.
The Kakeya conjecture asks whether every Kakeya set in $\R^n$ has Hausdorff dimension $n$.
In 2025 Hong Wang and Joshua Zahl posted a proof for $n = 3$.
Some authors do not require a Kakeya set to be compact.

## Explanations

### L1 human

A Kakeya set is a set in which a needle of length one can be found pointing in every direction.
A disc of diameter one works, since a diameter can point in any direction.
The surprise is that such a set can be so thin that it has area zero.
Its needles overlap in a very clever way, so that very little room is used.
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<document source="https://tamnd.github.io/math2/node/6PHR16TB0V.md" tag="6PHR16TB0V" kind="definition">
# Banach bundle

## Definition

Let $M$ be a [[Banach manifold|VB95VVVDM8]] of class $C^p$, with $p \ge 0$.
Let $E$ be a topological space and $\pi \colon E \to M$ a continuous surjection, and assume each fibre $E_x = \pi^{-1}(x)$ carries the structure of a real [[Banach space|0BDTTM24DP]].
For Banach spaces $X$ and $Y$, let $\mathcal{L}(X; Y)$ be the Banach space of [[bounded linear maps|0S3SMRH0Z5]] with the [[operator norm|0ZTEH055YF]].
The map $\pi$ is a *Banach bundle* of class $C^p$ if there are data of the following kind.
There are an open cover $(U_i)_{i \in I}$ of $M$, Banach spaces $X_i$ and maps $\tau_i \colon \pi^{-1}(U_i) \to U_i \times X_i$ such that:


1. Each $\tau_i$ is a [[homeomorphism|0QT00PB5G2]] with $\mathrm{pr}_{U_i} \circ \tau_i = \pi$.
2. For each $x \in U_i$, the induced map $\tau_{i,x} \colon E_x \to X_i$ is a continuous linear isomorphism.
3. For all $i, j \in I$, the map $U_i \cap U_j \to \mathcal{L}(X_i; X_j)$, $x \mapsto \tau_{j,x} \circ \tau_{i,x}^{-1}$, is of class $C^p$.

## Remarks

Example: for a Banach manifold $M$ and a Banach space $X$, the product $M \times X \to M$ with the single trivialisation $\tau = \id$ is a Banach bundle; the transition map is constant.
The tangent bundle of a $C^{p+1}$ Banach manifold is a Banach bundle of class $C^p$, with transition maps given by derivatives of chart changes.

Non-example: clause 3 asks for continuity in the operator norm, not only pointwise.
On $X = L^2(\R)$ let $T_s f(t) = f(t - s)$ be the translation by $s$.
Then $s \mapsto T_s f$ is continuous for each $f$, but $\norm{T_s - T_{s'}} = 2$ for $s \ne s'$.
So gluing two trivial bundles $(-\infty, 1) \times X$ and $(0, \infty) \times X$ over $M = \R$ by $x \mapsto T_{1/x}$ on $(0, 1)$ violates clause 3.

When all $X_i$ are finite-dimensional, this is the notion of a vector bundle of class $C^p$.
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# Stein's lemma

## Statement

Let $Z$ be a [[normal|05AQE0QR0D]] random variable with mean $\mu \in \R$ and variance $\sigma^2 > 0$.
Let $g \colon \R \to \R$ be [[absolutely continuous|ZT6F4C2YHK]] on every bounded interval, with derivative $g'$, and assume $E\abs{g'(Z)} < \infty$.
Then $E\abs{g(Z)(Z - \mu)} < \infty$ and
$$E\big[ g(Z) (Z - \mu) \big] = \sigma^2 \, E\big[ g'(Z) \big].$$

## Remarks

The normal density $\varphi$ with mean $\mu$ and variance $\sigma^2$ satisfies $\varphi'(z) = -\frac{z - \mu}{\sigma^2} \varphi(z)$.
So the left side is $-\sigma^2 \int g(z) \varphi'(z) \, dz$, and [[integration by parts|08Y1BBQTK9]] moves the derivative onto $g$.
The integrability hypothesis is what controls the boundary terms at $\pm\infty$; the careful proof writes $g(z) - g(\mu)$ as an integral of $g'$ and uses Fubini's theorem.

Example: with $\mu = 0$ and $g(z) = z^3$, the left side is $E[Z^4] = 3\sigma^4$.
The right side is $\sigma^2 E[3Z^2] = 3\sigma^4$, as it should be.

The multivariate form is used in estimation.
If $X$ is [[multivariate normal|0XN3K7RSZY]] with mean $\theta$ and covariance $\sigma^2 I_p$, then $E[(X_i - \theta_i) g(X)] = \sigma^2 E[\partial_i g(X)]$ for suitable $g \colon \R^p \to \R$.

## Explanations

### L1 human

For a bell-shaped random variable, the average of $g(Z)$ times the deviation of $Z$ from its mean equals the variance times the average slope of $g$.
It is an integration by parts in disguise.
It turns covariances with the data into averages of derivatives, which are often easier to handle.

### L2 human

The identity characterizes the normal distribution: if it holds for all bounded smooth $g$, then $Z$ is normal with mean $\mu$ and variance $\sigma^2$.
This is the starting point of Stein's method for proving normal approximations.
In decision theory, the multivariate form gives Stein's unbiased risk estimate and through it the risk of the James–Stein estimator.
</document>

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# Lefschetz number

## Definition

Let $X$ be a finite [[CW complex|5W95QFHXPR]], or more generally a space homotopy equivalent to one, and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
For each $k \ge 0$, let $f_{\ast,k} \colon H_k(X; \Q) \to H_k(X; \Q)$ be the induced linear map on [[homology with rational coefficients|XT7VWNTKGB]], a finite-dimensional vector space.
The *Lefschetz number* of $f$ is
$$L(f) = \sum_{k \ge 0} (-1)^k \tr\big(f_{\ast,k}\big),$$
where $\tr$ is the [[trace|0KZDNGA561]].
The sum is finite: if $X$ is homotopy equivalent to a finite CW complex of dimension $d$, then $H_k(X; \Q) = 0$ for $k > d$.

## Remarks

Homotopic maps induce the same maps on homology, so $L(f)$ depends only on the homotopy class of $f$.
For the identity map, $L(\id_X) = \sum_k (-1)^k \dim H_k(X; \Q) = \chi(X)$, the [[Euler characteristic|RNDDWDAFJ5]].
One may use $H_k(X; \Z)$ modulo torsion instead of rational homology and get the same number.

Example: let $X$ be connected and let $f$ be constant.
Then $f_{\ast,0}$ is the identity of $H_0(X; \Q) \cong \Q$ and $f_{\ast,k} = 0$ for $k > 0$, so $L(f) = 1$.

Example: the antipodal map $a$ of $S^n$ acts on $H_n(S^n; \Q)$ by its degree $(-1)^{n+1}$, so $L(a) = 1 + (-1)^n (-1)^{n+1} = 0$, in line with the fact that $a$ has no fixed point.

Non-example: for an infinite wedge of circles, $H_1(X; \Q)$ is infinite-dimensional, so the trace in the formula is not defined; the finiteness hypothesis on $X$ fails.
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# d'Alembert's functional equation

## Summary

d'Alembert's functional equation is the [[functional equation|SRAN4EKWS0]] $f(x + y) + f(x - y) = 2 f(x) f(y)$ for an unknown function $f \colon \R \to \R$.
It is the addition rule of the cosine, written as an equation.
Its continuous solutions are $f = 0$, $f(x) = \cos(ax)$ and $f(x) = \cosh(ax)$ with $a \in \R$.
Versions on groups, with $x - y$ replaced by $x y^{-1}$, are a main topic of the theory of functional equations on groups.

## Explanations

### L1 human

The [[cosine|08WFZP6P89]] satisfies $\cos(x + y) + \cos(x - y) = 2 \cos x \cos y$.
The equation turns this identity around and asks for all functions with the same property.
For example, $\cosh x = (e^x + e^{-x})/2$ also satisfies it.
Up to stretching the variable and the trivial solution $0$, the cosine and the hyperbolic cosine are the only continuous solutions.
</document>

<document source="https://tamnd.github.io/math2/node/6Q36P44PHZ.md" tag="6Q36P44PHZ" kind="statement">
# Reduction of a boundary value problem to a Fredholm equation

## Statement

Let $a < b$ and let $q, F \colon [a, b] \to \R$ be continuous.
Let $G \colon [a, b] \times [a, b] \to \R$ be the [[Green's function|SH6JY9RN2E]] of $y'' = 0$ with $y(a) = y(b) = 0$, namely
$$G(x, s) = \begin{cases} \dfrac{(x - a)(s - b)}{b - a}, & a \le x \le s \le b, \\[2mm] \dfrac{(s - a)(x - b)}{b - a}, & a \le s \le x \le b. \end{cases}$$
For a continuous function $y \colon [a, b] \to \R$, the following are equivalent.

1. $y \in C^2[a, b]$ and $y$ solves the [[boundary value problem|TM8VQJWGHZ]] $y'' + q(x) y = F(x)$, $y(a) = y(b) = 0$.
2. $y$ solves the [[Fredholm integral equation|526TCVMSHP]] of the second kind
$$y(x) = \int_a^b G(x, s) F(s) \, ds - \int_a^b G(x, s) q(s) y(s) \, ds, \quad x \in [a, b].$$

## Remarks

Proof idea: for continuous $g$, the function $w(x) = \int_a^b G(x, s) g(s) \, ds$ is the unique $C^2$ solution of $w'' = g$, $w(a) = w(b) = 0$; uniqueness holds because a linear function vanishing at $a$ and $b$ is zero.
If $y$ satisfies 1, apply this with $g = F - q y$ to get 2.
If $y$ satisfies 2, the right side is $w$ with the continuous function $g = F - q y$, so $y = w$ is $C^2$ with $y'' = F - q y$ and $y(a) = y(b) = 0$.

Example: on $[0, 1]$, $G(x, s) = x (s - 1)$ for $x \le s$ and $s (x - 1)$ for $x \ge s$.
The eigenvalue problem $y'' + \mu y = 0$, $y(0) = y(1) = 0$, becomes the homogeneous equation $y(x) = -\mu \int_0^1 G(x, s) y(s) \, ds$, whose kernel $-G$ is symmetric.
Its nonzero solutions occur for $\mu = k^2 \pi^2$, $k \ge 1$, with $y = \sin(k \pi x)$.

The integral equation has the continuous kernel $-G(x, s) q(s)$, so Fredholm theory applies.
So the boundary value problem has a unique solution for every $F$ if and only if the problem with $F = 0$ has only the zero solution.
The same method works for other separated boundary conditions and other second-order operators, provided the Green's function of the unperturbed problem exists.

## Explanations

### L1 human

A boundary value problem fixes the values of the solution at both ends of an interval.
The Green's function turns the differential equation into an integral equation in which the boundary values are already built in.
The resulting equation integrates over the whole interval, so it is a Fredholm equation rather than a Volterra equation.

### L2 human

This reduction is how Sturm–Liouville problems enter the theory of integral equations.
For self-adjoint problems the Green's function is a symmetric kernel, and the Hilbert–Schmidt theory of symmetric kernels then gives real eigenvalues and expansions in eigenfunctions.
In contrast, an initial value problem leads to a Volterra equation, which is always uniquely solvable.
A boundary value problem leads to a Fredholm equation, which may fail to be solvable at characteristic values.
</document>

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# Induced representation of a locally compact group

## Summary

Unitary induction builds a [[unitary representation|74BPYJNZTM]] of a [[locally compact group|AHV05ZBCYW]] $G$ from a unitary representation of a closed subgroup $H$.
The new space consists of square-integrable vector-valued functions on $G$ that transform under $H$ by the given representation, and $G$ acts by translation.
It extends the [[induced representation|JJP4JFH3C6]] of finite groups to subgroups of infinite index, such as lattices and parabolic subgroups.
Mackey's theory of induced representations uses it to build and classify irreducible representations, for example the principal series of semisimple Lie groups.

## Explanations

### L1 human

Suppose you know how a closed subgroup $H$ of $G$ acts on a Hilbert space $K$.
Look at functions on $G$ with values in $K$ whose behaviour along each coset $gH$ is dictated by the action of $H$.
Such a function is really a section of a bundle over the coset space $G/H$, and $G$ moves these sections around by translation.
For example, inducing the trivial representation of the trivial subgroup gives $L^2(G)$, where $G$ acts by left translation.
Inducing a character of $\Z$ up to $\R$ gives functions on the line that repeat up to a fixed phase factor after each step of length $1$.
</document>

<document source="https://tamnd.github.io/math2/node/6Q7H2FQ12Q.md" tag="6Q7H2FQ12Q" kind="example">
# Population balance equation

## Example

Let $n(v, t) \ge 0$ be the number density of particles of volume $v > 0$ at time $t$ in a well-mixed vessel.
Let $G(v)$ be the growth rate of a particle of volume $v$, and let $\beta(v, w) = \beta(w, v) \ge 0$ be the aggregation kernel.
Let $S(v) \ge 0$ be the breakage rate.
Let $b(v \mid w)$ be the density of fragments of volume $v$ produced when a particle of volume $w > v$ breaks.
A population balance equation with growth, aggregation and breakage is
$$\frac{\partial n}{\partial t} + \frac{\partial}{\partial v}\big(G(v) n\big) = \frac12 \int_0^v \beta(v - w, w) \, n(v - w, t) \, n(w, t) \, dw - n(v, t) \int_0^\infty \beta(v, w) \, n(w, t) \, dw + \int_v^\infty b(v \mid w) \, S(w) \, n(w, t) \, dw - S(v) \, n(v, t).$$
The left side is a transport equation in the internal coordinate $v$.
The right side lists birth and death of particles by aggregation and by breakage.
It contains partial derivatives in $t$ and $v$ and integrals of $n$ in $v$, so it is an [[integro-partial differential equation|TH4NGC4016]].

Check in two special cases.
With $G = 0$ and $S = 0$ only aggregation remains, and the equation is the [[Smoluchowski coagulation equation|9JRNCRZ97G]] in the volume variable.
With $\beta = 0$, $S = 0$ and constant $G = g$, the equation is $\partial_t n + g \, \partial_v n = 0$.
It is solved by $n(v, t) = n_0(v - g t)$ for any initial profile $n_0$ that vanishes for $v \le 0$.
Every particle grows at the same rate, and the profile moves to the right.

## Remarks

In general the internal coordinate can be a vector of several properties, such as size, composition and age, and the particles can also move in physical space.
Moment methods turn the equation into ordinary differential equations for the moments $\int v^k n \, dv$ when the kernels allow it.
Such equations are used for crystallization, emulsions, aerosols and cell populations.

## Explanations

### L1 human

A population balance keeps track of how many particles of each size there are.
Particles grow, which moves them to larger sizes; they stick together, which creates big particles from small ones; and they break, which creates small particles from big ones.
Growth gives a derivative in the size variable, while sticking and breaking give integrals over the sizes of the partners.
</document>

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# Monoid ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity and let $M$ be a [[monoid|0N6R902FG1]], written multiplicatively with identity $e$.
The *monoid ring* $R[M]$ is the set of functions $f \colon M \to R$ with $f(m) \ne 0$ for only finitely many $m \in M$.
Such a function is written as a finite formal sum $\sum_{m \in M} f(m) \, m$.
Addition is pointwise, and multiplication is the convolution
$$(f g)(m) = \sum_{\substack{x, y \in M \\ x y = m}} f(x) \, g(y).$$
The sum is finite because $f$ and $g$ have finite support.
With these operations $R[M]$ is a ring with identity $1 \cdot e$.

## Remarks

Example: for the monoid $M = \set{1, x, x^2, \dots}$ with $x^i x^j = x^{i+j}$, the convolution is the usual product of polynomials, so $R[M]$ is the [[polynomial ring|0SDC934R52]] $R[x]$.
For instance $(1 + x)(1 - x) = 1 \cdot 1 + (1 - 1) x - x^2 = 1 - x^2$.
For a [[group|G5S1M9]] $G$, $R[G]$ is the [[group ring|4NMTSX0XDS]], and for $M = (\Z, +)$ it is the [[Laurent polynomial ring|QMY5JTZZB8]].
Non-example: the set of all functions $\Z \to R$ with the same product is not a ring, because the finiteness condition fails and the convolution sum $\sum_{i + j = n} f(i) g(j)$ can be an infinite sum.
The ring $R[M]$ is commutative when $R$ and $M$ are commutative.
When $R$ is commutative, each monoid homomorphism from $M$ to the multiplicative monoid of an $R$-algebra $A$ extends uniquely to an $R$-algebra homomorphism $R[M] \to A$.
In additive notation for $M$, the same ring is often called the monoid algebra.
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# Aleph number

## Summary

The aleph numbers $\aleph_\alpha$, indexed by the [[ordinals|0YSCSX5YZC]], are the infinite [[cardinal numbers|0EBACQFAZW]] of well-orderable sets, listed in increasing order.
The first is [[aleph-null|0V2H5PGJ9B]], the size of the natural numbers, and $\aleph_1$ is the next infinite cardinal.
With the axiom of choice, every infinite set has an aleph as its cardinality, so the alephs measure all infinite sizes.
The [[precise definition|TEWH75TX8Y]] is by transfinite recursion.

## Explanations

### L1 human

The alephs are the sizes of infinity, lined up in order.
The smallest infinite size is $\aleph_0$, the size of $\set{0, 1, 2, \ldots}$.
The next one is $\aleph_1$, then $\aleph_2$, and so on, and after all $\aleph_n$ for finite $n$ comes $\aleph_\omega$.
The real numbers have size $2^{\aleph_0}$, which is at least $\aleph_1$.
Which aleph it is cannot be decided from the usual axioms; the [[continuum hypothesis|0MN2MER9WN]] says it is $\aleph_1$.
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<document source="https://tamnd.github.io/math2/node/6QAPNQVBEK.md" tag="6QAPNQVBEK" kind="definition">
# HOMFLY polynomial

## Definition

Let $\mathcal{L}$ be the set of isotopy classes of oriented tame [[links|1A5WRCNTM3]] in $S^3$.
Let $L_+$, $L_-$ and $L_0$ be oriented links whose diagrams agree outside a small disk.
Inside it, they have a positive crossing, a negative crossing, and the oriented resolution.
The *HOMFLY polynomial* is the unique map $P \colon \mathcal{L} \to \Z[a, a^{-1}, z, z^{-1}]$ such that:

1. $P(O) = 1$ for the [[unknot|1TQ06R4Q66]] $O$;
2. $a \, P(L_+) - a^{-1} P(L_-) = z \, P(L_0)$ for every such triple.

## Remarks

Several conventions for the variables are in use; they differ by substitutions.
Example: a one-crossing diagram of the unknot gives $a - a^{-1} = z \, P(O \sqcup O)$, so the two-component unlink has $P = (a - a^{-1}) z^{-1}$.
Non-example: the map $L \mapsto 1$ satisfies clause 1 but violates clause 2, since $a - a^{-1} \ne z$.
The substitution $a = t^{-1}$, $z = t^{1/2} - t^{-1/2}$ gives the [[Jones polynomial|4SPC17FPXE]].
The substitution $a = 1$, $z = t^{1/2} - t^{-1/2}$ gives the Conway-normalized [[Alexander polynomial|JKA25KDPQP]].
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# Jensen's functional equation

## Summary

Jensen's functional equation is the [[functional equation|SRAN4EKWS0]] $f\left(\frac{x + y}{2}\right) = \frac{f(x) + f(y)}{2}$ for an unknown function $f \colon \R \to \R$.
It says that the value at the midpoint of two points is the average of the values.
Its solutions are the functions $f = A + b$ with $A$ an [[additive function|KF719SZD5T]] and $b$ a constant, and the continuous ones are the affine functions $f(x) = ax + b$.
It is the equality case of midpoint convexity, which links it to [[convex functions|07Q7CNM5WF]].

## Explanations

### L1 human

Take two points on the graph and their midpoint on the $x$-axis.
The equation asks the graph to pass exactly through the middle of the chord between the two points.
A straight line does this: for $f(x) = 2x + 3$, the midpoint of $0$ and $2$ is $1$, and $f(1) = 5$ is the average of $f(0) = 3$ and $f(2) = 7$.
A curved graph such as $x^2$ fails, since it sags below its chords.
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# Finite Desarguesian planes are Pappian

## Statement

Let $\Pi$ be a finite [[projective plane|DANFNN8C4Y]] in which [[Desargues's theorem|5Z19Y02MME]] holds.
Then [[Pappus's hexagon theorem|B6C3J2GJWP]] holds in $\Pi$.
Moreover $\Pi$ is isomorphic to $PG(2, q)$, the projective plane over the [[finite field|0ZXZ82FTX2]] with $q$ elements, for a prime power $q$.

## Remarks

A [[Desarguesian plane|HPX85PA8NX]] is coordinatized by a [[division ring|0EA5BTA0CX]] $D$, in the sense that $\Pi \cong PG(2, D)$; see the [[coordinatization of projective planes|MAM2MDG0XG]].
If $\Pi$ is finite, $D$ is finite, so $D$ is a field by [[Wedderburn's little theorem|0RRNVX1VKA]].
A finite field has prime power order $q$, and in a plane over a field Pappus's theorem holds.

Finiteness is necessary.
The projective plane over the quaternions is Desarguesian, but Pappus's theorem fails in it, because the quaternions are not commutative.

In the other direction, Hessenberg's theorem says that Pappus's theorem implies Desargues's theorem in every projective plane.
So for finite planes the two configuration theorems are equivalent.

## Explanations

### L1 human

Desargues's theorem and Pappus's theorem are two classical facts about points and lines that may or may not hold in an abstract projective plane.
In general Pappus is stronger.
For finite planes they turn out to be the same, because of an algebraic fact: a finite skew field is commutative.

### L2 human

Desargues's theorem lets one build coordinates from a division ring, and Pappus's theorem holds exactly when that division ring is commutative.
Wedderburn's little theorem is therefore the algebraic input that makes every finite Desarguesian plane classical.
It also shows that the order of a finite Desarguesian plane is a prime power.
A finite plane that is not of the form $PG(2, q)$, such as the Hall plane of order $9$, must therefore fail Desargues's theorem.
</document>

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# Constructive function theory

## Definition

Let $f$ be a function on $[a, b]$, or a $2\pi$-periodic function, in a fixed normed space.
Let $E_n(f)$ be the [[error of best approximation|YDN0T4MEX0]] of $f$ by algebraic, respectively trigonometric, polynomials of degree at most $n$.
*Constructive function theory* is the part of [[approximation theory|YPJSH79SHQ]] that characterizes classes of functions by the behaviour of the sequence $(E_n(f))_{n \ge 0}$ as $n \to \infty$.
Its results are of two kinds.

1. Direct theorems: a smoothness condition on $f$, often a bound on its [[modulus of continuity|AXCJBSQZWF]], implies an upper bound for $E_n(f)$.
2. Inverse theorems: an upper bound for $E_n(f)$ implies a smoothness condition on $f$.

## Remarks

Example: for $2\pi$-periodic continuous $f$, the uniform norm, and $0 < \alpha < 1$, one has $E_n(f) = O(n^{-\alpha})$ if and only if $\abs{f(x) - f(y)} \le M \abs{x - y}^\alpha$ for some constant $M$ and all $x, y$.
The direction from smoothness to the rate is a direct theorem of Jackson type, and the converse is an inverse theorem of Bernstein type.

Non-example: the statement that $E_n(f) \to 0$ for every continuous $f$ is a result of approximation theory.
But it gives no rate and distinguishes no class of functions, so it is not of this kind.
</document>

<document source="https://tamnd.github.io/math2/node/6QF83CFK2G.md" tag="6QF83CFK2G" kind="concept">
# Dual matroid

## Summary

The dual of a [[matroid|PX31ZNYS3Z]] $M$ is the matroid $M^*$ on the same ground set whose [[bases|QTSQR7130G]] are the complements of the bases of $M$.
Duality is an involution, $M^{**} = M$, and it exchanges deletion and contraction of [[minors|4WCYBSH1P6]].
For a planar graph, the dual of its graphic matroid is the graphic matroid of the [[dual graph|1QHTD1DS5J]].
So matroid duality extends planar graph duality to all matroids.

## Explanations

### L1 human

In a matroid, a basis is a largest independent set.
Taking the complement of every basis gives the bases of a new matroid, the dual.
For a connected planar graph, the edges outside a spanning tree correspond to a spanning tree of the dual graph, and this is the same idea.
For the uniform matroid in which bases are the $k$-element subsets of an $n$-element set, the dual has the $(n-k)$-element subsets as bases.
</document>

<document source="https://tamnd.github.io/math2/node/6QFEVWKE25.md" tag="6QFEVWKE25" kind="definition">
# Veblen function

## Definition

The functions $\varphi_\gamma$, for ordinals $\gamma$, map [[ordinals|0RDJK7EQYY]] to ordinals.
They are defined by [[transfinite recursion|0ZHQMQ2C1N]] on $\gamma$:

1. $\varphi_0(\alpha) = \omega^\alpha$, with ordinal exponentiation from [[ordinal arithmetic|2C7MBG93AZ]].
2. $\varphi_{\gamma + 1}$ lists in increasing order the fixed points of $\varphi_\gamma$, the ordinals $\alpha$ with $\varphi_\gamma(\alpha) = \alpha$.
3. For a nonzero limit ordinal $\gamma$, $\varphi_\gamma$ lists in increasing order the common fixed points of all $\varphi_\delta$ with $\delta < \gamma$.

The two-variable function $(\gamma, \alpha) \mapsto \varphi_\gamma(\alpha)$, also written $\varphi(\gamma, \alpha)$, is the Veblen function.

## Remarks

Each $\varphi_\gamma$ is a [[normal function|PXTF18X433]], and this makes clauses 2 and 3 meaningful, since the fixed points of normal functions form a closed unbounded class.
Example: $\varphi_1(\alpha) = \varepsilon_\alpha$, the $\alpha$-th epsilon number, so $\varphi_1(0) = \varepsilon_0$ is the least $\alpha$ with $\omega^\alpha = \alpha$.
Check: $\varphi_1$ lists the fixed points of $\varphi_0$, and these are by definition the epsilon numbers.
Non-example: the function $\alpha \mapsto \omega^\alpha + 1$ cannot start the hierarchy, since it is not normal and has no fixed points.
The least ordinal $\alpha$ with $\varphi_\alpha(0) = \alpha$ is the Feferman–Schütte ordinal $\Gamma_0$.
More generally, one may start from any normal function $f$ in place of $\alpha \mapsto \omega^\alpha$.
</document>

<document source="https://tamnd.github.io/math2/node/6QG8QNHBF4.md" tag="6QG8QNHBF4" kind="statement">
# Double bubble theorem

## Statement

Let $V_1, V_2 > 0$.
Consider pairs $(E_1, E_2)$ of disjoint [[Caccioppoli sets|W6R7QNEAF1]] in $\R^3$ with volumes $\abs{E_1} = V_1$ and $\abs{E_2} = V_2$.
For such a pair, let $A(E_1, E_2) = \mathcal{H}^2(\partial^* E_1 \cup \partial^* E_2)$ be the total area of the interfaces.
Here $\partial^*$ is the reduced boundary and $\mathcal{H}^2$ is the $2$-dimensional [[Hausdorff measure|Q1TGZK8RN6]], so that a common wall is counted once.
Then $A$ is least for the standard double bubble with volumes $V_1$ and $V_2$.
Every minimizing pair is a standard double bubble, up to a rigid motion and up to sets of measure zero.

The *standard double bubble* is made of three spherical caps, or of two caps and a flat disc when $V_1 = V_2$.
The three pieces have a common boundary circle and meet along it at angles of $120^\circ$.

## Remarks

Michael Hutchings, Frank Morgan, Manuel Ritoré and Antonio Ros proved the theorem, published in 2002.
Existence and regularity of a minimizer come from geometric measure theory.
Hutchings showed that a minimizer is a surface of revolution about a line, with strong limits on the number of components of its regions.
The authors then excluded the remaining non-standard candidates by an instability argument based on rotating parts of the surface.
The case of equal volumes had been proved earlier by Joel Hass and Roger Schlafly with computer help.
The analogous result in the plane, with circular arcs in place of spherical caps, had been proved earlier by a group of undergraduates.
Ben Reichardt later extended the theorem to all dimensions.
For one region instead of two, the corresponding statement is the [[isoperimetric inequality|KBX8RBZ626]]: the ball encloses a given volume with least area.

## Explanations

### L1 human

Blow two soap bubbles that stick together.
They form two spherical pieces separated by a third spherical wall, all meeting along a circle at equal angles.
The theorem says that this familiar shape is the cheapest possible way, in surface area, to enclose and separate two given volumes of air.

### L2 human

The theorem is the simplest case of the problem of least-area clusters, and it shows how hard such problems are.
The single-bubble case is classical, but the two-bubble case needed new ideas in the 1990s.
The $120^\circ$ angles agree with Plateau's laws, which every minimizer must satisfy.
The analogous problem for three or more bubbles in $\R^3$ is much harder.
</document>

<document source="https://tamnd.github.io/math2/node/6QS7X5GVMY.md" tag="6QS7X5GVMY" kind="problem">
# Variational inequality

## Problem

Let $H$ be a real [[Hilbert space|HM2HHE3J51]] with inner product $\ip{\cdot}{\cdot}$.
Let $K \subseteq H$ be a nonempty closed [[convex set|02AS45T3H9]], and let $F \colon K \to H$ be a map.
Find $u \in K$ such that
$$\ip{F(u)}{v - u} \ge 0 \quad \text{for all } v \in K.$$

## Remarks

If $K = H$, then taking $v = u \pm w$ shows that the inequality holds exactly when $F(u) = 0$, so the problem contains the equation $F(u) = 0$.
If $F = \nabla f$ for a convex differentiable function $f$, then $u$ solves the problem exactly when $u$ minimizes $f$ on $K$.
Stampacchia's theorem treats the linear case.
Let $a$ be a bounded [[coercive bilinear form|86KRHS6X22]] on $H$ and let $\ell$ be a continuous linear functional on $H$.
Then there is exactly one $u \in K$ with $a(u, v - u) \ge \ell(v - u)$ for all $v \in K$.
When $a$ is symmetric, this $u$ minimizes $\tfrac12 a(v, v) - \ell(v)$ over $K$.
For $H = \R^n$, $K$ compact and $F$ continuous, a solution always exists, by the Hartman–Stampacchia theorem, whose proof uses the Brouwer fixed point theorem.
For a continuous [[monotone|Y1M03JQKKR]] $F$ the solution set is closed and convex, and a strictly monotone $F$ has at most one solution.

## Explanations

### L1 human

A variational inequality describes an equilibrium under a constraint.
Think of a ball resting on a curved surface inside a box.
If the lowest point lies inside the box, the force there vanishes; if it lies on a wall, the force need not vanish but must push into the wall.
The inequality $\ip{F(u)}{v - u} \ge 0$ says exactly this: from $u$, no allowed direction $v - u$ goes "downhill".
Obstacle problems in elasticity, contact problems and many equilibrium models in economics take this form.
</document>

<document source="https://tamnd.github.io/math2/node/6QW3MQC3SK.md" tag="6QW3MQC3SK" kind="definition">
# Holomorphic automorphisms of C^n

## Definition

Let $n \ge 1$.
A *holomorphic automorphism* of $\C^n$ is a bijective [[holomorphic map|3MQTWKBBBT]] $F \colon \C^n \to \C^n$ whose inverse $F^{-1}$ is also holomorphic.
In other words, it is a [[biholomorphism|FX92PP3HZR]] of $\C^n$ onto itself.
The set of these maps, with composition as the operation, is the group $\Aut(\C^n)$.

## Remarks

Example: for an entire function $g$ on $\C$, the shear $F(z, w) = (z, w + g(z))$ is an automorphism of $\C^2$.
It is holomorphic, and $G(z, w) = (z, w - g(z))$ is holomorphic with $G \circ F = F \circ G = \id$.
Invertible affine maps $z \mapsto Az + b$, with $A$ an invertible $n \times n$ complex matrix, are automorphisms as well.

Non-example: $F(z, w) = (z, zw)$ is holomorphic on $\C^2$ but not injective, since $F(0, w) = (0, 0)$ for all $w$.

For a holomorphic map, bijectivity already implies that the inverse is holomorphic, so the last condition is automatic.
</document>

<document source="https://tamnd.github.io/math2/node/6QXBY7J0M5.md" tag="6QXBY7J0M5" kind="definition">
# Axiom of regularity

## Definition

Work in first-order logic with the relation symbol $\in$.
The *axiom of regularity*, or *axiom of foundation*, is the axiom
$$\forall x\, \bigl( x \ne \emptyset \to \exists y\, ( y \in x \wedge y \cap x = \emptyset ) \bigr).$$
In words: every nonempty [[set|0SJ3RX9TXA]] $x$ has an [[element|0EZ758XE73]] $y$ such that no element of $y$ is an element of $x$.

## Remarks

Example: let $x = \set{\emptyset, \set{\emptyset}}$.
The element $y = \emptyset$ satisfies $y \cap x = \emptyset$, because the [[empty set|0ECVK59Y03]] has no elements.
Non-example: a set $a$ with $a = \set{a}$ would violate the axiom.
The only element of $x = \set{a}$ is $a$, and $a \cap x = x \cap x = x \ne \emptyset$.
So in [[ZF|0HE5WDQ9FJ]], no set is an element of itself.
The axiom says that $\in$ is a [[well-founded relation|0230ECX1Q5]] on every set.
Over the other axioms of ZF, it is equivalent to the statement that every set lies in some level $V_\alpha$ of the cumulative hierarchy.
With the axiom of dependent choice, it is equivalent to the statement that there is no sequence $(x_n)$ with $x_{n+1} \in x_n$ for all $n$.
In Mathlib, the statement for ZFC sets is `ZFSet.regularity`.
</document>

<document source="https://tamnd.github.io/math2/node/6QY76Q5WVS.md" tag="6QY76Q5WVS" kind="statement">
# Mahler's compactness theorem

## Statement

Let $n \ge 1$, and let $\mathcal{L}_n$ be the set of [[lattices|WN9V7WSJKF]] in $\R^n$.
Identify $\mathcal{L}_n$ with the [[quotient space|0BR3FY5Q8M]] $\mathrm{GL}_n(\R)/\mathrm{GL}_n(\Z)$, by sending $g$ to the lattice $g\Z^n$, and give it the quotient topology.
Let $A \subseteq \mathcal{L}_n$.
Then $A$ has [[compact|0R6EWFKRCF]] closure in $\mathcal{L}_n$ if and only if there are constants $C > 0$ and $\varepsilon > 0$ such that for every $\Lambda \in A$:

1. the covolume of $\Lambda$ is at most $C$;
2. every nonzero vector $v \in \Lambda$ has $\norm{v} \ge \varepsilon$.

## Remarks

The proof uses reduction theory.
Under the two bounds, each lattice of $A$ has a reduced basis whose vectors have length bounded above and below, with angles bounded away from $0$.
Such bases form a compact set of matrices, whose image in $\mathcal{L}_n$ contains $A$.
The converse holds because covolume and the length of a shortest nonzero vector are continuous on $\mathcal{L}_n$.

Example: the lattices $\Lambda_t = \Z (t, 0) \oplus \Z (0, 1/t)$ in $\R^2$ have covolume $1$.
Their shortest nonzero vector has length $t$ for $0 < t \le 1$.
As $t \to 0$ condition 2 fails, and indeed $\Lambda_t$ leaves every compact subset of $\mathcal{L}_2$.

Both conditions are needed.
The lattices $t\Z^2$ with $t \to \infty$ satisfy condition 2 but not condition 1, and they also leave every compact set.

## Explanations

### L1 human

A lattice in the plane is a regular grid of points.
The theorem says when a family of grids stays within a bounded region of the space of all grids.
The grids must not have too large a cell, and they must not have two points too close together.
A grid that degenerates, by squashing in one direction, always produces short vectors.

### L2 human

For unimodular lattices, condition 1 is automatic, so a set of unimodular lattices is relatively compact exactly when the lattices avoid a fixed ball around $0$.
This gives the basic criterion for divergence in the homogeneous space $\mathrm{SL}_n(\R)/\mathrm{SL}_n(\Z)$.
It is used in homogeneous dynamics, for example in the non-divergence results of Margulis and Dani for unipotent flows, and in Diophantine approximation through Dani's correspondence.
</document>

<document source="https://tamnd.github.io/math2/node/6R0CA4JWWY.md" tag="6R0CA4JWWY" kind="concept">
# Mal'cev condition

## Summary

A Mal'cev condition is a property of a [[variety|AJNZJD8442]] that asserts the existence of terms satisfying certain identities.
Many properties of the congruence lattices of all algebras in a variety are equivalent to Mal'cev conditions.
Examples are congruence permutability, given by a [[Mal'cev term|Y203PPTS61]], congruence distributivity, given by [[Jónsson terms|3M8Y76E4AG]], and congruence modularity, given by [[Day terms|B0YKA8401M]].
Mal'cev conditions turn questions about all algebras of a variety into questions about finitely many equations.

## Explanations

### L1 human

Some properties of a whole class of algebras come from a single formula that works in all of them.
In groups, the formula $x y^{-1} z$ returns $x$ when $y = z$ and returns $z$ when $x = y$.
The existence of such a formula already forces every group to have well-behaved congruences.
A Mal'cev condition is a statement of this kind: there are formulas that satisfy given equations.
</document>

<document source="https://tamnd.github.io/math2/node/6R2ZXY5VZW.md" tag="6R2ZXY5VZW" kind="concept">
# Hardy spaces of harmonic functions

## Summary

The harmonic Hardy space $h^p$ of the unit ball consists of [[harmonic functions|0DEB0JMN0W]].
Their $L^p$ means over the spheres of radius $r$ must stay bounded as $r \to 1$.
For $1 < p \le \infty$ these are exactly the [[Poisson integrals|TCYSWCTYYM]] of functions in $L^p$ of the sphere.
The space $h^1$ consists of the Poisson integrals of finite signed measures.
They are the harmonic counterparts of the holomorphic [[Hardy spaces|PNNNFJVF6W]] $H^p$ and are the natural classes for the Dirichlet problem with $L^p$ boundary data.

## Explanations

### L1 human

A harmonic function on a ball is in $h^p$ if it is not too large on average near the boundary sphere.
Such a function has boundary values, and it is recovered from them by the Poisson integral.
For instance, every bounded harmonic function lies in $h^\infty$.
The Poisson kernel with its pole at one boundary point is in $h^1$ but in no $h^p$ with $p > 1$, because its boundary values form a point mass.
</document>

<document source="https://tamnd.github.io/math2/node/6R57T5W7BV.md" tag="6R57T5W7BV" kind="definition">
# Ring spectrum

## Definition

Work in the [[stable homotopy category|26GTNMAF7B]], with smash product $\wedge$, unit the [[sphere spectrum|14CGGHA85T]] $S$, and twist isomorphism $\tau \colon E \wedge E \to E \wedge E$.
A *ring spectrum* is a [[spectrum|1GHF4EKBXS]] $E$ with morphisms
$$\mu \colon E \wedge E \to E, \qquad \eta \colon S \to E$$
such that:

1. (associativity) $\mu \circ (\mu \wedge \id_E) = \mu \circ (\id_E \wedge \mu)$ as morphisms $E \wedge E \wedge E \to E$;
2. (unit) $\mu \circ (\eta \wedge \id_E)$ and $\mu \circ (\id_E \wedge \eta)$ are the canonical isomorphisms $S \wedge E \to E$ and $E \wedge S \to E$.

The ring spectrum is *commutative* if moreover $\mu \circ \tau = \mu$.
In other words, a ring spectrum is a monoid in the stable homotopy category.

## Remarks

For a ring spectrum $E$, the coefficient group $\pi_*(E)$ is a graded ring.
For a space $X$, the [[generalized cohomology theory|H28PSZR7XG]] $E^*(X)$ is a graded ring under the cup product induced by $\mu$ and the diagonal of $X$.

Example: for a commutative ring $R$, the [[Eilenberg–MacLane spectrum|XJPYJT9Z1P]] $HR$ is a commutative ring spectrum, with $\mu$ induced by the multiplication of $R$.
It gives the usual cup product on $H^*(X; R)$.
The sphere spectrum, complex K-theory $KU$ and complex cobordism $MU$ are other commutative ring spectra.

Non-example: $H(\Q/\Z)$ is not a ring spectrum.
A unit and multiplication would make $\pi_0 = \Q/\Z$ a ring with identity $1$.
If $1$ has additive order $n$, then $n x = (n \cdot 1) x = 0$ for all $x$, which fails for an element of order $2n$.

These axioms hold only up to homotopy.
Ring spectra whose associativity and commutativity hold up to all higher coherent homotopies are called $A_\infty$ and $E_\infty$ ring spectra.
They are modelled as monoids in symmetric spectra or orthogonal spectra.
</document>

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# Mutual energy

## Definition

Let $n \ge 2$ and let $\mu$ and $\nu$ be finite positive [[Borel measures|DYR1QFNGNJ]] on $\R^n$ with compact [[supports|FQTXV6BQZG]].
Let $k$ be the [[Newtonian kernel|6TJTWQ5HCR]] $k(x) = \abs{x}^{2-n}$ when $n \ge 3$, and the [[logarithmic kernel|VDZ5FKYC2T]] $k(x) = \log(1/\abs{x})$ when $n = 2$, with $k(0) = +\infty$.
The *mutual energy* of $\mu$ and $\nu$ is
$$I(\mu, \nu) = \iint k(x - y)\, d\mu(x)\, d\nu(y) \in (-\infty, +\infty].$$

## Remarks

The mutual energy is symmetric, $I(\mu, \nu) = I(\nu, \mu)$, and $I(\mu, \mu)$ is the [[energy|SFSRQCM3EG]] $I(\mu)$.
It equals $\int U^\mu \, d\nu$, where $U^\mu$ is the [[potential|JHKSF3XFY2]] of $\mu$.
For a signed measure $\lambda = \mu - \nu$ with $I(\mu)$, $I(\nu)$ and $I(\mu, \nu)$ finite, one puts $I(\lambda) = I(\mu) - 2 I(\mu, \nu) + I(\nu)$.

Example: for Dirac measures $\delta_a$ and $\delta_b$ with $a \ne b$ in $\R^3$, $I(\delta_a, \delta_b) = 1/\abs{a - b}$.
For the uniform probability measure $\sigma$ on the unit sphere in $\R^3$ and $\delta_0$, $I(\sigma, \delta_0) = U^\sigma(0) = 1$.

Non-example: for $\mu = \nu = \delta_0$, the integrand is $k(0) = +\infty$ on the only point of the support, so $I(\mu, \nu) = +\infty$ is not a finite mutual energy.
</document>

<document source="https://tamnd.github.io/math2/node/6RDM3069KC.md" tag="6RDM3069KC" kind="statement">
# Birkhoff's representation theorem

## Statement

Let $L$ be a finite [[distributive lattice|0HM0HQCK6N]].
Let $J(L)$ be the set of [[join-irreducible elements|ZHRQ0AT8PM]] of $L$.
These are the elements $j$, other than the least element, such that $j = x \vee y$ implies $j = x$ or $j = y$.
Order $J(L)$ by the order of $L$, and let $\mathcal{O}(J(L))$ be the set of lower sets of the [[partially ordered set|0QV2JS9RJ3]] $J(L)$, ordered by inclusion.
Then the map
$$a \mapsto \set{j \in J(L) : j \le a}$$
is a lattice isomorphism from $L$ onto $\mathcal{O}(J(L))$, where meet and join in $\mathcal{O}(J(L))$ are intersection and union.

## Remarks

Every element $a$ of a finite lattice is the join of the join-irreducible elements below it, so the map is injective and order-preserving in both directions.
Distributivity is used for surjectivity: if $j \le x \vee y$ with $j$ join-irreducible, then $j = (j \wedge x) \vee (j \wedge y)$, so $j \le x$ or $j \le y$.
This shows that the map sends joins to unions, and so every lower set $D$ is the image of $\bigvee D$.

Example: the divisors of $12$ ordered by divisibility form a distributive lattice.
Its join-irreducible elements are the prime powers $2$, $4$ and $3$, with $2 \le 4$ and $3$ incomparable to both.
This poset has six lower sets, matching the six divisors $1, 2, 3, 4, 6, 12$; for example $6$ corresponds to $\set{2, 3}$.

Distributivity is necessary: in the [[diamond lattice M3|JQ6FJH2TP8]] all three middle elements are join-irreducible and pairwise incomparable, so the lower sets of $J(M_3)$ number $8$, while $M_3$ has $5$ elements.

## Explanations

### L1 human

Every finite distributive lattice looks like a family of sets closed under union and intersection.
The building blocks are the join-irreducible elements, which cannot be split as a join of smaller elements.
Each element is recorded by the set of building blocks below it.

### L2 human

The theorem gives a correspondence between finite posets and finite distributive lattices: a poset $P$ gives the lattice of its lower sets, and a lattice $L$ gives the poset $J(L)$.
This correspondence extends to a duality between the category of finite posets with order-preserving maps and the category of finite distributive lattices with bounded lattice homomorphisms.
For infinite distributive lattices, the analogue needs topology, as in [[Priestley duality|SA22SEC75K]].
In the infinite case, prime filters take the place of join-irreducible elements, and the [[prime ideal theorem for distributive lattices|ME8GM0VTWP]] guarantees that there are enough of them.
</document>

<document source="https://tamnd.github.io/math2/node/6RE2JMYMA0.md" tag="6RE2JMYMA0" kind="definition">
# Fukaya category

## Definition

This gives the simplest version, over the field $\F_2$, in the exact setting.
Let $(M, \omega)$ be an exact [[symplectic manifold|CAN490EPRT]], $\omega = d\lambda$, that is convex at infinity.
Fix a generic [[almost complex structure|9SYWRGVTRS]] $J$ compatible with $\omega$ and of contact type at infinity.

1. The objects are the closed exact [[Lagrangian submanifolds|H8ACPTMX08]] $L \subset M$, those for which $\lambda|_L$ is exact.
2. Let $L_0, L_1$ be objects meeting transversally. The morphism space $\hom(L_0, L_1)$ is the $\F_2$-vector space $CF(L_0, L_1)$ with basis the finite set $L_0 \cap L_1$. For non-transverse pairs one first moves $L_1$ by a chosen Hamiltonian perturbation.
3. For $k \ge 1$ and objects $L_0, \ldots, L_k$, there is a map
$$\mu^k \colon \hom(L_{k-1}, L_k) \otimes \cdots \otimes \hom(L_0, L_1) \to \hom(L_0, L_k)$$
It sends $x_k \otimes \cdots \otimes x_1$ to $\sum_y m(y; x_1, \ldots, x_k)\, y$. The number $m(y; x_1, \ldots, x_k) \in \F_2$ counts modulo $2$ certain [[pseudoholomorphic|N469E47V75]] maps $u$ from a disc with $k + 1$ boundary punctures to $M$. These maps send the boundary arcs in cyclic order to $L_0, L_1, \ldots, L_k$ and converge to $x_1, \ldots, x_k$ and $y$ at the punctures. Only maps in moduli spaces of dimension $0$ are counted.

The *Fukaya category* of $M$ is this collection of objects, morphism spaces and maps $\mu^k$.

## Remarks

The maps satisfy the $A_\infty$ relations: for every $k \ge 1$,
$$\sum_{m + n \le k} \mu^{k - m + 1}\left(x_k, \ldots, x_{n + m + 1}, \mu^m(x_{n + m}, \ldots, x_{n + 1}), x_n, \ldots, x_1\right) = 0,$$
with $m \ge 1$ and $n \ge 0$ (over $\F_2$ there are no signs).
For $k = 1$ this says $\mu^1 \circ \mu^1 = 0$, and $\mu^1$ is the Lagrangian [[Floer|NCKFPAGHXF]] differential.
For $k = 2$ it says that $\mu^2$ is a chain map, and for $k = 3$ that $\mu^2$ is associative up to the homotopy $\mu^3$.
So the cohomology groups $HF(L_0, L_1)$ with the product induced by $\mu^2$ form an ordinary category, the Donaldson–Fukaya category.

Example: in $T^*S^1 = \R \times S^1$ with $\lambda = p\,d\theta$, the zero section and the graph of $d f$ for a function $f$ on $S^1$ are exact Lagrangians.
A non-example of an object is a circle $\set{p = c}$ with $c \ne 0$: there $\lambda|_L = c\, d\theta$ is closed but not exact.

The construction for closed or non-exact manifolds needs much more work, because of disc bubbling; Fukaya, Oh, Ohta and Ono developed it with obstruction theory.
Working over $\Z$ or with a $\Z$-grading needs extra data on the Lagrangians, such as spin structures and gradings.
</document>

<document source="https://tamnd.github.io/math2/node/6RF8CB69KQ.md" tag="6RF8CB69KQ" kind="concept">
# Linear topology

## Summary

A linear topology on a ring or a module is a topology in which the neighbourhoods of $0$ are generated by [[ideals|096Z9MP2ME]] or [[submodules|05WDSN597C]].
It makes the ring a [[topological ring|TSF8RZDD5B]] whose small neighbourhoods of $0$ are closed under addition and under multiplication by ring elements.
The main examples are the [[I-adic topologies|2VF2TSJ3PY]], which lead to [[completions|KBN13NPYJQ]] such as the $p$-adic integers and formal power series rings.

## Explanations

### L1 human

In a linear topology, "being close to $0$" means "lying in a small ideal".
Two elements are close when their difference lies in a small ideal from a chosen family.
For the integers with the $p$-adic topology, the chosen ideals are $p^n \Z$.
So $p^{10}$ is closer to $0$ than $p$, and the numbers $1, 1 + p, 1 + p + p^2, \ldots$ get closer and closer to each other.
</document>

<document source="https://tamnd.github.io/math2/node/6RJVSY3PQ0.md" tag="6RJVSY3PQ0" kind="definition">
# Atom

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]] with a [[least element|0AANKKNHS4]] $0$.
An *atom* of $P$ is an element $a \in P$ such that:

1. $0 < a$;
2. there is no $x \in P$ with $0 < x < a$.

## Remarks

Equivalently, $a$ is an atom when $a$ [[covers|4HHDPZ4BDY]] $0$, or when $a$ is a minimal element of $P \setminus \set{0}$.
Example: in $\N_{>0}$ ordered by divisibility, with least element $1$, every prime $p$ is an atom.
Indeed $1 \ne p$, and the only divisors of $p$ are $1$ and $p$, so nothing lies strictly between them.
Non-example: $4$ is not an atom of this poset, since $1 < 2 < 4$ in the divisibility order, so clause 2 fails.
The interval $[0, 1]$ with its usual order has no atoms, because $0 < a / 2 < a$ for each $a > 0$.
The dual notion, an element just below a greatest element, is a *coatom*.
</document>

<document source="https://tamnd.github.io/math2/node/6RM4RWWSSX.md" tag="6RM4RWWSSX" kind="concept">
# Berkovich space dynamics

## Summary

Berkovich space dynamics studies the iteration of a rational map over a complete non-Archimedean field on the Berkovich projective line.
More generally, it studies endomorphisms of Berkovich analytic spaces.
The field itself, such as $\C_p$, is totally disconnected and not locally compact, which makes analysis of orbits awkward.
The Berkovich projective line adds extra points to $\mathbb{P}^1(\C_p)$ and is compact, Hausdorff and path-connected.
Every rational map extends to it continuously.
On this space one gets [[Julia sets|5M4S7P04GR]], [[Fatou sets|C0N0XGC6GK]] and invariant measures that behave much like those of [[complex dynamics|BQQZ0ZCGQF]].

## Explanations

### L1 human

Over the $p$-adic numbers, discs never overlap partially and the space falls apart into tiny pieces.
Berkovich added one new point for each closed disc, together with a few limit points, so that the discs are joined together into a tree-like space.
A rational map moves discs to discs, so it moves these new points too.
For the map $z \mapsto z^2$, the point that stands for the closed unit disc stays fixed.
This point is the whole Julia set, while on the ordinary $p$-adic points the Julia set is empty.
</document>

<document source="https://tamnd.github.io/math2/node/6RW68JGBHC.md" tag="6RW68JGBHC" kind="concept">
# Immersion

## Summary

An immersion is a smooth map between [[smooth manifolds|F45VQ47GPV]] whose [[differential|6P9PXQDSEN]] is injective at every point.
Locally it looks like the inclusion of $\R^m$ into $\R^n$, but globally it may cross itself.
Immersions are the maps along which a manifold can be drawn in another one without corners or folds.
Their classification up to regular homotopy is a model case of the homotopy principle.

## Explanations

### L1 human

Think of drawing a curve on paper with a pen that never stops moving.
The drawing may cross itself, but it never has a corner or a point where the pen stands still.
That is an immersion of a curve into the plane.
For example, the figure eight $t \mapsto (\sin 2t, \sin t)$ immerses the circle into $\R^2$: its velocity is never zero, but the image crosses itself at the origin.
An immersion that is one-to-one and a homeomorphism onto its image is called an embedding.
</document>

<document source="https://tamnd.github.io/math2/node/6RXEC9CT6M.md" tag="6RXEC9CT6M" kind="definition">
# Kummer theory

## Definition

Let $n \ge 1$ be an integer, and let $K$ be a [[field|04C5ZBG1WT]] whose characteristic does not divide $n$.
Assume that $K$ contains the group $\mu_n$ of $n$ distinct $n$-th [[roots of unity|0YV2CKP2AV]].
Fix an algebraic closure $\overline{K}$ of $K$.
For a subgroup $B$ of $K^\times$ that contains $K^{\times n} = \set{x^n : x \in K^\times}$, let $K(\sqrt[n]{B})$ be the subfield of $\overline{K}$ generated over $K$ by all $\beta \in \overline{K}$ with $\beta^n \in B$.
A *Kummer extension* of exponent $n$ of $K$ is an extension of the form $K(\sqrt[n]{B})$.
For such an extension $L = K(\sqrt[n]{B})$, the *Kummer pairing* is the map
$$\Gal(L/K) \times B/K^{\times n} \to \mu_n, \qquad \langle \sigma, b \rangle = \frac{\sigma(\beta)}{\beta},$$
where $\beta \in L$ is any element with $\beta^n = b$.
Kummer theory is the study of these extensions through this pairing.

## Remarks

The pairing is well defined: two choices of $\beta$ differ by a factor in $\mu_n \subseteq K$, which $\sigma$ fixes.
The main theorem of Kummer theory: $L/K$ is a [[Galois extension|0QMZAQ6QQ4]] with [[abelian|082GHNPJ53]] Galois group of exponent dividing $n$, and if $B/K^{\times n}$ is finite the pairing is perfect, so $\Gal(L/K) \cong \Hom(B/K^{\times n}, \mu_n)$.
The map $B \mapsto K(\sqrt[n]{B})$ is a bijection from the subgroups $B$ with $K^{\times n} \subseteq B \subseteq K^\times$ and $B/K^{\times n}$ finite onto the finite abelian extensions of $K$ in $\overline{K}$ of exponent dividing $n$.
In particular, for $a \in K^\times$, the extension $K(\sqrt[n]{a})/K$ is [[cyclic|0KVGH0J5B3]] of degree equal to the order of $a$ in $K^\times / K^{\times n}$.

Example: $K = \Q$, $n = 2$, $\mu_2 = \set{\pm 1}$, and $B$ generated by $\Q^{\times 2}$, $2$ and $3$.
Then $L = \Q(\sqrt{2}, \sqrt{3})$, and $B/\Q^{\times 2}$ has order $4$, so $\Gal(L/\Q) \cong (\Z/2\Z)^2$, as expected.

Non-example: $K = \Q$ and $n = 3$.
The field $\Q$ does not contain the three cube roots of unity, so the definition does not apply, and indeed $\Q(\sqrt[3]{2})/\Q$ is not even Galois.
</document>

<document source="https://tamnd.github.io/math2/node/6S4CM98021.md" tag="6S4CM98021" kind="definition">
# Hopfian group

## Definition

Let $G$ be a [[group|G5S1M9]].
$G$ is *Hopfian* if every [[surjective|01RVCMSN5N]] group [[homomorphism|0JYTG5K9Y4]] $\varphi \colon G \to G$ is [[injective|09NWETVTNY]].

## Remarks

Equivalently, if $N$ is a normal subgroup of $G$ and $G/N \cong G$, then $N = \set{1}$.

Example: $\Z$ is Hopfian.
A homomorphism $\varphi \colon \Z \to \Z$ is $n \mapsto kn$ with $k = \varphi(1)$.
If it is surjective, then $k = \pm 1$, and then it is injective.

Non-example: let $G = \bigoplus_{n \ge 1} \Z/2$, the group of sequences $(x_1, x_2, \ldots)$ in $\Z/2$ with only finitely many nonzero terms.
The shift $\varphi(x_1, x_2, x_3, \ldots) = (x_2, x_3, \ldots)$ is a surjective homomorphism.
It is not injective, since $\varphi(1, 0, 0, \ldots) = 0$.
A finitely generated non-Hopfian group is the [[Baumslag–Solitar group|429JJ5ZNBY]] $BS(2, 3)$.

The dual notion is a co-Hopfian group: every injective homomorphism from the group to itself is surjective.
</document>

<document source="https://tamnd.github.io/math2/node/6SK4PQPD3C.md" tag="6SK4PQPD3C" kind="statement">
# Gagliardo–Nirenberg interpolation inequality

## Statement

Let $n \ge 1$, let $j$ and $m$ be integers with $0 \le j < m$, and let $1 \le q, r \le \infty$.
Let $\theta$ be a real number with $j/m \le \theta \le 1$, and define $p$ by
$$\frac{1}{p} = \frac{j}{n} + \theta \Big( \frac{1}{r} - \frac{m}{n} \Big) + \frac{1 - \theta}{q},$$
assuming that $1 \le p \le \infty$.
For an integer $k \ge 0$ write $\abs{D^k u}$ for the Euclidean norm of the vector of all partial derivatives of $u$ of order $k$.
Assume that it is not the case that $1 < r < \infty$, $m - j - n/r$ is a nonnegative integer and $\theta = 1$.
Then there is a constant $C = C(n, m, j, q, r, \theta)$ such that
$$\norm{D^j u}_{L^p(\R^n)} \le C \, \norm{D^m u}_{L^r(\R^n)}^{\theta} \, \norm{u}_{L^q(\R^n)}^{1 - \theta}$$
for every smooth function $u \colon \R^n \to \R$ with compact support.

## Remarks

The relation defining $p$ is forced by scaling: under $u(x) \mapsto u(\lambda x)$ both sides must change by the same power of $\lambda$.
The proof combines the [[Sobolev inequality|B03CA692K3]] with one-dimensional bounds obtained by integration by parts, such as $\norm{u'}_{L^{p}(\R)}^2 \le C \norm{u}_{L^{q}(\R)} \norm{u''}_{L^{r}(\R)}$ for $\frac{2}{p} = \frac{1}{q} + \frac{1}{r}$.
The inequality extends by density to functions in $L^q(\R^n)$ whose [[weak derivatives|C3EC148VZD]] of order $m$ lie in $L^r(\R^n)$, with an extra condition at infinity when $j = 0$, $rm < n$ and $q = \infty$.

The excluded case is a real exception.
For $n = 2$, $m = 1$, $j = 0$ and $r = 2$, the choice $\theta = 1$ gives $p = \infty$, and the inequality $\norm{u}_{L^\infty} \le C \norm{\nabla u}_{L^2}$ fails on $\R^2$.

A common special case is Ladyzhenskaya's inequality in the plane: $\norm{u}_{L^4(\R^2)}^2 \le C \norm{u}_{L^2(\R^2)} \norm{\nabla u}_{L^2(\R^2)}$, which is the case $n = 2$, $j = 0$, $m = 1$, $q = r = 2$, $\theta = 1/2$.

## Explanations

### L1 human

The inequality bounds an intermediate quantity by two extreme ones: a lower derivative is controlled by the function and a higher derivative together.
It is like saying that if a curve is not too high and does not bend too much, then its slope cannot be too large.
The weights $\theta$ and $1 - \theta$ say how much each of the two extremes contributes.

### L2 human

In nonlinear evolution equations one usually controls the $L^q$ norm of a solution by a conservation law and some higher derivative by an energy estimate.
The Gagliardo–Nirenberg inequality then bounds the intermediate norms that appear in the nonlinear terms.
For $\theta = 1$ and $j = 0$, $m = 1$, it reduces to the [[Sobolev inequality|B03CA692K3]] on $\R^n$.
It is a basic tool for the Navier–Stokes and nonlinear Schrödinger equations.
</document>

<document source="https://tamnd.github.io/math2/node/6SNCVN8JZQ.md" tag="6SNCVN8JZQ" kind="definition">
# Einstein manifold

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $\operatorname{Ric}$ be its [[Ricci curvature|4GTNM0WHDH]] tensor.
The manifold $(M, g)$ is an *Einstein manifold*, and $g$ is an *Einstein metric*, if there is a real constant $\lambda$ such that
$$\operatorname{Ric} = \lambda g.$$
The constant $\lambda$ is the *Einstein constant*.

## Remarks

Example: the unit sphere $S^n \subset \R^{n+1}$ with $n \ge 2$ has sectional curvature $1$, so $\operatorname{Ric} = (n-1) g$, and it is Einstein with $\lambda = n - 1$.

Non-example: the product $S^2 \times S^3$ of unit spheres has $\operatorname{Ric} = g$ on vectors tangent to $S^2$ and $\operatorname{Ric} = 2g$ on vectors tangent to $S^3$.
No single constant $\lambda$ works, so it is not Einstein.

Taking the trace gives $\lambda = R/n$, where $R$ is the [[scalar curvature|ZHX2VYQDZ3]] and $n = \dim M$, so an Einstein metric has constant scalar curvature.
If $M$ is connected, $n \ge 3$ and $\operatorname{Ric} = f g$ for a smooth function $f$, the contracted Bianchi identity forces $f$ to be constant.
In dimension $2$ every metric satisfies $\operatorname{Ric} = K g$ with $K$ the Gaussian curvature, so a surface is Einstein exactly when its curvature is constant.
The same definition is used for pseudo-Riemannian metrics.
</document>

<document source="https://tamnd.github.io/math2/node/6SPMVHRPM2.md" tag="6SPMVHRPM2" kind="concept">
# Rigged Hilbert space

## Summary

A rigged Hilbert space, or Gelfand triple, is a chain $\Phi \subset H \subset \Phi'$.
Here $H$ is a Hilbert space, $\Phi$ is a dense subspace with a finer topology, and $\Phi'$ is the dual of $\Phi$.
It gives a rigorous home to objects such as the Dirac delta and plane waves, which are not vectors of $H$.
With a nuclear space $\Phi$, it carries the spectral theorem in terms of generalized eigenvectors.

## Explanations

### L1 human

Quantum mechanics likes to speak of states of exact position, but such states are not square-integrable functions.
A rigged Hilbert space adds a small space of very nice functions and a large space of generalized functions around the Hilbert space.
The usual example is Schwartz functions inside $L^2(\R)$ inside tempered distributions.
The delta function $\delta_a$ lives in the large space and is an eigenvector of the position operator there.
</document>

<document source="https://tamnd.github.io/math2/node/6SQXPE0JPN.md" tag="6SQXPE0JPN" kind="statement">
# Fundamental group of the circle

## Statement

Let $S^1 = \set{z \in \C : \abs{z} = 1}$ be the unit [[circle|060W5CBZP6]], with base point $1$.
For $n \in \Z$, let $\omega_n \colon [0, 1] \to S^1$ be the loop $\omega_n(s) = e^{2\pi i n s}$.
Then the map $\Z \to \pi_1(S^1, 1)$, $n \mapsto [\omega_n]$, is a [[group isomorphism|08MG7H8JW5]] from the additive group of [[integers|0C4JS9RF5M]] to the [[fundamental group|S1XK3HXHFX]] of $S^1$.
In particular $\pi_1(S^1, 1) \cong \Z$, generated by the class of the loop that goes once around the circle.

## Remarks

The proof uses the [[covering map|3XZRRE165N]] $p \colon \R \to S^1$, $p(t) = e^{2\pi i t}$.
By the [[path lifting property|HSWWF35BEX]], a loop $\gamma$ at $1$ lifts to a unique path $\tilde\gamma$ in $\R$ with $\tilde\gamma(0) = 0$, and $\tilde\gamma(1)$ is an integer, the degree of $\gamma$.
Homotopy lifting shows that path homotopic loops have the same degree, so the inverse map $[\gamma] \mapsto \tilde\gamma(1)$ is well defined.
The lift of $\omega_n$ is $s \mapsto ns$.
The lift of a concatenation is obtained by translating and joining lifts, which gives the homomorphism property.
The degree is the topological form of the [[winding number|0NQSV2EWDR]].
Consequences include the Brouwer fixed point theorem in dimension $2$ and the fundamental theorem of algebra.

## Explanations

### L1 human

Every loop on a circle that starts and ends at a fixed point winds around the circle some whole number of times, counting clockwise turns as negative.
Two such loops can be deformed into each other exactly when they wind the same number of times.
Going around $m$ times and then $n$ times is going around $m + n$ times, so the loops behave like the integers under addition.

### L2 human

This is the first nontrivial computation of a fundamental group, and the model for computations through covering spaces.
It shows that the circle is not contractible and that it is not a retract of the closed disk.
With the [[Seifert–van Kampen theorem|S17DY98VF2]] it gives the fundamental groups of wedges of circles, of surfaces and of many CW complexes.
It is also the case $n = 1$ of the computation $\pi_n(S^n) \cong \Z$.
</document>

<document source="https://tamnd.github.io/math2/node/6SVWAP8WP5.md" tag="6SVWAP8WP5" kind="concept">
# Descending chain condition

## Summary

A partially ordered set satisfies the descending chain condition when every decreasing sequence of its elements is eventually constant.
It is the dual of the [[ascending chain condition|0GZ6Y1H4RN]].
Applied to the ideals of a ring, it defines [[Artinian rings|20QEAM8AX2]]; applied to submodules, it defines [[Artinian modules|YQNZTQGP1X]].
For posets it is equivalent, under the axiom of dependent choice, to being a [[well-founded order|55WD0H03CZ]].

## Explanations

### L1 human

Picture going down a staircase in a poset, one step at a time, never going back up.
The descending chain condition says that you must stop moving after finitely many real steps.
The natural numbers satisfy it, since you cannot go down from $10$ more than ten times.
The integers do not: $0 > -1 > -2 > \cdots$ goes down forever.
</document>

<document source="https://tamnd.github.io/math2/node/6SX85ZKBNJ.md" tag="6SX85ZKBNJ" kind="definition">
# Algebraic K-theory

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.

1. $K_0(R)$ is the [[Grothendieck group|5FDMY7PDYY]] of the commutative monoid of isomorphism classes of finitely generated [[projective|BFFA03GE3V]] left $R$-modules under direct sum.
2. Let $GL(R) = \bigcup_{n \ge 1} GL_n(R)$, where $GL_n(R)$ is the [[general linear group|0FCHE902D7]] and $GL_n(R) \subseteq GL_{n+1}(R)$ through $g \mapsto \begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix}$.
Then $K_1(R) = GL(R)/[GL(R), GL(R)]$.
3. For $n \ge 1$, $K_n(R) = \pi_n(BGL(R)^+)$, the $n$-th [[homotopy group|P976WYEFMB]] of the plus construction of the [[classifying space|WA796ZCNRX]] $BGL(R)$ with respect to the commutator subgroup $[GL(R), GL(R)]$.

The *algebraic K-theory* of $R$ is the sequence of abelian groups $K_n(R)$, $n \ge 0$.

## Remarks

For $n = 1$ clause 3 gives back clause 2, since $\pi_1(BGL(R)^+) = GL(R)/[GL(R), GL(R)]$.
By Whitehead's lemma, $[GL(R), GL(R)]$ is the subgroup generated by elementary matrices.
Example: for a field $F$, every finitely generated module is free, so $K_0(F) \cong \Z$ by the dimension.
The [[determinant|00KDZRW4C0]] gives $K_1(F) \cong F^\times$, because the elementary matrices generate $SL(F)$.
Caution: replacing projective modules by all finitely generated modules gives a different group, $G_0(R)$.
For $R = k[\varepsilon]/(\varepsilon^2)$ over a field $k$, $K_0(R) \cong \Z$ is generated by $[R]$, $G_0(R) \cong \Z$ is generated by $[k]$, and $[R] = 2[k]$.
</document>

<document source="https://tamnd.github.io/math2/node/6SZDXD3KYH.md" tag="6SZDXD3KYH" kind="statement">
# Triangulability of analytic sets

## Statement

Let $X \subseteq \R^n$ be a closed [[subanalytic set|K1HTQF26MA]] and let $A_1, \dots, A_k$ be subanalytic subsets of $X$.
Then there are a locally finite [[simplicial complex|V8GZ0W1ZZ5]] $K$ and a [[homeomorphism|0QT00PB5G2]] $h \colon \abs{K} \to X$ from the polyhedron of $K$ onto $X$ with the following property.
Each $A_j$ is a union of sets $h(\sigma^\circ)$, where $\sigma^\circ$ is the interior of a simplex $\sigma$ of $K$.

## Remarks

Łojasiewicz proved the statement for [[semianalytic sets|E1V22QQHYJ]], and Hironaka extended it to subanalytic sets.
The proof first decomposes $X$ into finitely many pieces on which suitable coordinate projections are finite and well behaved, and then triangulates by induction on $n$.
Every closed real analytic subset of $\R^n$ is semianalytic, and every [[complex analytic set|GB7M582M5Y]] in $\C^n = \R^{2n}$ is a closed real analytic subset.
So both kinds of analytic sets are [[triangulable|YH170WBQ6M]], and a triangulation can be chosen compatible with the singular locus.
For instance, the cone $\set{x^2 + y^2 = z^2}$ in $\R^3$ can be triangulated with the vertex at the origin as a vertex of $K$.

## Explanations

### L1 human

An analytic set can have singular points, such as the tip of a cone or a crossing of two curves.
The theorem says that, however complicated it is, the set can be cut into pieces that are curved copies of points, segments, triangles and their higher analogues, glued along faces.
This makes analytic sets accessible to combinatorial topology.

### L2 human

Triangulability implies that a compact analytic set has finitely generated homology and a well defined Euler characteristic.
It also implies local conic structure: a small neighbourhood of a point of an analytic set is homeomorphic to the cone over its link.
The choice of a triangulation compatible with the subsets $A_j$ lets one treat a set together with a stratification of it.
</document>

<document source="https://tamnd.github.io/math2/node/6T3E3AXZT7.md" tag="6T3E3AXZT7" kind="statement">
# Bertrand's test

## Statement

Let $(a_n)_{n \ge 2}$ be a sequence of positive real numbers, and put
$$B_n = \ln n \left( n \left( \frac{a_n}{a_{n+1}} - 1 \right) - 1 \right),$$
where $\ln$ is the [[natural logarithm|0044CPHJBV]].

1. If there are $r > 1$ and $N$ with $B_n \ge r$ for all $n \ge N$, then $\sum a_n$ [[converges|0CVVKY7GY2]].
2. If there is $N$ with $B_n \le 1$ for all $n \ge N$, then $\sum a_n$ [[diverges|0NBRP6093V]].

In particular, if $B_n \to B$, the series converges when $B > 1$ and diverges when $B < 1$.

## Remarks

This is [[Kummer's test|9JQEWK5405]] with $c_n = n \ln n$.
A short computation gives $\rho_n = B_n - (n+1) \ln(1 + 1/n)$.
The inequality $\ln(1 + x) \ge x/(1 + x)$ gives $(n+1)\ln(1 + 1/n) \ge 1$, and this quantity tends to $1$.
So $B_n \ge r > 1$ gives $\rho_n \ge (r - 1)/2$ for large $n$, and $B_n \le 1$ gives $\rho_n \le 0$; moreover $\sum 1/(n \ln n)$ diverges.
Example: if $a_n / a_{n+1} = 1 + 1/n + O(1/n^s)$ with $s > 1$, then $B_n = \ln n \cdot O(n^{1-s}) \to 0$, so the series diverges.
This is the case that [[Raabe's test|CD6CAQ6XDN]] leaves open in [[Gauss's test|A7MJXJERNY]].
For $a_n = 1/(n (\ln n)^p)$ one finds $B_n \to p$, so the test decides these series for $p \ne 1$.

## Explanations

### L1 human

Bertrand's test is the next step after Raabe's test.
It is used when Raabe's quantity tends to $1$.
It compares the series with the slowly converging or diverging series $\sum 1/(n (\ln n)^p)$.

### L2 human

The ratio test, Raabe's test and Bertrand's test form a scale of finer and finer tests, all special cases of Kummer's test.
Each one looks at the next term in the expansion of $a_n / a_{n+1}$.
One use of Bertrand's test is the proof of Gauss's test in the boundary case $h = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/6T74XFKWRA.md" tag="6T74XFKWRA" kind="definition">
# Projective dimension

## Definition

Let $R$ be a [[ring|0B1261B5ED]] and let $M$ be a nonzero [[$R$-module|0NZ8JA25WK]].
The *projective dimension* $\operatorname{pd}_R M$ is the least integer $n \ge 0$ such that $M$ has a [[projective resolution|HK2AA1YSQ2]]
$$0 \to P_n \to P_{n-1} \to \cdots \to P_0 \to M \to 0$$
of length $n$.
If no such finite resolution exists, then $\operatorname{pd}_R M = \infty$.

## Remarks

One has $\operatorname{pd}_R M = 0$ if and only if $M$ is a [[projective module|BFFA03GE3V]].
Equivalently, $\operatorname{pd}_R M \le n$ if and only if $\operatorname{Ext}^{n+1}_R(M, N) = 0$ for every $R$-module $N$, where $\operatorname{Ext}$ is the [[Ext functor|4P7GBWJ4RR]].
Example: $\operatorname{pd}_\Z \Z/n\Z = 1$ for $n \ge 2$; the resolution $0 \to \Z \xrightarrow{\cdot n} \Z \to \Z/n\Z \to 0$ has length $1$, and $\Z/n\Z$ is not projective, because projective $\Z$-modules are free and hence have no nonzero torsion.
Non-example of finiteness: over $R = \Z/4\Z$, the module $\Z/2\Z$ has $\operatorname{pd}_R \Z/2\Z = \infty$.
Some authors set $\operatorname{pd}_R 0 = -\infty$.
</document>

<document source="https://tamnd.github.io/math2/node/6TF2JBTPPJ.md" tag="6TF2JBTPPJ" kind="object">
# Constructible universe

## Description

For a set $X$, let $\operatorname{Def}(X)$ be the set of all subsets of $X$ that are definable in the structure $(X, \in)$.
Here a definition is a first-order formula with parameters from $X$.
Define sets $L_\alpha$ by recursion on the [[ordinals|0YSCSX5YZC]] $\alpha$:
$$L_0 = \emptyset, \qquad L_{\alpha + 1} = \operatorname{Def}(L_\alpha), \qquad L_\lambda = \bigcup_{\alpha < \lambda} L_\alpha \text{ for a limit ordinal } \lambda.$$
The *constructible universe* is the [[class|Y48FA69563]] $L = \bigcup_{\alpha} L_\alpha$, where $\alpha$ runs over all ordinals.
A set is *constructible* when it lies in $L$.

The class $L$ is an [[inner model|G78W36EGMB]] of [[ZF|0HE5WDQ9FJ]].
Each $L_\alpha$ is transitive, so $L$ is transitive.
The ordinal $\alpha$ lies in $L_{\alpha + 1}$, because it is the set of ordinals of $L_\alpha$, and this set is definable over $L_\alpha$.
So $L$ contains every ordinal.
The axioms of ZF hold in $L$; the hardest case is the separation schema, which uses a reflection argument.

## Remarks

For finite $n$ one has $L_n = V_n$, so $L_\omega = V_\omega$ is the set of hereditarily finite sets.
At the next level they differ: $L_{\omega + 1}$ is countable, because there are only countably many formulas with finitely many parameters from the countable set $L_\omega$, while $V_{\omega + 1}$ is uncountable.
The class $L$ satisfies the [[axiom of constructibility|4QKRPRCSXH]] $V = L$, and it is contained in every inner model of ZF.
[[Gödel's theorem|Q9176JRJ5M]] says that the axiom of choice and the generalized continuum hypothesis hold in $L$.

## Explanations

### L1 human

The constructible universe is the smallest world of sets that one can build in stages using only what can be described.
At each stage one keeps exactly the subsets of the sets built so far that a formula can define.
Nothing is added by arbitrary choice, so the world is very thin and very orderly.
For example, at the finite stages every subset can be described, so all finite sets of finite sets appear.
At stage $\omega + 1$ only countably many sets of natural numbers appear, although there are uncountably many in all.
</document>

<document source="https://tamnd.github.io/math2/node/6TJ7YHP2HE.md" tag="6TJ7YHP2HE" kind="concept">
# Krylov–Bogolyubov averaging

## Summary

Krylov–Bogolyubov averaging is a method for differential equations with a small parameter and a fast periodic dependence on time.
It replaces the right-hand side by its average over one period and studies the simpler averaged equation.
The averaging theorem says that solutions of the two equations stay close over long time intervals, of length of order $1/\varepsilon$.
It is a basic tool for weakly nonlinear oscillations in mechanics and physics.
See the [[definition|NGYZK893VE]].

## Explanations

### L1 human

When a quantity changes slowly but is shaken quickly back and forth, the fast shaking mostly cancels out.
Only the average effect of each period of shaking matters over a long time.
For example, $x' = \varepsilon x \sin^2 t$ has the averaged equation $y' = \frac{1}{2} \varepsilon y$, because $\sin^2 t$ has average $\frac{1}{2}$.
The exact solution is $x_0 e^{\varepsilon (t/2 - \sin(2t)/4)}$, and it differs from $x_0 e^{\varepsilon t/2}$ only by the small wobbling factor $e^{-\varepsilon \sin(2t)/4}$.
</document>

<document source="https://tamnd.github.io/math2/node/6TJTWQ5HCR.md" tag="6TJTWQ5HCR" kind="concept">
# Newtonian kernel

## Summary

The Newtonian kernel on $\R^n$, for $n \ge 3$, is the function $k(x, y) = \abs{x - y}^{2-n}$ of two points.
In space it is $1/\abs{x - y}$, the potential at $x$ of a unit mass at $y$ in Newton's law of gravitation.
It is a constant multiple of the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] translated to the pole $y$.
Integrating it against a measure gives the Newtonian potential, the central object of classical potential theory in higher dimensions.

## Explanations

### L1 human

Put a point mass at $y$.
The Newtonian kernel tells you how strongly a point $x$ feels it.
The value is large when $x$ is close to $y$ and tends to $0$ when $x$ is far away.
In three dimensions the value is one over the distance, so doubling the distance halves the potential.
For a cloud of masses one adds up these kernels, one for each mass.
</document>

<document source="https://tamnd.github.io/math2/node/6TKGN8K8KT.md" tag="6TKGN8K8KT" kind="concept">
# Homology manifold

## Summary

A homology $n$-manifold is a space whose local homology at every point is that of a point in $\R^n$.
Every [[topological manifold|QQQ9B1DK4P]] is a homology manifold, but there are homology manifolds that are not manifolds.
Poincaré duality holds for compact oriented homology manifolds, so they are the natural setting for many results first proved for manifolds.

## Explanations

### L1 human

Near any point of an $n$-dimensional manifold, the space looks like $\R^n$.
Homology cannot see the whole shape of a neighbourhood, only some of its features, such as the holes left when the point is removed.
A homology manifold is a space where, near every point, these features are the same as in $\R^n$.
For example, take the cone over a space with the homology of a sphere but a nontrivial fundamental group.
It passes this test at its tip, though it is not a manifold there.
</document>

<document source="https://tamnd.github.io/math2/node/6TP0G4DKW2.md" tag="6TP0G4DKW2" kind="definition">
# Variogram

## Definition

Let $D = \R^d$ and let $(Z(s))_{s \in D}$ be a real random field with $E[Z(s)^2] < \infty$ for all $s$.
The field is *intrinsically stationary* if for every $h \in \R^d$:

1. $E[Z(s + h) - Z(s)] = 0$ for all $s$, and
2. $\operatorname{Var}(Z(s + h) - Z(s))$ does not depend on $s$.

For such a field the *variogram* is the function $2\gamma \colon \R^d \to [0, \infty)$ given by
$$2\gamma(h) = \operatorname{Var}(Z(s + h) - Z(s)) = E[(Z(s + h) - Z(s))^2],$$
and $\gamma$ is the *semivariogram*.

## Remarks

Every variogram satisfies $\gamma(0) = 0$ and $\gamma(-h) = \gamma(h)$.
It is conditionally negative definite: $\sum_{i,j} a_i a_j \gamma(s_i - s_j) \le 0$ whenever $\sum_i a_i = 0$.
If $Z$ is weakly stationary with [[covariance|0KBFAWYRTE]] function $C(h) = \operatorname{Cov}(Z(s + h), Z(s))$, then $\gamma(h) = C(0) - C(h)$.

Example: the standard [[Wiener process|22735FGJV6]] $(B(t))_{t \in \R}$, two-sided with $B(0) = 0$, has $B(t + h) - B(t)$ normal with mean $0$ and variance $\abs{h}$.
So it is intrinsically stationary with $2\gamma(h) = \abs{h}$, although its variance $\abs{t}$ is not constant in $t$.

Non-example: let $X$ be a standard normal random variable and $Z(s) = s^2 X$ on $\R$.
The increment $Z(s + h) - Z(s) = (2sh + h^2) X$ has mean $0$, but its variance $(2sh + h^2)^2$ depends on $s$, so clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/6TQSVJQDEH.md" tag="6TQSVJQDEH" kind="concept">
# Young measure

## Summary

A Young measure on a set $\Omega \subseteq \R^n$ is a measurable family $(\nu_x)_{x \in \Omega}$ of [[probability measures|059TJN0NZ1]] on $\R^m$.
Young measures describe the limits of oscillating sequences of functions: $\nu_x$ is the distribution of the values that the functions take near $x$.
They let one pass to the limit in nonlinear expressions $\varphi(u_k)$ when $u_k$ converges only weakly.
They are a standard tool in the calculus of variations and in nonlinear partial differential equations.

## Explanations

### L1 human

Take the function on $(0,1)$ that jumps between $-1$ and $1$ on $k$ equal pieces, and let $k$ grow.
The functions oscillate faster and faster, and their weak limit is $0$.
But the square of each function is $1$, so the weak limit forgets something.
The Young measure keeps it: at every point it is the measure giving weight $1/2$ to $-1$ and $1/2$ to $1$.
It records which values occur and how often, not where they occur.
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<document source="https://tamnd.github.io/math2/node/6TSMHGCXVM.md" tag="6TSMHGCXVM" kind="statement">
# Modica–Mortola theorem

## Statement

Let $\Omega \subseteq \R^n$ be a bounded open set with Lipschitz boundary, and let $W(t) = t^2 (1 - t)^2$ for $t \in \R$.
For $\varepsilon > 0$ define $F_\varepsilon \colon L^1(\Omega) \to [0, +\infty]$ by
$$F_\varepsilon(u) = \int_\Omega \Big( \varepsilon \abs{\nabla u}^2 + \frac{1}{\varepsilon} W(u) \Big) \, dx$$
if $u$ lies in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,2}(\Omega)$, and $F_\varepsilon(u) = +\infty$ otherwise.
Define $F \colon L^1(\Omega) \to [0, +\infty]$ by
$$F(u) = c_W \operatorname{Per}(E; \Omega), \qquad c_W = 2 \int_0^1 \sqrt{W(s)} \, ds = \frac13,$$
if $u = \mathbf{1}_E$ is the characteristic function of a set $E \subseteq \Omega$ of finite [[perimeter|Y68BHXZJBV]] $\operatorname{Per}(E; \Omega)$ in $\Omega$, and $F(u) = +\infty$ otherwise.
Then $F_\varepsilon$ [[Γ-converges|NW9MM370A5]] to $F$ in the [[Lp space|SASP14MA1Z]] $L^1(\Omega)$ as $\varepsilon \to 0$.

## Remarks

The lower bound comes from the inequality $a^2 + b^2 \ge 2ab$, which gives $\varepsilon \abs{\nabla u}^2 + W(u)/\varepsilon \ge 2 \sqrt{W(u)} \abs{\nabla u} = \abs{\nabla (\phi \circ u)}$ with $\phi(t) = 2 \int_0^t \sqrt{W(s)} \, ds$.
The lower semicontinuity of the total variation then bounds the limit from below by the perimeter of $E$ times $\phi(1) - \phi(0) = c_W$.
For the recovery sequence, one puts the optimal one-dimensional profile across $\partial E$.
Here $q(s) = 1/(1 + e^{-s})$ solves $q' = \sqrt{W(q)}$.
When $\partial E$ is smooth, $u_\varepsilon(x) = q(d(x)/\varepsilon)$ works, with $d$ the signed distance to $\partial E$.
The same result holds for other continuous double-well potentials $W \ge 0$ that vanish exactly at two points and grow suitably at infinity.
Then $c_W$ is the integral of $2\sqrt{W}$ between the two wells.

The theorem is often combined with a volume constraint $\int_\Omega u \, dx = m$.
Then the [[fundamental theorem of Γ-convergence|23RK8EZ20P]] shows that minimizers of $F_\varepsilon$ converge, along subsequences, to characteristic functions of sets of least perimeter with the given volume.

## Explanations

### L1 human

A mixture of two fluids, such as oil and water, can be described by a function $u$ that is near $0$ in one phase and near $1$ in the other.
The energy $F_\varepsilon$ punishes values between the phases through $W$ and punishes fast changes through the gradient term.
When the parameter $\varepsilon$ becomes small, the transition layer gets thin and its energy concentrates on the interface.
The theorem says that in the limit the energy is a constant times the area of the interface, so minimizing the energy means minimizing the area of the interface.

### L2 human

This is the model example of [[Γ-convergence|NW9MM370A5]] and the reason it became popular.
It connects the diffuse-interface energy behind the [[Allen–Cahn equation|7S813PEV69]] with the sharp-interface perimeter functional of geometric measure theory.
The constant $c_W$ is the energy per unit area of the optimal one-dimensional transition profile.
The result is due to Luciano Modica and Stefano Mortola, and Modica used it to study phase transitions with a volume constraint.
</document>

<document source="https://tamnd.github.io/math2/node/6TV7FR553H.md" tag="6TV7FR553H" kind="conjecture">
# Dynamical Mordell–Lang conjecture

## Statement

Let $X$ be a [[quasi-projective variety|EQSHJPE7H8]] over $\C$ and let $\Phi \colon X \to X$ be a [[morphism|CDS1VWB7A3]].
Write $\Phi^n$ for the $n$-th [[iterate|AYVTR2ES5B]] of $\Phi$, with $\Phi^0 = \id$.
Let $V \subseteq X$ be a closed [[subvariety|4YSCK1DF26]] and let $x \in X(\C)$.
Then the set
$$\set{n \in \N : \Phi^n(x) \in V}$$
is a finite union of sets of the form $\set{a k + b : k \in \N}$ with $a, b \in \N$.

## Remarks

A set $\set{ak + b : k \in \N}$ with $a = 0$ is the single point $\set{b}$, so finite sets are allowed.
An equivalent form concerns the orbit $\set{\Phi^n(x) : n \in \N}$ of $x$.
If the orbit meets $V$ in infinitely many points, then $V$ contains a positive-dimensional subvariety $W$ with $\Phi^a(W) \subseteq W$ for some $a \ge 1$ and $\Phi^b(x) \in W$ for some $b$.
The conjecture is open in general.
Bell, Ghioca and Tucker proved it when $\Phi$ is étale.
When $X = \C^N$ and $\Phi$ is linear, it is the Skolem–Mahler–Lech theorem on the zeros of a [[linear recurrence|0NDVHBVBXB]].
When $\Phi$ is a translation on a semiabelian variety, it follows from the Mordell–Lang theorem of Faltings and Vojta, which gives the name.
The main tool is $p$-adic.
On suitable residue classes of $n$, the map $n \mapsto \Phi^n(x)$ is interpolated by a $p$-adic analytic function.
A nonzero analytic function on $\Z_p$ has finitely many zeros.
Over fields of positive characteristic the statement fails.
For example, over $\F_p(t)$ let $\Phi(x, y) = (tx, (1 + t)y)$ on $\mathbb{G}_m^2$, let $V$ be the line $y = x + 1$ and start at $(1, 1)$.
Then $\Phi^n(1, 1) = (t^n, (1 + t)^n)$ lies in $V$ exactly when $(1 + t)^n = t^n + 1$, that is, when $n$ is a power of $p$.

## Explanations

### L1 human

Start at a point and apply the same polynomial map again and again.
Ask at which steps the point lands on a fixed curve or surface.
The conjecture predicts that these steps are never irregular: they form finitely many arithmetic progressions, plus finitely many extra steps.
So either the orbit hits the surface only a few times, or it hits it in a regular repeating pattern.
</document>

<document source="https://tamnd.github.io/math2/node/6V59T5KJBX.md" tag="6V59T5KJBX" kind="definition">
# Fourier transform

## Definition

Let $n \ge 1$ and let $f \colon \R^n \to \C$ be a Lebesgue [[integrable function|X7GTF3D98F]].
The *Fourier transform* of $f$ is the function $\hat f \colon \R^n \to \C$ given by the [[Lebesgue integral|DWJF7ABPR1]]
$$\hat f(\xi) = \int_{\R^n} f(x)\, e^{-2\pi i\, x \cdot \xi} \, dx, \qquad \xi \in \R^n,$$
where $x \cdot \xi = x_1 \xi_1 + \dots + x_n \xi_n$.
It is also written $\mathcal{F} f$.

## Remarks

The integral converges absolutely because $\abs{f(x) e^{-2\pi i x \cdot \xi}} = \abs{f(x)}$.
So $\abs{\hat f(\xi)} \le \int \abs{f}$, and $\hat f$ is continuous by dominated convergence.
By the Riemann–Lebesgue lemma, $\hat f(\xi) \to 0$ as $\abs{\xi} \to \infty$.
Other common conventions are $\int f(x) e^{-i x \cdot \xi}\, dx$ and $(2\pi)^{-n/2} \int f(x) e^{-i x \cdot \xi}\, dx$; they change the constants in the formulas below.

With the convention above, if $f$ and $\partial_j f$ are integrable, then $\widehat{\partial_j f}(\xi) = 2\pi i \xi_j \hat f(\xi)$.
If $\hat f$ is also integrable, the inversion formula $f(x) = \int \hat f(\xi) e^{2\pi i x \cdot \xi}\, d\xi$ holds for almost every $x$.
Plancherel's theorem extends $\mathcal F$ to a unitary map of $L^2(\R^n)$, and duality extends it to [[tempered distributions|AES2T5JPFP]].

Example: $f(x) = e^{-\pi \abs{x}^2}$ satisfies $\hat f = f$.
For $n = 1$, completing the square gives $\hat f(\xi) = e^{-\pi \xi^2} \int_\R e^{-\pi (x + i\xi)^2}\, dx$, and the last integral equals $\int_\R e^{-\pi x^2}\, dx = 1$ by a contour shift.

Non-example: the constant function $f = 1$ on $\R$ is not integrable, and the integral $\int_\R e^{-2\pi i x \xi}\, dx$ diverges for every $\xi$.
Its Fourier transform exists only as a tempered distribution, namely the Dirac delta at $0$.

## Explanations

### L1 human

The transform measures how strongly $f$ oscillates with each frequency $\xi$.
It is the continuous analogue of the list of coefficients of a Fourier series.
</document>

<document source="https://tamnd.github.io/math2/node/6V6QPT0B3Y.md" tag="6V6QPT0B3Y" kind="definition">
# Ideal of an algebraic set

## Definition

Let $k$ be a field, let $n \ge 0$, and let $X \subseteq k^n$ be any subset.
The *ideal of $X$* (or vanishing ideal of $X$) is
$$I(X) = \set{f \in k[x_1, \dots, x_n] : f(a) = 0 \text{ for all } a \in X},$$
where $k[x_1, \dots, x_n]$ is the [[polynomial ring|0SDC934R52]].
It is used mostly when $X$ is an [[algebraic set|VHFMQQYSTN]].

## Remarks

$I(X)$ is an [[ideal|096Z9MP2ME]], and it equals its own [[radical|437YRCNYRJ]]: if $f^m$ vanishes at a point $a$, then $f(a)^m = 0$, so $f(a) = 0$.

Example: for the parabola $X = V(y - x^2) \subseteq k^2$ with $k$ infinite, $I(X) = (y - x^2)$.
Check: dividing $f(x, y)$ by $y - x^2$ as a polynomial in $y$ gives $f = (y - x^2) q + r(x)$; if $f$ vanishes on $X$, then $r(t) = 0$ for all $t \in k$, so $r = 0$.

Non-example: the ideal $(x^2)$ in $k[x]$ is not the ideal of any set.
It is not radical, since $x \notin (x^2)$ while $x^2 \in (x^2)$; in fact $I(V(x^2)) = I(\set{0}) = (x)$.

For an algebraic set $X$ one has $V(I(X)) = X$, and $X \subseteq Y$ if and only if $I(Y) \subseteq I(X)$.
If $k$ is [[algebraically closed|0PF6TM8TWT]], [[Hilbert's Nullstellensatz|D85VZWS9QK]] gives $I(V(J)) = \sqrt{J}$ for every ideal $J$.
So $X \mapsto I(X)$ is then an inclusion-reversing bijection from algebraic sets in $k^n$ to radical ideals of $k[x_1, \dots, x_n]$.
</document>

<document source="https://tamnd.github.io/math2/node/6V6WDRHRTF.md" tag="6V6WDRHRTF" kind="definition">
# Kobayashi hyperbolic manifold

## Definition

Let $X$ be a connected [[complex manifold|JSGWAN2RJV]] with [[Kobayashi pseudodistance|DY8QQ5GTTX]] $k_X$.
The manifold $X$ is *Kobayashi hyperbolic* if $k_X(p, q) > 0$ for all $p, q \in X$ with $p \ne q$, that is, if $k_X$ is a [[distance|0VKD1TP7F3]] on $X$.

## Remarks

When $X$ is hyperbolic, $k_X$ induces the manifold topology of $X$, by a theorem of Barth.
Example: a bounded domain $\Omega \subseteq \C^n$ is hyperbolic.
Check: if $\abs{z} < R$ on $\Omega$ and $p \ne q$, choose a complex linear function $\ell$ with $\ell(p) \ne \ell(q)$ and $\abs{\ell(z)} < 1$ on $\Omega$; since $\ell \colon \Omega \to \mathbb{D}$ decreases Kobayashi distances, $k_\Omega(p, q) \ge k_{\mathbb{D}}(\ell(p), \ell(q)) > 0$.
Non-example: $\C^n$ is not hyperbolic, since $k_{\C^n} \equiv 0$.
More generally, if there is a nonconstant holomorphic map $f \colon \C \to X$, then $k_X(f(a), f(b)) \le k_\C(a, b) = 0$ with $f(a) \ne f(b)$ for some $a, b$, so $X$ is not hyperbolic.
Hence complex tori and complex projective spaces are not hyperbolic.
A Riemann surface is hyperbolic exactly when its universal covering is the unit disc.
</document>

<document source="https://tamnd.github.io/math2/node/6V75QVQCX6.md" tag="6V75QVQCX6" kind="definition">
# Robbins–Monro algorithm

## Definition

Let $M \colon \R \to \R$ be an unknown function and let $\alpha \in \R$ be a target value.
Assume that at each $\theta \in \R$ one can observe a [[random variable|0DKRR22AEQ]] $N(\theta)$ with [[expectation|0WQNRCNY3M]] $E[N(\theta)] = M(\theta)$.
Let $(a_n)_{n \ge 1}$ be positive step sizes.
The *Robbins–Monro algorithm* is this [[stochastic approximation|ZPCPW4W1Y9]] scheme:
$$\theta_{n+1} = \theta_n - a_n \left( N_n - \alpha \right), \qquad n \ge 1.$$
Here $\theta_1$ is an initial guess.
Given the past, $N_n$ is an observation of $N(\theta_n)$, made independently of the earlier observations.
Its aim is a root $\theta^*$ of the equation $M(\theta) = \alpha$.

## Remarks

Robbins and Monro proved this convergence result in 1951.
Assume that $N(\theta)$ is uniformly bounded, $M$ is nondecreasing, $M'(\theta^*)$ exists and is positive, and the step sizes satisfy $\sum_n a_n = \infty$ and $\sum_n a_n^2 < \infty$.
Then $\theta_n \to \theta^*$ in $L^2$, and hence [[in probability|0QA1HBD9Z4]].

Example: let $N(\theta) = \theta - Z$ with $Z$ uniform on $[0, 2]$, so $M(\theta) = \theta - 1$, and let $\alpha = 0$ and $a_n = 1/n$.
The step sizes satisfy both conditions, since $\sum 1/n = \infty$ and $\sum 1/n^2 < \infty$, and the root is $\theta^* = 1$.
With fresh draws $Z_n$, the recursion is $\theta_{n+1} = \theta_n - \frac1n (\theta_n - Z_n)$, so $\theta_{n+1} = (Z_1 + \cdots + Z_n)/n$, the sample mean, which tends to $1$ by the law of large numbers.

Non-example: constant step sizes $a_n = c > 0$ violate $\sum a_n^2 < \infty$; in the example, $\theta_{n+1} = (1 - c) \theta_n + c Z_n$ for $0 < c < 1$ keeps fluctuating with conditional variance $c^2 \operatorname{Var}(Z) > 0$ at each step, so it does not converge.
</document>

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# Compressed sensing

## Summary

Compressed sensing is the recovery of a sparse vector from far fewer linear measurements than its length.
Let $x \in \R^N$ have at most $s$ nonzero entries, and let $y = Ax$ for a suitable $m \times N$ matrix $A$ with $m$ much smaller than $N$.
Then $x$ can be found from $y$ by solving a convex problem.
The key sufficient condition on $A$ is the [[restricted isometry property|B0K18PWN2A]], which random matrices satisfy with high probability when $m$ is of order $s \log(N/s)$.
It was developed by Candès, Romberg, Tao and Donoho in the mid-2000s and is used in medical imaging and signal processing.

## Explanations

### L1 human

Usually, to know a vector of length $N$ one must measure $N$ numbers.
If one knows that only a few entries are nonzero, much fewer measurements can be enough, as long as they mix the entries well.
The reconstruction looks for the [[sparse|WC8P971XGW]] vector that agrees with the measurements.
For example, with the $2 \times 3$ matrix $A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$, any vector with one nonzero entry is determined by its two measurements $Ax$.
Among such vectors, the measurement $(2, 2)$ can only come from $(0, 0, 2)$, and the measurement $(3, 0)$ only from $(3, 0, 0)$.
</document>

<document source="https://tamnd.github.io/math2/node/6VC6HQTFSA.md" tag="6VC6HQTFSA" kind="concept">
# Sheaf of differentials

## Summary

The sheaf of relative differentials $\Omega_{X/Y}$ of a morphism of schemes $X \to Y$ is the sheaf version of the module of [[Kähler differentials|7HCFARHG7D]].
On an affine piece $\operatorname{Spec} B \to \operatorname{Spec} A$ it is given by the $B$-module $\Omega_{B/A}$.
For a smooth variety it is a vector bundle, the cotangent bundle, whose dual is the tangent bundle.
Its exterior powers give the sheaves of differential forms, and the top one is the [[canonical bundle|W05VYY4HBR]].

## Explanations

### L1 human

On a manifold, the cotangent bundle collects the differentials $df$ of functions at every point.
The sheaf of differentials does the same on a scheme, using only algebra: $d$ is linear and satisfies the product rule.
On the affine plane with coordinates $x, y$, every differential is $g \, dx + h \, dy$ with polynomial $g, h$.
On a curve defined by $f(x, y) = 0$, there is one extra relation $\partial_x f \, dx + \partial_y f \, dy = 0$.
At smooth points this leaves a module of rank one, the cotangent line of the curve.
</document>

<document source="https://tamnd.github.io/math2/node/6VC9JQD45M.md" tag="6VC9JQD45M" kind="definition">
# Cesàro summation

## Definition

Let $\sum_{n \ge 0} a_n$ be a series of complex numbers with [[partial sums|0NB643K60Q]] $s_n = a_0 + \cdots + a_n$.

1. The series is *Cesàro summable* to $s$, or $(C, 1)$ summable to $s$, if the [[Cesàro means|0P0DJHBGBB]] of the partial sums converge to $s$:
$$\sigma_n = \frac{s_0 + s_1 + \cdots + s_n}{n + 1} \to s \quad (n \to \infty).$$
2. More generally, let $\alpha > -1$ be real, and put $A^{\alpha}_0 = 1$ and $A^{\alpha}_m = \frac{(\alpha + 1)(\alpha + 2) \cdots (\alpha + m)}{m!}$ for $m \ge 1$.
The series is $(C, \alpha)$ summable to $s$ if
$$\sigma^{\alpha}_n = \frac{1}{A^{\alpha}_n} \sum_{k=0}^{n} A^{\alpha}_{n-k} a_k \to s \quad (n \to \infty).$$

## Remarks

For $\alpha = 0$ one has $\sigma^0_n = s_n$, which is ordinary convergence.
For $\alpha = 1$ one has $A^1_m = m + 1$, and $\sigma^1_n = \frac{1}{n+1} \sum_{k=0}^{n} (n - k + 1) a_k = \sigma_n$, so clause 2 extends clause 1.

Example: for $1 - 1 + 1 - \cdots$ the partial sums are $1, 0, 1, 0, \dots$.
So $\sigma_{2m} = \frac{m+1}{2m+1}$ and $\sigma_{2m+1} = \frac{1}{2}$, and both tend to $1/2$.
The series is $(C, 1)$ summable to $1/2$.

Non-example: for $1 - 2 + 3 - 4 + \cdots$ the partial sums are $1, -1, 2, -2, 3, \dots$.
Their sums are $1, 0, 2, 0, 3, 0, \dots$, so $\sigma_{2m} = \frac{m+1}{2m+1} \to \frac12$ and $\sigma_{2m+1} = 0$.
The means oscillate, so the series is not $(C, 1)$ summable.
It is $(C, 2)$ summable to $1/4$.

If a series is $(C, \alpha)$ summable to $s$, it is $(C, \beta)$ summable to $s$ for every $\beta > \alpha$.
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# Peter–Weyl theorem

## Statement

Let $G$ be a [[compact group|EVHCE5VR6Y]] with normalised [[Haar measure|VY0HXD3GC4]].
Let $\widehat{G}$ be a set of representatives of the isomorphism classes of continuous [[irreducible|FX3NEFYFFN]] [[unitary representations|74BPYJNZTM]] of $G$.
Each $\pi \in \widehat{G}$ is finite-dimensional; let $d_\pi$ be its dimension, and let $\pi_{ij}(g)$ be its matrix entries in an orthonormal basis.
A matrix coefficient of $\pi$ is a function $g \mapsto \ip{\pi(g)u}{v}$ for vectors $u, v$ of its space.

1. The matrix coefficients of the finite-dimensional continuous unitary representations of $G$ span a dense subspace of $C(G)$, with the supremum norm.
2. The functions $\sqrt{d_\pi}\, \pi_{ij}$, for $\pi \in \widehat{G}$ and $1 \le i, j \le d_\pi$, form an orthonormal basis of the [[Hilbert space|HM2HHE3J51]] $L^2(G)$.
3. Every continuous unitary representation of $G$ on a Hilbert space is a Hilbert direct sum of finite-dimensional irreducible representations.

## Remarks

The proof applies the spectral theorem to self-adjoint convolution operators $f \mapsto k * f$ on $L^2(G)$ with continuous kernels $k$.
Such operators are compact and commute with right translations, so their eigenspaces for nonzero eigenvalues are finite-dimensional invariant subspaces.
Letting $k$ run through an approximate identity shows that these subspaces span a dense subspace, which gives parts 1 and 2.
Part 3 follows by applying the same idea to a general representation.
Orthonormality in part 2 is the [[Schur orthogonality for compact groups|ETNDTECY97]].
Example: for $G = \mathbb{T}$ the irreducible representations are the characters $z \mapsto z^n$, $n \in \Z$, and part 2 says that the functions $e^{in\theta}$ form an orthonormal basis of $L^2(\mathbb{T})$, the completeness of Fourier series.
For a finite group, the theorem reduces to the decomposition of the regular representation, with $\sum_\pi d_\pi^2 = \abs{G}$.

## Explanations

### L1 human

On the circle, every reasonable function is a sum of waves $e^{in\theta}$.
The Peter–Weyl theorem says the same for every compact group, where the waves are replaced by the entries of the matrices of irreducible representations.
These matrix entries are like a complete set of perpendicular axes for the space of functions on the group.

### L2 human

Part 1 implies that every compact Lie group has a faithful finite-dimensional representation, so it is isomorphic to a closed subgroup of some unitary group $U(n)$.
Part 3 reduces the study of unitary representations of compact groups to finite-dimensional ones, where [[complete reducibility|N6JC6KNXJJ]] and characters are available.
In terms of the left and right regular actions, part 2 says that $L^2(G)$ is the Hilbert direct sum over $\pi \in \widehat{G}$ of the spaces $\End(V_\pi)$, each of dimension $d_\pi^2$.
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# Convex cone

## Summary

A convex cone is a subset of a real vector space that is closed under addition and under multiplication by positive scalars.
It is the same as a set that is both a cone and [[convex|02AS45T3H9]].
Convex cones describe the solution sets of homogeneous linear inequalities, orders on vector spaces, and the feasible regions of conic optimisation.
Examples are the nonnegative orthant and the cone of positive semidefinite matrices.

## Explanations

### L1 human

Picture a wedge or an ice cream cone with its tip at the origin, extending forever.
If two vectors lie in the wedge, so does their sum, and so does any stretched copy of one of them.
The first quadrant $\set{(x, y) : x \ge 0, y \ge 0}$ of the plane is a convex cone.
The union of the first and third quadrants is not, because $(1, 0) + (0, -1) = (1, -1)$ falls outside.
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# Embedding theorem for Stein manifolds

## Statement

Let $X$ be a [[Stein manifold|HMCD0JBFST]] of complex dimension $n$.
Then there is a [[proper|TM528Q96ZD]] injective [[holomorphic map|3MQTWKBBBT]] $F \colon X \to \C^{2n+1}$ whose differential is injective at every point.
So $X$ is biholomorphic to a closed complex submanifold of $\C^{2n+1}$.

## Remarks

The theorem is due to Remmert, Bishop and Narasimhan.
The proof starts from the many global holomorphic functions that Cartan's theorems A and B provide on a Stein manifold.
With them one builds a proper injective holomorphic map into some $\C^N$ with injective differential.
Then one lowers $N$ by generic linear projections, as in the proof of the Whitney embedding theorem.
The converse holds: every closed complex submanifold of $\C^N$ is Stein, because the coordinate functions separate points and give compact holomorphically convex hulls.
So the Stein manifolds are exactly the closed complex submanifolds of complex Euclidean spaces.
Example: a noncompact Riemann surface, which is Stein, embeds properly into $\C^3$.
The dimension $2n + 1$ is not optimal.
For $n \ge 2$, Eliashberg and Gromov, with a final step by Schürmann, showed that $\C^N$ with $N = \floor{3n/2} + 1$ suffices, and this value is optimal.

## Explanations

### L1 human

The theorem says that every Stein manifold can be drawn inside some $\C^N$ by holomorphic functions, without self-intersections.
Going off to infinity in the manifold then means going off to infinity in $\C^N$.
So the abstract definition of a Stein manifold, by conditions on its holomorphic functions, describes the same objects as the concrete closed submanifolds of $\C^N$.
It is the complex-analytic analogue of the Whitney embedding theorem for smooth manifolds.

### L2 human

Embeddings reduce questions about Stein manifolds to questions about submanifolds of $\C^N$.
For example, $\abs{F}^2$ is a smooth strictly plurisubharmonic exhaustion function, which gives the easy half of Grauert's characterization of Stein manifolds.
Whether every open Riemann surface embeds properly in $\C^2$ is an open problem.
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# Pseudo-Riemannian manifold

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, with [[tangent spaces|WY6K3P4YCG]] $T_pM$.
A *pseudo-Riemannian metric* on $M$ is a smooth symmetric $(0, 2)$-[[tensor field|WX302T93TW]] $g$ on $M$ such that:

1. for each $p \in M$, the map $g_p \colon T_pM \times T_pM \to \R$ is a [[symmetric bilinear form|0AXZNE5XNH]];
2. for each $p \in M$, the form $g_p$ is [[nondegenerate|01NHFZF2SK]]: if $g_p(v, w) = 0$ for all $w \in T_pM$, then $v = 0$;
3. the [[signature|07EZA80NEH]] of $g_p$ is the same for all $p \in M$.

A *pseudo-Riemannian manifold* is a pair $(M, g)$ of a smooth manifold $M$ and a pseudo-Riemannian metric $g$ on $M$.

## Remarks

Example: on $\R^2$ with coordinates $(t, x)$, let $g = -dt^2 + dx^2$.
At each point its matrix is $\operatorname{diag}(-1, 1)$, which is symmetric with nonzero determinant, so $g$ is nondegenerate with constant signature: $(\R^2, g)$ is pseudo-Riemannian.

Non-example: on $\R^2$, let $g = dt^2 + t \, dx^2$.
Its matrix $\operatorname{diag}(1, t)$ is degenerate where $t = 0$, so clause 2 fails on the line $t = 0$.

On a connected manifold, clause 3 follows from clause 2 and continuity, since the signature can only change where the determinant of $g$ vanishes.
A Riemannian metric is the case where $g_p$ is positive definite at every point.
In local coordinates $(x^1, \ldots, x^n)$ one writes $g = \sum_{i,j} g_{ij} \, dx^i \, dx^j$ with a symmetric invertible matrix $(g_{ij})$.
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<document source="https://tamnd.github.io/math2/node/6VFF2TTXYZ.md" tag="6VFF2TTXYZ" kind="example">
# The left cosets of a subgroup of order two in S3

## Example

In the [[symmetric group|0515A6D2Y1]] $S_3$, let $H = \{e, (1\,2)\}$. Compose permutations from right to left.
The [[left cosets|05FD317V2W]] of $H$ are $eH = \{e, (1\,2)\}$, $(1\,3)H = \{(1\,3), (1\,2\,3)\}$ and $(2\,3)H = \{(2\,3), (1\,3\,2)\}$.
Each of the $6$ elements of $S_3$ is in exactly one of these cosets.
Each coset has $2$ elements. Thus $6 = 3 \cdot 2$, as [[Lagrange's theorem|3C53TD]] states.

## Remarks

The right cosets of $H$ are different sets, because $H$ is not normal in $S_3$.
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# Kolmogorov's weak-type estimate for the conjugate function

## Statement

Let $\mathbb{T} = \R / 2\pi\Z$, identified with $[-\pi, \pi)$ with [[Lebesgue measure|2Z2QVWZ262]] $\abs{\cdot}$.
There is an absolute constant $C$ with the following property.
For every integrable function $f$ on $\mathbb{T}$, with [[conjugate function|A1H48DDW49]] $\tilde f$, and every $\lambda > 0$,
$$\abs{\set{\theta \in \mathbb{T} : \abs{\tilde f(\theta)} > \lambda}} \le \frac{C}{\lambda} \int_{\mathbb{T}} \abs{f(\theta)} \, d\theta.$$
That is, $f \mapsto \tilde f$ is of weak type $(1, 1)$.

## Remarks

A real-variable proof uses the [[Calderón–Zygmund lemma|K1YKXVPVDG]] at height $\lambda$ to write $f = g + b$.
The good part $g$ is bounded by $2\lambda$ and is handled by the $L^2$ bound and Chebyshev's inequality.
The bad part $b$ has mean zero on each interval $Q_k$.
So the smoothness of the kernel $\cot(t/2)$ away from $0$ makes $\tilde b$ small off the doubled intervals.
These have total length at most $C\lambda^{-1}\norm{f}_1$.

The strong bound $\norm{\tilde f}_{L^1} \le C\norm{f}_{L^1}$ fails.
For example, if $f$ is the Poisson kernel $P_r$ and $r \to 1^-$, then $\norm{P_r}_{L^1}$ stays bounded while $\tilde P_r$, the conjugate Poisson kernel, has $L^1$ norm comparable to $\log\frac{1}{1 - r}$.

So $\tilde f$ is finite almost everywhere for $f \in L^1$, and it lies in weak $L^1$, but it need not be integrable.

## Explanations

### L1 human

For an integrable function, the conjugate function can fail to be integrable, so one cannot bound its size in the $L^1$ norm.
Kolmogorov's estimate says that it is still not large on a big set: the set where it exceeds $\lambda$ has measure at most a constant times $\norm{f}_1/\lambda$.
This is the same bound that Chebyshev's inequality would give if $\tilde f$ were integrable.

### L2 human

Together with the $L^2$ bound from Parseval's identity, the Marcinkiewicz interpolation theorem turns this estimate into the $L^p$ bounds of the [[M. Riesz theorem|PJFH51Z48M]] for $1 < p \le 2$, and duality covers $2 \le p < \infty$.
It is the model for the weak type $(1, 1)$ bounds of all [[Calderón–Zygmund operators|3SCXCW4AH3]].
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# Feit–Thompson theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]] whose [[order|0B85S2V2DR]] is odd.
Then $G$ is [[solvable|0FXJ7YFMMX]].

## Remarks

An equivalent form: every finite non-abelian [[simple group|0YMYMNQ0QZ]] has even order.
Indeed, a finite group is solvable exactly when all its [[composition factors|0PHTC93M58]] are cyclic of prime order, and every composition factor of a group of odd order has odd order.
The proof, published by Walter Feit and John Thompson in 1963, fills a whole issue of a journal and combines local analysis of subgroups with character theory.
A formal proof was completed in the Coq proof assistant in 2012.
The hypothesis of odd order is necessary: the alternating group $A_5$ has order $60$ and is not solvable.
For example, every group of order $3 \cdot 5 \cdot 7 \cdot 11 = 1155$ is solvable, although [[Burnside's theorem|0FMAQSB1SC]] on groups of order $p^a q^b$ does not apply to it.

## Explanations

### L1 human

The theorem says that a finite group with an odd number of elements can always be broken down into abelian pieces.
So every interesting simple group must have an even number of elements, and hence contain an element of order $2$.

### L2 human

By [[Cauchy's theorem|0JM7PA288B]], a non-abelian finite simple group therefore contains involutions, that is, elements of order $2$.
This opened the way to studying simple groups through the [[centralisers|02YYV36BFM]] of their involutions, the main strategy of the [[classification of finite simple groups|06J6P03MG7]].
The [[Brauer–Fowler theorem|K5CQM8MYYG]] shows that the centraliser of an involution limits the size of the simple group.
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# Caputo fractional derivative

## Summary

The Caputo fractional derivative of order $\alpha$ first differentiates a function $n = \ceil{\alpha}$ times and then applies the [[Riemann–Liouville integral|JTKCZ44NN3]] of order $n - \alpha$.
This order of operations is the reverse of the Riemann–Liouville derivative.
Its main advantage is that the derivative of a constant is zero, and fractional differential equations with it take ordinary initial values $f(a), f'(a), \dots$.
It is a standard tool of [[fractional calculus|T73XK15SA2]] in applied models.

## Explanations

### L1 human

To take a derivative of order $1/2$, the Caputo rule takes one ordinary derivative and then a half integral.
Because the ordinary derivative comes first, constants are killed, just as in ordinary calculus.
For example, with lower terminal $0$, the Caputo derivative of order $1/2$ of $x$ is the half integral of $1$, which is $2\sqrt{x/\pi}$.
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# Mutual energy

## Summary

The mutual energy of two positive measures is the double integral of a potential-theoretic kernel against one measure in each variable.
It is the interaction energy of two charge distributions, and it equals the integral of the potential of one measure against the other.
The [[energy of a measure|SFSRQCM3EG]] is the mutual energy of the measure with itself.
So the mutual energy is the bilinear form whose quadratic form is the energy.
Its [[positive definiteness|PNZT35FFN2]] turns measures of finite energy into a pre-Hilbert space, which is the basis of the energy method in potential theory.

## Explanations

### L1 human

Take two clouds of charge.
Their mutual energy adds up the repulsion between each bit of the first cloud and each bit of the second.
For two point charges at $a$ and $b$ in space, it is $1/\abs{a - b}$, which grows as the charges approach each other.
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# Ring of invariants

## Summary

The ring of invariants of a group acting on a commutative ring is the subring of elements fixed by every element of the group.
For a group acting linearly on a vector space, it is the ring of polynomial functions that do not change under the group.
It is the central object of invariant theory.
Whether it is finitely generated is the subject of [[Hilbert's fourteenth problem|M5N08HD75V]].
Its ring-theoretic properties are the subject of results such as the [[Hochster–Roberts theorem|56107GMFYV]].
Geometrically, it is the coordinate ring of the quotient of the space by the group.

## Explanations

### L1 human

Let the symmetric group permute the variables $x_1, \ldots, x_n$.
The polynomials that do not change, such as $x_1 + \cdots + x_n$ and $x_1 x_2 \cdots x_n$, are the symmetric polynomials, and they form a ring.
Another example: over the rational numbers, the map $(x, y) \mapsto (-x, -y)$ fixes exactly the polynomials whose terms all have even degree.
That ring is generated by $x^2$, $xy$ and $y^2$, which satisfy the relation $x^2 \cdot y^2 = (xy)^2$.
Rings of invariants describe the "shape" of a space after points in the same orbit are glued together.
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# Markov random field

## Summary

A Markov random field is a family of random variables indexed by the vertices of an undirected [[graph|00C0BR7D7H]].
Each variable depends on the others only through its neighbours.
It is the undirected kind of [[graphical model|Q3FQAD1BXG]].
By the [[Hammersley–Clifford theorem|K07QH8EKD0]], positive Markov random fields are exactly the distributions that factor over the [[cliques|0BYX1D7C2E]] of the graph.
The Ising model and many models in image analysis are Markov random fields.

## Explanations

### L1 human

Think of the pixels of a black and white image, each joined to its four neighbours.
In a Markov random field, once the colours of the neighbours of a pixel are known, the rest of the image tells nothing more about that pixel.
This captures the idea that nearby pixels tend to agree while far pixels interact only through the pixels between them.
The same picture works for spins in a magnet, which is the Ising model.
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# Method of fundamental solutions

## Summary

The method of fundamental solutions approximates the solution of a linear homogeneous elliptic equation by a sum of [[fundamental solutions|K2NK9A8M5F]].
These are centred at source points outside the domain.
Each term solves the equation exactly inside the domain, so only the boundary condition has to be fitted.
It needs neither a mesh nor integrals, only the evaluation of the fundamental solution at boundary points.
It is used when the fundamental solution is known in closed form.
Examples are the Laplace equation $\Delta u = 0$, with $\Delta$ the [[Laplacian|0E0B6RA8K0]], the [[Helmholtz equation|MNXKTXJ35G]], and linear elasticity.
See the [[definition|XS0Q09SV2T]].

## Explanations

### L1 human

Place a few point charges outside a region.
The electric potential they create satisfies the Laplace equation everywhere inside the region.
By choosing the strengths of the charges well, one can make this potential match given values on the boundary of the region, at least approximately.
The method of fundamental solutions does exactly this, and the resulting sum of charge potentials is the approximate solution.
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# Genus of a curve

## Summary

The genus of a smooth projective connected curve is the dimension of its space of regular differential $1$-forms.
It is the main invariant of a curve.
Curves of genus $0$ are conics, and curves of genus $1$ with a rational point are elliptic curves.
Curves of genus at least $2$ have finitely many automorphisms.
Over $\C$ it equals the topological [[genus|NJ787GZS1N]] of the associated compact Riemann surface, the number of handles.
For a singular curve, one takes the genus of its normalization, called the geometric genus.

## Explanations

### L1 human

Over the complex numbers, a smooth projective connected curve is a closed surface, a sphere with some handles.
The number of handles is the genus.
The projective line is a sphere, of genus $0$.
A smooth plane cubic such as $y^2 = x^3 - x$ is a torus, of genus $1$.
The genus controls almost everything about the curve, for instance through the [[Riemann–Roch theorem|WMVAT9MBZ1]].
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# Theta correspondence

## Definition

Let $W$ be a finite-dimensional real [[symplectic vector space|PQ7JJY0VPH]] with [[symplectic group|X12FS2JAP0]] $\mathrm{Sp}(W)$ and [[metaplectic group|8B0JM2YDE2]] $p \colon \mathrm{Mp}(W) \to \mathrm{Sp}(W)$.
A *reductive dual pair* in $\mathrm{Sp}(W)$ is a pair $(G, G')$ of [[reductive|42ZCE14AD2]] subgroups of $\mathrm{Sp}(W)$ with two properties.
The centralizer of $G'$ in $\mathrm{Sp}(W)$ is $G$, and the centralizer of $G$ is $G'$.
Let $\tilde G = p^{-1}(G)$ and $\tilde G' = p^{-1}(G')$, and let $\omega^\infty$ be the space of smooth vectors of the [[oscillator representation|CCR7BAQ5N5]] of $\mathrm{Mp}(W)$, as a representation of $\tilde G \times \tilde G'$.
Let $\mathcal R(\tilde G)$ be the set of classes of [[irreducible|FX3NEFYFFN]] [[admissible representations|SZV8Y806Y7]] $\pi$ of $\tilde G$ that occur as quotients of $\omega^\infty$.
Define $\mathcal R(\tilde G')$ in the same way.
By Howe duality, for each $\pi \in \mathcal R(\tilde G)$ there is exactly one $\pi' \in \mathcal R(\tilde G')$ such that $\pi \otimes \pi'$ is a quotient of $\omega^\infty$, and $\pi \mapsto \pi'$ is a bijection $\mathcal R(\tilde G) \to \mathcal R(\tilde G')$.
This bijection is the *theta correspondence* for the pair $(G, G')$, written $\pi' = \theta(\pi)$.

## Remarks

The theorem used in the definition is Howe's duality theorem for real groups; the same definition over $p$-adic fields rests on the work of several later authors.

Example: let $W = \R^2$, so $\mathrm{Sp}(W) = \mathrm{SL}_2(\R)$, and let $G = \set{\pm 1}$, which is the orthogonal group $\mathrm{O}(1)$.
The centralizer of $\set{\pm 1}$ is all of $\mathrm{SL}_2(\R)$, and the centralizer of $\mathrm{SL}_2(\R)$ is its center $\set{\pm 1}$, so $(\mathrm{O}(1), \mathrm{SL}_2(\R))$ is a dual pair.
In the Schrödinger model on $L^2(\R)$, the element $-1$ acts by $\psi(x) \mapsto \psi(-x)$, up to a constant depending on the lift.
So the two characters of $\mathrm{O}(1)$ correspond to the two irreducible pieces of the oscillator representation of $\mathrm{Mp}_2(\R)$, on even and on odd functions.

Non-example: in $\mathrm{SL}_2(\R)$, let $A = \set{\operatorname{diag}(a, a^{-1}) : a > 0}$.
The centralizer of $A$ is $\set{\operatorname{diag}(a, a^{-1}) : a \ne 0}$, which is larger than $A$, so $(A, A)$ is not a dual pair.

The standard dual pairs include $(\mathrm{O}(p, q), \mathrm{Sp}_{2m}(\R))$ in $\mathrm{Sp}(\R^{p+q} \otimes \R^{2m})$ and $(\mathrm{U}(p, q), \mathrm{U}(r, s))$.
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# Holomorphic dynamics in several variables

## Summary

Holomorphic dynamics in several variables studies the iteration of holomorphic maps $f \colon X \to X$ of a [[complex manifold|JSGWAN2RJV]] $X$ of dimension at least $2$.
The two main families of examples are holomorphic endomorphisms of complex projective space $\mathbb{P}^k$ and polynomial automorphisms of $\C^2$, such as the [[Hénon maps|KS4FJ6H5JH]].
As in one-variable [[complex dynamics|BQQZ0ZCGQF]], space splits into a Fatou set, where the dynamics is stable, and a Julia set, where it is chaotic.
Many tools are new, notably plurisubharmonic functions, positive closed currents and invariant measures.

## Explanations

### L1 human

Take a holomorphic map of a space to itself and apply it again and again.
Each starting point traces out an orbit, and the question is how these orbits behave.
In one complex variable this gives Julia sets and the Mandelbrot set.
In two or more variables the same question has richer answers: orbits can be attracted to curves, strange sets appear, and maps can be invertible yet chaotic.
The map $(x, y) \mapsto (x^2 + c - y, x)$ of $\C^2$ is a simple example: it is invertible, and its orbits already show this rich behaviour.
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# Gamma function as Mellin transform of the exponential

## Example

Let $f(x) = e^{-x}$ for $x > 0$.
Its [[Mellin transform|N5PVGRJE1T]] is the [[gamma function|0H5SYBSED9]]:
$$\mathcal{M}f(s) = \int_0^\infty x^{s-1} e^{-x} \, dx = \Gamma(s), \qquad \operatorname{Re} s > 0.$$
The right side is [[Euler's integral|Z8D1CZFTWP]], which is the usual definition of $\Gamma$ on the half-plane $\operatorname{Re} s > 0$.

Check of [[Mellin convergence|SGZZ4Y0D36]]: let $\sigma = \operatorname{Re} s$.
On $(0, 1]$, $x^{\sigma - 1} e^{-x} \le x^{\sigma - 1}$, which is integrable exactly when $\sigma > 0$.
On $[1, \infty)$, $x^{\sigma - 1} e^{-x} \le C_\sigma e^{-x/2}$ for a constant $C_\sigma$, which is integrable for every $\sigma$.
For $\sigma \le 0$, $x^{\sigma - 1} e^{-x} \ge e^{-1} x^{\sigma - 1}$ on $(0, 1]$, which is not integrable.
So $f$ is Mellin convergent exactly on the half-plane $\operatorname{Re} s > 0$.

Integration by parts gives $\Gamma(s + 1) = s \Gamma(s)$, and $\Gamma(1) = \int_0^\infty e^{-x} \, dx = 1$, so $\Gamma(n + 1) = n!$.
By the dilation rule, $\int_0^\infty x^{s-1} e^{-ax} \, dx = a^{-s} \Gamma(s)$ for $a > 0$.
By the [[Mellin inversion theorem|XJWC13NY9E]], for every $c > 0$ and $x > 0$,
$$e^{-x} = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \Gamma(s) x^{-s} \, ds.$$

## Explanations

### L1 human

The gamma function, which extends the factorial, is the simplest Mellin transform.
One weighs the decaying exponential $e^{-x}$ against the powers $x^{s-1}$ and integrates.
At $s = 1, 2, 3, \dots$ the answers are $0!, 1!, 2!, \dots$.
The integral only makes sense when $\operatorname{Re} s > 0$, because otherwise the power blows up too fast at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/6WH08A4BTK.md" tag="6WH08A4BTK" kind="concept">
# Determinantal variety

## Summary

A determinantal variety is the set of matrices of a fixed size whose [[rank|0ZC9YTXZRY]] is at most a given number.
It is cut out by the vanishing of all [[minors|0BT9BYWVRY]] of one size larger, which generate a [[determinantal ideal|SXD7X0WQH1]].
These varieties are irreducible, normal and Cohen–Macaulay, and they are basic examples of singular varieties.
Many varieties in geometry, such as secant varieties and rational normal curves, are described by determinantal equations.

## Explanations

### L1 human

A $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ has rank at most $1$ exactly when $ad - bc = 0$.
This one equation defines a three-dimensional cone in the four-dimensional space of matrices, with its tip at the zero matrix.
For larger matrices, "rank at most $r$" is expressed by the vanishing of all $(r + 1) \times (r + 1)$ minors.
The resulting sets are the determinantal varieties, and their only singular points are the matrices of rank smaller than $r$.
</document>

<document source="https://tamnd.github.io/math2/node/6WTHKQG40F.md" tag="6WTHKQG40F" kind="definition">
# Contiguity of measures

## Definition

For each $n \ge 1$, let $(\Omega_n, \mathcal{F}_n)$ be a measurable space, with $\mathcal{F}_n$ a [[sigma-algebra|04WSW8T3SY]], and let $P_n$ and $Q_n$ be [[probability measures|059TJN0NZ1]] on it.
The sequence $(Q_n)$ is *contiguous with respect to* $(P_n)$, written $Q_n \triangleleft P_n$, if for every sequence of events $A_n \in \mathcal{F}_n$,
$$P_n(A_n) \to 0 \implies Q_n(A_n) \to 0.$$
The sequences are *mutually contiguous*, written $P_n \triangleleft \triangleright Q_n$, if $Q_n \triangleleft P_n$ and $P_n \triangleleft Q_n$.

## Remarks

Contiguity is the asymptotic analogue of [[absolute continuity|2G173YJC54]]; for constant sequences $P_n = P$ and $Q_n = Q$ it is the same as $Q \ll P$.
If $Q_n \triangleleft P_n$, then every sequence of statistics that tends to $0$ in $P_n$-probability also tends to $0$ in $Q_n$-probability.

Example: let $P_n = \mathcal{N}(0, 1)^{\otimes n}$ and $Q_n = \mathcal{N}(h/\sqrt{n}, 1)^{\otimes n}$, products of [[normal distributions|07RXBVTVZB]], for a fixed $h \in \R$.
Under $P_n$, the log-likelihood ratio $\log(dQ_n/dP_n) = h n^{-1/2}\sum_i X_i - h^2/2$ has the law $\mathcal{N}(-h^2/2, h^2)$ for every $n$.
By Le Cam's first lemma, a limit law $L$ of $\log(dQ_n/dP_n)$ under $P_n$ with $E e^{L} = 1$ implies $Q_n \triangleleft P_n$, and here $E e^{L} = e^{-h^2/2 + h^2/2} = 1$; by symmetry the sequences are mutually contiguous.

Non-example: let $P_n = \mathcal{N}(0, 1)^{\otimes n}$ and $Q_n = \mathcal{N}(1, 1)^{\otimes n}$, and let $A_n = \set{\bar{x}_n > 1/2}$, with $\bar{x}_n$ the mean of the coordinates.
Then $P_n(A_n) = P(Z > \sqrt{n}/2) \to 0$ but $Q_n(A_n) = P(Z > -\sqrt{n}/2) \to 1$ for $Z$ standard normal, so $Q_n$ is not contiguous with respect to $P_n$.
</document>

<document source="https://tamnd.github.io/math2/node/6WV71T5CK1.md" tag="6WV71T5CK1" kind="object">
# Klein quadric

## Description

Let $F$ be a field.
The *Klein quadric* is the [[quadric|Q3Q4FF23VG]] in $\mathbb{P}^5(F)$, with coordinates $[p_{01} : p_{02} : p_{03} : p_{12} : p_{13} : p_{23}]$, given by
$$p_{01} p_{23} - p_{02} p_{13} + p_{03} p_{12} = 0.$$
Sending a line of $\mathbb{P}^3(F)$ to its [[Plücker coordinates|B8PAMSJ81T]] is a bijection from the set of lines of $\mathbb{P}^3(F)$ onto the Klein quadric.
This bijection is the Klein correspondence.

Two lines of $\mathbb{P}^3(F)$ meet if and only if the line of $\mathbb{P}^5(F)$ joining their images lies entirely on the Klein quadric.
Check on an example: the lines $L_1$ through $e_0, e_1$ and $L_2$ through $e_0, e_2$ meet at $[e_0]$.
Their images are $[1 : 0 : 0 : 0 : 0 : 0]$ and $[0 : 1 : 0 : 0 : 0 : 0]$.
Every point $[s : t : 0 : 0 : 0 : 0]$ of the joining line satisfies the equation, since each product in it has a zero factor.

## Remarks

The quadratic form $p_{01} p_{23} - p_{02} p_{13} + p_{03} p_{12}$ is a sum of three hyperbolic planes, so the Klein quadric is a non-degenerate quadric of maximal Witt index in $\mathbb{P}^5(F)$.
Its largest linear subspaces are planes, and they come in two families.
The planes of one family correspond to the sets of lines through a fixed point of $\mathbb{P}^3(F)$.
The planes of the other family correspond to the sets of lines in a fixed plane of $\mathbb{P}^3(F)$.
Over a finite field $\F_q$ the Klein quadric has $(q^2 + 1)(q^2 + q + 1)$ points, the number of lines of $PG(3, q)$.

## Explanations

### L1 human

The lines of three-dimensional projective space form a four-dimensional family.
The Klein quadric packages this family as a single curved hypersurface in a five-dimensional projective space.
Each line becomes one point, and questions about lines, such as whether two lines meet, become questions about points on the quadric.
The coordinates are named after Julius Plücker and the quadric after Felix Klein.
</document>

<document source="https://tamnd.github.io/math2/node/6WVTB9Y3XD.md" tag="6WVTB9Y3XD" kind="definition">
# Kähler–Einstein metric

## Definition

Let $(M, J)$ be a complex manifold and let $g$ be a Kähler metric on $M$.
Thus $g$ is a [[Hermitian metric|CK1DCPC3D3]] whose fundamental form $\omega$ is closed.
Let $\operatorname{Ric}$ be the [[Ricci curvature|4GTNM0WHDH]] of $g$, and let $\rho(X, Y) = \operatorname{Ric}(JX, Y)$ be its Ricci form.
The metric $g$ is *Kähler–Einstein* if there is a real constant $\lambda$ such that $\operatorname{Ric} = \lambda g$, or equivalently $\rho = \lambda \omega$.

## Remarks

Example: the flat metric on a complex torus $\C^n / \Lambda$, for a lattice $\Lambda$ of rank $2n$.
It is Kähler with constant coefficients, its curvature vanishes, and so $\operatorname{Ric} = 0 = 0 \cdot g$.
The Fubini–Study metric on $\C P^n$ is Kähler–Einstein with $\lambda > 0$, and the Bergman metric on the unit ball of $\C^n$ is Kähler–Einstein with $\lambda < 0$.

Non-example: on $\C P^1 \times \Sigma$, with $\Sigma$ a compact Riemann surface of genus $2$, take the product of a round metric of curvature $1$ and a metric of curvature $-1$.
It is Kähler, but $\operatorname{Ric}$ equals $g$ on the first factor and $-g$ on the second, so no single $\lambda$ works.

On a compact Kähler manifold the Ricci form represents $2\pi c_1(M)$ in de Rham cohomology.
So a Kähler–Einstein metric with constant $\lambda$ forces $2\pi c_1(M) = \lambda [\omega]$: the first Chern class is negative, zero or positive according to the sign of $\lambda$.
A Kähler–Einstein metric is an [[Einstein metric|41S2EZ0KR3]] in the Riemannian sense.
</document>

<document source="https://tamnd.github.io/math2/node/6WYEMXSHD9.md" tag="6WYEMXSHD9" kind="object">
# Exotic sphere

## Description

An *exotic sphere* is a [[smooth manifold|F45VQ47GPV]] $\Sigma$ that is [[homeomorphic|0QT00PB5G2]] to the standard sphere $S^n$ but not [[diffeomorphic|0KZZ70ZHXP]] to it.
Equivalently, it is $S^n$ with a [[smooth structure|R2TQ82PCNB]] that is not diffeomorphic to the standard one.

Milnor found the first examples in dimension $7$.
He took total spaces of bundles over $S^4$ with fibre $S^3$.
For suitable bundles he built a Morse function with exactly two critical points, so the total space is homeomorphic to $S^7$.
He then defined an invariant of such a manifold from a compact $8$-manifold that it bounds, using the [[Hirzebruch signature theorem|TEG1ZX1AGJ]].
This invariant is zero for the standard $S^7$ and nonzero for some of the bundles, so these are not diffeomorphic to $S^7$.

## Remarks

Kervaire and Milnor showed that for $n \ge 5$ the oriented diffeomorphism classes of smooth homotopy $n$-spheres form a finite abelian group $\Theta_n$ under [[connected sum|Y0K0K5NGWW]].
For $n = 7$ this group is cyclic of order $28$.
For $n = 1, 2, 3, 5, 6$ there are no exotic spheres.
In dimension $4$ it is unknown whether exotic spheres exist; this is the [[smooth four-dimensional Poincaré conjecture|P70VGYNFS5]].

## Explanations

### L1 human

An exotic sphere is a space that is a sphere for a topologist who allows bending and stretching, but not for one who also demands smoothness.
There is no smooth way to straighten it into the round sphere.
The first one appears in dimension seven.
Up to orientation-preserving diffeomorphism, there are $28$ different smooth versions of the $7$-sphere.
</document>

<document source="https://tamnd.github.io/math2/node/6WYYMPH21Q.md" tag="6WYYMPH21Q" kind="statement">
# Fundamental recurrence formulas

## Statement

Let $b_0, b_1, b_2, \dots$ be complex numbers and let $a_1, a_2, \dots$ be nonzero complex numbers.
Consider the [[continued fraction|0BK6JBK58W]] $b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cdots}}$, and let $x_n$ be its $n$-th [[convergent|0SRWNR42BQ]], evaluated in the [[Riemann sphere|0CDMXGPSES]] with $a/0 = \infty$ and $a/\infty = 0$ for $a \ne 0$.
Define $A_{-1} = 1$, $A_0 = b_0$, $B_{-1} = 0$, $B_0 = 1$ and, for $n \ge 1$,
$$A_n = b_n A_{n-1} + a_n A_{n-2}, \qquad B_n = b_n B_{n-1} + a_n B_{n-2}.$$
Then for every $n \ge 0$:

1. $x_n = \dfrac{A_n}{B_n}$, where $A_n$ and $B_n$ are not both $0$;
2. $A_n B_{n-1} - A_{n-1} B_n = (-1)^{n-1} a_1 a_2 \cdots a_n$.

## Remarks

Proof idea: write $x_n = t_0 \circ t_1 \circ \cdots \circ t_n(0)$ with $t_0(w) = b_0 + w$ and $t_k(w) = \frac{a_k}{b_k + w}$.
These are [[Möbius transformations|0B346DDAM5]], and the product of their matrices is $\begin{pmatrix} A_{n-1} & A_n \\ B_{n-1} & B_n \end{pmatrix}$ by induction.
Taking determinants gives item 2, and applying the product to $0$ gives item 1; item 2 shows that $A_n$ and $B_n$ do not both vanish.
Example: for $1 + \cfrac{1}{2 + \cfrac{1}{2 + \cdots}}$ the recurrences give $A_n = 1, 3, 7, 17, \dots$ and $B_n = 1, 2, 5, 12, \dots$, the convergents of $\sqrt{2}$.
Item 2 gives $x_n - x_{n-1} = \dfrac{(-1)^{n-1} a_1 \cdots a_n}{B_n B_{n-1}}$, so the convergents are the partial sums of a series, and convergence questions become questions about series.

## Explanations

### L1 human

Computing a continued fraction from the inside out has to be redone each time a new level is added.
These formulas compute each new convergent from the two previous ones, with one multiplication and addition for its numerator and one for its denominator.

### L2 human

The numerators $A_n$ and denominators $B_n$ satisfy a three-term linear recurrence, the same one for both.
For a [[simple continued fraction|06XF7BCFFJ]] they reduce to the recurrences for $p_n$ and $q_n$, and item 2 becomes $p_n q_{n-1} - p_{n-1} q_n = (-1)^{n-1}$.
The difference formula in the remarks is the starting point of the [[Śleszyński–Pringsheim theorem|NEC48RHNN0]] and other convergence tests for continued fractions.
</document>

<document source="https://tamnd.github.io/math2/node/6X05N89XDP.md" tag="6X05N89XDP" kind="statement">
# Analyticity of the Laplace transform in its half-plane

## Statement

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$ with $T > 0$.
Assume that the [[Laplace integral|05XT5AVJBK]] $\int_0^\infty f(t) e^{-s_0 t}\, dt$ converges, as an improper integral, at some $s_0 \in \C$.
Let $\sigma_c$ be the [[abscissa of convergence|FTHFSJ30WJ]] of $f$.
Then:

1. The integral $F(s) = \int_0^\infty f(t) e^{-st}\, dt$ converges for every $s \in \C$ with $\operatorname{Re} s > \operatorname{Re} s_0$.
2. The function $F$ is [[holomorphic|0DG0Z72CQD]] on the half-plane $\set{s \in \C : \operatorname{Re} s > \sigma_c}$.
3. For every $n \in \N$ and every $s$ with $\operatorname{Re} s > \sigma_c$,
$$F^{(n)}(s) = \int_0^\infty (-t)^n f(t) e^{-st}\, dt,$$
and this integral converges.

## Remarks

For part 1, let $\varphi(t) = \int_0^t f(u) e^{-s_0 u}\, du$; it is continuous and bounded on $[0, \infty)$ because it has a finite limit.
Integration by parts gives $\int_0^T f(t) e^{-st}\, dt = \varphi(T) e^{-(s - s_0)T} + (s - s_0) \int_0^T \varphi(t) e^{-(s - s_0)t}\, dt$, and both terms converge when $\operatorname{Re}(s - s_0) > 0$.
The same formula shows that the truncated integrals converge locally uniformly on the half-plane $\operatorname{Re} s > \operatorname{Re} s_0$.
By part 1 the integral converges at real points $\sigma$ arbitrarily close to $\sigma_c$ from the right, so these half-planes cover $\operatorname{Re} s > \sigma_c$.
A locally uniform limit of holomorphic functions is holomorphic, which gives part 2.
Part 3 follows by differentiating under the integral sign on $[0, T]$ and passing to the limit.
Example: for $f = 1$, $F(s) = 1/s$ is holomorphic on $\operatorname{Re} s > 0$, and $F'(s) = -1/s^2 = \int_0^\infty (-t) e^{-st}\, dt$.
The half-plane is in general not the largest domain of holomorphy: $1/s$ extends to $\C \setminus \set{0}$, but the integral diverges for $\operatorname{Re} s \le 0$.

## Explanations

### L1 human

The Laplace transform of a function is a very smooth object: wherever the integral converges (off the boundary line), the transform is a holomorphic function of the complex variable $s$.
The region of convergence is always a right half-plane.
Derivatives of the transform are transforms of $f$ multiplied by powers of $-t$.

### L2 human

This theorem is why complex analysis applies to Laplace transforms.
Residues, contour shifts and analytic continuation are used to invert transforms, as in the Bromwich integral, and to read off the asymptotic behaviour of $f$ from the singularities of $F$.
Part 3 is the rule "multiplication by $-t$ corresponds to differentiation in $s$".
It is the analogue of the fact that a power series converges on a disc and is holomorphic inside it; Dirichlet series obey the same pattern.
</document>

<document source="https://tamnd.github.io/math2/node/6X0C4HNMAZ.md" tag="6X0C4HNMAZ" kind="concept">
# Wynn's epsilon algorithm

## Summary

Wynn's epsilon algorithm is a recursive scheme that computes the values of the [[Shanks transformation|C0TXDZZMH7]] without evaluating determinants.
Each new entry of its table is computed from three earlier entries by one subtraction, one division and one addition.
It is one of the most widely used nonlinear [[series acceleration|B1S2XJPGYD]] methods.

## Explanations

### L1 human

The Shanks transformation is defined by large determinants, which are expensive and unstable to compute.
Wynn found a short recursion that fills a triangular table column by column.
The even columns of the table contain the Shanks values; the odd columns are only intermediate results.
For example, the second column built from the partial sums $1, \frac23, \frac{13}{15}$ of the Leibniz series gives $\frac{19}{24}$, the first Shanks value.
</document>

<document source="https://tamnd.github.io/math2/node/6X2D0JDFQW.md" tag="6X2D0JDFQW" kind="statement">
# Invariance of domain

## Statement

Let $n \ge 1$, let $U$ be an [[open subset|02BH3MCBQB]] of [[$\R^n$|0XPR1E85W0]], and let $f \colon U \to \R^n$ be a [[continuous|0VGFHTM75Y]] [[injective|09NWETVTNY]] map.
Then $f(U)$ is open in $\R^n$, and $f$ is a [[homeomorphism|0QT00PB5G2]] from $U$ onto $f(U)$.

## Remarks

For $n = 1$ the claim follows from the intermediate value theorem, since $f$ is then strictly monotone on each interval in $U$; so let $n \ge 2$.
It suffices to show that, for every closed ball $B \subseteq U$, the image $f(B \setminus \partial B)$ of its interior is open.
The image $f(\partial B)$ is homeomorphic to the sphere $S^{n-1}$, so by the [[Jordan–Brouwer separation theorem|2Q008XWQDE]] its complement has two components, a bounded one and an unbounded one.
A homology argument shows that $f(B \setminus \partial B)$ is connected and disjoint from $f(\partial B)$ and that $\R^n \setminus f(B)$ is connected, which forces $f(B \setminus \partial B)$ to be the bounded component, an open set.
The second claim follows from the first, since $f$ then maps open subsets of $U$ to open sets.

Example: it implies invariance of dimension, that is, $\R^m$ and $\R^n$ are not homeomorphic if $m < n$.
Indeed, a homeomorphism $\R^n \to \R^m \subset \R^n$ would have an open image inside the hyperplane $\R^m \times \set{0}$.
But this hyperplane has empty interior in $\R^n$.

Both hypotheses are needed.
Without injectivity, $f(x) = x^2$ on $\R$ has image $[0, \infty)$, which is not open.
The target must have the same dimension: the map $x \mapsto (x, 0)$ from $\R$ to $\R^2$ is continuous and injective, but its image is not open in $\R^2$.

## Explanations

### L1 human

A continuous one-to-one map from an open piece of $n$-dimensional space into $n$-dimensional space cannot squash anything flat or tear it open.
Its image is again an open piece, and the map can be undone continuously.

### L2 human

The theorem shows that the dimension of a topological manifold is well defined.
It also shows that a continuous bijection between manifolds of the same dimension without boundary is a homeomorphism.
It is false in infinite dimensions: on a Hilbert space, the shift operator is a continuous injective linear map whose image is not open.
It is usually proved with homology, through the Jordan–Brouwer separation theorem, or with the Brouwer fixed point theorem.
</document>

<document source="https://tamnd.github.io/math2/node/6X94NPK782.md" tag="6X94NPK782" kind="definition">
# Abel–Poisson summation of Fourier series

## Definition

Let $(c_k)_{k \in \Z}$ be complex numbers, and consider the trigonometric series $\sum_{k \in \Z} c_k e^{ikx}$, summed symmetrically as $c_0 + \sum_{k \ge 1} (c_k e^{ikx} + c_{-k} e^{-ikx})$.
For $0 \le r < 1$ and real $x$ put
$$A_r(x) = \sum_{k \in \Z} r^{\abs{k}} c_k e^{ikx}$$
whenever this series converges.
The series is *Abel–Poisson summable* at $x$ to $s$ if $A_r(x)$ is defined for all $0 \le r < 1$ and $A_r(x) \to s$ as $r \to 1^-$.
This is [[Abel summation|EZXPY4A19C]] of the symmetric series at $x$.

## Remarks

If $f$ is a [[Riemann integrable|0N1JQXZXHC]] $2\pi$-[[periodic function|089KWVYD69]] and $c_k = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-ikt} \, dt$ are its [[Fourier coefficients|G12KXDHPPS]], then the $c_k$ are bounded, $A_r(x)$ converges absolutely, and
$$A_r(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} P_r(x - t) f(t) \, dt, \qquad P_r(\theta) = \sum_{k \in \Z} r^{\abs{k}} e^{ik\theta} = \frac{1 - r^2}{1 - 2r\cos\theta + r^2}.$$
Here $P_r$ is the [[Poisson kernel|TCYSWCTYYM]] of the unit disc.

Example: for $f(x) = \cos x$ one has $c_1 = c_{-1} = \frac12$ and $c_k = 0$ otherwise, so $A_r(x) = r \cos x \to \cos x$.

Non-example: for $c_k = 1$ for all $k$ and $x = 0$, one gets $A_r(0) = \sum_{k \in \Z} r^{\abs{k}} = \frac{1 + r}{1 - r} \to +\infty$.
So the series $\sum_{k \in \Z} e^{ikx}$ is not Abel–Poisson summable at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/6X9S10E3M2.md" tag="6X9S10E3M2" kind="conjecture">
# Hadamard conjecture

## Statement

For each positive integer $k$, there is a [[Hadamard matrix|DM59GHSMAJ]] of order $4k$.

## Remarks

A Hadamard matrix of order $n > 2$ can exist only when $n$ is a multiple of $4$, so the conjecture says that this necessary condition is also sufficient.
The conjecture is open.
Sylvester's construction by [[Kronecker products|0JK6M18QZH]] gives all orders $2^m$.
Paley's construction uses [[quadratic residues|0SQDB87Z3T]] in a [[finite field|08RTDHX1SK]] $\F_q$ and gives order $q + 1$ when $q \equiv 3 \pmod 4$ is a prime power.
Combining such constructions covers many multiples of $4$; the smallest multiple of $4$ for which no Hadamard matrix is known is $668$.

## Explanations

### L1 human

A Hadamard matrix is a square table of plus and minus signs in which any two rows agree in exactly half of the places.
Simple counting shows that the size must be a multiple of $4$, apart from sizes $1$ and $2$.
The conjecture says that every multiple of $4$ really is possible.
</document>

<document source="https://tamnd.github.io/math2/node/6XM4CCVTKE.md" tag="6XM4CCVTKE" kind="concept">
# Normal closure

## Summary

The normal closure of an algebraic extension $L/K$ is the smallest extension of $L$ that is [[normal|08S0NW8SES]] over $K$.
It is obtained by adjoining to $L$ all the roots of the [[minimal polynomials|0VX9EP2GS8]] over $K$ of the elements of $L$.
It exists and is unique up to an isomorphism over $L$.
When $L/K$ is finite and separable, its normal closure is the smallest [[Galois extension|0QMZAQ6QQ4]] of $K$ containing $L$.
This lets Galois theory study extensions that are not themselves Galois.

## Explanations

### L1 human

Some extensions contain one root of a polynomial but not the others.
The normal closure adds the missing roots, and nothing more.
For example, $\Q(\sqrt[3]{2})$ contains the real cube root of $2$ but not the two complex cube roots.
Adding them gives $\Q(\sqrt[3]{2}, \omega)$ with $\omega = e^{2\pi i/3}$, a field of degree $6$ over $\Q$.
This field is the normal closure of $\Q(\sqrt[3]{2})$ over $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/6XQYHACGT0.md" tag="6XQYHACGT0" kind="concept">
# Weighted polynomial approximation

## Summary

Weighted polynomial approximation measures the distance between a function and polynomials with a weight $w$, as in $\sup_x \abs{f(x) - p(x)} w(x)$ or in a weighted $L^p$ norm.
On unbounded sets such as $\R$ the weight must decay fast enough to make polynomials have finite norm.
A classical question, Bernstein's approximation problem, asks for which weights on $\R$ polynomials are dense in the corresponding space of continuous functions.
For the weights $e^{-\abs{x}^\alpha}$ the answer is yes exactly when $\alpha \ge 1$, and [[orthogonal polynomials|8TXASARQN1]] for such weights are a main tool.

## Explanations

### L1 human

On a closed bounded interval, every continuous function is a uniform limit of polynomials, by the [[Weierstrass approximation theorem|0JQE130DHW]].
On the whole real line this fails, because a nonconstant polynomial grows without bound.
A weight $w(x)$ that tends to $0$ quickly at infinity tames this growth, and one only asks that $(f - p) w$ be small.
For example, with $w(x) = e^{-x^2}$ every polynomial $p$ has $p(x) w(x) \to 0$ as $\abs{x} \to \infty$.
How fast the weight decays decides whether polynomials can approximate every function of the weighted space.
</document>

<document source="https://tamnd.github.io/math2/node/6Y0Y9JRXSK.md" tag="6Y0Y9JRXSK" kind="definition">
# Dimension of an algebraic variety

## Definition

Let $X$ be an [[algebraic variety|HV6YBS0YDH]] over an algebraically closed field $k$, with its Zariski topology.
The *dimension* $\dim X$ is the supremum of the integers $n$ such that there is a chain
$$Z_0 \subsetneq Z_1 \subsetneq \cdots \subsetneq Z_n$$
of [[irreducible|SY2MWC434Z]] [[closed subsets|0ZCK8DQYN1]] $Z_i$ of $X$.

## Remarks

For every variety this supremum is finite.
If $X$ is an [[affine variety|JERVQT4C12]], then $\dim X$ is the [[Krull dimension|MVGBS3J1P7]] of its [[coordinate ring|G9WB27VWP7]] $k[X]$.
The reason is that irreducible closed subsets of $X$ correspond to prime ideals of $k[X]$, with inclusions reversed.
For every variety, $\dim X$ equals the [[transcendence degree|D7TR0YWR9H]] of the [[function field|2D5SS6WM14]] $k(X)$ over $k$.
A nonempty open subset $U$ of $X$ has $\dim U = \dim X$.

Example: $\dim \mathbb{A}^n = \dim \mathbb{P}^n = n$, since $k(x_1, \dots, x_n)$ has transcendence degree $n$ over $k$.
The chain $\set{0} \subsetneq V(x_2, \dots, x_n) \subsetneq \cdots \subsetneq V(x_n) \subsetneq \mathbb{A}^n$ of linear subspaces has length $n$.

Example: if $f \in k[x_1, \dots, x_n]$ is irreducible and not constant, the hypersurface $V(f) \subseteq \mathbb{A}^n$ has dimension $n - 1$.

The dimension of a reducible algebraic set is the largest dimension of its irreducible components.
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<document source="https://tamnd.github.io/math2/node/6Y1RY5767R.md" tag="6Y1RY5767R" kind="concept">
# Degenerate kernel

## Summary

A degenerate kernel is a [[kernel|GSMBXT7FPK]] that is a finite sum of products $a_i(x) b_i(s)$ of functions of one variable.
It is also called a separable kernel or a kernel of finite rank, since its integral operator is a [[finite-rank operator|Z8FABSYQTS]].
An integral equation of the second kind with a degenerate kernel reduces to a finite [[system of linear equations|0GT4WWGWH2]].
Approximating a general kernel by degenerate ones is a main route to Fredholm theory and to numerical methods.

## Explanations

### L1 human

A degenerate kernel splits into finitely many pieces, each a function of $x$ times a function of $s$.
Then the integral in the equation only produces finitely many unknown numbers, the integrals $\int b_i \varphi$.
Solving for these numbers is plain linear algebra.
For example, $K(x, s) = 1 + x s$ is degenerate with two terms, while $e^{x s}$ is not degenerate.
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<document source="https://tamnd.github.io/math2/node/6Y451VYFV9.md" tag="6Y451VYFV9" kind="definition">
# Order statistic

## Definition

Let $X_1, \dots, X_n$ be real [[random variables|0DKRR22AEQ]] on a probability space $(\Omega, \mathcal{F}, P)$.
For $\omega \in \Omega$, list the numbers $X_1(\omega), \dots, X_n(\omega)$ in increasing order, with repetitions, as $X_{(1)}(\omega) \le X_{(2)}(\omega) \le \dots \le X_{(n)}(\omega)$.
For $1 \le k \le n$, the random variable $X_{(k)}$ is the *$k$-th order statistic* of the sample.

## Remarks

Each $X_{(k)}$ is a [[statistic|036756NFD6]], since sorting is a continuous map from $\R^n$ to $\R^n$.
Example: if $X_1(\omega) = 4$, $X_2(\omega) = 1$, $X_3(\omega) = 4$, then $X_{(1)}(\omega) = 1$ and $X_{(2)}(\omega) = X_{(3)}(\omega) = 4$.
Non-example: $X_2$ itself is not the second order statistic in general; in the example $X_2(\omega) = 1 \ne 4 = X_{(2)}(\omega)$.
By definition, $X_{(1)} = \min_i X_i$ and $X_{(n)} = \max_i X_i$.
If the $X_i$ are i.i.d. with cumulative distribution function $F$, then $P(X_{(n)} \le t) = F(t)^n$ and $P(X_{(1)} > t) = (1 - F(t))^n$.
If the $X_i$ are i.i.d. uniform on $[0, 1]$, then $X_{(k)}$ has the beta distribution $\mathrm{Beta}(k, n - k + 1)$.
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<document source="https://tamnd.github.io/math2/node/6Y50T68WKY.md" tag="6Y50T68WKY" kind="example">
# Schwarzschild metric

## Example

Fix a mass parameter $m > 0$, in units where the gravitational constant and the speed of light are $1$.
On $M = \R \times (2m, \infty) \times S^2$ with coordinates $(t, r, \theta, \phi)$, the *Schwarzschild metric* is
$$g = -\left(1 - \frac{2m}{r}\right) dt^2 + \left(1 - \frac{2m}{r}\right)^{-1} dr^2 + r^2 \left(d\theta^2 + \sin^2\theta \, d\phi^2\right).$$
Check that $(M, g)$ is a [[Lorentzian manifold|BY4QYYS788]]: for $r > 2m$ the number $1 - 2m/r$ is positive.
So the coefficient of $dt^2$ is negative and the coefficient of $dr^2$ is positive.
The last term is $r^2$ times the round metric of $S^2$, which is positive definite.
Hence $g$ has signature $(-, +, +, +)$ at every point.
A direct computation of the Christoffel symbols shows that the [[Ricci curvature|4GTNM0WHDH]] of $g$ is zero, so $g$ solves the vacuum Einstein equations.
It is static and spherically symmetric, and $g$ tends to the Minkowski metric as $r \to \infty$.

The slice $t = 0$ carries the Riemannian metric $(1 - 2m/r)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2)$, the spatial Schwarzschild metric.
It has zero scalar curvature, and in the coordinate $r$ extended to $r \ge 2m$ its boundary sphere $r = 2m$ is a minimal surface of area $16\pi m^2$.

## Explanations

### L1 human

The Schwarzschild metric describes space and time outside a single round, non-rotating, uncharged mass, such as a star or a black hole.
Far away it looks like flat spacetime, and near the mass time runs slower and space is stretched.
The sphere $r = 2m$ is the event horizon of the black hole: the coordinates above break down there, although the spacetime itself does not.
It was the first exact solution of Einstein's equations, found by Karl Schwarzschild in 1916.
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<document source="https://tamnd.github.io/math2/node/6Y7NJCXXTN.md" tag="6Y7NJCXXTN" kind="definition">
# Generalized convolution of an integral transform

## Definition

Let $V$ be a complex vector space of functions, and let $\Omega$ be a set.
Let $T$ be an [[integral transform|YCK709QB06]], a linear map from $V$ to the space of complex functions on $\Omega$.
A *convolution* of $T$ is a bilinear map $\circledast \colon V \times V \to V$ such that
$$T(f \circledast g)(\omega) = (Tf)(\omega) \, (Tg)(\omega)$$
for all $f, g \in V$ and all $\omega \in \Omega$.

## Remarks

The identity $T(f \circledast g) = Tf \cdot Tg$ is called the factorization identity or convolution theorem of $T$.
If $T$ is injective, then $T$ has at most one convolution, namely $f \circledast g = T^{-1}(Tf \cdot Tg)$, and then $\circledast$ is automatically commutative and associative.
Variants allow a fixed weight function $\gamma$ on $\Omega$, with $T(f \circledast g) = \gamma \cdot Tf \cdot Tg$, or mix several transforms.

Example: for the Mellin transform $(\mathcal{M} f)(s) = \int_0^\infty x^{s-1} f(x) \, dx$, the [[Mellin convolution|A2PMJG7J68]]
$$(f \circledast g)(x) = \int_0^\infty f(x/y) \, g(y) \, \frac{dy}{y}$$
is a convolution.
Indeed, the substitution $x = uy$ gives $\int_0^\infty x^{s-1} \int_0^\infty f(x/y) g(y) \frac{dy}{y} dx = \int_0^\infty u^{s-1} f(u) \, du \int_0^\infty y^{s-1} g(y) \, dy$ when the integrals converge absolutely.
The [[convolution on the line|ERP5737VZ1]] is a convolution of the Fourier transform, and the [[convolution on the half-line|FCKX8KFVGN]] is one of the Laplace transform.

Non-example: the pointwise product $fg$ is not a convolution of the Fourier transform $\hat f(\xi) = \int_\R f(x) e^{-i x \xi} \, dx$ on $L^1(\R) \cap L^2(\R)$.
For $f = g = \mathbf{1}_{[0,1]}$, the product is $\mathbf{1}_{[0,1]}$ with $\hat{f}(0) = 1$, but the factorization identity would need $\widehat{fg}(\xi) = \hat f(\xi)^2$, and $\hat f(\xi)^2 \ne \hat f(\xi)$ at $\xi = \pi$, where $\hat f(\pi) = 2/(i\pi) \ne 0$.
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<document source="https://tamnd.github.io/math2/node/6YCKECNR3T.md" tag="6YCKECNR3T" kind="definition">
# Rees algebra

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $t$ be a variable, with $I^0 = R$.
The *Rees algebra* of $I$ is the [[subring|0GNCRPYXYV]] of the [[polynomial ring|0SDC934R52]] $R[t]$
$$R[It] = \bigoplus_{n \ge 0} I^n t^n = \set{\textstyle\sum_{n} a_n t^n : a_n \in I^n \text{ for all } n}.$$

## Remarks

It is a subring because $1 \in I^0$ and $I^m I^n \subseteq I^{m+n}$, and it is a [[graded ring|8KB4SQGC0C]] with degree $n$ part $I^n t^n$.
If $I$ is generated by $a_1, \dots, a_s$, then $R[It]$ is generated as an $R$-algebra by $a_1 t, \dots, a_s t$; so if $R$ is Noetherian, then $R[It]$ is Noetherian.
Example: for $R = k[x]$, with $k$ a field, and $I = (x)$, one gets $R[It] = k[x, xt]$, and $x$ and $xt$ are algebraically independent, so this is a polynomial ring in two variables.
Non-example: for a proper ideal $I$, the whole ring $R[t]$ is not $R[It]$, because $t$ lies in $R[t]$ but its coefficient $1$ does not lie in $I^1 = I$.
The quotient $R[It]/I R[It]$ is the [[associated graded ring|8HFAEGFQT1]] $\bigoplus_{n} I^n/I^{n+1}$, and the extended Rees algebra $R[It, t^{-1}]$ is used as well.
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<document source="https://tamnd.github.io/math2/node/6YD7S3C7W7.md" tag="6YD7S3C7W7" kind="definition">
# p-adic gamma function

## Definition

Let $p$ be a prime number and let $\Z_p$ be the ring of [[p-adic integers|V6TJAFZ4MD]].
For a positive integer $n$, put
$$\Gamma_p(n) = (-1)^n \prod_{\substack{0 < i < n \\ p \nmid i}} i.$$
The *$p$-adic gamma function* is the unique continuous function $\Gamma_p \colon \Z_p \to \Z_p$ that takes these values at the positive integers.

## Remarks

The positive integers are dense in $\Z_p$, so such a continuous extension is unique.
It exists because $\Gamma_p(m)$ and $\Gamma_p(n)$ are $p$-adically close when $m$ and $n$ are; the proof uses a generalisation of Wilson's theorem.
The values of $\Gamma_p$ are units of $\Z_p$.

Example: for $p = 3$, $\Gamma_3(1) = -1$, $\Gamma_3(2) = 1$, $\Gamma_3(3) = -2$, $\Gamma_3(4) = 2$ and $\Gamma_3(5) = -8$.
Each value follows from the previous one by the rule
$$\Gamma_p(x + 1) = \begin{cases} -x\, \Gamma_p(x) & \text{if } x \in \Z_p^\times, \\ -\Gamma_p(x) & \text{if } x \in p\Z_p, \end{cases}$$
which holds for all $x \in \Z_p$; here $\Gamma_p(0) = 1$.

Non-example: the [[factorial|0XWG0JGDFW]] $n \mapsto n!$ has no continuous extension to $\Z_p$.
Indeed $1 + p^k \to 1$ in $\Z_p$ as $k \to \infty$, but $(1 + p^k)! = (1 + p^k) \cdot (p^k)!$ is divisible by $p^k$ and so tends to $0$, while $1! = 1$.
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<document source="https://tamnd.github.io/math2/node/6YDMMN6M9R.md" tag="6YDMMN6M9R" kind="definition">
# Barrelled space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a Hausdorff [[locally convex space|B6GY2KJMWW]] over $\mathbb{K}$.
A *barrel* in $E$ is a subset $T \subseteq E$ that is [[closed|0ZCK8DQYN1]], [[convex|02AS45T3H9]], [[balanced|2TDSHJJYS4]] and [[absorbing|NM1BCK3ZX9]].
$E$ is *barrelled* if every barrel in $E$ is a [[neighbourhood|0ZCY0Z80B6]] of $0$.

## Remarks

Some authors define barrelled spaces among all topological vector spaces, with the same condition.
Example: every Banach space $E$ is barrelled.
If $T$ is a barrel, then $E = \bigcup_{n \ge 1} nT$ since $T$ is absorbing.
By the Baire category theorem some $nT$, hence $T$, has an interior point.
Convexity and balance then give that $T$ contains a ball around $0$.
The same argument shows that every Fréchet space is barrelled.
Non-example: let $c_{00}$ be the space of finitely supported real sequences with the norm $\norm{x} = \sup_n \abs{x_n}$, and let $T = \set{x : \abs{x_n} \le 1/n \text{ for all } n}$.
$T$ is a barrel, but it is not a neighbourhood of $0$: for each $\varepsilon > 0$ the vector $\varepsilon e_n$ has norm $\varepsilon$ and lies outside $T$ once $n > 1 / \varepsilon$.
In a barrelled space, every family of continuous linear maps into a locally convex space that is bounded at each point is equicontinuous.
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<document source="https://tamnd.github.io/math2/node/6YJ77MYM0Q.md" tag="6YJ77MYM0Q" kind="definition">
# Topological monoid

## Definition

A *topological monoid* is a [[monoid|0N6R902FG1]] $M$ together with a [[topology|0CQHFPGJ45]] on $M$ such that the multiplication $M \times M \to M$, $(x, y) \mapsto xy$, is a [[continuous multiplication|ZZK2GM1V15]].
That is, the map is continuous when $M \times M$ carries the product topology.

## Remarks

Example: $(\R, \cdot)$ with the usual topology.
Multiplication is associative with identity $1$, and $(x, y) \mapsto xy$ is continuous on $\R^2$.
Non-example: the monoid $(\Z, +)$ with the cofinite topology.
It has identity $0$, but addition is not jointly continuous, because every nonempty open set $U \times V$ of $\Z \times \Z$ contains a pair $(x, -x)$, so the preimage of $\Z \setminus \set{0}$ is not open.
No condition on the identity element is needed beyond its existence.
A topological monoid in which every element is invertible and inversion is continuous is a topological group.
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<document source="https://tamnd.github.io/math2/node/6YSVDFEM39.md" tag="6YSVDFEM39" kind="concept">
# Lauricella hypergeometric series

## Summary

The Lauricella series $F_A^{(n)}, F_B^{(n)}, F_C^{(n)}, F_D^{(n)}$ are four multiple power series in $n$ variables that extend the [[Appell series|QG9VDK1HN9]] from two variables to any number of variables.
For $n = 2$ they are the Appell series $F_2, F_3, F_4, F_1$, and for $n = 1$ each becomes the Gauss [[hypergeometric series|428M453CR2]].
The series $F_D$ is the most used: it has a one-dimensional Euler-type integral and it describes periods of certain algebraic curves.

## Explanations

### L1 human

The Gauss hypergeometric series has one variable, the Appell series have two, and the Lauricella series allow any number $n$.
The coefficient of $x_1^{m_1} \cdots x_n^{m_n}$ is a ratio of rising factorials that depend either on the single indices $m_i$ or on their total $m_1 + \cdots + m_n$.
For example $F_D$ is what one gets by expanding $\int_0^1 t^{a - 1} (1 - t)^{c - a - 1} \prod_i (1 - x_i t)^{-b_i}\, dt$ in powers of the $x_i$.
Each factor $(1 - x_i t)^{-b_i}$ contributes one variable.
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<document source="https://tamnd.github.io/math2/node/6YT8ZM6B0F.md" tag="6YT8ZM6B0F" kind="statement">
# Abel's lemma for multiple power series

## Statement

Let $\sum_{\alpha \in \N^n} c_\alpha z^\alpha$ be a [[power series in several variables|R3QTCPTJK7]] with centre $0$.
Let $w \in \C^n$ and $M \ge 0$ be such that $\abs{c_\alpha w^\alpha} \le M$ for every $\alpha \in \N^n$.
Let $0 < \rho < 1$ and $K_\rho = \set{z \in \C^n : \abs{z_j} \le \rho \abs{w_j} \text{ for } j = 1, \dots, n}$.
Then $\sum_\alpha \abs{c_\alpha z^\alpha} \le M (1 - \rho)^{-n}$ for every $z \in K_\rho$, and the series converges absolutely and [[uniformly|08VZ57PT26]] on $K_\rho$.

## Remarks

For $z \in K_\rho$ one has $\abs{c_\alpha z^\alpha} \le \abs{c_\alpha w^\alpha} \rho^{\abs{\alpha}} \le M \rho^{\abs{\alpha}}$, and $\sum_{\alpha \in \N^n} \rho^{\abs{\alpha}} = (1 - \rho)^{-n}$ is a product of $n$ geometric series; the Weierstrass M-test concludes.
If all $w_j \ne 0$, every compact subset of the [[polydisc|DJBWP7D3M8]] $\set{\abs{z_j} < \abs{w_j}}$ lies in some $K_\rho$, so the series converges absolutely there, uniformly on compact subsets.
In particular, convergence at one point $w$ with all $w_j \ne 0$ already gives convergence on this whole open polydisc.
Example: for $\sum_{k \ge 0} (z_1 z_2)^k$ and $w = (2, 1/2)$, all terms have modulus $1$, so the series converges on $\set{\abs{z_1} < 2, \abs{z_2} < 1/2}$.
The same series also converges on $\set{\abs{z_1} < 1/2, \abs{z_2} < 2}$, so its domain of convergence $\set{\abs{z_1 z_2} < 1}$ is larger than any single polydisc given by the lemma.

## Explanations

### L1 human

If the terms of a power series stay bounded at one point, the series converges at every point that is strictly smaller in each coordinate.
The reason is that the terms are then dominated by a convergent geometric series.
This is the same idea as for power series in one variable, done coordinate by coordinate.

### L2 human

The lemma is the first step in describing where a power series converges.
It shows that the domain of convergence is a union of polydiscs centred at $0$, hence a complete Reinhardt domain.
Combined with Hölder's inequality it gives the [[logarithmic convexity of the domain of convergence|D8S8JWE45R]].
It also shows that the sum is a uniform limit of polynomials on compact sets, hence holomorphic.
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<document source="https://tamnd.github.io/math2/node/6YYPNKK6WZ.md" tag="6YYPNKK6WZ" kind="statement">
# Laurent phenomenon

## Statement

Let $(\mathbf{x}, B)$ be a seed, with cluster $\mathbf{x} = (x_1, \dots, x_n)$ and a skew-symmetrizable integer matrix $B$, and let $\mathcal{A}(B)$ be the coefficient-free [[cluster algebra|TKK0TV2MYC]] it generates.
Then every cluster variable of $\mathcal{A}(B)$ is a Laurent polynomial with integer coefficients in $x_1, \dots, x_n$, that is, an element of the [[Laurent polynomial ring|QMY5JTZZB8]] $\Z[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$.

## Remarks

The theorem is due to Fomin and Zelevinsky.
Each mutation divides by a cluster variable, so a priori the cluster variables are only rational functions of the initial ones.
The content of the theorem is that all these denominators are monomials.
The proof goes by induction on the length of a mutation sequence, and it uses that the exchange polynomials of neighbouring mutations are coprime.
Example: for $n = 2$ and $B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, the cluster variables satisfy $x_{m+1} x_{m-1} = x_m + 1$.
Starting from $x_1, x_2$ one gets $x_3 = (x_2 + 1)/x_1$, $x_4 = (x_1 + x_2 + 1)/(x_1 x_2)$, $x_5 = (x_1 + 1)/x_2$, and then $x_6 = x_1$, $x_7 = x_2$.
All five are Laurent polynomials in $x_1, x_2$.
The special form of the exchange relations matters: a general rational recurrence produces denominators that are not monomials.

## Explanations

### L1 human

Some recurrences divide at every step, yet the result never has a real denominator.
The Laurent phenomenon says this happens for all the variables of a cluster algebra: only powers of the starting variables ever appear in the denominators.
Setting all starting variables to $1$ then gives integers, which explains why some nonlinear integer sequences stay integral.

### L2 human

The Laurent phenomenon is one of the founding results of the theory of cluster algebras.
Several nonlinear recurrences, such as the Somos-4 and Somos-5 recurrences, can be realized by mutations.
So their terms are Laurent polynomials in the initial values, which explains the integrality of the [[Somos sequences|27K5X7S4XN]] for $k = 4, 5$.
A stronger statement, that the coefficients of these Laurent polynomials are nonnegative, was conjectured by Fomin and Zelevinsky and later proved for skew-symmetric matrices $B$.
</document>

<document source="https://tamnd.github.io/math2/node/6Z33W3D482.md" tag="6Z33W3D482" kind="definition">
# 6-j symbol

## Definition

Let $V_j$ be the irreducible representation of [[SU(2)|E5P8AG6CK7]] of dimension $2j + 1$, and let $j_1, j_2, j_3, J$ be spins.
Coupling uses the Clebsch–Gordan coefficients of the Condon–Shortley convention, as in the definition of the [[3-j symbol|YJRJVZ998D]].
In the [[tensor product|0BF2DE2SYK]] $V_{j_1} \otimes V_{j_2} \otimes V_{j_3}$, let $\lvert (j_1 j_2) j_{12}, j_3; J M \rangle$ be the orthonormal vectors obtained by coupling $j_1$ and $j_2$ to $j_{12}$, then $j_{12}$ and $j_3$ to $J$.
Let $\lvert j_1, (j_2 j_3) j_{23}; J M \rangle$ be obtained by coupling $j_2$ and $j_3$ to $j_{23}$ and then $j_1$ and $j_{23}$ to $J$.
The *6-j symbol* is the number $\begin{Bmatrix} j_1 & j_2 & j_{12} \\ j_3 & J & j_{23} \end{Bmatrix}$ such that
$$\langle (j_1 j_2) j_{12}, j_3; J M \mid j_1, (j_2 j_3) j_{23}; J M \rangle = (-1)^{j_1 + j_2 + j_3 + J} \sqrt{(2 j_{12} + 1)(2 j_{23} + 1)}\, \begin{Bmatrix} j_1 & j_2 & j_{12} \\ j_3 & J & j_{23} \end{Bmatrix}.$$
The left side does not depend on $M$.
The symbol is set to $0$ when one of the couplings is impossible.

## Remarks

A symbol $\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix}$ can be nonzero only if each of the triads $(j_1, j_2, j_3)$, $(j_1, j_5, j_6)$, $(j_4, j_2, j_6)$, $(j_4, j_5, j_3)$ satisfies the triangle inequality and has an integer sum.
It is unchanged by any permutation of its columns, and by exchanging the upper and lower entries in any two columns.

Example: take $j_3 = 0$, so $J = j_{12} = c$ and $j_{23} = j_2 = b$, with $j_1 = a$.
Coupling with the trivial representation changes nothing, so both vectors equal $\lvert (a b) c; c M \rangle$ and the overlap is $1$.
Hence $\begin{Bmatrix} a & b & c \\ 0 & c & b \end{Bmatrix} = \frac{(-1)^{a + b + c}}{\sqrt{(2b+1)(2c+1)}}$.

Non-example: $\begin{Bmatrix} 1 & 1 & 3 \\ 1 & 1 & 1 \end{Bmatrix} = 0$, because the triad $(1, 1, 3)$ fails the triangle inequality.

The Racah W-coefficient is the same number with another sign and order: $\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix} = (-1)^{j_1 + j_2 + j_4 + j_5} W(j_1 j_2 j_5 j_4; j_3 j_6)$.
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# Laplace transform of distributions

## Summary

The [[Laplace transform|0V49JEZW06]] extends from functions to distributions supported in $[0, \infty)$ that do not grow faster than an exponential.
The extension gives meaning to transforms of generalized functions such as the [[Dirac delta|0KPHENZ5S7]] and its derivatives: $\mathcal{L}\delta = 1$ and $\mathcal{L}\delta^{(k)} = s^k$.
It makes the rule "differentiation becomes multiplication by $s$" exact, with no initial-value terms, which is the natural setting for impulses in operational calculus.

## Explanations

### L1 human

Engineers like to feed a system an instantaneous kick, the Dirac delta, which is not a genuine function.
The ordinary Laplace transform cannot handle it, but the distributional version can, and it gives the delta the transform $1$.
The jump of a switched-on signal is then accounted for by a delta in its derivative.
For example, the unit step has transform $1/s$, and its derivative, the delta, has transform $s \cdot 1/s = 1$.
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<document source="https://tamnd.github.io/math2/node/6ZBNGS0P98.md" tag="6ZBNGS0P98" kind="concept">
# Homogeneous ideal

## Summary

A homogeneous ideal of a [[graded ring|8KB4SQGC0C]] is an [[ideal|096Z9MP2ME]] that contains all homogeneous components of each of its elements.
Equivalently, it is generated by homogeneous elements.
Homogeneous ideals of a polynomial ring are the ideals that define subsets of projective space.
The quotient by a homogeneous ideal is again graded.

## Explanations

### L1 human

A polynomial is a sum of homogeneous parts of different degrees.
An ideal is homogeneous when it never mixes degrees: whenever it contains a polynomial, it contains each homogeneous part separately.
The ideal generated by $x^2 - yz$ in $k[x, y, z]$ is homogeneous, while the ideal generated by $x - 1$ in $k[x]$ is not, since it contains $x - 1$ but not $x$.
The [[definition|3PNJYFPJ3P]] makes this precise.
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<document source="https://tamnd.github.io/math2/node/6ZDP261K7R.md" tag="6ZDP261K7R" kind="concept">
# Tauberian condition

## Summary

A Tauberian condition is an extra hypothesis on the terms of a series that lets one pass from summability back to ordinary convergence.
[[Abel's theorem|0S4EMDQS2F]] says that a convergent series is Abel summable to its sum, but the converse fails.
[[Tauber's theorem|8SK3GQ4S0M]] shows that the converse holds when $n a_n \to 0$, and this condition is the first example of a Tauberian condition.
Such conditions are the subject of [[Tauberian theory|D6XWK1NRY1]].

## Explanations

### L1 human

A [[summability method|8X25Q4XYMN]] can give a value to a series that does not converge, such as $1 - 1 + 1 - 1 + \cdots$.
So knowing that a series is summable does not tell you that it converges.
A Tauberian condition is a promise that the terms are small or tame enough to rule out such wild cases.
For example, if the terms satisfy $n a_n \to 0$ and the series is Abel summable, then it really converges.
The terms of $1 - 1 + 1 - \cdots$ do not shrink at all, so they fail every such condition.
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<document source="https://tamnd.github.io/math2/node/6ZDSVX9GAR.md" tag="6ZDSVX9GAR" kind="statement">
# Besicovitch covering theorem

## Statement

For each integer $n \ge 1$ there is an integer $N_n$, depending only on $n$, with the following property.
Let $\mathcal{F}$ be a family of [[closed balls|05DSV40EB9]] of positive radius in [[Euclidean space|0XPR1E85W0]] $\R^n$ with $\sup \set{\operatorname{diam} B : B \in \mathcal{F}} < \infty$.
Let $A$ be the set of centres of the balls of $\mathcal{F}$.
Then there are subfamilies $\mathcal{G}_1, \dots, \mathcal{G}_{N_n} \subseteq \mathcal{F}$ such that:

1. Each $\mathcal{G}_i$ is a countable family of pairwise disjoint balls.
2. $A \subseteq \bigcup_{i=1}^{N_n} \bigcup_{B \in \mathcal{G}_i} B$.

## Remarks

The proof chooses balls greedily, each time a ball of nearly maximal radius centred at a point not yet covered.
A geometric argument in $\R^n$ shows that each chosen ball meets only a bounded number of earlier chosen balls of comparable or larger size, with a bound depending on $n$.
The chosen balls are then sorted into $N_n$ disjoint subfamilies.

Unlike the Vitali covering lemma, no enlargement of the balls is needed and no measure enters the statement.
So its consequences hold for every Radon measure on $\R^n$, doubling or not.
A consequence concerns a locally finite Borel measure $\mu$ on $\R^n$ and a set $A$.
If every point of $A$ is the centre of balls of $\mathcal{F}$ of arbitrarily small radius, then some countable disjoint subfamily of $\mathcal{F}$ covers $\mu$-almost all of $A$.

The fact that the balls are centred at points of $A$ is essential.
The theorem does not extend to general metric spaces.

## Explanations

### L1 human

Every point of a set is the centre of a ball.
One can choose some of these balls so that they still cover all the centres, and so that no point is in too many of the chosen balls.
The bound on the overlap depends only on the dimension.

### L2 human

The theorem is the covering tool for measures that are not doubling.
It gives the differentiation theorem $\lim_{r \to 0} \nu(B(x,r)) / \mu(B(x,r)) = \frac{d\nu}{d\mu}(x)$ for Radon measures $\mu, \nu$ on $\R^n$ with $\nu \ll \mu$, for $\mu$-almost every $x$.
It is a basic tool of geometric measure theory.
</document>

<document source="https://tamnd.github.io/math2/node/6ZGF49H346.md" tag="6ZGF49H346" kind="definition">
# Hasse–Weil zeta function

## Definition

Let $X$ be a [[scheme|5A35TYZD02]] of finite type over $\Z$, of Krull dimension $d$.
Let $\abs{X}$ be its set of closed points; the residue field $\kappa(x)$ of each $x \in \abs{X}$ is a [[finite field|0ZXZ82FTX2]], and let $N(x) = \#\kappa(x)$.
The *Hasse–Weil zeta function* of $X$ is
$$\zeta_X(s) = \prod_{x \in \abs{X}} \frac{1}{1 - N(x)^{-s}},$$
an Euler product that converges absolutely for $\mathrm{Re}(s) > d$.

Now let $V$ be a smooth projective [[variety|HV6YBS0YDH]] over $\Q$.
Choose a finite set of primes $S$ and a model $X$ of $V$ that is smooth and projective over $\Z[1/S]$.
For $p \notin S$, let $X_p$ be the fibre of $X$ over $\F_p$, and let
$$Z(X_p, t) = \exp\left( \sum_{m \ge 1} \frac{\#X_p(\F_{p^m})}{m} \, t^m \right).$$
The zeta function of $V$ is the product of the local factors $Z(X_p, p^{-s})$ over $p \notin S$.
Changing the model changes only finitely many factors.

## Remarks

The two forms agree: the local factor $Z(X_p, p^{-s})$ is the product of $(1 - N(x)^{-s})^{-1}$ over the closed points $x$ of $X$ lying over $p$.

Example: for $X = \mathrm{Spec}\, \Z$, the closed points are the primes $p$ with $N(p) = p$, so $\zeta_X(s) = \prod_p (1 - p^{-s})^{-1} = \zeta(s)$, the [[Riemann zeta function|1XWBY4N0VT]], which converges for $\mathrm{Re}(s) > 1 = d$.
For $X = \mathbb{A}^1_{\Z}$, one finds $\zeta_X(s) = \zeta(s - 1)$, since $\#\mathbb{A}^1(\F_{p^m}) = p^m$.

Non-example: the [[Dirichlet series|8MWTEF6462]] $\sum_{n \ge 1} (-1)^{n} n^{-s}$ is not of this form, since its coefficients take negative values, while expanding an Euler product of factors $(1 - N(x)^{-s})^{-1}$ gives nonnegative integer coefficients.

For an elliptic curve $E$ over $\Q$ with a good model away from its conductor, the local factor at a good prime $p$ is $\dfrac{1 - a_p p^{-s} + p^{1 - 2s}}{(1 - p^{-s})(1 - p^{1-s})}$.
The Hasse–Weil conjecture predicts meromorphic continuation to $\C$ and a functional equation; it is known for elliptic curves over $\Q$ by the modularity theorem.
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<document source="https://tamnd.github.io/math2/node/6ZMQ1XZ44B.md" tag="6ZMQ1XZ44B" kind="definition">
# Riesz space

## Definition

Let $V$ be a real vector space with a partial order $\le$ such that $(V, \le)$ is an [[ordered vector space|X32V6Y5PYW]].
The space $V$ is a *Riesz space*, or *vector lattice*, if $(V, \le)$ is a [[lattice|WH4SFAACAS]].
That is, any two elements $x, y \in V$ have a least upper bound $x \vee y$ and a greatest lower bound $x \wedge y$.

## Remarks

Example: $\R^n$ with the componentwise order is a Riesz space, with $x \vee y$ the componentwise maximum.
Example: the space $C([0, 1])$ of [[continuous functions|0P27W9RW0G]] with the pointwise order is a Riesz space, since $\max(f, g)$ and $\min(f, g)$ are continuous.
Non-example: on $\R^2$, let $(a, b) \le (c, d)$ when $a \le c$ and $b = d$.
This makes $\R^2$ an ordered vector space, but $(0, 0)$ and $(0, 1)$ have no common upper bound, so it is not a lattice.
In a Riesz space one writes $x^+ = x \vee 0$, $x^- = (-x) \vee 0$ and $\abs{x} = x \vee (-x)$, and then $x = x^+ - x^-$ and $\abs{x} = x^+ + x^-$.
A Riesz space with a complete norm satisfying $\abs{x} \le \abs{y} \implies \norm{x} \le \norm{y}$ is a [[Banach lattice|G4CA22W2XX]].
</document>

<document source="https://tamnd.github.io/math2/node/6ZP4DJ7JMF.md" tag="6ZP4DJ7JMF" kind="example">
# Divergent asymptotic series

## Example

For $x > 0$ let
$$F(x) = \int_0^\infty \frac{e^{-t}}{1 + x t}\,dt.$$
Then, as $x \to 0^+$,
$$F(x) \sim \sum_{n \ge 0} (-1)^n n!\, x^n = 1 - x + 2x^2 - 6x^3 + \cdots,$$
and this series is a [[divergent series|0JWMJSTBSG]] for every $x \ne 0$.

Check of the expansion: for $N \ge 0$ and $t \ge 0$,
$$\frac{1}{1 + x t} = \sum_{n=0}^{N} (-x t)^n + \frac{(-x t)^{N+1}}{1 + x t}.$$
Integrating against $e^{-t}$ and using $\int_0^\infty t^n e^{-t}\,dt = n!$ gives
$$F(x) - \sum_{n=0}^{N} (-1)^n n!\, x^n = R_N(x), \qquad \abs{R_N(x)} \le x^{N+1} \int_0^\infty t^{N+1} e^{-t}\,dt = (N + 1)!\, x^{N+1}.$$
So the error is $O(x^{N+1})$ for each $N$, which is the condition of a [[Poincaré asymptotic series|2HVC5G0H5K]] at $0$.

Check of divergence: for $x \ne 0$ the ratio of consecutive terms is $(n + 1) \abs{x} \to \infty$, so the terms do not tend to $0$.

## Remarks

For a fixed small $x$ the terms $n!\, x^n$ first decrease and then grow, and the smallest one is near $n \approx 1/x$.
Stopping there, which is [[optimal truncation|1YR53GZEPT]], gives an error of order $\sqrt{2\pi/x}\, e^{-1/x}$ by Stirling's formula, which is extremely small for small $x$.
The function $F$ is the Borel sum of the series, and $F(x) = \frac1x e^{1/x} E_1(1/x)$, where $E_1$ is the [[exponential integral|EV6ZDQ0NDY]].

## Explanations

### L1 human

The series $1 - x + 2x^2 - 6x^3 + 24x^4 - \cdots$ has coefficients that grow like factorials, so it diverges for every nonzero $x$.
Yet for $x = 0.1$ its first ten or so terms give $F(0.1)$ to about three decimal places.
Each partial sum is a good approximation; only the infinite sum is meaningless.
This is typical of the series that come out of asymptotic analysis, and it is why one studies asymptotic series and not only convergent ones.
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<document source="https://tamnd.github.io/math2/node/6ZPKWE0VAZ.md" tag="6ZPKWE0VAZ" kind="concept">
# Schwarzian derivative

## Summary

The Schwarzian derivative of a locally injective holomorphic function $f$ is the expression $f'''/f' - \frac{3}{2}(f''/f')^2$.
It vanishes exactly for Möbius transformations and does not change when $f$ is composed on the left with a Möbius transformation.
So it measures how far $f$ is from being a Möbius map.
It appears in univalence criteria, in the theory of linear second order differential equations, and in Teichmüller theory.

## Explanations

### L1 human

The ordinary derivative is zero for constant functions, and the second derivative is zero for linear functions.
The Schwarzian derivative is a third order expression that is zero for all Möbius maps $z \mapsto (az + b)/(cz + d)$.
It ignores any Möbius map applied after $f$, so it only sees the part of $f$ that a Möbius map cannot explain.
For the exponential function it is the constant $-1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/6ZQRVRD8KP.md" tag="6ZQRVRD8KP" kind="definition">
# Volterra integro-differential equation

## Definition

Let $t_0 \in \R$, $T > 0$, $I = [t_0, t_0 + T]$ and $n \ge 1$.
Let $f \colon I \times \R^n \to \R^n$ and $K \colon I \times I \times \R^n \to \R^n$ be continuous, and let $x_0 \in \R^n$.
A *Volterra integro-differential equation* with initial value $x_0$ is the problem of finding a continuously differentiable $x \colon I \to \R^n$ with
$$x'(t) = f\big(t, x(t)\big) + \int_{t_0}^{t} K\big(t, s, x(s)\big) \, ds \quad \text{for } t \in I, \qquad x(t_0) = x_0.$$
It is *linear* when $f(t, x) = A(t) x + b(t)$ and $K(t, s, x) = B(t, s) x$ for continuous matrix functions $A$, $B$ and a continuous $b$.
Equations with higher derivatives of $x$, or with derivatives of $x$ inside the integral, are included in the same name.

## Remarks

Example: $x'(t) = \int_0^t x(s) \, ds$, $x(0) = 1$, with $f = 0$ and $K(t, s, x) = x$.
Differentiating gives $x'' = x$ with $x(0) = 1$ and $x'(0) = 0$, so $x(t) = \cosh t$.
Check: $\int_0^t \cosh s \, ds = \sinh t = x'(t)$.

Non-example: $x'(t) = \int_0^1 x(s) \, ds$ for $t \in [0, 1]$ has an integral over the fixed interval $[0, 1]$, including future times, so it is an integro-differential equation of Fredholm type, not of Volterra type.

Integrating from $t_0$ to $t$ turns the problem into the [[Volterra integral equation|DND8CBEVXT]]
$$x(t) = x_0 + \int_{t_0}^{t} \left[f\big(s, x(s)\big) + \int_{t_0}^{s} K\big(s, r, x(r)\big) \, dr\right] ds.$$
If $f$ and $K$ are Lipschitz in their last variable, uniformly in the others, successive approximations converge on all of $I$ and give a unique solution.
This generalizes the [[initial value problem|0FXDW992DS]] for ordinary differential equations, which is the case $K = 0$.
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<document source="https://tamnd.github.io/math2/node/6ZQYHVGE1S.md" tag="6ZQYHVGE1S" kind="definition">
# Balanced set

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $V$ be a [[vector space|05EGHEQZAW]] over $\mathbb{K}$.
A subset $A \subseteq V$ is *balanced* if $\lambda a \in A$ for every $a \in A$ and every $\lambda \in \mathbb{K}$ with $\abs{\lambda} \le 1$, where $\abs{\lambda}$ is the [[absolute value|00S7C68D51]].

## Remarks

In short, $\lambda A \subseteq A$ whenever $\abs{\lambda} \le 1$.
Balanced sets are also called *circled*.
A set that is both convex and balanced is called *absolutely convex*, or a *disk*.
Example: the closed unit ball $A = \set{x : \norm{x} \le 1}$ of a normed space is balanced, since $\norm{\lambda a} = \abs{\lambda} \norm{a} \le 1$ for $a \in A$ and $\abs{\lambda} \le 1$.
Non-example: in $\R$, the interval $[0, 1]$ is not balanced, since $1 \in [0, 1]$ but $(-1) \cdot 1 \notin [0, 1]$.
Over $\C$ the condition is stronger: the square $Q = \set{a + b i : \abs{a} \le 1, \abs{b} \le 1}$ is balanced in the real vector space $\C = \R^2$, but not in the complex vector space $\C$, since $1 + i \in Q$ while $e^{i \pi / 4} (1 + i) = \sqrt{2} \, i \notin Q$.
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<document source="https://tamnd.github.io/math2/node/6ZRRH96VZT.md" tag="6ZRRH96VZT" kind="statement">
# Nine lemma

## Statement

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], such as the category of modules over a ring.
Consider a [[commutative diagram|WV898P4X16]]
$$\begin{array}{ccccccccc} & & 0 & & 0 & & 0 & & \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & A_1 & \to & B_1 & \to & C_1 & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & A_2 & \to & B_2 & \to & C_2 & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 & \to & A_3 & \to & B_3 & \to & C_3 & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ & & 0 & & 0 & & 0 & & \end{array}$$
in $\mathcal{A}$ in which the three columns are [[short exact sequences|KJKNJMGKZZ]].

1. If the second and third rows are short exact, then the first row is short exact.
2. If the first and second rows are short exact, then the third row is short exact.

## Remarks

For part 1, apply the [[snake lemma|Q8AYGD93CW]] to the map from the second row to the third row given by the vertical morphisms.
Since the columns are exact, the kernels of these vertical morphisms are $A_1, B_1, C_1$ and their cokernels are $0$.
The exact sequence of the snake lemma becomes $0 \to A_1 \to B_1 \to C_1 \to 0$, which is the first row.
Part 2 is the dual statement and follows in the same way from the map from the first row to the second row, whose cokernels are $A_3, B_3, C_3$.
Example: let $A \subseteq B \subseteq C$ be modules.
Take the rows $0 \to A \to A \to 0 \to 0$, then $0 \to B \to C \to C/B \to 0$, then $0 \to B/A \to C/A \to C/B \to 0$, with the evident maps.
The columns $0 \to A \to B \to B/A \to 0$, $0 \to A \to C \to C/A \to 0$ and $0 \to 0 \to C/B \to C/B \to 0$ are short exact, and so are the first two rows.
Part 2 gives that the third row is short exact, which is the third isomorphism theorem $(C/A)/(B/A) \cong C/B$.

## Explanations

### L1 human

The nine lemma is about a three-by-three grid of objects in which every column is a short exact sequence.
If two of the rows are short exact as well, either the bottom two or the top two, then the remaining row is short exact too.
It is a bookkeeping tool: exactness in two directions propagates through the grid.

### L2 human

The lemma is also called the $3 \times 3$ lemma.
It is a quick consequence of the snake lemma, and it is used to compare filtrations and to build short exact sequences of quotients.
A typical use is the third isomorphism theorem in an abelian category.
It also appears when one shows that constructions such as the mapping cone or the total complex respect short exact sequences.
</document>

<document source="https://tamnd.github.io/math2/node/6ZS77J1873.md" tag="6ZS77J1873" kind="definition">
# Product

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $(X_i)_{i \in I}$ be a family of objects of $\mathcal{C}$ indexed by a set $I$.
A *product* of the family is an object $P$ with [[morphisms|859F0TFVA3]] $\pi_i \colon P \to X_i$ for $i \in I$, called *projections*, with the following property.
For every object $Y$ and every family of morphisms $f_i \colon Y \to X_i$, there is exactly one morphism $f \colon Y \to P$ with $\pi_i \circ f = f_i$ for all $i$.
The object $P$ is written $\prod_{i \in I} X_i$, or $X_1 \times X_2$ for two objects, and $f$ is written $(f_i)_{i \in I}$ or $\langle f_1, f_2 \rangle$.

## Remarks

A product is the [[limit|CHS5AFJGWE]] of the diagram whose shape is the set $I$ seen as a category with only identity morphisms.
So it is unique up to unique isomorphism, and the product of the empty family is a terminal object.

Example: in the category of sets, the [[Cartesian product|09CEDJYBKP]] $X_1 \times X_2$ with the coordinate maps is a product.
Given $f_1 \colon Y \to X_1$ and $f_2 \colon Y \to X_2$, the map $f(y) = (f_1(y), f_2(y))$ satisfies $\pi_i \circ f = f_i$, and any such $f$ must have these coordinates.
In the category of groups the product is the [[direct product|00HF4RJ5QN]], and in the category of topological spaces it is the Cartesian product with the product topology.

Non-example: in the category of fields, $\mathbb{F}_2$ and $\mathbb{F}_3$ have no product.
Field homomorphisms preserve the characteristic, so no field $P$ has morphisms to both $\mathbb{F}_2$ and $\mathbb{F}_3$.
</document>

<document source="https://tamnd.github.io/math2/node/702SNFQG0W.md" tag="702SNFQG0W" kind="concept">
# Orientability

## Summary

A manifold is orientable when one can choose an orientation at every point, varying continuously over the whole manifold.
An orientation is a sense of "clockwise" in dimension $2$, or of "right-handed" in dimension $3$.
The sphere and the torus are orientable; the Möbius strip, the Klein bottle and the real projective plane are not.
Orientability is needed to integrate differential forms of top degree, to define the fundamental class and to state Poincaré duality with integer coefficients.

## Explanations

### L1 human

Draw a small clockwise circle on a surface and slide it around along any closed path.
On a sphere, the circle always comes back clockwise.
On a [[Möbius strip|04AJE0M271]], one trip around the strip brings it back turning the other way.
A surface is orientable when such a reversal never happens, so that "clockwise" has one meaning everywhere.
In the same way, a $3$-dimensional manifold is orientable when a right-handed glove can never be carried around a loop and come back as a left-handed one.
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<document source="https://tamnd.github.io/math2/node/705CC9AZCG.md" tag="705CC9AZCG" kind="definition">
# Seifert fiber space

## Definition

Let $a \ge 1$ and $b$ be coprime integers, let $D^2$ be the closed unit disk and let $\rho \colon D^2 \to D^2$ be the rotation by the angle $2\pi b / a$.
The *fibered solid torus* $T(a, b)$ is the quotient of $D^2 \times [0, 1]$ by $(x, 0) \sim (\rho(x), 1)$, decomposed into the circles that are the images of the segments $\set{x} \times [0, 1]$.
Its central circle, the image of $\set{0} \times [0, 1]$, is called *regular* if $a = 1$ and *exceptional* if $a > 1$.

A *Seifert fiber space* is a closed [[3-manifold|R20TB2KEQQ]] $M$ with a decomposition of $M$ into disjoint circles, the *fibers*, such that the following holds.
Each fiber $C$ has a neighborhood $U$ that is a union of fibers.
There is a homeomorphism from $U$ to some $T(a, b)$ that maps fibers to fibers and $C$ to the central circle.

## Remarks

Example: for a closed surface $S$, the product $S \times S^1$, decomposed into the circles $\set{s} \times S^1$, is a Seifert fiber space.
Each fiber has the neighborhood $B \times S^1$ with $B$ a small disk around $s$, which is $T(1, 0)$.

Non-example: on the $3$-torus $T^3 = \R^3 / \Z^3$, take the decomposition into images of lines parallel to $(1, \sqrt{2}, 0)$.
These images are not circles, because $\sqrt{2}$ is irrational, so the clause that the pieces are circles fails.

In $T(a, b)$ every fiber other than the central one closes up after $a$ trips around, so a compact Seifert fiber space has only finitely many exceptional fibers.
Variants of the definition allow boundary, and, for non-orientable manifolds, fibered solid Klein bottles as neighborhoods.
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<document source="https://tamnd.github.io/math2/node/707DEWCCFX.md" tag="707DEWCCFX" kind="object">
# Gauge group

## Description

Let $G$ be a [[Lie group|7JBRGCVJRB]] and $P \to X$ a smooth [[principal $G$-bundle|8E37E74XVP]] over a smooth manifold $X$, with $G$ acting on $P$ on the right.
The *gauge group* $\mathcal{G}(P)$ is the set of diffeomorphisms $u \colon P \to P$ such that $\pi \circ u = \pi$ and $u(p g) = u(p) g$ for all $p \in P$ and $g \in G$, with composition as product.

It is a group: the identity map has both properties, and so does the composite of two such maps.
The inverse of such a map $u$ commutes with $\pi$ and with the right action, because $u$ does.

## Remarks

Equivalent forms: a gauge transformation $u$ is the same as a $G$-equivariant map $\sigma \colon P \to G$, with $\sigma(p g) = g^{-1} \sigma(p) g$, through $u(p) = p \, \sigma(p)$.
So $\mathcal{G}(P)$ is the group of sections of the bundle $P \times_G G$ associated with the conjugation action.
If $P = X \times G$ is trivial, then $\mathcal{G}(P)$ is the group $C^\infty(X, G)$ of smooth maps with pointwise product.

The gauge group acts on the space of [[connections|NJ21PN5ZGJ]] on $P$ by pullback.
Take a trivial bundle and a matrix group $G$, and write a connection as a $\mathfrak{g}$-valued $1$-form $A$.
Then a map $g \colon X \to G$ sends $A$ to $g^{-1} A g + g^{-1} dg$, and the [[curvature|ZAJFW1S078]] changes to $g^{-1} F_A g$.


## Explanations

### L1 human

A principal bundle attaches a copy of the group $G$ to each point of $X$, with no preferred identity element in each copy.
A gauge transformation changes the identification in each copy independently, point by point.
Physical quantities, such as the curvature of a connection up to conjugation, do not depend on these choices, so the gauge group is a symmetry group of gauge theory.
For a trivial bundle over $X$ it is simply the group of smooth maps from $X$ to $G$.
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<document source="https://tamnd.github.io/math2/node/709T196XCT.md" tag="709T196XCT" kind="concept">
# James–Stein estimator

## Summary

The James–Stein estimator estimates the mean vector of a multivariate normal distribution in dimension $p \ge 3$ by shrinking the observation toward a fixed point.
Under squared error loss its risk is strictly smaller than that of the observation itself, for every value of the mean.
It is the standard illustration of [[Stein's example|7SC9E65XDN]] and the prototype of [[shrinkage estimators|GPBSS1V3SS]].

## Explanations

### L1 human

Suppose one observes a noisy vector $X$ whose average is the unknown vector $\theta$.
The James–Stein estimator multiplies $X$ by a factor a bit smaller than $1$, pulling it toward the origin.
The factor is close to $1$ when $X$ is far from the origin, and smaller when $X$ is close to it.
For example, with $p = 3$, unit noise and observation $X = (1, 1, 1)$, the factor is $1 - 1/3$, and the estimate is $(2/3, 2/3, 2/3)$.
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<document source="https://tamnd.github.io/math2/node/70HJTK5VZH.md" tag="70HJTK5VZH" kind="concept">
# Summation of families over arbitrary index sets

## Summary

Let $(x_i)_{i \in I}$ be a family indexed by an arbitrary set $I$.
It is summable when its finite partial sums approach a limit as the finite set of indices grows.
This notion of sum uses no order on $I$, so it works for index sets such as $\N \times \N$, $\Z$ or uncountable sets.
For real or complex terms indexed by $\N$, summability is the same as [[absolute convergence|0YBDSJARWN]].
The exact definition is in [[summable family|Z4H7272S9S]].

## Explanations

### L1 human

An ordinary series adds its terms in a fixed order.
A summable family has a sum that does not depend on any order.
Once a large enough finite batch of terms is added, adding further terms keeps you close to the sum.
For example, the numbers $1/(2^m 3^n)$ for all pairs $(m, n)$ of natural numbers form a summable family with sum $3$.
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# Hurewicz fibration

## Definition

Let $E$ and $B$ be [[topological spaces|0CQHFPGJ45]] and let $p \colon E \to B$ be a [[continuous map|0VGFHTM75Y]].
Write $I = [0, 1]$.
The map $p$ is a *Hurewicz fibration* if it has the [[homotopy lifting property|G4QDM01P6J]] with respect to every topological space $X$.
This means the following.
Let $H \colon X \times I \to B$ be a [[homotopy|9YA6S6HD9V]] and let $\tilde{h}_0 \colon X \to E$ be continuous with $p \circ \tilde{h}_0 = H(-, 0)$.
Then there is a homotopy $\tilde{H} \colon X \times I \to E$ with
$$p \circ \tilde{H} = H \quad \text{and} \quad \tilde{H}(-, 0) = \tilde{h}_0.$$

## Remarks

Example: for spaces $B$ and $F$, the projection $p \colon B \times F \to B$ is a Hurewicz fibration.
Given $H$ and $\tilde{h}_0 = (H(-, 0), g)$, the lift $\tilde{H}(x, t) = (H(x, t), g(x))$ works.

Non-example: the inclusion $p \colon \set{0} \to I$ is not a fibration.
Take $X$ a point, $H(t) = t$ and $\tilde{h}_0 = 0$: a lift would satisfy $p(\tilde{H}(1)) = 1$, but $1$ is not in the image of $p$.

Covering maps and fibre bundles over paracompact Hausdorff bases are Hurewicz fibrations.
If $p$ is a Hurewicz fibration and $B$ is path-connected, all fibres $p^{-1}(b)$ are homotopy equivalent.
Every Hurewicz fibration is a [[Serre fibration|PB9JXQ2KC3]], where the lifting property is only required for $X$ a cube $I^n$ or a CW complex, but not conversely.
</document>

<document source="https://tamnd.github.io/math2/node/70M1VAPFN0.md" tag="70M1VAPFN0" kind="concept">
# Convergence of filters

## Summary

A [[filter|15PF73BKY6]] on a [[topological space|0CQHFPGJ45]] converges to a point $x$ when it contains every [[neighbourhood|0ZCY0Z80B6]] of $x$.
This notion extends the [[convergence of sequences|0W4CN9E2W5]] to all spaces.
Closures, [[continuity|0VGFHTM75Y]], [[compactness|0TVJZYB6SH]] and the [[Hausdorff property|0S1FJ8VEX9]] can all be described by convergent filters.

## Explanations

### L1 human

A filter is a family of "big" sets, and it converges to $x$ when every set around $x$, however small, counts as big.
For a sequence $x_1, x_2, \dots$, take as big sets those that contain all terms from some index on.
This filter converges to $x$ exactly when the sequence converges to $x$.
In spaces where sequences do not see the whole topology, filters still do.
</document>

<document source="https://tamnd.github.io/math2/node/70Q4YVF322.md" tag="70Q4YVF322" kind="definition">
# Building

## Definition

Let $(W, S)$ be a [[Coxeter system|G6FND7X69J]].
For $J \subseteq S$, let $W_J$ be the subgroup of $W$ generated by $J$.
The Coxeter complex $\Sigma(W, S)$ is the [[simplicial complex|V8GZ0W1ZZ5]] whose simplices are the cosets $w W_J$ with $w \in W$ and $J \subseteq S$.
A simplex $w W_J$ is a face of $w' W_{J'}$ when $w W_J \supseteq w' W_{J'}$.
A *building* is a simplicial complex $\Delta$ together with a family of subcomplexes, called *apartments*, such that:

1. Each apartment is isomorphic to a Coxeter complex.
2. For any two simplices $A$ and $B$ of $\Delta$, there is an apartment that contains both.
3. If two apartments $\Sigma$ and $\Sigma'$ both contain the simplices $A$ and $B$, then some isomorphism $\Sigma \to \Sigma'$ fixes every vertex of $A$ and of $B$.

The maximal simplices are the *chambers*.
The building is *thick* if each face of codimension one of a chamber lies in at least three chambers.

## Remarks

Example: let $\Delta$ be a [[tree|062D9KXJB1]] in which every vertex has degree at least $3$, and let the apartments be the bi-infinite paths in $\Delta$.
Each such path is the Coxeter complex of the infinite dihedral group, a line divided into edges.
Clause 2 holds because the path between two edges extends at both ends to a bi-infinite path, since no vertex has degree $1$.
Clause 3 holds because two bi-infinite paths that contain $A$ and $B$ both contain the unique path between them, and a shift or reflection of lines matches them along it.
Each vertex lies in at least three edges, so the building is thick.

Non-example: a finite path with at least one edge, with the same rule for apartments.
It contains no bi-infinite path, so clause 2 fails for every pair of edges.

For a field $F$, the projective line over $F$, as a set of points with all two-point subsets as apartments, is a thick building of rank $1$ for $\mathrm{GL}_2(F)$.
</document>

<document source="https://tamnd.github.io/math2/node/70QWWYQQG9.md" tag="70QWWYQQG9" kind="example">
# Jónsson–Tarski algebra

## Example

A *Jónsson–Tarski algebra* is an [[algebra|QJGPT2W0FT]] $(A, \cdot, L, R)$ with a binary operation $\cdot$ and two [[unary operations|Q2W26ZY10B]] $L$ and $R$ such that, for all $x, y \in A$:

1. $L(x \cdot y) = x$;
2. $R(x \cdot y) = y$;
3. $L(x) \cdot R(x) = x$.

These laws say exactly that $(x, y) \mapsto x \cdot y$ is a [[bijection|0GG9TVW90M]] from $A \times A$ to $A$, with inverse $x \mapsto (L(x), R(x))$.
Since the laws are identities, Jónsson–Tarski algebras form a [[variety|AJNZJD8442]].

A concrete instance is the set $A$ of infinite binary sequences $s = (s_0, s_1, s_2, \dots)$.
Let $s \cdot t = (s_0, t_0, s_1, t_1, \dots)$ interleave the two sequences.
Let $L(s) = (s_0, s_2, s_4, \dots)$ and $R(s) = (s_1, s_3, s_5, \dots)$.
Then $L(s \cdot t) = s$ and $R(s \cdot t) = t$ because the even places of $s \cdot t$ hold $s$ and the odd places hold $t$.
Also $L(s) \cdot R(s) = s$ because interleaving the even and odd places rebuilds $s$.

## Remarks

A Jónsson–Tarski algebra with more than one element is infinite.
Indeed, if $A$ is finite with $n$ elements, a bijection $A \times A \to A$ forces $n^2 = n$, so $n \le 1$.
Jónsson and Tarski showed that in this variety the [[free algebra|GTS4DRRMZR]] on one generator is [[isomorphic|PZMGQMDN25]] to the free algebra on $n$ generators for every finite $n \ge 1$.
So the number of free generators is not an invariant here, unlike for vector spaces.

## Explanations

### L1 human

A Jónsson–Tarski algebra is a set with a perfect way to pack two elements into one and unpack them again.
Packing $x$ and $y$ gives $x \cdot y$, and the maps $L$ and $R$ recover the left and right parts.
Every element arises from exactly one pair in this way.
Infinite binary sequences give an example: shuffle two sequences together like two decks of cards, and unshuffle to get them back.
No finite set with more than one element can do this, since it has more pairs than elements.
</document>

<document source="https://tamnd.github.io/math2/node/70QYCEWHVY.md" tag="70QYCEWHVY" kind="statement">
# Inverse iteration

## Statement

Let $A \in \C^{n \times n}$ be diagonalizable with [[eigenvalues|0BQAA2BK3Q]] $\lambda_1, \dots, \lambda_n$ and a basis of [[eigenvectors|0N6960W0BV]] $v_1, \dots, v_n$, $A v_i = \lambda_i v_i$.
Let $\mu \in \C$ be a shift that is not an eigenvalue, and suppose that there is an index $j$ with
$$\abs{\lambda_j - \mu} < \abs{\lambda_i - \mu} \quad \text{for all } i \text{ with } \lambda_i \ne \lambda_j.$$
Let $x_0 = \sum_i c_i v_i$ with $\sum_{i : \lambda_i = \lambda_j} c_i v_i \ne 0$, and define *inverse iteration* by
$$x_{k+1} = \frac{(A - \mu I)^{-1} x_k}{\norm{(A - \mu I)^{-1} x_k}}.$$
Then the angle between $x_k$ and the eigenspace of $\lambda_j$ tends to $0$, and its sine is $O(\rho^k)$ with
$$\rho = \frac{\abs{\lambda_j - \mu}}{\min_{\lambda_i \ne \lambda_j} \abs{\lambda_i - \mu}} < 1.$$

## Remarks

The matrix $(A - \mu I)^{-1}$ has the same eigenvectors as $A$, with eigenvalues $1/(\lambda_i - \mu)$.
The hypothesis makes $1/(\lambda_j - \mu)$ the eigenvalue of largest modulus, so inverse iteration is [[power iteration|H4MQD841BR]] applied to $(A - \mu I)^{-1}$.
Expanding $(A - \mu I)^{-k} x_0 = \sum_i c_i (\lambda_i - \mu)^{-k} v_i$ and dividing by $(\lambda_j - \mu)^{-k}$ shows that the other components shrink like $\rho^k$.

Example: for $A = \mathrm{diag}(1, 2, 4)$ and $\mu = 2.1$, the ratio is $\rho = 0.1/1.1 \approx 0.09$, so each step gains about one decimal digit toward $e_2$.
If two distinct eigenvalues are equally close to $\mu$, then $\rho = 1$ and the iteration need not converge to an eigenvector.
In practice one factors $A - \mu I = LU$ once and solves a linear system at each step instead of forming the inverse.
The near-singularity of $A - \mu I$ for a good shift does not harm the method: the error in the solve is concentrated in the direction of the wanted eigenvector.

## Explanations

### L1 human

Power iteration finds the eigenvector with the largest eigenvalue.
Inverse iteration finds the eigenvector whose eigenvalue is closest to a chosen number $\mu$, by running power iteration on the inverse of $A - \mu I$.
The closer $\mu$ is to the wanted eigenvalue, the faster it converges.

### L2 human

Inverse iteration is the standard way to compute an eigenvector once a good approximation of its eigenvalue is known.
Such an approximation may come from the [[QR algorithm|B6F8YWHYZ5]] or from bisection on a tridiagonal matrix.
Updating the shift at each step with the Rayleigh quotient gives [[Rayleigh quotient iteration|Y119KHA4VG]], which converges much faster.
The [[rate of convergence|YWZQC5E54B]] is linear with ratio $\rho$, which is why a shift near the eigenvalue matters.
</document>

<document source="https://tamnd.github.io/math2/node/70ZP456XG6.md" tag="70ZP456XG6" kind="concept">
# Virtual knot

## Summary

A virtual knot is an equivalence class of knot diagrams that may have, besides ordinary crossings, virtual crossings that are not really there.
Louis Kauffman introduced them as a generalization of classical [[knots|QYEAZTYGPT]].
Geometrically, virtual knots are knots in thickened surfaces, taken up to a suitable stabilization.
Classical knots form a part of virtual knot theory: two classical knots that are equivalent as virtual knots are equivalent as classical knots.

## Explanations

### L1 human

Some knot drawings cannot be made without extra crossings that come only from the flat paper.
A virtual crossing is such a fake crossing, drawn with a small circle around it.
Virtual knots allow these crossings and treat them as invisible: strands may slide freely across them.
They can be pictured as knots drawn on a surface with handles, such as a torus, where the fake crossings disappear.
</document>

<document source="https://tamnd.github.io/math2/node/710VV73909.md" tag="710VV73909" kind="concept">
# Kernel regression

## Summary

Kernel regression estimates the regression function $m(x) = E[Y \mid X = x]$ without assuming a parametric form for it.
The estimate at $x$ is a weighted average of the observed responses, with weights given by a kernel that favours observations with $X_i$ near $x$.
Its standard form is the Nadaraya–Watson estimator, a regression analogue of [[kernel density estimation|ZV65XMJ9A4]].

## Explanations

### L1 human

To guess the typical value of $Y$ at a point $x$, look at the data points whose $X$ is close to $x$ and average their $Y$ values.
Closer points get more weight, and the bandwidth decides how far "close" reaches.
For example, to estimate the typical weight of people of height $170$ cm, one averages the weights of people of nearby heights.
The result is a smooth curve through the cloud of data points.
</document>

<document source="https://tamnd.github.io/math2/node/71346N77AX.md" tag="71346N77AX" kind="definition">
# Entropic algebra

## Definition

Let $A$ be an [[algebra|QJGPT2W0FT]].
An $n$-ary operation $f$ and an $m$-ary operation $g$ of $A$ *commute* if
$$f\bigl(g(a_{11}, \dots, a_{1m}), \dots, g(a_{n1}, \dots, a_{nm})\bigr) = g\bigl(f(a_{11}, \dots, a_{n1}), \dots, f(a_{1m}, \dots, a_{nm})\bigr)$$
for all elements $a_{ij} \in A$, $1 \le i \le n$, $1 \le j \le m$.
The algebra $A$ is *entropic* if every two of its operations commute, including each operation with itself.

## Remarks

Equivalently, $A$ is entropic if and only if each $n$-ary operation $f \colon A^n \to A$ is a [[homomorphism|KBXSZC6BV1]] from the [[direct power|WJ73XYBNE4]] $A^n$ to $A$.
For an algebra with a single binary operation, the condition for $f = g$ is exactly the [[medial law|VZHN6GA3TA]] $(xy)(uv) = (xu)(yv)$.

Example: a module $M$ over a commutative ring $R$, with addition, the constant $0$, negation and one unary operation $x \mapsto rx$ for each $r \in R$.
Addition commutes with itself because $(a + b) + (c + d) = (a + c) + (b + d)$.
Scalar multiplication commutes with addition because $r(a + b) = ra + rb$, and two scalars commute because $r(sa) = (rs)a = (sr)a = s(ra)$.
The checks for $0$ and negation, such as $r0 = 0$ and $-(a + b) = (-a) + (-b)$, are similar.

Non-example: a nonabelian group $G$ under multiplication.
Multiplication commutes with itself only if $(xy)(uv) = (xu)(yv)$ for all $x, y, u, v$; with $x = v = 1$ this gives $yu = uy$, which fails for some $y, u \in G$.
</document>

<document source="https://tamnd.github.io/math2/node/713MRSZ9XA.md" tag="713MRSZ9XA" kind="concept">
# Ginzburg–Landau equation

## Summary

The Ginzburg–Landau equations describe a superconductor through a complex order parameter $\psi$ and a magnetic vector potential $A$.
They are the Euler–Lagrange equations of the Ginzburg–Landau energy, which penalises variations of $\psi$, deviations of $\abs{\psi}$ from $1$, and magnetic energy.
The quantity $\abs{\psi}^2$ measures the density of superconducting electrons.
Their solutions include vortices, points or lines around which $\psi$ winds and where $\abs{\psi}$ vanishes.

## Explanations

### L1 human

Below a critical temperature some metals conduct electricity with no resistance and push out magnetic fields.
Ginzburg and Landau described this state by a complex number $\psi$ at each point.
Where $\abs{\psi}$ is near $1$ the metal is superconducting, and where it is $0$ the metal is normal.
The energy prefers $\abs{\psi} = 1$, prefers $\psi$ to vary slowly, and charges for magnetic field.
The two simplest solutions are the fully superconducting state $\psi = 1$ with no field and the normal state $\psi = 0$ with a uniform field.
In between, a magnetic field can enter through thin vortex tubes.
</document>

<document source="https://tamnd.github.io/math2/node/7174H28FT6.md" tag="7174H28FT6" kind="definition">
# Dirichlet process

## Definition

Let $(X, \mathcal{A})$ be a measurable space, where $\mathcal{A}$ is a [[sigma-algebra|04WSW8T3SY]] on $X$.
Let $H$ be a [[probability measure|0VGMEBVW1M]] on $(X, \mathcal{A})$ and let $\alpha > 0$ be a real number.
A random probability measure $G$ on $(X, \mathcal{A})$ is a *Dirichlet process* with concentration parameter $\alpha$ and base measure $H$ if the following holds.
For every finite partition $A_1, \ldots, A_r$ of $X$ into sets of $\mathcal{A}$, the random vector $(G(A_1), \ldots, G(A_r))$ has the [[Dirichlet distribution|RX4CVSSRRJ]] with parameters $(\alpha H(A_1), \ldots, \alpha H(A_r))$.
Here a parameter equal to $0$ means that the corresponding coordinate is $0$ almost surely.
One writes $G \sim \mathrm{DP}(\alpha, H)$.

## Remarks

Example: take $A \in \mathcal{A}$ with $0 < H(A) < 1$, and the partition $A, X \setminus A$.
Then $G(A)$ has the [[Beta distribution|0EW824Z7G2]] with parameters $\alpha H(A)$ and $\alpha (1 - H(A))$.
Hence $E[G(A)] = H(A)$ and $\mathrm{Var}(G(A)) = H(A)(1 - H(A))/(\alpha + 1)$.

Non-example: let $H$ be uniform on $[0, 1]$ and let $G = H$ be constant.
Then $G$ is not a Dirichlet process for any $\alpha$.
For $A = [0, 1/2]$ the variance of $G(A)$ is $0$, while a Dirichlet process gives the variance $1/(4(\alpha + 1)) > 0$.

A Dirichlet process exists for every $\alpha$ and $H$ as above.
One construction is stick-breaking.
Let $V_1, V_2, \ldots$ be independent with the Beta distribution with parameters $1$ and $\alpha$, and let $\theta_1, \theta_2, \ldots$ be independent with law $H$ and independent of the $V_i$.
Put $w_k = V_k \prod_{i < k} (1 - V_i)$.
Then $G = \sum_{k \ge 1} w_k \delta_{\theta_k}$ is a Dirichlet process with parameters $\alpha$ and $H$, where $\delta_\theta$ is the [[Dirac measure|099SXAG014]] at $\theta$.
In particular a Dirichlet process is a discrete measure almost surely.

Conjugacy: suppose $G \sim \mathrm{DP}(\alpha, H)$ and, given $G$, the points $x_1, \ldots, x_n$ are independent with law $G$.
Then, given $x_1, \ldots, x_n$, the measure $G$ is a Dirichlet process with concentration $\alpha + n$ and base measure $(\alpha H + \sum_{i=1}^n \delta_{x_i})/(\alpha + n)$.
</document>

<document source="https://tamnd.github.io/math2/node/71AWVBG6KH.md" tag="71AWVBG6KH" kind="concept">
# Alexandrov space

## Summary

An Alexandrov space is a metric space with a lower bound on its curvature, expressed by comparing triangles with triangles of a model surface.
It generalizes a complete Riemannian manifold with sectional curvature bounded below.
For fixed $k$ and a fixed bound on the dimension, the class of compact Alexandrov spaces is closed under [[Gromov–Hausdorff limits|GT56RTG9X5]].
This makes it the natural home of such limits.
Some authors use the name also for spaces with curvature bounded above; here it means a lower bound.

## Explanations

### L1 human

In a space of curvature at least $0$, triangles are at least as fat as flat triangles with the same side lengths.
The surface of a convex body, like a cube or an egg, is a good picture: it has corners, but it curves outward everywhere.
The surface of a cube is an Alexandrov space of curvature at least $0$, although it is not smooth at the corners.
A saddle surface is not, because its triangles are thinner than flat ones.
</document>

<document source="https://tamnd.github.io/math2/node/71DMQHW6JZ.md" tag="71DMQHW6JZ" kind="concept">
# FK-space

## Summary

An FK-space is a [[sequence space|XYTN19EPF2]] with a complete metrizable locally convex topology in which each coordinate $x \mapsto x_n$ is continuous.
The letters stand for Fréchet and the German word Koordinate.
FK-spaces are the main tool of the functional analytic approach to summability: the domain of a matrix summability method is an FK-space.
A sequence space carries at most one FK topology, and inclusions between FK-spaces are automatically continuous.

## Explanations

### L1 human

A sequence space becomes much easier to study once it has a good notion of distance.
In an FK-space the distance is complete, like the distance on the real line, and it controls each term of a sequence.
So if two sequences are close in the space, then their first terms are close, their second terms are close, and so on.
The spaces $\ell^p$ with their usual norms are FK-spaces, and so is the space of all sequences with the topology of termwise convergence.
</document>

<document source="https://tamnd.github.io/math2/node/71K5G4GNNZ.md" tag="71K5G4GNNZ" kind="definition">
# Sparse matrix

## Definition

Let $A$ be an $m \times n$ [[matrix|0VD9ZSWWJG]], and let $\operatorname{nnz}(A)$ be the number of its nonzero entries.
The matrix $A$ is *sparse* when $\operatorname{nnz}(A)$ is small compared with $mn$, so that it pays to store and process only the nonzero entries.
For a family of $n \times n$ matrices with $n$ growing, a common precise form of this condition is $\operatorname{nnz}(A) = O(n)$.

## Remarks

The notion has no fixed threshold; it depends on the size of the matrix and on the algorithms in use.
The *density* of $A$ is $\operatorname{nnz}(A) / (mn)$, and a matrix that is not sparse is called *dense*.
Example: the $n \times n$ [[tridiagonal matrix|DARX1KZRHZ]] with $2$ on the diagonal and $-1$ beside it has $3n - 2$ nonzero entries, which is $O(n)$, so the family is sparse.
Non-example: the $n \times n$ matrix with all entries $1$ has $n^2$ nonzero entries, so it is dense.
Common storage formats are the coordinate list and the compressed sparse row format.
</document>

<document source="https://tamnd.github.io/math2/node/71M4GJKN0N.md" tag="71M4GJKN0N" kind="definition">
# Massey product

## Definition

Let $X$ be a space and let $R$ be a [[commutative ring|0310REJJY8]].
Let $C^*(X; R)$ be the singular cochain complex, with coboundary $\delta$ and [[cup product|JX47MS1R1H]] $\smile$, so that $\delta(x \smile y) = \delta x \smile y + (-1)^{k} x \smile \delta y$ for $x$ of degree $k$.
Let $u \in H^p(X; R)$, $v \in H^q(X; R)$ and $w \in H^r(X; R)$ be classes in [[singular cohomology|6N7BC70CX0]] with $u \smile v = 0$ and $v \smile w = 0$.

1. Choose cocycles $a, b, c$ representing $u, v, w$.
2. Choose cochains $s$ of degree $p + q - 1$ and $t$ of degree $q + r - 1$ with $\delta s = a \smile b$ and $\delta t = b \smile c$.
3. Then $m = s \smile c + (-1)^{p+1} a \smile t$ is a cocycle of degree $p + q + r - 1$.

The *Massey product* $\langle u, v, w \rangle$ is the set of all classes $[m] \in H^{p+q+r-1}(X; R)$ obtained from all such choices.

## Remarks

The cocycle condition is a direct check: $\delta(s \smile c) = a \smile b \smile c$ and $\delta(a \smile t) = (-1)^p a \smile b \smile c$, so $\delta m = 0$.
Changing $s$ or $t$ by a cocycle changes $[m]$ by an element of $H^{p+q-1}(X; R) \smile w$ or of $u \smile H^{q+r-1}(X; R)$.
So $\langle u, v, w \rangle$ is a coset of the subgroup $u \smile H^{q+r-1}(X; R) + H^{p+q-1}(X; R) \smile w$, called the indeterminacy.
The Massey product is said to vanish if it contains $0$.
Sign conventions differ between books; they change the set only by a sign.

Example: if $u = 0$, take $a = 0$ and $s = 0$; then $m = 0$, so $\langle 0, v, w \rangle$ contains $0$ and vanishes.

Non-example: on the torus $T^2$ with degree-one generators $x, y$, the triple $\langle x, y, x \rangle$ is not defined, because $x \smile y \neq 0$, so clause 2 has no solution $s$.

Higher Massey products $\langle u_1, \ldots, u_n \rangle$ are defined in the same way when the lower ones vanish compatibly.
With rational or real coefficients, all Massey products vanish on formal spaces, such as spheres and compact Kähler manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/71VK7M83WS.md" tag="71VK7M83WS" kind="definition">
# Polytope combinatorics

## Definition

A convex polytope is the [[convex hull|0FQSDC2EQ2]] $P$ of a finite set of points of $\R^d$.
A hyperplane $H$ supports $P$ if $P$ lies in one of the two closed half-spaces bounded by $H$.
A *face* of $P$ is $P$ itself, the empty set, or a set $P \cap H$ for a supporting hyperplane $H$.
The *face lattice* of $P$ is the set of its faces, [[partially ordered|0QV2JS9RJ3]] by inclusion.
Two polytopes are *combinatorially equivalent* if their face lattices are isomorphic as partially ordered sets.
*Polytope combinatorics* is the study of the properties of convex polytopes that depend only on their face lattice.

## Remarks

Each face is again a convex polytope, and its dimension is the dimension of its affine span; the empty face has dimension $-1$.
Faces of dimension $0$, $1$ and $\dim P - 1$ are called vertices, edges and facets.

Example: a square and a trapezium in $\R^2$ are combinatorially equivalent.
Each has $4$ vertices and $4$ edges.
Match the vertices of one with the vertices of the other in cyclic order; this matching preserves which vertices lie on which edges.

Non-example: a square and a triangle are not combinatorially equivalent, since their face lattices have $4$ and $3$ vertices.

Properties such as the number of faces of each dimension, or whether every facet is a simplex, belong to polytope combinatorics; volume and edge lengths do not.
</document>

<document source="https://tamnd.github.io/math2/node/71XBFB49HE.md" tag="71XBFB49HE" kind="statement">
# Kiefer–Wolfowitz equivalence theorem

## Statement

Let $\mathcal{X}$ be a compact metric space, the design region.
Let $f \colon \mathcal{X} \to \R^p$ be continuous, and consider the [[linear regression|41FVC4P5JB]] model $E[y(x)] = \theta^T f(x)$ with $\theta \in \R^p$ and uncorrelated errors of equal variance.
A design is a [[probability measure|059TJN0NZ1]] $\xi$ on the Borel sets of $\mathcal{X}$.
Its information matrix is $M(\xi) = \int_{\mathcal{X}} f(x) f(x)^T \, d\xi(x)$.
When $M(\xi)$ is nonsingular, its standardized variance function is $d(x, \xi) = f(x)^T M(\xi)^{-1} f(x)$.
Assume that some design has a nonsingular information matrix.
For a design $\xi^*$ with $M(\xi^*)$ nonsingular, the following three conditions are equivalent:

1. $\xi^*$ is D-optimal: it maximizes $\det M(\xi)$ over all designs $\xi$.
2. $\xi^*$ is G-optimal: it minimizes $\max_{x \in \mathcal{X}} d(x, \xi)$ over all designs $\xi$ with $M(\xi)$ nonsingular.
3. $\max_{x \in \mathcal{X}} d(x, \xi^*) = p$.

Moreover, all D-optimal designs have the same information matrix, and $d(x, \xi^*) = p$ at every support point $x$ of a D-optimal design $\xi^*$.

## Remarks

For every design with nonsingular information matrix, $\int d(x, \xi) \, d\xi(x) = \tr(M(\xi)^{-1} M(\xi)) = p$.
So $\max_x d(x, \xi) \ge p$ always holds, and this explains the number $p$ in condition 3.
The function $\xi \mapsto \log \det M(\xi)$ is concave on the convex set of designs, and its directional derivative at $\xi$ towards the point mass $\delta_x$ equals $d(x, \xi) - p$.
So condition 3 says that no direction increases $\log \det M$, which for a concave function means a maximum.
Example: for the line $f(x) = (1, x)^T$ on $\mathcal{X} = [-1, 1]$, the design $\xi^* = \frac12 \delta_{-1} + \frac12 \delta_{1}$ has $M(\xi^*) = I_2$.
Then $d(x, \xi^*) = 1 + x^2 \le 2 = p$, so $\xi^*$ is D-optimal and G-optimal.
The theorem was proved by Jack Kiefer and Jacob Wolfowitz in 1960.

## Explanations

### L1 human

An experimenter chooses where to take measurements in order to fit a regression model.
One goal is to make the joint uncertainty about the parameters small, measured by the [[determinant|00KDZRW4C0]] of the [[information matrix|G1Z3WSTE8Q]].
Another goal is to make the worst prediction variance over the whole region small.
The theorem says that these two goals give the same best designs, and it gives a simple test for being best.

### L2 human

The equivalence theorem turns a hard optimization over probability measures into a checkable condition: compute $d(x, \xi)$ and see whether it stays below $p$.
It is the basis of the Fedorov–Wynn type algorithms, which add weight at the point where $d(x, \xi)$ is largest.
Similar equivalence theorems hold for other concave criteria of [[optimal experimental design|PM5PQX6TVP]], with the variance function replaced by the corresponding directional derivative.
</document>

<document source="https://tamnd.github.io/math2/node/720FQEBT68.md" tag="720FQEBT68" kind="example">
# Table of Lie groups

## Example

The table lists common real [[Lie groups|7JBRGCVJRB]] with a few invariants.
These are the real dimension, [[compactness|0TVJZYB6SH]], the number of connected components, and the [[fundamental group|S1XK3HXHFX]] $\pi_1$ of the identity component.
Here $n \ge 3$, to keep the entries uniform.

| Group | Dimension | Compact | Components | $\pi_1$ |
|---|---|---|---|---|
| $\R^n$ | $n$ | no | $1$ | $0$ |
| $U(1)$ | $1$ | yes | $1$ | $\Z$ |
| $GL_n(\R)$ | $n^2$ | no | $2$ | $\Z/2$ |
| $SL_n(\R)$ | $n^2 - 1$ | no | $1$ | $\Z/2$ |
| $O(n)$ | $n(n-1)/2$ | yes | $2$ | $\Z/2$ |
| $SO(n)$ | $n(n-1)/2$ | yes | $1$ | $\Z/2$ |
| $\mathrm{Spin}(n)$ | $n(n-1)/2$ | yes | $1$ | $0$ |
| $U(n)$ | $n^2$ | yes | $1$ | $\Z$ |
| $SU(n)$ | $n^2 - 1$ | yes | $1$ | $0$ |
| $GL_n(\C)$ | $2n^2$ | no | $1$ | $\Z$ |
| $SL_n(\C)$ | $2n^2 - 2$ | no | $1$ | $0$ |
| $Sp(2n, \R)$ | $n(2n+1)$ | no | $1$ | $\Z$ |
| $Sp(n)$ | $n(2n+1)$ | yes | $1$ | $0$ |

The rows are the [[circle group|04EAFVFTF7]] $U(1)$, the [[general linear group|0FCHE902D7]] and the [[special linear group|0V9PMN0RBH]].
Then come the [[orthogonal group|0TKYAEJ8N7]], the [[special orthogonal group|0CGYNDCWRR]], the [[spin group|JJSXXJQGEH]], the [[unitary group|0R9R0Y1HDK]] and the [[special unitary group|CZCWFPNS4N]].
The last rows are the real [[symplectic group|X12FS2JAP0]] $Sp(2n, \R)$ of $2n \times 2n$ real matrices.
They also include the [[compact symplectic group|5NNPBFQA02]] $Sp(n)$ of quaternionic $n \times n$ matrices.

Check of the row $O(n)$.
Its Lie algebra is the space of real skew-symmetric $n \times n$ matrices, which has dimension $n(n-1)/2$.
The determinant of an orthogonal matrix is $\pm 1$, and both values occur, so $O(n)$ has at least two components.
Each fibre of $\det$ is a translate of the connected group $SO(n)$, so there are exactly two.
$O(n)$ is closed and bounded in $\R^{n^2}$, since $A^T A = I$ forces every entry into $[-1, 1]$.
So $O(n)$ is compact.

## Remarks

For $n = 2$ some entries change: $SO(2) \cong U(1)$ and the identity components of $GL_2(\R)$ and $SL_2(\R)$ have fundamental group $\Z$.
The $\pi_1$ entries for the noncompact groups use one fact.
A connected real Lie group deformation retracts onto a maximal compact subgroup, for example $SO(n)$ in $SL_n(\R)$ and $U(n)$ in $Sp(2n, \R)$.
For $n \ge 3$, $\mathrm{Spin}(n)$ is the universal cover of $SO(n)$, a double cover.

## Explanations

### L1 human

Most Lie groups met in practice are groups of matrices: invertible matrices, matrices of determinant one, rotations, unitary matrices and symplectic matrices.
The table records a few basic numbers for each one.
The dimension counts how many independent parameters describe an element, as the three angles of a rotation in space.
Compactness says whether the group is bounded, and the components and the fundamental group describe its shape as a space.

### L2 human

The invariants in the table are linked through maximal compact subgroups.
A connected Lie group has the homotopy type of its maximal compact subgroup, so $\pi_1$ of $SL_n(\R)$ is read off from $SO(n)$ and $\pi_1$ of $Sp(2n, \R)$ from $U(n)$.
Many rows pair a noncompact group with a compact form of the same complexified Lie algebra, such as $SL_n(\R)$ and $SU(n)$.
This explains why such pairs, like $Sp(2n, \R)$ and $Sp(n)$, have the same dimension.
</document>

<document source="https://tamnd.github.io/math2/node/720QQR553X.md" tag="720QQR553X" kind="concept">
# Immersed boundary method

## Summary

The immersed boundary method simulates elastic structures moving in a viscous incompressible fluid on a fixed Cartesian grid.
The fluid is described by the [[Navier–Stokes equations|18KXZHVF4M]] on the grid, and the structure by points that move with the fluid.
The two are coupled by smoothed versions of the [[Dirac delta function|0KPHENZ5S7]].
The structure spreads its elastic forces to the grid and takes its velocity from the grid.
It was introduced by Charles Peskin to model blood flow through the heart.
See the [[definition|E8FQ2R87VX]].

## Explanations

### L1 human

Picture a heart valve, a thin elastic sheet, waving in the blood.
Building a mesh that follows the valve as it moves is hard.
The immersed boundary method keeps a fixed square grid for the blood and treats the valve as a chain of points carried by the flow.
The points push on the nearby fluid with their elastic forces, and the fluid carries the points along.
</document>

<document source="https://tamnd.github.io/math2/node/727TZQJ4ZA.md" tag="727TZQJ4ZA" kind="concept">
# Polarity

## Summary

A polarity of a projective space is a correlation of order $2$.
It sends points to hyperplanes and hyperplanes to points, it reverses inclusion, and applying it twice gives the identity.
Polarities are the geometric form of non-degenerate reflexive forms, such as symmetric, alternating and Hermitian forms.
They underlie [[duality in projective geometry|DCMP1A0138]], the theory of pole and polar for conics, and [[polar spaces|Z7T0V2VAWJ]].

## Explanations

### L1 human

For a circle in the plane, each point outside it has a polar line, the line through the two points where the tangents from it touch the circle.
Each line has a pole, and going from point to line and back returns the starting point.
This exchange of points and lines that preserves incidence is a polarity.
In linear algebra, taking the orthogonal complement with respect to a dot product is the same kind of operation.
</document>

<document source="https://tamnd.github.io/math2/node/727ZEPVSTC.md" tag="727ZEPVSTC" kind="concept">
# Kontsevich integral

## Summary

The Kontsevich integral is a [[knot invariant|GZQ5ZE3X6G]] with values in a completed algebra of chord diagrams.
Maxim Kontsevich defined it by an iterated integral of logarithmic forms.
It is the universal rational [[finite type invariant|TWH3WVKFF1]]: every such invariant is obtained from it by a linear map.
It ties knot theory to quantum groups and to the theory of Drinfeld associators.

## Explanations

### L1 human

Place a knot in space and slice it by horizontal planes.
Each slice meets the knot in finitely many points, which move as the plane goes up.
The Kontsevich integral records how these points wind around each other, using chords that join pairs of them.
The output is not a number but a formal sum of chord diagrams, circles with chords drawn inside.
From this one object, every rational finite type invariant can be read off.
</document>

<document source="https://tamnd.github.io/math2/node/72JSAPJB96.md" tag="72JSAPJB96" kind="definition">
# Fully invariant congruence

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.
A [[congruence|DCSXXT4SER]] $\theta$ on $\mathbf{A}$ is *fully invariant* when, for every endomorphism $\varepsilon$ of $\mathbf{A}$ and all $a, b \in A$, $(a, b) \in \theta$ implies $(\varepsilon(a), \varepsilon(b)) \in \theta$.
Here an endomorphism is a [[homomorphism|KBXSZC6BV1]] from $\mathbf{A}$ to itself.

## Remarks

The congruences $\Delta_A$ and $\nabla_A$ are always fully invariant, and so is the intersection of any family of fully invariant congruences.

Example: on the group $\Z^2 = \Z \times \Z$, congruence modulo $2$ in both coordinates is fully invariant.
Every endomorphism of $\Z^2$ is given by an integer matrix $M$, and $a - b \in 2\Z^2$ implies $M a - M b = M(a - b) \in 2\Z^2$.

Non-example: on $\Z^2$, let $(a, b) \in \theta$ when the first coordinates of $a$ and $b$ are equal.
This is a congruence, but the swap $(x, y) \mapsto (y, x)$ is an endomorphism that sends the related pair $((0, 0), (0, 1))$ to the unrelated pair $((0, 0), (1, 0))$.

On the [[term algebra|3YYKGWJJND]] $\mathbf{T}(X)$ over a set of variables $X$, the endomorphisms are the substitutions.
So the fully invariant congruences on $\mathbf{T}(X)$ are the [[equational theories|F1M51B4MDG]] in the variables $X$.
</document>

<document source="https://tamnd.github.io/math2/node/72K5F9VPCR.md" tag="72K5F9VPCR" kind="definition">
# Global dimension

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]].
The *global dimension* of $R$ is
$$\operatorname{gl.dim} R = \sup \set{\operatorname{pd}_R M : M \text{ a nonzero } R\text{-module}},$$
where $\operatorname{pd}_R M$ is the [[projective dimension|MR9QCH1PNA]] of the [[$R$-module|0NZ8JA25WK]] $M$.
It is an integer $\ge 0$ or $\infty$.

## Remarks

By a theorem of Auslander, the supremum can be taken over the cyclic modules $R/I$ only, with $I$ an ideal.
For a ring that is not commutative, the same definition with left modules or right modules gives the left and right global dimensions, which can differ.
Example: $\operatorname{gl.dim} \Z = 1$; every subgroup of a free abelian group is free, so every module has a free resolution of length at most $1$, and $\Z/2\Z$ is not projective.
Example: a ring has global dimension $0$ if and only if every module is projective; for a field this holds since every vector space is free.
Non-example of finiteness: $\Z/4\Z$ has infinite global dimension, because the projective dimension of $\Z/2\Z$ over it is infinite.
For a Noetherian local ring $(R, \mathfrak{m}, k)$, the global dimension equals $\operatorname{pd}_R k$.
</document>

<document source="https://tamnd.github.io/math2/node/72PG6XCQ7D.md" tag="72PG6XCQ7D" kind="concept">
# Ternary discriminator

## Summary

The ternary discriminator on a set $A$ is the operation $t(x, y, z)$ that returns $z$ when $x = y$ and returns $x$ otherwise.
It tests whether two elements are equal and chooses its output accordingly.
Algebras in which this operation is a term operation are called quasi-primal when they are finite.
Classes of algebras sharing a term for it generate [[discriminator varieties|MC9MX40Z7S]], which have a very clear structure theory.

## Explanations

### L1 human

The discriminator is an "if equal, then this, else that" operation.
It looks at its first two arguments: if they are the same, it outputs the third, and if they differ, it outputs the first.
On $\set{0, 1, 2}$, for example, $t(1, 1, 2) = 2$ and $t(1, 0, 2) = 1$.
Many logical and algebraic structures can express this test with their own operations, and this has strong consequences.
</document>

<document source="https://tamnd.github.io/math2/node/72S3YSQFZ7.md" tag="72S3YSQFZ7" kind="concept">
# Equational logic

## Summary

Equational logic is the fragment of first-order logic whose statements are [[identities|PKD440G2XW]] between terms and whose proofs use only equational reasoning.
Its rules are reflexivity, symmetry, transitivity, replacement of equals inside an operation, and [[substitution|NCDZ969V7X]] of terms for variables.
By Birkhoff's completeness theorem these rules derive exactly the identities that follow semantically from a given set of identities.
It is the logic of algebraic manipulation, and it underlies term rewriting and automated equational reasoning.

## Explanations

### L1 human

Equational logic is the logic of high school algebra.
You start from some laws, such as $x + 0 \approx x$ and $x + y \approx y + x$.
You may plug any expressions into a law, replace equals by equals inside a bigger expression, and chain equalities together.
For example, from these two laws one derives $0 + x \approx x + 0$ by substituting into commutativity, and then $0 + x \approx x$ by transitivity.
</document>

<document source="https://tamnd.github.io/math2/node/72SN9MMH5Y.md" tag="72SN9MMH5Y" kind="definition">
# Absolute neighborhood retract

## Definition

Let $Y$ be a topological space and $A \subseteq Y$.
A *retraction* of $Y$ onto $A$ is a [[continuous|0VGFHTM75Y]] map $r \colon Y \to A$ with $r(a) = a$ for all $a \in A$.

Let $X$ be a [[metrizable space|091GJ3FYD7]].
$X$ is an *absolute neighborhood retract* (ANR) if the following holds for every metrizable space $Y$.
Let $h$ be any [[homeomorphism|0QT00PB5G2]] of $X$ onto a [[closed|0ZCK8DQYN1]] subset $h(X)$ of $Y$.
Then there is an [[open|02BH3MCBQB]] set $U \subseteq Y$ with $h(X) \subseteq U$ and a retraction of $U$ onto $h(X)$.
$X$ is an *absolute retract* (AR) if one can always take $U = Y$.

## Remarks

Some authors define these classes for compact metrizable spaces only, or for other classes of spaces; the definition above is for metrizable spaces.
Equivalently, $X$ is an ANR if every continuous map $f \colon A \to X$ from a closed subset $A$ of a metrizable space $Y$ extends to a continuous map on some open set $U \supseteq A$.

Example: $X = [0, 1]$ is an AR.
Given $h$ as above, the map $h^{-1} \colon h(X) \to [0, 1]$ is continuous on the closed set $h(X)$, and by the [[Tietze extension theorem|0X52TNMTCJ]] it extends to a continuous $g \colon Y \to [0, 1]$.
Then $r = h \circ g$ is a retraction of $Y$ onto $h(X)$.

Non-example: $X = \set{0} \cup \set{1/n : n \ge 1}$, a closed subset of $Y = \R$, with $h$ the inclusion.
Any open $U \supseteq X$ contains an interval $(-\varepsilon, \varepsilon)$.
A retraction $r \colon U \to X$ would map this connected interval onto a connected subset of $X$ that contains $0$ and $1/n$ for large $n$, which is impossible.
</document>

<document source="https://tamnd.github.io/math2/node/72TSY1Z4VA.md" tag="72TSY1Z4VA" kind="concept">
# Jacobi transform

## Summary

The Jacobi transform is a finite [[integral transform|YCK709QB06]] on $[-1, 1]$.
It sends a function to the sequence of its integrals against the [[Jacobi polynomials|TRD7DSF2KR]] $P_n^{(\alpha, \beta)}$ with their weight.
It is the coefficient map of the expansion of a function in Jacobi polynomials.
It turns the Jacobi differential operator into multiplication by $-n(n + \alpha + \beta + 1)$, so it is used to solve boundary value problems on an interval.
The Gegenbauer and Legendre transforms are special cases.

## Explanations

### L1 human

Jacobi polynomials are a family of polynomials that are orthogonal on $[-1, 1]$ for a weight that may favour one end of the interval.
Any reasonable function on $[-1, 1]$ can be written as a sum of these polynomials.
The Jacobi transform records the numbers that tell how much of each polynomial is needed.
For example, the transform of $P_2^{(\alpha, \beta)}$ itself is zero at every $n$ except $n = 2$.
A differential equation for the function becomes a simple algebraic equation for these numbers.
The same name is also used for a transform on the half-line with Jacobi functions as kernel; this node is about the finite transform.
</document>

<document source="https://tamnd.github.io/math2/node/72V1Z82M5F.md" tag="72V1Z82M5F" kind="concept">
# Bounded lattice

## Summary

A bounded lattice is a [[lattice|WH4SFAACAS]] with a [[greatest element|0XR2QK49V6]] $1$ and a [[least element|0AANKKNHS4]] $0$.
These elements are the identities for the meet and the join: $x \wedge 1 = x$ and $x \vee 0 = x$.
Every finite nonempty lattice is bounded, and bounded lattices are the setting for complements, atoms and Boolean algebras.

## Explanations

### L1 human

A bounded lattice has a bottom element below everything and a top element above everything.
In the lattice of subsets of a set $X$, the bottom is the empty set and the top is $X$ itself.
The interval $[0, 1]$ with its usual order is a bounded lattice, with bottom $0$ and top $1$.
The integers with their usual order form a lattice that is not bounded: there is no largest and no smallest integer.
</document>

<document source="https://tamnd.github.io/math2/node/72W4HXM98H.md" tag="72W4HXM98H" kind="definition">
# Splitting of prime ideals

## Definition

Let $L/K$ be a finite extension of [[algebraic number fields|MWBDFS2KJ5]] of [[degree|0EAJ2XXADC]] $n$, with rings of integers $\mathcal{O}_K \subseteq \mathcal{O}_L$.
Let $\mathfrak{p}$ be a nonzero [[prime ideal|00MHHZ9SNE]] of $\mathcal{O}_K$.
Since $\mathcal{O}_L$ is a [[Dedekind domain|7NPY2CN5KB]], the ideal $\mathfrak{p}\mathcal{O}_L$ factors uniquely as
$$\mathfrak{p}\mathcal{O}_L = \mathfrak{P}_1^{e_1} \cdots \mathfrak{P}_g^{e_g}$$
with distinct nonzero prime ideals $\mathfrak{P}_i$ of $\mathcal{O}_L$ and integers $e_i \ge 1$.
The $\mathfrak{P}_i$ are the primes of $L$ *lying over* $\mathfrak{p}$.
The number $e_i$ is the [[ramification index|H5RC67SD24]] of $\mathfrak{P}_i$ over $\mathfrak{p}$, and the *residue degree* (or inertia degree) is $f_i = [\mathcal{O}_L/\mathfrak{P}_i : \mathcal{O}_K/\mathfrak{p}]$.
The prime $\mathfrak{p}$ is said to:

1. *split completely* in $L$ if $g = n$, that is, $e_i = f_i = 1$ for all $i$;
2. be *inert* in $L$ if $\mathfrak{p}\mathcal{O}_L$ is a prime ideal, that is, $g = 1$ and $e_1 = 1$;
3. be *ramified* in $L$ if $e_i > 1$ for some $i$.

## Remarks

The integers satisfy $\sum_{i=1}^g e_i f_i = n$, the [[fundamental identity|REJDQ6BTQV]].
If $L/K$ is a [[Galois extension|0QMZAQ6QQ4]], the Galois group permutes the $\mathfrak{P}_i$ transitively, so all $e_i$ are equal to some $e$, all $f_i$ to some $f$, and $efg = n$.

Example: for $K = \Q$, $L = \Q(i)$ and $\mathfrak{p} = 5\Z$, one has $5\Z[i] = (2+i)(2-i)$ with two distinct primes, each with residue field $\F_5$.
So $g = 2 = n$ and $5$ splits completely.

Non-example: $3$ does not split completely in $\Q(i)$.
The ideal $3\Z[i]$ is prime, because $x^2 + 1$ is irreducible modulo $3$; so $g = 1$, $f = 2$, and $3$ is inert.
The prime $2$ is ramified: $2\Z[i] = (1+i)^2$, with $e = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/72WNRWJQC1.md" tag="72WNRWJQC1" kind="statement">
# Explicit formula

## Statement

Let $\psi$ be the second [[Chebyshev function|B8AKSVBE5Z]] and $\zeta$ the [[Riemann zeta function|1XWBY4N0VT]].
For $x > 1$, let $\psi_0(x) = \psi(x)$ if $x$ is not a prime power, and $\psi_0(x) = \psi(x) - \frac{1}{2} \log p$ if $x = p^k$ with $p$ prime and $k \ge 1$.
Let $\rho$ run over the nontrivial zeros of $\zeta$, those with $0 < \operatorname{Re} \rho < 1$, each counted with its multiplicity.
Then
$$\psi_0(x) = x - \lim_{T \to \infty} \sum_{\abs{\operatorname{Im} \rho} < T} \frac{x^{\rho}}{\rho} - \log(2\pi) - \frac{1}{2} \log\!\left(1 - x^{-2}\right).$$

## Remarks

This is the explicit formula of Riemann and von Mangoldt.
The proof starts from Perron's formula: for $c > 1$,
$$\psi_0(x) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \left(-\frac{\zeta'(s)}{\zeta(s)}\right) \frac{x^s}{s} \, ds,$$
because $-\zeta'(s)/\zeta(s) = \sum_n \Lambda(n) n^{-s}$ with the [[von Mangoldt function|0NK8JTE1NN]] $\Lambda$.
Moving the line of integration to the left picks up residues.
The pole at $s = 1$ gives $x$, and each nontrivial zero gives $-x^\rho/\rho$.
The point $s = 0$ gives $-\zeta'(0)/\zeta(0) = -\log(2\pi)$.
Each trivial zero $-2n$ gives $x^{-2n}/(2n)$, and these terms add up to $-\frac{1}{2}\log(1 - x^{-2})$.
The sum over zeros converges only conditionally, so the order of summation matters.
Heuristically, the [[prime number theorem|0EHBY5Y84N]] $\psi(x) \sim x$ reflects that each term $x^\rho/\rho$ is $o(x)$, since $\operatorname{Re} \rho < 1$.
A rigorous proof along these lines also needs a zero-free region to control the whole sum.

## Explanations

### L1 human

The formula writes a prime-counting function exactly as a main term $x$ plus a sum of waves, one for each zero of the zeta function.
The real part of a zero gives the size of its wave, and the imaginary part gives its frequency.
This is why the zeros of zeta control the distribution of the primes.

### L2 human

Under the [[Riemann hypothesis|VCGF9YR36X]] each term has size $\abs{x^\rho} = \sqrt{x}$, and a truncated form of the formula gives $\psi(x) = x + O(\sqrt{x} \log^2 x)$.
Weil generalized the formula to a symmetric identity between a sum over the zeros of an L-function and a sum over prime powers, for a class of test functions.
Similar formulas hold for [[Dirichlet L-functions|9D2Z543EJQ]], and they give the distribution of primes in arithmetic progressions.
</document>

<document source="https://tamnd.github.io/math2/node/734Q6Q51X7.md" tag="734Q6Q51X7" kind="concept">
# Randomization

## Summary

Randomization is the assignment of treatments to experimental units by a chance mechanism with known probabilities.
It balances, on average, all unit characteristics between the treatment groups, including those that nobody measured.
So differences in outcome can be attributed to the treatments.
It also provides the probability model behind randomization tests and, in large samples, justifies the usual tests of [[analysis of variance|M9AQ5HKGP4]].
It is one of the three principles of [[design of experiments|2TE42ZWJ3E]] and the basis of randomized controlled trials and of [[causal inference|V3QJFN5800]].

## Explanations

### L1 human

If a doctor chooses which patients get a new drug, she may give it to those who look strongest, and the drug will seem better than it is.
Tossing a coin for each patient removes that choice.
Then the two groups differ only by chance, in age, health and everything else.
With four patients and two places in each group, there are $6$ ways to split them, and the randomization picks one of them, each with probability $1/6$.
</document>

<document source="https://tamnd.github.io/math2/node/73ANTTKAH3.md" tag="73ANTTKAH3" kind="statement">
# Fredholm property of elliptic operators

## Statement

Let $M$ be a compact smooth manifold without boundary, with a smooth positive density.
Let $E$ and $F$ be complex vector bundles over $M$ with Hermitian metrics.
Let $P \colon \Gamma(E) \to \Gamma(F)$ be an [[elliptic operator|MT9P2X85V8]] of order $m$.
Let $P^* \colon \Gamma(F) \to \Gamma(E)$ be its formal adjoint.
It is the [[differential operator|CGBX1N9V8H]] with $\int_M \ip{Pu}{v} = \int_M \ip{u}{P^* v}$ for all smooth sections $u$ and $v$.
For $s \in \R$, let $H^s(E)$ and $H^s(F)$ be the [[Sobolev spaces|M4CEZ5K9HP]] of sections.

1. For every $s \in \R$, $P$ extends to a bounded operator $P_s \colon H^{s+m}(E) \to H^s(F)$, and $P_s$ is a [[Fredholm operator|8HWGKWD6ZM]].
2. The kernel of $P_s$ consists of smooth sections and equals the kernel of $P$ on $\Gamma(E)$.
In particular it is finite-dimensional and does not depend on $s$.
3. The kernel of $P^*$ is a finite-dimensional space of smooth sections.
The range of $P_s$ is the set of $f \in H^s(F)$ with $\int_M \ip{f}{v} = 0$ for all $v$ in the kernel of $P^*$.
4. The [[index|2V9Q4KHY1T]] of $P_s$ equals $\dim \ker P - \dim \ker P^*$ and does not depend on $s$.

## Remarks

The proof constructs a [[parametrix|Z97HVFD6JE]] $Q$, a pseudodifferential operator of order $-m$ built from the inverse of the principal symbol, with $QP - I$ and $PQ - I$ smoothing.
Smoothing operators map $H^s$ into $H^{s+1}$, and by the Rellich lemma the inclusion $H^{s+1} \to H^s$ is [[compact|M2NEP1K5AE]] on a compact manifold.
So $P_s$ is invertible modulo compact operators, which is Atkinson's characterisation of Fredholm operators.
The parametrix also gives [[elliptic regularity|2PVQY9PCT4]]: if $Pu$ is smooth then $u$ is smooth, which yields part 2.

Example: on the circle $S^1 = \R / 2\pi\Z$, let $P = -d^2/dx^2$.
Its kernel and the kernel of $P^* = P$ are the constant functions, so the index is $1 - 1 = 0$, and the range consists of the functions with average zero.

Compactness of $M$ is needed.
On $\R$, the operator $d/dx \colon H^1(\R) \to L^2(\R)$ is elliptic and injective, but its range is dense and not closed, so it is not Fredholm.

## Explanations

### L1 human

An elliptic operator on a closed manifold behaves almost like an invertible matrix.
It kills only a finite-dimensional space of sections, and it misses only a finite-dimensional space of targets.
The difference of these two dimensions, the index, is a stable number that does not change when the operator is deformed through elliptic operators.

### L2 human

This theorem is the analytic foundation of index theory.
The index depends only on the homotopy class of the principal symbol through invertible symbols, which is why it can be computed topologically by the Atiyah–Singer index theorem.
It also underlies Hodge theory: applied to the Hodge Laplacian, it gives finite-dimensional spaces of harmonic forms and the orthogonal decomposition of forms.
</document>

<document source="https://tamnd.github.io/math2/node/73DXSAH73B.md" tag="73DXSAH73B" kind="definition">
# Lattice of clones

## Definition

Let $A$ be a nonempty set and let $\mathcal{L}_A$ be the set of all [[clones|PTKR703G6A]] on $A$.
The *lattice of clones* on $A$ is the partially ordered set $(\mathcal{L}_A, \subseteq)$.
For a family $(C_i)_{i \in I}$ of clones, its meet is $\bigcap_{i \in I} C_i$ and its join is the clone generated by $\bigcup_{i \in I} C_i$.

## Remarks

The intersection of clones is a clone, since each clause of the definition of a clone is kept under intersections, so $\mathcal{L}_A$ is a [[complete lattice|D7A0ZGZT5S]].
Example: on $\set{0, 1}$, the meet of the clones generated by $\wedge$ and by $\vee$ is the clone of [[projections|769B35T6WG]].
Their join is the clone of all monotone operations that preserve both $0$ and $1$.
Check for the meet: the members of the first clone are the meets $x_{i_1} \wedge \cdots \wedge x_{i_k}$, and those of the second are joins.
A meet of $k \ge 2$ distinct variables differs from every join on some input, so only projections are in both.
Non-example: the join is not the union in general.
On $\set{0, 1}$, the union of the clones generated by $\wedge$ and by $\vee$ is not a clone, since it misses the majority operation $(x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$, which lies in their join.
The lattice of clones on any set is an algebraic lattice whose compact elements are the finitely generated clones.
</document>

<document source="https://tamnd.github.io/math2/node/73EBKQPP8Q.md" tag="73EBKQPP8Q" kind="example">
# Hawaiian earring

## Example

For $n \ge 1$ let $C_n \subseteq \R^2$ be the [[circle|060W5CBZP6]] with centre $(1/n, 0)$ and radius $1/n$.
The Hawaiian earring is $H = \bigcup_{n \ge 1} C_n$ with the subspace topology from $\R^2$.
All circles pass through the origin $0$.
The space $H$ is [[compact|0TVJZYB6SH]] and [[path-connected|0WB6DH499T]], but it is not [[semilocally simply connected|R3SHZX0NZR]] at $0$.

Check.
$H$ is closed and bounded in $\R^2$, so it is compact.
Each $C_n$ is path-connected and contains $0$, so $H$ is path-connected.
Fix $n$ and let $r_n \colon H \to C_n$ be the identity on $C_n$ and send every other circle to $0$.
This map is continuous, and it is a [[retraction|68PFSGP39B]] onto $C_n$.
Let $\gamma_n$ be the loop that goes once around $C_n$.
The map $r_n$ sends it to a generator of the [[fundamental group|S1XK3HXHFX]] $\pi_1(C_n) \cong \Z$, so $\gamma_n$ is not null-homotopic in $H$.
Every neighbourhood of $0$ contains $C_n$ for all large $n$, so it contains loops at $0$ that are not null-homotopic in $H$.

## Remarks

Since $H$ is not semilocally simply connected, it has no universal [[covering space|3XZRRE165N]].
The retractions $r_n$ together give a homomorphism from $\pi_1(H, 0)$ onto the product $\prod_{n \ge 1} \Z$.
It is onto: a loop can run $k_n$ times around $C_n$ during the time interval $[1 - 2^{1-n}, 1 - 2^{-n}]$.
This loop is continuous at time $1$ because the circles shrink.
So $\pi_1(H, 0)$ is uncountable.
The [[wedge sum|5JSWM1HBX2]] of countably many circles, with the CW topology, has a countable free fundamental group and is not homeomorphic to $H$.

## Explanations

### L1 human

The Hawaiian earring is an infinite family of circles in the plane, all touching at one point and getting smaller and smaller.
It looks like a bouquet of circles, but the small circles crowd near the common point.
However close you look at that point, you still see whole circles there that cannot be shrunk away.
This makes its loops far more complicated than those of a finite bouquet.
</document>

<document source="https://tamnd.github.io/math2/node/73RYYEVDKA.md" tag="73RYYEVDKA" kind="concept">
# Flat module

## Summary

A flat module is a module $M$ such that tensoring with $M$ keeps injective linear maps injective.
Equivalently, the functor $- \otimes_R M$ preserves [[exact sequences|JB7ZBFBA9C]].
Free and [[projective modules|BFFA03GE3V]] are flat, and so are [[localizations|4C1AE84N5N]] such as $\Q$ over $\Z$.
Flatness is the algebraic form of "varying continuously in a family".
Flat ring maps are the basic tool of descent and base change in algebraic geometry.

## Explanations

### L1 human

Tensoring with a module is a way to change the ring of scalars or to combine two modules.
In general this operation can collapse things: a nonzero map may become zero.
A flat module never causes such a collapse of injective maps.
For example, $\Z/2\Z$ is not flat over $\Z$: the injective map "multiply by $2$" on $\Z$ becomes the zero map on $\Z/2\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/73SAQBFWKQ.md" tag="73SAQBFWKQ" kind="definition">
# Elliptic surface

## Definition

Let $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field.
An *elliptic fibration* on $X$ is a surjective [[morphism|CDS1VWB7A3]] $f \colon X \to C$ onto a smooth projective [[curve|XJM2MT4S3Y]] $C$ such that:

1. every fibre of $f$ is connected, and
2. the fibre $f^{-1}(c)$ is a smooth curve of [[genus|6W8TDV9VVN]] $1$ for all but finitely many $c \in C$.

An *elliptic surface* is a surface $X$ together with an elliptic fibration on it.

## Remarks

Many authors also require that $f$ be relatively minimal, meaning that no fibre contains a smooth rational curve of self-intersection $-1$, and some require a section of $f$.
Example: for an elliptic curve $E$ and a smooth projective curve $C$, the projection $E \times C \to C$ has every fibre isomorphic to $E$, so it is an elliptic fibration.
Example: blowing up $\mathbb{P}^2$ at the nine base points of a general pencil of plane cubics gives a rational elliptic surface, whose fibres are the cubics of the pencil.
Non-example: the projection $\mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^1$ has all fibres of genus $0$, so it fails clause 2.
In characteristics $2$ and $3$ one also meets quasi-elliptic fibrations, whose general fibre is a cuspidal rational curve.
</document>

<document source="https://tamnd.github.io/math2/node/73WPYFAEWS.md" tag="73WPYFAEWS" kind="concept">
# Five-point stencil

## Summary

A five-point stencil is a set of five grid points used to approximate derivatives by finite differences.
On a line, it is a point with two neighbours on each side, and it gives fourth-order formulas for the first and second derivative.
In the plane, it is a point with its four nearest neighbours, and it gives the standard second-order approximation of the [[Laplacian|0E0B6RA8K0]].
The planar version is the basic discretisation of the Poisson equation in the [[finite difference method|X7HYEBA5SA]].

## Explanations

### L1 human

Picture a square grid and a point on it together with its neighbours to the left, right, above and below: these five points form a plus sign.
Adding the four neighbour values and subtracting four times the centre value measures how much the centre differs from the average around it.
For $u(x, y) = x^2 + y^2$ at the origin with spacing $1$, this gives $1 + 1 + 1 + 1 - 0 = 4$, which equals the Laplacian $u_{xx} + u_{yy} = 4$.
</document>

<document source="https://tamnd.github.io/math2/node/73X2JGSE0W.md" tag="73X2JGSE0W" kind="definition">
# Pick matrix

## Definition

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].
Let $n \ge 1$, let $z_1, \ldots, z_n \in \mathbb{D}$ be distinct, and let $w_1, \ldots, w_n \in \C$.
The *Pick matrix* of the data $(z_i, w_i)$ is the $n \times n$ [[matrix|0N1MY1T0W9]] $P$ with entries
$$P_{ij} = \frac{1 - w_i \overline{w_j}}{1 - z_i \overline{z_j}}, \qquad 1 \le i, j \le n.$$

## Remarks

The denominators are not $0$, because $\abs{z_i \overline{z_j}} < 1$.
One has $\overline{P_{ji}} = P_{ij}$, so $P$ is a [[Hermitian matrix|0EKZCX4K3P]].
By [[Nevanlinna–Pick interpolation|4VMSSYB5DN]], with all $w_i \in \mathbb{D}$, there is a holomorphic $f \colon \mathbb{D} \to \mathbb{D}$ with $f(z_i) = w_i$ if and only if $P$ is [[positive semidefinite|0880F9EQCP]].

Example: $n = 2$, $z_1 = 0$, $w_1 = 0$, $z_2 = a$, $w_2 = b$ with $a \ne 0$.
Then $P = \begin{pmatrix} 1 & 1 \\ 1 & \frac{1 - \abs{b}^2}{1 - \abs{a}^2} \end{pmatrix}$, and $\det P = \frac{\abs{a}^2 - \abs{b}^2}{1 - \abs{a}^2}$.
So $P$ is positive semidefinite exactly when $\abs{b} \le \abs{a}$, which matches the [[Schwarz lemma|0MB48XDBY3]].

Non-example: the data must have $z_i \in \mathbb{D}$.
For $z_1 = 1$ on the unit circle the entry $P_{11}$ would have denominator $1 - \abs{z_1}^2 = 0$, so no Pick matrix is defined.

The entries are the values $(1 - w_i \overline{w_j}) k(z_i, z_j)$, where $k(z, w) = \frac{1}{1 - z \bar w}$ is the reproducing kernel of the [[Hardy space|PNNNFJVF6W]] $H^2$.
</document>

<document source="https://tamnd.github.io/math2/node/73XN6AS0WX.md" tag="73XN6AS0WX" kind="example">
# Enneper surface

## Example

For $(u, v) \in \R^2$, let
$$x(u, v) = \left(u - \frac{u^3}{3} + u v^2, \; -v + \frac{v^3}{3} - u^2 v, \; u^2 - v^2\right).$$
Its image is the Enneper surface.
It is a [[minimal surface|41X0YW2R9N]].

Check.
The partial derivatives are $x_u = (1 - u^2 + v^2, \, -2uv, \, 2u)$ and $x_v = (2uv, \, -1 + v^2 - u^2, \, -2v)$.
A direct computation gives $E = G = (1 + u^2 + v^2)^2$ and $F = 0$, so $x$ is a conformal immersion.
The second derivatives are $x_{uu} = (-2u, -2v, 2)$ and $x_{vv} = (2u, 2v, -2)$, so $x_{uu} + x_{vv} = 0$.
For a conformal parametrization this means that the [[mean curvature|PA78NFEPCF]] vanishes.

## Remarks

In the [[Weierstrass–Enneper parameterization|YWV6J23AP6]], this surface has data $f = 2$ and $g(z) = z$ on $\C$, with $z = u + iv$.
The surface is complete, but it is not embedded: it intersects itself.
Its Gauss map is the inverse stereographic projection of $g(z) = z$, which covers the sphere once except for one point, so its total curvature is $-4\pi$.

## Explanations

### L1 human

Near the origin the Enneper surface looks like a saddle.
Farther out its edges fold over and pass through each other.
It is one of the simplest minimal surfaces after the plane, the catenoid and the helicoid, and its formula is just a polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/73XV4RP5PT.md" tag="73XV4RP5PT" kind="definition">
# Field trace

## Definition

Let $L/K$ be a [[finite extension|0NRYPM9Y27]] and let $\alpha \in L$.
Let $m_\alpha \colon L \to L$ be the map $x \mapsto \alpha x$; it is a [[$K$-linear map|0RJF668P1T]] of the finite-dimensional $K$-vector space $L$.
The *trace* of $\alpha$ from $L$ to $K$ is $\mathrm{Tr}_{L/K}(\alpha) = \tr(m_\alpha) \in K$, the [[trace|0RPDTWY2JS]] of $m_\alpha$.

## Remarks

The map $\mathrm{Tr}_{L/K} \colon L \to K$ is $K$-linear, and $\mathrm{Tr}_{L/K}(c) = [L:K]\, c$ for $c \in K$.
It is transitive in towers: $\mathrm{Tr}_{M/K} = \mathrm{Tr}_{L/K} \circ \mathrm{Tr}_{M/L}$.
If the [[minimal polynomial|0GK1WB9R98]] of $\alpha$ over $K$ is $x^d + a_{d-1}x^{d-1} + \cdots + a_0$, then $\mathrm{Tr}_{L/K}(\alpha) = -[L:K(\alpha)]\, a_{d-1}$.
If $L/K$ is separable and $\sigma_1, \ldots, \sigma_n$ are the embeddings of $L$ over $K$ into an algebraic closure, then $\mathrm{Tr}_{L/K}(\alpha) = \sigma_1(\alpha) + \cdots + \sigma_n(\alpha)$.
The map $\mathrm{Tr}_{L/K}$ is not identically zero if and only if $L/K$ is separable.

Example: $L = \Q(i)$, $K = \Q$ and $\alpha = a + bi$ with $a, b \in \Q$.
Check: the matrix of $m_\alpha$ in the basis $1, i$ is $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$, so $\mathrm{Tr}(\alpha) = 2a$.

Non-example: the real part $a + bi \mapsto a$ is $\Q$-linear but is not the trace of $\Q(i)/\Q$; it is half of it, since $\mathrm{Tr}(1) = [\Q(i):\Q] = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/744826NNN9.md" tag="744826NNN9" kind="definition">
# Hardy–Littlewood maximal function

## Definition

Let $n \ge 1$, let $\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$, and let $f \colon \R^n \to \R$ be locally [[integrable|X7GTF3D98F]].
For $x \in \R^n$ and $r > 0$, let $B(x, r)$ be the [[open ball|0YE5868HZB]] of centre $x$ and radius $r$.
The *Hardy–Littlewood maximal function* of $f$ is the function $Mf \colon \R^n \to [0, \infty]$ given by
$$Mf(x) = \sup_{r > 0} \frac{1}{\lambda(B(x, r))} \int_{B(x, r)} \abs{f(y)} \, d\lambda(y).$$

## Remarks

Example: let $n = 1$ and $f = \mathbf{1}_{[0, 1]}$, and let $x > 1$.
For $r \ge x$ the average is $1/(2r) \le 1/(2x)$.
For $x - 1 < r < x$ the average is $(r - x + 1)/(2r)$, which increases with $r$ and tends to $1/(2x)$.
For $r \le x - 1$ the average is $0$.
So $Mf(x) = 1/(2x)$.

Non-example of integrability: this shows that $Mf$ need not be integrable even when $f$ is, since $\int_1^\infty dx/(2x) = \infty$.
In fact $Mf$ is integrable on $\R^n$ only when $f = 0$ almost everywhere.

Variants use balls containing $x$ instead of balls centred at $x$ (the uncentred maximal function), or cubes instead of balls.
They are comparable to $Mf$ up to constants depending on $n$.
</document>

<document source="https://tamnd.github.io/math2/node/74BPYJNZTM.md" tag="74BPYJNZTM" kind="concept">
# Unitary representation

## Summary

A unitary representation of a [[topological group|92700QAB5Y]] $G$ is a homomorphism from $G$ to the [[unitary operators|MM0EJ2GDE4]] on a [[Hilbert space|HM2HHE3J51]].
It must be continuous in the strong operator topology.
It is the kind of [[group representation|WYGRXCG2RN]] used for infinite groups in analysis and physics, because unitary operators preserve lengths and angles.
Harmonic analysis on a locally compact group is largely the study of its unitary representations.
In quantum mechanics, symmetry groups act on state spaces in this way.

## Explanations

### L1 human

A unitary operator is a rotation of a possibly infinite-dimensional space: it keeps all lengths and angles.
A unitary representation assigns such a rotation to each element of a group, so that the product of two elements goes to the composition of their rotations.
For example, the real line acts on the space $L^2(\R)$ of square-integrable functions by shifting: the number $t$ sends $f$ to the function $x \mapsto f(x - t)$.
Shifting does not change the integral of $\abs{f}^2$, so each shift is unitary, and shifting by a small amount changes $f$ only a little.
</document>

<document source="https://tamnd.github.io/math2/node/74BXAPVSW8.md" tag="74BXAPVSW8" kind="concept">
# Linear canonical transformation

## Summary

A linear canonical transformation is an integral transform of functions on $\R$ attached to a real $2 \times 2$ matrix of determinant $1$.
It includes the [[Fourier transform|DPVM34ADZV]], the [[fractional Fourier transform|28VV812XWQ]], the Fresnel transform, scalings and multiplication by chirps as special cases.
Composing two such transforms gives, up to a sign, the transform of the product matrix.
So they form a projective unitary representation of $SL(2, \R)$, which is a genuine representation of the [[metaplectic group|8B0JM2YDE2]].
In optics it describes the propagation of light through lenses and free space in the paraxial approximation.

## Explanations

### L1 human

A signal can be pictured in a plane with time along one axis and frequency along the other.
Linear maps of this plane that keep areas and orientation are given by matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $ad - bc = 1$.
For each such map there is a transform of signals that moves their picture in the plane accordingly.
The Fourier transform is the quarter-turn rotation, a scaling of time is a squeeze of the plane, and multiplying by $e^{i\pi c t^2}$ is a shear.
</document>

<document source="https://tamnd.github.io/math2/node/74DV24XGKZ.md" tag="74DV24XGKZ" kind="definition">
# Euler–Poisson equation for higher derivatives

## Definition

Let $k \ge 1$ and $a < b$.
Let $L \colon [a, b] \times \R^{k+1} \to \R$, written $L(x, z_0, z_1, \dots, z_k)$, be a function of class $C^{k+1}$, and write $L_{z_j}$ for its [[partial derivative|09EGC6F8PT]] in $z_j$.
Consider the functional
$$J(y) = \int_a^b L\big(x, y(x), y'(x), \dots, y^{(k)}(x)\big) \, dx.$$
The *Euler–Poisson equation* of $J$ is the equation
$$\sum_{j=0}^{k} (-1)^j \frac{d^j}{dx^j} \Big( L_{z_j}\big(x, y(x), y'(x), \dots, y^{(k)}(x)\big) \Big) = 0, \qquad a \le x \le b,$$
for an unknown $y \in C^{2k}([a, b])$.

## Remarks

For $k = 1$ it is the [[Euler–Lagrange equation|PS0VV149ET]] $L_{z_0} - \frac{d}{dx} L_{z_1} = 0$ of an [[integral functional|JRQZ803Q5Y]].
A minimiser $y \in C^{2k}([a, b])$ of $J$ among functions with prescribed $y^{(j)}(a)$ and $y^{(j)}(b)$ for $0 \le j \le k - 1$ solves it.
To see this, take variations $\eta$ with $\eta^{(j)}(a) = \eta^{(j)}(b) = 0$ for $j < k$, integrate the term with $\eta^{(j)}$ by parts $j$ times, and apply the fundamental lemma of the calculus of variations.

Example: for $k = 2$ and $L = \frac{1}{2} z_2^2$, one has $L_{z_0} = L_{z_1} = 0$ and $L_{z_2} = z_2$, so the equation is $y'''' = 0$.
Every cubic polynomial solves it.
Non-example: $y(x) = x^4$ does not solve this equation, since $y'''' = 24$.
</document>

<document source="https://tamnd.github.io/math2/node/74JREVZVY3.md" tag="74JREVZVY3" kind="definition">
# Eulerian poset

## Definition

Let $P$ be a finite [[graded poset|1C7TKGTB2F]] with rank function $\rho$.
Assume that $P$ is [[bounded|B0QT3SM5DT]], with a least element $\hat 0$ and a greatest element $\hat 1$.
For $x \le y$ in $P$, let $[x, y] = \set{z \in P : x \le z \le y}$.
The poset $P$ is *Eulerian* when for all $x < y$ in $P$, the interval $[x, y]$ has as many elements of even rank as of odd rank.

## Remarks

Equivalently, the [[Möbius function|2Y75S8NG5B]] of $P$ is $\mu(x, y) = (-1)^{\rho(y) - \rho(x)}$ for all $x \le y$.

Example: the lattice of subsets of a finite set $S$, with $\rho(A) = \abs{A}$.
Check: for $A \subsetneq B$ with $\abs{B \setminus A} = k \ge 1$, the interval $[A, B]$ has $\sum_{j \text{ even}} \binom{k}{j} = 2^{k-1}$ elements of each parity of rank.

Non-example: the chain $0 < 1 < 2$ with $\rho(i) = i$ is graded and bounded, but the interval $[0, 2]$ has two elements of even rank and one of odd rank.

Every interval of an Eulerian poset is again Eulerian.
The face lattice of a convex polytope is Eulerian.
</document>

<document source="https://tamnd.github.io/math2/node/74TDTC7ETT.md" tag="74TDTC7ETT" kind="definition">
# Continuous multiplication

## Definition

Let $M$ be a [[topological space|0CQHFPGJ45]] with a binary operation $M \times M \to M$, written $(x, y) \mapsto xy$.
The operation is a *continuous multiplication* if the map $\mu \colon M \times M \to M$, $\mu(x, y) = xy$, is [[continuous|0VGFHTM75Y]].
Here $M \times M$ carries the [[product topology|0GBG8B49AA]].

## Remarks

Example: $\R$ with its usual topology and ordinary multiplication.
The map $(x, y) \mapsto xy$ is a polynomial in the coordinates, so it is continuous on $\R^2$.
Non-example: $\Z$ with the cofinite topology and addition.
Every nonempty open set $U \times V$ of $\Z \times \Z$ meets the set $\set{(x, -x) : x \in \Z}$, because $U$ and $-V$ are cofinite and so they intersect.
So the preimage of the open set $\Z \setminus \set{0}$ under addition contains no nonempty open set, and addition is not continuous.
Continuity of $\mu$ implies that every left translation $y \mapsto xy$ and every right translation $x \mapsto xy$ is continuous.
The converse fails: in the non-example above every translation is a bijection of $\Z$, hence a homeomorphism for the cofinite topology.
In additive notation the same condition is called continuous addition.
</document>

<document source="https://tamnd.github.io/math2/node/74VCDYQDGP.md" tag="74VCDYQDGP" kind="concept">
# NIP theory

## Summary

A NIP theory is a complete first-order theory in which no formula has the independence property.
A formula has the independence property when, with suitable parameters, it can cut out every subset of arbitrarily large finite sets.
NIP theories include both the [[stable theories|T7JW5QY6KR]] and the [[o-minimal theories|DTWHAW1N0V]].
The condition says that every definable family of sets has finite [[VC dimension|KCFXR9K3GD]], which links model theory with learning theory and combinatorics.

## Explanations

### L1 human

Fix a formula and think of it as a way to choose subsets, one subset for each value of the parameter.
A formula is "wild" if, for every $n$, there are $n$ points such that every one of the $2^n$ subsets of them can be chosen.
NIP means that no formula is wild in this way.
On the real line with its order, the formula $x < y$ only chooses initial segments, so among two points $a < b$ it can never choose $\set{b}$ alone.
In the random graph, the edge relation can choose every subset of any finite set of vertices, so its theory is not NIP.
</document>

<document source="https://tamnd.github.io/math2/node/74VMD07DRX.md" tag="74VMD07DRX" kind="statement">
# Lebesgue decomposition theorem

## Statement

Let $(X, \Sigma)$ be a measurable space and let $\mu$ and $\nu$ be [[σ-finite measures|WEBBT18D4X]] on $\Sigma$.
Then there are measures $\nu_a$ and $\nu_s$ on $\Sigma$ such that:

1. $\nu = \nu_a + \nu_s$.
2. $\nu_a$ is [[absolutely continuous|P0P21KDV6E]] with respect to $\mu$.
3. $\nu_s$ and $\mu$ are [[mutually singular|EBGECZNCV5]].

The measures $\nu_a$ and $\nu_s$ are unique.

## Remarks

One proof applies the Radon–Nikodym theorem to $\mu$ and $\mu + \nu$.
With $f = d\mu / d(\mu + \nu)$, the set $S = \{f = 0\}$ is $\mu$-null, and one puts $\nu_s(A) = \nu(A \cap S)$ and $\nu_a(A) = \nu(A \setminus S)$.
Uniqueness holds because a measure that is both absolutely continuous and singular with respect to $\mu$ is zero.

Example: let $\mu$ be Lebesgue measure on $\R$ and $\nu = \mu|_{[0,1]} + \delta_2$, where $\delta_2$ is the point mass at $2$.
Then $\nu_a = \mu|_{[0,1]}$ and $\nu_s = \delta_2$.

On the real line with $\mu$ Lebesgue measure, the singular part splits further into a part made of point masses and a continuous singular part.
The distribution of the Cantor function is an example of a continuous singular measure.

## Explanations

### L1 human

Any reasonable measure can be cut into two pieces relative to a reference measure.
One piece is smooth: it has a density with respect to the reference.
The other piece lives on a set that the reference does not see at all.

### L2 human

Combined with the Radon–Nikodym theorem, it says $\nu = f\mu + \nu_s$ with $f = d\nu_a / d\mu$ and $\nu_s \perp \mu$.
In probability this splits a distribution on $\R$ into a part with a density, a discrete part and a singular continuous part.
The same decomposition holds for finite signed and complex measures, applied to the total variation.
</document>

<document source="https://tamnd.github.io/math2/node/74X7JRG2J7.md" tag="74X7JRG2J7" kind="concept">
# Semilattice

## Summary

A join-semilattice is a [[partially ordered set|0QV2JS9RJ3]] in which every two elements have a [[join|J2T75ATRV5]].
A meet-semilattice is one in which every two elements have a meet.
Algebraically, a semilattice is a [[semigroup|0XMEM406JM]] whose operation is commutative and idempotent.
The two views are equivalent, and a lattice is a set that is both a join- and a meet-semilattice for the same order.
Semilattices appear in the theory of semigroups, in domain theory and in the analysis of programs.

## Explanations

### L1 human

A join-semilattice is an ordering in which any two things have a smallest common upper bound, but not always a largest common lower bound.
Picture a rooted tree drawn with its root at the top: any two nodes have a closest common ancestor above them, and this is their join.
For example, take two branches $a$ and $b$ that meet at a top element $1$.
Then $a \vee b = 1$, but $a$ and $b$ have no common lower bound at all.
So this is a join-semilattice but not a meet-semilattice.
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<document source="https://tamnd.github.io/math2/node/750MCA763X.md" tag="750MCA763X" kind="concept">
# Strong Cesàro summability

## Summary

Strong Cesàro summability asks that the partial sums of a series stay close to a value $s$ on average, measured by the mean of $\abs{s_k - s}^q$.
Ordinary [[Cesàro summation|N8ZGG7EW6J]] only asks that the mean of $s_k - s$ tends to $0$, which allows positive and negative deviations to cancel.
For exponent $q \ge 1$, strong Cesàro summability implies Cesàro summability to the same value.
It is used in the theory of Fourier series and is closely tied to [[statistical convergence|ZW6EHSMANH]].

## Explanations

### L1 human

Grandi's series $1 - 1 + 1 - \cdots$ has partial sums $1, 0, 1, 0, \dots$, whose average tends to $1/2$.
But each partial sum is at distance $1/2$ from $1/2$, so the partial sums are never close to the value.
Strong summability forbids this: the average distance from the value must itself tend to $0$.
So the partial sums must be near $s$ most of the time, not only in balance.
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<document source="https://tamnd.github.io/math2/node/752Z7DRT7P.md" tag="752Z7DRT7P" kind="concept">
# Finite difference coefficients

## Summary

Finite difference coefficients are the weights that turn values of a function on a stencil of points into an approximation of a derivative.
For a given derivative and stencil, they are found by matching Taylor expansions.
This leads to a Vandermonde system with a unique solution.
They are the building blocks of [[numerical differentiation|NVH3F8FTYS]] and of the [[finite difference method|X7HYEBA5SA]] for differential equations.
Tables of central, forward and backward coefficients of various orders are standard.

## Explanations

### L1 human

To estimate the second derivative of $f$ at $x$ from the values at $x - h$, $x$ and $x + h$, one uses the weights $1, -2, 1$ and divides by $h^2$.
For $f(x) = x^2$ at $x = 0$ with $h = 1$, this gives $1 - 0 + 1 = 2$, the true second derivative.
Other derivatives and other sets of points need other weights.
Finite difference coefficients are these weights, and there is a systematic way to compute them for any stencil.
</document>

<document source="https://tamnd.github.io/math2/node/758EYCN27B.md" tag="758EYCN27B" kind="concept">
# Turing machine

## Summary

A Turing machine is an abstract model of computation with a finite set of states and an unbounded tape of symbols.
A moving head reads and writes the symbols one cell at a time.
Alan Turing introduced it in 1936 to make the notion of an algorithm precise.
A function is computable when some Turing machine computes it, and by the [[Church–Turing thesis|RWRNBS3Z7B]] this matches the informal idea of an algorithm.
Turing machines are the standard model in [[computability theory|X14Y880PN5]] and in complexity theory.

## Explanations

### L1 human

Picture a long strip of paper divided into cells, each holding one symbol, and a reading head that sees one cell at a time.
The machine is in one of finitely many states, like moods.
At each step it reads the symbol under the head, writes a symbol, moves one cell left or right, and changes state, all by a fixed table of rules.
For example, a machine can scan a word of $0$s and $1$s from left to right and flip each digit, stopping at the first blank cell.
Despite this simplicity, every algorithm can be carried out by such a machine.
</document>

<document source="https://tamnd.github.io/math2/node/759C4C6NXZ.md" tag="759C4C6NXZ" kind="definition">
# Summation in topological groups

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] whose group operation $+$ is [[commutative|082GHNPJ53]].
Let $I$ be a set, and let $(x_i)_{i \in I}$ be a family of elements of $G$.
For a finite subset $F \subseteq I$ write $s_F = \sum_{i \in F} x_i$.
The family is *summable* with *sum* $s \in G$ if the following holds.
For every [[neighbourhood|0ZCY0Z80B6]] $U$ of $s$ there is a finite set $F_0 \subseteq I$ with $s_F \in U$ for every finite $F$ with $F_0 \subseteq F \subseteq I$.
Then one writes $\sum_{i \in I} x_i = s$.

## Remarks

Equivalently, the [[net|4DVJQJV0DG]] $(s_F)$, indexed by the finite subsets of $I$ ordered by inclusion, converges to $s$.
If $G$ is [[Hausdorff|0S1FJ8VEX9]], a family has at most one sum; in Mathlib this is `HasSum.unique`.
The definition does not use any order on $I$, so a bijection $\sigma \colon I \to I$ does not change summability or the sum.

Example: in $\R$, the family $(2^{-n})_{n \ge 0}$ has sum $2$.
For $U \supseteq (2 - \varepsilon, 2 + \varepsilon)$ choose $N$ with $2^{-N} < \varepsilon$ and $F_0 = \set{0, \dots, N}$.
For finite $F \supseteq F_0$ one has $2 - 2^{-N} \le s_F < 2$, so $s_F \in U$.

Non-example: in $\R$, the family $((-1)^{n+1}/n)_{n \ge 1}$ is not summable.
For any finite $F_0$, adding finitely many odd $n$ beyond $F_0$ to $F_0$ makes $s_F$ as large as one likes, because $\sum_{n \text{ odd}} 1/n$ diverges.

In $\R$ and $\C$, a family indexed by $\N$ is summable if and only if it is [[absolutely convergent|0YBDSJARWN]], and then the sum equals the sum of the series.
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<document source="https://tamnd.github.io/math2/node/759JMPAZCZ.md" tag="759JMPAZCZ" kind="definition">
# Advanced Z-transform

## Definition

Let $f \colon [0, \infty) \to \C$ be a function, let $T > 0$ be a sampling period, and let $m$ be a real number with $0 \le m \le T$.
The *advanced Z-transform* of $f$ is
$$F(z, m) = \sum_{k = 0}^{\infty} f(kT + m) \, z^{-k},$$
for the complex $z$ where the series converges absolutely.
So $F(\cdot, m)$ is the [[unilateral Z-transform|EPG6JPVJE7]] of the sequence $(f(kT + m))_{k \ge 0}$.

## Remarks

For $m = 0$ it is the ordinary [[Z-transform|5FY0JQGHYZ]] of the samples $f(kT)$.
Example: for $f(t) = e^{-at}$ with $a \in \C$, $F(z, m) = e^{-am} \sum_{k \ge 0} (e^{-aT} z^{-1})^k = e^{-am} \, z/(z - e^{-aT})$ for $\abs{z} > \abs{e^{-aT}}$.
For $f(t) = t$, $F(z, m) = \sum_{k \ge 0} (kT + m) z^{-k} = Tz/(z - 1)^2 + m z/(z - 1)$ for $\abs{z} > 1$.
Non-example: the sum $\sum_k f(k T + m) z^{-k}$ with $m$ outside $[0, T]$ is not covered; for $m = T$ it is not a new transform either, since $F(z, T) = z (F(z, 0) - f(0))$.
Some texts call a closely related transform, with a delay parameter in place of the advance, the modified Z-transform.
</document>

<document source="https://tamnd.github.io/math2/node/759KSC8XQD.md" tag="759KSC8XQD" kind="definition">
# Geometric lattice

## Definition

Let $L$ be a finite [[lattice|WH4SFAACAS]].
The lattice $L$ is *geometric* if:

1. $L$ is [[atomistic|99DVVV7BH2]]: every element of $L$ is the join of a set of [[atoms|TTVCTW6TNC]];
2. $L$ is [[semimodular|C1NET23GG0]] (upper semimodular).

## Remarks

Example: the Boolean lattice of all subsets of a finite set $S$.
The atoms are the singletons, and every subset is the union of its singletons.
If $A$ covers $A \cap B$, then $A = (A \cap B) \cup \set{s}$ with $s \notin B$, so $A \cup B = B \cup \set{s}$ covers $B$.

Non-example: the chain $0 < a < 1$ with three elements.
It is semimodular, but its only atom is $a$, and $1$ is not a join of atoms.

The lattice of flats of a finite [[matroid|PX31ZNYS3Z]] is geometric, and every geometric lattice is isomorphic to the lattice of flats of a simple matroid, unique up to isomorphism.
Some authors allow infinite geometric lattices, replacing finiteness by finite length or by compactness conditions.
</document>

<document source="https://tamnd.github.io/math2/node/75B8VHDGSA.md" tag="75B8VHDGSA" kind="concept">
# Iterated function system

## Summary

An iterated function system is a finite family of [[contractions|0QN69YA357]] of a complete metric space.
It has a unique nonempty compact set, its attractor, that is the union of its images under the maps of the family.
Many fractals, such as the [[Cantor set|0BXNDZFQME]] and the [[Sierpiński carpet|PCM3H6J3EE]], are attractors of iterated function systems.
The attractor can be drawn by applying the maps at random to a point, which is the chaos game.

## Explanations

### L1 human

Take a few rules, each of which shrinks the plane towards some point.
Apply all rules to a picture and put the results together, and repeat.
Whatever picture you start from, the results converge to one shape, which is made of shrunk copies of itself.
For the two rules $x \mapsto x/3$ and $x \mapsto x/3 + 2/3$ on the line, the shape is the Cantor set: two copies of itself, each one third as large.
</document>

<document source="https://tamnd.github.io/math2/node/75BPV95AAS.md" tag="75BPV95AAS" kind="concept">
# Complete Boolean algebra

## Summary

A complete Boolean algebra is a [[Boolean algebra|0XADJKM699]] in which every subset has a supremum, so that it is also a [[complete lattice|D7A0ZGZT5S]].
Power sets are complete, and so are the algebras of [[regular open sets|833XAA2Q65]] of topological spaces.
Every Boolean algebra embeds densely into a complete one, its completion.
Complete Boolean algebras are the basic tool of Boolean-valued models and of forcing in set theory.

## Explanations

### L1 human

In a Boolean algebra one can always take "or" of two or of finitely many elements.
In a complete Boolean algebra one can also take "or" of infinitely many elements at once, and the result is still in the algebra.
The subsets of a set form such an algebra: the "or" of any family of subsets is their union.
The finite and cofinite subsets of $\N$ do not.
The set of even numbers is a union of finite sets, but it is neither finite nor cofinite, and the family has no least upper bound in the algebra.
</document>

<document source="https://tamnd.github.io/math2/node/75EJ0HB40M.md" tag="75EJ0HB40M" kind="definition">
# Cofinal subset

## Definition

Let $(P, \le)$ be a [[preordered set|0VABYWJWH9]] and let $C \subseteq P$ be a [[subset|0BJR9QR08E]].
The subset $C$ is *cofinal* in $P$ when for every $x \in P$ there is $c \in C$ with $x \le c$.

## Remarks

Example: the set $\N$ is cofinal in $\R$ with the usual order.
Check: for $x \in \R$, the integer $\ceil{\abs{x}}$ is in $\N$ and is at least $x$.

Non-example: the set $(-\infty, 0]$ is not cofinal in $\R$, since no element of it is at least $1$.

If $P$ has a greatest element $m$, then a subset is cofinal exactly when it contains some $c$ with $m \le c$.
In a partial order, this means that the subset contains $m$.
Being cofinal is transitive: a cofinal subset of a cofinal subset of $P$ is cofinal in $P$.
The dual notion is a coinitial subset, in which every element of $P$ lies above some element of the subset.
</document>

<document source="https://tamnd.github.io/math2/node/75H4HY10Z4.md" tag="75H4HY10Z4" kind="concept">
# Classical group

## Summary

The classical groups are the [[general linear groups|0FCHE902D7]] and [[special linear groups|0V9PMN0RBH]], and the groups of linear maps that preserve a nondegenerate form.
For symmetric, alternating and Hermitian forms these are the [[orthogonal|0TKYAEJ8N7]], symplectic and [[unitary groups|0R9R0Y1HDK]].
The name goes back to Hermann Weyl's book on the classical groups.
Over $\C$ they give all the complex simple Lie groups except five exceptional ones, up to local isomorphism.
Over finite fields, their projective versions give several infinite families of finite simple groups.

## Explanations

### L1 human

A classical group is a group of matrices defined by one natural condition.
The general linear group keeps only invertibility, and the special linear group asks for determinant $1$.
The orthogonal group keeps lengths and angles, as rotations and reflections of space do.
The symplectic group keeps an area-like form, which appears in classical mechanics, and the unitary group keeps the length of complex vectors, as in quantum mechanics.
For example, the rotations of three-dimensional space form the special orthogonal group $\mathrm{SO}(3)$.
</document>

<document source="https://tamnd.github.io/math2/node/75P3EJPH81.md" tag="75P3EJPH81" kind="counterexample">
# Lindley's paradox

## Counterexample

Let $X_1, \dots, X_n$ be i.i.d. [[normal|07RXBVTVZB]] $\mathcal{N}(\theta, 1)$, so that $\bar{X}_n \sim \mathcal{N}(\theta, 1/n)$.
Test $H_0 \colon \theta = 0$ against $H_1 \colon \theta \ne 0$.
Suppose the data give $\sqrt{n}\,\bar{x}_n = z$ with $z = 1.96$, for every $n$.
The two-sided [[p-value|0EHYZFZ4PK]] is then $P(\abs{Z} \ge 1.96) \approx 0.05$ for $Z \sim \mathcal{N}(0, 1)$, whatever $n$ is.
So a test of level $0.05$ is at the edge of rejecting $H_0$ for every $n$.
For the Bayesian comparison, give $\theta$ the prior $\mathcal{N}(0, \tau^2)$ under $H_1$, with $\tau > 0$ fixed.
Under $H_0$ the density of $\bar{X}_n$ is that of $\mathcal{N}(0, 1/n)$, and under $H_1$ it is that of $\mathcal{N}(0, \tau^2 + 1/n)$.
A direct computation gives the [[Bayes factor|NSBYE143KS]] of $H_0$ against $H_1$:
$$B_{01} = \sqrt{1 + n\tau^2} \, \exp\left(-\frac{z^2}{2} \cdot \frac{n\tau^2}{1 + n\tau^2}\right).$$
As $n \to \infty$ with $z$ fixed, $B_{01} \to \infty$.
So for any prior probability $P(H_0) \in (0, 1)$, the posterior probability of $H_0$ tends to $1$, while the p-value stays at $0.05$.

## Remarks

The p-value and the posterior probability answer different questions, and for large samples they can point in opposite directions.
For large $n$, a fixed $z$ means that $\bar{x}_n$ is very close to $0$.
Such a value is much better explained by $\theta = 0$ than by a $\theta$ spread over a range of width about $\tau$.
The conflict also depends on the prior: as $\tau \to \infty$ with $n$ fixed, $B_{01} \to \infty$ as well, so a vague prior under $H_1$ favours $H_0$.

## Explanations

### L1 human

With a huge sample, a small but "significant" deviation from zero can appear.
A classical test then says "reject the null hypothesis at the $5\%$ level".
A Bayesian comparison of "the effect is exactly zero" with "the effect is some reasonable size" can say the opposite: the data favour exactly zero.
The paradox shows that significance and evidence are not the same thing.
</document>

<document source="https://tamnd.github.io/math2/node/75WGJ014GR.md" tag="75WGJ014GR" kind="concept">
# Real Hardy space H1

## Summary

The real Hardy space $H^1(\R^n)$ is the space of integrable functions $f$ on $\R^n$ whose Riesz transforms $R_1 f, \dots, R_n f$ are also integrable.
It is a proper subspace of $L^1(\R^n)$, and its elements have integral zero.
It is the right replacement for $L^1$ in harmonic analysis.
Singular integrals such as the Hilbert and Riesz transforms map $H^1$ to $L^1$, but not $L^1$ to $L^1$.
Its dual space is the space [[BMO|QJYTXFT8Y4]] of functions of bounded mean oscillation, and its elements are sums of simple building blocks called atoms.
For $n = 1$ it is the space of real parts of boundary values of functions in the holomorphic [[Hardy space|PNNNFJVF6W]] $H^1$ of the upper half-plane.

## Explanations

### L1 human

An integrable function can still behave badly under singular integrals, like the Hilbert transform, which can turn it into a non-integrable function.
The real Hardy space keeps exactly the integrable functions for which this does not happen.
Such functions must have some cancellation: their total integral is zero.
A typical element on $\R$ is $\mathbf{1}_{[0, 1]} - \mathbf{1}_{[-1, 0]}$, a positive bump next to a negative bump of the same size.
</document>

<document source="https://tamnd.github.io/math2/node/75XCD68557.md" tag="75XCD68557" kind="definition">
# Padé approximant

## Definition

Let $f(z) = \sum_{k \ge 0} c_k z^k$ be a [[formal power series|072W67KFYK]] with complex coefficients, and let $m, n \ge 0$ be integers.
Consider pairs $(P, Q)$ of polynomials with complex coefficients such that:

1. $P$ has [[degree|0DPF6WRZW6]] at most $m$;
2. $Q$ is not the zero polynomial and has degree at most $n$;
3. the series $Q(z) f(z) - P(z)$ has no terms of degree at most $m + n$, that is, $Q f - P = O(z^{m+n+1})$.

Such pairs exist, and the [[rational function|0WCZH4AFMT]] $P/Q$ is the same for all of them.
This rational function is the *$[m/n]$ Padé approximant* of $f$, written $[m/n]_f$.

## Remarks

Existence: condition 3 for the coefficients of $z^{m+1}, \dots, z^{m+n}$ is a system of $n$ homogeneous linear equations in the $n + 1$ coefficients of $Q$.
So a nonzero $Q$ exists, and then $P$ is the truncation of $Q f$ at degree $m$.
Uniqueness: for two pairs, $P Q' - P' Q = Q'(Q f - P) - Q(Q' f - P')$ is a polynomial of degree at most $m + n$ that is $O(z^{m+n+1})$, so it is $0$.
Example: for $f = e^z$, $m = n = 1$, take $Q = 1 - z/2$ and $P = 1 + z/2$.
Then $Q f = 1 + z/2 + 0 \cdot z^2 - z^3/12 + \cdots$, so $Q f - P = O(z^3)$ and $[1/1]_{e^z} = (2 + z)/(2 - z)$.
Non-example: $P = 1 + z$ and $Q = 1$ satisfy clauses 1 and 2 for $m = n = 1$, but $Q f - P = z^2/2 + \cdots$ fails clause 3.
For $n = 0$ the approximant $[m/0]_f$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $m$.
When $Q(0) \ne 0$ one usually normalizes $Q(0) = 1$, and then $f - P/Q = O(z^{m+n+1})$; when $Q(0) = 0$ the approximant may match fewer coefficients.
</document>

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# Dirichlet conditions

## Statement

Let $f \colon \R \to \C$ be a [[periodic function|0ATF6Q0633]] of period $2\pi$ that has [[bounded variation|A8251BHCMP]] on $[-\pi, \pi]$.
Let $S_N f$ be the partial sums of its [[Fourier series|G12KXDHPPS]].

1. For every $x \in \R$ the one-sided limits $f(x^+)$ and $f(x^-)$ exist, and
$$\lim_{N \to \infty} S_N f(x) = \frac{f(x^+) + f(x^-)}{2}.$$
2. If moreover $f$ is continuous on an open interval $I$, then $S_N f \to f$ uniformly on every closed bounded interval contained in $I$.

## Remarks

A function of bounded variation is a difference of two monotone functions, so it suffices to treat a monotone $f$.
One writes $S_N f(x) - s$ as an integral of $f(x + t) + f(x - t) - 2s$ against the [[Dirichlet kernel|3JY4BVE4DW]] over $[0, \pi]$, with $s = \frac{1}{2}(f(x^+) + f(x^-))$.
The second mean value theorem for integrals, together with the uniform boundedness of $\int_0^a \frac{\sin((N + 1/2) t)}{t} \, dt$ in $a$ and $N$, controls the part near $t = 0$, and the Riemann–Lebesgue lemma handles the rest.
Dirichlet's original conditions are that $f$ is bounded on a period, with finitely many maxima and minima and finitely many discontinuities there.
Such an $f$ has bounded variation, so Jordan's form above contains them.
Example: for the square wave, equal to $\operatorname{sgn} x$ on $(-\pi, \pi)$, the series converges to $0$ at the [[jumps|004FH91BN5]], the average of $-1$ and $1$, and to $f(x)$ elsewhere.
Bounded variation cannot be replaced by continuity: there are continuous functions whose Fourier series diverge at a point.

## Explanations

### L1 human

If a periodic function does not wiggle infinitely much, its Fourier series converges at every point.
At a point where the function jumps, the series picks the midpoint of the jump.
Where the function is continuous, it converges to the value of the function.

### L2 human

This is the convergence theorem used in most applications, since piecewise monotone and piecewise smooth functions have bounded variation.
It is a pointwise test, like the Dini test; the two tests are independent, and neither implies the other.
Part 2 explains why the convergence for the square wave is uniform away from the jumps but not near them, where the Gibbs phenomenon appears.
Camille Jordan stated the test in this form, for functions of bounded variation.
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<document source="https://tamnd.github.io/math2/node/762Y2NYA7W.md" tag="762Y2NYA7W" kind="statement">
# Egorov's theorem

## Statement

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] with $\mu(X) < \infty$.
Let $f_1, f_2, \ldots$ and $f$ be [[measurable functions|YYNCJSR0PJ]] from $X$ to $\R$ such that $f_n(x) \to f(x)$ for [[almost every|RARTMA0ZEC]] $x \in X$.
Then for every $\varepsilon > 0$ there is a set $E \in \Sigma$ with $\mu(E) < \varepsilon$ such that $f_n \to f$ [[uniformly|08VZ57PT26]] on $X \setminus E$.

## Remarks

For $k, n \ge 1$ let $E_{n,k} = \bigcup_{m \ge n} \set{x : \abs{f_m(x) - f(x)} \ge 1/k}$.
For fixed $k$ these sets decrease in $n$ to a null set, so continuity from above, which needs $\mu(X) < \infty$, gives $n_k$ with $\mu(E_{n_k, k}) < \varepsilon / 2^k$.
The set $E = \bigcup_k E_{n_k, k}$ works.
Example: $f_n(x) = x^n$ on $[0, 1]$ with Lebesgue measure tends to $0$ almost everywhere, not uniformly; it does converge uniformly on $[0, 1 - \delta]$ for each $\delta > 0$.
The finite measure hypothesis is necessary: on $\R$ with Lebesgue measure, $f_n = \mathbf{1}_{[n, \infty)}$ tends to $0$ everywhere.
Every set $E$ of finite measure misses points of $[n, \infty)$ for each $n$, so $\sup_{X \setminus E} f_n = 1$ and the convergence is never uniform there.

## Explanations

### L1 human

A sequence of functions may converge at every point but at very different speeds, so that it is not uniformly convergent.
Egorov's theorem says that on a space of finite measure this is a small problem.
After removing a set as small as one likes, the convergence becomes uniform on what is left.

### L2 human

This is the third of Littlewood's three principles: pointwise convergence is nearly uniform convergence.
It is used to prove convergence theorems for integrals, such as the bounded convergence theorem and the Vitali convergence theorem.
The convergence it gives, uniform outside sets of small measure, is called almost uniform convergence.
Almost uniform convergence implies [[convergence in measure|G5WT6QN8WR]], so on finite measure spaces almost everywhere convergence implies convergence in measure.
</document>

<document source="https://tamnd.github.io/math2/node/764GZHTGV8.md" tag="764GZHTGV8" kind="concept">
# Finitely presented algebra

## Summary

An algebra $B$ over a commutative ring $A$ is finitely presented if it can be described by finitely many generators and finitely many relations.
That is, $B \cong A[x_1, \ldots, x_n]/I$ with $I$ a finitely generated ideal.
Over a Noetherian ring every [[finite type algebra|AC5TFFQT9M]] is finitely presented, so the distinction matters only over non-Noetherian bases.
Finite presentation is the right finiteness condition for limit arguments and for étale and smooth morphisms in algebraic geometry.

## Explanations

### L1 human

To describe a ring by a finite amount of data, one lists some generators and some polynomial equations between them.
For example, $\Z[i]$ is $\Z[x]/(x^2 + 1)$: one generator and one relation.
Finite type asks only for finitely many generators, while finite presentation also asks that finitely many equations imply all the others.
Over a ring like $\Z$ or a field, the second demand is automatic.
</document>

<document source="https://tamnd.github.io/math2/node/765P54K9BR.md" tag="765P54K9BR" kind="definition">
# Trigonometric interpolation

## Definition

Let $n \ge 0$ and let $x_0, \ldots, x_{2n}$ be $2n + 1$ real numbers that are pairwise distinct modulo $2\pi$.
Let $y_0, \ldots, y_{2n}$ be complex numbers.
The *trigonometric interpolation polynomial* for these data is the unique [[trigonometric polynomial|J3NJQ5FZJ5]]
$$p(x) = \sum_{k=-n}^{n} c_k e^{ikx}$$
of degree at most $n$ such that $p(x_j) = y_j$ for $j = 0, \ldots, 2n$.
*Trigonometric interpolation* is the construction of $p$ from the data.

## Remarks

Existence and uniqueness: the conditions form a square linear system in $c_{-n}, \ldots, c_n$.
Let $p$ have degree at most $n$ and vanish at all $x_j$.
With $z = e^{ix}$, the function $z^n p$ is an ordinary polynomial of degree at most $2n$ in $z$.
It has the $2n + 1$ distinct roots $e^{ix_j}$, so $p = 0$.
Hence the system has a unique solution.
If all $y_j$ are real, then $c_{-k} = \overline{c_k}$ and $p$ is real-valued.

For equally spaced nodes $x_j = 2\pi j / (2n+1)$, the coefficients are
$$c_k = \frac{1}{2n+1} \sum_{j=0}^{2n} y_j e^{-ik x_j},$$
a [[discrete Fourier transform|TST0GH66A5]] of the data.

Example: $n = 1$, nodes $0, 2\pi/3, 4\pi/3$ and values $1, 0, 0$.
The formula gives $c_0 = c_1 = c_{-1} = 1/3$, so $p(x) = \frac{1}{3} + \frac{2}{3} \cos x$.
Check: $p(0) = 1$ and $p(2\pi/3) = p(4\pi/3) = \frac{1}{3} - \frac{1}{3} = 0$.

Non-example: the nodes $0$ and $2\pi$ are equal modulo $2\pi$, so they are not allowed.
The data $p(0) = 1$, $p(2\pi) = 0$ cannot be met by any $2\pi$-periodic function.

With an even number $2n$ of nodes one uses trigonometric polynomials of degree $n$ with a constraint on the top frequency, such as a cosine term only.
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# Splitting circle method

## Statement

Let $p$ be a polynomial of degree $n \ge 1$ with [[complex|09FTAET5Z4]] coefficients.
Let $\Gamma$ be the circle $\abs{z - c} = \rho$, traversed once counterclockwise, with no [[root|019MA1VTR2]] of $p$ on it.
Let $z_1, \dots, z_k$ be the roots of $p$ inside $\Gamma$, listed with multiplicity, and let $f(z) = \prod_{j=1}^k (z - z_j)$.

1. For every integer $m \ge 0$, the [[power sum|0ZNXZB4HNB]] $s_m = z_1^m + \cdots + z_k^m$ equals the [[contour integral|06ZGPYTWK8]]
$$s_m = \frac{1}{2\pi i} \oint_\Gamma z^m \, \frac{p'(z)}{p(z)} \, dz.$$
In particular $s_0 = k$.
2. The coefficients of $f$ are determined by $s_1, \dots, s_k$ through [[Newton's identities|0WG2A3HE0S]]. Moreover $p = f g$, where the roots of the polynomial $g$ are the roots of $p$ outside $\Gamma$.

## Remarks

Clause 1 follows from the [[residue theorem|032HKYS329]]: $p'/p = \sum_j 1/(z - z_j)$ over all roots of $p$, and $z^m/(z - z_j)$ has residue $z_j^m$ at $z_j$ when $z_j$ lies inside $\Gamma$, and no pole inside otherwise.
For $m = 0$ this is the [[argument principle|0PN2BXRS8G]].
The *splitting circle method* of Arnold Schönhage turns this into an algorithm.
It chooses a circle that splits the roots into two nonempty groups and stays well away from all roots, using estimates of the root moduli such as Graeffe squaring steps.
It approximates the integrals by the trapezoidal rule on the circle, which is computed with the fast Fourier transform.
It then refines the approximate factorization $p \approx f g$ by a Newton iteration, and it recurses on $f$ and $g$ until all factors are linear.
The method was designed to bound the cost of computing all roots of a polynomial to a given precision, measured in bit operations.

## Explanations

### L1 human

Draw a circle in the complex plane that has some roots of a polynomial inside and the rest outside.
Integrals around the circle count the roots inside and give the sums of their powers.
From these sums one can rebuild the factor of the polynomial that holds exactly the roots inside, and then split again until every root stands alone.

### L2 human

The method is a divide-and-conquer approach to the fundamental theorem of algebra made effective.
Its main uses are in complexity theory and in high-precision software, rather than in everyday root finding.
It shows how complex analysis, through the residue theorem, gives concrete numerical algorithms.
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# Projection operation

## Summary

A projection operation on a set $A$ is an operation $A^n \to A$ that returns one fixed argument and ignores the others.
The projections are the operations given by single variables, so they are the simplest term operations of any algebra.
Every [[clone|PTKR703G6A]] contains all projections, and the set of all projections is the smallest clone on $A$.

## Explanations

### L1 human

The projection $\pi^3_2$ takes three inputs and hands back the second one.
For example, $\pi^3_2(5, 7, 9) = 7$.
Such an operation does no real computation; it only selects a position.
Projections matter because composing operations needs a way to feed the same variable into several places, and projections do exactly this bookkeeping.
</document>

<document source="https://tamnd.github.io/math2/node/76BXV761BR.md" tag="76BXV761BR" kind="definition">
# Residually finite algebra

## Definition

Fix a signature and let $A$ be an algebra.
The algebra $A$ is *residually finite* if the following holds for all $a, b \in A$ with $a \ne b$.
There are a finite algebra $F$ of the same signature and a [[homomorphism|KBXSZC6BV1]] $f \colon A \to F$ with $f(a) \ne f(b)$.

## Remarks

Equivalently, $A$ is isomorphic to a [[subdirect product|88WN3AD2VQ]] of finite algebras.
Equivalently, the congruences of $A$ with finite quotient intersect to the trivial congruence.
For groups, this is the definition of a [[residually finite group|HEE922EHQ8]].

Example: the group $(\Z, +)$ is residually finite.
For $a \ne b$, let $n = \abs{a - b} + 1$; the reduction map $\Z \to \Z/n\Z$ sends $a$ and $b$ to different classes, since $n$ does not divide $a - b$.

Non-example: the group $(\Q, +)$ is not residually finite.
A homomorphic image of $\Q$ is divisible, and a finite divisible abelian group is trivial, since multiplication by its order is both zero and onto.
So every homomorphism from $\Q$ to a finite group sends $0$ and $1$ to the same element.
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<document source="https://tamnd.github.io/math2/node/76DF4G7J9S.md" tag="76DF4G7J9S" kind="concept">
# Cellular algebra

## Summary

A cellular algebra is an [[associative algebra|3CTW1P4HBP]] with a special basis, the cellular basis.
This basis is compatible with an anti-involution and with a partial order on its labels.
The notion was introduced by Graham and Lehrer.
The basis produces cell modules, which play the role of the Specht modules of the symmetric group.
The simple modules come out of the cell modules in a uniform way.
[[Hecke algebras|5J2P2AGY4N]] of type $A$, [[Temperley–Lieb algebras|TZNR514JQN]] and Brauer algebras are cellular.

## Explanations

### L1 human

Many algebras that come from combinatorics have a basis indexed by pairs of objects of the same shape, such as pairs of tableaux.
In a cellular algebra, multiplying a basis element on the left only changes the first object of the pair, up to terms of smaller shape.
A reversing map swaps the two objects of each pair.
The model is the matrix algebra: the unit matrix $E_{ij}$ is labelled by the pair $(i, j)$, left multiplication mixes the first index, and transposition swaps the indices.
This structure lets one build all the simple modules from a single recipe.
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# Rational approximation

## Definition

Let $a < b$ be real numbers and let $m, n \ge 0$ be integers.
Let $R_{m,n}$ be the set of [[rational functions|0WCZH4AFMT]] $r = p/q$ with real polynomials $p$ and $q$ such that $\deg p \le m$ and $\deg q \le n$.
In addition, $q$ has no zero on $[a, b]$.
Let $f \colon [a, b] \to \R$ be a [[continuous function|0X3WGX95FE]].
A *rational approximation* of type $(m, n)$ to $f$ is an element $r$ of $R_{m,n}$ used in place of $f$.
Its error is $\norm{f - r}_\infty = \max_{x \in [a, b]} \abs{f(x) - r(x)}$.
The *error of best rational approximation* is
$$E_{m,n}(f) = \inf_{r \in R_{m,n}} \norm{f - r}_\infty.$$

## Remarks

For $n = 0$ this is polynomial approximation, and $E_{m,0}(f)$ is the error of [[best approximation|7N4988HN3P]] by polynomials of degree at most $m$.
For continuous $f$ on $[a, b]$, a best rational approximation in $R_{m,n}$ exists and is unique, and Chebyshev's equioscillation theorem characterizes it.
$R_{m,n}$ is not a vector space for $n \ge 1$, since a sum of two of its elements may need numerator degree $m + n$ and denominator degree $2n$.
This is why the theory differs from the linear case.
Example: $r(x) = 1/(1 + x^2)$ lies in $R_{0,2}$ on any interval, and $E_{0,2}(r) = 0$.
Non-example: on $[-1, 1]$, $1/x$ is not an element of $R_{0,1}$, since its denominator vanishes at $0 \in [-1, 1]$.
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# Tychonoff space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
$X$ is *completely regular* when the following holds.
For every [[closed set|0ZCK8DQYN1]] $F \subseteq X$ and every point $x \in X \setminus F$, there is a [[continuous function|0X3WGX95FE]] $f \colon X \to [0, 1]$ with $f(x) = 0$ and $f(y) = 1$ for all $y \in F$.
$X$ is a *Tychonoff space* when it is completely regular and [[T1|0NW6CH38YR]].

## Remarks

In the definition, $T_1$ can be replaced by $T_0$, because a completely regular $T_0$ space is $T_1$.
A Tychonoff space is [[regular|03HMJASQBQ]] and Hausdorff: the sets $f^{-1}([0, 1/2))$ and $f^{-1}((1/2, 1])$ separate $x$ from $F$.
A space is Tychonoff if and only if it is homeomorphic to a subspace of a product $[0, 1]^I$.
By [[Urysohn's lemma|01EY3VR7ZD]], every [[normal|0NSR9V8TED]] $T_1$ space is Tychonoff.

Example: every metric space $(X, d)$ is Tychonoff.
For $x \notin F$ with $F$ closed, $d(x, F) > 0$, and $f(y) = \min\bigl(1, d(y, x) / d(x, F)\bigr)$ is continuous, equal to $0$ at $x$ and to $1$ on $F$.

Non-example: an infinite set $X$ with the [[cofinite topology|0B906ZBTMZ]] is $T_1$ but not completely regular.
Each continuous $f \colon X \to \R$ is constant: two distinct values would have disjoint open preimages, but two nonempty open sets of $X$ always meet.
So no $f$ separates a point $x$ from the closed set $\set{y}$ with $y \ne x$.

Notation: $T_{3\frac12}$.
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# Solution of recurrences by generating functions

## Definition

Let $K$ be a field and let $(u_n)_{n \ge 0}$ be a sequence in $K$ that satisfies a recurrence relation for $n \ge 0$.
The *generating function method* for this recurrence consists of these steps.

1. Form the [[generating function|0PHAC833JF]] $U(x) = \sum_{n \ge 0} u_n x^n$ in the ring $K[\![x]\!]$ of [[formal power series|033WGN00BF]].
2. Multiply the recurrence for index $n$ by a suitable power $x^{n+d}$ and sum over $n \ge 0$. This gives an equation for $U(x)$ that involves the initial values.
3. Solve this equation for $U(x)$.
4. Read off $u_n$ as the coefficient of $x^n$ in the solution.

For a [[linear recurrence with constant coefficients|081THG13JZ]] $u_{n+d} = c_{d-1} u_{n+d-1} + \cdots + c_0 u_n$, step 3 gives
$$U(x) = \frac{P(x)}{1 - c_{d-1} x - c_{d-2} x^2 - \cdots - c_0 x^d},$$
where $P$ is a polynomial of degree less than $d$ determined by $u_0, \dots, u_{d-1}$.

## Remarks

Example: the [[Fibonacci sequence|0GA9K57A39]] has $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$.
Multiplying by $x^{n+2}$ and summing gives $F(x) - x = x F(x) + x^2 F(x)$, so $F(x) = x/(1 - x - x^2)$.
Partial fractions with the roots of $1 - x - x^2$ then give $F_n = (\varphi^n - \psi^n)/\sqrt{5}$, where $\varphi, \psi$ are the roots of $X^2 - X - 1$.
Non-example: for the nonlinear recurrence $u_{n+1} = u_n^2$, step 2 does not give an equation in $U$ alone.
The series $\sum u_n^2 x^n$ is not a simple function of $U(x)$, so the method does not apply directly.
The denominator in the constant-coefficient case is $x^d \chi(1/x)$, where $\chi$ is the [[characteristic polynomial|0YA7X86K05]] of the recurrence.
Other generating functions, such as exponential ones, suit other recurrences, for example those with factors $n$ in the coefficients.
</document>

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# Positive operator

## Summary

A positive operator is a bounded self-adjoint operator $A$ on a Hilbert space with $\ip{Ax}{x} \ge 0$ for every vector $x$.
It is the operator version of a [[positive semidefinite matrix|01MMVW8CKC]] and of a nonnegative real number.
Positive operators define an order $A \le B$ on self-adjoint operators, they have [[square roots|3JY5HNFRMF]], and every operator of the form $T^*T$ is positive.

## Explanations

### L1 human

A positive operator never turns a vector to point against itself: the component of $Ax$ along $x$ is never negative.
On $\C^2$ the diagonal matrix with entries $1$ and $2$ is positive, since $\ip{Ax}{x} = \abs{x_1}^2 + 2\abs{x_2}^2 \ge 0$.
The diagonal matrix with entries $1$ and $-1$ is not, since it sends $(0, 1)$ to $(0, -1)$.
Positive operators play the role of the nonnegative numbers among operators.
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# Real closure

## Summary

A real closure of an [[ordered field|0P3VCBDEE2]] is a real closed algebraic extension whose order extends the given one.
A real closed field is a [[formally real field|X32TSWJSYT]] with no proper formally real algebraic extension, such as $\R$.
Every ordered field has a real closure, and it is unique up to a unique isomorphism over the field.
It plays for ordered fields the role that the [[algebraic closure|0X6AB5P3G1]] plays for fields.

## Explanations

### L1 human

The real numbers have the property that every positive number has a square root and every odd-degree polynomial has a root.
The real closure of an ordered field adds exactly what is needed to get these properties, and nothing more.
The real closure of $\Q$ is the field of real algebraic numbers, such as $\sqrt 2$ and the real cube root of $5$, but not $\pi$.
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# Model theory of algebraically closed fields

## Definition

Let $\mathcal{L} = \set{+, -, \cdot, 0, 1}$ be the language of rings.
The *theory of algebraically closed fields* $\mathrm{ACF}$ is the set of first-order $\mathcal{L}$-sentences that consists of:

1. the axioms of a [[field|04C5ZBG1WT]];
2. for each $n \ge 1$, the sentence $\forall a_0 \, \forall a_1 \cdots \forall a_{n-1} \, \exists x \; x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0$.

For a prime $p$, the theory $\mathrm{ACF}_p$ is $\mathrm{ACF}$ together with the sentence $1 + 1 + \cdots + 1 = 0$ with $p$ terms.
The theory $\mathrm{ACF}_0$ is $\mathrm{ACF}$ together with, for each prime $p$, the sentence $1 + 1 + \cdots + 1 \ne 0$ with $p$ terms.
The *model theory of algebraically closed fields* is the study of these [[theories|92SWXST1ZM]] and of their models.
The models of $\mathrm{ACF}_p$ are exactly the [[algebraically closed fields|0PF6TM8TWT]] of [[characteristic|1ZB13N78XW]] $p$, for $p$ prime or $0$.

## Remarks

The basic results are these:

1. $\mathrm{ACF}$ has [[quantifier elimination|7FDDXV3JK7]] in $\mathcal{L}$.
2. For $p$ prime or $0$, the theory $\mathrm{ACF}_p$ is [[complete|655K8Y7VS8]] and decidable.
3. For $p$ prime or $0$ and every uncountable cardinal $\kappa$, any two models of $\mathrm{ACF}_p$ of cardinality $\kappa$ are isomorphic, so $\mathrm{ACF}_p$ is [[categorical|1JJCGNX240]] in $\kappa$; it is not $\aleph_0$-categorical.

Statement 3 follows from [[Steinitz's theorem|CAEHMSX9RT]]: an algebraically closed field is determined up to isomorphism by its characteristic and its transcendence degree over the prime field.
See also the theory of algebraically closed fields as an example in model theory.

Example: $\C$ is a model of $\mathrm{ACF}_0$, and the algebraic closure of $\F_p$ is a model of $\mathrm{ACF}_p$.

Non-example: $\R$ satisfies clause 1 but not clause 2, since the sentence for $n = 2$ fails at $a_1 = 0$, $a_0 = 1$: there is no real $x$ with $x^2 + 1 = 0$.
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# Nerve of a category

## Definition

Let $\mathcal{C}$ be a small [[category|A2DE66937D]], and let $\Delta$ be the [[simplex category|FRFYHJGAN3]].
View each object $[n]$ of $\Delta$ as the category with objects $0, 1, \dots, n$ and exactly one morphism $i \to j$ when $i \le j$.
The *nerve* of $\mathcal{C}$ is the [[simplicial set|WW81ARB8K9]] $N\mathcal{C}$ with
$$N\mathcal{C}_n = \set{\text{functors } [n] \to \mathcal{C}},$$
where an order-preserving map $\theta \colon [m] \to [n]$ acts by precomposition, sending $F \colon [n] \to \mathcal{C}$ to $F \circ \theta$.

## Remarks

A [[functor|28NQ0KRHD3]] $[n] \to \mathcal{C}$ is the same as a string $c_0 \xrightarrow{f_1} c_1 \to \cdots \xrightarrow{f_n} c_n$ of composable morphisms.
So $N\mathcal{C}_0$ is the set of objects and $N\mathcal{C}_1$ the set of morphisms.
The face map $d_i$ for $0 < i < n$ composes $f_{i+1} \circ f_i$, while $d_0$ drops $f_1$ and $d_n$ drops $f_n$; the degeneracy $s_i$ inserts an identity at $c_i$.
Example: if $\mathcal{C}$ has one object and only the identity morphism, then $N\mathcal{C}_n$ has one element for each $n$, and $N\mathcal{C}$ is the point $\Delta^0$.
Non-example: the boundary $\partial \Delta^2$ consists of three edges forming a triangle, with no 2-simplex.
It is not the nerve of any category, because in a nerve each pair of composable edges is filled by a 2-simplex.
A simplicial set is isomorphic to a nerve exactly when every inner horn $\Lambda^n_k \to X$ with $0 < k < n$ has a unique filler.
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# Kelvin functions

## Summary

The Kelvin functions $\operatorname{ber}_\nu$, $\operatorname{bei}_\nu$, $\operatorname{ker}_\nu$ and $\operatorname{kei}_\nu$ are the real and imaginary parts of Bessel functions evaluated along the ray $\arg z = \frac{3\pi}{4}$ or $\frac{\pi}{4}$.
They are real functions of a positive real variable.
They describe alternating current in a round wire, where the current crowds toward the surface (the skin effect).

## Explanations

### L1 human

Some physical problems lead to Bessel's equation with a complex coefficient, such as $x^2 w'' + x w' - i x^2 w = 0$.
Its solutions are complex numbers, and one splits them into real and imaginary parts.
These parts have names: $\operatorname{ber}$ and $\operatorname{bei}$ for the solution that is finite at $0$, and $\operatorname{ker}$ and $\operatorname{kei}$ for the one that decays at infinity.
For example, $\operatorname{ber}_0(x) = 1 - \frac{(x/2)^4}{(2!)^2} + \frac{(x/2)^8}{(4!)^2} - \cdots$.
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# Universally measurable set

## Definition

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]].
For a [[probability measure|0VGMEBVW1M]] $\mu$ on $(X, \Sigma)$, let $\Sigma_\mu$ be the σ-algebra of the [[completion|KJNSKV75VQ]] of $\mu$.
A subset $A \subseteq X$ is *universally measurable* if $A \in \Sigma_\mu$ for every probability measure $\mu$ on $(X, \Sigma)$.

## Remarks

Every $A \in \Sigma$ is universally measurable, since $\Sigma \subseteq \Sigma_\mu$ for every $\mu$.
The universally measurable sets form a σ-algebra, the intersection of all the $\Sigma_\mu$.
Let $X = [0, 1]$ with its Borel σ-algebra, and let $\lambda$ be Lebesgue measure on it.
Then $\Sigma_\lambda$ is the σ-algebra of Lebesgue measurable subsets of $[0, 1]$, and a Vitali set $V \subseteq [0, 1]$ is not in it.
So $V$ is not universally measurable.
The definition is usually applied when $X$ is a Polish space and $\Sigma$ is its Borel σ-algebra; then every analytic set is universally measurable.
Replacing probability measures by σ-finite measures gives the same family of sets.
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# Point process

## Definition

Let $E$ be a complete separable metric space with its Borel $\sigma$-algebra $\mathcal{B}(E)$.
A measure $\mu$ on $\mathcal{B}(E)$ is a *locally finite counting measure* if $\mu(B) \in \set{0, 1, 2, \ldots}$ for every bounded Borel set $B$.
Let $\mathcal{N}$ be the set of these measures, with the smallest $\sigma$-algebra that makes each map $\mu \mapsto \mu(B)$ measurable.
A *point process* on $E$ is a measurable map $N$ from a [[probability space|0NFPJVJJHX]] $(\Omega, \mathcal{F}, P)$ to $\mathcal{N}$.

## Remarks

So each $N(B)$, the number of points in $B$, is a [[random variable|0PNRKMRC55]].
The point process is *simple* if $N(\set{x}) \le 1$ for all $x \in E$, almost surely.
Its intensity measure is $\Lambda(B) = \mathbb{E}[N(B)]$.
Example: if $U_1, \ldots, U_n$ are random points of $[0, 1]$, then $N = \sum_{i=1}^n \delta_{U_i}$ is a point process; $N(B) = \#\set{i : U_i \in B}$ is a random variable with values in $\set{0, \ldots, n}$.
Non-example: $\sum_{k \ge 1} \delta_{1/k}$ is not a locally finite counting measure on $\R$, since it gives infinite mass to the bounded set $[0, 1]$.
The [[Poisson point process|0MTNR28K60]] with intensity $\Lambda$ is the point process with $N(B)$ Poisson of mean $\Lambda(B)$ and independent counts on disjoint sets.
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# Coupled map lattice

## Summary

A coupled map lattice is a [[dynamical system|BBZP1R872N]] with discrete time, discrete space and continuous state.
At each site of a lattice sits a real variable.
It is updated by a nonlinear map such as the [[logistic map|QN9X9BNKTX]] and then mixed with the values at neighbouring sites.
Coupled map lattices were introduced by Kunihiko Kaneko in the 1980s as simple models of spatiotemporal chaos, turbulence and pattern formation.
They show a rich variety of behaviour, such as frozen patterns, travelling defects and fully developed spatiotemporal chaos.

## Explanations

### L1 human

Think of a row of cells, each holding a number.
At every tick of a clock, each cell first applies the same simple rule to its own number, and then averages a little with its two neighbours.
With a chaotic rule, each cell alone would behave erratically; the coupling makes the whole row form patterns that move, merge and break up.
It is a toy model for systems like a turbulent fluid, where chaos happens in space as well as in time.
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# Künneth theorem

## Statement

Let $R$ be a [[principal ideal domain|0CMD0BCH8T]].

1. Let $C_\bullet$ and $D_\bullet$ be [[chain complexes|WQ82C0M3QB]] of $R$-modules such that every $C_n$ is a free $R$-module. Then for every $n$ there is a natural short exact sequence
$$0 \to \bigoplus_{i+j=n} H_i(C) \otimes_R H_j(D) \to H_n(C \otimes_R D) \to \bigoplus_{i+j=n-1} \operatorname{Tor}_1^R(H_i(C), H_j(D)) \to 0,$$
and it splits, though not naturally.
2. Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]]. Then for every $n$ there is a natural short exact sequence, which splits but not naturally,
$$0 \to \bigoplus_{i+j=n} H_i(X; R) \otimes_R H_j(Y; R) \to H_n(X \times Y; R) \to \bigoplus_{i+j=n-1} \operatorname{Tor}_1^R(H_i(X; R), H_j(Y; R)) \to 0.$$

Here $H$ is the [[homology|2J4JJJYP22]] of a complex, and $H_n(-; R)$ is [[singular homology|M0Q5CFJ7ZH]] with coefficients in $R$.
The symbol $\otimes_R$ is the [[tensor product|F0HQXQ0HCD]] and $\operatorname{Tor}_1^R$ is the [[Tor functor|DDQRERAQCV]].

## Remarks

For the algebraic part, note that the cycles $Z_\bullet$ and boundaries $B_\bullet$ of $C$ are free, since $R$ is a principal ideal domain.
So $0 \to Z_\bullet \to C_\bullet \to B_{\bullet - 1} \to 0$ is a split sequence of free complexes.
The long exact sequence after tensoring with $D$ gives the result.
The topological part follows from the algebraic one and the Eilenberg–Zilber theorem.
That theorem gives a chain homotopy equivalence between the singular chains of $X \times Y$ and the tensor product of the singular chains of $X$ and of $Y$.
If $R$ is a field, all Tor terms vanish and $H_n(X \times Y; R) \cong \bigoplus_{i+j=n} H_i(X; R) \otimes_R H_j(Y; R)$.
Over $\Z$ the Tor term matters.
For the real projective plane $P$, one has $H_0(P) = \Z$, $H_1(P) = \Z/2\Z$ and $H_2(P) = 0$.
So the tensor part in degree $3$ is zero, and the formula gives $H_3(P \times P; \Z) \cong \operatorname{Tor}_1^\Z(\Z/2\Z, \Z/2\Z) \cong \Z/2\Z$.

## Explanations

### L1 human

The Künneth theorem computes the holes of a product space from the holes of its factors.
With coefficients in a field, the answer is the simplest one: the homology of the product is the tensor product of the homologies.
For the torus, the product of two circles, this gives one class in degree $0$, two in degree $1$ and one in degree $2$.
With integer coefficients there is a correction term built from torsion.

### L2 human

The theorem is a direct application of the derived functor Tor: the failure of $H(C \otimes D) \cong H(C) \otimes H(D)$ is measured by $\operatorname{Tor}_1$.
Taking $D$ to be a module $M$ in degree $0$ gives the homology form of the [[universal coefficient theorem|GTA8ZFGZ3N]].
There are also cohomology versions, with a cross product into $H^n(X \times Y; R)$, and spectral sequence versions over rings that are not principal ideal domains.
</document>

<document source="https://tamnd.github.io/math2/node/77N4TVRM15.md" tag="77N4TVRM15" kind="example">
# Orders of the elements of the integers modulo 6

## Example

Let $\mathbb{Z}/6\mathbb{Z} = \{0, 1, 2, 3, 4, 5\}$ be the [[integers modulo $6$|03VFZ2WETT]] with addition.
The element $2$ has [[order|0JPYSD5E0B]] $3$, because $2 + 2 + 2 = 6 = 0$ and $2 + 2 = 4 \ne 0$.
The element $3$ has order $2$, and the elements $1$ and $5$ have order $6$. The element $4$ has order $3$.
Each of the orders $1$, $2$, $3$ and $6$ divides $6$, as the [[result on element orders|08V7JHS3QZ]] states.

## Remarks

The group is abelian, but not each element has order $6$.
</document>

<document source="https://tamnd.github.io/math2/node/77NG85NDQM.md" tag="77NG85NDQM" kind="definition">
# Multivariate approximation

## Definition

Let $d \ge 2$, let $\Omega \subseteq \R^d$, and let $X$ be a [[normed space|03J5RAKB30]] of real functions on $\Omega$, such as $C(\Omega)$ for compact $\Omega$ or $L^p(\Omega)$.
Let $V_1, V_2, \dots$ be subsets of $X$, called approximating families, usually finite-dimensional subspaces or sets described by $n$ parameters.
*Multivariate approximation* is the approximation of functions $f \in X$ of the $d$ variables by elements of the families $V_n$.
Its quality is measured by the [[error of best approximation|YDN0T4MEX0]]
$$E(f, V_n) = \inf_{v \in V_n} \norm{f - v}_X.$$
The subject studies how $E(f, V_n)$ depends on $n$, on $d$ and on the smoothness of $f$.

## Remarks

Example: $\Omega = [0, 1]^d$, $X = C(\Omega)$, and $V_n$ the [[polynomials in several variables|07GNX8GCYP]] of total degree at most $n$, a space of dimension $\binom{n + d}{d}$.
By the [[Stone–Weierstrass theorem|1P8D3MK79H]], $E(f, V_n) \to 0$ for every continuous $f$.

Non-example: for $d = 1$ the same question is univariate approximation, the classical case covered by the [[Weierstrass approximation theorem|0JQE130DHW]].

A typical rate for a fixed dimension: if $f$ has bounded derivatives of order $s$ and $V_n$ has dimension $N$, the best linear methods give errors of order $N^{-s/d}$.
This rate degrades as $d$ grows.
</document>

<document source="https://tamnd.github.io/math2/node/77PV0ZAVXT.md" tag="77PV0ZAVXT" kind="concept">
# Product measure

## Summary

The product measure $\mu \times \nu$ of two [[σ-finite|WEBBT18D4X]] measures is the measure on the [[product σ-algebra|6HVCTJGB61]] that gives each rectangle $A \times B$ the measure $\mu(A) \nu(B)$.
It exists and is unique under σ-finiteness.
It is the setting of Fubini's and Tonelli's theorems, which compute integrals over the product as iterated integrals.
In probability it describes pairs of independent random variables.

## Explanations

### L1 human

The area of a rectangle is width times height.
The product measure extends this rule from rectangles to all reasonable subsets of a product space.
Lebesgue measure on the plane is the product of Lebesgue measure on the line with itself.
If one throws a fair coin and a fair die independently, the probability of "heads and a six" is $\tfrac12 \cdot \tfrac16 = \tfrac{1}{12}$, a value of the product measure.
</document>

<document source="https://tamnd.github.io/math2/node/77VNNV4VET.md" tag="77VNNV4VET" kind="statement">
# Capacity and Hausdorff dimension

## Statement

Let $K \subseteq \R^n$ be a [[compact set|0R6EWFKRCF]] and let $0 < s < n$.
For a Borel [[probability measure|059TJN0NZ1]] $\mu$ on $K$, let
$$I_s(\mu) = \int_K \int_K \abs{x - y}^{-s} \, d\mu(x) \, d\mu(y)$$
be its [[Riesz energy|835MPDJXMA]] of order $s$.
The Riesz $s$-capacity of $K$ is $C_s(K) = \sup_\mu 1 / I_s(\mu)$, with $1/\infty = 0$.
Let $\mathcal{H}^s$ be the $s$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] and $\dim_H$ the [[Hausdorff dimension|SSNFTHZ6YJ]].

1. If $\mathcal{H}^s(K) < \infty$, then $C_s(K) = 0$.
2. If $\dim_H K > s$, then $C_s(K) > 0$.

Hence $\dim_H K = \sup \set{ s \in (0, n) : C_s(K) > 0 } = \inf \set{ s \in (0, n) : C_s(K) = 0 }$ whenever $0 < \dim_H K < n$.

## Remarks

Proof idea for part 2: choose $t$ with $s < t < \dim_H K$.
Then $\mathcal{H}^t(K) > 0$, and the [[Frostman lemma|FYNFT90NT2]] gives a nonzero measure $\mu$ on $K$ with $\mu(B(x, r)) \le r^t$ for all balls.
Write $\int \abs{x - y}^{-s} \, d\mu(y) = \int_0^\infty \mu(B(x, \rho^{-1/s})) \, d\rho$.
The bound on balls shows that this integral is bounded in $x$, so $I_s(\mu) < \infty$ after normalizing $\mu$.
For part 1, a measure with finite $s$-energy satisfies $\mu(B(x, r)) / r^s \to 0$ as $r \to 0$ for $\mu$-almost every $x$.
A density theorem for Hausdorff measure shows that this is impossible when $\mathcal{H}^s(K) < \infty$ and $\mu \ne 0$.

Example: for $n \ge 3$ and $s = n - 2$, the capacity $C_{n-2}$ has the same null sets as the Newtonian capacity.
So a compact set with $\mathcal{H}^{n-2}(K) < \infty$, such as a segment in $\R^3$, has Newtonian capacity zero.
A compact set of Hausdorff dimension greater than $n - 2$, such as a flat disc in $\R^3$, has positive Newtonian capacity.

Neither part can be reversed in general: a set with $\dim_H K = s$ may have $C_s(K) = 0$ or $C_s(K) > 0$, depending on finer properties of the set.

## Explanations

### L1 human

Capacity and Hausdorff dimension both measure how thin a set is.
The theorem says they draw the line in the same place.
A set of dimension bigger than $s$ can carry a charge with finite $s$-energy, and a set with finite $s$-dimensional measure cannot.
In space, Newtonian capacity corresponds to $s = 1$: smooth curves are too thin, smooth surfaces are thick enough.

### L2 human

The result turns questions about capacity into questions about dimension and back.
It is the basis of the energy method for lower bounds of Hausdorff dimension, used in the study of projections and of Brownian paths.
In potential theory it shows that removable and exceptional sets, which are defined through capacity, are always small in the sense of dimension.
In the plane the logarithmic capacity sits at the level $s = 0$: every compact set of positive Hausdorff dimension has positive logarithmic capacity.
</document>

<document source="https://tamnd.github.io/math2/node/78GT1VFFPR.md" tag="78GT1VFFPR" kind="definition">
# Complete local ring

## Definition

Let $(R, \mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]].
The ring $R$ is *complete* if the natural map $R \to \hat R = \varprojlim_n R/\mathfrak m^n$ to its [[$\mathfrak m$-adic completion|NS84D2920R]] is an isomorphism.

## Remarks

Equivalently, $R$ is Hausdorff for the [[$\mathfrak m$-adic topology|2VF2TSJ3PY]] and every Cauchy sequence for this topology converges; for Noetherian local rings the Hausdorff condition always holds by the Krull intersection theorem.
Example: the ring $k[\![x]\!]$ of [[formal power series|033WGN00BF]] over a field $k$ is complete: a compatible sequence of truncations modulo $x^n$ defines a unique series.
Non-example: $R = \Q[x]_{(x)}$ is a Noetherian local ring that is not complete; its completion $\Q[\![x]\!]$ is uncountable while $R$ is countable, so $R \to \hat R$ is not surjective.
The completion $\hat R$ of any Noetherian local ring is a complete Noetherian local ring with maximal ideal $\mathfrak m \hat R$.
</document>

<document source="https://tamnd.github.io/math2/node/78JRW8KTH8.md" tag="78JRW8KTH8" kind="statement">
# Fundamental theorem of Riemannian geometry

## Statement

Let $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]].
Then there is exactly one [[affine connection|JX65KSVHSJ]] $\nabla$ on $M$ with the following two properties, for all smooth vector fields $X, Y, Z$ on $M$:

1. $\nabla$ is torsion-free: $\nabla_X Y - \nabla_Y X = [X, Y]$;
2. $\nabla$ is compatible with $g$: $X\bigl(g(Y, Z)\bigr) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$.

This connection is the [[Levi-Civita connection|CAZYGE1T4E]] of $(M, g)$.

## Remarks

The proof writes the compatibility identity three times with $X, Y, Z$ permuted cyclically and combines them using torsion-freeness.
This gives the Koszul formula
$$2 g(\nabla_X Y, Z) = X g(Y, Z) + Y g(X, Z) - Z g(X, Y) + g([X, Y], Z) - g([X, Z], Y) - g([Y, Z], X).$$
Since $g$ is nondegenerate, the right side determines $\nabla_X Y$, which gives uniqueness.
For existence, one checks that the vector field defined by the Koszul formula is an affine connection with both properties.
In coordinates the formula gives the [[Christoffel symbols|FZBA1HA9ZM]] $\Gamma^k_{ij} = \tfrac{1}{2} \sum_l g^{kl}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij})$.
Both conditions are needed for uniqueness: there are many metric-compatible connections with [[torsion|JZNQAN00ZJ]], and many torsion-free connections that do not preserve $g$.

## Explanations

### L1 human

A metric measures lengths and angles at each point, but it does not by itself say how to differentiate vector fields.
This theorem says that once one asks the differentiation rule to keep lengths and angles and not to twist, there is no choice left: exactly one rule works.
So every Riemannian manifold comes with its own canonical way of differentiating.

### L2 human

The theorem is why geodesics, parallel transport and curvature are invariants of the metric alone.
The Koszul formula is also a practical tool: it computes the connection of a left-invariant metric on a Lie group or of a warped product directly from brackets.
The same proof works for any nondegenerate metric, so the result holds for Lorentzian manifolds as used in general relativity.
For a surface in $\R^3$, the Levi-Civita connection is the tangential part of the ordinary derivative.
This is one way to see that Gaussian curvature depends only on the first fundamental form.
</document>

<document source="https://tamnd.github.io/math2/node/78N6AXP9VV.md" tag="78N6AXP9VV" kind="concept">
# Calderón–Zygmund kernel

## Summary

A Calderón–Zygmund kernel is a function $K(x, y)$ on $\R^n \times \R^n$ off the diagonal.
It is at most of size $\abs{x - y}^{-n}$, and it is smooth in a quantitative way away from the diagonal.
These are the kernels of the [[singular integral operators|HTMCEARGDD]] of Calderón–Zygmund theory, such as the Hilbert and [[Riesz transforms|FRNDFJ9WQ5]].
The size and smoothness conditions are what turn an $L^2$ bound into $L^p$ bounds for $1 < p < \infty$.

## Explanations

### L1 human

Many important operators act by integrating a function against a kernel $K(x, y)$ that blows up when $y$ comes close to $x$.
The blow-up is of the borderline size $\abs{x - y}^{-n}$, just too big to integrate.
A Calderón–Zygmund kernel has this size and, in addition, changes slowly when $x$ or $y$ moves by a small amount compared to their distance.
The model example on the line is $K(x, y) = \frac{1}{\pi(x - y)}$, the kernel of the Hilbert transform.
</document>

<document source="https://tamnd.github.io/math2/node/78R2TV41D4.md" tag="78R2TV41D4" kind="statement">
# Jordan's theorem on primitive groups containing a prime cycle

## Statement

Let $n$ be a positive integer, and let $G$ be a [[primitive permutation group|Y10S0751KG]] on a set $\Omega$ with $n$ elements.
Let $p$ be a [[prime number|01826KMG2H]] with $p \le n - 3$.
Suppose that $G$ contains a [[cycle|07CT9RDY7Q]] of length $p$.
Then $G$ is the [[alternating group|0Z5905Q73D]] $A_n$ or the [[symmetric group|0515A6D2Y1]] $S_n$ on $\Omega$.

## Remarks

The proof uses that the $p$-cycle fixes at least three points.
Combining this with primitivity, one shows that $G$ is [[multiply transitive|BDJ1AX4XBB]], and then that $G$ contains $A_n$.

The bound $p \le n - 3$ is necessary.
The cyclic group generated by $(1\,2\,3\,4\,5)$ is primitive on $5$ points and contains a $5$-cycle, so $p = n$, but it is neither $A_5$ nor $S_5$.
The group $\mathrm{PGL}_2(\F_5)$ of order $120$ acts primitively on the $6$ points of the projective line over $\F_5$.
There the map $x \mapsto x + 1$ is a $5$-cycle, so $p = n - 1$.
But $120$ is neither $360$ nor $720$, so the group is neither $A_6$ nor $S_6$.
The group $\mathrm{PSL}_2(\F_8)$ of order $504$ acts $2$-transitively, hence primitively, on the $9$ points of the projective line over $\F_8$.
For a generator $a$ of $\F_8^\times$, the map $x \mapsto ax$ fixes $0$ and $\infty$ and is a $7$-cycle, so $p = n - 2$.
But $504$ is smaller than $\abs{A_9}$, so the group is neither $A_9$ nor $S_9$.

The theorem gives a practical test for a permutation group of degree $n$ to contain $A_n$.
It is enough that the group is primitive and contains one cycle of prime length at most $n - 3$.

## Explanations

### L1 human

Primitive groups shuffle a set without respecting any grouping of its points.
Jordan's theorem says that such a group is very large as soon as it contains one special element.
This element is a cycle of prime length that leaves at least three points in place.
Then the group contains all even permutations.
So a single element, together with primitivity, can force a group to be huge.

### L2 human

The theorem is a classical tool to prove that a given permutation group is the full symmetric or alternating group.
It is used, for example, to show that the Galois group of a polynomial is $S_n$.
An irreducible polynomial gives a transitive Galois group, and one then looks for suitable cycles.
Similar results say that a primitive group containing a transposition is $S_n$, and a primitive group containing a $3$-cycle contains $A_n$.
The proof works with [[multiply transitive group actions|BDJ1AX4XBB]] and the sets of fixed points of elements.
</document>

<document source="https://tamnd.github.io/math2/node/7938EKMMVR.md" tag="7938EKMMVR" kind="statement">
# Artin reciprocity law

## Statement

Let $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]] of [[algebraic number fields|MWBDFS2KJ5]] with [[abelian|082GHNPJ53]] Galois group $G$.
Let $\mathfrak{m}$ be a modulus of $K$, as in [[class field theory|AA943HVK6Z]].
Assume that $\mathfrak{m}$ is divisible by every finite prime and every real place of $K$ that [[ramifies|H5RC67SD24]] in $L$.
Let $I_K^{\mathfrak{m}}$ be the group of [[fractional ideals|YB4JTB4WJN]] of $K$ coprime to $\mathfrak{m}$, and let $P_K^{\mathfrak{m}}$ be its subgroup of principal ideals $(\alpha)$ with $\alpha \equiv 1 \bmod^\times \mathfrak{m}$.
The *Artin map* $\psi_{L/K} \colon I_K^{\mathfrak{m}} \to G$ is the homomorphism that sends each prime ideal $\mathfrak{p}$ coprime to $\mathfrak{m}$ to its [[Frobenius element|H5145REFMW]] $\mathrm{Frob}_{\mathfrak{p}}$.
Then:

1. $\psi_{L/K}$ is surjective.
2. Assume that the finite part of $\mathfrak{m}$ is divisible by high enough powers of the ramified primes.
Then the kernel of $\psi_{L/K}$ is $P_K^{\mathfrak{m}} \, N_{L/K}(I_L^{\mathfrak{m}})$, where $N_{L/K}$ is the ideal norm.
Hence
$$I_K^{\mathfrak{m}} \big/ P_K^{\mathfrak{m}} N_{L/K}(I_L^{\mathfrak{m}}) \cong \Gal(L/K).$$

## Remarks

The smallest modulus for which part 2 holds is the conductor of $L/K$.
In the language of [[ideles|9GHD9MXNYV]], there is a surjective homomorphism from the idele class group of $K$ onto $G$.
Its kernel is the image of the norm from the idele class group of $L$.
Artin proved the law using Chebotarev's density theorem and a reduction to cyclotomic extensions.

Example: $K = \Q$, $L = \Q(\zeta_n)$ a [[cyclotomic field|NN77VYSGXR]], $\mathfrak{m} = n\infty$.
For a prime $p \nmid n$, $\psi(p\Z)$ is the automorphism $\zeta_n \mapsto \zeta_n^p$.
So $\psi$ sends the ideal generated by a positive integer $a$ coprime to $n$ to $\zeta_n \mapsto \zeta_n^a$.
Its kernel is the set of ideals $(a/b)$ with $a, b > 0$ coprime to $n$ and $a \equiv b \pmod n$, which is $P_{\Q}^{\mathfrak{m}}$.
The theorem becomes $\Gal(\Q(\zeta_n)/\Q) \cong (\Z/n\Z)^\times$.

## Explanations

### L1 human

For an abelian extension, the Frobenius element of an unramified prime records how that prime splits.
Artin reciprocity says that the Frobenius element depends only on the class of the prime in a suitable generalised ideal class group of the base field.
So the splitting of primes in abelian extensions follows congruence rules.

### L2 human

The law contains [[quadratic reciprocity|085MJS6907]] as a special case.
Let $q$ be an odd prime, $q^* = (-1)^{(q-1)/2} q$ and $L = \Q(\sqrt{q^*})$.
The Frobenius of an odd prime $p \ne q$ in $L$ is given by the Legendre symbol $(q^*/p)$.
Since $L \subset \Q(\zeta_q)$, it also depends only on $p$ modulo $q$.
It also contains the higher power reciprocity laws.
For $\mathfrak{m} = 1$ and the [[Hilbert class field|WWCJ7CR2R3]] it gives an isomorphism between the class group of $K$ and the Galois group.
</document>

<document source="https://tamnd.github.io/math2/node/793SSEV9ED.md" tag="793SSEV9ED" kind="definition">
# Riemann curvature tensor

## Definition

Let $M$ be a smooth manifold with an [[affine connection|JX65KSVHSJ]] $\nabla$, and let $\mathfrak{X}(M)$ be the space of smooth vector fields on $M$.
The *curvature tensor* of $\nabla$ is the map $R : \mathfrak{X}(M)^3 \to \mathfrak{X}(M)$ given by
$$R(X, Y) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z,$$
where $[X, Y]$ is the [[Lie bracket|68FGQEBD6J]].
When $(M, g)$ is a [[Riemannian manifold|7CBT4SM83Y]] and $\nabla$ is its [[Levi-Civita connection|CAZYGE1T4E]], $R$ is the *Riemann curvature tensor* of $(M, g)$.

## Remarks

$R$ is $C^\infty(M)$-linear in $X$, $Y$ and $Z$, so it is a [[tensor field|WX302T93TW]] of type $(1, 3)$, and $R(Y, X) = -R(X, Y)$.
Some books use the opposite sign; the convention here gives positive sectional curvature $g(R(X, Y) Y, X) > 0$ on the sphere.
Example: for the flat connection of $\R^n$, $\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z = \nabla_{[X, Y]} Z$ by symmetry of second derivatives, so $R = 0$.
For the round sphere of radius $r$, $R(X, Y) Z = r^{-2} \big( g(Y, Z) X - g(X, Z) Y \big)$.
Non-example of a tensor: the expression $\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z$ alone is not $C^\infty(M)$-linear in $X$, as replacing $X$ by $fX$ adds $-(Yf) \nabla_X Z$; the term $\nabla_{[X, Y]} Z$ removes this.
In a chart, $R(\partial_i, \partial_j) \partial_k = \sum_l R^l{}_{kij} \partial_l$ with $R^l{}_{kij} = \partial_i \Gamma^l_{jk} - \partial_j \Gamma^l_{ik} + \sum_m (\Gamma^l_{im} \Gamma^m_{jk} - \Gamma^l_{jm} \Gamma^m_{ik})$.
For the Levi-Civita connection, $R_{abcd} = g(R(X_c, X_d) X_b, X_a)$ is antisymmetric in $(a, b)$ and in $(c, d)$ and symmetric under $(a, b) \leftrightarrow (c, d)$.
</document>

<document source="https://tamnd.github.io/math2/node/794N251WY8.md" tag="794N251WY8" kind="concept">
# Annihilator method

## Summary

The annihilator method finds a particular solution of a [[linear differential equation|0QQRP17Z7S]] $L y = f$ with constant coefficients.
It applies when the right side $f$ itself solves some homogeneous constant-coefficient equation $A f = 0$.
Applying $A$ turns the problem into the homogeneous equation $A L y = 0$, whose solutions are known, and this tells exactly which trial functions to use.
It is the systematic form of the [[method of undetermined coefficients|0XRB469Z32]].

## Explanations

### L1 human

Some right-hand sides, such as $e^{2t}$, $t^3$ or $\cos t$, can be wiped out by a simple differential operator: $(D - 2) e^{2t} = 0$, $D^4 t^3 = 0$, $(D^2 + 1)\cos t = 0$.
Applying this wiping operator to both sides of the equation leaves an equation with right side zero.
Its solutions show the exact shape of a particular solution, and only some constants remain to be found.
</document>

<document source="https://tamnd.github.io/math2/node/799DX4375M.md" tag="799DX4375M" kind="concept">
# Summability field of a matrix method

## Summary

The summability field of an infinite matrix $A$ is the set of all sequences $x$ that $A$ transforms into a convergent sequence $Ax$.
It is the domain of the [[matrix summability method|M6311DWJ1E]] given by $A$, and it is a [[sequence space|XYTN19EPF2]].
Many questions of summability theory are questions about summability fields.
For instance, one asks whether a summability field contains the space $c$ of convergent sequences or whether two of them are equal.
With a natural topology, the summability field is an [[FK-space|71DMQHW6JZ]], which brings functional analysis into summability.

## Explanations

### L1 human

A matrix method takes a sequence, forms weighted averages of its terms, and looks at whether these averages converge.
The summability field is the collection of all sequences for which this works.
For the identity matrix it is just the convergent sequences.
For the Cesàro averages it is bigger: it also contains $1, 0, 1, 0, \dots$, whose averages tend to $1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/79DH4GBCZ5.md" tag="79DH4GBCZ5" kind="definition">
# Gelfand–Naimark–Segal construction

## Definition

Let $A$ be a unital C*-algebra and let $\rho$ be a [[state|F7MB2WREBJ]] on $A$, a positive linear functional with $\rho(1) = 1$.
The *GNS construction* for $\rho$ is the triple $(H_\rho, \pi_\rho, \xi_\rho)$ defined as follows.

1. The set $N_\rho = \set{a \in A : \rho(a^*a) = 0}$ is a closed left ideal of $A$, and $\ip{a + N_\rho}{b + N_\rho} = \rho(b^*a)$ is an inner product on $A/N_\rho$.
2. $H_\rho$ is the [[Hilbert space|HM2HHE3J51]] completion of $A/N_\rho$ for this inner product.
3. For $a \in A$, $\pi_\rho(a)$ is the bounded operator on $H_\rho$ that extends $b + N_\rho \mapsto ab + N_\rho$.
4. $\xi_\rho = 1 + N_\rho$.

## Remarks

Then $\pi_\rho$ is a $*$-representation of $A$, the vector $\xi_\rho$ is cyclic, that is, $\pi_\rho(A)\xi_\rho$ is dense in $H_\rho$, and $\rho(a) = \ip{\pi_\rho(a)\xi_\rho}{\xi_\rho}$ for all $a \in A$.
Any other cyclic representation with a cyclic vector $\xi$ satisfying $\rho(a) = \ip{\pi(a)\xi}{\xi}$ is unitarily equivalent to $\pi_\rho$ by a unitary sending $\xi$ to $\xi_\rho$.
The boundedness in clause 3 comes from $\rho(b^*a^*ab) \le \norm{a}^2 \rho(b^*b)$.

Example: $A = C[0, 1]$ and $\rho(f) = \int_0^1 f(x) \, dx$.
Then $N_\rho = \set{0}$, since $\int_0^1 \abs{f}^2 \, dx = 0$ forces a continuous $f$ to vanish, and the inner product is $\int_0^1 f \overline{g} \, dx$.
So $H_\rho = L^2[0, 1]$, $\pi_\rho(f)$ is multiplication by $f$ and $\xi_\rho$ is the constant function $1$.

Non-example: the functional $\sigma(f) = -f(0)$ on $C[0, 1]$ is linear but not a state, and the construction fails at clause 1.
The form $\ip{a}{b} = \sigma(b^*a)$ gives $\ip{1}{1} = -1 < 0$, so it is not an inner product.

For a non-unital C\*-algebra the construction works in the same way for positive functionals, with the cyclic vector obtained from an approximate unit.
</document>

<document source="https://tamnd.github.io/math2/node/79M1TG6F5N.md" tag="79M1TG6F5N" kind="definition">
# Falling and rising factorials

## Definition

Let $x$ be an element of a commutative ring with identity, for example a real number, and let $n$ be a natural number.
The *falling factorial* of $x$ of order $n$ is
$$x^{\underline{n}} = x (x - 1) (x - 2) \cdots (x - n + 1) = \prod_{i=0}^{n-1} (x - i).$$
The *rising factorial* of $x$ of order $n$ is
$$x^{\overline{n}} = x (x + 1) (x + 2) \cdots (x + n - 1) = \prod_{i=0}^{n-1} (x + i).$$
For $n = 0$ both products are empty, and $x^{\underline{0}} = x^{\overline{0}} = 1$.

## Remarks

Example: for $x = 4$ and $n = 2$, $4^{\underline{2}} = 4 \cdot 3 = 12$ and $4^{\overline{2}} = 4 \cdot 5 = 20$.
Non-example: the product $4 \cdot 2 = 8$ is not $4^{\underline{2}}$, because its factors do not decrease by steps of $1$.
For a natural number $m \ge n$, $m^{\underline{n}} = m!/(m-n)!$, and $m^{\underline{n}} = 0$ when $n > m$; see [[falling factorial|0N5M924N3P]] and [[factorial|0WFCGVMRRC]].
For a natural number $m \ge 1$, $m^{\overline{n}} = (m + n - 1)!/(m - 1)!$, and $1^{\overline{n}} = n^{\underline{n}} = n!$.
The two are linked by $x^{\overline{n}} = (x + n - 1)^{\underline{n}} = (-1)^n (-x)^{\underline{n}}$.
For real $x > 0$, $x^{\overline{n}} = \Gamma(x + n)/\Gamma(x)$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
Other notations: $(x)_n$ is used for the falling factorial in combinatorics and for the rising factorial (the Pochhammer symbol) in special functions; the rising factorial is also written $x^{(n)}$.
The [[q-Pochhammer symbol|Y1X8ZZQ7VM]] is a $q$-analogue of the rising factorial.
</document>

<document source="https://tamnd.github.io/math2/node/79X8DJZRBD.md" tag="79X8DJZRBD" kind="definition">
# Space of functions vanishing at infinity

## Definition

Let $X$ be a [[locally compact|0FE2SB8WVJ]] [[Hausdorff space|0S1FJ8VEX9]] and let $\mathbb{K}$ be $\R$ or $\C$.
A [[continuous function|0X3WGX95FE]] $f \colon X \to \mathbb{K}$ *vanishes at infinity* if, for every $\varepsilon > 0$, the set $\set{x \in X : \abs{f(x)} \ge \varepsilon}$ is [[compact|0TVJZYB6SH]].
The *space of functions vanishing at infinity* $C_0(X)$ is the set of these functions, with pointwise operations and the norm $\norm{f}_\infty = \sup_{x \in X} \abs{f(x)}$.

## Remarks

Example: $f(x) = e^{-x^2}$ on $\R$.
For $0 < \varepsilon \le 1$, the set $\set{\abs{f} \ge \varepsilon}$ is the closed bounded interval $\abs{x} \le \sqrt{\log(1/\varepsilon)}$, which is compact.
For $\varepsilon > 1$ it is empty.

Non-example: the constant function $1$ on $\R$.
For $\varepsilon = 1/2$, the set $\set{\abs{f} \ge 1/2}$ is all of $\R$, which is not compact.

Each $f \in C_0(X)$ is bounded: $\abs{f} < 1$ off the compact set $\set{\abs{f} \ge 1}$, and $\abs{f}$ attains a maximum on that set if it is nonempty.
So $C_0(X)$ is a subspace of the [[space of bounded continuous functions|MG3X8Y1VT4]], and it is closed there, so it is a Banach space.
If $X$ is compact, then $C_0(X) = C(X)$.
For $X = \N$ with the discrete topology, $C_0(\N)$ is the sequence space $c_0$.
The compactly supported continuous functions are dense in $C_0(X)$.
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<document source="https://tamnd.github.io/math2/node/7A55AWN7EV.md" tag="7A55AWN7EV" kind="definition">
# M-estimator

## Definition

Let $X_1, \dots, X_n$ be observations with values in a set $\mathcal{X}$, and let $\Theta$ be a parameter set.
Let $\rho \colon \mathcal{X} \times \Theta \to \R$ be a function.
An *M-estimator* with loss $\rho$ is an [[estimator|0TTNRP5PFQ]] $\hat{\theta}_n$ that minimizes
$$\theta \mapsto \sum_{i=1}^n \rho(X_i, \theta)$$
over $\Theta$.

## Remarks

When $\Theta \subseteq \R^k$ is open and $\rho$ is differentiable in $\theta$ with gradient $\psi(x, \theta) = \nabla_\theta \rho(x, \theta)$, an interior minimizer solves $\sum_{i=1}^n \psi(X_i, \theta) = 0$.
An estimator defined as a root of such an equation, for a given function $\psi$, is also called an M-estimator (of $\psi$-type), or a Z-estimator.

Example: with $\rho(x, \theta) = (x - \theta)^2$ on $\R$, the sum is minimized at the [[sample mean|0QNRBHWV35]], since its derivative $-2\sum_i (X_i - \theta)$ vanishes only at $\theta = \bar{X}$.
With $\rho(x, \theta) = \abs{x - \theta}$, the minimizers are the sample medians.
With $\rho(x, \theta) = -\log f(x; \theta)$ for a family of densities, the M-estimator is the [[maximum likelihood estimator|0FREPFKNS7]].

Non-example: the midrange $(\min_i X_i + \max_i X_i)/2$ minimizes $\max_i \abs{X_i - \theta}$, a maximum over the observations rather than a sum of terms $\rho(X_i, \theta)$, so this description does not exhibit it as an M-estimator.

Under regularity conditions, $\sqrt{n}(\hat{\theta}_n - \theta_0)$ is asymptotically normal with the sandwich covariance $A^{-1} B A^{-T}$, where $A = E[\nabla_\theta \psi(X, \theta_0)]$ and $B = E[\psi(X, \theta_0)\psi(X, \theta_0)^T]$, and $\theta_0$ minimizes $E[\rho(X, \theta)]$.
</document>

<document source="https://tamnd.github.io/math2/node/7A67N3PVQG.md" tag="7A67N3PVQG" kind="definition">
# Kohn–Rossi cohomology

## Definition

Let $M$ be a smooth [[CR manifold|ZC78JKZX9S]] with CR bundle $T^{1,0}M \subset \C TM$, and let $T^{0,1}M = \overline{T^{1,0}M}$.
For $q \ge 0$, a $(0,q)$-form on $M$ is a smooth section $\varphi$ of the bundle $\Lambda^q (T^{0,1}M)^*$.
Equivalently, $\varphi$ assigns a smooth function to $q$ smooth sections of $T^{0,1}M$, and it is $C^\infty(M)$-linear and alternating.
Write $\mathcal{E}^{0,q}(M)$ for the space of $(0,q)$-forms.
Let $\bar Z_0, \dots, \bar Z_q$ be sections of $T^{0,1}M$.
A hat marks an omitted argument, and $[\cdot, \cdot]$ is the [[Lie bracket of vector fields|68FGQEBD6J]].
The tangential Cauchy–Riemann operator $\bar\partial_b \colon \mathcal{E}^{0,q}(M) \to \mathcal{E}^{0,q+1}(M)$ is given by
$$
\begin{aligned}
(\bar\partial_b \varphi)(\bar Z_0, \dots, \bar Z_q) ={} & \sum_{j=0}^{q} (-1)^j\, \bar Z_j \bigl(\varphi(\bar Z_0, \dots, \widehat{\bar Z_j}, \dots, \bar Z_q)\bigr) \\
& + \sum_{0 \le j < k \le q} (-1)^{j+k}\, \varphi\bigl([\bar Z_j, \bar Z_k], \bar Z_0, \dots, \widehat{\bar Z_j}, \dots, \widehat{\bar Z_k}, \dots, \bar Z_q\bigr)
\end{aligned}
$$
It satisfies $\bar\partial_b \circ \bar\partial_b = 0$.
The *Kohn–Rossi cohomology* groups of $M$ are
$$H^{0,q}_b(M) = \frac{\ker\bigl(\bar\partial_b \colon \mathcal{E}^{0,q}(M) \to \mathcal{E}^{0,q+1}(M)\bigr)}{\operatorname{im}\bigl(\bar\partial_b \colon \mathcal{E}^{0,q-1}(M) \to \mathcal{E}^{0,q}(M)\bigr)}, \qquad q \ge 0,$$
with $\mathcal{E}^{0,-1}(M) = 0$.

## Remarks

The formula makes sense because $T^{0,1}M$ is closed under the Lie bracket, which is the integrability condition in the definition of a CR manifold; this also gives $\bar\partial_b^2 = 0$.
It is the formula for the exterior derivative of a [[differential form|0E3G0GRPZ6]], applied only to vectors in $T^{0,1}M$.
For $q = 0$, $\bar\partial_b u$ is the restriction of $du$ to $T^{0,1}M$, so $H^{0,0}_b(M)$ is the space of smooth CR functions on $M$.
Example: on $M = \C^n$ with $T^{1,0}M$ the usual holomorphic tangent bundle, $\bar\partial_b = \bar\partial$, and $H^{0,q}_b(\C^n)$ is the Dolbeault cohomology $H^{0,q}(\C^n)$, which is $0$ for $q \ge 1$.
Non-example: on the unit sphere $S^3 \subset \C^2$, with $\bar L = z_2 \partial / \partial \bar z_1 - z_1 \partial / \partial \bar z_2$, the function $u = \bar z_1$ satisfies $\bar\partial_b u (\bar L) = \bar L \bar z_1 = z_2 \ne 0$, so it is not a cocycle and gives no element of $H^{0,0}_b(S^3)$.
More generally one defines groups $H^{p,q}_b(M)$ using forms of type $(p,q)$.
For a compact strongly pseudoconvex CR manifold of hypersurface type and real dimension $2n + 1$, Kohn proved that $H^{0,q}_b(M)$ is finite-dimensional for $1 \le q \le n - 1$.
</document>

<document source="https://tamnd.github.io/math2/node/7A68BZR6QS.md" tag="7A68BZR6QS" kind="concept">
# Rogers–Ramanujan continued fraction

## Summary

The Rogers–Ramanujan continued fraction is $R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^2}{1 + \cdots}}}$ for $\abs{q} < 1$.
It equals $q^{1/5}$ times the quotient of the two series of the [[Rogers–Ramanujan identities|M24H4GJPPW]], hence an infinite product.
As a function of $\tau$ with $q = e^{2\pi i \tau}$ it is a [[modular function|63TX60RTRQ]] for the group $\Gamma(5)$.
Ramanujan found closed forms of its values at special points, such as $R(e^{-2\pi}) = \sqrt{\frac{5 + \sqrt5}{2}} - \frac{1 + \sqrt5}{2}$.

## Explanations

### L1 human

A [[continued fraction|04P2RJ4WPS]] is a fraction whose denominator contains a fraction, whose denominator contains a fraction, and so on.
In this one the numerators are the powers $q, q^2, q^3, \dots$ and the extra factor $q^{1/5}$ in front makes the function behave well under symmetries.
At $q = e^{-2\pi}$ the value is an explicit number built from $\sqrt5$, related to the [[golden ratio|0SC8T7548D]].
Ramanujan sent such evaluations to Hardy in his famous letters.
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<document source="https://tamnd.github.io/math2/node/7A9WVPD9DG.md" tag="7A9WVPD9DG" kind="definition">
# Restricted isometry property

## Definition

Let $A$ be a real or complex $m \times N$ [[matrix|0VD9ZSWWJG]] and let $1 \le s \le N$ be an integer.
A vector $x$ is *$s$-sparse* if it has at most $s$ nonzero entries; $\norm{\cdot}_2$ is the Euclidean norm.
The *restricted isometry constant* $\delta_s = \delta_s(A)$ is the smallest $\delta \ge 0$ such that
$$(1 - \delta) \norm{x}_2^2 \le \norm{Ax}_2^2 \le (1 + \delta) \norm{x}_2^2 \quad \text{for every $s$-sparse } x \in \mathbb{K}^N ,$$
where $\mathbb{K}$ is $\R$ or $\C$.
The matrix $A$ has the *restricted isometry property* of order $s$ with constant $\delta \in (0, 1)$ if $\delta_s(A) \le \delta$.

## Remarks

Equivalently, $\delta_s = \max_S \norm{A_S^* A_S - I}_{2 \to 2}$, where $S$ runs over the sets of at most $s$ column indices, $A_S$ keeps the columns in $S$, and the norm is the spectral [[matrix norm|54T6Y6DSY4]].
So $\delta_s$ bounds how far the eigenvalues of all the Gram matrices $A_S^* A_S$ are from $1$.
Example: if the columns of $A$ are orthonormal, then $\norm{Ax}_2 = \norm{x}_2$ for all $x$, so $\delta_s = 0$ for every $s$.
Non-example: suppose the first two columns of $A$ are equal.
Then $x = e_1 - e_2$ is $2$-sparse with $Ax = 0$, so the left inequality needs $\delta \ge 1$, and $A$ has no restricted isometry property of order $2$.
The constants increase with the order: $\delta_1 \le \delta_2 \le \cdots \le \delta_N$.
If $\delta_{2s} < 1$, then distinct $s$-sparse vectors have distinct images, since their difference is $2s$-sparse.
</document>

<document source="https://tamnd.github.io/math2/node/7ABT36371X.md" tag="7ABT36371X" kind="definition">
# Witt group

## Definition

Let $F$ be a field whose [[characteristic|0F6ES3BRYP]] is not $2$.
A *quadratic space* over $F$ is a finite-dimensional vector space with a nondegenerate [[quadratic form|0RBQS1JJCR]].
The *hyperbolic plane* $H$ is $F^2$ with the form $q(x, y) = xy$.
For quadratic spaces $q_1$ and $q_2$, write $q_1 \perp q_2$ for their orthogonal sum, the form $(v_1, v_2) \mapsto q_1(v_1) + q_2(v_2)$ on the direct sum.
Write $mH$ for the orthogonal sum of $m$ copies of $H$.
Two quadratic spaces $q_1, q_2$ are *Witt equivalent* when $q_1 \perp mH$ and $q_2 \perp nH$ are isometric for some integers $m, n \ge 0$.
The *Witt group* $W(F)$ is the set of Witt equivalence classes of quadratic spaces over $F$, with the operation $[q_1] + [q_2] = [q_1 \perp q_2]$.

## Remarks

This operation is well defined, and $W(F)$ is an [[abelian group|082GHNPJ53]].
The zero is the class of $H$ and of the zero space, and $-[q] = [-q]$, because $q \perp (-q)$ is a sum of hyperbolic planes.
[[Witt's theorem|CCB8VNQCV0]] gives that each class contains exactly one anisotropic form up to isometry.
Example: $W(\C) \cong \Z/2\Z$, given by the dimension modulo $2$, because over $\C$ the form $x^2 + y^2$ is hyperbolic.
Example: $W(\R) \cong \Z$, given by the signature.
Non-example: the class of $x^2$ in $W(\R)$ is not zero, because $x^2$ is anisotropic and the zero class contains only the zero space as anisotropic form.
With the tensor product of forms, $W(F)$ is a commutative ring, the Witt ring.
</document>

<document source="https://tamnd.github.io/math2/node/7ADX0MGPNV.md" tag="7ADX0MGPNV" kind="definition">
# Weak dimension

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity and let $M$ be a left $R$-[[module|0NZ8JA25WK]].
A *flat resolution* of length $n$ of $M$ is an [[exact sequence|JB7ZBFBA9C]]
$$0 \to F_n \to F_{n-1} \to \cdots \to F_0 \to M \to 0$$
in which every $F_i$ is a [[flat|73RYYEVDKA]] left $R$-module.
The *weak dimension* (or *flat dimension*) $\operatorname{wd}(M)$ is the least $n \ge 0$ such that $M$ has a flat resolution of length $n$.
If there is no such $n$, then $\operatorname{wd}(M) = \infty$.
The *weak global dimension* of $R$ is the supremum of $\operatorname{wd}(M)$ over all left $R$-modules $M$.

## Remarks

Equivalently, $\operatorname{wd}(M) \le n$ if and only if $\operatorname{Tor}^R_{n+1}(N, M) = 0$ for every right $R$-module $N$, with [[Tor|DDQRERAQCV]] the derived functors of the tensor product.
So the weak global dimension of $R$ is the supremum of the $n$ with $\operatorname{Tor}^R_n(N, M) \ne 0$ for some right module $N$ and left module $M$.
It is the same for left and right modules.
Example: over $R = \Z$, the module $\Z/2$ has weak dimension $1$.
The sequence $0 \to \Z \xrightarrow{2} \Z \to \Z/2 \to 0$ is a flat resolution of length $1$.
The module $\Z/2$ is not flat, since tensoring the injective map $\Z \xrightarrow{2} \Z$ with $\Z/2$ gives the zero map on $\Z/2$, which is not injective.
Non-example of finite dimension: over $R = \Z/4$, the module $\Z/2$ has infinite weak dimension.
Its resolution $\cdots \to \Z/4 \xrightarrow{2} \Z/4 \xrightarrow{2} \Z/4 \to \Z/2 \to 0$ gives $\operatorname{Tor}^R_n(\Z/2, \Z/2) \cong \Z/2 \ne 0$ for every $n \ge 1$.
The weak dimension of a module is at most its [[projective dimension|MR9QCH1PNA]], so the weak global dimension is at most the left and the right [[global dimension|GWA0T1WKCN]].
For a left Noetherian ring it equals the left global dimension.
A ring has weak global dimension $0$ if and only if it is von Neumann regular.
</document>

<document source="https://tamnd.github.io/math2/node/7AMQ8GS78Y.md" tag="7AMQ8GS78Y" kind="concept">
# Morse–Smale system

## Summary

A Morse–Smale system is a smooth dynamical system whose non-wandering set consists of finitely many hyperbolic periodic orbits, with stable and unstable manifolds meeting transversally.
These are the dynamical systems with the simplest possible recurrent behaviour.
They are [[structurally stable|D4MGFAHVMT]], and on a compact manifold they have zero topological entropy.
The gradient flow of a [[Morse function|KPQ96640ZJ]] for a generic metric is Morse–Smale, which links them to Morse theory and Morse homology.

## Explanations

### L1 human

Think of water flowing on a landscape with finitely many peaks, passes and pits.
Every drop starts at one of these rest points and ends at another.
A Morse–Smale system is a motion of this kind.
All its rest points and periodic orbits are of the clean, non-degenerate type, and the paths between them cross in general position.
For example, on a round sphere, the flow from the north pole down to the south pole is Morse–Smale.
See the [[definition|10B1TS96VA]].
</document>

<document source="https://tamnd.github.io/math2/node/7AWZJNQGT2.md" tag="7AWZJNQGT2" kind="statement">
# Oka–Weil theorem

## Statement

Let $K \subseteq \C^n$ be a compact [[polynomially convex set|AD3474PVHR]].
Let $f$ be a [[holomorphic function|507C34W0ZT]] on an open set containing $K$.
Then for every $\varepsilon > 0$ there is a [[polynomial|07GNX8GCYP]] $p$ in $z_1, \dots, z_n$ with $\abs{f(z) - p(z)} < \varepsilon$ for all $z \in K$.
That is, $f$ is a [[uniform limit|08VZ57PT26]] of polynomials on $K$.

## Remarks

The proof first finds polynomials $p_1, \dots, p_N$ such that the polynomial polyhedron $P = \set{z : \abs{z_j} \le R, \ \abs{p_k(z)} \le 1}$ contains $K$ and lies in the open set where $f$ is defined; this uses polynomial convexity of $K$.
The map $z \mapsto (z, p_1(z), \dots, p_N(z))$ embeds a neighbourhood of $P$ as a closed complex submanifold of a polydisc in $\C^{n+N}$.
Oka's extension theorem extends $f$ from this submanifold to a holomorphic function on the polydisc, and the Taylor series of the extension gives polynomials converging to $f$ uniformly on $K$.

Example: for $n = 1$ the theorem is the polynomial case of [[Runge's theorem|6DG4CTGXJD]]: if $\C \setminus K$ is connected, every function holomorphic near $K$ is a uniform limit of polynomials on $K$.

Polynomial convexity cannot be dropped.
On the unit circle $K$, which is not polynomially convex, the function $1/z$ is not a uniform limit of polynomials: polynomials $p$ satisfy $\int_K p(z) \, dz = 0$, while $\int_K z^{-1} \, dz = 2\pi i$.

## Explanations

### L1 human

On a compact set that equals its own polynomial hull, every holomorphic function can be approximated as closely as one likes by polynomials.
This is the several-variable version of Runge's approximation theorem.
The condition on the set rules out holes that would trap functions such as $1/z$.

### L2 human

The theorem is a basic approximation result of several complex variables.
It generalises to holomorphically convex compact subsets of Stein manifolds and of domains of holomorphy, with polynomials replaced by holomorphic functions on the whole domain.
The method of its proof, embedding a polynomial polyhedron into a polydisc, reduces global questions to questions on polydiscs.
It is also used to show that convex domains and balls are Runge domains.
</document>

<document source="https://tamnd.github.io/math2/node/7B03TB64N9.md" tag="7B03TB64N9" kind="concept">
# Oval

## Summary

An oval in a projective plane is a set of points that behaves like a conic.
No line meets it in more than two points, and at each of its points there is exactly one tangent line.
In a finite plane of order $n$, an oval is the same as a set of $n + 1$ points with no three on a line.
Non-degenerate [[projective conics|R3TMVHTP3A]] are ovals.
[[Segre's theorem|SPFP3GHVYJ]] says that in $PG(2, q)$ with $q$ odd there are no others.
For $q$ even, ovals extend to [[hyperovals|RNG2A8BFVZ]], and for $q \ge 8$ there are ovals that are not conics.

## Explanations

### L1 human

A circle meets any line in at most two points, and at each of its points it has a single tangent.
An oval keeps these two properties, stated only in terms of points and lines, so it makes sense in any projective plane.
In the Fano plane, the four points off a fixed line form a set with no three on a line; any three of them form an oval.
Ovals are the starting point for the study of arcs and caps in finite geometry.
</document>

<document source="https://tamnd.github.io/math2/node/7B53ZGXXEQ.md" tag="7B53ZGXXEQ" kind="concept">
# Brouwer–Heyting–Kolmogorov interpretation

## Summary

The Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, explains the meaning of the logical connectives in [[intuitionistic logic|Q11N8X99FK]].
It says what counts as a proof of a compound statement in terms of proofs of its parts.
It is an informal explanation, not a formal semantics, and it expresses the view of [[constructivism|G958EQ8CWW]].
Realizability and the Curry–Howard correspondence are precise versions of it.

## Explanations

### L1 human

In classical logic, a statement is true or false, and the connectives act on truth values.
In the BHK interpretation, one asks instead what a proof of the statement is.
A proof of "$A$ and $B$" is a pair of proofs, one for each.
A proof of "if $A$ then $B$" is a method that turns each proof of $A$ into a proof of $B$.
So a proof of "$A$ or not $A$" would have to decide which one holds, and in general no such method is known.
</document>

<document source="https://tamnd.github.io/math2/node/7BGRWJGYWB.md" tag="7BGRWJGYWB" kind="definition">
# Euler transform

## Definition

Let $(a_n)_{n \ge 0}$ be a sequence of real or complex numbers, and consider the [[alternating series|017BR1M98J]] $\sum_{n \ge 0} (-1)^n a_n$.
The forward difference is $\Delta a_n = a_{n+1} - a_n$, and its iterates satisfy
$$\Delta^n a_0 = \sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} a_k,$$
with the [[binomial coefficients|092GEF38YD]] $\binom{n}{k}$.
The *Euler transform* of the series is the [[series|0JH4MN4K1C]]
$$\sum_{n \ge 0} \frac{(-1)^n \Delta^n a_0}{2^{n+1}}.$$
If $\sum (-1)^n a_n$ converges, then its Euler transform converges to the same sum.

## Remarks

Example: for $a_n = \frac{1}{n+1}$ one has $(-1)^n \Delta^n a_0 = \frac{1}{n+1}$, so the alternating harmonic series becomes $\sum_{n \ge 0} \frac{1}{(n+1) 2^{n+1}}$, which equals $\ln 2$.
For $a_n = 1$ the series $1 - 1 + 1 - \cdots$ diverges, but $\Delta^n a_0 = 0$ for $n \ge 1$ and the transform is $\frac12$; this is how Euler summation assigns the value $\frac12$ to it.
Non-example: for $a_n = 3^n$ one gets $\Delta^n a_0 = 2^n$, and the transform $\sum_{n \ge 0} \frac{(-1)^n}{2}$ diverges, so the transform does not always produce a convergent series.

## Explanations

### L1 human

The transform replaces the terms by weighted differences of the first terms.
For smooth terms the high differences are small, and the weights $2^{-n-1}$ make the new series converge fast.
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<document source="https://tamnd.github.io/math2/node/7BJEH1GCTE.md" tag="7BJEH1GCTE" kind="concept">
# Darboux function

## Summary

A Darboux function is a real function on an interval that has the intermediate value property.
It never skips a value between two of its values.
Every [[continuous function|0P27W9RW0G]] on an interval is a Darboux function by the [[intermediate value theorem|00KAFBR32E]].
By [[Darboux's theorem|0Q5A47J48D]], every derivative is a Darboux function too, even when it is not continuous.
So the class of Darboux functions is much larger than the class of continuous functions.

## Explanations

### L1 human

Imagine driving along a road whose height is given by the function.
A Darboux function cannot get from height $0$ to height $1$ without passing every height in between.
Continuous functions behave like this, but so do some wild functions.
For example, $\sin(1/x)$ with value $0$ at $0$ wiggles infinitely often near $0$ and is not continuous there, yet it skips no values.
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<document source="https://tamnd.github.io/math2/node/7BKQT71FYZ.md" tag="7BKQT71FYZ" kind="concept">
# p-harmonic function

## Summary

A $p$-harmonic function is a solution of the [[p-Laplace equation|XJMYGY0ZZF]] $\operatorname{div}(\abs{\nabla u}^{p-2} \nabla u) = 0$, where $1 < p < \infty$.
For $p = 2$ these are the classical [[harmonic functions|0DEB0JMN0W]].
They are the minimizers of the energy $\int \abs{\nabla u}^p \, dx$ with given boundary values, and they are the central objects of [[nonlinear potential theory|M923VKQVZM]].
The equation is nonlinear for $p \ne 2$, so sums of $p$-harmonic functions are in general not $p$-harmonic.

## Explanations

### L1 human

A harmonic function makes the energy $\int \abs{\nabla u}^2$ as small as possible among functions with the same boundary values.
A $p$-harmonic function does the same for the energy $\int \abs{\nabla u}^p$.
Changing the exponent changes how strongly steep slopes are penalized.
Linear functions are $p$-harmonic for every $p$, because their gradient is constant.
Many properties of harmonic functions survive, such as the maximum principle and a Harnack inequality, but the mean value property does not.
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<document source="https://tamnd.github.io/math2/node/7BPSC4T4M3.md" tag="7BPSC4T4M3" kind="concept">
# Steiner system

## Summary

A Steiner system $S(t, k, n)$ is a set of $n$ points with a family of $k$-element blocks.
Every $t$ points lie in exactly one block.
It is a [[block design|4JCSS2Z1BA]] with $\lambda = 1$.
Steiner triple systems, finite projective planes and the large Witt design $S(5, 8, 24)$ are the best-known examples.
The node [[Steiner system|QHPW53E7YA]] gives the precise definition.

## Explanations

### L1 human

A Steiner system is a way to group points into blocks of equal size so that any choice of $t$ points determines exactly one block.
With $t = 2$, two points always lie together in exactly one block, like two points on exactly one line.
The [[Fano plane|QQ8GW1ASZM]] is a Steiner system $S(2, 3, 7)$: $7$ points, $7$ blocks of size $3$, and every pair of points in one block.
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# Fixed-point iteration

## Definition

Let $X$ be a set, let $f \colon X \to X$ be a map, and let $x_0 \in X$.
The *fixed-point iteration* of $f$ starting at $x_0$ is the [[sequence|0QE8STGJH2]] $(x_n)_{n \ge 0}$ defined by
$$x_{n+1} = f(x_n) \quad \text{for } n \ge 0.$$
A *fixed point* of $f$ is a point $x^* \in X$ with $f(x^*) = x^*$.

## Remarks

If $X$ is a metric space, $f$ is [[continuous|0P27W9RW0G]], and the iteration [[converges|06JYPJ4WHH]] to $x^*$, then $x^*$ is a fixed point, since $f(x^*) = \lim f(x_n) = \lim x_{n+1} = x^*$.
Example: for $f(x) = x/2 + 1/x$ on $(0, \infty)$ and $x_0 = 1$, the iterates are $1, 1.5, 1.41\overline{6}, \ldots$, and they converge to $\sqrt{2}$, the fixed point, since $x = x/2 + 1/x$ means $x^2 = 2$.
Non-example of convergence: for $f(x) = 2x$ on $\R$ and $x_0 = 1$, the iterates $2^n$ diverge, although $0$ is a fixed point.
The [[Banach fixed point theorem|0VK6VEH345]] says the iteration converges to the unique fixed point when $f$ is a [[contraction|0QN69YA357]] of a nonempty complete metric space.
If $q < 1$ is the contraction constant, the error after $n$ steps is at most $q^n d(x_1, x_0) / (1 - q)$.
The sequence is also written $x_n = f^n(x_0)$, the $n$-th iterate of $f$ applied to $x_0$.
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# Harnack inequality for parabolic equations

## Statement

Let $U \subseteq \R^n$ be open, let $T > 0$, and let $0 < \lambda \le \Lambda$.
Let $A = (a_{ij})$ be a measurable function on $U \times (0, T]$ with values in the symmetric $n \times n$ matrices such that $\lambda \abs{\eta}^2 \le \sum_{i,j} a_{ij}(x, t)\, \eta_i \eta_j \le \Lambda \abs{\eta}^2$ for all $\eta \in \R^n$ and all $(x, t)$.
Let $u \ge 0$ be a [[weak solution|SJZX81A783]] of the [[parabolic equation|1CCTH8X2FT]]
$$\partial_t u = \sum_{i,j=1}^{n} \partial_{x_i}\big(a_{ij}\, \partial_{x_j} u\big) \quad \text{in } U \times (0, T].$$
Let $V$ be a connected open set with compact closure $\overline{V} \subset U$, and let $0 < t_1 < t_2 \le T$.
Then there is a constant $C$, depending only on $n$, $\lambda$, $\Lambda$, $U$, $V$, $t_1$ and $t_2$, such that
$$\sup_{x \in V} u(x, t_1) \le C \inf_{x \in V} u(x, t_2).$$

## Remarks

This is Moser's parabolic Harnack inequality of 1964; the solution is Hölder continuous by the De Giorgi–Nash–Moser theory, so the pointwise values make sense.
Moser's proof bounds $\sup u$ and $\inf u$ on small cylinders by integral means of powers of $u$, using iterated energy estimates.
An estimate for $\log u$ then links the two bounds.
A chain of overlapping cylinders then gives the statement on $V$, which is why $V$ must be connected.
The time lag $t_1 < t_2$ is necessary.
For the [[heat equation|VP2HN86VAG]] $u_t = \Delta u$ on $\R^n$, the solutions $u_y(x, t) = (4\pi t)^{-n/2} e^{-\abs{x - y}^2/(4t)}$ with $y \in \R^n$ satisfy $u_y(x_1, t)/u_y(x_2, t) = e^{(\abs{x_2 - y}^2 - \abs{x_1 - y}^2)/(4t)}$, which is unbounded over $y$ for fixed distinct $x_1, x_2$.
So no bound $\sup_V u(\cdot, t) \le C \inf_V u(\cdot, t)$ at a single time can hold for all nonnegative solutions.
The hypothesis $u \ge 0$ is also necessary, as for [[Harnack's inequality|YNH832D50W]] for harmonic functions.

## Explanations

### L1 human

A nonnegative temperature distribution cannot be very hot at one place now and almost zero at a nearby place a little later.
Heat that is present now spreads, so it raises the temperature nearby at later times.
The inequality makes this precise: the largest value at an earlier time is controlled by the smallest value at a later time.
Looking at the same time is not enough, because a very concentrated source far away can make two nearby points have very different temperatures.

### L2 human

The parabolic Harnack inequality is the time-dependent analogue of [[Harnack's inequality|YNH832D50W]] for nonnegative harmonic functions.
It holds for equations with merely measurable coefficients, which makes it a central tool in the regularity theory of nonlinear parabolic equations.
From it one derives Hölder continuity of solutions and a strong maximum principle.
For the heat equation on Riemannian manifolds with nonnegative Ricci curvature, Li and Yau proved a sharp differential form, which integrates to a Harnack inequality with explicit constants.
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# Einstein field equations

## Definition

Let $M$ be a four-dimensional manifold, let $\Lambda \in \R$ be a constant, the cosmological constant, and let $\kappa > 0$ be a constant.
Let $T$ be a given symmetric $(0,2)$-[[tensor field|WX302T93TW]] on $M$, the stress–energy tensor.
A [[Lorentzian metric|BY4QYYS788]] $g$ on $M$ satisfies the *Einstein field equations* if
$$\mathrm{Ric}_g - \tfrac{1}{2} R_g \, g + \Lambda g = \kappa T.$$
Here $\mathrm{Ric}_g$ is the [[Ricci curvature|4GTNM0WHDH]] and $R_g$ is the [[scalar curvature|ZHX2VYQDZ3]] of $g$.
In physical units $\kappa = 8 \pi G / c^4$, with $G$ the gravitational constant and $c$ the speed of light.
The *vacuum Einstein equations* are the case $T = 0$ and $\Lambda = 0$.

## Remarks

Taking the trace with respect to $g$ in dimension four shows that the vacuum equations are equivalent to $\mathrm{Ric}_g = 0$.
In local coordinates the equations read $R_{\mu\nu} - \tfrac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu}$, ten equations for the ten components of $g$.

Example: Minkowski space, $\R^4$ with $g = -dt^2 + dx^2 + dy^2 + dz^2$, is flat, so $\mathrm{Ric}_g = 0$ and it solves the vacuum equations.

Non-example: de Sitter space has $\mathrm{Ric}_g = \Lambda_0 g$ with a constant $\Lambda_0 > 0$, so it does not solve the vacuum equations.
It solves the equations with $T = 0$ and cosmological constant $\Lambda = \Lambda_0$.

The left side has zero divergence by the contracted Bianchi identity, which forces the conservation law $\nabla^\mu T_{\mu\nu} = 0$.
Yvonne Choquet-Bruhat proved that the vacuum equations have a well-posed initial value problem for suitable initial data.
The Riemannian analogue of $\mathrm{Ric}_g = \lambda g$ defines an [[Einstein manifold|41S2EZ0KR3]].
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# Nonlinear difference equation

## Summary

A nonlinear difference equation is a [[recurrence relation|0FYX44GP92]] in which the next term is not an affine function of the earlier terms.
Products, powers, quotients or other nonlinear functions of the earlier terms may appear.
Unlike linear recurrences, such equations rarely have closed-form solutions, and their solutions can converge, oscillate, or behave chaotically.
They model population dynamics, iterative numerical methods and many discrete-time systems.

## Explanations

### L1 human

In a linear recurrence, the next term is a weighted sum of earlier terms plus a known amount.
In a nonlinear one, the rule may square terms, multiply them, or divide by them.
The [[logistic map|QN9X9BNKTX]] $x_{n+1} = r x_n (1 - x_n)$ is a famous example.
For $r = 2$ and $x_0 = 0.1$ the values settle at $1/2$, but for $r = 4$ they jump around without any pattern.
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# Chebyshev polynomials of the second kind

## Summary

The Chebyshev polynomials of the second kind $U_n$ are the polynomials with $U_n(\cos \theta) \sin \theta = \sin((n + 1)\theta)$.
They are the companions of the [[Chebyshev polynomials|M6EAR1JHPP]] of the first kind $T_n$, and satisfy the same three-term recurrence with a different start.
They are orthogonal on $[-1, 1]$ for the weight $\sqrt{1 - x^2}$, and they are the [[Gegenbauer polynomials|5TRFA6F79B]] with parameter $1$.

## Explanations

### L1 human

The ratio $\sin((n + 1)\theta) / \sin \theta$ can always be written as a polynomial in $\cos \theta$.
That polynomial is $U_n$.
For example, $\sin(3\theta) / \sin \theta = 4 \cos^2 \theta - 1$, so $U_2(x) = 4x^2 - 1$.
They show up as derivatives of the first-kind polynomials, as determinants of tridiagonal matrices, and in the semicircle law of random matrices.
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# Riemannian manifold

## Summary

A Riemannian manifold is a smooth manifold together with a [[Riemannian metric|Q61VFHCH2J]].
It is the setting of Riemannian geometry, where one studies lengths, angles, geodesics, volume and curvature on curved spaces.
Euclidean space, spheres and hyperbolic spaces are the basic examples, and every submanifold of Euclidean space inherits a Riemannian structure.
Replacing positive definiteness by a Lorentzian signature gives the [[pseudo-Riemannian manifolds|GH6CXWFNDJ]] that model spacetime in general relativity.

## Explanations

### L1 human

A Riemannian manifold is a curved space where you can measure.
Near each point it looks like a small piece of ordinary space, and you know how to measure lengths and angles there.
From this you can find the shortest paths, compute areas, and detect how curved the space is.
The surface of the Earth, with distances measured along the surface, is a two-dimensional example.
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# Modified Richardson iteration

## Statement

Let $A$ be a real symmetric [[positive definite|0KX66CZMBB]] $n \times n$ matrix, let $b \in \R^n$, and let $\omega > 0$.
The *modified Richardson iteration* for the [[linear system|0GT4WWGWH2]] $A x = b$ starts from any $x^{(0)} \in \R^n$ and sets
$$x^{(k+1)} = x^{(k)} + \omega \left( b - A x^{(k)} \right).$$
Let $\lambda_{\min}$ and $\lambda_{\max}$ be the least and greatest [[eigenvalues|0BQAA2BK3Q]] of $A$.

1. The iterates converge to $A^{-1} b$ for every $b$ and every $x^{(0)}$ if and only if $0 < \omega < 2 / \lambda_{\max}$.
2. The [[spectral radius|9V7YQA85PC]] of the iteration matrix $I - \omega A$ is least for $\omega = 2 / (\lambda_{\min} + \lambda_{\max})$. Its value there is $(\kappa - 1) / (\kappa + 1)$, where $\kappa = \lambda_{\max} / \lambda_{\min}$.

## Remarks

The error satisfies $e_{k+1} = (I - \omega A) e_k$, and $I - \omega A$ has the eigenvalues $1 - \omega \lambda$ for the eigenvalues $\lambda$ of $A$.
All of them lie in $(-1, 1)$ exactly when $0 < \omega \lambda_{\max} < 2$, which gives clause 1.
The largest of $\abs{1 - \omega \lambda_{\min}}$ and $\abs{1 - \omega \lambda_{\max}}$ is least when the two are equal, which gives clause 2.
Example: if $A$ has eigenvalues $1$ and $3$, the best choice is $\omega = 1/2$, and the error shrinks by the factor $1/2$ at each step.
The number $\kappa$ is the [[condition number|AB3S6MHJ8X]] of $A$, so the method is slow when $A$ is ill-conditioned.

## Explanations

### L1 human

The residual $b - A x$ tells how far the current guess is from solving the system.
Richardson's method moves the guess a fixed fraction $\omega$ of the residual and repeats.
The step must not be too large, or the guess overshoots and the error grows.

### L2 human

Richardson iteration is the simplest stationary [[iterative method|2RPE77ACER]], and the Jacobi method is Richardson iteration applied to the system scaled by the diagonal.
With a [[preconditioner|KE98SW3SBD]] $P$ it becomes $x^{(k+1)} = x^{(k)} + \omega P^{-1} (b - A x^{(k)})$, and the same analysis applies to $P^{-1} A$.
For a symmetric positive definite $A$ it is gradient descent with a fixed step on $\frac12 x^T A x - b^T x$.
Letting $\omega$ vary with $k$ and choosing the steps as reciprocal Chebyshev nodes leads to Chebyshev iteration.
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# Subalgebra

## Summary

Let $A$ be an [[associative algebra|3CTW1P4HBP]] over a commutative ring $K$.
A subalgebra of $A$ is a subset that contains $1$ and is itself a $K$-algebra under the same operations.
It is a [[subring|0GNCRPYXYV]] of $A$ that is also a [[submodule|05WDSN597C]] for the scalar multiplication by $K$.
Subalgebras are the natural subobjects of algebras.
Images of algebra homomorphisms are subalgebras, and so are the [[subalgebras generated|2TQD8NMG9P]] by a few elements, such as $K[a] \subseteq A$.
It is the [[substructure|676H7EGTKJ]] for the operations of a $K$-algebra, with $1$ as a constant.

## Explanations

### L1 human

Take an algebra, a place where you can add, multiply and scale by numbers.
A subalgebra is a smaller piece that you never leave when you do these operations, and that contains $1$.
For example, in the algebra of all $2 \times 2$ real matrices, the upper triangular matrices form a subalgebra.
Adding, multiplying or scaling upper triangular matrices gives an upper triangular matrix, and the identity matrix is upper triangular.
The polynomials with integer coefficients do not form a subalgebra of the real polynomials $\R[x]$, because scaling $1$ by $1/2$ leaves the set.
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# Theta function

## Definition

Let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]].
The *Jacobi theta function* is the function $\vartheta \colon \C \times \mathbb{H} \to \C$ given by
$$\vartheta(z, \tau) = \sum_{n = -\infty}^{\infty} \exp(\pi i n^2 \tau + 2 \pi i n z).$$
Its value at $z = 0$ is the theta constant $\theta(\tau) = \vartheta(0, \tau) = \sum_{n \in \Z} e^{\pi i n^2 \tau}$.

## Remarks

For $\operatorname{Im} \tau > 0$ the terms decay like $e^{-\pi n^2 \operatorname{Im} \tau}$ times a factor locally bounded in $z$, so the series converges absolutely and locally uniformly.
So $\vartheta$ is [[holomorphic|0NAHXR1FRH]] in both variables.
It satisfies $\vartheta(z + 1, \tau) = \vartheta(z, \tau)$ and $\vartheta(z + \tau, \tau) = \exp(-\pi i \tau - 2 \pi i z)\, \vartheta(z, \tau)$.
Check of the second rule: replace $n$ by $m - 1$ and use $n^2 + 2n = (n + 1)^2 - 1$.
The theta constant satisfies $\theta(-1/\tau) = \sqrt{-i\tau}\, \theta(\tau)$, by the Poisson summation formula.
Example: $\theta(\tau)^4 = \sum_{n \ge 0} r_4(n) e^{\pi i n \tau}$, where $r_4(n)$ counts the ways to write $n$ as a sum of four squares.
Non-example: for real $\tau$ the series diverges, because every term has absolute value $1$ when $z$ is also real; this is why $\tau$ must lie in $\mathbb{H}$.
Other authors use the nome $q = e^{\pi i \tau}$ and the four functions $\vartheta_1, \ldots, \vartheta_4$; the function above is then $\vartheta_3$.
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# Ehresmann's fibration theorem

## Statement

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]].
Let $f \colon M \to N$ be a smooth map that is a surjective [[submersion|BMY55EW3RF]] and a [[proper map|15XEBGNGHC]].
Then $f$ is a locally trivial [[fiber bundle|CX4M8K61AB]].
That is, for every $q \in N$ there are an open neighbourhood $V$ of $q$ and a [[diffeomorphism|0KZZ70ZHXP]] $\varphi \colon f^{-1}(V) \to V \times f^{-1}(q)$ such that $\mathrm{pr}_1 \circ \varphi = f$ on $f^{-1}(V)$.

## Remarks

The idea of the proof: choose a complement of the kernel of $df$ in $TM$, for instance the orthogonal complement for a Riemannian metric; this is an [[Ehresmann connection|TD6VEP3SCE]].
Over a coordinate ball $V$ around $q$, lift the coordinate vector fields of $V$ to horizontal vector fields on $f^{-1}(V)$.
Properness makes their [[flows|06738GGWWK]] exist as long as the flow downstairs stays in $V$.
Composing these flows moves the fiber $f^{-1}(q)$ onto every nearby fiber, and this gives $\varphi$.

Example: if $M$ is compact and $N$ is connected, every submersion $f \colon M \to N$ with $M$ nonempty is proper and surjective, so all fibers of $f$ are diffeomorphic.

Properness is needed.
Let $M = \R^2 \setminus \set{0}$, $N = \R$ and $f(x, y) = x$.
Then $f$ is a surjective submersion, but its fiber over $0$ is two disjoint half-lines while its fiber over $1$ is one line.
These fibers are not diffeomorphic, so $f$ is not locally trivial near $0$.
Here $f$ is not proper: the preimage of $[-1, 1]$ is not compact.

## Explanations

### L1 human

A submersion is a smooth map with no critical points, so near each point it looks like a projection.
The theorem says that if, in addition, preimages of compact sets are compact, then the map looks like a projection near each whole fiber, not only near each point.
In particular all fibers over a connected base are the same manifold.

### L2 human

The theorem is a main source of fiber bundles in geometry and topology.
For instance, a proper family of smooth hypersurfaces with no singular member has diffeomorphic members, which is the start of the study of monodromy and of the Milnor fibration.
It is the smooth counterpart of the fact that a proper local homeomorphism between connected manifolds is a covering map.
For non-proper maps, local triviality can still hold, but one must control the behaviour at infinity by other means.
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# A priori estimate

## Summary

An a priori estimate is a bound on a solution of a differential equation in terms of the data.
It is proved under the assumption that the solution already exists and is regular enough.
It is proved before, and independently of, any existence result.
A priori estimates are the main input of existence proofs by continuity, by fixed points, or by compactness, and they also give uniqueness and stability.

## Explanations

### L1 human

Suppose one wants to show that an equation has a solution, but cannot write it down.
A common strategy is to first assume a solution exists and show it cannot be too large.
This bound is the a priori estimate: "a priori" because it comes before knowing that a solution exists.
For example, a solution of $-u'' = f$ on $(0, 1)$ with $u(0) = u(1) = 0$ always satisfies $\max \abs{u} \le \frac{1}{8} \max \abs{f}$.
With such bounds in hand, approximate solutions stay under control and a limit gives a true solution.
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# Semimodular lattice

## Definition

Let $L$ be a [[lattice|WH4SFAACAS]] with [[join and meet|J2T75ATRV5]] $\vee$ and $\wedge$.
Write $x \lessdot y$ when $y$ [[covers|4HHDPZ4BDY]] $x$, that is, $x < y$ and no $z \in L$ satisfies $x < z < y$.
The lattice $L$ is *semimodular*, or *upper semimodular*, if for all $a, b \in L$,
$$a \wedge b \lessdot a \implies b \lessdot a \vee b.$$
Dually, $L$ is *lower semimodular* if $b \lessdot a \vee b$ implies $a \wedge b \lessdot a$ for all $a, b \in L$.

## Remarks

Every [[modular lattice|QX7WZ92QAK]] is both upper and lower semimodular, because in a modular lattice the intervals $[a \wedge b, a]$ and $[b, a \vee b]$ are isomorphic.

Example: the [[lattice of partitions|9RP82GD0Q4]] of a finite set, ordered by refinement with the coarser partition above, is upper semimodular.
If a partition $a$ covers $a \wedge b$, then $a$ arises from $a \wedge b$ by merging two blocks.
Then $a \vee b$ arises from $b$ by merging the two blocks of $b$ that contain these two blocks.
These blocks of $b$ are different, since otherwise $a \le b$ and $a = a \wedge b$; so $b \lessdot a \vee b$.
For a set of $4$ or more elements, this lattice is not modular.

Non-example: the [[pentagon lattice N5|DHC8TZXGYS]] with $0 < a < c < 1$ and $b$ incomparable to $a$ and $c$.
Apply the condition to the pair $(b, a)$: $b \wedge a = 0$ and $0 \lessdot b$, but $a < c < 1 = a \vee b$, so $a \lessdot a \vee b$ fails.

For a finite lattice, upper semimodularity holds if and only if the lattice is graded with a rank function $r$ satisfying $r(a) + r(b) \ge r(a \wedge b) + r(a \vee b)$ for all $a, b$.
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# Arithmetical hierarchy

## Summary

The arithmetical hierarchy classifies sets of natural numbers by the number of alternating blocks of unbounded quantifiers needed to define them in first-order arithmetic.
Its levels are the classes $\Sigma^0_n$, $\Pi^0_n$ and $\Delta^0_n$ for $n \in \N$.
The first levels are familiar: $\Delta^0_1$ sets are the [[computable sets|PPGD5KW8JC]] and $\Sigma^0_1$ sets are the [[computably enumerable sets|TN9G24VABN]].
[[Post's theorem|NYJX5QXX1T]] links each level to computability relative to iterated [[Turing jumps|60P2XWY5QX]].

## Explanations

### L1 human

Some properties of a number $e$ can be checked by a search that ends when a witness is found: "there is a $c$ such that ...".
These are the $\Sigma^0_1$ properties, for example "program $e$ halts on input $e$".
Other properties need two alternating quantifiers.
An example is "for every $x$ there is a $c$ such that program $e$ halts on $x$ within $c$ steps", which says that program $e$ is total.
Each extra alternation of "there is" and "for all" can define strictly more sets.
The hierarchy measures how unsolvable a problem is.
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# Frobenius method

## Definition

Let $z_0$ be a [[regular singular point|H96HXG6CA0]] of $y'' + p(z) y' + q(z) y = 0$.
Write $(z - z_0)\, p(z) = \sum_{k \ge 0} p_k (z - z_0)^k$ and $(z - z_0)^2 q(z) = \sum_{k \ge 0} q_k (z - z_0)^k$ as convergent [[power series|0MZB06ST2B]].
The *Frobenius method* seeks solutions of the form
$$y = (z - z_0)^r \sum_{k \ge 0} a_k (z - z_0)^k, \qquad a_0 \ne 0,$$
with an exponent $r \in \C$ and coefficients $a_k \in \C$, as follows.

1. Put $F(s) = s (s - 1) + p_0 s + q_0$. The exponent $r$ must be a root of the *indicial equation* $F(r) = 0$.
2. The coefficients satisfy, for $k \ge 1$, the recurrence
$$F(r + k)\, a_k = - \sum_{j=0}^{k-1} \left( (r + j)\, p_{k-j} + q_{k-j} \right) a_j.$$
3. When $F(r + k) \ne 0$ for all $k \ge 1$, the recurrence determines all $a_k$ from $a_0$.

## Remarks

The rules come from inserting the series into $(z - z_0)^2 y'' + (z - z_0) \cdot (z - z_0) p\, y' + (z - z_0)^2 q\, y = 0$ and comparing the coefficients of $(z - z_0)^{r + k}$.
The roots of the indicial equation are the *exponents* of the equation at $z_0$.
Fuchs' theorem says when the series converges and how to get a second solution.

Example: for $z^2 y'' + z y' + (z^2 - \frac{1}{4}) y = 0$ at $z_0 = 0$, one has $p_0 = 1$, $q_0 = -\frac{1}{4}$, $q_2 = 1$, and all other $p_k, q_k$ vanish.
So $F(s) = s^2 - \frac{1}{4}$, with roots $r = \pm \frac{1}{2}$.
For $r = \frac{1}{2}$ the recurrence is $k (k + 1) a_k = - a_{k-2}$, with $a_1 = 0$, which gives $y = z^{-1/2} \sin z$ when $a_0 = 1$.

Non-example: at an irregular singular point the method can fail.
For $z^2 y'' + y' = 0$ at $0$, the only solutions of the form above are constants, while $y = \int e^{1/z}\, dz$ is a second solution whose derivative has an essential singularity at $0$.
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# Torus integral

## Definition

Let $c = (c_1, \dots, c_n) \in \C^n$ and $r = (r_1, \dots, r_n)$ with each $r_j > 0$.
Let $T(c, r)$ be the [[distinguished boundary|A6HN207Y2T]] $\set{\zeta : \abs{\zeta_j - c_j} = r_j \text{ for all } j}$.
For $\theta = (\theta_1, \dots, \theta_n) \in [0, 2\pi]^n$, put $\gamma(\theta) = (c_1 + r_1 e^{i\theta_1}, \dots, c_n + r_n e^{i\theta_n})$.
Let $f \colon T(c, r) \to \C$ be a function such that $\theta \mapsto f(\gamma(\theta))$ is [[Lebesgue integrable|DWJF7ABPR1]] on $[0, 2\pi]^n$, for example a continuous function.
The *torus integral* of $f$ over $T(c, r)$ is
$$\int_{T(c, r)} f(\zeta) \, d\zeta_1 \cdots d\zeta_n = \int_{[0, 2\pi]^n} f(\gamma(\theta)) \prod_{j=1}^n \left( i r_j e^{i\theta_j} \right) d\theta_1 \cdots d\theta_n.$$

## Remarks

The factor $i r_j e^{i\theta_j} \, d\theta_j$ is $d\zeta_j$ along the circle $\zeta_j = c_j + r_j e^{i\theta_j}$, so for $n = 1$ this is the [[contour integral|06ZGPYTWK8]] over the positively oriented circle.
For continuous $f$, Fubini's theorem makes it equal to the iterated integral of $n$ one-variable contour integrals, in any order.
Example: for $f(\zeta) = 1/(\zeta_1 \zeta_2)$ on $T(0, (1, 1))$, the integrand is $e^{-i\theta_1} e^{-i\theta_2} \cdot i e^{i\theta_1} \cdot i e^{i\theta_2} = -1$, so the integral is $-(2\pi)^2 = (2\pi i)^2$.
Non-example: $f(\zeta) = 1/(\zeta_1 - 1)$ is not defined at the points of $T(0, (1, 1))$ with $\zeta_1 = 1$.
Near $\theta_1 = 0$, $\abs{f(\gamma(\theta))}$ behaves like $1/\abs{\theta_1}$, so it is not integrable and the torus integral is not defined.
Other notation: $\oint_{T(c, r)} f(\zeta) \, d\zeta$.
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# Runge–Kutta methods

## Summary

Runge–Kutta methods are one-step methods for the numerical solution of an [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$.
Each step evaluates $f$ at several intermediate points, the stages, and combines these slopes to advance the solution.
They reach high order without derivatives of $f$, and they include the [[Euler method|0JARTXX4XP]], the classical fourth-order method, and implicit methods for [[stiff equations|RR5FCG92PN]].
A method is described by its coefficients, collected in a [[Butcher tableau|RH8BD7PW5F]].

## Explanations

### L1 human

The Euler method follows the slope at the start of each step, which is crude.
A Runge–Kutta method first probes the slope at a few points inside the step and then takes a weighted average of them.
This is like checking the road ahead before deciding how to steer.
The classical fourth-order method uses four slopes per step; halving the step size makes its error about sixteen times smaller.
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# Integral scheme

## Definition

A [[scheme|5A35TYZD02]] $X$ is *integral* if $X$ is nonempty and for every nonempty open set $U \subseteq X$ the ring $\mathcal{O}_X(U)$ is an [[integral domain|0QVD7WBFGN]].

## Remarks

A scheme is integral if and only if it is [[reduced|ZXSN8D4MCB]] and its underlying space is irreducible.
An affine scheme $\operatorname{Spec} A$ is integral if and only if $A$ is an integral domain.
Then its [[generic point|3A19C4YXGE]] is $(0)$, and the local ring there is the [[field of fractions|0F2PTGX1A2]] of $A$, the function field.
Example: $\operatorname{Spec} \Z$ is integral, since $\Z$ is a domain and each $\mathcal{O}(D(n)) = \Z[1/n]$ is a domain.
Non-example: $\operatorname{Spec} k[x, y]/(xy)$ for a field $k$ is reduced but not integral, since $x y = 0$ with $x \ne 0$ and $y \ne 0$.
The open set $U = \emptyset$ is excluded because $\mathcal{O}_X(\emptyset)$ is the zero ring, which is not a domain.
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# Dixon elliptic functions

## Definition

Consider the system of [[ordinary differential equations|0NT64JV96A]]
$$\operatorname{sm}'(z) = \operatorname{cm}(z)^2, \qquad \operatorname{cm}'(z) = -\operatorname{sm}(z)^2,$$
with initial values $\operatorname{sm}(0) = 0$ and $\operatorname{cm}(0) = 1$.
The *Dixon elliptic functions* $\operatorname{sm}$ and $\operatorname{cm}$ are its unique solution near $z = 0$, continued by [[analytic continuation|0M3YQJ81NB]] to meromorphic functions on $\C$.

## Remarks

The derivative of $\operatorname{sm}^3 + \operatorname{cm}^3$ is $3\operatorname{sm}^2\operatorname{cm}^2 - 3\operatorname{cm}^2\operatorname{sm}^2 = 0$, so $\operatorname{sm}^3 + \operatorname{cm}^3 = 1$ everywhere: the point $(\operatorname{cm} z, \operatorname{sm} z)$ lies on the Fermat cubic $x^3 + y^3 = 1$.
Hence $\operatorname{sm}' = (1 - \operatorname{sm}^3)^{2/3}$ near $0$, and $\operatorname{sm}$ is the inverse of $x \mapsto \int_0^x dt/(1 - t^3)^{2/3}$ near $0$.
Both functions are [[elliptic|FF9ZPCZ59K]], with real period
$$\pi_3 = 3\int_0^1 \frac{dt}{(1 - t^3)^{2/3}} = B(1/3, 1/3) \approx 5.29992.$$
Let $\omega = e^{2\pi i/3}$.
Example: $f(z) = \omega^{-1}\operatorname{sm}(\omega z)$ and $g(z) = \operatorname{cm}(\omega z)$ satisfy $f' = g^2$, $g' = -\omega^3 f^2 = -f^2$, $f(0) = 0$, $g(0) = 1$; by uniqueness $\operatorname{sm}(\omega z) = \omega\operatorname{sm}(z)$ and $\operatorname{cm}(\omega z) = \operatorname{cm}(z)$.
So $\omega\pi_3$ is also a period, and the period lattice contains $\pi_3(\Z + \Z\omega)$.
Non-example: $\sin$ and $\cos$ satisfy $\sin' = \cos$, not $\cos^2$, and $\sin^3 + \cos^3$ is not constant (it is $1$ at $0$ and $2^{-1/2}$ at $\pi/4$).
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# Kunen's inconsistency theorem

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
For an ordinal $\gamma$, let $V_\gamma$ be the set of sets of rank less than $\gamma$, as in the [[von Neumann universe|3SZADQXVZJ]].

1. For every ordinal $\delta$, there is no [[elementary embedding|CGH8C8EA9N]] $j \colon V_{\delta + 2} \to V_{\delta + 2}$ other than the identity.
2. There is no elementary embedding $j \colon V \to V$ other than the identity, where $j$ is a class in the sense of von Neumann–Bernays–Gödel set theory.

## Remarks

Let $j$ be a nontrivial embedding with critical point $\kappa_0$, let $\kappa_{n+1} = j(\kappa_n)$, and let $\lambda = \sup_n \kappa_n$, so that $j(\lambda) = \lambda$.
The proof uses the axiom of choice to find a function on the countable subsets of $\lambda$ that $j$ cannot respect.
One such tool is an $\omega$-Jónsson function, from a theorem of Erdős and Hajnal.
Part 2 follows from part 1 applied to $\delta = \lambda$, after restricting $j$ to $V_{\lambda + 2}$.
The theorem sets an upper limit to large cardinal axioms built from embeddings: the axioms just below it ask for elementary embeddings of $V_\lambda$ or $V_{\lambda + 1}$ into itself.
It is an open problem whether part 2 holds without the axiom of choice.

## Explanations

### L1 human

Many large cardinals are defined by maps from the universe of sets into a smaller copy of itself that keep all truths.
One might hope for the strongest possible version, a map from the universe into itself that is not the identity.
Kunen proved that, with the axiom of choice, such a map cannot exist.
So there is a ceiling for this way of building large cardinals.

### L2 human

The embeddings $j \colon V \to M$ of measurable and supercompact cardinals become stronger as $M$ gets closer to $V$.
Kunen's theorem says that $M$ cannot be all of $V$, and more precisely that $M$ cannot contain the set $j[\lambda]$ for the $\lambda$ above.
The strongest large cardinal axioms in use, such as elementary embeddings $j \colon V_{\lambda + 1} \to V_{\lambda + 1}$, sit just below the limit given by part 1.
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# Congruence generated by a relation

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$ and let $R \subseteq A \times A$.
The *congruence generated by* $R$ is
$$\operatorname{Cg}(R) = \bigcap \set{\theta : \theta \text{ is a congruence on } \mathbf{A} \text{ and } R \subseteq \theta}.$$
For $a, b \in A$ one writes $\operatorname{Cg}(a, b)$ for $\operatorname{Cg}(\set{(a, b)})$, the principal congruence of $a$ and $b$.

## Remarks

The family in the intersection contains $A \times A$, so it is nonempty.
An intersection of [[congruences|DCSXXT4SER]] is a congruence, so $\operatorname{Cg}(R)$ is the least congruence containing $R$.

A description from below: $(a, b) \in \operatorname{Cg}(R)$ if and only if there are elements $a = z_0, z_1, \dots, z_n = b$ with the following property.
For each $i < n$ there are $(c_i, d_i) \in R$ and a unary [[polynomial operation|XP67WCPQ59]] $p_i$ with $\set{z_i, z_{i+1}} = \set{p_i(c_i), p_i(d_i)}$.

Example: in the group $(\Z, +, -, 0)$, $\operatorname{Cg}(0, 6)$ is congruence modulo $6$.
Congruence modulo $6$ contains $(0, 6)$, and every congruence $\theta$ containing $(0, 6)$ contains $(a, a + 6k)$ for all $a, k \in \Z$ by compatibility with $+$ and $-$.

Non-example: the equivalence relation with classes $\set{0, 6}$ and singletons elsewhere contains $(0, 6)$ but is not $\operatorname{Cg}(0, 6)$.
It is not a congruence, since $0 + 1 = 1$ and $6 + 1 = 7$ are not related.
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# Thin set

## Summary

A set $E \subseteq \R^n$ is thin at a point $x$ if $E$ is very sparse near $x$.
Precisely, some [[superharmonic function|X4V2S56YQQ]] is strictly larger along $E$ near $x$ than at $x$ itself.
Thinness measures how visible a set is to Brownian motion and to harmonic functions near a point.
A boundary point of a domain is irregular for the Dirichlet problem exactly when the complement of the domain is thin there.
[[Wiener's criterion|STMT1K1865]] decides thinness by a series of capacities.

## Explanations

### L1 human

A set can come arbitrarily close to a point and still be almost invisible from it.
Thinness makes this precise with superharmonic functions, which may jump up along a thin set but not along a fat one.
A sequence of points converging to $x$ in space $\R^3$ is thin at $x$, because points have capacity zero.
A solid ball is never thin at a point of its boundary.
In $\R^3$, a sufficiently sharp spike, as in Lebesgue's example, is thin at its tip even though it reaches that point.
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# Wreath product

## Definition

Let $A$ and $G$ be [[groups|G5S1M9]], and let $G$ [[act|0WCY42EV9V]] on a set $\Omega$, written $(g, \omega) \mapsto g\omega$.
Let $A^\Omega$ be the group of all functions $f \colon \Omega \to A$ with the pointwise product, that is, the [[direct product|00HF4RJ5QN]] of $\abs{\Omega}$ copies of $A$.
For $g \in G$ and $f \in A^\Omega$, let $g \cdot f \in A^\Omega$ be the function $(g \cdot f)(\omega) = f(g^{-1}\omega)$.
This defines an action of $G$ on $A^\Omega$ by automorphisms.
The *wreath product* $A \wr_\Omega G$ is the [[semidirect product|0H3WZXCZNF]] $A^\Omega \rtimes G$ for this action.
Its elements are the pairs $(f, g)$ with $f \in A^\Omega$ and $g \in G$, and the product is
$$(f, g)(f', g') = (f \, (g \cdot f'), \; g g').$$
The subgroup $A^\Omega$ is the *base group*.
When $\Omega = G$ with the action by left multiplication, the group is the *regular wreath product*, written $A \wr G$.

## Remarks

The order of a finite wreath product is $\abs{A \wr_\Omega G} = \abs{A}^{\abs{\Omega}} \abs{G}$.

Example: let $A = C_2$ and let $G = C_2$ act on $\Omega = \set{1, 2}$ by swapping.
Then $A \wr_\Omega G$ has order $2^2 \cdot 2 = 8$.
It is isomorphic to the dihedral group of order $8$, the symmetry group of a square.
The base group $C_2 \times C_2$ consists of the identity, the half-turn and the reflections in the two diagonals.
The swap is the reflection in an axis through the midpoints of two opposite sides.

Non-example: the direct product $A^\Omega \times G$ has the same underlying set, but there $G$ does not move the functions at all.
Its multiplication differs from that of the wreath product when $A \ne 1$ and $G$ moves some point, because then $g \cdot f \ne f$ for some $g$ and $f$.

For $G = S_m$ acting on $\set{1, \dots, m}$, the group $S_k \wr S_m$ is the subgroup of $S_{km}$ that preserves a fixed partition of $km$ points into $m$ parts of size $k$.
The group $C_2 \wr S_n$ is the group of signed permutations, of order $2^n n!$.
For an infinite set $\Omega$, the group defined above is the *unrestricted* wreath product, built from all functions $\Omega \to A$.
The *restricted* wreath product uses only the functions with $f(\omega) = 1$ for all but finitely many $\omega$.
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# Purely transcendental extension

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
The extension $L/K$ is *purely transcendental* if $L = K(S)$ for a subset $S \subseteq L$ that is [[algebraically independent|KV4PT6KW10]] over $K$.

## Remarks

Then $L$ is isomorphic over $K$ to the [[field of rational functions|4V81611P9N]] $K(x_s : s \in S)$, by sending $x_s$ to $s$.
Example: $\Q(\pi)/\Q$ is purely transcendental, with $S = \set{\pi}$.
Non-example: $\Q(\sqrt 2, t)/\Q$ with $t$ transcendental is not purely transcendental.
Indeed, $K$ is algebraically closed in $K(S)$ for independent $S$, but $\sqrt 2$ is algebraic over $\Q$ and not in $\Q$.
By [[Lüroth's theorem|AVCZ6HVBA7]], every field strictly between $K$ and $K(t)$ is again purely transcendental over $K$.
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# Freudenthal suspension theorem

## Statement

Let $n \ge 1$, and let $X$ be a pointed [[CW complex|5W95QFHXPR]] that is [[$(n-1)$-connected|1A943BWBQT]].
Let $\Sigma X$ be the reduced [[suspension|Q7RFNJH76Q]] of $X$.
The suspension homomorphism $E \colon \pi_i(X) \to \pi_{i+1}(\Sigma X)$, $[f] \mapsto [\Sigma f]$, on [[homotopy groups|P976WYEFMB]] has the following properties:

1. it is an isomorphism for $i < 2n - 1$;
2. it is surjective for $i = 2n - 1$.

## Remarks

The usual proof applies the [[Blakers–Massey theorem|QF764XKBA3]] to the two cones that form $\Sigma X$, whose intersection is a copy of $X$.
For spheres, $S^n$ is $(n-1)$-connected and $\Sigma S^n = S^{n+1}$, so $\pi_i(S^n) \to \pi_{i+1}(S^{n+1})$ is an isomorphism for $i < 2n - 1$.
Hence $\pi_{n+k}(S^n)$ does not depend on $n$ once $n \ge k + 2$, which defines the [[stable homotopy groups of spheres|HGZAYPQBXW]].
The bound is sharp: for $n = 2$ and $i = 3$, the map $\pi_3(S^2) \cong \Z \to \pi_4(S^3) \cong \Z/2$ is onto but not injective.
The theorem also gives $\pi_n(S^n) \cong \Z$ for all $n \ge 1$ by induction from $\pi_1(S^1) \cong \Z$, using $\pi_2(S^2) \cong \Z$ from the Hopf fibration at the first step.

## Explanations

### L1 human

Suspending a space places it as the equator of a larger space made of two cones.
Every map of spheres into the space suspends to a map one dimension higher.
The theorem says that, in a range of dimensions, this loses no information and creates nothing new.

### L2 human

The theorem is the starting point of stable homotopy theory.
It shows that the suspension sequence $\pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1}) \to \cdots$ eventually becomes constant, so stable groups and spectra make sense.
In the stable range, homotopy groups behave like a homology theory, since excision holds there.
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# Optimal truncation

## Definition

Let $f(x) \sim \sum_{n \ge 0} a_n \varphi_n(x)$ be an [[asymptotic expansion|BJQTEPEZ9G]] of $f$ as $x \to x_0$, and fix a value of $x$ near $x_0$.
Assume that the sequence $\abs{a_n \varphi_n(x)}$, $n \ge 0$, attains a minimum, and let $N(x)$ be the smallest index $n$ at which it does.
The *optimally truncated* sum at $x$ is
$$S_{\mathrm{opt}}(x) = \sum_{n=0}^{N(x) - 1} a_n \varphi_n(x),$$
the partial sum that stops just before the smallest term.
*Optimal truncation* is the approximation of $f(x)$ by $S_{\mathrm{opt}}(x)$.

## Remarks

Some authors include the smallest term, or half of it; these choices change the error only by a fraction of the smallest term.
The index $N(x)$ depends on $x$ and usually tends to infinity as $x \to x_0$.

Example: $F(x) = \int_0^\infty \frac{e^{-t}}{1 + xt}\,dt \sim \sum_{n \ge 0} (-1)^n n!\, x^n$ as $x \to 0^+$.
For $x = 0.1$ the terms $n!\, (0.1)^n$ decrease while $n + 1 < 10$, the terms with $n = 9$ and $n = 10$ are equal, and then they increase.
So $N(0.1) = 9$, and the error of the optimally truncated sum is at most the first omitted term $9!\, (0.1)^9 \approx 3.6 \cdot 10^{-4}$, since the remainder after $N$ terms is bounded by the next term.

Non-example: for a convergent series such as $\frac{1}{1 - x} = \sum x^n$ at $x = \frac12$ the terms decrease forever, no minimum is attained, and optimal truncation is not defined; one simply sums the series.
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# Haken manifold

## Summary

A Haken manifold is a compact irreducible [[3-manifold|R20TB2KEQQ]] that contains a two-sided [[incompressible surface|6N9GW98MR6]].
Such a manifold can be cut along incompressible surfaces again and again until only balls remain, a process called a Haken hierarchy.
This makes Haken manifolds the class where induction arguments work.
Many results, such as Waldhausen's topological rigidity and Thurston's hyperbolization, were first proved for them.

## Explanations

### L1 human

A Haken manifold is a $3$-dimensional space that is big enough to contain an essential surface.
Cutting along that surface gives a simpler space, and one can keep cutting until only solid balls are left.
This lets one prove facts about the whole space step by step, from the balls back up.
The $3$-torus, a cube with opposite faces glued, is Haken: a torus parallel to two opposite faces is an incompressible surface.
The $3$-sphere is not Haken, because every closed surface in it of positive genus can be compressed.
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# Manifold with boundary

## Summary

A manifold with boundary is a space that looks near each point either like $\R^n$ or like the closed half-space of $\R^n$.
The points of the second kind form the boundary, which is itself a manifold of dimension $n - 1$ without boundary.
Closed disks, closed intervals and solid balls are the first examples, and cobordism and surgery theory are built on them.

## Explanations

### L1 human

A closed disk is like a flat sheet with an edge.
Around a point inside, it looks like an open piece of the plane.
Around a point on the edge circle, it looks like a piece of a half-plane, with the edge as a border line.
The edge of the disk is the circle, and the circle has no edge of its own.
A closed interval $[0, 1]$ is a $1$-dimensional example whose boundary is the two end points.
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# Elementary class

## Definition

Let $\sigma$ be a first-order [[signature|WDVVEYD6AT]].
Let $K$ be a class of [[$\sigma$-structures|983GZBQYT4]].
Then $K$ is an *elementary class* if there is a first-order [[theory|92SWXST1ZM]] $T$ in $\sigma$ whose [[models|MVCH6N9SNW]] are exactly the members of $K$.

## Remarks

If $T$ can be chosen finite, or equivalently a single sentence, $K$ is called a *basic elementary class*.
Some authors write $\mathrm{EC}$ for basic elementary classes and $\mathrm{EC}_\Delta$ for elementary classes.

Example: groups, in the signature $(\cdot, {}^{-1}, e)$, form a basic elementary class, axiomatized by the three group axioms.
Fields of characteristic $0$ form an elementary class, using the infinitely many axioms $1 + \cdots + 1 \ne 0$; it is not a basic elementary class.

Non-example: torsion groups do not form an elementary class.
Suppose $T$ axiomatized them.
Add a new constant $c$ and the sentences $c^n \ne e$ for all $n \ge 1$.
Each finite subset of this theory has a model, namely a finite cyclic group of large order with $c$ a generator.
By the [[compactness theorem|1NR3HAWG9D]] the whole theory has a model, which is a model of $T$ with an element of infinite order, a contradiction.
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# Quadratic field

## Definition

A *quadratic field* is an [[algebraic number field|MWBDFS2KJ5]] $K$ with [[degree|0EAJ2XXADC]] $[K : \Q] = 2$.

## Remarks

Every quadratic field is $\Q(\sqrt{d}) = \set{a + b\sqrt{d} : a, b \in \Q}$ for exactly one [[square-free integer|08X0VCR2NW]] $d$ with $d \ne 0, 1$.
It is called real if $d > 0$ and imaginary if $d < 0$.
Example: $\Q(\sqrt{2})$; the element $\sqrt{2}$ is a root of the irreducible polynomial $x^2 - 2$, so $1, \sqrt{2}$ is a basis and the degree is $2$.
Non-example: $\Q(\sqrt[3]{2})$ has degree $3$, because $x^3 - 2$ is irreducible over $\Q$; so it is a number field but not a quadratic field.
The ring of integers of $\Q(\sqrt{d})$ is $\Z[(1 + \sqrt{d})/2]$ if $d \equiv 1 \pmod 4$, and $\Z[\sqrt{d}]$ if $d \equiv 2$ or $3 \pmod 4$.
The discriminant is $d$ in the first case and $4d$ in the second.
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# Barnes G-function

## Summary

The Barnes G-function is an [[entire function|0G4F0XB9X2]] $G$ with $G(1) = 1$ and $G(z+1) = \Gamma(z)\, G(z)$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
It plays for the gamma function the role that the gamma function plays for the identity function.
At positive integers it gives the superfactorials $G(n) = 0!\,1!\cdots(n-2)!$.
It is, up to an elementary factor, a [[multiple gamma function|FM0ZZS1PY2]] of order two, and it appears in random matrix theory and in determinant formulas.

## Explanations

### L1 human

The factorial multiplies the numbers $1, 2, \ldots, n$.
The superfactorial multiplies the factorials $0!, 1!, \ldots$ in the same way.
The Barnes G-function fills in values of the superfactorial between the whole numbers, as the gamma function does for the factorial.
For example, $G(5) = 0! \cdot 1! \cdot 2! \cdot 3! = 12$.
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# Artin–Mazur zeta function

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a map.
Assume that, for every $n \ge 1$, the set $\operatorname{Fix}(f^n)$ of [[fixed points|R6QVS6P21N]] of the $n$-th iterate $f^n$ is finite.
Write $N_n = \abs{\operatorname{Fix}(f^n)}$.
The *Artin–Mazur zeta function* of $f$ is the [[formal power series|033WGN00BF]]
$$\zeta_f(z) = \exp\left( \sum_{n=1}^{\infty} \frac{N_n}{n} z^n \right).$$
It has rational coefficients.

## Remarks

Example: let $f(z) = z^2$ on the unit circle in $\C$.
The fixed points of $f^n$ are the solutions of $z^{2^n} = z$ with $\abs{z} = 1$, that is, the $(2^n - 1)$-th roots of unity, so $N_n = 2^n - 1$.
Then
$$\zeta_f(z) = \exp\left( \sum_{n \ge 1} \frac{(2z)^n}{n} - \sum_{n \ge 1} \frac{z^n}{n} \right) = \frac{1 - z}{1 - 2z}.$$

Non-example: the identity map of the circle has infinitely many fixed points, so $N_1$ is not finite and the definition does not apply.

If $N_n \le C a^n$ for constants $C$ and $a$, the series has a positive [[radius of convergence|0N288P2TB5]] and $\zeta_f$ is a holomorphic function near $0$.
For the shift on a subshift of finite type with transition matrix $A$, $N_n = \tr(A^n)$ and $\zeta_f(z) = 1 / \det(I - zA)$.
For an Axiom A diffeomorphism of a compact manifold, the zeta function is rational (Manning).
The function has the product form $\zeta_f(z) = \prod_{\gamma} (1 - z^{\abs{\gamma}})^{-1}$, a product over the periodic orbits $\gamma$ of $f$ with $\abs{\gamma}$ the number of points of $\gamma$.
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# Simple polytope

## Summary

A simple polytope is a [[convex polytope|300K6CB1KH]] of dimension $d$ in which every [[vertex|R7V3S9V891]] lies on exactly $d$ edges.
Near each vertex it then looks like the corner of a cube.
Simple polytopes are the duals of [[simplicial polytopes|YVJ15PGKDM]], so results about one class translate into results about the other.
Polytopes given by inequalities in general position are simple, which makes them the generic case in linear programming.

## Explanations

### L1 human

At each corner of a cube exactly three edges meet, and the cube lives in three dimensions.
A polytope with this property at every corner, in any dimension, is called simple.
The square pyramid is not simple: four edges meet at its apex.
If you slightly shift the walls of any polytope in a random way, the result is simple.
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# Quasi-isomorphism

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $f \colon C_\bullet \to D_\bullet$ be a [[chain map|HTK9WQZJ1G]] of [[chain complexes|WQ82C0M3QB]] in $\mathcal{A}$.
The map $f$ is a *quasi-isomorphism* if the induced morphism on [[homology|2J4JJJYP22]]
$$H_n(f) \colon H_n(C) \to H_n(D)$$
is an isomorphism for every integer $n$.

## Remarks

Every chain homotopy equivalence is a quasi-isomorphism, because [[chain homotopic|4EBHAWJJ8C]] maps induce the same maps on homology.
Quasi-isomorphisms are closed under composition, but in general a quasi-isomorphism has no chain map inverse up to homotopy.

Example: let $C_\bullet$ have $C_1 = C_0 = \Z$ with $d_1$ multiplication by $2$, and let $D_\bullet$ be $\Z/2\Z$ in degree $0$.
Let $f_0 \colon \Z \to \Z/2\Z$ be the reduction and $f_1 = 0$; this is a chain map since $f_0(2x) = 0$.
Both complexes have homology $\Z/2\Z$ in degree $0$ and $0$ elsewhere, and $H_0(f)$ sends the class of $1$ to $1$, so $f$ is a quasi-isomorphism.
It is not a chain homotopy equivalence: every chain map $D_\bullet \to C_\bullet$ is $0$, because $\Z/2\Z$ has no nonzero map to $\Z$.

Non-example: the zero chain map from $\Z$ in degree $0$ to itself is not a quasi-isomorphism, since $H_0$ of it is the zero map $\Z \to \Z$.
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# Neutral differential equation

## Definition

Let $\tau > 0$ and $n \ge 1$, and let $f \colon \R \times \R^n \times \R^n \times \R^n \to \R^n$ be a function.
A *neutral differential equation* with delay $\tau$ has the form
$$x'(t) = f\bigl(t, x(t), x(t - \tau), x'(t - \tau)\bigr).$$
Here $x$ is an unknown function with values in $\R^n$, and $f$ depends on its last argument.
An initial value problem prescribes a history $\phi$ on $[-\tau, 0]$ that is continuously differentiable.

## Remarks

Example: $x'(t) = \frac{1}{2} x'(t - 1) - x(t)$, with $f(t, u, v, w) = \frac{1}{2} w - u$, which depends on $w$.

Non-example: $x'(t) = -x(t - 1)$ is a [[delay differential equation|39PCDRDB01]] of retarded type, not a neutral one, since $f(t, u, v, w) = -v$ does not depend on $w$.

A more general form, due to Hale, is $\frac{d}{dt} D(t, x_t) = F(t, x_t)$, where $x_t$ is the segment of $x$ on $[t - \tau, t]$ and $D(t, \psi) = \psi(0) - g(t, \psi)$ for a suitable function $g$ of the past.
The linear case $\frac{d}{dt}\bigl(x(t) - c\, x(t - \tau)\bigr) = a\, x(t) + b\, x(t - \tau)$ is the standard test example.

## Explanations

### L1 human

A neutral equation lets past derivatives, not only past values, drive the present derivative.
This feature makes its solutions keep the roughness of their initial data.
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# Reaction–diffusion system

## Definition

Let $n, m \ge 1$, let $\Omega \subseteq \R^n$ be an open set, and let $T > 0$.
Let $D = \operatorname{diag}(d_1, \ldots, d_m)$ with diffusion coefficients $d_i > 0$, and let $R \colon \R^m \to \R^m$ be a smooth map, the reaction term.
The unknown is a map $u = (u_1, \ldots, u_m) \colon \Omega \times (0, T) \to \R^m$.
The *reaction–diffusion system* is the [[parabolic|1CCTH8X2FT]] system
$$\partial_t u = D \Delta u + R(u) \quad \text{in } \Omega \times (0, T).$$
Here the [[Laplacian|0E0B6RA8K0]] $\Delta$ acts on each component.
So $\partial_t u_i = d_i \Delta u_i + R_i(u_1, \ldots, u_m)$ for $i = 1, \ldots, m$.

## Remarks

For $R = 0$ and $m = 1$ it is the [[heat equation|VP2HN86VAG]], up to the constant $d_1$.
Example: $m = 1$, $d_1 = 1$ and $R(u) = u(1 - u)$ give the Fisher–KPP equation $\partial_t u = \Delta u + u(1 - u)$; the constant $u = 1$ is a solution, since both sides are $0$ there.
Non-example: Burgers' equation $\partial_t u = \nu \partial_x^2 u - u \, \partial_x u$ is not of this form, since its nonlinear term involves the derivative $\partial_x u$ and not only the value $u$.
Some authors allow $d_i = 0$ for some components, or a reaction term that depends on $x$ and $t$.
The system is usually completed by initial values and, when $\Omega \ne \R^n$, by boundary conditions such as zero normal derivative.
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# Jacobson ring

## Summary

A Jacobson ring, also called a Hilbert ring, is a [[commutative ring|0310REJJY8]] in which every [[prime ideal|00MHHZ9SNE]] is an intersection of [[maximal ideals|0PV74TR5FP]].
In such a ring, the maximal ideals alone determine all radical ideals, as in [[Hilbert's Nullstellensatz|D85VZWS9QK]].
Fields, $\Z$, and every finitely generated algebra over a Jacobson ring are Jacobson rings; this last fact is a general form of the Nullstellensatz.
In geometry, the closed points of the spectrum of a Jacobson ring are dense in every closed subset.

## Explanations

### L1 human

In the ring of polynomials over $\C$, a curve is determined by its points, and each point is a maximal ideal.
A Jacobson ring is a ring where every prime ideal is determined in this way by the maximal ideals above it.
The integers are an example: the prime ideal $0$ is the intersection of all ideals $p\Z$, since no nonzero integer is divisible by every prime.
A local domain that is not a field, such as $\Z_{(p)}$, is not Jacobson.
Its only maximal ideal is $p\Z_{(p)}$, and this ideal cannot recover the prime $0$.
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# Whittaker function

## Summary

The Whittaker functions $M_{\kappa, \mu}(z)$ and $W_{\kappa, \mu}(z)$ are solutions of Whittaker's equation $w'' + \left( -\frac14 + \frac{\kappa}{z} + \frac{\frac14 - \mu^2}{z^2} \right) w = 0$.
They are the [[confluent hypergeometric functions|6AZ4ZBN1M8]] rewritten so that the equation has no first derivative term.
This normal form is convenient in physics and in asymptotic analysis, and it makes the symmetry $\mu \mapsto -\mu$ visible.

## Explanations

### L1 human

Kummer's equation has a first derivative term; multiplying the unknown by $e^{-z/2} z^{b/2}$ removes it.
The result is Whittaker's equation, which looks like a Schrödinger equation with a constant, a Coulomb-type term $\kappa / z$ and a centrifugal-type term $(\frac14 - \mu^2)/z^2$.
The function $M_{\kappa, \mu}$ is the solution that behaves like $z^{\mu + 1/2}$ near $0$, and $W_{\kappa, \mu}$ is the solution that decays like $e^{-z/2} z^{\kappa}$ at $+\infty$.
A simple case is $W_{0, 1/2}(z) = e^{-z/2}$, which indeed satisfies $w'' - \frac14 w = 0$.
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# Quantum calculus

## Summary

Quantum calculus is calculus without limits.
It replaces the derivative by a difference quotient taken at a fixed scale.
The [[q-derivative|PMJT3FZE0Y]] compares $f(qx)$ with $f(x)$, and the $h$-derivative compares $f(x + h)$ with $f(x)$.
The matching antiderivatives are the [[Jackson integral|7EMPA6M8S7]] and finite sums.
It is the natural language for [[q-analogs|GFSDMJHHVA]] in combinatorics, for basic hypergeometric series and for [[q-difference equations|FQPYGXJJN0]].
It recovers ordinary calculus in the limit $q \to 1$ or $h \to 0$.

## Explanations

### L1 human

In ordinary calculus the slope of $f$ at $x$ is the limit of $(f(x + h) - f(x))/h$ as $h$ shrinks to zero.
In quantum calculus one stops before the limit and keeps a fixed step.
With the multiplicative step $x \mapsto qx$, the function $x^n$ has $q$-derivative $[n]_q x^{n-1}$, where $[n]_q = 1 + q + \cdots + q^{n-1}$.
This number $[n]_q$ is the $q$-analog of $n$, and it becomes $n$ when $q = 1$.
Many formulas of calculus keep a $q$-version, which often counts objects with an extra weight.
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# Jackson integral

## Summary

The Jackson integral, or $q$-integral, is a [[q-analog|GFSDMJHHVA]] of the [[Riemann integral|0W29TBTY12]].
It replaces the integral over $[0, a]$ by a weighted sum of the values of the function at the geometric sequence of points $a, aq, aq^2, \dots$.
It is the inverse of the [[q-derivative|PMJT3FZE0Y]] in the same way as an integral is the inverse of a derivative.
Many identities for basic hypergeometric series, such as $q$-beta integrals, are stated with it.

## Explanations

### L1 human

A Riemann sum cuts $[0, a]$ into pieces of equal length.
The Jackson integral instead cuts it at the points $a q^k$, so the pieces become shorter and shorter near $0$.
On each piece $[a q^{k+1}, a q^k]$ it takes the value at the right end times the length $(1 - q) a q^k$.
For example, $\int_0^1 x \, d_q x = \frac{1}{1 + q}$, which tends to $\frac12$ as $q \to 1$.
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# Partially ordered ring

## Summary

A partially ordered ring is a [[ring|0B1261B5ED]] with a [[partial order|0QV2JS9RJ3]] that is preserved by addition and for which products of nonnegative elements are nonnegative.
Rings of real functions with the pointwise order and rings of real matrices with the entrywise order are typical examples.
[[Ordered rings|H6WXC7Q71A]] are the case of a total order, and [[f-rings|VHHMZPNVEA]] are an important class of lattice-ordered rings.

## Explanations

### L1 human

Real functions on a set can be added, multiplied and compared point by point.
If $f \le g$, then $f + h \le g + h$, and the product of two nonnegative functions is nonnegative.
But two functions need not be comparable: one may be larger at some points and smaller at others.
A partially ordered ring is a ring with an order of this kind.
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# Finite morphism

## Definition

A [[morphism of schemes|JXRKH6P1Z4]] $f \colon X \to Y$ is *finite* if there is a cover of $Y$ by open [[affine|B3V3XJ2W8G]] subschemes $V_i = \operatorname{Spec} B_i$ such that, for each $i$:

1. $f^{-1}(V_i)$ is an affine scheme $\operatorname{Spec} A_i$;
2. $A_i$, made into a $B_i$-module by the ring map $B_i \to A_i$ of $f$, is a [[finitely generated module|05QY1VEBYD]].

## Remarks

If $f$ is finite, then clauses 1 and 2 hold for every open affine $V \subseteq Y$.
A morphism $\operatorname{Spec} A \to \operatorname{Spec} B$ is finite if and only if $B \to A$ is a [[finite morphism of rings|1NBACAEC2V]].
A finite morphism is [[affine|AFC6JMYWEH]] and of finite type.
Example: for a field $k$, the morphism $\mathbb{A}^1_k \to \mathbb{A}^1_k$ given by $k[x] \to k[t]$, $x \mapsto t^2$, is finite, since $k[t] = k[x] \cdot 1 + k[x] \cdot t$.
Non-example: the open immersion $\operatorname{Spec} k[x, x^{-1}] \to \operatorname{Spec} k[x]$ satisfies clause 1 but not clause 2.
If $k[x, x^{-1}]$ were a finite $k[x]$-module, then $x^{-1}$ would be integral over $k[x]$.
Since $k[x]$ is an [[integrally closed domain|H20P600HQS]], this would give $x^{-1} \in k[x]$, which is false.
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# Oscillatory integral

## Definition

Let $n \ge 1$.
Let $\phi \colon \R^n \to \R$ be a [[smooth function|0V33BNKZ9S]], the *phase*.
Let $\psi \colon \R^n \to \C$ be a smooth function with compact support, the *amplitude*.
The *oscillatory integral* with phase $\phi$ and amplitude $\psi$ is the function of the real parameter $\lambda$ given by
$$I(\lambda) = \int_{\R^n} e^{i\lambda\phi(x)} \psi(x) \, dx.$$
A point $x$ with $\nabla\phi(x) = 0$ is a *critical point*, or stationary point, of the phase.

## Remarks

Example: $n = 1$, $\phi(x) = x$ and $\psi$ smooth with compact support.
Then $I(\lambda) = \int e^{i\lambda x}\psi(x) \, dx$ is the Fourier transform of $\psi$ at $-\lambda/(2\pi)$, and repeated integration by parts gives $\abs{I(\lambda)} \le C_N \abs{\lambda}^{-N}$ for every $N$.
More generally, if $\nabla\phi \ne 0$ on the support of $\psi$, then $I(\lambda)$ decays faster than any power of $\lambda$.

Non-example: $\phi(x) = x$ and $\psi \equiv 1$ on $\R$.
The amplitude does not have compact support, and $\int_\R e^{i\lambda x} \, dx$ does not converge as a [[Lebesgue integral|DWJF7ABPR1]].
Such integrals are still given a meaning as distributions, which is the setting of the more general oscillatory integrals of Hörmander.

If $x_0$ is the only critical point in the support of $\psi$ and the Hessian of $\phi$ at $x_0$ is invertible, the method of stationary phase gives $I(\lambda) = O(\lambda^{-n/2})$ as $\lambda \to \infty$.
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# Mittag-Leffler star

## Definition

Let $a \in \C$, and let $f(z) = \sum_{k \ge 0} c_k (z - a)^k$ be a [[power series|0VP56B4EMX]] with positive [[radius of convergence|0660QX1VPR]].
For $z \in \C$, let $\gamma_z \colon [0, 1] \to \C$ be the segment $\gamma_z(t) = a + t(z - a)$.
The *Mittag-Leffler star* of $f$ with respect to $a$ is the set of all $z \in \C$ with the following property.
$f$ has an [[analytic continuation|0M3YQJ81NB]] along the [[path|0JDE56HWQ8]] $\gamma_z$.

## Remarks

Example: for $f(z) = 1/(1 - z) = \sum_k z^k$ and $a = 0$, the point $z = i$ lies in the star, because the segment from $0$ to $i$ avoids $1$ and $1/(1-z)$ is holomorphic on $\C \setminus \set{1}$.
The star is $\C \setminus [1, \infty)$.
Non-example: $z = 2$ is not in the star, because the segment from $0$ to $2$ passes through the pole $1$, where no continuation exists.
The star is open, it contains the disc of convergence, and with each point $z$ it contains the whole segment from $a$ to $z$.
For an entire function the star is all of $\C$.
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# Ornstein–Uhlenbeck process

## Definition

Let $(W_t)_{t \ge 0}$ be a [[Wiener process|552AWX3QXN]], let $X_0$ be a real random variable independent of $W$, and let $\theta > 0$, $\sigma > 0$ and $\mu \in \R$.
The *Ornstein–Uhlenbeck process* with these parameters is the solution $(X_t)_{t \ge 0}$ of the stochastic differential equation
$$dX_t = \theta (\mu - X_t) \, dt + \sigma \, dW_t$$
with initial value $X_0$.
Explicitly,
$$X_t = \mu + (X_0 - \mu) e^{-\theta t} + \sigma \int_0^t e^{-\theta (t - s)} \, dW_s.$$

## Remarks

The explicit formula solves the equation: Itô's lemma applied to $e^{\theta t} X_t$ gives $d(e^{\theta t} X_t) = \theta \mu e^{\theta t} \, dt + \sigma e^{\theta t} \, dW_t$, which integrates to the formula.
If $X_0$ is constant, $X_t$ is [[normal|05AQE0QR0D]] with mean $\mu + (X_0 - \mu) e^{-\theta t}$ and variance $\frac{\sigma^2}{2\theta}(1 - e^{-2\theta t})$.
If $X_0$ is normal with mean $\mu$ and variance $\sigma^2 / (2\theta)$, the process is [[stationary|DT0QXJE705]], with covariance $\frac{\sigma^2}{2\theta} e^{-\theta \abs{t - s}}$.
For $\theta = 0$, which the definition excludes, the equation gives $X_0 + \sigma W_t$, a Brownian motion with no mean reversion and no stationary law.
Up to linear changes of space and time variables, it is the only non-trivial process that is stationary, Gaussian and Markov.
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# First variation

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]], let $A \subseteq V$, and let $J \colon A \to \R$ be a [[functional|1EQM0DJ4NC]].
Let $y \in A$ and $\eta \in V$, and suppose that $y + t\eta \in A$ for all real $t$ with $\abs{t}$ small enough.
The *first variation* of $J$ at $y$ in the direction $\eta$ is
$$\delta J(y; \eta) = \frac{d}{dt} J(y + t\eta) \Big|_{t = 0} = \lim_{t \to 0} \frac{J(y + t\eta) - J(y)}{t},$$
when this limit exists.

## Remarks

It is also called the Gâteaux variation; it is a [[directional derivative|06KWGHDN2J]] of $J$.
When the [[Gateaux derivative|EAZF9NJW12]] of $J$ at $y$ exists, $\delta J(y; \eta)$ is its value at $\eta$.
If $V$ is normed and $J$ is Fréchet differentiable at $y$, then $\delta J(y; \eta) = DJ(y)\eta$ for all $\eta$.

Example: on $V = C([0, 1])$ let $J(y) = \int_0^1 y^2 \, dx$.
Then $J(y + t\eta) = J(y) + 2t \int_0^1 y\eta \, dx + t^2 \int_0^1 \eta^2 \, dx$, so $\delta J(y; \eta) = 2 \int_0^1 y \eta \, dx$.

Non-example: for $J(y) = \int_0^1 \abs{y} \, dx$, $y = 0$ and $\eta = 1$, the quotient $(J(t\eta) - J(0))/t = \abs{t}/t$ has no limit as $t \to 0$, so the first variation does not exist.

For an [[integral functional|JRQZ803Q5Y]] $J(y) = \int_a^b L(x, y, y') \, dx$ with $L$ of class $C^1$, and $y, \eta \in C^1([a, b])$, differentiation under the integral sign gives
$$\delta J(y; \eta) = \int_a^b \big( L_z(x, y, y') \, \eta + L_p(x, y, y') \, \eta' \big) \, dx,$$
where $L_z$ and $L_p$ are the partial derivatives of $L(x, z, p)$.
If $y$ is a local minimum of $J$ along the line $t \mapsto y + t\eta$ and $\delta J(y; \eta)$ exists, then $\delta J(y; \eta) = 0$.
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# Markov theorem

## Statement

For $n \ge 1$, let $B_n$ be the [[braid group|BNR4H5GCTP]] with generators $\sigma_1, \ldots, \sigma_{n-1}$.
View $B_n$ as a subgroup of $B_{n+1}$ by adding a straight strand on the right.
For a braid $\beta$, let $\hat\beta$ be its closure, an oriented [[link|1A5WRCNTM3]] in $\R^3$.
The two Markov moves on braids are:

1. Conjugation: replace $\beta \in B_n$ by $\gamma \beta \gamma^{-1}$ for some $\gamma \in B_n$, or the converse.
2. Stabilization: replace $\beta \in B_n$ by $\beta \sigma_n$ or by $\beta \sigma_n^{-1}$ in $B_{n+1}$, or the converse.

Let $\beta \in B_n$ and $\beta' \in B_m$.
Then $\hat\beta$ and $\hat\beta'$ are ambient isotopic as oriented links if and only if a finite sequence of Markov moves turns $\beta$ into $\beta'$.

## Remarks

One direction is a direct check.
Closing up $\gamma \beta \gamma^{-1}$ lets $\gamma$ slide around the closure and cancel $\gamma^{-1}$, so the closure does not change.
Closing up $\beta \sigma_n^{\pm 1}$ adds one strand with a single kink, which a first [[Reidemeister move|T3HAGWYKK7]] removes.
The hard direction follows an isotopy of closed braids step by step and shows that each step is a combination of Markov moves.
For example, $\sigma_1 \in B_2$ and the identity braid in $B_1$ both close up to the unknot, and they differ by one stabilization.
Conjugation alone does not suffice, since conjugation never changes the number of strands, while braids with different numbers of strands can close to the same link.

## Explanations

### L1 human

Many different braids close up to the same knot.
The theorem lists two simple changes, and says these are the only ones you ever need.
You may slide a piece of the braid around the closure to the other end, and you may add or remove an extra strand with one twist.

### L2 human

By [[Alexander's theorem|PRWP1JF486]], every oriented link is a closed braid.
The Markov theorem then says that oriented links up to isotopy are the same as braids in all $B_n$ up to the two Markov moves.
So a family of functions $f_n$ on $B_n$ gives a link invariant when $f_n$ is constant on [[conjugacy classes|0DEBXAJ5SA]] and $f_{n+1}(\beta \sigma_n^{\pm 1})$ equals $f_n(\beta)$.
Traces on the Temperley–Lieb and Hecke algebras with this property give the [[Jones polynomial|4SPC17FPXE]] and the [[HOMFLY polynomial|QBQ0BH589C]].
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# Moufang plane

## Summary

A Moufang plane is a [[projective plane|DANFNN8C4Y]] in which the little Desargues theorem holds.
This is the special case of [[Desargues's theorem|5Z19Y02MME]] where the centre of perspective lies on the axis.
Equivalently, every line is a translation line: the plane has as many elations with that axis as possible.
Moufang planes are exactly the planes coordinatized by alternative division rings.
Every [[Desarguesian plane|HPX85PA8NX]] is a Moufang plane, and the [[octonionic projective plane|S26GYRVDR5]] is a Moufang plane that is not Desarguesian.

## Explanations

### L1 human

Desargues's theorem says that two triangles in perspective from a point are also in perspective from a line.
A Moufang plane is only asked to satisfy this when the point lies on the line.
This weaker rule still gives the plane many symmetries.
In coordinates, it allows multiplication that is not associative, as long as it is alternative, as for the octonions.
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# Liouville property of graphs

## Definition

Let $G = (V, E)$ be a connected infinite [[graph|00C0BR7D7H]] in which every vertex has finitely many neighbours, with symmetric positive edge weights $c$.
The graph $G$ has the *Liouville property* if every bounded function $f \colon V \to \R$ that is [[harmonic|WRY0M7SW59]] on all of $V$ is constant.
It has the *strong Liouville property* if every positive function that is harmonic on all of $V$ is constant.
Without weights, the edge weights are taken equal to $1$, which corresponds to the simple [[random walk|04VZE3FZXF]].

## Remarks

Example: $\Z^d$ with unit weights has the Liouville property for every $d \ge 1$.
For $d = 1$ this is direct, since the harmonic functions on $\Z$ are $f(n) = an + b$, and such a function is bounded only when $a = 0$.
Non-example: the $3$-regular [[tree|062D9KXJB1]] $T_3$ fails it.
Fix an edge $\set{o, o'}$, and let $h(x)$ be the probability that the simple random walk from $x$ eventually stays in the component of $T_3$ minus this edge that contains $o'$.
The walk on $T_3$ is transient, so it crosses the edge finitely often, and $h$ is harmonic by conditioning on the first step.
Its values lie in $[0, 1]$, and it is not constant.
Indeed $h(x)$ tends to $1$ as $x$ moves away from the edge on the side of $o'$, and to $0$ on the side of $o$.
The strong Liouville property implies the Liouville property, since a bounded harmonic $f$ gives the positive harmonic function $f - \inf f + 1$.
</document>

<document source="https://tamnd.github.io/math2/node/7FCN2KNWC0.md" tag="7FCN2KNWC0" kind="example">
# The integers modulo 2 and the group of signs are isomorphic

## Example

Let $\mathbb{Z}/2\mathbb{Z} = \{0, 1\}$ be the [[integers modulo $2$|03VFZ2WETT]] with addition. Let $\{1, -1\}$ be a group under multiplication.
Define $f(0) = 1$ and $f(1) = -1$. Then $f$ is a bijection.
We check the four products: $f(0 + 0) = 1 = 1 \cdot 1$, $f(0 + 1) = -1 = 1 \cdot (-1)$, $f(1 + 0) = -1 = (-1) \cdot 1$ and $f(1 + 1) = f(0) = 1 = (-1) \cdot (-1)$.
Thus $f$ is a homomorphism and a bijection, so $f$ is an [[isomorphism|085PND4KNT]].

## Remarks

Both groups are cyclic of order $2$. This agrees with the result that [[cyclic groups of the same order are isomorphic|0736AXH2ME]].
</document>

<document source="https://tamnd.github.io/math2/node/7FDDXV3JK7.md" tag="7FDDXV3JK7" kind="concept">
# Quantifier elimination

## Summary

A theory has quantifier elimination when every formula is equivalent, modulo the theory, to a formula without quantifiers.
Then the definable sets of its models are simple to describe, and questions about the theory reduce to questions about quantifier-free formulas.
Quantifier elimination often gives completeness and decidability.
Classical examples are dense linear orders without endpoints, [[algebraically closed fields|MT8TDVPBXB]], and [[real closed fields|DDY3Q3PJ6W]] in the language of ordered rings.

## Explanations

### L1 human

Quantifiers are the words "for all" and "there exists".
A theory has quantifier elimination if every statement that uses them can be rewritten as a statement that does not use them.
The rewriting must keep the meaning in every model of the theory.
For example, over the real numbers, "there exists $x$ with $x^2 + bx + c = 0$" means the same as "$b^2 - 4c \ge 0$".
After the rewriting, one only has to check equations and inequalities about the given numbers.
</document>

<document source="https://tamnd.github.io/math2/node/7FNVA0Q6YQ.md" tag="7FNVA0Q6YQ" kind="concept">
# Point spectrum

## Summary

The point spectrum of an operator is its set of eigenvalues.
It is the part of the [[spectrum|BMKATJ802A]] where $T - \lambda I$ fails to be injective.
For matrices the point spectrum is the whole spectrum, but in infinite dimensions it can be a small part of it, or empty.

## Explanations

### L1 human

An eigenvalue $\lambda$ is a number for which some nonzero vector is just multiplied by $\lambda$.
The point spectrum collects these numbers.
For the left shift $(x_1, x_2, \ldots) \mapsto (x_2, x_3, \ldots)$ on square-summable sequences, the sequence $(1, \lambda, \lambda^2, \ldots)$ is an eigenvector for every $\abs{\lambda} < 1$.
For the right shift, which only moves entries further out, there are no eigenvalues at all.
</document>

<document source="https://tamnd.github.io/math2/node/7FR2PZ141W.md" tag="7FR2PZ141W" kind="definition">
# Purely inseparable extension

## Definition

Let $L/K$ be an [[algebraic extension|09EYMKMSB9]].
If $K$ has [[characteristic|0P7CH1CND5]] $0$, the extension $L/K$ is *purely inseparable* if $L = K$.
If $K$ has characteristic $p > 0$, the extension $L/K$ is *purely inseparable* if for every $\alpha \in L$ there is an integer $n \ge 0$ with $\alpha^{p^n} \in K$.

## Remarks

In characteristic $p > 0$, the following are equivalent for an algebraic extension $L/K$:

1. $L/K$ is purely inseparable;
2. the [[minimal polynomial|0GK1WB9R98]] over $K$ of each $\alpha \in L$ has the form $x^{p^n} - a$ with $a \in K$ and $n \ge 0$;
3. every element of $L$ that is separable over $K$ lies in $K$.

Example: let $K = \F_p(t)$ and $L = K(s)$ with $s^p = t$.
Check: every element of $L$ is $\sum_{i=0}^{p-1} c_i s^i$ with $c_i \in K$, and its $p$-th power is $\sum c_i^p t^i \in K$.

Non-example: $\F_{p^2}/\F_p$ is not purely inseparable.
For $\alpha \in \F_{p^2} \setminus \F_p$, the powers $\alpha^{p^n}$ are $\alpha$ or its conjugate $\alpha^p$, and neither lies in $\F_p$.
</document>

<document source="https://tamnd.github.io/math2/node/7FTW5Q9GHC.md" tag="7FTW5Q9GHC" kind="concept">
# Schwarz–Christoffel mapping

## Summary

A Schwarz–Christoffel mapping is a conformal map from the upper half-plane onto the interior of a polygon.
Such maps have an explicit integral form, in which the derivative is a product of powers of linear factors, one for each vertex.
The exponents come from the angles of the polygon, while the positions of the preimages of the vertices must usually be found numerically.
The formula is one of the few cases where the Riemann mapping theorem comes with a concrete map.
It is much used in engineering and in numerical conformal mapping.

## Explanations

### L1 human

A conformal map from the half-plane must bend the straight boundary line into the edges of a polygon.
Along the real axis, the direction of the image changes only at a few points, where it turns by the exterior angle of a corner.
A factor $(z - x_k)^{\alpha_k - 1}$ in the derivative does exactly this: it turns the direction by a fixed angle when $z$ passes $x_k$.
For instance, $z \mapsto z^{1/2}$ maps the upper half-plane onto the quarter-plane, a polygon with one corner of angle $\pi/2$ at $0$ and one at infinity.
</document>

<document source="https://tamnd.github.io/math2/node/7FWX24X5MW.md" tag="7FWX24X5MW" kind="concept">
# Attractor

## Summary

An attractor of a dynamical system is an [[invariant set|WWZTW11664]] towards which all nearby states evolve.
It describes what one actually observes after a long time, since transients die out and the state settles near the attractor.
Attractors can be single points, closed orbits, tori, or sets with fractal structure, the [[strange attractors|9WKCZGG906]].
Several precise definitions are in use; they agree on simple examples and differ in how they treat attraction and indecomposability.
The [[global attractor|4SH2ZB984E]] of a dissipative system is a related notion that attracts every bounded set.

## Explanations

### L1 human

Drop a marble into a bowl: wherever it starts, it rolls down and comes to rest at the bottom.
The bottom of the bowl is an attractor of the motion.
A swinging pendulum with friction is attracted to the resting position, while a pendulum clock with a driving spring settles into a steady swing, an attracting closed loop.
For the map $x \mapsto x/2$ on the real line, every orbit tends to $0$, so $\set{0}$ is an attractor.
</document>

<document source="https://tamnd.github.io/math2/node/7FYKA5F3S6.md" tag="7FYKA5F3S6" kind="counterexample">
# Fort space

## Counterexample

Let $X$ be an infinite set and let $p \in X$.
Declare $U \subseteq X$ open if $p \notin U$, or if $p \in U$ and $X \setminus U$ is finite.
This is the Fort space on $X$ with point $p$.
It is [[compact|0TVJZYB6SH]] and [[Hausdorff|0S1FJ8VEX9]].
If $X$ is [[uncountable|0W7FBYS1F1]], it is not [[first-countable|0CP5NDD21C]], and so not [[metrizable|091GJ3FYD7]].

## Remarks

Compactness: in an open cover, one member $U$ contains $p$, and the finitely many points outside $U$ need finitely many more members.
Hausdorff: two points $x, y \ne p$ are separated by $\set{x}$ and $\set{y}$, and $x \ne p$ is separated from $p$ by $\set{x}$ and $X \setminus \set{x}$.
Let $X$ be uncountable and let $U_1, U_2, \dots$ be open sets containing $p$.
Their intersection has countable complement, so it contains a point $x \ne p$.
Then $X \setminus \set{x}$ is an open set containing $p$ that contains no $U_n$, so $p$ has no countable neighbourhood base.
The Fort space is the [[one-point compactification|0GWN3CMRK8]] of the discrete space $X \setminus \set{p}$.
If $X$ is countably infinite, it is homeomorphic to $\set{0} \cup \set{1/n : n \ge 1}$ with the usual topology, which is metrizable.

## Explanations

### L1 human

Take a set of isolated points and add one extra point $p$.
A set around $p$ is open when it leaves out only finitely many points.
This makes the space compact and keeps points apart.
With uncountably many points, no countable list of open sets can pin down $p$, so no distance function gives this topology.
</document>

<document source="https://tamnd.github.io/math2/node/7FYMK2XH1V.md" tag="7FYMK2XH1V" kind="definition">
# Radical extension

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
It is a *radical extension* if there are elements $\alpha_1, \dots, \alpha_m \in L$ and positive integers $n_1, \dots, n_m$ such that:

1. $L = K(\alpha_1, \dots, \alpha_m)$;
2. $\alpha_1^{n_1} \in K$;
3. $\alpha_i^{n_i} \in K(\alpha_1, \dots, \alpha_{i-1})$ for each $i$ with $2 \le i \le m$.

Equivalently, there is a tower $K = K_0 \subseteq K_1 \subseteq \dots \subseteq K_m = L$ in which each $K_i = K_{i-1}(\alpha_i)$ is a [[simple extension|0P6SBR7YM2]] with $\alpha_i^{n_i} \in K_{i-1}$.

## Remarks

Example: $L = \Q(\sqrt{2}, \sqrt[3]{1 + \sqrt{2}})$ is radical over $\Q$, with $\alpha_1 = \sqrt{2}$, $n_1 = 2$, $\alpha_2 = \sqrt[3]{1 + \sqrt{2}}$, $n_2 = 3$, and $\alpha_2^3 = 1 + \sqrt{2} \in \Q(\alpha_1)$.
Non-example: let $\theta = \cos(2\pi/7)$, a root of the irreducible cubic $8x^3 + 4x^2 - 4x - 1$.
The field $\Q(\theta)$ is real, and its three conjugates $\cos(2\pi k/7)$, $k = 1, 2, 3$, are real, so by the [[casus irreducibilis|MJ864PBRRJ]] $\theta$ lies in no radical extension of $\Q$ contained in $\R$.
In particular $\Q(\theta)/\Q$ is not radical, although $\Q(\theta)$ lies inside the radical extension $\Q(\zeta_7)$, with $\zeta_7 = e^{2\pi i/7}$.
A radical extension is finite, but it need not be normal or separable: $\Q(\sqrt[3]{2})/\Q$ is radical and not normal.
</document>

<document source="https://tamnd.github.io/math2/node/7FZB7YYBXA.md" tag="7FZB7YYBXA" kind="concept">
# k-vertex-connected graph

## Summary

A graph is $k$-vertex-connected, or $k$-connected, if it has more than $k$ vertices and stays [[connected|0KWFFMDF8T]] after the removal of any fewer than $k$ vertices.
The largest such $k$ is the vertex connectivity $\kappa(G)$, a measure of how robust a network is against the failure of nodes.
[[Menger's theorem|E6GYGZDBS7]] gives another view.
A graph with more than $k$ vertices is $k$-connected if and only if every two vertices are joined by $k$ internally disjoint paths.
The node [[k-vertex-connected graph|JJN72VZCZE]] gives the precise definition.

## Explanations

### L1 human

Think of a computer network in which some machines may break down.
The network is $2$-connected if the rest of the network can still talk after any single machine fails.
A ring of machines is $2$-connected: removing one machine leaves a chain, which is still in one piece.
A chain is only $1$-connected, since removing a machine in the middle cuts it in two.
</document>

<document source="https://tamnd.github.io/math2/node/7G1PCS2RPG.md" tag="7G1PCS2RPG" kind="statement">
# Thom–Mather isotopy lemmas

## Statement

Let $M$, $N$ and $P$ be [[smooth manifolds|F45VQ47GPV]].
A map between stratified sets is *stratum-preserving* if it maps each stratum into a stratum.

1. (First isotopy lemma.) Let $X \subseteq M$ be a closed subset with a [[Whitney stratification|3F84AMVJNV]], and let $f \colon M \to P$ be a smooth map.
Assume that $f|_X \colon X \to P$ is [[proper|15XEBGNGHC]] and that $f|_S \colon S \to P$ is a [[submersion|GTMT5ZFV00]] for every stratum $S$ of $X$.
Then $f|_X$ is a locally trivial fibration with stratum-preserving trivialisations: every $p \in P$ has an open neighbourhood $U$ and a stratum-preserving [[homeomorphism|0QT00PB5G2]] $h \colon f^{-1}(U) \cap X \to (f^{-1}(p) \cap X) \times U$ with $\mathrm{pr}_U \circ h = f$.
2. (Second isotopy lemma.) Let $X \subseteq M$ and $Y \subseteq N$ be closed Whitney stratified subsets, and let $f \colon M \to N$ and $g \colon N \to P$ be smooth maps with $f(X) \subseteq Y$.
Assume that $f|_X$ and $g|_Y$ are proper, and that $f$ maps each stratum of $X$ submersively onto a stratum of $Y$.
Assume that $f|_X$ satisfies Thom's condition $a_f$, and that $g|_S$ is a submersion for every stratum $S$ of $Y$.
Then $f|_X$ is locally trivial over $P$, in this sense.
Every $p \in P$ has an open neighbourhood $U$ and stratum-preserving homeomorphisms $X \cap (g f)^{-1}(U) \to (X \cap (g f)^{-1}(p)) \times U$ and $Y \cap g^{-1}(U) \to (Y \cap g^{-1}(p)) \times U$.
These homeomorphisms commute with the projections to $U$ and with $f$.

## Remarks

The proof uses the control data of a [[Thom–Mather stratified space|P9R3777ZJF]], which Mather constructed on every Whitney stratified set.
One lifts the coordinate vector fields of $U$ to controlled vector fields on $X$, smooth on each stratum and compatible with the tubular projections and distance functions.
Controlled vector fields have continuous flows that preserve each stratum, and properness makes the flows defined for the needed times.
Integrating them gives the trivialisation.
The flows are only continuous across strata, so the trivialisations are homeomorphisms, not diffeomorphisms.

Example: in $M = \R^3$ let $X$ be the union of the cylinder $\set{x^2 + y^2 = 1}$ and the $t$-axis $\set{x = y = 0}$, with these two strata, and let $f(x, y, t) = t$.
Each stratum is mapped submersively onto $\R$, and $f|_X$ is proper since $X \cap \set{\abs{t} \le c}$ is compact.
The first lemma gives a homeomorphism of $X$ with $(\text{circle} \sqcup \text{point}) \times \R$ that keeps the two strata apart.

Properness is needed: the inclusion of $M = X = \R \setminus \set{0}$ into $P = \R$, with one stratum, is a submersion.
But its fibres are points over $t \ne 0$ and empty over $0$, so it is not locally trivial near $0$.

## Explanations

### L1 human

A stratified space is a space made of smooth pieces of different dimensions, such as a cone made of its tip and its smooth part.
The first lemma is a version of Ehresmann's theorem for such spaces.
If a proper map has no critical points on any piece, then its fibres all look the same, up to a homeomorphism that respects the pieces.
The second lemma does the same for a map between two stratified spaces that moves in a family.

### L2 human

For smooth manifolds, Ehresmann's fibration theorem says that a proper submersion is a locally trivial fibration.
The first isotopy lemma extends this to Whitney stratified sets, at the price of getting only homeomorphisms.
It shows, for instance, that a Whitney stratified set is locally topologically trivial along each stratum.
So the local topology near a point stays the same as the point moves within a connected stratum.
The second lemma is a key tool in Mather's proof that topologically stable maps are dense.
Thom's condition $a_f$ is the hypothesis that makes the two families of trivialisations compatible.
</document>

<document source="https://tamnd.github.io/math2/node/7G7CSM2NZ9.md" tag="7G7CSM2NZ9" kind="statement">
# Kodaira embedding theorem

## Statement

Let $X$ be a compact [[complex manifold|JSGWAN2RJV]] and let $L$ be a [[holomorphic line bundle|QAC7BYK9KX]] on $X$.
Suppose that $L$ is [[positive|J45KWFHENP]]: it has a Hermitian metric whose Chern curvature form $\frac{i}{2\pi}\Theta$ is a positive $(1,1)$-form.
Then there is an integer $k_0$ with the following property.
For every $k \ge k_0$, the holomorphic sections of $L^{\otimes k}$ define a holomorphic embedding of $X$ into a complex [[projective space|ZT9A6JK8QF]] $\mathbb{CP}^N$.
In particular, $X$ is projective.

## Remarks

Proof idea: choose a basis $s_0, \ldots, s_N$ of the space of holomorphic sections of $L^{\otimes k}$ and consider $x \mapsto [s_0(x) : \cdots : s_N(x)]$.
This is an embedding as soon as the sections separate points and tangent vectors.
To get sections with prescribed values at one or two points, one blows up these points.
Then the [[Kodaira vanishing theorem|NY77DQ2AX8]] on the blow-up kills the obstruction in $H^1$; for large $k$ the needed positivity holds.

Equivalent form: a compact complex manifold is projective if and only if it has a [[Kähler|X60JG9AEQJ]] form $\omega$ with a rational cohomology class.
Rational means that $[\omega]$ lies in the image of $H^2(X; \Q)$; after scaling, one may take it integral.
Such $X$ are called Hodge manifolds.
Then $L$ is the line bundle with first Chern class $[\omega]$.
By [[Chow's theorem|6EFEG5VJEF]], $X$ is then the zero set of homogeneous polynomials in $\mathbb{CP}^N$.

Example: on $\mathbb{CP}^n$ the hyperplane bundle $\mathcal{O}(1)$ is positive, with the Fubini–Study metric, and already $k = 1$ gives the identity embedding.

The positivity hypothesis cannot be dropped.
A generic complex torus $\C^2 / \Lambda$ is Kähler, but its Kähler classes are not rational.
It has no positive line bundle and it is not projective; it even has no nonconstant meromorphic functions.

## Explanations

### L1 human

The theorem gives a test for when a compact complex manifold can be described by polynomial equations in projective space.
The test is the existence of a line bundle with positive curvature.
If there is one, high powers of it have enough holomorphic sections to serve as projective coordinates and to embed the manifold.

### L2 human

The theorem is the bridge between complex differential geometry and algebraic geometry: a curvature condition implies an algebraic conclusion.
Combined with the theory of Kähler manifolds it says that projective manifolds are exactly the compact Kähler manifolds with an integral Kähler class.
A positive line bundle is thus [[ample|RRK8P6TXG2]], and the converse also holds.
It shows, for instance, that every compact Riemann surface is a projective algebraic curve.
Via the Riemann bilinear relations, it also tells which complex tori are abelian varieties.
</document>

<document source="https://tamnd.github.io/math2/node/7GEXM87S0H.md" tag="7GEXM87S0H" kind="example">
# The exponential map from the additive reals to the positive reals

## Example

Let $(\mathbb{R}, +)$ be the [[real numbers|08J2M1JYAG]] with addition, and let $(\mathbb{R}_{>0}, \times)$ be the positive real numbers with multiplication.
The [[exponential function|0RMVS7RXZX]] gives a map $\exp : \mathbb{R} \to \mathbb{R}_{>0}$ with $\exp(x + y) = \exp(x)\exp(y)$. Thus it is a homomorphism.
Its kernel is $\{0\}$, because $\exp(x) = 1$ only for $x = 0$. Thus the map is [[injective|0N2P0XRTDF]].
Each $y > 0$ is $\exp(\ln y)$, so the map is onto. Thus $\exp$ is an [[isomorphism|085PND4KNT]].

## Remarks

The inverse isomorphism is the natural logarithm.
</document>

<document source="https://tamnd.github.io/math2/node/7GG0XYHB0T.md" tag="7GG0XYHB0T" kind="problem">
# Serre's problem on projective modules

## Problem

Let $k$ be a field, let $n \ge 1$, and let $R = k[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]] in $n$ variables over $k$.
Is every [[finitely generated|05QY1VEBYD]] [[projective|BFFA03GE3V]] $R$-module [[free|0DZ9DKAKZR]]?

## Remarks

The problem is solved: the answer is yes, by the [[Quillen–Suslin theorem|7N50GA46W1]].
For $n = 1$, the ring $k[x]$ is a principal ideal domain, so the answer is yes by the structure theorem for modules over such rings.
Serre showed that every finitely generated projective $R$-module $P$ is stably free: $P \oplus R^m$ is free for some $m \ge 0$.

## History

Jean-Pierre Serre raised the question in his 1955 paper on coherent algebraic sheaves.
C. S. Seshadri proved the case $n = 2$ in 1958.
Daniel Quillen and Andrei Suslin solved the general case independently in 1976.

## Explanations

### L1 human

Serre asked whether every vector bundle on affine space is trivial, in the language of algebra.
The question is easy to state: does every finitely generated projective module over a polynomial ring have a basis?
It was known to be true for one and two variables, but the general case stayed open for about twenty years.
Quillen and Suslin both proved that the answer is yes, in 1976.
</document>

<document source="https://tamnd.github.io/math2/node/7GJ4STCNNM.md" tag="7GJ4STCNNM" kind="example">
# Complex projective space

## Example

Let $n \ge 1$.
*Complex projective space* $\mathbb{CP}^n$ is the [[quotient|0BR3FY5Q8M]] of $\C^{n+1} \setminus \set{0}$ by the relation $z \sim \lambda z$ for $\lambda \in \C \setminus \set{0}$; the class of $z = (z_0, \ldots, z_n)$ is written $[z_0 : \cdots : z_n]$.
So $\mathbb{CP}^n$ is the set of complex lines through $0$ in $\C^{n+1}$.

It is a [[complex manifold|JSGWAN2RJV]] of dimension $n$.
For $i = 0, \ldots, n$, let $U_i = \set{[z] : z_i \ne 0}$ and $\varphi_i[z] = (z_0/z_i, \ldots, \widehat{z_i/z_i}, \ldots, z_n/z_i) \in \C^n$, where the hat means that the entry is left out.
Each $\varphi_i$ is a well-defined homeomorphism of $U_i$ onto $\C^n$, and the $U_i$ cover $\mathbb{CP}^n$.
On $\varphi_i(U_i \cap U_j)$ the coordinate $w$ that stands for $z_j/z_i$ is nonzero.
The components of the transition map $\varphi_j \circ \varphi_i^{-1}$ are $1/w$ and the other coordinates divided by $w$, so it is holomorphic.
The space is Hausdorff and [[compact|0TVJZYB6SH]]: it is the continuous image of the unit sphere $S^{2n+1} \subset \C^{n+1}$, since every line meets the sphere.

## Remarks

For $n = 1$ one gets the [[Riemann sphere|0J78KBPFZB]], and for $n = 2$ the [[complex projective plane|58WSWYBCWA]].
The map $S^{2n+1} \to \mathbb{CP}^n$ is a circle bundle, the [[Hopf fibration|M5AKNYC4QQ]] when $n = 1$.
The Fubini–Study metric makes $\mathbb{CP}^n$ a [[Kähler manifold|X60JG9AEQJ]].
By [[Chow's theorem|6EFEG5VJEF]], every closed complex submanifold of $\mathbb{CP}^n$ is a [[projective variety|A4H4CZ2V35]].
By the [[Kodaira embedding theorem|7G7CSM2NZ9]], a compact complex manifold embeds in some $\mathbb{CP}^N$ if and only if it carries a positive line bundle.
Since $\mathbb{CP}^n$ is compact and connected, its only holomorphic functions are the [[constants|2SM64J81YC]].

## Explanations

### L1 human

Complex projective space collects all complex lines through the origin of $\C^{n+1}$ into one space, each line becoming a single point.
Near any point it looks like $\C^n$: one coordinate is nonzero, and dividing by it gives $n$ complex numbers.
The points where that coordinate vanishes are the "points at infinity", and they make the space compact.
For $n = 1$ this is the Riemann sphere: the complex plane plus one point at infinity.
</document>

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# Minkowski sum of polytopes

## Summary

The Minkowski sum of two convex polytopes $P$ and $Q$ is the set of all sums $p + q$ with $p \in P$ and $q \in Q$.
It is again a [[convex polytope|300K6CB1KH]], and it is the special case of [[Minkowski addition|SXS6D87PHE]] for polytopes.
Its faces are sums of faces of $P$ and $Q$.
Its [[normal fan|XJ83PTS3H0]] is the common refinement of the normal fans of $P$ and $Q$.
Minkowski sums lead to [[zonotopes|Z5SA6A67KF]], to [[mixed volumes|QB3Q5FBJ08]] and to the Newton polytopes of products of polynomials.

## Explanations

### L1 human

Slide a copy of $Q$ so that a chosen point of it visits every point of $P$, and collect everything that $Q$ sweeps out.
The swept region is the Minkowski sum $P + Q$.
For example, the sum of a horizontal unit segment and a vertical unit segment is the unit square.
The sum of a triangle and the same triangle turned upside down is a hexagon.
</document>

<document source="https://tamnd.github.io/math2/node/7GKCGQ7215.md" tag="7GKCGQ7215" kind="concept">
# Hausdorff topological group

## Summary

A Hausdorff topological group is a [[topological group|92700QAB5Y]] whose topology is [[Hausdorff|0S1FJ8VEX9]].
For topological groups this is a mild condition: it holds as soon as the one-point set of the identity is closed.
Many authors include it in the definition of a topological group.
Every topological group has a largest Hausdorff quotient, obtained by dividing by the closure of the identity.

## Explanations

### L1 human

In a Hausdorff space two different points can be put in two disjoint open sets.
For a topological group, this follows already from a much weaker looking condition: the identity alone forms a closed set.
The real line under addition and every matrix group are Hausdorff.
A group with the indiscrete topology, where the only open sets are the empty set and the whole group, is not Hausdorff once it has two elements.
</document>

<document source="https://tamnd.github.io/math2/node/7GRSF20H15.md" tag="7GRSF20H15" kind="statement">
# Sphere theorem

## Statement

Let $(M, g)$ be a connected, simply connected, complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \ge 2$.
Suppose that every [[sectional curvature|1X42JAA349]] $K$ of $M$ satisfies
$$\frac{1}{4} < K \le 1.$$
Then:

1. $M$ is [[homeomorphic|0QT00PB5G2]] to the [[sphere|T62369RFH6]] $S^n$ (Berger and Klingenberg).
2. $M$ is [[diffeomorphic|0KZZ70ZHXP]] to $S^n$ (Brendle and Schoen).

## Remarks

By Myers's theorem, $M$ is compact.
The idea of the topological proof: Klingenberg's estimate shows that the injectivity radius of $M$ is at least $\pi$.
With Toponogov's comparison theorem one then shows that $M$ is the union of two embedded open balls centred at two points at maximal distance.
A closed manifold that is the union of two embedded open discs is homeomorphic to a sphere.
The differentiable version uses the Ricci flow: Brendle and Schoen showed that the flow, normalized, takes the metric to one of constant curvature.

The constant $1/4$ is sharp: take the complex projective space $\C P^m$ with $m \ge 2$ and the Fubini–Study metric scaled so that $1/4 \le K \le 1$.
It is simply connected and compact but not homeomorphic to a sphere.
Simple connectedness is needed: real projective space $\R P^n$ with the round metric has $K = 1$ and is not a sphere.

## Explanations

### L1 human

Suppose a simply connected space bends positively.
If the bending in its most curved direction is less than four times that in its least curved direction, then the space has the shape of a sphere.
Not only roughly but also smoothly: it can be deformed into a round sphere.

### L2 human

The topological sphere theorem was proved by Berger and Klingenberg around 1960, after a version with a weaker constant by Rauch.
The smooth version, proved by Brendle and Schoen in 2009, answered the question whether exotic spheres can carry quarter-pinched metrics.
Their method also gives the result under a pointwise pinching condition.
</document>

<document source="https://tamnd.github.io/math2/node/7GZ908QZ04.md" tag="7GZ908QZ04" kind="statement">
# Identity theorem in several variables

## Statement

Let $D \subseteq \C^n$ be a domain, that is, a nonempty connected open set.
Let $f$ and $g$ be [[holomorphic functions|507C34W0ZT]] on $D$.
If $f = g$ on a nonempty open subset of $D$, then $f = g$ on all of $D$.

## Remarks

Put $h = f - g$ and let $E$ be the set of points of $D$ near which $h$ vanishes identically.
The set $E$ is open by definition and nonempty by hypothesis.
It is also closed in $D$.
At a limit point $a \in D$ of $E$, all partial derivatives of $h$ vanish by continuity, so the power series of $h$ at $a$ is zero and $h = 0$ near $a$.
Since $D$ is connected, $E = D$.
The one-variable condition "agreement on a set with a limit point" is not enough when $n \ge 2$: $f(z_1, z_2) = z_1$ vanishes on the whole line $\set{z_1 = 0}$ but is not zero.
Connectedness is needed.
Let $D = \set{\abs{z_1} < 1} \cup \set{\abs{z_1 - 3} < 1}$ in $\C$, or its product with $\C^{n-1}$.
The function equal to $0$ on the first piece and $1$ on the second is holomorphic and vanishes on an open set without being zero.

## Explanations

### L1 human

A holomorphic function on a connected domain is fixed everywhere by its values on any small ball.
So two such functions that agree on a small ball agree everywhere.
This is the rigidity that separates holomorphic functions from smooth real functions.

### L2 human

This is the several-variable form of the [[identity theorem|0YBVKMYX0W]].
The proof uses only the local power series expansion and connectedness, exactly as in one variable.
What changes is the size of the set needed.
Zero sets of holomorphic functions in $\C^n$, $n \ge 2$, are never isolated points, so agreement on a discrete set or even on a complex hypersurface says little.
It is enough that $f$ and $g$ agree on a set that is not contained in any proper analytic subset of $D$.
An example of such a set is a nonempty open subset of $\R^n \cap D$, when $D$ meets $\R^n$.
The theorem underlies uniqueness of analytic continuation in several variables.
</document>

<document source="https://tamnd.github.io/math2/node/7H0E1ZBWTB.md" tag="7H0E1ZBWTB" kind="definition">
# Normal fan

## Definition

Let $P$ be a nonempty [[convex polytope|300K6CB1KH]] in $\R^n$, and write $\ip{u}{x}$ for the standard inner product.
For a nonempty [[face|KY7SB854VH]] $F$ of $P$, the *normal cone* of $F$ is
$$N_F = \set{u \in \R^n : \ip{u}{x} \ge \ip{u}{y} \text{ for all } x \in F \text{ and } y \in P}.$$
So $N_F$ is the set of vectors $u$ such that the linear functional $\ip{u}{\cdot}$ attains its maximum over $P$ at every point of $F$.
The *normal fan* of $P$ is the collection $\mathcal{N}(P) = \set{N_F : F \text{ a nonempty face of } P}$.

## Remarks

Each $N_F$ is a polyhedral [[convex cone|6VE4KA12VD]], and $\dim N_F = n - \dim F$.
The collection $\mathcal{N}(P)$ is a complete fan:

1. every face of a cone of $\mathcal{N}(P)$ is a cone of $\mathcal{N}(P)$;
2. the intersection of two cones of $\mathcal{N}(P)$ is a face of each of them;
3. the union of all cones of $\mathcal{N}(P)$ is $\R^n$.

Inclusion is reversed: $F \subseteq G$ holds exactly when $N_G \subseteq N_F$.
Each $N_F$ contains the orthogonal complement of the direction space of the affine hull of $P$.
So the cones are pointed exactly when $P$ is full-dimensional, and then the vertices of $P$ give the $n$-dimensional cones and the facets give the rays.

Example: let $P = [-1,1]^2$.
The vertex $(1,1)$ maximizes $\ip{u}{\cdot}$ exactly when $u_1 \ge 0$ and $u_2 \ge 0$, so its normal cone is the closed first quadrant.
The edge $\set{1} \times [-1,1]$ is maximal exactly for $u_1 \ge 0$ and $u_2 = 0$, so its normal cone is the ray spanned by $(1,0)$.
The face $P$ itself has normal cone $\set{0}$.
So $\mathcal{N}(P)$ consists of the four closed quadrants, the four half-axes and the origin.

Non-example: the closed upper half-plane and the closed right half-plane do not form a fan, even with all their faces added.
Their intersection is the first quadrant, which is not a face of either half-plane, so clause 2 fails.

Some authors use the functionals that are minimal on $F$ instead; this replaces each cone by its negative.
The normal fan of a Minkowski sum $P + Q$ is the common refinement of the normal fans of $P$ and $Q$.
</document>

<document source="https://tamnd.github.io/math2/node/7H2639JPXJ.md" tag="7H2639JPXJ" kind="definition">
# Kernel regression

## Definition

Let $(X_1, Y_1), \dots, (X_n, Y_n)$ be [[i.i.d.|0AJN7WHBJZ]] pairs of real random variables with $E\abs{Y_1} < \infty$, and with regression function $m(x) = E[Y_1 \mid X_1 = x]$, a [[conditional expectation|0TTDGTEFMN]].
Let $K \colon \R \to [0, \infty)$ be a kernel as in [[kernel density estimation|ZV65XMJ9A4]], and let $h > 0$ be a bandwidth.
The *Nadaraya–Watson kernel regression estimator* of $m$ is
$$\hat m_h(x) = \frac{\sum_{i=1}^{n} K\big((x - X_i)/h\big)\, Y_i}{\sum_{i=1}^{n} K\big((x - X_i)/h\big)},$$
defined at each $x$ where the denominator is not $0$.

## Remarks

The weights $K((x - X_i)/h) / \sum_j K((x - X_j)/h)$ are nonnegative and sum to $1$, so $\hat m_h(x)$ is a weighted average of the $Y_i$.
Dividing the numerator and the denominator by $nh$ shows the denominator is the kernel density estimate of the law of $X$ at $x$.
Example: with the box kernel $K = \frac{1}{2} \mathbf{1}_{[-1, 1]}$, $\hat m_h(x)$ is the plain average of the $Y_i$ with $\abs{X_i - x} \le h$.
Non-example: the numerator alone, divided by $nh$, estimates $m(x) f_X(x)$ rather than $m(x)$, where $f_X$ is the density of $X$.
Local polynomial regression generalizes this estimator; the Nadaraya–Watson estimator is the local constant case.
</document>

<document source="https://tamnd.github.io/math2/node/7H3YSSDEBK.md" tag="7H3YSSDEBK" kind="statement">
# Schaefer's fixed point theorem

## Statement

Let $X$ be a real [[Banach space|06GZ956657]] and let $A \colon X \to X$ be a [[continuous|0X3WGX95FE]] map.
Assume that $A$ is compact: it maps each bounded subset of $X$ into a relatively compact set.
Assume that the set
$$\set{u \in X : u = \lambda A(u) \text{ for some } \lambda \in [0, 1]}$$
is bounded.
Then $A$ has a fixed point: there is $u \in X$ with $A(u) = u$.

## Remarks

The proof uses the [[Schauder fixed-point theorem|DCPFVVZZ54]].
Choose $M$ larger than the norm of every $u$ in the bounded set of the hypothesis, and let $B$ be the closed ball of radius $M$.
On $B$, define $\tilde A(u) = A(u)$ if $\norm{A(u)} \le M$ and $\tilde A(u) = M A(u)/\norm{A(u)}$ otherwise.
Then $\tilde A$ maps $B$ continuously into a compact subset of $B$, so it has a fixed point $u$.
If $\norm{A(u)} > M$, then $u = \lambda A(u)$ with $\lambda = M/\norm{A(u)} < 1$ and $\norm{u} = M$, which contradicts the choice of $M$; so $A(u) = u$.

The hypothesis is an a priori bound: one shows that all solutions of the family of problems $u = \lambda A(u)$ lie in a fixed ball, without knowing in advance that any exists.

Example: on $X = \R$, let $A(u) = 1 + \frac{1}{2}\sin u$.
If $u = \lambda A(u)$ with $\lambda \in [0, 1]$, then $\abs{u} \le 3/2$, so the set is bounded, and $A$ has a fixed point.

The a priori bound is necessary: $A(u) = u + 1$ on $\R$ is continuous and compact, but has no fixed point, and the solutions $u = \lambda/(1 - \lambda)$ of $u = \lambda A(u)$ are unbounded as $\lambda \to 1$.

## Explanations

### L1 human

To show that an equation $u = A(u)$ has a solution, it is enough to deform it into a trivial problem, $u = 0$, through the family $u = \lambda A(u)$.
If all the solutions along the way stay in a bounded region, a solution survives at the end.
This turns an existence question into the often easier task of proving estimates.

### L2 human

The theorem is a standard tool for nonlinear elliptic and parabolic equations, as in Evans's book on partial differential equations.
One writes the equation as $u = A(u)$, where $A$ solves a linear problem with the nonlinearity frozen, and gets compactness of $A$ from elliptic regularity and a compact Sobolev embedding.
The a priori bound is then proved by energy estimates or the maximum principle.
It is a special case of the Leray–Schauder continuation principle, which is based on degree theory.
</document>

<document source="https://tamnd.github.io/math2/node/7H44JD9NQD.md" tag="7H44JD9NQD" kind="concept">
# Holonomic function

## Summary

A holonomic function, also called a D-finite function, is a function or [[formal power series|033WGN00BF]] that satisfies a [[linear differential equation|0QQRP17Z7S]] whose coefficients are polynomials.
Many [[special functions|HWY72ZK273]] are holonomic, for example the exponential, the sine, the logarithm, the [[hypergeometric function|463X5V31D9]] and the [[Bessel functions|CZCMRDWDYW]].
The class is closed under sums and products, and the coefficients of a holonomic power series satisfy a [[recurrence|0FYX44GP92]] with polynomial coefficients.
For this reason such a differential equation is a finite data structure for the function, which computer algebra uses to prove identities automatically.

## Explanations

### L1 human

Some functions can be described by a short rule that ties the function to its own derivatives.
A holonomic function is one for which this rule is linear and has polynomials as coefficients.
For example $f(x) = e^{x^2}$ satisfies $f'(x) - 2x f(x) = 0$, so it is holonomic.
Knowing the equation and a few starting values pins down the function completely.
The [[gamma function|0H5SYBSED9]] is a well known function that is not holonomic.
</document>

<document source="https://tamnd.github.io/math2/node/7H6T24G8BM.md" tag="7H6T24G8BM" kind="definition">
# Enriques surface

## Definition

Let $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over $\C$, with structure sheaf $\mathcal{O}_X$ and [[canonical divisor|W05VYY4HBR]] $K_X$.
$X$ is an *Enriques surface* if:

1. $H^1(X, \mathcal{O}_X) = 0$, that is, the irregularity $q$ is $0$;
2. $H^0(X, \mathcal{O}_X(K_X)) = 0$, that is, the geometric genus $p_g$ is $0$; and
3. $2 K_X$ is linearly equivalent to $0$.

## Remarks

Clauses 2 and 3 together force $K_X$ to be a nonzero element of order $2$ in the Picard group, since $K_X \sim 0$ would give $p_g = 1$.
The double cover $\tilde X \to X$ defined by $K_X$ is unramified, and $\tilde X$ is a [[K3 surface|Q3V566R6K2]]; conversely, the quotient of a K3 surface by a fixed-point-free involution is an Enriques surface.
The fundamental group of an Enriques surface is $\Z/2$, and its Euler characteristic is $12$.
Example: in $\mathbb{P}^5$ with coordinates $x_0, x_1, x_2, y_0, y_1, y_2$, let $Y$ be a general intersection of three quadrics $Q_i(x) + R_i(y) = 0$, $i = 1, 2, 3$.
Then $Y$ is a K3 surface, and the involution $(x, y) \mapsto (x, -y)$ acts on $Y$ without fixed points.
The quotient of $Y$ by this involution is an Enriques surface.
Non-example: a K3 surface satisfies clauses 1 and 3, but it has $K \sim 0$ and so $p_g = 1$, which fails clause 2.
In characteristic $2$ the definition is modified, and there are three kinds of Enriques surfaces.
</document>

<document source="https://tamnd.github.io/math2/node/7H7CT6H9AC.md" tag="7H7CT6H9AC" kind="definition">
# Modular representation theory

## Definition

Let $G$ be a finite group and let $K$ be a field whose [[characteristic|0F6ES3BRYP]] is a prime $p$ that divides $\abs{G}$.
A *modular representation* of $G$ over $K$ is a [[representation|WYGRXCG2RN]] $\rho \colon G \to GL(V)$ on a finite-dimensional vector space $V$ over $K$.
Equivalently, it is a finitely generated [[module|0NZ8JA25WK]] over the [[group algebra|MNP2HBWB5Q]] $KG$.
*Modular representation theory* is the study of these representations and modules.

## Remarks

Example: $G = \Z/p\Z$ with generator $g$, $K = \F_p$, and $\rho(g) = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.
This is a representation because $\rho(g)^p = \begin{pmatrix} 1 & p \\ 0 & 1 \end{pmatrix} = I$ in characteristic $p$.
The line $K e_1$ is invariant, but any other line $K(a e_1 + e_2)$ is sent to $K((a+1) e_1 + e_2)$, a different line, so $V$ is not a direct sum of invariant lines.
Non-example: representations of $G$ over $\C$, or over a [[finite field|0ZXZ82FTX2]] whose characteristic does not divide $\abs{G}$, are ordinary representations; there Maschke's theorem makes every representation semisimple.
Let $K$ be a splitting field for $G$.
Brauer showed that the number of simple $KG$-modules is the number of conjugacy classes of elements of $G$ of order prime to $p$.
</document>

<document source="https://tamnd.github.io/math2/node/7H7SACRYEF.md" tag="7H7SACRYEF" kind="concept">
# Rogers polynomials

## Summary

The Rogers polynomials $C_n(x; \beta \mid q)$, also called the continuous $q$-ultraspherical polynomials, are a family of orthogonal polynomials with a parameter $\beta$ and a base $q$.
They are the $q$-analogue of the [[Gegenbauer polynomials|5TRFA6F79B]]: with $\beta = q^\lambda$ they tend to $C_n^{(\lambda)}(x)$ as $q \to 1$.
L. J. Rogers used them in his work that led to the [[Rogers–Ramanujan identities|M24H4GJPPW]].
They are special cases of the [[Askey–Wilson polynomials|RSNZNG0WA1]] and contain the [[continuous q-Hermite polynomials|XFKVZJ55Q5]] as the case $\beta = 0$.

## Explanations

### L1 human

The Rogers polynomials are defined by one generating function, a ratio of four infinite products.
Changing the parameter $\beta$ moves between several known families.
At $\beta = q$ they are the Chebyshev polynomials of the second kind, and at $\beta = 0$ they are the continuous $q$-Hermite polynomials up to a factor.
For example, $C_1(x; \beta \mid q) = \frac{2(1 - \beta)}{1 - q}\, x$.
</document>

<document source="https://tamnd.github.io/math2/node/7HA6EEDR9S.md" tag="7HA6EEDR9S" kind="statement">
# Mairhuber–Curtis theorem

## Statement

Let $d \ge 2$ and let $\Omega \subseteq \R^d$ have an [[interior point|0RZH6W98HK]].
Let $n \ge 2$ and let $u_1, \dots, u_n \colon \Omega \to \R$ be continuous.
Then some nonzero linear combination of $u_1, \dots, u_n$ has at least $n$ zeros in $\Omega$.
So $(u_1, \dots, u_n)$ is not a [[Chebyshev system|NY5AK617TR]] on any set containing these zeros, and $\Omega$ carries no Haar subspace of dimension $n \ge 2$.

## Remarks

The idea of the proof: suppose $(u_1, \dots, u_n)$ is a Chebyshev system, so $D(x_1, \dots, x_n) = \det \big( u_i(x_j) \big)$ is nonzero for distinct points.
Choose distinct $x_1, \dots, x_n$ in a small open ball inside $\Omega$.
Because $d \ge 2$, one can move $x_1$ and $x_2$ continuously inside the ball so that they exchange places, without colliding with each other or with the other points.
Along this motion $D$ changes continuously from $D(x_1, x_2, \dots)$ to $D(x_2, x_1, \dots) = -D(x_1, x_2, \dots)$, so by the intermediate value theorem it vanishes somewhere, a contradiction.

Example: the polynomials of degree at most $1$ in two variables are spanned by $1, x, y$.
They are not a Haar space on the unit square: the nonzero function $x - y$ vanishes on the whole diagonal.

In dimension $d = 1$ the argument fails, because two points of an interval cannot exchange places without meeting.
Indeed the polynomials of degree at most $n - 1$ form a Haar subspace there.

## Explanations

### L1 human

On an interval you can always fit a unique polynomial of degree $n - 1$ through $n$ points, whatever the points are.
In the plane or in space nothing of this kind is possible for a fixed family of functions.
For some placement of the points the interpolation problem breaks down.
The reason is that two points in the plane can swap places by walking around each other.

### L2 human

The theorem is why multivariate interpolation and uniform approximation behave so differently from the one-variable theory.
Uniqueness of best uniform approximation and unisolvence for all point sets cannot hold for a fixed space of functions when $d \ge 2$.
Practical methods respond by letting the space depend on the data points, as in [[radial basis function interpolation|XZ33P3HBB9]], where the basis functions are centred at the data points.
The theorem holds more generally for compact Hausdorff spaces: a Haar subspace of dimension at least $2$ exists only if the space embeds in a circle.
</document>

<document source="https://tamnd.github.io/math2/node/7HACJC0JE2.md" tag="7HACJC0JE2" kind="concept">
# Characteristic class

## Summary

A characteristic class assigns to each bundle of a given type over a space $X$ a cohomology class of $X$, compatibly with pullbacks.
It measures how far a bundle is from being trivial: the trivial bundle has only the trivial classes.
The main families are the [[Stiefel–Whitney classes|JQCK7A210W]], the [[Chern classes|PAD37GKDPD]], the [[Pontryagin classes|DN11FKWN63]] and the [[Euler class|BJX6QS0ANT]].
Through the [[classifying space|WSZ40H5E4T]], characteristic classes of principal $G$-bundles correspond to cohomology classes of $BG$.

## Explanations

### L1 human

Two bundles over the same space can look alike locally but be twisted in different ways globally.
A characteristic class is a label in the cohomology of the base that detects such twisting.
The labels behave well under change of base: pulling a bundle back along a map pulls the label back too.
For example, the first Stiefel–Whitney class of the Möbius band over the circle is nonzero, while that of the cylinder is zero, so the two bundles differ.
</document>

<document source="https://tamnd.github.io/math2/node/7HCFARHG7D.md" tag="7HCFARHG7D" kind="concept">
# Kähler differentials

## Summary

The module of Kähler differentials $\Omega_{A/R}$ of an algebra $A$ over a commutative ring $R$ is the algebraic version of the module of differential $1$-forms.
It is generated by symbols $da$ for $a \in A$, subject only to the rules of differentiation.
It is the universal target of [[derivations|3J89REV9WQ]] of $A$ over $R$.
It is the basic tool to define [[smooth|286G0P1B69]], [[unramified|GY3QA6NH0E]] and [[étale|FTRW0679E2]] maps of rings without any analysis.

## Explanations

### L1 human

On a curve or surface given by polynomial equations, one can differentiate polynomial functions without limits, using only the product rule and linearity.
Kähler differentials collect all such formal differentials $df$ in one module.
For the polynomial ring $k[x, y]$, every differential is a combination $g \, dx + h \, dy$, and $d(x^2 y) = 2xy \, dx + x^2 \, dy$.
On the curve $y^2 = x^3$, one also has the relation $2y \, dy = 3x^2 \, dx$, which comes from differentiating the equation.
At the cusp $x = y = 0$ this relation becomes $0 = 0$, and this signals the singular point.
</document>

<document source="https://tamnd.github.io/math2/node/7HWZ06SRMF.md" tag="7HWZ06SRMF" kind="definition">
# Jacobson radical

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
The *Jacobson radical* of $R$ is the intersection of all [[maximal ideals|09J6KGT65G]] of $R$:
$$J(R) = \bigcap_{M \text{ maximal}} M.$$

## Remarks

Equivalent form: $x \in J(R)$ if and only if $1 - xy$ is a [[unit|019NFGFPR5]] of $R$ for every $y \in R$.
Example: in $\Z/12\Z$ the maximal ideals are $2\Z/12\Z$ and $3\Z/12\Z$, so $J(\Z/12\Z) = 6\Z/12\Z = \set{0, 6}$; for $y \in \Z/12\Z$, $1 - 6y$ is $1$ or $7$, both units.
Non-example: $2 \notin J(\Z)$, since $2 \notin 3\Z$; with the equivalent form, $1 - 2 \cdot 2 = -3$ is not a unit of $\Z$.
The [[nilradical|QQDN6EYQNJ]] is contained in $J(R)$, since every maximal ideal is prime.
The inclusion can be strict: for $\Z_{(p)}$, the integers localized at a prime $p$, the nilradical is $0$ while $J = p\Z_{(p)}$.
$R$ is a [[local ring|MZ02BKPHJA]] exactly when $J(R)$ is a maximal ideal.
Other notations are $\operatorname{Jac}(R)$ and $\operatorname{rad}(R)$.
For an ideal $I$, the Jacobson radical of $I$ is the intersection of the maximal ideals that contain $I$.
</document>

<document source="https://tamnd.github.io/math2/node/7HYF6YG0RS.md" tag="7HYF6YG0RS" kind="concept">
# Numerical integration

## Summary

Numerical integration is the computation of approximate values of [[definite integrals|04TMP0XGMJ]] from finitely many values of the integrand.
It is needed when an antiderivative has no closed form or when the integrand is only known at sample points.
It is also needed for integrals over domains of many dimensions.
The main tools are [[quadrature rules|ADSVMCY77A]], weighted sums of function values, together with error estimates based on the smoothness of the integrand.

## Explanations

### L1 human

The integral of a function is the area under its graph, but often there is no formula for it.
Numerical integration estimates that area by measuring the height of the graph at some points and adding up suitable weighted heights.
The simplest example cuts the interval into strips and replaces each strip by a trapezoid.
For $\int_0^1 e^{-x^2}\, dx$, which has no elementary antiderivative, such methods give $0.7468\ldots$ to any accuracy one wants.
</document>

<document source="https://tamnd.github.io/math2/node/7J3P1W042Y.md" tag="7J3P1W042Y" kind="statement">
# Mertens' theorems

## Statement

Let $p$ run over the [[prime numbers|01826KMG2H]], let $\log$ be the [[natural logarithm|04DXAFN2ES]], and let $x \ge 2$ be real.

1. $\displaystyle \sum_{p \le x} \frac{\log p}{p} = \log x + O(1)$.
2. There is a constant $M$ such that $\displaystyle \sum_{p \le x} \frac{1}{p} = \log \log x + M + O\!\left(\frac{1}{\log x}\right)$.
3. $\displaystyle \prod_{p \le x} \left(1 - \frac{1}{p}\right) \sim \frac{e^{-\gamma}}{\log x}$ as $x \to \infty$, where $\gamma$ is the [[Euler–Mascheroni constant|00C5P9JZBK]].

Here $O$ is [[big O notation|0TGV80E6A4]] as $x \to \infty$, and $\sim$ is [[asymptotic equivalence|0QR92XBV6H]].

## Remarks

For the first theorem, compare two expressions of $\log(n!)$.
Legendre's formula gives $\log(n!) = \sum_{p^k \le n} \floor{n/p^k} \log p = n \sum_{p \le n} \frac{\log p}{p} + O(n)$, using the bound $\psi(n) = O(n)$ of Chebyshev for the [[Chebyshev function|B8AKSVBE5Z]] $\psi$.
[[Stirling's formula|0KC0F51T6A]] gives $\log(n!) = n \log n + O(n)$, and dividing by $n$ gives the result.
The second theorem follows from the first by partial summation.
The third follows from the second by taking logarithms; identifying the constant $e^{-\gamma}$ needs the behaviour of the [[Riemann zeta function|1XWBY4N0VT]] near $s = 1$.
The constant $M \approx 0.2615$ is the Meissel–Mertens constant.
The theorems do not need the [[prime number theorem|0EHBY5Y84N]]; they are elementary.

## Explanations

### L1 human

The sum of the reciprocals of the primes diverges, but very slowly, like $\log \log x$.
Mertens' theorems give the precise size of this sum and of two related quantities.
The product $\prod_{p \le x} (1 - 1/p)$ is the proportion of integers that have no prime factor up to $x$, in a heuristic sense, and it decays like $e^{-\gamma}/\log x$.

### L2 human

The theorems are typical of elementary prime number theory: they use only Chebyshev bounds and the factorization of $n!$.
The third theorem is used in sieve methods, where products over primes such as $\prod_{p \le z} (1 - 1/p)$ give the main term.
A naive sieve heuristic with $z = \sqrt{x}$ would predict about $2e^{-\gamma} x/\log x \approx 1.12\, x/\log x$ primes up to $x$, while the prime number theorem gives $x/\log x$.
The difference shows that divisibility by different large primes is not independent.
</document>

<document source="https://tamnd.github.io/math2/node/7JA7WH92K0.md" tag="7JA7WH92K0" kind="concept">
# Representation of a Lie group

## Summary

A representation of a [[Lie group|7JBRGCVJRB]] $G$ is a continuous homomorphism from $G$ to the group of invertible linear maps of a vector space.
It lets the group act on vectors by linear maps, so that group theory can be studied with linear algebra.
Each finite-dimensional representation has a derivative, a representation of the Lie algebra, which often determines it.
Representations of Lie groups are central in harmonic analysis, number theory and quantum physics.

## Explanations

### L1 human

A representation turns each element of a group into a matrix, so that composing elements becomes multiplying matrices.
The rotation group of space acts on vectors of $\R^3$ by rotation matrices, and this is a representation.
It also acts on functions on the sphere, by rotating the function's graph, and this is a larger representation.
The two must be continuous: rotations by nearby angles give nearby matrices.
Breaking a representation into irreducible pieces is like the Fourier series of a function.
</document>

<document source="https://tamnd.github.io/math2/node/7JBRGCVJRB.md" tag="7JBRGCVJRB" kind="concept">
# Lie group

## Summary

A Lie group is a group that is also a smooth manifold, in such a way that multiplication and inversion are smooth maps.
It is the natural setting for continuous symmetry: rotations, translations, matrix groups and the symmetry groups of physics are all Lie groups.
Its structure near the identity is captured by a finite-dimensional Lie algebra, which turns many questions about the group into linear algebra.

## Explanations

### L1 human

A Lie group is a family of symmetries that can be varied smoothly, like the rotations of a plane through any angle.
Each element is a point of a smooth space, and you can move from one element to a nearby one by a small change.
The circle of complex numbers of absolute value $1$, with multiplication, is a simple example: it is a smooth curve, and multiplying two points of it just adds their angles.
The invertible $n \times n$ real matrices form another example, a group that is an open subset of the space of all matrices.
</document>

<document source="https://tamnd.github.io/math2/node/7JCF2TE3V5.md" tag="7JCF2TE3V5" kind="concept">
# Invertible sheaf

## Summary

An invertible sheaf on a ringed space is a sheaf of modules that is locally isomorphic to the structure sheaf.
It is the algebraic version of a line bundle, a family of one-dimensional vector spaces over the points of the space.
The name comes from the fact that, on a scheme, such sheaves are exactly the sheaves of modules with an inverse for the tensor product.
Their isomorphism classes form the Picard group.

## Explanations

### L1 human

A line bundle attaches a line to each point of a space, and these lines vary smoothly.
Locally it looks like the space times a line, but globally it can twist, like a Möbius band over a circle.
An invertible sheaf records the sections of a line bundle: over small open sets they are just functions, but globally they can be glued with a twist.
On projective space, the twisting sheaves $\mathcal{O}(n)$ are the basic examples, and for $n \ge 0$ the global sections of $\mathcal{O}(n)$ are the homogeneous polynomials of degree $n$.
</document>

<document source="https://tamnd.github.io/math2/node/7JE39BAHQG.md" tag="7JE39BAHQG" kind="definition">
# Pseudospectrum

## Definition

Let $A \in \C^{n \times n}$, let $\Lambda(A)$ be its set of [[eigenvalues|0BQAA2BK3Q]], and let $\varepsilon > 0$.
Let $\norm{\cdot}$ be the Euclidean norm on $\C^n$ with its [[operator norm|06EKW1AKXF]] on matrices.
The *$\varepsilon$-pseudospectrum* of $A$ is
$$\Lambda_\varepsilon(A) = \Lambda(A) \cup \set{z \in \C \setminus \Lambda(A) : \norm{(zI - A)^{-1}} > 1/\varepsilon}.$$

## Remarks

The following sets are equal to $\Lambda_\varepsilon(A)$:

1. $\set{z \in \C : z \in \Lambda(A + E) \text{ for some } E \in \C^{n \times n} \text{ with } \norm{E} < \varepsilon}$;
2. $\set{z \in \C : \norm{(zI - A) v} < \varepsilon \text{ for some unit vector } v}$;
3. $\set{z \in \C : \sigma_{\min}(zI - A) < \varepsilon}$, where $\sigma_{\min}$ is the smallest [[singular value|09VMDSRXWJ]].

The convention $\norm{(zI - A)^{-1}} = \infty$ for $z \in \Lambda(A)$ makes the definition read $\norm{(zI - A)^{-1}} > 1/\varepsilon$ for all $z$.

Example: if $A$ is normal, then $\norm{(zI - A)^{-1}} = 1/\mathrm{dist}(z, \Lambda(A))$, so $\Lambda_\varepsilon(A)$ is the union of the open discs of radius $\varepsilon$ about the eigenvalues.
Example: for $J = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $z \ne 0$, $(zI - J)^{-1} = \begin{pmatrix} 1/z & 1/z^2 \\ 0 & 1/z \end{pmatrix}$, whose norm is at least $1/\abs{z}^2$; so every $z$ with $0 < \abs{z} < \sqrt{\varepsilon}$ lies in $\Lambda_\varepsilon(J)$.
Non-example: for $A = 0 \in \C^{1 \times 1}$ and $\varepsilon = 0.1$, the point $z = 0.2$ is not in $\Lambda_\varepsilon(A)$, since $\norm{(zI - A)^{-1}} = 5 \le 10$.
The sets are open, they increase with $\varepsilon$, and their intersection over all $\varepsilon > 0$ is $\Lambda(A)$.
The same definition makes sense for a bounded operator on a Banach space, with its [[spectrum|BMKATJ802A]] in place of $\Lambda(A)$.

## Explanations

### L1 human

A point is in the $\varepsilon$-pseudospectrum when it is "almost an eigenvalue": some vector is mapped by $A - z$ to a vector shorter than $\varepsilon$.
These sets are drawn as contour plots of the resolvent norm in the complex plane.
</document>

<document source="https://tamnd.github.io/math2/node/7JN0ADA86C.md" tag="7JN0ADA86C" kind="definition">
# Smoothing spline

## Definition

Let $n \ge 2$, let $a \le x_1 < x_2 < \cdots < x_n \le b$ be real nodes, let $y_1, \dots, y_n$ be real numbers, and let $\lambda > 0$.
Let $W$ be the space of functions $f \colon [a, b] \to \R$ such that $f'$ is absolutely continuous and $f''$ is square integrable.
The *smoothing spline* for these data is the function $f \in W$ that minimizes
$$J_\lambda(f) = \sum_{i=1}^{n} \big(y_i - f(x_i)\big)^2 + \lambda \int_a^b f''(x)^2 \, dx.$$

## Remarks

The minimizer exists and is unique, and it is a natural [[cubic spline|V42H0R3EKX]] with knots $x_1, \dots, x_n$: a cubic on each $[x_i, x_{i+1}]$, of class $C^2$, and linear on $[a, x_1]$ and $[x_n, b]$.
Uniqueness uses $n \ge 2$: then a function with $f'' = 0$ that vanishes at all nodes is zero.
Example: if the data lie on a line, $y_i = \alpha + \beta x_i$, then $f(x) = \alpha + \beta x$ gives $J_\lambda(f) = 0$, so it is the smoothing spline for every $\lambda$.
Non-example: take $n \ge 3$ data not on a line, and let $g$ be the natural cubic spline through them.
Its [[second derivative|0Y9D2FJRQ4]] penalty is positive, and moving $g$ slightly toward a line lowers $J_\lambda$, so $g$ is not the minimizer.
As $\lambda \to 0$ the smoothing spline tends to the natural interpolating cubic spline, and as $\lambda \to \infty$ it tends to the [[least squares|0BYTQ4X1HC]] regression line.
In practice $\lambda$ is often chosen by generalized cross-validation.
</document>

<document source="https://tamnd.github.io/math2/node/7JQS6VKFEZ.md" tag="7JQS6VKFEZ" kind="concept">
# Curvature of a curve

## Summary

The curvature of a curve measures how fast its direction turns per unit of length travelled.
For a unit-speed curve it is the length of the acceleration vector.
A straight line has curvature $0$ and a circle of radius $r$ has curvature $1/r$.
Curvature, together with torsion, determines a space curve up to a rigid motion.

## Explanations

### L1 human

Drive a car along a road at constant speed and watch the steering wheel.
On a straight road the wheel is centred, and on a sharp bend it is turned far.
Curvature is the number that records how far the wheel is turned at each point of the road.
A circle of radius $2$ metres has curvature $1/2$ per metre at every point, and a larger circle bends more gently and has smaller curvature.
</document>

<document source="https://tamnd.github.io/math2/node/7JS0JFR89E.md" tag="7JS0JFR89E" kind="concept">
# Diffeology

## Summary

A diffeology on a set declares which maps from open subsets of Euclidean spaces into the set count as smooth; these maps are called plots.
A set with a diffeology is a diffeological space.
Every [[smooth manifold|F45VQ47GPV]] is a diffeological space, but so are quotients, subsets and spaces of smooth maps, which are often not manifolds.
Diffeological spaces form a [[cartesian closed category|BFBNQ36QS8]] with all limits and colimits, so constructions that leave the world of manifolds stay inside this one.

## Explanations

### L1 human

On a manifold, one knows which curves and surfaces inside it are smooth.
A diffeology takes this knowledge as the definition: a space is "smooth" once one says which parametrized families of points in it are smooth.
For instance, the quotient of the torus by a line of irrational slope has a useless topology, since its only open sets are the empty set and everything.
Yet its diffeology still remembers a lot of geometry.
This makes diffeology a good language for singular and infinite-dimensional spaces.
</document>

<document source="https://tamnd.github.io/math2/node/7JSQZ126BP.md" tag="7JSQZ126BP" kind="concept">
# Finite difference

## Summary

A finite difference of a function is the change of its value between two points at a fixed distance, such as $f(x + h) - f(x)$.
Divided by the step $h$, it becomes a [[difference quotient|AZBDB5F4M4]], which approximates the derivative when $h$ is small.
Finite differences are the basic operation of difference equations, of interpolation and of numerical methods for differential equations.

## Explanations

### L1 human

A derivative measures change over an infinitely small step; a finite difference measures change over a step of fixed size.
Think of reading a thermometer once every hour: the hourly changes are finite differences of the temperature.
For $f(x) = x^2$ and step $1$, the difference is $(x + 1)^2 - x^2 = 2x + 1$.
The three usual kinds look forward, backward, or symmetrically around the point.
</document>

<document source="https://tamnd.github.io/math2/node/7JVVQ2ZZY2.md" tag="7JVVQ2ZZY2" kind="definition">
# Local homeomorphism

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]] and let $f \colon X \to Y$ be a map.
The map $f$ is a *local homeomorphism* if every $x \in X$ has an [[open|02BH3MCBQB]] neighbourhood $U$ with the following two properties.
The set $f(U)$ is open in $Y$.
The restriction $f|_U \colon U \to f(U)$ is a [[homeomorphism|0QT00PB5G2]], where $U$ and $f(U)$ carry the [[subspace topology|0WS9P12FQR]].

## Remarks

A local homeomorphism is [[continuous|0VGFHTM75Y]] and [[open|0YCG648KDD]].
Every homeomorphism is a local homeomorphism, and so is the inclusion of an open subset.

Example: $p \colon \R \to S^1$, $p(t) = (\cos 2\pi t, \sin 2\pi t)$.
For $x \in \R$, take $U = (x - 1/2, x + 1/2)$.
Then $p(U)$ is the circle minus one point, which is open, and $p|_U$ is a continuous bijection onto it with a continuous inverse given by a branch of the angle.

Non-example: $f \colon \C \to \C$, $f(z) = z^2$.
Every neighbourhood $U$ of $0$ contains some $\varepsilon \ne 0$ and $-\varepsilon$, with $f(\varepsilon) = f(-\varepsilon)$, so $f|_U$ is not injective.
At every $z \ne 0$, however, $f$ is a local homeomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/7JZVT385PC.md" tag="7JZVT385PC" kind="object">
# Thue–Morse sequence

## Description

The *Thue–Morse sequence* is the binary [[sequence|0G9EKM0E15]] $(t_n)_{n \ge 0}$ given by $t_0 = 0$, $t_{2n} = t_n$ and $t_{2n+1} = 1 - t_n$.
Equivalently, $t_n$ is $0$ if the binary expansion of $n$ has an even number of digits $1$, and $1$ otherwise.
Its first sixteen terms are
$$0110\,1001\,1001\,0110.$$

## Remarks

Check: $t_5 = 0$ because $5 = 101_2$ has two digits $1$, and $t_7 = 1$ because $7 = 111_2$ has three.

The sequence is the fixed point, starting with $0$, of the morphism $\mu$ of words with $\mu(0) = 01$ and $\mu(1) = 10$.
So the first $2^{m+1}$ terms are the first $2^m$ terms followed by their complement, where $0$ and $1$ are swapped.

The sequence is overlap-free: it has no factor of the form $a x a x a$, where $a$ is a letter and $x$ is a word.
In particular it has no factor of the form $u u u$ with $u$ nonempty.
Thue proved overlap-freeness, and it makes the sequence a basic example in [[combinatorics on words|YJ961M051W]].
It is not periodic, and it is a $2$-automatic sequence, computed by a finite automaton that reads $n$ in base $2$.

## Explanations

### L1 human

Start with $0$, and at each stage append the complement of what you have.
This gives $0$, then $01$, then $0110$, then $01101001$, and so on.
The result never repeats any block three times in a row.
It is also used as a fair order of turns: two players who pick in the order $A B B A B A A B \ldots$ share the advantage of going first.
</document>

<document source="https://tamnd.github.io/math2/node/7KA78K00QJ.md" tag="7KA78K00QJ" kind="definition">
# Bessel–Clifford function

## Definition

Let $n \in \C$ and let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
The *Bessel–Clifford function* of order $n$ is $\mathcal{C}_n \colon \C \to \C$,
$$\mathcal{C}_n(z) = \sum_{k=0}^{\infty} \frac{1}{\Gamma(n + k + 1)}\, \frac{z^k}{k!},$$
where $\frac{1}{\Gamma}$ is the [[reciprocal gamma function|PY6K9AWR33]], equal to $0$ at the non-positive integers.

## Remarks

The series converges for every $z$, so $\mathcal{C}_n$ is entire in $z$.
It solves $z y'' + (n + 1) y' - y = 0$: with $a_k = \frac{1}{k!\, \Gamma(n + k + 1)}$, the coefficient of $z^k$ in $z y'' + (n+1) y'$ is $(k + 1)(n + k + 1)\, a_{k+1} = a_k$.
Example: $\mathcal{C}_{-1/2}(z) = \frac{\cosh(2\sqrt{z})}{\sqrt{\pi}}$, since $k!\, \Gamma\left( k + \frac{1}{2} \right) = \frac{(2k)!\sqrt{\pi}}{4^k}$.
For the [[Bessel function of the first kind|CZCMRDWDYW]] and the [[modified Bessel function|2ET068BZZH]], $J_n(x) = \left( \frac{x}{2} \right)^n \mathcal{C}_n\left( -\frac{x^2}{4} \right)$ and $I_n(x) = \left( \frac{x}{2} \right)^n \mathcal{C}_n\left( \frac{x^2}{4} \right)$.
A non-example: $J_0$ is not $\mathcal{C}_0$, since $J_0(x) = \mathcal{C}_0(-x^2/4)$ and, for instance, $\mathcal{C}_0(1) = \sum_k \frac{1}{(k!)^2} > 2 > 1 \ge \abs{J_0(1)}$.
In hypergeometric notation, $\mathcal{C}_n(z) = \frac{1}{\Gamma(n+1)}\, {}_0F_1(; n + 1; z)$ when $n$ is not a negative integer.
</document>

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# Wu's formula

## Statement

Let $M$ be a closed [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $[M] \in H_n(M; \Z/2\Z)$ be its mod $2$ [[fundamental class|ZEA0PEM3B2]].
For each $k \ge 0$, let $v_k \in H^k(M; \Z/2\Z)$ be the unique class such that
$$\langle v_k \smile x, [M] \rangle = \langle \operatorname{Sq}^k x, [M] \rangle \quad \text{for all } x \in H^{n-k}(M; \Z/2\Z),$$
where $\operatorname{Sq}^k$ is the $k$-th [[Steenrod square|VR7RHP41M0]] and $\smile$ is the [[cup product|JX47MS1R1H]].
Let $v = 1 + v_1 + v_2 + \cdots$ be the total Wu class.
Then the total [[Stiefel–Whitney class|JQCK7A210W]] of the tangent bundle of $M$ is
$$w(M) = \operatorname{Sq}(v), \quad \text{that is,} \quad w_k(M) = \sum_{i=0}^{k} \operatorname{Sq}^{i}(v_{k-i}).$$

## Remarks

The classes $v_k$ exist and are unique by Poincaré duality with $\Z/2\Z$ coefficients.
The proof uses the Thom class of the tangent bundle of $M$, viewed as the normal bundle of the diagonal in $M \times M$.
Steenrod squares of a Thom class give the Stiefel–Whitney classes of the bundle.

Example: for $M = \R P^2$, $H^*(M; \Z/2\Z) = (\Z/2\Z)[a]/(a^3)$.
Since $\operatorname{Sq}^1 a = a^2$, one gets $v_1 = a$, and $v_2 = 0$ because $\operatorname{Sq}^2$ vanishes on $H^0$.
So $w(\R P^2) = \operatorname{Sq}(1 + a) = 1 + a + a^2$, which agrees with $(1 + a)^3$.

Since $\operatorname{Sq}^k x = 0$ for $\deg x < k$, the class $v_k$ is $0$ for $k > n/2$.

## Explanations

### L1 human

Stiefel–Whitney classes are defined from the tangent bundle, which uses the smooth structure of the manifold.
Wu's formula computes them from the mod $2$ cohomology ring and its Steenrod squares alone.
So these classes depend only on the homotopy type of the manifold.

### L2 human

The formula has two steps: Poincaré duality turns the Steenrod squares into the Wu classes $v_k$, and then $w = \operatorname{Sq}(v)$.
It implies that Stiefel–Whitney numbers are homotopy invariants of closed manifolds, which is relevant for [[Thom's cobordism theorem|HM23BV561Z]] since these numbers decide unoriented cobordism.
For example, $w_1(M) = v_1$, so $M$ is orientable exactly when $\operatorname{Sq}^1 \colon H^{n-1}(M; \Z/2\Z) \to H^n(M; \Z/2\Z)$ is zero.
</document>

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# Lp space

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]], let $\mathbb{K}$ be $\R$ or $\C$, and let $1 \le p < \infty$.
Let $\mathcal{L}^p(\mu)$ be the set of [[measurable functions|YYNCJSR0PJ]] $f \colon X \to \mathbb{K}$ with $\int_X \abs{f}^p \, d\mu < \infty$, an [[integral of a non-negative function|DG76G5Z6XW]].
Call $f \sim g$ when $f = g$ [[almost everywhere|RARTMA0ZEC]].
The space $L^p(\mu)$ is the set of [[equivalence classes|082RMRSY26]] of $\mathcal{L}^p(\mu)$ under $\sim$, with pointwise operations on representatives.
Its [[norm|08NYZ2FS6B]] is
$$\norm{f}_p = \Big( \int_X \abs{f}^p \, d\mu \Big)^{1/p}.$$

## Remarks

Example: on $\R$ with Lebesgue measure, $f(x) = e^{-x^2}$ lies in $L^p$ for every $p$, since $\int e^{-p x^2} \, dx = \sqrt{\pi / p} < \infty$.

Non-example: $g(x) = 1 / (1 + \abs{x})$ on $\R$ is in $L^2$ but not in $L^1$.
Indeed $\int_\R (1 + \abs{x})^{-1} \, dx = \infty$, while $\int_\R (1 + \abs{x})^{-2} \, dx = 2$.

The operations and the norm do not depend on the representatives.
The map $\norm{\cdot}_p$ is a norm by the [[Minkowski inequality|N4QZS912MT]], and $\norm{f}_p = 0$ forces $f = 0$ almost everywhere.
With this norm $L^p(\mu)$ is a Banach space, by the [[Riesz–Fischer theorem|WVV4MFB2SE]].
For $p = \infty$ one uses the essential supremum instead, which gives the space [[L-infinity|ZT01G7BTSV]].
With the counting measure on $\N$, $L^p$ is the sequence space $\ell^p$.
</document>

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# Dirac comb

## Example

Let $T > 0$.
The *Dirac comb* of period $T$ is the sum of [[Dirac deltas|0KPHENZ5S7]] at the points $kT$, $k \in \Z$:
$$\operatorname{III}_T(\varphi) = \sum_{k \in \Z} \varphi(kT) \quad \text{for } \varphi \in \mathcal{S}(\R),$$
where $\mathcal{S}(\R)$ is the [[Schwartz space|CG9JEATW3Q]].
It is a [[tempered distribution|AES2T5JPFP]].
Indeed, $\abs{\varphi(kT)} \le (1 + k^2 T^2)^{-1} \sup_x (1 + x^2) \abs{\varphi(x)}$, so
$$\abs{\operatorname{III}_T(\varphi)} \le \Big( \sum_{k \in \Z} \frac{1}{1 + k^2 T^2} \Big) \big( p_{0,0}(\varphi) + p_{2,0}(\varphi) \big),$$
where $p_{a,b}(\varphi) = \sup_x \abs{x^a \varphi^{(b)}(x)}$.
This is a bound by finitely many Schwartz seminorms.

## Remarks

Its [[Fourier transform|5PVSQXCTCT]], with the convention $\hat\varphi(\xi) = \int \varphi(x) e^{-2\pi i x \xi}\, dx$, is again a Dirac comb:
$$\widehat{\operatorname{III}_T} = \frac{1}{T} \operatorname{III}_{1/T}.$$
This identity is a restatement of the [[Poisson summation formula|GARGJYSGFG]] $\sum_k \varphi(kT) = \frac1T \sum_k \hat\varphi(k/T)$, applied to $\hat\varphi$ and using $\hat{\hat\varphi}(x) = \varphi(-x)$.
For $T = 1$ the comb is its own Fourier transform.

Multiplying a function by $\operatorname{III}_T$ models sampling at spacing $T$.
By the identity above, sampling makes the spectrum periodic with period $1/T$, which is the mechanism behind aliasing and the sampling theorem.

## Explanations

### L1 human

The Dirac comb is an infinite row of unit spikes placed at equal distance $T$.
Its spectrum is again a row of spikes, at distance $1/T$.
Spikes close together in time give spikes far apart in frequency, and the reverse.
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# Regularity structures

## Definition

A *regularity structure* is a triple $\mathscr{T} = (A, T, G)$ such that:

1. $A \subset \R$ is an index set that is bounded below, locally finite, and contains $0$.
2. $T = \bigoplus_{\alpha \in A} T_\alpha$ is a graded vector space, each $T_\alpha$ is a [[Banach space|06GZ956657]], and $T_0 \cong \R$ is spanned by an element written $\mathbf{1}$.
3. $G$ is a [[group|G5S1M9]] of continuous [[linear maps|0NRYHJPQG2]] $T \to T$ such that for all $\Gamma \in G$, $\alpha \in A$ and $\tau \in T_\alpha$,
$$\Gamma \tau - \tau \in \bigoplus_{\beta \in A, \ \beta < \alpha} T_\beta,$$
and $\Gamma \mathbf{1} = \mathbf{1}$.

The set $A$ is the set of homogeneities, $T$ is the model space, and $G$ is the structure group.

## Remarks

Example: the polynomial regularity structure on $\R^d$.
Take $A = \N$, let $T_k$ be the space of homogeneous polynomials of degree $k$ in abstract variables $X = (X_1, \ldots, X_d)$, and let $G = \set{\Gamma_h : h \in \R^d}$ with $(\Gamma_h P)(X) = P(X + h)$.
The check: $A$ contains $0$ and is locally finite, $T_0$ is spanned by the constant $\mathbf{1}$, $\Gamma_h \Gamma_{h'} = \Gamma_{h + h'}$, and $P(X + h) - P(X)$ has only terms of degree less than $\deg P$.
Non-example: adding the scaling $P(X) \mapsto P(2X)$ to $G$ breaks clause 3, since it sends $X_1$ to $2X_1$ and $2X_1 - X_1 = X_1$ has degree $1$, not less than $1$.
To use a structure one adds a model, which realises each symbol as a concrete distribution near each point.
The reconstruction theorem then turns local expansions into a global distribution.
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# Simple algebra

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.
Let $\Delta_A = \set{(a, a) : a \in A}$ and $\nabla_A = A \times A$.
The algebra $\mathbf{A}$ is *simple* when $A$ has at least two elements and its only [[congruences|DCSXXT4SER]] are $\Delta_A$ and $\nabla_A$.

## Remarks

Equivalently, the [[congruence lattice|REE07CSXHR]] of $\mathbf{A}$ is the two-element chain.
By the correspondence theorem, a quotient $\mathbf{A}/\theta$ is simple exactly when $\theta$ is a maximal element of $\operatorname{Con} \mathbf{A} \setminus \set{\nabla_A}$.

Example: the group $\Z/p\Z$ with $p$ prime is simple.
Its congruences come from its subgroups, and by Lagrange's theorem these are only $\set{0}$ and $\Z/p\Z$.

Non-example: the ring $\Z$ is not simple.
Congruence modulo $2$ is a congruence, and it is neither $\Delta_\Z$, since $(0, 2)$ belongs to it, nor $\nabla_\Z$, since $(0, 1)$ does not.

A field, viewed as a ring, is simple, since its only ideals are $\set{0}$ and the whole field.
The one-element algebra is excluded, because there $\Delta_A = \nabla_A$.
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# Bifurcation from a characteristic value

## Definition

Let $X$ be a real [[Banach space|06GZ956657]], let $K \colon X \to X$ be a [[compact linear operator|M2NEP1K5AE]], and let $R \colon X \to X$ be a continuous map with $\norm{R(\varphi)} / \norm{\varphi} \to 0$ as $\varphi \to 0$.
Consider the equation
$$\varphi = \lambda\bigl(K \varphi + R(\varphi)\bigr), \quad \lambda \in \R,\ \varphi \in X,$$
which has the trivial solution $\varphi = 0$ for every $\lambda$.
A number $\lambda_0$ is a *bifurcation point* of the equation if every neighbourhood of $(\lambda_0, 0)$ in $\R \times X$ contains a solution $(\lambda, \varphi)$ with $\varphi \ne 0$.
One then says that nonzero solutions *bifurcate from* $\lambda_0$.

## Remarks

Every bifurcation point is a [[characteristic value|DC7HMHVREP]] of $K$: a number $\lambda_0$ with $\varphi = \lambda_0 K \varphi$ for some $\varphi \ne 0$.
Indeed, if $\lambda_0$ is not one, then $I - \lambda K$ is invertible with uniformly bounded inverse for $\lambda$ near $\lambda_0$, and $\norm{\varphi} \le C \abs{\lambda} \norm{R(\varphi)}$ forces $\varphi = 0$ for small $\varphi$.
Krasnosel'skii's theorem gives a converse: if $R$ is also compact, every characteristic value of $K$ of odd [[algebraic multiplicity|0FY42N02DV]] is a bifurcation point.

Example: $X = \R$, $K = 1$ and $R(\varphi) = -\varphi^3$.
The characteristic value is $\lambda_0 = 1$, and for $\lambda > 1$ the nonzero solutions are $\varphi = \pm\sqrt{1 - 1/\lambda}$, which tend to $0$ as $\lambda \to 1$; so $1$ is a bifurcation point, a [[pitchfork bifurcation|P88X34MSVV]].

Non-example: $X = \R^2$, $K = I$ and $R(x, y) = (-y^3, x^3)$.
The characteristic value $1$ has multiplicity $2$, which is even.
The equations are $(1 - \lambda) x + \lambda y^3 = 0$ and $(1 - \lambda) y - \lambda x^3 = 0$; multiplying the first by $y$, the second by $x$ and subtracting gives $\lambda(x^4 + y^4) = 0$.
So for $\lambda$ near $1$ the only solution is $x = y = 0$, and $1$ is not a bifurcation point.
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# von Neumann's inequality

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $T \colon H \to H$ be a bounded linear operator with $\norm{T} \le 1$.
Then for every [[polynomial|1G0B73WPX8]] $p$ with complex coefficients,
$$\norm{p(T)} \le \sup \set{\abs{p(z)} : z \in \C, \ \abs{z} \le 1}.$$

## Remarks

Proof idea: by [[Sz.-Nagy's dilation theorem|YB7Q3EK0ND]] there is a unitary $U$ on a larger space $K \supseteq H$ with $T^n = P_H U^n|_H$ for $n \ge 0$, so $p(T) = P_H \, p(U)|_H$.
Hence $\norm{p(T)} \le \norm{p(U)}$.
Since $U$ is unitary, its spectrum lies on the unit circle.
So $\norm{p(U)}$ is the maximum of $\abs{p}$ on the spectrum, which is at most the maximum of $\abs{p}$ on the closed unit disc.

Example: for the nilpotent contraction $T = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $p(z) = 1 + z$, one has $p(T) = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ with norm $\tfrac{1 + \sqrt 5}{2} \approx 1.618$, which is at most $\max_{\abs{z} \le 1} \abs{1 + z} = 2$.

The Hilbert space structure matters.
On a Banach space a contraction need not satisfy the inequality.
The inequality extends to two commuting contractions (Andô's theorem) but fails in general for three commuting contractions.

## Explanations

### L1 human

If an operator never stretches vectors, then a polynomial in the operator is no bigger than the same polynomial is on the unit disc of the complex plane.
So a contraction behaves, for polynomial expressions, no worse than a complex number of modulus at most $1$.

### L2 human

The inequality lets one define $f(T)$ for every $f$ in the disc algebra, the uniform closure of the polynomials on the closed disc, with $\norm{f(T)} \le \norm{f}_\infty$.
This is the start of the holomorphic functional calculus of contractions developed by Sz.-Nagy and Foias.
Its failure for three commuting contractions shows that multivariable operator theory differs from the one-variable theory.
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# Crystallographic group

## Summary

A crystallographic group is a discrete group of isometries of [[Euclidean space|0XPR1E85W0]] $\R^n$ whose quotient space is compact.
These groups describe the symmetries of periodic patterns, such as crystals in dimension $3$ and wallpapers in dimension $2$.
By the [[Bieberbach theorems|YE87R5CDJY]], each one contains a lattice of translations of finite index.
In each dimension there are only finitely many of them up to isomorphism.
In dimension $3$ they are the space groups of crystallography, and in dimension $2$ the [[wallpaper groups|MY3CBHARYQ]].

## Explanations

### L1 human

Imagine a pattern that fills all of space and repeats in every direction, such as the atoms of a salt crystal.
The rigid motions that carry the pattern onto itself form a crystallographic group.
The translations that shift the pattern by one period are always among them.
For the square grid of integer points in the plane, the group contains the shifts by integer vectors and the quarter turns about the grid points.
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# Level-set method

## Summary

The level-set method tracks a moving curve or surface as the zero [[level set|0B2RZ0778N]] of a function $\phi(x, t)$ defined on a fixed grid.
The motion of the interface becomes a [[Hamilton–Jacobi equation|5TRXQKP7DY]] for $\phi$, which is solved with finite difference schemes.
It was introduced by Stanley Osher and James Sethian.
Its main strength is that merging and splitting of interfaces need no special treatment, and normals and curvature are computed from derivatives of $\phi$.
See the [[definition|KT0380H39M]].

## Explanations

### L1 human

Think of a landscape whose shoreline is the interface you want to follow.
Instead of moving points along the shoreline, you raise or lower the whole landscape, and the shoreline moves by itself.
If two islands grow and touch, the shoreline simply changes its shape, with no bookkeeping.
For a circle of radius $r_0$ that grows at unit speed, the function $\phi(x, t) = \abs{x} - (r_0 + t)$ works: its zero set at time $t$ is the circle of radius $r_0 + t$.
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# KdV equation as an integrable system

## Example

The [[Korteweg–De Vries equation|KTT75QC72Z]], written here as
$$u_t + u_{xxx} + 6 u u_x = 0$$
for a real function $u(x, t)$ that decays fast as $\abs{x} \to \infty$, is an infinite-dimensional [[integrable system|N0N7CMRCMH]].

It is Hamiltonian: with
$$\mathcal{H}(u) = \int_{\R} \left( \tfrac{1}{2} u_x^2 - u^3 \right) dx,$$
the variational derivative is $\delta \mathcal{H} / \delta u = -u_{xx} - 3 u^2$, and the equation reads $u_t = \partial_x \left( \delta \mathcal{H} / \delta u \right)$.

It has infinitely many conserved quantities.
The first ones are $\int u \, dx$, $\int u^2 \, dx$ and $\int \left( u^3 - \tfrac{1}{2} u_x^2 \right) dx$.
For the third one, integration by parts gives $\frac{d}{dt} \int u^3 \, dx = -3 \int u_x^3 \, dx$ and $\frac{d}{dt} \int \tfrac{1}{2} u_x^2 \, dx = -3 \int u_x^3 \, dx$, so the difference is constant.

It has a [[Lax pair|G3A4S7CSB3]]: with the [[Schrödinger operator|W6QS1ME39J]] $L = -\partial_x^2 - u$, there is a third-order differential operator $B$ such that the equation is equivalent to $L_t = [B, L]$.
Hence the spectrum of $L$ does not change in time.

## Remarks

Gardner, Greene, Kruskal and Miura solved the equation in 1967 by the inverse scattering transform.
The scattering data of $L$ evolve by explicit linear equations, and $u$ is recovered from them.
Each eigenvalue of $L$ corresponds to a [[soliton|63D4AXYCT4]], and a general decaying solution splits for large time into solitons and a dispersing tail.
Zakharov and Faddeev showed in 1971 that the scattering data give action-angle variables, in analogy with the [[Liouville–Arnold theorem|PNQCW85HTJ]].

## Explanations

### L1 human

The KdV equation describes long waves in shallow water.
It behaves like a mechanical system with infinitely many degrees of freedom, and it has infinitely many conserved quantities.
That is why it can be solved exactly, and why its solitary waves pass through each other and keep their shape.
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# Bernstein–Sato polynomial

## Summary

The Bernstein–Sato polynomial, or $b$-function, of a polynomial $f$ is a monic polynomial $b(s)$, see the [[definition|R2AV47AX0Y]].
It is the one of least degree for which a differential operator turns $f^{s+1}$ into $b(s) f^s$.
Joseph Bernstein proved its existence for polynomials.
Mikio Sato introduced it independently in his theory of prehomogeneous vector spaces.
Masaki Kashiwara proved that its roots are negative rational numbers.
It is a fine invariant of the singularities of $f^{-1}(0)$.
Its roots are linked with the eigenvalues of the [[monodromy|DQP63REKEK]] and with the log canonical threshold.
It also gives the meromorphic continuation of the distribution $\abs{f}^s$ in $s$.

## Explanations

### L1 human

For $f(x) = x$, differentiating $x^{s+1}$ gives $(s+1) x^s$.
So one derivative turns $x^{s+1}$ into $x^s$ times the polynomial $s + 1$.
Bernstein showed that for every polynomial $f$ in any number of variables there is a differential operator doing the same job, up to a factor $b(s)$.
The best possible factor is the Bernstein–Sato polynomial, and its roots record how singular $f$ is.
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# Method of lines

## Definition

Consider an evolution equation $u_t = \mathcal{A}(u)$ for an unknown $u(x, t)$, $x \in \Omega \subseteq \R^m$, $t \ge 0$.
Here $\mathcal{A}$ is a differential operator in the space variable $x$.
Boundary conditions and an initial value $u(\cdot, 0) = u_0$ are given.
The *method of lines* computes an approximate solution in two steps.

1. Semi-discretisation: choose a spatial discretisation with $N$ unknowns, such as [[finite differences|X7HYEBA5SA]] on a grid. Choose an operator $A_N \colon \R^N \to \R^N$ that approximates $\mathcal{A}$ with the boundary conditions built in. This gives the system of ordinary differential equations $U'(t) = A_N(U(t))$, with $U(0)$ an approximation of $u_0$.
2. Time integration: solve this [[initial value problem|0FXDW992DS]] with a [[numerical method for ordinary differential equations|558XADG64M]].

## Remarks

Example: for the [[heat equation|VP2HN86VAG]] $u_t = u_{xx}$ on $(0, 1)$ with zero boundary values and grid $x_j = j \Delta x$, $\Delta x = 1/(N + 1)$, step 1 gives $U_j' = (U_{j+1} - 2U_j + U_{j-1}) / \Delta x^2$ for $1 \le j \le N$, with $U_0 = U_{N+1} = 0$.
Step 2 with the [[Euler method|0JARTXX4XP]] and step size $\Delta t$ gives the explicit scheme $U_j^{n+1} = U_j^n + r (U_{j+1}^n - 2U_j^n + U_{j-1}^n)$ with $r = \Delta t / \Delta x^2$.
The eigenvalues of the semi-discrete matrix lie in $(-4/\Delta x^2, 0)$, and the Euler method is stable for $\Delta t \lambda \in [-2, 0]$, so the scheme is stable when $r \le 1/2$.
Non-example: Rothe's method discretises time first, which gives a sequence of elliptic boundary value problems in $x$; it fails clause 1, since no system of ODEs is formed.
The semi-discrete systems are often stiff, because their eigenvalues grow like $\Delta x^{-2}$ for diffusion, so implicit or stabilised time integrators are used.

## Explanations

### L1 human

Discretise space, keep time continuous, and hand the resulting ODE system to an ODE solver.
The two choices, in space and in time, can then be made and analysed separately.
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# Omitting types theorem

## Statement

Let $L$ be a countable signature, let $T$ be a satisfiable $L$-[[theory|92SWXST1ZM]], and let $p(x_1, \dots, x_n)$ be an $n$-[[type|9V71Q3SD1G]] of $T$.
Call $p$ isolated if there is an $L$-formula $\varphi(x_1, \dots, x_n)$ such that $T \cup \set{\varphi}$ is satisfiable and $T \models \forall \bar{x} \, (\varphi(\bar{x}) \to \psi(\bar{x}))$ for every $\psi \in p$.
If $p$ is not isolated, then $T$ has a [[countable|01TQB88H4C]] [[model|MVCH6N9SNW]] that omits $p$.

## Remarks

The proof is a Henkin construction.
One adds countably many new constants, lists all tuples of constants, and builds a complete theory with witnesses step by step.
At the step for a tuple $\bar{c}$, one adds the negation of some formula of $p$ at $\bar{c}$.
This is possible because no formula consistent with what has been built so far implies all of $p$.
The canonical model of the final theory is countable and no tuple in it realizes $p$.
The same argument omits countably many non-isolated types at once.

Example: let $T$ be the theory of $(\N, 0, 1, +, \cdot, <)$ and $p(x) = \set{x \ne 0, x \ne 1, x \ne 1 + 1, \dots}$.
The type $p$ is not isolated, and $\N$ is a countable model of $T$ that omits it.

If $T$ is complete, an isolated type is realized in every model of $T$, so the hypothesis cannot be dropped.
Countability of $L$ is also needed.
Take constants $c_\alpha$ for $\alpha < \omega_1$ and $d_k$ for $k \in \N$, let $T$ say that the $c_\alpha$ are pairwise distinct, and let $p(x) = \set{x \ne d_k : k \in \N}$.
Then $p$ is not isolated, but every model of $T$ is uncountable and so has an element different from all $d_k$.

## Explanations

### L1 human

A type is a wish list of properties for an element.
Some wish lists are forced: a single property, consistent with the theory, already implies the whole list.
For a complete theory, every model then has an element that meets the whole list.
The theorem says that, in a countable language, every list that is not forced in this way can be avoided by some countable model.

### L2 human

The omitting types theorem is the main tool for building countable models with prescribed properties.
It shows that a complete theory in a countable language has an atomic model, that is, a prime model, when the isolated types are dense.
It is used in the proof of the Ryll-Nardzewski theorem and in the study of the number of countable models, the subject of the Vaught conjecture.
It is a counterpart of the compactness theorem: compactness produces models that realize types, and this theorem produces models that omit them.
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# Morse homology

## Definition

Let $M$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary, let $f \colon M \to \R$ be a [[Morse function|KPQ96640ZJ]], and let $g$ be a [[Riemannian metric|Q61VFHCH2J]] on $M$.
Let $p$ be a critical point, and consider the flow lines of the vector field $-\nabla f$.
The *unstable manifold* $W^u(p)$ is the set of points whose flow line tends to $p$ as time goes to $-\infty$.
The *stable manifold* $W^s(p)$ is the set of points whose flow line tends to $p$ as time goes to $+\infty$.
Assume that the pair $(f, g)$ is *Morse–Smale*: $W^u(p)$ and $W^s(q)$ are [[transverse|A2PDZ8GA9C]] for all critical points $p, q$.
For $k \ge 0$, let $C_k$ be the $\Z/2$-vector space with basis the critical points of $f$ of index $k$.
Let $p$ be a critical point of index $k$ and $q$ one of index $k - 1$.
Let $n(p, q) \in \Z/2$ be the number, modulo $2$, of flow lines of $-\nabla f$ from $p$ to $q$, counted up to time shift; it is finite.
Let $\partial \colon C_k \to C_{k-1}$ be the linear map with $\partial p = \sum_q n(p, q) \, q$.
Then $\partial \circ \partial = 0$, and the *Morse homology* of $(f, g)$ is the homology of the [[chain complex|WQ82C0M3QB]] $(C_*, \partial)$.

## Remarks

The Morse homology is isomorphic to $H_*(M; \Z/2)$, so it does not depend on $f$ or $g$.
With a choice of orientations of the unstable manifolds, one can count flow lines with signs and get a chain complex over $\Z$ whose homology is $H_*(M; \Z)$.
The identity $\partial^2 = 0$ comes from broken flow lines $p \to q \to r$ with index drop $2$.
They are exactly the ends of the one-dimensional spaces of flow lines from $p$ to $r$, and these ends come in pairs.
Example: the height function on the round sphere $S^n$, $n \ge 1$, has one critical point of index $0$ and one of index $n$.
For $n \ge 2$ the indices differ by more than one, so $\partial = 0$.
For $n = 1$ there are two flow lines from the maximum to the minimum, so again $\partial = 0$ modulo $2$.
In both cases the homology is $\Z/2$ in degrees $0$ and $n$.
Non-example: take a metric on the upright torus in which the flow line leaving the upper saddle runs straight into the lower saddle.
Then $W^u$ of the upper saddle and $W^s$ of the lower saddle meet along a curve, so they are not transverse.
The pair is not Morse–Smale and the definition does not apply; a small change of the metric removes the problem.
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# Ideal norm

## Definition

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\mathcal{O}_K$, and let $\mathfrak{a}$ be a nonzero [[ideal|096Z9MP2ME]] of $\mathcal{O}_K$.
The *norm* of $\mathfrak{a}$ is the [[cardinality|02CK6QMXYD]] of the [[quotient ring|0076HWHGHQ]]:
$$N(\mathfrak{a}) = \#(\mathcal{O}_K / \mathfrak{a}).$$
This number is finite.

## Remarks

Example: in $\Z[i]$, the ideal $(1 + i)$ has the classes of $0$ and $1$, so $N((1 + i)) = 2$, which equals the [[norm of the Gaussian integer|0HYRJF4WNQ]] $1 + i$, namely $1^2 + 1^2$.
Non-example: the zero ideal is excluded, since $\mathcal{O}_K / (0) \cong \mathcal{O}_K$ is infinite.
The norm is multiplicative: $N(\mathfrak{a}\mathfrak{b}) = N(\mathfrak{a}) N(\mathfrak{b})$.
For a nonzero $\alpha \in \mathcal{O}_K$, $N(\alpha \mathcal{O}_K) = \abs{N_{K/\Q}(\alpha)}$, the absolute value of the field norm.
By multiplicativity the norm extends to a group homomorphism from the nonzero [[fractional ideals|YB4JTB4WJN]] of $\mathcal{O}_K$ to $\Q_{>0}^{\times}$.
For a nonzero prime ideal $\mathfrak{p}$ lying over the prime $p$, $N(\mathfrak{p}) = p^f$, where $f$ is the residue degree.
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# Braided monoidal category

## Definition

Let $(\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)$ be a [[monoidal category|M3MSQ3NKAE]].
A *braiding* on $\mathcal{C}$ is a natural isomorphism $\gamma$ with components $\gamma_{A,B} \colon A \otimes B \to B \otimes A$, natural in $A$ and $B$.
It must satisfy the two hexagon identities for all objects $A, B, C$:

1. $\alpha_{B,C,A} \circ \gamma_{A, B \otimes C} \circ \alpha_{A,B,C} = (\id_B \otimes \gamma_{A,C}) \circ \alpha_{B,A,C} \circ (\gamma_{A,B} \otimes \id_C)$ as morphisms $(A \otimes B) \otimes C \to B \otimes (C \otimes A)$;
2. $\alpha^{-1}_{C,A,B} \circ \gamma_{A \otimes B, C} \circ \alpha^{-1}_{A,B,C} = (\gamma_{A,C} \otimes \id_B) \circ \alpha^{-1}_{A,C,B} \circ (\id_A \otimes \gamma_{B,C})$ as morphisms $A \otimes (B \otimes C) \to (C \otimes A) \otimes B$.

A *braided monoidal category* is a monoidal category with a braiding.

## Remarks

Example: $\mathbf{Vect}_k$ with the usual tensor product and $\gamma_{V,W}(v \otimes w) = w \otimes v$.
On $(u \otimes v) \otimes w$ both sides of the first hexagon give $v \otimes (w \otimes u)$, and both sides of the second hexagon send $u \otimes (v \otimes w)$ to $(w \otimes u) \otimes v$.
Non-example: on $\mathbf{Vect}_k$ with $\operatorname{char} k \ne 2$, the natural isomorphism $\gamma_{V,W}(v \otimes w) = -\, w \otimes v$ is not a braiding.
In each hexagon the left side contains one copy of $\gamma$ and the right side two, so for nonzero spaces the two sides differ by the sign $-1$.
The first hexagon says that moving $A$ past $B \otimes C$ in one step equals moving it past $B$ and then past $C$.
The second says the same for moving $C$ past $A \otimes B$.
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# Quadratic functional equation

## Summary

The quadratic functional equation is the [[functional equation|SRAN4EKWS0]] $f(x + y) + f(x - y) = 2 f(x) + 2 f(y)$.
The unknown $f$ maps an abelian group, usually a real vector space, to the real numbers.
Every [[quadratic form|0RBQS1JJCR]] solves it, and on $\R$ the continuous solutions are the functions $f(x) = cx^2$.
It is the functional form of the [[parallelogram law|026HPGYGS4]], and it lies behind the [[Jordan–von Neumann theorem|46WGVM5VS6]].

## Explanations

### L1 human

For $f(x) = x^2$ the equation is the identity $(x + y)^2 + (x - y)^2 = 2x^2 + 2y^2$.
For the squared length of vectors it is the parallelogram law: the squares of the two diagonals add up to the squares of the four sides.
The equation asks which functions behave like a square in this sense.
Its solutions are exactly the functions of the form $f(x) = B(x, x)$ for a symmetric biadditive map $B$.
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# Rectifiable set

## Summary

An $m$-rectifiable set is a subset of $\R^n$ covered by countably many Lipschitz images of $\R^m$.
The cover may miss a set of $m$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] zero.
Such sets are the measure-theoretic version of $m$-dimensional surfaces, and they may have corners, cusps and countably many pieces.
They are the basic objects of geometric measure theory, where they replace smooth submanifolds in problems such as the Plateau problem.
Every set of finite $m$-dimensional Hausdorff measure splits into a rectifiable part and a purely unrectifiable part, and the two parts behave very differently.

## Explanations

### L1 human

Think of a set that is built from countably many bent and crumpled copies of a flat $m$-dimensional piece.
It may look rough, but at almost every point it has an approximate tangent plane, much as a smooth surface does.
In the plane, a polygon, a circle, and a countable union of line segments are all $1$-rectifiable.
A filled square is not $1$-rectifiable, because countably many curves cover only a set of area zero.
Some fractal dust sets of dimension one, such as the four-corner Cantor set, are not rectifiable at all: they meet every curve in a set of length zero.
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# Best approximation

## Summary

A best approximation of a point $x$ from a set $A$ in a normed space is a point of $A$ nearest to $x$.
It is the optimal answer to an approximation problem.
Existence, uniqueness and characterization of best approximations are the first questions of approximation theory.

## Explanations

### L1 human

Think of a point in the plane and a line: the best approximation is the foot of the perpendicular, the point of the line nearest to the given point.
In approximation theory the "points" are functions and the set is, for example, all polynomials of degree at most $n$.
For $f(x) = x^2$ on $[0, 1]$ and the uniform distance, the best straight line is $x - 1/8$, which is never further than $1/8$ from the parabola.
A best approximation may fail to exist, for instance when the set has a hole where the nearest point should be.
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# Quillen–Suslin theorem

## Statement

Let $k$ be a field, let $n \ge 1$, and let $R = k[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]] in $n$ variables over $k$.
Then every [[finitely generated|05QY1VEBYD]] [[projective|BFFA03GE3V]] $R$-module is [[free|0DZ9DKAKZR]].

## Remarks

The theorem answers [[Serre's problem|7GG0XYHB0T]] and was proved independently by Daniel Quillen and Andrei Suslin in 1976.
Quillen's proof combines a local-global principle, which reduces the question to polynomial rings over local rings, with Horrocks' theorem on projective modules over $A[x]$ for a local ring $A$.
Suslin's proof works with unimodular rows and shows that they can be completed to invertible matrices.
Quillen's argument gives the result more generally for polynomial rings over a [[principal ideal domain|0CMD0BCH8T]].

Example: if $f, g \in R$ satisfy $af + bg = 1$ for some $a, b \in R$, the kernel $P$ of $R^2 \to R$, $(u, v) \mapsto uf + vg$, is projective, because the map splits.
The theorem says that $P$ is free; here one can see it directly, since $P$ is generated by $(-g, f)$.

The hypothesis on the ring cannot be dropped.
Over $A = \R[x, y, z]/(x^2 + y^2 + z^2 - 1)$, the kernel of $A^3 \to A$, $(u, v, w) \mapsto ux + vy + wz$, is projective but not free; this module corresponds to the tangent bundle of the $2$-sphere.

## Explanations

### L1 human

Projective modules are the algebraic version of vector bundles, families of vector spaces glued over a space.
Over affine space, a space without holes, one expects every such family to be trivial.
The theorem confirms this: over a polynomial ring over a field, every finitely generated projective module has a basis.
It took about twenty years from the question to the answer.

### L2 human

The theorem says that algebraic vector bundles on affine space $\mathbb{A}^n_k$ are trivial, the algebraic analogue of the contractibility of $\R^n$.
It gives $K_0(k[x_1, \ldots, x_n]) \cong \Z$ in a strong form: every class is represented by a free module.
In matrix terms, every unimodular row over $k[x_1, \ldots, x_n]$ is the first row of an invertible matrix.
The local-global principle in Quillen's proof is itself a widely used tool in the study of projective modules.
</document>

<document source="https://tamnd.github.io/math2/node/7N5NDD77MS.md" tag="7N5NDD77MS" kind="definition">
# Near-unanimity term

## Definition

Let $\mathcal{V}$ be a [[variety of algebras|AJNZJD8442]] and let $n \ge 3$ be an integer.
A *near-unanimity term* of arity $n$ for $\mathcal{V}$ is a [[term|N9M2N27END]] $t(x_1, \dots, x_n)$ such that $\mathcal{V}$ satisfies the [[identities|PKD440G2XW]]
$$t(y, x, x, \dots, x) \approx t(x, y, x, \dots, x) \approx \cdots \approx t(x, x, \dots, x, y) \approx x.$$
Here the $i$-th term in the chain has $y$ in position $i$ and $x$ in all other positions.
An algebra $A$ has a near-unanimity term when the variety it generates has one, that is, when the identities hold in $A$.

## Remarks

Example: in the variety of lattices, $t(x, y, z) = (x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$ is a near-unanimity term of arity $3$.
Check: $t(y, x, x) = (y \wedge x) \vee x \vee (y \wedge x) = x$ by absorption, and the other two identities follow in the same way by symmetry of $t$.
A near-unanimity term of arity $3$ is called a [[majority term|9879ZMW5AA]].
Non-example: the variety of groups has no near-unanimity term of any arity.
The reason is that a variety with a near-unanimity term is congruence-distributive, while the group $\Z_2 \times \Z_2$ has a non-distributive lattice of normal subgroups.
If $t$ is a near-unanimity term of arity $n$, then $s(x_1, \dots, x_{n + 1}) = t(x_1, \dots, x_n)$ is one of arity $n + 1$.
The abbreviation NU term is common.
</document>

<document source="https://tamnd.github.io/math2/node/7N8E1BR67D.md" tag="7N8E1BR67D" kind="statement">
# Peano kernel theorem

## Statement

Let $a < b$ and $k \ge 0$ be given, and write $(t)_+^k$ for $t^k$ when $t \ge 0$ and $0$ when $t < 0$.
Let $\alpha_0, \dots, \alpha_k$ be piecewise continuous functions on $[a, b]$, and let $m_0, \dots, m_k \ge 0$ be integers.
For $0 \le j \le k$ and $1 \le i \le m_j$, let $\beta_{ij}$ be a real number and $x_{ij}$ a point of $[a, b]$.
Let $L$ be the linear functional on $C^{k+1}[a, b]$ given by
$$L(f) = \sum_{j=0}^{k} \Big( \int_a^b \alpha_j(x) f^{(j)}(x)\,dx + \sum_{i=1}^{m_j} \beta_{ij} f^{(j)}(x_{ij}) \Big).$$
Assume that $L(p) = 0$ for every polynomial $p$ of degree at most $k$.
Then for every $f \in C^{k+1}[a, b]$,
$$L(f) = \int_a^b K(\theta)\, f^{(k+1)}(\theta)\,d\theta, \qquad K(\theta) = \frac{1}{k!}\, L_x\big[(x - \theta)_+^k\big],$$
where $L_x$ means that $L$ acts on $(x - \theta)_+^k$ as a function of $x$ for fixed $\theta$.
The function $K$ is the *Peano kernel* of $L$.

## Remarks

Proof idea: write $f$ by [[Taylor's theorem with the integral remainder|0WJGR81HBD]] at $a$, as the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ plus $\frac{1}{k!} \int_a^b (x - \theta)_+^k f^{(k+1)}(\theta)\,d\theta$.
The functional kills the polynomial part.
The special form of $L$ allows $L$ to be moved inside the integral over $\theta$, which gives the formula.

If $K$ does not change sign on $[a, b]$, the mean value theorem for integrals gives $L(f) = \frac{f^{(k+1)}(\xi)}{(k+1)!}\, L\big(x^{k+1}\big)$ for some $\xi \in [a, b]$.
Also $\abs{L(f)} \le \norm{K}_{L^1} \, \norm{f^{(k+1)}}_\infty$ in all cases.

Example: the error of the [[trapezoidal rule|FH0538XCZR]], $L(f) = \int_a^b f - \frac{b - a}{2} \big( f(a) + f(b) \big)$, with $k = 1$.
Here $K(\theta) = \int_\theta^b (x - \theta)\,dx - \frac{b - a}{2} (b - \theta) = -\frac12 (\theta - a)(b - \theta)$, which is $\le 0$.
Since $\int_a^b K = -\frac{(b - a)^3}{12}$, one gets $L(f) = -\frac{(b - a)^3}{12} f''(\xi)$ for some $\xi$.

The hypothesis $L(p) = 0$ for degree at most $k$ is needed: for $L(f) = f(0)$ on $[0, 1]$ and $k = 0$, the formula would give $f(0) = \int_0^1 K f'$, which fails for $f = 1$.

## Explanations

### L1 human

Many approximation formulas, such as quadrature rules, are exact for all polynomials up to some degree.
The theorem says that their error is then an average of a high derivative of the function, with a fixed weight called the Peano kernel.
So the error is small when that derivative is small.
The kernel depends only on the formula, not on the function.

### L2 human

The theorem turns the question "how large is the error of this rule" into a computation with one explicit function $K$.
It is the standard way to derive the error terms of Newton–Cotes rules, of Gaussian rules and of interpolation and differentiation formulas.
When $K$ keeps one sign, the error has the familiar form "constant times $f^{(k+1)}(\xi)$", and the constant is found by testing the rule on $x^{k+1}$.
When $K$ changes sign, one still has the bound by $\norm{K}_{L^1} \norm{f^{(k+1)}}_\infty$, which is sharp over the class of functions with bounded $(k+1)$-st derivative.
The [[degree of exactness|ZYT1JNT1Y0]] of a rule fixes the largest $k$ for which the theorem applies.
</document>

<document source="https://tamnd.github.io/math2/node/7NA22G672W.md" tag="7NA22G672W" kind="concept">
# Semihereditary ring

## Summary

A left semihereditary ring is a [[ring|0B1261B5ED]] in which every finitely generated left ideal is a [[projective module|BFFA03GE3V]].
Every [[hereditary ring|89VBQH8HNS]] is semihereditary, and the two notions agree for left Noetherian rings.
Commutative semihereditary domains are exactly the [[Prüfer domains|2RDAA4PCK8]], and every [[von Neumann regular ring|DXX7GNY3HE]] is semihereditary.

## Explanations

### L1 human

Semihereditary rings ask for less than hereditary rings: only the finitely generated ideals need to be projective.
This weaker condition matters for rings that are not Noetherian, where some ideals need infinitely many generators.
The ring of all algebraic integers is a good example.
Every finitely generated ideal in it is principal, hence projective, so it is semihereditary.
It is not hereditary, because it has an ideal that is not finitely generated and not projective.
</document>

<document source="https://tamnd.github.io/math2/node/7NPY2CN5KB.md" tag="7NPY2CN5KB" kind="concept">
# Dedekind domain

## Summary

A Dedekind domain is an integral domain in which every nonzero proper ideal is, in a unique way, a product of prime ideals.
Rings of integers of [[number fields|MWBDFS2KJ5]] are Dedekind domains, and so are all [[principal ideal domains|0CMD0BCH8T]].
In such rings unique factorisation of elements may fail, but unique factorisation of ideals always holds.
The [[ideal class group|WQ46BD3ARY]] measures the gap between the two.

## Explanations

### L1 human

In the integers every number splits into primes in only one way.
In larger number rings this can fail: in $\Z[\sqrt{-5}]$, $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ are two different factorisations into irreducible elements.
Dedekind's repair was to factor ideals instead of numbers.
In a Dedekind domain every nonzero proper ideal splits into prime ideals in exactly one way, and the two factorisations of $6$ come from regrouping the same four prime ideals.
</document>

<document source="https://tamnd.github.io/math2/node/7P72CCAMGZ.md" tag="7P72CCAMGZ" kind="definition">
# Sturm–Liouville theory

## Definition

Let $a < b$ be real numbers.
Let $p \in C^1([a, b])$ and $q, w \in C([a, b])$ be real functions with $p(x) > 0$ and $w(x) > 0$ for all $x \in [a, b]$.
Let $\alpha_1, \alpha_2, \beta_1, \beta_2$ be real numbers with $(\alpha_1, \alpha_2) \ne (0, 0)$ and $(\beta_1, \beta_2) \ne (0, 0)$.
The regular Sturm–Liouville problem with these data is the [[boundary value problem|TM8VQJWGHZ]]
$$-(p y')' + q y = \lambda w y \quad \text{on } [a, b],$$
$$\alpha_1 y(a) + \alpha_2 y'(a) = 0, \qquad \beta_1 y(b) + \beta_2 y'(b) = 0,$$
where $\lambda$ is a complex parameter.
A number $\lambda$ is an eigenvalue of the problem if the problem has a solution $y \in C^2([a, b])$ that is not identically zero.
Such a $y$ is an eigenfunction for $\lambda$.
Sturm–Liouville theory is the study of the eigenvalues and eigenfunctions of such problems.
It also studies singular versions, where the interval is unbounded or the coefficients are singular at an end.

## Remarks

Example: $p = w = 1$, $q = 0$ on $[0, \pi]$ with $y(0) = y(\pi) = 0$.
Then $y = \sin(nx)$ satisfies $-y'' = n^2 y$ and both boundary conditions, so $n^2$ is an eigenvalue for each $n \ge 1$.
Non-example: the problem $-y'' = \lambda y$ with $y(0) = y(2\pi)$, $y'(0) = y'(2\pi)$ is not a regular Sturm–Liouville problem in this sense.
Its [[periodic boundary conditions|3FW4J270FV]] are not separated, and the eigenvalue $1$ then has the two eigenfunctions $\cos x$ and $\sin x$.
With the inner product $\langle f, g \rangle = \int_a^b f \overline{g}\, w \, dx$, the operator $y \mapsto \frac{1}{w}(-(p y')' + q y)$ with these boundary conditions is symmetric.
This is why the eigenvalues are real and eigenfunctions for distinct eigenvalues are orthogonal.

## Explanations

### L1 human

A Sturm–Liouville problem is a second-order linear equation with a free number $\lambda$ and one condition at each end.
For most values of $\lambda$ only the zero function fits; the special values where a nonzero solution fits are the eigenvalues.
</document>

<document source="https://tamnd.github.io/math2/node/7PCHXZM5YE.md" tag="7PCHXZM5YE" kind="definition">
# L-infinity space

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]], and let $\mathbb{K}$ be $\R$ or $\C$.
A [[measurable function|YYNCJSR0PJ]] $f \colon X \to \mathbb{K}$ is *essentially bounded* if there is $M \ge 0$ with $\mu(\set{x \in X : \abs{f(x)} > M}) = 0$.
Its norm is the [[essential supremum|WPC2XRMB9Y]]
$$\norm{f}_\infty = \inf \set{M \ge 0 : \mu(\set{x : \abs{f(x)} > M}) = 0}.$$
The space $L^\infty(\mu)$ is the set of essentially bounded measurable functions modulo equality [[almost everywhere|RARTMA0ZEC]], with the norm $\norm{\cdot}_\infty$.

## Remarks

Example: on $[0, 1]$ with Lebesgue measure, let $f(x) = x$ for rational $x$ and $f(x) = 0$ otherwise.
The set where $f \ne 0$ is countable, so it has measure $0$, and $\norm{f}_\infty = 0$.
So $f$ is the zero element of $L^\infty$.

Non-example: $f(x) = 1/x$ on $(0, 1]$ is not essentially bounded.
For each $M > 0$, the set $\set{f > M}$ contains the interval $(0, \min(1, 1/M))$, which has positive measure.

The infimum in the definition is attained: $\abs{f} \le \norm{f}_\infty$ almost everywhere.
With the counting measure on a set $I$, $L^\infty$ is the space $\ell^\infty(I)$ of bounded families with the supremum norm.
</document>

<document source="https://tamnd.github.io/math2/node/7PDWB550C9.md" tag="7PDWB550C9" kind="concept">
# Irreducible ideal

## Summary

An irreducible ideal is a proper ideal that is not the intersection of two strictly larger ideals.
In a [[Noetherian ring|0670T59HGE]], every proper ideal is a finite intersection of irreducible ideals, and every irreducible ideal is [[primary|TMTPWVETQK]].
This is the usual route to the existence part of the [[Lasker–Noether theorem|5H8Z56KXMS]].

## Explanations

### L1 human

An irreducible ideal cannot be split as the overlap of two bigger ideals.
In $\Z$, the ideal $(6)$ splits as $(2) \cap (3)$, so it is not irreducible.
The ideal $(8)$ cannot be split, because every strictly larger ideal is $(4)$, $(2)$ or $\Z$, and these form a chain whose intersections are never $(8)$.
</document>

<document source="https://tamnd.github.io/math2/node/7PEE48WBQF.md" tag="7PEE48WBQF" kind="concept">
# Combinatorial species

## Summary

A combinatorial species is a rule that assigns to each finite set the structures of a given kind on it, see the [[definition|K9VZKJACKT]].
The rule must be compatible with renaming the elements, which makes it a [[functor|28NQ0KRHD3]] on finite sets and bijections.
Each species has an [[exponential generating function|0JHXPVFKT0]], and operations on species match operations on these series.
The theory was introduced by André Joyal and gives a conceptual basis for the [[symbolic method|JP11D90EJC]] for labelled structures.

## Explanations

### L1 human

Think of a kind of structure, such as "a linear order", "a graph" or "a rooted tree", that you can put on any set of labels.
The species records, for each set of labels, all such structures on it.
Renaming the labels moves structures to structures, in a consistent way.
For example, the species of linear orders gives $n!$ structures on a set of $n$ labels, and the species of graphs gives $2^{\binom{n}{2}}$.
</document>

<document source="https://tamnd.github.io/math2/node/7PJ8KVZZCQ.md" tag="7PJ8KVZZCQ" kind="concept">
# Germ of a holomorphic function

## Summary

A germ of a holomorphic function at a point $a \in \C^n$ is a [[holomorphic function|507C34W0ZT]] defined near $a$.
Two such functions are identified when they agree on some neighbourhood of $a$.
It is the special case of a [[germ|VZXF7RNFKS]] for holomorphic functions.
The germs at $a$ form a ring $\mathcal{O}_{n,a}$, which is isomorphic to the [[ring of convergent power series|EVPZ7ZG1HB]] in $n$ variables.
Local analytic geometry studies this ring and its ideals.

## Explanations

### L1 human

A germ keeps only what a function does arbitrarily close to one point.
For holomorphic functions this is a lot of information.
By the identity theorem, the germ at $a$ determines the function on every connected open set containing $a$ where it is defined.
For example, $1/(1 - z)$ on $\C \setminus \set{1}$ and the series $\sum_{k \ge 0} z^k$ on the unit disc have the same germ at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/7PN328ANQA.md" tag="7PN328ANQA" kind="definition">
# Riesz kernel energy

## Definition

Let $d \ge 1$ be an integer and let $s > 0$.
The *Riesz $s$-kernel* on $\R^d$ is $k_s(x, y) = \abs{x - y}^{-s}$ for $x \ne y$, and $k_s(x, x) = +\infty$, where $\abs{x - y}$ is the [[Euclidean distance|0NPPGHQNXQ]].
Let $\mu$ be a finite positive [[Borel measure|DYR1QFNGNJ]] on $\R^d$ with compact support.
The *Riesz $s$-energy* of $\mu$ is
$$I_s(\mu) = \int \int \frac{d\mu(x)\, d\mu(y)}{\abs{x - y}^{s}} \in [0, +\infty].$$
The measure $\mu$ has *finite $s$-energy* if $I_s(\mu) < \infty$.
For distinct points $x_1, \dots, x_N \in \R^d$, the *discrete Riesz $s$-energy* is
$$E_s(x_1, \dots, x_N) = \sum_{i \ne j} \frac{1}{\abs{x_i - x_j}^{s}}.$$

## Remarks

The inner integral $\int \abs{x - y}^{-s}\, d\mu(y)$ is, up to a constant factor when $0 < s < d$, the [[Riesz potential|1V2N8MFSHC]] of $\mu$ at $x$.
So $I_s(\mu)$ is the integral of that potential against $\mu$.
Example: let $d = 3$, $s = 1$ and let $\sigma$ be the uniform probability measure on the unit sphere.
By the shell theorem, $\int \abs{x - y}^{-1}\, d\sigma(y) = 1$ for every $x$ on the sphere, so $I_1(\sigma) = 1$.
Non-example of finite energy: a point mass $\delta_a$ has $I_s(\delta_a) = k_s(a, a) = +\infty$ for every $s > 0$.
For $s \ge d$, every nonzero measure as above has infinite $s$-energy.
The Riesz $s$-capacity of a compact set $K$ is $1 / \inf I_s(\mu)$, the infimum over Borel probability measures $\mu$ on $K$.
</document>

<document source="https://tamnd.github.io/math2/node/7PN9MD1PQB.md" tag="7PN9MD1PQB" kind="concept">
# Whitney C-infinity topology

## Summary

The Whitney $C^\infty$ topology, also called the strong $C^\infty$ topology, is a topology on the set $C^\infty(M, N)$ of smooth maps between [[smooth manifolds|F45VQ47GPV]].
Two maps are close in it when their values and derivatives up to some order are close at every point.
The allowed error may shrink towards infinity.
It is the standard topology in which one says that a property of maps is generic, stable or open.
With it, $C^\infty(M, N)$ is a [[Baire space|0ZH5AJ2ED7]], which makes transversality arguments work.

## Explanations

### L1 human

Think of a smooth function on the real line and draw a thin tube around its graph.
The tube may get thinner and thinner far out, like the function $1/(1 + x^2)$.
In the Whitney topology, the functions whose graphs stay inside such a tube form a neighbourhood, and one can ask the same of the derivatives.
So the constant function $1/n$ is never close to $0$ in this topology, however large $n$ is: far out, the tube around $0$ is thinner than $1/n$.
This strict control at infinity is what makes many sets of nice maps open.
</document>

<document source="https://tamnd.github.io/math2/node/7PY20WMZDZ.md" tag="7PY20WMZDZ" kind="definition">
# Group C*-algebra

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] with left Haar measure $\mu$, and let $L^1(G)$ be its [[group algebra|6GD25S53ND]].
For a [[unitary representation|74BPYJNZTM]] $\pi$ of $G$ on a Hilbert space $H$ and $f \in L^1(G)$, let $\pi(f)$ be the bounded operator on $H$ given by
$$\ip{\pi(f)u}{v} = \int_G f(x) \ip{\pi(x)u}{v} \, d\mu(x) \quad (u, v \in H).$$
For $f \in L^1(G)$ put
$$\norm{f}_* = \sup_\pi \norm{\pi(f)},$$
where $\pi$ runs over all unitary representations of $G$.
The *group C\*-algebra* $C^*(G)$ is the completion of $L^1(G)$ for the norm $\norm{\cdot}_*$.
The *reduced group C\*-algebra* $C^*_r(G)$ is the closure of $\set{\lambda(f) : f \in L^1(G)}$ in the bounded operators on $L^2(G)$, where $\lambda$ is the left regular representation.

## Remarks

The supremum is finite, since $\norm{\pi(f)} \le \norm{f}_1$, and $\norm{\cdot}_*$ is a norm on $L^1(G)$, since the left regular representation is injective on $L^1(G)$.
Both $C^*(G)$ and $C^*_r(G)$ are C*-algebras.

Example: for $G = \Z$, the Fourier transform $f \mapsto \sum_n f(n) z^n$ identifies $C^*(\Z)$ and $C^*_r(\Z)$ with the algebra $C(\mathbb{T})$ of continuous functions on the circle with the supremum norm.

Non-example: $L^1(G)$ with its own norm $\norm{\cdot}_1$ is in general not a C\*-algebra.
In $\ell^1(\Z)$, let $f = \delta_0 + \delta_1 - \delta_2$.
Then $f^* * f = 3\delta_0 - \delta_2 - \delta_{-2}$, so $\norm{f^* * f}_1 = 5$, while $\norm{f}_1^2 = 9$, and the C\*-identity $\norm{f^* f} = \norm{f}^2$ fails.

There is always a surjective homomorphism $C^*(G) \to C^*_r(G)$; it is an isomorphism exactly when $G$ is amenable.
</document>

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# Fitting subgroup

## Definition

Let $G$ be a finite [[group|G5S1M9]].
The *Fitting subgroup* $F(G)$ is the subgroup [[generated|0RP7VWQ0YN]] by all [[normal subgroups|0CB0DX5BVV]] of $G$ that are [[nilpotent|0Q7MMQHC1P]].

## Remarks

Fitting's theorem says that if $M$ and $N$ are nilpotent normal subgroups of $G$, then $MN$ is a nilpotent normal subgroup.
Hence $F(G)$ is itself nilpotent and normal, and it contains every nilpotent normal subgroup of $G$: it is the unique largest one.
$F(G)$ is a [[characteristic subgroup|0KABPMRHM2]], and it is the product over the primes $p$ of the largest normal $p$-subgroups $O_p(G)$.
If $G$ is [[solvable|0FXJ7YFMMX]], then the [[centraliser|02YYV36BFM]] of $F(G)$ in $G$ is contained in $F(G)$.

Example: $G = S_4$.
Its normal subgroups are $\set{e}$, $V = \set{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)}$, $A_4$ and $S_4$.
The group $V$ is abelian, hence nilpotent, and it is the [[Klein four-group|0PMRQJVTME]].
The groups $A_4$ and $S_4$ are not nilpotent, because their [[centres|0DZ93493VN]] are trivial.
So $F(S_4) = V$.

Non-example: $A_4$ is a normal subgroup of $S_4$ that is not nilpotent, so it is not contained in the generating family, and it is not $F(S_4)$.

If $G$ is a nonabelian [[simple group|0YMYMNQ0QZ]], then $F(G) = \set{e}$.
If $G$ is nilpotent, then $F(G) = G$.
</document>

<document source="https://tamnd.github.io/math2/node/7Q60BB01YG.md" tag="7Q60BB01YG" kind="concept">
# Irreducible matrix

## Summary

A square matrix is irreducible when no simultaneous permutation of its rows and columns puts it into block upper triangular form.
For a nonnegative matrix this means that its directed graph is strongly connected: every index can reach every other index.
Irreducibility is the main hypothesis of the [[Perron–Frobenius theorem|62NS7PV1M9]] for nonnegative matrices that are not positive.

## Explanations

### L1 human

Read an $n \times n$ matrix as a network: there is an arrow from $i$ to $j$ when the entry $a_{ij}$ is not zero.
The matrix is irreducible when you can travel along arrows from any node to any other node.
Then the matrix does not split into a part that never feeds back into the rest.
For example, $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ has arrows $1 \to 2$ and $2 \to 1$, so it is irreducible.
The matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ has no arrow from $2$ to $1$, so it is reducible.
</document>

<document source="https://tamnd.github.io/math2/node/7Q6KK8WJ02.md" tag="7Q6KK8WJ02" kind="definition">
# Ore condition

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
Let $S \subseteq R$ be a [[multiplicative set|AN9SKEZ3VY]]: $1 \in S$ and $st \in S$ for all $s, t \in S$.
The set $S$ satisfies the *right Ore condition* if for all $a \in R$ and $s \in S$ there exist $b \in R$ and $t \in S$ with
$$a t = s b.$$
Equivalently, $aS \cap sR \ne \emptyset$ for all $a \in R$ and $s \in S$.
The *left Ore condition* asks instead for $b \in R$ and $t \in S$ with $t a = b s$.
A *right Ore domain* is a domain $R$ such that $R \setminus \set{0}$ satisfies the right Ore condition.

## Remarks

Example: if $R$ is commutative, every multiplicative set $S$ satisfies both Ore conditions: take $t = s$ and $b = a$.
Every right Noetherian domain is a right Ore domain.

Non-example: let $k$ be a field and $R = k\langle x, y \rangle$ the free algebra, with $S = R \setminus \set{0}$.
Each nonzero element of $xS$ is a combination of words beginning with $x$, and each nonzero element of $yR$ is a combination of words beginning with $y$.
So $xS \cap yR = \set{0}$, while $0 \notin xS$ since $R$ is a domain.
Hence $xS \cap yR = \emptyset$ and the right Ore condition fails for $a = x$ and $s = y$.

The right Ore condition says that a left fraction $s^{-1} a$ can be rewritten as a right fraction $b t^{-1}$, since $a t = s b$ gives $s^{-1} a = b t^{-1}$ once $s$ and $t$ are invertible.
A multiplicative set that satisfies the right Ore condition and is right reversible (if $s a = 0$ with $s \in S$, then $a t = 0$ for some $t \in S$) is a *right denominator set*.
If $S$ contains no [[zero divisors|0NNRDW10RZ]], reversibility is automatic.
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<document source="https://tamnd.github.io/math2/node/7Q9YSCM8YY.md" tag="7Q9YSCM8YY" kind="definition">
# Weak lower semicontinuity

## Definition

Let $X$ be a real [[Banach space|06GZ956657]] and let $J \colon X \to (-\infty, +\infty]$ be a [[functional|1EQM0DJ4NC]].
A sequence $(u_k)$ in $X$ converges *weakly* to $u$ if $\ell(u_k) \to \ell(u)$ for every continuous linear functional $\ell$ on $X$.
The functional $J$ is *sequentially weakly lower semicontinuous* if
$$J(u) \le \liminf_{k \to \infty} J(u_k)$$
for every $u \in X$ and every sequence $(u_k)$ that converges weakly to $u$.

## Remarks

Replacing sequences by the [[weak topology|P6KHVSTPZV]] itself gives weak lower [[semicontinuity|RN4G8SGP6P]] in the topological sense, which implies the sequential form.
In the calculus of variations the sequential form is the one that is used.

Example: the norm $J(u) = \norm{u}$ is weakly lower semicontinuous.
By the Hahn–Banach theorem $\norm{u} = \sup \abs{\ell(u)}$ over continuous linear $\ell$ with $\norm{\ell} \le 1$.
For each such $\ell$, $\abs{\ell(u)} = \lim_k \abs{\ell(u_k)} \le \liminf_k \norm{u_k}$.

Non-example: on $X = L^2(0, 2\pi)$ let $J(u) = -\norm{u}^2$ and $u_k(x) = \sin(kx)$.
Then $u_k \to 0$ weakly, by the Riemann–Lebesgue lemma, and $J(u_k) = -\pi$ for all $k$.
But $J(0) = 0 > -\pi$, so $J$ is not weakly lower semicontinuous, although it is continuous for the norm.

By Mazur's lemma, a convex functional that is lower semicontinuous for the norm is weakly lower semicontinuous.
</document>

<document source="https://tamnd.github.io/math2/node/7QERXM0C52.md" tag="7QERXM0C52" kind="concept">
# Vanishing moments of a wavelet

## Summary

A [[wavelet|4PZ35C0GV3]] $\psi$ has $p$ vanishing moments if it is orthogonal to all polynomials of degree less than $p$, that is, $\int x^k \psi(x) \, dx = 0$ for $k = 0, \dots, p - 1$.
Then the wavelet coefficients of a function that is smooth near a point are very small at fine scales near that point.
This is why wavelets with many vanishing moments compress smooth signals well.
The [[Daubechies wavelets|90V8S5CXE6]] have the shortest support among orthonormal wavelets with a given number of vanishing moments.

## Explanations

### L1 human

A wavelet with vanishing moments cannot see polynomials of low degree: its inner product with them is zero.
A smooth function looks like a polynomial on a small interval, by Taylor's formula.
So a narrow wavelet with many vanishing moments sees almost nothing of a smooth function, and its coefficient is close to zero.
Large coefficients appear only near jumps and other rough spots, which is what makes compression possible.
</document>

<document source="https://tamnd.github.io/math2/node/7QFKD5JXH6.md" tag="7QFKD5JXH6" kind="statement">
# Stone's theorem on paracompactness of metric spaces

## Statement

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] with the topology induced by $d$.
Then $X$ is [[paracompact|8R29G6NG9Q]].
That is, every [[open cover|049PXYHW1T]] of $X$ has an open refinement that is a [[locally finite family|DZ297MMGV8]].

## Remarks

The theorem is due to A. H. Stone.
A short proof by Mary Ellen Rudin well-orders the members of the cover.
For each $n$ and each member $U$, it takes the union of balls of radius $2^{-n}$ around suitable points whose first containing member is $U$.
The radii are chosen so that each point has a ball meeting only finitely many of the new sets.
The proof uses the axiom of choice through the well-ordering.

Example at work: every metric space is therefore [[normal|0NSR9V8TED]] and admits partitions of unity subordinate to any open cover.

The metric is essential only through the topology it induces.
So every metrizable space is paracompact.

## Explanations

### L1 human

Metric spaces can be very large and far from compact.
Still, every open cover of a metric space can be replaced by a finer cover that is finite near each point.
This makes metric spaces good spaces for gluing local objects.

### L2 human

Stone's theorem shows that paracompactness is a necessary condition for metrizability.
Metrization theorems such as the [[Nagata–Smirnov theorem|KB11T6389G]] go in the other direction.
Combined with [[Dieudonné's theorem|B6Y8PEY72W]], it gives another proof that metric spaces are normal.
It is also why paracompactness is automatic for metrizable manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/7QFW771QMA.md" tag="7QFW771QMA" kind="statement">
# Kummer's 24 solutions

## Statement

Let $a, b, c \in \C$ be such that none of $c$, $c - a - b$, $a - b$ is an integer, and consider the [[hypergeometric differential equation|W8VE25R5A7]] with these parameters.
Let ${}_2F_1$ be the [[hypergeometric function|463X5V31D9]].

1. The following six functions are solutions, each on a suitable simply connected domain:
$$\begin{aligned} w_1 &= {}_2F_1(a, b; c; z), & w_2 &= z^{1-c} \, {}_2F_1(a - c + 1, b - c + 1; 2 - c; z), \\ w_3 &= {}_2F_1(a, b; a + b - c + 1; 1 - z), & w_4 &= (1 - z)^{c-a-b} \, {}_2F_1(c - a, c - b; c - a - b + 1; 1 - z), \\ w_5 &= z^{-a} \, {}_2F_1(a, a - c + 1; a - b + 1; 1/z), & w_6 &= z^{-b} \, {}_2F_1(b, b - c + 1; b - a + 1; 1/z). \end{aligned}$$
2. Applying the [[Pfaff and Euler transformations|KAAJ0JJ2GF]] to each $w_i$ gives four expressions $z^{\lambda} (1 - z)^{\mu} \, {}_2F_1(a', b'; c'; \xi)$.
In each, $\xi$ is one of $z$, $1 - z$, $1/z$, $1/(1 - z)$, $z/(z - 1)$, $(z - 1)/z$.
This gives $24$ expressions of solutions, falling into six groups of four equal functions.

## Remarks

The six variables are the images of $z$ under the six [[Möbius transformations|0SG74WBFA2]] that permute $\set{0, 1, \infty}$.
Each such map carries the equation to a hypergeometric equation with permuted local exponents.
The solutions $w_1, \ldots, w_6$ are the pairs of local solutions at $0$, $1$ and $\infty$, each with a pure power behaviour there.
The hypothesis that $c$, $c - a - b$ and $a - b$ are not integers makes the two exponents at each singular point differ by a non-integer, so each pair is a basis.
Without it, some of the six formulas are undefined or coincide, and logarithmic solutions appear.

Example: the group of $w_1$ consists of
$${}_2F_1(a, b; c; z) = (1 - z)^{-a} \, {}_2F_1\left(a, c - b; c; \tfrac{z}{z - 1}\right) = (1 - z)^{-b} \, {}_2F_1\left(c - a, b; c; \tfrac{z}{z - 1}\right) = (1 - z)^{c - a - b} \, {}_2F_1(c - a, c - b; c; z).$$

Since the solution space is two-dimensional, any three of the $w_i$ are linearly dependent.
For instance, on $\C \setminus \big((-\infty, 0] \cup [1, \infty)\big)$,
$$w_1 = \frac{\Gamma(c) \Gamma(c - a - b)}{\Gamma(c - a) \Gamma(c - b)} \, w_3 + \frac{\Gamma(c) \Gamma(a + b - c)}{\Gamma(a) \Gamma(b)} \, w_4,$$
and the first coefficient is the value given by [[Gauss's hypergeometric theorem|1FC0T0GCCX]].

## Explanations

### L1 human

The hypergeometric equation has three special points, $0$, $1$ and $\infty$, and near each of them there are two natural solutions that behave like powers.
Each of these six solutions can be written as a hypergeometric function in four different ways.
That gives Kummer's list of $24$ formulas.

### L2 human

The list is the tool for continuing ${}_2F_1$ from one singular point to another.
Linear relations between solutions at different points, the connection formulas, follow from Gauss's theorem and give the monodromy explicitly.
</document>

<document source="https://tamnd.github.io/math2/node/7QM2KNZ358.md" tag="7QM2KNZ358" kind="definition">
# Sheaf

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A *presheaf* $\mathcal F$ of sets on $X$ consists of the following data.
For each [[open set|02BH3MCBQB]] $U \subseteq X$ there is a set $\mathcal F(U)$.
For each pair of open sets $V \subseteq U$ there is a map $\rho_{UV} \colon \mathcal F(U) \to \mathcal F(V)$, written $s \mapsto s|_V$.
These satisfy $\rho_{UU} = \id$ and $\rho_{VW} \circ \rho_{UV} = \rho_{UW}$ whenever $W \subseteq V \subseteq U$.
The elements of $\mathcal F(U)$ are the *sections* of $\mathcal F$ over $U$.
A presheaf $\mathcal F$ is a *sheaf* if, for every open set $U$ and every open cover $U = \bigcup_{i \in I} U_i$:

1. (Locality) if $s, t \in \mathcal F(U)$ satisfy $s|_{U_i} = t|_{U_i}$ for all $i$, then $s = t$;
2. (Gluing) if $s_i \in \mathcal F(U_i)$ satisfy $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$ for all $i, j$, then there is $s \in \mathcal F(U)$ with $s|_{U_i} = s_i$ for all $i$.

A sheaf of [[abelian groups|082GHNPJ53]] is a sheaf in which each $\mathcal F(U)$ is an abelian group and each restriction map is a homomorphism.
A sheaf of [[commutative rings|0310REJJY8]] is defined in the same way.

## Remarks

Applied to the empty cover of $U = \emptyset$, the two conditions say that $\mathcal F(\emptyset)$ has exactly one element.

Example: for $U \subseteq \R$ open, let $\mathcal C(U)$ be the ring of [[continuous|0VGFHTM75Y]] functions $U \to \R$, with the usual restriction.
Locality holds because a function is determined by its values.
Gluing holds because the glued function agrees near each point with some continuous $s_i$, so it is continuous.
So $\mathcal C$ is a sheaf of commutative rings.

Non-example: let $\mathcal B(U)$ be the set of bounded continuous functions on $U$.
Gluing fails for $U = \R$ with the cover by the intervals $U_m = (-m, m)$ and the sections $s_m(x) = x$ on $U_m$: they agree on overlaps, but their only possible gluing, $x \mapsto x$, is not bounded.

Equivalently, conditions 1 and 2 say that $\mathcal F(U) \to \prod_i \mathcal F(U_i) \rightrightarrows \prod_{i, j} \mathcal F(U_i \cap U_j)$ is an equalizer diagram.
Sheaves of abelian groups are the coefficients of [[sheaf cohomology|KB76923A1M]].
</document>

<document source="https://tamnd.github.io/math2/node/7QNRC9SJVV.md" tag="7QNRC9SJVV" kind="concept">
# Harmonic analysis

## Summary

Harmonic analysis studies functions by writing them as superpositions of basic waves, and studies operators through their action on these waves.
Its classical core is the theory of [[Fourier series|G12KXDHPPS]] on the circle and of the [[Fourier transform|DPVM34ADZV]] on $\R^n$.
On a general locally compact group the waves are replaced by characters or irreducible representations, which gives [[harmonic analysis on groups|5E6KB3V8R0]].
The subject supplies tools for partial differential equations, number theory, probability and signal processing.

## Explanations

### L1 human

A chord played on a piano is a sum of single notes, and a trained ear can hear which notes are present.
Harmonic analysis does the same for functions: it splits a function into pure oscillations and records how much of each frequency it contains.
Many questions become easier on the side of the frequencies, because differentiation and translation act very simply on a pure wave.
For example, the derivative of $e^{inx}$ is $in \, e^{inx}$, so differentiation becomes multiplication by $in$.
The hard part is to put the pieces back together and control in which sense the sum of the waves gives back the function.
</document>

<document source="https://tamnd.github.io/math2/node/7R0F4956PX.md" tag="7R0F4956PX" kind="definition">
# Trapezoidal rule

## Definition

Let $a < b$ and let $f \colon [a, b] \to \R$ be continuous.
The *trapezoidal rule* is the [[quadrature rule|ADSVMCY77A]]
$$T(f) = \frac{b - a}{2} \big( f(a) + f(b) \big).$$
For an integer $N \ge 1$, let $h = (b - a)/N$ and $x_i = a + i h$ for $i = 0, \dots, N$.
The *composite trapezoidal rule* with $N$ subintervals is
$$T_N(f) = h \left( \frac{f(x_0)}{2} + f(x_1) + \cdots + f(x_{N-1}) + \frac{f(x_N)}{2} \right),$$
the sum of $T$ over the subintervals $[x_{i-1}, x_i]$.

## Remarks

$T$ is the closed [[Newton–Cotes formula|BQPQAY1FDJ]] with $n = 1$, and its [[degree of exactness|ZYT1JNT1Y0]] is $1$.
Example: for $f(x) = x^2$ on $[0, 1]$ and $N = 2$, $T_2(f) = \tfrac{1}{2}\big(0 + \tfrac{1}{4} + \tfrac{1}{2}\big) = \tfrac{3}{8}$, while $\int_0^1 x^2\, dx = \tfrac{1}{3}$.
Non-example: the left Riemann sum $h \big( f(x_0) + \cdots + f(x_{N-1}) \big)$ is not the composite trapezoidal rule.
It gives full weight to $f(x_0)$ and none to $f(x_N)$; for $f(x) = x$ on $[0, 1]$ and $N = 1$ it gives $0$ instead of $1/2$.
The composite rule is the average of the left and right Riemann sums.
For a continuous function of period $b - a$, $T_N(f) = h \sum_{i=0}^{N-1} f(x_i)$, and for smooth periodic $f$ the error decreases faster than any power of $h$.
</document>

<document source="https://tamnd.github.io/math2/node/7R8W704VGW.md" tag="7R8W704VGW" kind="statement">
# Smale–Hirsch theorem

## Statement

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] of dimensions $m$ and $n$.
Assume that $m < n$, or that $m = n$ and no connected component of $M$ is compact.
Let $\mathrm{Imm}(M, N)$ be the space of [[immersions|6RW68JGBHC]] $M \to N$ with the $C^1$ topology.
Let $\mathrm{Mono}(TM, TN)$ be the space of continuous bundle maps $TM \to TN$ between [[tangent bundles|WY6K3P4YCG]] that are injective on each fibre, with the compact-open topology.
Then the map $\mathrm{Imm}(M, N) \to \mathrm{Mono}(TM, TN)$, $f \mapsto df$, which sends an immersion to its [[differential|6P9PXQDSEN]], is a weak homotopy equivalence.
That is, it induces a bijection on path components and isomorphisms on all [[homotopy groups|P976WYEFMB]].

## Remarks

On path components the theorem says: two immersions are regularly homotopic, that is homotopic through immersions, if and only if their differentials are homotopic through fibrewise injective bundle maps.
It also says that $M$ immerses in $N$ as soon as there is a fibrewise injective bundle map $TM \to TN$.
Smale first proved the case of spheres in Euclidean space.
For $m < n$, regular homotopy classes of immersions $S^m \to \R^n$ correspond to $\pi_m(V_{n, m})$.
Here $V_{n, m}$ is the Stiefel manifold of orthonormal $m$-frames in $\R^n$.
Hirsch extended the result to general $M$ and $N$.
Example: $\pi_2(V_{3, 2}) = \pi_2(SO(3)) = 0$, so all immersions $S^2 \to \R^3$ are regularly homotopic, which gives sphere eversion.
The hypothesis cannot be dropped: for $m = n = 1$ and $M = S^1$, $N = \R$, the set $\mathrm{Mono}(TS^1, T\R)$ is not empty, but there is no immersion $S^1 \to \R$.

## Explanations

### L1 human

Can one manifold be drawn smoothly inside another, possibly with self-crossings?
To decide this, one only needs to match up tangent directions continuously, without integrating them into an actual map.
The same is true for deciding when two such drawings can be deformed into each other.
This turns a hard geometric question into a question of algebraic topology.

### L2 human

The theorem is the first major instance of the homotopy principle.
Its proof builds the regular homotopy cell by cell over a handle decomposition.
It uses a covering homotopy property of the restriction map to a smaller submanifold.
The extra room needed comes from the dimension condition or from the openness of $M$.
Combined with computations of homotopy groups of Stiefel manifolds, it classifies immersions of spheres, and with obstruction theory it gives existence results for immersions of general manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/7RAH6V0C5R.md" tag="7RAH6V0C5R" kind="definition">
# Regular prime

## Definition

Let $p$ be an odd [[prime number|01826KMG2H]], and let $\zeta_p$ be a primitive $p$-th [[root of unity|0YV2CKP2AV]].
The prime $p$ is *regular* if $p$ does not divide the [[class number|KTKK9B1X41]] of the [[cyclotomic field|NN77VYSGXR]] $\Q(\zeta_p)$.
Otherwise $p$ is *irregular*.

## Remarks

Kummer's criterion gives an equivalent form: $p$ is regular if and only if $p$ does not divide the numerator of any of the [[Bernoulli numbers|0KC00K7KY4]] $B_k$ for $k = 2, 4, \ldots, p - 3$.
Kummer proved that if $p$ is regular, then $x^p + y^p = z^p$ has no solution in positive integers, the regular case of [[Fermat's Last Theorem|0Y5X3SDBCZ]].
It is known that there are infinitely many irregular primes.
Whether there are infinitely many regular primes is an open question.

Example: $p = 5$ is regular.
By the criterion, only $B_2 = \frac{1}{6}$ needs checking, and $5$ does not divide its numerator $1$; in fact $\Q(\zeta_5)$ has class number $1$.

Non-example: $p = 37$ is irregular.
The numerator of $B_{32}$ is divisible by $37$, and $32 \le 37 - 3$; equivalently, $37$ divides the class number of $\Q(\zeta_{37})$.
</document>

<document source="https://tamnd.github.io/math2/node/7RC9CN2Q49.md" tag="7RC9CN2Q49" kind="concept">
# Chern connection

## Summary

The Chern connection is a canonical [[connection|5ZYRRVE8BW]] on a holomorphic vector bundle with a Hermitian metric.
It is the unique connection that preserves the metric and whose $(0,1)$-part is the $\bar\partial$-operator of the bundle.
It turns the pair (holomorphic structure, metric) into curvature, a $(1,1)$-form with values in endomorphisms.
Its curvature is the basis of positivity notions, of vanishing theorems such as the [[Kodaira vanishing theorem|NY77DQ2AX8]], and of Chern–Weil representatives of [[Chern classes|PAD37GKDPD]].

## Explanations

### L1 human

On a bundle with a metric there are many ways to differentiate sections that keep lengths constant under parallel transport.
On a holomorphic bundle there is also a natural way to take the "antiholomorphic part" of a derivative.
The Chern connection is the only one that does both jobs at once.
For a line bundle with a local holomorphic section $e$ of squared length $H = \abs{e}^2$, it is given by $\nabla e = \partial \log H \otimes e$.
So the metric alone, written in a holomorphic frame, determines the connection and its curvature.
</document>

<document source="https://tamnd.github.io/math2/node/7RHW0KA5RY.md" tag="7RHW0KA5RY" kind="problem">
# Hilbert's twenty-first problem

## Problem

Let $a_1, \ldots, a_m$ be distinct points of the Riemann sphere, let $U$ be the sphere with these points removed, and fix $z_0 \in U$.
Let $\rho \colon \pi_1(U, z_0) \to \mathrm{GL}_n(\C)$ be a group homomorphism.
Does there exist a linear differential equation of Fuchsian class whose singular points are $a_1, \ldots, a_m$ and whose [[monodromy representation|C8HD62AB6Q]] is $\rho$?

## Remarks

In the form used today, the question is asked for first-order systems $Y' = A(z) Y$ of Fuchsian type, $A(z) = \sum_{i} \frac{A_i}{z - a_i}$ with constant matrices $A_i$, which have simple poles only.
If one only asks for a system with [[regular singular points|H96HXG6CA0]], the answer is positive; this was shown by Plemelj in 1908.
Plemelj's work was long taken as a positive solution for Fuchsian systems.
In 1989 Bolibrukh found a counterexample: a representation that is not the monodromy of any Fuchsian system with the given singular points.
The answer is positive when $\rho$ is irreducible, a result of Bolibrukh and of Kostov.
For scalar equations of order $n$, extra apparent singular points are needed in general.
Plemelj's approach reduces the question to a matrix [[Riemann–Hilbert problem|RY8YQ9JZF9]] on a curve through the singular points.

## Explanations

### L1 human

A linear differential equation with a few singular points has a monodromy: a rule that says how its solutions get mixed when one walks around each singular point.
Hilbert asked for the converse.
Given the points and any prescribed mixing rule, is there a tame (Fuchsian) equation that produces exactly this rule?
For a long time the answer was believed to be yes, but in 1989 it turned out to be no in general.
</document>

<document source="https://tamnd.github.io/math2/node/7RP8XW5YH3.md" tag="7RP8XW5YH3" kind="example">
# Eisenstein series as a double series

## Example

Let $\tau$ be a point of the [[upper half-plane|X6XGGH5FPW]] and let $k \ge 3$ be an integer.
The [[Eisenstein series|EYREVY31BN]] $G_k(\tau) = \sum_{(m,n) \ne (0,0)} (m\tau + n)^{-k}$ is a [[double series|PAXCH4FFSV]] indexed by $\Z^2 \setminus \set{(0,0)}$.
It converges [[absolutely|0W3K3QPEWT]].
Check: since $1$ and $\tau$ are linearly independent over $\R$, there is $c > 0$ with $\abs{x\tau + y} \ge c \max(\abs{x}, \abs{y})$ for all real $x, y$.
For each integer $r \ge 1$ there are exactly $8r$ pairs $(m, n)$ with $\max(\abs{m}, \abs{n}) = r$.
So $\sum \abs{m\tau + n}^{-k} \le \sum_{r \ge 1} 8r \, (cr)^{-k} = 8 c^{-k} \sum_{r \ge 1} r^{1-k}$, which is finite because $k - 1 \ge 2$.
By absolute convergence, the sum does not depend on the order of the terms.
Summing by rows, by columns or over growing squares gives the same value.
For odd $k$ the terms for $(m,n)$ and $(-m,-n)$ cancel and $G_k = 0$, so the interesting cases are the even $k \ge 4$.

For $k = 2$ the same count gives the divergent bound $\sum 8r(cr)^{-2}$, and indeed the double series is not absolutely convergent.
Summing first over $n$ and then over $m$ still gives a value, but this function is not a modular form.
The order of summation matters here.

## Explanations

### L1 human

The Eisenstein series adds up one term for each point of a two-dimensional grid, so it is naturally a double series.
Counting the grid points on squares of growing size shows that the terms decay fast enough when $k \ge 3$.
Then the order in which the terms are added does not matter, and that is what makes the function well behaved.
</document>

<document source="https://tamnd.github.io/math2/node/7RP93F2W47.md" tag="7RP93F2W47" kind="definition">
# Gödel numbering

## Definition

Let $\Sigma$ be a countable alphabet whose symbols are listed effectively, and let $\Sigma^*$ be the set of finite strings over $\Sigma$.
A *Gödel numbering* of $\Sigma^*$ is a map $\ulcorner \cdot \urcorner \colon \Sigma^* \to \N$ such that:

1. $\ulcorner \cdot \urcorner$ is injective;
2. $\ulcorner \cdot \urcorner$ is computable;
3. the set of codes $\set{\ulcorner w \urcorner : w \in \Sigma^*}$ is a [[computable set|PPGD5KW8JC]], and the inverse map from codes to strings is computable.

For a formal language, a Gödel numbering of its [[formulas|HYJ10AF6RT]] or proofs is the restriction of a Gödel numbering of its strings of symbols.

## Remarks

Example: give the symbols injective codes $c(s) \ge 1$, and let $p_1 = 2, p_2 = 3, p_3 = 5, \ldots$ be the primes.
Set $\ulcorner s_1 s_2 \cdots s_n \urcorner = p_1^{c(s_1)} p_2^{c(s_2)} \cdots p_n^{c(s_n)}$.
By unique factorization the map is injective, and the exponents of a number can be found by trial division, so all three clauses hold.

Non-example: the map that sends each string to its length is computable but fails clause 1, since different strings of the same length get the same number.

The code of a formula $\varphi$ is written $\ulcorner \varphi \urcorner$, and in a theory of arithmetic its numeral is also written $\ulcorner \varphi \urcorner$.
Any two Gödel numberings are related by a computable bijection between their sets of codes.
</document>

<document source="https://tamnd.github.io/math2/node/7RPEDHAT7R.md" tag="7RPEDHAT7R" kind="statement">
# Delsarte linear programming bound

## Statement

Let $n \ge 2$.
For $k \ge 0$, let $G_k$ be the polynomial of degree $k$ with $G_k(1) = 1$ such that $G_0, G_1, G_2, \dots$ are orthogonal on $[-1, 1]$ for the weight $(1 - t^2)^{(n-3)/2}$.
These are the Gegenbauer polynomials for the unit sphere $S^{n-1} \subset \R^n$.
Let $-1 \le s < 1$, and let $f$ be a real polynomial with $f = \sum_{k=0}^{d} f_k G_k$ such that:

1. $f_0 > 0$ and $f_k \ge 0$ for $k = 1, \dots, d$;
2. $f(t) \le 0$ for all $t \in [-1, s]$.

Let $C \subset S^{n-1}$ be a [[spherical code|SDCX4WDP4F]] such that $x \cdot y \le s$ for all distinct $x, y \in C$, where $x \cdot y$ is the [[dot product|0S52TNNTH9]].
Then
$$\abs{C} \le \frac{f(1)}{f_0}.$$

## Remarks

The key fact is that $\sum_{x, y \in C} G_k(x \cdot y) \ge 0$ for every $k$, because $G_k(x \cdot y)$ is a positive definite kernel on the sphere, a consequence of the addition theorem for spherical harmonics.
Then $\sum_{x, y \in C} f(x \cdot y) = \sum_k f_k \sum_{x, y} G_k(x \cdot y) \ge f_0 \abs{C}^2$, since the term $k = 0$ equals $f_0 \abs{C}^2$.
On the other side, the diagonal terms give $\abs{C} f(1)$ and the off-diagonal terms are at most $0$ by clause 2.
So $f_0 \abs{C}^2 \le \abs{C} f(1)$.
The bound was introduced by Philippe Delsarte for codes in Hamming space, and by Delsarte, Goethals and Seidel for spherical codes in 1977.
With $s = 1/2$ it bounds the [[kissing number|B408FZYXF9]]; Odlyzko and Sloane, and independently Levenshtein, found polynomials giving $\tau(8) \le 240$ and $\tau(24) \le 196560$, which are sharp.

## Explanations

### L1 human

To show that a set of points on a sphere cannot be large, one picks a clever polynomial and sums its values over all pairs of points.
Positivity of certain building blocks makes the sum large, while the spacing of the points makes it small.
Comparing the two gives a bound on the number of points.

### L2 human

Finding the best polynomial of a given degree is a linear program, hence the name.
The method gives exact answers in dimensions $8$ and $24$, where the extremal codes are the shortest vectors of the [[E8 lattice|2CXH65EKY7]] and of the [[Leech lattice|XETV3657DG]].
In dimension $3$ it gives only $\tau(3) \le 13$.
Cohn and Elkies adapted the idea to sphere packings in $\R^n$, which led to the [[solution in dimensions 8 and 24|YJ48S89BZA]].
</document>

<document source="https://tamnd.github.io/math2/node/7RPV2T29CF.md" tag="7RPV2T29CF" kind="statement">
# Pontryagin's maximum principle

## Statement

Let $U \subseteq \R^m$ be a nonempty set, and let $f \colon \R^n \times U \to \R^n$ and $L \colon \R^n \times U \to \R$ be continuous, with partial derivatives $\partial f / \partial x$ and $\partial L / \partial x$ that are continuous on $\R^n \times U$.
Let $T > 0$ and $x_0, x_1 \in \R^n$ be fixed.
Consider the [[optimal control|T3V6ZNAMDN]] problem: minimize
$$J(u) = \int_0^T L(x(t), u(t)) \, dt$$
over the measurable bounded controls $u \colon [0, T] \to U$ whose trajectory of the [[control system|ZPDTVZQN5K]] $x' = f(x, u)$ with $x(0) = x_0$ satisfies $x(T) = x_1$.
For $\lambda_0 \in \R$ and $p \in \R^n$, let $H(x, u, p, \lambda_0) = p \cdot f(x, u) - \lambda_0 L(x, u)$.
Let $u^*$ be an optimal control with trajectory $x^*$.
Then there are a number $\lambda_0 \in \set{0, 1}$ and an [[absolutely continuous|ZT6F4C2YHK]] function $p \colon [0, T] \to \R^n$, with $(\lambda_0, p(t)) \ne (0, 0)$ for all $t$, such that:

1. $p'(t) = -\dfrac{\partial H}{\partial x}(x^*(t), u^*(t), p(t), \lambda_0)$ for almost every $t \in [0, T]$;
2. $H(x^*(t), u^*(t), p(t), \lambda_0) = \max_{v \in U} H(x^*(t), v, p(t), \lambda_0)$ for almost every $t \in [0, T]$;
3. the function $t \mapsto H(x^*(t), u^*(t), p(t), \lambda_0)$ is equal almost everywhere to a constant.

## Remarks

The proof compares the optimal control with needle variations, which replace $u^*$ by a constant value $v \in U$ on a short interval.
The first-order effects of these variations on the endpoint and the cost form a convex cone.
Optimality means that this cone does not meet a certain direction.
A separating hyperplane gives $(\lambda_0, p(T))$, and $p$ is then transported backward by the adjoint equation of condition 1.
Example: minimize $\int_0^1 \frac{1}{2} u^2 \, dt$ for $x' = u$, $x(0) = 0$, $x(1) = 1$, $U = \R$.
Here $H = p u - \lambda_0 u^2 / 2$ and $p' = 0$, so $p$ is constant; if $\lambda_0 = 0$, condition 2 forces $p = 0$, which is excluded, so $\lambda_0 = 1$ and $u = p$ is constant, which gives $u \equiv 1$ and $x(t) = t$.
The theorem gives necessary conditions only: an extremal that satisfies them need not be optimal.
The case $\lambda_0 = 0$, called abnormal, can occur, and then the conditions do not involve the cost at all.

## Explanations

### L1 human

To steer a system in the cheapest way, one cannot test every possible control.
The maximum principle says that an optimal control must, at each moment, maximize a single function $H$ over the allowed control values.
The function $H$ combines the dynamics and the cost with a vector $p(t)$ of shadow prices that solves its own differential equation.
This turns an optimization over functions into a pointwise maximization plus a boundary value problem.

### L2 human

The maximum principle extends the Euler–Lagrange equations of the calculus of variations to controls with values in a constraint set $U$, such as $[-1, 1]$.
When $H$ is linear in $u$ and $U$ is a box, condition 2 forces $u$ to take extreme values, which gives [[bang–bang controls|GK1Z3Q1WRZ]] except on singular arcs.
The state equation $x' = \partial H / \partial p$ and the adjoint equation form a Hamiltonian system with boundary conditions at both ends, usually solved by shooting methods.
Variants with free final time add the condition that the constant in condition 3 is zero.
Variants with a free endpoint and no terminal cost add the transversality condition $p(T) = 0$.
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<document source="https://tamnd.github.io/math2/node/7RRB4PCRJB.md" tag="7RRB4PCRJB" kind="statement">
# Lanczos algorithm

## Statement

Let $A \in \R^{n \times n}$ be [[symmetric|0VR671PHE5]], and let $q_1 \in \R^n$ be a unit vector.
The *Lanczos algorithm* sets $q_0 = 0$, $\beta_0 = 0$, and for $j = 1, 2, \dots$ computes
$$w_j = A q_j - \beta_{j-1} q_{j-1}, \quad \alpha_j = q_j^T w_j, \quad w_j' = w_j - \alpha_j q_j, \quad \beta_j = \norm{w_j'}, \quad q_{j+1} = w_j' / \beta_j,$$
stopping when $\beta_j = 0$.
Suppose that in exact arithmetic $\beta_1, \dots, \beta_{m-1}$ are nonzero, and let $Q_m = (q_1 \mid \cdots \mid q_m)$.
Then:

1. the columns of $Q_m$ form an [[orthonormal basis|0S9GGTR0EX]] of the [[Krylov subspace|QB45PCGQPV]] $\mathcal{K}_m(A, q_1) = \mathrm{span}\set{q_1, A q_1, \dots, A^{m-1} q_1}$;
2. $Q_m^T A Q_m = T_m$, the symmetric [[tridiagonal|DARX1KZRHZ]] matrix with diagonal $\alpha_1, \dots, \alpha_m$ and off-diagonal $\beta_1, \dots, \beta_{m-1}$;
3. $A Q_m = Q_m T_m + \beta_m q_{m+1} e_m^T$, where $e_m$ is the last standard basis vector of $\R^m$.

If $\beta_m = 0$, then $\mathcal{K}_m(A, q_1)$ is invariant under $A$ and every eigenvalue of $T_m$ is an [[eigenvalue|0BQAA2BK3Q]] of $A$.

## Remarks

The proof is by induction on $j$.
The vector $A q_j$ is orthogonal to $q_1, \dots, q_{j-2}$ because $q_i^T A q_j = (A q_i)^T q_j$ and $A q_i \in \mathcal{K}_{i+1}$, which is orthogonal to $q_j$ for $i + 1 < j$.
So only the two most recent vectors must be subtracted, and the short recurrence suffices; this is the [[Arnoldi iteration|CJ5EC820S7]] specialized to symmetric matrices.

The eigenvalues of $T_m$, called Ritz values, approximate eigenvalues of $A$, and the extreme ones converge first, often long before $m$ reaches $n$.
Example: for $A = \mathrm{diag}(1, 2, 3)$ and $q_1 = (1, 1, 1)/\sqrt{3}$, one gets $\alpha_1 = 2$ and $w_1' = (-1, 0, 1)/\sqrt{3}$, so $\beta_1 = \sqrt{2/3}$ and the Ritz values of $T_2$ are $2 \pm \sqrt{2/3}$, approaching $1$ and $3$.
In floating point arithmetic the vectors $q_j$ lose orthogonality once a Ritz value converges, and copies of converged eigenvalues appear.
Paige's analysis explains this, and practical codes use selective or full reorthogonalization.

## Explanations

### L1 human

The Lanczos algorithm squeezes a huge symmetric matrix into a small tridiagonal one by looking at it only through repeated multiplications with a starting vector.
The eigenvalues of the small matrix approximate the largest and smallest eigenvalues of the big one.
It needs only matrix-vector products, so it works for matrices too large to store as a full array.

### L2 human

The algorithm builds an orthonormal basis of a Krylov subspace with a three-term recurrence, which is cheap in memory and time.
Applied to linear systems with a symmetric positive definite matrix, the same recurrence underlies the [[conjugate gradient method|AFXZGQB4EN]].
Its convergence for extreme eigenvalues is described by the Kaniel–Paige theory using Chebyshev polynomials, and the gap between eigenvalues controls the speed.
For nonsymmetric matrices one uses the Arnoldi iteration or a two-sided Lanczos process.
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<document source="https://tamnd.github.io/math2/node/7S813PEV69.md" tag="7S813PEV69" kind="concept">
# Allen–Cahn equation

## Summary

The Allen–Cahn equation is the [[reaction–diffusion equation|TAAHWE917M]] $u_t = \Delta u - \varepsilon^{-2}(u^3 - u)$.
It models phase separation in a binary alloy, where $u = \pm 1$ are the two pure phases.
It is the gradient flow in $L^2$ of the Ginzburg–Landau energy.
As $\varepsilon \to 0$, the thin interfaces between the phases move approximately by [[mean curvature flow|2YYV01C7QQ]].

## Explanations

### L1 human

Picture a material that wants to be in one of two states, labelled $+1$ and $-1$.
The cubic term pushes each point toward one of the two states, and the Laplacian smooths out the differences between neighbours.
After a short time the material splits into regions of the two phases, separated by thin transition layers.
In one dimension a stationary layer is $u(x) = \tanh\!\big(x / (\sqrt{2}\,\varepsilon)\big)$, which goes from $-1$ to $+1$.
</document>

<document source="https://tamnd.github.io/math2/node/7SB03M850M.md" tag="7SB03M850M" kind="concept">
# Unitization

## Summary

The unitization of an algebra $A$ is the algebra $\mathbb{K} \oplus A$ with a product that makes $(1, 0)$ a unit.
It contains $A$ as an ideal of codimension one.
It lets one apply tools that need a unit, such as invertibility and the spectrum, to algebras without one.
For normed algebras and Banach algebras it carries a natural norm, and for C*-algebras a natural C*-norm.

## Explanations

### L1 human

Some algebras have no element that acts like the number $1$, for example the continuous functions on $\R$ that tend to $0$ at infinity.
The unitization adds such an element by hand.
Its elements are formal sums $\lambda + a$ of a scalar and an element of the algebra, multiplied by expanding the brackets.
For the functions vanishing at infinity, adding the constants gives the continuous functions on $\R$ that have the same limit at $+\infty$ and at $-\infty$.
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<document source="https://tamnd.github.io/math2/node/7SC9E65XDN.md" tag="7SC9E65XDN" kind="counterexample">
# Stein's example

## Counterexample

Let $p \ge 3$ and let $X$ have the [[multivariate normal distribution|0XN3K7RSZY]] on $\R^p$ with unknown mean $\theta \in \R^p$ and covariance matrix the identity $I_p$.
Estimate $\theta$ under the loss $L(\theta, a) = \norm{a - \theta}^2$, the squared Euclidean distance.

The estimator $\delta_0(X) = X$ has [[risk|N26TSVRH2S]] $R(\theta, \delta_0) = E_\theta \norm{X - \theta}^2 = p$ for every $\theta$.
It is not [[admissible|DSTF6X8DRJ]].
The [[James–Stein estimator|709T196XCT]]
$$\delta_{JS}(X) = \Big( 1 - \frac{p - 2}{\norm{X}^2} \Big) X$$
has risk
$$R(\theta, \delta_{JS}) = p - (p - 2)^2 \, E_\theta\Big[ \frac{1}{\norm{X}^2} \Big].$$
For $p \ge 3$ the expectation is finite and positive, so $R(\theta, \delta_{JS}) < p$ for every $\theta$.
Thus $\delta_{JS}$ dominates $X$.

Check of the risk formula: write $\delta_{JS}(X) = X + g(X)$ with $g(x) = -(p - 2) x / \norm{x}^2$.
Then $\norm{g(x)}^2 = (p - 2)^2 / \norm{x}^2$ and $\sum_i \partial_i g_i(x) = -(p - 2)^2 / \norm{x}^2$.
[[Stein's unbiased risk estimate|30XVCMK916]] gives $R(\theta, \delta_{JS}) = E_\theta\big[p + \norm{g(X)}^2 + 2 \sum_i \partial_i g_i(X)\big]$, which is the formula above.

## Remarks

For $p = 1$ and $p = 2$, the estimator $X$ is admissible, so the dimension bound is sharp.
The improvement is largest near $\theta = 0$, the point toward which $\delta_{JS}$ shrinks: there $E_0[1/\norm{X}^2] = 1/(p - 2)$, so the risk is $2$ instead of $p$.
Any fixed point can replace $0$ as the target of the shrinkage.

## Explanations

### L1 human

To estimate three or more unrelated averages at once, the obvious method is to use each observed value as its own estimate.
Stein showed that this obvious method can be beaten for every possible truth, by pulling all estimates a little toward a common point.
The total squared error goes down, even though the quantities have nothing to do with each other.
This surprising fact changed how statisticians think about estimating many parameters together.
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<document source="https://tamnd.github.io/math2/node/7SCA4TQ1Y8.md" tag="7SCA4TQ1Y8" kind="definition">
# Homotopy equivalence

## Definition

Let $X$ and $Y$ be topological spaces.
A [[continuous map|0VGFHTM75Y]] $f \colon X \to Y$ is a *homotopy equivalence* if there is a continuous map $g \colon Y \to X$ with these two properties.
The map $g \circ f$ is [[homotopic|F2F143KQ2M]] to $\id_X$, and $f \circ g$ is homotopic to $\id_Y$.
Such a map $g$ is a *homotopy inverse* of $f$.

## Remarks

Every homeomorphism is a homotopy equivalence, with its inverse as homotopy inverse.
Example: let $i \colon S^1 \to \R^2 \setminus \set{0}$ be the inclusion and $r(x) = x / \norm{x}$.
Then $r \circ i = \id_{S^1}$, and $H(x, t) = (1 - t) x + t x / \norm{x}$ is a homotopy in $\R^2 \setminus \set{0}$ from $\id$ to $i \circ r$, so $i$ is a homotopy equivalence.
Non-example: the constant map $S^1 \to \set{p}$ is not a homotopy equivalence, since the [[fundamental group|S1XK3HXHFX]] of $S^1$ is $\Z$ and that of a point is trivial.
Composites of homotopy equivalences are homotopy equivalences.
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<document source="https://tamnd.github.io/math2/node/7SHV2SPYP6.md" tag="7SHV2SPYP6" kind="concept">
# Sum-free set

## Summary

A sum-free set is a subset $A$ of an abelian group that contains no solution of $x + y = z$ with $x, y, z \in A$.
Equivalently, the [[sumset|XNBN31HQWB]] $A + A$ is disjoint from $A$.
Sum-free sets are central in [[additive combinatorics|3NM362A2JG]] and [[Ramsey theory|G5TWXV565S]].
[[Schur's theorem|V6XJQXZ7E8]] says that the positive integers cannot be split into finitely many sum-free sets.

## Explanations

### L1 human

A set of numbers is sum-free if adding two of its members, possibly the same one twice, never gives a member.
The odd numbers are sum-free, because two odd numbers add up to an even number.
The set $\set{6, 7, 8, 9, 10}$ is sum-free too, because every sum of two of its members is at least $12$.
The set $\set{1, 2, 3}$ is not, since $1 + 2 = 3$.
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<document source="https://tamnd.github.io/math2/node/7SP03R55HW.md" tag="7SP03R55HW" kind="definition">
# Mal'cev condition

## Definition

A *strong Mal'cev condition* is given by finitely many operation symbols $g_1, \dots, g_k$ with arities and a finite set $\Sigma$ of [[identities|PKD440G2XW]] in these symbols.
A [[variety|AJNZJD8442]] $\mathcal{V}$ *satisfies* it if $\mathcal{V}$ has terms $t_1, \dots, t_k$ of the same arities with the following property.
Each identity of $\Sigma$ holds in $\mathcal{V}$ after each $g_j$ is replaced by $t_j$.
A *Mal'cev condition* is a sequence $U_1, U_2, \dots$ of strong Mal'cev conditions such that every variety that satisfies $U_n$ also satisfies $U_{n+1}$, for each $n$.
A variety satisfies the Mal'cev condition if it satisfies $U_n$ for some $n$.
A property of varieties is *Mal'cev definable* if some Mal'cev condition is satisfied by exactly the varieties with that property.

## Remarks

Example: the existence of a [[Mal'cev term|Y203PPTS61]] is a strong Mal'cev condition, with one ternary symbol $p$ and $\Sigma = \set{p(x, y, y) = x, \ p(x, x, y) = y}$.
The variety of groups satisfies it with $t(x, y, z) = x y^{-1} z$, since $x y^{-1} y = x$ and $x x^{-1} z = z$.
The existence of [[Jónsson terms|3M8Y76E4AG]] for some $n$ is a Mal'cev condition, with one strong Mal'cev condition $U_n$ for each $n$.
Non-example: local finiteness of a variety is not Mal'cev definable.
The variety of sets, with no operations, is locally finite, and its terms are the projections.
If the projections satisfy $\Sigma$, then the projections of every variety satisfy $\Sigma$, so every variety would satisfy the condition.
But the variety of groups is not locally finite.

## Explanations

### L1 human

A Mal'cev condition says that a variety has some terms that satisfy a fixed list of equations, possibly with the length of the list allowed to grow.
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<document source="https://tamnd.github.io/math2/node/7SQS69T17R.md" tag="7SQS69T17R" kind="concept">
# Reverse mathematics

## Summary

Reverse mathematics is a program in mathematical logic that asks which axioms are needed to prove the theorems of ordinary mathematics.
It works in subsystems of [[second-order arithmetic|DC60FYCSPM]] over a weak base theory, and proves theorems equivalent to set existence axioms.
A striking finding is that many classical theorems are equivalent to one of five subsystems, often called the big five.

## Explanations

### L1 human

Usually one goes from axioms to theorems.
Reverse mathematics goes backwards: it starts from a theorem and finds the weakest axioms that prove it.
Often the theorem turns out to be equivalent to those axioms, so it needs exactly them.
For example, the Bolzano–Weierstrass theorem, that every bounded sequence of reals has a convergent subsequence, is equivalent to the axiom system called $\mathsf{ACA}_0$ over the base theory.
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<document source="https://tamnd.github.io/math2/node/7SSKBQYFW1.md" tag="7SSKBQYFW1" kind="statement">
# Sylvester's determinant identity

## Statement

Let $K$ be a [[commutative ring|0310REJJY8]], let $1 \le k < n$, and let $A = (a_{ij})$ be an $n \times n$ matrix over $K$.
Let $A_0$ be the $k \times k$ submatrix of $A$ formed by its first $k$ rows and its first $k$ columns.
For $k < i, j \le n$, let $b_{ij}$ be the [[minor|0BT9BYWVRY]] of $A$ with rows $1, \dots, k, i$ and columns $1, \dots, k, j$.
Let $B$ be the $(n-k) \times (n-k)$ matrix $(b_{ij})_{k < i, j \le n}$.
Then
$$\det B = (\det A_0)^{n-k-1} \det A,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

First assume that $A_0$ is invertible, and write $A$ in blocks with $A_0$ in the top left corner.
By the [[Schur complement formula|0KA7A4ENVJ]], $\det A = \det A_0 \cdot \det S$, where $S = (s_{ij})$ is the Schur complement of $A_0$ in $A$.
The same formula on each bordered submatrix gives $b_{ij} = \det A_0 \cdot s_{ij}$, so $B = (\det A_0) S$ and the identity follows.
Both sides are polynomials in the entries of $A$ with integer coefficients, so the identity holds over every commutative ring.

For $k = n - 1$, the matrix $B$ is $1 \times 1$ with entry $\det A$, and the identity is trivial.
For $k = n - 2$, it is a form of the Desnanot–Jacobi identity, which is the basis of [[Dodgson condensation|RQ44FKS5QM]].
It should not be confused with [[Sylvester's determinant theorem|0G24VQ1088]], which says $\det(I_m + AB) = \det(I_n + BA)$.

## Explanations

### L1 human

Fix the top left corner block of a square matrix.
Build a smaller matrix whose entries are determinants of this corner block bordered by one more row and one more column.
The determinant of the new matrix equals the determinant of the original matrix times a power of the determinant of the corner block.
For a $3 \times 3$ matrix and $k = 1$, the four entries of $B$ are $2 \times 2$ minors that all contain the entry $a_{11}$, and $\det B = a_{11} \det A$.

### L2 human

The identity explains why fraction-free Gaussian elimination works over the integers.
The intermediate entries of the Bareiss algorithm are such bordered minors, and the division by the previous pivot is exact.
It sits next to the [[determinant via the Schur complement|0KA7A4ENVJ]] and to the [[determinant of a block triangular matrix|0J9TVTNB4K]].
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<document source="https://tamnd.github.io/math2/node/7SV4WA92SM.md" tag="7SV4WA92SM" kind="concept">
# Quadratic field

## Summary

A quadratic field is an [[algebraic number field|MWBDFS2KJ5]] of degree $2$ over $\Q$.
Each one is $\Q(\sqrt{d})$ for a unique square-free integer $d \ne 0, 1$.
Quadratic fields are the simplest number fields beyond $\Q$, and their arithmetic is closely tied to [[binary quadratic forms|SBDSYA037Q]] and to [[Pell's equation|0AF3DANJ76]].

## Explanations

### L1 human

Take the rational numbers and add one square root, such as $\sqrt{2}$ or $\sqrt{-5}$.
The new numbers have the form $a + b\sqrt{d}$ with $a, b$ rational, and they can be added, multiplied and divided like ordinary fractions.
When $d > 0$ the field sits inside the real numbers and is called real; when $d < 0$ it is called imaginary.
Already here unique factorisation can fail: in $\Q(\sqrt{-5})$ one has $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$.
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<document source="https://tamnd.github.io/math2/node/7SWSRGDT4A.md" tag="7SWSRGDT4A" kind="concept">
# Germ of a smooth function

## Summary

The germ of a [[smooth function|0V33BNKZ9S]] at a point $p$ of a [[smooth manifold|F45VQ47GPV]] is the function seen only on arbitrarily small neighbourhoods of $p$.
It is the special case of a [[germ|VZXF7RNFKS]] for smooth real functions.
The germs at $p$ form a [[local ring|MZ02BKPHJA]], and this ring carries all local information of smooth calculus at $p$.
Tangent vectors, jets and the classification of singularities are all built on germs.

## Explanations

### L1 human

Two smooth functions have the same germ at $p$ when they agree near $p$, however far apart they are elsewhere.
Think of a germ as what a function looks like under a microscope pointed at $p$.
For example, on $\R$ the function $f(x) = x$ and a smooth function $g$ that equals $x$ on $(-1, 1)$ and equals $0$ for $\abs{x} > 2$ have the same germ at $0$.
Germs can be added and multiplied, just like functions.
A germ is invertible exactly when its value at $p$ is not zero.
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<document source="https://tamnd.github.io/math2/node/7SWYFFHDBF.md" tag="7SWYFFHDBF" kind="definition">
# Censoring

## Definition

Let $T$ be a real [[random variable|0PNRKMRC55]] of interest.
An observation of $T$ is *censored* if it reveals only that $T$ lies in a set, rather than the value of $T$.
The standard types are:

1. *Right censoring*: for a censoring time $C$, one observes $\tilde T = \min(T, C)$ and $\delta = \mathbf{1}\set{T \le C}$; when $\delta = 0$ one only learns $T > C$.
2. *Left censoring*: one only learns $T \le c$ for a known $c$.
3. *Interval censoring*: one only learns $L < T \le R$ for known $L < R$.

## Remarks

Right censoring is called *independent* when $T$ and $C$ are [[independent|0GBJACNAP6]]; most standard methods of survival analysis assume this or a closely related non-informative censoring condition.

Example: a patient enters a study at month $0$, the study closes at month $24$, and the patient has had no relapse by then.
With $T$ the relapse time and $C = 24$, the record is $\tilde T = 24$, $\delta = 0$, which says exactly that $T > 24$.

Non-example: truncation is different.
Suppose a study only includes people whose event time exceeds $5$.
Then those with $T \le 5$ never appear in the data, so nothing about them is observed, not even a bound.
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<document source="https://tamnd.github.io/math2/node/7SXKJG1Q9B.md" tag="7SXKJG1Q9B" kind="concept">
# Direct limit of algebras

## Summary

The direct limit of a directed system of [[algebras|QJGPT2W0FT]] is the algebra obtained by gluing the algebras of the system along its [[homomorphisms|KBXSZC6BV1]].
It is the [[direct limit|X9HEQ29Q49]] in the category of algebras of a fixed signature, and it is the union when the maps are inclusions.
Varieties are closed under direct limits, and every algebra is the direct limit of its finitely generated subalgebras.
The dual construction is the [[inverse limit of algebras|DE2KEQ85ZT]].

## Explanations

### L1 human

Imagine a growing sequence of algebras, each mapped into the next one.
The direct limit is what you get "at the end of the sequence".
An element lives in some stage, and two elements are equal if they become equal at some later stage.
For example, the groups $\Z/p\Z \to \Z/p^2\Z \to \Z/p^3\Z \to \cdots$, with each map $x \mapsto px$, have as direct limit the Prüfer $p$-group, the group of all complex roots of unity of $p$-power order.
The rationals $\Q$ under addition are the direct limit of the groups $\frac{1}{n!}\Z$, which is simply their union.
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<document source="https://tamnd.github.io/math2/node/7SZF58P6Z7.md" tag="7SZF58P6Z7" kind="definition">
# Class VII surface

## Definition

Let $X$ be a compact, connected [[complex manifold|JSGWAN2RJV]] of dimension $2$.
Then $X$ is a *surface of class VII* if:

1. its [[Kodaira dimension|ZW00Q0SAGR]] is $\kappa(X) = -\infty$, and
2. its first [[Betti number|ST71ZV3FES]] is $b_1(X) = 1$.

A class VII surface is in *class $\mathrm{VII}_0$* if it is minimal, that is, it contains no smooth rational curve of self-intersection $-1$.

## Remarks

Example: a [[Hopf surface|PNFNGBXRM3]] is a class VII surface.
For the diagonal Hopf surface, $b_1 = 1$ because it is diffeomorphic to $S^1 \times S^3$, and no power of the canonical bundle has a nonzero holomorphic section.
[[Inoue surfaces|GPSVJK3R4X]] are class VII surfaces with $b_2 = 0$ and no curves.

Non-example: $\mathbb{CP}^2$ has $\kappa = -\infty$, but $b_1 = 0$, so clause 2 fails; it is a rational surface.

Since $b_1$ is odd, a class VII surface is not Kähler, and in particular not projective.
Blowing up a point keeps $\kappa$ and $b_1$ and raises $b_2$ by $1$, so every class VII surface is a blow-up of a minimal one.
A theorem of Bogomolov states that every class $\mathrm{VII}_0$ surface with $b_2 = 0$ is a Hopf surface or an Inoue surface.
Li, Yau and Zheng, and later Teleman, gave complete proofs.
For $b_2 > 0$ the classification is open; the global spherical shell conjecture predicts that every class $\mathrm{VII}_0$ surface with $b_2 > 0$ contains a global spherical shell.
</document>

<document source="https://tamnd.github.io/math2/node/7T2RGR9XTZ.md" tag="7T2RGR9XTZ" kind="definition">
# Regularity theory

## Definition

Let $\Omega \subseteq \R^n$ be an open set and consider a [[partial differential equation|GC5BDZ6QBC]] $P(u) = f$ in $\Omega$.
Let $X$ be a space of functions on $\Omega$ in which solutions are first sought, such as a [[Sobolev space|M4CEZ5K9HP]] for [[weak solutions|SJZX81A783]].
Let $Y \subseteq X$ be a smaller space of smoother functions.
A *regularity result* for the equation is a statement of this form.
If $u \in X$ solves $P(u) = f$, and $f$ and the coefficients of $P$ have a given smoothness, then $u \in Y$.
The conclusion may hold on all of $\Omega$ or on every compact subset of $\Omega$.
*Regularity theory* is the study of such results and of the estimates of the norm of $u$ in $Y$ that go with them.

## Remarks

Example: let $u \in H^1_{\mathrm{loc}}(\Omega)$ be a weak solution of $-\Delta u = f$ with $f \in L^2_{\mathrm{loc}}(\Omega)$.
Then $u \in H^2_{\mathrm{loc}}(\Omega)$, and more generally $f \in H^k_{\mathrm{loc}}$ gives $u \in H^{k+2}_{\mathrm{loc}}$; this is [[elliptic regularity|2PVQY9PCT4]] with $X = H^1_{\mathrm{loc}}$ and $Y = H^2_{\mathrm{loc}}$.

Non-example: for the transport equation $u_t + u_x = 0$ in $\R^2$, the function $u(x, t) = g(x - t)$ is a weak solution for every $g \in L^2_{\mathrm{loc}}(\R)$.
So smooth data, here $f = 0$, do not force smooth solutions, and no regularity result of this kind holds; singularities travel along characteristics.

Interior regularity concerns compact subsets of $\Omega$; regularity up to the boundary also needs a smooth boundary and smooth boundary data.

## Explanations

### L1 human

Regularity theory asks how smooth a solution must be, once it is known to exist in a weak sense.
Elliptic equations improve smoothness, while hyperbolic equations only carry the roughness of the data along.
</document>

<document source="https://tamnd.github.io/math2/node/7T98FQFGYY.md" tag="7T98FQFGYY" kind="definition">
# Bayes estimator

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a statistical model with a [[loss function|3BTG6VVCGZ]] $L$, and let $\pi$ be a [[prior|00NBBSXRQS]] on $\Theta$.
For an [[estimator|0SSS6ERQ0S]] $\delta$, let $r(\pi, \delta)$ be its [[Bayes risk|VEBDNCJJTN]], the [[risk|N26TSVRH2S]] $R(\theta, \delta)$ averaged over $\pi$.
An estimator $\delta_\pi$ is a *Bayes estimator* with respect to $\pi$ if $r(\pi, \delta_\pi) < \infty$ and
$$r(\pi, \delta_\pi) = \inf_{\delta} r(\pi, \delta),$$
where the infimum is over all estimators $\delta$.

## Remarks

If $r(\pi) < \infty$, an estimator that minimizes the [[posterior|0KJ7ZS8AJA]] expected loss $E[L(\theta, \delta(x)) \mid X = x]$ for almost every $x$ is a Bayes estimator.
Under squared error loss $L(\theta, a) = (a - \theta)^2$, the Bayes estimator is the posterior mean $E[\theta \mid X]$.
Under absolute error loss, it is a posterior median.

Example: let $X$ be [[binomial|0WCWA1KT3M]] with parameters $n$ and $\theta$, and let $\pi$ be the [[beta distribution|0EW824Z7G2]] with parameters $a, b > 0$.
The posterior is the beta distribution with parameters $a + X$ and $b + n - X$.
So under squared error loss the Bayes estimator is its mean $(a + X)/(a + b + n)$.

Non-example: for the uniform prior, $a = b = 1$, the Bayes estimator is $(X + 1)/(n + 2)$.
The proportion $X/n$ differs from it whenever $X \ne n/2$, and for $n \ge 1$ its Bayes risk $1/(6n)$ is larger, so $X/n$ is not the Bayes estimator for this prior.
</document>

<document source="https://tamnd.github.io/math2/node/7T9BZV552E.md" tag="7T9BZV552E" kind="statement">
# Paley–Wiener theorem for Volterra resolvents

## Statement

Let $n \ge 1$ and let $k \in L^1(\R_+; \C^{n \times n})$ be an integrable matrix-valued kernel on $\R_+ = [0, \infty)$.
Write $(k * r)(t) = \int_0^t k(t - s) \, r(s) \, ds$.
Let $r \in L^1_{\mathrm{loc}}(\R_+; \C^{n \times n})$ be the resolvent of $k$, that is, the unique locally integrable solution of
$$r + k * r = k \quad \text{and} \quad r + r * k = k.$$
Let $\hat k(z) = \int_0^\infty e^{-z t} k(t) \, dt$ be the [[Laplace transform|0V49JEZW06]] of $k$, defined for $\operatorname{Re} z \ge 0$.
Then $r \in L^1(\R_+; \C^{n \times n})$ if and only if
$$\det\big(I + \hat k(z)\big) \ne 0 \quad \text{for all } z \in \C \text{ with } \operatorname{Re} z \ge 0.$$

## Remarks

The resolvent gives the solution of the [[Volterra equation|DND8CBEVXT]] $x + k * x = f$ as $x = f - r * f$.
So the theorem decides when the equation is [[stable|8M012XRPZM]] in the sense that the resolvent is integrable.

Proof idea.
If $r$ is integrable, taking Laplace transforms gives $(I + \hat k(z))(I - \hat r(z)) = I$ for $\operatorname{Re} z \ge 0$, so $I + \hat k(z)$ is invertible.
The converse is the deep part.
It is a Wiener-type theorem for the convolution algebra $L^1(\R_+)$ with a unit adjoined.
In that algebra, an element whose transform never vanishes on the closed right half-plane, including at infinity, is invertible.
The point at infinity causes no trouble, because $\hat k(z) \to 0$ as $\abs{z} \to \infty$ in the closed half-plane.
Compare [[Wiener's lemma|NC768TWZ3D]] for absolutely convergent Fourier series.

Example: let $n = 1$ and $k(t) = a e^{-t}$ with $a \in \C$.
Then $\hat k(z) = a / (z + 1)$, and $1 + \hat k(z) = 0$ only for $z = -1 - a$.
The theorem says that $r$ is integrable if and only if $\operatorname{Re} a > -1$.
Indeed $r(t) = a e^{-(1 + a) t}$, as one checks from $(k * r)(t) = a e^{-t} - a e^{-(1 + a) t}$.

## Explanations

### L1 human

A Volterra equation of convolution type feeds the past of the unknown back into the present through a kernel $k$.
Its solution is obtained from the data by a second kernel, the resolvent.
The theorem tells when the resolvent is integrable, which means that bounded or integrable data give bounded or integrable solutions.
The test is a single condition on the Laplace transform of $k$: the matrix $I + \hat k(z)$ must stay invertible on the closed right half-plane.

### L2 human

The theorem is due to Paley and Wiener and is the basic tool for the stability of linear [[Volterra equations of convolution type|SNFYTC1TTA]].
It is the integral equation analogue of the test for linear ordinary differential equations $x' = A x$, where integrability of $e^{tA}$ holds exactly when all eigenvalues of $A$ have negative real part.
With $r$ integrable, Young's inequality shows that the solution map $f \mapsto f - r * f$ is bounded on $L^p(\R_+)$ for $1 \le p \le \infty$, see [[Lp space|SASP14MA1Z]].
Through linearization, the integrability of the [[resolvent kernel|ARTJV1XCRX]] is also the starting point for the stability of nonlinear Volterra equations.
</document>

<document source="https://tamnd.github.io/math2/node/7T9ZS932SG.md" tag="7T9ZS932SG" kind="definition">
# Fredholm determinant

## Definition

Let $a < b$ and let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]].
For $n \ge 1$ and points $x_1, \dots, x_n \in [a, b]$, let $\det[K(x_i, x_j)]_{i,j=1}^{n}$ be the [[determinant|00KDZRW4C0]] of the $n \times n$ matrix with entries $K(x_i, x_j)$.
For $n \ge 1$ let
$$c_n = \frac{1}{n!} \int_a^b \cdots \int_a^b \det[K(x_i, x_j)]_{i,j=1}^{n} \, dx_1 \cdots dx_n,$$
and let $c_0 = 1$.
The *Fredholm determinant* of $K$ is the function $D \colon \C \to \C$ given by
$$D(\lambda) = \sum_{n=0}^{\infty} (-1)^n c_n \lambda^n.$$
The series converges for every $\lambda \in \C$, so $D$ is an [[entire function|0G4F0XB9X2]].

## Remarks

Example: for $K(x, y) = xy$ on $[0, 1]$, $c_1 = \int_0^1 x^2 \, dx = 1/3$.
For $n \ge 2$ the matrix $[x_i x_j]$ has rank at most $1$, so its determinant is $0$ and $c_n = 0$.
Hence $D(\lambda) = 1 - \lambda/3$.

Non-example: the definition does not apply when $K$ is only square-integrable, because the diagonal values $K(x, x)$ enter $c_1 = \int_a^b K(x, x) \, dx$ and are not defined for an element of $L^2$.
For such kernels one uses a modified determinant in which the diagonal terms are removed.

The convergence for all $\lambda$ follows from Hadamard's inequality: if $\abs{K} \le M$, then $\abs{c_n} \le n^{n/2} M^n (b - a)^n / n!$.
The coefficient $c_1$ is the integral $\int_a^b K(x, x) \, dx$, the trace of the kernel, and $D(\lambda) = 1 - c_1 \lambda + \cdots$.
For a degenerate kernel $K(x, y) = \sum_{i=1}^{m} a_i(x) b_i(y)$, one has $D(\lambda) = \det(I_m - \lambda A)$, where $A_{ij} = \int_a^b b_i(y) a_j(y) \, dy$.
</document>

<document source="https://tamnd.github.io/math2/node/7TA23MSPZ0.md" tag="7TA23MSPZ0" kind="definition">
# X-ray transform

## Definition

Let $n \ge 2$ and let $S^{n-1}$ be the unit [[sphere|P9CM4GQ2T0]] in $\R^n$.
Each line in $\R^n$ is $\set{x + t\theta : t \in \R}$ for a direction $\theta \in S^{n-1}$ and a point $x \in \theta^\perp = \set{y : y \cdot \theta = 0}$.
Let $f \colon \R^n \to \C$ be continuous and [[integrable|X7GTF3D98F]] on every line.
The *X-ray transform* of $f$ is the function
$$(P f)(\theta, x) = \int_\R f(x + t\theta)\, dt, \qquad \theta \in S^{n-1},\ x \in \theta^\perp.$$

## Remarks

The same line is given by $(\theta, x)$ and $(-\theta, x)$, and $P f(-\theta, x) = P f(\theta, x)$.
For $f \in L^1(\R^n)$ the transform is defined for almost every $x \in \theta^\perp$, for each $\theta$.
By Fubini's theorem, $\int_{\theta^\perp} P f(\theta, x)\, e^{-i x \cdot \xi}\, dx = \hat f(\xi)$ for $\xi \in \theta^\perp$, where $\hat f(\xi) = \int f(y)\, e^{-i y \cdot \xi}\, dy$; this is the projection-slice theorem for lines.

Example: $f(y) = e^{-\abs{y}^2}$.
For $x \perp \theta$, $\abs{x + t\theta}^2 = \abs{x}^2 + t^2$, so $P f(\theta, x) = \sqrt{\pi}\, e^{-\abs{x}^2}$.

Non-example: $f(y) = (1 + \abs{y})^{-1}$ is continuous and bounded but has no X-ray transform, since $\int_\R (1 + \abs{x + t\theta})^{-1}\, dt$ diverges like $\int dt / \abs{t}$.

For $n = 2$, let $\theta'$ be $\theta$ rotated by a right angle.
Then $P f(\theta, s\theta')$ equals the [[Radon transform|VJNZJ352PC]] $R f(\theta', s)$, since both integrate $f$ over the line $y \cdot \theta' = s$.
</document>

<document source="https://tamnd.github.io/math2/node/7TCB6AGE72.md" tag="7TCB6AGE72" kind="definition">
# Path algebra

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $Q = (Q_0, Q_1, s, t)$ be a [[quiver|RE9YFFS3TS]], with vertex set $Q_0$, arrow set $Q_1$, and source and target maps $s, t \colon Q_1 \to Q_0$.
A *path* of length $m \ge 1$ is a sequence $p = \alpha_1 \alpha_2 \cdots \alpha_m$ of arrows with $t(\alpha_i) = s(\alpha_{i+1})$ for $1 \le i < m$; it has source $s(p) = s(\alpha_1)$ and target $t(p) = t(\alpha_m)$.
For each vertex $v$ there is also a path $e_v$ of length $0$ with $s(e_v) = t(e_v) = v$.
The *path algebra* $kQ$ is the [[vector space|000AYPA9AJ]] over $k$ with [[basis|0KCQ9Y3QKA]] the set of all paths.
The product of two paths $p$ and $q$ is the concatenation $pq$ if $t(p) = s(q)$ and $0$ otherwise.
Here $e_v$ acts as an identity on paths through $v$, and the product on $kQ$ is the bilinear extension.

## Remarks

The product is associative, so $kQ$ is an [[associative algebra|3CTW1P4HBP]]; it has an identity $\sum_{v \in Q_0} e_v$ when $Q_0$ is finite.
The $e_v$ are orthogonal idempotents: $e_v e_w = e_v$ if $v = w$ and $0$ otherwise.
Example: let $Q$ have vertices $1, 2$ and one arrow $\alpha \colon 1 \to 2$.
The paths are $e_1, e_2, \alpha$, and the nonzero products of basis elements are $e_1 e_1 = e_1$, $e_2 e_2 = e_2$ and $e_1 \alpha = \alpha = \alpha e_2$.
So $kQ$ has dimension $3$, and it is isomorphic to the algebra of upper triangular $2 \times 2$ matrices over $k$, via $e_1 \mapsto E_{11}$, $e_2 \mapsto E_{22}$, $\alpha \mapsto E_{12}$.
Non-example: for the quiver with one vertex and two loops $a, b$, the paths $ab$ and $ba$ are different basis elements.
So $kQ$ is the [[free algebra|2P3Y2FRCSZ]] $k\langle a, b \rangle$ and not the commutative ring $k[a, b]$.
$kQ$ is finite-dimensional if and only if $Q$ is finite and has no oriented cycles.
Some books compose paths in the opposite order, which replaces $kQ$ by its opposite algebra.
</document>

<document source="https://tamnd.github.io/math2/node/7THJSH9SFP.md" tag="7THJSH9SFP" kind="concept">
# LF-space

## Summary

An LF-space is a [[locally convex space|B6GY2KJMWW]] obtained as the inductive limit of an increasing sequence of [[Fréchet spaces|3GAA614673]].
The letters stand for "limit of Fréchet spaces".
The main example is the space $C_c^\infty(\Omega)$ of test functions on an open set $\Omega \subseteq \R^d$.
It is the union, over compact sets $K \subseteq \Omega$, of the Fréchet spaces of smooth functions supported in $K$.
Its dual is the space of distributions.
LF-spaces are [[barrelled|ZE6DXEJB8H]] and [[bornological|4R6SQD60FN]], although they are usually not metrizable.

## Explanations

### L1 human

Some spaces are natural unions of a growing chain of nicer spaces.
Smooth functions with compact support are the union of the smooth functions supported in $[-1, 1]$, in $[-2, 2]$, in $[-3, 3]$, and so on.
Each piece is a Fréchet space with a good topology.
The LF topology glues these topologies together in the best convex way.
For test functions, a sequence converges when all its terms lie in one piece and it converges there.
In particular the supports must stay in a fixed compact set.
</document>

<document source="https://tamnd.github.io/math2/node/7THS519D4A.md" tag="7THS519D4A" kind="statement">
# Budan's theorem

## Statement

Let $p$ be a nonzero polynomial with real coefficients, and let $l < r$ be real numbers.
For $c \in \R$, let $V_c$ be the number of sign changes in the sequence of nonzero coefficients of the polynomial $p(x + c)$.
Let $N$ be the number of [[roots|00EHX6FWC0]] of $p$ in the half-open interval $(l, r]$, counted with [[multiplicity|03XVP8Q4GY]].
Then $N \le V_l - V_r$, and $V_l - V_r - N$ is even.

## Remarks

The roots of $p(x + c)$ are the roots of $p$ shifted by $-c$.
So [[Descartes' rule of signs|T6DQQ36JZ5]] applied to $p(x + l)$ and $p(x + r)$ bounds the roots of $p$ greater than $l$ and greater than $r$.
The theorem says that the difference of the two bounds is again a bound, with the same parity rule.
Example: take $p = x^2 - 2$, $l = 0$ and $r = 2$.
Then $p(x) = x^2 - 2$ has $V_0 = 1$, and $p(x + 2) = x^2 + 4x + 2$ has $V_2 = 0$.
So there is exactly one root in $(0, 2]$, namely $\sqrt 2$.
Take $l = 0$ and $r$ larger than the real part of every complex root of $p$.
Then all coefficients of $p(x + r)$ have the same sign, so $V_r = 0$, and one gets back Descartes' rule.

## Explanations

### L1 human

Budan's theorem bounds the number of roots of a polynomial in an interval by counting sign changes of coefficients.
One shifts the polynomial so that each end of the interval moves to $0$, and then compares the two counts.
Like Descartes' rule, it can overcount, but only by an even number.

### L2 human

The theorem is used in root isolation: an interval with $V_l - V_r = 0$ has no root, and one with $V_l - V_r = 1$ has exactly one.
An equivalent form, due to Fourier, counts sign changes in the sequence $p(c), p'(c), \dots, p^{(n)}(c)$.
By Taylor's formula, these numbers are the coefficients of $p(x + c)$ up to positive factors.
Compared with [[Sturm's theorem|R44CV0KD9T]], it is cheaper to compute but gives only a bound.
</document>

<document source="https://tamnd.github.io/math2/node/7TNH4MR0WP.md" tag="7TNH4MR0WP" kind="definition">
# Distributive property

## Definition

Let $S$ be a set with two [[binary operations|0M418FE7BB]] $\ast$ and $+$.

1. The operation $\ast$ is *left distributive* over $+$ if $a \ast (b + c) = (a \ast b) + (a \ast c)$ for all $a, b, c \in S$.
2. The operation $\ast$ is *right distributive* over $+$ if $(b + c) \ast a = (b \ast a) + (c \ast a)$ for all $a, b, c \in S$.
3. The operation $\ast$ is *distributive* over $+$ if it is both left and right distributive over $+$.

## Remarks

Example: matrix multiplication on the set of $n \times n$ real matrices is distributive over matrix addition, since $A(B + C) = AB + AC$ and $(B + C)A = BA + CA$ by the entrywise formulas.

Non-example: on the set $\R_{>0}$ of positive reals, take ordinary multiplication $\cdot$ in place of $+$, and let $a \ast b = a^b$.
Then $\ast$ is right distributive over $\cdot$, since $(bc)^a = b^a c^a$.
It is not left distributive, since $2^{1 \cdot 1} = 2$ while $2^1 \cdot 2^1 = 4$.

If $\ast$ is commutative, left and right distributivity are equivalent.
In a ring, multiplication is distributive over addition; in a Boolean algebra, each of $\wedge$ and $\vee$ is distributive over the other.
</document>

<document source="https://tamnd.github.io/math2/node/7TP5M906HB.md" tag="7TP5M906HB" kind="definition">
# Supersolvable group

## Definition

Let $G$ be a [[group|G5S1M9]].
The group $G$ is *supersolvable* when it has a chain of subgroups $\{e\} = G_0 \le G_1 \le \cdots \le G_n = G$ with this property.
Each $G_i$ is a [[normal subgroup|0CB0DX5BVV]] of $G$ and each [[quotient group|0KYX2N69J0]] $G_i/G_{i-1}$ is [[cyclic|0KVGH0J5B3]].

## Remarks

Example: $S_3$ is supersolvable with the series $\{e\} \le A_3 \le S_3$.
Here $A_3$ and $S_3$ are normal in $S_3$, $A_3 \cong \Z/3\Z$ and $S_3/A_3 \cong \Z/2\Z$.
Non-example: $A_4$ is not supersolvable.
Its only normal subgroups are $\{e\}$, the Klein four-group $V$ and $A_4$.
So the first term of such a series that differs from $\{e\}$ is $V$ or $A_4$, and the factor at that step is $V$ or $A_4$, which is not cyclic.
Every finitely generated [[nilpotent group|0Q7MMQHC1P]] is supersolvable, and every supersolvable group is [[polycyclic|9RBA56NXX3]], hence [[solvable|0FXJ7YFMMX]].
Some authors write supersoluble.

## Explanations

### L1 human

A group is supersolvable when it can be built from cyclic layers, with every partial stack normal in the whole group.
</document>

<document source="https://tamnd.github.io/math2/node/7TPBQPPN97.md" tag="7TPBQPPN97" kind="concept">
# Cone

## Summary

A cone over a [[diagram|HJK75HDWFE]] $D \colon J \to \mathcal{C}$ is an object $N$, the apex, with a morphism from $N$ to each object $D(j)$ of the diagram.
All triangles formed with the morphisms of the diagram must commute.
Dually, a cocone has morphisms from the objects of the diagram into a common object.
Limits are the universal cones and colimits are the universal cocones.

## Explanations

### L1 human

Picture the diagram as a figure drawn on the floor and the apex $N$ as a point above it, with a string from $N$ down to each object.
The strings must be consistent with the arrows $D(j) \to D(k)$ of the figure.
Going down to $D(j)$ and then along the arrow gives the same result as going down directly to $D(k)$.
For two sets $X$ and $Y$ with no arrows between them, a cone is just a set $N$ with two functions $N \to X$ and $N \to Y$.
</document>

<document source="https://tamnd.github.io/math2/node/7TSXP95QN6.md" tag="7TSXP95QN6" kind="concept">
# Directed-complete partial order

## Summary

A directed-complete partial order, or dcpo, is a partially ordered set in which every directed subset has a supremum.
Dcpos are the basic objects of [[domain theory|Q2P5D74Q1V]], where elements are read as pieces of information and directed sets as consistent ways of collecting information.
Many authors also ask for a least element and call the result a pointed dcpo or a cpo.

## Explanations

### L1 human

Think of the elements as partial results of a computation, ordered by "is less defined than".
A directed set is a collection of partial results in which any two have a common improvement inside the collection.
In a dcpo, such a collection always has a best combined result, its supremum.
Every complete lattice is a dcpo.
The partial functions from $\N$ to $\N$, ordered by extension, form a dcpo: the supremum of a directed family of partial functions is their union.
</document>

<document source="https://tamnd.github.io/math2/node/7TVFY6NZDB.md" tag="7TVFY6NZDB" kind="definition">
# Weak Lp space

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $0 < p < \infty$.
For a [[measurable function|YYNCJSR0PJ]] $f$ on $X$ with real or complex values, let
$$\norm{f}_{p,\infty} = \sup_{t > 0} \, t \, \mu(\set{x \in X : \abs{f(x)} > t})^{1/p}.$$
The *weak $L^p$ space* $L^{p,\infty}(\mu)$ is the set of measurable functions $f$ with $\norm{f}_{p,\infty} < \infty$, modulo equality [[almost everywhere|RARTMA0ZEC]].

## Remarks

Example: on $\R^n$ with Lebesgue measure, $f(x) = \abs{x}^{-n/p}$ lies in $L^{p,\infty}$.
The set $\set{\abs{f} > t}$ is the open ball of radius $t^{-p/n}$, with measure $\omega_n t^{-p}$, where $\omega_n$ is the volume of the unit ball.
So $\norm{f}_{p,\infty} = \omega_n^{1/p}$, while $\int \abs{f}^p \, dx = \int \abs{x}^{-n} \, dx = \infty$.

Non-example: on $(0, 1)$ with Lebesgue measure, $g(x) = x^{-2}$ is not in $L^{1,\infty}$.
For $t > 1$, $\set{g > t} = (0, t^{-1/2})$, so $t \, \mu(\set{g > t}) = t^{1/2}$, which is unbounded.

Chebyshev's inequality $t^p \mu(\set{\abs{f} > t}) \le \int \abs{f}^p \, d\mu$ gives $\norm{f}_{p,\infty} \le \norm{f}_p$, so $L^p \subseteq L^{p,\infty}$.
The map $\norm{\cdot}_{p,\infty}$ is a quasinorm, not a norm.
The triangle inequality holds only with a constant: $\norm{f + g}_{p,\infty} \le \max(2, 2^{1/p}) (\norm{f}_{p,\infty} + \norm{g}_{p,\infty})$.
It follows from $\set{\abs{f + g} > t} \subseteq \set{\abs{f} > t/2} \cup \set{\abs{g} > t/2}$.
</document>

<document source="https://tamnd.github.io/math2/node/7TXCCCA17A.md" tag="7TXCCCA17A" kind="statement">
# Log-modulus of a holomorphic function is subharmonic

## Statement

Let $\Omega \subseteq \C$ be an [[open set|02BH3MCBQB]] and let $f$ be [[holomorphic|0NAHXR1FRH]] on $\Omega$, not identically zero on any connected component of $\Omega$.
Then the function $u = \log\abs{f}$, with $u(z) = -\infty$ where $f(z) = 0$, is [[subharmonic|Z1MHMGFNFW]] on $\Omega$.
Moreover $u$ is [[harmonic|0DEB0JMN0W]] on the open set $\set{z \in \Omega : f(z) \ne 0}$.

## Remarks

Proof idea: let $a$ be a point with $f(a) \ne 0$.
On a small disc about $a$ there is a holomorphic $g$ with $f = e^{g}$, so $u = \operatorname{Re} g$ there.
The [[real part of a holomorphic function is harmonic|031BG625Y3]], so $u$ is harmonic near $a$.
At a zero $a$ of $f$, $u(a) = -\infty$, and $-\infty$ is at most any mean of $u$, so the sub-mean inequality holds at $a$.
The function $u$ is continuous as a map into $[-\infty, \infty)$, so it is upper semicontinuous.

Example: for $f(z) = z$, $u(z) = \log\abs{z}$ is harmonic off $0$ and equals $-\infty$ at $0$.
Its mean over the circle of radius $r$ about $0$ is $\log r > -\infty = u(0)$.
Composing with the convex increasing function $t \mapsto e^{pt}$ shows that $\abs{f}^p$ is subharmonic for every $p > 0$, by [[convex functions of subharmonic functions|E820CTMJVM]].
The function $\log\abs{f}$ is not harmonic on all of $\Omega$ when $f$ has zeros.
Its Laplacian in the sense of distributions is $2\pi \sum_a m_a \delta_a$, where $a$ runs over the zeros of $f$ and $m_a$ is the multiplicity of $a$.

## Explanations

### L1 human

Take a holomorphic function, look at the size $\abs{f}$ of its values, and take the logarithm.
The result is harmonic, except at the zeros of $f$, where it dives down to $-\infty$.
Those dives only push values below the average, so the function is subharmonic.

### L2 human

This is the main bridge from complex analysis to potential theory.
Applied to $\log\abs{f}$, the maximum principle for subharmonic functions gives the maximum modulus principle, and the convexity results for subharmonic functions give Hadamard's three-circle theorem.
Jensen's formula is the mean value formula for $\log\abs{f}$, with the zeros of $f$ as the correction term.
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# Incomplete Cholesky factorization

## Definition

Let $A$ be a real symmetric [[positive definite|0KX66CZMBB]] $n \times n$ [[sparse matrix|JGACZYEEY7]].
Let $S \subseteq \set{(i, j) : 1 \le j \le i \le n}$ be a set of positions that contains every diagonal position $(i, i)$.
An *incomplete Cholesky factorization* of $A$ with pattern $S$ is a [[lower triangular|GTG8MJGDKC]] matrix $L$ with positive diagonal such that:

1. $l_{ij} = 0$ for every $i \ge j$ with $(i, j) \notin S$;
2. $(L L^T)_{ij} = a_{ij}$ for every $(i, j) \in S$.

When $S$ is the set of positions $(i, j)$ with $i \ge j$ and $a_{ij} \ne 0$, the factorization is called *IC(0)*.

## Remarks

If $S$ contains all positions with $i \ge j$, the incomplete factor is the exact [[Cholesky factor|0WKG8CYY6A]].
Example: let $A$ be tridiagonal and symmetric positive definite.
Its exact Cholesky factor is lower bidiagonal, so it vanishes outside the pattern of $A$.
Hence the IC(0) factor exists and is the exact factor.
Non-example: for the $2 \times 2$ identity matrix, the matrix $L$ with rows $(1, 0)$ and $(1, 1)$ and the pattern of all positions satisfies clause 1.
It fails clause 2, since $(L L^T)_{21} = 1 \ne 0 = a_{21}$.
The factor is computed by the Cholesky recurrences with each update outside $S$ discarded.
For a general symmetric positive definite $A$, the process can meet a nonpositive pivot, and then no such $L$ is produced.
It exists for every pattern when $A$ is a symmetric [[M-matrix|DNKXHMGQY1]], a result of Meijerink and van der Vorst.
Variants choose $S$ by levels of fill or drop entries below a threshold.
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# Hadwiger's theorem

## Statement

Let $n \ge 1$ and let $\mathcal{K}^n$ be the set of nonempty compact convex subsets of $\R^n$, with the [[Hausdorff distance|F79WYYC07C]].
Let $\varphi \colon \mathcal{K}^n \to \R$ be a [[valuation|V0RBKHYTV7]] that is continuous and invariant under the [[rigid motions|0H3YY8CPKH]] of $\R^n$.
Then there are real numbers $c_0, c_1, \dots, c_n$ such that
$$\varphi(K) = \sum_{j=0}^n c_j V_j(K) \quad \text{for all } K \in \mathcal{K}^n,$$
where $V_0, \dots, V_n$ are the [[intrinsic volumes|C2CK6KSSTV]].

## Remarks

Conversely, each $V_j$ is a continuous valuation invariant under rigid motions, so these valuations form a real vector space of dimension $n + 1$ with basis $V_0, \dots, V_n$.
A short proof by Klain rests on a lemma about simple valuations.
The lemma says: a continuous valuation invariant under rigid motions that vanishes on all sets of dimension below $n$ is a multiple of volume.
Induction on $n$ then handles the restriction of $\varphi$ to lower-dimensional sets.

Example: surface area in $\R^n$ is a continuous valuation invariant under rigid motions, and the theorem identifies it as $2 V_{n-1}$.

Invariance under rotations is needed.
In $\R^2$ the width in the direction of the first axis, $\varphi(K) = \max_{x \in K} x_1 - \min_{x \in K} x_1$, is a continuous valuation invariant under translations.
It gives $1$ to the segment $[0,1] \times \set{0}$ and $0$ to the congruent segment $\set{0} \times [0, 1]$, so it is not a combination of intrinsic volumes.

## Explanations

### L1 human

There are many ways to measure the size of a convex shape.
Suppose a measure is additive like an area, does not change when the shape is moved, and changes little when the shape changes little.
Hadwiger's theorem says that such a measure is built from a short list of basic measures.
In space the list has four entries: volume, surface area, mean width and the Euler characteristic.

### L2 human

The theorem is the foundation of integral geometry in Euclidean space.
It gives quick proofs of the principal kinematic formula and of Crofton-type formulas: one checks that both sides are such valuations and compares them on balls.
It also explains why intrinsic volumes appear in probability, for example in the expected Euler characteristic of random sets.
Its classification has been extended by Alesker and others to valuations invariant under smaller groups.
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# Kakutani fixed-point theorem

## Statement

Let $n \ge 1$ and let $S \subseteq \R^n$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].
Let $\varphi$ be a set-valued map that assigns to each $x \in S$ a subset $\varphi(x) \subseteq S$, such that:

1. $\varphi(x)$ is nonempty and convex for every $x \in S$;
2. the graph $\set{(x, y) \in S \times S : y \in \varphi(x)}$ is a [[closed set|0ZCK8DQYN1]].

Then $\varphi$ has a fixed point: there is $x \in S$ with $x \in \varphi(x)$.

## Remarks

When each $\varphi(x)$ is a single point $f(x)$, the graph is closed exactly when $f$ is continuous, and the theorem becomes the [[Brouwer fixed-point theorem|N7AAJ2MW5T]].
The usual proof approximates $\varphi$ by continuous maps on finer and finer triangulations of a simplex containing $S$.
It applies Brouwer's theorem to these maps and passes to a limit using the closed graph and the convex values.

Convex values are needed.
On $S = [0, 1]$, let $\varphi(x) = \set{1}$ for $x < 1/2$, $\varphi(1/2) = \set{0, 1}$ and $\varphi(x) = \set{0}$ for $x > 1/2$.
The graph is $[0, 1/2] \times \set{1} \cup [1/2, 1] \times \set{0}$, which is closed, but no $x$ lies in $\varphi(x)$; the value $\varphi(1/2)$ is not convex.

Example: consider a two-player game with mixed strategy sets $S_1$ and $S_2$, which are simplices.
Let $B_1(x_2) \subseteq S_1$ be the set of best replies of player 1 to the strategy $x_2$ of player 2, and define $B_2(x_1) \subseteq S_2$ in the same way.
On $S = S_1 \times S_2$, let $\varphi(x_1, x_2) = B_1(x_2) \times B_2(x_1)$.
Its values are nonempty and convex and its graph is closed, so it has a fixed point, which is a Nash equilibrium.

## Explanations

### L1 human

Brouwer's theorem is about a map that sends each point to one point.
Kakutani's theorem allows each point to be sent to a whole convex set of points, and still finds a point that lies in its own image.
This is what one needs when a choice is not unique, such as the best reply to an opponent's move in a game.

### L2 human

Nash used the theorem in his proof of the existence of equilibria in finite games, and it is basic in general equilibrium theory in economics.
The closed graph condition is a form of upper hemicontinuity, which is equivalent to it when the values are closed in a compact $S$.
The Kakutani–Glicksberg–Fan theorem extends the result to compact convex subsets of locally convex spaces, just as Schauder and Tychonoff extend Brouwer's theorem.
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# Lebesgue constant

## Summary

The Lebesgue constant of a set of interpolation nodes measures how much [[polynomial interpolation|SS1YMS2NPC]] at these nodes can amplify errors.
It is the [[operator norm|06EKW1AKXF]] of the interpolation map on continuous functions with the uniform norm.
It compares the interpolation error with the error of best approximation, and it measures the sensitivity of interpolation to errors in the data.
For equally spaced nodes it grows exponentially with the degree, while for [[Chebyshev nodes|HYK8J56PSH]] it grows only like a logarithm.

## Explanations

### L1 human

Suppose the data values at the nodes are each wrong by at most $\varepsilon$.
The interpolating polynomial may then be wrong by up to $\Lambda \varepsilon$ somewhere, where $\Lambda$ is the Lebesgue constant.
For the nodes $-1, 0, 1$ the constant is $5/4$, so the amplification is mild.
For $20$ equally spaced nodes it is already in the thousands, which is one cause of Runge's phenomenon.
</document>

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# Measure

## Definition

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]], and let $[0, \infty]$ be the non-negative part of the [[extended real number line|0GSZ7B0CGR]].
A *measure* on $(X, \Sigma)$ is a function $\mu \colon \Sigma \to [0, \infty]$ such that:

1. $\mu(\varnothing) = 0$.
2. $\mu$ is [[countably additive|D7KB9HFCGZ]]: for every sequence $A_1, A_2, \ldots$ of pairwise disjoint sets in $\Sigma$,
$$\mu\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} \mu(A_n).$$

## Remarks

The [[series|0NM626FH6Q]] in clause 2 has non-negative terms, so it has a value in $[0, \infty]$.
For $x \in X$, the [[Dirac measure|0ACRGYNHSR]] $\delta_x$, with $\delta_x(A) = 1$ if $x \in A$ and $\delta_x(A) = 0$ otherwise, is a measure.
For clause 2, the point $x$ lies in at most one of the disjoint sets $A_n$, so both sides are $1$ or both are $0$.
On the power set of $\N$, the function with $\nu(\varnothing) = 0$ and $\nu(A) = 1$ for $A \ne \varnothing$ is not a measure.
Clause 2 fails: $\nu(\set{1} \cup \set{2}) = 1$, while $\nu(\set{1}) + \nu(\set{2}) = 2$.
A measure is monotone, $\mu(A) \le \mu(B)$ when $A \subseteq B$, and countably subadditive, $\mu(\bigcup_n A_n) \le \sum_n \mu(A_n)$.
A measure with $\mu(X) = 1$ is a probability measure.
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# Symbolic power

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]], let $P$ be a [[prime ideal|06HC6HK0Z8]] of $R$, and let $n \ge 1$ be an integer.
The $n$-th *symbolic power* of $P$ is the ideal
$$P^{(n)} = \set{r \in R : sr \in P^n \text{ for some } s \in R \setminus P}.$$

## Remarks

Equivalently, $P^{(n)}$ is the preimage in $R$ of the ideal $P^n R_P$ of the [[localization|X6JD4AG973]] $R_P$.
One always has $P^n \subseteq P^{(n)} \subseteq P$, and $P^{(1)} = P$.
When $R$ is Noetherian, $P^{(n)}$ is the $P$-primary component of $P^n$, and it is the smallest $P$-[[primary ideal|V026AAV8E6]] that contains $P^n$.
Example: in $R = k[x, y, z]/(xy - z^2)$ over a field $k$, with $P = (x, z)$, one has $y \notin P$ and $yx = z^2 \in P^2$, so $x \in P^{(2)}$; but $x \notin P^2$, as $P^2$ is generated by elements of degree $2$.
Non-example: in the same ring, $y \notin P^{(2)}$, since $y \notin P$ and $P^{(2)} \subseteq P$.
For a maximal ideal $\mathfrak{m}$, the power $\mathfrak{m}^n$ is already primary, so $\mathfrak{m}^{(n)} = \mathfrak{m}^n$.

## Explanations

### L1 human

The symbolic power keeps the elements that lie in $P^n$ once one is allowed to clear away factors outside $P$.
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# Affine space as an algebra

## Example

Let $F$ be a [[field|04C5ZBG1WT]] and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
The [[affine space|09XNAPTJQ7]] of $V$, seen as an algebra in the sense of universal algebra, is the set $V$ with these operations:

1. the [[ternary operation|WC47MFR6GW]] $p(x, y, z) = x - y + z$;
2. for each $r \in F$, the binary operation $b_r(x, y) = (1 - r)x + ry$.

There is no constant and no addition of two points.
Each operation is an [[affine combination|06HYY6DK6X]], since its coefficients sum to $1$.

Check that every affine combination is a term in these operations.
Take $x_1, \dots, x_n \in V$ and $\lambda_1, \dots, \lambda_n \in F$ with $\lambda_1 + \cdots + \lambda_n = 1$.
Then $\sum_i \lambda_i x_i = x_1 + \sum_{i \ge 2} \lambda_i (x_i - x_1)$.
Each point $b_{\lambda_i}(x_1, x_i) = x_1 + \lambda_i (x_i - x_1)$ is a term.
For points $x_1 + a$ and $x_1 + c$, the term $p(x_1 + a, x_1, x_1 + c)$ equals $x_1 + a + c$.
Applying $p$ repeatedly therefore gives $x_1 + \sum_{i \ge 2} \lambda_i (x_i - x_1)$.
So the subalgebras are exactly the subsets closed under affine combinations, that is, the affine subspaces and the empty set.

## Remarks

Every translation $x \mapsto x + v$ is an automorphism of this algebra, because each operation is an affine combination.
A translation by $v \ne 0$ does not fix $0$, so it is not a linear map of $V$.
This shows that the algebra forgets the origin of $V$.
The term $p$ satisfies $p(x, x, y) = y$ and $p(x, y, y) = x$, so it is a [[Mal'cev term|Y203PPTS61]].

## Explanations

### L1 human

A vector space has a special point, the zero vector, but a plane in space has no special point.
The affine space keeps only the operations that do not depend on where zero is.
One may take weighted averages of points, such as the midpoint, but one may not add two points.
For example, in $\R^2$ the midpoint of $(0, 0)$ and $(2, 4)$ is $b_{1/2}((0,0), (2,4)) = (1, 2)$.
Moving every point by the same vector keeps all these operations, so such a move is a symmetry of the algebra.
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# Bisection method

## Statement

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a [[continuous function|0X3WGX95FE]] with $f(a) f(b) < 0$.
The *bisection method* sets $a_0 = a$ and $b_0 = b$.
For $k \ge 0$ it lets $m_k = (a_k + b_k)/2$ and proceeds as follows.
If $f(m_k) = 0$ it stops with the zero $m_k$; if $f(a_k) f(m_k) < 0$ it sets $(a_{k+1}, b_{k+1}) = (a_k, m_k)$; otherwise it sets $(a_{k+1}, b_{k+1}) = (m_k, b_k)$.
If the method does not stop, then for every $k \ge 0$:

1. $f(a_k) f(b_k) < 0$, so $[a_k, b_k]$ contains a zero of $f$;
2. $b_k - a_k = (b - a)/2^k$;
3. the sequences $(a_k)$, $(b_k)$ and $(m_k)$ converge to the same zero $r$ of $f$, and $\abs{m_k - r} \le (b - a)/2^{k+1}$.

## Remarks

Clause 1 holds by induction, since the new interval keeps an endpoint where $f$ is negative and one where it is positive.
The [[intermediate value theorem|00KAFBR32E]] then gives a zero in it.
Clause 2 is immediate, and the [[nested intervals theorem|MF56GK7QY2]] gives a common limit $r$ of $a_k$ and $b_k$; by continuity $f(r)^2 = \lim f(a_k) f(b_k) \le 0$, so $f(r) = 0$.
Example: for $f(x) = x^2 - 2$ on $[1, 2]$, the midpoints are $1.5, 1.25, 1.375, 1.4375, \ldots$, and $20$ steps locate $\sqrt{2}$ within $2^{-21} < 5 \times 10^{-7}$.
Continuity is needed: $f(x) = 1/x$ for $x \ne 0$ and $f(0) = 1$ changes sign on $[-1, 1]$, and bisection gives $[-1, 0], [-1/2, 0], \ldots$ and converges to $0$, which is not a zero.
The error bound halves at each step, so the method converges linearly with ratio $1/2$ in the sense of the error bound, whatever $f$ is.

## Explanations

### L1 human

If a continuous function is negative at one end of an interval and positive at the other, it must cross zero somewhere in between.
Check the middle: the sign there tells which half still has a crossing.
Keep halving.
After $k$ steps the zero is trapped in an interval $2^k$ times shorter than at the start.

### L2 human

Bisection is the simplest [[root-finding algorithm|TR4CNS5PMJ]] and the most robust: it needs only continuity and a sign change, and its error bound is known in advance.
It gains one binary digit per step, so it is slow compared with [[Newton's method|07EXVNF9VP]] or the [[secant method|9ZC177PVPC]].
Practical methods such as [[Brent's method|5F76XA1ABC]] combine a fast open method with bisection steps to keep the guarantee.
It is also the constructive content of the intermediate value theorem.
</document>

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# Face of a convex set

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]] and let $C \subseteq V$ be a [[convex set|02AS45T3H9]].
A *face* of $C$ is a subset $F \subseteq C$ such that:

1. $F$ is convex;
2. $F$ is an [[extreme set|WD3RJZC5A7]] of $C$: if $x_1, x_2 \in C$ and the [[open segment|CCZA91NE5X]] $(x_1, x_2)$ meets $F$, then $x_1, x_2 \in F$.

## Remarks

The empty set and $C$ are faces of $C$.
A one-point set $\set{x}$ is a face exactly when $x$ is an [[extreme point|3MSGFJM2NX]] of $C$.
A face of a face of $C$ is a face of $C$.

Example: for the square $C = [0, 1]^2$, the edge $F = [0, 1] \times \set{1}$ is a face.
It is convex.
If a point of $F$ lies strictly between $x_1, x_2 \in C$, the second coordinates of $x_1, x_2$ are at most $1$ and average to $1$ with positive weights.
So both equal $1$, and $x_1, x_2 \in F$.

Non-example: the lower half $F = [0, 1] \times [0, 1/2]$ of the same square is convex, but the point $(1/2, 1/2) \in F$ lies strictly between $(1/2, 0)$ and $(1/2, 1)$, and $(1/2, 1) \notin F$, so clause 2 fails.

If $H$ is a [[supporting hyperplane|K7N565NKCV]] of $C$, then $C \cap H$ is a face, called an exposed face.
Not every face is exposed: in the stadium, the convex hull of two unit discs centred at $(\pm 1, 0)$, the point $(1, 1)$ is a face but not an exposed face.
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# Irreducible analytic germ

## Summary

An irreducible analytic germ is a nonempty [[germ of an analytic set|Q9B74BYZ8C]] that is not the union of two strictly smaller analytic germs.
Equivalently, its ideal in the ring of holomorphic germs is a prime ideal.
Since this ring is Noetherian, every analytic germ is a finite union of irreducible germs.
The irreducible germs are the building blocks of local analytic geometry.

## Explanations

### L1 human

Near a point, an analytic set may consist of several pieces that only meet at the point, like two lines crossing.
An irreducible germ is one that cannot be split in this way.
The cusp $\set{z_1^2 = z_2^3}$ is irreducible at the origin, although it is singular there.
The node $\set{z_2^2 = z_1^2(1 + z_1)}$ is irreducible as a global curve, but its germ at the origin is the union of two smooth branches, so it is reducible there.
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# Stably finite ring

## Summary

A ring is stably finite when all its matrix rings are [[Dedekind-finite|GV1HKEAWVN]].
Being Dedekind-finite does not pass to matrix rings in general, so this is a stronger condition.
Stably finite nonzero rings have invariant basis number, so the rank of a finitely generated free module over them is well defined.

## Explanations

### L1 human

A ring is Dedekind-finite when a one-sided inverse is always two-sided.
Stable finiteness asks the same thing for square matrices of every size over the ring.
Commutative rings are stably finite: for square matrices over them, $AB = I$ gives $\det(A)\det(B) = 1$, and then $A$ is invertible.
So the rule "a left inverse of a square matrix is a right inverse" holds there in every size.
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# Normed algebra

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$.
A *normed algebra* over $\mathbb{K}$ is an [[associative algebra|3CTW1P4HBP]] $A$ over $\mathbb{K}$ with a [[norm|0V2WP9B1MF]] $\norm{\cdot}$ on the vector space $A$ such that
$$\norm{x y} \le \norm{x} \norm{y} \quad \text{for all } x, y \in A.$$
If $A$ has a unit $1$, many authors also ask $\norm{1} = 1$.

## Remarks

The inequality makes the product $A \times A \to A$ continuous.
A normed algebra that is complete as a [[normed vector space|03J5RAKB30]] is a [[Banach algebra|9KYWN94QDN]].

Example: the algebra $C([0,1])$ of continuous real functions with $\norm{f}_\infty = \sup_t \abs{f(t)}$.
For each $t$, $\abs{f(t) g(t)} \le \norm{f}_\infty \norm{g}_\infty$, so $\norm{fg}_\infty \le \norm{f}_\infty \norm{g}_\infty$.
The polynomial functions form a subalgebra, which is a normed algebra that is not complete.

Non-example: $M_2(\R)$ with the norm $\norm{a} = \max_{i,j} \abs{a_{ij}}$.
For the matrix $a$ with all entries $1$, $a^2$ has all entries $2$, so $\norm{a^2} = 2 > 1 = \norm{a}^2$.
The operator norm on $M_2(\R)$, in contrast, does make it a normed algebra.
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# Waring's problem

## Problem

Let $k \ge 2$ be an [[integer|0C4JS9RF5M]].
Let $g(k)$ be the least $s$ such that every positive integer is a sum of at most $s$ $k$-th powers of positive integers.
Let $G(k)$ be the least integer $s$ such that every sufficiently large integer is a sum of at most $s$ $k$-th powers of positive integers.
Waring's problem asks:

1. whether $g(k)$ exists for every $k \ge 2$;
2. what the values of $g(k)$ and $G(k)$ are.

## Remarks

Hilbert proved in 1909 that $g(k)$ exists for every $k$ (the Hilbert–Waring theorem).
By definition $G(k) \le g(k)$.
[[Lagrange's four-square theorem|0KYYSMEVHH]] gives $g(2) = 4$, and $G(2) = 4$ because no integer $n \equiv 7 \pmod 8$ is a sum of three squares.
Known values include $g(3) = 9$ and $g(4) = 19$; the numbers $23$ and $79$ need $9$ cubes and $19$ fourth powers.
The lower bound $g(k) \ge 2^k + \floor{(3/2)^k} - 2$ comes from the integer $n = 2^k \floor{(3/2)^k} - 1$, which is less than $3^k$ and so must be written with the powers $1^k$ and $2^k$ only.
Equality is known for all $k$ outside a possible finite set, and the possible exceptions have been excluded far beyond any small $k$.
For $G(k)$ much less is known: $G(4) = 16$ (Davenport), and $4 \le G(3) \le 7$, the upper bound being due to Linnik.
The [[Hardy–Littlewood circle method|ZQFBEX62D0]] gives an asymptotic formula for the number of representations when the number of powers is large compared with $k$.

## Explanations

### L1 human

Every positive integer is a sum of four squares, for example $7 = 4 + 1 + 1 + 1$.
Waring asked whether the same is true for cubes, fourth powers and so on, with some fixed number of terms.
The answer is yes: for instance, every positive integer is a sum of at most nine cubes.
The hard part is to find the smallest number of terms that always suffices, or that suffices for all large numbers.
</document>

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# Ring with involution

## Summary

A ring with involution, or $*$-ring, is a [[ring|0B1261B5ED]] with an additive map $a \mapsto a^*$ that reverses products and is its own inverse.
It abstracts complex conjugation of numbers and the conjugate transpose of matrices.
When the ring is also an algebra over a commutative $*$-ring and the involution is conjugate-linear, it is a $*$-algebra.
These structures underlie C*-algebras, where the involution is tied to a norm, and the study of Hermitian and unitary elements.

## Explanations

### L1 human

An [[involution|GP56NBN9RY]] is an operation that undoes itself when applied twice.
In a ring with involution this operation also reverses the order of products, as taking inverses does in a group: $(ab)^* = b^* a^*$.
Complex conjugation on $\C$ is the simplest example, and the conjugate transpose of complex matrices is the most important one.
Elements with $a^* = a$, such as real numbers or Hermitian matrices, play the role of "real" elements.
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# Superposition of solutions

## Statement

Let $K$ be a field, let $d \ge 1$, and let $a_0, \dots, a_{d-1} \colon \N \to K$ be sequences.
For a sequence $u \colon \N \to K$, let $Lu$ be the sequence
$$(Lu)(n) = u_{n+d} + a_{d-1}(n)\, u_{n+d-1} + \cdots + a_0(n)\, u_n .$$
Then the following hold.

1. If $Lu = 0$ and $Lv = 0$, then $L(\alpha u + \beta v) = 0$ for all $\alpha, \beta \in K$.
2. If $Lu = g_1$ and $Lv = g_2$, then $L(\alpha u + \beta v) = \alpha g_1 + \beta g_2$ for all $\alpha, \beta \in K$.
3. If $Lp = g$, then a sequence $u$ satisfies $Lu = g$ if and only if $u = p + h$ for a sequence $h$ with $Lh = 0$.

## Remarks

All three parts follow from the linearity of $L$: $L(\alpha u + \beta v) = \alpha Lu + \beta Lv$, which one checks term by term.
For part 3, if $Lu = g$ then $L(u - p) = g - g = 0$, and conversely $L(p + h) = g + 0 = g$.
Example: for $u_{n+1} - 2 u_n = 1$, the constant $p_n = -1$ is a particular solution, and the [[homogeneous|8Y28JQB0Q5]] solutions are $c\, 2^n$, so all solutions are $c\, 2^n - 1$.
Linearity is necessary: for $u_{n+1} = u_n^2$, the sequence $u_n = 1$ is a solution, but $2u$ is not, since $2 \ne 4$.
The same principle holds for linear differential equations; see the [[superposition principle|0F2VBPXNP9]].

## Explanations

### L1 human

For a linear recurrence, you can add solutions and scale them, and the result is again a solution of the matching equation.
So to solve an inhomogeneous recurrence, find one solution of it by any trick, then add all solutions of the homogeneous recurrence.

### L2 human

Part 1 says that the solutions of a homogeneous linear recurrence form a vector space over $K$.
Since the leading coefficient is $1$, the first $d$ terms determine a solution, by the same induction as in [[uniqueness given initial values|WREYNFJHH2]]; so this space has dimension $d$.
Part 3 says that the solutions of the inhomogeneous recurrence form an affine space, a translate of that vector space.
This is the base of the method of undetermined coefficients and of [[variation of parameters|QQFYC988SJ]] for difference equations.
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# Comodule

## Summary

A comodule over a [[coalgebra|A06RA8KB0T]] $C$ is a vector space $M$ with a coaction $\rho \colon M \to M \otimes C$, the dual notion of a [[module|0NZ8JA25WK]] over an algebra.
Comodules over the coordinate ring of an affine algebraic group are the same as its rational representations.
Comodules over the group algebra $KG$ with its standard coalgebra structure are the same as $G$-graded vector spaces.
Comodules are as basic for coalgebras and [[Hopf algebras|7XFYP0N84H]] as modules are for rings.

## Explanations

### L1 human

A module lets algebra elements act on vectors.
A comodule goes the other way: each vector is expanded as a sum of vectors tagged by coalgebra elements.
Take the coalgebra with basis a set $S$ and $\Delta(s) = s \otimes s$.
A comodule over it is just a vector space split into pieces $M_s$, one for each $s$, and the coaction tags each vector of $M_s$ with $s$.
This is how gradings, such as the degree of polynomials, appear as comodule structures.
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<document source="https://tamnd.github.io/math2/node/7WFKPAMQFG.md" tag="7WFKPAMQFG" kind="concept">
# Mixed-type equation

## Summary

A mixed-type equation is a partial differential equation that is elliptic in one part of its domain and hyperbolic in another part.
The two regions are separated by a set where the equation degenerates, often a parabolic line.
The basic example is the [[Tricomi equation|21DD2334NR]].
Such equations arise in the study of transonic flow, where the speed of a gas crosses the speed of sound.
See the [[definition|GKFDKRHEP4]].

## Explanations

### L1 human

The type of a second-order equation can change from point to point when its coefficients vary.
In the Tricomi equation $u_{yy} + y \, u_{xx} = 0$, the upper half plane $y > 0$ is elliptic and the lower half plane $y < 0$ is hyperbolic.
So in one region the equation behaves like the Laplace equation and in the other like the wave equation.
For gas flow past an airplane wing, the flow is slower than sound in one region and faster in another, and this is the physical origin of such equations.
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# Simple function

## Summary

A simple function is a [[measurable function|YYNCJSR0PJ]] that takes only finitely many values.
It is a finite combination of [[indicator functions|0RCQZ36MME]] of measurable sets, so its integral is a finite sum.
The Lebesgue integral of a nonnegative function is defined by approximating it from below by simple functions.

## Explanations

### L1 human

A simple function is like a staircase, except that each step may sit on a complicated set instead of an interval.
On each set it has a constant value, and there are only finitely many sets.
For example, the function on $[0, 1]$ that is $2$ on the rational numbers and $5$ on the irrational numbers is simple.
Its integral is $2$ times the measure of the rationals plus $5$ times the measure of the irrationals, which is $0 + 5 = 5$.
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# Tame Fréchet space

## Summary

A tame Fréchet space is a graded [[Fréchet space|3GAA614673]] of a special kind.
Up to tame linear isomorphism, it is a direct summand of a space of exponentially decreasing sequences in a Banach space.

The notion is due to Richard Hamilton.
Tame spaces carry smoothing operators, and this is what the [[Nash–Moser theorem|9ZCCXT3KKP]] needs to run Newton's method with loss of derivatives.
The basic example is the space of smooth functions on a compact manifold, graded by the $C^n$ norms.

## Explanations

### L1 human

A space of smooth functions is measured by a whole ladder of norms: the size of the function, of its first derivatives, of its second derivatives, and so on.
Many natural operators, such as differential operators, cost a fixed number of rungs of this ladder.
A tame Fréchet space is one in which the ladder behaves like the ladder of weights $e^{nk}$ on a sequence $(f_k)$.
Then one can smooth an element, trading rungs in a controlled way, and this makes hard implicit function theorems possible.
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# Saturated model

## Definition

Let $L$ be a signature, let $M$ be an $L$-structure, and let $\kappa$ be an infinite [[cardinal|0EBACQFAZW]].
The structure $M$ is *$\kappa$-saturated* if the following holds for every set $A$ of elements of $M$ with $\abs{A} < \kappa$.
Every complete $1$-[[type|9V71Q3SD1G]] over $A$ is realized in $M$.
The structure $M$ is *saturated* if it is infinite and $\abs{M}$-saturated, where $\abs{M}$ is the [[cardinality|0G0ASKNA6Z]] of its domain.

## Remarks

A $\kappa$-saturated structure also realizes every complete $n$-type over fewer than $\kappa$ parameters, for every $n \ge 1$.

Example: $(\Q, <)$ is saturated.
A complete $1$-type over a finite set $A \subseteq \Q$ says where $x$ lies relative to the elements of $A$.
Either $x$ equals some $a \in A$, or $x$ lies in one of the open intervals cut out by $A$, including the two unbounded ones.
Each of these intervals contains a rational number, so the type is realized.

Non-example: the field $\overline{\Q}$ of algebraic numbers is a model of the theory of algebraically closed fields of characteristic $0$, but it is not $\aleph_0$-saturated.
The set $\set{f(x) \ne 0 : f \in \Z[x], f \ne 0}$ is consistent with its theory over $A = \emptyset$, as $\C$ shows, and it extends to a complete type.
No element of $\overline{\Q}$ realizes it, since every algebraic number is a root of a nonzero integer polynomial.

Every structure has a $\kappa$-saturated elementary extension for each $\kappa$.
Saturated models of the same complete theory and the same cardinality are isomorphic.
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# Sturm–Liouville theory

## Summary

Sturm–Liouville theory studies the eigenvalue problems $-(p y')' + q y = \lambda w y$ on an interval, with boundary conditions at the ends.
For a regular problem with separated boundary conditions, the eigenvalues are real and simple and form a sequence $\lambda_1 < \lambda_2 < \cdots$ that tends to $+\infty$.
The $n$-th eigenfunction has exactly $n - 1$ zeros inside the interval, and the eigenfunctions form an orthogonal basis of the weighted space $L^2((a, b), w)$.
The theory explains Fourier series and many classical special functions, and it is the model case of the spectral theory of [[self-adjoint operators|0NEGN5CRSC]].

## Explanations

### L1 human

A vibrating string of varying thickness can only vibrate at certain frequencies.
Sturm–Liouville theory finds these frequencies as the [[eigenvalues|0BQAA2BK3Q]] of a second-order differential equation with conditions at the two ends of the string.
For a uniform string on $[0, \pi]$ fixed at both ends, the equation is $-y'' = \lambda y$ with $y(0) = y(\pi) = 0$.
The eigenvalues are $1, 4, 9, \ldots$ with the shapes $\sin x, \sin 2x, \sin 3x, \ldots$.
The $n$-th shape crosses zero $n - 1$ times between the ends, and the shapes are building blocks for any other shape.
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# Parallelizable manifold

## Summary

A smooth manifold is parallelizable if it carries a global frame.
This is a family of [[vector fields|0299VCW1TH]] that form a basis of the tangent space at every point.
Equivalently, its [[tangent bundle|WY6K3P4YCG]] is trivial.
Lie groups are parallelizable, while the $2$-sphere is not, by the hairy ball theorem.
Among the spheres, exactly $S^0$, $S^1$, $S^3$ and $S^7$ are parallelizable, a fact tied to the real division algebras.

## Explanations

### L1 human

On a parallelizable manifold you can set up "north" and "east" (and further directions) at every point at once, smoothly, and with no point where they collapse.
The plane and the torus allow this: use the two coordinate directions everywhere.
The round $2$-sphere does not: by the hairy ball theorem every tangent vector field on it vanishes somewhere, so not even one direction can be chosen everywhere.
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# Interior algebra

## Definition

Let $B$ be a [[Boolean algebra|06GKBTAA5E]] with meet $\wedge$, greatest element $1$ and order $\le$.
An *interior operator* on $B$ is a map $I \colon B \to B$ such that for all $x, y \in B$:

1. $I(1) = 1$;
2. $I(x) \le x$;
3. $I(I(x)) = I(x)$;
4. $I(x \wedge y) = I(x) \wedge I(y)$.

An *interior algebra* is a pair $(B, I)$ of a Boolean algebra $B$ and an interior operator $I$ on $B$.

## Remarks

Example: let $X$ be a [[topological space|0CNQXVBHCX]] and let $B$ be the power set of $X$.
Let $I(A)$ be the [[interior|0C38S0KMJK]] of $A$.
The interior of $X$ is $X$, and the interior of $A$ lies in $A$.
The interior of an open set is the set itself, and $I(A \cap A') = I(A) \cap I(A')$.
So the four clauses hold.
Non-example: on a Boolean algebra with $0 \ne 1$, the map $I(x) = 0$ for all $x$ satisfies clauses 2, 3 and 4, but fails clause 1.
Clause 4 implies that $I$ is monotone: if $x \le y$, then $I(x) = I(x \wedge y) = I(x) \wedge I(y) \le I(y)$.
The map $C(x) = \neg I(\neg x)$ is the dual closure operator; a Boolean algebra with such a closure operator is a *closure algebra*.
Interior algebras are the algebraic semantics of the [[modal logic|G8VGTW9P4F]] S4, with $I$ interpreting the necessity operator $\square$.
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# Decomposition group

## Summary

Let $L/K$ be a Galois extension of number fields, and let $\mathfrak{P}$ be a prime of $L$.
The decomposition group of $\mathfrak{P}$ is the subgroup of the [[Galois group|0XTY86BZKY]] that maps $\mathfrak{P}$ to itself.
It surjects onto the Galois group of the residue field extension, and the kernel of this map is the inertia group.
The decomposition group describes how $\mathfrak{P}$ splits, and it is the home of the [[Frobenius element|H5145REFMW]].
It is the Galois group of the local extension obtained by completing at $\mathfrak{P}$.

## Explanations

### L1 human

The Galois group of $L/K$ moves the prime ideals of $L$ that lie over a given prime of $K$ around among themselves.
The decomposition group of one of them is the set of symmetries that leave it in place.
If the prime splits into many pieces, few symmetries fix a given piece, and the decomposition group is small.
In $\Q(i)/\Q$, complex conjugation swaps the two primes $(2 + i)$ and $(2 - i)$ over $5$, so their decomposition groups are trivial.
It fixes the prime $(3)$, so the decomposition group of $(3)$ is the whole Galois group.
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# Quasilinear PDE

## Summary

A quasilinear PDE is a [[partial differential equation|GC5BDZ6QBC]] that is linear in the highest-order derivatives of the unknown.
Its coefficients may depend on the unknown and its lower-order derivatives.
This class contains the equations of fluid dynamics, the [[minimal surface equation|3GV2E2YTZX]] and the [[p-Laplace equation|XJMYGY0ZZF]].
It sits between [[semilinear|RCNY89A4NY]] and [[fully nonlinear|2D3N1ZJWBW]] equations.
See the [[definition|DV92N2YWZJ]].

## Explanations

### L1 human

In a quasilinear equation the highest derivatives still enter one at a time, multiplied by coefficients.
But these coefficients may depend on the solution itself.
In [[Burgers' equation|NEGHXN8RE1]] $\partial_t u + u \, \partial_x u = 0$, the speed at which a wave moves is the value $u$ of the wave.
So high parts of the wave overtake low parts, and the solution can form a shock in finite time.
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# Method of characteristics

## Summary

The method of characteristics solves a first-order [[partial differential equation|GC5BDZ6QBC]] by reducing it to a system of [[ordinary differential equations|0NT64JV96A]].
Along special curves, the characteristics, the equation says how the unknown changes, so one integrates along these curves from the initial data.
It gives explicit solutions of transport, Hamilton–Jacobi and conservation law equations, and it explains why their solutions may break down in finite time.
See the [[definition|VDWKE1KK4H]].

## Explanations

### L1 human

A first-order equation such as $u_t + c\, u_x = 0$ says that $u$ does not change when one moves with speed $c$.
So one follows the lines $x = x_0 + ct$, and along each line $u$ keeps its initial value $u(x_0, 0)$.
This gives the solution $u(x, t) = u(x - ct, 0)$ at once.
In general the curves are not straight and $u$ may change along them, but one still only solves ordinary differential equations.
When two characteristic curves carrying different values cross, the solution cannot stay smooth, and a shock forms.
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# Dirichlet character

## Summary

A Dirichlet character modulo $q$ is a function $\chi \colon \Z \to \C$ that comes from a homomorphism $(\Z/q\Z)^\times \to \C^\times$.
It is periodic with period $q$ and completely multiplicative.
There are exactly $\varphi(q)$ characters modulo $q$, and their orthogonality relations isolate a single residue class.
Dirichlet introduced them to prove [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]], through the [[Dirichlet L-functions|9D2Z543EJQ]] $L(s, \chi)$.

## Explanations

### L1 human

A Dirichlet character tags each integer with a complex number that depends only on its remainder modulo $q$.
Integers that share a factor with $q$ get $0$, and the others get a [[root of unity|0YV2CKP2AV]].
The tags multiply: the tag of a product is the product of the tags.
For $q = 4$, the character gives $0$ to even numbers, $1$ to numbers $1 \bmod 4$, and $-1$ to numbers $3 \bmod 4$.
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# Hopf algebra

## Summary

A Hopf algebra is a [[bialgebra|Q9HHVN606G]] with an [[antipode|PG3ECG5D0G]], a linear map that plays the role of taking inverses in a group.
Group algebras, coordinate rings of affine algebraic groups and universal enveloping algebras of Lie algebras are the main classical examples.
Quantum groups are Hopf algebras that deform these examples.
Hopf algebras also appear in algebraic topology, as the cohomology of Lie groups and H-spaces, and in combinatorics and renormalization.

## Explanations

### L1 human

A group has a product, an identity and inverses.
A Hopf algebra copies all three in linear form: a multiplication, a comultiplication that splits elements, and an antipode that imitates $g \mapsto g^{-1}$.
In the group algebra $KG$, the antipode sends each group element $g$ to $g^{-1}$, and the axiom $S(g)g = 1$ is just $g^{-1}g = 1$.
The extra structure lets one tensor representations together and take duals of representations, as one does for groups.
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# Algebraic structure

## Definition

Let $\sigma$ be a [[signature|WDVVEYD6AT]] whose symbols are all function symbols, each with an arity $n \ge 0$.
An *algebraic structure* of signature $\sigma$, or $\sigma$-algebra, is a pair $\mathbf{A} = (A, (f^{\mathbf{A}})_{f \in \sigma})$ such that:

1. $A$ is a nonempty set, the universe of $\mathbf{A}$;
2. for each symbol $f \in \sigma$ of arity $n$, $f^{\mathbf{A}} \colon A^n \to A$ is an $n$-ary [[finitary operation|EJ8YJVM9BW]] on $A$.

An operation of arity $0$ is identified with an element of $A$, a constant.
So a $\sigma$-algebra is the same as a [[structure|983GZBQYT4]] for the purely functional signature $\sigma$.

## Remarks

Example: $(\Z, +, -, 0)$ is an algebraic structure of signature $(2, 1, 0)$.
Addition is a map $\Z^2 \to \Z$, negation is a map $\Z \to \Z$, and $0 \in \Z$ is a constant.

Non-example: $(\N, +, -, 0)$ with $-$ the map $x \mapsto -x$ is not an algebraic structure, because $-1 \notin \N$, so clause 2 fails for the unary symbol.

A *class* of algebras, such as the class of groups, is given by a signature together with axioms that the operations must satisfy.
The axioms are not part of the definition of an algebraic structure.
Some authors allow the empty set as universe when $\sigma$ has no constants.
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# Classifying space of a group

## Definition

Let $G$ be a [[group|G5S1M9]], viewed as a discrete group.
A *classifying space* of $G$ is a path-connected [[CW complex|5W95QFHXPR]] $X$ such that:

1. The [[fundamental group|S1XK3HXHFX]] $\pi_1(X)$ is isomorphic to $G$.
2. The [[homotopy group|P976WYEFMB]] $\pi_n(X)$ is trivial for every $n \ge 2$.

Such a space is written $BG$.

## Remarks

A classifying space of $G$ is the same as an [[Eilenberg–MacLane space|AJHEYJE5XY]] $K(G, 1)$.
Any two classifying spaces of $G$ are [[homotopy equivalent|XZG6MTSWCN]].
Equivalently, $X$ is path-connected, $\pi_1(X) \cong G$, and its [[universal cover|Y1YPE6PM52]] is [[contractible|R5WP8Z69Q8]]; then $G$ acts freely on the universal cover and $X$ is the quotient.

Example: the circle $S^1$ is a classifying space of $\Z$.
Its fundamental group is $\Z$, and its universal cover $\R$ is contractible, so $\pi_n(S^1) = 0$ for $n \ge 2$.
In the same way, the torus is a classifying space of $\Z^2$, and the infinite real projective space $\R P^\infty$ is one of $\Z/2\Z$.

Non-example: the real projective plane $\R P^2$ has fundamental group $\Z/2\Z$, but its universal cover is the sphere $S^2$, so $\pi_2(\R P^2) \cong \pi_2(S^2) \cong \Z$.
Clause 2 fails, so $\R P^2$ is not a classifying space of $\Z/2\Z$.
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# Comparison theorem

## Statement

Let $D \subseteq \R^2$ be open and let $f : D \to \R$ be continuous and locally Lipschitz in its second variable.
Let $a < b$, let $u : [a, b] \to \R$ be a [[solution|04SW7WECS8]] of $u' = f(t, u)$, and let $v : [a, b] \to \R$ be differentiable, with the graphs of $u$ and $v$ in $D$.
Assume that $v'(t) \le f(t, v(t))$ for each $t \in [a, b]$ and that $v(a) \le u(a)$.
Then $v(t) \le u(t)$ for each $t \in [a, b]$.

## Remarks

Proof idea: suppose $v(t_1) > u(t_1)$ and let $s$ be the last time before $t_1$ with $v(s) \le u(s)$, so $v(s) = u(s)$.
Just after $s$ the difference $w = v - u$ is positive and satisfies $w' \le f(t, v) - f(t, u) \le K w$ by the [[Lipschitz condition|055VVSN00V]].
Then $w(t) e^{-K(t - s)}$ is non-increasing and $w(s) = 0$, so $w \le 0$ there, a contradiction; this is the differential form of [[Grönwall's inequality|07EJHQYR9A]].
Example: $v(t) = 0$ satisfies $v' \le 1 + v^2$, so the solution $u = \tan t$ of $u' = 1 + u^2$, $u(0) = 0$, is $\ge 0$ on $[0, \pi/2)$; with $v(t) = t$, which satisfies $v' = 1 \le 1 + v^2$, it gives $\tan t \ge t$ there.
The Lipschitz hypothesis is needed: for $f(t, x) = 2\sqrt{\abs{x}}$, the solution $u = 0$ and the function $v = t^2$ satisfy $v' = 2\sqrt{v}$ and $v(0) = u(0)$, but $v > u$ for $t > 0$.
Without Lipschitz conditions the conclusion holds when $u$ is replaced by the maximal solution of the initial value problem.

## Explanations

### L1 human

If a quantity always grows at most as fast as the rule $u' = f(t, u)$ allows, and starts no higher, then it stays below the solution of that rule.
One can bound solutions of a hard equation by solutions of an easier one.

### L2 human

Comparison arguments give a priori bounds, blow-up times and global existence results for scalar equations.
For systems, one applies the theorem to a scalar quantity such as $\norm{x(t)}^2$ or a [[Lyapunov function|04P7B5XC22]].
The [[Bihari–LaSalle inequality|9KMJSKNZ72]] and Grönwall's inequality are comparison results in integral form.
The [[Sturm comparison theorem|W7VXNYZQN2]] is a different result, about zeros of solutions of second-order linear equations.
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# Kervaire invariant

## Definition

Let $k \ge 0$ and let $M$ be a closed [[smooth manifold|F45VQ47GPV]] of dimension $4k + 2$ with a framing of its stable normal bundle.
Let $V = H^{2k+1}(M; \F_2)$ be its middle [[singular cohomology|6N7BC70CX0]] with coefficients in $\F_2$.
The [[cup product|JX47MS1R1H]] followed by evaluation on the fundamental class gives a nondegenerate bilinear form $x \cdot y$ on $V$, and $x \cdot x = 0$ for all $x$.
The framing determines a function $q \colon V \to \F_2$ such that
$$q(x + y) = q(x) + q(y) + x \cdot y \quad \text{for all } x, y \in V.$$
Choose a basis $a_1, \dots, a_m, b_1, \dots, b_m$ of $V$ with $a_i \cdot b_j = \delta_{ij}$ and $a_i \cdot a_j = b_i \cdot b_j = 0$.
The *Kervaire invariant* of $M$ is the Arf invariant of the [[quadratic form|0RBQS1JJCR]] $q$:
$$\Phi(M) = \sum_{i=1}^{m} q(a_i)\, q(b_i) \in \F_2.$$

## Remarks

The Arf invariant does not depend on the choice of the symplectic basis.
It is the value that $q$ takes on more than half of the elements of $V$.

Example: let $M = S^1 \times S^1$ with the framing given by its Lie group structure, so $k = 0$.
Here $V = \F_2^2$ with basis $a$, $b$ dual to the two circle factors, and $a \cdot b = 1$.
For this framing $q(a) = q(b) = 1$, so $\Phi(M) = 1$.

Non-example: a closed framed manifold of dimension $4$ has no Kervaire invariant, since $4$ is not of the form $4k + 2$.
The construction needs an odd middle degree $2k + 1$.

The Kervaire invariant is a framed cobordism invariant.
By the Pontryagin–Thom construction it defines a homomorphism $\pi^s_{4k+2} \to \Z/2$ on the stable homotopy groups of spheres.
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# Hasse's theorem on elliptic curves

## Statement

Let $\F_q$ be a [[finite field|0ZXZ82FTX2]] with $q$ elements, and let $E$ be an [[elliptic curve|427N6S6QQW]] over $\F_q$.
Let $N = \# E(\F_q)$ be the number of points of $E$ over $\F_q$, including the point at infinity.
Then
$$\abs{N - (q + 1)} \le 2\sqrt{q}.$$

## Remarks

One proof uses the $q$-power Frobenius endomorphism $\phi$ of $E$.
The points of $E(\F_q)$ form the kernel of $1 - \phi$, which is separable, so $N = \deg(1 - \phi)$.
The degree is a positive definite quadratic form on the endomorphism ring, with $\deg \phi = q$, and the Cauchy–Schwarz inequality for this form gives the bound.
Example: $E \colon y^2 = x^3 + 1$ over $\F_5$.
The cubes modulo $5$ run through all residues, so $x \mapsto x^3 + 1$ takes each value once.
The value $0$ gives one point, the squares $1$ and $4$ give two points each, and the non-squares give none.
With the point at infinity, $N = 1 + 2 + 2 + 1 = 6 = q + 1$.
The number $a_q = q + 1 - N$ is the trace of Frobenius, and the theorem says $\abs{a_q} \le 2\sqrt{q}$.

## Explanations

### L1 human

An elliptic curve over a finite field with $q$ elements has about $q + 1$ points.
Hasse's theorem says the error is at most $2\sqrt{q}$, which is small compared to $q$.

### L2 human

The theorem is the Riemann hypothesis for the zeta function of an elliptic curve over a finite field: the roots of $1 - a_q T + q T^2$ have absolute value $q^{-1/2}$.
Weil generalized it to curves of any genus $g$, with bound $2g\sqrt{q}$, and Deligne to all smooth projective varieties.
The bound defines the angles in the [[Sato–Tate conjecture|TRGBJYPBXH]] and is used to estimate the running time of algorithms in elliptic curve cryptography.
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# Adapted process

## Definition

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in T}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] indexed by a totally ordered set $T$.
Let $(S, \mathcal{S})$ be a [[measurable|YYNCJSR0PJ]] space.
A [[stochastic process|0QH5ENKZM1]] $X = (X_t)_{t \in T}$ with values in $S$ is *adapted* to $(\mathcal{F}_t)$ if the following holds.
For every $t \in T$, the map $X_t \colon \Omega \to S$ is measurable with respect to $\mathcal{F}_t$ and $\mathcal{S}$.

## Remarks

Every process is adapted to its natural filtration $\mathcal{F}_t^X = \sigma(X_s : s \le t)$, and this is the smallest filtration to which it is adapted.

Example: let $Y_1, Y_2, \dots$ be random variables with natural filtration $\mathcal{F}_n = \sigma(Y_1, \dots, Y_n)$.
The partial sums $S_n = Y_1 + \cdots + Y_n$ form an adapted process, since $S_n$ is a measurable function of $Y_1, \dots, Y_n$.

Non-example: let the $Y_k$ be independent fair signs, equal to $1$ or $-1$, with the same filtration, and let $X_n = Y_{n+1}$.
Then $X_n$ is independent of $\mathcal{F}_n$ and not almost surely constant, so it is not $\mathcal{F}_n$-measurable; the process $X$ looks one step into the future.

In continuous time a stronger notion is often needed: $X$ is *progressively measurable* if, for every $t$, the map $(s, \omega) \mapsto X_s(\omega)$ on $[0, t] \times \Omega$ is measurable with respect to $\mathcal{B}([0, t]) \otimes \mathcal{F}_t$.
An adapted process with right-continuous or left-continuous paths is progressively measurable.
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# Error function

## Summary

The error function is $\operatorname{erf} x = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}\, dt$.
It is the integral of the Gaussian bell curve, scaled so that $\operatorname{erf} x \to 1$ as $x \to \infty$.
It gives the [[cumulative distribution function|0H8H7ZTHW2]] of the [[normal distribution|05AQE0QR0D]], and it appears in solutions of the [[heat equation|VP2HN86VAG]].
It has no expression in terms of elementary functions.

## Explanations

### L1 human

The bell curve $e^{-t^2}$ is the shape of the normal distribution.
The error function measures the area under this curve between $0$ and $x$, scaled so that the total area on the positive side is $1$.
It starts at $0$, rises quickly, and levels off near $1$ after $x \approx 2$.
For example, $\operatorname{erf}(1/\sqrt{2}) \approx 0.683$: a normal random value lies within one standard deviation of the mean with probability about $68\%$.
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# Locally ringed space

## Definition

Let $(X, \mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]], that is, a topological space $X$ with a [[sheaf|TB6TM6R4MD]] $\mathcal{O}_X$ of commutative rings on $X$.
It is a *locally ringed space* if for every point $x \in X$ the [[stalk|JTAKAPTT8H]] $\mathcal{O}_{X,x}$ is a [[local ring|MZ02BKPHJA]].
The maximal ideal of $\mathcal{O}_{X,x}$ is written $\mathfrak{m}_x$.

Let $(X, \mathcal{O}_X)$ and $(Y, \mathcal{O}_Y)$ be locally ringed spaces.
A *morphism of locally ringed spaces* $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is a pair $(f, f^\#)$ such that:

1. $f \colon X \to Y$ is a [[continuous map|0VGFHTM75Y]].
2. $f^\# \colon \mathcal{O}_Y \to f_* \mathcal{O}_X$ is a morphism of sheaves of rings on $Y$.
3. For every $x \in X$, the induced ring map $f^\#_x \colon \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X,x}$ is a [[local homomorphism|P8YTMGEN9R]], that is, $(f^\#_x)^{-1}(\mathfrak{m}_x) = \mathfrak{m}_{f(x)}$.

## Remarks

Example: let $M$ be a smooth manifold and let $\mathcal{O}_M(U)$ be the ring of smooth real functions on the open set $U$.
A germ $g$ at $p$ is invertible in $\mathcal{O}_{M,p}$ exactly when $g(p) \ne 0$.
So the non-invertible germs form the ideal $\set{g : g(p) = 0}$, and $\mathcal{O}_{M,p}$ is local.
Non-example: the one-point space with the ring $\Z$ as its sheaf is a ringed space.
Its only stalk $\Z$ is not local, since $(2)$ and $(3)$ are two different maximal ideals.
Clause 3 is not automatic: a morphism of ringed spaces between locally ringed spaces need not satisfy it.
It is the clause that makes morphisms respect "vanishing at a point".
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# Laplacian matrix

## Definition

Let $G$ be a finite simple [[graph|0JD5C82QMB]] with vertices $v_1, \dots, v_n$.
Let $D$ be the diagonal matrix with $D_{ii} = \deg(v_i)$, the [[degree|0T0F6CCS9W]] of $v_i$, and let $A$ be the [[adjacency matrix|0NZ8MDBN6Q]] of $G$.
The Laplacian matrix of $G$ is the $n \times n$ matrix $L = D - A$.
So $L_{ii} = \deg(v_i)$, $L_{ij} = -1$ if $i \ne j$ and $v_i v_j$ is an edge, and $L_{ij} = 0$ otherwise.

## Remarks

For every real vector $x$, $x^{T} L x = \sum (x_i - x_j)^2$, the sum running over the edges $v_i v_j$.
Hence $L$ is [[symmetric|0N28QN4CYA]] and [[positive semidefinite|0880F9EQCP]], the all-ones vector lies in its kernel, and the multiplicity of the eigenvalue $0$ equals the number of connected components of $G$.
Example: for the path $v_1 v_2 v_3$,
$$L = \begin{pmatrix} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{pmatrix},$$
whose eigenvalues are $0$, $1$ and $3$, with eigenvectors $(1,1,1)$, $(1,0,-1)$ and $(1,-2,1)$.
Non-example: the matrix $D + A$ has the right diagonal but the wrong sign off the diagonal.
It is the signless Laplacian, and it has the same spectrum as $L$ when $G$ is bipartite, but not in general.
The normalized Laplacian $D^{-1/2} L D^{-1/2}$, defined when no vertex is isolated, is also common.
Mathlib defines the Laplacian matrix as `SimpleGraph.lapMatrix`.
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# Basis of a matroid

## Definition

Let $M = (E, \mathcal{I})$ be a [[matroid|PX31ZNYS3Z]].
A *basis* of $M$ is a set $B \in \mathcal{I}$ such that no set $A \in \mathcal{I}$ satisfies $B \subsetneq A$.

## Remarks

By the augmentation axiom, all bases of $M$ have the same number of elements.
The bases satisfy the exchange property: if $B_1, B_2$ are bases and $x \in B_1 \setminus B_2$, then there is $y \in B_2 \setminus B_1$ such that $(B_1 \setminus \set{x}) \cup \set{y}$ is a basis.
Conversely, a nonempty family of subsets of $E$ with this exchange property is the family of bases of exactly one matroid, whose independent sets are the subsets of bases.

Example: let $U_{2,3}$ be the uniform matroid on $\set{1, 2, 3}$, whose independent sets are the subsets with at most $2$ elements.
The set $\set{1, 2}$ is a basis of $U_{2,3}$.
It is independent, and its only proper superset $\set{1, 2, 3}$ has $3$ elements, so it is dependent.

Non-example: in $U_{2,3}$ the set $\set{1}$ is independent but not a basis, since $\set{1, 2}$ is a larger independent set.

In the graphic matroid of a connected graph, the bases are the [[spanning trees|0S4ZEZM7KE]].
In the matroid of the columns of a matrix, the bases are the sets of columns that form a basis of the column space.
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# Extraspecial group

## Summary

An extraspecial group is a finite $p$-group whose centre has order $p$ and whose quotient by the centre is a nontrivial elementary abelian $p$-group.
They are non-abelian $p$-groups that are as close to abelian as possible, and their order is always an odd power of $p$.
They occur often inside larger groups, for example in centralisers of involutions in sporadic simple groups.

## Explanations

### L1 human

An extraspecial group is almost abelian.
All commutators lie in a tiny central subgroup of order $p$, and the group modulo that subgroup is abelian with every element of order $p$.
The quaternion group $Q_8$ and the dihedral group of order $8$ are the two smallest examples, both with $p = 2$.
For an odd prime $p$, the group of $3 \times 3$ upper unitriangular matrices over the field $\F_p$ is an example of order $p^3$.
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# Barrier

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]], and let $\xi$ be a point of its [[boundary|05D5EGMFYV]] $\partial\Omega$.
A *barrier* at $\xi$ is a function $w \colon \overline{\Omega} \to \R$ such that:

1. $w$ is [[continuous|0X3WGX95FE]] on $\overline{\Omega}$;
2. $w$ is [[superharmonic|X4V2S56YQQ]] on $\Omega$;
3. $w(\xi) = 0$ and $w(x) > 0$ for every $x \in \overline{\Omega} \setminus \set{\xi}$.

## Remarks

Example: suppose there is a closed ball $\overline{B}(y, R)$ with $\overline{B}(y, R) \cap \overline{\Omega} = \set{\xi}$, and let $n \ge 3$.
Then $w(x) = R^{2-n} - \abs{x - y}^{2-n}$ is a barrier at $\xi$.
It is continuous on $\overline{\Omega}$ and harmonic on $\Omega$, since $y \notin \overline{\Omega}$; it vanishes at $\xi$ because $\abs{\xi - y} = R$; and $\abs{x - y} > R$ for the other points $x \in \overline{\Omega}$, which gives $w(x) > 0$.
For $n = 2$ the function $w(x) = \log(\abs{x - y}/R)$ plays the same role.
Non-example: $w(x) = \abs{x - \xi}^2$ satisfies clauses 1 and 3, but $\Delta w = 2n > 0$, so it is strictly subharmonic and fails clause 2.
A *local barrier* is a function with these properties on $\overline{\Omega} \cap N$ for some neighbourhood $N$ of $\xi$.
A local barrier yields a barrier, so the two notions give the same regular points.
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# Topology optimization

## Problem

Let $D \subseteq \R^d$ be a bounded open design region with Lipschitz boundary, and let $0 < V < \abs{D}$ be a volume bound.
Let $A$ be the elasticity tensor of a given isotropic elastic material, and let $f \in L^2(D; \R^d)$ be a load.
For a measurable set $\omega \subseteq D$ let $\chi_\omega$ be its indicator function, and fix a small $\varepsilon > 0$.
Let $u_\omega \in H^1(D; \R^d)$ be the displacement that solves the linear elasticity system $-\operatorname{div}\big((\chi_\omega + \varepsilon (1 - \chi_\omega)) A \, e(u)\big) = f$ in $D$, where $e(u) = \frac12 (\nabla u + \nabla u^{T})$.
The boundary conditions are $u = 0$ on a part of $\partial D$ of positive surface measure and zero traction on the rest.
The *minimum compliance topology optimization problem* is to find a set $\omega$ with $\abs{\omega} \le V$ that minimizes the compliance $\int_D f \cdot u_\omega \, dx$.

## Remarks

The material occupies $\omega$, and a very weak material of stiffness $\varepsilon$ fills the rest of $D$ so that the equation stays elliptic.
Unlike classical [[shape optimization|WWQJQXJECK]], no topology of $\omega$ is fixed in advance: holes may appear or disappear.
In general this problem has no minimizer: minimizing sequences develop finer and finer mixtures of material and void.
Bendsøe and Kikuchi replaced the sets $\omega$ by composite materials with a microstructure, given by homogenization, and this relaxed problem has solutions.
A widely used simpler method replaces $\chi_\omega$ by a density $\rho \colon D \to [0, 1]$ and the stiffness by $\rho^p A$ with $p > 1$, which pushes $\rho$ toward the values $0$ and $1$.
The resulting discrete problems are solved by gradient methods with adjoint sensitivities, as in [[optimal control of partial differential equations|S2BBZPAFDR]].

## Explanations

### L1 human

Topology optimization asks where to put material inside a box so that a structure is as stiff as possible for a given amount of material.
The answer may have holes, struts and branches that no designer fixed in advance.
Computers solve such problems by giving each small cell of the box a density between $0$ and $1$ and improving the densities step by step.
The results often look like bones or trees, and they are used to design light parts for aircraft and cars.
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# Artin–Schreier theory

## Summary

Artin–Schreier theory describes the [[cyclic extensions|VFE5B71E7Y]] of degree $p$ of a field of characteristic $p > 0$.
Each of them is generated by a root of a polynomial $x^p - x - a$.
It plays the role that [[Kummer theory|XZQENYV15Y]] plays in degree $n$ prime to the characteristic, where $x^n - a$ is used instead.

## Explanations

### L1 human

In characteristic $p$, adjoining a $p$-th root of an element $a$ that is not a $p$-th power does not give a Galois extension.
The reason is that $x^p - a$ has only one root, repeated $p$ times.
The fix is the polynomial $x^p - x - a$.
If $\alpha$ is one of its roots, then $\alpha + 1, \alpha + 2, \dots, \alpha + p - 1$ are the others, because $(\alpha + 1)^p = \alpha^p + 1$ in characteristic $p$.
So the symmetries of the extension shift the roots by constants, and they form a cyclic group of order $p$.
Over $\F_2$, the polynomial $x^2 + x + 1$ is of this form, and its root field is $\F_4$.
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# Tropical geometry

## Summary

Tropical geometry studies polynomials over the tropical semiring, where addition is replaced by minimum (or maximum) and multiplication by ordinary addition, see [[max-plus algebra|64G0VSC5F4]].
Tropical polynomials are piecewise linear functions, and their zero sets, the [[tropical varieties|FT2GJEHWY5]], are polyhedral complexes.
Through a [[valuation|B85T5VKR3M]], an algebraic variety degenerates to a tropical variety that keeps much of its combinatorial information, such as degrees and intersection numbers.
This makes problems of [[enumerative geometry|ZPZA1FKZZC]] and of [[toric varieties|YWB981MEF2]] accessible to combinatorics.

## Explanations

### L1 human

In tropical arithmetic, $3 \oplus 5 = \min(3, 5) = 3$ and $3 \odot 5 = 3 + 5 = 8$.
A polynomial in this arithmetic becomes a minimum of finitely many linear functions, whose graph is made of flat pieces.
The tropical curve of such a polynomial is the set of corners, where the minimum is attained twice.
A tropical line in the plane, for example, is a point with three rays coming out of it.
These piecewise linear pictures are shadows of ordinary curves, and counting them can answer classical questions such as how many curves pass through given points.
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# Zernike polynomials

## Summary

The Zernike polynomials are an orthogonal family of polynomials on the closed unit disk.
In polar coordinates each one is a radial polynomial times $\cos(m\varphi)$ or $\sin(m\varphi)$.
They are the standard basis in optics for describing the aberrations of a wavefront through a circular lens or mirror.
Their radial parts are [[Jacobi polynomials|TRD7DSF2KR]] in disguise.

## Explanations

### L1 human

The light passing through a round lens arrives with small delays that vary across the disk of the lens.
Optical engineers write this delay map as a sum of standard patterns on the disk, the Zernike polynomials.
Each pattern has a name: $2\rho^2 - 1$ is "defocus", a bowl shape, and $\rho \cos \varphi$ is "tilt", a plane sloping in one direction.
Because the patterns are orthogonal, the size of each defect can be read off separately.
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# Étale cohomology

## Summary

Étale cohomology is a cohomology theory for schemes, built from sheaves on the étale site instead of the Zariski topology.
Grothendieck and Artin introduced it to obtain, for varieties over any field, groups that behave like the singular cohomology of a complex variety.
It contains [[Galois cohomology|9MPCZ3EAJM]] as the case of a point.
It is the setting of [[ℓ-adic cohomology|E1DVHN06CR]] and of Deligne's proof of the [[Weil conjectures|81TXP32VDM]].

## Explanations

### L1 human

The Zariski topology of a variety has too few open sets: on an irreducible variety, any two nonempty open sets meet.
So sheaf cohomology with constant coefficients in that topology sees almost nothing.
Étale cohomology allows "open sets" that are étale maps $U \to X$, which look like local isomorphisms, such as finite unramified covers.
These play the role of small discs in topology.
For a smooth projective complex curve of genus $g$, the first étale cohomology with coefficients $\Z/n\Z$ is $(\Z/n\Z)^{2g}$, as in topology.
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# Matroid minor

## Definition

Let $M = (E, \mathcal{I})$ be a [[matroid|PX31ZNYS3Z]] with [[rank function|SYRCTH853H]] $r$, and let $X \subseteq E$.

1. The *deletion* $M \setminus X$ is the matroid on $E \setminus X$ whose independent sets are the sets $I \in \mathcal{I}$ with $I \subseteq E \setminus X$.
2. The *contraction* $M / X$ is the matroid on $E \setminus X$ with rank function $r_{M/X}(A) = r(A \cup X) - r(X)$ for $A \subseteq E \setminus X$.
3. A matroid $N$ is a *minor* of $M$ if $N = (M \setminus X) / Y$ for some disjoint sets $X, Y \subseteq E$.

## Remarks

If $J$ is a maximal independent subset of $X$, then a set $I \subseteq E \setminus X$ is independent in $M / X$ if and only if $I \cup J \in \mathcal{I}$.
Deletion and contraction commute, and $M / X = (M^* \setminus X)^*$, where $M^*$ is the [[dual matroid|6QF83CFK2G]].
Many authors call $N$ a minor when it is isomorphic to a matroid of the form in clause 3.
For the graphic matroid of a graph, deleting or contracting an edge gives the graphic matroid of the graph with that edge deleted or contracted.

Example: let $M = U_{2,4}$, the matroid on $\set{1, 2, 3, 4}$ whose independent sets are the subsets with at most $2$ elements.
Then $M \setminus \set{4} = U_{2,3}$, and $M / \set{4} = U_{1,3}$, since $r(A \cup \set{4}) - 1 = \min(\abs{A} + 1, 2) - 1 = \min(\abs{A}, 1)$.

Non-example: $U_{2,4}$ is not a minor of $U_{1,5}$, because deletion and contraction never raise the rank, and $U_{2,4}$ has rank $2$ while $U_{1,5}$ has rank $1$.

A theorem of Tutte states that a matroid is representable over the field with two elements if and only if it has no minor isomorphic to $U_{2,4}$.
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# Maximal subfield

## Definition

Let $K$ be a field and let $D$ be a [[division ring|0EA5BTA0CX]] that is a [[central simple algebra|5HRKJBAY2V]] over $K$.
A [[subfield|0GFGHRT2FJ]] of $D$ is a subring $L$ of $D$ that is a field.
A *maximal subfield* of $D$ is a subfield $L$ with $K \subseteq L$ that is not properly contained in any other subfield of $D$.

## Remarks

Every maximal subfield $L$ satisfies $[L : K] = \deg D$, the [[degree|NNXSP5KKAC]] of $D$, and $L$ [[splits|NVD7R0T835]] $D$.
Indeed, maximality gives $C_D(L) = L$, and the double centralizer theorem gives $[L : K]^2 = \dim_K D$.
Conversely, a subfield $L \supseteq K$ with $[L : K] = \deg D$ is maximal.
For a central simple algebra $A$ that is not a division algebra, a subfield $L \supseteq K$ of $A$ with $[L : K] = \deg A$ is also called a maximal subfield.

Example: in $\mathbb{H}$ over $\R$, the subalgebra $L = \R + \R i$ is a field isomorphic to $\C$, and $[L : \R] = 2 = \deg \mathbb{H}$, so $L$ is a maximal subfield.

Non-example: the subfield $\R$ of $\mathbb{H}$ contains $\R$ but is properly contained in $\R + \R i$, so it is not maximal.
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# Noether normalization lemma

## Statement

Let $k$ be a [[field|04C5ZBG1WT]] and let $A$ be a nonzero [[finitely generated|AC5TFFQT9M]] $k$-algebra.
Then there are an integer $d \ge 0$ and elements $y_1, \ldots, y_d \in A$ such that:

1. $y_1, \ldots, y_d$ are algebraically independent over $k$, so the subalgebra $k[y_1, \ldots, y_d]$ is a [[polynomial ring|0SDC934R52]] in $d$ variables.
2. $A$ is a [[finite|1NBACAEC2V]] $k[y_1, \ldots, y_d]$-algebra, that is, a finitely generated module over it.

## Remarks

The proof is by induction on the number of generators of $A$.
Suppose the generators $x_1, \ldots, x_n$ satisfy a nonzero polynomial relation.
The substitution $x_i' = x_i - x_n^{r^i}$ for $i < n$, with $r$ large, turns the relation into one that is monic in $x_n$ up to a nonzero constant.
Then $A$ is finite over $k[x_1', \ldots, x_{n-1}']$, and one continues with this smaller algebra.
When $k$ is infinite, linear substitutions $x_i' = x_i - c_i x_n$ with suitable $c_i \in k$ suffice.

The number $d$ is the [[Krull dimension|MVGBS3J1P7]] of $A$, by [[going-up|GHKYR5PBQC]] and [[incomparability|Q2XBHJ96GV]].
When $A$ is a domain, $d$ is also the [[transcendence degree|D7TR0YWR9H]] of its field of fractions over $k$.

Example: let $A = k[x, y]/(xy - 1)$ and $t = x + y$.
In $A$, $x(t - x) = xy = 1$, so $x^2 - t x + 1 = 0$ and likewise for $y$; hence $A$ is generated by $1$ and $x$ as a $k[t]$-module.
The element $t$ is transcendental over $k$, since $A$ is not finite-dimensional over $k$, so $d = 1$.
Note that $x$ alone does not work: $A = k[x, x^{-1}]$ is not finite over $k[x]$.

## Explanations

### L1 human

Every finitely generated algebra over a field sits finitely over a polynomial ring.
Geometrically, every affine variety of dimension $d$ maps onto the $d$-dimensional affine space by a map with finite fibres.
So complicated varieties can be studied as finite covers of the simplest ones.

### L2 human

Noether normalization is the main tool to compute the dimension of finitely generated algebras over a field and to prove that it equals the transcendence degree.
It gives a short proof of [[Zariski's lemma|YBJ2VQHFDQ]] and hence of the [[weak Nullstellensatz|ZWS0YWGF17]].
If $A$ is a field, then $k[y_1, \ldots, y_d]$ is a field by [[integral extensions of fields|5GCANV7337]], so $d = 0$.
It is also used to show that finitely generated domains over a field are catenary.
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# Zonoid

## Definition

Let $n \ge 1$.
A *zonoid* in $\R^n$ is a [[convex body|437FSG3419]] $Z$ that is the limit of a sequence of [[zonotopes|Z5SA6A67KF]] in $\R^n$.
The limit is taken in the [[Hausdorff distance|F79WYYC07C]].
Here a zonotope is a Minkowski sum $[a_1, b_1] + \cdots + [a_m, b_m]$ of finitely many closed segments.

## Remarks

Equivalently, the [[support function|1H92CYA4ZC]] of $Z$ has the form $h_Z(x) = x \cdot a + \int_{S^{n-1}} \abs{x \cdot v} \, d\mu(v)$.
Here $a \in \R^n$, and $\mu$ is a finite even Borel measure on the unit sphere $S^{n-1}$.
Example: the closed unit ball $B$ of $\R^n$ is a zonoid.
Its support function is $h_B(x) = \abs{x}$, and by rotation invariance $\int_{S^{n-1}} \abs{x \cdot v} \, d\sigma(v) = c_n \abs{x}$ for the normalized surface measure $\sigma$ and a constant $c_n > 0$, so $\mu = \sigma / c_n$ works with $a = 0$.
Non-example: a triangle in $\R^2$ is not a zonoid.
Each zonotope is symmetric about a point, this property passes to Hausdorff limits, and a triangle has no centre of symmetry.
In $\R^2$ every convex body with a centre of symmetry is a zonoid; in $\R^3$ the regular octahedron is a centrally symmetric convex body that is not a zonoid.
Some authors also call lower-dimensional Hausdorff limits of zonotopes, such as single segments, zonoids; here a zonoid is required to have nonempty interior, like every convex body.
</document>

<document source="https://tamnd.github.io/math2/node/7Z57ARX3MF.md" tag="7Z57ARX3MF" kind="definition">
# FBI transform

## Definition

Let $n \ge 1$ and let $f \colon \R^n \to \C$ be [[integrable|X7GTF3D98F]].
The *FBI transform* of $f$ is the function on $\R^n \times (\R^n \setminus \set{0})$ given by
$$Tf(x, \xi) = \int_{\R^n} e^{i \xi \cdot (x - y) - \frac{1}{2} \abs{\xi} \abs{x - y}^2}\, f(y)\, dy.$$
The same formula, read as a pairing, defines $Tf$ for a distribution $f$ with compact support.

## Remarks

The factor $e^{-\frac12 \abs{\xi} \abs{x - y}^2}$ is a Gaussian window centred at $x$ of width of order $\abs{\xi}^{-1/2}$, and $e^{i\xi \cdot (x - y)}$ selects the frequency $\xi$.
Authors use different normalizations of the window and of the constant in front; the qualitative properties do not depend on these choices.
The main property concerns the analytic [[wave front set|3T5Q9WVQX8]].
Cut $f$ off near a point $x_0$, and let $\xi_0 \ne 0$.
Then $(x_0, \xi_0)$ lies outside the analytic wave front set of $f$ exactly when $Tf(x, \lambda \xi)$ decays like $e^{-c\lambda}$ as $\lambda \to \infty$.
Here $c > 0$ is a constant and the bound is uniform for $(x, \xi)$ near $(x_0, \xi_0)$.

Example: $n = 1$ and $f(y) = e^{-y^2/2}$.
For $\xi > 0$ the exponent is a quadratic polynomial in $y$.
The Gaussian integral gives
$$Tf(x, \xi) = \sqrt{\frac{2\pi}{1 + \xi}}\, \exp\Big( i\xi x - \frac{\xi x^2}{2} + \frac{(\xi x - i\xi)^2}{2(1 + \xi)} \Big),$$
whose modulus is $\sqrt{2\pi/(1+\xi)}\, \exp\big(-\frac{\xi (x^2 + \xi)}{2(1 + \xi)}\big)$; it decays like $e^{-\xi/2}$ as $\xi \to \infty$, as it must for a real analytic function.

Non-example: $f(y) = e^{\abs{y}^3}$ is not integrable, and the integral defining $Tf(x, \xi)$ diverges for every $(x, \xi)$, because $e^{\abs{y}^3}$ grows faster than any Gaussian decays.
</document>

<document source="https://tamnd.github.io/math2/node/7Z5ZS7KNXS.md" tag="7Z5ZS7KNXS" kind="object">
# Out(F_n)

## Description

Let $n \ge 1$ and let $F_n$ be the [[free group|0KTW26C80J]] of rank $n$.
The group $\mathrm{Out}(F_n) = \Aut(F_n) / \mathrm{Inn}(F_n)$ is the [[outer automorphism group|0Q8Y862DTZ]] of $F_n$.
It is the quotient of the [[automorphism group|0K1SCPBNKJ]] of $F_n$ by its normal subgroup of inner automorphisms $x \mapsto g x g^{-1}$.

## Remarks

For $n = 1$, $F_1 = \Z$ has the two automorphisms $x \mapsto \pm x$ and no nontrivial inner automorphisms, so $\mathrm{Out}(F_1) \cong \Z/2\Z$.
Every automorphism of $F_n$ induces an automorphism of the abelianisation $\Z^n$, and inner automorphisms induce the identity.
This gives a homomorphism $\mathrm{Out}(F_n) \to \mathrm{GL}_n(\Z)$, into the [[general linear group|0FCHE902D7]] over $\Z$, which is surjective, because elementary matrices and sign changes lift to automorphisms of $F_n$.
For $n = 2$ this map is an isomorphism, so $\mathrm{Out}(F_2) \cong \mathrm{GL}_2(\Z)$.
For $n \ge 3$ its kernel is infinite.

$\mathrm{Out}(F_n)$ is finitely presented.
Culler and Vogtmann built a contractible space, Outer space, on which $\mathrm{Out}(F_n)$ acts properly; it plays the role that Teichmüller space plays for the [[mapping class group|JV5NYG913V]] of a surface.
$\mathrm{Out}(F_n)$ satisfies the [[Tits alternative|QYXGVES1XM]].

## Explanations

### L1 human

The free group on $n$ letters has many symmetries: for example, one may swap two letters, invert a letter, or replace $a$ by $ab$.
Some symmetries are boring, namely conjugation by a fixed element; dividing them out gives $\mathrm{Out}(F_n)$.
It is the free-group analogue of the group $\mathrm{GL}_n(\Z)$ of integer matrices with integer inverses, and for $n = 2$ the two groups coincide.
It is a central object of [[geometric group theory|2T3218RXTD]].
</document>

<document source="https://tamnd.github.io/math2/node/7ZB7MRKPQ9.md" tag="7ZB7MRKPQ9" kind="concept">
# Transformation semigroup

## Summary

A transformation semigroup is a set of maps from a set $X$ to itself that is closed under [[composition|0THBK9V825]].
It is to [[semigroups|0XMEM406JM]] what a [[permutation group|ZDJQRKPWGA]] is to groups.
Every semigroup is isomorphic to a transformation semigroup, by an analogue of Cayley's theorem.
Transformation semigroups model the actions of finite automata, which is the starting point of [[Krohn–Rhodes theory|3WDZY1J002]].

## Explanations

### L1 human

Think of a machine with a finite set of states, and buttons that each move every state to some state.
Pressing buttons one after another composes these moves, and the set of all composite moves is a transformation semigroup.
Unlike the symmetries of a group, the moves need not be reversible: a reset button that sends every state to one start state loses information.
</document>

<document source="https://tamnd.github.io/math2/node/7ZC93QSXNT.md" tag="7ZC93QSXNT" kind="definition">
# Ramanujan tau function

## Definition

Let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]], and for $z \in \mathbb{H}$ let $q = e^{2 \pi i z}$.
The *Ramanujan tau function* $\tau \colon \Z_{\ge 1} \to \Z$ is defined by the expansion
$$q \prod_{m = 1}^{\infty} (1 - q^m)^{24} = \sum_{n = 1}^{\infty} \tau(n) q^n.$$
The left side is the [[cusp form|A50Y012RBP]] $\Delta(z)$ of weight $12$.

## Remarks

Example: the coefficient of $q^2$ comes only from $q \cdot (1 - q)^{24}$, which gives $-24$; so $\tau(2) = -24$.
The function is multiplicative: $\tau(mn) = \tau(m)\tau(n)$ when $\gcd(m, n) = 1$.
For primes $p$ and $r \ge 1$, $\tau(p^{r+1}) = \tau(p)\tau(p^r) - p^{11} \tau(p^{r-1})$.
Non-example: $\tau$ is not completely multiplicative, since $\tau(4) = \tau(2)^2 - 2^{11} = -1472 \neq 576 = \tau(2)^2$.
Ramanujan's congruence: $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$, where $\sigma_{11}(n) = \sum_{d \mid n} d^{11}$.
For $n = 2$: $\sigma_{11}(2) - \tau(2) = 2049 + 24 = 2073 = 3 \cdot 691$.
Whether $\tau(n) \neq 0$ for all $n$ is an open question, known as Lehmer's conjecture.
</document>

<document source="https://tamnd.github.io/math2/node/7ZEFP7W1XZ.md" tag="7ZEFP7W1XZ" kind="definition">
# Jet bundle

## Definition

Let $\pi \colon E \to M$ be a smooth [[fiber bundle|CX4M8K61AB]] over a [[smooth manifold|F45VQ47GPV]] $M$, and let $k \ge 0$ be an integer.
A local section of $\pi$ is a smooth map $s \colon U \to E$, with $U \subseteq M$ open, such that $\pi \circ s = \id_U$.
For $p \in U$, let $j^k_p s$ be the [[$k$-jet|1QE0Z0756D]] of $s$ at $p$, viewed as a map from $M$ to $E$.
The *$k$-th jet manifold* of $\pi$ is the set
$$J^k \pi = \set{ j^k_p s : p \in M,\ s \text{ a local section of } \pi \text{ defined near } p }.$$
It comes with the maps $\pi_k \colon J^k \pi \to M$, $j^k_p s \mapsto p$, and $\pi_{k,0} \colon J^k\pi \to E$, $j^k_p s \mapsto s(p)$.

Let $(x^1, \dots, x^n, u^1, \dots, u^m)$ be adapted coordinates on an open set of $E$, so that $x^i$ are coordinates on $M$ and $u^\alpha$ are fiber coordinates.
They induce coordinates $(x^i, u^\alpha, u^\alpha_I)$ on $J^k\pi$, where $I$ runs over multi-indices with $1 \le \abs{I} \le k$ and
$$u^\alpha_I(j^k_p s) = \frac{\partial^{\abs{I}} s^\alpha}{\partial x^I}(p).$$
These charts make $J^k\pi$ a smooth manifold.
With this structure, $\pi_k \colon J^k\pi \to M$ is called the *$k$-th jet bundle* of $\pi$.

## Remarks

The maps $\pi_k$ and $\pi_{k,0}$, and more generally the maps $J^k\pi \to J^l\pi$ for $l < k$ that forget the higher derivatives, are smooth fiber bundles.
If $E$ is a [[vector bundle|P4DZM975BV]], then $J^k\pi \to M$ is again a vector bundle.

Example: for the trivial bundle $\pi \colon \R \times \R \to \R$, $(x, u) \mapsto x$, the local sections are graphs of functions $u = s(x)$.
The first jet manifold has the global coordinates $(x, u, u_x)$, so $J^1\pi \cong \R^3$, and $j^1_p s = (p, s(p), s'(p))$.

Non-example: the $k$-jets of all smooth maps $M \to E$, not only of sections, form the larger space $J^k(M, E)$.
A jet of a map $f$ with $\pi \circ f \ne \id$ near $p$ is not in $J^k\pi$, because the definition admits only sections.
</document>

<document source="https://tamnd.github.io/math2/node/7ZGQKWYKQS.md" tag="7ZGQKWYKQS" kind="statement">
# Splitting lemma

## Statement

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ be a [[short exact sequence|KJKNJMGKZZ]] in $\mathcal{A}$.
Let $A \oplus C$ be the [[biproduct|BZEJKTQ76T]], with inclusion $i_A$ and projection $p_C$.
The following are equivalent:

1. there is a morphism $t \colon B \to A$ with $t \circ f = 1_A$;
2. there is a morphism $u \colon C \to B$ with $g \circ u = 1_C$;
3. the sequence is [[split|1ZPMDE4ARK]]: there is an isomorphism $h \colon B \to A \oplus C$ with $h \circ f = i_A$ and $p_C \circ h = g$.

## Remarks

From (3), the morphisms $t = p_A \circ h$ and $u = h^{-1} \circ i_C$ give (1) and (2).
From (1), the morphism $h = (t, g) \colon B \to A \oplus C$ satisfies $h \circ f = i_A$ and $p_C \circ h = g$, and it is an isomorphism by the [[five lemma|W6AVEAW2WR]] applied to the map from the given sequence to $0 \to A \to A \oplus C \to C \to 0$.
From (2), one argues dually, using $B \cong A \oplus C$ through $f$ and $u$.
Example: in $0 \to \Z/2\Z \to \Z/4\Z \to \Z/2\Z \to 0$, the quotient map has no right inverse.
A right inverse would send the generator of $\Z/2\Z$ to an element of $\Z/4\Z$ of order at most $2$, that is, to $0$ or $2$, and both map to $0$.
So the sequence does not split, in agreement with $\Z/4\Z \not\cong \Z/2\Z \oplus \Z/2\Z$.
The lemma fails for non-abelian groups: for the symmetric group, $1 \to A_3 \to S_3 \to \Z/2\Z \to 1$ has a right inverse of the quotient map, but $S_3$ is not isomorphic to $A_3 \times \Z/2\Z$.

## Explanations

### L1 human

A short exact sequence builds $B$ from a subobject $A$ and a quotient $C$.
The lemma says that $B$ falls apart as $A \oplus C$ as soon as one can undo either one of the two maps.
A way to project $B$ back onto $A$, or a way to lift $C$ back into $B$, is already enough.

### L2 human

The lemma is the standard test for splitting.
For example, every short exact sequence ending in a projective object splits, since the identity of $C$ lifts along $g$.
Dually, every short exact sequence starting with an injective object splits.
For vector spaces every short exact sequence splits, because a basis of $C$ can be lifted to $B$.
In the theory of extensions, the split sequence is the zero element of $\mathrm{Ext}^1(C, A)$.
For groups, condition (2) alone gives a semidirect product, and condition (1) gives a direct product.
</document>

<document source="https://tamnd.github.io/math2/node/7ZKBV18BFG.md" tag="7ZKBV18BFG" kind="problem">
# Word problem for an algebra

## Problem

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of finite signature, and let $A = \langle X \mid R \rangle_{\mathcal{V}}$ be an algebra given by a finite [[presentation|43M1WFPYGQ]], with $X$ and $R$ finite.
The *word problem* for this presentation asks for an algorithm, that is, a [[computable function|KZSB5E471K]], with the following task.
For any two terms $s$ and $t$ over $X$, it decides whether $s$ and $t$ denote the same element of $A$.
The problem is *solvable* for $A$ if such an algorithm exists.
The *uniform word problem* for $\mathcal{V}$ asks for one algorithm that takes the finite presentation as part of its input.

## Remarks

Solvability does not depend on the choice of the finite presentation of $A$.
Suppose that $\mathcal{V}$ is finitely based.
Then the set of pairs $(s, t)$ with $s = t$ in $A$ is computably enumerable, by listing all derivations from $R$ and the identities of $\mathcal{V}$.
The difficulty is to certify that $s \neq t$.
The problem is unsolvable in general.
Markov and Post independently gave finitely presented semigroups with unsolvable word problem in 1947.
Novikov and Boone gave finitely presented groups with unsolvable [[word problem for groups|PBGTTAVMMT]].
It is solvable in many classes, for example for finitely presented abelian groups and for one-relator groups, the latter by a theorem of Magnus.
When the relations together with the identities of $\mathcal{V}$ can be oriented into a finite convergent rewriting system, normal forms solve it.

## Explanations

### L1 human

An algebra given by generators and relations is like a puzzle: different expressions may name the same element because the relations say so.
The word problem asks for a mechanical procedure that always tells whether two given expressions are equal.
For many familiar algebras such a procedure exists.
A famous discovery of the twentieth century is that for some finitely presented semigroups and groups no procedure can exist at all.
</document>

<document source="https://tamnd.github.io/math2/node/7ZMWEDNCVQ.md" tag="7ZMWEDNCVQ" kind="statement">
# Sunada's theorem

## Statement

Let $M$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]], and let $G$ be a finite group that acts freely on $M$ by isometries.
Let $H_1$ and $H_2$ be subgroups of $G$ that are *almost conjugate*: every conjugacy class $C$ of $G$ satisfies $\abs{C \cap H_1} = \abs{C \cap H_2}$.
Give the quotients $M_1 = M/H_1$ and $M_2 = M/H_2$ the Riemannian metrics for which the projections from $M$ are local isometries.
Then $M_1$ and $M_2$ are [[isospectral|18MQEGN13R]]: their [[Laplace–Beltrami operators|F4NWZNTRMD]] have the same [[eigenvalues with the same multiplicities|B8TYKV7AA1]].

## Remarks

The idea of the proof is to compare the spaces of eigenfunctions.
An eigenfunction on $M/H$ with eigenvalue $\lambda$ is the same as an $H$-invariant eigenfunction on $M$.
So the multiplicity of $\lambda$ on $M/H$ is the dimension of the space of $H$-fixed vectors in the representation of $G$ on the $\lambda$-eigenspace of $M$.
Almost conjugate subgroups give permutation representations $\C[G/H_1]$ and $\C[G/H_2]$ with equal characters, and by Frobenius reciprocity these dimensions agree for every representation of $G$.
If $H_1$ and $H_2$ are conjugate in $G$, then $M_1$ and $M_2$ are isometric, so the theorem is of interest for almost conjugate subgroups that are not conjugate.
The projections $M \to M_i$ are [[covering maps|3XZRRE165N]], so $M_1$ and $M_2$ have a common finite cover.
This method gave many pairs of isospectral manifolds that are not isometric, including pairs of Riemann surfaces and, by later work, planar domains that are isospectral for the Dirichlet problem.

## Explanations

### L1 human

Two shapes can sound the same, that is, have the same frequencies of vibration, without being the same shape.
Sunada's theorem is a recipe that builds such pairs.
Start with one big symmetric space and fold it up in two different ways by two groups of symmetries.
If the two groups are balanced in a precise counting sense, the two folded spaces have the same frequencies.

### L2 human

The theorem turns a question about the spectrum of the Laplacian into a question in finite group theory.
The algebraic input is the same one that produces number fields with the same Dedekind zeta function, which is where almost conjugate subgroups first appeared.
Before this method, examples of isospectral non-isometric manifolds were sporadic, such as [[Milnor's flat tori|JNR9YJTT54]] in dimension $16$.
The theorem shows that the spectrum of the Laplacian does not determine the metric, so spectral geometry asks which properties the spectrum does determine.
</document>

<document source="https://tamnd.github.io/math2/node/7ZQ9FWZMP1.md" tag="7ZQ9FWZMP1" kind="statement">
# Bishop's theorem on analytic sets

## Statement

Let $X$ be an open subset of $\C^n$, let $E$ be an [[analytic subset|VMDTGTMN9P]] of $X$, and let $k \ge 1$ be an integer.
Let $A$ be an analytic subset of $X \setminus E$ of [[pure dimension|C2R5Q2Q717]] $k$.
Suppose that every point of $E$ has a neighbourhood $W$ in $X$ such that $\mathcal{H}^{2k}(A \cap W) < \infty$, where $\mathcal{H}^{2k}$ is the $2k$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] on $\C^n = \R^{2n}$.
Then the closure $\overline{A}$ of $A$ in $X$ is an analytic subset of $X$, of pure dimension $k$.

## Remarks

No condition on the dimension of $E$ is needed; it is replaced by the condition that $A$ has locally finite $2k$-dimensional volume near $E$.
The idea of the proof is to use the volume bound to control generic linear projections to $\C^k$.
Near each point of $E$, the closure $\overline{A}$ then projects properly, with a bounded number of sheets.
The symmetric functions of the sheets are bounded holomorphic functions outside a thin set, and they extend.
Example: let $E = \set{z_1 = 0}$ in $\C^2$ and $A = \set{z_2 = 0,\ z_1 \ne 0}$; the area of $A$ in each ball is finite, and indeed $\overline{A}$ is the $z_1$-axis.
The volume hypothesis is needed: for $A = \set{(z_1, e^{1/z_1}) : z_1 \ne 0}$ and the same $E$, the closure $A \cup E$ is not analytic, so by the theorem $A$ has infinite area near every point of $E$.
The theorem also holds with $X$ a complex manifold, since the statement is local.

## Explanations

### L1 human

An analytic set known outside an exceptional analytic set extends across it, as long as it has finite area, or finite volume in the right dimension, near the exceptional set.
A curve spiralling wildly towards a line, like the graph of $e^{1/z}$ near $z = 0$, has infinite area and does not extend.
Finite volume rules out this kind of wild behaviour.

### L2 human

Bishop's theorem strengthens the Remmert–Stein theorem, in which the exceptional set must have smaller dimension than $A$.
Volume bounds of this kind come naturally from Wirtinger's inequality and Stokes' theorem, which makes the theorem useful for limits of analytic cycles.
It is a main step in proving that limits of complex analytic cycles with bounded volume are analytic.
</document>

<document source="https://tamnd.github.io/math2/node/7ZR66SH0MB.md" tag="7ZR66SH0MB" kind="definition">
# Flat module

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].
The module $M$ is *flat* if the following holds for every injective $R$-[[linear map|0NRYHJPQG2]] $f \colon N' \to N$ of $R$-modules.
The map
$$f \otimes \id_M \colon N' \otimes_R M \to N \otimes_R M$$
is injective, where $\otimes_R$ is the [[tensor product|F0HQXQ0HCD]].

## Remarks

Since $- \otimes_R M$ is always right exact, $M$ is flat exactly when $- \otimes_R M$ takes [[exact sequences|JB7ZBFBA9C]] to exact sequences.
It is enough to check the condition for the inclusions $I \to R$ of finitely generated [[ideals|096Z9MP2ME]]: $M$ is flat if and only if $I \otimes_R M \to M$ is injective for each such $I$.
Over a [[principal ideal domain|0CMD0BCH8T]], a module is flat exactly when it is torsion-free.
Projective modules are flat, and a finitely presented flat module is projective.

Example: $\Q$ is a flat $\Z$-module.
For any $\Z$-module $N$, the module $N \otimes_\Z \Q$ is the localization $S^{-1}N$ with $S = \Z \setminus \set{0}$, and localization preserves injective maps.

Non-example: $\Z/2\Z$ is not a flat $\Z$-module.
The map $f \colon \Z \to \Z$, $x \mapsto 2x$, is injective, but $f \otimes \id$ is multiplication by $2$ on $\Z \otimes_\Z \Z/2\Z \cong \Z/2\Z$, which is the zero map on a nonzero module.
</document>

<document source="https://tamnd.github.io/math2/node/7ZR88SCSRF.md" tag="7ZR88SCSRF" kind="statement">
# Mazur's lemma

## Statement

Let $X$ be a [[normed vector space|03J5RAKB30]] over $\R$ or $\C$.
Let $(x_n)_{n \ge 1}$ be a sequence in $X$ that converges to $x \in X$ in the [[weak topology|P6KHVSTPZV]].
For each $n \ge 1$ there are an integer $N(n) \ge n$ and real numbers $\lambda_{n,k} \ge 0$ for $n \le k \le N(n)$, with $\sum_{k=n}^{N(n)} \lambda_{n,k} = 1$, with the following property.
The convex combinations
$$y_n = \sum_{k=n}^{N(n)} \lambda_{n,k}\, x_k$$
converge to $x$ in norm.

## Remarks

Fix $n$ and let $C_n$ be the norm closure of the convex hull of $\set{x_k : k \ge n}$.
By the [[Hahn–Banach separation theorem|6GWMGFGFJJ]], a norm-closed convex set is weakly closed, so $C_n$ is weakly closed.
Since $x_k \to x$ weakly and $x_k \in C_n$ for $k \ge n$, the limit $x$ lies in $C_n$.
Hence some convex combination $y_n$ of finitely many $x_k$ with $k \ge n$ satisfies $\norm{y_n - x} < 1/n$.

In the sequence space $\ell^2$, the unit vectors $e_n$ converge weakly to $0$ but $\norm{e_n} = 1$ for all $n$.
The averages $y_n = \frac{1}{n}(e_n + e_{n+1} + \cdots + e_{2n-1})$ have $\norm{y_n} = 1/\sqrt{n}$, so they tend to $0$ in norm, as the lemma predicts.

## Explanations

### L1 human

Weak convergence is a loose kind of convergence, and a weakly convergent sequence may stay far from its limit.
The lemma says that averaging repairs this.
Suitable averages of the later terms of the sequence do converge in the usual sense of distance.

### L2 human

The lemma is a quantitative form of the fact that a convex set has the same closure in the norm topology and in the weak topology.
It is used in the calculus of variations and in partial differential equations: one obtains a weak limit from compactness.
Then one uses the lemma to pass to strong limits of averages, which interact well with convex functionals.
For example, it shows that a convex lower semicontinuous functional on a Banach space is weakly lower semicontinuous.
</document>

<document source="https://tamnd.github.io/math2/node/7ZYKHQRKSJ.md" tag="7ZYKHQRKSJ" kind="statement">
# Hyperbolic law of cosines

## Statement

Let $ABC$ be a [[hyperbolic triangle|GCZ8EEDCHS]] in the hyperbolic plane of constant curvature $-1$.
Let $a$, $b$, $c$ be the lengths of the sides opposite $A$, $B$, $C$, and let $\gamma$ be the angle at $C$.
Then
$$\cosh c = \cosh a \cosh b - \sinh a \sinh b \cos \gamma,$$
where $\cosh$ and $\sinh$ are the [[hyperbolic cosine|036696EMG2]] and [[hyperbolic sine|0H1JFKE7H4]].

## Remarks

Proof idea: work in the [[hyperboloid model|809288G50Y]] with the form $B(x, y) = x_0 y_0 - x_1 y_1 - x_2 y_2$, where $\cosh d(x, y) = B(x, y)$.
After an isometry, $C = (1, 0, 0)$, $A = (\cosh b, \sinh b, 0)$ and $B = (\cosh a, \sinh a \cos \gamma, \sinh a \sin \gamma)$.
Then $\cosh c = B(A, B) = \cosh a \cosh b - \sinh a \sinh b \cos \gamma$.
For $\gamma = \pi/2$ it gives the hyperbolic Pythagorean theorem $\cosh c = \cosh a \cosh b$.
For small sides, $\cosh x \approx 1 + x^2/2$ and $\sinh x \approx x$, and the formula reduces to the Euclidean [[law of cosines|060ZVZBV3E]] $c^2 = a^2 + b^2 - 2ab \cos \gamma$.
In curvature $-k^2$ with $k > 0$, the same formula holds with $a, b, c$ replaced by $ka, kb, kc$.
There is also a dual law, $\cos \gamma = -\cos \alpha \cos \beta + \sin \alpha \sin \beta \cosh c$, which shows that the angles determine the triangle.

## Explanations

### L1 human

If you know two sides of a hyperbolic triangle and the angle between them, this formula gives the third side.
It plays the role of the school law of cosines, with hyperbolic functions in place of squares.
For a right triangle it says $\cosh c = \cosh a \cosh b$, the hyperbolic form of Pythagoras.

### L2 human

The law of cosines is the basic tool of hyperbolic trigonometry: the law of sines, the formulas for right triangles and the angle of parallelism all follow from it.
Since the right-hand side grows like $e^{a + b}/4 \cdot (1 - \cos \gamma)$ for long sides, it shows that two geodesics leaving a point at a fixed angle diverge exponentially, a key feature of negative curvature.
It is the hyperbolic member of a family with the spherical law $\cos c = \cos a \cos b + \sin a \sin b \cos \gamma$ and the Euclidean law as the flat limit.
</document>

<document source="https://tamnd.github.io/math2/node/802QSF9NM3.md" tag="802QSF9NM3" kind="concept">
# Sine-Gordon equation

## Summary

The sine-Gordon equation is the nonlinear wave equation $u_{tt} - u_{xx} + \sin u = 0$ for a real function $u(x,t)$.
Its name is a pun on the [[Klein–Gordon equation|AYCCDQWCJF]] $u_{tt} - u_{xx} + u = 0$, which is its linearisation at $u = 0$.
It first appeared in the geometry of surfaces of constant negative Gaussian curvature.
It is completely integrable and has kink solitons that connect the rest states $0$ and $2\pi$.

## Explanations

### L1 human

Picture a long row of pendulums hanging from a horizontal wire, with neighbours joined by springs.
The angle $u$ of the pendulums along the row obeys the sine-Gordon equation: $\sin u$ is the pull of gravity and $u_{xx}$ is the pull of the springs.
A kink is a place where the pendulums make one full turn, and this twist can travel along the row without changing shape.
The static kink is $u(x) = 4 \arctan(e^x)$, which goes from $0$ at $x = -\infty$ to $2\pi$ at $x = +\infty$.
</document>

<document source="https://tamnd.github.io/math2/node/8050R3Z6HA.md" tag="8050R3Z6HA" kind="concept">
# Operational calculus

## Summary

Operational calculus is a method that turns problems of analysis, above all linear differential and integral equations, into algebraic problems.
The differentiation operator is treated as an algebraic quantity, so an equation for the unknown function becomes an equation in a ring or field.
It is solved there by algebra, and the solution is translated back into a function.
The classical forms are the [[Heaviside operational calculus|QQ98C2AXZP]], its justification by the [[Laplace transform|0V49JEZW06]], and the [[Mikusiński operational calculus|TD5XK5AHNV]], which avoids integral transforms.

## Explanations

### L1 human

The idea is to treat "take the derivative" like multiplication by a number $s$.
Then a differential equation such as $y' - 2y = 0$ with $y(0) = 1$ becomes the algebra problem $sY - 1 - 2Y = 0$, so $Y = 1/(s - 2)$.
A table of translations says that $1/(s - 2)$ stands for the function $e^{2t}$, which is indeed the solution.
Engineers used this kind of symbolic shortcut long before mathematicians made it rigorous.
</document>

<document source="https://tamnd.github.io/math2/node/807S91H4RY.md" tag="807S91H4RY" kind="definition">
# Pure weights

## Definition

Let $E$ be a [[quadratic field|7E23FBM9NQ]] that is real. Let $S_\infty$ be the set of its infinite [[places|BT5V3TSNY6]], that is, its embeddings into the real numbers. Let $G_E$ be the group $\mathrm{GL}_3$ over $E$, and let $\mu$ be a highest weight for $G_E$.

Such a weight is a family $\mu=(\mu_\tau)_{\tau \in S_\infty}$. Each $\mu_\tau=(m^+_\tau,m^-_\tau,v_\tau)$ is a dominant weight of the diagonal torus of $\mathrm{GL}_3$.

The weight $\mu$ is called **pure** when both clauses hold:

1. The entry $v_\tau$ is the same for every $\tau \in S_\infty$.
2. The equality $m^+_\tau=m^-_\tau$ holds for every $\tau \in S_\infty$.

The **pure weights** are the highest weights for $G_E$ that satisfy both clauses.
</document>

<document source="https://tamnd.github.io/math2/node/8085SD9EYY.md" tag="8085SD9EYY" kind="definition">
# Score test

## Definition

Let $X$ be data with log-likelihood $\ell(\theta) = \log L(\theta \mid X)$, differentiable in $\theta$ in an open set $\Theta \subseteq \R^k$, and let $\theta_0 \in \Theta$.
Let $U(\theta) = \nabla \ell(\theta)$ be the [[score|CP97GRVTQ0]] and let $I(\theta)$ be the [[Fisher information|G1Z3WSTE8Q]] of $X$, assumed invertible at $\theta_0$.
The *score statistic* for $H_0 \colon \theta = \theta_0$ is
$$S = U(\theta_0)^{T} I(\theta_0)^{-1} U(\theta_0).$$
The *score test* at level $\alpha$ rejects $H_0$ when $S$ exceeds the $1 - \alpha$ quantile of the [[chi-squared distribution|0RFMMBHFTV]] with $k$ degrees of freedom.

## Remarks

Under $H_0$ and the usual regularity conditions for i.i.d. samples, $S$ converges in distribution to $\chi^2_k$, so the test has asymptotic level $\alpha$.
For a composite $H_0$ with nuisance parameters, one evaluates the score and the information at the maximum likelihood estimate under $H_0$.
Example: for $X$ binomial with parameters $n$ and $p$, one has $U(p) = (X - np)/(p(1 - p))$ and $I(p) = n/(p(1 - p))$.
So $S = (X - np_0)^2/(np_0(1 - p_0))$, the square of the usual z-statistic for a proportion.
Non-example: the statistic $U(\theta_0)^T U(\theta_0)$, without the factor $I(\theta_0)^{-1}$, does not have a $\chi^2_k$ limit in general, so it is not the score statistic.
</document>

<document source="https://tamnd.github.io/math2/node/809288G50Y.md" tag="809288G50Y" kind="example">
# Hyperboloid model

## Example

Let $n \ge 1$ and give $\R^{n+1}$, with coordinates $x = (x_0, x_1, \ldots, x_n)$, the symmetric [[bilinear form|0SQZMXB6GP]]
$$B(x, y) = x_0 y_0 - x_1 y_1 - \cdots - x_n y_n.$$
The *hyperboloid model* of hyperbolic $n$-space is the upper sheet of a hyperboloid of two sheets,
$$H^n = \set{x \in \R^{n+1} : B(x, x) = 1, \ x_0 > 0},$$
with the distance $d(x, y) = \operatorname{arcosh} B(x, y)$.
On each tangent space of $H^n$ the form $-B$ is positive definite, and the resulting [[Riemannian manifold|7CBT4SM83Y]] has constant sectional curvature $-1$.
Its geodesics, the lines of [[hyperbolic geometry|GN4N94GQ0Z]], are the nonempty intersections of $H^n$ with 2-dimensional linear subspaces of $\R^{n+1}$.

Check for $n = 1$: the curve $\gamma(t) = (\cosh t, \sinh t)$ lies on $H^1$, since $\cosh^2 t - \sinh^2 t = 1$ and $\cosh t > 0$.
Its velocity is $\gamma'(t) = (\sinh t, \cosh t)$, and $-B(\gamma', \gamma') = \cosh^2 t - \sinh^2 t = 1$, so $t$ is arc length.
The formula gives $d(\gamma(0), \gamma(t)) = \operatorname{arcosh}(\cosh t) = \abs{t}$, in agreement.

## Remarks

The isometries of $H^n$ are the linear maps of $\R^{n+1}$ that preserve $B$ and map the upper sheet to itself, the subgroup $O^+(1, n)$ of the [[indefinite orthogonal group|M2900DMEW6]].
So isometries act by matrices, which makes computations in this model linear algebra.
The central projection $x \mapsto (x_1/x_0, \ldots, x_n/x_0)$ maps $H^n$ onto the [[Beltrami–Klein model|1WN9B18H86]].
The projection $x \mapsto (x_1, \ldots, x_n)/(1 + x_0)$ from the point $(-1, 0, \ldots, 0)$ maps $H^n$ onto the [[Poincaré disk model|85MM2EA0TK]] when $n = 2$, and onto the Poincaré ball model in general.
For $n = 3$, $\R^4$ with the form $B$ is Minkowski spacetime, and $H^3$ is the set of unit future-pointing timelike vectors.

## Explanations

### L1 human

Take the bowl-shaped sheet $x_0 = \sqrt{1 + x_1^2 + x_2^2}$ in space, and measure lengths on it with a modified rule in which the vertical direction counts negatively.
With this rule the bowl becomes the hyperbolic plane.
Hyperbolic lines are the curves cut out of the bowl by planes through the origin, and rotations and Lorentz boosts move the bowl onto itself.
This is the hyperbolic analogue of the round sphere, which is the set of points at distance $1$ from the origin for the usual form $x_0^2 + x_1^2 + x_2^2$.
</document>

<document source="https://tamnd.github.io/math2/node/80GQXWM7BV.md" tag="80GQXWM7BV" kind="concept">
# Moore family

## Summary

A Moore family on a set $X$ is a collection of subsets of $X$ that contains $X$ and is closed under intersections of nonempty subfamilies.
Such a collection is also called a closure system.
Moore families on $X$ are exactly the families of closed sets of [[closure operators|17FVAN6HXX]] on the subsets of $X$.
Ordered by inclusion, a Moore family is a complete lattice.

## Explanations

### L1 human

Many kinds of substructures are stable under intersection.
The intersection of any number of subgroups of a group is a subgroup, and the whole group is a subgroup.
So the subgroups of a group form a Moore family.
The same holds for the subspaces of a vector space and the closed sets of a topological space.
</document>

<document source="https://tamnd.github.io/math2/node/80MTNXWW80.md" tag="80MTNXWW80" kind="concept">
# Levi form

## Summary

The Levi form is a [[Hermitian form|48P4A1CX85]] built from the mixed second derivatives $\partial^2 \rho / \partial z_j \partial \bar z_k$ of a real function $\rho$ on a complex domain.
For the boundary of a domain in $\C^n$ one restricts it to the complex tangent directions of the boundary.
Its sign decides the convexity notions of [[several complex variables|BH57D2FBNR]]: pseudoconvexity, strong pseudoconvexity and plurisubharmonicity.

## Explanations

### L1 human

Ordinary convexity of a boundary is measured by the real second derivatives of a function that cuts out the domain.
In complex analysis only the part of this second derivative that is unchanged by holomorphic changes of coordinates matters, and this part is the Levi form.
One also looks only along the directions of the boundary that are complex lines, because these are the directions a holomorphic disc can move in.
For the unit ball in $\C^n$, cut out by $\rho(z) = \abs{z_1}^2 + \cdots + \abs{z_n}^2 - 1$, the Levi form is $\abs{w_1}^2 + \cdots + \abs{w_n}^2$, which is positive.
So the sphere curves "the right way" in every complex direction.
</document>

<document source="https://tamnd.github.io/math2/node/80QZ05NFCB.md" tag="80QZ05NFCB" kind="definition">
# Invariant set

## Definition

Let $U$ be an open subset of $\R^n$, let $f \colon U \to \R^n$ be locally Lipschitz, and let $\varphi$ be the [[flow|02903SSA48]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.
For $x \in U$ let $I(x)$ be the maximal interval of existence of the solution through $x$.
A set $M \subseteq U$ is invariant if $\varphi_t(x) \in M$ for every $x \in M$ and every $t \in I(x)$.
It is positively invariant if $\varphi_t(x) \in M$ for every $x \in M$ and every $t \in I(x)$ with $t \ge 0$, and negatively invariant if the same holds for $t \le 0$.

## Remarks

A set is invariant exactly when it is a union of [[trajectories|07TZ2Q4C1M]].
Example: for $x' = -y$, $y' = x$ on $\R^2$, the flow is rotation by the angle $t$, so it preserves $x^2 + y^2$; hence each disc $\set{x^2 + y^2 \le r^2}$ is invariant.
Non-example: for $x' = -x$ on $\R$, the interval $[0, 1]$ is positively invariant, since $\varphi_t(x) = e^{-t} x$ stays in $[0, 1]$ for $t \ge 0$.
It is not invariant, since $\varphi_{-1}(1) = e > 1$.
An equilibrium point and a periodic orbit are invariant sets.
For a continuously differentiable function $V \colon U \to \R$ with $\nabla V \cdot f \le 0$ on $U$, each sublevel set $\set{x \in U : V(x) \le c}$ is positively invariant.
For a general flow $\varphi$ on a set $X$ the same definition reads $\varphi_t(M) \subseteq M$ for all $t$.

## Explanations

### L1 human

An invariant set is a trap with no way out in either direction of time.
A positively invariant set is a trap that solutions can enter but never leave.
</document>

<document source="https://tamnd.github.io/math2/node/81035XQ5HE.md" tag="81035XQ5HE" kind="problem">
# Hilbert's third problem

## Problem

Let $P$ and $Q$ be two [[polyhedra|0325SGSJAR]] in $\R^3$ of equal volume.
Can $P$ always be cut into finitely many polyhedral pieces that can be reassembled by isometries into $Q$?

## Remarks

The answer is no.
Max Dehn proved in 1900 that the regular tetrahedron and the cube of the same volume cannot be cut into each other in this way.
His tool was the [[Dehn invariant|S0TGRX3ARS]], which does not change under cutting and reassembling: it is $0$ for the cube and nonzero for the regular tetrahedron.
Jean-Pierre Sydler proved in 1965 that two polyhedra are scissors-congruent if and only if they have the same volume and the same Dehn invariant.
In the plane the analogous question has a positive answer, by the [[Wallace–Bolyai–Gerwien theorem|5RQGCEQYK5]].

## Explanations

### L1 human

Two polygons with the same area can always be cut into pieces and rearranged into each other.
Hilbert asked whether the same is true for solids with the same volume.
It is not: no matter how you cut a regular tetrahedron into finitely many flat-faced pieces, you cannot rebuild a cube from them.
This was the first of Hilbert's 23 problems to be solved.
</document>

<document source="https://tamnd.github.io/math2/node/811NW2WEJH.md" tag="811NW2WEJH" kind="problem">
# Hauptvermutung

## Problem

Let $K$ and $L$ be [[simplicial complexes|V8GZ0W1ZZ5]] whose geometric realizations $\abs{K}$ and $\abs{L}$ are [[homeomorphic|0QT00PB5G2]].
Do $K$ and $L$ always have subdivisions $K'$ and $L'$ that are isomorphic as simplicial complexes?

Equivalently: are any two [[triangulations|YH170WBQ6M]] of the same space related by a piecewise linear homeomorphism?

## Remarks

The name is German for "main conjecture".
A positive answer would make the combinatorial invariants of a triangulation, such as simplicial homology, automatically topological invariants; homology was later shown to be a topological invariant by other means.
The answer is yes for complexes of dimension at most $2$, and for $3$-manifolds by [[Moise's theorem|SV0816DJP9]]; for surfaces it goes with [[Radó's theorem|C9X05EDS5F]].
The answer is no in general: John Milnor gave counterexamples in 1961 among finite simplicial complexes that are not manifolds.
For manifolds the answer is also no.
In dimensions at least $5$, the theory behind the [[Kirby–Siebenmann class|E4NJER1RFQ]] gives topological manifolds with several PL structures.
The corresponding triangulations are not related by any piecewise linear homeomorphism.

## Explanations

### L1 human

The same shape can be cut into triangles, or into higher-dimensional simplices, in many ways.
The question is whether any two such cuttings can be refined until they become the same cutting.
For shapes of dimension up to three this works, but in higher dimensions there are shapes where it fails.
</document>

<document source="https://tamnd.github.io/math2/node/812THX456R.md" tag="812THX456R" kind="statement">
# Euler's integral representation of the hypergeometric function

## Statement

Let $a, b, c \in \C$ with $\operatorname{Re} c > \operatorname{Re} b > 0$, and let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
Then for every $z \in \C \setminus [1, \infty)$,
$${}_2F_1(a, b; c; z) = \frac{\Gamma(c)}{\Gamma(b) \, \Gamma(c - b)} \int_0^1 t^{b-1} (1 - t)^{c - b - 1} (1 - zt)^{-a} \, dt,$$
where ${}_2F_1$ is the [[hypergeometric function|463X5V31D9]] and $(1 - zt)^{-a}$ is the principal power.

## Remarks

For $\abs{z} < 1$, expand $(1 - zt)^{-a} = \sum_n \frac{(a)_n}{n!} z^n t^n$ by the binomial series and integrate term by term.
The $n$-th term gives the [[beta function|QHNWVSC6AQ]] value $B(b + n, c - b) = \frac{\Gamma(b + n) \Gamma(c - b)}{\Gamma(c + n)}$, and $\Gamma(b + n)/\Gamma(b) = (b)_n$, $\Gamma(c + n)/\Gamma(c) = (c)_n$, which gives the series of ${}_2F_1$.
The integral is holomorphic in $z \in \C \setminus [1, \infty)$, because $1 - zt$ never lies on $(-\infty, 0]$ there, so the identity extends by analytic continuation.

Example: with $a = 1$, $b = 1$, $c = 2$ the integral is $\int_0^1 \frac{dt}{1 - zt} = -\frac{\log(1 - z)}{z}$, which is ${}_2F_1(1, 1; 2; z)$.

The hypothesis $\operatorname{Re} c > \operatorname{Re} b > 0$ is needed for convergence of the integral at $t = 0$ and $t = 1$.
For $b = c$ the factor $(1 - t)^{-1}$ makes the integral diverge at $t = 1$, although ${}_2F_1(a, b; b; z) = (1 - z)^{-a}$ is perfectly defined.
A Pochhammer contour integral removes this restriction.

## Explanations

### L1 human

The hypergeometric function is first defined by a power series that only converges for $\abs{z} < 1$.
Euler's formula writes it as an integral over $[0, 1]$ of an explicit function.
The integral makes sense for every $z$ off the cut $[1, \infty)$, so it shows directly how to extend the function there.

### L2 human

This representation is the main tool for proving identities for ${}_2F_1$.
Setting $z = 1$ gives [[Gauss's hypergeometric theorem|1FC0T0GCCX]] by the beta integral, and the substitution $t \mapsto 1 - t$ gives the [[Pfaff transformation|KAAJ0JJ2GF]].
It also exhibits ${}_2F_1$ as a period integral of a twisted differential form, which leads to the monodromy and to generalizations such as Appell and Lauricella functions.
</document>

<document source="https://tamnd.github.io/math2/node/816AHR3DRT.md" tag="816AHR3DRT" kind="statement">
# Hard Lefschetz theorem

## Statement

Let $X \subset \mathbb{P}^N_{\C}$ be a smooth complex [[projective variety|A4H4CZ2V35]] of dimension $n$.
Let $\omega \in H^2(X, \Q)$ be the class of a hyperplane section $X \cap H$, in [[singular cohomology|6N7BC70CX0]].
Let $L \colon H^j(X, \Q) \to H^{j+2}(X, \Q)$ be the map $\alpha \mapsto \omega \smile \alpha$, given by the [[cup product|JX47MS1R1H]].
Then for each integer $k$ with $0 \le k \le n$, the map
$$L^k \colon H^{n-k}(X, \Q) \to H^{n+k}(X, \Q)$$
is an isomorphism.

## Remarks

The classical proof uses [[Hodge theory|32YJXG30MF]] on the [[Kähler manifold|X60JG9AEQJ]] $X(\C)$.
The operator $L$, its adjoint $\Lambda$ and the degree operator form a representation of $\mathfrak{sl}_2$ on harmonic forms, and $L^k$ is an isomorphism by the representation theory of $\mathfrak{sl}_2$.
For $X = \mathbb{P}^n$, $H^{2i}(X, \Q) = \Q \, \omega^i$ for $0 \le i \le n$ and the odd groups vanish, so $L^k$ sends $\omega^{(n-k)/2}$ to $\omega^{(n+k)/2}$, which is a basis.
The theorem shows that $L \colon H^j \to H^{j+2}$ is injective for $j < n$, so the Betti numbers satisfy $b_j \le b_{j+2}$ for $j < n$, and also $b_j = b_{2n-j}$.
Over a field of positive characteristic, Deligne proved the analogue for $\ell$-adic cohomology.
The hypothesis that $\omega$ comes from an ample class matters: for a class $\omega$ with $\omega^n = 0$, such as $0$, the map $L^n \colon H^0 \to H^{2n}$ is zero.

## Explanations

### L1 human

A smooth projective variety of dimension $n$ has cohomology groups in degrees $0$ to $2n$.
Cutting by a hyperplane again and again gives a map from degree $n-k$ to degree $n+k$.
The theorem says that this map loses nothing and adds nothing: it is a bijection.
So the cohomology is symmetric around the middle degree in a very rigid way.

### L2 human

The theorem gives the Lefschetz decomposition of $H^*(X, \Q)$ into primitive parts, and together with the Hodge–Riemann bilinear relations it shows that the cohomology carries a polarized Hodge structure.
It is one of the standard tools that restrict which manifolds can be projective varieties: for example, the sequences $b_0, b_2, b_4, \ldots$ and $b_1, b_3, b_5, \ldots$ cannot decrease before the middle degree.
It complements the [[Lefschetz hyperplane theorem|EX6YSYN7B4]], which compares the cohomology of $X$ with that of a hyperplane section below the middle degree.
Grothendieck's Lefschetz-type standard conjecture asks that the inverse of $L^k$ come from an algebraic cycle.
</document>

<document source="https://tamnd.github.io/math2/node/818832PMSV.md" tag="818832PMSV" kind="statement">
# Donsker's theorem

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|00MRN0FARV]] real random variables with $E[X_1] = 0$ and $\sigma^2 = E[X_1^2] \in (0, \infty)$.
Let $S_0 = 0$ and $S_k = X_1 + \cdots + X_k$.
For $n \ge 1$ and $t \in [0, 1]$, let
$$W_n(t) = \frac{1}{\sigma \sqrt{n}} \left( S_{\floor{nt}} + (nt - \floor{nt}) \, X_{\floor{nt} + 1} \right),$$
the linear interpolation of the rescaled [[random walk|04VZE3FZXF]].
Let $C[0, 1]$ be the space of [[continuous|0X3WGX95FE]] functions on $[0, 1]$ with the supremum norm.
Then $W_n$ [[converges in distribution|0SRYJGXKR4]], as random elements of $C[0, 1]$, to a standard [[Wiener process|22735FGJV6]] $(W(t))_{t \in [0, 1]}$.

## Remarks

The proof has two steps.
The [[central limit theorem|0EPZC4Z1FC]], applied to the independent increments, gives the convergence of the finite-dimensional distributions of $W_n$ to those of $W$.
A maximal inequality then shows that the laws of the $W_n$ are tight in $C[0, 1]$, and tightness together with convergence of finite-dimensional distributions gives [[weak convergence|MBW9JCGZC1]].

Example: the supremum map $f \mapsto \max_{t \in [0, 1]} f(t)$ is continuous on $C[0, 1]$.
By the [[continuous mapping theorem|CFRN0G7GDB]], $\max_{k \le n} S_k / (\sigma \sqrt{n})$ converges in distribution to $\max_{t \le 1} W(t)$.
By the [[reflection principle|K6R7GHB3VS]], this limit has the law of $\abs{W(1)}$.
So the limit does not depend on the law of $X_1$, which is why the result is called an invariance principle.

A second form of the theorem concerns the [[empirical process|XKFG1676CG]].
If $U_1, U_2, \dots$ are i.i.d. uniform on $[0, 1]$ with empirical distribution function $F_n$, then $\sqrt{n}(F_n(t) - t)$ converges in distribution to a [[Brownian bridge|3MCVMBAGD1]], in a suitable space of functions with jumps.

## Explanations

### L1 human

Draw the path of a random walk with many small steps and shrink it so that it fits in a unit box.
As the steps get smaller, the picture looks more and more like the path of Brownian motion.
This holds for any step law with mean zero and finite variance, so Brownian motion is a universal limit.

### L2 human

Donsker's theorem is the functional form of the central limit theorem.
With the continuous mapping theorem, it gives the limit laws of functionals of random walks, computed once for Brownian motion.
Examples are the maximum, the time spent above zero and the time of the last zero.
In statistics, the empirical-process form gives the limit law of the Kolmogorov–Smirnov statistic $\sqrt{n} \sup_x \abs{F_n(x) - F(x)}$ for a continuous $F$.
</document>

<document source="https://tamnd.github.io/math2/node/81AQ28RQDP.md" tag="81AQ28RQDP" kind="definition">
# Weierstrass transform

## Definition

Let $f \colon \R \to \C$ be measurable with $\int_\R \abs{f(y)}\, e^{-c y^2}\, dy < \infty$ for every $c > 0$.
The *Weierstrass transform* of $f$ is the function $Wf \colon \R \to \C$ given by
$$(Wf)(x) = \frac{1}{\sqrt{4\pi}} \int_\R f(y)\, e^{-(x - y)^2/4}\, dy.$$

## Remarks

The kernel $\frac{1}{\sqrt{4\pi}} e^{-u^2/4}$ is the density of the [[normal distribution|05AQE0QR0D]] with mean $0$ and variance $2$.
So $Wf(x) = \mathbb{E}[f(x + Y)]$ for a random variable $Y$ with that law.
It is the [[heat kernel|KD404MF042]] $\frac{1}{\sqrt{4\pi t}} e^{-u^2/(4t)}$ at $t = 1$, so $Wf = u(\cdot, 1)$ where $u$ solves the [[heat equation|VP2HN86VAG]] $u_t = u_{xx}$ with $u(\cdot, 0) = f$.
For integrable $f$ and $\hat f(\xi) = \int f(x) e^{-i x \xi}\, dx$, one has $\widehat{Wf}(\xi) = e^{-\xi^2} \hat f(\xi)$.
The growth condition can be weakened to $\int \abs{f(y)} e^{-c y^2}\, dy < \infty$ for one $c < \frac14$, which is enough for the integral to converge for every $x$.

Example: $f(x) = x^2$.
With $Y$ of mean $0$ and variance $2$, $Wf(x) = \mathbb{E}[(x + Y)^2] = x^2 + 2x\, \mathbb{E}[Y] + \mathbb{E}[Y^2] = x^2 + 2$.
Likewise $W[e^{a x}](x) = e^{a x} \mathbb{E}[e^{aY}] = e^{a^2} e^{a x}$ for every real $a$.

Non-example: $f(x) = e^{x^2/4}$ fails the condition for $c < \frac14$, and indeed $\int_\R e^{y^2/4} e^{-(x-y)^2/4}\, dy = e^{-x^2/4} \int_\R e^{x y/2}\, dy$ diverges for every $x$.
</document>

<document source="https://tamnd.github.io/math2/node/81JHJAVRV4.md" tag="81JHJAVRV4" kind="example">
# Riemann–Liouville fractional integral equation

## Example

Let $b > 0$ and $0 < \alpha < 1$.
The [[Riemann–Liouville integral|JTKCZ44NN3]] of order $\alpha$ of a continuous function $\varphi$ on $[0, b]$ is
$$(I^\alpha \varphi)(x) = \frac{1}{\Gamma(\alpha)} \int_0^x (x - t)^{\alpha - 1} \varphi(t) \, dt,$$
where $\Gamma$ is the [[Gamma function|0H5SYBSED9]].
The *Riemann–Liouville fractional integral equation* of order $\alpha$ is $I^\alpha \varphi = f$ for a given $f$ on $[0, b]$.
It is a [[Volterra equation of the first kind|H6ZGEDM4VV]] of [[convolution type|SNFYTC1TTA]], with the weakly singular kernel $k(u) = u^{\alpha - 1} / \Gamma(\alpha)$.

Check on power functions: let $\beta > -1$.
The substitution $t = xu$ gives
$$I^\alpha (t^\beta)(x) = \frac{x^{\alpha + \beta}}{\Gamma(\alpha)} \int_0^1 (1 - u)^{\alpha - 1} u^\beta \, du = \frac{x^{\alpha + \beta} B(\alpha, \beta + 1)}{\Gamma(\alpha)} = \frac{\Gamma(\beta + 1)}{\Gamma(\alpha + \beta + 1)} \, x^{\alpha + \beta},$$
using the [[Beta function|QHNWVSC6AQ]].
So for $\mu > \alpha - 1$, the equation $I^\alpha \varphi = x^\mu$ is solved by
$$\varphi(t) = \frac{\Gamma(\mu + 1)}{\Gamma(\mu + 1 - \alpha)} \, t^{\mu - \alpha}.$$
For instance, $\alpha = \tfrac12$ and $\mu = 1$ give $I^{1/2} \varphi = x$ with $\varphi(t) = \frac{\Gamma(2)}{\Gamma(3/2)} t^{1/2} = \frac{2}{\sqrt{\pi}} \sqrt{t}$.

## Remarks

When a continuous solution exists, it is $\varphi = \frac{d}{dx} I^{1 - \alpha} f$, the Riemann–Liouville fractional derivative of order $\alpha$ of $f$.
This follows from the semigroup law $I^{1 - \alpha} I^\alpha = I^1$, and it is the [[Abel inversion formula|TDMXFY06A2]] in other notation.
For $\alpha = \tfrac12$, the equation is [[Abel's integral equation|G4GWBYTW7B]] divided by $\Gamma(\tfrac12) = \sqrt{\pi}$.

## Explanations

### L1 human

Integrating $n$ times can be written as one integral with the kernel $(x - t)^{n - 1}/(n - 1)!$.
Replacing $n$ by a fraction $\alpha$ gives the fractional integral $I^\alpha$.
Solving $I^\alpha \varphi = f$ means taking a derivative of order $\alpha$ of $f$.
The power functions show how this works: integrating to order $\alpha$ raises the exponent by $\alpha$, and solving lowers it by $\alpha$.
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# Subnormal operator

## Summary

A subnormal operator is a bounded operator on a Hilbert space that is the restriction of a normal operator to an invariant subspace.
Every normal operator is subnormal, and every subnormal operator is [[hyponormal|35XKRHEC3Q]].
The unilateral shift is the basic example, and multiplication by $z$ on spaces of analytic functions gives many more.
Scott Brown showed that subnormal operators have nontrivial invariant subspaces.

## Explanations

### L1 human

A subnormal operator is a normal operator seen only on part of the space that it maps into itself.
Multiplication by $z$ on square-integrable functions on the unit circle is normal.
On the smaller space of functions that extend analytically into the disc it is no longer normal, but it is subnormal.
In finite dimensions a subnormal operator is already normal, so the notion only matters in infinite dimensions.
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# Euler summation

## Summary

Euler summation is a [[summability method|8X25Q4XYMN]] that replaces a series by a new series built from binomial combinations of its terms.
The value is the sum of the new series.
It goes back to the [[Euler transform|WT0HACKQV7]] of alternating series, which speeds up their convergence.
The method gives the usual sum to every convergent series and sums many divergent alternating series.
It belongs to the family of Hausdorff methods, together with the Cesàro methods.

## Explanations

### L1 human

Euler summation mixes neighbouring terms of a series with binomial weights, as in the rows of Pascal's triangle.
For an alternating series, large terms of opposite sign then cancel each other.
For $1 - 1 + 1 - 1 + \cdots$, every mixed term after the first is $0$, and the new series is just $\frac12$.
So Euler summation gives the value $1/2$, as Cesàro and Abel summation do.
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# Weil conjectures

## Statement

Let $q$ be a prime power and let $X$ be a smooth, geometrically irreducible [[projective variety|A4H4CZ2V35]] of dimension $n$ over the [[finite field|04SGPNT8R9]] $\F_q$.
For $m \ge 1$ let $N_m$ be the number of [[rational points|SWJYG87QQH]] of $X$ over $\F_{q^m}$.
The zeta function of $X$ is the power series
$$Z(X, t) = \exp\Big(\sum_{m \ge 1} \frac{N_m}{m} t^m\Big),$$
in the ring of formal power series over $\Q$.
Then the following hold.

1. Rationality: $Z(X, t) = \dfrac{P_1(t) P_3(t) \cdots P_{2n-1}(t)}{P_0(t) P_2(t) \cdots P_{2n}(t)}$ with $P_i(t) \in \Z[t]$, $P_0(t) = 1 - t$ and $P_{2n}(t) = 1 - q^n t$.
2. Functional equation: $Z\big(X, \tfrac{1}{q^n t}\big) = \pm\, q^{nE/2} \, t^{E} \, Z(X, t)$, where $E$ is the self-intersection number of the diagonal in $X \times X$.
3. Riemann hypothesis: for $1 \le i \le 2n - 1$, $P_i(t) = \prod_j (1 - \alpha_{ij} t)$, where the $\alpha_{ij}$ are algebraic integers with $\abs{\alpha_{ij}} = q^{i/2}$ for every complex absolute value.
4. Betti numbers: suppose $X$ is the good reduction modulo a prime of a smooth projective variety $Y$ over a number field $K \subset \C$.
   Then $\deg P_i$ is the $i$-th Betti number of the complex manifold $Y(\C)$.

## Remarks

In the proof one takes $P_i(t) = \det(1 - tF \mid H^i(\bar{X}, \Q_\ell))$, where $F$ is the Frobenius acting on [[ℓ-adic cohomology|E1DVHN06CR]].
The [[Grothendieck–Lefschetz trace formula|P6QD6TKAGS]] then gives rationality, Poincaré duality gives the functional equation, and $E$ is the [[Euler characteristic|RNDDWDAFJ5]] of $\bar{X}$.
Dwork proved rationality in 1960 by $p$-adic methods, Grothendieck proved rationality and the functional equation with étale cohomology, and Deligne proved the Riemann hypothesis in 1974.
Example: for an elliptic curve $X$ over $\F_q$, $Z(X, t) = \dfrac{1 - a t + q t^2}{(1 - t)(1 - q t)}$ with $a = q + 1 - N_1$.
The Riemann hypothesis says that the two roots of $1 - a t + q t^2$ have inverses of absolute value $\sqrt{q}$, which is [[Hasse's theorem|7Y26317GJ0]] $\abs{a} \le 2\sqrt{q}$.

## Explanations

### L1 human

Take a system of polynomial equations with integer coefficients and count its solutions in the finite fields with $q$, $q^2$, $q^3, \ldots$ elements.
Weil predicted that these counts are governed by finitely many numbers $\alpha$, with $N_m$ a signed sum of their $m$-th powers.
He also predicted the sizes of these numbers and their link with the shape of the corresponding complex variety.
All four predictions are now theorems.

### L2 human

Point (3) is the analogue of the [[Riemann hypothesis|VCGF9YR36X]] for the zeta function of a variety over a finite field.
It gives sharp estimates of point counts that are used throughout number theory.
Weil proved the conjectures for curves and abelian varieties, and they motivated the creation of étale and $\ell$-adic cohomology.
The local zeta functions $Z(X, t)$ at the primes of a number field are the factors of the [[Hasse–Weil zeta function|37TNNBQ4PK]].
Grothendieck proposed his standard conjectures as a route to point (3), and they would give a second proof of it.
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# Identity

## Definition

Let $\sigma$ be a signature with only function symbols and let $X$ be a set of variables.
An *identity*, or *equation*, over $\sigma$ and $X$ is an ordered pair $(s, t)$ of [[terms|6D3WA7N2W4]] over $\sigma$ and $X$, written $s \approx t$.
An algebra $A$ of signature $\sigma$ *satisfies* $s \approx t$, written $A \models s \approx t$, if $s^A[\alpha] = t^A[\alpha]$ for every map $\alpha \colon X \to A$, where $t^A[\alpha]$ is the [[value|DH30WTA26T]] of $t$.
A class $\mathcal{K}$ of algebras satisfies $s \approx t$ if every member of $\mathcal{K}$ does.

## Remarks

The symbol $\approx$ separates the formal pair from the equality $=$ of elements.
As a first-order sentence, $s \approx t$ stands for the universal closure $\forall x_1 \cdots \forall x_n \, (s = t)$, where $x_1, \ldots, x_n$ are the variables of $s$ and $t$.

Example: the group $(\Z, +, -, 0)$ satisfies $x + y \approx y + x$, since $a + b = b + a$ for all integers $a, b$.

Non-example: the same group does not satisfy $x + x \approx x$.
At $\alpha(x) = 1$ the two sides take the values $2$ and $1$.
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# Gelfand transform

## Definition

Let $A$ be a commutative complex [[Banach algebra|9KYWN94QDN]], and let $\Delta(A)$ be its [[character space|M7JVFJ3MMX]].
For $a \in A$, the *Gelfand transform* of $a$ is the function
$$\hat a \colon \Delta(A) \to \C, \qquad \hat a(\varphi) = \varphi(a).$$
The *Gelfand transform* (or Gelfand representation) of $A$ is the map $\Gamma \colon A \to C_0(\Delta(A))$, $a \mapsto \hat a$, into the [[continuous functions vanishing at infinity|N59BW1QFD1]] on $\Delta(A)$.

## Remarks

Each $\hat a$ is continuous by the definition of the weak-* topology, and it vanishes at infinity.
The map $\Gamma$ is an algebra homomorphism with $\norm{\hat a}_\infty \le \norm{a}$.
If $A$ has a unit, then $\hat a(\Delta(A)) = \sigma(a)$, the spectrum of $a$, so $\norm{\hat a}_\infty$ is the spectral radius of $a$.
The kernel of $\Gamma$ is the Jacobson radical of $A$.

Example: let $A = \ell^1(\Z)$ with convolution $(a * b)_n = \sum_k a_k b_{n-k}$.
For $z$ in the unit circle, $\varphi_z(a) = \sum_n a_n z^n$ is linear, nonzero, and multiplicative by the Cauchy product formula, so it is a character.
Every character has this form, and $z \mapsto \varphi_z$ identifies $\Delta(A)$ with the circle.
The Gelfand transform of $a$ is the absolutely convergent Fourier series $\hat a(z) = \sum_n a_n z^n$.

Non-example: the construction needs $A$ to be commutative.
The algebra $M_2(\C)$ has no characters at all.
A character vanishes on commutators, and the commutators span the trace-zero matrices.
So its kernel is a nonzero two-sided ideal of the simple algebra $M_2(\C)$, hence everything.
So for $M_2(\C)$ the "Gelfand transform" would map into functions on the empty set and lose all information.
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# Kolmogorov's normability criterion

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a [[Hausdorff|0S1FJ8VEX9]] [[topological vector space|T32W553W95]] over $\mathbb{K}$.
Then some [[norm|0V2WP9B1MF]] on $E$ induces the topology of $E$ if and only if $0$ has a [[convex|02AS45T3H9]] [[von Neumann bounded|3BAJ8ZTVPX]] [[neighbourhood|0ZCY0Z80B6]] in $E$.

## Remarks

If the topology comes from a norm, the open unit ball is a convex bounded neighbourhood of $0$.
Conversely, a convex bounded neighbourhood of $0$ contains a convex balanced open neighbourhood $U$ of $0$, which is still bounded.
Its [[Minkowski functional|PAX9SHDHDW]] $p_U$ is a seminorm, and boundedness of $U$ shows that the sets $t U$, $t > 0$, form a neighbourhood base at $0$, so $p_U$ induces the topology.
The Hausdorff property then forces $p_U(x) > 0$ for $x \ne 0$, so $p_U$ is a norm.
Example: the Fréchet space $C(\R)$ with the seminorms $p_n(f) = \sup_{\abs{t} \le n} \abs{f(t)}$ is not normable, because no neighbourhood of $0$ is bounded.
The space $L^p([0, 1])$ with $0 < p < 1$ is not normable either: it has bounded neighbourhoods of $0$ but no convex ones other than the whole space.
The Hausdorff hypothesis is needed: $\R$ with the indiscrete topology is a topological vector space.
In it, $\R$ itself is a convex bounded neighbourhood of $0$, but no norm induces this topology.

## Explanations

### L1 human

The theorem tells exactly when the topology of a topological vector space comes from a single norm.
Two features are needed near the origin: some neighbourhood must be convex, and it must also be small, in the sense of being bounded.
A norm ball has both features, and the theorem says that a set with both features can always be turned into a norm ball.

### L2 human

Local convexity means that the topology is generated by seminorms.
So the criterion says that a locally convex space is normable precisely when one seminorm suffices, up to equivalence.
It shows that spaces such as $C^\infty(\Omega)$, $H(\Omega)$ or the Schwartz space are not normable.
In each of them a bounded neighbourhood of $0$ would be relatively compact, which happens only in finite dimension.
A topological vector space with a bounded neighbourhood of $0$ is called locally bounded, so the criterion reads: normable means Hausdorff, locally convex and locally bounded.
The construction of the norm is the same Minkowski functional argument that links convex sets and seminorms in the theory of [[locally convex spaces|B6GY2KJMWW]].
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# Arithmetic geometry

## Summary

Arithmetic geometry applies the methods of algebraic geometry to problems of number theory.
Its central objects are varieties and [[schemes|5A35TYZD02]] over [[number fields|MWBDFS2KJ5]], finite fields, local fields and rings of integers, and their [[rational points|SWJYG87QQH]].
Questions about [[Diophantine equations|08B1RTW7VZ]] become questions about the geometry of the variety that the equations define.
Landmark results include [[Faltings's theorem|H0DB07WM20]], the [[Mordell–Weil theorem|MXRCPAH09H]] and the [[Weil conjectures|81TXP32VDM]].

## Explanations

### L1 human

The equation $x^2 + y^2 = 1$ describes a circle, and its solutions in rational numbers are rational points on that circle.
Arithmetic geometry studies such solutions through the shape of the curve or higher-dimensional object that the equations define.
A striking lesson is that the shape controls the arithmetic.
Over $\Q$, a smooth curve of genus $0$ with one rational point, like the circle, has infinitely many rational points.
A smooth curve of genus at least $2$ has only finitely many.
The field combines algebra, geometry and number theory, and it is the setting of Wiles's proof of Fermat's Last Theorem.
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# Wave front of a Legendrian map

## Summary

Let a [[contact manifold|HBQ8BZ7XB5]] be fibred over a base by a Legendrian fibration.
The wave front of a Legendrian map is the image of a [[Legendrian submanifold|Q2DMSTWZYZ]] in this base.
In the model case of the space of $1$-jets $J^1(Q) = T^*Q \times \R$, it is the projection of the Legendrian submanifold to $Q \times \R$.
The front is usually a hypersurface with singularities such as cusps.
At its smooth points the front determines the Legendrian submanifold through its tangent hyperplanes.
Wave fronts are the contact counterpart of [[caustics|GMRYG4S3G0]], and they describe the moving fronts of waves in geometric optics.

## Explanations

### L1 human

Drop a stone in a pond with a curved bank: the ripple reflected from the bank is a curve that moves and changes shape.
At some moments this curve develops sharp corners called cusps.
The ripple is a wave front, and it is the shadow of a smooth curve one dimension up.
That smooth object records, at each point of the front, also the direction in which the front is facing.
For a function $f$ of one variable, the front of its $1$-jet curve $(q, f'(q), f(q))$ is just the graph of $f$.
The semicubical cusp $(t^2, \tfrac23 t^3)$ is a front that is not a graph near $0$.
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# Triangle center

## Definition

Let $T$ be the set of triples $(a, b, c)$ of positive real numbers with $a < b + c$, $b < c + a$ and $c < a + b$.
These are the side lengths of the [[triangles|0CTNFGYMM2]] of the Euclidean plane.
A *triangle center function* is a real function $f$ on $T$, not identically zero, such that:

1. $f$ is homogeneous: there is a real $n$ with $f(ta, tb, tc) = t^n f(a, b, c)$ for all $t > 0$ and $(a, b, c) \in T$.
2. $f$ is bisymmetric: $f(a, c, b) = f(a, b, c)$ for all $(a, b, c) \in T$.

Let $ABC$ be a triangle with $a = BC$, $b = CA$ and $c = AB$.
The *triangle center* given by $f$ is the point of the plane with trilinear coordinates
$$f(a, b, c) : f(b, c, a) : f(c, a, b),$$
whenever these are not all zero.

## Remarks

Trilinear coordinates $\alpha : \beta : \gamma$ describe the point whose signed distances to the lines $BC$, $CA$, $AB$ are in the ratio $\alpha : \beta : \gamma$.
Homogeneity makes the point move with the triangle under scaling, and bisymmetry makes it independent of the labelling of the vertices.

Example: $f(a, b, c) = 1$ is homogeneous of degree $0$ and bisymmetric.
Its center $1 : 1 : 1$ is equidistant from the three sides, so it is the [[incentre|0M9MX8YW0A]].
Similarly $f(a, b, c) = 1/a$ gives the [[centroid|0X7EEWES5G]].

Non-example: $f(a, b, c) = c/b$ is homogeneous but not bisymmetric, since $f(a, c, b) = b/c$.
The point $c/b : a/c : b/a$ is the first Brocard point, which changes into the second Brocard point when two vertices are swapped, so it is not a triangle center.
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# Envelope

## Summary

The envelope of a one-parameter family of plane curves is a curve that touches every member of the family.
At each of its points it is tangent to one of the curves.
Envelopes give the [[singular solutions|0Q2MDD13G4]] of first-order differential equations, as for [[Clairaut's equation|0T6D4PE51C]], and they appear in optics as caustics.
See the [[definition|HB14PWFCS6]].

## Explanations

### L1 human

Draw many straight lines, each one slightly turned from the previous one.
Although no curve is drawn, the eye sees a curve outlined by the lines: this is the envelope.
For example, the lines at distance $1$ from the origin, in all directions, outline the unit circle.
Each line touches the circle at exactly one point.
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# Levin-type transformations

## Definition

Let $\sum_{j \ge 0} a_j$ be a [[series|0JH4MN4K1C]] with [[partial sums|0NB643K60Q]] $s_n = \sum_{j=0}^{n} a_j$.
Let $(\omega_n)$ be a sequence of nonzero numbers, the remainder estimates, let $\beta > 0$, and let $k \ge 1$ be an integer.
The *Levin transformation* is
$$\mathcal{L}_k^{(n)} = \frac{\displaystyle\sum_{j=0}^{k} (-1)^j \binom{k}{j} (\beta + n + j)^{k-1} \frac{s_{n+j}}{\omega_{n+j}}}{\displaystyle\sum_{j=0}^{k} (-1)^j \binom{k}{j} (\beta + n + j)^{k-1} \frac{1}{\omega_{n+j}}},$$
defined when the denominator is not $0$.
The choice $\omega_n = a_n$ gives Levin's $t$-transformation, and $\omega_n = (\beta + n) a_n$ gives Levin's $u$-transformation.
Transformations of the same construction with other choices of remainder estimates or of the factors $(\beta + n + j)^{k-1}$ are called *Levin-type transformations*.

## Remarks

The formula is exact for the model sequences
$$s_n = s + \omega_n \sum_{i=0}^{k-1} \frac{c_i}{(\beta + n)^i}.$$
Indeed, $(\beta + n)^{k-1}(s_n - s)/\omega_n$ is then a polynomial in $n$ of degree at most $k - 1$, so its $k$-th difference vanishes.
Solving this for $s$ gives $\mathcal{L}_k^{(n)} = s$.
Example: for $a_j = x^j$ with $x \ne 0, 1$, one has $s_n - \frac{1}{1-x} = a_n \cdot \frac{-x}{1 - x}$, so the $t$-transformation with $k = 1$ returns $\frac{1}{1-x}$ for every $n$.
Non-example: if some term $a_{n+j}$ is $0$, the remainder estimate $\omega_{n+j} = a_{n+j}$ is $0$ and the $t$-transformation is not defined there.

## Explanations

### L1 human

The transformation divides out a guess of the error, removes a polynomial correction by taking differences, and reads off the sum.
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# Sublattice

## Definition

Let $(L, \vee, \wedge)$ be a [[lattice|WH4SFAACAS]].
A *sublattice* of $L$ is a nonempty subset $M \subseteq L$ such that $x \vee y \in M$ and $x \wedge y \in M$ for all $x, y \in M$.

## Remarks

A sublattice $M$ is a lattice with the restricted operations, and its order is the order of $L$ restricted to $M$.
Some authors allow the empty subset as a sublattice.
Example: in the power set of $\set{1, 2, 3}$ ordered by $\subseteq$, the set $M = \set{\emptyset, \set{1}, \set{2}, \set{1, 2}}$ is a sublattice.
The union and the intersection of two members of $M$ are subsets of $\set{1, 2}$, and all subsets of $\set{1, 2}$ are in $M$.
Non-example: $N = \set{\emptyset, \set{1}, \set{2}, \set{1, 2, 3}}$ is a lattice in the induced order, where the join of $\set{1}$ and $\set{2}$ is $\set{1, 2, 3}$.
It is not a sublattice, because the join of $\set{1}$ and $\set{2}$ in the power set is $\set{1, 2} \notin N$.
Every chain in a lattice is a sublattice, since $x \vee y$ and $x \wedge y$ are $x$ or $y$ when $x$ and $y$ are comparable.
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# Bishop–Gromov inequality

## Statement

Let $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \ge 2$, and let $k$ be a real number.
Assume that the [[Ricci curvature|4GTNM0WHDH]] satisfies $\operatorname{Ric}(v, v) \ge (n-1) k \, g(v, v)$ for every tangent vector $v$.
For $r > 0$ let $V_k(r)$ be the volume of a ball of radius $r$ in the simply connected $n$-dimensional [[space form|1VM1AQT956]] of curvature $k$.
Let $p \in M$ and let $B(p, r)$ be the ball of radius $r$ about $p$ for the [[Riemannian distance|B4QR306D8G]].
Let $R = \pi/\sqrt{k}$ if $k > 0$ and $R = \infty$ if $k \le 0$.
Then the function
$$r \mapsto \frac{\operatorname{vol} B(p, r)}{V_k(r)}$$
is nonincreasing on $(0, R)$.
Since it tends to $1$ as $r \to 0$, it follows that $\operatorname{vol} B(p, r) \le V_k(r)$ for $0 < r < R$.

## Remarks

The idea of the proof: write the volume in geodesic polar coordinates about $p$, with density $\mathcal{A}(t, \theta)$ along the geodesic in direction $\theta$, set to $0$ beyond the cut locus.
The Ricci bound and the Riccati equation for the mean curvature of distance spheres give that $\mathcal{A}(t, \theta) / \bar{\mathcal{A}}_k(t)$ is nonincreasing in $t$, where $\bar{\mathcal{A}}_k$ is the density of the model.
Integrating over $\theta$ and $t$ gives the monotonicity of the ratio of volumes.
The inequality $\operatorname{vol} B(p, r) \le V_k(r)$ is due to Bishop; the monotonicity is due to Gromov.

Example: for $\operatorname{Ric} \ge 0$ the comparison is with $\R^n$, so $\operatorname{vol} B(p, r) / (\omega_n r^n)$ is nonincreasing, where $\omega_n$ is the volume of the unit ball of $\R^n$.
For $M = \R^n$ the ratio is constant equal to $1$.

## Explanations

### L1 human

A lower bound on Ricci curvature means volumes grow no faster than in the model space.
Balls in $M$ are never larger than balls of the same radius in the model, and the ratio of the two volumes can only decrease as the radius grows.

### L2 human

The inequality is the basic volume estimate under a Ricci lower bound.
Its monotonicity gives the doubling property $\operatorname{vol} B(p, 2r) \le C \operatorname{vol} B(p, r)$ with $C$ depending only on $n$, $k$ and an upper bound for $r$.
This is the input for Gromov's precompactness theorem for the Gromov–Hausdorff distance and for many estimates in geometric analysis.
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# Følner condition

## Summary

A group satisfies the Følner condition when it has finite subsets that are almost invariant under translation.
Moving such a set by a given element changes only a small fraction of it.
A sequence of such sets is a Følner sequence.
By Følner's theorem, a discrete group satisfies the condition exactly when it is [[amenable|CEAT9BESKK]].
Følner sequences are the natural sets over which to take averages in ergodic theorems for group actions.

## Explanations

### L1 human

In the integers, the block $\set{0, 1, \ldots, n-1}$ is almost unchanged when it is shifted by $1$: only one point leaves and one point enters.
As $n$ grows, the fraction of points that change goes to $0$.
Groups with such almost invariant finite sets are said to satisfy the Følner condition.
In the grid $\Z^2$, large squares work, since their boundary is small compared with their area.
In a free group, every finite set has a boundary comparable to its size, so the condition fails.
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<document source="https://tamnd.github.io/math2/node/82YAXZMMQ1.md" tag="82YAXZMMQ1" kind="concept">
# Mikusiński field of operators

## Summary

The Mikusiński field of operators is the [[field of fractions|0F2PTGX1A2]] of the ring of continuous functions on $[0, \infty)$ under addition and [[convolution on the half-line|FCKX8KFVGN]].
It exists because this ring has [[no zero divisors|E9QNNPQD2P]].
Its elements are called operators.
They include all such functions, all complex numbers, the Dirac delta as the unit $1$, and the differentiation operator $s$, the inverse of integration.

## Explanations

### L1 human

The rational numbers are built from the integers by allowing fractions $a/b$.
In the same way, Mikusiński built a field from continuous functions, with convolution as multiplication, by allowing fractions $f/g$.
Integration from $0$ is convolution with the constant function $1$, called $l$.
Its reciprocal $s = 1/l$ is not a function, but in this field it behaves like differentiation.
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<document source="https://tamnd.github.io/math2/node/82YGJ392SJ.md" tag="82YGJ392SJ" kind="statement">
# Moser's trick

## Statement

Let $M$ be a closed (compact, without boundary) smooth manifold.
Let $(\omega_t)_{t \in [0, 1]}$ be a family of [[symplectic forms|CAN490EPRT]] on $M$ that depends smoothly on $t$.
Suppose that the [[de Rham class|0ZV14XRNF0]] $[\omega_t] \in H^2_{dR}(M)$ does not depend on $t$.
Then there is a smooth family of [[diffeomorphisms|0KZZ70ZHXP]] $\phi_t \colon M \to M$, $t \in [0, 1]$, with $\phi_0 = \id$ and $\phi_t^* \omega_t = \omega_0$ for all $t$.

## Remarks

Since the class is constant, $\tfrac{d}{dt}\omega_t$ is exact, and one can choose $1$-forms $\sigma_t$ depending smoothly on $t$ with $\tfrac{d}{dt}\omega_t = d\sigma_t$.
Nondegeneracy gives a unique time-dependent vector field $X_t$ with $\iota_{X_t} \omega_t = -\sigma_t$.
Let $\phi_t$ be its [[flow|06738GGWWK]], which exists for all $t \in [0, 1]$ because $M$ is compact.
By Cartan's formula for the [[Lie derivative|PJWK0PKN6Q]],
$$\frac{d}{dt} \phi_t^* \omega_t = \phi_t^* \Big( \mathcal{L}_{X_t} \omega_t + \frac{d}{dt}\omega_t \Big) = \phi_t^* \big( d\iota_{X_t}\omega_t + d\sigma_t \big) = 0.$$
The trick is to look for the diffeomorphisms as the flow of a vector field, which turns a nonlinear problem into a linear equation for $X_t$.

Example of use: two area forms on a closed connected oriented surface, both positive for the orientation and with the same total area, are related by a diffeomorphism.
Apply the theorem to $\omega_t = (1 - t)\omega_0 + t\omega_1$, which stays nondegenerate because both forms are positive for the orientation, so at each point one is a positive multiple of the other.

Compactness is needed for the flow to exist up to time $1$; on open manifolds one adds conditions, as in local versions near a point or a submanifold.

## Explanations

### L1 human

If one moves a symplectic form continuously without changing its cohomology class, nothing really changes.
The theorem gives a family of diffeomorphisms that carries the moving form back to the starting one.
So on a closed manifold, symplectic forms in a fixed class that can be joined by a path are all the same up to diffeomorphism.

### L2 human

The same argument proves Darboux's theorem, the Weinstein neighbourhood theorem for [[Lagrangian submanifolds|H8ACPTMX08]], and Moser's theorem on volume forms.
In each case one joins two structures by a path that agrees on a set, solves $\iota_{X_t}\omega_t = -\sigma_t$, and integrates.
The method shows that symplectic structures have no local invariants, in contrast with Riemannian metrics, which have curvature.
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<document source="https://tamnd.github.io/math2/node/82ZPRD39J7.md" tag="82ZPRD39J7" kind="concept">
# Chern–Weil homomorphism

## Summary

The Chern–Weil homomorphism turns invariant polynomials on the Lie algebra of a group $G$ into de Rham cohomology classes.
These classes live on the base of a principal $G$-bundle.
One chooses a connection, plugs its [[curvature form|ZAJFW1S078]] into the polynomial, and obtains a closed form whose cohomology class does not depend on the connection.
The classes obtained are the real [[characteristic classes|7HACJC0JE2]] of the bundle, such as Chern and Pontryagin classes.
It links local differential geometry, through curvature, with global topology.

## Explanations

### L1 human

A bundle can be twisted in ways that no local measurement detects directly.
Curvature, however, is local and can be computed point by point.
Chern–Weil theory takes a symmetric expression in the curvature, such as a trace or a determinant, and integrates it or takes its cohomology class.
The answer is the same for every connection, so it measures the twisting of the bundle itself.
The Gauss–Bonnet theorem, where the integral of Gaussian curvature over a closed surface is $2\pi$ times the Euler characteristic, is the simplest instance.
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<document source="https://tamnd.github.io/math2/node/82ZR6BCJXV.md" tag="82ZR6BCJXV" kind="concept">
# Brill–Noether theory

## Summary

Brill–Noether theory studies the linear systems on an [[algebraic curve|XJM2MT4S3Y]].
For a given degree $d$ and dimension $r$, it asks which line bundles of degree $d$ have at least $r + 1$ independent sections.
The answer for a general curve of genus $g$ is governed by the Brill–Noether number $\rho = g - (r + 1)(g - d + r)$.
Such line bundles exist on every curve when $\rho \ge 0$, and they do not exist on a general curve when $\rho < 0$.
The theory organizes the maps of curves to projective spaces and is a central tool in the study of moduli of curves.

## Explanations

### L1 human

A curve can be mapped to a line, a plane or a higher projective space in many ways, and each map comes from a line bundle with enough sections.
Brill–Noether theory asks which maps of a given degree into a given space exist.
A simple count of conditions predicts the answer, and for a typical curve the prediction is right.
For instance, the count says that a typical curve of genus $3$ has no map of degree $2$ to the line, so it is not hyperelliptic.
</document>

<document source="https://tamnd.github.io/math2/node/831H5X5JS4.md" tag="831H5X5JS4" kind="definition">
# Möbius plane

## Definition

Let $(\mathcal{P}, \mathcal{Z}, \in)$ be an [[incidence structure|FGKNAFNHA3]] whose elements of $\mathcal{P}$ are called points.
The elements of $\mathcal{Z}$ are subsets of $\mathcal{P}$, called circles.
The structure is a *Möbius plane* if:

1. any three distinct points lie on exactly one circle;
2. for every circle $z$, every point $P \in z$ and every point $Q \notin z$, there is exactly one circle $z'$ with $P, Q \in z'$ and $z \cap z' = \set{P}$;
3. every circle has at least three points, there is at least one circle, and there are four points not on a common circle.

## Remarks

Clause 2 is the touching axiom: $z'$ touches $z$ at $P$.

Example: the classical real Möbius plane has as points $\R^2 \cup \set{\infty}$, the [[Riemann sphere|0J78KBPFZB]].
Its circles are the Euclidean [[circles|060W5CBZP6]] and the sets $g \cup \set{\infty}$ for lines $g$.
Three distinct points lie on exactly one such circle: on a line with $\infty$ if they are collinear or one is $\infty$, and on the circumcircle otherwise.
Given $z$, $P \in z$ and $Q \notin z$, the circles through $P$ tangent to $z$ at $P$ form a pencil, and exactly one of them passes through $Q$.

Non-example: the Euclidean plane with only its circles as blocks fails clause 1, since three collinear points lie on no circle.

The plane sections of an [[ovoid|HNTE5FPS14]] in a three-dimensional projective space form a Möbius plane, called ovoidal; the classical plane comes from a sphere.
A finite Möbius plane has an order $n$: every circle has $n + 1$ points and there are $n^2 + 1$ points.
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<document source="https://tamnd.github.io/math2/node/831N2Q4GCB.md" tag="831N2Q4GCB" kind="concept">
# Reduced complex space

## Summary

A reduced complex space is a [[complex analytic space|HXSZA44J5J]] whose structure sheaf has no nonzero nilpotent germs.
On such a space, a function is determined by its values at points, so the structure sheaf is a sheaf of genuine complex-valued functions.
Reduced complex spaces are the same as complex analytic varieties with their sheaves of holomorphic functions.
Every complex space has a reduction, a reduced complex space with the same underlying topological space.

## Explanations

### L1 human

A complex space may carry extra infinitesimal information, such as the double point given by $z^2 = 0$, where $z$ is a nonzero function that vanishes at the only point.
A reduced complex space has none of this extra information.
Its functions are ordinary functions, and a function that is zero at every point is zero.
The cross $\set{z_1 z_2 = 0}$ in $\C^2$ is reduced, while the doubled line $\set{z_1^2 = 0}$ is not.
</document>

<document source="https://tamnd.github.io/math2/node/833XAA2Q65.md" tag="833XAA2Q65" kind="concept">
# Regular open set

## Summary

A regular open set in a [[topological space|0CQHFPGJ45]] is an [[open set|02BH3MCBQB]] that equals the [[interior|0RZH6W98HK]] of its [[closure|0QEAJB91GZ]].
Such a set has no "cracks": it does not miss single points or thin pieces inside its own closure.
The regular open sets of any space form a complete [[Boolean algebra|0XADJKM699]], which makes them useful in set theory and in pointless topology.

## Explanations

### L1 human

Take an open set, fill in its boundary by taking the closure, and then remove the boundary again by taking the interior.
If you get back the same set, the set is regular open.
An open interval $(0, 1)$ in $\R$ is regular open: closing gives $[0, 1]$, and the interior of that is $(0, 1)$ again.
The set $(0, 1) \cup (1, 2)$ is not, because the process fills the gap at $1$ and returns $(0, 2)$.
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<document source="https://tamnd.github.io/math2/node/833XVEEZCJ.md" tag="833XVEEZCJ" kind="statement">
# Kingman's subadditive ergodic theorem

## Statement

Let $(X, \mathcal{A}, \mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \colon X \to X$ be a [[measure-preserving transformation|556WBDAHNP]].
Let $(f_n)_{n \ge 1}$ be [[integrable|X7GTF3D98F]] functions $f_n \colon X \to \R$ that are subadditive along $T$:
$$f_{m+n} \le f_m + f_n \circ T^m \quad \text{almost everywhere, for all } m, n \ge 1.$$
Let $\gamma = \inf_{n \ge 1} \frac{1}{n} \int_X f_n \, d\mu$, an element of $[-\infty, \infty)$.
Then:

1. $f_n / n$ converges [[almost everywhere|RARTMA0ZEC]] to a function $g \colon X \to [-\infty, \infty)$ with $g \circ T = g$ almost everywhere;
2. $\int_X g \, d\mu = \gamma$, and if $\gamma > -\infty$, then $f_n / n \to g$ in $L^1(\mu)$;
3. if $T$ is [[ergodic|AZRPHS478B]], then $g = \gamma$ almost everywhere.

## Remarks

If $f_n = \sum_{k=0}^{n-1} f \circ T^k$ for an integrable $f$, then equality holds in the subadditivity condition, and the theorem reduces to [[Birkhoff's ergodic theorem|J6NMBX3E55]].
The integrals $a_n = \int_X f_n \, d\mu$ form a subadditive sequence, $a_{m+n} \le a_m + a_n$, so $a_n / n \to \gamma$ by [[Fekete's lemma|8A4GGX1MEY]]; the theorem is the pointwise version of this fact.
In the proof, the function $\liminf_n f_n / n$ is shown to be invariant.
A covering argument cuts a long orbit segment into blocks on which $f_k / k$ is close to this $\liminf$.
Subadditivity then bounds $f_N$ by the sum over the blocks.

Example: let $A \colon X \to GL_d(\R)$ be measurable with $\log^+ \norm{A}$ and $\log^+ \norm{A^{-1}}$ integrable, and let $f_n(x) = \log \norm{A(T^{n-1}(x)) \cdots A(T(x)) A(x)}$.
Submultiplicativity of the operator norm gives $f_{m+n} \le f_m + f_n \circ T^m$, and the two integrability conditions make each $f_n$ integrable.
For ergodic $T$, the theorem gives the almost sure limit of $f_n / n$, the top [[Lyapunov exponent|NJ1FJSNK6A]] of the matrix cocycle; this is the Furstenberg–Kesten theorem.

## Explanations

### L1 human

Sometimes the cost of a long journey is at most the cost of its first part plus the cost of the rest, as for the length of a shortest path.
The theorem says that, in a random environment that looks the same everywhere, the cost per unit of length of such a journey settles down to a limit.
For an ergodic environment this limit is a fixed number, the same for almost every starting point.

### L2 human

Kingman proved the theorem in 1968.
It extends Birkhoff's theorem from additive to subadditive sequences.
It is the main tool for the existence of Lyapunov exponents, as in the Furstenberg–Kesten theorem and in the proof of the Oseledets multiplicative ergodic theorem.
Another standard use is the time constant of first-passage percolation, where travel times between points are subadditive but not additive.
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<document source="https://tamnd.github.io/math2/node/835MPDJXMA.md" tag="835MPDJXMA" kind="concept">
# Riesz kernel energy

## Summary

The Riesz $s$-energy of a positive measure $\mu$ on $\R^d$ is the double integral of the kernel $\abs{x - y}^{-s}$ against $\mu \times \mu$.
For $s = d - 2$ with $d \ge 3$ it is the Newtonian [[energy of a measure|SFSRQCM3EG]], and for other $s$ it is the energy attached to the [[Riesz potential|1V2N8MFSHC]].
Minimising this energy over probability measures on a compact set defines the Riesz capacity of the set.
The discrete version, a sum over pairs of points, is the energy in problems on optimal point configurations such as the [[Thomson problem|3C7V5X5FY8]].

## Explanations

### L1 human

Spread a unit of charge over a set, and let two small pieces of charge at distance $r$ repel each other with strength $r^{-s}$.
The Riesz $s$-energy adds up this repulsion over all pairs of pieces.
Charge piled up at a single point gives infinite energy, so a set can carry a measure of finite energy only if it is large enough.
For example, the uniform probability measure on the unit sphere of $\R^3$ has $1$-energy equal to $1$.
The exponent $s$ tunes how strongly close pairs are penalised.
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<document source="https://tamnd.github.io/math2/node/839P1ZNRD7.md" tag="839P1ZNRD7" kind="concept">
# Down-closure

## Summary

The down-closure of a subset $S$ of a [[partially ordered set|011C29Q1S1]] is the set of all elements that lie below some element of $S$.
It is the smallest [[lower set|BWYFKCWRW5]] that contains $S$.
The down-closure of a single element $x$ is the principal down-set $\downarrow x$ of all elements below $x$.

## Explanations

### L1 human

To take the down-closure of a set, add everything that lies below one of its elements.
In the real numbers, the down-closure of $\set{1, 5}$ is the half-line $(-\infty, 5]$.
In the positive integers ordered by divisibility, the down-closure of $\set{12}$ is the set of divisors of $12$.
The result is closed downwards, and it is the least such set that contains the original one.
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<document source="https://tamnd.github.io/math2/node/83DSSH4HZY.md" tag="83DSSH4HZY" kind="definition">
# Fuzzy measure

## Definition

Let $X$ be a set and let $\mathcal{C}$ be a family of subsets of $X$ with $\emptyset \in \mathcal{C}$.
A *fuzzy measure* on $(X, \mathcal{C})$ is a map $g \colon \mathcal{C} \to [0, \infty]$ such that:

1. $g(\emptyset) = 0$.
2. If $E, F \in \mathcal{C}$ and $E \subseteq F$, then $g(E) \le g(F)$.

If $X \in \mathcal{C}$ and $g(X) = 1$, the fuzzy measure is *normalised*.

## Remarks

Often $\mathcal{C}$ is a $\sigma$-algebra or the set of all subsets of a finite set $X$.
Example: let $\mu$ be a probability measure on a $\sigma$-algebra $\mathcal{C}$ and $g(E) = \mu(E)^2$.
Then $g(\emptyset) = 0$, and $g$ is monotone because $\mu$ is monotone and squaring is increasing on $[0, 1]$.
It is not additive: if $\mu(E) = \mu(F) = 1/2$ with $E \cap F = \emptyset$, then $g(E) + g(F) = 1/2$ but $g(E \cup F) = 1$.
Non-example: $g(E) = \mu(E) - \mu(E)^2$ vanishes on $\emptyset$, but it is not monotone: $g(E) = 1/4$ for $\mu(E) = 1/2$, while $g(X) = 0$.
Every [[measure|PR11BZQQPG]] is a fuzzy measure.
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<document source="https://tamnd.github.io/math2/node/83H3AMTXST.md" tag="83H3AMTXST" kind="definition">
# Poincaré–Lindstedt method

## Definition

Let $\omega_0 > 0$, let $f$ be smooth, and consider the [[ordinary differential equation|0NT64JV96A]]
$$x'' + \omega_0^2 x = \varepsilon f(x, x'),$$
with a small parameter $\varepsilon$, for which one seeks [[periodic solutions|FHNDX22WMK]].
The *Poincaré–Lindstedt method* proceeds as follows.

1. Introduce the scaled time $\tau = \omega t$, where the unknown frequency is expanded as $\omega = \omega_0 + \varepsilon \omega_1 + \varepsilon^2 \omega_2 + \cdots$.
2. Seek $x$ as a $2\pi$-periodic function of $\tau$, expanded as $x = x_0(\tau) + \varepsilon x_1(\tau) + \varepsilon^2 x_2(\tau) + \cdots$.
3. Insert both expansions in the equation, written as $\omega^2 \frac{d^2 x}{d\tau^2} + \omega_0^2 x = \varepsilon f\bigl(x, \omega \frac{dx}{d\tau}\bigr)$, and collect powers of $\varepsilon$.
4. At each order $k \ge 1$, the equation has the form $\omega_0^2 (x_k'' + x_k) = g_k(\tau)$.
   Choose $\omega_k$, and the free constants of earlier orders, so that $g_k$ has no Fourier component in $\cos \tau$ and $\sin \tau$.

Step 4 removes *secular terms* such as $\tau \sin \tau$, so every $x_k$ is periodic.
The result is an [[asymptotic expansion|Q26QN6GYZX]] of the periodic solution and its frequency.

## Remarks

Example: the Duffing equation $x'' + x + \varepsilon x^3 = 0$, with $\omega_0 = 1$, $x(0) = a$, $x'(0) = 0$.
Order $0$ gives $x_0 = a \cos \tau$.
Order $1$ gives $x_1'' + x_1 = 2 \omega_1 a \cos \tau - a^3 \cos^3 \tau = \bigl(2 \omega_1 a - \tfrac{3}{4} a^3\bigr) \cos \tau - \tfrac{1}{4} a^3 \cos 3\tau$, using $\cos^3 \tau = \frac{3}{4} \cos \tau + \frac{1}{4} \cos 3 \tau$.
Removing the $\cos \tau$ term gives $\omega_1 = \frac{3}{8} a^2$, so $\omega = 1 + \frac{3}{8} \varepsilon a^2 + O(\varepsilon^2)$.

Non-example: the naive expansion $x = x_0(t) + \varepsilon x_1(t)$ in the original time, step 1 omitted, gives $x_1$ containing $-\frac{3}{8} a^3 t \sin t$.
This term grows without bound, so the expansion is not uniformly valid for $t$ of order $1/\varepsilon$.

## Explanations

### L1 human

The method expands both the motion and its frequency, and it fixes the frequency so that no term grows forever.
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<document source="https://tamnd.github.io/math2/node/83PJCPB9VA.md" tag="83PJCPB9VA" kind="concept">
# Domino tiling

## Summary

A domino tiling of a region made of unit squares is a covering of it by non-overlapping $1 \times 2$ and $2 \times 1$ rectangles.
Domino tilings are the same as [[perfect matchings|0NHK9MFFW9]] of the grid graph of the cells.
Counting them is a classical problem of combinatorics and statistical physics, solved for rectangles by Kasteleyn and by Temperley and Fisher in 1961.

## Explanations

### L1 human

Take a chessboard and a box of dominoes, each covering exactly two neighbouring squares.
A domino tiling places dominoes so that every square is covered exactly once.
A $2 \times 2$ board has two tilings: two horizontal dominoes or two vertical ones.
The ordinary $8 \times 8$ chessboard has $12{,}988{,}816$ domino tilings.
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<document source="https://tamnd.github.io/math2/node/83TW9SVG0A.md" tag="83TW9SVG0A" kind="definition">
# Basic hypergeometric series

## Definition

Let $q \in \C$ with $0 < \abs{q} < 1$, let $r, s \ge 0$ be integers, and let $a_1, \dots, a_r, b_1, \dots, b_s \in \C$ with $b_j \ne q^{-m}$ for all $j$ and all integers $m \ge 0$.
Write $(c_1, \dots, c_k; q)_n = (c_1; q)_n \cdots (c_k; q)_n$ with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].
The *basic hypergeometric series* is the [[power series|0MZB06ST2B]] in $z$
$${}_r\phi_s \left( \begin{matrix} a_1, \dots, a_r \\ b_1, \dots, b_s \end{matrix} ; q, z \right) = \sum_{n=0}^{\infty} \frac{(a_1, \dots, a_r; q)_n}{(b_1, \dots, b_s, q; q)_n} \left( (-1)^n q^{\binom{n}{2}} \right)^{1 + s - r} z^n.$$
The *bilateral basic hypergeometric series* is
$${}_r\psi_s \left( \begin{matrix} a_1, \dots, a_r \\ b_1, \dots, b_s \end{matrix} ; q, z \right) = \sum_{n=-\infty}^{\infty} \frac{(a_1, \dots, a_r; q)_n}{(b_1, \dots, b_s; q)_n} \left( (-1)^n q^{\binom{n}{2}} \right)^{s - r} z^n,$$
where $(a; q)_{-n} = 1 / (a q^{-n}; q)_n$ for $n \ge 1$.

## Remarks

For $r = s + 1$ the series ${}_r\phi_s$ converges for $\abs{z} < 1$; for $r \le s$ it converges for all $z \in \C$.
If some $a_i = q^{-N}$ with an integer $N \ge 0$, then $(a_i; q)_n = 0$ for $n > N$, and the series terminates.
The factor $\left( (-1)^n q^{\binom{n}{2}} \right)^{1+s-r}$ makes the limit $a_r \to \infty$ of ${}_r\phi_s(\dots; q, z/a_r)$ a series ${}_{r-1}\phi_s$.

Example: ${}_1\phi_0(a; -; q, z) = \sum_{n \ge 0} \frac{(a; q)_n}{(q; q)_n} z^n$, and here the ratio of the terms of index $n+1$ and $n$ is $\frac{1 - a q^n}{1 - q^{n+1}}\, z$, a rational function of $q^n$.
By the [[q-binomial theorem|VVT27RB46B]] it equals $\frac{(az; q)_\infty}{(z; q)_\infty}$ for $\abs{z} < 1$.

Non-example: $\sum_{n \ge 0} \frac{z^n}{n!}$ is a hypergeometric series but not a basic one, since the ratio $\frac{z}{n+1}$ is not a rational function of $q^n$.

As $q \to 1$, $\frac{(q^a; q)_n}{(1 - q)^n}$ tends to the Pochhammer symbol $(a)_n$, so ${}_{s+1}\phi_s(q^{a_1}, \dots; q^{b_1}, \dots; q, z)$ tends to ${}_{s+1}F_s(a_1, \dots; b_1, \dots; z)$.
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# Euclid's Elements

## Description

The *Elements* is a mathematical treatise in thirteen books, written in Greek by Euclid of Alexandria around 300 BC.
Books I to VI treat plane geometry, and Books VII to IX treat the arithmetic of whole numbers.
Book X treats incommensurable magnitudes, and Books XI to XIII treat solid geometry.
Book I starts from definitions, five postulates and a list of common notions, and derives every proposition from these and from earlier propositions.
The fifth postulate of Book I is the [[parallel postulate|2PC3XY0PYZ]].
Book I ends with the [[Pythagorean theorem|0VP3A1290K]] and its converse.
Book VII contains the [[Euclidean algorithm|0TRNT4X7J0]], Book IX contains [[Euclid's theorem|0292EPPPT0]] that there are infinitely many primes, and Book XIII constructs the five [[Platonic solids|017E5PRS1Q]].

## Remarks

The book is the model of the axiomatic method: a short list of first principles and a chain of proofs.
Its axioms are not complete by modern standards; for example, the proof of its first proposition assumes that two circles meet without an axiom that says so.
Modern axiom systems such as [[Hilbert's axioms|KM1FB8XZ2R]] fill these gaps.

## Explanations

### L1 human

The *Elements* is the oldest mathematics textbook that is still read.
It starts from a few simple rules, such as "a straight line can be drawn between any two points", and proves hundreds of facts about figures and numbers from them.
For two thousand years, students learned geometry and the idea of a proof from it.
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<document source="https://tamnd.github.io/math2/node/83ZA89PAGK.md" tag="83ZA89PAGK" kind="definition">
# Cotton tensor

## Definition

Let $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] of dimension $n \ge 3$, with [[Levi-Civita connection|CAZYGE1T4E]] $\nabla$, [[Ricci curvature|4GTNM0WHDH]] $\operatorname{Ric}$ and [[scalar curvature|ZHX2VYQDZ3]] $S$.
Let $P = \frac{1}{n - 2} \left( \operatorname{Ric} - \frac{S}{2(n - 1)} g \right)$ be the Schouten tensor.
The *Cotton tensor* of $(M, g)$ is the $(0,3)$-tensor
$$C(X, Y, Z) = (\nabla_Z P)(X, Y) - (\nabla_Y P)(X, Z),$$
for vector fields $X, Y, Z$; in index notation $C_{ijk} = \nabla_k P_{ij} - \nabla_j P_{ik}$.

## Remarks

Example: if $(M, g)$ is Einstein with $\operatorname{Ric} = \lambda g$ for a constant $\lambda$ and $n \ge 3$, then $S = n\lambda$ is constant and $P$ is a constant multiple of $g$.
Since $\nabla g = 0$, also $\nabla P = 0$, so $C = 0$.

Non-example: for $n = 2$ the factor $1/(n - 2)$ is not defined, so there is no Cotton tensor of a surface in this sense.

$C$ is antisymmetric in its last two arguments, and its trace over the first two arguments vanishes by the contracted Bianchi identity.
For $n = 3$, $C = 0$ if and only if $g$ is locally conformally flat.
For $n \ge 4$, the divergence of the [[Weyl tensor|SBHY31HESP]] is $(n - 3)$ times $C$, up to a sign fixed by the conventions.
Some texts use the opposite sign or a different order of indices.
</document>

<document source="https://tamnd.github.io/math2/node/841C9J1QTC.md" tag="841C9J1QTC" kind="definition">
# Turing jump

## Definition

Let $M_0, M_1, \ldots$ be a fixed effective list of all [[oracle machines|Z5CCVBME2V]].
Let $A \subseteq \N$.
The *Turing jump* of $A$ is the set
$$A' = \set{e \in \N : M_e^A \text{ halts on input } e}.$$
The $n$-th jump $A^{(n)}$ is defined by $A^{(0)} = A$ and $A^{(n+1)} = (A^{(n)})'$.
For a [[Turing degree|P5N2ND3VCG]] $\mathbf{a} = \deg(A)$, the jump is $\mathbf{a}' = \deg(A')$.

## Remarks

Example: for $A = \emptyset$, the machine $M_e^{\emptyset}$ never gets a "yes" answer, so it behaves like an ordinary Turing machine.
Hence $\emptyset'$ is the set $\set{e : M_e \text{ halts on input } e}$, a form of the [[halting problem|Z7TN2B35B0]].

The definition of $\mathbf{a}'$ is legitimate: if $A \le_T B$, then $A' \le_T B'$, so $A \equiv_T B$ implies $A' \equiv_T B'$.
For every $A$, one has $A <_T A'$, that is, $A \le_T A'$ and $A' \not\le_T A$; the proof is a diagonal argument as for the halting problem.
The jump of $A$ does depend on the chosen list of machines, but only up to a computable permutation, so its degree does not.
By [[Post's theorem|NYJX5QXX1T]], $\emptyset^{(n)}$ is complete for the $\Sigma^0_n$ sets of the [[arithmetical hierarchy|7CVJCQKY41]] for $n \ge 1$.
</document>

<document source="https://tamnd.github.io/math2/node/844NHRQQN3.md" tag="844NHRQQN3" kind="definition">
# Quasi-ergodicity

## Definition

Let $(X, \mathcal{A}, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $T \colon X \to X$ be a [[quasi-measure-preserving map|F608XX51KA]]: $T$ is measurable and $\mu(T^{-1}(A)) = 0$ for every $A \in \mathcal{A}$ with $\mu(A) = 0$.
The map $T$ is *quasi-ergodic* if every $A \in \mathcal{A}$ with $\mu(T^{-1}(A) \mathbin{\triangle} A) = 0$ satisfies $\mu(A) = 0$ or $\mu(X \setminus A) = 0$.

## Remarks

Here $\triangle$ is the symmetric difference, so the condition $\mu(T^{-1}(A) \mathbin{\triangle} A) = 0$ says that $A$ is invariant up to a [[null set|VQQM4Y550T]].
If $T$ preserves $\mu$ and $\mu$ is a probability measure, quasi-ergodicity is the same as being an [[ergodic transformation|AZRPHS478B]].
Example: let $\mu$ be a probability measure on $[0,1)$ with a strictly positive density $\rho$ with respect to Lebesgue measure $\lambda$, and let $T(x) = x + \alpha \bmod 1$ with $\alpha$ irrational.
The measures $\mu$ and $\lambda$ have the same null sets, and $T$ preserves $\lambda$, so $T$ is quasi-measure-preserving for $\mu$.
An almost invariant set for $\mu$ is almost invariant for $\lambda$, so it has $\lambda$-measure $0$ or $1$ by ergodicity of the rotation, hence $\mu$-measure $0$ or $1$.
Non-example: the identity map of $[0,1]$ with Lebesgue measure is quasi-measure-preserving, but $A = [0, 1/2]$ is invariant with $0 < \lambda(A) < 1$.
In older physics texts the "quasi-ergodic hypothesis" means something else: that a trajectory comes arbitrarily close to every point of its energy surface.

## Explanations

### L1 human

A quasi-ergodic map has no invariant pieces except the trivial ones, up to sets of measure zero.
The map is not asked to preserve the measure, only to respect which sets have measure zero.
</document>

<document source="https://tamnd.github.io/math2/node/8452SY027Z.md" tag="8452SY027Z" kind="counterexample">
# Counterexample to the Markus–Yamabe conjecture

## Counterexample

The Markus–Yamabe conjecture asked the following.
Let $f \colon \R^n \to \R^n$ be $C^1$ with $f(0) = 0$.
Let the [[Jacobian matrix|047NDTEPCV]] $Df(x)$ be [[Hurwitz-stable|9A3TBBHRFA]] for every $x \in \R^n$, that is, all its [[eigenvalues|0BQAA2BK3Q]] have negative real part.
Then the [[equilibrium point|0N8NGKDZDZ]] $0$ of $\dot x = f(x)$ is globally asymptotically stable.

It fails for $n = 3$.
Let $f \colon \R^3 \to \R^3$ be
$$f(x, y, z) = \bigl(-x + z (x + y z)^2,\ -y - (x + y z)^2,\ -z\bigr).$$
Write $w = x + y z$.
The Jacobian matrix is
$$Df = \begin{pmatrix} -1 + 2 z w & 2 z^2 w & w^2 + 2 y z w \\ -2 w & -1 - 2 z w & -2 y w \\ 0 & 0 & -1 \end{pmatrix}.$$
The last row shows that $-1$ is an eigenvalue, and the other two are the eigenvalues of the upper left $2 \times 2$ block.
That block has trace $-2$ and determinant $(-1 + 2 z w)(-1 - 2 z w) + 4 z^2 w^2 = 1$.
So its characteristic polynomial is $(\lambda + 1)^2$, and all eigenvalues of $Df(x, y, z)$ equal $-1$ at every point.
Also $f(0, 0, 0) = 0$.

But the curve $x(t) = 18 e^t$, $y(t) = -12 e^{2t}$, $z(t) = e^{-t}$ is a solution.
Indeed $w = 18 e^t - 12 e^t = 6 e^t$, so the first component of $f$ is $-18 e^t + 36 e^t = 18 e^t = \dot x$, the second is $12 e^{2t} - 36 e^{2t} = -24 e^{2t} = \dot y$, and the third is $-e^{-t} = \dot z$.
This solution is unbounded, so $0$ is not globally asymptotically stable.

## Remarks

The example is due to Cima, van den Essen, Gasull, Hubbers and Mañosas (1997).
Adding the equations $\dot u_i = -u_i$ for extra coordinates gives counterexamples in every dimension $n \ge 3$.
For $n = 1$ the conjecture is easy, and for $n = 2$ it is a theorem, proved in the 1990s.
The condition at every point implies [[asymptotic stability|0NB689782W]] of $0$, by linearisation; only the global part fails.

## Explanations

### L1 human

If a system pulls inward at every single point when you look very closely, one might expect every orbit to fall to the rest point.
This example shows that in three dimensions it is false.
Each local picture is a contraction, but the nonlinear terms carry one orbit off to infinity.
</document>

<document source="https://tamnd.github.io/math2/node/849ASYBYJ6.md" tag="849ASYBYJ6" kind="definition">
# Multivariate analysis of variance

## Definition

Let $g \ge 2$ and $p \ge 1$ be integers.
For $i = 1, \ldots, g$ and $j = 1, \ldots, n_i$, let $X_{ij}$ be independent random vectors in $\R^p$ with $X_{ij} \sim N_p(\mu_i, \Sigma)$, a [[multivariate normal distribution|0XN3K7RSZY]] with unknown $\mu_i$ and unknown common $\Sigma$.
Let $N = n_1 + \cdots + n_g$, let $\bar X_i$ be the mean of group $i$ and $\bar X$ the mean of all $N$ observations.
The between-group and within-group matrices are
$$B = \sum_{i=1}^g n_i (\bar X_i - \bar X)(\bar X_i - \bar X)^T, \qquad W = \sum_{i=1}^g \sum_{j=1}^{n_i} (X_{ij} - \bar X_i)(X_{ij} - \bar X_i)^T.$$
*One-way multivariate analysis of variance* is the [[test|087P03NEND]] of $H_0 \colon \mu_1 = \cdots = \mu_g$ by a statistic that depends on the eigenvalues of $W^{-1} B$.
The standard statistics are:

1. Wilks' lambda $\Lambda = \det W / \det(W + B)$, which rejects for small values.
2. Pillai's trace $\tr(B (B + W)^{-1})$.
3. The Hotelling–Lawley trace $\tr(W^{-1} B)$.
4. Roy's largest root, the largest eigenvalue of $W^{-1} B$.

## Remarks

The matrix $W$ is [[invertible|0EB9N0S33C]] with probability $1$ when $N - g \ge p$ and $\Sigma$ is positive definite.
Example: for $p = 1$, $B$ and $W$ are the between and within sums of squares of the one-way [[analysis of variance|M9AQ5HKGP4]].
Then $\Lambda = W/(W + B)$, a decreasing function of the usual $F$ statistic, so MANOVA reduces to ANOVA.
Non-example: with $p = 5$ responses and $g = 3$ groups of $2$ observations each, $N - g = 3 < p$, so $W$ has rank at most $3$, is not invertible, and $\Lambda = 0$ whatever the data.
For $g = 2$ the four statistics are equivalent to Hotelling's $T^2$ statistic.
</document>

<document source="https://tamnd.github.io/math2/node/84BR7X6HET.md" tag="84BR7X6HET" kind="concept">
# Sparse grid

## Summary

A sparse grid is a set of points in a cube in $\R^d$ built by Smolyak's construction from one-dimensional rules.
It keeps only those products of one-dimensional levels whose total level is small, instead of the full tensor grid.
For functions with bounded mixed derivatives, sparse grids give nearly the accuracy of full grids with far fewer points.
They are used for integration and interpolation in moderately high dimensions, where full grids suffer from the [[curse of dimensionality|YS424FTY4E]].

## Explanations

### L1 human

A full grid with $m$ points in each of $d$ directions has $m^d$ points, which is far too many when $d$ is large.
A sparse grid allows fine resolution in one direction only when the other directions are coarse.
The picture in two dimensions looks like a cross or a star: dense along the axes and thin near the corners.
For example, from the levels $\set{0}$ and $\set{-1, 0, 1}$ in each direction, the sparse grid of the next level in $d$ dimensions has $2d + 1$ points, while the full grid has $3^d$.
</document>

<document source="https://tamnd.github.io/math2/node/84MVNJCYR4.md" tag="84MVNJCYR4" kind="concept">
# p-adic gamma function

## Summary

The $p$-adic gamma function $\Gamma_p$ of Morita is a continuous function on the [[p-adic integers|V6TJAFZ4MD]] that plays the role of the [[gamma function|0H5SYBSED9]].
It interpolates a modified [[factorial|0XWG0JGDFW]] in which the factors divisible by $p$ are left out.
It satisfies a functional equation like $\Gamma(x+1) = x\,\Gamma(x)$, and it appears in the Gross–Koblitz formula for Gauss sums.

## Explanations

### L1 human

The ordinary factorial $n!$ cannot be extended continuously in the $p$-adic sense, because $n!$ becomes divisible by high powers of $p$ and so $p$-adically small.
The fix is to skip the factors that are multiples of $p$.
The resulting products, with a sign, change only a little when $n$ changes by a high power of $p$.
So they extend to a continuous function on all $p$-adic integers.
For example, for $p = 3$ the value at $5$ is $-(1 \cdot 2 \cdot 4) = -8$, since the factor $3$ is skipped.
</document>

<document source="https://tamnd.github.io/math2/node/84TJ8KQ926.md" tag="84TJ8KQ926" kind="concept">
# Spline

## Summary

A spline is a function made of polynomial pieces on consecutive intervals.
The pieces are joined at points called knots, so that the whole function is as smooth as the pieces allow.
A spline of degree $n$ is usually required to have $n - 1$ continuous derivatives.
Splines combine the flexibility of piecewise functions with the simplicity of polynomials, and they avoid the oscillations of high-degree polynomial interpolation.
They are central in [[spline interpolation|D59HBKKRP9]], in approximation theory and in computer-aided design, where [[B-splines|62AFX32JGT]] give a convenient basis.

## Explanations

### L1 human

The name comes from the thin flexible strips of wood that draftsmen bent through fixed pegs to draw smooth curves.
A mathematical spline does the same with polynomial pieces: each piece is simple, and the pieces meet without corners or jumps in bending.
The simplest example is a broken line, a spline of degree $1$: straight pieces joined continuously.
Cubic splines use pieces of degree $3$ joined so that position, slope and curvature all match.
The [[definition|KTVJR95GSF]] makes the conditions precise.
</document>

<document source="https://tamnd.github.io/math2/node/84XXPH6XJ8.md" tag="84XXPH6XJ8" kind="definition">
# Reduced product

## Definition

Let $I$ be a nonempty set and let $F$ be a [[filter|15PF73BKY6]] on $I$.
Let $(\mathbf{A}_i)_{i \in I}$ be a family of [[algebras|QJGPT2W0FT]] of the same signature, and let $\mathbf{P} = \prod_{i \in I} \mathbf{A}_i$ be their [[direct product|WJ73XYBNE4]].
Define $\theta_F$ on the universe of $\mathbf{P}$ by $(a, b) \in \theta_F$ if and only if $\set{i \in I : a(i) = b(i)} \in F$.
The *reduced product* $\prod_{i \in I} \mathbf{A}_i / F$ is the [[quotient algebra|N5KE7FH6CX]] $\mathbf{P}/\theta_F$.

## Remarks

The relation $\theta_F$ is a [[congruence|DCSXXT4SER]] on $\mathbf{P}$.
Reflexivity uses $I \in F$, and transitivity uses that $F$ is closed under finite intersections and supersets.
Compatibility holds because $\set{i : f(a_1, \dots, a_n)(i) = f(b_1, \dots, b_n)(i)}$ contains the intersection of the sets $\set{i : a_k(i) = b_k(i)}$.

Example: if $F = \set{I}$, then $\theta_F$ is equality and the reduced product is the direct product itself.
If $F = \set{X \subseteq I : J \subseteq X}$ for some nonempty $J \subseteq I$, the reduced product is isomorphic to $\prod_{j \in J} \mathbf{A}_j$.

Non-example: on $I = \set{1, 2, 3}$, the family $G = \set{\set{1, 2}, \set{2, 3}, I}$ is not a filter, since $\set{1, 2} \cap \set{2, 3} = \set{2} \notin G$.
With it, the analogous relation on $\Z^3$ is not transitive: $(0, 0, 0)$ is related to $(0, 0, 1)$, which is related to $(1, 0, 1)$, but $(0, 0, 0)$ and $(1, 0, 1)$ agree only on $\set{2}$.

When $F$ is an ultrafilter, the reduced product is the [[ultraproduct|AX9Q86GB8M]] $\prod_{i \in I} \mathbf{A}_i / F$.
</document>

<document source="https://tamnd.github.io/math2/node/850Y7GFENR.md" tag="850Y7GFENR" kind="example">
# Divisibility poset

## Example

Let $\Z_{>0}$ be the set of positive integers, and write $a \mid b$ when $a$ [[divides|0PG2H7WKGG]] $b$.
The relation $\mid$ is a [[partial order|0QV2JS9RJ3]] on $\Z_{>0}$.
It is reflexive since $a = 1 \cdot a$.
It is antisymmetric since $a \mid b$ and $b \mid a$ give $a \le b \le a$ for positive integers.
It is transitive since $b = ka$ and $c = lb$ give $c = (kl) a$.

It is a [[lattice|WH4SFAACAS]]: the meet of $a$ and $b$ is their [[greatest common divisor|0C8WYW4KYQ]], and the join is their [[least common multiple|007NPQX34K]].
It is a [[distributive lattice|0HM0HQCK6N]].
By the [[fundamental theorem of arithmetic|084HH1NDCB]], write $a$, $b$, $c$ as products of primes.
For each prime $p$, gcd takes the minimum and lcm takes the maximum of the exponents of $p$.
The identity $\min(x, \max(y, z)) = \max(\min(x, y), \min(x, z))$ for integers then gives $\gcd(a, \operatorname{lcm}(b, c)) = \operatorname{lcm}(\gcd(a, b), \gcd(a, c))$.

## Remarks

The element $1$ is the least element, and there is no greatest element in $\Z_{>0}$.
If $0$ is added, it becomes the greatest element, since every integer divides $0$.
The positive divisors of a fixed $n$ form a finite sublattice; for $n = p_1^{e_1} \cdots p_k^{e_k}$ it is isomorphic to a product of chains with $e_1 + 1, \ldots, e_k + 1$ elements.
The order is not total: $2$ and $3$ are incomparable.

## Explanations

### L1 human

Order the positive integers by "divides" instead of "is smaller than".
Then $2$ is below $6$, but $2$ and $3$ are side by side, and the primes sit just above $1$.
The greatest common divisor and the least common multiple play the roles of "meet" and "join".
</document>

<document source="https://tamnd.github.io/math2/node/856B6C7EE4.md" tag="856B6C7EE4" kind="statement">
# Nichols–Zoeller theorem

## Statement

Let $K$ be a field and let $H$ be a finite-dimensional [[Hopf algebra|7XFYP0N84H]] over $K$.
Let $B \subseteq H$ be a Hopf subalgebra, that is a subalgebra with $\Delta(B) \subseteq B \otimes B$ and $S(B) \subseteq B$.
Then $H$ is a [[free|0DZ9DKAKZR]] left $B$-module and a free right $B$-module.
In particular, $\dim_K B$ divides $\dim_K H$.

## Remarks

The proof views $H$ as a relative Hopf module: a right $B$-module under multiplication with the compatible right $H$-coaction $\Delta$.
One shows that a direct sum of finitely many copies of such a module is free over $B$.
A Krull–Schmidt argument for finite-dimensional modules then shows that $H$ itself is free.
Example: for a finite group $G$ and a subgroup $L$, the group algebra $KG$ is a free right $KL$-module with basis a set of representatives of the left cosets $gL$.
Then $\dim KL = \abs{L}$ divides $\dim KG = \abs{G}$, which is [[Lagrange's theorem|3C53TD]].
The divisibility statement follows from freeness: if $H \cong B^n$ as $B$-modules, then $\dim H = n \dim B$.

## Explanations

### L1 human

Lagrange's theorem says that the size of a subgroup divides the size of a finite group.
The Nichols–Zoeller theorem is the version for finite-dimensional Hopf algebras: the dimension of a Hopf subalgebra divides the dimension of the whole algebra.
The stronger statement is that the big algebra is a free module over the small one, just as a group is a disjoint union of cosets of a subgroup.

### L2 human

The theorem is a basic tool in the classification of finite-dimensional Hopf algebras of small dimension.
For example, the group-like elements of $H$ span a Hopf subalgebra $KG(H)$, so the order of the group of group-likes divides $\dim H$.
The [[index|0DTXBQPA2S]] analogue is the integer $\dim H / \dim B$, the rank of $H$ as a free $B$-module.
</document>

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# Morphism

## Summary

A morphism is an arrow $f \colon A \to B$ of a [[category|A2DE66937D]], from its domain $A$ to its codomain $B$.
Morphisms abstract structure-preserving maps such as [[group homomorphisms|0JYTG5K9Y4]], [[linear maps|0NRYHJPQG2]] and [[continuous maps|0VGFHTM75Y]].
Isomorphisms, endomorphisms and automorphisms are special morphisms defined only through domains, codomains and composition.
Much of [[category theory|Q1RZT7DV0F]] consists of describing objects by the morphisms into them and out of them.

## Explanations

### L1 human

A morphism is an arrow from one object to another that can be chained with other arrows.
In most examples it is a map that respects the structure, but a category does not need to say what the arrow "does" to elements.
An isomorphism is an arrow that can be undone by another arrow.
For instance, in the category of vector spaces over $\R$, an invertible $2 \times 2$ matrix gives an isomorphism $\R^2 \to \R^2$.
A matrix with determinant $0$ gives a morphism that is not an isomorphism.
</document>

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# Injective module

## Summary

An injective module is a module $E$ such that every linear map into $E$ from a submodule extends to the whole module.
It is the dual notion of a [[projective module|BFFA03GE3V]], with the arrows reversed.
Every module embeds into an injective module, and the smallest such embedding is the [[injective hull|QCJN05GN64]].
Injective modules are used to build injective resolutions, which compute derived functors such as [[Ext|4P7GBWJ4RR]] and local cohomology.

## Explanations

### L1 human

Think of $E$ as a target that is big and flexible enough to receive maps.
If a linear map into $E$ is defined only on part of a module, it can always be extended to the whole module.
The rational numbers $\Q$ form an injective $\Z$-module, because you can divide in $\Q$ by any nonzero integer.
The integers $\Z$ are not injective: the map $2\Z \to \Z$ sending $2$ to $1$ cannot be extended to $\Z$, since then $1 = 2 \cdot (\text{image of } 1)$.
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<document source="https://tamnd.github.io/math2/node/85KK79FCN5.md" tag="85KK79FCN5" kind="statement">
# Lax–Milgram theorem

## Statement

Let $H$ be a real [[Hilbert space|HM2HHE3J51]] with norm $\norm{\cdot}$.
Let $B \colon H \times H \to \R$ be a [[bilinear form|0SQZMXB6GP]] with these two properties:

1. $B$ is bounded: there is $M \ge 0$ with $\abs{B(u, v)} \le M \norm{u} \norm{v}$ for all $u, v \in H$.
2. $B$ is [[coercive|86KRHS6X22]]: there is $\alpha > 0$ with $B(u, u) \ge \alpha \norm{u}^2$ for all $u \in H$.

Then for every bounded linear functional $f$ on $H$, an element of the [[continuous dual space|4A82S5FVRT]] $H'$, there is a unique $u \in H$ such that
$$B(u, v) = f(v) \quad \text{for all } v \in H.$$
Moreover $\norm{u} \le \norm{f}_{H'} / \alpha$.

## Remarks

The form $B$ need not be symmetric.
By the [[Riesz representation theorem|TTC40N49M6]], there are a bounded linear operator $A$ on $H$ with $B(u, v) = \ip{Au}{v}$ and an element $w$ with $f(v) = \ip{w}{v}$.
Coercivity gives $\alpha \norm{u} \le \norm{Au}$, so $A$ is injective with closed range.
If $v$ is orthogonal to the range, then $B(v, v) = \ip{Av}{v} = 0$, so $v = 0$; hence $A$ is onto and $u = A^{-1} w$.
The bound follows from $\alpha \norm{u}^2 \le B(u, u) = f(u) \le \norm{f}_{H'} \norm{u}$.
When $B$ is symmetric, the theorem is the Riesz representation theorem for the inner product $B$.

Example: let $U \subset \R^n$ be a bounded open set and $g \in L^2(U)$.
On $H = H^1_0(U)$, the [[Sobolev space|M4CEZ5K9HP]], take $B(u, v) = \int_U \nabla u \cdot \nabla v \, dx$ and $f(v) = \int_U g v \, dx$.
The [[Poincaré inequality|JS2CCR3RNS]] shows that $B$ is coercive, so there is a unique [[weak solution|SJZX81A783]] $u \in H^1_0(U)$ of $-\Delta u = g$ with $u = 0$ on $\partial U$.

Coercivity cannot be dropped.
On $H = \ell^2$, let $S$ be the right shift and $B(u, v) = \ip{Su}{v}$, which is bounded.
For $f(v) = \ip{e_1}{v}$, a solution would satisfy $S u = e_1$, which is impossible since the first coordinate of $Su$ is $0$.

## Explanations

### L1 human

Many boundary value problems can be written as "find $u$ such that $B(u, v) = f(v)$ for every test vector $v$".
The theorem says that this problem has exactly one solution as soon as $B$ is bounded and positive in a strong sense.
It also says that the solution depends continuously on the data $f$.

### L2 human

This is the standard existence theorem for weak solutions of linear elliptic equations in divergence form.
One checks boundedness from bounds on the coefficients and coercivity from uniform ellipticity together with the Poincaré inequality, possibly after adding a multiple of $u$ to the equation.
The proof is also the basis of finite element error analysis.
Restricting the problem to a finite-dimensional subspace gives the Galerkin approximation, and the constants $M$ and $\alpha$ control its error.
The Babuška version replaces coercivity by an inf-sup condition and allows different trial and test spaces.
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# Siegel modular form

## Definition

Let $g \ge 1$ and $k$ be integers, and let $\mathbb{H}_g$ be the [[Siegel upper half-space|5PTYE97PF6]] of degree $g$.
Let $\Gamma_g = Sp(2g, \Z)$ be the group of integer matrices in the [[symplectic group|X12FS2JAP0]] $Sp(2g, \R)$.
A *Siegel modular form* of degree $g$ and weight $k$ is a function $f \colon \mathbb{H}_g \to \C$ such that:

1. $f$ is [[holomorphic|507C34W0ZT]] in the $g(g+1)/2$ entries $Z_{ij}$, $i \le j$, of $Z$;
2. $f\big((AZ + B)(CZ + D)^{-1}\big) = \det(CZ + D)^k f(Z)$ for all $Z \in \mathbb{H}_g$ and all $\begin{pmatrix} A & B \\ C & D \end{pmatrix} \in \Gamma_g$;
3. if $g = 1$, $f$ is holomorphic at the cusp: $f$ is bounded on $\set{Z : \operatorname{Im} Z \ge 1}$.

## Remarks

For $g = 1$ this is the definition of a [[modular form|GWKN5VG78A]] of weight $k$ for $SL_2(\Z)$.
For $g \ge 2$ the Koecher principle shows that conditions 1 and 2 imply boundedness on every set $\set{Z : \operatorname{Im} Z \ge c I_g}$ with $c > 0$, so no condition at the cusp is needed.
Every Siegel modular form has a Fourier expansion $f(Z) = \sum_T a(T) e^{2\pi i \operatorname{tr}(TZ)}$, over half-integral symmetric matrices $T \ge 0$.
Forms of weight $k$ for subgroups of finite index of $\Gamma_g$, and vector-valued forms, are defined in the same way.

Example: a constant function $f = c$ is a Siegel modular form of weight $0$, since condition 2 reads $c = \det(CZ + D)^0 c$.
Example: let $S$ be a positive definite even unimodular integer matrix of size $m$.
The theta series $\theta_S(Z) = \sum_{X} e^{\pi i \operatorname{tr}(X^T S X Z)}$, over all integer $m \times g$ matrices $X$, is a Siegel modular form of weight $m/2$.

Non-example: the $j$-invariant $j(z)$ is holomorphic on $\mathbb{H}_1$ and satisfies condition 2 with $k = 0$.
But $j(z) = e^{-2\pi i z} + 744 + \cdots$ is unbounded on $\operatorname{Im} z \ge 1$, so condition 3 fails.
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# Poincaré disk model

## Description

The *Poincaré disk model* of the hyperbolic plane is the [[unit disc|8G5ABXRK85]] $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ with the Riemannian metric
$$ds = \frac{2\, \abs{dz}}{1 - \abs{z}^2}.$$
The distance between two points is the infimum of the lengths of piecewise smooth curves joining them, which makes $\mathbb{D}$ a [[metric space|0VKD1TP7F3]].
The distance from $0$ to $z$ is
$$d(0, z) = \int_0^{\abs{z}} \frac{2\, dt}{1 - t^2} = \log\frac{1 + \abs{z}}{1 - \abs{z}},$$
attained along the radius, so points near the boundary circle are infinitely far from $0$.
In general, $d(z, w) = 2 \operatorname{artanh} \abs{\dfrac{z - w}{1 - \bar w z}}$.

## Remarks

The metric has constant Gaussian curvature $-1$.
The geodesics are the diameters of $\mathbb{D}$ and the arcs of circles that meet the unit circle at right angles.
The orientation-preserving isometries are the [[Möbius transformations|0SG74WBFA2]] $z \mapsto e^{i\theta}(z - a)/(1 - \bar a z)$ with $\abs{a} < 1$, which are the conformal automorphisms of $\mathbb{D}$.
The model is conformal: hyperbolic angles equal Euclidean angles.
The [[Schwarz–Pick theorem|V1QYA9RW73]] says that every holomorphic self-map of $\mathbb{D}$ does not increase this distance.
The Cayley map $z \mapsto (z - i)/(z + i)$ is an isometry from the Poincaré half-plane model onto the disk model.

## Explanations

### L1 human

The Poincaré disk is a picture of a geometry where the parallel postulate fails.
The whole infinite hyperbolic plane is squeezed into a round disc, and distances grow without bound toward the rim.
Straight lines of this geometry appear as diameters or as circular arcs that hit the rim at right angles.
Through a point off such a line there are infinitely many lines that never meet it.
M. C. Escher's Circle Limit prints show tilings of this disc by congruent figures that look smaller and smaller toward the edge.
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# Hermite functions

## Definition

Let $H_n$ be the (physicists') [[Hermite polynomial|MGACY98E9H]] of degree $n \ge 0$.
The *Hermite function* of degree $n$ is the function $\psi_n \colon \R \to \R$ given by
$$\psi_n(x) = \left( 2^n \, n! \, \sqrt{\pi} \right)^{-1/2} H_n(x) \, e^{-x^2/2}.$$

## Remarks

Example: $\psi_0(x) = \pi^{-1/4} e^{-x^2/2}$, and $\int_{\R} \psi_0(x)^2 \, dx = \pi^{-1/2} \int_{\R} e^{-x^2} \, dx = 1$.
The orthogonality of the Hermite polynomials for the weight $e^{-x^2}$ gives $\int_{\R} \psi_m \psi_n \, dx = \delta_{mn}$.
The family $(\psi_n)_{n \ge 0}$ is complete, so it is an orthonormal basis of [[$L^2(\R)$|SASP14MA1Z]].
Each $\psi_n$ solves $-\psi_n'' + x^2 \psi_n = (2n + 1) \psi_n$.
With the Fourier transform $\hat{f}(\xi) = (2\pi)^{-1/2} \int_{\R} f(x) e^{-i x \xi} \, dx$, one has $\hat{\psi}_n = (-i)^n \psi_n$.
Non-example: the polynomial $H_0 = 1$ without the Gaussian factor is not square integrable on $\R$, so it is not an element of $L^2(\R)$ at all.
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# Homogeneous space

## Summary

A homogeneous space for a group $G$ is a space on which $G$ acts [[transitively|02GK6HZYBT]], so that all points look alike.
In algebraic geometry, it is a variety with a transitive [[action of an algebraic group|RW85C34GQW]].
Such a variety is a quotient $G / H$ by the [[stabilizer|0AYMP41THN]] $H$ of a point, at least in characteristic $0$.
Projective spaces, Grassmannians and [[flag varieties|NHRWCQQKTH]] are homogeneous spaces.

## Explanations

### L1 human

On a sphere, any point can be moved to any other by a rotation, so the sphere looks the same everywhere.
A homogeneous space is a space with this property for some group of symmetries.
In projective space every point is like every other, because invertible matrices move any line through the origin to any other.
Once one point and its stabilizer are known, the whole space is described as the group modulo the stabilizer.
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# Unitization

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $\mathbb{K}$, with or without a unit.
The *unitization* of $A$ is the vector space $A^+ = \mathbb{K} \times A$ with the product
$$(\lambda, a)(\mu, b) = (\lambda \mu,\ \lambda b + \mu a + a b).$$
It is an associative algebra over $\mathbb{K}$ with unit $(1, 0)$.
The map $a \mapsto (0, a)$ embeds $A$ in $A^+$ as a [[two-sided ideal|0CD87SFM2N]] of codimension $1$.

## Remarks

One writes $(\lambda, a)$ as $\lambda 1 + a$.
Every algebra homomorphism $\varphi$ from $A$ to a unital algebra $B$ extends uniquely to a unital homomorphism $A^+ \to B$, by $\lambda 1 + a \mapsto \lambda 1_B + \varphi(a)$.
If $A$ is a [[normed algebra|4DJ4V5VBSH]], then $\norm{(\lambda, a)} = \abs{\lambda} + \norm{a}$ makes $A^+$ a normed algebra, complete when $A$ is complete.
Indeed $\abs{\lambda \mu} + \norm{\lambda b + \mu a + a b} \le (\abs{\lambda} + \norm{a})(\abs{\mu} + \norm{b})$.
For a C*-algebra $A$ one uses another norm on $A^+$, which makes it a C*-algebra.
The construction adds a new unit even when $A$ already has one.

Example: for $A = C_0(\R)$, the complex continuous functions that tend to $0$ at infinity, $A^+$ is isomorphic to the functions $f + \lambda$ with $f \in A$ and $\lambda \in \C$.
These are the continuous functions on $\R$ with one common limit at $+\infty$ and $-\infty$.
They are the continuous functions on the one-point compactification of $\R$.

Non-example: the componentwise product $(\lambda, a)(\mu, b) = (\lambda \mu, a b)$ on $\mathbb{K} \times A$ gives no unitization when $A \neq 0$.
For it, $(1, 0)(\mu, b) = (\mu, 0)$, so $(1, 0)$ is not a unit.
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# Ring of convergent power series

## Definition

Let $n \ge 0$ and let $\C[\![z_1, \dots, z_n]\!]$ be the ring of [[formal power series|KCBFS516DE]] in $n$ variables over $\C$.
For a multi-index $\alpha = (\alpha_1, \dots, \alpha_n) \in \N^n$ write $z^\alpha = z_1^{\alpha_1} \cdots z_n^{\alpha_n}$ and $\abs{\alpha} = \alpha_1 + \cdots + \alpha_n$.
A [[power series|R3QTCPTJK7]] $f = \sum_{\alpha} c_\alpha z^\alpha$ is *convergent* if there is a real number $r > 0$ with
$$\sum_{\alpha \in \N^n} \abs{c_\alpha}\, r^{\abs{\alpha}} < \infty.$$
The *ring of convergent power series* $\C\{z_1, \dots, z_n\}$ is the set of convergent power series, with the addition and multiplication of formal power series.

## Remarks

The condition says that the series converges absolutely on the closed polydisc of radius $r$ about $0$; it is equivalent to convergence on some neighbourhood of $0$.
Sums and products of convergent series are convergent, so $\C\{z_1, \dots, z_n\}$ is a subring of $\C[\![z_1, \dots, z_n]\!]$.
Example: $f = \sum_{k \ge 0} z_1^k z_2^k$ is convergent: with $r = 1/2$, $\sum_k r^{2k} = 4/3 < \infty$.
Non-example: $g = \sum_{k \ge 0} k!\, z_1^k$ is not convergent, since $k!\, r^k \to \infty$ for every $r > 0$.
For $n = 0$ the ring is $\C$.
A convergent series is a unit if and only if its constant term is nonzero, so the ring is local with maximal ideal $(z_1, \dots, z_n)$.
Sending a series to its sum near $0$ gives an isomorphism with the ring $\mathcal{O}_{n,0}$ of germs of holomorphic functions at $0$.
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# Classifying topos

## Summary

A classifying topos of a geometric theory $\mathbb{T}$ is a [[Grothendieck topos|S3YVE24AQK]] that contains a generic model of $\mathbb{T}$.
Models of $\mathbb{T}$ in any Grothendieck topos $\mathcal{E}$ correspond to [[geometric morphisms|N5R1WAEPFB]] from $\mathcal{E}$ to the classifying topos.
Every geometric theory has a classifying topos, unique up to equivalence.
This turns questions about theories into questions about toposes, and lets one read a topos as a theory.

## Explanations

### L1 human

A classifying topos is a universe that contains one generic model of a theory.
Every model of the theory in another topos is obtained by pulling back this generic model along a map of toposes.
It plays the role that a classifying space plays for vector bundles in topology.
For example, the topos of functors from finite sets to sets classifies "objects": a map into it from a topos $\mathcal{E}$ is the same as one object of $\mathcal{E}$.
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# Abstract data type as an algebra

## Summary

In the algebraic approach to programming, an abstract data type is described by its operations and their equations.
It does not depend on a concrete memory layout.
Such a description is an algebraic specification: a [[many-sorted signature|WAQJJ5GX68]] together with a set of equations.
The data type itself is the [[initial algebra|R76EC56V9R]] of the specification, which is unique up to isomorphism.
So any two correct implementations behave in the same way.
This view, worked out in the 1970s, makes the correctness of an implementation a question about homomorphisms of algebras.

## Explanations

### L1 human

A programmer who uses a stack only needs to know what push, pop and top do, not how the stack is stored.
The algebraic view writes down these operations and the rules between them, such as "pop after push gives back the old stack".
Many different data structures obey these rules, and some obey extra rules by accident.
The intended data type is the one that obeys the rules and nothing more, and where every value is built by the operations.
This is the [[initial algebra|HVF403PQ4H]] of the rules, and it is the same up to renaming for every honest implementation.
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# Single layer potential

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]] whose [[boundary|05D5EGMFYV]] $\partial\Omega$ is a $C^1$ hypersurface, with surface measure $\sigma$.
Let $\Phi$ be the [[fundamental solution of the Laplacian|BSJ8X2XRN9]],
$$\Phi(x) = -\frac{1}{2\pi} \log\abs{x} \ (n = 2), \qquad \Phi(x) = \frac{1}{(n-2)\,\omega_n}\, \abs{x}^{2-n} \ (n \ge 3),$$
where $\omega_n$ is the area of the unit sphere in $\R^n$, so that $-\Delta \Phi = \delta_0$.
Let $\varphi \colon \partial\Omega \to \R$ be continuous.
The *single layer potential* with density $\varphi$ is the function
$$S\varphi(x) = \int_{\partial\Omega} \Phi(x - y)\, \varphi(y)\, d\sigma(y), \qquad x \in \R^n.$$

## Remarks

The integral is a [[surface integral|065HE5V98E]].
For $x \notin \partial\Omega$ the integrand is smooth in $x$, and $S\varphi$ is [[harmonic|0DEB0JMN0W]] on $\R^n \setminus \partial\Omega$.
For $x \in \partial\Omega$ the singularity of $\Phi(x - y)$ is integrable on the $(n-1)$-dimensional surface, so $S\varphi(x)$ is still defined.
It is the [[potential of the measure|JHKSF3XFY2]] $\varphi\,\sigma$ carried by $\partial\Omega$.

Example: let $\Omega$ be the unit ball in $\R^3$ and $\varphi \equiv 1$.
Then $\Phi(x) = 1/(4\pi\abs{x})$ and the mean of $1/\abs{x - y}$ over the unit sphere in $y$ is $1/\max(\abs{x}, 1)$.
Since the sphere has area $4\pi$, $S\varphi(x) = 1$ for $\abs{x} \le 1$ and $S\varphi(x) = 1/\abs{x}$ for $\abs{x} \ge 1$.
It is continuous across the sphere, while its radial derivative jumps from $0$ to $-1$.

Non-example: the integral $\int_\Omega \Phi(x - y) f(y)\, dy$ over the open set itself is a volume potential, the [[Newtonian potential|DCMFQ01QGS]] of $f$, and not a single layer potential.
It is a potential of a charge spread through a solid, not over a surface.
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# Chebyshev series

## Definition

Let $T_k$ be the [[Chebyshev polynomials|M6EAR1JHPP]] of the first kind.
Let $f \colon [-1, 1] \to \R$ be measurable with $\int_{-1}^{1} \abs{f(x)} (1 - x^2)^{-1/2} \, dx < \infty$.
The *Chebyshev coefficients* of $f$ are
$$a_k = \frac{2}{\pi} \int_{-1}^{1} \frac{f(x) \, T_k(x)}{\sqrt{1 - x^2}} \, dx, \qquad k = 0, 1, 2, \dots$$
The *Chebyshev series* of $f$ is the formal series
$$\frac{a_0}{2} + \sum_{k=1}^{\infty} a_k T_k(x).$$

## Remarks

Example: for $f(x) = x^2$, one finds $a_0 = 1$, $a_2 = 1/2$ and $a_k = 0$ otherwise.
Check: $\frac{2}{\pi} \int_{-1}^{1} x^2 (1 - x^2)^{-1/2} \, dx = \frac{2}{\pi} \cdot \frac{\pi}{2} = 1$, and $\frac{1}{2} + \frac{1}{2}(2x^2 - 1) = x^2$.

Non-example: $f(x) = (1 - x^2)^{-1/2}$ has no Chebyshev series, since $\int_{-1}^{1} (1 - x^2)^{-1} \, dx$ diverges.

With $x = \cos \theta$ and $g(\theta) = f(\cos \theta)$, one gets $a_k = \frac{2}{\pi} \int_0^{\pi} g(\theta) \cos(k\theta) \, d\theta$.
So the Chebyshev series of $f$ is the [[Fourier series|G12KXDHPPS]] of the even function $g$, and convergence results transfer.
For example, if $f$ is Lipschitz on $[-1, 1]$, the series converges uniformly to $f$.
The $T_k$ are [[orthogonal polynomials|8TXASARQN1]] for the weight $(1 - x^2)^{-1/2}$, and, when $\int_{-1}^{1} f(x)^2 (1 - x^2)^{-1/2} \, dx < \infty$, the partial sums are the least squares approximations of $f$ for this weight.
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# Reduction of an initial value problem to a Volterra equation

## Statement

Let $a < b$, let $n \ge 1$, let $p_1, \ldots, p_n, F \colon [a, b] \to \R$ be continuous, and let $c_0, \ldots, c_{n-1} \in \R$.
Consider the [[initial value problem|0FXDW992DS]] for the [[linear differential equation of order n|0QQRP17Z7S]]
$$y^{(n)} + p_1(x) y^{(n-1)} + \cdots + p_n(x) y = F(x), \qquad y^{(j)}(a) = c_j \text{ for } 0 \le j \le n - 1.$$
Define, for $x, t \in [a, b]$,
$$K(x, t) = -\sum_{i=1}^{n} p_i(x) \frac{(x - t)^{i-1}}{(i-1)!}, \qquad f(x) = F(x) - \sum_{i=1}^{n} p_i(x) \sum_{j=n-i}^{n-1} c_j \frac{(x - a)^{j - n + i}}{(j - n + i)!}.$$
Then the following hold.

1. If $y \in C^n[a, b]$ solves the initial value problem, then $u = y^{(n)}$ is a continuous solution of the [[Volterra integral equation|DND8CBEVXT]] of the second kind
$$u(x) = f(x) + \int_a^x K(x, t) u(t) \, dt, \quad x \in [a, b].$$
2. If $u \in C[a, b]$ solves this Volterra equation, then
$$y(x) = \sum_{j=0}^{n-1} c_j \frac{(x - a)^j}{j!} + \int_a^x \frac{(x - t)^{n-1}}{(n-1)!} u(t) \, dt$$
is in $C^n[a, b]$, satisfies $y^{(n)} = u$, and solves the initial value problem.

## Remarks

Proof idea: for $y \in C^n[a, b]$ with the given initial values, Taylor's formula with integral remainder, a repeated use of the [[fundamental theorem of calculus|0NHGY0N0JB]], gives for $0 \le k \le n - 1$
$$y^{(k)}(x) = \sum_{j=k}^{n-1} c_j \frac{(x - a)^{j-k}}{(j-k)!} + \int_a^x \frac{(x - t)^{n-k-1}}{(n-k-1)!} y^{(n)}(t) \, dt.$$
Substituting these for $y^{(n-i)}$, $1 \le i \le n$, in the differential equation gives the Volterra equation.
Conversely, differentiating the formula for $y$ gives $y^{(n)} = u$ and the initial values, and the Volterra equation turns back into the differential equation.

Example: for $y'' + y = 0$, $y(0) = 0$, $y'(0) = 1$, one has $p_1 = 0$, $p_2 = 1$, $K(x, t) = -(x - t)$ and $f(x) = -x$.
The equation $u(x) = -x - \int_0^x (x - t) u(t) \, dt$ has the solution $u = -\sin x$, and indeed $y = \sin x$ has $y'' = -\sin x$.

The kernel $K$ is continuous, so the Volterra equation has a unique continuous solution; this gives the existence and uniqueness theorem for linear initial value problems.
For the first-order nonlinear problem $y' = g(x, y)$, $y(a) = c_0$, the analogous reduction is the [[integral equation form of an initial value problem|015KK993C5]].

## Explanations

### L1 human

A differential equation with initial values can be traded for an integral equation.
The unknown becomes the highest derivative $y^{(n)}$, and the lower derivatives are recovered from it by integration.
The initial values are built into the equation, so no side conditions remain.

### L2 human

This reduction is the bridge between linear ordinary differential equations and the theory of Volterra equations.
Volterra equations of the second kind with continuous kernels are uniquely solvable for every right side.
So the reduction proves that a linear initial value problem with continuous coefficients has exactly one solution on the whole interval $[a, b]$.
For a first-order equation, taking $y$ itself as the unknown gives the integral form used in Picard iteration.
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# Adjoint of an unbounded operator

## Summary

The adjoint of a densely defined operator $T$ between Hilbert spaces is an operator $T^*$ in the opposite direction.
Its domain consists of the vectors $y$ for which $x \mapsto \ip{Tx}{y}$ is bounded on the domain of $T$.
It extends the [[Hilbert space adjoint|6HSJBJ8TTQ]] of bounded operators, and the domain of $T^*$ is part of the answer.
The adjoint is needed to define self-adjoint unbounded operators, the observables of quantum mechanics.

## Explanations

### L1 human

For bounded operators, the adjoint moves the operator from one side of the inner product to the other.
For an unbounded operator this can only be done for some vectors $y$, and the adjoint is defined exactly on those.
The domain of $T$ must be dense; otherwise the vector $T^* y$ would not be determined.
For the diagonal operator $(x_n) \mapsto (n x_n)$ on its natural domain in $\ell^2$, the adjoint is the same operator on the same domain.
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# F-space

## Summary

An F-space is a [[topological vector space|T32W553W95]] whose topology comes from a complete translation-invariant metric.
F-spaces are the natural setting of the Baire category arguments of functional analysis: the open mapping theorem and the closed graph theorem hold for F-spaces.
Every Banach space and every [[Fréchet space|3GAA614673]] is an F-space, but F-spaces need not be locally convex.

## Explanations

### L1 human

An F-space is a vector space with a notion of distance that behaves well under shifting and has no missing limit points.
Shifting two vectors by the same amount does not change their distance.
Every Cauchy sequence converges.
The space $L^p([0, 1])$ with $0 < p < 1$ and distance $d(f, g) = \int_0^1 \abs{f - g}^p$ is an F-space.
Its only open convex set containing $0$ is the whole space, so it is not locally convex, and in particular it is not a Banach space.
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# Coercive bilinear form

## Summary

A coercive bilinear form on a real Hilbert space is a [[bilinear form|0SQZMXB6GP]] $a$ with $a(u, u) \ge \alpha \norm{u}^2$ for a fixed $\alpha > 0$.
Coercivity is the key hypothesis of the Lax–Milgram theorem.
It turns the weak form of many elliptic partial differential equations into a problem with exactly one solution.

## Explanations

### L1 human

A bilinear form takes two vectors and returns a number, like the dot product.
Coercive means that pairing a vector with itself always gives a value at least as large as a fixed fraction of its squared length.
So no nonzero vector is "invisible" to the form, and long vectors give large values.
For example, $a(u, v) = 2 u_1 v_1 + 3 u_2 v_2$ on $\R^2$ is coercive, since $a(u, u) \ge 2 (u_1^2 + u_2^2)$.
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# Reduced function

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]], and let $u \ge 0$ be a [[superharmonic function|X4V2S56YQQ]] on $\Omega$.
Let $E \subseteq \Omega$.
The *reduced function* of $u$ on $E$ is the function $R^E_u \colon \Omega \to [0, +\infty]$ given by
$$R^E_u(x) = \inf \set{v(x) : v \ge 0 \text{ is superharmonic on } \Omega \text{ and } v \ge u \text{ on } E}.$$
The *regularised reduced function* $\widehat{R}^E_u$ is its lower semicontinuous regularisation:
$$\widehat{R}^E_u(x) = \min\left(R^E_u(x), \liminf_{y \to x} R^E_u(y)\right).$$

## Remarks

The function $u$ itself is admissible, so $0 \le \widehat{R}^E_u \le R^E_u \le u$, and $R^E_u = u$ on $E$.
By the [[fundamental convergence theorem|RJK6ZKNF6R]], $\widehat{R}^E_u$ is superharmonic and differs from $R^E_u$ only on a [[polar set|C95QJ7YH5T]] contained in $E$.
The function $\widehat{R}^E_u$ is harmonic on $\Omega \setminus \overline{E}$.
Example: let $\Omega = \R^3$, $u = 1$ and $E = \set{x : \abs{x} \le 1}$.
The function $v(x) = \min(1, 1/\abs{x})$ is positive, superharmonic and equal to $1$ on $E$.
Any admissible $v$ satisfies $v \ge 1/\abs{x}$ on $\abs{x} > 1$ by the minimum principle, so $R^E_1 = \widehat{R}^E_1 = \min(1, 1/\abs{x})$.
Non-example of an admissible function: in the same example, the indicator function of the closed ball $E$ equals $u = 1$ on $E$.
It is not superharmonic, since it is not lower semicontinuous at the points of the unit sphere.
Other notations are $R_u^E$, $\operatorname{red}_E u$, and the French name réduite.
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# Gradient flow on a manifold

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $f \colon M \to \R$ be a [[smooth function|0V33BNKZ9S]].
The [[gradient|0ABK9V9Y64]] of $f$ is the vector field $\nabla f$ on $M$ with $g(\nabla f, X) = df(X)$ for every tangent vector $X$.
The *(negative) gradient flow* of $f$ is the [[flow|06738GGWWK]] $\varphi$ of $-\nabla f$.
So it is the solution of
$$\frac{\partial}{\partial t} \varphi_t(x) = -\nabla f\big(\varphi_t(x)\big), \qquad \varphi_0(x) = x,$$
defined on its maximal domain.

## Remarks

Along a flow line, $\frac{d}{dt} f(\varphi_t(x)) = -\abs{\nabla f}^2 \le 0$, with equality exactly at [[critical points|0YTCMFB4HC]] of $f$.
So $f$ is a Lyapunov function for the flow, and the rest points of the flow are the critical points of $f$.
If $M$ is compact, the flow is defined for all $t \in \R$.

Example: on $\R^2$ with the Euclidean metric, let $f(x, y) = x^2 - y^2$.
Then $-\nabla f = (-2x, 2y)$ and $\varphi_t(x, y) = (e^{-2t} x, e^{2t} y)$.
The origin is the only rest point, and the flow has the familiar saddle picture.

Non-example: on $\R^2$, the rotation field $X(x, y) = (-y, x)$ is not $-\nabla f$ for any smooth $f$.
Its flow has closed orbits, but $f$ would strictly decrease along a nonconstant flow line of a gradient flow, so it could not return to its starting point.

The flow depends on the metric $g$, not only on $f$.
A vector field $X$ with $df(X) < 0$ away from the critical points, and of a standard form near them, is called gradient-like for $f$.
Such fields serve the same purposes in Morse theory.
For a Morse function and a generic metric, the gradient flow is a [[Morse–Smale system|7AMQ8GS78Y]].
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# Elliptic complex

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]], and let $E_0, E_1, \ldots, E_N$ be smooth [[vector bundles|P4DZM975BV]] over $M$.
For $0 \le k < N$, let $P_k \colon \Gamma(E_k) \to \Gamma(E_{k+1})$ be a [[differential operator|CGBX1N9V8H]] of order $m_k$, with [[principal symbol|S8GC34D5ZG]] $\sigma(P_k) = \sigma_{m_k}(P_k)$.
The sequence
$$0 \to \Gamma(E_0) \xrightarrow{P_0} \Gamma(E_1) \xrightarrow{P_1} \cdots \xrightarrow{P_{N-1}} \Gamma(E_N) \to 0$$
is an *elliptic complex* if both of the following hold.

1. $P_{k+1} \circ P_k = 0$ for $0 \le k < N - 1$.
2. For every $x \in M$ and every nonzero $\xi \in T^*_x M$, the sequence of linear maps
$$0 \to E_{0,x} \xrightarrow{\sigma(P_0)(x, \xi)} E_{1,x} \to \cdots \xrightarrow{\sigma(P_{N-1})(x, \xi)} E_{N,x} \to 0$$
is exact.

## Remarks

For $N = 1$ the complex is a single operator $P_0$, and it is elliptic exactly when $P_0$ is an [[elliptic operator|MT9P2X85V8]].
When $M$ is compact without boundary, the cohomology groups $\ker P_k / \operatorname{im} P_{k-1}$ are finite-dimensional, and the index of the complex is $\sum_k (-1)^k \dim(\ker P_k / \operatorname{im} P_{k-1})$.
Suppose that all $P_k$ have the same order and that $M$ and the bundles carry metrics, and write $P = \sum_k P_k$ and $P^* = \sum_k P_k^*$ for the formal adjoint.
Then $P + P^* \colon \Gamma(\bigoplus_k E_{2k}) \to \Gamma(\bigoplus_k E_{2k+1})$ is elliptic and has the same index as the complex.

Example: the de Rham complex $0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \cdots \xrightarrow{d} \Omega^n(M) \to 0$ of a manifold of dimension $n$, with the [[exterior derivative|0ZHSTAG8ZY]].
It satisfies $d \circ d = 0$, and the symbol of $d$ at $\xi$ is $\omega \mapsto \xi \wedge \omega$.
For $\xi \ne 0$, complete $\xi$ to a basis $\xi = e_1, e_2, \ldots, e_n$ of $T^*_x M$.
A form $\omega$ with $\xi \wedge \omega = 0$ is then a sum of terms containing $e_1$.
So it equals $\xi \wedge \eta$ for some $\eta$, which proves exactness.
Its cohomology is the [[de Rham cohomology|GX74C07T0G]], and its index is the [[Euler characteristic|RNDDWDAFJ5]] of $M$.

Non-example: on a manifold of dimension $n \ge 2$, the two-term complex $0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \to 0$ satisfies clause 1 but not clause 2.
The symbol sequence $0 \to \R \to T^*_x M \to 0$ is not exact at $T^*_x M$, because the image of $c \mapsto c \, \xi$ is a line in a space of dimension $n$.

## Explanations

### L1 human

An elliptic complex is a chain of differential operators that compose to zero and that, at top order in each nonzero direction, fit together exactly.
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<document source="https://tamnd.github.io/math2/node/872BAKDQQK.md" tag="872BAKDQQK" kind="concept">
# f-vector

## Summary

The f-vector of a [[convex polytope|300K6CB1KH]] lists how many faces it has in each dimension.
For a $d$-polytope it is $(f_0, f_1, \dots, f_{d-1})$, where $f_i$ is the number of $i$-dimensional faces.
The same notion applies to [[simplicial complexes|V8GZ0W1ZZ5]], and it is the starting point for the h-vector; see [[f-vector and h-vector|DCK64GFPD7]].
Describing which vectors are f-vectors is a main problem of [[polytope combinatorics|879SBTVE7V]].

## Explanations

### L1 human

Count the corners, edges, faces and so on of a shape, and write the counts in a row.
That row is the f-vector.
A cube gives $(8, 12, 6)$: $8$ corners, $12$ edges and $6$ square faces.
The numbers in an f-vector are not arbitrary; for solids in space the corners minus edges plus faces is always $2$.
</document>

<document source="https://tamnd.github.io/math2/node/878B9A4D9B.md" tag="878B9A4D9B" kind="definition">
# Tricolorability

## Definition

Let $D$ be a diagram of a [[knot|QYEAZTYGPT]] $K$.
The *arcs* of $D$ are the pieces into which $D$ falls when each under-strand is cut at each crossing.
A *tricoloring* of $D$ is a map from the set of arcs of $D$ to a set of three colors such that:

1. at each crossing, the over-arc and the two under-arcs have either one color or three distinct colors;
2. at least two colors are used.

The diagram $D$ is *tricolorable* if it has a tricoloring.
The knot $K$ is *tricolorable* if some diagram of $K$ is tricolorable.

## Remarks

Each [[Reidemeister move|T3HAGWYKK7]] turns a tricoloring into a tricoloring, so if one diagram of $K$ is tricolorable then every diagram of $K$ is.
Example: the standard diagram of the [[trefoil knot|VYTYSVHN4A]] has three arcs and three crossings, and each crossing meets all three arcs; three distinct colors satisfy both clauses.
Non-example: the round diagram of the [[unknot|1TQ06R4Q66]] has one arc, so clause 2 fails; hence the trefoil is not the unknot.
The [[figure-eight knot|C01XBGAHC6]] is not tricolorable.
With the colors read as elements of $\Z/3$, clause 1 says $2a \equiv b + c \pmod 3$ at a crossing with over-arc $a$ and under-arcs $b, c$; this is Fox $3$-coloring.
A knot is tricolorable if and only if $3$ divides its determinant $\abs{\Delta_K(-1)}$, where $\Delta_K$ is the [[Alexander polynomial|JKA25KDPQP]].
</document>

<document source="https://tamnd.github.io/math2/node/879SBTVE7V.md" tag="879SBTVE7V" kind="concept">
# Polytope combinatorics

## Summary

Polytope combinatorics, also called polyhedral combinatorics, studies convex polytopes through their faces.
It asks how many faces of each dimension a polytope has and how the faces fit together.
Its typical questions ask which [[f-vectors|DCK64GFPD7]] are possible, as in [[Euler's polyhedron formula|09HT3P35YP]], the [[Dehn–Sommerville equations|4WKHF5BCTA]] and the [[g-theorem|4CZMWQVC30]].
The same name covers the study of the facets of polytopes defined by combinatorial optimization problems, a tool of integer programming.

## Explanations

### L1 human

A cube has $8$ vertices, $12$ edges and $6$ square faces, and any squashed cube with flat faces has the same pattern.
Polytope combinatorics forgets lengths and angles and keeps only this pattern of vertices, edges and faces.
It asks, for example, which lists of face numbers can occur, and in every dimension, not only in three.
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<document source="https://tamnd.github.io/math2/node/87AWY7NQS0.md" tag="87AWY7NQS0" kind="concept">
# Hyperkähler manifold

## Summary

A hyperkähler manifold is a Riemannian manifold that is Kähler with respect to three complex structures $I$, $J$, $K$ that multiply like the quaternion units.
Its dimension is a multiple of $4$, and its holonomy group lies in the compact symplectic group $Sp(n)$.
Hyperkähler manifolds are Ricci-flat, and they are one of the special holonomy cases in Berger's list.
Examples include K3 surfaces with their Ricci-flat metrics and many moduli spaces from gauge theory.

## Explanations

### L1 human

The quaternions extend the complex numbers with three square roots of $-1$, called $i$, $j$ and $k$, with $ij = k$.
A hyperkähler manifold lets one multiply tangent vectors by all three, in a way that respects lengths and stays the same when moved along curves.
The simplest example is the space $\mathbb{H}^n$ of quaternion vectors with its flat metric.
So each such manifold is a complex manifold in many ways at once, one for each unit imaginary quaternion.
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<document source="https://tamnd.github.io/math2/node/87DAYKCZ5M.md" tag="87DAYKCZ5M" kind="concept">
# Heun function

## Summary

Heun functions are solutions of Heun's equation, the general second order linear [[Fuchsian equation|ZBVH2YE50B]] with four [[regular singular points|H96HXG6CA0]], placed at $0, 1, a$ and $\infty$.
The local Heun function is the solution that is analytic at $0$ with value $1$.
Heun's equation is the next case after the [[hypergeometric differential equation|W8VE25R5A7]], which has three regular singular points.
It contains the Lamé, Mathieu and spheroidal wave equations as special or confluent cases.

## Explanations

### L1 human

The hypergeometric equation is fixed, up to its parameters, by the positions of its three singular points and the exponents there.
With four singular points this is no longer true: besides the exponents there is one extra number $q$, the accessory parameter.
This extra freedom makes Heun functions much harder to handle than hypergeometric functions.
They come up in quantum mechanics and in general relativity, among other places.
A trivial example: when $q = 0$ and $\alpha = 0$, the constant function $1$ solves Heun's equation.
</document>

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# Landweber iteration

## Statement

Let $X, Y$ be [[Hilbert spaces|HM2HHE3J51]], let $A \colon X \to Y$ be a bounded linear operator with adjoint $A^*$, and let $A^\dagger$ be its [[Moore–Penrose generalized inverse|0YY8VX28JR]].
Let $0 < \omega < 2/\norm{A}^2$.
For $z \in Y$, the *Landweber iteration* is $x_0(z) = 0$ and
$$x_{k+1}(z) = x_k(z) + \omega A^* \bigl( z - A x_k(z) \bigr), \qquad k \ge 0.$$

1. If $y \in \mathcal{R}(A) + \mathcal{R}(A)^\perp$, then $x_k(y) \to A^\dagger y$ as $k \to \infty$.
2. If moreover $\omega \le 1/\norm{A}^2$, then $\norm{x_k(y^\delta) - x_k(y)} \le \sqrt{\omega k} \, \delta$ for all $y^\delta \in Y$ with $\norm{y^\delta - y} \le \delta$.
3. Consequently, if $k(\delta) \to \infty$ and $\delta \sqrt{k(\delta)} \to 0$ as $\delta \to 0$, then $x_{k(\delta)}(y^\delta) \to A^\dagger y$ uniformly over all $y^\delta$ with $\norm{y^\delta - y} \le \delta$.

## Remarks

Write $x^\dagger = A^\dagger y$; then $A^* y = A^* A x^\dagger$.
The error satisfies $x_k(y) - x^\dagger = -(I - \omega A^* A)^k x^\dagger$.
The self-adjoint operator $A^* A$ has spectrum in $[0, \norm{A}^2]$, and $\abs{1 - \omega t} < 1$ for $0 < t \le \norm{A}^2$.
Since $x^\dagger$ is orthogonal to $\ker A$, the spectral theorem gives $(I - \omega A^* A)^k x^\dagger \to 0$, which is part 1.
For part 2, $x_k(z) = \sum_{j < k} (I - \omega A^* A)^j \omega A^* z$, and a spectral bound on $\sqrt{\omega t} \sum_{j<k} (1 - \omega t)^j$ gives the factor $\sqrt{\omega k}$.

The iteration is gradient descent with fixed step $\omega$ for $\tfrac12 \norm{A x - z}^2$.
When $\mathcal{R}(A)$ is not closed, $x_k(y^\delta)$ first approaches $x^\dagger$ and then diverges, a behavior called semi-convergence.
So the iteration index plays the role of the [[regularization|WTY54BFC46]] parameter, and one must stop early, for example by the discrepancy principle.
Without the condition $\omega < 2/\norm{A}^2$, components along large singular values are amplified and the iteration diverges.

## Explanations

### L1 human

Landweber iteration solves $A x = y$ by repeatedly correcting the guess in the direction that reduces the misfit $\norm{A x - y}$.
With exact data it converges to the best solution.
With noisy data it first improves and later starts to fit the noise, so one stops it at the right moment.

### L2 human

Landweber iteration is the simplest iterative regularization method, and it is the model for others such as conjugate gradients on the normal equations.
Stopping after $k$ steps acts like a filter on the singular values: components with $\omega \sigma^2 k \gg 1$ are recovered and the others are damped.
It converges slowly, but it extends directly to nonlinear operators and to constraints by adding a projection.
It was introduced by Louis Landweber in 1951.
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# Constant-mean-curvature surface

## Definition

Let $S \subset \R^3$ be an orientable [[regular surface|6080X3FAD4]] with a chosen unit normal, and let $H \colon S \to \R$ be its [[mean curvature|PA78NFEPCF]].
The surface $S$ is a *constant-mean-curvature surface* if there is a real number $H_0$ with $H(p) = H_0$ for every $p \in S$.

## Remarks

The sphere of radius $r$ is a CMC surface: for the inward normal both principal curvatures are $1/r$, so $H = 1/r$ everywhere.
The paraboloid $z = x^2 + y^2$ is not: its mean curvature is $2$ at the origin and tends to $0$ far from the origin.

The case $H_0 = 0$ gives the [[minimal surfaces|41X0YW2R9N]].
Changing the normal changes the sign of $H$, so the condition does not depend on the choice of normal.
The same definition applies to immersed surfaces, where self-intersections are allowed.
By a theorem of Alexandrov, a compact embedded CMC surface is a round sphere, but compact immersed CMC tori exist, such as the [[Wente torus|E5NNY7XM44]].

## Explanations

### L1 human

A CMC surface is a surface whose mean curvature is the same number at every point.
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# Nuclear space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a Hausdorff [[locally convex space|B6GY2KJMWW]] over $\mathbb{K}$.
For a continuous [[seminorm|FP5GVCN8GV]] $p$ on $E$, let $E_p$ be the Banach space obtained by completing the quotient $E / \set{x : p(x) = 0}$ with the norm induced by $p$.
If $q \ge p$ are continuous seminorms, the identity of $E$ induces a continuous linear map $\iota_{q, p} \colon E_q \to E_p$ of norm at most $1$.
$E$ is *nuclear* if the following holds.
For every continuous seminorm $p$ on $E$ there is a continuous seminorm $q \ge p$ such that $\iota_{q, p}$ is a [[nuclear operator|XPANM7C1D3]].

## Remarks

Example: the Schwartz space $\mathcal{S}(\R^n)$, the space $C^\infty(\Omega)$ of smooth functions on an open set $\Omega \subseteq \R^n$, and the space $H(\Omega)$ of holomorphic functions on an open set $\Omega \subseteq \C$ are nuclear.
Non-example: an infinite-dimensional normed space $E$ is not nuclear.
Take $p$ to be the norm; then every continuous seminorm $q \ge p$ is equivalent to the norm.
So $\iota_{q, p}$ is an isomorphism of infinite-dimensional Banach spaces, which is not compact and hence not nuclear.
Grothendieck's original definition is equivalent: $E$ is nuclear if and only if, for every Banach space $F$, the projective and injective tensor topologies on $E \otimes F$ agree.
Every nuclear Fréchet space is a Montel space.
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<document source="https://tamnd.github.io/math2/node/87VMQ892VW.md" tag="87VMQ892VW" kind="concept">
# Mixed finite element method

## Summary

A mixed finite element method approximates two or more unknown fields of a boundary value problem at once, each in its own finite element space.
It starts from a saddle-point formulation, such as velocity and pressure for Stokes flow, or flux and potential for the Poisson equation.
The spaces cannot be chosen independently: stability requires the [[Ladyzhenskaya–Babuška–Brezzi condition|SC5FV3GHV5]].
Mixed methods give accurate fluxes that satisfy conservation element by element, which matters for flow in porous media.
See the [[definition|DBVE636BXN]].

## Explanations

### L1 human

In many physical problems one quantity is of main interest, but another one comes with it, such as the flow of heat together with the temperature.
The standard finite element method computes the temperature and gets the heat flow by differentiating, which loses accuracy.
A mixed method computes both at the same time, as separate unknowns linked by the equations.
For the Poisson equation $-\Delta u = f$, one introduces the flux $\sigma = \nabla u$ and solves for the pair $(\sigma, u)$ with $\nabla \cdot \sigma = -f$.
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<document source="https://tamnd.github.io/math2/node/87WH65FJB8.md" tag="87WH65FJB8" kind="concept">
# Height of a prime ideal

## Summary

The height of a prime ideal $P$ is the largest length of a strictly increasing chain of prime ideals that ends at $P$.
It equals the [[Krull dimension|MVGBS3J1P7]] of the [[localization|X6JD4AG973]] at $P$, and it is the algebraic notion of codimension.
[[Krull's principal ideal theorem|SJZW2HN1SC]] and [[Krull's height theorem|TBRTZMHFTV]] bound the height by the number of generators of an ideal.

## Explanations

### L1 human

The dimension of a ring counts the longest staircase of prime ideals anywhere.
The height of a prime $P$ counts only the staircases that end at $P$, looking down from $P$.
In $k[x, y, z]$, the prime $(x)$ is a surface and has height $1$, while $(x, y)$ is a line and has height $2$: it has codimension $2$ in space.
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<document source="https://tamnd.github.io/math2/node/8831C5ARPB.md" tag="8831C5ARPB" kind="definition">
# Faithfully flat module

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].
The module $M$ is *faithfully flat* if both conditions hold:

1. $M$ is [[flat|73RYYEVDKA]].
2. For every $R$-module $N$, if $N \otimes_R M = 0$, then $N = 0$.

## Remarks

The following are equivalent for an $R$-module $M$:

1. $M$ is faithfully flat.
2. A sequence $N' \to N \to N''$ of $R$-modules is [[exact|JB7ZBFBA9C]] if and only if $N' \otimes_R M \to N \otimes_R M \to N'' \otimes_R M$ is exact.
3. $M$ is flat and $\mathfrak{m} M \ne M$ for every [[maximal ideal|0PV74TR5FP]] $\mathfrak{m}$ of $R$.

Example: a nonzero free module $M = R^{(I)}$ is faithfully flat.
It is flat, and $N \otimes_R R^{(I)} \cong N^{(I)}$, which is zero only when $N = 0$, since $I$ is not empty.

Non-example: $\Q$ is a flat $\Z$-module, so clause 1 holds.
But $\Z/2\Z \otimes_\Z \Q = 0$ while $\Z/2\Z \ne 0$, since $1 \otimes q = 2 \otimes \frac{q}{2} = 0 \otimes \frac{q}{2} = 0$; so clause 2 fails.
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# Radon's theorem

## Statement

Let $d \ge 1$ and let $X \subseteq \R^d$ be a finite set with at least $d + 2$ points.
Then $X$ can be [[partitioned|08QMKTWF9T]] into two disjoint sets $A$ and $B$ whose [[convex hulls|0FQSDC2EQ2]] intersect:
$$\operatorname{conv}(A) \cap \operatorname{conv}(B) \ne \emptyset.$$

## Remarks

Proof idea: take $d + 2$ points $x_1, \dots, x_{d+2}$ of $X$.
The $d + 2$ vectors $(x_i, 1) \in \R^{d+1}$ are linearly dependent, so there are reals $\lambda_i$, not all $0$, with $\sum \lambda_i x_i = 0$ and $\sum \lambda_i = 0$.
Let $A$ hold the points with $\lambda_i > 0$ and let $B$ hold the rest, and put $s = \sum_{\lambda_i > 0} \lambda_i > 0$.
Then $\sum_{\lambda_i > 0} (\lambda_i / s) x_i = \sum_{\lambda_i \le 0} (-\lambda_i / s) x_i$ is a point of both hulls; the remaining points of $X$ can be added to either part.

Example: take four points in the plane.
If one of them lies in the convex hull of the other three, that point against the other three is a Radon partition.
Otherwise they are the vertices of a convex quadrilateral, and its two diagonals cross.

The number $d + 2$ is sharp: the $d + 1$ vertices of a simplex in $\R^d$ are affinely independent, and no partition of them has intersecting hulls.

## Explanations

### L1 human

Take any $d + 2$ points in $d$-dimensional space, for example four points in the plane.
You can always colour them red and blue so that the red shape and the blue shape overlap.
With three points in the plane this can fail: the corners of a triangle cannot be split this way.

### L2 human

Radon's theorem is the key step in a short proof of Helly's theorem.
It also shows that half-spaces in $\R^d$ have VC dimension at most $d + 1$ in learning theory, since no set of $d + 2$ points can be shattered.
Its generalisation to partitions into more than two parts is Tverberg's theorem, and a topological version replaces convex hulls by images of faces of a simplex under continuous maps.
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# Space of connections modulo gauge

## Description

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, let $X$ be a compact manifold, and let $P \to X$ be a principal $G$-bundle.
The space $\mathcal{A}(P)$ of [[connections|NJ21PN5ZGJ]] on $P$ is an affine space over the vector space $\Omega^1(X; \mathrm{ad} P)$ of $1$-forms with values in the adjoint bundle.
The [[gauge group|707DEWCCFX]] $\mathcal{G}(P)$ acts on $\mathcal{A}(P)$, and the *space of connections modulo gauge* is the [[quotient space|0BR3FY5Q8M]]
$$\mathcal{B}(P) = \mathcal{A}(P) / \mathcal{G}(P).$$

## Remarks

The simplest case is $G = U(1)$, $X = S^1 = \R / 2\pi \Z$ and $P$ trivial.
A connection is $A = i a(t) \, dt$ with $a$ a smooth real function, and a gauge transformation $g = e^{i \phi}$, with $\phi \colon \R \to \R$ smooth and $\phi(t + 2\pi) - \phi(t) \in 2\pi \Z$, sends $A$ to $A + i \, d\phi$.
So the class of $A$ is determined by $\int_0^{2\pi} a(t) \, dt$ modulo $2\pi$, and $\mathcal{B}(P)$ is a circle; the map is the holonomy $A \mapsto \exp(-i \int_0^{2\pi} a \, dt) \in U(1)$.

In general $\mathcal{B}(P)$ is not a manifold, because the [[stabilisers|0RGGF8ZS6D]] of connections vary.
The stabiliser of $A$ always contains the centre of $G$, acting by constant gauge transformations.
The connection $A$ is called irreducible when its stabiliser is exactly the centre.

After completing in suitable Sobolev norms, the irreducible connections modulo gauge form a Hilbert manifold, with local charts given by the Coulomb slice $\set{A + a : d_A^* a = 0, \ a \text{ small}}$.

## Explanations

### L1 human

Two connections that differ only by a change of gauge describe the same physical field, so one wants to count them once.
The space of connections modulo gauge is the result: one point for each field, not for each description of it.
On a circle, a $U(1)$ connection modulo gauge is described by a single angle, its holonomy around the circle.
Moduli spaces of instantons and of other special connections sit inside this quotient space.
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# Meixner polynomials

## Definition

Let $\beta > 0$ and $0 < c < 1$ be real numbers, and let $n \ge 0$ be an integer.
Write $(a)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]].
The *Meixner polynomial* of degree $n$ is
$$M_n(x; \beta, c) = \sum_{k=0}^{n} \frac{(-n)_k \, (-x)_k}{(\beta)_k \, k!} \left( 1 - \frac{1}{c} \right)^k,$$
that is, $M_n(x; \beta, c) = {}_2F_1\left( -n, -x; \beta; 1 - \frac{1}{c} \right)$, a terminating [[hypergeometric function|463X5V31D9]].

## Remarks

They are orthogonal on the nonnegative integers:
$$\sum_{x=0}^{\infty} \frac{(\beta)_x}{x!} c^x M_m(x; \beta, c) M_n(x; \beta, c) = \frac{c^{-n} \, n!}{(\beta)_n (1 - c)^\beta} \delta_{mn}.$$
Example: $M_1(x; \beta, c) = 1 - \frac{(1 - c) x}{\beta c}$.
Check: the weights sum to $(1 - c)^{-\beta}$, and their mean value of $x$ is $\frac{\beta c}{1 - c}$, so the weighted sum of $M_1$ is $(1 - c)^{-\beta} (1 - 1) = 0$.
Non-example: for $c \ge 1$ the clause $0 < c < 1$ fails, and the weights $\frac{(\beta)_x}{x!} c^x$ do not have a finite sum, so the orthogonality relation has no meaning.
Limit to Laguerre: $\lim_{c \to 1} M_n\left( \frac{x}{1 - c}; \alpha + 1, c \right) = \frac{L_n^{(\alpha)}(x)}{L_n^{(\alpha)}(0)}$.
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# Gaussian curvature

## Definition

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]], let $p \in S$, and choose a smooth unit normal field near $p$.
Let $S_p$ be the [[shape operator|DQCD11X1VM]] at $p$ and let $k_1, k_2$ be the [[principal curvatures|FA8QE1SGP4]] at $p$.
The *Gaussian curvature* of $S$ at $p$ is
$$K(p) = \det S_p = k_1 k_2.$$
It does not depend on the choice of the unit normal.

## Remarks

In a local parametrisation, with $E, F, G$ the coefficients of the [[first fundamental form|MAJ3DH7MZS]] and $e, f, g$ those of the [[second fundamental form|9KHQR16P1B]],
$$K = \frac{e g - f^2}{E G - F^2}.$$
Example: on the sphere of radius $r$, $S_p = \pm \frac{1}{r} \id$, so $K = 1/r^2$.
On a cylinder $k_1 = 1/r$ and $k_2 = 0$, so $K = 0$; on the saddle $z = xy$ at the origin $k_1 = 1$ and $k_2 = -1$, so $K = -1$.
Non-example: on that saddle, the normal curvatures in the orthogonal directions $(1, 0, 0)$ and $(0, 1, 0)$ are both $0$.
Their product $0$ is not $K = -1$, because these directions are not principal directions.
For the graph $z = h(x, y)$ of a smooth function, $K = (h_{xx} h_{yy} - h_{xy}^2) / (1 + h_x^2 + h_y^2)^2$.
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# Subdirect product

## Summary

A subdirect product of a family of [[algebras|QJGPT2W0FT]] is a [[subalgebra|7CHG6SD365]] of their [[direct product|WJ73XYBNE4]] that still projects onto each factor.
It can be much smaller than the full product, yet it remembers each factor completely.
By [[Birkhoff's subdirect representation theorem|CZF5696TPY]], every algebra is isomorphic to a subdirect product of [[subdirectly irreducible algebras|DASF939439]].

## Explanations

### L1 human

The full product of two algebras contains every pair of elements.
A subdirect product keeps only some pairs, but enough so that every element of each factor still appears in some kept pair.
For example, the diagonal $\set{(x, x)}$ in $\mathbf{A} \times \mathbf{A}$ is a subdirect product of two copies of $\mathbf{A}$.
The integers embed in the product of all $\Z/p\Z$ for primes $p$ by sending $n$ to its list of remainders, and this is a subdirect product too.
</document>

<document source="https://tamnd.github.io/math2/node/88XF0NVBVG.md" tag="88XF0NVBVG" kind="definition">
# Diagram

## Definition

Let $J$ and $\mathcal{C}$ be [[categories|A2DE66937D]].
A *diagram of shape $J$* in $\mathcal{C}$ is a [[functor|28NQ0KRHD3]] $D \colon J \to \mathcal{C}$.
The category $J$ is called the *index category* or *shape* of the diagram.
The diagram is *small* if $J$ is small, and *finite* if $J$ has finitely many objects and morphisms.

## Remarks

The objects $D(j)$ and morphisms $D(u)$ for $u$ in $J$ form the picture that the diagram draws in $\mathcal{C}$.
Because $D$ is a functor, $D(v \circ u) = D(v) \circ D(u)$, so every composite in $J$ is sent to the corresponding composite in $\mathcal{C}$.

Example: let $J$ have three objects $1, 2, 3$ and, besides identities, morphisms $u \colon 1 \to 3$ and $v \colon 2 \to 3$.
A diagram of shape $J$ in the category of sets is a pair of functions $f = D(u) \colon X \to Z$ and $g = D(v) \colon Y \to Z$; its limit is the pullback of $f$ and $g$.

Non-example: let $J$ have one object $\ast$ and one non-identity morphism $e$ with $e \circ e = e$.
Sending $\ast$ to $\Z$ and $e$ to the map $n \mapsto -n$ is not a diagram of shape $J$ in the category of sets.
The image of $e \circ e$ would have to be $n \mapsto -n$, but the composite of $n \mapsto -n$ with itself is the identity.
</document>

<document source="https://tamnd.github.io/math2/node/8943THTS6W.md" tag="8943THTS6W" kind="definition">
# Convolution power

## Definition

Let $d \ge 1$ and let $f \in L^1(\R^d)$.
For each integer $k \ge 1$, the *$k$-th convolution power* $f^{*k} \in L^1(\R^d)$ is defined by recursion:
$$f^{*1} = f, \qquad f^{*(k+1)} = f^{*k} * f,$$
where $*$ is the [[convolution|2Z29JN3VES]].

## Remarks

By [[associativity of convolution|HZ5J8GGS16]], $f^{*k}$ is the convolution of $k$ copies of $f$ in any bracketing, and $f^{*j} * f^{*k} = f^{*(j+k)}$.
Each $f^{*k}$ lies in [[$L^1(\R^d)$|SASP14MA1Z]] with $\norm{f^{*k}}_1 \le \norm{f}_1^k$.
For finite Borel measures $\mu$ the same recursion applies, and one sets $\mu^{*0} = \delta_0$, the [[Dirac measure|0ACRGYNHSR]] at $0$, which is the unit for convolution.

Example: for $\lambda > 0$ let $f(x) = \lambda e^{-\lambda x}$ for $x \ge 0$ and $f(x) = 0$ for $x < 0$, the exponential density.
Then $f^{*k}(x) = \lambda^k x^{k-1} e^{-\lambda x} / (k-1)!$ for $x \ge 0$.
The induction step is
$$\int_0^x \frac{\lambda^k y^{k-1} e^{-\lambda y}}{(k-1)!} \, \lambda e^{-\lambda (x - y)} \, dy = \frac{\lambda^{k+1} e^{-\lambda x}}{(k-1)!} \int_0^x y^{k-1} \, dy = \frac{\lambda^{k+1} x^k e^{-\lambda x}}{k!}.$$
This is the Erlang density of the sum of $k$ independent exponential waiting times.

Non-example: the constant $f = 1$ on $\R$ is not in $L^1(\R)$, and indeed $f * f$ is defined nowhere, since $\int_\R 1 \, dy = \infty$.
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<document source="https://tamnd.github.io/math2/node/89AA6XJY5K.md" tag="89AA6XJY5K" kind="concept">
# Schrödinger operator

## Summary

A Schrödinger operator is a differential operator of the form $-\Delta + V$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $V$ is a real function called the potential.
In one dimension it is the operator $-\frac{d^2}{dx^2} + V(x)$.
Its eigenvalue equation is the time-independent Schrödinger equation of quantum mechanics, and its eigenvalues are the allowed energies of a particle in the potential $V$.
In the theory of ordinary differential equations it is the main example of a Sturm–Liouville operator.

## Explanations

### L1 human

The operator takes a function, measures how much it bends with the second derivative, and adds the function times the potential.
In physics the bending term is the kinetic energy and $V$ is the potential energy.
For $V(x) = x^2$ the operator $-y'' + x^2 y$ is the quantum harmonic oscillator.
Its eigenvalues are $1, 3, 5, \ldots$, and the first eigenfunction is $e^{-x^2/2}$.
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<document source="https://tamnd.github.io/math2/node/89CHZ3ZMCH.md" tag="89CHZ3ZMCH" kind="statement">
# Mutual absolute continuity of harmonic measures

## Statement

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be a bounded domain, and for $x \in \Omega$ let $\omega^x$ be the [[harmonic measure|CFD7BPM51W]] of $\Omega$ at $x$.
Let $x, y \in \Omega$.
Then there is a constant $C \ge 1$, depending only on $x$, $y$ and $\Omega$, such that
$$C^{-1}\, \omega^y(E) \le \omega^x(E) \le C\, \omega^y(E) \qquad \text{for every Borel set } E \subseteq \partial\Omega.$$
In particular $\omega^x$ and $\omega^y$ are mutually [[absolutely continuous|P0P21KDV6E]].

## Remarks

Here $\omega^x(E)$ is the value at $x$ of the [[Perron–Wiener–Brelot solution|R12KQGVGN6]] of the Dirichlet problem with boundary data the indicator function of $E$.
Idea of the proof: for a fixed Borel set $E$, the function $z \mapsto \omega^z(E)$ is harmonic on $\Omega$ with values in $[0, 1]$.
Join $x$ to $y$ by a path in $\Omega$ and cover it by finitely many balls whose doubles lie in $\Omega$.
[[Harnack's inequality|YNH832D50W]] on each ball compares the values of a nonnegative harmonic function at nearby points.
The chain of balls then gives a constant $C$ that does not depend on $E$.
Example: for the unit disc, $d\omega^z = \frac{1 - \abs{z}^2}{\abs{e^{i\theta} - z}^2} \frac{d\theta}{2\pi}$.
With $y = 0$ and $\abs{x} = r$, the density lies between $\frac{1 - r}{1 + r}$ and $\frac{1 + r}{1 - r}$, so one can take $C = \frac{1 + r}{1 - r}$.
Connectedness is needed: on a disjoint union of two discs, the harmonic measures at points of different discs are carried by disjoint circles.

## Explanations

### L1 human

Harmonic measure at $x$ records where a random path from $x$ leaves the domain.
Starting from a different point changes the probabilities, but only by a bounded factor.
So the sets that are never hit are the same for every starting point.

### L2 human

Because of this result one speaks of sets of harmonic measure zero without naming the starting point.
This is how the class of null sets of harmonic measure enters statements such as Fatou-type theorems, Dahlberg's theorem and Makarov's theorem.
The constant $C$ blows up as $x$ or $y$ approaches the boundary, which is why boundary estimates need finer tools such as the boundary Harnack principle.
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<document source="https://tamnd.github.io/math2/node/89P5GTKW87.md" tag="89P5GTKW87" kind="concept">
# Diagonally dominant matrix

## Summary

A square matrix is diagonally dominant when each row has a large diagonal entry.
In absolute value, it is at least the sum of the absolute values of the other entries of its row.
It is strictly diagonally dominant when this inequality is strict in every row.
A strictly diagonally dominant matrix is invertible, by the [[Levy–Desplanques theorem|PWKN6ANA0G]].
For strictly diagonally dominant matrices, Gaussian elimination needs no pivoting, and the Jacobi and Gauss–Seidel iterations converge.

## Explanations

### L1 human

In a diagonally dominant matrix, each row is controlled by its diagonal entry: the other entries together cannot outweigh it.
For example, $\begin{pmatrix} 4 & 1 & -2 \\ 1 & 3 & 1 \\ 0 & 2 & 5 \end{pmatrix}$ is diagonally dominant, since $4 \ge 3$, $3 \ge 2$ and $5 \ge 2$.
Such a matrix behaves much like a diagonal matrix, which is easy to invert.
This makes it easy to see invertibility and to solve systems with it reliably.
</document>

<document source="https://tamnd.github.io/math2/node/89S5KGQEMJ.md" tag="89S5KGQEMJ" kind="statement">
# Gauss–Hermite quadrature

## Statement

Let $n \ge 1$ and let $H_n$ be the physicists' [[Hermite polynomial|MGACY98E9H]] of degree $n$.
Let $x_1, \ldots, x_n$ be the zeros of $H_n$; they are real and simple.
Set
$$w_i = \frac{2^{n-1} \, n! \, \sqrt{\pi}}{n^2 \, H_{n-1}(x_i)^2}, \qquad i = 1, \ldots, n.$$
Then $w_i > 0$ for each $i$, and for every polynomial $p$ of degree at most $2n - 1$,
$$\int_{-\infty}^{\infty} e^{-x^2} p(x) \, dx = \sum_{i=1}^{n} w_i \, p(x_i).$$

## Remarks

This is the [[Gaussian quadrature|FP37BTP93K]] rule for the weight $e^{-x^2}$ on $\R$, whose orthogonal polynomials are the $H_n$.
For $p$ of degree at most $2n - 1$, write $p = q H_n + r$ with $\deg q, \deg r \le n - 1$.
Orthogonality kills $\int e^{-x^2} q H_n$, and $r$ is integrated exactly by the interpolatory weights.
The closed form of $w_i$ comes from the Christoffel–Darboux formula.
Example: for $n = 2$, $H_2 = 4x^2 - 2$ has zeros $\pm 1/\sqrt{2}$, and $H_1(x) = 2x$ gives $w_1 = w_2 = \frac{4\sqrt{\pi}}{4 \cdot 2} = \frac{\sqrt{\pi}}{2}$.
Check with $p(x) = x^2$: the rule gives $2 \cdot \frac{\sqrt{\pi}}{2} \cdot \frac12 = \frac{\sqrt{\pi}}{2} = \int_{\R} x^2 e^{-x^2} dx$.
For $f \in C^{2n}(\R)$ with suitable growth, the error is $\frac{n! \sqrt{\pi}}{2^n} \frac{f^{(2n)}(\xi)}{(2n)!}$ for some real $\xi$.
The degree $2n - 1$ is the best possible: $p = H_n^2$ has degree $2n$, the rule gives $0$, but the integral is positive.

## Explanations

### L1 human

Many integrals over the whole real line carry a bell-shaped factor $e^{-x^2}$.
Gauss–Hermite quadrature replaces such an integral by a weighted sum of a few function values at special points.
With $n$ points it is exact for every polynomial up to degree $2n - 1$, and very accurate for smooth functions that do not grow too fast.

### L2 human

The rule computes expectations under a normal distribution: if $Z$ is standard normal, then $\mathbb{E}[g(Z)] = \frac{1}{\sqrt{\pi}} \int_{\R} e^{-x^2} g(\sqrt{2} x) \, dx$.
It is used in statistics, in quantum mechanics with harmonic oscillator states, and in uncertainty quantification.
For functions that are not smooth or that grow fast, the nodes spread out and accuracy can be poor, and a change of variable may help.
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<document source="https://tamnd.github.io/math2/node/89VBQH8HNS.md" tag="89VBQH8HNS" kind="concept">
# Hereditary ring

## Summary

A left hereditary ring is a [[ring|0B1261B5ED]] in which every left ideal is a [[projective module|BFFA03GE3V]].
Equivalently, every submodule of a projective left module is projective, so the left [[global dimension|GWA0T1WKCN]] is at most $1$.
Principal ideal domains, Dedekind domains and path algebras over a field of finite quivers without oriented cycles are hereditary.
These rings are the next simplest case after semisimple rings, where every module is projective.

## Explanations

### L1 human

Projective modules are the well-behaved modules, the ones that look like pieces of free modules.
In a hereditary ring this good behaviour is inherited by submodules: a submodule of a projective module is again projective.
The ring of integers $\Z$ is the standard example.
Every subgroup of a free abelian group is free, so subgroups of projective $\Z$-modules are projective.
The ring $\Z/4\Z$ is not hereditary, because its ideal $2\Z/4\Z$ is not projective.
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<document source="https://tamnd.github.io/math2/node/89ZY5YRETE.md" tag="89ZY5YRETE" kind="concept">
# Tracy–Widom distribution

## Summary

The Tracy–Widom distributions are the limit laws of the largest eigenvalue of large random Hermitian or symmetric matrices, after centring and scaling.
There are three of them, $F_1$, $F_2$ and $F_4$, for the orthogonal, unitary and symplectic Gaussian ensembles.
They are given by a solution of the Painlevé II equation, or by a Fredholm determinant with the Airy kernel.
They also appear far from matrices, for example for the longest increasing subsequence of a random permutation and for random growth models.

## Explanations

### L1 human

The eigenvalues of a large random matrix fill an interval, by the [[semicircle law|3QTT95DMFH]].
The largest one sits near the right end of that interval, with small random fluctuations.
After zooming in by the right factor, these fluctuations follow a fixed law, the Tracy–Widom law.
It is not the normal law: it is asymmetric, with a left tail and a right tail that decay at different speeds.
</document>

<document source="https://tamnd.github.io/math2/node/8A00Y84SW7.md" tag="8A00Y84SW7" kind="counterexample">
# Amitsur's noncrossed product

## Counterexample

Let $k = \Q$ and let $n \ge 2$.
Let $X$ and $Y$ be two $n \times n$ matrices whose entries are independent commuting variables over $k$.
The $k$-algebra generated by $X$ and $Y$ is the [[generic matrix algebra|GJGEVMZ828]]; it has no zero divisors.
Its ring of central quotients $\mathrm{UD}(k, n)$ is the *generic division algebra*.
It is a [[division ring|0EA5BTA0CX]] and a [[central simple algebra|5HRKJBAY2V]] of [[degree|NNXSP5KKAC]] $n$ over its center.
Amitsur proved that $\mathrm{UD}(k, n)$ is not a [[crossed product algebra|G21MZQCME9]] when $n$ is divisible by $8$ or by $p^2$ for an odd prime $p$.
That is, for such $n$ it contains no maximal subfield that is a Galois extension of its center.

## Remarks

The idea of the proof is to specialize.
A crossed product structure on $\mathrm{UD}(k, n)$ with group $G$ would pass to every division algebra of degree $n$ over suitable fields of iterated Laurent series.
Amitsur exhibited two such division algebras that are crossed products only for groups of incompatible types, so no single group $G$ works.
For small degrees the opposite holds: every central division algebra of degree $2$, $3$, $4$ or $6$ is a crossed product.
Over local and global fields every central simple algebra is cyclic, so noncrossed products can only appear over more complicated fields.

## Explanations

### L1 human

For a long time it was open whether every finite-dimensional central division algebra comes from the crossed product recipe.
Amitsur found division algebras where this fails.
They are built from two "generic" square matrices of variables, so that they carry no special structure that a crossed product would need.
</document>

<document source="https://tamnd.github.io/math2/node/8A4GGX1MEY.md" tag="8A4GGX1MEY" kind="statement">
# Fekete's lemma

## Statement

Let $(a_n)_{n \ge 1}$ be a [[real sequence|0CTV783HYT]] that is subadditive, that is, $a_{m+n} \le a_m + a_n$ for all integers $m, n \ge 1$.
Then the [[limit|05M5SWSZZD]] of $a_n / n$ as $n \to \infty$ exists in $[-\infty, \infty)$, and
$$\lim_{n \to \infty} \frac{a_n}{n} = \inf_{n \ge 1} \frac{a_n}{n},$$
where $\inf$ is the [[infimum|04KFTA34C5]].

## Remarks

Fix $m \ge 1$ and write each $n$ as $n = qm + r$ with integers $q \ge 0$ and $1 \le r \le m$.
Subadditivity gives $a_n \le q a_m + a_r$, so $a_n / n \le (q m / n)(a_m / m) + a_r / n$.
As $n \to \infty$ the last term tends to $0$ and $qm/n$ tends to $1$, so $\limsup a_n / n \le a_m / m$.
Taking the infimum over $m$ gives $\limsup a_n/n \le \inf_m a_m/m \le \liminf a_n/n$, which proves the claim.
The limit can be $-\infty$: the sequence $a_n = -n^2$ is subadditive, since $-(m+n)^2 \le -m^2 - n^2$, and $a_n / n = -n$.
A typical use: if $c_n > 0$ and $c_{m+n} \le c_m c_n$ for all $m, n$, then $a_n = \log c_n$ is subadditive, so $c_n^{1/n}$ converges to $\inf_n c_n^{1/n}$.
For a bounded linear operator $A$ on a Banach space with no power equal to $0$, this applies to $c_n = \norm{A^n}$.
The same lemma is often called the subadditivity lemma.

## Explanations

### L1 human

A sequence is subadditive when the value at $m + n$ is at most the sum of the values at $m$ and at $n$.
Fekete's lemma says that the average growth $a_n / n$ then settles down.
Its limit is the smallest of all the averages $a_n / n$.

### L2 human

The lemma turns a crude inequality into the existence of a growth rate, without any formula for the terms.
It is used to define constants such as the connective constant of self-avoiding walks, entropies, and the spectral radius $\lim_n \norm{A^n}^{1/n}$.
Kingman's subadditive ergodic theorem is a random version of the same idea.
For a superadditive sequence, with $a_{m+n} \ge a_m + a_n$, apply the lemma to $-a_n$: then $a_n / n$ tends to $\sup_n a_n / n$.
</document>

<document source="https://tamnd.github.io/math2/node/8A7YXMW727.md" tag="8A7YXMW727" kind="concept">
# Dual object

## Summary

A dual of an object $A$ in a [[monoidal category|M3MSQ3NKAE]] is an object $A^*$ with an evaluation $A^* \otimes A \to I$ and a coevaluation $I \to A \otimes A^*$.
These two maps must satisfy the zigzag identities.
It abstracts the relation between a finite-dimensional vector space and its [[dual space|07JBSZE0V4]].
A dual, when it exists, is unique up to unique isomorphism.
Duals give traces and dimensions in monoidal categories and correspond to bending strings in string diagrams.

## Explanations

### L1 human

A finite-dimensional vector space $V$ has a dual $V^*$ of linear functionals.
One can pair a functional with a vector to get a number, and one can produce a canonical tensor $\sum_i e_i \otimes e^i$ out of nothing.
Doing one after the other, in the right order, gives back the identity: these are the zigzag identities.
A dual object is an object with these two maps and these two identities, in any monoidal category.
In pictures, the two maps are a cap and a cup, and the identities say that a zigzag in a string can be pulled straight.
</document>

<document source="https://tamnd.github.io/math2/node/8A8FBH4C7Y.md" tag="8A8FBH4C7Y" kind="concept">
# Regularity of minimizers

## Summary

Regularity of minimisers is the study of how smooth the minimisers of a variational problem are.
The direct method produces minimisers only in a Sobolev space, so it is a separate question whether they are continuous, differentiable, smooth or analytic.
It is the subject of [[Hilbert's nineteenth problem|QPMM2ZPWBW]], solved for scalar problems by the [[De Giorgi–Nash–Moser theorem|32C3D8FG9V]].
For vector-valued problems minimisers can be singular, and the theory proves partial regularity: smoothness outside a small closed set.
It is a central part of [[regularity theory|N383CX7WKS]] for elliptic equations.

## Explanations

### L1 human

Existence proofs in the calculus of variations give a minimiser that might have kinks, jumps in its derivative, or worse.
Regularity theory asks whether these defects really occur.
For the Dirichlet energy $\int \abs{\nabla u}^2 \, dx$ the minimisers are harmonic functions, which are smooth, even analytic, inside the domain.
For many energies of maps with several components, singular minimisers do exist; then one shows that the bad points form a small set.
</document>

<document source="https://tamnd.github.io/math2/node/8AD5RQTTK9.md" tag="8AD5RQTTK9" kind="definition">
# Lusin area integral

## Definition

Let $n \ge 1$ and let $\R^{n+1}_+ = \set{(y, t) : y \in \R^n, \ t > 0}$ be the upper half-space.
Let $f$ be in the [[Lp space|SASP14MA1Z]] $L^p(\R^n)$ for some $1 \le p \le \infty$.
Let $u(y, t) = (P_t * f)(y)$ be its Poisson integral, where $P_t$ is the [[Poisson kernel|TCYSWCTYYM]] of the half-space, so that $u$ is [[harmonic|0DEB0JMN0W]] on $\R^{n+1}_+$.
Let $\nabla u = (\partial_{y_1} u, \dots, \partial_{y_n} u, \partial_t u)$ be its full gradient.
For $x \in \R^n$ let $\Gamma(x) = \set{(y, t) \in \R^{n+1}_+ : \abs{y - x} < t}$ be the cone with vertex $x$.
The *Lusin area integral* of $f$ is the function $S(f) \colon \R^n \to [0, \infty]$ given by
$$S(f)(x) = \left( \iint_{\Gamma(x)} \abs{\nabla u(y, t)}^2 \, t^{1 - n} \, dy \, dt \right)^{1/2}.$$

## Remarks

The power $t^{1 - n}$ makes $S(f)$ invariant under dilations: $S(f(\delta \cdot))(x) = S(f)(\delta x)$.
For $1 < p < \infty$ one has $c \norm{f}_p \le \norm{S(f)}_p \le C \norm{f}_p$ with constants depending only on $n$ and $p$.
Replacing the cone by the vertical half-line over $x$, with the weight $t$, gives the Littlewood–Paley $g$-function.

Example: the $L^2$ identity.
Integrating over $x$ first, $\int_{\R^n} S(f)(x)^2 \, dx = \omega_n \iint_{\R^{n+1}_+} t \abs{\nabla u(y, t)}^2 \, dy \, dt$, where $\omega_n$ is the volume of the unit ball of $\R^n$.
With $\hat f(\xi) = \int f(x) e^{-2\pi i x \cdot \xi} dx$, one has $\hat u(\xi, t) = e^{-2\pi t \abs{\xi}} \hat f(\xi)$, so $\int \abs{\nabla u(y, t)}^2 dy = 8\pi^2 \int \abs{\xi}^2 e^{-4\pi t \abs{\xi}} \abs{\hat f(\xi)}^2 \, d\xi$.
Since $\int_0^\infty t e^{-4\pi t \abs{\xi}} \, dt = (4\pi\abs{\xi})^{-2}$, this gives $\norm{S(f)}_2^2 = \frac{\omega_n}{2} \norm{f}_2^2$.

Non-example: $f = 1$ is bounded but in no $L^p$ with $p < \infty$.
Its Poisson integral is $u = 1$, so $\nabla u = 0$ and $S(f) = 0$ although $f \neq 0$.
So the lower bound $c \norm{f}_p \le \norm{S(f)}_p$ cannot hold for $p = \infty$.
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<document source="https://tamnd.github.io/math2/node/8AGSK9S92W.md" tag="8AGSK9S92W" kind="definition">
# Harmonic map

## Definition

Let $(M, g)$ and $(N, h)$ be [[Riemannian manifolds|7CBT4SM83Y]] of dimensions $m$ and $n$, and let $u \colon M \to N$ be a smooth map.
Use local coordinates $x^i$ on $M$ and $y^\alpha$ on $N$.
The *tension field* of $u$ is
$$\tau(u)^\gamma = \Delta_g u^\gamma + \sum_{i,j,\alpha,\beta} g^{ij} \, \Gamma^\gamma_{\alpha\beta}(u) \, \partial_i u^\alpha \, \partial_j u^\beta, \qquad \gamma = 1, \dots, n,$$
where $\Delta_g$ is the [[Laplace–Beltrami operator|F4NWZNTRMD]] of $g$ and $\Gamma^\gamma_{\alpha\beta}$ are the [[Christoffel symbols|FZBA1HA9ZM]] of $h$.
The map $u$ is *harmonic* if $\tau(u) = 0$ on $M$.

## Remarks

The tension field is a section of $u^* TN$ and does not depend on the coordinates; it is the trace with respect to $g$ of the covariant derivative of $du$.
Let $M$ be compact.
Then $u$ is harmonic if and only if it is a critical point of the energy
$$E(u) = \frac{1}{2} \int_M \abs{du}^2 \, dV_g, \qquad \abs{du}^2 = \sum g^{ij} h_{\alpha\beta}(u) \, \partial_i u^\alpha \, \partial_j u^\beta,$$
which generalizes the [[Dirichlet energy|YG3W8CBB3E]]; indeed $-\tau(u)$ is the gradient of $E$.

Example: let $N = \R$; then $\Gamma = 0$ and $u$ is harmonic if and only if $\Delta_g u = 0$.
Let $M$ be an interval with its standard metric; then $\tau(u) = 0$ is the geodesic equation, so harmonic maps from an interval are constant-speed geodesics.

Non-example: let $M = N = \R / 2\pi\Z$ with the standard metric and $u(\theta) = \theta + \frac{1}{2}\sin\theta$.
Here $\Gamma = 0$ and $\tau(u) = u''(\theta) = -\frac{1}{2}\sin\theta \ne 0$, so $u$ is not harmonic, although it is a diffeomorphism.
The identity map $\theta \mapsto \theta$ is harmonic.
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<document source="https://tamnd.github.io/math2/node/8AJEMCZ29B.md" tag="8AJEMCZ29B" kind="definition">
# Holomorphic map between Riemann surfaces

## Definition

Let $X$ and $Y$ be [[Riemann surfaces|5K0N8QMV6Q]].
A map $f \colon X \to Y$ is *holomorphic* if:

1. $f$ is [[continuous|0P27W9RW0G]], and
2. for every chart $\varphi \colon U \to \C$ of $X$ and every chart $\psi \colon V \to \C$ of $Y$, the map
$$\psi \circ f \circ \varphi^{-1} \colon \varphi(U \cap f^{-1}(V)) \to \C$$
is [[holomorphic|0DG0Z72CQD]].

A holomorphic map is a *biholomorphism* if it is bijective; its inverse is then also holomorphic.

## Remarks

Continuity makes $U \cap f^{-1}(V)$ open, so clause 2 makes sense.
It suffices to check clause 2 for charts from one atlas of $X$ and one atlas of $Y$.
When $X$ and $Y$ are open subsets of $\C$ with the identity chart, this is the usual notion of a holomorphic function.

Example: $f \colon \hat\C \to \hat\C$ with $f(z) = z^2$ for $z \in \C$ and $f(\infty) = \infty$.
In the chart $z$ near finite points it is $z \mapsto z^2$.
Near $\infty$, in the chart $w = 1/z$ on both sides, it reads $w \mapsto 1/(1/w)^2 = w^2$, which is holomorphic.

Non-example: complex conjugation $z \mapsto \bar z$ on $\C$ is continuous, but in the identity chart it is [[not holomorphic|0Z8ATBPM5W]], so clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/8APFT1RPM4.md" tag="8APFT1RPM4" kind="statement">
# McDiarmid's inequality

## Statement

Let $X_1, \dots, X_n$ be [[independent|0GQ938GDE0]] random variables, with $X_i$ taking values in a measurable space $\mathcal{X}_i$.
Let $f \colon \mathcal{X}_1 \times \dots \times \mathcal{X}_n \to \R$ be measurable, and let $c_1, \dots, c_n \ge 0$ be constants with the bounded differences property: $\abs{f(x) - f(x')} \le c_i$ whenever $x$ and $x'$ differ only in the $i$-th coordinate.
Assume $c_1^2 + \dots + c_n^2 > 0$, and let $Y = f(X_1, \dots, X_n)$.
Then for every $t > 0$,
$$P(Y - E[Y] \ge t) \le \exp\left(-\frac{2 t^2}{\sum_{i=1}^n c_i^2}\right),$$
and the same bound holds for $P(Y - E[Y] \le -t)$.

## Remarks

Proof idea: let $Z_k = E[Y \mid X_1, \dots, X_k]$, the [[Doob martingale|Z57CG6W9S6]] of $Y$.
By independence and the bounded differences property, given $X_1, \dots, X_{k-1}$ the increment $Z_k - Z_{k-1}$ lies in an interval of length $c_k$.
Hoeffding's lemma then bounds its conditional moment-generating function by $e^{\lambda^2 c_k^2 / 8}$, and the argument of [[Azuma's inequality|BDA5TTB1F0]] finishes the proof.
Example: throw $m$ balls independently into $N$ bins and let $Y$ be the number of empty bins. Moving one ball changes $Y$ by at most $1$, so $P(\abs{Y - E[Y]} \ge t) \le 2 e^{-2t^2 / m}$.
With $f(x) = x_1 + \dots + x_n$ and $X_i \in [a_i, b_i]$, one gets [[Hoeffding's inequality|0WPDNXGSSN]].
Independence is needed: if $X_1 = \dots = X_n$ is a fair $0/1$ coin and $f$ is the sum, then $c_i = 1$ but $Y \in \set{0, n}$, so $P(Y - E[Y] \ge n/2) = \frac{1}{2}$, far above $e^{-n/2}$.

## Explanations

### L1 human

If a quantity depends on many independent random inputs, and no single input can change it much, then the quantity is almost constant.
McDiarmid's inequality makes this precise: the chance of a deviation $t$ decays like $e^{-2t^2 / \sum_i c_i^2}$.

### L2 human

This is the method of bounded differences.
It is often the quickest way to prove concentration in random graphs, randomized algorithms and learning theory, for instance for generalization bounds through Rademacher complexity.
It only gives concentration around the mean; estimating the mean itself needs other arguments.
Sharper results for convex Lipschitz functions follow from [[Talagrand's concentration inequality|GGEAGW4RRV]].
</document>

<document source="https://tamnd.github.io/math2/node/8APGCB1CKH.md" tag="8APGCB1CKH" kind="definition">
# Specialization preorder

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
For $y \in X$, write $\overline{\set{y}}$ for the [[closure|0QEAJB91GZ]] of $\set{y}$.
The *specialization preorder* on $X$ is the relation $\le$ defined by
$$x \le y \iff x \in \overline{\set{y}}, \qquad x, y \in X.$$

## Remarks

The relation is a [[preorder|0VABYWJWH9]]: $x \in \overline{\set{x}}$, and if $x \in \overline{\set{y}}$ and $y \in \overline{\set{z}}$ then $\overline{\set{y}} \subseteq \overline{\set{z}}$, so $x \in \overline{\set{z}}$.
Equivalently, $x \le y$ if and only if every open set that contains $x$ also contains $y$.

The preorder is antisymmetric, hence a partial order, if and only if $X$ is a [[T0 space|07SG12TQH3]].
It is the equality relation if and only if every point of $X$ is closed, that is, $X$ is T1.
Every continuous map is monotone for the specialization preorders.

Example: in the spectrum of a commutative ring with the Zariski topology, the closure of a prime $\mathfrak q$ is the set of primes that contain $\mathfrak q$.
So $\mathfrak p \le \mathfrak q$ holds exactly when $\mathfrak q \subseteq \mathfrak p$.

Non-example: on $X = \set{a, b}$ with the indiscrete topology, $a \le b$ and $b \le a$ but $a \ne b$.
So the specialization preorder is not a partial order, and indeed $X$ is not T0.

Some authors use the opposite convention, with $x \le y$ when $y \in \overline{\set{x}}$.
</document>

<document source="https://tamnd.github.io/math2/node/8APSJK9GSK.md" tag="8APSJK9GSK" kind="definition">
# Fokker–Planck equation

## Definition

Let $n, d \ge 1$.
Let $\mu \colon \R^n \times [0, \infty) \to \R^n$ and $\sigma \colon \R^n \times [0, \infty) \to \R^{n \times d}$ be smooth.
Let $D = \frac{1}{2}\sigma \sigma^T$, a symmetric positive semidefinite $n \times n$ matrix at each point.
The *Fokker–Planck equation* with drift $\mu$ and diffusion matrix $D$ is the equation for $p(x, t)$:
$$\partial_t p = -\sum_{i=1}^{n} \partial_{x_i}\big(\mu_i\, p\big) + \sum_{i,j=1}^{n} \partial_{x_i} \partial_{x_j}\big(D_{ij}\, p\big),$$
where $\partial$ denotes [[partial derivatives|09EGC6F8PT]].
It is associated with the [[stochastic differential equation|B219G9SHJA]] $dX_t = \mu(X_t, t)\, dt + \sigma(X_t, t)\, dW_t$, where $W$ is a $d$-dimensional [[Wiener process|22735FGJV6]].

## Remarks

Under suitable regularity of $\mu$ and $\sigma$, if $X_t$ has a smooth [[probability density|0ZY9DX8FBF]] $p(\cdot, t)$, then $p$ solves this equation.
Example: for $n = d = 1$, $\mu = 0$ and $\sigma = 1$, the equation is $p_t = \frac{1}{2} p_{xx}$.
Its solution $p(x, t) = (2\pi t)^{-1/2} e^{-x^2/(2t)}$ is the density of $W_t$; one checks $p_t = \frac{1}{2}p_{xx}$ by direct differentiation.
Non-example: $p_t = -\frac{1}{2} p_{xx}$ is not a Fokker–Planck equation, since it would need $D = -\frac{1}{2}$, which is not positive semidefinite.
The equation is also called the forward Kolmogorov equation.
In one dimension it reads $p_t = -(\mu p)_x + (D p)_{xx}$.
</document>

<document source="https://tamnd.github.io/math2/node/8AX4XREEPX.md" tag="8AX4XREEPX" kind="statement">
# Schlömilch's generalization of the condensation test

## Statement

Let $(a_n)_{n \ge 1}$ be a [[decreasing|0KN22AAW5R]] real sequence with $a_n \ge 0$ for all $n$.
Let $u_0 < u_1 < u_2 < \cdots$ be positive integers, and put $\Delta u_k = u_{k+1} - u_k$.
Assume that there is a constant $M$ with $\Delta u_{k} \le M \, \Delta u_{k-1}$ for all $k \ge 1$.
Then $\sum_{n=1}^{\infty} a_n$ [[converges|0CVVKY7GY2]] if and only if $\sum_{k=0}^{\infty} \Delta u_k \, a_{u_k}$ converges.

## Remarks

The proof cuts the series into blocks $u_k \le n < u_{k+1}$ of $\Delta u_k$ terms each.
Since $(a_n)$ decreases, the block sum lies between $\Delta u_k \, a_{u_{k+1}}$ and $\Delta u_k \, a_{u_k}$.
The upper bound shows that convergence of the condensed series gives convergence of $\sum a_n$.
The lower bound and $\Delta u_{k+1} \le M \Delta u_k$ show that $\Delta u_{k+1} a_{u_{k+1}}$ is at most $M$ times the block sum, which gives the converse.
With $u_k = 2^k$ this is the [[Cauchy condensation test|0B96KMDDD1]].
With $u_k = k^2$ for $k \ge 1$ the condensed series is $\sum (2k+1) a_{k^2}$, and the condition holds since $(2k+3)/(2k+1) \le 3$.
For $a_n = 1/n^p$ with $p > 0$ this gives $\sum (2k+1)/k^{2p}$, which converges exactly when $2p - 1 > 1$, that is $p > 1$.

## Explanations

### L1 human

For a decreasing series, you may keep only a sparse set of terms, provided you weight each kept term by the length of the gap after it.
The gaps may grow, but not too fast from one gap to the next.
The sparse weighted series and the original series then converge or diverge together.

### L2 human

The Cauchy condensation test uses the gaps $2^k$.
Schlömilch's version lets you choose gaps adapted to the series, such as the squares, which helps when $a_{k^2}$ has a simpler form than $a_{2^k}$.
The hypothesis that consecutive gaps have bounded ratio is what makes the lower bound work.
The proof is the same comparison of block sums as in the Cauchy case.
</document>

<document source="https://tamnd.github.io/math2/node/8B0JM2YDE2.md" tag="8B0JM2YDE2" kind="example">
# Metaplectic group

## Example

Let $n \ge 1$ and let $\mathrm{Sp}(2n, \R)$ be the real [[symplectic group|X12FS2JAP0]].
It is a connected Lie group, and $U(n)$ is a maximal compact subgroup of it, so its [[fundamental group|S1XK3HXHFX]] is isomorphic to $\pi_1(U(n)) \cong \Z$.
The group $\Z$ has exactly one subgroup of index $2$, so $\mathrm{Sp}(2n, \R)$ has exactly one connected two-sheeted [[covering|3XZRRE165N]], up to isomorphism.
That covering space carries a unique Lie group structure for which the covering map is a homomorphism.

The *metaplectic group* $\mathrm{Mp}(2n, \R)$ is this connected double cover of $\mathrm{Sp}(2n, \R)$.
So there is an exact sequence
$$1 \to \set{\pm 1} \to \mathrm{Mp}(2n, \R) \to \mathrm{Sp}(2n, \R) \to 1.$$
For $n = 1$, $\mathrm{Sp}(2, \R) =$ [[SL(2,R)|Q0GBCB9RA4]], and $\mathrm{Mp}(2, \R)$ is the connected double cover of $SL(2, \R)$.

## Remarks

$\mathrm{Mp}(2n, \R)$ is not a matrix group: it has no faithful finite-dimensional representation.
It does have a faithful unitary representation on the infinite-dimensional Hilbert space $L^2(\R^n)$, the oscillator representation, also called the Segal–Shale–Weil representation.
This representation does not come from a representation of $\mathrm{Sp}(2n, \R)$; it is defined only up to sign on $\mathrm{Sp}(2n, \R)$.
The group $\mathrm{Mp}(2, \R)$ is the setting for modular forms of half-integral weight, such as the Jacobi theta function.

## Explanations

### L1 human

The metaplectic group is a twofold cover of the real symplectic group, just as the spin group is a twofold cover of the rotation group.
Each symplectic matrix corresponds to two elements of the metaplectic group, which differ by a sign.
It appears because some natural constructions, such as the quantization of linear symplectic maps in quantum mechanics, are only defined up to a sign on the symplectic group.
Passing to the metaplectic group removes this sign ambiguity.
</document>

<document source="https://tamnd.github.io/math2/node/8B57B7EGK4.md" tag="8B57B7EGK4" kind="concept">
# Separably closed field

## Summary

A separably closed field is a field with no proper finite [[separable extension|0WCX7NB4N7]].
Equivalently, every [[separable polynomial|0RZ89QWTSJ]] over it of positive degree has a root in it.
Every [[algebraically closed field|0PF6TM8TWT]] is separably closed, and a field is algebraically closed if and only if it is separably closed and [[perfect|VWBVKT794W]].
Separably closed fields have trivial absolute Galois group, and the [[separable closure|JJ91A1V8NQ]] of a field is the separably closed field built from it.

## Explanations

### L1 human

An algebraically closed field contains a root of every polynomial.
A separably closed field only promises a root for polynomials without repeated roots.
In characteristic $0$ the two notions agree.
In characteristic $p$ they differ: the separable closure of $\F_p(t)$ is separably closed, but it has no root of $x^p - t$.
So separably closed fields are as closed as Galois theory can detect.
</document>

<document source="https://tamnd.github.io/math2/node/8BBSFVN1CS.md" tag="8BBSFVN1CS" kind="definition">
# Cauchy-type integral

## Definition

Let $\Gamma$ be an oriented [[rectifiable curve|0HATW2VWAZ]] in $\C$, closed or not, and let $\varphi \colon \Gamma \to \C$ be continuous.
The *Cauchy-type integral* of $\varphi$ over $\Gamma$ is the function
$$\Phi(z) = \frac{1}{2\pi i} \int_\Gamma \frac{\varphi(\tau)}{\tau - z} \, d\tau, \quad z \in \C \setminus \Gamma,$$
where the integral is a [[contour integral|06ZGPYTWK8]].
The function $\varphi$ is its *density*.

## Remarks

The function $\Phi$ is [[holomorphic|0NAHXR1FRH]] on $\C \setminus \Gamma$, and $\Phi(z) \to 0$ as $z \to \infty$.
Example: let $\Gamma$ be the unit circle, oriented counterclockwise, and $\varphi(\tau) = \tau^{-1}$.
For $\abs{z} < 1$, the integrand $\frac{1}{\tau(\tau - z)} = \frac{1}{z} \left( \frac{1}{\tau - z} - \frac{1}{\tau} \right)$ gives $\Phi(z) = \frac{1}{z}(1 - 1) = 0$; for $\abs{z} > 1$ it gives $\Phi(z) = \frac{1}{z}(0 - 1) = -\frac{1}{z}$.
So the boundary values from inside and outside are $0$ and $-\tau^{-1}$, and their difference is $\varphi(\tau)$.
Non-example: for $z \in \Gamma$ the formula is not a convergent integral in general.
The reason is that $\frac{1}{\tau - z}$ is not integrable along $\Gamma$ near $\tau = z$.
On $\Gamma$ one uses instead the [[Cauchy principal value|0SYR10HD4R]], as in [[singular integral equations|E30EA6ETZ4]].
Let $\Gamma$ be a closed curve bounding a domain $D$, and let $\varphi$ be the boundary value of a function $F$ holomorphic on $D$ and continuous on its closure.
Then $\Phi = F$ on $D$ and $\Phi = 0$ outside $\overline{D}$, by [[Cauchy's integral formula|0DG10TVYPN]] and Cauchy's theorem.
The boundary values of $\Phi$ for Hölder continuous $\varphi$ are given by the [[Sokhotski–Plemelj formulae|DRT1E6WY6K]].
</document>

<document source="https://tamnd.github.io/math2/node/8BEMNJW0B9.md" tag="8BEMNJW0B9" kind="concept">
# Wishart distribution

## Summary

The Wishart distribution is a probability distribution on symmetric positive semidefinite matrices.
It is the distribution of the scatter matrix $\sum_i X_i X_i^T$ of independent centred multivariate normal vectors.
It is the matrix version of the chi-squared distribution, and it is the sampling distribution of the sample covariance matrix of normal data.
In Bayesian statistics it is the conjugate prior of the precision matrix of a multivariate normal distribution.

## Explanations

### L1 human

Draw $n$ random points in $p$-dimensional space from a normal distribution centred at the origin.
Add up the matrices $X_i X_i^T$; the result records how the cloud of points spreads in every direction.
This random matrix has the Wishart distribution.
In dimension $p = 1$ the matrix is the number $\sum_i X_i^2$, so the Wishart distribution is a scaled chi-squared distribution.
</document>

<document source="https://tamnd.github.io/math2/node/8BGJ6J9PYF.md" tag="8BGJ6J9PYF" kind="concept">
# Bar resolution

## Summary

The bar resolution of a group $G$ is an explicit free resolution of the trivial module $\Z$ over the [[group ring|MNP2HBWB5Q]] $\Z[G]$.
Its basis elements in degree $n$ are the symbols $[g_1 | \cdots | g_n]$, one for each list of $n$ elements of $G$.
It turns [[group cohomology|YXJXA0Q52R]] into a concrete computation with functions of $n$ group elements, which gives the classical cocycle descriptions of $H^1$ and $H^2$.
The same construction exists for augmented algebras, where it computes Hochschild homology and Tor.

## Explanations

### L1 human

To compute group cohomology one needs a resolution, and the bar resolution is one that works for every group without any choices.
Its pieces are free modules with a basis vector for every list of group elements.
The boundary map multiplies neighbouring entries of the list, with alternating signs, and drops the first or last entry at the ends.
It is large and wasteful for computing, but its uniformity makes it the right tool for proofs and for writing down cocycles.
</document>

<document source="https://tamnd.github.io/math2/node/8BJ5M76FSX.md" tag="8BJ5M76FSX" kind="concept">
# Epigraph

## Summary

The epigraph of a function is the set of points lying on or above its graph.
It turns statements about functions into statements about sets.
A function is [[convex|07Q7CNM5WF]] exactly when its epigraph is a convex set, and lower [[semicontinuous|RN4G8SGP6P]] exactly when its epigraph is closed.
This dictionary is the starting point of convex analysis and variational analysis.
The [[definition|5Q9BNSZ5XZ]] gives the precise set.

## Explanations

### L1 human

Draw the graph of a function and shade everything above it: the shaded region is the epigraph.
For $f(x) = x^2$ it is the inside of the upward parabola, together with the parabola itself.
A bowl-shaped function has a shaded region with no dents, which is what convexity of the set means.
A jump in the graph shows up as a missing or present edge of the region, which is how semicontinuity is seen.
</document>

<document source="https://tamnd.github.io/math2/node/8BMM456H7G.md" tag="8BMM456H7G" kind="concept">
# Steiner symmetrization

## Summary

Steiner symmetrization is an operation that turns a [[convex body|437FSG3419]] into a convex body that is symmetric about a chosen hyperplane.
It slides each chord of the body orthogonal to the hyperplane along its line until the hyperplane cuts the chord in its midpoint.
The operation keeps the volume, and it does not increase the surface area or the diameter.
Repeating it in well chosen directions makes a body converge to a ball, which gives proofs of the [[isoperimetric inequality|KBX8RBZ626]] and of related extremal results.

## Explanations

### L1 human

Think of a convex body as a bundle of straight sticks, all parallel to one fixed direction.
Slide every stick along its own line until its midpoint lies on one fixed plane.
The sticks keep their lengths, so the volume does not change.
The new body is a mirror image of itself across the plane, and it is "rounder" than before.
For example, the right triangle with vertices $(0, 0)$, $(2, 0)$, $(0, 2)$, symmetrized about the $x$-axis, becomes the triangle with vertices $(0, 1)$, $(0, -1)$, $(2, 0)$, which has the same area $2$.
</document>

<document source="https://tamnd.github.io/math2/node/8BMQ90EJPR.md" tag="8BMQ90EJPR" kind="statement">
# Leray spectral sequence

## Statement

Let $f \colon X \to Y$ be a continuous map of topological spaces and let $\mathcal{F}$ be a [[sheaf|TB6TM6R4MD]] of abelian groups on $X$.
For $q \ge 0$, let $R^q f_* \mathcal{F}$ be the $q$-th higher direct image, that is, the $q$-th right [[derived functor|NPMC3K6J2F]] of the direct image $f_*$.
It is the sheaf on $Y$ associated with the presheaf $V \mapsto H^q(f^{-1}(V); \mathcal{F})$.
Then there is a first-quadrant [[spectral sequence|GG8AYV847V]] with
$$E_2^{p,q} = H^p(Y; R^q f_* \mathcal{F}) \Longrightarrow H^{p+q}(X; \mathcal{F}),$$
where $H^*$ denotes [[sheaf cohomology|KB76923A1M]], and the differentials are $d_r \colon E_r^{p,q} \to E_r^{p+r, q-r+1}$.

## Remarks

Since $\Gamma(Y, f_* \mathcal{F}) = \Gamma(X, \mathcal{F})$ and $f_*$ sends injective sheaves to flasque sheaves, which are acyclic for $\Gamma(Y, -)$, this is the [[Grothendieck spectral sequence|ZV7VGE2SQ2]] of the composite functor $\Gamma(Y, -) \circ f_*$.

Example: if $R^q f_* \mathcal{F} = 0$ for all $q > 0$, the spectral sequence collapses and $H^p(X; \mathcal{F}) \cong H^p(Y; f_* \mathcal{F})$ for all $p$.
This applies when $f$ is the inclusion of a closed subspace, because then $f_*$ is exact.

Let $f$ be a fibre bundle with fibre $F$ over a nice base, and let $\mathcal{F}$ be a constant sheaf.
Then $R^q f_* \mathcal{F}$ is locally constant with stalks $H^q(F)$, and one recovers a form of the Serre spectral sequence.

## Explanations

### L1 human

The Leray spectral sequence computes the cohomology of a space $X$ by looking at it through a map $f$ to another space $Y$.
Over each small open set of $Y$, one first computes the cohomology of the part of $X$ above it.
These local answers form sheaves on $Y$, and the cohomology of $Y$ with values in them leads to the cohomology of $X$.

### L2 human

Leray introduced sheaves and spectral sequences together for this purpose.
In algebraic geometry the same spectral sequence, for a morphism of schemes and étale or coherent cohomology, is a basic tool.
It is the prototype of the Grothendieck spectral sequence for a composite of derived functors.
</document>

<document source="https://tamnd.github.io/math2/node/8BPMHA5MT4.md" tag="8BPMHA5MT4" kind="statement">
# Thomson's principle

## Statement

Let $(G, c)$ be a finite [[electrical network|T7KRE7J3B4]] with vertex set $V$ and resistances $r = 1/c$, and let $a, b \in V$ with $a \ne b$.
A *unit flow* from $a$ to $b$ is a function $\theta$ on ordered pairs of neighbours with $\theta(y, x) = -\theta(x, y)$ such that:
1. $\sum_y \theta(x, y) = 0$ for every $x \notin \set{a, b}$;
2. $\sum_y \theta(a, y) = 1$.

Its energy is $\mathcal{E}(\theta) = \sum_{\set{x, y} \in E} r(x, y)\, \theta(x, y)^2$.
Then
$$R_{\mathrm{eff}}(a, b) = \min \set{\mathcal{E}(\theta) : \theta \text{ a unit flow from } a \text{ to } b},$$
where $R_{\mathrm{eff}}$ is the [[effective resistance|XAE5YJ3H68]].
The minimum is attained exactly at the unit current flow.
This is the current of strength $1$ from $a$ to $b$ given by a voltage that is harmonic off $\set{a, b}$.

## Remarks

Let $i$ be the unit current flow, with voltage $v$, so that $r(x, y)\, i(x, y) = v(x) - v(y)$.
For any unit flow $\theta$, the difference $\theta - i$ has net flow $0$ at every vertex.
Summing by parts gives $\sum_{\set{x, y} \in E} r(x, y)\, i(x, y)\big(\theta - i\big)(x, y) = \sum_x v(x) \sum_y (\theta - i)(x, y) = 0$.
Hence $\mathcal{E}(\theta) = \mathcal{E}(i) + \mathcal{E}(\theta - i)$, which is at least $\mathcal{E}(i)$ with equality only for $\theta = i$.
Finally $\mathcal{E}(i) = \sum_x v(x) \sum_y i(x, y) = v(a) - v(b) = R_{\mathrm{eff}}(a, b)$.
Example: let the network consist of two paths from $a$ to $b$ that share only their endpoints, with total resistances $r_1$ and $r_2$.
A unit flow carries a constant amount $t$ along the first path and $1 - t$ along the second, so its energy is $r_1 t^2 + r_2 (1 - t)^2$.
Minimizing over $t$ gives $r_1 r_2/(r_1 + r_2)$, the parallel law.
The dual statement, Dirichlet's principle, says that $1/R_{\mathrm{eff}}(a, b)$ is the minimum of $\sum_{\set{x, y} \in E} c(x, y)\big(f(x) - f(y)\big)^2$ over all $f$ with $f(a) = 1$ and $f(b) = 0$.

## Explanations

### L1 human

Send one unit of current through a network from $a$ to $b$.
There are many ways to split the current among the wires, and each way wastes some energy as heat.
The way that nature chooses, given by Ohm's law, wastes the least energy.
That least energy equals the effective resistance between $a$ and $b$.

### L2 human

The principle turns effective resistance into a minimum, so any unit flow gives an upper bound on $R_{\mathrm{eff}}(a, b)$.
Dually, any test function in Dirichlet's principle gives a lower bound.
The Rayleigh monotonicity law follows at once, since raising resistances raises the energy of every flow.
On infinite graphs, a unit flow of finite energy to infinity exists exactly when the random walk is transient.
This gives a proof of the transience of $\Z^3$ by exhibiting such a flow.
</document>

<document source="https://tamnd.github.io/math2/node/8BVNQ3R3YP.md" tag="8BVNQ3R3YP" kind="definition">
# Bar resolution

## Definition

Let $G$ be a [[group|G5S1M9]] and let $\Z[G]$ be its [[group ring|MNP2HBWB5Q]].
For $n \ge 0$, let $B_n$ be the [[free|0DZ9DKAKZR]] left $\Z[G]$-module with basis the symbols $[g_1 | \cdots | g_n]$ with $g_1, \dots, g_n \in G$; the basis of $B_0$ is the single symbol $[\,]$.
For $n \ge 1$, the $\Z[G]$-linear map $d_n \colon B_n \to B_{n-1}$ is given on the basis by
$$d_n [g_1 | \cdots | g_n] = g_1 [g_2 | \cdots | g_n] + \sum_{i=1}^{n-1} (-1)^i [g_1 | \cdots | g_i g_{i+1} | \cdots | g_n] + (-1)^n [g_1 | \cdots | g_{n-1}].$$
The augmentation $\varepsilon \colon B_0 \to \Z$ sends $[\,]$ to $1$, where every element of $G$ acts on $\Z$ as the identity.
The *bar resolution* of $G$ is the complex $\cdots \to B_2 \xrightarrow{d_2} B_1 \xrightarrow{d_1} B_0 \xrightarrow{\varepsilon} \Z \to 0$.

## Remarks

It is a free [[resolution|HK2AA1YSQ2]] of $\Z$ over $\Z[G]$: one checks $d \circ d = 0$, and a contracting homotopy over $\Z$ is given by $s(g_0 [g_1 | \cdots | g_n]) = [g_0 | g_1 | \cdots | g_n]$.
Check of $d \circ d = 0$ in low degree: $d_1 [g] = g[\,] - [\,]$ and $d_2 [g | h] = g[h] - [gh] + [g]$, so $d_1 d_2 [g | h] = (gh - g)[\,] - (gh - 1)[\,] + (g - 1)[\,] = 0$.
Applying $\Hom_{\Z[G]}(-, M)$ identifies $\Hom_{\Z[G]}(B_n, M)$ with the set of functions $G^n \to M$, the inhomogeneous cochains of group cohomology.
The *normalized* bar resolution uses only symbols with all $g_i \ne 1$; it is also a free resolution of $\Z$.
Non-example: for $G = \Z/2\Z = \set{1, t}$, the periodic complex $\cdots \to \Z[G] \xrightarrow{1 - t} \Z[G] \xrightarrow{1 + t} \Z[G] \xrightarrow{1 - t} \Z[G] \to \Z \to 0$ is a free resolution of $\Z$, but it is not the bar resolution, whose term $B_n$ has rank $2^n$.
</document>

<document source="https://tamnd.github.io/math2/node/8BXFDDSSG7.md" tag="8BXFDDSSG7" kind="concept">
# Kantorovich polynomials

## Summary

Kantorovich polynomials are a variant of [[Bernstein polynomials|0JT92VK307]] that use averages of $f$ over small intervals instead of point values.
Because they only need integrals, they are defined for every integrable function on $[0, 1]$.
They converge to $f$ in the $L^p$ norm for $1 \le p < \infty$.
This makes sense for elements of $L^p$, where point values are not defined.

## Explanations

### L1 human

A Bernstein polynomial looks at $f$ only at the points $k/n$.
If $f$ is only known up to its values on sets of measure zero, those point values mean nothing.
The Kantorovich polynomial replaces the value at a point by the average of $f$ over a small interval around it.
For the constant $1$, every average is $1$, and the Kantorovich polynomial is again $1$.
</document>

<document source="https://tamnd.github.io/math2/node/8C1JYDJBYS.md" tag="8C1JYDJBYS" kind="statement">
# van der Waerden conjecture on permanents

## Statement

Let $n \ge 1$, and let $A$ be an $n \times n$ [[doubly stochastic matrix|JJA90EB2PS]].
This means that its entries are nonnegative real numbers and that each row and each column sums to $1$.
Then
$$\operatorname{perm}(A) \ge \frac{n!}{n^n},$$
where $\operatorname{perm}$ is the [[permanent|X9B1XETBPF]].
Equality holds if and only if every entry of $A$ equals $1/n$.

## Remarks

The statement was conjectured by B. L. van der Waerden and proved independently by G. P. Egorychev and D. I. Falikman.
Egorychev's proof uses the Alexandrov–Fenchel inequality for mixed discriminants.
A later short proof by L. Gurvits uses real stable polynomials.

Example: for $n = 2$, every doubly stochastic matrix has the form $\begin{pmatrix} t & 1 - t \\ 1 - t & t \end{pmatrix}$ with $0 \le t \le 1$.
Its permanent is $t^2 + (1-t)^2$, whose minimum on $[0, 1]$ is $1/2 = 2!/2^2$, reached only at $t = 1/2$.

The value $n!/n^n$ is the permanent of the matrix with all entries $1/n$.
By Stirling's formula it is about $e^{-n} \sqrt{2 \pi n}$.

## Explanations

### L1 human

A doubly stochastic matrix spreads one unit of mass in each row and each column.
The theorem says that its permanent cannot be very small, and that the most evenly spread matrix gives the smallest value.

### L2 human

For a $k$-regular bipartite graph with $n$ vertices on each side, the biadjacency matrix divided by $k$ is doubly stochastic.
So the theorem gives at least $k^n n!/n^n$ perfect matchings.
In particular, every doubly stochastic matrix has a positive permanent, so it has a permutation pattern of positive entries.
</document>

<document source="https://tamnd.github.io/math2/node/8CFTJP1WN8.md" tag="8CFTJP1WN8" kind="concept">
# Modality of a singularity

## Summary

The modality of a critical point of a function counts continuous parameters, called moduli.
It is the number of moduli needed to describe all nearby critical points up to a change of coordinates.
Germs of modality $0$ are the simple singularities: only finitely many types occur near them.
Arnold introduced the notion and classified the germs of modality $0$, $1$ and $2$.
Modality measures how far the classification of singularities is from being a finite list.

## Explanations

### L1 human

Near some critical points there are only finitely many shapes of critical point, like a short catalogue.
Near others the shapes vary continuously, like the angles in a family of triangles, and no finite catalogue suffices.
The modality is the number of such continuous dials.
For example, the function $x^4 + y^4$ on $\C^2$ vanishes on four lines through $0$.
Four lines through a point have a cross-ratio, which no change of coordinates can alter.
So nearby functions $x^4 + y^4 + a x^2 y^2$ give infinitely many different types, and the modality is $1$.
</document>

<document source="https://tamnd.github.io/math2/node/8CKDVESXT6.md" tag="8CKDVESXT6" kind="statement">
# Lévy's characterization of Brownian motion

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_t)_{t \ge 0}$.
Let $(X_t)_{t \ge 0}$ be a real [[adapted|YRNFE016FF]] process with continuous paths and $X_0 = 0$.
Assume that $(X_t)$ and $(X_t^2 - t)$ are both [[local martingales|ZWJ9G89W3T]].
Then $(X_t)$ is a [[Wiener process|552AWX3QXN]], and for all $0 \le s < t$ the increment $X_t - X_s$ is independent of $\mathcal{F}_s$.

## Remarks

The second hypothesis says that the [[quadratic variation|ZBYC3MFW89]] of $X$ is $[X]_t = t$.
For a fixed $u \in \R$, Itô's formula shows that $Z_t = \exp(i u X_t + u^2 t / 2)$ is a local martingale; it is bounded on bounded time intervals, hence a martingale.
So $E[\exp(i u (X_t - X_s)) \mid \mathcal{F}_s] = \exp(-u^2 (t - s)/2)$, which identifies the conditional law of the increment as normal with mean $0$ and variance $t - s$.
Continuity is necessary: for a Poisson process $N$ of rate $1$, both $N_t - t$ and $(N_t - t)^2 - t$ are martingales, but $N_t - t$ is not a Wiener process.
In dimension $d$, a continuous process $X$ with $X_0 = 0$ is a $d$-dimensional Wiener process if each $X^i$ and each $X^i X^j - \delta_{ij} t$ is a local martingale.

## Explanations

### L1 human

Brownian motion is the only continuous fair game whose squared size grows on average exactly like time.
Nothing about normal distributions or independence has to be assumed: they come for free.
So these two martingale properties pin down Brownian motion completely.

### L2 human

The theorem is the standard way to recognise a Brownian motion inside stochastic calculus.
It gives the Dambis–Dubins–Schwarz theorem: a continuous local martingale $M$ with $[M]_\infty = \infty$ is a time-changed Brownian motion, $M_t = B_{[M]_t}$.
It is also a step in the proof of the [[Girsanov theorem|Z76HD1BZ29]], and it shows that $\int_0^t \sgn(W_s) \, dW_s$ is a Wiener process, which is used with [[Tanaka's formula|GEM0ERJ5ZB]].
</document>

<document source="https://tamnd.github.io/math2/node/8CKE0V6ANK.md" tag="8CKE0V6ANK" kind="concept">
# Approximation property

## Summary

A Banach space has the approximation property when the identity operator can be approximated, uniformly on each compact set, by bounded operators of finite rank.
Grothendieck showed that this is equivalent to a statement on operators.
For every Banach space $Y$, every compact operator from $Y$ into the space is a norm limit of finite-rank operators.
Spaces with a Schauder basis, such as $\ell^p$ for $1 \le p < \infty$ and $c_0$, have the property.
[[Enflo's counterexample|ZKP5KGFWCK]] shows that some separable Banach spaces do not.

## Explanations

### L1 human

Finite-rank operators are the simplest operators: their values lie in a finite-dimensional space, so they are matrices in disguise.
The approximation property says that the identity map can be imitated by such operators on any compact set, up to any small error.
In $\ell^2$, the map that keeps the first $n$ coordinates of a sequence and replaces the rest by $0$ does this for $n$ large.
</document>

<document source="https://tamnd.github.io/math2/node/8CV1KW9AJ8.md" tag="8CV1KW9AJ8" kind="definition">
# Geometric integrator

## Definition

Let $U \subseteq \R^d$ be open, let $f \colon U \to \R^d$ be smooth, and let $\varphi_t$ be the [[flow|06738GGWWK]] of the equation $y' = f(y)$.
Let $\mathcal{P}$ be a property of maps $U \to \R^d$ that every $\varphi_t$ has.
Standard choices of $\mathcal{P}$ are:

1. symplecticity, when the equation is a [[Hamiltonian system|0EDKGA2FAK]];
2. the conservation of a given [[first integral|0262JP0NV9]] $I$, that is, $I \circ \Phi = I$;
3. the preservation of volume, when $f$ has divergence zero;
4. reversibility with respect to a linear involution $\rho$, that is, $\rho \circ \Phi = \Phi^{-1} \circ \rho$, when $f \circ \rho = -\rho \circ f$.

Let $\Phi_h$ be a one-step numerical method with step size $h$.
It is a *geometric integrator* for $\mathcal{P}$ if $\Phi_h$ has the property $\mathcal{P}$ for every equation of the class and every small enough $h > 0$.

## Remarks

Example: take the harmonic oscillator $q' = p$, $p' = -q$.
The symplectic Euler method $p_{n+1} = p_n - h q_n$, $q_{n+1} = q_n + h p_{n+1}$ is the linear map
$$\begin{pmatrix} q_{n+1} \\ p_{n+1} \end{pmatrix} = \begin{pmatrix} 1 - h^2 & h \\ -h & 1 \end{pmatrix} \begin{pmatrix} q_n \\ p_n \end{pmatrix}.$$
Its determinant is $1$, and in dimension $2$ this means that the map is symplectic.
So the method is a geometric integrator for property 1.
Non-example: the explicit Euler method $q_{n+1} = q_n + h p_n$, $p_{n+1} = p_n - h q_n$ has matrix determinant $1 + h^2 \ne 1$, so it is not symplectic for any $h > 0$.
A method may be geometric for one property and not another.
The implicit midpoint rule is symplectic and conserves all quadratic first integrals.
But no Runge–Kutta method conserves all polynomial first integrals of degree $3$ or more.

## Explanations

### L1 human

A geometric integrator copies a structural feature of the true motion exactly, not only approximately.
The feature is chosen in advance: a conserved quantity, area in phase space, or symmetry under reversing time.
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<document source="https://tamnd.github.io/math2/node/8D277VG29T.md" tag="8D277VG29T" kind="definition">
# Resultant

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]], and let $m, n \ge 0$ be integers with $m + n \ge 1$.
Let $f = a_m x^m + \dots + a_0$ and $g = b_n x^n + \dots + b_0$ be elements of $R[x]$.
The Sylvester matrix $S(f, g)$ is the $(m + n) \times (m + n)$ matrix whose rows are:

1. for $i = 1, \dots, n$, the row $(a_m, a_{m-1}, \dots, a_0)$ preceded by $i - 1$ zeros and filled up with zeros on the right;
2. for $j = 1, \dots, m$, the row $(b_n, b_{n-1}, \dots, b_0)$ preceded by $j - 1$ zeros and filled up with zeros on the right.

The resultant of $f$ and $g$ is $\operatorname{Res}(f, g) = \det S(f, g)$, the [[determinant|0C3CFHDN6R]] of this matrix.

## Remarks

Example: for $f = x - a$ and $g = x - b$, the matrix is $\begin{pmatrix} 1 & -a \\ 1 & -b \end{pmatrix}$, and $\operatorname{Res}(f, g) = a - b$.
Let $R = K$ be a field with $a_m \ne 0$ and $b_n \ne 0$, and let $\alpha_1, \dots, \alpha_m$ and $\beta_1, \dots, \beta_n$ be the roots of $f$ and $g$ in a [[splitting field|02NZZ1J0FH]] of $f g$.
Then
$$\operatorname{Res}(f, g) = a_m^n \, b_n^m \prod_{i=1}^{m} \prod_{j=1}^{n} (\alpha_i - \beta_j).$$
So $\operatorname{Res}(f, g) = 0$ if and only if $f$ and $g$ have a common root, if and only if they have a common factor of positive degree.
For instance, $\operatorname{Res}(x^2 - 1, x - 1) = 0$, because of the common root $1$.
The value depends on the chosen formal degrees $m$ and $n$: if $a_m = b_n = 0$, the first column is zero and the resultant is $0$ even without a common root.
For $f$ of degree $n$ with leading coefficient $a_n$, $\operatorname{Res}(f, f') = (-1)^{n(n-1)/2} a_n \operatorname{Disc}(f)$, where $\operatorname{Disc}$ is the [[discriminant|0CCPN09EMJ]].

## Explanations

### L1 human

The resultant is the determinant of a square table filled with shifted copies of the two coefficient lists.
It is zero exactly when the two polynomials share a root, as long as both leading coefficients are nonzero.
</document>

<document source="https://tamnd.github.io/math2/node/8D455SA1D4.md" tag="8D455SA1D4" kind="definition">
# Weierstrass excess function

## Definition

Let $a < b$, let $n \ge 1$ and let $L \colon [a, b] \times \R^n \times \R^n \to \R$, written $L(x, y, p)$, be of class $C^1$.
Write $L_p$ for the vector of [[partial derivatives|05AXG3QRKR]] of $L$ in the variables $p$.
The *Weierstrass excess function* of $L$ is the function $E \colon [a, b] \times \R^n \times \R^n \times \R^n \to \R$ given by
$$E(x, y, p, q) = L(x, y, q) - L(x, y, p) - (q - p) \cdot L_p(x, y, p).$$

## Remarks

Example: for $L(x, y, p) = \sqrt{1 + p^2}$ with $n = 1$, the function $q \mapsto L(x, y, q)$ is convex.
A convex $C^1$ function lies above each of its tangent lines, so $E \ge 0$ everywhere.

Non-example of nonnegativity: for $L = (p^2 - 1)^2$ with $n = 1$, at $p = 0$ one gets $E(x, y, 0, q) = (q^2 - 1)^2 - 1$.
This is negative for $0 < \abs{q} < \sqrt{2}$, since $L$ is not convex in $p$.

The Weierstrass necessary condition reads as follows.
Let $y_0 \in C^1([a, b]; \R^n)$ be a strong local minimiser of $J(y) = \int_a^b L(x, y, y') \, dx$ with fixed end values, see [[weak and strong extremum|WMPCR723MJ]].
Then $E(x, y_0(x), y_0'(x), q) \ge 0$ for all $x \in [a, b]$ and all $q \in \R^n$.

For $C^2$ functions $L$, Taylor's formula gives $E(x, y, p, q) = \frac{1}{2} (q - p)^T L_{pp}(x, y, \bar p) (q - p)$ for some $\bar p$ on the segment from $p$ to $q$.
So the Weierstrass condition implies the Legendre condition $L_{pp}(x, y_0, y_0') \ge 0$.
</document>

<document source="https://tamnd.github.io/math2/node/8D56GQ4AG8.md" tag="8D56GQ4AG8" kind="problem">
# Dirichlet problem

## Problem

Let $\Omega$ be a bounded [[domain|0EWMDD6C14]] in $\R^n$, for example in $\C = \R^2$, with [[boundary|05D5EGMFYV]] $\partial\Omega$.
Let $\varphi \colon \partial\Omega \to \R$ be a [[continuous function|0X3WGX95FE]].
Find a continuous function $u \colon \overline{\Omega} \to \R$ that is [[harmonic|0DEB0JMN0W]] in $\Omega$, that is, its [[Laplacian|0E0B6RA8K0]] $\Delta u$ vanishes there, with $u = \varphi$ on $\partial\Omega$.

## Remarks

A solution, if it exists, is unique, by the maximum principle for harmonic functions.
For the unit disc, the solution exists for every $\varphi$ and is the Poisson integral of $\varphi$ against the [[Poisson kernel|TCYSWCTYYM]].
For a general bounded domain, Perron's method produces a candidate solution as the upper envelope of [[subharmonic functions|Z1MHMGFNFW]] lying below $\varphi$ on the boundary.
The candidate takes the boundary value $\varphi$ at every regular boundary point.
The problem is solvable for every continuous $\varphi$ if and only if every boundary point is regular.
Every boundary point of a planar domain bounded by finitely many Jordan curves is regular.
Example of failure: let $\Omega = \set{z \in \C : 0 < \abs{z} < 1}$, with $\varphi = 0$ on $\abs{z} = 1$ and $\varphi(0) = 1$.
A solution would be bounded and harmonic near $0$, so it would extend to a harmonic function on the whole disc.
This function has boundary values $0$, so it is $0$ everywhere, which contradicts $u(0) = 1$.

## Explanations

### L1 human

Fix the temperature along the edge of a metal plate and wait until the temperature stops changing.
The Dirichlet problem asks for the temperature inside the plate.
The equilibrium temperature is a harmonic function, so the problem is to find a harmonic function with given values on the boundary.
For nice domains there is always exactly one answer, but for domains with very thin pieces of boundary, such as an isolated point, there may be none.
</document>

<document source="https://tamnd.github.io/math2/node/8D5Y3JNF09.md" tag="8D5Y3JNF09" kind="definition">
# Holomorphy on Banach spaces

## Definition

Let $X$ and $Y$ be complex [[Banach spaces|06GZ956657]], let $U \subseteq X$ be open and let $f \colon U \to Y$.
The map $f$ is *holomorphic* on $U$ if, for each $a \in U$, there is a continuous complex-linear map $L \colon X \to Y$ with
$$\lim_{h \to 0} \frac{\norm{f(a + h) - f(a) - L h}}{\norm{h}} = 0.$$
That is, $f$ has a complex [[Fréchet derivative|06EH97V2A7]] $Df(a) = L$ at each point $a$ of $U$.

## Remarks

The following conditions on $f \colon U \to Y$ are equivalent:

1. $f$ is holomorphic.
2. $f$ is [[analytic|H1KHRWNH8B]] on $U$ over $\C$.
3. $f$ is continuous, and for all $a \in U$ and $h \in X$, the map $z \mapsto f(a + z h)$ is a [[holomorphic function|0NAHXR1FRH]] on the open set of $z \in \C$ with $a + z h \in U$.

A map with the property on complex lines in 3, without continuity, is called Gateaux holomorphic, after the [[Gateaux derivative|EAZF9NJW12]].

Example: let $B$ be a complex Banach algebra and $f(x) = x^2$.
Then $f(a + h) - f(a) = a h + h a + h^2$, and $\norm{h^2} \le \norm{h}^2$, so $Df(a) h = a h + h a$, which is complex-linear.

Non-example: on $\C$, the map $f(z) = \overline{z}$ has the real derivative $h \mapsto \overline{h}$, which is not complex-linear.
So $f$ is not holomorphic, although it is real Fréchet differentiable.

A discontinuous linear form on an infinite-dimensional complex Banach space is holomorphic on every complex line but is not holomorphic, since it is not continuous.
This shows that continuity in clause 3 cannot be dropped.
</document>

<document source="https://tamnd.github.io/math2/node/8DEH72QHEA.md" tag="8DEH72QHEA" kind="definition">
# f-vector and h-vector

## Definition

Let $\Delta$ be a finite [[simplicial complex|V8GZ0W1ZZ5]] of dimension $d - 1$, with $d \ge 1$.
For $-1 \le i \le d - 1$, let $f_i$ be the number of faces of $\Delta$ of dimension $i$, that is, with $i + 1$ vertices; so $f_{-1} = 1$ counts the empty face.
The *f-vector* of $\Delta$ is $(f_{-1}, f_0, \dots, f_{d-1})$.
The *h-vector* of $\Delta$ is the sequence $(h_0, h_1, \dots, h_d)$ defined by
$$\sum_{i=0}^{d} h_i \, t^{d-i} = \sum_{i=0}^{d} f_{i-1} \, (t - 1)^{d-i}.$$

## Remarks

Comparing coefficients gives $h_k = \sum_{i=0}^{k} (-1)^{k-i} \binom{d-i}{k-i} f_{i-1}$.
In particular $h_0 = 1$, $h_1 = f_0 - d$, and $h_0 + h_1 + \dots + h_d = f_{d-1}$.
The f-vector can be recovered from the h-vector, so both carry the same information.
For a simplicial polytope of dimension $d$ one uses the f-vector and h-vector of its boundary complex, which has dimension $d - 1$.

Example: the boundary of an octahedron has $d = 3$ and $f = (1, 6, 12, 8)$.
Then $(t-1)^3 + 6(t-1)^2 + 12(t-1) + 8 = t^3 + 3t^2 + 3t + 1$, so $h = (1, 3, 3, 1)$, and indeed $1 + 3 + 3 + 1 = 8 = f_2$.

Non-example of a nice property: the h-vector can have negative entries.
For two disjoint edges, $d = 2$ and $f = (1, 4, 2)$, and $(t-1)^2 + 4(t-1) + 2 = t^2 + 2t - 1$ gives $h = (1, 2, -1)$.
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<document source="https://tamnd.github.io/math2/node/8DJPSYSHGJ.md" tag="8DJPSYSHGJ" kind="concept">
# Borel summation

## Summary

Borel summation is a method that assigns a value to many [[divergent series|0NBRP6093V]], in particular to series whose coefficients grow like factorials.
It divides the $k$-th coefficient by $k!$, sums the resulting convergent series, and recovers the lost factorials with a [[Laplace transform|05XT5AVJBK]].
The method gives the usual sum for every convergent series.
It is a basic tool for the asymptotic series of perturbation theory and of differential equations.

## Explanations

### L1 human

Some series, such as $1 - 1! + 2! - 3! + \cdots$, grow too fast to have a sum in the usual sense.
The identity $k! = \int_0^\infty t^k e^{-t} \, dt$ suggests a trick: replace $k!$ by this integral and exchange sum and integral.
After dividing by $k!$, the series becomes a convergent one, and the integral then gives a finite number.
For this example the Borel sum is $\int_0^\infty e^{-t}/(1+t) \, dt$, about $0.596$.
</document>

<document source="https://tamnd.github.io/math2/node/8DVYNQAA1C.md" tag="8DVYNQAA1C" kind="definition">
# Free product

## Definition

Let $(G_i)_{i \in I}$ be a family of [[groups|G5S1M9]], with $G_i \cap G_j = \emptyset$ for $i \ne j$ after relabelling.
A *reduced word* is a finite sequence $(g_1, \ldots, g_m)$, with $m \ge 0$, such that:

1. each $g_k$ is a non-identity element of some $G_{i_k}$;
2. consecutive letters lie in different groups: $i_k \ne i_{k+1}$ for $1 \le k < m$.

The *free product* $\mathop{\ast}_{i \in I} G_i$ is the set of reduced words.
The product of two words is their concatenation followed by reduction.
To reduce, replace two adjacent letters of the same $G_i$ by their product in $G_i$, and delete the letter if it is the identity.
Repeat until the word is reduced.
For two groups it is written $G * H$.

## Remarks

The product is well defined and associative, the empty word is the identity, and each $G_i$ embeds as the words of length at most $1$.
Equivalently, $\mathop{\ast}_i G_i$ is the [[coproduct|4FJ0S71P58]] of the $G_i$ in the category of groups.
That is, for every group $K$ and [[homomorphisms|0JYTG5K9Y4]] $\varphi_i \colon G_i \to K$, there is a unique homomorphism $\mathop{\ast}_i G_i \to K$ restricting to each $\varphi_i$.
Example: $\Z * \Z$ is the free group of rank $2$.
With generators $a$ of the first factor and $b$ of the second, its reduced words are the reduced words in $a^{\pm 1}, b^{\pm 1}$.
Another example: $\Z/2 * \Z/3$ is isomorphic to the [[modular group|VEKV1EM77B]] $PSL_2(\Z)$.
Non-example: the direct product $\Z \times \Z$ is not the free product $\Z * \Z$.
In $\Z \times \Z$ the element $a b a^{-1} b^{-1}$ is the identity, while in $\Z * \Z$ it is a reduced word of length $4$.
Given presentations $G = \langle S \mid R \rangle$ and $H = \langle T \mid Q \rangle$ with $S \cap T = \emptyset$, one has $G * H = \langle S \cup T \mid R \cup Q \rangle$.
</document>

<document source="https://tamnd.github.io/math2/node/8DWRYNAMFT.md" tag="8DWRYNAMFT" kind="statement">
# Asymptotic normality of the maximum likelihood estimator

## Statement

Let $\set{f(\cdot; \theta) : \theta \in \Theta}$ be a family of probability densities with respect to a measure $\nu$, with $\Theta \subseteq \R^k$, and let $X_1, X_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] with density $f(\cdot; \theta_0)$.
Assume:

1. $\theta_0$ lies in the [[interior|0C38S0KMJK]] of $\Theta$;
2. on a neighbourhood $N$ of $\theta_0$, $f(x; \theta) > 0$ and $\theta \mapsto f(x; \theta)$ is twice continuously differentiable, for every $x$;
3. $\int \sup_{\theta \in N} \norm{\nabla_\theta f(x; \theta)} \, d\nu(x) < \infty$ and $\int \sup_{\theta \in N} \norm{\nabla^2_\theta f(x; \theta)} \, d\nu(x) < \infty$;
4. the [[Fisher information|G1Z3WSTE8Q]] $I(\theta_0) = E_{\theta_0}\big[s(\theta_0; X_1)\, s(\theta_0; X_1)^T\big]$ exists and is nonsingular, where $s(\theta; x) = \nabla_\theta \log f(x; \theta)$ is the [[score|CP97GRVTQ0]];
5. $E_{\theta_0}\big[\sup_{\theta \in N} \norm{\nabla^2_\theta \log f(X_1; \theta)}\big] < \infty$.

Let $\hat{\theta}_n$ be a [[maximum likelihood estimator|0FREPFKNS7]] with $\hat{\theta}_n \to \theta_0$ in probability.
Then
$$\sqrt{n}\,(\hat{\theta}_n - \theta_0) \xrightarrow{d} \mathcal{N}_k\big(0, I(\theta_0)^{-1}\big),$$
where $\xrightarrow{d}$ is [[convergence in distribution|0JDAQHWA9N]] and $\mathcal{N}_k$ is the [[multivariate normal distribution|0G558GJQMW]].

## Remarks

Idea of proof: with probability tending to one, $\hat{\theta}_n$ is an interior maximizer, so the score equation $\sum_i s(\hat{\theta}_n; X_i) = 0$ holds.
A Taylor expansion around $\theta_0$ gives $0 = \sum_i s(\theta_0; X_i) + H_n (\hat{\theta}_n - \theta_0)$, where $H_n$ is the [[Hessian|0693Y2METT]] of the log-likelihood at intermediate points.
Conditions 2 and 3 allow differentiation under the integral, so the score has mean $0$ and covariance $I(\theta_0)$, and the [[central limit theorem|0EPZC4Z1FC]] gives $n^{-1/2}\sum_i s(\theta_0; X_i) \xrightarrow{d} \mathcal{N}_k(0, I(\theta_0))$.
Condition 5 and consistency give $H_n/n \to -I(\theta_0)$ in probability, and solving for $\sqrt{n}(\hat{\theta}_n - \theta_0)$ gives the result.
Consistency itself is provided, for example, by the [[consistency theorem|54387XJ9QW]].

Example: for i.i.d. Exponential($\lambda$) data, $\hat{\lambda}_n = 1/\bar{X}_n$ and $I(\lambda) = 1/\lambda^2$, so $\sqrt{n}(\hat{\lambda}_n - \lambda) \xrightarrow{d} \mathcal{N}(0, \lambda^2)$.

The interior condition is needed.
For $\mathcal{N}(\mu, 1)$ data with $\Theta = [0, \infty)$ and $\mu_0 = 0$, the MLE is $\max(\bar{X}_n, 0)$.
Then $\sqrt{n}\,\hat{\mu}_n$ converges to the law of $\max(Z, 0)$ with $Z$ standard normal, which is not normal.
Smoothness in $\theta$ with a fixed support is also needed: for Uniform$(0, \theta)$ data the MLE is $\max_i X_i$, and $n(\theta_0 - \hat{\theta}_n)$ has an exponential limit.

## Explanations

### L1 human

For large samples, the maximum likelihood estimate is approximately normally distributed around the true value.
Its spread shrinks like $1/\sqrt{n}$, and the constant is the inverse of the Fisher information, which measures how sharply the likelihood peaks.
This is the smallest spread allowed by the Cramér–Rao bound, so maximum likelihood is asymptotically efficient.

### L2 human

The theorem justifies the usual Wald confidence intervals $\hat{\theta}_n \pm 1.96\, (n\, I(\hat{\theta}_n))^{-1/2}$ and the Wald test.
Together with the delta method it gives limit laws for smooth functions of the MLE.
The same expansion of the log-likelihood leads to Wilks' theorem for the likelihood-ratio statistic and to the score test.
The modern view puts the result in the frame of local asymptotic normality, where the bound $I(\theta_0)^{-1}$ appears in the Hájek–Le Cam convolution theorem.
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# Projective hierarchy

## Summary

The projective hierarchy sorts the projective subsets of a [[Polish space|584GXXY73M]] into levels $\Sigma^1_n$, $\Pi^1_n$ and $\Delta^1_n$ for $n \ge 1$.
It starts with the [[analytic sets|3RWWZYPAFV]], and alternates two operations: taking complements and taking projections along the [[Baire space|P2WS16HKD7]].
The first level $\Delta^1_1$ consists exactly of the Borel sets, by [[Suslin's theorem|9418TGREZX]].
ZFC does not decide whether the higher levels behave well, for example whether all their sets are Lebesgue measurable.
They do under [[projective determinacy|2GY3131ZPS]], and they do not under the [[axiom of constructibility|4QKRPRCSXH]].

## Explanations

### L1 human

Start with the shadows of closed sets: these are the analytic sets.
Take complements of these, then their shadows, then complements again, and so on.
Each step gives a new level of complexity, and the sets reached after finitely many steps are the projective sets.
Most sets of real numbers that appear in ordinary mathematics are projective, often at a low level.
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# Forward difference operator

## Summary

The forward difference operator $\Delta_h$ sends a function $f$ to the function $x \mapsto f(x + h) - f(x)$.
It is the discrete analogue of differentiation, and difference equations are often written with it.
It equals $E_h - I$, where $E_h$ is the [[shift operator|MBK6AE6734]] and $I$ is the identity.

## Explanations

### L1 human

The forward difference looks one step ahead and records how much the function changes.
For a sequence $u_0, u_1, u_2, \dots$ with step $1$ it gives the sequence of gaps $u_1 - u_0, u_2 - u_1, \dots$.
For the squares $0, 1, 4, 9, 16, \dots$ the gaps are the odd numbers $1, 3, 5, 7, \dots$.
Doing it again gives the constant gaps $2, 2, 2, \dots$, just as the second derivative of $x^2$ is the constant $2$.
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# Principal bundle

## Summary

A principal bundle is a fiber bundle whose fibers are copies of a group $G$.
The group $G$ acts on the total space and moves points within each fiber.
Each fiber is like the group itself, but without a chosen identity element.
Frame bundles of manifolds and the Hopf fibration are standard examples.
Connections on principal bundles are the language of gauge theory, and every vector bundle can be built from a principal bundle.

## Explanations

### L1 human

Over each point of a manifold, consider all possible coordinate frames for the tangent space.
No frame is preferred, but any two differ by exactly one invertible matrix.
So the frames over one point look like the group of invertible matrices, with no special element.
Collecting these sets over all points gives a principal bundle, the frame bundle.
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# Minimax estimator

## Summary

A minimax estimator is an estimator whose worst-case [[risk|N26TSVRH2S]], over all values of the unknown parameter, is as small as possible.
The smallest achievable worst-case risk is the minimax risk of the problem.
The minimax principle gives a way to choose among estimators without a prior, and it is a basic benchmark in decision theory and nonparametric statistics.

## Explanations

### L1 human

Imagine that an opponent picks the true parameter after seeing which estimator is used, so as to make it look as bad as possible.
A minimax estimator is the best choice against such an opponent.
It may be a little worse than others for typical parameters, but it guards against the worst case.
For a coin tossed $n$ times, the minimax estimator of the heads probability under squared error is $(X + \sqrt{n}/2)/(n + \sqrt{n})$, not the plain proportion $X/n$.
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# Order complex

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
The *order complex* $\Delta(P)$ of $P$ is the abstract [[simplicial complex|V8GZ0W1ZZ5]] with vertex set $P$ whose faces are the finite [[chains|0S8FE7ZMFM]] of $P$.
So the faces are the finite subsets $\set{x_0, x_1, \ldots, x_k} \subseteq P$ with $x_0 < x_1 < \cdots < x_k$.

## Remarks

$\Delta(P)$ is a simplicial complex because every subset of a chain is a chain.
A chain with $k + 1$ elements is a face of dimension $k$, so the dimension of $\Delta(P)$ is the length of the longest chain of $P$, when it is finite.
Topological properties of $P$, such as its homotopy type or homology, mean those of the geometric realization of $\Delta(P)$.
Example: let $P = \set{a, b, c, d}$ with $a < c$, $a < d$, $b < c$, $b < d$ and no other strict relations.
Its nonempty chains are the four points and the four pairs $\set{a, c}$, $\set{a, d}$, $\set{b, c}$, $\set{b, d}$.
So $\Delta(P)$ is a cycle of four edges, and its realization is a circle.
Non-example: for this $P$, the family of all subsets of $P$ is a simplicial complex on the vertex set $P$.
It is not $\Delta(P)$, since it contains $\set{a, b}$, which is not a chain.
The order complex of a chain with $n + 1$ elements is a full $n$-simplex.
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# Painlevé property

## Summary

An ordinary differential equation in the complex domain has the Painlevé property if its movable singularities are poles.
Here a singularity of a solution is movable when its position depends on the initial data.
Linear equations have it, since their solutions can be continued along every path that avoids the singular points of the coefficients.
Nonlinear equations with the property are rare, and Painlevé and his school classified those of second order of a certain rational type.
Six new equations appeared in this classification, the Painlevé equations, whose solutions define new special functions.
See the [[definition|H2E8KW6DB5]].

## Explanations

### L1 human

Solutions of nonlinear equations can blow up at places that move when the starting value changes.
For $y' = y^2$, the solution $y = \frac{1}{c - z}$ has a pole at $z = c$, and $c$ depends on the starting value.
A pole is harmless: the solution stays single-valued around it.
For $y' = y^3$, the solutions behave like $(c - z)^{-1/2}$, a square root, which branches at a moving point.
The Painlevé property forbids such moving branch points, so the solutions can be continued as single-valued meromorphic functions on suitable domains.
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# Central polynomial

## Definition

Let $k$ be a [[field|04C5ZBG1WT]], let $n \ge 1$, and let $M_n(k)$ be the ring of $n \times n$ [[matrices|0JXZKX526R]] over $k$.
Let $f \in k\langle x_1, \ldots, x_m \rangle$ be an element of the [[free algebra|2P3Y2FRCSZ]] with zero constant term.
The polynomial $f$ is a *central polynomial* for $M_n(k)$ if:

1. $f(a_1, \ldots, a_m)$ is a scalar matrix $\lambda I$, with $\lambda \in k$, for all $a_1, \ldots, a_m \in M_n(k)$;
2. $f$ is not a [[polynomial identity|C0EDT81EGH]] of $M_n(k)$: $f(a_1, \ldots, a_m) \ne 0$ for some $a_1, \ldots, a_m \in M_n(k)$.

## Remarks

Clause 1 says that all values of $f$ lie in the [[center|MVT64B3B6H]] $k I$ of $M_n(k)$.

Example: $f = (x_1 x_2 - x_2 x_1)^2$ is a central polynomial for $M_2(k)$.
For $a, b \in M_2(k)$ the matrix $c = ab - ba$ has trace $0$, so the [[Cayley–Hamilton theorem|0F765KFC2J]] gives $c^2 = -\det(c) \, I$, a scalar matrix.
For $a = e_{12}$ and $b = e_{21}$ one has $c = e_{11} - e_{22}$ and $c^2 = I \ne 0$, so clause 2 holds.

Non-example: $g = x_1 x_2 - x_2 x_1$ fails clause 1 for $M_2(k)$, since $g(e_{12}, e_{21}) = e_{11} - e_{22}$ is not scalar.
Non-example: $s_4$, the standard polynomial of degree $4$, satisfies clause 1 for $M_2(k)$ only because all its values are $0$, so it fails clause 2.

For $n = 1$ every polynomial with a nonzero value on $k$ is central, so the notion is of interest for $n \ge 2$.
Central polynomials exist for every $n \ge 1$, a result of Formanek and of Razmyslov.
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# Frobenius algebra

## Summary

A Frobenius algebra is a finite-dimensional [[associative algebra|3CTW1P4HBP]] over a field with a [[nondegenerate bilinear form|01NHFZF2SK]] that is compatible with the multiplication.
Equivalently, the algebra is isomorphic to its own dual as a left module over itself.
Group algebras of finite groups and matrix algebras are Frobenius algebras.
Frobenius algebras are [[self-injective|KQKJENEMSN]], and over them projective and injective modules coincide.
Commutative Frobenius algebras also describe two-dimensional topological field theories.

## Explanations

### L1 human

A Frobenius algebra carries a way to pair any two of its elements to a number, and the pairing respects the multiplication.
The matrix algebra is the model: pair two matrices $a$ and $b$ by taking the trace of $ab$.
No nonzero matrix pairs to zero with every other matrix, which is the nondegeneracy.
For the group algebra of a finite group, pair $a$ and $b$ by taking the coefficient of the identity element in $ab$.
This pairing gives a perfect symmetry between the algebra and its dual space.
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# Bessel potential space

## Definition

Let $n \ge 1$, let $s \in \R$ and let $1 < p < \infty$.
Let $\mathcal{F}$ denote the [[Fourier transform|DPVM34ADZV]] on the space $\mathcal{S}'(\R^n)$ of [[tempered distributions|AES2T5JPFP]].
The *Bessel potential space* $H^{s,p}(\R^n)$ is the set of all $f \in \mathcal{S}'(\R^n)$ such that
$$J^s f = \mathcal{F}^{-1}\big[ (1 + \abs{\xi}^2)^{s/2} \, \mathcal{F} f \big]$$
is a function in the [[Lebesgue space|SASP14MA1Z]] $L^p(\R^n)$.
Its norm is $\norm{f}_{H^{s,p}} = \norm{J^s f}_{L^p}$.

## Remarks

The operator $J^{-s} = (I - \Delta)^{-s/2}$ is the *Bessel potential* of order $s$.
For $s > 0$ it is convolution with an integrable kernel $G_s$ whose Fourier transform is $(1 + \abs{\xi}^2)^{-s/2}$.
So for $s > 0$, $H^{s,p}(\R^n)$ is the set of functions $G_s * g$ with $g \in L^p(\R^n)$, and $\norm{G_s * g}_{H^{s,p}} = \norm{g}_{L^p}$.

Example: for $s = 0$ the multiplier is $1$, so $J^0 f = f$ and $H^{0,p}(\R^n) = L^p(\R^n)$ with the same norm.

Non-example: the constant function $f = 1$ is a tempered distribution, but it is not in any $H^{s,p}(\R^n)$.
Its Fourier transform is a multiple of the Dirac mass at $0$, and the multiplier equals $1$ at $\xi = 0$.
So $J^s 1 = 1$, which is not in $L^p(\R^n)$ for $p < \infty$.

For an integer $k \ge 0$ and $1 < p < \infty$, $H^{k,p}(\R^n)$ equals the [[Sobolev space|M4CEZ5K9HP]] $W^{k,p}(\R^n)$ with an equivalent norm.
The proof of this uses the boundedness of the Riesz transforms on $L^p$.
For $p = 2$, Plancherel's theorem gives $H^{s,2}(\R^n) = H^s(\R^n)$, the [[fractional Sobolev space|N6NX1VT837]].
Other notations are $L^p_s$, $H^s_p$ and $W^{s,p}$.
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# Primitive ring

## Summary

A left primitive ring is a ring that has a faithful simple left module.
By the [[Jacobson density theorem|EK0T7FRKCY]], such a ring acts densely on a vector space over a division ring.
So it is close to a full ring of linear maps.
Every left primitive ring is [[prime|Z9TJSZZYG1]], and a commutative ring is primitive exactly when it is a field.
Primitive rings are the building blocks of Jacobson's structure theory: every [[semiprimitive ring|1B2DBKV1C9]] is a subdirect product of primitive rings.

## Explanations

### L1 human

A module is a set on which the ring acts by something like linear maps.
A ring is primitive if it can act on one irreducible module without losing information, that is, with no nonzero element acting as zero.
The typical example is the ring of all linear maps of a vector space $V$, acting on $V$ itself.
The integers are not primitive: each simple $\Z$-module $\Z/p\Z$ is killed by $p$.
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# Directed set

## Definition

Let $D$ be a set with a [[preorder|0VABYWJWH9]] $\le$.
The set $D$ is *directed* (upward directed) when:

1. $D$ is nonempty.
2. For all $a, b \in D$ there is $c \in D$ with $a \le c$ and $b \le c$, that is, every two elements have an upper bound in $D$.

## Remarks

Example: the set of neighbourhoods of a point $x$ in a topological space, ordered by reverse inclusion ($U \le V$ when $V \subseteq U$), is directed.
Check: $x$ has the neighbourhood consisting of the whole space, and for neighbourhoods $U, V$ the set $U \cap V$ is a neighbourhood with $U \le U \cap V$ and $V \le U \cap V$.

Non-example: the set $\set{a, b}$ with the trivial order, where each element is only below itself, fails clause 2, since $a$ and $b$ have no common upper bound.

By induction, every finite subset of a directed set has an upper bound in it.
A downward directed set is defined dually, with lower bounds.
Some authors, and the Mathlib predicate `IsDirected`, drop clause 1.
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# Fusion category

## Definition

Let $k$ be a [[field|04C5ZBG1WT]].
A *fusion category* over $k$ is a [[monoidal category|M3MSQ3NKAE]] $(\mathcal{C}, \otimes, I)$ such that:

1. $\mathcal{C}$ is $k$-linear: every $\Hom(A, B)$ is a $k$-vector space, composition is $k$-bilinear, and $\otimes$ is $k$-bilinear on morphisms;
2. every $\Hom(A, B)$ is finite-dimensional;
3. $\mathcal{C}$ is semisimple: it is an [[abelian category|Y9CZ3MTNF1]] in which every object is a finite direct sum of simple objects;
4. $\mathcal{C}$ has only finitely many isomorphism classes of simple objects;
5. $\mathcal{C}$ is [[rigid|A10WW8GEKR]];
6. the unit object $I$ is simple.

Here an object is simple if it is nonzero and has no subobjects other than $0$ and itself.
Many authors assume that $k$ is algebraically closed.

## Remarks

Example: for a finite group $G$, the category $\mathrm{Rep}(G)$ of finite-dimensional complex [[representations|WYGRXCG2RN]] of $G$ is a fusion category over $\C$.
It is semisimple by Maschke's theorem, has finitely many irreducible representations up to isomorphism, the dual of a representation gives rigidity, and the trivial representation is simple.
Non-example: the category of finite-dimensional complex representations of the group $\Z$ satisfies clauses 1, 2, 5 and 6.
It is not semisimple, and it has infinitely many simple objects, one for each $\lambda \in \C^\times$.
Non-example: for a field $k$ of characteristic $p$ and a group $G$ of order divisible by $p$, $\mathrm{Rep}_k(G)$ fails clause 3 by the failure of Maschke's theorem.
If $X_1, \ldots, X_n$ are the simple objects, the fusion rules are the integers $N_{ij}^m$ with $X_i \otimes X_j \cong \bigoplus_m N_{ij}^m X_m$.
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# WZ theory

## Summary

WZ theory, named after Herbert Wilf and Doron Zeilberger, is a method to prove identities of the form $\sum_k F(n, k) = 1$ by computer, see the [[definition|4HP429BZQ5]].
The proof consists of one rational function, the WZ certificate.
The certificate turns $F$ into a [[Wilf–Zeilberger pair|WC0QNABPZ4]], whose defining equation can be checked by algebra alone.
It applies to sums of hypergeometric terms, such as most identities for [[binomial coefficients|0GBZM0MT0F]], and it relates to [[Zeilberger's algorithm|VA7MH05Z3B]] and [[Gosper's algorithm|NHMJQBS04M]].

## Explanations

### L1 human

Many identities for sums of binomial coefficients have tricky proofs by hand.
WZ theory replaces the trick by a mechanical certificate.
A computer finds a rational function, and anyone can check by simple algebra that it satisfies one equation.
That equation forces the sum to be the same for all $n$, so it is enough to check the case $n = 0$.
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# Anosov flow

## Summary

An Anosov flow is a flow on a compact manifold with a splitting of the tangent bundle into three parts.
They are the flow direction, a uniformly contracted direction and a uniformly expanded direction.
It is the continuous-time analogue of an [[Anosov diffeomorphism|RCQH8P6DPQ]].
The [[geodesic flow|VF43FTDA1H]] of a compact Riemannian manifold of negative sectional curvature is an Anosov flow, a fact proved by Anosov.
Anosov flows are structurally stable up to orbit equivalence, and those of class $C^2$ that preserve a smooth volume are ergodic.

## Explanations

### L1 human

Imagine particles moving along a family of curves that fill a closed space.
Nearby curves either approach each other exponentially fast or separate exponentially fast, except for the trivial shift along the curve itself.
This is an Anosov flow.
The motion of a free particle sliding on a surface shaped everywhere like a saddle is the classical example.
See the [[definition|EN4A9MM7RP]].
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# Logarithmically convex function

## Summary

A logarithmically convex function is a positive function whose logarithm is a [[convex function|07Q7CNM5WF]].
Every such function is itself convex, but the converse fails.
The best known example is the [[gamma function|0H5SYBSED9]] on the positive reals, and log-convexity is the key hypothesis of the [[Bohr–Mollerup theorem|QFF87AXWN2]].
The [[definition|E97P413DKM]] gives the precise condition.

## Explanations

### L1 human

Take a positive function and look at the graph of its logarithm.
If that graph bends upward everywhere, the function is log-convex.
This is a strong form of convexity: between any two points, the graph lies below the exponential curve through the two end points.
For example $e^{x^2}$ is log-convex, since its logarithm is $x^2$.
The function $x^2$ on $(0, \infty)$ is convex but not log-convex, since its logarithm $2 \ln x$ bends downward.
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# Borel determinacy theorem

## Statement

Let $A$ be a nonempty set with the discrete topology, and give $A^{\N}$ the product topology.
Let $X \subseteq A^{\N}$ be a Borel set, that is, a member of the [[Borel sigma-algebra|0WQXY32Q8D]] of $A^{\N}$.
Then the game $G(A, X)$ is [[determined|QXECDPAYTN]]: one of the two players has a winning strategy.

## Remarks

The theorem was proved by Donald A. Martin in [[ZFC|0HE5WDQ9FJ]].
The base case is the Gale–Stewart theorem: games with an open or a closed payoff set are determined.
Martin's proof "unravels" a Borel game: it builds an auxiliary game on a larger set of moves in which the payoff set becomes clopen.
Winning strategies of the auxiliary game then give winning strategies of the original game.
The induction runs along the levels of the Borel hierarchy, and each step uses power sets of the sets of moves of the previous step.

The hypothesis that $X$ is Borel cannot simply be dropped: with the axiom of choice one can build a set $X \subseteq \set{0, 1}^{\N}$ that is not determined.
Harvey Friedman showed that the theorem cannot be proved in second-order arithmetic, so the use of large sets in the proof is unavoidable.

## Explanations

### L1 human

Imagine an infinite game where two players take turns choosing numbers, and the winner is decided by whether the infinite sequence of moves lands in a set $X$.
Suppose $X$ can be built from simple sets by countable unions, countable intersections and complements.
Then one of the two players has a strategy that always wins.

### L2 human

The theorem is a basic tool of descriptive set theory.
From it one gets, for instance, Wadge's lemma for Borel subsets $B$ and $C$ of the Baire space $\N^{\N}$.
It says that $B = f^{-1}(C)$ for some continuous map $f$, or $C = g^{-1}(\N^{\N} \setminus B)$ for some continuous map $g$.
Going beyond Borel sets needs more than ZFC: determinacy of analytic games follows from a measurable cardinal, and projective determinacy follows from infinitely many Woodin cardinals.
The theorem is also a famous example of a statement about countable objects whose proof requires uncountably many iterations of the power set operation.
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# Boolean-valued model

## Definition

Let $B$ be a complete [[Boolean algebra|0XADJKM699]], that is, one in which every subset has a supremum and an infimum.
Let $\mathcal{L}$ be a first-order language.
A *$B$-valued model* $\mathcal{M}$ of $\mathcal{L}$ consists of a nonempty set $M$ and of the following data:

1. For each $n$-ary relation symbol $R$ (including equality), a map $M^n \to B$, written $(a_1, \ldots, a_n) \mapsto \norm{R(a_1, \ldots, a_n)}$.
2. For each $n$-ary function symbol $f$, a map $f^{\mathcal{M}} \colon M^n \to M$.

These data must satisfy, for all elements of $M$:

3. $\norm{a = a} = 1$ and $\norm{a = b} = \norm{b = a}$.
4. $\norm{a = b} \wedge \norm{b = c} \le \norm{a = c}$.
5. $\norm{a_1 = b_1} \wedge \cdots \wedge \norm{a_n = b_n} \wedge \norm{R(a_1, \ldots, a_n)} \le \norm{R(b_1, \ldots, b_n)}$ for each relation symbol $R$.
6. $\norm{a_1 = b_1} \wedge \cdots \wedge \norm{a_n = b_n} \le \norm{f^{\mathcal{M}}(a_1, \ldots, a_n) = f^{\mathcal{M}}(b_1, \ldots, b_n)}$ for each function symbol $f$.

The truth value of a formula with parameters from $M$ is defined by recursion: $\norm{\neg \varphi} = \neg \norm{\varphi}$, $\norm{\varphi \wedge \psi} = \norm{\varphi} \wedge \norm{\psi}$, and
$$\norm{\exists x \, \varphi(x)} = \sup_{a \in M} \norm{\varphi(a)}, \qquad \norm{\forall x \, \varphi(x)} = \inf_{a \in M} \norm{\varphi(a)}.$$

## Remarks

Example: if $B = \set{0, 1}$ and $\norm{a = b} = 1$ exactly when $a = b$, then a $B$-valued model is an ordinary [[structure|983GZBQYT4]], where $R(a_1, \ldots, a_n)$ holds when its value is $1$.
Non-example: if $B$ is not complete, a supremum over $M$ may not exist, and then the value of $\exists x \, \varphi(x)$ is not defined.
This is why the definition asks for a complete algebra.
For a complete Boolean algebra $B$, the Boolean-valued model $V^B$ of set theory gives every axiom of ZFC the value $1$.
Some authors also give function symbols values in $B$, by treating them as relation symbols.
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# Box–Muller transform

## Statement

Let $U_1$ and $U_2$ be independent random variables, each with the [[uniform distribution|0AREFQ9Z65]] on $(0, 1)$.
Let
$$Z_1 = \sqrt{-2 \ln U_1}\, \cos(2\pi U_2), \qquad Z_2 = \sqrt{-2 \ln U_1}\, \sin(2\pi U_2).$$
Then $Z_1$ and $Z_2$ are [[independent|0GQ938GDE0]], and each has the standard [[normal distribution|05AQE0QR0D]], with mean $0$ and variance $1$.

## Remarks

Let $R = \sqrt{-2 \ln U_1}$ and $\Theta = 2\pi U_2$, so that $(Z_1, Z_2)$ has [[polar coordinates|0BSW4T7THS]] $(R, \Theta)$.
For $t \ge 0$, $P(R^2 > t) = P(U_1 < e^{-t/2}) = e^{-t/2}$, so $R^2$ has the [[exponential distribution|0E99F51GVB]] with mean $2$, and $R$ has density $r e^{-r^2/2}$ on $(0, \infty)$.
Also $\Theta$ is uniform on $(0, 2\pi)$ and independent of $R$.
By the [[change of variables formula|00N5FHP1CV]] with Jacobian $r$, the pair $(Z_1, Z_2)$ has density $\frac{1}{2\pi} e^{-r^2/2} = \frac{1}{\sqrt{2\pi}} e^{-z_1^2/2} \cdot \frac{1}{\sqrt{2\pi}} e^{-z_2^2/2}$, which is the product of two standard normal densities.
Example: $U_1 = e^{-2}$ and $U_2 = 1/8$ give $R = 2$, $\Theta = \pi/4$, and $(Z_1, Z_2) = (\sqrt{2}, \sqrt{2})$.
Independence of $U_1$ and $U_2$ is necessary: with $U_2 = U_1$ the point $(Z_1, Z_2)$ lies on one curve in the plane, so the pair is not a standard normal pair.
To obtain a normal variable with mean $\mu$ and standard deviation $s$, use $\mu + s Z_1$.

## Explanations

### L1 human

The transform turns two uniform random numbers into two independent bell-curve random numbers.
The first uniform number decides the distance from the origin, through a logarithm, and the second decides the direction.
The trick works because the two-dimensional standard normal distribution is the same in every direction, and its squared radius has a simple exponential law.

### L2 human

One cannot invert the normal distribution function in closed form, so plain inverse transform sampling does not give normal variables cheaply; going to two dimensions and polar coordinates solves this.
Marsaglia's polar method avoids the sine and cosine by sampling a uniform point in the unit disc by rejection and rescaling it.
In floating point, $U_1$ must avoid $0$, and the tail of $Z_1$ is truncated by the smallest representable $U_1$.
For speed, many libraries now prefer the ziggurat algorithm, which uses mostly one multiplication and one comparison per sample.
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# Commuting matrices

## Summary

Two square matrices $A$ and $B$ commute when $AB = BA$.
[[Matrix multiplication|032Q9FB8MT]] is [[not commutative|0P506SPZP2]] in general, so commuting pairs are special.
Commuting matrices behave much like numbers: for example, $e^{A + B} = e^{A} e^{B}$ for [[commuting matrices|04BD23PSK5]].
Diagonalisable matrices can be diagonalised by one common basis exactly when they commute, by [[simultaneous diagonalisation|0FK17N4DKA]].

## Explanations

### L1 human

For numbers, the order of a product never matters, but for matrices it usually does.
Matrices that commute are the pairs for which the order does not matter.
As maps, applying $A$ then $B$ gives the same result as applying $B$ then $A$.
For example, every matrix commutes with itself, with its powers, and with every [[scalar matrix|067NKSBEWT]] $cI$.
The matrices $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ do not commute.
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# Doubling measure

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.
For $x \in X$ and $r > 0$, let $B(x, r)$ be the [[open ball|0YE5868HZB]] of centre $x$ and radius $r$.
The measure $\mu$ is *doubling* if:

1. $0 < \mu(B(x, r)) < \infty$ for all $x \in X$ and $r > 0$.
2. There is a constant $C \ge 1$ with $\mu(B(x, 2r)) \le C \, \mu(B(x, r))$ for all $x \in X$ and $r > 0$.

## Remarks

Example: Lebesgue measure $\lambda$ on $\R^n$ is doubling with $C = 2^n$.
Indeed $\lambda(B(x, r)) = \omega_n r^n$ with $\omega_n$ the volume of the unit ball, so the ratio is exactly $2^n$.

Non-example: the standard Gaussian measure $\gamma$ on $\R$ satisfies clause 1 but not clause 2.
Let $\varphi(t) = e^{-t^2/2}/\sqrt{2\pi}$ be its density, which decreases on $[0, \infty)$, and let $x > 2$.
Then $\gamma(B(x, 1)) \le 2 \varphi(x - 1)$ and $\gamma(B(x, 2)) \ge \frac12 \varphi(x - \frac32)$.
So $\gamma(B(x,2)) / \gamma(B(x,1)) \ge \frac14 e^{(x - 5/4)/2}$, which is unbounded as $x \to \infty$.

Some authors drop clause 1 and only ask for clause 2 for balls of positive finite measure.
A metric space with a doubling measure is a doubling metric space: each ball is covered by a bounded number of balls of half the radius.
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# Unit disc

## Description

The *open unit disc* is $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$, and the *closed unit disc* is $\overline{\mathbb{D}} = \set{z \in \C : \abs{z} \le 1}$.
The open disc is open, since it is the preimage of $(-\infty, 1)$ under the continuous map $z \mapsto \abs{z}$, and it is connected, since it is convex.
So it is a [[domain in the complex plane|0EWMDD6C14]].
It is star-shaped about $0$, so every closed curve in it contracts to $0$ along straight segments, and it is a [[simply connected domain|0QW64ARSHB]].
Its boundary is the unit circle $\mathbb{T} = \set{z \in \C : \abs{z} = 1}$.

## Remarks

By the [[Riemann mapping theorem|02WHMEG5ZR]], every simply connected domain other than $\C$ is conformally equivalent to $\mathbb{D}$.
The Cayley map $z \mapsto (z - i)/(z + i)$ is a conformal map of the upper half-plane onto $\mathbb{D}$.
The conformal automorphisms of $\mathbb{D}$ are the [[Möbius transformations|0SG74WBFA2]] $z \mapsto e^{i\theta}(z - a)/(1 - \bar a z)$ with $\abs{a} < 1$ and real $\theta$.
With the hyperbolic metric it becomes the [[Poincaré disk model|85MM2EA0TK]] of the hyperbolic plane.

## Explanations

### L1 human

The unit disc is the set of complex numbers at distance less than $1$ from $0$: a round region without its rim.
It is the standard domain of complex analysis, because by the Riemann mapping theorem many other regions can be turned into it without distorting small angles.
Function spaces such as the [[Hardy spaces|PNNNFJVF6W]] and the [[Bergman spaces|2D619WTPYC]] are usually first defined on the unit disc.
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# Rational homotopy theory

## Summary

Rational homotopy theory studies spaces up to maps that induce isomorphisms on rational homotopy groups, so it ignores all torsion.
For simply connected spaces this coarser theory is completely algebraic.
By the work of Quillen and Sullivan, rational homotopy types correspond to commutative differential graded algebras over $\Q$, or to differential graded Lie algebras.
Concrete invariants such as [[Sullivan minimal models|3QQ2PBGY3N]] make many questions computable that are out of reach integrally.

## Explanations

### L1 human

The homotopy groups of even a sphere are very hard to compute, mostly because of their finite parts.
If one tensors everything with the rationals, the finite parts vanish and what remains is much simpler.
For example, $S^3$ rationally has only one homotopy group, $\Q$ in degree $3$, while $S^2$ rationally has $\Q$ in degrees $2$ and $3$.
Rational homotopy theory describes such answers with algebra: graded algebras with a differential, in place of spaces.
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# Admissible control

## Definition

Let $x' = f(x, u)$ be a [[control system|SBQVGQPXT1]] on $\R^n$ with control set $U \subseteq \R^m$.
Fix a time horizon $T > 0$, an initial state $x_0 \in \R^n$ and a closed target set $S \subseteq \R^n$.
An *admissible control* for $(x_0, T, S)$ is a function $u \colon [0, T] \to \R^m$ such that:

1. $u$ is [[measurable|YYNCJSR0PJ]] and essentially bounded, and $u(t) \in U$ for almost every $t \in [0, T]$;
2. there is an [[absolutely continuous|ZT6F4C2YHK]] trajectory $x \colon [0, T] \to \R^n$ with $x(0) = x_0$ and $x'(t) = f(x(t), u(t))$ for almost every $t$;
3. this trajectory satisfies $x(T) \in S$.

## Remarks

By the local Lipschitz condition on $f$, the trajectory in clause 2 is unique when it exists.
Clause 2 can fail because the solution blows up before time $T$.
Many authors call any control satisfying clause 1 admissible, as in the [[control system|SBQVGQPXT1]] definition, and add clauses 2 and 3 as constraints of the problem.
Problems with free final time, state constraints or other control classes, such as piecewise continuous controls, change the clauses accordingly.

Example: $x' = u$ on $\R$ with $U = [-1, 1]$, $x_0 = 0$, $T = 1$ and $S = \set{\tfrac12}$.
The constant control $u = \tfrac12$ is admissible: it is bounded with values in $U$, and $x(t) = t/2$ gives $x(1) = \tfrac12$.

Non-example: the constant control $u = 1$ in the same setting satisfies clauses 1 and 2, with $x(t) = t$.
It fails clause 3, since $x(1) = 1 \notin S$.
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# Empirical Bayes method

## Summary

An empirical Bayes method estimates the [[prior distribution|00NBBSXRQS]] from the data and then uses it as if it were known.
It applies when many similar parameters are estimated at once, so that the data on all of them show how the parameters are spread.
It sits between full [[Bayesian inference|0JWRJX2RMP]] and frequentist estimation, and it explains shrinkage estimators such as the [[James–Stein estimator|709T196XCT]].

## Explanations

### L1 human

Suppose you rate a thousand baseball players, each from a short season.
Each player's own average is noisy, but the thousand averages together show how batting skill is spread in the league.
Empirical Bayes uses that spread as a prior and pulls each player's average a bit towards the league average.
The pull is strong for players with little data and weak for players with a lot of data.
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# Gabor transform

## Summary

The Gabor transform is the [[short-time Fourier transform|3DHNP5NEWH]] with a Gaussian window.
Dennis Gabor proposed it in 1946 for the analysis of signals.
The Gaussian is the window that is best concentrated in time and frequency at once, in the sense of the Heisenberg [[uncertainty principle|S1FB2K14NV]].
This makes the Gabor transform a natural choice for a time–frequency picture.

## Explanations

### L1 human

Look at a signal through a soft window shaped like a bell curve, centred at a time $\tau$.
Then measure the frequencies of what is seen.
Unlike a window with sharp edges, the bell curve introduces no sharp jumps, so it creates no spurious high frequencies.
Moving $\tau$ along the time axis gives a picture of how the frequency content of the signal changes in time.
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# Multidimensional Z-transform

## Definition

Let $n \ge 1$ and let $x \colon \Z^n \to \C$ be a sequence, written $x[k]$ for $k = (k_1, \dots, k_n)$.
The *$n$-dimensional Z-transform* of $x$ is the function
$$X(z_1, \dots, z_n) = \sum_{k \in \Z^n} x[k]\, z_1^{-k_1} \cdots z_n^{-k_n},$$
defined at those $(z_1, \dots, z_n) \in (\C \setminus \set{0})^n$ where the series converges absolutely.
This set is its [[region of convergence|1P8WX3EGHF]].

## Remarks

For $n = 1$ this is the bilateral [[Z-transform|5FY0JQGHYZ]].
The series is a [[Laurent series|0GXMPDNYX3]] in several variables.
Absolute convergence at $(z_1, \dots, z_n)$ depends only on $(\abs{z_1}, \dots, \abs{z_n})$.

Example: $x[k_1, k_2] = a^{k_1} b^{k_2}$ for $k_1, k_2 \ge 0$ and $x[k] = 0$ otherwise, with $a, b \in \C$.
The sum is a product of two geometric series, and $X(z_1, z_2) = \frac{1}{(1 - a z_1^{-1})(1 - b z_2^{-1})}$ for $\abs{z_1} > \abs{a}$ and $\abs{z_2} > \abs{b}$.

Non-example: $x[k] = 1$ for all $k \in \Z^2$ has an empty region of convergence.
For every $z$, choose signs $\epsilon_1, \epsilon_2 \in \set{1, -1}$ with $\abs{z_j}^{-\epsilon_j} \ge 1$ for $j = 1, 2$.
Then the infinitely many terms with $k_j = \epsilon_j m_j$, $m_1, m_2 \ge 0$, all have absolute value at least $1$, so the series diverges.
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# Bonferroni correction

## Summary

The Bonferroni correction is the simplest way to test many hypotheses at once: test each of $m$ hypotheses at level $\alpha/m$.
By [[Boole's inequality|0EGSEE9J9R]], the [[family-wise error rate|GQCY1N8JZT]] is then at most $\alpha$.
It needs no assumption on the dependence between the tests, but it loses power when $m$ is large.

## Explanations

### L1 human

If you look at many results, some will look surprising just by chance.
The Bonferroni correction makes each single test stricter by dividing the threshold by the number of tests.
With $20$ tests and an overall level of $5\%$, each p-value must be below $0.0025$ to count.
Then the chance of even one false alarm in the whole family is at most $5\%$.
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# Legendre's relation

## Statement

Let $0 < k < 1$ and $k' = \sqrt{1 - k^2}$.
Let $K = K(k)$ and $K' = K(k')$ be [[complete elliptic integrals of the first kind|RM54CPPE12]], and let $E = E(k)$ and $E' = E(k')$ be [[complete elliptic integrals of the second kind|VS2PSX14C8]].
Then
$$E K' + E' K - K K' = \frac{\pi}{2}.$$

## Remarks

Proof idea: differentiation under the integral sign gives $\frac{dE}{dk} = \frac{E - K}{k}$ and $\frac{dK}{dk} = \frac{E - k'^2 K}{k k'^2}$.
With these rules, and the same rules for $E'$ and $K'$ via $dk'/dk = -k/k'$, the derivative of the left side in $k$ is $0$, so the left side is constant.
To find the constant, let $k \to 0$: then $K, E \to \pi/2$ and $E' \to 1$, while $K'$ grows only like $\log(4/k)$ and $E - K = O(k^2)$, so $(E - K)K' \to 0$ and the left side tends to $\pi/2$.
Example: at $k = 1/\sqrt{2}$ one has $k = k'$, so $K = K'$ and $E = E'$, and the relation becomes $2EK - K^2 = \pi/2$.
There is a parallel form for the periods $\omega_1, \omega_2$ and quasi-periods $\eta_1, \eta_2$ of the [[Weierstrass zeta function|QATW7046PD]]: $\eta_1\omega_2 - \eta_2\omega_1 = 2\pi i$ when $\operatorname{Im}(\omega_2/\omega_1) > 0$.

## Explanations

### L1 human

The four complete elliptic integrals $K, E, K', E'$ are transcendental functions of $k$, and none of them has a simple closed form.
Yet this particular combination of them is always exactly $\pi/2$, whatever the modulus.
It is one of the few exact links between these integrals and the number $\pi$.

### L2 human

Legendre's relation is the key identity behind the Gauss–Legendre algorithm for $\pi$, also called the Brent–Salamin algorithm.
That algorithm combines it with the [[arithmetic–geometric mean|E0V3N93DX8]] formula for $K$ and a similar formula for $E$.
For an elliptic curve it is the same fact as the quasi-period relation for the Weierstrass zeta function, read for a curve with real periods.
It is also used to compute $E$ from $K$ at special moduli, for example at $k = 1/\sqrt 2$, where it gives $E(1/\sqrt2) = \frac{\pi/2 + K^2}{2K}$ with $K = K(1/\sqrt2)$, a value tied to the [[lemniscate constant|V5A2S8QYQQ]].
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# Mann–Whitney U test

## Definition

Let $X_1, \dots, X_m$ be [[i.i.d.|0AJN7WHBJZ]] with continuous [[distribution function|0MXMS3AE13]] $F$, and let $Y_1, \dots, Y_n$ be i.i.d. with continuous distribution function $G$, independent of the $X_i$.
The *Mann–Whitney statistic* is
$$U = \#\set{(i, j) : 1 \le i \le m, \ 1 \le j \le n, \ X_i > Y_j}.$$
The *Mann–Whitney U test* of $H_0 \colon F = G$ rejects $H_0$ when $U$ is far from $mn/2$.
For a one-sided alternative it rejects only when $U$ is large, or only when $U$ is small.
Under $H_0$ the law of $U$ depends only on $m$ and $n$.

## Remarks

Let $R_X$ be the sum of the ranks of $X_1, \dots, X_m$ in the pooled sample of size $m + n$.
Then $U = R_X - m(m+1)/2$, so the test is the same as the Wilcoxon rank-sum test.
Under $H_0$, $E[U] = mn/2$ and $\operatorname{Var}(U) = mn(m + n + 1)/12$, and $U$ is asymptotically normal.
The ratio $U/(mn)$ is an unbiased estimator of $P(X_1 > Y_1)$, and $U$ is a two-sample [[U-statistic|NTVQB1DC56]].
With ties, each tied pair counts $1/2$.

Example: $X = (3, 5)$ and $Y = (1, 4)$.
The pairs with $X_i > Y_j$ are $(3, 1)$, $(5, 1)$ and $(5, 4)$, so $U = 3$.
In the pooled order $1, 3, 4, 5$ the $X$ values have ranks $2$ and $4$, and $R_X - 3 = 6 - 3 = 3$ agrees.

Non-example: measurements of the same subjects before and after a treatment are not independent samples.
The clause on independence fails, and the [[Wilcoxon signed-rank test|38SETDT2HN]] on the differences is used instead.
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# Haefliger structure

## Summary

A Haefliger structure is a generalisation of a [[foliation|98HTB7MSPX]] in which the local defining maps need not be submersions.
It is given by local maps to $\R^q$ glued by [[germs|VZXF7RNFKS]] of local diffeomorphisms.
In other words, it is a cocycle with values in the groupoid of such germs.
Unlike foliations, Haefliger structures can be pulled back along any continuous map.
Up to homotopy, they are classified by maps into a [[classifying space|WSZ40H5E4T]] $B\Gamma_q$.
This homotopy theory is the basis of the existence theory of foliations.

## Explanations

### L1 human

A foliation of codimension $q$ is locally described by a submersion to $\R^q$, whose level sets are the pieces of leaves.
On overlaps the local submersions differ by local diffeomorphisms of $\R^q$.
A Haefliger structure keeps the gluing data but drops the requirement that the local maps be submersions.
A constant map to $\R^q$ already gives one, even though it does not foliate anything.
This flexibility makes Haefliger structures behave like vector bundles: they pull back along maps and have a classifying space.
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# Directional statistics

## Summary

Directional statistics is the part of statistics whose observations are directions, angles or rotations.
Such data live on the circle, on a [[sphere|T62369RFH6]] or on other manifolds, not on a line, so the usual mean and variance can mislead.
Means are taken of unit vectors instead of angles, and models such as the [[von Mises–Fisher distribution|V8ZZ7A2FMF]] replace the normal distribution.
Applications include wind directions, animal navigation, times of day and orientations of rocks.

## Explanations

### L1 human

Two wind directions of $350$ and $10$ degrees both point almost due north.
Their ordinary average is $180$ degrees, due south, which is plainly wrong.
Directional statistics treats each direction as an arrow of length $1$ and adds the arrows.
The sum of these two arrows points due north, the sensible answer.
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# Lévy–Prokhorov metric

## Summary

The Lévy–Prokhorov metric is a distance between Borel [[probability measures|059TJN0NZ1]] on a [[metric space|0VKD1TP7F3]].
Two measures are close when each gives to every set little more mass than the other gives to a slight enlargement of it.
On a separable metric space, convergence in this metric is the same as [[weak convergence|MBW9JCGZC1]].
So weak convergence of probability measures comes from a metric.

## Explanations

### L1 human

Moving mass a short distance should count as a small change, and so should changing a small amount of mass.
The Lévy–Prokhorov distance allows both kinds of error at once, with the same tolerance $\varepsilon$.
For two point masses at $x$ and $y$, the distance is the smaller of $d(x, y)$ and $1$.
So point masses at $1/n$ and at $0$ are at distance $1/n$, which tends to $0$.
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# Ornstein isomorphism theorem

## Statement

Let $p = (p_1, \dots, p_k)$ and $q = (q_1, \dots, q_l)$ be probability vectors, and let $B(p)$ and $B(q)$ be the two-sided [[Bernoulli schemes|D33JK3AXRZ]] they define.
Two invertible [[measure-preserving transformations|556WBDAHNP]] $T$ on $(X, \mu)$ and $S$ on $(Y, \nu)$ are isomorphic if the following holds.
There are sets $X_0 \subseteq X$, $Y_0 \subseteq Y$ of full measure and a measurable bijection $\varphi \colon X_0 \to Y_0$ with measurable inverse.
It satisfies $\nu(\varphi(A)) = \mu(A)$ for measurable $A \subseteq X_0$, and $\varphi \circ T = S \circ \varphi$ on $X_0$.
Then $B(p)$ and $B(q)$ are isomorphic if and only if they have the same [[Kolmogorov–Sinai entropy|X1D1C7VH6P]], that is,
$$-\sum_{i=1}^{k} p_i \log p_i = -\sum_{j=1}^{l} q_j \log q_j.$$

## Remarks

The direction "isomorphic implies equal entropy" is due to Kolmogorov, who introduced entropy as an isomorphism invariant in 1958.
The converse is Ornstein's theorem of 1970.
Its proof builds the isomorphism as a limit of better and better finite codes, using the finitely determined property, which every Bernoulli scheme has.
Example: $B(1/4, 1/4, 1/4, 1/4)$ and $B(1/2, 1/8, 1/8, 1/8, 1/8)$ both have entropy $\log 4$, so they are isomorphic, although their alphabets have different sizes.
Meshalkin had found an explicit isomorphism in this case before Ornstein's theorem.
On the other hand $B(1/2, 1/2)$ and $B(1/3, 1/3, 1/3)$ have entropies $\log 2 \ne \log 3$, so they are not isomorphic, even though they are spectrally isomorphic.
Ornstein's theory extends the theorem to Bernoulli schemes on countable alphabets and to Bernoulli flows.

## Explanations

### L1 human

Two machines that produce random symbols independently, one symbol per second, might look very different, for instance with different numbers of symbols.
The theorem says that one can be perfectly recoded into the other, in a way that respects time, exactly when both produce the same amount of randomness per second.
Entropy, a single number, is the complete answer.

### L2 human

Before Ornstein, it was not known whether entropy was the only obstruction.
The theorem makes entropy a complete invariant for Bernoulli schemes.
Ornstein's methods then showed that many smooth systems are isomorphic to Bernoulli schemes.
Examples are ergodic toral automorphisms and geodesic flows on surfaces of negative curvature.
Every Bernoulli scheme is a [[Kolmogorov automorphism|YRHJA7C7HS]], but Ornstein also constructed K-automorphisms that are not isomorphic to any Bernoulli scheme.
Compare with Sinai's factor theorem, which gives Bernoulli factors of every system of positive entropy.
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# 4-manifold

## Summary

A $4$-manifold is a [[topological manifold|QQQ9B1DK4P]] of dimension $4$.
Dimension $4$ is special: the methods that classify manifolds of dimension at least $5$ fail, and topological and smooth $4$-manifolds behave very differently.
Freedman classified simply connected closed topological $4$-manifolds through their [[intersection form|WK8GKRKAVD]], together with the Kirby–Siebenmann invariant when the form is odd.
[[Donaldson's theorem|34B7K77P31]] and Seiberg–Witten theory show that many of them have no smooth structure, or many smooth structures.
Even $\R^4$ has [[exotic smooth structures|X44PNC38E2]], which happens in no other dimension.

## Explanations

### L1 human

A $4$-manifold is a space that looks like $4$-dimensional space near each point.
Examples are the $4$-sphere, the product $S^2 \times S^2$ of two spheres, and the complex projective plane.
Dimension $4$ is the hardest one in topology.
There is too little room to untangle things as in higher dimensions, and too much room for the tools of dimensions $2$ and $3$.
Ideas from physics, such as gauge theory, gave the main tools for smooth $4$-manifolds.
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<document source="https://tamnd.github.io/math2/node/8HA39P7ZFN.md" tag="8HA39P7ZFN" kind="definition">
# Richardson extrapolation

## Definition

Let $A^*$ be a number and let $A(h)$ be approximations of $A^*$, defined for small $h > 0$.
Assume that there are a constant $c$ and exponents $0 < k < k'$ such that
$$A(h) = A^* + c\, h^k + O(h^{k'}) \quad (h \to 0),$$
in [[big O notation|0RYTGBJAGN]].
For a fixed ratio $t > 1$, the *Richardson extrapolation* of $A$ is
$$R(h) = \frac{t^k A(h/t) - A(h)}{t^k - 1}.$$
Then $R(h) = A^* + O(h^{k'})$.

## Remarks

The last claim is a direct computation: in $t^k A(h/t) - A(h)$ the terms $t^k c (h/t)^k$ and $c h^k$ cancel, and $t^k A^* - A^* = (t^k - 1)A^*$.
Example: for a function $f$ with five continuous derivatives, the central difference $A(h) = \frac{f(x+h) - f(x-h)}{2h}$ satisfies $A(h) = f'(x) + \frac{f'''(x)}{6} h^2 + O(h^4)$.
With $k = 2$ and $t = 2$, the extrapolation $\frac{4A(h/2) - A(h)}{3}$ approximates the [[derivative|0F1WR9C83K]] $f'(x)$ with error $O(h^4)$.
Non-example: if the true error is $A(h) = A^* + h$ but one uses $k = 2$, then $R(h) = A^* + \frac{h}{3}$, which is still of first order; the exponent $k$ must be the true leading exponent.
The procedure can be repeated with the next exponent of the error expansion, which gives a triangular table of extrapolated values.

## Explanations

### L1 human

Compute with two step sizes, and use the known shape of the error to remove its main part.
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<document source="https://tamnd.github.io/math2/node/8HDWPN2FXS.md" tag="8HDWPN2FXS" kind="definition">
# Determinant line bundle

## Definition

Let $H_0$ and $H_1$ be complex [[Hilbert spaces|HM2HHE3J51]], and let $Y$ be a topological space.
Let $y \mapsto T_y$ be a map from $Y$ to the [[Fredholm operators|8HWGKWD6ZM]] $H_0 \to H_1$ that is continuous in the operator norm.
For a finite-dimensional space $V$ write $\det V = \Lambda^{\dim V} V$, with $\det 0 = \C$.
The *determinant line* of $T_y$ is
$$\det T_y = (\det \ker T_y)^* \otimes \det \operatorname{coker} T_y.$$
The *determinant line bundle* is the set $\det T = \bigsqcup_{y \in Y} \det T_y$ with the following topology.
Near a point $y_0$, choose a finite-dimensional subspace $V \subseteq H_1$ with $\operatorname{im} T_y + V = H_1$ for all $y$ near $y_0$.
Then the spaces $T_y^{-1}(V)$ form a [[vector bundle|P4DZM975BV]] over a neighbourhood of $y_0$.
The exact sequence $0 \to \ker T_y \to T_y^{-1}(V) \to V \to \operatorname{coker} T_y \to 0$ gives a canonical isomorphism $\det T_y \cong (\det T_y^{-1}(V))^* \otimes \det V$.
The right side is a line bundle near $y_0$.
The topology of $\det T$ is the one that makes these isomorphisms continuous, and it does not depend on the choice of $V$.

## Remarks

Some authors use the dual line $\det \ker T_y \otimes (\det \operatorname{coker} T_y)^*$; the two conventions give dual line bundles.

Example: if every $T_y$ is invertible, then $\ker T_y = 0$ and $\operatorname{coker} T_y = 0$, so each fibre is $\C$ and $\det T$ is the trivial line bundle $Y \times \C$.

Non-example: the family $T_y = y \cdot \id$ on an infinite-dimensional Hilbert space, for $y \in [0, 1]$, is not a family of Fredholm operators, since $T_0 = 0$ has infinite-dimensional kernel.
So $\det T_0$ is not defined.
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<document source="https://tamnd.github.io/math2/node/8HEZ9TW1NK.md" tag="8HEZ9TW1NK" kind="concept">
# BK-space

## Summary

A BK-space is a [[sequence space|XYTN19EPF2]] with a norm that makes it a [[Banach space|06GZ956657]] in which each coordinate $x \mapsto x_n$ is continuous.
The letters stand for Banach and the German word Koordinate.
BK-spaces are exactly the [[FK-spaces|71DMQHW6JZ]] whose topology comes from a norm.
The spaces $\ell^p$, $c$ and $c_0$ with their usual norms are the basic examples.

## Explanations

### L1 human

A BK-space is a space of sequences with a way of measuring size, a norm, that behaves well.
The space is complete, so there are no "missing" limits.
And a small norm forces every single term of the sequence to be small.
For example, in the space of bounded sequences the norm is the largest absolute value of the terms, so each term is at most the norm.
</document>

<document source="https://tamnd.github.io/math2/node/8HFAEGFQT1.md" tag="8HFAEGFQT1" kind="concept">
# Associated graded ring

## Summary

The associated graded ring of a [[commutative ring|0310REJJY8]] $R$ with respect to an [[ideal|096Z9MP2ME]] $I$ is the [[graded ring|8KB4SQGC0C]] $\operatorname{gr}_I(R) = \bigoplus_{n \ge 0} I^n/I^{n+1}$.
It keeps the leading terms of elements and forgets the rest.
For a local ring and its maximal ideal it describes the tangent cone of the point.
Since it is graded, tools for graded rings such as the [[Hilbert polynomial|RMH4FBCFDA]] apply to local rings through it.

## Explanations

### L1 human

Near a point, a curve looks to first approximation like its tangent lines.
The associated graded ring records only these first approximations, by keeping the lowest-degree part of each function.
For the integers and a prime $p$, the piece $p^n\Z/p^{n+1}\Z$ records one digit in base $p$, and these pieces together form the polynomial ring $\F_p[t]$.
The [[definition|XBZ99KH3RP]] makes this precise.
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# Crossed product

## Summary

The crossed product $A \rtimes_\alpha G$ is a C*-algebra built from a C\*-algebra $A$ and a group $G$.
The group acts on $A$ by automorphisms through a [[group action|0WCY42EV9V]] $\alpha$.
When $A$ is unital and $G$ is discrete, it contains $A$ and unitaries $u_g$ for $g \in G$.
In it the action becomes conjugation: $u_g a u_g^* = \alpha_g(a)$.
It turns a dynamical system into a single algebra, and many examples, such as the noncommutative tori, arise this way.
There is a parallel construction for von Neumann algebras.

## Explanations

### L1 human

Suppose a group moves the points of a space around.
The crossed product puts the functions on the space and the group elements into one algebra.
Moving a function by the group then becomes the algebraic operation $a \mapsto u_g a u_g^{*}$.
For the group $\Z$ rotating a circle by an irrational angle, the crossed product is the noncommutative torus.
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<document source="https://tamnd.github.io/math2/node/8HHY1K3XDC.md" tag="8HHY1K3XDC" kind="definition">
# Field norm

## Definition

Let $L/K$ be a [[finite extension|0NRYPM9Y27]] and let $\alpha \in L$.
Let $m_\alpha \colon L \to L$ be the map $x \mapsto \alpha x$; it is a [[$K$-linear map|0RJF668P1T]] of the finite-dimensional $K$-vector space $L$.
The *norm* of $\alpha$ from $L$ to $K$ is $N_{L/K}(\alpha) = \det(m_\alpha) \in K$, the [[determinant|03F5KP94K0]] of $m_\alpha$.

## Remarks

The norm is multiplicative: $N_{L/K}(\alpha\beta) = N_{L/K}(\alpha) N_{L/K}(\beta)$, and $N_{L/K}(c) = c^{[L:K]}$ for $c \in K$.
It is transitive in towers: $N_{M/K} = N_{L/K} \circ N_{M/L}$.
If the [[minimal polynomial|0GK1WB9R98]] of $\alpha$ over $K$ is $x^d + a_{d-1}x^{d-1} + \cdots + a_0$, then $N_{L/K}(\alpha) = \left((-1)^d a_0\right)^{[L:K(\alpha)]}$.
If $L/K$ is separable and $\sigma_1, \ldots, \sigma_n$ are the embeddings of $L$ over $K$ into an algebraic closure, then $N_{L/K}(\alpha) = \sigma_1(\alpha) \cdots \sigma_n(\alpha)$.

Example: $L = \Q(i)$, $K = \Q$ and $\alpha = a + bi$ with $a, b \in \Q$.
Check: in the basis $1, i$, the map $m_\alpha$ sends $1 \mapsto a + bi$ and $i \mapsto -b + ai$, so its matrix is $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$ and $N(\alpha) = a^2 + b^2$.

Non-example: the complex absolute value $a + bi \mapsto \sqrt{a^2 + b^2}$ is not the norm of $\Q(i)/\Q$, since it does not take values in $\Q$; for example $\abs{1 + i} = \sqrt{2}$.
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# Central difference

## Definition

Let $f \colon \R \to \R$ be a function and let $h > 0$.
The *central difference* of $f$ with step $h$ is the function $\delta_h f \colon \R \to \R$ given by
$$\delta_h f(x) = f\left(x + \tfrac{h}{2}\right) - f\left(x - \tfrac{h}{2}\right).$$

## Remarks

It is the [[finite difference|7JSQZ126BP]] with offsets $-\tfrac{h}{2}$ and $\tfrac{h}{2}$.
Example: for $f(x) = x^3$, $\delta_h f(x) = 3hx^2 + \tfrac{h^3}{4}$, by expanding both cubes.
Non-example: $f(x + h) - f(x)$ is not centred at $x$, so it is the forward difference and not the central difference.
For sequences one often uses the version $f(x + h) - f(x - h)$ with step $2h$, which avoids half-integer points.
If $f$ is three times continuously differentiable, Taylor's theorem gives $\frac{f(x + h) - f(x - h)}{2h} = f'(x) + O(h^2)$ as $h \to 0$.
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# Almost convergence

## Summary

A bounded real sequence is almost convergent to $L$ when every [[Banach limit|SYF2ND6J97]] gives it the value $L$.
Every convergent sequence is almost convergent to its limit, but some divergent sequences, such as $1, 0, 1, 0, \dots$, are almost convergent too.
[[Lorentz's characterization|P8TMYNPSYD]] describes almost convergence by averages over long blocks of consecutive terms.

## Explanations

### L1 human

Banach limits are ways to assign a "limit" to every bounded sequence, and there are many of them.
For most divergent sequences they disagree with each other.
A sequence is almost convergent when all of them agree on one value.
For $1, 0, 1, 0, \dots$ they all give $1/2$, which is the average value the sequence takes in the long run, uniformly along the sequence.
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<document source="https://tamnd.github.io/math2/node/8HWGKWD6ZM.md" tag="8HWGKWD6ZM" kind="concept">
# Fredholm operator

## Summary

A Fredholm operator is a bounded linear map between Banach spaces whose kernel and cokernel are both finite-dimensional.
Such an operator is invertible up to finite-dimensional errors, so linear equations with it behave almost like square linear systems.
Fredholm operators carry an integer invariant, the index, which is stable under small and compact perturbations and links operator theory with topology.

## Explanations

### L1 human

For a square matrix, the equation $Ax = b$ either has a unique solution for every $b$, or it fails in a finite number of directions.
A Fredholm operator has the same kind of behaviour on infinite-dimensional spaces.
Only finitely many independent vectors are sent to zero, and only finitely many independent directions are missed by the image.
For example, the shift $(x_1, x_2, \ldots) \mapsto (0, x_1, x_2, \ldots)$ on square-summable sequences sends no nonzero vector to zero and misses only the direction of the first coordinate.
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<document source="https://tamnd.github.io/math2/node/8J3B7N8CAT.md" tag="8J3B7N8CAT" kind="concept">
# Multivariate analysis of variance

## Summary

Multivariate analysis of variance, or MANOVA, tests whether several groups have the same mean vector when each observation consists of several measurements.
It extends the [[analysis of variance|M9AQ5HKGP4]] from one response variable to a vector of responses.
It compares a between-group and a within-group matrix of sums of squares and cross-products.
Common test statistics are Wilks' lambda, Pillai's trace, the Hotelling–Lawley trace and Roy's largest root.

## Explanations

### L1 human

Three teaching methods may be compared by both reading scores and writing scores of the pupils.
Running two separate comparisons ignores that the two scores are correlated.
MANOVA asks one question about both scores at once: do the groups differ in their average pair of scores?
It weighs how far apart the group averages are against how much the pupils vary inside each group.
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<document source="https://tamnd.github.io/math2/node/8J8AW0TG9M.md" tag="8J8AW0TG9M" kind="statement">
# Besicovitch–Federer projection theorem

## Statement

Let $1 \le m < n$ be integers, and let $\mathcal{H}^m$ be the $m$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] on $\R^n$.
Let $G(n, m)$ be the Grassmannian of $m$-dimensional linear subspaces of $\R^n$, with its unique probability measure $\gamma_{n,m}$ that is invariant under the orthogonal group.
For $V \in G(n, m)$, let $P_V \colon \R^n \to V$ be the orthogonal projection onto $V$.
Let $E \subseteq \R^n$ be $\mathcal{H}^m$-measurable with $\mathcal{H}^m(E) < \infty$.

1. $E$ is purely $m$-unrectifiable if and only if $\mathcal{H}^m(P_V(E)) = 0$ for $\gamma_{n,m}$-almost every $V \in G(n, m)$.
2. If $E$ is [[$m$-rectifiable|7N0GFMW05R]] and $\mathcal{H}^m(E) > 0$, then $\mathcal{H}^m(P_V(E)) > 0$ for $\gamma_{n,m}$-almost every $V \in G(n, m)$.

## Remarks

Part 2 is the easier half.
A rectifiable set of positive measure has an approximate tangent plane $W$ at almost every point, and for almost every $V$ the projection $P_V$ is injective on $W$.
Then $P_V$ is bi-Lipschitz on a piece of $E$ of positive measure, so the projection has positive measure.
Part 1 is the deep half; Besicovitch proved it for $m = 1$ and $n = 2$, and Federer proved the general case.
Example: the four-corner Cantor set in $\R^2$ has positive and finite $\mathcal{H}^1$ measure and is purely $1$-unrectifiable, so almost all of its projections onto lines have length zero.

## Explanations

### L1 human

Shine light on a set and look at its shadows on lines or planes in all directions.
If the set is made of curve-like or surface-like pieces, almost every shadow has positive size.
If the set is a fractal dust with no such pieces, almost every shadow has size zero, even though the set itself has positive size.
So the shadows tell whether a set is rectifiable.

### L2 human

A set $E$ of finite measure splits as $E = R \cup P$ with $R$ rectifiable and $P$ purely unrectifiable.
The theorem says that almost every projection of $E$ sees only $R$.
It is the model for many later results that characterise rectifiability by densities, tangent measures or singular integrals.
It is used in the study of removable sets for bounded analytic functions, through the [[analytic capacity|QGQPHRANWZ]] of planar sets.
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# Invertible sheaf

## Definition

Let $(X, \mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]].
An *invertible sheaf* on $X$ is a [[sheaf of $\mathcal{O}_X$-modules|S33B2H3MYD]] $\mathcal{L}$ with the following property.
Every point of $X$ has an open neighbourhood $U$ such that $\mathcal{L}|_U \cong \mathcal{O}_X|_U$ as $\mathcal{O}_U$-modules.

## Remarks

The structure sheaf $\mathcal{O}_X$ is invertible, with $U = X$ for every point.
On $X = \mathbb{A}^2_k = \operatorname{Spec} k[x, y]$ over a field $k$, the ideal sheaf $\mathcal{I}$ of the origin, given by the ideal $(x, y)$, is not invertible.
Its stalk at the origin is the maximal ideal of $k[x, y]_{(x, y)}$, which needs two generators, so it is not free of rank $1$.
In other words, an invertible sheaf is a locally free sheaf of rank $1$.
For invertible sheaves $\mathcal{L}$ and $\mathcal{M}$, the tensor product $\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{M}$ is invertible.
The dual $\mathcal{L}^\vee = \mathcal{H}om(\mathcal{L}, \mathcal{O}_X)$ satisfies $\mathcal{L} \otimes \mathcal{L}^\vee \cong \mathcal{O}_X$, which explains the name.
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<document source="https://tamnd.github.io/math2/node/8JMZDBJNZR.md" tag="8JMZDBJNZR" kind="definition">
# Additive energy

## Definition

Let $G$ be an [[abelian group|082GHNPJ53]], and let $A$ and $B$ be finite subsets of $G$.
The *additive energy* of $A$ and $B$ is the number
$$E(A, B) = \abs{\set{(a_1, a_2, b_1, b_2) \in A \times A \times B \times B : a_1 + b_1 = a_2 + b_2}}.$$
One writes $E(A) = E(A, A)$.

## Remarks

Let $r(x)$ be the number of pairs $(a, b) \in A \times B$ with $a + b = x$.
Then $E(A, B) = \sum_{x} r(x)^2$, summed over the [[sumset|XNBN31HQWB]] $A + B$.
For $A = \set{0, 1, 2}$ in $\Z$, the values of $r$ on $0, 1, 2, 3, 4$ are $1, 2, 3, 2, 1$, so $E(A) = 1 + 4 + 9 + 4 + 1 = 19$.
A quadruple with $a_1 + b_1 \ne a_2 + b_2$ is not counted; for example $(0, 1, 0, 0)$ is not counted for $A = B = \set{0, 1, 2}$, since $0 + 0 \ne 1 + 0$.
The trivial quadruples with $a_1 = a_2$ and $b_1 = b_2$ always count, so $E(A, B) \ge \abs{A}\abs{B}$.
Also $E(A, B) \le \min(\abs{A}^2\abs{B}, \abs{A}\abs{B}^2)$.
The Cauchy–Schwarz inequality gives $E(A, B) \ge \abs{A}^2 \abs{B}^2 / \abs{A + B}$.
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# Minimum description length

## Summary

The minimum description length (MDL) principle selects the model that gives the shortest description of the data.
The description counts the bits to state the model and the bits to encode the data with the help of the model.
It turns model selection into data compression: a good model captures regularities and so compresses the data.
It is a computable relative of [[Kolmogorov complexity|8PY3K6NCKF]], and Jorma Rissanen introduced it.

## Explanations

### L1 human

To send a long list of numbers to a friend, you may first send a rule, such as "a straight line plus small errors", and then the errors.
A simple rule is short to send but may leave large errors, which are long to send.
A very complex rule leaves tiny errors but is itself long.
MDL chooses the rule for which the total message is shortest.
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# Robinson–Schensted–Knuth correspondence

## Statement

Let $m, n \ge 1$.
Let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] with nonnegative integer entries.
List the pairs $(i, j)$ in lexicographic order, each pair repeated $a_{ij}$ times.
Insert the second coordinates $j$ in this order into the empty tableau by row insertion, and let $P(A)$ be the result.
Let $Q(A)$ be the tableau that has entry $i$ in the cell created when the pair $(i, j)$ is inserted.

1. $P(A)$ and $Q(A)$ are semistandard [[Young tableaux|Q6TH9PH2R7]] of the same shape.
2. The entry $j$ appears in $P(A)$ exactly $\sum_i a_{ij}$ times, and the entry $i$ appears in $Q(A)$ exactly $\sum_j a_{ij}$ times.
3. The map $A \mapsto (P(A), Q(A))$ is a bijection onto the set of pairs $(P, Q)$ of semistandard tableaux of the same shape. Here $P$ has entries in $\set{1, \dots, n}$ and $Q$ has entries in $\set{1, \dots, m}$.
4. $P(A^T) = Q(A)$ and $Q(A^T) = P(A)$.

## Remarks

The insertion rule is the one of the [[Robinson–Schensted correspondence|1SCF8X47G1]]: an inserted entry $x$ bumps the leftmost entry of the row that is strictly larger than $x$.
Equal first coordinates are listed with increasing second coordinates, so the recording tableau stays semistandard.
For a permutation matrix the construction gives back the Robinson–Schensted correspondence.

Example: for $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ the pairs are $(1,1), (1,2), (2,2)$.
Inserting $1, 2, 2$ gives $P = \begin{smallmatrix} 1 & 2 & 2 \end{smallmatrix}$, and $Q = \begin{smallmatrix} 1 & 1 & 2 \end{smallmatrix}$.
The column sums $1, 2$ of $A$ match the content of $P$, and the row sums $2, 1$ match the content of $Q$.

## Explanations

### L1 human

The correspondence turns a table of counts into a pair of semistandard tableaux of the same shape, and back.
The column totals of the table become the numbers of $1$s, $2$s, and so on in one tableau.
The row totals become the same numbers for the other tableau.

### L2 human

Summing over all matrices with weight $\prod_{i,j} (x_i y_j)^{a_{ij}}$ gives the Cauchy identity
$$\prod_{i=1}^{m} \prod_{j=1}^{n} \frac{1}{1 - x_i y_j} = \sum_{\lambda} s_\lambda(x_1, \dots, x_m) \, s_\lambda(y_1, \dots, y_n),$$
where $s_\lambda$ is the [[Schur polynomial|43VFMJCDYJ]].
Part 4 applied to symmetric matrices gives similar identities for $\sum_\lambda s_\lambda$.
The correspondence is due to Knuth, and it is the main bijective tool in the theory of symmetric functions.
</document>

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# Root-finding algorithm

## Definition

Let $D \subseteq \R^n$ and let $f \colon D \to \R^n$ be a [[continuous function|0X3WGX95FE]]; the case $n = 1$ is the most common, and $\C$ may replace $\R$.
A *zero* (or *root*) of $f$ is a point $r \in D$ with $f(r) = 0$.
A *root-finding algorithm* is a procedure that takes $f$ and starting data.
It produces a [[sequence|04CC6K5VDT]] $(x_k)$ in $D$, or a sequence of nested sets containing a zero.
It uses $f$ only through values of $f$ and possibly of its derivatives.
It is designed so that $(x_k)$ has a zero of $f$ as its [[limit|05M5SWSZZD]], or so that the sets shrink to a zero.

## Remarks

Example: the iteration $x_{k+1} = x_k - (x_k^2 - 2)/(2x_k)$ from $x_0 = 1$ gives $1, 1.5, 1.41667, 1.414216, \ldots$, which tends to the zero $\sqrt{2}$ of $f(x) = x^2 - 2$.
Non-example: the iteration $x_{k+1} = 2/x_k$ from $x_0 = 1$ oscillates between $1$ and $2$, so it does not tend to a zero of $x^2 - 2$, although $\sqrt{2}$ is its fixed point.
An algorithm is *globally convergent* on a class of functions if it converges from every admissible start, and *locally convergent* if it converges from starts close enough to a zero.
Its speed is measured by its order of convergence, see [[rate of convergence|YWZQC5E54B]].
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# Little q-Jacobi polynomials

## Definition

Let $0 < q < 1$, let $a, b$ be real numbers, and let $n \ge 0$ be an integer.
Write $(c; q)_k$ for the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].
Assume $aq^{j} \ne 1$ for $j = 1, \ldots, n$.
The *little $q$-Jacobi polynomial* of degree $n$ is
$$p_n(x; a, b \mid q) = {}_2\phi_1\!\left( \begin{matrix} q^{-n}, \; abq^{n+1} \\ aq \end{matrix} ; q, \, qx \right) = \sum_{k=0}^{n} \frac{(q^{-n}; q)_k \, (abq^{n+1}; q)_k}{(aq; q)_k \, (q; q)_k} (qx)^k,$$
where ${}_2\phi_1$ is the [[basic hypergeometric series|EB4N9RQR51]].
The series stops at $k = n$, because $(q^{-n}; q)_k = 0$ for $k > n$.

## Remarks

Orthogonality: if $0 < aq < 1$ and $bq < 1$, then for $m \ne n$
$$\sum_{k=0}^{\infty} \frac{(bq; q)_k}{(q; q)_k} (aq)^k \, p_m(q^k) \, p_n(q^k) = 0,$$
and the weights $\frac{(bq; q)_k}{(q; q)_k} (aq)^k$ are positive.

Example: for $n = 1$ the sum has two terms, and $(1 - q^{-1}) q / (1 - q) = -1$ gives
$$p_1(x) = 1 - \frac{1 - abq^2}{1 - aq} \, x.$$
Check of orthogonality to $p_0 = 1$ when $b = 0$: the weights are $w_k = (aq)^k / (q; q)_k$, and the $q$-binomial theorem gives $\sum_k w_k = 1/(aq; q)_\infty$ and $\sum_k w_k q^k = 1/(aq^2; q)_\infty$.
So $\sum_k w_k p_1(q^k) = \frac{1}{(aq; q)_\infty} - \frac{1}{(1 - aq)(aq^2; q)_\infty} = 0$, since $(aq; q)_\infty = (1 - aq)(aq^2; q)_\infty$.

Non-example: for $b = 0$ the monomial $x$ is not the little $q$-Jacobi polynomial of degree $1$, since $x$ is not orthogonal to $p_0 = 1$ for these positive weights: $\sum_k w_k q^k > 0$.

Limit: $\lim_{q \to 1} p_n(x; q^\alpha, q^\beta \mid q) = P_n^{(\alpha, \beta)}(1 - 2x) / P_n^{(\alpha, \beta)}(1)$, where $P_n^{(\alpha,\beta)}$ is the [[Jacobi polynomial|TRD7DSF2KR]].
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<document source="https://tamnd.github.io/math2/node/8K5FT58EK5.md" tag="8K5FT58EK5" kind="concept">
# Orthocentric system

## Summary

An orthocentric system is a set of four points of the plane.
In it, one point is the [[orthocentre|0WB619CA0V]] of the triangle formed by the other three.
Then each of the four points is the orthocentre of the triangle formed by the other three.
The four triangles share the same [[nine-point circle|033H8MQRC3]], and their circumcircles all have the same radius.

## Explanations

### L1 human

Take a triangle that has no right angle and mark its orthocentre, where the altitudes meet.
Now you have four points.
Pick any three of them: their orthocentre is the fourth point.
So the four points play symmetric roles, and none of them is special.
For example, $(0, 2)$, $(-1, 0)$, $(1, 0)$ and $(0, \frac{1}{2})$ form such a system.
</document>

<document source="https://tamnd.github.io/math2/node/8K9YQ1YGVS.md" tag="8K9YQ1YGVS" kind="definition">
# Beta function

## Definition

The *beta function* is the function $B \colon (0, \infty) \times (0, \infty) \to \R$ given by the [[improper integral|0VCTBJMEJF]]
$$B(x, y) = \int_0^1 t^{x - 1} (1 - t)^{y - 1}\, dt.$$
The integral converges for all real $x > 0$ and $y > 0$.

## Remarks

Example: $B(\frac12, \frac12) = \int_0^1 \frac{dt}{\sqrt{t(1-t)}}$.
The substitution $t = \sin^2 \theta$ gives $dt = 2 \sin\theta \cos\theta\, d\theta$ and $\sqrt{t(1-t)} = \sin\theta\cos\theta$, so $B(\frac12, \frac12) = \int_0^{\pi/2} 2\, d\theta = \pi$.

Non-example of a valid argument: for $x = 0$ and $y = 1$ the integral is $\int_0^1 t^{-1}\, dt = \infty$, so $(0, 1)$ is outside the domain.

The beta function is symmetric, $B(x, y) = B(y, x)$, and it satisfies $B(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x + y)}$, where $\Gamma$ is the [[gamma function|0V3FZMVB4K]].
For positive integers $m, n$ this gives $B(m, n) = \frac{(m-1)!\,(n-1)!}{(m + n - 1)!}$.
The same integral defines $B(x, y)$ for complex $x, y$ with positive real parts.
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<document source="https://tamnd.github.io/math2/node/8KB4SQGC0C.md" tag="8KB4SQGC0C" kind="concept">
# Graded ring

## Summary

A graded ring is a [[ring|0B1261B5ED]] that splits as a direct sum of pieces $R_0, R_1, R_2, \dots$.
The product of an element of $R_m$ and an element of $R_n$ must lie in $R_{m+n}$.
The main example is a polynomial ring, split by degree.
Graded rings are the algebraic side of projective geometry, and they appear in commutative algebra through [[associated graded rings|8HFAEGFQT1]] and [[Rees algebras|W8ZAH8TTHS]].

## Explanations

### L1 human

In a polynomial ring every polynomial is a sum of homogeneous parts, one for each degree.
Multiplying two homogeneous parts adds their degrees.
A graded ring is any ring with this kind of bookkeeping by degree.
For example, $x^2 + 3xy + 5$ in $k[x, y]$ has a part $x^2 + 3xy$ of degree $2$ and a part $5$ of degree $0$.
The [[definition|CAPQBQDRA7]] makes this precise.
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<document source="https://tamnd.github.io/math2/node/8KCB5KJKNT.md" tag="8KCB5KJKNT" kind="concept">
# Symplectic capacity

## Summary

A symplectic capacity is a number attached to symplectic manifolds, or to subsets of $\R^{2n}$, that measures their size.
It is compatible with symplectic embeddings.
It cannot grow under symplectic embeddings and is normalized so that a ball and a cylinder of the same radius have the same capacity.
The existence of capacities is equivalent in strength to [[Gromov's non-squeezing theorem|RDX55HQ3XT]], and capacities are a main tool of quantitative symplectic topology.

## Explanations

### L1 human

Volume is a natural size for a region of space, but in symplectic geometry it misses much.
A thin infinite cylinder has infinite volume, yet a large ball cannot be fitted into it symplectically.
A capacity is a size that sees this: it behaves like an area, measured in a two-dimensional symplectic direction.
The simplest one is the Gromov width, the area $\pi r^2$ of the largest ball of radius $r$ that fits into a set by a symplectic embedding.
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<document source="https://tamnd.github.io/math2/node/8KDV43WZ3Z.md" tag="8KDV43WZ3Z" kind="statement">
# Generalized Poincaré conjecture

## Statement

Let $n \ge 1$ and let $M$ be a [[compact|0TVJZYB6SH]] [[topological manifold|QQQ9B1DK4P]] of dimension $n$ with empty boundary.
If $M$ is [[homotopy equivalent|XZG6MTSWCN]] to the [[$n$-sphere|T62369RFH6]] $S^n$, then $M$ is [[homeomorphic|0QT00PB5G2]] to $S^n$.

## Remarks

For $n \le 2$ the statement follows from the classification of curves and surfaces.
For $n \ge 5$ it was proved by Stephen Smale for smooth manifolds, through handle decompositions and the h-cobordism theorem.
Max Newman extended it to topological manifolds, and John Stallings and Christopher Zeeman gave other proofs.
For $n = 4$ it was proved by Michael Freedman in 1982, and for $n = 3$ it is the Poincaré conjecture, proved by Grigori Perelman with Ricci flow.
The analogous statement with "diffeomorphic" in place of "homeomorphic" is false: John Milnor found smooth manifolds homeomorphic but not diffeomorphic to $S^7$, the exotic spheres.
The smooth version for $n = 4$ is an open problem.
For $n \ge 2$, a closed $n$-manifold is homotopy equivalent to $S^n$ as soon as it is simply connected and has the integer homology of $S^n$, by the Hurewicz and Whitehead theorems.

## Explanations

### L1 human

If a closed manifold cannot be told apart from a sphere by deforming loops, surfaces and higher spheres inside it, then it really is a sphere.
The statement is true in every dimension, but each dimension needed its own ideas.
Strangely, dimensions five and higher were settled before dimensions three and four, and dimension three was done last.

### L2 human

The theorem is the starting point of the classification of manifolds up to homeomorphism.
In dimensions at least five, it comes out of the h-cobordism theorem, which turns homotopy information into homeomorphisms using handle moves.
In the smooth category the question becomes the study of exotic spheres, which Kervaire and Milnor organized into groups of homotopy spheres under connected sum.
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<document source="https://tamnd.github.io/math2/node/8KJ9P9KQ5H.md" tag="8KJ9P9KQ5H" kind="concept">
# Matching pursuit

## Summary

Matching pursuit is a greedy algorithm that builds an approximation of a vector from a dictionary of unit vectors, one element at a time.
At each step it picks the dictionary element most correlated with the current residual and removes the projection of the residual on that element.
It gives a cheap way to compute [[n-term approximations|GEK0NP5WQM]] and [[sparse approximations|WC8P971XGW]] when the exact problem is too hard.
A variant, orthogonal matching pursuit, re-projects onto all chosen elements at each step.

## Explanations

### L1 human

Imagine explaining a sound with a few notes from a large catalogue.
Matching pursuit first picks the note that matches the sound best, subtracts it, and then repeats with what is left.
Each step makes the leftover part shorter, because a component of it is removed.
In $\R^2$ with the dictionary $e_1$, $e_2$ and $u = (e_1 + e_2)/\sqrt{2}$, the vector $f = (2, 1)$ first picks $u$, since $\ip{f}{u} = 3/\sqrt{2}$ is the largest correlation.
The leftover $(1/2, -1/2)$ then needs both $e_1$ and $e_2$, so the greedy method uses three elements, while $f = 2e_1 + e_2$ needs only two.
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<document source="https://tamnd.github.io/math2/node/8KJCN49V64.md" tag="8KJCN49V64" kind="concept">
# Integral equation of the second kind

## Summary

An integral equation of the second kind is a [[linear integral equation|AYZCKG34QK]] of the form $\varphi(x) = f(x) + \lambda \int K(x, s) \varphi(s) \, ds$, where the unknown also appears outside the integral.
In operator form it reads $(I - \lambda K) \varphi = f$.
For reasonable kernels this equation is well behaved: Fredholm theory shows that it is uniquely solvable for all $f$ unless $\lambda$ is a characteristic value.
For small $\abs{\lambda}$ the solution is given by the [[Liouville–Neumann series|H1HWSWME6B]].

## Explanations

### L1 human

The unknown equals a known function plus a correction that depends on the unknown itself.
When the correction is small, one can solve the equation by starting with $f$ and correcting again and again.
For example, $\varphi(x) = 1 + \int_0^x \varphi(s) \, ds$ is of the second kind.
Starting with $1$ and correcting gives $1$, $1 + x$, $1 + x + x^2/2$, and so on, which builds up $e^x$.
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<document source="https://tamnd.github.io/math2/node/8KK1BQSPJT.md" tag="8KK1BQSPJT" kind="definition">
# Krull domain

## Definition

Let $A$ be an [[integral domain|09S3PFBEPM]] with [[field of fractions|0VF11ZTTMD]] $K$.
A *discrete valuation* of $K$ is a surjective map $v \colon K^\times \to \Z$ with $v(xy) = v(x) + v(y)$ and $v(x + y) \ge \min(v(x), v(y))$ whenever $x, y, x + y \in K^\times$; its ring is $R_v = \set{0} \cup \set{x \in K^\times : v(x) \ge 0}$, a [[discrete valuation ring|MGM3CNJW9V]].
$A$ is a *Krull domain* if there is a family $(v_i)_{i \in I}$ of discrete valuations of $K$ such that:

1. $A = \bigcap_{i \in I} R_{v_i}$;
2. for every $x \in K^\times$, $v_i(x) \ne 0$ for only finitely many $i \in I$.

## Remarks

Example: $\Z$ is a Krull domain, with the $p$-adic valuations $v_p$ of $\Q$ for all primes $p$.
A rational number $x$ lies in $\Z$ exactly when $v_p(x) \ge 0$ for all $p$, and a nonzero rational number involves only finitely many primes.
More generally every [[unique factorisation domain|06YMXJMXYS]] and every Noetherian [[integrally closed domain|H20P600HQS]] is a Krull domain.
Non-example: $\Z[\sqrt{-3}]$ is not a Krull domain.
An intersection of valuation rings is integrally closed, but $(1 + \sqrt{-3})/2$ is a root of $x^2 - x + 1$ and does not lie in $\Z[\sqrt{-3}]$.
So clause 1 fails for every family.
The family can be chosen as the valuations of the localizations $A_P$ at the height-one primes $P$ of $A$.
By the [[Mori–Nagata theorem|J8YYYP5CDN]], the integral closure of a Noetherian domain is a Krull domain.
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<document source="https://tamnd.github.io/math2/node/8KPDNVKX7B.md" tag="8KPDNVKX7B" kind="concept">
# Formally smooth map

## Summary

A map of commutative rings $R \to A$ is formally smooth when it has a lifting property.
Every map from $A$ to a quotient $C/J$ by an ideal $J$ with $J^2 = 0$ must lift to a map to $C$.
This is the algebraic form of an idea from geometry.
On a smooth space, there is no obstruction to extending a point to a small thickening.
Together with finite presentation it gives the notion of a [[smooth morphism of rings|286G0P1B69]], and with uniqueness of the lift it gives formally étale maps.

## Explanations

### L1 human

Think of a map $A \to C/J$ as a point of the space of $A$ with coordinates in $C/J$.
Since $J^2 = 0$, the ring $C$ is only a little bigger than $C/J$, like adding a first-order neighbourhood.
Formal smoothness says that every such point extends to $C$.
A polynomial ring has this property, because one may lift the images of the variables freely.
The ring $k[x]/(x^2)$ does not: the point $x \mapsto x$ of $k[x]/(x^2)$ does not extend to $k[x]/(x^3)$, since no lift squares to $0$.
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<document source="https://tamnd.github.io/math2/node/8KYEBXE16P.md" tag="8KYEBXE16P" kind="definition">
# Homotopy lifting property

## Definition

Let $E$, $B$ and $X$ be [[topological spaces|0CQHFPGJ45]], let $p \colon E \to B$ be a [[continuous map|0X3WGX95FE]], and let $I = [0, 1]$.
The map $p$ has the *homotopy lifting property* with respect to $X$ if the following holds.
For every continuous map $\tilde f \colon X \to E$ and every [[homotopy|9YA6S6HD9V]] $H \colon X \times I \to B$ with $H(x, 0) = p(\tilde f(x))$ for all $x \in X$, there is a homotopy $\tilde H \colon X \times I \to E$ such that:

1. $p \circ \tilde H = H$.
2. $\tilde H(x, 0) = \tilde f(x)$ for all $x \in X$.

## Remarks

Example: every [[covering map|3XZRRE165N]] has the homotopy lifting property with respect to every space, and the lift is then unique.
For $X$ a point this is the path lifting property: for $p \colon \R \to S^1$, $p(t) = e^{2\pi i t}$, the loop $H(t) = e^{2\pi i t}$ with starting point $0$ lifts to $\tilde H(t) = t$.

Non-example: let $E = \set{0, 1}$ with the discrete topology, $B = [0, 1]$ and $p$ the inclusion.
For $X$ a point, $\tilde f = 0$ and $H(t) = t$, clauses 1 and 2 would force $\tilde H$ to be a path in $E$ from $0$ to $1$, but $E$ is not path-connected.

In diagram form: every commutative square with $X \to E$ on top, $X \times \set{0} \hookrightarrow X \times I$ on the left and $p$ on the right has a diagonal $X \times I \to E$ making both triangles commute.
A map with this property for all spaces $X$ is a [[Hurewicz fibration|Y773B7EQA7]]; for all cubes $I^n$, a [[Serre fibration|PB9JXQ2KC3]].
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<document source="https://tamnd.github.io/math2/node/8M012XRPZM.md" tag="8M012XRPZM" kind="concept">
# Stability of Volterra equations

## Summary

Stability of a [[Volterra integral equation|DND8CBEVXT]] concerns the behaviour of its solutions on the whole half-line $t \ge 0$.
For a linear equation of convolution type it asks whether bounded or integrable data always give bounded or integrable solutions.
This holds when the resolvent kernel of the equation is integrable, which is the standard notion of stability for such equations.
Stability results are used for models with memory, such as viscoelastic materials, feedback systems with delay and population dynamics.

## Explanations

### L1 human

In a Volterra equation the present value of the unknown depends on its whole past through a kernel.
Such a memory can damp disturbances or amplify them.
The equation is stable when every bounded input produces a bounded output, so that small causes have small effects for all time.
For example, $x(t) + \int_0^t a e^{-(t - s)} x(s) \, ds = f(t)$ is stable exactly when the real part of $a$ is larger than $-1$.
For $a = -1$ and $f = 1$ the solution is $x(t) = 1 + t$, which grows without bound.
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<document source="https://tamnd.github.io/math2/node/8M04CT7CCM.md" tag="8M04CT7CCM" kind="definition">
# Tricomi equation

## Definition

The *Tricomi equation* is the [[partial differential equation|GC5BDZ6QBC]]
$$y \, u_{xx} + u_{yy} = 0.$$
The unknown is a real function $u(x, y)$ on an open subset of $\R^2$.
Subscripts denote [[partial derivatives|09EGC6F8PT]].

## Remarks

The equation is [[elliptic|B4TTNHSTRX]] where $y > 0$, [[hyperbolic|92C3S03AQZ]] where $y < 0$, and parabolic on the line $y = 0$, so it is of [[mixed type|7WFKPAMQFG]].
In the half-plane $y < 0$ its characteristic curves are $3x \pm 2(-y)^{3/2} = \text{const}$.
Some authors write it as $u_{xx} = x \, u_{yy}$, which is the same equation after renaming and reflecting the variables.

Example: $u(x, y) = x^2 - \tfrac{1}{3} y^3$ is a solution, because $y \, u_{xx} + u_{yy} = 2y - 2y = 0$.

Non-example: $u(x, y) = x^2 + y^2$ is not a solution, because $y \, u_{xx} + u_{yy} = 2y + 2$ is not identically $0$.
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<document source="https://tamnd.github.io/math2/node/8M0QH4Z60S.md" tag="8M0QH4Z60S" kind="definition">
# Regular open set

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
For $S \subseteq X$, write $\overline{S}$ for the [[closure|0QEAJB91GZ]] of $S$ and $\operatorname{int} S$ for its [[interior|0RZH6W98HK]].
A subset $U \subseteq X$ is *regular open* if
$$U = \operatorname{int} \overline{U}.$$

## Remarks

A regular open set is [[open|02BH3MCBQB]], since an interior is always open.
For every open $U$, one has $U \subseteq \operatorname{int} \overline{U}$, and the set $\operatorname{int} \overline{U}$ is regular open.

Example: in $\R$, let $U = (0, 1)$.
Then $\overline{U} = [0, 1]$ and $\operatorname{int} [0, 1] = (0, 1) = U$, so $U$ is regular open.

Non-example: in $\R$, let $U = (0, 1) \cup (1, 2)$.
Then $\overline{U} = [0, 2]$ and $\operatorname{int} \overline{U} = (0, 2) \ne U$, so $U$ is open but not regular open.

Dually, a set $F$ is *regular closed* if $F = \overline{\operatorname{int} F}$.
A set is regular open if and only if its complement is regular closed.
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<document source="https://tamnd.github.io/math2/node/8M31QKB0GW.md" tag="8M31QKB0GW" kind="concept">
# Admissible control

## Summary

An admissible control is an input function that is allowed in a given control problem.
It takes values in the prescribed control set, and it is regular enough for the state equation to make sense.
It also steers the state from the initial point to the target within the time horizon.
The cost of an optimal control problem is minimized over the admissible controls only.
The [[definition|8GETTS41P6]] gives the conditions for a [[control system|ZPDTVZQN5K]] driven by an ordinary differential equation.

## Explanations

### L1 human

A control problem fixes the rules of the game before asking for the best strategy.
The rules say which values the control may take, for example a throttle between $0$ and $1$.
They also say which goal must be reached, for example arriving at a given place at a given time.
A control that obeys all the rules is admissible.
For $x' = u$ with $\abs{u} \le 1$, starting at $x(0) = 0$ and required to reach $x(1) = \tfrac12$, the constant control $u = \tfrac12$ is admissible, while $u = 1$ overshoots and is not.
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<document source="https://tamnd.github.io/math2/node/8M4N38VGEE.md" tag="8M4N38VGEE" kind="definition">
# Quasi-analytic function

## Definition

Let $a < b$ and let $M = (M_k)_{k \ge 0}$ be a sequence of positive real numbers.
The class $C^M([a, b])$ is the set of [[smooth functions|0BE3QDH73A]] $f \colon [a, b] \to \R$ for which there are constants $A, B > 0$ with
$$\abs{f^{(k)}(x)} \le A B^k M_k \quad \text{for all } k \ge 0 \text{ and all } x \in [a, b].$$
The class $C^M([a, b])$ is *quasi-analytic* if the following holds.
If $f \in C^M([a, b])$ and $f^{(k)}(x_0) = 0$ for some $x_0 \in [a, b]$ and all $k \ge 0$, then $f = 0$ on $[a, b]$.
A *quasi-analytic function* is a function in a quasi-analytic class.

## Remarks

Equivalently, two functions in a quasi-analytic class that have the same derivatives of all orders at one point are equal.
At the endpoints, the [[derivatives|084NGSTWM7]] are one-sided.

Example: $M_k = k!$.
A function with $\abs{f^{(k)}} \le A B^k k!$ on $[a, b]$ is [[analytic|0JD5ZNW0F4]], with Taylor series of radius at least $1/B$ around each point.
If all its derivatives vanish at $x_0$, then it vanishes near $x_0$, and by moving along $[a, b]$ in steps it vanishes everywhere.
So $C^M([a, b])$ is quasi-analytic.

Non-example: $M_k = (k!)^2$.
Here $\sum_{k \ge 1} M_{k-1}/M_k = \sum_{k \ge 1} 1/k^2$ converges, so by the [[Denjoy–Carleman theorem|KV0GW9K7JS]] the class is not quasi-analytic.
It contains a nonzero function that vanishes on a subinterval, and all its derivatives vanish at the points of that subinterval.
</document>

<document source="https://tamnd.github.io/math2/node/8MADRXG8PQ.md" tag="8MADRXG8PQ" kind="proof">
# Proof that cyclic groups are abelian

## Proof

1. Let $G$ be a [[cyclic group|0B85Q41RP6]] with generator $g$.
2. Let $x$ and $y$ be elements of $G$. Then there are integers $m$ and $n$ with $x = g^m$ and $y = g^n$.
3. By the [[laws of exponents|00DRK90Z6K]], $xy = g^m g^n = g^{m+n}$.
4. In the same way, $yx = g^n g^m = g^{n+m}$.
5. Since $m + n = n + m$, $xy = yx$. Thus $G$ is [[abelian|071DDFVG4J]].

## Idea

Every element is a power of one generator, and powers of the same element commute.
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# Short exact sequence

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], for example the category of modules over a ring.
A *short exact sequence* in $\mathcal{A}$ is a diagram
$$0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$$
of objects and morphisms of $\mathcal{A}$ such that:

1. $f$ is a [[monomorphism|Y2NP1JFR86]];
2. $g$ is an [[epimorphism|RVXDSY04KG]];
3. the image of $f$ equals the [[kernel|S417YNEJ6M]] of $g$.

## Remarks

Equivalently, the sequence is [[exact|JB7ZBFBA9C]] at $A$, at $B$ and at $C$.
Equivalently again, $g \circ f = 0$, $f$ is a kernel of $g$, and $g$ is a cokernel of $f$.
Then $C$ is isomorphic to the quotient $B / f(A)$, and $B$ is called an *extension* of $C$ by $A$.

Example: $0 \to \Z \xrightarrow{4} \Z \xrightarrow{\pi} \Z/4\Z \to 0$ is short exact, with $\pi$ the reduction modulo $4$.
Multiplication by $4$ is injective, $\pi$ is onto, and the image $4\Z$ of the first map is the kernel of $\pi$.

Non-example: $0 \to 2\Z \xrightarrow{\iota} \Z \xrightarrow{\pi} \Z/4\Z \to 0$, with $\iota$ the inclusion, fails clause 3.
Clauses 1 and 2 hold, but the image of $\iota$ is $2\Z$ while the kernel of $\pi$ is $4\Z$.
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<document source="https://tamnd.github.io/math2/node/8MD8DJE30F.md" tag="8MD8DJE30F" kind="statement">
# Uniform convergence of Bernstein approximations

## Statement

Let $f \colon [0, 1] \to \R$ be continuous, and for $n \ge 1$ let $B_n(f)$ be its $n$-th [[Bernstein polynomial|0C4VWBFY73]],
$$B_n(f)(x) = \sum_{k=0}^{n} f\Big( \frac{k}{n} \Big) \binom{n}{k} x^k (1 - x)^{n-k}.$$
Then $B_n(f) \to f$ [[uniformly|08FPABDQ4C]] on $[0, 1]$ as $n \to \infty$.

## Remarks

The idea of the proof: the weights $\binom{n}{k} x^k (1 - x)^{n-k}$ are nonnegative and sum to $1$, so $B_n(f)(x) - f(x)$ is an average of $f(k/n) - f(x)$.
By [[uniform continuity|0Q4DR37DVE]], the terms with $\abs{k/n - x} < \delta$ contribute less than $\varepsilon$.
The weights are binomial probabilities with mean $x$ and variance $x(1 - x)/n \le 1/(4n)$, so by Chebyshev's inequality the remaining terms carry total weight at most $1/(4 n \delta^2)$, and they contribute at most $2 \norm{f}_\infty / (4 n \delta^2)$.

Example: $B_n$ reproduces $1$ and $x$, and $B_n(x^2) = x^2 + x(1 - x)/n$, so $\norm{B_n(x^2) - x^2}_\infty = 1/(4n)$.
This shows that the convergence is slow: of order $1/n$ even for a polynomial.
More generally $\norm{B_n(f) - f}_\infty$ is bounded by a constant multiple of the [[modulus of continuity|AXCJBSQZWF]] $\omega(f, n^{-1/2})$.

Continuity is needed: for the step function $f = 0$ on $[0, 1/2]$ and $f = 1$ on $(1/2, 1]$, each $B_n(f)$ is continuous, so the convergence cannot be uniform.
The result gives a constructive proof of the [[Weierstrass approximation theorem|0JQE130DHW]].

## Explanations

### L1 human

The Bernstein polynomial of $f$ samples $f$ at the points $0, 1/n, 2/n, \dots, 1$ and blends these values with smooth weights.
The theorem says that as you take more sample points, the blended curve hugs the graph of $f$ everywhere on $[0, 1]$.
This gives an explicit polynomial close to any continuous function.

### L2 human

The proof is the model for the theory of positive linear operators.
The operator $B_n$ is linear and maps nonnegative functions to nonnegative functions.
It is exact on $1$ and $x$, with a small error on $x^2$.
Korovkin's theorem shows that these three facts alone force uniform convergence.
Bernstein polynomials also preserve shape: if $f$ is monotone or convex, so is $B_n(f)$, which is why they appear in computer graphics as Bézier curves.
The price is the slow rate, which cannot beat order $1/n$ for nonlinear smooth $f$.
</document>

<document source="https://tamnd.github.io/math2/node/8MEDVAEERQ.md" tag="8MEDVAEERQ" kind="example">
# Loop group

## Example

Let $G$ be a [[Lie group|7JBRGCVJRB]] and let $S^1$ be the circle.
The *loop group* of $G$ is the set
$$LG = \set{\gamma \colon S^1 \to G : \gamma \text{ smooth}},$$
with pointwise multiplication $(\gamma\delta)(z) = \gamma(z)\delta(z)$.
It is a [[group|G5S1M9]].
The pointwise product and the pointwise inverse $z \mapsto \gamma(z)^{-1}$ of smooth loops are smooth, because multiplication and inversion of $G$ are smooth.
The identity element is the constant loop at $e$, and associativity holds pointwise.

For compact $G$, $LG$ is an infinite-dimensional Lie group and a [[Fréchet manifold|EBMCFQT0JN]].
Its Lie algebra is the loop algebra $L\mathfrak{g}$ of smooth maps $S^1 \to \mathfrak{g}$ with the pointwise bracket.

## Remarks

The constant loops form a copy of $G$ inside $LG$.
For $G$ the [[circle group|04EAFVFTF7]] $U(1)$, a loop $\gamma \colon S^1 \to U(1)$ has a winding number, and $LU(1)$ has one connected component for each integer.
Central extensions of loop groups of compact Lie groups are closely tied to affine Lie algebras and their representations.

## Explanations

### L1 human

A loop group consists of all smooth closed paths in a Lie group, where two paths are multiplied by multiplying their values at each moment.
In the circle group, take a loop that runs once around and a loop that runs once around the other way.
Their product is a loop of winding number $0$.
Loop groups are among the best understood infinite-dimensional groups, and they appear in conformal field theory and the theory of integrable systems.
</document>

<document source="https://tamnd.github.io/math2/node/8MGC7M64GW.md" tag="8MGC7M64GW" kind="concept">
# F-sigma set

## Summary

An $F_\sigma$ set in a [[topological space|0CQHFPGJ45]] is a union of countably many [[closed sets|0ZCK8DQYN1]].
The name comes from the French words for closed set and sum.
The complements of $F_\sigma$ sets are the [[G-delta sets|V1TBNP8R3S]], and in a [[metric space|0VKD1TP7F3]] every open set is $F_\sigma$.

## Explanations

### L1 human

A single point is closed, so any countable set is a countable union of closed sets.
For example, the set $\Q$ of rational numbers is $F_\sigma$ in $\R$.
An open interval is also $F_\sigma$, since $(0, 1)$ is the union of the closed intervals $[1/n, 1 - 1/n]$.
The irrational numbers, however, are not $F_\sigma$ in $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/8MGCGZP0E6.md" tag="8MGCGZP0E6" kind="definition">
# Hyperoval

## Definition

Let $\Pi$ be a finite [[projective plane|DANFNN8C4Y]] of [[order|K2YVMTGTBB]] $n$.
A *hyperoval* of $\Pi$ is a set of $n + 2$ points of $\Pi$ such that no three of them lie on a line.

## Remarks

Every line meets a hyperoval $H$ in $0$ or $2$ points.
Suppose a line met $H$ in exactly one point $P$.
The other $n$ lines through $P$ would then cover the remaining $n + 1$ points of $H$, each at most one of them, which is impossible.

Example: in the [[Fano plane|QQ8GW1ASZM]], of order $2$, let $H$ be the four points off a fixed line $\ell$.
Any other line meets $\ell$ in one point, so it has exactly $2$ points in $H$; hence no three points of $H$ are collinear, and $\abs{H} = 4 = 2 + 2$.

Non-example: $PG(2, 3)$ has no hyperoval.
More generally, a plane of odd order $n$ has none.
Suppose $S$ were a hyperoval, and take a point $P$ outside $S$.
Every line through $P$ meets $S$ in $0$ or $2$ points, so these lines split $S$ into pairs.
Then $\abs{S}$ is even, while $n + 2$ is odd.

For $n$ even, the tangents of an [[oval|7B03TB64N9]] all pass through one point, its nucleus, and adding the nucleus gives a hyperoval.
</document>

<document source="https://tamnd.github.io/math2/node/8MGH8PGJWR.md" tag="8MGH8PGJWR" kind="definition">
# Total ring of fractions

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $S$ be the set of elements of $R$ that are not [[zero divisors|0NNRDW10RZ]].
So $s \in S$ means: $sa = 0$ implies $a = 0$, for all $a \in R$.
Then $S$ is a [[multiplicative set|AN9SKEZ3VY]].
The *total ring of fractions* of $R$ is the [[localization|4C1AE84N5N]]
$$Q(R) = S^{-1}R.$$

## Remarks

The canonical map $R \to Q(R)$, $a \mapsto a/1$, is injective, because no element of $S$ kills a nonzero element.
If $S'$ is a multiplicative set and $R \to S'^{-1}R$ is injective, then $S' \subseteq S$.

Example: if $R$ is an [[integral domain|0QVD7WBFGN]], then $S = R \setminus \set{0}$ and $Q(R)$ is the [[field of fractions|0F2PTGX1A2]] of $R$.
For $R = \Z \times \Z$, the non-zero-divisors are the pairs $(a, b)$ with $a \ne 0$ and $b \ne 0$, and $Q(R) \cong \Q \times \Q$.
For $R = \Z/6\Z$, the non-zero-divisors are $1$ and $5$, which are already units, so $Q(R) = R$.

Non-example: in $R = \Z \times \Z$, the set $T = \set{(1,0)^n : n \ge 0} = \set{(1,1), (1,0)}$ is multiplicative, but $(1, 0)$ is a zero divisor.
The localization $T^{-1}R \cong \Z$ is not the total ring of fractions, and $R \to T^{-1}R$ kills $(0, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/8MNQDK5X5W.md" tag="8MNQDK5X5W" kind="object">
# 600-cell

## Description

The *600-cell* is the convex [[regular polytope|F60PD912D5]] in $\R^4$ with [[Schläfli symbol|MAE6BF4DFK]] $\set{3, 3, 5}$.
Its cells are $600$ regular tetrahedra, five around each edge, and its vertex figure is a regular icosahedron.
It has $120$ vertices, $720$ edges, $1200$ triangular faces and $600$ cells.

A model: let $\varphi = (1 + \sqrt5)/2$ be the [[golden ratio|0SC8T7548D]].
Take the $8$ points obtained from $(\pm1, 0, 0, 0)$ by permuting coordinates, and the $16$ points $\frac12(\pm1, \pm1, \pm1, \pm1)$.
Add the $96$ points obtained from $\frac12(\pm\varphi, \pm1, \pm\varphi^{-1}, 0)$ by even permutations of the coordinates.
These $8 + 16 + 96 = 120$ points lie on the unit sphere, and their convex hull is a 600-cell with edge length $\varphi^{-1}$.

Check of the counts: the icosahedral vertex figure has $12$ vertices, so each vertex lies on $12$ edges and there are $120 \cdot 12 / 2 = 720$ edges.
Each tetrahedron has $4$ triangles, each lying in $2$ cells, so $600 \cdot 4 / 2 = 1200$ faces.
Then $120 - 720 + 1200 - 600 = 0$.

## Remarks

The 600-cell is the dual of the [[120-cell|3RQEK50CP0]], and its symmetry group is the reflection group $H_4$ of order $14400$.
Read as unit quaternions, the $120$ vertices of the model above form a group under multiplication, the binary icosahedral group.
It is the four-dimensional analogue of the icosahedron $\set{3, 5}$.

## Explanations

### L1 human

The 600-cell is a four-dimensional solid made of $600$ regular tetrahedra, with $120$ corners.
Around each corner the tetrahedra are arranged like the triangles of an icosahedron.
It is the four-dimensional cousin of the icosahedron, and its corners have coordinates built from the golden ratio.
</document>

<document source="https://tamnd.github.io/math2/node/8MRPZ4Q4NM.md" tag="8MRPZ4Q4NM" kind="definition">
# Word metric

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$ that generates $G$.
The *word length* $\abs{g}_S$ of $g \in G$ is the least integer $n \ge 0$ such that $g = s_1^{\varepsilon_1} s_2^{\varepsilon_2} \cdots s_n^{\varepsilon_n}$ for some $s_1, \ldots, s_n \in S$ and $\varepsilon_1, \ldots, \varepsilon_n \in \set{1, -1}$.
The *word metric* on $G$ with respect to $S$ is the map $d_S \colon G \times G \to \Z_{\ge 0}$ given by
$$d_S(g, h) = \abs{g^{-1} h}_S.$$

## Remarks

$d_S$ is a [[metric|0VKD1TP7F3]].
It is symmetric because $\abs{g^{-1}}_S = \abs{g}_S$, and the triangle inequality holds because $\abs{gh}_S \le \abs{g}_S + \abs{h}_S$.
It is invariant under left multiplication: $d_S(kg, kh) = d_S(g, h)$ for all $k \in G$.
It equals the path distance between $g$ and $h$ in the [[Cayley graph|0GYQK9NDJF]] of $G$ with respect to $S$.
If $S$ and $T$ are finite generating sets, there is a constant $C \ge 1$ with $d_T \le C d_S$ and $d_S \le C d_T$.

Example: let $G = \Z$ and $S = \set{2, 3}$.
Then $\abs{1}_S = 2$: the number $1$ is not in $\set{\pm 2, \pm 3}$, so $\abs{1}_S \ge 2$, and $1 = 3 + (-2)$.
So $d_S(0, 1) = 2$, while $d_{\set{1}}(0, 1) = 1$.

Non-example: in $\Z$, the set $S = \set{2}$ does not generate $\Z$.
The number $1$ is not a sum of terms $\pm 2$, so $\abs{1}_S$ is not defined, and $d_S(0, 1)$ is not a real number.
</document>

<document source="https://tamnd.github.io/math2/node/8MS51FASMX.md" tag="8MS51FASMX" kind="concept">
# Essential submodule

## Summary

An essential submodule of a [[module|0NZ8JA25WK]] $M$ is a [[submodule|05WDSN597C]] that meets every nonzero submodule of $M$ in a nonzero element.
One then says that $M$ is an essential extension of it.
Essential submodules are "large" in a lattice sense, even when they are small in size.
Essential extensions are the basis of the theory of injective hulls, uniform dimension and Goldie's theorem.

## Explanations

### L1 human

A submodule is essential when it cannot be avoided: every nonzero piece of the big module overlaps it.
The even integers $2\Z$ are essential in $\Z$, since any nonzero subgroup $a\Z$ contains the nonzero even number $2a$.
In $\Z \oplus \Z$, the first axis $\Z \oplus 0$ is not essential, since the second axis meets it only in zero.
The rational numbers $\Q$ are an essential extension of $\Z$, and $\Q$ is the injective hull of $\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/8MTVCMAATD.md" tag="8MTVCMAATD" kind="concept">
# F-test

## Summary

An F-test is a [[hypothesis test|087P03NEND]] whose statistic has an [[F-distribution|PBT8VRJADK]] when the null hypothesis holds.
The statistic is a ratio of two independent variance estimates, each scaled by its degrees of freedom.
It compares group means in [[analysis of variance|M9AQ5HKGP4]] and nested [[linear regression|41FVC4P5JB]] models.
It also compares the variances of two normal samples.

## Explanations

### L1 human

An F-test asks whether one source of variation is large compared with another.
In a comparison of several groups, it compares the spread between the group means with the spread inside the groups.
If the group means differ only by chance, the ratio is near $1$.
A ratio much larger than $1$ suggests that the groups really differ.
</document>

<document source="https://tamnd.github.io/math2/node/8MWTEF6462.md" tag="8MWTEF6462" kind="concept">
# Dirichlet series

## Summary

A Dirichlet series is a series $\sum_{n \ge 1} a(n) n^{-s}$, where $a$ is an arithmetic function and $s$ is a complex variable.
It converges in a right half-plane and defines a holomorphic function there.
Multiplying two Dirichlet series corresponds to the [[Dirichlet convolution|037F589D1D]] of their coefficients, which makes them the natural generating functions of multiplicative number theory.
The [[Riemann zeta function|1XWBY4N0VT]] and the [[Dirichlet L-functions|9D2Z543EJQ]] are the main examples.

## Explanations

### L1 human

A power series $\sum a_n z^n$ packs a sequence into a function in a way suited to adding indices.
A Dirichlet series $\sum a(n) n^{-s}$ does the same in a way suited to multiplying indices, since $m^{-s} n^{-s} = (mn)^{-s}$.
For the constant sequence $a(n) = 1$ one obtains $\zeta(s) = 1 + 2^{-s} + 3^{-s} + \cdots$.
Facts about divisors and primes then become facts about products and quotients of such functions.
</document>

<document source="https://tamnd.github.io/math2/node/8N5CYXWXDV.md" tag="8N5CYXWXDV" kind="definition">
# Pseudospectral optimal control

## Definition

Consider a [[trajectory optimization|58895A7T4H]] problem on $[t_0, t_f]$ with dynamics $x' = f(x, u, t)$, cost $\phi(x(t_f)) + \int_{t_0}^{t_f} L(x, u, t) \, dt$, and boundary conditions.
Map the interval to $[-1, 1]$ by $t = \frac{t_f - t_0}{2} \tau + \frac{t_f + t_0}{2}$.
Let $N \ge 2$ and let $-1 = \tau_0 < \tau_1 < \cdots < \tau_N = 1$ be the Legendre–Gauss–Lobatto nodes, where $\tau_1, \dots, \tau_{N-1}$ are the zeros of $P_N'$ and $P_N$ is the [[Legendre polynomial|QFYGJVA996]] of degree $N$.
Let $w_0, \dots, w_N$ be the corresponding quadrature weights, and let $\ell_0, \dots, \ell_N$ be the [[Lagrange basis polynomials|237KR6FRCK]] of the nodes.
Let $D$ be the $(N + 1) \times (N + 1)$ matrix with $D_{kj} = \ell_j'(\tau_k)$.
The *Legendre pseudospectral method* replaces the problem by the nonlinear program with unknowns $x_0, \dots, x_N \in \R^n$ and $u_0, \dots, u_N \in \R^m$:

1. minimise $\phi(x_N) + \frac{t_f - t_0}{2} \sum_{k=0}^N w_k L(x_k, u_k, t(\tau_k))$;
2. subject to $\sum_{j=0}^N D_{kj} x_j = \frac{t_f - t_0}{2} f(x_k, u_k, t(\tau_k))$ for $k = 0, \dots, N$;
3. and to the boundary conditions on $x_0$ and $x_N$ and the path constraints at the nodes.

The state is then approximated by $x(\tau) \approx \sum_j x_j \ell_j(\tau)$.
*Pseudospectral optimal control* is the family of such methods, with Legendre, Chebyshev or other Gaussian nodes.

## Remarks

It is [[direct collocation|T58W34D60G]] with one interval and the nodes of a Gaussian rule as collocation points.
The Lobatto rule with $N + 1$ nodes integrates polynomials of degree up to $2N - 1$ exactly.
Variants use Legendre–Gauss nodes, which omit both endpoints, or Legendre–Gauss–Radau nodes, which omit one endpoint.

Example: $N = 2$ gives $P_2'(\tau) = 3\tau$, so the nodes are $-1, 0, 1$, with weights $\frac{1}{3}, \frac{4}{3}, \frac{1}{3}$ (Simpson's rule).
The differentiation matrix is
$$D = \begin{pmatrix} -3/2 & 2 & -1/2 \\ -1/2 & 0 & 1/2 \\ 1/2 & -2 & 3/2 \end{pmatrix}.$$
For instance, the first row gives the derivative at $-1$ of the parabola through $(-1, a)$, $(0, b)$, $(1, c)$, which is $(-3a + 4b - c) / 2$.

Non-example: collocation at equally spaced nodes with one global polynomial of high degree is not a pseudospectral method, and it suffers from the Runge phenomenon.

## Explanations

### L1 human

The method approximates the whole trajectory by one polynomial through special nodes and enforces the dynamics at those nodes.
</document>

<document source="https://tamnd.github.io/math2/node/8NAXJ0JH71.md" tag="8NAXJ0JH71" kind="concept">
# Homotopy group with coefficients

## Summary

A homotopy group with coefficients in $\Z/k$ is the set of homotopy classes of maps from a mod $k$ [[Moore space|68MA0RM5P6]] into a space.
It plays the role for [[homotopy groups|P976WYEFMB]] that homology with coefficients plays for homology.
These groups sit in a short exact sequence with the ordinary homotopy groups and are used to study $p$-primary phenomena in homotopy theory.

## Explanations

### L1 human

An ordinary homotopy group records maps from a sphere into a space.
A homotopy group with coefficients replaces the sphere by a space that is a sphere with a disk glued on by a degree $k$ map.
Maps from that space see the parts of homotopy that are killed by $k$ and the classes modulo $k$.
For example, for the sphere $S^n$ with $n \ge 3$, the group $\pi_n(S^n; \Z/k)$ is cyclic of order $k$.
</document>

<document source="https://tamnd.github.io/math2/node/8NCGCJ2YBH.md" tag="8NCGCJ2YBH" kind="definition">
# Benz plane

## Definition

A *Benz plane* is an incidence structure of points and circles that is a [[Möbius plane|DED008EJVH]], a [[Laguerre plane|F0JQ1TGKB0]] or a [[Minkowski plane|VB2D03PD9Y]].

## Remarks

The three types differ in their parallelism of points.
A Möbius plane has no parallel points, a Laguerre plane has one parallelism, and a Minkowski plane has two.
In each type, three pairwise non-parallel points lie on exactly one circle, and a touching axiom holds.

Example: the real plane with a point $\infty$ added, with circles and lines through $\infty$ as circles, is a Möbius plane, hence a Benz plane.

Non-example: a projective plane, with its lines as circles, is not a Benz plane.
Three non-collinear points lie on no line, so the axiom on three points fails for each of the three types.

The classical Benz planes are the geometries of plane sections of a sphere, a cylinder and a hyperboloid of one sheet.
</document>

<document source="https://tamnd.github.io/math2/node/8NE9Q44FEG.md" tag="8NE9Q44FEG" kind="statement">
# Kaplansky's theorem on PI rings

## Statement

Let $R$ be a [[primitive ring|8F6EHDVJ62]] (right or left).
Assume that $R$ satisfies a monic [[polynomial identity|C0EDT81EGH]] of degree $d$.
Let $Z$ be the [[center|MVT64B3B6H]] of $R$.
Then $Z$ is a field, $R$ is a [[central simple algebra|5HRKJBAY2V]] over $Z$, and
$$\dim_Z R = m^2 \quad \text{with} \quad m \le d/2.$$

## Remarks

By the [[Jacobson density theorem|EK0T7FRKCY]], $R$ is a dense ring of linear maps of a vector space $V$ over a [[division ring|0EA5BTA0CX]] $D$.
If $\dim_D V$ were large, a subring of $R$ would map onto a ring of matrices $M_s(D)$ with $s$ large.
Such a matrix ring fails every monic identity of degree less than $2s$, so $\dim_D V$ is finite and $R \cong M_s(D)$.
Then $D$ is a PI division ring, and a maximal subfield argument shows that $D$ is finite-dimensional over its centre.
The bound $m \le d/2$ comes from comparing with matrix rings over a splitting field, using part 2 of the [[Amitsur–Levitzki theorem|YDH60KPTWT]].

Example: $R = M_n(F)$ for a field $F$ is primitive, satisfies the standard identity $s_{2n}$ of degree $d = 2n$, and has $\dim_F R = n^2$ with $n = d/2$.
So the bound is sharp.

The primitive hypothesis is needed: the polynomial ring $\Z[x]$ satisfies the identity $x_1 x_2 - x_2 x_1$ of degree $2$, but its centre $\Z[x]$ is not a field.

## Explanations

### L1 human

Primitive rings are the rings that act faithfully and irreducibly on some module, and in general they can be huge, such as rings of operators on infinite-dimensional spaces.
Kaplansky's theorem says that once such a ring satisfies a polynomial identity, it must be small.
It is a matrix ring over a division ring that is finite-dimensional over its centre.
The degree of the identity bounds the size of the matrices.

### L2 human

The theorem is the first structure theorem of PI theory.
Because every semiprimitive ring is a subdirect product of primitive rings, it lets one embed semiprimitive PI rings into products of central simple algebras of bounded degree.
It extends to prime PI rings through Posner's theorem, and it is one of the tools behind the positive answer to the Kurosh problem for PI algebras.
</document>

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# Hermitian form

## Definition

Let $V$ be a vector space over $\C$.
Let $h \colon V \times V \to \C$ be a [[sesquilinear form|054YEQ0Y8M]], linear in the first argument and conjugate-linear in the second.
The form $h$ is a *Hermitian form* on $V$ when
$$h(v, u) = \overline{h(u, v)} \qquad \text{for all } u, v \in V,$$
where the bar is the [[complex conjugate|0FTH2E9TAS]].

## Remarks

Example: on $\C^2$, let $h(x, y) = x_1 \overline{y_1} + 2 x_2 \overline{y_2}$.
It is sesquilinear, and $h(y, x) = y_1 \overline{x_1} + 2 y_2 \overline{x_2} = \overline{h(x, y)}$, so it is Hermitian.
Non-example: $h(x, y) = x_1 \overline{y_2}$ on $\C^2$ is sesquilinear but not Hermitian, because $h(e_1, e_2) = 1$ while $h(e_2, e_1) = 0$.
Taking $u = v$ gives $h(v, v) = \overline{h(v, v)}$, so $h(v, v)$ is real for every $v$.
On $\C^n$, every Hermitian form is $h(x, y) = y^* M x$ for one [[Hermitian matrix|0P17NGQQMQ]] $M$, that is, $M^* = M$.
Some authors take the form conjugate-linear in the first argument instead; the condition $h(v, u) = \overline{h(u, v)}$ is the same.
</document>

<document source="https://tamnd.github.io/math2/node/8NMGMXNF02.md" tag="8NMGMXNF02" kind="definition">
# Classical ring of quotients

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
An element $c \in R$ is *regular* if $c$ is not a [[zero divisor|0NNRDW10RZ]] on either side: $c a = 0$ or $a c = 0$ implies $a = 0$.
Let $C$ be the set of regular elements of $R$.
A *right classical ring of quotients* of $R$ is a ring $Q$ containing $R$ as a [[subring|0GNCRPYXYV]] such that:

1. every regular element of $R$ is a [[unit|0XVEZDE9J8]] of $Q$;
2. every element of $Q$ has the form $a c^{-1}$ with $a \in R$ and $c \in C$.

The *left classical ring of quotients* is defined in the same way with elements $c^{-1} a$.
If $R$ has a right and a left classical ring of quotients, they coincide, and this ring is the *classical ring of quotients* of $R$.

## Remarks

The right classical ring of quotients is the [[Ore localization|SG7AQ9T8HD]] $R C^{-1}$.
It exists if and only if $C$ satisfies the right [[Ore condition|TJP4ARC2X7]]; reversibility holds automatically, since elements of $C$ are not zero divisors.
It is then unique up to an isomorphism that fixes $R$.

Example: for a commutative integral domain $R$, the set $C$ is $R \setminus \set{0}$, and the classical ring of quotients is the field of fractions.
For $R = \Z$ it is $\Q$: each nonzero integer is invertible in $\Q$, and each rational number is $a c^{-1}$ with $a, c \in \Z$ and $c \ne 0$.

Non-example: the free algebra $k\langle x, y \rangle$ over a field $k$ is a domain, so $C$ consists of all nonzero elements.
It has no right classical ring of quotients, because $C$ fails the right Ore condition: $x C \cap y R = \emptyset$.

A ring $R$ with a right classical ring of quotients $Q$ is called a *right order* in $Q$.
A right Ore domain has a right classical ring of quotients that is a division ring.
</document>

<document source="https://tamnd.github.io/math2/node/8NMH42GJRK.md" tag="8NMH42GJRK" kind="definition">
# Riemann–Stieltjes integral

## Definition

Let $a < b$ be real numbers and let $f, g \colon [a, b] \to \R$ be functions.
A tagged [[partition|069P20KYQT]] $(P, c)$ of $[a, b]$ consists of points $a = x_0 < x_1 < \cdots < x_n = b$ and tags $c_i \in [x_{i-1}, x_i]$.
Its mesh is $\max_i (x_i - x_{i-1})$, and its Riemann–Stieltjes sum is
$$S(P, c, f, g) = \sum_{i=1}^{n} f(c_i) \bigl(g(x_i) - g(x_{i-1})\bigr).$$
The function $f$ is *Riemann–Stieltjes integrable* with respect to $g$, with integral $A \in \R$, if the following holds.
For every $\varepsilon > 0$ there is $\delta > 0$ such that $\abs{S(P, c, f, g) - A} < \varepsilon$ for every tagged partition $(P, c)$ of mesh less than $\delta$.
One writes $A = \int_a^b f(x)\, dg(x)$.

## Remarks

Example: if $f$ is continuous and $g$ is continuously differentiable on $[a, b]$, then $\int_a^b f \, dg = \int_a^b f(x) g'(x)\, dx$.
By the mean value theorem, $g(x_i) - g(x_{i-1}) = g'(d_i)(x_i - x_{i-1})$ for some $d_i \in (x_{i-1}, x_i)$.
Uniform continuity of $g'$ lets one replace $d_i$ by $c_i$ with a small error, which leaves a [[Riemann sum|0HBY48G8G1]] of $f g'$.
With $g(x) = x$ the definition is that of the [[Riemann integral|0QM4TXT3DR]].

Non-example: on $[-1, 1]$ let $f = g$ be $0$ on $[-1, 0]$ and $1$ on $(0, 1]$.
If $0 = x_k$ is a partition point, the only nonzero term is $f(c_{k+1})$.
It is $0$ for the tag $c_{k+1} = 0$ and $1$ for any other tag.
So sums of every small mesh take both values $0$ and $1$, and $f$ is not integrable with respect to $g$.

If $f$ is continuous and $g$ has [[bounded variation|A8251BHCMP]], then $\int_a^b f \, dg$ exists.
Some authors use the limit over refinements of partitions instead of the mesh, which gives a slightly larger class of integrable pairs.
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# G-delta set

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A subset $A \subseteq X$ is a *$G_\delta$ set* if there are [[open sets|02BH3MCBQB]] $U_1, U_2, U_3, \dots$ of $X$ with
$$A = \bigcap_{n=1}^{\infty} U_n.$$

## Remarks

Every open set is $G_\delta$: take $U_n = A$ for all $n$.
In a [[metric space|0VKD1TP7F3]] with metric $d$, every [[closed set|0ZCK8DQYN1]] $F$ is $G_\delta$, since $F = \bigcap_n \set{x : d(x, F) < 1/n}$.

Example: the set of irrational numbers is $\bigcap_{q \in \Q} (\R \setminus \set{q})$.
Each $\R \setminus \set{q}$ is open and $\Q$ is [[countable|01TQB88H4C]], so the irrationals form a $G_\delta$ set.

Non-example: $\Q$ is not a $G_\delta$ subset of $\R$.
If $\Q = \bigcap_n U_n$ with $U_n$ open, each $U_n$ is open and dense, and so are the sets $\R \setminus \set{q}$ for $q \in \Q$.
These countably many dense open sets have empty intersection, which contradicts the [[Baire category theorem|0A7KGAYHGV]].

A set is $G_\delta$ if and only if its complement is an [[F-sigma set|8MGC7M64GW]].
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# Lévy process

## Summary

A Lévy process is a stochastic process that starts at $0$ and has independent and stationary increments, with paths continuous in probability.
It is the continuous-time analogue of a random walk.
Brownian motion and the Poisson process are the two basic examples, and every Lévy process combines a drift, a Brownian part and jumps.
The law of each $X_t$ is [[infinitely divisible|BX3V3NYM1Y]], and the [[Lévy–Khintchine representation|Q82B4H4Z4J]] describes all of them.

## Explanations

### L1 human

A Lévy process moves by random increments that do not remember the past and do not depend on the clock.
What happens between time $1$ and time $2$ has the same law as what happens between time $5$ and time $6$, and the two pieces are independent.
The path may move continuously, like Brownian motion, or jump, like a counter of random clicks.
For example, the number of radioactive decays up to time $t$ is a Poisson process, a Lévy process that jumps by $1$.
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# Density theorem

## Statement

Let $\mathcal{C}$ be a small [[category|A2DE66937D]], let $\widehat{\mathcal{C}} = [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]$ be the [[functor category|1X0WKFQX8K]] of [[presheaves|AND2WHZC8P]] on $\mathcal{C}$, and let $y \colon \mathcal{C} \to \widehat{\mathcal{C}}$ be the [[Yoneda embedding|4DKB6AQ8J7]].
Let $P$ be a presheaf on $\mathcal{C}$, and let $\int P$ be its [[category of elements|S9EJKKRHW1]], with objects the pairs $(c, x)$ such that $x \in P(c)$.
Let $\pi \colon \int P \to \mathcal{C}$ be the projection $(c, x) \mapsto c$.
Then $P$ is the [[colimit|BJS1CGTKQ5]] of the diagram $y \circ \pi \colon \int P \to \widehat{\mathcal{C}}$.
The colimit cocone has components $y(c) \to P$ given by the natural transformations that correspond to $x \in P(c)$ under the Yoneda lemma.

## Remarks

Proof idea: colimits in $\widehat{\mathcal{C}}$ are computed objectwise in $\mathbf{Set}$.
At an object $d$, the colimit of the sets $\Hom(d, c)$ over $(c, x) \in \int P$ is a quotient of the set of pairs $(g \colon d \to c, x \in P(c))$, and each pair is identified with $(\id_d, P(g)(x))$.
So the colimit at $d$ is $P(d)$, and the identifications are compatible with the maps of $P$.

Example: if $P = y(a)$ is itself [[representable|KCCHH87H6E]], then $\int P$ is the slice category over $a$, which has the terminal object $(a, \id_a)$, and the colimit is just $y(a)$.

In short: every presheaf is a colimit of representable presheaves.
Smallness of $\mathcal{C}$ makes $\int P$ small, so the colimit is a small colimit.

## Explanations

### L1 human

Representable presheaves are the building blocks of all presheaves.
Every presheaf can be glued together from copies of representable ones, one copy for each element, along the maps between those elements.
This is like saying that every set is the union of its one-point subsets, but in a setting with arrows.

### L2 human

The theorem is used to extend a functor $F \colon \mathcal{C} \to \mathcal{E}$ to all presheaves when $\mathcal{E}$ has colimits: send $P$ to the colimit of $F \circ \pi$.
This extension is the left Kan extension of $F$ along $y$, and it is left adjoint to the functor $E \mapsto \Hom_{\mathcal{E}}(F(-), E)$.
Geometric realisation of simplicial sets is built this way, from the standard simplices.
The theorem also shows that $\widehat{\mathcal{C}}$ is the free cocompletion of $\mathcal{C}$.
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# Spectral element method

## Definition

Let $\Omega \subset \R^d$ be partitioned into elements $K_e$, each the image of the reference cube $\hat K = [-1, 1]^d$ under a smooth map $F_e$.
Fix a polynomial degree $N \ge 1$.
Let $\xi_0 = -1 < \xi_1 < \cdots < \xi_N = 1$ be the Gauss–Lobatto–Legendre points.
These are $\pm 1$ and the roots of $P_N'$, where $P_N$ is the [[Legendre polynomial|QFYGJVA996]] of degree $N$.
The *spectral element method* is the [[Galerkin method|Y5B0PEBKE8]] with three features:

1. the discrete space consists of continuous functions whose pullback to $\hat K$ on each element is a polynomial of degree at most $N$ in each variable;
2. the basis on $\hat K$ is the tensor product of the [[Lagrange polynomials|Y7KSGE97GK]] at the points $\xi_i$;
3. all integrals are computed with the Gauss–Lobatto–Legendre [[quadrature|FP37BTP93K]] rule at the same points.

Convergence is obtained mainly by increasing $N$ on a fixed mesh.

## Remarks

The Gauss–Lobatto–Legendre rule with $N + 1$ points is exact for polynomials of degree at most $2N - 1$ in one variable.
Since the nodal basis functions vanish at all nodes but one, the quadrature makes the mass matrix diagonal.
For smooth solutions the error decreases faster than any power of $N$, as for [[spectral methods|MYV657YRDW]].

Example: in one dimension with $N = 2$, the points on $[-1, 1]$ are $-1, 0, 1$, the roots of $P_2'(\xi) = 3\xi$ together with $\pm 1$.
The quadrature weights are $\tfrac13, \tfrac43, \tfrac13$, which is Simpson's rule.
The basis on each element is the three quadratic Lagrange polynomials at these points.

Non-example: high-degree Lagrange polynomials at equally spaced points on each element fail clause 2.
Their interpolation is badly conditioned for large $N$, which is the Runge phenomenon, and with exact integration the mass matrix is not diagonal.
This is a [[finite element|98GZ9DVHTA]] method of high order but not a spectral element method.
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# Symbolic dynamics

## Definition

Let $\mathcal{A}$ be a finite alphabet and let $\sigma(x)_i = x_{i+1}$ be the shift map on $\mathcal{A}^{\Z}$.
A *symbolic dynamical system* is a pair $(X, \sigma_X)$ where $X \subseteq \mathcal{A}^{\Z}$ is a [[shift space|A1XE59ZHNM]] and $\sigma_X$ is the restriction of $\sigma$ to $X$.
One-sided versions use $\mathcal{A}^{\N}$.
*Symbolic dynamics* is the part of [[topological dynamics|5X6E97EMFR]] that studies symbolic dynamical systems and the maps between them that commute with the shifts.
It also studies codings: given a [[continuous map|0VGFHTM75Y]] $T \colon M \to M$ and a partition $M = P_0 \cup \cdots \cup P_{k-1}$, the *itinerary* of $x \in M$ is the sequence $(a_n)_{n \ge 0}$ with $T^n(x) \in P_{a_n}$.

## Remarks

The itinerary map $x \mapsto (a_n)$ turns $T$ into the shift: the itinerary of $T(x)$ is the shifted itinerary of $x$.
A coding is rarely continuous, since a connected space cannot be split into finitely many disjoint nonempty open pieces.
Instead one often has a continuous surjection $\pi$ from a shift space onto $M$ with $\pi \circ \sigma = T \circ \pi$, a [[semiconjugacy|AEXJNRBASV]] that sends a sequence to a point with that itinerary.
Example: for the doubling map $T(x) = 2x \bmod 1$ on $[0, 1)$ with $P_0 = [0, 1/2)$ and $P_1 = [1/2, 1)$, the itinerary of $x$ is its binary expansion $x = \sum_{n \ge 0} a_n 2^{-n-1}$ that does not end in all $1$s.
Indeed $T$ deletes the first binary digit, and the first digit is $0$ exactly when $x \in P_0$.
Here $\pi(a) = \sum_{n \ge 0} a_n 2^{-n-1}$ is a semiconjugacy from the one-sided full shift on $\set{0, 1}$ onto the circle $\R / \Z$.
Non-example: a coding with the single region $P_0 = M$ gives the constant itinerary $000\ldots$ for every point, so it records nothing about the dynamics.

## Explanations

### L1 human

Symbolic dynamics studies systems whose states are infinite words, moved by deleting or shifting letters, and uses them as codes for other systems.
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# Kolmogorov complexity

## Summary

The Kolmogorov complexity of a finite string is the length of a shortest program that prints the string.
It measures how much information the string contains, or how far it can be compressed.
By the invariance theorem the choice of programming language changes the complexity only by a bounded amount.
It is the basis of algorithmic information theory and of [[algorithmic randomness|KW3V44GPPC]].

## Explanations

### L1 human

A string like $0101010101\ldots01$ with a million digits has a short description: "print $01$ half a million times".
A string of a million coin flips usually has no description much shorter than the string itself.
Kolmogorov complexity makes this precise: it is the length of the shortest program that outputs the string.
Simple, patterned strings have low complexity, and typical random strings have complexity close to their length.
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# Ramanujan's series for 1/pi

## Statement

The [[series|0AF7X08PCR]] below converges, and
$$\frac{1}{\pi} = \frac{2\sqrt2}{9801}\sum_{k=0}^{\infty} \frac{(4k)!\,(1103 + 26390k)}{(k!)^4\,396^{4k}},$$
where $\pi$ is the number [[pi|0C1VYVNV2N]] and $n!$ is the [[factorial|0XWG0JGDFW]].

## Remarks

Convergence: by Stirling's formula $(4k)!/(k!)^4$ grows like a constant times $256^k k^{-3/2}$, while $396^4 \approx 2.46 \times 10^{10}$, so the ratio of successive terms tends to $256/396^4 \approx 1.04 \times 10^{-8}$.
So each term adds about eight correct decimal digits.
Example: the $k = 0$ term alone gives $\pi \approx 9801/(2\sqrt2 \cdot 1103) = 3.14159273\ldots$, against $\pi = 3.14159265\ldots$.
Ramanujan published this series in 1914 among several series of the same shape, without a complete proof.
Proof idea: the series comes from the theory of [[complete elliptic integrals|RM54CPPE12]] and modular equations.
It uses a singular value of the modulus attached to the imaginary quadratic field $\Q(\sqrt{-58})$, which is reflected in $e^{\pi\sqrt{58}} \approx 396^4 - 104$.
Identities such as [[Legendre's relation|8GSKDNZ4TK]] convert the resulting relations between elliptic integrals into a formula for $1/\pi$.
A complete proof was given later by Jonathan and Peter Borwein.

## Explanations

### L1 human

This is a formula that writes $1/\pi$ as an infinite sum of fractions built from factorials and powers of $396$.
Its terms shrink so fast that each new term gives about eight more digits of $\pi$.
The first term alone already gives $\pi$ to six decimal places.

### L2 human

The series is the most famous member of the family of Ramanujan–Sato series, in which $1/\pi$ equals a sum of hypergeometric-type terms $\frac{(4k)!}{(k!)^4}(a + bk)x^k$ or similar.
Each such series comes from a singular value of the modulus, where the elliptic curve has complex multiplication.
The same idea with the $j$-invariant and the field $\Q(\sqrt{-163})$ gives the [[Chudnovsky algorithm|JKCKXDT5B5]], which gains about fourteen digits per term.
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# Volterra equation of the first kind

## Definition

Let $a < b$, let $\Delta = \set{(t, s) : a \le s \le t \le b}$, let $K \colon \Delta \to \C$ be a [[kernel|GSMBXT7FPK]], and let $f \colon [a, b] \to \C$ be a given function.
A *linear Volterra equation of the first kind* is the equation
$$\int_a^t K(t, s) \, x(s) \, ds = f(t), \quad t \in [a, b],$$
for an unknown function $x$ on $[a, b]$.
The unknown appears only inside the [[integral|0W29TBTY12]], and the upper limit of integration is the variable $t$.

## Remarks

Example: $\int_0^t (t - s) \, x(s) \, ds = t^3$ on $[0, 1]$ is of the first kind with $K(t, s) = t - s$.
Differentiating twice gives $x(t) = 6t$, and indeed $\int_0^t (t - s) \, 6s \, ds = 3t^3 - 2t^3 = t^3$.
Non-example: $x(t) - \int_0^t x(s) \, ds = 1$ is not of the first kind, since $x$ also appears outside the integral; it is a [[Volterra equation of the second kind|ATPSAHYG7C]].
A necessary condition for a continuous solution when $K$ is continuous is $f(a) = 0$, since the integral vanishes at $t = a$.
Often $K$ is required to be continuous on $\Delta$, but weakly singular kernels such as $(t - s)^{-1/2}$ are also allowed, as in [[Abel's integral equation|G4GWBYTW7B]].
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# Total variation

## Summary

The total variation of a real function on an interval measures how much its values change in total, counting each rise and each fall.
It is the supremum of the sums of the absolute changes over all partitions of the interval.
A function has [[bounded variation|A8251BHCMP]] exactly when its total variation is finite.
The same idea gives the total variation of a signed measure and of functions of several variables.

## Explanations

### L1 human

Think of the graph of a function as a mountain path and measure only the height you climb plus the height you descend.
The total variation is this sum of heights.
For $f(x) = \sin x$ on $[0, 2\pi]$ you go up by $1$, down by $2$, and up by $1$, so the total variation is $4$.
For a monotone function the total variation is just the difference between the end values.
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# Principal curvature

## Definition

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]] with a smooth unit normal field $N$ near a point $p \in S$.
Let $S_p$ be the [[shape operator|DQCD11X1VM]] at $p$.
Since $S_p$ is [[self-adjoint|0MB8HQW495]], it has two real [[eigenvalues|0ZND7NQ970]] $k_1 \ge k_2$, counted with multiplicity.
These are the *principal curvatures* of $S$ at $p$.
A unit eigenvector of $S_p$ is a *principal direction* at $p$.

## Remarks

The principal directions can be chosen orthonormal, by the spectral theorem.
Equivalently, $k_1$ and $k_2$ are the maximum and the minimum of the [[second fundamental form|9KHQR16P1B]] $I\!I_p(w)$ over unit vectors $w \in T_p S$.
Replacing $N$ by $-N$ changes $k_1, k_2$ into $-k_2, -k_1$.
Example: for the saddle $z = xy$ at the origin, with $N = (0, 0, 1)$, one has $I\!I_0(a, b) = 2ab$ on $T_0 S = \R^2 \times \set{0}$.
So $k_1 = 1$ and $k_2 = -1$, with principal directions $(1, 1, 0)/\sqrt{2}$ and $(1, -1, 0)/\sqrt{2}$.
Non-example: on the same saddle, the normal curvature in the direction $(1, 0, 0)$ is $I\!I_0(1, 0) = 0$, which is neither $k_1$ nor $k_2$, so $(1, 0, 0)$ is not a principal direction.
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# Artin group

## Definition

Let $I$ be a set and let $M = (m_{ij})_{i, j \in I}$ be a Coxeter matrix, as in the definition of a [[Coxeter group|G6FND7X69J]].
For letters $x, y$ and an integer $m \ge 2$, let $\Pi(x, y; m) = x y x y \cdots$ be the alternating word of length $m$ that begins with $x$.
The *Artin group* of $M$ is the group with the [[presentation|0JGEQXQ4AC]]
$$A(M) = \langle s_i \ (i \in I) \mid \Pi(s_i, s_j; m_{ij}) = \Pi(s_j, s_i; m_{ij}) \text{ for all } i \ne j \text{ with } m_{ij} < \infty \rangle.$$
A group is an *Artin group*, or *Artin–Tits group*, if it is [[isomorphic|08MG7H8JW5]] to $A(M)$ for some Coxeter matrix $M$.

## Remarks

For $m_{ij} = 2$ the relation is $s_i s_j = s_j s_i$, and for $m_{ij} = 3$ it is $s_i s_j s_i = s_j s_i s_j$.
The quotient of $A(M)$ by the relations $s_i^2 = 1$ is the Coxeter group $W(M)$.
$A(M)$ is of *finite type*, or spherical, when $W(M)$ is finite.

Example: the [[braid group|BNR4H5GCTP]] $B_n$ is the Artin group with $I = \set{1, \ldots, n-1}$, $m_{i, i+1} = 3$ and $m_{ij} = 2$ for $\abs{i - j} \ge 2$.
These are exactly Artin's relations $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ and $\sigma_i \sigma_j = \sigma_j \sigma_i$.
If all $m_{ij} = \infty$, there are no relations and $A(M)$ is the [[free group|0KTW26C80J]] on $I$; if all $m_{ij} = 2$ for $i \ne j$, then $A(M)$ is free abelian.

Non-example: the group $\Z/2$ is not an Artin group.
In the abelianization of $A(M)$, a relation with $m_{ij}$ even becomes trivial, and a relation with $m_{ij}$ odd becomes $s_i = s_j$.
So the abelianization of an Artin group is free abelian, but $\Z/2$ is abelian and not free abelian.
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# Neumann series

## Statement

Let $X$ be a [[Banach space|06GZ956657]], and let $T \colon X \to X$ be a [[continuous linear map|03TM135G0N]] with [[operator norm|06EKW1AKXF]] $\norm{T} < 1$.
Write $I$ for the identity map of $X$ and $T^0 = I$.
Then the following hold.

1. The series $\sum_{n=0}^{\infty} T^n$ converges in operator norm.
2. The operator $I - T$ is [[invertible|CPAD5W1CGW]], and
$$(I - T)^{-1} = \sum_{n=0}^{\infty} T^n.$$
3. $\norm{(I - T)^{-1}} \le \dfrac{1}{1 - \norm{T}}$.

## Remarks

Proof idea: $\norm{T^n} \le \norm{T}^n$, so the series is dominated by a [[geometric series|01X5CVV1Q5]] and converges absolutely in the Banach space of bounded operators on $X$.
See [[absolutely convergent series in a Banach space converge|082MGNWNA8]].
The partial sums $S_N = \sum_{n=0}^{N} T^n$ satisfy $(I - T) S_N = S_N (I - T) = I - T^{N+1}$, and $T^{N+1} \to 0$.
Letting $N \to \infty$ gives both inverse identities, and the norm bound follows from the geometric series.

Example: on $\ell^2$, let $T(x_1, x_2, \ldots) = (0, \tfrac12 x_1, \tfrac12 x_2, \ldots)$, half the right shift.
Then $\norm{T} = \tfrac12$, so $I - T$ is invertible with $\norm{(I - T)^{-1}} \le 2$.

Completeness is needed.
On the space of finitely supported sequences with the $\ell^2$ norm, $T$ as above has norm $\tfrac12$.
But the solution of $(I - T)x = (1, 0, 0, \ldots)$ is $x = (1, \tfrac12, \tfrac14, \ldots)$, which is not finitely supported, so $I - T$ is not surjective there.

A consequence: if $A$ is invertible and $\norm{B - A} < 1 / \norm{A^{-1}}$, then $B$ is invertible, so the invertible operators form an open set.
The series also converges, with the same sum, whenever $\norm{T^k} < 1$ for some $k \ge 1$.

## Explanations

### L1 human

For numbers, $1/(1 - t) = 1 + t + t^2 + \cdots$ when $\abs{t} < 1$.
The Neumann series says the same formula works for operators, with the absolute value replaced by the operator norm.
So an operator that is a small perturbation of the identity can always be undone, and the inverse is computed by a convergent series.

### L2 human

The Neumann series is the main tool to prove that invertibility is stable under small perturbations.
It shows that the set of invertible operators is open and that inversion is continuous, indeed analytic, on it.
In spectral theory it shows that the spectrum of $T$ lies in the closed disc of radius $\norm{T}$, and that the resolvent is analytic.
For integral equations $x - Kx = y$ with a small kernel $K$, the series gives the solution by iteration.
The series is named after Carl Neumann, who used it in potential theory.
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# Locally finite variety

## Summary

A locally finite variety is a [[variety|AJNZJD8442]] of algebras in which every finitely generated member is finite.
Equivalently, its free algebras on finitely many generators are finite.
Many structural results on finite algebras, such as those of tame congruence theory, are stated for locally finite varieties.

## Explanations

### L1 human

In some kinds of algebra, a few elements can only produce finitely many new elements by the operations.
A variety is locally finite when this happens for every finite set of starting elements in every algebra of the variety.
Boolean algebras are an example: $n$ elements generate a Boolean algebra with at most $2^{2^n}$ elements.
Groups are not an example, since one element of infinite order already generates the infinite group $\Z$.
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# Solving linear difference equations by the Z-transform

## Definition

Let $k \ge 1$, let $a_0, \ldots, a_k$ be complex numbers with $a_0 \ne 0$, let $(f_n)_{n \ge 0}$ be a sequence with a [[unilateral Z-transform|EPG6JPVJE7]] $F$, and let $c_0, \ldots, c_{k-1}$ be complex numbers.
Consider the [[linear recurrence with constant coefficients|0NDVHBVBXB]]
$$a_0 y_{n+k} + a_1 y_{n+k-1} + \cdots + a_k y_n = f_n \quad (n \ge 0), \qquad y_j = c_j \ (0 \le j < k).$$
The *Z-transform method* solves it in three steps.

1. Apply the unilateral Z-transform and the [[shift property|JWH2KYCGBQ]], $\mathcal{Z}\{(y_{n+m})_{n \ge 0}\} = z^m \left( Y(z) - \sum_{j<m} y_j z^{-j} \right)$, to get $P(z) Y(z) = F(z) + Q(z)$, where $P(z) = \sum_{m=0}^{k} a_{k-m} z^m$ and $Q$ is a polynomial built from $c_0, \ldots, c_{k-1}$.
2. Solve: $Y(z) = (F(z) + Q(z)) / P(z)$.
3. Recover $(y_n)$ from $Y$ by the [[inverse Z-transform|PH61T3B7W7]], in practice by a [[partial fraction decomposition|030VFEREKG]] and known transform pairs.

## Remarks

Example: the Fibonacci recurrence $y_{n+2} = y_{n+1} + y_n$ with $y_0 = 0$, $y_1 = 1$.
Step 1 gives $z^2 (Y - z^{-1}) = z Y + Y$, so $Y(z) = z/(z^2 - z - 1)$.
With $\varphi = (1 + \sqrt 5)/2$ and $\psi = (1 - \sqrt 5)/2$, one has $Y(z) = \frac{1}{\sqrt 5}\left( \frac{z}{z - \varphi} - \frac{z}{z - \psi} \right)$, and the pair $a^n \leftrightarrow z/(z - a)$ gives $y_n = (\varphi^n - \psi^n)/\sqrt 5$.
Non-example: the recurrence $y_{n+1} = y_n^2$ is not linear; the transform of $(y_n^2)$ is not a function of $Y(z)$ alone, so step 1 gives no algebraic equation for $Y$.
The polynomial $P$ is the characteristic polynomial of the recurrence, so the poles of $Y$ that do not come from $F$ lie among its roots.
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# Cauchy interlacing theorem

## Statement

Let $1 \le m \le n$, and let $A$ be an $n \times n$ [[Hermitian matrix|0P17NGQQMQ]] with [[eigenvalues|0BQAA2BK3Q]] $\alpha_1 \ge \dots \ge \alpha_n$.
Let $P$ be an $n \times m$ complex matrix with $P^* P = I_m$, and let $B = P^* A P$, with eigenvalues $\beta_1 \ge \dots \ge \beta_m$.
Then, for $j = 1, \dots, m$,
$$\alpha_j \ge \beta_j \ge \alpha_{j + n - m}.$$
In particular, if $B$ is obtained from $A$ by deleting row $i$ and column $i$ for some $i$, then
$$\alpha_1 \ge \beta_1 \ge \alpha_2 \ge \beta_2 \ge \dots \ge \alpha_{n-1} \ge \beta_{n-1} \ge \alpha_n.$$

## Remarks

Both inequalities follow from the [[Courant–Fischer min-max theorem|CJK2HS2508]].
A subspace $W$ of $\C^m$ of dimension $j$ maps to the subspace $PW$ of $\C^n$ of dimension $j$, and $P$ preserves the Rayleigh quotient: $R_B(w) = R_A(Pw)$.
So the max-min for $\beta_j$ runs over fewer subspaces than the one for $\alpha_j$, which gives $\beta_j \le \alpha_j$.
The min-max form gives the other inequality.

Example: $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ has eigenvalues $3$ and $1$.
Deleting the second row and column leaves $(2)$, and indeed $3 \ge 2 \ge 1$.

## Explanations

### L1 human

Remove one row and the matching column from a symmetric matrix.
The eigenvalues of the smaller matrix fit between those of the larger one, one in each gap.
So a principal submatrix cannot have eigenvalues far from those of the whole matrix.

### L2 human

Interlacing is used to bound eigenvalues of graphs: the adjacency matrix of an induced subgraph is a principal submatrix.
Applied repeatedly, it compares the eigenvalues of $A$ with those of each of its principal submatrices.
The converse holds too: any two interlacing sequences arise from a Hermitian matrix and a principal submatrix of size one less.
</document>

<document source="https://tamnd.github.io/math2/node/8QTP2EFA92.md" tag="8QTP2EFA92" kind="definition">
# Bernstein–Gelfand–Ponomarev reflection functors

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $Q$ be a finite [[quiver|RE9YFFS3TS]].
A vertex $i$ is a *sink* if no arrow has source $i$, and a *source* if no arrow has target $i$.
Let $\sigma_i Q$ be the quiver obtained from $Q$ by reversing every arrow incident to $i$; write $a^*$ for the reverse of $a$.
Let $i$ be a sink of $Q$ and $V$ a [[representation|TXVWFWP6VQ]] of $Q$ over $k$.
The representation $S_i^+ V$ of $\sigma_i Q$ is defined by:

1. $(S_i^+ V)_j = V_j$ and $(S_i^+ V)_b = V_b$ for vertices $j \ne i$ and arrows $b$ not incident to $i$;
2. $(S_i^+ V)_i = \ker \big( \bigoplus_{a \colon j \to i} V_j \xrightarrow{(V_a)} V_i \big)$, the sum taken over the arrows $a$ ending at $i$;
3. for an arrow $a \colon j \to i$, the map $(S_i^+ V)_{a^*} \colon (S_i^+ V)_i \to V_j$ is the inclusion of the kernel followed by the projection onto the summand $V_j$ of index $a$.

With the evident action on morphisms, $S_i^+$ is a [[functor|28NQ0KRHD3]].
Dually, for a source $i$, $S_i^-$ replaces $V_i$ by the cokernel of $V_i \to \bigoplus_{a \colon i \to j} V_j$.

## Remarks

Let $S_i$ be the simple representation at $i$.
If $V$ is [[indecomposable|E2P29GCD9T]] and $V \not\cong S_i$, then $S_i^+ V$ is indecomposable, $S_i^- S_i^+ V \cong V$, and the dimension vector changes by $\dim (S_i^+ V)_i = \sum_{a \colon j \to i} \dim V_j - \dim V_i$.
Also $S_i^+ S_i = 0$.
Example: for $Q = (1 \xrightarrow{a} 2)$, the vertex $2$ is a sink.
For $V = (k \xrightarrow{\id} k)$, the kernel of $\id \colon k \to k$ is $0$, so $S_2^+ V$ is the representation $k \leftarrow 0$ of $1 \leftarrow 2$.
Its dimension vector $(1, 0)$ is the image of $(1, 1)$ under the reflection $x_2 \mapsto x_1 - x_2$.
Non-example: $S_1^+$ is not defined on $Q$, because vertex $1$ is not a sink.
</document>

<document source="https://tamnd.github.io/math2/node/8QYVM1JTMF.md" tag="8QYVM1JTMF" kind="statement">
# Hartogs–Bochner extension theorem

## Statement

Let $n \ge 2$ and let $D \subseteq \C^n$ be a [[bounded|09QSZVVM26]] open set with $C^1$ boundary $\partial D$ such that $\C^n \setminus \bar D$ is [[connected|0SPE8WNWFV]].
Let $f \colon \partial D \to \C$ be a $C^1$ function that satisfies the [[tangential Cauchy–Riemann equations|X7GP9BECD8]] on $\partial D$.
Then there is a continuous function $F$ on $\bar D$ that is [[holomorphic|507C34W0ZT]] on $D$ and satisfies $F = f$ on $\partial D$.

## Remarks

The proof uses the [[Bochner–Martinelli formula|YMHYAEJQ80]].
The Bochner–Martinelli integral of $f$ over $\partial D$ defines a function $F_-$ on $D$ and a function $F_+$ on $\C^n \setminus \bar D$.
The tangential Cauchy–Riemann equations make both functions holomorphic, and the jump formula gives $F_- - F_+ = f$ on $\partial D$.
The function $F_+$ tends to $0$ at infinity.
On each complex line that misses $\bar D$ it is an entire function of one variable tending to $0$, hence zero by [[Liouville's theorem|0EHKVRF4J6]].
These lines fill a nonempty open set.
Since $\C^n \setminus \bar D$ is connected, $F_+ = 0$ by the identity theorem, so $F = F_-$ has boundary values $f$.

Example: if $f$ is the restriction to $\partial D$ of a function holomorphic near $\bar D$, then $f$ satisfies the tangential Cauchy–Riemann equations and $F$ is that function.

The hypothesis $n \ge 2$ is necessary.
For $n = 1$ the tangential condition is empty.
The function $f(z) = \bar z$ on the unit circle has no holomorphic extension to the disc that is continuous up to the boundary.
Indeed $\int_{\abs{z} = 1} \bar z \, dz = 2\pi i \ne 0$, whereas Cauchy's theorem would make the integral of such an extension zero.

## Explanations

### L1 human

On the boundary of a domain in $\C^n$ with $n \ge 2$, a function may already satisfy the part of the Cauchy–Riemann equations that only involves directions along the boundary.
The theorem says that such a function is always the boundary value of a holomorphic function inside.
So, in several variables, holomorphic functions on a domain are determined by, and can be built from, suitable functions on its boundary.

### L2 human

The theorem is a boundary version of [[Hartogs's extension theorem|FSGES670AZ]].
Applied to the restriction of a holomorphic function defined near $\partial D$, it gives the extension of that function to $D$.
It is the starting point of the theory of CR functions on real hypersurfaces and of their holomorphic extension.
</document>

<document source="https://tamnd.github.io/math2/node/8R29G6NG9Q.md" tag="8R29G6NG9Q" kind="concept">
# Paracompact space

## Summary

A paracompact space is a [[topological space|0CQHFPGJ45]] in which every [[open cover|049PXYHW1T]] has an open refinement that is a [[locally finite family|DZ297MMGV8]].
Every [[compact space|0TVJZYB6SH]] and every [[metric space|0VKD1TP7F3]] is paracompact.
Paracompact Hausdorff spaces are [[normal|0NSR9V8TED]], and they admit partitions of unity subordinate to any open cover.
This is why paracompactness is a standard assumption on manifolds: it lets one glue local constructions into global ones.

## Explanations

### L1 human

Compactness asks that every open cover can be cut down to finitely many sets.
Paracompactness asks less: every open cover can be replaced by smaller open sets that are only "finitely many near each point".
On the real line, any open cover can be refined to small intervals that pile up only finitely often around each point.
This local finiteness is enough to add up infinitely many local pieces without trouble.
</document>

<document source="https://tamnd.github.io/math2/node/8R4CVVV8WT.md" tag="8R4CVVV8WT" kind="definition">
# B-spline

## Definition

Let $t_0 \le t_1 \le \cdots \le t_m$ be real numbers, called *knots*.
The *B-splines* $B_{i,p}$ of degree $p \ge 0$ for these knots are the functions $\R \to \R$ defined by recursion on $p$:

1. for $0 \le i \le m - 1$, $B_{i,0}(x) = 1$ if $t_i \le x < t_{i+1}$ and $B_{i,0}(x) = 0$ otherwise;
2. for $p \ge 1$ and $0 \le i \le m - p - 1$,
$$B_{i,p}(x) = \frac{x - t_i}{t_{i+p} - t_i} \, B_{i,p-1}(x) + \frac{t_{i+p+1} - x}{t_{i+p+1} - t_{i+1}} \, B_{i+1,p-1}(x),$$
where a term whose denominator is $0$ is taken to be $0$.

## Remarks

$B_{i,p}$ is a [[spline|84TJ8KQ926]] of degree $p$ with knots $t_i, \dots, t_{i+p+1}$; if $t_i < t_{i+p+1}$, it is positive on $(t_i, t_{i+p+1})$ and zero outside $[t_i, t_{i+p+1})$.
On $[t_p, t_{m-p})$ the B-splines of degree $p$ sum to $1$, so they form a [[partition of unity|NH739XXVXK]] there.
At a knot of multiplicity $r \le p$, the B-splines of degree $p$ have $p - r$ continuous derivatives.
An equivalent definition writes $B_{i,p}$ as $(t_{i+p+1} - t_i)$ times a divided difference of the truncated power $(\cdot - x)_+^p$ at $t_i, \dots, t_{i+p+1}$.

Example: knots $0, 1, 2$ and $p = 1$.
Then $B_{0,1}(x) = x \, B_{0,0}(x) + (2 - x) \, B_{1,0}(x)$, which is $x$ on $[0, 1)$, $2 - x$ on $[1, 2)$ and $0$ elsewhere: the hat function.

Non-example: let $g$ be $x$ on $[0, 1)$, $1$ on $[1, 2)$ and $0$ elsewhere.
It is piecewise linear on the same knots, but it is not $B_{0,1}$.
It differs from the hat function on $(1, 2)$, and it jumps at $2$.
</document>

<document source="https://tamnd.github.io/math2/node/8R4H8GW1CD.md" tag="8R4H8GW1CD" kind="example">
# Ricker model

## Example

Let $r > 0$ and $K > 0$ be constants.
The *Ricker model* is the [[nonlinear difference equation|7C7FBZGERM]]
$$N_{t+1} = f(N_t), \qquad f(N) = N\, e^{r (1 - N/K)},$$
for a population size $N_t \ge 0$.
Its [[equilibria|D62PEKHRV5]] are the solutions of $f(N) = N$, namely $N = 0$ and $N = K$.
The derivative is $f'(N) = e^{r(1 - N/K)} (1 - r N / K)$, so $f'(0) = e^r > 1$ and $f'(K) = 1 - r$.
Hence the equilibrium $0$ is unstable, and the equilibrium $K$ is asymptotically [[stable|CHEQ235HW9]] when $\abs{1 - r} < 1$, that is, when $0 < r < 2$.
At $r = 2$ one has $f'(K) = -1$, and as $r$ increases past $2$ the solutions go through a [[period-doubling cascade|NQDTTZ59X8]] towards chaotic behaviour.

## Explanations

### L1 human

The Ricker model describes a population, such as a fish stock, that reproduces once per season.
When the population is far below $K$, it grows by about the factor $e^r$; above $K$, the exponential factor is less than $1$ and the population shrinks.
For a small growth rate $r$, the population settles at $K$.
For larger $r$, it overshoots and undershoots, first in a regular cycle of period $2$, then $4$, and finally erratically.
</document>

<document source="https://tamnd.github.io/math2/node/8RA7AMWSAF.md" tag="8RA7AMWSAF" kind="concept">
# Conservation law

## Summary

A conservation law is a [[partial differential equation|GC5BDZ6QBC]] of the form $u_t + \nabla \cdot F(u) = 0$.
It says that the amount of $u$ in a region changes only by the flux $F(u)$ through its boundary.
Mass, momentum and energy in continuum mechanics obey such laws.
Nonlinear conservation laws develop discontinuities in finite time, so their theory is built on weak solutions, [[shock waves|JSDMGGTG7G]] and entropy conditions.
See the [[definition|XYWMB0G011]].

## Explanations

### L1 human

Think of cars on a long road, with $u(x, t)$ the number of cars per kilometre.
Cars are neither created nor destroyed, so the number of cars between two points changes only because cars drive in at one end and out at the other.
The rate at which cars pass a point is the flux $F(u)$, which depends on how dense the traffic is.
Writing this bookkeeping for very short stretches of road gives the equation $u_t + F(u)_x = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/8RDSR3JXP2.md" tag="8RDSR3JXP2" kind="statement">
# Alternating-direction implicit method

## Statement

Consider the [[heat equation|VP2HN86VAG]] $u_t = u_{xx} + u_{yy}$ on the square $(0, 1)^2$, with $u = 0$ on the boundary and a given initial value.
Let $M \ge 1$, $h = 1/(M + 1)$, $\Delta t > 0$, and let $u^n_{jk}$ approximate $u(jh, kh, n \Delta t)$ for $1 \le j, k \le M$, with zero values on the boundary.
Let $\delta_x^2$ and $\delta_y^2$ be the second differences $(\delta_x^2 v)_{jk} = (v_{j+1,k} - 2 v_{jk} + v_{j-1,k}) / h^2$ and $(\delta_y^2 v)_{jk} = (v_{j,k+1} - 2 v_{jk} + v_{j,k-1}) / h^2$.
The Peaceman–Rachford alternating-direction implicit method computes
$$\left(I - \tfrac{\Delta t}{2} \delta_x^2\right) u^{*} = \left(I + \tfrac{\Delta t}{2} \delta_y^2\right) u^n, \qquad \left(I - \tfrac{\Delta t}{2} \delta_y^2\right) u^{n+1} = \left(I + \tfrac{\Delta t}{2} \delta_x^2\right) u^{*}.$$

1. Each half step consists of independent tridiagonal systems, one for each grid line in the $x$ direction or in the $y$ direction.
2. The method is unconditionally stable: $\norm{u^{n+1}}_2 \le \norm{u^n}_2$ for all $\Delta t > 0$ and $h$, where $\norm{v}_2^2 = h^2 \sum_{j,k} v_{jk}^2$.
3. For smooth solutions, the method is second-order accurate: its local truncation error is $O(\Delta t^2 + h^2)$.

## Remarks

The matrices $\delta_x^2$ and $\delta_y^2$ are symmetric, negative definite and commute, with common eigenvectors $\sin(p \pi j h) \sin(q \pi k h)$.
On such an eigenvector, with eigenvalues $-\alpha$ and $-\beta$ ($\alpha, \beta > 0$), one step multiplies by
$$g = \frac{1 - \frac{\Delta t}{2} \alpha}{1 + \frac{\Delta t}{2} \alpha} \cdot \frac{1 - \frac{\Delta t}{2} \beta}{1 + \frac{\Delta t}{2} \beta},$$
and each factor has absolute value less than $1$; this gives clause 2.
Eliminating $u^*$ shows that the method equals the two-dimensional [[Crank–Nicolson method|M92S0Y15N1]] plus the term $\frac{\Delta t^2}{4} \delta_x^2 \delta_y^2 (u^{n+1} - u^n)$, which is $O(\Delta t^3)$ for smooth solutions; this gives clause 3.
Each tridiagonal system is solved in linear time by the [[tridiagonal matrix algorithm|F1QWA1PEQQ]], so a step costs $O(M^2)$ operations, against a sparse system with $M^2$ unknowns for Crank–Nicolson.

## Explanations

### L1 human

Implicit methods for the heat equation in two dimensions lead to a large linear system at every step.
The alternating-direction implicit method splits each step in two halves.
In the first half it is implicit only along the rows of the grid, in the second only along the columns.
So it solves many small tridiagonal systems instead of one large one, and it keeps the stability and the second-order accuracy of Crank–Nicolson.

### L2 human

The method is an operator splitting of $\delta_x^2 + \delta_y^2$, and its analysis uses that the two pieces commute; for variable coefficients or non-rectangular domains this fails and the analysis is harder.
The same iteration, used with varying parameters, is an iterative solver for the discrete Poisson equation and for Lyapunov and Sylvester matrix equations.
In three dimensions one usually uses the Douglas form of the method instead.
</document>

<document source="https://tamnd.github.io/math2/node/8REBFQD6E0.md" tag="8REBFQD6E0" kind="problem">
# Catenary problem

## Problem

A uniform, perfectly flexible and inextensible chain of length $\ell$ hangs under constant gravity from two fixed points $(a, A)$ and $(b, B)$.
Here $a < b$, and $\ell$ is larger than the distance between the points.
What shape does the chain take?

As a variational problem: among $y \in C^1([a, b])$ with $y(a) = A$, $y(b) = B$ and [[arc length|0TH78S7GBZ]]
$$\int_a^b \sqrt{1 + y'(x)^2} \, dx = \ell,$$
find one that minimises the potential energy, which is proportional to
$$U(y) = \int_a^b y(x) \sqrt{1 + y'(x)^2} \, dx.$$

## Remarks

This is an [[isoperimetric problem|6FY0KHMK5A]].
With a multiplier $\lambda$, the Lagrangian is $(y - \lambda) \sqrt{1 + y'^2}$, which does not depend on $x$.
The [[Beltrami identity|F7EKTEESQ5]] gives $(y - \lambda) / \sqrt{1 + y'^2} = c$, and the solutions with $c > 0$ are the catenaries
$$y(x) = \lambda + c \cosh\Big(\frac{x - d}{c}\Big),$$
where the constants $c$, $d$, $\lambda$ are fixed by the two end conditions and the length.
Here $\cosh$ is the [[hyperbolic cosine|036696EMG2]].
Jacob Bernoulli posed the problem in Acta Eruditorum, and solutions by Leibniz, Huygens and Johann Bernoulli were published in 1691.

## Explanations

### L1 human

Hold a chain or a necklace by its two ends and let it hang.
The curve it forms looks like a parabola but is a different curve, the catenary, given by the hyperbolic cosine.
The chain settles in the shape that puts its centre of mass as low as possible, given its length.
</document>

<document source="https://tamnd.github.io/math2/node/8RT3FGZPZY.md" tag="8RT3FGZPZY" kind="definition">
# Shape operator

## Definition

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]], let $W \subseteq S$ be open in $S$, and let $N \colon W \to \R^3$ be a smooth unit [[normal vector|0BJDQ2HBH1]] field.
For $p \in W$, the differential $dN_p$ maps the tangent plane $T_p S$ into itself, because $\ip{N}{N} = 1$.
The *shape operator* of $S$ at $p$ is the linear map
$$S_p = -dN_p \colon T_p S \to T_p S.$$

## Remarks

The map $S_p$ is [[self-adjoint|0MB8HQW495]] for the inner product of $T_p S$, and $\ip{S_p v}{w} = I\!I_p(v, w)$, where $I\!I_p$ is the polar form of the [[second fundamental form|9KHQR16P1B]].
In a local parametrisation, the matrix of $S_p$ in the basis $x_u, x_v$ is
$$\begin{pmatrix} E & F \\ F & G \end{pmatrix}^{-1} \begin{pmatrix} e & f \\ f & g \end{pmatrix},$$
with $E, F, G$ from the [[first fundamental form|MAJ3DH7MZS]] and $e, f, g$ from the second.
Example: on the sphere of radius $r$ centred at $0$ with the outer normal $N(p) = p / r$, one has $dN_p = \frac{1}{r} \id$, so $S_p = -\frac{1}{r} \id$.
With the inner normal, $S_p = \frac{1}{r} \id$.
Non-example: on a Möbius strip there is no global continuous unit normal, so the shape operator is defined only locally, up to sign.
</document>

<document source="https://tamnd.github.io/math2/node/8RWDHW0CXC.md" tag="8RWDHW0CXC" kind="concept">
# Bergman kernel

## Summary

The Bergman kernel of a domain is the reproducing kernel of its [[Bergman space|2D619WTPYC]] $A^2$ of square-integrable [[holomorphic functions|0NAHXR1FRH]].
It is a function $K(z, w)$, holomorphic in $z$ and antiholomorphic in $w$, such that every $f \in A^2$ satisfies $f(w) = \int f(z)\overline{K(z, w)}\, dA(z)$.
For the unit disc it is $K(z, w) = \frac{1}{\pi (1 - z\bar w)^2}$.
It gives the orthogonal projection from $L^2$ onto $A^2$, it transforms simply under biholomorphic maps, and in several variables it defines the Bergman metric.

## Explanations

### L1 human

In the Bergman space, the value of a function at a point is an inner product with one special function attached to that point.
The Bergman kernel collects all these special functions into one formula.
On the unit disc the kernel at the point $w = 0$ is the constant $1/\pi$, which says that $f(0)$ is the average of $f$ over the disc.
The kernel encodes the shape of the domain, which is why it is used to study conformal and biholomorphic maps.
</document>

<document source="https://tamnd.github.io/math2/node/8S129JX919.md" tag="8S129JX919" kind="statement">
# Remez inequality

## Statement

Let $n \ge 0$ be an integer and let $\sigma > 0$.
Let $p$ be a real polynomial of [[degree|0DPF6WRZW6]] at most $n$.
Let $E \subseteq [-1, 1 + \sigma]$ be a Lebesgue measurable set of measure at least $2$ such that $\abs{p(x)} \le 1$ for all $x \in E$.
Then
$$\max_{-1 \le x \le 1 + \sigma} \abs{p(x)} \le T_n(1 + \sigma),$$
where $T_n$ is the [[Chebyshev polynomial|M6EAR1JHPP]] of the first kind of degree $n$.

## Remarks

The bound is sharp: $p = T_n$ and $E = [-1, 1]$ give equality, since $T_n$ is increasing on $[1, \infty)$.
Proof idea: compare $p$ with a rescaled Chebyshev polynomial whose set where it is at most $1$ in absolute value has measure $2$.
A count of sign changes, as in the minimal deviation property of $T_n$, shows that $p$ cannot be larger.

Example: for $n = 1$, a line with $\abs{p} \le 1$ on a set of measure $2$ in $[-1, 1 + \sigma]$ is at most $1 + \sigma = T_1(1 + \sigma)$ in absolute value on the whole interval.

By an affine change of variable, for an interval $I$ and a measurable $E \subseteq I$ of positive measure,
$$\sup_{I} \abs{p} \le T_n\left(\frac{2 \abs{I}}{\abs{E}} - 1\right) \sup_{E} \abs{p} \le \left(\frac{4 \abs{I}}{\abs{E}}\right)^n \sup_{E} \abs{p}.$$
Without a lower bound on the measure of $E$ there is no bound: $p(x) = c x$ is small on a tiny interval around $0$ for every $c$.

## Explanations

### L1 human

A polynomial of degree $n$ that is small on a large part of an interval cannot be huge on the rest of it.
The Remez inequality gives the exact limit, and the Chebyshev polynomial is the worst case.

### L2 human

The inequality is a quantitative form of the fact that a polynomial is determined by its values on any set of positive measure.
It extends Chebyshev's extremal property, which concerns the growth of $T_n$ outside $[-1, 1]$, to arbitrary measurable sets.
It is used to bound the size of sublevel sets of polynomials and has versions for several variables and for exponential sums.
</document>

<document source="https://tamnd.github.io/math2/node/8S2SJ4SC2E.md" tag="8S2SJ4SC2E" kind="definition">
# Kurosh–Amitsur radical theory

## Definition

Work with associative [[rings that need not have an identity|MBK7NFE74S]].
A class $\mathcal{R}$ of such rings, closed under isomorphism, is a *radical class* in the sense of Kurosh and Amitsur if the following hold.

1. If $A \in \mathcal{R}$ and $B$ is a homomorphic image of $A$, then $B \in \mathcal{R}$.
2. Every ring $A$ has a [[two-sided ideal|0DDA6CY7GJ]] $\mathcal{R}(A)$ that belongs to $\mathcal{R}$ and contains every ideal of $A$ that belongs to $\mathcal{R}$.
3. For every ring $A$, the quotient $A/\mathcal{R}(A)$ has no nonzero ideal in $\mathcal{R}$, that is, $\mathcal{R}(A/\mathcal{R}(A)) = 0$.

The ideal $\mathcal{R}(A)$ is the *$\mathcal{R}$-radical* of $A$.
A ring $A$ is *$\mathcal{R}$-semisimple* if $\mathcal{R}(A) = 0$.

## Remarks

Example: the class of nil rings is a radical class.
Homomorphic images of nil rings are nil.
The sum of all nil ideals of $A$ is the largest nil ideal, the [[upper nilradical|3G2JQEYBEK]] $N$.
If $I/N$ is a nil ideal of $A/N$, then every $x \in I$ has a power in $N$, so $I$ is nil and $I/N = 0$.
Non-example: the class of nilpotent rings satisfies clause 1 but not clause 2, since a ring can have an infinite family of nilpotent ideals whose sum is not nilpotent.
Other radical classes are the Jacobson radical rings (rings equal to their [[Jacobson radical|XA0DCNH7WY]]), the locally nilpotent rings, which give the [[Levitzki radical|9DD8M9R868]], and the class giving the [[prime radical|WF117EJ2HF]].
</document>

<document source="https://tamnd.github.io/math2/node/8SETG842RX.md" tag="8SETG842RX" kind="example">
# Apollonian gasket

## Example

Start with four [[circles|060W5CBZP6]], one enclosing the other three, that are pairwise tangent at six distinct points.
The three inner circles and the outer circle bound four curvilinear triangles.
Into each such curvilinear triangle, insert the unique circle inside it tangent to its three sides.
This creates three new curvilinear triangles.
Repeat forever.
The *Apollonian gasket* is the family of all circles obtained, or the closure of their union.
Apart from the outer circle, which encloses the others, the circles have disjoint interiors, so they form a [[circle packing|S360JBEYKD]].
For the curvatures, take the outer circle with negative sign.
By Descartes' circle theorem, four pairwise tangent circles with curvatures $k_1, k_2, k_3, k_4$ satisfy $(k_1 + k_2 + k_3 + k_4)^2 = 2(k_1^2 + k_2^2 + k_3^2 + k_4^2)$.
Check with the start $(-1, 2, 2, 3)$: an outer circle of radius $1$, two circles of radius $\tfrac12$ and one of radius $\tfrac13$; then $6^2 = 36 = 2(1 + 4 + 4 + 9)$.
Each new circle replaces one curvature $k_4$ by $2(k_1 + k_2 + k_3) - k_4$, so in this gasket all curvatures are integers.

## Remarks

The leftover set, the complement in the outer disc of all open discs, has area zero.
Its [[Hausdorff dimension|SSNFTHZ6YJ]] is about $1.3057$, strictly between $1$ and $2$, so it is a fractal.

## Explanations

### L1 human

Draw three coins that touch each other in pairs, and a large circle around them touching all three.
Fill every curved triangular gap with the largest circle that fits, then fill the smaller gaps, and keep going forever.
The picture becomes a lace of ever smaller circles, the Apollonian gasket.
With a good start, such as curvatures $-1, 2, 2, 3$, every circle in the picture has a whole-number curvature.
</document>

<document source="https://tamnd.github.io/math2/node/8SFZYANHNC.md" tag="8SFZYANHNC" kind="definition">
# Kolmogorov automorphism

## Definition

Let $(X, \mathcal{B}, \mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \colon X \to X$ be an invertible [[measure-preserving transformation|556WBDAHNP]] whose inverse is also measure-preserving.
For a [[sub-σ-algebra|0SV8K0TDGY]] $\mathcal{K} \subseteq \mathcal{B}$ and $n \in \Z$, let $T^n \mathcal{K} = \set{T^n A : A \in \mathcal{K}}$.
The map $T$ is a *Kolmogorov automorphism*, or *K-automorphism*, if there is a sub-σ-algebra $\mathcal{K}$ such that:

1. $T \mathcal{K} \subseteq \mathcal{K}$.
2. The σ-algebra generated by $\bigcup_{n \ge 0} T^{-n} \mathcal{K}$ is $\mathcal{B}$, up to sets of measure $0$.
3. Every set in $\bigcap_{n \ge 0} T^{n} \mathcal{K}$ has measure $0$ or $1$.

## Remarks

Example: let $X = \set{0, 1}^{\Z}$ with the product of the measures giving weight $1/2$ to each symbol, and $T$ the left shift, $(Tx)_n = x_{n+1}$.
Let $\mathcal{K}$ be generated by the coordinates $x_m$ with $m \le 0$.
Then $T^{n} \mathcal{K}$ is generated by the $x_m$ with $m \le -n$, for every $n \in \Z$.
So clause 1 holds, the $T^{-n}\mathcal{K}$ for $n \ge 0$ together generate all coordinates (clause 2), and clause 3 is [[Kolmogorov's zero–one law|073Q8Z94KG]] for the independent coordinates.

Non-example: an irrational rotation $x \mapsto x + \alpha \bmod 1$ is not a K-automorphism.
A K-automorphism has positive [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] unless the space is trivial, and the rotation has entropy $0$.

On a standard probability space, $T$ is a K-automorphism if and only if every nontrivial factor of $T$ has positive entropy.
</document>

<document source="https://tamnd.github.io/math2/node/8SH7WETCKY.md" tag="8SH7WETCKY" kind="statement">
# Singular cardinals hypothesis

## Statement

Work in [[ZFC|0HE5WDQ9FJ]].
For an infinite [[cardinal|0EBACQFAZW]] $\kappa$, let $\operatorname{cf}(\kappa)$ be its [[cofinality|BAWHE1W6MH]], and call $\kappa$ singular when $\operatorname{cf}(\kappa) < \kappa$.
Let $\kappa^+$ be the least cardinal greater than $\kappa$.
The *singular cardinals hypothesis* (SCH) is the assertion: for every singular cardinal $\kappa$ with $2^{\operatorname{cf}(\kappa)} < \kappa$, one has $\kappa^{\operatorname{cf}(\kappa)} = \kappa^+$.

## Remarks

An infinite cardinal that is not singular is a [[regular cardinal|13ZV2F8JAR]].
SCH is equivalent to the statement that $2^\kappa = \kappa^+$ for every singular strong limit cardinal $\kappa$, where strong limit means $2^\lambda < \kappa$ for all $\lambda < \kappa$.
The generalized form of the [[continuum hypothesis|0MN2MER9WN]], $2^\lambda = \lambda^+$ for every infinite $\lambda$, implies SCH.
So SCH is consistent with ZFC whenever ZFC is consistent.
SCH is not provable in ZFC if suitable large cardinals are consistent: Magidor built models where it fails at $\aleph_\omega$ from large cardinal hypotheses.
Its failure needs large cardinals: by Jensen's covering lemma, if $0^\sharp$ does not exist, then SCH holds.
Gitik and Mitchell showed that the failure of SCH is equiconsistent with a measurable cardinal $\kappa$ of Mitchell order $\kappa^{++}$.
Solovay proved that SCH holds at all singular cardinals above a strongly compact cardinal.
Silver proved that if $\kappa$ is singular of uncountable cofinality and $2^\lambda = \lambda^+$ for all infinite $\lambda < \kappa$, then $2^\kappa = \kappa^+$.

## Explanations

### L1 human

Here $2^\kappa$ is the number of subsets of a set of size $\kappa$.
And $\kappa^\lambda$ is the number of functions from a set of size $\lambda$ to a set of size $\kappa$.
For regular cardinals, ZFC says very little about the size of $2^\kappa$: almost any reasonable value is consistent.
Singular cardinals, such as $\aleph_\omega$, behave differently, and ZFC proves real restrictions on their powers.
The singular cardinals hypothesis says that these powers take the smallest values that the restrictions allow.
It is a weak form of the generalized continuum hypothesis that concerns only singular cardinals.

### L2 human

In ZFC, cardinal exponentiation $\kappa^\lambda$ is governed by the continuum function $\lambda \mapsto 2^\lambda$ on regular cardinals together with the values $\kappa^{\operatorname{cf}(\kappa)}$ at singular cardinals.
Under SCH, the second part is determined by the first, so all of cardinal arithmetic is computed from the values of $2^\lambda$ at regular $\lambda$.
The study of how SCH can fail led to Shelah's pcf theory, which gives bounds in ZFC alone.
A famous such bound is that if $\aleph_\omega$ is a strong limit, then $2^{\aleph_\omega} < \aleph_{\omega_4}$.
</document>

<document source="https://tamnd.github.io/math2/node/8SK3GQ4S0M.md" tag="8SK3GQ4S0M" kind="statement">
# Tauber's theorem

## Statement

Let $(a_n)_{n \ge 0}$ be a sequence of complex numbers such that the [[power series|0VP56B4EMX]] $f(x) = \sum_{n=0}^{\infty} a_n x^n$ converges for $0 \le x < 1$.
Assume that the [[one-sided limit|0909PWZR62]] $s = \lim_{x \to 1^-} f(x)$ exists.
Assume also that $n a_n \to 0$ as $n \to \infty$.
Then the series $\sum_{n=0}^{\infty} a_n$ [[converges|0CVVKY7GY2]], and its sum is $s$.

## Remarks

The idea of the proof is to compare the partial sum $s_N = \sum_{n \le N} a_n$ with $f(x)$ for $x = 1 - 1/N$.
The difference is at most $\sum_{n \le N} \abs{a_n} (1 - x^n) + \sum_{n > N} \abs{a_n} x^n$.
Use $1 - x^n \le n(1 - x)$ in the first sum, which is then at most the average of $n \abs{a_n}$ over $n \le N$, and this tends to $0$.
In the second sum, $\abs{a_n} \le \varepsilon / n \le \varepsilon / N$ for large $N$, so it is at most $\varepsilon / (N(1 - x)) = \varepsilon$.

The theorem is a partial converse of [[Abel's theorem|0S4EMDQS2F]], which gives $\lim_{x \to 1^-} f(x) = \sum a_n$ whenever the series converges.
The condition on $n a_n$ cannot be dropped.
For $a_n = (-1)^n$, one has $f(x) = 1/(1 + x) \to 1/2$, but $\sum (-1)^n$ diverges, and $n a_n$ does not tend to $0$.
Littlewood showed that the weaker condition $n a_n = O(1)$ is enough; see [[big O notation|0RYTGBJAGN]].

## Explanations

### L1 human

Abel's theorem says that a convergent series can be summed by plugging $x \to 1$ into its power series.
The converse is false in general: the power series can have a limit even when the series diverges.
Tauber's theorem says that the converse does hold when the terms shrink faster than $1/n$.

### L2 human

This is the first Tauberian theorem, and it gave its name to the whole family of [[Abelian and Tauberian theorems|D6XWK1NRY1]].
An Abelian theorem passes from ordinary convergence to convergence of a summation method; a Tauberian theorem goes back, under an extra size condition on the terms.
Later results of this type, such as the Hardy–Littlewood and Wiener Tauberian theorems, are central tools in analytic number theory.
</document>

<document source="https://tamnd.github.io/math2/node/8SQK7RTB63.md" tag="8SQK7RTB63" kind="concept">
# Zariski topology

## Summary

The Zariski topology is a topology on the [[spectrum|FB45BHHA2J]] of a [[commutative ring|0310REJJY8]].
Its closed sets are the sets of [[prime ideals|00MHHZ9SNE]] containing a given set of elements.
On an affine variety it restricts to the topology whose closed sets are the zero sets of polynomials.
It has few open sets: it is [[quasi-compact|J4YW23MD6N]] and usually not [[Hausdorff|0S1FJ8VEX9]].
It is the topology used throughout algebraic geometry.

## Explanations

### L1 human

In the Zariski topology a closed set is a set of common zeros of some functions.
On the line $\C$, the zero set of a nonzero polynomial is finite, so the closed subsets of the line are the finite sets and the whole line.
Any two nonempty open sets then meet, which is very different from the usual topology.
On $\operatorname{Spec} R$ one says that $f$ "vanishes" at a prime $\mathfrak p$ when $f \in \mathfrak p$.
The [[definition|EXJAXVXW51]] uses this to define the closed sets.
</document>

<document source="https://tamnd.github.io/math2/node/8ST7CTZ2DE.md" tag="8ST7CTZ2DE" kind="definition">
# Residue field

## Definition

Let $(R, \mathfrak m)$ be a [[local ring|MZ02BKPHJA]].
The *residue field* of $R$ is the [[quotient ring|0076HWHGHQ]] $k = R/\mathfrak m$.
It is a [[field|04C5ZBG1WT]] because $\mathfrak m$ is a [[maximal ideal|0PV74TR5FP]].

Let $R$ be a [[commutative ring|0310REJJY8]] and $\mathfrak p$ a [[prime ideal|00MHHZ9SNE]].
The *residue field of $R$ at $\mathfrak p$* is the residue field
$$\kappa(\mathfrak p) = R_{\mathfrak p} / \mathfrak p R_{\mathfrak p}$$
of the [[localization|X6JD4AG973]] of $R$ at $\mathfrak p$.

## Remarks

The field $\kappa(\mathfrak p)$ is isomorphic to the [[field of fractions|0F2PTGX1A2]] of the [[integral domain|0QVD7WBFGN]] $R/\mathfrak p$.

Example: the residue field of $\Z_{(p)}$ is $\Z/p\Z$, since $a/b \mapsto a b^{-1} \bmod p$ is a surjection with kernel $p\Z_{(p)}$.
The residue field of $k[\![x]\!]$ is $k$, by taking the constant term.
For $R = \Z$, the residue field at $(0)$ is $\Q$ and the residue field at $p\Z$ is $\Z/p\Z$.

Non-example: in the local ring $\Z_{(p)}$, the quotient $\Z_{(p)}/p^2\Z_{(p)} \cong \Z/p^2\Z$ is not a field, because $p \ne 0$ there and $p \cdot p = 0$.
Only the quotient by the maximal ideal is a field.
</document>

<document source="https://tamnd.github.io/math2/node/8T34EC16NH.md" tag="8T34EC16NH" kind="definition">
# Elementary abelian group

## Definition

Let $p$ be a [[prime|01826KMG2H]].
A group $A$ is an *elementary abelian $p$-group* if $A$ is [[abelian|082GHNPJ53]] and every element $a \neq e$ of $A$ has [[order|0WE2NHZJ7B]] $p$.
A group is *elementary abelian* if it is an elementary abelian $p$-group for some prime $p$.

## Remarks

Example: $(\Z/3\Z)^2$ is elementary abelian: it is abelian, and $3(a, b) = (3a, 3b) = (0, 0)$, so every nonzero element has order $3$.

Non-example: $\Z/4\Z$ is abelian but not elementary abelian, since $1$ has order $4$.
Non-example: the symmetric group $S_3$ is not elementary abelian: it is not abelian, and it has elements of orders $2$ and $3$.

Writing $A$ additively, the scalar multiplication $(k + p\Z) \cdot a = ka$ is well defined because $pa = 0$, and it makes $A$ a [[vector space|000AYPA9AJ]] over the [[finite field|0ZXZ82FTX2]] $\F_p$; the subgroups are the subspaces.
Hence a finite elementary abelian $p$-group is isomorphic to $(\Z/p\Z)^n$, where $n$ is its dimension, and its order is $p^n$.
Its automorphism group is the [[general linear group|0FCHE902D7]] $\mathrm{GL}_n(\F_p)$.
The trivial group is elementary abelian for every $p$, with $n = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/8T4M7RN9AC.md" tag="8T4M7RN9AC" kind="concept">
# MV-algebra

## Summary

An MV-algebra is an algebraic structure with a truncated addition $\oplus$, a negation $\neg$ and a constant $0$.
It is modelled on the unit interval $[0, 1]$ with $x \oplus y = \min(1, x + y)$ and $\neg x = 1 - x$.
Chen Chung Chang introduced MV-algebras in 1958 to give an algebraic proof of the completeness of Łukasiewicz infinite-valued logic, a [[many-valued logic|DWYCKAKXZD]].
Every [[Boolean algebra|0XADJKM699]] is an MV-algebra, and MV-algebras play for Łukasiewicz logic the role that Boolean algebras play for classical logic.
Daniele Mundici showed that MV-algebras correspond to lattice-ordered abelian groups with a strong unit.

## Explanations

### L1 human

In Łukasiewicz logic a statement can be partly true, with a truth value anywhere between $0$ (false) and $1$ (true).
"Not $x$" has truth value $1 - x$, and a kind of "or" adds the truth values but stops at $1$.
For example, $0.3 \oplus 0.5 = 0.8$, while $0.7 \oplus 0.5 = 1$.
An MV-algebra is any structure that obeys the same equations as these operations on $[0, 1]$.
</document>

<document source="https://tamnd.github.io/math2/node/8T7XY797SV.md" tag="8T7XY797SV" kind="definition">
# Contractible space

## Definition

A topological space $X$ is *contractible* if the map from $X$ to a one-point space is a [[homotopy equivalence|XZG6MTSWCN]].

## Remarks

Equivalently, $X$ is nonempty and the identity map $\id_X$ is [[null-homotopic|SMEV5BSQ5S]].
A homotopy from $\id_X$ to a constant map is called a *contraction* of $X$.
The empty space is not contractible, since there is no map from a point to it.
Example: a convex subset $C \neq \emptyset$ of $\R^n$ is contractible; for a fixed $c \in C$, the map $H(x, t) = (1 - t) x + t c$ stays in $C$ by convexity and contracts $C$ to $c$.
Non-example: the circle $S^1$ is not contractible, since its [[fundamental group|S1XK3HXHFX]] is $\Z$, not trivial.
Every contractible space is [[path-connected|0WB6DH499T]].
</document>

<document source="https://tamnd.github.io/math2/node/8T9S6F9HRW.md" tag="8T9S6F9HRW" kind="example">
# Fence

## Example

Let $n \ge 1$.
The fence $Z_n$ is the [[partially ordered set|0QV2JS9RJ3]] on $x_1, \ldots, x_n$ with these covering relations:
$$x_1 < x_2 > x_3 < x_4 > \cdots.$$
That is, $x_i < x_{i+1}$ for odd $i$ and $x_i > x_{i+1}$ for even $i$, with $1 \le i < n$.
It is also called a zigzag poset.
Its Hasse diagram is a path that goes up and down in turn.

A [[linear extension|W8DZ5KGGHN]] of $Z_n$ is a bijection $f \colon \set{x_1, \ldots, x_n} \to \set{1, \ldots, n}$ with $f(x_1) < f(x_2) > f(x_3) < \cdots$.
So the linear extensions of $Z_n$ correspond to the alternating permutations of $\set{1, \ldots, n}$, and their number is the Euler zigzag number $E_n$.
Check for $n = 3$: the poset $x_1 < x_2 > x_3$ has the two linear extensions $x_1, x_3, x_2$ and $x_3, x_1, x_2$, and $E_3 = 2$.

## Remarks

The number of down-sets of $Z_n$ is the Fibonacci number $F_{n+2}$, with $F_1 = F_2 = 1$.
For $n = 3$ the down-sets are $\varnothing$, $\set{x_1}$, $\set{x_3}$, $\set{x_1, x_3}$ and $\set{x_1, x_2, x_3}$, which makes $5 = F_5$.
Every fence has height at most $2$.

## Explanations

### L1 human

A fence is a zigzag: up, down, up, down.
Putting its elements in a row that respects the zigzag is the same as writing the numbers $1$ to $n$ so that they go up, down, up, down.
This links a simple poset with a classical counting problem.
</document>

<document source="https://tamnd.github.io/math2/node/8TD2KCVH5N.md" tag="8TD2KCVH5N" kind="concept">
# Birkhoff sum and average

## Summary

The Birkhoff sum of a function along an orbit adds up its values at the first $n$ points of the orbit.
The Birkhoff average divides this sum by $n$.
The average is the time average of an observable: the mean value seen by watching the system for $n$ steps.
The ergodic theorems of Birkhoff and von Neumann say when these averages converge and identify the limit.
For an [[ergodic transformation|AZRPHS478B]] and an integrable function, the limit is the space average of the function for almost every starting point.
Birkhoff sums also appear in the definitions of Lyapunov exponents, cocycles and the thermodynamic formalism.

## Explanations

### L1 human

Imagine a system moving step by step and a quantity, such as temperature, that one can read off at each state.
The Birkhoff sum adds up the readings over the first $n$ steps, and the Birkhoff average is their mean.
For example, let $f$ be $1$ on a region $A$ and $0$ elsewhere.
Then the Birkhoff sum counts how many of the first $n$ steps the orbit spends in $A$, and the average is the fraction of time spent in $A$.
</document>

<document source="https://tamnd.github.io/math2/node/8TEV7CSX2H.md" tag="8TEV7CSX2H" kind="example">
# Standard map

## Example

Let $K \in \R$ be a parameter.
The *standard map*, or Chirikov–Taylor map, acts on points $(\theta, p)$ of the cylinder, where $\theta$ is taken modulo $2\pi$.
On the torus, $p$ is also taken modulo $2\pi$.
The map is
$$p_{n+1} = p_n + K \sin \theta_n, \qquad \theta_{n+1} = \theta_n + p_{n+1}.$$
Its Jacobian matrix with respect to $(p_n, \theta_n)$ is
$$\begin{pmatrix} 1 & K \cos \theta_n \\ 1 & 1 + K \cos \theta_n \end{pmatrix},$$
whose determinant is $1 + K \cos \theta_n - K \cos \theta_n = 1$.
So the map preserves area and orientation, and it is a [[symplectomorphism|A8TS2SNX9Z]] of the plane form $dp \wedge d\theta$.

## Remarks

For $K = 0$ the map is integrable: $p$ is constant and each [[circle|060W5CBZP6]] $p = c$ is invariant, with rotation by the angle $c$.
For small $K \ne 0$, [[KAM theory|3ZZSXKM16Z]] for twist maps shows that the invariant circles with sufficiently irrational rotation numbers survive, slightly deformed.
These circles wrap around the cylinder and prevent $p$ from growing without bound.
As $K$ grows the circles break; numerically, the last one disappears near $K \approx 0.9716$, found by Greene, and beyond that $p$ can drift without bound.
For large $K$ most orbits look [[chaotic|DBKXCGDKTD]], with islands of stable motion around periodic orbits.

The map models a kicked rotor: a rotor that receives a periodic kick whose strength depends on its angle.

## Explanations

### L1 human

The standard map is a simple rule that takes a point of a cylinder to a new point, with one parameter $K$.
For $K = 0$ nothing interesting happens: each point slides along its circle.
As $K$ grows, the picture of its orbits changes from neat curves to a mixture of curves, islands and chaotic seas.
It is the standard laboratory for seeing the transition from order to chaos in Hamiltonian systems.
</document>

<document source="https://tamnd.github.io/math2/node/8TK1JNG87X.md" tag="8TK1JNG87X" kind="definition">
# Tempered distribution

## Definition

Let $n \ge 1$ and let $\mathcal{S}(\R^n)$ be the [[Schwartz space|CG9JEATW3Q]] of smooth functions $\varphi \colon \R^n \to \C$ such that
$$p_{\alpha,\beta}(\varphi) = \sup_{x \in \R^n} \abs{x^\alpha \partial^\beta \varphi(x)} < \infty$$
for all multi-indices $\alpha$ and $\beta$.
A *tempered distribution* on $\R^n$ is a linear map $T \colon \mathcal{S}(\R^n) \to \C$ for which there are a constant $C \ge 0$ and an integer $N \ge 0$ such that
$$\abs{T(\varphi)} \le C \sum_{\abs{\alpha} \le N, \, \abs{\beta} \le N} p_{\alpha,\beta}(\varphi) \quad \text{for all } \varphi \in \mathcal{S}(\R^n).$$
The space of tempered distributions is written $\mathcal{S}'(\R^n)$.

## Remarks

The inequality says that $T$ is a [[continuous linear map|03TM135G0N]] for the Fréchet topology of $\mathcal{S}(\R^n)$ given by the seminorms $p_{\alpha,\beta}$.
Equivalently, $T(\varphi_k) \to 0$ whenever $p_{\alpha,\beta}(\varphi_k) \to 0$ for all $\alpha, \beta$.

Example: the Dirac delta $\delta(\varphi) = \varphi(0)$ is tempered.
It is linear, and $\abs{\delta(\varphi)} \le p_{0,0}(\varphi)$, so the condition holds with $C = 1$ and $N = 0$.
More generally, if $f$ is measurable and $\int \abs{f(x)} (1 + \abs{x})^{-m} \, dx < \infty$ for some $m \ge 0$, then $\varphi \mapsto \int f \varphi \, dx$ is tempered.

Non-example: on $\R$, the function $\varphi(x) = e^{-\sqrt{1 + x^2}}$ lies in $\mathcal{S}(\R)$, but $\int e^x \varphi(x) \, dx = \infty$.
So the formula $\varphi \mapsto \int e^x \varphi \, dx$ does not define a linear map on all of $\mathcal{S}(\R)$, and $e^x$ gives no tempered distribution.
It still defines a distribution on compactly supported test functions.

Derivatives act by $(\partial^\beta T)(\varphi) = (-1)^{\abs{\beta}} T(\partial^\beta \varphi)$, and the Fourier transform acts by $\hat{T}(\varphi) = T(\hat{\varphi})$.
</document>

<document source="https://tamnd.github.io/math2/node/8TPR96GV7F.md" tag="8TPR96GV7F" kind="definition">
# 2-category

## Definition

A *2-category* (or strict 2-category) is a category [[enriched|BPH9SJ30Y0]] over the [[category of small categories|ZY6R4H65P8]] $\mathbf{Cat}$ with the Cartesian product as tensor product.
Explicitly, a 2-category $\mathcal{K}$ consists of:

1. a class of objects;
2. for each pair of objects $a, b$, a [[category|A2DE66937D]] $\mathcal{K}(a, b)$ of 1-morphisms $f \colon a \to b$ and 2-morphisms $\theta \colon f \Rightarrow g$, whose composition is called vertical;
3. for each triple $a, b, c$, a [[functor|28NQ0KRHD3]] $\circ \colon \mathcal{K}(b, c) \times \mathcal{K}(a, b) \to \mathcal{K}(a, c)$, called horizontal composition;
4. for each object $a$, an identity 1-morphism $1_a$;

such that horizontal composition is strictly associative and $1_a$ and its identity 2-morphism are strict units for it.

## Remarks

That horizontal composition is a functor is the interchange law: $(\psi' \cdot \psi) \circ (\theta' \cdot \theta) = (\psi' \circ \theta') \cdot (\psi \circ \theta)$, where $\cdot$ is vertical composition.
Example: $\mathbf{Cat}$, with small categories as objects, functors as 1-morphisms and [[natural transformations|KYCPCC57FT]] as 2-morphisms, is a 2-category; composition of functors is strictly associative.
Non-example: rings, bimodules and bimodule maps, with composition of bimodules given by the tensor product, do not form a 2-category.
Take an $R$-$S$-bimodule $M$, an $S$-$T$-bimodule $N$ and a $T$-$U$-bimodule $P$.
The bimodules $(M \otimes_S N) \otimes_T P$ and $M \otimes_S (N \otimes_T P)$ are isomorphic but not equal in general, so strict associativity fails; they form a bicategory instead.
</document>

<document source="https://tamnd.github.io/math2/node/8TPW8B9G48.md" tag="8TPW8B9G48" kind="definition">
# Cap product

## Definition

Let $X$ be a topological space and let $R$ be a [[commutative ring|0310REJJY8]] with $1$.
For a singular simplex $\sigma \colon \Delta^n \to X$ with vertices $v_0, \ldots, v_n$ and a cochain $\varphi \in C^k(X; R)$ with $k \le n$, the *cap product* is the chain
$$\sigma \frown \varphi = \varphi\big(\sigma|_{[v_0, \ldots, v_k]}\big) \, \sigma|_{[v_k, \ldots, v_n]} \in C_{n-k}(X; R).$$
It extends linearly to a bilinear map $C_n(X; R) \times C^k(X; R) \to C_{n-k}(X; R)$.
It satisfies $\partial(\sigma \frown \varphi) = (-1)^k \big(\partial\sigma \frown \varphi - \sigma \frown \delta\varphi\big)$.
Hence it induces a bilinear map $\frown \colon H_n(X; R) \times H^k(X; R) \to H_{n-k}(X; R)$ between [[singular homology|M0Q5CFJ7ZH]] and [[singular cohomology|6N7BC70CX0]], also called the cap product.

## Remarks

Cap and [[cup product|JX47MS1R1H]] are related by $\psi(\sigma \frown \varphi) = (\varphi \smile \psi)(\sigma)$ for $\varphi \in C^k$ and $\psi \in C^{n-k}$.
On homology, $(c \frown \varphi) \frown \psi = c \frown (\varphi \smile \psi)$, so $H_\bullet(X; R)$ is a module over $H^\bullet(X; R)$.
It is natural in the form $f_*(c \frown f^*\varphi) = f_*(c) \frown \varphi$ for a continuous map $f$.

Example: for $k = n$, $\sigma \frown \varphi = \varphi(\sigma)\, \sigma(v_n)$ is a $0$-chain.
Its image under the augmentation $C_0(X; R) \to R$ is $\varphi(\sigma)$, so the cap product in top degree followed by augmentation is the Kronecker pairing of cohomology with homology.

Non-example: for $k > n$ the formula does not apply, since an $n$-simplex has no face of dimension $k$; the product $H_n \times H^k \to H_{n-k}$ is only defined for $k \le n$.
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<document source="https://tamnd.github.io/math2/node/8TV0DJBBAY.md" tag="8TV0DJBBAY" kind="statement">
# Toeplitz–Hausdorff theorem

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $T \colon H \to H$ be a bounded linear operator.
Then the [[numerical range|XWXZ0DGW3Z]]
$$W(T) = \set{\ip{Tx}{x} : x \in H, \ \norm{x} = 1}$$
is a [[convex subset|02AS45T3H9]] of $\C$.

## Remarks

Proof idea: take two points $\ip{Tx}{x}$ and $\ip{Ty}{y}$ of $W(T)$ and let $M$ be the span of the unit vectors $x$ and $y$.
The numerical range of the compression $P_M T|_M$ of $T$ to $M$ is contained in $W(T)$ and contains both points, so it is enough to treat a $2 \times 2$ matrix.
For a $2 \times 2$ matrix, a direct computation shows that the numerical range is a closed elliptical disc with foci at the two eigenvalues.
The disc may degenerate to a segment or a point.

Example: for $T = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ on $\C^2$ and a unit vector $x = (x_1, x_2)$, one gets $\ip{Tx}{x} = \abs{x_1}^2 - \abs{x_2}^2$, so $W(T) = [-1, 1]$, a convex set.
The set $\set{1, -1}$ of eigenvalues is not convex, but $W(T)$ fills in the segment between them.

## Explanations

### L1 human

Feed a unit vector $x$ into $T$ and measure how much of $Tx$ points back along $x$; that complex number is $\ip{Tx}{x}$.
The set of all these numbers is the numerical range.
The theorem says that this set is convex: the segment between any two of its points lies inside it.

### L2 human

The numerical range contains the eigenvalues of $T$, and its closure contains the [[spectrum|BMKATJ802A]].
With convexity, the closure of $W(T)$ contains the convex hull of the spectrum, with equality for normal operators.
The numerical radius $w(T) = \sup \set{\abs{z} : z \in W(T)}$ satisfies $\tfrac12 \norm{T} \le w(T) \le \norm{T}$, so it is a norm on bounded operators equivalent to the operator norm.
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# Markov chain mixing time

## Summary

The mixing time of a finite [[Markov chain|0G48PYM96Q]] is the number of steps it needs to come close to its [[stationary distribution|0KRH5157WP]].
This must hold whatever the starting state.
Closeness is measured in total variation distance, usually with threshold $1/4$.
Mixing times measure the cost of Markov chain Monte Carlo and of card shuffling.
Their study is a large part of modern Markov chain theory.

## Explanations

### L1 human

Shuffle a deck of cards again and again: how many shuffles until the order is close to random?
The mixing time answers this kind of question for any chain.
It is the first time at which, from the worst starting state, no event has a probability more than $1/4$ away from its probability under the equilibrium law.
For a chain that jumps to a uniformly random state at each step, one step is enough, so the mixing time is $1$.
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# Orthogonal polynomials

## Summary

A sequence of orthogonal polynomials is a sequence $p_0, p_1, p_2, \dots$ with $\deg p_n = n$ that are pairwise orthogonal.
Orthogonality refers to the inner product $\int p q \, d\mu$ of a measure $\mu$ on the real line.
They arise from the monomials $1, x, x^2, \dots$ by Gram–Schmidt orthogonalization, and they are unique up to constant factors.
The classical families of Hermite, Laguerre, Jacobi, Legendre and Chebyshev polynomials are examples.
They satisfy a three-term recurrence, their zeros are real and interlace, and they are used in approximation, numerical integration, spectral theory and random matrices.

## Explanations

### L1 human

Two polynomials are orthogonal for a weight if the integral of their product against the weight is zero.
Start with $1, x, x^2, \dots$ and straighten them one at a time so that each is orthogonal to all earlier ones.
The result is a list of polynomials, one of each degree, that act like perpendicular axes in a space of functions.
For the plain weight on $[-1, 1]$ one gets the Legendre polynomials $1$, $x$, $\frac12(3x^2 - 1)$, and so on.
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# Freyd–Mitchell embedding theorem

## Statement

Let $\mathcal{A}$ be a small [[abelian category|Y9CZ3MTNF1]], that is, an abelian category whose objects form a set.
Then there exist a [[ring|0B1261B5ED]] $R$ with identity and a [[functor|28NQ0KRHD3]] $F \colon \mathcal{A} \to R\text{-}\mathrm{Mod}$ into the [[category of left $R$-modules|JBSSXP4FFY]] such that:

1. $F$ is [[full and faithful|GRG8WEBSH5]];
2. $F$ is an [[exact functor|S1G61TDW0K]].

## Remarks

The idea of the proof is to embed $\mathcal{A}$, by a Yoneda-type construction, into a category of left exact functors from $\mathcal{A}$ to abelian groups.
That category is abelian and has enough injectives and a generator, which leads to an abelian category with a projective generator $P$ containing a copy of $\mathcal{A}$.
One then takes $R$ to be the endomorphism ring of a suitable such generator and uses the functor $\Hom(P, -)$.
Since $F$ is full, faithful and exact, a diagram in $\mathcal{A}$ commutes, or a sequence in $\mathcal{A}$ is exact, exactly when its image under $F$ does.
So a diagram lemma such as the five lemma or the snake lemma can be proved in an arbitrary abelian category by chasing elements of modules.
For a large abelian category, one applies the theorem to a small full abelian subcategory that contains the given diagram and is closed under kernels and cokernels.

## Explanations

### L1 human

Abelian categories are an abstract setting for kernels, cokernels and exact sequences.
The theorem says that a small abelian category is, up to a faithful copy, just a part of a category of modules over some ring.
So nothing new can happen in an abstract abelian category that does not already happen for modules.

### L2 human

The main use of the theorem is to justify element chasing.
A statement about a finite diagram of objects and morphisms in an abelian category can be transferred to modules, proved there with elements, and transferred back.
This is why many texts prove diagram lemmas only for modules.
An alternative that stays inside the category is the method of [[pseudo-elements|1WV2THA6NY]].
The [[Gabriel–Popescu theorem|3MP41FQHZ5]] is a related embedding result for Grothendieck categories, where the functor is not exact but has an exact left adjoint.
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# Rational singularity

## Definition

Let $k$ be a field of characteristic $0$ and let $X$ be a normal [[algebraic variety|HV6YBS0YDH]] over $k$ (see [[normal scheme|EK261G1XP9]]).
Let $f \colon Y \to X$ be a [[resolution of singularities|MB52NDA6WN]], that is, a [[proper|9537RKBZPP]] [[birational|KA5890R9YM]] morphism from a nonsingular variety $Y$.
The variety $X$ has *rational singularities* if
$$R^i f_* \mathcal{O}_Y = 0 \quad \text{for all } i > 0,$$
where $R^i f_*$ are the higher direct images, computed by [[sheaf cohomology|PQ6Q143ARJ]] on preimages of open sets.
A point $x \in X$ is a *rational singularity* if some open neighbourhood of $x$ has rational singularities.

## Remarks

The condition does not depend on the chosen resolution.
Since $X$ is normal and $f$ is proper and birational, one also has $f_* \mathcal{O}_Y = \mathcal{O}_X$, by Zariski's main theorem.
Every nonsingular variety has rational singularities, with $f = \id$.
Example: for the cone $X \colon xy = z^2$ in $\mathbb{A}^3$, the blow-up of the vertex is a resolution whose exceptional curve is $E \cong \mathbb{P}^1$ with normal bundle $\mathcal{O}(-2)$.
Here $Y$ is the total space of the line bundle $\mathcal{O}(-2)$ on $E$, and $X$ is affine, so $R^1 f_* \mathcal{O}_Y$ is determined by $H^1(Y, \mathcal{O}_Y) = \bigoplus_{m \ge 0} H^1(\mathbb{P}^1, \mathcal{O}(2m)) = 0$.
So the singularity is rational.
Non-example: for the cone $X \colon x^3 + y^3 + z^3 = 0$ over a smooth plane cubic $E$, the blow-up of the vertex is the total space of $\mathcal{O}_E(-1)$, and the same computation gives $H^1(Y, \mathcal{O}_Y) = \bigoplus_{m \ge 0} H^1(E, \mathcal{O}_E(m))$.
The term $m = 0$ is $H^1(E, \mathcal{O}_E) \cong k$, so $R^1 f_* \mathcal{O}_Y \ne 0$ and the singularity is not rational.
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# Abstract Volterra equation

## Definition

Let $X$ be a complex [[Banach space|06GZ956657]] and let $A \colon D(A) \subseteq X \to X$ be a [[closed linear operator|VM7B38PA87]].
Let $a \in L^1_{\mathrm{loc}}(\R_+)$ be a scalar [[locally integrable|X7GTF3D98F]] kernel on $\R_+ = [0, \infty)$, let $\tau > 0$, $J = [0, \tau]$, and let $f \in C(J; X)$.
The *abstract Volterra equation* with kernel $a$, operator $A$ and forcing $f$ is
$$u(t) = f(t) + \int_0^t a(t - s) \, A u(s) \, ds, \qquad t \in J.$$
A *strong solution* is a function $u \in C(J; X)$ such that $u(t) \in D(A)$ for all $t \in J$, $A u \in C(J; X)$, and the equation holds for all $t \in J$.

## Remarks

A *mild solution* is a function $u \in C(J; X)$ such that $\int_0^t a(t - s) u(s) \, ds \in D(A)$ for all $t \in J$ and $u(t) = f(t) + A \int_0^t a(t - s) u(s) \, ds$.
Every strong solution is a mild solution, since $A$ is closed; compare the [[mild solution|3VBRS94V1G]] of an abstract Cauchy problem.

Special cases.
For $a = 1$, a strong solution with $f \in C^1(J; X)$ satisfies $u' = A u + f'$ and $u(0) = f(0)$.
For $a(t) = t$, differentiating twice gives $u'' = A u + f''$ with $u(0) = f(0)$ and $u'(0) = f'(0)$, when $f \in C^2(J; X)$.

Example: $X = \C$, $A = 1$, $a = 1$ and $f = 1$.
The function $u(t) = e^t$ is a strong solution, since $1 + \int_0^t e^s \, ds = e^t$.

Non-example: in the same setting, $u(t) = 1 + t$ is not a solution, since $1 + \int_0^t (1 + s) \, ds = 1 + t + t^2/2$.

## Explanations

### L1 human

It is a Volterra equation in which numbers are replaced by vectors of a Banach space and multiplication by a fixed number is replaced by an operator $A$.
The kernel $a$ weights the past, and the operator $A$ usually acts in the space variables, for instance as a Laplacian.
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# Carleson embedding theorem

## Statement

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]], let $0 < p < \infty$, and let $H^p$ be the [[Hardy space|PNNNFJVF6W]] of $\mathbb{D}$ with norm $\norm{\cdot}_{H^p}$.
Let $\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\mathbb{D}$.
For an arc $I$ of the unit circle, let $\ell(I)$ be its length and $S(I)$ its Carleson box.
Then the following are equivalent:

1. $\mu$ is a [[Carleson measure|SQHGJ9C0QE]], that is, $\mu(S(I)) \le C \ell(I)$ for some $C$ and every arc $I$.
2. There is a constant $C_p$ such that
$$\int_{\mathbb{D}} \abs{f}^p \, d\mu \le C_p \norm{f}_{H^p}^p \quad \text{for all } f \in H^p.$$

## Remarks

To show that 1 implies 2, let $f^*$ be the nontangential maximal function of $f$ on the circle.
For $\lambda > 0$, the open set $\set{f^* > \lambda}$ of the circle is a union of disjoint arcs $I_k$.
Up to a fixed change of aperture, the set $\set{z \in \mathbb{D} : \abs{f(z)} > \lambda}$ is covered by the Carleson boxes over the enlarged arcs $3 I_k$.
Hence $\mu(\set{\abs{f} > \lambda}) \le C \abs{\set{f^* > \lambda}}$, and integrating in $\lambda$ gives $\int \abs{f}^p \, d\mu \le C \norm{f^*}_{L^p}^p \le C' \norm{f}_{H^p}^p$.
To show that 2 implies 1, test the inequality on $f(z) = \left( (1 - \abs{a}^2)/(1 - \bar a z)^2 \right)^{1/p}$ for $a \in \mathbb{D}$, which has $H^p$ norm $1$ and is large on the Carleson box near $a/\abs{a}$ of size $1 - \abs{a}$.

The theorem is due to Carleson, who used it in his solution of the interpolation problem for bounded holomorphic functions and in his proof of the corona theorem.
The condition does not depend on $p$, so a measure that works for one $p$ works for all $p$.

Example: area measure on $\mathbb{D}$ is a Carleson measure, so $\int_{\mathbb{D}} \abs{f}^p \, dA \le C_p \norm{f}_{H^p}^p$, a weak form of the fact that $H^p$ is contained in the Bergman space $A^p$.

## Explanations

### L1 human

A Carleson measure spreads its mass near the boundary circle in a controlled way.
A box of side $\ell$ touching the circle carries at most a constant times $\ell$ of mass.
The theorem says that exactly these measures can be used to measure the size of Hardy space functions inside the disc.
So the boundary behaviour of $f$ controls its size against such measures.

### L2 human

The theorem turns a statement about holomorphic functions into a geometric condition on boxes, which is often easy to check.
Let $(z_k)$ be a sequence of points of $\mathbb{D}$.
The measure $\sum (1 - \abs{z_k}) \delta_{z_k}$ is a Carleson measure exactly when the map $f \mapsto ((1 - \abs{z_k})^{1/p} f(z_k))_k$ is bounded from $H^p$ to $\ell^p$.
This is a key step in interpolation problems.
In the upper half-space of $\R^{n+1}$ the analogous theorem holds for Poisson integrals of $L^p$ functions, $1 < p < \infty$.
Carleson measures are also linked with BMO.
A function $b$ is in BMO exactly when $t \abs{\nabla u(x, t)}^2 \, dx \, dt$ is a Carleson measure, where $u$ is the Poisson integral of $b$.
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# Singular integral operator

## Definition

Let $n \ge 1$, let $S^{n-1}$ be the unit sphere of $\R^n$ with its surface measure $d\sigma$, and let $\Omega \colon S^{n-1} \to \C$ be an integrable function with
$$\int_{S^{n-1}} \Omega(\omega) \, d\sigma(\omega) = 0.$$
Let $K(x) = \Omega(x/\abs{x})/\abs{x}^n$ for $x \ne 0$.
The *singular integral operator* with kernel $K$ is
$$Tf(x) = \lim_{\varepsilon \to 0^+} \int_{\abs{y} > \varepsilon} K(y) f(x - y) \, dy,$$
defined first for smooth functions $f$ with compact support, where the [[limit|05M5SWSZZD]] exists for every $x$.

## Remarks

The limit is a principal value: the mean zero condition makes $\int_{\varepsilon < \abs{y} < 1} K(y) \, dy = 0$, so $f(x - y)$ may be replaced by $f(x - y) - f(x) = O(\abs{y})$ near $y = 0$, which is integrable against $\abs{y}^{-n}$.
When $\Omega$ is Lipschitz continuous, $T$ extends to a bounded operator on $L^p(\R^n)$ for $1 < p < \infty$, and it is of weak type $(1, 1)$.
The more general [[Calderón–Zygmund operators|3SCXCW4AH3]] have kernels $K(x, y)$ that need not be of convolution type.

Example: for $n = 1$, $S^0 = \set{-1, 1}$, and $\Omega(\pm 1) = \pm 1/\pi$ has mean zero.
Then $K(x) = 1/(\pi x)$ and $T$ is the Hilbert transform.
For $n \ge 2$, $\Omega_j(\omega) = c_n\omega_j$ has mean zero by symmetry, and gives the [[Riesz transforms|FRNDFJ9WQ5]].

Non-example: $\Omega \equiv 1$ fails the mean zero condition.
For $f$ equal to $1$ near $x$, the integral $\int_{\varepsilon < \abs{y} < 1} \abs{y}^{-n} \, dy$ is a constant times $\log(1/\varepsilon)$, so the limit does not exist.
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# Efficiency of an estimator

## Summary

The efficiency of an unbiased estimator measures how close its variance comes to the smallest possible variance, the Cramér–Rao lower bound.
An estimator whose variance attains the bound is called efficient.
The relative efficiency of two estimators is the ratio of their variances.
It tells how many more observations the worse estimator needs to match the better one.
Asymptotic versions compare the variances of the limit distributions as the sample size grows.

## Explanations

### L1 human

Two estimators can both be right on average, but one may scatter less around the truth.
The one with smaller scatter uses the information in the data better; it is more efficient.
For normal data, the sample mean and the sample median both estimate the centre, but the median has about $1/0.64 \approx 1.57$ times the variance of the mean in large samples.
So the median needs about $57\%$ more data to be as precise as the mean.
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# Polygon

## Summary

A polygon is a plane figure bounded by a closed chain of finitely many straight segments.
The segments are its edges and their endpoints are its vertices.
Triangles, quadrilaterals and [[regular polygons|01KJJ9A400]] are polygons.
Polygons are the basic building blocks for area, triangulation and much of discrete geometry.

## Explanations

### L1 human

Put some dots on a sheet of paper and join them in order by straight lines, coming back to the first dot at the end.
The closed path you draw is a polygon.
A square with corners $(0, 0)$, $(1, 0)$, $(1, 1)$, $(0, 1)$ is a polygon with four edges.
If the path never crosses itself, the polygon encloses a single region of the plane.
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# Liouville's theorem (differential algebra)

## Statement

Let $(F, \partial)$ be a [[differential field|GA5PTQGC6G]] of characteristic $0$, with [[derivation|3J89REV9WQ]] $\partial$, and write $u' = \partial u$.
A differential [[field extension|0EQWVHF3W5]] $E = F(t_1, \dots, t_m)$ of $F$ is *elementary* if, for each $i$, with $F_{i-1} = F(t_1, \dots, t_{i-1})$, one of the following holds:

1. $t_i$ is algebraic over $F_{i-1}$;
2. $t_i' = u'/u$ for some nonzero $u \in F_{i-1}$ (a logarithm);
3. $t_i' = u' t_i$ for some $u \in F_{i-1}$ (an exponential).

Let $\alpha \in F$.
Suppose that some elementary extension $E$ of $F$ with the same field of constants as $F$ contains an element $y$ with $y' = \alpha$.
Then there are constants $c_1, \dots, c_n$ of $F$, nonzero elements $u_1, \dots, u_n \in F$, and an element $v \in F$ such that
$$\alpha = v' + \sum_{i=1}^{n} c_i \frac{u_i'}{u_i}.$$

## Remarks

The proof goes by induction on the length of the tower $F \subseteq F_1 \subseteq \cdots \subseteq E$.
At each step one writes the antiderivative in terms of the last generator $t_i$ and compares coefficients.
This works because derivatives of rational functions in a logarithm or an exponential have a controlled form.
An algebraic step is removed by taking traces, which is where the characteristic $0$ is used.

Example: over $F = \C(x)$ with $\partial = d/dx$, the element $\alpha = 1/x$ has the antiderivative $\log x$, and indeed $\alpha = c_1 u_1'/u_1$ with $c_1 = 1$, $u_1 = x$, $v = 0$.
The theorem is the tool that proves that $e^{-x^2}$, $e^x/x$ and $1/\log x$ have no [[antiderivative|0BYWRWFT4F]] built from elementary functions.

The hypothesis on the constants cannot be dropped.
Over $F = \R(x)$, the element $\alpha = 1/(1 + x^2)$ has the antiderivative $\arctan x = \frac{1}{2i} \log \frac{x - i}{x + i}$, which lies in an elementary extension that contains the new constant $i$.
But $\alpha$ has no expression of the form above with real constants and $u_i, v \in \R(x)$.

## Explanations

### L1 human

Some functions, such as $e^{-x^2}$, have no antiderivative that can be written with the usual functions: powers, roots, exponentials, logarithms and their combinations.
Liouville's theorem explains why.
If an elementary antiderivative exists, it must have a very special shape: a function from the same field plus a sum of constant multiples of logarithms of such functions.
One can then check that no function of this shape works.

### L2 human

The theorem turns the question "is this integral elementary?" into an algebraic question about the field $F$.
For $F = \C(x, e^{g})$ with $g$ a nonconstant polynomial, it reduces to a first-order linear differential equation over $\C(x)$.
This gives a classical criterion for rational $f$: the integral $\int f e^{g} \, dx$ is elementary exactly when $R' + g' R = f$ has a solution $R \in \C(x)$.
The [[Risch algorithm|VJ51B7J125]] makes this reduction systematic and decides elementary integrability in towers of logarithms, exponentials and algebraic functions.
The theorem is a basic result of differential algebra, alongside the theory of [[Picard–Vessiot extensions|TB53Y5NMJD]].
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# Polyconvexity

## Summary

A function of an $m \times n$ matrix is polyconvex if it is a convex function of the matrix together with all its minors.
For $3 \times 3$ matrices the minors are the entries, the entries of the [[cofactor|0XPKR18APA]] matrix and the [[determinant|00KDZRW4C0]].
Polyconvexity implies [[Morrey quasiconvexity|DVJ17KVWXP]], and it is much easier to check.
It allows energies that blow up as the determinant tends to $0$, so it is the standard hypothesis on stored energies in nonlinear elasticity.

## Explanations

### L1 human

Convexity in the full gradient is too strong for elasticity, because it is incompatible with the fact that compressing a material to zero volume costs infinite energy.
Polyconvexity is a way around this.
One allows the energy to depend convexly on lengths, areas and volumes separately, that is, on the gradient, its cofactor and its determinant.
For example, for $2 \times 2$ matrices, $f(\xi) = \abs{\xi}^2 + (\det \xi)^2$ is polyconvex, since it is a convex function of $\xi$ and $\det \xi$.
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# Simple ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with $1$.
The ring $R$ is *simple* if $R \ne 0$ and the only [[two-sided ideals|0CD87SFM2N]] of $R$ are $0$ and $R$.

## Remarks

Example: let $F$ be a field and $R = M_2(F)$.
Let $I$ be a nonzero two-sided ideal and $A \in I$ with an entry $a_{ij} \ne 0$.
With the matrix units $E_{kl}$, one has $E_{ki} A E_{jl} = a_{ij} E_{kl}$, so every $E_{kl}$ lies in $I$, hence $I = R$.

Non-example: $\Z$ is not simple, since $2\Z$ is a two-sided ideal different from $0$ and $\Z$.

A commutative ring is simple if and only if it is a field.
A simple ring need not be Artinian: the Weyl algebra over a field of characteristic $0$ is simple and has no zero divisors, but it is not a division ring.
The center of a simple ring is a field.
Some authors require a simple ring to be Artinian as well; here no such condition is imposed.
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# Weil divisor

## Definition

Let $X$ be a [[Noetherian|ND6XEY68P4]] [[integral|17NVR4XBHK]] [[separated|X30G4V7VAJ]] scheme that is regular in codimension one.
This means that the local ring of $X$ at every point of codimension one is a regular local ring of dimension one.
A *prime divisor* on $X$ is a closed integral subscheme $Y \subseteq X$ of codimension one.
A *Weil divisor* on $X$ is an element of the free [[abelian group|082GHNPJ53]] $\operatorname{Div}(X)$ generated by the prime divisors.
So it is a finite formal sum $D = \sum_i n_i Y_i$ with $n_i \in \Z$ and prime divisors $Y_i$.

## Remarks

Let $Y$ be a prime divisor with generic point $\eta_Y$.
The local ring $\mathcal{O}_{X, \eta_Y}$ is a [[discrete valuation ring|MGM3CNJW9V]], with valuation $v_Y$ on the function field $K$ of $X$.
For $f \in K^\times$, the *principal divisor* of $f$ is $\operatorname{div}(f) = \sum_Y v_Y(f) Y$, which is a finite sum.
On $\mathbb{P}^1_k$ over a field $k$, the function $f = x^2/(x - 1)$ has $\operatorname{div}(f) = 2[0] - [1] - [\infty]$.
A closed point of $\mathbb{A}^2_k$ is not a prime divisor, since it has codimension two, not one.
The quotient of $\operatorname{Div}(X)$ by the principal divisors is the divisor class group $\operatorname{Cl}(X)$.
For example, $\operatorname{Cl}(\mathbb{P}^n_k) \cong \Z$ by the degree, for $n \ge 1$.
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# Inversive geometry

## Definition

Let $\hat{\C} = \C \cup \set{\infty}$ be the extended plane, the [[Riemann sphere|0J78KBPFZB]].
A *generalized circle* of $\hat{\C}$ is either a Euclidean [[circle|060W5CBZP6]] or a set $g \cup \set{\infty}$ with $g$ a line.
Extend each [[inversion in a circle|YXST7NGMTC]] to $\hat{\C}$ by swapping its center with $\infty$, and each [[reflection|06BRDF3F0E]] in a line by fixing $\infty$.
The *inversive group* is the group of bijections of $\hat{\C}$ generated by these maps.
*Inversive geometry* is the study of the properties of subsets of $\hat{\C}$ that are invariant under the inversive group.

## Remarks

In the language of the [[Erlangen program|9QW65PGEPW]], inversive geometry is the geometry of $\hat{\C}$ with the inversive group as its group of transformations.
The elements of the inversive group that preserve orientation are the [[Möbius transformations|0SG74WBFA2]] $z \mapsto (az + b)/(cz + d)$ with $ad - bc \ne 0$.
The others are the maps $z \mapsto (a\bar{z} + b)/(c\bar{z} + d)$ with $ad - bc \ne 0$.
Every element maps generalized circles to generalized circles and preserves the size of angles between curves.

Example: the property "these four points lie on a common generalized circle" is inversive, since the inversive group maps generalized circles to generalized circles.

Non-example: the property "this circle has radius $1$" is not inversive; the inversion $z \mapsto 4 / \bar{z}$ maps the circle $\abs{z} = 1$ to the circle $\abs{z} = 4$.

The same construction in $\R^n \cup \set{\infty}$, with inversions in spheres and reflections in hyperplanes, gives inversive geometry in dimension $n$.
</document>

<document source="https://tamnd.github.io/math2/node/8WETP2V4MB.md" tag="8WETP2V4MB" kind="definition">
# Descriptive geometry

## Definition

Let $E$ be Euclidean 3-space with [[Cartesian coordinates|0RWKYX8339]] $(x, y, z)$.
Let $\pi_1$ be the horizontal [[plane|0V3XAMF3C1]] $z = 0$ and $\pi_2$ the vertical plane $y = 0$; they meet in the ground line $x_{12}$, the $x$-axis.
The *Monge projection* of a point $P = (x, y, z)$ is the pair of its orthogonal projections, the top view $P' = (x, y, 0) \in \pi_1$ and the front view $P'' = (x, 0, z) \in \pi_2$.
Rotate $\pi_2$ about $x_{12}$ onto $\pi_1$, so that both views lie in one drawing plane.
Then $P'$ and $P''$ lie on a common line perpendicular to $x_{12}$.
*Descriptive geometry* is the study of figures of $E$ through such pairs of views, or through more views on further image planes.
Its aim is to solve spatial problems by plane constructions on the views.

## Remarks

The map $P \mapsto (P', P'')$ is injective, since the two views give $x$, $y$ and $z$.
So a pair of views on a common perpendicular to $x_{12}$ determines a unique point.
Example: for a segment $PQ$ the true length follows from the views, since $\abs{PQ}^2 = \abs{P'Q'}^2 + (z_P - z_Q)^2$, where $\abs{P'Q'}$ is measured in the top view and $z_P - z_Q$ in the front view.
For $P = (0, 0, 0)$ and $Q = (3, 4, 12)$, one gets $\abs{P'Q'} = 5$ and $\abs{PQ} = \sqrt{25 + 144} = 13$.
Non-example: a single view does not determine the point: the points $(1, 2, 0)$ and $(1, 2, 5)$ have the same top view.
The [[Euclidean space|0XPR1E85W0]] can also be shown by central projections, as in a [[perspective projection|YHKSDF59HK]].
</document>

<document source="https://tamnd.github.io/math2/node/8WJPSS7TE9.md" tag="8WJPSS7TE9" kind="example">
# The even integers form a subgroup of the integers

## Example

Let $G = (\mathbb{Z}, +)$ be the [[integers|0C4JS9RF5M]] with addition, and let $2\mathbb{Z}$ be the set of even integers.
The identity $0$ is even.
The sum of two even integers is even, and the negative of an even integer is even.
Thus $2\mathbb{Z}$ is a [[subgroup|0N0PXCTYA7]] of $G$.
$G$ is abelian, so $g + 2\mathbb{Z} = 2\mathbb{Z} + g$ for each $g$. Thus $2\mathbb{Z}$ is [[normal|0W77CVBCHQ]] in $G$.

## Remarks

In the same way, $n\mathbb{Z}$ is a subgroup of $\mathbb{Z}$ for each $n \ge 0$.
</document>

<document source="https://tamnd.github.io/math2/node/8WMX3GRGCZ.md" tag="8WMX3GRGCZ" kind="example">
# Computed tomography as an inverse problem

## Example

Let $\mu \colon \R^2 \to [0, \infty)$ be the X-ray attenuation coefficient of a slice of a body, a function with compact support.
An X-ray beam that crosses the slice along a [[line|0MP9BCF6DP]] $L$ enters with intensity $I_0$ and leaves with intensity $I$.
By the Beer–Lambert law of absorption, $I = I_0 \exp\left(-\int_L \mu \, ds\right)$, where $ds$ is arc length on $L$.
So each measurement gives the number
$$-\log \frac{I}{I_0} = \int_L \mu \, ds = (R\mu)(L),$$
the value of the [[Radon transform|VJNZJ352PC]] $R\mu$ at the line $L$.
A scanner measures $R\mu(L)$ for many lines $L$, and the task is to recover $\mu$ from these values.

This is an [[inverse problem|PCMW7KN5P4]]: the forward map $\mu \mapsto R\mu$ is known and linear, and one must invert it from data.
When $R\mu(L)$ is known for all lines, $\mu$ is determined, and the [[Radon inversion formula|HN93ST5CBT]] reconstructs it.
The inversion is unstable, because it involves a derivative, a "ramp filter" $\abs{\omega}$ in the Fourier variable.
So small errors at high frequencies in the data become large errors in $\mu$.
With data from only a limited range of directions the instability is much worse.

## Explanations

### L1 human

A CT scanner cannot look inside the body directly.
It only measures how much each X-ray beam is weakened, and that number is the total density along the beam.
From thousands of such totals along lines in many directions, a computer rebuilds the density at each point.
This "rebuild the cause from its effects" is a typical inverse problem.
Noise in the measurements is the main enemy, so practical reconstruction methods filter the data before they invert it.
</document>

<document source="https://tamnd.github.io/math2/node/8WWTMFR278.md" tag="8WWTMFR278" kind="statement">
# Cartan's theorem on fine continuity of superharmonic functions

## Statement

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be an open set, let $u$ be a [[superharmonic function|X4V2S56YQQ]] on $\Omega$, and let $x \in \Omega$.
Then there is a set $E \subseteq \Omega \setminus \set{x}$ that is [[thin|7DGF6MMEGX]] at $x$ such that
$$\lim_{y \to x,\ y \in \Omega \setminus E} u(y) = u(x).$$

## Remarks

Equivalently, $u$ is continuous at $x$ for the [[fine topology|W2ZD4XP74R]], once the finely open sets are described as the sets whose complement is thin at each of their points.
Idea of the proof: if $u(x) = +\infty$, lower semicontinuity already gives $u(y) \to +\infty$ as $y \to x$, and one takes $E = \emptyset$.
If $u(x) < \infty$, then for each $\varepsilon > 0$ the set $\set{y : u(y) > u(x) + \varepsilon}$ is thin at $x$, with $u$ itself as the witness.
These sets, cut down to suitable shrinking balls about $x$, combine into a single set $E$ that is still thin at $x$.
This step uses the description of thinness by capacity.
Example: in $\R^3$, the potential $u(y) = \sum_{k \ge 1} 4^{-k} \abs{y - 2^{-k} e_1}^{-1}$ is superharmonic with $u(0) = 1$ and $u = +\infty$ at the points $2^{-k} e_1$.
So $u$ is not continuous at $0$.
It tends to $1$ as $y \to 0$ outside the union $E$ of the balls of radius $k^2 4^{-k}$ about the points $2^{-k} e_1$, by dominated convergence.
This $E$ is thin at $0$ by Wiener's criterion, since $\sum_k 2^k k^2 4^{-k} < \infty$.
The exceptional set cannot be dropped in general, as this example shows.

## Explanations

### L1 human

A superharmonic function can jump up at points arbitrarily close to $x$.
The theorem says that these jumps happen only on a set that is negligible near $x$ in the sense of potential theory.
Outside that set, the function behaves continuously at $x$.

### L2 human

This is the result that makes the fine topology useful.
It shows that the topology defined abstractly by superharmonic functions has a concrete description by thin sets, so fine limits can be checked with capacities through Wiener's criterion.
Fine limits then appear in the boundary theory of harmonic functions, for instance in the fine form of Fatou-type theorems on the Martin boundary.
</document>

<document source="https://tamnd.github.io/math2/node/8WX57M9ZVA.md" tag="8WX57M9ZVA" kind="example">
# Excluded point topology

## Example

Let $X$ be a set and let $p \in X$.
The *excluded point topology* on $X$ declares a set $U \subseteq X$ [[open|02BH3MCBQB]] if $U = X$ or $p \notin U$.
This is a topology: $\emptyset$ and $X$ are open.
A union of open sets is either $X$ or avoids $p$, and so is a finite intersection of open sets.
So $X$ is a [[topological space|0CQHFPGJ45]].

- The [[closed sets|0ZCK8DQYN1]] are $\emptyset$ and the sets that contain $p$.
- $X$ is [[compact|0TVJZYB6SH]]: the only open set that contains $p$ is $X$, so every open cover has $X$ as a member.
  Then $\set{X}$ is a finite subcover.
- $X$ is [[connected|0V032VWTHD]]: a clopen set that contains $p$ is open, hence equal to $X$.
  A clopen set that misses $p$ is closed, hence empty.
- $X$ is a [[T0 space|07SG12TQH3]]: for $x \ne p$, the open set $\set{x}$ contains $x$ but not $p$.
  For distinct $x, y \ne p$, the open set $\set{x}$ separates them.
- If $X$ has at least two points, $X$ is not a [[T1 space|0NW6CH38YR]], since every open set that contains $p$ contains all points.

The closed sets of this topology are exactly the open sets of the [[particular point topology|4VSZ22TX4B]] at $p$.

## Explanations

### L1 human

Here one point $p$ is kept out: a set is open when it avoids $p$, and the whole space is open too.
Every point other than $p$ is open by itself, so away from $p$ the space looks discrete.
The point $p$ can only be surrounded by the whole space, which is why one member of any open cover already covers the space, and the space is compact.
It is the mirror image of the particular point topology.
</document>

<document source="https://tamnd.github.io/math2/node/8WZF9SBP39.md" tag="8WZF9SBP39" kind="concept">
# Elementary substructure

## Summary

An elementary substructure of a [[structure|983GZBQYT4]] is a [[substructure|676H7EGTKJ]] that agrees with the big structure on first-order formulas.
Each formula with parameters from the substructure has the same truth value in both.
The big structure is then an elementary extension of the small one.
Elementary substructures are the substructures that first-order logic cannot distinguish from the whole, even when it may name their elements.

## Explanations

### L1 human

A substructure can look different from the whole structure even when it is isomorphic to it.
The positive integers sit inside the natural numbers with the usual order.
In the positive integers the number $1$ is the least element, but in the natural numbers it is not, since $0 < 1$.
An elementary substructure forbids such differences: every first-order property of its elements is the same in the small and in the big structure.
The rational numbers are an elementary substructure of the real numbers, as ordered sets.
</document>

<document source="https://tamnd.github.io/math2/node/8X1ZRYBZ46.md" tag="8X1ZRYBZ46" kind="definition">
# Property (T)

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]], and let $\pi$ be a [[unitary representation|74BPYJNZTM]] of $G$ on a [[Hilbert space|HM2HHE3J51]] $\mathcal{H}$.
For a [[compact|0TVJZYB6SH]] set $Q \subseteq G$ and $\varepsilon > 0$, a unit vector $\xi \in \mathcal{H}$ is *$(Q, \varepsilon)$-invariant* if $\norm{\pi(g) \xi - \xi} < \varepsilon$ for all $g \in Q$.
A vector $\xi$ is *invariant* if $\pi(g) \xi = \xi$ for all $g \in G$.
The group $G$ has *property (T)* if there exist a compact set $Q \subseteq G$ and $\varepsilon > 0$ with the following property.
Every unitary representation of $G$ with a $(Q, \varepsilon)$-invariant unit vector has a nonzero invariant vector.

## Remarks

Example: every compact group $G$ has property (T), with $Q = G$ and $\varepsilon = \sqrt{2}$.
If $\norm{\pi(g)\xi - \xi} < \sqrt{2}$ for all $g$, then $\mathrm{Re} \ip{\pi(g)\xi}{\xi} > 0$ for all $g$.
The average $\eta = \int_G \pi(g) \xi \, dg$ over the normalised Haar measure is invariant, and $\mathrm{Re} \ip{\eta}{\xi} > 0$, so $\eta \neq 0$.

Non-example: $\Z$ does not have property (T).
Here compact sets are finite.
In the regular representation on $\ell^2(\Z)$, let $\xi_N = N^{-1/2} \mathbf{1}_{\set{0, \ldots, N-1}}$.
Then $\norm{\pi(k)\xi_N - \xi_N} \le \sqrt{2\abs{k}/N}$, which is small on any finite $Q$ when $N$ is large.
But a nonzero vector of $\ell^2(\Z)$ invariant under all shifts would have constant coordinates and infinite norm.

Equivalently, the trivial representation is an isolated point of the unitary dual of $G$ in the Fell topology.
For a discrete group, $Q$ can be taken finite, and a discrete group with property (T) is finitely generated.
The groups $\mathrm{SL}_n(\R)$ and $\mathrm{SL}_n(\Z)$ have property (T) for $n \ge 3$, but not for $n = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/8X25Q4XYMN.md" tag="8X25Q4XYMN" kind="concept">
# Summability method

## Summary

A summability method is a rule that assigns a value to some sequences, including some that do not converge.
Applied to partial sums, it assigns values to some series.
Ordinary convergence is the simplest such rule; [[Cesàro summation|N8ZGG7EW6J]], [[Abel summation|EZXPY4A19C]] and [[Borel summation|8DJPSYSHGJ]] are classical ones that extend it.
Summability methods give useful values to [[divergent series|0JWMJSTBSG]] in analysis, number theory and physics.
They are studied through properties such as being [[regular|29C7C41A52]], [[linear and stable|N6Z14G81VF]].

## Explanations

### L1 human

The series $1 - 1 + 1 - 1 + \cdots$ has no sum, because its partial sums jump between $1$ and $0$.
A summability method is a different way of reading off a number from such a sequence of partial sums.
Taking running averages, for example, gives $\frac12$ for this series.
Different methods can sum different series, and a good method should give the usual sum whenever the series converges.
</document>

<document source="https://tamnd.github.io/math2/node/8X7293647G.md" tag="8X7293647G" kind="definition">
# Score

## Definition

Let $\set{f(\cdot; \theta) : \theta \in \Theta}$ be a family of probability densities with respect to a measure $\nu$, with $\Theta$ an open subset of $\R^k$.
Let $x$ be a point with $f(x; \theta) > 0$ near a parameter $\theta$, and assume that $\theta \mapsto \log f(x; \theta)$ is differentiable at $\theta$.
The *score* of $\theta$ at $x$ is the [[gradient|0NS2QYVC4Y]] of the log-[[likelihood|05FSTXM92W]]:
$$s(\theta; x) = \nabla_\theta \log f(x; \theta) = \left(\frac{\partial \log f(x; \theta)}{\partial \theta_1}, \dots, \frac{\partial \log f(x; \theta)}{\partial \theta_k}\right)^T.$$
For i.i.d. data $x_1, \dots, x_n$, the score of the sample is $\sum_{i=1}^n s(\theta; x_i)$.

## Remarks

If differentiation under the integral sign is allowed in $\int f(x; \theta)\, d\nu(x) = 1$, then $E_\theta[s(\theta; X)] = 0$, where $E_\theta$ is the [[expectation|0DCT0QZVQ1]] when $X$ has density $f(\cdot; \theta)$.
Its covariance matrix $E_\theta[s(\theta; X) s(\theta; X)^T]$ is then the Fisher information.

Example: for $f(x; \mu) = (2\pi)^{-1/2} e^{-(x - \mu)^2/2}$, $\log f(x; \mu) = -\frac{1}{2}\log(2\pi) - \frac{1}{2}(x - \mu)^2$, so $s(\mu; x) = x - \mu$, which has mean $0$ and variance $1$ when $X \sim \mathcal{N}(\mu, 1)$.

Non-example: for the uniform densities $f(x; \theta) = \theta^{-1}\mathbf{1}[0 < x < \theta]$ with $\theta > 0$, the map $\theta \mapsto \log f(x; \theta)$ is not differentiable (it jumps from $-\infty$) at $\theta = x$.
Where it is defined, the score is $-1/\theta$, whose mean is not $0$: the support depends on $\theta$, and differentiation under the integral fails.
</document>

<document source="https://tamnd.github.io/math2/node/8X7ZESGYBW.md" tag="8X7ZESGYBW" kind="definition">
# Khinchin integral

## Definition

Let $a < b$ be real numbers.
Let $F \colon [a, b] \to \R$ be a function and let $E \subseteq [a, b]$.
The function $F$ is *absolutely continuous on $E$* if the following holds.
For every $\varepsilon > 0$ there is $\delta > 0$ such that for every finite family of non-overlapping intervals $[a_1, b_1], \dots, [a_n, b_n]$ with all endpoints in $E$,
$$\sum_{k=1}^{n} (b_k - a_k) < \delta \quad \text{implies} \quad \sum_{k=1}^{n} \abs{F(b_k) - F(a_k)} < \varepsilon.$$
The function $F$ is *generalized absolutely continuous*, or ACG, on $[a, b]$ if two conditions hold.
First, $F$ is [[continuous|0P27W9RW0G]] on $[a, b]$.
Second, $[a, b]$ is a countable union of sets on each of which $F$ is absolutely continuous.
Let $f \colon [a, b] \to \R$ be a function.
It is *Khinchin integrable* if some ACG function $F$ on $[a, b]$ has [[approximate derivative|4W31MKE01V]] equal to $f(x)$ at [[almost every|RARTMA0ZEC]] $x \in [a, b]$.
The *Khinchin integral* of $f$ is then $\int_a^b f = F(b) - F(a)$.

## Remarks

The integral is well defined: two ACG functions with the same approximate derivative almost everywhere differ by a constant.

Example: let $f$ be Lebesgue integrable on $[a, b]$.
Then $F(x) = \int_a^x f$ is [[absolutely continuous|ZT6F4C2YHK]] on all of $[a, b]$, hence ACG, and $F' = f$ almost everywhere.
So the Khinchin and Lebesgue integrals of $f$ agree.

Non-example of a primitive: the [[Cantor function|NPBYYK63FE]] $c$ is continuous and $c'(x) = 0$ for almost every $x$, but $c$ is not ACG.
If it were, the uniqueness above would force $c$ to be constant, while $c(1) - c(0) = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/8X8Y61MGHW.md" tag="8X8Y61MGHW" kind="statement">
# Tikhonov's theorem on slow-fast systems

## Statement

Let $f \colon \R^n \times \R^m \to \R^n$ and $g \colon \R^n \times \R^m \to \R^m$ be continuously differentiable.
For $\varepsilon > 0$ consider the [[singularly perturbed|G6J21TGRMM]] [[initial value problem|0FXDW992DS]] for a system of [[ordinary differential equations|0NT64JV96A]]
$$\dot x = f(x, y), \quad \varepsilon \dot y = g(x, y), \quad x(0) = x^0, \quad y(0) = y^0.$$
Let $T > 0$, and assume the following.

1. There are an open set $U \subseteq \R^n$ and a continuously differentiable $\varphi \colon U \to \R^m$ with $g(x, \varphi(x)) = 0$ for all $x \in U$.
2. For each $x \in U$, all eigenvalues of the matrix $\partial_y g(x, \varphi(x))$ have negative real part.
3. The *reduced problem* $\dot{\bar x} = f(\bar x, \varphi(\bar x))$, $\bar x(0) = x^0$, has a solution $\bar x$ on $[0, T]$ with values in $U$.
4. The solution of the *boundary layer problem* $\frac{d\hat y}{d\tau} = g(x^0, \hat y)$, $\hat y(0) = y^0$, exists for all $\tau \ge 0$ and tends to $\varphi(x^0)$ as $\tau \to \infty$.

Then there is $\varepsilon_1 > 0$ such that for $0 < \varepsilon < \varepsilon_1$ the solution $(x_\varepsilon, y_\varepsilon)$ exists on $[0, T]$.
Moreover, as $\varepsilon \to 0$, $x_\varepsilon \to \bar x$ uniformly on $[0, T]$, and $y_\varepsilon \to \varphi(\bar x)$ uniformly on $[t_0, T]$ for every $t_0 \in (0, T]$.

## Remarks

In the fast time $\tau = t / \varepsilon$, the variable $x$ hardly moves, and $y$ follows the boundary layer equation, in which $\varphi(x)$ is an [[asymptotically stable|0NB689782W]] [[equilibrium point|0N8NGKDZDZ]] by hypothesis 2.
So $y$ first jumps quickly to near the slow manifold $y = \varphi(x)$, and then the system moves slowly along it according to the reduced problem.
The proof turns this into estimates with a Lyapunov function for the fast subsystem, uniform in $x$ on a compact neighbourhood of the reduced trajectory.

Example: $\dot x = -x + y$, $\varepsilon \dot y = x - 2y$, with $x(0) = 1$, $y(0) = 5$.
Here $\varphi(x) = x / 2$ and $\partial_y g = -2 < 0$, and the reduced problem $\dot{\bar x} = -\bar x / 2$ gives $\bar x(t) = e^{-t/2}$.
The theorem gives $x_\varepsilon(t) \to e^{-t/2}$ and $y_\varepsilon(t) \to e^{-t/2} / 2$ for $t > 0$, while $y_\varepsilon(0) = 5$, so $y$ has a [[boundary layer|YCFPD0GE5G]] at $t = 0$.

Stability of the slow manifold is needed.
For $\dot x = 0$, $\varepsilon \dot y = y$ with $y(0) = y^0 \neq 0$, the root is $\varphi = 0$ but $\partial_y g = 1 > 0$, and $y_\varepsilon(t) = y^0 e^{t/\varepsilon}$ does not converge.

## Explanations

### L1 human

Some systems have a fast part and a slow part.
If the fast part always settles quickly to a rest state, one can replace it by that rest state and only study the slow part.
The theorem says this replacement gives the right answer as the speed gap becomes infinite, except for a short initial jump.

### L2 human

Tikhonov's theorem justifies the quasi-steady-state approximation of chemical kinetics, such as the Michaelis–Menten law in enzyme kinetics.
It is a result on finite time intervals; Fenichel's theorem, the basis of geometric singular perturbation theory, gives the invariant slow manifold for $\varepsilon > 0$.
The approximation can be refined to all orders by the method of matched asymptotic expansions, under extra smoothness.
</document>

<document source="https://tamnd.github.io/math2/node/8XAT77QQTN.md" tag="8XAT77QQTN" kind="definition">
# Cohomology of Lie groups

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] and let $k$ be a field.
The *cohomology* of $G$ with coefficients in $k$ is the graded $k$-algebra $H^*(G; k) = \bigoplus_{n \ge 0} H^n(G; k)$.
Here $H^n(G; k)$ is the [[singular cohomology|6N7BC70CX0]] of the underlying manifold of $G$, and the product is the [[cup product|JX47MS1R1H]].
When $G$ is connected and each $H^n(G; k)$ is finite-dimensional, the multiplication $\mu \colon G \times G \to G$ also gives a coproduct $\mu^*$, which makes $H^*(G; k)$ a [[Hopf algebra|WJXK0V9NE9]].

## Remarks

Example: $G = S^1$, the circle group.
Then $H^0(S^1; \R) = \R$, $H^1(S^1; \R) = \R x$, and all other groups vanish.
So $H^*(S^1; \R)$ is the exterior algebra on one generator $x$ of degree $1$, with $\mu^*(x) = x \otimes 1 + 1 \otimes x$.

Non-example: the cohomology of $G$ as an abstract group, $H^*(BG^\delta; k)$, or the continuous cohomology of $G$, is a different object.
For $G = S^1$ the continuous cohomology with coefficients in $\R$ vanishes in positive degrees, while $H^1(S^1; \R) \ne 0$.

For $k = \R$, the de Rham theorem identifies $H^*(G; \R)$ with the [[de Rham cohomology|GX74C07T0G]] of $G$.
If $G$ is compact and connected, averaging over $G$ shows that each class has a left- and right-invariant representative.
This gives an isomorphism $H^*(G; \R) \cong H^*(\mathfrak{g}; \R)$ with the Lie algebra cohomology of the Lie algebra $\mathfrak{g}$.
</document>

<document source="https://tamnd.github.io/math2/node/8XDYNK6TV0.md" tag="8XDYNK6TV0" kind="definition">
# Non-Archimedean absolute value

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
A *non-Archimedean absolute value* on $K$ is a map $\abs{\cdot} \colon K \to \R$ such that:

1. $\abs{x} \ge 0$ for all $x \in K$, and $\abs{x} = 0$ if and only if $x = 0$.
2. $\abs{xy} = \abs{x} \abs{y}$ for all $x, y \in K$.
3. $\abs{x + y} \le \max(\abs{x}, \abs{y})$ for all $x, y \in K$.

Clause 3 is the *strong triangle inequality*.

## Remarks

Clause 3 implies the triangle inequality, so a non-Archimedean absolute value is an absolute value, and $(K, \abs{\cdot})$ is a [[normed field|ZFW9Q3WTAQ]].
Example: for a prime $p$, the [[p-adic absolute value|WGN82TCEF2]] on $\Q$, given by $\abs{x}_p = p^{-v_p(x)}$ and $\abs{0}_p = 0$, is non-Archimedean.
Clause 2 is $p^{-v_p(xy)} = p^{-v_p(x)} p^{-v_p(y)}$, and clause 3 follows from $v_p(x + y) \ge \min(v_p(x), v_p(y))$.
Non-example: the usual absolute value on $\Q$ satisfies clauses 1 and 2 but fails clause 3, since $\abs{1 + 1} = 2 > 1 = \max(\abs{1}, \abs{1})$.
An absolute value on $K$ is non-Archimedean if and only if $\abs{n \cdot 1} \le 1$ for all integers $n$.
If $\abs{x} \ne \abs{y}$, then $\abs{x + y} = \max(\abs{x}, \abs{y})$.
For $0 < c < 1$, the rule $\abs{x} = c^{v(x)}$ turns a [[valuation|B85T5VKR3M]] $v$ with values in $\R$ into a non-Archimedean absolute value, and every non-Archimedean absolute value arises this way.
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<document source="https://tamnd.github.io/math2/node/8XS5T7BEJP.md" tag="8XS5T7BEJP" kind="definition">
# Statistical convergence

## Definition

Let $(x_n)_{n \ge 1}$ be a [[sequence|0QE8STGJH2]] of real or complex numbers and let $L$ be a number.
The sequence *converges statistically* to $L$ if for every $\varepsilon > 0$ the set
$$\set{n \ge 1 : \abs{x_n - L} \ge \varepsilon}$$
has [[natural density|RVG9Z8NYA6]] $0$.

## Remarks

Every [[convergent sequence|0W4CN9E2W5]] with limit $L$ converges statistically to $L$, since each of the sets above is finite and finite sets have density $0$.
The statistical limit is unique when it exists.

Example: let $x_n = 1$ if $n$ is a perfect square and $x_n = 0$ otherwise.
For $0 < \varepsilon \le 1$, the set of $n$ with $\abs{x_n} \ge \varepsilon$ is the set of squares, which has at most $\sqrt{N}$ elements up to $N$, so its density is $0$.
Hence $(x_n)$ converges statistically to $0$, although it does not converge.

Non-example: $x = (1, 0, 1, 0, \dots)$ does not converge statistically.
For any $L$, one of $\abs{1 - L}$ and $\abs{L}$ is at least $1/2$.
So for $\varepsilon = 1/2$ the set above contains all odd or all even $n$, and its density is not $0$.

A sequence converges statistically to $L$ if and only if there is a set $K$ of density $1$ such that $x_n \to L$ as $n \to \infty$ through $K$.
A bounded sequence converges statistically to $L$ if and only if $\frac{1}{n} \sum_{k=1}^{n} \abs{x_k - L} \to 0$, which is the condition of [[strong Cesàro summability|750MCA763X]] with exponent $1$ applied to the sequence $(x_n)$.
</document>

<document source="https://tamnd.github.io/math2/node/8Y0M6X8Q2A.md" tag="8Y0M6X8Q2A" kind="definition">
# Galois ring

## Definition

Let $p$ be a prime number and let $n \ge 1$ and $r \ge 1$ be integers.
Let $f \in (\Z/p^n\Z)[x]$ be a [[monic polynomial|0C3QBCR46M]] of degree $r$ whose reduction modulo $p$ is [[irreducible|03FKX7Z5DC]] in $\F_p[x]$.
The *Galois ring* of characteristic $p^n$ and rank $r$ is the [[quotient ring|0076HWHGHQ]]
$$\mathrm{GR}(p^n, r) = (\Z/p^n\Z)[x] / (f).$$
Here $\Z/p^n\Z$ is the ring of [[integers modulo|03VFZ2WETT]] $p^n$.
Up to isomorphism, this ring depends only on $p$, $n$ and $r$, and not on the choice of $f$.

## Remarks

Take $p = 2$, $n = 2$, $r = 2$ and $f = x^2 + x + 1$.
The polynomial $f$ is monic, and modulo $2$ it has no root in $\F_2$, so it is irreducible of degree $2$ over $\F_2$.
Hence $\mathrm{GR}(4, 2) = (\Z/4\Z)[x]/(x^2 + x + 1)$ is a Galois ring with $4^2 = 16$ elements.
The ring $(\Z/4\Z)[x]/(x^2 + 1)$ is not obtained this way, because $x^2 + 1 \equiv (x + 1)^2$ modulo $2$ is reducible.
The ring $\mathrm{GR}(p^n, r)$ has $p^{nr}$ elements and characteristic $p^n$.
It is a local ring with maximal ideal $p \, \mathrm{GR}(p^n, r)$ and residue field the [[finite field|0ZXZ82FTX2]] $\F_{p^r}$.
The special cases are $\mathrm{GR}(p, r) = \F_{p^r}$ and $\mathrm{GR}(p^n, 1) = \Z/p^n\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/8Y28JQB0Q5.md" tag="8Y28JQB0Q5" kind="concept">
# Homogeneous and inhomogeneous linear recurrences

## Summary

A linear recurrence expresses a term of a sequence as a linear combination of earlier terms.
It may also add an extra term that depends only on the index.
It is homogeneous when this extra term is zero, and inhomogeneous otherwise.
The solutions of a homogeneous recurrence form a vector space.
The solutions of an inhomogeneous one form a translate of that space.
So solving an inhomogeneous recurrence reduces to finding one particular solution and the solutions of the homogeneous one.

## Explanations

### L1 human

The rule $u_{n+2} = u_{n+1} + u_n$ only mixes earlier terms, so it is homogeneous.
The rule $u_{n+1} = 2 u_n + 1$ adds a fixed amount at each step, so it is inhomogeneous.
For a homogeneous rule, the all-zero sequence is always a solution, and sums and multiples of solutions are again solutions.
For an inhomogeneous rule this fails: the zero sequence does not satisfy $u_{n+1} = 2 u_n + 1$.
Its solutions are $u_n = c\, 2^n - 1$: one particular solution $-1$, plus any solution $c\, 2^n$ of $u_{n+1} = 2 u_n$.
</document>

<document source="https://tamnd.github.io/math2/node/8Y4C6N09YQ.md" tag="8Y4C6N09YQ" kind="definition">
# Arbitrary-precision arithmetic

## Definition

Fix an integer base $B \ge 2$, in practice a power of $2$ such as $2^{64}$.
In *arbitrary-precision arithmetic* an [[integer|0C4JS9RF5M]] $a$ is stored as a sign and a list of digits $(a_0, \ldots, a_{k-1})$.
Here $0 \le a_i < B$ and
$$\abs{a} = \sum_{i=0}^{k-1} a_i B^i,$$
where the length $k$ is not fixed in advance and grows as needed.
Arithmetic operations on such lists return the exact integer result in the same form.
Rational numbers are stored as pairs of such integers.
Real numbers are stored as [[floating-point|RTKRSC3KPH]] numbers $m \cdot 2^{e}$, where $m$ and $e$ are such integers and the precision $p$ is chosen by the user.

## Remarks

Example: with $B = 10$, the integer $123456780$ is the list $(0, 8, 7, 6, 5, 4, 3, 2, 1)$; adding $1$ changes only $a_0$, while adding $999999999$ produces a list of length $10$.
Non-example: IEEE binary64 arithmetic is not arbitrary-precision; its precision is fixed at $53$ bits, so $2^{53} + 1$ cannot be stored exactly.
Addition of two $k$-digit numbers costs $O(k)$ digit operations, schoolbook multiplication costs $O(k^2)$, and Karatsuba multiplication costs $O(k^{\log_2 3})$.
</document>

<document source="https://tamnd.github.io/math2/node/8YQJQXQHVE.md" tag="8YQJQXQHVE" kind="statement">
# Nash embedding theorems

## Statement

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] of dimension $m$.
Write $\ip{\cdot}{\cdot}$ for the Euclidean inner product on $\R^n$.
An *isometric embedding* of $(M, g)$ into $\R^n$ is an embedding $f \colon M \to \R^n$ with $\ip{df_p(v)}{df_p(w)} = g_p(v, w)$ for all $p \in M$ and $v, w \in T_pM$.

1. ($C^1$ theorem, Nash and Kuiper.) Let $n \ge m + 1$ and let $f \colon M \to \R^n$ be a smooth embedding that is short, that is $\abs{df_p(v)}^2 < g_p(v, v)$ for all $p$ and all nonzero $v \in T_pM$. Then for every $\varepsilon > 0$ there is a $C^1$ embedding $\tilde f \colon M \to \R^n$ that is isometric for $g$ and satisfies $\abs{\tilde f(p) - f(p)} < \varepsilon$ for all $p \in M$.
2. ($C^k$ theorem, Nash.) Let $3 \le k \le \infty$ and let $g$ be of class $C^k$. Then $(M, g)$ has an isometric embedding of class $C^k$ into $\R^n$. Here $n = m(3m + 11)/2$ if $M$ is compact, and $n = m(m + 1)(3m + 11)/2$ otherwise.

## Remarks

Part 2 is proved by solving the nonlinear system $f^*\ip{\cdot}{\cdot} = g$ with an iteration that combines Newton's method with smoothing operators, the Nash–Moser technique.
Part 1 is proved by convex integration: one adds many small fast oscillations that increase lengths in a controlled way.
An example of part 1: the flat square torus $\R^2 / \Z^2$ has a $C^1$ isometric embedding into $\R^3$.
It has no $C^2$ isometric embedding there, because a compact surface in $\R^3$ has a point of positive [[Gaussian curvature|SQVQ5DQT6W]], while the flat torus has curvature $0$.
So the low codimension in part 1 really needs the low regularity $C^1$.
Nash proved part 1 with $n \ge m + 2$, and Kuiper extended it to $n \ge m + 1$.

## Explanations

### L1 human

Every abstract curved space described by a Riemannian metric can be realised as a surface inside some ordinary Euclidean space, with lengths measured in the usual way.
So the abstract definition of a Riemannian manifold gives nothing new compared with curved submanifolds of $\R^n$.
If one allows the surface to be only once differentiable, very little room is needed, and strange crumpled embeddings appear.

### L2 human

The [[Whitney embedding theorem|2GSRBDT96M]] embeds a manifold in $\R^{2m}$ but ignores the metric; Nash's theorems make the embedding respect the metric at the cost of more dimensions or less smoothness.
The $C^1$ theorem is an early instance of the h-principle: a short embedding can be deformed into an isometric one by arbitrarily small moves.
The bounds on $n$ in part 2 have been lowered by later work, but the existence statement is what is used in practice.
</document>

<document source="https://tamnd.github.io/math2/node/8YYDZZCFY6.md" tag="8YYDZZCFY6" kind="definition">
# Dual object

## Definition

Let $(\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)$ be a [[monoidal category|M3MSQ3NKAE]] and let $A$ be an object of $\mathcal{C}$.
A *dual* of $A$ is an object $A^*$ together with morphisms $\mathrm{ev} \colon A^* \otimes A \to I$ and $\mathrm{coev} \colon I \to A \otimes A^*$ such that:

1. $\rho_A \circ (\id_A \otimes \mathrm{ev}) \circ \alpha_{A,A^*,A} \circ (\mathrm{coev} \otimes \id_A) \circ \lambda_A^{-1} = \id_A$;
2. $\lambda_{A^*} \circ (\mathrm{ev} \otimes \id_{A^*}) \circ \alpha^{-1}_{A^*,A,A^*} \circ (\id_{A^*} \otimes \mathrm{coev}) \circ \rho_{A^*}^{-1} = \id_{A^*}$.

These are the zigzag (or triangle) identities.
In this situation $A^*$ is called a right dual of $A$, and $A$ a left dual of $A^*$.
Some authors exchange the words left and right.

## Remarks

Example: in $\mathbf{Vect}_k$, a finite-dimensional space $V$ with basis $e_1, \ldots, e_n$ has the [[dual space|07JBSZE0V4]] $V^*$ with dual basis $e^1, \ldots, e^n$ as dual.
Take $\mathrm{ev}(f \otimes v) = f(v)$ and $\mathrm{coev}(1) = \sum_i e_i \otimes e^i$.
The first identity sends $v$ to $\sum_i e_i \, e^i(v) = v$, and the second sends $f$ to $\sum_i f(e_i)\, e^i = f$.
Non-example: an infinite-dimensional space $V$ has no dual in $\mathbf{Vect}_k$.
The element $\mathrm{coev}(1)$ is a finite sum $\sum_{i=1}^m v_i \otimes w_i$, so the first identity would express every $v \in V$ as a combination of $v_1, \ldots, v_m$, which is impossible.
Duals are unique up to unique isomorphism compatible with $\mathrm{ev}$ and $\mathrm{coev}$.
In a symmetric monoidal category left and right duals coincide, and the trace of $f \colon A \to A$ is $\mathrm{ev} \circ \gamma_{A,A^*} \circ (f \otimes \id_{A^*}) \circ \mathrm{coev} \colon I \to I$.
</document>

<document source="https://tamnd.github.io/math2/node/8YYJBVP68F.md" tag="8YYJBVP68F" kind="statement">
# Loop theorem

## Statement

Let $M$ be a [[3-manifold|R20TB2KEQQ]] with boundary $\partial M$, and let $F \subseteq \partial M$ be a connected surface.
Let $N$ be a [[normal subgroup|0CB0DX5BVV]] of the [[fundamental group|S1XK3HXHFX]] $\pi_1(F)$.
Assume that the kernel of the map $\pi_1(F) \to \pi_1(M)$ induced by inclusion is not contained in $N$.
Then there is an embedding $g \colon (D^2, \partial D^2) \to (M, F)$ of the closed disk such that the loop $g|_{\partial D^2}$ represents an element of $\pi_1(F)$ that is not in $N$.

## Remarks

The version with $N$ is due to Stallings; Papakyriakopoulos proved the case $N = \set{1}$ in 1957.
The proof uses the same tower of double coverings as the proof of [[Dehn's lemma|FH5WSPD79F]].
Take $N = \set{1}$, and suppose some loop that is essential in $F$ becomes null-homotopic in $M$.
Then some embedded essential loop in $F$ bounds an embedded disk in $M$.
For example, if $M$ is a solid torus $D^2 \times S^1$ and $F$ is its boundary torus, the meridian disk $D^2 \times \set{1}$ is such a disk.

## Explanations

### L1 human

If a loop on the boundary of a $3$-dimensional space can be shrunk inside the space but not on the boundary, then there is an embedded disk that shows it.
The disk has its edge on the boundary and does not cross itself.
So an algebraic fact about loops produces a concrete geometric object.

### L2 human

The loop theorem is the reason why "compressible" can be tested with fundamental groups.
A two-sided surface in a $3$-manifold is incompressible exactly when its fundamental group injects into that of the manifold.
Together with Dehn's lemma and the sphere theorem, it is a basic step in the theory of Haken manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/8Z18NXTHT4.md" tag="8Z18NXTHT4" kind="definition">
# Cayley–Menger determinant

## Definition

Let $n \ge 1$ and $N \ge 1$, and let $p_0, \ldots, p_n$ be points of $\R^N$.
Write $d_{ij} = \norm{p_i - p_j}$ for their [[Euclidean distances|0C95TG69SG]].
The *Cayley–Menger determinant* of the points is the [[determinant|0C3CFHDN6R]] of the $(n + 2) \times (n + 2)$ matrix
$$\operatorname{CM}(p_0, \ldots, p_n) = \det \begin{pmatrix} 0 & 1 & 1 & \cdots & 1 \\ 1 & 0 & d_{01}^2 & \cdots & d_{0n}^2 \\ 1 & d_{10}^2 & 0 & \cdots & d_{1n}^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & d_{n0}^2 & d_{n1}^2 & \cdots & 0 \end{pmatrix}.$$

## Remarks

The $n$-dimensional volume $V_n$ of the simplex with vertices $p_0, \ldots, p_n$ satisfies
$$V_n^2 = \frac{(-1)^{n+1}}{2^n (n!)^2} \operatorname{CM}(p_0, \ldots, p_n).$$

Example: for $n = 1$ and two points at distance $d$, the matrix is $\begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & d^2 \\ 1 & d^2 & 0 \end{pmatrix}$.
Its determinant is $2d^2$, and the formula gives $V_1^2 = \frac{1}{2} \cdot 2d^2 = d^2$, the squared length.

Example: for $n = 2$ and an equilateral triangle of side $1$, the matrix is $J - I$ with $J$ the $4 \times 4$ all-ones matrix.
Its eigenvalues are $3, -1, -1, -1$, so the determinant is $-3$.
The formula gives $V_2^2 = \frac{-1}{16} \cdot (-3) = \frac{3}{16}$, the square of the area $\frac{\sqrt{3}}{4}$.

Degenerate case: if the points are affinely dependent, such as three points on a line, the determinant is $0$.
The border of ones is essential: the determinant of the matrix of squared distances alone does not give the volume.
</document>

<document source="https://tamnd.github.io/math2/node/8Z55NXZWD5.md" tag="8Z55NXZWD5" kind="problem">
# Bolza problem

## Problem

Let $U \subseteq \R^m$ be a nonempty set, and let $t_0 < t_1$ be fixed times.
Let $f \colon [t_0, t_1] \times \R^n \times U \to \R^n$ and $L \colon [t_0, t_1] \times \R^n \times U \to \R$ be continuous.
Let $\varphi \colon \R^n \times \R^n \to \R$ be continuous, and let $S \subseteq \R^n \times \R^n$ be a closed set of allowed endpoint pairs.
An [[admissible control|8M31QKB0GW]] is a measurable map $u \colon [t_0, t_1] \to U$ whose trajectory $x$ of the [[control system|ZPDTVZQN5K]] $x'(t) = f(t, x(t), u(t))$ is absolutely continuous on $[t_0, t_1]$ and satisfies $(x(t_0), x(t_1)) \in S$.
The *Bolza problem* asks to minimize the [[functional|1EQM0DJ4NC]]
$$J(u) = \varphi(x(t_0), x(t_1)) + \int_{t_0}^{t_1} L(t, x(t), u(t)) \, dt$$
over all admissible controls $u$.

## Remarks

The cost has two parts: an endpoint term $\varphi$ and a running term given by the integral of $L$.
The case $L = 0$ is the Mayer problem, and the case $\varphi = 0$ is the Lagrange problem; see [[Mayer and Lagrange problems|1J8TYT7C49]].
The three forms are equivalent: adding the state $y' = L(t, x, u)$ with $y(t_0) = 0$ turns the Bolza cost into the endpoint cost $\varphi(x(t_0), x(t_1)) + y(t_1)$.
In the classical calculus of variations the control is the velocity, $f(t, x, u) = u$, and the problem is to minimize $\varphi(x(t_0), x(t_1)) + \int_{t_0}^{t_1} L(t, x, x') \, dt$ with possible side conditions.
Necessary conditions for a minimizer are given by [[Pontryagin's maximum principle|7RPV2T29CF]], with transversality conditions coming from $\varphi$ and $S$.
Existence of a minimizer needs extra hypotheses, such as compactness of $U$ and convexity of the set of velocities and costs; see [[Filippov's existence theorem|XEX7QZV1BS]].

## Explanations

### L1 human

A Bolza problem asks for the best way to steer a system when one pays both along the way and at the end.
The running cost might be fuel spent, and the end cost might be a penalty for missing a target.
Many optimal control problems in engineering and economics take this form.
</document>

<document source="https://tamnd.github.io/math2/node/8Z55VMJSEX.md" tag="8Z55VMJSEX" kind="statement">
# Cantor–Lebesgue theorem

## Statement

Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be real numbers.
Let $E \subset \R$ be a measurable set of positive [[Lebesgue measure|2Z2QVWZ262]].
Assume that $a_n \cos nx + b_n \sin nx \to 0$ as $n \to \infty$ for every $x \in E$.
Then $a_n \to 0$ and $b_n \to 0$.

## Remarks

Proof idea: write $a_n \cos nx + b_n \sin nx = \rho_n \cos(nx + \varphi_n)$ with $\rho_n = \sqrt{a_n^2 + b_n^2}$.
If $\rho_n \not\to 0$, then $\rho_{n_j} \ge \varepsilon > 0$ along a subsequence, so $\cos^2(n_j x + \varphi_{n_j}) \to 0$ on $E$.
Replace $E$ by a bounded subset of positive measure.
By [[dominated convergence|BYNV84PF86]], $\int_E \cos^2(n_j x + \varphi_{n_j}) \, dx \to 0$.
But this integral equals $\frac{1}{2} \abs{E} + \frac{1}{2} \int_E \cos(2 n_j x + 2 \varphi_{n_j}) \, dx$, and the last integral tends to $0$ by the [[Riemann–Lebesgue lemma|WVKDC011KP]], a contradiction.

In complex form: if $c_n e^{inx} + c_{-n} e^{-inx} \to 0$ for every $x$ in a set of positive measure, then $c_n \to 0$ as $\abs{n} \to \infty$.
So a [[trigonometric series|FZK2M8KGXQ]] that converges on a set of positive measure has coefficients tending to $0$.

Example: if $\sum_n (a_n \cos nx + b_n \sin nx)$ converges for every $x$ in $[0, 1]$, its terms tend to $0$ there, so $a_n, b_n \to 0$.

Positive measure is needed: at the single point $x = 0$ the terms $n \sin nx$ are all $0$, but $b_n = n \not\to 0$.

## Explanations

### L1 human

If a sequence of waves with amplitudes $\rho_n$ dies out at every point of a set of positive length, then the amplitudes themselves must go to $0$.
A wave of large amplitude cannot be small on a large set, because it oscillates and spends a fixed fraction of the time far from $0$.

### L2 human

The theorem is the first step in the proof of [[Cantor's uniqueness theorem|NSS3XW4BYZ]].
It reduces questions about convergent trigonometric series to series whose coefficients tend to $0$, where Riemann's methods apply.
Its proof is a typical use of the Riemann–Lebesgue lemma together with a convergence theorem for integrals.
</document>

<document source="https://tamnd.github.io/math2/node/8Z8PE69SJS.md" tag="8Z8PE69SJS" kind="statement">
# Hall's theorem on solvable groups

## Statement

Let $G$ be a finite [[solvable group|0FXJ7YFMMX]], and let $\pi$ be a set of [[prime numbers|01826KMG2H]].
A $\pi$-subgroup is a subgroup whose order has all its prime factors in $\pi$.

1. $G$ has a [[Hall $\pi$-subgroup|KPWW9X14V2]].
2. Any two Hall $\pi$-subgroups of $G$ are [[conjugate|0H11ZPX4ND]] in $G$.
3. Every $\pi$-subgroup of $G$ is contained in some Hall $\pi$-subgroup of $G$.

## Remarks

The proof is by induction on $\abs{G}$.
A minimal normal subgroup $N$ of a solvable group is an elementary abelian $p$-group.
One applies induction to $G/N$ and takes the preimage $K$ of a Hall $\pi$-subgroup of $G/N$.
If $p \in \pi$, then $K$ is a Hall $\pi$-subgroup of $G$.
If $p \notin \pi$, then a complement of $N$ in $K$, given by the [[Schur–Zassenhaus theorem|64K1A69C80]], is one.

Example: $G = S_3 \times \Z/5\Z$ of order $30$ is solvable, and $S_3 \times \set{0}$ is a Hall $\set{2, 3}$-subgroup, of order $6$ and index $5$.

Solvability is necessary.
The [[simple group|0YMYMNQ0QZ]] $A_5$ of order $60$ has no subgroup of order $15$, so it has no Hall $\set{3, 5}$-subgroup.
In the simple group $\mathrm{PSL}_2(\F_7)$ of order $168$ there are two conjugacy classes of subgroups isomorphic to $S_4$; these are Hall $\set{2, 3}$-subgroups, of index $7$, that are not conjugate.

Philip Hall also proved a converse.
Let $p'$ denote the set of all primes other than $p$.
If a finite group has a Hall $p'$-subgroup for every prime $p$, then it is solvable.

## Explanations

### L1 human

In any finite group, the Sylow theorems give subgroups that collect all factors of one prime in the order.
Hall's theorem says that in a solvable group one can do the same for any set of primes at once.
For example, a solvable group of order $60$ has a subgroup of order $12$, one of order $15$, and one of order $20$.
Such subgroups exist for every choice of primes, and they are unique up to conjugation.

### L2 human

The theorem extends the [[Sylow theorems|FESTJ0K9TX]] from one prime to sets of primes, but only for solvable groups.
Together with the converse, it characterises solvable groups by the existence of Hall subgroups.
Hall $p'$-subgroups are also called Sylow complements, since a Hall $p'$-subgroup and a Sylow $p$-subgroup together make up the whole order of the group.
</document>

<document source="https://tamnd.github.io/math2/node/8ZAPAPYJFG.md" tag="8ZAPAPYJFG" kind="concept">
# Derived set

## Summary

The derived set of a subset $A$ of a [[topological space|0CQHFPGJ45]] is the set of all [[limit points|0E3X7TXS67]] of $A$.
It is written $A'$.
The [[closure|0QEAJB91GZ]] of $A$ is $A \cup A'$, and repeating the operation $A \mapsto A'$ leads to the Cantor–Bendixson analysis of closed sets.

## Explanations

### L1 human

A limit point of $A$ is a point that has points of $A$, other than itself, arbitrarily close to it.
The derived set collects all such points, whether they lie in $A$ or not.
For the set $A = \set{1, 1/2, 1/3, \dots}$ in $\R$, every point of $A$ is alone in a small interval, but the points pile up at $0$.
So the derived set of $A$ is $\set{0}$.
</document>

<document source="https://tamnd.github.io/math2/node/8ZQ9TA4Q2Q.md" tag="8ZQ9TA4Q2Q" kind="example">
# Poincaré group

## Example

Let $O(1, 3)$ be the [[Lorentz group|QWK3BYVJ9W]], acting on Minkowski space $\R^{1,3} = \R^4$.
The *Poincaré group* is the set of pairs $(\Lambda, a)$ with $\Lambda \in O(1, 3)$ and $a \in \R^4$, acting on $\R^4$ by
$$(\Lambda, a) \cdot x = \Lambda x + a,$$
a Lorentz transformation followed by a [[translation|087VJM5Q77]].
Composing two such maps gives the group law
$$(\Lambda, a)(\Lambda', a') = (\Lambda \Lambda', \ \Lambda a' + a).$$
Check: $(\Lambda, a) \cdot ((\Lambda', a') \cdot x) = \Lambda(\Lambda' x + a') + a = \Lambda\Lambda' x + (\Lambda a' + a)$.
The identity is $(I, 0)$ and the inverse of $(\Lambda, a)$ is $(\Lambda^{-1}, -\Lambda^{-1} a)$.

So the Poincaré group is the [[semidirect product|0H3WZXCZNF]] $\R^4 \rtimes O(1, 3)$, a [[Lie group|7JBRGCVJRB]] of dimension $4 + 6 = 10$.

## Remarks

The Poincaré group is the group of all isometries of Minkowski space, that is, the bijections $f$ of $\R^4$ with $Q(f(x) - f(y)) = Q(x - y)$ for the Minkowski form $Q$.
The translations form an abelian normal subgroup, so the Poincaré group is not semisimple.
Often the name refers to the identity component $\R^4 \rtimes SO^+(1, 3)$.
Its irreducible unitary representations, studied by Wigner, are labelled by mass and spin in physics.

## Explanations

### L1 human

The Poincaré group is the full symmetry group of special relativity.
Besides rotations and boosts, it contains the shifts of space and time.
Its $10$ dimensions count $4$ translations, $3$ rotations and $3$ boosts.
The laws of physics in flat space-time have the same form in every frame related by an element of this group.
</document>

<document source="https://tamnd.github.io/math2/node/8ZSFY9BWME.md" tag="8ZSFY9BWME" kind="definition">
# Frölicher space

## Definition

Let $X$ be a set, let $\mathcal{C}$ be a set of maps $c \colon \R \to X$ and let $\mathcal{F}$ be a set of maps $f \colon X \to \R$.
The triple $(X, \mathcal{C}, \mathcal{F})$ is a *Frölicher space* if both conditions hold.

1. A map $f \colon X \to \R$ lies in $\mathcal{F}$ if and only if $f \circ c$ is a [[smooth function|0V33BNKZ9S]] $\R \to \R$ for every $c \in \mathcal{C}$.
2. A map $c \colon \R \to X$ lies in $\mathcal{C}$ if and only if $f \circ c$ is a smooth function $\R \to \R$ for every $f \in \mathcal{F}$.

The elements of $\mathcal{C}$ are the smooth curves and those of $\mathcal{F}$ the smooth functions.
A map $\Phi \colon X \to Y$ of Frölicher spaces is *smooth* if $\Phi \circ c \in \mathcal{C}_Y$ for every $c \in \mathcal{C}_X$; equivalently, $f \circ \Phi \in \mathcal{F}_X$ for every $f \in \mathcal{F}_Y$.

## Remarks

Example: $X = \R^n$ with $\mathcal{C}$ the smooth curves and $\mathcal{F}$ the smooth functions.
Condition 2 holds, since a curve is smooth when its coordinate functions are.
Condition 1 is a theorem of Boman: a function on $\R^n$ is smooth if it is smooth along every smooth curve.
The same works for every finite-dimensional [[smooth manifold|F45VQ47GPV]].

Non-example: $X = \R$, $\mathcal{C}$ the smooth curves and $\mathcal{F}$ the polynomial functions.
Condition 1 fails: $\sin \circ c$ is smooth for every smooth curve $c$, but $\sin$ is not a polynomial.

Every Frölicher space carries a [[diffeology|7JS0JFR89E]]: its plots are the maps $p$ from open subsets of Euclidean spaces with $f \circ p$ smooth for every $f \in \mathcal{F}$.
Frölicher spaces and smooth maps form a [[cartesian closed category|BFBNQ36QS8]].
</document>

<document source="https://tamnd.github.io/math2/node/8ZW9P9JRQD.md" tag="8ZW9P9JRQD" kind="concept">
# Moishezon manifold

## Summary

A Moishezon manifold is a compact connected [[complex manifold|JSGWAN2RJV]] with as many independent meromorphic functions as its dimension.
In other words, its [[algebraic dimension|F0GF0JEZTG]] equals its dimension.
Every projective manifold is Moishezon.
Moishezon proved that every Moishezon manifold becomes projective after finitely many blow-ups along smooth centres.
He also proved that a Moishezon manifold is projective if and only if it is [[Kähler|X60JG9AEQJ]].
So Moishezon manifolds are the compact complex manifolds closest to algebraic varieties; they are the manifolds that correspond to proper algebraic spaces.

## Explanations

### L1 human

A Moishezon manifold carries enough meromorphic functions to serve as coordinates almost everywhere.
Projective varieties, the spaces cut out by polynomial equations in projective space, are the main examples.
There are also Moishezon manifolds that are not projective, first found by Hironaka in dimension $3$, but they differ from projective ones only by a birational change.
In dimension $2$ every Moishezon manifold is projective.
</document>

<document source="https://tamnd.github.io/math2/node/901CCKQHVY.md" tag="901CCKQHVY" kind="concept">
# Exterior differential system

## Summary

An exterior differential system on a [[smooth manifold|F45VQ47GPV]] is an [[ideal|096Z9MP2ME]] of [[differential forms|0E3G0GRPZ6]] that is closed under the [[exterior derivative|0ZHSTAG8ZY]].
Its integral manifolds are the submanifolds on which all forms of the ideal vanish.
Every system of partial differential equations can be written as an exterior differential system, so that solutions become integral manifolds.
The theory, largely due to Élie Cartan, studies existence and size of integral manifolds in a way that does not depend on coordinates.

## Explanations

### L1 human

A differential equation can be rewritten as a set of conditions "this form must vanish" on a space of variables and derivatives.
For example, the conditions $du - p\, dx = 0$ and $dp - u\, dx = 0$ on the space with coordinates $(x, u, p)$ describe curves with $p = du/dx$ and $dp/dx = u$, that is, solutions of $u'' = u$.
The curves on which these forms vanish are the integral manifolds.
Working with forms instead of equations makes changes of variables painless and shows the geometry of the solution set.
</document>

<document source="https://tamnd.github.io/math2/node/902X7EHFSC.md" tag="902X7EHFSC" kind="statement">
# Hopf degree theorem

## Statement

Let $n \ge 1$ and let $M$ be a compact connected [[oriented|913WEQERBE]] [[smooth manifold|F45VQ47GPV]] of dimension $n$ without boundary.
Let $f, g \colon M \to S^n$ be continuous maps to the [[$n$-sphere|T62369RFH6]].
Then $f$ and $g$ are [[homotopic|F2F143KQ2M]] if and only if $\deg f = \deg g$.
Moreover, every integer is the [[degree|TMFG2P912R]] of some continuous map $M \to S^n$.

## Remarks

One direction is the homotopy invariance of degree.
For the other, one may take $f$ and $g$ smooth and use a regular value.
By the Pontryagin–Thom construction, the homotopy class of a map $M \to S^n$ is determined by the framed cobordism class of the preimage of a regular value.
This preimage is a finite set of signed points, and in dimension $0$ such a class is determined by the sum of the signs.
In other words, the degree gives a bijection from the set $[M, S^n]$ of homotopy classes to $\Z$.

Example: for $M = S^1$ and $n = 1$, the theorem says that loops in the circle are classified up to homotopy by their winding number, which recovers $\pi_1(S^1) \cong \Z$.

The dimensions must agree: the Hopf map $S^3 \to S^2$ is not null-homotopic, though $H_3(S^2; \Z) = 0$ so that no degree can distinguish it from a constant map.
If $M$ is not orientable, the same holds with the degree replaced by the degree modulo $2$.

## Explanations

### L1 human

Take two maps from a closed oriented manifold to a sphere of the same dimension, for example from a closed surface to $S^2$.
One can be deformed into the other exactly when both wrap around the sphere the same number of times.
So a single integer tells all about such maps up to deformation.

### L2 human

The theorem computes $[M, S^n] \cong \Z \cong H^n(M; \Z)$ for closed connected oriented $n$-manifolds.
It is a first case of the representability of cohomology by Eilenberg–MacLane spaces.
The reason is that a map $S^n \to K(\Z, n)$ representing a generator induces isomorphisms on $\pi_i$ for $i \le n$, which is enough for maps out of an $n$-dimensional $M$.
It is used to define indices of vector fields and to prove the Poincaré–Hopf theorem, and it is the base case of obstruction theory for maps into spheres.
</document>

<document source="https://tamnd.github.io/math2/node/909B1WYQ4Q.md" tag="909B1WYQ4Q" kind="concept">
# Reduced function

## Summary

Let $u$ be a positive [[superharmonic function|X4V2S56YQQ]] and $E$ a set.
The reduced function, or réduite, of $u$ on $E$ is the lower envelope of all positive superharmonic functions that dominate $u$ on $E$.
Its lower semicontinuous regularisation is again superharmonic and equals $u$ on $E$ up to a [[polar set|C95QJ7YH5T]].
When $u$ is the potential of a measure, this regularisation is the potential of the [[balayage|NTFY23YDEJ]] of that measure onto $E$.
Reduced functions give capacities, harmonic measures and thinness in one framework.

## Explanations

### L1 human

Start with a positive superharmonic function $u$ and a set $E$.
Among all positive superharmonic functions that are at least $u$ on $E$, take the smallest possible value at each point.
The result keeps $u$ on $E$ and is as small as superharmonicity allows elsewhere.
For example, in $\R^3$ with $u = 1$ and $E$ the closed unit ball, the reduced function is $\min(1, 1/\abs{x})$, the potential of a charged metal ball.
</document>

<document source="https://tamnd.github.io/math2/node/90ANMX55DP.md" tag="90ANMX55DP" kind="definition">
# Series acceleration

## Definition

A *sequence transformation* $T$ assigns to a sequence $(s_n)$ of real or complex numbers a new sequence $(t_n) = T(s_n)$.
Each $t_n$ depends on finitely many terms of $(s_n)$.
Let $(s_n)$ [[converge|0XKWSSY8P2]] to $s$, with $s_n \ne s$ for all large $n$.
The transformation $T$ *accelerates the convergence* of $(s_n)$ if $(t_n)$ converges to $s$ and
$$\lim_{n \to \infty} \frac{t_n - s}{s_n - s} = 0.$$
*Series acceleration* means applying such a transformation to the sequence of [[partial sums|0NB643K60Q]] of a convergent [[series|0JH4MN4K1C]].

## Remarks

Example: for $s_n = 2^{-n}$ with limit $0$, the transformation $t_n = s_n s_{n+1} = 2^{-2n-1}$ gives $t_n / s_n = 2^{-n-1} \to 0$, so it accelerates this sequence.
Non-example: the shift $t_n = s_{n+1}$ gives $t_n / s_n = \frac12$ for the same sequence.
The error is halved, but the ratio does not tend to $0$, so this is not acceleration in the sense above.
A given transformation accelerates some sequences and not others.

## Explanations

### L1 human

The new sequence must have the same limit and its error must become negligible compared with the old error.
Gaining a fixed factor is not enough.
</document>

<document source="https://tamnd.github.io/math2/node/90ASENKFMH.md" tag="90ASENKFMH" kind="definition">
# Fractional ideal

## Definition

Let $R$ be an [[integral domain|0QVD7WBFGN]] and $K$ its [[field of fractions|0F2PTGX1A2]].
A *fractional ideal* of $R$ is a subset $I \subseteq K$ such that:

1. $I$ is an $R$-[[submodule|05WDSN597C]] of $K$;
2. $I \ne \set{0}$;
3. there is a nonzero $r \in R$ with $rI \subseteq R$.

The product of fractional ideals $I$ and $J$ is the set $IJ$ of finite sums $\sum x_k y_k$ with $x_k \in I$ and $y_k \in J$.
A fractional ideal $I$ is *invertible* if $IJ = R$ for some fractional ideal $J$.

## Remarks

Example: $I = \tfrac12 \Z \subseteq \Q$ for $R = \Z$; it is a nonzero $\Z$-submodule, and $2I = \Z \subseteq \Z$, so it is a fractional ideal, with inverse $2\Z$.
Non-example: $\Z[\tfrac12] = \set{a/2^k : a \in \Z, k \ge 0}$ is a nonzero $\Z$-submodule of $\Q$, but it fails clause 3: for any nonzero integer $r$, choose $k$ with $2^k > \abs{r}$; then $r/2^k \in r\Z[\tfrac12]$ is not an integer.
Every nonzero [[ideal|096Z9MP2ME]] of $R$ is a fractional ideal, with $r = 1$; these are the integral ideals.
For $x \in K^{\times}$, the set $xR$ is a principal fractional ideal, with inverse $x^{-1}R$.
When $R$ is Noetherian, clause 3 is equivalent to $I$ being a finitely generated $R$-module.
Some authors, and Mathlib, also count $\set{0}$ as a fractional ideal.
</document>

<document source="https://tamnd.github.io/math2/node/90F3NWRHVD.md" tag="90F3NWRHVD" kind="concept">
# Geometric optics

## Summary

In PDE theory, geometric optics studies high-frequency solutions of wave equations.
It uses the ansatz $u = e^{i\varphi/\varepsilon}\big(a_0 + \varepsilon a_1 + \cdots\big)$ with a small wavelength $\varepsilon$.
The phase $\varphi$ solves a first-order nonlinear equation, the [[eikonal equation|HGMX1QGTAZ]], whose characteristics are the light rays.
The amplitudes $a_k$ solve [[transport equations|PVD7VZJZ9B]] along these rays.
It explains how the ray picture of light emerges from the wave picture, and it underlies the WKB method and microlocal analysis.

## Explanations

### L1 human

Light is a wave, yet in daily life it seems to travel in straight lines called rays.
This is because its wavelength is tiny compared with the objects it meets.
Geometric optics makes this precise: for very short waves, the crests of the wave move along rays, and the strength of the wave is carried along each ray.
In a medium where the speed is constant, the rays are straight lines; where the speed varies, as in a mirage, the rays bend.
</document>

<document source="https://tamnd.github.io/math2/node/90MF8M915N.md" tag="90MF8M915N" kind="definition">
# Logarithm of a matrix

## Definition

Let $n \ge 1$ and let $A$ be an $n \times n$ complex [[matrix|0JXZKX526R]].
A *logarithm* of $A$ is an $n \times n$ complex matrix $B$ such that $e^B = A$, where $e^B$ is the [[matrix exponential|020M04FN1H]] of $B$.

## Remarks

Example: $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ has the logarithm $B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$.
Check: $B^2 = 0$, so $e^B = I + B = A$.
Non-example: the zero matrix has no logarithm, because $\det e^B = e^{\tr B} \ne 0$ for every $B$, by the [[determinant of a matrix exponential|0A1QQN333A]].
A complex matrix has a logarithm if and only if it is [[invertible|0EB9N0S33C]].
If $A$ has no eigenvalue in $(-\infty, 0]$, there is a unique logarithm whose eigenvalues have imaginary part in $(-\pi, \pi)$; it is the *principal logarithm*, written $\log A$.
If $\norm{A - I} < 1$ for a submultiplicative norm, then $\log A = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} (A - I)^k$.
</document>

<document source="https://tamnd.github.io/math2/node/90PB1H41A6.md" tag="90PB1H41A6" kind="definition">
# Clopen set

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A subset $A \subseteq X$ is *clopen* if $A$ is [[open|02BH3MCBQB]] in $X$ and $A$ is [[closed|0ZCK8DQYN1]] in $X$.
Equivalently, $A$ and its complement $X \setminus A$ are both open.

## Remarks

Example: let $X = [0, 1] \cup [2, 3]$ with the topology from $\R$.
The set $A = [0, 1]$ equals $X \cap (-1, 3/2)$, so it is open in $X$.
Its complement $[2, 3]$ equals $X \cap (3/2, 4)$, so it is open as well, and $A$ is closed.

Example: in the rationals $\Q$ with the topology from $\R$, the set $\set{q \in \Q : q < \sqrt{2}}$ is clopen, since $\sqrt{2} \notin \Q$.

Non-example: in $\R$, the set $[0, 1)$ is not open, because no interval around $0$ lies inside it.
It is not closed either.

In every space, $\emptyset$ and $X$ are clopen.
A space $X$ is [[connected|0V032VWTHD]] if and only if these are its only clopen subsets.
</document>

<document source="https://tamnd.github.io/math2/node/90RA6TAT7H.md" tag="90RA6TAT7H" kind="conjecture">
# 11/8 conjecture

## Statement

Let $M$ be a closed, oriented, [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]] that admits a [[spin structure|RSPZ9TCFQ8]].
Let $b_2(M)$ be the rank of $H_2(M; \Z)$ and $\sigma(M)$ the [[signature|07EZA80NEH]] of its [[intersection form|WK8GKRKAVD]].
Then
$$b_2(M) \ge \frac{11}{8} \abs{\sigma(M)}.$$

## Remarks

The conjecture is open.
The K3 surface has $b_2 = 22$ and $\sigma = -16$, so it attains equality.
For a simply connected $M$, the intersection form is even, and by [[Donaldson's theorem|34B7K77P31]] it is indefinite unless it is zero.
So it is a sum of copies of $\pm E_8$ and of the hyperbolic plane.
The conjecture then says that each pair of $E_8$ summands needs at least three hyperbolic summands, as in the K3 surface.
Mikio Furuta proved the $10/8$ theorem: if $\sigma(M) \ne 0$, then $b_2(M) \ge \frac{10}{8} \abs{\sigma(M)} + 2$.
Later work has improved the constant term of this bound.
Without smoothness the inequality fails: the [[$E_8$ manifold|BNDRN0G9BR]] is a closed topological manifold with an even form, $b_2 = 8$ and $\sigma = 8$.

## Explanations

### L1 human

For a smooth four-dimensional space of spin type, the conjecture compares two numbers read off from how surfaces cross inside it.
It predicts that the total number of independent surfaces is at least $11/8$ times the size of the signature.
The K3 surface, a famous complex surface, shows that $11/8$ cannot be improved.
</document>

<document source="https://tamnd.github.io/math2/node/90V8S5CXE6.md" tag="90V8S5CXE6" kind="example">
# Daubechies wavelet

## Example

For each integer $N \ge 1$, Ingrid Daubechies constructed an orthonormal [[wavelet|4PZ35C0GV3]] $\psi$ from a [[multiresolution analysis|1GHQMBR7DE]].
Its [[scaling function|YH4BERB952]] $\varphi$ satisfies a refinement equation $\varphi(x) = \sqrt2 \sum_{k=0}^{2N-1} h_k \varphi(2x - k)$ with $2N$ real coefficients.
Both $\varphi$ and $\psi$ have compact support of length $2N - 1$, and $\psi$ has $N$ [[vanishing moments|7QERXM0C52]].
For $N = 1$ this is the Haar wavelet.

The case $N = 2$ has the filter
$$h_0 = \frac{1 + \sqrt3}{4\sqrt2}, \quad h_1 = \frac{3 + \sqrt3}{4\sqrt2}, \quad h_2 = \frac{3 - \sqrt3}{4\sqrt2}, \quad h_3 = \frac{1 - \sqrt3}{4\sqrt2}.$$
Check of the filter conditions.
The sum is $\frac{8}{4\sqrt2} = \sqrt2$, as needed for $\int \varphi = 1$.
The sum of squares is $\frac{(4 + 2\sqrt3) + (12 + 6\sqrt3) + (12 - 6\sqrt3) + (4 - 2\sqrt3)}{32} = 1$.
And $h_0 h_2 + h_1 h_3 = \frac{(1 + \sqrt3)(3 - \sqrt3) + (3 + \sqrt3)(1 - \sqrt3)}{32} = \frac{2\sqrt3 - 2\sqrt3}{32} = 0$.
So the filter is orthogonal to its shift by $2$, which reflects the orthogonality of the translates $\varphi(\cdot - k)$.

The wavelet is $\psi(x) = \sqrt2 \sum_k g_k \varphi(2x - k)$ with $g_k = (-1)^k h_{1-k}$, so its coefficients are $g_{-2} = h_3$, $g_{-1} = -h_2$, $g_0 = h_1$, $g_1 = -h_0$.
Check of two vanishing moments: $\sum_k g_k = h_3 - h_2 + h_1 - h_0 = 0$ and $\sum_k k g_k = -2h_3 + h_2 - h_0 = \frac{-2 + 2\sqrt3 + 3 - \sqrt3 - 1 - \sqrt3}{4\sqrt2} = 0$.
These two discrete conditions give $\int \psi = 0$ and $\int x \psi(x) \, dx = 0$.
The scaling function of this case is continuous but not smooth, and it has no closed formula; it is computed from the refinement equation.

## Explanations

### L1 human

Daubechies wavelets are a family of wavelets that live on short intervals and are smoother as the index grows.
Each one is fixed by a short list of numbers, its filter, and the discrete wavelet transform uses only these numbers.
The four numbers above give the simplest one after the Haar wavelet.
Their shape looks jagged, almost like a fractal, but they are widely used in signal and image compression.
</document>

<document source="https://tamnd.github.io/math2/node/913WEQERBE.md" tag="913WEQERBE" kind="concept">
# Orientation of a manifold

## Summary

An orientation of an $n$-dimensional [[manifold|QQQ9B1DK4P]] is a consistent choice, at every point, of one of the two local directions of rotation, or "handedness".
In algebraic topology it is a continuous choice of generator of the local homology group $H_n(M, M \setminus \set{x}) \cong \Z$ at each point $x$.
For smooth manifolds it is the same as a continuous choice of [[orientation|0QJJAQ6ZYZ]] of every tangent space.
Orientations are needed for fundamental classes, Poincaré duality with integer coefficients, and integration of differential forms.

## Explanations

### L1 human

On a small piece of surface you can always decide which way is clockwise.
A surface is orientable if you can make this decision everywhere so that nearby choices agree.
A sphere is orientable: clockwise as seen from outside works everywhere.
A Möbius band is not: carry a small clockwise circle once around the band and it comes back counterclockwise.
An [[orientable surface|0TD904MG06]] is a surface where such a global choice exists.
</document>

<document source="https://tamnd.github.io/math2/node/917PGHVZPM.md" tag="917PGHVZPM" kind="concept">
# Henselization

## Summary

The Henselization of a [[local ring|MZ02BKPHJA]] $R$ is the smallest [[Henselian local ring|RNXPV3KC5J]] that receives a local map from $R$.
It is characterized by a universal property among local homomorphisms from $R$ to Henselian local rings.
It keeps the residue field of $R$ and, for a Noetherian $R$, it has the same completion as $R$.
It is a substitute for the completion that stays algebraic over $R$, and it is used in étale cohomology and in approximation theorems.

## Explanations

### L1 human

[[Hensel's lemma|010QVB71T2]] lets one lift a simple root of a polynomial modulo the maximal ideal to a true root.
Many local rings do not have this property, and the Henselization adds just enough elements to get it.
The completion also has the property, but it adds far more elements, most of them transcendental.
For the ring $\Z_{(p)}$ of fractions with denominator prime to $p$, the Henselization is the ring of $p$-adic integers that are algebraic over $\Q$.
So it lies between $\Z_{(p)}$ and the full ring of $p$-adic integers.
</document>

<document source="https://tamnd.github.io/math2/node/9186DYXE86.md" tag="9186DYXE86" kind="concept">
# Stanley–Reisner ring

## Summary

The Stanley–Reisner ring of a finite simplicial complex is the quotient of a polynomial ring by the ideal of products of variables over the non-faces.
It turns a combinatorial object into a commutative ring.
Algebraic invariants of the ring, such as the Hilbert series and the Krull dimension, then encode counts of faces.
Its ideal is a squarefree [[monomial ideal|4G25S7PJ22]], and every squarefree monomial ideal arises this way.
The construction is a main tool in the proof of the upper bound theorem for simplicial spheres.

## Explanations

### L1 human

A simplicial complex is a collection of vertices, edges, triangles and higher simplices glued along faces.
Give each vertex a variable.
In the Stanley–Reisner ring, a product of distinct variables is zero exactly when those vertices do not span a face.
For the boundary of a triangle with vertices $1, 2, 3$, every pair of vertices is an edge, but the whole triangle is missing, so the ring is $k[x_1, x_2, x_3]/(x_1 x_2 x_3)$.
Counting the monomials that survive in each degree recovers the numbers of vertices, edges and so on.
</document>

<document source="https://tamnd.github.io/math2/node/91DW8V0PET.md" tag="91DW8V0PET" kind="statement">
# Maximum modulus principle in several variables

## Statement

Let $D \subseteq \C^n$ be a nonempty connected open set, and let $f$ be a [[holomorphic function|507C34W0ZT]] on $D$.
If there is a point $a \in D$ with $\abs{f(z)} \le \abs{f(a)}$ for all $z \in D$, then $f$ is constant on $D$.

## Remarks

For $z$ near $a$, the map $t \mapsto f(a + t(z - a))$ is a holomorphic function of one variable on a disc around $t = 0$ whose modulus has a maximum at $t = 0$.
By the one-variable [[maximum modulus principle|048F2P90WR]] it is constant, so $f$ is constant near $a$, and the [[identity theorem|7GZ908QZ04]] makes $f$ constant on $D$.
Consequence: if $D$ is bounded and $f$ is continuous on the closure $\overline{D}$ and holomorphic on $D$, then $\max_{\overline{D}} \abs{f}$ is attained on the boundary of $D$.
Connectedness is needed: on two disjoint balls, a function equal to $0$ on one and $1$ on the other attains its maximum modulus without being constant.
Example: on the unit polydisc of $\C^2$, $f(z) = z_1 z_2$ satisfies $\abs{f} < 1$ inside and reaches $\abs{f} = 1$ only where $\abs{z_1} = \abs{z_2} = 1$, on the distinguished boundary.

## Explanations

### L1 human

The size $\abs{f}$ of a non-constant holomorphic function never has a highest point inside its domain.
So on a bounded domain the largest value of $\abs{f}$ is found on the edge.
It is the same principle as in one variable, and it is proved by restricting to complex lines.

### L2 human

The proof is a typical way to pass from one to several variables: restrict to complex lines through a point and use the one-variable result.
For special domains the maximum is found on a much smaller part of the boundary.
On a polydisc it is the [[distinguished boundary|A6HN207Y2T]], an $n$-dimensional torus.
The study of these smallest sets leads to the Shilov boundary and to peak points.
</document>

<document source="https://tamnd.github.io/math2/node/91GE4C1BC5.md" tag="91GE4C1BC5" kind="concept">
# Supercompact cardinal

## Summary

A supercompact cardinal is a cardinal $\kappa$ that is the critical point of many [[elementary embeddings|CGH8C8EA9N]] $j \colon V \to M$.
For each length, one of them has a target $M$ closed under sequences of that length.
It strengthens the embedding form of [[measurability|AKBWT323RQ]], which asks only for closure under sequences of length $\kappa$.
Supercompact cardinals are [[Woodin cardinals|CMC2C1D53W]] and lie high in the hierarchy of [[large cardinals|X1FG7PSCTY]].
They are used to prove the consistency of strong axioms such as the [[proper forcing axiom|35XTXNEWQ9]].

## Explanations

### L1 human

A measurable cardinal gives an embedding of the universe into a copy of itself that keeps sequences of length $\kappa$.
A supercompact cardinal gives such embeddings that keep sequences of every length, as long as one fixes the length in advance.
So the copy looks more and more like the whole universe.
This makes supercompact cardinals very strong, and many consistency results use them.
</document>

<document source="https://tamnd.github.io/math2/node/91KJT4TXG8.md" tag="91KJT4TXG8" kind="concept">
# Maslov index

## Summary

The Maslov index is an integer attached to a loop of Lagrangian subspaces of the symplectic vector space $\R^{2n}$.
It is the class of the loop in the [[fundamental group|S1XK3HXHFX]] of the Lagrangian Grassmannian, which is isomorphic to $\Z$.
It can be applied to the tangent planes of a [[Lagrangian submanifold|H8ACPTMX08]] along a loop.
Then it counts with signs how often the loop crosses the [[Lagrangian singularities|QKPMR9MC9W]] of the projection to the base.
It enters the quantization rules of semiclassical mechanics, the index theory of Hamiltonian systems, and Floer homology.

## Explanations

### L1 human

In the plane $\R^2$, every line through the origin is a Lagrangian subspace.
Turn such a line around the origin until it comes back to itself; this takes half a turn.
The Maslov index counts how many such half turns a loop of lines makes, with a sign for the direction.
In higher dimensions the subspaces are $n$-dimensional, and the index adds up the turning of all the directions at once.
For a light beam or a quantum wave, each unit of Maslov index along a closed path shifts the phase by a fixed amount, which corrects the naive quantization rule.
</document>

<document source="https://tamnd.github.io/math2/node/91N7YTXJT2.md" tag="91N7YTXJT2" kind="definition">
# Formal group law in topology

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]].
A *(one-dimensional, commutative) formal group law* over $R$ is a [[formal power series|072W67KFYK]] $F(x, y) \in R[\![x, y]\!]$ such that:

1. $F(x, y) \equiv x + y$ modulo terms of total degree at least $2$.
2. $F(x, F(y, z)) = F(F(x, y), z)$ in $R[\![x, y, z]\!]$.
3. $F(x, y) = F(y, x)$.

Let $E$ be a multiplicative [[generalized cohomology theory|H28PSZR7XG]] with coefficient ring $E^* = E^*(\mathrm{pt})$.
A *complex orientation* of $E$ is a class $x \in \tilde E^2(\C P^\infty)$ whose restriction to $\tilde E^2(\C P^1) \cong E^0(\mathrm{pt})$ is the unit $1$.
Then $E^*(\C P^\infty) = E^*[\![x]\!]$ and $E^*(\C P^\infty \times \C P^\infty) = E^*[\![x_1, x_2]\!]$, with $x_i$ pulled back from the $i$-th factor.
Let $m \colon \C P^\infty \times \C P^\infty \to \C P^\infty$ classify the tensor product of the two universal line bundles.
The *formal group law of $(E, x)$* is the power series $F$ over $E^*$ with $m^*(x) = F(x_1, x_2)$.
It satisfies the three conditions above, and it is homogeneous of degree $2$ when $x$, $x_1$ and $x_2$ have degree $2$.

## Remarks

Example: in ordinary cohomology $H^*(-; \Z)$ with $x$ the first Chern class of the universal line bundle, $c_1(L_1 \otimes L_2) = c_1(L_1) + c_1(L_2)$, so $F(x, y) = x + y$.
The three conditions are clear for this additive law.

Example: in complex K-theory with Bott class $\beta \in K^{-2}(\mathrm{pt})$, take $x = \beta^{-1}([L] - 1)$ for the universal line bundle $L$.
Since $[L_1 L_2] - 1 = ([L_1] - 1) + ([L_2] - 1) + ([L_1] - 1)([L_2] - 1)$, the law is $F(x, y) = x + y + \beta x y$, the multiplicative formal group law.

Non-example: $G(x, y) = x + y + x^2 y$ over $\Z$ satisfies clause 1 but fails clause 3, since $G(y, x) = x + y + x y^2$.

There is a universal formal group law over the Lazard ring $L$: formal group laws over $R$ correspond to ring maps $L \to R$.
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<document source="https://tamnd.github.io/math2/node/91PCJMS5VC.md" tag="91PCJMS5VC" kind="definition">
# F-distribution

## Definition

Let $d_1$ and $d_2$ be positive integers.
Let $U_1$ and $U_2$ be [[independent|0E9502NT92]] random variables with [[chi-squared distributions|0RFMMBHFTV]] with $d_1$ and $d_2$ degrees of freedom.
The distribution of
$$X = \frac{U_1 / d_1}{U_2 / d_2}$$
is the *F-distribution* with $d_1$ and $d_2$ degrees of freedom, written $F(d_1, d_2)$.

## Remarks

It has the [[probability density function|04YG690N5V]]
$$f(x) = \frac{1}{B(d_1/2, d_2/2)} \left(\frac{d_1}{d_2}\right)^{d_1/2} x^{d_1/2 - 1} \left(1 + \frac{d_1}{d_2} x\right)^{-(d_1 + d_2)/2}, \quad x > 0,$$
where $B$ is the beta function.
The same formula with real $d_1, d_2 > 0$ defines $F(d_1, d_2)$ for non-integer parameters.
Its mean is $d_2/(d_2 - 2)$ when $d_2 > 2$.
If $X \sim F(d_1, d_2)$, then $1/X \sim F(d_2, d_1)$.

Example: let $T$ have the [[Student t-distribution|0NQ5798SB1]] with $\nu$ degrees of freedom, so $T = Z/\sqrt{V/\nu}$ with $Z$ standard normal and $V$ chi-squared with $\nu$ degrees of freedom, independent.
Then $T^2 = (Z^2/1)/(V/\nu)$, and $Z^2$ is chi-squared with $1$ degree of freedom, so $T^2 \sim F(1, \nu)$.

Non-example: the independence clause matters.
If $d_1 = d_2$ and $U_2 = U_1$, then $X = 1$ always, which is not an F-distribution.
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<document source="https://tamnd.github.io/math2/node/91QMF4R1CY.md" tag="91QMF4R1CY" kind="definition">
# Shift operator

## Definition

Let $G$ be an additive abelian group, let $Y$ be a set, and let $h \in G$.
The *shift operator* $E_h$ with step $h$ sends a [[function|0XHDTE5YMZ]] $f \colon G \to Y$ to the function
$$E_h f \colon G \to Y, \qquad E_h f(x) = f(x + h).$$
For a [[sequence|0QE8STGJH2]] $u = (u_n)_{n \ge 0}$ in $Y$, the (left) shift $E$ is defined by $(E u)_n = u_{n+1}$ for $n \ge 0$.

## Remarks

Example: for $f(x) = \sin x$ and $h = \pi/2$, $E_h f(x) = \sin(x + \pi/2) = \cos x$.
Non-example: $f \mapsto f(2x)$ is not a shift operator; it rescales the argument instead of translating it.
The shifts compose by $E_h E_k = E_{h+k}$, and $E_0$ is the identity.
When $Y$ is an abelian group, $E_h$ is additive, and the [[forward difference operator|8E0EBA8X07]] is $\Delta_h = E_h - I$.
Other notations are $T_h$ and $\tau_h$; on sequences the left shift is also called the backward shift.
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<document source="https://tamnd.github.io/math2/node/91RFB97TNV.md" tag="91RFB97TNV" kind="concept">
# Singular control

## Summary

A singular control is an optimal control, or a candidate for one, on a time interval where the maximum principle gives no information.
There the maximum condition of [[Pontryagin's maximum principle|7RPV2T29CF]] does not determine the control value.
This happens when the [[Hamiltonian|RACVWG5QTK]] is affine in the control.
The coefficient of the control, called the switching function, then vanishes on a whole interval.
On such an interval the control usually takes interior values, unlike a [[bang–bang control|GK1Z3Q1WRZ]].
Singular arcs are found by differentiating the switching function in time until the control appears.

## Explanations

### L1 human

For many problems the best control is at full power in one direction or the other, depending on the sign of a switching function.
Sometimes this switching function stays at zero for a while, and then the rule gives no answer.
The control on such a stretch is called singular, and it is usually some intermediate value.
For example, to keep a quantity at zero with a control between $-1$ and $1$, the best control after reaching zero is $0$, which is neither extreme.
</document>

<document source="https://tamnd.github.io/math2/node/91RZ5YARTM.md" tag="91RZ5YARTM" kind="concept">
# Hörmander's condition

## Summary

Hörmander's condition is a property of a family of [[vector fields|0299VCW1TH]].
It says that the fields and their iterated [[Lie brackets|68FGQEBD6J]] span the tangent space at every point.
Hörmander's theorem says that under this condition the second-order operator $\sum_j X_j^2 + X_0$ is [[hypoelliptic|K9DSR21DQC]], even though it may be far from elliptic.
The condition is central in sub-Riemannian geometry and in stochastic analysis, where it gives smooth densities for solutions of stochastic differential equations.

## Explanations

### L1 human

Suppose you can only move in a few allowed directions, such as driving a car, which cannot slide sideways.
By combining moves, such as going forward while turning and then backing up, you can still reach every nearby position; the Lie brackets record these combined moves.
Hörmander's condition says that the allowed directions, together with their brackets, reach every direction.
Then diffusion along the allowed directions still spreads in all directions and smooths everything out.
In the plane, the fields $\partial_x$ and $x \partial_y$ satisfy it, because their bracket is $\partial_y$.
</document>

<document source="https://tamnd.github.io/math2/node/91WG6QWRQ1.md" tag="91WG6QWRQ1" kind="definition">
# Open segment

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]] and let $x, y \in V$.
The *open segment* between $x$ and $y$ is the set
$$(x, y) = \set{(1 - t) x + t y : t \in \R,\ 0 < t < 1}.$$

## Remarks

Example: on the real line, $(0, 1)$ in this sense is the open interval of numbers strictly between $0$ and $1$; the point $1/4$ is in it with $t = 1/4$.

Non-example: the endpoint $x$ of a [[line segment|VQ4Q2V45RP]] $[x, y]$ with $x \ne y$ is not in $(x, y)$, since $(1 - t) x + t y = x$ forces $t (y - x) = 0$, so $t = 0$, which is not allowed.

If $x \ne y$, then $(x, y) = [x, y] \setminus \set{x, y}$.
If $x = y$, then $(x, y) = \set{x}$, so the open segment is not empty.
</document>

<document source="https://tamnd.github.io/math2/node/91YF82M8AA.md" tag="91YF82M8AA" kind="counterexample">
# Conway base 13 function

## Counterexample

Use base $13$ with the thirteen digits $0, 1, \ldots, 9, \mathtt{+}, \mathtt{-}, \mathtt{.}$, in this order.
For $x \in \R$, take the base-$13$ expansion of the fractional part of $x$ that does not end in an infinite string of the last digit.
Suppose that from some position on, this expansion has the form
$$\mathtt{\pm}\, a_1 a_2 \ldots a_k \,\mathtt{.}\, b_1 b_2 b_3 \ldots$$
This tail consists of a sign symbol and finitely many digits $a_i \in \set{0, \ldots, 9}$.
Then comes the point symbol, and after it only digits $b_j \in \set{0, \ldots, 9}$.
Then set $f(x) = \pm\, a_1 a_2 \ldots a_k . b_1 b_2 b_3 \ldots$, read as an ordinary decimal number.
Otherwise set $f(x) = 0$.

Each open interval $J$ contains all numbers with a fixed integer part whose expansion begins with a suitable finite block of digits.
After that block, any tail can be appended.
So for each real $y$, some $x \in J$ has a tail that encodes $y$, and $f(J) = \R$.

## Remarks

The function $f$ maps every interval onto $\R$, so it is a [[Darboux function|7BJEH1GCTE]]: it has the intermediate value property.
But $f$ is [[continuous|0P27W9RW0G]] at no point, since its values on every interval are unbounded.
So the converse of the [[intermediate value theorem|00KAFBR32E]] fails badly.
Almost every real number has a base-$13$ expansion with infinitely many point symbols, so $f = 0$ [[almost everywhere|RARTMA0ZEC]].

## Explanations

### L1 human

Conway's function hides a decimal number in the end of the base-$13$ digits of $x$.
Every tiny interval contains numbers with every possible ending, so the function takes every real value on every tiny interval.
It never skips a value, yet it is wildly discontinuous everywhere.
So "never skips values" is much weaker than "continuous".
</document>

<document source="https://tamnd.github.io/math2/node/9240K052CS.md" tag="9240K052CS" kind="concept">
# Quotient of a normed space

## Summary

The quotient of a normed space $X$ by a closed subspace $M$ is the [[quotient vector space|0BR3FY5Q8M]] $X/M$.
Its norm measures the distance of a vector to $M$.
It is a normed space, and it is a [[Banach space|06GZ956657]] when $X$ is.
Quotients appear whenever one ignores a subspace.
Examples are the open mapping theorem, the study of the kernel of an operator, and duality, where $(X/M)^*$ is identified with the annihilator of $M$.

## Explanations

### L1 human

To form the quotient, one declares two vectors equal when their difference lies in $M$.
The size of a class is the shortest length of a vector in that class, that is, the distance from the class to the origin.
In the plane with the usual length, take $M$ to be the horizontal axis.
Each class is a horizontal line, and its size is its height above or below the axis.
</document>

<document source="https://tamnd.github.io/math2/node/92433H25TC.md" tag="92433H25TC" kind="concept">
# Borel functional calculus

## Summary

The Borel functional calculus assigns an operator $f(N)$ to a normal or self-adjoint operator $N$ and a Borel function $f$ on its spectrum.
It is defined by integrating $f$ against the projection-valued measure given by the spectral theorem.
It extends the continuous functional calculus to discontinuous functions, such as indicator functions, which produce the spectral projections of $N$.

## Explanations

### L1 human

For a diagonal matrix, applying a function to the matrix means applying it to each diagonal entry.
The spectral theorem says that normal operators are diagonal in a generalized sense, with a measure in place of a list of eigenvalues.
The Borel functional calculus applies a function to an operator in the same way, entry by entry along the spectrum.
For the matrix $\operatorname{diag}(1, 2)$ and the indicator function $f$ of the set $\set{2}$, the result is $f(N) = \operatorname{diag}(0, 1)$, the projection onto the eigenspace of $2$.
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<document source="https://tamnd.github.io/math2/node/92700QAB5Y.md" tag="92700QAB5Y" kind="concept">
# Topological group

## Summary

A topological group is a [[group|G5S1M9]] with a [[topology|0CQHFPGJ45]] in which multiplication and inversion are continuous.
It is the basic object where algebra and topology work together.
It is the setting of abstract harmonic analysis and of the theory of Lie groups.
Translations by group elements are homeomorphisms, so the topology near any point looks like the topology near the identity.
The real line under addition, the circle group, matrix groups and the $p$-adic integers are all topological groups.

## Explanations

### L1 human

A topological group is a group where it also makes sense to say that two elements are close.
The group law must respect closeness: if $x$ is near $x'$ and $y$ is near $y'$, then $xy$ is near $x'y'$, and $x^{-1}$ is near $x'^{-1}$.
Picture the real line under addition: shifting the whole line by a number does not tear it.
Another example is the circle of complex numbers of absolute value $1$ under multiplication, where multiplying means rotating.
</document>

<document source="https://tamnd.github.io/math2/node/928JSGQT9C.md" tag="928JSGQT9C" kind="definition">
# Lax pair

## Definition

Consider an evolution equation for an unknown $u(t)$.
Let $L(t)$ and $P(t)$ be families of linear operators on a common space, depending on $u(t)$.
They form a *Lax pair* for the equation if the equation holds if and only if
$$\frac{dL}{dt} = [P, L] = PL - LP.$$
The equation $\frac{dL}{dt} = [P, L]$ is the *Lax equation*.

## Remarks

Example: for $u(x, t)$ smooth, put $L = -\partial_x^2 + u$ and $P = -4\partial_x^3 + 6u\partial_x + 3u_x$, operators acting on functions of $x$.
A direct computation gives $[P, L] = 6 u u_x - u_{xxx}$, as a multiplication operator, while $\frac{dL}{dt} = u_t$.
So $(L, P)$ is a Lax pair for the [[Korteweg–De Vries equation|KTT75QC72Z]] $u_t - 6 u u_x + u_{xxx} = 0$.

Non-example: if $P = 0$, the Lax equation says that $L$ is constant in $t$.
When $L$ determines $u$, as $L = -\partial_x^2 + u$ does, a pair $(L, 0)$ is a Lax pair only for the trivial equation $u_t = 0$.

If $P(t)$ is skew-adjoint and $U(t)$ solves $U' = P U$ with $U(0) = I$, then $L(t) = U(t) L(0) U(t)^{-1}$.
So the [[eigenvalues|0BQAA2BK3Q]] of $L(t)$ do not depend on $t$: the flow is *isospectral*.
For matrices, $\tr L(t)^k$ is then constant for each $k$.

## Explanations

### L1 human

A Lax pair rewrites an equation as the statement that an operator $L$ changes only by conjugation.
Conjugation does not change eigenvalues, so the equation has the eigenvalues of $L$ as conserved quantities.
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<document source="https://tamnd.github.io/math2/node/92C3S03AQZ.md" tag="92C3S03AQZ" kind="concept">
# Hyperbolic partial differential equation

## Summary

A hyperbolic partial differential equation is an evolution equation whose Cauchy problem is well posed for data on suitable hypersurfaces.
Disturbances travel at finite speed.
The model case is the [[wave equation|5EHD2XBQDG]].
Hyperbolic equations describe waves, vibrations and the transport of conserved quantities, and singularities of the data propagate along characteristics instead of being smoothed out.
See the [[definition|XQG2VCBMKQ]].

## Explanations

### L1 human

Pluck a guitar string: the disturbance moves along the string at a fixed speed and keeps its shape.
The equation of the vibrating string, $\partial_t^2 u = c^2 \partial_x^2 u$, is the basic hyperbolic equation.
Its solutions are sums $F(x - ct) + G(x + ct)$ of a wave moving right and a wave moving left.
A kink in the initial shape is carried along, not smoothed, which is the opposite of what the heat equation does.
</document>

<document source="https://tamnd.github.io/math2/node/92GVPZH9FT.md" tag="92GVPZH9FT" kind="concept">
# Lyapunov equation

## Summary

The Lyapunov equation is the linear matrix equation $AX + XA^* + Q = 0$, or its discrete form $AXA^* - X + Q = 0$, for an unknown matrix $X$.
Its solutions give quadratic [[Lyapunov functions|01REYCJ371]] for the linear system $x' = A^* x$ or $x_{k+1} = A^* x_k$.
Every eigenvalue of $A$ has negative real part exactly when the continuous equation with a positive definite $Q$ has a positive definite solution.
So the equation turns [[Lyapunov stability|0F2SYCKDT2]] into linear algebra.
The continuous form is a special case of the [[Sylvester equation|M9SW4NVP2H]].

## Explanations

### L1 human

To show that a system comes to rest, one looks for an energy that always decreases along its motion.
For a linear system the natural energies are quadratic, $x^* X x$ for a matrix $X$.
The Lyapunov equation is the condition on $X$ that makes this energy decrease at a prescribed rate.
For one variable $x' = ax$ with $a < 0$, the energy $x^2$ decreases, and the equation $2aX + q = 0$ has the positive solution $X = -q/(2a)$ for $q > 0$.
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<document source="https://tamnd.github.io/math2/node/92JCZQZSST.md" tag="92JCZQZSST" kind="definition">
# Finite field arithmetic

## Definition

Let $p$ be a prime, let $n \ge 1$, and let $f \in \F_p[x]$ be a monic [[irreducible polynomial|00X2A9G9W3]] of [[degree|0DPF6WRZW6]] $n$.
The [[quotient ring|0VFW4Y69BT]] $\F_p[x]/(f)$ of the [[polynomial ring|0Q9EZS90GY]] is a [[finite field|08RTDHX1SK]] with $p^n$ elements.
*Finite field arithmetic* in this representation computes with the polynomials $a(x)$ of degree less than $n$ over $\F_p$, one for each element, as follows.

1. The sum $a + b$ is the coefficientwise sum modulo $p$.
2. The product $a \cdot b$ is the remainder of the polynomial product $a(x) b(x)$ under [[division|D6D0SR1FK3]] by $f$.
3. The inverse of $a \ne 0$ is the polynomial $u$ of degree less than $n$ with $u a + v f = 1$ for some $v \in \F_p[x]$, found by the extended [[Euclidean algorithm|0TRNT4X7J0]].

## Remarks

Take $p = 2$ and $f = x^3 + x + 1$, which has no root in $\F_2$ and so is irreducible.
Then $(x^2 + 1)(x + 1) = x^3 + x^2 + x + 1$, and its remainder modulo $f$ is $x^2$.
The inverse of $x$ is $x^2 + 1$, since $x(x^2 + 1) = x^3 + x = f + 1$.
The polynomial $x^2 + 1$ over $\F_2$ cannot serve as $f$, since $x^2 + 1 = (x + 1)^2$, and then $x + 1$ has no inverse.
Over $\F_2$, an element is stored as the bit string of its coefficients, and addition is the bitwise exclusive or.
The inverse can also be computed as $a^{p^n - 2}$, since $a^{p^n - 1} = 1$ for $a \ne 0$.
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# Multidimensional Mellin transform

## Summary

The multidimensional Mellin transform takes a function on the open orthant $(0, \infty)^n$ to a function of $n$ complex variables, by integrating against $x_1^{s_1 - 1} \cdots x_n^{s_n - 1}$.
It is the [[Mellin transform|N5PVGRJE1T]] applied in each variable in turn.
It turns dilations in each variable into multiplication by powers, and it is used for the asymptotics of multiple integrals and sums.

## Explanations

### L1 human

The one-variable Mellin transform measures how a function behaves under stretching of its variable.
In several variables each variable may be stretched separately, and one complex exponent is attached to each.
For example, $e^{-x_1 - x_2}$ has the two-variable Mellin transform $\Gamma(s_1)\Gamma(s_2)$, the product of two gamma functions.
The logarithmic change of variables $x_j = e^{-t_j}$ turns it into a multidimensional Laplace or Fourier transform.
</document>

<document source="https://tamnd.github.io/math2/node/92RQG73SC5.md" tag="92RQG73SC5" kind="concept">
# Dynkin quiver

## Summary

A Dynkin quiver is a [[quiver|RE9YFFS3TS]] whose underlying undirected graph is one of the simply laced Dynkin diagrams $A_n$, $D_n$, $E_6$, $E_7$ or $E_8$.
By [[Gabriel's theorem|FC78F2E87E]], these are exactly the connected quivers with only finitely many indecomposable representations up to isomorphism.
The orientation of the arrows does not matter for this property.

## Explanations

### L1 human

Forget the directions of the arrows of a quiver and look at the graph that remains.
If that graph is a line, a line with a fork at one end, or one of three special trees with a long tail, the quiver is a Dynkin quiver.
For example, $1 \to 2 \leftarrow 3$ is a Dynkin quiver of type $A_3$, since its graph is a line with three vertices.
Dynkin quivers are the quivers whose representations can be listed completely.
The same diagrams classify simple Lie algebras, which is a striking coincidence explained by root systems.
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<document source="https://tamnd.github.io/math2/node/92SQ10PN6J.md" tag="92SQ10PN6J" kind="concept">
# Pseudo algebraically closed field

## Summary

A field $K$ is pseudo algebraically closed, or PAC, if every absolutely irreducible [[algebraic variety|HV6YBS0YDH]] defined over $K$ has a point with coordinates in $K$.
[[Algebraically closed fields|0PF6TM8TWT]] are PAC, but many PAC fields are far from algebraically closed.
PAC fields are a central notion of field arithmetic, and their theory links Galois theory with model theory through the [[absolute Galois group|PGKWBJDTKX]].

## Explanations

### L1 human

An algebraically closed field contains a root of every polynomial equation in one unknown.
A PAC field asks for something different.
Every system of polynomial equations in several unknowns whose solution set is geometrically "one piece" must have a solution in the field.
The field of rational numbers is not PAC, since $x^2 + y^2 = -1$ has no rational solution although its solution set over $\C$ is one irreducible curve.
Examples of PAC fields include the infinite algebraic extensions of a finite field.
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<document source="https://tamnd.github.io/math2/node/92SWXST1ZM.md" tag="92SWXST1ZM" kind="concept">
# Theory

## Summary

A theory in a first-order language is a set of sentences of that language.
Its sentences play the role of axioms, such as the axioms of groups or of fields.
Theories are the syntactic side of model theory, and the structures in which all their sentences hold are their models.

## Explanations

### L1 human

A theory is a list of rules written in a formal language.
For example, the three group axioms, written with the symbols $\cdot$, $e$ and ${}^{-1}$, form the theory of groups.
A theory does not say which objects satisfy the rules.
The objects that satisfy all the rules, here all groups, are the models of the theory.
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<document source="https://tamnd.github.io/math2/node/9328WEVJEK.md" tag="9328WEVJEK" kind="concept">
# Quasi-category

## Summary

A quasi-category is a [[simplicial set|WW81ARB8K9]] in which every inner horn has a filler.
Quasi-categories are the most used model of [[(∞,1)-categories|YFY1FWA1MY]], developed by Joyal and Lurie.
They include both ordinary categories, through their [[nerves|RV0MKJ3CQ6]], and [[Kan complexes|FDFYCM3YYC]], which model $\infty$-groupoids.
Much of ordinary category theory, including limits, adjunctions and the Yoneda lemma, extends to quasi-categories.

## Explanations

### L1 human

A simplicial set has points, arrows between points, triangles, tetrahedra and so on.
Think of points as objects and arrows as morphisms; a triangle with edges $f$, $g$ and $h$ says that $h$ is a composite of $f$ and $g$.
In a quasi-category, any two arrows that meet end to start can be composed: there is a triangle that fills them in.
The composite is not unique, but any two choices are linked by higher simplices, so it is unique up to a contractible space of choices.
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<document source="https://tamnd.github.io/math2/node/93343FN5PF.md" tag="93343FN5PF" kind="statement">
# Feuerbach's theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane that is not equilateral.
Then its [[nine-point circle|033H8MQRC3]] is internally tangent to its [[incircle|07DFG5JYPT]].
It is also externally tangent to each of its three excircles.
An excircle is a circle outside the triangle tangent to one side and to the extensions of the other two sides.

## Remarks

Let $R$ be the circumradius, $r$ the inradius, and $N$ and $I$ the centres of the nine-point circle and of the incircle.
The nine-point circle has radius $R/2$, and $N$ is the midpoint of the circumcentre $O$ and the [[orthocentre|0WB619CA0V]] $H$.
Write $a, b, c$ for the side lengths, and $A, B, C$ also for the angles.
Two known formulas give $OH^2 = 9R^2 - (a^2 + b^2 + c^2)$ and $IH^2 = 2r^2 - 4R^2 \cos A \cos B \cos C$.
With these and [[Euler's theorem|RFCQT6TKZF]] for $OI$, [[Apollonius's theorem|0NAKHQ9XYY]] in the triangle $OIH$ gives $IN = R/2 - r$.
This says the two circles are internally tangent; a similar computation gives $I_a N = R/2 + r_a$ for each excentre $I_a$ with exradius $r_a$.

In an equilateral triangle the nine-point circle and the incircle coincide, since $N = I$ and $R/2 = r$.
The point of tangency with the incircle is the Feuerbach point of the triangle.

## Explanations

### L1 human

Every triangle has a special circle through nine notable points, the nine-point circle.
It also has an inscribed circle and three circles that touch its sides from outside.
Feuerbach's theorem says the nine-point circle touches all four of these circles.
This is a surprising coincidence that holds for every triangle.

### L2 human

The theorem is a highlight of classical triangle geometry, published by Karl Wilhelm Feuerbach in 1822.
It ties together the [[circumcircle|03VZJE6C8C]], the incircle and the nine-point circle through the radii $R$, $r$ and $R/2$.
</document>

<document source="https://tamnd.github.io/math2/node/938SXZQS8G.md" tag="938SXZQS8G" kind="concept">
# Dual Radon transform

## Summary

The dual Radon transform, or back-projection, acts on functions $g$ on the set of hyperplanes.
Its value at a point $x$ of $\R^n$ is the integral of $g$ over all hyperplanes through $x$.
It is the formal adjoint of the [[Radon transform|VJNZJ352PC]] for the natural measures.
It is not the inverse of the Radon transform, but the [[Radon inversion formula|HN93ST5CBT]] writes the inverse as the dual transform composed with a filter.
In tomography this is the back-projection step of [[filtered back-projection|SJQDDY9FMX]].

## Explanations

### L1 human

Each line measurement of a scanner says how much material lies along that line, but not where on the line.
Back-projection smears each measured value back uniformly along its line.
Adding the smears of all lines through a point gives the value of the dual transform there.
For example, if every line carries the value $1$, then every point receives the same total.
The resulting picture is a blurred version of the true one, and a filter removes the blur.
</document>

<document source="https://tamnd.github.io/math2/node/93FKWZP2AZ.md" tag="93FKWZP2AZ" kind="definition">
# Fixed-point property

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
$X$ has the *fixed-point property* if for every [[continuous map|0X3WGX95FE]] $f \colon X \to X$ there is a point $x \in X$ with $f(x) = x$.

## Remarks

Example: the interval $[0, 1]$ has the fixed-point property.
For continuous $f \colon [0, 1] \to [0, 1]$, the function $g(x) = f(x) - x$ satisfies $g(0) \ge 0$ and $g(1) \le 0$, so $g$ has a zero by the [[intermediate value theorem|00KAFBR32E]].
Non-example: the [[circle|060W5CBZP6]] $S^1 \subseteq \C$ fails it, because the rotation $z \mapsto -z$ is continuous and has no fixed point.
If $X$ has the fixed-point property and $Y$ is homeomorphic to $X$, then $Y$ has it as well.
A [[retract|68PFSGP39B]] of a space with the fixed-point property also has it.
The [[Brouwer fixed-point theorem|N7AAJ2MW5T]] states that the closed unit ball of $\R^n$ has the fixed-point property.
</document>

<document source="https://tamnd.github.io/math2/node/93GK2SPBKV.md" tag="93GK2SPBKV" kind="concept">
# Nelson–Aalen estimator

## Summary

The Nelson–Aalen estimator is a nonparametric estimator of the cumulative [[hazard function|Y959NKQ3XW]] from right-[[censored|MY8ZZP84NP]] data.
At each observed event time it adds the number of events divided by the number of subjects at risk.
It is the natural estimator in the counting process approach to survival analysis, where its large-sample theory follows from martingale results.
Taking $e^{-\hat H(t)}$ gives an estimator of the survival function close to the [[Kaplan–Meier estimator|B7AJ8RWZRC]].

## Explanations

### L1 human

The cumulative hazard adds up the risk of the event over time.
At each moment when someone has the event, the Nelson–Aalen estimator adds "events now divided by people still at risk".
If one person of four dies, it adds $1/4$; if later one person of two dies, it adds $1/2$.
The total grows faster when events are frequent among those still at risk.
Plotting it shows whether the hazard is constant (a straight line) or changes with time.
</document>

<document source="https://tamnd.github.io/math2/node/93MFWKENQ7.md" tag="93MFWKENQ7" kind="definition">
# Global analysis

## Definition

The setting is the category of [[smooth manifolds|F45VQ47GPV]], smooth maps between them and smooth [[vector bundles|P4DZM975BV]] over them.
*Global analysis* is the branch of mathematics that studies, in this setting, the following objects.

1. [[Differential equations|GC5BDZ6QBC]] and differential operators whose unknowns are maps between manifolds or sections of bundles.
2. Variational problems on such maps and sections, and their critical points.
3. Spaces of maps between manifolds, seen as infinite-dimensional manifolds.

Its results concern the whole manifold, as opposed to results that hold in one coordinate chart.

## Remarks

The Hodge theorem is a result of global analysis.
It says that on a compact oriented Riemannian manifold each de Rham cohomology class has exactly one harmonic representative.
This is a statement about a differential operator, the Hodge Laplacian, and about the topology of the whole manifold.

A result that is not global is the local existence of solutions of an ordinary differential equation near a point.
It holds in one chart and says nothing about the whole manifold.

The name is also used for the analysis on infinite-dimensional manifolds of maps, which grew out of the same questions.
</document>

<document source="https://tamnd.github.io/math2/node/93RQRG9R1P.md" tag="93RQRG9R1P" kind="statement">
# Serre's theorem on regular local rings

## Statement

Let $(R, \mathfrak{m}, k)$ be a commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] with [[residue field|NJ3APQ8BHH]] $k$.
The following are equivalent:

1. $R$ is a [[regular local ring|63XXTF6B5M]].
2. $R$ has finite [[global dimension|GWA0T1WKCN]].
3. The [[projective dimension|MR9QCH1PNA]] $\operatorname{pd}_R k$ is finite.

When they hold, $\operatorname{gl.dim} R = \operatorname{pd}_R k = \dim R$, where $\dim R$ is the [[Krull dimension|MVGBS3J1P7]].

## Remarks

If $R$ is regular, then $\mathfrak{m}$ is generated by a regular sequence of length $\dim R$, and its [[Koszul complex|YZY11W08SD]] is a free resolution of $k$ of that length.
For a Noetherian local ring, the global dimension equals $\operatorname{pd}_R k$, which gives the implication from 1 to 2.
The converse is the deep part: a finite resolution of $k$ forces the minimal number of generators of $\mathfrak{m}$ to equal $\dim R$.
Example: $R = K[x]/(x^2)$ is not regular, and indeed $\cdots \to R \xrightarrow{x} R \xrightarrow{x} R \to K \to 0$ is a minimal resolution of $K$ that never stops.
A major consequence is that every localization of a regular local ring at a prime ideal is again regular.

## Explanations

### L1 human

Regular local rings are the algebraic version of smooth points of a space.
The theorem says that smoothness can be detected by homological algebra alone: a local ring is regular exactly when every module has a finite resolution.
It is enough to test one module, the residue field.

### L2 human

The theorem was proved by Serre and, independently, by Auslander and Buchsbaum.
The homological criterion gives a short proof that a localization of a regular local ring is regular, since finite global dimension passes to localizations.
It also gives [[Hilbert's syzygy theorem|R0DSZPRG6T]] in the local setting, and, combined with the [[Auslander–Buchsbaum formula|SQB74J5E8Q]], it controls the resolutions of all finitely generated modules over a regular local ring.
</document>

<document source="https://tamnd.github.io/math2/node/93T4M1R5CT.md" tag="93T4M1R5CT" kind="definition">
# Initial and terminal objects

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
An object $I$ of $\mathcal{C}$ is *initial* if for every object $X$ of $\mathcal{C}$ there is exactly one [[morphism|859F0TFVA3]] $I \to X$.
An object $T$ of $\mathcal{C}$ is *terminal* if for every object $X$ of $\mathcal{C}$ there is exactly one morphism $X \to T$.

## Remarks

If $I$ and $I'$ are both initial, the unique morphisms $I \to I'$ and $I' \to I$ compose to the unique endomorphisms, which are the identities, so they are inverse isomorphisms.
So an initial object is unique up to unique isomorphism, and the same holds for terminal objects.
A terminal object of $\mathcal{C}$ is an initial object of $\mathcal{C}^{\mathrm{op}}$.
Common notations are $0$ or $\emptyset$ for an initial object and $1$ or $*$ for a terminal object.

Example: in the category of sets, the [[empty set|0ECVK59Y03]] is initial, since the empty function is the only function $\emptyset \to X$.
Every one-point set $\{*\}$ is terminal, since the constant map is the only function $X \to \{*\}$.
In the category of rings with identity, $\Z$ is initial and the zero ring is terminal.

Non-example: in the category of sets, the two-point set $\{0, 1\}$ is not terminal, since a non-empty set $X$ has $2^{\abs{X}} \ge 2$ functions to it.
The category of fields has no initial object, since a field homomorphism preserves the characteristic, so no field maps to both $\mathbb{F}_2$ and $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/93VXAB8JYS.md" tag="93VXAB8JYS" kind="statement">
# Period three implies chaos

## Statement

Let $J \subseteq \R$ be an [[interval|0Q1D85B8G6]] and let $F \colon J \to J$ be a [[continuous map|0VGFHTM75Y]].
Suppose there is a point $a \in J$ such that $b = F(a)$, $c = F^2(a)$ and $d = F^3(a)$ satisfy $d \le a < b < c$ or $d \ge a > b > c$.
Then:

1. For every integer $k \ge 1$ there is a [[periodic point|T75CZ5H47N]] in $J$ of minimal period $k$.
2. There is an uncountable set $S \subseteq J$ that contains no periodic point and satisfies:
   - for all $p, q \in S$ with $p \neq q$, $\limsup_{n \to \infty} \abs{F^n(p) - F^n(q)} > 0$ and $\liminf_{n \to \infty} \abs{F^n(p) - F^n(q)} = 0$;
   - for every $p \in S$ and every periodic point $q \in J$, $\limsup_{n \to \infty} \abs{F^n(p) - F^n(q)} > 0$.

## Remarks

This is the theorem of Li and Yorke, from their paper "Period three implies chaos".
A point of minimal period $3$ satisfies the hypothesis after a suitable choice of $a$ on its orbit: one of the three points can serve as $a$, and then $d = a$.
The proof of part 1 uses a covering form of the intermediate value theorem.
If $F(K) \supseteq L$ for compact intervals $K, L$, then $K$ contains a compact interval mapped onto $L$, and chains of such intervals give periodic points.
Part 1 is also a special case of [[Sharkovskii's theorem|YBCZ9VY7SX]], since $3$ is first in the Sharkovskii ordering.
Example: the logistic map $F(x) = 4x(1 - x)$ on $[0, 1]$ has the period-$3$ orbit of $\sin^2(\pi/7)$, $\sin^2(2\pi/7)$, $\sin^2(4\pi/7)$, so it has points of all periods.
The interval hypothesis is necessary: the rotation of a circle by one third has period $3$ points but no fixed point.

## Explanations

### L1 human

For a continuous map of a segment of the line, one orbit of period three is enough to produce orbits of every period.
It also produces an uncountable set of orbits that wander forever without settling, coming close to each other and moving apart again and again.
The title of the paper of Li and Yorke made "chaos" a common word in mathematics.

### L2 human

The result gives an easy test for complicated dynamics in one-dimensional maps: find one point with the stated order pattern.
The set $S$ is called a scrambled set, and maps with an uncountable scrambled set are called chaotic in the sense of Li and Yorke.
The theorem says nothing about how large $S$ is in measure; for some maps with period three, almost every orbit tends to a stable periodic orbit.
</document>

<document source="https://tamnd.github.io/math2/node/93WX6PZC8R.md" tag="93WX6PZC8R" kind="statement">
# Aoki–Rassias theorem

## Statement

Let $E_1$ be a real [[normed vector space|03J5RAKB30]] and let $E_2$ be a real [[Banach space|06GZ956657]].
Let $\theta \ge 0$ and $0 \le p < 1$.
Let $f \colon E_1 \to E_2$ satisfy
$$\norm{f(x + y) - f(x) - f(y)} \le \theta \left( \norm{x}^p + \norm{y}^p \right) \quad \text{for all } x, y \in E_1.$$
Then there is a unique [[additive map|KF719SZD5T]] $A \colon E_1 \to E_2$ such that
$$\norm{f(x) - A(x)} \le \frac{2\theta}{2 - 2^p} \norm{x}^p \quad \text{for all } x \in E_1,$$
where $\norm{0}^0$ is read as $1$.
It is given by $A(x) = \lim_{n \to \infty} 2^{-n} f(2^n x)$.
If moreover, for each $x \in E_1$, the map $t \mapsto f(tx)$ from $\R$ to $E_2$ is continuous, then $A$ is [[linear|0NRYHJPQG2]].

## Remarks

The proof follows [[Hyers' theorem|VFRJ0P2P15]].
Setting $x = y$ gives $\norm{f(2x) - 2f(x)} \le 2\theta \norm{x}^p$, so consecutive terms of $2^{-n} f(2^n x)$ differ by at most $\theta \, 2^{n(p-1)} \norm{x}^p$.
Because $p < 1$, these bounds form a convergent geometric series with sum $\frac{2\theta}{2 - 2^p}\norm{x}^p$.
For $p = 0$ the hypothesis is a constant bound $2\theta$, and the theorem gives back Hyers' theorem with $\delta = 2\theta$.
Example: $f(x) = x + \sqrt{\abs{x}}$ on $\R$ satisfies the hypothesis with $\theta = 1$ and $p = 1/2$, because $0 \le \sqrt{\abs{x}} + \sqrt{\abs{y}} - \sqrt{\abs{x + y}} \le \sqrt{\abs{x}} + \sqrt{\abs{y}}$; here $A(x) = x$.
The hypothesis $p < 1$ cannot be dropped: Z. Gajda showed that the statement fails for $p = 1$, already for maps $\R \to \R$.
T. Aoki proved the additive case in 1950, and Th. M. Rassias proved the linear case in 1978.

## Explanations

### L1 human

Hyers' theorem allows a fixed error in the rule $f(x + y) = f(x) + f(y)$.
This theorem allows an error that grows with the size of $x$ and $y$, but more slowly than linearly.
The conclusion is the same in spirit: $f$ is close to an exact additive map, with a distance that grows like $\norm{x}^p$.

### L2 human

The theorem started the study of stability with a control function, often called Hyers–Ulam–Rassias stability.
The same rescaling argument works for $p > 1$ with $A(x) = \lim_n 2^n f(2^{-n} x)$, and the excluded case $p = 1$ marks the boundary where the geometric series diverges.
Similar results hold for the Jensen, quadratic and many other [[functional equations|SRAN4EKWS0]].
</document>

<document source="https://tamnd.github.io/math2/node/9418TGREZX.md" tag="9418TGREZX" kind="statement">
# Suslin's theorem

## Statement

Let $X$ be a [[Polish space|584GXXY73M]] and let $A \subseteq X$.
Then $A$ belongs to the [[Borel sigma-algebra|0WQXY32Q8D]] of $X$ if and only if both $A$ and its [[complement|0RKF1ZT5JJ]] $X \setminus A$ are [[analytic sets|3RWWZYPAFV]].

## Remarks

One direction is easy: every Borel set is analytic, and the complement of a Borel set is Borel.
For the other direction, the Lusin separation theorem says that two disjoint analytic sets $A$ and $B$ can be separated by a Borel set $C$, that is, $A \subseteq C$ and $C \cap B = \emptyset$.
Applied to $A$ and $X \setminus A$, it gives a Borel set $C$ with $A \subseteq C \subseteq A$, so $C = A$.

The hypothesis on the complement cannot be dropped.
Every uncountable Polish space contains an analytic set that is not Borel, and the complement of such a set is not analytic.

In the notation of the projective hierarchy, the theorem says $\boldsymbol{\Delta}^1_1 = \boldsymbol{\Sigma}^1_1 \cap \boldsymbol{\Pi}^1_1$ is exactly the class of Borel sets.

## Explanations

### L1 human

Analytic sets are the images of Borel sets under continuous maps, and they can be more complicated than Borel sets.
The theorem says when an analytic set is Borel after all: exactly when its complement is analytic too.
So "Borel" means "simple to describe from both sides".

### L2 human

The theorem is the standard way to prove that a set is Borel without writing it explicitly from open sets by countable unions and complements.
One shows that the set and its complement are both projections of Borel sets, and the theorem does the rest.
A common use is the result that a bijective Borel map between Polish spaces has a Borel inverse.
</document>

<document source="https://tamnd.github.io/math2/node/942BKZYGQS.md" tag="942BKZYGQS" kind="definition">
# P-adic analysis

## Definition

Let $p$ be a prime number.
Let $\Q_p$ be the field of [[$p$-adic numbers|BT6T702C28]], the completion of $\Q$ for the [[$p$-adic absolute value|WGN82TCEF2]] $\abs{x}_p = p^{-v_p(x)}$, where $v_p$ is the [[$p$-adic valuation|03HDF50VN4]].
*$p$-adic analysis* is the study of functions defined on subsets of $\Q_p$ or of a complete field extension of $\Q_p$.
The values lie in such a field or in $\R$ or $\C$.
It treats limits, series, continuity, differentiability, power series and integration for the topology given by $\abs{\cdot}_p$.

## Remarks

The absolute value $\abs{\cdot}_p$ satisfies the strong triangle inequality $\abs{x + y}_p \le \max(\abs{x}_p, \abs{y}_p)$.
Hence a series $\sum_n a_n$ in $\Q_p$ converges if and only if $a_n \to 0$.

Example: $\abs{p^n}_p = p^{-n} \to 0$, so $\sum_{n \ge 0} p^n$ converges in $\Q_p$, and its sum is $\frac{1}{1 - p}$, since $(1 - p) \sum_{n=0}^{N} p^n = 1 - p^{N+1} \to 1$.

Non-example: the same series $\sum_{n \ge 0} p^n$ diverges for the usual absolute value on $\R$; its study is part of real analysis, not of $p$-adic analysis.

Two typical facts.
The ring $\Z_p$ of [[$p$-adic integers|V6TJAFZ4MD]] is compact and totally disconnected.
The exponential series $\sum_n x^n / n!$ converges in $\Q_p$ exactly when $\abs{x}_p < p^{-1/(p-1)}$.
</document>

<document source="https://tamnd.github.io/math2/node/942FK83S09.md" tag="942FK83S09" kind="statement">
# Finitely generated projective modules over local rings are free

## Statement

Let $R$ be a [[local ring|MZ02BKPHJA]] with maximal ideal $\mathfrak{m}$ and [[residue field|NJ3APQ8BHH]] $k = R/\mathfrak{m}$.
Let $P$ be a [[finitely generated|05QY1VEBYD]] [[projective|BFFA03GE3V]] $R$-module.
Then $P$ is [[free|0DZ9DKAKZR]], of rank $n = \dim_k P/\mathfrak{m}P$.

## Remarks

Choose $x_1, \ldots, x_n \in P$ whose classes form a basis of the $k$-vector space $P/\mathfrak{m}P$.
By [[Nakayama's lemma|15JHBYR6KH]], they generate $P$, so there is a surjection $\pi \colon R^n \to P$; it splits because $P$ is projective, so $R^n \cong P \oplus K$ with $K = \ker \pi$.
Reducing modulo $\mathfrak{m}$ gives $k^n \cong k^n \oplus K/\mathfrak{m}K$, so $K/\mathfrak{m}K = 0$.
As a direct summand of $R^n$, $K$ is finitely generated, and Nakayama's lemma gives $K = 0$; thus $P \cong R^n$.

The local hypothesis is necessary.
Over $R = \Z/6\Z$, the module $\Z/2\Z$ is a direct summand of $R$, hence projective, but it has $2$ elements, while every finitely generated free $R$-module has $6^n$ elements for some $n \ge 0$.
Kaplansky's theorem removes the finiteness hypothesis: every projective module over a local ring is free.
More generally, a finitely presented flat module over a local ring is free.

## Explanations

### L1 human

Projective modules are pieces of free modules, and in general they can be twisted.
Over a local ring there is no room for twisting: every finitely generated projective module has a basis.
Its size is read off from the vector space you get by reducing modulo the maximal ideal.

### L2 human

This is why finitely generated projective modules over any commutative ring are called locally free: their localizations at prime ideals are free.
The rank at each prime gives a locally constant function on the spectrum, and rank one projectives form the [[Picard group|DZW58R6T2Q]].
Geometrically, finitely generated projective modules are [[vector bundles|P4DZM975BV]] on $\operatorname{Spec} R$, and the theorem says that a vector bundle is trivial near each point.
Whether such modules are globally free over polynomial rings is the content of the [[Quillen–Suslin theorem|7N50GA46W1]].
</document>

<document source="https://tamnd.github.io/math2/node/948A39DZDE.md" tag="948A39DZDE" kind="statement">
# Adaptive quadrature

## Statement

Let $f \colon [a, b] \to \R$ be integrable and let $\tau > 0$ be a tolerance.
For each closed subinterval $J$ of $[a, b]$, let $A(J)$ be a computable approximation of $I(J) = \int_J f(x) \, dx$, and let $E(J) \ge 0$ be a computable error estimate.
The *adaptive quadrature* procedure $\mathrm{ADAPT}(J, \tau_J)$ returns $A(J)$ if $E(J) \le \tau_J$; otherwise it splits $J$ at its midpoint into $J_1$ and $J_2$ and returns $\mathrm{ADAPT}(J_1, \tau_J / 2) + \mathrm{ADAPT}(J_2, \tau_J / 2)$.
Suppose that $\mathrm{ADAPT}([a, b], \tau)$ terminates, and let $J_1, \ldots, J_m$ be the intervals at which it accepted $A(J_i)$ with tolerance $\tau_i$.
Then:

1. the intervals $J_1, \ldots, J_m$ cover $[a, b]$ and overlap only at endpoints, and $\tau_1 + \cdots + \tau_m = \tau$;
2. if $\abs{I(J_i) - A(J_i)} \le E(J_i)$ for each $i$, then the result $S = \sum_{i=1}^m A(J_i)$ satisfies $\abs{\int_a^b f(x)\,dx - S} \le \tau$.

## Remarks

Clause 1 holds by induction on the recursion: a split replaces one interval with tolerance $\tau_J$ by its two halves with tolerances adding up to $\tau_J$.
Clause 2 then follows from the triangle inequality, since $\abs{\int_a^b f - S} \le \sum_i \abs{I(J_i) - A(J_i)} \le \sum_i \tau_i$.
The usual example is adaptive [[Simpson's rule|MNXC26G3BQ]]: with $S(J)$ the Simpson value on $J$, one takes $A(J) = S(J_1) + S(J_2)$ and $E(J) = \abs{S(J_1) + S(J_2) - S(J)}/15$.
This estimate is justified for $f \in C^4$ and small $J$: the Simpson error on an interval of length $\ell$ is $-\frac{\ell^5}{2880} f^{(4)}(\xi)$, so halving the interval divides the error by about $16$.
The hypothesis of clause 2 is essential, and the estimate $E$ can be fooled.
For $f(x) = \big(x (x - \tfrac14)(x - \tfrac12)(x - \tfrac34)(x - 1)\big)^2$ on $[0, 1]$, all the Simpson sample points of the first test are zeros of $f$, so $E([0,1]) = 0$ and the procedure returns $0$, while $\int_0^1 f > 0$.
Real programs also bound the depth of the recursion, since termination is not guaranteed for every $f$.

## Explanations

### L1 human

An adaptive integrator spends its effort where the function is hard.
It estimates the error on an interval; if the error is too big, it cuts the interval in half and treats each half with half of the allowed error.
Smooth regions are finished after a few steps, and only the difficult spots get cut many times.

### L2 human

Compared with a fixed composite rule, adaptivity can save many function evaluations when $f$ has a peak, a kink or an endpoint singularity.
The error estimate comes from comparing two rules on the same interval, such as Simpson's rule on one and on two panels, or a Gauss rule with its [[Gauss–Kronrod extension|E8QJJVC1KV]].
The guarantee is only as good as the estimate: it is a heuristic, not a proof, for a function known only through samples.
Global variants keep a list of all intervals and always split the one with the largest estimated error.
</document>

<document source="https://tamnd.github.io/math2/node/94AKWRHND0.md" tag="94AKWRHND0" kind="statement">
# Pascal's theorem

## Statement

Let $k$ be a [[field|04C5ZBG1WT]], and work in the [[projective plane|DANFNN8C4Y]] $\mathbb{P}^2(k)$.
Let $\mathcal{C}$ be a nondegenerate [[conic|00HJCMC2CD]]: the zero set of a homogeneous quadratic polynomial $F$ in three variables.
Here $F$ is not a product of linear factors over any extension of $k$.
Let $A, B, C, D, E, F$ be six distinct points of $\mathcal{C}$.
Then the three points
$$AB \cap DE, \quad BC \cap EF, \quad CD \cap FA$$
are [[collinear|0AFNNTRTJE]].

## Remarks

Here $AB$ denotes the line through $A$ and $B$.
No three points of a nondegenerate conic are collinear, so the lines in each intersection are distinct and the three points are well defined.
The line through them is the Pascal line of the hexagon $ABCDEF$.

One proof over $\C$, which extends to any field by passing to an algebraic closure, uses cubic curves.
The cubics $AB \cdot CD \cdot EF$ and $BC \cdot DE \cdot FA$, products of three lines each, meet in the six points and in the three points of the statement.
A suitable combination of them vanishes on a seventh point of $\mathcal{C}$, so by Bézout's theorem it contains $\mathcal{C}$, and the remaining factor is a line through the three points.

Example: on the unit circle $x^2 + y^2 = 1$ in the real plane, take a regular hexagon.
Opposite sides are then parallel, so the three points lie on the line at infinity.

When the conic degenerates into two lines, the statement becomes [[Pappus's hexagon theorem|B6C3J2GJWP]].
The dual statement is [[Brianchon's theorem|3THE3Q96KK]].

## Explanations

### L1 human

Pick six points on an ellipse, number them around the curve or in any order, and join them into a closed hexagon.
Extend the three pairs of opposite sides until each pair crosses.
The three crossing points always lie on one straight line.

### L2 human

Pascal's theorem is a basic tool for constructions with conics: given five points of a conic, it lets one construct any number of further points with a straightedge alone.
Its converse also holds: if the three points are collinear, the six points lie on a conic, which may be degenerate.
It is the projective form of a statement that, in the real plane, covers ellipses, parabolas and hyperbolas at once, since these are all the same in projective geometry.
Through [[duality|DCMP1A0138]] it gives [[Brianchon's theorem|3THE3Q96KK]].
</document>

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# Ionescu-Tulcea theorem

## Statement

For each $i \in \set{0, 1, 2, \ldots}$ let $(\Omega_i, \mathcal{F}_i)$ be a [[measurable space|N4KDT83V3S]], and for $n \ge 0$ put $\Omega^n = \Omega_0 \times \cdots \times \Omega_n$ with the [[product σ-algebra|6HVCTJGB61]].
Let $P_0$ be a [[probability measure|059TJN0NZ1]] on $\Omega_0$.
For each $n \ge 1$ let $\kappa_n$ be a [[Markov kernel|2AVNDP8BJP]] from $\Omega^{n-1}$ to $\Omega_n$.
Then there is a unique probability measure $P$ on $\prod_{i=0}^{\infty} \Omega_i$ with its product σ-algebra such that, for all $n \ge 0$ and all $A_i \in \mathcal{F}_i$,
$$P\Big( A_0 \times \cdots \times A_n \times \prod_{i > n} \Omega_i \Big) = \int_{A_0} \int_{A_1} \cdots \int_{A_n} \kappa_n(\omega_0, \ldots, \omega_{n-1}, d\omega_n) \cdots \kappa_1(\omega_0, d\omega_1) \, P_0(d\omega_0).$$

## Remarks

The right side defines a consistent family of probability measures $P^n$ on the spaces $\Omega^n$, so it gives a finitely additive set function on cylinder sets.
Countable additivity is proved by contradiction.
Suppose a decreasing sequence of cylinder sets has measures bounded below by $\varepsilon > 0$ and empty intersection.
Using the kernels, one chooses coordinates $\omega_0, \omega_1, \ldots$ one at a time and gets a point in all of the sets.
[[Carathéodory's extension theorem|Q8EXG7EKYW]] then gives $P$.

Example: let $\Omega_i = S$ for a measurable space $S$, let $P_0$ be an initial law, and let $\kappa_n(\omega_0, \ldots, \omega_{n-1}, \cdot) = K(\omega_{n-1}, \cdot)$ for one Markov kernel $K$ on $S$.
Then $P$ is the law of a Markov chain on $S$ with initial law $P_0$ and transition kernel $K$.

Unlike the [[Kolmogorov extension theorem|WR4X7MMTH2]], this theorem needs no topological assumption on the spaces $\Omega_i$.
The price is that the data must come as kernels, that is, as conditional laws given the past, and the index set is countable.

## Explanations

### L1 human

Think of a random process that runs step by step.
At each step, the law of the next value may depend on everything that happened before.
The theorem says that such step-by-step rules always define a single probability measure on the space of whole infinite histories.

### L2 human

The theorem is the general existence result for discrete-time processes, in particular Markov chains on general state spaces and controlled processes in dynamic programming.
With kernels that ignore the past it gives the countable [[infinite product measure|2HWVFHM13R]].
It complements the Kolmogorov extension theorem: Kolmogorov needs Polish spaces and handles any index set, while Ionescu-Tulcea needs kernels and handles arbitrary measurable spaces with a countable sequence of steps.
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<document source="https://tamnd.github.io/math2/node/94BWTKKGJM.md" tag="94BWTKKGJM" kind="concept">
# Morse homology

## Summary

Morse homology is the homology of a chain complex built from a [[Morse function|KPQ96640ZJ]] on a closed manifold.
The chains are generated by the critical points, graded by their index.
The boundary map counts gradient flow lines between critical points whose indices differ by one.
Its homology is isomorphic to the [[singular homology|M0Q5CFJ7ZH]] of the manifold, whatever Morse function and metric are used.
This point of view is the model for [[Floer homology|NCKFPAGHXF]] in infinite-dimensional settings.

## Explanations

### L1 human

Pour water on a landscape and watch it flow downhill from passes to pits, or from peaks to passes.
Morse homology records which critical points are joined by such flow lines and in how many ways.
From this purely local bookkeeping one recovers the global holes of the space.
For example, on an upright torus there are two flow lines from the top down to each saddle, and they cancel in pairs.
The resulting homology has one generator in degree $0$, two in degree $1$ and one in degree $2$, as it should.
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<document source="https://tamnd.github.io/math2/node/94GSMEWPTN.md" tag="94GSMEWPTN" kind="problem">
# Gauss variational problem

## Problem

Let $n \ge 3$, let $K \subseteq \R^n$ be a [[compact set|0R6EWFKRCF]] of positive [[capacity|4YZVFGJVSZ]], and let $f \colon K \to \R$ be a continuous function.
For a positive Borel measure $\mu$ on $K$, let $U^\mu(x) = \int_K \abs{x - y}^{2-n} \, d\mu(y)$ be its [[Newtonian potential|DCMFQ01QGS]].
The *Gauss integral* of $\mu$ is
$$G_f(\mu) = \int_K \left( U^\mu - 2 f \right) d\mu,$$
that is, the [[energy|SFSRQCM3EG]] of $\mu$ minus twice the integral of $f$.
The problem asks for a positive Borel measure $\mu$ on $K$ with $\mu(K) = 1$ that makes $G_f(\mu)$ as small as possible.

## Remarks

Under these hypotheses a minimizing measure exists and is unique.
Existence follows from the weak-star compactness of the probability measures on $K$ and the lower semicontinuity of the energy.
Uniqueness follows from the strict positive definiteness of the Newtonian energy on signed measures of finite energy.
For $f = 0$ the problem is to minimize the energy among unit measures on $K$, and its solution is the [[equilibrium measure|464NN2K4FZ]] of $K$.
The minimizer $\mu$ is characterized by two inequalities with one constant $c$: $U^\mu - f \ge c$ on $K$ outside a set of capacity zero, and $U^\mu - f \le c$ on the support of $\mu$.
Gauss treated the problem for forces that follow the inverse square law.
Rigorous proofs of existence rely on the measure theory that was developed later.

## Explanations

### L1 human

Put a fixed amount of charge on a conductor $K$ that also feels an outside electric field $f$.
The charges repel each other but are pulled by the field.
The Gauss problem asks where the charge ends up when the total energy is as small as possible.
Without an outside field the answer is the equilibrium distribution, whose total energy measures the capacity of $K$.
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<document source="https://tamnd.github.io/math2/node/94GVY3APPQ.md" tag="94GVY3APPQ" kind="concept">
# Quasi-identity

## Summary

A quasi-identity is a law of the form "if these finitely many equations between terms hold, then this further equation holds".
It is required for all values of the variables.
It generalizes an [[identity|PKD440G2XW]], which is the case with no hypotheses.
Quasi-identities are exactly the axioms of [[quasivarieties|WRXEKT4NQS]], just as identities are the axioms of varieties.
They are the strict universal [[Horn clauses|QR92EZDTKF]] of an algebraic language.

## Explanations

### L1 human

An identity such as $x \cdot y = y \cdot x$ must hold for every choice of $x$ and $y$.
A quasi-identity only demands its conclusion when some equations already hold, like a conditional rule.
The cancellation law "if $x z = y z$ then $x = y$" is a typical example.
It holds in every group, since one can multiply by $z^{-1}$ on the right.
It fails in the integers under multiplication, because $1 \cdot 0 = 2 \cdot 0$ while $1 \ne 2$.
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<document source="https://tamnd.github.io/math2/node/94M0BPV1W4.md" tag="94M0BPV1W4" kind="statement">
# Krull–Schmidt theorem

## Statement

Let $R$ be a ring and let $M$ be a nonzero left [[$R$-module|09VDS2W1E8]] of [[finite length|V6JBKT168T]].

1. $M$ is a direct sum $M = M_1 \oplus \cdots \oplus M_m$ of finitely many [[indecomposable|E2P29GCD9T]] submodules.
2. If also $M = N_1 \oplus \cdots \oplus N_n$ with each $N_j$ indecomposable, then $m = n$ and there is a permutation $\sigma$ of $\set{1, \dots, m}$ with $M_i \cong N_{\sigma(i)}$ for all $i$.

## Remarks

Existence follows by induction on the length: a module that is not indecomposable splits into two summands of smaller length.
For uniqueness, [[Fitting's lemma|F52B2JMEHY]] shows that the endomorphism ring of each indecomposable summand is local.
An exchange argument then matches $M_1$ with some $N_j$ and replaces one by the other, and induction on $m$ finishes.
Example: a finite abelian group is a $\Z$-module of finite length, and its decomposition into cyclic groups of prime power order is unique up to order and isomorphism.
Uniqueness can fail without finite length: in the ring of integers $\mathcal{O}$ of a number field with nontrivial class group, a non-principal ideal $I$ satisfies $I \oplus I^{-1} \cong \mathcal{O} \oplus \mathcal{O}$, though $I \not\cong \mathcal{O}$.

## Explanations

### L1 human

A module of finite length breaks into pieces that cannot be broken further.
The theorem says that the list of pieces is determined by the module, up to reordering and isomorphism, just as the prime factors of an integer are.

### L2 human

The theorem reduces the study of modules of finite length to that of indecomposable modules.
For finite-dimensional algebras over a field, every finite-dimensional module has finite length, so the theorem applies; this is the base of representation theory of quivers and algebras.
The same proof works in any additive category in which idempotents split and endomorphism rings of indecomposable objects are local.
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<document source="https://tamnd.github.io/math2/node/94SXH8EDPS.md" tag="94SXH8EDPS" kind="definition">
# Cartan matrix of an algebra

## Definition

Let $K$ be a field and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $K$.
Let $P_1, \dots, P_n$ be representatives of the isomorphism classes of [[indecomposable|E2P29GCD9T]] [[projective|BFFA03GE3V]] left $A$-modules.
Let $S_i = P_i / JP_i$, where $J$ is the [[Jacobson radical|XA0DCNH7WY]] of $A$; then $S_1, \dots, S_n$ represent the isomorphism classes of [[simple|G00C22AEDH]] left $A$-modules.
The *Cartan matrix* of $A$ is the $n \times n$ integer matrix $C = (c_{ij})$.
Here $c_{ij}$ is the number of times $S_j$ occurs among the [[composition factors|0PHTC93M58]] of $P_i$.

## Remarks

Example: $A = K[x]/(x^2)$ has the single indecomposable projective $P_1 = A$, with composition series $0 \subset (x) \subset A$.
Both factors are the simple module $K$, so $C = (2)$.
Example: for the algebra $T_2(K)$ of upper triangular $2 \times 2$ matrices, $P_1 = K E_{11}$ is simple and $P_2 = K E_{12} + K E_{22}$ has the factors $S_2$ on top and $S_1$ below, so $C = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$.
Non-example: the polynomial ring $K[x]$ is not finite-dimensional, and its projective module $K[x]$ has no composition series, so it has no Cartan matrix in this sense.
Some authors use the transpose of $C$.
If $K$ is a splitting field for $A$, for example algebraically closed, then $c_{ij} = \dim_K \Hom_A(P_j, P_i)$.
For the group algebra $KG$ of a finite group over a splitting field of characteristic $p$, Brauer showed that $C = D^{T} D$, where $D$ is the decomposition matrix, so $C$ is symmetric.
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<document source="https://tamnd.github.io/math2/node/94TNBAAJYX.md" tag="94TNBAAJYX" kind="concept">
# Weakly singular kernel

## Summary

A weakly singular kernel is a [[kernel|GSMBXT7FPK]] that may become infinite on the diagonal $x = s$.
It may grow there only like a power $\abs{x - s}^{-\alpha}$ with exponent $\alpha$ below the dimension.
Such a singularity is integrable, so the integral operator still makes sense on continuous functions and is compact there.
So the Fredholm theory for continuous kernels carries over.
Kernels of this type arise in potential theory and in Abel's integral equation.

## Explanations

### L1 human

The kernel blows up where the two points meet, but slowly enough that the area under it stays finite.
On an interval this means a blow-up like $1/\abs{x - s}^{\alpha}$ with $\alpha < 1$.
For example, $1/\sqrt{\abs{x - s}}$ and $\log \abs{x - s}$ are weakly singular on $[0, 1]$.
The kernel $1/(x - s)$ is not, because $\int_0^1 ds / \abs{x - s}$ is infinite.
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<document source="https://tamnd.github.io/math2/node/9534R6VMY0.md" tag="9534R6VMY0" kind="concept">
# Weak-type estimate

## Summary

A weak-type estimate bounds the measure of the set where $\abs{Tf}$ is larger than $\lambda$, instead of bounding a norm of $Tf$.
An operator $T$ is of weak type $(p, q)$ if this measure is at most $(C\norm{f}_p/\lambda)^q$.
Equivalently, $T$ maps $L^p$ boundedly into the [[weak Lp space|H4C888M5AZ]] $L^{q,\infty}$.
Such estimates are the natural substitute for $L^1$ bounds, which often fail, for maximal functions and singular integrals.
By the [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]], two weak-type estimates give strong $L^p$ bounds for all exponents in between.

## Explanations

### L1 human

Bounding $\int \abs{Tf}$ asks for control of the total size of $Tf$.
A weak-type bound asks for less: it only controls how much room the large values of $Tf$ take up.
For example, the function $1/x$ on $(0, \infty)$ has infinite integral, but the set where it exceeds $\lambda$ is $(0, 1/\lambda)$, of length exactly $1/\lambda$.
That is the typical shape of a weak $L^1$ function, and the Hilbert transform and the maximal function produce such tails.
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<document source="https://tamnd.github.io/math2/node/9537RKBZPP.md" tag="9537RKBZPP" kind="concept">
# Proper morphism

## Summary

A proper morphism of schemes is a morphism that is separated, of finite type and universally closed.
It is the algebraic analogue of a proper map of topological spaces, and of a compact space when the target is a point.
Many finiteness results, such as the finiteness of cohomology of coherent sheaves and the existence of the Stein factorization, hold for proper morphisms.

## Explanations

### L1 human

A proper morphism is a family of spaces in which no fibre has points "missing at infinity".
Projective space over a field is proper over that field, while the affine line is not.
The affine line misses the point at infinity: a point moving off to infinity has no limit in it.
Properness rules out this kind of escape, and so it plays the role of compactness in algebraic geometry.
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<document source="https://tamnd.github.io/math2/node/9538W1K1EX.md" tag="9538W1K1EX" kind="concept">
# j-invariant of an elliptic curve

## Summary

The $j$-invariant of an [[elliptic curve|427N6S6QQW]] is a number computed from the coefficients of a [[Weierstrass equation|Z3C8B6QTS6]] of the curve.
It does not change under changes of coordinates.
Over an algebraically closed field, two elliptic curves are isomorphic exactly when they have the same $j$-invariant.
So the $j$-invariant is a coordinate on the set of isomorphism classes of elliptic curves.
Over $\C$ it agrees with the value of the modular [[j-invariant|FEC2XH822R]] at the period ratio $\tau$ of the curve.

## Explanations

### L1 human

Many different equations describe the same elliptic curve after a change of variables.
The $j$-invariant is a single number that sees through all these changes.
For $y^2 = x^3 + ax + b$ it is $1728 \cdot 4a^3 / (4a^3 + 27b^2)$.
For instance, every curve $y^2 = x^3 + ax$ with $a \ne 0$ has $j = 1728$, and every curve $y^2 = x^3 + b$ with $b \ne 0$ has $j = 0$.
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<document source="https://tamnd.github.io/math2/node/955ZNEXEF0.md" tag="955ZNEXEF0" kind="definition">
# Subnormal number

## Definition

Consider a [[floating-point|RTKRSC3KPH]] system with base $\beta$, precision $p$ and minimal exponent $e_{\min}$.
The smallest positive normalized number is $\beta^{e_{\min}}$.
A *subnormal number* of the system is a real number
$$x = \pm\, (0 . d_1 d_2 \ldots d_{p-1})_\beta \times \beta^{e_{\min}},$$
where the digits $d_i$ are integers in $\set{0, \ldots, \beta - 1}$, not all zero.
Equivalently, $x = \pm k \, \beta^{e_{\min} - p + 1}$ with $k$ an integer and $1 \le k \le \beta^{p-1} - 1$.

## Remarks

Example: in IEEE binary64, $\beta = 2$, $p = 53$ and $e_{\min} = -1022$, so the subnormal numbers are $\pm k \, 2^{-1074}$ with $1 \le k \le 2^{52} - 1$; the smallest is $2^{-1074} \approx 4.9 \times 10^{-324}$.
Non-example: $2^{-1022}$ is not subnormal in binary64; it is the smallest positive normalized number, with leading digit $1$.
All subnormal numbers are multiples of the same spacing $\beta^{e_{\min} - p + 1}$, which is also the spacing of the normalized numbers in $[\beta^{e_{\min}}, \beta^{e_{\min}+1})$.
A subnormal number has fewer than $p$ significant digits, so the relative rounding error in the subnormal range can exceed the unit roundoff.
Other names are *denormal* and *denormalized* numbers.
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<document source="https://tamnd.github.io/math2/node/956E1C2VE6.md" tag="956E1C2VE6" kind="definition">
# Equalizer

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $f, g \colon X \to Y$ be two [[morphisms|859F0TFVA3]] of $\mathcal{C}$.
An *equalizer* of $f$ and $g$ is an object $E$ with a morphism $e \colon E \to X$ such that:

1. $f \circ e = g \circ e$;
2. for every object $Z$ and every morphism $k \colon Z \to X$ with $f \circ k = g \circ k$, there is exactly one morphism $u \colon Z \to E$ with $e \circ u = k$.

## Remarks

An equalizer is the [[limit|CHS5AFJGWE]] of the diagram $X \rightrightarrows Y$ formed by $f$ and $g$, so it is unique up to unique isomorphism.
It is a monomorphism: if $e \circ a = e \circ b$, then $a$ and $b$ both solve clause 2 for $k = e \circ a$, so $a = b$.

Example: in the category of sets, let $E = \set{x \in X : f(x) = g(x)}$ and let $e$ be the inclusion.
Clause 1 holds by the choice of $E$.
If $f \circ k = g \circ k$, every $k(z)$ lies in $E$, so $u(z) = k(z)$ works, and it is the only choice since $e$ is injective.
In the category of abelian groups the equalizer of $f$ and $g$ is the [[kernel|0W88CEGQP1]] of $f - g$ with its inclusion.

Non-example: for $f, g \colon \R \to \R$ with $f(x) = x^2$ and $g(x) = x$, the inclusion $\{0\} \to \R$ satisfies clause 1 but not clause 2.
The map $k \colon \{*\} \to \R$ with $k(*) = 1$ has $f \circ k = g \circ k$ but does not factor through $\{0\}$.
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<document source="https://tamnd.github.io/math2/node/958TMJZ51Y.md" tag="958TMJZ51Y" kind="statement">
# Schur's inequality

## Statement

Let $x, y, z \ge 0$ and $t > 0$ be real numbers.
Then
$$x^t (x - y)(x - z) + y^t (y - z)(y - x) + z^t (z - x)(z - y) \ge 0.$$
Equality holds if and only if $x = y = z$, or two of $x, y, z$ are equal and the third is $0$.

## Remarks

Proof idea: the expression is symmetric, so assume $x \ge y \ge z$.
Group the first two terms as $(x - y)\big(x^t (x - z) - y^t (y - z)\big)$, which is $\ge 0$ because $x^t \ge y^t \ge 0$ and $x - z \ge y - z \ge 0$.
The last term $z^t (x - z)(y - z)$ is also $\ge 0$.

Example: for $t = 1$, expanding gives
$$x^3 + y^3 + z^3 + 3xyz \ge xy(x + y) + yz(y + z) + zx(z + x).$$
In terms of the [[elementary symmetric polynomials|0ZMY8441GW]] $p = x + y + z$, $q = xy + yz + zx$ and $r = xyz$, this reads $p^3 - 4pq + 9r \ge 0$.

## Explanations

### L1 human

Schur's inequality is a sum of three terms, one for each of three nonnegative numbers.
Some of the terms can be negative, but the total never is.
The case $t = 1$ is a cubic inequality that is often the missing step in symmetric problems.

### L2 human

Schur's inequality is a degree $t + 2$ symmetric inequality that does not follow from [[Muirhead's inequality|PBPFVY20WT]] alone, because it has negative coefficients.
It is a standard companion of Muirhead's inequality in proofs of symmetric inequalities in three variables.
Its equality cases, which include $(1, 1, 0)$, show that it is sharp at the boundary as well as at the centre.
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<document source="https://tamnd.github.io/math2/node/95BSKT0S6M.md" tag="95BSKT0S6M" kind="concept">
# Semi-continuity

## Summary

A real function is upper semicontinuous at a point if its values near the point cannot jump far above its value at the point.
It is lower semicontinuous if they cannot jump far below.
A function is [[continuous|0P27W9RW0G]] exactly when it is both.
Semicontinuity is the right condition for minimum and maximum problems: a lower semicontinuous function on a nonempty compact space attains its minimum.

## Explanations

### L1 human

Think of the graph of a function with a jump.
Upper semicontinuity allows a jump only if the value at the jump point sits on the upper side.
The floor function $\floor{x}$ jumps up at each integer $n$, and its value there is $n$, the upper value.
So the floor function is upper semicontinuous, but it is not lower semicontinuous.
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<document source="https://tamnd.github.io/math2/node/95MTQMX59H.md" tag="95MTQMX59H" kind="concept">
# Cauchy matrix

## Summary

A Cauchy matrix has entries $1/(x_i - y_j)$ for two lists of numbers $x_1, \dots, x_m$ and $y_1, \dots, y_n$ with $x_i \ne y_j$.
Its determinant has an explicit product formula, the Cauchy determinant.
As a result, a square Cauchy matrix is invertible when the $x_i$ are distinct and the $y_j$ are distinct.
The [[Hilbert matrix|2DTWB6F5TA]] is the best known Cauchy matrix.

## Explanations

### L1 human

Pick two lists of numbers that share no value.
Fill a table with one over the difference of a number from the first list and a number from the second.
For example, with $x = (1, 2)$ and $y = (0, -1)$ one gets $\begin{pmatrix} 1 & \tfrac12 \\ \tfrac12 & \tfrac13 \end{pmatrix}$.
Such tables appear in rational interpolation and in coding theory.
Their determinants factor completely, much like the Vandermonde determinant.
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<document source="https://tamnd.github.io/math2/node/95N3QVHAQN.md" tag="95N3QVHAQN" kind="definition">
# Rational equivalence

## Definition

Let $X$ be an [[algebraic variety|HV6YBS0YDH]] over a field, and let $Z_r(X)$ be the group of $r$-dimensional [[algebraic cycles|9VFWPZEVGN]] on $X$.
Let $W \subset X$ be an irreducible closed subvariety of dimension $r + 1$, and let $f$ be a nonzero element of its [[function field|2D5SS6WM14]] $k(W)$.
For each irreducible closed subvariety $V \subset W$ of dimension $r$, the local ring $A = \mathcal{O}_{W, V}$ is a one-dimensional Noetherian local domain with fraction field $k(W)$.
Write $f = a/b$ with $a, b \in A \setminus \set{0}$.
Set $\operatorname{ord}_V(f) = \ell_A(A/aA) - \ell_A(A/bA)$, where $\ell_A$ is the [[length|V6JBKT168T]] of an $A$-module.
This does not depend on the choice of $a$ and $b$.
The *divisor* of $f$ is the $r$-cycle $\operatorname{div}(f) = \sum_V \operatorname{ord}_V(f) \, [V]$, a finite sum.
Let $\operatorname{Rat}_r(X) \subset Z_r(X)$ be the subgroup generated by all such $\operatorname{div}(f)$, for all $W$ and $f$.
Two $r$-cycles $Z, Z'$ are *rationally equivalent* if $Z - Z' \in \operatorname{Rat}_r(X)$.

## Remarks

When $W$ is normal along $V$, $\operatorname{ord}_V$ is the discrete valuation of $k(W)$ at $V$.
Example: on $X = \mathbb{P}^1$ over a field $k$, take $W = X$ and $f = (x - a)/(x - b)$ with $a \ne b$ in $k$.
Then $\operatorname{div}(f) = [a] - [b]$, so the points $a$ and $b$ are rationally equivalent.
Non-example: on an elliptic curve $E$ over an algebraically closed field, two distinct points $P$ and $Q$ are not rationally equivalent.
If they were, then $[P] - [Q] = \operatorname{div}(f)$ for some $f \in k(E)^{\times}$, since $W = E$ is the only choice on a curve.
Then $f$ would be a rational function of degree $1$, hence an isomorphism $E \to \mathbb{P}^1$, which is impossible because $E$ has genus $1$.
Equivalently, $Z - Z' = \sum_i \big([V_i(0)] - [V_i(\infty)]\big)$ for finitely many subvarieties $V_i \subset X \times \mathbb{P}^1$ of dimension $r + 1$ that map onto $\mathbb{P}^1$, where $V_i(a)$ is the fiber of $V_i$ over $a$, viewed in $X$.
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# Unitary dual

## Summary

The unitary dual $\widehat{G}$ of a [[locally compact group|AHV05ZBCYW]] $G$ is the set of equivalence classes of irreducible [[unitary representations|74BPYJNZTM]] of $G$.
It usually carries the Fell topology.
It is the space on which a "Fourier transform" of functions on $G$ lives.
For abelian groups it is the Pontryagin dual, a group again.
For nonabelian groups it is only a topological space, often non-Hausdorff.
Determining it for a given group is a central problem of representation theory.

## Explanations

### L1 human

For the circle, the irreducible unitary representations are the characters $z \mapsto z^n$, one for each integer $n$, so the unitary dual of the circle is $\Z$.
This is why functions on the circle have Fourier series indexed by integers.
For the real line the characters are $x \mapsto e^{ixy}$, one for each real $y$, which gives the Fourier transform.
For a general group the unitary dual plays the same role: it lists the basic frequencies in which functions on the group can be decomposed.
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<document source="https://tamnd.github.io/math2/node/95QBBPRJ9E.md" tag="95QBBPRJ9E" kind="concept">
# Besov space

## Summary

A Besov space $B^s_{p,q}(\R^n)$ is a space of functions or [[tempered distributions|AES2T5JPFP]] that measures smoothness of order $s$ in the $L^p$ sense.
A third index $q$ gives a fine tuning.
It is defined by cutting the [[Fourier transform|DPVM34ADZV]] of a function into dyadic frequency shells and measuring each piece in $L^p$.
Besov spaces contain many classical spaces, such as Hölder spaces and fractional Sobolev spaces, and they are stable under real [[interpolation|VX1R6FN6PS]].
They appear naturally as spaces of boundary values of Sobolev functions and in the study of nonlinear evolution equations.

## Explanations

### L1 human

A function can be split into pieces that oscillate at frequencies of size about $1, 2, 4, 8, \ldots$.
Smoothness means that the high-frequency pieces are small.
The Besov norm asks that the piece at frequency $2^j$ be smaller than $2^{-js}$ in the $L^p$ sense, in a summable way controlled by $q$.
For $p = q = 2$ this is the familiar condition $\int (1 + \abs{\xi}^2)^s \abs{\hat f(\xi)}^2 \, d\xi < \infty$ on the Fourier transform.
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<document source="https://tamnd.github.io/math2/node/95QTEPMDA4.md" tag="95QTEPMDA4" kind="concept">
# System of parameters

## Summary

A system of parameters of a Noetherian [[local ring|MZ02BKPHJA]] of [[Krull dimension|MVGBS3J1P7]] $d$ is a list of $d$ elements of the maximal ideal $\mathfrak m$.
The ideal that they generate must be primary to $\mathfrak m$.
The [[dimension theorem for local rings|KW3HY77V6D]] says that such a list always exists and that no shorter list can do the same job.
Systems of parameters play the role of local coordinates, and they are the starting point of the theory of Cohen–Macaulay rings and multiplicities.

## Explanations

### L1 human

Near a point of a $d$-dimensional space, $d$ equations are needed to cut the space down to that single point.
A system of parameters is such a list of $d$ equations, read in the local ring of the point.
For the plane at the origin, the coordinates $x$ and $y$ work, and so do $x^2$ and $y^3$, since $x^2 = y^3 = 0$ still only allows the origin.
The [[definition|XSZ9C68SPG]] makes this precise.
</document>

<document source="https://tamnd.github.io/math2/node/963QWX2E2Z.md" tag="963QWX2E2Z" kind="concept">
# Fiber of a morphism

## Summary

The fiber of a [[morphism of schemes|JXRKH6P1Z4]] $f \colon X \to Y$ over a point $y$ is a scheme $X_y$ over the [[residue field|G51E8MW7J1]] $\kappa(y)$.
Its underlying space is the preimage $f^{-1}(y)$, but it also carries a scheme structure, which may be non-reduced.
A morphism is often thought of as a family of schemes, its fibers, parametrized by the points of $Y$.

## Explanations

### L1 human

The fiber over $y$ is "everything in $X$ that lies above $y$", made into a scheme of its own.
Take $\operatorname{Spec} \Z[i] \to \operatorname{Spec} \Z$.
Over the prime $5$ the fiber has two points, because $x^2 + 1$ factors modulo $5$.
Over the prime $3$ it has one point with residue field the field of $9$ elements.
Over the prime $2$ it has one point, but the fiber is $\operatorname{Spec} \F_2[x]/((x+1)^2)$, which has a nonzero nilpotent: this records that $2$ ramifies.
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<document source="https://tamnd.github.io/math2/node/966GNVGB8A.md" tag="966GNVGB8A" kind="concept">
# Purely inseparable extension

## Summary

A purely inseparable extension is an [[algebraic extension|0PTBM7K0R2]] in which every element has some $p$-power, $p^n$, in the base field, where $p$ is the [[characteristic|0F6ES3BRYP]].
It is the opposite of a [[separable extension|0WCX7NB4N7]]: the only elements that are both separable and purely inseparable over $K$ are those of $K$.
Every algebraic extension splits into a separable part followed by a purely inseparable part, so the two kinds together describe all algebraic extensions.
In characteristic $0$ the only purely inseparable extension of $K$ is $K$ itself.

## Explanations

### L1 human

In characteristic $p$, the map $x \mapsto x^p$ respects sums and products.
A purely inseparable extension adds elements that become old elements after you raise them to a $p$-th power a few times.
For example, over the field $\F_p(t)$ of rational functions, you may add an element $s$ with $s^p = t$.
The polynomial $x^p - t$ then becomes $(x - s)^p$, so it has a single root repeated $p$ times.
This repeated root is what "inseparable" refers to.
</document>

<document source="https://tamnd.github.io/math2/node/9689SFPVE9.md" tag="9689SFPVE9" kind="concept">
# Magma

## Summary

A magma is a set with one [[binary operation|0M418FE7BB]] and no further axioms.
It is the most basic [[algebraic structure|QJGPT2W0FT]] with a binary operation: [[semigroups|0XMEM406JM]], [[monoids|0N6R902FG1]], [[groups|G5S1M9]] and quasigroups are magmas with extra conditions.
Older texts call a magma a groupoid, a name that now usually means something else in category theory.

## Explanations

### L1 human

A magma is a set where any two elements can be combined into a third, in a fixed order.
No rule is asked for: the result may depend on how you place brackets and on the order of the two elements.
Subtraction on the integers is an example: $5 - 3$ is an integer, but $(5 - 3) - 1 \ne 5 - (3 - 1)$.
Many familiar structures are magmas with additional rules, so facts about magmas hold for all of them.
</document>

<document source="https://tamnd.github.io/math2/node/969F3AWBKC.md" tag="969F3AWBKC" kind="definition">
# Statistical decision theory

## Definition

A *statistical decision problem* consists of the following data.

1. A [[statistical model|K3AMMSY8DD]] $\set{P_\theta : \theta \in \Theta}$ on a sample space $\mathcal{X}$, from which an observation $X$ is drawn.
2. A measurable space $\mathcal{A}$ of actions.
3. A [[loss function|3BTG6VVCGZ]] $L \colon \Theta \times \mathcal{A} \to [0, \infty)$.

A decision rule is a measurable map $\delta \colon \mathcal{X} \to \mathcal{A}$, and its performance is measured by its [[risk function|N26TSVRH2S]] $R(\theta, \delta) = E_\theta[L(\theta, \delta(X))]$.
*Statistical decision theory* is the study of decision problems through the comparison of the risk functions of decision rules.

## Remarks

A rule $\delta_1$ is at least as good as $\delta_2$ if $R(\theta, \delta_1) \le R(\theta, \delta_2)$ for all $\theta$.
This is only a partial order, so the theory adds criteria: admissibility, the minimax principle and the Bayes principle.
Randomized rules, which choose an action at random given the data, are often allowed; they make the set of risk functions convex.

Example: point estimation of a normal mean is the decision problem with $P_\theta$ the law of $n$ independent normal variables with mean $\theta$ and variance $1$.
Here $\Theta = \mathcal{A} = \R$ and $L(\theta, a) = (a - \theta)^2$.
The rule $\delta(X) = \bar X$ is a measurable map from $\R^n$ to $\mathcal{A}$, with risk $1/n$.

Non-example: a model and a set of estimators without a loss function do not form a decision problem.
Clause 3 is missing, so risks are undefined and rules cannot be compared.
</document>

<document source="https://tamnd.github.io/math2/node/96AZPZSQKR.md" tag="96AZPZSQKR" kind="definition">
# Quasiregular map

## Definition

Let $n \ge 2$, let $\Omega \subset \R^n$ be open, and let $K \ge 1$.
A map $f \colon \Omega \to \R^n$ is *$K$-quasiregular* if:

1. $f$ is continuous.
2. $f$ lies in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,n}_{\mathrm{loc}}(\Omega; \R^n)$, that is, its coordinates have locally $n$-th power integrable [[weak derivatives|C3EC148VZD]].
3. $\norm{Df(x)}^n \le K\, J_f(x)$ for almost every $x \in \Omega$.

Here $Df(x)$ is the [[Jacobian matrix|0YF0VXKXGB]] of weak derivatives, $\norm{Df(x)}$ is its operator norm, and $J_f(x) = \det Df(x)$ is the [[Jacobian determinant|0ZCSMYHTBP]].
A map is *quasiregular* if it is $K$-quasiregular for some $K \ge 1$.

## Remarks

Example: for $n = 2$, a holomorphic function $f$ on $\Omega \subset \C = \R^2$ is $1$-quasiregular.
Its Jacobian matrix is a rotation times $\abs{f'(z)}$, so $\norm{Df} = \abs{f'}$ and $J_f = \abs{f'}^2$.

Non-example: complex conjugation $f(z) = \bar z$ fails clause 3.
Here $\norm{Df} = 1$ but $J_f = -1$, so $1 \le -K$ is false.

A quasiregular homeomorphism onto its image is a [[quasiconformal map|SB8KG46NHP]].
By a theorem of Reshetnyak, a nonconstant quasiregular map is open and discrete.
</document>

<document source="https://tamnd.github.io/math2/node/96B4E2GPKJ.md" tag="96B4E2GPKJ" kind="concept">
# Lie group action

## Summary

A Lie group action is a [[group action|0WCY42EV9V]] of a [[Lie group|7JBRGCVJRB]] on a [[smooth manifold|F45VQ47GPV]] such that the action map is smooth.
It is the precise form of continuous symmetry of a geometric space.
Orbits of such actions are immersed submanifolds, and stabilisers are closed subgroups.
Differentiating the action turns elements of the Lie algebra into vector fields on the manifold.

## Explanations

### L1 human

A Lie group action lets each element of a Lie group move the points of a smooth space.
The motion depends smoothly on both the element and the point.
Rotations of the plane about the origin are an example: rotating by $\theta$ moves each point smoothly, and rotating twice adds the angles.
The orbits are the circles centred at the origin, together with the origin itself, which is fixed by every rotation.
</document>

<document source="https://tamnd.github.io/math2/node/96CHDZKGQ0.md" tag="96CHDZKGQ0" kind="concept">
# Polylogarithm

## Summary

The polylogarithm $\operatorname{Li}_s(z)$ is the power series $\sum_{k \ge 1} \frac{z^k}{k^s}$ and its [[analytic continuation|0M3YQJ81NB]].
For $s = 1$ it is $-\ln(1 - z)$, which explains the name, and for $\operatorname{Re} s > 1$ its value at $z = 1$ is the [[Riemann zeta function|1XWBY4N0VT]] $\zeta(s)$.
The case $s = 2$ is the [[dilogarithm|G0X0NWSDB7]].
Polylogarithms appear in number theory, in algebraic K-theory, in quantum field theory and in the statistics of Fermi and Bose gases.

## Explanations

### L1 human

The series $z + \frac{z^2}{2} + \frac{z^3}{3} + \cdots$ adds up to $-\ln(1 - z)$.
Dividing the $k$-th term by $k$ once more, or $s$ times in all, gives the polylogarithm of order $s$.
Each step is the same as integrating the previous function after dividing by $z$.
For $z = 1$ and $s = 2$ one gets $1 + \frac14 + \frac19 + \cdots = \frac{\pi^2}{6}$.
</document>

<document source="https://tamnd.github.io/math2/node/96MQARP0N4.md" tag="96MQARP0N4" kind="definition">
# Δ-complex

## Definition

For $n \ge 0$ let $\Delta^n$ be the standard $n$-simplex with ordered vertices $e_0, \dots, e_n$, and let $\mathring{\Delta}^n$ be its interior.
For $0 \le i \le n$ with $n \ge 1$, let $\delta^i \colon \Delta^{n-1} \to \Delta^n$ be the affine map onto the $i$-th face that sends $e_0, \dots, e_{n-1}$ in order to the vertices $e_0, \dots, \widehat{e_i}, \dots, e_n$.
Let $X$ be a [[topological space|0CNQXVBHCX]].
A *$\Delta$-complex structure* on $X$ is a family of [[continuous maps|0SY9ECQ19Q]] $\sigma_\alpha \colon \Delta^{n_\alpha} \to X$ such that:

1. Each $\sigma_\alpha$ is injective on $\mathring{\Delta}^{n_\alpha}$, and each point of $X$ lies in $\sigma_\alpha(\mathring{\Delta}^{n_\alpha})$ for exactly one $\alpha$.
2. For each $\alpha$ with $n_\alpha \ge 1$ and each $i$, the map $\sigma_\alpha \circ \delta^i$ is $\sigma_\beta$ for some $\beta$ with $n_\beta = n_\alpha - 1$.
3. A subset $A \subseteq X$ is open if and only if $\sigma_\alpha^{-1}(A)$ is open in $\Delta^{n_\alpha}$ for every $\alpha$.

A *$\Delta$-complex* is a space with a $\Delta$-complex structure.

## Remarks

Every $\Delta$-complex is a CW complex whose cells are the open simplices $\sigma_\alpha(\mathring{\Delta}^{n_\alpha})$.
The combinatorial data of a $\Delta$-complex, the sets of $n$-simplices with the face maps $d_i$ given by clause 2, form a semi-simplicial set.

Example: the circle $S^1$ with one $0$-simplex $\sigma_0 \colon \Delta^0 \to S^1$ onto a point $p$ and one $1$-simplex $\sigma_1 \colon \Delta^1 \to S^1$ that wraps once around, sending both ends to $p$.
Clause 1 holds since $\sigma_1$ is injective on the open interval and misses only $p$.
Clause 2 holds since both faces of $\sigma_1$ equal $\sigma_0$, and clause 3 holds since $S^1$ is a quotient of $\Delta^1$.

Non-example: the circle with one $0$-simplex at $p$ and one $1$-simplex that wraps twice around fails clause 1, since the map is not injective on the open interval.
</document>

<document source="https://tamnd.github.io/math2/node/972X7QM34D.md" tag="972X7QM34D" kind="definition">
# Mackey machine

## Definition

Let $G = N \rtimes H$ be a second countable [[locally compact group|AHV05ZBCYW]], with $N$ closed, normal and abelian, and $H$ closed.
Let $\widehat{N}$ be the group of continuous characters $\chi \colon N \to \set{z \in \C : \abs{z} = 1}$.
The group $H$ acts on $\widehat{N}$ by $(h \cdot \chi)(n) = \chi(h^{-1} n h)$.
For $\chi \in \widehat{N}$, let $H_\chi$ be its stabiliser in $H$, and let $\sigma$ be an irreducible [[unitary representation|74BPYJNZTM]] of $H_\chi$.
The formula $(\chi \otimes \sigma)(n h) = \chi(n) \sigma(h)$ for $n \in N$ and $h \in H_\chi$ defines a unitary representation of $N \rtimes H_\chi$.
The *Mackey machine* assigns to the pair $(\chi, \sigma)$ the [[induced representation|6Q7AHAZ6Q4]]
$$\pi_{\chi, \sigma} = \operatorname{Ind}_{N \rtimes H_\chi}^{G} (\chi \otimes \sigma).$$

## Remarks

Mackey's theorem: suppose the action of $H$ on $\widehat{N}$ is regular, meaning that the [[orbit|0YQPZB5Q1G]] space $\widehat{N}/H$ is countably separated.
Then each $\pi_{\chi, \sigma}$ is irreducible, and every irreducible unitary representation of $G$ is equivalent to some $\pi_{\chi, \sigma}$.
Two of them are equivalent exactly when the characters lie in one orbit and the representations $\sigma$ correspond under conjugation.

Example: the $ax + b$ group $G = \R \rtimes \R_{>0}$, where $a > 0$ acts on $\R$ by $b \mapsto ab$.
Here $\widehat{N} \cong \R$ through $\chi_\xi(b) = e^{i \xi b}$, and $a$ sends $\chi_\xi$ to $\chi_{\xi / a}$.
There are three orbits: $\set{0}$, $(0, \infty)$ and $(-\infty, 0)$.
The orbit $\set{0}$ has stabiliser $\R_{>0}$ and gives the one-dimensional characters of $\R_{>0}$, trivial on $N$.
The other two orbits have trivial stabiliser and give two infinite-dimensional irreducible representations $\operatorname{Ind}_N^G \chi_{\pm 1}$.

Non-example: let $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ act on $N = \Z^2$, and let $G = \Z^2 \rtimes_A \Z$.
Then $\widehat{N}$ is the torus $\R^2 / \Z^2$, and the dual action of $A$ on it is ergodic, so the orbit space is not countably separated.
The regularity hypothesis fails, and some irreducible unitary representations of $G$ are not of the form $\pi_{\chi, \sigma}$.
</document>

<document source="https://tamnd.github.io/math2/node/975PMY4FG8.md" tag="975PMY4FG8" kind="concept">
# Filtered category

## Summary

A filtered category is a [[category|A2DE66937D]] in which every finite diagram has a cocone.
It generalizes a [[directed set|D9RY4667K6]]: any two objects map to a common object.
Also, any two parallel arrows become equal after composing with a further arrow.
Colimits over filtered categories behave like unions, and in sets they commute with finite limits.
They are the basis of the theory of locally presentable categories.

## Explanations

### L1 human

Think of a process that runs forward without end, where any two stages are followed by a later stage that contains both.
A filtered category is such a process, with the extra rule that two different ways of going forward can always be made to agree later on.
A directed set, such as the natural numbers with their usual order, is the simplest example.
A category with a terminal object is also filtered, since everything maps to the terminal object.
</document>

<document source="https://tamnd.github.io/math2/node/97CGS5KKRW.md" tag="97CGS5KKRW" kind="statement">
# Uhlenbeck compactness theorem

## Statement

Let $X$ be a compact [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$, let $G$ be a compact Lie group, and let $P \to X$ be a [[principal $G$-bundle|8E37E74XVP]].
Let $p > n/2$, and let $(A_i)$ be a sequence of [[connections|NJ21PN5ZGJ]] on $P$ of [[Sobolev class|M4CEZ5K9HP]] $L^p_1$ whose [[curvatures|ZAJFW1S078]] are bounded in $L^p$:
$$\sup_i \norm{F_{A_i}}_{L^p} < \infty.$$
Then there are a subsequence $(A_{i_j})$, [[gauge transformations|707DEWCCFX]] $u_j$ of class $L^p_2$, and a connection $A$ of class $L^p_1$ with this property.
The connections $u_j^* A_{i_j}$ converge to $A$ weakly in $L^p_1$.

## Remarks

The key step is Uhlenbeck's gauge fixing lemma: on a ball, a connection with small enough curvature in $L^{n/2}$ is gauge equivalent to one in Coulomb gauge, $d^* A = 0$, with $\norm{A}_{L^p_1} \le C \norm{F_A}_{L^p}$.
For $p > n/2$, Hölder's inequality turns the $L^p$ bound into a small $L^{n/2}$ bound on balls of a fixed small radius.
So one can fix the gauge on each ball of a finite cover, extract weakly convergent subsequences, and patch the local gauges together.
The condition $p > n/2$ is needed.
In dimension $4$ with only an $L^2$ bound on curvature, the scale-invariant case $p = n/2$, the curvature of instantons can concentrate at points.
Then a sequence may have no subsequence that converges, after gauge, on the whole of $X$.
For a sequence of instantons on a bundle of fixed charge, a refinement of the argument gives convergence after gauge on the complement of finitely many points.
By Uhlenbeck's removable singularity theorem the limit is a connection on a possibly different bundle; this is the bubbling picture.

## Explanations

### L1 human

A connection can be described in many ways, one for each choice of gauge, and in a bad gauge a harmless sequence can look as if it diverges.
The theorem says that if the curvature is bounded in a strong enough sense, then one can choose good gauges in which the sequence has a convergent subsequence.
So a bound on curvature, which is gauge invariant, controls the connections themselves.

### L2 human

The theorem plays for connections the role that the Rellich and Arzelà–Ascoli compactness theorems play for functions.
In dimension $4$ and at the critical exponent it underlies the Uhlenbeck compactification of instanton moduli spaces.
There, limits of instantons are instantons on a bundle with smaller charge, together with points where curvature has concentrated.

This compactification is a central tool in Donaldson theory.
</document>

<document source="https://tamnd.github.io/math2/node/97DDM0ZBYG.md" tag="97DDM0ZBYG" kind="definition">
# Difference quotient

## Definition

Let $D \subseteq \R$ and let $f \colon D \to \R$ be a [[function|0XHDTE5YMZ]].
Let $x \in D$ and let $h \ne 0$ be a real number with $x + h \in D$.
The *difference quotient* of $f$ at $x$ with increment $h$ is
$$\frac{f(x + h) - f(x)}{h}.$$
Equivalently, for $a \ne b$ in $D$, the difference quotient of $f$ between $a$ and $b$ is $\dfrac{f(b) - f(a)}{b - a}$.

## Remarks

The function $f$ has [[derivative|0F1WR9C83K]] $f'(x)$ at an interior point $x$ of $D$ exactly when the difference quotient has the [[limit|0TXTSEMHR5]] $f'(x)$ as $h \to 0$.
The [[mean value theorem|0BARQY52N5]] says that, for $f$ continuous on $[a, b]$ and differentiable on $(a, b)$, the difference quotient between $a$ and $b$ equals $f'(c)$ for some $c \in (a, b)$.

Example: for $f(x) = x^2$, the difference quotient is $\frac{(x + h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$.
It tends to $2x$ as $h \to 0$, which is $f'(x)$.

Non-example: the increment $h = 0$ is not allowed, since the expression $\frac{f(x) - f(x)}{0}$ has no meaning.
This is why the derivative is a limit and not a value of the quotient.

In numerical analysis, the quotient is also called the forward difference quotient, and $\frac{f(x + h) - f(x - h)}{2h}$ is the central difference quotient.
The symbol $\frac{\Delta y}{\Delta x}$ is common notation for it.
</document>

<document source="https://tamnd.github.io/math2/node/97GRC6YGF5.md" tag="97GRC6YGF5" kind="definition">
# Principal bundle

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] and let $M$ be a [[smooth manifold|F45VQ47GPV]].
A *principal $G$-bundle* over $M$ is a smooth [[fiber bundle|CX4M8K61AB]] $\pi \colon P \to M$ together with a smooth right [[action|0WCY42EV9V]] $P \times G \to P$, $(u, g) \mapsto u \cdot g$, such that:

1. the action preserves fibers: $\pi(u \cdot g) = \pi(u)$ for all $u \in P$ and $g \in G$;
2. every point of $M$ has an open neighbourhood $U$ and a diffeomorphism $\varphi \colon \pi^{-1}(U) \to U \times G$ with $\operatorname{pr}_1 \circ \varphi = \pi$ and $\varphi(u \cdot g) = \varphi(u) \cdot g$, where $G$ acts on $U \times G$ by $(x, h) \cdot g = (x, hg)$.

## Remarks

The product $M \times G$ with $(x, h) \cdot g = (x, hg)$ is a principal $G$-bundle, the trivial one.
The map $\R \to S^1$, $t \mapsto e^{2\pi i t}$, with $\Z$ acting by $t \cdot n = t + n$, is a principal $\Z$-bundle.
The Hopf map $S^3 \to S^2$ is a principal $U(1)$-bundle.

Non-example: the circle group acts on $S^2$ by rotations about the vertical axis.
This action preserves the fibers of the identity map $S^2 \to S^2$, but these fibers are points, not copies of the circle.
More generally, an action with a fixed point cannot be part of a principal bundle structure, because clause 2 forces $G$ to act freely.

Clause 2 implies that $G$ acts freely and transitively on each fiber.
Conversely, for a free and proper smooth action of a Lie group $G$ on a manifold $P$, the quotient map $P \to P/G$ is a principal $G$-bundle.
The same definition with topological groups and homeomorphisms gives topological principal bundles.

## Explanations

### L1 human

A principal bundle is a fiber bundle whose fibers are copies of a group, on which the group acts by right multiplication in a way compatible with local trivializations.
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<document source="https://tamnd.github.io/math2/node/97HMV9BT2W.md" tag="97HMV9BT2W" kind="statement">
# Cauchy–Kovalevskaya theorem

## Statement

Let $k \ge 1$ and $N \ge 1$ be integers, and write the variables in $\R \times \R^n$ as $(t, x)$.
Consider a system of $N$ [[partial differential equations|GC5BDZ6QBC]] for an unknown $u = (u_1, \dots, u_N)$ of the following form.
$$\partial_t^k u = F\left(t, x, \left(\partial_t^j \partial_x^\alpha u\right)_{j < k, \ j + \abs{\alpha} \le k}\right).$$
Here $\alpha$ runs over [[multi-indices|0PJXBR6QA3]] in $\N^n$.
Let $x_0 \in \R^n$, and let $\varphi_0, \dots, \varphi_{k-1}$ be $\R^N$-valued [[real-analytic functions|081WKEDK6C]] on a neighbourhood of $x_0$.
Assume that $F$ is real-analytic on a neighbourhood of the point $\left(0, x_0, (\partial_x^\alpha \varphi_j(x_0))_{j < k, \ j + \abs{\alpha} \le k}\right)$.
Then there is a neighbourhood $W$ of $(0, x_0)$ in $\R \times \R^n$ and a real-analytic function $u$ on $W$ that solves the system in $W$.
Moreover $u$ satisfies the initial conditions $\partial_t^j u(0, x) = \varphi_j(x)$ for $j = 0, \dots, k - 1$ and all $x$ with $(0, x) \in W$.
Any two real-analytic solutions with these properties agree on a neighbourhood of $(0, x_0)$.

## Remarks

The equation and the initial conditions determine all derivatives of $u$ at $(0, x_0)$, so the Taylor series of a real-analytic solution is fixed; this gives uniqueness.
For existence, the proof shows that this Taylor series converges by the method of majorants: one compares with an explicit system whose analytic solution dominates the coefficients.
The form of the equation matters: the equation must be solved for $\partial_t^k u$, and the right side may contain neither $\partial_t^k u$ nor any derivative of total order greater than $k$.
Without this, the conclusion fails: for the [[heat equation|VP2HN86VAG]] $\partial_t u = \partial_x^2 u$ with $u(0, x) = 1/(1 - x)$, the formal power series solution is $\sum_{j \ge 0} \frac{(2j)!}{j!} \frac{t^j}{(1 - x)^{2j + 1}}$, which diverges for every $t \ne 0$.
Analyticity of the data is also needed; for smooth data, existence can fail even for a linear equation, as Lewy's example of an operator without [[local solvability|P6E9RZNRBZ]] shows.
Uniqueness among all smooth solutions, not only analytic ones, holds for linear equations by [[Holmgren's uniqueness theorem|1VVKF0Y19B]].

## Explanations

### L1 human

Suppose that an equation and its starting values are given by convergent power series.
The theorem says that then there is a solution near the starting surface, again given by a convergent power series.
It is the basic existence theorem for partial differential equations.
It needs strong hypotheses: everything must be analytic, and the equation must be solved for the highest time derivative.

### L2 human

The theorem gives local existence in the analytic class for any noncharacteristic Cauchy problem, after a change of variables that makes the initial surface $t = 0$.
It does not say that the problem is well posed: [[Hadamard's example|SZ6MEVM89M]] shows that for the Laplace equation the analytic solution exists but depends discontinuously on the data.
So for evolution equations one looks for theories in Sobolev or $C^\infty$ spaces, such as the energy method for hyperbolic equations.
For ordinary differential equations, with $n = 0$, the theorem reduces to the existence of analytic solutions to analytic initial value problems.
</document>

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# Compact operator

## Definition

Let $X$ and $Y$ be [[normed spaces|03J5RAKB30]] over $\R$ or $\C$ and let $T \colon X \to Y$ be a [[linear map|0RJF668P1T]].
Let $B_X = \set{x \in X : \norm{x} \le 1}$ be the closed unit ball of $X$.
The map $T$ is *compact* when the [[closure|0QEAJB91GZ]] of $T(B_X)$ is a [[compact subset|0R6EWFKRCF]] of $Y$.

## Remarks

Equivalent form: $T$ is compact if and only if for every bounded sequence $(x_n)$ in $X$, the sequence $(Tx_n)$ has a convergent subsequence in $Y$.
A compact operator is bounded, since a compact set is bounded.

Example: on $\ell^2$ let $T$ be the diagonal operator $T(x_1, x_2, \ldots) = (x_1, x_2/2, x_3/3, \ldots)$.
The finite-rank operators $T_N$ that keep the first $N$ entries of $Tx$ and set the rest to $0$ satisfy $\norm{T - T_N} \le 1/(N+1)$.
A norm limit of finite-rank operators into a Banach space is compact, because the image of the unit ball is then totally bounded; so $T$ is compact.

Non-example: the identity of $\ell^2$ is not compact.
The basis vectors $e_n$ lie in the unit ball, and $\norm{e_n - e_m} = \sqrt{2}$ for $n \ne m$, so $(e_n)$ has no convergent subsequence.

The set of compact operators on a Banach space is a closed two-sided ideal of the algebra of bounded operators, the [[ideal of compact operators|2WAZF1RV2Y]] in the Hilbert space case.
</document>

<document source="https://tamnd.github.io/math2/node/97MYM5X7N5.md" tag="97MYM5X7N5" kind="concept">
# Čech cohomology

## Summary

Čech cohomology computes cohomology of a sheaf from an open cover.
It uses the sections of the sheaf on the open sets of the cover and on their finite intersections.
It is defined by an explicit cochain complex, so it is often the practical way to compute [[sheaf cohomology|PQ6Q143ARJ]].
For a quasi-coherent sheaf on a separated scheme and an affine open cover, Čech cohomology equals sheaf cohomology.
The same idea applied to the nerve of a cover gives Čech cohomology of a topological space.

## Explanations

### L1 human

Cover a space by open sets on which everything is simple.
Record data on each piece, compare the data on each overlap of two pieces, and check compatibility on overlaps of three pieces.
Cohomology in degree $1$ counts ways to give compatible differences on overlaps that do not come from data on the pieces.
For the circle covered by two arcs that overlap in two small arcs, with integer constants as data, this gives one free integer: the cohomology group $\Z$ in degree $1$.
This number detects the hole in the circle.
</document>

<document source="https://tamnd.github.io/math2/node/97NPWM3S6E.md" tag="97NPWM3S6E" kind="concept">
# Comonad

## Summary

A comonad on a category is the dual of a [[monad|FF6XXA6D1A]].
It is an endofunctor $G$ with a counit $\varepsilon \colon G \Rightarrow \mathrm{Id}$ and a comultiplication $\delta \colon G \Rightarrow G G$ that satisfy coassociativity and counit laws.
Every adjunction $F \dashv U$ gives a comonad $F U$ on the domain of $U$.
Comonads govern coalgebraic structure, descent theory and comonadic resolutions in homological algebra.

## Explanations

### L1 human

A monad builds things up, while a comonad unfolds things.
The counit extracts a value, and the comultiplication duplicates the context around it.
For a fixed set $A$, the functor $X \mapsto A \times X$ on sets is a comonad.
The counit forgets the label $a$ in $(a, x)$, and the comultiplication copies the label, giving $(a, (a, x))$.
</document>

<document source="https://tamnd.github.io/math2/node/97NS6JRVNE.md" tag="97NS6JRVNE" kind="definition">
# Final topology

## Definition

Let $Y$ be a set, let $I$ be a set, and for each $i \in I$ let $X_i$ be a [[topological space|0CQHFPGJ45]] and $f_i \colon X_i \to Y$ a map.
The *final topology* on $Y$ for the family $(f_i)_{i \in I}$ is the set of all $U \subseteq Y$ with the following property.
For every $i \in I$, the [[preimage|01AHG4G9RT]] $f_i^{-1}(U)$ is [[open|02BH3MCBQB]] in $X_i$.

## Remarks

This family of sets is a topology, because preimages commute with unions and intersections.
It is the [[finest|072J68M6HX]] topology on $Y$ for which every $f_i$ is [[continuous|0VGFHTM75Y]].
Universal property: for a topological space $Z$, a map $g \colon Y \to Z$ is continuous for the final topology if and only if $g \circ f_i$ is continuous for every $i$.

Example: for a surjection $q \colon X \to Y$ from a space $X$, the final topology for $q$ is the [[quotient topology|0NQ1H2X495]].
Example: for the injections of the pieces into a disjoint union, the final topology is the [[disjoint union topology|ZXJKSB30WF]].

Non-example: let $q \colon \R \to \R$ be the identity, with the usual topology on the source.
The trivial topology $\set{\varnothing, \R}$ on the target makes $q$ continuous but is not the final topology.
The set $(0, 1)$ has an open preimage, yet it is not open in the trivial topology.

The final topology is also called the strong topology or the coinduced topology for the family.
</document>

<document source="https://tamnd.github.io/math2/node/97NYHNBZ2D.md" tag="97NYHNBZ2D" kind="concept">
# Sub-Riemannian manifold

## Summary

A sub-Riemannian manifold is a [[smooth manifold|F45VQ47GPV]] where motion is allowed only along chosen horizontal directions.
These form a subbundle of the tangent bundle, and they carry an inner product.
Distance is the infimum of lengths of curves that are tangent to the horizontal directions at every time.
Assume that the horizontal directions and their [[Lie brackets|68FGQEBD6J]] span all directions.
Then, by the Chow–Rashevskii theorem, any two points of a connected manifold are joined by a horizontal curve, and the distance is finite.
Sub-Riemannian geometry appears in control theory, in the study of hypoelliptic operators and in the geometry of the Heisenberg group.

## Explanations

### L1 human

Think of a car: it can drive forward and backward and turn its wheels, but it cannot move sideways.
Still, by combining these moves, as in parallel parking, it can reach every position.
A sub-Riemannian manifold models this: only some directions of motion are allowed, but combining them reaches everywhere.
The distance between two places is then the length of the shortest allowed route, which can be much longer than in the unrestricted space.
</document>

<document source="https://tamnd.github.io/math2/node/97PBMQCPPM.md" tag="97PBMQCPPM" kind="concept">
# Exponential sum

## Summary

An exponential sum is a finite sum of numbers $e^{2\pi i f(n)}$ on the unit circle, where $f$ is a real function on the integers.
The trivial bound for such a sum is its number of terms, and the aim is to show cancellation, that is, a much smaller bound.
Such bounds are central in analytic number theory: they control [[equidistribution modulo one|XM6DH57BWY]], the zeros of zeta functions, and the circle method.
Weyl sums, [[Gauss sums|EQ6A3ZN48K]] and [[Kloosterman sums|E1SQWQ2PAV]] are important special cases.

## Explanations

### L1 human

Each term $e^{2\pi i f(n)}$ is a point on the unit circle, like an arrow of length $1$ pointing in the direction $f(n)$.
Adding the arrows is like a random walk: if the directions vary enough, the arrows cancel and the sum stays small.
For example, with $f(n) = n/4$, the arrows point to $i, -1, -i, 1$ and the sum of each block of four is $0$.
Showing that such cancellation happens is the main problem of the theory.
</document>

<document source="https://tamnd.github.io/math2/node/97QF3KHSBY.md" tag="97QF3KHSBY" kind="statement">
# Ackermann function is not primitive recursive

## Statement

Let $A \colon \N \times \N \to \N$ be the [[Ackermann function|N470JF7ZJX]], given by $A(0, n) = n + 1$, $A(m + 1, 0) = A(m, 1)$ and $A(m + 1, n + 1) = A(m, A(m + 1, n))$.
Then $A$ is a total [[computable function|KZSB5E471K]] that is not [[primitive recursive|5NVA2JRHPV]].

## Remarks

The key lemma is a domination result.
For every primitive recursive $f \colon \N^k \to \N$ there is $m \in \N$ such that $f(\bar x) < A(m, \max \bar x)$ for all $\bar x \in \N^k$.
It is proved by induction on the construction of $f$, using that $A$ is increasing in each argument.
If $A$ were primitive recursive, so would be $g(n) = A(n, n)$.
Then some $m$ would give $A(n, n) < A(m, n)$ for all $n$, which fails at $n = m$.

The theorem shows that the class of primitive recursive functions is strictly smaller than the class of total computable functions.
Computability of $A$ follows from the fact that the recursion decreases the pair $(m, n)$ in the lexicographic order.

## Explanations

### L1 human

Primitive recursive functions are those computed by programs whose loops have bounds fixed before they start.
The Ackermann function can be computed, but it grows so fast that no such program keeps up with it.
So bounded loops are not enough to compute everything computable.

### L2 human

Each primitive recursive function is dominated by one level $n \mapsto A(m, n)$ of the Ackermann function.
The diagonal $n \mapsto A(n, n)$ escapes every level, so it is outside the class.
This is a diagonal argument, like the one that shows that no effective list of total computable functions contains all of them.
To reach all computable functions one adds unbounded search, which gives the [[general recursive functions|HBK5QHC1GC]].
</document>

<document source="https://tamnd.github.io/math2/node/97SRYBF41G.md" tag="97SRYBF41G" kind="statement">
# Baker–Pixley theorem

## Statement

Let $A$ be a finite algebra and let $d \ge 2$ be an integer.
Assume that $A$ has a [[near-unanimity term|4CRH8NNFEF]] of arity $d + 1$.
Let $n \ge 1$ and let $f \colon A^n \to A$ be an operation.
Then $f$ is a [[term operation|N9M2N27END]] of $A$ if and only if every [[subalgebra|7CHG6SD365]] of the [[direct power|WJ73XYBNE4]] $A^d$ is closed under $f$, applied coordinatewise.

## Remarks

Every term operation preserves every subalgebra of every power, so only the converse needs work.
The idea is interpolation.
The operation $f$ agrees with a term operation on each set of at most $d$ points of $A^n$.
The reason is that $f$ preserves the subalgebra of $A^d$ generated by the coordinate tuples of those points.
The near-unanimity term then glues such local interpolations, by induction on the number $k > d$ of points.
Given $k$ points, pick $d + 1$ of them, $p_1, \dots, p_{d + 1}$, and by induction take $d + 1$ terms, the $i$-th agreeing with $f$ on all $k$ points except possibly $p_i$.
At each point at most one of these terms disagrees with $f$, so the near-unanimity term applied to them agrees with $f$ on all $k$ points.
Since $A^n$ is finite, one reaches a term that agrees with $f$ on all of $A^n$.
For a [[majority term|9879ZMW5AA]], so $d = 2$, the theorem says that the term operations are the operations that preserve all subalgebras of $A^2$.
For example, a finite lattice $L$ has the majority term $(x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$, so an operation on $L$ is a lattice term operation exactly when it preserves all sublattices of $L^2$.
For an infinite algebra the same argument only shows that $f$ agrees with some term operation on each finite subset of $A^n$.

## Explanations

### L1 human

To test whether an operation can be written with the operations of an algebra, one would like a finite list of conditions to check.
The theorem gives such a list when the algebra has a near-unanimity term.
It is enough to check that the operation respects every subalgebra of a fixed small power $A^d$.

### L2 human

The theorem is a strong form of the statement that the clone of term operations of $A$ equals the set of polymorphisms of the subalgebras of $A^d$.
In the language of the Galois connection between operations and relations, the term clone of $A$ is determined by relations of arity at most $d$.
It also shows that two finite algebras on the same set with near-unanimity terms have the same term operations once they have the same subalgebras of $A^d$.
In an arithmetical variety a Pixley term $p$ gives the majority term $p(x, p(x, y, z), z)$, so the theorem applies there with $d = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/97VNBS15CD.md" tag="97VNBS15CD" kind="statement">
# Schwarz alternating method

## Statement

Let $V$ be a real [[Hilbert space|HM2HHE3J51]] with inner product $a(\cdot, \cdot)$ and norm $\norm{v}_a = a(v, v)^{1/2}$.
Let $\ell$ be a bounded linear functional on $V$, and let $u \in V$ be the solution of $a(u, v) = \ell(v)$ for all $v \in V$.
Let $V_1, V_2$ be closed subspaces of $V$ with $V_1 + V_2 = V$.
The *alternating Schwarz method* starts from $u^0 \in V$ and repeats two half steps.

1. Find $w_1 \in V_1$ with $a(w_1, v) = \ell(v) - a(u^n, v)$ for all $v \in V_1$, and set $u^{n+1/2} = u^n + w_1$.
2. Find $w_2 \in V_2$ with $a(w_2, v) = \ell(v) - a(u^{n+1/2}, v)$ for all $v \in V_2$, and set $u^{n+1} = u^{n+1/2} + w_2$.

Then there is a constant $\rho < 1$, depending only on $V_1$ and $V_2$, such that
$$\norm{u^n - u}_a \le \rho^n \norm{u^0 - u}_a \quad \text{for all } n \ge 0.$$

## Remarks

Let $P_i$ be the $a$-[[orthogonal projection|0B8KZRT28M]] onto $V_i$.
The errors satisfy $u^{n+1} - u = (I - P_2)(I - P_1)(u^n - u)$, a product of the projections onto $V_1^\perp$ and $V_2^\perp$.
Since $V_1 + V_2 = V$ is closed, $V_1^\perp \cap V_2^\perp = \set{0}$ and the angle between $V_1^\perp$ and $V_2^\perp$ is positive, which gives $\rho = \norm{(I - P_2)(I - P_1)}_a < 1$.
If $V_1 + V_2$ is only dense in $V$, von Neumann's theorem on alternating projections still gives $u^n \to u$, but possibly without a geometric rate.

The original setting is the Dirichlet problem $-\Delta u = f$ in $\Omega$, $u = 0$ on $\partial \Omega$, where $\Omega = \Omega_1 \cup \Omega_2$ is a union of two overlapping domains.
Take $V = H^1_0(\Omega)$ (a [[Sobolev space|M4CEZ5K9HP]]), $a(u, v) = \int_\Omega \nabla u \cdot \nabla v \, dx$, and $V_i = H^1_0(\Omega_i)$ extended by zero.
Then each half step solves the Dirichlet problem on $\Omega_i$ with boundary values taken from the current iterate on $\partial \Omega_i \cap \Omega$.
A partition of unity subordinate to the overlap shows $V_1 + V_2 = V$ when the overlap has positive width.
The rate $\rho$ tends to $1$ as the overlap shrinks.

## Explanations

### L1 human

Hermann Schwarz wanted to solve a boundary value problem on a region made of two overlapping simple pieces, such as a disc and a rectangle.
He solved on the disc, used that answer as boundary data on the part of the rectangle's edge inside the disc, solved on the rectangle, and went back and forth.
The theorem says this back-and-forth converges to the solution on the whole region, and the error shrinks by a fixed factor each round.

### L2 human

The method is the prototype of [[domain decomposition methods|AN054RHT1K]]: a large problem is split into problems on subdomains that are cheaper to solve.
In the abstract form it is a block Gauss–Seidel iteration on the splitting $V = V_1 + V_2$, and the error operator is a product of orthogonal projections.
The [[additive Schwarz method|X68MFYW014]] is the Jacobi-type variant, which solves on all pieces at once and is used as a preconditioner.
The convergence rate depends on the overlap, which is why practical methods add a coarse space.
</document>

<document source="https://tamnd.github.io/math2/node/97ZY7H816E.md" tag="97ZY7H816E" kind="definition">
# Multicategory

## Definition

A *multicategory* $\mathcal{M}$ consists of the following data.

1. A class of objects.
2. For each $n \ge 0$, each list of objects $x_1, \ldots, x_n$ and each object $y$, a set $\mathcal{M}(x_1, \ldots, x_n; y)$ of *multimorphisms* from $(x_1, \ldots, x_n)$ to $y$.
3. For each object $x$, an identity $1_x \in \mathcal{M}(x; x)$.
4. Composition maps
$$\mathcal{M}(y_1, \ldots, y_n; z) \times \prod_{i=1}^{n} \mathcal{M}(x_{i1}, \ldots, x_{ik_i}; y_i) \to \mathcal{M}(x_{11}, \ldots, x_{1k_1}, \ldots, x_{n1}, \ldots, x_{nk_n}; z),$$
written $(g, f_1, \ldots, f_n) \mapsto g \circ (f_1, \ldots, f_n)$.

These data satisfy two axioms.

1. (associativity) $\left(h \circ (g_1, \ldots, g_m)\right) \circ (\mathbf{f}_1, \ldots, \mathbf{f}_m) = h \circ \left(g_1 \circ \mathbf{f}_1, \ldots, g_m \circ \mathbf{f}_m\right)$, where $\mathbf{f}_i$ is the list of multimorphisms that are plugged into the inputs of $g_i$.
2. (unit) $1_y \circ (f) = f$ and $f \circ (1_{x_1}, \ldots, 1_{x_n}) = f$ for every $f \in \mathcal{M}(x_1, \ldots, x_n; y)$.

## Remarks

Example: the objects are the [[vector spaces|000AYPA9AJ]] over a field $K$, and $\mathcal{M}(V_1, \ldots, V_n; W)$ is the set of [[multilinear maps|0443XNARDV]] $V_1 \times \cdots \times V_n \to W$; a map with $n = 0$ inputs is an element of $W$.
Composition is substitution of maps into the arguments.
A multilinear map with multilinear maps substituted in its arguments is again multilinear in the combined list of arguments, and substitution is associative, so the axioms hold.

Non-example: keep the same objects but allow only maps with $n \ge 2$ inputs.
Composition is still defined, but there is no identity in $\mathcal{M}(V; V)$, so the unit clause fails.

A multicategory with exactly one object is a non-symmetric operad.
A *symmetric* multicategory has in addition actions of the symmetric groups that permute the inputs, compatible with composition.
Restricting to multimorphisms with exactly one input gives an ordinary category.
</document>

<document source="https://tamnd.github.io/math2/node/983GZBQYT4.md" tag="983GZBQYT4" kind="concept">
# Structure

## Summary

A structure for a [[signature|WDVVEYD6AT]] is a set together with a meaning for each symbol of the signature.
Each symbol becomes a function, a relation or an element of the set.
Structures are the objects in which first-order formulas are true or false.
Groups, rings, ordered sets and graphs are all structures for suitable signatures.

## Explanations

### L1 human

A signature is a list of names, such as "$+$", "$0$" and "$<$", with the number of inputs of each name.
A structure picks a set and says what each name means on that set.
For example, the integers with ordinary addition, the number zero and the usual order form a structure for the signature with $+$, $0$ and $<$.
The same signature can have many different structures, such as the rational numbers with the same symbols.
</document>

<document source="https://tamnd.github.io/math2/node/984525NJS0.md" tag="984525NJS0" kind="definition">
# Prime ideal of a lattice

## Definition

Let $(L, \vee, \wedge)$ be a [[lattice|WH4SFAACAS]].
A *prime ideal* of $L$ is an [[ideal|5GW2T2AM1A]] $I$ of $L$ such that:

1. $I \ne L$;
2. for all $x, y \in L$, if $x \wedge y \in I$, then $x \in I$ or $y \in I$.

## Remarks

A proper ideal $I$ is prime exactly when its complement $L \setminus I$ is a [[filter|Z15FXM9WM1]]; such a filter is called a prime filter.
Example: let $X$ be a set, $p \in X$, and $I = \set{A \subseteq X : p \notin A}$ in the power set of $X$.
$I$ is an ideal, it is proper since $X \notin I$, and if $p \notin A \cap B$ then $p \notin A$ or $p \notin B$.
Non-example: in the power set of $\set{1, 2, 3}$, the principal ideal $\set{\emptyset, \set{1}}$ is proper but not prime.
Indeed $\set{2} \cap \set{3} = \emptyset$ lies in it, while neither $\set{2}$ nor $\set{3}$ does.
In a [[distributive lattice|0HM0HQCK6N]], if an ideal $I$ and a filter $F$ are disjoint, then some prime ideal contains $I$ and is disjoint from $F$.
This is the prime ideal theorem for distributive lattices, and its proof uses a form of the axiom of choice.
</document>

<document source="https://tamnd.github.io/math2/node/984ENA3M5Q.md" tag="984ENA3M5Q" kind="definition">
# Sublinear function

## Definition

Let $X$ be a [[vector space|000AYPA9AJ]] over $\R$.
A *sublinear function* on $X$ is a map $p \colon X \to \R$ such that:

1. $p(tx) = t\, p(x)$ for all real $t > 0$ and all $x \in X$ (positive homogeneity).
2. $p(x + y) \le p(x) + p(y)$ for all $x, y \in X$ (subadditivity).

## Remarks

Every norm on $X$ is sublinear.
For example, $p(x) = \abs{x}$ on $\R$ satisfies $\abs{tx} = t\abs{x}$ for $t > 0$ and $\abs{x + y} \le \abs{x} + \abs{y}$.
Every linear functional is sublinear, and so is $p(x_1, x_2) = \max(x_1, x_2)$ on $\R^2$.
The map $p(x) = x^2$ on $\R$ is not sublinear: $p(2 \cdot 1) = 4$ but $2\, p(1) = 2$, so clause 1 fails.
Clause 1 with $t = 2$ and $x = 0$ gives $p(0) = 2p(0)$, so $p(0) = 0$; hence clause 1 also holds for $t = 0$.
A sublinear function $p$ with $p(-x) = p(x)$ for all $x$ is exactly a [[seminorm|FP5GVCN8GV]].
</document>

<document source="https://tamnd.github.io/math2/node/9879ZMW5AA.md" tag="9879ZMW5AA" kind="concept">
# Majority term

## Summary

A majority term is a ternary term $m(x, y, z)$ that returns the value shared by at least two of its three arguments.
It is the [[near-unanimity term|4CRH8NNFEF]] of the smallest possible arity.
Lattices and Boolean algebras have a majority term, and every variety with a majority term is congruence-distributive.
For a finite algebra with a majority term, the term operations are exactly the operations that preserve the subalgebras of its square.

## Explanations

### L1 human

Three people vote, and the majority term reports the opinion of at least two of them.
On the set $\set{0, 1}$ the majority function is $m(x, y, z) = 1$ exactly when at least two of $x, y, z$ equal $1$.
For example $m(1, 0, 1) = 1$ and $m(0, 0, 1) = 0$.
In a lattice the same role is played by $(x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$, the median of three elements in a chain.
</document>

<document source="https://tamnd.github.io/math2/node/98F5NBDR0V.md" tag="98F5NBDR0V" kind="definition">
# F-algebra

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $F \colon \mathcal{C} \to \mathcal{C}$ be a [[functor|28NQ0KRHD3]].
An *$F$-algebra* is a pair $(A, \alpha)$ of an object $A$ of $\mathcal{C}$ and a morphism $\alpha \colon F A \to A$.
A *homomorphism* of $F$-algebras $(A, \alpha) \to (B, \beta)$ is a morphism $h \colon A \to B$ with $h \circ \alpha = \beta \circ F h$.
With these morphisms, $F$-algebras form a category.

## Remarks

Example: on $\mathbf{Set}$, let $F X = \set{\ast} \sqcup X$.
The natural numbers with $\alpha(\ast) = 0$ and $\alpha(n) = n + 1$ form an $F$-algebra $(\N, \alpha)$.
More generally, for an algebraic signature $\sigma$ with $\sigma_n$ the set of $n$-ary symbols, the $F_\sigma$-algebras for $F_\sigma X = \bigsqcup_n \sigma_n \times X^n$ are exactly the $\sigma$-algebras, and their homomorphisms are the usual ones.

Non-example: a morphism $\gamma \colon A \to F A$ in the other direction does not make $A$ an $F$-algebra; such a pair $(A, \gamma)$ is an [[F-coalgebra|6BS0300CMV]].
For $F X = \set{\ast} \sqcup X$ on $\mathbf{Set}$, a coalgebra describes a system in which each state either stops or has one next state.

If $(T, \eta, \mu)$ is a monad, every algebra over the monad is a $T$-algebra in this sense, but not conversely, since no unit or associativity law is required here.
</document>

<document source="https://tamnd.github.io/math2/node/98GZ9DVHTA.md" tag="98GZ9DVHTA" kind="concept">
# Finite element method

## Summary

The finite element method is a way to compute approximate solutions of boundary value problems for [[partial differential equations|GC5BDZ6QBC]].
It writes the problem in [[weak form|SJZX81A783]] and splits the domain into small pieces such as triangles.
Then it looks for the best approximation among functions that are polynomials on each piece.
It is a [[Galerkin method|Y5B0PEBKE8]] whose basis functions are supported on a few pieces, so it leads to a large sparse linear system.
It handles complicated geometries well and is the standard tool of engineering simulation in structural mechanics, heat flow and electromagnetism.

## Explanations

### L1 human

Cover the region with a mesh of small triangles.
On each triangle, approximate the unknown function by a flat (linear) piece, and glue the pieces so that the result is continuous.
Each such function is fixed by its values at the corners of the triangles, so the problem becomes finding finitely many numbers.
The equation is not imposed point by point, but in an averaged sense against every function of the same kind.
For the equation $-u'' = f$ on an interval with "hat" functions on a uniform grid, this gives the tridiagonal system with rows $(-1, 2, -1)/h$.
</document>

<document source="https://tamnd.github.io/math2/node/98HTB7MSPX.md" tag="98HTB7MSPX" kind="concept">
# Foliation

## Summary

A foliation of a manifold is a decomposition of it into disjoint connected submanifolds of one fixed dimension, the leaves.
Locally the leaves fit together like the parallel slices $\R^p \times \set{y}$ of $\R^p \times \R^q$.
Foliations arise from integrable plane fields by the [[Frobenius theorem|5Q4BHBFXJY]] and from submersions.
They also arise from locally free group actions and from the orbits of flows without fixed points.
They are a basic object of differential topology and dynamics, because individual leaves can be dense or wind around in complicated ways.

## Explanations

### L1 human

Think of the pages of a book, or the layers of an onion without its centre.
Near any point the space looks like a stack of parallel sheets, but globally a single sheet can wander through the whole space.
On a torus, take all lines of a fixed slope.
If the slope is rational, each line closes into a circle; if it is irrational, each line wraps around forever and comes close to every point.
Both families are foliations of the torus by curves.
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<document source="https://tamnd.github.io/math2/node/98MVMS1NRR.md" tag="98MVMS1NRR" kind="statement">
# Thom–Sebastiani theorem

## Statement

Let $f(x_0, \dots, x_n)$ and $g(y_0, \dots, y_m)$ be holomorphic functions near the origin, each with an [[isolated singularity|VTGYEES3VS]] at $0$.
Let $f \oplus g$ be the function $(x, y) \mapsto f(x) + g(y)$ near $0$ in $\C^{n+1} \times \C^{m+1}$; it also has an isolated singularity at $0$.
Let $F_f$, $F_g$ and $F_{f \oplus g}$ be the Milnor fibres of the [[Milnor fibrations|KTFG1S2SMG]] at $0$, with [[monodromies|C8HD62AB6Q]] $T_f$, $T_g$ and $T_{f \oplus g}$ on [[reduced homology|NJQMWVK3QK]].
Then there is an isomorphism
$$\tilde H_{n+m+1}(F_{f \oplus g}; \Z) \cong \tilde H_n(F_f; \Z) \otimes \tilde H_m(F_g; \Z)$$
under which $T_{f \oplus g} = T_f \otimes T_g$.

## Remarks

Sebastiani and Thom proved the theorem in 1971.
The idea is that $F_{f \oplus g}$ has the homotopy type of the join $F_f * F_g$.
The reduced homology of a join is the tensor product of the reduced homologies, shifted by one degree.
The monodromy of $f \oplus g$ is the join of the two monodromies.

Taking ranks gives $\mu(f \oplus g) = \mu(f) \, \mu(g)$ for the [[Milnor numbers|1ZE51P7GRQ]].
This also follows from algebra, since the local algebra of $f \oplus g$ is the tensor product of the local algebras of $f$ and $g$.

Example: $f(x) = x^p$ and $g(y) = y^q$.
The Milnor fibre of $x^p$ is $p$ points, cyclically permuted by the monodromy, whose eigenvalues on reduced $H_0$ are the $p$-th roots of unity other than $1$.
So the monodromy of $x^p + y^q$ on $H_1$ has as eigenvalues the products $\zeta \eta$ with $\zeta^p = 1 \ne \zeta$ and $\eta^q = 1 \ne \eta$, and $\mu(x^p + y^q) = (p-1)(q-1)$.

## Explanations

### L1 human

If a function is a sum of two functions in separate sets of variables, its singularity is built from the two singularities.
The theorem says how: the topology of the sum is a product, in a precise sense, of the topologies of the summands.
For instance, the Milnor number of the sum is the product of the two Milnor numbers.

### L2 human

The theorem reduces many computations to one-variable functions.
Iterating it, for a Brieskorn–Pham polynomial $z_0^{a_0} + \cdots + z_n^{a_n}$, the monodromy is diagonalizable with eigenvalues $\zeta_0 \cdots \zeta_n$, where $\zeta_j^{a_j} = 1 \ne \zeta_j$, and $\mu = \prod_j (a_j - 1)$.
This is the computation behind the topology of [[Brieskorn manifolds|5E9E5R9RYZ]].
Adding a square $z^2$ in a new variable does not change $\mu$, and it changes the monodromy only by a sign.
This explains the stable behaviour of singularities under adding squares.
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<document source="https://tamnd.github.io/math2/node/98QASBAMEB.md" tag="98QASBAMEB" kind="concept">
# Rational singularity

## Summary

A rational singularity is a singular point that looks, from the point of view of sheaf cohomology, like a smooth point.
It is defined by the vanishing of the higher direct images of the structure sheaf under a [[resolution|MB52NDA6WN]].
Quotient singularities, toric singularities and the [[ADE singularities|HQNXQBZ6C7]] are rational.
Rational singularities play a central role in the [[minimal model program|Q8CTDCZH01]], since the log terminal singularities it uses are rational.

## Explanations

### L1 human

When a singular point is resolved, the new smooth variety may carry extra cohomology above the point.
A singularity is rational when no such extra cohomology appears in the structure sheaf.
The tip of the cone $x^2 + y^2 = z^2$ is rational: its resolution adds a single projective line, which has no first cohomology.
The tip of the cone over a smooth plane cubic is not: its resolution adds an elliptic curve, whose first cohomology is one-dimensional.
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<document source="https://tamnd.github.io/math2/node/98QE7KY55W.md" tag="98QE7KY55W" kind="concept">
# Weyl algebra

## Summary

The $n$-th Weyl algebra $A_n(k)$ over a field $k$ is the algebra generated by $x_1, \dots, x_n$ and $y_1, \dots, y_n$ with the relations $y_i x_i - x_i y_i = 1$.
All other pairs of generators commute.
In characteristic zero it is the [[ring of differential operators|GBWQF3HZG0]] with polynomial coefficients, where $x_i$ acts by multiplication and $y_i$ by $\partial / \partial x_i$.
In characteristic zero it is the basic example of a simple Noetherian domain that is not a division ring.
It also models the canonical commutation relation of quantum mechanics.

## Explanations

### L1 human

Take polynomials in $x$ and the operator $D = d/dx$ that differentiates.
By the product rule, $D(x f) = x D(f) + f$, so the operators satisfy $D x - x D = 1$.
The first Weyl algebra is the algebra built from two symbols with exactly this rule.
Every element can be written uniquely as a finite sum $\sum a_{ij} x^i D^j$, like a polynomial in two variables that do not commute.
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<document source="https://tamnd.github.io/math2/node/98TJTT6GZN.md" tag="98TJTT6GZN" kind="definition">
# Positive harmonic function

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]].
A *positive harmonic function* on $\Omega$ is a [[harmonic function|0DEB0JMN0W]] $u \colon \Omega \to \R$ such that $u(x) > 0$ for every $x \in \Omega$.

## Remarks

Example: on the upper half-plane $\set{(x, y) : y > 0}$, the function $u(x, y) = y$ is positive harmonic, since $\Delta u = 0$ and $y > 0$ there.
Non-example: $u(x, y) = x$ on the same half-plane is harmonic but takes negative values, for instance at $(-1, 1)$.
Some authors call a harmonic function with $u \ge 0$ nonnegative harmonic; on a connected open set, the minimum principle shows that such a function is either identically $0$ or positive everywhere.
On a ball, positive harmonic functions satisfy [[Harnack's inequality|YNH832D50W]].
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# Group action on a variety

## Definition

Let $k$ be a field, and let $G$ be an [[algebraic group|GH81SYAPPN]] over $k$ with multiplication $m$ and unit $e$.
Let $X$ be a [[variety|HV6YBS0YDH]] over $k$.
An *action* of $G$ on $X$ is a [[morphism|CDS1VWB7A3]] $\sigma \colon G \times_k X \to X$ such that:

1. $\sigma \circ (e \times \id_X)$ is the identity of $X$, where $\operatorname{Spec} k \times_k X$ is identified with $X$;
2. $\sigma \circ (m \times \id_X) = \sigma \circ (\id_G \times \sigma)$ as morphisms $G \times_k G \times_k X \to X$.

## Remarks

On points, with $g \cdot x = \sigma(g, x)$, the clauses say $e \cdot x = x$ and $(gh) \cdot x = g \cdot (h \cdot x)$.
So for each $k$-algebra $R$ one gets a [[group action|0WCY42EV9V]] of $G(R)$ on $X(R)$.

Example: $\mathrm{GL}_n$ acts on $\mathbb{A}^n$ by $\sigma(g, v) = g v$.
The coordinates of $g v$ are polynomials in the entries of $g$ and $v$, the identity matrix fixes every $v$, and $(gh) v = g (h v)$.

Non-example: the morphism $\sigma \colon \mathbb{G}_m \times \mathbb{A}^1 \to \mathbb{A}^1$, $\sigma(t, x) = x + t - 1$, satisfies clause 1.
It fails clause 2, since $\sigma(st, x) = x + st - 1$ while $\sigma(s, \sigma(t, x)) = x + s + t - 2$.
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<document source="https://tamnd.github.io/math2/node/993HPHG9VF.md" tag="993HPHG9VF" kind="definition">
# Admissible function

## Definition

Let $a < b$ and $A, B \in \R$, and let $J(y) = \int_a^b L(x, y, y') \, dx$ be an [[integral functional|JRQZ803Q5Y]] with a continuous Lagrangian $L \colon [a, b] \times \R \times \R \to \R$.
In the fixed-endpoint problem for $J$, a function $y \colon [a, b] \to \R$ is *admissible* if:

1. $y \in C^1([a, b])$, that is, $y$ is [[continuously differentiable|0VW90BBSJ4]] on $[a, b]$;
2. $y(a) = A$;
3. $y(b) = B$.

The set $\mathcal{A}$ of admissible functions is the *admissible class*.
More generally, the admissible class of a variational problem is the set of functions in the domain of the functional that satisfy all its constraints.
Its elements are the admissible functions.

## Remarks

Example: for $a = 0$, $b = 1$, $A = 0$, $B = 1$, the function $y(x) = x^2$ is admissible: it is $C^1$, $y(0) = 0$ and $y(1) = 1$.
Non-example: for $a = 0$, $b = 1$, $A = B = 1$, the function $y(x) = \abs{x - 1/2} + 1/2$ satisfies $y(0) = y(1) = 1$ but is not differentiable at $1/2$, so it fails clause 1.
Many texts use the larger class of piecewise $C^1$ functions, which admits such broken curves.
If $y$ is admissible and $\eta \in C^1([a, b])$ has $\eta(a) = \eta(b) = 0$, then $y + t\eta$ is admissible for every $t \in \R$; such $\eta$ are the admissible variations.
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# Esakia space

## Summary

An Esakia space is a [[Priestley space|ZGBV8V3823]] in which the down-set of every clopen set is again clopen.
Esakia spaces are the spaces dual to [[Heyting algebras|NG943XNQWH]], by [[Esakia duality|CD9Y7YP4EJ]].
The Heyting algebra of an Esakia space is its algebra of clopen up-sets.
They connect the algebraic semantics of [[intuitionistic logic|Q11N8X99FK]] with its [[Kripke semantics|NGHWZ9Y7ZV]].

## Explanations

### L1 human

An Esakia space is a compact space whose points are also partially ordered, in a way that fits the topology well.
One can think of the points as states of knowledge, where moving up in the order means learning more.
The extra rule says that the set of states below a clopen set is again clopen.
This rule makes the implication of two clopen up-sets again a clopen up-set.
Every finite partially ordered set, with the discrete topology, is an Esakia space.
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<document source="https://tamnd.github.io/math2/node/99709K2NJM.md" tag="99709K2NJM" kind="concept">
# Riesz space

## Summary

A Riesz space, or vector lattice, is an [[ordered vector space|HHMS3WH9T7]] whose order is a [[lattice|WH4SFAACAS]].
It is at the same time a real vector space and a [[lattice-ordered group|J0DHYQ0MDE]].
Spaces of real functions with the pointwise order, such as spaces of continuous functions and $L^p$ spaces, are Riesz spaces.
Riesz spaces are the setting for positive operators and for [[Banach lattices|HDJ1GKG0NA]] in functional analysis.

## Explanations

### L1 human

In the space of real functions on an interval, you can add functions, scale them, and compare them point by point.
For two functions you can also take their pointwise maximum and minimum.
A Riesz space is a vector space with an order where these maxima and minima always exist and behave well with addition and scaling.
For instance, the absolute value $\abs{f} = \max(f, -f)$ of a function makes sense in any Riesz space.
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<document source="https://tamnd.github.io/math2/node/99879Z5Y1K.md" tag="99879Z5Y1K" kind="example">
# Chandrasekhar's H-function

## Example

Let $\Psi$ be an even polynomial in $\mu$, the characteristic function, with $\Psi \ge 0$ on $[0, 1]$ and $\int_0^1 \Psi(\mu)\, d\mu \le \frac{1}{2}$.
*Chandrasekhar's H-function* is the solution $H$ on $[0, 1]$ of
$$H(\mu) = 1 + \mu\, H(\mu) \int_0^1 \frac{\Psi(\mu')\, H(\mu')}{\mu + \mu'}\, d\mu'.$$
One takes the solution that is analytic in $\mu$ in the right half-plane and continuous on $[0, 1]$.
The simplest case is isotropic scattering, $\Psi = \frac{\omega_0}{2}$ with albedo $0 < \omega_0 \le 1$.
The unknown appears in a product with itself, so this is a [[nonlinear integral equation|BSCYBP6AS6]].

Check of a moment identity.
Let $I = \int_0^1 \Psi(\mu) H(\mu)\, d\mu$.
Multiply the equation by $\Psi(\mu)$ and integrate over $[0, 1]$.
The double integral $\int_0^1 \int_0^1 \frac{\mu}{\mu + \mu'}\, \Psi(\mu) H(\mu)\, \Psi(\mu') H(\mu')\, d\mu'\, d\mu$ does not change when $\mu$ and $\mu'$ swap, so it equals half the integral with $\frac{\mu + \mu'}{\mu + \mu'} = 1$, that is $\frac{1}{2} I^2$.
So $I = \int_0^1 \Psi\, d\mu + \frac{1}{2} I^2$, and the physical root is
$$\int_0^1 \Psi(\mu) H(\mu)\, d\mu = 1 - \sqrt{1 - 2 \int_0^1 \Psi(\mu)\, d\mu}.$$
In the conservative case $\Psi = \frac{1}{2}$ this gives $\int_0^1 H(\mu)\, d\mu = 2$.
Also $H(0) = 1$, directly from the equation.

## Remarks

The function $H$ is tied to a Wiener–Hopf factorization.
For $z$ off the segment $[-1, 1]$, with the analytic continuation of $H$, one has
$$\frac{1}{H(z)\, H(-z)} = 1 - 2 z^2 \int_0^1 \frac{\Psi(\mu)}{z^2 - \mu^2}\, d\mu.$$
The right side is the dispersion function of the [[radiative transfer equation|EX9YYRMAJA]] in a half-space; $H(-z)$ is the factor holomorphic for $\operatorname{Re} z < 0$ and $H(z)$ the factor holomorphic for $\operatorname{Re} z > 0$.
As $z \to \infty$, both sides tend to $1 - 2\int_0^1 \Psi\, d\mu$, consistent with $H(\infty) = 1/(1 - I)$ from the equation.
In the conservative case, $H$ gives the exact angular distribution of the light leaving a grey atmosphere, the companion of the [[Milne problem|KKT4Y0TBRZ]].

## Explanations

### L1 human

Light that enters a thick cloud or a stellar atmosphere is scattered many times before some of it comes back out.
Chandrasekhar showed that the outgoing light is described by a single function $H$ of the cosine $\mu$ of the exit angle.
This function solves an equation in which it appears twice, once outside and once inside an integral.
Simple integral identities, such as $\int_0^1 H\, d\mu = 2$ for pure scattering, follow by symmetry without solving the equation.
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# Injective hull

## Definition

Let $R$ be a [[ring|0B1261B5ED]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].
An *essential extension* of $M$ is an injective homomorphism $i \colon M \to E$ of $R$-modules such that every nonzero [[submodule|05WDSN597C]] $N$ of $E$ satisfies $N \cap i(M) \ne 0$.
An *injective hull* (or injective envelope) of $M$ is an essential extension $i \colon M \to E$ such that $E$ is an [[injective module|85H7PRHDPZ]].
It is written $E(M)$ or $E_R(M)$.

## Remarks

Every module has an injective hull, and two injective hulls $i \colon M \to E$ and $i' \colon M \to E'$ are related by an isomorphism $f \colon E \to E'$ with $f \circ i = i'$.
Equivalently, an injective hull is a maximal essential extension, or a minimal injective module containing $M$.
Example: over $R = \Z$, the inclusion $\Z \to \Q$ is an injective hull; $\Q$ is divisible, hence injective, and a nonzero subgroup $N$ of $\Q$ containing $a/b \ne 0$ also contains $b \cdot (a/b) = a \in N \cap \Z$.
Non-example: the inclusion $\Z \to \Q \oplus \Q/\Z$, $n \mapsto (n, 0)$, has an injective target but is not essential, since the submodule $0 \oplus \Q/\Z$ meets the image only in $0$.
For a prime $p$, the injective hull of $\Z/p\Z$ over $\Z$ is the Prüfer group $\Z[1/p]/\Z$.
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<document source="https://tamnd.github.io/math2/node/99B3B7AAVT.md" tag="99B3B7AAVT" kind="definition">
# Weierstrass product

## Definition

For an integer $p \ge 0$, the *elementary factor* $E_p$ is the entire function
$$E_0(z) = 1 - z, \qquad E_p(z) = (1 - z) \exp\left(\sum_{k=1}^{p} \frac{z^k}{k}\right) \quad (p \ge 1).$$
Let $(a_n)_{n \ge 1}$ be a sequence of nonzero complex numbers with $\abs{a_n} \to \infty$, and let $(p_n)_{n \ge 1}$ be integers $p_n \ge 0$ such that
$$\sum_{n \ge 1} \left(\frac{r}{\abs{a_n}}\right)^{p_n + 1} < \infty \quad \text{for every } r > 0.$$
The *Weierstrass product* with zeros $(a_n)$ and indices $(p_n)$ is the function
$$P(z) = \prod_{n \ge 1} E_{p_n}\!\left(\frac{z}{a_n}\right).$$

## Remarks

Since $\abs{1 - E_p(z)} \le \abs{z}^{p+1}$ for $\abs{z} \le 1$, the product converges [[uniformly|08FPABDQ4C]] on compact subsets of $\C$.
So $P$ is an [[entire function|0419794R0T]], and its [[zeros|0CEGEASY49]] are exactly the $a_n$, with multiplicity the number of indices $n$ with $a_n = a$.
The choice $p_n = n - 1$ always satisfies the condition.
Example: $a_n = n$ and $p_n = 1$; then $\sum_n (r/n)^2 < \infty$, so $\prod_{n \ge 1} (1 - z/n) e^{z/n}$ is a Weierstrass product, with zeros at the positive integers.
Non-example: $a_n = n$ and $p_n = 0$ fail the condition, since $\sum_n r/n$ diverges; indeed $\prod_n (1 - z/n)$ diverges at $z = -1$.
When all $p_n$ equal the least integer $p \ge 0$ with $\sum_n \abs{a_n}^{-(p+1)} < \infty$, the product is called the canonical product, and $p$ is its genus.
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# Atomistic lattice

## Summary

A lattice with a least element is atomistic if every element is the join of the [[atoms|TTVCTW6TNC]] below it.
So the whole lattice is generated from its atoms by joins, as every subset of a set is the union of its one-element subsets.
Every atomistic lattice is [[atomic|5SE8EMSDNY]], but not conversely.
Geometric lattices, such as the lattice of subspaces of a vector space, are atomistic.

## Explanations

### L1 human

In an atomistic lattice the atoms are building blocks, and every element is put together from the atoms it contains.
In the subsets of a set, every set is the union of its points.
In the subspaces of a finite-dimensional vector space, the atoms are the lines through the origin, and every subspace is spanned by the lines it contains.
A chain $0 < 1 < 2$ is atomic but not atomistic: its only atom is $1$, and $2$ is not a join of atoms.
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<document source="https://tamnd.github.io/math2/node/99GNAS98PV.md" tag="99GNAS98PV" kind="definition">
# Lebesgue covering dimension

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
An [[open cover|049PXYHW1T]] $\mathcal{V}$ of $X$ *refines* an open cover $\mathcal{U}$ if each member of $\mathcal{V}$ lies in some member of $\mathcal{U}$.
The *order* of a family $\mathcal{V}$ is the largest $k$ such that some point of $X$ lies in $k + 1$ members of $\mathcal{V}$.
For an integer $n \ge -1$, write $\dim X \le n$ if every finite open cover of $X$ has a finite open refinement of order at most $n$.
The *Lebesgue covering dimension* $\dim X$ is the least $n$ with $\dim X \le n$, and $\dim X = \infty$ if there is no such $n$.

## Remarks

The empty space has $\dim \emptyset = -1$, and a nonempty finite discrete space has dimension $0$.
The interval $[0, 1]$ has $\dim [0, 1] = 1$.
Given a finite open cover, the [[Lebesgue number lemma|040CFNJEAP]] gives a number $\delta > 0$.
Refine the cover by finitely many relatively open intervals of length less than $\delta$ in which only neighbours overlap.
This refinement has order at most $1$.
The cover $\set{[0, 2/3), (1/3, 1]}$ has no open refinement of order $0$.
Such a refinement would be a cover by disjoint open sets, and by connectedness one of them would be all of $[0, 1]$.
But $[0, 1]$ lies in no member of the cover.
So $\dim [0, 1] \le 0$ fails.
Some authors use all open covers instead of finite ones; the two versions agree for compact spaces.
For every $n$, $\dim \R^n = n$.

## Explanations

### L1 human

Take the smallest number $k$ such that every finite open cover can be made finer with each point in at most $k$ of the sets.
The covering dimension of the space is $k - 1$.
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# Polynomial greatest common divisor

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f, g \in K[x]$ be polynomials that are not both $0$.
A greatest common divisor of $f$ and $g$ is a polynomial $d \in K[x]$ such that:

1. $d$ divides $f$ and $d$ divides $g$;
2. every polynomial that divides both $f$ and $g$ divides $d$.

The greatest common divisor $\gcd(f, g)$ is the unique [[monic|0BDX4MBKT4]] polynomial with these two properties.

## Remarks

Two greatest common divisors differ by a nonzero constant factor, so exactly one of them is monic.
Example: $\gcd(x^2 - 1, x^2 - 3x + 2) = x - 1$.
Check: $x - 1$ divides both, and a common divisor divides $(x^2 - 1) - (x^2 - 3x + 2) = 3x - 3$, hence divides $x - 1$.
Non-example: $2x - 2$ satisfies both clauses, but it is not monic, so it is a greatest common divisor and not $\gcd(f, g)$.
The [[Euclidean algorithm|023VY493K0]] computes $\gcd(f, g)$, and [[Bézout's identity|0VY3WPP9Y3]] gives $a, b \in K[x]$ with $a f + b g = \gcd(f, g)$.
Over a [[unique factorisation domain|06YMXJMXYS]] such as $\Z$, greatest common divisors in $\Z[x]$ also exist, but only up to units, see [[greatest common divisor in an integral domain|0F7KNJ1ZYN]].

## Explanations

### L1 human

The greatest common divisor is the largest polynomial that goes into both.
Asking for leading coefficient $1$ picks one answer among its constant multiples.
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# Isometry of Riemannian manifolds

## Definition

Let $(M, g)$ and $(N, h)$ be [[Riemannian manifolds|7CBT4SM83Y]].
Let $\varphi \colon M \to N$ be a [[diffeomorphism|0KZZ70ZHXP]], and let $d\varphi_p$ be its [[differential|6P9PXQDSEN]] at $p \in M$.
The map $\varphi$ is an *isometry* from $(M, g)$ to $(N, h)$ if, for all $p \in M$ and all $v, w \in T_pM$,
$$h_{\varphi(p)}\bigl(d\varphi_p(v), d\varphi_p(w)\bigr) = g_p(v, w).$$
Equivalently, $\varphi^* h = g$.

## Remarks

Example: let $A \in O(n + 1)$ be an orthogonal matrix.
The map $x \mapsto Ax$ restricts to a diffeomorphism of the unit sphere $S^n$ with differential $v \mapsto Av$.
Since $\ip{Av}{Aw} = \ip{v}{w}$, it is an isometry of the round sphere.

Non-example: the map $x \mapsto 2x$ of $\R^n$ is a diffeomorphism, but $\ip{2v}{2w} = 4 \ip{v}{w}$, so it does not preserve the metric.

Let $M$ and $N$ have the same dimension.
A smooth map $\varphi$ with $\varphi^* h = g$, not required to be a diffeomorphism, is a local isometry.
The covering map $\R \to S^1$, $t \mapsto (\cos t, \sin t)$, is an example.
An isometry preserves the [[Riemannian distance|B4QR306D8G]]; conversely, by the [[Myers–Steenrod theorem|Z3VRMW8ETQ]], a distance-preserving bijection between connected Riemannian manifolds is an isometry in this sense.
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# de Boor's algorithm

## Definition

Let $k \ge 1$ be an order and let $(t_i)$ be a nondecreasing knot sequence.
Let $s = \sum_i c_i B_{i,k}$, where $B_{i,k}$ are the [[B-splines|62AFX32JGT]] of order $k$ and the $c_i$ are real numbers or points of $\R^d$.
Let $x$ be a point with $t_j \le x < t_{j+1}$.
*De Boor's algorithm* computes $s(x)$ as follows.

1. Set $c_i^{[0]} = c_i$ for $i = j - k + 1, \dots, j$.
2. For $r = 1, \dots, k - 1$ and $i = j - k + 1 + r, \dots, j$, set
$$\alpha_{i,r} = \frac{x - t_i}{t_{i+k-r} - t_i}, \qquad c_i^{[r]} = (1 - \alpha_{i,r})\, c_{i-1}^{[r-1]} + \alpha_{i,r}\, c_i^{[r-1]}.$$
3. Return $c_j^{[k-1]}$, which equals $s(x)$.

## Remarks

The denominators are positive, since $t_i \le t_j < t_{j+1} \le t_{i+k-r}$ in the stated range, and $\alpha_{i,r} \in [0, 1]$, so each step is a [[convex combination|21SXYJYC59]].
Correctness follows from the [[Cox–de Boor recursion|WVD069C7QF]]: one step rewrites $\sum_i c_i B_{i,k}(x)$ as $\sum_i c_i^{[1]} B_{i,k-1}(x)$, and at order $1$ only $B_{j,1}(x) = 1$ survives.
Example: $k = 2$, knots $t_0 = 0, t_1 = 1, t_2 = 2$, $x = 1/2$, so $j = 0$; then $\alpha_{0,1} = 1/2$ and $s(1/2) = (c_{-1} + c_0)/2$, which is the piecewise linear interpolant at the midpoint.
Non-example: for $x$ outside $[t_j, t_{j+1})$ the same formulas with this $j$ do not give $s(x)$ in general, because other B-splines are active there.
The cost is $O(k^2)$ operations per point.
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# Central simple algebra

## Definition

Let $K$ be a [[field|04C5ZBG1WT]].
A *central simple algebra* over $K$ is an [[associative algebra|3CTW1P4HBP]] $A$ over $K$ with these three properties:

1. $A$ is finite-dimensional as a vector space over $K$.
2. $A$ is a [[simple ring|62M0NC3E2T]]: $A \ne 0$ and its only two-sided ideals are $0$ and $A$.
3. The [[center|MVT64B3B6H]] of $A$ is $K \cdot 1$.

## Remarks

Example: $M_n(K)$ is a central simple algebra over $K$ for $n \ge 1$.
It has dimension $n^2$, its only two-sided ideals are $0$ and itself, and a matrix that commutes with all matrix units $E_{ij}$ is a scalar matrix.

Non-example: $\C$ as an algebra over $\R$ is finite-dimensional and simple, but its center is $\C \ne \R$, so clause 3 fails.
Non-example: $K \times K$ is commutative of dimension $2$ and has the ideal $K \times 0$, so clause 2 fails.

Hamilton's quaternions $\mathbb{H}$ are a central simple algebra over $\R$ of dimension $4$.
The dimension of a central simple algebra is always a square $n^2$, and $n$ is its degree.
If $A$ and $B$ are central simple over $K$, so is $A \otimes_K B$; and $A \otimes_K A^{\mathrm{op}} \cong M_{n^2}(K)$ when $\dim_K A = n^2$.
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# Superadditive function

## Summary

A superadditive function is a function $f$ with $f(x + y) \ge f(x) + f(y)$ for all $x, y$ in its domain.
It is the mirror image of a [[subadditive function|3X9DD9J3CV]]: $f$ is superadditive exactly when $-f$ is subadditive.
For superadditive sequences, [[Fekete's lemma|8A4GGX1MEY]] gives that $a_n / n$ converges to its supremum, which may be $+\infty$.
Superadditivity describes situations where joining parts gains something, as for convex functions vanishing at $0$.

## Explanations

### L1 human

A function is superadditive when the whole is worth at least the sum of its parts.
The square function on $[0, \infty)$ is an example: $(3 + 4)^2 = 49$ is more than $3^2 + 4^2 = 25$.
The number of pairs that can be formed from a group of people is another: two groups merged have all the old pairs and some new ones.
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<document source="https://tamnd.github.io/math2/node/99ZVSM254K.md" tag="99ZVSM254K" kind="example">
# Category of abelian groups

## Example

The *category of abelian groups* $\mathbf{Ab}$ has the [[abelian groups|082GHNPJ53]] as objects and the [[group homomorphisms|0JYTG5K9Y4]] between them as morphisms, with composition of maps.
It is an [[abelian category|Y9CZ3MTNF1]].
Each set $\Hom(A, B)$ is an abelian group under pointwise addition $(f + g)(a) = f(a) + g(a)$, and composition is bilinear.
The trivial group is a [[zero object|AQET9YHH6F]], and $A \oplus B$ is both the product and the coproduct of $A$ and $B$.
Every homomorphism $f \colon A \to B$ has a [[kernel|S417YNEJ6M]] $\set{a \in A : f(a) = 0}$ and a [[cokernel|H8W38933NC]] $B / f(A)$.
An injective homomorphism $f$ is the kernel of the projection $B \to B/f(A)$.
A surjective homomorphism is the cokernel of the inclusion of its kernel.
So every monomorphism is a kernel and every epimorphism is a cokernel.

## Remarks

$\mathbf{Ab}$ is the model case of an abelian category: the [[Freyd–Mitchell embedding theorem|8V296XR2M3]] says that each small abelian category embeds fully and exactly into a category of modules.
It is also a full subcategory of the [[category of groups|2CX7RPPXA9]], but the coproducts differ: in $\mathbf{Ab}$ the coproduct of $\Z$ and $\Z$ is $\Z \oplus \Z$, not the free product.
The group $\Z$ is a [[generator|CK08JW32WB]], and $\Q/\Z$ is an [[injective object|5QRFTBF4PG]].
$\mathbf{Ab}$ is the same as the [[category of modules|JBSSXP4FFY]] over the ring $\Z$.

## Explanations

### L1 human

In the category of abelian groups, maps between two groups can themselves be added, because the groups are commutative.
This extra structure lets one talk about kernels, images and exact sequences, which is the starting point of homological algebra.
For example, multiplication by $2$ on $\Z$ is a morphism with kernel $0$ and cokernel $\Z/2\Z$.
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# Hurwitz-stable matrix

## Summary

A Hurwitz-stable matrix is a [[square matrix|0JXZKX526R]] whose [[eigenvalues|0BQAA2BK3Q]] all have negative real part.
These are exactly the matrices $A$ for which every solution of $x' = A x$ tends to $0$.
Through linearisation they also decide the [[exponential stability|VY180V8QY2]] of equilibria of nonlinear systems.
They are also called stable matrices or Hurwitz matrices.
See the [[definition|R3VWS51XY5]].

## Explanations

### L1 human

Each eigenvalue $\lambda$ of $A$ contributes terms like $e^{\lambda t}$ to the solutions of $x' = Ax$.
When the real part of $\lambda$ is negative, such a term dies out, possibly while rotating.
So a Hurwitz-stable matrix is one whose linear system always returns to rest.
For example, $\begin{pmatrix} -1 & 2 \\ 0 & -3 \end{pmatrix}$ is Hurwitz-stable, with eigenvalues $-1$ and $-3$.
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# Universal Horn class

## Summary

A universal Horn class is the class of all structures of a signature that satisfy a given set of universal Horn sentences.
Such a sentence is a universally quantified [[Horn clause|QR92EZDTKF]] built from first-order atomic formulas.
Universal Horn classes are [[universal classes|ZSTEZ3PRT9]], and in addition they are closed under direct products of nonempty families.
In an algebraic signature, the classes axiomatized by universal Horn sentences with a positive literal are exactly the [[quasivarieties|WRXEKT4NQS]].

## Explanations

### L1 human

A universal Horn sentence says: for all elements, if these basic facts hold, then that basic fact holds too, or else that these facts never hold together.
Partial orders form a universal Horn class: reflexivity, antisymmetry and transitivity all have this shape.
Total orders do not, because "$x \le y$ or $y \le x$" offers a choice between two facts.
Indeed the product of two 2-element chains is a square, in which two corners are incomparable.
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# Pettis integral

## Summary

The Pettis integral of a function with values in a Banach space is a vector in that space.
Each continuous linear functional maps it to the integral of the corresponding scalar function.
It is a weak notion of integral, defined through the dual space.
Every Bochner integrable function is Pettis integrable with the same integral, but in infinite dimensions there are Pettis integrable functions that are not Bochner integrable.
It is also called the Gelfand–Pettis integral.

## Explanations

### L1 human

To integrate a vector-valued function, one can integrate each coordinate.
In an infinite-dimensional space, the role of coordinates is played by continuous linear functionals.
The Pettis integral is a vector that has the right value in every such coordinate, for every measurable set.
For a function into $\R^2$, it is the vector whose two coordinates are the integrals of the two components of the function.
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# Discrete series representation

## Summary

A discrete series representation of a locally compact group $G$ is an irreducible unitary representation.
It occurs as a direct summand of the regular representation on $L^2(G)$.
These are the representations that contribute point masses to the Plancherel measure.
Let $G$ be a connected semisimple Lie group with finite center.
Harish-Chandra showed that $G$ has discrete series exactly when it has a compact Cartan subgroup, and he classified them.
They are the building blocks from which the [[Langlands classification|YHFP9HGBTT]] constructs all irreducible admissible representations.

## Explanations

### L1 human

The functions on a group can be broken into pieces on which the group acts irreducibly, much as a sound breaks into pure tones.
For the circle or the real line, all pieces come in continuous families, like frequencies.
Some groups, such as $\mathrm{SL}_2(\R)$, also have isolated pieces that appear on their own, with positive "weight".
These isolated pieces are the discrete series.
For $\mathrm{SL}_2(\R)$, they are realised on spaces of holomorphic functions on the upper half-plane and are closely linked to modular forms.
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# Jacobi field

## Summary

A Jacobi field is a vector field along a [[geodesic|NSYMQMHPDN]] that solves the Jacobi equation, a linear second-order equation involving the [[Riemann curvature tensor|JA29ETCM02]].
Jacobi fields are exactly the variation fields of families of geodesics, so they describe how nearby geodesics spread apart or come together.
They connect curvature with the behaviour of geodesics and lead to [[conjugate points|ZPXE5VSZ35]], comparison theorems and the second variation of length.

## Explanations

### L1 human

Take a family of geodesics that start close together, like lines of longitude leaving the north pole.
The small gap between neighbouring geodesics, as one moves along them, is a Jacobi field.
In a flat plane, such gaps grow at a constant rate, linearly.
On a sphere, positive curvature pulls the geodesics back together: the gap between two meridians grows like $\sin t$ and closes again at the south pole.
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# Prime radical

## Definition

Let $R$ be a [[ring|0894J980RK]].
The *prime radical* of $R$ is the intersection of all [[prime ideals|E8K2MN5V1E]] of $R$:
$$\operatorname{Nil}_*(R) = \bigcap_{P \text{ prime}} P.$$
It is a [[two-sided ideal|0DDA6CY7GJ]] of $R$.

## Remarks

Example: let $R$ be the ring of upper triangular $2 \times 2$ matrices over a field $k$ and $N$ its strictly upper triangular matrices.
Then $R/N \cong k \times k$, the prime ideals of $R$ are the kernels of the two maps $R \to k$ that take a diagonal entry, and their intersection is $N$.
Non-example: the matrix unit $e_{11}$ is not in the prime radical of $R$, since it is a nonzero idempotent, while every element of the prime radical is nilpotent.
Equivalent forms: $\operatorname{Nil}_*(R)$ is the smallest semiprime ideal of $R$, and it is the set of strongly nilpotent elements of $R$.
It is also called the lower nilradical or the Baer radical, and it is contained in the [[upper nilradical|3G2JQEYBEK]].
For a commutative ring it equals the [[nilradical|QQDN6EYQNJ]].
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# Brunn–Minkowski theorem

## Statement

Let $n \ge 1$, and let $\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$.
Let $A, B \subseteq \R^n$ be nonempty [[compact sets|0R6EWFKRCF]], and let $A + B = \set{a + b : a \in A, \ b \in B}$ be their Minkowski sum.
Then
$$\lambda(A + B)^{1/n} \ge \lambda(A)^{1/n} + \lambda(B)^{1/n}.$$

## Remarks

For compact $A$ and $B$ the set $A + B$ is compact, hence measurable; for general measurable sets $A + B$ need not be measurable.
One proof first checks boxes, where the inequality is the AM–GM inequality for the side ratios.
It then passes to finite unions of disjoint boxes by induction on the number of boxes, cutting both sets by a coordinate hyperplane in matching proportions.
Compact sets are then approximated from outside by such unions.
Another proof applies the [[Prékopa–Leindler inequality|J9CJX9F42Y]] to indicator functions.
For compact convex sets of positive volume, equality holds if and only if $A$ and $B$ are homothetic.
Example: for balls of radii $r$ and $s$, $A + B$ is a ball of radius $r + s$, and both sides equal $\omega_n^{1/n} (r + s)$, where $\omega_n$ is the volume of the unit ball.

## Explanations

### L1 human

Add two shapes by adding every point of one to every point of the other.
The theorem says that the volume of the sum is at least as large as a natural guess.
After taking $n$-th roots, volume behaves at least additively under this addition.
This one inequality hides many facts about volume and shape.

### L2 human

The function $t \mapsto \lambda((1-t) A + t B)^{1/n}$ is concave on $[0, 1]$ for convex $A$ and $B$; this concavity form goes back to Brunn.
Applied to $A$ and a small ball, the theorem gives the [[isoperimetric inequality|KBX8RBZ626]].
It is the starting point of the theory of mixed volumes, and it has a functional form, the Prékopa–Leindler inequality, used in probability and convex geometry.
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# Projective plane

## Definition

A *projective plane* is an [[incidence structure|FGKNAFNHA3]] of points and lines such that:

1. Any two distinct points lie on exactly one common line.
2. Any two distinct lines have exactly one common point.
3. There are four points, no three of which are [[collinear|0AFNNTRTJE]].

## Remarks

Example: for a field $k$, the projective plane $\mathbb{P}^2(k)$ has as points the one-dimensional subspaces of $k^3$ and as lines the two-dimensional subspaces, with incidence given by inclusion.
Check: two distinct one-dimensional subspaces span exactly one plane.
Two distinct planes in $k^3$ meet in a line through $0$, by the dimension formula $2 + 2 - 3 = 1$.
The points $[1:0:0]$, $[0:1:0]$, $[0:0:1]$, $[1:1:1]$ have no three collinear.
For $k = \F_2$ this is the Fano plane, with $7$ points and $7$ lines.

Non-example: an [[affine plane|EC7XT2ZM5S]] satisfies clauses 1 and 3 but not clause 2, since it has parallel lines.

Clause 3 excludes degenerate cases, such as a single line with all points on it.
The axioms are self-dual: exchanging points and lines turns each clause into a statement that also holds, so the dual of a projective plane is a projective plane.
In a [[finite projective plane|2HWB7P1DZN]] every line has $n + 1$ points for some $n \ge 2$, the order of the plane.
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# Thurston norm

## Summary

The Thurston norm is a [[seminorm|FP5GVCN8GV]] on the second homology of a compact orientable [[3-manifold|R20TB2KEQQ]].
It measures the least complexity, in terms of [[Euler characteristic|RNDDWDAFJ5]], of an embedded surface representing a given class.
Thurston showed that its unit ball is a polyhedron.
The classes represented by fibers of fibrations over the circle fill the cones over certain top-dimensional faces.
It connects the topology of surfaces in $3$-manifolds with fibrations over the circle.

## Explanations

### L1 human

Every $2$-dimensional homology class of a $3$-manifold is represented by embedded surfaces, often in many ways.
The Thurston norm of the class is the smallest complexity of such a surface, where a surface of genus $g \ge 1$ counts $2g - 2$ and spheres and disks count nothing.
For the complement of a nontrivial knot, the norm of the generator is $2g - 1$ with $g$ the genus of the knot, so the norm detects the simplest Seifert surface.
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# Class number

## Definition

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] and $\operatorname{Cl}(K)$ its [[ideal class group|WQ46BD3ARY]].
The *class number* of $K$ is the [[order|0B85S2V2DR]] of this group:
$$h_K = \#\operatorname{Cl}(K).$$
It is a positive integer by the [[finiteness of the class number|F99RM1T02C]].

## Remarks

Example: for $K = \Q(i)$, the ring of integers $\Z[i]$ is a [[Euclidean domain|0PN13BB7BT]], hence a principal ideal domain, so $\operatorname{Cl}(K)$ is trivial and $h_K = 1$.
Example: for $K = \Q(\sqrt{-5})$, $\operatorname{Cl}(K) \cong \Z/2\Z$, so $h_K = 2$.
Non-example: for a general [[Dedekind domain|7NPY2CN5KB]] the class group can be infinite, and then there is no finite class number.
For instance, let $E$ be the elliptic curve $y^2 = x^3 - x$ over $\C$.
The class group of its affine coordinate ring $\C[x, y]/(y^2 - x^3 + x)$ is isomorphic to the infinite group $E(\C)$.
By the [[Stark–Heegner theorem|KZTFH832H0]], the imaginary quadratic fields $\Q(\sqrt{d})$ with $h = 1$ are those with $d = -1, -2, -3, -7, -11, -19, -43, -67, -163$.
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# Mehler kernel

## Statement

Let $H_n$ be the (physicists') [[Hermite polynomial|MGACY98E9H]] of degree $n$.
Let $\rho \in \C$ with $\abs{\rho} < 1$, and let $x, y \in \R$.
Then (Mehler's formula)
$$\sum_{n=0}^{\infty} \frac{H_n(x) H_n(y)}{n!} \left( \frac{\rho}{2} \right)^n = \frac{1}{\sqrt{1 - \rho^2}} \exp\left( \frac{2\rho x y - \rho^2 (x^2 + y^2)}{1 - \rho^2} \right),$$
where the square root is the principal one.
The right-hand side, as a function of $(x, y)$, is the *Mehler kernel*.

## Remarks

Proof idea: insert the integral formula $H_n(x) = \frac{2^n}{\sqrt{\pi}} \int_{\R} (x + it)^n e^{-t^2} \, dt$ for both factors.
The sum over $n$ becomes an exponential, and the remaining double integral is Gaussian and can be computed in closed form.
Check at $x = y = 0$: $H_0(0) = 1$, $H_2(0) = -2$ and $H_4(0) = 12$ give $1 + \frac{\rho^2}{2} + \frac{3\rho^4}{8} + \cdots$, which is the expansion of $(1 - \rho^2)^{-1/2}$.
In terms of the [[Hermite functions|52VS07H6GK]] $\psi_n$ the formula reads
$$\sum_{n=0}^{\infty} \rho^n \psi_n(x) \psi_n(y) = \frac{1}{\sqrt{\pi (1 - \rho^2)}} \exp\left( -\frac{(1 + \rho^2)(x^2 + y^2) - 4\rho x y}{2 (1 - \rho^2)} \right).$$
So the Mehler kernel is the integral kernel of the operator that multiplies the $n$-th Hermite function by $\rho^n$.
The hypothesis $\abs{\rho} < 1$ is needed: at $\rho = 1$ the right side is undefined and the series does not converge to a function.

## Explanations

### L1 human

Mehler's formula adds up an infinite series of products of Hermite polynomials and gets a single Gaussian expression.
It is the Hermite analogue of summing a geometric series.
The resulting kernel describes how a quantum harmonic oscillator evolves in time.
It also describes a particle pulled back to the origin by a spring and shaken by noise.

### L2 human

Write $N$ for the operator with $N \psi_n = n \psi_n$, the number operator of the harmonic oscillator.
Then the Mehler kernel with $\rho = e^{-t}$, $t > 0$, is the kernel of $e^{-tN}$, a smoothing semigroup on $L^2(\R)$.
In probability, the probabilists' version of the same formula gives the transition density of the Ornstein–Uhlenbeck process.
Letting $\rho$ tend to a point $e^{-i\alpha}$ of the unit circle leads to the kernel of the fractional Fourier transform.
The formula is a model for bilinear generating functions of other orthogonal polynomials, such as the Hille–Hardy formula for Laguerre polynomials.
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# Height of a prime ideal

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $P$ be a [[prime ideal|06HC6HK0Z8]] of $R$.
The *height* $\operatorname{ht} P$ of $P$ is the supremum of the integers $n$ for which there is a chain of prime ideals $P_0 \subsetneq P_1 \subsetneq \cdots \subsetneq P_n = P$ of $R$.

## Remarks

The height is an element of $\set{0, 1, 2, \dots} \cup \set{\infty}$, and $\operatorname{ht} P = \dim R_P$, the [[Krull dimension|1B4G153ZGE]] of the localization at $P$.
Example: in $k[x, y]$ over a field $k$, $\operatorname{ht}(x, y) = 2$, because of the chain $(0) \subsetneq (x) \subsetneq (x, y)$, and the dimension of $k[x, y]$ is $2$.
Non-example of a chain ending at $(x, y)$: $(x^2) \subsetneq (x) \subsetneq (x, y)$ does not give height $3$, because $(x^2)$ is not prime.
The prime ideals of height $0$ are the [[minimal prime ideals|S2D8823P8B]] of $R$.
For any ideal $I \ne R$, the height of $I$ is the least height of a prime ideal that contains $I$.
One has $\operatorname{ht} P + \dim R/P \le \dim R$.

## Explanations

### L1 human

The height of a prime ideal is the length of the longest strictly increasing chain of prime ideals that ends at it.
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# Orderable group

## Summary

An orderable group, or bi-orderable group, is a [[group|G5S1M9]] that admits a total order preserved by multiplication on both sides.
In other words, it is a group that can be made into a [[linearly ordered group|P2E8BNYC7B]].
Orderability is a property of the group, not extra structure, and a group can have many different orders.
Torsion-free abelian groups and free groups are orderable, while groups with nontrivial elements of finite order are not.

## Explanations

### L1 human

Some groups can have their elements lined up so that the line is respected by the group operation.
The integers and the rationals under addition are examples.
A group with an element $g \ne e$ and $g^n = e$ cannot be lined up like this.
If $g$ were larger than $e$, then $g^2$, $g^3$, and so on would be larger still, and $g^n = e$ could not happen.
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# Axiom schema of replacement

## Definition

Work in first-order logic with the relation symbol $\in$.
Let $\varphi(x, y, w_1, \dots, w_n)$ be a formula whose free variables are among $x, y, w_1, \dots, w_n$, and in which $A$ and $B$ do not occur.
The *instance of replacement* for $\varphi$ is the axiom
$$\forall w_1 \cdots \forall w_n\, \forall A\, \Bigl( \forall x \in A\, \exists! y\, \varphi(x, y, w) \to \exists B\, \forall y\, \bigl( y \in B \leftrightarrow \exists x \in A\, \varphi(x, y, w) \bigr) \Bigr).$$
The *axiom schema of replacement* is the set of these axioms, one for each formula $\varphi$.

In words: if $\varphi$ defines a [[function|0XHDTE5YMZ]] on the [[set|0SJ3RX9TXA]] $A$, then the image of $A$ under this function is a set.

## Remarks

Example: let $\varphi(x, y)$ say "$x$ is a natural number and $y$ is the set $\omega + x$".
For $A = \omega$ the hypothesis holds, so replacement gives the set $\set{\omega + n : n \in \omega}$, whose union is the ordinal $\omega + \omega$.
Non-example: the set $V_{\omega + \omega}$ of all sets of rank below $\omega + \omega$ satisfies all axioms of Zermelo set theory, but not replacement.
In it, the instance above fails: each $\omega + n$ lies in $V_{\omega + \omega}$, but the set $\set{\omega + n : n \in \omega}$ has rank $\omega + \omega$ and does not.
Replacement implies the separation scheme, given the other axioms of [[ZF|0HE5WDQ9FJ]].
It is used to justify definitions by transfinite recursion on the ordinals.
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# Stability of a difference equation

## Definition

Let $D \subseteq \R^m$, let $\norm{\cdot}$ be a [[norm|0V2WP9B1MF]] on $\R^m$, and let $f \colon D \to D$ be a map.
Let $x^* \in D$ be an [[equilibrium|D62PEKHRV5]] of $x_{n+1} = f(x_n)$, so $f(x^*) = x^*$.
Write $f^n$ for the $n$-th iterate of $f$.

1. The equilibrium $x^*$ is *stable* if for every $\varepsilon > 0$ there is $\delta > 0$ such that $\norm{x_0 - x^*} < \delta$ and $x_0 \in D$ imply $\norm{f^n(x_0) - x^*} < \varepsilon$ for all $n \ge 0$.
2. It is *attracting* if there is $\eta > 0$ such that $\norm{x_0 - x^*} < \eta$ and $x_0 \in D$ imply $f^n(x_0) \to x^*$ as $n \to \infty$.
3. It is *asymptotically stable* if it is stable and attracting.
4. It is *unstable* if it is not stable.

## Remarks

Example: for $f(x) = x/2$ on $\R$, the equilibrium $0$ is asymptotically stable: take $\delta = \varepsilon$, since $\abs{f^n(x_0)} = \abs{x_0}/2^n \le \abs{x_0}$, and $f^n(x_0) \to 0$ for every $x_0$.
Non-example: for $f(x) = 2x$, the equilibrium $0$ is not stable, since $\abs{f^n(x_0)} = 2^n \abs{x_0}$ exceeds any $\varepsilon$ for $x_0 \ne 0$.
For $f(x) = -x$, the equilibrium $0$ is stable but not attracting, as solutions jump between $x_0$ and $-x_0$.
Since all norms on $\R^m$ are equivalent, the notions do not depend on the choice of norm.
For a higher-order equation, the same definitions apply to the equivalent first-order system on $\R^{mk}$.
They are the discrete analogues of the definitions of [[Lyapunov stability|092S6VBECQ]] and [[asymptotic stability|05Y1JAM3Y5]] for flows.
</document>

<document source="https://tamnd.github.io/math2/node/9BS2PM0227.md" tag="9BS2PM0227" kind="definition">
# Coercive bilinear form

## Definition

Let $H$ be a real [[Hilbert space|HM2HHE3J51]] with norm $\norm{\cdot}$.
A [[bilinear form|0SQZMXB6GP]] $a \colon H \times H \to \R$ is *coercive* if there is a constant $\alpha > 0$ such that
$$a(u, u) \ge \alpha \norm{u}^2 \quad \text{for all } u \in H.$$

## Remarks

Example: the inner product $a(u, v) = \ip{u}{v}$ is coercive with $\alpha = 1$, since $a(u, u) = \norm{u}^2$.
Non-example: on $\R^2$, the form $a(u, v) = u_1 v_1$ is not coercive, because $a(u, u) = 0$ for $u = (0, 1)$, while $\norm{u}^2 = 1$.
The same definition is used on a real normed space.
On a complex Hilbert space, a sesquilinear form $a$ is called coercive if $\operatorname{Re} a(u, u) \ge \alpha \norm{u}^2$ for all $u$.
Some authors call a coercive form elliptic.
In the Lax–Milgram theorem the form is also assumed bounded: $\abs{a(u, v)} \le M \norm{u} \norm{v}$ for some constant $M$ and all $u, v \in H$.
</document>

<document source="https://tamnd.github.io/math2/node/9BWWD4QK8Y.md" tag="9BWWD4QK8Y" kind="concept">
# Maxwell's equations

## Summary

Maxwell's equations are four partial differential equations that govern the electric field $E$ and the magnetic field $B$.
They relate the fields to their sources, the electric charge density and the current density.
They unify electricity, magnetism and light, since in vacuum each field component solves the [[wave equation|5EHD2XBQDG]] with the speed of light.
They form a linear first-order system and are a basic model for hyperbolic systems in mathematical physics.

## Explanations

### L1 human

Electric charges create electric fields, and there are no magnetic charges.
A magnetic field that changes in time creates a circulating electric field, which is how a generator works.
An electric current, or an electric field that changes in time, creates a circulating magnetic field.
These four rules feed each other, so a disturbance in the fields travels through empty space as a wave: this wave is light.
</document>

<document source="https://tamnd.github.io/math2/node/9C1WSBNQX9.md" tag="9C1WSBNQX9" kind="concept">
# Polish group

## Summary

A Polish group is a [[topological group|92700QAB5Y]] whose topology is a [[Polish space|584GXXY73M]], that is, separable and completely metrizable.
Polish groups are the natural setting for descriptive set theory of group actions.
Every second-countable locally compact Hausdorff group is Polish, and so is the group of all permutations of $\N$.
Another example is the unitary group of a separable Hilbert space with the strong operator topology.
Their theory combines the Baire category theorem with the group structure, for example through automatic continuity results.

## Explanations

### L1 human

A Polish group is a group that has a good notion of distance, with no holes, and a countable set of points that comes close to everything.
The real line under addition is one, and so is the circle.
A larger example is the group of all rearrangements of the natural numbers, where two rearrangements are close when they agree on a long initial list of numbers.
These groups are big, but their topology is tame enough for strong theorems.
</document>

<document source="https://tamnd.github.io/math2/node/9C4MPVMTA7.md" tag="9C4MPVMTA7" kind="statement">
# Edgeworth expansion

## Statement

Let $X_1, X_2, \dots$ be [[independent and identically distributed|0AJN7WHBJZ]] real random variables with mean $\mu$, variance $\sigma^2 > 0$ and $E\abs{X_1}^3 < \infty$.
Let $\mu_3 = E\big((X_1 - \mu)^3\big)$ be the third [[central moment|0ZT24EC90K]].
Assume that the distribution of $X_1$ is not a lattice distribution.
That is, it is not concentrated on a set of the form $\set{a + kh : k \in \Z}$ with $h > 0$.
Let $F_n$ be the distribution function of $S_n = \big(\sum_{i=1}^n X_i - n\mu\big)/(\sigma\sqrt{n})$, and let $\Phi$ and $\varphi$ be the distribution function and the density of the standard [[normal distribution|07RXBVTVZB]].
Then
$$F_n(x) = \Phi(x) - \frac{\mu_3}{6 \sigma^3 \sqrt{n}} (x^2 - 1)\, \varphi(x) + o\!\left(\frac{1}{\sqrt{n}}\right)$$
uniformly in $x \in \R$.

## Remarks

The proof compares characteristic functions.
The characteristic function of $S_n$ is $e^{-t^2/2}\big(1 + \frac{\mu_3}{6\sigma^3\sqrt{n}}(it)^3\big) + o(n^{-1/2})$ for $t$ in ranges of order $\sqrt{n}$, and the correction term is the Fourier transform of the correction in the statement.
A smoothing inequality turns closeness of characteristic functions into closeness of distribution functions; the non-lattice hypothesis controls the characteristic function away from $0$.

The [[central limit theorem|0EPZC4Z1FC]] says $F_n(x) \to \Phi(x)$; the expansion gives the first correction, of order $1/\sqrt{n}$, which comes from skewness.
For a symmetric distribution, $\mu_3 = 0$ and the error of the normal approximation is $o(1/\sqrt{n})$.
Under stronger moment conditions and Cramér's condition $\limsup_{\abs{t} \to \infty} \abs{E e^{itX_1}} < 1$, one gets further terms in powers of $1/\sqrt{n}$ with Hermite polynomials and higher cumulants.

The non-lattice hypothesis cannot be dropped.
For Bernoulli($1/2$) variables, $F_n$ has jumps of size of order $1/\sqrt{n}$, while the right side is continuous in $x$, so the error cannot be $o(1/\sqrt{n})$ uniformly.

## Explanations

### L1 human

The central limit theorem says that a standardized sum is approximately normal.
The Edgeworth expansion says how the first error looks.
If the summands are skewed, the distribution of the sum is a little skewed too, by an amount that shrinks like $1/\sqrt{n}$.
The formula gives the shape of this correction.

### L2 human

The Berry–Esseen theorem bounds the error of the normal approximation by $C\, E\abs{X_1 - \mu}^3/(\sigma^3\sqrt{n})$; the Edgeworth expansion identifies the leading term of that error.
Expansions of this kind explain why the bootstrap is often more accurate than the normal approximation for studentized statistics.
They also give corrected quantiles, through the Cornish–Fisher expansion, and corrected confidence intervals.
</document>

<document source="https://tamnd.github.io/math2/node/9C5WXYYQEK.md" tag="9C5WXYYQEK" kind="concept">
# Subderivative

## Summary

Take a [[convex function|07Q7CNM5WF]] and a point of its graph.
A subderivative there is the slope of a line through that point that stays below the whole graph.
At every interior point a convex function has at least one subderivative, even where it has no [[derivative|0V1EJCT8G6]].
The set of all subderivatives at a point is the subdifferential, and it replaces the derivative in the optimality conditions of nonsmooth convex optimization.
The [[definition|A289X08MR1]] gives the precise condition.

## Explanations

### L1 human

At a smooth point of a convex graph there is one tangent line, and it lies below the graph.
At a corner there are many lines that touch the graph there and stay below it.
Their slopes are the subderivatives.
For $\abs{x}$ at $0$, every slope between $-1$ and $1$ works, so the subdifferential is $[-1, 1]$.
A point is a minimum of a convex function exactly when $0$ is one of the subderivatives there.
</document>

<document source="https://tamnd.github.io/math2/node/9C8SS4522J.md" tag="9C8SS4522J" kind="statement">
# Cahen–Mellin integral

## Statement

Let $c > 0$ and let $y \in \C$ with $\operatorname{Re} y > 0$.
Let $y^{-s} = e^{-s \log y}$, with the principal branch of $\log y$.
Then
$$e^{-y} = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \Gamma(s) \, y^{-s} \, ds,$$
where $\Gamma$ is the [[gamma function|0H5SYBSED9]] and the integral converges absolutely.

## Remarks

For real $y > 0$ this is the [[inverse Mellin transform|CHESR1W1CY]] of $\Gamma = \mathcal{M}[e^{-x}]$, so it follows from the Mellin inversion theorem.
For complex $y$, put $\abs{\arg y} = \theta < \pi/2$; then $\abs{y^{-c - it}} = \abs{y}^{-c} e^{t \arg y} \le \abs{y}^{-c} e^{\theta \abs{t}}$, while $\abs{\Gamma(c + it)}$ decays like $\abs{t}^{c - 1/2} e^{-\pi \abs{t}/2}$ by Stirling's formula.
So the integral converges absolutely and locally uniformly, it is holomorphic in $y$, and the identity extends from $y > 0$ by the identity theorem.
A direct proof moves the line of integration to the left: by the [[residue theorem|032HKYS329]], the poles of $\Gamma$ at $s = -k$ with residues $(-1)^k / k!$ give $\sum_{k \ge 0} (-1)^k y^k / k! = e^{-y}$.
The hypothesis $c > 0$ is needed: for $-1 < c < 0$ the line passes to the left of the pole at $0$, and the integral equals $e^{-y} - 1$.

## Explanations

### L1 human

The exponential function $e^{-y}$ can be written as an integral of powers $y^{-s}$ against the gamma function along a vertical line.
It is the inverse of the fact that the gamma function is the Mellin transform of $e^{-x}$.

### L2 human

This integral is the prototype of [[Mellin–Barnes integrals|1PDGBGEHM2]] and of the method of shifting contours.
In analytic number theory it is applied to $y = n x$ and summed over $n$: the sum $\sum_n a(n) e^{-n x}$ then becomes $\frac{1}{2\pi i} \int \Gamma(s) F(s) x^{-s} \, ds$ with $F$ the Dirichlet series of the $a(n)$.
Shifting the line past the poles of $\Gamma(s) F(s)$ gives asymptotic expansions of the exponential sum as $x \to 0^+$.
It is the smooth counterpart of Perron's formula, which uses the kernel $x^{s}/s$ instead of $\Gamma(s) x^{-s}$ and has worse convergence.
</document>

<document source="https://tamnd.github.io/math2/node/9CCSBY0JQW.md" tag="9CCSBY0JQW" kind="definition">
# Betweenness

## Definition

Let $F$ be an [[ordered field|0P3VCBDEE2]], let $V$ be a vector space over $F$, and let $P$ be an [[affine space|09XNAPTJQ7]] over $V$.
Let $x, y, z \in P$.
The point $y$ is *weakly between* $x$ and $z$ if $y = x + t(z - x)$ for some $t \in F$ with $0 \le t \le 1$.
The point $y$ is *strictly between* $x$ and $z$ if it is weakly between them and $y \ne x$ and $y \ne z$.

## Remarks

Example: in $\R^2$, the point $(1, 1)$ is strictly between $(0, 0)$ and $(2, 2)$, with $t = 1/2$.
Non-example: $(3, 3)$ is on the line through $(0, 0)$ and $(2, 2)$, but it is not between them, since it needs $t = 3/2 > 1$.
Weak betweenness is symmetric: if $y$ is weakly between $x$ and $z$, then it is weakly between $z$ and $x$, with $1 - t$ in place of $t$.
The set of points weakly between $x$ and $z$ is the closed segment $[x, z]$.
In axiomatic geometry, betweenness is not defined but taken as a primitive relation that satisfies the axioms of order.
</document>

<document source="https://tamnd.github.io/math2/node/9CD4SGERVP.md" tag="9CD4SGERVP" kind="definition">
# Ovoid

## Definition

Let $\mathcal{P}$ be a [[projective space|ZT9A6JK8QF]] of dimension at least $3$.
A line $g$ is *tangent* to a set $O$ of points at $P \in O$ if $g \cap O = \set{P}$.
An *ovoid* of $\mathcal{P}$ is a set $O$ of points such that:

1. every line meets $O$ in at most two points;
2. for every point $P \in O$, the union of the tangent lines to $O$ at $P$ is exactly a hyperplane through $P$.

## Remarks

The hyperplane in clause 2 is the tangent hyperplane of $O$ at $P$.
In a projective plane the same two conditions define an [[oval|7B03TB64N9]].

Example: in $\mathrm{PG}(3, q)$ over the [[finite field|0ZXZ82FTX2]] $\F_q$, an elliptic [[quadric|Q3Q4FF23VG]] is an ovoid.
It contains no line, so a line meets it in at most two points.
At each point, the tangent lines are the lines through that point in its tangent plane.
It has $q^2 + 1$ points.
In real projective $3$-space, a sphere is an ovoid.

Non-example: a hyperbolic quadric $x_0 x_1 = x_2 x_3$ contains whole lines, such as $x_0 = x_2 = 0$, so clause 1 fails.

In $\mathrm{PG}(3, q)$ every ovoid has $q^2 + 1$ points.
For $q = 2^{2e+1}$ with $e \ge 1$, Tits found ovoids that are not quadrics.
In finite projective spaces of dimension at least $4$ there are no ovoids.
</document>

<document source="https://tamnd.github.io/math2/node/9CE11G48JY.md" tag="9CE11G48JY" kind="conjecture">
# Restriction conjecture

## Statement

Let $n \ge 2$, let $S^{n-1} \subseteq \R^n$ be the unit [[sphere|T62369RFH6]], and let $\sigma$ be its surface measure.
Let $\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$.
Let $1 \le p < \frac{2n}{n+1}$, let $p'$ be given by $1/p + 1/p' = 1$, and let $1 \le q \le \frac{n-1}{n+1} p'$.
Then there is a constant $C$ such that
$$\Big( \int_{S^{n-1}} \abs{\hat f(\xi)}^q \, d\sigma(\xi) \Big)^{1/q} \le C \norm{f}_{L^p(\R^n)}$$
for every [[Schwartz function|CG9JEATW3Q]] $f$ on $\R^n$.

## Remarks

The conditions on $p$ and $q$ are necessary.
The condition $p < 2n/(n+1)$ comes from the function $\widehat{d\sigma}$.
It decays exactly like $\abs{x}^{-(n-1)/2}$, so it lies in $L^{p'}$ only for $p' > 2n/(n-1)$.
By duality, the conjecture requires $\widehat{d\sigma}$ to lie in $L^{p'}$.
The condition $q \le \frac{n-1}{n+1}p'$ comes from the Knapp example, a function whose Fourier transform is concentrated on a small cap of the sphere.

Status: the conjecture holds for $n = 2$ and is open for every $n \ge 3$.
The [[Stein–Tomas restriction theorem|ZZVWHNFY6F]] proves the case $q = 2$, where the range is $p \le 2(n+1)/(n+3)$.
The conjecture implies the Kakeya conjecture that every [[Kakeya set|6PEN7AKC7M]] in $\R^n$ has Hausdorff dimension $n$.

## Explanations

### L1 human

The conjecture says exactly when the Fourier transform of a function in $L^p$ can be restricted to the sphere, with a bound in an $L^q$ norm on the sphere.
The two conditions on $p$ and $q$ are known to be needed, and the question is whether they are enough.
It is one of the main open problems of harmonic analysis, linked to the geometry of thin tubes in Kakeya sets.
</document>

<document source="https://tamnd.github.io/math2/node/9CJ0VSYZ81.md" tag="9CJ0VSYZ81" kind="definition">
# Topological vector space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$ with its usual topology.
A *topological vector space* over $\mathbb{K}$ is a [[vector space|000AYPA9AJ]] $E$ over $\mathbb{K}$ together with a [[topology|0CQHFPGJ45]] on $E$ such that:

1. the addition $E \times E \to E$, $(x, y) \mapsto x + y$, is [[continuous|0VGFHTM75Y]];
2. the scalar multiplication $\mathbb{K} \times E \to E$, $(\lambda, x) \mapsto \lambda x$, is continuous.

Both products carry the [[product topology|0GBG8B49AA]].

## Remarks

Example: $\R^n$ with the Euclidean topology.
Addition and scalar multiplication are given by polynomials in the coordinates, so both are continuous.
Non-example: $\R$ as a vector space over $\R$ with the discrete topology.
Addition is continuous, but scalar multiplication fails clause 2.
If it were continuous, then the map $\lambda \mapsto \lambda \cdot 1$ from $\mathbb{K}$ to $E$ would be continuous.
Then the preimage $\set{1}$ of the open set $\set{1}$ would be open in $\mathbb{K}$, which is false.
Many authors also require the topology to be Hausdorff.
The definition extends to vector spaces over any topological field.
</document>

<document source="https://tamnd.github.io/math2/node/9CJCXNYNK8.md" tag="9CJCXNYNK8" kind="concept">
# Simple algebra

## Summary

A simple algebra is a nontrivial [[algebra|QJGPT2W0FT]] whose only [[congruences|DCSXXT4SER]] are the two trivial ones.
So it has no proper nontrivial [[quotient algebras|N5KE7FH6CX]]: every homomorphism out of it is injective or constant.
The notion covers [[simple groups|0YMYMNQ0QZ]] and simple rings, and it is the basic building block in the study of varieties.

## Explanations

### L1 human

A simple algebra cannot be shrunk by gluing elements together, except by gluing everything into one point.
For a group, gluing is controlled by normal subgroups, so a simple group is a group with no normal subgroups other than the trivial one and itself.
For example, $\Z/5\Z$ is a simple group, while $\Z/6\Z$ is not, since it can be glued down to $\Z/2\Z$.
Every field is simple as a ring, because its only ideals are $\set{0}$ and the whole field.
</document>

<document source="https://tamnd.github.io/math2/node/9CMAJWWBNZ.md" tag="9CMAJWWBNZ" kind="definition">
# Skew group ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity and let $G$ be a [[group|G5S1M9]].
Let $\alpha \colon G \to \Aut(R)$ be a group homomorphism, written $g \mapsto \alpha_g$; it is an [[action|0WCY42EV9V]] of $G$ on $R$ by ring automorphisms.
The *skew group ring* $R * G$, also written $R \rtimes_\alpha G$, is the set of finite formal sums $\sum_{g \in G} r_g g$ with $r_g \in R$.
Addition is termwise, and multiplication is the biadditive extension of
$$(r g)(s h) = r \, \alpha_g(s) \, gh \qquad (r, s \in R, \ g, h \in G).$$

## Remarks

The product is associative because $\alpha_{gh} = \alpha_g \circ \alpha_h$: both $((rg)(sh))(tk)$ and $(rg)((sh)(tk))$ equal $r \alpha_g(s) \alpha_{gh}(t) \, ghk$.
The identity is $1 \cdot e$, where $e$ is the identity of $G$, and $R$ is a subring through $r \mapsto r e$.
If $\alpha$ is trivial, $R * G$ is the [[group ring|4NMTSX0XDS]] $R[G]$.
Example: let $G = \set{e, c}$ act on $\C$ by complex conjugation, $\alpha_c(z) = \bar z$.
Then $R * G = \C * G$ has dimension $4$ over $\R$.
The map sending $z$ to multiplication by $z$ and $c$ to conjugation, both as $\R$-linear maps of $\C = \R^2$, is an isomorphism onto the [[matrix ring|0FHVPRAE60]] $M_2(\R)$.
Non-example: if $\alpha \colon G \to \Aut(R)$ is a map that is not a homomorphism, the same formula fails associativity, since $(1g)((1h)(s e)) = \alpha_g(\alpha_h(s)) gh$ while $((1g)(1h))(s e) = \alpha_{gh}(s) gh$.
When a twisting cocycle is also allowed, $g h$ is replaced by a scalar multiple of $gh$, and one gets a crossed product.
</document>

<document source="https://tamnd.github.io/math2/node/9CNSVFQ801.md" tag="9CNSVFQ801" kind="concept">
# Solvable Lie group

## Summary

A solvable Lie group is a connected [[Lie group|7JBRGCVJRB]] that is a [[solvable group|0FXJ7YFMMX]]: its derived series reaches the trivial group after finitely many steps.
Equivalently, its Lie algebra is a solvable Lie algebra.
Solvable Lie groups are the counterpart of semisimple ones.
By the Levi decomposition, every connected Lie group is built from a solvable normal subgroup and a semisimple part.
Typical examples are groups of invertible upper triangular matrices.

## Explanations

### L1 human

A solvable Lie group is one that can be taken apart, in finitely many steps, into commutative layers.
Take the commutators of the group, then the commutators of those, and so on: in a solvable group this process ends with the identity alone.
The group of maps $x \mapsto a x + b$ of the real line with $a > 0$ is an example.
Its commutators are the translations $x \mapsto x + b$, and these commute with each other, so the process stops after two steps.
</document>

<document source="https://tamnd.github.io/math2/node/9CWM3F4M04.md" tag="9CWM3F4M04" kind="statement">
# Weyl's lemma

## Statement

Let $U \subseteq \R^n$ be an open set and let $u$ be a locally integrable function on $U$.
Assume that
$$\int_U u(x) \, \Delta\varphi(x) \, dx = 0$$
for every infinitely differentiable function $\varphi$ with compact support in $U$ (a [[test function|HS65DW0HXH]]).
Then there is a [[harmonic function|0DEB0JMN0W]] $h$ on $U$ with $u = h$ almost everywhere.

## Remarks

The hypothesis says that $\Delta u = 0$ in the sense of distributions, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
Proof idea: let $\eta_\varepsilon$ be a smooth radial mollifier supported in $B(0, \varepsilon)$.
Let $u_\varepsilon = u * \eta_\varepsilon$, a [[convolution|2Z29JN3VES]] defined at the points at distance more than $\varepsilon$ from the complement of $U$.
Then $\Delta u_\varepsilon = u * \Delta\eta_\varepsilon = 0$ by the hypothesis, so $u_\varepsilon$ is smooth and harmonic.
By the [[mean value property|0HYPR64JM2]], $u_\varepsilon * \eta_\delta = u_\varepsilon$, so $u_\varepsilon = u_\varepsilon * \eta_\delta = u_\delta * \eta_\varepsilon = u_\delta$ where both are defined.
Thus the $u_\varepsilon$ fit together into one harmonic function $h$, and $u_\varepsilon \to u$ almost everywhere as $\varepsilon \to 0$ gives $u = h$ almost everywhere.
The same holds for distributions: a distribution $T$ on $U$ with $\Delta T = 0$ is given by a harmonic function.
Example: $u(x) = \abs{x}^{2-n}$ on $\R^n$, $n \ge 3$, is locally integrable and harmonic off $0$.
But $\Delta u$ is a nonzero multiple of the Dirac delta, so the hypothesis fails on $U = \R^n$, and indeed $u$ has no harmonic version there.

## Explanations

### L1 human

Weyl's lemma says that it does not matter how weakly one solves Laplace's equation.
Suppose a function is only known to satisfy it "on average", tested against smooth bumps.
Then it is a genuine smooth harmonic function after a change on a set of measure zero.

### L2 human

The lemma is the starting point of elliptic regularity: weak solutions of elliptic equations are as smooth as the data allow.
It is used, for instance, in Hilbert space methods for the Dirichlet problem, where Dirichlet's principle produces a weak solution, and Weyl's lemma upgrades it to a classical harmonic function.
In complex analysis it shows that a distributional solution of the Cauchy–Riemann equations is holomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/9CWYNB9003.md" tag="9CWYNB9003" kind="concept">
# Sequential analysis

## Summary

Sequential analysis is the part of statistics in which the sample size is not fixed in advance.
Observations arrive one at a time, and after each one a rule decides whether to stop and make a decision or to take another observation.
Abraham Wald developed the theory during the Second World War and published the book *Sequential Analysis* in 1947.
Sequential procedures often reach a decision with fewer observations on average than the best fixed-sample procedures with the same error rates.

## Explanations

### L1 human

Imagine testing whether a coin is fair by tossing it.
If the first twenty tosses are all heads, there is little point in tossing it eighty more times.
A sequential procedure looks at the data after each toss and stops as soon as the evidence is strong enough either way.
This saves time and cost, which matters in clinical trials and quality control.
</document>

<document source="https://tamnd.github.io/math2/node/9CZVKQMCZP.md" tag="9CZVKQMCZP" kind="definition">
# Choquet integral

## Definition

Let $(X, \mathcal{F})$ be a [[measurable space|N4KDT83V3S]], and let $\nu \colon \mathcal{F} \to [0, \infty)$ be a [[fuzzy measure|CRFQEGJDA9]].
Let $f \colon X \to [0, \infty)$ be [[measurable|YYNCJSR0PJ]].
The *Choquet integral* of $f$ with respect to $\nu$ is
$$(C)\!\int f \, d\nu = \int_0^\infty \nu(\set{x \in X : f(x) \ge t}) \, dt.$$
The function $t \mapsto \nu(\set{f \ge t})$ is non-increasing, so the right side is a well-defined Lebesgue integral with value in $[0, \infty]$.

## Remarks

If $f$ takes the values $0 = a_0 < a_1 < \cdots < a_k$, then $(C)\!\int f \, d\nu = \sum_{i=1}^{k} (a_i - a_{i-1}) \, \nu(\set{f \ge a_i})$.
If $\nu$ is a measure, the Choquet integral equals the Lebesgue integral, by the [[layer cake representation|H78RXJC1K1]].
Example: let $X = \set{1, 2}$ with all subsets, $\nu(\set{1}) = \nu(\set{2}) = 0$ and $\nu(X) = 1$.
Then $\nu$ is monotone with $\nu(\emptyset) = 0$, and the indicators $1_{\set{1}}$ and $1_{\set{2}}$ have Choquet integral $0$.
Their sum $1_X$ has Choquet integral $\int_0^1 \nu(X) \, dt = 1$, so the integral is not additive.
Non-example: if $f$ is not measurable, some set $\set{f \ge t}$ lies outside $\mathcal{F}$, so $\nu$ is not defined on it and the formula makes no sense.
Some authors use $\set{f > t}$; this changes the integrand at only countably many $t$, so the integral is the same.
</document>

<document source="https://tamnd.github.io/math2/node/9D26WKTFPT.md" tag="9D26WKTFPT" kind="definition">
# Scott topology

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
A subset $U \subseteq P$ is *Scott-open* if:

1. $U$ is an [[upper set|ETCE287E2R]]: if $x \in U$ and $x \le y$, then $y \in U$;
2. $U$ is inaccessible by directed suprema: for every [[directed subset|D9RY4667K6]] $D \subseteq P$ that has a [[supremum|0R3H5PANRQ]] with $\bigvee D \in U$, the set $D \cap U$ is nonempty.

The Scott-open sets form a topology on $P$, the *Scott topology*.

## Remarks

The Scott topology makes $P$ a [[topological space|0CQHFPGJ45]] that is $T_0$, and its [[specialization preorder|CMYVK9CEKZ]] is the order $\le$; the closure of $\set{x}$ is $\set{y : y \le x}$.
It is $T_1$ only when the order is trivial.
Example: in $\N \cup \set{\infty}$, the set $U = \set{n \in \N : n \ge 3} \cup \set{\infty}$ is Scott-open.
It is an upper set, and a directed $D$ with $\bigvee D \in U$ either contains its supremum, or is infinite and so contains some $n \ge 3$.
Non-example: in the same poset, $\set{\infty}$ is an upper set but is not Scott-open, since $D = \N$ has supremum $\infty$ and $D \cap \set{\infty} = \varnothing$.
Closed sets of the Scott topology are the down-sets closed under suprema of directed subsets.
</document>

<document source="https://tamnd.github.io/math2/node/9D2Z543EJQ.md" tag="9D2Z543EJQ" kind="concept">
# Dirichlet L-function

## Summary

A Dirichlet L-function is the [[Dirichlet series|8MWTEF6462]] $L(s, \chi) = \sum_{n \ge 1} \chi(n) n^{-s}$ of a [[Dirichlet character|7XB5VJKDPG]] $\chi$, together with its analytic continuation.
It has an Euler product over the primes and generalizes the [[Riemann zeta function|1XWBY4N0VT]], which is the case of the trivial character modulo $1$.
The fact that $L(1, \chi) \ne 0$ for every nonprincipal $\chi$ is the heart of [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]].

## Explanations

### L1 human

The zeta function adds $1/n^s$ over all positive integers.
A Dirichlet L-function adds the same terms with the weights $\chi(n)$, which depend on $n$ modulo $q$.
These weights let one separate the primes according to their remainder modulo $q$.
For the character modulo $4$ with values $1, 0, -1, 0$, the value at $s = 1$ is $1 - 1/3 + 1/5 - 1/7 + \cdots = \pi/4$.
</document>

<document source="https://tamnd.github.io/math2/node/9D47D1Q89E.md" tag="9D47D1Q89E" kind="definition">
# Minimax estimator

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a statistical model with a loss function, and let $R(\theta, \delta)$ be the [[risk function|N26TSVRH2S]] of an [[estimator|0SSS6ERQ0S]] $\delta$.
The *minimax risk* of the problem is
$$R^* = \inf_{\delta} \sup_{\theta \in \Theta} R(\theta, \delta),$$
where the infimum is over all estimators.
An estimator $\delta^*$ is *minimax* if
$$\sup_{\theta \in \Theta} R(\theta, \delta^*) = R^*.$$

## Remarks

A minimax estimator need not exist, and it need not be unique.
A Bayes estimator whose risk function is constant in $\theta$ is minimax.

Example: let $X$ be [[binomial|0WCWA1KT3M]] with parameters $n$ and $\theta \in [0, 1]$, with squared error loss.
Let $\delta^*(X) = (X + \sqrt{n}/2)/(n + \sqrt{n})$.
A direct computation of variance plus squared bias gives $R(\theta, \delta^*) = \frac{1}{4(1 + \sqrt{n})^2}$ for every $\theta$.
This $\delta^*$ is the Bayes estimator for the beta prior with parameters $\sqrt{n}/2$ and $\sqrt{n}/2$, and its risk is constant, so it is minimax.

Non-example: $\delta(X) = X/n$ in the same problem.
Its risk is $\theta(1 - \theta)/n$, with maximum $\frac{1}{4n}$ at $\theta = \frac{1}{2}$.
Since $\frac{1}{4n} > \frac{1}{4(1 + \sqrt{n})^2}$, its worst-case risk exceeds $R^*$, so it is not minimax.
</document>

<document source="https://tamnd.github.io/math2/node/9D5NT00H6G.md" tag="9D5NT00H6G" kind="statement">
# Crofton formula

## Statement

For $\theta \in [0, \pi)$ and $p \in \R$, let $L(\theta, p)$ be the [[line|0MP9BCF6DP]] $\set{x \in \R^2 : x_1 \cos \theta + x_2 \sin \theta = p}$.
Let $\gamma \colon [a, b] \to \R^2$ be a rectifiable [[plane curve|EWS4WFTA7W]] of [[length|0TH78S7GBZ]] $\ell(\gamma)$.
Let $n_\gamma(\theta, p) \in \set{0, 1, 2, \ldots} \cup \set{\infty}$ be the number of $t \in [a, b]$ with $\gamma(t) \in L(\theta, p)$.
Then $n_\gamma$ is measurable and
$$\ell(\gamma) = \frac{1}{2} \int_0^\pi \int_{-\infty}^{\infty} n_\gamma(\theta, p) \, dp \, d\theta.$$

## Remarks

Both sides are additive when the curve is cut into pieces, and both are invariant under rigid motions, since $dp \, d\theta$ is the invariant measure of [[integral geometry|TAN5G5EQ96]].
Take a segment of length $\ell$ in the direction $\varphi$, and fix $\theta$.
The lines $L(\theta, p)$ meeting it form an interval of $p$ of length $\ell \abs{\cos(\theta - \varphi)}$, and $\int_0^\pi \ell \abs{\cos(\theta - \varphi)} \, d\theta = 2 \ell$.
So the formula holds for polygonal curves, and the general case follows by approximating $\gamma$ by inscribed polygons and passing to the limit.
Example: a circle of radius $r$ meets $L(\theta, p)$ twice when $\abs{p} < r$, so the right side is $\frac{1}{2} \cdot \pi \cdot 2r \cdot 2 = 2 \pi r$.

## Explanations

### L1 human

The length of a curve equals, up to a fixed factor, the average number of times a random line crosses it.
Long or wiggly curves are crossed more often, and the formula makes this exact.

### L2 human

For the boundary of a plane convex body, almost every line that meets the body crosses the boundary twice.
So the perimeter equals $\int_0^\pi w(\theta) \, d\theta$, where $w(\theta)$ is the width in the direction $\theta$.
This is Cauchy's formula, and it gives Barbier's theorem for curves of constant width.
The formula explains [[Buffon's needle problem|06X5PF6YFP]].
It is the model for Crofton formulas in higher dimensions, which recover intrinsic volumes of convex bodies from intersections with random affine subspaces.
</document>

<document source="https://tamnd.github.io/math2/node/9D7WZEN64H.md" tag="9D7WZEN64H" kind="definition">
# Finite morphism of rings

## Definition

Let $A$ and $B$ be commutative rings and let $f \colon A \to B$ be a [[ring homomorphism|0GGE0Z0JQ1]].
The ring $B$ is an [[$A$-module|0NZ8JA25WK]] by $a \cdot b = f(a) b$.
The homomorphism $f$ is *finite*, and $B$ is a *finite $A$-algebra*, if $B$ is a [[finitely generated|05QY1VEBYD]] $A$-module.
That is, there are $b_1, \ldots, b_n \in B$ such that every $b \in B$ can be written as $b = f(a_1) b_1 + \cdots + f(a_n) b_n$ with $a_i \in A$.

## Remarks

Example: $\Z \to \Z[\sqrt{2}]$ is finite, generated as a module by $1$ and $\sqrt{2}$.
Every surjective homomorphism $A \to A/I$ is finite, generated by $1$.

Non-example: $\Z \to \Q$ is not finite.
If $q_1, \ldots, q_n$ generated $\Q$ as a $\Z$-module, every rational number would have a denominator dividing the product of their denominators, which fails for $1/p$ with $p$ a large prime.

A homomorphism is finite if and only if it is [[integral|B7X0KFY1S1]] and of [[finite type|AC5TFFQT9M]].
A composition of finite homomorphisms is finite.
</document>

<document source="https://tamnd.github.io/math2/node/9DD8M9R868.md" tag="9DD8M9R868" kind="concept">
# Levitzki radical

## Summary

The Levitzki radical of a ring is its largest locally nilpotent ideal.
An ideal is locally nilpotent when every finite subset of it generates a nilpotent subring.
It lies between the [[prime radical|WF117EJ2HF]] and the [[upper nilradical|3G2JQEYBEK]].
It is one of the classical radicals of the [[Kurosh–Amitsur radical theory|A72J7ZVNR5]].

## Explanations

### L1 human

A nilpotent ideal is one where any long enough product of its elements is zero.
A locally nilpotent ideal asks less: only products of elements chosen from a fixed finite set must vanish when they are long enough.
The Levitzki radical collects all such ideals.
In a commutative ring every nilpotent element generates a nilpotent ideal, and the Levitzki radical is the set of nilpotent elements.
</document>

<document source="https://tamnd.github.io/math2/node/9DEG0WRDHR.md" tag="9DEG0WRDHR" kind="concept">
# Ehrhart polynomial

## Summary

The Ehrhart polynomial of a [[lattice polytope|F39N9NBFPZ]] $P$ counts the integer points in the dilates $tP$ for positive integers $t$.
Ehrhart's theorem says that this count is a polynomial in $t$ of degree $\dim P$.
Its leading coefficient is the volume of $P$ when $P$ is full-dimensional, and its constant term is $1$.
It is a discrete analogue of volume, and it satisfies [[Ehrhart–Macdonald reciprocity|ZDBER9FJ6T]].

## Explanations

### L1 human

Blow up a lattice polygon by a factor $t$ and count the grid points inside it, including those on the boundary.
For the unit square the count is $(t+1)^2$: a $t$ by $t$ square holds $t+1$ points along each side.
For every lattice polytope the count follows a polynomial formula in $t$ like this one.
For a full-dimensional polytope in $\R^n$ and large $t$, the count is close to the volume times $t^n$, as one expects from area.
</document>

<document source="https://tamnd.github.io/math2/node/9DNDF7AX59.md" tag="9DNDF7AX59" kind="definition">
# Zeros and poles

## Definition

Let $U \subseteq \C$ be an open set and let $f$ be a [[meromorphic function|0NSD1TPHR0]] on $U$.
So there is a closed discrete set $P \subset U$ such that $f$ is [[holomorphic|0DG0Z72CQD]] on $U \setminus P$ and each point of $P$ is a pole of $f$.

1. A *zero* of $f$ is a point $a \in U \setminus P$ with $f(a) = 0$, as in the definition of a [[zero of a holomorphic function|0CEGEASY49]].
2. A *pole* of $f$ is a point $a \in P$, that is, an isolated singularity with $\abs{f(z)} \to \infty$ as $z \to a$, as in the definition of a [[pole|0WPQYW8MTE]].

## Remarks

Example: $f(z) = \dfrac{z - 1}{z^2 + 1}$ on $\C$ has one zero, at $1$, and two poles, at $i$ and $-i$, where the denominator vanishes and the numerator does not.
Non-example: $e^{1/z}$ is not meromorphic on $\C$, and $0$ is neither a zero nor a pole of it.
The point $0$ is an essential singularity: near $0$, $e^{1/z}$ is neither bounded nor tends to $\infty$.
If $U$ is connected and $f$ is not identically $0$, the zeros of $f$ are isolated and the poles of $f$ are the zeros of $1/f$.
The definition extends to meromorphic functions on the Riemann sphere or on a Riemann surface, using local coordinates.
A nonzero rational function, seen on the Riemann sphere, has as many zeros as poles when both are counted with their [[orders|K2S5TAW2BF]].
</document>

<document source="https://tamnd.github.io/math2/node/9DNFN8M83X.md" tag="9DNFN8M83X" kind="concept">
# Incidence matrix

## Summary

The incidence matrix of a graph records which vertices lie on which edges.
It has a row for each vertex, a column for each edge, and a nonzero entry where the vertex is an end of the edge.
In the oriented version the two ends get opposite signs, and the product of the matrix with its transpose is the [[Laplacian matrix|TN6M6DVZRP]].
More generally, an incidence matrix describes any incidence relation between two finite sets, such as points and blocks of a design.

## Explanations

### L1 human

Make a table with the vertices of a graph down the side and the edges across the top.
Put a $1$ in a cell when the vertex is an end of the edge, and $0$ otherwise.
Each column then has exactly two $1$s, one for each end of the edge, and the row of a vertex has as many $1$s as the vertex has edges.
This table is the incidence matrix.
</document>

<document source="https://tamnd.github.io/math2/node/9DQB442K3Y.md" tag="9DQB442K3Y" kind="concept">
# Lie supergroup

## Summary

A Lie supergroup is a group object in the category of [[supermanifolds|Z0SB9HQBAJ]]: a supermanifold with multiplication, inverse and unit morphisms satisfying the group laws.
It is the analogue of a [[Lie group|7JBRGCVJRB]] in which some coordinates are anticommuting, or odd.
Its infinitesimal version is a Lie superalgebra, and Lie supergroups describe supersymmetry in physics.

## Explanations

### L1 human

In a supermanifold, besides ordinary coordinates there are odd coordinates $\theta$ that anticommute, so $\theta_1\theta_2 = -\theta_2\theta_1$ and $\theta^2 = 0$.
A Lie supergroup is such a space with a smooth group law.
A simple example has one ordinary coordinate $t$ and one odd coordinate $\theta$, with product $(t, \theta)(t', \theta') = (t + t' + \theta\theta', \theta + \theta')$.
Forgetting the odd coordinates leaves an ordinary Lie group, here the real line under addition.
</document>

<document source="https://tamnd.github.io/math2/node/9DQD6HTTA5.md" tag="9DQD6HTTA5" kind="definition">
# Ginzburg–Landau equation

## Definition

Let $\Omega \subseteq \R^3$ be an open set and let $\kappa > 0$ be a constant, the Ginzburg–Landau parameter.
For a function $\psi \colon \Omega \to \C$ and a [[vector field|0299VCW1TH]] $A \colon \Omega \to \R^3$, write $\nabla_A \psi = \nabla \psi - i A \psi$, where $\nabla$ is the [[gradient|0ABK9V9Y64]].
The *Ginzburg–Landau equations* are the system
$$- \nabla_A \cdot \nabla_A \psi = \kappa^2 \big( 1 - \abs{\psi}^2 \big) \psi, \qquad \nabla \times \nabla \times A = \mathrm{Im}\big( \overline{\psi} \, \nabla_A \psi \big),$$
for the unknowns $\psi$, the order parameter, and $A$, the magnetic potential.
Here $\nabla_A \cdot V = \nabla \cdot V - i A \cdot V$ for a vector field $V$ with values in $\C^3$, and $\nabla \times$ is the [[curl|0NF6GQNXMY]].

## Remarks

The equations are the Euler–Lagrange equations of the Ginzburg–Landau energy
$$\mathcal{E}(\psi, A) = \int_\Omega \Big( \tfrac{1}{2} \abs{\nabla_A \psi}^2 + \tfrac{\kappa^2}{4} \big( 1 - \abs{\psi}^2 \big)^2 + \tfrac{1}{2} \abs{\nabla \times A}^2 \Big) dx,$$
written in dimensionless units.
On a bounded domain the natural boundary condition is $\nabla_A \psi \cdot n = 0$.
The energy and the equations are invariant under the gauge change $\psi \mapsto e^{i\chi} \psi$, $A \mapsto A + \nabla \chi$.

Example: $\psi = 1$ and $A = 0$ is a solution: both sides of each equation are $0$.

Non-example: $\psi = 1/2$ and $A = 0$ is not a solution, because the left side of the first equation is $0$ while the right side is $\tfrac{3}{8} \kappa^2 \ne 0$.

Without magnetic field, setting $A = 0$ and dropping the second equation gives the simplified equation $-\Delta \psi = \kappa^2 (1 - \abs{\psi}^2) \psi$.
The name Ginzburg–Landau equation is also used for the time-dependent complex equation $\partial_t \psi = (1 + i\alpha) \Delta \psi + \psi - (1 + i\beta) \abs{\psi}^2 \psi$ of pattern formation.
</document>

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# Nested dissection

## Statement

Let $A$ be a real symmetric [[positive definite|0KX66CZMBB]] [[sparse matrix|JGACZYEEY7]] whose [[graph|00C0BR7D7H]] $G$ has vertex set $V$.
A *separator* of $G$ is a set $S \subseteq V$ such that $V \setminus S$ splits into two parts $V_1$ and $V_2$.
No edge of $G$ joins $V_1$ and $V_2$.
The *nested dissection ordering* is defined recursively.
Choose a separator $S$; order the vertices of $V_1$ first, then those of $V_2$, and those of $S$ last.
Order $V_1$ and $V_2$ by the same rule, applied to the subgraphs they induce, until the parts are small.

Let $A$ be the matrix of the five-point [[finite difference|X7HYEBA5SA]] Laplacian on a $k \times k$ grid, so $n = k^2$.
Let each separator be a grid line that splits the current rectangle into two halves.
Then the [[Cholesky factor|0WKG8CYY6A]] of $A$ in this ordering has $O(n \log n)$ nonzero entries.
The factorization takes $O(n^{3/2})$ arithmetic operations.

## Remarks

Since the vertices of $S$ come last, no elimination step inside $V_1$ can create fill linking $V_1$ to $V_2$.
So the factor has a block structure with two independent blocks and a dense final block for $S$.
On the grid the separators have size $O(\sqrt{m})$ for a subgrid of $m$ points, and summing the dense separator blocks over the levels of the recursion gives the two bounds.
For comparison, a banded ordering of the same grid gives a factor with about $n^{3/2}$ nonzeros and costs about $n^2$ operations.
The same bounds hold for any matrix whose graph is [[planar|0NC9R64TXP]], by the planar separator theorem of Lipton and Tarjan.
On the grid, no ordering beats the operation count $O(n^{3/2})$ by more than a constant factor.

## Explanations

### L1 human

Cut the problem into two halves with a thin wall, solve the two halves separately, and handle the wall last.
Then cut each half the same way, and so on.
Because the two halves never talk to each other directly, elimination cannot spread nonzeros from one half to the other.
This divide-and-conquer order makes direct solvers for two-dimensional grids much cheaper.

### L2 human

Nested dissection is the main fill-reducing ordering for matrices from discretized partial differential equations.
Its recursive structure also exposes parallelism: the independent halves can be factored at the same time.
In three dimensions the separators are larger, and the cost on a $k \times k \times k$ grid becomes $O(n^2)$ operations with $n = k^3$.
</document>

<document source="https://tamnd.github.io/math2/node/9DYE6V628H.md" tag="9DYE6V628H" kind="definition">
# Whittaker function

## Definition

Let $\kappa$ and $\mu$ be complex numbers, let $z \in \C \setminus (-\infty, 0]$, and use principal branches of powers of $z$.
Let $M(a, b, z)$ be [[Kummer's function|6AZ4ZBN1M8]] and $U(a, b, z)$ the [[Tricomi function|234F5DNAJH]].
If $2\mu \notin \set{-1, -2, -3, \dots}$, the *Whittaker function* $M_{\kappa, \mu}$ is
$$M_{\kappa, \mu}(z) = e^{-z/2} z^{\mu + \frac12}\, M\left( \mu - \kappa + \tfrac12,\ 1 + 2\mu,\ z \right).$$
For all $\kappa, \mu$, the *Whittaker function* $W_{\kappa, \mu}$ is
$$W_{\kappa, \mu}(z) = e^{-z/2} z^{\mu + \frac12}\, U\left( \mu - \kappa + \tfrac12,\ 1 + 2\mu,\ z \right).$$

## Remarks

Both functions solve *Whittaker's equation*
$$\frac{d^2 w}{dz^2} + \left( -\frac14 + \frac{\kappa}{z} + \frac{\frac14 - \mu^2}{z^2} \right) w = 0.$$
This follows from Kummer's equation for $M$ and $U$ by the substitution $w = e^{-z/2} z^{b/2} y$ with $b = 1 + 2\mu$.
Example: for $\kappa = 0$ and $\mu = \frac12$, one has $U(1, 2, z) = z^{-1}$, so $W_{0, 1/2}(z) = e^{-z/2}$.
Check: $w = e^{-z/2}$ gives $w'' = \frac14 w$, and the equation becomes $\left( \frac14 - \frac14 + 0 + 0 \right) w = 0$.
Non-example: for $\kappa = 1$ and $\mu = \frac12$, the function $e^{-z/2}$ is not a solution, since the equation leaves the term $\frac{1}{z} e^{-z/2} \ne 0$.
As $z \to \infty$ with $\abs{\arg z} < \frac{3\pi}{2} - \delta$, one has $W_{\kappa, \mu}(z) \sim e^{-z/2} z^{\kappa}$.
The equation depends on $\mu$ only through $\mu^2$, and indeed $W_{\kappa, -\mu} = W_{\kappa, \mu}$.
</document>

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# Intersection theory

## Definition

Let $X$ be a smooth, irreducible [[quasi-projective variety|EQSHJPE7H8]] of dimension $n$ over a field, and let $CH^j(X)$ be its [[Chow group|EBEF6PEK4N]] of codimension $j$ cycles.
Let $V$ and $W$ be irreducible subvarieties of codimensions $p$ and $q$.
They *meet properly* if every component $Z$ of $V \cap W$ has codimension $p + q$.
In that case set $V \cdot W = \sum_Z i(Z; V \cdot W) \, [Z]$, where $i(Z; V \cdot W)$ is the [[intersection multiplicity|GPQF9GK4T9]].
Extend this bilinearly to [[cycles|9VFWPZEVGN]] whose components meet properly.
The *intersection product* is the map $CH^p(X) \times CH^q(X) \to CH^{p+q}(X)$, $(\alpha, \beta) \mapsto \alpha \cdot \beta$, defined as follows.
Choose cycles $V$ in $\alpha$ and $W$ in $\beta$ that meet properly, and let $\alpha \cdot \beta$ be the class of $V \cdot W$.
With this product, $CH^\bullet(X) = \bigoplus_{j=0}^n CH^j(X)$ is the *Chow ring* of $X$.
*Intersection theory* is the study of this ring and of the maps between such rings.

## Remarks

Chow's moving lemma says that such representatives $V$ and $W$ always exist, and the class of $V \cdot W$ does not depend on their choice.
The product makes $CH^\bullet(X)$ a commutative, associative graded ring with unit $[X]$.
For $X$ projective, the degree of a $0$-cycle gives intersection numbers $\deg(\alpha \cdot \beta)$ for $p + q = n$.
Example: for $X = \mathbb{P}^n$ with $h$ the class of a hyperplane, $CH^\bullet(\mathbb{P}^n) \cong \Z[h]/(h^{n+1})$, and a subvariety of codimension $j$ and degree $d$ has class $d h^j$.
Non-example: in $\mathbb{P}^2$, a line $L$ does not meet itself properly, so $L \cdot L$ is not given by the formula with $V = W = L$.
One moves one copy to another line $L'$, and $[L] \cdot [L] = [L \cap L']$ is the class of a point.
Fulton and MacPherson gave a construction of the product, by deformation to the normal cone, that works without the moving lemma and in greater generality.
</document>

<document source="https://tamnd.github.io/math2/node/9EA7KJ50AE.md" tag="9EA7KJ50AE" kind="statement">
# Sullivan conjecture

## Statement

Let $G$ be a finite group, let $BG$ be its [[classifying space|7XRBR7AV3C]], and let $X$ be a finite-dimensional [[CW complex|5W95QFHXPR]] with a basepoint.
Then the space $\operatorname{Map}_*(BG, X)$ of [[pointed|T9DG8DDE75]] continuous maps from $BG$ to $X$ is weakly contractible.
That is, the map from it to a point is a [[weak homotopy equivalence|4Q3WY1RX91]].

## Remarks

This form was proved by Haynes Miller in 1984, and is often called Miller's theorem.
The proof uses the category of unstable modules over the Steenrod algebra: one shows that the relevant Ext groups vanish and uses the unstable Adams spectral sequence.
The hypothesis that $X$ is finite-dimensional cannot be dropped: if $G \ne 1$, the identity map of $BG$ is not null-homotopic, since $\pi_1(BG) \cong G$ is nontrivial.

At work: every map from $B\Z/p$ to a finite-dimensional complex, such as a sphere $S^n$, is null-homotopic.
Sullivan's conjecture also has a fixed-point form, proved independently by Carlsson, by Lannes, and by Dwyer, Miller and Neisendorfer.
It says that, for a finite $p$-group acting on a finite complex, the fixed points agree with the homotopy fixed points after $p$-completion, under suitable hypotheses.

## Explanations

### L1 human

The classifying space of a finite group is an infinite-dimensional space with complicated cohomology.
The theorem says that it cannot be mapped in an essential way into any finite-dimensional complex.
Every pointed map can be deformed to the constant map, and in a coherent way.
So finite groups are invisible to finite complexes in this homotopical sense.

### L2 human

Miller's theorem was the starting point of the theory of Lannes' $T$ functor and unstable modules over the Steenrod algebra.
These methods give a homotopy-theoretic description of maps out of $BG$ in many cases.
Its fixed-point form compares the fixed points $X^G$ of a group action with the homotopy fixed points $X^{hG}$, and is parallel to the [[Segal conjecture|DE29ZC399Z]] for spectra.
</document>

<document source="https://tamnd.github.io/math2/node/9EABA9ZDZS.md" tag="9EABA9ZDZS" kind="statement">
# Bourbaki–Witt theorem

## Statement

Let $(P, \le)$ be a nonempty [[partially ordered set|0QV2JS9RJ3]] in which every [[chain|0S8FE7ZMFM]] has a [[supremum|0R3H5PANRQ]].
Let $f \colon P \to P$ be a map with $x \le f(x)$ for every $x \in P$.
Then $f$ has a fixed point: there is $x \in P$ with $f(x) = x$.

## Remarks

The map $f$ need not be monotone; the hypothesis $x \le f(x)$ says that $f$ is *inflationary*.
Proof idea: fix $a \in P$ and let $T$ be the smallest subset of $P$ that contains $a$, is closed under $f$, and contains the supremum of each of its nonempty chains.
A careful induction shows that $T$ is a chain, so it has a supremum $s$, and $s \in T$.
Then $f(s) \in T$, so $f(s) \le s \le f(s)$ and $f(s) = s$.
The proof uses no form of the axiom of choice.
Combined with the axiom of choice, the theorem shows that such a poset has a maximal element.
Otherwise a choice of some $f(x) > x$ for each $x$ would be an inflationary map with no fixed point.
This is a standard step in the proof of [[Zorn's lemma|04RXTQA5ZA]] from the axiom of choice.
The hypothesis $x \le f(x)$ is needed: on the chain $\set{0, 1}$, where every chain has a supremum, the map swapping $0$ and $1$ has no fixed point.
The completeness hypothesis is needed too: on $\N$, the inflationary map $n \mapsto n + 1$ has no fixed point, and the chain $\N$ has no supremum.

## Explanations

### L1 human

Suppose a rule never moves any point downward, and every chain of points has a top.
Then some point is not moved at all.
The idea is to start anywhere and keep applying the rule, jumping to the top of the chain built so far whenever the steps run out.
This climb cannot go on forever, so it stops at a fixed point.

### L2 human

The theorem is a choice-free fixed-point principle for chain-complete posets.
It is a standard route to [[Zorn's lemma|04RXTQA5ZA]] from the axiom of choice, and it is used to build objects by transfinite recursion without naming ordinals.
Unlike the [[Knaster–Tarski theorem|RHXP3HA64M]], it does not need monotonicity, but it says nothing about least fixed points.
</document>

<document source="https://tamnd.github.io/math2/node/9EAHDX44B2.md" tag="9EAHDX44B2" kind="definition">
# Internal set theory

## Definition

*Internal set theory* (IST) is the first-order theory in the language of set theory with one extra unary predicate $\operatorname{st}$, read "is standard".
A formula is *internal* if it does not contain $\operatorname{st}$, and *external* otherwise.
Write $\forall^{\mathrm{st}} x$ for "for all standard $x$", $\exists^{\mathrm{st}} x$ for "there is a standard $x$", and $\forall^{\mathrm{st\,fin}} z$ for "for all standard [[finite|0BBTH3YSZA]] sets $z$".
The axioms of IST are the following.

1. The axioms of [[ZFC|0HE5WDQ9FJ]], where the schemes of separation and replacement are taken only for internal formulas.
2. Idealization: for each internal formula $B(x, y)$, possibly with other free variables,
$$\forall^{\mathrm{st\,fin}} z \, \exists x \, \forall y \in z \; B(x, y) \;\leftrightarrow\; \exists x \, \forall^{\mathrm{st}} y \; B(x, y).$$
3. Standardization: for each formula $C(z)$, internal or external, possibly with other free variables,
$$\forall^{\mathrm{st}} x \, \exists^{\mathrm{st}} y \, \forall^{\mathrm{st}} z \, \bigl(z \in y \leftrightarrow (z \in x \wedge C(z))\bigr).$$
4. Transfer: for each internal formula $A(x, t_1, \dots, t_k)$ with no other free variables,
$$\forall^{\mathrm{st}} t_1 \cdots \forall^{\mathrm{st}} t_k \, \bigl(\forall^{\mathrm{st}} x \, A(x, t_1, \dots, t_k) \to \forall x \, A(x, t_1, \dots, t_k)\bigr).$$

## Remarks

Example: there is a nonstandard natural number.
Apply idealization to the internal formula $B(x, y)$: "$x \in \N$, and if $y \in \N$ then $y < x$".
The left side holds, since every finite set $z$ has only finitely many natural numbers, and some natural number exceeds them all.
So some $\nu \in \N$ is larger than every standard natural number, and $\nu$ is not standard.

Non-example: the collection of standard natural numbers is not given by separation, because the formula "$n$ is standard" is external, so clause 1 does not apply to it.
In fact no such set exists: its complement in $\N$ would have a least element $m > 0$, and then $m - 1$ would be standard, so $m$ would be standard by transfer.

By transfer, every object uniquely defined by an internal formula without parameters, such as $0$, $\N$, $\R$ or $\pi$, is standard.
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<document source="https://tamnd.github.io/math2/node/9EE43J2W16.md" tag="9EE43J2W16" kind="definition">
# Way-below relation

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]] and let $x, y \in P$.
The element $x$ is *way below* $y$, written $x \ll y$, if the following holds.
For every [[directed subset|D9RY4667K6]] $D \subseteq P$ with a [[supremum|0R3H5PANRQ]] such that $y \le \bigvee D$, there is $d \in D$ with $x \le d$.

## Remarks

If $x \ll y$, then $x \le y$, by taking $D = \set{y}$.
If $w \le x \ll y \le z$, then $w \ll z$.
An element $c$ is [[compact|9F7TYES4KE]] if and only if $c \ll c$.
Example: in $[0, 1]$ with the usual order, $x \ll y$ holds if and only if $x = 0$ or $x < y$; for $x < y \le \bigvee D$, some $d \in D$ exceeds $x$ by the definition of the supremum.
Non-example: in $[0, 1]$, $1/2 \not\ll 1/2$, because $D = \set{1/2 - 1/n : n \ge 3}$ is directed with supremum $1/2$ and no member is at least $1/2$.
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<document source="https://tamnd.github.io/math2/node/9EG3A21ET1.md" tag="9EG3A21ET1" kind="concept">
# Geometric invariant theory

## Summary

Geometric invariant theory, or GIT, is a method for building quotients of algebraic varieties by actions of [[reductive groups|2XS84PWE3S]].
It replaces the orbit space, which is often badly behaved, by the spectrum or the Proj of a [[ring of invariants|6VTR5EBNYC]].
To get a good quotient one throws away the unstable points and keeps the semistable ones.
David Mumford developed the theory to construct [[moduli spaces|FJZRW9P0K3]], such as the moduli of curves and of vector bundles.

## Explanations

### L1 human

Suppose a group moves the points of a space around and you want a space whose points are the orbits.
Some orbits are too small or too close to others, and the naive orbit space is then not even Hausdorff.
GIT uses invariant polynomials to decide which orbits to keep: two points are identified when no invariant can separate them.
For example, the group of nonzero scalars acts on $\C^{n+1}$ by scaling.
With a suitable linearization, the origin is the only unstable point, and removing it and taking orbits gives the projective space $\mathbb{P}^n$.
</document>

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# Continuous functional calculus

## Definition

Let $A$ be a C*-algebra with unit $1$, and let $a \in A$ be normal, that is, $a^* a = a a^*$.
Let $\sigma(a) \subseteq \C$ be the [[spectrum|EH0CZ19RKP]] of $a$, and let $C(\sigma(a))$ be the C\*-algebra of [[continuous|0X3WGX95FE]] functions $\sigma(a) \to \C$ with the sup norm.
There is a unique $*$-homomorphism $\Phi_a \colon C(\sigma(a)) \to A$ with $\Phi_a(1) = 1$ and $\Phi_a(\iota) = a$, where $\iota(z) = z$ is the inclusion of $\sigma(a)$ into $\C$.
The map $\Phi_a$ is the *continuous functional calculus* of $a$, and for $f \in C(\sigma(a))$ one writes $f(a) = \Phi_a(f)$.

## Remarks

Existence comes from the [[commutative Gelfand–Naimark theorem|NT564Y315M]] applied to the commutative C\*-algebra $C^*(a, 1)$ generated by $a$ and $1$, whose character space is homeomorphic to $\sigma(a)$.
Uniqueness holds because the polynomials in $z$ and $\overline{z}$ are dense in $C(\sigma(a))$, by the Stone–Weierstrass theorem.
The map $\Phi_a$ is isometric, its image is $C^*(a, 1)$, and $\sigma(f(a)) = f(\sigma(a))$.

Example: $A = M_n(\C)$ and $a = U D U^*$ with $U$ unitary and $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$.
Here $\sigma(a) = \set{\lambda_1, \ldots, \lambda_n}$ and $f(a) = U \operatorname{diag}(f(\lambda_1), \ldots, f(\lambda_n)) U^*$.
This map is linear, multiplicative, preserves adjoints, sends $1$ to $I$ and $\iota$ to $a$.

Non-example: $a = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ in $M_2(\C)$ is not normal.
Its spectrum is $\set{0}$, so $\iota$ is the zero function on $\sigma(a)$, and any linear map sends it to $0 \ne a$.
So no such $\Phi_a$ exists, and normality cannot be dropped.
</document>

<document source="https://tamnd.github.io/math2/node/9F0CPB14FQ.md" tag="9F0CPB14FQ" kind="definition">
# Permutation group

## Definition

Let $\Omega$ be a set, and let $\operatorname{Sym}(\Omega)$ be the [[symmetric group|0515A6D2Y1]] of $\Omega$: the group of all bijections $\Omega \to \Omega$ under composition.
A *permutation group* on $\Omega$ is a [[subgroup|0X1JXT47GA]] $G$ of $\operatorname{Sym}(\Omega)$.
The *degree* of $G$ is the cardinality $\abs{\Omega}$.

## Remarks

Example: label the corners of a square $1, 2, 3, 4$ in cyclic order.
The eight symmetries of the square give the set $G = \set{e, (1\,2\,3\,4), (1\,3)(2\,4), (1\,4\,3\,2), (1\,3), (2\,4), (1\,2)(3\,4), (1\,4)(2\,3)}$.
It is closed under composition and inverses, because the composite of two symmetries of the square is again one.
So $G$ is a permutation group of degree $4$, isomorphic to the [[dihedral group|0JZ27DMJ5D]] of order $8$.

Non-example: the set $\set{e, (1\,2), (1\,3), (2\,3)}$ in $S_3$.
It contains $(1\,2)$ and $(1\,3)$ but not their product, which is a $3$-cycle, so it is not a subgroup.

A [[group action|0WCY42EV9V]] of a group $H$ on $\Omega$ is the same as a homomorphism $H \to \operatorname{Sym}(\Omega)$, and its image is a permutation group on $\Omega$.
The image is isomorphic to $H$ exactly when the action is [[faithful|0P7G9NGG2G]].
By [[Cayley's theorem|0151VYVX27]], the action of a group $H$ on itself by left multiplication is faithful, so $H$ is isomorphic to a permutation group on $H$.
</document>

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# Cumulant

## Summary

The cumulants $\kappa_1, \kappa_2, \dots$ of a real random variable are the Taylor coefficients at $0$ of its [[cumulant generating function|0N2SP6K9ZC]], the logarithm of the [[moment-generating function|0WQVXP7PDB]].
The first cumulant is the [[mean|0WQNRCNY3M]] and the second is the [[variance|0477R9PYAE]].
Cumulants of a sum of [[independent|0GQ938GDE0]] random variables are the sums of their cumulants, which makes them easier to use than [[moments|01XAHPADBA]].
The [[normal distribution|05AQE0QR0D]] is the law whose cumulants of order $3$ and higher all vanish.

## Explanations

### L1 human

Moments of a sum of independent variables mix in a complicated way, but cumulants just add.
The first cumulant tells where the distribution sits, and the second how spread out it is.
The third measures how lopsided it is, and the fourth how heavy its tails are compared with a normal law.
For a [[Poisson|0XB9GQTN9G]] variable with mean $\lambda$, every cumulant equals $\lambda$.
For a normal variable, only the first two cumulants are nonzero.
</document>

<document source="https://tamnd.github.io/math2/node/9F6B05QAWH.md" tag="9F6B05QAWH" kind="example">
# Multiplication operator

## Example

Let $(\Omega, \mu)$ be a $\sigma$-finite measure space and let $\varphi \colon \Omega \to \C$ be an essentially bounded measurable function.
The *multiplication operator* $M_\varphi$ on the [[Lebesgue space|SASP14MA1Z]] $L^2(\Omega, \mu)$ is
$$(M_\varphi f)(x) = \varphi(x) f(x).$$

Check of the main properties:

1. $M_\varphi$ is bounded with $\norm{M_\varphi} = \norm{\varphi}_\infty$, since $\int \abs{\varphi f}^2 \, d\mu \le \norm{\varphi}_\infty^2 \int \abs{f}^2 \, d\mu$, and $\sigma$-finiteness gives sets of finite positive measure on which $\abs{\varphi}$ is close to $\norm{\varphi}_\infty$.
2. Its adjoint is $M_{\bar\varphi}$, because $\int \varphi f \bar g \, d\mu = \int f \, \overline{\bar\varphi g} \, d\mu$.
3. It is a normal operator, since $M_\varphi M_{\bar\varphi} = M_{\abs{\varphi}^2} = M_{\bar\varphi} M_\varphi$.
4. Its [[spectrum|BMKATJ802A]] is the essential range of $\varphi$.

A concrete case: on $L^2[0, 1]$ with Lebesgue measure, take $\varphi(x) = x$.
Then $M_\varphi$ is self-adjoint and positive, its spectrum is $[0, 1]$, and it has no eigenvalues: $x f(x) = \lambda f(x)$ almost everywhere forces $f = 0$ almost everywhere.

## Remarks

The spectral theorem for bounded normal operators says that every bounded normal operator on a separable Hilbert space is unitarily equivalent to a multiplication operator of this form.
So multiplication operators are, up to unitary equivalence, all the normal operators.

## Explanations

### L1 human

A multiplication operator multiplies every function by one fixed function $\varphi$.
It is the infinite-dimensional version of a diagonal matrix, with the values of $\varphi$ playing the role of the diagonal entries.
Multiplying by $x$ on $[0, 1]$ has a whole interval as its spectrum, yet no function is an eigenvector, which shows how the spectrum can differ from the set of eigenvalues.
</document>

<document source="https://tamnd.github.io/math2/node/9F7TYES4KE.md" tag="9F7TYES4KE" kind="concept">
# Compact element

## Summary

A compact element of a partially ordered set is an element with this property.
Whenever it lies below the supremum of a directed set, it already lies below some member of that set.
In a complete lattice this means: whenever it lies below a join, it lies below the join of finitely many of the joined elements.
Compact elements play the role of finite or finitely generated objects, and they are the building blocks of [[algebraic lattices|3J0BTXSCVN]] and of [[Scott domains|1SAAZG46DG]].

## Explanations

### L1 human

The name comes from compact sets in topology, where every open cover has a finite subcover.
In the lattice of subsets of a set, an element is compact exactly when it is a finite set.
A finite set inside a union of sets is already inside the union of finitely many of them.
In the lattice of subgroups of a group, the compact elements are the finitely generated subgroups.
So compactness is the order-theoretic way to say "finite" without counting.
</document>

<document source="https://tamnd.github.io/math2/node/9FB0AZSEGM.md" tag="9FB0AZSEGM" kind="concept">
# Drazin inverse

## Summary

The Drazin inverse of a square matrix $A$ is a matrix $X$ that commutes with $A$ and satisfies $XAX = X$ and $A^{k+1}X = A^k$ for some $k$.
It exists and is unique for every square matrix over a field.
It inverts $A$ on the part of the space where $A$ acts invertibly, and it is $0$ on the rest.

## Explanations

### L1 human

A square matrix splits the space into two pieces that it keeps: one where it acts invertibly and one where some power of it is zero.
The Drazin inverse inverts the matrix on the first piece and is zero on the second.
For example, for $A = \operatorname{diag}(2, 0)$, the Drazin inverse is $\operatorname{diag}(1/2, 0)$.
Unlike the [[generalized inverse|FPHNJ5AAYF]], it is defined through powers of $A$, so it fits well with polynomials in $A$.
</document>

<document source="https://tamnd.github.io/math2/node/9FBJED00CP.md" tag="9FBJED00CP" kind="concept">
# Calabi flow

## Summary

The Calabi flow is an evolution of Kähler metrics on a compact complex manifold that stays inside one Kähler class.
It moves the Kähler potential with speed equal to the difference between the [[scalar curvature|ZHX2VYQDZ3]] and its average.
Eugenio Calabi introduced it in his work on extremal Kähler metrics.
Along the flow the Calabi energy, the squared $L^2$ norm of that difference, does not increase.
Its fixed points are the Kähler metrics of constant scalar curvature, so it is a tool for finding such metrics.

## Explanations

### L1 human

A Kähler metric is a way to measure lengths on a complex manifold that fits well with the complex structure.
The Calabi flow changes such a metric step by step, pushing it where the scalar curvature is above average and pulling it where it is below average.
The aim is to reach a metric whose scalar curvature is the same at every point.
The equation is of fourth order in the space variables, so it is harder to study than second-order flows such as the [[Ricci flow|CR14D9YQTY]].
A flat metric on a complex torus does not move at all, since its scalar curvature is $0$ everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/9FN0PFZZXZ.md" tag="9FN0PFZZXZ" kind="statement">
# Derivative of a convolution

## Statement

Let $n \ge 1$, and let $f \colon \R^n \to \C$ be locally integrable, that is, integrable on every compact set.
Let $g \colon \R^n \to \C$ be a [[continuously differentiable function|0VW90BBSJ4]] with compact support.
Then the [[convolution|2Z29JN3VES]] $f * g$ is defined everywhere and is continuously differentiable, and for $j = 1, \dots, n$
$$\frac{\partial}{\partial x_j} (f * g) = f * \frac{\partial g}{\partial x_j}.$$

## Remarks

Write $(f * g)(x) = \int f(y) g(x - y) \, dy$ and [[differentiate under the integral sign|0AR06W1E1B]].
Let $x$ range over a bounded set and let $0 < \abs{t} \le 1$.
By the mean value theorem, the difference quotients $(g(x + t e_j - y) - g(x - y))/t$ are bounded by $\sup \abs{\partial_j g}$.
They vanish for $y$ outside a fixed compact set $K$.
Since $f$ is integrable on $K$, the [[dominated convergence theorem|BYNV84PF86]] lets the limit pass inside the integral.
Continuity of $f * \partial_j g$ follows in the same way from the uniform continuity of $\partial_j g$.

Example: on $\R$, take $f = \mathbf{1}_{[0, \infty)}$ and $g$ a continuously differentiable function with compact support.
Then $(f * g)(x) = \int_{-\infty}^x g(u) \, du$, and the formula gives $(f * g)' = f * g' = g$, as the fundamental theorem of calculus predicts.

The derivative falls on the smooth factor; $f$ itself need not be differentiable or even continuous.
Without compact support (or some decay) of $g$ the integral need not converge: for $f = 1$ and $g(x) = 1 + x^2$ the convolution is defined nowhere.

## Explanations

### L1 human

To differentiate a convolution, it is enough to differentiate one of the two factors.
You may choose the nicer one.
So a rough function convolved with a smooth one has a derivative, even if the rough function does not.

### L2 human

This is the basic step behind [[smoothing by convolution|4P7KPA3Y7R]]: by induction, $f * g$ is as differentiable as $g$.
It also explains why convolution commutes with constant-coefficient differential operators $P(\partial)$: $P(\partial)(f * g) = f * P(\partial) g$.
In the theory of distributions the same formula holds with $\partial(f * g) = (\partial f) * g = f * (\partial g)$ for any distributions $f, g$ one of which has compact support.
</document>

<document source="https://tamnd.github.io/math2/node/9FR2ZRAVCE.md" tag="9FR2ZRAVCE" kind="definition">
# Spherical trigonometry

## Definition

Let $S^2 \subset \R^3$ be the unit [[sphere|T62369RFH6]] with the great circles as lines, as in [[spherical geometry|SQ9RQVANMG]].
Let $A$, $B$, $C \in S^2$ be linearly independent vectors of $\R^3$.
The *spherical triangle* $ABC$ is the region of $S^2$ bounded by the three shorter great-circle arcs $BC$, $CA$, $AB$.
Its sides are the arc lengths $a = \arccos(B \cdot C)$, $b = \arccos(C \cdot A)$, $c = \arccos(A \cdot B)$, each in $(0, \pi)$.
Its angle $\gamma$ at $C$ is the angle between the planes of the arcs $CA$ and $CB$, measured inside the triangle.
The angles $\alpha$ at $A$ and $\beta$ at $B$ are defined in the same way.
*Spherical trigonometry* is the study of the relations among these six quantities $a, b, c, \alpha, \beta, \gamma$.

## Remarks

Example: for $A = (1, 0, 0)$, $B = (0, 1, 0)$, $C = (0, 0, 1)$, all sides and all angles equal $\pi/2$.
The spherical law of cosines $\cos c = \cos a \cos b + \sin a \sin b \cos \gamma$ checks: both sides are $0$.
Non-example: the vectors $A = (1, 0, 0)$, $B = (-1, 0, 0)$ and $C = (0, 0, 1)$ are linearly dependent.
The points $A$ and $B$ are antipodal, there is no unique shorter arc $AB$, and $A, B, C$ do not form a spherical triangle.
The standard relations are the law of cosines for sides above, the law of cosines for angles $\cos \gamma = -\cos \alpha \cos \beta + \sin \alpha \sin \beta \cos c$, and the law of sines $\dfrac{\sin a}{\sin \alpha} = \dfrac{\sin b}{\sin \beta} = \dfrac{\sin c}{\sin \gamma}$.
On a sphere of radius $R$, the sides are the arc lengths divided by $R$.
</document>

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# Idempotence

## Summary

Idempotence is the property of something that gives the same result when it is applied twice as when it is applied once.
An element $x$ is idempotent for a [[binary operation|0M418FE7BB]] $\ast$ when $x \ast x = x$, and a function $f$ is idempotent when $f \circ f = f$.
Lattice operations, projections and closure operators are idempotent.
The [[idempotent elements of a ring|0N5NJZKKN3]] are the idempotents for its multiplication.

## Explanations

### L1 human

Pressing the "on" button of a machine that is already on changes nothing; that is idempotence.
In numbers, $\max(x, x) = x$ for every $x$, so the operation $\max$ is idempotent.
Rounding down is an idempotent function: rounding a number that is already an integer leaves it unchanged, so rounding twice is the same as rounding once.
Addition of numbers is not idempotent, because $1 + 1 \ne 1$.
</document>

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# Gelfand–Naimark–Segal construction

## Summary

The Gelfand–Naimark–Segal (GNS) construction starts from a [[state|F7MB2WREBJ]] $\rho$ on a C*-algebra $A$.
It builds a Hilbert space $H_\rho$, a representation $\pi_\rho$ of $A$ on it and a cyclic vector $\xi_\rho$ with $\rho(a) = \ip{\pi_\rho(a)\xi_\rho}{\xi_\rho}$.
It shows that states and cyclic representations are two views of the same thing.
Applied to enough states, it proves that every C\*-algebra is isomorphic to an algebra of operators on a Hilbert space.
The same idea turns a positive-definite function on a group into a unitary representation.

## Explanations

### L1 human

A state assigns a number to each element of an algebra, like the average value of an observable in physics.
The GNS construction uses the state to define an inner product on the algebra itself, by $\ip{a}{b} = \rho(b^* a)$.
After removing elements of length zero and filling in limits, the algebra acts on this space by multiplication on the left.
For the algebra of continuous functions on $[0, 1]$ with the state "integrate against $dx$", the result is the space $L^2[0, 1]$, on which functions act by multiplication.
</document>

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# Monoid object

## Summary

A monoid object in a [[monoidal category|M3MSQ3NKAE]] is an object $M$ with a multiplication $M \otimes M \to M$ and a unit $I \to M$.
These must satisfy the associativity and unit laws, written as equations between morphisms.
It generalizes the notion of a [[monoid|0N6R902FG1]] to any monoidal category.
Monoids in sets are ordinary monoids, monoids in abelian groups are [[rings|0B1261B5ED]], monoids in vector spaces are associative algebras, and monoids among endofunctors are [[monads|FF6XXA6D1A]].
Many algebraic structures are thus one notion read in different monoidal categories.

## Explanations

### L1 human

A monoid is a set with an associative multiplication and a unit, like the whole numbers under addition.
A monoid object asks for the same thing, but the "set" may be any object of a monoidal category and the multiplication is a morphism.
The laws are written as equalities of composite morphisms instead of equations between elements.
For example, a ring is an abelian group $R$ with a multiplication map $R \otimes R \to R$ that is associative and has a unit $1$: a monoid object in abelian groups.
</document>

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# Laplace's method

## Summary

Laplace's method approximates integrals of the form $\int_a^b e^{M f(x)}\, dx$ when the parameter $M$ is large.
For large $M$ almost all of the integral comes from a small neighbourhood of the point where $f$ is largest.
There $f$ is close to a downward parabola.
Replacing $f$ by this parabola turns the integral into a Gaussian integral, which gives the leading term of an [[asymptotic equivalence|00SB5EA3VW]].
[[Stirling's formula|0KC0F51T6A]] is the classical application.

## Explanations

### L1 human

Raise $e$ to the power $M f(x)$ with a huge $M$: the values of $x$ where $f$ is largest become far more important than all others.
The graph of $e^{M f(x)}$ turns into a tall, thin bump above the maximum point.
Near its top the bump looks like a bell curve, whose area is known exactly.
So the integral is about the height of the bump times a width proportional to $1/\sqrt{M}$.
</document>

<document source="https://tamnd.github.io/math2/node/9G82Z17DKA.md" tag="9G82Z17DKA" kind="statement">
# Summation by parts

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]], let $(f_k)$ and $(g_k)$ be sequences in $R$, and let $m \le n$ be integers.
Write $\Delta f_k = f_{k+1} - f_k$ and $\Delta g_k = g_{k+1} - g_k$ for the [[forward differences|TXTR68V4R1]].
Then
$$\sum_{k=m}^{n} f_k\, \Delta g_k = f_{n+1} g_{n+1} - f_m g_m - \sum_{k=m}^{n} g_{k+1}\, \Delta f_k.$$

## Remarks

The product rule for differences is $\Delta(f g)_k = f_k \Delta g_k + g_{k+1} \Delta f_k$, which one checks by expanding.
Summing it from $k = m$ to $n$, the left side telescopes to $f_{n+1} g_{n+1} - f_m g_m$, and this gives the formula.
A common form, often called the Abel transformation, uses partial sums $A_k = a_1 + \dots + a_k$:
$$\sum_{k=1}^{n} a_k b_k = A_n b_n - \sum_{k=1}^{n-1} A_k (b_{k+1} - b_k).$$
Example: with $f_k = k$ and $g_k = 2^k$, $\Delta g_k = 2^k$ and the formula gives $\sum_{k=0}^{n} k\, 2^k = (n+1) 2^{n+1} - \sum_{k=0}^{n} 2^{k+1} = (n - 1) 2^{n+1} + 2$.
For $n = 2$ both sides equal $10$.

## Explanations

### L1 human

Summation by parts moves a difference from one factor of a product to the other, at the price of boundary terms.
It is the discrete form of [[integration by parts|08Y1BBQTK9]].
It helps when one factor is easy to sum and the other changes slowly.

### L2 human

The Abel form is the key step in [[Dirichlet's test|0V9ME8GQE7]] and [[Abel's test|0CN97GADE5]] for the convergence of series $\sum a_k b_k$.
In Dirichlet's test, bounded partial sums $A_k$ and monotone $b_k$ tending to $0$ make the right side converge.
Its continuous-variable version is [[Abel's summation formula|V5EPT2PX9Z]], used in analytic number theory.
In the [[calculus of finite differences|FT9JQA58KJ]] it is proved from the [[fundamental theorem of the calculus of finite differences|MNSSV4QHD1]] applied to the product $f g$.
</document>

<document source="https://tamnd.github.io/math2/node/9G8XXBG9F3.md" tag="9G8XXBG9F3" kind="definition">
# Series-parallel partial order

## Definition

The class of *series-parallel partial orders* is the smallest class $\mathcal{S}$ of finite [[partially ordered sets|0QV2JS9RJ3]] such that:

1. every one-element poset is in $\mathcal{S}$;
2. if $P$ and $Q$ are in $\mathcal{S}$, then their [[ordinal sum|6AJ7QQCTQQ]] $P \oplus Q$ is in $\mathcal{S}$;
3. if $P$ and $Q$ are in $\mathcal{S}$, then their [[disjoint union|TZWMEM7GKA]] $P + Q$ is in $\mathcal{S}$;
4. every poset isomorphic to a member of $\mathcal{S}$ is in $\mathcal{S}$.

## Remarks

Example: the poset with $a < b < d$, $a < c < d$ and $b$, $c$ incomparable is $\set{a} \oplus (\set{b} + \set{c}) \oplus \set{d}$, so it is series-parallel.
Non-example: the poset $N$ on $\set{a, b, c, d}$ with $a < c$, $b < c$, $b < d$ as the only strict relations is not series-parallel.
It is connected, so it is not a disjoint union of two nonempty posets.
It is not an ordinal sum $A \oplus B$ of two nonempty posets either.
Nothing lies below $a$, so $a$ would be in $A$, and every element of $B$ would be above $a$.
The only element above $a$ is $c$, so $B = \set{c}$, but $d$ would then be below $c$, which fails.
A finite poset is series-parallel if and only if it has no subposet isomorphic to $N$.
Total orders and antichains are series-parallel.
</document>

<document source="https://tamnd.github.io/math2/node/9G91CJJ0JQ.md" tag="9G91CJJ0JQ" kind="definition">
# Algebraic surface

## Definition

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]].
An *algebraic surface* over $k$ is an [[algebraic variety|HV6YBS0YDH]] over $k$ of [[dimension|36X2WTHYCB]] $2$.

## Remarks

In the classification theory, "surface" usually means a smooth projective surface, and these extra conditions are then stated.
Example: the projective plane $\mathbb{P}^2$ and the product $\mathbb{P}^1 \times \mathbb{P}^1$ are algebraic surfaces.
Example: the zero set in $\mathbb{P}^3$ of one nonconstant irreducible homogeneous polynomial is an algebraic surface, since a hypersurface in $\mathbb{P}^3$ has dimension $3 - 1 = 2$.
Non-example: a plane curve such as $y^2 = x^3 - x$ in $\mathbb{A}^2$ is an algebraic variety of dimension $1$, not $2$.
Non-example: the affine space $\mathbb{A}^3$ is a variety of dimension $3$.
Over $\C$, by Chow's theorem, a smooth projective algebraic surface is the same as a compact complex surface that embeds in projective space.
Not every compact complex surface is of this kind.
</document>

<document source="https://tamnd.github.io/math2/node/9G9G8ED97D.md" tag="9G9G8ED97D" kind="counterexample">
# Hausdorff paradox

## Counterexample

Let $S^2 = \set{x \in \R^3 : \norm{x} = 1}$ be the unit sphere.
Assume the [[axiom of choice|0HTGB13FE2]].
There are a [[countable set|01TQB88H4C]] $D \subseteq S^2$ and pairwise disjoint sets $A$, $B$, $C$ with $S^2 \setminus D = A \cup B \cup C$.
They can be chosen so that the four sets $A$, $B$, $C$ and $B \cup C$ are congruent to each other by [[rotations|0HFREYRSZP]] of $\R^3$.
Hence there is no finitely additive, rotation-invariant function $\mu$ on all subsets of $S^2$ with values in $[0, \infty)$ and $\mu(S^2) = 1$.

## Remarks

The proof uses two rotations $\varphi$ and $\psi$ in the [[special orthogonal group|0CGYNDCWRR]] $SO(3)$ with $\varphi^2 = \psi^3 = \id$.
They can be chosen to generate a free product of a group of order $2$ and a group of order $3$.
The set $D$ collects the fixed points of the non-identity elements of this group, and the group acts freely on the rest of the sphere.
Choosing one point in each orbit transfers a paradoxical splitting of the group to the sphere.
For the consequence, finite additivity and invariance give $\mu(A) = \mu(B) = \mu(C) = \mu(B) + \mu(C)$.
So all three are $0$, and $\mu(D) = 1$.
But for a countable $D$ there is a rotation $\rho$ such that the sets $D, \rho(D), \rho^2(D), \ldots$ are pairwise disjoint, which forces $\mu(D) \le 1/n$ for every $n$.
Hausdorff used this to show that the measure problem has a negative answer in dimension $3$ and higher; the Banach–Tarski paradox later strengthened the result.

## Explanations

### L1 human

Apart from a countable set of points, the sphere can be cut into three pieces $A$, $B$, $C$ with a strange property.
Rotating $A$ gives $B$, rotating it again gives $C$, and some rotation also turns $A$ into $B$ and $C$ together.
So $A$ looks like one third of the sphere and also like one half of it.
This shows that no notion of area can be defined for all subsets of the sphere, if it adds up correctly and does not change under rotation.
The pieces cannot be drawn; their existence uses the axiom of choice.
</document>

<document source="https://tamnd.github.io/math2/node/9G9S92T1VJ.md" tag="9G9S92T1VJ" kind="concept">
# Loewner order

## Summary

The Loewner order is the [[partial order|0QV2JS9RJ3]] on [[Hermitian matrices|0P17NGQQMQ]] in which $A \succeq B$ means that $A - B$ is [[positive semidefinite|01MMVW8CKC]].
It extends the usual order of real numbers to matrices, but it is not a total order.
It is the order used in semidefinite programming, matrix inequalities and the theory of operator monotone functions.

## Explanations

### L1 human

For numbers, $a \ge b$ means that $a - b$ is not negative.
For Hermitian matrices, $A \succeq B$ means that $A - B$ is not negative in the matrix sense: $x^* (A - B) x \ge 0$ for every vector $x$.
So $A$ is bigger than $B$ in every direction.
Two matrices can be incomparable: $\operatorname{diag}(1, 0)$ is bigger in the first direction and $\operatorname{diag}(0, 1)$ in the second.
</document>

<document source="https://tamnd.github.io/math2/node/9GHD9MXNYV.md" tag="9GHD9MXNYV" kind="concept">
# Idele class group

## Summary

The idele class group of a [[number field|MWBDFS2KJ5]] $K$ is the quotient $C_K = I_K / K^\times$ of the idele group $I_K$ by the nonzero elements of $K$.
The idele group $I_K$ is the group of units of the [[adele ring|SEH68BA1QQ]] of $K$, with its own restricted product topology.
In [[class field theory|BEKCEYHCHX]], the [[Artin reciprocity map|7938EKMMVR]] goes from $C_K$ onto the Galois group of the maximal abelian extension of $K$.
The finite abelian extensions of $K$ then match the open subgroups of finite index of $C_K$.

## Explanations

### L1 human

An idele is a list with one nonzero entry for each completion of $K$, one entry for every [[place|BT5V3TSNY6]].
Almost all entries must be units, so the list is "small" at almost every prime.
The number field $K$ sits inside diagonally: an element $a$ gives the list $(a, a, a, \ldots)$.
Dividing out by these lists gives the idele class group.
For $K = \Q$, every idele is a rational number times an idele with a positive real entry and unit entries at all primes.
So $C_\Q$ is the product of the positive reals and the group $\prod_p \Z_p^\times$.
</document>

<document source="https://tamnd.github.io/math2/node/9GKRK5B2B5.md" tag="9GKRK5B2B5" kind="definition">
# Monotone likelihood ratio

## Definition

Let $\Theta \subseteq \R$ and let $\set{f_\theta : \theta \in \Theta}$ be a family of [[densities|04YG690N5V]] (or probability mass functions) on a sample space $\mathcal{X}$.
Let $T \colon \mathcal{X} \to \R$ be a [[statistic|036756NFD6]].
The family has a *monotone likelihood ratio* in $T$ if, for all $\theta_1 < \theta_2$ in $\Theta$:

1. $f_{\theta_1} \ne f_{\theta_2}$;
2. there is a nondecreasing function $g \colon \R \to [0, \infty]$ with $f_{\theta_2}(x)/f_{\theta_1}(x) = g(T(x))$ for every $x$ with $f_{\theta_1}(x) > 0$ or $f_{\theta_2}(x) > 0$.

Here $c/0 = \infty$ for $c > 0$.

## Remarks

The ratio in clause 2 is the [[likelihood ratio|05FSTXM92W]] of $\theta_2$ to $\theta_1$ at the data $x$.
Example: let $f_\theta(x) = h(x) \exp\big(\eta(\theta) T(x) - A(\theta)\big)$ be a one-parameter exponential family with $\eta$ strictly increasing.
Then $f_{\theta_2}(x)/f_{\theta_1}(x) = \exp\big((\eta(\theta_2) - \eta(\theta_1)) T(x) - A(\theta_2) + A(\theta_1)\big)$, which is increasing in $T(x)$.
This covers the binomial law in $p$ with $T(x) = x$, and $\mathcal{N}(\theta, 1)$ with $T(x) = x$.
Non-example: the Cauchy location family $f_\theta(x) = 1/(\pi(1 + (x - \theta)^2))$ fails clause 2 with $T(x) = x$.
For $\theta_1 = 0$ and $\theta_2 = 1$, the ratio $(1 + x^2)/(1 + (x - 1)^2)$ tends to $1$ as $x \to \pm\infty$ and is larger than $1$ at $x = 1$, so it is not monotone.
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<document source="https://tamnd.github.io/math2/node/9GKS3RBZ8V.md" tag="9GKS3RBZ8V" kind="statement">
# Δ-system lemma

## Statement

A family $\mathcal{A}$ of [[sets|0SJ3RX9TXA]] is a Δ-system if there is a set $r$, the root, with $a \cap b = r$ for all distinct $a, b \in \mathcal{A}$.
Let $\mathcal{A}$ be an [[uncountable|0X55M9XET4]] family of finite sets.
Then there is an uncountable subfamily $\mathcal{B} \subseteq \mathcal{A}$ that is a Δ-system.

## Remarks

The proof first passes to an uncountable subfamily whose sets all have the same size $n$.
This is possible since $\mathcal{A}$ is the union of [[countably|0ZG61TFBES]] many subfamilies, one for each size.
Then it inducts on $n$.
If some element lies in uncountably many sets, remove it, apply the induction hypothesis to those sets, and add it back to the root.
Otherwise each element lies in countably many sets, and a recursion of length $\omega_1$ picks uncountably many pairwise disjoint sets, a Δ-system with empty root.
Example: the family $\set{\set{0, \alpha} : 0 < \alpha < \omega_1}$ is a Δ-system with root $\set{0}$.
Finiteness matters: the countable ordinals $\alpha < \omega_1$ are countable sets, and they have no uncountable Δ-subsystem.
Indeed $\alpha \cap \beta = \min(\alpha, \beta)$, and this set is not the same for all pairs in an uncountable subfamily.
A general form: let $\kappa$ be an infinite cardinal and $\theta > \kappa$ a regular cardinal with $\abs{\alpha}^{<\kappa} < \theta$ for all $\alpha < \theta$.
Then each family of at least $\theta$ sets, each of size less than $\kappa$, has a Δ-subsystem of size $\theta$.
The finite analogue is the [[sunflower lemma|N6TH42TD7F]] of Erdős and Rado, where a Δ-system is called a sunflower.

## Explanations

### L1 human

A Δ-system is a family of sets that all overlap in exactly the same common part.
The lemma says that from uncountably many finite sets one can always pick uncountably many that form such a pattern.
So, after throwing some sets away, the overlaps become as simple as possible.

### L2 human

The Δ-system lemma is a standard tool in forcing.
It is the usual way to show that a forcing notion satisfies the countable chain condition, for example the forcing that adds Cohen reals.
Given uncountably many finite conditions, one finds uncountably many whose domains form a Δ-system, and then two of them agree on the root and are compatible.
</document>

<document source="https://tamnd.github.io/math2/node/9GNAAHQJY6.md" tag="9GNAAHQJY6" kind="definition">
# Convergence of filters

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]], let $\mathcal{F}$ be a [[filter|15PF73BKY6]] on $X$, and let $x \in X$.
Let $\mathcal{N}(x)$ be the filter of [[neighbourhoods|0ZCY0Z80B6]] of $x$.
The filter $\mathcal{F}$ *converges* to $x$, written $\mathcal{F} \to x$, if $\mathcal{N}(x) \subseteq \mathcal{F}$.
Then $x$ is a *limit* of $\mathcal{F}$.

## Remarks

For a map $f \colon S \to X$ from a set $S$ and a filter $\mathcal{G}$ on $S$, the *image filter* is $f(\mathcal{G}) = \set{B \subseteq X : f^{-1}(B) \in \mathcal{G}}$.
One says that $f(s) \to x$ along $\mathcal{G}$ if $f(\mathcal{G}) \to x$.

Example: let $(x_n)$ be a sequence in $X$, and let $\mathcal{F}$ be the family of sets $B \subseteq X$ that contain $x_n$ for all large $n$.
Then $\mathcal{F}$ is a filter, and $\mathcal{F} \to x$ if and only if the sequence [[converges|0W4CN9E2W5]] to $x$, since both mean that each neighbourhood of $x$ contains $x_n$ for all large $n$.

Example: the neighbourhood filter $\mathcal{N}(x)$ converges to $x$, and so does the principal filter $\set{B : x \in B}$.

Non-example: in $\R$, let $\mathcal{F}$ be the filter of sets that contain an interval $[n, \infty)$.
For each $x \in \R$, the neighbourhood $(x - 1, x + 1)$ contains no such interval, so $\mathcal{F}$ does not converge to $x$.

A space is [[Hausdorff|0S1FJ8VEX9]] if and only if every filter on it has at most one limit.
</document>

<document source="https://tamnd.github.io/math2/node/9GQQ7TK36T.md" tag="9GQQ7TK36T" kind="statement">
# Keller's conjecture

## Statement

Let $n \ge 1$.
Call a [[tiling|GKJK293DXS]] of $\R^n$ a *cube tiling* if each tile is a [[translate|087VJM5Q77]] $[0, 1]^n + v$ of the unit cube, with $v \in \R^n$.
Then the following are equivalent.

1. Every cube tiling of $\R^n$ contains two cubes that meet in a common $(n-1)$-dimensional face.
2. $n \le 7$.

## Remarks

Ott-Heinrich Keller conjectured in 1930 that clause 1 holds in every dimension.
Oskar Perron proved it for $n \le 6$ in 1940.
Jeffrey Lagarias and Peter Shor showed in 1992 that it fails for $n \ge 10$, and John Mackey showed in 2002 that it fails for $n = 8$, hence also for $n = 9$.
The case $n = 7$ was settled positively in 2020 by Joshua Brakensiek, Marijn Heule, John Mackey and David Narváez with a large computer search based on a SAT solver.
The counterexamples come from cliques in the *Keller graphs*, finite graphs whose large cliques correspond to cube tilings without face-to-face pairs.
For $n = 2$ the statement is easy to see.
A tiling of the plane by unit squares splits into rows or into columns, and two neighbouring squares in a row or column share an edge.

## Explanations

### L1 human

Fill space with equal cubes, all facing the same way, with no gaps and no overlaps.
Keller guessed that some two cubes must always touch along a whole face.
This is true in dimensions up to $7$ but false from dimension $8$ on.

### L2 human

In the plane the conjecture is easy, and in three dimensions it can still be checked by hand.
Hajós turned the problem into a question about factorizations of finite abelian groups, and later the problem was reduced to finite graphs.
These reductions allowed computer searches, and the final case, dimension $7$, needed a computer-assisted proof.
The lattice version, where the translation vectors form a lattice, is Minkowski's conjecture, proved by Hajós in 1941 in all dimensions.
</document>

<document source="https://tamnd.github.io/math2/node/9GS8XQMZED.md" tag="9GS8XQMZED" kind="definition">
# Jacobian variety

## Definition

Let $X$ be a compact connected [[Riemann surface|5K0N8QMV6Q]] of [[genus|NJ787GZS1N]] $g \ge 1$.
Let $\Omega(X)$ be the complex vector space of holomorphic [[Abelian differentials|P014C13K5H]] on $X$, which has dimension $g$, and let $\Omega(X)^*$ be its dual space.
Each class $\gamma \in H_1(X, \Z)$ gives the linear form $\omega \mapsto \int_\gamma \omega$ on $\Omega(X)$, and this defines a map $H_1(X, \Z) \to \Omega(X)^*$ whose image $\Lambda$ is a [[lattice|WN9V7WSJKF]] of rank $2g$.
The *Jacobian variety* of $X$ is the [[complex torus|Y7MA6KBE9M]]
$$J(X) = \Omega(X)^* / \Lambda.$$

## Remarks

Choosing a basis $\omega_1, \ldots, \omega_g$ of $\Omega(X)$ identifies $J(X)$ with $\C^g / \Lambda$, where $\Lambda$ is spanned by the period vectors $\left( \int_\gamma \omega_1, \ldots, \int_\gamma \omega_g \right)$.
The Riemann bilinear relations give $J(X)$ a principal polarization, so it is an abelian variety.
By the Abel–Jacobi theorem, $J(X)$ is isomorphic to the group $\operatorname{Pic}^0(X)$ of divisor classes of degree $0$; this description defines the Jacobian of a smooth projective curve over any field.
Example: for $X = \C / \Lambda_0$ with $g = 1$ and $\omega_1 = dz$, the periods of $dz$ are the elements of $\Lambda_0$, so $J(X) \cong \C / \Lambda_0 \cong X$.
Non-example: a general complex torus $\C^2 / \Lambda$ of dimension $2$ admits no nonconstant meromorphic functions, so it is not an abelian variety and is not the Jacobian of any curve.
Torelli's theorem says that $X$ is determined by $J(X)$ together with its principal polarization.
</document>

<document source="https://tamnd.github.io/math2/node/9GVQ7JXC25.md" tag="9GVQ7JXC25" kind="definition">
# Cahn–Hilliard equation

## Definition

Let $\Omega \subseteq \R^n$ be open and let $\gamma > 0$.
The *Cahn–Hilliard equation* is the [[parabolic equation|1CCTH8X2FT]] of fourth order for $c \colon \Omega \times (0, T) \to \R$:
$$c_t = \Delta \mu, \qquad \mu = c^3 - c - \gamma\, \Delta c,$$
where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] in $x$.
The function $\mu$ is the chemical potential.
On a bounded $\Omega$ one usually imposes $\partial_\nu c = \partial_\nu \mu = 0$ on $\partial \Omega$, where $\nu$ is the outer normal.

## Remarks

Example: every constant $c \equiv c_0$ is a solution, since then $\mu$ is constant and $\Delta \mu = 0$.
Non-example: the [[Allen–Cahn equation|7S813PEV69]] $c_t = \gamma \Delta c - (c^3 - c)$ is of second order and does not conserve mass, so it is not of this form.
With the boundary conditions above, $\frac{d}{dt}\int_\Omega c\, dx = \int_{\partial\Omega} \partial_\nu \mu \, dS = 0$, so the mass is conserved.
The energy $E(c) = \int_\Omega \big(\frac{\gamma}{2}\abs{\nabla c}^2 + \frac{1}{4}(1 - c^2)^2\big)\, dx$ does not increase along smooth solutions.
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<document source="https://tamnd.github.io/math2/node/9GWGW4YRD9.md" tag="9GWGW4YRD9" kind="definition">
# Cobweb plot

## Definition

Let $I \subseteq \R$ be an interval, let $f \colon I \to I$ be a map, and let $x_0 \in I$.
Let $x_n = f^n(x_0)$ for $n \ge 0$, where $f^n$ is the $n$-th [[iterate|AYVTR2ES5B]] of $f$, so that $x_{n+1} = f(x_n)$.
Consider the points $P_{2n} = (x_n, x_n)$ and $P_{2n+1} = (x_n, x_{n+1})$ for $n \ge 0$.
The *cobweb plot* of the orbit of $x_0$ is the polygonal path $P_0 P_1 P_2 \cdots$.
It is drawn together with the graph of $f$ and the diagonal $\set{(x, x) : x \in I}$.

## Remarks

The points $(x_n, x_{n+1})$ lie on the graph of $f$ and the points $(x_n, x_n)$ lie on the diagonal.
So the path alternates vertical segments to the graph and horizontal segments to the diagonal.
A [[fixed point|R6QVS6P21N]] $p$ of $f$ is a point where the graph meets the diagonal, and the path starting at $x_0 = p$ stays at $(p, p)$.

Example: let $f(x) = x/2 + 1$ on $\R$ and $x_0 = 0$.
The orbit is $0, 1, 3/2, 7/4, \dots$, with $x_n = 2 - 2^{1-n}$ for $n \ge 0$.
The cobweb path goes from $(0, 0)$ up to $(0, 1)$, across to $(1, 1)$, up to $(1, 3/2)$, and so on, a staircase that climbs to the fixed point $(2, 2)$.

Non-example: the plot of the points $(n, x_n)$ against the time $n$ is a time series plot.
It is not a cobweb plot, because it does not use the graph of $f$ and the diagonal.
</document>

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# Congruence-distributive variety

## Summary

A congruence-distributive variety is a [[variety|AJNZJD8442]] in which the [[congruence lattice|REE07CSXHR]] of every algebra is a [[distributive lattice|0HM0HQCK6N]].
Lattices, Boolean algebras and Heyting algebras form congruence-distributive varieties, while groups do not.
In such a variety, Jónsson's lemma locates all subdirectly irreducible algebras, which makes the variety much easier to describe.

## Explanations

### L1 human

The congruences of an algebra form a lattice, which records all the ways to collapse the algebra in a manner compatible with its operations.
In a congruence-distributive variety this lattice always obeys the distributive law, like the lattice of subsets of a set.
For lattices themselves this holds, so every lattice has a distributive lattice of congruences.
For groups it fails: the Klein four-group has three subgroups of order two, and they do not distribute.
</document>

<document source="https://tamnd.github.io/math2/node/9H61HXXJ11.md" tag="9H61HXXJ11" kind="concept">
# Join-prime element

## Summary

An element $a$ of a [[lattice|WH4SFAACAS]] is join-prime if it is not the bottom element and has the following property.
Whenever $a$ lies below a join $b \vee c$, it lies below $b$ or below $c$.
The definition copies that of a [[prime element|01VH21EV6F]] of a ring, with "divides" replaced by "lies below" and the product replaced by the [[join|J2T75ATRV5]].
Every join-prime element is [[join-irreducible|ZHRQ0AT8PM]], and in a [[distributive lattice|0HM0HQCK6N]] the two notions coincide.

## Explanations

### L1 human

A join-prime element cannot hide inside a join without hiding inside one of the two parts.
In the subsets of a set, a singleton $\set{x}$ is join-prime: if $x$ is in $B \cup C$, then $x$ is in $B$ or in $C$.
In the lattice $\set{0, a, b, c, 1}$ with three incomparable middle elements, $a$ is join-irreducible but not join-prime.
It lies below $b \vee c = 1$ but not below $b$ or $c$.
</document>

<document source="https://tamnd.github.io/math2/node/9H65YDXMGJ.md" tag="9H65YDXMGJ" kind="example">
# Laver table

## Example

For $n \ge 0$, the *Laver table* $A_n$ is the set $\set{1, 2, \dots, 2^n}$ with the unique binary operation $\star$ such that:

1. $p \star 1 = p + 1$ for $p < 2^n$, and $2^n \star 1 = 1$;
2. $p \star (q \star r) = (p \star q) \star (p \star r)$ for all $p, q, r$.

So $A_n$ is a [[shelf|PDFC0MRYR8]]: clause 2 is [[left self-distributivity|VEFE5WZMS3]].
The table can be computed row by row from the bottom, where row $2^n$ is $2^n \star q = q$.
For $p < 2^n$ one uses $p \star (q + 1) = (p \star q) \star (p + 1)$, where $p \star q > p$ lies in a row already known.

For $n = 2$ this gives the table of $A_2$, with $p$ indexing rows and $q$ indexing columns:

| $\star$ | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 2 | 4 | 2 | 4 |
| 2 | 3 | 4 | 3 | 4 |
| 3 | 4 | 4 | 4 | 4 |
| 4 | 1 | 2 | 3 | 4 |

Check of one instance of clause 2: $1 \star (2 \star 1) = 1 \star 3 = 2$, and $(1 \star 2) \star (1 \star 1) = 4 \star 2 = 2$.
All $64$ instances of clause 2 hold in $A_2$.
The shelf $A_2$ is not a rack, since the row of $1$ takes only the values $2$ and $4$.

## Remarks

Each row of $A_n$ is periodic, and its period is a power of $2$.
Laver proved that, if a rank-into-rank [[large cardinal|X1FG7PSCTY]] exists, then the period of the row of $1$ in $A_n$ tends to infinity as $n$ grows.
No proof of this statement in ZFC alone is known.
The period of the row of $1$ grows extremely slowly: it is $2$ in $A_2$ and $4$ in $A_3$.
Dougherty showed that the least $n$ for which it reaches $32$, if there is one, is enormous.

## Explanations

### L1 human

A Laver table is a small multiplication table on the numbers $1$ to $2^n$.
It is fixed by two rules: multiplying by $1$ on the right adds one, and the operation distributes over itself.
These two rules leave no freedom, so each size has exactly one table.
The tables look like simple finite puzzles, but a basic fact about how their rows repeat has only been proved using axioms of infinity far beyond ordinary mathematics.
</document>

<document source="https://tamnd.github.io/math2/node/9HBDQ2W9CF.md" tag="9HBDQ2W9CF" kind="concept">
# Completely normal space

## Summary

A completely normal space is a topological space in which every subspace is [[normal|0NSR9V8TED]].
Equivalently, any two separated sets, where neither meets the closure of the other, lie in disjoint open sets.
Normality is not inherited by subspaces in general, so this is a real strengthening.
A completely normal $T_1$ space is called a $T_5$ space.

## Explanations

### L1 human

Normal spaces let you pull apart two closed sets that do not meet.
In a completely normal space, you can do more: you can pull apart two sets that merely keep away from each other's closures.
For example, on the real line, the open intervals $(0, 1)$ and $(1, 2)$ touch at the point $1$, yet neither contains a limit point of the other.
They lie in the disjoint open sets $(0, 1)$ and $(1, 2)$.
Every metric space is completely normal.
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<document source="https://tamnd.github.io/math2/node/9HD1MEV7M4.md" tag="9HD1MEV7M4" kind="definition">
# Meijer G-function transforms

## Definition

Fix integers $0 \le m \le q$ and $0 \le n \le p$, and complex parameters $a_1, \dots, a_p$ and $b_1, \dots, b_q$.
Let $G(z) = G^{m,n}_{p,q}(z \mid a_1, \dots, a_p; b_1, \dots, b_q)$ be the [[Meijer G-function|19JNNN5KV7]] with these parameters.
Let $f \colon (0, \infty) \to \C$ be such that the integral below converges for $y$ in a set $Y \subseteq (0, \infty)$.
The *G-transform* of $f$ with these parameters is the function
$$g(y) = \int_0^\infty G(xy)\, f(x)\, dx, \qquad y \in Y.$$

## Remarks

Example: $G^{1,0}_{0,1}(z \mid -; 0) = e^{-z}$, so with $m = q = 1$, $n = p = 0$ and $b_1 = 0$ the G-transform is the [[Laplace transform|0V49JEZW06]] $g(y) = \int_0^\infty e^{-xy} f(x)\, dx$.
Example: $G^{1,1}_{1,1}(z \mid 0; 0) = (1 + z)^{-1}$, so another choice gives the transform $\int_0^\infty f(x) (1 + xy)^{-1}\, dx$, which is a form of the [[Stieltjes transform|RG1HFAYD98]].
Example: $G^{1,0}_{0,2}(z \mid -; \nu/2, -\nu/2) = J_\nu(2\sqrt{z})$, a Bessel function kernel.

Non-example: for the Laplace case, $f(x) = e^{x^2}$ gives a divergent integral for every $y > 0$, so it has no G-transform with these parameters.

The [[Mellin transform|N5PVGRJE1T]] turns the G-transform into a product: if $\mathcal{M}$ denotes the Mellin transform, then formally $\mathcal{M} g(s) = \mathcal{M} G(s)\, \mathcal{M} f(1 - s)$, and
$$\mathcal{M} G(s) = \frac{\prod_{j=1}^{m} \Gamma(b_j + s) \prod_{j=1}^{n} \Gamma(1 - a_j - s)}{\prod_{j=m+1}^{q} \Gamma(1 - b_j - s) \prod_{j=n+1}^{p} \Gamma(a_j + s)}.$$
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<document source="https://tamnd.github.io/math2/node/9HD9A3YWTN.md" tag="9HD9A3YWTN" kind="definition">
# Dawson function

## Definition

The *Dawson function* is the function $F \colon \R \to \R$ given by
$$F(x) = e^{-x^2} \int_0^x e^{t^2}\, dt,$$
where $e$ denotes the [[exponential function|0J0FCWV32D]] and the integral is a [[definite integral|04TMP0XGMJ]].

## Remarks

Example: $F(0) = 0$, and $F$ is odd because $e^{t^2}$ is even.
The product rule gives the differential equation $F'(x) = 1 - 2x F(x)$, with $F(0) = 0$.

Non-example: $x \mapsto e^{x^2} \int_0^x e^{-t^2}\, dt$ is not the Dawson function; the signs of the exponents are swapped, and this function grows without bound.

With the imaginary error function $\operatorname{erfi} x = \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2}\, dt$ one has $F(x) = \frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi} x$.
For large $x$, $F(x) \sim \frac{1}{2x}$.
Some texts call this $D_+(x)$ and also use $D_-(x) = e^{x^2} \int_0^x e^{-t^2}\, dt$, which is the non-example above.
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<document source="https://tamnd.github.io/math2/node/9HDQ094ADQ.md" tag="9HDQ094ADQ" kind="concept">
# Diffusion process

## Summary

A diffusion process is a Markov process with continuous paths, such as Brownian motion with a drift and a variable volatility.
Its local behaviour is described by a drift vector and a diffusion matrix.
Its [[generator|VPA41PWRR1]] is a second-order differential operator, so diffusions connect probability with parabolic and elliptic partial differential equations.
Most diffusions are built as solutions of stochastic differential equations.

## Explanations

### L1 human

Picture a particle of dust in a moving fluid.
The fluid carries it along, which is the drift.
Collisions with molecules shake it at random, which is the diffusion part.
The path is continuous but very rough, like the path of Brownian motion.
The [[Ornstein–Uhlenbeck process|HZM2ASCDY4]], pulled back towards $0$ while being shaken, is a standard example.
</document>

<document source="https://tamnd.github.io/math2/node/9HHEJVCS2Q.md" tag="9HHEJVCS2Q" kind="definition">
# Strongly pseudoconvex domain

## Definition

Let $\Omega \subseteq \C^n$ be a domain with $C^2$ boundary, and let $\rho$ be a defining function of $\Omega$.
For $p \in \partial\Omega$, let $H_p$ be the complex tangent space of $\partial\Omega$ at $p$.
The domain $\Omega$ is *strongly pseudoconvex* if the [[Levi form|80MTNXWW80]] of $\partial\Omega$ is positive definite at every boundary point:
$$\sum_{j,k=1}^{n} \frac{\partial^2 \rho}{\partial z_j \partial \bar z_k}(p)\, w_j \bar w_k > 0 \quad \text{for all } p \in \partial\Omega \text{ and all } w \in H_p \setminus \set{0}.$$
A boundary point where this holds is a *strongly pseudoconvex point*.

## Remarks

A strongly pseudoconvex domain is [[Levi pseudoconvex|PETEQQCS0B]].
Many authors also ask $\Omega$ to be bounded.
Example: the unit ball, with $\rho(z) = \abs{z}^2 - 1$, has Levi form $\abs{w}^2$, which is positive for $w \ne 0$.
Non-example: the domain $\set{z \in \C^2 : \abs{z_1}^2 + \abs{z_2}^4 < 1}$ is Levi pseudoconvex but not strongly pseudoconvex.
At $p = (1, 0)$, with $\rho(z) = \abs{z_1}^2 + \abs{z_2}^4 - 1$, the space $H_p$ is $\set{w : w_1 = 0}$ and $L_\rho(p; (0, w_2)) = 4 \abs{p_2}^2 \abs{w_2}^2 = 0$.
A bounded domain $\Omega$ with $C^2$ boundary is strongly pseudoconvex if and only if it has a defining function that is strictly plurisubharmonic near $\partial\Omega$.
This means that $L_\rho(p; \cdot)$ is positive definite on all of $\C^n$, not only on $H_p$.
</document>

<document source="https://tamnd.github.io/math2/node/9HNTG5C2XX.md" tag="9HNTG5C2XX" kind="definition">
# Perturbation theory of linear operators

## Definition

Let $X$ be a [[Banach space|06GZ956657]] and let $T$ be a linear [[operator|KRJX5RA0PW]] in $X$ with domain $D(T)$.
A linear operator $V$ in $X$ is *relatively bounded with respect to $T$*, or *$T$-bounded*, if:

1. $D(T) \subseteq D(V)$, and
2. there are constants $a, b \ge 0$ with $\norm{Vx} \le a \norm{x} + b \norm{Tx}$ for all $x \in D(T)$.

The infimum of the numbers $b$ for which some $a$ makes clause 2 hold is the *$T$-bound* of $V$.
The operator $T + V$ with domain $D(T)$ is a *perturbation* of $T$.
Perturbation theory of linear operators studies which properties of $T$ pass to $T + V$, or to $T + \kappa V$ for small $\kappa$.

## Remarks

Example: every bounded operator $V$ on $X$ is $T$-bounded with $T$-bound $0$, since $\norm{Vx} \le \norm{V}\norm{x}$.
Non-example: on $L^2(\R)$, let $T = -i \, d/dx$ with domain the Sobolev space $H^1(\R)$, and let $V = -d^2/dx^2$ with domain $H^2(\R)$.
Clause 1 fails, since $H^1(\R)$ is not contained in $H^2(\R)$.

A basic result is the Kato–Rellich theorem.
If $T$ is [[self-adjoint|0NEGN5CRSC]] on a Hilbert space and $V$ is symmetric and $T$-bounded with $T$-bound less than $1$, then $T + V$ is self-adjoint on $D(T)$.
Another is [[Weyl's theorem|X9XRGMDCQZ]]: a compact perturbation does not change the [[essential spectrum|FRZG0E15SK]] of a self-adjoint operator.
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# Nullary operation

## Summary

A nullary operation on a set $A$ is a [[finitary operation|EJ8YJVM9BW]] of [[arity|MSXX2P86CY]] $0$, that is, a map from the one-element set $A^0$ to $A$.
Such a map is the same as the choice of one element of $A$, so nullary operations are the constants of an [[algebraic structure|QJGPT2W0FT]].
Treating constants as operations lets identity elements, such as $0$ in a ring or $e$ in a [[group|G5S1M9]], be part of the signature.

## Explanations

### L1 human

An operation with no inputs cannot depend on anything, so it always gives the same element.
That element is a constant.
For example, in the integers with addition, the number $0$ can be seen as a nullary operation.
When $0$ is part of the signature, a [[homomorphism|KBXSZC6BV1]] must send the constant $0$ to the constant $0$ of the target.
</document>

<document source="https://tamnd.github.io/math2/node/9HTB5KVS72.md" tag="9HTB5KVS72" kind="definition">
# Locally finite poset

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
For $x, y \in P$, the closed interval from $x$ to $y$ is $[x, y] = \set{z \in P : x \le z \le y}$.
The poset $P$ is *locally finite* when $[x, y]$ is a [[finite set|070R982G4B]] for all $x, y \in P$.

## Remarks

Example: $\N^k$ with the product order is locally finite.
Check: if $x \le y$, the interval $[x, y]$ is the box $\prod_{i=1}^k \set{x_i, \ldots, y_i}$, which has $\prod_i (y_i - x_i + 1)$ elements; if $x \not\le y$, it is empty.

Non-example: $\Q$ with the usual order is not locally finite, since $[0, 1]$ is infinite.

Every finite poset is locally finite.
The interval $[x, y]$ is empty unless $x \le y$, so only comparable pairs matter.
In Mathlib the class `LocallyFiniteOrder` provides the intervals as finite sets.
</document>

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# Cauchy's logarithmic equation

## Summary

Cauchy's logarithmic equation is the [[functional equation|SRAN4EKWS0]] $f(xy) = f(x) + f(y)$ for an unknown function $f \colon (0, \infty) \to \R$.
It asks which functions turn products into sums.
Its continuous solutions are the functions $f(x) = c \ln x$ with $c \in \R$, the multiples of the [[natural logarithm|04DXAFN2ES]].
It is the equation behind slide rules, decibels and other logarithmic scales.

## Explanations

### L1 human

A logarithm turns multiplication into addition.
This equation takes that rule as the starting point and asks which functions obey it.
For example, $f(x) = \log_{10} x$ gives $\log_{10}(100 \cdot 1000) = 5 = 2 + 3$.
Up to a constant factor, the logarithm is the only reasonable function with this property.
</document>

<document source="https://tamnd.github.io/math2/node/9HWTPQHHZK.md" tag="9HWTPQHHZK" kind="definition">
# Moving least squares

## Definition

Let $d \ge 1$ and $m \ge 0$, and let $\Pi_m$ be the space of [[polynomials in $d$ variables|07GNX8GCYP]] of total degree at most $m$.
Let $x_1, \dots, x_N \in \R^d$ be pairwise distinct, let $f_1, \dots, f_N \in \R$, and let $\theta \colon [0, \infty) \to [0, \infty)$ be a weight function.
For $x \in \R^d$, let $p_x \in \Pi_m$ minimise the [[weighted least squares|H5CGZHBR58]] error
$$\sum_{i=1}^{N} \theta(\norm{x - x_i}) \, \bigl(p(x_i) - f_i\bigr)^2$$
over $p \in \Pi_m$, where $\norm{\cdot}$ is the Euclidean norm.
The *moving least squares approximation* of the data is the function $\hat f(x) = p_x(x)$, defined at every $x$ where this minimiser is unique.

## Remarks

The minimiser is unique when the points $x_i$ with $\theta(\norm{x - x_i}) > 0$ include a [[unisolvent set|5276MDSDQV]] for $\Pi_m$.
In general $\hat f$ only approximates the data; singular weights, such as in the example below, make it interpolate.
If the data come from a polynomial in $\Pi_m$, then $\hat f$ reproduces that polynomial.

Example: $m = 0$ and $\theta(r) = r^{-p}$ for $r > 0$, with $p > 0$, at a point $x$ that is not a data point.
Then $p_x$ is the constant $\sum_i \theta_i f_i / \sum_i \theta_i$, where $\theta_i = \theta(\norm{x - x_i})$.
Extended by the data values at the data points, this is [[inverse distance weighting|QHN67MSH8S]].

Non-example: with constant weights $\theta = 1$ the fit is an ordinary least squares fit.
It gives the same polynomial $p_x = p$ at every $x$, so the result is one global polynomial.
The dependence of the weights on $x$, the "moving" feature, is lost.
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# Pontryagin–Thom construction

## Summary

The Pontryagin–Thom construction starts from a closed submanifold with a tubular neighbourhood.
It gives a map from the ambient manifold to the [[Thom space|5MNYT3AFPH]] of the normal bundle.
It collapses everything outside the tubular neighbourhood to a point.
Pontryagin used it to identify framed [[cobordism|T8BVGPPSZP]] classes of submanifolds of spheres with [[homotopy groups of spheres|YD6VAXXSBM]].
Thom used it to compute cobordism groups as homotopy groups of Thom spaces, see [[Thom's cobordism theorem|HM23BV561Z]].

## Explanations

### L1 human

Picture a closed curve sitting inside a sphere, with a thin tube around it.
Squash everything outside the tube to a single point.
What remains is the tube with its outer wall glued to that point, and this only depends on the curve and how the tube is twisted.
Running the construction backwards, a map to a sphere gives back a submanifold as the preimage of a point.
For a single point in $S^n$ with a small ball around it, the construction gives a map $S^n \to S^n$ of degree $\pm 1$.
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# Rader's FFT algorithm

## Statement

Let $p$ be a prime, let $\omega = e^{-2\pi i/p}$, and let $g$ be a [[primitive root modulo $p$|0F33KX9Q6F]].
For an integer $j$ write $g^{j}$ and $g^{-j}$ for the residues in $\set{1, \ldots, p-1}$ of these powers modulo $p$.
Let $x \in \C^p$ and let $X$ be its [[discrete Fourier transform|TST0GH66A5]], $X_k = \sum_{n=0}^{p-1} x_n \omega^{nk}$.
Define sequences of period $p - 1$ by $a_m = x_{g^m}$ and $c_m = \omega^{g^{-m}}$.
Then $X_0 = \sum_{n=0}^{p-1} x_n$, and for $0 \le q \le p-2$,
$$X_{g^{-q}} = x_0 + \sum_{m=0}^{p-2} a_m\, c_{q-m}.$$
So a DFT of prime length $p$ reduces to a cyclic [[convolution|2Z29JN3VES]] of length $p - 1$ and $O(p)$ further additions.

## Remarks

Since $g$ generates the multiplicative group modulo $p$, the maps $m \mapsto g^m$ and $q \mapsto g^{-q}$ are bijections from $\set{0, \ldots, p-2}$ onto $\set{1, \ldots, p-1}$.
For $n = g^m$ and $k = g^{-q}$ one has $nk \equiv g^{-(q-m)} \pmod p$, and $\omega^{nk}$ depends only on $nk$ modulo $p$; this gives the formula.
The cyclic convolution of length $p-1$ can be computed by DFTs of length $p - 1$, which is composite for $p > 3$.
These can use the [[Cooley–Tukey algorithm|9N8D9E5TB2]], possibly after zero padding to a power of $2$.
Example: $p = 5$ and $g = 2$, so $(g^0, g^1, g^2, g^3) = (1, 2, 4, 3)$ and $(g^0, g^{-1}, g^{-2}, g^{-3}) = (1, 3, 4, 2)$.
The hypothesis that $p$ is prime gives a primitive root, and makes the nonzero indices a group under multiplication.
For composite lengths the nonzero indices do not form such a group.

## Explanations

### L1 human

Fast Fourier transforms usually split the length into factors, which fails when the length is prime.
Rader noticed that for a prime length the nonzero indices can be listed as powers of one number $g$.
In that order, the transform becomes a convolution, and convolutions can be done fast with other FFTs.

### L2 human

The algorithm uses the structure of $(\Z/p\Z)^\times$, which is cyclic of order $p - 1$.
Permuting inputs by $n = g^m$ and outputs by $k = g^{-q}$ turns the $(p-1) \times (p-1)$ block of the DFT matrix into a circulant matrix.
Circulant matrices are diagonalised by the DFT, which is why the convolution can be done in $O(p \log p)$ operations.
Winograd's short-length FFT modules are built on the same reduction.
Bluestein's algorithm is an alternative that works for every length and does not need a primitive root.
</document>

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# Spectral sequence

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $a \ge 0$ be an integer.
A *cohomological spectral sequence* in $\mathcal{A}$ starting at page $a$ consists of:

1. for every $r \ge a$ and all integers $p, q$, an object $E_r^{p,q}$ of $\mathcal{A}$;
2. for every $r \ge a$ and all $p, q$, a morphism $d_r^{p,q} \colon E_r^{p,q} \to E_r^{p+r, q-r+1}$ such that $d_r^{p+r, q-r+1} \circ d_r^{p,q} = 0$;
3. for every $r \ge a$ and all $p, q$, an isomorphism of $E_{r+1}^{p,q}$ with the [[homology|2J4JJJYP22]] $\ker d_r^{p,q} / \im d_r^{p-r, q+r-1}$.

The family $(E_r^{p,q})_{p,q}$ is the *$r$-th page*.

## Remarks

A spectral sequence is *first quadrant* if $E_a^{p,q} = 0$ unless $p \ge 0$ and $q \ge 0$.
Then for fixed $(p, q)$ the differentials into and out of $E_r^{p,q}$ vanish for $r$ large, so $E_r^{p,q}$ is eventually constant; its value is $E_\infty^{p,q}$.
The sequence *converges* to objects $H^n$, written $E_a^{p,q} \Rightarrow H^{p+q}$, if each $H^n$ has a finite [[filtration|HNK1Y62G7V]] $H^n = F^0 H^n \supseteq F^1 H^n \supseteq \cdots \supseteq F^{n+1} H^n = 0$ with $F^p H^n / F^{p+1} H^n \cong E_\infty^{p, n-p}$.
Example: if $E_2^{p,q} = 0$ for $q \ne 0$, then every $d_r$ with $r \ge 2$ starts or ends at a zero object, because it changes $q$ by $1 - r \ne 0$; so $E_2 = E_\infty$ and $H^n \cong E_2^{n,0}$.
Non-example: let $E_2^{0,1} = E_2^{2,0} = \Z/2\Z$, all other $E_2^{p,q} = 0$, let $d_2^{0,1}$ be the identity of $\Z/2\Z$, and put $E_3 = E_2$.
Clause 3 fails, since the homology of $d_2$ is zero at $(0,1)$ and at $(2,0)$, while $E_3^{0,1} = \Z/2\Z$.
Homological spectral sequences are defined in the same way with $d^r \colon E^r_{p,q} \to E^r_{p-r, q+r-1}$.
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# Continuous group homomorphism

## Summary

A continuous group homomorphism is a map between [[topological groups|92700QAB5Y]] that respects the group law and is continuous.
These maps are the morphisms of the category of topological groups.
Because translations are homeomorphisms, a homomorphism is continuous as soon as it is continuous at the identity.
Characters of locally compact abelian groups and finite-dimensional representations of Lie groups are continuous homomorphisms.

## Explanations

### L1 human

A group homomorphism turns products into products.
In topological groups one also wants the map to send nearby elements to nearby elements.
For example, the map $t \mapsto e^{2\pi i t}$ wraps the real line around the circle.
It turns sums of real numbers into products of points on the circle, and it is continuous, so it is a continuous group homomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/9J7GFJ1GPF.md" tag="9J7GFJ1GPF" kind="concept">
# Beltrami equation

## Summary

The Beltrami equation is the partial differential equation $f_{\bar z} = \mu f_z$ for a function $f$ of one complex variable.
The coefficient $\mu$ is a given measurable function with $\norm{\mu}_\infty < 1$.
For $\mu = 0$ it is the Cauchy–Riemann equation, whose solutions are the [[holomorphic functions|0NAHXR1FRH]].
Its homeomorphic solutions are exactly the [[quasiconformal mappings|SB8KG46NHP]] with complex dilatation $\mu$.
The [[measurable Riemann mapping theorem|TZHD2WKNSD]] says that it always has such a solution.

## Explanations

### L1 human

The coefficient $\mu$ prescribes, at each point, an ellipse field: a direction and a stretching ratio.
A solution $f$ of the Beltrami equation is a map that sends these tiny ellipses to tiny circles.
So solving the equation means finding coordinates in which a given distorted geometry looks round.
For example, with the constant coefficient $\mu = 1/3$ the map $f(z) = z + \frac{1}{3}\bar z$ solves the equation, since $f_z = 1$ and $f_{\bar z} = 1/3$.
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<document source="https://tamnd.github.io/math2/node/9JA1N3CH80.md" tag="9JA1N3CH80" kind="definition">
# Homology of a chain complex

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], for example the category of modules over a ring, and let $(C_\bullet, d)$ be a [[chain complex|WQ82C0M3QB]] in $\mathcal{A}$.
For each integer $n$, let $Z_n = \ker d_n$ be the [[kernel|S417YNEJ6M]] of $d_n$, called the object of *$n$-cycles*.
Since $d_n \circ d_{n+1} = 0$, the morphism $d_{n+1}$ factors through a morphism $\bar d_{n+1} \colon C_{n+1} \to Z_n$.
Its image $B_n = \im \bar d_{n+1}$ is a subobject of $Z_n$, called the object of *$n$-boundaries*.
The *$n$-th homology* of $C_\bullet$ is the [[cokernel|H8W38933NC]] of $\bar d_{n+1}$:
$$H_n(C) = Z_n / B_n = \ker d_n / \im d_{n+1}.$$

## Remarks

The complex is exact at $C_n$ if and only if $H_n(C) = 0$.
A [[chain map|HTK9WQZJ1G]] $f \colon C_\bullet \to D_\bullet$ sends cycles to cycles and boundaries to boundaries, so it induces morphisms $H_n(f) \colon H_n(C) \to H_n(D)$, and $H_n$ is a functor.
For a cochain complex $(C^\bullet, d)$ the same construction gives the *cohomology* $H^n(C) = \ker d^n / \im d^{n-1}$.

Example: let $C_1 = C_0 = \Z$ with $d_1$ multiplication by $2$, and $C_n = 0$ otherwise.
Then $Z_1 = \ker d_1 = 0$, so $H_1(C) = 0$; and $Z_0 = \Z$, $B_0 = 2\Z$, so $H_0(C) = \Z/2\Z$.

Non-example: for the sequence $\Z \xrightarrow{1} \Z \xrightarrow{1} \Z$, the image $\Z$ of the first map is not contained in the kernel $0$ of the second.
So $\ker / \im$ is not defined; the condition $d^2 = 0$ is what makes homology make sense.
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# Sample mean and sample covariance

## Definition

Let $n \ge 2$ and $k \ge 1$, and let $X_1, \dots, X_n$ be random vectors in $\R^k$, written as columns.
The *sample mean* is the [[statistic|036756NFD6]]
$$\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \in \R^k.$$
The *sample covariance matrix* is the $k \times k$ matrix
$$Q = \frac{1}{n - 1} \sum_{i=1}^{n} \left(X_i - \bar{X}\right)\left(X_i - \bar{X}\right)^T.$$
Its entry $Q_{jl}$ is the sample covariance of the $j$-th and the $l$-th coordinates.

## Remarks

Example: for the observations $(0, 0)^T$ and $(2, 2)^T$, the mean is $(1, 1)^T$ and the deviations are $(-1, -1)^T$ and $(1, 1)^T$.
Each deviation gives the matrix with all entries $1$, so $Q$ has all four entries equal to $2$.
For $k = 1$, $\bar{X}$ is the [[sample mean|0QNRBHWV35]] and $Q$ is the [[sample variance|0JN9P4A944]].
If the $X_i$ are i.i.d. with mean vector $\mu$ and finite [[covariance matrix|02CBHV4ZZQ]] $\Sigma$, then $E[\bar{X}] = \mu$ and $E[Q] = \Sigma$.
Non-example: the matrix with divisor $n$ in place of $n - 1$ is also used, but it is not unbiased; its expectation is $\frac{n-1}{n}\Sigma$.
The matrix $Q$ is symmetric and positive semidefinite, and it is singular when $n \le k$.
</document>

<document source="https://tamnd.github.io/math2/node/9JQEWK5405.md" tag="9JQEWK5405" kind="statement">
# Kummer's test

## Statement

Let $(a_n)$ and $(c_n)$ be sequences of positive real numbers.
For each $n$ put
$$\rho_n = c_n \frac{a_n}{a_{n+1}} - c_{n+1}.$$

1. If there are $\delta > 0$ and $N$ with $\rho_n \ge \delta$ for all $n \ge N$, then $\sum a_n$ [[converges|0CVVKY7GY2]].
2. If $\sum 1/c_n$ diverges and there is $N$ with $\rho_n \le 0$ for all $n \ge N$, then $\sum a_n$ [[diverges|0NBRP6093V]].

## Remarks

For part 1, multiply $\rho_n \ge \delta$ by $a_{n+1}$ to get $c_n a_n - c_{n+1} a_{n+1} \ge \delta a_{n+1}$ for $n \ge N$.
Summing from $N$ to $m$ telescopes, so $\delta \sum_{n=N+1}^{m+1} a_n \le c_N a_N$, and the partial sums are bounded.
For part 2, the sequence $c_n a_n$ increases from $n = N$ on, so $a_n \ge c_N a_N / c_n$, and the [[comparison test|04Z5G0THN6]] with $\sum 1/c_n$ gives divergence.
The choice $c_n = 1$ gives the [[ratio test|0QNY6ZBXPN]], the choice $c_n = n$ gives [[Raabe's test|CD6CAQ6XDN]], and the choice $c_n = n \ln n$ gives [[Bertrand's test|6T3E3AXZT7]].

## Explanations

### L1 human

Kummer's test is a family of tests, one for each choice of comparison weights $c_n$.
It compares the decay of the terms $a_n$ with the growth of the weights.
The familiar ratio test is the simplest member of the family.

### L2 human

When the ratio test gives the limit $1$, one uses Kummer's test with weights that grow, such as $n$ or $n \ln n$.
Each such choice gives a finer test: Raabe's test catches the $p$-series, and Bertrand's test catches series like $\sum 1/(n (\ln n)^p)$.
The proof only uses telescoping sums and the comparison test.
</document>

<document source="https://tamnd.github.io/math2/node/9JRNCRZ97G.md" tag="9JRNCRZ97G" kind="example">
# Smoluchowski coagulation equation

## Example

Let $n(x, t) \ge 0$ be the number density of clusters of mass $x > 0$ at time $t \ge 0$.
Let the coagulation kernel $K(x, y) = K(y, x) \ge 0$ be the rate at which clusters of masses $x$ and $y$ merge.
The continuous Smoluchowski coagulation equation is
$$\frac{\partial n(x, t)}{\partial t} = \frac12 \int_0^x K(x - y, y) \, n(x - y, t) \, n(y, t) \, dy - n(x, t) \int_0^\infty K(x, y) \, n(y, t) \, dy.$$
The first term counts the clusters of mass $x$ formed from two smaller ones, with the factor $\frac12$ because each pair is counted twice.
The second term counts the clusters of mass $x$ lost by merging with any other cluster.
It is a nonlinear [[integro-differential equation|YRKW2KX2CW]]: a time derivative of $n$ and integrals of $n$ over mass.

Check for the constant kernel $K = k > 0$: let $N(t) = \int_0^\infty n(x, t) \, dx$ be the total number of clusters.
Integrating over $x$ and exchanging the order of integration in the first term gives $N'(t) = \frac{k}{2} N^2 - k N^2 = -\frac{k}{2} N^2$.
Hence $N(t) = \frac{N(0)}{1 + k N(0) t / 2}$, so the number of clusters decreases like $\frac{2}{kt}$.

## Remarks

For bounded kernels the total mass $\int_0^\infty x \, n(x, t) \, dx$ is conserved.
For the product kernel $K(x, y) = xy$ the mass is conserved only up to a finite time, after which mass escapes into an infinitely large cluster; this is called gelation.
The discrete version replaces the integrals by sums over integer cluster sizes.

## Explanations

### L1 human

Small particles in a smoke or a polymer solution collide and stick together, forming bigger clusters.
The equation describes how the distribution of cluster sizes changes in time.
A cluster of a given size is created when two smaller ones meet, which gives an integral over the ways to split it.
It is destroyed when it meets any other cluster.
</document>

<document source="https://tamnd.github.io/math2/node/9JVJKJTXDE.md" tag="9JVJKJTXDE" kind="definition">
# Gibbs phenomenon

## Definition

Let $f \colon \R \to \R$ be a $2\pi$-periodic function that is piecewise continuously differentiable.
Let $x_0$ be a point of [[jump discontinuity|004FH91BN5]] of $f$, with jump $J = f(x_0^+) - f(x_0^-) \ne 0$.
Let $S_N f$ be the partial sums of the [[Fourier series|G12KXDHPPS]] of $f$.
The *Gibbs phenomenon* is the following behaviour of $S_N f$ near $x_0$:
$$\lim_{N \to \infty} S_N f\left(x_0 + \frac{\pi}{N}\right) = f(x_0^+) + J \left( \frac{1}{\pi} \int_0^{\pi} \frac{\sin t}{t} \, dt - \frac{1}{2} \right).$$
The symmetric formula holds at $x_0 - \frac{\pi}{N}$, where $f(x_0^-)$ and $-J$ replace $f(x_0^+)$ and $J$.

## Remarks

The constant $\frac{1}{\pi} \int_0^\pi \frac{\sin t}{t} \, dt - \frac{1}{2}$ is about $0.0895$, so the partial sums overshoot the jump by about $9\%$ of $\abs{J}$ on each side.
So $\sup_{x_0 < x < x_0 + \delta} S_N f(x)$ stays away from $f(x_0^+)$ for every $\delta > 0$, and the [[convergence|V6Q24CHJG8]] is not uniform on any interval with $x_0$ as an endpoint, although $S_N f(x) \to f(x)$ at each point of continuity.
Example: for the square wave equal to $\operatorname{sgn} x$ on $(-\pi, \pi)$, with $J = 2$ at $x_0 = 0$, one gets $S_N f(\pi/N) \to \frac{2}{\pi} \int_0^\pi \frac{\sin t}{t} \, dt \approx 1.179$.
Non-example: for a continuous piecewise smooth function such as $\abs{x}$ on $[-\pi, \pi]$, there is no jump, the Fourier series converges uniformly, and there is no overshoot of this kind.
The Fejér means of the Fourier series show no Gibbs phenomenon.
They are averages against a nonnegative kernel of mass $1$, so they stay between the infimum and supremum of $f$.
</document>

<document source="https://tamnd.github.io/math2/node/9K11FXYX5S.md" tag="9K11FXYX5S" kind="definition">
# Hilbert C*-module

## Definition

Let $A$ be a C*-algebra.
Let $E$ be a complex vector space that is a right $A$-module, with $\lambda (x a) = (\lambda x) a = x (\lambda a)$ for $\lambda \in \C$, $x \in E$ and $a \in A$.
An *$A$-valued inner product* on $E$ is a map $\ip{\cdot}{\cdot} \colon E \times E \to A$ such that for all $x, y, z \in E$, $a \in A$ and $\lambda \in \C$:

1. $\ip{x}{y + \lambda z} = \ip{x}{y} + \lambda \ip{x}{z}$.
2. $\ip{x}{y a} = \ip{x}{y} a$.
3. $\ip{y}{x} = \ip{x}{y}^*$.
4. $\ip{x}{x}$ is a [[positive element|NA8DF5X3M4]] of $A$, and $\ip{x}{x} = 0$ only for $x = 0$.

Then $\norm{x} = \norm{\ip{x}{x}}^{1/2}$ is a norm on $E$.
A *Hilbert C*-module* over $A$ is such a module $E$ that is complete for this norm.

## Remarks

The inner product is linear in the second argument and conjugate-linear in the first, as is usual in this theory.
A Hilbert $\C$-module is the same as a complex [[Hilbert space|HM2HHE3J51]] with this convention.

Example: $E = A$ with $\ip{a}{b} = a^* b$.
Clauses 1 to 3 follow from the algebra rules and $(a^* b)^* = b^* a$; the element $a^* a$ is positive, and $a^* a = 0$ forces $a = 0$.
The norm is $\norm{a^* a}^{1/2} = \norm{a}$ by the C* identity, so $E$ is complete.

Another example is $A^n$ with $\ip{x}{y} = \sum_{i=1}^n x_i^* y_i$.

Non-example: $E = \C$ over $A = \C$ with $\ip{x}{y} = -\overline{x} y$.
Clauses 1 to 3 hold, but $\ip{1}{1} = -1$ is not positive, so clause 4 fails.
</document>

<document source="https://tamnd.github.io/math2/node/9K15AMGBV9.md" tag="9K15AMGBV9" kind="concept">
# Banach manifold

## Summary

A Banach manifold is a space that looks locally like an open subset of a Banach space, with smooth changes of coordinates.
It extends the notion of a smooth manifold to infinite dimension.
Spaces of maps, of curves or of operators often carry such a structure.
Calculus on Banach manifolds works much as in finite dimension, because the inverse function theorem holds in Banach spaces.

## Explanations

### L1 human

A surface is built from patches, each of which looks like a piece of the plane.
A Banach manifold is built from patches that look like pieces of a Banach space, which may be a space of functions.
The patches are glued by maps that can be differentiated.
For example, the unit sphere of the Hilbert space $\ell^2$ is a Banach manifold, and its patches are pieces of a hyperplane in $\ell^2$.
</document>

<document source="https://tamnd.github.io/math2/node/9K1W9Q49NR.md" tag="9K1W9Q49NR" kind="concept">
# Favard constant

## Summary

The Favard constants $K_0, K_1, K_2, \dots$ are explicit numbers given by alternating series of odd powers.
They are the sharp constants in Jackson's inequality for periodic functions whose $r$-th derivative is bounded by $1$.
They begin with $K_0 = 1$, $K_1 = \pi/2$ and $K_2 = \pi^2/8$, and they tend to $4/\pi$.

## Explanations

### L1 human

If the $r$-th derivative of a periodic function is at most $1$ in size, trigonometric polynomials of degree below $n$ can approximate it within $K_r / n^r$.
The Favard constant $K_r$ is the best number that works here for every such function.
For $r = 1$, a function with slope at most $1$ is within $\pi/(2n)$ of some trigonometric polynomial of degree at most $n - 1$.
</document>

<document source="https://tamnd.github.io/math2/node/9KAQ1REPE8.md" tag="9KAQ1REPE8" kind="concept">
# Nevanlinna theory of difference equations

## Summary

Nevanlinna theory of difference equations applies the value distribution theory of [[meromorphic functions|0ZZ7ZYDR8W]] to equations such as $f(z + 1) = R(z, f(z))$ in the complex plane.
Its central tool is a difference analogue of the lemma on the logarithmic derivative.
Let $f$ be meromorphic of finite order and let $c \in \C$.
Then the proximity function of $f(z + c)/f(z)$ is small compared with the [[Nevanlinna characteristic|YT6Z0GWS0W]] of $f$.
With this, the classical methods of [[Nevanlinna theory|1EHH63VNAD]] for complex differential equations carry over to difference equations.
It has been proposed as a test of integrability.
Equations with enough meromorphic solutions of finite order are expected to be discrete analogues of the Painlevé equations.

## Explanations

### L1 human

Nevanlinna theory measures how often a meromorphic function takes each value, using a growth function $T(r, f)$.
For differential equations, it is known that derivatives cannot make a function much bigger in this sense.
The difference version shows the same for shifts: $f(z + 1)$ is not much bigger than $f(z)$ on average, as long as $f$ grows at most like $e^{\abs{z}^\rho}$ for some $\rho$.
This lets one rule out meromorphic solutions of many difference equations and single out the special, integrable ones.
</document>

<document source="https://tamnd.github.io/math2/node/9KD9R5TGD3.md" tag="9KD9R5TGD3" kind="definition">
# Tightness of measures

## Definition

Let $E$ be a topological space with its [[Borel σ-algebra|0WQXY32Q8D]], and let $\mathcal{M}$ be a set of [[probability measures|059TJN0NZ1]] on $E$.
$\mathcal{M}$ is *tight* if for every $\varepsilon > 0$ there is a [[compact set|0R6EWFKRCF]] $K \subseteq E$ such that
$$\mu(E \setminus K) < \varepsilon \qquad \text{for all } \mu \in \mathcal{M}.$$
A sequence of random variables with values in $E$ is tight if the set of their laws is tight.

## Remarks

On $\R^d$, a family $\mathcal{M}$ is tight if and only if $\sup_{\mu \in \mathcal{M}} \mu(\set{x : \abs{x} > R}) \to 0$ as $R \to \infty$, because the compact sets are the closed bounded sets.
Example: the [[normal|05AQE0QR0D]] laws $N(m, 1)$ with $\abs{m} \le 1$ form a tight family. Given $\varepsilon$, choose $r$ with $P(\abs{Z} > r) < \varepsilon$ for $Z$ standard normal; then $K = [-1 - r, 1 + r]$ works, since $N(m, 1)(\R \setminus K) \le P(\abs{Z} > r)$.
Non-example: the point masses $\delta_n$, $n \in \N$, on $\R$. Each compact $K$ lies in some $[-R, R]$, and then $\delta_n(\R \setminus K) = 1$ for $n > R$.
Also non-tight: the normal laws $N(0, n)$, $n \ge 1$, whose mass spreads out.
A single Borel probability measure on a [[Polish space|584GXXY73M]] is tight, and so is every finite family.
On a Polish space, [[Prokhorov's theorem|D6FMEQYA76]] says that $\mathcal{M}$ is tight if and only if it is relatively compact for [[weak convergence|MBW9JCGZC1]].
Every sequence of real random variables that converges in distribution is tight.
</document>

<document source="https://tamnd.github.io/math2/node/9KDC2DQ72A.md" tag="9KDC2DQ72A" kind="definition">
# Induced representation

## Definition

Let $G$ be a group, let $H$ be a [[subgroup|0X1JXT47GA]] of finite [[index|0DTXBQPA2S]] $n = [G : H]$, and let $F$ be a field.
Let $\theta \colon H \to GL(W)$ be a [[representation|WYGRXCG2RN]] of $H$ on a vector space $W$ over $F$.
Choose representatives $t_1, \ldots, t_n$ of the [[left cosets|03DEH43F4H]] of $H$ in $G$.
For $g \in G$ and each $i$ there are a unique $j$ and a unique $h_i \in H$ with $g t_i = t_j h_i$.
The *induced representation* $\operatorname{Ind}_H^G \theta$ acts on $V = \bigoplus_{i=1}^{n} t_i W$, a direct sum of $n$ copies of $W$, by
$$g \cdot (t_i w) = t_j \, (\theta(h_i) w) \quad (w \in W).$$

## Remarks

Up to isomorphism, the result does not depend on the choice of the $t_i$.
Equivalently, $\operatorname{Ind}_H^G W = F[G] \otimes_{F[H]} W$, using the [[group algebra|MNP2HBWB5Q]].
Its degree is $[G : H] \cdot \dim W$.
For finite $G$, its character is
$$\operatorname{Ind}_H^G \chi(g) = \frac{1}{\abs{H}} \sum_{x \in G,\ x^{-1} g x \in H} \chi(x^{-1} g x).$$
Example: let $G = S_3$, $H = A_3$ and $\theta$ trivial of degree $1$.
Then $V$ has dimension $2$, and the formula gives the values $2$ at $1$, $0$ at transpositions and $2$ at $3$-cycles.
This is the sum of the trivial and sign characters.
Non-example: for $H = \set{0}$ in $G = \Z$, the index is infinite, so the definition does not apply; the space $\bigoplus_{t \in \Z} W$ with the shift action is a different construction, sometimes called compact induction.
</document>

<document source="https://tamnd.github.io/math2/node/9KE5E84M3M.md" tag="9KE5E84M3M" kind="definition">
# Serre fibration

## Definition

Let $E$ and $B$ be [[topological spaces|0CQHFPGJ45]] and let $p \colon E \to B$ be a [[continuous map|0X3WGX95FE]].
Let $I = [0, 1]$.
The map $p$ is a *Serre fibration* if it has the [[homotopy lifting property|G4QDM01P6J]] with respect to the cube $I^n$ for every integer $n \ge 0$.
That is, let $n \ge 0$, let $\tilde f \colon I^n \to E$ be continuous, and let $H \colon I^n \times I \to B$ be continuous with $H(x, 0) = p(\tilde f(x))$ for all $x$.
Then there is a continuous $\tilde H \colon I^n \times I \to E$ with $p \circ \tilde H = H$ and $\tilde H(x, 0) = \tilde f(x)$ for all $x \in I^n$.

## Remarks

Example: for spaces $B$ and $F$, the projection $p \colon B \times F \to B$ is a Serre fibration.
Given $\tilde f = (f_1, f_2)$ and $H$, the map $\tilde H(x, t) = (H(x, t), f_2(x))$ is a lift with the right start.

Non-example: the inclusion $p \colon \set{0} \to [0, 1]$ is not a Serre fibration.
Take $n = 0$, $\tilde f$ the map to $0$ and $H(t) = t$; any lift would satisfy $p(\tilde H(1)) = 1$, but the only point of $\set{0}$ maps to $0$.

Equivalently, $p$ has the homotopy lifting property with respect to all CW complexes.
Every [[Hurewicz fibration|Y773B7EQA7]] is a Serre fibration, and every [[fiber bundle|CX4M8K61AB]] is a Serre fibration.
</document>

<document source="https://tamnd.github.io/math2/node/9KGA2BZC1M.md" tag="9KGA2BZC1M" kind="definition">
# Rng

## Definition

A *rng* is a set $R$ with two binary operations $+$ and $\cdot$ such that:

1. $(R, +)$ is an [[abelian group|071DDFVG4J]];
2. $(ab)c = a(bc)$ for all $a, b, c \in R$;
3. $a(b + c) = ab + ac$ and $(a + b)c = ac + bc$ for all $a, b, c \in R$.

## Remarks

No multiplicative identity is required; a ring is a rng that has one.
Other names are *non-unital ring* and *pseudo-ring*.
Example: the even integers $2\Z$ with the usual operations form a rng; the clauses hold because they hold in $\Z$ and $2\Z$ is closed under both operations.
It has no identity, since $e \cdot 2 = 2$ forces $e = 1 \notin 2\Z$.
Non-example: $\R^3$ with the cross product as multiplication fails clause 2.
With the standard basis $i, j, k$, one has $(i \times i) \times j = 0$ but $i \times (i \times j) = i \times k = -j$.
Every rng $R$ embeds as an ideal in the ring $\Z \times R$ with the product $(m, a)(n, b) = (mn, mb + na + ab)$, whose identity is $(1, 0)$.
</document>

<document source="https://tamnd.github.io/math2/node/9KHQR16P1B.md" tag="9KHQR16P1B" kind="concept">
# Second fundamental form

## Summary

The second fundamental form of a surface in $\R^3$ is a quadratic form on each tangent plane.
It measures how the surface bends away from that plane.
In a local parametrisation it is given by three functions, often written $e$, $f$, $g$ or $L$, $M$, $N$.
Together with the [[first fundamental form|MAJ3DH7MZS]] it determines the principal, Gaussian and mean curvatures.

## Explanations

### L1 human

Put a flat sheet of glass on a surface so that it touches at one point.
Near that point the surface pulls away from the glass, faster in some directions and slower in others.
The second fundamental form records, for each direction, how fast the surface pulls away.
On a sphere it pulls away equally in all directions, and on a saddle it rises in some directions and falls in others.
</document>

<document source="https://tamnd.github.io/math2/node/9KMJSKNZ72.md" tag="9KMJSKNZ72" kind="statement">
# Bihari–LaSalle inequality

## Statement

Let $a < b$, let $u, f : [a, b] \to [0, \infty)$ be continuous, and let $c > 0$.
Let $w : [0, \infty) \to [0, \infty)$ be continuous and non-decreasing, with $w(r) > 0$ for $r > 0$.
Define $G(r) = \int_c^r \frac{ds}{w(s)}$ for $r > 0$, and let $G(\infty) = \lim_{r \to \infty} G(r) \in (0, \infty]$.
Assume that
$$u(t) \le c + \int_a^t f(s)\, w(u(s))\, ds \quad \text{for each } t \in [a, b].$$
Then for each $t \in [a, b]$ with $\int_a^t f(s)\, ds < G(\infty)$,
$$u(t) \le G^{-1}\left( \int_a^t f(s)\, ds \right).$$

## Remarks

The function $G$ is continuous and strictly increasing with $G(c) = 0$, so $G^{-1}$ is defined on $[0, G(\infty))$.
Proof idea: let $v(t) = c + \int_a^t f(s)\, w(u(s))\, ds$.
Then $v \ge c > 0$, $u \le v$, and $v' = f\, w(u) \le f\, w(v)$ since $w$ is non-decreasing.
So $\frac{d}{dt} G(v(t)) = v'(t) / w(v(t)) \le f(t)$, and integration gives $G(v(t)) \le \int_a^t f$.
For $w(r) = r$ one gets $G(r) = \ln(r/c)$ and the bound $u(t) \le c\, \exp\left(\int_a^t f\right)$, which is [[Grönwall's inequality|07EJHQYR9A]] with constant $\alpha = c$.
For $w(r) = r^2$ and $f = 1$ one gets $u(t) \le c / (1 - c(t - a))$ while $c(t - a) < 1$; the restriction on $t$ matches the blow-up of the solution of $u' = u^2$.

## Explanations

### L1 human

Grönwall's inequality controls a quantity whose growth rate is at most proportional to its size.
The Bihari–LaSalle inequality does the same when the growth rate is a more general increasing function of the size.
The bound is the solution of the comparison equation $v' = f\, w(v)$, $v(a) = c$.

### L2 human

The inequality is the nonlinear version of the integral form of [[Grönwall's inequality|07EJHQYR9A]].
It gives a priori bounds for solutions of nonlinear equations and estimates of the time of existence.
With $c \to 0$ it gives the [[Osgood uniqueness criterion|2AHNVSH76M]], which relaxes the Lipschitz condition in uniqueness theorems.
</document>

<document source="https://tamnd.github.io/math2/node/9KP392QCGV.md" tag="9KP392QCGV" kind="definition">
# Distinguished triangle

## Definition

Let $(\mathcal{T}, \Sigma)$ be a [[triangulated category|T2V7GJJ2EN]].
A *triangle* in $\mathcal{T}$ is a diagram $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} \Sigma X$ of objects and morphisms of $\mathcal{T}$.
A *distinguished triangle*, or *exact triangle*, is a triangle in the class of distinguished triangles that is part of the structure of $\mathcal{T}$.

## Remarks

In a distinguished triangle, consecutive maps compose to zero: $v u = 0$, $w v = 0$ and $(\Sigma u) w = 0$.
For each object $W$, applying $\Hom(W, -)$ gives a long exact sequence $\cdots \to \Hom(W, X) \to \Hom(W, Y) \to \Hom(W, Z) \to \Hom(W, \Sigma X) \to \cdots$.
By TR1 and TR3, the third object $Z$ is determined by $u$ up to a non-unique isomorphism; it is called a cone of $u$.
Example: $X \xrightarrow{\id} X \to 0 \to \Sigma X$ is distinguished for every $X$, by the axiom TR1.
Non-example: if $X \ne 0$, then $X \to 0 \to 0 \to \Sigma X$ is not distinguished.
Indeed, the long exact sequence for $W = \Sigma X$ would give $\Hom(\Sigma X, 0) \to \Hom(\Sigma X, \Sigma X) \to \Hom(\Sigma X, \Sigma 0)$ exact, which forces $\Hom(\Sigma X, \Sigma X) = 0$, but $\id_{\Sigma X} \ne 0$.
In the homotopy category of complexes, the distinguished triangles are those isomorphic to $X \xrightarrow{f} Y \to \operatorname{Cone}(f) \to X[1]$ for a chain map $f$.
</document>

<document source="https://tamnd.github.io/math2/node/9KYWN94QDN.md" tag="9KYWN94QDN" kind="concept">
# Banach algebra

## Summary

A Banach algebra is an associative algebra over $\R$ or $\C$ with a complete norm that satisfies $\norm{xy} \le \norm{x}\norm{y}$.
It is a [[normed algebra|4DJ4V5VBSH]] that is also a [[Banach space|06GZ956657]].
Completeness makes power series converge, so in a unital Banach algebra $1 - x$ is invertible when $\norm{x} < 1$.
In a nonzero complex unital Banach algebra the spectrum of every element is a nonempty compact set.
Algebras of bounded operators, continuous functions and integrable functions on groups are the main examples, and C*-algebras are the most studied special class.

## Explanations

### L1 human

A Banach algebra is a space where one can add, multiply and take limits, with sizes measured by a norm.
The norm of a product is at most the product of the norms, so multiplication cannot blow up sizes.
The continuous functions on $[0, 1]$, with the largest absolute value as norm, form a Banach algebra.
The $n \times n$ complex matrices with the operator norm form another, and there the algebra is not commutative.
The point of the theory is to study invertibility and spectra of elements, as for matrices, in infinite dimensions.
</document>

<document source="https://tamnd.github.io/math2/node/9M0NNNFJGM.md" tag="9M0NNNFJGM" kind="concept">
# Kernel of an integral transform

## Summary

The kernel of an [[integral transform|YCK709QB06]] is the function $K(t, u)$ of two variables that is integrated against the input function.
It determines the transform completely: different transforms differ only in their kernels and in their domains of integration.
Properties of the kernel, such as symmetry or decay, decide which functions can be transformed and whether the transform can be inverted.

## Explanations

### L1 human

The kernel plays the role of a matrix with continuously many rows and columns.
A matrix acts on a vector by $y_u = \sum_t K_{t u} x_t$, and a kernel acts on a function by $(Tf)(u) = \int f(t) K(t, u)\, dt$.
The Laplace transform has kernel $e^{-ut}$ and the Fourier transform has kernel $e^{-2\pi i t u}$.
The word "kernel" here has nothing to do with the kernel of a linear map, which is the set of vectors sent to zero.
</document>

<document source="https://tamnd.github.io/math2/node/9M8XJX4DQ7.md" tag="9M8XJX4DQ7" kind="statement">
# Minkowski's first inequality

## Statement

Let $n \ge 1$ and let $K, L$ be nonempty compact convex subsets of $\R^n$ ([[convex bodies|437FSG3419]]).
Let $V$ be the [[mixed volume|QB3Q5FBJ08]] and write $V_1(K, L) = V(K, \dots, K, L)$, with $K$ repeated $n - 1$ times.
Then
$$V_1(K, L)^n \ge \operatorname{vol}(K)^{n-1} \operatorname{vol}(L),$$
where $\operatorname{vol}$ is [[Lebesgue measure|2Z2QVWZ262]].
If $K$ and $L$ have nonempty interiors, equality holds if and only if $K$ and $L$ are homothetic.

## Remarks

One derivation uses the [[Brunn–Minkowski theorem|9B02RPN7XD]].
If $\operatorname{vol}(K) = 0$ the inequality is trivial, so assume $\operatorname{vol}(K) > 0$.
The function $f(t) = \operatorname{vol}(K + tL)^{1/n}$ is concave on $[0, \infty)$, so $f'(0) \ge f(1) - f(0) \ge \operatorname{vol}(L)^{1/n}$.
By the definition of mixed volumes, $\operatorname{vol}(K + tL) = \operatorname{vol}(K) + n V_1(K, L)\, t + O(t^2)$, so $f'(0) = V_1(K, L) \operatorname{vol}(K)^{(1-n)/n}$, which gives the inequality.
It also follows from the [[Alexandrov–Fenchel inequality|D3AVHVYQW9]].

Example: take $L = B$, the closed unit ball, of volume $\kappa_n$.
Then $n V_1(K, B)$ is the surface area $S(K)$, and the inequality becomes $S(K)^n \ge n^n \kappa_n \operatorname{vol}(K)^{n-1}$, the [[isoperimetric inequality|KBX8RBZ626]] for convex bodies.
For $n = 2$ this is $\operatorname{per}(K)^2 \ge 4 \pi \operatorname{area}(K)$.

## Explanations

### L1 human

The first mixed volume $V_1(K, L)$ measures how fast the volume of $K$ grows when a small multiple of $L$ is added to it.
Minkowski's first inequality bounds this growth rate from below by the sizes of $K$ and $L$.
With $L$ a ball it is the isoperimetric inequality: among convex bodies of given volume, the ball has the least surface area.

### L2 human

The inequality is a key step in the uniqueness part of the Minkowski problem: two convex bodies with the same surface area measure are translates of each other.
It is the case $K_3 = \cdots = K_n = K$ of a chain of inequalities obtained from the [[Alexandrov–Fenchel inequality|D3AVHVYQW9]].
Minkowski's second inequality, $V_1(K, L)^2 \ge \operatorname{vol}(K)\, V(K, \dots, K, L, L)$, is another case of that chain.
</document>

<document source="https://tamnd.github.io/math2/node/9MB4JD02YF.md" tag="9MB4JD02YF" kind="definition">
# Nuclear C*-algebra

## Definition

Let $A$ be a C*-algebra.
For a C\*-algebra $B$, let $A \odot B$ be the algebraic [[tensor product|1735VGRT2P]] of $A$ and $B$, a $*$-algebra with $(a \otimes b)^* = a^* \otimes b^*$.
A *C\*-norm* on $A \odot B$ is a norm $\gamma$ with $\gamma(xy) \le \gamma(x) \gamma(y)$, $\gamma(x^*) = \gamma(x)$ and $\gamma(x^* x) = \gamma(x)^2$.
The algebra $A$ is *nuclear* if, for every C\*-algebra $B$, there is exactly one C\*-norm on $A \odot B$.

## Remarks

There is always a smallest C\*-norm $\norm{\cdot}_{\min}$ and a largest $\norm{\cdot}_{\max}$, so $A$ is nuclear when $\norm{\cdot}_{\min} = \norm{\cdot}_{\max}$ on $A \odot B$ for all $B$.
By theorems of Choi–Effros and Kirchberg, $A$ is nuclear if and only if it has the completely positive approximation property.
This means that there are nets of [[completely positive|CGY299QMC3]] contractions $\varphi_\lambda \colon A \to M_{n_\lambda}(\C)$ and $\psi_\lambda \colon M_{n_\lambda}(\C) \to A$ with $\psi_\lambda(\varphi_\lambda(a)) \to a$ in norm for every $a \in A$.
A C\*-algebra is nuclear if and only if it is an [[amenable Banach algebra|6P9AE21Q7X]].

Example: $M_n(\C)$ is nuclear.
Take $\varphi_\lambda = \psi_\lambda$ equal to the identity map of $M_n(\C)$, which is completely positive and contractive.
Then $\psi_\lambda(\varphi_\lambda(a)) = a$, so the completely positive approximation property holds.
Commutative C\*-algebras and [[AF algebras|4Y7VENNBVK]] are also nuclear.

Non-example: let $\mathbb{F}_2$ be the free group on two generators.
For a discrete group $G$, the reduced group C*-algebra $C^*_r(G)$ is nuclear if and only if $G$ is amenable; since $\mathbb{F}_2$ is not amenable, $C^*_r(\mathbb{F}_2)$ is not nuclear.
The algebra $B(H)$ of all bounded operators on an infinite-dimensional Hilbert space is not nuclear either.
</document>

<document source="https://tamnd.github.io/math2/node/9MMG0335JE.md" tag="9MMG0335JE" kind="statement">
# Fishburn's theorem on interval orders

## Statement

Let $(P, <)$ be a finite [[strict partial order|60FNZ35H6R]].
Then $P$ is an [[interval order|VSNQV29KFJ]] if and only if $P$ contains no $\mathbf{2} + \mathbf{2}$.
Being an interval order means that there are real intervals $[l_x, r_x]$, for $x \in P$, with $x < y$ if and only if $r_x < l_y$.
Here $P$ contains a $\mathbf{2} + \mathbf{2}$ when there are $a < b$ and $c < d$ in $P$ with $a$ incomparable to $d$ and $c$ incomparable to $b$.

## Remarks

One direction is direct.
If intervals represent $P$ and $a < b$, $c < d$, then $r_a < l_b$ and $r_c < l_d$; if also $a \not< d$ and $c \not< b$, then $l_d \le r_a$ and $l_b \le r_c$, so $r_a < l_b \le r_c < l_d \le r_a$, a contradiction.
For the converse, the absence of $\mathbf{2} + \mathbf{2}$ means that the sets of predecessors $\set{y : y < x}$ are totally ordered by inclusion, and so are the sets of successors.
List the distinct predecessor sets as $D_1 \subsetneq D_2 \subsetneq \cdots \subsetneq D_k$.
Let $l_x$ be the index of the predecessor set of $x$, and let $r_x$ be the largest $j$ with $x \notin D_j$.
Then $x < y$ means $x \in D_{l_y}$, which happens exactly when $r_x < l_y$.

Example: the four elements $a < b$, $c < d$ with no other relations form a $\mathbf{2} + \mathbf{2}$, so they cannot be represented by intervals.
The order on three intervals $[0, 1]$, $[2, 3]$, $[0, 3]$ has the single relation $[0,1] < [2,3]$ and contains no $\mathbf{2} + \mathbf{2}$.

## Explanations

### L1 human

Think of events that each take up a time interval, with one event before another when it ends before the other starts.
The orders that arise this way are the interval orders.
Fishburn's theorem says that one small pattern is the only obstacle: two pairs $a$ before $b$ and $c$ before $d$, where $a$ overlaps $d$ and $c$ overlaps $b$.
Intervals on a line cannot do this, and every order that avoids it can be drawn with intervals.

### L2 human

The theorem is a forbidden-subposet characterization, like the characterization of semiorders by the absence of $\mathbf{2} + \mathbf{2}$ and $\mathbf{3} + \mathbf{1}$.
It shows that interval orders are exactly the posets whose strict down-sets form a chain under inclusion, which gives a fast test for being an interval order.
Interval orders are the orders whose incomparability graphs are interval graphs, so the theorem connects order theory with the theory of interval graphs.
</document>

<document source="https://tamnd.github.io/math2/node/9MPCZ3EAJM.md" tag="9MPCZ3EAJM" kind="concept">
# Galois cohomology

## Summary

Galois cohomology is the [[group cohomology|YXJXA0Q52R]] of Galois groups, see the [[definition|EESNTXFWHZ]].
The main case is the [[absolute Galois group|PGKWBJDTKX]] of a field, acting continuously on a module.
Many arithmetic invariants of a field are Galois cohomology groups.
[[Hilbert's Theorem 90|YFF8B5A0QG]] says that $H^1$ of the multiplicative group vanishes.
[[Kummer theory|XZQENYV15Y]] computes $H^1$ of the roots of unity, and $H^2$ of the multiplicative group is the [[Brauer group|BEW48TA9WZ]].
It is a central language of algebraic number theory, class field theory and arithmetic geometry.

## Explanations

### L1 human

When a group of symmetries acts on some objects, cohomology measures the ways in which "local" choices fail to fit together into one symmetric choice.
For a field $K$, the symmetries are the automorphisms of a large extension field that fix $K$.
The group $H^0$ is just the set of objects fixed by all symmetries.
The group $H^1$ counts, for example, the different forms over $K$ of an object that become the same over the large field.
For instance, over $\Q$ the field extensions of degree $2$ correspond to the nonzero elements of $H^1$ with coefficients $\set{\pm 1}$.
</document>

<document source="https://tamnd.github.io/math2/node/9MPM6E51WV.md" tag="9MPM6E51WV" kind="definition">
# Autoregressive integrated moving average

## Definition

Let $B$ be the backshift operator, $B X_t = X_{t-1}$, and let $\nabla = 1 - B$ be the difference operator, so $\nabla X_t = X_t - X_{t-1}$.
Let $p, d, q \ge 0$ be integers.
A [[time series|EA3MF4VTEH]] $(X_t)$ is an *ARIMA($p$, $d$, $q$) process* if the differenced series $Y_t = \nabla^d X_t$ is a causal [[ARMA($p$, $q$) process|B1HNX0YT0Z]].
Equivalently, $\phi(B) (1 - B)^d X_t = c + \theta(B) \varepsilon_t$ for a white noise $(\varepsilon_t)$, a constant $c$, and polynomials $\phi$ of degree $p$ and $\theta$ of degree $q$.
Here $\phi(0) = \theta(0) = 1$, and $\phi$ has no root in the closed unit disk.

## Remarks

For $d = 0$ this is a causal ARMA($p$, $q$) process.
For $d \ge 1$ the process is not stationary in general.
It is determined by $Y$ only up to $d$ starting values, since $\nabla^d$ kills polynomials in $t$ of degree less than $d$.
A series whose $d$-th difference is stationary but whose $(d-1)$-th difference is not is called integrated of order $d$, written I($d$).

Example: the [[random walk|04VZE3FZXF]] $X_t = X_{t-1} + \varepsilon_t$ is ARIMA($0$, $1$, $0$), because $\nabla X_t = \varepsilon_t$ is a white noise, which is ARMA($0$, $0$).
The model $X_t = X_{t-1} + 0.5 (X_{t-1} - X_{t-2}) + \varepsilon_t$ is ARIMA($1$, $1$, $0$), since $\nabla X_t = 0.5 \nabla X_{t-1} + \varepsilon_t$.

Non-example: the explosive series with $X_0 = 0$ and $X_t = 1.5 X_{t-1} + \varepsilon_t$ for $t \ge 1$ is not ARIMA for any $d$.
Its autoregressive polynomial $1 - 1.5 z$ has the root $2/3$ inside the unit disk, so the variance of $X_t$ grows geometrically.
Differencing does not remove this explosive root, because $\nabla^d$ only contributes the root $1$, so $\nabla^d X_t$ is never stationary.

Seasonal ARIMA models also apply seasonal differences $1 - B^s$ for a period $s$, such as $s = 12$ for monthly data.
</document>

<document source="https://tamnd.github.io/math2/node/9MXPNHHMT1.md" tag="9MXPNHHMT1" kind="definition">
# Frobenius algebra

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $k$ with identity.
The algebra $A$ is a *Frobenius algebra* if there is a [[bilinear form|0SQZMXB6GP]] $\sigma \colon A \times A \to k$ such that:

1. $\sigma$ is [[nondegenerate|01NHFZF2SK]];
2. $\sigma(ab, c) = \sigma(a, bc)$ for all $a, b, c \in A$.

## Remarks

Example: the matrix algebra $M_n(k)$ with $\sigma(a, b) = \tr(ab)$.
Clause 2 holds because both sides equal $\tr(abc)$.
Clause 1 holds because $\tr(a E_{ji}) = a_{ij}$ for the matrix unit $E_{ji}$, so $\sigma(a, \cdot) = 0$ forces $a = 0$.
Another example is the group algebra $k[G]$ of a finite group $G$, with $\sigma(a, b)$ the coefficient of the identity element of $G$ in $ab$.
Non-example: $A = k[x, y]/(x^2, xy, y^2)$, with basis $1, x, y$, is not Frobenius.
If $\sigma$ satisfies clause 2, then $\sigma(a, b) = \lambda(ab)$ for the linear form $\lambda = \sigma(1, \cdot)$.
The plane spanned by $x$ and $y$ meets $\ker \lambda$ in some nonzero $u = \alpha x + \beta y$.
Since $ub \in k u$ for every $b \in A$, $\sigma(u, b) = 0$ for all $b$, and clause 1 fails.
Equivalent forms: $A$ is Frobenius if and only if some linear form $\lambda \colon A \to k$ has a kernel that contains no nonzero left ideal.
This holds if and only if $A \cong \Hom_k(A, k)$ as left $A$-modules.
If moreover $\sigma$ is symmetric, $A$ is called a symmetric algebra.
</document>

<document source="https://tamnd.github.io/math2/node/9MZFTX77CD.md" tag="9MZFTX77CD" kind="concept">
# Jackson kernel

## Summary

The Jackson kernel is a nonnegative trigonometric polynomial, the normalised square of the [[Fejér kernel|DCCA2T9GKH]].
Convolution with it gives trigonometric polynomials that approximate a continuous periodic function with error of order $\omega(f, 1/n)$.
It is the standard tool in the proof of Jackson's inequality.

## Explanations

### L1 human

Averaging a function against a narrow bump smooths it and keeps it close to the original.
If the bump is itself a trigonometric polynomial, the average is a trigonometric polynomial too.
The Fejér kernel is such a bump, but its tails are too heavy for the best error bounds.
Squaring it makes the tails much thinner, which is exactly what Jackson's proof needs.
</document>

<document source="https://tamnd.github.io/math2/node/9N027RT5R6.md" tag="9N027RT5R6" kind="statement">
# Nilpotence theorem

## Statement

Let $R$ be a homotopy associative [[ring spectrum|BAW0T68GWK]], and let $MU$ be the spectrum of [[complex cobordism|5CZHFNY9F3]].
Let $h \colon \pi_* R \to MU_* R$ be the Hurewicz homomorphism, which sends $\alpha$ to the image of $\alpha$ under the unit map $R \to MU \wedge R$.
Then every element of the kernel of $h$ is nilpotent.

## Remarks

The theorem was proved by Devinatz, Hopkins and Smith in 1988.
The proof filters $MU$ by a sequence of Thom spectra $X(n)$ built from $\Omega SU(n)$ and shows inductively that detection of nilpotence passes from $X(n+1)$ to $X(n)$.
For $R$ the sphere spectrum, $MU_* = \pi_* MU$ is torsion-free while every element of positive degree in $\pi_* S$ is torsion.
So the Hurewicz image of such an element is zero, and the theorem gives back the [[Nishida nilpotence theorem|FQ048Z65F9]].
Equivalent forms state that a self-map $\Sigma^d X \to X$ of a finite spectrum is nilpotent when its $MU$-homology is nilpotent.

## Explanations

### L1 human

Complex cobordism $MU$ is a homology theory that sees a lot of stable homotopy theory.
The theorem says that whatever $MU$ cannot see in a ring spectrum must be nilpotent.
So $MU$ detects all the elements that can be iterated forever.

### L2 human

The nilpotence theorem is the foundation of chromatic homotopy theory.
Together with the work of Hopkins and Smith it yields the thick subcategory theorem and the periodicity theorem, which classify finite spectra by their Morava $K$-theory type.
These results proved most of the Ravenel conjectures.
</document>

<document source="https://tamnd.github.io/math2/node/9N6R5ADJXT.md" tag="9N6R5ADJXT" kind="concept">
# Blaschke product

## Summary

A Blaschke product is a bounded [[holomorphic function|0DG0Z72CQD]] on the [[unit disc|8G5ABXRK85]] built as a product of disc automorphisms, one for each prescribed zero.
It exists exactly when the zeros satisfy the Blaschke condition $\sum_n (1 - \abs{a_n}) < \infty$.
Blaschke products are the basic [[inner functions|NN8XH5RJYB]], and every nonzero function in a [[Hardy space|PNNNFJVF6W]] factors through the Blaschke product of its zeros.

## Explanations

### L1 human

Each factor $\frac{\abs{a}}{a} \frac{a - z}{1 - \bar a z}$ is a [[Möbius transformation|0B346DDAM5]] that maps the disc onto itself and vanishes only at $a$.
Multiplying such factors gives a function with exactly the chosen zeros and with modulus at most $1$.
With infinitely many zeros, the product converges only if the zeros approach the circle fast enough.
For example, the zeros $a_n = 1 - 2^{-n}$ are allowed, since $\sum 2^{-n} < \infty$.
The zeros $a_n = 1 - 1/n$ are not allowed, since $\sum 1/n = \infty$.
</document>

<document source="https://tamnd.github.io/math2/node/9N8D9E5TB2.md" tag="9N8D9E5TB2" kind="statement">
# Cooley–Tukey FFT algorithm

## Statement

For an integer $M \ge 1$ let $\omega_M = e^{-2\pi i/M}$, a [[primitive $M$-th root of unity|0EHM5ZP5QH]].
Let $N = N_1 N_2$ with integers $N_1, N_2 \ge 1$, let $x = (x_0, \ldots, x_{N-1}) \in \C^N$, and let $X$ be its [[discrete Fourier transform|TST0GH66A5]], $X_k = \sum_{n=0}^{N-1} x_n \omega_N^{nk}$.
Then for all $0 \le k_1 < N_1$ and $0 \le k_2 < N_2$,
$$X_{k_1 + N_1 k_2} = \sum_{n_2=0}^{N_2-1} \omega_{N_2}^{n_2 k_2} \left[ \omega_N^{n_2 k_1} \sum_{n_1=0}^{N_1-1} x_{N_2 n_1 + n_2}\, \omega_{N_1}^{n_1 k_1} \right].$$
Thus a DFT of length $N$ is obtained in three stages.
First come $N_2$ DFTs of length $N_1$, then $N$ multiplications by the twiddle factors $\omega_N^{n_2 k_1}$, then $N_1$ DFTs of length $N_2$.
In particular, if $N = 2^m$, applying this with $N_1 = 2$ (or $N_2 = 2$) recursively computes $X$ with $O(N \log N)$ arithmetic operations.

## Remarks

The proof is one line: write $n = N_2 n_1 + n_2$ and $k = k_1 + N_1 k_2$ and expand $\omega_N^{nk}$.
The factor $\omega_N^{N_2 N_1 n_1 k_2}$ equals $1$, while $\omega_N^{N_2 n_1 k_1} = \omega_{N_1}^{n_1 k_1}$ and $\omega_N^{N_1 n_2 k_2} = \omega_{N_2}^{n_2 k_2}$.
For $N = 2M$ the radix-2 form reads $X_k = E_k + \omega_N^k O_k$ and $X_{k+M} = E_k - \omega_N^k O_k$ for $0 \le k < M$, where $E$ and $O$ are the DFTs of length $M$ of the even-indexed and odd-indexed entries.
The cost then satisfies $T(N) = 2T(N/2) + O(N)$, so $T(N) = O(N \log N)$.
Direct evaluation of the defining sums costs about $N^2$ multiplications; for $N = 2^{20}$ the ratio is about $5 \times 10^4$.
The splitting is useless when $N$ is prime, since then one factor is $1$.

## Explanations

### L1 human

The discrete Fourier transform of $N$ numbers looks like it needs $N^2$ multiplications.
The Cooley–Tukey idea is to split the input into interleaved pieces, transform each piece, and glue the results with a few extra multiplications.
For $N = 8$: transform the entries with even index and those with odd index separately, then combine.
Repeating the split when $N$ is a power of $2$ brings the cost down to a multiple of $N \log N$.

### L2 human

This is the most common fast Fourier transform.
The index maps $n = N_2 n_1 + n_2$ and $k = k_1 + N_1 k_2$ turn a one-dimensional transform into a two-dimensional one, up to the twiddle factors.
Radix-2, radix-4, split-radix and mixed-radix versions differ only in how $N$ is factored at each step.
When $N_1$ and $N_2$ are coprime, the prime-factor algorithm uses other index maps and avoids the twiddle factors.
Prime lengths need a different reduction, such as Rader's or Bluestein's algorithm.
</document>

<document source="https://tamnd.github.io/math2/node/9NM376YWF2.md" tag="9NM376YWF2" kind="concept">
# Condensed set

## Summary

A condensed set is a [[sheaf|RT8A8FJ566]] of sets on the category of profinite sets ([[Stone spaces|ZV38HA5PAQ]]).
The coverings are the finite families of maps that are jointly surjective.
Condensed sets were introduced by Dustin Clausen and Peter Scholze as a replacement for topological spaces in algebra.
Condensed abelian groups form an [[abelian category|Y9CZ3MTNF1]] with good properties, while topological abelian groups do not.
Each topological space $X$ gives a condensed set $S \mapsto C(S, X)$, and for compactly generated spaces no information is lost.

## Explanations

### L1 human

A topological space can be probed by mapping small test spaces into it.
A condensed set keeps only this probing data.
For each profinite set $S$, such as a finite set or the Cantor set, it gives the set of "maps from $S$".
It also says how these maps restrict along maps of test spaces.
This is enough to recover most spaces of interest, and it behaves much better with algebra.
For example, the quotient $\R / \Q$ is a useless indiscrete space, but as a condensed group it keeps real information.
</document>

<document source="https://tamnd.github.io/math2/node/9NMMRYKBNG.md" tag="9NMMRYKBNG" kind="definition">
# Proximal operator

## Definition

Let $H$ be a real [[Hilbert space|HM2HHE3J51]] with norm $\norm{\cdot}$.
Let $f \colon H \to (-\infty, +\infty]$ be a [[convex|07Q7CNM5WF]], [[lower semicontinuous|RN4G8SGP6P]] function that is not identically $+\infty$.
The *proximal operator* of $f$ is the map $\operatorname{prox}_f \colon H \to H$ given by
$$\operatorname{prox}_f(v) = \operatorname*{arg\,min}_{x \in H} \left( f(x) + \tfrac12 \norm{x - v}^2 \right).$$

## Remarks

The minimizer exists and is unique, since the function being minimized is lower semicontinuous, strongly convex and tends to $+\infty$ as $\norm{x} \to \infty$.
Equivalently, $p = \operatorname{prox}_f(v)$ exactly when $v - p$ lies in the subdifferential $\partial f(p)$.
Example: let $C \subseteq H$ be nonempty, closed and convex, and let $f$ be $0$ on $C$ and $+\infty$ outside.
Then $f(x) + \tfrac12 \norm{x - v}^2$ is finite only on $C$, so $\operatorname{prox}_f(v)$ is the point of $C$ nearest to $v$, the projection of $v$ onto $C$.
Example: for $f(x) = \lambda \abs{x}$ on $\R$ with $\lambda > 0$, one finds $\operatorname{prox}_f(v) = \sgn(v) \max(\abs{v} - \lambda, 0)$, the soft thresholding map.
Non-example: let $f$ be $0$ on $\set{-1, 1}$ and $+\infty$ elsewhere on $\R$, so $f$ is not convex.
At $v = 0$ both $-1$ and $1$ minimize $f(x) + \tfrac12 x^2$, so no single proximal point is defined.
</document>

<document source="https://tamnd.github.io/math2/node/9NPY7BF6ZJ.md" tag="9NPY7BF6ZJ" kind="definition">
# Natural boundary condition

## Definition

Let $a < b$ and $A \in \R$, and let $L \colon [a, b] \times \R \times \R \to \R$, written $L(x, z, p)$, be of class $C^1$, with [[partial derivative|09EGC6F8PT]] $L_p$ in $p$.
Consider the [[integral functional|JRQZ803Q5Y]] $J(y) = \int_a^b L(x, y, y') \, dx$ on the class of [[admissible functions|G7T008BEWY]] $y \in C^1([a, b])$ with $y(a) = A$ and no condition on $y(b)$.
The *natural boundary condition* at the free end $b$ is
$$L_p\big(b, y(b), y'(b)\big) = 0.$$
The natural boundary condition at a free end $a$ is defined in the same way, with $a$ in place of $b$.

## Remarks

Let $L$ be of class $C^2$ and let $y \in C^2([a, b])$ minimise $J$ on this class.
For $\eta \in C^1([a, b])$ with $\eta(a) = 0$, integration by parts in the [[first variation|5B8NFYB2XQ]] gives
$$0 = \delta J(y; \eta) = \int_a^b \Big( L_z - \frac{d}{dx} L_p \Big) \eta \, dx + L_p\big(b, y(b), y'(b)\big) \, \eta(b).$$
Variations with $\eta(b) = 0$ give the [[Euler–Lagrange equation|PS0VV149ET]], so the integral vanishes; a variation with $\eta(b) = 1$ then gives the natural boundary condition.

Example: let $J(y) = \int_0^1 \big( \frac{1}{2} y'^2 - y \big) \, dx$ with $y(0) = 0$ and $y(1)$ free.
The Euler–Lagrange equation is $y'' = -1$ and the natural boundary condition is $y'(1) = 0$.
The function $y(x) = x - x^2/2$ satisfies both: $y'' = -1$, $y(0) = 0$ and $y'(1) = 1 - 1 = 0$.
Non-example: $y(x) = x$ has $y(0) = 0$, but $L_p = y' = 1$ at $x = 1$, so it fails the natural boundary condition.

In several variables, for $J(u) = \int_\Omega \big( \frac{1}{2} \abs{\nabla u}^2 - f u \big) \, dx$ with no condition on $\partial \Omega$, the natural boundary condition is $\partial u / \partial \nu = 0$, where $\nu$ is the outer normal.
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# Kalman rank condition

## Statement

Let $n, m \ge 1$, let $A$ be a real $n \times n$ matrix and let $B$ be a real $n \times m$ matrix.
Consider the linear [[control system|ZPDTVZQN5K]] $x' = A x + B u$ with control set $\R^m$.
The system is [[controllable|A9WFM77Z4P]] if and only if the $n \times nm$ matrix
$$\mathcal{C} = \begin{pmatrix} B & AB & A^2 B & \cdots & A^{n-1} B \end{pmatrix}$$
has [[rank|0ZC9YTXZRY]] $n$.
In that case every state can be steered to every other state in any prescribed time $T > 0$.

## Remarks

The proof uses the controllability Gramian $W_T = \int_0^T e^{-As} B B^T e^{-A^T s} \, ds$, built from the [[matrix exponential|020M04FN1H]].
A vector $v$ lies in the kernel of $W_T$ exactly when $v^T e^{-As} B = 0$ for all $s \in [0, T]$, and by Taylor expansion and the [[Cayley–Hamilton theorem|0F765KFC2J]] this holds exactly when $v^T A^k B = 0$ for $k = 0, \ldots, n - 1$.
So $W_T$ is invertible if and only if $\rank \mathcal{C} = n$.
If $W_T$ is invertible, the control $u(s) = B^T e^{-A^T s} W_T^{-1} (e^{-AT} x_1 - x_0)$ steers $x_0$ to $x_1$ in time $T$.
If $\rank \mathcal{C} < n$, every trajectory from $0$ stays in the column space of $\mathcal{C}$, so the states outside it are not reachable from $0$.
Example: for the double integrator, $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ give $\mathcal{C} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, of rank $2$, so the system is controllable.

## Explanations

### L1 human

The control pushes the state in the directions of the columns of $B$.
The dynamics then turns these directions by $A$, then by $A$ again, and so on.
The test says that the system can be steered anywhere exactly when all these directions together span the whole state space.
It turns a question about all possible control functions into the rank of one matrix.

### L2 human

The condition is purely algebraic and easy to check by hand or by computer.
By duality, the system $x' = A x$, $y = C x$ is [[observable|C4HN3EAQWM]] if and only if the matrix with block rows $C, CA, \ldots, CA^{n-1}$ has rank $n$.
This is the Kalman condition for the pair $(A^T, C^T)$.
When the condition fails, a linear change of coordinates splits the state into a controllable part and a part on which the control has no effect.
Controllability of $(A, B)$ is also equivalent to the possibility of placing the eigenvalues of $A + BK$ anywhere, symmetric under complex conjugation, by a choice of the feedback matrix $K$.
</document>

<document source="https://tamnd.github.io/math2/node/9NZ9CXC8FY.md" tag="9NZ9CXC8FY" kind="statement">
# Hadamard's inequality in Fredholm theory

## Statement

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]], and let $M = \max \abs{K}$.
Then for every $n \ge 1$ and all $x_1, \dots, x_n, y_1, \dots, y_n \in [a, b]$,
$$\abs{\det[K(x_i, y_j)]_{i,j=1}^{n}} \le n^{n/2} M^n.$$
Consequently, the $n$-th coefficient $c_n = \frac{1}{n!} \int_{[a, b]^n} \det[K(x_i, x_j)] \, dx$ of the [[Fredholm determinant|PPX0AK81QV]] satisfies
$$\abs{c_n} \le \frac{n^{n/2} \, M^n (b - a)^n}{n!},$$
and the series $\sum_n (-1)^n c_n \lambda^n$ converges for every $\lambda \in \C$.

## Remarks

The first bound is [[Hadamard's inequality|CGDN1X32NQ]] for the matrix $[K(x_i, y_j)]$: each column has $n$ entries of modulus at most $M$, so its Euclidean norm is at most $\sqrt{n} M$.
Integrating over $[a, b]^n$ gives the bound on $c_n$.
By Stirling's formula $n! \ge (n/e)^n$, so $\abs{c_n \lambda^n} \le \left(e M (b - a) \abs{\lambda}\right)^n / n^{n/2}$, and the $n$-th root of this tends to $0$.
Hence the radius of convergence is infinite, and $D$ is an [[entire function|0G4F0XB9X2]].

The same argument, applied to matrices of size $n + k$, shows that the series defining the [[Fredholm minors|HP5VJ9EBST]] converge for every $\lambda$, uniformly in the fixed points.

The crude bound $\abs{\det} \le n! \, M^n$, from expanding the determinant into $n!$ products, is not enough: it only gives $\abs{c_n} \le (M (b - a))^n$ and hence convergence for $\abs{\lambda} < 1 / (M (b - a))$.
The factor $n^{n/2}$ instead of $n!$ is what makes the whole theory work for every $\lambda$.

## Explanations

### L1 human

A determinant of size $n$ is a sum of $n!$ products, and that number grows very fast.
Hadamard's inequality shows that the determinant itself grows much more slowly, like $n^{n/2}$.
This is why the Fredholm determinant, a power series built from such determinants divided by $n!$, converges for every value of the parameter.

### L2 human

Fredholm needed exactly this estimate to show that his determinant and minors are entire functions of $\lambda$.
Without it, his formulas would only work near $\lambda = 0$, where the Neumann series already does the job.
</document>

<document source="https://tamnd.github.io/math2/node/9NZ9FY1S7C.md" tag="9NZ9FY1S7C" kind="concept">
# Leavitt path algebra

## Summary

Let $E$ be a [[directed graph|0KFYGSHHF5]] and $K$ a field.
The Leavitt path algebra $L_K(E)$ is generated by the vertices and edges of $E$ and by a "ghost" edge $e^*$ for each edge $e$.
The generators satisfy the Cuntz–Krieger relations.
It contains the [[path algebra|2PYANBGXFD]] of $E$ and adds partial inverses of the edges.
These algebras include the matrix rings $M_n(K)$, the Laurent polynomial ring $K[x, x^{-1}]$ and Leavitt's algebras $L(1, n)$ without invariant basis number.
They are the algebraic counterparts of graph C*-algebras, and many ring-theoretic properties of $L_K(E)$ can be read off from the graph.

## Explanations

### L1 human

Start with a directed graph and walk along its edges.
For each edge $e$ add a ghost edge $e^*$ that walks back the opposite way.
Walking back along $e^*$ and then forward along $e$ returns you to where you started, and at each vertex the ways out add up to standing still.
For one vertex with one loop, the loop becomes an invertible variable $x$ with $x^* = x^{-1}$, and the algebra is $K[x, x^{-1}]$.
</document>

<document source="https://tamnd.github.io/math2/node/9P839VB8S7.md" tag="9P839VB8S7" kind="statement">
# Euler–Maclaurin formula

## Statement

Let $m < n$ be integers, let $p \ge 1$ be an integer, and let $f \colon [m, n] \to \R$ have a continuous $p$-th derivative.
Let $B_k$ be the [[Bernoulli numbers|0RAS24E1R5]] with $B_2 = \frac16$, and let $P_p(x) = B_p(x - \floor{x})$, where $B_p(t)$ is the [[Bernoulli polynomial|BXQKBJJGFG]] of degree $p$.
Then
$$\sum_{i=m}^{n} f(i) = \int_m^n f(x)\, dx + \frac{f(m) + f(n)}{2} + \sum_{k=1}^{\floor{p/2}} \frac{B_{2k}}{(2k)!} \bigl(f^{(2k-1)}(n) - f^{(2k-1)}(m)\bigr) + R_p,$$
where
$$R_p = (-1)^{p+1} \int_m^n f^{(p)}(x) \, \frac{P_p(x)}{p!}\, dx.$$

## Remarks

The proof is repeated integration by parts on each interval $[i, i + 1]$, using $B_k'(t) = k B_{k-1}(t)$ and the fact that $B_k(0) = B_k(1)$ for $k \ge 2$.
For $p = 1$ the formula reads $\sum_{i=m}^{n} f(i) = \int_m^n f + \frac{f(m) + f(n)}{2} + \int_m^n f'(x) \bigl(x - \floor{x} - \frac12\bigr)\, dx$.
Example: for $f(x) = x^2$ on $[0, n]$ with $p = 3$, the remainder vanishes since $f''' = 0$, and the formula gives $\sum_{i=0}^{n} i^2 = \frac{n^3}{3} + \frac{n^2}{2} + \frac{1}{12} \cdot 2n = \frac{n(n+1)(2n+1)}{6}$.
The infinite series obtained by letting $p \to \infty$ usually diverges, so the formula is used with a fixed $p$ and an estimate of $R_p$.

## Explanations

### L1 human

The formula compares a sum of values $f(m) + \cdots + f(n)$ with the integral of $f$.
The difference is given by the end values of $f$ and its derivatives, with the Bernoulli numbers as coefficients, plus a small remainder.

### L2 human

In one direction it estimates sums by integrals: it gives [[Stirling's formula|0KC0F51T6A]] for $\ln n!$, asymptotics of harmonic numbers, and the values of power sums $\sum i^k$ as polynomials in $n$.
In the other direction it estimates integrals by sums: it describes the error of the trapezoidal rule, and it explains why that rule is very accurate for smooth periodic functions.
It is a quantitative link between [[series|0JH4MN4K1C]] and the [[Riemann integral|0QM4TXT3DR]].
</document>

<document source="https://tamnd.github.io/math2/node/9P83YFRWDJ.md" tag="9P83YFRWDJ" kind="definition">
# Profunctor

## Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be [[categories|A2DE66937D]], and let $\mathcal{D}^{\mathrm{op}}$ be the [[opposite category|DAMK8XE1FR]] of $\mathcal{D}$.
A *profunctor* $P$ from $\mathcal{C}$ to $\mathcal{D}$, written $P \colon \mathcal{C} \nrightarrow \mathcal{D}$, is a [[functor|28NQ0KRHD3]]
$$P \colon \mathcal{D}^{\mathrm{op}} \times \mathcal{C} \to \mathbf{Set}$$
on the [[product category|AFNE9C4NH6]].
A morphism of profunctors is a natural transformation between such functors.

## Remarks

Conventions differ: some authors call the same functor a profunctor from $\mathcal{D}$ to $\mathcal{C}$.
For small $\mathcal{D}$ it is the same as a functor $\mathcal{C} \to [\mathcal{D}^{\mathrm{op}}, \mathbf{Set}]$, from $\mathcal{C}$ to presheaves on $\mathcal{D}$.
Example: the [[hom functor|VJXS8AK67E]] $\Hom_{\mathcal{C}} \colon \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathbf{Set}$ is a profunctor from $\mathcal{C}$ to $\mathcal{C}$, the identity profunctor.
Example: a functor $F \colon \mathcal{C} \to \mathcal{D}$ gives the profunctor $(d, c) \mapsto \Hom_{\mathcal{D}}(d, F(c))$ from $\mathcal{C}$ to $\mathcal{D}$.
Non-example: for the same $F$, the functor $(d, c) \mapsto \Hom_{\mathcal{D}}(F(c), d)$ is covariant in $d$ and contravariant in $c$.
So it is not a profunctor from $\mathcal{C}$ to $\mathcal{D}$; it is a profunctor from $\mathcal{D}$ to $\mathcal{C}$.
For $P \colon \mathcal{C} \nrightarrow \mathcal{D}$ and $Q \colon \mathcal{D} \nrightarrow \mathcal{E}$ with $\mathcal{D}$ small, the composite is given by the [[coend|W9QTFV0Z8D]] $(Q \circ P)(e, c) = \int^{d \in \mathcal{D}} Q(e, d) \times P(d, c)$.
Composition is associative and unital only up to natural isomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/9P87VQZ0P7.md" tag="9P87VQZ0P7" kind="concept">
# Anger function

## Summary

The Anger function $\mathbf{J}_\nu$ is the function obtained by using Bessel's integral formula for an arbitrary order $\nu$.
For integer $\nu$ it equals the [[Bessel function of the first kind|CZCMRDWDYW]] $J_\nu$; for other $\nu$ it does not.
Its companion is the Weber function $\mathbf{E}_\nu$, defined with a sine in place of the cosine.
Both solve inhomogeneous forms of Bessel's equation.

## Explanations

### L1 human

Bessel found that $J_n(z)$ is the average of $\cos(n\tau - z\sin\tau)$ over $0 \le \tau \le \pi$ when $n$ is a whole number.
One can compute the same average when $n$ is replaced by any number $\nu$.
The result is the Anger function.
When $\nu$ is not a whole number, it is no longer a Bessel function, which shows that Bessel's integral is special to integer orders.
</document>

<document source="https://tamnd.github.io/math2/node/9P8D8YAHQS.md" tag="9P8D8YAHQS" kind="concept">
# Exponential type

## Summary

An [[entire function|0419794R0T]] is of exponential type if its modulus grows at most like $e^{\tau \abs{z}}$ for some constant $\tau$.
The least such $\tau$ is the type of the function.
Fourier and Laplace transforms of functions or measures with bounded support are of exponential type, and the Paley–Wiener theorem gives a converse.
They are the natural setting of the [[Paley–Wiener theorem|THY12GPZ8C]] and of [[Carlson's theorem|6F5B8KPEGH]].

## Explanations

### L1 human

The exponential $e^{\tau z}$ grows like $e^{\tau \abs{z}}$ in the direction where $z$ is positive real.
A function of exponential type never grows faster than this in any direction, for some fixed $\tau$.
Polynomials, $\sin z$, $\cos z$ and $e^{3z}$ are of exponential type, with types $0$, $1$, $1$ and $3$.
The function $e^{z^2}$ grows like $e^{\abs{z}^2}$ along the real axis and is not of exponential type.
</document>

<document source="https://tamnd.github.io/math2/node/9P8VK3H4YZ.md" tag="9P8VK3H4YZ" kind="statement">
# Cartan's classification of symmetric spaces

## Statement

1. Every simply connected Riemannian [[symmetric space|PAA93R4WYD]] is isometric to a Riemannian product $\R^k \times M_1 \times \cdots \times M_r$ of a Euclidean space and irreducible simply connected symmetric spaces $M_i$.
2. Every irreducible simply connected symmetric space other than $\R$ is of one of two types.
   Those of *compact type* are compact and have nonnegative sectional curvature.
   Those of *noncompact type* have nonpositive sectional curvature and are diffeomorphic to a Euclidean space.
3. There is a duality between irreducible simply connected symmetric spaces of compact type and of noncompact type.
   On Lie algebras it replaces $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ by $\mathfrak{k} \oplus i\mathfrak{p}$.
4. Up to scaling, the irreducible symmetric spaces of noncompact type are the quotients $G/K$.
   Here $G$ is a connected noncompact simple Lie group with trivial center and $K$ is a maximal compact subgroup.
   These spaces correspond bijectively to the isomorphism classes of noncompact real simple Lie algebras.

## Remarks

Statement 1 is the de Rham decomposition applied to symmetric spaces.
The rest reduces geometry to algebra: the symmetric space is determined by its symmetric Lie algebra $(\mathfrak{g}, \sigma)$, where $\sigma$ is the involution coming from the geodesic symmetry.
Using Cartan's classification of real simple Lie algebras, the list of irreducible noncompact type spaces consists of two kinds.
The first kind is $G_{\C}/G_u$ for a complex simple Lie group $G_{\C}$ with compact real form $G_u$.
The second kind comes from the noncompact real forms of the complex simple Lie algebras: seven infinite families labelled AI, AII, AIII, BDI, DIII, CI, CII and twelve exceptional spaces.

Examples: the hyperbolic space $H^n = SO_0(n, 1)/SO(n)$ is of noncompact type, and its dual is the sphere $S^n = SO(n+1)/SO(n)$.
The space $SL(n, \R)/SO(n)$ of positive definite symmetric matrices of determinant $1$ is of type AI.

## Explanations

### L1 human

Symmetric spaces look the same from every point, and there are surprisingly few of them.
Apart from flat space, every one is built from pieces that are either curved like a sphere or curved like hyperbolic space.
These pieces come in pairs and are listed completely by a finite table plus a few infinite families.

### L2 human

The classification turns a question in Riemannian geometry into the classification of simple Lie algebras and their involutions.
Symmetric spaces of noncompact type are the spaces on which arithmetic groups act, as in locally symmetric spaces $\Gamma \backslash G/K$.
The compact type includes the Grassmannians and the compact simple Lie groups with a bi-invariant metric.
For a simply connected symmetric space $G/K$ without Euclidean factor, the [[holonomy|WHDZ3MK56B]] group is $K$ acting on the tangent space.
This is why symmetric spaces appear as a separate case in Berger's list.
</document>

<document source="https://tamnd.github.io/math2/node/9PA8CVSWPD.md" tag="9PA8CVSWPD" kind="concept">
# Elimination theory

## Summary

Elimination theory studies how to remove some variables from a system of polynomial equations, so as to obtain the equations satisfied by the remaining variables.
Its basic object is the elimination ideal, the set of consequences of the system that involve only the remaining variables.
Classical tools are resultants; the modern tool is a [[Gröbner basis|C83KPT47R0]] for a lexicographic order.
Geometrically, elimination computes the closure of the projection of the solution set to fewer coordinates.

## Explanations

### L1 human

From $x = t$ and $y = t^2$, one can eliminate $t$ and get $y = x^2$, the equation of a parabola.
Elimination theory makes this process systematic for any system of polynomial equations.
It answers questions such as: which values of some variables extend to a solution of the whole system?
This is the algebraic form of solving a system by substitution, as one does in school for linear equations.
</document>

<document source="https://tamnd.github.io/math2/node/9PBJF44KBX.md" tag="9PBJF44KBX" kind="statement">
# Quadratic convergence of Newton's method

## Statement

Let $I$ be an open interval, let $f \colon I \to \R$ be [[twice continuously differentiable|0VW90BBSJ4]], and let $r \in I$ with $f(r) = 0$ and $f'(r) \ne 0$.
For $\delta > 0$ with $J = [r - \delta, r + \delta] \subseteq I$ and $f' \ne 0$ on $J$, let
$$M = \frac{\max_{x \in J} \abs{f''(x)}}{2 \min_{x \in J} \abs{f'(x)}}.$$
If $M \delta < 1$, then for every $x_0 \in J$ the iterates of [[Newton's method|07EXVNF9VP]], $x_{k+1} = x_k - f(x_k)/f'(x_k)$, are defined, lie in $J$, converge to $r$, and satisfy
$$\abs{x_{k+1} - r} \le M \abs{x_k - r}^2 \quad \text{for all } k \ge 0.$$
Such a $\delta$ always exists.
If moreover $x_k \ne r$ for all $k$, then
$$\lim_{k \to \infty} \frac{x_{k+1} - r}{(x_k - r)^2} = \frac{f''(r)}{2 f'(r)}.$$

## Remarks

[[Taylor's theorem with the Lagrange remainder|0H5ZAZCX2T]] at $x_k$ gives $0 = f(r) = f(x_k) + f'(x_k)(r - x_k) + \tfrac12 f''(\xi_k)(r - x_k)^2$ for some $\xi_k$ between $x_k$ and $r$.
Dividing by $f'(x_k)$ and using the definition of $x_{k+1}$ gives $x_{k+1} - r = \frac{f''(\xi_k)}{2 f'(x_k)} (x_k - r)^2$, so $\abs{x_{k+1} - r} \le M \abs{x_k - r}^2 \le M\delta \abs{x_k - r}$; this keeps the iterates in $J$ and makes them converge, and the limit follows from $\xi_k, x_k \to r$.
Since $f'(r) \ne 0$ and $f'$ and $f''$ are continuous, $M$ stays bounded as $\delta \to 0$, so a small $\delta$ gives $M\delta < 1$.
Example: for $f(x) = x^2 - 2$ from $x_0 = 1$, the errors are about $4.1 \times 10^{-1}$, $8.6 \times 10^{-2}$, $2.5 \times 10^{-3}$, $2.1 \times 10^{-6}$, $1.6 \times 10^{-12}$; each error is about $0.35$ times the square of the previous one, and $f''(r)/(2f'(r)) = 1/(2\sqrt{2}) \approx 0.35$.
The hypothesis $f'(r) \ne 0$ is needed: for $f(x) = x^2$ the iteration is $x_{k+1} = x_k/2$, which converges only linearly.
Differentiability at $r$ is needed as well: for $f(x) = x^{1/3}$ the iteration is $x_{k+1} = -2x_k$, which diverges from every $x_0 \ne 0$.

## Explanations

### L1 human

Near a simple zero, Newton's method roughly doubles the number of correct digits at each step.
If a guess is right to $3$ digits, the next one is right to about $6$, then $12$.
This holds only once the guess is close enough and the graph crosses the axis at a nonzero slope.

### L2 human

This is the basic local convergence theorem for [[Newton's method|07EXVNF9VP]]: order $2$ in the sense of [[rate of convergence|YWZQC5E54B]], with asymptotic constant $f''(r)/(2f'(r))$.
The same proof works for systems $F(x) = 0$ in $\R^n$ with an invertible Jacobian at the zero, where it is a special case of the Newton–Kantorovich theory.
At a zero of multiplicity $m \ge 2$ the convergence becomes linear with ratio $(m - 1)/m$, and the modified step $x_{k+1} = x_k - m f(x_k)/f'(x_k)$ restores quadratic convergence.
Global convergence is not guaranteed, which is why practical solvers combine Newton steps with bisection or line searches.
</document>

<document source="https://tamnd.github.io/math2/node/9PK608W8MS.md" tag="9PK608W8MS" kind="example">
# Lattès map

## Example

Let $\Lambda \subset \C$ be a [[lattice|WN9V7WSJKF]], let $T = \C / \Lambda$ be the [[complex torus|Y7MA6KBE9M]], and let $\wp$ be the Weierstrass [[elliptic function|FF9ZPCZ59K]] of $\Lambda$, with invariants $g_2, g_3$.
The duplication formula expresses $\wp(2u)$ as a [[rational function|0XDK9CGYKJ]] of $\wp(u)$:
$$\wp(2u) = L(\wp(u)), \qquad L(x) = \frac{x^4 + \frac{g_2}{2} x^2 + 2 g_3 x + \frac{g_2^2}{16}}{4x^3 - g_2 x - g_3}.$$
The map $L$ of degree $4$ is a *Lattès map*.
It is the shadow of the doubling map $u \mapsto 2u$ on $T$ under the projection $\wp \colon T \to \hat{\C}$.

For a lattice with $g_2 = 4$ and $g_3 = 0$, which exists because $g_2^3 - 27 g_3^2 = 64 \ne 0$, the formula gives
$$L(x) = \frac{(x^2 + 1)^2}{4x(x^2 - 1)}.$$

Its [[Julia set|5M4S7P04GR]] is the whole Riemann sphere.
The reason: the doubling map on $T$ multiplies small distances by $2$.
So the image under $u \mapsto 2^n u$ of any small disc in $T$ covers $T$ for $n$ large.
Applying $\wp$, for every nonempty open set $U \subseteq \hat{\C}$ some iterate $L^n$ maps $U$ onto $\hat{\C}$.
The Fatou set is invariant and the Julia set is nonempty, so an open subset of the Fatou set never maps onto $\hat{\C}$.
Hence the Fatou set of $L$ is empty.

## Remarks

Lattès maps are the standard examples of rational maps whose Julia set is the whole sphere.
In general, a Lattès map is a rational map induced by an affine self-map $u \mapsto a u + b$ of a complex torus, with $\abs{a}^2 \ge 2$ an integer.
The map passes to the sphere through the quotient of the torus by a finite group of automorphisms.

## Explanations

### L1 human

On a doughnut-shaped torus, doubling every point stretches each small patch until it wraps around the whole torus.
The Weierstrass function folds the torus onto the sphere, and the doubling becomes a rational map of the sphere.
Every small region of the sphere gets spread over the whole sphere after a few steps, so the map is chaotic everywhere.
There is no calm region at all: the Julia set is the entire sphere.
</document>

<document source="https://tamnd.github.io/math2/node/9PM8Q2E30Q.md" tag="9PM8Q2E30Q" kind="concept">
# Potential theory

## Summary

Potential theory is the study of [[harmonic functions|0DEB0JMN0W]], [[subharmonic functions|Z1MHMGFNFW]] and the potentials generated by distributions of mass or charge.
It began with the gravitational and electrostatic potentials of physics, which satisfy the Laplace equation away from the masses.
Its main topics include the [[Dirichlet problem|8D56GQ4AG8]], [[Green's functions|M36SD56MWQ]], [[capacity|DBQXRTC0VN]], [[harmonic measure|CFD7BPM51W]] and the boundary behaviour of harmonic functions.
In the plane it is tied closely to complex analysis, since the real part of a [[holomorphic function|0NAHXR1FRH]] is harmonic and $\log\abs{f}$ is subharmonic.

## Explanations

### L1 human

The gravitational pull of a planet or the electric field of a charge can be described by a single number at each point, the potential.
Away from the masses and charges, this potential satisfies the Laplace equation, so it is a harmonic function.
Potential theory studies such functions: how they are determined by boundary values, how they behave near the edge of a region, and how charges spread out to minimize energy.
For example, the equilibrium temperature in a plate with fixed edge temperatures is a harmonic function, and finding it is a central problem of the theory.
</document>

<document source="https://tamnd.github.io/math2/node/9PN6QZKFCB.md" tag="9PN6QZKFCB" kind="definition">
# Mixed volume

## Definition

Let $n \ge 1$ and let $\mathcal{K}^n$ be the set of nonempty compact convex subsets of $\R^n$ (the [[convex bodies|437FSG3419]] in the broad sense).
Let $\operatorname{vol}$ be [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$.
There is a unique function $V \colon (\mathcal{K}^n)^n \to \R$ such that:

1. $V$ is symmetric in its $n$ arguments;
2. for every $r \ge 1$, all $K_1, \dots, K_r \in \mathcal{K}^n$ and all $\lambda_1, \dots, \lambda_r \ge 0$,
$$\operatorname{vol}(\lambda_1 K_1 + \cdots + \lambda_r K_r) = \sum_{i_1 = 1}^{r} \cdots \sum_{i_n = 1}^{r} V(K_{i_1}, \dots, K_{i_n}) \, \lambda_{i_1} \cdots \lambda_{i_n},$$
where the sum on the left is the [[Minkowski sum|SXS6D87PHE]].

The number $V(K_1, \dots, K_n)$ is the *mixed volume* of $K_1, \dots, K_n$.

## Remarks

The left side of clause 2 is a [[homogeneous polynomial|2C3PAQ7H50]] of degree $n$ in $\lambda_1, \dots, \lambda_r$; this is the theorem of Minkowski that makes the definition possible.
An explicit formula is
$$V(K_1, \dots, K_n) = \frac{1}{n!} \sum_{\emptyset \ne J \subseteq \set{1, \dots, n}} (-1)^{n - \abs{J}} \operatorname{vol}\Big(\sum_{j \in J} K_j\Big).$$
Mixed volumes are nonnegative, invariant under translating each argument, linear in each argument for Minkowski addition with nonnegative scalars, and monotone under inclusion.
Also $V(K, \dots, K) = \operatorname{vol}(K)$.

Example: $n = 2$, $K$ a compact convex set of area $A$ and perimeter $L$, and $D$ the closed unit disk.
The area of $K + tD$ is $A + L t + \pi t^2$, while clause 2 gives $A + 2 V(K, D)\, t + \pi t^2$.
So $V(K, D) = L / 2$.

Non-example: without clause 1 the coefficients are not unique.
For $n = 2$ one may move all of the coefficient $2 V(K_1, K_2)$ to the term $(i_1, i_2) = (1, 2)$ and put $0$ at $(2, 1)$, which still satisfies clause 2.
</document>

<document source="https://tamnd.github.io/math2/node/9PT772R94A.md" tag="9PT772R94A" kind="concept">
# Cauchy–Riemann equations in several variables

## Summary

The Cauchy–Riemann equations in several variables are a system of $2n$ real first-order partial differential equations.
They are $n$ pairs of [[Cauchy–Riemann equations|0T71SSYZ39]], one pair for each complex coordinate $z_j = x_j + i y_j$.
In complex form they read $\partial f / \partial \bar z_j = 0$ for $j = 1, \dots, n$.
A continuously differentiable function on an open subset of $\C^n$ is [[holomorphic|507C34W0ZT]] exactly when it satisfies them.
For $n \ge 2$ the system is overdetermined: $n$ complex equations for one unknown function, which is a source of the rigidity of the theory.

## Explanations

### L1 human

A function of one complex variable is holomorphic when its real and imaginary parts satisfy two linked equations.
In several variables, the same two equations must hold in each variable separately.
So there are two equations for each coordinate, $2n$ in all.
For example, $f(z_1, z_2) = z_1 z_2$ satisfies all of them, while $f(z_1, z_2) = z_1 \bar z_2$ fails the pair for $z_2$.
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# Riemann solver

## Definition

Let $\Omega \subseteq \R^m$ be open and let $F \colon \Omega \to \R^m$ be the $C^1$ flux of a [[hyperbolic conservation law|XN081NCPF5]] $u_t + F(u)_x = 0$.
A *Riemann solver* for $F$ is a map $H \colon \Omega \times \Omega \to \R^m$.
Its value $H(u_l, u_r)$ is the numerical flux at an interface between a left state $u_l$ and a right state $u_r$.
It is obtained from an exact or approximate solution of the [[Riemann problem|GJWKX22VVE]] with data $u_l, u_r$.
It is required to be:

1. consistent: $H(u, u) = F(u)$ for all $u \in \Omega$;
2. locally Lipschitz continuous in both arguments.

The *exact Riemann solver* is $H(u_l, u_r) = F(w(0; u_l, u_r))$, where $w(x/t; u_l, u_r)$ is the self-similar [[entropy solution|1A01A3EJRX]] of the Riemann problem, when it exists and is unique.

## Remarks

Example: the HLL solver uses two estimates $S_L < S_R$ of the slowest and fastest wave speeds.
It sets $H = F(u_l)$ if $0 \le S_L$, $H = F(u_r)$ if $S_R \le 0$, and otherwise
$$H(u_l, u_r) = \frac{S_R F(u_l) - S_L F(u_r) + S_L S_R (u_r - u_l)}{S_R - S_L}.$$
For $u_l = u_r = u$ every case gives $F(u)$, so this solver is consistent.

Non-example: $H(u_l, u_r) = F(u_l) + F(u_r)$ fails clause 1, since $H(u, u) = 2F(u)$.
A scheme with such a flux approximates the wrong equation.

For the scalar equation $u_t + a u_x = 0$, the exact solver gives $H(u_l, u_r) = a u_l$ if $a > 0$ and $a u_r$ if $a < 0$, which is the upwind flux.
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# Harmonic function on a ball

## Example

Let $n \ge 2$, let $B = \set{x \in \R^n : \abs{x} < 1}$ be the unit [[open ball|0YE5868HZB]], and fix a point $\zeta$ with $\abs{\zeta} = 1$.
Consider the function
$$u(x) = \frac{1 - \abs{x}^2}{\abs{x - \zeta}^n}, \qquad x \in B,$$
which is the [[Poisson kernel|TCYSWCTYYM]] of the ball with pole $\zeta$, up to a constant factor.
It is positive on $B$, because $\abs{x} < 1$ and $x \ne \zeta$.
To check that it is harmonic, write $f(x) = 1 - \abs{x}^2$, $g(x) = \abs{x - \zeta}^{-n}$ and $r = \abs{x - \zeta}$.
Then $\Delta f = -2n$, $\nabla f = -2x$, $\nabla g = -n r^{-n-2} (x - \zeta)$ and $\Delta g = 2n r^{-n-2}$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
The product rule $\Delta(fg) = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f$ gives
$$\Delta u = 2n r^{-n-2} \left( 1 - \abs{x}^2 + 2 x \cdot (x - \zeta) - r^2 \right).$$
Since $r^2 = \abs{x}^2 - 2 x \cdot \zeta + 1$, the bracket equals $0$, so $\Delta u = 0$.

## Remarks

As $x$ tends to a boundary point $\eta \ne \zeta$, the numerator tends to $0$ and the denominator stays away from $0$, so $u(x) \to 0$.
As $x \to \zeta$ along the radius, $u(t\zeta) = (1 + t)/(1 - t)^{n-1}$ tends to $+\infty$.
So $u$ is a positive harmonic function on the ball that vanishes on the whole boundary sphere except at one point.

## Explanations

### L1 human

This function is the steady temperature in a ball when all the heat enters through one point $\zeta$ of the surface.
It is zero near the rest of the surface and blows up at $\zeta$.
Every positive harmonic function on the ball is an average of such functions with different poles.
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# Discrete-time algebraic Riccati equation

## Definition

Let $n, m \ge 1$.
Let $A$ be a real $n \times n$ [[matrix|0VD9ZSWWJG]] and let $B$ be a real $n \times m$ matrix.
Let $Q$ be a real [[symmetric|0VR671PHE5]] [[positive semidefinite|01MMVW8CKC]] $n \times n$ matrix, and let $R$ be a real symmetric [[positive definite|0KX66CZMBB]] $m \times m$ matrix.
Write $M^T$ for the [[transpose|09DXWYRD43]] of a matrix $M$.
The *discrete-time algebraic Riccati equation* is
$$X = A^T X A - A^T X B \left(R + B^T X B\right)^{-1} B^T X A + Q$$
for an unknown real symmetric positive semidefinite $n \times n$ matrix $X$.

## Remarks

For such $X$ the matrix $R + B^T X B$ is positive definite, hence [[invertible|0EB9N0S33C]], so the equation makes sense.
A solution $X$ is *stabilising* if every eigenvalue of $A - BK$ has absolute value less than $1$, where $K = (R + B^T X B)^{-1} B^T X A$.
If the pair $(A, B)$ is stabilisable and the pair $(Q^{1/2}, A)$ is detectable, the equation has a unique positive semidefinite solution, and it is stabilising.
The optimal control of the infinite-horizon linear-quadratic regulator is then $u_k = -K x_k$.
Example: for $n = m = 1$ and $A = B = Q = R = 1$, the equation reads $X = X - X^2/(1 + X) + 1$, that is, $X^2 - X - 1 = 0$.
Its nonnegative solution is $X = (1 + \sqrt{5})/2$, and then $K = X/(1 + X) = 1/X$ and $A - BK = 1 - 1/X \approx 0.38$, so it is stabilising.
Non-example: the other root $X = (1 - \sqrt{5})/2$ satisfies the matrix identity but is negative, so it is not an admissible solution.
For $B = 0$ the equation becomes the linear [[Stein equation|FM9Y4Y6D8F]] $X - A^T X A = Q$.
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# Equational basis

## Summary

An equational basis of a class of algebras is a set of [[identities|PKD440G2XW]] from which all identities of the class follow.
It is a set of axioms for the [[equational theory|F1M51B4MDG]] of the class, and it defines the [[variety|AJNZJD8442]] generated by the class.
The usual axioms of groups, rings or lattices are equational bases, and a central question is whether a given algebra has a finite one.

## Explanations

### L1 human

An algebra satisfies infinitely many laws, and an equational basis is a list of laws from which all the others can be derived.
For groups, written with multiplication, inverse and identity element, three short laws suffice: associativity, $x e \approx x$ and $x x^{-1} \approx e$.
Every other law that holds in all groups, such as $(x y)^{-1} \approx y^{-1} x^{-1}$, can be derived from these.
A basis is useful because it is much shorter than the full list of laws.
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# Kolmogorov–Nagumo characterization of quasi-arithmetic means

## Statement

Let $I$ be a real [[interval|0Q1D85B8G6]].
For each integer $n \ge 1$, let $M_n \colon I^n \to \R$ be a function.
Assume that the sequence $(M_n)$ has these four properties for all $n \ge 1$.

1. $M_n$ is [[continuous|0X3WGX95FE]] and [[strictly increasing|011EJQWWJS]] in each variable.
2. $M_n$ is symmetric: its value does not change when its arguments are permuted.
3. $M_n(x, x, \dots, x) = x$ for all $x \in I$.
4. For all $1 \le k \le n$ and $x_1, \dots, x_n \in I$, with $m = M_k(x_1, \dots, x_k)$, one has $M_n(x_1, \dots, x_k, x_{k+1}, \dots, x_n) = M_n(m, \dots, m, x_{k+1}, \dots, x_n)$, where $m$ is repeated $k$ times.

Then there is a continuous strictly monotone function $\varphi \colon I \to \R$ such that, for all $n$, $M_n$ is the [[quasi-arithmetic mean|SG5C5TVGRB]] generated by $\varphi$:
$$M_n(x_1, \dots, x_n) = \varphi^{-1}\left( \frac{1}{n} \sum_{i=1}^{n} \varphi(x_i) \right).$$
Conversely, every quasi-arithmetic mean has properties 1 to 4.

## Remarks

The converse is a direct check: property 4 holds because replacing $x_1, \dots, x_k$ by $m$ does not change the sum $\sum \varphi(x_i)$.
For the main direction, one fixes two points $a < b$ of $I$ and defines $\varphi$ on the numbers $M_n(a, \dots, a, b, \dots, b)$ with $k$ copies of $b$ out of $n$ as $k/n$.
Properties 1 to 4 show that this is consistent and strictly increasing on a dense set of points, and continuity extends it.
The theorem was proved independently by Kolmogorov and by Nagumo in 1930.
Strict monotonicity cannot be weakened to monotonicity: $M_n(x_1, \dots, x_n) = \min_i x_i$ is continuous, increasing in each variable and has properties 2 to 4, but it is not a quasi-arithmetic mean.

## Explanations

### L1 human

There are many ways to average numbers, and this theorem explains why most of them share one shape.
A good average should not care about the order of the numbers, should return $x$ when all numbers equal $x$, and should grow when one of the numbers grows.
It should also allow replacing a group of numbers by their own average without changing the total result.
Any average with these properties is the ordinary average computed on some hidden scale $\varphi$.

### L2 human

The theorem turns a question about means into a [[functional equation|SRAN4EKWS0]] for the generator $\varphi$, and the solution is unique up to $\varphi \mapsto a \varphi + b$.
It is used to justify the choice of particular means in statistics, economics and information theory.
For example, adding the requirement $M_n(x_1 + c, \dots, x_n + c) = M_n(x_1, \dots, x_n) + c$ narrows the generators to $\varphi(x) = x$ and $\varphi(x) = e^{ax}$ up to affine changes.
That narrowing is the route from quasi-arithmetic means to Rényi entropies.
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# Napoleon's theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane.
On each side of $ABC$ erect an equilateral triangle, either all three outside $ABC$ or all three overlapping $ABC$ inwards.
Let $X$, $Y$ and $Z$ be the [[centroids|0X7EEWES5G]] of the equilateral triangles on $BC$, $CA$ and $AB$.
Then $XYZ$ is an equilateral triangle.

## Remarks

A short proof uses rotations.
For the outer case, the rotations by $120^\circ$ about $Z$, $X$ and $Y$, in the same sense, send $A$ to $B$, $B$ to $C$ and $C$ to $A$.
Their composition has total angle $360^\circ$, so it is a translation, and it fixes $A$, so it is the identity.
Three rotations by $120^\circ$ whose composition is the identity have centres that form an equilateral triangle.
The triangle $XYZ$ is called the outer or the inner Napoleon triangle.
Both Napoleon triangles have the same centroid as $ABC$, and the area of the outer one minus the area of the inner one equals the area of $ABC$.
For an equilateral triangle $ABC$ the inner Napoleon triangle degenerates to a single point, its centre.

## Explanations

### L1 human

Take any triangle, even a very lopsided one.
Draw an equilateral triangle on each of its sides, pointing outwards, and mark the centre of each.
The three centres always form an equilateral triangle.
The irregular shape of the starting triangle disappears completely.

### L2 human

The theorem is a standard exercise for complex numbers or for compositions of rotations in the plane.
The same equilateral triangles appear in the construction of the [[Fermat point|QM0CGSQ4TF]].
The lines joining each vertex to the apex of the outer equilateral triangle on the opposite side meet in one point.
It is one of the first examples where a [[regular polygon|01KJJ9A400]] is produced from an arbitrary triangle by a fixed recipe.
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# Iterated kernel

## Summary

The iterated kernels of a [[kernel|GSMBXT7FPK]] $K$ are the kernels $K_1 = K$, $K_2$, $K_3$, ... of the powers of its integral operator.
They are defined by $K_{n}(x, s) = \int_a^b K(x, t) K_{n-1}(t, s) \, dt$, the continuous analogue of matrix powers.
They are the terms of the [[Liouville–Neumann series|H1HWSWME6B]] and of the [[resolvent kernel|ARTJV1XCRX]].
Their growth in $n$ decides for which $\lambda$ these series converge.

## Explanations

### L1 human

Applying the integral operator twice is again an integral operator, and its kernel is $K_2$.
To compute $K_2(x, s)$, sum over all intermediate points $t$ the product of the influence of $t$ on $x$ and of $s$ on $t$.
It works like multiplying a matrix by itself, with the sum over the middle index replaced by an integral.
For example, for $K(x, s) = x s$ on $[0, 1]$ one gets $K_2(x, s) = \int_0^1 x t \cdot t s \, dt = x s / 3$.
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# Field with one element

## Summary

The field with one element, written $\F_1$, is a hoped-for object that would behave like a [[finite field|0ZXZ82FTX2]] $\F_q$ in the limit $q \to 1$.
No field has one element, since a [[field|04C5ZBG1WT]] needs $0 \ne 1$, so $\F_1$ is not a field in the usual sense.
Many counting formulas over $\F_q$ become classical combinatorial formulas at $q = 1$, and $\F_1$ is meant to explain this.
Several competing theories of geometry over $\F_1$ exist, partly motivated by the hope of treating $\operatorname{Spec} \Z$ like a curve over a finite field.

## Explanations

### L1 human

Over a finite field $\F_q$, the number of lines through the origin in the plane $\F_q^2$ is $q + 1$.
At $q = 1$ this gives $2$, the number of one-element subsets of a two-element set.
More generally, the number of $k$-dimensional subspaces of $\F_q^n$ is a polynomial in $q$ whose value at $q = 1$ is the binomial coefficient $\binom{n}{k}$.
So a finite set behaves like a vector space over a field with one element, and the symmetric group $S_n$ plays the role of $GL_n(\F_1)$.
There is no single agreed definition, and $\F_1$ remains a guiding idea rather than one fixed object.
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# Nemytskii operator

## Definition

Let $(\Omega, \mathcal{A}, m)$ be a $\sigma$-finite [[measure space|P15VQS5DJ9]] and let $g \colon \Omega \times \R \to \R$ be a *Carathéodory function*:

1. for every $u \in \R$, the map $x \mapsto g(x, u)$ is [[measurable|YYNCJSR0PJ]];
2. for almost every $x \in \Omega$, the map $u \mapsto g(x, u)$ is continuous.

The *Nemytskii operator* of $g$ sends each measurable function $u \colon \Omega \to \R$ to the function $N_g u \colon \Omega \to \R$ given by
$$(N_g u)(x) = g\bigl(x, u(x)\bigr).$$
Under the two conditions, $N_g u$ is again measurable.

## Remarks

A theorem of Krasnosel'skii describes the [[$L^p$ spaces|SASP14MA1Z]] case.
Let $\Omega \subseteq \R^m$ be measurable with Lebesgue measure, and let $1 \le p, q < \infty$.
Then $N_g$ maps $L^p(\Omega)$ into $L^q(\Omega)$ if and only if there are $a \in L^q(\Omega)$ and $b \ge 0$ with $\abs{g(x, u)} \le a(x) + b \abs{u}^{p/q}$ for almost every $x$ and all $u$.
In that case $N_g \colon L^p(\Omega) \to L^q(\Omega)$ is continuous and maps bounded sets to bounded sets.

Example: $\Omega = [0, 1]$ with Lebesgue measure and $g(x, u) = u^2$.
Both conditions hold, and $\abs{g(x, u)} \le \abs{u}^{4/2}$, so $N_g$ maps $L^4[0, 1]$ into $L^2[0, 1]$; indeed $\int_0^1 (u^2)^2 = \int_0^1 u^4 < \infty$.

Non-example: the map $u \mapsto \bigl(x \mapsto \int_0^1 u(t)\, dt\bigr)$ is not a Nemytskii operator.
The value at $x$ depends on all values of $u$, not only on $u(x)$.

Other names are superposition operator and, for $g$ independent of $x$, composition operator.
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# Erlangen program

## Summary

The Erlangen program is a view of geometry that Felix Klein stated in 1872.
In it, a geometry is the study of the properties of a space that are invariant under a given [[group|G5S1M9]] of transformations.
Euclidean, affine and projective geometry then differ only in their groups: the [[Euclidean group|0H3YY8CPKH]], the [[affine group|G6GMSEZDB1]] and the projective group.
A larger group has fewer invariants, so projective geometry has fewer theorems than affine geometry, which has fewer than [[Euclidean geometry|KRZD96GSVC]].

## Explanations

### L1 human

Klein asked what makes a fact "geometric".
His answer: a fact is geometric if it survives the moves that the geometry allows.
In Euclidean geometry the moves are rigid motions, so lengths and angles count.
In affine geometry the moves may also stretch and shear, so lengths are lost, but parallel lines stay parallel and midpoints stay midpoints.
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# Overdetermined system

## Definition

Let $F$ be a [[field|04C5ZBG1WT]].
Let $A$ be an $m \times n$ matrix over $F$ and let $b \in F^m$.
The [[system of linear equations|0GT4WWGWH2]] $Ax = b$ in the unknown $x \in F^n$ is *overdetermined* if $m > n$, that is, if it has more equations than unknowns.

## Remarks

Example: the system $x = 1$, $y = 2$, $x + y = 3$ over $\R$ has $m = 3$ equations and $n = 2$ unknowns, so it is overdetermined; it is consistent, with the solution $(x, y) = (1, 2)$.
Non-example: the system $x + y = 1$, $x - y = 0$ has $m = n = 2$, so it is not overdetermined.
If $m > n$, then the [[column space|03AJKXH1AV]] of $A$ has dimension at most $n < m$, so it is a proper subspace of $F^m$.
Hence there are right-hand sides $b$ for which the system is [[inconsistent|0X6V9B1V10]].
Over $\R$, the consistent right-hand sides form a proper subspace of $\R^m$, so a randomly chosen $b$ almost never gives a consistent system.
Over $\R$, the [[least squares|0BYTQ4X1HC]] solutions are the solutions of the normal equations $A^{T} A x = A^{T} b$, which are always consistent.
A system with fewer equations than unknowns is called *underdetermined*.
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# Injective object

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
An object $I$ of $\mathcal{C}$ is *injective* if, for every [[monomorphism|Y2NP1JFR86]] $m \colon A \to B$ and every [[morphism|859F0TFVA3]] $f \colon A \to I$, there is a morphism $g \colon B \to I$ with $g \circ m = f$.

## Remarks

Equivalently, the contravariant [[hom functor|VJXS8AK67E]] $\Hom(-, I)$ sends monomorphisms to surjections of sets.
In an [[abelian category|Y9CZ3MTNF1]], $I$ is injective if and only if $\Hom(-, I)$ is exact, and if and only if every short exact sequence $0 \to I \to B \to C \to 0$ splits.
In the category of modules over a ring, the injective objects are the [[injective modules|85H7PRHDPZ]].
An injective object of $\mathcal{C}$ is the same as a projective object of the opposite category.

Example: $\Q$ is injective in the category of abelian groups.
It is a [[divisible group|ZXVBEPMTCN]], and by Baer's criterion a divisible abelian group is injective: a map $f \colon n\Z \to \Q$ extends to $\Z$ by sending $1$ to $f(n)/n$.

Non-example: $\Z$ is not injective in the category of abelian groups.
For $m \colon \Z \to \Z$, $m(x) = 2x$, and $f = 1_{\Z}$, an extension $g$ would need $2 g(1) = g(m(1)) = f(1) = 1$, which has no solution in $\Z$.
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# Polycyclic group

## Summary

A polycyclic group is a group with a subnormal series whose factors are all cyclic.
Equivalently, it is a solvable group in which every subgroup is finitely generated.
Polycyclic groups form a well-behaved class of infinite groups: they are finitely presented, and many algorithmic questions about them can be decided.

## Explanations

### L1 human

A polycyclic group is built by stacking cyclic groups one on top of another.
Each layer is generated by a single element, so the whole group is described by finitely many elements and the way they act on each other.
The group $\Z^2$ is polycyclic: the series $0 \le \Z \times 0 \le \Z^2$ has factors $\Z$ and $\Z$.
Every finite solvable group is polycyclic, but the additive group of the rational numbers is not, since it is not finitely generated.
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# Paraproduct

## Summary

A paraproduct is the part of a product $fg$ in which the high frequencies of $f$ are multiplied by the low frequencies of $g$.
Using the [[Littlewood–Paley decomposition|D1PAT0K446]], every product splits as $fg = \Pi_g f + \Pi_f g + R(f, g)$, two paraproducts and a remainder in which both factors have comparable frequencies.
Paraproducts are bilinear operators that behave much better than the full product in many function spaces.
They appear in Bony's paradifferential calculus for nonlinear PDE and in the T(1) theorem, where they are the model non-convolution Calderón–Zygmund operators.

## Explanations

### L1 human

When two functions are multiplied, their frequencies interact.
If $f$ oscillates fast and $g$ varies slowly, the product $fg$ oscillates at about the same frequencies as $f$, with an amplitude modulated by $g$.
The paraproduct $\Pi_g f$ collects exactly these interactions, frequency band by frequency band.
For example, if $g$ is a constant $c$, then the paraproduct $\Pi_g f$ is just $c f$.
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# Cyclic homology

## Definition

Let $k$ be a field of characteristic $0$ and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$ with identity.
Tensor products are over $k$.
For $n \ge 0$ let $t \colon A^{\otimes (n+1)} \to A^{\otimes (n+1)}$ be the cyclic operator
$$t(a_0 \otimes a_1 \otimes \cdots \otimes a_n) = (-1)^n \, a_n \otimes a_0 \otimes \cdots \otimes a_{n-1},$$
and let $C^\lambda_n(A) = A^{\otimes (n+1)} / \im(1 - t)$.
The Hochschild boundary $b$ of the complex $C_\bullet(A, A)$ of [[Hochschild homology|S5H153HFWC]] maps $\im(1 - t)$ into $\im(1 - t)$, so it induces a boundary on $C^\lambda_\bullet(A)$.
The *cyclic homology* of $A$ is $HC_n(A) = H_n(C^\lambda_\bullet(A), b)$, the homology of this Connes complex.

## Remarks

Example: in degree $0$, $C^\lambda_0(A) = A$ and $b(a_0 \otimes a_1) = a_0 a_1 - a_1 a_0$, so $HC_0(A) = A/[A, A] = HH_0(A)$.
Example: for $A = k$, the space $C^\lambda_n(k)$ is $k$ for $n$ even and $0$ for $n$ odd, since then $t = -1$ and $1 - t = 2$ is invertible.
The boundary vanishes on it, so $HC_n(k) = k$ for even $n \ge 0$ and $HC_n(k) = 0$ for odd $n$.
Over a general ground ring, cyclic homology is defined from Connes' bicomplex instead, and the two definitions agree when $k$ contains $\Q$.
There is a long exact sequence of Connes relating the two theories:
$$\cdots \to HH_n(A) \to HC_n(A) \to HC_{n-2}(A) \to HH_{n-1}(A) \to \cdots.$$
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# Borel–Kolmogorov paradox

## Counterexample

Let $P$ be the uniform [[probability measure|059TJN0NZ1]] on the unit [[sphere|P9CM4GQ2T0]] $S^2 \subset \R^3$.
Describe a point by its longitude $\lambda \in (-\pi, \pi]$ and its latitude $\varphi \in [-\pi/2, \pi/2]$, as in [[spherical coordinates|01QFQAQ2PA]].
The area element of $S^2$ is $\cos \varphi \, d\lambda \, d\varphi$, so under $P$ the pair $(\lambda, \varphi)$ has the [[density|0ZY9DX8FBF]] $\frac{1}{4\pi} \cos \varphi$.
Hence $\lambda$ and $\varphi$ are independent, $\lambda$ is uniform on $(-\pi, \pi]$, and $\varphi$ has density $\frac{1}{2} \cos \varphi$.

Condition on the event $\{\varphi = 0\}$, the equator.
The [[conditional distribution|0ZN9EZTEM6]] of $\lambda$ given $\varphi = 0$ is the law of $\lambda$, which is uniform.
So the point is uniform on the equator.

Now let $\Theta = \lambda \bmod \pi \in [0, \pi)$ and condition on the event $\{\Theta = 0\}$, the great circle through the two poles.
Since $\Theta$ is independent of $\varphi$, the conditional distribution of $\varphi$ given $\Theta = 0$ still has density $\frac{1}{2} \cos \varphi$.
So the point is not uniform on this great circle: it is rarely near the poles.

A rotation of the sphere maps the equator onto the great circle through the poles and preserves $P$.
Yet the two conditional laws on these two great circles are not images of each other under the rotation.

## Remarks

Both events have probability $0$.
The conditional law given a null event depends on the random variable whose level set it is, not on the event alone.
A [[regular conditional probability|PHHT5W3BEJ]] given $\varphi$, or given $\Theta$, is defined only up to null sets of values, so it fixes no answer for a single null event.

## Explanations

### L1 human

Pick a point at random on the surface of the Earth, and suppose you are told that it lies on a given great circle.
One may expect the point to be uniform on that circle.
If the circle is seen as a line of constant latitude (the equator), the answer is uniform.
If the same kind of circle is seen as a line of constant longitude (a meridian circle), points near the poles are less likely, because meridians crowd together there.
The lesson is that one cannot condition on an event of probability zero by itself.
One must say which family of events it belongs to, and this is what a [[regular conditional probability|PHHT5W3BEJ]] does.
</document>

<document source="https://tamnd.github.io/math2/node/9RN0KYXNT1.md" tag="9RN0KYXNT1" kind="statement">
# Łoś–Tarski preservation theorem

## Statement

Let $\sigma$ be a first-order signature and let $T$ be a first-order [[theory|92SWXST1ZM]] in $\sigma$.
Then the following are equivalent.

1. The class of [[models|MVCH6N9SNW]] of $T$ is closed under [[substructures|676H7EGTKJ]]: if $B \models T$ and $A$ is a substructure of $B$, then $A \models T$.
2. There is a set $U$ of universal sentences of $\sigma$ with the same models as $T$; so the models of $T$ form a [[universal class|ZSTEZ3PRT9]].

## Remarks

The direction from 2 to 1 holds because universal sentences pass from a structure to its substructures.
For the other direction, let $U$ be the set of universal sentences that follow from $T$, and let $A \models U$.
Expand the signature by a constant for each element of $A$, and let $D$ be the set of quantifier-free sentences true in $A$, its diagram.
If $T \cup D$ had no model, then by the [[compactness theorem|1NR3HAWG9D]] $T$ would prove the negation of a finite conjunction from $D$.
Replacing the new constants by variables gives a universal sentence in $U$ that fails in $A$.
So $T \cup D$ has a model $B$, into which $A$ embeds, and condition 1 gives $A \models T$.

A version for formulas holds too: a formula is preserved by substructures between models of $T$ exactly when it is equivalent modulo $T$ to a universal formula.
Dually, the class of models of $T$ is closed under extensions exactly when $T$ is equivalent to a set of existential sentences.

Example: the theory of groups in the signature $(\cdot, {}^{-1}, e)$ is closed under substructures, and the group axioms are indeed universal.
In the signature $(\cdot)$ alone, a substructure of a group is only a subsemigroup, such as $\N_{>0}$ in $(\Z, +)$, and the axioms of groups in this signature need existential quantifiers.

The theorem uses compactness, and it fails when only finite structures are considered.

## Explanations

### L1 human

A universal axiom says that something holds for all elements, without asking for any element to exist.
Such axioms survive passing to a part of a structure.
The theorem says the converse: if a first-order axiom system always survives passing to parts, then it can be rewritten with universal axioms only.
So a semantic property, closure under substructures, matches a syntactic one, the shape of the axioms.

### L2 human

The Łoś–Tarski theorem is the first of the preservation theorems of model theory.
Others in the same spirit are the Chang–Łoś–Suszko theorem, for unions of chains and $\forall\exists$ sentences, and Lyndon's theorem, for homomorphic images and positive sentences.
In universal algebra it explains why an elementary class of algebras closed under subalgebras has a universal axiomatization.
This is a first step towards the descriptions of quasivarieties and varieties.
The proof method, realizing the diagram of a structure in a model of $T$, is the method of diagrams used throughout model theory.
</document>

<document source="https://tamnd.github.io/math2/node/9RNQX1X11T.md" tag="9RNQX1X11T" kind="definition">
# Jacobi transform

## Definition

Fix real numbers $\alpha > -1$ and $\beta > -1$, and let $w(x) = (1 - x)^\alpha (1 + x)^\beta$ for $-1 < x < 1$.
Let $P_n^{(\alpha, \beta)}$ be the [[Jacobi polynomial|TRD7DSF2KR]] of degree $n$.
Let $F \colon (-1, 1) \to \C$ be a function such that $w F$ is integrable on $(-1, 1)$.
The *Jacobi transform* of $F$ is the sequence
$$f(n) = \int_{-1}^{1} w(x)\, P_n^{(\alpha, \beta)}(x)\, F(x)\, dx, \qquad n = 0, 1, 2, \dots$$

## Remarks

Let $h_n = \int_{-1}^{1} w(x)\, P_n^{(\alpha, \beta)}(x)^2\, dx > 0$.
For $F$ in the weighted space $L^2(w)$, the inverse transform is the expansion $F = \sum_{n \ge 0} \frac{f(n)}{h_n} P_n^{(\alpha, \beta)}$, which converges in $L^2(w)$.

Let $R[F] = w^{-1} \frac{d}{dx}\left( (1 - x)^{\alpha + 1} (1 + x)^{\beta + 1} F' \right)$.
If $F$ is smooth enough and the boundary terms vanish, two integrations by parts and the Jacobi differential equation give that the transform of $R[F]$ is $-n(n + \alpha + \beta + 1) f(n)$.

Example: $F = P_m^{(\alpha, \beta)}$ has $f(n) = 0$ for $n \ne m$ and $f(m) = h_m$, by orthogonality.

Non-example: for $\alpha = -1$ the weight $(1 - x)^{-1}(1 + x)^\beta$ is not integrable near $x = 1$, so already $F = 1$ has no transform; this is why $\alpha, \beta > -1$ is required.

For $\alpha = \beta = \lambda - 1/2$ one gets, up to constant factors, the [[Gegenbauer transform|TMNNRRRGAS]].
</document>

<document source="https://tamnd.github.io/math2/node/9RP82GD0Q4.md" tag="9RP82GD0Q4" kind="example">
# Lattice of partitions

## Example

Let $n \ge 1$ and let $\Pi_n$ be the set of all [[partitions|08QMKTWF9T]] of $\set{1, \ldots, n}$.
Order $\Pi_n$ by [[refinement|0SPZ1H3N9V]]: $\sigma \le \tau$ when each block of $\sigma$ lies inside some block of $\tau$.
This is a partial order, and $\Pi_n$ is a [[lattice|WH4SFAACAS]].
The meet $\sigma \wedge \tau$ has as blocks the nonempty intersections $B \cap C$ of a block $B$ of $\sigma$ and a block $C$ of $\tau$.
The join $\sigma \vee \tau$ is the finest partition coarser than both.
In it, two points share a block when a chain of blocks of $\sigma$ and $\tau$, each meeting the next, connects them.
The least element is the partition into singletons, and the greatest element is the partition with one block.
The number of elements of $\Pi_n$ is the [[Bell number|0TKP3TDD02]] $B_n$.

Check for $n = 3$.
The five partitions are $1|2|3$, $12|3$, $13|2$, $23|1$ and $123$.
The three partitions with one pair lie strictly between the bottom and the top, and no two of them are comparable.
The meet of any two of them is $1|2|3$ and their join is $123$, so $\Pi_3$ is the [[diamond lattice|JQ6FJH2TP8]] $M_3$.

## Remarks

The lattice $\Pi_n$ is graded: the rank of a partition with $k$ blocks is $n - k$.
It is a [[geometric lattice|VDF7CGPC0P]], but for $n \ge 3$ it is not distributive, since it contains $M_3$.
The Möbius function of $\Pi_n$ from the bottom to the top equals $(-1)^{n-1} (n-1)!$.

## Explanations

### L1 human

Think of ways to split a group of $n$ people into teams.
One splitting is below another when you get the second by merging some teams of the first.
Any two splittings have a best common refinement and a coarsest common merge, so the splittings form a lattice.
For three people there are five splittings, and the three ways to form one pair sit side by side in the middle.
</document>

<document source="https://tamnd.github.io/math2/node/9RPYB7S4HM.md" tag="9RPYB7S4HM" kind="definition">
# Wilcoxon signed-rank test

## Definition

Let $Z_1, \dots, Z_n$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with a continuous distribution that is symmetric about a number $\theta$, that is, $Z_1 - \theta$ and $\theta - Z_1$ have the same law.
Let $R_i$ be the rank of $\abs{Z_i}$ among $\abs{Z_1}, \dots, \abs{Z_n}$, so that the smallest has rank $1$.
The *signed-rank statistic* is
$$W^+ = \sum_{i \,:\, Z_i > 0} R_i.$$
The *Wilcoxon signed-rank test* of $H_0 \colon \theta = 0$ rejects $H_0$ when $W^+$ is too large, too small, or either, according to the alternative.
Under $H_0$, $W^+$ has the law of $\sum_{k=1}^{n} k B_k$, where $B_1, \dots, B_n$ are independent [[Bernoulli|0J97P5KZY0]] variables with parameter $1/2$.

## Remarks

Under $H_0$, $E[W^+] = n(n+1)/4$ and $\operatorname{Var}(W^+) = n(n+1)(2n+1)/24$, and $W^+$ is asymptotically normal.
For matched pairs $(X_i, Y_i)$ one takes $Z_i = Y_i - X_i$.

Example: $Z = (1.2, -0.4, 2.5)$.
The absolute values $1.2, 0.4, 2.5$ have ranks $2, 1, 3$, so $W^+ = 2 + 3 = 5$.
Under $H_0$ each of the $8$ sign patterns has probability $1/8$, and only the sets of ranks $\set{2, 3}$ and $\set{1, 2, 3}$ give a sum at least $5$.
So the one-sided p-value is $P(W^+ \ge 5) = 2/8 = 1/4$.

Non-example: if the $Z_i$ have median $0$ but a skewed law, the clause on symmetry fails.
Then the null law above is wrong, and the test can reject often even though the median is $0$; the [[sign test|NBK6G2VB7A]] is valid in that case.

Inverting the test gives the [[Hodges–Lehmann estimator|T6KRQAQQ15]] of $\theta$.
</document>

<document source="https://tamnd.github.io/math2/node/9RS2YPTPJY.md" tag="9RS2YPTPJY" kind="definition">
# Deligne–Lusztig theory

## Definition

Let $p$ be a prime, let $q$ be a power of $p$, and let $\overline{\F}_q$ be an [[algebraic closure|0X6AB5P3G1]] of the [[finite field|0ZXZ82FTX2]] $\F_q$.
Let $\mathbf{G}$ be a connected [[reductive group|2XS84PWE3S]] over $\overline{\F}_q$ with a Frobenius endomorphism $F$ for an $\F_q$-structure.
Let $G = \mathbf{G}^F$ be the finite group of fixed points of $F$.
Let $\mathbf{T}$ be an $F$-stable [[maximal torus|3JVGZVVTB0]] of $\mathbf{G}$, and let $T = \mathbf{T}^F$.
Let $\mathbf{B}$ be a [[Borel subgroup|5JSYA1JNCC]] containing $\mathbf{T}$, not necessarily $F$-stable, with unipotent radical $\mathbf{U}$.
The Deligne–Lusztig variety is
$$\tilde{X} = \set{g \in \mathbf{G} : g^{-1} F(g) \in F(\mathbf{U})}.$$
The group $G$ acts on $\tilde{X}$ by left multiplication and $T = \mathbf{T}^F$ acts by right multiplication.
Let $\ell \ne p$ be a prime and let $\theta \colon T \to \overline{\Q}_\ell^\times$ be a character.
Let $H^i_c(\tilde{X})_\theta$ be the subspace of the $\ell$-adic cohomology with compact support $H^i_c(\tilde{X}, \overline{\Q}_\ell)$ on which $T$ acts through $\theta$.
The *Deligne–Lusztig virtual character* is the function on $G$
$$R_{\mathbf{T}}^\theta(g) = \sum_{i \ge 0} (-1)^i \tr\big(g, H^i_c(\tilde{X})_\theta\big).$$
*Deligne–Lusztig theory* is the study of the representations of $G$ through the virtual characters $R_{\mathbf{T}}^\theta$.

## Remarks

The virtual character $R_{\mathbf{T}}^\theta$ does not depend on the choice of $\mathbf{B}$.
Every irreducible [[character|RMWMYBY8TC]] of $G$ occurs with nonzero multiplicity in some $R_{\mathbf{T}}^\theta$.
Its degree is $R_{\mathbf{T}}^\theta(1) = \pm [G : T]_{p'}$, the part of the index prime to $p$, up to a sign.
Example: let $\mathbf{G} = GL_2$ with $F$ raising matrix entries to the $q$-th power, let $\mathbf{T}$ be the diagonal torus and $\mathbf{B}$ the upper triangular matrices, both $F$-stable.
Then $R_{\mathbf{T}}^\theta$ is induced from the upper triangular subgroup $B = \mathbf{B}^F$ of $GL_2(\F_q)$, by the character of $B$ that equals $\theta$ on $T$ and is $1$ on $\mathbf{U}^F$.
For $\theta = 1$ it has degree $q + 1 = [G : B]$, and it is the sum of the trivial and the Steinberg character.
Second example: for $\mathbf{G} = GL_2$ there is also an $F$-stable maximal torus $\mathbf{T}$ with $T \cong \F_{q^2}^\times$, contained in no $F$-stable Borel subgroup.
For a character $\theta$ of $\F_{q^2}^\times$ with $\theta \ne \theta^q$, the function $-R_{\mathbf{T}}^\theta$ is an irreducible cuspidal character of degree $q - 1$, which is not induced from any proper parabolic subgroup.
This shows why the cohomological construction is needed.
</document>

<document source="https://tamnd.github.io/math2/node/9RVKQJQ5JJ.md" tag="9RVKQJQ5JJ" kind="example">
# Contact process

## Example

Let $d \ge 1$ and $\lambda > 0$.
For $x \in \Z^d$ and $\eta \in \set{0, 1}^{\Z^d}$, let
$$c(x, \eta) = \begin{cases} 1 & \text{if } \eta(x) = 1, \\ \lambda \, \#\set{y \in \Z^d : \abs{y - x} = 1, \ \eta(y) = 1} & \text{if } \eta(x) = 0. \end{cases}$$
The *contact process* on $\Z^d$ with infection rate $\lambda$ is the spin system on $\set{0, 1}^{\Z^d}$ with these flip rates.
Here $\eta(x) = 1$ means that site $x$ is infected.
An infected site recovers at rate $1$, and a healthy site is infected at rate $\lambda$ times its number of infected neighbours.
The rates are bounded by $\max(1, 2d\lambda)$ and each depends on $2d + 1$ sites, so it is an [[interacting particle system|GR49AM3PSZ]].

## Remarks

The configuration with no infected site is absorbing.
There is a critical value $\lambda_c \in (0, \infty)$: started from one infected site, the infection dies out almost surely for $\lambda < \lambda_c$ and survives forever with positive probability for $\lambda > \lambda_c$.
A lower bound is $\lambda_c \ge 1/(2d)$.
Indeed, the number of infected sites is dominated by a branching process in which each individual dies at rate $1$ and gives birth at rate $2d\lambda$.
This branching process dies out almost surely when $2d\lambda \le 1$.

## Explanations

### L1 human

It is a model of an epidemic on a grid.
Sick people heal at a fixed rate and pass the illness to healthy neighbours at rate $\lambda$ each.
If $\lambda$ is small the epidemic dies out; if $\lambda$ is large it may last forever.
</document>

<document source="https://tamnd.github.io/math2/node/9RWS7TZHXH.md" tag="9RWS7TZHXH" kind="concept">
# Subfactor

## Summary

A subfactor is an inclusion $N \subseteq M$ of factors, von Neumann algebras with trivial center, that share the same identity.
For factors of type $\mathrm{II}_1$, Vaughan Jones attached to such an inclusion a number $[M:N]$, the index.
It measures the relative size of $M$ over $N$.
The [[Jones index theorem|31N0FH22ZT]] shows that the possible index values are surprisingly restricted.
Subfactor theory connects operator algebras with knot invariants, quantum groups, tensor categories and conformal field theory.

## Explanations

### L1 human

A subfactor is like a field extension or a subgroup, but for operator algebras.
For a subgroup $H$ of a finite group $G$, the index $[G:H]$ counts how many copies of $H$ fit into $G$.
For a subfactor, the index is a real number that plays the same role.
For example, let $N$ be a $\mathrm{II}_1$ factor and look at the $k \times k$ matrices with entries in $N$.
The diagonal matrices with all diagonal entries equal form a copy of $N$ inside them, and the index is $k^2$.
The surprise is that the index need not be an integer.
</document>

<document source="https://tamnd.github.io/math2/node/9S18DSRFND.md" tag="9S18DSRFND" kind="statement">
# Uniqueness of best approximation in strictly convex spaces

## Statement

Let $X$ be a [[strictly convex|GDTKW9KPH8]] normed space, let $C \subseteq X$ be a nonempty [[convex set|02AS45T3H9]], and let $x \in X$.
Then $x$ has at most one [[best approximation|7N4988HN3P]] from $C$.
In particular, if $V \subseteq X$ is a finite-dimensional linear subspace, every $x \in X$ has exactly one best approximation from $V$.

## Remarks

The idea of the proof: let $d = \inf_{c \in C} \norm{x - c}$ and suppose $c_1 \ne c_2$ in $C$ both have $\norm{x - c_i} = d$.
Then $d > 0$, since otherwise $c_1 = x = c_2$.
The midpoint $m = (c_1 + c_2)/2$ lies in $C$, and $x - m$ is the midpoint of $x - c_1$ and $x - c_2$, two different vectors of norm $d$.
Strict convexity gives $\norm{x - m} < d$, which contradicts the definition of $d$.
The second sentence follows with the existence of best approximations from finite-dimensional subspaces.

Example: every Hilbert space and every $L^p$ space with $1 < p < \infty$ is strictly convex, so least squares and $L^p$ polynomial approximations are unique.

Strict convexity cannot be dropped.
In $\R^2$ with the maximum norm, every point $(t, 0)$ with $\abs{t} \le 1$ is a best approximation of $(0, 1)$ from the horizontal axis.
The space $C[a, b]$ with the supremum norm is not strictly convex either; there uniqueness of best polynomial approximation needs the Haar condition instead.

Convexity of $C$ is also needed: from the centre of a circle in the Euclidean plane, all points of the circle are nearest.

## Explanations

### L1 human

If the unit ball of a space has no flat pieces on its boundary, then from any point there cannot be two equally near points in a convex set.
If there were, the point halfway between them would be strictly nearer.
This is why least squares problems have unique solutions.

### L2 human

The theorem separates the geometry of the norm from the approximating set.
For finite-dimensional subspaces, it says that strictly convex spaces make every such subspace a Chebyshev set.
In non-strictly convex spaces such as $C(K)$ and $L^1$, uniqueness depends on the subspace, and Haar's theorem characterizes the finite-dimensional subspaces of $C(K)$ with unique best approximations.
</document>

<document source="https://tamnd.github.io/math2/node/9SDYXHB8P6.md" tag="9SDYXHB8P6" kind="definition">
# Brelot space

## Definition

Let $X$ be a connected, locally connected, [[locally compact|0FE2SB8WVJ]] Hausdorff space.
Let $\mathcal{H}$ assign to each open $U \subseteq X$ a linear subspace $\mathcal{H}(U)$ of the real [[continuous functions|0X3WGX95FE]] on $U$.
Call an open set *regular* as in a [[harmonic space|DWN0XEQHHK]].
The pair $(X, \mathcal{H})$ is a *Brelot space* if:

1. $\mathcal{H}$ is a [[sheaf|TB6TM6R4MD]] of functions.
2. The regular domains, that is, the connected regular open sets, form a base of the topology of $X$.
3. Let $U \subseteq X$ be a domain and $(h_n)$ an increasing sequence in $\mathcal{H}(U)$. Then $\lim_n h_n$ either lies in $\mathcal{H}(U)$ or is $+\infty$ at every point of $U$.

## Remarks

Clause 3 implies the convergence axiom of a harmonic space: a locally bounded limit cannot be identically $+\infty$ on any component.
Example: $\R^n$ with the classical harmonic functions is a Brelot space.
Balls are regular domains, by the Poisson integral, and clause 3 is [[Harnack's principle|P15K8RKHDV]].
Non-example: $\R \times \R$ with the solutions of the [[heat equation|VP2HN86VAG]] $\partial_t u = \partial_x^2 u$ fails clause 3.
Let $\Phi(x, t) = (4\pi t)^{-1/2} e^{-x^2/(4t)}$ for $t > 0$ and $\Phi(x, t) = 0$ for $t \le 0$, the [[heat kernel|KD404MF042]] extended by $0$.
It solves the heat equation on the domain $U = \R^2 \setminus \set{(0, 0)}$, and $h_n = n \Phi$ increases to $+\infty$ where $t > 0$ and to $0$ where $t \le 0$.
The limit is neither a solution on $U$ nor identically $+\infty$.
</document>

<document source="https://tamnd.github.io/math2/node/9SEF1CWT6X.md" tag="9SEF1CWT6X" kind="definition">
# Topological semigroup

## Definition

A *topological semigroup* is a [[semigroup|0XMEM406JM]] $S$ together with a [[topology|0CQHFPGJ45]] on $S$ such that the multiplication $S \times S \to S$, $(x, y) \mapsto xy$, is a [[continuous multiplication|ZZK2GM1V15]].
That is, the map is continuous when $S \times S$ carries the product topology.

## Remarks

Example: $[0, 1]$ with ordinary multiplication and the topology from $\R$.
The product is associative, it maps $[0,1] \times [0,1]$ into $[0,1]$, and it is continuous as the restriction of a polynomial.
Non-example: the semigroup $(\Z, +)$ with the cofinite topology.
Addition is associative, but it is not jointly continuous: the preimage of $\Z \setminus \set{0}$ is not open, since every nonempty basic open set $U \times V$ contains a pair $(x, -x)$.
If one asks only that each translation $y \mapsto xy$ and $y \mapsto yx$ be continuous, the weaker structure is called a semitopological semigroup.
</document>

<document source="https://tamnd.github.io/math2/node/9SNTXYEEBQ.md" tag="9SNTXYEEBQ" kind="concept">
# Equivalence of varieties

## Summary

Two [[varieties|AJNZJD8442]] are equivalent, or term equivalent, when each can be interpreted in the other and the two translations undo each other.
Equivalent varieties have the same algebras up to a change of the basic operations, so they have the same subalgebras, congruences and term operations.
The standard example is the equivalence between Boolean algebras and Boolean rings.

## Explanations

### L1 human

The same structure can often be described with different basic operations.
A Boolean algebra uses "or", "and" and "not", while a Boolean ring uses "exclusive or" and "and".
Each set of operations can be written in terms of the other, and translating there and back gives the original operations.
So the two descriptions are equivalent: they are two names for one kind of structure.
</document>

<document source="https://tamnd.github.io/math2/node/9SYWRGVTRS.md" tag="9SYWRGVTRS" kind="concept">
# Almost complex manifold

## Summary

An almost complex manifold is a smooth manifold whose tangent spaces carry a smoothly varying multiplication by $i$.
Every complex manifold is one, but the converse fails in general.
The [[Newlander–Nirenberg theorem|VCT848WV1T]] says exactly when an almost complex structure comes from complex coordinates.
Almost complex structures are basic in Hermitian, Kähler and symplectic geometry.

## Explanations

### L1 human

In the complex plane, multiplying by $i$ turns each arrow by a quarter turn.
An almost complex structure gives such a quarter turn $J$ on every tangent plane of a manifold, with $J$ applied twice giving minus the identity.
On $\R^2$ the rule $J(a, b) = (-b, a)$ is the quarter turn.
Only manifolds of even dimension can carry such a structure.
</document>

<document source="https://tamnd.github.io/math2/node/9SYZS6S4NY.md" tag="9SYZS6S4NY" kind="definition">
# Response surface methodology

## Definition

Let $x = (x_1, \dots, x_k)$ be the settings of $k$ controllable factors in a region $R \subseteq \R^k$.
Assume that the observed response is $y = \eta(x) + \varepsilon$, where $\eta \colon R \to \R$ is an unknown smooth function and $\varepsilon$ is a random error with mean $0$.
*Response surface methodology* is the sequential strategy for locating an optimum of $\eta$ that repeats these steps:

1. Run an experiment, planned by the methods of [[design of experiments|2TE42ZWJ3E]], in a small neighbourhood of the current point.
2. While curvature is negligible, fit by [[least squares|0BYTQ4X1HC]] a first-order model $\hat y = b_0 + \sum_i b_i x_i$. Then move the current point along the path of steepest ascent, the direction $(b_1, \dots, b_k)$.
3. When the first-order model shows lack of fit, fit a second-order model $\hat y = b_0 + \sum_i b_i x_i + \sum_{i \le j} b_{ij} x_i x_j$, and locate and classify its stationary point.

## Remarks

Example: a fitted second-order model can be written $\hat y = b_0 + x^T b + x^T B x$ with a symmetric $k \times k$ matrix $B$.
When $B$ is invertible, its stationary point is $x_s = -\frac12 B^{-1} b$, since the gradient $b + 2 B x$ vanishes there.
The [[Hessian matrix|0693Y2METT]] of $\hat y$ is $2B$, so if all eigenvalues of $B$ are negative, $x_s$ is the maximum of the fitted surface.
This eigenvalue check is called canonical analysis.

Non-example: a single [[factorial experiment|EHJ3R589ZX]] analysed once, with no move to new settings, is not response surface methodology, because step 2 is missing.

Typical designs are two-level factorial designs with centre points for the first-order stage, and central composite designs or Box–Behnken designs for the second-order stage.
</document>

<document source="https://tamnd.github.io/math2/node/9T2XVJNBBY.md" tag="9T2XVJNBBY" kind="concept">
# Entropic algebra

## Summary

An entropic algebra is an [[algebra|QJGPT2W0FT]] in which any two operations commute with each other.
Equivalently, every operation $A^n \to A$ is a [[homomorphism|KBXSZC6BV1]] of algebras from the power $A^n$ to $A$.
For an algebra with one binary operation this is the [[medial law|VZHN6GA3TA]].
Modules over a commutative ring, semilattices and affine spaces are entropic.
In an entropic variety, the homomorphisms from one algebra $A$ to another algebra $B$ form a subalgebra of the power $B^A$.

## Explanations

### L1 human

Fill a rectangle with elements, $n$ rows and $m$ columns.
Take two operations, one with $n$ arguments and one with $m$ arguments.
You can apply the second operation along each row and then the first to the results.
Or you can apply the first down each column and then the second to the results.
An algebra is entropic when these two ways always agree.
For a vector space this says, for instance, that $r(a + b) = ra + rb$ and $r(sa) = s(ra)$.
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<document source="https://tamnd.github.io/math2/node/9T6ZEV4125.md" tag="9T6ZEV4125" kind="concept">
# Single layer potential

## Summary

A single layer potential is the potential produced by a charge spread over a surface with a given density.
It is the integral of the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] against the density times surface measure on the boundary of a domain.
It is [[harmonic|0DEB0JMN0W]] off the surface and continuous across it, while its normal derivative jumps by the density, as the [[jump relations|J58YA8QNNX]] say.
Together with the [[double layer potential|EG8Z3FWV9A]], it turns boundary value problems for the Laplace equation into integral equations on the boundary.

## Explanations

### L1 human

Paint electric charge onto a thin shell, with more charge in some places and less in others.
The electric potential that this painted charge creates in space is a single layer potential.
Away from the shell it is a smooth harmonic function, and when you cross the shell its value does not jump, but its slope does.
For a sphere in space with charge spread evenly, the potential is constant inside the sphere and falls off like $1/\abs{x}$ outside it.
</document>

<document source="https://tamnd.github.io/math2/node/9T7D9ZRWYY.md" tag="9T7D9ZRWYY" kind="statement">
# Liouville's theorem on phase-space volume

## Statement

Let $U \subseteq \R^{2n}$ be open, with coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$, and let $H \colon U \to \R$ be twice continuously differentiable.
Let $\varphi_t$ be the [[flow|06738GGWWK]] of the [[Hamiltonian system|0EDKGA2FAK]]
$$q_i' = \frac{\partial H}{\partial p_i}, \qquad p_i' = -\frac{\partial H}{\partial q_i}, \qquad i = 1, \ldots, n.$$
Let $D \subseteq U$ be a measurable set and $t \in \R$ such that $\varphi_t(x)$ is defined for every $x \in D$.
Then $\varphi_t(D)$ has the same [[Lebesgue measure|2Z2QVWZ262]] as $D$.

## Remarks

The vector field $X = (\partial H / \partial p, -\partial H / \partial q)$ has [[divergence|0MWV6ZTK6S]]
$$\sum_{i=1}^n \left( \frac{\partial^2 H}{\partial q_i \partial p_i} - \frac{\partial^2 H}{\partial p_i \partial q_i} \right) = 0.$$
The derivative in $t$ of the volume of $\varphi_t(D)$ is the integral of this divergence over $\varphi_t(D)$, so the volume is constant.
In symplectic language, the flow of a [[Hamiltonian vector field|C5EH3B4BEP]] preserves the symplectic form $\omega$, hence also the volume form $\omega^n / n!$.

Example: for $H = \frac{1}{2}(q^2 + p^2)$ the flow rotates the plane, and rotations preserve area.
For a damped oscillator $q' = p$, $p' = -q - p$ the divergence is $-1$, so areas shrink like $e^{-t}$; this system is not Hamiltonian.

An equivalent form concerns a density $\rho(t, q, p)$ carried by the flow: it satisfies $\frac{\partial \rho}{\partial t} + \{\rho, H\} = 0$, where $\{\cdot, \cdot\}$ is the [[Poisson bracket|2S0MXDT4YH]].

## Explanations

### L1 human

Take a blob of possible initial states of a frictionless mechanical system and let each state move by the equations of motion.
The blob changes shape, often wildly, but its volume in phase space never changes.
This is the reason why a Hamiltonian system cannot have an attracting fixed point.

### L2 human

The theorem is the first sign that Hamiltonian flows preserve geometric structure: they preserve the symplectic form, and volume is its top power.
It is the base of statistical mechanics, since it shows that the Liouville measure is invariant.
With the Poincaré recurrence theorem it shows that if the motion stays in a region of finite volume, then almost every state returns arbitrarily close to where it started.
In dimension $2n \ge 4$, preserving volume is much weaker than preserving $\omega$; [[Gromov's non-squeezing theorem|RDX55HQ3XT]] shows the difference.
</document>

<document source="https://tamnd.github.io/math2/node/9TBVX1RHDB.md" tag="9TBVX1RHDB" kind="definition">
# Non-uniform rational B-spline

## Definition

Let $k \ge 1$ be an order, let $t_1 \le t_2 \le \cdots \le t_{n+k}$ be a knot vector with $t_k < t_{n+1}$, and let $B_{1,k}, \dots, B_{n,k}$ be the [[B-splines|62AFX32JGT]] of order $k$ for these knots.
Let $P_1, \dots, P_n \in \R^d$ be control points and let $w_1, \dots, w_n > 0$ be weights.
The *NURBS curve* with these data is the map $C \colon [t_k, t_{n+1}] \to \R^d$,
$$C(u) = \frac{\sum_{i=1}^{n} w_i\, B_{i,k}(u)\, P_i}{\sum_{i=1}^{n} w_i\, B_{i,k}(u)}.$$
A *NURBS surface* is defined in the same way with tensor products $B_{i,k}(u) B_{j,l}(v)$, control points $P_{ij}$ and positive weights $w_{ij}$.

## Remarks

The denominator is positive on $[t_k, t_{n+1})$, because the B-splines are nonnegative and do not all vanish there; so $C(u)$ is a convex combination of the control points.
At $u = t_{n+1}$, where the right-continuous B-splines may all vanish, $C$ is defined by its limit from the left, as usual.
In [[homogeneous coordinates|FS9FRJ17D2]], $C$ is the central projection of the polynomial B-spline curve with control points $(w_i P_i, w_i)$ in $\R^{d+1}$.
If all weights are equal, $C$ is the ordinary B-spline curve $\sum_i B_{i,k}(u) P_i$.
Example: take $k = 3$, knots $0, 0, 0, 1, 1, 1$, points $(1, 0)$, $(1, 1)$, $(0, 1)$ and weights $1$, $1/\sqrt{2}$, $1$.
Then $C$ is a rational quadratic [[Bézier curve|NEXBTGYYRG]], and $C(1/2) = (1/\sqrt{2}, 1/\sqrt{2})$.
In fact $C$ traces the quarter of the unit circle in the first quadrant.
Non-example: the same data with weight $w_2 = -1$ fail the positivity clause, and then the denominator $(1-u)^2 - 2u(1-u) + u^2 = (2u - 1)^2$ vanishes at $u = 1/2$, so $C$ is not defined on all of $[0, 1]$.
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<document source="https://tamnd.github.io/math2/node/9TFC6ADC1C.md" tag="9TFC6ADC1C" kind="concept">
# Compact Lie group

## Summary

A compact Lie group is a [[Lie group|7JBRGCVJRB]] whose underlying manifold is compact.
Examples are the circle, the tori, the orthogonal groups $\mathrm{O}(n)$ and the unitary groups $\mathrm{U}(n)$.
These groups carry a finite invariant measure, so averaging over the group works as for finite groups.
As a result their representation theory is as clean as that of finite groups.
Their structure is described by [[maximal tori|3JVGZVVTB0]], root systems and Weyl groups.

## Explanations

### L1 human

A compact Lie group is a group of symmetries that is also a smooth, closed and bounded shape.
The rotations of the plane form a circle, and the circle is the simplest example.
The rotations of space form the group $\mathrm{SO}(3)$, a compact space of dimension $3$.
Because the group is bounded, one can take the average of a function over all its elements.
This averaging trick makes many questions about these groups as easy as for finite groups.
</document>

<document source="https://tamnd.github.io/math2/node/9TPS3W497D.md" tag="9TPS3W497D" kind="statement">
# Schmidt subspace theorem

## Statement

Let $n \ge 2$, and let $L_1, \dots, L_n$ be [[linearly independent|0CRMGPAH83]] [[linear forms|07YQ1C78G0]] in $n$ variables whose coefficients are real [[algebraic numbers|0MNF0BYFQC]].
Let $\varepsilon > 0$.
For $x = (x_1, \dots, x_n) \in \Z^n$ write $\norm{x} = \max_i \abs{x_i}$.
Then the set of nonzero $x \in \Z^n$ with
$$\abs{L_1(x) \, L_2(x) \cdots L_n(x)} < \norm{x}^{-\varepsilon}$$
is contained in the union of finitely many proper linear subspaces of $\Q^n$.

## Remarks

For $n = 2$, take $L_1(x, y) = \alpha y - x$ and $L_2(x, y) = y$ with $\alpha$ a real algebraic irrational.
For $x/y$ close to $\alpha$, the norm $\norm{(x, y)}$ is at most a constant times $\abs{y}$.
Up to this constant, the inequality then reads $\abs{\alpha - x/y} < \abs{y}^{-2-\varepsilon}$, and the theorem gives the [[Thue–Siegel–Roth theorem|R9JZA3G2NX]].
The proof extends Roth's method with an auxiliary polynomial and uses the geometry of numbers.
The conclusion speaks of finitely many subspaces, not finitely many points: in the example above, all the points $(x, 0)$ satisfy the inequality, because $L_2$ vanishes on them.
Like Roth's theorem, the result is not effective.

## Explanations

### L1 human

Roth's theorem says that an algebraic number has few very good rational approximations.
The subspace theorem says the same for several algebraic linear forms at once.
The integer points where the product of the forms is very small do not spread out: they all lie on finitely many planes.

### L2 human

The theorem is the main tool for finiteness results on Diophantine equations in several variables, such as $S$-unit equations $x_1 + \dots + x_n = 1$.
It also gives transcendence results for numbers with special digit expansions.
Versions with several places, the $p$-adic subspace theorem, are used for these applications.
</document>

<document source="https://tamnd.github.io/math2/node/9TTJRYP841.md" tag="9TTJRYP841" kind="definition">
# Positive line bundle

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] and let $L \to X$ be a [[holomorphic line bundle|QAC7BYK9KX]].
For a [[Hermitian metric|Q6X6W2N29G]] $h$ on $L$, let $\Theta_h$ be the curvature of its [[Chern connection|7RC9CN2Q49]], a $(1,1)$-form on $X$.
In local coordinates $z_1, \dots, z_n$ write $i \Theta_h = i \sum_{j,k} a_{jk}\, dz_j \wedge d\bar z_k$; the matrix $(a_{jk})$ is Hermitian.
The line bundle $L$ is *positive* if some Hermitian metric $h$ on $L$ has $(a_{jk}(x))$ [[positive definite|0KX66CZMBB]] at every point $x \in X$.

## Remarks

If $e$ is a local holomorphic frame of $L$ and $\abs{e}_h^2 = e^{-\varphi}$, then $i \Theta_h = i \partial \bar\partial \varphi$.
So $L$ is positive exactly when some metric has local weights $\varphi$ that are strictly plurisubharmonic.
The form $\frac{i}{2\pi} \Theta_h$ represents the first Chern class $c_1(L)$, so positivity implies that $c_1(L)$ contains a Kähler form.

Example: $\mathcal{O}(1)$ on $\mathbb{P}^n$ with the metric dual to the standard metric on $\mathcal{O}(-1)$.
On $\set{z_0 \ne 0}$ with affine coordinates $z$, the dual frame $e^*$ has $\abs{e^*}^2 = (1 + \abs{z}^2)^{-1}$, so $\varphi = \log(1 + \abs{z}^2)$.
Then $i \partial \bar\partial \varphi$ is the Fubini–Study form, which is positive definite at each point.

Non-example: the trivial line bundle on a compact connected complex manifold $X$ of positive dimension.
Any metric is $\abs{1}_h^2 = e^{-\varphi}$ for a global smooth function $\varphi$.
At a maximum point of $\varphi$ the complex Hessian $(\partial^2 \varphi / \partial z_j \partial \bar z_k)$ is negative semidefinite, so it is not positive definite there.
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<document source="https://tamnd.github.io/math2/node/9V11N8VVTP.md" tag="9V11N8VVTP" kind="concept">
# Handle decomposition

## Summary

A handle decomposition builds a manifold from a ball, or from a given boundary, by attaching handles $D^k \times D^{n-k}$ one after another.
Each handle is glued along part of its boundary.
It is the smooth analogue of a CW structure: a $k$-handle plays the role of a $k$-cell, thickened to full dimension.
Every compact smooth manifold has a handle decomposition, obtained from a [[Morse function|KPQ96640ZJ]], with one $k$-handle for each critical point of index $k$.
Handle decompositions are the main tool in the proof of the h-cobordism theorem and in surgery theory.
In dimensions $3$ and $4$ they underlie Kirby calculus.

## Explanations

### L1 human

Build a surface out of clay: start with a disc, glue on flat strips along their two ends, and cap off with more discs.
The discs at the start are $0$-handles, the strips are $1$-handles, and the final caps are $2$-handles.
A torus can be built from one disc, two strips and one cap.
In higher dimensions the pieces are thickened discs of the same dimension as the manifold, glued along part of their boundary.
</document>

<document source="https://tamnd.github.io/math2/node/9V1GF5WMKA.md" tag="9V1GF5WMKA" kind="concept">
# Seifert fiber space

## Summary

A Seifert fiber space is a [[3-manifold|R20TB2KEQQ]] decomposed into disjoint circles, called fibers, that fit together locally like a standard fibered solid torus.
Almost all fibers are regular, and around each of the finitely many exceptional fibers the nearby fibers wind several times.
The space of fibers is a $2$-dimensional [[orbifold|RPAVY0W0XS]], and the manifold is classified by this base and a few integer invariants.
Seifert fiber spaces account for six of the eight Thurston geometries and are one of the two kinds of pieces in the JSJ decomposition.

## Explanations

### L1 human

Picture a $3$-dimensional space filled by circles, like a bundle of loops of string that never touch.
Near most circles, the neighbors run parallel to it, as in a product of a disk and a circle.
Near a few special circles, the neighbors wind around several times before closing up, like the strands of a twisted rope.
The product $S^2 \times S^1$ and the $3$-sphere, filled by the circles of the Hopf fibration, are examples.
</document>

<document source="https://tamnd.github.io/math2/node/9V3QA0KTW8.md" tag="9V3QA0KTW8" kind="concept">
# Shifted Legendre polynomials

## Summary

The shifted Legendre polynomials are the [[Legendre polynomials|QFYGJVA996]] moved from the interval $[-1, 1]$ to the interval $[0, 1]$.
They are given by $\tilde{P}_n(x) = P_n(2x - 1)$.
They are orthogonal on $[0, 1]$ for the constant weight.
So they are the natural polynomial basis for least squares and Gaussian quadrature on the unit interval.

## Explanations

### L1 human

Legendre polynomials live on the interval from $-1$ to $1$.
Many problems live on the interval from $0$ to $1$ instead, for example in probability.
The substitution $x \mapsto 2x - 1$ stretches and moves $[0, 1]$ onto $[-1, 1]$, and composing with it gives the shifted polynomials.
For example, $P_1(x) = x$ becomes $\tilde{P}_1(x) = 2x - 1$, which has average zero on $[0, 1]$.
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<document source="https://tamnd.github.io/math2/node/9V4PR1RVXW.md" tag="9V4PR1RVXW" kind="concept">
# Neighborly polytope

## Summary

A [[convex polytope|300K6CB1KH]] is $k$-neighborly when every set of $k$ of its [[vertices|R7V3S9V891]] spans a face.
A $d$-polytope is called neighborly when it is $\floor{d/2}$-neighborly, the largest value possible for a polytope that is not a simplex.
Neighborly polytopes exist in every dimension and with any number of vertices beyond $d$.
Neighborly simplicial polytopes have the largest possible numbers of faces, which is why they appear in the upper bound theorem.

## Explanations

### L1 human

In a polygon or a solid in space, two corners are usually not joined by an edge.
In four or more dimensions this can change: there are four-dimensional solids where every pair of corners is joined by an edge.
Such shapes are called neighborly, because every corner is a neighbour of every other.
In higher dimensions the same happens with triples, quadruples and so on of corners.
</document>

<document source="https://tamnd.github.io/math2/node/9V6PRD1Z90.md" tag="9V6PRD1Z90" kind="example">
# Representation theory of SU(2)

## Example

Let $G = \mathrm{SU}(2)$ be the [[group|E5P8AG6CK7]] of complex $2 \times 2$ unitary matrices of determinant $1$.
For each integer $n \ge 0$, let $V_n$ be the space of [[homogeneous polynomials|2C3PAQ7H50]] of degree $n$ in two complex variables $z = (z_1, z_2)$, of dimension $n + 1$.
Put $(\pi_n(g) P)(z) = P(g^{-1} z)$ for $g \in G$ and $P \in V_n$.
This is a continuous homomorphism $G \to \mathrm{GL}(V_n)$, so it is a [[representation|7JA7WH92K0]] of $G$.

The classification: each $V_n$ is [[irreducible|FX3NEFYFFN]], the $V_n$ are pairwise non-isomorphic, and each irreducible finite-dimensional complex representation of $G$ is isomorphic to exactly one $V_n$.

Check of the character on $t_\theta = \operatorname{diag}(e^{i\theta}, e^{-i\theta})$, with $\theta \notin \pi\Z$.
The monomial $z_1^a z_2^b$ with $a + b = n$ is an eigenvector with eigenvalue $e^{i(b - a)\theta}$, so
$$\tr \pi_n(t_\theta) = \sum_{j=0}^{n} e^{i(n - 2j)\theta} = \frac{\sin\big((n+1)\theta\big)}{\sin \theta}.$$
For $n = 1$ this gives $2 \cos \theta$, the trace of $t_\theta$ itself.
Since each element of $G$ is conjugate to some $t_\theta$, these characters determine the representations.

## Remarks

Tensor products decompose by the Clebsch–Gordan rule: $V_m \otimes V_n \cong V_{m+n} \oplus V_{m+n-2} \oplus \cdots \oplus V_{\abs{m-n}}$.
The element $-I$ acts on $V_n$ by $(-1)^n$, so $V_n$ is a representation of $\mathrm{SO}(3) = \mathrm{SU}(2)/\set{\pm I}$ exactly when $n$ is even.
In physics $V_n$ is the spin $n/2$ representation; odd $n$ gives the half-integer spins of electrons and other fermions.
On the Lie algebra, $V_n$ is the irreducible representation of $\mathfrak{sl}_2(\C)$ of highest weight $n$.

## Explanations

### L1 human

The group $\mathrm{SU}(2)$ is the group of rotations of quantum spin.
Its basic ways of acting on vector spaces are labelled by one number $n = 0, 1, 2, \dots$, and the $n$-th one has dimension $n + 1$.
They are built from polynomials of degree $n$ in two variables.
Every other finite-dimensional representation is a sum of these basic ones.
</document>

<document source="https://tamnd.github.io/math2/node/9V71Q3SD1G.md" tag="9V71Q3SD1G" kind="concept">
# Type

## Summary

In model theory, a type is a set of formulas in some free variables that is consistent with a theory.
It describes the possible behaviour of a tuple of elements.
A complete type gives a full first-order description of a tuple: every formula or its negation belongs to it.
A type is realized in a model when some tuple of the model satisfies all its formulas, and omitted otherwise.
Types are the basic tool for comparing models, and counting them leads to [[saturated models|RMTZ5J64FX]], categoricity and stability.

## Explanations

### L1 human

A type is a list of properties that an element could have, written in the language of the theory.
For example, take the ordered field of real numbers and the list "$x > 1$, $x > 2$, $x > 3$, and so on".
It is a type, because every finite part of it is satisfied by some real number.
No real number satisfies the whole list, so $\R$ omits this type.
In a nonstandard model of the reals the type is realized by an infinitely large element.
So types record which kinds of elements a model has and which it is missing.
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<document source="https://tamnd.github.io/math2/node/9V7YQA85PC.md" tag="9V7YQA85PC" kind="concept">
# Spectral radius

## Summary

The spectral radius of a square complex matrix is the largest absolute value of its [[eigenvalues|0BQAA2BK3Q]].
It controls the long-run growth of the powers $A^k$: they tend to $0$ exactly when the spectral radius is less than $1$.
It is bounded by every operator norm, and [[Gelfand's formula|3KX0KQ7V05]] recovers it from the norms of the powers.

## Explanations

### L1 human

Draw the eigenvalues of a matrix in the complex plane.
The spectral radius is the radius of the smallest disc around $0$ that contains all of them.
For $\begin{pmatrix} 2 & 0 \\ 0 & -3 \end{pmatrix}$ the eigenvalues are $2$ and $-3$, so the spectral radius is $3$.
Applying the matrix many times makes vectors grow roughly like $3^k$.
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<document source="https://tamnd.github.io/math2/node/9V800QGBQH.md" tag="9V800QGBQH" kind="definition">
# Category of elements

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $F \colon \mathcal{C} \to \mathbf{Set}$ be a [[functor|28NQ0KRHD3]] to the category of sets.
The *category of elements* $\int F$, also written $\mathrm{el}(F)$, is defined as follows:

1. its objects are the pairs $(A, x)$ with $A$ an object of $\mathcal{C}$ and $x \in F(A)$;
2. a morphism $(A, x) \to (B, y)$ is a morphism $f \colon A \to B$ of $\mathcal{C}$ with $F(f)(x) = y$;
3. composition and identities are those of $\mathcal{C}$.

## Remarks

Composition is well defined because $F(g \circ f)(x) = F(g)(F(f)(x))$.
It is the [[comma category|PC6HG1JEGM]] $(\ast \downarrow F)$, where $\ast$ is a one-point set, and the map $(A, x) \mapsto A$ is a projection functor $\int F \to \mathcal{C}$.
Example: let $G$ be a [[group|G5S1M9]] seen as a category with one object, and let $F$ be a set $X$ with a [[group action|0WCY42EV9V]] of $G$.
Then $\int F$ has the elements of $X$ as objects and a morphism $g \colon x \to y$ for each $g \in G$ with $g \cdot x = y$; it is the action groupoid.
Non-example: if $g \cdot x \ne y$, then $g$ is not a morphism from $x$ to $y$ in $\int F$, although it is a morphism of $G$.
For a contravariant functor $P \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$, the same construction is usually written with morphisms $f \colon A \to B$ such that $P(f)(y) = x$.
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<document source="https://tamnd.github.io/math2/node/9VA6P5KQQT.md" tag="9VA6P5KQQT" kind="concept">
# Matrix difference equation

## Summary

A matrix difference equation describes how a vector of quantities changes in discrete time by multiplication by a matrix, possibly plus an input term.
It is the discrete analogue of a system of linear differential equations.
Every linear recurrence of higher order can be rewritten as a first-order matrix difference equation for a state vector.
With a constant matrix $A$, the homogeneous solutions are $x(n) = A^n x(0)$, so their behaviour is governed by the eigenvalues of $A$.

## Explanations

### L1 human

Think of a state, such as the numbers of young and old animals in a population, recorded once a year.
A matrix difference equation says that next year's state is a fixed matrix times this year's state.
For the Fibonacci numbers, the pair $(F_{n+1}, F_n)$ is obtained from $(F_n, F_{n-1})$ by the matrix with rows $(1, 1)$ and $(1, 0)$.
So repeated multiplication by one matrix generates the whole sequence.
</document>

<document source="https://tamnd.github.io/math2/node/9VA95Q3K6G.md" tag="9VA95Q3K6G" kind="definition">
# Approximate identity

## Definition

Let $A$ be a [[Banach algebra|9KYWN94QDN]], not necessarily with a unit.
A *left approximate identity* of $A$ is a [[net|4DVJQJV0DG]] $(e_\lambda)_{\lambda \in \Lambda}$ in $A$ such that $\norm{e_\lambda a - a} \to 0$ for every $a \in A$.
A *right approximate identity* is defined the same way with $a e_\lambda$ in place of $e_\lambda a$.
An *approximate identity* (or two-sided approximate identity) is a net that is both a left and a right approximate identity.
It is *bounded* if $\sup_\lambda \norm{e_\lambda} < \infty$.

## Remarks

Example: let $A = C_0(\R)$, the continuous functions on $\R$ that tend to $0$ at infinity, with pointwise product and the sup norm.
For $n \ge 1$ let $e_n \in A$ be continuous with $0 \le e_n \le 1$, $e_n = 1$ on $[-n, n]$ and $e_n = 0$ outside $[-n-1, n+1]$.
For $f \in A$, $\norm{e_n f - f}_\infty \le \sup_{\abs{x} > n} \abs{f(x)} \to 0$.
So $(e_n)$ is a bounded approximate identity, with $\norm{e_n} = 1$, although $A$ has no unit.

Non-example: in $L^1(\R)$ with convolution, let $e_n = \tfrac{1}{2n} \mathbf{1}_{[-n, n]}$.
For $f \in L^1(\R)$, $\abs{(e_n * f)(x)} \le \norm{f}_1 / (2n)$, so $e_n * f \to 0$ uniformly.
If also $e_n * f \to f$ in $L^1$, a subsequence would converge to $f$ almost everywhere, so $f = 0$.
Thus the convergence condition fails for every $f \ne 0$.
Spikes $u_n = \tfrac{n}{2} \mathbf{1}_{[-1/n, 1/n]}$ do give a bounded approximate identity of $L^1(\R)$.

A unit $1$ of $A$ gives the constant approximate identity $e_\lambda = 1$.
Every C*-algebra has an approximate identity of positive elements of norm at most $1$.
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<document source="https://tamnd.github.io/math2/node/9VFWPZEVGN.md" tag="9VFWPZEVGN" kind="concept">
# Algebraic cycle

## Summary

An algebraic cycle on a variety $X$ is a finite formal sum, with integer coefficients, of irreducible closed subvarieties of $X$.
Cycles of codimension one are the [[Weil divisors|Z9DN23DAE2]].
Cycles modulo [[rational equivalence|K76N026JFK]] form the [[Chow groups|EBEF6PEK4N]], the basic objects of [[intersection theory|4HV4WTAX9P]].
The question of which cohomology classes come from cycles is the subject of the [[Hodge conjecture|XAEEDWV3PM]] and of the [[standard conjectures|D8GA2R3QT5]].

## Explanations

### L1 human

A cycle is a list of subvarieties, each with a whole number attached, such as "two copies of this curve minus one copy of that curve".
Cycles let one add and subtract subvarieties as if they were numbers.
For example, a line in the plane usually meets a cubic curve in the cycle $[P] + [Q] + [R]$ of three points.
If the line is tangent to the cubic at $P$, the intersection is $2[P] + [R]$.
The coefficients record multiplicities in this way.
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# Symmetric derivative

## Definition

Let $I \subseteq \R$ be an open interval, let $f \colon I \to \R$ be a [[function|0XHDTE5YMZ]], and let $x \in I$.
The function $f$ is *symmetrically differentiable* at $x$ if the [[limit|0TXTSEMHR5]]
$$f_s(x) = \lim_{h \to 0} \frac{f(x + h) - f(x - h)}{2h}$$
exists and is a real number.
The number $f_s(x)$ is the *symmetric derivative* of $f$ at $x$.

## Remarks

If $f$ is differentiable at $x$, then $\frac{f(x + h) - f(x - h)}{2h}$ is the average of two [[difference quotients|AZBDB5F4M4]] with increments $h$ and $-h$.
Both tend to $f'(x)$, so $f_s(x) = f'(x)$.
The symmetric derivative does not use the value $f(x)$.
For example, the indicator function of $\set{0}$ has symmetric derivative $0$ at $0$, although it is not continuous there.

Example: $f(x) = \abs{x}$ at $x = 0$.
For $h \ne 0$, the quotient is $\frac{\abs{h} - \abs{-h}}{2h} = 0$, so $f_s(0) = 0$, while $f'(0)$ does not exist.

Non-example: $f(x) = \sqrt[3]{x}$ at $x = 0$.
The quotient is $\frac{h^{1/3} + h^{1/3}}{2h} = h^{-2/3}$, which tends to $+\infty$, so the limit is not a real number.
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# Symmetric space

## Definition

Let $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]].
The manifold $M$ is a *(Riemannian globally) symmetric space* if each point $p \in M$ has an [[isometry|R9YWQS15AA]] $s_p \colon M \to M$ with two properties.
The map $s_p$ satisfies $s_p(p) = p$, and its [[differential|6P9PXQDSEN]] $(ds_p)_p \colon T_p M \to T_p M$ equals $-\id$.
The map $s_p$ is the *geodesic symmetry* at $p$.

## Remarks

Example: the unit sphere $S^n \subset \R^{n+1}$ with the induced metric.
For $p \in S^n$ put $s_p(x) = 2 \ip{x}{p} p - x$.
It is the restriction of a linear orthogonal map, so it is an isometry of $S^n$; it fixes $p$; and on $T_p S^n = p^\perp$ it acts as $v \mapsto -v$.

Non-example: a closed surface of genus $2$ with a metric of constant curvature $-1$.
Its isometry group is finite, so it does not act transitively on the surface.
A symmetric space is homogeneous, since $s_q$, for $q$ the midpoint of a shortest geodesic from $p$ to $p'$, swaps $p$ and $p'$; so the surface is not symmetric.
It is *locally* symmetric: its curvature tensor is parallel.

A symmetric space is complete, and $s_p$ is unique: it maps $\exp_p(v)$ to $\exp_p(-v)$.
Let $G$ be the identity component of the isometry group and $K$ the stabilizer of a point; then $M = G/K$.
Conjugation by $s_p$ gives an involution $\sigma$ of $G$, and its fixed point group satisfies $(G^\sigma)_0 \subseteq K \subseteq G^\sigma$.
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# Constructive analysis

## Definition

*Constructive analysis* is real and functional analysis developed within [[constructivism|F1P5SZ1XPH]], with intuitionistic logic.
In Bishop's approach, the basic notions are these.

1. A *real number* is a sequence $(x_n)_{n \ge 1}$ of rational numbers that is *regular*: $\abs{x_m - x_n} \le \frac{1}{m} + \frac{1}{n}$ for all $m, n \ge 1$.
2. Two real numbers $(x_n)$ and $(y_n)$ are *equal* if $\abs{x_n - y_n} \le \frac{2}{n}$ for all $n \ge 1$.
3. A real number $(x_n)$ is *positive* if $x_n > \frac{1}{n}$ for some $n$, and $x < y$ means that $y - x$ is positive.

Functions, continuity and other notions are then defined so that each existence statement comes with a method of construction.

## Remarks

A regular sequence is a [[Cauchy sequence|0QZMTN5AXX]] with an explicit modulus of convergence, and it defines a classical [[real number|08J2M1JYAG]].
Example: the sequence $x_n = \sum_{k=0}^{n} \frac{1}{k!}$ is regular, since $\abs{x_m - x_n} \le \sum_{k > n} \frac{1}{k!} \le \frac{1}{n}$ for $m > n \ge 1$; it defines $e$.
Non-example: the statement "for every real $x$, either $x = 0$ or $x \ne 0$" is not a theorem of constructive analysis.
A proof would give a method to decide, for each regular sequence, whether all its terms stay within $\frac{2}{n}$ of $0$.
Such a method would need the [[law of excluded middle|0RYYK8PG4S]] for statements about infinitely many terms.
The [[intermediate value theorem|00KAFBR32E]] holds in an approximate form: if $f$ is continuous on $[0, 1]$ with $f(0) < 0 < f(1)$, then for each $\varepsilon > 0$ there is $x$ with $\abs{f(x)} < \varepsilon$.
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# Sheafification

## Definition

Let $(\mathcal{C}, J)$ be a small [[site|NYE0Z94MME]] and let $P$ be a [[presheaf|AND2WHZC8P]] of sets on $\mathcal{C}$.
A *sheafification* of $P$ is a [[sheaf|RT8A8FJ566]] $aP$ for $J$ together with a morphism of presheaves $\eta \colon P \to aP$ with this property.
For every sheaf $F$ and every morphism $\varphi \colon P \to F$, there is exactly one morphism of sheaves $\psi \colon aP \to F$ with $\psi \circ \eta = \varphi$.

## Remarks

A sheafification exists for every presheaf on a small site and is unique up to unique isomorphism.
So $P \mapsto aP$ is a functor $\mathrm{PSh}(\mathcal{C}) \to \mathrm{Sh}(\mathcal{C}, J)$, [[left adjoint|TZHF6ZM92S]] to the inclusion, and it preserves finite limits.
One construction is $aP = (P^+)^+$, where $P^+(c)$ is the colimit over covering sieves $S \in J(c)$ of the sets of matching families for $S$.
If $P$ is already a sheaf, then $\eta$ is an isomorphism.
Example: on the site of open sets of a space $X$, the sheafification of the constant presheaf $U \mapsto A$ is the sheaf of locally constant functions $U \to A$.
Non-example: let $A = \set{0, 1}$.
The constant presheaf itself, with $\eta = \id$, is not a sheafification, since it is not a sheaf.
Indeed, the empty sieve covers $\emptyset$, and its unique (empty) matching family has two amalgamations in $P(\emptyset) = A$.
The sheafification repairs this: the sheaf of locally constant functions has exactly one section over $\emptyset$.
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# Poincaré duality

## Statement

Let $M$ be a closed [[topological manifold|QQQ9B1DK4P]] of dimension $n$, that is, compact and without boundary.
Let $R$ be a commutative ring with $1$, and assume that $M$ is $R$-[[oriented|913WEQERBE]], with [[fundamental class|ZEA0PEM3B2]] $[M] \in H_n(M; R)$.
Then for every $k$ the map
$$D \colon H^k(M; R) \to H_{n-k}(M; R), \qquad D(a) = [M] \frown a,$$
given by the [[cap product|N7SXBQ7WK0]], is an isomorphism from [[singular cohomology|6N7BC70CX0]] to [[singular homology|M0Q5CFJ7ZH]].

## Remarks

The proof first shows a version with compactly supported cohomology for every $R$-oriented manifold, not only closed ones.
It holds for open subsets of $\R^n$ that are convex, and it passes to unions by a [[Mayer–Vietoris|X8BWXFGK53]] argument and to increasing unions by a limit argument.
For a closed manifold, compactly supported cohomology is ordinary cohomology.

Every closed manifold is $\Z/2$-oriented, so the theorem always holds with $\Z/2$ coefficients.
With a field $F$ of coefficients, it implies that the pairing $H^k(M; F) \times H^{n-k}(M; F) \to F$, $(a, b) \mapsto \langle a \smile b, [M] \rangle$, is nondegenerate.
In particular, the [[Betti numbers|ST71ZV3FES]] satisfy $b_k(M) = b_{n-k}(M)$ for closed orientable $M$, and the [[Euler characteristic|RNDDWDAFJ5]] of a closed manifold of odd dimension is $0$.

The hypotheses are needed.
Without orientability: for the [[real projective plane|1DDYB6VVSR]], $H^0(\R P^2; \Z) = \Z$ but $H_2(\R P^2; \Z) = 0$.
Without compactness: $H^0(\R^n; \Z) = \Z$ but $H_n(\R^n; \Z) = 0$ for $n \ge 1$.

## Explanations

### L1 human

On a closed oriented manifold of dimension $n$, the holes of dimension $k$ and the holes of dimension $n - k$ come in matching pairs.
For a closed orientable surface, the number of independent loops is even, because each loop has a partner loop that it crosses once.
The duality comes from cutting the manifold into cells and also into the dual cells, where each $k$-cell matches an $(n - k)$-cell crossing it.

### L2 human

The theorem identifies cohomology and homology of closed oriented manifolds with a shift of degree $n$.
It turns the cup product into intersection of cycles, which gives the intersection form of a $4$-manifold and its unimodularity.
It yields the symmetry of Betti numbers and the vanishing of the Euler characteristic in odd dimensions.
Lefschetz duality extends it to compact manifolds with boundary, and Alexander duality is a consequence for complements of subsets of spheres.
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# Z-group

## Summary

A Z-group is a finite group in which every Sylow subgroup is cyclic.
These groups are well understood: each one is solvable, and in fact metacyclic, that is, an extension of a cyclic group by a cyclic group.
Every group of square-free order is a Z-group.
The name Z-group is also used for some unrelated classes of groups; this node is about the finite groups with cyclic Sylow subgroups.

## Explanations

### L1 human

For each prime $p$, a finite group has Sylow $p$-subgroups, the largest subgroups whose order is a power of $p$.
In a Z-group, each of these is generated by a single element.
For example, the symmetric group $S_3$ has order $6 = 2 \cdot 3$, its Sylow subgroups have orders $2$ and $3$, and groups of prime order are cyclic.
So $S_3$ is a Z-group, while the Klein four-group is not, since it is its own Sylow $2$-subgroup and it is not cyclic.
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# Commuting probability

## Definition

Let $G$ be a finite [[group|G5S1M9]].
The *commuting probability* of $G$ is
$$\mathrm{cp}(G) = \frac{\abs{\set{(x, y) \in G \times G : xy = yx}}}{\abs{G}^2}.$$

## Remarks

Counting the pairs by their first entry gives $\mathrm{cp}(G) = \frac{1}{\abs{G}^2} \sum_{x \in G} \abs{C_G(x)}$, where $C_G(x)$ is the [[centraliser|02YYV36BFM]] of $x$.
Since the [[conjugacy class|0DEBXAJ5SA]] of $x$ has $\abs{G}/\abs{C_G(x)}$ elements, this sum gives $\mathrm{cp}(G) = k(G)/\abs{G}$, where $k(G)$ is the number of conjugacy classes of $G$.
Example: $S_3$ has $3$ conjugacy classes, so $\mathrm{cp}(S_3) = 3/6 = 1/2$.
Example: the [[quaternion group|0NRJGW864D]] $Q_8$ has $5$ conjugacy classes, $\{1\}$, $\{-1\}$, $\{\pm i\}$, $\{\pm j\}$, $\{\pm k\}$, so $\mathrm{cp}(Q_8) = 5/8$.
The value $\mathrm{cp}(G) = 1$ holds exactly when $G$ is [[abelian|082GHNPJ53]]; if $G$ is not abelian, then $\mathrm{cp}(G) \le 5/8$.

## Explanations

### L1 human

The commuting probability is the chance that two random elements of the group commute.
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# Pariah groups

## Definition

Let $\mathbb{M}$ be the [[Monster group|C9VY20TAPT]].
Let $H$ be a [[subgroup|0X1JXT47GA]] of $\mathbb{M}$ and let $N$ be a normal subgroup of $H$.
A group isomorphic to such a [[quotient group|0KYX2N69J0]] $H/N$ is a subquotient of $\mathbb{M}$.
A *pariah group* is a [[sporadic group|PZ49Z4CBFZ]] that is not a subquotient of $\mathbb{M}$.

## Remarks

There are exactly six pariah groups: $J_1$, $J_3$, $J_4$, the O'Nan group $\mathrm{O'N}$, the Rudvalis group $\mathrm{Ru}$ and the Lyons group $\mathrm{Ly}$.
Example: the [[Janko group|PSEC8D7CX0]] $J_1$ is a pariah.
Non-example: the [[Conway group|366D4ETGQH]] $\mathrm{Co}_1$ is not a pariah.
The Monster contains an involution whose centraliser $H$ has a normal subgroup $N$ of order $2^{25}$ with $H/N \cong \mathrm{Co}_1$, so $\mathrm{Co}_1$ is a subquotient of $\mathbb{M}$.

## Explanations

### L1 human

A pariah is a sporadic simple group that cannot be found inside the Monster, even as a quotient of a subgroup.
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# Newton polynomial

## Summary

The Newton polynomial is the [[interpolation polynomial|SS1YMS2NPC]] of given data written in the Newton form.
The basis is $1$, $(x - x_0)$, $(x - x_0)(x - x_1)$, and so on, and the coefficients are [[divided differences|N5ZHV4HWDC]].
Adding one more data point adds one more term and leaves the earlier terms unchanged.
This makes the Newton form convenient for computation and for deriving error terms and finite difference formulas.

## Explanations

### L1 human

Start with the constant through the first point.
Add a multiple of $(x - x_0)$ to also pass through the second point; this does not change the value at $x_0$.
Then add a multiple of $(x - x_0)(x - x_1)$ to pass through the third point, without disturbing the first two.
For the points $(0, 1)$, $(1, 3)$, $(2, 7)$ this gives $1 + 2x + x(x - 1) = x^2 + x + 1$.
The [[definition|155K06F4JF]] gives the general formula.
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# Tammes problem

## Problem

Let $N \ge 2$ be an integer and let $S^2 \subset \R^3$ be the unit sphere.
Find the largest number $d_N$ such that there are $N$ points on $S^2$ with pairwise distances at least $d_N$.
Find also the configurations of $N$ points that attain it.

## Remarks

Equivalently, find the largest radius of $N$ equal non-overlapping circular caps on a sphere, or the best [[spherical code|SDCX4WDP4F]] of size $N$ in $\R^3$.
The botanist Tammes posed it in 1930 while studying the distribution of pores on pollen grains.
For $N = 4$, $6$ and $12$ the answers are the vertices of the regular tetrahedron, octahedron and icosahedron.
The problem is solved for $N \le 14$ and for $N = 24$.
Oleg Musin and Alexey Tarasov solved the case $N = 13$.
It is linked with the [[kissing number problem in three dimensions|Q1H6BCKTBE]]: thirteen unit balls cannot touch a central one, so the optimal angle for $13$ points is below $60^\circ$.

## Explanations

### L1 human

How should $N$ people stand on a round planet so that the two closest people are as far apart as possible?
For small $N$ the answer is often a familiar solid, such as the corners of a tetrahedron for four people.
For most $N$ the answer is unknown, and even small cases needed long proofs.
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# Strange attractor

## Summary

A strange attractor is an [[attractor|7FWX24X5MW]] with a fractal structure, unlike a point, a closed curve or a smooth surface.
The term was introduced by Ruelle and Takens in their 1971 work on turbulence.
Strange attractors usually carry chaotic dynamics: nearby orbits on the attractor separate exponentially fast, which is the [[butterfly effect|G0ZT1SDQGS]].
The [[Lorenz system|WYRC41CQN9]] and the Hénon map are the classical examples.

## Explanations

### L1 human

A strange attractor is the shape that the long-time motion of a chaotic system traces out.
Orbits are pulled onto it, yet on it they never settle into a rest point or a repeating loop.
The set is built by stretching and folding again and again, which gives it layers inside layers, like a puff pastry.
Its dimension, measured as for fractals, is often not a whole number: the Lorenz attractor is a little thicker than a surface but much thinner than a solid.
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# Bromwich integral

## Definition

Let $a$ be a real number, and let $F$ be a [[holomorphic function|0DG0Z72CQD]] on the half-plane $\{s \in \C : \operatorname{Re} s > a\}$.
Let $\gamma > a$ be real and let $t$ be real.
The *Bromwich integral* of $F$ along the line $\operatorname{Re} s = \gamma$, at time $t$, is
$$\frac{1}{2 \pi i} \int_{\gamma - i \infty}^{\gamma + i \infty} e^{s t} F(s) \, ds = \frac{1}{2 \pi i} \lim_{T \to \infty} \int_{\gamma - i T}^{\gamma + i T} e^{s t} F(s) \, ds.$$
Each integral on the right is the [[contour integral|00FWNP9TXY]] along the vertical segment from $\gamma - i T$ to $\gamma + i T$.
The Bromwich integral exists at $t$ when this limit exists.

## Remarks

Example: let $F(s) = 1/s$ on $\operatorname{Re} s > 0$, and $\gamma > 0$.
For $t > 0$ one closes the segment by a large arc to the left, whose contribution tends to $0$.
The residue of $e^{s t}/s$ at $0$ is $1$, so the integral is $1$.
For $t < 0$ one closes to the right, where there are no poles, and gets $0$; for $t = 0$ the symmetric limit is $\frac{1}{2 \pi i} \lim_{T \to \infty} \left( \log(\gamma + i T) - \log(\gamma - i T) \right) = \tfrac{1}{2}$.
Non-example: for $F(s) = 1$ and $t \ne 0$, the integral over the segment equals $e^{\gamma t} \sin(T t) / (\pi t)$, which has no limit as $T \to \infty$; so the Bromwich integral does not exist.
The limit is a symmetric one; it can exist even when $e^{s t} F(s)$ is not absolutely integrable on the line.
The integral is also called the Fourier–Mellin integral.
When $F$ is the [[Laplace transform|05XT5AVJBK]] of a suitable function $f$, the integral equals $f(t)$ at each $t > 0$ where $f$ is continuous; this is Mellin's inverse formula.
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# Logarithmic capacity

## Definition

Let $K$ be a [[compact set|0R6EWFKRCF]] in $\C$.
For a Borel [[probability measure|059TJN0NZ1]] $\mu$ on $K$, its *logarithmic energy* is
$$I(\mu) = \int_K \int_K \log\frac{1}{\abs{z - w}}\, d\mu(z)\, d\mu(w) \in (-\infty, \infty].$$
The *Robin constant* of $K$ is $V(K) = \inf_\mu I(\mu)$, the infimum over all Borel probability measures $\mu$ on $K$.
The *logarithmic capacity* of $K$ is $\operatorname{cap}(K) = e^{-V(K)}$, with $\operatorname{cap}(K) = 0$ when $V(K) = \infty$.

## Remarks

Example: for the closed disc or the circle of radius $r$, the minimizing measure is normalized arc length on the circle of radius $r$, and $\operatorname{cap} = r$.
For a segment of length $L$, $\operatorname{cap} = L/4$.
A finite set $K$ has $\operatorname{cap}(K) = 0$: every probability measure on $K$ has an atom, and an atom of mass $m > 0$ at $a$ contributes $m^2 \log(1/\abs{a - a}) = \infty$ to $I(\mu)$.
Let $K$ be connected with more than one point, and let its complement in the Riemann sphere $\hat\C$ be simply connected.
Let $\varphi$ be the conformal map of $\hat\C \setminus K$ onto $\set{\abs{z} > 1} \cup \set{\infty}$ with $\varphi(\infty) = \infty$.
Near $\infty$ it has the form $\varphi(z) = z/c + O(1)$ with $c > 0$, and then $\operatorname{cap}(K) = c$, the outer conformal radius of $K$.
By theorems of Fekete and Szegő, the capacity also equals the transfinite diameter of $K$ and its Chebyshev constant.
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# Deduction theorem

## Statement

Consider the standard [[Hilbert system|1MGMQX43B8]] for propositional logic, with modus ponens as its only rule.
Let $\Gamma$ be a set of formulas, and let $\varphi$ and $\psi$ be formulas.
Then $\Gamma \cup \set{\varphi} \vdash \psi$ if and only if $\Gamma \vdash \varphi \to \psi$.

## Remarks

The direction from right to left is one step of modus ponens.
For the other direction, take a derivation $\psi_1, \dots, \psi_n = \psi$ from $\Gamma \cup \set{\varphi}$.
Show by induction on $i$ that $\Gamma \vdash \varphi \to \psi_i$.
If $\psi_i$ is an axiom or in $\Gamma$, use axiom 1, $\psi_i \to (\varphi \to \psi_i)$; if $\psi_i = \varphi$, use the derivation of $\varphi \to \varphi$.
If $\psi_i$ comes by modus ponens from $\psi_j$ and $\psi_j \to \psi_i$, use axiom 2 and modus ponens twice.
For example, from $\set{p \to q, \; q \to r, \; p} \vdash r$, which takes two steps of modus ponens, the theorem gives $\set{p \to q, \; q \to r} \vdash p \to r$.
In first-order logic with the rule of generalization, the theorem needs a hypothesis.
It holds when $\varphi$ is a sentence, or more generally when the derivation never generalizes on a variable free in $\varphi$.
Without this, $\set{P(x)} \vdash \forall x \, P(x)$ holds, but $P(x) \to \forall x \, P(x)$ is not valid.

## Explanations

### L1 human

The theorem says that "assume $\varphi$ and prove $\psi$" is a correct way to prove "if $\varphi$ then $\psi$".
Mathematicians use this all the time without thinking.
The theorem shows that a Hilbert system, which has no rule for assumptions, still allows it.

### L2 human

The deduction theorem connects the turnstile $\vdash$ with the connective $\to$.
It is the reason that proofs in a [[Hilbert system|1MGMQX43B8]] can be built from proofs under hypotheses, which keeps them manageable.
In [[natural deduction|6KT3BJZ7D1]] the same principle is a primitive rule, implication introduction.
Its semantic counterpart is that $\Gamma \cup \set{\varphi} \models \psi$ if and only if $\Gamma \models \varphi \to \psi$, for [[logical consequence|1HK0VYWTV7]].
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# Graded manifold

## Definition

Let $n, m \ge 0$ be integers, let $Z$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $C^\infty_Z$ be its sheaf of real [[smooth functions|0V33BNKZ9S]].
Let $\Lambda_m = \Lambda(\R^m)$ be the real [[exterior algebra|0MDMZ965VE]] on $m$ generators, with its $\Z/2$-grading into even and odd parts.
A *graded manifold* of dimension $(n, m)$ with body $Z$ is a [[ringed space|KNDXDE7K8M]] $(Z, \mathcal{A})$ such that:

1. $\mathcal{A}$ is a [[sheaf|TB6TM6R4MD]] of $\Z/2$-graded commutative real algebras on $Z$.
2. There is a surjective morphism of sheaves of algebras $\mathcal{A} \to C^\infty_Z$ whose kernel is the ideal $\mathcal{N}$ of [[nilpotent|0ZB2JG1M9C]] sections.
3. Every point of $Z$ has an open neighbourhood $U$ such that $\mathcal{A}|_U$ is isomorphic, as a sheaf of graded algebras, to the sheaf $C^\infty_U \otimes_\R \Lambda_m$.

Here $\Z/2$-graded commutative means $ab = (-1)^{\abs{a}\abs{b}} ba$ for homogeneous elements $a, b$ of degrees $\abs{a}, \abs{b} \in \set{0, 1}$.

## Remarks

Example: let $E \to Z$ be a smooth real vector bundle of rank $m$, and let $\mathcal{A}$ be the sheaf of smooth sections of the exterior algebra bundle $\Lambda E^*$.
Over an open set $U$ where $E$ is trivial, a frame gives $\mathcal{A}|_U \cong C^\infty_U \otimes \Lambda_m$, so clause 3 holds.
The kernel of the map $\mathcal{A} \to C^\infty_Z$ that keeps the degree-zero component is generated by sections of $E^*$, which are nilpotent, so clause 2 holds.
For $E = T^*Z$ this is the graded manifold of differential forms on $Z$.
Batchelor's theorem says that every smooth graded manifold is isomorphic to one of this form, though not canonically.

Non-example: let $\mathcal{A} = C^\infty_Z[\varepsilon] / (\varepsilon^2)$, with $\varepsilon$ declared even, so that all of $\mathcal{A}$ is even.
Clauses 1 and 2 hold, with nilpotent ideal $\varepsilon C^\infty_Z$.
Clause 3 fails: for $m = 0$ the local model has no non-zero nilpotents, and for $m \ge 1$ it has non-zero odd elements, while $\mathcal{A}$ has nilpotents and no odd part.

The local odd coordinates are the images $\theta_1, \dots, \theta_m$ of a basis of $\R^m$ under clause 3; they satisfy $\theta_i \theta_j = -\theta_j \theta_i$.
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# Bernstein's theorem on analytic functions

## Statement

Let $f \colon [-1, 1] \to \R$ be continuous and let $E_n(f)$ be the uniform distance on $[-1, 1]$ from $f$ to the polynomials of degree at most $n$.
For $\rho > 1$, let $\mathcal{E}_\rho$ be the open region bounded by the [[ellipse|0VKCCEXBJ1]] with foci $\pm 1$ whose semi-axes have sum $\rho$.
Then $f$ extends to a [[holomorphic function|0NAHXR1FRH]] on $\mathcal{E}_\rho$ if and only if
$$\limsup_{n \to \infty} E_n(f)^{1/n} \le \frac{1}{\rho}.$$

## Remarks

The region $\mathcal{E}_\rho$ is the image of the annulus $1 < \abs{z} < \rho$ under the map $z \mapsto (z + z^{-1})/2$, together with $[-1, 1]$.
Proof idea for the direct half: for $1 < s < \rho$, the coefficients of the Chebyshev series of $f$ are $O(s^{-n})$, by a contour integral over the image of $\abs{z} = s$.
Truncating the series gives $E_n(f) = O(s^{-n})$.
For the converse, write $f = p_0 + \sum (p_{n+1} - p_n)$ with best approximations $p_n$ and bound each difference on $\mathcal{E}_s$ for $s < \rho$ by its size on $[-1, 1]$.

Example: for $a > 1$ and $f(x) = 1/(x - a)$, the pole $a$ lies on the boundary of $\mathcal{E}_\rho$ with $\rho = a + \sqrt{a^2 - 1}$.
So $E_n(f)$ decays like $(a + \sqrt{a^2 - 1})^{-n}$, up to factors that grow slower than any exponential.
For the Runge function $1/(1 + 25x^2)$, the poles $\pm i/5$ give $\rho = (1 + \sqrt{26})/5 \approx 1.22$.

## Explanations

### L1 human

For an analytic function on $[-1, 1]$, the error of the best polynomial approximation shrinks geometrically with the degree.
The rate is set by how far the function extends into the complex plane, measured by the largest ellipse with foci $\pm 1$ on which it stays analytic.

### L2 human

This explains why Chebyshev methods converge so fast for analytic functions, and why nearby complex singularities slow them down even when $f$ looks harmless on the real line.
It is an inverse and a direct theorem at once: geometric decay of $E_n(f)$ and analyticity in an ellipse are equivalent.
The ellipses $\partial \mathcal{E}_\rho$ are the level curves of the Green function of the complement of $[-1, 1]$, which leads to the general theory of Walsh for other sets.
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# Elementary embedding

## Definition

Let $L$ be a signature and let $\mathcal{M}$ and $\mathcal{N}$ be $L$-[[structures|983GZBQYT4]] with domains $M$ and $N$.
A [[map|0EPAT51DXQ]] $j \colon M \to N$ is an elementary embedding of $\mathcal{M}$ into $\mathcal{N}$ when, for every $L$-[[formula|HYJ10AF6RT]] $\varphi(x_1, \dots, x_n)$ and all $a_1, \dots, a_n \in M$,
$$\mathcal{M} \models \varphi(a_1, \dots, a_n) \iff \mathcal{N} \models \varphi(j(a_1), \dots, j(a_n)).$$

## Remarks

An elementary embedding is injective: apply the condition to the formula $x_1 = x_2$.
It is an embedding of structures: apply the condition to the atomic formulas.
Its image is the domain of an [[elementary substructure|8WZF9SBP39]] of $\mathcal{N}$.

Example: the inclusion $\Q \to \R$ is an elementary embedding of $(\Q, <)$ into $(\R, <)$, because $(\Q, <)$ is an elementary substructure of $(\R, <)$.

Non-example: the map $j(n) = n + 1$ from $(\N, <)$ to $(\N, <)$ preserves $<$ and is injective.
It is not elementary: for $\varphi(x) = \exists y \, (y < x)$, $(\N, <) \models \neg\varphi(0)$ but $(\N, <) \models \varphi(j(0))$.

## Explanations

### L1 human

An elementary embedding places a copy of one structure inside another so that no first-order statement changes its truth value.
Each elementary embedding is an isomorphism onto an elementary substructure.
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<document source="https://tamnd.github.io/math2/node/9X6KG770GV.md" tag="9X6KG770GV" kind="statement">
# Pfaff–Saalschütz theorem

## Statement

Let $n$ be a nonnegative integer and let $a, b, c$ be complex numbers.
Assume that neither $c$ nor $1 + a + b - c - n$ lies in $\set{0, -1, \dots, 1 - n}$.
Write $(x)_k = x (x + 1) \cdots (x + k - 1)$ for the [[Pochhammer symbol|YZHM3HSM0F]].
Then the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_3F_2$ at $1$ is
$${}_3F_2\left( \begin{matrix} -n,\ a,\ b \\ c,\ 1 + a + b - c - n \end{matrix} ; 1 \right) = \frac{(c - a)_n\, (c - b)_n}{(c)_n\, (c - a - b)_n}.$$

## Remarks

The series is *balanced* (Saalschützian): the sum of the lower parameters exceeds the sum of the upper parameters by $1$.
A standard proof starts from Euler's transformation $(1 - x)^{a + b - c}\, {}_2F_1(a, b; c; x) = {}_2F_1(c - a, c - b; c; x)$.
Comparing the coefficients of $x^n$ on both sides gives the identity after a rearrangement of Pochhammer symbols.
For $n = 1$ the left side is $1 - \frac{ab}{c(a + b - c)}$.
This equals $\frac{c(a + b - c) - ab}{c(a + b - c)} = \frac{(c - a)(c - b)}{c(c - a - b)}$, which is the right side.
The hypothesis on $c$ and $1 + a + b - c - n$ keeps the denominators of the terms of the series nonzero; it also gives $(c - a - b)_n \ne 0$, since $(c - a - b)_n = (-1)^n (1 + a + b - c - n)_n$.

## Explanations

### L1 human

A hypergeometric series with a negative integer $-n$ on top stops after $n + 1$ terms, so it is a finite sum.
In general such a sum at $1$ has no simple closed form.
The theorem says that when the parameters are balanced, the sum collapses to a single product of rising factorials.

### L2 human

The theorem is the ${}_3F_2$ analogue of the Chu–Vandermonde identity for terminating ${}_2F_1$ sums at $1$.
It is one of the basic summation formulas used to evaluate sums of products of binomial coefficients.
It has a $q$-analogue in the theory of basic hypergeometric series.
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# Holomorphic functional calculus

## Definition

Let $A$ be a complex [[Banach algebra|9KYWN94QDN]] with unit $e$, and let $a \in A$ with [[spectrum|EH0CZ19RKP]] $\sigma(a)$.
Let $U \subseteq \C$ be an open set with $\sigma(a) \subseteq U$, and let $f \colon U \to \C$ be [[holomorphic|0NAHXR1FRH]].
Let $\Gamma$ be a finite union of closed piecewise smooth curves in $U \setminus \sigma(a)$.
Assume that the winding number of $\Gamma$ is $1$ around each point of $\sigma(a)$ and $0$ around each point of $\C \setminus U$.
The element $f(a) \in A$ is the [[contour integral|06ZGPYTWK8]]
$$f(a) = \frac{1}{2\pi i} \int_\Gamma f(\lambda) \, (\lambda e - a)^{-1} \, d\lambda .$$
The map $f \mapsto f(a)$, from the algebra of holomorphic functions on $U$ to $A$, is the *holomorphic functional calculus* of $a$.

## Remarks

The integrand is continuous, since $\lambda \mapsto (\lambda e - a)^{-1}$, the [[resolvent|J5YNNKSB9P]], is defined and holomorphic off $\sigma(a)$.
By Cauchy's theorem for $A$-valued holomorphic functions, $f(a)$ does not depend on the choice of $\Gamma$.
Example: for $f(\lambda) = \lambda^n$ and $\Gamma$ a large circle $\abs{\lambda} = R$ with $R > \norm{a}$, expand $(\lambda e - a)^{-1} = \sum_{k \ge 0} \lambda^{-k-1} a^k$; only the term $k = n$ survives the integral, so $f(a) = a^n$.
Non-example: $f(\lambda) = \bar\lambda$ is not holomorphic, so the definition does not apply; for non-normal operators there is no reasonable way to define $\bar a$ by this method.
The calculus is an algebra homomorphism, it is continuous for locally uniform convergence of $f$, and $\sigma(f(a)) = f(\sigma(a))$.
For a normal operator on a Hilbert space it agrees with the [[continuous functional calculus|JB7S6WPZ77]] on holomorphic functions.
When $f$ is $1$ near one closed and open part of $\sigma(a)$ and $0$ near the rest, $f(a)$ is the [[Riesz projector|1AR19ZQQM2]] of that part.
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# Shape derivative

## Summary

The shape derivative measures how a quantity that depends on a domain changes when the domain is deformed a little.
One moves each point $x$ of the domain to $x + \theta(x)$ for a small vector field $\theta$.
The shape derivative is the derivative of the quantity with respect to $\theta$.
For smooth domains it usually depends only on the normal component of $\theta$ on the boundary.
This fact is known as the Hadamard structure theorem.
It is the gradient that drives numerical methods of [[shape optimization|WWQJQXJECK]].

## Explanations

### L1 human

Push the boundary of a region outward a little, in some places more than in others.
The area of the region grows, and to first order the growth is the outward push integrated along the boundary.
That first-order rate of change is the shape derivative of the area.
For other quantities, such as the drag on a wing, the shape derivative tells an engineer where to push the boundary to improve the design.
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# Maximum principle for subharmonic functions

## Statement

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be a connected [[open set|02BH3MCBQB]], and let $u$ be [[subharmonic|Z1MHMGFNFW]] on $\Omega$.

1. If there is a point $a \in \Omega$ with $u(a) = \sup_{\Omega} u$, then $u$ is constant on $\Omega$.
2. If $\Omega$ is bounded and $\limsup_{x \to \zeta,\, x \in \Omega} u(x) \le 0$ for every $\zeta$ in the [[boundary|05D5EGMFYV]] $\partial\Omega$, then $u \le 0$ on $\Omega$.

## Remarks

Proof idea for part 1: let $M = \sup_\Omega u$ and $A = \set{x \in \Omega : u(x) = M}$.
The set $A = \set{u \ge M}$ is closed in $\Omega$ because $u$ is [[upper semicontinuous|RN4G8SGP6P]].
Let $b \in A$.
The sub-mean inequality on small spheres about $b$ and $u \le M$ force $u = M$ almost everywhere on a small ball about $b$.
Since $A$ is closed, $u = M$ on that whole ball, so $A$ is open.
Connectedness gives $A = \Omega$ when $A$ is nonempty.
For part 2, extend $u$ to $\overline{\Omega}$ by its [[limit superior|0T3RZQXDYD]] at boundary points.
The extension is upper semicontinuous on a compact set, so it attains its maximum, and part 1 rules out a positive maximum inside.

Boundedness of $\Omega$ is needed in part 2.
On the half-space $\set{x \in \R^n : x_1 > 0}$, the function $u(x) = x_1$ is harmonic, hence subharmonic, and tends to $0$ at every boundary point, but it is positive everywhere.
Connectedness is needed in part 1.
On two disjoint discs, the function equal to $0$ on one disc and $1$ on the other is subharmonic and attains its supremum, but it is not constant.
Applied to $u$ and $-u$ for a harmonic $u$, it gives the maximum and minimum principles for harmonic functions.

## Explanations

### L1 human

A subharmonic function cannot have a peak inside its domain unless it is flat.
So on a bounded region its largest values are found at the edge.
If it is at most $0$ near the edge, it is at most $0$ everywhere.

### L2 human

This is the tool behind uniqueness for the [[Dirichlet problem|8D56GQ4AG8]] and behind the Perron method, where subharmonic functions are compared with harmonic ones.
It also gives the comparison principle: if $u$ is subharmonic, $v$ is superharmonic, and $\limsup (u - v) \le 0$ at the boundary of a bounded domain, then $u \le v$ inside.
Unbounded domains need extra growth conditions, which is the subject of Phragmén–Lindelöf type results.
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# Grothendieck construction

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]], let $\mathbf{Cat}$ be the category of small categories, and let $F \colon \mathcal{C} \to \mathbf{Cat}$ be a [[functor|28NQ0KRHD3]].
The *Grothendieck construction* of $F$ is the category $\int F$ given as follows.

1. An object is a pair $(c, x)$ with $c$ an object of $\mathcal{C}$ and $x$ an object of $F(c)$.
2. A morphism $(c, x) \to (c', x')$ is a pair $(f, \varphi)$ with $f \colon c \to c'$ in $\mathcal{C}$ and $\varphi \colon F(f)(x) \to x'$ in $F(c')$.
3. The composite of $(f, \varphi) \colon (c, x) \to (c', x')$ and $(g, \psi) \colon (c', x') \to (c'', x'')$ is $(g \circ f, \ \psi \circ F(g)(\varphi))$.
4. The identity of $(c, x)$ is $(\id_c, \id_x)$.

The projection functor $p \colon \int F \to \mathcal{C}$ sends $(c, x)$ to $c$ and $(f, \varphi)$ to $f$.

## Remarks

Example: if $F$ is the constant functor with value a small category $\mathcal{D}$, then $F(f) = \id_{\mathcal{D}}$ for every $f$.
A morphism $(c, x) \to (c', x')$ is then a pair $(f, \varphi)$ with $\varphi \colon x \to x'$ in $\mathcal{D}$, so $\int F$ is the [[product category|AFNE9C4NH6]] $\mathcal{C} \times \mathcal{D}$.

Non-example: the disjoint union of the categories $F(c)$, with no morphisms between different fibres, is not $\int F$ when $\mathcal{C}$ has a non-identity morphism $f \colon c \to c'$ and $F(c)$ is non-empty.
Clause 2 gives the morphism $(f, \id_{F(f)(x)}) \colon (c, x) \to (c', F(f)(x))$ between two fibres.

When each $F(c)$ is a set seen as a category with only identity morphisms, $\int F$ is the category of elements of $F$.
For a functor $F \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Cat}$ the dual construction takes morphisms $(f, \varphi)$ with $f \colon c \to c'$ and $\varphi \colon x \to F(f)(x')$ in $F(c)$; its projection is a fibration.
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<document source="https://tamnd.github.io/math2/node/9Y01DS79PR.md" tag="9Y01DS79PR" kind="definition">
# Fundamental solution of the Laplacian

## Definition

Let $n \ge 2$ and let $\omega_n$ be the volume of the unit ball of $\R^n$.
The *fundamental solution of the Laplacian* on $\R^n$ is the function $\Phi \colon \R^n \setminus \set{0} \to \R$ given by
$$\Phi(x) = -\frac{1}{2\pi} \log \abs{x} \quad (n = 2), \qquad \Phi(x) = \frac{1}{n(n-2)\omega_n} \, \frac{1}{\abs{x}^{n-2}} \quad (n \ge 3).$$

## Remarks

The function $\Phi$ is locally integrable on $\R^n$, and in the sense of distributions it satisfies $-\Delta \Phi = \delta$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $\delta$ is the [[Dirac delta|0KPHENZ5S7]] at $0$.
So $-\Phi$ is a [[fundamental solution|K2NK9A8M5F]] of $\Delta$ in the general sense; many texts use this sign convention so that $\Phi > 0$ near the origin.
Example: for $n = 3$, $\omega_3 = \frac{4\pi}{3}$ and $\Phi(x) = \frac{1}{4\pi\abs{x}}$.
For a radial function $f(r)$ with $r = \abs{x}$ one has $\Delta f = f'' + \frac{n-1}{r} f'$, and $f = r^{2-n}$ gives $(2-n)(1-n) r^{-n} + (n-1)(2-n) r^{-n} = 0$, so $\Phi$ is [[harmonic|0DEB0JMN0W]] on $\R^n \setminus \set{0}$.
Non-example: $\abs{x}^{-1}$ on $\R^4$ is not a multiple of $\Phi$, and it is not harmonic there, since $\Delta \abs{x}^{-1} = -\abs{x}^{-3}$.
The constant is chosen so that the flux of $-\nabla \Phi$ out of every sphere about $0$ equals $1$; here $n\omega_n$ is the area of the unit sphere.
Other normalisations, such as $\abs{x}^{2-n}$ or $\log(1/\abs{x})$ without constants, are common in potential theory.
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# Homotopy category of chain complexes

## Summary

Let $\mathcal{A}$ be an additive category.
The homotopy category of chain complexes $K(\mathcal{A})$ has the [[chain complexes|WQ82C0M3QB]] as objects and the [[chain maps|HTK9WQZJ1G]] modulo [[chain homotopy|4EBHAWJJ8C]] as morphisms.
In it, chain homotopy equivalences become isomorphisms.
It is a [[triangulated category|T2V7GJJ2EN]], and the [[derived category|5ST7NVJFSM]] of an abelian category is obtained from it by inverting the quasi-isomorphisms.

## Explanations

### L1 human

Two chain maps that are chain homotopic induce the same map on homology, so for many purposes they are the same.
The homotopy category identifies them.
A complex that can be contracted to zero, such as $0 \to \Z \xrightarrow{\id} \Z \to 0$, becomes isomorphic to the zero complex.
It is the algebraic analogue of the homotopy category of spaces, where homotopic maps are identified.
</document>

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# Homotopy

## Summary

A homotopy between two [[continuous maps|0VGFHTM75Y]] $f, g \colon X \to Y$ is a continuous deformation of $f$ into $g$, parametrised by a time $t \in [0, 1]$.
Formally it is a continuous map $H \colon X \times [0, 1] \to Y$ with $H(\cdot, 0) = f$ and $H(\cdot, 1) = g$.
Homotopy is the basic notion of algebraic topology.
Homotopic maps induce the same maps on homology and cohomology, and homotopies that fix a base point give the same maps on homotopy groups.

## Explanations

### L1 human

Think of $f$ as a picture drawn in $Y$, and of the homotopy as a film that moves this picture continuously until it becomes $g$.
At each moment $t$ the frame $H(\cdot, t)$ is again a continuous map.
For example, in the plane $\R^2$ any two maps $f, g \colon X \to \R^2$ are joined by the straight-line homotopy $H(x, t) = (1 - t) f(x) + t g(x)$.
In a space with a hole, such as the plane without the origin, a loop around the hole cannot be moved to a constant loop.
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# Space of functions of bounded variation

## Summary

The space $BV([a, b])$ collects the real functions on $[a, b]$ that have [[bounded variation|A8251BHCMP]].
With a norm built from the [[total variation|8QA5DEWNYD]], it is a [[Banach space|06GZ956657]].
Its elements are exactly the differences of two monotone functions, and they serve as integrators in the Riemann–Stieltjes integral.
In several variables, $BV(\Omega)$ is the space of integrable functions whose distributional gradient is a finite measure.
It plays a role in the calculus of variations and in image processing.

## Explanations

### L1 human

A function has bounded variation when the total distance its values travel up and down is finite.
These functions form a vector space, because the variation of a sum is at most the sum of the variations.
The size of a function in this space is its starting value plus the total distance travelled.
For example, $x \mapsto x^2$ on $[-1, 1]$ goes down by $1$ and then up by $1$, so its total variation is $2$.
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# Gromov–Witten invariant

## Definition

Let $X$ be a smooth projective variety over $\C$ of dimension $r$.
Let $g, n \ge 0$ be integers and let $\beta \in H_2(X, \Z)$.
A *stable map* of genus $g$ with $n$ marked points and class $\beta$ is a tuple $(C, x_1, \dots, x_n, f)$ such that:

1. $C$ is a connected projective curve of arithmetic genus $g$ whose only singularities are nodes.
2. $x_1, \dots, x_n$ are distinct smooth points of $C$.
3. $f \colon C \to X$ is a morphism with $f_*[C] = \beta$.
4. The tuple has finitely many automorphisms.

Let $\overline{M}_{g,n}(X, \beta)$ be the [[moduli stack|FJZRW9P0K3]] of such stable maps, with evaluation maps $\mathrm{ev}_i \colon \overline{M}_{g,n}(X, \beta) \to X$, $\mathrm{ev}_i(C, x, f) = f(x_i)$.
It carries a virtual fundamental class $[\overline{M}_{g,n}(X, \beta)]^{\mathrm{vir}}$ of degree
$$(1 - g)(r - 3) + \int_\beta c_1(T_X) + n.$$
For $\gamma_1, \dots, \gamma_n \in H^*(X, \Q)$, the *Gromov–Witten invariant* is
$$\langle \gamma_1, \dots, \gamma_n \rangle_{g, \beta} = \int_{[\overline{M}_{g,n}(X, \beta)]^{\mathrm{vir}}} \mathrm{ev}_1^* \gamma_1 \cup \cdots \cup \mathrm{ev}_n^* \gamma_n \in \Q.$$

## Remarks

Example: let $X = \mathbb{P}^2$, $g = 0$, $\beta = d$ times the class of a line, and $n = 3d - 1$.
The virtual dimension is $-1 + 3d + (3d - 1) = 6d - 2$, and each point class has complex degree $2$, so $3d - 1$ point classes fill the dimension exactly.
The invariant $N_d = \langle \mathrm{pt}, \dots, \mathrm{pt} \rangle_{0, d}$ is the number of rational plane curves of degree $d$ through $3d - 1$ general points, with $N_1 = 1$, $N_2 = 1$ and $N_3 = 12$.

Non-example: let $C$ have a genus $0$ component that $f$ contracts to a point.
If that component carries only two special points, nodes or marked points, the map is not stable.
That component has infinitely many automorphisms fixing its two special points, so clause 4 fails.

The same construction works for a compact symplectic manifold with a compatible almost complex structure, with [[pseudoholomorphic curves|N469E47V75]] in place of algebraic maps.
The invariant is zero unless the degrees of the $\gamma_i$ add up to the virtual dimension.
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# Hermite interpolation

## Summary

Hermite interpolation finds a polynomial that matches, at each node, a prescribed value and prescribed derivatives up to some order.
It extends [[polynomial interpolation|SS1YMS2NPC]], which prescribes values only, and Taylor polynomials, which prescribe derivatives at a single point.
If $N$ conditions are imposed in total, there is exactly one polynomial of degree at most $N - 1$ that satisfies them.
It can be computed with the Newton form using [[divided differences|N5ZHV4HWDC]] with repeated nodes, and it underlies cubic Hermite splines and Gaussian quadrature.

## Explanations

### L1 human

Ordinary interpolation asks a curve to pass through given points.
Hermite interpolation also asks it to have given slopes there, and possibly given curvature and more.
Passing through $(0, 0)$ and $(1, 1)$ with slope $0$ at both points gives four conditions, and the cubic $3x^2 - 2x^3$ meets all of them.
This curve leaves and arrives flat, which is why it is used to join pieces of curves smoothly.
The [[definition|DZQ4EW6907]] states the conditions.
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# Difference kernel

## Definition

Let $J \subseteq \R$ be an interval, possibly unbounded.
A *difference kernel* on $J$ is a [[kernel|GSMBXT7FPK]] $K \colon J \times J \to \C$ for which there is a function $k \colon \set{x - s : x, s \in J} \to \C$ such that
$$K(x, s) = k(x - s) \quad \text{for all } x, s \in J.$$

## Remarks

Example: $K(x, s) = \cos(x - s)$ with $k = \cos$.

Non-example: $K(x, s) = x s$ on $[0, 1]$.
If $x s = k(x - s)$, then putting $s = x$ gives $x^2 = k(0)$ for all $x \in [0, 1]$, which is impossible.

On $J = \R$, the integral operator is the [[convolution|2Z29JN3VES]] $\int_{\R} k(x - s) \varphi(s) \, ds = (k * \varphi)(x)$.
On $J = [0, \infty)$ with upper limit $x$, it is the one-sided convolution $\int_0^x k(x - s) \varphi(s) \, ds$, which the Laplace transform turns into a product.
Equations with a difference kernel are called *convolution equations*.
The equation $\varphi(x) = f(x) + \int_0^{\infty} k(x - s) \varphi(s) \, ds$ for $x \ge 0$ is the Wiener–Hopf equation.
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# Herglotz representation theorem

## Statement

Let $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ and $\mathbb{T} = \set{\zeta \in \C : \abs{\zeta} = 1}$.

1. Let $u \colon \mathbb{D} \to \R$ be a [[harmonic function|0DEB0JMN0W]] with $u \ge 0$. Then there is a unique finite positive Borel measure $\mu$ on $\mathbb{T}$ such that for every $z \in \mathbb{D}$
$$u(z) = \int_{\mathbb{T}} \frac{1 - \abs{z}^2}{\abs{\zeta - z}^2}\, d\mu(\zeta).$$
Its total mass is $\mu(\mathbb{T}) = u(0)$.
2. Let $f$ be a [[holomorphic function|0NAHXR1FRH]] on $\mathbb{D}$ with $\operatorname{Re} f \ge 0$. Then, with $\mu$ the measure of part 1 for $u = \operatorname{Re} f$, for every $z \in \mathbb{D}$
$$f(z) = \int_{\mathbb{T}} \frac{\zeta + z}{\zeta - z}\, d\mu(\zeta) + i \operatorname{Im} f(0).$$

## Remarks

The integrand in part 1 is the [[Poisson kernel|W19567AVYG]].
For $0 < r < 1$, the measures $\mu_r = u(r\zeta)\, \frac{\abs{d\zeta}}{2\pi}$ on $\mathbb{T}$ are positive with total mass $u(0)$, by the mean value property.
The Poisson formula on the disc of radius $r$ expresses $u(rz)$ through $\mu_r$; a weak-star limit of the $\mu_r$ as $r \to 1$ gives $\mu$.
Part 2 follows from part 1 as in the [[Schwarz integral formula|61WT6CE08W]].
Example: for the Dirac measure at $1$, the formula of part 1 gives $u(z) = (1 - \abs{z}^2)/\abs{1 - z}^2$, a positive harmonic function that is unbounded near $1$, and part 2 gives $f(z) = (1 + z)/(1 - z)$.
The hypothesis $u \ge 0$ is needed.
The function $u(z) = \operatorname{Re} \frac{(1 + z)^2}{(1 - z)^2}$ is harmonic but takes both signs.
It is not the Poisson integral of any finite signed measure, because it grows too fast near $1$.

## Explanations

### L1 human

A nonnegative harmonic function on the disc is a weighted average of the simplest ones, the Poisson kernels with their pole at a single boundary point.
The weights form a measure on the boundary circle.
So every such function is built from a mass distribution on the circle, and the mass distribution is unique.

### L2 human

The theorem identifies the cone of nonnegative harmonic functions on the disc with the cone of finite positive measures on the circle.
In the holomorphic form it describes all functions with nonnegative real part, the Carathéodory class, which is used in coefficient problems for [[univalent functions|R91E83SDBD]].
It also underlies spectral representations in operator theory, such as the moment problem and the spectral theorem for unitary operators.
Together with [[Harnack's inequality|YNH832D50W]] it controls the size of positive harmonic functions.
</document>

<document source="https://tamnd.github.io/math2/node/9ZBM46J88T.md" tag="9ZBM46J88T" kind="statement">
# Artinian reduced rings are semisimple

## Statement

Let $R$ be a left [[Artinian ring|20QEAM8AX2]] with identity.
Assume that $R$ is [[reduced|SZHY975FFX]]: its only [[nilpotent element|0ZB2JG1M9C]] is $0$.
Then $R$ is a [[semisimple ring|NY25HXA478]].
More precisely, $R \cong D_1 \times \cdots \times D_k$ for some [[division rings|0EA5BTA0CX]] $D_1, \dots, D_k$.
If $R$ is moreover commutative, then $R$ is a finite product of [[fields|04C5ZBG1WT]].

## Remarks

The [[Jacobson radical|XA0DCNH7WY]] $J$ of a left Artinian ring is [[nilpotent|H7RAXJV9CP]], so each element of $J$ is nilpotent.
Since $R$ is reduced, $J = 0$.
A left Artinian ring with zero Jacobson radical is semisimple, and the [[Artin–Wedderburn theorem|R1F271180N]] gives $R \cong M_{n_1}(D_1) \times \cdots \times M_{n_k}(D_k)$.
If some $n_i \ge 2$, the matrix unit $e_{12}$ in that factor is a nonzero element with square $0$, which is impossible; so every $n_i = 1$.

Example: $\Z/6\Z \cong \Z/2\Z \times \Z/3\Z$ is Artinian and reduced, and it is a product of two fields.
The Artinian hypothesis is needed: $\Z$ is reduced but not semisimple, since the ideal $2\Z$ has no complement in $\Z$.
Reducedness is needed: $\Z/4\Z$ is Artinian, but $2$ is a nonzero nilpotent element, and $\Z/4\Z$ is not semisimple.

## Explanations

### L1 human

An Artinian ring is a ring with a strong finiteness condition on its ideals.
If such a ring has no nonzero element whose power is zero, then it breaks up into finitely many pieces, and each piece is a division ring.
In the commutative case, each piece is a field.

### L2 human

The nilpotent elements are the only obstruction to semisimplicity for Artinian rings, because the radical of an Artinian ring is a nilpotent ideal.
So for Artinian rings, "reduced" implies "semiprimitive", which implies "semisimple".
In commutative algebra this is the statement that a reduced Artinian ring is a finite product of fields.
It is used, for example, to describe reduced zero-dimensional Noetherian rings and finite étale algebras over a field.
</document>

<document source="https://tamnd.github.io/math2/node/9ZBWK5WSC7.md" tag="9ZBWK5WSC7" kind="definition">
# Euler equations

## Definition

Let $\Omega \subseteq \R^3$ be an open set and let $T > 0$.
The *incompressible Euler equations* are the system
$$\partial_t u + (u \cdot \nabla) u = - \nabla p, \qquad \nabla \cdot u = 0,$$
for an unknown [[vector field|0299VCW1TH]] $u \colon \Omega \times (0, T) \to \R^3$, the velocity, and an unknown function $p \colon \Omega \times (0, T) \to \R$, the pressure.
Here $\nabla p$ is the [[gradient|0ABK9V9Y64]] of $p$ in $x$, $\nabla \cdot u$ is the [[divergence|0MWV6ZTK6S]] of $u$, and $(u \cdot \nabla) u$ has components $\sum_{j=1}^3 u_j \partial_j u_i$.
The density is normalised to $1$.
On a bounded domain one adds the boundary condition $u \cdot n = 0$, where $n$ is the outer unit normal.

## Remarks

Example: on $\Omega = \R^3$, the rigid rotation $u = (-y, x, 0)$ with pressure $p = (x^2 + y^2)/2$ is a solution.
Indeed $\nabla \cdot u = 0$, $(u \cdot \nabla) u = (-x, -y, 0)$ and $-\nabla p = (-x, -y, 0)$.

Non-example: $u = (x, 0, 0)$ is not a solution for any pressure, because $\nabla \cdot u = 1 \ne 0$.

The *compressible Euler equations* have the density $\rho$ and the pressure as further unknowns:
$$\partial_t \rho + \nabla \cdot (\rho u) = 0, \qquad \partial_t (\rho u) + \nabla \cdot (\rho u \otimes u) + \nabla p = 0,$$
together with an energy equation or an equation of state that relates $p$ to $\rho$.
They form a system of hyperbolic conservation laws, whose solutions can develop shock waves.
</document>

<document source="https://tamnd.github.io/math2/node/9ZC177PVPC.md" tag="9ZC177PVPC" kind="statement">
# Secant method

## Statement

Let $I$ be an open interval, let $f \colon I \to \R$ be [[twice continuously differentiable|0VW90BBSJ4]], and let $r \in I$ with $f(r) = 0$ and [[derivative|0V1EJCT8G6]] $f'(r) \ne 0$.
The *secant method* starts from two points $x_0 \ne x_1$ in $I$ and sets
$$x_{k+1} = x_k - f(x_k) \, \frac{x_k - x_{k-1}}{f(x_k) - f(x_{k-1})}, \qquad k \ge 1,$$
stopping if $f(x_k) = 0$.
Then there is $\delta > 0$ with $J = [r - \delta, r + \delta] \subseteq I$ such that for all distinct $x_0, x_1 \in J$:

1. all iterates are defined and lie in $J$, and $x_k \to r$;
2. the errors $e_k = x_k - r$ satisfy
$$e_{k+1} = \frac{f''(\xi_k)}{2 f'(\eta_k)} \, e_k \, e_{k-1}$$
for some points $\xi_k, \eta_k$ in the smallest interval containing $x_{k-1}$, $x_k$ and $r$.

## Remarks

The error formula comes from writing $x_{k+1} - r$ with divided differences: $e_{k+1} = e_k e_{k-1} f[x_{k-1}, x_k, r] / f[x_{k-1}, x_k]$, and the mean value theorem for divided differences turns these into $f''(\xi_k)/2$ and $f'(\eta_k)$.
Let $M$ be the maximum of $\abs{f''}$ over $J$ divided by twice the minimum of $\abs{f'}$ over $J$.
Choose $\delta$ so that $f' \ne 0$ on $J$ and $M \delta < 1$; then $\abs{e_{k+1}} \le M \delta \abs{e_k}$, which gives clause 1.
Example: for $f(x) = x^2 - 2$, $x_0 = 1$, $x_1 = 2$, the iterates are $4/3$, $1.4$, $1.41463$, $1.41421$, approaching $\sqrt{2}$.
The condition $f'(r) \ne 0$ matters.
At a double zero such as that of $f(x) = x^2$, the method still converges from nearby points on the same side of $0$, but only linearly, with ratio tending to $(\sqrt{5} - 1)/2$.
Far from a zero the method can fail, for example when $f(x_k) = f(x_{k-1})$ and the step is undefined.

## Explanations

### L1 human

The secant method is Newton's method without derivatives.
Instead of the tangent line at one point, it draws the line through the last two points of the graph.
Its crossing with the axis is the next guess.
Near a simple zero the guesses improve very fast, and each step needs only one new value of $f$.

### L2 human

The error recursion $e_{k+1} \approx C e_k e_{k-1}$ with $C = f''(r)/(2f'(r))$ is the source of the [[superlinear convergence of the secant method|GP9E4AWJ8W]], with order the golden ratio $(1 + \sqrt{5})/2 \approx 1.618$.
[[Newton's method|07EXVNF9VP]] has order $2$ but needs $f'$ at each step; per function evaluation, the secant method is often faster.
Keeping a bracket instead of the last two points gives [[regula falsi|462AEY7MWF]], which is safe but can be slow.
</document>

<document source="https://tamnd.github.io/math2/node/9ZCCXT3KKP.md" tag="9ZCCXT3KKP" kind="statement">
# Nash–Moser theorem

## Statement

A *graded* [[Fréchet space|3GAA614673]] is a Fréchet space $F$ with a fixed increasing sequence of seminorms $\norm{\cdot}_0 \le \norm{\cdot}_1 \le \norm{\cdot}_2 \le \cdots$ that defines its topology.
Let $F$ and $G$ be graded Fréchet spaces and let $U \subseteq F$ be open.
A continuous map $P \colon U \to G$ is *tame* if each point of $U$ has a neighbourhood $V$ with this property.
There are integers $r, b \ge 0$ and constants $C_n$ with $\norm{P(f)}_n \le C_n(1 + \norm{f}_{n+r})$ for all $f \in V$ and all $n \ge b$.
It is *smooth tame* if it is smooth and all its derivatives are tame.
A graded Fréchet space is *tame* if it is a tame direct summand of a space of exponentially decreasing sequences in a Banach space.
For a compact manifold $M$, the space $C^\infty(M)$ graded by the $C^n$ norms is tame.

Let $F$ and $G$ be tame Fréchet spaces, let $U \subseteq F$ be open, and let $P \colon U \to G$ be a smooth tame map.
Suppose that for every $f \in U$ the [[derivative|06EH97V2A7]] $DP(f) \colon F \to G$ is bijective, and that the family of inverses $VP \colon U \times G \to F$, $VP(f, k) = DP(f)^{-1} k$, is a smooth tame map.
Then $P$ is locally invertible: each $f_0 \in U$ has an open neighbourhood $U_0$ such that $P(U_0)$ is open and $P \colon U_0 \to P(U_0)$ is a bijection.
Moreover, each such local inverse is a smooth tame map.

## Remarks

This is the form of the theorem given by Hamilton; Nash introduced the method to prove his embedding theorem for Riemannian manifolds, and Moser developed it as an abstract iteration.
The [[inverse function theorem in Banach spaces|M218MWZ62X]] does not apply, because in typical applications $DP(f)^{-1}$ loses derivatives: it maps $C^{n+r}$ data only to $C^{n}$ solutions.
The proof runs a Newton iteration $f_{j+1} = f_j - S_j\, VP(f_j, P(f_j) - g)$, where $S_j$ are smoothing operators that cut off high frequencies.
The quadratic convergence of Newton's method beats the loss of derivatives, and the tame estimates make this precise.

Example: in Kolmogorov–Arnold–Moser theory and in isometric embeddings of Riemannian manifolds, the linearised equations lose derivatives.
There the theorem gives solutions of the nonlinear equations near a known one.

## Explanations

### L1 human

The usual inverse function theorem says that if the derivative of a map is invertible, then the map itself is invertible near the point.
For maps between spaces of smooth functions this can fail when the inverse of the derivative makes functions rougher.
The Nash–Moser theorem restores the conclusion when every map involved obeys polynomial-type bounds that lose only a fixed number of derivatives.

### L2 human

The theorem is a tool for nonlinear PDEs whose linearisation is solvable only with a loss of derivatives, such as small divisor problems and isometric embeddings.
To apply it, one first checks that the nonlinear map is smooth tame, which is usually routine for differential operators.
One then checks that the linearised problem is solvable at every point near the base point, with tame estimates.
The second check is the real work, and it is often done with energy estimates or with Fourier analysis.
</document>

<document source="https://tamnd.github.io/math2/node/9ZG6HNM84Q.md" tag="9ZG6HNM84Q" kind="statement">
# Poisson–Jensen formula

## Statement

Let $0 < R < \infty$ and let $f$ be a [[meromorphic function|0ZZ7ZYDR8W]] on a neighbourhood of the closed disc $\abs{z} \le R$, not identically zero.
Assume that $f$ has no zeros and no poles on the circle $\abs{z} = R$.
Let $a_1, \dots, a_m$ be the zeros and $b_1, \dots, b_n$ the poles of $f$ in $\abs{z} < R$, each listed as often as its multiplicity.
Then, for every $z$ with $\abs{z} < R$ that is neither a zero nor a pole,
$$\log\abs{f(z)} = \frac{1}{2\pi} \int_0^{2\pi} \frac{R^2 - \abs{z}^2}{\abs{R e^{i\theta} - z}^2} \log\abs{f(R e^{i\theta})} \, d\theta - \sum_{\mu = 1}^{m} \log\abs{\frac{R^2 - \bar a_\mu z}{R (z - a_\mu)}} + \sum_{\nu = 1}^{n} \log\abs{\frac{R^2 - \bar b_\nu z}{R (z - b_\nu)}}.$$

## Remarks

Proof idea: each factor $B_a(z) = R(z - a)/(R^2 - \bar a z)$ is a [[Blaschke factor|9N6R5ADJXT]] for the disc of radius $R$: it vanishes only at $a$ and has modulus $1$ on $\abs{z} = R$.
So $g = f \prod_\nu B_{b_\nu} / \prod_\mu B_{a_\mu}$ is holomorphic and zero-free near the closed disc, and $\log\abs{g}$ is [[harmonic|0DEB0JMN0W]].
The Poisson formula, with the [[Poisson kernel|TCYSWCTYYM]] as in the integral, represents $\log\abs{g}$, and $\log\abs{g} = \log\abs{f}$ on the circle.
At $z = 0$ (if $f(0) \ne 0, \infty$) the formula becomes [[Jensen's formula|497K99BQ4Y]]: $\log\abs{f(0)}$ equals the [[circle average|Y7A1TP40MB]] of $\log\abs{f}$ minus $\sum_\mu \log(R/\abs{a_\mu})$ plus $\sum_\nu \log(R/\abs{b_\nu})$.
Example: $f(z) = z$ and $R = 1$; the integral vanishes and the only zero is $a_1 = 0$, so the right side is $-\log\abs{1/z} = \log\abs{z}$.
The formula still holds when $f$ has zeros or poles on the circle, since $\log\abs{f}$ remains integrable there, but the proof needs an extra limit argument.

## Explanations

### L1 human

If one knows the size of a meromorphic function on a circle and where its zeros and poles are inside, one knows its size everywhere inside.
The integral is the harmonic part, and each zero pulls $\log\abs{f}$ down while each pole pushes it up.

### L2 human

The formula is the basic identity of Nevanlinna theory, where it is used to bound the number of zeros of a function by its growth.
It shows that $\log\abs{f}$ for holomorphic $f$ is a harmonic function minus a sum of Green's functions of the disc, one for each zero.
This is a special case of the Riesz decomposition of the subharmonic function $\log\abs{f}$, whose Riesz measure is the sum of point masses at the zeros.
</document>

<document source="https://tamnd.github.io/math2/node/9ZK07AKD4M.md" tag="9ZK07AKD4M" kind="definition">
# Kolmogorov epsilon-entropy

## Definition

Let $(X, \rho)$ be a [[metric space|0VKD1TP7F3]], let $K \subseteq X$ be a nonempty [[totally bounded set|020Q06TYYH]], and let $\varepsilon > 0$.
Let $N_\varepsilon(K)$ be the least number $N$ such that $K \subseteq A_1 \cup \cdots \cup A_N$ for some subsets $A_1, \dots, A_N$ of $X$ of diameter at most $2\varepsilon$.
The *Kolmogorov $\varepsilon$-entropy* of $K$ is
$$H_\varepsilon(K) = \log_2 N_\varepsilon(K).$$

## Remarks

The number $N_\varepsilon(K)$ is finite because $K$ is totally bounded.
Covering by closed balls of radius $\varepsilon$ instead gives the covering number $N'_\varepsilon(K)$, and $N_\varepsilon(K) \le N'_\varepsilon(K) \le N_{\varepsilon/2}(K)$.
So the two versions have the same growth up to a change of $\varepsilon$ by a factor $2$.
Kolmogorov and Tikhomirov also used the $\varepsilon$-capacity, the logarithm of the largest number of points of $K$ with pairwise distances greater than $\varepsilon$.

Example: $K = [0, 1]$ in $\R$.
A set of diameter at most $2\varepsilon$ lies in an interval of length $2\varepsilon$, so at least $1/(2\varepsilon)$ of them are needed.
On the other hand, $\ceil{1/(2\varepsilon)}$ consecutive closed intervals of length $2\varepsilon$ suffice.
So $N_\varepsilon([0, 1]) = \ceil{1/(2\varepsilon)}$ and $H_\varepsilon([0, 1]) = \log_2 \ceil{1/(2\varepsilon)}$.

Non-example: $K = [0, \infty)$ is not totally bounded, and no finite family of sets of diameter $2\varepsilon$ covers it, so $H_\varepsilon(K)$ is not defined, or is $+\infty$.

For the class of functions $f$ on $[0, 1]$ with $\abs{f} \le 1$ and Lipschitz constant at most $1$, in the supremum norm, $H_\varepsilon$ is of order $1/\varepsilon$ as $\varepsilon \to 0$.
</document>

<document source="https://tamnd.github.io/math2/node/9ZP4ER2N65.md" tag="9ZP4ER2N65" kind="definition">
# Product measure

## Definition

Let $(X, \Sigma, \mu)$ and $(Y, T, \nu)$ be measure spaces, and assume that $\mu$ and $\nu$ are [[σ-finite|WEBBT18D4X]].
Let $\Sigma \otimes T$ be the [[product σ-algebra|6HVCTJGB61]] on $X \times Y$.
The *product measure* $\mu \times \nu$ is the unique [[measure|PR11BZQQPG]] on $\Sigma \otimes T$ with
$$(\mu \times \nu)(A \times B) = \mu(A) \, \nu(B) \quad \text{for all } A \in \Sigma, \ B \in T,$$
with the convention $0 \cdot \infty = 0$.

## Remarks

Existence follows from [[Carathéodory's extension theorem|Q8EXG7EKYW]] applied to the rectangles, and uniqueness from σ-finiteness and the π-λ theorem.
For $E \in \Sigma \otimes T$ one has $(\mu \times \nu)(E) = \int_X \nu(E_x) \, d\mu(x) = \int_Y \mu(E^y) \, d\nu(y)$, where $E_x$ and $E^y$ are the sections of $E$.

Example: let $\mu$ and $\nu$ be Lebesgue measure on the Borel sets of $[0, 1]$.
Then $\mu \times \nu$ is two-dimensional Lebesgue measure on the Borel sets of $[0, 1]^2$: both measures give each rectangle its area, so they agree by uniqueness.

Non-example: let $\delta$ be the image of Lebesgue measure $\lambda$ on $[0, 1]$ under $x \mapsto (x, x)$, a measure on the diagonal of $[0, 1]^2$.
For $A = [0, \tfrac12]$ and $B = [\tfrac12, 1]$, $\delta(A \times B) = \lambda(A \cap B) = 0$, but $\lambda(A) \lambda(B) = \tfrac14$.
So $\delta$ is not $\lambda \times \lambda$.

Without σ-finiteness, uniqueness can fail.
For Lebesgue measure on $[0, 1]$ and counting measure on $[0, 1]$, there are different measures on the product σ-algebra with the rectangle property.
The product of Lebesgue measures is often completed to give Lebesgue measure on $\R^{m+n}$.
</document>

<document source="https://tamnd.github.io/math2/node/9ZQC7ARWTT.md" tag="9ZQC7ARWTT" kind="concept">
# Critical point theory

## Summary

Critical point theory studies the [[critical points|0YTCMFB4HC]] of a differentiable function on a manifold.
Its tool is the topology of the manifold and of the sublevel sets of the function.
Its main branches are Morse theory, which uses [[nondegenerate critical points|MGPPCDFQDG]] and their [[indices|MAX2ZYG6H9]], and [[Lusternik–Schnirelmann theory|QARH15D2PN]], which uses minimax values.
In infinite dimensions it needs a compactness hypothesis such as the [[Palais–Smale condition|4724KRNHAT]].
It then gives existence results for geodesics, harmonic maps and nonlinear elliptic equations.

## Explanations

### L1 human

A critical point of a function is a point where its derivative vanishes, such as a peak, a pit or a saddle of a landscape.
Critical point theory turns topology into existence theorems: if the shape of the space is complicated, the function must have many critical points.
For example, any smooth function on a circle has at least two critical points, a maximum and a minimum.
On a torus it has at least three, and a Morse function on a torus has at least four.
</document>

<document source="https://tamnd.github.io/math2/node/9ZSMF6NYKM.md" tag="9ZSMF6NYKM" kind="concept">
# Differential inclusion

## Summary

A differential inclusion is a relation $x'(t) \in F(t, x(t))$, where $F$ is a [[multivalued function|03CPD1HGF3]] that gives a set of allowed velocities at each time and state.
It generalizes the [[ordinary differential equation|0NT64JV96A]] $x' = f(t, x)$, which is the case where each set $F(t, x)$ has one element.
Differential inclusions describe control systems, equations with discontinuous right side, and systems with uncertain data.

## Explanations

### L1 human

An ordinary differential equation prescribes one velocity at each point.
A differential inclusion only prescribes a set of possible velocities, and a solution may choose any of them at each moment.
So there are usually many solutions from one starting point, like the many paths of a driver who may choose any speed below a limit.
For example, the solutions of $x'(t) \in [-1, 1]$ are the functions whose slope never exceeds $1$ in absolute value, such as $x(t) = \sin t$ or $x(t) = \abs{t}$.
</document>

<document source="https://tamnd.github.io/math2/node/9ZV3T5C6B7.md" tag="9ZV3T5C6B7" kind="definition">
# Subharmonic function

## Definition

Let $\Omega$ be an [[open set|02BH3MCBQB]] in $\C$.
A function $u \colon \Omega \to [-\infty, \infty)$ is *subharmonic* if:

1. $u$ is upper semicontinuous, that is, $\set{z \in \Omega : u(z) < c}$ is open for every real $c$.
2. $u$ satisfies the sub-mean value inequality: for every $a \in \Omega$ and $r > 0$ with $\set{z : \abs{z - a} \le r} \subseteq \Omega$,
$$u(a) \le \frac{1}{2\pi}\int_0^{2\pi} u(a + r e^{it})\, dt.$$

## Remarks

Some authors also require that $u$ is not identically $-\infty$ on any connected component of $\Omega$.
The same definition, with averages over spheres, gives subharmonic functions on open sets of $\R^n$.
It is equivalent to require the inequality only for all small enough $r$ at each point.
Example: $u(z) = \abs{z}^2$ is continuous, and its mean over the circle of radius $r$ about $a$ is $\abs{a}^2 + r^2 \ge \abs{a}^2$, so $u$ is subharmonic.
Non-example: $u(z) = -\abs{z}^2$ is continuous, but its mean over the circle of radius $r$ about $0$ is $-r^2 < 0 = u(0)$, so clause 2 fails.
A function $u$ of class $C^2$ is subharmonic if and only if its [[Laplacian|0E0B6RA8K0]] satisfies $\Delta u \ge 0$.
A function is [[harmonic|0DEB0JMN0W]] if and only if both $u$ and $-u$ are subharmonic and $u$ is finite.
For a holomorphic function $f$ on $\Omega$, the function $\log\abs{f}$ is subharmonic, with value $-\infty$ at the zeros of $f$.
</document>

<document source="https://tamnd.github.io/math2/node/9ZVCJWYPB2.md" tag="9ZVCJWYPB2" kind="definition">
# Moore family

## Definition

Let $X$ be a set and let $\mathcal{M}$ be a family of subsets of $X$, that is, a subset of the [[power set|0BC7KX51Z8]] of $X$.
The family $\mathcal{M}$ is a *Moore family* on $X$ when both conditions hold.

1. $X \in \mathcal{M}$.
2. For each nonempty subfamily $\mathcal{N} \subseteq \mathcal{M}$, the [[intersection|0839XNQXDK]] $\bigcap \mathcal{N}$ lies in $\mathcal{M}$.

## Remarks

The closed sets of a topological space $X$ form a Moore family: $X$ is closed, and any intersection of closed sets is closed.
The open sets of $\R$ do not: the open intervals $(-1/n, 1/n)$ for $n \ge 1$ have intersection $\set{0}$, which is not open.
If $\mathcal{M}$ is a Moore family, then $c(A) = \bigcap \set{M \in \mathcal{M} : A \subseteq M}$ is a [[closure operator|17FVAN6HXX]] on the subsets of $X$.
Its closed sets are exactly the members of $\mathcal{M}$.
Conversely, the closed sets of any closure operator on the subsets of $X$ form a Moore family.
With the convention that the empty intersection is $X$, the two conditions merge into one: $\mathcal{M}$ is closed under all intersections.
Other names are closure system and intersection-closed family.

## Explanations

### L1 human

A Moore family is a collection of subsets that contains the whole set and survives taking intersections.
Its members are the "closed" sets of a closure operator.
</document>

<document source="https://tamnd.github.io/math2/node/9ZW7BCJMDT.md" tag="9ZW7BCJMDT" kind="statement">
# Combinatorial Nullstellensatz

## Statement

Let $F$ be a [[field|04C5ZBG1WT]] and let $f \in F[x_1, \dots, x_n]$ be a [[polynomial in $n$ variables|07GNX8GCYP]].
Let $S_1, \dots, S_n$ be nonempty finite subsets of $F$ and let $g_i(x_i) = \prod_{s \in S_i} (x_i - s)$.

1. If $f(s_1, \dots, s_n) = 0$ for all $(s_1, \dots, s_n) \in S_1 \times \dots \times S_n$, then there are polynomials $h_1, \dots, h_n \in F[x_1, \dots, x_n]$ with $\deg h_i \le \deg f - \deg g_i$ and $f = \sum_{i=1}^n h_i g_i$.
2. Let $t_1, \dots, t_n \ge 0$ be integers with $\abs{S_i} > t_i$ for each $i$. Suppose that the [[degree|0P4JRMSWNH]] of $f$ is $t_1 + \dots + t_n$ and that the coefficient of $x_1^{t_1} \cdots x_n^{t_n}$ in $f$ is nonzero. Then there is $(s_1, \dots, s_n) \in S_1 \times \dots \times S_n$ with $f(s_1, \dots, s_n) \ne 0$.

## Remarks

The result and its name are due to Alon.
For part 1, reduce $f$ by replacing each power $x_i^{e}$ with $e \ge \abs{S_i}$ using $g_i(x_i) = 0$; the remainder has degree less than $\abs{S_i}$ in each $x_i$ and still vanishes on $S_1 \times \dots \times S_n$.
A polynomial of that kind which vanishes on the whole grid is zero.
This follows by induction on $n$, since a nonzero polynomial in one variable of degree $d$ has at most $d$ roots.
Part 2 follows from part 1.
If $f$ vanished on the grid, then $f = \sum_i h_i g_i$ with the stated degree bounds.
Every monomial of top degree in $h_i g_i$ has $x_i$-degree at least $\abs{S_i} > t_i$, so $x_1^{t_1} \cdots x_n^{t_n}$ could not appear in $f$.

The size condition is needed: for $n = 1$, $f = x_1(x_1 - 1)$, $t_1 = 2$ and $S_1 = \set{0, 1}$, we have $\abs{S_1} = t_1$ and $f$ vanishes on $S_1$.

Example: the [[Cauchy–Davenport theorem|F7K7XFACSG]] follows from part 2.
For a prime $p$ and nonempty $A, B \subseteq \Z/p\Z$ with $\abs{A} + \abs{B} - 1 \le p$, suppose the [[sumset|XNBN31HQWB]] $A + B$ lies in a set $C$ of size $\abs{A} + \abs{B} - 2$.
Apply part 2 to $f(x, y) = \prod_{c \in C} (x + y - c)$ with $t_1 = \abs{A} - 1$ and $t_2 = \abs{B} - 1$: the coefficient of $x^{t_1} y^{t_2}$ is $\binom{t_1 + t_2}{t_1}$, which is nonzero modulo $p$, yet $f$ vanishes on $A \times B$, a contradiction.

## Explanations

### L1 human

A polynomial that is not too big cannot vanish on a large grid of points.
The theorem makes "not too big" precise: look at one top-degree term and compare its exponents with the sizes of the sets along each axis.
If each set is larger than the matching exponent, the polynomial is nonzero somewhere on the grid.

### L2 human

The method is to encode a combinatorial property as the vanishing of a polynomial on a grid, then exhibit a nonzero top coefficient.
It gives short proofs of results in additive number theory, such as Cauchy–Davenport, and of results on graph colouring and on hyperplane coverings of the cube.
It is a sharp version, for grids, of the fact that a nonzero polynomial cannot vanish everywhere on $F^n$ when $F$ is infinite.
</document>

<document source="https://tamnd.github.io/math2/node/A00DAT34F9.md" tag="A00DAT34F9" kind="definition">
# Lifting scheme

## Definition

Let $x = (x_n)_{n \in \Z}$ be a sequence of real numbers.
One *lifting stage* with operators $P$ and $U$ on sequences, usually linear filters, consists of three steps:

1. Split: $e_k = x_{2k}$ and $o_k = x_{2k+1}$ for $k \in \Z$.
2. Predict: $d = o - P(e)$.
3. Update: $s = e + U(d)$.

The output is the pair $(s, d)$ of approximation and detail sequences.
The *lifting scheme* computes a wavelet transform as a composition of such stages.
A stage may contain several predict and update steps in alternation.
At the end, $s$ and $d$ may be multiplied by constants.
The inverse of one stage is $e = s - U(d)$, then $o = d + P(e)$, then the merging of $e$ and $o$ into $x$.

## Remarks

The inverse formulas hold for every choice of $P$ and $U$, since each step adds to one half of the data a function of the other half only.
So invertibility does not depend on the operators, and $P$ and $U$ may even be nonlinear, for instance include rounding to integers.

Example (Haar): take $P(e)_k = e_k$ and $U(d)_k = d_k/2$.
Then $d_k = x_{2k+1} - x_{2k}$ and $s_k = x_{2k} + (x_{2k+1} - x_{2k})/2 = (x_{2k} + x_{2k+1})/2$.
This is the Haar transform up to constant factors: the orthonormal Haar step gives $\sqrt{2}\, s_k$ and $-d_k/\sqrt{2}$.

Example (CDF 5/3): take $P(e)_k = (e_k + e_{k+1})/2$ and $U(d)_k = (d_{k-1} + d_k)/4$.
This computes the 5/3 biorthogonal wavelet transform; with rounding inside $P$ and $U$ it maps integers to integers and is the lossless transform of JPEG 2000.

Non-example: a step $o \mapsto o - P(o)$ that modifies the odd samples using the odd samples themselves is not a lifting step.
It need not be invertible: with $P = \id$ it sends every odd part $o$ to $0$, so $o$ cannot be recovered.

Every [[discrete wavelet transform|P3H4MMC4ZH]] given by a two-channel perfect-reconstruction filter bank with finite filters can be factored into finitely many lifting steps and a scaling.

## Explanations

### L1 human

A lifting stage splits a signal into even and odd samples, records how badly the evens predict the odds, and then corrects the evens.
Each step only adds something to one half, computed from the other half, so it can always be undone.
</document>

<document source="https://tamnd.github.io/math2/node/A01GEGWKVB.md" tag="A01GEGWKVB" kind="concept">
# Variance reduction

## Summary

Variance reduction is the family of techniques that make a [[Monte Carlo method|HWP11G3HTN]] more accurate for the same computing effort.
The error of a plain Monte Carlo estimate with $n$ samples is about $\sigma/\sqrt{n}$, where $\sigma^2$ is the [[variance|0477R9PYAE]] of one sample.
The rate $n^{-1/2}$ is fixed for plain random sampling.
So the gains come from replacing the sampled quantity by another with the same mean and a smaller variance.
Halving $\sigma$ saves a factor of four in the number of samples.

## Explanations

### L1 human

A Monte Carlo estimate is noisy, and the noise shrinks slowly as more samples are taken.
Variance reduction attacks the noise at its source: it changes what is sampled so that each sample is less scattered while the average stays the same.
For example, to estimate the average of $f(U) = U$ for uniform $U$ in $(0, 1)$, the pair average $\frac12 (U + (1 - U))$ is always exactly $\frac12$, so it has no scatter at all.
Real problems do not give such perfect cancellation, but gains by factors of ten or a hundred are common.
</document>

<document source="https://tamnd.github.io/math2/node/A02K3MKQ3Q.md" tag="A02K3MKQ3Q" kind="definition">
# Metropolis–Hastings algorithm

## Definition

Let $S$ be a countable set and let $\pi$ be a probability distribution on $S$ with $\pi(x) > 0$ for all $x \in S$.
Let $Q = (Q(x, y))_{x, y \in S}$ be a [[transition matrix|0F462493EA]], the *proposal*, such that $Q(x, y) > 0$ if and only if $Q(y, x) > 0$.
For $x \ne y$ with $Q(x, y) > 0$, the *acceptance probability* is
$$\alpha(x, y) = \min\left(1, \frac{\pi(y) \, Q(y, x)}{\pi(x) \, Q(x, y)}\right).$$
The *Metropolis–Hastings chain* is the [[Markov chain|033TJBZDG0]] $(X_n)$ built as follows.
Given $X_n = x$, draw $Y$ from $Q(x, \cdot)$.
With probability $\alpha(x, Y)$ set $X_{n+1} = Y$; otherwise set $X_{n+1} = x$.
Its transition matrix has $P(x, y) = Q(x, y) \, \alpha(x, y)$ for $y \ne x$ and $P(x, x) = 1 - \sum_{y \ne x} P(x, y)$.

## Remarks

For $x \ne y$, $\pi(x) P(x, y) = \min\left(\pi(x) Q(x, y), \pi(y) Q(y, x)\right)$, which is symmetric in $x$ and $y$.
So $P$ satisfies [[detailed balance|CTGN6CK1ZS]] with respect to $\pi$, and $\pi$ is a [[stationary distribution|0KRH5157WP]].
Only the ratio $\pi(y)/\pi(x)$ is used, so $\pi$ may be known up to a constant factor.
Example: if $Q$ is symmetric, $\alpha(x, y) = \min(1, \pi(y)/\pi(x))$; this special case is the Metropolis algorithm.
Non-example: the chain that always accepts the proposal has transition matrix $Q$, whose stationary law is in general not $\pi$; it skips the acceptance step.
The chain is a [[Markov chain Monte Carlo|ZGNBR2SPZ0]] method when it is also irreducible.
The same construction works on general state spaces with densities in place of $\pi$ and $Q$.
</document>

<document source="https://tamnd.github.io/math2/node/A05WR6TEK5.md" tag="A05WR6TEK5" kind="concept">
# Empirical distribution function

## Summary

The empirical distribution function of a sample is the [[cumulative distribution function|0H8H7ZTHW2]] of the distribution that puts mass $1/n$ on each observation.
At each point $t$ it gives the fraction of observations that are at most $t$.
It estimates the true distribution function without any parametric assumption, and for an i.i.d. sample the [[Glivenko–Cantelli theorem|E7B78X780X]] shows that it converges to it uniformly, almost surely.
Goodness-of-fit tests such as the [[Kolmogorov–Smirnov test|YZYHWC6C5F]] compare it with a hypothesised distribution function.

## Explanations

### L1 human

Sort your data and draw a staircase that starts at height $0$ and jumps up by $1/n$ at each data point.
This staircase is the empirical distribution function.
For the data $1, 2, 2, 5$, it is $0$ before $1$, $1/4$ on $[1, 2)$, $3/4$ on $[2, 5)$ and $1$ from $5$ on.
With more and more data, the staircase hugs the true distribution function more and more closely.
</document>

<document source="https://tamnd.github.io/math2/node/A06RA8KB0T.md" tag="A06RA8KB0T" kind="concept">
# Coalgebra

## Summary

A coalgebra is a vector space $C$ over a field $K$ with a comultiplication $\Delta \colon C \to C \otimes C$ and a counit $\varepsilon \colon C \to K$.
Its axioms come from reversing the arrows in the definition of an [[associative algebra|3CTW1P4HBP]].
The dual space of a coalgebra is an algebra, and the dual of a finite-dimensional algebra is a coalgebra.
Coalgebras are the other half of [[bialgebras|Q9HHVN606G]] and [[Hopf algebras|7XFYP0N84H]], and they model ways of splitting an object into two pieces.

## Explanations

### L1 human

Multiplication takes two elements and returns one; comultiplication takes one element and returns a sum of pairs.
It lists the ways to split an element into two parts.
For the polynomial ring $K[x]$, one can split $x^n$ as $\Delta(x^n) = \sum_{k} \binom{n}{k} x^k \otimes x^{n-k}$, like choosing which of $n$ objects go to the left.
The counit picks out the part where nothing is split off.
Coassociativity says that splitting into three parts gives the same result whichever way one splits first.
</document>

<document source="https://tamnd.github.io/math2/node/A06SGFCSCE.md" tag="A06SGFCSCE" kind="concept">
# General linear model

## Summary

The general linear model writes a matrix of responses as a known design matrix times an unknown coefficient matrix, plus random errors with mean zero.
It contains multiple [[linear regression|41FVC4P5JB]], [[analysis of variance|M9AQ5HKGP4]], analysis of covariance, [[multivariate analysis of variance|8J3B7N8CAT]] and the $t$-test and [[F-test|8MTVCMAATD]] as special cases.
Its theory rests on [[ordinary least squares|YM708KHV2W]] and on projections in $\R^n$.
It should not be confused with the [[generalized linear model|K6C48KMRKC]], which allows non-normal responses and a link function.

## Explanations

### L1 human

Many classical methods look different but share one shape: response equals a linear combination of known columns plus noise.
The known columns can be measured quantities, like height or dose, or indicator variables that say which group an observation belongs to.
For example, comparing the means of two groups is a regression on two indicator columns, one for each group.
Writing all these methods as one model lets a single theory of estimation and testing cover them all.
</document>

<document source="https://tamnd.github.io/math2/node/A0926EFKVP.md" tag="A0926EFKVP" kind="definition">
# Hausdorff measure

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]], and let $\operatorname{diam} U$ be the [[diameter|0KW3PN547K]] of a set $U \subseteq X$.
Let $s \ge 0$, with the convention $(\operatorname{diam} U)^0 = 1$ for every $U$.
For $A \subseteq X$ and $\delta > 0$, let
$$\mathcal{H}^s_\delta(A) = \inf \set{ \sum_{i=1}^\infty (\operatorname{diam} U_i)^s : A \subseteq \bigcup_{i=1}^\infty U_i, \ \operatorname{diam} U_i \le \delta },$$
where the [[infimum|0QF1MZ9FRG]] runs over all countable covers of $A$ by sets $U_i \subseteq X$ of diameter at most $\delta$.
The *$s$-dimensional Hausdorff measure* of $A$ is
$$\mathcal{H}^s(A) = \lim_{\delta \to 0} \mathcal{H}^s_\delta(A) = \sup_{\delta > 0} \mathcal{H}^s_\delta(A).$$

## Remarks

The limit exists in $[0, \infty]$ because $\mathcal{H}^s_\delta(A)$ grows as $\delta$ decreases.
The map $\mathcal{H}^s$ is a [[metric outer measure|3T6WG315GC]], so all Borel sets are measurable and its restriction to them is a measure.
Some authors multiply by a constant so that $\mathcal{H}^n$ equals [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$; without it, $\mathcal{H}^n$ is a constant multiple of Lebesgue measure.

Example: $\mathcal{H}^0$ is counting measure.
On $\R$, $\mathcal{H}^1$ is Lebesgue measure: covers by intervals realize the infimum, and the length of an interval is its diameter.

Non-example: the quantity without the limit $\delta \to 0$, the Hausdorff content $\mathcal{H}^s_\infty$ with no bound on diameters, is not a measure on Borel sets.
In $\R^2$ with $s = 1$, the segments $A = [0, 1] \times \set{0}$ and $B = [0, 1] \times \set{1}$ each have content $1$.
But $A \cup B$ lies in the unit square, of diameter $\sqrt 2$, so its content is at most $\sqrt 2 < 2$.

## Explanations

### L1 human

Cover the set by small pieces, add the $s$-th powers of their sizes, and take the best cover.
Letting the pieces become arbitrarily small gives the $s$-dimensional size of the set.
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<document source="https://tamnd.github.io/math2/node/A0E47AYDPX.md" tag="A0E47AYDPX" kind="definition">
# Max-plus algebra

## Definition

Let $\R_{\max} = \R \cup \set{-\infty}$.
For $a, b \in \R_{\max}$, define
$$a \oplus b = \max(a, b), \qquad a \otimes b = a + b,$$
with the conventions $\max(a, -\infty) = a$ and $a + (-\infty) = -\infty$.
The *max-plus semiring* is $(\R_{\max}, \oplus, \otimes)$.
The *max-plus algebra* is the algebra of vectors and [[matrices|0VD9ZSWWJG]] with entries in $\R_{\max}$, with the operations
$$(A \oplus B)_{ij} = a_{ij} \oplus b_{ij}, \qquad (A \otimes B)_{ij} = \bigoplus_{k} a_{ik} \otimes b_{kj} = \max_k (a_{ik} + b_{kj}).$$

## Remarks

$(\R_{\max}, \oplus, \otimes)$ is a commutative semiring with zero element $-\infty$ and unit element $0$.
It is idempotent: $a \oplus a = a$ for all $a$.
Example: $2 \oplus 7 = 7$, $2 \otimes 7 = 9$, and $2 \otimes (3 \oplus 7) = 9 = (2 \otimes 3) \oplus (2 \otimes 7)$, which checks distributivity in one case.
Non-example: $\R_{\max}$ is not a ring, since $5$ has no additive inverse: $\max(5, b) = -\infty$ has no solution $b$.
Let $A$ be an $n \times n$ matrix, with $a_{ij}$ the weight of an arc from $i$ to $j$.
Then $(A^{\otimes k})_{ij}$ is the largest weight of a path with $k$ arcs from $i$ to $j$.
The min-plus semiring $(\R \cup \set{+\infty}, \min, +)$ is isomorphic to it by $x \mapsto -x$.
</document>

<document source="https://tamnd.github.io/math2/node/A0F09CMEJM.md" tag="A0F09CMEJM" kind="definition">
# Poisson's equation

## Definition

Let $U \subseteq \R^n$ be an [[open set|02BH3MCBQB]] and let $f \colon U \to \R$ be a given function.
*Poisson's equation* on $U$ with right-hand side $f$ is the equation
$$\Delta u = f$$
for an unknown function $u \colon U \to \R$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
A *classical solution* is a twice continuously differentiable function $u$ on $U$ with $\Delta u(x) = f(x)$ for every $x \in U$.

## Remarks

Example: on $U = \R^n$ with $f = 1$, the function $u(x) = \abs{x}^2 / (2n)$ is a classical solution, since each term $\partial_i^2 (x_i^2 / (2n))$ equals $1/n$.
Non-example: $u(x) = \abs{x}^2 / 2$ on $\R^2$ is not a solution for $f = 1$, since $\Delta u = 2$.
With $f = 0$ the equation is [[Laplace's equation|S8RG6N1PRT]].
If $u_1$ and $u_2$ solve the same Poisson equation, then $u_1 - u_2$ is [[harmonic|0DEB0JMN0W]].
Many texts write the equation as $-\Delta u = f$, the sign that makes the operator positive.
If $f$ is twice continuously differentiable with compact support in $\R^n$, $n \ge 2$, then $u = \Phi * f$ solves $-\Delta u = f$ on $\R^n$, where $\Phi$ is the [[fundamental solution of the Laplacian|BSJ8X2XRN9]].
Weak and distributional solutions are defined for less regular $f$.
</document>

<document source="https://tamnd.github.io/math2/node/A0K9EZRM7E.md" tag="A0K9EZRM7E" kind="concept">
# Polymorphism of a relation

## Summary

An operation $f \colon A^n \to A$ is a polymorphism of a relation $R \subseteq A^m$ when applying $f$ coordinatewise to tuples of $R$ always gives a tuple of $R$.
One also says that $f$ preserves $R$, or that $R$ is invariant under $f$.
The polymorphisms of a set of relations form a [[clone|PTKR703G6A]], and this links operations to relations through a Galois connection.
Polymorphisms are the main tool of the algebraic approach to constraint satisfaction problems.

## Explanations

### L1 human

Write the tuples of a relation as the rows of a table, and pick $n$ rows.
Apply $f$ to each column of the chosen rows, and you get a new row.
The operation $f$ is a polymorphism if this new row is always again in the table.
For example, the operation $\min$ on $\R$ is a polymorphism of the order $\le$: if $a \le b$ and $c \le d$, then $\min(a, c) \le \min(b, d)$.
</document>

<document source="https://tamnd.github.io/math2/node/A0MXE8YWZA.md" tag="A0MXE8YWZA" kind="statement">
# Calderón–Vaillancourt theorem

## Statement

Let $n \ge 1$ and let $a \colon \R^n \times \R^n \to \C$ be a smooth function such that every derivative of $a$ is bounded: for all multi-indices $\alpha, \beta$,
$$\sup_{x, \xi \in \R^n} \abs{\partial_x^\alpha \partial_\xi^\beta a(x, \xi)} < \infty.$$
That is, $a$ lies in the [[symbol class|AM4RNWVV4Q]] $S^0_{0,0}$.
Let $a(x, D)$ be the [[pseudo-differential operator|FF27N0BWTP]] with symbol $a$, defined on Schwartz functions $u$ by
$$a(x, D) u(x) = (2\pi)^{-n} \int_{\R^n} e^{i x \cdot \xi} a(x, \xi) \, \hat u(\xi) \, d\xi,$$
where $\hat u$ is the [[Fourier transform|DPVM34ADZV]] of $u$.
Then $a(x, D)$ extends to a bounded operator on $L^2(\R^n)$.
Moreover there are a constant $C$ and an integer $N$, depending only on $n$, with
$$\norm{a(x, D)}_{L^2 \to L^2} \le C \max_{\abs{\alpha} \le N,\ \abs{\beta} \le N} \ \sup_{x, \xi} \abs{\partial_x^\alpha \partial_\xi^\beta a(x, \xi)}.$$

## Remarks

One standard proof cuts the symbol into pieces $a_{jk}$ localized near the points $(j, k)$ of the lattice $\Z^n \times \Z^n$ by a smooth partition of unity.
Each piece gives an operator with norm bounded by finitely many derivatives of $a$.
Integration by parts shows that the products $T_{jk}^* T_{j'k'}$ and $T_{jk} T_{j'k'}^*$ decay fast in $\abs{j - j'}$ and $\abs{k - k'}$.
The [[Cotlar–Stein lemma|XBJEDC18B1]] then sums the pieces.

Example: if $a(x, \xi) = m(\xi)$ does not depend on $x$, then $a(x, D)$ is the Fourier multiplier with symbol $m$, and Plancherel's theorem gives the bound $\sup \abs{m}$ directly.
The theorem says that bounded smooth dependence on $x$ costs only finitely many derivatives.
The same conclusion holds for symbols in the classes $S^0_{\rho, \rho}$ with $0 \le \rho < 1$.

For the class $S^0_{1,1}$, which allows $x$-derivatives to grow like $\abs{\xi}$, the operators are in general not bounded on $L^2$.

## Explanations

### L1 human

A pseudo-differential operator acts on a function by changing the strength of each frequency $\xi$, by an amount $a(x, \xi)$ that may also depend on the position $x$.
When $a$ depends only on $\xi$, the operator is bounded on $L^2$ as long as $a$ is bounded.
The Calderón–Vaillancourt theorem says the operator stays bounded on $L^2$ when $a$ depends on both variables, provided $a$ and its derivatives are all bounded.

### L2 human

This is the basic $L^2$ boundedness result of the calculus of pseudo-differential operators.
It shows that operators of order $0$ in the classes $S^0_{\rho,\delta}$ with $\delta \le \rho$ and $\delta < 1$ are bounded on $L^2$.
By composition with $(I - \Delta)^{s/2}$, operators of order $m$ then map the Sobolev space $H^{s+m}$ to $H^s$.
Its proof is a model use of almost orthogonality.
The class $S^0_{0,0}$ is invariant under translations in both $x$ and $\xi$, which makes it natural in phase-space and semiclassical analysis.
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<document source="https://tamnd.github.io/math2/node/A0NP6HQNPX.md" tag="A0NP6HQNPX" kind="definition">
# Partially ordered ring

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $\le$ be a [[partial order|011C29Q1S1]] on $R$.
The pair $(R, \le)$ is a *partially ordered ring* if for all $a, b, c \in R$:

1. if $a \le b$, then $a + c \le b + c$;
2. if $0 \le a$ and $0 \le b$, then $0 \le ab$.

## Remarks

Example: $\R^n$ with componentwise addition and multiplication and the componentwise order is a partially ordered ring.
Clause 1 holds coordinatewise, and the product of nonnegative real numbers is nonnegative, which gives clause 2.
Non-example: $\R$ with the reversed order $a \preceq b \iff a \ge b$ satisfies clause 1 but not clause 2.
Indeed $0 \preceq -1$, but $(-1)(-1) = 1$ and $0 \not\preceq 1$.
The set $P = \set{a \in R : 0 \le a}$ is the *positive cone*.
It satisfies $P + P \subseteq P$, $PP \subseteq P$ and $P \cap (-P) = \set{0}$, and the order is recovered as $a \le b \iff b - a \in P$.
</document>

<document source="https://tamnd.github.io/math2/node/A0V7ZAEQBW.md" tag="A0V7ZAEQBW" kind="statement">
# Landen's transformation

## Statement

Let $0 < k < 1$, let $k' = \sqrt{1 - k^2}$, and let
$$k_1 = \frac{1 - k'}{1 + k'}.$$
Let $F(\varphi, k)$ be the incomplete [[elliptic integral|NXMCYGCB59]] of the first kind and $K(k) = F(\pi/2, k)$ the [[complete one|RM54CPPE12]].

1. For real $\varphi$, put $\varphi_1 = \varphi + \arctan(k'\tan\varphi)$, where the branch of $\arctan$ is chosen so that $\varphi_1$ depends continuously on $\varphi$ and $\varphi_1 = 0$ at $\varphi = 0$. Then
$$F(\varphi, k) = \frac{1 + k_1}{2} \, F(\varphi_1, k_1).$$
2. In particular $K(k) = (1 + k_1) \, K(k_1)$.

## Remarks

Part 2 is part 1 at $\varphi = \pi/2$, where $\varphi_1 = \pi$ and $F(\pi, k_1) = 2K(k_1)$.
Proof idea for part 1: differentiate both sides in $\varphi$ and check, with $\tan(\varphi_1 - \varphi) = k'\tan\varphi$, that the integrands agree; both sides vanish at $\varphi = 0$.
Check at small $k$: then $k' \approx 1$ and $k_1 \approx k^2/4 \approx 0$, so $\varphi_1 \approx 2\varphi$ and the right side is about $\frac12 \cdot 2\varphi = \varphi = F(\varphi, 0)$.
The modulus shrinks fast: $k_1 < k^2$ for $0 < k < 1$, since $k_1 = k^2/(1 + k')^2$.
Iterating part 2 sends the modulus to $0$ quadratically and gives $K(k) = \frac{\pi}{2}\prod_{n \ge 1} (1 + k_n)$, where $k_{n+1}$ is obtained from $k_n$ as $k_1$ from $k$.
In terms of the integral $I(a, b) = \int_0^{\pi/2} d\theta / \sqrt{a^2\cos^2\theta + b^2\sin^2\theta}$, the transformation reads $I(a, b) = I\big(\frac{a + b}{2}, \sqrt{ab}\big)$, which is the step behind the [[arithmetic–geometric mean formula|E0V3N93DX8]].

## Explanations

### L1 human

Landen's transformation changes the modulus of an elliptic integral into a much smaller one, at the cost of a simple factor and a new angle.
Doing it again and again makes the modulus tend to $0$, where the integral is just the angle itself.
This is an efficient way to compute elliptic integrals.

### L2 human

The complete form $K(k) = (1 + k_1)K(k_1)$ is equivalent to the identity $K(k) = \pi / (2M(1, k'))$ with the arithmetic–geometric mean $M$.
Indeed $k_1' = 2\sqrt{k'}/(1 + k')$, and the homogeneity $M(\lambda a, \lambda b) = \lambda M(a, b)$ gives $M(1, k_1') = \frac{2}{1 + k'} M(1, k')$.
Read backwards, with $k$ in place of $k_1$, it is the ascending Landen transformation, which pushes the modulus toward $1$.
For the theta-function side, Landen's transformation corresponds to replacing the nome $q$ by $q^2$, which doubles the ratio $K'/K$.
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<document source="https://tamnd.github.io/math2/node/A10WW8GEKR.md" tag="A10WW8GEKR" kind="concept">
# Rigid monoidal category

## Summary

A rigid monoidal category, also called an autonomous category, is a [[monoidal category|M3MSQ3NKAE]] in which every object has a left and a right [[dual object|8A7YXMW727]].
The main example is the category of finite-dimensional vector spaces, and more generally of finite-dimensional representations of a group.
Rigidity gives every morphism a transpose, and in the braided case it allows traces and dimensions.
Fusion categories and Tannakian categories are rigid by definition.

## Explanations

### L1 human

Every finite-dimensional vector space has a dual space, and every linear map has a transpose between the duals.
A rigid monoidal category is one where every object has such a dual.
In string diagrams, this means that every string can be bent up or down, turning inputs into outputs.
Infinite-dimensional vector spaces spoil this, so the category of all vector spaces is not rigid.
</document>

<document source="https://tamnd.github.io/math2/node/A127WFVYKW.md" tag="A127WFVYKW" kind="statement">
# Hodge theorem

## Statement

Let $(M, g)$ be a compact [[oriented|913WEQERBE]] [[Riemannian manifold|7CBT4SM83Y]] without boundary, of dimension $n$.
For $0 \le k \le n$, let $\Omega^k(M)$ be the space of smooth $k$-[[forms|0E3G0GRPZ6]] and $d$ the [[exterior derivative|0ZHSTAG8ZY]].
Let $\delta = (-1)^{n(k+1)+1} \star d \star$ be the codifferential on $k$-forms, where $\star$ is the [[Hodge star|1Y4WX4XQWJ]].
Let $\Delta = d\delta + \delta d$ be the Hodge Laplacian.
Let $\mathcal{H}^k = \set{\omega \in \Omega^k(M) : \Delta\omega = 0}$ be the space of harmonic $k$-forms.
Then:

1. $\mathcal{H}^k$ is finite-dimensional.
2. There is a decomposition, orthogonal for the $L^2$ inner product $(\alpha, \beta) = \int_M \alpha \wedge \star\beta$:
$$\Omega^k(M) = \mathcal{H}^k \oplus d\,\Omega^{k-1}(M) \oplus \delta\,\Omega^{k+1}(M).$$
3. Each class of the [[de Rham cohomology|GX74C07T0G]] group $H^k_{dR}(M)$ contains exactly one harmonic form, so $\mathcal{H}^k \cong H^k_{dR}(M)$.

## Remarks

The idea of the proof: $\delta$ is the $L^2$ adjoint of $d$, so $(\Delta\omega, \omega) = \norm{d\omega}^2 + \norm{\delta\omega}^2$, and $\omega$ is harmonic if and only if $d\omega = 0$ and $\delta\omega = 0$.
The operator $\Delta$ is elliptic, and elliptic regularity with the Rellich compactness theorem gives part 1 and the decomposition $\Omega^k = \mathcal{H}^k \oplus \Delta(\Omega^k)$, from which part 2 follows.
For part 3, a closed form is orthogonal to $\delta\Omega^{k+1}$, so it is its harmonic part plus an exact form.

Example: on the flat torus $\R^n / \Z^n$ the harmonic $k$-forms are the forms with constant coefficients in $dx^{i_1} \wedge \dots \wedge dx^{i_k}$.
So $\dim H^k_{dR} = \binom{n}{k}$.

Compactness is needed: on $\R$, every function $a x + b$ is harmonic, and the space of harmonic $0$-forms is $2$-dimensional while $H^0_{dR}(\R)$ is $1$-dimensional.

## Explanations

### L1 human

Each cohomology class, a kind of "hole" of the manifold, has many representing forms.
The Hodge theorem picks a best one: the unique harmonic form, which minimizes the energy in its class.
So topology is captured by solutions of a differential equation.

### L2 human

Since $\star$ maps harmonic $k$-forms to harmonic $(n-k)$-forms, the theorem gives Poincaré duality $b_k = b_{n-k}$ for the Betti numbers.
Combined with Weitzenböck identities, it turns curvature conditions into vanishing of cohomology, as in Bochner's theorem.
On compact Kähler manifolds a refinement gives the Hodge decomposition of complex cohomology into $(p, q)$-types.
</document>

<document source="https://tamnd.github.io/math2/node/A15BF6QHDR.md" tag="A15BF6QHDR" kind="concept">
# Riemann domain over C^n

## Summary

A Riemann domain over $\C^n$ is a connected space that lies over $\C^n$ through a [[local homeomorphism|MDH71ZZ8XX]], possibly with several sheets over the same point.
It is the several-variable analogue of a Riemann surface spread over the plane.
Riemann domains are needed because analytic continuation of a holomorphic function can lead to several values over one point.
The envelope of holomorphy of a domain in $\C^n$ is in general a Riemann domain and not a subset of $\C^n$.

## Explanations

### L1 human

Imagine several transparent copies of a region of $\C^n$, stacked on top of each other and glued along some cuts, like a spiral staircase.
Over each point of $\C^n$ there may be several points of the staircase, but near each of its points the staircase looks exactly like an open set of $\C^n$.
For example, the map $(w, z_2) \mapsto (e^w, z_2)$ from $\C^2$ to $\C^2$ makes $\C^2$ into a Riemann domain over $\C^2$ with infinitely many sheets over each point with $z_1 \ne 0$.
On this domain the logarithm of $z_1$ becomes a single-valued function, namely $w$.
</document>

<document source="https://tamnd.github.io/math2/node/A167XK4YTF.md" tag="A167XK4YTF" kind="concept">
# Graph theory

## Summary

Graph theory is the part of [[combinatorics|0JKW7W7XFR]] that studies [[graphs|00C0BR7D7H]]: sets of vertices joined in pairs by edges.
It asks about paths and cycles, connectivity, colourings, matchings, embeddings in surfaces and the spectra of graph matrices.
Graphs model networks of every kind, so the subject is central in computer science, operations research, chemistry and physics.

## Explanations

### L1 human

A graph is a set of dots with lines between some pairs of dots.
Road maps, social networks and molecules can all be drawn this way.
Graph theory asks questions such as: can one walk over every road exactly once and return home?
Euler answered that question for the bridges of Königsberg in 1736, and this is often seen as the start of graph theory.
</document>

<document source="https://tamnd.github.io/math2/node/A16HP3P271.md" tag="A16HP3P271" kind="definition">
# Nonlinear regression

## Definition

Let $\Theta \subseteq \R^p$ be a parameter set, let $x_1, \dots, x_n$ be known regressor values in a set $\mathcal{X}$, and let $f \colon \mathcal{X} \times \Theta \to \R$ be a known function.
Assume that $\theta \mapsto f(x, \theta)$ is not affine for some $x$.
A *nonlinear regression model* states that
$$y_i = f(x_i, \theta) + \varepsilon_i, \quad i = 1, \dots, n,$$
for an unknown $\theta \in \Theta$ and random errors $\varepsilon_i$ with mean $0$.
The *nonlinear least squares estimate* is a minimiser over $\theta \in \Theta$ of
$$S(\theta) = \sum_{i=1}^n \big( y_i - f(x_i, \theta) \big)^2.$$

## Remarks

The estimate is computed iteratively.
The Gauss–Newton method replaces $f(x_i, \theta)$ near the current value $\theta^{(k)}$ by its first-order Taylor expansion, solves the resulting linear [[least squares|0BYTQ4X1HC]] problem, and repeats.
It is an approximation of [[Newton's method|07EXVNF9VP]] for minimising $S$.
The minimiser may not be unique, and the iteration may converge to a local minimum.

Example: the exponential decay model $f(x, \theta) = \theta_1 e^{-\theta_2 x}$ with $\theta_1, \theta_2 > 0$.
The map $\theta \mapsto f(x, \theta)$ is not affine, since $\partial f / \partial \theta_1 = e^{-\theta_2 x}$ depends on $\theta_2$ for $x \ne 0$.

Non-example: $f(x, \theta) = \theta_0 + \theta_1 x + \theta_2 x^2$ is affine in $\theta$ for each $x$, so this model is [[linear regression|41FVC4P5JB]], although the curve is a parabola.

If the errors are independent $N(0, \sigma^2)$, the least squares estimate is the maximum likelihood estimate.
</document>

<document source="https://tamnd.github.io/math2/node/A17Q3X6285.md" tag="A17Q3X6285" kind="example">
# Bruhat order

## Example

Let $(W, S)$ be a [[Coxeter system|G6FND7X69J]], let $\ell(w)$ be the length of $w \in W$ with respect to $S$, and let $T = \set{w s w^{-1} : w \in W, s \in S}$ be the set of reflections.
For $u, w \in W$, write $u \to w$ when $w = u t$ for some $t \in T$ and $\ell(u) < \ell(w)$.
The *Bruhat order* is the reflexive and transitive closure of $\to$: $u \le w$ when there is a sequence $u = u_0 \to u_1 \to \cdots \to u_k = w$ with $k \ge 0$.
It is a partial order on $W$, graded by the length $\ell$.
By the subword property, $u \le w$ if and only if some [[reduced word|0KBTP8Q2FS]] for $w$ has a subword that is a reduced word for $u$.

Check in the [[symmetric group|0515A6D2Y1]] $S_3$ with $S = \set{s_1, s_2}$, the adjacent transpositions.
Its elements are $e$ of length $0$, $s_1, s_2$ of length $1$, $s_1 s_2, s_2 s_1$ of length $2$, and $s_1 s_2 s_1$ of length $3$.
Each element of length $2$ has a reduced word that contains both $s_1$ and $s_2$ as subwords.
So both $s_1$ and $s_2$ lie below both $s_1 s_2$ and $s_2 s_1$.
So the Bruhat order on $S_3$ has the bottom $e$, the top $s_1 s_2 s_1$, and all four relations between the two middle levels.

## Remarks

The Bruhat order contains the [[weak order|C8SEGT1AZS]]: in $S_3$ the weak order has only two of the four relations between the middle levels.
For $S_n$, the reflections are the transpositions, so $u \to w$ when $w$ comes from $u$ by swapping two entries and the number of inversions goes up.

## Explanations

### L1 human

Elements of a Coxeter group, such as permutations, can be written as words in simple generators.
One element is below another in the Bruhat order when you can get it by deleting letters from a shortest word of the other, and still have a shortest word.
In $S_3$ this gives one element at the bottom, two on each middle level, and one at the top.
Every element of level one lies below every element of level two.
The order describes how Schubert cells sit in the closures of other Schubert cells, which is why it appears in geometry.
</document>

<document source="https://tamnd.github.io/math2/node/A18HPFPC4D.md" tag="A18HPFPC4D" kind="definition">
# Orthogonal polynomials

## Definition

Let $\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\R$ such that:

1. every moment $\int_\R \abs{x}^k \, d\mu(x)$, $k = 0, 1, 2, \dots$, is finite;
2. the [[support|FQTXV6BQZG]] of $\mu$ is an infinite set.

A sequence $(p_n)_{n \ge 0}$ of real [[polynomials|1G0B73WPX8]] is a sequence of *orthogonal polynomials* for $\mu$ if $\deg p_n = n$ for every $n$ and
$$\int_\R p_m(x) \, p_n(x) \, d\mu(x) = 0 \quad \text{for all } m \ne n.$$

## Remarks

Condition 2 ensures that $\int p^2 \, d\mu > 0$ for every nonzero polynomial $p$, so the integral is an inner product on the polynomials.
Gram–Schmidt orthogonalization of $1, x, x^2, \dots$ then gives such a sequence, and each $p_n$ is determined up to a nonzero constant factor.
The normalization $p_n(x) = x^n + \cdots$ gives the *monic* orthogonal polynomials, and $\int p_n^2 \, d\mu = 1$ gives the *orthonormal* ones.
Often $d\mu = w(x) \, dx$ for a weight function $w \ge 0$ on an interval.

Example: $d\mu = dx$ on $[-1, 1]$ and $p_0 = 1$, $p_1 = x$, $p_2 = \frac12(3x^2 - 1)$, the first Legendre polynomials.
The integrals of $p_0 p_1 = x$ and of $p_1 p_2$ vanish because these are odd functions.
And $\int_{-1}^{1} \frac12(3x^2 - 1) \, dx = \frac12 (2 - 2) = 0$.

Non-example: the monomials $1, x, x^2$ for the same measure are not orthogonal, since $\int_{-1}^{1} 1 \cdot x^2 \, dx = \frac23 \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/A1E8TDEVJ9.md" tag="A1E8TDEVJ9" kind="concept">
# Satellite knot

## Summary

A satellite knot is a [[knot|QYEAZTYGPT]] that lies inside a thickened copy of another nontrivial knot, the companion, in an essential way.
Equivalently, its complement contains an incompressible torus that is not parallel to the boundary.
Satellite knots form one of the three classes in Thurston's division of knots, with [[torus knots|NS60VV786W]] and [[hyperbolic knots|K47G2Q24DT]].
Every [[knot sum|AGZSZW7GYE]] of two nontrivial knots is a satellite knot.

## Explanations

### L1 human

Take a nontrivial knot and thicken it into a knotted tube.
Now draw a new knot inside the tube, one that winds through the tube and cannot be pulled out of it or shrunk to a small ball inside it.
The new knot is a satellite, and the old knot is its companion.
Cables, where the new knot winds around the tube several times, are a simple example.
The granny knot, the sum of two trefoils, is a satellite with a trefoil as companion.
</document>

<document source="https://tamnd.github.io/math2/node/A1G25R99PY.md" tag="A1G25R99PY" kind="statement">
# Spectral theorem for compact self-adjoint operators

## Statement

Let $H$ be a real or complex [[Hilbert space|HM2HHE3J51]].
Let $T$ be a [[compact operator|M2NEP1K5AE]] on $H$ with $T = T^*$, where $T^*$ is the [[adjoint|6HSJBJ8TTQ]] of $T$.
Then there are a finite or countable orthonormal family $(e_n)_{n \in N}$ in $H$ and nonzero real numbers $(\lambda_n)_{n \in N}$ such that:

1. $T e_n = \lambda_n e_n$ for every $n \in N$;
2. $\abs{\lambda_1} \ge \abs{\lambda_2} \ge \cdots$, and $\lambda_n \to 0$ if $N$ is infinite;
3. $T x = \sum_{n \in N} \lambda_n \ip{x}{e_n} e_n$ for every $x \in H$;
4. $H$ is the orthogonal direct sum of the [[kernel|0S23E6YE83]] $\ker T$ and the closed span of $(e_n)$.

## Remarks

The proof starts from the fact that $\norm{T} = \sup_{\norm{x} = 1} \abs{\ip{Tx}{x}}$ for self-adjoint $T$.
Compactness makes this supremum attained, at an [[eigenvector|0N6960W0BV]] with eigenvalue $\pm \norm{T}$.
The orthogonal complement of this eigenvector is invariant under $T$, and one repeats the argument there; the eigenvalues found tend to $0$ by the [[spectral theory of compact operators|JEPDM5B5FC]].
Adding an [[orthonormal basis|0S9GGTR0EX]] of $\ker T$ to $(e_n)$ gives an orthonormal basis of $H$ made of eigenvectors of $T$.
The nonzero [[eigenvalues|0BQAA2BK3Q]] of $T$ are exactly the numbers $\lambda_n$, and each appears as many times as the dimension of its eigenspace, which is finite.
Example: the operator $(Tf)(x) = \int_0^1 \min(x, y) f(y) \, dy$ on $L^2([0,1])$ is compact and self-adjoint, with eigenvalues $\big((n - \tfrac{1}{2}) \pi\big)^{-2}$ and eigenfunctions $\sqrt{2} \sin\big((n - \tfrac{1}{2}) \pi x\big)$, $n \ge 1$.
Without self-adjointness the theorem fails: the Volterra operator $(Vf)(x) = \int_0^x f(y) \, dy$ on $L^2([0,1])$ is compact but has no eigenvalue at all.
In finite dimension, this is the [[spectral theorem|06J4PXRG22]] for self-adjoint linear maps.

## Explanations

### L1 human

A symmetric matrix can be diagonalized in an orthonormal basis of eigenvectors.
This theorem says the same for compact self-adjoint operators on infinite-dimensional spaces.
The operator acts by stretching along infinitely many perpendicular axes, by factors that shrink to zero.

### L2 human

The theorem is the main tool for solving linear integral equations with symmetric kernels, and for boundary value problems such as the Sturm–Liouville problem through their Green's operators.
It shows that the eigenfunctions of such problems form an orthonormal basis, which justifies expansions in Fourier-type series.
Applied to $T^* T$ for an arbitrary compact operator $T$, it gives the singular values and the Schmidt decomposition of $T$.
For noncompact self-adjoint operators, eigenvectors may not exist, and one needs the spectral theorem in its measure form.
</document>

<document source="https://tamnd.github.io/math2/node/A1H48DDW49.md" tag="A1H48DDW49" kind="concept">
# Conjugate function

## Summary

Let $f$ be an integrable function on the circle, and extend it to a harmonic function in the unit disc.
The conjugate function $\tilde f$ is the boundary value of the [[harmonic conjugate|050F2FMTKF]] of this extension.
On Fourier series it acts by $e^{ik\theta} \mapsto -i \sgn(k) e^{ik\theta}$, so it turns $\cos k\theta$ into $\sin k\theta$.
It is the circle version of the [[Hilbert transform|NSCEBXM7ZM]].
Its $L^p$ bounds control the convergence of [[Fourier series|G12KXDHPPS]] in $L^p$.

## Explanations

### L1 human

Every real function $f$ on the circle is the real part of the boundary values of some holomorphic function in the disc.
The imaginary part of these boundary values is the conjugate function $\tilde f$.
In terms of waves, it shifts every cosine wave into the matching sine wave, and every sine wave into minus the cosine wave.
For example, the holomorphic function $z = re^{i\theta}$ equals $r\cos\theta + i r\sin\theta$.
On the circle $r = 1$ its real part is $\cos\theta$ and its imaginary part is $\sin\theta$, so the conjugate of $\cos\theta$ is $\sin\theta$.
</document>

<document source="https://tamnd.github.io/math2/node/A1KWT817JX.md" tag="A1KWT817JX" kind="concept">
# Variational capacity

## Summary

Let $K$ be a compact set inside an open set $\Omega \subseteq \R^n$, and let $1 < p < \infty$.
The variational capacity, or $p$-capacity, of $K$ is the least $p$-Dirichlet energy $\int_\Omega \abs{\nabla u}^p \, dx$ of a suitable function $u$.
Here $u$ is smooth, at least $1$ on $K$, and zero near the boundary of $\Omega$.
For $p = 2$ it is the classical [[condenser capacity|NTEFMRR3HR]].
It is the natural measure of size in nonlinear potential theory and for the $p$-Laplace equation.
Sets of $p$-capacity zero are the sets that [[Sobolev functions|M4CEZ5K9HP]] with $p$-integrable gradient cannot see.

## Explanations

### L1 human

Think of $\Omega$ as a large room and $K$ as an object inside it.
A test function must climb from $0$ near the walls to at least $1$ on $K$, and the capacity is the cheapest cost of this climb, measured by $\int \abs{\nabla u}^p$.
A large object close to the walls forces steep slopes and has a large capacity.
A single point in $\R^n$ has $p$-capacity zero when $p \le n$: the climb can be made very cheap by spreading it over many scales.
For $p > n$ even a point has positive $p$-capacity, which matches the fact that such Sobolev functions are continuous.
</document>

<document source="https://tamnd.github.io/math2/node/A1MRBS3Z5C.md" tag="A1MRBS3Z5C" kind="definition">
# Polymorphism of a relation

## Definition

Let $A$ be a set, let $m, n \ge 1$, and let $R \subseteq A^m$ be an $m$-ary relation on $A$.
An $n$-ary [[operation|EJ8YJVM9BW]] $f \colon A^n \to A$ is a *polymorphism* of $R$ if for all tuples
$$(a_{1 1}, \dots, a_{m 1}), \; \dots, \; (a_{1 n}, \dots, a_{m n}) \in R$$
the tuple
$$\bigl(f(a_{1 1}, \dots, a_{1 n}), \dots, f(a_{m 1}, \dots, a_{m n})\bigr)$$
is also in $R$.
For a set $\Gamma$ of finitary relations on $A$, $\operatorname{Pol}(\Gamma)$ denotes the set of operations that are polymorphisms of every $R \in \Gamma$.
For a set $F$ of finitary operations on $A$, $\operatorname{Inv}(F)$ denotes the set of finitary relations on $A$ of which every $f \in F$ is a polymorphism.

## Remarks

Example: on $\set{0, 1}$, the majority operation $m$ is a polymorphism of the [[binary relation|05AVARPXA2]] $R = \set{(0, 1), (1, 0)}$.
Check: given three pairs $(a_j, 1 - a_j)$ in $R$, the second coordinates are the complements of the first, so $m(1 - a_1, 1 - a_2, 1 - a_3) = 1 - m(a_1, a_2, a_3)$ and the result lies in $R$.
Non-example: the operation $\wedge$ on $\set{0, 1}$ is not a polymorphism of the same $R$, since applying it to $(0, 1)$ and $(1, 0)$ gives $(0, 0) \notin R$.
For a unary relation $B \subseteq A$, the polymorphisms are the operations under which $B$ is closed, so $B$ is a subuniverse.
For an equivalence relation, the polymorphisms are the operations compatible with it, as in the definition of a congruence.
If $A$ carries a relational structure, its polymorphisms are the homomorphisms from its $n$-th power to itself.
</document>

<document source="https://tamnd.github.io/math2/node/A1PANCDYTG.md" tag="A1PANCDYTG" kind="statement">
# Courant–Fischer characterization of characteristic values

## Statement

Let $a < b$ and let $K \in L^2([a, b] \times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]] that is [[Hermitian|2ZB3NFZ94T]].
Let $R_K$ be its [[Rayleigh quotient|HRCQ3JHYY0]] on $L^2[a, b]$.
List the positive [[characteristic values|DC7HMHVREP]] of $K$ as $0 < \lambda_1 \le \lambda_2 \le \cdots$.
Each $\lambda$ is repeated as often as the dimension of the space of solutions of $\varphi = \lambda T_K \varphi$.
Let $k \ge 1$ be such that there are at least $k$ of them.
Then
$$\frac{1}{\lambda_k} = \max_{\dim V = k} \; \min_{\varphi \in V \setminus \set{0}} R_K(\varphi) = \min_{\psi_1, \dots, \psi_{k-1} \in L^2[a, b]} \; \sup \set{R_K(\varphi) : \varphi \ne 0, \ \ip{\varphi}{\psi_j} = 0 \text{ for } j < k},$$
where $V$ runs over the linear subspaces of $L^2[a, b]$ of dimension $k$.

## Remarks

Let $\varphi_1, \varphi_2, \dots$ be orthonormal [[eigenfunctions|4NSACSWRJZ]] with $T_K \varphi_j = \varphi_j / \lambda_j$, from the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]].
On the span of $\varphi_1, \dots, \varphi_k$, the quotient is at least $1/\lambda_k$; so the max-min is at least $1/\lambda_k$.
Any $k$-dimensional $V$ contains a nonzero $\varphi$ orthogonal to $\varphi_1, \dots, \varphi_{k-1}$, and on that orthogonal complement $R_K \le 1/\lambda_k$; so the max-min is exactly $1/\lambda_k$.
The min-max form is proved in the same way, with the choice $\psi_j = \varphi_j$ attaining the minimum.

Example: for $K(x, y) = \min(x, y)$ on $[0, 1]$, $\lambda_k = (k - \tfrac12)^2 \pi^2$.
For $k = 1$ the theorem says $\int_0^1 \int_0^1 \min(x, y) \varphi(x) \varphi(y) \, dx \, dy \le \frac{4}{\pi^2} \int_0^1 \varphi^2$ for real $\varphi$, with equality for $\varphi(x) = \sin(\pi x / 2)$.

The symmetry of $K$ is needed: without it the quotient is not real, and the characteristic values need not be real or exist at all.
The negative characteristic values satisfy the same formulas applied to the kernel $-K$.

## Explanations

### L1 human

The characteristic values of a symmetric kernel can be found by optimization.
The smallest positive one corresponds to the largest stretching of a single function by the integral operator.
The next ones are the best stretching that one can guarantee when some directions are forbidden.

### L2 human

This is the integral form of the [[Courant–Fischer min-max theorem|CJK2HS2508]] for Hermitian matrices.
It allows one to compare characteristic values of two kernels without computing them: if $R_{K_1} \le R_{K_2}$ everywhere, then the $k$-th positive characteristic value of $K_1$ is at least that of $K_2$.
It is the basis of the Rayleigh–Ritz method, which gives bounds for characteristic values by restricting the quotient to finite-dimensional trial spaces.
</document>

<document source="https://tamnd.github.io/math2/node/A1X6T9ZD9E.md" tag="A1X6T9ZD9E" kind="definition">
# Fourier transform as Laplace transform on the imaginary axis

## Definition

Let $f \colon \R \to \C$ be measurable.
Its two-sided Laplace transform is $\mathcal{B}f(s) = \int_{\R} f(x) e^{-sx} \, dx$, for $s \in \C$ at which $x \mapsto f(x) e^{-sx}$ is [[integrable|X7GTF3D98F]].
Let $\sigma \in \R$ be such that $x \mapsto e^{-\sigma x} f(x)$ is integrable.
The *restriction of $\mathcal{B}f$ to the line $\operatorname{Re} s = \sigma$* is the function $\xi \mapsto \mathcal{B}f(\sigma + 2\pi i \xi)$ on $\R$.
For $\sigma = 0$, that is, for $f$ integrable, this is the restriction of $\mathcal{B}f$ to the imaginary axis, and it equals the [[Fourier transform|DPVM34ADZV]]:
$$\hat f(\xi) = \int_{\R} f(x) e^{-2\pi i x \xi} \, dx = \mathcal{B}f(2\pi i \xi), \qquad \xi \in \R.$$

## Remarks

In general $\mathcal{B}f(\sigma + 2\pi i \xi) = \widehat{f_\sigma}(\xi)$ with $f_\sigma(x) = e^{-\sigma x} f(x)$, by writing $e^{-(\sigma + 2\pi i \xi) x} = e^{-\sigma x} e^{-2\pi i x \xi}$.
Example: $f(x) = e^{-\abs{x}}$ is integrable, and $\mathcal{B}f(s) = \frac{1}{1 + s} + \frac{1}{1 - s} = \frac{2}{1 - s^2}$ for $\abs{\operatorname{Re} s} < 1$.
At $s = 2\pi i \xi$ this gives $\hat f(\xi) = 2/(1 + 4\pi^2 \xi^2)$, the known Fourier transform of $e^{-\abs{x}}$.
Non-example: $f(x) = e^{x}$ for $x \ge 0$ and $f(x) = 0$ for $x < 0$ is not integrable, so the imaginary axis is not in its region of convergence.
Here $\mathcal{B}f(s) = 1/(s - 1)$ for $\operatorname{Re} s > 1$, and this function is not the Fourier transform of $f$, which does not exist as an integral.
With the convention $\int f(x) e^{-i \omega x} \, dx$, the same relation reads $\hat f(\omega) = \mathcal{B}f(i\omega)$.
</document>

<document source="https://tamnd.github.io/math2/node/A1XE59ZHNM.md" tag="A1XE59ZHNM" kind="concept">
# Shift space

## Summary

A shift space, or subshift, is a closed set of infinite sequences over a finite alphabet that is invariant under the shift by one place.
Equivalently, it is the set of all sequences that avoid a given list of forbidden finite words.
Shift spaces with the shift map are the objects of symbolic dynamics, and they serve as models for coding more general dynamical systems.
The full shift, the [[subshifts of finite type|624XDXX8FN]] and the sofic shifts are the main classes.

## Explanations

### L1 human

Take an alphabet such as $\set{0, 1}$ and consider infinite words in it.
Forbid some finite patterns, for example the pattern $11$.
The set of all infinite words that never contain a forbidden pattern is a shift space: here, the words in which no two $1$s are adjacent.
Shifting such a word by one place gives again such a word, so the set is mapped to itself by the shift.
Shift spaces are the simplest systems in which complicated dynamics can be described exactly.
</document>

<document source="https://tamnd.github.io/math2/node/A289X08MR1.md" tag="A289X08MR1" kind="definition">
# Subderivative

## Definition

Let $I$ be an open [[interval|0TK8Y3J5RW]] of $\R$, let $f \colon I \to \R$ be a [[convex function|0VV0HSW1ZS]], and let $x_0 \in I$.
A real number $c$ is a *subderivative* of $f$ at $x_0$ if
$$f(x) - f(x_0) \ge c \, (x - x_0) \quad \text{for all } x \in I.$$
The *subdifferential* $\partial f(x_0)$ is the set of all subderivatives of $f$ at $x_0$.

## Remarks

The subdifferential is the closed interval $\partial f(x_0) = [f'_-(x_0), f'_+(x_0)]$, where $f'_-$ and $f'_+$ are the one-sided derivatives, which exist for a convex function on an open interval.
In particular it is never empty.
It is the single point $\set{f'(x_0)}$ exactly when $f$ has a [[derivative|0F1WR9C83K]] at $x_0$.
For a convex function on $\R^n$, a vector $g$ with $f(x) - f(x_0) \ge \ip{g}{x - x_0}$ for all $x$ is called a *subgradient*.

Example: $f(x) = \abs{x}$ on $\R$ and $x_0 = 0$.
For $c \in [-1, 1]$ one has $\abs{x} \ge c x$ for all $x$, so $\partial f(0) = [-1, 1]$.

Non-example: $c = 2$ for the same $f$ and $x_0$.
At $x = 1$ the condition reads $1 \ge 2$, which fails.
</document>

<document source="https://tamnd.github.io/math2/node/A2DE66937D.md" tag="A2DE66937D" kind="concept">
# Category

## Summary

A category is made of objects, of morphisms (arrows) between pairs of objects, and of a composition of morphisms that is associative and has identities.
It is the basic notion of [[category theory|Q1RZT7DV0F]].
Sets with [[functions|0EPAT51DXQ]], [[groups|G5S1M9]] with [[group homomorphisms|0JYTG5K9Y4]], and [[topological spaces|0CQHFPGJ45]] with [[continuous maps|0VGFHTM75Y]] are categories.
A category can also be a single structure: a [[monoid|0N6R902FG1]] is a category with one object.
A [[preorder|0VABYWJWH9]] is a category with at most one morphism between any two objects.

## Explanations

### L1 human

Picture dots for the objects and arrows between them for the morphisms.
Whenever one arrow ends where another starts, the two can be chained into one arrow.
Chaining is associative, and each dot carries an identity arrow that changes nothing when chained.
For example, take as dots the numbers $1, 2, 3$, and draw one arrow from $a$ to $b$ when $a \le b$.
Arrows chain because $a \le b$ and $b \le c$ give $a \le c$, and the identity arrows come from $a \le a$.
</document>

<document source="https://tamnd.github.io/math2/node/A2KKZAV29W.md" tag="A2KKZAV29W" kind="concept">
# Discrete valuation

## Summary

A discrete valuation is a [[valuation|B85T5VKR3M]] on a field whose values on nonzero elements are exactly the integers, see the [[definition|ZYK5F0GBK6]].
It counts the power of a fixed "prime" element in each nonzero element.
The [[p-adic valuation|03HDF50VN4]] on $\Q$ and the order of vanishing of a rational function at a point are discrete valuations.
The elements of nonnegative value form a [[discrete valuation ring|MGM3CNJW9V]], one of the basic rings of number theory and algebraic geometry.

## Explanations

### L1 human

A discrete valuation gives each nonzero element an integer, its order, much as the exponent of $p$ in a fraction.
Picture a rational function $f(x)$ near the point $0$: the valuation is the order of the zero of $f$ at $0$, or minus the order of the pole.
For example $x^3/(1 + x)$ has value $3$, and $1/x^2$ has value $-2$.
The values are whole numbers and every integer occurs, which is what "discrete" means here.
</document>

<document source="https://tamnd.github.io/math2/node/A2KQ49Y4NF.md" tag="A2KQ49Y4NF" kind="concept">
# Poisson manifold

## Summary

A Poisson manifold is a [[smooth manifold|F45VQ47GPV]] with a bracket on its smooth functions.
The bracket has the algebraic properties of the [[Poisson bracket|2S0MXDT4YH]] of a [[symplectic manifold|CAN490EPRT]].
Every symplectic manifold is a Poisson manifold, but the bracket of a Poisson manifold may be degenerate.
Poisson manifolds are the phase spaces of mechanics with constraints or symmetry, and they are the classical limits that deformation quantization starts from.

## Explanations

### L1 human

In mechanics the Poisson bracket of two observables tells how one changes along the motion generated by the other.
A Poisson manifold keeps only the rules this bracket obeys: it is antisymmetric, satisfies the Jacobi identity, and behaves like a derivative in each slot.
Such a manifold splits into pieces, called symplectic leaves, each of which is an ordinary symplectic manifold.
For example, on $\R^3$ the bracket with $\{x, y\} = z$, $\{y, z\} = x$, $\{z, x\} = y$ comes from the rotation group, and its leaves are the spheres centered at the origin and the origin itself.
</document>

<document source="https://tamnd.github.io/math2/node/A2MX36J5GD.md" tag="A2MX36J5GD" kind="definition">
# Crisis

## Definition

Let $(\phi_\mu)$ be a one-parameter family of dynamical systems, flows or maps, with $\mu \in \R$.
Assume that for $\mu$ near $\mu_0$ the system has a chaotic [[attractor|7FWX24X5MW]] $A_\mu$.
A *crisis* at $\mu_0$ is a bifurcation at which $A_\mu$ changes discontinuously as $\mu$ crosses $\mu_0$.
The attractor is suddenly destroyed or suddenly jumps in size.
At a crisis, $A_{\mu_0}$ touches an unstable [[periodic orbit|0AA97XA70K]] or its stable manifold.
The crisis is a *boundary crisis* if this orbit lies on the boundary of the basin of $A_\mu$.
It is an *interior crisis* if this orbit lies inside the basin.

## Remarks

Example: the [[logistic map|QN9X9BNKTX]] $f_r(x) = r x (1 - x)$ at $r = 4$.
For $r = 4$, the interval $[0, 1]$ is invariant and carries a chaotic attractor.
The attractor contains the repelling fixed point $0$ and its preimage $1$, which lie on the boundary of the basin.
For $r > 4$, points near $1/2$ are mapped above $1$ and then escape to $-\infty$, so the attractor is destroyed: a boundary crisis.
After the crisis, typical orbits stay near the old attractor for a long time before escaping, a chaotic transient.
Non-example: a [[period-doubling bifurcation|NQDTTZ59X8]] of a periodic attractor is not a crisis, since the attractor changes continuously and is not chaotic.
Crises are a basic route by which [[strange attractors|9WKCZGG906]] appear or vanish in [[bifurcation theory|D9TW46P19Y]].
</document>

<document source="https://tamnd.github.io/math2/node/A2PDZ8GA9C.md" tag="A2PDZ8GA9C" kind="concept">
# Transversality

## Summary

Transversality is the condition that two [[submanifolds|VKFW27WXEZ]], or a smooth map and a submanifold, meet in general position.
At each meeting point their tangent spaces together span the tangent space of the ambient manifold.
Transverse intersections are again submanifolds, of the expected dimension, and they persist under small perturbations.
Transversality is the basic notion of general position in differential topology and underlies intersection numbers, degree theory and cobordism.

## Explanations

### L1 human

Two lines in the plane that cross at an angle meet transversally, while two lines that touch along a common direction do not.
A small wiggle cannot destroy a transverse crossing, but it can easily separate two curves that only touch.
In space, a plane and a line meet transversally when the line pierces the plane rather than lying in it.
Two curves in $\R^3$ meet transversally only if they do not meet at all, because two lines cannot span three dimensions.
</document>

<document source="https://tamnd.github.io/math2/node/A2PMJG7J68.md" tag="A2PMJG7J68" kind="concept">
# Mellin convolution

## Summary

The Mellin convolution of two functions $f$ and $g$ on $(0, \infty)$ is $(f \star g)(x) = \int_0^\infty f(x/y) g(y) \, \frac{dy}{y}$.
It is the [[convolution|2Z29JN3VES]] of the multiplicative group $(0, \infty)$ with its invariant measure $dy/y$.
The [[Mellin transform|N5PVGRJE1T]] turns it into a product: $\mathcal{M}(f \star g) = \mathcal{M}f \cdot \mathcal{M}g$ where both transforms converge absolutely.
It describes, for example, the law of a product of independent positive random variables and many integral operators with homogeneous kernels.

## Explanations

### L1 human

Ordinary convolution averages $f(x - y)$ against $g(y)$, mixing shifts.
Mellin convolution averages $f(x/y)$ against $g(y)$, mixing rescalings instead.
The weight $dy/y$ treats the intervals $[1, 2]$ and $[10, 20]$ as equally large, as a logarithmic scale does.
For example, the indicator function of $(0, 1]$ convolved with itself gives $-\log x$ on $(0, 1]$ and $0$ beyond.
</document>

<document source="https://tamnd.github.io/math2/node/A2Q4RXD162.md" tag="A2Q4RXD162" kind="counterexample">
# Comb space

## Counterexample

Let $K = \set{0} \cup \set{1/n : n \ge 1} \subseteq \R$.
The *comb space* is the subspace of $\R^2$
$$C = ([0, 1] \times \set{0}) \cup (K \times [0, 1]).$$
It consists of a base segment, the tooth $\set{0} \times [0, 1]$, and the teeth $\set{1/n} \times [0, 1]$.

The space $C$ is [[path-connected|0WB6DH499T]].
Each point can be joined to $(0, 0)$ by going down its tooth and then along the base.

The space $C$ is not [[locally connected|FDVAHRJKF3]].
Take $p = (0, 1)$ and the open set $U = C \cap B(p, 1/2)$.
For large $n$, the set $U$ contains a piece of the tooth $\set{1/n} \times [0, 1]$ near height $1$.
Any open set $V$ with $p \in V \subseteq U$ contains such pieces, but $U$ contains no part of the base.
The vertical line $x = c$ with $c$ strictly between two teeth splits $V$ into two disjoint open parts, one of them containing $p$.
So no open neighbourhood of $p$ inside $U$ is connected.

## Remarks

So connectedness, even path-connectedness, does not imply local connectedness.
For the same reason, $C$ is not [[locally path-connected|1ZVR8BRK65]].
The comb space is contractible, so it also shows that contractible spaces need not be locally connected.

## Explanations

### L1 human

Picture a hair comb with infinitely many teeth crowding toward the left edge.
You can walk from any point to any other along the teeth and the base, so the comb is in one piece.
But near the top of the left edge, every small window shows many separate bits of teeth.
So the space is connected overall, yet not connected in the small.
</document>

<document source="https://tamnd.github.io/math2/node/A2QG6D75KR.md" tag="A2QG6D75KR" kind="counterexample">
# Failure of Aizerman's conjecture

## Counterexample

Aizerman's conjecture concerns systems of the form
$$\dot x = P x + q\, f(e), \qquad e = r^{T} x,$$
where $x \in \R^n$, $P$ is a constant real $n \times n$ matrix, $q, r \in \R^n$ are constant vectors, and $f \colon \R \to \R$ satisfies $f(0) = 0$.
Assume that $k_1 < f(e)/e < k_2$ for all $e \ne 0$, and that for every $k \in (k_1, k_2)$ the [[linear system|037ZNY7NAG]] $\dot x = (P + k\, q r^{T}) x$ is [[asymptotically stable|0NB689782W]].
The conjecture claims that then the [[equilibrium|0N8NGKDZDZ]] $x = 0$ is globally asymptotically stable.

The claim is false in general.
Let $P$, $q$, $r$, $f$ be a [[counterexample to Kalman's conjecture|5MP7DM0V5V]].
So $f$ is differentiable, $f(0) = 0$, $k_1 < f' < k_2$, and all the linear systems above are asymptotically stable.
Yet the system has a non-constant periodic solution.
For $e \ne 0$ the [[mean value theorem|0BARQY52N5]] gives $\xi$ between $0$ and $e$ with $f(e)/e = f'(\xi)$, so $k_1 < f(e)/e < k_2$.
Hence the same system satisfies the hypotheses of Aizerman's conjecture.
Its periodic solution does not tend to $0$, so the conclusion fails.

## Remarks

Aizerman posed the problem in 1949.
The hypothesis of Aizerman's conjecture bounds the secant slopes $f(e)/e$, while Kalman's conjecture bounds the slopes $f'$; the second hypothesis is stronger, so Kalman's conjecture is the weaker claim.
This is why a counterexample to Kalman's conjecture is also one to Aizerman's.
In the counterexamples constructed by Leonov, Kuznetsov and coauthors, a stable periodic orbit coexists with the stable equilibrium, a so-called hidden oscillation.

## Explanations

### L1 human

Suppose a feedback loop is stable whenever its nonlinearity is replaced by a straight line through the origin with a slope in a given range.
The conjecture says that the loop is then also stable with the nonlinearity itself.
This sounds plausible, since the nonlinearity always lies between such lines.
It is false: some of these systems oscillate forever, although every linear comparison system settles down.
</document>

<document source="https://tamnd.github.io/math2/node/A2VXXBYX6A.md" tag="A2VXXBYX6A" kind="definition">
# CR manifold

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] and let $T_\C M = TM \otimes_\R \C$ be its complexified [[tangent bundle|WY6K3P4YCG]].
A *CR structure* on $M$ is a complex [[vector subbundle|P4DZM975BV]] $L \subseteq T_\C M$ such that:

1. $L \cap \overline{L} = 0$, where $\overline{L}$ is the complex conjugate subbundle;
2. $L$ is involutive: for all smooth sections $X$, $Y$ of $L$, the [[Lie bracket|68FGQEBD6J]] $[X, Y]$ is a section of $L$.

A *CR manifold* is a smooth manifold with a CR structure.
The complex rank of $L$ is its *CR dimension*, and $\dim_\R M - 2 \operatorname{rank}_\C L$ is its *CR codimension*.

## Remarks

Example: let $M$ be a smooth real hypersurface of $\C^n$, and let $L_p = T^{1,0}_p \C^n \cap (T_p M \otimes \C)$ for $p \in M$, where $T^{1,0}\C^n$ is spanned by the $\partial / \partial z_j$.
This is a subbundle of rank $n - 1$.
Clause 1 holds because $T^{1,0} \C^n \cap T^{0,1} \C^n = 0$.
Clause 2 holds because $T^{1,0} \C^n$ is involutive and vector fields tangent to $M$ have brackets tangent to $M$.
So $M$ is a CR manifold of CR dimension $n - 1$ and CR codimension $1$.

Non-example: on $M = \R$, the complex line bundle $L$ spanned by $\partial / \partial x$ is involutive but satisfies $\overline{L} = L$, so clause 1 fails.

When $\operatorname{rank}_\C L = 1$, clause 2 holds automatically, since $[fX, gX] = (f \, Xg - g \, Xf) X$.
When $\dim_\R M = 2 \operatorname{rank}_\C L$, a CR structure is the same as an integrable almost complex structure, by the Newlander–Nirenberg theorem.
Some authors call a structure that satisfies only clause 1 an almost CR structure.
</document>

<document source="https://tamnd.github.io/math2/node/A2W5SNSC05.md" tag="A2W5SNSC05" kind="definition">
# Backward error analysis

## Definition

Let $X$ and $Y$ be normed spaces with [[norms|0V2WP9B1MF]] $\norm{\cdot}$, and let $f \colon X \to Y$ be a problem.
Let $x \in X$ with $x \ne 0$, and let $\hat y \in Y$ be a computed approximation of $y = f(x)$.

1. The *forward error* of $\hat y$ is $\norm{\hat y - y}$, and its relative forward error is $\norm{\hat y - y}/\norm{y}$ when $y \ne 0$.
2. The *relative backward error* of $\hat y$ is
$$\eta(\hat y) = \inf \set{ \frac{\norm{\Delta x}}{\norm{x}} : f(x + \Delta x) = \hat y },$$
with $\eta(\hat y) = \infty$ when no such $\Delta x$ exists.
3. *Backward error analysis* of an algorithm is a bound on $\eta(\hat y)$ for its computed outputs $\hat y$, usually by a modest multiple of the unit roundoff.

## Remarks

Example: in floating-point arithmetic the computed sum is $\hat s = \mathrm{fl}(a + b) = (a + b)(1 + \delta)$ with $\abs{\delta} \le u$.
It equals the exact sum of $a(1 + \delta)$ and $b(1 + \delta)$.
So for the problem $(a, b) \mapsto a + b$ with the max norm, the relative backward error is at most $u$.
Non-example: a bound on $\norm{\hat y - y}$ alone, with no statement about a perturbed input, is a forward error analysis.
If $f$ is differentiable with relative [[condition number|AB3S6MHJ8X]] $\kappa$ at $x$, then to first order the relative forward error is at most $\kappa \, \eta(\hat y)$.
Componentwise backward errors, which bound each $\abs{\Delta x_i}/\abs{x_i}$ separately, are also common.
</document>

<document source="https://tamnd.github.io/math2/node/A2WHCVP6JN.md" tag="A2WHCVP6JN" kind="definition">
# Germ of an analytic set

## Definition

Let $a \in \C^n$.
Consider pairs $(U, A)$ where $U$ is an open [[neighbourhood|0ZCY0Z80B6]] of $a$ in $\C^n$ and $A$ is an [[analytic subset|VMDTGTMN9P]] of $U$.
Two pairs $(U, A)$ and $(V, B)$ are equivalent if there is an open neighbourhood $W \subseteq U \cap V$ of $a$ with $A \cap W = B \cap W$.
A *germ of an analytic set* at $a$ is an equivalence class for this relation; the class of $(U, A)$ is written $A_a$.

## Remarks

This is the notion of [[germ|Y98MD3WVTG]] applied to analytic sets.
Inclusion, finite union and finite intersection of germs are defined through representatives on a common neighbourhood.
To a germ $A_a$ one attaches the ideal $\mathcal{I}(A_a)$ of all [[holomorphic germs|2TGPZ8VW2D]] $f_a \in \mathcal{O}_{n,a}$ that have a representative vanishing on a representative of $A_a$; it is a radical ideal.
Conversely, an ideal $I \subseteq \mathcal{O}_{n,a}$ with generators $f_{1,a}, \dots, f_{k,a}$ defines the germ $V(I)$ of the common zero set of representatives $f_1, \dots, f_k$.
Example: the sets $\set{z_1 z_2 = 0}$ and $\set{z_1 z_2 (z_1 - 1) = 0}$ in $\C^2$ have the same germ at $0$, since they agree on the ball of radius $1$ about $0$.
Non-example: the sets $\set{z_1 = 0}$ and $\set{z_1 = z_2^2}$ have different germs at $0$.
Indeed, every neighbourhood of $0$ contains a point $(0, t)$ with $t \ne 0$, which lies in the first set but not the second.
The germ of $A$ at a point $a \notin A$ is the empty germ.
</document>

<document source="https://tamnd.github.io/math2/node/A2YTPG6CQF.md" tag="A2YTPG6CQF" kind="definition">
# Homotopy limit

## Definition

Let $\mathcal{M}$ be a [[model category|SDTZVZ6B4Q]] and let $I$ be a small category.
Assume that the [[functor category|1X0WKFQX8K]] $\mathcal{M}^I$ carries the injective model structure.
In it, [[weak equivalences|E2870TEDG7]] and cofibrations are defined objectwise; it exists, for instance, when $\mathcal{M}$ is combinatorial.
The [[limit|CHS5AFJGWE]] functor $\lim \colon \mathcal{M}^I \to \mathcal{M}$ is right [[adjoint|TZHF6ZM92S]] to the constant diagram functor, which preserves cofibrations and trivial cofibrations, so $\lim$ is a right [[Quillen functor|SY38F954FE]].
The *homotopy limit* is its total right derived functor
$$\operatorname{holim} = \mathbf{R}\lim \colon \mathrm{Ho}(\mathcal{M}^I) \to \mathrm{Ho}(\mathcal{M}),$$
between the [[homotopy categories|48QQ7T98Z9]].
On a diagram $D$ it is computed as $\lim R D$, where $D \to RD$ is a weak equivalence to a [[fibrant|HSV1J7BJXX]] diagram in the injective model structure.

## Remarks

The homotopy limit sends objectwise weak equivalences of diagrams to weak equivalences, and it comes with a natural map $\lim D \to \operatorname{holim} D$.

Example: for a diagram $X \xrightarrow{f} Z \xleftarrow{g} Y$ of topological spaces, the homotopy limit is the homotopy pullback
$$X \times_Z Z^{[0,1]} \times_Z Y = \set{(x, \omega, y) : \omega \colon [0, 1] \to Z,\ \omega(0) = f(x),\ \omega(1) = g(y)}.$$
When $f$ or $g$ is a Serre fibration, the map from the ordinary pullback to the homotopy pullback is a weak equivalence.

Non-example: the ordinary limit is not homotopy invariant.
There is a map from the diagram $\set{0} \to \R \leftarrow \set{1}$ to the diagram $\set{0} \to \R \leftarrow \set{0}$ that is the identity on $\set{0}$, sends $1$ to $0$, and is $t \mapsto t^2 - t$ on $\R$.
It commutes with the maps of the diagrams and is a weak equivalence at each object.
The pullback of the first is empty and that of the second is a point, while both homotopy pullbacks are contractible path spaces.

For a diagram $D$ of Kan complexes there is an explicit model, the Bousfield–Kan formula $\operatorname{holim} D = \int_{i \in I} \mathrm{Map}(N(I/i), D(i))$, where $N(I/i)$ is the nerve of the over category.
Homotopy colimits are defined dually, as the left derived functor of the colimit, using the projective model structure.
</document>

<document source="https://tamnd.github.io/math2/node/A2ZXW6QTEA.md" tag="A2ZXW6QTEA" kind="definition">
# Dandelin spheres

## Definition

Let $K$ be a right circular [[cone|7TPBQPPN97]] in $\R^3$ with vertex $V$.
Let $\pi$ be a plane not through $V$ that meets $K$ in a [[conic section|00HJCMC2CD]] $C$.
A *Dandelin sphere* of $K$ and $\pi$ is a [[sphere|P9CM4GQ2T0]] $\Sigma$ such that:

1. $\Sigma$ is tangent to $K$ along a circle, that is, $\Sigma$ is inscribed in $K$;
2. $\Sigma$ is tangent to the plane $\pi$ at one point.

## Remarks

When $C$ is an [[ellipse|0VKCCEXBJ1]] there are exactly two Dandelin spheres, one on each side of $\pi$.
For a hyperbola there are two, one in each nappe, and for a parabola there is one.
Check of the focal property for an ellipse: let $\Sigma_1, \Sigma_2$ touch $\pi$ at $F_1, F_2$ and touch $K$ along circles $c_1, c_2$.
For $P \in C$, the generator line $VP$ meets $c_1$ and $c_2$ at points $Q_1, Q_2$.
Two tangent segments from $P$ to the same sphere have the same length, so $PF_1 = PQ_1$ and $PF_2 = PQ_2$.
Hence $PF_1 + PF_2 = Q_1 Q_2$, which is the distance between the parallel circles $c_1$ and $c_2$ measured along a generator, the same for all $P$; so $C$ is an ellipse with foci $F_1, F_2$.
When $C$ is not a circle, the plane of each circle $c_i$ meets $\pi$ in a line, which is the directrix belonging to the focus $F_i$.
Non-example: take a small sphere inscribed in $K$ near the vertex, on the vertex side of $\pi$ and far from it.
It is tangent to $K$ along a circle but does not touch $\pi$, so clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/A30TFC9Z92.md" tag="A30TFC9Z92" kind="definition">
# Multiple gamma function

## Definition

Let $N \ge 1$ be an integer, let $a = (a_1, \ldots, a_N)$ with real $a_i > 0$, and let $w$ be a complex number with $\operatorname{Re} w > 0$.
The *Barnes zeta function* is
$$\zeta_N(s, w \mid a) = \sum_{n_1, \ldots, n_N \ge 0} \frac{1}{(w + n_1 a_1 + \cdots + n_N a_N)^s}, \qquad \operatorname{Re} s > N,$$
with the principal branch of the power.
It has a meromorphic [[analytic continuation|0M3YQJ81NB]] in $s$ to $\C$ that is holomorphic at $s = 0$.
The *multiple gamma function* of order $N$ is
$$\Gamma_N(w \mid a) = \exp\left(\left.\frac{\partial}{\partial s} \zeta_N(s, w \mid a)\right|_{s = 0}\right).$$

## Remarks

Example: for $N = 1$ and $a_1 = 1$, $\zeta_1(s, w \mid 1)$ is the [[Hurwitz zeta function|G4SW7HJG88]] $\zeta(s, w)$.
Lerch's formula $\frac{\partial}{\partial s}\zeta(s, w)\big|_{s=0} = \log \Gamma(w) - \frac12 \log 2\pi$ gives $\Gamma_1(w \mid 1) = \Gamma(w)/\sqrt{2\pi}$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]].

The identity $\zeta_N(s, w \mid a) - \zeta_N(s, w + a_j \mid a) = \zeta_{N-1}(s, w \mid a^{(j)})$, where $a^{(j)}$ omits $a_j$, gives the functional equation
$$\Gamma_N(w + a_j \mid a) = \frac{\Gamma_N(w \mid a)}{\Gamma_{N-1}(w \mid a^{(j)})},$$
with the convention $\Gamma_0(w) = 1/w$.
For $N = 1$ it reads $\Gamma_1(w + 1 \mid 1) = w\, \Gamma_1(w \mid 1)$, the functional equation of $\Gamma$.

Non-example of a valid parameter: if $a_1 = 0$, each term is repeated for every value of $n_1$, the series diverges for every $s$, and $\Gamma_N$ is not defined.

Several normalisations are in use; they differ by factors of the form $\exp(\text{polynomial in } w)$.
The function $1/\Gamma_N(w \mid a)$ extends to an entire function of $w$.
</document>

<document source="https://tamnd.github.io/math2/node/A3413S0RTM.md" tag="A3413S0RTM" kind="definition">
# Monge–Ampère equation

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be an open set, and let $f \colon \Omega \times \R \times \R^n \to \R$ be a given function.
The *Monge–Ampère equation* is the [[fully nonlinear PDE|2D3N1ZJWBW]]
$$\det D^2 u(x) = f\big(x, u(x), \nabla u(x)\big), \qquad x \in \Omega,$$
for a twice differentiable function $u \colon \Omega \to \R$, where $D^2 u$ is the [[Hessian matrix|0693Y2METT]] of $u$.
The most studied case is $\det D^2 u = f(x)$ with $f > 0$, for an unknown [[convex function|07GR3TNW4H]] $u$.

## Remarks

Example: $u(x) = \tfrac12 \abs{x}^2$ solves $\det D^2 u = 1$ on $\R^n$, since $D^2 u$ is the identity matrix.
Non-example: in $\R^2$, $u(x) = \tfrac12 x_1^2$ does not solve $\det D^2 u = 1$, since $D^2 u = \operatorname{diag}(1, 0)$ has determinant $0$.
The equation is elliptic at a solution $u$ when $D^2 u$ is positive definite, which is why one looks for convex solutions with $f > 0$.
For $n = 2$ it reads $u_{xx} u_{yy} - u_{xy}^2 = f$.
The graph of $u$ has Gauss curvature $K = \det D^2 u / (1 + \abs{\nabla u}^2)^{(n+2)/2}$, so prescribing $K$ is a Monge–Ampère equation.
For convex functions that are not twice differentiable, the equation is understood in the sense of Aleksandrov or in the viscosity sense.
</document>

<document source="https://tamnd.github.io/math2/node/A34VR4FMRE.md" tag="A34VR4FMRE" kind="concept">
# Topological homogeneity

## Summary

A topological space is homogeneous when for any two of its points some [[homeomorphism|0QT00PB5G2]] of the space carries the first point to the second.
So the space looks the same near every point.
Every [[topological group|92700QAB5Y]] is homogeneous, because translations are homeomorphisms.
Homogeneity is therefore a first test of whether a space can carry the structure of a topological group.

## Explanations

### L1 human

A homogeneous space has no special points: whatever you can say about the space near one point, you can say near any other.
The circle is homogeneous, since rotating it moves any point to any other.
A closed interval $[0, 1]$ is not homogeneous: an endpoint is different from an inner point.
Removing an endpoint leaves the interval connected, while removing an inner point cuts it in two.
So a closed interval cannot be made into a topological group.
</document>

<document source="https://tamnd.github.io/math2/node/A38B1QFK20.md" tag="A38B1QFK20" kind="concept">
# Clifford algebra

## Summary

Let $V$ be a vector space with a [[quadratic form|0RBQS1JJCR]] $Q$.
The Clifford algebra of $(V, Q)$ is the associative algebra generated by $V$ in which each vector $v$ squares to the scalar $Q(v)$.
It contains the complex numbers, the quaternions and the [[exterior algebra|0MDMZ965VE]] as special cases.
It is used to build spinors and spin groups, and it gives an algebraic language for rotations and reflections in geometry and physics.

## Explanations

### L1 human

Start with vectors and allow to multiply them, with one rule: a vector times itself is the number $Q(v)$.
On the line $\R$ with $Q(x) = -x^2$, the basis vector $e$ satisfies $e^2 = -1$, so it behaves like the imaginary unit $i$.
In the plane with $Q(x, y) = -(x^2 + y^2)$, the basis vectors $e_1, e_2$ square to $-1$ and anticommute.
Then $e_1$, $e_2$ and $e_1 e_2$ behave like the quaternion units $i$, $j$ and $k$.
</document>

<document source="https://tamnd.github.io/math2/node/A399EBX2YD.md" tag="A399EBX2YD" kind="definition">
# Real form of a Lie group

## Definition

Let $G_\C$ be a complex [[Lie group|7JBRGCVJRB]] with complex Lie algebra $\mathfrak{g}_\C$.
A *real form* of $G_\C$ is a closed real [[Lie subgroup|NM8XCPS1Q3]] $G$ of $G_\C$ whose Lie algebra $\mathfrak{g} \subseteq \mathfrak{g}_\C$ is a real form of $\mathfrak{g}_\C$.
That is, $\mathfrak{g}_\C = \mathfrak{g} \oplus i \mathfrak{g}$ as real vector spaces.

## Remarks

Example: $G = \mathrm{SU}(n)$ in $G_\C = \mathrm{SL}_n(\C)$.
Here $\mathfrak{g} = \mathfrak{su}(n)$ is the space of skew-Hermitian matrices of trace $0$, and $i \mathfrak{g}$ is the space of Hermitian matrices of trace $0$.
Each traceless complex matrix $Z$ is uniquely $\frac{1}{2}(Z - Z^*) + \frac{1}{2}(Z + Z^*)$, a skew-Hermitian plus a Hermitian traceless matrix, so $\mathfrak{sl}_n(\C) = \mathfrak{g} \oplus i\mathfrak{g}$.

Non-example: $G = \mathrm{SO}(n)$ in $\mathrm{SL}_n(\C)$ for $n \ge 2$.
Its Lie algebra $\mathfrak{so}(n)$ of real antisymmetric matrices satisfies $\mathfrak{so}(n) \cap i\,\mathfrak{so}(n) = 0$, but $\mathfrak{so}(n) \oplus i\,\mathfrak{so}(n)$ is the space of complex antisymmetric matrices, which misses the diagonal matrices of trace $0$.

Often a real form is given as the fixed point group of an antiholomorphic involution $\sigma$ of $G_\C$, or as an open subgroup of it.
For $\sigma(g) = \bar g$ on $\mathrm{SL}_n(\C)$, the fixed points form $\mathrm{SL}_n(\R)$; for $\sigma(g) = (g^*)^{-1}$, they form $\mathrm{SU}(n)$.
A real form that is compact is a compact real form; for a connected complex semisimple group it exists and is unique up to conjugation.
</document>

<document source="https://tamnd.github.io/math2/node/A39XQHCVNW.md" tag="A39XQHCVNW" kind="concept">
# Proj construction

## Summary

The Proj construction makes a [[scheme|5A35TYZD02]] $\operatorname{Proj} S$ out of a [[graded ring|8KB4SQGC0C]] $S$.
Its points are the homogeneous prime ideals that do not contain all elements of positive degree.
It plays for projective geometry the role that $\operatorname{Spec}$ plays for affine geometry: $\operatorname{Proj}$ of a polynomial ring is projective space.

## Explanations

### L1 human

A point of projective space is given by coordinates $[x_0 : \cdots : x_n]$ that matter only up to a common factor.
So only homogeneous polynomials, whose zero sets do not depend on the factor, make sense on projective space.
The Proj construction turns this into algebra: it uses the homogeneous prime ideals of a graded ring.
It is covered by affine pieces, one for each homogeneous element $f$ of positive degree, on which $f \ne 0$.
For $S = k[x_0, \ldots, x_n]$ and $f = x_0$, that piece is the affine space with coordinates $x_1/x_0, \ldots, x_n/x_0$.
</document>

<document source="https://tamnd.github.io/math2/node/A3AP0P0D4M.md" tag="A3AP0P0D4M" kind="statement">
# Hopf invariant one problem

## Statement

Let $n \ge 2$.
There is a continuous map $f \colon S^{2n-1} \to S^n$ with [[Hopf invariant|TEGTSY7XMX]] $H(f) = \pm 1$ if and only if $n \in \set{2, 4, 8}$.

## Remarks

The maps for $n = 2, 4, 8$ are the Hopf fibrations built from the complex numbers, the quaternions and the octonions.
For odd $n$, every Hopf invariant is $0$, and for even $n$, the Whitehead product $[\iota_n, \iota_n]$ has Hopf invariant $\pm 2$, so only the value $\pm 1$ is in question.
An argument with Steenrod squares shows that $n$ must be a power of $2$.
J. F. Adams excluded all remaining $n \ge 16$ with secondary cohomology operations, which is [[Adams' theorem on Hopf invariant one|ZJVKZ8D64P]].
Adams and Atiyah later gave a short proof with Adams operations in topological K-theory.
Consequences: $\R^n$ carries the structure of a real division algebra, with a bilinear product and no zero divisors, only for $n = 1, 2, 4, 8$, and $S^{n-1}$ is an H-space only for $n = 1, 2, 4, 8$.

## Explanations

### L1 human

The Hopf map wraps the $3$-sphere around the $2$-sphere with the simplest possible twist, measured by Hopf invariant one.
The question is in which dimensions such a simplest twist exists.
The answer is only in dimensions $2$, $4$ and $8$, the dimensions of the complex numbers, the quaternions and the octonions.

### L2 human

The theorem explains why the list $\R, \C, \mathbb{H}, \mathbb{O}$ of division algebras stops.
A bilinear product without zero divisors on $\R^n$ makes $S^{n-1}$ an H-space, and the Hopf construction turns this multiplication into a map $S^{2n-1} \to S^n$ of Hopf invariant $\pm 1$.
So the topological theorem implies the algebraic one.
The problem also led to the Adams spectral sequence and to the use of K-theory operations in homotopy theory.
</document>

<document source="https://tamnd.github.io/math2/node/A3DP43YC3Z.md" tag="A3DP43YC3Z" kind="definition">
# Linear topology

## Definition

Let $A$ be a [[commutative ring|0310REJJY8]].
A [[topology|0CQHFPGJ45]] on $A$ is *linear* if it satisfies both conditions:

1. With this topology, $A$ is a [[topological ring|TSF8RZDD5B]].
2. The open [[ideals|096Z9MP2ME]] of $A$ form a fundamental system of neighbourhoods of $0$: every neighbourhood of $0$ contains an open ideal.

More generally, let $M$ be an [[$A$-module|0NZ8JA25WK]] and give $A$ a linear topology.
A topology on $M$ is *linear* if $M$ is a topological group under addition and the scalar multiplication $A \times M \to M$ is continuous.
In addition, the open [[submodules|05WDSN597C]] of $M$ must form a fundamental system of neighbourhoods of $0$.

## Remarks

Equivalently, a linear topology on $A$ is given by a family $\mathcal{F}$ of ideals such that each intersection $I \cap J$ of two members contains a member.
A set $U$ is then open when for each $x \in U$ there is $I \in \mathcal{F}$ with $x + I \subseteq U$.

Example: for an ideal $I$ of $A$, the [[I-adic topology|2VF2TSJ3PY]] has the powers $I^n$ as a fundamental system of neighbourhoods of $0$.
Each $I^n$ is an open ideal, so clause 2 holds; addition and multiplication are continuous because $(x + I^n) + (y + I^n) = x + y + I^n$ and $(x + I^n)(y + I^n) \subseteq xy + I^n$.
The discrete topology is linear, with the zero ideal open.

Non-example: the usual topology on $\R$ makes $\R$ a topological ring, so clause 1 holds.
But the only ideals of the field $\R$ are $0$ and $\R$.
The set $\set{0}$ is not open, and the interval $(-1, 1)$ does not contain $\R$; so clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/A3FR6PNEPB.md" tag="A3FR6PNEPB" kind="statement">
# Generalized minimal residual method

## Statement

Let $A$ be an [[invertible|0EB9N0S33C]] complex $n \times n$ matrix, let $b \in \C^n$, let $x_0 \in \C^n$, and let $r_0 = b - A x_0 \ne 0$.
For $k \ge 1$, the $k$-th iterate of the *generalized minimal residual method* (GMRES) is the vector $x_k \in x_0 + \mathcal{K}_k(A, r_0)$ that minimizes $\norm{b - A x}_2$ there.
Here $\mathcal{K}_k$ is the [[Krylov subspace|QB45PCGQPV]].

1. $x_k$ exists and is unique, and $\norm{r_k}_2 \le \norm{r_{k-1}}_2$, where $r_k = b - A x_k$.
2. Run $k$ steps of the [[Arnoldi iteration|CJ5EC820S7]] from $r_0$, giving $Q_{k+1}$ and $\tilde{H}_k$, and let $\beta = \norm{r_0}_2$. Then $x_k = x_0 + Q_k y_k$, where $y_k$ minimizes $\norm{\beta e_1 - \tilde{H}_k y}_2$ over $y \in \C^k$.
3. Some $r_k$ with $k \le n$ is zero, so GMRES finds $A^{-1} b$ in at most $n$ steps in exact arithmetic.
4. Let $P_k$ be the set of polynomials $p$ of degree at most $k$ with $p(0) = 1$. If $A = V \Lambda V^{-1}$ with $\Lambda$ diagonal, then $$\frac{\norm{r_k}_2}{\norm{r_0}_2} \le \kappa_2(V) \min_{p \in P_k} \max_{\lambda} \abs{p(\lambda)},$$ where $\lambda$ runs over the eigenvalues of $A$.

## Remarks

Every $x \in x_0 + \mathcal{K}_k(A, r_0)$ has residual $p(A) r_0$ for a polynomial $p$ of degree at most $k$ with $p(0) = 1$, which gives clause 4.
Clause 1 holds because the search spaces grow, and clause 2 follows from $A Q_k = Q_{k+1} \tilde{H}_k$ and the orthonormality of the columns of $Q_{k+1}$.
For clause 3, if $\mathcal{K}_k(A, r_0)$ becomes invariant under $A$, then it contains $A^{-1} r_0$ and the residual vanishes.
Clause 2 turns the large minimization into a small [[least squares|0BYTQ4X1HC]] problem with a Hessenberg matrix, solved by Givens rotations.
Example: if $A = I + N$ with $N^2 = 0$, then $p(t) = (1 - t)^2$ vanishes on $A$, so $r_2 = 0$ for any $b$.
The constant $\kappa_2(V)$ is the [[condition number|AB3S6MHJ8X]] of the eigenvector matrix; for normal $A$ it equals $1$.
For highly non-normal $A$, the eigenvalues alone do not decide the convergence.

## Explanations

### L1 human

GMRES looks at all vectors it can build from the starting residual by repeated multiplication with $A$.
Among them it picks the one whose residual is as small as possible.
The residual can never grow, and with exact arithmetic the method reaches the solution after at most $n$ steps.
It works for any invertible matrix, symmetric or not.

### L2 human

GMRES is the residual-minimizing [[iterative method|2RPE77ACER]] for general nonsymmetric systems, while the [[conjugate gradient method|AFXZGQB4EN]] minimizes the energy norm for symmetric positive definite ones.
Since Arnoldi needs all basis vectors, the cost per step grows; restarted GMRES($m$) throws them away every $m$ steps, and it may then stagnate.
It is almost always used with a [[preconditioner|KE98SW3SBD]], often an incomplete LU factorization.
The method is due to Saad and Schultz.
</document>

<document source="https://tamnd.github.io/math2/node/A3FTHW4NES.md" tag="A3FTHW4NES" kind="statement">
# Blichfeldt's theorem

## Statement

Let $n \ge 1$ and let $\Lambda \subset \R^n$ be a [[lattice|WN9V7WSJKF]] with covolume $\det \Lambda$.
Let $S \subseteq \R^n$ be a [[Lebesgue measurable|2Z2QVWZ262]] set with volume $\operatorname{vol}(S) > \det \Lambda$.
Then there are two distinct points $x, y \in S$ such that $x - y \in \Lambda$.

## Remarks

Let $P$ be a fundamental parallelepiped of $\Lambda$, so that the translates $P + v$ with $v \in \Lambda$ are pairwise disjoint and cover $\R^n$.
Cut $S$ into the pieces $S \cap (P + v)$ and translate each piece by $-v$ into $P$.
The translated pieces have total volume $\operatorname{vol}(S) > \operatorname{vol}(P) = \det \Lambda$, so two of them overlap; a common point gives $x - v = y - w$ with $x \ne y$ in $S$ and $v \ne w$ in $\Lambda$.
[[Minkowski's theorem|0K387S2PKQ]] follows: if $K$ is convex, symmetric about $0$, and $\operatorname{vol}(K) > 2^n \det \Lambda$, apply the theorem to $S = \tfrac12 K$.
It gives $x \ne y$ in $S$, and $x - y = \tfrac12(2x) + \tfrac12(-2y)$ is a nonzero point of $K \cap \Lambda$.
The bound is sharp: the half-open square $[0, 1)^2$ has area $1 = \det \Z^2$, and no two of its distinct points differ by an integer vector.
More generally, if $\operatorname{vol}(S) > k \det \Lambda$ for a positive integer $k$, the same argument gives $k + 1$ distinct points of $S$ whose pairwise differences lie in $\Lambda$.

## Explanations

### L1 human

Move every point of a region of the plane by whole steps left, right, up or down until it lands in the unit square.
If the region has area larger than $1$, the moved pieces cannot all fit in the square without overlapping.
So the region contains two points that differ by a whole-number shift.
The theorem says the same for any lattice in any dimension, with the volume of a lattice cell in place of $1$.

### L2 human

Blichfeldt's theorem is the measure-theoretic pigeonhole principle behind the geometry of numbers.
It gives a short proof of Minkowski's convex body theorem, and through it of results such as the two-square and four-square theorems.
It holds for any measurable set, convex or not, which makes it useful when the region of interest is not convex.
</document>

<document source="https://tamnd.github.io/math2/node/A3R53S95KP.md" tag="A3R53S95KP" kind="concept">
# Runge domain

## Summary

A Runge domain is an open set in $\C^n$ on which every [[holomorphic function|507C34W0ZT]] can be approximated by polynomials, uniformly on each compact subset.
The name comes from [[Runge's theorem|6DG4CTGXJD]], which describes such open sets in the plane: they are the open sets whose connected components are simply connected.
In several variables, convex domains and balls are Runge domains, and the [[Oka–Weil theorem|7AWZJNQGT2]] is the main tool to recognise them.

## Explanations

### L1 human

Polynomials are the simplest holomorphic functions, so one asks on which regions they are enough to approximate all holomorphic functions.
On a disc, Taylor polynomials do the job.
On an annulus they do not: the function $1/z$ cannot be approximated by polynomials on a circle around the hole.
A Runge domain is a region where polynomials always suffice, at least on each compact part.
</document>

<document source="https://tamnd.github.io/math2/node/A3W7RGXXK9.md" tag="A3W7RGXXK9" kind="concept">
# Lebesgue covering dimension

## Summary

The Lebesgue covering dimension of a topological space is the least $n$ with the following property.
Every finite open cover can be refined to an open cover in which no point lies in more than $n + 1$ sets.
It is a topological invariant, and it gives $n$ for the Euclidean space $\R^n$.
It is one of the main notions of dimension theory, together with the [[inductive dimensions|MKS9H497AM]].

## Explanations

### L1 human

Cover a line segment by short overlapping open intervals; one can arrange that each point lies in at most two of them, but not always in only one.
Cover a square by small open tiles; one can arrange that each point lies in at most three of them, like bricks in a wall, but not in fewer.
The covering dimension turns this count into a definition: a space has dimension $n$ when the best one can do is $n + 1$.
So a segment has dimension $1$ and a square has dimension $2$.
</document>

<document source="https://tamnd.github.io/math2/node/A3ZHAPT767.md" tag="A3ZHAPT767" kind="definition">
# Tate cohomology

## Definition

Let $G$ be a finite group and let $M$ be a $G$-module, a [[module|0NZ8JA25WK]] over the [[group ring|4NMTSX0XDS]] $\Z[G]$.
Let $M^G$ be the submodule of elements fixed by $G$, and let $M_G = M / I_G M$ be the module of coinvariants, where $I_G M$ is generated by all $gm - m$.
The norm map $N \colon M \to M$ is $N(m) = \sum_{g \in G} gm$; it induces a map $\bar N \colon M_G \to M^G$.
The *Tate cohomology groups* $\hat H^n(G, M)$, for $n \in \Z$, are:

1. $\hat H^n(G, M) = H^n(G, M)$ for $n \ge 1$, the [[group cohomology|YXJXA0Q52R]].
2. $\hat H^0(G, M) = M^G / N(M)$.
3. $\hat H^{-1}(G, M) = \ker(\bar N \colon M_G \to M^G)$.
4. $\hat H^{-n-1}(G, M) = H_n(G, M)$ for $n \ge 1$, the [[group homology|CFAG3VRXEA]].

## Remarks

Equivalently, $\hat H^n(G, M)$ is the cohomology of $\Hom_G(P_\bullet, M)$ for a complete resolution $P_\bullet$.
A complete resolution is an exact complex of projective $\Z[G]$-modules indexed by $\Z$ that agrees with a projective resolution of $\Z$ in nonnegative degrees.

Example: let $G$ be [[cyclic|0KVGH0J5B3]] of order $m$ and let $M = \Z$ with trivial action.
Then $M^G = M_G = \Z$ and $N$ is multiplication by $m$.
So $\hat H^0(G, \Z) = \Z / m\Z$ and $\hat H^{-1}(G, \Z) = \ker(m \colon \Z \to \Z) = 0$.
For cyclic $G$, $\hat H^n(G, M) \cong \hat H^{n+2}(G, M)$ for every $n$ and every $M$.

Example: for $M = \Z[G]$, all groups $\hat H^n(G, \Z[G])$ vanish.
In degree $0$, both $M^G$ and $N(M)$ equal $\Z \cdot \sum_{g \in G} g$.

Non-example: for an infinite group $G$, the sum $\sum_{g \in G} gm$ is not defined, so the definition needs $G$ finite.
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<document source="https://tamnd.github.io/math2/node/A486VR787Z.md" tag="A486VR787Z" kind="definition">
# Disjoint union topology

## Definition

Let $I$ be a set and let $(X_i)_{i \in I}$ be a family of [[topological spaces|0CQHFPGJ45]].
The disjoint union of the family is the set
$$X = \bigsqcup_{i \in I} X_i = \set{(i, x) : i \in I, \ x \in X_i}.$$
For each $i \in I$, the canonical injection $\iota_i \colon X_i \to X$ is $\iota_i(x) = (i, x)$.
The *disjoint union topology* on $X$ is the set of all $U \subseteq X$ such that $\iota_i^{-1}(U)$ is [[open|02BH3MCBQB]] in $X_i$ for every $i \in I$.

## Remarks

It is the [[final topology|6ERJRRZK32]] on $X$ for the family of maps $\iota_i$.
Each $\iota_i$ is an open [[topological embedding|00ZG9Y9TR4]], and the image of each $X_i$ is open and closed in $X$.
Universal property: a map $f \colon X \to Y$ to a topological space $Y$ is [[continuous|0VGFHTM75Y]] if and only if $f \circ \iota_i$ is continuous for every $i$.

Example: if each $X_i$ is a one-point space, then every subset of $X$ is open, so $X$ is the discrete space on $I$.
Indeed, the preimage of any subset under each $\iota_i$ is empty or a point, and both are open.

Non-example: let $X_0$ and $X_1$ be two copies of a two-point space with the trivial topology, and let $X = X_0 \sqcup X_1$.
The trivial topology on the four-point set $X$ is not the disjoint union topology.
The set $\iota_0(X_0)$ has open preimages in both pieces, but it is not open in the trivial topology.

Notation: $\bigsqcup_i X_i$ or $\coprod_i X_i$; for two spaces, $X_1 \sqcup X_2$.
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<document source="https://tamnd.github.io/math2/node/A49BATWQEH.md" tag="A49BATWQEH" kind="definition">
# Ordered group

## Definition

Let $G$ be a [[group|G5S1M9]] with identity $e$, and let $\le$ be a [[partial order|011C29Q1S1]] on $G$.
The pair $(G, \le)$ is an *ordered group*, or *partially ordered group*, if for all $a, b, c \in G$,
$$a \le b \implies ca \le cb \text{ and } ac \le bc.$$

## Remarks

Example: $\Z^2$ under addition, with $(a, b) \le (c, d)$ when $a \le c$ and $b \le d$, is an ordered group.
If $a \le c$ and $b \le d$, then $a + x \le c + x$ and $b + y \le d + y$ for each $(x, y) \in \Z^2$.
Non-example: on $\Z$, let $a \preceq b$ when $a = b$ or $(a, b) = (0, 1)$.
This is a partial order, but $0 \preceq 1$ while $1 \not\preceq 2$, so adding $1$ breaks the condition.
The *positive cone* of $G$ is $P = \set{g \in G : e \le g}$.
It satisfies $PP \subseteq P$, $P \cap P^{-1} = \set{e}$ and $gPg^{-1} = P$ for each $g \in G$.
Conversely, each subset $P$ with these three properties defines an ordered group by $a \le b \iff a^{-1} b \in P$.
For an abelian group the condition is written additively: $a \le b$ implies $a + c \le b + c$.
</document>

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# Skeleton of a category

## Summary

A skeleton of a [[category|A2DE66937D]] $\mathcal{C}$ is a [[full subcategory|V3BFGKBJF7]] that contains exactly one object from each isomorphism class of objects of $\mathcal{C}$.
Under the axiom of choice every category has a skeleton, and the inclusion of the skeleton is an [[equivalence of categories|K7A04ZBD92]].
So a skeleton is a smallest model of the category up to equivalence, and two categories are equivalent exactly when their skeletons are isomorphic.

## Explanations

### L1 human

Many objects of a category are copies of each other up to isomorphism.
A skeleton keeps one representative of each kind and throws the copies away, while keeping all arrows between the representatives.
For finite sets, the sets $\set{1, \dots, n}$, one for each $n \ge 0$, form a skeleton.
Nothing essential is lost, because every finite set is in bijection with exactly one of them.
</document>

<document source="https://tamnd.github.io/math2/node/A4E24P47KP.md" tag="A4E24P47KP" kind="definition">
# Hausdorff topological group

## Definition

A *Hausdorff topological group* is a [[topological group|92700QAB5Y]] $G$ whose topology is [[Hausdorff|0S1FJ8VEX9]].
That is, for any two distinct $x, y \in G$ there are disjoint open sets $U \ni x$ and $V \ni y$.

## Remarks

For a topological group $G$ with identity $e$, the following are equivalent: $G$ is Hausdorff; $G$ is $T_1$; $G$ is $T_0$; the set $\set{e}$ is [[closed|0ZCK8DQYN1]].
They are also equivalent to: the intersection of all neighbourhoods of $e$ is $\set{e}$.
Example: $(\R, +)$ with the usual topology, which comes from a metric and so is Hausdorff.
Non-example: $\Z/2\Z$ with the indiscrete topology is a topological group, but its two points have no disjoint open neighbourhoods.
In fact a topological group is Hausdorff if and only if it is a Tychonoff space.
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<document source="https://tamnd.github.io/math2/node/A4E7SZ935M.md" tag="A4E7SZ935M" kind="definition">
# Almgren–Pitts min-max theory

## Definition

Let $(M, g)$ be a closed [[Riemannian manifold|7CBT4SM83Y]] of dimension $n + 1 \ge 3$.
Let $\mathcal{Z}_n(M; \Z_2)$ be the space of $n$-dimensional mod $2$ flat cycles in $M$ that are boundaries, with the flat topology.
These are the mod $2$ integral [[currents|Z6QXZX1THM]] $T = \partial U$, where $U$ is an $(n+1)$-dimensional mod $2$ integral current.
Let $\mathbf{M}(T)$ be the mass of $T$.

1. A *sweepout* of $M$ is a map $\Phi \colon [0, 1] \to \mathcal{Z}_n(M; \Z_2)$ with $\Phi(0) = \Phi(1) = 0$, continuous in the flat topology and with no concentration of mass.
It must represent the nontrivial class of $\pi_1(\mathcal{Z}_n(M; \Z_2), 0)$.
This group is $\Z_2$, since the Almgren isomorphism identifies it with $H_{n+1}(M; \Z_2)$.
2. The *width* of $M$ is
$$W(M) = \inf_{\Phi} \ \sup_{t \in [0, 1]} \mathbf{M}\big(\Phi(t)\big),$$
the infimum over all sweepouts.

*Almgren–Pitts min-max theory* is the theory of such min-max values, and of the analogous values over $k$-parameter families of cycles.
Its main theorem states that $W(M) > 0$.
Moreover, $W(M)$ is the area of a closed embedded [[minimal hypersurface|41X0YW2R9N]], counted with integer multiplicities.
This hypersurface is smooth when $3 \le n + 1 \le 7$.
In general its singular set has [[Hausdorff dimension|SSNFTHZ6YJ]] at most $n - 7$.

## Remarks

Pitts proved the smoothness for $3 \le n + 1 \le 6$, and the case $n + 1 = 7$ and the higher-dimensional statement follow from the regularity theory of Schoen and Simon.
The positivity of the width follows from an isoperimetric inequality: cycles of small mass bound regions of small volume, so a sweepout of small mass would be trivial.

Example: for the round unit sphere $S^3$, the level sets of a height function $x_4$ form a sweepout, $t \mapsto \partial\set{x_4 \le 2t - 1}$, whose largest member is the equatorial $2$-sphere, of area $4\pi$.
Here the width equals $4\pi$.

Non-example: the family $t \mapsto \partial B_{r(t)}(p)$ of boundaries of small geodesic balls, with $r(t) = \varepsilon \sin \pi t$ and $\varepsilon$ small, starts and ends at $0$ but is not a sweepout.
It is contractible in $\mathcal{Z}_n(M; \Z_2)$ relative to $0$, because the enclosed regions never fill $M$; so it represents the trivial class and fails clause 1.
</document>

<document source="https://tamnd.github.io/math2/node/A4H4CZ2V35.md" tag="A4H4CZ2V35" kind="concept">
# Projective variety

## Summary

A projective variety is an irreducible subset of projective space $\mathbb{P}^n$ over an algebraically closed field cut out by homogeneous polynomial equations.
Projective varieties are the algebraic analogue of compact spaces.
Over $\C$ they are compact in the usual topology, and images of projective varieties under morphisms are closed.
Most classical algebraic geometry, such as plane curves, Bézout's theorem and intersection theory, takes place on projective varieties.

## Explanations

### L1 human

In projective space a point has homogeneous coordinates that are only defined up to scaling.
So an equation makes sense on points only if scaling does not change whether it holds, which is true for homogeneous polynomials such as $x^2 + y^2 - z^2$.
The zero set of $x^2 + y^2 - z^2$ in the projective plane is a conic.
In the chart $z = 1$ it is the circle $x^2 + y^2 = 1$, and the projective version adds no points at infinity over $\R$ but two over $\C$.
</document>

<document source="https://tamnd.github.io/math2/node/A4P47SGE4S.md" tag="A4P47SGE4S" kind="definition">
# Prolongation of a section to a jet bundle

## Definition

Let $\pi \colon E \to M$ be a smooth [[fiber bundle|CX4M8K61AB]], let $k \ge 0$, and let $\pi_k \colon J^k\pi \to M$ be its $k$-th [[jet bundle|HC3VBBCJQP]].
Let $s \colon U \to E$ be a smooth local section of $\pi$ over an open set $U \subseteq M$.
The *$k$-th prolongation* of $s$ is the map
$$j^k s \colon U \to J^k\pi, \qquad p \mapsto j^k_p s,$$
where $j^k_p s$ is the [[$k$-jet|1QE0Z0756D]] of $s$ at $p$.
It is a smooth local section of $\pi_k$.

A local section $\sigma$ of $\pi_k$ is *holonomic* if $\sigma = j^k s$ for some local section $s$ of $\pi$.

## Remarks

In adapted coordinates $(x^i, u^\alpha, u^\alpha_I)$, if $s$ has components $s^\alpha(x)$, then
$$j^k s(x) = \Big(x^i,\ s^\alpha(x),\ \frac{\partial^{\abs{I}} s^\alpha}{\partial x^I}(x)\Big)_{1 \le \abs{I} \le k}.$$
If $\sigma = j^k s$, then $s = \pi_{k,0} \circ \sigma$, so $s$ is determined by $\sigma$.

Example: for $\pi \colon \R \times \R \to \R$ and $s(x) = x^2$, the first prolongation is $j^1 s(x) = (x, x^2, 2x)$.

Non-example: the section $\sigma(x) = (x, 0, 1)$ of $J^1\pi \to \R$ for the same bundle is not holonomic.
If $\sigma = j^1 s$, then $s = \pi_{1,0} \circ \sigma = 0$, so $j^1 s(x) = (x, 0, 0) \ne \sigma(x)$.

A [[partial differential equation|GC5BDZ6QBC]] of order $k$ for sections of $\pi$ can be given as a subset $\mathcal{R} \subseteq J^k\pi$.
A local section $s$ is a solution exactly when $j^k s$ takes its values in $\mathcal{R}$.
</document>

<document source="https://tamnd.github.io/math2/node/A4SS2SN0MB.md" tag="A4SS2SN0MB" kind="definition">
# Moduli space

## Definition

Let $S$ be a base scheme, and let $F \colon (\mathrm{Sch}/S)^{\mathrm{op}} \to \mathrm{Set}$ be a contravariant [[functor|28NQ0KRHD3]], the *moduli functor*.
The set $F(T)$ consists of the isomorphism classes of families of the objects to be classified, parametrized by the [[scheme|5A35TYZD02]] $T$.
The functor $F$ acts on morphisms by pullback of families.
For an $S$-scheme $M$, write $h_M = \Hom_S(-, M)$ for its [[functor of points|SWWW9X4A3N]].

1. A *fine moduli space* for $F$ is an $S$-scheme $M$ with an isomorphism of functors $F \cong h_M$, that is, $M$ [[represents|KCCHH87H6E]] $F$.
   The family $\mathcal{U} \in F(M)$ that corresponds to $\id_M$ is the *universal family*.
2. A *coarse moduli space* for $F$ is an $S$-scheme $M$ with a natural transformation $\Phi \colon F \to h_M$ such that:
   1. for every [[algebraically closed field|02SJG1HC32]] $\Omega$ and morphism $\operatorname{Spec} \Omega \to S$, the map $\Phi \colon F(\operatorname{Spec} \Omega) \to M(\Omega)$ is a bijection;
   2. for every $S$-scheme $N$ and natural transformation $\Psi \colon F \to h_N$, there is a unique morphism $f \colon M \to N$ with $\Psi = h_f \circ \Phi$.

## Remarks

A fine moduli space is also a coarse one, and every family over $T$ is the pullback of $\mathcal{U}$ along a unique map $T \to M$.
Example: let $F(T)$ be the set of line subbundles $L \subset \mathcal{O}_T^{n+1}$ with locally free quotient.
Then $\mathbb{P}^n_S$ is a fine moduli space for $F$, with universal family the tautological line bundle $\mathcal{O}(-1) \subset \mathcal{O}^{n+1}$.
Non-example: elliptic curves over an algebraically closed field of characteristic $0$ have no fine moduli space.
Every elliptic curve has the automorphism $-1$, and this gives nontrivial families whose fibers are all isomorphic.
The affine line, through the [[j-invariant|9538W1K1EX]], is a coarse moduli space for them.
Automorphisms are the usual obstruction to fine moduli spaces, and moduli stacks are a way around it.
</document>

<document source="https://tamnd.github.io/math2/node/A4TRMA5V4V.md" tag="A4TRMA5V4V" kind="statement">
# Multiplication formula for the Fourier transform

## Statement

Let $n \ge 1$ and let $f, g \in L^1(\R^n)$.
Let $\hat h(\xi) = \int_{\R^n} h(x) e^{-2\pi i x \cdot \xi}\, dx$ denote the [[Fourier transform|DPVM34ADZV]] of $h \in L^1(\R^n)$.
Then
$$\int_{\R^n} \hat f(y)\, g(y)\, dy = \int_{\R^n} f(y)\, \hat g(y)\, dy,$$
and both integrals converge absolutely.

## Remarks

Both sides equal the double integral $\int\int f(x) g(y) e^{-2\pi i x \cdot y}\, dx\, dy$.
This double integral converges absolutely, with bound $\norm{f}_1 \norm{g}_1$, so [[Fubini's theorem|0R0T1VWH5X]] allows either order of integration.
Absolute convergence of each side also follows from $\abs{\hat f} \le \norm{f}_1$ and $g \in$ [[$L^1$|SASP14MA1Z]].

Example: take $n = 1$, $f = \mathbf{1}_{[-1/2, 1/2]}$ and $g(y) = e^{-\pi y^2}$, so that $\hat g = g$.
The right side is $\int_{-1/2}^{1/2} e^{-\pi y^2}\, dy$.
The left side is $\int_\R \frac{\sin \pi y}{\pi y} e^{-\pi y^2}\, dy$, and the formula says the two numbers are equal.

The formula is the basis of the definition of the Fourier transform of a tempered distribution $T$, namely $\hat T(\varphi) = T(\hat \varphi)$.
By the formula, this agrees with the transform of $f$ when $T$ is integration against $f$.

## Explanations

### L1 human

The transform can be moved from one factor to the other inside an integral of a product.
It is a symmetry of the Fourier transform with respect to the pairing $\int f g$.

### L2 human

The formula is the main tool in proofs of the Fourier inversion theorem: one pairs $\hat f$ with a Gaussian of shrinking width and moves the transform onto the Gaussian.
It is also what makes the extension of the transform to tempered distributions consistent with the integral definition.
With complex conjugates it leads to the Parseval form $\int f \bar g = \int \hat f \overline{\hat g}$ on $L^1 \cap L^2$.
</document>

<document source="https://tamnd.github.io/math2/node/A50Y012RBP.md" tag="A50Y012RBP" kind="concept">
# Cusp form

## Summary

A cusp form is a [[modular form|GWKN5VG78A]] that vanishes at every cusp.
For the full [[modular group|VEKV1EM77B]], this means that the constant term of its $q$-expansion is $0$.
The cusp forms of weight $k$ form a subspace $S_k$ of $M_k$, on which the [[Petersson inner product|EMZTGD63V3]] is defined.
The coefficients of cusp forms grow more slowly than those of Eisenstein series, and they carry deep arithmetic information.

## Explanations

### L1 human

A modular form has an expansion $a_0 + a_1 q + a_2 q^2 + \cdots$ with $q = e^{2 \pi i \tau}$.
As $\tau$ goes up to infinity, $q$ goes to $0$, and the value tends to $a_0$.
A cusp form is one with $a_0 = 0$, so it dies off at infinity.
The first example is the discriminant $\Delta = q \prod_{n \ge 1} (1 - q^n)^{24}$ of weight $12$.
Its coefficients are the values of the [[Ramanujan tau function|6DNZAGCSQS]].
</document>

<document source="https://tamnd.github.io/math2/node/A51SQRPM5T.md" tag="A51SQRPM5T" kind="concept">
# Brelot space

## Summary

A Brelot space is a connected, locally connected space with a sheaf of harmonic functions, as in a [[harmonic space|DWN0XEQHHK]].
In it, an increasing sequence of harmonic functions on a domain tends either to a harmonic function or to $+\infty$ everywhere.
Its axioms, due to Marcel Brelot, model elliptic equations such as the Laplace equation.
In a Brelot space a Harnack inequality holds, as for classical [[harmonic functions|0DEB0JMN0W]].
The heat equation gives a harmonic space that is not a Brelot space.

## Explanations

### L1 human

Brelot's axioms capture the behaviour of harmonic functions in Euclidean space without using derivatives.
The key axiom says that an increasing sequence of harmonic functions on a connected open set cannot blow up in one place and stay finite in another.
This reflects the fact that positive harmonic functions cannot be very large at one point and small at a nearby point.
For temperatures that change in time this fails: heat at a later time does not control heat at an earlier time.
</document>

<document source="https://tamnd.github.io/math2/node/A54T86KS63.md" tag="A54T86KS63" kind="concept">
# Uniform space

## Summary

A uniform space is a set with a uniform structure, a family of [[entourages|FVFFKNKEDF]] that say when two points are close to a given degree.
It is the natural setting for [[uniform continuity|0M4JJT780X]], Cauchy filters and completeness, without a metric.
Metric spaces and topological groups carry uniform structures, and every uniform space has an underlying topology.

## Explanations

### L1 human

In a metric space one can say that $x$ and $y$ are as close as $u$ and $v$: both distances are below $\varepsilon$.
A topology alone cannot compare closeness at different places, but a uniform structure can.
It lists the relations "being $\varepsilon$-close" as sets of pairs of points, without numbers.
For example, on $\R$ the sets $\set{(x, y) : \abs{x - y} < \varepsilon}$ for $\varepsilon > 0$ generate the usual uniform structure.
</document>

<document source="https://tamnd.github.io/math2/node/A56086N8XQ.md" tag="A56086N8XQ" kind="statement">
# Gauss–Lucas theorem

## Statement

Let $P$ be a nonconstant polynomial with coefficients in the [[complex numbers|09FTAET5Z4]], and let $P'$ be its derivative.
Then every [[root|00EHX6FWC0]] of $P'$ lies in the [[convex hull|0TCHAS04WK]] of the set of roots of $P$, viewed as points of the plane $\C$.

## Remarks

Proof idea: write $P(z) = c \prod_{i=1}^{n} (z - z_i)$, so that $\frac{P'(z)}{P(z)} = \sum_{i=1}^{n} \frac{1}{z - z_i}$ when $P(z) \ne 0$.
Let $w$ be a root of $P'$; if $P(w) = 0$ there is nothing to prove.
Otherwise $\sum_i \frac{1}{w - z_i} = 0$; taking complex conjugates gives $\sum_i \frac{w - z_i}{\abs{w - z_i}^2} = 0$.
So $w = \sum_i \lambda_i z_i$ with $\lambda_i = \abs{w - z_i}^{-2} \big/ \sum_j \abs{w - z_j}^{-2}$, a convex combination of the $z_i$.

Example: $P(z) = z^3 - 1$ has roots the three cube roots of unity, and $P'(z) = 3z^2$ has the root $0$, the centre of the triangle they form.

Take a real polynomial whose roots are all real.
Then the theorem says that the roots of $P'$ lie between the smallest and the largest root of $P$, which also follows from [[Rolle's theorem|0194DY35BG]].

## Explanations

### L1 human

Mark the roots of a polynomial as points in the plane and stretch a rubber band around them.
The roots of the derivative all lie inside the band.
Differentiating never moves the roots outside the region spanned by the original ones.

### L2 human

The theorem is a complex analogue of Rolle's theorem.
Applied repeatedly, it shows that the roots of all derivatives $P^{(k)}$, $k < \deg P$, lie in the convex hull of the roots of $P$.
It is a basic tool in the geometry of polynomials and is used, for example, in a short proof of [[Bernstein's inequality for polynomials|F0YGC89S9R]].
</document>

<document source="https://tamnd.github.io/math2/node/A56P5S5D6Z.md" tag="A56P5S5D6Z" kind="problem">
# Tautochrone problem

## Problem

A bead slides without friction along a curve in a vertical plane, under a constant downward gravitational acceleration $g > 0$, and starts at rest.
Find the curves with a lowest point that have the following property.
The time the bead needs to reach the lowest point does not depend on its starting point on the curve.

## Remarks

The problem is solved: the answer is an arc of a cycloid with its cusps pointing up.
For the cycloid traced by a circle of radius $a$, the descent time from every starting point is $\pi \sqrt{a / g}$.
Huygens gave a geometric solution and used it for the design of pendulum clocks, in his book Horologium Oscillatorium of 1673.

In the nineteenth century Abel treated the problem through an integral equation.
Let $y$ be the height above the lowest point and $s(y)$ the arc length from the lowest point, and let $\varphi(y) = s'(y)$.
By conservation of energy, the speed at height $y$ of a bead released at height $h$ is $\sqrt{2 g (h - y)}$, so the descent time is
$$T(h) = \frac{1}{\sqrt{2g}} \int_0^h \frac{\varphi(y)}{\sqrt{h - y}} \, dy.$$
Asking that $T(h) = T$ for all $h$ is [[Abel's integral equation|G4GWBYTW7B]] with constant right side $\sqrt{2g} \, T$.
The [[Abel inversion formula|TDMXFY06A2]] gives $\varphi(y) = \frac{\sqrt{2g} \, T}{\pi \sqrt{y}}$, and the curves with $s'(y)$ proportional to $y^{-1/2}$ are exactly the cycloids, with $a = g T^2 / \pi^2$.
Check: on the cycloid $s^2 = 8 a y$, energy conservation gives $s'' = -\frac{g}{4a} s$, a harmonic oscillation whose quarter period $\frac{\pi}{2} \sqrt{4a/g} = \pi \sqrt{a/g}$ does not depend on the amplitude.

## Explanations

### L1 human

On most slides, a ball released higher up takes longer to reach the bottom.
On a cycloid, the steeper upper part speeds up a ball released high just enough to compensate for the longer way.
All balls arrive at the bottom at the same moment, wherever they start.
The problem is a famous early example in which an unknown curve is found from a condition on travel times.
Abel's treatment of it is often seen as a starting point of the theory of integral equations.
</document>

<document source="https://tamnd.github.io/math2/node/A58Z1DQ593.md" tag="A58Z1DQ593" kind="statement">
# Frullani integral

## Statement

Let $f \colon [0, \infty) \to \R$ be [[continuous|0P27W9RW0G]] and assume that the limit $f(\infty) = \lim_{x \to \infty} f(x)$ exists in $\R$.
Let $a, b > 0$.
Then the [[improper integral|0VCTBJMEJF]] below converges and
$$\int_0^\infty \frac{f(ax) - f(bx)}{x}\, dx = \bigl(f(\infty) - f(0)\bigr) \ln \frac{a}{b},$$
where $\ln$ is the [[natural logarithm|0044CPHJBV]].

## Remarks

For $0 < \varepsilon < R$, the substitutions $t = ax$ and $t = bx$ give
$$\int_\varepsilon^R \frac{f(ax) - f(bx)}{x}\, dx = \int_{a\varepsilon}^{b\varepsilon} \frac{f(t)}{t}\, dt - \int_{aR}^{bR} \frac{f(t)}{t}\, dt.$$
As $\varepsilon \to 0$ the first integral tends to $f(0) \ln(b/a)$ by continuity at $0$, and as $R \to \infty$ the second tends to $f(\infty) \ln(b/a)$.
Example: with $f(x) = e^{-x}$ one gets $\int_0^\infty \frac{e^{-ax} - e^{-bx}}{x}\, dx = \ln \frac{b}{a}$.
The limit at infinity matters: for $f(x) = \sin x$ the right side has no meaning, although the integral still converges.

## Explanations

### L1 human

The formula evaluates a family of integrals that look hard but depend only on the values of $f$ at $0$ and at infinity.
Comparing $f$ at two different scales $ax$ and $bx$ leaves only the behaviour at the two ends, weighted by $\ln(a/b)$.

### L2 human

Frullani integrals appear in the computation of integral representations of logarithms, such as $\ln(b/a) = \int_0^\infty \frac{e^{-ax} - e^{-bx}}{x}\, dx$.
The proof is a change of variables in an [[improper integral|0VCTBJMEJF]] together with continuity at the two ends.
Variants replace continuity by local integrability with suitable mean values of $f$ near $0$ and near infinity.
</document>

<document source="https://tamnd.github.io/math2/node/A5D94C38HZ.md" tag="A5D94C38HZ" kind="definition">
# Conditionally complete lattice

## Definition

Let $(L, \le)$ be a [[lattice|WH4SFAACAS]].
It is a *conditionally complete lattice* if:

1. every nonempty subset of $L$ that has an [[upper bound|0S60DVMVPN]] in $L$ has a [[supremum|0R3H5PANRQ]] in $L$, and
2. every nonempty subset of $L$ that has a lower bound in $L$ has an [[infimum|0QF1MZ9FRG]] in $L$.

## Remarks

Clause 2 follows from clause 1.
Indeed, the infimum of a nonempty set $S$ bounded below is the supremum of the set of lower bounds of $S$.
That set is nonempty and bounded above by any element of $S$.
Example: the [[real numbers|08J2M1JYAG]] with the usual order, by the completeness axiom.
Non-example: $\Q$, since $\set{q \in \Q : q^2 < 2}$ is nonempty and bounded above by $2$ but has no supremum in $\Q$.
Every [[complete lattice|D7A0ZGZT5S]] is conditionally complete, and a conditionally complete lattice with a least and a greatest element is complete.
A totally ordered set with property 1 is said to have the least-upper-bound property.
</document>

<document source="https://tamnd.github.io/math2/node/A5GDAAYNM0.md" tag="A5GDAAYNM0" kind="definition">
# Shock wave

## Definition

Let $F \colon \R \to \R$ be a $C^1$ function and consider the scalar [[conservation law|8RA7AMWSAF]] $u_t + F(u)_x = 0$ on $\R \times (0, \infty)$.
Let $u$ be a [[weak solution|SJZX81A783]] and let $C = \set{(s(t), t) : t \in I}$ be a curve, with $I$ an open interval and $s \colon I \to \R$ of class $C^1$.
Suppose that $u$ is $C^1$ on each side of $C$ near $C$.
Let $u_l(t)$ and $u_r(t)$ be its one-sided limits from the left and from the right of $(s(t), t)$.
The curve $C$ is a *shock* of $u$ if $u_l(t) \ne u_r(t)$ for all $t \in I$, so that $u$ jumps across $C$.
The shock is *admissible*, or satisfies the *Lax entropy condition*, if
$$F'(u_l(t)) > s'(t) > F'(u_r(t)) \qquad \text{for all } t \in I.$$

## Remarks

Because $u$ is a weak solution, the shock speed is forced by the [[Rankine–Hugoniot condition|S4XMYJ3V42]] $s'(t) = \dfrac{F(u_l) - F(u_r)}{u_l - u_r}$.
The entropy condition says that characteristics run into the shock from both sides, so the shock does not create information.
For uniformly convex $F$ it is equivalent to $u_l > u_r$.
For systems a shock of the $k$-th family is admissible when $\lambda_k(u_l) > s' > \lambda_k(u_r)$ together with conditions on the other eigenvalues.

Example: for [[Burgers' equation|NEGHXN8RE1]], $F(u) = u^2/2$, the function $u = 1$ for $x < t/2$ and $u = 0$ for $x > t/2$ has an admissible shock along $s(t) = t/2$.
Check: the speed $1/2 = (\tfrac12 - 0)/(1 - 0)$ satisfies the Rankine–Hugoniot condition, which makes $u$ a weak solution, and $F'(1) = 1 > \tfrac12 > 0 = F'(0)$.

Non-example: the function $u = 0$ for $x < t/2$ and $u = 1$ for $x > t/2$ is also a weak solution of Burgers' equation with a jump along $s(t) = t/2$.
It fails the entropy condition, since $F'(0) = 0 < \tfrac12$.
The admissible solution with these initial values is a [[rarefaction wave|MFN4Q5SH8H]] instead.
</document>

<document source="https://tamnd.github.io/math2/node/A5HTJ565DN.md" tag="A5HTJ565DN" kind="concept">
# Ideal completion

## Summary

The ideal completion of a partially ordered set is the set of its order ideals, the nonempty directed down-closed subsets, ordered by inclusion.
Each element $x$ becomes the principal ideal of all elements below $x$.
The ideal completion is a [[directed-complete partial order|7TSXP95QN6]] in which every element is a directed union of principal ideals.
It is the free way to add suprema of directed sets.
It is a basic construction of [[domain theory|Q2P5D74Q1V]].

## Explanations

### L1 human

An order ideal is a set of elements that is closed downward and in which any two elements have a common bound inside the set.
Below each element $x$ there is the ideal of all elements below $x$, but there may be other ideals that do not have a top.
Adding these ideals as new elements fills in the missing limits of increasing sequences.
For the natural numbers, the only new ideal is $\N$ itself, so the completion is $\N$ with one point $\infty$ added on top.
</document>

<document source="https://tamnd.github.io/math2/node/A5JB9JQKR0.md" tag="A5JB9JQKR0" kind="statement">
# Cartan decomposition of a Lie group

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center, with Lie algebra $\mathfrak{g}$ and Killing form $B$.
Let $\theta$ be a Cartan involution of $\mathfrak{g}$: an automorphism with $\theta^2 = \id$ such that $(X, Y) \mapsto -B(X, \theta Y)$ is positive definite.
Let $\mathfrak{k}$ and $\mathfrak{p}$ be the eigenspaces of $\theta$ for the eigenvalues $1$ and $-1$, so that $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$.
Let $K$ be the connected Lie subgroup of $G$ with Lie algebra $\mathfrak{k}$.
Then:

1. $K$ is a compact subgroup of $G$, and it is a [[maximal compact subgroup|T2EB8R9HEV]];
2. the map $K \times \mathfrak{p} \to G$, $(k, X) \mapsto k \exp X$, is a diffeomorphism, where $\exp$ is the [[exponential map|H3QX1ZFC8K]].

In particular $G = K \exp(\mathfrak{p})$.

## Remarks

A Cartan involution exists and is unique up to inner automorphisms; this is the Cartan decomposition of the Lie algebra.
The proof of 2 lets $G$ act on the symmetric space $G/K$, which has nonpositive curvature.
Then the map $X \mapsto \exp(X) K$ is a diffeomorphism from $\mathfrak{p}$ onto $G/K$.

Example: $G = \mathrm{SL}_n(\R)$ with $\theta(X) = -X^{T}$.
Then $\mathfrak{k} = \mathfrak{so}(n)$, $\mathfrak{p}$ is the space of symmetric matrices of trace $0$, and $K = \mathrm{SO}(n)$.
The theorem is the polar decomposition: each $g \in \mathrm{SL}_n(\R)$ is uniquely $g = k e^{X}$ with $k$ orthogonal and $e^X$ symmetric positive definite.

Without a finite center, part 2 still holds with $K$ closed, but $K$ is no longer compact.
For the universal cover of $\mathrm{SL}_2(\R)$, the subgroup $K$ is isomorphic to $\R$.

## Explanations

### L1 human

Every element of a semisimple Lie group splits as a rotation part times a stretching part.
The rotation part lies in a compact subgroup $K$.
The stretching part is the exponential of an element of a vector space $\mathfrak{p}$.
For invertible matrices this is the familiar fact that a matrix is a rotation times a positive symmetric matrix.

### L2 human

The theorem shows that $G$ is diffeomorphic to $K \times \R^{d}$ with $d = \dim \mathfrak{p}$.
So the topology of $G$, for example its fundamental group, is the topology of the compact group $K$.
The quotient $G/K$ is a Riemannian symmetric space of noncompact type, diffeomorphic to $\mathfrak{p}$.
The Iwasawa decomposition $G = KAN$ and the decomposition $G = KAK$ refine this splitting.
</document>

<document source="https://tamnd.github.io/math2/node/A5RCGG79JX.md" tag="A5RCGG79JX" kind="definition">
# Discrete-time dynamical system

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a [[map|0EPAT51DXQ]].
The *discrete-time dynamical system* generated by $f$ is the pair $(X, f)$, viewed as the [[dynamical system|BBZP1R872N]] $\Phi \colon \N \times X \to X$ with $\Phi(n, x) = f^n(x)$.
Here $f^n$ is the $n$-th [[iterate|AYVTR2ES5B]] of $f$, with $f^0 = \id_X$.
If $f$ is a bijection, the system is *invertible*, and $\Phi(n, x) = f^n(x)$ is defined for all $n \in \Z$.
Here $f^{-n}$ is the $n$-th iterate of the [[inverse map|0J2WAR83WP]] $f^{-1}$.

## Remarks

The forward orbit of $x$ is the sequence $x, f(x), f^2(x), \dots$.
Every dynamical system with time $\N$ arises in this way, with $f = \Phi(1, \cdot)$, since clause 2 of the definition of a dynamical system forces $\Phi(n, \cdot) = f^n$.

Example: $X = \R$ and $f(x) = 3x$.
Then $\Phi(n, x) = 3^n x$, and since $f$ is a bijection the system is invertible with $\Phi(-n, x) = 3^{-n} x$.

Non-example of an invertible system: $f(x) = x^2$ on $\R$ generates a discrete-time system, but it is not invertible, because $f(-1) = f(1)$.

Usually $X$ is a topological space and $f$ is continuous, or $X$ is a measure space and $f$ is measurable.
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<document source="https://tamnd.github.io/math2/node/A5T30ZHVGH.md" tag="A5T30ZHVGH" kind="statement">
# Minimum total potential energy principle

## Statement

Let $\Omega \subset \R^3$ be a bounded connected open set with Lipschitz boundary $\partial\Omega = \overline{\Gamma_D} \cup \overline{\Gamma_N}$, where $\Gamma_D$ and $\Gamma_N$ are disjoint and $\Gamma_D$ has positive surface measure.
Let $\varepsilon(v) = \tfrac12(\nabla v + \nabla v^T)$ be the linearised strain of a displacement $v \colon \Omega \to \R^3$.
Let $\mathsf{C}$ be a bounded measurable elasticity tensor field that is symmetric and uniformly [[positive definite|0KX66CZMBB]] on symmetric $3 \times 3$ matrices.
Let $f \in L^2(\Omega)^3$ be the body force and $g \in L^2(\Gamma_N)^3$ the surface traction.
Let $V = \set{v \in H^1(\Omega)^3 : v = 0 \text{ on } \Gamma_D}$, the admissible displacements, and define the total potential energy
$$\Pi(v) = \frac12 \int_\Omega \mathsf{C}\varepsilon(v) : \varepsilon(v)\, dx - \int_\Omega f \cdot v \, dx - \int_{\Gamma_N} g \cdot v \, dS.$$
Then:

1. $\Pi$ has exactly one minimiser $u$ on $V$.
2. A displacement $u \in V$ minimises $\Pi$ on $V$ if and only if $u$ is a [[weak solution|SJZX81A783]] of the equilibrium equations below.
   Here $\nu$ is the outward unit normal.
   $$-\operatorname{div}(\mathsf{C}\varepsilon(u)) = f \text{ in } \Omega, \quad u = 0 \text{ on } \Gamma_D, \quad (\mathsf{C}\varepsilon(u))\nu = g \text{ on } \Gamma_N.$$

## Remarks

The functional $\Pi$ is a strictly convex quadratic functional on the [[Sobolev space|M4CEZ5K9HP]] $V$.
Korn's inequality, together with $\abs{\Gamma_D} > 0$, shows that the strain energy controls the $H^1$ norm on $V$, so the [[Lax–Milgram theorem|85KK79FCN5]] gives existence and uniqueness.
For a convex quadratic functional, minimisers are exactly the points where the first variation vanishes, and the first variation of $\Pi$ is the weak form of equilibrium.
Without $\abs{\Gamma_D} > 0$, the body can move rigidly: $\Pi$ is unchanged along rigid motions when the loads are balanced, so uniqueness fails, and $\Pi$ is unbounded below when they are not.
Example in one dimension: a bar $[0, \ell]$ with stiffness $k > 0$, clamped at $0$ and pulled by a force $P$ at $\ell$, has $\Pi(v) = \tfrac{k}{2}\int_0^\ell v'^2\,dx - P v(\ell)$, whose minimiser is $u(x) = Px/k$.
The same idea, that stable equilibria minimise the total potential energy, holds for conservative systems more generally.
It is the basis of the finite element method in structural mechanics.

## Explanations

### L1 human

An elastic body at rest under loads takes the shape that makes its total potential energy as small as possible.
This energy is the energy stored by deforming the body minus the work done by the loads.
A ball rolls to the bottom of a bowl for the same reason.

### L2 human

The principle is the variational form of linear elasticity, and it is a model case of the direct method of the [[calculus of variations|PQ25NHV0KS]].
It replaces a system of partial differential equations by the minimisation of a convex functional.
Restricting the minimisation to a finite-dimensional space of trial displacements gives the Ritz and finite element approximations, and the convexity gives their convergence.
</document>

<document source="https://tamnd.github.io/math2/node/A5TDM8F4KB.md" tag="A5TDM8F4KB" kind="definition">
# Dirac operator on a spin manifold

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with a [[spin structure|RSPZ9TCFQ8]].
Let $S \to M$ be the complex [[spinor bundle|PYYQQKAZ0C]], with its Clifford multiplication $T M \otimes S \to S$, $v \otimes \psi \mapsto v \cdot \psi$, which satisfies $v \cdot v \cdot \psi = -g(v, v) \, \psi$, as in the [[Clifford algebra|A38B1QFK20]].
Let $\nabla^S$ be the spin connection on $S$, the connection induced by the [[Levi-Civita connection|CAZYGE1T4E]].
The *Dirac operator* of $M$ is the operator $D \colon \Gamma(S) \to \Gamma(S)$ given by
$$D \psi = \sum_{i=1}^{n} e_i \cdot \nabla^S_{e_i} \psi,$$
where $e_1, \ldots, e_n$ is any local orthonormal frame of $TM$.

## Remarks

The right side does not depend on the frame, because it is the contraction of $\nabla^S \psi \in \Gamma(T^*M \otimes S)$ with Clifford multiplication, after identifying $T^*M$ with $TM$ by the metric.
The principal symbol of $D$ at $\xi$ is Clifford multiplication by the vector dual to $\xi$, up to a factor $i$ in some conventions, and its square is $-\abs{\xi}^2$.
So $D$ is an [[elliptic operator|MT9P2X85V8]] and a [[Dirac operator|3DXGJTEQZH]] in the general sense.
It is formally self-adjoint for the Hermitian metric on $S$.
In even dimension $S = S^+ \oplus S^-$, and $D$ maps $\Gamma(S^\pm)$ to $\Gamma(S^\mp)$.

Example: on $\R^n$ with the flat metric and the trivial spin structure, $S = \R^n \times \C^N$, the spin connection is the ordinary derivative, and $D = \sum_j \gamma_j \partial_j$, where $\gamma_j$ are complex $N \times N$ matrices with $\gamma_j \gamma_k + \gamma_k \gamma_j = -2 \delta_{jk} I$.
For $n = 1$ one can take $N = 1$ and $\gamma_1 = i$, which gives $D = i \, d/dx$ and $D^2 = -d^2/dx^2$.

Non-example: the complex projective plane $\C P^2$ with the Fubini–Study metric has no spin structure, since its second Stiefel–Whitney class is nonzero.
So it has no spinor bundle, and the definition does not apply; one uses a spin$^c$ structure instead.

## Explanations

### L1 human

The Dirac operator of a spin manifold differentiates a spinor field in each direction of an orthonormal frame and multiplies by the matching Clifford element, then sums.
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<document source="https://tamnd.github.io/math2/node/A625F93ANV.md" tag="A625F93ANV" kind="definition">
# Fourier transform on locally compact abelian groups

## Definition

Let $G$ be a [[locally compact abelian group|ZR09S9R3AW]] with a [[Haar measure|VY0HXD3GC4]] $\mu$, and let $\widehat{G}$ be its [[Pontryagin dual|FMW57WNPDB]].
Let $f \in L^1(G, \mu)$, the [[group algebra|6GD25S53ND]] of $\mu$-integrable functions.
The *Fourier transform* of $f$ is the function $\widehat{f} \colon \widehat{G} \to \C$ given by
$$\widehat{f}(\chi) = \int_G f(x) \overline{\chi(x)} \, d\mu(x).$$

## Remarks

Example: for $G = \R$ with Lebesgue measure, the continuous characters are $\chi_\xi(x) = e^{2\pi i \xi x}$ for $\xi \in \R$, and the formula gives $\widehat{f}(\chi_\xi) = \int_\R f(x) e^{-2\pi i \xi x} \, dx$, the classical Fourier transform.
For $G = \mathbb{T}$ with normalised arc length and $\chi_n(z) = z^n$, it gives the Fourier coefficients of $f$.
Non-example: for $f(x) = 1$ on $\R$, the integral does not converge because $f \notin L^1(\R)$; the transform of such $f$ only makes sense as a distribution.
The integral converges for $f \in L^1$ because $\abs{\chi(x)} = 1$, and $\abs{\widehat{f}(\chi)} \le \norm{f}_1$.
The function $\widehat{f}$ is continuous on $\widehat{G}$ and vanishes at infinity, and $\widehat{f * g} = \widehat{f}\,\widehat{g}$ for the convolution $f * g$.
Some authors use $\chi(x)$ instead of $\overline{\chi(x)}$; this changes $\widehat{f}(\chi)$ into $\widehat{f}(\overline{\chi})$.
</document>

<document source="https://tamnd.github.io/math2/node/A66VNJ7BED.md" tag="A66VNJ7BED" kind="definition">
# Arc in a finite plane

## Definition

Let $\Pi$ be a finite [[projective plane|DANFNN8C4Y]] and let $k \ge 1$ be an integer.
A *$k$-arc* of $\Pi$ is a set of $k$ points of $\Pi$ such that no three of them lie on a common line.

## Remarks

Let $\Pi$ have [[order|K2YVMTGTBB]] $n$, so each point lies on $n + 1$ lines.
If $P$ is a point of a $k$-arc $K$, the lines joining $P$ to the other $k - 1$ points of $K$ are distinct.
Hence $k - 1 \le n + 1$, that is, $k \le n + 2$.
When $n$ is odd, $k \le n + 1$.
An arc with $n + 1$ points is called an oval, and one with $n + 2$ points a hyperoval.

Example: in the [[plane $\mathrm{PG}(2, q)$|CSMB49BDPT]], let $K$ consist of the points $(1 : t : t^2)$ for $t \in \F_q$ and the point $(0 : 0 : 1)$.
Three points $(1 : s : s^2)$, $(1 : t : t^2)$, $(1 : u : u^2)$ with distinct $s, t, u$ are not collinear, because their Vandermonde determinant $(t - s)(u - s)(u - t)$ is nonzero.
For $(1 : s : s^2)$, $(1 : t : t^2)$ and $(0 : 0 : 1)$ the determinant is $t - s \ne 0$.
So $K$ is a $(q + 1)$-arc.

Non-example: the $n + 1$ points of a line, for $n \ge 2$, contain three collinear points, so they do not form an arc.

More generally, a $(k, d)$-arc is a set of $k$ points with at most $d$ on any line, so that a $k$-arc is a $(k, 2)$-arc.
</document>

<document source="https://tamnd.github.io/math2/node/A67HV2KCHR.md" tag="A67HV2KCHR" kind="definition">
# Birkhoff's axioms

## Definition

Let $\Pi$ be a set of points with a family of subsets called lines and a distance function $d \colon \Pi \times \Pi \to \R$.
For a point $O$, the rays from $O$ carry a measure: $\angle AOB$ denotes the angle of the rays $OA$ and $OB$.
*Birkhoff's axioms* are the following four postulates.

1. Line measure: the points of each [[line|0MP9BCF6DP]] $\ell$ can be put in bijection with the [[real numbers|08J2M1JYAG]], $A \mapsto x_A$, so that $\abs{x_A - x_B} = d(A, B)$ for all $A, B \in \ell$.
2. Point and line: through two distinct points there is exactly one line.
3. Angle measure: the rays from each point $O$ can be put in bijection with $\R / 2\pi\Z$. Then $\angle AOB$ is the difference of the numbers of $OA$ and $OB$. Moreover, as $B$ moves continuously on a line not through $O$, the number of the ray $OB$ varies continuously.
4. Similarity: let $ABC$ and $A'B'C'$ be [[triangles|0CTNFGYMM2]] and $k > 0$ such that $d(A', B') = k\, d(A, B)$, $d(A', C') = k\, d(A, C)$ and $\angle B'A'C' = \pm \angle BAC$. Then $d(B', C') = k\, d(B, C)$, $\angle C'B'A' = \pm \angle CBA$ and $\angle A'C'B' = \pm \angle ACB$.

## Remarks

Example: $\R^2$ with the usual distance and the polar angle of rays satisfies all four postulates.
The similarity postulate holds there because a similarity of ratio $k$ maps one triangle onto the other.
Non-example: the sphere with great circles as lines and the great-circle distance fails postulate 2, since there are infinitely many great circles through two antipodal points.
The axioms of [[angles|0GJAJC6TNZ]] are those of a protractor, and the axioms of lines are those of a ruler.
The four postulates determine the Euclidean plane up to isometry.
</document>

<document source="https://tamnd.github.io/math2/node/A68WQ4ZAF0.md" tag="A68WQ4ZAF0" kind="statement">
# Collar neighbourhood theorem

## Statement

Let $M$ be a smooth [[manifold with boundary|7DS6B2BFK8]], and let $\partial M$ be its boundary.
Then there is an open neighbourhood $U$ of $\partial M$ in $M$ and a [[diffeomorphism|0KZZ70ZHXP]] $c \colon \partial M \times [0, 1) \to U$ such that $c(x, 0) = x$ for every $x \in \partial M$.

## Remarks

The map $c$, or its image $U$, is a *collar* of $\partial M$.
The idea of the proof is to choose a smooth vector field near $\partial M$ that points into $M$ along $\partial M$.
Such a field is glued from local choices with a partition of unity.
Its flow, started at the boundary points and run for short times, gives the collar.
Example: for the closed unit disc $D^2$, the map $c(x, t) = (1 - t/2) x$ is a collar $S^1 \times [0, 1) \to D^2$ of the boundary circle.
The statement also holds for topological manifolds with boundary, a result of Morton Brown.

## Explanations

### L1 human

Near its boundary, every smooth manifold with boundary looks like a product of the boundary with a short interval.
A disc near its rim looks like a thin ring, a circle times an interval.
This lets one glue manifolds along common boundary pieces and get a smooth manifold again.

### L2 human

Collars are used to glue manifolds along boundary components, for example in the composition of cobordisms and in handle attachments.
They show that $M$ is homotopy equivalent to its interior, and that $M$ with an outer collar $\partial M \times [0, 1]$ attached is diffeomorphic to $M$.
The collar theorem is the boundary analogue of the tubular neighbourhood theorem: the inward normal bundle of $\partial M$ is a trivial half-line bundle.
</document>

<document source="https://tamnd.github.io/math2/node/A69F1SM1AC.md" tag="A69F1SM1AC" kind="definition">
# Continuous-time Markov chain

## Definition

Let $S$ be a countable set.
A *time-homogeneous continuous-time Markov chain* on $S$ is a [[stochastic process|0QH5ENKZM1]] $(X_t)_{t \ge 0}$ with values in $S$ and right-continuous paths that has the following property.
There are [[transition matrices|0F462493EA]] $P(t) = (p_{ij}(t))_{i, j \in S}$, $t \ge 0$, such that, for all $n \ge 0$, all times $0 \le t_0 < t_1 < \cdots < t_{n+1}$ and all states $i_0, \ldots, i_{n+1} \in S$ with $P(X_{t_0} = i_0, \ldots, X_{t_n} = i_n) > 0$,
$$P(X_{t_{n+1}} = i_{n+1} \mid X_{t_0} = i_0, \ldots, X_{t_n} = i_n) = p_{i_n i_{n+1}}(t_{n+1} - t_n).$$

## Remarks

The condition is the [[Markov property|0B77KZCQ51]] in continuous time, and it implies the [[Chapman–Kolmogorov equation|01APZ8GDQH]] $P(s + t) = P(s) P(t)$.
For a finite $S$ there is a rate matrix $Q = P'(0)$, the [[generator|VPA41PWRR1]], with $q_{ij} \ge 0$ for $i \ne j$ and zero row sums, and $P(t) = e^{tQ}$ is a [[matrix exponential|020M04FN1H]].
The chain stays in state $i$ for an [[exponential|0E99F51GVB]] time with rate $q_i = -q_{ii}$, then jumps to $j \ne i$ with probability $q_{ij}/q_i$, when $q_i > 0$.
Example: the Poisson process of rate $\lambda > 0$ is a chain on $\N$ with $q_{i, i+1} = \lambda$ and $q_{ii} = -\lambda$, and $p_{ij}(t) = e^{-\lambda t} (\lambda t)^{j-i} / (j-i)!$ for $j \ge i$.
Non-example: a process on $\Z$ that moves up by $1$ at the times $1, 2, 3, \ldots$ is not time-homogeneous.
Indeed $P(X_{t+1/2} = X_t + 1)$ is $1$ for $t = 3/4$ and $0$ for $t = 1/4$, so no single $P(1/2)$ works.
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<document source="https://tamnd.github.io/math2/node/A6BW0KPXNC.md" tag="A6BW0KPXNC" kind="definition">
# Directional statistics

## Definition

Let $p \ge 2$ and let $S^{p-1} = \set{x \in \R^p : \abs{x} = 1}$ be the unit [[sphere|T62369RFH6]].
*Directional statistics* is the statistics of data $x_1, \ldots, x_n$ with values in $S^{p-1}$.
More generally the data may lie in a compact manifold such as a rotation group.
For $p = 2$ it is called circular statistics.
Its basic [[statistics|0Q5CG2NEQ8]] come from the resultant vector $R = x_1 + \cdots + x_n$.

1. The *mean resultant length* is $\bar R = \abs{R}/n$, a number in $[0, 1]$.
2. If $R \ne 0$, the *mean direction* is $R / \abs{R} \in S^{p-1}$.
3. For $p = 2$, the *circular variance* is $1 - \bar R$.

## Remarks

Example: the angles $350°$ and $10°$ give the unit vectors $(\cos 10°, -\sin 10°)$ and $(\cos 10°, \sin 10°)$.
Their sum is $R = (2 \cos 10°, 0)$, so the mean direction is $0°$ and $\bar R = \cos 10° \approx 0.985$.
Non-example: for the angles $0°$ and $180°$, $R = 0$, so the mean direction is not defined; then $\bar R = 0$ and the circular variance is $1$.
The value $\bar R$ is close to $1$ when the directions are concentrated, and close to $0$ when they are spread out or balanced.
The mean direction is the point $m \in S^{p-1}$ that maximises $\sum_i m^T x_i$, so it does not depend on the choice of a zero angle.
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<document source="https://tamnd.github.io/math2/node/A6DPR620B5.md" tag="A6DPR620B5" kind="concept">
# Ultrametric space

## Summary

An ultrametric space is a [[metric space|0VKD1TP7F3]] whose distance satisfies the strong triangle inequality $d(x, z) \le \max(d(x, y), d(y, z))$.
The main examples are the $p$-adic numbers and spaces of infinite sequences, where two sequences are close when they agree for a long time.
Ultrametric geometry is strange at first: every triangle is isosceles and every point of a ball is a centre of that ball.
Balls are both open and closed, so ultrametric spaces are [[zero-dimensional|JJ7D0CYECZ]].

## Explanations

### L1 human

Think of a family tree, and measure the distance between two people by how far back their nearest common ancestor lives.
To go from $x$ to $z$ you never need a longer jump than the longest of the two jumps through $y$.
On $\Q$, the $3$-adic distance makes $9$ and $0$ closer than $1$ and $0$, since $9$ is divisible by a higher power of $3$.
</document>

<document source="https://tamnd.github.io/math2/node/A6EXFAP5EQ.md" tag="A6EXFAP5EQ" kind="definition">
# Irregular singularity

## Definition

Let $p_1, \ldots, p_n$ be meromorphic functions near $z_0 \in \C$, and consider the [[linear equation|0QQRP17Z7S]]
$$y^{(n)} + p_1(z)\, y^{(n-1)} + \cdots + p_n(z)\, y = 0.$$
The point $z_0$ is an *irregular singular point* (or irregular singularity) if it is a [[singular point|QK2J038VB4]] but not a [[regular singular point|H96HXG6CA0]].
Equivalently, $(z - z_0)^k\, p_k(z)$ is not analytic at $z_0$ for at least one $k \in \set{1, \ldots, n}$.
The point $\infty$ is irregular singular if $w = 0$ is irregular singular for the equation obtained by the substitution $z = 1/w$.

## Remarks

Example: $z^2 y'' + y' = 0$ has $p_1 = \frac{1}{z^2}$, and $z\, p_1 = \frac{1}{z}$ is not analytic at $0$, so $0$ is an irregular singular point.
Its solutions are the constants and $\int e^{1/z}\, dz$, and the derivative $e^{1/z}$ of the latter has an essential singularity at $0$.

Non-example: $z^2 y'' + z y' - y = 0$ has $z p_1 = 1$ and $z^2 p_2 = -1$, both analytic, so $0$ is a regular singular point and not irregular; its solutions are $z$ and $z^{-1}$.

For $n = 2$, the point $\infty$ is irregular for $y'' + p y' + q y = 0$ exactly when $z\, p(z)$ or $z^2 q(z)$ is not analytic at $\infty$.
For example, $\infty$ is irregular for the Bessel equation, the Airy equation $y'' = z y$ and the equation $y'' = y$, whose solutions $e^{\pm z}$ have essential singularities at $\infty$.
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<document source="https://tamnd.github.io/math2/node/A6GYWEFFTY.md" tag="A6GYWEFFTY" kind="statement">
# Marden's theorem

## Statement

Let $p$ be a polynomial of degree $3$ with complex coefficients whose three [[roots|019MA1VTR2]] $z_1, z_2, z_3$ are not collinear in the plane $\C$.
Let $T$ be the [[triangle|0CTNFGYMM2]] with vertices $z_1, z_2, z_3$.
Then there is a unique [[ellipse|0VKCCEXBJ1]] inscribed in $T$ and tangent to its three sides at their midpoints.
The foci of this ellipse are the two roots of the derivative $p'$.

## Remarks

The ellipse is the Steiner inellipse of $T$.
If $p$ is monic, then $p'(z) = (z - z_1)(z - z_2) + (z - z_2)(z - z_3) + (z - z_3)(z - z_1)$.
One proof computes the foci of the Steiner inellipse and checks, with the reflection property of the tangent lines of an ellipse, that they are the roots of this quadratic.
For an equilateral triangle the inellipse is the incircle, and for $p(z) = z^3 - 1$ its centre $0$ is the double root of $p'(z) = 3z^2$.
Example: for $p(z) = z(z - 1)(z - i)$, one gets $p'(z) = 3z^2 - 2(1 + i)z + i$, whose roots are the foci of the inellipse of the triangle $0, 1, i$.
The theorem refines the Gauss–Lucas theorem for cubics: the critical points lie not only in the triangle but at the foci of a specific ellipse inside it.

## Explanations

### L1 human

A cubic polynomial has three roots, which form a triangle in the plane.
There is exactly one ellipse that fits inside the triangle and touches each side at its midpoint.
Marden's theorem says that the two foci of this ellipse are exactly the roots of the derivative.
It is a striking link between algebra and classical geometry.

### L2 human

The roots of $p'$ are the [[critical points|0YTCMFB4HC]] of $p$, so the theorem locates the critical points of a cubic from its roots alone.
It is a classical result of the geometry of polynomials, which studies where the zeros of a polynomial and of its derivatives lie.
Together with [[Sendov's conjecture|101H2VSX2R]], it belongs to the study of how the zeros of $p'$ sit relative to the zeros of $p$.
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<document source="https://tamnd.github.io/math2/node/A6HN207Y2T.md" tag="A6HN207Y2T" kind="concept">
# Distinguished boundary of a polydisc

## Summary

The distinguished boundary of a [[polydisc|DJBWP7D3M8]] $P(a, r)$ in $\C^n$ is the torus of points with $\abs{z_j - a_j} = r_j$ for every $j$.
It is the set over which the [[Cauchy integral formula on a polydisc|FG5448NZ18]] integrates.
A function holomorphic on the polydisc and continuous on its closure attains its largest modulus on this torus.
It is the [[Shilov boundary|JHQNT58FEH]] of the algebra of such functions, and for $n \ge 2$ it is a small part of the topological boundary.

## Explanations

### L1 human

The boundary of the unit polydisc in $\C^2$ is the set where $\abs{z_1} = 1$ or $\abs{z_2} = 1$.
The distinguished boundary is the smaller set where both $\abs{z_1} = 1$ and $\abs{z_2} = 1$.
It is a product of two circles, a torus of real dimension $2$, while the full boundary has real dimension $3$.
For example, $f(z_1, z_2) = z_1 z_2$ has modulus at most $1$ on the closed polydisc, and it reaches $1$ exactly on the distinguished boundary.
So, for holomorphic functions, this thin torus already sees the maximum.
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<document source="https://tamnd.github.io/math2/node/A6J731Y193.md" tag="A6J731Y193" kind="concept">
# Term rewriting for equational theories

## Summary

Term rewriting replaces the symmetric equations of an equational theory by one-way rules $l \to r$ that rewrite terms.
Two terms are equal in the theory exactly when they are connected by a chain of rule applications in either direction.
When the rules are terminating and confluent, every term has a unique normal form.
Then two terms are equal in the theory exactly when their normal forms coincide.
This turns many word problems for free algebras into a mechanical computation, and the Knuth–Bendix procedure tries to find such rules.

## Explanations

### L1 human

An equation like $0 + x = x$ can be read as an instruction: wherever $0 + t$ appears, replace it by $t$.
Applying such instructions until none applies simplifies a term to a normal form.
If the instructions always stop and the final answer never depends on the order of the steps, then comparing normal forms decides equality.
For example, with the rules $0 + x \to x$ and $s(x) + y \to s(x + y)$, the term $s(0) + s(0)$ rewrites to $s(0 + s(0))$ and then to $s(s(0))$.
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<document source="https://tamnd.github.io/math2/node/A6M3GTHM40.md" tag="A6M3GTHM40" kind="concept">
# Pythagorean field

## Summary

A Pythagorean field is a [[field|04C5ZBG1WT]] in which every sum of two squares is again a square.
The name comes from the [[Pythagorean theorem|0VP3A1290K]].
In such a field, a right triangle with legs in the field has a hypotenuse in the field.
Real closed fields and algebraically closed fields are Pythagorean, but $\Q$ is not.

## Explanations

### L1 human

In the rational numbers, the right triangle with legs $1$ and $1$ has hypotenuse $\sqrt 2$, which is not rational.
A Pythagorean field is a field where this never happens: $\sqrt{a^2 + b^2}$ always lies in the field.
The real numbers have this property, and so do the complex numbers.
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<document source="https://tamnd.github.io/math2/node/A6YS3A9DNP.md" tag="A6YS3A9DNP" kind="definition">
# Trigonometric polynomial

## Definition

Let $N \ge 0$ be an integer.
A *trigonometric polynomial* of degree at most $N$ is a function $T \colon \R \to \C$ with
$$T(x) = \sum_{n=-N}^{N} c_n e^{inx},$$
for complex numbers $c_{-N}, \dots, c_N$, where $e^{inx}$ is the [[complex exponential|07GJV36896]].
Its *degree* is the largest $\abs{n}$ with $c_n \ne 0$.

## Remarks

By Euler's formula the same functions are the sums $a_0 + \sum_{n=1}^{N} (a_n \cos nx + b_n \sin nx)$ with complex $a_n, b_n$, built from [[sine and cosine|08WFZP6P89]].
The two forms are related by $a_0 = c_0$, $a_n = c_n + c_{-n}$ and $b_n = i(c_n - c_{-n})$ for $n \ge 1$.
$T$ is real-valued exactly when $c_{-n} = \overline{c_n}$ for all $n$, that is, when all $a_n$ and $b_n$ are real.
Example: $\cos^2 x = \frac{1}{4} e^{-2ix} + \frac{1}{2} + \frac{1}{4} e^{2ix}$, so $\cos^2 x$ is a trigonometric polynomial of degree $2$.
Non-example: $\sin(x/2)$ is not a trigonometric polynomial, since it has period $4\pi$ and not $2\pi$, while every trigonometric polynomial is $2\pi$-periodic.
The coefficients are unique: $c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} T(x) e^{-inx} \, dx$.
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<document source="https://tamnd.github.io/math2/node/A6Z3GR2RWP.md" tag="A6Z3GR2RWP" kind="definition">
# James–Stein estimator

## Definition

Let $p \ge 3$ and let $X$ have the [[multivariate normal distribution|0XN3K7RSZY]] on $\R^p$ with unknown mean $\theta \in \R^p$ and covariance matrix $\sigma^2 I_p$, where $\sigma^2 > 0$ is known.
The *James–Stein estimator* of $\theta$ is the [[estimator|0SSS6ERQ0S]]
$$\delta_{JS}(X) = \Big( 1 - \frac{(p - 2) \sigma^2}{\norm{X}^2} \Big) X,$$
where $\norm{\cdot}$ is the Euclidean norm.

## Remarks

The estimator is defined with probability $1$, since $X \ne 0$ almost surely.
It is a [[shrinkage estimator|GPBSS1V3SS]] toward $0$; shrinking toward a fixed point $\nu \in \R^p$ gives $\nu + \big(1 - (p - 2)\sigma^2 / \norm{X - \nu}^2\big)(X - \nu)$.
When $\norm{X}^2 < (p - 2)\sigma^2$ the factor is negative; the *positive-part* estimator replaces the factor by its positive part, and it has even smaller risk.

Under the loss $\norm{a - \theta}^2$, the estimators $(1 - c\sigma^2/\norm{X}^2) X$ have [[risk|N26TSVRH2S]] $p\sigma^2 - (2c(p - 2) - c^2) \sigma^4 E_\theta[1/\norm{X}^2]$.
This is below $p\sigma^2$, the risk of $X$, exactly when $0 < c < 2(p - 2)$, and it is smallest for $c = p - 2$, the James–Stein choice.

Example: $p = 3$, $\sigma = 1$ and the observation $X = (1, 1, 1)$.
Then $\norm{X}^2 = 3$, the factor is $1 - 1/3 = 2/3$, and $\delta_{JS}(X) = (2/3, 2/3, 2/3)$.

Non-example: the same formula with $c = 2(p - 2)$ in place of $p - 2$.
It also shrinks toward $0$, but by the risk formula above its risk equals $p\sigma^2$ for every $\theta$, so it gives no improvement over $X$.
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<document source="https://tamnd.github.io/math2/node/A6ZJ8MSCAY.md" tag="A6ZJ8MSCAY" kind="concept">
# Core model

## Summary

A core model is an [[inner model|G78W36EGMB]], usually written $K$, built from fine-structural pieces like the [[constructible universe|6TF2JBTPPJ]].
Unlike $L$, it is large enough to contain some large cardinal structure.
It is defined under an assumption that rules out large cardinals beyond a certain level.
An example of such an assumption is that there is no inner model with a [[Woodin cardinal|CMC2C1D53W]].
Under such an assumption the core model is close to the universe $V$, in the sense of covering properties like [[Jensen's covering theorem|RJE1W5NRE3]] for $L$.
The theory of the core model links large cardinals with the strength of combinatorial statements.

## Explanations

### L1 human

The constructible universe is a very thin and well understood world of sets.
If [[zero sharp|HCTNBZZQ0W]] does not exist, the constructible universe is close to the full universe, and $L$ plays the role of the core model.
If larger objects exist, one needs a bigger inner model that still has the good structure of $L$; this is the core model.
It is the largest inner model of its kind that can be built from below, and it approximates the full universe well.
Its exact definition depends on which large cardinals are assumed not to exist.
</document>

<document source="https://tamnd.github.io/math2/node/A6ZREX2DMW.md" tag="A6ZREX2DMW" kind="definition">
# Selmer group

## Definition

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] with absolute [[Galois group|0XTY86BZKY]] $G_K$.
Let $\phi \colon E \to E'$ be an isogeny of [[elliptic curves|427N6S6QQW]] over $K$.
That is, $\phi$ is a nonconstant morphism over $K$ that is also a [[group homomorphism|0JYTG5K9Y4]].
Let $E[\phi] = \ker(\phi \colon E(\overline{K}) \to E'(\overline{K}))$, a finite $G_K$-module.
For each [[place|BT5V3TSNY6]] $v$ of $K$, let $\delta_v \colon E'(K_v) \to H^1(K_v, E[\phi])$ be the connecting map of the [[Galois cohomology|9MPCZ3EAJM]] sequence of $0 \to E[\phi] \to E \xrightarrow{\phi} E' \to 0$ over the completion $K_v$.
The *$\phi$-Selmer group* of $E$ over $K$ is
$$\mathrm{Sel}^{(\phi)}(E/K) = \set{\xi \in H^1(K, E[\phi]) : \mathrm{res}_v(\xi) \in \delta_v(E'(K_v)) \text{ for every place } v \text{ of } K},$$
where $\mathrm{res}_v \colon H^1(K, E[\phi]) \to H^1(K_v, E[\phi])$ is restriction.
For an integer $n \ge 2$, the *$n$-Selmer group* $\mathrm{Sel}^{(n)}(E/K)$ is the case $\phi = [n] \colon E \to E$.

## Remarks

Equivalently, $\mathrm{Sel}^{(\phi)}(E/K)$ is the kernel of $H^1(K, E[\phi]) \to \prod_v H^1(K_v, E)$.
There is an exact sequence
$$0 \to E'(K)/\phi(E(K)) \to \mathrm{Sel}^{(\phi)}(E/K) \to \text{Ш}(E/K)[\phi] \to 0,$$
where $\text{Ш}(E/K)$ is the [[Tate–Shafarevich group|BK9WBMF56G]].
The Selmer group is finite and effectively computable, which bounds the rank of $E(K)$.

Example: for $E \colon y^2 = x^3 - x$ over $\Q$ and $n = 2$, one has $E(\Q) = \set{O, (0, 0), (1, 0), (-1, 0)}$ and $2E(\Q) = \set{O}$.
So $E(\Q)/2E(\Q)$ has order $4$, and by the exact sequence $\mathrm{Sel}^{(2)}(E/\Q)$ has order at least $4$.

Non-example: a class of $H^1(\Q, E[2])$ whose restriction to $H^1(\R, E[2])$ does not come from $E(\R)$ fails the local condition at the real place, so it is not in $\mathrm{Sel}^{(2)}(E/\Q)$.
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<document source="https://tamnd.github.io/math2/node/A71EWRZBZV.md" tag="A71EWRZBZV" kind="definition">
# Shift space

## Definition

Let $\mathcal{A}$ be a finite set, the alphabet, with the discrete topology, and give $\mathcal{A}^{\Z}$ the [[product topology|0GBG8B49AA]].
The *shift map* $\sigma \colon \mathcal{A}^{\Z} \to \mathcal{A}^{\Z}$ is $\sigma(x)_i = x_{i+1}$ for $x = (x_i)_{i \in \Z}$.
A *shift space* (or *subshift*) over $\mathcal{A}$ is a subset $X \subseteq \mathcal{A}^{\Z}$ such that:

1. $X$ is closed in $\mathcal{A}^{\Z}$.
2. $\sigma(X) = X$.

## Remarks

The space $\mathcal{A}^{\Z}$ is [[compact|0TVJZYB6SH]], with the metric $d(x, y) = 2^{-\min\set{\abs{i} : x_i \neq y_i}}$ for $x \neq y$, and $\sigma$ is a [[homeomorphism|0QT00PB5G2]]; so $\sigma$ restricts to a homeomorphism of each shift space.
The whole space $\mathcal{A}^{\Z}$ is the *full shift*.
Equivalent form: $X$ is a shift space if and only if it is defined by a set $\mathcal{F}$ of forbidden words.
Here $\mathcal{F}$ is a set of finite words, elements of the [[free monoid|CV013KHVJG]] on $\mathcal{A}$.
Then $X$ is the set of sequences in which no word of $\mathcal{F}$ occurs as a block of consecutive symbols.
One-sided shift spaces are defined in the same way in $\mathcal{A}^{\N}$, with $\sigma(X) \subseteq X$.
Example: the golden mean shift $X \subseteq \set{0, 1}^{\Z}$, the sequences with no block $11$.
A limit of such sequences has no block $11$, since a block $11$ at positions $i, i+1$ would appear in the sequences close to it.
So $X$ is closed, and shifting does not create or remove blocks, so $\sigma(X) = X$.
Non-example: the set of sequences in $\set{0, 1}^{\Z}$ with exactly one symbol $1$ satisfies clause 2, but not clause 1.
Its elements with the $1$ at position $n$ tend to the all-$0$ sequence as $n \to \infty$, which is not in the set.

## Explanations

### L1 human

A shift space is a closed, shift-invariant set of two-sided infinite sequences over a finite alphabet.
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# Kurosh–Amitsur radical theory

## Summary

Kurosh–Amitsur radical theory is the general theory of radicals of rings.
A radical in this sense is a class of rings closed under homomorphic images.
Every ring must have a largest ideal in the class, and the quotient by that ideal must have no nonzero ideal in the class.
The [[Jacobson radical|XA0DCNH7WY]], the [[prime radical|WF117EJ2HF]], the [[Levitzki radical|9DD8M9R868]], the [[upper nilradical|3G2JQEYBEK]] and the [[Brown–McCoy radical|EWWP84BGSC]] are all examples.
The theory lets one prove facts about all these radicals at once, and it extends to other algebraic structures.

## Explanations

### L1 human

Many constructions in ring theory pick out a "bad part" of a ring, an ideal one would like to remove.
Removing it should leave a ring with no bad part left.
Radical theory turns this pattern into axioms.
For instance, the class of nil rings, the rings in which every element is nilpotent, is a radical class, and its radical is the largest nil ideal.
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<document source="https://tamnd.github.io/math2/node/A72XTVF2GK.md" tag="A72XTVF2GK" kind="concept">
# Schur-convex function

## Summary

A Schur-convex function is a function of $n$ real variables that preserves the order of [[majorization|5K7VT2VKZT]].
If $x$ is more spread out than $y$ and has the same sum, then $f(x) \ge f(y)$ for a Schur-convex $f$.
Every symmetric convex function is Schur-convex, and many classical inequalities, such as [[Karamata's inequality|FWKSHEJ9RP]], are statements that some function is Schur-convex.
The [[definition|24CEMRAGJ8]] gives the precise condition.

## Explanations

### L1 human

Think of a vector of $n$ numbers as a way to share a fixed total among $n$ people.
One sharing majorizes another when it is less equal.
A Schur-convex function measures inequality: it grows when the sharing becomes less equal.
For example the largest entry $\max_i x_i$ is Schur-convex: $(3, 0)$ is less equal than $(2, 1)$, and $3 \ge 2$.
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<document source="https://tamnd.github.io/math2/node/A7549DJW1N.md" tag="A7549DJW1N" kind="definition">
# Darboux function

## Definition

Let $I \subseteq \R$ be an [[interval|0TK8Y3J5RW]] and let $f \colon I \to \R$ be a [[function|0XHDTE5YMZ]].
The function $f$ is a *Darboux function*, or has the *intermediate value property*, if the following holds.
For all $a < b$ in $I$ and each real $y$ between $f(a)$ and $f(b)$, there is $c \in [a, b]$ with $f(c) = y$.

## Remarks

Equivalently, $f$ maps every subinterval of $I$ onto an interval.
Each [[continuous function|0P27W9RW0G]] on $I$ is a Darboux function, by the [[intermediate value theorem|00KAFBR32E]].
Each derivative on an open interval is a Darboux function, by [[Darboux's theorem|0Q5A47J48D]].

Example: let $f(x) = \sin(1/x)$ for $x \ne 0$ and $f(0) = 0$, on $\R$.
On a subinterval not containing $0$, $f$ is continuous, so its image is an interval.
A subinterval with more than one point that contains $0$ contains a full period of $\sin(1/x)$ on one side of $0$, so its image is $[-1, 1]$.
So $f$ is a Darboux function, although it is not continuous at $0$.

Non-example: the floor function $\floor{x}$ on $\R$ has $\floor{0} = 0$ and $\floor{1} = 1$, but it never takes the value $1/2$.

The [[Conway base 13 function|91YF82M8AA]] is a Darboux function that is continuous at no point.
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<document source="https://tamnd.github.io/math2/node/A755BDRP1A.md" tag="A755BDRP1A" kind="concept">
# Melnikov method

## Summary

The Melnikov method is a test for whether a [[homoclinic orbit|6DBSQKZK4W]] of a planar system survives a small periodic perturbation.
It reduces the question to the zeros of one explicit integral, the Melnikov function.
A simple zero of this function means that the stable and unstable manifolds of the perturbed saddle cross transversally.
Such a crossing forces chaotic dynamics nearby, so the method is a standard way to prove chaos in forced oscillators.
See the [[definition|BW7NGTGXAY]].

## Explanations

### L1 human

Think of an unforced system with a saddle point and a loop that leaves the saddle and comes back to it.
A small periodic push splits this loop into two curves: the set of points that go to the saddle and the set that come from it.
The Melnikov function measures, to first order in the size of the push, the signed gap between the two curves.
If the gap changes sign, the curves cross, and the crossings create very complicated motion.
For the forced, damped [[Duffing equation|SGQQGP261M]], the method says that crossings occur when the forcing is large enough compared with the damping.
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<document source="https://tamnd.github.io/math2/node/A77EHY9DAN.md" tag="A77EHY9DAN" kind="definition">
# Signature operator

## Definition

Let $(M, g)$ be a closed oriented [[Riemannian manifold|7CBT4SM83Y]] of dimension $4k$.
Let $\Omega^*(M) = \bigoplus_p \Omega^p(M)$ be the space of smooth complex-valued [[differential forms|0E3G0GRPZ6]], with [[exterior derivative|0ZHSTAG8ZY]] $d$, [[Hodge star|1Y4WX4XQWJ]] $\star$ and codifferential $\delta$, the formal adjoint of $d$.
Define $\tau \colon \Omega^*(M) \to \Omega^*(M)$ by
$$\tau \omega = i^{p(p-1) + 2k} \star \omega \quad \text{for } \omega \in \Omega^p(M).$$
Then $\tau^2 = \id$ and $\tau (d + \delta) = -(d + \delta) \tau$.
Let $\Omega^\pm(M)$ be the $\pm 1$ eigenspaces of $\tau$.
The *signature operator* of $M$ is the restriction
$$D^+ = d + \delta \colon \Omega^+(M) \to \Omega^-(M).$$

## Remarks

The operator $d + \delta$ is elliptic with square the [[Hodge Laplacian|PWG3GKZG80]], so $D^+$ is an [[elliptic operator|MT9P2X85V8]] of order $1$.
Its index equals the signature $\sigma(M)$ of the [[intersection form|WK8GKRKAVD]] on $H^{2k}(M; \R)$.
Indeed, the harmonic forms of degrees $p \ne 2k$ contribute equally to the kernel and the cokernel and cancel.
On harmonic $2k$-forms $\tau = \star$, whose $\pm 1$ eigenspaces are the self-dual and anti-self-dual forms.
The Atiyah–Singer index theorem applied to $D^+$ gives the [[Hirzebruch signature theorem|TEG1ZX1AGJ]].

Example: on $2k$-forms, $\tau = i^{2k(2k-1) + 2k} \star = i^{4k^2} \star = \star$.
For $M = \C P^2$ ($k = 1$), the harmonic $2$-forms are the multiples of the Kähler form $\omega_{FS}$, which is self-dual, so $\sigma(\C P^2) = 1$ and the index of $D^+$ is $1$.

Non-example: the operator $d + \delta \colon \Omega^{\mathrm{even}}(M) \to \Omega^{\mathrm{odd}}(M)$ uses the splitting by degree instead of the splitting by $\tau$.
It is the de Rham or Euler operator, and its index is the Euler characteristic, not the signature; for $\C P^2$ it is $3$.

## Explanations

### L1 human

The signature operator is $d + d^*$ on forms, split by a twisted Hodge star, and its index is the signature of the manifold.
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<document source="https://tamnd.github.io/math2/node/A77FER1Q5H.md" tag="A77FER1Q5H" kind="concept">
# Curry–Howard correspondence

## Summary

The Curry–Howard correspondence is the match between proofs in [[intuitionistic logic|Q11N8X99FK]] and programs in typed [[lambda calculus|BD756KR8FM]].
A proposition corresponds to a type, and a proof of the proposition corresponds to a program of that type.
Simplifying a proof corresponds to running the program.
It is the basis of proof assistants built on [[type theory|Q48KGV33AH]], where checking a proof means checking the type of a program.

## Explanations

### L1 human

The slogan is "propositions are types, and proofs are programs".
A proof of "if $A$ then $B$" is a function that turns any proof of $A$ into a proof of $B$.
A proof of "$A$ and $B$" is a pair made of a proof of $A$ and a proof of $B$.
For example, the program that takes $x$ and $y$ and returns $x$ has type $A \to (B \to A)$, and it is also a proof of that formula.
Curry noticed such a match for combinators, and Howard extended it to natural deduction.
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<document source="https://tamnd.github.io/math2/node/A79PHYDXMW.md" tag="A79PHYDXMW" kind="definition">
# Modulus of continuity

## Definition

Let $D \subseteq \R$ and let $f \colon D \to \R$ be a [[function|0XHDTE5YMZ]].
A *modulus of continuity* is a function $\omega \colon [0, \infty) \to [0, \infty]$ that is nondecreasing, satisfies $\omega(0) = 0$, and satisfies $\omega(\delta) \to 0$ as $\delta \to 0^+$.
The function $\omega$ is a modulus of continuity *for* $f$ if
$$\abs{f(x) - f(y)} \le \omega(\abs{x - y}) \quad \text{for all } x, y \in D.$$
The *optimal modulus of continuity* of $f$ is the function $\omega_f \colon [0, \infty) \to [0, \infty]$ given by
$$\omega_f(\delta) = \sup \set{\abs{f(x) - f(y)} : x, y \in D, \ \abs{x - y} \le \delta},$$
where the [[supremum|0BTMQ04556]] is taken in the [[extended real line|0GSZ7B0CGR]].

## Remarks

The function $f$ has a modulus of continuity if and only if $f$ is [[uniformly continuous|0Q4DR37DVE]], and then $\omega_f$ is the smallest one.
The same definitions work for a map between metric spaces, with distances in place of $\abs{x - y}$ and $\abs{f(x) - f(y)}$.

Example: let $f(x) = \sqrt x$ on $D = [0, \infty)$.
For $0 \le y \le x$, one has $\sqrt x - \sqrt y \le \sqrt{x - y}$, since squaring gives $x - 2\sqrt{xy} + y \le x - y$, that is $y \le \sqrt{xy}$.
Equality holds for $y = 0$, so $\omega_f(\delta) = \sqrt \delta$, and this tends to $0$.

Non-example: let $f(x) = x^2$ on $\R$.
For $\delta > 0$, one has $f(x + \delta) - f(x) = 2x\delta + \delta^2$, which is unbounded in $x$.
So $\omega_f(\delta) = \infty$ for each $\delta > 0$, and $\omega_f$ does not tend to $0$.

The modulus $\omega(\delta) = L\delta$ gives the [[Lipschitz condition|0QNVQ4VD9T]] with constant $L$.
The modulus $\omega(\delta) = C\delta^\alpha$ with $0 < \alpha \le 1$ gives the [[Hölder condition|0WZTM0EFR6]] of exponent $\alpha$.
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<document source="https://tamnd.github.io/math2/node/A7MJXJERNY.md" tag="A7MJXJERNY" kind="statement">
# Gauss's test

## Statement

Let $(a_n)$ be a sequence of positive real numbers.
Assume that there are a real number $h$, a real number $s > 1$ and a bounded real sequence $(\beta_n)$ with
$$\frac{a_n}{a_{n+1}} = 1 + \frac{h}{n} + \frac{\beta_n}{n^s}$$
for all $n \ge 1$.
Then $\sum a_n$ [[converges|0CVVKY7GY2]] if $h > 1$, and [[diverges|0NBRP6093V]] if $h \le 1$.

## Remarks

The quantity of [[Raabe's test|CD6CAQ6XDN]] is $n(a_n/a_{n+1} - 1) = h + \beta_n n^{1-s}$, which tends to $h$.
So Raabe's test settles the cases $h > 1$ and $h < 1$.
For $h = 1$, the quantity of [[Bertrand's test|6T3E3AXZT7]] is $\beta_n n^{1-s} \ln n$, which tends to $0$, so the series diverges.
Example: let $a, b, c > 0$ and let $a_n$ be the $n$-th term of the hypergeometric series $\sum_n \frac{(a)_n (b)_n}{(c)_n \, n!}$, where $(x)_n = x(x+1)\cdots(x+n-1)$.
Then $a_n / a_{n+1} = \frac{(n + c)(n + 1)}{(n + a)(n + b)} = 1 + \frac{c + 1 - a - b}{n} + O(1/n^2)$.
So the series converges exactly when $c > a + b$.

## Explanations

### L1 human

Gauss's test reads off convergence from the expansion of the ratio $a_n / a_{n+1}$ in powers of $1/n$.
Only the coefficient $h$ of $1/n$ matters: above $1$ the series converges, otherwise it diverges.
It is the natural test for terms that are products of linear factors in $n$.

### L2 human

Gauss's test packages Raabe's test and Bertrand's test into one statement for ratios with an asymptotic expansion.
A classical application is the convergence of the hypergeometric series at $x = 1$, as in the example above.
The error term must be $O(1/n^s)$ with $s > 1$; an error like $1/(n \ln n)$ would hide the boundary case.
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# Persistent homology

## Definition

Let $\F$ be a [[field|04C5ZBG1WT]] and let $K$ be a finite [[simplicial complex|V8GZ0W1ZZ5]] with a filtration by subcomplexes
$$K_0 \subseteq K_1 \subseteq \cdots \subseteq K_m = K.$$
For $i \le j$ and $p \ge 0$, the inclusion $K_i \subseteq K_j$ induces a linear map $f_p^{i,j} \colon H_p(K_i; \F) \to H_p(K_j; \F)$ of [[simplicial homology|C6J40QF82R]] groups, which are [[vector spaces|000AYPA9AJ]] over $\F$.
The *$p$-th persistent homology group* from $i$ to $j$ is the image
$$H_p^{i,j} = \im f_p^{i,j} \subseteq H_p(K_j; \F),$$
and its dimension $\beta_p^{i,j}$ is the *persistent Betti number*.
The family of all $H_p(K_i; \F)$ with all maps $f_p^{i,j}$ is the *$p$-th persistence module* of the filtration.

## Remarks

A class is *born* at $i$ if it lies in $H_p(K_i; \F)$ but not in the image of $f_p^{i-1,i}$.
It *dies* at $j$ if it merges with an older class or becomes $0$ when passing from $K_{j-1}$ to $K_j$.
The persistence module splits, uniquely up to order, as a direct sum of interval modules.
The list of their intervals $[b, d)$ is the *barcode*.
The same data, as points $(b, d)$ in the plane, form the *persistence diagram*.

Example: let $K_0$ consist of two vertices $a, b$ and let $K_1$ add the edge $ab$.
Then $H_0(K_0; \F) \cong \F^2$, $H_0(K_1; \F) \cong \F$, and $f_0^{0,1}$ is onto, so $\beta_0^{0,1} = 1$.
The barcode in degree $0$ is $\set{[0, \infty), [0, 1)}$: one component lives forever and one dies when the edge joins them.

Non-example: a family of complexes $K_0, K_1$ where $K_1$ does not contain $K_0$ is not a filtration, and no maps $f_p^{0,1}$ are induced.
Zigzag persistence treats such cases with maps in both directions.

For a finite set of points in a metric space, the standard filtrations are the Vietoris–Rips and Čech complexes at increasing scale.
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# Multidimensional Z-transform

## Summary

The multidimensional Z-transform sends a sequence indexed by $\Z^n$ to a Laurent series in $n$ complex variables $z_1, \dots, z_n$.
It is the discrete analogue of the [[multidimensional Laplace transform|QT3M720TCV]] and the $n$-variable form of the [[Z-transform|5FY0JQGHYZ]].
It is used to analyze multidimensional digital filters, for example filters acting on images.
Its region of convergence depends only on the moduli $\abs{z_j}$.
Its image under $z \mapsto (\log\abs{z_1}, \dots, \log\abs{z_n})$ is convex, and it plays the role of the annulus of the one-variable case.

## Explanations

### L1 human

A digital image is a table of numbers $x[k_1, k_2]$ indexed by two integers.
The two-dimensional Z-transform packs the whole table into one function of two complex variables, with $x[k_1, k_2]$ as the coefficient of $z_1^{-k_1} z_2^{-k_2}$.
Shifting the image becomes multiplying the function by a monomial, and filtering becomes multiplying two functions.
For example, the table with a single $1$ at position $(1, 2)$ has the transform $z_1^{-1} z_2^{-2}$.
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# Fractional Sobolev space

## Definition

Let $U \subseteq \R^n$ be open, let $1 \le p < \infty$, and let $0 < \sigma < 1$.
The *fractional Sobolev space* $W^{\sigma,p}(U)$, or Sobolev–Slobodeckij space, is the set of $u \in L^p(U)$ with finite Gagliardo seminorm
$$[u]_{\sigma,p,U} = \Big( \int_U \int_U \frac{\abs{u(x) - u(y)}^p}{\abs{x - y}^{n + \sigma p}} \, dx \, dy \Big)^{1/p}.$$
Here $L^p(U)$ is the [[Lp space|SASP14MA1Z]].
It carries the norm $\norm{u}_{W^{\sigma,p}(U)} = \big( \norm{u}_{L^p(U)}^p + [u]_{\sigma,p,U}^p \big)^{1/p}$.
Let $s = k + \sigma$ with $k \ge 1$ an integer and $0 < \sigma < 1$.
The space $W^{s,p}(U)$ is the set of $u$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{k,p}(U)$ with $D^\alpha u \in W^{\sigma,p}(U)$ whenever $\abs{\alpha} = k$.

## Remarks

Each $W^{s,p}(U)$ is a Banach space.
On $\R^n$ and for every real $s \ge 0$, one also defines $H^s(\R^n)$ as the set of $u \in L^2(\R^n)$ with $(1 + \abs{\xi}^2)^{s/2} \hat{u} \in L^2(\R^n)$, where $\hat u$ is the [[Fourier transform|DPVM34ADZV]].
For $0 < s$ not an integer, $W^{s,2}(\R^n) = H^s(\R^n)$ with equivalent norms; for $0 < \sigma < 1$ this follows from Plancherel's theorem, since $[u]_{\sigma,2,\R^n}^2 = c \int \abs{\xi}^{2\sigma} \abs{\hat u(\xi)}^2 \, d\xi$ for a constant $c = c(n, \sigma) > 0$.
For $s$ not an integer, $W^{s,p}(\R^n)$ is the [[Besov space|95QBBPRJ9E]] $B^s_{p,p}(\R^n)$.
Negative orders are defined by duality, as spaces of [[tempered distributions|AES2T5JPFP]] or of distributions.

Example: on $U = (-1, 1)$, let $u = 1$ on $(0, 1)$ and $u = 0$ on $(-1, 0)$, and let $0 < \sigma < 1/2$.
The double integral is $2 \int_0^1 \int_{-1}^0 \abs{x - y}^{-1 - 2\sigma} \, dy \, dx$, which is finite because $\int_0^1 \int_0^1 (a + b)^{-1 - 2\sigma} \, da \, db < \infty$ when $2\sigma < 1$.
So $u \in W^{\sigma,2}(-1, 1)$.

Non-example: the same $u$ is not in $W^{1/2,2}(-1, 1)$.
For $\sigma = 1/2$ the inner integral $\int_{-1}^{0} (x - y)^{-2} \, dy$ equals $\frac{1}{x} - \frac{1}{x + 1} \ge \frac{1}{x} - 1$, and $\int_0^1 (\frac{1}{x} - 1) \, dx = \infty$.
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# Infinite loop space

## Summary

An infinite loop space is a space $X$ that is the [[loop space|2JFF5F3EGF]] of a space.
That space is again a loop space, and so on without end.
Infinite loop spaces are exactly the zeroth spaces of $\Omega$-spectra, so they are the unstable shadows of [[spectra|1GHF4EKBXS]] and of generalized cohomology theories.
Recognition principles, such as the [[May recognition principle|5SMK5KEBQE]], detect them by a homotopy-coherent commutative multiplication.

## Explanations

### L1 human

Loops in a space can be composed, so a loop space has a multiplication that is associative up to homotopy.
If the space is a double loop space, the multiplication is also commutative up to homotopy.
An infinite loop space has infinitely many such layers of structure, which make its multiplication as commutative as homotopy theory allows.
For an abelian group $A$, the Eilenberg–MacLane spaces $K(A, n)$ satisfy $K(A, n) \simeq \Omega K(A, n+1)$, so each of them is an infinite loop space.
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<document source="https://tamnd.github.io/math2/node/A7W468BQSM.md" tag="A7W468BQSM" kind="concept">
# Grothendieck category

## Summary

A Grothendieck category is an [[abelian category|Y9CZ3MTNF1]] with all small colimits, in which filtered colimits are exact, and which has a generator.
Grothendieck introduced these conditions in his 1957 Tôhoku paper and proved that such categories have enough injective objects.
This made it possible to define sheaf cohomology as a derived functor.
Module categories and categories of sheaves of abelian groups are Grothendieck categories.
By the [[Gabriel–Popescu theorem|3MP41FQHZ5]], every Grothendieck category is a localization of a module category.

## Explanations

### L1 human

Abelian categories are a good setting for homological algebra, but some need extra properties to construct resolutions.
A Grothendieck category asks for three things that modules have.
One can glue infinitely many objects together, and gluing along a directed system does not destroy exactness.
There is also one object, like the ring itself, from which all others can be probed.
With these, every object embeds in an injective object, so derived functors such as sheaf cohomology can be defined.
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# Newtonian kernel

## Definition

Let $n \ge 3$.
The *Newtonian kernel* on $\R^n$ is the function $k \colon \R^n \times \R^n \to (0, +\infty]$ given by
$$k(x, y) = \frac{1}{\abs{x - y}^{n-2}} \quad (x \ne y), \qquad k(x, x) = +\infty.$$

## Remarks

Example: for $n = 3$, $k(x, y) = 1/\abs{x - y}$; for $x = (1, 0, 0)$ and $y = 0$ it equals $1$.
For fixed $y$, the function $x \mapsto k(x, y)$ is [[harmonic|0DEB0JMN0W]] on $\R^n \setminus \set{y}$, since $\Delta r^{2-n} = (2-n)(1-n) r^{-n} + (n-1)(2-n) r^{-n} = 0$ for $r = \abs{x - y}$.
Non-example: on $\R^4$, the function $1/\abs{x - y}$ is not the Newtonian kernel; its Laplacian in $x$ is $-\abs{x - y}^{-3} \ne 0$, so it is not harmonic off the pole.
The kernel is symmetric, $k(x, y) = k(y, x)$, lower semicontinuous, and positive.
With $\sigma_n$ the area of the unit sphere of $\R^n$, one has $k(x, y) = (n-2)\sigma_n \, \Phi(x - y)$, where $\Phi$ is the [[fundamental solution of the Laplacian|BSJ8X2XRN9]].
Some authors include this constant in the kernel; the definition above is the usual one in potential theory.
In the plane its role is played by the logarithmic kernel $\log(1/\abs{x - y})$.
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# Bounded variation

## Summary

A real function on a closed interval has bounded variation when the total up-and-down movement of its values is finite.
This total movement is the [[total variation|8QA5DEWNYD]] of the function.
Functions of bounded variation are exactly the differences of two [[monotone functions|0WFF6AZPBP]], by the [[Jordan decomposition|KDT3E0VE0K]].
They are the natural integrators for the [[Riemann–Stieltjes integral|JT9AFHG0NA]] and they are differentiable almost everywhere.

## Explanations

### L1 human

Walk along the graph of a function from left to right and count how far the value goes up and down in total.
If this distance is finite, the function has bounded variation.
Every monotone function on $[a, b]$ has bounded variation, since its values move in one direction only.
The function $x \sin(1/x)$ on $[0, 1]$, with value $0$ at $0$, does not.
It swings up and down infinitely often near $0$, and the sizes of the swings add up to infinity.
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# Virtual knot

## Definition

A *virtual knot diagram* is a generic immersion of an oriented circle in $\R^2$ with marked double points.
Each double point is marked either as a classical crossing, with over- and under-strand, or as a virtual crossing.
Two virtual knot diagrams are *equivalent* if they are related by planar isotopy and a finite sequence of the following moves:

1. the three classical [[Reidemeister moves|T3HAGWYKK7]];
2. the three virtual Reidemeister moves, which are the Reidemeister moves with all crossings virtual;
3. the mixed move, which slides a strand with two virtual crossings past a classical crossing.

A *virtual knot* is an equivalence class of virtual knot diagrams.

## Remarks

The moves of type 3 that slide a strand with classical crossings past a virtual crossing are forbidden.
Example: the virtual trefoil has two classical crossings and one virtual crossing; it is a nontrivial virtual knot that is not classical.
Non-example: the forbidden moves are not equivalences, and allowing both of them makes every virtual knot equivalent to the unknot.
A diagram with no virtual crossings is a diagram of a classical [[knot|QYEAZTYGPT]].
Goussarov, Polyak and Viro showed that classical knots equivalent as virtual knots are isotopic.
Kuperberg showed that virtual knots correspond to knots in thickened oriented surfaces $\Sigma \times [0,1]$, up to stabilization, with a unique irreducible representative.
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# Absolute summability

## Summary

Absolute summability is a strengthening of summability by a [[matrix summability method|M6311DWJ1E]].
A series is absolutely summable by a method when the transformed sequence not only converges but has bounded variation.
It plays for summability methods the role that [[absolute convergence|0YBDSJARWN]] plays for ordinary convergence.

## Explanations

### L1 human

A summability method turns the partial sums of a series into a new sequence, for example their running averages.
Ordinary summability asks that this new sequence settles down to a limit.
Absolute summability asks more: the total of all the jumps between consecutive terms of the new sequence must be finite.
For the identity method, where the new sequence is the sequence of partial sums itself, the jumps are the terms $a_n$, and one gets back absolute convergence.
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# Mittag-Leffler summation

## Definition

Let $\sum_{k \ge 0} a_k$ be a [[series|0JH4MN4K1C]] of complex numbers, and let $\Gamma$ be the [[gamma function|0V3FZMVB4K]].
For $\alpha > 0$ put
$$M(\alpha) = \sum_{k \ge 0} \frac{a_k}{\Gamma(1 + \alpha k)}$$
whenever this series converges.
The series $\sum a_k$ is *Mittag-Leffler summable* to $s$ if $M(\alpha)$ is defined for all small $\alpha > 0$ and the [[one-sided limit|0909PWZR62]] $\lim_{\alpha \to 0^+} M(\alpha)$ equals $s$.

## Remarks

For $a_k = z^k$ one gets $M(\alpha) = \sum_{k \ge 0} \frac{z^k}{\Gamma(1 + \alpha k)}$, the Mittag-Leffler function $E_\alpha(z)$.

Example: let $\sum a_k$ converge absolutely to $s$.
Since $\Gamma(x) \ge \frac12$ for all $x \ge 1$, one has $\abs{a_k / \Gamma(1 + \alpha k)} \le 2 \abs{a_k}$ for all $\alpha$ and $k$.
Each term tends to $a_k$ as $\alpha \to 0^+$, so by dominated convergence for series $M(\alpha) \to s$.

Non-example: for $1 + 1 + 1 + \cdots$, the terms $1/\Gamma(1 + \alpha k)$ are positive and tend to $1$ for each $k$.
So for each $N$, $M(\alpha) \ge \sum_{k=0}^{N} \frac{1}{\Gamma(1 + \alpha k)} \to N + 1$ as $\alpha \to 0^+$, and $M(\alpha)$ has no finite limit.
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# Determinant trick

## Statement

Let $R$ be a [[commutative ring|0310REJJY8]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module with $IM = M$.
Then there is an element $r \in R$ with $r - 1 \in I$ and $rM = 0$.

## Remarks

Apply the [[Cayley–Hamilton theorem for modules|SH3R5VTMZH]] to $\varphi = \id_M$, which satisfies $\varphi(M) = M = IM$.
It gives $a_1, \ldots, a_n \in I$ with $(1 + a_1 + \cdots + a_n) \id_M = 0$, so $r = 1 + a_1 + \cdots + a_n$ works.
The name comes from the determinant of the matrix $I_n - (a_{ij})$ used in that proof.
If $I$ lies in the [[Jacobson radical|XA0DCNH7WY]] of $R$, then $r$ is a unit, so $M = 0$; this is [[Nakayama's lemma|15JHBYR6KH]].
The hypothesis that $M$ is finitely generated is needed: for $R = \Z$, $I = 2\Z$ and $M = \Q$ one has $IM = M$, but the only $r$ with $r\Q = 0$ is $r = 0$, and $0 - 1 \notin 2\Z$.

## Explanations

### L1 human

If multiplying a finitely generated module by an ideal does not make it smaller, then the module is killed by some element that is congruent to $1$ modulo the ideal.
It turns a statement about modules into a statement about a single ring element.

### L2 human

This is the most flexible form of [[Nakayama's lemma|15JHBYR6KH]].
It gives Nakayama's lemma for local rings and for the Jacobson radical at once.
It also shows that a finitely generated ideal $I$ with $I^2 = I$ is generated by an idempotent.
Indeed, there is $r \in 1 + I$ with $rI = 0$.
Then $e = 1 - r \in I$ satisfies $e x = x$ for $x \in I$, so $e^2 = e$ and $I = eR$.
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# Nijenhuis tensor

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]].
Let $J \colon TM \to TM$ be a smooth [[tensor field|WX302T93TW]] of type $(1,1)$, that is, a smooth field of linear endomorphisms of the tangent spaces.
Let $[\cdot, \cdot]$ be the [[Lie bracket of vector fields|68FGQEBD6J]].
The *Nijenhuis tensor* of $J$ is the map $N_J$ that sends vector fields $X, Y$ to the vector field
$$N_J(X, Y) = [JX, JY] - J[JX, Y] - J[X, JY] + J^2 [X, Y].$$

## Remarks

The map $N_J$ is $C^\infty(M)$-bilinear and antisymmetric, so it is a tensor field of type $(1,2)$, a $2$-form with values in $TM$.
It equals $\frac{1}{2}[J, J]$, where $[J, J]$ is the [[Frölicher–Nijenhuis bracket|D46QECMC10]] of $J$ with itself.
For an [[almost complex structure|9SYWRGVTRS]], where $J^2 = -\id$, the formula reads $N_J(X, Y) = [JX, JY] - J[JX, Y] - J[X, JY] - [X, Y]$.
Some authors use a different constant factor.

Example: on $\R^2$ let $J \partial_x = \partial_y$ and $J \partial_y = -\partial_x$.
For $X, Y$ among $\partial_x, \partial_y$, all of $X, Y, JX, JY$ are constant combinations of $\partial_x$ and $\partial_y$, so all brackets vanish.
By bilinearity over functions, $N_J = 0$.
More generally, for an almost complex structure one has $N_J(JX, Y) = -J N_J(X, Y)$, so $N_J(JX, X) = 0$; on a $2$-manifold $X$ and $JX$ span each tangent space where $X \ne 0$, hence $N_J = 0$ on every surface.

Non-example: the expression $[JX, JY] - [X, Y]$ alone is not a tensor.
Replacing $X$ by $fX$ for a function $f$ gives $f([JX, JY] - [X, Y]) - (JY)(f)\, JX + Y(f)\, X$, since $[fA, B] = f[A, B] - B(f) A$.
The last two terms are not zero in general, so the expression is not $C^\infty(M)$-linear in $X$; the two middle terms of $N_J$ cancel them.
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# Ideal of compact operators

## Definition

Let $H$ be a [[Hilbert space|HM2HHE3J51]] and let $\mathcal{B}(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$ with the operator norm.
The *ideal of compact operators* on $H$ is the set
$$\mathcal{K}(H) = \set{T \in \mathcal{B}(H) : T \text{ is a compact operator}},$$
with compactness in the sense of [[compact operators|M2NEP1K5AE]].
It is a norm-closed [[two-sided ideal|0CD87SFM2N]] of $\mathcal{B}(H)$ and is closed under taking adjoints.

## Remarks

Example: on $H = \ell^2$, the diagonal operator with entries $1, 1/2, 1/3, \ldots$ belongs to $\mathcal{K}(H)$, since it is the norm limit of its finite-rank truncations.
For any bounded $A$, the products $AT$ and $TA$ are again in $\mathcal{K}(H)$.
Indeed, a bounded operator maps bounded sets to bounded sets and relatively compact sets to relatively compact sets.

Non-example: if $H$ is infinite-dimensional, the identity $I$ is not in $\mathcal{K}(H)$, since an orthonormal sequence has no convergent subsequence; so $\mathcal{K}(H)$ is a proper ideal.
If $H$ is finite-dimensional, every operator is compact and $\mathcal{K}(H) = \mathcal{B}(H)$.

Other notations are $\mathcal{K}$ and $\mathcal{B}_0(H)$.
On a Hilbert space, $\mathcal{K}(H)$ equals the norm closure of the finite-rank operators.
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# Mortar methods

## Definition

Let $\Omega \subset \R^d$ be a bounded polygonal domain, divided into non-overlapping polygonal subdomains $\Omega_1, \dots, \Omega_N$.
On each $\Omega_k$ let $X_k \subset H^1(\Omega_k)$ be a conforming [[finite element|98GZ9DVHTA]] space on a mesh of $\Omega_k$.
The meshes of neighboring subdomains need not match on their common boundary.
Let $\Gamma_1, \dots, \Gamma_M$ be the interfaces, each a common side of two subdomains.
For each $\Gamma_m$ choose one side as the *slave* side $s(m)$ and the other as the *master* side $\mu(m)$.
Let $M_h(\Gamma_m)$ be a space of functions on $\Gamma_m$ (the *mortar* or *multiplier space*) built on the mesh of the slave side.
Its dimension equals the number of slave degrees of freedom interior to $\Gamma_m$.
The *mortar finite element space* is
$$V_h = \Bigl\{ v : v|_{\Omega_k} \in X_k \text{ for all } k, \ \int_{\Gamma_m} \bigl( v|_{\Omega_{\mu(m)}} - v|_{\Omega_{s(m)}} \bigr) \psi \, ds = 0 \text{ for all } \psi \in M_h(\Gamma_m) \text{ and all } m \Bigr\},$$
where $v|_{\Omega_k}$ on $\Gamma_m$ means the [[trace|Z2T5R88QDB]] of the restriction, together with the boundary conditions of the problem.
A *mortar method* for a problem $a(u, v) = \ell(v)$ is the Galerkin method in $V_h$ with the broken form $a_h(u, v) = \sum_k a_{\Omega_k}(u, v)$.

## Remarks

Example: take $-\Delta u = f$ on two squares meeting along a segment $\Gamma$, with piecewise linear elements on independent triangulations.
The standard $M_h(\Gamma)$ consists of the continuous piecewise linear functions on the slave mesh of $\Gamma$ that are constant on the two end elements.
The number of these functions equals the number of slave nodes interior to $\Gamma$, so the condition determines the slave interior values from the master trace.
Non-example: requiring the slave nodal values to equal the master trace at the slave nodes is pointwise interpolation, not a mortar method.
The gluing is not by orthogonality to a multiplier space, and it loses the optimal error order on non-matching meshes.
The functions of $V_h$ are in general discontinuous across interfaces, so the method is nonconforming, and $a_h$ replaces the global form.
Equivalently, the constraint can be kept as an equation with a Lagrange multiplier on each interface, which gives a saddle point formulation.

## Explanations

### L1 human

Each subdomain gets its own mesh and its own finite elements.
Along an interface, the two sides may not agree at every point, but their difference must average to zero against every test function in the mortar space.
This weak gluing is enough to keep the method as accurate as a method on one matching mesh.
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# Symplectomorphism

## Summary

A symplectomorphism is a diffeomorphism between symplectic manifolds that carries one symplectic form to the other.
These maps are the isomorphisms of symplectic geometry.
In mechanics they are the canonical transformations, and the time-$t$ maps of Hamiltonian flows are symplectomorphisms.
Unlike isometries of a Riemannian manifold, the symplectomorphisms of a symplectic manifold form an infinite-dimensional group.

## Explanations

### L1 human

In the plane, a symplectomorphism is a smooth change of shape that keeps every area the same, with orientation.
A rotation keeps areas, and so does a shear that slides each vertical line up or down.
Stretching by $2$ in one direction and keeping the other direction fixed doubles areas, so it is not a symplectomorphism.
In higher dimensions a symplectomorphism keeps a whole family of areas, one for each pair of coordinates, not only volume.
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# Separable algebra

## Definition

Let $K$ be a commutative ring and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $K$.
Let $A^{\mathrm{op}}$ be the [[opposite ring|M8Q24XB0AF]] of $A$ and let $A^e = A \otimes_K A^{\mathrm{op}}$ be the [[tensor product of algebras|1735VGRT2P]].
An $A$-[[bimodule|EM7FHAFD3N]] is the same as a left $A^e$-module, and $A$ itself is one.
Let $\mu \colon A^e \to A$ be the map $a \otimes b \mapsto ab$, which is a homomorphism of $A^e$-modules.
The algebra $A$ is *separable* over $K$ if there is a homomorphism of $A^e$-modules $s \colon A \to A^e$ with $\mu \circ s = \id_A$.

## Remarks

Equivalently, $A$ is a projective $A^e$-module.
Equivalently, there is a *separability idempotent*: an element $e = \sum_i x_i \otimes y_i$ of $A^e$ with $\sum_i x_i y_i = 1$ and $\sum_i a x_i \otimes y_i = \sum_i x_i \otimes y_i a$ for all $a \in A$; take $e = s(1)$.

Example: $A = M_n(K)$ is separable over $K$.
With matrix units $E_{ij}$, the element $e = \sum_{i} E_{i1} \otimes E_{1i}$ satisfies $\sum_i E_{i1} E_{1i} = \sum_i E_{ii} = 1$, and for all $k, l$ the sums $\sum_i E_{kl} E_{i1} \otimes E_{1i}$ and $\sum_i E_{i1} \otimes E_{1i} E_{kl}$ both equal $E_{k1} \otimes E_{1l}$.

Non-example: let $K = \F_p(t)$ and $A = K(t^{1/p})$, a purely inseparable field extension of degree $p$.
Then $A \otimes_K A \cong A[x]/(x - t^{1/p})^p$ has nonzero nilpotent elements, so it is not semisimple, and $A$ is not separable over $K$.

When $K$ is a field, a finite-dimensional algebra $A$ is separable if and only if $A \otimes_K L$ is [[semisimple|NY25HXA478]] for every field extension $L$ of $K$.
Equivalently, $A$ is semisimple and the center of each simple factor of $A$ is a [[separable field extension|0WCX7NB4N7]] of $K$.
A finite field extension $L/K$ is a separable algebra exactly when it is a separable extension.
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# Numerical integration

## Definition

Let $a < b$ be real numbers and let $F$ be a class of [[integrable functions|RSAK8YHMXF]] $f \colon [a, b] \to \R$, with the [[definite integral|021WEM680W]] $I(f) = \int_a^b f(x)\, dx$.
A *numerical integration method* on $F$ is a rule that assigns to each $f \in F$ a number $Q(f)$.
The number $Q(f)$ is computed from finitely many values $f(x_1), \dots, f(x_n)$ at points $x_i \in [a, b]$, and possibly from finitely many values of derivatives of $f$.
Its *error* at $f$ is $E(f) = I(f) - Q(f)$.
*Numerical integration* is the design and analysis of such methods, so that $\abs{E(f)}$ is small for the functions of interest.

## Remarks

The same definition applies to integrals over regions of $\R^d$ or with a weight function; in several dimensions one often speaks of cubature.
Example: the midpoint method $Q(f) = (b - a) f\big(\tfrac{a + b}{2}\big)$ on continuous functions uses one value; for $f(x) = x^2$ on $[0, 1]$ it gives $1/4$, with error $1/3 - 1/4 = 1/12$.
Non-example: $Q(f) = F(b) - F(a)$ with an antiderivative $F$ is exact, but it is not a numerical integration method in this sense, because it needs an antiderivative, not finitely many values of $f$.
The points $x_i$ may also be chosen adaptively, depending on earlier values, as in adaptive quadrature, or at random, as in Monte Carlo integration.
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# Van Wijngaarden transformation

## Definition

Let $\sum_{k \ge 0} (-1)^k a_k$ be an [[alternating series|017BR1M98J]] with [[partial sums|0NB643K60Q]] $s_k = \sum_{i=0}^{k} (-1)^i a_i$.
The *van Wijngaarden transformation* builds the table $(s_{j,k})_{j, k \ge 0}$ by
$$s_{0,k} = s_k, \qquad s_{j+1,k} = \frac{s_{j,k} + s_{j,k+1}}{2}.$$
The entries $s_{j,k}$ with $j \ge 1$ serve as approximations of the sum of the series.

## Remarks

By induction, $s_{j,k} = 2^{-j} \sum_{i=0}^{j} \binom{j}{i} s_{k+i}$, a binomially weighted mean of $j + 1$ consecutive partial sums.
Example: for the Leibniz series, $s_0, \dots, s_3$ are $1, 0.6667, 0.8667, 0.7238$.
The next rows are $0.8333, 0.7667, 0.7952$, then $0.8000, 0.7810$, then $0.7905$, which is within $0.006$ of $\frac{\pi}{4} \approx 0.7854$.
Non-example: the series $\sum 2^{-k}$ is not alternating, and its partial sums $1, 1.5, 1.75$ give the averages $1.25, 1.625$.
These are worse than the last partial sum; the method relies on the alternation of signs.

## Explanations

### L1 human

Write the partial sums in a row, then write the averages of neighbours below them, and repeat.
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# Heine's transformation

## Statement

Let $a, b, c, q, z \in \C$ with $0 < \abs{q} < 1$, $\abs{z} < 1$, $0 < \abs{b} < 1$, and with $c$ and $az$ not of the form $q^{-m}$ for an integer $m \ge 0$.
Then
$${}_2\phi_1 \left( \begin{matrix} a, b \\ c \end{matrix} ; q, z \right) = \frac{(b; q)_\infty\, (az; q)_\infty}{(c; q)_\infty\, (z; q)_\infty}\; {}_2\phi_1 \left( \begin{matrix} c/b, z \\ az \end{matrix} ; q, b \right),$$
where ${}_2\phi_1$ is the [[basic hypergeometric series|EB4N9RQR51]] and $(\cdot\,; q)_\infty$ is the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].

## Remarks

Proof idea: write $\frac{(b; q)_n}{(c; q)_n} = \frac{(b; q)_\infty}{(c; q)_\infty} \cdot \frac{(c q^n; q)_\infty}{(b q^n; q)_\infty}$.
Expand $\frac{(c q^n; q)_\infty}{(b q^n; q)_\infty}$ by the [[q-binomial theorem|VVT27RB46B]] as a series in $b q^n$, exchange the two sums, and sum over $n$ with the $q$-binomial theorem again.
Example: for $c = b$ the left side is ${}_1\phi_0(a; -; q, z) = \frac{(az; q)_\infty}{(z; q)_\infty}$, and on the right ${}_2\phi_1(1, z; az; q, b) = 1$ since $(1; q)_n = 0$ for $n \ge 1$; both sides agree.
Iterating the transformation gives two more forms, and the three are often called Heine's first, second and third transformations.
The third form is ${}_2\phi_1(a, b; c; q, z) = \frac{(abz/c; q)_\infty}{(z; q)_\infty}\, {}_2\phi_1(c/a, c/b; c; q, abz/c)$, valid when both series converge.

## Explanations

### L1 human

Heine's transformation rewrites one basic hypergeometric series as an infinite product times another basic hypergeometric series with permuted parameters.
The roles of the variable $z$ and the parameter $b$ are exchanged.
Such a change can turn a slowly converging series into a fast one, or a hard sum into an easy one.

### L2 human

It is the first and most used transformation of the ${}_2\phi_1$ series, and its third form is the $q$-analogue of Euler's transformation of the [[hypergeometric series|428M453CR2]] ${}_2F_1$.
Setting $z = c/(ab)$ and summing the right side with the $q$-binomial theorem gives the $q$-Gauss sum ${}_2\phi_1(a, b; c; q, c/(ab)) = \frac{(c/a, c/b; q)_\infty}{(c, c/(ab); q)_\infty}$ for $\abs{c/(ab)} < 1$.
Its terminating case leads to the [[q-Vandermonde identity|MBRSCQRGGH]].
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# Second fundamental form

## Definition

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]], let $x \colon U \to S$ be a local parametrisation, and let $N = x_u \times x_v / \norm{x_u \times x_v}$ be the unit [[normal vector|0BJDQ2HBH1]].
The coefficients of the *second fundamental form* of $S$ with respect to $x$ are
$$e = \ip{x_{uu}}{N}, \qquad f = \ip{x_{uv}}{N}, \qquad g = \ip{x_{vv}}{N}.$$
At $p = x(q)$, the second fundamental form is the quadratic form $I\!I_p$ on the tangent plane $T_p S$ with $I\!I_p(a x_u + b x_v) = e a^2 + 2 f ab + g b^2$.

## Remarks

One writes $I\!I = e \, du^2 + 2 f \, du \, dv + g \, dv^2$.
Equivalently, $I\!I_p(w) = -\ip{dN_p(w)}{w}$, where $dN_p$ is the differential of the unit normal field; this shows that $I\!I_p$ depends only on $S$ and the choice of $N$.
Replacing $N$ by $-N$ changes the sign of $I\!I$.
Example: for the cylinder $x(\theta, z) = (r \cos \theta, r \sin \theta, z)$ with the inner normal $N = -(\cos \theta, \sin \theta, 0)$, one gets $e = r$, $f = 0$ and $g = 0$.
For the plane, $x_{uu}$, $x_{uv}$ and $x_{vv}$ vanish for a linear parametrisation, so $I\!I = 0$.
Non-example: on a Möbius strip there is no continuous unit normal on the whole surface, so the second fundamental form is defined only locally, up to sign.
For $w \in T_p S$ with $I_p(w) = 1$, the number $I\!I_p(w)$ is the normal curvature of $S$ in the direction $w$, where $I_p$ is the [[first fundamental form|MAJ3DH7MZS]].
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# Order polytope

## Definition

Let $(P, \le)$ be a finite [[partially ordered set|0QV2JS9RJ3]], and let $\R^P$ be the real vector space of functions $f \colon P \to \R$.
The *order polytope* of $P$ is the set $\mathcal{O}(P)$ of all $f \in \R^P$ such that:

1. $0 \le f(x) \le 1$ for all $x \in P$.
2. $f(x) \le f(y)$ for all $x, y \in P$ with $x \le y$.

## Remarks

Clause 2 says that $f$ is [[order-preserving|0A7KV2WB7A]].
The vertices of $\mathcal{O}(P)$ are the indicator functions of the [[upper sets|ETCE287E2R]] of $P$.
Its volume is $e(P)/p!$, where $p = \abs{P}$ and $e(P)$ is the number of [[linear extensions|W8DZ5KGGHN]].
For an integer $m \ge 1$, the dilate $m \, \mathcal{O}(P)$ has $\Omega(P, m + 1)$ points with integer coordinates, where $\Omega$ is the [[order polynomial|VDBDJZWPEA]].

Example: for the chain $x < y$, $\mathcal{O}(P)$ is the triangle $0 \le f(x) \le f(y) \le 1$.
Check: its vertices $(0, 0)$, $(0, 1)$, $(1, 1)$ are the indicators of the upper sets $\emptyset$, $\set{y}$, $\set{x, y}$, and its area is $1/2 = e(P)/2!$.

Non-example: the point $f(x) = 1$, $f(y) = 0$ satisfies clause 1 but not clause 2, so it is not in $\mathcal{O}(P)$.

For an antichain of $p$ elements, $\mathcal{O}(P)$ is the unit cube $[0, 1]^p$.
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# Coxeter group

## Definition

Let $I$ be a set.
A *Coxeter matrix* on $I$ is a family $M = (m_{ij})_{i, j \in I}$ with values in $\set{1, 2, 3, \ldots} \cup \set{\infty}$ such that:

1. $m_{ii} = 1$ for all $i \in I$;
2. $m_{ij} = m_{ji}$ for all $i, j \in I$;
3. $m_{ij} \ge 2$ for all $i \ne j$.

The *Coxeter group* of $M$ is the group with the [[presentation|0JGEQXQ4AC]]
$$W(M) = \langle s_i \ (i \in I) \mid (s_i s_j)^{m_{ij}} = 1 \text{ for all } i, j \in I \text{ with } m_{ij} < \infty \rangle.$$
A group is a *Coxeter group* if it is [[isomorphic|08MG7H8JW5]] to $W(M)$ for some Coxeter matrix $M$.
The pair $(W(M), \set{s_i})$ is a *Coxeter system*.

## Remarks

The relation for $i = j$ says $s_i^2 = 1$.
For $m_{ij} = 2$ the relation says that $s_i$ and $s_j$ commute.
In $W(M)$ each $s_i$ has order exactly $2$, and $s_i s_j$ has order exactly $m_{ij}$.
The matrix is often drawn as a Coxeter diagram: a vertex for each $i$, and an edge labelled $m_{ij}$ when $m_{ij} \ge 3$.

Example: the [[symmetric group|0515A6D2Y1]] $S_n$ is a Coxeter group.
Take $s_i = (i \ i{+}1)$ for $1 \le i \le n - 1$, with $m_{i, i+1} = 3$ and $m_{ij} = 2$ for $\abs{i - j} \ge 2$.
The relations hold: $(i \ i{+}1)(i{+}1 \ i{+}2)$ is a $3$-cycle, and disjoint transpositions commute.
A theorem of Moore says that these relations define $S_n$.

Non-example: the cyclic group $\Z/4$ is not a Coxeter group.
A Coxeter group is generated by elements of order $2$, so in its abelianization every element has order at most $2$.
But $\Z/4$ is abelian, so it equals its abelianization, and it has an element of order $4$.
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<document source="https://tamnd.github.io/math2/node/A9B7Z5SA9K.md" tag="A9B7Z5SA9K" kind="example">
# Rectangular function

## Example

The *rectangular function* is $\operatorname{rect} \colon \R \to \R$ with $\operatorname{rect}(x) = 1$ for $\abs{x} < \frac12$, $\operatorname{rect}(x) = \frac12$ for $\abs{x} = \frac12$, and $\operatorname{rect}(x) = 0$ for $\abs{x} > \frac12$.
It is integrable, so its [[Fourier transform|DPVM34ADZV]] is defined by the integral.
For $\xi \ne 0$,
$$\widehat{\operatorname{rect}}(\xi) = \int_{-1/2}^{1/2} e^{-2\pi i x \xi}\, dx = \frac{e^{\pi i \xi} - e^{-\pi i \xi}}{2\pi i \xi} = \frac{\sin(\pi \xi)}{\pi \xi},$$
and $\widehat{\operatorname{rect}}(0) = 1$.
This function is the normalized sinc function, $\operatorname{sinc}(\xi) = \sin(\pi\xi)/(\pi\xi)$.

## Remarks

The transform is continuous and tends to $0$ at infinity, as for every integrable function.
It decays only like $1/\abs{\xi}$ and is not integrable, which reflects the jumps of $\operatorname{rect}$.
So the inversion formula does not hold as an absolutely convergent integral.
It holds in $L^2(\R)$.
As a symmetric improper integral it gives $\frac12$ at $x = \pm\frac12$, the average of the two one-sided limits, which is why the value $\frac12$ is used there.

By the [[convolution theorem|QY8ETVDA1Z]], the triangle function $\operatorname{rect} * \operatorname{rect}(x) = \max(1 - \abs{x}, 0)$ has transform $\operatorname{sinc}^2$, which is integrable.
Read in the other direction, $\operatorname{sinc}$ is a [[band-limited function|P3T104G564]] whose transform is $\operatorname{rect}$.

## Explanations

### L1 human

The rectangular function is a box: it switches on at $-\frac12$ and off at $\frac12$.
Its frequency content is the sinc wave $\sin(\pi\xi)/(\pi\xi)$, a central peak with slowly shrinking ripples.
Sharp edges in time need many high frequencies, which is why the ripples die out so slowly.
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# Packing density

## Definition

Let $n \ge 1$ and let $\operatorname{vol}$ be the [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$.
Let $B_R$ be the [[closed ball|0Q0H13G8ZB]] of radius $R$ centred at $0$.
A *packing* in $\R^n$ is a countable family $\mathcal P = (K_i)_{i \in I}$ of compact sets with pairwise disjoint interiors.
The *upper density* of $\mathcal P$ is
$$\delta(\mathcal P) = \limsup_{R \to \infty} \frac{\sum_{i \in I} \operatorname{vol}(K_i \cap B_R)}{\operatorname{vol}(B_R)},$$
with the [[limit superior|0T3RZQXDYD]]; when the limit exists it is the *density* of $\mathcal P$.
For a compact set $K$ with nonempty interior, the *packing density* of $K$ is $\delta(K) = \sup \delta(\mathcal P)$, over all packings $\mathcal P$ of congruent copies of $K$.

## Remarks

The value lies in $[0, 1]$.
Example: the packing of discs of radius $\tfrac12$ centred at the points of $\Z^2$.
Each disc lies in its unit square, the squares tile the plane, so the ratio tends to $\pi/4$, the share of each square covered by its disc.
Non-example: discs of radius $1$ centred at $\Z^2$ do not form a packing, since neighbouring discs overlap in a set of positive area.
For them the same ratio tends to $\pi > 1$.
For a lattice packing $K + \Lambda$ of a body $K$, the density exists and equals $\operatorname{vol}(K)/\det \Lambda$.
Restricting to lattice packings gives the lattice packing density $\delta_L(K) \le \delta(K)$.
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# Banach limit

## Definition

Let $\ell^\infty$ be the vector space of bounded real sequences $x = (x_n)_{n \ge 1}$, as in the [[spaces ℓ^p|08P6T2WH6R]].
Let $S \colon \ell^\infty \to \ell^\infty$ be the shift, $S(x_1, x_2, x_3, \dots) = (x_2, x_3, \dots)$, and let $\mathbf{1} = (1, 1, 1, \dots)$.
A *Banach limit* is a linear map $\varphi \colon \ell^\infty \to \R$ such that:

1. $\varphi(x) \ge 0$ for every $x$ with $x_n \ge 0$ for all $n$.
2. $\varphi(Sx) = \varphi(x)$ for every $x \in \ell^\infty$.
3. $\varphi(\mathbf{1}) = 1$.

## Remarks

Every Banach limit satisfies $\liminf_n x_n \le \varphi(x) \le \limsup_n x_n$.
Check of the left inequality: if $x_n \ge c$ for all $n \ge N$, then $S^N x - c \mathbf{1}$ has nonnegative terms, so $\varphi(x) = \varphi(S^N x) \ge c$.
Hence $\varphi(x) = \lim_n x_n$ for every convergent $x$, and $\abs{\varphi(x)} \le \sup_n \abs{x_n}$, so $\varphi$ is continuous of norm $1$.
Banach limits exist by the [[Hahn–Banach theorem|QB5AE3VYZZ]], applied to the sublinear functional $p(x) = \limsup_{m} \frac{1}{m} \sum_{k=1}^{m} x_k$ or a similar one.

Non-example: $\varphi(x) = \limsup_n x_n$ satisfies clauses 1, 2 and 3, but it is not linear.
For $x = (1, 0, 1, 0, \dots)$ and $y = Sx$ one has $\varphi(x) + \varphi(y) = 2$ but $\varphi(x + y) = \varphi(\mathbf{1}) = 1$.

Non-example: $\varphi(x) = x_1$ is linear and positive with $\varphi(\mathbf{1}) = 1$, but it fails clause 2.
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# Quasiconformal mapping

## Definition

Let $U$ and $V$ be [[open sets|02BH3MCBQB]] in $\C$ and let $K \ge 1$.
Write $f_z = \frac{1}{2}(f_x - i f_y)$ and $f_{\bar z} = \frac{1}{2}(f_x + i f_y)$ for the [[Wirtinger derivatives|WMXDVF36ZT]] of a function $f$, where $f_x$ and $f_y$ are its partial derivatives.
A map $f \colon U \to V$ is *$K$-quasiconformal* if:

1. $f$ is an orientation-preserving [[homeomorphism|0QT00PB5G2]] of $U$ onto $V$.
2. $f$ has locally square-integrable weak first partial derivatives on $U$, that is, $f$ lies in the Sobolev space $W^{1,2}_{\mathrm{loc}}(U)$.
3. $\abs{f_{\bar z}} \le k \abs{f_z}$ almost everywhere on $U$, where $k = (K - 1)/(K + 1)$.

A map is *quasiconformal* if it is $K$-quasiconformal for some $K \ge 1$.

## Remarks

Example: for $0 \le k < 1$, the real-linear map $f(z) = z + k \bar z$ has $f_z = 1$ and $f_{\bar z} = k$.
Its Jacobian determinant is $\abs{f_z}^2 - \abs{f_{\bar z}}^2 = 1 - k^2 > 0$, so it is an orientation-preserving homeomorphism of $\C$, and it is $K$-quasiconformal with $K = (1 + k)/(1 - k)$.
It maps the unit circle onto an ellipse with axis ratio $(1 + k)/(1 - k)$.
Non-example: complex conjugation $f(z) = \bar z$ is a homeomorphism, but $f_z = 0$ and $f_{\bar z} = 1$, so clause 3 fails for every $K$; it reverses orientation.
The quotient $\mu_f = f_{\bar z} / f_z$ is the complex dilatation of $f$, and $K_f = (1 + \norm{\mu_f}_\infty)/(1 - \norm{\mu_f}_\infty)$ is its maximal dilatation.
A map is $1$-quasiconformal if and only if it is conformal.
An equivalent geometric definition uses the moduli of quadrilaterals or annuli: $f$ is $K$-quasiconformal when it distorts each [[conformal modulus|2FQAKBVBX5]] by a factor of at most $K$.
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# Thom–Boardman singularities

## Summary

The Thom–Boardman singularities sort the [[singular points|1JG3HFWX3T]] of a smooth map by a sequence of numbers $(i_1, i_2, \ldots)$.
The first number is the dimension of the kernel of the differential.
The second is the same dimension for the restriction of the map to the set of points with that first number, and so on.
Thom introduced the first steps, and Boardman made the construction work in general by defining the corresponding subsets of jet space.
For a generic map the sets $\Sigma^{i_1, i_2, \ldots}(f)$ are submanifolds of known codimension, and they give a first coarse classification of singularities.

## Explanations

### L1 human

For a typical map between manifolds, first look at how many directions the map crushes at each point; that number is $i_1$.
The points where exactly $i_1$ directions are crushed form a curve, a surface, or some other smaller manifold.
Now restrict the map to this smaller manifold and ask again how many directions it crushes there; that number is $i_2$.
For a map of the plane to the plane, a fold point has symbol $(1, 0)$: one direction is crushed, but the map is regular along the fold curve.
A cusp point has symbol $(1, 1, 0)$: the map also crushes the direction along the fold curve at that point.
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# Controllability

## Summary

A [[control system|ZPDTVZQN5K]] is controllable if every state can be steered to every other state by a suitable choice of the control.
It is the first question of control theory: before asking how to steer well, one asks whether steering is possible at all.
For linear systems $x' = A x + B u$ it is decided by the [[Kalman rank condition|9NSCZ2KBEB]], and for systems without drift by the [[Chow–Rashevskii theorem|VQPN24VDH1]].

## Explanations

### L1 human

Imagine a cart on a rail that can be pushed forward or backward.
By pushing at the right times, one can bring it to any position with any velocity, so the cart is controllable.
If instead the push only acts on a second cart that is not attached to the first, the first cart cannot be moved, and the system is not controllable.
Controllability asks whether the inputs reach every direction of the state, directly or through the dynamics.
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# Littlewood–Paley theory

## Summary

Littlewood–Paley theory splits a function on $\R^n$ into pieces whose [[Fourier transforms|DPVM34ADZV]] live in dyadic annuli $\set{2^{j-1} < \abs{\xi} < 2^{j+1}}$, by the [[Littlewood–Paley decomposition|D1PAT0K446]].
Its main result is the [[Littlewood–Paley inequality|6GYQNB2862]].
It says that, for $1 < p < \infty$, the $L^p$ norm of $f$ is comparable to that of its [[square function|H4E1PW67Z1]] $\left( \sum_j \abs{\Delta_j f}^2 \right)^{1/2}$.
So the dyadic pieces behave in $L^p$ almost like orthogonal pieces in $L^2$.
The theory underlies multiplier theorems, the description of function spaces such as Sobolev, Besov and Triebel–Lizorkin spaces, and paraproduct decompositions of products.

## Explanations

### L1 human

Think of a sound signal split into octaves: low notes, then notes one octave higher, then the next octave, and so on.
In $L^2$, the energy of the signal is the sum of the energies of the octaves, by Plancherel's theorem.
Littlewood–Paley theory says that a similar statement survives for the $L^p$ norms with $1 < p < \infty$.
One only has to combine the octaves through a square root of a sum of squares.
For example, changing the signs of the octave pieces in any way only changes the $L^p$ norm by a bounded factor.
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# Linear difference equation

## Summary

A linear difference equation is a recurrence in which the unknown sequence appears only linearly: $a_k(n) u_{n+k} + \dots + a_0(n) u_n = b(n)$.
The coefficients may depend on $n$; when they are constant one gets a [[linear recurrence with constant coefficients|0NDVHBVBXB]].
Its solutions have the same structure as those of a linear differential equation.
The solutions of the homogeneous equation form a vector space, and the general solution is one particular solution plus a homogeneous solution.

## Explanations

### L1 human

In a linear difference equation each term of the sequence is a weighted sum of earlier terms, plus perhaps an outside input.
No products or powers of the unknown terms appear.
The Fibonacci rule $u_{n+2} = u_{n+1} + u_n$ is linear with constant weights.
The rule $u_{n+1} = (n + 1) u_n$, which produces the factorials from $u_0 = 1$, is linear with a weight that changes with $n$.
The rule $u_{n+1} = u_n^2$ is not linear.
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# Class field theory

## Definition

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\mathcal{O}_K$.
A *modulus* of $K$ is a formal product $\mathfrak{m} = \mathfrak{m}_0 \mathfrak{m}_\infty$ of a nonzero ideal $\mathfrak{m}_0$ of $\mathcal{O}_K$ and a set $\mathfrak{m}_\infty$ of real embeddings of $K$.
Let $I_K^{\mathfrak{m}}$ be the group of [[fractional ideals|YB4JTB4WJN]] of $K$ coprime to $\mathfrak{m}_0$.
Let $P_K^{\mathfrak{m}}$ be its [[subgroup|0X1JXT47GA]] of principal ideals $(\alpha)$ with $\alpha \in K^\times$ such that $v_{\mathfrak{p}}(\alpha - 1) \ge v_{\mathfrak{p}}(\mathfrak{m}_0)$ for every prime $\mathfrak{p} \mid \mathfrak{m}_0$ and $\sigma(\alpha) > 0$ for every $\sigma \in \mathfrak{m}_\infty$.
A *congruence subgroup* modulo $\mathfrak{m}$ is a subgroup $H$ with $P_K^{\mathfrak{m}} \subseteq H \subseteq I_K^{\mathfrak{m}}$; the quotient $I_K^{\mathfrak{m}}/P_K^{\mathfrak{m}}$ is the *ray class group* modulo $\mathfrak{m}$.
Let $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]].
It is the *class field* of $H$ if the following holds for every nonzero prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ coprime to $\mathfrak{m}_0$.
The prime $\mathfrak{p}$ [[splits completely|B7KNZCBFYT]] in $L$ if and only if $\mathfrak{p} \in H$.
Class field theory is the theory of class fields of congruence subgroups.

## Remarks

The main theorems are as follows.
Each congruence subgroup $H$ has a class field $L$, unique inside a fixed algebraic closure.
This $L/K$ is [[abelian|082GHNPJ53]] and unramified at the places not dividing $\mathfrak{m}$.
The [[Artin map|7938EKMMVR]] gives $\Gal(L/K) \cong I_K^{\mathfrak{m}}/H$.
Every finite abelian extension of $K$ is the class field of some congruence subgroup.
The ray class group is finite, and its order is a multiple of the class number.

Example: $K = \Q$, $\mathfrak{m} = n\infty$ with $n \ge 3$, and $H = P_{\Q}^{\mathfrak{m}}$.
For a prime $p \nmid n$, the ideal $(p)$ has the unique positive generator $p$, so $(p) \in H$ if and only if $p \equiv 1 \pmod n$.
This is exactly the condition for $p$ to split completely in the [[cyclotomic field|NN77VYSGXR]] $\Q(\zeta_n)$, so $\Q(\zeta_n)$ is the class field of $H$.

Non-example: for $\mathfrak{m} = 5$ without the infinite place, $(p) = (-p)$ lies in $P_{\Q}^{\mathfrak{m}}$ when $p \equiv \pm 1 \pmod 5$.
So $\Q(\zeta_5)$ is not the class field of $P_{\Q}^{\mathfrak{m}}$: the prime $19 \equiv -1 \pmod 5$ gives $(19) \in P_{\Q}^{\mathfrak{m}}$, but $19$ does not split completely in $\Q(\zeta_5)$.
The class field of this group is the real subfield $\Q(\zeta_5 + \zeta_5^{-1}) = \Q(\sqrt{5})$.
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# Geometry of numbers

## Summary

The geometry of numbers studies [[lattices|WN9V7WSJKF]] in $\R^n$ together with convex bodies and other regions, and uses volume arguments to find lattice points in them.
It was founded by Hermann Minkowski, whose [[convex body theorem|0K387S2PKQ]] says that a large symmetric convex set contains a nonzero lattice point.
Its main tools include [[Blichfeldt's theorem|A3FTHW4NES]], [[successive minima|YDMBKD8FP2]] and [[Minkowski's second theorem|5H4XZVF9B9]].
It gives results in algebraic number theory, Diophantine approximation and quadratic forms, and it is closely tied to sphere packing.

## Explanations

### L1 human

Many questions about whole numbers can be turned into questions about points of a grid.
The geometry of numbers answers them by looking at shapes and their areas or volumes.
If a shape is large enough and symmetric about the origin, it must contain a grid point other than the origin.
For instance, this idea shows that every prime of the form $4k + 1$ is a sum of two squares.
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<document source="https://tamnd.github.io/math2/node/AABCR83PXG.md" tag="AABCR83PXG" kind="example">
# Laplacian eigenvalues of the round sphere

## Example

Let $S^n \subset \R^{n+1}$ be the unit [[sphere|P9CM4GQ2T0]] with its round metric, $n \ge 1$, and let $\Delta = -\operatorname{div} \operatorname{grad}$ be its [[Laplace–Beltrami operator|F4NWZNTRMD]].
Its eigenvalues are
$$\lambda_k = k(k + n - 1), \quad k = 0, 1, 2, \ldots.$$
The $\lambda_k$-eigenspace consists of the restrictions to $S^n$ of the homogeneous harmonic polynomials of degree $k$ on $\R^{n+1}$, the [[spherical harmonics|3F32Y21HP4]] of degree $k$.
Its dimension is
$$\binom{n + k}{n} - \binom{n + k - 2}{n},$$
with the convention that the second term is $0$ when $k < 2$.

Check of the eigenvalue: write a homogeneous harmonic polynomial of degree $k$ as $P = r^k Y$ with $r = \abs{x}$ and $Y$ a function on $S^n$.
In polar coordinates the Euclidean Laplacian is $\partial_r^2 + \frac{n}{r} \partial_r - \frac{1}{r^2} \Delta$.
Applied to $P$ it gives $r^{k-2} \big( k(k-1) + nk \big) Y - r^{k-2} \Delta Y = 0$, so $\Delta Y = k(k + n - 1) Y$.
These eigenfunctions span a dense subspace of $L^2(S^n)$, so there are no other eigenvalues.

For $S^2$ the eigenvalues are $k(k+1)$ with multiplicity $2k + 1$.
For $k = 1$ the eigenfunctions are the restrictions of the coordinate functions $x_1, \ldots, x_{n+1}$, with eigenvalue $n$.

## Remarks

Since the unit sphere has Ricci curvature $(n - 1)$ times the metric, the first nonzero eigenvalue $n$ shows that the bound in the [[Lichnerowicz–Obata theorem|TK1PT9FDV0]] is attained.
The multiplicities grow like a constant times $k^{n-1}$, which agrees with [[Weyl's law|2JK9YMZV2B]].

## Explanations

### L1 human

The sphere vibrates only at a discrete list of frequencies, and the list can be written down exactly.
The $k$-th frequency comes from polynomials of degree $k$, such as $x$ for $k = 1$ or $xy$ for $k = 2$.
On the ordinary sphere $S^2$ the values are $0, 2, 6, 12, \ldots$, with $1, 3, 5, 7, \ldots$ independent vibration patterns.
These patterns are the spherical harmonics used for orbitals in chemistry and for maps of the Earth's gravity.
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<document source="https://tamnd.github.io/math2/node/AADNYSB6TC.md" tag="AADNYSB6TC" kind="definition">
# Primitive permutation group

## Definition

Let a group $G$ [[act|0WCY42EV9V]] on a nonempty set $\Omega$.
The action is *primitive* if:

1. it is [[transitive|02GK6HZYBT]], and
2. the only [[block systems|SF738NKWJ0]] of $\Omega$ for $G$ are the partition into singletons and the partition $\set{\Omega}$ with one part.

A [[permutation group|ZDJQRKPWGA]] $G \le \operatorname{Sym}(\Omega)$ is *primitive* if its natural action on $\Omega$ is primitive.

## Remarks

Condition 2 says that every block $B$ of $G$ has $\abs{B} = 1$ or $B = \Omega$.
If $\abs{\Omega} \ge 2$ and $G$ is transitive, then $G$ is primitive if and only if the [[stabiliser|0AYMP41THN]] $G_\omega$ of a point $\omega$ is a maximal subgroup of $G$.

Example: a transitive group $G$ of prime degree $p$.
For a transitive group, all parts of a block system have the same size, which divides $p$, so each block system is trivial.
For instance, the [[cyclic group|0KVGH0J5B3]] generated by $(1\,2\,3\,4\,5)$ is primitive on $\set{1, \dots, 5}$.
Every $2$-transitive group is primitive (see [[multiply transitive group action|BDJ1AX4XBB]]).

Non-example: the eight symmetries of a square acting on its corners $1, 2, 3, 4$ in cyclic order.
The action is transitive, but the partition $\set{\set{1, 3}, \set{2, 4}}$ into the two diagonals is a nontrivial block system, so clause 2 fails.

If $G$ acts primitively and [[faithfully|0P7G9NGG2G]], then every [[normal subgroup|0CB0DX5BVV]] $N \ne 1$ of $G$ is transitive on $\Omega$, because the orbits of $N$ form a block system.
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<document source="https://tamnd.github.io/math2/node/AAECHE30BD.md" tag="AAECHE30BD" kind="concept">
# Fourier series on the torus

## Summary

A Fourier series on the torus $\mathbb{T}^n = \R^n / \Z^n$ expands a function that is periodic in each of $n$ variables into waves.
There is one wave $e^{2\pi i k \cdot x}$ for each integer vector $k \in \Z^n$.
For $n = 1$ it is the classical [[Fourier series|G12KXDHPPS]] of a periodic function.
The coefficients form a function on $\Z^n$.
The passage from $f$ to its coefficients is a case of the [[Fourier transform on locally compact abelian groups|VHN5VW5Q92]], for the compact group $\mathbb{T}^n$.
Convergence in several variables depends on how the partial sums are formed, for example over cubes or over balls in $\Z^n$.
See the [[definition|T4JEMK1549]].

## Explanations

### L1 human

Think of a function on a square whose values on opposite edges match, like a wallpaper pattern that tiles the plane.
Such a pattern can be written as a sum of plane waves, each oscillating a whole number of times across the square in each direction.
The wave with frequency vector $k = (k_1, k_2)$ is $e^{2\pi i (k_1 x_1 + k_2 x_2)}$.
For instance, $\cos(2\pi x_1) \cos(2\pi x_2)$ is the sum of the four waves with $k = (\pm 1, \pm 1)$, each with coefficient $\frac14$.
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<document source="https://tamnd.github.io/math2/node/AAN6HWAC4A.md" tag="AAN6HWAC4A" kind="definition">
# Legendre functions

## Definition

Let $\nu, \mu \in \C$.
The *general Legendre equation* is the [[ordinary differential equation|0NT64JV96A]]
$$(1 - z^2) w'' - 2z w' + \left( \nu(\nu + 1) - \frac{\mu^2}{1 - z^2} \right) w = 0.$$
A *Legendre function* of degree $\nu$ and order $\mu$ is a solution $w$ of this equation on a domain of $\C \setminus \set{-1, 1}$.
The *Legendre function of the first kind* of degree $\nu$ and order $0$ is
$$P_\nu(z) = {}_2F_1\left( -\nu, \nu + 1; 1; \frac{1 - z}{2} \right),$$
where ${}_2F_1$ is the [[hypergeometric function|463X5V31D9]].
The series converges for $\abs{1 - z} < 2$, and $P_\nu$ extends analytically to $\C \setminus (-\infty, -1]$.

## Remarks

For $\nu = n$ a nonnegative integer, the factor $(-n)_k$ vanishes for $k > n$, the series stops, and $P_n$ is the [[Legendre polynomial|QFYGJVA996]] of degree $n$.
Check for $n = 1$: $P_1(z) = 1 + \frac{(-1)(2)}{1} \cdot \frac{1 - z}{2} = z$.
For $\mu = 0$ the equation is [[Legendre's equation|W6VE30V10Y]] with $\lambda = \nu(\nu + 1)$.
A second, independent solution is the Legendre function of the second kind $Q_\nu$; for example $Q_0(x) = \frac12 \ln \frac{1 + x}{1 - x}$ for $-1 < x < 1$.
Non-example: $w(z) = z^2$ is not a Legendre function of order $0$ for any $\nu$, since the equation would require $2 - 6z^2 + \nu(\nu + 1) z^2 = 0$ for all $z$, which fails at $z = 0$.
For general order $\mu$ the functions $P_\nu^\mu$ and $Q_\nu^\mu$ are also given by hypergeometric functions, with a factor $\left( \frac{z + 1}{z - 1} \right)^{\mu/2}$ in front.
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<document source="https://tamnd.github.io/math2/node/AAQAF5GGES.md" tag="AAQAF5GGES" kind="definition">
# Kriging

## Definition

Let $D \subseteq \R^d$ and let $(Z(s))_{s \in D}$ be a real random field of the form $Z(s) = \mu + \delta(s)$.
Here $\mu$ is an unknown constant and $\delta$ is intrinsically stationary with [[variogram|ZF8AA2JVKV]] $2\gamma$ and $E[\delta(s)] = 0$.
Let $s_1, \dots, s_n, s_0 \in D$, and write $\gamma_{ij} = \gamma(s_i - s_j)$ for $0 \le i, j \le n$.

The *ordinary kriging predictor* of $Z(s_0)$ is $\hat Z(s_0) = \sum_{i=1}^{n} \lambda_i Z(s_i)$, where the weights minimise the [[mean squared error|05NZ1HHQMV]] $E[(Z(s_0) - \hat Z(s_0))^2]$ subject to $\sum_{i=1}^{n} \lambda_i = 1$.
Equivalently, $\lambda_1, \dots, \lambda_n$ and a number $m$ solve the linear system
$$\sum_{j=1}^{n} \lambda_j \gamma_{ij} + m = \gamma_{i0} \quad (1 \le i \le n), \qquad \sum_{j=1}^{n} \lambda_j = 1.$$

## Remarks

The constraint $\sum_i \lambda_i = 1$ makes the predictor unbiased, $E[\hat Z(s_0)] = \mu = E[Z(s_0)]$, whatever the value of $\mu$.
The minimal mean squared error, the kriging variance, is $\sum_{i=1}^{n} \lambda_i \gamma_{i0} + m$.
Simple kriging assumes $\mu$ known and drops the constraint; universal kriging replaces $\mu$ by a linear combination of known functions of $s$.

Example: with one observation, $n = 1$, the constraint forces $\lambda_1 = 1$, so $\hat Z(s_0) = Z(s_1)$.
The first equation then gives $m = \gamma_{10} - \gamma_{11} = \gamma(s_1 - s_0)$, since $\gamma(0) = 0$, so the kriging variance is $2\gamma(s_1 - s_0)$, which is indeed $E[(Z(s_0) - Z(s_1))^2]$.

Non-example: inverse distance weighting also predicts by a weighted average with weights summing to $1$.
Its weights are fixed by the distances alone and do not minimise the mean squared error for the given variogram, so it is not kriging.
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<document source="https://tamnd.github.io/math2/node/AAT8PVDW23.md" tag="AAT8PVDW23" kind="definition">
# Regular conditional probability

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]] and let $\mathcal{G} \subseteq \mathcal{F}$ be a sub-[[σ-algebra|0SV8K0TDGY]].
A *regular conditional probability* given $\mathcal{G}$ is a map $\nu \colon \Omega \times \mathcal{F} \to [0, 1]$ such that:

1. for each $\omega \in \Omega$, the map $A \mapsto \nu(\omega, A)$ is a [[probability measure|059TJN0NZ1]] on $(\Omega, \mathcal{F})$;
2. for each $A \in \mathcal{F}$, the map $\omega \mapsto \nu(\omega, A)$ is $\mathcal{G}$-measurable;
3. for each $A \in \mathcal{F}$ and each $B \in \mathcal{G}$, $\int_B \nu(\omega, A) \, P(d\omega) = P(A \cap B)$.

## Remarks

Clauses 2 and 3 say that $\nu(\cdot, A)$ is a version of the [[conditional probability|0990W5PN13]] $P(A \mid \mathcal{G}) = E[\mathbf{1}_A \mid \mathcal{G}]$ for each $A$.
Then $E[Z \mid \mathcal{G}](\omega) = \int Z(\omega') \, \nu(\omega, d\omega')$ almost surely for every integrable $Z$.
A regular conditional probability exists when $(\Omega, \mathcal{F})$ is a standard Borel space, for example a [[Polish space|584GXXY73M]] with its Borel σ-algebra.
It is then unique up to a $P$-null set of $\omega$.
It need not exist in general.
A related notion is the regular conditional distribution of a random variable $Y$ given $X$.
It is a [[Markov kernel|2AVNDP8BJP]] $\kappa$ with $P(X \in A, Y \in B) = \int_A \kappa(x, B) \, P_X(dx)$.
It exists whenever $Y$ takes values in a standard Borel space.
Example: let $\Omega = [0, 1]^2$ with Lebesgue measure, and let $\mathcal{G}$ be generated by the first coordinate. Then $\nu((x, y), A) = \lambda(\set{y' : (x, y') \in A})$, with $\lambda$ the Lebesgue measure on $[0, 1]$, is a regular conditional probability; clause 3 is [[Fubini's theorem|0R0T1VWH5X]].
Non-example: if $\mathcal{G} = \set{\emptyset, \Omega}$ and $0 < P(A) < 1$ for some $A$, then $\nu(\omega, A) = \mathbf{1}_A(\omega)$ satisfies clause 1 but fails clause 2, since $\mathbf{1}_A$ is not constant.
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<document source="https://tamnd.github.io/math2/node/AATQKCWAAC.md" tag="AATQKCWAAC" kind="statement">
# du Bois-Reymond lemma

## Statement

Let $a < b$ and let $f \colon [a, b] \to \R$ be [[continuous|0X3WGX95FE]].
Suppose that
$$\int_a^b f(x) \, \eta'(x) \, dx = 0$$
for every [[continuously differentiable|0VW90BBSJ4]] $\eta \colon [a, b] \to \R$ with $\eta(a) = \eta(b) = 0$.
Then $f$ is constant on $[a, b]$.

## Remarks

Let $c = \frac{1}{b - a} \int_a^b f \, dx$ be the mean of $f$, and set $\eta(x) = \int_a^x (f(s) - c) \, ds$.
Then $\eta$ is $C^1$, $\eta(a) = \eta(b) = 0$ and $\eta' = f - c$.
The hypothesis gives $\int_a^b f (f - c) \, dx = 0$, and $\int_a^b c (f - c) \, dx = 0$ by the choice of $c$.
Subtracting, $\int_a^b (f - c)^2 \, dx = 0$, so $f = c$ by continuity.

A common extension: if $f, g$ are continuous and $\int_a^b (g \eta + f \eta') \, dx = 0$ for all such $\eta$, then $f$ is $C^1$ and $f' = g$.
To see it, apply the lemma to $f - G$, where $G(x) = \int_a^x g(s) \, ds$, after an integration by parts of $\int_a^b g \eta \, dx$.

Example: for $f(x) = x$ on $[0, 1]$ and $\eta(x) = x(1 - x)$, one finds $\int_0^1 x (1 - 2x) \, dx = -1/6 \ne 0$, as the lemma predicts for a non-constant $f$.

## Explanations

### L1 human

If $f$ gives zero against the slope of every function that starts and ends at zero, then $f$ cannot vary.
The reason is that one can build a test function whose slope is exactly $f$ minus its average.

### L2 human

The extension above gives the Euler–Lagrange equation in integrated form, $L_p(x, y, y') = \int_a^x L_z(s, y, y') \, ds + C$, for a $C^1$ minimiser $y$, without assuming that $y$ has a second derivative.
It follows that $x \mapsto L_p(x, y(x), y'(x))$ is $C^1$, which is the start of regularity theory in one variable.
The [[fundamental lemma|JNDZ86CV3Y]] is the version where $\eta$ itself, instead of $\eta'$, appears in the integral.
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<document source="https://tamnd.github.io/math2/node/AAV343QP5E.md" tag="AAV343QP5E" kind="definition">
# Presentation of an algebra

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]], let $X$ be a set, and let $F = F_{\mathcal{V}}(X)$ be the [[free algebra|GTS4DRRMZR]] of $\mathcal{V}$ on $X$.
Let $R$ be a set of pairs $(s, t)$ of elements of $F$, called *relations* and written $s = t$.
Let $\theta_R$ be the [[congruence generated|QP06QH2Q7C]] by $R$ on $F$.
The algebra *presented* by $X$ and $R$ in $\mathcal{V}$ is the [[quotient algebra|N5KE7FH6CX]]
$$\langle X \mid R \rangle_{\mathcal{V}} = F / \theta_R.$$
A *presentation* of an algebra $A \in \mathcal{V}$ is a pair $(X, R)$ with an [[isomorphism|PZMGQMDN25]] $\langle X \mid R \rangle_{\mathcal{V}} \to A$.
The algebra $A$ is *finitely presented* in $\mathcal{V}$ if it has a presentation with $X$ and $R$ finite.

## Remarks

The presented algebra has this universal property.
Let $B \in \mathcal{V}$ and let $f \colon X \to B$ be a map under which each relation $s = t$ holds in $B$.
Then there is exactly one homomorphism $\langle X \mid R \rangle_{\mathcal{V}} \to B$ that extends $f$.
Every $A \in \mathcal{V}$ has a presentation: take $X = A$ and $R$ the kernel of the homomorphism $F_{\mathcal{V}}(A) \to A$ that extends the identity map.
Example: in the variety of commutative monoids, $\langle a \mid a^2 = a^3 \rangle$ is the monoid $\set{1, a, a^2}$ with $a^3 = a^2$.
The relation holds there, and the universal property is checked by sending $a$ to any element $b$ with $b^2 = b^3$.
Non-example: the same pair $(\set{a}, \set{a^2 = a^3})$ does not present $\Z / 2\Z$ as a commutative monoid, since in $\Z / 2\Z = \set{1, a}$ one has $a^2 = 1$ and $a^3 = a$, so the relation $a^2 = a^3$ fails.
The presented algebra depends on the variety: $\langle a, b \mid \ \rangle$ is $\N^2$ in commutative monoids and the free monoid on two letters in monoids.
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<document source="https://tamnd.github.io/math2/node/AAYEQE.md" tag="AAYEQE" kind="computation">
# The subgroups of the symmetric group S4

## Description

The program in GAP lists each subgroup of $S_4$, the [[group|G5S1M9]] of the permutations of $\{1, 2, 3, 4\}$. It writes each subgroup with the sorted list of its elements to the certificate, and it prints the number of subgroups of each order.

The checker is a short Python program that does not use GAP. It finds the subgroups of $S_4$ again with its own search. It starts with the trivial subgroup and adds the subgroup that a known subgroup and one more element generate, until it finds no new subgroup. It accepts the certificate only when the lines of the certificate are these subgroups, each one time, and the order of each one divides 24.

## Remarks

The orders divide 24, as [[Lagrange's theorem|3C53TD]] says. The converse of the theorem is not true for each finite group, but it is true for $S_4$. For each divisor $d$ of 24, $S_4$ has a subgroup of order $d$. The alternating group $A_4$ of order 12 has no subgroup of order 6, so the converse is false for $A_4$.
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<document source="https://tamnd.github.io/math2/node/AB22FWBQ0M.md" tag="AB22FWBQ0M" kind="problem">
# Tarski's finite basis problem

## Problem

A finite algebra $A$ of finite signature is given by the finite tables of its operations.
It is [[finitely based|YGYZAQ12MZ]] if it has a finite [[equational basis|9Q008GMZSW]].
That is, some finite set of [[identities|PKD440G2XW]] true in $A$ implies every identity true in $A$.
Tarski's finite basis problem asks for an algorithm, that is, a [[computable function|KZSB5E471K]], with the following task.
Given a finite algebra of finite signature, it decides whether the algebra is finitely based.

## Remarks

The problem was solved negatively by Ralph McKenzie in 1996: no such algorithm exists.
Earlier work showed that the answer to "is every finite algebra finitely based?" is no.
Lyndon showed that every two-element algebra of finite signature is finitely based, and he also gave a seven-element algebra that is not.
Many classes of finite algebras are known to consist of finitely based algebras, among them finite groups, by the [[Oates–Powell theorem|5RFPT46DXX]], and finite lattices.
[[Baker's finite basis theorem|35G4HQG2FN]] covers all finite algebras of finite signature that generate a congruence-distributive variety.
The decision problem restricted to finite semigroups is still open.

## Explanations

### L1 human

The identities true in a finite algebra form an infinite list.
Sometimes finitely many of them imply all the others, and sometimes they do not.
Tarski asked whether a computer could always tell which case happens, just by looking at the tables of the operations.
McKenzie proved that no computer program can do this for all finite algebras.
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<document source="https://tamnd.github.io/math2/node/AB3ATTS48V.md" tag="AB3ATTS48V" kind="statement">
# Log-convexity of the gamma function

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]] on $(0, \infty)$.
Then $\Gamma$ is [[logarithmically convex|E97P413DKM]]: the function $x \mapsto \ln \Gamma(x)$ is convex on $(0, \infty)$.
Equivalently, for all $x, y > 0$ and $\lambda \in [0, 1]$,
$$\Gamma(\lambda x + (1 - \lambda) y) \le \Gamma(x)^{\lambda} \, \Gamma(y)^{1 - \lambda}.$$

## Remarks

Write $t^{\lambda x + (1-\lambda) y - 1} e^{-t} = \big(t^{x-1}e^{-t}\big)^{\lambda}\big(t^{y-1}e^{-t}\big)^{1-\lambda}$ and apply [[Hölder's inequality|JQ9WXC729D]] with exponents $1/\lambda$ and $1/(1-\lambda)$ to Euler's integral; this gives the inequality.
A second proof: the second derivative is $(\ln\Gamma)''(x) = \sum_{n \ge 0} \frac{1}{(x + n)^2} > 0$, so $\ln \Gamma$ is even strictly convex.
Example: with $x = 1$, $y = 3$ and $\lambda = \tfrac12$ the inequality says $\Gamma(2) = 1 \le \sqrt{\Gamma(1)\Gamma(3)} = \sqrt{2}$.
Log-convexity implies ordinary convexity of $\Gamma$, so $\Gamma$ has a single minimum on $(0, \infty)$, near $x \approx 1.4616$.
In Mathlib the statement is `Real.convexOn_log_Gamma`.

## Explanations

### L1 human

The graph of $\ln \Gamma$ on the positive axis bends upward everywhere.
This is a strong form of convexity for $\Gamma$ itself.

### L2 human

Log-convexity is the property that singles out $\Gamma$ among all functions with $f(1) = 1$ and $f(x + 1) = x f(x)$: this is the [[Bohr–Mollerup theorem|QFF87AXWN2]].
It gives inequalities such as Gautschi's bounds for ratios $\Gamma(x + s)/\Gamma(x + 1)$.
The positivity of $(\ln\Gamma)''$ says that the [[digamma function|HWRRXB0YXS]] is strictly increasing on $(0, \infty)$.
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<document source="https://tamnd.github.io/math2/node/AB3S6MHJ8X.md" tag="AB3S6MHJ8X" kind="concept">
# Condition number

## Summary

A condition number measures how much the output of a problem can change, relative to the size of a small relative change in its input.
For an [[invertible matrix|0EB9N0S33C]] $A$ and a [[matrix norm|54T6Y6DSY4]], it is $\kappa(A) = \norm{A} \norm{A^{-1}}$.
It bounds the relative error in the solution of $Ax = b$ caused by a relative error in $b$.
A problem with a large condition number is ill-conditioned: rounding errors in the data can be strongly amplified, whatever algorithm is used.

## Explanations

### L1 human

Some problems are sensitive: a tiny change in the data gives a big change in the answer.
The condition number puts a number on this sensitivity.
For a linear system $Ax = b$, a condition number of $1000$ means that a $0.1\%$ error in $b$ can cause up to a $100\%$ error in $x$.
For example, the diagonal matrix with entries $1$ and $0.001$ has condition number $1000$ in the Euclidean norm.
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<document source="https://tamnd.github.io/math2/node/AB7W6DBA4P.md" tag="AB7W6DBA4P" kind="definition">
# Singular integral equation

## Definition

Let $\Gamma$ be a finite union of disjoint smooth closed curves in $\C$.
Let $a$, $b$ and $f$ be [[Hölder continuous|012PBA0FMD]] functions on $\Gamma$, and let $k \colon \Gamma \times \Gamma \to \C$ be Hölder continuous.
A *singular integral equation* with Cauchy kernel on $\Gamma$ is an equation
$$a(t) \, \varphi(t) + \frac{b(t)}{\pi i} \, \mathrm{p.v.} \int_\Gamma \frac{\varphi(\tau)}{\tau - t} \, d\tau + \int_\Gamma k(t, \tau) \, \varphi(\tau) \, d\tau = f(t), \quad t \in \Gamma,$$
for an unknown Hölder continuous function $\varphi$ on $\Gamma$.
Here $\mathrm{p.v.}$ is the [[Cauchy principal value|0SYR10HD4R]]: the limit as $\varepsilon \to 0$ of the integral over the part of $\Gamma$ outside the disc $\abs{\tau - t} < \varepsilon$.
The equation is *of normal type* when $a(t)^2 - b(t)^2 \ne 0$ for all $t \in \Gamma$.
The equation with $k = 0$ is its *dominant* or *characteristic* equation.

## Remarks

Example: on the unit circle, with $a = 0$, $b = 1$ and $k = 0$, the equation $\frac{1}{\pi i} \, \mathrm{p.v.} \int_\Gamma \frac{\varphi(\tau)}{\tau - t} \, d\tau = f(t)$ is of normal type, since $a^2 - b^2 = -1$.
For $\varphi(\tau) = \tau^n$ with $n \ge 0$ the left side is $t^n$, and for $n < 0$ it is $-t^n$; so $f(t) = t^2$ is solved by $\varphi(\tau) = \tau^2$.
Non-example: if the kernel is $k(t, \tau)$ alone, continuous, with $b = 0$ and $a = 1$, the equation is a Fredholm equation of the second kind and is not singular.
The term $\frac{1}{\tau - t}$ makes the integral divergent as an ordinary integral at $\tau = t$ unless $\varphi(t) = 0$; the Hölder condition makes the principal value exist.
More generally, the name covers equations with other kernels that are integrated as principal values, including kernels on open arcs and in several variables.
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<document source="https://tamnd.github.io/math2/node/AB9FNZD18B.md" tag="AB9FNZD18B" kind="statement">
# Theorem of the three geodesics

## Statement

Let $g$ be a smooth [[Riemannian metric|Q61VFHCH2J]] on the two-dimensional [[sphere|P9CM4GQ2T0]] $S^2$.
Then $(S^2, g)$ has at least three geometrically distinct simple [[closed geodesics|4MCCP8GG4J]], where simple means without self-intersections.

## Remarks

The idea goes back to Lyusternik and Schnirelmann.
Consider the space of unoriented simple closed curves on $S^2$, together with the point curves.
Its topology is rich enough to force three minimax levels, in the spirit of [[Lusternik–Schnirelmann theory|QARH15D2PN]].
The difficulty is to deform curves by a length-decreasing process that keeps them simple; this caused gaps in the early arguments.
One complete proof uses the [[curve-shortening flow|DVGFRRYK10]], which preserves simplicity and, by a theorem of Grayson, either shrinks a curve to a point or converges to a closed geodesic.

Example: on a round sphere every great circle is a simple closed geodesic, so there are infinitely many.
On an ellipsoid in $\R^3$ with three distinct axes, the three principal sections in the coordinate planes are simple closed geodesics.
The bound three cannot be improved in general.

## Explanations

### L1 human

However one deforms a sphere smoothly, it still carries at least three different closed loops that go straight ahead and never cross themselves.
On a round sphere there are infinitely many such loops, the great circles.
On an egg-shaped surface with three different axes there are typically exactly three, the waists around the three axes.

### L2 human

The theorem is a sharp lower bound for simple closed geodesics.
When self-intersections are allowed, the theorem of [[Bangert and Franks|X93AEE88PX]] gives infinitely many closed geodesics on any Riemannian $S^2$.
Its proof is a model of Lusternik–Schnirelmann theory on a space of curves: three independent families of curves sweep out the sphere, and each family yields a critical level.
The first step, a single closed geodesic on $S^2$, is the [[Birkhoff min-max method|1R5X9HW6D3]].
</document>

<document source="https://tamnd.github.io/math2/node/ABE41ZYD78.md" tag="ABE41ZYD78" kind="definition">
# Lie sphere geometry

## Definition

Let $n \ge 1$ and give $\R^{n+3}$, with coordinates $(x_1, \ldots, x_{n+3})$, the symmetric [[bilinear form|0SQZMXB6GP]] of signature $(n+1, 2)$
$$\ip{x, y} = -x_1 y_1 + x_2 y_2 + \cdots + x_{n+2} y_{n+2} - x_{n+3} y_{n+3}.$$
The *Lie quadric* is the [[quadric|Q3Q4FF23VG]] $Q = \set{[x] \in \mathbb{RP}^{n+2} : \ip{x, x} = 0}$ in the [[projective space|ZT9A6JK8QF]] $\mathbb{RP}^{n+2}$.
The points of $Q$ correspond bijectively to the following objects of $\R^n$:

1. the oriented [[sphere|P9CM4GQ2T0]] with centre $p \in \R^n$ and signed radius $r \ne 0$, and the point $p$ for $r = 0$, to $\bigl[ \tfrac{1 + \abs{p}^2 - r^2}{2}, \tfrac{1 - \abs{p}^2 + r^2}{2}, p, r \bigr]$;
2. the oriented hyperplane $\set{x : x \cdot N = h}$ with unit normal $N$ and $h \in \R$, to $[h, -h, N, 1]$;
3. the point at infinity, to $[1, -1, 0, 0]$.

Two such objects are in *oriented contact* when their points $[x], [y] \in Q$ satisfy $\ip{x, y} = 0$.
A *Lie sphere transformation* is a projective transformation of $\mathbb{RP}^{n+2}$ that maps $Q$ onto itself.
*Lie sphere geometry* is the study of the properties of these objects that are invariant under Lie sphere transformations.

## Remarks

For two oriented spheres with centres $p_1, p_2$ and signed radii $r_1, r_2$, let $k_1, k_2$ be their vectors as in clause 1; a computation gives
$$\ip{k_1, k_2} = -\tfrac{1}{2} \bigl( \abs{p_1 - p_2}^2 - (r_1 - r_2)^2 \bigr).$$
So they are in oriented contact exactly when $\abs{p_1 - p_2} = \abs{r_1 - r_2}$, that is when they are tangent with matching orientations at the point of contact.
Example: in $\R^2$, take the circle with centre $(0, 0)$ and radius $2$ and the circle with centre $(1, 0)$ and radius $1$.
They have $\abs{p_1 - p_2} = 1 = \abs{2 - 1}$, so they are in oriented contact; indeed they touch at $(2, 0)$.
Non-example: the circles of radius $1$ with centres $(0, 0)$ and $(1, 0)$ have $\abs{p_1 - p_2} = 1 \ne 0 = \abs{r_1 - r_2}$; they cross at two points and are not in oriented contact.
A point ($r = 0$) is in oriented contact with a sphere exactly when it lies on it, and every hyperplane is in oriented contact with the point at infinity.
The group of Lie sphere transformations is $O(n+1, 2) / \set{\pm 1}$.
The Möbius transformations of $\R^n \cup \set{\infty}$ are the Lie sphere transformations that map points to points.
</document>

<document source="https://tamnd.github.io/math2/node/ABEFWF4JHH.md" tag="ABEFWF4JHH" kind="concept">
# Standard Borel space

## Summary

A standard Borel space is a measurable space whose measurable sets are the Borel sets of some Polish topology.
These are the measurable spaces met in practice, such as the real line, $[0, 1]$ and spaces of sequences.
By a theorem of Kuratowski, two standard Borel spaces are isomorphic exactly when they have the same cardinality.

## Explanations

### L1 human

Many different-looking spaces carry the same measurable structure.
The real line, the unit interval and the plane look different as topological spaces, but as measurable spaces with their Borel sets they are all the same.
A standard Borel space is a measurable space that comes from a nice topological space in this way.
Up to isomorphism there are only few of them: the finite ones, the countably infinite one and the one of the real line.
</document>

<document source="https://tamnd.github.io/math2/node/ABJ88GR4Z0.md" tag="ABJ88GR4Z0" kind="definition">
# Dunkl operator

## Definition

Let $\R^n$ have the standard inner product $\langle \cdot, \cdot \rangle$.
For $\alpha \in \R^n \setminus \set{0}$, let $\sigma_\alpha(x) = x - 2 \frac{\langle \alpha, x \rangle}{\langle \alpha, \alpha \rangle} \alpha$ be the [[reflection|06BRDF3F0E]] in the hyperplane orthogonal to $\alpha$.
Let $R \subset \R^n \setminus \set{0}$ be a root system: a finite set such that $\sigma_\alpha(R) = R$ and $R \cap \R \alpha = \set{\alpha, -\alpha}$ for every $\alpha \in R$.
Let $W$ be the finite [[reflection group|G6FND7X69J]] generated by the $\sigma_\alpha$, and let $R_+ \subset R$ contain exactly one of $\alpha, -\alpha$ for each $\alpha \in R$.
Let $k \colon R \to \C$ be a multiplicity function, that is, $k(w \alpha) = k(\alpha)$ for all $w \in W$ and $\alpha \in R$.
For $\xi \in \R^n$, the *Dunkl operator* $T_\xi$ acts on smooth functions $f$ on $\R^n$ by
$$T_\xi f(x) = \partial_\xi f(x) + \sum_{\alpha \in R_+} k(\alpha)\, \langle \alpha, \xi \rangle\, \frac{f(x) - f(\sigma_\alpha x)}{\langle \alpha, x \rangle},$$
where $\partial_\xi$ is the [[directional derivative|06KWGHDN2J]] along $\xi$.

## Remarks

Each term of the sum is the same for $\alpha$ and $-\alpha$, so $T_\xi$ does not depend on the choice of $R_+$.
The difference quotient extends smoothly across the hyperplane $\langle \alpha, x \rangle = 0$, so $T_\xi$ maps homogeneous polynomials of degree $d$ to homogeneous polynomials of degree $d - 1$.
Dunkl's theorem states that $T_\xi T_\eta = T_\eta T_\xi$ for all $\xi, \eta \in \R^n$.
For $k = 0$, $T_\xi = \partial_\xi$.

Example: $n = 1$, $R = \set{1, -1}$, $W = \set{x \mapsto x, x \mapsto -x}$ and $k(\pm 1) = k$.
Then $T f(x) = f'(x) + k \frac{f(x) - f(-x)}{x}$.
For $f(x) = x^m$, the quotient is $(1 - (-1)^m) x^{m-1}$, so $T x^m = m x^{m-1}$ for even $m$ and $T x^m = (m + 2k) x^{m-1}$ for odd $m$.

Non-example: the operator $f \mapsto f'(x) + k \frac{f(x)}{x}$ on $\R$ is not a Dunkl operator for $k \ne 0$.
It lacks the reflected term $f(-x)$, and it does not even map the constant $1$ to a polynomial.

The Dunkl Laplacian $\sum_{i=1}^n T_{e_i}^2$ reduces to the Laplacian when $k = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/ABKND96G8P.md" tag="ABKND96G8P" kind="definition">
# Quotient of a topological group

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Let $G/H = \set{gH : g \in G}$ be the set of [[left cosets|03DEH43F4H]] of $H$, and let $\pi \colon G \to G/H$, $\pi(g) = gH$.
The *quotient* of $G$ by $H$ is the set $G/H$ with the [[quotient topology|0NQ1H2X495]] of $\pi$.
So a set $U \subseteq G/H$ is open if and only if $\pi^{-1}(U)$ is open in $G$.
If $H$ is a [[normal subgroup|0CB0DX5BVV]], then $G/H$ is also given the structure of the [[quotient group|0KYX2N69J0]].

## Remarks

The projection $\pi$ is continuous and an [[open map|0YCG648KDD]]: for $U$ open in $G$, the set $\pi^{-1}(\pi(U)) = UH$ is a union of the open sets $Uh$.
If $H$ is normal, then $G/H$ is a topological group, and $\pi$ is a continuous homomorphism.
The space $G/H$ is [[Hausdorff|0S1FJ8VEX9]] if and only if $H$ is closed in $G$.
Example: $\R/\Z$ is a compact Hausdorff topological group, isomorphic to the circle group through $x + \Z \mapsto e^{2\pi i x}$.
Non-example of the Hausdorff property: $\R/\Q$ carries the indiscrete topology, because $\Q$ is dense, so every nonempty open set of $\R$ meets every coset of $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/ABN5PTXEHY.md" tag="ABN5PTXEHY" kind="statement">
# Cantor's isomorphism theorem

## Statement

Let $(A, <)$ and $(B, <)$ be [[strict total orders|C9GDH7264N]] with these properties.

1. $A$ and $B$ are nonempty and [[countable|0ZG61TFBES]].
2. Both orders are [[dense|JB9HJ9CPEN]].
3. Neither $A$ nor $B$ has a least element or a greatest element.

Then there is an [[order isomorphism|0Y74KN99A5]] from $A$ onto $B$.
In particular, each such order is order isomorphic to $(\Q, <)$.

## Remarks

A standard proof is the back-and-forth method.
List $A = \set{a_0, a_1, \dots}$ and $B = \set{b_0, b_1, \dots}$, and build finite order-preserving partial maps step by step.
At even steps, the first element of $A$ not yet in the domain gets an image.
At odd steps, the first element of $B$ not yet in the range gets a preimage.
Density and the lack of endpoints guarantee that a suitable partner exists at each step.
The union of the finite maps is the isomorphism.
For example, the dyadic rationals $\set{m / 2^k : m \in \Z, k \in \N}$ and the rationals in the open interval $(0, 1)$ are both order isomorphic to $\Q$.
Countability is needed: $\R$ satisfies the other hypotheses but is not order isomorphic to $\Q$, because $\R$ is uncountable.
The condition on endpoints is needed: $\Q \cap [0, 1]$ has a least element and $\Q$ has none.
Density is needed: $\Z$ is countable without endpoints but not dense.
In model theory, the theorem says that the theory of dense linear orders without endpoints is $\aleph_0$-categorical.

## Explanations

### L1 human

The rationals have a very particular shape: countably many points, no ends, and always something between two points.
The theorem says this shape is unique.
Any other countable line with no ends and no gaps of the "next element" kind can be matched point by point with the rationals, keeping the order.
So the rationals between $0$ and $1$ and all the rationals have the same order, even though one sits inside the other.

### L2 human

The theorem is the basic example of the back-and-forth method, which is used widely in model theory.
It shows that the [[order type|28PE4GRFD7]] $\eta$ of $\Q$ is the unique countable dense order type without endpoints.
Every countable total order embeds into $\Q$; the proof is the "forth" half of the same argument.
The analogous question for uncountable orders is much harder; [[Suslin's problem|D1T39KHPDN]] is a famous case.
</document>

<document source="https://tamnd.github.io/math2/node/ABPK3RW5MW.md" tag="ABPK3RW5MW" kind="concept">
# Choquet theory

## Summary

Choquet theory studies how a point of a compact [[convex set|02AS45T3H9]] is an average of its [[extreme points|3MSGFJM2NX]].
The average is taken with respect to a [[probability measure|059TJN0NZ1]] on the set, called a representing measure.
It refines the [[Krein–Milman theorem|RSRAY0Q68S]], which only says that the set is the closed convex hull of its extreme points.
The theory applies to sets of measures, of states and of positive functions, where extreme points are the simplest objects.

## Explanations

### L1 human

In a triangle, every point is a weighted average of the three corners, with unique weights.
Choquet theory extends this picture to infinite-dimensional convex sets, where a point may need infinitely many extreme points and a measure replaces the finite weights.
In a square, the centre is the average of two opposite corners in two different ways.
So the weights need not be unique, and the sets with unique weights, the simplices, play a special role.
</document>

<document source="https://tamnd.github.io/math2/node/ABS17CB1D2.md" tag="ABS17CB1D2" kind="statement">
# Atkinson's theorem

## Statement

Let $X$ and $Y$ be [[Banach spaces|06GZ956657]], and let $T \colon X \to Y$ be a [[bounded linear map|03TM135G0N]].
Then $T$ is a [[Fredholm operator|8HWGKWD6ZM]] if and only if there is a bounded linear map $S \colon Y \to X$ such that $ST - I_X$ and $TS - I_Y$ are [[compact operators|M2NEP1K5AE]].

## Remarks

If $T$ is Fredholm, choose closed complements $X = \ker T \oplus X_0$ and $Y = T(X) \oplus Y_0$, with $Y_0$ finite-dimensional.
Then $T$ maps $X_0$ bijectively onto $T(X)$, and $S$ can be taken as its inverse on $T(X)$ and $0$ on $Y_0$.
With this choice, $I_X - ST$ and $I_Y - TS$ are projections onto $\ker T$ and $Y_0$, so they are even [[finite-rank operators|Z8FABSYQTS]].
Conversely, if $ST = I_X + K$ with $K$ compact, then $\ker T \subseteq \ker(I_X + K)$ is finite-dimensional, and $TS = I_Y + K'$ shows that $T(X) \supseteq (I_Y + K')(Y)$ has finite codimension.
For a Hilbert space $H$, the theorem says that $T \in B(H)$ is Fredholm if and only if its image is invertible in the [[Calkin algebra|6FE7JZ2VF1]] $B(H)/K(H)$.
Example: the right shift $S$ on $\ell^2$ and the left shift $S^*$ satisfy $S^* S = I$ and $S S^* = I - P$, where $P$ is the rank-one projection onto the first coordinate, so $S$ is Fredholm.

## Explanations

### L1 human

A Fredholm operator is one that can be inverted up to a compact error.
The theorem says that this is exactly the condition of finite-dimensional kernel and cokernel.
So Fredholm operators are the operators that are invertible once one ignores compact operators.

### L2 human

The theorem turns Fredholm theory into algebra: Fredholm operators are the preimages of the invertible elements of a quotient algebra.
From this view, it follows at once that compact perturbations of Fredholm operators are Fredholm, and that products of Fredholm operators are Fredholm.
It also leads to the essential spectrum of an operator $T$ on a Hilbert space, the spectrum of its image in the Calkin algebra.
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<document source="https://tamnd.github.io/math2/node/ABSRB2KXA6.md" tag="ABSRB2KXA6" kind="concept">
# Quasi-Monte Carlo method

## Summary

A quasi-Monte Carlo method approximates an integral over the unit cube $[0, 1]^s$ by an average of the integrand at deterministic points.
The points are spread evenly: they come from a [[low-discrepancy sequence|XSPX1KW9AQ]].
It keeps the simple form of [[Monte Carlo integration|DAR3QJ5H7Y]], an equal-weight average, but replaces random points by structured ones.
For integrands of bounded variation, the Koksma–Hlawka inequality bounds the error by the variation of the integrand times the discrepancy of the points.
This gives an error of order $(\log N)^s / N$ instead of $N^{-1/2}$.
Quasi-Monte Carlo methods are widely used in computational finance and in high-dimensional integration.

## Explanations

### L1 human

Monte Carlo integration averages a function at random points, and random points leave gaps and clusters.
Quasi-Monte Carlo uses points that are placed deliberately to avoid gaps, like a carefully raked lawn instead of scattered seeds.
For example, to integrate $f$ over $[0, 1]$ one can average $f$ at $\frac12, \frac14, \frac34, \frac18, \frac58, \ldots$, the van der Corput points.
For smooth functions of not too many variables, the error then shrinks almost like $1/N$ rather than $1/\sqrt N$.
The price is the loss of a simple statistical error estimate, which randomized versions of the method restore.
</document>

<document source="https://tamnd.github.io/math2/node/ABVBHCA9VW.md" tag="ABVBHCA9VW" kind="concept">
# Gluing of schemes

## Summary

Gluing builds a new [[scheme|5A35TYZD02]] from a family of schemes by identifying open pieces of them along isomorphisms.
The isomorphisms must agree on triple overlaps, a condition called the cocycle condition.
Gluing is how most schemes beyond affine ones are made, for example projective space.
It is also used to prove that constructions such as fiber products exist for all schemes.

## Explanations

### L1 human

Imagine making a sphere from two discs by sewing them along their edges.
Gluing of schemes does the same with algebraic pieces: one says which open parts of the pieces are to be identified, and by which maps.
The projective line comes from two copies of the affine line, with coordinates $t$ and $s$.
The parts where $t \ne 0$ and $s \ne 0$ are identified by the rule $s = 1/t$.
The point $t = 0$ and the point $s = 0$ stay separate, so the result is a line with one extra point at infinity.
</document>

<document source="https://tamnd.github.io/math2/node/ABXD13M340.md" tag="ABXD13M340" kind="concept">
# Principal component analysis

## Summary

Principal component analysis replaces correlated variables by new uncorrelated variables, the principal components, ordered by decreasing variance.
The components are the coordinates along the [[eigenvectors|0N6960W0BV]] of the [[covariance matrix|0V7G6WMFK0]].
Keeping only the first few components gives the best low-dimensional linear summary of the data in the least squares sense.
It is the most common method of dimension reduction.

## Explanations

### L1 human

Picture a cloud of data points in the plane shaped like a tilted cigar.
The first principal component is the direction along the cigar, where the points spread the most.
The second is the direction across it, at a right angle to the first.
If two test scores have variances $2$ and $2$ and covariance $1$, the first direction is $(1, 1)/\sqrt{2}$, along which the variance is $3$.
Describing each point by its position along the cigar alone loses little information.
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<document source="https://tamnd.github.io/math2/node/ABYAPP6Q4C.md" tag="ABYAPP6Q4C" kind="statement">
# Smith conjecture

## Statement

Let $f \colon S^3 \to S^3$ be an orientation-preserving diffeomorphism of finite order $n \ge 2$.
Suppose that the fixed point set $\set{x \in S^3 : f(x) = x}$ is homeomorphic to a circle.
Then this fixed circle is the [[unknot|1TQ06R4Q66]].

## Remarks

P. A. Smith showed that the fixed point set of such a map of prime order is empty or a circle, and asked whether the circle can be knotted.
The proof combined work of several authors, including Thurston on hyperbolic structures, Meeks and Yau on minimal surfaces, Bass on group theory, and Gordon and Litherland.
One case uses incompressible surfaces in the knot complement; the other uses a hyperbolic structure given by Thurston's work.
The orientation hypothesis matters: a reflection of $S^3$ has order $2$ and fixes a $2$-sphere.
The analogue in higher dimensions fails: in $S^4$, some knotted $2$-spheres are fixed sets of periodic diffeomorphisms.
Example: the rotation by angle $2\pi/n$ about the unknotted great circle $\set{(z, 0)} \subset S^3 \subset \C^2$, given by $(z, w) \mapsto (z, e^{2\pi i/n} w)$, fixes exactly that circle.

## Explanations

### L1 human

Imagine a rotation of three-dimensional space that, repeated a few times, comes back to where it started.
The points that do not move form an axis.
In ordinary space this axis is a straight line, and in the $3$-sphere it is an unknotted circle.
The theorem says that no smooth symmetry of finite order can have a knotted axis.
So a nontrivial [[knot|QYEAZTYGPT]] can never be the axis of such a rotation.

### L2 human

Equivalently, a nontrivial knot $K$ is never the branch set of a cyclic branched cover $S^3 \to S^3$.
The theorem was a test case for Thurston's geometrization program: the decisive input came from hyperbolic geometry and minimal surfaces.
It is a precursor of the orbifold theorem, which handles more general finite group actions on $3$-manifolds.
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# Landau's constants

## Summary

Landau's constants are three numbers that measure how large a disc the image of a normalised holomorphic function on the unit disc must contain.
They are the Bloch constant $B$, the Landau constant $L$ and the univalent Landau constant $A$, and they satisfy $B \le L \le A$.
They are positive by [[Bloch's theorem|KW31KDX7TB]], but their exact values are open problems.

## Explanations

### L1 human

Take a holomorphic function $f$ on the unit disc with $f(0) = 0$ and $f'(0) = 1$.
Near $0$ it looks like the identity, so its image should contain some disc.
Landau's constant $L$ is the largest radius that works for every such $f$, the disc being anywhere in the image.
The Bloch constant $B$ asks in addition that the disc be covered exactly once, and $A$ restricts to injective $f$.
For the identity $f(z) = z$, the image is the unit disc itself, which contains discs of every radius below $1$.
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<document source="https://tamnd.github.io/math2/node/AC5CB7RH6Z.md" tag="AC5CB7RH6Z" kind="statement">
# Kantorovich duality

## Statement

Let $X$ and $Y$ be [[Polish spaces|584GXXY73M]], let $\mu$ be a Borel [[probability measure|059TJN0NZ1]] on $X$ and $\nu$ a Borel probability measure on $Y$.
Let $c \colon X \times Y \to [0, \infty]$ be a lower semicontinuous cost function.
Let $\Pi(\mu, \nu)$ be the set of [[couplings|HTR7J0W66V]] of $\mu$ and $\nu$, the Borel probability measures on $X \times Y$ with marginals $\mu$ and $\nu$.
Then
$$\min_{\pi \in \Pi(\mu, \nu)} \int_{X \times Y} c \, d\pi = \sup \set{ \int_X \varphi \, d\mu + \int_Y \psi \, d\nu : \varphi \in L^1(\mu), \ \psi \in L^1(\nu), \ \varphi(x) + \psi(y) \le c(x, y) \text{ for all } x, y },$$
and the minimum on the left is attained.
The supremum does not change if $\varphi$ and $\psi$ are required to be bounded and continuous.

## Remarks

The left side is the Kantorovich form of the [[optimal transport problem|K8EBASR5VG]], and the right side is its dual.
The inequality $\ge$ is easy: if $\varphi(x) + \psi(y) \le c(x, y)$ and $\pi \in \Pi(\mu, \nu)$, integrating against $\pi$ gives $\int \varphi \, d\mu + \int \psi \, d\nu \le \int c \, d\pi$.
The reverse inequality is a minimax argument.
For compact spaces and continuous $c$ it follows from a duality theorem of convex analysis, such as the Fenchel–Rockafellar theorem.
The general case follows by approximation.
Example: if $\mu = \delta_a$ is a Dirac mass, the only coupling is $\delta_a \otimes \nu$, so the left side equals $\int c(a, y) \, d\nu(y)$.
When $X = Y$ and $c$ is a metric $d$, the dual can be taken over $\psi = -\varphi$ with $\varphi$ $1$-Lipschitz, which is the [[Kantorovich–Rubinstein duality|SYXSZD8EFK]].
Lower semicontinuity of $c$ is used to show that the minimum is attained, through the weak compactness of $\Pi(\mu, \nu)$.

## Explanations

### L1 human

Think of a shipping company that moves goods from factories with distribution $\mu$ to shops with distribution $\nu$, at cost $c(x, y)$ per unit.
Instead of paying for each route, one may sell the job to a contractor who charges a loading price $\varphi(x)$ and an unloading price $\psi(y)$.
The contractor's prices are competitive if $\varphi(x) + \psi(y)$ never exceeds the direct cost $c(x, y)$.
The theorem says that the best competitive income of the contractor equals the least shipping cost.

### L2 human

Kantorovich duality is the linear-programming duality of optimal transport, extended to infinite spaces.
Under mild integrability conditions on $c$, optimal dual pairs $(\varphi, \psi)$ exist and can be chosen $c$-concave, and an optimal plan is concentrated on the set where $\varphi(x) + \psi(y) = c(x, y)$.
This is the starting point for Brenier's theorem, where the optimal $\varphi$ gives the optimal map, and for the theory of Wasserstein distances.
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# Finite type algebra

## Summary

An [[algebra|DPBJ04JMH8]] $B$ over a commutative ring $A$ is of finite type if finitely many elements of $B$ generate it as an $A$-algebra.
Equivalently, $B$ is a quotient of a [[polynomial ring|0SDC934R52]] $A[x_1, \ldots, x_n]$ in finitely many variables.
Finitely generated algebras over a field are the coordinate rings of affine varieties, the main objects of classical algebraic geometry.
Over a Noetherian base they are Noetherian, by [[Hilbert's basis theorem|0YKA9J3XEE]].

## Explanations

### L1 human

An algebra is of finite type if every element can be written as a polynomial, with coefficients in $A$, in a fixed finite list of elements.
The ring $\C[x, y]/(x^2 + y^2 - 1)$ of polynomial functions on a circle is generated over $\C$ by the two coordinate functions.
Finite type is weaker than finite: $k[x]$ is of finite type over $k$, generated by $x$, but it is not a finitely generated $k$-module.
The ring $\Q$ is not of finite type over $\Z$: finitely many fractions only produce finitely many prime denominators.
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<document source="https://tamnd.github.io/math2/node/AC6WNX8F9Z.md" tag="AC6WNX8F9Z" kind="concept">
# Yamabe flow

## Summary

The Yamabe flow is an evolution equation for a [[Riemannian metric|Q61VFHCH2J]].
The metric changes in the direction of minus its [[scalar curvature|ZHX2VYQDZ3]] times itself.
Richard Hamilton introduced it as a tool for the Yamabe problem.
This problem asks for a metric of constant scalar curvature in a given conformal class.
The flow keeps the metric in its conformal class.
Its fixed points, after normalizing the volume, are the metrics of constant scalar curvature.

## Explanations

### L1 human

A metric tells how to measure lengths on a curved space.
The Yamabe flow rescales lengths at each point: it shrinks them where the scalar curvature is positive and stretches them where it is negative.
Angles never change, because the change at each point is a pure rescaling.
The hope is that the curvature evens out over time.
For example, the round sphere of dimension $n$ and radius $1$ shrinks: its metric at time $t$ is $1 - n(n-1)t$ times the original one, until it collapses at $t = 1/(n(n-1))$.
</document>

<document source="https://tamnd.github.io/math2/node/AC9PY6FBYF.md" tag="AC9PY6FBYF" kind="statement">
# Multigrid method

## Statement

Let $A_h$ be an [[invertible|0EB9N0S33C]] real $n \times n$ matrix (the fine-grid matrix) and $A_H$ an invertible real $N \times N$ matrix with $N < n$ (the coarse-grid matrix).
Let $P$ be an $n \times N$ matrix (prolongation) and $R$ an $N \times n$ matrix (restriction).
Let $W$ be an invertible $n \times n$ matrix, and let $S = I - W^{-1} A_h$ be the iteration matrix of the smoother $x \mapsto x + W^{-1}(b - A_h x)$.
One step of the *two-grid method* for $A_h x = b$ applies $\nu \ge 1$ smoothing steps to $x$ and then the coarse-grid correction $x \mapsto x + P A_H^{-1} R (b - A_h x)$.

1. The error $e = x - A_h^{-1} b$ is mapped to $M e$, where $M = (I - P A_H^{-1} R A_h) S^{\nu}$.
2. (Hackbusch) Let $\norm{\cdot}$ be an operator norm, and let $C > 0$ and $\eta(\nu) \ge 0$ be numbers with $\norm{A_h^{-1} - P A_H^{-1} R} \le C / \norm{A_h}$ (approximation property) and $\norm{A_h S^{\nu}} \le \eta(\nu) \norm{A_h}$ (smoothing property). Then $\norm{M} \le C \, \eta(\nu)$.

## Remarks

Clause 1 follows by composing the error maps $e \mapsto S e$ of the smoother and $e \mapsto (I - P A_H^{-1} R A_h) e$ of the correction.
Clause 2 follows from the factorization $M = (A_h^{-1} - P A_H^{-1} R) \, A_h S^{\nu}$ and the two bounds.
Consider a family of [[finite difference|X7HYEBA5SA]] or finite element discretizations of an [[elliptic equation|B4TTNHSTRX]], with mesh widths $h$ and $H = 2h$.
Use damped [[Jacobi|B8FVHNXAN1]] smoothing and standard transfer matrices.
Under elliptic regularity assumptions, both properties hold with $C$ and $\eta$ independent of $h$, and $\eta(\nu) \to 0$.
Then for $\nu$ large enough the two-grid method contracts the error by a factor less than $1$ that does not depend on $h$.
Example: for $A_h$ symmetric positive definite and $S = I - A_h / \norm{A_h}_2$ in the [[matrix norm|54T6Y6DSY4]] $\norm{\cdot}_2$, one has $\norm{A_h S^\nu}_2 \le \max_{0 \le t \le 1} t (1 - t)^\nu \norm{A_h}_2 \le \norm{A_h}_2 / (\nu + 1)$.
Solving the coarse problem recursively by the same method gives the V-cycle and the W-cycle.

## Explanations

### L1 human

Simple methods such as Jacobi or [[Gauss–Seidel|G0EDYEDG4S]] remove the rough, oscillating part of the error quickly, but the smooth part only very slowly.
A smooth error can be represented well on a coarser grid with far fewer unknowns, where it is cheap to remove.
Multigrid alternates the two: smooth on the fine grid, correct on the coarse grid, and repeat over a hierarchy of grids.

### L2 human

For elliptic problems, multigrid reaches a fixed accuracy in $O(n)$ operations, which is optimal up to a constant.
The analysis splits into the smoothing property, a statement about the smoother alone, and the approximation property, a statement about the discretizations on the two grids.
Algebraic multigrid builds the coarse levels from the matrix alone, without a geometric grid.
Multigrid is also used as a [[preconditioner|KE98SW3SBD]] for the [[conjugate gradient method|AFXZGQB4EN]], and it is among the fastest [[iterative methods|2RPE77ACER]] for such problems.
</document>

<document source="https://tamnd.github.io/math2/node/ACBJ8RW42F.md" tag="ACBJ8RW42F" kind="definition">
# Summability kernel

## Definition

A *summability kernel* is a sequence $(k_n)_{n \ge 1}$ of [[continuous|0X3WGX95FE]] $2\pi$-[[periodic functions|0ATF6Q0633]] $k_n \colon \R \to \C$ such that:

1. $\frac{1}{2\pi} \int_{-\pi}^{\pi} k_n(t) \, dt = 1$ for every $n$.
2. There is a constant $C$ with $\frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{k_n(t)} \, dt \le C$ for every $n$.
3. For every $\delta$ with $0 < \delta < \pi$, $\int_{\delta \le \abs{t} \le \pi} \abs{k_n(t)} \, dt \to 0$ as $n \to \infty$.

## Remarks

For an integrable $2\pi$-periodic $f$, the convolutions $k_n * f(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} k_n(t) f(x - t) \, dt$ tend to $f$ in $L^1$.
If $f$ is continuous, they tend to $f$ uniformly.
A kernel with $k_n \ge 0$ is called positive; then clause 2 follows from clause 1 with $C = 1$.

Example: the [[Fejér kernel|DCCA2T9GKH]] $F_N(t) = \sum_{\abs{j} < N} \left(1 - \frac{\abs{j}}{N}\right) e^{ijt} = \frac{1}{N} \left( \frac{\sin(Nt/2)}{\sin(t/2)} \right)^2$.
Its mean is the coefficient of $e^{0}$, which is $1$.
It is nonnegative, so clause 2 holds with $C = 1$.
For $\delta \le \abs{t} \le \pi$ it is at most $\frac{1}{N \sin^2(\delta/2)}$, which gives clause 3.

Non-example: the [[Dirichlet kernel|3JY4BVE4DW]] $D_n(t) = \sum_{\abs{j} \le n} e^{ijt}$ satisfies clause 1, but $\frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{D_n}$ grows like a constant times $\log n$, so clause 2 fails.

Kernels indexed by a continuous parameter are treated the same way; the [[Poisson kernel|TCYSWCTYYM]] $P_r$ with $r \to 1^-$ is the main example.
A summability kernel is a bounded [[approximate identity|CDYXTW3PJF]] of the convolution algebra $L^1(\mathbb{T})$.
</document>

<document source="https://tamnd.github.io/math2/node/ACDEK1GD9G.md" tag="ACDEK1GD9G" kind="concept">
# GCD domain

## Summary

A GCD domain is an [[integral domain|09S3PFBEPM]] in which any two elements have a [[greatest common divisor|0F7KNJ1ZYN]].
Every [[unique factorisation domain|06YMXJMXYS]] and every [[Bézout domain|HGEX40P0Z0]] is a GCD domain.
In a GCD domain every [[irreducible element|052SDEPZH7]] is [[prime|0MNHXBSX3M]], and the ring is [[integrally closed|H20P600HQS]].
GCD domains need not have factorizations into irreducibles, so the notion is strictly weaker than unique factorization.

## Explanations

### L1 human

In the integers, any two numbers have a greatest common divisor, such as $\gcd(12, 18) = 6$.
A GCD domain is a ring where this keeps working: among the common divisors of $a$ and $b$ there is one that every other common divisor divides.
The ring $\Z[\sqrt{-5}]$ is not a GCD domain: the elements $6$ and $2(1 + \sqrt{-5})$ have the common divisors $2$ and $1 + \sqrt{-5}$, but no common divisor is divisible by both.
</document>

<document source="https://tamnd.github.io/math2/node/ACN92H5M6D.md" tag="ACN92H5M6D" kind="concept">
# Bogdanov–Takens bifurcation

## Summary

A Bogdanov–Takens bifurcation is a bifurcation of an [[equilibrium point|0N8NGKDZDZ]] in a family with two parameters, at which the linearisation has a double zero eigenvalue.
It is a point where a [[saddle-node bifurcation|MAP5J6HK0T]] curve and a [[Hopf bifurcation|F5H8GW1XYS]] curve meet.
Near it, the family also has a curve of homoclinic bifurcations.
It is one of the basic codimension-two bifurcations.
See the [[definition|H3GWX277EF]].

## Explanations

### L1 human

Most bifurcations happen when one parameter reaches a critical value.
A Bogdanov–Takens point needs two parameters to be tuned at once.
In the plane of the two parameters, three curves start from this point.
On one of them two equilibria collide, on another a small oscillation is born, and on the third this oscillation grows until it touches a saddle and disappears.
So a single point organizes several kinds of change in the behaviour of the system.
</document>

<document source="https://tamnd.github.io/math2/node/ACQ7A58XY7.md" tag="ACQ7A58XY7" kind="statement">
# Sinai's factor theorem

## Statement

Let $(X, \mathcal{B}, \mu)$ be a [[standard probability space|QDDRDW9PXS]] and let $T \colon X \to X$ be an invertible [[ergodic|AZRPHS478B]] [[measure-preserving transformation|556WBDAHNP]] with measurable inverse.
Assume that its [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] $h_\mu(T)$ is positive.
Let $B$ be a two-sided [[Bernoulli scheme|D33JK3AXRZ]] on a finite alphabet, acting on $(Y, \nu)$, with entropy $h(B) \le h_\mu(T)$.
Then $B$ is a factor of $T$: there is a measurable map $\varphi \colon X \to Y$ with $\mu(\varphi^{-1}C) = \nu(C)$ for every measurable $C \subseteq Y$ and $\varphi \circ T = B \circ \varphi$ almost everywhere.

## Remarks

A factor map can only lower entropy, so the hypothesis $h(B) \le h_\mu(T)$ is necessary.
Ergodicity is needed too.
Let $T$ be the disjoint union, with weight $1/2$ each, of a Bernoulli scheme $B'$ and an identity map, so that $h_\mu(T) = h(B')/2 > 0$.
Let $B$ be any Bernoulli scheme with at least two symbols of positive weight.
A factor map $\varphi$ onto $B$ satisfies $\varphi(x) = B(\varphi(x))$ on the identity part.
So it sends a set of measure $1/2$ into the fixed points of $B$, which form a null set.
So $T$ has no such factor, whatever the entropy of $B$.
The proof constructs the factor map as a limit of finite codings, using the Rokhlin lemma to build towers and the Shannon–McMillan–Breiman theorem to count typical names.
Example: an irrational rotation has entropy $0$, so the theorem says nothing about it.
Indeed it has no nontrivial Bernoulli factor, since every factor of a zero-entropy system has entropy $0$.
Example: every ergodic toral automorphism of positive entropy has Bernoulli factors of every entropy up to its own.

## Explanations

### L1 human

Any ergodic system with some randomness, measured by positive entropy, contains a perfectly random process inside it.
More precisely, by watching the system through a suitable lens one sees a sequence of independent die rolls.
The die can be any loaded die whose entropy does not exceed the entropy of the system.
So positive entropy always means genuine randomness of the coin-tossing kind.

### L2 human

It was a step toward the Ornstein isomorphism theorem, whose proof sharpens the same kind of coding argument from factor maps to isomorphisms.
It shows that positive entropy, a number defined by counting, forces a concrete structure: an independent process sitting inside the system.
</document>

<document source="https://tamnd.github.io/math2/node/ACSF58HYDH.md" tag="ACSF58HYDH" kind="concept">
# Weierstrass product

## Summary

A Weierstrass product is an infinite product of elementary factors $E_{p_n}(z/a_n)$ that converges to an [[entire function|0419794R0T]] with zeros exactly at the points $a_n$.
Each elementary factor is $1 - z/a_n$ times an exponential correction that makes the product converge.
Weierstrass products show that any sequence of nonzero complex numbers tending to infinity is the zero set of an entire function.
They are the building blocks of the [[Weierstrass factorization theorem|11VC51YHQJ]] and the [[Hadamard factorization theorem|KYSR7NHNFQ]].

## Explanations

### L1 human

To build a polynomial with given roots, multiply the factors $1 - z/a$.
With infinitely many roots, this product may diverge, as for $\prod_{n \ge 1} (1 - z/n)$.
Weierstrass multiplies each factor by $\exp(z/a + z^2/(2a^2) + \cdots + z^p/(p a^p))$, which equals $1$ at $z = 0$ and has no zeros.
This correction removes the first terms of $\log(1 - z/a)$ and makes the product converge.
For the roots $n = 1, 2, 3, \dots$, the product $\prod_n (1 - z/n) e^{z/n}$ converges, and it is closely related to the gamma function.
</document>

<document source="https://tamnd.github.io/math2/node/AD0G065GK8.md" tag="AD0G065GK8" kind="example">
# Pompeiu derivative

## Example

Let $h$ be a function that is differentiable at every point.
Its [[derivative|0V1EJCT8G6]] $h'$ is a Pompeiu derivative if $h'$ vanishes on a [[dense set|0BSJSX7ME2]] but is not identically zero.
Here is a construction.
Let $q_1, q_2, \dots$ list the rational numbers in $[0, 1]$, and write $u^{1/3}$ for the real cube root.
For $x \in [0, 1]$ set
$$g(x) = \sum_{n=1}^\infty 2^{-n} (x - q_n)^{1/3}.$$
The series converges uniformly, and each term is increasing, so $g$ is continuous and [[strictly increasing|011EJQWWJS]].

Each difference quotient of $u \mapsto u^{1/3}$ at a point $u \ne 0$ is positive and at most $\frac{4}{3} \abs{u}^{-2/3}$, since $a^2 + ab + b^2 \ge \frac{3}{4} a^2$ for real $a, b$.
Use this bound with dominated convergence where the series below is finite, and Fatou's lemma where it is infinite.
It follows that $g$ has at each $x$ a derivative in $(0, \infty]$, namely
$$g'(x) = \sum_{n=1}^\infty \frac{2^{-n}}{3} \abs{x - q_n}^{-2/3},$$
with $g'(q_n) = \infty$ for every $n$ (one-sided at the end points).
Since $\abs{x - q_n} \le 1$, every term is at least $2^{-n}/3$, so $g'(x) \ge 1/3$.

Let $h = g^{-1}$ on the interval $[g(0), g(1)]$.
By the rule for the derivative of an inverse function, $h$ is differentiable everywhere, with $h'(y) = 1/g'(h(y))$, read as $0$ when $g'(h(y)) = \infty$.
So $0 \le h' \le 3$.
The derivative $h'$ vanishes at every point $g(q_n)$, and these points are dense in $[g(0), g(1)]$ because $g$ is a homeomorphism.
But $h'$ is not identically zero, since $h$ is strictly increasing and a function with zero derivative is constant.
So $h'$ is a Pompeiu derivative.

## Explanations

### L1 human

One might think that a derivative which is zero on a dense set of points must be zero everywhere.
This example shows that this is false.
The function $g$ has vertical tangents at all rational points, so its inverse $h$ has horizontal tangents at a dense set of points, and yet $h$ keeps increasing.
Like every derivative, $h'$ still has the intermediate value property, so it is a [[Darboux function|7BJEH1GCTE]].
</document>

<document source="https://tamnd.github.io/math2/node/AD3474PVHR.md" tag="AD3474PVHR" kind="concept">
# Polynomially convex set

## Summary

A compact set $K \subseteq \C^n$ is polynomially convex if every point outside $K$ can be separated from $K$ by a polynomial.
This means that some polynomial is larger in absolute value at the point than everywhere on $K$.
They are the natural sets for polynomial approximation.
By the [[Oka–Weil theorem|7AWZJNQGT2]], every function holomorphic near such a set is a uniform limit of polynomials on it.
In $\C$ they are the compact sets with connected complement.
For $n \ge 2$ the condition is much subtler, and it is not determined by the topology of $K$ alone.

## Explanations

### L1 human

The hull of a compact set $K$ collects the points where no polynomial beats its maximum over $K$.
The set is polynomially convex if the hull adds nothing.
A closed disc in $\C$ is polynomially convex, but a circle is not: by the maximum principle, its hull is the whole closed disc.
Every compact convex set in $\C^n$ is polynomially convex, and so is every compact subset of $\R^n \subseteq \C^n$.
</document>

<document source="https://tamnd.github.io/math2/node/AD9GD9EM9R.md" tag="AD9GD9EM9R" kind="conjecture">
# Green–Griffiths–Lang conjecture

## Statement

Let $X$ be a complex [[projective variety|A4H4CZ2V35]] of general type, that is, one whose [[Kodaira dimension|ZW00Q0SAGR]] equals its dimension.
Then there is a proper algebraic subvariety $Y \subsetneq X$ such that every nonconstant holomorphic map $f \colon \C \to X$ has image contained in $Y$.

## Remarks

Green and Griffiths posed the problem for entire curves in varieties of general type.
Serge Lang stated it as part of a group of conjectures that link hyperbolicity with the arithmetic of varieties.
The conjecture is open in general.
For curves it holds: a smooth projective curve of general type has genus at least $2$, so its universal covering is the unit disc.
Hence every holomorphic map from $\C$ to it is constant.
Michael McQuillan proved it for surfaces of general type with $c_1^2 > c_2$.
It is also known for very general hypersurfaces of high degree in projective space.
A related conjecture of Lang says that a projective variety is [[Kobayashi hyperbolic|HRPAX8V7HA]] if and only if all its subvarieties are of general type.

## Explanations

### L1 human

A variety of general type is, in a precise sense, a variety with a lot of negative curvature.
The conjecture predicts that holomorphic copies of the complex plane in such a variety cannot spread everywhere: they all lie inside one smaller algebraic subvariety.
In other words, such a variety is [[Brody hyperbolic|Y7W36GJDJD]] outside a small exceptional set.
</document>

<document source="https://tamnd.github.io/math2/node/AD9GZCC2PB.md" tag="AD9GZCC2PB" kind="definition">
# Linear multistep method

## Definition

Let $f \colon \R \times \R^n \to \R^n$, and consider the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$.
Let $k \ge 1$ be an integer, let $h > 0$, let $t_j = t_0 + j h$, and write $f_j = f(t_j, y_j)$.
Let $\alpha_0, \ldots, \alpha_k$ and $\beta_0, \ldots, \beta_k$ be real numbers with $\alpha_k = 1$ and $(\alpha_0, \beta_0) \ne (0, 0)$.
The *linear $k$-step method* with these coefficients determines $y_{j+k}$ from $y_j, \ldots, y_{j+k-1}$ by
$$\sum_{i=0}^{k} \alpha_i \, y_{j+i} = h \sum_{i=0}^{k} \beta_i \, f_{j+i}, \qquad j = 0, 1, 2, \ldots$$
The method is *explicit* if $\beta_k = 0$ and *implicit* otherwise.

## Remarks

The characteristic polynomials of the method are $\rho(\zeta) = \sum_i \alpha_i \zeta^i$ and $\sigma(\zeta) = \sum_i \beta_i \zeta^i$.
The method is consistent if $\rho(1) = 0$ and $\rho'(1) = \sigma(1)$, and zero-stable if every root of $\rho$ satisfies $\abs{\zeta} \le 1$ and the roots with $\abs{\zeta} = 1$ are simple.
Dahlquist's equivalence theorem says that a consistent method converges for all smooth problems with consistent starting values if and only if it is zero-stable.
Example: the two-step Adams–Bashforth method $y_{j+2} - y_{j+1} = h \left( \frac32 f_{j+1} - \frac12 f_j \right)$ has $\rho(\zeta) = \zeta^2 - \zeta$ and $\sigma(\zeta) = \frac32 \zeta - \frac12$, so $\rho(1) = 0$ and $\rho'(1) = 1 = \sigma(1)$; it is explicit, consistent and zero-stable.
The [[Euler method|0JARTXX4XP]] is the case $k = 1$, $\alpha_0 = -1$, $\beta_0 = 1$, $\beta_1 = 0$.
Non-example: the classical fourth-order [[Runge–Kutta method|7DBPJKYC6C]] is not a linear multistep method, since its update uses values of $f$ at intermediate points that are not of the form $f_{j+i}$.
The explicit method $y_{j+2} + 4 y_{j+1} - 5 y_j = h (4 f_{j+1} + 2 f_j)$ is consistent of order $3$, but $\rho(\zeta) = (\zeta - 1)(\zeta + 5)$ has the root $-5$, so it is not zero-stable and its errors grow without bound.
</document>

<document source="https://tamnd.github.io/math2/node/ADBEEPPKRR.md" tag="ADBEEPPKRR" kind="example">
# Toda lattice

## Example

Let $n \ge 2$.
The open *Toda lattice* is the [[Hamiltonian system|0EDKGA2FAK]] on $\R^{2n}$ with positions $q_1, \ldots, q_n$, momenta $p_1, \ldots, p_n$ and Hamiltonian
$$H(q, p) = \sum_{k=1}^{n} \frac{p_k^2}{2} + \sum_{k=1}^{n-1} e^{-(q_{k+1} - q_k)}.$$
It describes $n$ particles on a line with exponential forces between neighbours.

It is an [[integrable system|N0N7CMRCMH]].
Flaschka's variables are $a_k = \frac{1}{2} e^{-(q_{k+1} - q_k)/2}$ for $1 \le k \le n - 1$ and $b_k = -\frac{1}{2} p_k$ for $1 \le k \le n$.
Let $L$ be the symmetric tridiagonal $n \times n$ matrix with diagonal entries $b_1, \ldots, b_n$ and off-diagonal entries $a_1, \ldots, a_{n-1}$.
The equations of motion can be written as a [[Lax pair|G3A4S7CSB3]] equation $L' = [B, L]$, where $B$ is an antisymmetric tridiagonal matrix whose off-diagonal entries are $\pm a_k$.
So $L(t)$ is conjugate to $L(0)$ for all $t$, and the $n$ eigenvalues of $L$, or the traces $\tr L^m$ for $m = 1, \ldots, n$, are first integrals.
These integrals are independent and Poisson-commute, which gives Liouville integrability.

A check of the scaling: $\tr L^2 = \sum_k b_k^2 + 2 \sum_k a_k^2 = \frac{1}{4} \sum_k p_k^2 + \frac{1}{2} \sum_k e^{-(q_{k+1} - q_k)} = \frac{1}{2} H$.
Also $\tr L = -\frac{1}{2} \sum_k p_k$ is a multiple of the total momentum.

## Remarks

Morikazu Toda introduced the lattice in 1967.
Its integrability was shown by Hénon and by Flaschka in 1974.
There is also a periodic Toda lattice, with $q_{n+1} = q_1$, which is integrable as well.

## Explanations

### L1 human

The Toda lattice is a row of particles pushed apart by forces that grow exponentially as neighbours come close.
Although the forces are nonlinear, the system has as many conserved quantities as particles, so its motion is completely regular.
The conserved quantities are hidden in a matrix built from positions and momenta: as time goes on the matrix changes, but its eigenvalues do not.
</document>

<document source="https://tamnd.github.io/math2/node/ADDS6HE5SV.md" tag="ADDS6HE5SV" kind="concept">
# Relative homotopy

## Summary

A homotopy relative to a subspace $A$ is a [[homotopy|9YA6S6HD9V]] that keeps every point of $A$ fixed during the whole deformation.
It lets one deform maps while keeping a boundary condition, such as the end points of a path.
Path homotopy, the fundamental group and deformation retractions are all defined with relative homotopies.

## Explanations

### L1 human

Imagine a rubber band pinned to a board at some points.
You may move the band, but the pins never move.
A homotopy relative to $A$ is such a movement, where $A$ is the set of pinned points.
For example, two paths in the plane with the same end points can be deformed into each other with both end points held fixed.
</document>

<document source="https://tamnd.github.io/math2/node/ADEXY830S8.md" tag="ADEXY830S8" kind="example">
# Birkhoff polytope

## Example

Let $n \ge 1$.
The *Birkhoff polytope* $B_n$ is the set of $n \times n$ [[doubly stochastic matrices|JJA90EB2PS]], seen as points of $\R^{n \times n}$:
$$B_n = \set{X \in \R^{n \times n} : x_{ij} \ge 0, \ \textstyle\sum_{j} x_{ij} = 1 \text{ for all } i, \ \sum_{i} x_{ij} = 1 \text{ for all } j}.$$
It is bounded, since $0 \le x_{ij} \le 1$, and is cut out by finitely many linear inequalities and equations, so it is a [[convex polytope|300K6CB1KH]].
By the [[Birkhoff–von Neumann theorem|3KY1RMEDMJ]], $B_n$ is the convex hull of the $n!$ [[permutation matrices|08SP3DF8NB]], and these are its [[vertices|R7V3S9V891]].
The $2n$ equations have rank $2n - 1$, so $\dim B_n = n^2 - (2n - 1) = (n-1)^2$.
For $n \ge 3$ the [[facets|DPDXN2FJC4]] are the $n^2$ sets $\set{X \in B_n : x_{ij} = 0}$.

## Remarks

For $n = 2$, $B_2$ is the segment of matrices $\begin{pmatrix} t & 1-t \\ 1-t & t \end{pmatrix}$, $0 \le t \le 1$, with the two permutation matrices as endpoints.
For $n = 3$, $B_3$ is a $4$-dimensional polytope with $6$ vertices and $9$ facets.
Its role in optimisation is that a linear function on $B_n$ attains its maximum at a permutation matrix, which solves the assignment problem by linear programming.

## Explanations

### L1 human

A doubly stochastic matrix is a square table of nonnegative numbers in which every row and every column adds up to $1$.
All such tables together form a solid shape, the Birkhoff polytope.
Its corners are the permutation matrices, tables of zeros with a single $1$ in each row and each column.
Every doubly stochastic matrix is therefore an average of permutation matrices.
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<document source="https://tamnd.github.io/math2/node/ADJZ99KY2Q.md" tag="ADJZ99KY2Q" kind="definition">
# Fredholm index

## Definition

Let $X$ and $Y$ be Banach spaces, and let $T \colon X \to Y$ be a [[Fredholm operator|8HWGKWD6ZM]].
The *index* of $T$ is the integer $\operatorname{ind} T = \dim \ker T - \dim \operatorname{coker} T$.
Here $\ker T$ is the [[kernel|0S23E6YE83]] of $T$.
The cokernel $\operatorname{coker} T = Y / T(X)$ is the [[quotient|0BR3FY5Q8M]] of $Y$ by the [[image|0WSFKKT6FF]] of $T$.

## Remarks

Example: the right shift $S$ on $\ell^2$ has $\ker S = \set{0}$ and $\operatorname{coker} S$ of dimension $1$, so $\operatorname{ind} S = -1$.
The left shift $S^*(x_1, x_2, \ldots) = (x_2, x_3, \ldots)$ is onto with a kernel of dimension $1$, so $\operatorname{ind} S^* = 1$.
Non-example: the diagonal operator $D(x_n) = (x_n / n)$ on $\ell^2$ is not Fredholm, so it has no index.
For a linear map $T \colon X \to Y$ between finite-dimensional spaces, the rank–nullity theorem gives $\operatorname{ind} T = \dim X - \dim Y$.
For Fredholm operators $T \colon X \to Y$ and $S \colon Y \to Z$, the product $ST$ is Fredholm and $\operatorname{ind}(ST) = \operatorname{ind} S + \operatorname{ind} T$.
If $K \colon X \to Y$ is a [[compact operator|M2NEP1K5AE]], then $T + K$ is Fredholm and $\operatorname{ind}(T + K) = \operatorname{ind} T$.
The Fredholm operators form an open subset of the bounded operators, and the index is constant on its connected components.
</document>

<document source="https://tamnd.github.io/math2/node/ADKBEJCTZE.md" tag="ADKBEJCTZE" kind="definition">
# 3-j symbol

## Definition

For $j \in \set{0, \frac12, 1, \dots}$, let $V_j$ be the [[irreducible representation|FX3NEFYFFN]] of [[SU(2)|E5P8AG6CK7]] of dimension $2j+1$, with the standard [[orthonormal basis|0S9GGTR0EX]] $\lvert j m \rangle$, $m = j, j-1, \dots, -j$, of the Condon–Shortley convention.
Let $j_1, j_2, J$ be such that $V_J$ occurs in the [[tensor product|0BF2DE2SYK]] $V_{j_1} \otimes V_{j_2}$, that is, $\abs{j_1 - j_2} \le J \le j_1 + j_2$ and $j_1 + j_2 + J \in \Z$.
Let $\lvert J M \rangle$ be the basis of a copy of $V_J$ in $V_{j_1} \otimes V_{j_2}$.
The *Clebsch–Gordan coefficients* $\langle j_1 m_1, j_2 m_2 \mid J M \rangle$ are its coordinates:
$$\lvert J M \rangle = \sum_{m_1, m_2} \langle j_1 m_1, j_2 m_2 \mid J M \rangle\, \lvert j_1 m_1 \rangle \otimes \lvert j_2 m_2 \rangle,$$
where the copy is fixed by the convention that all coefficients are real and $\langle j_1 j_1, j_2 (J - j_1) \mid J J \rangle > 0$.
The *3-j symbol* is
$$\begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix} = \frac{(-1)^{j_1 - j_2 - m_3}}{\sqrt{2 j_3 + 1}}\, \langle j_1 m_1, j_2 m_2 \mid j_3, -m_3 \rangle.$$
It is set to $0$ when $j_1, j_2, j_3$ fail the conditions above or some $\abs{m_i} > j_i$.

## Remarks

The symbol is $0$ unless $m_1 + m_2 + m_3 = 0$, since $J_z$ is additive on tensor products.
It is unchanged by an even permutation of its columns and is multiplied by $(-1)^{j_1 + j_2 + j_3}$ by an odd permutation.
The orthogonality relation reads $\sum_{m_1, m_2} (2 j_3 + 1) \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j_3' \\ m_1 & m_2 & m_3' \end{pmatrix} = \delta_{j_3 j_3'} \delta_{m_3 m_3'}$.

Example: for $j_1 = j_2 = \frac12$ and $J = 0$, the invariant vector is $\lvert 0 0 \rangle = \frac{1}{\sqrt2}\left(\lvert \tfrac12 \tfrac12 \rangle \otimes \lvert \tfrac12, -\tfrac12 \rangle - \lvert \tfrac12, -\tfrac12 \rangle \otimes \lvert \tfrac12 \tfrac12 \rangle\right)$.
So $\begin{pmatrix} \frac12 & \frac12 & 0 \\ \frac12 & -\frac12 & 0 \end{pmatrix} = \frac{(-1)^{0}}{1} \cdot \frac{1}{\sqrt2} = \frac{1}{\sqrt2}$.
This agrees with the general value $\begin{pmatrix} j & j & 0 \\ m & -m & 0 \end{pmatrix} = \frac{(-1)^{j-m}}{\sqrt{2j+1}}$.

Non-example: $\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \end{pmatrix} = 0$, because $m_1 + m_2 + m_3 = 2 \ne 0$.

Physicists write the coefficients $\langle j_1 m_1, j_2 m_2 \mid J M \rangle$ also as $C^{J M}_{j_1 m_1 j_2 m_2}$.
</document>

<document source="https://tamnd.github.io/math2/node/ADKK56VZHW.md" tag="ADKK56VZHW" kind="concept">
# Distinguished triangle

## Summary

A distinguished triangle is a sequence $X \to Y \to Z \to X[1]$ of morphisms in a [[triangulated category|T2V7GJJ2EN]] that belongs to the chosen class of exact triangles.
It is the analogue of a short exact sequence: $Z$ plays the role of a cokernel of $X \to Y$, and $X[1]$ records the connecting map.
In the [[derived category|5ST7NVJFSM]], every short exact sequence of complexes gives a distinguished triangle.
In the homotopy category of chain complexes the standard example is built from the [[mapping cone|6CXPGB13WD]].

## Explanations

### L1 human

A distinguished triangle is three objects and three maps that go around in a circle, with a shift on the way back.
It says that $Z$ is what is left of $Y$ after removing the image of $X$, in a homotopy-invariant sense.
Unlike a cokernel, $Z$ also remembers what $X \to Y$ kills, shifted by one degree.
For complexes, the triangle $X \to Y \to \operatorname{Cone}(f) \to X[1]$ of a chain map $f$ is the basic example.
</document>

<document source="https://tamnd.github.io/math2/node/ADNHVGY2W4.md" tag="ADNHVGY2W4" kind="concept">
# Tautological one-form

## Summary

The tautological one-form is a canonical $1$-form on the cotangent bundle of any smooth manifold.
In coordinates $(q, p)$ it is $\sum_i p_i \, dq^i$, and it needs no choice of coordinates or metric.
Its exterior derivative gives the canonical [[symplectic form|CAN490EPRT]] of the cotangent bundle, which makes every cotangent bundle a symplectic manifold.
It is also called the Liouville form or the canonical one-form.

## Explanations

### L1 human

A point of the cotangent bundle is a pair: a position $q$ in the manifold and a covector $p$ at $q$, which in mechanics is a momentum.
The tautological form takes a small motion of such a pair, looks only at how the position moves, and feeds that motion to $p$.
So it uses the point itself as the instructions for what to measure, which is why it is called tautological.
On the cotangent bundle of the line, with coordinates $(q, p)$, it is $p \, dq$.
</document>

<document source="https://tamnd.github.io/math2/node/ADSVMCY77A.md" tag="ADSVMCY77A" kind="concept">
# Quadrature rule

## Summary

A quadrature rule approximates a [[definite integral|04TMP0XGMJ]] $\int_a^b f$ by a weighted sum $\sum_i w_i f(x_i)$ of values of $f$ at fixed nodes.
It is the basic object of [[numerical integration|7HYF6YG0RS]] in one dimension.
Rules are compared by the polynomials they integrate exactly and by their error for smooth functions.
Newton–Cotes rules use equally spaced nodes, and Gaussian rules use optimally placed nodes.

## Explanations

### L1 human

To estimate an area under a curve, one can measure the curve's height at a few chosen spots and combine the heights with fixed multipliers.
The spots are the nodes, the multipliers are the weights, and the recipe is the quadrature rule.
The same recipe is then used for every function.
For example, the midpoint rule measures the height once, at the centre, and multiplies it by the length of the interval.
</document>

<document source="https://tamnd.github.io/math2/node/ADTYF8FG6C.md" tag="ADTYF8FG6C" kind="concept">
# Symmetric kernel

## Summary

A symmetric kernel is a real-valued [[kernel|GSMBXT7FPK]] with $K(x, s) = K(s, x)$ for all $x$ and $s$.
It is the continuous analogue of a [[symmetric matrix|0VR671PHE5]], and the real case of a [[Hermitian kernel|2ZB3NFZ94T]].
A nonzero square-integrable symmetric kernel has real characteristic values and an orthonormal system of eigenfunctions, as the Hilbert–Schmidt theory shows.
Green's functions of self-adjoint boundary value problems are symmetric kernels.

## Explanations

### L1 human

The influence of the point $s$ on the point $x$ is the same as the influence of $x$ on $s$.
If one draws the kernel as a surface over the square, the surface is mirror symmetric across the diagonal $x = s$.
For example, $K(x, s) = \min(x, s)$ and $K(x, s) = e^{-\abs{x - s}}$ are symmetric.
The kernel $K(x, s) = x$ is not, because $K(1, 0) = 1$ but $K(0, 1) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/ADZAXRA6QT.md" tag="ADZAXRA6QT" kind="concept">
# Quasicircle

## Summary

A quasicircle is the image of a circle under a [[quasiconformal mapping|SB8KG46NHP]] of the plane.
Quasicircles are Jordan curves, but they can be very rough, such as the Koch snowflake.
Ahlfors characterized them geometrically: a Jordan curve is a quasicircle exactly when it has no narrow necks or cusps, in a precise metric sense.
They appear as limit sets of quasi-Fuchsian groups and as Julia sets of some polynomials.

## Explanations

### L1 human

Take a round circle and deform the whole plane by a map that never squashes tiny circles too much.
The image of the circle is a quasicircle.
Such a curve may wiggle a lot at every scale, but it never comes back close to itself after a long detour.
For example, the boundary of a square is a quasicircle, while a curve with a sharp inward cusp is not.
</document>

<document source="https://tamnd.github.io/math2/node/AE08JXN7CA.md" tag="AE08JXN7CA" kind="definition">
# Birkhoff interpolation

## Definition

Let $m \ge 1$ and $n \ge 0$, and let $x_1, \dots, x_m \in \R$ be pairwise distinct.
An *incidence matrix* is an $m \times (n + 1)$ matrix $E = (e_{ij})$, with $1 \le i \le m$ and $0 \le j \le n$.
Its entries are $0$ or $1$, and exactly $n + 1$ of them are equal to $1$.
Given real numbers $y_{ij}$ for the pairs with $e_{ij} = 1$, the *Birkhoff interpolation problem* asks for a polynomial $p$ with real coefficients such that:

1. $p = 0$ or $\deg p \le n$;
2. $p^{(j)}(x_i) = y_{ij}$ for every pair $(i, j)$ with $e_{ij} = 1$, where $p^{(j)}$ is the $j$-th [[derivative|0V1EJCT8G6]].

The pair of $E$ and the nodes is *poised* if this problem has exactly one solution for every choice of the data $y_{ij}$.

## Remarks

The conditions form a square linear system in the $n + 1$ coefficients of $p$, so poisedness means that its determinant is nonzero.
If in each row of $E$ the ones form an initial block $e_{i0} = \cdots = e_{i,k_i - 1} = 1$, the problem is [[Hermite interpolation|9Z50M2WPTP]], which is always poised.
If moreover each row has a single one in column $0$, it is [[polynomial interpolation|SS1YMS2NPC]].

Example: nodes $x_1 = 0$, $x_2 = 1$, $n = 1$, conditions $p(0) = a$ and $p'(1) = b$.
For $p = c_0 + c_1 x$ these give $c_0 = a$ and $c_1 = b$, so the problem is poised.

Non-example: nodes $-1, 0, 1$, $n = 2$, conditions $p(-1) = a$, $p'(0) = b$, $p(1) = c$.
For $p = c_0 + c_1 x + c_2 x^2$ the system has the matrix with rows $(1, -1, 1)$, $(0, 1, 0)$, $(1, 1, 1)$, whose determinant is $0$.
For instance $p = x^2 - 1$ is a nonzero solution with zero data, so the problem is not poised.
</document>

<document source="https://tamnd.github.io/math2/node/AE0EZQH4ZM.md" tag="AE0EZQH4ZM" kind="statement">
# Marcinkiewicz–Zygmund inequality

## Statement

Let $1 \le p < \infty$.
There are constants $0 < A_p \le B_p < \infty$, depending only on $p$, with the following property.
Let $N \ge 1$ and let $X_1, \dots, X_N$ be [[independent|0GQ938GDE0]] real or complex [[random variables|0PNRKMRC55]] with $\mathbb{E}[X_i] = 0$ and $\mathbb{E}[\abs{X_i}^p] < \infty$ for each $i$, where $\mathbb{E}$ is the [[expected value|0WQNRCNY3M]].
Then
$$A_p \, \mathbb{E}\left[ \left( \sum_{i=1}^N \abs{X_i}^2 \right)^{p/2} \right] \le \mathbb{E}\left[ \Big\vert \sum_{i=1}^N X_i \Big\vert^p \right] \le B_p \, \mathbb{E}\left[ \left( \sum_{i=1}^N \abs{X_i}^2 \right)^{p/2} \right].$$

## Remarks

The proof uses symmetrization and the [[Khintchine inequality|AGWMZ50SSZ]].
Let $(\varepsilon_i)$ be independent random signs, independent of the $X_i$.
Because the $X_i$ have mean zero, $\mathbb{E}\abs{\sum X_i}^p$ is comparable, with constants depending only on $p$, to $\mathbb{E}\abs{\sum \varepsilon_i X_i}^p$.
Conditioning on the $X_i$ and applying Khintchine's inequality to the signs gives the comparison with $\mathbb{E}(\sum \abs{X_i}^2)^{p/2}$.

For $p = 2$ the inequality is an identity with $A_2 = B_2 = 1$: independence and mean zero give $\mathbb{E}\abs{\sum X_i}^2 = \sum \mathbb{E}\abs{X_i}^2$.
The mean zero hypothesis is needed: if every $X_i = 1$, the middle term is $N^p$ while the outer terms are of order $N^{p/2}$, so the upper bound fails for large $N$.

## Explanations

### L1 human

A sum of independent random quantities with average zero has much cancellation.
The inequality says that its size, measured by a $p$-th moment, is comparable to the size of the square root of the sum of the squares of the terms.
So the sum behaves as if the terms were orthogonal, as in the Pythagorean theorem.

### L2 human

The inequality is the probabilistic counterpart of the [[Khintchine inequality|AGWMZ50SSZ]], which is the special case of random signs times constants.
It is a model for the Littlewood–Paley inequality: the dyadic pieces of a function play the role of independent terms, and the square function plays the role of $\left( \sum \abs{X_i}^2 \right)^{1/2}$.
Burkholder extended it to martingale differences for $1 < p < \infty$, which links it further with Littlewood–Paley theory.
It is used to bound moments of sums of independent variables, for example in proofs of laws of large numbers with rates.
</document>

<document source="https://tamnd.github.io/math2/node/AE0VPNDDXZ.md" tag="AE0VPNDDXZ" kind="definition">
# Signed measure

## Definition

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]].
A *signed measure* on $(X, \Sigma)$ is a map $\nu \colon \Sigma \to [-\infty, \infty]$, with values in the [[extended real line|0GSZ7B0CGR]], such that:

1. $\nu(\emptyset) = 0$;
2. $\nu$ takes at most one of the two values $+\infty$ and $-\infty$;
3. $\nu$ is [[countably additive|D7KB9HFCGZ]]: for pairwise disjoint $E_1, E_2, \ldots$ in $\Sigma$,
$$\nu\Big( \bigcup_{n} E_n \Big) = \sum_{n} \nu(E_n),$$
where the series converges absolutely when the left side is finite.

## Remarks

Example: let $\mu$ be a measure on $(X, \Sigma)$ and $f \in L^1(\mu)$ real-valued.
Then $\nu(E) = \int_E f \, d\mu$ is a signed measure with finite values.
Countable additivity follows from the dominated convergence theorem, and absolute convergence from $\sum_n \abs{\nu(E_n)} \le \int \abs{f} \, d\mu$.
More generally, $\mu_1 - \mu_2$ is a signed measure whenever $\mu_1, \mu_2$ are measures and at least one of them is finite.

Non-example: on $X = \set{a, b}$ with all subsets measurable, no signed measure has $\nu(\set{a}) = +\infty$ and $\nu(\set{b}) = -\infty$.
Clause 2 excludes it, and indeed $\nu(\set{a, b}) = \nu(\set{a}) + \nu(\set{b})$ would be $\infty - \infty$.

A *finite* signed measure is one with values in $\R$.
Every measure is a signed measure.
</document>

<document source="https://tamnd.github.io/math2/node/AE1HJ8D8GV.md" tag="AE1HJ8D8GV" kind="statement">
# Calderón–Zygmund theorem

## Statement

Let $n \ge 1$ and let $K$ be a locally integrable function on $\R^n \setminus \set{0}$.
Assume that there is a constant $A$ such that:

1. $\abs{K(x)} \le A\abs{x}^{-n}$ for all $x \ne 0$;
2. $\int_{\abs{x} \ge 2\abs{y}} \abs{K(x - y) - K(x)} \, dx \le A$ for all $y \ne 0$;
3. $\int_{r < \abs{x} < R} K(x) \, dx = 0$ for all $0 < r < R$.

For $\varepsilon > 0$, let $T_\varepsilon f(x) = \int_{\abs{y} > \varepsilon} K(y) f(x - y) \, dy$.
Then for each $1 < p < \infty$ there is a constant $C$, depending only on $n$ and $p$, such that
$$\norm{T_\varepsilon f}_{L^p(\R^n)} \le C A \norm{f}_{L^p(\R^n)} \quad \text{for all } f \in L^p(\R^n) \text{ and } \varepsilon > 0.$$
Moreover, the operators $T_\varepsilon$ are of weak type $(1, 1)$, uniformly in $\varepsilon$.

## Remarks

The proof has three steps.
The three conditions together give a uniform bound on the Fourier transforms of the truncated kernels, hence a uniform $L^2$ bound by the Plancherel theorem.
The [[Calderón–Zygmund lemma|K1YKXVPVDG]] splits $f$ into a bounded good part and a bad part with mean zero on disjoint cubes.
Condition 2 controls the operator on the bad part away from the cubes, and this gives the weak type $(1, 1)$ bound.
The [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]] then gives $1 < p \le 2$, and duality gives $2 \le p < \infty$.

Example: on $\R$, $K(x) = \frac{1}{\pi x}$ satisfies condition 1 with $A = 1/\pi$, and condition 3 because $K$ is odd.
For condition 2, $\abs{K(x - y) - K(x)} = \frac{\abs{y}}{\pi \abs{x}\abs{x - y}} \le \frac{2\abs{y}}{\pi x^2}$ when $\abs{x} \ge 2\abs{y}$, whose integral is $2/\pi$.
So the truncated Hilbert transforms are bounded on $L^p(\R)$ uniformly in $\varepsilon$, which recovers the [[M. Riesz theorem|PJFH51Z48M]].

The cancellation condition 3 is needed.
For $K(x) = \abs{x}^{-n}$, conditions 1 and 2 hold, but $T_\varepsilon f(x)$ grows like $\log(1/\varepsilon)$ for $f$ equal to $1$ near $x$.
The conclusion fails for $p = 1$ and $p = \infty$, as already for the Hilbert transform.

## Explanations

### L1 human

Integrating a function against a kernel of size $\abs{x}^{-n}$ is dangerous, since this kernel is not integrable near $0$ or near infinity.
Suppose the kernel is smooth on average, and its positive and negative values balance on every shell.
The theorem says that the integral then gives an operator that preserves $L^p$ size, for $1 < p < \infty$.
The Hilbert transform and the Riesz transforms are the main examples.

### L2 human

This is the central theorem of the theory of [[singular integral operators|HTMCEARGDD]].
Its method, an $L^2$ bound, the Calderón–Zygmund decomposition and interpolation, also proves the Hörmander–Mikhlin multiplier theorem and the bounds for maximal functions.
It extends to non-convolution operators, the [[Calderón–Zygmund operators|3SCXCW4AH3]], once an $L^2$ bound is known; the [[T(1) theorem|3FBF628XEF]] gives a test for that $L^2$ bound.
</document>

<document source="https://tamnd.github.io/math2/node/AE1VC00NZN.md" tag="AE1VC00NZN" kind="definition">
# Right-angled Artin group

## Definition

Let $\Gamma$ be a finite simple [[graph|00C0BR7D7H]] with vertex set $V$ and edge set $E$.
The *right-angled Artin group* $A(\Gamma)$ is the group with [[presentation|0JGEQXQ4AC]]
$$A(\Gamma) = \langle V \mid uv = vu \text{ for each edge } \set{u, v} \in E \rangle.$$
A group is a right-angled Artin group if it is isomorphic to $A(\Gamma)$ for some finite simple graph $\Gamma$.

## Remarks

Example: if $\Gamma$ has $n$ vertices and no edges, $A(\Gamma)$ is the [[free group|0KTW26C80J]] $F_n$; if $\Gamma$ is complete, $A(\Gamma)$ is the [[free abelian group|0WJX8MGG37]] $\Z^n$.
Example: if $\Gamma$ is the path $a - b - c$, the relations say that $b$ commutes with $a$ and $c$, and $a, c$ generate a free group, so $A(\Gamma) \cong F_2 \times \Z$.

Non-example: the braid group $B_3 = \langle a, b \mid aba = bab \rangle$ is an Artin group but not a right-angled one.
Its abelianisation is $\Z$, since the relation forces $a = b$ there; the only right-angled Artin group with abelianisation $\Z$ is $\Z$ itself, and $B_3$ is not abelian.

The abelianisation of $A(\Gamma)$ is $\Z^{V}$.
Two right-angled Artin groups $A(\Gamma)$ and $A(\Gamma')$ are isomorphic if and only if the graphs $\Gamma$ and $\Gamma'$ are isomorphic.
They are also called graph groups or partially commutative groups.
</document>

<document source="https://tamnd.github.io/math2/node/AEABS105PS.md" tag="AEABS105PS" kind="definition">
# Direct and inverse theorems of approximation

## Definition

Let $X$ be a normed space of functions and let $V_0 \subseteq V_1 \subseteq \cdots$ be subspaces of $X$.
For $f \in X$ let $E_n(f) = \inf_{v \in V_n} \norm{f - v}$ be the [[error of best approximation|YDN0T4MEX0]].
Let $\Omega(f, \delta)$, for $\delta > 0$, be a measure of smoothness of $f$, such as a [[modulus of continuity|AXCJBSQZWF]].

1. A *direct theorem* is an inequality $E_n(f) \le C \, \Omega(f, 1/n)$ for all $f \in X$ and $n \ge 1$, with $C$ independent of $f$ and $n$.
2. An *inverse theorem* is an implication that bounds $\Omega(f, \delta)$ in terms of the sequence $(E_n(f))_{n \ge 0}$.

## Remarks

Example: let $X$ be the continuous $2\pi$-periodic functions with the uniform norm, and $V_n$ the [[trigonometric polynomials|J3NJQ5FZJ5]] of degree at most $n$.
Jackson's inequality $E_n(f) \le C \, \omega(f, 1/n)$, with $\omega$ the modulus of continuity, is a direct theorem.
Bernstein's theorem, that $E_n(f) \le M n^{-\alpha}$ for all $n \ge 1$ with $0 < \alpha < 1$ implies $\omega(f, \delta) \le M' \delta^{\alpha}$, is an inverse theorem.
Combined, they say that $E_n(f) = O(n^{-\alpha})$ if and only if $f$ is [[Hölder continuous|012PBA0FMD]] with exponent $\alpha$.

Non-example: the bound $E_n(f) \le \norm{f}$, from $0 \in V_n$, is not a direct theorem.
It does not involve any smoothness of $f$ and gives no rate as $n \to \infty$.

Inverse theorems usually rest on inequalities of Bernstein type for the derivatives of elements of $V_n$.
Direct theorems usually rest on good approximating operators, such as convolution with the Jackson kernel.
</document>

<document source="https://tamnd.github.io/math2/node/AEDA02M5NZ.md" tag="AEDA02M5NZ" kind="definition">
# Identifiability

## Definition

Let $\mathcal{P} = \set{P_\theta : \theta \in \Theta}$ be a [[statistical model|JZ6R7J36K5]] indexed by a parameter space $\Theta$.
The model is *identifiable* if the map $\theta \mapsto P_\theta$ from $\Theta$ to the set of probability measures is [[injective|0PVG1CRAJN]].
That is, for all $\theta_1, \theta_2 \in \Theta$, if $P_{\theta_1} = P_{\theta_2}$, then $\theta_1 = \theta_2$.

## Remarks

Example: the normal model $P_{(\mu, \sigma^2)} = \mathcal{N}(\mu, \sigma^2)$ with $\mu \in \R$ and $\sigma^2 > 0$ is identifiable, because the mean and the variance of the law recover $\mu$ and $\sigma^2$.
Non-example: let $P_{(a, b)} = \mathcal{N}(a + b, 1)$ with $(a, b) \in \R^2$.
Then $P_{(1, 0)} = P_{(0, 1)}$ while $(1, 0) \ne (0, 1)$, so the model is not identifiable.
A function $\tau(\theta)$ is identifiable when $P_{\theta_1} = P_{\theta_2}$ implies $\tau(\theta_1) = \tau(\theta_2)$; in the non-example, $\tau(a, b) = a + b$ is identifiable.
</document>

<document source="https://tamnd.github.io/math2/node/AEF4CBRAMN.md" tag="AEF4CBRAMN" kind="definition">
# Abelian category

## Definition

An *abelian category* is an [[additive category|B2VP96NJSK]] $\mathcal{A}$ such that:

1. every morphism of $\mathcal{A}$ has a [[kernel|S417YNEJ6M]] and a [[cokernel|H8W38933NC]];
2. every [[monomorphism|Y2NP1JFR86]] of $\mathcal{A}$ is the kernel of some morphism;
3. every [[epimorphism|RVXDSY04KG]] of $\mathcal{A}$ is the cokernel of some morphism.

## Remarks

Clauses 2 and 3 are equivalent, given clause 1, to the condition that for every morphism $f$ the canonical morphism $\operatorname{coim} f \to \operatorname{im} f$ is an isomorphism, where $\operatorname{coim} f = \operatorname{coker}(\ker f)$ and $\operatorname{im} f = \ker(\operatorname{coker} f)$.
In an abelian category, a morphism that is both a monomorphism and an epimorphism is an isomorphism.

Example: the category of left [[modules|0NZ8JA25WK]] over a ring $R$ is abelian.
A monomorphism $m \colon M \to N$ is injective, and it is the kernel of the quotient map $N \to N / m(M)$.
An epimorphism $e \colon N \to P$ is surjective, and it is the cokernel of the inclusion of its kernel.

Non-example: the category of [[free abelian groups|0WJX8MGG37]] is additive but not abelian.
Multiplication by $2$ on $\Z$ is an epimorphism there: if $u \circ 2 = v \circ 2$ with torsion-free target, then $2(u - v) = 0$ forces $u = v$.
It is not the cokernel of any morphism $k$: from $2 \circ k = 0$ follows $k = 0$, whose cokernel is the identity of $\Z$, not multiplication by $2$.
So clause 3 fails.
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<document source="https://tamnd.github.io/math2/node/AEG6QWEFRG.md" tag="AEG6QWEFRG" kind="problem">
# Minimal surface of revolution

## Problem

Let $a < b$ and $A, B > 0$.
Among curves $y \in C^1([a, b])$ with $y > 0$, $y(a) = A$ and $y(b) = B$, find one whose [[surface of revolution|NE09EDK19V]] about the $x$-axis has the least [[surface area|0S29Y4ZW0Y]]
$$S(y) = 2\pi \int_a^b y(x) \sqrt{1 + y'(x)^2} \, dx.$$

## Remarks

The Lagrangian $y\sqrt{1 + y'^2}$ does not depend on $x$, so the [[Beltrami identity|F7EKTEESQ5]] gives $y / \sqrt{1 + y'^2} = c$ for a constant $c > 0$.
The solutions are the catenaries $y(x) = c \cosh\big((x - d)/c\big)$, and their surfaces of revolution are [[catenoids|6NN1HP54GB]].
Depending on the positions of the two end points, there are two, one or no catenaries through them.
When the end circles are far apart compared with their radii, no catenary joins the points and there is no minimiser among such smooth curves.
Then the infimum of $S$ is $\pi(A^2 + B^2)$, the area of the two end discs, approached by curves that run close to the axis; this limit is the Goldschmidt solution.
For some positions where catenaries do exist, the two discs still have smaller total area than every catenoid through the points, and again no smooth curve attains the infimum.

## Explanations

### L1 human

Dip two parallel rings into soapy water and pull them apart a little.
The soap film between them takes the shape of a catenoid, a surface of revolution whose area cannot be lowered by small changes.
If the rings are pulled too far apart, the film breaks and leaves a flat film on each ring.
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<document source="https://tamnd.github.io/math2/node/AES2T5JPFP.md" tag="AES2T5JPFP" kind="concept">
# Tempered distribution

## Summary

A tempered distribution on $\R^n$ is a continuous linear functional on the [[Schwartz space|CG9JEATW3Q]] of smooth, rapidly decreasing functions.
Tempered distributions include all $L^p$ functions, all polynomials, finite measures and the Dirac delta.
They form the natural domain of the [[Fourier transform|DPVM34ADZV]], which maps the space of tempered distributions onto itself.
For this reason they are the standard setting for linear partial differential equations with constant coefficients and for function spaces defined by Fourier analysis.

## Explanations

### L1 human

A tempered distribution is a "generalized function" that is known only through its averages against very nice test functions.
The test functions are smooth and decay faster than any power of $\abs{x}$, so the generalized function may grow, but at most like a polynomial.
A function $f$ of polynomial growth gives the tempered distribution $\varphi \mapsto \int f \varphi \, dx$.
The Dirac delta, $\varphi \mapsto \varphi(0)$, is a tempered distribution that is not a function at all.
The function $e^x$ on $\R$ grows too fast, and it is not a tempered distribution.
</document>

<document source="https://tamnd.github.io/math2/node/AEX1QP3178.md" tag="AEX1QP3178" kind="definition">
# Baire function

## Definition

Let $X$ be a [[metric space|063W5JMY85]], for example a subset of $\R$.
Let $\omega_1$ be the first uncountable [[ordinal|0RDJK7EQYY]].
For each ordinal $\alpha < \omega_1$, define a set $B_\alpha$ of functions $X \to \R$ as follows.

1. $B_0$ is the set of [[continuous functions|0P27W9RW0G]] $X \to \R$.
2. For $0 < \alpha < \omega_1$, $B_\alpha$ is the set of functions $f \colon X \to \R$ that are [[pointwise limits|056FSKG32J]] $f(x) = \lim_{n \to \infty} f_n(x)$ of sequences $(f_n)$ in $\bigcup_{\beta < \alpha} B_\beta$.

A function of *Baire class* $\alpha$ is an element of $B_\alpha$.
A *Baire function* is a function that lies in $B_\alpha$ for some $\alpha < \omega_1$.

## Remarks

The classes increase: $B_\beta \subseteq B_\alpha$ for $\beta \le \alpha$, since a function is the limit of a constant sequence.
The set of Baire functions is the smallest set of functions $X \to \R$ that contains the continuous functions and is closed under pointwise limits of sequences.
For $X = \R$, the Baire functions are exactly the functions that are measurable for the [[Borel sigma-algebra|0BVHZXWKCD]].

Example: if $F \colon \R \to \R$ is differentiable, then $F'$ is of Baire class $1$.
Indeed $F'(x) = \lim_{n \to \infty} n \big(F(x + 1/n) - F(x)\big)$, and each function on the right is continuous.

Non-example: the [[Dirichlet function|0T4EJ8ZJM9]] $\mathbf 1_\Q$ is not of Baire class $1$.
A function of Baire class $1$ on $\R$ is continuous at the points of a dense set, while $\mathbf 1_\Q$ is continuous nowhere.
It is of Baire class $2$, since $\mathbf 1_\Q(x) = \lim_{m \to \infty} \lim_{n \to \infty} \cos(m! \pi x)^{2n}$.
</document>

<document source="https://tamnd.github.io/math2/node/AEXJNRBASV.md" tag="AEXJNRBASV" kind="concept">
# Semiconjugacy and factor map

## Summary

A semiconjugacy from one dynamical system to another is a continuous surjection that intertwines the two maps.
The target system is then called a factor of the source, and the source an extension of the target.
The map itself is a factor map.
A factor is a simplified picture of the system that may forget some information.
Still, every orbit of the factor is the image of an orbit of the extension.
A semiconjugacy that is a homeomorphism is a [[topological conjugacy|HNXXAPQKG3]]; the same idea with measure-preserving maps gives factors in ergodic theory.

## Explanations

### L1 human

A factor map is like a blurry camera filming a system: it may merge different states, but it is faithful to the motion.
Watching the blurry film step by step is the same as filming each step of the real system.
For example, write a number in $[0, 1)$ in binary and shift the digits one place to the left.
Reading the digits as a number turns this shift into the doubling map $x \mapsto 2x \bmod 1$, so the doubling map is a factor of the shift.
</document>

<document source="https://tamnd.github.io/math2/node/AEZ7PPCNQS.md" tag="AEZ7PPCNQS" kind="concept">
# Perron method

## Summary

The Perron method solves the [[Dirichlet problem|8D56GQ4AG8]] for the Laplacian on an arbitrary bounded open set.
It takes the upper envelope of all [[subharmonic functions|Z1MHMGFNFW]] that lie below the boundary data at the boundary.
This envelope is always harmonic; the only question is whether it takes the right boundary values.
That question is local and leads to barriers and regular boundary points.

## Explanations

### L1 human

You want a harmonic function with given values on the boundary.
Look at all subharmonic functions that stay below these values near the boundary, and at each point take the largest value any of them reaches.
Subharmonic functions sag below harmonic ones, so this supremum is the best harmonic candidate.
On the unit disc with boundary values $\cos\theta$, the supremum is the harmonic function $x$.
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<document source="https://tamnd.github.io/math2/node/AEZE6B2XXY.md" tag="AEZE6B2XXY" kind="concept">
# Collectively compact operator approximation

## Summary

A sequence of operators $K_n$ is a collectively compact approximation of an operator $K$ if $K_n \to K$ pointwise.
In addition, the images of the unit ball under all $K_n$ together must form a relatively compact set.
This weaker notion replaces norm convergence, which often fails for numerical methods such as the Nyström method.
It still gives the main conclusion when $I - K$ is invertible.
Then $I - K_n$ is invertible for large $n$, with uniformly bounded inverses, and the approximate solutions converge.
The theory was developed by P. M. Anselone for the numerical solution of integral equations.

## Explanations

### L1 human

In the Nyström method one replaces an integral by a weighted sum over finitely many points.
The resulting operators do not get close to the integral operator in the strongest sense, because a function can be large between the quadrature points.
But all of them smooth functions in the same uniform way, and that is enough for the approximate equations to have solutions close to the true one.
A simple example: with the rectangle rule, $(K_n\varphi)(x) = \frac{1}{n} \sum_{j=1}^n k(x, \frac{j}{n}) \varphi(\frac{j}{n})$ approximates $\int_0^1 k(x, t) \varphi(t) \, dt$.
</document>

<document source="https://tamnd.github.io/math2/node/AEZNCY3N3K.md" tag="AEZNCY3N3K" kind="definition">
# Abelian and Tauberian theorems

## Definition

Let $M$ be a [[summability method|8X25Q4XYMN]] that assigns values to some [[series|0JH4MN4K1C]] of complex numbers.

1. An *Abelian theorem* for $M$ is a theorem stating that every [[convergent series|0CVVKY7GY2]] is $M$-summable to its sum.
More generally, an Abelian theorem for two methods $M$ and $M'$ states that every $M$-summable series is $M'$-summable to the same value.
2. A *Tauberian theorem* for $M$ is a theorem stating that an $M$-summable series converges if its terms satisfy a stated additional condition.
Its sum is then its $M$-value.
The additional condition is called a [[Tauberian condition|6ZDP261K7R]].

The same terms are used for methods that act on sequences, functions or measures, with "convergent" replaced by the corresponding ordinary limit.

## Remarks

Example of an Abelian theorem: Abel's theorem on power series says that if $\sum a_n$ converges to $s$, then $\sum a_n x^n \to s$ as $x \to 1^-$.
Example of a Tauberian theorem: Tauber's theorem says that if $\sum a_n x^n \to s$ as $x \to 1^-$ and $n a_n \to 0$, then $\sum a_n$ converges to $s$.

Non-example: the statement "if $\sum a_n x^n \to s$ as $x \to 1^-$, then $\sum a_n$ converges" has no Tauberian condition, and it is false.
For $a_n = (-1)^n$ one has $\sum a_n x^n = \frac{1}{1+x} \to \frac12$, but $\sum (-1)^n$ diverges.

The names honour Abel and Tauber.
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<document source="https://tamnd.github.io/math2/node/AF9NY73QHJ.md" tag="AF9NY73QHJ" kind="concept">
# Lifting scheme

## Summary

The lifting scheme is a way to build and compute wavelet transforms by a sequence of simple, invertible steps.
A signal is split into its even and odd samples.
The odd samples are predicted from the even ones, and the even ones are updated from the prediction errors.
Every step can be undone by reversing its sign, so perfect reconstruction is automatic.
The scheme can cut the cost of many [[fast wavelet transforms|PDVYQAW887]] by up to about one half.
It works in place, gives integer-to-integer transforms, and extends to irregular grids.

## Explanations

### L1 human

Imagine a signal as two interleaved lists: the even-numbered and the odd-numbered samples.
If the signal is smooth, each odd sample is close to its even neighbours, so it can be guessed from them.
Store only the error of the guess; for smooth signals these errors are small.
Then adjust the even samples slightly so that they keep the average of the signal.
For example, with $x = (2, 4, 6, 8)$ and the guess "odd equals the even sample before it", the errors are $(2, 2)$ and the updated evens are $(3, 7)$, the pairwise averages.
Running the steps backwards gives back the original signal exactly.
</document>

<document source="https://tamnd.github.io/math2/node/AFBY76MAX7.md" tag="AFBY76MAX7" kind="definition">
# Porous medium equation

## Definition

Let $\Omega \subseteq \R^n$ be open and let $m > 1$ be a real number.
The *porous medium equation* is the [[parabolic equation|1CCTH8X2FT]] for a function $u \colon \Omega \times (0, T) \to [0, \infty)$:
$$u_t = \Delta (u^m),$$
where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] in $x$.

## Remarks

Example: the Barenblatt solution
$$U(x, t) = t^{-\alpha} \Big( C - k \abs{x}^2 t^{-2\alpha/n} \Big)_+^{1/(m-1)}, \qquad \alpha = \frac{n}{n(m - 1) + 2}, \quad k = \frac{\alpha (m - 1)}{2 m n},$$
with a constant $C > 0$ and $s_+ = \max(s, 0)$, solves the equation on $\R^n \times (0, \infty)$ in the weak sense.
For each $t$ its support is the ball $\abs{x} \le (C/k)^{1/2}\, t^{\alpha/n}$, so it spreads at finite speed.
Non-example: for $m = 1$ the equation is the [[heat equation|VP2HN86VAG]], which is excluded; its solutions with nonnegative nonzero data are positive everywhere for $t > 0$.
Writing $\Delta(u^m) = \nabla \cdot (m u^{m-1} \nabla u)$ shows that the equation is degenerate parabolic where $u = 0$.
The case $0 < m < 1$ is called the fast diffusion equation.
</document>

<document source="https://tamnd.github.io/math2/node/AFC6JMYWEH.md" tag="AFC6JMYWEH" kind="concept">
# Affine morphism

## Summary

An affine morphism $X \to Y$ of schemes is one for which the preimage of every open affine subscheme of $Y$ is affine.
Affine morphisms over $Y$ correspond to quasi-coherent sheaves of algebras on $Y$, through the relative spectrum construction.
They include closed immersions, finite morphisms and vector bundles.

## Explanations

### L1 human

An affine morphism is a family of affine schemes parametrized by the points of $Y$, glued in a coherent way.
Over each affine piece $\operatorname{Spec} B$ of $Y$, the part of $X$ above it is the spectrum of a single $B$-algebra.
Every morphism between affine schemes is affine.
The map from the plane minus the origin to a point is not affine, since the plane minus the origin is not an affine scheme.
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<document source="https://tamnd.github.io/math2/node/AFMBQT06ZV.md" tag="AFMBQT06ZV" kind="statement">
# Kepler conjecture

## Statement

Every [[sphere packing|HF32CBN5MA]] of equal balls in $\R^3$ has upper [[packing density|4P5PSWADJD]] at most
$$\frac{\pi}{\sqrt{18}} = \frac{\pi}{3\sqrt 2} \approx 0.7405.$$
The face-centred cubic packing attains this density.

## Remarks

Johannes Kepler stated the claim in 1611.
Gauss proved it for lattice packings in 1831.
Thomas Hales announced a proof in 1998, completed with Samuel Ferguson.
It reduces the problem to the minimization of a function over finitely many local configurations, and checks these by computer with linear programming bounds.
The Flyspeck project, led by Hales, completed a formal proof checked by the proof assistants HOL Light and Isabelle in 2014.
The bound is attained by infinitely many packings.
Any stacking of hexagonal layers, each layer resting in the hollows of the one below, has density $\pi/\sqrt{18}$.
The face-centred cubic and the hexagonal close packings are two of them.

## Explanations

### L1 human

The usual way grocers stack oranges, in layers where each orange sits in a dip of the layer below, fills about $74\%$ of space.
Kepler guessed that no arrangement of equal balls fills more.
This took almost four hundred years to prove, and the proof needed extensive computer checks.

### L2 human

The Kepler conjecture is the three-dimensional case of the sphere packing problem; the two-dimensional case is [[Thue's theorem|3P246RHXMF]].
Unlike in the plane, the densest local cluster of balls around one ball does not extend to a packing of space, so local arguments alone fail.
Hales's proof uses a carefully chosen partition of space into cells around the balls and a weighted volume score that handles this.
The other dimensions where the answer is known are $8$ and $24$, by the work of [[Viazovska and her coauthors|YJ48S89BZA]].
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<document source="https://tamnd.github.io/math2/node/AFNE9C4NH6.md" tag="AFNE9C4NH6" kind="concept">
# Product category

## Summary

The product category $\mathcal{C} \times \mathcal{D}$ of two [[categories|A2DE66937D]] has pairs of objects as objects and pairs of morphisms as morphisms, composed in each coordinate.
It is the [[product|S5744QZV4N]] of $\mathcal{C}$ and $\mathcal{D}$ in the category of small categories.
It is needed to speak of [[functors|28NQ0KRHD3]] of two variables, such as the hom functor $\Hom(-, -)$ on $\mathcal{C}^{\mathrm{op}} \times \mathcal{C}$ or the tensor product of modules.

## Explanations

### L1 human

The product category is to categories what the [[Cartesian product|09CEDJYBKP]] is to sets.
An object is a pair, one object from each category, and an arrow is a pair of arrows, one in each category.
Two pairs of arrows chain by chaining each coordinate separately.
For example, if $\mathcal{C}$ and $\mathcal{D}$ are both the order $0 \le 1$, then $\mathcal{C} \times \mathcal{D}$ is the order on the four pairs $(a, b)$ with $(a, b) \le (a', b')$ when $a \le a'$ and $b \le b'$.
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<document source="https://tamnd.github.io/math2/node/AFQ3HQYEPS.md" tag="AFQ3HQYEPS" kind="definition">
# Branched covering

## Definition

Let $X$ and $Y$ be [[Riemann surfaces|5K0N8QMV6Q]].
A map $f \colon X \to Y$ is a *(holomorphic) branched covering* if:

1. $f$ is a [[holomorphic map|T19P7J53A1]] that is not constant, and
2. $f$ is proper, that is, $f^{-1}(K)$ is [[compact|00R0X1WGPX]] for every compact $K \subseteq Y$.

A point $x \in X$ is a *ramification point* of $f$ if $f$ is not injective on any neighbourhood of $x$; its image is a *branch point*.

## Remarks

Every branched covering $f$ is surjective, and its ramification points form a closed discrete set $R \subseteq X$.
There is an integer $d \ge 1$, the degree, such that $f$ restricts to a $d$-sheeted [[covering map|3XZRRE165N]] from $X \setminus f^{-1}(f(R))$ onto $Y \setminus f(R)$.
Every fibre has exactly $d$ points when counted with multiplicity, where the multiplicity at $x$ is the $k$ such that $f$ looks like $z \mapsto z^k$ near $x$ in suitable charts.
If $X$ is compact, every non-constant holomorphic map $X \to Y$ is a branched covering.
In topology, a branched covering of surfaces is defined in the same way with continuous maps that look locally like $z \mapsto z^k$.

Example: $f \colon \C \to \C$, $f(z) = z^2$.
It is holomorphic and non-constant, and $f^{-1}(K)$ is closed and bounded for compact $K$, since $\abs{z}^2 = \abs{f(z)}$.
Its degree is $2$, its only ramification point is $0$, and the only [[branch point|XP5JC1VXKZ]] is $0$.

Non-example: $\exp \colon \C \to \C$ is holomorphic and non-constant but not proper.
The preimage of the compact set $\set{1}$ is $2\pi i \Z$, which is not compact; clause 2 fails.
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<document source="https://tamnd.github.io/math2/node/AFQGK1NE0X.md" tag="AFQGK1NE0X" kind="definition">
# Dirichlet energy

## Definition

Let $n \ge 1$ and let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]].
Let $u \colon \Omega \to \R$ be continuously differentiable, or more generally weakly differentiable with locally integrable gradient $\nabla u$.
The *Dirichlet energy* of $u$ is
$$E[u] = \frac{1}{2} \int_\Omega \abs{\nabla u(x)}^2 \, dx \in [0, \infty].$$

## Remarks

Example: on the unit square $(0, 1)^2$, $u(x, y) = x^2$ has $\nabla u = (2x, 0)$, so $E[u] = \frac12 \int_0^1 4x^2 \, dx = \frac23$.
Non-example of finiteness: on the interval $(0, 1)$, $u(x) = x^{1/4}$ is continuously differentiable, but $E[u] = \frac12 \int_0^1 \frac{1}{16} x^{-3/2} \, dx = \infty$.
The functions in $L^2(\Omega)$ with finite energy form the [[Sobolev space|M4CEZ5K9HP]] $H^1(\Omega) = W^{1,2}(\Omega)$.
Some authors omit the factor $\frac12$; the minimizers are the same.
$E[u] = 0$ for a function on a connected open set if and only if $u$ is constant.
The energy is invariant under conformal maps in dimension $2$.
</document>

<document source="https://tamnd.github.io/math2/node/AFRP1AAPV9.md" tag="AFRP1AAPV9" kind="concept">
# Gradient flow on a manifold

## Summary

The gradient flow of a smooth function $f$ on a [[Riemannian manifold|7CBT4SM83Y]] is the [[flow|06738GGWWK]] of the vector field $-\nabla f$.
Some authors use $\nabla f$ instead.
Along its flow lines the function decreases, strictly away from the [[critical points|0YTCMFB4HC]], which are exactly the rest points of the flow.
It is the basic tool of Morse theory and of critical point theory: it deforms sublevel sets of $f$ and connects critical points.

## Explanations

### L1 human

Think of $f$ as the height of a landscape and let water run downhill along the steepest path.
The path that a drop follows is a flow line of the negative gradient flow.
Drops stop only where the ground is flat, that is, at critical points of $f$.
For example, for $f(x, y) = x^2 + y^2$ on the plane, the flow lines are the rays $t \mapsto e^{-2t}(x_0, y_0)$, which run into the minimum at the origin.
</document>

<document source="https://tamnd.github.io/math2/node/AFS2DCWCD3.md" tag="AFS2DCWCD3" kind="concept">
# Bochner–Riesz means

## Summary

The Bochner–Riesz means of a function $f$ on $\R^n$ are the operators $S^\delta_R f$ that multiply the [[Fourier transform|DPVM34ADZV]] of $f$ by $(1 - \abs{\xi}^2/R^2)_+^\delta$.
They are a smoothed version of the spherical partial sums of the Fourier integral, with smoothing order $\delta \ge 0$.
The central question is for which $\delta$ and $p$ they are bounded on $L^p(\R^n)$, which decides whether $S^\delta_R f \to f$ in $L^p$ as $R \to \infty$.
This question is tied to the restriction and Kakeya problems.

## Explanations

### L1 human

To rebuild a function from its frequencies, one may add up all frequencies of size less than $R$ and let $R$ grow.
A sharp cut at $\abs{\xi} = R$ behaves badly in higher dimensions, so one fades the frequencies out gently near the edge of the ball instead.
The exponent $\delta$ says how gentle the fade is: $\delta = 0$ is the sharp cut, and larger $\delta$ gives a smoother fade.
In one dimension with $\delta = 1$ and the factor $1 - \abs{\xi}/R$ instead, one gets the familiar Fejér means; the Bochner–Riesz means are the round analogue in $\R^n$.
</document>

<document source="https://tamnd.github.io/math2/node/AFXZGQB4EN.md" tag="AFXZGQB4EN" kind="statement">
# Conjugate gradient method

## Statement

Let $A$ be a real symmetric [[positive definite|0KX66CZMBB]] $n \times n$ matrix, let $b \in \R^n$, and let $x^* = A^{-1} b$.
For $v \in \R^n$ write $\norm{v}_A = \sqrt{v^T A v}$.
The *conjugate gradient method* starts from $x_0 \in \R^n$ with $r_0 = b - A x_0$ and $p_0 = r_0$, and for $k = 0, 1, \dots$ while $r_k \ne 0$ sets
$$\alpha_k = \frac{r_k^T r_k}{p_k^T A p_k}, \quad x_{k+1} = x_k + \alpha_k p_k, \quad r_{k+1} = r_k - \alpha_k A p_k,$$
$$\beta_k = \frac{r_{k+1}^T r_{k+1}}{r_k^T r_k}, \quad p_{k+1} = r_{k+1} + \beta_k p_k.$$
In exact arithmetic the following hold.

1. $r_k = b - A x_k$, and $x_k$ is the unique minimizer of $\norm{x - x^*}_A$ over $x \in x_0 + \mathcal{K}_k(A, r_0)$, where $\mathcal{K}_k$ is the [[Krylov subspace|QB45PCGQPV]].
2. The residuals are mutually orthogonal, $r_i^T r_j = 0$ for $i \ne j$, and the directions are $A$-conjugate, $p_i^T A p_j = 0$ for $i \ne j$.
3. Some $r_k$ with $k \le n$ is zero, so the method finds $x^*$ in at most $n$ steps.
4. With $\kappa = \lambda_{\max} / \lambda_{\min}$ the ratio of the extreme eigenvalues of $A$, for every $k$, $$\norm{x_k - x^*}_A \le 2 \left( \frac{\sqrt{\kappa} - 1}{\sqrt{\kappa} + 1} \right)^k \norm{x_0 - x^*}_A.$$

## Remarks

Clause 1 implies $x_k - x^* = q(A)(x_0 - x^*)$ for the polynomial $q$ of degree at most $k$ with $q(0) = 1$ that minimizes the $A$-norm.
So $\norm{x_k - x^*}_A \le \max_\lambda \abs{q(\lambda)} \norm{x_0 - x^*}_A$ for every such $q$, with $\lambda$ over the eigenvalues of $A$.
Choosing the scaled Chebyshev polynomial of [[Chebyshev iteration|PE5H892XVS]] gives clause 4.
Choosing $q$ that vanishes at all eigenvalues shows more: if $A$ has only $m$ distinct eigenvalues, the method stops after at most $m$ steps.
Example: for $A$ with rows $(2, 1)$ and $(1, 2)$ and $b = (3, 3)$, start at $x_0 = 0$.
Then $r_0 = (3, 3)$ is an eigenvector of $A$ for the eigenvalue $3$, so $\alpha_0 = 1/3$ and $x_1 = (1, 1) = x^*$ after one step.
Without symmetry or positive definiteness the coefficients can break down, and the minimization property is lost.

## Explanations

### L1 human

Solving $A x = b$ for a symmetric positive definite $A$ is the same as finding the bottom of the bowl $f(x) = \frac12 x^T A x - b^T x$.
Plain steepest descent zigzags down a narrow valley.
The conjugate gradient method picks each new direction so that it never spoils the progress made along the old ones.
Each step needs one matrix-vector product and a few vector operations, and memory for only four vectors.

### L2 human

Conjugate gradients is the standard [[iterative method|2RPE77ACER]] for large sparse symmetric positive definite systems, such as those from discretized elliptic equations.
In floating point the finite termination of clause 3 is lost, and the method is used as an iterative method with the rate of clause 4.
The rate depends on $\sqrt{\kappa}$, where $\kappa$ is the [[condition number|AB3S6MHJ8X]], so one usually applies it to a preconditioned system.
With a symmetric positive definite [[preconditioner|KE98SW3SBD]] $P$, such as an [[incomplete Cholesky factorization|QYYCFQ8E5D]], the same bounds hold with $P^{-1} A$ in place of $A$.
The method is due to Hestenes and Stiefel.
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<document source="https://tamnd.github.io/math2/node/AG1GMQDTDW.md" tag="AG1GMQDTDW" kind="concept">
# Invariant basis number

## Summary

A ring has invariant basis number (IBN) if any two bases of a finitely generated [[free module|0DZ9DKAKZR]] over it have the same size.
For such a ring the rank of a free module is well defined, just as the dimension of a vector space is.
All nonzero commutative rings and all division rings have this property, but some noncommutative rings do not.

## Explanations

### L1 human

Over a field, every basis of a vector space has the same number of elements; this number is the dimension.
For modules over a general ring this can fail: a free module can have a basis of $1$ element and also a basis of $2$ elements.
A ring has invariant basis number when this strange behaviour does not happen.
For example, $\Z$ has invariant basis number, so $\Z^m$ and $\Z^n$ are isomorphic only when $m = n$.
</document>

<document source="https://tamnd.github.io/math2/node/AG4FF4DNYQ.md" tag="AG4FF4DNYQ" kind="statement">
# Hilbert–Burch theorem

## Statement

Let $R$ be a commutative [[Noetherian ring|0670T59HGE]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $n \ge 1$.
Suppose that
$$0 \to R^n \xrightarrow{A} R^{n+1} \xrightarrow{B} R \to R/I \to 0$$
is a [[free resolution|HK2AA1YSQ2]] of $R/I$, where $A$ is an $(n+1) \times n$ matrix.
For $i = 1, \ldots, n+1$, let $\Delta_i$ be the determinant of the matrix obtained from $A$ by deleting row $i$.
Let $I_n(A) = (\Delta_1, \ldots, \Delta_{n+1})$ be the [[ideal of maximal minors|SXD7X0WQH1]].
Then:

1. There is a non-[[zero divisor|0NNRDW10RZ]] $a \in R$ such that the entries of $B$ are $a \Delta_1, \ldots, a \Delta_{n+1}$ up to alternating signs; in particular $I = a I_n(A)$.
2. If the [[depth|M2RADBFW99]] of $I$ on $R$ is at least $2$, then $a$ can be taken to be $1$, so $I = I_n(A)$.
3. Conversely, let $A$ be any $(n+1) \times n$ matrix over $R$ such that the depth of $I_n(A)$ on $R$ is at least $2$. Let $B$ be the row of the signed maximal minors of $A$. Then the complex above is a free resolution of $R/I_n(A)$.

## Remarks

The proof compares the resolution with the complex built from the signed minors.
That complex is exact by the Buchsbaum–Eisenbud acyclicity criterion when the minors generate an ideal of depth at least $2$.
Example: for $R = K[x, y]$ and $A = (-y, x)^T$, the maximal minors are $x$ and $-y$, and $0 \to R \xrightarrow{A} R^2 \xrightarrow{(x \;\; y)} R \to R/(x, y) \to 0$ is a free resolution.
A classical application: the ideal of the twisted cubic curve in $K[x_0, x_1, x_2, x_3]$ is generated by the three $2 \times 2$ minors of a $3 \times 2$ matrix of linear forms.
The theorem describes all perfect ideals of depth $2$, that is, all ideals $I$ of depth $2$ with $\operatorname{pd}_R R/I = 2$.

## Explanations

### L1 human

If a quotient ring $R/I$ has a free resolution of length $2$, then the whole resolution is controlled by one matrix.
The generators of $I$ are, up to a common factor, the maximal minors of that matrix.
So these ideals are determined by a single rectangular matrix.

### L2 human

The theorem is the first case of the structure theory of finite free resolutions.
It explains, for example, why the ideal of an arithmetically Cohen–Macaulay subscheme of codimension $2$ in a projective space is generated by the maximal minors of a matrix.
The analogous result for Gorenstein ideals of depth $3$ is the Buchsbaum–Eisenbud structure theorem with Pfaffians.
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# Binet's formulas for the log-gamma function

## Statement

Let $\operatorname{Log}\Gamma$ be the [[log-gamma function|JT95B44G4G]] and $\operatorname{Log}$ the principal [[complex logarithm|02QGGD5G87]].
For $\operatorname{Re} z > 0$ put
$$\mu(z) = \operatorname{Log}\Gamma(z) - \left(z - \tfrac12\right)\operatorname{Log} z + z - \tfrac12 \ln(2\pi).$$
Then for every $z$ with $\operatorname{Re} z > 0$:

1. (Binet's first formula) $$\mu(z) = \int_0^\infty \left(\frac12 - \frac1t + \frac{1}{e^t - 1}\right)\frac{e^{-zt}}{t}\, dt.$$
2. (Binet's second formula) $$\mu(z) = 2\int_0^\infty \frac{\arctan(t/z)}{e^{2\pi t} - 1}\, dt,$$ where $\arctan(t/z) = \frac{1}{2i}\operatorname{Log}\dfrac{1 + it/z}{1 - it/z}$ is the principal branch.

## Remarks

The integrand of the first formula is bounded near $t = 0$, since $\frac12 - \frac1t + \frac{1}{e^t - 1} = \frac{t}{12} + O(t^3)$.
The first formula comes from integrating the integral representation of the [[digamma function|HWRRXB0YXS]].
The second comes from the Abel–Plana summation formula, or from the first by a partial fraction expansion of $\frac{1}{e^t - 1}$.
The function $\mu$ is called Binet's function; it is positive and decreasing on $(0, \infty)$ and tends to $0$ as $x \to \infty$.
This limit is [[Stirling's formula for the gamma function|53750WSJ6K]] on the real axis, and expanding $\frac12 - \frac1t + \frac{1}{e^t - 1}$ in powers of $t$ gives [[Stirling's series|K29QYWB6A5]] term by term.
For example, the first term $\frac{t}{12}$ contributes $\int_0^\infty \frac{1}{12} e^{-zt}\,dt = \frac{1}{12z}$, the first correction in Stirling's series.

## Explanations

### L1 human

Stirling's formula approximates $\ln\Gamma(z)$ by a simple expression.
Binet's formulas give the exact error of this approximation as an integral.
From the integrals one can read off how small the error is and in which direction it goes.

### L2 human

Binet's first formula is a Laplace transform, so Watson's lemma turns the Taylor series of its integrand directly into the asymptotic expansion of $\mu(z)$.
The second formula is convenient for error bounds in the right half-plane, because the kernel $\frac{1}{e^{2\pi t} - 1}$ is explicit and positive.
Both show that $\mu(x) \to 0$ monotonically, which gives sharp bounds such as $0 < \mu(x) < \frac{1}{12x}$ for $x > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/AGMBTZQC97.md" tag="AGMBTZQC97" kind="definition">
# Matrix difference equation

## Definition

Let $K$ be a field and let $m \ge 1$.
Let $A(n)$, for $n \in \N$, be $m \times m$ [[matrices|0VD9ZSWWJG]] over $K$, and let $b(n)$, for $n \in \N$, be vectors in $K^m$.
A *matrix difference equation* (of first order) is the equation
$$x(n+1) = A(n)\, x(n) + b(n) \qquad (n \in \N)$$
for a [[sequence|0G9EKM0E15]] of vectors $x \colon \N \to K^m$.

1. It is *homogeneous* if $b(n) = 0$ for all $n$.
2. It has *constant coefficients* if $A(n) = A$ does not depend on $n$.

A *solution* is a sequence $x$ that satisfies the equation for all $n \in \N$.

## Remarks

Example: with $m = 2$, $A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ and $b = 0$, the sequence $x(n) = (F_{n+1}, F_n)^T$ of [[Fibonacci numbers|0GA9K57A39]] is a solution, since $F_{n+2} = F_{n+1} + F_n$ and the second entry just shifts.
Non-example: $x(n+1) = x(n)^2$ for $x(n) \in K$ is not a matrix difference equation, since the right side is not of the form $A(n) x(n) + b(n)$.
In the homogeneous constant-coefficient case the solutions are $x(n) = A^n x(0)$.
Equations of higher order, such as $x(n+1) = A_0 x(n) + A_1 x(n-1)$, reduce to first order by stacking $(x(n), x(n-1))$ into one vector of size $2m$.
In this way a scalar linear recurrence of order $d$ becomes a first-order system with a [[companion matrix|0ZCWPNZTQ3]].
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# Riesz–Thorin theorem

## Statement

Let $(X, \mu)$ and $(Y, \nu)$ be [[σ-finite|WEBBT18D4X]] measure spaces, and consider [[Lp spaces|SASP14MA1Z]] of complex-valued functions.
Let $1 \le p_0, p_1, q_0, q_1 \le \infty$.
Let $T$ be a linear map from $L^{p_0}(\mu) + L^{p_1}(\mu)$ to $L^{q_0}(\nu) + L^{q_1}(\nu)$.
Assume that there are constants $M_0, M_1 \ge 0$ with
$$\norm{T f}_{q_0} \le M_0 \norm{f}_{p_0} \text{ for } f \in L^{p_0}(\mu), \qquad \norm{T f}_{q_1} \le M_1 \norm{f}_{p_1} \text{ for } f \in L^{p_1}(\mu).$$
Let $0 < \theta < 1$, and define $p$ and $q$ by
$$\frac{1}{p} = \frac{1 - \theta}{p_0} + \frac{\theta}{p_1}, \qquad \frac{1}{q} = \frac{1 - \theta}{q_0} + \frac{\theta}{q_1}.$$
Then $T$ maps $L^p(\mu)$ into $L^q(\nu)$, and
$$\norm{T f}_q \le M_0^{1 - \theta} M_1^{\theta} \norm{f}_p \quad \text{for all } f \in L^p(\mu).$$

## Remarks

The proof is Thorin's complex-variable argument.
By [[density of simple functions|B6RYM7QTYZ]] and duality, it suffices to bound $\int (T f) g \, d\nu$ for simple $f$ and $g$.
One embeds $f$ and $g$ in analytic families $f_z$, $g_z$ for $z$ in the strip $0 \le \operatorname{Re} z \le 1$, with $f_\theta = f$ and $g_\theta = g$.
The function $z \mapsto \int (T f_z) g_z \, d\nu$ is bounded and analytic, and the [[Hadamard three-lines theorem|WHS58R43N0]] interpolates the bounds on the two edges.

Example: the Fourier transform satisfies $\norm{\hat f}_\infty \le \norm{f}_1$ and, by Plancherel, $\norm{\hat f}_2 = \norm{f}_2$ on $\R^n$ with suitable normalisation.
Riesz–Thorin with $p_0 = 1$, $q_0 = \infty$, $p_1 = q_1 = 2$ gives the Hausdorff–Young inequality $\norm{\hat f}_{p'} \le \norm{f}_p$ for $1 \le p \le 2$, where $1/p + 1/p' = 1$.

Complex scalars matter for the constant.
For spaces of real-valued functions the conclusion still holds, but the constant $M_0^{1-\theta} M_1^\theta$ may need an extra factor of at most $2$.

## Explanations

### L1 human

Suppose an operation on functions is under control in two ways of measuring size, say $L^1$ and $L^2$.
The theorem says that it is then also under control in every way of measuring size in between, such as $L^{4/3}$.
The bound in between is a weighted geometric mean of the two end bounds.

### L2 human

Riesz–Thorin is the basic tool for proving $L^p$ bounds from two easy endpoint bounds.
Typical uses are the Hausdorff–Young inequality, [[Young's convolution inequality|JXDKJCBXJ8]] and bounds for integral operators.
It needs linear operators and strong-type bounds at the endpoints.
The [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]] allows sublinear operators and weak-type endpoint bounds, at the cost of a worse constant and, in its basic form, only on the diagonal $p = q$.
Both are the starting points of abstract [[interpolation space|VX1R6FN6PS]] theory, the complex and the real method.
</document>

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# Khintchine inequality

## Statement

Let $0 < p < \infty$.
Let $\varepsilon_1, \varepsilon_2, \ldots$ be independent [[random variables|0PNRKMRC55]] with $\Pr(\varepsilon_k = 1) = \Pr(\varepsilon_k = -1) = 1/2$.
Then there are constants $A_p > 0$ and $B_p > 0$, depending only on $p$, with the following property.
For every $N \ge 1$ and all real numbers $a_1, \ldots, a_N$,
$$A_p \Big( \sum_{k=1}^{N} a_k^2 \Big)^{1/2} \le \Big( \mathbb{E} \Big\lvert \sum_{k=1}^{N} a_k \varepsilon_k \Big\rvert^p \Big)^{1/p} \le B_p \Big( \sum_{k=1}^{N} a_k^2 \Big)^{1/2},$$
where $\mathbb{E}$ is the [[expected value|0WQNRCNY3M]].

## Remarks

For $p = 2$ the middle term equals $\left(\sum a_k^2\right)^{1/2}$ exactly, since $\mathbb{E}[\varepsilon_j \varepsilon_k] = 0$ for $j \ne k$.
So one can take $A_p = 1$ for $p \ge 2$ and $B_p = 1$ for $p \le 2$, by monotonicity of $L^p$ norms on a probability space.

Proof idea for the upper bound: normalize to $\sum a_k^2 = 1$.
Since $\cosh t \le e^{t^2/2}$, independence gives $\mathbb{E} \exp\left(t \sum a_k \varepsilon_k\right) \le e^{t^2/2}$ for all real $t$.
This gives the tail bound $\Pr\left(\abs{\sum a_k \varepsilon_k} > s\right) \le 2 e^{-s^2/2}$, and integrating the tail bounds every moment.
The lower bound for $p < 2$ follows from the upper bound for some $q > 2$ by Hölder's inequality, which places $L^2$ between $L^p$ and $L^q$.

Example: for $N = 2$, $a_1 = a_2 = 1$ and $p = 1$, the sum $\varepsilon_1 + \varepsilon_2$ is $\pm 2$ with probability $1/2$ and $0$ otherwise, so $\mathbb{E}\abs{\varepsilon_1 + \varepsilon_2} = 1 = \frac{1}{\sqrt 2} \cdot \sqrt 2$.
This shows that $A_1 \le 1/\sqrt 2$; Szarek proved that $A_1 = 1/\sqrt 2$ is the best constant.

## Explanations

### L1 human

Flip $N$ fair coins and add the numbers $a_k$ with random signs.
The typical size of the result is the square root of the sum of the squares of the $a_k$, as in Pythagoras' theorem.
Khintchine's inequality says this holds for every way of measuring "typical size" by a $p$-th moment, up to a constant factor.
So random sign sums behave like Gaussian variables with the same variance.

### L2 human

In harmonic analysis the $\varepsilon_k$ are realized as the Rademacher functions $r_k(t) = \operatorname{sgn} \sin(2^k \pi t)$ on $[0, 1]$, and the inequality says that all $L^p$ norms are equivalent on their span.
Combined with Fubini's theorem it turns square-function estimates into estimates for random sums, a basic step in Littlewood–Paley theory.
In Banach space theory it is the starting point of the notions of type and cotype.
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# Pontryagin class

## Definition

Let $E \to B$ be a real vector bundle of rank $n$ over a paracompact space $B$.
Let $E \otimes \C$ be its complexification, the complex vector bundle of rank $n$ with fibers $E_b \otimes_\R \C$.
For $i \ge 0$, the $i$-th *Pontryagin class* of $E$ is
$$p_i(E) = (-1)^i c_{2i}(E \otimes \C) \in H^{4i}(B; \Z),$$
where $c_{2i}$ is the $2i$-th [[Chern class|PAD37GKDPD]].
The total Pontryagin class is $p(E) = 1 + p_1(E) + p_2(E) + \cdots$.

## Remarks

The odd Chern classes of $E \otimes \C$ satisfy $2 c_{2i+1}(E \otimes \C) = 0$, because $E \otimes \C$ is isomorphic to its complex conjugate; this is why only the even ones are used.
Pontryagin classes are natural, so they are [[characteristic classes|7HACJC0JE2]], and $p_i(E) = 0$ for $4i > 2n$, that is, for $i > n/2$.
The Whitney sum formula holds modulo $2$-torsion: $2\bigl(p(E \oplus E') - p(E)\, p(E')\bigr) = 0$.

Example: for the complex projective space $\C P^m$ with generator $a \in H^2(\C P^m; \Z)$, the tangent bundle has $p(T\C P^m) = (1 + a^2)^{m+1}$.
For $m = 2$ this gives $p_1(T\C P^2) = 3a^2$.

Non-example: Pontryagin classes do not detect every nontrivial bundle.
The Möbius line bundle over $S^1$ is not trivial, but $p_1$ lies in $H^4(S^1; \Z) = 0$, so all its Pontryagin classes beyond $p_0$ vanish.

For a smooth manifold $M$, $p_i(M) = p_i(TM)$.
</document>

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# Riesz mean

## Summary

A Riesz mean of a series $\sum a_n$ is a weighted partial sum in which the term $a_n$ is damped by a factor $(1 - \lambda_n/\omega)^{\delta}$.
Here $\lambda_n$ is a given increasing sequence and $\omega$ is a cut-off that grows to infinity.
The series is Riesz summable when these means converge.
With $\lambda_n = n$ and $\delta = 1$ the Riesz means at integer cut-offs are the [[Cesàro means|08R2V9GZF7]] of the partial sums.
Riesz means with $\lambda_n = \ln n$ are a standard tool for [[Dirichlet series|8MWTEF6462]] in analytic number theory.

## Explanations

### L1 human

A partial sum cuts a series off sharply: terms before the cut-off count fully and the rest not at all.
A Riesz mean cuts it off smoothly: terms well before the cut-off count almost fully, and terms near it count only a little.
The smooth cut-off removes much of the oscillation that makes a series diverge.
The parameter $\delta$ controls how smooth the cut-off is.
</document>

<document source="https://tamnd.github.io/math2/node/AGYBSTD7BG.md" tag="AGYBSTD7BG" kind="statement">
# Łoś's theorem

## Statement

Let $L$ be a signature, let $\mathcal{U}$ be an [[ultrafilter|J64Y3JC0RE]] on a nonempty set $I$, and let $(M_i)_{i \in I}$ be $L$-structures with nonempty domains.
Let $M = \prod_{i \in I} M_i / \mathcal{U}$ be their [[ultraproduct|AX9Q86GB8M]].
Let $\varphi(x_1, \dots, x_n)$ be a first-order [[$L$-formula|HYJ10AF6RT]], and let $a_1, \dots, a_n \in \prod_{i \in I} M_i$.
Then
$$M \models \varphi([a_1], \dots, [a_n]) \iff \set{i \in I : M_i \models \varphi(a_1(i), \dots, a_n(i))} \in \mathcal{U}.$$

## Remarks

The proof is an induction on the formula.
Atomic formulas hold by the definition of the ultraproduct, and conjunction uses that $\mathcal{U}$ is closed under finite intersections.
Negation uses that $\mathcal{U}$ contains exactly one of each set and its complement, which is where the ultrafilter property enters.
For $\exists y \, \psi$, the axiom of choice picks a witness $b(i)$ in each factor where one exists.

In particular, a sentence holds in an ultraproduct exactly when the set of factors where it holds lies in $\mathcal{U}$.
For an ultrapower $M^I / \mathcal{U}$, the diagonal map $m \mapsto [(m, m, \dots)]$ is an elementary embedding of $M$.
For example, every ultrapower of $\R$ is a real closed ordered field, since these properties are first-order.
The theorem also gives a proof of the compactness theorem.
If every finite subset of a theory $T$ has a model, an ultraproduct of these models over a suitable ultrafilter is a model of $T$.

The ultrafilter is necessary: for a filter that is not an ultrafilter, the corresponding reduced product of fields can fail to be a field.

## Explanations

### L1 human

An ultraproduct glues many structures into one, and an ultrafilter decides which sets of indices count as "most".
Łoś's theorem says that a first-order statement is true in the glued structure exactly when it is true in "most" of the pieces.
So the glued structure keeps every first-order property that most pieces share.

### L2 human

Łoś's theorem is the reason ultraproducts are useful in model theory.
It gives the compactness theorem without any proof system, and it shows that a class of structures closed under ultraproducts and elementary equivalence is axiomatisable.
With a non-principal ultrafilter on $\N$, the ultrapower of $\R$ gives the hyperreal numbers, and the theorem becomes the transfer principle of nonstandard analysis.
Ultraproducts of the finite fields $\F_p$, indexed by the primes $p$, over a non-principal ultrafilter are fields of characteristic $0$.
This links statements about large primes with statements in characteristic $0$.
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# Knot sum

## Summary

The knot sum, or connected sum, of two oriented [[knots|QYEAZTYGPT]] is a new knot.
It is obtained by cutting each knot open at one point and joining the loose ends.
It is the version for knots of the [[connected sum|Y0K0K5NGWW]] of manifolds.
The sum is well defined, commutative and associative, with the [[unknot|1TQ06R4Q66]] as neutral element.
It lets one break knots into [[prime knots|GJ6FJ4G5VC]], much as integers break into primes.

## Explanations

### L1 human

Take two knotted loops of string lying far apart.
Cut each loop once and tie the four loose ends together in pairs, so that you get one loop with both knots on it.
The result is the knot sum.
For example, the sum of a [[trefoil knot|VYTYSVHN4A]] with itself is the granny knot.
The sum of a trefoil with its mirror image is the square knot, which is a different knot.
</document>

<document source="https://tamnd.github.io/math2/node/AH059Y44NF.md" tag="AH059Y44NF" kind="definition">
# Morse index

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]], let $f \colon M \to \R$ be a [[smooth function|0V33BNKZ9S]], and let $p$ be a [[critical point|0YTCMFB4HC]] of $f$.
Let $H = \operatorname{Hess}_p f$ be its Hessian, the [[symmetric bilinear form|0AXZNE5XNH]] on $T_p M$ from the definition of a [[nondegenerate critical point|MGPPCDFQDG]].
The *Morse index* of $f$ at $p$ is the largest dimension of a linear subspace $V \subseteq T_p M$ such that $H(v, v) < 0$ for every nonzero $v \in V$.
The *nullity* of $f$ at $p$ is the dimension of the kernel $\set{v \in T_p M : H(v, w) = 0 \text{ for all } w \in T_p M}$.

## Remarks

In a chart, the index is the number of negative eigenvalues of the Hessian matrix, counted with multiplicity, and the nullity is the dimension of its kernel.
By Sylvester's law of inertia both are independent of the chart.
The point $p$ is nondegenerate exactly when the nullity is $0$, and the term Morse index is mostly used in that case.

Example: $f(x, y, z) = x^2 - y^2 - 3z^2$ on $\R^3$ has Hessian matrix $\operatorname{diag}(2, -2, -6)$ at the origin.
It is negative definite on the $yz$-plane and not on any $3$-dimensional subspace, so the index is $2$ and the nullity is $0$.

Non-example: for $g(x) = x^3$ on $\R$, the critical point $0$ has index $0$ and nullity $1$.
Index $0$ does not make $0$ a local minimum here, since the point is degenerate; this is why the index describes the local shape only at nondegenerate critical points.

The same definition applies on a Hilbert manifold, where the index may be infinite.
For the [[energy functional on paths|Q12AB1MJ64]], the index at a geodesic is computed by the [[Morse index theorem for geodesics|YZ4268SK8G]].
</document>

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# Kneser's construction of real analytic tetration

## Statement

Let $\exp$ be the [[exponential function|0J0FCWV32D]].
There is a [[real-analytic|081WKEDK6C]], strictly increasing function $F \colon (-2, \infty) \to \R$ such that
$$F(0) = 1 \quad \text{and} \quad F(x + 1) = \exp(F(x)) \quad \text{for all } x > -2.$$
So $F$ extends [[tetration|NT65WHM5VV]] to base $e$ from integer heights to real heights: $F(n) = {}^{n}e$ for all integers $n \ge 1$.

## Remarks

Kneser's construction starts at a non-real fixed point $L$ of $\exp$ in $\C$, with $L \approx 0.318 + 1.337i$.
There $\exp'(L) = L$ has modulus greater than $1$, so $\exp$ has a holomorphic Schröder function near $L$, and this gives a complex solution of the [[Abel equation|68SHB7Z721]].
This solution is not real on the real axis, and Kneser corrects it by a conformal change of variables, built with the Riemann mapping theorem, to obtain a real-analytic one.
From $F$ one gets a real-analytic [[functional square root|E7890PHMGP]] of $\exp$ on $\R$: $g(x) = F\big(F^{-1}(x) + \tfrac12\big)$ satisfies $g(g(x)) = \exp(x)$.
The values at integers are forced: $F(-1) = \log F(0) = 0$, $F(1) = e$, $F(2) = e^e$, and $F(x) \to -\infty$ as $x \to -2$ from the right.
The equation $F(x + 1) = \exp(F(x))$ with $F(0) = 1$ has many solutions, even smooth ones, so extra conditions are needed to single out Kneser's solution.

## Explanations

### L1 human

Tetration ${}^{n}e$ is a tower of $n$ copies of $e$, and it is easy to define when $n$ is a whole number.
Kneser showed that one can fill in the values between the whole numbers by a smooth curve, in fact an analytic one.
The curve still obeys the rule "one more floor means one more exponential".
This also gives a function that, applied twice, is the exponential function.

### L2 human

The equation $F(x + 1) = \exp(F(x))$ is an Abel-type equation for the inverse of $F$, and its solutions are easy to build piece by piece on intervals of length $1$.
Such solutions are usually not analytic, and the difficulty is to find a solution that is analytic everywhere.
Kneser's method combines the theory of linearization near a repelling fixed point, the Schröder and Koenigs theory, with the Riemann mapping theorem.
It is the standard reference point for later work on tetration to other bases and on uniqueness of analytic solutions.
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# Separable Banach space

## Summary

A separable Banach space is a [[Banach space|06GZ956657]] that has a countable dense subset.
Most spaces met in analysis are of this kind, for example $\ell^p$ and $L^p[0, 1]$ for $1 \le p < \infty$, $c_0$ and $C[0, 1]$.
Separability allows arguments by sequences and diagonal extraction.
For instance, the closed unit ball of the dual of a separable Banach space is metrizable in the weak* topology.
The [[Banach–Mazur theorem|QC5MB5WM8F]] says that every separable Banach space is isometric to a closed subspace of $C[0, 1]$.

## Explanations

### L1 human

A space is separable if a countable list of its points comes as close as you like to every point.
The rationals play this role in the real line.
In the space $C[0, 1]$ of continuous functions with the sup norm, the polynomials with rational coefficients form such a list.
The space $\ell^\infty$ of bounded sequences is too big: it has uncountably many points at distance $1$ from each other.
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# Generalized inverse

## Definition

Let $F$ be a [[field|04C5ZBG1WT]] and let $A$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $F$.
A *generalized inverse* of $A$ is an $n \times m$ matrix $G$ over $F$ such that
$$A G A = A.$$

## Remarks

Example: for $A = \begin{pmatrix} 1 & 1 \end{pmatrix}$, the column $G = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ is a generalized inverse, because $AG = (1)$ and so $AGA = A$.
Non-example: for $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, the matrix $G = \begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix}$ is not one, because $AGA = \begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix} \ne A$.
If $A$ is square and [[invertible|0EB9N0S33C]], then $A^{-1}$ is its only generalized inverse.
If $Ax = b$ has a solution and $G$ is a generalized inverse, then $x = Gb$ is a solution, since $AGb = AGAx = Ax = b$.
A generalized inverse is often written $A^-$ or $A^{(1)}$; it is also called a $\{1\}$-inverse.
A matrix $G$ that satisfies also $GAG = G$ is a reflexive generalized inverse.
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# Quadric

## Definition

Let $F$ be a [[field|04C5ZBG1WT]], let $V$ be a [[vector space|000AYPA9AJ]] over $F$ of finite dimension $n + 1 \ge 2$, and let $\mathbb{P}(V)$ be its [[projective space|ZT9A6JK8QF]].
Let $Q$ be a nonzero [[quadratic form|0RBQS1JJCR]] on $V$.
For a nonzero vector $v$, write $\langle v \rangle$ for the point of $\mathbb{P}(V)$ that it spans.
The *(projective) quadric* of $Q$ is the set
$$\set{\langle v \rangle \in \mathbb{P}(V) : Q(v) = 0}.$$
It is well defined because $Q(\lambda v) = \lambda^2 Q(v)$.
An *affine quadric* in $F^n$ is the set of zeros of a polynomial of degree exactly $2$ in $n$ variables with coefficients in $F$.

## Remarks

Example: in $\mathbb{P}^3(F)$, the form $Q(x) = x_0 x_1 + x_2 x_3$ defines a quadric that contains the line of points $[s : 0 : t : 0]$, since $Q(s, 0, t, 0) = 0$ for all $s, t$.
Example: the sphere $x^2 + y^2 + z^2 = 1$ in $\R^3$ is an affine quadric, and $x^2 + y^2 + z^2 - w^2 = 0$ is its projective closure.

Non-example: the zero set of $y - x^3$ in $F^2$ is not an affine quadric, because the polynomial has degree $3$.

For $n = 2$ the projective quadrics are the [[projective conics|R3TMVHTP3A]].
Different forms can define the same point set.
For example, over $\R$ every positive definite form defines the empty quadric.
For this reason the quadric is often taken together with its form, up to a nonzero scalar.
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# Graphon

## Definition

A *graphon* is a [[measurable function|YYNCJSR0PJ]] $W \colon [0, 1]^2 \to [0, 1]$, for the [[Lebesgue measure|2Z2QVWZ262]] on $[0, 1]^2$, such that $W(x, y) = W(y, x)$ for all $x, y \in [0, 1]$.

## Remarks

Example: let $W(x, y) = 1$ if exactly one of $x, y$ is less than $1/2$, and $W(x, y) = 0$ otherwise.
It takes values in $\set{0, 1}$, and it is measurable because it is constant on four rectangles.
It is symmetric, since swapping $x$ and $y$ does not change whether exactly one of them is below $1/2$.
It is the limit of the complete bipartite graphs $K_{n,n}$.

Non-example: $W(x, y) = x$ is measurable with values in $[0, 1]$, but it is not symmetric, since $W(1, 0) = 1 \ne 0 = W(0, 1)$.

A graph $G$ on $\set{1, \ldots, n}$ gives a step graphon $W_G$: $W_G(x, y) = 1$ if $\ceil{nx}$ and $\ceil{ny}$ are adjacent in $G$, and $0$ otherwise (with $\ceil{0}$ read as $1$).
For a finite graph $F$ on $\set{1, \ldots, k}$, the homomorphism density of $F$ in $W$ is
$$t(F, W) = \int_{[0,1]^k} \prod_{\set{i, j} \in E(F)} W(x_i, x_j) \, dx_1 \cdots dx_k.$$
Graphons that differ on a set of measure zero, or by a measure-preserving change of variables, are usually identified.
Some authors allow any standard probability space in place of $[0, 1]$.
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# Semifield

## Summary

A semifield is an algebraic structure that satisfies all the axioms of a [[division ring|0EA5BTA0CX]] except possibly the associativity of multiplication.
Semifields arise in finite geometry, where each semifield coordinatizes a [[projective plane|DANFNN8C4Y]], called a semifield plane.
Finite semifields that are not fields give non-Desarguesian planes.
In other areas, a semifield is a semiring whose nonzero elements form a group under multiplication.
This node is about the first meaning.

## Explanations

### L1 human

In a field one can add, subtract, multiply and divide by any nonzero element, and multiplication is associative and commutative.
A semifield keeps addition, subtraction and division, and the two distributive laws, but drops the rule $(ab)c = a(bc)$.
Every field and every division ring is a semifield.
The [[octonions|1HRMT3DSX2]], an eight-dimensional algebra over the real numbers, are a semifield whose multiplication is not associative.
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<document source="https://tamnd.github.io/math2/node/AHKJ361FDB.md" tag="AHKJ361FDB" kind="statement">
# Long exact sequence in homology

## Statement

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], such as the category of modules over a ring.
Let $0 \to A_\bullet \xrightarrow{f} B_\bullet \xrightarrow{g} C_\bullet \to 0$ be a sequence of [[chain complexes|WQ82C0M3QB]] and [[chain maps|HTK9WQZJ1G]] in $\mathcal{A}$ such that $0 \to A_n \to B_n \to C_n \to 0$ is a [[short exact sequence|KJKNJMGKZZ]] for every $n$.
Then there are morphisms $\partial_n \colon H_n(C) \to H_{n-1}(A)$, called *connecting morphisms*, such that the sequence of [[homology|2J4JJJYP22]] objects
$$\cdots \to H_n(A) \xrightarrow{H_n(f)} H_n(B) \xrightarrow{H_n(g)} H_n(C) \xrightarrow{\partial_n} H_{n-1}(A) \xrightarrow{H_{n-1}(f)} H_{n-1}(B) \to \cdots$$
is [[exact|JB7ZBFBA9C]].
The connecting morphisms are natural with respect to morphisms of such short exact sequences of complexes.

## Remarks

For modules, $\partial_n$ is computed as follows.
Lift a cycle $z \in C_n$ to $y \in B_n$.
Then $d y$ maps to $d z = 0$ in $C_{n-1}$, so $d y = f(x)$ for a unique $x \in A_{n-1}$, which is a cycle, and $\partial_n [z] = [x]$.
The proof applies the [[snake lemma|Q8AYGD93CW]] to the map $\operatorname{coker} d^A_{n+1} \to \ker d^A_{n-1}$ induced by $d$, and the same for $B$ and $C$.
The kernels and cokernels of these maps are $H_n$ and $H_{n-1}$, which yields the six-term pieces of the sequence.
Example: let $A_\bullet = \Z$ in degree $0$, let $B_\bullet$ have $B_1 = B_0 = \Z$ with $d_1 = 1_{\Z}$, and let $C_\bullet = \Z$ in degree $1$, with the evident inclusion and projection.
Then $H(B) = 0$, so exactness of $0 = H_1(B) \to H_1(C) \to H_0(A) \to H_0(B) = 0$ makes $\partial_1 \colon \Z \to \Z$ an isomorphism.

## Explanations

### L1 human

A short exact sequence of complexes builds the middle complex from a subcomplex and a quotient complex.
Homology does not keep the sequence short exact, but the theorem says that the failure is controlled.
The three homology sequences link up into one long exact sequence, with a connecting map that lowers the degree by one.

### L2 human

This is the most used tool for computing homology.
In topology, the short exact sequences of singular chains for a pair $(X, A)$ give the long exact sequence of a pair.
For an open cover by two sets they give the Mayer–Vietoris sequence.
In algebra, applying it to a short exact sequence of resolutions gives the long exact sequences of derived functors such as $\mathrm{Ext}$ and $\mathrm{Tor}$.
Together with the [[five lemma|W6AVEAW2WR]], it lets one prove that a map is a quasi-isomorphism by checking pieces.
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<document source="https://tamnd.github.io/math2/node/AHTE5DVX61.md" tag="AHTE5DVX61" kind="concept">
# Morphism of finite type

## Summary

A morphism of finite type $X \to Y$ is a [[morphism of schemes|JXRKH6P1Z4]] described by finitely many coordinates over $Y$.
Over each affine piece $\operatorname{Spec} B$ of $Y$, the source is covered by finitely many spectra of finitely generated $B$-algebras.
It is the relative version of a [[finitely generated algebra|AC5TFFQT9M]].
Varieties over a field $k$ are schemes of finite type over $\operatorname{Spec} k$.

## Explanations

### L1 human

A scheme of finite type over a field is one that can be described by finitely many coordinates and equations on each of finitely many pieces.
The plane curve $y^2 = x^3 - x$ over $\Q$ is of finite type: it is $\operatorname{Spec} \Q[x, y]/(y^2 - x^3 + x)$.
The spectrum of the field $\Q(x)$ of rational functions is not of finite type over $\Q$, because no finite set of elements generates $\Q(x)$ as an algebra.
Most of classical algebraic geometry deals with schemes of finite type over a field.
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# Locally compact group

## Summary

A locally compact group is a Hausdorff [[topological group|92700QAB5Y]] whose topology is [[locally compact|0FE2SB8WVJ]].
These groups carry a [[Haar measure|VY0HXD3GC4]], an invariant way to measure and integrate, which makes harmonic analysis on them possible.
They include all Lie groups, all discrete groups, all compact groups and the groups of points of algebraic groups over local fields such as $\Q_p$.

## Explanations

### L1 human

A space is locally compact if every point has a small compact patch around it, like a closed and bounded interval around a point of the line.
The real line, the circle and the $p$-adic numbers all have this property, while an infinite-dimensional Hilbert space does not.
A locally compact group is a group whose space of elements is of this kind and whose operations are continuous.
For example, the group of invertible $n \times n$ real matrices is locally compact, since it is an open subset of $\R^{n^2}$.
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<document source="https://tamnd.github.io/math2/node/AHW4B5480K.md" tag="AHW4B5480K" kind="statement">
# Serre spectral sequence

## Statement

Let $\pi \colon E \to B$ be a [[Serre fibration|PB9JXQ2KC3]] with $B$ path-connected, let $b_0 \in B$, and let $F = \pi^{-1}(b_0)$ be the fibre.
Let $R$ be a commutative ring.
Then there is a first-quadrant cohomological [[spectral sequence|GG8AYV847V]] $(E_r^{p,q}, d_r)_{r \ge 2}$, with $d_r \colon E_r^{p,q} \to E_r^{p+r, q-r+1}$, such that:

1. $E_2^{p,q} = H^p(B; \mathcal{H}^q(F; R))$, the [[singular cohomology|6N7BC70CX0]] of $B$ with [[local coefficients|M58WRZ7PFM]] in the system $b \mapsto H^q(\pi^{-1}(b); R)$;
2. the spectral sequence converges to $H^{p+q}(E; R)$: there is a filtration of $H^n(E; R)$ whose successive quotients are the groups $E_\infty^{p, n-p}$;
3. it is multiplicative: each $E_r$ is a bigraded ring and $d_r$ is a derivation.
On $E_2^{p,q} \otimes E_2^{s,t}$ the product is the cup product times $(-1)^{qs}$.

If the [[fundamental group|S1XK3HXHFX]] $\pi_1(B, b_0)$ acts as the identity on $H^*(F; R)$, for instance if $B$ is simply connected, then $E_2^{p,q} = H^p(B; H^q(F; R))$ with constant coefficients.

## Remarks

One proof filters the singular cochains of $E$ by the preimages of the skeleta of a CW approximation of $B$.
Another proof filters the singular simplices of $E$ by how they project to $B$.
There is a homological version with $E^2_{p,q} = H_p(B; \mathcal{H}_q(F; R))$ converging to $H_{p+q}(E; R)$.

Example: for $n \ge 2$, the [[path space fibration|D3BN7MBKTR]] $\Omega S^n \to P S^n \to S^n$ has contractible total space.
So all classes except $E_2^{0,0}$ must die, which forces $d_n \colon E_n^{0, k(n-1)} \to E_n^{n, (k-1)(n-1)}$ to be isomorphisms.
It follows that $H^i(\Omega S^n; \Z) \cong \Z$ when $n - 1$ divides $i$ and $0$ otherwise.

## Explanations

### L1 human

A fibration is a space $E$ that looks locally like a product of a base $B$ and a fibre $F$.
The Serre spectral sequence starts from the cohomology of $B$ with coefficients in the cohomology of $F$, as if $E$ were a product.
Then it corrects this guess step by step, and the end result is the cohomology of $E$.

### L2 human

The Serre spectral sequence is the main tool for computing cohomology of fibre bundles, loop spaces, homogeneous spaces and Eilenberg–MacLane spaces.
It gives the Gysin and Wang sequences as special cases when the fibre or the base is a sphere.
Serre used it to prove that the homotopy groups of spheres are finitely generated, and finite except in a few degrees.
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<document source="https://tamnd.github.io/math2/node/AHYMBCMW11.md" tag="AHYMBCMW11" kind="definition">
# Huber loss

## Definition

Let $\delta > 0$.
The *Huber loss* with threshold $\delta$ is the function $L_\delta \colon \R \to \R$ given by
$$L_\delta(a) = \begin{cases} \dfrac{a^2}{2} & \text{if } \abs{a} \le \delta, \\[1ex] \delta\Big(\abs{a} - \dfrac{\delta}{2}\Big) & \text{if } \abs{a} > \delta. \end{cases}$$

## Remarks

The two pieces agree at $\abs{a} = \delta$, where both equal $\delta^2/2$, and so do their derivatives.
So $L_\delta$ is [[convex|0VV0HSW1ZS]] and continuously differentiable, with derivative $\psi_\delta(a) = \max(-\delta, \min(\delta, a))$, the residual clipped to $[-\delta, \delta]$.
Huber's [[M-estimator|4503R8BVF8]] of location minimises $\sum_{i=1}^{n} L_\delta(x_i - \mu)$ over $\mu \in \R$, or equivalently solves $\sum_i \psi_\delta(x_i - \mu) = 0$.
As $\delta \to \infty$ it tends to the mean, and as $\delta \to 0$ it tends to the median.

Example: with $\delta = 1$, $L_1(0.5) = 0.125$, as for squared loss, and $L_1(3) = 3 - 0.5 = 2.5$, much less than $3^2/2 = 4.5$.

Non-example: the function $a \mapsto \abs{a}$ grows linearly but is not differentiable at $0$.
It is not quadratic on any interval around $0$, so it is not a Huber loss for any $\delta$.
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<document source="https://tamnd.github.io/math2/node/AHYZMSBBNH.md" tag="AHYZMSBBNH" kind="concept">
# Pseudoconvexity

## Summary

Pseudoconvexity is the notion of convexity adapted to several complex variables.
An open set in $\C^n$ is pseudoconvex if it admits a continuous [[plurisubharmonic|FF2P7ER3QT]] exhaustion function.
Such a function tends to $+\infty$ at the boundary and, for unbounded sets, at infinity.
Unlike holomorphic convexity, it is a local condition at the boundary, and for smooth boundaries it can be checked with the Levi form.
By the solution of the Levi problem, the pseudoconvex domains in $\C^n$ are exactly the domains of holomorphy.

## Explanations

### L1 human

Ordinary convexity can be tested with convex functions: a convex region carries a convex function that blows up at its edge.
Pseudoconvexity replaces convex functions by plurisubharmonic ones, which only need to be subharmonic along each complex line.
This condition is much weaker than convexity and is preserved by holomorphic changes of coordinates.
Every convex open set and every open set in $\C$ is pseudoconvex, but the punctured ball $\set{0 < \abs{z} < 1}$ in $\C^2$ is not.
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<document source="https://tamnd.github.io/math2/node/AJ4FM5BT7Q.md" tag="AJ4FM5BT7Q" kind="definition">
# Periodic sequence

## Definition

Let $S$ be a set and let $(x_n)_{n \ge 0}$ be a [[sequence|0G9EKM0E15]] of elements of $S$.
The sequence is *periodic* if there is an integer $p \ge 1$ such that
$$x_{n+p} = x_n \quad \text{for all } n \ge 0.$$
Such a $p$ is a *period* of the sequence, and the smallest such $p$ is its *least period*.

## Remarks

Example: $x_n = (-1)^n$ in $\R$ satisfies $x_{n+2} = (-1)^n (-1)^2 = x_n$ for all $n$, so it is periodic, with least period $2$ since $x_1 \ne x_0$.

Non-example: $x_n = n$ is not periodic, since $x_{n+p} = n + p \ne x_n$ for every $p \ge 1$.

The sequence is *eventually periodic* if the condition holds for all $n \ge n_0$ for some $n_0$; for example $0, 1, 1, 1, \dots$ is eventually periodic but not periodic.
Every period is a multiple of the least period.
The same definition applies to two-sided sequences $(x_n)_{n \in \Z}$, with the condition required for all $n \in \Z$.
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# Milman–Pettis theorem

## Statement

Let $X$ be a [[Banach space|06GZ956657]] over $\R$ or $\C$.
If $X$ is [[uniformly convex|BEAMC5011M]], then $X$ is [[reflexive|NN8RWPC8WG]].

## Remarks

Let $J \colon X \to X^{**}$ be the canonical embedding, and let $\xi \in X^{**}$ with $\norm{\xi} = 1$.
By the [[Goldstine theorem|K4S9G3AQBM]], $\xi$ is a weak* limit of a net $J(x_\alpha)$ with $\norm{x_\alpha} \le 1$.
Given $\varepsilon > 0$, let $\delta$ be as in the definition of uniform convexity, and choose $f \in X^*$ of norm $1$ with $\operatorname{Re} \xi(f) > 1 - \delta$.
For late indices $\alpha, \beta$ one has $\operatorname{Re} f(x_\alpha + x_\beta) > 2 - 2\delta$, so $\norm{(x_\alpha + x_\beta)/2} > 1 - \delta$, and uniform convexity gives $\norm{x_\alpha - x_\beta} < \varepsilon$.
This shows that $\xi$ is within any given $\varepsilon$ of the closed set $J(X)$, so $\xi \in J(X)$.

Example: $L^p$ spaces for $1 < p < \infty$ are uniformly convex by [[Clarkson's inequalities|1V11QM5C07]], so they are reflexive.
Completeness is needed.
The space of finitely supported sequences with the norm of $\ell^2$ is an inner product space, hence uniformly convex, but it is not complete and therefore not reflexive.
The converse fails: there are reflexive spaces with no equivalent uniformly convex norm.

## Explanations

### L1 human

Uniform convexity is a geometric condition on the shape of the unit ball: it is round in a uniform way.
Reflexivity is an abstract condition about the double dual.
The theorem says that the geometric condition is enough to give the abstract one.

### L2 human

The theorem is the usual quick route to the reflexivity of $L^p$ for $1 < p < \infty$, without computing the dual twice.
In a uniformly convex space, weak convergence plus convergence of norms implies norm convergence, and closed convex sets have unique nearest points.
These facts and reflexivity make such spaces convenient in variational problems.
Being reflexive is invariant under isomorphism, so every Banach space isomorphic to a uniformly convex one is reflexive.
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# Spin(7)-manifold

## Definition

Let $\varphi_0$ be the standard $3$-form of $G_2$ on $\R^7$, and let $*_7$ be the Hodge star of $\R^7$.
The Cayley $4$-form on $\R^8 = \R \oplus \R^7$ is $\Phi_0 = dx_0 \wedge \varphi_0 + *_7 \varphi_0$.
The group $\operatorname{Spin}(7)$ is identified with the group of linear maps $A \in GL(8, \R)$ with $A^* \Phi_0 = \Phi_0$.
It is a compact Lie group of dimension $21$ contained in $SO(8)$.

Let $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $8$.
Identify the tangent space at a point with $\R^8$ by a suitable linear isomorphism.
The manifold $M$ is a *Spin(7)-manifold* if its [[holonomy|WHDZ3MK56B]] group is then contained in $\operatorname{Spin}(7)$.

## Remarks

Equivalently, $M$ carries a [[$4$-form|0E3G0GRPZ6]] $\Phi$ that agrees with $\Phi_0$ at each point in suitable linear coordinates and is parallel for the [[Levi-Civita connection|CAZYGE1T4E]]; such a $\Phi$ determines $g$.

Example: $\R^8$ with $\Phi = \Phi_0$, whose constant coefficients make it parallel.
If $(N, \varphi)$ is a [[$G_2$ manifold|P1TFW9QWAD]], then $N \times S^1$ with $\Phi = dt \wedge \varphi + *\varphi$ is a Spin(7)-manifold, with holonomy contained in $G_2$.

Non-example: the round sphere $S^8$ is not a Spin(7)-manifold, since a metric with holonomy in $\operatorname{Spin}(7)$ is Ricci-flat and the round metric is not.

Compact manifolds with holonomy exactly $\operatorname{Spin}(7)$ were first constructed by Joyce.
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# Eilenberg–MacLane space

## Summary

An Eilenberg–MacLane space $K(G, n)$ is a path-connected space whose only nonzero [[homotopy group|P976WYEFMB]] is $G$ in degree $n$.
Such spaces exist for every group $G$ when $n = 1$ and every abelian group $G$ when $n \ge 2$, and they are unique up to weak homotopy equivalence.
They are the building blocks of all spaces in the sense of Postnikov towers.
They represent ordinary cohomology: for an abelian group $G$ and a CW complex $X$, $H^n(X; G)$ is the set of homotopy classes of maps $X \to K(G, n)$.

## Explanations

### L1 human

Homotopy groups measure holes of each dimension in a space.
An Eilenberg–MacLane space has holes of exactly one dimension, described by the group $G$.
The circle is $K(\Z, 1)$: it has a one-dimensional hole, and its higher homotopy groups vanish because its universal cover, the line, is contractible.
Infinite-dimensional complex projective space is $K(\Z, 2)$, and infinite-dimensional real projective space is $K(\Z/2, 1)$.
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<document source="https://tamnd.github.io/math2/node/AJNZJD8442.md" tag="AJNZJD8442" kind="concept">
# Variety of algebras

## Summary

A variety of algebras is the class of all algebras of a fixed signature that satisfy a given set of identities.
Groups, rings, lattices and modules over a fixed ring are the standard examples.
Varieties are the basic objects of [[universal algebra|W82R6N0GM9]], because many constructions, such as free algebras and presentations, work in every variety at once.
By Birkhoff's theorem, a class of algebras is a variety exactly when it is closed under homomorphic images, subalgebras and direct products.

## Explanations

### L1 human

Pick some operations and some equations that the operations must obey.
The variety is the collection of every algebra, of any size, in which these equations hold.
For example, the equations $x(yz) = (xy)z$, $ex = xe = x$ and $x x^{-1} = x^{-1} x = e$ in the operations $\cdot$, $e$ and ${}^{-1}$ define the variety of groups.
A field is not described this way, because the inverse of $0$ is undefined, and indeed fields do not form a variety.
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<document source="https://tamnd.github.io/math2/node/AJXFNKX4AZ.md" tag="AJXFNKX4AZ" kind="definition">
# Riemann–Liouville integral

## Definition

Let $a < b$ be real numbers, possibly $b = \infty$, and let $f$ be a locally integrable real function on $[a, b)$.
Let $\alpha > 0$ be real.
The *Riemann–Liouville integral* of order $\alpha$ of $f$, with lower terminal $a$, is the function
$$I_a^\alpha f(x) = \frac{1}{\Gamma(\alpha)} \int_a^x (x - t)^{\alpha - 1} f(t) \, dt, \qquad a < x < b,$$
where $\Gamma$ is the [[gamma function|0H5SYBSED9]].

## Remarks

The integral converges for almost every $x$, and for every $x$ when $f$ is continuous on $[a, b)$, because $(x - t)^{\alpha - 1}$ is integrable near $t = x$ for $\alpha > 0$.
Example: for $\beta > -1$ and $a = 0$, the substitution $t = xs$ and the beta integral give $I_0^\alpha t^\beta = \frac{\Gamma(\beta + 1)}{\Gamma(\alpha + \beta + 1)} x^{\alpha + \beta}$.
For $\beta = 0$ this is $x^\alpha / \Gamma(\alpha + 1)$.
Non-example: for $\alpha = -\tfrac{1}{2}$ and $f = 1$, the integrand $(x - t)^{-3/2}$ is not integrable near $t = x$, so the formula defines nothing; negative orders need a derivative, as in [[fractional calculus|T73XK15SA2]].
The operators satisfy the semigroup law $I_a^\alpha I_a^\beta f = I_a^{\alpha + \beta} f$ for $\alpha, \beta > 0$.
For $\alpha = 1$, $I_a^1 f(x) = \int_a^x f(t) \, dt$.

## Explanations

### L1 human

The definition averages the past values of $f$ with the weight $(x - t)^{\alpha - 1}$.
When $\alpha$ is a whole number, this is the usual repeated integral.
When $\alpha$ is not a whole number, for example $\alpha = \tfrac{1}{2}$, the same formula gives an integral of fractional order.
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<document source="https://tamnd.github.io/math2/node/AK1ZRCXR32.md" tag="AK1ZRCXR32" kind="concept">
# Noetherian module

## Summary

A Noetherian module is a module in which every increasing chain of submodules stops after finitely many steps.
Equivalently, every submodule is finitely generated.
A ring is a [[Noetherian ring|0670T59HGE]] exactly when it is a Noetherian module over itself.
This finiteness condition makes induction arguments work and underlies most of commutative algebra.

## Explanations

### L1 human

Think of building bigger and bigger submodules, one step at a time.
In a Noetherian module you cannot keep growing forever: the chain must become constant.
For example, in $\Z$ seen as a module over itself, the chain $(12) \subset (6) \subset (3) \subset \Z$ cannot be made infinitely long upward.
By contrast, in $\Q$ seen as a $\Z$-module, the chain $\Z \subset \frac{1}{2}\Z \subset \frac{1}{4}\Z \subset \cdots$ goes on forever.
</document>

<document source="https://tamnd.github.io/math2/node/AK2XQC04PC.md" tag="AK2XQC04PC" kind="definition">
# Exact functor

## Definition

Let $\mathcal{A}$ and $\mathcal{B}$ be [[abelian categories|Y9CZ3MTNF1]] and let $F \colon \mathcal{A} \to \mathcal{B}$ be an [[additive functor|KAN07YC9T2]].
Let $0 \to A' \to A \to A'' \to 0$ denote an arbitrary [[short exact sequence|KJKNJMGKZZ]] in $\mathcal{A}$.

1. $F$ is *left exact* if $0 \to F(A') \to F(A) \to F(A'')$ is exact for every such sequence.
2. $F$ is *right exact* if $F(A') \to F(A) \to F(A'') \to 0$ is exact for every such sequence.
3. $F$ is *exact* if $0 \to F(A') \to F(A) \to F(A'') \to 0$ is exact for every such sequence, that is, if $F$ is both left exact and right exact.

## Remarks

An exact functor preserves all exact sequences, not only the short ones.
A functor is left exact if and only if it preserves kernels, and right exact if and only if it preserves cokernels.
For a contravariant functor the same words are used after replacing $\mathcal{A}$ by its opposite category.

Example: for a ring $R$ and an $R$-module $M$, the functor $\Hom_R(M, -)$ is left exact, and $M \otimes_R -$ is right exact.
The functor $\Q \otimes_{\Z} -$ on abelian groups is exact, because $\Q$ is a flat $\Z$-module.

Non-example: $F = \Z/2\Z \otimes_{\Z} -$ is right exact but not exact.
Apply it to $0 \to \Z \xrightarrow{2} \Z \to \Z/2\Z \to 0$.
The first map becomes multiplication by $2$ on $\Z/2\Z$, which is zero and so not injective.
</document>

<document source="https://tamnd.github.io/math2/node/AK51SYB1Y9.md" tag="AK51SYB1Y9" kind="definition">
# Infinite loop space

## Definition

Let $X$ be a [[pointed space|T9DG8DDE75]], and write $\Omega Y$ for the [[loop space|2JFF5F3EGF]] of a pointed space $Y$.
The space $X$ is an *infinite loop space* if there are pointed spaces $X_0, X_1, X_2, \ldots$ and [[weak homotopy equivalences|4Q3WY1RX91]]
$$X \simeq X_0, \qquad X_n \to \Omega X_{n+1} \quad \text{for all } n \ge 0.$$

## Remarks

The sequence $(X_n)$ with the adjoint maps $\Sigma X_n \to X_{n+1}$ is an $\Omega$-[[spectrum|1GHF4EKBXS]] $E$, and $X$ is weakly equivalent to its zeroth space $\Omega^\infty E$.
Conversely, the zeroth space of every $\Omega$-spectrum is an infinite loop space.

Example: for an abelian group $A$, the Eilenberg–MacLane spaces satisfy $K(A, n) \simeq \Omega K(A, n+1)$, so $K(A, 0) = A$ with the discrete topology is an infinite loop space with $X_n = K(A, n)$.
The check: $\pi_k \Omega K(A, n+1) = \pi_{k+1} K(A, n+1)$, which is $A$ for $k = n$ and $0$ otherwise.

Example: for a pointed CW complex $Y$, the space $QY = \operatorname{colim}_n \Omega^n \Sigma^n Y$ is an infinite loop space, with $X_n = Q \Sigma^n Y$.
By Bott periodicity, $\Z \times BU$ is an infinite loop space too.

Non-example: the sphere $S^2$ is not an infinite loop space, and not even a loop space.
A loop space is an [[H-space|EASHTX8EG5]], and $S^2$ is not one: a cup product computation in $H^*(S^2 \times S^2; \Z)$ rules out any H-space multiplication on $S^2$.
</document>

<document source="https://tamnd.github.io/math2/node/AK9J4RNZ51.md" tag="AK9J4RNZ51" kind="statement">
# Bauer–Fike theorem

## Statement

Let $A$ be an $n \times n$ complex matrix that is [[diagonalisable|07SQCNV233]], with $A = V \Lambda V^{-1}$ for an invertible matrix $V$ and a diagonal matrix $\Lambda$.
Let $1 \le p \le \infty$, let $\norm{\cdot}_p$ be the [[operator norm|06EKW1AKXF]] induced by the $p$-norm on $\C^n$, and let $\kappa_p(V) = \norm{V}_p \norm{V^{-1}}_p$ be the [[condition number|AB3S6MHJ8X]] of $V$.
Let $E$ be an $n \times n$ complex matrix, and let $\mu$ be an [[eigenvalue|0BQAA2BK3Q]] of $A + E$.
Then there is an eigenvalue $\lambda$ of $A$ with
$$\abs{\lambda - \mu} \le \kappa_p(V) \norm{E}_p.$$

## Remarks

If $\mu$ is an eigenvalue of $A$, there is nothing to prove.
Otherwise $\Lambda - \mu I$ is invertible, and $A + E - \mu I = V(\Lambda - \mu I)\big(I + (\Lambda - \mu I)^{-1} V^{-1} E V\big) V^{-1}$ is singular.
So $1 \le \norm{(\Lambda - \mu I)^{-1} V^{-1} E V}_p \le \kappa_p(V) \norm{E}_p / \min_\lambda \abs{\lambda - \mu}$, which is the claim.

If $A$ is [[normal|0MR0BYF0K2]], $V$ can be chosen unitary, so $\kappa_2(V) = 1$ and each eigenvalue of $A + E$ is within $\norm{E}_2$ of an eigenvalue of $A$.

The diagonalisability hypothesis is needed for a bound linear in $\norm{E}$.
For $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $E = \begin{pmatrix} 0 & 0 \\ \varepsilon & 0 \end{pmatrix}$ with $\varepsilon > 0$, the eigenvalues of $A + E$ are $\pm \sqrt{\varepsilon}$, much larger than $\varepsilon$ when $\varepsilon$ is small.

## Explanations

### L1 human

If a diagonalisable matrix is changed by a small amount, its eigenvalues move by a small amount too.
The theorem gives the size of the move: the size of the change times a factor that measures how far the eigenvectors are from being orthogonal.

### L2 human

The theorem is a basic tool of [[eigenvalue perturbation|VR89YHC0TX]] theory and of the error analysis of eigenvalue algorithms.
It shows that the eigenvalue problem of a normal matrix is well conditioned.
When the eigenvectors are nearly parallel, $\kappa_p(V)$ is large and the eigenvalues can be very sensitive.
</document>

<document source="https://tamnd.github.io/math2/node/AKBWT323RQ.md" tag="AKBWT323RQ" kind="concept">
# Measurable cardinal

## Summary

A measurable cardinal is an uncountable cardinal $\kappa$ that carries a nonprincipal [[ultrafilter|J64Y3JC0RE]] closed under intersections of fewer than $\kappa$ sets.
Equivalently, $\kappa$ is the critical point of an [[elementary embedding|CGH8C8EA9N]] $j \colon V \to M$ of the universe into a transitive class.
Measurable cardinals are [[weakly compact|1RHTVMH34G]], and their existence contradicts the [[axiom of constructibility|4QKRPRCSXH]].
The name comes from the measure problem of Banach and Ulam: a two-valued measure on all subsets of $\kappa$.

## Explanations

### L1 human

An ultrafilter on a set decides, for each subset, whether it is "large" or "small".
On a measurable cardinal there is such a decision that is very robust: the intersection of fewer than $\kappa$ large sets is still large, and single points are small.
On the natural numbers one only gets this for finitely many sets at a time.
This robust notion of "large" makes $\kappa$ so big that its existence changes the whole universe; for example, not every set can be constructible.
</document>

<document source="https://tamnd.github.io/math2/node/AKDR5YBBE2.md" tag="AKDR5YBBE2" kind="statement">
# Rice's theorem

## Statement

Fix a standard [[Gödel numbering|MFJXPAF0R0]] of programs, and let $\varphi_e$ be the partial [[computable function|KZSB5E471K]] of one argument computed by program $e$.
Let $\mathcal P$ be a set of partial computable functions $\N \rightharpoonup \N$ that is neither empty nor the set of all partial computable functions.
Then the index set $\set{e \in \N : \varphi_e \in \mathcal P}$ is not a [[computable set|PPGD5KW8JC]].

## Remarks

Replacing $\mathcal P$ by its complement if needed, assume the empty function is not in $\mathcal P$, and pick $\psi \in \mathcal P$.
For each $e$, let $g(e)$ be a program that on input $y$ first runs program $e$ on input $e$, and then, if that halts, computes $\psi(y)$.
By the [[smn theorem|YWX4FCFPAP]], $g$ is total computable.
Then $\varphi_{g(e)} = \psi$ if program $e$ halts on $e$, and $\varphi_{g(e)}$ is the empty function otherwise.
So a decision procedure for the index set would decide the [[halting problem|Z7TN2B35B0]].

Example: the set of programs that compute a total function, and the set of programs that output $0$ on input $0$, are not computable.

Both conditions on $\mathcal P$ are needed: if $\mathcal P$ is empty or contains all partial computable functions, the index set is $\emptyset$ or $\N$, which are computable.
The theorem concerns properties of the computed function, not of the program text.
A property such as "the program has fewer than $10$ instructions" is decidable.

## Explanations

### L1 human

Rice's theorem says that no algorithm can decide any interesting property of what programs do.
"Interesting" means that some programs have the property and some do not, and that it depends only on the input-output behaviour.
So no tool can decide, for every program, whether it ever prints $0$ or whether it computes the squaring function.

### L2 human

The theorem generalises the undecidability of the [[halting problem|Z7TN2B35B0]] to all non-trivial semantic properties.
Its proof is a many-one reduction from the halting set, built with the smn theorem.
The Rice–Shapiro theorem goes further and describes which index sets are computably enumerable.
In practice, program analysis tools work around the theorem by giving approximate answers.
</document>

<document source="https://tamnd.github.io/math2/node/AKEDFDGCDY.md" tag="AKEDFDGCDY" kind="definition">
# Riemann P-symbol

## Definition

Let $a, b, c$ be distinct points of the Riemann sphere, and let $\alpha, \alpha', \beta, \beta', \gamma, \gamma' \in \C$ with $\alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1$.
The *Riemann P-symbol* is the table
$$P\begin{Bmatrix} a & b & c & \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' & \end{Bmatrix}.$$
It denotes the set of solutions $w(z)$ of [[Riemann's differential equation|RS8Z48E784]] with singular points $a, b, c$.
The exponents are $\alpha, \alpha'$ at $a$, $\beta, \beta'$ at $b$ and $\gamma, \gamma'$ at $c$.
When one of the points is $\infty$, the equation is the limiting form in which the factors involving that point are dropped.

## Remarks

The order of the columns, and the order of the two exponents within a column, do not matter.
Two basic rules hold.

1. If $T$ is a [[Möbius transformation|0SG74WBFA2]], then the P-symbol with points $a, b, c$ and variable $z$ equals the P-symbol with points $T(a), T(b), T(c)$, the same exponents, and variable $T(z)$.
2. If $a, b, c$ are finite, then for all $k, l \in \C$,
$$\left(\frac{z - a}{z - b}\right)^{k} \left(\frac{z - c}{z - b}\right)^{l} P\begin{Bmatrix} a & b & c & \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' & \end{Bmatrix} = P\begin{Bmatrix} a & b & c & \\ \alpha + k & \beta - k - l & \gamma + l & z \\ \alpha' + k & \beta' - k - l & \gamma' + l & \end{Bmatrix}.$$

Example: the [[hypergeometric function|463X5V31D9]] satisfies
$${}_2F_1(a, b; c; z) \in P\begin{Bmatrix} 0 & \infty & 1 & \\ 0 & a & 0 & z \\ 1 - c & b & c - a - b & \end{Bmatrix},$$
and the exponents add up to $0 + (1 - c) + a + b + 0 + (c - a - b) = 1$, as required.

Non-example: a table whose six exponents do not add up to $1$ is not a valid P-symbol.
No second-order Fuchsian equation has exactly those three singular points with those exponents.

Rule 2 is consistent with the sum condition: the shifts $k$, $-k-l$, $l$ appear twice each and add up to $0$.
</document>

<document source="https://tamnd.github.io/math2/node/AKMASKR0ZG.md" tag="AKMASKR0ZG" kind="definition">
# Quasivariety

## Definition

Let $\sigma$ be an algebraic signature.
Let $K$ be a class of $\sigma$-algebras.
Then $K$ is a *quasivariety* if there is a set $\Sigma$ of [[quasi-identities|94GVY3APPQ]] of type $\sigma$ with the following property.
An algebra of type $\sigma$ is in $K$ exactly when it satisfies every member of $\Sigma$.

## Remarks

Example: the class of cancellative semigroups is a quasivariety.
It is defined by the associative law, which is a quasi-identity with no hypotheses, and by the two quasi-identities $z x \approx z y \rightarrow x \approx y$ and $x z \approx y z \rightarrow x \approx y$.

Non-example: the class of fields, in the signature $(+, -, \cdot, 0, 1)$, is not a quasivariety.
Every quasivariety contains the one-element algebra, since all terms take the same value there, so every conclusion $s \approx t$ holds.
But a field needs $0 \ne 1$.
Also, every quasivariety is closed under direct products, and $\Q \times \Q$ is not a field.

Each [[variety|AJNZJD8442]] is a quasivariety, defined by identities, which are quasi-identities with no hypotheses.
The smallest quasivariety containing a class $K$ is denoted $Q(K)$ or $\mathbf{ISPP_U}(K)$, by [[Mal'cev's characterization|DFNM3BGMAY]].
</document>

<document source="https://tamnd.github.io/math2/node/AM07TESJED.md" tag="AM07TESJED" kind="definition">
# Quaternion algebra

## Definition

Let $F$ be a [[field|04C5ZBG1WT]].
A *quaternion algebra* over $F$ is a [[central simple algebra|5HRKJBAY2V]] over $F$ of dimension $4$.

Let the [[characteristic|0F6ES3BRYP]] of $F$ be different from $2$, and let $a, b \in F$ be nonzero.
The quaternion algebra $\left(\frac{a, b}{F}\right)$ is the [[associative algebra|3CTW1P4HBP]] over $F$ with basis $1, i, j, k$ and multiplication determined by
$$i^2 = a, \qquad j^2 = b, \qquad ij = -ji = k.$$

## Remarks

When the characteristic of $F$ is not $2$, every quaternion algebra over $F$ is isomorphic to $\left(\frac{a, b}{F}\right)$ for some nonzero $a, b$.
In this algebra $k^2 = -ab$.

Example: $\left(\frac{-1, -1}{\R}\right)$ is Hamilton's algebra $\mathbb{H}$ of quaternions.
The element $q = x + yi + zj + wk$ has $q \bar{q} = x^2 + y^2 + z^2 + w^2$ for $\bar{q} = x - yi - zj - wk$, so every $q \ne 0$ is invertible, and $\mathbb{H}$ is a division algebra.

Example: $\left(\frac{1, 1}{F}\right) \cong M_2(F)$, through $i \mapsto \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ and $j \mapsto \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.

Non-example: the algebra $F[x, y]/(x^2 - a, y^2 - b, xy - yx)$ also has dimension $4$, but it is commutative, so its center is not $F$ and it is not a quaternion algebra.

Every quaternion algebra is either a division algebra or isomorphic to $M_2(F)$.
The algebra $\left(\frac{a, b}{F}\right)$ is isomorphic to $M_2(F)$ if and only if the equation $a x^2 + b y^2 = 1$ has a solution with $x, y \in F$.
</document>

<document source="https://tamnd.github.io/math2/node/AM4RNWVV4Q.md" tag="AM4RNWVV4Q" kind="concept">
# Symbol class

## Summary

A symbol class is a space of smooth functions $a(x, \xi)$ of a position $x$ and a frequency $\xi$.
Its elements grow or decay like a fixed power of $\abs{\xi}$, and their derivatives in $\xi$ decay faster.
The standard classes are the Hörmander classes $S^m_{\rho, \delta}$, where $m$ is the order.
Symbols generalise the polynomials $\sum_\alpha a_\alpha(x) \xi^\alpha$ attached to differential operators.
They are the building blocks of pseudo-differential operators and of Fourier integral operators.

## Explanations

### L1 human

A differential operator such as $\partial_x^2 + 1$ acts on a wave $e^{ix\xi}$ by multiplying it by the polynomial $1 - \xi^2$, its symbol.
Allowing more general functions of $\xi$ in place of polynomials leads to more general operators, for example a square root of $1 - \Delta$.
To keep a good calculus one asks that the symbol behaves like a power $\abs{\xi}^m$ for large $\xi$, and that each derivative in $\xi$ lowers this power by one.
For instance $(1 + \abs{\xi}^2)^{s/2}$ is a symbol of order $s$.
</document>

<document source="https://tamnd.github.io/math2/node/AM7YJP8AR6.md" tag="AM7YJP8AR6" kind="concept">
# Simple Lie group

## Summary

A simple Lie group is a connected [[Lie group|7JBRGCVJRB]] whose Lie algebra is a simple Lie algebra.
Such a Lie algebra is not abelian and has no ideals other than $0$ and itself.
Simple Lie groups are the building blocks of the [[semisimple Lie groups|XJS07AJ50Y]], and through them of a large part of Lie theory.
Up to coverings and central quotients, they are classified by the simple Lie algebras.
These form the classical families $A_n, B_n, C_n, D_n$ and five exceptional types.

## Explanations

### L1 human

A simple Lie group is a continuous group of symmetries that cannot be broken into smaller continuous pieces.
Think of the rotations of space: there is no smaller family of rotations, closed under conjugation and depending continuously on parameters, other than the trivial one.
Simplicity is checked on the infinitesimal level, on the Lie algebra, so a simple Lie group may still have a finite centre.
For example, $SU(2)$ is a simple Lie group even though its centre $\set{I, -I}$ is a normal subgroup.
The rotation group $SO(3)$, the groups $SL(n, \R)$ for $n \ge 2$ and the exceptional group $E_8$ are other examples.
</document>

<document source="https://tamnd.github.io/math2/node/AMC3JCQ497.md" tag="AMC3JCQ497" kind="definition">
# Voronoi diagram

## Definition

Let $n \ge 1$ and let $S = \set{p_1, \dots, p_k}$ be a finite set of $k \ge 1$ distinct points of [[Euclidean space|0XPR1E85W0]] $\R^n$, called *sites*.
Let $\norm{x - y}$ denote the [[Euclidean distance|0NPPGHQNXQ]].
The *Voronoi cell* of the site $p_i$ is
$$V(p_i) = \set{x \in \R^n : \norm{x - p_i} \le \norm{x - p_j} \text{ for all } j = 1, \dots, k}.$$
The *Voronoi diagram* of $S$ is the family of cells $V(p_1), \dots, V(p_k)$.

## Remarks

Example: for the two sites $p_1 = (-1, 0)$ and $p_2 = (1, 0)$ in $\R^2$, the condition $\norm{x - p_1} \le \norm{x - p_2}$ reduces to $x_1 \le 0$.
So $V(p_1)$ is the closed half-plane $x_1 \le 0$, $V(p_2)$ is the half-plane $x_1 \ge 0$, and they meet along the perpendicular bisector $x_1 = 0$.

Non-example: the cells $\set{x : x_1 \le -1}$ and $\set{x : x_1 \ge -1}$ are not the Voronoi diagram of these two sites, since the point $(-1/2, 0)$ is closer to $p_1$ but lies in the second set only.

Each condition $\norm{x - p_i} \le \norm{x - p_j}$ defines a closed half-space, so each cell is a convex polyhedron.
The cells cover $\R^n$, and two distinct cells have disjoint interiors, so the Voronoi diagram is a tiling of $\R^n$.
The Voronoi cell of a point of a lattice, with all lattice points as sites, is a [[Voronoi cell|V1R1R88GV6]] in the sense of the geometry of numbers.
</document>

<document source="https://tamnd.github.io/math2/node/AMDN6PK5RA.md" tag="AMDN6PK5RA" kind="definition">
# Steenrod reduced power

## Definition

Let $p$ be an odd prime, and write $H^n(X)$ for the [[singular cohomology|6N7BC70CX0]] group $H^n(X; \F_p)$ of a space $X$.
The *Steenrod reduced powers* are the unique family of homomorphisms
$$P^i \colon H^n(X) \to H^{n + 2i(p-1)}(X), \qquad i \ge 0,\ n \ge 0,$$
defined for all spaces $X$, with the following properties.

1. Each $P^i$ is natural: $f^* \circ P^i = P^i \circ f^*$ for every continuous map $f \colon Y \to X$.
2. Each $P^i$ commutes with the suspension isomorphism $H^n(X) \cong H^{n+1}(SX)$ in reduced cohomology.
3. $P^0$ is the identity.
4. If $x \in H^{2i}(X)$, then $P^i x = x^p$, the $p$-th [[cup product|JX47MS1R1H]] power of $x$.
5. If $x \in H^n(X)$ and $2i > n$, then $P^i x = 0$.
6. (Cartan formula) $P^k(x \smile y) = \sum_{i + j = k} P^i x \smile P^j y$ for all $x, y \in H^*(X)$.

## Remarks

Existence of such a family is a theorem of Steenrod; uniqueness follows from the axioms by evaluating on products of Eilenberg–MacLane spaces.
The operations also exist on cohomology of pairs and commute with the coboundary maps of pairs.

Example: on $X = \C P^\infty$, with $H^*(X) = \F_p[x]$ and $\deg x = 2$, clauses 4 and 6 give $P^i(x^m) = \binom{m}{i} x^{m + i(p-1)}$.
For instance $P^1 x = x^p$, which is nonzero.

Non-example: let $Q^0$ be the identity and $Q^i = 0$ for $i > 0$.
This family satisfies clauses 1, 2, 3, 5 and 6.
It fails clause 4: on $\C P^\infty$ one has $Q^1 x = 0$, while $x^p \neq 0$.

The operation $P^i$ is a [[cohomology operation|K8T15QHN3W]] of type $(\F_p, n; \F_p, n + 2i(p-1))$.
For $p = 2$ the analogous operations are the Steenrod squares $Sq^i$, with $Sq^{2i}$ playing the role of $P^i$.
</document>

<document source="https://tamnd.github.io/math2/node/AME1ZE5HBX.md" tag="AME1ZE5HBX" kind="definition">
# Exterior differential system

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] and let $\Omega(M) = \bigoplus_{k \ge 0} \Omega^k(M)$ be the algebra of smooth [[differential forms|0E3G0GRPZ6]] with the [[wedge product|0GRAATMGP1]].
An *exterior differential system* on $M$ is a subset $\mathcal{I} \subseteq \Omega(M)$ with these properties.

1. $\mathcal{I}$ is a two-sided [[ideal|096Z9MP2ME]]: it is closed under addition, and $\alpha \wedge \varphi \in \mathcal{I}$ for all $\alpha \in \Omega(M)$ and $\varphi \in \mathcal{I}$.
2. $\mathcal{I}$ is graded: if $\varphi \in \mathcal{I}$, then each homogeneous component of $\varphi$ is in $\mathcal{I}$.
3. $\mathcal{I}$ is closed under the [[exterior derivative|0ZHSTAG8ZY]]: $d\varphi \in \mathcal{I}$ for all $\varphi \in \mathcal{I}$.

An ideal with these properties is also called a differential ideal.
An *integral manifold* of $\mathcal{I}$ is an [[immersion|6RW68JGBHC]] $\iota \colon N \to M$ with $\iota^* \varphi = 0$ for all $\varphi \in \mathcal{I}$.

## Remarks

Usually $\mathcal{I}$ is given by finitely many generators $\varphi_1, \dots, \varphi_r$.
The differential ideal they generate is the algebraic ideal generated by the $\varphi_a$ and the $d\varphi_a$.
Adding the $d\varphi_a$ changes no integral manifold, because $\iota^* d\varphi_a = d(\iota^* \varphi_a) = 0$ whenever $\iota^* \varphi_a = 0$.

Example: on $M = \R^3$ with coordinates $(x, u, p)$, let $\mathcal{I}$ be generated by $\theta = du - p\, dx$ and $d\theta = dx \wedge dp$.
For every smooth function $f$, the curve $x \mapsto (x, f(x), f'(x))$ is an integral manifold: $\theta$ pulls back to $(f' - f')\, dx = 0$, and every $2$-form pulls back to $0$ on a curve.

Non-example: the algebraic ideal generated by $\theta$ alone fails clause 3.
Every element of it has the form $\alpha \wedge \theta$, so every $2$-form in it is zero after wedging with $\theta$.
But $\theta \wedge d\theta = du \wedge dx \wedge dp \ne 0$, so $d\theta$ is not in that ideal.
</document>

<document source="https://tamnd.github.io/math2/node/AMP0CRE14P.md" tag="AMP0CRE14P" kind="statement">
# Existence of best approximations from finite-dimensional subspaces

## Statement

Let $X$ be a real or complex [[normed vector space|03J5RAKB30]], and let $V \subseteq X$ be a [[linear subspace|00YC1D652K]] of finite [[dimension|03GSZ8RTZJ]].
Then every $x \in X$ has a [[best approximation|7N4988HN3P]] from $V$: there is $v^* \in V$ with
$$\norm{x - v^*} = \inf_{v \in V} \norm{x - v}.$$

## Remarks

The idea of the proof: the point $0 \in V$ has distance $\norm{x}$ from $x$, so it suffices to minimize over $B = \set{v \in V : \norm{v} \le 2 \norm{x}}$, since every $v \notin B$ has $\norm{x - v} \ge \norm{v} - \norm{x} > \norm{x}$.
In a finite-dimensional normed space closed bounded sets are compact, so $B$ is compact.
The continuous function $v \mapsto \norm{x - v}$ attains its minimum on $B$.

Example: for $f \in C[a, b]$ with the supremum norm and $V$ the polynomials of degree at most $n$, a best uniform approximation by polynomials of degree at most $n$ exists.
The same holds for trigonometric polynomials, for splines with fixed knots, and in every $L^p$ norm.

Finite dimension is needed.
In the space $c_0$ of null sequences with the supremum norm, the functional $\varphi(y) = \sum_{k \ge 1} 2^{-k} y_k$ has norm $1$ but $\abs{\varphi(y)} < \norm{y}_\infty$ for every $y \ne 0$.
Its kernel is a closed subspace, and no point outside the kernel has a best approximation from it.

The best approximation need not be unique: in $\R^2$ with the maximum norm, all points $(t, 0)$ with $\abs{t} \le 1$ are best approximations of $(0, 1)$ from the horizontal axis.

## Explanations

### L1 human

When the approximating family depends on finitely many parameters, such as the coefficients of a polynomial of degree $n$, a best choice always exists.
The reason is compactness: only a bounded region of parameters is worth searching, and in finitely many dimensions a continuous function attains its minimum on a closed bounded region.

### L2 human

This is the basic existence result of linear approximation theory, and every concrete problem with a fixed degree, such as minimax polynomial approximation, starts from it.
The questions that follow are uniqueness, which needs extra structure such as strict convexity or the Haar condition, and characterization, given by the Kolmogorov criterion and the equioscillation theorem.
In a Hilbert space the result holds for every closed subspace, by the projection theorem, but in general Banach spaces infinite-dimensional closed subspaces may fail to have best approximations.
</document>

<document source="https://tamnd.github.io/math2/node/AMVHH8JT14.md" tag="AMVHH8JT14" kind="definition">
# Sigma-compact space

## Definition

A [[topological space|0CQHFPGJ45]] $X$ is *σ-compact* when there are [[compact|0R6EWFKRCF]] subsets $K_1, K_2, K_3, \ldots$ of $X$ with
$$X = \bigcup_{n \ge 1} K_n.$$

## Remarks

The family may be finite or [[countable|01TQB88H4C]]; a finite family can be padded by repeating a set.
Every σ-compact space is [[Lindelöf|SW0SN1NEJE]]: an open cover has a finite subcover of each $K_n$, and the union of these is a countable subcover.
A closed subspace and a continuous image of a σ-compact space are σ-compact.

Example: $\R^n$ is σ-compact.
It is the union of the closed balls $\set{x : \norm{x} \le m}$ for $m \ge 1$, which are compact by the Heine–Borel theorem.
Example: $\Q$ is σ-compact, as the countable union of its one-point subsets.

Non-example: an uncountable set $X$ with the discrete topology is not σ-compact.
Its compact subsets are finite, and a countable union of finite sets is countable.

Notation: σ-compact, also written sigma-compact.
</document>

<document source="https://tamnd.github.io/math2/node/AMXXM288Y0.md" tag="AMXXM288Y0" kind="definition">
# Minimal sufficient statistic

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a [[statistical model|JZ6R7J36K5]] on $(S, \mathcal{S})$ and let $X$ be the data.
A statistic $T = T(X)$ is *minimal sufficient* if:

1. $T$ is [[sufficient|QNMR5F525C]] for $\theta$, and
2. for every sufficient statistic $U = U(X)$ there is a measurable function $\varphi$ with $T = \varphi(U)$ $P_\theta$-almost surely for every $\theta \in \Theta$.

## Remarks

Example: for $X_1, \dots, X_n$ i.i.d. Bernoulli($p$), $p \in (0, 1)$, the sum $T = \sum_i X_i$ is minimal sufficient.
Indeed, the likelihood ratio $P_p(X = x)/P_p(X = y) = (p/(1 - p))^{\sum_i x_i - \sum_i y_i}$ is free of $p$ only when $\sum_i x_i = \sum_i y_i$, and by the criterion below this makes $T$ minimal.
Non-example: for $n \ge 2$, the whole sample $X$ is sufficient but not minimal.
Clause 2 fails with $U = T$: $X$ is not a function of $T$, since $(1, 0, \dots, 0)$ and $(0, 1, 0, \dots, 0)$ have the same sum.
A common criterion, for a model with densities $f_\theta$: if $T(x) = T(y)$ holds exactly when $f_\theta(x)/f_\theta(y)$ does not depend on $\theta$, then $T$ is minimal sufficient.
A minimal sufficient statistic is unique up to one-to-one measurable functions.
</document>

<document source="https://tamnd.github.io/math2/node/AN054RHT1K.md" tag="AN054RHT1K" kind="concept">
# Domain decomposition methods

## Summary

Domain decomposition methods solve a [[boundary value problem|TM8VQJWGHZ]] on a domain by splitting the domain into smaller subdomains.
They solve problems on the subdomains, often in parallel, and couple the local solutions through the interfaces until they fit together.
They are mainly used as [[preconditioners|KE98SW3SBD]] for Krylov methods on large parallel computers.

## Explanations

### L1 human

A big problem on a complicated region is hard to solve at once.
Cut the region into pieces, give each piece to a different processor, and solve each small problem there.
The pieces then exchange information along their common boundaries and solve again, until the pieces agree.
The classical example, due to Schwarz, solves a problem on the union of a disc and a rectangle by alternating between the two shapes.
</document>

<document source="https://tamnd.github.io/math2/node/AN0BYRKNRA.md" tag="AN0BYRKNRA" kind="concept">
# Dimension of an analytic set

## Summary

The dimension of a [[complex analytic variety|GB7M582M5Y]] at a point measures how many complex parameters are needed to describe it near that point.
At a [[regular point|ZW5A88PCPK]] it is the dimension of the set as a complex manifold.
At a singular point it is the largest dimension of the manifold pieces nearby.
It agrees with the Krull dimension of the local ring of the set, and with the number $d$ of the local parametrization theorem.

## Explanations

### L1 human

A curve has dimension $1$ and a surface dimension $2$, counted in complex parameters.
An analytic set can mix pieces of different dimensions.
In $\C^3$, take the union of the plane $\set{z_3 = 0}$ and the line $\set{z_1 = z_2 = 0}$.
It has dimension $2$ at the origin, where the plane passes, and dimension $1$ at the points of the line away from the plane.
</document>

<document source="https://tamnd.github.io/math2/node/AN7QEWTNBT.md" tag="AN7QEWTNBT" kind="definition">
# Arnold diffusion

## Definition

Let $n \ge 1$, let $D \subseteq \R^n$ be open and let $\mathbb{T}^n = \R^n / 2\pi \Z^n$.
Use [[action-angle coordinates|M3T81YNSKC]] $(I, \theta)$ on $D \times \mathbb{T}^n$.
Consider the family of [[Hamiltonian systems|0EDKGA2FAK]] with Hamiltonians
$$H_\varepsilon(I, \theta, t) = H_0(I) + \varepsilon H_1(I, \theta, t),$$
where $H_1$ is smooth and may depend periodically on time $t$.
The family exhibits *Arnold diffusion* if there is a constant $\delta > 0$ with the following property.
For every sufficiently small $\varepsilon > 0$ there are a solution $(I(t), \theta(t))$ of the system of $H_\varepsilon$ and a time $T > 0$ with
$$\abs{I(T) - I(0)} \ge \delta.$$

## Remarks

The point is that $\delta$ does not depend on $\varepsilon$: the actions drift by an amount of order one, while for $\varepsilon = 0$ they are constant.
The time $T$ needed tends to infinity as $\varepsilon \to 0$.

Example: Arnold's 1964 example is the time-periodic Hamiltonian
$$H(I, \theta, t) = \tfrac{1}{2}(I_1^2 + I_2^2) + \varepsilon (\cos \theta_1 - 1)\big(1 + \mu (\sin \theta_2 + \cos t)\big)$$
with two angles and periodic time.
Arnold proved that for suitable small $\mu$ depending on $\varepsilon$ there are orbits along which $I_2$ changes by an amount of order one.

Non-example: an autonomous real-analytic system with $n = 2$ degrees of freedom and isoenergetically nondegenerate $H_0$ has no Arnold diffusion.
By the [[KAM theorem|3ZZSXKM16Z]], each $3$-dimensional energy level contains invariant $2$-tori that cut it into regions.
The action of an orbit is trapped between two such tori, so the drift tends to $0$ with $\varepsilon$.

For real-analytic $H_\varepsilon$ with steep $H_0$, the [[Nekhoroshev estimates|PYEN6SFS9P]] show that the drift time $T$ grows at least exponentially in a power of $1/\varepsilon$.
</document>

<document source="https://tamnd.github.io/math2/node/AN9SKEZ3VY.md" tag="AN9SKEZ3VY" kind="concept">
# Multiplicative set

## Summary

A multiplicative set in a [[commutative ring|0310REJJY8]] is a subset that contains $1$ and is closed under multiplication.
These sets are exactly the sets of elements that one may turn into units by [[localization|4C1AE84N5N]].
The complement of a [[prime ideal|00MHHZ9SNE]] and the set of powers of one element are the two main examples.

## Explanations

### L1 human

Think of a multiplicative set as a list of allowed denominators.
If you may divide by $s$ and by $t$, then you may also divide by $st$, so the list must be closed under products.
You may always divide by $1$, so $1$ is on the list.
In the integers, the powers $1, 2, 4, 8, \ldots$ form a multiplicative set, and allowing them as denominators gives the fractions $a / 2^k$.
The [[definition|PCH3G8BJ9Y]] gives the precise conditions.
</document>

<document source="https://tamnd.github.io/math2/node/AND2WHZC8P.md" tag="AND2WHZC8P" kind="concept">
# Presheaf

## Summary

A presheaf on a [[category|A2DE66937D]] $\mathcal{C}$ is a contravariant [[functor|28NQ0KRHD3]] from $\mathcal{C}$ to the category of sets.
Presheaves on $\mathcal{C}$ and the natural transformations between them form a category that contains $\mathcal{C}$ and has all limits and colimits.
The name comes from topology, where a presheaf on a space assigns data to open sets with restriction maps to smaller open sets.

## Explanations

### L1 human

A presheaf attaches a set to each object.
For each arrow $X \to Y$ it gives a way to pull elements back from the set at $Y$ to the set at $X$.
The model case is a [[topological space|0CQHFPGJ45]]: to each open set $U$ attach the set of continuous real functions on $U$.
If $V \subseteq U$, a function on $U$ restricts to a function on $V$, so the maps go from big sets to small sets, against the inclusion.
Restricting twice is the same as restricting once, which is the functor rule.
</document>

<document source="https://tamnd.github.io/math2/node/ANEFY3FQZR.md" tag="ANEFY3FQZR" kind="definition">
# Quotient of a normed space

## Definition

Let $X$ be a [[normed vector space|03J5RAKB30]] over $\R$ or $\C$, and let $M$ be a [[linear subspace|00YC1D652K]] of $X$ that is [[closed|0ZCK8DQYN1]].
The *quotient normed space* $X/M$ is the [[quotient vector space|0BR3FY5Q8M]] $X/M = \set{x + M : x \in X}$ with the norm
$$\norm{x + M} = \inf_{m \in M} \norm{x + m}.$$
The map $\pi \colon X \to X/M$, $\pi(x) = x + M$, is the quotient map.

## Remarks

The number $\norm{x + M}$ is the distance from $x$ to $M$.
Closedness of $M$ is what makes this a norm: $\norm{x + M} = 0$ means that $x$ is in the closure of $M$, hence in $M$.
The quotient map satisfies $\norm{\pi(x)} \le \norm{x}$ and maps the open unit ball of $X$ onto the open unit ball of $X/M$.
If $X$ is a Banach space, then so is $X/M$.
Example: in $\R^2$ with the Euclidean norm, let $M = \set{(a, 0) : a \in \R}$; then $\norm{(a, b) + M} = \inf_{c} \sqrt{(a + c)^2 + b^2} = \abs{b}$, which is a norm on $\R^2/M$.
Example: in $\ell^\infty$ with $M = c_0$, one has $\norm{x + c_0} = \limsup_{n} \abs{x_n}$.
Non-example: in $c_0$, the subspace $c_{00}$ of finitely supported sequences is dense but not closed; every $x \in c_0$ has $\norm{x + c_{00}} = 0$, so the formula gives only a seminorm on $c_0/c_{00}$.
</document>

<document source="https://tamnd.github.io/math2/node/ANH16GQJZQ.md" tag="ANH16GQJZQ" kind="concept">
# Tensor product of fields

## Summary

Let $E$ and $F$ be extensions of a field $K$.
Their tensor product $E \otimes_K F$ is the [[tensor product of vector spaces|0BF2DE2SYK]] over $K$, with the product $(a \otimes b)(c \otimes d) = ac \otimes bd$.
It is usually not a field.
Each way to put $E$ and $F$ inside one field gives a quotient of $E \otimes_K F$ by a prime ideal.
The field of fractions of this quotient is the [[compositum|T1S29NN8PE]], so the tensor product records all these ways at once.
For a finite separable extension $E = K(\alpha)$ it is a product of fields, one for each irreducible factor over $F$ of the minimal polynomial of $\alpha$.

## Explanations

### L1 human

Given two extensions of $K$, there may be several ways to fit them together in a larger field.
The tensor product keeps track of all of them at once.
For example, $\C \otimes_\R \C$ is isomorphic to $\C \times \C$.
The two factors correspond to the two ways of identifying the second copy of $\C$ with the first: the identity and complex conjugation.
</document>

<document source="https://tamnd.github.io/math2/node/ANRXAG227H.md" tag="ANRXAG227H" kind="statement">
# Dedekind's characterization of modular lattices

## Statement

Let $L$ be a [[lattice|WH4SFAACAS]].
Let $N_5$ be the [[pentagon lattice|DHC8TZXGYS]], with elements $0 < a < c < 1$ and an element $b$ that is incomparable to $a$ and $c$, with $b \wedge c = 0$ and $a \vee b = 1$.
Then $L$ is a [[modular lattice|QX7WZ92QAK]] if and only if $L$ has no [[sublattice|PV90EW00XK]] isomorphic to $N_5$.

## Remarks

One direction is easy: $N_5$ is not modular, and the modular law holds in every sublattice of a modular lattice.
For the other direction, suppose $a \le b$ and $x$ violate the law, and put $u = a \vee (x \wedge b)$ and $v = (a \vee x) \wedge b$, so $u < v$.
A direct computation gives $x \wedge u = x \wedge v = x \wedge b$ and $x \vee u = x \vee v = a \vee x$.
So the five elements $x \wedge b$, $u$, $v$, $x$ and $a \vee x$ form a sublattice isomorphic to $N_5$.

Example: the lattice of subspaces of a vector space is modular, so it contains no copy of $N_5$.
Together with the [[M3–N5 characterization of distributivity|C6GH4VHAMQ]], it shows that the pentagon and the diamond are the only obstructions to distributivity.

## Explanations

### L1 human

The modular law is an equation that must hold in all of a lattice.
The theorem says that it fails only when a five-element picture, the pentagon, appears inside the lattice.
So to test modularity, one looks for a pentagon.

### L2 human

The theorem turns an equational condition into a forbidden-substructure condition.
It is used to show that a given lattice is not modular by exhibiting a pentagon, and to prove modularity of lattices built from modular pieces.
The same style of result, with the diamond $M_3$ added, characterizes [[distributive lattices|0HM0HQCK6N]].
The proof is short and gives the pentagon explicitly from any failure of the modular law.
</document>

<document source="https://tamnd.github.io/math2/node/ANRZZMH2Q0.md" tag="ANRZZMH2Q0" kind="concept">
# Lane–Emden equation

## Summary

The Lane–Emden equation is the [[ordinary differential equation|0NT64JV96A]] $\xi^{-2} (\xi^2 \theta')' + \theta^n = 0$, which describes the density of a self-gravitating ball of gas in equilibrium, called a polytrope.
It is the equation $\Delta u + u^n = 0$ for radial functions in $\R^3$.
In PDE the name also refers to the semilinear elliptic equation $-\Delta u = u^p$ in any dimension, whose solutions depend on how $p$ compares with critical exponents.
It is a standard test case for existence, nonexistence and Liouville-type theorems.

## Explanations

### L1 human

A star made of gas is held together by its own gravity and pushed apart by its pressure.
If the pressure is a fixed power of the density, the balance of the two forces becomes the Lane–Emden equation.
The unknown $\theta$ measures the density as a function of the scaled distance $\xi$ from the centre, starting at $\theta(0) = 1$ with zero slope.
For $n = 1$ the solution is $\theta(\xi) = \sin \xi / \xi$, and its first zero $\xi = \pi$ marks the surface of the star.
</document>

<document source="https://tamnd.github.io/math2/node/AP1KY4EHMA.md" tag="AP1KY4EHMA" kind="concept">
# Handlebody

## Summary

In 3-manifold topology, a handlebody of genus $g$ is the compact orientable $3$-manifold obtained from a $3$-ball by attaching $g$ one-handles.
It is a solid version of the closed orientable surface of [[genus|NJ787GZS1N]] $g$, which is its boundary.
It is a ball for $g = 0$, a solid torus for $g = 1$, and a solid double torus for $g = 2$.
Every closed orientable $3$-manifold is obtained by gluing two handlebodies of the same genus along their boundaries, which is a [[Heegaard splitting|EDE9919B5R]].
More generally, the word handlebody is also used for any manifold with a given [[handle decomposition|9V11N8VVTP]].

## Explanations

### L1 human

A handlebody is the solid inside of a surface with holes, like a solid doughnut or a solid pretzel.
Take a ball of clay and attach $g$ solid handles to it, like the handle of a cup; the result is a handlebody of genus $g$.
A solid doughnut is the handlebody of genus $1$, and its surface is the torus.
A thickened version of any connected graph drawn in space is also a handlebody.
</document>

<document source="https://tamnd.github.io/math2/node/AP2ZDXANKC.md" tag="AP2ZDXANKC" kind="concept">
# Quantum group

## Summary

Quantum group is a common name for several families of [[Hopf algebras|7XFYP0N84H]] that deform classical objects attached to Lie groups and Lie algebras.
The main families are the [[quantum enveloping algebras|1V8H34F6N9]] $U_q(\mathfrak{g})$ of Drinfeld and Jimbo, their duals the quantized coordinate algebras $\mathcal{O}_q(G)$, and Woronowicz's [[compact quantum groups|FHQKT6HPW2]].
They are neither commutative nor cocommutative in general, and many carry a [[quasitriangular|NCRPWS9W5R]] structure.
This structure gives solutions of the [[Yang–Baxter equation|663BP8ZHS5]] and invariants of knots and links.

## Explanations

### L1 human

A group can be studied through the algebra of functions on it.
That algebra is commutative.
A quantum group is what one gets by allowing this algebra of functions to become noncommutative while keeping a rule for "multiplying points", the coproduct.
A typical example depends on a parameter $q$: at $q = 1$ it is the coordinate ring of $SL_2$, and for other $q$ the coordinates $a, b$ satisfy $ab = q\, ba$ instead of $ab = ba$.
There is no actual group behind it, only the algebra, but much of the theory of groups and their representations survives.
</document>

<document source="https://tamnd.github.io/math2/node/AP4A0GX0V2.md" tag="AP4A0GX0V2" kind="statement">
# Eells–Sampson theorem

## Statement

Let $(M, g)$ and $(N, h)$ be compact [[Riemannian manifolds|7CBT4SM83Y]] without boundary, and assume that $N$ has nonpositive [[sectional curvature|1X42JAA349]].
Let $u_0 \colon M \to N$ be a smooth map.
Then the [[harmonic map heat flow|MD99PYBCE6]] with initial map $u_0$ has a smooth solution $u$ on $M \times [0, \infty)$.
There is a sequence $t_k \to \infty$ such that $u(\cdot, t_k)$ converges smoothly to a [[harmonic map|KZJYG9DS3M]] $u_\infty \colon M \to N$, and $u_\infty$ is [[homotopic|9YA6S6HD9V]] to $u_0$.
In particular, every continuous map from $M$ to $N$ is homotopic to a harmonic map.

## Remarks

The key estimate is a Bochner formula for the energy density $e = \frac{1}{2} \abs{du}^2$ along the flow: $(\partial_t - \Delta_g) e \le C e$, where $C$ depends on a lower bound for the Ricci curvature of $M$.
The curvature term of $N$ enters with a sign that helps because the sectional curvature of $N$ is nonpositive.
Together with the decrease of the energy, this gives uniform bounds on $e$, hence on all derivatives by parabolic regularity, so the flow cannot blow up.
Since $\int_0^\infty \int_M \abs{\tau(u)}^2 \, dV_g \, dt \le E(u_0)$, the tension tends to zero along a sequence of times, and compactness gives the harmonic limit.

Example: for $M = S^1$ with its standard metric, harmonic maps are closed geodesics parametrized proportionally to arc length, or constant maps.
So the theorem shows that every free homotopy class of loops in a compact manifold of nonpositive curvature contains a closed geodesic or a constant loop.

The curvature hypothesis cannot be dropped: Eells and Wood showed that, for any metrics, there is no harmonic map of degree $\pm 1$ from a $2$-torus to a $2$-sphere.

## Explanations

### L1 human

Take a map into a space that is curved like a saddle everywhere.
The theorem says that it can be relaxed, by continuously lowering its energy, into a harmonic map, the best possible map in its class.
The relaxing process is a heat flow for maps.
Nonpositive curvature of the target prevents the map from tearing or concentrating energy at a point.

### L2 human

This was the first general existence theorem for harmonic maps, and it introduced the heat flow method into geometry.
Hartman later showed that the whole flow converges, not only a sequence, and proved uniqueness results for harmonic maps in a homotopy class under negative curvature.
Without the curvature condition, the flow can form singularities in finite or infinite time, and two-dimensional domains show bubbling of harmonic spheres.
Harmonic maps from this theorem are a basic tool in rigidity theory, for example in proofs of Mostow-type rigidity and in non-abelian Hodge theory.
</document>

<document source="https://tamnd.github.io/math2/node/APA7GHAYT2.md" tag="APA7GHAYT2" kind="definition">
# Semiring of sets

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]].
A *semiring of sets* on $X$ is a family $\mathcal{S}$ of subsets of $X$ such that:

1. $\varnothing \in \mathcal{S}$.
2. If $A, B \in \mathcal{S}$, then the [[intersection|0HBRW4E6G1]] $A \cap B$ is in $\mathcal{S}$.
3. If $A, B \in \mathcal{S}$, then there are finitely many pairwise disjoint sets $C_1, \ldots, C_n \in \mathcal{S}$ such that $A \setminus B$ is their [[union|07KVB4JSC7]] $C_1 \cup \cdots \cup C_n$.

## Remarks

The half-open intervals $[a, b)$ with $a, b \in \R$ form a semiring on $\R$, where $[a, b) = \varnothing$ when $b \le a$.
For clause 2, $[a, b) \cap [c, d) = [\max(a, c), \min(b, d))$.
For clause 3, when $c < d$, $[a, b) \setminus [c, d) = [a, \min(b, c)) \cup [\max(a, d), b)$, a union of two disjoint half-open intervals.
The open intervals $(a, b)$, together with $\varnothing$, do not form a semiring, since clause 3 fails.
Indeed $(0, 3) \setminus (1, 2) = (0, 1] \cup [2, 3)$ is not open, while a finite union of open intervals is open.
By clause 2 a semiring is a [[π-system|4W9NM803BB]].
The finite disjoint unions of members of a semiring form the smallest ring of sets that contains it.
</document>

<document source="https://tamnd.github.io/math2/node/APD4NYG9TG.md" tag="APD4NYG9TG" kind="definition">
# Order dimension

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
A *realizer* of $P$ is a set $\mathcal{R}$ of [[linear extensions|W8DZ5KGGHN]] of $\le$ with the following property.
For all $x, y \in P$, $x \le y$ holds if and only if $x \preceq y$ for every $\preceq$ in $\mathcal{R}$.
The *order dimension* $\dim P$ is the least cardinality of a realizer of $P$.

## Remarks

The set of all linear extensions is a realizer, by the [[Szpilrajn extension theorem|T9CJHV6JW3]], so the dimension is defined.
Example: an antichain $\set{a, b, c}$ has dimension $2$.
The orders $a \prec b \prec c$ and $c \prec b \prec a$ form a realizer, since each pair of distinct elements appears in both directions, and one linear order alone would make two elements comparable.
Non-example of a realizer: for the same antichain, the set $\set{a \prec b \prec c,\ b \prec a \prec c}$ is not a realizer, since $a$ precedes $c$ in both orders but $a \not\le c$.
A poset has dimension $1$ exactly when it is a total order.
Equivalently, $\dim P$ is the least $d$ such that $P$ embeds in a product of $d$ chains.
</document>

<document source="https://tamnd.github.io/math2/node/APGRBE5DW9.md" tag="APGRBE5DW9" kind="statement">
# Jean Taylor's theorem on soap film singularities

## Statement

Let $S \subseteq \R^3$ be a closed set such that $\mathcal{H}^2(S \cap K) < \infty$ for every compact set $K$, where $\mathcal{H}^2$ is the $2$-dimensional [[Hausdorff measure|Q1TGZK8RN6]].
Let $\delta > 0$, $C > 0$ and $\alpha > 0$.
Suppose that $S$ is almost minimizing in the sense of Almgren: for every Lipschitz map $\phi \colon \R^3 \to \R^3$, with $W = \set{x : \phi(x) \ne x}$ and $r = \operatorname{diam}(W \cup \phi(W)) < \delta$,
$$\mathcal{H}^2(S \cap W) \le (1 + C r^\alpha) \, \mathcal{H}^2(\phi(S \cap W)).$$
Then every point of $S$ has a neighbourhood $U$ with the following property.
The set $S \cap U$ is the image, under a $C^{1,\beta}$ diffeomorphism with $\beta > 0$, of an open piece of one of these three cones:

1. a plane;
2. the cone $Y$: three closed half-planes with a common boundary line, meeting along it at angles of $120^\circ$;
3. the cone $T$ over the six edges of a regular [[tetrahedron|JFX8ASQS3E]] centred at the origin, whose six sectors meet in threes along four rays.

## Remarks

Jean Taylor proved the theorem in 1976, using the regularity theory of Almgren.
The main step is to classify the minimizing cones.
A tangent cone of $S$ at a point is a cone over a network of arcs of great circles on the unit sphere.
The minimizing condition leaves only the three cones above.
Then an epiperimetric inequality shows that $S$ approaches its tangent cone at a rate that is a power of the radius, which gives the $C^{1,\beta}$ parametrization.
Soap films and clusters of soap bubbles are modelled by such almost-minimizing sets, so the theorem proves [[Plateau's laws|381RB634WS]] within this model.
For example, take a soap film on a wire frame shaped like the edges of a regular tetrahedron.
It consists of six flat triangles meeting at the centre, a piece of the cone $T$.

## Explanations

### L1 human

Soap films can meet along edges and at corners, but only in a few ways.
Take any surface that cannot reduce its area by small deformations.
Taylor's theorem says that near each point it looks like a bent copy of one of three models.
The models are a plane, three sheets meeting along an edge at $120^\circ$, and the tetrahedral corner where four such edges meet.
This turns the old experimental rules of Plateau into a theorem.

### L2 human

The theorem is a model case of the classification of singularities in geometric variational problems.
Its ingredients, the tangent cone classification and the epiperimetric inequality, have since been used for other problems, such as the regularity of free boundaries.
The almost-minimizing condition is flexible: it covers films spanning wires, bubbles enclosing fixed volumes, and films with small external forces.
</document>

<document source="https://tamnd.github.io/math2/node/APRFMTZ10K.md" tag="APRFMTZ10K" kind="concept">
# Nakayama algebra

## Summary

A Nakayama algebra is a finite-dimensional algebra whose indecomposable projective modules, on both sides, are [[uniserial|Z1045A0V29]]: their submodules form a single chain.
Nakayama studied them under the name generalized uniserial algebras, and they are now named after him.
Every indecomposable module over a Nakayama algebra is uniserial, so the algebra has only finitely many indecomposable modules up to isomorphism.
They are one of the few classes of algebras whose modules can be listed completely, and they serve as a test case in representation theory.

## Explanations

### L1 human

A module is uniserial when its submodules line up in one chain, like the rungs of a ladder.
A Nakayama algebra is an algebra whose basic projective building blocks all have this ladder shape.
The algebra $K[x]/(x^n)$ is the simplest example: its ideals are $(x^0) \supset (x^1) \supset \dots \supset (x^n) = 0$, one chain.
The path algebra of a quiver that is a single directed line $1 \to 2 \to \dots \to n$ is another example.
Because everything is a ladder, every indecomposable module is determined by its top layer and its length.
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<document source="https://tamnd.github.io/math2/node/APX4ZA01DK.md" tag="APX4ZA01DK" kind="definition">
# Algebraic Riccati equation

## Definition

Let $n, m \ge 1$.
Let $A$ be a real $n \times n$ [[matrix|0VD9ZSWWJG]] and $B$ a real $n \times m$ matrix.
Let $Q$ be a real [[symmetric|0VR671PHE5]] [[positive semidefinite|01MMVW8CKC]] $n \times n$ matrix.
Let $R$ be a real symmetric [[positive definite|0KX66CZMBB]] $m \times m$ matrix.
Write $M^T$ for the [[transpose|09DXWYRD43]] of $M$.

1. The *continuous-time algebraic Riccati equation* is
$$A^T X + XA - XBR^{-1}B^T X + Q = 0$$
for an unknown real symmetric $n \times n$ matrix $X$.
2. The *discrete-time algebraic Riccati equation* is
$$X = A^T X A - A^T X B (R + B^T X B)^{-1} B^T X A + Q$$
for an unknown real symmetric positive semidefinite $n \times n$ matrix $X$.

## Remarks

A solution $X$ of the continuous equation is *stabilising* if every eigenvalue of $A - BR^{-1}B^T X$ has negative real part.
If the pair $(A, B)$ is stabilisable and the pair $(Q^{1/2}, A)$ is detectable, the continuous equation has a unique positive semidefinite solution, and it is stabilising.
The optimal feedback of the linear-quadratic regulator is then $u = -R^{-1}B^T X x$.
Example: for $n = m = 1$, $A = 0$ and $B = Q = R = 1$, the continuous equation is $1 - X^2 = 0$, with solutions $X = 1$ and $X = -1$.
Only $X = 1$ is stabilising, since $A - BR^{-1}B^T X = -X$.
For $B = 0$ the continuous equation becomes the [[Lyapunov equation|92GVPZH9FT]] $A^T X + XA + Q = 0$, which is linear and not of Riccati type.
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<document source="https://tamnd.github.io/math2/node/AQ15060AAA.md" tag="AQ15060AAA" kind="statement">
# Atomic decomposition of Hardy spaces

## Statement

Let $n \ge 1$.
For a ball $B$, let $\abs{B}$ be its [[Lebesgue measure|2Z2QVWZ262]].
An *atom* is a measurable function $a$ on $\R^n$ with a ball $B$ such that $a = 0$ outside $B$, $\abs{a} \le 1/\abs{B}$, and $\int_B a \, dx = 0$.
Let $H^1(\R^n)$ be the [[real Hardy space|75WGJ014GR]].
Then a measurable $f$ lies in $H^1(\R^n)$ exactly when there are atoms $a_k$ and complex numbers $\lambda_k$ with
$$\sum_{k} \abs{\lambda_k} < \infty \quad \text{and} \quad f = \sum_k \lambda_k a_k,$$
the series converging in $L^1(\R^n)$.
Moreover there are constants $0 < c \le C$, depending only on $n$, such that
$$c \norm{f}_{H^1} \le \inf \sum_k \abs{\lambda_k} \le C \norm{f}_{H^1},$$
where the infimum is over all such representations of $f$.

## Remarks

Each atom has $\norm{a}_{L^1} \le 1$, and a computation with the Riesz transforms shows $\norm{a}_{H^1} \le C$; this gives the easy direction.
The hard direction starts from the maximal function characterization of $H^1$.
For each $k \in \Z$, the level set $\set{f^* > 2^k}$ of a suitable maximal function $f^*$ is decomposed into [[Whitney cubes|FBERT0KQXN]].
A Calderón–Zygmund decomposition at height $2^k$ produces functions supported on these cubes.
The differences between consecutive levels split into pieces that live on slightly enlarged Whitney cubes $Q$, have integral zero and are bounded by $C 2^k$.
Each such piece is a multiple $\lambda$ of an atom with $\abs{\lambda} \le C 2^k \abs{Q}$, and summing over $k$ gives $\sum \abs{\lambda} \le C \sum_k 2^k \abs{\set{f^* > 2^k}} \le C \int f^* \le C \norm{f}_{H^1}$.

The theorem is due to Coifman for $n = 1$ and to Latter in all dimensions.
It reduces many questions about $H^1$ to atoms.
For example, let $T$ be a Calderón–Zygmund operator and $a$ an atom on a ball $B$.
Then $Ta$ is controlled on the double ball $2B$ by the $L^2$ bound of $T$.
Off $2B$ it is controlled by the cancellation of $a$ and the smoothness of the kernel.
This gives $\norm{Ta}_{L^1} \le C$ for all atoms, and hence $T$ is bounded from $H^1$ to $L^1$.
Care is needed in such arguments, since for a general operator a uniform bound on atoms does not by itself imply boundedness on all of $H^1$.
For Calderón–Zygmund operators the passage from atoms to $H^1$ can be justified.

## Explanations

### L1 human

Every function in the Hardy space $H^1$ can be built from simple pieces, called atoms.
An atom lives on a ball, is not too large there, and has integral zero.
The theorem says that $H^1$ consists exactly of absolutely summable combinations of atoms.
So, to show that an operator behaves well on $H^1$, it is often enough to check it on a single atom.

### L2 human

The atomic decomposition is the most used description of $H^1(\R^n)$.
It gives a short proof of the duality between $H^1$ and BMO: pairing an atom $a$ on a ball $B$ with a BMO function $b$ gives $\abs{\int a b} = \abs{\int_B a (b - b_B)} \le \frac{1}{\abs{B}}\int_B \abs{b - b_B}$.
It extends to the spaces $H^p(\R^n)$ for $0 < p < 1$, where atoms must have vanishing moments up to an order depending on $p$.
Similar decompositions exist on spaces of homogeneous type, where the Fourier transform is not available.
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<document source="https://tamnd.github.io/math2/node/AQ4CNCXAJ0.md" tag="AQ4CNCXAJ0" kind="concept">
# Isogeometric analysis

## Summary

Isogeometric analysis is a [[Galerkin method|Y5B0PEBKE8]] built on spline functions, such as [[B-splines|62AFX32JGT]] and [[NURBS|Y8B2N9ERPE]].
The same functions describe the geometry and approximate the solution.
It was proposed by Thomas Hughes and coworkers to connect computer-aided design directly with simulation.
Geometries given by NURBS, such as circles and cylinders, are represented exactly, with no meshing error.
The basis functions can be smoother than standard finite elements, which helps for higher-order equations.
See the [[definition|VF17XWCK8Z]].

## Explanations

### L1 human

Engineers design shapes on computers with smooth spline curves and surfaces.
Classical finite element methods first replace the shape by a mesh of flat triangles or boxes, which loses the exact geometry.
Isogeometric analysis skips this step and computes directly with the spline functions of the design.
For example, a quarter of a circle is exactly a quadratic NURBS curve, so a ring-shaped part is described without error.
</document>

<document source="https://tamnd.github.io/math2/node/AQ5RN6FFT0.md" tag="AQ5RN6FFT0" kind="concept">
# Superharmonic potential

## Summary

A potential, in the sense of classical potential theory, is a nonnegative [[superharmonic function|X4V2S56YQQ]] whose greatest harmonic minorant is $0$.
It is the part of a superharmonic function that comes from charge, with no harmonic background left over.
By the [[Riesz decomposition theorem|CBZ20SS8RQ]], on a domain with a Green function the potentials are exactly the Green potentials of positive measures.
The Newtonian kernel $\abs{x}^{2-n}$ on $\R^n$ for $n \ge 3$ is the basic example.

## Explanations

### L1 human

A positive superharmonic function can be split into a harmonic piece and a piece produced by charges.
A potential is a function made only of the charge piece.
Its harmonic piece is zero, which roughly means it fades away at the edge of the domain.
The potential $1/\abs{x}$ of a unit point charge in space is a potential in this sense, while a positive constant is not.
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<document source="https://tamnd.github.io/math2/node/AQ75M2JTB2.md" tag="AQ75M2JTB2" kind="concept">
# Schramm–Loewner evolution

## Summary

The Schramm–Loewner evolution with parameter $\kappa > 0$, written $\mathrm{SLE}_\kappa$, is a random curve in a planar domain.
It is obtained from the [[Loewner differential equation|51G4QJZ345]] driven by a multiple of a Brownian motion.
Oded Schramm introduced it in 2000 as the candidate scaling limit of interfaces in planar models of statistical physics that are conformally invariant.
It is characterised by conformal invariance together with a domain Markov property, and the single parameter $\kappa$ governs the shape of the curve.

## Explanations

### L1 human

Many random paths on a fine lattice, such as the boundary of a cluster in critical percolation, look the same at every scale.
In the limit, they are unchanged by conformal maps.
Schramm found that any such limiting random curve must be described by the Loewner equation with a driving function $\sqrt{\kappa}\, B_t$, where $B_t$ is a Brownian motion.
Small $\kappa$ gives curves that are nearly straight, and larger $\kappa$ gives curves that wiggle more, touch themselves, and for $\kappa \ge 8$ fill the whole domain.
For example $\kappa = 6$ describes the limit of percolation interfaces on the triangular lattice.
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<document source="https://tamnd.github.io/math2/node/AQDXWH4812.md" tag="AQDXWH4812" kind="concept">
# Positive linear operator

## Summary

A positive linear operator is a [[linear map|0NRYHJPQG2]] between ordered vector spaces that sends nonnegative elements to nonnegative elements.
On spaces of functions, it maps nonnegative functions to nonnegative functions.
Bernstein polynomials, convolution with nonnegative kernels and integration are all positive linear operators.
Their positivity gives strong convergence tests, such as Korovkin's theorem.

## Explanations

### L1 human

A positive operator never turns something nonnegative into something negative.
Averaging is the typical example: an average of nonnegative numbers is nonnegative.
So any operator that replaces $f(x)$ by a weighted average of values of $f$, with nonnegative weights, is positive.
Differentiation is not positive: $1 - x$ is nonnegative on $[0, 1]$, but its derivative is $-1$.
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<document source="https://tamnd.github.io/math2/node/AQET9YHH6F.md" tag="AQET9YHH6F" kind="concept">
# Zero object

## Summary

A zero object of a [[category|A2DE66937D]] is an object that is both [[initial and terminal|EK85VSSRMR]].
Categories with a zero object, such as groups, modules and pointed sets, have a distinguished zero morphism between any two objects.
This is the starting point for kernels, cokernels and exact sequences in homological algebra.

## Explanations

### L1 human

A zero object has exactly one arrow to every object and exactly one arrow from every object.
The [[trivial group|0BNPC9850Y]] $\{e\}$ is a zero object among groups.
There is one homomorphism from it to any group, sending $e$ to the identity.
There is one homomorphism to it from any group, sending everything to $e$.
The zero vector space plays the same role among vector spaces.
Among sets there is no zero object: the empty set is initial but not terminal, and a one-point set is terminal but not initial.
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<document source="https://tamnd.github.io/math2/node/AQK50V3NGJ.md" tag="AQK50V3NGJ" kind="definition">
# Peak function

## Definition

Let $\Omega \subseteq \C^n$ be a bounded domain and let $p \in \partial\Omega$.
A *peak function* for $\Omega$ at $p$ is a function $f \colon \overline{\Omega} \to \C$ such that:

1. $f$ is continuous on $\overline{\Omega}$ and [[holomorphic|507C34W0ZT]] on $\Omega$.
2. $f(p) = 1$.
3. $\abs{f(z)} < 1$ for every $z \in \overline{\Omega} \setminus \set{p}$.

The point $p$ is a *peak point* of $\Omega$ if a peak function at $p$ exists.

## Remarks

Example: on the unit ball $B \subseteq \C^n$, let $e_1 = (1, 0, \dots, 0)$ and $f(z) = (1 + z_1)/2$.
Then $f(e_1) = 1$, and $\abs{f(z)} \le (1 + \abs{z_1})/2 \le 1$ on $\overline{B}$, with equality only if $z_1 = 1$, which forces $z = e_1$.
Non-example: on the bidisc $\Delta^2$, the point $p = (1, 0)$ is not a peak point.
Suppose $f$ satisfies the three clauses.
The function $w \mapsto f(1, w)$ is holomorphic on the open unit disc, as a locally uniform limit of the functions $w \mapsto f(r, w)$ for $r \to 1$.
It has modulus at most $1$ and reaches the modulus $1$ at the interior point $w = 0$, so it is the constant $1$ by the maximum modulus principle, against clause 3.
Every boundary point of a bounded [[strongly pseudoconvex domain|Z7RVXBA2KB]] with smooth boundary is a peak point.
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<document source="https://tamnd.github.io/math2/node/AQN6ZN66NK.md" tag="AQN6ZN66NK" kind="definition">
# Pi-system

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]].
A *π-system* on $X$ is a nonempty family $\mathcal{P}$ of subsets of $X$ such that $A \cap B \in \mathcal{P}$ for all $A, B \in \mathcal{P}$, where $A \cap B$ is the [[intersection|0HBRW4E6G1]].

## Remarks

The family $\set{(-\infty, a] : a \in \R}$ is a π-system on $\R$, since $(-\infty, a] \cap (-\infty, b] = (-\infty, \min(a, b)]$.
The family $\set{[0, 1], [1, 2]}$ is not a π-system, since $[0, 1] \cap [1, 2] = \set{1}$ is not in it.
Every [[σ-algebra|04WSW8T3SY]] is a π-system.
By induction, a π-system is closed under all finite intersections.
The name is also written $\pi$-system.
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<document source="https://tamnd.github.io/math2/node/AQVHM3FP0D.md" tag="AQVHM3FP0D" kind="concept">
# Knot group

## Summary

The knot group of a [[knot|QYEAZTYGPT]] is the [[fundamental group|S1XK3HXHFX]] of its complement in $\R^3$.
It is a [[knot invariant|GZQ5ZE3X6G]]: equivalent knots have isomorphic knot groups.
It is a powerful one, since it distinguishes the unknot from every other knot.
A diagram of the knot gives an explicit presentation of it, the [[Wirtinger presentation|P6GR8ZQCK3]].

## Explanations

### L1 human

Imagine the knot as a fixed wire in space, and consider loops of string that start and end at a fixed point and may wind around the wire.
Two loops count as the same if one can be slid to the other without crossing the wire.
Following one loop and then another gives a way to combine loops, and this makes them a group.
For the unknot, a loop is described by how many times it winds through the circle, so the group is the integers $\Z$.
For the trefoil, the group is not commutative, which proves that the trefoil cannot be untied.
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<document source="https://tamnd.github.io/math2/node/AR14VMVHNG.md" tag="AR14VMVHNG" kind="problem">
# Word problem for free algebras of a variety

## Problem

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of finite signature with a finite [[equational basis|9Q008GMZSW]] $\Sigma$.
Let $X$ be a countably infinite set of variables, and let $F_{\mathcal{V}}(X)$ be the [[free algebra|GTS4DRRMZR]] of $\mathcal{V}$ on $X$.
The problem asks for an algorithm, that is, a [[computable function|KZSB5E471K]], with the following task.
For any two terms $s$ and $t$ over $X$, it decides whether $s$ and $t$ denote the same element of $F_{\mathcal{V}}(X)$.
Equivalently, it asks whether the [[equational theory|F1M51B4MDG]] of $\mathcal{V}$ is decidable.
That is, can one decide whether the [[identity|PKD440G2XW]] $s \approx t$ holds in every algebra of $\mathcal{V}$?

## Remarks

Since $\Sigma$ is finite, the identities of $\mathcal{V}$ are computably enumerable by listing all derivations in equational logic; the difficulty is to certify that an identity fails.
The problem is solvable for many classical varieties.
In groups, every element of a free group has a unique reduced word.
In semigroups and monoids, the free algebras are the algebras of words, so equality is literal equality of words.
In Boolean algebras, an identity holds exactly when it holds in the two-element Boolean algebra, which can be checked by truth tables.
It is unsolvable in general: Tarski showed that the equational theory of relation algebras, which form a finitely based variety, is undecidable.

## Explanations

### L1 human

Fix a list of laws, such as the group laws.
The question is whether there is a mechanical test that decides if an equation between two expressions follows from those laws.
For groups the test is to cancel $x x^{-1}$ until nothing cancels and compare the results.
For some finite lists of laws, no such test exists.
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<document source="https://tamnd.github.io/math2/node/AR52AWP2XE.md" tag="AR52AWP2XE" kind="definition">
# Leray–Schauder method

## Definition

Let $X$ be a real [[Banach space|06GZ956657]].
A [[continuous|0X3WGX95FE]] map $T \colon [0, 1] \times X \to X$ is *compact* if it maps bounded sets to sets with [[compact|0R6EWFKRCF]] closure.
The *Leray–Schauder method* for an equation $u = F(u)$ in $X$ consists of the following steps.

1. Choose a compact map $T \colon [0, 1] \times X \to X$ with $T(1, \cdot) = F$ and $T(0, \cdot) = 0$.
2. Prove an a priori bound: there is $R > 0$ such that every solution of $u = T(\lambda, u)$ with $\lambda \in [0, 1]$ satisfies $\norm{u} < R$.
3. Conclude by the homotopy invariance of the [[Leray–Schauder degree|3VJC5Q9MYA]] on the ball $B_R = \set{u : \norm{u} < R}$:
$$\deg\bigl(I - T(1, \cdot), B_R, 0\bigr) = \deg\bigl(I - T(0, \cdot), B_R, 0\bigr) = \deg(I, B_R, 0) = 1,$$
so $u = F(u)$ has a solution in $B_R$.

## Remarks

Step 2 guarantees that no solution lies on the boundary of $B_R$ for any $\lambda$, which is what homotopy invariance needs.
The form with $T(\lambda, u) = \lambda F(u)$ is the Leray–Schauder alternative, also known as [[Schaefer's fixed point theorem|7H3YSSDEBK]]: either the set of solutions of $u = \lambda F(u)$, $0 \le \lambda \le 1$, is unbounded, or $u = F(u)$ has a solution.

Example: in $X = C[0, 1]$, let $F(u)(x) = \int_0^1 k(x, t)\, \arctan u(t)\, dt$ with $k$ continuous.
Then $F$ is compact by the Arzelà–Ascoli theorem.
Any solution of $u = \lambda F(u)$ with $0 \le \lambda \le 1$ satisfies $\norm{u}_\infty \le \frac{\pi}{2} \max \abs{k}$, since $\abs{\arctan} < \frac{\pi}{2}$.
So $R = \frac{\pi}{2} \max \abs{k} + 1$ works, and $u = F(u)$ has a continuous solution.

Non-example: in an infinite-dimensional space, without compactness, step 3 has no meaning.
For instance, $T(\lambda, u) = \lambda u$ on the unit ball of $\ell^2$ is not compact, and the Leray–Schauder degree of $I - T(\lambda, \cdot)$ is not defined.
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<document source="https://tamnd.github.io/math2/node/AR8SP73SXX.md" tag="AR8SP73SXX" kind="concept">
# Simple homotopy equivalence

## Summary

A simple homotopy equivalence between finite CW complexes is a homotopy equivalence of a special kind.
Up to homotopy, it is a finite sequence of elementary expansions and collapses, which add or remove a pair of cells in a trivial way.
Not every homotopy equivalence is simple; the obstruction is the Whitehead torsion, which lies in the Whitehead group of the fundamental group.
Simple homotopy type is finer than homotopy type and is the right notion for the s-cobordism theorem and for distinguishing lens spaces.

## Explanations

### L1 human

Imagine a shape made of cells, like a model built of balls and disks.
An elementary expansion glues on a new solid ball along half of its boundary.
This adds two cells but changes nothing up to deformation, since the ball can be squashed back flat.
An elementary collapse is the reverse move.
Two complexes are simple homotopy equivalent when one can pass from one to the other by finitely many such moves.
A solid square and a single point are simple homotopy equivalent: collapse the square onto an edge, then the edge onto a point.
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<document source="https://tamnd.github.io/math2/node/ARAEEM8CVF.md" tag="ARAEEM8CVF" kind="concept">
# Near-ring

## Summary

A near-ring is a set with an addition and a multiplication that satisfies most of the [[ring|0B1261B5ED]] axioms.
Addition need not be commutative, and only one distributive law is required.
The standard example is the set of all maps from a [[group|G5S1M9]] to itself, with pointwise addition and composition.
Near-rings stand to arbitrary maps of groups as rings stand to endomorphisms of abelian groups.
They appear in the study of nonlinear maps, of permutation groups and of combinatorial designs.

## Explanations

### L1 human

For an abelian group, its endomorphisms can be added and composed, and they form a ring.
If one takes all maps of the group to itself, not only homomorphisms, one can still add them pointwise and compose them.
Composition then distributes over addition from one side only: $(f + g) \circ h = f \circ h + g \circ h$ always holds, but $f \circ (g + h) = f \circ g + f \circ h$ needs $f$ to respect addition.
A near-ring is any structure with exactly the rules that survive.
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<document source="https://tamnd.github.io/math2/node/ARCYEPTR4Q.md" tag="ARCYEPTR4Q" kind="definition">
# Subalgebra lattice

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.
A *subuniverse* of $\mathbf{A}$ is a subset of $A$ closed under all fundamental operations, as for a [[subalgebra|7CHG6SD365]].
Let $\operatorname{Sub} \mathbf{A}$ be the set of all subuniverses of $\mathbf{A}$.
The *subalgebra lattice* of $\mathbf{A}$ is the partially ordered set $(\operatorname{Sub} \mathbf{A}, \subseteq)$.
Its meet and join are
$$B \wedge C = B \cap C, \qquad B \vee C = \operatorname{Sg}(B \cup C),$$
where $\operatorname{Sg}$ is the [[generated subuniverse|2TQD8NMG9P]].

## Remarks

The intersection of any family of subuniverses is a subuniverse, so $\operatorname{Sub} \mathbf{A}$ is a [[complete lattice|D7A0ZGZT5S]].
It is an [[algebraic lattice|3J0BTXSCVN]].
Its [[compact elements|9F7TYES4KE]] are the subuniverses $\operatorname{Sg}(X)$ with $X$ finite, that is, the [[finitely generated|WR4WF1K2KW]] ones.

Example: for the group $(\Z, +, -, 0)$ the subuniverses are the sets $n\Z$ with $n \ge 0$.
Here $m\Z \subseteq n\Z$ if and only if $n$ divides $m$, and $m\Z \vee n\Z = d\Z$ with $d = \gcd(m, n)$.

Non-example: the lattice of all subsets of $\Z$ is not the subalgebra lattice of this group.
The subset $\set{1}$ is not closed under addition, so it is not a subuniverse.

When the signature has no constant symbols, the empty set is a subuniverse and is the least element of $\operatorname{Sub} \mathbf{A}$.
Some authors exclude it and call only nonempty subuniverses subalgebras.
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<document source="https://tamnd.github.io/math2/node/ARG1F8W9NZ.md" tag="ARG1F8W9NZ" kind="definition">
# Mackey topology

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $(X, Y, \langle \cdot, \cdot \rangle)$ be a [[dual pair|R68HNB5SVR]] over $\mathbb{K}$.
Let $\mathcal{K}$ be the family of subsets $K \subseteq Y$ that are [[convex|02AS45T3H9]], [[balanced|2TDSHJJYS4]] and [[compact|0R6EWFKRCF]] for the [[weak topology|P6KHVSTPZV]] $\sigma(Y, X)$.
For $K \in \mathcal{K}$, put $p_K(x) = \sup_{y \in K} \abs{\langle x, y \rangle}$ for $x \in X$.
The *Mackey topology* $\tau(X, Y)$ is the [[topology generated by the seminorms|ZVFXQ7S6FY]] $p_K$, $K \in \mathcal{K}$.

## Remarks

Each $p_K$ is finite, because $y \mapsto \langle x, y \rangle$ is $\sigma(Y, X)$-continuous and $K$ is $\sigma(Y, X)$-compact.
For a locally convex space $E$, the Mackey topology of $E$ is $\tau(E, E')$, and $E$ is called a *Mackey space* if its topology equals $\tau(E, E')$.
Example: every metrizable locally convex space, in particular every normed space, is a Mackey space.
Check for a Banach space $E$: the closed unit ball $K$ of $E'$ is convex, balanced and weak-\* compact by the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], and $p_K(x) = \norm{x}$ by the Hahn–Banach theorem.
So $\tau(E, E')$ is at least as fine as the norm topology.
Conversely, each weak-\* compact $K \subseteq E'$ is bounded at each point of $E$, hence norm bounded by the uniform boundedness principle, say by $r$.
Then $p_K(x) \le r \norm{x}$, so $\tau(E, E')$ is no finer than the norm topology.
Non-example: on an infinite-dimensional normed space $E$, the weak topology $\sigma(E, E')$ has the same continuous dual $E'$ but is strictly coarser than $\tau(E, E')$, so it is not the Mackey topology.
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<document source="https://tamnd.github.io/math2/node/ARGNTXTAST.md" tag="ARGNTXTAST" kind="definition">
# Generic matrix algebra

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $m, n \ge 1$ be integers.
Let $P = k\big[x^{(l)}_{ij} : 1 \le l \le m, \ 1 \le i, j \le n\big]$ be the [[polynomial ring|0SDC934R52]] in $m n^2$ commuting variables.
For $1 \le l \le m$, the *generic matrix* $X_l \in M_n(P)$ is the [[matrix|0JXZKX526R]] whose $(i, j)$ entry is $x^{(l)}_{ij}$.
The *generic matrix algebra* $k\{X_1, \ldots, X_m\}$ is the $k$-subalgebra of $M_n(P)$ generated by $X_1, \ldots, X_m$ and the identity matrix.

## Remarks

The algebra $k\{X_1, \ldots, X_m\}$ is an [[associative algebra|3CTW1P4HBP]] over $k$, and it is a subring of $M_n(P)$, so it satisfies every [[polynomial identity|C0EDT81EGH]] of $M_n(P)$.
Let $\pi \colon k\langle x_1, \ldots, x_m \rangle \to k\{X_1, \ldots, X_m\}$ be the homomorphism from the [[free algebra|2P3Y2FRCSZ]] with $\pi(x_l) = X_l$.
If $k$ is infinite, the kernel of $\pi$ is the set of polynomial identities of $M_n(k)$ in $m$ variables.
Indeed, $f(X_1, \ldots, X_m) = 0$ means that each entry of the matrix $f(X_1, \ldots, X_m)$ is the zero polynomial.
Over an infinite field, a polynomial is zero exactly when it vanishes at all points, that is, when $f(a_1, \ldots, a_m) = 0$ for all $a_l \in M_n(k)$.

Example: for $n = 1$ the generic matrices are the variables $x^{(l)}_{11}$, and $k\{X_1, \ldots, X_m\}$ is the polynomial ring in $m$ variables.

Non-example: for $n \ge 2$, the scalar matrix $x^{(1)}_{11} I$ lies in $M_n(P)$ but not in $k\{X_1, \ldots, X_m\}$.
The entries of $f(X_1, \ldots, X_m)$ have degree-$1$ part equal to the corresponding entry of $\sum_l c_l X_l$, where $c_l$ is the coefficient of $x_l$ in $f$.
If $f(X_1, \ldots, X_m) = x^{(1)}_{11} I$, comparing the $(1, 1)$ entries gives $c_1 = 1$, and comparing the $(2, 2)$ entries gives $\sum_l c_l x^{(l)}_{22} = x^{(1)}_{11}$, which is impossible.
So the subalgebra generated by the $X_l$ over the commutative ring $P$ is strictly larger than the generic matrix algebra.

For $n \ge 2$ and $m \ge 2$, the generic matrix algebra is a noncommutative domain.
</document>

<document source="https://tamnd.github.io/math2/node/ARGZ549BQT.md" tag="ARGZ549BQT" kind="definition">
# Möbius function of a poset

## Definition

Let $(P, \le)$ be a [[locally finite poset|TV1V4JDRDD]], and write $x < y$ when $x \le y$ and $x \ne y$.
The *Möbius function* of $P$ is the function $\mu \colon P \times P \to \Z$ given by:

1. $\mu(x, x) = 1$ for all $x \in P$.
2. $\mu(x, y) = -\sum_{x \le z < y} \mu(x, z)$ for $x < y$.
3. $\mu(x, y) = 0$ when $x \not\le y$.

## Remarks

Clause 2 is a recursion over the finite interval $[x, y]$, so $\mu$ is well defined.
Equivalently, $\mu$ is the inverse of the zeta function $\zeta$ in the [[incidence algebra|1H9SFWHYEB]] of $P$, where $\zeta(x, y) = 1$ for $x \le y$ and $0$ otherwise.
From this one also gets $\mu(x, y) = -\sum_{x < z \le y} \mu(z, y)$ for $x < y$.

Example: in the lattice of subsets of a finite set, $\mu(S, T) = (-1)^{\abs{T \setminus S}}$ for $S \subseteq T$.
Check: for $\abs{T \setminus S} = k \ge 1$, the sum $\sum_{S \subseteq U \subseteq T} (-1)^{\abs{U \setminus S}} = \sum_{j=0}^{k} \binom{k}{j} (-1)^j = (1 - 1)^k = 0$, as clauses 1 and 2 require.

Non-example: the function $\nu(x, y) = (-1)^{y - x}$ for $x \le y$ on the chain $\N$ is not the Möbius function.
It satisfies clause 1, but clause 2 fails at $(0, 2)$: there $\mu(0, 2) = -(1 - 1) = 0$, while $\nu(0, 2) = 1$.

For the positive integers ordered by divisibility, $\mu(d, n)$ is the number-theoretic Möbius function of $n / d$.
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<document source="https://tamnd.github.io/math2/node/ARTJV1XCRX.md" tag="ARTJV1XCRX" kind="concept">
# Resolvent kernel

## Summary

The resolvent kernel $R(x, s; \lambda)$ of a [[kernel|GSMBXT7FPK]] $K$ gives the solution of the [[integral equation of the second kind|8KJCN49V64]] $\varphi = f + \lambda K \varphi$.
The solution is $\varphi(x) = f(x) + \lambda \int_a^b R(x, s; \lambda) f(s) \, ds$.
For small $\abs{\lambda}$ it is the sum $\sum_{n \ge 1} \lambda^{n-1} K_n$ of the [[iterated kernels|9QNBFERMHD]].
It thus inverts the operator $I - \lambda K$ once and for all, for every right side $f$.
For a continuous kernel it extends to a meromorphic function of $\lambda$, whose poles are characteristic values of $K$.

## Explanations

### L1 human

Solving the equation for each new right side separately is wasteful.
The resolvent kernel is a single formula that turns any right side directly into the solution.
It plays the role of the inverse matrix in linear algebra.
For example, for $K(x, s) = x s$ on $[0, 1]$ the resolvent kernel is $R(x, s; \lambda) = 3 x s / (3 - \lambda)$.
</document>

<document source="https://tamnd.github.io/math2/node/AS00YM4X7F.md" tag="AS00YM4X7F" kind="concept">
# Arbitrary-precision arithmetic

## Summary

Arbitrary-precision arithmetic computes with numbers whose number of digits is limited only by the memory of the computer, not by a fixed hardware format.
Integers and rational numbers can then be handled exactly, and real numbers with as many correct digits as one asks for.
Unlike fixed-size [[floating-point arithmetic|RTKRSC3KPH]], each operation costs more as the numbers grow, so fast multiplication algorithms matter.
It is used in computer algebra, number theory, cryptography and to check numerical results with higher precision.

## Explanations

### L1 human

A calculator shows perhaps $12$ digits, and a computer's built-in numbers hold about $16$ decimal digits.
Arbitrary-precision arithmetic stores a number as a list of digits that can be as long as needed, just as one writes long numbers on paper.
It can compute $2^{1000}$, a number with $302$ digits, exactly, or $\pi$ to a million decimals.
The price is speed: a product of two numbers with $n$ digits each takes far more time than one machine multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/AS18AWFXSB.md" tag="AS18AWFXSB" kind="statement">
# Halmos–von Neumann theorem

## Statement

Let $T$ on $(X, \mu)$ and $S$ on $(Y, \nu)$ be invertible [[ergodic|AZRPHS478B]] [[measure-preserving transformations|556WBDAHNP]] of [[standard probability spaces|QDDRDW9PXS]], with measurable inverses.
Assume both have discrete spectrum, in the sense of the [[spectral theory of dynamical systems|Z7GZ6SPFA5]].
This means that $L^2$ has an orthonormal basis of [[eigenvectors|0BQAA2BK3Q]] of the [[Koopman operator|37TRG5DBQC]].
Let $\Lambda(T) \subseteq \C$ be the set of eigenvalues of the Koopman operator $U_T$ of $T$.

1. $T$ and $S$ are isomorphic if and only if $\Lambda(T) = \Lambda(S)$.
2. $\Lambda(T)$ is a countable subgroup of the unit circle. Moreover, $T$ is isomorphic to a rotation $x \mapsto x + a$ of a compact metrizable abelian group $K$ with its [[Haar measure|VY0HXD3GC4]], where $\set{na : n \in \Z}$ is dense in $K$.

## Remarks

Here isomorphic means conjugate by a measure-preserving bijection defined up to null sets.
For part 1, ergodicity makes every eigenvalue simple and every eigenfunction of constant modulus, so a product of eigenfunctions is again an eigenfunction and $\Lambda(T)$ is a group.
Choose for each $\lambda \in \Lambda(T)$ an eigenfunction $f_\lambda$ of modulus $1$; the products $f_\lambda f_{\lambda'}$ and $f_{\lambda \lambda'}$ differ by constants.
The proof adjusts these constants, which is possible because the circle group is divisible.
It then matches the eigenfunctions of $T$ and $S$ to build a unitary map that preserves products, and such a map comes from an isomorphism of the spaces.
For part 2, $K$ is the Pontryagin dual of $\Lambda(T)$ with the discrete topology, and $a$ is the inclusion character.
Example: for an irrational $\alpha$, the rotation $x \mapsto x + \alpha \bmod 1$ has $\Lambda = \set{e^{2\pi i k \alpha} : k \in \Z}$.
So two irrational rotations by $\alpha$ and $\beta$ are isomorphic exactly when $\beta = \pm \alpha \bmod 1$.
Without the discrete spectrum hypothesis the eigenvalues do not determine the system: all weakly mixing systems have $\Lambda = \set{1}$.

## Explanations

### L1 human

Some systems behave like a collection of clocks running at fixed frequencies, such as rotations of a circle or a torus.
For them, the list of frequencies tells everything: two such systems are the same exactly when they have the same frequencies.
Each of them is in fact a rotation of a compact group.

### L2 human

This is the first complete classification theorem in ergodic theory, obtained by Halmos and von Neumann in 1942.
It shows the power of spectral methods in the class where the spectrum is discrete, and it is the model case for the [[spectral theory of dynamical systems|Z7GZ6SPFA5]].
For systems with continuous spectrum spectral invariants are far from complete, which motivated the search for other invariants such as the Kolmogorov–Sinai entropy.
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<document source="https://tamnd.github.io/math2/node/AS4WVMYEDZ.md" tag="AS4WVMYEDZ" kind="concept">
# Boolean matrix

## Summary

A Boolean matrix, also called a logical matrix or a $(0,1)$-matrix, is a [[matrix|0VD9ZSWWJG]] whose entries are $0$ and $1$, read as false and true.
It records a [[binary relation|05AVARPXA2]] between two finite sets.
With Boolean arithmetic, the product of two such matrices records the [[composition of the relations|0PYSR20HGF]].
The [[adjacency matrix|08B9CX109G]] of a [[directed graph|0KFYGSHHF5]] is the main example.

## Explanations

### L1 human

Write the elements of one set along the rows and the elements of another set along the columns.
Put a $1$ in a cell when the row element is related to the column element, and a $0$ otherwise.
For the relation "divides" on $\set{1, 2, 3}$, the first row is $1, 1, 1$, because $1$ divides every number.
The second row is $0, 1, 0$ and the third row is $0, 0, 1$.
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<document source="https://tamnd.github.io/math2/node/AS7ZA3P37H.md" tag="AS7ZA3P37H" kind="definition">
# Character space

## Definition

Let $A$ be a commutative complex [[Banach algebra|9KYWN94QDN]].
A *character* of $A$ is a nonzero linear map $\varphi \colon A \to \C$ with $\varphi(ab) = \varphi(a) \varphi(b)$ for all $a, b \in A$.
The *character space* $\Delta(A)$ of $A$ is the set of characters of $A$.
Each character is continuous with $\norm{\varphi} \le 1$, so $\Delta(A)$ is a subset of the [[continuous dual space|4A82S5FVRT]] $A^*$.
The topology on $\Delta(A)$ is the relative weak-* topology, the [[weak topology|P6KHVSTPZV]] of the maps $\varphi \mapsto \varphi(a)$, $a \in A$.

## Remarks

Other names are the *maximal ideal space* and the *Gelfand spectrum*, written $\sigma(A)$ or $\mathfrak{M}(A)$.
The space $\Delta(A)$ is locally compact and Hausdorff.
If $A$ has a unit, then $\varphi(1) = 1$ for every character, $\Delta(A)$ is weak-* closed in the unit ball of $A^*$, and so it is compact by the [[Banach–Alaoglu theorem|1BXEJXQ2YT]].
For unital $A$, the map $\varphi \mapsto \ker \varphi$ is a bijection from $\Delta(A)$ onto the set of [[maximal ideals|0PV74TR5FP]] of $A$; the [[Gelfand–Mazur theorem|3D6XH41A3B]] shows that each quotient $A / \mathfrak{m}$ is $\C$.

Example: let $X$ be a compact Hausdorff space and $A = C(X)$ with the sup norm.
For $x \in X$, the evaluation $\varphi_x(f) = f(x)$ is linear and multiplicative, and $\varphi_x(1) = 1$, so it is a character.
Every character of $C(X)$ has this form, and $x \mapsto \varphi_x$ is a homeomorphism from $X$ onto $\Delta(A)$.

Non-example: on $C([0,1])$, the map $\varphi(f) = \int_0^1 f(t) \, dt$ is linear and nonzero but not multiplicative.
For $f(t) = t$ one has $\varphi(f^2) = 1/3$, while $\varphi(f)^2 = 1/4$.
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# Biharmonic equation

## Summary

The biharmonic equation is the fourth-order equation $\Delta^2 u = 0$, where $\Delta^2 = \Delta \Delta$ is the [[Laplacian|0E0B6RA8K0]] applied twice.
Its solutions are called biharmonic functions.
It models the small bending of a thin elastic plate and, in two dimensions, the stream function of slow viscous flow.
It is the basic example of a higher-order elliptic equation, for which tools such as the maximum principle no longer hold.

## Explanations

### L1 human

A [[harmonic function|0DEB0JMN0W]] has zero Laplacian.
A biharmonic function only needs its Laplacian to be harmonic.
So every harmonic function is biharmonic, but there are many more biharmonic functions.
For example, $u(x, y) = x^2 + y^2$ has Laplacian $4$, which is constant and so harmonic; hence $u$ is biharmonic, although it is not harmonic.
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# P-adic analysis

## Summary

$p$-adic analysis is the analysis of functions of [[$p$-adic numbers|BT6T702C28]], where nearness is measured by the [[$p$-adic absolute value|WGN82TCEF2]] instead of the usual one.
This absolute value is ultrametric, which makes many things simpler than in real analysis: a series converges as soon as its terms tend to $0$.
Other things behave differently: the $p$-adic integers form a compact, totally disconnected space, and the exponential series converges only on a small disc.
The subject is a main tool of modern number theory, for example in $p$-adic $L$-functions and in the study of congruences.

## Explanations

### L1 human

In the $p$-adic world, a number is small when it is divisible by a high power of the prime $p$.
So $p, p^2, p^3, \ldots$ tend to $0$, and the series $1 + p + p^2 + \cdots$ converges, to $1/(1 - p)$.
One can then do calculus with limits, series, continuity and derivatives, as over the reals.
Much of it works the same way, but some facts change: for example, a function with derivative $0$ everywhere on $\Z_p$ need not be constant.
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# Minimax approximation

## Summary

Fix a continuous function $f$ on an interval and a degree $n$.
A minimax approximation of $f$ is a polynomial $p$ of degree at most $n$ that makes the largest error $\max \abs{f - p}$ as small as possible.
It is the [[best approximation|7N4988HN3P]] for the uniform norm, so it controls the error at every point and not only on average.
Minimax polynomials are used to build the routines that compute $\sin$, $\exp$ and other functions in software libraries.

## Explanations

### L1 human

Draw the graph of $f$ and a thin band around it of the same width $\varepsilon$ everywhere.
A polynomial approximates $f$ within $\varepsilon$ when its graph stays inside the band.
The minimax polynomial is the polynomial of the given degree that fits inside the thinnest possible band.
For example, the best constant approximation of $f(x) = x$ on $[0, 1]$ is the constant $1/2$, with largest error $1/2$.
Any other constant $c$ is farther than $1/2$ from $0$ or from $1$.
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<document source="https://tamnd.github.io/math2/node/ASXD4S15Z4.md" tag="ASXD4S15Z4" kind="example">
# Non-surjective epimorphism of algebras

## Example

Work in the [[concrete category|ZD0GTPSJS1]] $\mathbf{Ring}$ of [[rings|0B1261B5ED]] with identity and [[ring homomorphisms|0GGE0Z0JQ1]] that preserve $1$.
The inclusion $i \colon \Z \to \Q$ of the [[integers|0C4JS9RF5M]] into the [[rationals|0NCJ5WXN6Z]] is an [[epimorphism|RVXDSY04KG]] in $\mathbf{Ring}$, but it is not surjective.

Check: let $R$ be a ring and let $f, g \colon \Q \to R$ be ring homomorphisms with $f \circ i = g \circ i$, so $f(n) = g(n)$ for every integer $n$.
For $n \ne 0$, $f(1/n) \, f(n) = f(1) = 1$ and $f(n) \, f(1/n) = 1$, so $f(n)$ is a unit of $R$ with inverse $f(1/n)$.
In the same way $g(1/n)$ is the inverse of $g(n) = f(n)$.
Inverses are unique, so $f(1/n) = g(1/n)$, and then $f(m/n) = f(m) f(1/n) = g(m) g(1/n) = g(m/n)$ for all $m, n$.
Hence $f = g$, so $i$ is right-cancellable, that is, an epimorphism.
It is not surjective, since $1/2 \notin \Z$.

## Remarks

In a concrete category of algebras every surjective homomorphism is an epimorphism, but the converse can fail.
It holds in the categories of groups and of abelian groups, where every epimorphism is surjective.
Other non-surjective epimorphisms are the inclusion $(\N, +) \to (\Z, +)$ in the category of monoids, and more generally every localization map $R \to S^{-1} R$ of commutative rings.

## Explanations

### L1 human

In category theory, an epimorphism is an arrow that can be cancelled on the right, which for functions means being onto.
For rings this goes wrong.
Once a homomorphism out of $\Q$ is known on the integers, it is already known on all fractions, because $1/n$ must go to the inverse of the image of $n$.
So the integers "control" the rationals, even though they are a small part of them.
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# Irregular singularity

## Summary

An irregular singularity of a linear differential equation is a [[singular point|QK2J038VB4]] that is not a [[regular singular point|H96HXG6CA0]].
Near it, solutions can grow or decay faster than any power, typically like $e^{c / (z - z_0)^k}$ or $e^{c z^k}$ near $\infty$.
The Frobenius method fails there, and the solutions are described by asymptotic expansions valid only in sectors.
This leads to the [[Stokes phenomenon|TTGF8KCA8C]].
See the [[definition|A6EXFAP5EQ]].

## Explanations

### L1 human

At a regular singular point solutions behave like powers of the distance to the point.
At an irregular one they can behave like exponentials of a negative power, which is much wilder.
For $y' = \frac{y}{z^2}$, the solution is $y = e^{-1/z}$, which has an essential singularity at $0$.
The Bessel and Airy equations have an irregular singular point at $\infty$, where their solutions oscillate or grow exponentially.
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# Riesz potential

## Definition

Let $n \ge 1$ and $0 < \alpha < n$.
Let $\Gamma$ be the [[gamma function|0H5SYBSED9]] and let
$$c_\alpha = \pi^{n/2} \, 2^\alpha \, \frac{\Gamma(\alpha/2)}{\Gamma((n - \alpha)/2)}.$$
For a measurable function $f$ on $\R^n$, the *Riesz potential* of order $\alpha$ of $f$ is
$$I_\alpha f(x) = \frac{1}{c_\alpha} \int_{\R^n} \frac{f(y)}{\abs{x - y}^{n - \alpha}} \, dy,$$
at each $x \in \R^n$ where the integral converges absolutely.

## Remarks

So $I_\alpha f$ is the [[convolution|2Z29JN3VES]] of $f$ with the kernel $\abs{x}^{\alpha - n} / c_\alpha$.
If $f \in L^p(\R^n)$ with $1 \le p < n/\alpha$, the integral converges absolutely for almost every $x$.
With the [[Fourier transform|DPVM34ADZV]] $\hat f(\xi) = \int f(x) e^{-2\pi i x \cdot \xi} \, dx$, the constant $c_\alpha$ is chosen so that $\widehat{I_\alpha f}(\xi) = (2\pi \abs{\xi})^{-\alpha} \hat f(\xi)$ for Schwartz functions $f$, that is, $I_\alpha = (-\Delta)^{-\alpha/2}$.
The semigroup law $I_\alpha I_\beta = I_{\alpha + \beta}$ holds for $\alpha + \beta < n$.

Example: for $n \ge 3$ and $\alpha = 2$, one has $c_2 = 4 \pi^{n/2} / \Gamma(n/2 - 1) = (n - 2) \sigma_{n-1}$, where $\sigma_{n-1} = 2 \pi^{n/2} / \Gamma(n/2)$ is the area of the unit sphere.
Indeed $\Gamma(n/2) = (n/2 - 1) \Gamma(n/2 - 1)$, so $(n - 2) \cdot 2 \pi^{n/2} / \Gamma(n/2) = 4 \pi^{n/2} / \Gamma(n/2 - 1)$.
So $I_2 f(x) = \frac{1}{(n-2) \sigma_{n-1}} \int f(y) \abs{x - y}^{2-n} \, dy$, and $-\Delta I_2 f = f$ for smooth $f$ with compact support.

Non-example: the constant function $f = 1$ has no Riesz potential.
For every $x$ the integral $\int \abs{x - y}^{\alpha - n} \, dy$ diverges, since $\abs{y}^{\alpha - n}$ is not integrable at infinity when $\alpha > 0$.
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# Hamiltonian path

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]].
A Hamiltonian path in $G$ is a [[path|0JDE56HWQ8]] in $G$ that contains every vertex of $G$.

## Remarks

Since a path visits each vertex at most once, a Hamiltonian path of a graph with $n$ vertices visits each vertex exactly once and has length $n - 1$.
Deleting one edge from a [[Hamiltonian cycle|0HKXNG2RS1]] gives a Hamiltonian path, so every [[Hamiltonian graph|090WPKFABX]] has one.
The converse fails: the [[Petersen graph|0NR95CTFCZ]] has a Hamiltonian path but no Hamiltonian cycle.
Example: in the [[cycle graph|0QQ0F12962]] $C_4$ with vertices $1, 2, 3, 4$ in cyclic order, the path $(1, 2, 3, 4)$ contains all four vertices, so it is a Hamiltonian path.
Non-example: the star $K_{1,3}$ with centre $c$ and leaves $a, b, d$ has no Hamiltonian path.
Each edge contains $c$, so a path alternates between $c$ and leaves, and since it visits $c$ at most once, it contains at most two leaves.
A graph with a Hamiltonian path is connected.
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# Subcategory

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
A *subcategory* $\mathcal{D}$ of $\mathcal{C}$ consists of a subclass $\mathrm{Ob}(\mathcal{D}) \subseteq \mathrm{Ob}(\mathcal{C})$ and, for all $A, B \in \mathrm{Ob}(\mathcal{D})$, a subclass $\Hom_{\mathcal{D}}(A, B) \subseteq \Hom_{\mathcal{C}}(A, B)$ of [[morphisms|859F0TFVA3]], such that:

1. $1_A \in \Hom_{\mathcal{D}}(A, A)$ for every $A \in \mathrm{Ob}(\mathcal{D})$;
2. $g \circ f \in \Hom_{\mathcal{D}}(A, C)$ for all $f \in \Hom_{\mathcal{D}}(A, B)$ and $g \in \Hom_{\mathcal{D}}(B, C)$.

## Remarks

With the composition and identities of $\mathcal{C}$, a subcategory is a category.
It is *wide* if $\mathrm{Ob}(\mathcal{D}) = \mathrm{Ob}(\mathcal{C})$, and *full* if $\Hom_{\mathcal{D}}(A, B) = \Hom_{\mathcal{C}}(A, B)$ for all its objects $A$, $B$.
Example: inside the category of [[groups|G5S1M9]], the [[abelian groups|082GHNPJ53]] with all homomorphisms between them form a subcategory.
The identity of an abelian group is a homomorphism, and a composite of homomorphisms is a homomorphism.
Non-example: in the category of sets, keep all sets but only the functions whose image has at most one point.
Composites of such functions are again of this kind, so clause 2 holds, but the identity of $\set{0, 1}$ is missing, so clause 1 fails.
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# Bartle–Dunford–Schwartz integral

## Definition

Let $(\Omega, \Sigma)$ be a measurable space, let $E$ be a real [[Banach space|06GZ956657]], and let $\mu \colon \Sigma \to E$ be a countably additive [[vector measure|PS28BSAJKJ]].
A set $N \in \Sigma$ is $\mu$-null if $\mu(A) = 0$ for every $A \in \Sigma$ with $A \subseteq N$.

1. For a [[simple function|7WMBX7M4P6]] $s = \sum_{i=1}^{k} a_i \mathbf{1}_{A_i}$ with $a_i \in \R$ and $A_i \in \Sigma$, and for $B \in \Sigma$, put $\int_B s \, d\mu = \sum_{i=1}^{k} a_i \, \mu(A_i \cap B)$.
2. A [[measurable|YYNCJSR0PJ]] $f \colon \Omega \to \R$ is $\mu$-integrable if there are simple functions $s_n \to f$, pointwise outside a $\mu$-null set, such that $\int_B s_n \, d\mu$ converges in $E$ for every $B \in \Sigma$.
3. For such $f$, the *Bartle–Dunford–Schwartz integral* is $\int_B f \, d\mu = \lim_{n \to \infty} \int_B s_n \, d\mu$ for $B \in \Sigma$.

The limit in clause 3 does not depend on the choice of the sequence $(s_n)$.

## Remarks

Example: every bounded measurable $f$ is $\mu$-integrable.
Take simple $s_n$ with $\abs{s_n - f} \le 1/n$ everywhere.
For a simple $s$ with $\abs{s} \le c$ one has $\norm{\int_B s \, d\mu} \le c \, \norm{\mu}(\Omega)$.
Here $\norm{\mu}(\Omega)$ is the supremum of $\norm{\sum_i a_i \mu(A_i)}$ over finite measurable partitions $(A_i)$ of $\Omega$ and reals $\abs{a_i} \le 1$; it is finite when $\mu$ is countably additive.
Apply this to $s = s_n - s_m$.
So $\int_B s_n \, d\mu$ is a Cauchy sequence in $E$.

Non-example: let $\mu$ be Lebesgue measure on $(0, 1]$, viewed as a vector measure with values in $\R$, and $f(x) = 1/x$.
For a real-valued finite measure, integrability in the sense of clause 2 is the same as Lebesgue integrability.
Since $\int_0^1 dx / x = \infty$, no sequence as in clause 2 exists, and $f$ is not $\mu$-integrable.

When $E = \R$ and $\mu$ is a finite signed measure, the integral is the Lebesgue integral against $\mu$.
For each continuous linear functional $x^*$ on $E$, $x^* \big( \int_B f \, d\mu \big) = \int_B f \, d(x^* \circ \mu)$.
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# Projection-valued measure

## Summary

A projection-valued measure assigns to each measurable set an orthogonal projection on a Hilbert space, in a way that is additive like a measure.
It is the object that the spectral theorem attaches to a self-adjoint or normal operator, and it allows one to integrate functions against an operator.
In quantum mechanics, projection-valued measures describe sharp measurements: the projection of a set corresponds to the event that the measured value lies in that set.

## Explanations

### L1 human

An ordinary measure gives each set a number, its size.
A projection-valued measure gives each set a projection, a way to cut out a piece of the Hilbert space.
Disjoint sets give perpendicular pieces, and the whole set gives the whole space.
For example, on $\C^2$ with the standard basis $e_1, e_2$, send a subset $B$ of $\R$ to the projection onto the span of those $e_k$ with $k \in B$.
This splits $\C^2$ into the pieces attached to the points $1$ and $2$, as the eigenspaces of the matrix $\operatorname{diag}(1, 2)$ do.
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# Abstract polytope

## Definition

Let $n \ge -1$ be an integer and let $(\mathcal{P}, \le)$ be a [[partially ordered set|0QV2JS9RJ3]], whose elements are called faces.
A *flag* of $\mathcal{P}$ is a maximal totally ordered subset.
Two flags are *adjacent* if they differ in exactly one face.
$\mathcal{P}$ is an *abstract polytope of rank $n$* if:

1. $\mathcal{P}$ has a least face $F_{-1}$ and a greatest face $F_n$;
2. every flag of $\mathcal{P}$ has exactly $n + 2$ faces;
3. $\mathcal{P}$ is strongly flag-connected: any two flags $\Phi$ and $\Psi$ are joined by flags $\Phi = \Phi_0, \Phi_1, \dots, \Phi_k = \Psi$.
   Here each $\Phi_{i-1}$ is adjacent to $\Phi_i$, and $\Phi \cap \Psi \subseteq \Phi_i$ for each $i$;
4. the diamond condition holds: if $F < G$ are faces whose ranks differ by $2$, then there are exactly two faces $H$ with $F < H < G$.

## Remarks

Conditions 1 and 2 make $\mathcal{P}$ a [[graded poset|1C7TKGTB2F]] with ranks $-1, 0, \dots, n$.
Faces of rank $0$ are vertices, faces of rank $1$ are edges, and faces of rank $n - 1$ are facets.

Example: the faces of a triangle with vertices $a, b, c$, together with $\varnothing$ as $F_{-1}$ and the triangle $T$ as $F_2$, ordered by inclusion, form an abstract polytope of rank $2$.
Every flag has the form $\varnothing < a < ab < T$, with $4 = 2 + 2$ faces.
Between $\varnothing$ and an edge there are exactly two vertices, and between a vertex and $T$ there are exactly two edges, so the diamond condition holds.
More generally, the faces of any [[convex polytope|300K6CB1KH]] of dimension $n$, with $\varnothing$ added, form an abstract polytope of rank $n$.

Non-example: take two triangles that share one vertex $c$, with all their vertices and edges, a least face, and one greatest face of rank $2$.
The vertex $c$ lies on four edges, so between $c$ and the greatest face there are four faces of rank $1$, and the diamond condition fails.

The hemicube is obtained from the cube by identifying antipodal faces.
It is an abstract polytope of rank $3$ with $4$ vertices, $6$ edges and $3$ square faces, and it is not the face lattice of any convex polytope.
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# Alexandrov topology

## Summary

An Alexandrov topology is a topology in which the intersection of any family of open sets, finite or infinite, is open.
Equivalently, every point has a smallest open neighbourhood.
Alexandrov topologies on a set correspond exactly to [[preorders|0VABYWJWH9]] on it: the open sets are the [[upper sets|ETCE287E2R]] of the preorder.
Every finite topological space is of this kind, so these spaces are the bridge between finite topology and order theory.

## Explanations

### L1 human

In most spaces, such as the real line, a point has smaller and smaller neighbourhoods with no smallest one.
In an Alexandrov space each point $x$ has a smallest open set around it, and this set says which points are "above" $x$.
So the whole topology is just a way of writing down an order.
For example, on $\set{1, 2, 3}$ ordered by $1 \le 2 \le 3$, the upper sets $\emptyset$, $\set{3}$, $\set{2, 3}$ and $\set{1, 2, 3}$ form an Alexandrov topology.
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# Volterra equation of the second kind

## Summary

A Volterra equation of the second kind is an equation $x(t) = f(t) + \lambda \int_a^t K(t, s) \, x(s) \, ds$, in which the unknown appears both outside and inside the integral.
It is the [[Volterra integral equation|DND8CBEVXT]] case of an [[integral equation of the second kind|8KJCN49V64]].
Unlike the Fredholm case, it has exactly one continuous solution for every value of $\lambda$ when the kernel and the function $f$ are continuous.
Initial value problems for linear ordinary differential equations lead to equations of this kind.

## Explanations

### L1 human

The value of the unknown at time $t$ is a known input plus a weighted memory of its own past values.
So one can compute the solution forward in time, starting from $x(a) = f(a)$.
For example, $x(t) = 1 + \int_0^t x(s) \, ds$ means $x' = x$ with $x(0) = 1$, so $x(t) = e^t$.
Because only the past enters, there is no resonance, and the equation can always be solved.
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# Point estimation

## Summary

Point estimation is the part of statistical inference that computes from data a single best guess of an unknown quantity.
A typical unknown quantity is a parameter of a statistical model.
The rule that turns data into the guess is an [[estimator|0SSS6ERQ0S]], and its value on the observed data is the estimate.
The theory compares estimators by bias, variance, mean squared error, consistency and efficiency.
It also gives general methods to build them, such as maximum likelihood and the method of moments.
It is contrasted with interval estimation, which gives a range of plausible values.

## Explanations

### L1 human

You flip a coin $100$ times and see $58$ heads; your guess for the probability of heads is $0.58$.
That single number is a point estimate, and the rule "number of heads divided by number of flips" is the estimator.
A different sample would give a different guess, so an estimator is itself random.
Point estimation studies which rules give guesses that are close to the truth most of the time.
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# Sphere bundle

## Definition

Let $k \ge 0$ be an integer.
A *sphere bundle* with fiber $S^k$ is a [[fiber bundle|CX4M8K61AB]] $\pi \colon E \to B$ whose fiber is the [[$k$-sphere|T62369RFH6]] $S^k$.

## Remarks

Many authors also require the structure group to be the orthogonal group $O(k+1)$ acting on $S^k$, so that the transition functions are rotations and reflections.
The main example: let $V \to B$ be a real vector bundle of rank $k + 1$ with a continuous inner product.
Then the unit sphere bundle $S(V) = \set{v \in V : \abs{v} = 1}$ is a sphere bundle with fiber $S^k$ and structure group $O(k+1)$.
A sphere bundle with fiber $S^1$ is called a circle bundle.

Example: the unit tangent bundle of $S^2$, the set of pairs $(x, v)$ with $x, v \in S^2$ and $v \perp x$, is a circle bundle over $S^2$.
Its total space is homeomorphic to $\R P^3$.

Non-example: the Möbius band over the circle is a fiber bundle, but its fiber is a closed interval, not a sphere, so it is not a sphere bundle.
Its boundary, however, is a sphere bundle with fiber $S^0$: it is the connected double cover of the circle.
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# Whittle likelihood

## Definition

Let $x_1, \dots, x_n$ be observations of a weakly [[stationary process|DT0QXJE705]] with mean $0$.
Let $\set{f_\theta : \theta \in \Theta}$ be a parametric family of spectral densities on $[-\pi, \pi]$, with $f_\theta > 0$.
Let $I(\omega) = \frac{1}{2\pi n} \abs{\sum_{t=1}^n x_t e^{-i t \omega}}^2$ be the [[periodogram|3ZYM0HXA3E]], scaled to estimate the spectral density, and let $\omega_j = 2\pi j / n$ for $j = 1, \dots, m$ with $m = \floor{(n-1)/2}$.
The *Whittle log-likelihood* is
$$\ell_W(\theta) = -\sum_{j=1}^{m} \left( \log f_\theta(\omega_j) + \frac{I(\omega_j)}{f_\theta(\omega_j)} \right).$$
The *Whittle estimator* is a maximizer $\hat\theta$ of $\ell_W$ over $\Theta$.

## Remarks

The idea is that, for a Gaussian process with smooth spectral density, the values $I(\omega_j)$ are approximately independent and exponentially distributed with means $f_\theta(\omega_j)$.
The log [[likelihood|0VW9E4QXR6]] of independent exponential variables with these means is exactly $\ell_W(\theta)$.
A version with an integral, $-\frac{n}{4\pi} \int_{-\pi}^{\pi} \left( \log f_\theta(\omega) + I(\omega) / f_\theta(\omega) \right) d\omega$, is also common.
Under standard conditions the Whittle estimator is consistent and asymptotically normal with the same limit as the Gaussian [[maximum likelihood|0FREPFKNS7]] estimator.

Example: for white noise, $f_\theta(\omega) = \theta / (2\pi)$ with $\theta = \sigma^2 > 0$.
Setting the derivative of $\ell_W(\theta) = -\sum_j \left( \log \frac{\theta}{2\pi} + \frac{2\pi I(\omega_j)}{\theta} \right)$ to zero gives $m/\theta = 2\pi \sum_j I(\omega_j) / \theta^2$, so $\hat\theta = \frac{2\pi}{m} \sum_{j=1}^{m} I(\omega_j)$.

Non-example: let $\Gamma_n = [\gamma_\theta(s - t)]_{s,t=1}^n$ be the covariance matrix of the data.
The exact Gaussian log-likelihood $-\frac12 \log \det \Gamma_n - \frac12 x^T \Gamma_n^{-1} x$, up to a constant, is not the Whittle likelihood.
The Whittle likelihood approximates it by treating $\Gamma_n$ as if it were diagonal in the Fourier basis.
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# Edge-of-the-wedge theorem

## Statement

Let $C \subseteq \R^n$ be a nonempty open [[convex cone|6VE4KA12VD]] with vertex $0$, let $E \subseteq \R^n$ be a nonempty [[open set|02BH3MCBQB]], and let $\delta > 0$.
Write $\R^n \subseteq \C^n$ as the set of points with zero imaginary part, and define the two wedges
$$W_\pm = \set{x + iy \in \C^n : x \in E, \ \pm y \in C, \ \abs{y} < \delta}.$$
Let $f_+$ and $f_-$ be [[holomorphic functions|507C34W0ZT]] on $W_+$ and $W_-$.
Suppose that $f_+$ extends continuously to $W_+ \cup E$, that $f_-$ extends continuously to $W_- \cup E$, and that the two extensions agree on $E$.
Then there are an open set $G \subseteq \C^n$ with $E \subseteq G$ and a holomorphic function $F$ on $G$ such that $F = f_+$ on $G \cap W_+$ and $F = f_-$ on $G \cap W_-$.

## Remarks

For $n = 1$ the wedges are a rectangle above and a rectangle below the interval $E$.
Then the theorem follows from Morera's theorem: the glued function is continuous, and its integrals over small triangles vanish.
In several variables the edge $E$ has only real dimension $n$ inside $\C^n$, and a direct Morera argument is not available.
Proofs use Cauchy integrals over suitable families of analytic discs, or Fourier analysis.
The theorem also holds when the continuity hypothesis is replaced by the condition that $f_+$ and $f_-$ have the same boundary value on $E$ in the sense of distributions.

The agreement on $E$ is necessary: $f_+ = 0$ and $f_- = 1$ satisfy all other hypotheses but cannot be glued to one holomorphic function.

## Explanations

### L1 human

Take two wedge-shaped regions of $\C^n$ that meet only along a common real edge.
Suppose a holomorphic function on each wedge has boundary values on the edge, and the two sets of boundary values agree.
Then the two functions are pieces of a single holomorphic function on a full neighbourhood of the edge.
The edge acts like a seam that disappears once the two sides match.

### L2 human

In one variable this is the familiar gluing of holomorphic functions across a real interval.
In several variables it is surprising, because the edge has only half the real dimension of the space, yet it still suffices for gluing.
The theorem arose in quantum field theory, where it is used to prove analyticity properties of correlation functions.
It is also a basic tool in the theory of hyperfunctions, where generalised functions on $\R^n$ are represented by boundary values of holomorphic functions on wedges.
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# Atiyah–Bott theorem on Yang–Mills over Riemann surfaces

## Statement

Let $M$ be a closed [[Riemann surface|5K0N8QMV6Q]] of genus $g$ with a compatible Riemannian metric and area form $\omega$.
Let $P \to M$ be a [[principal $U(n)$-bundle|8E37E74XVP]] of degree $d$, let $\mathcal{A}$ be the space of [[connections|NJ21PN5ZGJ]] on $P$, and let $\mathcal{G}$ be its [[gauge group|707DEWCCFX]].
Each $A \in \mathcal{A}$ defines a [[holomorphic structure|TKA958Z4KZ]] $E_A$ on the associated vector bundle $E$ of rank $n$.

1. A connection $A$ satisfies the [[Yang–Mills equations|JJWETW5AT8]] if and only if $E$ splits in the following way.
There is an orthogonal direct sum $E = E_1 \oplus \cdots \oplus E_r$ of subbundles preserved by $A$.
On each $E_j$ the [[curvature|ZAJFW1S078]] of $A$ equals $c_j \, \mathrm{id}_{E_j} \otimes \omega$ for a constant $c_j$.
2. The stratification of $\mathcal{A}$ by the Harder–Narasimhan type of $E_A$ is $\mathcal{G}$-equivariantly perfect for rational coefficients.
So the [[Yang–Mills functional|CKGDR9P9PA]] behaves on $\mathcal{A}$ like an equivariantly perfect Morse function.
3. Let $n$ and $d$ be coprime.
The [[moduli space|FJZRW9P0K3]] of [[stable bundles|RQRJ5H4Y3E]] of rank $n$ and degree $d$ on $M$ has integral cohomology without torsion.
Its Poincaré polynomial is determined by an explicit recursion over Harder–Narasimhan types.

## Remarks

On a surface, the Yang–Mills equation $d_A^* F_A = 0$ says that the function $\star F_A$, a section of the adjoint bundle, is covariantly constant.
Its eigenvalues are therefore constant and its eigenspaces give the parallel splitting of part 1.
By the Narasimhan–Seshadri theorem, the connections of constant central curvature correspond to polystable holomorphic bundles, so the critical sets are described by stable bundles of lower rank.

For part 2 one compares the [[equivariant cohomology|C7MZ3SFGYF]] of $\mathcal{A}$, which is the cohomology of the classifying space of $\mathcal{G}$ because $\mathcal{A}$ is contractible, with that of the strata.
Perfection means that the Thom–Gysin sequences of the strata split into short exact sequences.
The open stratum consists of the semistable bundles, and in the coprime case the gauge group acts on it with constant stabiliser $U(1)$, which gives part 3.

## Explanations

### L1 human

Atiyah and Bott treated the Yang–Mills functional over a surface as a height function on an infinite-dimensional space.
Its critical points can be listed completely, and they correspond to bundles that split into pieces of constant curvature.
The way the space breaks into pieces flowing to each critical point is as efficient as possible.
This lets one compute the topology of the space of stable vector bundles on a curve.

### L2 human

The work links gauge theory on surfaces with algebraic geometry of vector bundles on curves through the Narasimhan–Seshadri theorem.
It is a model of infinite-dimensional Morse theory with a symmetry group, carried out through equivariant cohomology.
The Harder–Narasimhan stratification it uses also appears in moment map geometry, where the curvature plays the role of a moment map for the gauge group.
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# Del Pezzo surface

## Summary

A del Pezzo surface is a smooth projective [[algebraic surface|KFTECAATKG]] whose anticanonical divisor is ample.
These are exactly the [[Fano varieties|FMHA649HJ9]] of dimension two.
Over an algebraically closed field they are $\mathbb{P}^1 \times \mathbb{P}^1$ and the blow-ups of the plane in at most eight points in general position.
So they are [[rational surfaces|55A4A53TRF]].
Their degree $K^2$ runs from $1$ to $9$, and the curves of self-intersection $-1$ on them are governed by root systems of type $E$.

## Explanations

### L1 human

Del Pezzo surfaces are the most positively curved surfaces in the algebraic sense.
One gets them by taking the projective plane and blowing up a few points in sufficiently general position.
Each blown-up point is replaced by a line, and new lines appear from the lines and conics through the chosen points.
The smooth cubic surfaces in space, with their $27$ lines, are the del Pezzo surfaces of degree $3$.
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# Projective connection

## Definition

Let $n \ge 2$, let $G = \mathrm{PGL}(n + 1, \R)$ act on the real projective space $\R P^n$, and let $H \subset G$ be the stabilizer of a point $o \in \R P^n$, so that $G / H \cong \R P^n$.
Let $M$ be a smooth manifold of dimension $n$.
A *projective connection* on $M$ is a [[Cartan connection|QPR7DCHXPG]] on $M$ modelled on $(G, H)$.

## Remarks

Classical description: two torsion-free [[affine connections|JX65KSVHSJ]] $\nabla, \nabla'$ on $M$ are *projectively equivalent* if they have the same [[geodesics|NSYMQMHPDN]] up to reparametrization.
This holds if and only if $\nabla'_X Y = \nabla_X Y + \alpha(X) Y + \alpha(Y) X$ for some $1$-form $\alpha$ on $M$.
A projective equivalence class of torsion-free affine connections is called a projective structure.
Cartan showed that it determines a projective connection in the sense above, unique once a normalization of the curvature is imposed.
Example: the model $G \to \R P^n$ with the Maurer–Cartan form of $G$ is a projective connection, the flat one; its unparametrized geodesics are the projective lines.
For the flat connection of $\R^n$ and $\alpha = dx^1$, the connection $\nabla'$ above has new geodesics such as $t \mapsto (f(t), 0, \dots, 0)$ with $f'' + 2 (f')^2 = 0$, which are straight lines traversed at a non-constant speed.
Non-example: let $\nabla$ be the flat connection of $\R^n$, let $g$ be the Euclidean metric, and let $V$ be a fixed nonzero vector.
Put $\nabla'_X Y = \nabla_X Y + g(X, Y) V$.
This changes the geodesics as curves, so $\nabla'$ is not projectively equivalent to $\nabla$.
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# Lüroth's theorem

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $K(t)$ be the [[field of rational functions|4V81611P9N]] in one variable $t$ over $K$.
Let $E$ be a field with $K \subsetneq E \subseteq K(t)$.
Then $E = K(u)$ for some [[rational function|0WCZH4AFMT]] $u \in K(t) \setminus K$.

## Remarks

Pick $u \in E \setminus K$ and write $u = f(t)/g(t)$ with $f, g$ coprime polynomials.
Then $t$ is a root of $f(x) - u\, g(x)$ over $K(u)$, and one shows that $[K(t) : K(u)] = \max(\deg f, \deg g)$.
The proof then takes the minimal polynomial of $t$ over $E$, chooses $u$ among its coefficients not in $K$, and compares degrees to get $E = K(u)$.
Example: $E = K(t^2 + t^{-2})$ lies between $K$ and $K(t)$, and the theorem says it is already of the form $K(u)$; here $u = t^2 + t^{-2}$ works, and $[K(t) : E] = 4$.
The analogue in two variables concerns fields $E \subseteq K(x, y)$ with $K(x, y)$ finite over $E$.
It holds over algebraically closed fields of characteristic $0$ by Castelnuovo's theorem, and it fails in general in three or more variables.

## Explanations

### L1 human

Every field between $K$ and the rational functions in one variable is again a field of rational functions in one variable.
So one cannot find an exotic field squeezed in there.
Geometrically, a curve that is the image of the line under a rational map is itself rational.

### L2 human

The theorem says that a subfield of a [[purely transcendental extension|G5PVJMM37Z]] of transcendence degree $1$ is purely transcendental.
In geometry, it gives that a unirational curve is rational.
The degree formula $[K(t) : K(u)] = \max(\deg f, \deg g)$ also computes the degree of a rational map from the line to itself, which is the [[degree|0GS74FGFEK]] of the field extension $K(t)/K(u)$.
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# Probabilistic method

## Definition

Let $\Omega$ be a nonempty finite set of objects and let $P$ be a property that objects in $\Omega$ may have.
A proof by the *probabilistic method* that some element of $\Omega$ has $P$ does the following:

1. It puts a probability measure $\Pr$ on $\Omega$, making it a [[probability space|0NFPJVJJHX]].
2. It shows that the [[event|031TPJVS74]] $\set{\omega \in \Omega : \omega \text{ has } P}$ has positive probability.
3. It concludes that this event is nonempty, so some element of $\Omega$ has $P$.

A common form of step 2 uses a random variable $X$ on $\Omega$: if the [[expected value|0WQNRCNY3M]] satisfies $\mathbb{E}[X] \ge t$, then $X(\omega) \ge t$ for some $\omega$.

## Remarks

Erdős' bound for diagonal Ramsey numbers is an example.
Let $k \ge 3$ and $n = \floor{2^{k/2}}$, and colour each edge of the complete graph $K_n$ red or blue independently with probability $1/2$.
The expected number of monochromatic copies of $K_k$ is $\binom{n}{k} 2^{1 - \binom{k}{2}}$, and this is less than $1$.
So some colouring has no monochromatic $K_k$, and the [[Ramsey number|0K9QVPJYJF]] satisfies $R(k, k) > \floor{2^{k/2}}$.
An argument that only shows that the event has probability at least $0$ is not a proof by this method.
Step 2 fails there, and an event of probability $0$ may be empty.
The method also applies to infinite probability spaces, with the same conclusion that an event of positive probability is nonempty.
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# Kirby calculus

## Summary

Kirby calculus is a set of moves on framed link diagrams in the $3$-sphere that relate any two diagrams describing the same $3$-manifold.
A framed [[link|1A5WRCNTM3]] describes a $4$-manifold, obtained by attaching $2$-handles to the $4$-ball along the link, and its boundary $3$-manifold, obtained by [[Dehn surgery|ZA405B0Z1S]].
Robion Kirby proved that two framed links give the same oriented $3$-manifold if and only if they are related by his moves.
It is a basic working tool of $3$- and $4$-manifold topology, used to construct manifolds and to prove that two of them are diffeomorphic.

## Explanations

### L1 human

Every closed orientable $3$-dimensional space can be drawn as a picture of knotted loops in ordinary space, each labelled by an integer.
Many different pictures describe the same space.
Kirby calculus is the list of allowed picture changes, a little like the rules for simplifying an algebraic expression.
For example, a single unknotted loop labelled $+1$ or $-1$, far away from the other loops, can be added or erased.
This does not change the space, and the loop alone describes the $3$-sphere.
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# Clausen function

## Summary

The Clausen function $\operatorname{Cl}_2$ is the real function $\theta \mapsto -\int_0^\theta \log \abs{2 \sin(t/2)} \, dt$, equal to the [[Fourier series|G12KXDHPPS]] $\sum_{k \ge 1} \sin(k\theta) / k^2$.
It is the imaginary part of the [[dilogarithm|G0X0NWSDB7]] on the unit circle.
It appears in evaluations of log-sine integrals, in values such as Catalan's constant, and in volumes of hyperbolic polyhedra.

## Explanations

### L1 human

Take the dilogarithm series $\sum z^k / k^2$ and put $z$ on the unit circle, $z = e^{i\theta}$.
The real part is a quadratic polynomial in $\theta$ on $[0, 2\pi]$, but the imaginary part, $\sum \sin(k\theta)/k^2$, is a new function: the Clausen function.
Its graph looks like a tilted sine wave, odd and periodic with period $2\pi$, with its maximum at $\theta = \pi/3$.
At $\theta = \pi/2$ the series is $1 - \frac{1}{9} + \frac{1}{25} - \cdots$, which is Catalan's constant $G \approx 0.91597$.
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# Poisson–Mellin–Newton cycle

## Definition

Let $(f_n)_{n \ge 0}$ be a sequence of complex numbers.
The *Poisson–Mellin–Newton cycle* of $(f_n)$ is the following chain of three objects.

1. The *Poisson generating function* $g(t) = e^{-t} \sum_{n \ge 0} f_n \frac{t^n}{n!}$, for $t > 0$ where the series converges.
2. The *Newton coefficients* $b_m = \sum_{k=0}^{m} \binom{m}{k} (-1)^k f_k$, which satisfy $g(t) = \sum_{m \ge 0} b_m \frac{(-t)^m}{m!}$ and $f_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k b_k$.
3. The [[Mellin transform|N5PVGRJE1T]] $\varphi(s) = \int_0^\infty g(t)\, t^{s-1}\, dt$, where it converges.

Assume that $b_m = b(m)$ for a function $b$ that is [[holomorphic|0NAHXR1FRH]] and of moderate growth in a half-plane.
Then the cycle closes by the relations
$$\varphi(s) = \Gamma(s)\, b(-s), \qquad f_n = \frac{(-1)^n}{2\pi i} \oint_{C} b(s)\, \frac{n!}{s (s - 1) \cdots (s - n)}\, ds,$$
where $C$ is a positively oriented contour that encircles the points $0, 1, \dots, n$ and no singularity of $b$.

## Remarks

The two identities in clause 2 come from multiplying the power series of $e^{-t}$ and $\sum f_n t^n / n!$ and comparing coefficients.
The map $(f_n) \mapsto (b_m)$ is the binomial transform, and it is its own inverse.
The first closing relation is an instance of [[Ramanujan's master theorem|ZN4NMX1DB4]], which needs growth conditions on $b$.
The second is the Nörlund–Rice integral: the residue of $n!\,/\,(s (s-1) \cdots (s-n))$ at $s = k$ is $(-1)^{n-k} \binom{n}{k}$, so the residue theorem gives back $f_n = \sum_k \binom{n}{k} (-1)^k b(k)$.
Here $\Gamma$ is the [[gamma function|0H5SYBSED9]].

Example: $f_n = (1/2)^n$.
Then $g(t) = e^{-t/2}$, the Newton coefficients are $b_m = (1 - 1/2)^m = 2^{-m}$, so $b(s) = 2^{-s}$.
Indeed $\varphi(s) = 2^{s}\, \Gamma(s) = \Gamma(s)\, b(-s)$ for $\operatorname{Re} s > 0$.

Non-example: for $f_n = (n!)^2$ the series $\sum_n f_n t^n / n!$ diverges for every $t > 0$, so there is no Poisson generating function and the cycle does not start.
</document>

<document source="https://tamnd.github.io/math2/node/AVTXQM3CNZ.md" tag="AVTXQM3CNZ" kind="definition">
# Domino tiling

## Definition

A *cell* is a unit square $[a, a+1] \times [b, b+1]$ with $a, b \in \Z$.
A *domino* is the union of two cells that share an edge, so it is a $1 \times 2$ or $2 \times 1$ rectangle with integer corners.
Let $R \subseteq \R^2$ be a region that is a union of finitely many cells.
A *domino tiling* of $R$ is a finite set $D$ of dominoes such that:

1. every domino of $D$ is contained in $R$;
2. the union of the dominoes of $D$ is $R$;
3. two distinct dominoes of $D$ have disjoint interiors.

## Remarks

Example: the $2 \times 3$ rectangle $[0, 3] \times [0, 2]$ has exactly three domino tilings.
They are: three vertical dominoes; one vertical domino on the left and two horizontal ones on the right; and the mirror image of the second.
More generally, the $2 \times n$ rectangle has $F_{n+1}$ domino tilings, where $F_1 = F_2 = 1$ and $F_{k+1} = F_k + F_{k-1}$ are the [[Fibonacci numbers|0GA9K57A39]].

Non-example: remove two opposite corner cells from the $8 \times 8$ board.
Colour the cells like a chessboard; each domino covers one black and one white cell, but the two removed cells have the same colour.
So the remaining $62$ cells have $32$ of one colour and $30$ of the other, and no set of dominoes satisfies clauses 2 and 3.

Equivalently, a domino tiling of $R$ is a [[perfect matching|0NHK9MFFW9]] of the graph whose vertices are the cells of $R$, two cells being adjacent when they share an edge.
</document>

<document source="https://tamnd.github.io/math2/node/AVXARAXHKY.md" tag="AVXARAXHKY" kind="statement">
# Tensor-hom adjunction

## Statement

Let $R$ be a [[commutative ring|0310REJJY8]] and let $M$, $N$, $P$ be [[$R$-modules|0NZ8JA25WK]].
Let $\Hom_R(N, P)$ be the $R$-module of $R$-[[linear maps|0NRYHJPQG2]] $N \to P$, with $(r\psi)(n) = r\psi(n)$.
Then the map
$$\Phi \colon \Hom_R(M \otimes_R N, P) \to \Hom_R(M, \Hom_R(N, P)), \qquad \Phi(\varphi)(m)(n) = \varphi(m \otimes n),$$
is an isomorphism of $R$-modules, natural in $M$, $N$ and $P$.
Thus the functor $- \otimes_R N$ is [[left adjoint|TZHF6ZM92S]] to the [[hom functor|VJXS8AK67E]] $\Hom_R(N, -)$.

## Remarks

The inverse sends $\psi \colon M \to \Hom_R(N, P)$ to the linear map $M \otimes_R N \to P$ induced by the bilinear map $(m, n) \mapsto \psi(m)(n)$.
This is exactly the universal property of the [[tensor product|F0HQXQ0HCD]]: linear maps out of $M \otimes_R N$ are bilinear maps on $M \times N$.
In turn, a bilinear map is a linear map from $M$ to linear maps on $N$.

The adjunction has consequences for [[exact sequences|JB7ZBFBA9C]].
The functor $\Hom_R(N, -)$ is left exact: if $0 \to A \to B \to C$ is exact, then so is $0 \to \Hom_R(N, A) \to \Hom_R(N, B) \to \Hom_R(N, C)$.
As a left adjoint, $- \otimes_R N$ is right exact: if $A \to B \to C \to 0$ is exact, then so is $A \otimes_R N \to B \otimes_R N \to C \otimes_R N \to 0$.
Neither functor is exact in general.
For $N = \Z/2\Z$ over $\Z$, the surjection $\Z \to \Z/2\Z$ gives $\Hom(\Z/2\Z, \Z) = 0 \to \Hom(\Z/2\Z, \Z/2\Z) \cong \Z/2\Z$, which is not surjective; and the injection $\Z \xrightarrow{2} \Z$ becomes the zero map after $- \otimes_\Z \Z/2\Z$.

## Explanations

### L1 human

A function of two variables can be seen as a function of the first variable whose values are functions of the second.
The tensor-hom adjunction is this idea for linear algebra over a ring.
A linear map out of $M \otimes N$ is the same thing as a linear map from $M$ into the maps out of $N$.
It lets one move a factor from one side of $\Hom$ to the other.

### L2 human

This is the prototype of an adjunction, and it explains at once why tensor products are right exact and $\Hom$ is left exact.
The failure of exactness is measured by the derived functors $\operatorname{Tor}$ and $\operatorname{Ext}$.
[[Flat modules|73RYYEVDKA]] are those $N$ for which $- \otimes_R N$ is exact, and [[projective modules|BFFA03GE3V]] those for which $\Hom_R(N, -)$ is exact.
The same adjunction holds for bimodules over noncommutative rings, and for sheaves of modules.
</document>

<document source="https://tamnd.github.io/math2/node/AW00HPFB54.md" tag="AW00HPFB54" kind="statement">
# Kronecker–Weber theorem

## Statement

Let $L$ be a [[finite extension|0C1Y04SZ1W]] of $\Q$ that is a [[Galois extension|0QMZAQ6QQ4]] with [[abelian|082GHNPJ53]] Galois group.
Then there is a positive integer $n$ such that $L$ is contained in the [[cyclotomic field|NN77VYSGXR]] $\Q(\zeta_n)$.
Here $\zeta_n$ is a primitive $n$-th [[root of unity|0YV2CKP2AV]].

## Remarks

The converse holds as well: $\Gal(\Q(\zeta_n)/\Q) \cong (\Z/n\Z)^\times$ is abelian, so every subfield of $\Q(\zeta_n)$ is abelian over $\Q$.
One standard proof reduces to cyclic extensions of prime power degree and studies their ramification groups at the ramified primes.
A key input is that $\Q$ has no nontrivial extension unramified at all primes, a consequence of the [[Hermite–Minkowski theorem|KMVW45B0TT]].
Another proof comes from [[class field theory|BEKCEYHCHX]]: the ray class field of $\Q$ for the modulus $n\infty$ is $\Q(\zeta_n)$.

Example: every [[quadratic field|7SV4WA92SM]] is abelian over $\Q$, so it lies in a cyclotomic field.
For instance $\sqrt{2} = \zeta_8 + \zeta_8^{-1}$ lies in $\Q(\zeta_8)$, and $\sqrt{5} = \zeta_5 - \zeta_5^2 - \zeta_5^3 + \zeta_5^4$ lies in $\Q(\zeta_5)$.

The abelian hypothesis is needed.
The field $\Q(\sqrt[3]{2})$ is not contained in any $\Q(\zeta_n)$: it is not Galois over $\Q$, while every subfield of $\Q(\zeta_n)$ is Galois over $\Q$.

## Explanations

### L1 human

Roots of unity are the solutions of $x^n = 1$.
The theorem says that every number field with a commutative Galois group over $\Q$ can be built from roots of unity.
So the abelian extensions of $\Q$ have a complete and explicit description.

### L2 human

The maximal abelian extension of $\Q$ is the field generated by all roots of unity.
Its Galois group is $\widehat{\Z}^\times$, the unit group of the profinite completion $\widehat{\Z}$ of $\Z$.
Hilbert's twelfth problem asks for similar explicit generators for the abelian extensions of other number fields.
For imaginary quadratic fields this is achieved by [[complex multiplication|D3HEJTVY7G]], using values of elliptic and modular functions.
</document>

<document source="https://tamnd.github.io/math2/node/AW16MYJV5Q.md" tag="AW16MYJV5Q" kind="example">
# Menshov's set of multiplicity of measure zero

## Example

Let $\mathbb{T} = \R / 2\pi\Z$.
In 1916 Menshov constructed a closed set $E \subset \mathbb{T}$ of Lebesgue measure zero that is a set of multiplicity.
So there is a [[trigonometric series|FZK2M8KGXQ]] $\sum_{n \in \Z} c_n e^{inx}$, with coefficients not all $0$, whose symmetric partial sums converge to $0$ at every point outside $E$.

The construction gives a nonzero positive [[Borel measure|DYR1QFNGNJ]] $\mu$ supported on a perfect [[null set|VQQM4Y550T]] $E$ whose Fourier–Stieltjes coefficients $\hat\mu(n) = \frac{1}{2\pi} \int e^{-inx} \, d\mu(x)$ tend to $0$ as $\abs{n} \to \infty$.
The series $\sum_n \hat\mu(n) e^{inx}$ then converges to $0$ at every point outside the support of $\mu$.
This follows from Riemann's principle of localization for series with coefficients tending to $0$.
Its coefficients are not all $0$, since $\hat\mu(0) = \mu(\mathbb{T}) / 2\pi > 0$.
Hence $E$ is not a [[set of uniqueness|K7CNWXF9SN]].

## Remarks

The example shows that measure zero is necessary but not sufficient for a closed set to be a set of uniqueness.
A measure whose Fourier coefficients tend to $0$ is called a Rajchman measure, and its closed support is always a set of multiplicity by the same argument.
Whether a set of measure zero is a set of uniqueness can depend on fine arithmetic properties of the set.

## Explanations

### L1 human

Cantor showed that a trigonometric series that is $0$ everywhere must be the zero series, and later this was extended to series that are $0$ outside a countable set.
One might guess that any exceptional set of total length zero is harmless.
Menshov showed that this guess is wrong: he built a set of length zero and a nonzero series that is $0$ everywhere outside it.
The series hides all of its non-zero behaviour on a tiny, dust-like set.
</document>

<document source="https://tamnd.github.io/math2/node/AW3J9AQSA0.md" tag="AW3J9AQSA0" kind="concept">
# Totally disconnected group

## Summary

A totally disconnected group is a [[topological group|92700QAB5Y]] whose underlying space is [[totally disconnected|0RJSBR9GW1]].
Equivalently, its identity component is the trivial subgroup.
Profinite groups, $p$-adic groups and discrete groups are examples.
For a totally disconnected locally compact group, van Dantzig's theorem gives a base of compact open subgroups at the identity.
This is the starting point of the structure theory of such groups.

## Explanations

### L1 human

A totally disconnected group has no connected pieces except single points.
Any two distinct elements can be separated, so the group looks like dust rather than like a curve or a surface.
The $p$-adic integers are the main example.
Two $p$-adic integers are close when their difference is divisible by a high power of $p$, and the space falls apart into small separate balls.
Every group with the discrete topology is also totally disconnected.
</document>

<document source="https://tamnd.github.io/math2/node/AW4T3NQYVF.md" tag="AW4T3NQYVF" kind="concept">
# Fréchet algebra

## Summary

A Fréchet algebra is an associative algebra that is also a [[Fréchet space|3GAA614673]], with a jointly continuous multiplication.
It generalizes a [[Banach algebra|9KYWN94QDN]]: the topology comes from a countable family of [[seminorms|FP5GVCN8GV]] instead of a single norm.
Algebras of smooth functions, of holomorphic functions, and of continuous functions on the real line are Fréchet algebras but not Banach algebras.

## Explanations

### L1 human

Sometimes no single norm can measure the functions of interest, for example all continuous functions on the real line, which may be unbounded.
Instead one measures a function by its maximum on $[-1, 1]$, on $[-2, 2]$, on $[-3, 3]$, and so on.
A sequence converges when it converges uniformly on each of these intervals.
With this notion of convergence, the continuous functions on $\R$ form a Fréchet algebra.
</document>

<document source="https://tamnd.github.io/math2/node/AWFXHJXGKE.md" tag="AWFXHJXGKE" kind="definition">
# Lebesgue point

## Definition

Let $n \ge 1$, let $\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$, and let $f \colon \R^n \to \R$ be locally [[integrable|X7GTF3D98F]].
For $x \in \R^n$ and $r > 0$, let $B(x, r)$ be the [[open ball|0YE5868HZB]] of centre $x$ and radius $r$.
A point $x \in \R^n$ is a *Lebesgue point* of $f$ if
$$\lim_{r \to 0^+} \frac{1}{\lambda(B(x, r))} \int_{B(x, r)} \abs{f(y) - f(x)} \, d\lambda(y) = 0.$$

## Remarks

Example: if $f$ is continuous at $x$, then $x$ is a Lebesgue point.
Given $\varepsilon > 0$, choose $r_0$ with $\abs{f(y) - f(x)} < \varepsilon$ on $B(x, r_0)$; then every average with $r < r_0$ is at most $\varepsilon$.

Non-example: let $n = 1$ and $f = \mathbf{1}_{[0, \infty)}$.
At $x = 0$, the integrand equals $1$ on $(-r, 0)$ and $0$ on $[0, r)$, so the average is $1/2$ for every $r$, and $0$ is not a Lebesgue point.

The set of Lebesgue points depends on the chosen function, not only on its class modulo null sets.
Changing $f$ at one point $x$ can destroy the Lebesgue point property at $x$ but at no other point.
</document>

<document source="https://tamnd.github.io/math2/node/AWK2MBQ94K.md" tag="AWK2MBQ94K" kind="definition">
# Canonical correlation

## Definition

Let $X$ be a random vector in $\R^p$ and $Y$ a random vector in $\R^q$, with finite second moments.
Assume that the [[covariance matrices|0V7G6WMFK0]] $\Sigma_{XX}$ and $\Sigma_{YY}$ are [[positive definite|0KX66CZMBB]], and let $\Sigma_{XY} = \operatorname{Cov}(X, Y)$.
Let $m = \min(p, q)$.
For $i = 1, \ldots, m$, the $i$-th pair of *canonical variates* $(a_i^T X, b_i^T Y)$ is defined inductively:
$a_i \in \R^p$ and $b_i \in \R^q$ maximise the [[correlation|0DNN9B6Q2F]] $\operatorname{Corr}(a^T X, b^T Y)$ under these constraints:

1. $\operatorname{Var}(a^T X) = \operatorname{Var}(b^T Y) = 1$;
2. $\operatorname{Cov}(a^T X, a_j^T X) = 0$ for all $j < i$;
3. $\operatorname{Cov}(b^T Y, b_j^T Y) = 0$ for all $j < i$.

The maximal value $\rho_i = \operatorname{Corr}(a_i^T X, b_i^T Y)$ is the $i$-th *canonical correlation*.

## Remarks

One has $1 \ge \rho_1 \ge \rho_2 \ge \cdots \ge \rho_m \ge 0$.
The numbers $\rho_1^2, \ldots, \rho_m^2$ are the $m$ largest [[eigenvalues|0BQAA2BK3Q]] of $\Sigma_{XX}^{-1} \Sigma_{XY} \Sigma_{YY}^{-1} \Sigma_{YX}$.
Example: for $p = q = 1$, the only canonical correlation is $\rho_1 = \abs{\operatorname{Corr}(X, Y)}$, attained with $b_1 = \pm 1/\sqrt{\operatorname{Var}(Y)}$ chosen with the sign of the covariance.
Non-example: for $X = (X_1, X_2)$ with $X_1 = Y$ and $X_2$ independent of $Y$, the pair $a = (0, 1)/\sqrt{\operatorname{Var}(X_2)}$ gives correlation $0$, which is not maximal, since $a = (1, 0)/\sqrt{\operatorname{Var}(Y)}$ gives correlation $1$.
The sample version replaces the covariance matrices by sample covariance matrices.
</document>

<document source="https://tamnd.github.io/math2/node/AWQA2Y30N4.md" tag="AWQA2Y30N4" kind="concept">
# Numerical methods for linear least squares

## Summary

Numerical methods for linear least squares are algorithms for the [[least squares|0BYTQ4X1HC]] problem.
They compute a vector $x$ minimizing $\norm{Ax - b}_2$ for a given matrix $A$ and vector $b$.
The main methods solve the normal equations by a [[Cholesky decomposition|0WKG8CYY6A]], use a [[QR decomposition|0ERWG4NXQ9]] of $A$, or use the [[singular value decomposition|000NPTMX6J]].
They differ in cost and in how rounding errors are amplified, which depends on the [[condition number|AB3S6MHJ8X]] of $A$.

## Explanations

### L1 human

Fitting a straight line through many measured points gives more equations than unknowns, and usually no line passes through all points.
Least squares picks the line that makes the sum of squared misses smallest.
On paper one can solve the normal equations $A^T A x = A^T b$, but on a computer this can lose twice as many digits as needed.
For example, if $A$ has condition number $10^8$, then $A^T A$ has condition number $10^{16}$.
In double precision the normal equations may then give no correct digits, while a QR-based method still gives about eight when the data are nearly consistent.
</document>

<document source="https://tamnd.github.io/math2/node/AWS7XTZ49C.md" tag="AWS7XTZ49C" kind="statement">
# Hua's identity

## Statement

Let $D$ be a [[division ring|0WZXK05ME7]], and let $a, b \in D$ with $a \ne 0$, $b \ne 0$ and $ab \ne 1$.
Then $a^{-1} + (b^{-1} - a)^{-1}$ is invertible, and
$$a - \left(a^{-1} + (b^{-1} - a)^{-1}\right)^{-1} = aba.$$

## Remarks

The element $b^{-1} - a$ is nonzero because $ab \ne 1$.
The proof is a direct factorization: $a^{-1} + (b^{-1} - a)^{-1} = a^{-1}\left((b^{-1} - a) + a\right)(b^{-1} - a)^{-1} = a^{-1} b^{-1} (b^{-1} - a)^{-1}$.
Its inverse is $(b^{-1} - a) b a = a - aba$, and subtracting it from $a$ gives $aba$.
For example, in $\Q$ with $a = 2$ and $b = 3$: $1/2 + (1/3 - 2)^{-1} = 1/2 - 3/5 = -1/10$, and $2 - (-10) = 12 = 2 \cdot 3 \cdot 2$.
The hypothesis $ab \ne 1$ is needed, since for $b = a^{-1}$ the term $(b^{-1} - a)^{-1}$ is not defined.

## Explanations

### L1 human

The identity writes the product $aba$ using only addition, subtraction and inverses.
So a map that respects sums, inverses and $1$ automatically respects these "sandwich" products.

### L2 human

Hua used the identity to show that a map $\sigma$ between division rings that is additive, sends $1$ to $1$ and satisfies $\sigma(x^{-1}) = \sigma(x)^{-1}$ preserves $aba$.
From this one deduces Hua's theorem: such a map is a ring homomorphism or an antihomomorphism.
The identity also underlies the use of the product $aba$ in Jordan algebras, where $aba$ plays the role of the quadratic operator.
</document>

<document source="https://tamnd.github.io/math2/node/AWSBZWWMG9.md" tag="AWSBZWWMG9" kind="concept">
# Endomorphism ring

## Summary

The endomorphism ring of a module is the set of all module homomorphisms from the module to itself, with pointwise addition and composition as multiplication.
It is the main source of noncommutative rings: every ring is a subring of the endomorphism ring of its own additive group.
Properties of a module are often read off its endomorphism ring.
For example, a nonzero module is indecomposable exactly when its endomorphism ring has no idempotents other than $0$ and $1$.
For a vector space it is the [[algebra of endomorphisms|0E3X2G1ZQB]], and for $K^n$ it is the [[matrix ring|0FHVPRAE60]] $M_n(K)$.

## Explanations

### L1 human

The structure-preserving maps from an object to itself can be added and composed.
Composition plays the role of multiplication, and the identity map plays the role of $1$.
For the integers as an abelian group, every such map is multiplication by a fixed integer, so the endomorphism ring is $\Z$ again.
For the plane $\R^2$ as a real vector space, the endomorphisms are the linear maps, and they form the ring of $2 \times 2$ real matrices.
</document>

<document source="https://tamnd.github.io/math2/node/AWWM9RWETX.md" tag="AWWM9RWETX" kind="concept">
# Supersolvable group

## Summary

A supersolvable group is a group with a finite chain of normal subgroups, all normal in the whole group, whose successive quotients are cyclic.
The condition is stronger than solvability and weaker than nilpotency for finite groups.
Supersolvable groups share many good properties of nilpotent groups, and they appear naturally in the study of finite solvable groups and their representations.

## Explanations

### L1 human

A solvable group can be built from abelian layers.
A supersolvable group can be built from cyclic layers, and each partial stack of layers is normal in the whole group.
The symmetric group $S_3$ is supersolvable: it has the normal subgroup $A_3$, which is cyclic of order $3$, and the quotient $S_3/A_3$ is cyclic of order $2$.
The alternating group $A_4$ is solvable but not supersolvable.
</document>

<document source="https://tamnd.github.io/math2/node/AWZHWFBN53.md" tag="AWZHWFBN53" kind="definition">
# Tensor field

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, let $T_pM$ be the tangent space at $p \in M$, and let $T_p^*M$ be its dual space.
Let $k, l \ge 0$ be integers.
A *tensor of type $(k, l)$ at $p$* is an element of the [[tensor product|0BF2DE2SYK]]
$$T^k_l(T_pM) = \underbrace{T_pM \otimes \cdots \otimes T_pM}_{k} \otimes \underbrace{T_p^*M \otimes \cdots \otimes T_p^*M}_{l}.$$
These spaces form a smooth vector bundle $T^k_l M \to M$, built from the [[tangent bundle|WY6K3P4YCG]] as $TM$ is.
A *tensor field of type $(k, l)$* on $M$ is a smooth [[section|WWX7H5W1RP]] of $T^k_l M$.
In a chart with coordinates $x^1, \dots, x^n$, a tensor field $T$ is written as follows, with summation over repeated indices.
$$T = T^{i_1 \cdots i_k}_{j_1 \cdots j_l} \, \partial_{i_1} \otimes \cdots \otimes \partial_{i_k} \otimes dx^{j_1} \otimes \cdots \otimes dx^{j_l}.$$
Smoothness means that the component functions $T^{i_1 \cdots i_k}_{j_1 \cdots j_l}$ are smooth in every chart.

## Remarks

Example: on $\R^2$, $g = dx \otimes dx + dy \otimes dy$ is a tensor field of type $(0, 2)$; its components are the constants $g_{11} = g_{22} = 1$ and $g_{12} = g_{21} = 0$, which are smooth.
A tensor field of type $(1, 0)$ is a vector field, and a tensor field of type $(0, 0)$ is a smooth function.
Non-example: the Christoffel symbols of a connection are not the components of a tensor field.
Under a change of chart they pick up a term with second derivatives of the coordinate change.
A tensor field of type $(0, l)$ is the same as a map $(X_1, \dots, X_l) \mapsto T(X_1, \dots, X_l)$ from $l$ vector fields to smooth functions that is $C^\infty(M)$-linear in each argument.
This criterion is the standard way to check that an expression is tensorial.
</document>

<document source="https://tamnd.github.io/math2/node/AX46SNQCAA.md" tag="AX46SNQCAA" kind="definition">
# Kuranishi family

## Definition

Let $X$ be a compact [[complex manifold|JSGWAN2RJV]].
A deformation of $X$ over a pointed [[complex analytic space|HXSZA44J5J]] $(S, s_0)$ consists of two pieces of data.
The first is a proper flat holomorphic map $\varpi \colon \mathcal{X} \to S$ whose fibres are compact complex manifolds.
The second is a biholomorphism $X \cong \varpi^{-1}(s_0)$.
This extends the notion of [[deformation|VKQSM3BCE4]] over manifolds.
A *Kuranishi family* (semi-universal deformation) of $X$ is a deformation $\varpi \colon \mathcal{X} \to (S, s_0)$ of $X$ such that:

1. It is complete: let $\mathcal{Y} \to (T, t_0)$ be any deformation of $X$.
   Then there are a neighbourhood $T'$ of $t_0$ and a holomorphic map $f \colon (T', t_0) \to (S, s_0)$ such that $\mathcal{Y}|_{T'} \cong f^* \mathcal{X}$.
   The isomorphism is required to be compatible with the identifications of the central fibres with $X$.
2. The differential $df_{t_0} \colon T_{t_0} T \to T_{s_0} S$ of any such $f$ is uniquely determined by the deformation $\mathcal{Y}$.

## Remarks

Here $T_{s_0} S$ is the [[Zariski tangent space|E7A5DHFT01]] of $S$ at $s_0$.
Kuranishi's theorem states that every compact complex manifold has a Kuranishi family, unique up to a non-unique isomorphism of germs.
Its [[Kodaira–Spencer map|EWZ1G10MAY]] identifies $T_{s_0} S$ with $H^1(X, \Theta_X)$, and the germ $(S, s_0)$ is the zero set of a holomorphic map from a neighbourhood of $0$ in $H^1(X, \Theta_X)$ to $H^2(X, \Theta_X)$.
If $H^0(X, \Theta_X) = 0$, the map $f$ in clause 1 is itself unique, and the family is universal.

Example: $X = \mathbb{P}^n$ has $H^1(X, \Theta_X) = 0$, so its Kuranishi family lives over a single point: $\mathbb{P}^n$ is rigid.

Non-example: for an elliptic curve $E$, the trivial family $E \times \Delta \to \Delta$ over a disc is a deformation of $E$ but not complete.
Every pullback of it is again trivial, while the family $\C/(\Z + \tau\Z)$ near $\tau_0$ contains fibres not biholomorphic to $E$, so it is not such a pullback.
</document>

<document source="https://tamnd.github.io/math2/node/AX4ZM9BV3Z.md" tag="AX4ZM9BV3Z" kind="statement">
# Long exact sequence of a pair

## Statement

Let $X$ be a topological space and let $A \subseteq X$ be a subspace.
Let $i \colon A \to X$ be the inclusion and $j \colon (X, \emptyset) \to (X, A)$ the identity map seen as a map of pairs.
For $n \ge 1$ let $\partial \colon H_n(X, A) \to H_{n-1}(A)$ send the class of a relative cycle $c$ to the class of the cycle $\partial c$ in $A$.
Then the sequence of [[singular homology|M0Q5CFJ7ZH]] and [[relative homology|4BVWH0BN28]] groups
$$\cdots \to H_n(A) \xrightarrow{i_*} H_n(X) \xrightarrow{j_*} H_n(X, A) \xrightarrow{\partial} H_{n-1}(A) \xrightarrow{i_*} \cdots \to H_0(X) \xrightarrow{j_*} H_0(X, A) \to 0$$
is [[exact|JB7ZBFBA9C]].
It is natural with respect to maps of pairs.

## Remarks

The chain complexes form a [[short exact sequence|KJKNJMGKZZ]] $0 \to C_\bullet(A) \to C_\bullet(X) \to C_\bullet(X, A) \to 0$.
The [[long exact sequence in homology|AHKJ361FDB]] of a short exact sequence of chain complexes gives the result, and its connecting map is the $\partial$ above.
If $A$ is nonempty, the same sequence holds with [[reduced homology|NJQMWVK3QK]] $\tilde H_n(A)$ and $\tilde H_n(X)$ in place of $H_n(A)$ and $H_n(X)$.

Example: for $(D^n, S^{n-1})$ with $n \ge 1$, the disc has $\tilde H_k(D^n) = 0$ for all $k$.
Exactness of the reduced sequence then forces $\partial \colon H_k(D^n, S^{n-1}) \to \tilde H_{k-1}(S^{n-1})$ to be an isomorphism for every $k$.

## Explanations

### L1 human

The homology of a space, of a subspace, and of the space relative to the subspace are tied together in one long chain of maps.
At each spot, what one map kills is exactly what the previous map produces.
So knowing two of the three kinds of groups often determines the third.

### L2 human

This is the exactness axiom of the Eilenberg–Steenrod axioms.
Combined with excision, it computes the homology of spheres by induction on dimension, and it gives the long exact sequences behind cellular homology.
The same algebra gives a long exact sequence for a triple $B \subseteq A \subseteq X$ and, dually, for singular cohomology with the arrows reversed.
</document>

<document source="https://tamnd.github.io/math2/node/AX9Q86GB8M.md" tag="AX9Q86GB8M" kind="concept">
# Ultraproduct

## Summary

An ultraproduct combines a family of structures of the same language into one new structure.
An [[ultrafilter|J64Y3JC0RE]] on the index set decides which properties hold in it.
A property holds in the ultraproduct when it holds in a "large" set of factors, where "large" means "in the ultrafilter".
By [[Łoś's theorem|AGYBSTD7BG]], this rule works for every first-order formula.
Ultraproducts give a short proof of the [[compactness theorem|1NR3HAWG9D]] and build nonstandard models such as the [[hyperreal numbers|421QMTFG9T]].

## Explanations

### L1 human

Take infinitely many copies of a structure, say the real numbers, one for each natural number.
An element of the new structure is a sequence, one entry from each copy.
Two sequences count as equal when they agree on a large set of indices, and an ultrafilter says which sets are large.
For example, with a non-principal ultrafilter on $\N$, the sequence $(1, 2, 3, \dots)$ becomes a number larger than every ordinary real number.
This is how infinitely large and infinitely small numbers can be built rigorously.
</document>

<document source="https://tamnd.github.io/math2/node/AXCH7Z9FKH.md" tag="AXCH7Z9FKH" kind="conjecture">
# 1/3–2/3 conjecture

## Statement

Let $(P, \le)$ be a finite [[partially ordered set|0QV2JS9RJ3]] that is not a [[total order|0FRXBVVR03]].
For $x, y \in P$, let $p(x, y)$ be the fraction of the [[linear extensions|W8DZ5KGGHN]] of $P$ in which $x$ comes before $y$.
Then there are elements $x, y \in P$ with
$$\frac{1}{3} \le p(x, y) \le \frac{2}{3}.$$

## Remarks

The conjecture is open.
The bounds cannot be improved: for $P = \set{a, b, c}$ with $a < b$ as the only relation, the linear extensions are $cab$, $acb$ and $abc$, so $p(a, c) = 2/3$, $p(c, b) = 2/3$ and $p(a, b) = 1$.
Kahn and Saks proved that a pair with $3/11 \le p(x, y) \le 8/11$ always exists.
Brightwell, Felsner and Trotter improved the constant $3/11$ to $(5 - \sqrt{5})/10 \approx 0.276$.
Linial proved the conjecture for posets of width $2$.
The conjecture was later stated again, independently, by Michael Fredman and by Nathan Linial.

## Explanations

### L1 human

Suppose that some comparisons between items are already known, and the goal is to sort them completely.
The conjecture says that one can always find a comparison whose answer is not too predictable: each answer occurs in at least one third of the remaining possible orders.
Such a comparison cuts the number of possible orders by a fixed factor, which is why the question matters for sorting with partial information.
</document>

<document source="https://tamnd.github.io/math2/node/AXCJBSQZWF.md" tag="AXCJBSQZWF" kind="concept">
# Modulus of continuity

## Summary

A modulus of continuity of a function $f$ is a function $\omega$ that bounds how much $f$ can change: $\abs{f(x) - f(y)} \le \omega(\abs{x - y})$.
The best such bound is the function $\delta \mapsto \sup \set{\abs{f(x) - f(y)} : \abs{x - y} \le \delta}$.
A function is [[uniformly continuous|0Q4DR37DVE]] exactly when this bound tends to $0$ with $\delta$.
Moduli of continuity measure the quality of continuity, and [[Lipschitz|0QNVQ4VD9T]] and [[Hölder|0WZTM0EFR6]] conditions are moduli of special forms.

## Explanations

### L1 human

Continuity says that small changes of the input give small changes of the output.
A modulus of continuity says how small, in numbers.
If two inputs are at distance $\delta$, the outputs differ by at most $\omega(\delta)$.
For $f(x) = 3x$ one can take $\omega(\delta) = 3\delta$, and for $f(x) = \sqrt x$ on $[0, \infty)$ one can take $\omega(\delta) = \sqrt \delta$.
</document>

<document source="https://tamnd.github.io/math2/node/AXDKKVD18N.md" tag="AXDKKVD18N" kind="concept">
# Stereographic projection

## Summary

Stereographic projection maps the unit sphere minus one point, its north pole, bijectively onto a plane, see the [[definition|RN8RBM2A8A]].
Sending the north pole to a point at infinity makes the sphere a model of the [[Riemann sphere|0J78KBPFZB]] $\C \cup \set{\infty}$.
The map preserves angles and sends circles on the sphere to circles or lines in the plane.
It is the bridge between the geometry of the sphere and complex analysis on the extended plane.

## Explanations

### L1 human

Put a ball on a flat floor so that it touches the floor at its south pole.
Shine a light from the north pole: each other point of the ball casts a shadow on the floor.
The map from a point to its shadow is the stereographic projection.
The south pole goes to itself, the equator goes to a circle, and points near the north pole go far away.
So the whole floor, together with one extra point at infinity, is a copy of the sphere.
</document>

<document source="https://tamnd.github.io/math2/node/AXND5JRQAZ.md" tag="AXND5JRQAZ" kind="definition">
# Stiff equation

## Definition

Stiffness is not a single precise condition; the definition below is the standard working one.
Let $I$ be an interval and consider an [[ordinary differential equation|0P97V20RW7]] $y' = f(t, y)$ with $f$ continuously differentiable, on $I$.
The equation is *stiff* on $I$ if explicit methods such as the [[Euler method|0JARTXX4XP]] need much smaller steps than accuracy requires.
The reason is that larger steps make the numerical solution unstable.
Consider a linear system $y' = A y + g(t)$ with a constant $n \times n$ matrix $A$.
Assume its [[eigenvalues|0BQAA2BK3Q]] $\lambda_1, \dots, \lambda_n$ all have $\operatorname{Re} \lambda_i < 0$.
A common measure is then the stiffness ratio
$$\frac{\max_i \abs{\operatorname{Re} \lambda_i}}{\min_i \abs{\operatorname{Re} \lambda_i}},$$
and the system is called stiff when this ratio is large.

## Remarks

Example: $y' = -1000\,(y - \cos t)$.
On the test equation $y' = \lambda y$, the Euler method gives $y_{k+1} = (1 + h\lambda)\, y_k$, which stays bounded only if $\abs{1 + h\lambda} \le 1$.
For $\lambda = -1000$ this forces $h \le 0.002$, although after a short time every solution is close to the smooth particular solution $\frac{10^6}{10^6 + 1} \cos t + \frac{10^3}{10^6 + 1} \sin t$, which is nearly $\cos t$.
The implicit Euler method $y_{k+1} = (1 - h\lambda)^{-1} y_k$ is stable for all $h > 0$ when $\lambda < 0$.
Non-example: $y' = -y$ is not stiff, since its stability bound $h \le 2$ is far larger than the steps needed for accuracy.
For nonlinear equations one uses the eigenvalues of the [[Jacobian matrix|047NDTEPCV]] $\partial f / \partial y$ along the solution.
</document>

<document source="https://tamnd.github.io/math2/node/AXS7YYF1YY.md" tag="AXS7YYF1YY" kind="definition">
# Polygon

## Definition

Let $n \ge 3$ be an integer.
A *polygon* with $n$ vertices in the [[Euclidean plane|0XW7V9ZZ05]] is a cyclic sequence of points $P_1, \ldots, P_n$, with indices taken modulo $n$, such that:

1. $P_i \ne P_{i+1}$ for every $i$.
2. $P_{i-1}$, $P_i$, $P_{i+1}$ are not [[collinear|0AFNNTRTJE]] for every $i$.

The points $P_i$ are its *vertices* and the segments $P_i P_{i+1}$ are its *edges*.
The polygon is *simple* if two edges meet only when they are consecutive, and then only in their common vertex.

## Remarks

By the [[Jordan curve theorem|687RGCHBTP]], a simple polygon bounds one bounded region of the plane, its interior.
Many authors use the word polygon for this closed region rather than for the boundary.
A polygon with $3$ vertices is a [[triangle|0CTNFGYMM2]], and one with $4$ vertices is a quadrilateral.
A simple polygon is convex if its interior is a [[convex set|02AS45T3H9]].

Example: $(0, 0), (1, 0), (1, 1), (0, 1)$ is a simple polygon, the unit square.
Consecutive vertices differ, and no three consecutive ones lie on a line.

Non-example: $(0, 0), (1, 0), (2, 0), (1, 1)$ fails clause 2.
The three consecutive vertices $(0, 0), (1, 0), (2, 0)$ are collinear, so the figure is a triangle with an extra point on one edge.
</document>

<document source="https://tamnd.github.io/math2/node/AXT498MW87.md" tag="AXT498MW87" kind="definition">
# Hopf invariant

## Definition

Let $n \ge 2$, and let $f \colon S^{2n-1} \to S^n$ be a continuous map.
Let $C_f = S^n \cup_f D^{2n}$ be the [[CW complex|5W95QFHXPR]] obtained by attaching a $2n$-cell to $S^n$ along $f$.
Its [[singular cohomology|6N7BC70CX0]] with integer coefficients is $H^k(C_f; \Z) \cong \Z$ for $k = 0, n, 2n$ and is zero otherwise.
Let $\alpha \in H^n(C_f; \Z)$ and $\beta \in H^{2n}(C_f; \Z)$ be the generators given by fixed orientations of $S^n$ and $D^{2n}$.
The *Hopf invariant* of $f$ is the integer $H(f)$ such that
$$\alpha \smile \alpha = H(f) \, \beta,$$
where $\smile$ is the [[cup product|JX47MS1R1H]].

## Remarks

The integer $H(f)$ depends only on the homotopy class of $f$, and $H$ is a homomorphism $\pi_{2n-1}(S^n) \to \Z$ on [[homotopy groups|P976WYEFMB]].
A change of the generator $\beta$ changes only the sign of $H(f)$.

Example: for the Hopf map $h \colon S^3 \to S^2$, the space $C_h$ is $\C P^2$.
The square of a generator of $H^2(\C P^2; \Z)$ generates $H^4(\C P^2; \Z)$, so $H(h) = \pm 1$.

Non-example: for $n = 1$ the construction breaks down.
A map $S^1 \to S^1$ of degree $d$ gives $H^2(C_f; \Z) \cong \Z/d$, which is not $\Z$ when $d \ne 0$, so no generator $\beta$ as above exists.

If $n$ is odd, then $\alpha \smile \alpha = -\alpha \smile \alpha$ by graded commutativity, and since $H^{2n}(C_f; \Z) \cong \Z$ has no torsion, $H(f) = 0$.
If $n$ is even, the Whitehead product $[\iota_n, \iota_n]$ of the identity of $S^n$ with itself has Hopf invariant $\pm 2$.
</document>

<document source="https://tamnd.github.io/math2/node/AXV2PH7FEZ.md" tag="AXV2PH7FEZ" kind="concept">
# Fisher's equation

## Summary

Fisher's equation, also called the KPP–Fisher equation, is the scalar [[reaction–diffusion equation|TAAHWE917M]] $u_t = D u_{xx} + r u(1 - u)$.
It couples diffusion with logistic growth.
It was proposed by Fisher in 1937 for the spread of an advantageous gene, and studied in the same year by Kolmogorov, Petrovsky and Piskunov.
It is the model example of an equation with [[travelling wave|M37M411KP7]] fronts that invade an unstable state.

## Explanations

### L1 human

Think of a population that spreads by random motion and grows until it reaches the carrying capacity $1$.
Where $u$ is near $0$ the population grows, and diffusion pushes it into empty space.
The result is a front that moves at constant speed, with the full state $u = 1$ behind it and the empty state $u = 0$ ahead.
With $D = r = 1$, the slowest such front has speed $2$.
</document>

<document source="https://tamnd.github.io/math2/node/AY0B20QC04.md" tag="AY0B20QC04" kind="statement">
# Schur orthogonality relations

## Statement

Let $G$ be a finite [[group|G5S1M9]].
For functions $\alpha, \beta \colon G \to \C$ write
$$\ip{\alpha}{\beta} = \frac{1}{\abs{G}} \sum_{g \in G} \alpha(g) \overline{\beta(g)},$$
where the bar is the [[complex conjugate|0FTH2E9TAS]].

1. Let $\rho$ and $\sigma$ be [[irreducible|FX3NEFYFFN]] complex representations of $G$ by [[unitary matrices|0AR68JTMDT]], of degrees $d_\rho$ and $d_\sigma$. If $\rho$ and $\sigma$ are not isomorphic, then $\ip{\rho_{ij}}{\sigma_{kl}} = 0$ for all indices. For all indices $i, j, k, l$ one has $\ip{\rho_{ij}}{\rho_{kl}} = \delta_{ik}\delta_{jl} / d_\rho$.
2. Let $\chi$ and $\psi$ be irreducible complex [[characters|RMWMYBY8TC]] of $G$. Then $\ip{\chi}{\psi} = 1$ if $\chi = \psi$ and $\ip{\chi}{\psi} = 0$ otherwise.
3. Let $g, h \in G$, and let $C_G(g)$ be the [[centraliser|02YYV36BFM]] of $g$. Then $\sum_{\chi} \chi(g) \overline{\chi(h)}$, summed over all irreducible complex characters $\chi$ of $G$, equals $\abs{C_G(g)}$ if $g$ and $h$ are [[conjugate|0DEBXAJ5SA]], and $0$ otherwise.

## Remarks

The proof of the first relation applies [[Schur's lemma|X4M2FFKQJ6]] to the averaged map $\sum_{g} \rho(g) A \sigma(g)^{-1}$, which commutes with the action for every matrix $A$.
Taking traces in the first relation gives the second.
The second relation shows that the irreducible characters are orthonormal in the space of class functions; they span this space, and the third relation follows from that.
For example, for $G = \Z/2\Z$ the two irreducible characters are $(1, 1)$ and $(1, -1)$, and $\ip{\chi}{\psi} = \tfrac{1}{2}(1 - 1) = 0$.
A consequence: a complex representation with character $\chi$ is irreducible if and only if $\ip{\chi}{\chi} = 1$.
Every complex representation of a finite group is isomorphic to a unitary one, so the first relation applies to all irreducible representations after a change of basis.

## Explanations

### L1 human

The irreducible characters of a finite group behave like perpendicular unit vectors.
If one averages the product of two different irreducible characters over the group, the result is zero.
This makes it easy to decompose any representation: the number of times an irreducible piece occurs is an inner product, like a coordinate along an axis.

### L2 human

The relations are the main computational tool of [[character tables|P16AFZT3K0]].
Row orthogonality (relation 2) gives the multiplicity $\ip{\chi_V}{\chi}$ of an irreducible $\chi$ in a representation $V$.
Column orthogonality (relation 3) with $g = h = 1$ gives $\sum_\chi \chi(1)^2 = \abs{G}$.
Together they show that the number of irreducible characters equals the number of conjugacy classes, and they let one fill in missing entries of a character table.
They are the finite group case of the Peter–Weyl theory for compact groups, where the average becomes an integral against Haar measure.
</document>

<document source="https://tamnd.github.io/math2/node/AY60KQR8RT.md" tag="AY60KQR8RT" kind="proof">
# Proof that the kernel is a normal subgroup

## Proof

1. Let $\varphi : G \to H$ be a group homomorphism, and let $K = \ker \varphi$ be its [[kernel|0Z424HK8CQ]].
2. [[$\varphi(e) = e$ and $\varphi(b^{-1}) = \varphi(b)^{-1}$|08FKQD7NJ3]]. Thus $e \in K$, and $K$ is not empty.
3. Let $a, b \in K$. Then $\varphi(ab^{-1}) = \varphi(a)\varphi(b)^{-1} = e e^{-1} = e$, so $ab^{-1} \in K$. Thus $K$ is a subgroup of $G$.
4. Let $k \in K$ and $g \in G$. Then $\varphi(gkg^{-1}) = \varphi(g)\, e\, \varphi(g)^{-1} = e$.
5. Thus $gKg^{-1} \subseteq K$ for all $g \in G$, and $K$ is a [[normal subgroup|0W77CVBCHQ]] of $G$.

## Idea

A homomorphism sends the identity to the identity and respects inverses. Thus the elements sent to the identity are closed under the group operations and under conjugation.
</document>

<document source="https://tamnd.github.io/math2/node/AYCCDQWCJF.md" tag="AYCCDQWCJF" kind="concept">
# Klein–Gordon equation

## Summary

The Klein–Gordon equation $u_{tt} - \Delta u + m^2 u = 0$ is the [[wave equation|5EHD2XBQDG]] with a mass term.
It is the relativistic wave equation of a free scalar particle of mass $m$, written in units with $c = \hbar = 1$.
It is hyperbolic, so signals travel no faster than light, and for $m > 0$ it is also dispersive: waves of different frequencies travel at different speeds.
Its nonlinear versions, such as $u_{tt} - \Delta u + m^2 u = \pm \abs{u}^{p-1}u$, are model problems in nonlinear wave theory.

## Explanations

### L1 human

Take a string that vibrates and is also held to its rest position by springs along its length.
The springs add the term $m^2 u$, which pulls the string back toward zero.
A plane wave $e^{i(kx - \omega t)}$ solves the equation exactly when $\omega^2 = k^2 + m^2$.
So short waves move almost at speed $1$, while long waves move more slowly, and a bump spreads out as it travels.
</document>

<document source="https://tamnd.github.io/math2/node/AYJ1CM1XT8.md" tag="AYJ1CM1XT8" kind="object">
# Fischer groups

## Description

The Fischer groups $\mathrm{Fi}_{22}$, $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{24}'$ are three [[sporadic groups|PZ49Z4CBFZ]], found by Bernd Fischer.
Their [[orders|0B85S2V2DR]] are:

1. $\abs{\mathrm{Fi}_{22}} = 64561751654400 = 2^{17} \cdot 3^9 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13$;
2. $\abs{\mathrm{Fi}_{23}} = 4089470473293004800 = 2^{18} \cdot 3^{13} \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 23$;
3. $\abs{\mathrm{Fi}_{24}'} = 1255205709190661721292800 = 2^{21} \cdot 3^{16} \cdot 5^2 \cdot 7^3 \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 29$.

Each of them is a finite [[simple group|0YMYMNQ0QZ]].

## Remarks

Fischer found these groups while classifying 3-transposition groups.
A 3-transposition group is a group generated by a [[conjugacy class|0DEBXAJ5SA]] $D$ of involutions such that the product of any two elements of $D$ has order $1$, $2$ or $3$.
The symmetric groups, with $D$ the class of transpositions, are the basic examples.
The group $\mathrm{Fi}_{24}$ generated by Fischer's class of 3-transpositions is not simple; its [[commutator subgroup|0ASGSTXSBW]] $\mathrm{Fi}_{24}'$ has index $2$ and is simple.
All three Fischer groups are subquotients of the [[Monster group|C9VY20TAPT]].

## Explanations

### L1 human

In a symmetric group, two transpositions multiply to an element of order $1$, $2$ or $3$.
Fischer asked which groups are generated by a class of elements of order $2$ with this property.
Besides symmetric groups and some matrix groups, the answer contains three new simple groups, the Fischer groups.
</document>

<document source="https://tamnd.github.io/math2/node/AYMJ2NFR4W.md" tag="AYMJ2NFR4W" kind="concept">
# Dispersive partial differential equation

## Summary

A dispersive partial differential equation is an evolution equation whose plane waves travel at speeds that depend on their frequency.
A localized solution, made of many frequencies, therefore spreads out in space and decays in amplitude while its energy is conserved.
The main examples are the [[Schrödinger equation|EPZR6H2WJN]], the [[Korteweg–De Vries equation|KTT75QC72Z]] and the [[Klein–Gordon equation|AYCCDQWCJF]].
Dispersion competes with nonlinear effects, and the balance between the two produces phenomena such as [[solitons|63D4AXYCT4]].

## Explanations

### L1 human

Throw a stone into a pond: the ring of waves spreads into a train of ripples, because long and short ripples travel at different speeds.
This separation by frequency is called dispersion.
In a dispersive equation, a bump does not keep its shape; it spreads out and gets lower.
For the Schrödinger equation $i u_t + u_{xx} = 0$, a plane wave $e^{i(kx - k^2 t)}$ has frequency $k^2$, and a packet of such waves near $k$ moves with speed $2k$, so fast and slow parts separate.
</document>

<document source="https://tamnd.github.io/math2/node/AYQ3VQ4B0V.md" tag="AYQ3VQ4B0V" kind="definition">
# Essentially surjective functor

## Definition

Let $F \colon \mathcal{C} \to \mathcal{D}$ be a [[functor|28NQ0KRHD3]].
The functor $F$ is *essentially surjective* if, for every object $D$ of $\mathcal{D}$, there is an object $A$ of $\mathcal{C}$ with $F(A) \cong D$.
Here $\cong$ means that there is an [[isomorphism|859F0TFVA3]] in $\mathcal{D}$.

## Remarks

Example: let $K$ be a field and let $\mathcal{C}$ be the [[full subcategory|V3BFGKBJF7]] of finite-dimensional [[vector spaces|000AYPA9AJ]] over $K$ spanned by the spaces $K^n$, $n \ge 0$.
Its inclusion into all finite-dimensional vector spaces over $K$ is essentially surjective, since a space of dimension $n$ is isomorphic to $K^n$.
Non-example: the inclusion of the full subcategory of finite sets into the category of sets is not essentially surjective.
An infinite set is not isomorphic, that is, not in bijection with, any finite set.
Every functor that is surjective on objects is essentially surjective.
</document>

<document source="https://tamnd.github.io/math2/node/AYRZ98Y0NW.md" tag="AYRZ98Y0NW" kind="statement">
# Cut-elimination theorem

## Statement

Let $\Gamma \vdash \Delta$ be a sequent of first-order logic.
If $\Gamma \vdash \Delta$ has a derivation in the [[sequent calculus|PZX9J3VEMT]] LK, then it has a derivation in LK that does not use the cut rule.
The same holds for the intuitionistic calculus LJ.

## Remarks

Gentzen proved the theorem in 1934, and called it the *Hauptsatz*.
The proof is a double induction.
The outer induction is on the complexity of the cut formula, and the inner one is on the heights of the derivations above the cut.
Each step pushes a topmost cut upward or replaces it by cuts on smaller formulas.
The cost is large: removing cuts can make a derivation super-exponentially longer.

In a cut-free derivation, every formula is a subformula of a formula in the final sequent, up to the substitution of terms for variables.
This *subformula property* gives quick consequences.
For example, the empty sequent $\vdash$ has no cut-free derivation, since no rule other than cut can produce it.
So LK is consistent.

## Explanations

### L1 human

The cut rule lets a proof use a lemma: prove $A$, then use $A$ to prove the goal.
The theorem says that lemmas are never needed in pure logic.
Every proof can be rewritten so that it only builds up the formulas of the conclusion, step by step.
The price is that the new proof can be enormously longer.

### L2 human

Cut elimination is the central result of structural [[proof theory|N4EDJNMTEF]].
It gives the subformula property, the consistency of pure logic, and Craig interpolation and Herbrand's theorem by short arguments.
For arithmetic, cuts cannot be fully removed in the presence of induction.
Gentzen's consistency proof for Peano arithmetic instead removes cuts partly, with a transfinite induction up to the ordinal $\varepsilon_0$ to show that the process ends.
Through the Curry–Howard correspondence, the removal of cuts is related to the evaluation of programs.
</document>

<document source="https://tamnd.github.io/math2/node/AYTEC1KET8.md" tag="AYTEC1KET8" kind="definition">
# Logistic regression

## Definition

Let $x_1, \dots, x_n \in \R^p$ be known regressor vectors.
The *logistic regression model* for responses $y_1, \dots, y_n \in \set{0, 1}$ states three things.
The $y_i$ are independent, $y_i$ has the [[Bernoulli distribution|0386Y0Z7Y7]] with parameter $p_i$, and
$$\log \frac{p_i}{1 - p_i} = x_i^T \beta$$
for an unknown $\beta \in \R^p$.
Equivalently, $p_i = \sigma(x_i^T \beta)$, where $\sigma(t) = 1/(1 + e^{-t})$ is the logistic function.

## Remarks

It is the [[generalized linear model|K6C48KMRKC]] with Bernoulli responses and the logit link $g(\mu) = \log(\mu / (1 - \mu))$, which is the canonical link.
The log-likelihood is $\ell(\beta) = \sum_i \big( y_i x_i^T \beta - \log(1 + e^{x_i^T \beta}) \big)$, a concave function.
The [[maximum likelihood|0FREPFKNS7]] estimate solves $\sum_i (y_i - p_i) x_i = 0$ and has no closed formula in general.
For a regressor $x_{ij}$, the number $e^{\beta_j}$ is the odds ratio for a unit increase of $x_{ij}$ with the other regressors fixed.

Example: with $x_i = (1, t_i)$ and $\beta = (-2, 0.5)$, at $t = 4$ the log-odds is $0$, so $p = \sigma(0) = 1/2$; at $t = 6$ it is $1$, so $p = e/(1 + e) \approx 0.73$.

Non-example: suppose the data are perfectly separated, that is, some $\gamma \in \R^p$ has $x_i^T \gamma > 0$ whenever $y_i = 1$ and $x_i^T \gamma < 0$ whenever $y_i = 0$.
Then $\ell(c\gamma)$ increases to $0$ as $c \to \infty$, while $\ell < 0$ everywhere, so the maximum likelihood estimate does not exist.
The model is still defined, but this fitting method fails.
</document>

<document source="https://tamnd.github.io/math2/node/AYVTR2ES5B.md" tag="AYVTR2ES5B" kind="concept">
# Iterated function

## Summary

An iterated function is a map of a set to itself composed with itself a given number of times.
The $n$-th iterate $f^n$ applies $f$ exactly $n$ times.
Iterates are the basic objects of discrete-time dynamics: the state after $n$ steps is $f^n$ of the initial state.
They also appear in [[fixed-point iteration|K6ZF91051B]], fractals and the study of functional equations.

## Explanations

### L1 human

Apply a function, then apply it again to the result, and keep going.
The $n$-th iterate is what one gets after $n$ rounds.
For $f(x) = x + 3$, the iterate $f^n$ is $x \mapsto x + 3n$.
For $f(x) = 2x$, the iterate $f^n$ is $x \mapsto 2^n x$.
Pressing the cosine key of a calculator many times, in radian mode, computes iterates of $\cos$, and the display settles near $0.739$.
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<document source="https://tamnd.github.io/math2/node/AYZCKG34QK.md" tag="AYZCKG34QK" kind="concept">
# Linear integral equation

## Summary

A linear integral equation is an [[integral equation|R3SG1SN7JQ]] in which the unknown function enters linearly, both outside and inside the integral.
Its general form is $a(x) \varphi(x) = f(x) + \lambda \int K(x, s) \varphi(s) \, ds$ with a given [[kernel|GSMBXT7FPK]] $K$.
The solutions form an affine space, so linear algebra and operator theory apply.
Fredholm and Volterra equations are the main classes.

## Explanations

### L1 human

Linear means that sums and multiples of solutions of the equation with $f = 0$ are again solutions.
The integral acts on the unknown like a matrix acts on a vector.
So a linear integral equation is a continuous version of a system of linear equations.
For example, $\varphi(x) = x + \int_0^1 x s \, \varphi(s) \, ds$ is linear, while an equation with $\varphi(s)^2$ under the integral is not.
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<document source="https://tamnd.github.io/math2/node/AZ17JCMRKB.md" tag="AZ17JCMRKB" kind="concept">
# Painlevé transcendents

## Summary

Painlevé transcendents are solutions of the six Painlevé equations $\mathrm{P_I}$ to $\mathrm{P_{VI}}$, nonlinear second order differential equations with the [[Painlevé property|8EV1G2Y6CC]].
Consider equations $y'' = R(t, y, y')$ with $R$ rational in $y$ and $y'$ that have this property.
Up to changes of variables, the six equations are the only ones among them that are not solvable by earlier known functions.
Their general solutions are new transcendental functions.
They play for nonlinear problems the role that classical special functions play for linear ones, for instance in random matrix theory and integrable systems.

## Explanations

### L1 human

Solutions of nonlinear equations usually have moving singularities whose type depends on the initial values, such as branch points.
Painlevé and his school looked for second order equations whose only moving singularities are poles, so that solutions stay single-valued around them.
Besides equations solvable by known functions, they found exactly six new families.
The simplest is $\mathrm{P_I}$: $y'' = 6y^2 + t$.
Its solutions are [[meromorphic functions|0ZZ7ZYDR8W]] on the whole plane that cannot be written in terms of classical functions.
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<document source="https://tamnd.github.io/math2/node/AZ6H4ZKK55.md" tag="AZ6H4ZKK55" kind="statement">
# Carathéodory's theorem on boundary extension

## Statement

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].
Let $J \subset \C$ be a Jordan curve, that is, a subset homeomorphic to a circle, and let $\Omega$ be the bounded component of $\C \setminus J$.
Let $f \colon \mathbb{D} \to \Omega$ be a [[conformal map|0VHNKP1N3K]] of $\mathbb{D}$ onto $\Omega$.
Then $f$ extends to a [[homeomorphism|05GZ11PWRY]] of the [[closure|0DRKT1QJR7]] $\overline{\mathbb{D}}$ onto $\overline{\Omega} = \Omega \cup J$.

## Remarks

Such an $f$ exists by the [[Riemann mapping theorem|02WHMEG5ZR]], since $\Omega$ is simply connected by the [[Jordan curve theorem|687RGCHBTP]].
The proof rests on a length–area estimate.
Let $\zeta$ be a point of the unit circle and let $r$ be small.
By the Cauchy–Schwarz inequality and the finite area of $\Omega$, some arc $\set{\abs{z - \zeta} = \rho} \cap \mathbb{D}$ with $r^2 < \rho < r$ has an image of length at most $C/\sqrt{\log(1/r)}$.
Because $J$ is a Jordan curve, a short crosscut of $\Omega$ cuts off a piece of small diameter, and this gives the continuity of $f$ at $\zeta$.
The Jordan property also gives injectivity on the circle.
The converse holds: if $f$ extends to such a homeomorphism, the boundary of $\Omega$ is the image of the circle, hence a Jordan curve.
A more general form says that $f$ extends continuously to $\overline{\mathbb{D}}$ if and only if the boundary of $\Omega$ is locally connected.
Example: the boundary of the slit disc $\mathbb{D} \setminus [0, 1)$ is not a Jordan curve.
A conformal map from $\mathbb{D}$ onto it extends continuously, but two points of the circle go to each inner point of the slit, so the extension is not injective.

## Explanations

### L1 human

The Riemann mapping theorem gives a conformal map from the disc onto any proper simply connected region, but it says nothing about the edge.
Carathéodory's theorem says that if the edge of the region is a simple closed curve, the map behaves well up to the edge.
The circle is then mapped continuously and one-to-one onto the boundary curve.

### L2 human

The theorem lets one transfer boundary value problems, such as the [[Dirichlet problem|8D56GQ4AG8]], from the disc to any Jordan domain.
It also shows that a conformal map between two Jordan domains extends to a homeomorphism of their closures, by composing two Riemann maps.
For domains with worse boundaries, the right replacement for boundary points is the notion of a [[prime end|GNWE3XR0ZG]].
Carathéodory's prime end theorem gives a bijection between the circle and the prime ends.
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<document source="https://tamnd.github.io/math2/node/AZBDB5F4M4.md" tag="AZBDB5F4M4" kind="concept">
# Difference quotient

## Summary

The difference quotient of a function between two points is the change of the value divided by the change of the input.
It is the slope of the secant line through two points of the graph.
The [[derivative|0F1WR9C83K]] is the limit of the difference quotient as the two points merge.
Difference quotients also appear on their own in the mean value theorem and in numerical approximation of derivatives.

## Explanations

### L1 human

If you drive $120$ kilometres in $2$ hours, your average speed is $60$ kilometres per hour.
That average speed is a difference quotient: change of position over change of time.
On a graph, it is the slope of the straight line joining two points of the curve.
For $f(x) = x^2$ between $x = 1$ and $x = 3$, the difference quotient is $(9 - 1)/(3 - 1) = 4$.
</document>

<document source="https://tamnd.github.io/math2/node/AZBTMF196E.md" tag="AZBTMF196E" kind="example">
# Lens space

## Example

Let $p \ge 2$ and $q$ be coprime integers, and let $\zeta = e^{2\pi i/p}$.
View the $3$-sphere as $S^3 = \set{(z_1, z_2) \in \C^2 : \abs{z_1}^2 + \abs{z_2}^2 = 1}$.
The map $g(z_1, z_2) = (\zeta z_1, \zeta^q z_2)$ generates an action of the cyclic group $\Z/p$ on $S^3$.
The lens space $L(p, q)$ is the quotient $S^3 / (\Z/p)$.

The action is free.
Indeed, suppose $g^k$ fixes $(z_1, z_2)$ with $0 < k < p$.
Then $\zeta^k \ne 1$ forces $z_1 = 0$, and $\zeta^{kq} \ne 1$ (because $q$ is prime to $p$) forces $z_2 = 0$, which is impossible on $S^3$.
So the quotient map is a [[covering map|3XZRRE165N]] and $L(p, q)$ is a closed orientable [[3-manifold|R20TB2KEQQ]].
Its [[fundamental group|S1XK3HXHFX]] is $\Z/p$, since $S^3$ is simply connected.

## Remarks

The integral homology of $L(p, q)$ is $H_0 = \Z$, $H_1 = \Z/p$, $H_2 = 0$ and $H_3 = \Z$.
For $p = 2$ one gets $L(2, 1)$, the [[real projective space|1DDYB6VVSR]] $\mathbb{RP}^3$.
A lens space is also the union of two solid tori glued along their boundary tori, so it has a [[Heegaard splitting|EDE9919B5R]] of genus $1$.
The spaces $L(p, q)$ and $L(p, q')$ are homeomorphic if and only if $q' \equiv \pm q^{\pm 1} \pmod p$.
They are homotopy equivalent if and only if $q q' \equiv \pm n^2 \pmod p$ for some integer $n$.
So $L(7, 1)$ and $L(7, 2)$ are homotopy equivalent, because $2 \equiv 3^2 \pmod 7$, but they are not homeomorphic, because $2 \not\equiv \pm 1 \pmod 7$.

## Explanations

### L1 human

A lens space is what one gets by folding the $3$-sphere onto itself by a rotation of order $p$ that moves every point.
Two lens spaces can have the same fundamental group and the same homology, and even be homotopy equivalent, while still being different manifolds.
This makes them the first test case for any new invariant of $3$-manifolds.
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<document source="https://tamnd.github.io/math2/node/AZES1C6VMT.md" tag="AZES1C6VMT" kind="definition">
# Order embedding

## Definition

Let $(P, \le_P)$ and $(Q, \le_Q)$ be [[partially ordered sets|011C29Q1S1]].
A map $f \colon P \to Q$ is an *order embedding* when, for all $x, y \in P$,
$$x \le_P y \iff f(x) \le_Q f(y).$$

## Remarks

The inclusion $\Q \to \R$ is an order embedding, since the order of $\Q$ is the restriction of the order of $\R$.
Let $P = \set{a, b}$ with $a$ and $b$ incomparable, and let $Q = \set{0, 1}$ with $0 < 1$.
The map $f(a) = 0$, $f(b) = 1$ is [[monotone|0WFF6AZPBP]], but it is not an order embedding: $f(a) \le f(b)$ while $a \not\le b$.
An order embedding is injective: $f(x) = f(y)$ gives $x \le y$ and $y \le x$, so $x = y$ by antisymmetry.
So $f$ is an [[order isomorphism|0Y74KN99A5]] from $P$ onto its image $f(P)$ with the induced order.
For total orders, a strictly increasing map is the same as an order embedding.

## Explanations

### L1 human

An order embedding keeps every comparison and creates no new ones.
It shows one ordered set as an ordered part of another.
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<document source="https://tamnd.github.io/math2/node/AZF9TXV5C0.md" tag="AZF9TXV5C0" kind="statement">
# Ebin's slice theorem

## Statement

Let $M$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary.
Let $\mathcal{M}$ be the space of smooth [[Riemannian metrics|Q61VFHCH2J]] on $M$, an open subset of the Fréchet space of smooth symmetric $2$-tensors.
Let $\mathcal{D}$ be the [[diffeomorphism group|WTCNKST134]] of $M$, acting on $\mathcal{M}$ on the right by pullback, $(\gamma, \eta) \mapsto \eta^* \gamma$.
For $\gamma \in \mathcal{M}$, let $I_\gamma \subseteq \mathcal{D}$ be its [[isometry|02HPP5VCZH]] group, which is the [[stabiliser|0RGGF8ZS6D]] of $\gamma$.
Then there is a submanifold $S_\gamma$ of $\mathcal{M}$ containing $\gamma$, the *slice* at $\gamma$, such that:

1. If $\eta \in I_\gamma$, then $\eta^* S_\gamma = S_\gamma$.
2. If $\eta \in \mathcal{D}$ and $\eta^* S_\gamma \cap S_\gamma \neq \emptyset$, then $\eta \in I_\gamma$.
3. There are a neighbourhood $U$ of the identity coset in $\mathcal{D} / I_\gamma$ and a local section $\chi \colon U \to \mathcal{D}$ of the projection $\mathcal{D} \to \mathcal{D} / I_\gamma$.
For these, the map $U \times S_\gamma \to \mathcal{M}$, $(u, s) \mapsto \chi(u)^* s$, is a homeomorphism onto a neighbourhood of $\gamma$.


## Remarks

The theorem is due to Ebin.
The tangent space to the orbit of $\gamma$ consists of the Lie derivatives $L_X \gamma$ of $\gamma$ along vector fields $X$.
Its $L^2$-orthogonal complement is the space of symmetric $2$-tensors $h$ with $\delta_\gamma h = 0$, where $\delta_\gamma$ is the divergence.
The slice is the image of a small ball in this complement under the exponential map of an $I_\gamma$-invariant metric on $\mathcal{M}$.
The analysis is carried out on Sobolev completions of $\mathcal{M}$ and $\mathcal{D}$, which are Hilbert manifolds, and then passed to the smooth case.
Compactness of $I_\gamma$, which is a compact Lie group because $M$ is compact, is used to average and to make the slice invariant.
It is the infinite-dimensional analogue of the [[slice theorem|VQB2N60ARE]] for actions of compact Lie groups on finite-dimensional manifolds.

Example: suppose that $\gamma$ has trivial isometry group.
Then near $\gamma$ the action is free, and a neighbourhood of the class of $\gamma$ in $\mathcal{M} / \mathcal{D}$ is homeomorphic to the slice $S_\gamma$.

## Explanations

### L1 human

Two Riemannian metrics that differ by a change of coordinates describe the same geometry.
So the space of geometries on $M$ is the space of metrics divided by the action of diffeomorphisms.
Ebin's theorem says that near any metric this division can be done cleanly.
There is a transversal family of metrics, the slice, that meets each nearby orbit in an orbit of the finite-dimensional isometry group.

### L2 human

The slice theorem is the basic tool for describing the moduli space of Riemannian metrics, sometimes called superspace in general relativity.
It shows that this space splits into pieces according to the isometry group of the metric.
Near a metric whose only isometry is the identity, the space is locally modelled on the slice.
The same pattern reappears for connections modulo gauge: an infinite-dimensional group acts with a slice built from an elliptic gauge condition.
There the Coulomb gauge plays the role of the divergence-free condition.
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<document source="https://tamnd.github.io/math2/node/AZK4RD8XV0.md" tag="AZK4RD8XV0" kind="concept">
# Lifting property

## Summary

A morphism $i$ has the left lifting property with respect to a morphism $p$ when every suitable commutative square has a diagonal filler.
The square has $i$ on its left side and $p$ on its right side.
Equivalently, $p$ has the right lifting property with respect to $i$.
Lifting properties define fibrations and cofibrations in homotopy theory and in model categories, and they describe many classes of maps by a single test.
When the filler is always unique, $i$ and $p$ are called orthogonal.

## Explanations

### L1 human

Picture a square of four maps: $u$ along the top, $v$ along the bottom, $i$ on the left and $p$ on the right, with the square commuting.
A lift is a diagonal map from the bottom left corner to the top right corner that keeps both triangles commuting.
In topology, the [[homotopy lifting property|G4QDM01P6J]] is of this kind: a homotopy in the base lifts to the total space once its start is lifted.
In sets, a surjection has the right lifting property with respect to the inclusion of the empty set into a point: every point of the target has a preimage.
</document>

<document source="https://tamnd.github.io/math2/node/AZPKCHND9T.md" tag="AZPKCHND9T" kind="statement">
# Smoothness and analyticity of harmonic functions

## Statement

Let $U \subseteq \R^n$ be an open set and let $u \colon U \to \R$ be [[harmonic|0DEB0JMN0W]].

1. The function $u$ is [[infinitely differentiable|0V33BNKZ9S]] on $U$, and every partial derivative of $u$ is harmonic on $U$.
2. The function $u$ is real [[analytic|081WKEDK6C]] on $U$: each $a \in U$ has a neighbourhood on which $u$ is a convergent power series in $x - a$.

## Remarks

Proof idea for part 1: by the [[mean value property|0HYPR64JM2]], $u$ equals its convolution with a smooth radial mollifier away from the boundary of $U$.
A convolution with a smooth function is smooth.
Since $\Delta$ commutes with partial derivatives, $\Delta(\partial^\alpha u) = \partial^\alpha(\Delta u) = 0$.
Proof idea for part 2: the [[gradient estimates|N76R7JWMNJ]] give $\abs{\partial^\alpha u(a)} \le C^{k+1} k^k \sup_{B(a, r)} \abs{u} / r^{k}$ for $\abs{\alpha} = k$, with $C$ depending only on $n$.
This growth makes the Taylor remainder tend to $0$ on a small ball.
Example: $u(x, y) = e^x \cos y$ is harmonic on $\R^2$, and its Taylor series at $0$ converges to it everywhere.
Harmonicity is needed and $C^2$ is not enough: $\abs{x}^3$ on $\R$ is twice continuously differentiable but not three times differentiable at $0$.
In contrast, a merely continuous function with the mean value property is already harmonic, hence analytic.

## Explanations

### L1 human

Laplace's equation forces its solutions to be extremely smooth.
Even if one only asks for two derivatives, a harmonic function automatically has derivatives of all orders, and it is given locally by a power series.
So a harmonic function cannot have a hidden corner or kink anywhere.

### L2 human

This is the simplest case of elliptic regularity: solutions of elliptic equations with smooth or analytic coefficients are as regular as the coefficients allow.
Analyticity implies a unique continuation property: a harmonic function on a connected open set that vanishes on a nonempty open subset vanishes everywhere.
The heat equation shares smoothness but not analyticity in time, and the wave equation has neither, so this property marks the elliptic case.
</document>

<document source="https://tamnd.github.io/math2/node/AZRPHS478B.md" tag="AZRPHS478B" kind="concept">
# Ergodic transformation

## Summary

A [[measure-preserving transformation|556WBDAHNP]] of a probability space is ergodic if it has no invariant measurable set other than null sets and sets of full measure.
Ergodic maps cannot be split into two separate systems of positive measure.
By Birkhoff's ergodic theorem, for an ergodic map the time average of an integrable function along almost every orbit equals its space average.
This is the mathematical form of the ergodic hypothesis of statistical mechanics.

## Explanations

### L1 human

Stir a container of fluid again and again.
The stirring is ergodic if it mixes everything into one piece: no region of positive volume is trapped and kept apart from the rest.
A rotation of a circle by an irrational fraction of a turn is ergodic, since each orbit fills the circle evenly.
A rotation by half a turn is not ergodic: the union of two opposite quarter arcs is moved onto itself.
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<document source="https://tamnd.github.io/math2/node/AZVRS57S7H.md" tag="AZVRS57S7H" kind="statement">
# Green's function and the Riemann map

## Statement

Let $\Omega \subsetneq \C$ be a [[simply connected domain|0QW64ARSHB]] and let $a \in \Omega$.
Let $\varphi \colon \Omega \to \mathbb{D}$ be a [[conformal map|0YYT36R4YD]] of $\Omega$ onto the unit disc $\mathbb{D}$ with $\varphi(a) = 0$; it exists by the [[Riemann mapping theorem|02WHMEG5ZR]].
Then the [[Green's function|M36SD56MWQ]] of $\Omega$ with pole at $a$ is
$$g(z, a) = -\log\abs{\varphi(z)}, \qquad z \in \Omega \setminus \set{a}.$$

## Remarks

Proof idea: since $\varphi$ is injective, $\varphi'(a) \ne 0$, so $\varphi(z) = (z - a) k(z)$ with $k$ holomorphic and zero-free on $\Omega$.
Hence $-\log\abs{\varphi(z)} = -\log\abs{z - a} - \log\abs{k(z)}$, and $\log\abs{k}$ is harmonic on $\Omega$.
Away from $a$, $\log\abs{\varphi}$ is harmonic because $\varphi$ is holomorphic and has no zero.
A conformal map onto $\mathbb{D}$ is proper, so $\abs{\varphi(z)} \to 1$ as $z$ tends to the boundary, and $g(z, a) \to 0$ there.
These are the defining properties, and the Green's function is unique by the maximum principle.
Example: for $\Omega = \mathbb{D}$ and $\varphi(z) = (z - a)/(1 - \bar a z)$, one gets $g(z, a) = \log\abs{(1 - \bar a z)/(z - a)}$.
Near the pole, $g(z, a) = -\log\abs{z - a} + \log\big(1/\abs{\varphi'(a)}\big) + o(1)$, and $1/\abs{\varphi'(a)}$ is the [[conformal radius|GYEX4BH5WM]] of $\Omega$ at $a$.
The hypothesis $\Omega \ne \C$ is needed: the plane has no Green's function, since a positive harmonic function on $\C \setminus \set{a}$ with a logarithmic pole at $a$ does not exist.

## Explanations

### L1 human

For a simply connected region, the Green's function and the Riemann map carry the same information.
The level curves of the Green's function are the images of the circles $\abs{w} = r$ under the inverse of the Riemann map.

### L2 human

The formula can also be read backwards.
If $g^*$ is a harmonic conjugate of $g(\cdot, a)$ on $\Omega \setminus \set{a}$, then $\exp(-(g + i g^*))$ is single-valued and gives the Riemann map.
This is one classical route to the Riemann mapping theorem via potential theory, through the solution of a Dirichlet problem.
On multiply connected domains the conjugate $g^*$ has periods, so $\exp(-(g + i g^*))$ is no longer single-valued.
</document>

<document source="https://tamnd.github.io/math2/node/B0223N93XS.md" tag="B0223N93XS" kind="definition">
# Support of a module

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an $R$-[[module|0NZ8JA25WK]].
For a [[prime ideal|00MHHZ9SNE]] $\mathfrak p$, let $M_{\mathfrak p}$ be the [[localization|68E0F8N94E]] of $M$ at $\mathfrak p$.
The *support* of $M$ is the following subset of the [[spectrum|FB45BHHA2J]] $\operatorname{Spec} R$:
$$\operatorname{Supp} M = \set{\mathfrak p \in \operatorname{Spec} R : M_{\mathfrak p} \ne 0}.$$

## Remarks

If $M$ is [[finitely generated|05QY1VEBYD]], then $\operatorname{Supp} M = V(\operatorname{Ann} M)$, the set of primes that contain the [[annihilator|HB4XDKJTW8]] $\operatorname{Ann} M = \set{a \in R : aM = 0}$.
In particular the support of a finitely generated module is closed in the [[Zariski topology|8SQK7RTB63]].
By the [[local property|3S58E70Z7Z]] of vanishing, $\operatorname{Supp} M = \emptyset$ if and only if $M = 0$.

Example: let $R = \Z$ and $M = \Z/6\Z$.
For a prime $p \ne 2, 3$, the integer $6$ lies outside $p\Z$, and $m/s = 6m/(6s) = 0$, so $M_{(p)} = 0$.
Also $M_{(0)} = 0$ since $6 \notin (0)$.
For $p = 2$, the element $1/1$ is nonzero in $M_{(2)}$, since $u \cdot 1 \ne 0$ in $\Z/6\Z$ for every odd $u$.
The same holds for $p = 3$, so $\operatorname{Supp} M = \set{2\Z, 3\Z} = V(6\Z)$.

Non-example of the formula with the annihilator: the $\Z$-module $\Q/\Z$ is not finitely generated, and $\operatorname{Ann}(\Q/\Z) = 0$, so $V(\operatorname{Ann}) = \operatorname{Spec} \Z$.
But $(\Q/\Z)_{(0)} = 0$, since every element is killed by a nonzero integer.
So $(0) \notin \operatorname{Supp}(\Q/\Z)$, and the support is not $V(\operatorname{Ann}(\Q/\Z))$.
</document>

<document source="https://tamnd.github.io/math2/node/B03CA692K3.md" tag="B03CA692K3" kind="statement">
# Sobolev inequality

## Statement

Let $n \ge 2$, let $1 \le p < n$, and let $p^* = \frac{np}{n - p}$.

1. There is a constant $C = C(n, p)$ such that $\norm{u}_{L^{p^*}(\R^n)} \le C \norm{\nabla u}_{L^p(\R^n)}$ for every continuously differentiable $u \colon \R^n \to \R$ with compact support.
2. Let $U \subset \R^n$ be a bounded open set with $C^1$ boundary. Then $W^{1,p}(U) \subseteq L^{p^*}(U)$, and there is a constant $C = C(n, p, U)$ with $\norm{u}_{L^{p^*}(U)} \le C \norm{u}_{W^{1,p}(U)}$ for all $u \in W^{1,p}(U)$.
3. Let $U$ be as in part 2, let $k \ge 1$ be an integer with $kp < n$, and let $\frac{1}{q} = \frac{1}{p} - \frac{k}{n}$. Then $W^{k,p}(U) \subseteq L^{q}(U)$, and $\norm{u}_{L^{q}(U)} \le C \norm{u}_{W^{k,p}(U)}$ for all $u \in W^{k,p}(U)$, with $C = C(n, k, p, U)$.

Here $W^{k,p}(U)$ is the [[Sobolev space|M4CEZ5K9HP]] and $L^q(U)$ is the [[Lp space|SASP14MA1Z]].

## Remarks

Part 1 is the Gagliardo–Nirenberg–Sobolev inequality.
For $p = 1$, the proof writes $\abs{u(x)} \le \int_{-\infty}^{x_i} \abs{\partial_i u} \, dt$ in each coordinate direction, multiplies the $n$ bounds, and applies a generalized [[Hölder's inequality|JQ9WXC729D]] one variable at a time.
The case $p > 1$ follows by applying the case $p = 1$ to $\abs{u}^\gamma$ for a suitable $\gamma > 1$.
Part 1 extends to all $u \in W^{1,p}(\R^n)$ by density, with the [[weak derivative|C3EC148VZD]] in place of $\nabla u$.
Part 2 follows from part 1 by extending $u$ to a compactly supported function on $\R^n$, and part 3 follows from part 2 by induction on $k$.

The exponent $p^*$ is forced by scaling.
Replacing $u(x)$ by $u(\lambda x)$ multiplies the two sides of part 1 by $\lambda^{-n/p^*}$ and $\lambda^{1 - n/p}$, and these agree for all $\lambda > 0$ only when $p^* = np/(n - p)$.
The inequality fails for $p = n$ with $p^* = \infty$: for $n \ge 2$ there are unbounded functions in $W^{1,n}$ of a ball, such as $\log \log (1 / \abs{x})$ near $0$ in the plane.
For $p > n$ the corresponding result is Morrey's inequality.
The [[Sobolev embedding theorem|GZTT6FT18G]] combines both cases.

## Explanations

### L1 human

If the slope of a function is small in an average sense, then the function itself cannot be too large on a big set.
The Sobolev inequality makes this precise: control of $\nabla u$ in $L^p$ gives control of $u$ in the better space $L^{p^*}$.
In dimension $3$ with $p = 2$, finite energy $\int \abs{\nabla u}^2$ forces $u$ to lie in $L^6$.

### L2 human

The inequality is the basic tool to pass from energy estimates, which control derivatives, to estimates of the solution itself.
It is used to show that the energy of a nonlinear elliptic problem is well defined, for example for terms like $\int \abs{u}^q$ with $q \le p^*$.
Both sides of part 1 scale in the same way under dilations $u(x) \mapsto u(\lambda x)$, which makes the critical exponent $p^*$ special.
The embedding into $L^{p^*}$ is not compact, while the embeddings into $L^q$ for $q < p^*$ on bounded sets are compact by the Rellich–Kondrachov theorem.
</document>

<document source="https://tamnd.github.io/math2/node/B0587TVWZY.md" tag="B0587TVWZY" kind="definition">
# Hecke algebra

## Definition

This is the Iwahori–Hecke algebra.
Let $(W, S)$ be a [[Coxeter system|G6FND7X69J]], with Coxeter matrix $(m_{st})_{s,t \in S}$, so that $m_{st}$ is the order of $st$ in $W$.
Let $R$ be a [[commutative ring|0310REJJY8]] with identity and let $q \in R$.
The *Hecke algebra* $H_q(W)$ is the [[associative algebra|3CTW1P4HBP]] over $R$ generated by elements $T_s$ for $s \in S$, subject to the relations:

1. $(T_s - q)(T_s + 1) = 0$ for every $s \in S$.
2. $T_s T_t T_s \cdots = T_t T_s T_t \cdots$, with $m_{st}$ factors on each side, for all $s \ne t$ in $S$ with $m_{st} < \infty$.

## Remarks

For $w \in W$ with a reduced expression $w = s_1 \cdots s_k$, put $T_w = T_{s_1} \cdots T_{s_k}$.
By the braid relations and Matsumoto's theorem $T_w$ does not depend on the reduced expression, and the $T_w$ for $w \in W$ form a basis of $H_q(W)$ as a [[free|0DZ9DKAKZR]] $R$-module.
The multiplication is $T_s T_w = T_{sw}$ if $\ell(sw) > \ell(w)$ and $T_s T_w = (q - 1) T_w + q T_{sw}$ if $\ell(sw) < \ell(w)$, where $\ell$ is the length function.
At $q = 1$ the relations become $T_s^2 = 1$ and the braid relations, which present the group algebra $R[W]$.
Example: $W = \set{1, s}$ of order $2$ gives $H_q(W) = R[T]/((T - q)(T + 1))$, free of rank $2$ with basis $1, T$.
Non-example: for $W = S_3$ with $S = \set{s, t}$ and $m_{st} = 3$, dropping relation 2 changes the algebra.
At $q = 1$ the result is the group algebra of the infinite dihedral group $\langle s, t \mid s^2 = t^2 = 1 \rangle$, which has infinite rank, while $H_1(S_3) = R[S_3]$ has rank $6$.
Many authors use the parameter $q^{1/2}$ or $v$ with $q = v^2$ and rescale the generators.
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<document source="https://tamnd.github.io/math2/node/B0K18PWN2A.md" tag="B0K18PWN2A" kind="concept">
# Restricted isometry property

## Summary

A matrix has the restricted isometry property of order $s$ when it nearly preserves the length of every vector with at most $s$ nonzero entries.
The size of the distortion is measured by the restricted isometry constant $\delta_s$.
If $\delta_{2s}$ is small, $s$-sparse vectors can be recovered stably from their images by $\ell^1$ minimization.
This is the basis of compressed sensing.
Computing $\delta_s$ exactly is hard, but random Gaussian matrices have small constants with high probability.

## Explanations

### L1 human

A wide [[matrix|0VD9ZSWWJG]] must squash some vectors to zero, because it maps a big space into a small one.
The restricted isometry property says that it still keeps the lengths of all sparse vectors almost unchanged.
If this holds for all $2s$-sparse vectors, then two different $s$-sparse vectors cannot have the same image.
They even stay well separated after the map.
For example, the identity matrix keeps every length exactly, so $\delta_s = 0$ for every $s$.
A matrix with two equal columns sends $e_1 - e_2$, which has only two nonzero entries, to $0$, so it fails the property of order $2$.
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<document source="https://tamnd.github.io/math2/node/B0K8MDA7CJ.md" tag="B0K8MDA7CJ" kind="concept">
# Cobweb plot

## Summary

A cobweb plot is a picture of an orbit of a map of the real line.
It is drawn on top of the graph of the map and the diagonal $y = x$.
The orbit is traced by a path that goes alternately vertically to the graph and horizontally to the diagonal.
It shows at a glance whether an orbit converges to a [[fixed point|R6QVS6P21N]], cycles around a [[periodic point|T75CZ5H47N]], or wanders.
Cobweb plots are a standard first tool for maps of an interval such as the logistic map.

## Explanations

### L1 human

Draw the graph of the map $f$ and the line $y = x$ in the same picture.
Start at a value $x_0$ on the line, and go up or down to the graph to read off $f(x_0)$.
Then go sideways to the line to make $f(x_0)$ the new starting value.
Repeating these two moves draws a staircase or a spiral.
For $f(x) = \cos x$, the path spirals in to the point where the graph of $\cos$ crosses the diagonal, near $0.739$.
A staircase means the orbit approaches the fixed point from one side, and a spiral means it jumps from side to side.
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<document source="https://tamnd.github.io/math2/node/B0QT3SM5DT.md" tag="B0QT3SM5DT" kind="concept">
# Bounded poset

## Summary

A bounded poset is a [[partially ordered set|011C29Q1S1]] that has both a [[greatest element|0Z7HDDXX3X]] and a [[least element|0T2NTJCV83]].
These are called top and bottom and written $\top$ and $\bot$, or $1$ and $0$.
Bounded posets are the setting for complements in [[lattices|WH4SFAACAS]].

## Explanations

### L1 human

A bounded poset has a top element above everything and a bottom element below everything.
The subsets of a set, ordered by inclusion, form a bounded poset: the empty set is the bottom and the whole set is the top.
The closed interval $[0, 1]$ with its usual order is bounded, with bottom $0$ and top $1$.
The natural numbers are not bounded: they have a bottom, $0$, but no top.
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<document source="https://tamnd.github.io/math2/node/B0YKA8401M.md" tag="B0YKA8401M" kind="concept">
# Day terms

## Summary

Day terms are a finite chain of quaternary terms of a [[variety|AJNZJD8442]] that satisfy certain identities.
By Day's theorem, a variety has Day terms if and only if it is a [[congruence-modular variety|QAQAS3JWXW]].
They give a Mal'cev condition for congruence modularity, in the same way that [[Jónsson terms|3M8Y76E4AG]] do for congruence distributivity.

## Explanations

### L1 human

Day terms are a list of formulas with four inputs that step from the first input to the last one.
Each formula returns $x$ when its inputs have the pattern $x, y, y, x$.
Neighbouring formulas agree on inputs of certain patterns with repeated values.
Their existence is exactly what makes all congruence lattices in the variety modular.
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<document source="https://tamnd.github.io/math2/node/B138JA6W20.md" tag="B138JA6W20" kind="definition">
# Elementary theory of the category of sets

## Definition

The *Elementary Theory of the Category of Sets* (ETCS) is the first-order theory of a [[category|A2DE66937D]] $\mathcal{S}$ with the following axioms.

1. $\mathcal{S}$ is an elementary [[topos|6837RS7N82]].
2. $\mathcal{S}$ is nondegenerate: its [[terminal object|EK85VSSRMR]] $1$ is not an initial object.
3. $\mathcal{S}$ is well-pointed: for every two distinct morphisms $f, g \colon X \to Y$ there is a morphism $x \colon 1 \to X$ with $f \circ x \ne g \circ x$.
4. $\mathcal{S}$ has a natural numbers object $(N, z, s)$, as described below.
5. Every [[epimorphism|RVXDSY04KG]] $e \colon X \to Y$ in $\mathcal{S}$ has a section, a morphism $m \colon Y \to X$ with $e \circ m = \id_Y$.

A natural numbers object is an object $N$ with morphisms $z \colon 1 \to N$ and $s \colon N \to N$ with this property.
For every object $X$ with morphisms $a \colon 1 \to X$ and $h \colon X \to X$ there is a unique $u \colon N \to X$ with $u \circ z = a$ and $u \circ s = h \circ u$.

## Remarks

Example: the category $\mathbf{Set}$ of sets and functions, built in a model of ZFC, satisfies all five axioms.
Its natural numbers object is $\N$ with $0$ and $n \mapsto n + 1$, and the sections come from the axiom of choice.
Non-example: the category of sets with an action of the group $\Z/2$ is a nondegenerate topos with a natural numbers object, but it is not well-pointed.
Its terminal object is a one-point set, and a morphism $1 \to X$ is a fixed point of $X$.
On $X = \Z/2$ with the action by translation there are no fixed points, so the identity and the swap of $X$ are distinct but agree on all morphisms $1 \to X$.
In ETCS an element of $X$ is a morphism $1 \to X$, and the subsets of $X$ are classified by maps into the [[subobject classifier|NKP5TPQ79X]], which turns out to be $1 + 1$.
ETCS has no axiom schema of replacement.
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<document source="https://tamnd.github.io/math2/node/B1E1ES1BYD.md" tag="B1E1ES1BYD" kind="statement">
# Convergence to equilibrium of ergodic Markov chains

## Statement

Let $(X_n)_{n \ge 0}$ be a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable set $S$ with [[transition matrix|0F462493EA]] $P = (p_{ij})$, and write $p^{(n)}_{ij}$ for the entries of $P^n$.
Assume that:

1. the chain is irreducible: for all $i, j \in S$ there is $n \ge 0$ with $p^{(n)}_{ij} > 0$;
2. the chain is aperiodic: for each $i \in S$, the greatest common divisor of $\set{n \ge 1 : p^{(n)}_{ii} > 0}$ is $1$;
3. the chain has a [[stationary distribution|0KRH5157WP]] $\pi$.

Then, for every law of $X_0$,
$$\lim_{n \to \infty} \sum_{j \in S} \abs{P(X_n = j) - \pi_j} = 0.$$
In particular $P(X_n = j) \to \pi_j$ for each $j \in S$.

## Remarks

The classical proof is by [[coupling|HTR7J0W66V]].
Run a second independent copy $(Y_n)$ started from $\pi$; aperiodicity makes the pair $(X_n, Y_n)$ an irreducible chain with a stationary distribution.
Hence the pair is [[recurrent|PBZTBWZ4GS]] and hits the diagonal at a finite time $\tau$, and $\sum_j \abs{P(X_n = j) - \pi_j} \le 2 P(\tau > n) \to 0$.
Example: on $S = \set{1, 2}$ let $p_{12} = a$ and $p_{21} = b$ with $0 < a, b \le 1$ and $a + b < 2$.
Then $\pi = (b, a)/(a + b)$ and $P(X_n = 1) - \pi_1 = (1 - a - b)^n \left(P(X_0 = 1) - \pi_1\right)$, which tends to $0$.
Aperiodicity is needed: for $a = b = 1$ the chain alternates, and $P(X_n = 1)$ oscillates between $0$ and $1$ when $X_0 = 1$.
Irreducibility is needed: with two absorbing states the limit depends on the starting state.

## Explanations

### L1 human

A chain that can reach every state from every other and does not move in a fixed cycle forgets where it started.
After a long time, the chance of being in each state is close to its value under the stationary distribution, whatever the start.
This is why shuffling a deck long enough makes every order of the cards almost equally likely.

### L2 human

The theorem justifies Markov chain Monte Carlo: running an ergodic chain produces samples whose law approaches $\pi$.
The speed of convergence is measured by the [[mixing time|8TV1YW8EXC]], and for finite chains it is geometric, governed by the second largest eigenvalue modulus of $P$.
Without aperiodicity, the averages $\frac1n \sum_{k=1}^{n} P(X_k = j)$ still converge to $\pi_j$.
For a finite state space, condition 3 follows from condition 1.
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# Autoregressive–moving-average model

## Summary

An autoregressive–moving-average model, or ARMA($p$, $q$) model, combines an [[autoregressive model|2JC96FMGA1]] of order $p$ with a [[moving-average model|PEB9DN0AHA]] of order $q$.
Each value is a linear combination of $p$ past values plus the current shock and $q$ past shocks.
ARMA models describe a wide range of stationary time series with few parameters, and they are the core of the Box–Jenkins approach to forecasting.

## Explanations

### L1 human

An autoregressive part lets the series remember its own past values, and a moving-average part lets recent shocks echo for a few steps.
Using both parts often describes the data with fewer numbers than using either part alone.
For example, $X_t = 0.5 X_{t-1} + \varepsilon_t + 0.3 \varepsilon_{t-1}$ is an ARMA($1$, $1$) model.
Forecasts from it decay towards the mean at the rate $0.5$ per step, after a first step that also uses the last shock.
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<document source="https://tamnd.github.io/math2/node/B1Q5F5VNSK.md" tag="B1Q5F5VNSK" kind="problem">
# Kurosh problem

## Problem

Let $k$ be a [[field|04C5ZBG1WT]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$ that is generated by finitely many elements.
Assume that $A$ is algebraic: every element of $A$ is [[algebraic|0H628RSGWX]] over $k$, that is, a root of a nonzero polynomial with coefficients in $k$.
Must $A$ be finite-dimensional over $k$?

## Remarks

The problem is a ring-theoretic analogue of the [[Burnside problem|XWEMJ7T2XV]] for periodic groups.
An important special case asks whether a finitely generated nil algebra, in which every element is nilpotent, must be nilpotent and hence finite-dimensional.

The answer is negative in general.
Golod used the [[Golod–Shafarevich theorem|RC5A5T7YY2]] in 1964 to construct a finitely generated infinite-dimensional nil algebra.
The same construction gave infinite finitely generated periodic groups, a negative answer to the general Burnside problem.

The answer is positive for algebras that satisfy a polynomial identity: a finitely generated algebraic [[polynomial identity algebra|C0EDT81EGH]] is finite-dimensional.
In particular it is positive when the elements of $A$ are algebraic of bounded degree.

A variant asks the same question for [[division rings|0EA5BTA0CX]] that are algebraic over their centre and finitely generated over it.

## Explanations

### L1 human

Each element of an algebraic algebra satisfies some polynomial equation, so each single element generates only a finite-dimensional piece.
Kurosh asked whether finitely many such elements must then generate a finite-dimensional algebra.
Surprisingly, the answer is no: there are algebras generated by a few elements, each element is even nilpotent, and yet the algebra is infinite-dimensional.
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<document source="https://tamnd.github.io/math2/node/B1S2XJPGYD.md" tag="B1S2XJPGYD" kind="concept">
# Series acceleration

## Summary

Series acceleration uses sequence transformations to speed up a slowly converging sequence, such as the [[partial sums|09RJ859N7A]] of a series.
The new sequence has the same limit but converges faster.
It matters in numerical work, where a series may need millions of terms for a few correct digits.
Classical methods are [[Aitken's delta-squared process|WF08BMN6Q6]], the [[Euler transform|WT0HACKQV7]], [[Richardson extrapolation|WMTZG71K2Y]], the [[Shanks transformation|C0TXDZZMH7]] and the [[Levin-type transformations|1P47PCRVTZ]].

## Explanations

### L1 human

A slowly converging series reaches its sum only after very many terms.
An acceleration method looks at the pattern of the first few partial sums and guesses where they are heading.
For the [[Leibniz series|037FD5NBRT]] $1 - \frac13 + \frac15 - \cdots = \frac{\pi}{4}$, the partial sums jump around $\frac{\pi}{4}$, and averaging neighbours already gives much better values.
No single method works for every sequence; each one is exact for some model of how the error behaves.
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<document source="https://tamnd.github.io/math2/node/B1S7A7J8H4.md" tag="B1S7A7J8H4" kind="definition">
# Infinitesimal stability

## Definition

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] and let $f \colon M \to N$ be a smooth map.
A *vector field along $f$* is a smooth map $w \colon M \to TN$ with $w(p) \in T_{f(p)} N$ for all $p \in M$, where $TN$ is the [[tangent bundle|WY6K3P4YCG]].
Let $df \colon TM \to TN$ be the [[differential|6P9PXQDSEN]] of $f$.
The map $f$ is *infinitesimally stable* if every vector field $w$ along $f$ has the following form.
There are a smooth vector field $\xi$ on $M$ and a smooth vector field $\eta$ on $N$ with
$$w = df \circ \xi + \eta \circ f.$$

## Remarks

The two terms are the derivatives at $t = 0$ of $f \circ \varphi_t$ and $\psi_t \circ f$, where $\varphi_t$ and $\psi_t$ are the flows of $\xi$ and $\eta$.
So the condition says that every first-order deformation of $f$ is tangent to the orbit of $f$ under changes of coordinates.
Example: a submersion $f$ is infinitesimally stable.
Indeed, each $df_p$ is surjective, so a choice of complement of $\ker df_p$ depending smoothly on $p$ gives a vector field $\xi$ with $df \circ \xi = w$, and one takes $\eta = 0$.
Non-example: a constant map $f \colon S^1 \to \R$ with value $c$ is not infinitesimally stable.
Here $df = 0$ and $\eta \circ f$ is the constant $\eta(c)$, so a nonconstant function $w$ on $S^1$, viewed as a vector field along $f$, is not of the required form.
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# Additive function

## Definition

Let $A$ and $B$ be [[abelian groups|082GHNPJ53]], written additively.
A [[function|0EPAT51DXQ]] $f \colon A \to B$ is *additive* if
$$f(x + y) = f(x) + f(y) \quad \text{for all } x, y \in A.$$
The most common case is $A = B = \R$.

## Remarks

Example: $f \colon \R \to \R$, $f(x) = -5x$, is additive, since $-5(x + y) = -5x + (-5y)$.

Non-example: $f(x) = x^2$ is not additive, since $f(1 + 1) = 4$ while $f(1) + f(1) = 2$.

Every additive function satisfies $f(0) = 0$, $f(-x) = -f(x)$ and $f(nx) = n f(x)$ for $n \in \Z$.
Between real vector spaces, an additive function satisfies $f(qx) = q f(x)$ for every rational $q$, so it is $\Q$-linear.
For arbitrary groups, the same condition defines a group homomorphism.
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<document source="https://tamnd.github.io/math2/node/B1TEQ98JZD.md" tag="B1TEQ98JZD" kind="concept">
# Tate algebra

## Summary

The Tate algebra in $n$ variables over a complete non-archimedean field $K$ is the algebra of power series whose coefficients tend to $0$.
These are exactly the power series that converge on the closed unit polydisc over an algebraic closure of $K$.
With the Gauss norm it is a [[Banach algebra|9KYWN94QDN]], and it is a [[Noetherian|0670T59HGE]] unique factorization domain.
Its quotients, the affinoid algebras, are the local building blocks of [[rigid analytic spaces|YAH6MZVG7K]], as polynomial rings are for algebraic varieties.

## Explanations

### L1 human

Over the [[$p$-adic numbers|BT6T702C28]], a series $\sum a_k$ converges exactly when its terms tend to $0$.
So a power series $\sum a_k x^k$ converges at every $x$ with $\abs{x} \le 1$ exactly when $\abs{a_k} \to 0$.
The Tate algebra collects these power series, in one or several variables.
For example $\sum_k p^k x^k$ belongs to it, while $\sum_k x^k$ does not, since it diverges at $x = 1$.
It plays the role that the ring of holomorphic functions on a closed polydisc plays in complex analysis.
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# Eilenberg–Moore category

## Definition

Let $(T, \eta, \mu)$ be a monad on a [[category|A2DE66937D]] $\mathcal{C}$.
The *Eilenberg–Moore category* $\mathcal{C}^T$ has these parts.

1. Its objects are the [[$T$-algebras|QG1ZXRC8EH]] $(A, a)$.
2. Its morphisms $(A, a) \to (B, b)$ are the morphisms of $T$-algebras, that is, the $f \colon A \to B$ with $f \circ a = b \circ T(f)$.
3. Composition and identities are those of $\mathcal{C}$.

The forgetful functor $U^T \colon \mathcal{C}^T \to \mathcal{C}$ sends $(A, a)$ to $A$.
The free functor $F^T \colon \mathcal{C} \to \mathcal{C}^T$ sends $X$ to $(T(X), \mu_X)$ and $f$ to $T(f)$.

## Remarks

The functor $F^T$ is [[left adjoint|TZHF6ZM92S]] to $U^T$, and $U^T F^T = T$.
The unit of the adjunction is $\eta$, and the monad of the adjunction is $(T, \eta, \mu)$.

Example: for the list monad on sets, $\mathcal{C}^T$ is isomorphic to the category of [[monoids|0N6R902FG1]] and monoid homomorphisms.
A morphism of algebras $f$ satisfies $f(m_1 \cdots m_n) = f(m_1) \cdots f(m_n)$, which is exactly the condition to be a monoid homomorphism.

Non-example: the category whose objects are monoids and whose morphisms are all functions between them is not $\mathcal{C}^T$.
A function such as $n \mapsto n + 1$ on $(\N, +)$ does not satisfy clause 2, since it does not send $0 + 0$ to $1 + 1$.

The notation $\mathcal{C}^T$ is standard.
The full subcategory of free algebras is equivalent to the Kleisli category of $T$.
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# Set-valued measure

## Definition

Let $(\Omega, \mathcal{F})$ be a [[measurable space|N4KDT83V3S]] and let $n \ge 1$.
A *set-valued measure* on $(\Omega, \mathcal{F})$ with values in $\R^n$ is a map $M$ from $\mathcal{F}$ to the nonempty subsets of $\R^n$ such that:

1. $M(\emptyset) = \set{0}$.
2. For every sequence of pairwise disjoint sets $A_1, A_2, \dots$ in $\mathcal{F}$ with union $A$,
$$M(A) = \set{\sum_{i=1}^{\infty} x_i : x_i \in M(A_i) \text{ for all } i, \ \sum_{i=1}^{\infty} \abs{x_i} < \infty}.$$

## Remarks

Many authors also require the values to be closed, compact or convex, and some work in a Banach space instead of $\R^n$.
Example: let $\mu$ be a finite measure on $\mathcal{F}$, let $K \subseteq \R^n$ be nonempty, compact and convex, and let $M(A) = \mu(A) K$.
Then $M(\emptyset) = \set{0}$.
Given $x_i \in \mu(A_i) K$, the series converges absolutely since $K$ is bounded, and its sum lies in $\mu(A) K$ because $K$ is convex and closed.
Conversely $\mu(A) k = \sum_i \mu(A_i) k$ for $k \in K$, so clause 2 holds.
Non-example: let $\Omega = \set{a, b}$ with counting measure $\mu$ and $K = \set{0, 1}$, which is not convex.
Then $M(\Omega) = 2K = \set{0, 2}$, while the sums from $M(\set{a})$ and $M(\set{b})$ form $\set{0, 1, 2}$, so clause 2 fails.
A [[vector measure|PS28BSAJKJ]] $\nu$ gives the set-valued measure $A \mapsto \set{\nu(A)}$.
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# Fenchel conjugate

## Definition

Let $X$ be a real [[normed space|03J5RAKB30]] with continuous [[dual space|07JBSZE0V4]] $X^*$, and write $\ip{y}{x} = y(x)$ for $y \in X^*$ and $x \in X$.
Let $f \colon X \to [-\infty, +\infty]$ be a function.
The *Fenchel conjugate* (or *convex conjugate*) of $f$ is the function $f^* \colon X^* \to [-\infty, +\infty]$ given by
$$f^*(y) = \sup_{x \in X} \big(\ip{y}{x} - f(x)\big),$$
where $\sup$ is the [[supremum|0R3H5PANRQ]] in the [[extended real line|0GSZ7B0CGR]].
For $X = \R^n$, one identifies $X^*$ with $\R^n$ through the dot product.

## Remarks

Example: $f(x) = \frac12 x^2$ on $\R$.
For fixed $y$, the function $x \mapsto yx - \frac12 x^2$ is maximal at $x = y$, so $f^*(y) = \frac12 y^2$.

Example without differentiability: $f(x) = \abs{x}$ on $\R$.
If $\abs{y} \le 1$ then $yx - \abs{x} \le 0$ with equality at $x = 0$, so $f^*(y) = 0$; if $\abs{y} > 1$ then $yx - \abs{x} \to +\infty$ along $x = t \sgn(y)$, so $f^*(y) = +\infty$.
The classical formula of the [[Legendre transformation|ZQYMWVT1TT]] does not apply to this $f$, because $f$ is not differentiable at $0$.

Non-example of the identity $f^{**} = f$: for $f(x) = -x^2$ on $\R$, every $f^*(y)$ is $+\infty$, so $f^{**} \equiv -\infty \neq f$; this $f$ is not convex.

Every conjugate $f^*$ is convex and lower semicontinuous, as a supremum of affine continuous functions.
The Fenchel–Young inequality $f(x) + f^*(y) \ge \ip{y}{x}$ holds whenever the left side is defined.
The Fenchel–Moreau theorem states that $f^{**} = f$ on $X$ when $f$ is proper, convex and lower semicontinuous.
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# Monad in nonstandard topology

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let ${}^*X$ be its extension in a model of [[nonstandard analysis|WM62GKCBWN]] that is an enlargement.
For an [[open set|02BH3MCBQB]] $U \subseteq X$, write ${}^*U \subseteq {}^*X$ for its extension.
The *monad* of a point $x \in X$ is
$$\mu(x) = \bigcap \set{{}^*U : U \subseteq X \text{ open}, \ x \in U}.$$
A point $y \in {}^*X$ is *infinitely close* to $x$ if $y \in \mu(x)$.

## Remarks

Example: for $X = \R$ with the usual topology, ${}^*X$ is the field of [[hyperreal numbers|421QMTFG9T]].
A hyperreal $y$ lies in $\mu(x)$ exactly when $\abs{y - x} < 1/n$ for every positive integer $n$, that is, when $y - x$ is infinitesimal; so $x + \varepsilon \in \mu(x)$ for every infinitesimal $\varepsilon$.
Non-example: for $x = 0$ and a positive infinitesimal $\varepsilon$, the hyperreal $1 + \varepsilon$ is not in $\mu(0)$, since it is not in ${}^*(-1/2, 1/2)$.
The [[standard part|1ZBY4W4E81]] of a finite hyperreal $y$ is the unique real $x$ with $y \in \mu(x)$.
Standard characterizations in an enlargement: $X$ is [[Hausdorff|0S1FJ8VEX9]] if and only if distinct points have disjoint monads.
$X$ is [[compact|0TVJZYB6SH]] if and only if every point of ${}^*X$ lies in the monad of some point of $X$.
A map $f \colon X \to Y$ is [[continuous|0X3WGX95FE]] at $x$ if and only if ${}^*f(\mu(x)) \subseteq \mu(f(x))$.
</document>

<document source="https://tamnd.github.io/math2/node/B1Z1C8AHVD.md" tag="B1Z1C8AHVD" kind="statement">
# Fáry–Milnor theorem

## Statement

Let $\gamma \colon [a, b] \to \R^3$ be a regular simple [[closed curve|0D032NSZKY]] of class $C^2$ whose derivatives up to order $2$ also agree at $a$ and $b$.
If the [[total curvature|QP9DY5K8QW]] of $\gamma$ is at most $4\pi$, then $\gamma$ is unknotted, that is, its image is a trivial [[knot|QYEAZTYGPT]].
Equivalently, every knotted curve of this kind has total curvature greater than $4\pi$.

## Remarks

Milnor's proof uses height functions.
For almost every unit vector $v$, the function $t \mapsto \ip{\gamma(t)}{v}$ has finitely many local maxima, and the total curvature equals $2\pi$ times the average number of these maxima over all $v$.
If the total curvature is less than $4\pi$, some direction $v$ gives exactly one local maximum and one local minimum.
Then each plane orthogonal to $v$ meets the curve in at most two points, and joining these pairs of points gives an embedded disc bounded by the curve.
So the curve is unknotted; the case of total curvature exactly $4\pi$ needs a further argument.
Example: a standard trefoil knot must have total curvature greater than $4\pi$, while a round circle, with $2\pi$, is unknotted.
István Fáry and John Milnor proved the theorem independently, around 1949 and 1950.

## Explanations

### L1 human

A knotted loop of wire must bend a lot.
In total, it must turn by more than two full turns.
So if a closed loop turns by at most two full turns, it can be pulled into a round circle without cutting.

### L2 human

The theorem refines [[Fenchel's theorem|F0BJ2BHSZG]], which gives the bound $2\pi$ for every closed curve.
The bound $4\pi$ is sharp: there are knotted curves with total curvature as close to $4\pi$ as one likes.
Milnor's averaging formula for the total curvature is a model of later integral-geometric methods, such as the Crofton formula.
</document>

<document source="https://tamnd.github.io/math2/node/B219G9SHJA.md" tag="B219G9SHJA" kind="concept">
# Stochastic differential equation

## Summary

A stochastic differential equation describes a quantity whose change in a short time is a drift plus a random kick.
The kick is proportional to the increment of a [[Wiener process|22735FGJV6]].
It is written $dX_t = b(t, X_t) \, dt + \sigma(t, X_t) \, dW_t$ and is made precise as an integral equation with an Itô integral from [[Itô calculus|JA95NH206E]].
Such equations model stock prices, noisy physical systems, and population sizes under random effects, and their solutions are diffusion processes.

## Explanations

### L1 human

An ordinary differential equation tells a particle exactly where to go next.
A stochastic differential equation adds a random push at each moment, built from the jittery path of Brownian motion.
The drift $b$ says where the particle tends to move, and the coefficient $\sigma$ says how strong the noise is.
For example, the solution of $dX_t = X_t \, dW_t$ with $X_0 = 1$ is $X_t = e^{W_t - t/2}$, not $e^{W_t}$, because the rules of calculus change for such rough paths.
</document>

<document source="https://tamnd.github.io/math2/node/B21YEPS76G.md" tag="B21YEPS76G" kind="concept">
# Sasakian manifold

## Summary

A Sasakian manifold is an odd-dimensional Riemannian manifold whose metric cone is a Kähler manifold.
It is the odd-dimensional partner of Kähler geometry, and it carries a natural [[contact structure|HBQ8BZ7XB5]] and a [[Reeb vector field|GGM63KXCF2]].
The round odd-dimensional spheres are the basic examples.
Sasaki–Einstein manifolds are of interest in the study of Einstein metrics and in physics.

## Explanations

### L1 human

Take a manifold $M$ and build a cone over it, like a cone over a circle in the plane.
The cone has one dimension more than $M$, and its metric grows with the distance $r$ from the tip.
$M$ is Sasakian when this cone is a Kähler manifold, a complex manifold with a compatible metric whose fundamental $2$-form is closed.
The cone over the unit sphere $S^{2n+1}$ is $\C^{n+1}$ with the origin removed, so the sphere is Sasakian.
</document>

<document source="https://tamnd.github.io/math2/node/B232EZ1FMB.md" tag="B232EZ1FMB" kind="concept">
# Invariant measure

## Summary

An invariant measure of a map is a [[measure|PR11BZQQPG]] that the map does not change: every set has the same measure as its preimage.
With respect to such a measure the map is a [[measure-preserving transformation|556WBDAHNP]], and the tools of ergodic theory apply.
Invariant probability measures describe the statistics of orbits, such as how often an orbit visits a region.
On a compact metric space every continuous map has at least one invariant probability measure, by the [[Krylov–Bogolyubov theorem|1DHZ81S0YE]].

## Explanations

### L1 human

Think of a measure as a way of spreading sand over the space.
Apply the map to every grain: the measure is invariant if the sand ends up spread exactly as before.
Lebesgue measure on the circle is invariant under every rotation, since rotating does not change lengths of arcs.
For the map $x \mapsto x/2$ on the real line, the point mass at $0$ is invariant, while Lebesgue measure is not, since the map squeezes everything towards $0$.
</document>

<document source="https://tamnd.github.io/math2/node/B23B0QEMN5.md" tag="B23B0QEMN5" kind="statement">
# Szemerédi's theorem

## Statement

Let $A$ be a set of positive integers with positive upper density, that is,
$$\limsup_{N \to \infty} \frac{\abs{A \cap \set{1, \dots, N}}}{N} > 0,$$
where $\limsup$ is the [[limit superior|0T3RZQXDYD]].
Then for every integer $k \ge 1$, the set $A$ contains an arithmetic progression $a, a + d, \dots, a + (k-1)d$ of length $k$ with $d \ge 1$.

## Remarks

An equivalent finite form reads as follows.
For every $k \ge 1$ and $\delta > 0$ there is $N_0$ such that the following holds for all $N \ge N_0$.
Each subset of $\set{1, \dots, N}$ with at least $\delta N$ elements contains an arithmetic progression of length $k$.
Szemerédi's original proof is combinatorial and introduced the [[Szemerédi regularity lemma|ETMAPNQY7T]].
Other proofs use ergodic theory (Furstenberg's multiple recurrence theorem), higher-order Fourier analysis with [[Gowers norms|ZBFZGA1J5G]], and hypergraph regularity.
The case $k = 3$ is [[Roth's theorem on arithmetic progressions|E7H8NWXESF]].
The density hypothesis cannot simply be dropped: the infinite set of powers of $2$ has density zero and contains no three-term arithmetic progression.
The primes also have density zero, so the theorem does not apply to them directly; the [[Green–Tao theorem|H8BAHWSEMC]] handles them separately.

## Explanations

### L1 human

If a set of whole numbers takes up a fixed positive fraction of the numbers, then it contains evenly spaced patterns of every length.
For example, it must contain three numbers like $10, 17, 24$, four numbers like $3, 8, 13, 18$, and so on.
This holds even if the set looks very irregular.

### L2 human

Szemerédi's theorem is the density version of [[van der Waerden's theorem|V96FRZ2ME1]]: in a finite colouring one colour class has positive upper density, so it contains long progressions.
It is a central theorem of [[additive combinatorics|3NM362A2JG]], and the search for good quantitative bounds led to Gowers norms and much of the modern theory.
The ergodic proof opened the field of ergodic Ramsey theory, with polynomial and multidimensional extensions.
</document>

<document source="https://tamnd.github.io/math2/node/B24T6DQCDR.md" tag="B24T6DQCDR" kind="concept">
# Stochastic optimal control

## Summary

Stochastic optimal control is [[optimal control|T3V6ZNAMDN]] of systems driven by noise, usually modelled by a [[stochastic differential equation|B219G9SHJA]].
The controller chooses its action at each time from the information observed so far and minimises an expected cost.
Dynamic programming leads to a second-order [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] for the value function.
The theory is used in engineering, in economics and in mathematical finance, for example in portfolio choice.

## Explanations

### L1 human

Think of steering a boat in gusty wind.
You cannot plan the whole path in advance, because the wind keeps pushing you off course.
Instead you decide at each moment what to do given where you are now.
A simple model is a position $X_t$ that moves with the chosen velocity $u_t$ plus a random [[Brownian|22735FGJV6]] kick.
The goal is to end near $0$ without spending too much effort.
The best rule turns out to be a feedback: push back towards $0$ in proportion to the current position.
</document>

<document source="https://tamnd.github.io/math2/node/B27HB43K86.md" tag="B27HB43K86" kind="statement">
# Geometrization conjecture

## Statement

There are eight model geometries, the simply connected homogeneous [[Riemannian manifolds|7CBT4SM83Y]]
$$S^3, \quad \R^3, \quad \mathbb{H}^3, \quad S^2 \times \R, \quad \mathbb{H}^2 \times \R, \quad \widetilde{SL}(2, \R), \quad \mathrm{Nil}, \quad \mathrm{Sol},$$
each with its maximal compatible group of isometries.
A manifold has a *geometric structure* modelled on such a geometry $X$ if it is a quotient $X / \Gamma$.
Here $\Gamma$ is a discrete group of isometries of $X$ acting freely.

Let $M$ be a closed connected [[orientable|702SNFQG0W]] prime [[3-manifold|R20TB2KEQQ]].
Then $M$ contains a finite collection of disjoint embedded [[incompressible|6N9GW98MR6]] tori with the following property.
The interior of each component of $M$ cut along these tori has a geometric structure of finite volume.

## Remarks

Thurston stated the conjecture around 1982 and proved it for Haken manifolds.
Perelman proved the general case in 2002 and 2003 with Hamilton's Ricci flow with surgery.
Except for some manifolds with Sol geometry, the tori can be taken to be those of the [[JSJ decomposition|TX4MSZ0SRP]], applied after the [[prime decomposition|FZH5T5340Q]].
For closed manifolds with finite fundamental group the conclusion is that $M = S^3 / \Gamma$ for a finite group $\Gamma$ acting freely by isometries, which contains the [[Poincaré conjecture|3R064KFF68]] as the case $\Gamma = 1$.
Examples: [[lens spaces|AZBTMF196E]] are spherical, the $3$-torus is Euclidean, and the complement of the figure-eight knot is hyperbolic.

## Explanations

### L1 human

Every closed orientable $3$-dimensional space can be cut along spheres and tori into pieces, each of which carries one of eight standard geometries.
The most common geometry, by far, is hyperbolic geometry.
So topology in dimension $3$ is controlled by geometry.

### L2 human

The theorem reduces many questions about $3$-manifolds to questions about the eight geometries.
Closed manifolds with six of the geometries are Seifert fiber spaces.
Closed Sol manifolds are torus bundles over the circle up to a double cover.
The hyperbolic pieces are the rich remaining case.
Consequences include the Poincaré conjecture, the classification of finite groups acting freely on $S^3$, and, with later work of Agol and Wise, the virtually Haken conjecture.
</document>

<document source="https://tamnd.github.io/math2/node/B2CYR292J6.md" tag="B2CYR292J6" kind="concept">
# Catenary ring

## Summary

A catenary ring is a [[commutative ring|0310REJJY8]] in which, between any two nested [[prime ideals|00MHHZ9SNE]], all maximal chains of primes have the same finite length.
In such a ring the distance between two primes is well defined.
Most rings met in practice are catenary, such as finitely generated algebras over a field and Cohen–Macaulay rings.
Nagata showed that some Noetherian local domains are not catenary.

## Explanations

### L1 human

Think of the prime ideals as floors of a building, ordered by inclusion.
A chain of primes is a staircase, and a maximal chain is one with no floor skipped.
The ring is catenary when every staircase between two given floors has the same number of steps.
For example, in $\Z$ the only chains from $(0)$ to a prime $(p)$ are $(0) \subsetneq (p)$, all of length $1$.
The [[definition|K6PPGRT5E1]] makes this precise.
</document>

<document source="https://tamnd.github.io/math2/node/B2EMR1MA1H.md" tag="B2EMR1MA1H" kind="definition">
# Eulerian number

## Definition

Let $n \ge 1$ be an integer and let $\pi$ be a [[permutation|0BQMWCNS8S]] of $\set{1, \ldots, n}$.
A *descent* of $\pi$ is an index $i$ with $1 \le i \le n - 1$ and $\pi(i) > \pi(i + 1)$.
For an integer $k$, the *Eulerian number* $A(n, k)$ is the number of permutations of $\set{1, \ldots, n}$ with exactly $k$ descents.

## Remarks

Example: for $n = 3$, the permutations in one-line notation are $123$, $132$, $213$, $231$, $312$ and $321$.
Their numbers of descents are $0, 1, 1, 1, 1, 2$, so $A(3, 0) = 1$, $A(3, 1) = 4$ and $A(3, 2) = 1$.

Non-example: the index $i = 3$ is not a descent of $\pi = 2\,4\,1\,3$, because $\pi(3) = 1 < 3 = \pi(4)$.
The descents of $\pi$ are only $i = 2$, so $\pi$ is counted by $A(4, 1)$.

Other notations are $\left\langle {n \atop k} \right\rangle$ and $E(n, k)$.
The numbers satisfy $A(n, k) = A(n, n - 1 - k)$ and $\sum_{k} A(n, k) = n!$.
They obey the recurrence $A(n, k) = (k + 1) A(n - 1, k) + (n - k) A(n - 1, k - 1)$.
An explicit formula is
$$A(n, k) = \sum_{j=0}^{k+1} (-1)^j \binom{n + 1}{j} (k + 1 - j)^n.$$
Worpitzky's identity states that $x^n = \sum_{k=0}^{n-1} A(n, k) \binom{x + k}{n}$ for $n \ge 1$.
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<document source="https://tamnd.github.io/math2/node/B2TRERFWEH.md" tag="B2TRERFWEH" kind="statement">
# Magnus's Freiheitssatz

## Statement

Let $F$ be the [[free group|0KTW26C80J]] on a set $X$, and let $r \in F$ be a cyclically reduced word.
Let $G = \langle X \mid r \rangle$ be the [[one-relator group|68AFPHR4GN]] it defines.
Let $Y$ be a subset of $X$ that does not contain some generator that occurs in $r$.
Then the [[subgroup|0X1JXT47GA]] of $G$ generated by the images of the elements of $Y$ is free, and these images form a free basis of it.

## Remarks

Magnus proved the theorem by induction on the length of $r$.
After a change of generators, $G$ is written as an [[HNN extension|YEYERRY37K]] or an amalgamated product of one-relator groups with shorter relators, and the induction applies to them.
The same method shows that the [[word problem|PBGTTAVMMT]] of one-relator groups is solvable.

Example: in $\Z^2 = \langle a, b \mid a b a^{-1} b^{-1} \rangle$, the theorem says that $a$ generates an infinite cyclic subgroup, a free group of rank $1$.
In the [[Baumslag–Solitar group|429JJ5ZNBY]] $\langle a, b \mid b a b^{-1} a^{-2} \rangle$, both $a$ and $b$ have infinite order.

The word $r$ must be cyclically reduced.
For $r = b a b^{-1}$, which contains $b$, the group $\langle a, b \mid b a b^{-1} \rangle$ is infinite cyclic generated by $b$, and $a = 1$ in it.
So $\set{a}$ omits $b$, but $a$ does not generate a free group of rank $1$.
The cyclic reduction of $r$ is $a$, which does not contain $b$.

## Explanations

### L1 human

Freiheitssatz is German for "freeness theorem".
A one-relator group is a free group in which one rule is imposed.
The theorem says that the rule cannot affect any set of generators that leaves out a letter of the rule.
Those generators stay as free as they were before.

### L2 human

The theorem is the starting point of the theory of one-relator groups.
Magnus's method of writing a one-relator group as an HNN extension of a group with a shorter relator is the main tool of the area.
Suppose that $r$ is cyclically reduced and that every generator in $X$ occurs in $r$.
Then every proper subset of $X$ is a free basis of a free subgroup of $G$.
</document>

<document source="https://tamnd.github.io/math2/node/B2VP96NJSK.md" tag="B2VP96NJSK" kind="concept">
# Additive category

## Summary

An additive category is a [[preadditive category|SC6FD5F4J0]] with a zero object in which any two objects have a [[biproduct|BZEJKTQ76T]].
Its morphisms between finite direct sums can be written as matrices of morphisms, composed by matrix multiplication.
Categories of modules, of free modules and of vector bundles are additive.
Additive categories are the natural home of chain complexes and of the homotopy category of chain complexes.

## Explanations

### L1 human

In an additive category one can add maps, and one can form direct sums of objects.
A map from $A \oplus B$ to $C \oplus D$ is then a $2 \times 2$ matrix whose entries are maps from $A$ or $B$ to $C$ or $D$.
This is exactly how one computes with linear maps between spaces split into pieces.
For example, finite-dimensional vector spaces over a field form an additive category, and a linear map $K^2 \to K^3$ is a $3 \times 2$ matrix.
</document>

<document source="https://tamnd.github.io/math2/node/B2YBWQQMHA.md" tag="B2YBWQQMHA" kind="statement">
# Poincaré–Lelong formula

## Statement

Let $X$ be a connected [[complex manifold|JSGWAN2RJV]] and let $f$ be a [[holomorphic function|507C34W0ZT]] on $X$ that is not identically $0$.
Let $Z_1, Z_2, \dots$ be the irreducible components of the hypersurface $f^{-1}(0)$.
Let $m_j \ge 1$ be the order of vanishing of $f$ at a general point of $Z_j$.
Let $[Z_j]$ be the current of integration over the regular part of $Z_j$, and let $[Z_f] = \sum_j m_j [Z_j]$.
Then, in the sense of [[currents|360H7MD9RD]] on $X$,
$$\frac{i}{\pi} \partial\bar\partial \log\abs{f} = [Z_f] .$$

## Remarks

Proof idea: off $f^{-1}(0)$ the function $\log\abs{f}$ is pluriharmonic, so both sides vanish there.
Near a regular point of $f^{-1}(0)$ one can choose coordinates with $f = u \, z_1^{m}$ for a holomorphic unit $u$.
Then $\log\abs{u}$ is pluriharmonic, and the claim reduces to the one-variable identity $\Delta \log\abs{z_1} = 2\pi \delta_0$ on each line $z' = \text{const}$.
The difference of the two sides is then a closed current of order zero supported in the singular set of $f^{-1}(0)$.
This set has codimension at least $2$, so such a current of bidegree $(1, 1)$ vanishes by Federer's support theorem.
Example: for $X = \C$ and $f(z) = z^2 (z - 1)$ the formula says $\frac{1}{2\pi} \Delta \log\abs{f} = 2 \delta_0 + \delta_1$, the classical statement that $\log\abs{f}$ has Riesz measure given by the zeros with multiplicity.
Consequence: $[Z_f]$ is $dd^c$ of a [[plurisubharmonic function|FF2P7ER3QT]], so it is a closed [[positive current|BYJHP6A906]].
Let $s$ be a holomorphic section of a holomorphic line bundle with a smooth Hermitian metric.
The same argument shows that $\frac{i}{\pi} \partial\bar\partial \log\abs{s}$ equals $[Z_s]$ minus a smooth closed form that represents the first Chern class of the bundle.

## Explanations

### L1 human

In one complex variable, $\log\abs{f}$ for a polynomial $f$ is harmonic except at the zeros, where it behaves like $m \log\abs{z - a}$.
Its Laplacian is a sum of point masses at the zeros, weighted by their multiplicities.
The Poincaré–Lelong formula says the same in several variables.
The complex second derivative of $\log\abs{f}$ is concentrated on the zero set of $f$, and it equals integration over it, counted with multiplicity.

### L2 human

The formula is the bridge between divisors and positive currents.
It shows that every effective divisor gives a closed positive $(1,1)$-current, and it lets one compute with zero sets through the plurisubharmonic function $\log\abs{f}$.
It is used in Nevanlinna theory in several variables and in the theory of [[Lelong numbers|WVB60RKC3Z]].
For example, the Lelong number of $\log\abs{f}$ at a point is the multiplicity of $f$ there.
The line bundle version links the zero divisor of a section with the curvature of the bundle, which is the starting point of many vanishing and extension theorems.
</document>

<document source="https://tamnd.github.io/math2/node/B2YPDYMR7E.md" tag="B2YPDYMR7E" kind="definition">
# Differentially closed field

## Definition

Let $(K, \partial)$ be a [[differential field|GA5PTQGC6G]] of characteristic $0$ with one [[derivation|3J89REV9WQ]] $\partial$.
Let $K\{y\}$ be the ring of differential polynomials over $K$.
It is the polynomial ring $K[y, y', y'', \dots]$ in the variables $y^{(k)}$, $k \ge 0$, with $\partial$ extended by $\partial(y^{(k)}) = y^{(k+1)}$.
The *order* of $f \in K\{y\}$ is the largest $k$ such that $y^{(k)}$ occurs in $f$; a nonzero element of $K$ has order $-1$.
For $a \in K$, $f(a)$ is the element obtained by replacing each $y^{(k)}$ by $\partial^k a$.
Let $f, g \in K\{y\}$ be nonzero, with the order of $g$ less than the order of $f$.
The field $K$ is *differentially closed* if, for all such $f$ and $g$, there is $a \in K$ with $f(a) = 0$ and $g(a) \ne 0$.

## Remarks

Taking $f$ of order $0$, that is $f \in K[y]$ nonconstant, and $g = 1$ shows that a differentially closed field is [[algebraically closed|0PF6TM8TWT]].
Its field of constants $\set{c \in K : \partial c = 0}$ is also algebraically closed.
Every differential field of characteristic $0$ is contained in a differentially closed field, and it has a differential closure, which is unique up to isomorphism over it.
The theory $\mathrm{DCF}_0$ of differentially closed fields of characteristic $0$ is complete, has elimination of quantifiers and is $\omega$-stable.

Example: by the existence result above, the differential closure of $(\Q, 0)$ is a differentially closed field.
Differentially closed fields are known through such existence results rather than by explicit construction.

Non-example: $\C(x)$ with $\partial = d/dx$ is not differentially closed.
Take $f = x y' - 1$, of order $1$, and $g = 1$, of order $-1$.
A solution would be $a \in \C(x)$ with $a' = 1/x$, and no rational function has this derivative.

Non-example: a field with the zero derivation is not differentially closed, since $f = y' - 1$ has no zero in it.
</document>

<document source="https://tamnd.github.io/math2/node/B2ZJW0YBQ7.md" tag="B2ZJW0YBQ7" kind="statement">
# Varignon's theorem

## Statement

Let $ABCD$ be a simple quadrilateral, that is a simple [[polygon|8VEFWD3EJB]] with four vertices, in the Euclidean plane.
Let $K$, $L$, $M$, $N$ be the [[midpoints|0S706JBZBG]] of $AB$, $BC$, $CD$, $DA$.
Then $KLMN$ is a parallelogram.
Its sides are parallel to the diagonals $AC$ and $BD$ and half as long, and its area is half the area of $ABCD$.

## Remarks

With vectors, $L - K = \frac{1}{2}(C - A) = M - N$, so $KL$ and $NM$ are parallel and equal, which makes $KLMN$ a parallelogram.
In the same way $KN$ and $LM$ equal $\frac{1}{2}(D - B)$.
The area of $ABCD$ is $\frac{1}{2}\abs{(C - A) \times (D - B)}$, and the area of $KLMN$ is $\abs{\frac{1}{2}(C - A) \times \frac{1}{2}(D - B)}$, which is half of it.
Example: for the square with vertices $(0, 0), (2, 0), (2, 2), (0, 2)$, the midpoints form a square of side $\sqrt 2$ and area $2$, half of $4$.
The parallelogram part needs no simplicity; for a self-crossing quadrilateral the area relation fails, and the parallelogram can even degenerate.

## Explanations

### L1 human

Draw any four-sided figure and mark the middle of each side.
Joining these four midpoints always gives a parallelogram.
This works even for very irregular quadrilaterals, and the parallelogram covers exactly half of the area.

### L2 human

The proof is the midpoint theorem for triangles applied twice, once in each triangle cut off by a diagonal.
It is a good first example of how vectors turn a geometric statement into one line of algebra.
</document>

<document source="https://tamnd.github.io/math2/node/B30ACDNS76.md" tag="B30ACDNS76" kind="statement">
# Uhlenbeck's removable singularity theorem

## Statement

Let $B \subseteq \R^4$ be the open unit ball with a smooth Riemannian metric, and let $G$ be a compact Lie group.
Let $P \to B \setminus \set{0}$ be a [[principal $G$-bundle|8E37E74XVP]].
Let $A$ be a smooth [[connection|NJ21PN5ZGJ]] on $P$ that satisfies the [[Yang–Mills equations|JJWETW5AT8]] $d_A^* F_A = 0$ on $B \setminus \set{0}$.
Assume that $A$ has finite energy:
$$\int_{B \setminus \set{0}} \abs{F_A}^2 < \infty,$$
where $F_A$ is the [[curvature|ZAJFW1S078]] of $A$.
Then $P$ extends to a principal $G$-bundle $\bar{P}$ over $B$.
There is a [[gauge transformation|707DEWCCFX]] $u$ on $B \setminus \set{0}$ such that $u^* A$ extends to a smooth Yang–Mills connection on $\bar{P}$.

## Remarks

The proof uses the conformal invariance of the energy in dimension $4$.
On dyadic annuli around $0$ the energy tends to $0$, so Uhlenbeck's gauge fixing gives good gauges on each annulus.
A decay estimate shows that $\abs{F_A}$ is in fact bounded near $0$.
Then a Coulomb gauge on the whole ball and elliptic regularity give the smooth extension.
The finite energy hypothesis cannot be dropped.
A connection that extends smoothly over $0$ has bounded curvature near $0$, and the energy is gauge invariant.
So a Yang–Mills connection with infinite energy on every punctured ball around $0$ never extends over $0$ in any gauge.
Example: a finite-energy anti-self-dual connection on $\R^4$ pulls back, via stereographic projection, to an instanton on $S^4$ minus a point.
The theorem extends it over that point, so it comes from a bundle over $S^4$.
Its energy is a fixed constant times the absolute value of the second Chern number of that bundle.

## Explanations

### L1 human

A Yang–Mills field defined everywhere except at one point of four-dimensional space might be badly singular there.
The theorem says that if the total energy of the field is finite, then the singularity is only apparent.
After a change of gauge, the field extends smoothly across the point.

### L2 human

The theorem is the reason why finite-energy instantons on $\R^4$ correspond to instantons on the sphere $S^4$, and why their energy is quantised.
It is also the second ingredient, next to the Uhlenbeck compactness theorem, in the description of limits of sequences of instantons.
The limit connection away from the bubble points has finite energy, so it extends over those points to a connection on a new bundle.
</document>

<document source="https://tamnd.github.io/math2/node/B33WNK1SVN.md" tag="B33WNK1SVN" kind="definition">
# Harish-Chandra module

## Definition

Let $G$ be a [[real reductive Lie group|42ZCE14AD2]] with Lie algebra $\mathfrak g_0$, and let $\mathfrak g = \mathfrak g_0 \otimes_\R \C$ be its complexified Lie algebra.
Let $K$ be a [[maximal compact subgroup|T2EB8R9HEV]] of $G$, with Lie algebra $\mathfrak k_0 \subseteq \mathfrak g_0$, acting on $\mathfrak g$ by the [[adjoint representation|P6ZP0TE07M]] $\mathrm{Ad}$.
A $(\mathfrak g, K)$-module is a complex vector space $V$ with a representation of $\mathfrak g$, written $X \cdot v$, and a representation $\pi$ of $K$, such that:

1. each $v \in V$ lies in a finite-dimensional $\pi(K)$-invariant subspace on which $K$ acts continuously;
2. $\frac{d}{dt}\big|_{t=0} \pi(\exp tX) v = X \cdot v$ for all $X \in \mathfrak k_0$ and $v \in V$;
3. $\pi(k)(X \cdot v) = (\mathrm{Ad}(k) X) \cdot (\pi(k) v)$ for all $k \in K$, $X \in \mathfrak g$ and $v \in V$.

A *Harish-Chandra module* is a $(\mathfrak g, K)$-module $V$ such that:

4. $V$ is finitely generated as a module over the [[universal enveloping algebra|SCZ8PRVY00]] $U(\mathfrak g)$;
5. each [[irreducible representation|FX3NEFYFFN]] of $K$ occurs in $V$ with finite multiplicity.

## Remarks

The derivative in clause 2 makes sense by clause 1, since it is taken inside a finite-dimensional space.

Example: let $\pi$ be a finite-dimensional continuous representation of $G$ on $V$, with differential $d\pi$ extended complex-linearly to $\mathfrak g$.
Then $V$ with $X \cdot v = d\pi(X) v$ and $\pi|_K$ is a Harish-Chandra module.
Clause 1 holds because $V$ itself is finite-dimensional, clauses 2 and 3 hold by the definition of $d\pi$ and by $\pi(k) \pi(\exp tX) \pi(k)^{-1} = \pi(\exp t \mathrm{Ad}(k) X)$, and clauses 4 and 5 hold because $\dim V < \infty$.

Non-example: an infinite direct sum of copies of the trivial one-dimensional module, with $\mathfrak g$ acting by $0$ and each $k \in K$ acting as the identity, is a $(\mathfrak g, K)$-module.
It is not a Harish-Chandra module, because the trivial representation of $K$ occurs infinitely often and clause 5 fails.

If $\pi$ is an [[admissible representation|SZV8Y806Y7]] of $G$ on a Hilbert space $H$, its $K$-finite vectors are smooth and form a Harish-Chandra module when $\pi$ has finite length.
Two such representations are called infinitesimally equivalent when their Harish-Chandra modules are isomorphic.
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# Malmquist's theorem

## Statement

Let $R(z, w)$ be a [[rational function|0XDK9CGYKJ]] of two complex variables.
Suppose the [[ordinary differential equation|0NT64JV96A]]
$$w'(z) = R\big(z, w(z)\big)$$
has a transcendental [[meromorphic|0ZZ7ZYDR8W]] solution on $\C$.
Here transcendental means that the solution is not a rational function.
Then there are rational functions $a_0, a_1, a_2$ of $z$ such that
$$R(z, w) = a_0(z) + a_1(z) w + a_2(z) w^2.$$
So the equation is a [[Riccati equation|0VT2QK6X6Q]] or a linear equation.

## Remarks

A proof with Nevanlinna theory compares the growth of both sides: if $R$ had degree $d$ in $w$, then $T(r, R(z, w)) = d\,T(r, w) + O(\log r)$, while $T(r, w')$ is at most $2T(r, w)$ plus a small error.
This bounds the degree, and a study of the poles of $w$ excludes nonpolynomial dependence on $w$.
Example: $w' = 1 + w^2$ has the transcendental meromorphic solution $w = \tan z$, and its right side is a polynomial of degree $2$ in $w$, as the theorem requires.
In contrast, the solutions of $w' = w^3$ are $w = \pm (c - 2z)^{-1/2}$ and $w = 0$; they have branch points, as the theorem predicts, so no transcendental meromorphic solution exists.
The hypothesis that the solution is transcendental is needed: $w = z$ solves $w' = 1 + (w - z)^3$, whose right side has degree $3$ in $w$.

## Explanations

### L1 human

Most first-order differential equations have solutions with branch points, which are not single-valued in the whole plane.
Take a first-order equation whose right side is rational.
Malmquist's theorem says that if it has a non-rational solution that is single-valued with only poles, then it must be a Riccati equation.
So, apart from Riccati equations, nonlinear first-order equations with rational right side have no such nice global solutions.

### L2 human

The theorem is the first-order prototype of the classification behind the Painlevé equations, where the same question is asked for second-order equations.
It also has an analogue for difference equations $w(z + 1) = R(z, w(z))$, studied in [[Nevanlinna theory of difference equations|9KAQ1REPE8]]: transcendental meromorphic solutions of finite order force strong restrictions on $R$.
Results of this kind are called Malmquist-type theorems.
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# Conley index

## Summary

The Conley index is a topological invariant attached to an isolated [[invariant set|WWZTW11664]] of a flow.
It is the [[homotopy type|SK0ZJGWXDP]] of a pointed space obtained from a compact neighbourhood by collapsing the part through which orbits leave.
It extends the Morse index of a nondegenerate critical point to invariant sets that may be large and complicated.
The index is unchanged under small perturbations of the flow, and a nontrivial index proves that the invariant set is nonempty.

## Explanations

### L1 human

Draw a box around a region where something interesting happens in a flow, such that nothing interesting touches the walls of the box.
Mark the part of the wall where the flow exits the box, and glue that exit part to a single point.
The shape of what results, up to continuous deformation, is the Conley index.
Take a saddle point in the plane, with one direction in and one direction out.
The box is a square, the exit is two opposite sides, and gluing them to a point gives a circle.
If the index is not the shape of a single point, there must be something invariant hidden inside the box.
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# Bombieri–Vinogradov theorem

## Statement

Let $q \ge 1$ and $a$ be integers, and let $y \ge 2$ be real.
Let $\psi(y; q, a)$ be the sum of the [[von Mangoldt function|0NK8JTE1NN]] $\Lambda(n)$ over $n \le y$ with $n \equiv a \pmod q$.
Let $\varphi$ be [[Euler's totient function|0DAMHCT17R]].
For every $A > 0$ there is $B = B(A) > 0$ such that, for all $x \ge 2$ and $Q = x^{1/2} (\log x)^{-B}$,
$$\sum_{q \le Q} \; \max_{\gcd(a, q) = 1} \; \max_{2 \le y \le x} \left| \psi(y; q, a) - \frac{y}{\varphi(q)} \right| \le C_A \frac{x}{(\log x)^{A}},$$
where $C_A$ depends only on $A$.

## Remarks

The proof combines the [[large sieve|N15YK7XG1H]] inequality for Dirichlet characters with a decomposition of $\Lambda$ into bilinear sums, such as Vaughan's identity; small moduli are handled by the [[Siegel–Walfisz theorem|XEPW251A97]].
The [[generalized Riemann hypothesis|DNKA4WBKDS]] would give $\abs{\psi(y; q, a) - y/\varphi(q)} \le C\, y^{1/2} (\log y)^2$ for each $q \le y$, which yields the same bound for $Q$ up to about $x^{1/2} (\log x)^{-A-2}$.
So the theorem says that GRH holds "on average over moduli" up to $\sqrt{x}$.
One says that the primes have level of distribution $1/2$.
The Elliott–Halberstam conjecture states that the same bound holds with $Q = x^{\theta}$ for every $\theta < 1$.

## Explanations

### L1 human

Primes are expected to split evenly among the residue classes modulo $q$ that they can reach.
For a single large $q$ nobody can prove a good error bound.
This theorem proves that, for almost all $q$ up to about $\sqrt{x}$, the error is small, which is as good as what the generalized Riemann hypothesis would give on average.

### L2 human

The Bombieri–Vinogradov theorem is the usual replacement for GRH in sieve theory.
It controls the remainder terms of a sieve applied to the primes, as in the proof of [[Chen's theorem|WE0RQTR68H]].
It is also the main input of the Goldston–Pintz–Yıldırım and Maynard methods for [[bounded gaps between primes|XMXFDWYYMD]].
Zhang's proof of bounded gaps needed a variant that goes slightly beyond the level $1/2$ for smooth moduli.
Compare with the [[Siegel–Walfisz theorem|XEPW251A97]], which is uniform in each single modulus but only for $q$ up to a power of $\log x$.
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# Krawtchouk polynomials

## Definition

Let $N \ge 0$ be an integer, let $0 < p < 1$, and let $n$ be an integer with $0 \le n \le N$.
Write $(a)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]].
The *Krawtchouk polynomial* of degree $n$ is
$$K_n(x; p, N) = \sum_{k=0}^{n} \frac{(-n)_k \, (-x)_k}{(-N)_k \, k!} \, p^{-k},$$
that is, $K_n(x; p, N) = {}_2F_1\left( -n, -x; -N; \frac{1}{p} \right)$.

## Remarks

The denominators $(-N)_k$ are nonzero because $k \le n \le N$.
They are orthogonal for the [[binomial distribution|0WCWA1KT3M]]:
$$\sum_{x=0}^{N} \binom{N}{x} p^x (1 - p)^{N - x} K_m(x; p, N) K_n(x; p, N) = \frac{\delta_{mn}}{\binom{N}{n}} \left( \frac{1 - p}{p} \right)^n.$$
Example: $K_1(x; p, N) = 1 - \frac{x}{Np}$.
Check: if $X$ is binomial with mean $Np$ and variance $Np(1 - p)$, then $E[K_1(X)] = 0$ and $E[K_1(X)^2] = \frac{Np(1 - p)}{N^2 p^2} = \frac{1}{N} \cdot \frac{1 - p}{p}$.
Non-example: the clause $n \le N$ matters; for $n > N$ the formula breaks down at $k = N + 1$, where $(-N)_k = 0$.
In coding theory one often uses the version $\mathcal{K}_k(x) = \sum_{j=0}^{k} (-1)^j (q - 1)^{k - j} \binom{x}{j} \binom{N - x}{k - j}$ for an alphabet of size $q$, which is a constant multiple of $K_k(x; 1 - 1/q, N)$.
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# Affine scheme

## Summary

An affine scheme is the [[spectrum|FB45BHHA2J]] $\operatorname{Spec} A$ of a [[commutative ring|0310REJJY8]] $A$, made into a [[locally ringed space|B7FPT8JTXR]] by its structure sheaf.
Every commutative ring gives one, and the ring can be read back as the ring of global functions.
Affine schemes are the local building blocks of all [[schemes|5A35TYZD02]], just as open balls are the local pieces of manifolds.

## Explanations

### L1 human

An affine scheme turns a ring into a geometric object.
Its points are the prime ideals of the ring, and the elements of the ring act as functions on these points.
For the polynomial ring $\C[x, y]$, the maximal ideals $(x - a, y - b)$ are the points $(a, b)$ of the complex plane, and other primes, such as $(y - x^2)$, stand for whole curves.
The structure sheaf records which fractions are allowed as functions on each open set.
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# Kissing number

## Summary

The kissing number $\tau(n)$ is the largest number of non-overlapping unit balls in $\R^n$ that can all touch one central unit ball.
It is known exactly only in dimensions $1$, $2$, $3$, $4$, $8$ and $24$, where it equals $2$, $6$, $12$, $24$, $240$ and $196560$.
The problem is a close relative of [[sphere packing|HF32CBN5MA]] and of the theory of [[spherical codes|SDCX4WDP4F]].

## Explanations

### L1 human

Put one coin on a table and ask how many equal coins can touch it without overlapping each other.
The answer is six, and they fit perfectly around it.
For balls in space the answer is twelve, but here there is some room to spare, which made the question hard.
In higher dimensions the numbers grow very quickly and are mostly unknown.
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# Symmetric derivative

## Summary

The symmetric derivative of a function at a point is a limit of slopes of secant lines.
The two points of each secant line lie symmetrically on both sides of the point.
When the ordinary [[derivative|0F1WR9C83K]] exists, the symmetric derivative exists and has the same value.
The converse fails, so the symmetric derivative is a genuine generalization.
Its quotient is also the central difference used to approximate derivatives numerically.

## Explanations

### L1 human

Instead of comparing $f(x + h)$ with $f(x)$, compare $f(x + h)$ with $f(x - h)$, and divide by the distance $2h$.
This balances the left and the right side.
For $\abs{x}$ at $0$, the two sides cancel: $\abs{h} - \abs{-h} = 0$.
So $\abs{x}$ has symmetric derivative $0$ at $0$, although it has a corner there.
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# Butterfly catastrophe

## Example

The *butterfly catastrophe* is the family of functions of one variable $x \in \R$
$$V(x; a, b, c, d) = x^6 + a x^4 + b x^3 + c x^2 + d x,$$
with four control parameters $(a, b, c, d) \in \R^4$.
For $a = b = c = d = 0$ the function is $x^6$, whose [[critical point|0YTCMFB4HC]] at $0$ is degenerate: the first five derivatives vanish there.

The derivative $V'(x) = 6x^5 + 4a x^3 + 3b x^2 + 2c x + d$ has degree $5$, so $V$ has at most five critical points.
Since $V(x) \to +\infty$ as $x \to \pm\infty$, nondegenerate critical points alternate minimum, maximum, minimum, maximum, minimum.
So $V$ can have three local minima, which the fold, cusp and swallowtail cannot.

Check with $a = -15/2$, $b = 0$, $c = 12$, $d = 0$.
Then $V(x) = x^6 - \tfrac{15}{2} x^4 + 12 x^2$ and
$$V'(x) = 6x^5 - 30x^3 + 24x = 6x(x^2 - 1)(x^2 - 4).$$
The critical points are $-2, -1, 0, 1, 2$.
The sign of $V'$ is negative on $(-\infty, -2)$ and changes at each of the five simple roots.
So $-2, 0, 2$ are local minima and $-1, 1$ are local maxima.
The region of parameters with three minima is the "pocket" of the butterfly, bounded by the [[bifurcation set|JV45S0KQE6]].

## Explanations

### L1 human

The butterfly is the elementary catastrophe of one state variable with four control knobs.
Its potential $x^6 + \cdots$ can have up to three valleys at once.
For the settings in the example, the valleys sit at $x = -2$, $0$ and $2$, with hills at $\pm 1$ between them.
A system that rests in the middle valley can be pushed into either side valley as the knobs move.
That extra middle state is what the butterfly adds to the simpler cusp.
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# Stallings–Swan theorem

## Statement

Let $G$ be a group.
Then $G$ has [[cohomological dimension|EF4ZWACHD8]] at most $1$ if and only if $G$ is a [[free group|0KTW26C80J]].

## Remarks

Stallings proved the theorem for finitely generated groups, and Swan extended it to all groups.

One direction is direct: if $G$ is free on a set $X$, then the augmentation ideal of the [[group ring|4NMTSX0XDS]] $\Z[G]$ is a free $\Z[G]$-module with basis $\set{x - 1 : x \in X}$.
This gives a free resolution $0 \to \bigoplus_{x \in X} \Z[G] \to \Z[G] \to \Z \to 0$ of length $1$.
For the converse with $G$ finitely generated and nontrivial, a group of dimension $1$ is torsion-free and has more than one end.
By [[Stallings theorem about ends of groups|GWZN3EKEWD]], it splits as a free product or HNN extension over a finite subgroup, which must be trivial.
Induction on the number of generators finishes the proof.

Example: every subgroup $H$ of a free group $F$ satisfies $\mathrm{cd}(H) \le \mathrm{cd}(F) \le 1$, so $H$ is free.
This recovers the [[Nielsen–Schreier theorem|W97X7WX5G8]].

Example: $\mathrm{cd}(\Z^2) = 2$, so $\Z^2$ is not free.

## Explanations

### L1 human

Free groups are the groups with no relations among their generators.
The theorem says they are also exactly the groups whose cohomology dies above degree one.
So an algebraic condition on cohomology detects a combinatorial property of the group.

### L2 human

Serre proved that a torsion-free group has the same cohomological dimension as its subgroups of finite index.
With this, the theorem shows that a torsion-free group with a free subgroup of finite index is itself free.
In topology, it says that every group of cohomological dimension at most $1$ has a graph as a [[classifying space|WA796ZCNRX]].
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# Coefficient of determination

## Summary

The coefficient of determination, written $R^2$, measures how much of the variation of a response is explained by a fitted regression model.
It compares the residual sum of squares with the total sum of squares around the mean.
For [[ordinary least squares|YM708KHV2W]] with an intercept, it lies between $0$ and $1$ and equals the squared correlation between the observed and the fitted values.
It is the most common summary of goodness of fit in [[linear regression|41FVC4P5JB]].

## Explanations

### L1 human

Imagine predicting each data point by the overall average; the squared errors add up to the total variation.
A regression line usually does better, and $R^2$ is the share of that total variation that the line removes.
An $R^2$ of $0.9$ means the fitted model leaves only a tenth of the original variation unexplained.
For the points $(0,0)$, $(1,1)$, $(2,3)$, the least squares line gives $R^2 = 27/28$.
A high $R^2$ does not prove that the model is correct or that one variable causes the other.
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# Integral geometry

## Definition

Let $G$ be a group acting on a space $X$.
*Integral geometry* is the study of $G$-invariant measures on spaces of geometric objects in $X$.
It also studies the integrals of geometric quantities against these measures.
In the classical case, $X$ is the [[Euclidean space|0XPR1E85W0]] $\R^n$ and $G$ is the [[Euclidean group|0H3YY8CPKH]] of rigid motions.
The objects are then the affine subspaces of a fixed dimension.

The basic example is the measure on [[lines|0MP9BCF6DP]] in $\R^2$.
For $\theta \in [0, \pi)$ and $p \in \R$, let $L(\theta, p) = \set{x \in \R^2 : x_1 \cos \theta + x_2 \sin \theta = p}$; every line is $L(\theta, p)$ for exactly one pair $(\theta, p)$.
Transport the [[measure|PR11BZQQPG]] $dp \, d\theta$ from $[0, \pi) \times \R$ to the set of lines by this parametrization.
The result is invariant under all rigid motions of $\R^2$, and up to a constant factor it is the only such measure.

## Remarks

Example: the lines that meet the closed disc of radius $r$ centred at the origin are the $L(\theta, p)$ with $\abs{p} \le r$, a set of measure $2 r \pi$.
A translation by a vector $a$ maps $L(\theta, p)$ to $L(\theta, p + a_1 \cos \theta + a_2 \sin \theta)$ and a rotation by $\alpha$ shifts $\theta$ by $\alpha$, so both preserve $dp \, d\theta$.
Non-example: writing non-vertical lines as $y = m x + c$, the measure $dm \, dc$ is not invariant under rotations.
In these coordinates the invariant measure is $(1 + m^2)^{-3/2} \, dm \, dc$.
The same ideas give invariant measures on affine $k$-planes of $\R^n$, and lead to the Crofton and kinematic formulas.
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# Symplectic integrator

## Summary

A symplectic integrator is a numerical method for [[Hamiltonian systems|0EDKGA2FAK]] whose one-step map is a symplectic map, like the exact flow.
Because it keeps this geometric structure, it avoids the slow drift of energy that ordinary methods show.
Its energy error stays small over very long times.
Examples are the symplectic Euler method, the [[Störmer–Verlet method|ESWE8Y3JD4]] or [[leapfrog method|3HGJCQBWVY]], and the Gauss–Legendre [[Runge–Kutta methods|7DBPJKYC6C]].
They are the standard tools in celestial mechanics and molecular dynamics.

## Explanations

### L1 human

Simulate a planet around the sun with the simple Euler method, and the planet slowly spirals outward, gaining energy it should not have.
A symplectic integrator is designed so that each step preserves areas in the space of positions and momenta, just as the true motion does.
The computed orbit then stays close to a true orbit of a slightly changed system, and the energy only wobbles.
For the harmonic oscillator, the Euler method multiplies $q^2 + p^2$ by $1 + h^2$ at each step, while the symplectic Euler method keeps a nearby quadratic quantity exactly.
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# Multilevel Monte Carlo method

## Statement

Let $P$ be a random variable with finite [[expected value|0WQNRCNY3M]].
Let $P_0, P_1, P_2, \ldots$ be random variables on the same probability space, the approximations of $P$ at levels $\ell = 0, 1, 2, \ldots$, and set $P_{-1} = 0$.
For each level $\ell$ let $Y_\ell$ be an estimator of $\mathbb E[P_\ell - P_{\ell - 1}]$ built from $N_\ell$ independent samples, with total cost $C_\ell$.
Let the $Y_\ell$ be independent.
Suppose there are positive constants $\alpha, \beta, \gamma, c_1, c_2, c_3$ with $\alpha \ge \frac12 \min(\beta, \gamma)$ such that for all $\ell \ge 0$:

1. $\abs{\mathbb E[P_\ell - P]} \le c_1 2^{-\alpha \ell}$;
2. $\mathbb E Y_\ell = \mathbb E[P_\ell - P_{\ell - 1}]$;
3. $\operatorname{Var} Y_\ell \le c_2 N_\ell^{-1} 2^{-\beta \ell}$;
4. $\mathbb E C_\ell \le c_3 N_\ell 2^{\gamma \ell}$.

Then there is a constant $c_4$ with the following property.
For every $0 < \varepsilon < e^{-1}$ there are a level $L$ and sample sizes $N_0, \ldots, N_L$ such that the estimator $Y = \sum_{\ell = 0}^{L} Y_\ell$ satisfies the following.
Its [[mean squared error|05NZ1HHQMV]] is $\mathbb E[(Y - \mathbb E P)^2] < \varepsilon^2$, and its expected total cost is
$$\mathbb E \sum_{\ell=0}^{L} C_\ell \le \begin{cases} c_4\, \varepsilon^{-2} & \text{if } \beta > \gamma, \\ c_4\, \varepsilon^{-2} (\ln \varepsilon)^2 & \text{if } \beta = \gamma, \\ c_4\, \varepsilon^{-2 - (\gamma - \beta)/\alpha} & \text{if } \beta < \gamma. \end{cases}$$

## Remarks

The telescoping sum $\mathbb E P_L = \sum_{\ell=0}^{L} \mathbb E[P_\ell - P_{\ell-1}]$ shows that $Y$ estimates $\mathbb E P_L$ without bias, so its mean squared error is the squared bias $(\mathbb E[P_L - P])^2$ plus $\sum_\ell \operatorname{Var} Y_\ell$.
One chooses $L$ by clause 1 so that the bias is below $\varepsilon/\sqrt2$.
The $N_\ell$ then minimise the cost under the constraint $\sum_\ell \operatorname{Var} Y_\ell \le \varepsilon^2/2$.
A Lagrange multiplier gives $N_\ell$ proportional to $\sqrt{V_\ell / c_\ell}$, where $V_\ell$ and $c_\ell$ are the variance and cost of one sample on level $\ell$.
The variance decay in clause 3 comes from coupling: $P_\ell$ and $P_{\ell-1}$ are computed from the same random input, so their difference is small.
Example: consider $\mathbb E\, \varphi(S_T)$, where $S$ solves a stochastic differential equation and $\varphi$ is Lipschitz.
Under standard conditions, the Euler–Maruyama scheme with $2^\ell$ time steps gives $\alpha = 1$, $\beta = 1$ and $\gamma = 1$.
So the cost is $O(\varepsilon^{-2} (\ln \varepsilon)^2)$, against $O(\varepsilon^{-3})$ for plain Monte Carlo with the same scheme.
In the case $\beta > \gamma$ the cost has the same order $\varepsilon^{-2}$ as plain Monte Carlo with exact samples of $P$ at unit cost.

## Explanations

### L1 human

Plain Monte Carlo with an approximate model must use a fine and expensive model for every sample.
The multilevel method uses many cheap samples of a coarse model and only a few expensive samples of the fine model, to correct the coarse answer.
The corrections have small variance, because the fine and coarse models are run with the same random input, so few samples are needed for them.
Overall the work can drop from $\varepsilon^{-3}$ to about $\varepsilon^{-2}$ for accuracy $\varepsilon$.

### L2 human

The method, introduced by Giles for stochastic differential equations, is related to earlier multilevel work of Heinrich on parametric integration.
It is a recursive [[variance reduction|A01GEGWKVB]]: each level acts as a control variate for the next.
The three regimes of the theorem say whether the cost is dominated by the coarsest levels ($\beta > \gamma$), spread evenly ($\beta = \gamma$), or dominated by the finest level ($\beta < \gamma$).
The Milstein scheme improves the strong convergence and gives $\beta = 2$ in many scalar problems, which moves the Euler–Maruyama case $\beta = \gamma$ to the optimal regime $\beta > \gamma$.
The same framework applies to partial differential equations with random coefficients, where the levels are finite element meshes.
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# Codimension of a singularity

## Summary

The codimension of a singularity of a smooth function counts how many independent conditions a function must satisfy to have that singularity.
Equivalently, it is the number of parameters a family of functions needs before the singularity occurs in it in a persistent way.
Persistent means that small perturbations of the family do not remove it.
It is computed from the Jacobian ideal of the [[germ|7SWSRGDT4A]].
It is also the least number of parameters of a [[versal unfolding|KSAFWZVB8K]], not counting the constant term.
In [[catastrophe theory|6FKJ6SXP3G]] it sorts the elementary catastrophes: the fold has codimension $1$ and the cusp has codimension $2$.

## Explanations

### L1 human

A nondegenerate minimum, like the bottom of the bowl $x^2$, is typical: it survives any small change of the function, so its codimension is $0$.
A flat inflection like $x^3$ at $0$ is rare: a small change $x^3 + a x$ destroys it unless $a = 0$, so it costs one condition, and its codimension is $1$.
It appears unavoidably only in families with at least one parameter.
The flatter point $x^4$ needs two parameters, $x^4 + a x^2 + b x$, so its codimension is $2$.
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<document source="https://tamnd.github.io/math2/node/B4QR306D8G.md" tag="B4QR306D8G" kind="concept">
# Riemannian distance

## Summary

The Riemannian distance between two points of a connected [[Riemannian manifold|7CBT4SM83Y]] is the infimum of the lengths of the curves that join them.
It turns the manifold into a [[metric space|0VKD1TP7F3]] whose topology is the manifold topology.
This distance is the bridge between the infinitesimal data of the [[Riemannian metric|Q61VFHCH2J]] and global questions such as completeness, diameter and shortest paths.

## Explanations

### L1 human

A Riemannian metric tells how long a tiny step is at each point.
Adding up the tiny steps gives the length of a whole path.
The distance between two points is the length of the shortest way between them, or the limit of shorter and shorter ways when no shortest one exists.
On the round unit sphere, the distance between the north pole and a point on the equator is $\pi/2$, the length of a quarter of a great circle.
Remove one point from the plane, and take two points on opposite sides of the hole.
Their distance is the length of the straight segment, but no path in the punctured plane has this length.
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<document source="https://tamnd.github.io/math2/node/B4TTNHSTRX.md" tag="B4TTNHSTRX" kind="concept">
# Elliptic partial differential equation

## Summary

An elliptic partial differential equation is one whose [[principal symbol|F31T650DKY]] vanishes for no nonzero covector.
For second-order equations this means that the matrix of second-order coefficients is definite.
Elliptic equations model equilibrium states, such as steady temperatures, electrostatic potentials and membranes at rest.
Their solutions are as smooth as the data allow, and boundary value problems are the natural problems for them.
See the [[definition|3BYHP9E9VF]].

## Explanations

### L1 human

The model elliptic equation is the [[Laplace equation|S8RG6N1PRT]] $\partial_x^2 u + \partial_y^2 u = 0$.
Its solutions, the [[harmonic functions|0DEB0JMN0W]], have the property that the value at a point is the average of the values on any circle around it.
So an elliptic equation averages things out: there are no preferred directions, no time, and no travelling waves.
A soap film stretched over a wire frame is a picture of a solution: its shape inside is determined by the frame.
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<document source="https://tamnd.github.io/math2/node/B4TYSQQC38.md" tag="B4TYSQQC38" kind="definition">
# Band-limited function

## Definition

Let $f \in L^2(\R)$, and let $\hat f \in L^2(\R)$ be its [[Fourier transform|DPVM34ADZV]], normalized so that $\hat f(\xi) = \int_\R f(x) e^{-2\pi i x \xi}\, dx$ for integrable $f$.
Let $B > 0$.
The function $f$ is *band-limited* with bandwidth $B$ if $\hat f(\xi) = 0$ for almost every $\xi$ with $\abs{\xi} > B$.
It is *band-limited* if it is band-limited with some bandwidth $B > 0$.

## Remarks

If $f$ is band-limited with bandwidth $B$, then $\hat f$ lies in $L^2[-B, B] \subset L^1[-B, B]$, and $f$ agrees almost everywhere with the continuous function $x \mapsto \int_{-B}^{B} \hat f(\xi) e^{2\pi i x \xi}\, d\xi$; one always uses this continuous representative.

Example: $f(x) = \sin(\pi x)/(\pi x)$, with $f(0) = 1$, lies in [[$L^2(\R)$|SASP14MA1Z]], and $\int_{-1/2}^{1/2} e^{2\pi i x \xi}\, d\xi = f(x)$.
So $\hat f$ is the indicator of $[-\frac12, \frac12]$, and $f$ is band-limited with bandwidth $\frac12$.

Non-example: $f(x) = e^{-\pi x^2}$ has $\hat f(\xi) = e^{-\pi \xi^2}$, which is positive everywhere, so $f$ is not band-limited.

By the [[Paley–Wiener theorem|THY12GPZ8C]], the band-limited functions with bandwidth $B$ are exactly the restrictions to $\R$ of certain entire functions.
These are the entire functions of exponential type at most $2\pi B$ that are square-integrable on $\R$.
In particular a nonzero band-limited function cannot vanish on an interval.
The same definition is used for tempered distributions, with the condition that the support of the distribution $\hat f$ lies in $[-B, B]$.
In this wider sense $\sin(2\pi B x)$ is band-limited, though not in $L^2$.
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<document source="https://tamnd.github.io/math2/node/B4VK33PCXZ.md" tag="B4VK33PCXZ" kind="concept">
# Regular prime

## Summary

An odd [[prime number|01826KMG2H]] $p$ is regular if it does not divide the [[class number|KTKK9B1X41]] of the [[cyclotomic field|NN77VYSGXR]] $\Q(\zeta_p)$.
Kummer introduced regular primes and proved [[Fermat's Last Theorem|0Y5X3SDBCZ]] for every regular prime exponent.
By Kummer's criterion, $p$ is regular exactly when $p$ divides none of the numerators of the [[Bernoulli numbers|0KC00K7KY4]] $B_2, B_4, \ldots, B_{p-3}$.
The smallest irregular prime is $37$.

## Explanations

### L1 human

Kummer tried to prove Fermat's Last Theorem by factoring $x^p + y^p$ in the ring generated by a $p$-th root of unity.
That ring does not always have unique factorisation, and the class number measures the failure.
When $p$ does not divide this class number, the failure is mild enough for his argument to work; such primes are called regular.
All odd primes below $37$ are regular, while $37$ is not, because $37$ divides the numerator of $B_{32}$.
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<document source="https://tamnd.github.io/math2/node/B4YD1ZXRVH.md" tag="B4YD1ZXRVH" kind="concept">
# Curve of constant width

## Summary

A curve of constant width is the boundary of a plane [[convex body|437FSG3419]] with this property.
Every two parallel supporting lines of the body are at the same distance, called the width.
The circle is the simplest example, but there are many others, such as the Reuleaux triangle.
Curves of constant width are a classical source of extremal problems in plane convex geometry.
All of them with a given width have the same length, but their areas differ.

## Explanations

### L1 human

Put the shape between two parallel rulers that touch it on opposite sides.
If the distance between the rulers is the same however the shape is turned, the boundary of the shape is a curve of constant width.
A coin with this property could roll smoothly between two boards, which is why such shapes can serve as rollers.
A circle of radius $1$ has constant width $2$.
A square does not: its width is $1$ across the sides and $\sqrt{2}$ across the corners.
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<document source="https://tamnd.github.io/math2/node/B4Z8C36HEB.md" tag="B4Z8C36HEB" kind="definition">
# Linear system of divisors

## Definition

Let $X$ be a normal [[projective variety|A4H4CZ2V35]] over an algebraically closed field $k$, and let $D$ be a [[Cartier divisor|FHV05ZK99F]] on $X$.
Two Cartier divisors are *linearly equivalent* if their difference is principal.
The *complete linear system* $\abs{D}$ is the set of effective Cartier divisors on $X$ that are linearly equivalent to $D$.
Let $V = H^0(X, \mathcal{O}_X(D))$, a finite-dimensional vector space, where $\mathcal{O}_X(D)$ is the [[invertible sheaf|7JCF2TE3V5]] of $D$.
Each nonzero $s \in V$ has a divisor of zeros $(s)_0 \in \abs{D}$, and $s \mapsto (s)_0$ gives a bijection between $(V \setminus \set{0})/k^*$ and $\abs{D}$.
A *linear system* is a subset of $\abs{D}$ of the form $\set{(s)_0 : s \in W \setminus \set{0}}$ for a linear subspace $W \subseteq V$.
Its dimension is $\dim W - 1$.

## Remarks

The *base locus* of a linear system is the intersection of the supports of all its divisors; the system is *base-point-free* if the base locus is empty.

Example: on $X = \mathbb{P}^n_k$ with $D$ a hyperplane, $V$ is the space of linear forms in $n+1$ variables.
So $\abs{D}$ is the set of all hyperplanes, a linear system of dimension $n$, and it is base-point-free.

Non-example: on $\mathbb{P}^1_k$, the set of divisors $\set{p, \; p + q}$ for two points $p, q$ is not a linear system.
The two divisors have degrees $1$ and $2$, so they are not linearly equivalent and do not lie in one $\abs{D}$.

A base-point-free linear system of dimension $r$, with a basis $s_0, \dots, s_r$ of $W$, gives a morphism $X \to \mathbb{P}^r_k$, $x \mapsto [s_0(x) : \dots : s_r(x)]$.
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<document source="https://tamnd.github.io/math2/node/B505P3MKTZ.md" tag="B505P3MKTZ" kind="concept">
# Lindelöf summation

## Summary

Lindelöf summation is a [[summability method|8X25Q4XYMN]] that damps the $n$-th term of a series by the factor $n^{-hn}$ and lets $h$ decrease to $0$.
The damping factors decay faster than any geometric sequence, so the damped series converges for many divergent series.
The method is regular: it gives the usual sum to every convergent series.
Applied to the Taylor series of an analytic function, it sums the series in the whole [[Mittag-Leffler star|C21ECQ1H84]] of the function.

## Explanations

### L1 human

Multiply the $n$-th term of a series by $n^{-hn}$ for a small positive number $h$.
For large $n$ this factor is tiny, so even a series with fast-growing terms can converge after damping.
As $h$ shrinks to $0$, every factor tends to $1$ and the damping disappears.
If the damped sums approach a limit, that limit is the Lindelöf sum.
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<document source="https://tamnd.github.io/math2/node/B51J3BC610.md" tag="B51J3BC610" kind="statement">
# Exterior cone condition

## Statement

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]], and let $\xi \in \partial\Omega$ be a [[boundary|05D5EGMFYV]] point.
Suppose there is a truncated closed circular cone $C$ with vertex $\xi$ and nonempty interior such that $C \cap \Omega = \emptyset$.
Then $\xi$ is a [[regular boundary point|B7948M365K]] of $\Omega$.
Suppose every boundary point of $\Omega$ satisfies this condition.
Then the Dirichlet problem on $\Omega$ has a solution continuous on $\overline{\Omega}$ for every continuous boundary function.

## Remarks

Proof idea: regularity at $\xi$ only gets easier when $\Omega$ shrinks, so it suffices to treat the complement of the cone near $\xi$.
For that set one can build a barrier of the form $\abs{x - \xi}^{\alpha} \phi(\theta)$, where $\theta$ is the angle from the axis of the cone.
Here $\phi$ solves an eigenvalue problem on the sphere.
For $n \ge 3$ one can instead use [[Wiener's criterion|STMT1K1865]].
The cone puts in each dyadic shell a ball of radius comparable to the shell, so each term of Wiener's series is bounded below.
Example: every bounded open set with Lipschitz boundary, such as a cube or a convex polytope, satisfies the condition at each boundary point.
The condition is sufficient but not necessary: in $\R^3$ a tip whose cross-sections shrink like a power of the distance is still regular, although no cone fits.
Without such a condition regularity can fail: the centre of a punctured ball has no exterior cone and is irregular.

## Explanations

### L1 human

If you can place a small solid ice-cream cone outside the set with its tip at a boundary point, then that point is good for the Dirichlet problem.
The solution then takes the prescribed value there.
Points on smooth boundaries and corners of polygons and polyhedra all pass this test.

### L2 human

The exterior cone condition is the usual practical way to check that a domain is regular.
It generalizes the exterior sphere condition, where a ball outside the set touches the boundary at the point, and it covers all Lipschitz domains.
It is a consequence of Wiener's criterion, and its barrier gives quantitative control of how fast the solution approaches its boundary values.
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<document source="https://tamnd.github.io/math2/node/B538T38WDS.md" tag="B538T38WDS" kind="definition">
# Analytic function between Banach spaces

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\mathbb{K}$.
For $n \ge 1$, let $L^n(X, Y)$ be the space of continuous [[multilinear maps|0443XNARDV]] $A \colon X^n \to Y$, with the norm $\norm{A} = \sup \set{\norm{A(h_1, \ldots, h_n)} : \norm{h_1}, \ldots, \norm{h_n} \le 1}$.
Let $L^0(X, Y) = Y$.
Let $U \subseteq X$ be open, let $f \colon U \to Y$ and let $a \in U$.
The map $f$ is *analytic at $a$* if there are $A_n \in L^n(X, Y)$ for $n \ge 0$ and a real $r > 0$ such that:

1. $\sup_n \norm{A_n} r^n < \infty$.
2. $f(a + h) = \sum_{n=0}^{\infty} A_n(h, \ldots, h)$ for all $h \in X$ with $\norm{h} < r$.

The map $f$ is *analytic on $U$* if it is analytic at each point of $U$.

## Remarks

Clause 1 makes the series in clause 2 converge absolutely, since $\norm{A_n(h, \ldots, h)} \le \norm{A_n} \norm{h}^n$.
The maps $h \mapsto A_n(h, \ldots, h)$ are the continuous homogeneous polynomials of degree $n$; one may take each $A_n$ symmetric.
For $X = Y = \R$, this is the usual [[analytic function|081WKEDK6C]].
An analytic map is $C^\infty$ in the sense of the Fréchet derivative.

Example: let $B$ be a Banach algebra with unit $1$ and $\norm{xy} \le \norm{x} \norm{y}$, and let $f(x) = (1 - x)^{-1}$ for $\norm{x} < 1$.
Take $A_n(h_1, \ldots, h_n) = h_1 h_2 \cdots h_n$, so $\norm{A_n} \le 1$ for $n \ge 1$, and $r = 1$.
The Neumann series $\sum_n x^n$ converges and equals $(1 - x)^{-1}$, so $f$ is analytic at $0$.

Non-example: the [[smooth non-analytic function|09RTYYHXKA]] $f(t) = e^{-1/t^2}$, $f(0) = 0$, on $\R$ is not analytic at $0$.
All its derivatives at $0$ vanish, so any series as in clause 2 would be $0$, but $f(t) \neq 0$ for $t \neq 0$.
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<document source="https://tamnd.github.io/math2/node/B53THC8F79.md" tag="B53THC8F79" kind="definition">
# Asymptotic expansion

## Definition

Let $D$ be a subset of $\R$ or of $\C$, and let $x_0$ be a limit point of $D$, possibly $\infty$.
An *asymptotic sequence* at $x_0$ is a sequence of functions $\varphi_0, \varphi_1, \varphi_2, \ldots$ on $D$ such that
$$\varphi_{n+1}(x) = o(\varphi_n(x)) \quad \text{as } x \to x_0 \text{ in } D$$
for each $n \ge 0$, where $o$ is [[little o notation|04TSHB9H8D]].
Let $f$ be a function on $D$ and let $a_0, a_1, a_2, \ldots$ be complex numbers.
The formal series $\sum_{n \ge 0} a_n \varphi_n$ is an *asymptotic expansion* of $f$ at $x_0$ if for each $N \ge 0$
$$f(x) - \sum_{n=0}^{N} a_n \varphi_n(x) = o(\varphi_N(x)) \quad \text{as } x \to x_0 \text{ in } D.$$
This is written $f(x) \sim \sum_{n \ge 0} a_n \varphi_n(x)$ as $x \to x_0$.

## Remarks

Example: on $D = (1, \infty)$ with $x_0 = \infty$ and $\varphi_n(x) = x^{-n-1}$, one has $\frac{1}{x-1} \sim \sum_{n \ge 0} x^{-n-1}$.
Check: $\frac{1}{x-1} - \sum_{n=0}^{N} x^{-n-1} = \frac{x^{-N-1}}{x-1}$, and this is $o(x^{-N-1})$ because $\frac{1}{x-1} \to 0$.
Non-example: $\varphi_0(x) = x^{-1}$, $\varphi_1(x) = x^{-1}$ is not an asymptotic sequence at $\infty$, because $\varphi_1 = o(\varphi_0)$ fails.
For a fixed asymptotic sequence, the coefficients are unique: $a_N$ is the limit of $\big(f(x) - \sum_{n<N} a_n \varphi_n(x)\big)/\varphi_N(x)$ as $x \to x_0$, whenever $\varphi_N$ does not vanish near $x_0$.
Different functions can have the same expansion: $e^{-x}$ and $0$ both have the zero expansion in powers of $x^{-1}$ as $x \to +\infty$.
The most common case is the power scale $\varphi_n(x) = x^{-n}$ at $\infty$ or $\varphi_n(z) = z^n$ at $0$; an equivalent condition is then that the error after $N$ terms is $O(x^{-N-1})$, respectively $O(z^{N+1})$.
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# Linear PDE

## Definition

Let $\Omega$ be an open set in $\R^n$ and let $m \ge 1$.
A *linear partial differential equation* of [[order|E7JDFDPM1X]] $m$ in $\Omega$ is a [[partial differential equation|GC5BDZ6QBC]] of the following form.
$$\sum_{\abs{\alpha} \le m} a_\alpha(x) \, \partial^\alpha u(x) = f(x), \quad x \in \Omega.$$
Here the sum runs over [[multi-indices|0PJXBR6QA3]] $\alpha$.
The coefficients $a_\alpha$ and the right-hand side $f$ are given functions on $\Omega$, and $a_\alpha$ is not identically zero for some $\alpha$ with $\abs{\alpha} = m$.
The map $P = \sum_{\abs{\alpha} \le m} a_\alpha \partial^\alpha$ is the associated *linear differential operator*.
The equation is *homogeneous* if $f = 0$ and *inhomogeneous* otherwise.
It has *constant coefficients* if every $a_\alpha$ is constant.

## Remarks

Example: $\partial_x^2 u + x^2 \partial_y^2 u + e^y u = \sin x$ on $\R^2$ is linear of order $2$, with coefficients $1$, $x^2$ and $e^y$.
Non-example: $\partial_x^2 u + u^3 = 0$ is not linear, because the term $u^3$ is not a coefficient times $u$.
For a homogeneous linear equation the solutions in $C^m(\Omega)$ form a vector space.
For an inhomogeneous equation, any two solutions differ by a solution of the homogeneous equation.
In short notation the equation reads $P u = f$.

## Explanations

### L1 human

A linear PDE is a weighted sum of derivatives of the unknown, with weights that depend only on the point, set equal to a known function.
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# Carleman's inequality

## Statement

Let $a_1, a_2, \dots$ be nonnegative real numbers.
Then
$$\sum_{n=1}^{\infty} (a_1 a_2 \cdots a_n)^{1/n} \le e \sum_{n=1}^{\infty} a_n,$$
where both [[series|0CVVKY7GY2]] take values in $[0, \infty]$.

## Remarks

Proof idea: let $c_k = (k + 1)^k / k^{k-1}$, so that $c_1 c_2 \cdots c_n = (n + 1)^n$.
By the [[AM–GM inequality|0DRF0F3XC9]],
$$(a_1 \cdots a_n)^{1/n} = \frac{(c_1 a_1 \cdots c_n a_n)^{1/n}}{n + 1} \le \frac{1}{n(n+1)} \sum_{k=1}^{n} c_k a_k.$$
Summing over $n$ and using $\sum_{n \ge k} \frac{1}{n(n+1)} = \frac{1}{k}$ gives the bound $\sum_k (1 + 1/k)^k a_k$, and $(1 + 1/k)^k < e$.

The constant $e$ cannot be replaced by a smaller number.
If $0 < \sum a_n < \infty$, the inequality is strict.

## Explanations

### L1 human

Replace each term of a series of nonnegative numbers by the geometric mean of the terms up to it.
Carleman's inequality says the new series is at most $e$ times the old one.
In particular, if the old series converges, so does the new one.

### L2 human

Carleman's inequality is a limiting case of the discrete [[Hardy's inequality|S7A8YFGG8X]].
Apply Hardy's inequality to $a_n^{1/p}$ and let $p \to \infty$: the power means tend to geometric means, and $\big(\frac{p}{p-1}\big)^p \to e$.
The direct proof above shows how a well chosen weighting makes the AM–GM inequality sharp enough.
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# Difference kernel

## Summary

A difference kernel is a [[kernel|GSMBXT7FPK]] of the form $K(x, s) = k(x - s)$, which depends only on the difference of its two variables.
Its integral operator is a [[convolution|2Z29JN3VES]] with $k$, so it commutes with translations.
Integral equations with difference kernels are called convolution equations.
Transforms that turn convolutions into products, such as the [[Laplace transform|0V49JEZW06]] and the Fourier transform, reduce them to algebraic equations.

## Explanations

### L1 human

The influence of the point $s$ on the point $x$ depends only on how far apart they are, and in which direction.
Shifting both points by the same amount does not change the kernel.
For example, $K(x, s) = e^{-\abs{x - s}}$ is a difference kernel with $k(t) = e^{-\abs{t}}$.
The Volterra equation $\varphi(x) = f(x) + \int_0^x (x - s) \varphi(s) \, ds$ has the difference kernel with $k(t) = t$.
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# Period and index

## Summary

The period and the index are two numbers attached to a central simple algebra $A$ over a field $K$.
The index is the [[degree|NNXSP5KKAC]] of the division algebra $D$ with $A \cong M_m(D)$.
The period is the order of the class of $A$ in the [[Brauer group|BEW48TA9WZ]].
The period divides the index, and the two numbers have the same prime divisors.
How far apart they can be is the period-index problem, a central question about Brauer groups of fields.

## Explanations

### L1 human

Every central simple algebra is a matrix algebra over a division algebra, and the index measures the size of that division algebra.
The period counts how many copies of the algebra must be tensored together before the result becomes a plain matrix algebra.
For Hamilton's quaternions over $\R$, both numbers are $2$: the quaternions have degree $2$, and $\mathbb{H} \otimes_\R \mathbb{H}$ is a matrix algebra.
Over number fields the two numbers are always equal, but over other fields the index can be larger.
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# Peirce decomposition

## Statement

Let $R$ be a [[ring|0894J980RK]] and let $e_1, \dots, e_n$ be a [[complete set of orthogonal idempotents|SS8T32TMEX]] of $R$.
For $1 \le i, j \le n$ put $R_{ij} = e_i R e_j = \set{e_i r e_j : r \in R}$.
Then:

1. $R = \bigoplus_{i,j=1}^{n} R_{ij}$ as an abelian group, and the component of $r \in R$ in $R_{ij}$ is $e_i r e_j$.
2. $R_{ij} R_{kl} \subseteq R_{il}$ if $j = k$, and $R_{ij} R_{kl} = 0$ if $j \ne k$.
3. Each $R_{ii}$ is a ring with unit element $e_i$, under the addition and multiplication of $R$.

## Remarks

The proof writes $r = 1 \cdot r \cdot 1 = \sum_{i,j} e_i r e_j$, which gives the sum.
The sum is direct: if $\sum_{i,j} e_i r_{ij} e_j = 0$, then multiplying by $e_k$ on the left and $e_l$ on the right leaves $e_k r_{kl} e_l = 0$, by the [[idempotent|0X4JKZXJ86]] and orthogonality relations.
Part 2 follows from $e_j e_k = 0$ for $j \ne k$.
For a single idempotent $e$ and $f = 1 - e$ this reads $R = eRe \oplus eRf \oplus fRe \oplus fRf$.
Example: in $M_2(K)$ with $e = E_{11}$, the four pieces are the matrices supported on one of the four entries, and the decomposition is the splitting of a matrix into its entries.
In general $R$ is isomorphic to the ring of $n \times n$ matrices $(a_{ij})$ with $a_{ij} \in R_{ij}$, with the usual matrix operations.

## Explanations

### L1 human

Idempotents act like projections, and a complete set of them splits the unit element into pieces.
Cutting every element of the ring on both sides by these pieces writes the ring as a "matrix" of smaller parts.
For a matrix ring with the diagonal matrix units, this is just splitting a matrix into its entries.

### L2 human

The Peirce decomposition is the basic tool for reducing questions about a ring to the corner rings $e R e$.
It shows that a ring with a complete set of orthogonal idempotents looks like a generalized matrix ring.
If the idempotents are central, the off-diagonal pieces vanish and $R$ is the direct product $R_{11} \times \cdots \times R_{nn}$.
It is used in the structure theory of Artinian rings and when idempotents are lifted modulo an ideal.
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# Jacobian conjecture

## Statement

Let $k$ be a field of [[characteristic|0F6ES3BRYP]] $0$ and let $n \ge 1$.
Let $F_1, \ldots, F_n \in k[x_1, \ldots, x_n]$ be polynomials, and let $F = (F_1, \ldots, F_n) \colon k^n \to k^n$.
Suppose that the [[Jacobian determinant|0ZCSMYHTBP]] $\det \left( \frac{\partial F_i}{\partial x_j} \right)$ is a nonzero constant in $k$.
Then there are polynomials $G_1, \ldots, G_n \in k[x_1, \ldots, x_n]$ such that $G = (G_1, \ldots, G_n)$ satisfies $G \circ F = F \circ G = \id$.

## Remarks

The converse holds: if $F$ has a polynomial inverse $G$, the chain rule gives $\det J_F \cdot (\det J_G \circ F) = 1$, so $\det J_F$ is a unit of the [[polynomial ring|0SDC934R52]], that is, a nonzero constant.
The conjecture is open for every $n \ge 2$; the case $n = 1$ is easy, since a polynomial with constant nonzero derivative has degree $1$.
The question was posed by Ott-Heinrich Keller in 1939.
Characteristic $0$ is needed: over $\F_p$, the polynomial $F(x) = x - x^p$ has derivative $1$, but it has degree $p$ and is not invertible by a polynomial.
It is known that the conjecture holds when all $F_i$ have degree at most $2$.
Bass, Connell and Wright showed that it suffices to prove it, for all $n$, for maps of the form $F = x + H$ with $H$ homogeneous of degree $3$.

## Explanations

### L1 human

A polynomial map whose Jacobian determinant is a nonzero constant is invertible near each point of $\R^n$ or $\C^n$, by the inverse function theorem.
The conjecture says that it is then invertible everywhere, and that the inverse is again given by polynomials.
For one variable this is clear, but already for two variables nobody knows a proof or a counterexample.
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<document source="https://tamnd.github.io/math2/node/B5N9WJH5F0.md" tag="B5N9WJH5F0" kind="conjecture">
# Homological mirror symmetry

## Statement

Let $X$ be a compact [[Calabi–Yau manifold|STYAB4PFDW]], viewed as a symplectic manifold.
Let $\check{X}$ be a Calabi–Yau manifold that is mirror to $X$ in the sense of [[mirror symmetry|YK3DMN2NSK]].
Then the [[derived category|5ST7NVJFSM]] of coherent sheaves $D^b\mathrm{Coh}(\check{X})$ is equivalent, as a triangulated category, to the split-closed derived [[Fukaya category|4HS60JABC5]] $D^\pi \mathcal{F}(X)$.
The same holds with the roles of $X$ and $\check{X}$ exchanged.

## Remarks

Kontsevich proposed the conjecture in his address to the International Congress of Mathematicians in Zürich in 1994, as a categorical explanation of mirror symmetry.
The conjecture is not a single precise statement, because "mirror" is not defined in general.
It is proved in many cases where a mirror pair is given by an explicit construction.
Polishchuk and Zaslow proved it for elliptic curves, Seidel for the quartic surface, Abouzaid and Smith for the four-dimensional torus, and Sheridan for Calabi–Yau hypersurfaces in projective space.
Versions for Fano manifolds, whose mirrors are Landau–Ginzburg models, replace the derived category on one side by a category of matrix factorizations or a Fukaya–Seidel category.

## Explanations

### L1 human

Mirror symmetry is a duality from string theory that pairs two very different spaces.
On one side one looks at the symplectic geometry of a space, through its Lagrangian submanifolds.
On the other side one looks at the algebraic geometry of the mirror space, through its coherent sheaves.
Homological mirror symmetry predicts that these two worlds have the same algebraic structure, packaged as a category.
For an elliptic curve, this means that closed curves on a torus correspond to vector bundles and sheaves on the mirror elliptic curve.
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<document source="https://tamnd.github.io/math2/node/B5YP8CY8DQ.md" tag="B5YP8CY8DQ" kind="definition">
# Symplectic manifold

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]].
A *symplectic form* on $M$ is a [[differential $2$-form|0E3G0GRPZ6]] $\omega$ on $M$ such that:

1. $\omega$ is [[closed|0ZV14XRNF0]]: $d\omega = 0$, where $d$ is the [[exterior derivative|004FRCXH6M]];
2. $\omega$ is nondegenerate: for each $p \in M$ and each nonzero $v \in T_p M$ there is $w \in T_p M$ with $\omega_p(v, w) \ne 0$.

A *symplectic manifold* is a pair $(M, \omega)$ of a smooth manifold and a symplectic form on it.

## Remarks

Example: $\R^{2n}$ with coordinates $(x_1, \ldots, x_n, y_1, \ldots, y_n)$ and $\omega_0 = \sum_{k=1}^n dx_k \wedge dy_k$.
It has constant coefficients, so $d\omega_0 = 0$.
If $v \ne 0$ has a nonzero component $v_{x_k}$, then $\omega_0(v, \partial_{y_k}) = v_{x_k} \ne 0$, and similarly if $v_{y_k} \ne 0$ one pairs with $\partial_{x_k}$; so $\omega_0$ is nondegenerate.

Non-example: the sphere $S^4$ has no symplectic form.
On a closed manifold of dimension $2n$, $\omega^n$ is a volume form, so the class $[\omega]^n$ is nonzero and hence $[\omega] \ne 0$ in $H^2_{dR}(M)$; but $H^2_{dR}(S^4) = 0$.

Clause 2 says that each $(T_p M, \omega_p)$ is a [[symplectic vector space|PQ7JJY0VPH]], so $\dim M$ is even.
Other standard examples are oriented surfaces with an area form and cotangent bundles $T^* Q$ with the form $-d\theta$ built from the tautological one-form $\theta$.
Darboux's theorem says that near each point there are coordinates in which $\omega = \omega_0$.
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<document source="https://tamnd.github.io/math2/node/B5ZWZHMD12.md" tag="B5ZWZHMD12" kind="definition">
# Differentiation operator p

## Definition

Let $C^1[0, \infty)$ be the space of continuously differentiable functions $f \colon [0, \infty) \to \C$, with the one-sided [[derivative|0V1EJCT8G6]] at $0$.
The *differentiation operator* $p \colon C^1[0, \infty) \to C[0, \infty)$ is $p f = f'$.
Its companion, the integration operator $p^{-1} \colon C[0, \infty) \to C^1[0, \infty)$, is
$$(p^{-1} f)(t) = \int_0^t f(\tau) \, d\tau.$$

## Remarks

The two operators satisfy
$$p \, p^{-1} f = f \quad (f \in C[0, \infty)), \qquad p^{-1} p f = f - f(0) \quad (f \in C^1[0, \infty)).$$
The first identity is the fundamental theorem of calculus: $p^{-1} f$ is the [[antiderivative|0BYWRWFT4F]] of $f$ that vanishes at $0$.
So $p^{-1}$ is a right inverse of $p$ but not a left inverse.

Example: for $f(t) = e^{at}$, $p f = a f$, and $p^{-1} f = (e^{at} - 1)/a$ for $a \ne 0$; then $p \, p^{-1} f = e^{at} = f$.

Non-example of a two-sided inverse: for the constant $f = 1$, $p^{-1} p f = p^{-1} 0 = 0 \ne 1$; the clause $p^{-1} p = \id$ fails exactly because $f(0) \ne 0$.

In the Mikusiński calculus, the integration operator is convolution with the constant $1$, and the corresponding element $s = 1/l$ of the field of operators satisfies $s f = f' + f(0)$, which repairs the defect above.
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# Consistency of summability methods

## Definition

Let $A$ and $B$ be [[summability methods|8X25Q4XYMN]] for series of real or complex numbers.
For a series $\sum a_n$ that is $A$-summable, write $A\text{-}\sum a_n$ for the value that $A$ assigns to it, and similarly for $B$.
The methods $A$ and $B$ are *consistent* if for every series $\sum a_n$ that is both $A$-summable and $B$-summable,
$$A\text{-}\sum a_n = B\text{-}\sum a_n.$$

## Remarks

Example: Cesàro summation and Abel summation are consistent.
Indeed, if $\sum a_n$ is Cesàro summable to $s$, then it is Abel summable to the same $s$, so whenever both methods apply their values agree.

Non-example: let $A$ assign to a series the limit of its even [[partial sums|0NB643K60Q]] $s_0, s_2, s_4, \dots$, and let $B$ assign the limit of its odd partial sums $s_1, s_3, s_5, \dots$.
Both methods give the usual sum to every convergent series.
For $1 - 1 + 1 - 1 + \cdots$, the even partial sums are all $1$ and the odd ones are all $0$.
So both methods sum this series, with values $1$ and $0$, and $A$ and $B$ are not consistent.

The same notion applies to methods that act on sequences rather than series, by comparing the values they assign to a common summable sequence.
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# Presburger arithmetic

## Example

*Presburger arithmetic* is the first-order theory of the [[natural numbers|0MWKJ44V5A]] with $0$, $1$ and addition.
It is the set of all sentences in the signature $\set{0, 1, +}$ that are true in $(\N, 0, 1, +)$.
It can be axiomatized by the axioms $x + 1 \ne 0$, $x + 1 = y + 1 \to x = y$, $x + 0 = x$, $x + (y + 1) = (x + y) + 1$, and the induction scheme for all formulas of this signature.

It does not have quantifier elimination in $\set{0, 1, +}$.
It has quantifier elimination after adding $<$ and, for each $n \ge 2$, a unary predicate $D_n(x)$ for "$n$ divides $x$".
Check on one formula: in $\N$, the formula $\exists y \, (x = y + y)$ says that $x$ is even.
It is equivalent to the quantifier-free formula $D_2(x)$.
It is not equivalent to any quantifier-free formula in $\set{0, 1, +}$, since those define finite or cofinite subsets of $\N$.
The set of even numbers is neither finite nor cofinite.

## Remarks

Quantifier elimination gives an algorithm that decides, for every sentence of the signature, whether it is true in $\N$.
So Presburger arithmetic is [[complete|655K8Y7VS8]] and decidable.
Mojżesz Presburger proved this in 1929.
In contrast, the theory of $(\N, 0, 1, +, \cdot)$ is undecidable, and the [[Peano axioms|0QFT7Z76N5]] do not decide all its sentences, by Gödel's incompleteness theorems.
Multiplication of two variables is therefore not definable from addition in $\N$.
The same method applies to the [[integers|0C4JS9RF5M]] with addition and order, where divisibility by $n$ is [[congruence modulo $n$|0WAY5WX8NY]] to $0$.

## Explanations

### L1 human

Presburger arithmetic is arithmetic of the natural numbers with addition but without multiplication.
Every statement in this language can be decided by a computer, unlike full arithmetic with multiplication.
The trick is that "there exists" can be removed if one also allows the order and statements such as "$x$ is divisible by $3$".
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# Pappus's hexagon theorem

## Statement

Let $k$ be a [[field|04C5ZBG1WT]], and work in the [[projective plane|DANFNN8C4Y]] $\mathbb{P}^2(k)$.
Let $g$ and $h$ be two distinct lines meeting at a point $O$.
Let $A, B, C$ be three distinct points of $g$ and $a, b, c$ three distinct points of $h$, all different from $O$.
Then the three points
$$X = Ab \cap aB, \quad Y = Ac \cap aC, \quad Z = Bc \cap bC$$
are [[collinear|0AFNNTRTJE]].

## Remarks

Here $Ab$ denotes the line through $A$ and $b$; the two lines in each intersection are distinct, so $X$, $Y$, $Z$ are well defined.
One proof chooses coordinates in which $g$ and $h$ are coordinate lines and $O$ is a coordinate point.
It then computes $X$, $Y$, $Z$ and checks that the $3 \times 3$ determinant of their coordinates vanishes, using $xy = yx$ for scalars.
Another proof views $g \cup h$ as a degenerate conic and applies [[Pascal's theorem|94AKWRHND0]] in the limiting case.

The commutativity of $k$ is necessary: over a noncommutative division ring $D$, Pappus's theorem fails in $\mathbb{P}^2(D)$ for suitable points.
In fact a projective plane satisfies Pappus's theorem exactly when it is isomorphic to $\mathbb{P}^2(k)$ for a field $k$.

Example: in the real affine plane let $g$ be $y = 0$ and $h$ be $y = 1$, which meet at infinity.
Take $A = (0,0)$, $B = (1,0)$, $C = (2,0)$ and $a = (0,1)$, $b = (1,1)$, $c = (2,1)$.
Then $X = (1/2, 1/2)$, $Y = (1, 1/2)$, $Z = (3/2, 1/2)$, and all three lie on $y = 1/2$.

## Explanations

### L1 human

Put three points on one line and three points on another line.
Connect them crosswise in pairs, like the laces of a shoe, and mark where the crossing lines meet.
The three marks always lie on a straight line.
The proof is a small piece of algebra, and the key step is that multiplication of numbers is commutative.

### L2 human

Pappus's theorem is the configuration theorem that encodes commutativity of the coordinates.
A plane in which it holds is called Pappian, and by Hessenberg's theorem every Pappian plane is Desarguesian, so it also satisfies [[Desargues's theorem|5Z19Y02MME]].
For finite planes the two notions agree, because a finite division ring is commutative by Wedderburn's little theorem.
It is the special case of [[Pascal's theorem|94AKWRHND0]] in which the conic degenerates into the pair of lines $g$ and $h$.
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# P-adic dynamics

## Summary

$p$-adic dynamics studies the iteration of maps defined over the [[$p$-adic numbers|BT6T702C28]] and their extensions, such as rational functions and convergent power series.
It is a [[discrete-time dynamical system|5AA9AJEARJ]] theory in which distances come from a [[non-Archimedean absolute value|1H5F1BP6XS]].
The strong triangle inequality makes the behaviour of orbits very different from the real or complex case.
It is a basic tool of [[arithmetic dynamics|BE4X0QG8XW]], where $p$-adic methods control orbits of points over number fields.

## Explanations

### L1 human

In $p$-adic numbers two numbers are close when their difference is divisible by a high power of $p$.
Iterating a map in this world means following a point step by step and watching how its $p$-adic size changes.
Discs either nest inside each other or do not meet, so orbits often move in very rigid patterns.
For example, the map $x \mapsto x + 1$ on the $p$-adic integers sends $0$ to $1, 2, 3, \dots$, and these points come arbitrarily close to every $p$-adic integer.
</document>

<document source="https://tamnd.github.io/math2/node/B6E54YTR64.md" tag="B6E54YTR64" kind="concept">
# Numerical analysis

## Summary

Numerical analysis is the study of algorithms that compute approximate solutions to problems of continuous mathematics.
It designs methods for tasks such as solving equations, computing integrals, finding eigenvalues and solving differential equations.
It also proves how large the error of such a method can be, and how fast the method converges.
Two typical sources of error are the [[truncation error|4AKCQERM89]] of a finite method and the [[round-off error|CPN6SD3BTV]] of [[floating-point arithmetic|RTKRSC3KPH]].

## Explanations

### L1 human

Many equations have no solution that one can write down with a formula, such as $\cos x = x$.
Numerical analysis gives step-by-step recipes that produce a number as close to the true answer as one wishes.
A computer works with a finite number of digits, so each step can introduce a tiny error.
Part of the subject is to check that these tiny errors stay small and do not grow out of control.
For example, the [[bisection method|7VMYC7MB3Q]] halves an interval around the solution again and again.
After $20$ steps the solution is known to within about one millionth of the starting length.
</document>

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# QR algorithm

## Summary

The QR algorithm computes all the [[eigenvalues|0BQAA2BK3Q]] of a square matrix.
It repeatedly factors the matrix as $QR$ by the [[QR decomposition|0ERWG4NXQ9]] and multiplies the factors in the reverse order, $RQ$.
Each step is a [[similarity|0E89HQATM8]] by an [[orthogonal matrix|0W02QESPJQ]], so the eigenvalues do not change.
The matrices tend to an [[upper triangular|0N4XY7Z3NN]] form with the eigenvalues on the diagonal.
With shifts and a preliminary reduction to [[Hessenberg form|YCEEWJTX56]], it is the standard method for dense eigenvalue problems.

## Explanations

### L1 human

The algorithm takes a matrix and splits it into an orthogonal part $Q$, which preserves lengths and angles, and a triangular part $R$.
Then it puts them back together the other way round.
The new matrix has the same eigenvalues as the old one.
Repeating this pushes the entries below the diagonal towards zero, so the eigenvalues appear on the diagonal.
For $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$, one step gives $\begin{pmatrix} 2.8 & 0.6 \\ 0.6 & 1.2 \end{pmatrix}$, already closer to the eigenvalues $3$ and $1$ on the diagonal.
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<document source="https://tamnd.github.io/math2/node/B6GY2KJMWW.md" tag="B6GY2KJMWW" kind="concept">
# Locally convex space

## Summary

A locally convex space is a [[topological vector space|T32W553W95]] in which every neighbourhood of $0$ contains a convex neighbourhood of $0$.
Equivalently, its topology is [[generated by a family of seminorms|ZVFXQ7S6FY]].
Normed spaces, [[Fréchet spaces|3GAA614673]], spaces of smooth functions and spaces of distributions are all locally convex.
Local convexity is what makes the Hahn–Banach theorem work, so these spaces have many continuous linear forms and a rich duality theory.

## Explanations

### L1 human

In a normed space the small balls around $0$ are convex: the segment between two points of a ball stays in the ball.
A locally convex space keeps this feature, but the small neighbourhoods need not come from one norm.
They may come from many seminorms used together.
For example, the space of all real sequences, with convergence meaning convergence of each coordinate, is locally convex but has no norm giving that convergence.
Some natural spaces fail local convexity, such as $L^p([0, 1])$ for $0 < p < 1$.
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<document source="https://tamnd.github.io/math2/node/B6H4EY6R4Q.md" tag="B6H4EY6R4Q" kind="definition">
# Topological homogeneity

## Definition

A [[topological space|0CQHFPGJ45]] $X$ is *homogeneous* if for all $x, y \in X$ there is a [[homeomorphism|0QT00PB5G2]] $h \colon X \to X$ with $h(x) = y$.

## Remarks

Example: every [[topological group|92700QAB5Y]] $G$ is homogeneous.
For $x, y \in G$, left translation by $y x^{-1}$ is a homeomorphism with inverse the translation by $x y^{-1}$, and it sends $x$ to $y$.
Non-example: the closed interval $[0, 1]$.
A homeomorphism $h$ with $h(0) = 1/2$ would restrict to a homeomorphism from $(0, 1]$, which is connected, onto $[0, 1] \setminus \set{1/2}$, which is not connected.
In the language of group actions, $X$ is homogeneous when its homeomorphism group acts transitively on $X$.
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<document source="https://tamnd.github.io/math2/node/B6K478R9XE.md" tag="B6K478R9XE" kind="definition">
# Yang–Mills functional

## Definition

Let $(M, g)$ be a compact oriented [[Riemannian manifold|7CBT4SM83Y]] without boundary, with volume form $\mathrm{vol}_g$.
Let $G$ be a compact [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{g}$.
Fix an inner product on $\mathfrak{g}$ that is invariant under the adjoint action of $G$.
Let $P \to M$ be a [[principal $G$-bundle|8E37E74XVP]], and let $\operatorname{ad} P = P \times_G \mathfrak{g}$ be its adjoint bundle.
For a [[connection|NJ21PN5ZGJ]] $A$ on $P$, let $F_A$ be its [[curvature|ZAJFW1S078]], a $2$-form on $M$ with values in $\operatorname{ad} P$.
The metric $g$ and the inner product on $\mathfrak{g}$ give a pointwise norm $\abs{F_A}$.
The *Yang–Mills functional* is the map from the space of connections on $P$ to $[0, \infty)$ given by
$$\mathrm{YM}(A) = \frac{1}{2} \int_M \abs{F_A}^2 \, \mathrm{vol}_g.$$

## Remarks

Many authors omit the factor $\frac{1}{2}$; this does not change the critical points.
The functional is invariant under gauge transformations, since $F_{u^* A} = \operatorname{Ad}(u^{-1}) F_A$ and the inner product is invariant.
Its Euler–Lagrange equation is $d_A^* F_A = 0$, the [[Yang–Mills equations|JJWETW5AT8]].

Example: let $G = U(1)$ with Lie algebra $i\R$ and inner product $\ip{ia}{ib} = ab$, and let $P = M \times U(1)$ be the trivial bundle.
A connection is $A = d + i\alpha$ for a real $1$-form $\alpha$, and its curvature is $F_A = i \, d\alpha$.
So $\mathrm{YM}(A) = \frac{1}{2} \int_M \abs{d\alpha}^2 \, \mathrm{vol}_g$, the energy of the electromagnetic field with potential $\alpha$.
It vanishes exactly when $d\alpha = 0$, that is, when $A$ is flat.

Non-example: the integral $\int_M \tr(F_A \wedge F_A)$ for a matrix group $G$ on a $4$-manifold is not the Yang–Mills functional.
By Chern–Weil theory its value depends only on the bundle $P$ and not on $A$, so it has no variational content.

## Explanations

### L1 human

The functional measures the total squared curvature of a connection, so flat connections have the smallest possible value, zero.
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# Density of simple functions in Lp

## Statement

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $1 \le p < \infty$.
Let $S$ be the set of [[simple functions|7WMBX7M4P6]] $s = \sum_{i=1}^{n} a_i \mathbf{1}_{A_i}$ with $A_i \in \Sigma$ and $\mu(A_i) < \infty$ for each $i$.
Then $S$ is a [[dense|0BSJSX7ME2]] subset of [[$L^p(\mu)$|SASP14MA1Z]].

## Remarks

For $f \ge 0$ in $L^p$, take simple functions $0 \le s_n \le f$ that increase pointwise to $f$.
Each $s_n$ vanishes outside a set of finite measure, because $s_n^p \le f^p$ is integrable.
Since $\abs{f - s_n}^p \le f^p$, the [[dominated convergence theorem|BYNV84PF86]] gives $\norm{f - s_n}_p \to 0$.
A general $f$ is split into the positive and negative parts of its real and imaginary parts.

The hypothesis $p < \infty$ is necessary.
On $\R$ with [[Lebesgue measure|2Z2QVWZ262]], the constant function $1$ lies in $L^\infty$, but every $s \in S$ vanishes outside a set of finite measure, so $\norm{1 - s}_\infty \ge 1$.
In $L^\infty(\mu)$ the set of all simple functions, without the condition $\mu(A_i) < \infty$, is dense.

On $\R^n$ the result is the first step towards the density of step functions and of continuous functions with compact support in $L^p$, $1 \le p < \infty$.

## Explanations

### L1 human

Every function in $L^p$ can be approximated as well as one wants by a staircase function that takes only finitely many values.
The steps of the staircase sit on sets of finite measure.
This lets one prove a fact about all of $L^p$ by checking it on indicator functions of sets and then passing to the limit.

### L2 human

The standard way to prove an inequality or an identity on $L^p$ is to check it on a dense class and extend it by continuity.
Simple functions are the first such class; they are used in the proof of the [[dual of Lp|Q1V7ZNS332]] and in interpolation theorems such as the [[Riesz–Thorin theorem|AGRKRG1TMJ]].
For $p = \infty$ the density of all simple functions is a uniform approximation, which is why the finite measure condition must then be dropped.
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# Rigid monoidal category

## Definition

A *rigid monoidal category* is a [[monoidal category|M3MSQ3NKAE]] $\mathcal{C}$ in which every object $A$ has both a right [[dual|8A7YXMW727]] $A^*$ and a left dual ${}^*A$.
That is, there is an object $A^*$ with morphisms $A^* \otimes A \to I$ and $I \to A \otimes A^*$ satisfying the zigzag identities.
There is also an object ${}^*A$ with morphisms $A \otimes {}^*A \to I$ and $I \to {}^*A \otimes A$ satisfying the zigzag identities.

## Remarks

Example: the category $\mathbf{FinVect}_k$ of finite-dimensional vector spaces over a field $k$ is rigid.
For $V$ both duals can be taken to be the dual space $V^*$, with evaluation and the coevaluation $1 \mapsto \sum_i e_i \otimes e^i$ or its flip.
Non-example: the category $\mathbf{Vect}_k$ of all vector spaces is not rigid, because an infinite-dimensional space has no dual.
In a rigid category, a morphism $f \colon A \to B$ has a transpose $f^* \colon B^* \to A^*$, and $A \mapsto A^*$ becomes a contravariant functor.
In a braided monoidal category, an object has a right dual if and only if it has a left dual, so asking for one kind suffices.
Rigid monoidal categories are also called autonomous categories; a symmetric rigid monoidal category is called compact closed.
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# Dieudonné's theorem on paracompact Hausdorff spaces

## Statement

Let $X$ be a [[paracompact|8R29G6NG9Q]] [[Hausdorff space|0S1FJ8VEX9]].
Then $X$ is [[normal|0NSR9V8TED]].
That is, for any two disjoint closed sets $A, B \subseteq X$, there are disjoint open sets $U \supseteq A$ and $V \supseteq B$.

## Remarks

The proof has two steps of the same shape.
First, $X$ is [[regular|03HMJASQBQ]]: for a closed set $C$ and a point $x \notin C$, cover $C$ by open sets whose closures miss $x$, and add $X \setminus C$.
Take a [[locally finite|DZ297MMGV8]] open refinement.
The union of the closures of the members that meet $C$ is closed, by local finiteness, and it misses $x$.
The second step repeats this argument with a closed set in place of the point $x$, using regularity.

Example at work: every [[compact|0TVJZYB6SH]] Hausdorff space is paracompact, so it is normal.

The Hausdorff hypothesis cannot be dropped.
Take $X = \set{a, b, c}$ with open sets $\emptyset$, $\set{a}$, $\set{a, b}$, $\set{a, c}$ and $X$.
It is finite, so paracompact.
The closed sets $\set{b}$ and $\set{c}$ are disjoint, but every open set containing either of them contains $a$.

## Explanations

### L1 human

A normal space is one where two disjoint closed sets can be put in disjoint open boxes.
Dieudonné's theorem says that paracompactness plus the Hausdorff condition already gives this.
It extends the classical fact that compact Hausdorff spaces are normal.

### L2 human

Normality is what makes [[Urysohn's lemma|01EY3VR7ZD]] work.
So on a paracompact Hausdorff space, there are many continuous real functions.
This is the first step toward partitions of unity subordinate to an open cover.
In fact, a Hausdorff space is paracompact if and only if every open cover has a subordinate partition of unity.
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# Double sequence

## Summary

A double sequence is a family $(x_{m,n})$ indexed by pairs of natural numbers, like an infinite matrix.
It has several notions of limit.
There is the joint limit as $m$ and $n$ both grow, and there are two iterated limits taken one index at a time.
These limits can differ, and theorems such as the [[Moore–Osgood theorem|QN2210XCST]] say when they agree.
The exact definition is in [[double sequence|WK5V372TG3]].

## Explanations

### L1 human

A double sequence is an infinite table of numbers with rows and columns, one entry for each pair $(m, n)$.
For example, $x_{m,n} = m + n$ gives the table with rows $0, 1, 2, \ldots$ and $1, 2, 3, \ldots$ and so on.
One can let the row number grow first, or the column number first, or both together, and the outcomes need not be the same.
</document>

<document source="https://tamnd.github.io/math2/node/B75H4VEARE.md" tag="B75H4VEARE" kind="definition">
# Triangulated category

## Definition

Let $\mathcal{T}$ be an [[additive category|B2VP96NJSK]] and let $\Sigma \colon \mathcal{T} \to \mathcal{T}$ be an [[additive functor|KAN07YC9T2]] that is an automorphism, the *shift*, also written $X \mapsto X[1]$.
A *triangle* is a diagram $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} \Sigma X$.
A morphism of triangles is a triple of morphisms $(f, g, h)$ between the first, second and third objects.
It must make the three squares commute, the last one with $\Sigma f$.
A *triangulated category* is such a pair $(\mathcal{T}, \Sigma)$ with a class of triangles, the *distinguished triangles*, such that:

1. (TR1) for each object $X$, the triangle $X \xrightarrow{\id} X \to 0 \to \Sigma X$ is distinguished. Each triangle isomorphic to a distinguished triangle is distinguished. Each morphism $u \colon X \to Y$ is the first morphism of a distinguished triangle $X \xrightarrow{u} Y \to Z \to \Sigma X$;
2. (TR2) a triangle $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} \Sigma X$ is distinguished if and only if $Y \xrightarrow{v} Z \xrightarrow{w} \Sigma X \xrightarrow{-\Sigma u} \Sigma Y$ is distinguished;
3. (TR3) for distinguished triangles $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} \Sigma X$ and $X' \xrightarrow{u'} Y' \xrightarrow{v'} Z' \xrightarrow{w'} \Sigma X'$ and morphisms $f \colon X \to X'$, $g \colon Y \to Y'$ with $g u = u' f$, there is $h \colon Z \to Z'$ such that $(f, g, h)$ is a morphism of triangles;
4. (TR4) the [[octahedral axiom|P8PFMNCB8Y]] holds.

## Remarks

Example: for an additive category $\mathcal{A}$, the homotopy category $K(\mathcal{A})$ with the shift of complexes is triangulated, the distinguished triangles being those isomorphic to $X \xrightarrow{f} Y \to \operatorname{Cone}(f) \to X[1]$ for chain maps $f$.
For a distinguished triangle and an object $W$, the sequences obtained by applying $\Hom(W, -)$ and $\Hom(-, W)$ are long exact.
Non-example: the category of abelian groups has no triangulated structure, for any shift and any class of triangles.
In a triangulated category every monomorphism $u$ is split, since in a triangle $\Sigma^{-1} Z \xrightarrow{t} X \xrightarrow{u} Y \to Z$ one has $u t = 0$, so $t = 0$, and a triangle with a zero map splits.
But the monomorphism $\Z \xrightarrow{\cdot 2} \Z$ does not split.
A category satisfying TR1 to TR3 only is called *pretriangulated*.
</document>

<document source="https://tamnd.github.io/math2/node/B763QH5521.md" tag="B763QH5521" kind="concept">
# Topological mixing

## Summary

A continuous map is topologically mixing when every nonempty open set, after some time, meets every other nonempty open set at all later times.
It strengthens [[topological transitivity|FR49Y9GQHR]], which only asks for one such time.
It is the topological counterpart of strong [[mixing|S3RJQ9CX2P]] in ergodic theory.
Expanding maps such as the doubling map of the circle are mixing, while rotations are never mixing.

## Explanations

### L1 human

Put a drop of ink in a glass of water and stir with a fixed rule.
Mixing means that after enough stirring the ink reaches every small region of the glass, and stays present there at every later moment.
The doubling map $x \mapsto 2x \bmod 1$ of the circle is mixing: it stretches every small arc until it covers the whole circle.
An irrational rotation is transitive but not mixing: a small arc just moves around rigidly and keeps missing a given region at many times.
</document>

<document source="https://tamnd.github.io/math2/node/B76G1KR3SB.md" tag="B76G1KR3SB" kind="definition">
# Motive

## Definition

Let $k$ be a field, and let $\mathcal{V}$ be the class of smooth [[projective varieties|A4H4CZ2V35]] over $k$.
Let $X, Y \in \mathcal{V}$ with $X$ irreducible of dimension $d$, and let $r \in \Z$.
The group of *correspondences of degree $r$* is $\operatorname{Corr}^r(X, Y) = CH^{d + r}(X \times Y) \otimes \Q$,
where $CH^\bullet$ is the [[Chow group|EBEF6PEK4N]] graded by codimension, that is, [[algebraic cycles|9VFWPZEVGN]] modulo [[rational equivalence|K76N026JFK]].
For general $X$, sum over the irreducible components of $X$.
The composition of $f \in \operatorname{Corr}^r(X, Y)$ and $g \in \operatorname{Corr}^s(Y, Z)$ is
$$g \circ f = (p_{XZ})_*\big(p_{XY}^* f \cdot p_{YZ}^* g\big) \in \operatorname{Corr}^{r+s}(X, Z),$$
where the $p$ are the projections from $X \times Y \times Z$ and the product is the [[intersection product|4HV4WTAX9P]].

The category of *Chow motives* over $k$ with rational coefficients is defined as follows.

1. Its objects, the *motives*, are triples $(X, p, m)$ with $X \in \mathcal{V}$, $p \in \operatorname{Corr}^0(X, X)$ such that $p \circ p = p$, and $m \in \Z$.
2. Its morphisms are $\Hom\big((X, p, m), (Y, q, n)\big) = q \circ \operatorname{Corr}^{n - m}(X, Y) \circ p$.
3. Composition is the composition of correspondences.

The *motive of* $X$ is $h(X) = (X, \Delta_X, 0)$, where $\Delta_X$ is the class of the diagonal.

## Remarks

The class $\Delta_X$ is the identity of $\operatorname{Corr}^0(X, X)$.
The *unit motive* is $\mathbb{1} = h(\operatorname{Spec} k)$ and the *Lefschetz motive* is $\mathbb{L} = (\operatorname{Spec} k, \id, -1)$.
Example: for $X = \mathbb{P}^1$, let $e_0 = [\mathrm{pt} \times \mathbb{P}^1]$ and $e_2 = [\mathbb{P}^1 \times \mathrm{pt}]$ in $\operatorname{Corr}^0(X, X) = CH^1(X \times X)_\Q$.
They are orthogonal idempotents, and $\Delta_X = e_0 + e_2$ because the diagonal and $e_0 + e_2$ both have bidegree $(1, 1)$ in $CH^1(\mathbb{P}^1 \times \mathbb{P}^1) \cong \Z^2$.
This gives $h(\mathbb{P}^1) = (X, e_0, 0) \oplus (X, e_2, 0) \cong \mathbb{1} \oplus \mathbb{L}$.
Non-example: $(X, 2\Delta_X, 0)$ is not a motive when $X$ is nonempty, since $2\Delta_X \circ 2\Delta_X = 4\Delta_X \ne 2\Delta_X$, so clause 1 fails.
Replacing rational equivalence by homological or numerical equivalence gives other categories of pure motives.
</document>

<document source="https://tamnd.github.io/math2/node/B7948M365K.md" tag="B7948M365K" kind="concept">
# Regular boundary point

## Summary

A boundary point of a bounded open set is regular if the generalized solution of the [[Dirichlet problem|8D56GQ4AG8]] tends to the prescribed value there.
This must hold for every continuous boundary function.
The Dirichlet problem is solvable for all continuous data exactly when every boundary point is regular.
Regularity is a local property: it holds at $\xi$ if and only if there is a [[barrier|ED0SZS1VED]] at $\xi$.
It is characterized by Wiener's criterion in terms of capacity, and the irregular points always form a polar set.

## Explanations

### L1 human

At most boundary points the solution of the Dirichlet problem behaves well and takes the boundary value you prescribe.
Some points are so small or so thin that the solution does not notice them.
The centre of a punctured disc is the simplest such irregular point: whatever value you prescribe there, the solution ignores it.
A point on a smooth piece of boundary, or the corner of a square, is regular.
</document>

<document source="https://tamnd.github.io/math2/node/B7AJ8RWZRC.md" tag="B7AJ8RWZRC" kind="concept">
# Kaplan–Meier estimator

## Summary

The Kaplan–Meier estimator is a nonparametric estimator of the [[survival function|QTM1HZM53Q]] from right-[[censored|MY8ZZP84NP]] data.
It multiplies, over the observed event times, the fractions of subjects at risk who survive each event time.
Without censoring it reduces to one minus the [[empirical distribution function|A05WR6TEK5]].
It was introduced by Edward Kaplan and Paul Meier, and its step-shaped plot is the most common figure in medical survival studies.

## Explanations

### L1 human

Follow four patients: deaths at months $2$, $5$ and $7$, and one patient who leaves the study at month $3$.
At month $2$ one of four patients at risk dies, so survival drops to $3/4$.
The patient who leaves at month $3$ is simply removed from the group at risk, without counting a death.
At month $5$ one of the two remaining patients dies, so survival drops to $3/4 \cdot 1/2 = 3/8$.
The censored patient helped in the first step, and it does not distort the later ones.
</document>

<document source="https://tamnd.github.io/math2/node/B7CV5DZPRK.md" tag="B7CV5DZPRK" kind="definition">
# Multiplicity of a local ring

## Definition

Let $(R, \mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] of [[Krull dimension|MVGBS3J1P7]] $d$, and let $I$ be an [[$\mathfrak m$-primary ideal|TMTPWVETQK]].
Let $P$ be the polynomial that agrees with the [[Hilbert–Samuel function|BNQVS83GZV]] $n \mapsto \ell(R/I^n)$ for all large $n$.
It has degree $d$, and its leading coefficient has the form $e/d!$ for a positive integer $e$.
The integer $e = e(I, R)$ is the *multiplicity* of $I$.
The *multiplicity of $R$* is $e(R) = e(\mathfrak m, R)$.

## Remarks

Example: let $R = k[x, y]_{(x, y)}/(y^2 - x^3)$, with $k$ a field.
For $n \ge 2$, $\ell(R/\mathfrak m^n) = 2n - 1$, since the monomials of degree $< n$ in $k[x, y]$ number $n(n+1)/2$ and those divisible by $y^2$ number $(n-2)(n-1)/2$.
So $d = 1$, the leading coefficient is $2 = e/1!$, and $e(R) = 2$.
Non-example: the function $n \mapsto \ell(R/I^n)$ for an ideal $I$ that is not $\mathfrak m$-primary, such as $I = (y)$ in $k[x, y]_{(x, y)}$, takes infinite values, so it has no multiplicity in this sense.
A regular local ring has multiplicity $1$, and an Artinian local ring $R$ has $e(R) = \ell(R)$.
</document>

<document source="https://tamnd.github.io/math2/node/B7EFVHCEQV.md" tag="B7EFVHCEQV" kind="concept">
# Asymptotic expansion from poles of a Mellin transform

## Summary

The poles of the [[Mellin transform|N5PVGRJE1T]] of a function $f$ control the behaviour of $f(x)$ as $x \to 0^+$ and as $x \to \infty$.
A pole at $s = \rho$ left of the [[fundamental strip|1JAA21FHHQ]] gives a term $x^{-\rho}$ in the [[asymptotic expansion|BJQTEPEZ9G]] of $f$ near $0$.
A multiple pole adds powers of $\log x$ to this term.
Poles to the right of the strip give the expansion of $f$ near $\infty$ in the same way.
The method turns asymptotic questions about sums and integrals into the study of a [[meromorphic function|0ZZ7ZYDR8W]].
It is a basic tool in analytic number theory and in the analysis of algorithms.

## Explanations

### L1 human

The Mellin transform sends a function of $x > 0$ to a function of a complex variable $s$.
Each singular point of the transform is a fingerprint of one term of $f$: a simple pole at $s = \rho$ with residue $c$ corresponds to a term $c\, x^{-\rho}$.
To find these terms one moves the line of the inverse Mellin integral across the poles and collects the [[residues|0PHN07KVQT]].
For example, the Mellin transform of $e^{-x}$ is the [[gamma function|0H5SYBSED9]] $\Gamma(s)$.
Its poles at $s = -n$ have residues $(-1)^n / n!$, and they give back the Taylor series $e^{-x} = \sum_{n \ge 0} (-1)^n x^n / n!$.
</document>

<document source="https://tamnd.github.io/math2/node/B7FPT8JTXR.md" tag="B7FPT8JTXR" kind="concept">
# Locally ringed space

## Summary

A locally ringed space is a [[ringed space|KNDXDE7K8M]] in which the ring of germs of functions at each point is a [[local ring|MZ02BKPHJA]].
The unique maximal ideal of that local ring consists of the germs that "vanish at the point".
Schemes, smooth manifolds and complex manifolds are all locally ringed spaces.
This is the common frame in which scheme theory is built.

## Explanations

### L1 human

Think of a space together with a rule that says which functions are allowed on each open set.
At a single point, one looks at the allowed functions near the point, where two functions count as the same if they agree near the point.
For a smooth manifold and a point $p$, such a germ $g$ is invertible exactly when $g(p) \ne 0$.
So the germs that are not invertible are those with $g(p) = 0$, and they form one ideal; this is what "local ring" means.
A locally ringed space is a space where this holds at every point.
</document>

<document source="https://tamnd.github.io/math2/node/B7GCKZZJE3.md" tag="B7GCKZZJE3" kind="definition">
# Proper cone

## Definition

Let $n \ge 1$.
A *proper cone* in $\R^n$ is a [[convex cone|6VE4KA12VD]] $K \subseteq \R^n$ such that:

1. $K$ is nonempty and [[closed|0ZCK8DQYN1]];
2. $K$ has nonempty [[interior|0RZH6W98HK]];
3. $K$ is salient: $K \cap (-K) = \set{0}$, so $K$ contains no line through the origin.

## Remarks

A nonempty closed convex cone contains $0 = \lim_{t \to 0^+} t x$, so a proper cone is a [[pointed cone|J9HA54ACME]].

Example: the nonnegative orthant $\R^n_{\ge 0}$.
It is a closed convex cone, its interior is the set of vectors with all coordinates positive, and if $x$ and $-x$ both have nonnegative coordinates then $x = 0$.

Non-example: the closed half-plane $\set{(x, y) \in \R^2 : y \ge 0}$ satisfies clauses 1 and 2, but it contains $(1, 0)$ and $(-1, 0)$, so clause 3 fails.

A proper cone $K$ gives a partial order on $\R^n$ by $x \preceq_K y$ if $y - x \in K$.
Conventions vary: some sources, Mathlib among them, call any closed pointed cone proper and drop clauses 2 and 3.
</document>

<document source="https://tamnd.github.io/math2/node/B7HPB2Z6GC.md" tag="B7HPB2Z6GC" kind="definition">
# Cramér–von Mises criterion

## Definition

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with [[empirical distribution function|A05WR6TEK5]] $F_n$.
Let $F_0$ be a fixed continuous [[cumulative distribution function|0MXMS3AE13]].
The *Cramér–von Mises statistic* is
$$T = n \int_{-\infty}^{\infty} \big(F_n(x) - F_0(x)\big)^2 \, dF_0(x).$$
The *Cramér–von Mises test* of $H_0 \colon$ "the $X_i$ have distribution function $F_0$" rejects $H_0$ when $T$ is large.
The critical value comes from the null law of $T$.

## Remarks

With the [[order statistics|58NHGZDSRE]] $X_{(1)} \le \dots \le X_{(n)}$, the integral can be computed exactly:
$$T = \frac{1}{12 n} + \sum_{i=1}^{n} \Big( \frac{2i - 1}{2n} - F_0(X_{(i)}) \Big)^2.$$
Under $H_0$ the law of $T$ does not depend on $F_0$, since $F_0(X_i)$ is uniform on $[0, 1]$.
As $n \to \infty$ it converges to the law of $\int_0^1 B(t)^2 \, dt$ for a [[Brownian bridge|3MCVMBAGD1]] $B$.

Example: $n = 1$, $X_1 = 0.5$ and $F_0(x) = x$ on $[0, 1]$.
The integral is $\int_0^{1/2} x^2 \, dx + \int_{1/2}^{1} (1 - x)^2 \, dx = 1/24 + 1/24 = 1/12$.
The formula gives the same value: $1/12 + (1/2 - 1/2)^2 = 1/12$.

Non-example: $n \int (F_n - F_0)^2 \, dx$, with Lebesgue measure in place of $dF_0$, is not the criterion.
Its null law depends on $F_0$, for example on its scale, so it gives no distribution-free test.

The Anderson–Darling statistic is a weighted variant of $T$.
</document>

<document source="https://tamnd.github.io/math2/node/B7K7NY8NTJ.md" tag="B7K7NY8NTJ" kind="conjecture">
# Selberg's 1/4 conjecture

## Statement

Let $\Gamma$ be a [[congruence subgroup|1H6R856P3C]] of the [[modular group|VEKV1EM77B]] $\mathrm{SL}_2(\Z)$, acting on the [[upper half-plane|X6XGGH5FPW]] $\mathbb H$.
Let $\Delta = -y^2 (\partial_x^2 + \partial_y^2)$ be the hyperbolic [[Laplace–Beltrami operator|F4NWZNTRMD]] on $L^2(\Gamma \backslash \mathbb H)$.
The conjecture states that every nonzero [[eigenvalue|0BQAA2BK3Q]] $\lambda$ of $\Delta$ on $L^2(\Gamma \backslash \mathbb H)$ satisfies $\lambda \ge \tfrac14$.

## Remarks

The eigenfunctions in question are the [[Maass cusp forms|3R959YSNYP]] for $\Gamma$, since the continuous spectrum is $[\tfrac14, \infty)$.
Selberg proved the bound $\lambda \ge \tfrac{3}{16}$.
The best bound known is $\lambda \ge \tfrac{975}{4096}$, due to Kim and Sarnak.
The conjecture is known for $\mathrm{SL}_2(\Z)$ itself and for congruence subgroups of small level.
It is the archimedean case of the generalised Ramanujan conjecture for $\mathrm{GL}_2$: an eigenvalue $\lambda = \tfrac14 - s^2$ with $0 < s < \tfrac12$ would give a non-tempered local component at the real place.
The arithmetic hypothesis matters: for general cofinite Fuchsian groups, eigenvalues below $\tfrac14$ can occur.

## Explanations

### L1 human

The quotient of the upper half-plane by a congruence group is a surface with a few cusps, and it can vibrate at certain frequencies.
Selberg predicted that, apart from the trivial vibration, all frequencies are at least $\tfrac14$, the bottom of the continuous spectrum.
This means that these arithmetic surfaces are very well connected, with no slow vibration, and it has consequences for counting problems and for expander graphs.
</document>

<document source="https://tamnd.github.io/math2/node/B7KNZCBFYT.md" tag="B7KNZCBFYT" kind="concept">
# Splitting of prime ideals

## Summary

Let $L/K$ be an extension of [[algebraic number fields|MWBDFS2KJ5]].
A [[prime ideal|00MHHZ9SNE]] of the ring of integers of $K$ need not stay prime in the ring of integers of $L$.
It factors into a product of prime ideals of $L$.
The factorisation is described by the number of factors, their exponents, called [[ramification|H5RC67SD24]] indices, and their residue degrees.
These numbers satisfy the [[fundamental identity|REJDQ6BTQV]] $\sum_i e_i f_i = [L : K]$.
In a Galois extension the splitting is controlled by the [[decomposition group|7X92G8CQ33]] and the [[Frobenius element|H5145REFMW]].
This leads to [[Chebotarev's density theorem|X8VKW5Y1CZ]] and to [[class field theory|BEKCEYHCHX]].

## Explanations

### L1 human

A prime number can stop being prime in a bigger ring of integers.
In the Gaussian integers $\Z[i]$, the prime $5$ splits as $(2 + i)(2 - i)$, the prime $3$ stays prime, and $2 = -i(1 + i)^2$ becomes a square up to a unit.
These three behaviours are called splitting, staying inert and ramifying.
Which one happens depends on $p$ modulo $4$: odd primes $p \equiv 1 \pmod 4$ split and primes $p \equiv 3 \pmod 4$ stay inert.
Such rules for general extensions are a central topic of algebraic number theory.
</document>

<document source="https://tamnd.github.io/math2/node/B7PZV00FP4.md" tag="B7PZV00FP4" kind="definition">
# Uniform structure of a topological group

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] with identity $e$, and let $\mathcal{N}$ be the set of [[neighbourhoods|0ZCY0Z80B6]] of $e$.
For $U \in \mathcal{N}$ put
$$L_U = \set{(x, y) \in G \times G : x^{-1} y \in U}, \qquad R_U = \set{(x, y) \in G \times G : y x^{-1} \in U}.$$
The *left uniform structure* of $G$ is the [[uniform structure|A54T86KS63]] on $G$ whose [[entourages|FVFFKNKEDF]] are the subsets of $G \times G$ that contain some $L_U$ with $U \in \mathcal{N}$.
The *right uniform structure* of $G$ is defined in the same way with the sets $R_U$.

## Remarks

Both uniform structures induce the given topology of $G$: the neighbourhoods of $x$ are the sets $xU$ for the left one and $Ux$ for the right one, with $U \in \mathcal{N}$.
Example: for $(\R, +)$, the set $L_U$ with $U = (-\varepsilon, \varepsilon)$ is $\set{(x, y) : \abs{y - x} < \varepsilon}$, so both uniformities are the uniformity of the usual metric.
If $G$ is abelian, then $L_U = R_U$ for every $U$, so the two structures coincide.
In general they differ; the inversion map $x \mapsto x^{-1}$ carries one to the other.
Some authors also use the two-sided uniformity, generated by the sets $L_U \cap R_U$.
</document>

<document source="https://tamnd.github.io/math2/node/B7X0KFY1S1.md" tag="B7X0KFY1S1" kind="concept">
# Integral extension

## Summary

A ring extension $A \subseteq B$ is integral if every element of $B$ is [[integral|1EHN4KPSFS]] over $A$.
It is the ring analogue of an [[algebraic extension|0PTBM7K0R2]] of fields.
Integral extensions keep the prime ideals close to those of the base.
The lying over, going-up and incomparability theorems hold, and the two rings have the same Krull dimension.
Every [[finite|1NBACAEC2V]] ring extension is integral.
In geometry an integral extension gives a closed map of spectra whose fibres have dimension zero.

## Explanations

### L1 human

An integral extension adds to a ring only elements that satisfy monic equations over it.
The Gaussian integers $\Z[i]$ are an integral extension of $\Z$, since every $a + bi$ is a root of $x^2 - 2a x + (a^2 + b^2)$.
The rational numbers are not an integral extension of $\Z$, since $1/2$ is not integral.
A picture: the parabola $y^2 = x$ projected to the $x$-axis corresponds to $k[x] \subseteq k[x, y]/(y^2 - x)$, an integral extension, and every point of the axis has one or two points above it.
</document>

<document source="https://tamnd.github.io/math2/node/B7X9M15Q92.md" tag="B7X9M15Q92" kind="definition">
# Lorentz space

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $f$ be a real or complex [[measurable function|YYNCJSR0PJ]] on $X$.
Its [[distribution function|HQVXKEMN5N]] is $d_f(s) = \mu(\set{x : \abs{f(x)} > s})$ for $s \ge 0$.
Its [[decreasing rearrangement|BWS042NC2P]] is $f^*(t) = \inf \set{s \ge 0 : d_f(s) \le t}$ for $t \ge 0$.
Let $0 < p < \infty$ and $0 < q \le \infty$, and put
$$\norm{f}_{p,q} = \left( \int_0^\infty \left( t^{1/p} f^*(t) \right)^q \frac{dt}{t} \right)^{1/q} \text{ if } q < \infty, \qquad \norm{f}_{p,\infty} = \sup_{t > 0} t^{1/p} f^*(t).$$
The *Lorentz space* $L^{p,q}(\mu)$ is the set of measurable $f$ with $\norm{f}_{p,q} < \infty$, modulo equality [[almost everywhere|RARTMA0ZEC]].

## Remarks

Example: for $q = p$, $\norm{f}_{p,p}^p = \int_0^\infty f^*(t)^p \, dt = \int_X \abs{f}^p \, d\mu$, because $f^*$ and $\abs{f}$ have the same distribution function.
So $L^{p,p}(\mu) = L^p(\mu)$ with equal norms.

Non-example: on $(0, \infty)$ with Lebesgue measure, $f(x) = x^{-1/p}$ is not in $L^{p,q}$ for $q < \infty$.
Here $f^* = f$, so $t^{1/p} f^*(t) = 1$ and the integral $\int_0^\infty dt / t$ diverges.
It does lie in $L^{p,\infty}$, with $\norm{f}_{p,\infty} = 1$.

The space $L^{p,\infty}$ is the weak $L^p$ space, since $\sup_t t^{1/p} f^*(t) = \sup_s s \, d_f(s)^{1/p}$.
For fixed $p$, the spaces increase with $q$: $L^{p,q_1} \subseteq L^{p,q_2}$ when $q_1 < q_2$.
In general $\norm{\cdot}_{p,q}$ is only a quasinorm.
</document>

<document source="https://tamnd.github.io/math2/node/B7XDAW51Z1.md" tag="B7XDAW51Z1" kind="definition">
# Spectral geometry

## Definition

Let $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]], possibly with boundary and a boundary condition.
Let $\operatorname{spec}(M, g)$ be the [[spectrum of the Laplace–Beltrami operator|B8TYKV7AA1]], the sequence of its eigenvalues with multiplicities.
More generally, one may take the spectrum of another natural elliptic operator, such as the Hodge Laplacian on forms.
*Spectral geometry* is the study of the relations between $\operatorname{spec}(M, g)$ and the geometric invariants of $(M, g)$.
It has two directions:

1. direct problems: estimate or compute the eigenvalues from geometric data;
2. inverse problems: recover geometric data from the spectrum.

## Remarks

Example of a direct result: the [[Lichnerowicz–Obata theorem|TK1PT9FDV0]] bounds the first nonzero eigenvalue below in terms of a positive lower bound on the Ricci curvature.
Example of an inverse result: by [[Weyl's law|2JK9YMZV2B]] the spectrum of a closed manifold determines its dimension and its volume.
The answer to the inverse problem is not complete in general: there are non-isometric manifolds with the same spectrum.

Non-example: the study of the length of a single closed geodesic, without reference to any spectrum, is Riemannian geometry but not spectral geometry.
Geodesic lengths enter spectral geometry only through the length spectrum and trace formulas.
</document>

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# Cubic spline

## Definition

Let $k \ge 1$ and let $a = t_0 < t_1 < \cdots < t_k = b$ be knots.
A *cubic spline* with these knots is a function $s \colon [a, b] \to \R$ such that:

1. for each $i = 1, \dots, k$, the restriction of $s$ to $[t_{i-1}, t_i]$ is a [[polynomial function|1G0B73WPX8]] of degree at most $3$;
2. $s$ is twice continuously differentiable on $[a, b]$.

A cubic spline is *natural* if moreover $s''(a) = s''(b) = 0$.

## Remarks

This is the [[spline|84TJ8KQ926]] of degree $3$.
Writing $s_i$ for the piece on $[t_{i-1}, t_i]$, clause 2 says that $s_i(t_i) = s_{i+1}(t_i)$, $s_i'(t_i) = s_{i+1}'(t_i)$ and $s_i''(t_i) = s_{i+1}''(t_i)$ for $1 \le i \le k - 1$, where $'$ is the [[derivative|0V1EJCT8G6]].
The cubic splines with fixed knots form a vector space of dimension $k + 3$.
Given values at all knots, there is exactly one natural cubic spline taking them.

Example: knots $-1 < 0 < 1$ and $s(x) = 0$ for $x \le 0$, $s(x) = x^3$ for $x \ge 0$.
At $0$ both pieces have value, first derivative and second derivative $0$, so $s$ is a cubic spline.
It is not natural, since $s''(1) = 6 \ne 0$.

Non-example: $s(x) = \abs{x}$ on $[-1, 1]$ with the knots $-1 < 0 < 1$ satisfies clause 1, since its pieces $-x$ and $x$ are polynomials.
It fails clause 2: its first derivative jumps from $-1$ to $1$ at $0$.
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<document source="https://tamnd.github.io/math2/node/B83KZW70C8.md" tag="B83KZW70C8" kind="definition">
# Distance geometry

## Definition

Let $G = (V, E)$ be a finite [[graph|0JD5C82QMB]], let $\delta \colon E \to [0, \infty)$ be a weight on its edges, and let $k \ge 1$ be an integer.
A *realization* of $(G, \delta)$ in $\R^k$ is a map $p \colon V \to \R^k$ such that
$$\norm{p(u) - p(v)} = \delta(uv) \quad \text{for every edge } uv \in E.$$
Here $\norm{p(u) - p(v)}$ is the [[Euclidean distance|0C95TG69SG]] of the two points.
The *distance geometry problem* asks, for given $(G, \delta)$ and $k$, whether a realization in $\R^k$ exists, and to find one.
Distance geometry is the study of such realizations and of the point sets they describe.

## Remarks

Example: let $G$ be a triangle with edge weights $3$, $4$ and $5$, and $k = 2$.
The points $(0, 0)$, $(3, 0)$ and $(3, 4)$ form a realization, since their pairwise distances are $3$, $4$ and $5$.

Non-example: the same triangle with weights $1$, $1$ and $3$ has no realization in any $\R^k$, since $3 > 1 + 1$.

When $G$ is a complete graph, a realization exists in some $\R^k$ if and only if the squared weights form a [[Euclidean distance matrix|1BCJ7A73KS]].
A realization is never unique: an isometry of $\R^k$ moves it to another one.
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# Fuzzy algebraic structure

## Definition

Let $A$ be an [[algebra|QJGPT2W0FT]] in which every operation has arity at least $1$.
A *fuzzy subset* of $A$ is a [[function|0EPAT51DXQ]] $\mu \colon A \to [0, 1]$; the number $\mu(a)$ is the degree of membership of $a$.
A fuzzy subset $\mu$ is a *fuzzy [[subalgebra|7CHG6SD365]]* of $A$ if
$$\mu(f(a_1, \dots, a_n)) \ge \min\set{\mu(a_1), \dots, \mu(a_n)}$$
for every operation $f$ of $A$ of arity $n$ and all $a_1, \dots, a_n \in A$.
For a [[group|G5S1M9]] $G$ with the operations $(x, y) \mapsto xy$ and $x \mapsto x^{-1}$, a fuzzy subalgebra is a *fuzzy subgroup*: $\mu(xy) \ge \min\set{\mu(x), \mu(y)}$ and $\mu(x^{-1}) \ge \mu(x)$ for all $x, y \in G$.

## Remarks

A fuzzy subset $\mu$ is a fuzzy subalgebra if and only if each level set $\mu_t = \set{a \in A : \mu(a) \ge t}$, for $t \in [0, 1]$, is empty or a subalgebra.
Indeed, if $a_1, \dots, a_n \in \mu_t$ then $\min_i \mu(a_i) \ge t$; conversely, apply the condition on $\mu_t$ with $t = \min_i \mu(a_i)$.
A fuzzy subset that takes only the values $0$ and $1$ is a fuzzy subalgebra exactly when $\set{a : \mu(a) = 1}$ is empty or a subalgebra.

Example: on the group $(\Z, +)$ let $\mu(n) = 1$ for even $n$ and $\mu(n) = 1/2$ for odd $n$.
If $m$ or $n$ is odd, the minimum is $1/2$ and $\mu(m + n) \ge 1/2$ holds; if both are even, $m + n$ is even and $\mu(m + n) = 1$.
Also $-n$ has the parity of $n$, so $\mu(-n) = \mu(n)$.
So $\mu$ is a fuzzy subgroup.

Non-example: on $(\Z, +)$ let $\mu(1) = 1$ and $\mu(n) = 0$ for $n \ne 1$.
Then $\mu(1 + 1) = 0 < 1 = \min\set{\mu(1), \mu(1)}$, so $\mu$ is not a fuzzy subgroup.
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# Valuation

## Summary

A valuation on a [[field|04C5ZBG1WT]] measures how divisible an element is, with values in a totally ordered abelian group, see the [[definition|SVY882HKSJ]].
The value of a product is the sum of the values.
The value of a sum is at least the smaller of the two values.
The [[p-adic valuation|03HDF50VN4]] on the rationals is the model case.
Valuations give [[valuation rings|QSX12D8RQM]], [[non-Archimedean absolute values|1H5F1BP6XS]] and the whole theory of valued fields.

## Explanations

### L1 human

A valuation asks how many times an element can be divided by a fixed "small" quantity.
For a prime $p$ and a nonzero rational number $x$, write $x = p^n a / b$ with $a$ and $b$ prime to $p$; the valuation of $x$ is $n$.
So $12 = 2^2 \cdot 3$ has $2$-adic valuation $2$, and $3/8$ has $2$-adic valuation $-3$.
Multiplying numbers adds their valuations, just as with exponents.
Adding two numbers can only keep or raise the smaller valuation: $4 + 8 = 12$ has $2$-adic valuation $2$, the smaller of $2$ and $3$.
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# Infinite compositions of functions

## Summary

An infinite composition is the limit of the finite compositions $f_1 \circ f_2 \circ \cdots \circ f_n$, or of $f_n \circ \cdots \circ f_1$, of a sequence of functions.
It extends [[function composition|0THBK9V825]] to infinitely many factors, in the same way as a series extends a finite sum.
Continued fractions, nested radicals and power towers are all infinite compositions.
Iteration of a single function is the special case where all $f_n$ are equal.

## Explanations

### L1 human

Feed a starting value into $f_n$, feed the result into $f_{n-1}$, and so on down to $f_1$.
Doing this for larger and larger $n$ gives a sequence of numbers, and the infinite composition is its limit when it exists.
For example, the continued fraction $\cfrac{1}{2 + \cfrac{1}{2 + \cdots}}$ comes from composing the maps $w \mapsto \frac{1}{2 + w}$ and evaluating at $0$.
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# Chebyshev functions

## Summary

The Chebyshev functions $\vartheta(x)$ and $\psi(x)$ count the [[prime numbers|01826KMG2H]] up to $x$ with the weight $\log p$.
The first sums $\log p$ over primes $p \le x$, and the second also counts the prime powers $p^k \le x$.
They are easier to handle analytically than the [[prime-counting function|0RQSNVTFVZ]] $\pi(x)$, and the [[prime number theorem|0EHBY5Y84N]] is equivalent to $\psi(x) \sim x$.

## Explanations

### L1 human

Instead of counting each prime as $1$, give the prime $p$ the weight $\log p$.
Then large primes count more, in the right amount to make the total grow like $x$ itself.
For $x = 10$, the primes $2, 3, 5, 7$ give $\vartheta(10) = \log 210 \approx 5.35$.
The function $\psi$ also counts $4, 8, 9$, and $\psi(10) = \log 2520 \approx 7.83$, the logarithm of the least common multiple of $1, \dots, 10$.
</document>

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# Non-wandering set

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
A point $x \in X$ is *non-wandering* for $f$ if for every open neighbourhood $U$ of $x$ there is an integer $n \ge 1$ with $f^n(U) \cap U \neq \emptyset$.
The *non-wandering set* $\Omega(f)$ is the set of non-wandering points.

## Remarks

So $x$ is non-wandering exactly when it is not a wandering point in the sense of [[wandering sets|VNW5FHCQ2M]].
The set $\Omega(f)$ is closed, and $f(\Omega(f)) \subseteq \Omega(f)$.
If $X$ is compact and nonempty, then $\Omega(f)$ is nonempty.
Every [[recurrent point|4SQHHRPKMV]], and so every point on a [[periodic orbit|0AA97XA70K]], is non-wandering.
A non-wandering point need not be recurrent.
Example: for $f(x) = x/2$ on $\R$, the point $0$ is fixed, so $0 \in f^n(U) \cap U$ for every neighbourhood $U$ of $0$.
Non-example: for the same map, a point $x > 0$ has the neighbourhood $U = (2x/3, 4x/3)$, and $f^n(U) \subseteq (0, 2x/3]$ for $n \ge 1$, so $f^n(U) \cap U = \emptyset$.
In fact $\Omega(f) = \set{0}$ for this map.

## Explanations

### L1 human

A point is non-wandering when every small region around it comes back, at some later time, to overlap with itself.
The non-wandering set collects these points.
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# Jacobi method

## Statement

Let $A$ be a real $n \times n$ matrix with nonzero diagonal entries, and let $b \in \R^n$.
Write $A = D + R$, where $D$ is the diagonal part of $A$ and $R = A - D$.
The *Jacobi method* for the [[linear system|0GT4WWGWH2]] $A x = b$ starts from any $x^{(0)} \in \R^n$ and sets
$$x^{(k+1)} = D^{-1} \left( b - R x^{(k)} \right), \quad \text{that is,} \quad x^{(k+1)}_i = \frac{1}{a_{ii}} \Big( b_i - \sum_{j \ne i} a_{ij} x^{(k)}_j \Big).$$
Suppose that $A$ is strictly [[diagonally dominant|89P5GTKW87]] by rows.
Then $A$ is invertible, and for every $b$ and every $x^{(0)}$ the iterates converge to $A^{-1} b$.

## Remarks

The error $e_k = x^{(k)} - A^{-1} b$ satisfies $e_{k+1} = G e_k$ with $G = -D^{-1} R$.
The $i$-th absolute row sum of $G$ is $\sum_{j \ne i} \abs{a_{ij}} / \abs{a_{ii}}$, which is less than $1$ by diagonal dominance.
So the [[matrix norm|54T6Y6DSY4]] $\norm{G}_\infty$ is less than $1$, and $\norm{e_k}_\infty \le \norm{G}_\infty^k \norm{e_0}_\infty \to 0$.
In general, the method converges for every $x^{(0)}$ if and only if the [[spectral radius|9V7YQA85PC]] of $G$ is less than $1$.
Example: for $A$ with rows $(2, 1)$ and $(1, 2)$ and $b = (3, 3)$, starting at $0$ gives $(1.5, 1.5)$, $(0.75, 0.75)$, $(1.125, 1.125)$, tending to $(1, 1)$.
The error is multiplied by $-1/2$ at each step.
Without diagonal dominance the method can diverge.
For $A$ with rows $(1, 2)$ and $(2, 1)$, the matrix $G$ has eigenvalues $\pm 2$.

## Explanations

### L1 human

Each equation of the system is solved for its own unknown, using the old values of all the other unknowns.
All unknowns are updated at once, and the process repeats.
When each diagonal entry outweighs the rest of its row, every equation is dominated by its own unknown.
Then the updates shrink the error at each step.

### L2 human

The Jacobi method is the simplest [[iterative method|2RPE77ACER]] built from a splitting $A = M - N$, here with $M = D$.
Each step costs one matrix-vector product, and all components can be computed in parallel.
It converges slowly on discretized elliptic problems, where the spectral radius tends to $1$ as the mesh is refined.
A damped version is still widely used as a smoother in multigrid methods.
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# Plücker coordinates

## Summary

Plücker coordinates describe a line of three-dimensional projective space by six homogeneous numbers, the $2 \times 2$ minors of a matrix whose rows span the line.
They turn the set of lines of $\mathbb{P}^3$ into the points of a quadric in $\mathbb{P}^5$, the [[Klein quadric|6WV71T5CK1]].
More generally, the same minors embed every [[Grassmannian|CYPA882RB9]] into a projective space.

## Explanations

### L1 human

A point of projective space has coordinates, but a line is a bigger object, so how does one give it coordinates?
Pick two points on the line and write their coordinates as the two rows of a $2 \times 4$ matrix.
The six $2 \times 2$ minors of that matrix do not depend, up to a common factor, on which two points were picked.
These six numbers are the Plücker coordinates of the line.
For the line through $[1 : 0 : 0 : 0]$ and $[0 : 1 : 0 : 0]$, only one minor is nonzero.
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# Spectrum of the Laplace–Beltrami operator

## Summary

On a closed Riemannian manifold the [[Laplace–Beltrami operator|F4NWZNTRMD]] has a discrete spectrum $0 = \lambda_0 < \lambda_1 \le \lambda_2 \le \cdots \to \infty$ of eigenvalues of finite multiplicity.
The eigenfunctions are smooth and form an orthonormal basis of $L^2(M)$.
This sequence is the basic object of [[spectral geometry|QKAFRDPGE4]]: it controls the heat equation, the wave equation and the vibrations of the manifold.

## Explanations

### L1 human

Think of the manifold as a drum without boundary.
Its pure tones are the eigenfunctions of the Laplacian, and the squares of their frequencies are the eigenvalues.
The lowest eigenvalue is $0$, for the constant function, which does not vibrate.
On the circle of length $2\pi$ the eigenfunctions are $\cos k\theta$ and $\sin k\theta$, with eigenvalues $k^2$.
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# Numerical linear algebra

## Summary

Numerical linear algebra is the study of algorithms that solve problems of linear algebra on a computer, with floating-point numbers.
Its main problems are linear systems, least squares, [[eigenvalues|0BQAA2BK3Q]] and [[singular values|09VMDSRXWJ]].
It asks how much work an algorithm needs and how rounding errors affect the answer, using notions such as the [[condition number|AB3S6MHJ8X]] and backward stability.
Most computations in science and engineering end in one of its problems.

## Explanations

### L1 human

On paper, linear algebra is exact; on a computer, every number is rounded to a fixed number of digits.
Numerical linear algebra designs methods that stay accurate despite these small errors and that stay fast for matrices with millions of rows.
For example, [[Gaussian elimination|0JVJ2WCXZH]] with row exchanges solves a linear system reliably.
On the other hand, eigenvalues are almost never computed as roots of the [[characteristic polynomial|0VSSPAFM2C]], because those roots react strongly to tiny errors in the coefficients.
Methods such as [[LU decomposition|0RKG5YW66Y]], the [[QR algorithm|B6F8YWHYZ5]] and [[power iteration|H4MQD841BR]] belong to this field.
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# Möbius inversion on posets

## Statement

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]] in which every set $\set{y \in P : y \le x}$, for $x \in P$, is finite.
Let $\mu$ be the Möbius function of $P$: $\mu(x, x) = 1$, $\mu(x, y) = -\sum_{x \le z < y} \mu(x, z)$ for $x < y$, and $\mu(x, y) = 0$ when $x \not\le y$.
Let $K$ be a [[commutative ring|0310REJJY8]] and let $f, g \colon P \to K$.
Then
$$g(x) = \sum_{y \le x} f(y) \quad \text{for all } x \in P$$
if and only if
$$f(x) = \sum_{y \le x} \mu(y, x) \, g(y) \quad \text{for all } x \in P.$$

## Remarks

In the [[incidence algebra|1H9SFWHYEB]] the Möbius function is the inverse of the zeta function $\zeta$, which is $1$ on all pairs $y \le x$.
The first equation says $g = f \zeta$ and the second says $f = g \mu$, where a function on $P$ is multiplied by an element of the incidence algebra on the right.
Since $\zeta \mu = \mu \zeta = \delta$, each equation implies the other.
There is a dual form with sums over $y \ge x$, under the hypothesis that every set $\set{y : y \ge x}$ is finite.

Example: for the subsets of a finite set ordered by inclusion, $\mu(S, T) = (-1)^{\abs{T \setminus S}}$, and the theorem becomes the inclusion–exclusion principle.
For the positive integers ordered by divisibility, $\mu(d, n)$ is the number-theoretic Möbius function of $n/d$, and the theorem becomes the classical [[Möbius inversion formula|07Q4TBTFVB]].
For the chain $0 < 1 < 2 < \dots$, $\mu(x, x+1) = -1$ and $\mu(x, y) = 0$ for $y \ge x + 2$, so $f(x) = g(x) - g(x-1)$: a difference undoes a partial sum.

The finiteness hypothesis is needed for the sums to make sense: on $\Z$ with its usual order, $\sum_{y \le x} f(y)$ is an infinite sum.

## Explanations

### L1 human

If you know the totals $g(x)$ obtained by adding $f$ over everything below $x$, you can recover $f$.
The recipe uses signed weights $\mu(y, x)$ that depend only on the shape of the poset.
For ordinary numbers in a line it just says: take the difference of two consecutive running totals.

### L2 human

The theorem was put in this general form by Rota, and it unifies many counting techniques.
To use it, compute $\mu$ for the poset at hand, often with product formulas or with the recursion.
For the lattice of partitions of a set, $\mu$ of the whole interval is $(-1)^{n-1}(n-1)!$, and inversion counts connected structures from all structures.
In the lattice of subspaces of $\F_q^n$, $\mu$ involves powers of $q$, and inversion counts subspaces avoiding given conditions.
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# Ravenel conjectures

## Statement

Fix a [[prime number|01826KMG2H]] $p$.
Let $K(n)$ be the $n$-th Morava K-theory, with $K(0) = H\Q$.
Let $L_n$ be Bousfield localization with respect to the Johnson–Wilson spectrum $E(n)$, as in [[chromatic homotopy theory|300V40FC1B]].
A finite $p$-local [[spectrum|1GHF4EKBXS]] $X$ has *type* $n$ if $K(n)_* X \ne 0$ and $K(m)_* X = 0$ for all $m < n$.
The following conjectures of Ravenel are theorems.

1. (Nilpotence) A self-map $f \colon \Sigma^d X \to X$ of a finite $p$-local spectrum is nilpotent if $K(n)_*(f)$ is nilpotent for every $n \ge 0$.
   This is a form of the [[nilpotence theorem|9N027RT5R6]].
2. (Periodicity) If $X$ has type $n \ge 1$, there is a map $v \colon \Sigma^d X \to X$ such that $K(n)_*(v)$ is an isomorphism and $K(m)_*(v)$ is nilpotent for $m \ne n$.
3. (Thick subcategories) Let $\mathcal{C}_n$ be the category of finite $p$-local spectra of type at least $n$.
   Every thick subcategory of $\mathcal{C}_0$ is either some $\mathcal{C}_n$ or the zero subcategory.
4. (Smash product) For every spectrum $X$, the natural map $X \wedge L_n S \to L_n X$ is an equivalence, where $S$ is the $p$-local sphere spectrum.
5. (Chromatic convergence) For every finite $p$-local spectrum $X$, the map $X \to \operatorname{holim}_n L_n X$ is an equivalence.

## Remarks

Ravenel posed these conjectures in 1984, together with the telescope conjecture.
Devinatz, Hopkins and Smith proved the nilpotence theorem, and Hopkins and Smith deduced the periodicity and thick subcategory theorems from it.
Hopkins and Ravenel proved the smash product and chromatic convergence theorems.
The [[telescope conjecture|68650624FF]] is the exception: it holds at heights $0$ and $1$ but fails at every height $n \ge 2$.

At work: the sphere has type $0$, and the mod $p$ Moore spectrum has type $1$.
For $p$ odd, the periodicity theorem is realised by Adams' map $v_1 \colon \Sigma^{2p-2} S/p \to S/p$, whose iterates give infinite families in the stable homotopy groups of spheres.

## Explanations

### L1 human

Ravenel predicted a precise structure for the stable homotopy category, organized by the Morava K-theories.
Every finite spectrum has a height, and at its height it admits a periodic self-map.
The only ways to cut the category into closed pieces are by height.
Almost all of these predictions are now theorems.

### L2 human

The thick subcategory theorem lets one prove a property for all finite spectra of type at least $n$ by checking it on a single example.
The periodicity theorem produces the $v_n$-periodic families in the homotopy of spheres that chromatic homotopy theory studies.
The smash product theorem says that $L_n$ is a smashing localization, so it is determined by $L_n S$.
Chromatic convergence says that nothing is lost by studying a finite spectrum layer by layer.
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# Kesten's theorem on planar bond percolation

## Statement

Let $\Z^2$ be the square lattice.
Its vertices are the points of $\R^2$ with integer coordinates, and its edges join points at distance $1$.
Then the [[bond percolation threshold|MP8VSZK19A]] of $\Z^2$ is
$$p_c(\Z^2) = \frac12.$$

## Remarks

The key tool is self-duality.
The dual lattice, with a vertex at the centre of each square face, is a translate of $\Z^2$.
A dual edge is declared open when the primal edge it crosses is closed.
At $p = 1/2$ the primal and dual configurations have the same law.
A box crossing argument then shows that a suitable rectangle is crossed by an open path with probability exactly $1/2$.
Harris proved in 1960 that $\theta(1/2) = 0$, which gives $p_c \ge 1/2$; Kesten proved in 1980 that $p_c \le 1/2$.
So there is no infinite open cluster at $p = 1/2$ itself, while for every $p > 1/2$ there is one almost surely.

## Explanations

### L1 human

Keep each edge of the infinite square grid with probability $p$, independently.
The theorem says that an infinite connected piece appears exactly when $p$ is larger than $1/2$.
The number $1/2$ comes from a symmetry: the open edges and the closed edges of the grid play the same role in the plane.

### L2 human

It is the model case of an exact percolation threshold computed by planar duality.
The same duality idea gives $p_c = 1/2$ for site percolation on the triangular lattice.
Estimates on the probability that open paths cross long rectangles are the main tool of the proof.
In dimensions $d \ge 3$ no exact value of $p_c(\Z^d)$ is known.
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# Borel measure

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $\mathcal{B}(X)$ be its [[Borel σ-algebra|0WQXY32Q8D]], the smallest σ-algebra on $X$ that contains every open set.
A *Borel measure* on $X$ is a [[measure|PR11BZQQPG]] $\mu \colon \mathcal{B}(X) \to [0, \infty]$.

## Remarks

Example: the Dirac measure $\delta_p$ at a point $p \in X$, with $\delta_p(A) = 1$ if $p \in A$ and $0$ otherwise, is a Borel measure.
It is defined on all subsets of $X$, hence on $\mathcal{B}(X)$, and it is countably additive because $p$ lies in at most one set of a disjoint family.

Non-example: on $\R$, let $\Sigma = \set{A \subseteq \R : A \text{ or } \R \setminus A \text{ is countable}}$ and let $\mu$ be $0$ on countable sets and $1$ on the others.
Then $\mu$ is a measure on $\Sigma$, but $\Sigma$ does not contain the interval $(0, 1)$, so $\mu$ is not defined on all Borel sets and is not a Borel measure.

Conventions vary.
Some authors call a measure on a σ-algebra containing $\mathcal{B}(X)$, such as Lebesgue measure on the Lebesgue sets, a Borel measure.
Some authors require in addition that $\mu(K) < \infty$ for every compact set $K$; then counting measure on $\R$ is not a Borel measure.
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# Intersection multiplicity

## Definition

Let $X$ be a smooth [[algebraic variety|HV6YBS0YDH]] of [[dimension|36X2WTHYCB]] $n$ over a field.
Let $V, W \subset X$ be irreducible closed subvarieties that meet *properly*: every irreducible component of $V \cap W$ has dimension $\dim V + \dim W - n$.
Let $Z$ be an irreducible component of $V \cap W$, and let $A = \mathcal{O}_{X, Z}$ be the local ring of $X$ at the generic point of $Z$.
Let $I_V, I_W \subset A$ be the ideals of $V$ and $W$.
The *intersection multiplicity* of $V$ and $W$ along $Z$ is
$$i(Z; V \cdot W) = \sum_{j \ge 0} (-1)^j \, \ell_A\big(\operatorname{Tor}_j^A(A/I_V, A/I_W)\big),$$
where $\operatorname{Tor}$ is the [[Tor functor|QAGW4V0S62]] and $\ell_A$ is the [[length|V6JBKT168T]] of an $A$-module.

## Remarks

The sum is finite because $A$ is a [[regular local ring|63XXTF6B5M]], and each term has finite length because $Z$ is a component of $V \cap W$.
This is Serre's Tor formula, and $i(Z; V \cdot W) \ge 1$.
When $A/I_V$ and $A/I_W$ are Cohen–Macaulay, the higher Tor terms vanish and $i(Z; V \cdot W) = \ell_A\big(A/(I_V + I_W)\big)$.
For two plane curves $f = 0$ and $g = 0$ without common component, meeting at a point $P$, this gives $i_P(f, g) = \dim_k \mathcal{O}_{\mathbb{A}^2, P}/(f, g)$.
Example: for $f = y - x^2$ and $g = y$ at $P = (0, 0)$, $\mathcal{O}_{\mathbb{A}^2, P}/(y - x^2, y) \cong k[x]_{(x)}/(x^2)$, of dimension $2$.
Non-example: if $V = W$ is a line in $X = \mathbb{A}^2$, then $V \cap W$ has dimension $1$, not $1 + 1 - 2 = 0$, so the intersection is not proper and the multiplicity is not defined.
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<document source="https://tamnd.github.io/math2/node/B99049JPY6.md" tag="B99049JPY6" kind="statement">
# Vizing's theorem

## Statement

Let $G$ be a finite simple [[graph|0JD5C82QMB]] with [[maximum degree|0KNAN9F27P]] $\Delta$.
Let $\chi'(G)$ be the chromatic index of $G$, the least number of colours in a proper [[edge colouring|02CDYWSWSY]] of $G$.
Then
$$\Delta \le \chi'(G) \le \Delta + 1.$$

## Remarks

The lower bound is clear: the $\Delta$ edges at a vertex of maximum degree need distinct colours.
The upper bound is proved by colouring the edges one at a time with $\Delta + 1$ colours.
When a new edge cannot be coloured, a fan of edges at one end and a path in two colours are recoloured to free a colour.
Graphs with $\chi' = \Delta$ are of class 1, and graphs with $\chi' = \Delta + 1$ are of class 2.
An odd cycle has $\Delta = 2$ and needs $3$ colours, so it is of class 2; every bipartite graph is of class 1, by a theorem of Kőnig.
The hypothesis that $G$ is simple is needed: the multigraph on three vertices with each pair joined by $k$ parallel edges has $\Delta = 2k$ but needs $3k$ colours.
Deciding whether a graph is of class 1 is NP-complete.

## Explanations

### L1 human

Colour the edges of a graph so that edges at the same vertex get different colours.
If some vertex has $\Delta$ edges, you need at least $\Delta$ colours.
Vizing's theorem says that one extra colour is always enough.

### L2 human

For vertex colouring, the chromatic number can be far above any simple lower bound, but for edge colouring Vizing's theorem pins the answer to two values.
The proof is constructive and gives an efficient algorithm for a colouring with $\Delta + 1$ colours.
For multigraphs, Vizing also proved $\chi' \le \Delta + \mu$, where $\mu$ is the largest number of parallel edges.
</document>

<document source="https://tamnd.github.io/math2/node/B9FNGB5EBM.md" tag="B9FNGB5EBM" kind="definition">
# Willmore energy

## Definition

Let $S \subset \R^3$ be a [[compact|0TVJZYB6SH]] orientable [[regular surface|6080X3FAD4]] without boundary.
Let $H$ be its [[mean curvature|PA78NFEPCF]], $H = (k_1 + k_2)/2$ for the principal curvatures $k_1, k_2$, and let $dA$ be its area element.
The *Willmore energy* of $S$ is
$$\mathcal{W}(S) = \int_S H^2 \, dA.$$

## Remarks

For the sphere of radius $r$, one has $H^2 = 1/r^2$ and area $4\pi r^2$, so $\mathcal{W} = 4\pi$, for every $r$.
For the cylinder $x^2 + y^2 = r^2$, which is not compact, $H^2 = 1/(4r^2)$ is constant and the area is infinite, so the integral diverges.

The same formula defines $\mathcal{W}$ for a smooth immersion of a compact surface into $\R^3$; the value does not depend on the orientation, since $H^2$ does not.
Since $H^2 - K = (k_1 - k_2)^2/4$ and $\int_S K \, dA = 2\pi\chi(S)$ by the [[Gauss–Bonnet theorem|ESGEHKPD60]],
$$\mathcal{W}(S) = 2\pi\chi(S) + \frac{1}{4} \int_S (k_1 - k_2)^2 \, dA.$$
Some authors call $\int_S (H^2 - K)\, dA$ the Willmore energy; it differs from $\mathcal{W}$ by the topological constant $2\pi\chi(S)$.
For every such $S$, $\mathcal{W}(S) \ge 4\pi$, with equality exactly for round spheres.

## Explanations

### L1 human

The Willmore energy is the total of the squared mean curvature over a closed surface.
</document>

<document source="https://tamnd.github.io/math2/node/B9NEZNTRPT.md" tag="B9NEZNTRPT" kind="concept">
# Constructive function theory

## Summary

Constructive function theory is the part of [[approximation theory|YPJSH79SHQ]] that describes a function through the speed of its approximation by polynomials.
Its central results show that smoothness of $f$ and fast decay of the [[error of best approximation|YDN0T4MEX0]] $E_n(f)$ go together.
Sergei Bernstein introduced the name for this subject.

## Explanations

### L1 human

A smooth function is easy to imitate with polynomials of low degree, and a rough function needs high degree.
Constructive function theory turns this impression into exact statements.
For example, for a $2\pi$-periodic function and $0 < \alpha < 1$, the best trigonometric approximation of degree $n$ has error of order $n^{-\alpha}$ exactly when the function is Hölder continuous with exponent $\alpha$.
So you can read off the smoothness of a function from how well it can be approximated, and the other way round.
</document>

<document source="https://tamnd.github.io/math2/node/B9T33D5MJK.md" tag="B9T33D5MJK" kind="statement">
# Stone duality

## Statement

Let $\mathbf{Bool}$ be the category of [[Boolean algebras|0XADJKM699]] with Boolean algebra homomorphisms.
Let $\mathbf{Stone}$ be the category of [[Stone spaces|ZV38HA5PAQ]] with continuous maps.
Then $\mathbf{Bool}$ is [[equivalent|K7A04ZBD92]] to the opposite category of $\mathbf{Stone}$, by the following functors.

1. A Boolean algebra $B$ goes to its space $S(B)$ of [[ultrafilters|CQM84EW10G]], with basic open sets $\set{U : a \in U}$ for $a \in B$.
A homomorphism $h \colon A \to B$ goes to the continuous map $S(B) \to S(A)$, $U \mapsto h^{-1}(U)$.
2. A Stone space $X$ goes to the Boolean algebra $\operatorname{Clop}(X)$ of its [[clopen subsets|DQ4E55T3BW]]; a continuous map $f \colon X \to Y$ goes to the homomorphism $\operatorname{Clop}(Y) \to \operatorname{Clop}(X)$, $C \mapsto f^{-1}(C)$.
3. The maps $B \to \operatorname{Clop}(S(B))$, $a \mapsto \set{U : a \in U}$, and $X \to S(\operatorname{Clop}(X))$, $x \mapsto \set{C : x \in C}$, are natural isomorphisms.

## Remarks

The first isomorphism in clause 3 is [[Stone's representation theorem|S4QJ395PA5]].
The second one holds for this reason.
In a compact Hausdorff space with a basis of clopen sets, the members of an ultrafilter of clopen sets have exactly one common point.
Under the duality, injective homomorphisms correspond to surjective continuous maps, and quotients of $B$ correspond to closed subspaces of $S(B)$.
For example, the free Boolean algebra on a set $I$ corresponds to the product space $\set{0, 1}^I$, and the finite Boolean algebras correspond to the finite discrete spaces.

## Explanations

### L1 human

Boolean algebras and a certain kind of compact space are two pictures of the same thing.
Each algebra gives a space, each space gives an algebra, and going there and back returns what one started with.
Maps between algebras become maps between spaces, going in the opposite direction.

### L2 human

Stone duality lets one prove facts about Boolean algebras by topology and facts about Stone spaces by algebra.
For instance, a Boolean algebra is complete exactly when its Stone space is extremally disconnected, and an atom of the algebra corresponds to an isolated point of the space.
It is the model for many later dualities: [[Priestley duality|SA22SEC75K]] for distributive lattices, [[Esakia duality|CD9Y7YP4EJ]] for Heyting algebras, and the duality between spatial frames and sober spaces.
</document>

<document source="https://tamnd.github.io/math2/node/B9X31R5EGG.md" tag="B9X31R5EGG" kind="definition">
# Chern connection

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]], let $E \to X$ be a [[holomorphic vector bundle|TKA958Z4KZ]], and let $h$ be a [[Hermitian metric|Q6X6W2N29G]] on $E$.
Let $\bar\partial_E$ be the [[$\bar\partial$-operator|M1VKCW3RFW]] of $E$, which sends a smooth section to an $E$-valued $(0,1)$-form and kills exactly the holomorphic sections.
The *Chern connection* of $(E, h)$ is the [[connection|5ZYRRVE8BW]] $\nabla$ on $E$ such that:

1. $\nabla$ is compatible with $h$: $d\, h(s, t) = h(\nabla s, t) + h(s, \nabla t)$ for all smooth sections $s, t$.
2. The $(0,1)$-part of $\nabla$ is $\bar\partial_E$: $\nabla^{0,1} = \bar\partial_E$.

Such a connection exists and is unique.

## Remarks

Write $\nabla = \nabla^{1,0} + \nabla^{0,1}$ for the splitting by type.
In a local holomorphic frame $e_1, \dots, e_r$, let $H$ be the matrix with entries $H_{ij} = h(e_j, e_i)$.
Then the connection matrix is $\theta = H^{-1} \partial H$ in a suitable index convention, and the curvature is $\Theta = \bar\partial \theta$, a $(1,1)$-form with values in $\operatorname{End} E$.

Example: let $L$ be a holomorphic line bundle with local holomorphic frame $e$, and let $H = h(e, e) > 0$.
Put $\nabla e = \partial \log H \otimes e$.
Clause 2 holds because $\bar\partial_E e = 0$ and $\partial \log H$ has type $(1,0)$.
Clause 1 holds because $h(\nabla e, e) + h(e, \nabla e) = (\partial \log H + \bar\partial \log H) H = dH$.
The curvature is $\Theta = d \partial \log H = -\partial \bar\partial \log H$.

Non-example: on the trivial line bundle over $\C$ with the standard metric, let $\nabla s = ds + i s\, dx$, where $x = \operatorname{Re} z$.
Since $i\, dx$ is purely imaginary, clause 1 holds.
But $i\, dx = \tfrac{i}{2}(dz + d\bar z)$, so $\nabla^{0,1} = \bar\partial + \tfrac{i}{2} d\bar z \ne \bar\partial$, and clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/B9ZSFFTYRR.md" tag="B9ZSFFTYRR" kind="definition">
# Z-transform as a generating function

## Definition

Let $(x_n)_{n \ge 0}$ be a sequence of complex numbers.
Its ordinary [[generating function|0PHAC833JF]] is the power series $G(w) = \sum_{n \ge 0} x_n w^n$, with [[radius of convergence|0N288P2TB5]] $\rho \in [0, \infty]$.
Its [[unilateral Z-transform|EPG6JPVJE7]] is $X(z) = \sum_{n \ge 0} x_n z^{-n}$.
The *generating-function form of the Z-transform* is the identity
$$X(z) = G(1/z) \quad \text{for } \abs{z} > 1/\rho,$$
with the conventions $1/0 = \infty$ and $1/\infty = 0$.

## Remarks

Example: $x_n = a^n$ with $a \ne 0$ has $G(w) = 1/(1 - a w)$ with $\rho = 1/\abs{a}$, and $G(1/z) = z/(z - a)$ for $\abs{z} > \abs{a}$, which is its Z-transform.
Non-example: a two-sided sequence with some $x_n \ne 0$ for $n < 0$ has a bilateral Z-transform with positive powers of $z$.
These become negative powers of $w$, so the result is a Laurent series in $w$ and not an ordinary generating function.
Under $w = 1/z$, the disc of convergence $\abs{w} < \rho$ of $G$ becomes the outside $\abs{z} > 1/\rho$ of a circle.
The shift rule $\mathcal{Z}\{(x_{n+1})\} = z(X(z) - x_0)$ corresponds to $(G(w) - x_0)/w$ for generating functions.
</document>

<document source="https://tamnd.github.io/math2/node/BA2S40MM0Y.md" tag="BA2S40MM0Y" kind="statement">
# Margulis normal subgroup theorem

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite centre and no compact factors, of real rank at least $2$.
Let $\Gamma$ be an irreducible [[lattice|BTYF5Y5PFY]] in $G$.
Let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $\Gamma$.
Then either $N$ is contained in the [[centre|0DZ93493VN]] of $G$, and so is finite, or $N$ has finite [[index|0DTXBQPA2S]] in $\Gamma$.

## Remarks

The proof has two halves, for a non-central $N$.
One half shows that $\Gamma/N$ is amenable, by an analysis of the measurable quotients of the boundary $G/P$, with $P$ a minimal parabolic subgroup.
The other half shows that $\Gamma/N$ has property (T) when $G$ has it; when $G$ does not, a different argument replaces this half.
A discrete group that is amenable and has property (T) is finite.

Example: $\Gamma = \mathrm{SL}_3(\Z)$.
The centre of $\mathrm{SL}_3(\R)$ is trivial, so every nontrivial normal subgroup of $\mathrm{SL}_3(\Z)$ has finite index.
The congruence subgroups, the kernels of reduction modulo $m$, are such normal subgroups.

The rank condition is necessary.
The principal congruence subgroup $\Gamma(3)$ of $\mathrm{SL}_2(\Z)$ is normal, of finite index and free of rank at least $2$.
Its commutator subgroup $N$ is characteristic in $\Gamma(3)$, so it is normal in $\mathrm{SL}_2(\Z)$.
But $N$ is infinite and has infinite index, since $\Gamma(3)/N$ is free abelian of positive rank.

## Explanations

### L1 human

A normal subgroup lets one form a quotient group.
For lattices in higher rank groups, such as $\mathrm{SL}_3(\Z)$, every quotient is either finite or almost the whole group.
So these groups are almost simple in an infinite sense: they cannot be mapped onto an infinite smaller group.

### L2 human

The theorem shows that higher rank lattices are just infinite up to the centre.
It is a strong contrast to lattices in $\mathrm{SL}_2(\R)$, which are often virtually free and have a huge supply of normal subgroups.
It works together with the [[Margulis superrigidity theorem|DSKV65NHZZ]] and the [[Margulis arithmeticity theorem|FEJK846N5X]] to describe these lattices completely.
</document>

<document source="https://tamnd.github.io/math2/node/BA9HBM8DFR.md" tag="BA9HBM8DFR" kind="concept">
# Besicovitch almost periodic function

## Summary

A Besicovitch almost periodic function is a locally integrable function on the real line with the following property.
It can be approximated in a mean sense by trigonometric polynomials with real frequencies.
The mean is the upper average of $\abs{f}^p$ over long intervals $[-T, T]$.
This class, introduced by Abram Besicovitch, is much larger than the class of [[Bohr almost periodic functions|ED69H4C3PK]].
For $p = 2$ it carries a complete Fourier theory with a Parseval identity, and it is used for signals with discrete but irregular spectrum.

## Explanations

### L1 human

Bohr's almost periodic functions must be uniformly close to sums of waves everywhere on the line.
Besicovitch only asks that they be close on average over long stretches of the line.
So a function may have jumps, or even large values on a bounded region, since a bounded region contributes nothing to a long-run average.
For instance, the indicator function of $[0, 1]$ is Besicovitch almost periodic, because its long-run average size is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/BAADB7WHWZ.md" tag="BAADB7WHWZ" kind="concept">
# Sidon set

## Summary

A Sidon set is a set of integers, or of elements of an abelian group, in which all sums of two elements are different.
Here sums that differ only in the order of the summands count as the same.
For sets of integers this is the same as saying that all differences of two distinct elements are different.
Sidon sets are the extreme opposite of sets with additive structure: their [[additive energy|MAQKH72P6V]] is as small as possible.
The largest Sidon subset of $\set{1, \dots, N}$ has size asymptotic to $\sqrt{N}$.

## Explanations

### L1 human

Pick numbers so that no two pairs of them have the same sum.
The powers of two $1, 2, 4, 8, 16$ work, because binary expansions show each sum of two of them only once.
Such sets cannot be very dense: among the numbers up to $N$, a Sidon set has only on the order of $\sqrt{N}$ elements.
The reason is that its roughly $n^2/2$ pairwise sums must all be different and at most $2N$.
</document>

<document source="https://tamnd.github.io/math2/node/BAGCJR5EQF.md" tag="BAGCJR5EQF" kind="definition">
# Rayleigh–Ritz method

## Definition

Let $H$ be a real or complex [[Hilbert space|HM2HHE3J51]] with inner product $\ip{\cdot}{\cdot}$.
Let $A$ be a [[self-adjoint operator|0NEGN5CRSC]] in $H$ that is bounded below.
Let $Q$ be its closed quadratic form with form domain $D(Q)$, so that $Q(u, v) = \ip{Au}{v}$ for $u$ in the domain of $A$.
Let $m \ge 1$ and let $\varphi_1, \dots, \varphi_m \in D(Q)$ be linearly independent trial vectors.
Let $K$ and $M$ be the $m \times m$ matrices with entries $K_{ij} = Q(\varphi_j, \varphi_i)$ and $M_{ij} = \ip{\varphi_j}{\varphi_i}$.
The *Rayleigh–Ritz method* computes the solutions of the generalised eigenvalue problem
$$K c = \mu M c, \qquad c \in \C^m \setminus \set{0}.$$
Its eigenvalues $\mu_1 \le \cdots \le \mu_m$ are the *Ritz values*, and the vectors $\sum_j c_j \varphi_j$ for the eigenvectors $c$ are the *Ritz vectors*.

## Remarks

The Ritz values are the stationary values of the [[Rayleigh quotient|06WHP8SE98]] $Q(u, u) / \ip{u}{u}$ on the span $V$ of the trial vectors.
If the spectrum of $A$ below the essential spectrum consists of [[eigenvalues|0BQAA2BK3Q]] $\lambda_1 \le \lambda_2 \le \cdots$, the [[min-max theorem|G11NE3AT6E]] gives $\lambda_k \le \mu_k$ for $k = 1, \dots, m$.
For a Hermitian $n \times n$ matrix $A$ and a matrix $W$ with $m$ orthonormal columns, the Ritz values are the eigenvalues of $W^* A W$.

Example: $H = L^2(0, 1)$ and $A = -\frac{d^2}{dx^2}$ with Dirichlet conditions, so $D(Q) = H^1_0(0, 1)$, $Q(u, u) = \int_0^1 \abs{u'}^2 \, dx$ and $\lambda_1 = \pi^2$.
With $m = 1$ and $\varphi_1(x) = x (1 - x)$, which vanishes at $0$ and $1$, one gets $K = \int_0^1 (1 - 2x)^2 \, dx = \frac{1}{3}$ and $M = \int_0^1 x^2 (1 - x)^2 \, dx = \frac{1}{30}$.
So $\mu_1 = 10$, an upper bound for $\pi^2 \approx 9.8696$.

Non-example: the trial function $\varphi_1 = 1$ is not in $H^1_0(0, 1)$, because it does not vanish at the ends.
Its Rayleigh quotient is $0 < \pi^2$, so the upper bound fails when the trial vectors leave the form domain.

## Explanations

### L1 human

The method restricts the eigenvalue problem to a few trial vectors and solves the resulting small matrix problem, which gives upper bounds for the lowest eigenvalues.
</document>

<document source="https://tamnd.github.io/math2/node/BAH89SW049.md" tag="BAH89SW049" kind="definition">
# Sheaf on a site

## Definition

Let $(\mathcal{C}, J)$ be a [[site|NYE0Z94MME]] and let $P \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ be a [[presheaf|AND2WHZC8P]].
For $x \in P(c)$ and $f \colon d \to c$, write $x|_f = P(f)(x)$.
Let $S$ be a [[sieve|CNCXNWKC6R]] on $c$.
A *matching family* for $S$ is a choice of an element $x_f \in P(d)$ for each $f \colon d \to c$ in $S$ such that $x_{f \circ g} = x_f|_g$ for every morphism $g \colon e \to d$.
An *amalgamation* of the family is an element $x \in P(c)$ with $x|_f = x_f$ for every $f \in S$.
The presheaf $P$ is a *sheaf* for $J$ when the following holds.
For every object $c$ and every covering sieve $S \in J(c)$, each matching family for $S$ has exactly one amalgamation.

## Remarks

Equivalently, $P$ is a sheaf when, for every covering sieve $S \subseteq \Hom(-, c)$, every natural transformation $S \to P$ extends uniquely to $\Hom(-, c) \to P$.
A morphism of sheaves is a natural transformation; sheaves on $(\mathcal{C}, J)$ form a full subcategory $\mathrm{Sh}(\mathcal{C}, J)$ of presheaves.
Example: on the site of open sets of a space $X$, the presheaf $P(U) = C(U, \R)$ of continuous functions is a sheaf.
Functions on the members of a covering sieve that agree on overlaps define one function on $U$, which is continuous because continuity is local.
Non-example: on the same site, the constant presheaf $P(U) = \set{0, 1}$ fails, because the empty cover of $U = \emptyset$ has a unique (empty) matching family but two amalgamations.
For this site the definition gives back the [[sheaves on a topological space|TB6TM6R4MD]].
</document>

<document source="https://tamnd.github.io/math2/node/BAN4VASZF0.md" tag="BAN4VASZF0" kind="concept">
# Substitution tiling

## Summary

A substitution tiling is a tiling of Euclidean space built from finitely many tile shapes.
The rule inflates each tile and cuts the result into tiles of the original shapes.
Repeating the rule produces larger and larger patches, and the tilings whose patches all arise in this way are the substitution tilings.
They give the standard examples of aperiodic order, such as the Penrose tilings, and they model quasicrystals.
The tilings of one substitution form a compact space on which translations act, and this [[dynamical system|BBZP1R872N]] carries much of their structure.

## Explanations

### L1 human

Start with a few shapes of tiles.
Blow each tile up by a fixed factor, then cut the big tile into tiles of the original shapes.
Do it again and again, and the patches grow to cover the whole plane.
A unit square, blown up by $2$ and cut into four unit squares, gives the ordinary square grid.
Cleverer rules, like the one for the Penrose tilings, give tilings that never repeat periodically.
See the [[definition|T4403SAMB2]].
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<document source="https://tamnd.github.io/math2/node/BANY6GN2GD.md" tag="BANY6GN2GD" kind="definition">
# Hyperbolic triangle

## Definition

Let $\mathbb{H}^2$ be the [[hyperbolic plane|GN4N94GQ0Z]] of constant curvature $-1$.
Let $A$, $B$, $C$ be three points of $\mathbb{H}^2$ that do not lie on one [[geodesic|NSYMQMHPDN]] line.
The *hyperbolic triangle* $ABC$ is the union of the three geodesic segments $AB$, $BC$ and $CA$, together with the region they bound.
Its sides are the hyperbolic lengths $a = d(B, C)$, $b = d(C, A)$, $c = d(A, B)$.
Its angles $\alpha$, $\beta$, $\gamma$ at $A$, $B$, $C$ are the angles between the tangent vectors of the two sides at each vertex.

## Remarks

Example: in the [[Poincaré disk model|85MM2EA0TK]], take $A = 0$, $B = 1/2$ and $C = i/2$.
The only geodesic through $0$ and $1/2$ is the real diameter, which does not contain $i/2$, so $A, B, C$ span a hyperbolic triangle.
Its angle at $A$ is $\pi/2$, since the model is conformal and the sides at $A$ are perpendicular diameters.
Non-example: the points $0$, $1/4$, $1/2$ lie on the real diameter, which is one geodesic line, so they do not form a hyperbolic triangle.
In curvature $-1$, the angles satisfy $\alpha + \beta + \gamma < \pi$, and the area is $\pi - (\alpha + \beta + \gamma)$.
One also allows vertices at infinity, on the ideal boundary; a triangle with all three vertices ideal is an *ideal triangle*, with all angles $0$ and area $\pi$.
The notion is the hyperbolic counterpart of the Euclidean [[triangle|0CTNFGYMM2]].
</document>

<document source="https://tamnd.github.io/math2/node/BANY9YR82A.md" tag="BANY9YR82A" kind="statement">
# Lagrange inversion theorem

## Statement

Let $K$ be a field of characteristic $0$.
Let $\phi(w) \in K[\![w]\!]$ be a [[formal power series|033WGN00BF]] with $\phi(0) \ne 0$.
Then there is a unique formal power series $T(z) \in K[\![z]\!]$ with $T(0) = 0$ and $T(z) = z\, \phi(T(z))$.
For every $H(w) \in K[\![w]\!]$ and every integer $n \ge 1$,
$$[z^n]\, H(T(z)) = \frac{1}{n} [w^{n-1}]\, H'(w)\, \phi(w)^n,$$
where $[z^n] F$ is the [[coefficient|0R0D0XY4HT]] of $z^n$ in $F$.
In particular, $[z^n]\, T(z) = \frac{1}{n} [w^{n-1}]\, \phi(w)^n$.

## Remarks

Proof idea: work with formal Laurent series, where the coefficient of $w^{-1}$ in a derivative is always $0$.
Change the variable from $z$ to $w = T(z)$, so $z = w / \phi(w)$.
The coefficient of $z^{-1}$ in $z^{-n-1} H(T(z))$ becomes a coefficient of $w^{-1}$, and an integration by parts gives the formula.

Example: rooted labelled trees have the exponential [[generating function|0PHAC833JF]] $T(z) = z e^{T(z)}$.
With $\phi(w) = e^w$, the theorem gives $[z^n] T = \frac{1}{n} \cdot \frac{n^{n-1}}{(n-1)!} = \frac{n^{n-1}}{n!}$.
So there are $n^{n-1}$ [[rooted trees|0Y0JGC2K9T]] on $n$ labelled vertices.

Example: with $\phi(w) = (1 + w)^2$, the series $T$ counts binary trees by internal nodes.
The theorem gives $[z^n] T = \frac{1}{n} \binom{2n}{n-1}$, which is the [[Catalan number|0YWZJEABB2]] $C_n$.

The hypothesis $\phi(0) \ne 0$ makes the problem meaningful.
If $\phi(0) = 0$, then $T = 0$ is the only solution with $T(0) = 0$, and the equation counts nothing.
Characteristic $0$ is used to divide by $n$.
There is also an analytic version for holomorphic $\phi$ near $0$, which then gives the Taylor series of the inverse function.

## Explanations

### L1 human

Many counting problems lead to an equation of the form $T = z\, \phi(T)$ for a generating function $T$.
Such an equation is usually hard to solve for $T$ in closed form.
Lagrange inversion gives the coefficients of $T$ directly, without solving the equation.
One only has to expand the power $\phi(w)^n$ and read off one coefficient.

### L2 human

The theorem is the main tool for counting trees and tree-like structures.
A family of rooted trees defined by the allowed numbers of children has a generating function with $T = z\, \phi(T)$, where $\phi$ records the allowed numbers of children.
In the [[symbolic method|JP11D90EJC]] such equations arise at once from the recursive description of the trees.
The form with $H$ counts forests, since a sequence of $k$ trees has generating function $T^k$.
Equivalently, the theorem computes the coefficients of the compositional inverse of $z / \phi(z)$.
</document>

<document source="https://tamnd.github.io/math2/node/BAVME420YM.md" tag="BAVME420YM" kind="definition">
# Rademacher functions

## Definition

Let $k \ge 1$ be an integer.
The *$k$-th Rademacher function* $r_k \colon [0, 1) \to \set{1, -1}$ is given by
$$r_k(t) = \sgn\big(\sin(2^k \pi t)\big)$$
where $\sin(2^k \pi t) \ne 0$.
At the finitely many other points of $[0,1)$, its value is fixed so that $r_k$ is right-continuous.
Equivalently, $r_k(t) = (-1)^{t_k}$, where $t = \sum_{i \ge 1} t_i 2^{-i}$ is the binary expansion of $t$ that does not end in an infinite run of $1$s.
The family $(r_k)_{k \ge 1}$ is the *Rademacher system*.

## Remarks

The Rademacher system is orthonormal in $L^2([0, 1))$ (see [[Lp space|SASP14MA1Z]]).
It is not complete: the product $r_1 r_2$ has norm $1$ and is orthogonal to every $r_k$.
Under [[Lebesgue measure|2Z2QVWZ262]] on $[0, 1)$, the $r_k$ are independent random variables, each equal to $1$ or $-1$ with probability $\tfrac12$.

Example: $r_2$ equals $1$ on $[0, \tfrac14)$ and $[\tfrac12, \tfrac34)$ and $-1$ on $[\tfrac14, \tfrac12)$ and $[\tfrac34, 1)$.
Check: $\sin(4 \pi t) > 0$ exactly for $t \in (0, \tfrac14) \cup (\tfrac12, \tfrac34)$.

Non-example: the function $\sgn(\sin(3 \pi t))$ is a $\pm 1$ square wave on $[0, 1)$.
Its sign changes are at $\tfrac13$ and $\tfrac23$, not at dyadic points, so it is not a Rademacher function.
</document>

<document source="https://tamnd.github.io/math2/node/BAW0T68GWK.md" tag="BAW0T68GWK" kind="concept">
# Ring spectrum

## Summary

A ring spectrum is a [[spectrum|1GHF4EKBXS]] with a multiplication and a unit that satisfy the ring axioms up to homotopy.
It is the stable analogue of a ring, with the [[sphere spectrum|14CGGHA85T]] in the role of the integers.
The cohomology theory represented by a ring spectrum has cup products, so its values on spaces are graded rings.

## Explanations

### L1 human

A ring is a set with a multiplication that has a unit and is associative.
A ring spectrum is the same idea one level up: a spectrum $E$ with a map $E \wedge E \to E$ and a unit map from the sphere spectrum.
Ordinary cohomology with coefficients in a ring, complex K-theory and complex cobordism all come from ring spectra.
This is why in each of them one can multiply two cohomology classes.
</document>

<document source="https://tamnd.github.io/math2/node/BAWHE1W6MH.md" tag="BAWHE1W6MH" kind="concept">
# Cofinality

## Summary

The cofinality of a limit [[ordinal|0YSCSX5YZC]] is the least length of a sequence of smaller ordinals that climbs all the way up to it.
The [[precise definition|NTE5C260S1]] uses cofinal subsets, and it applies also to partially ordered sets.
Cofinality splits infinite cardinals into [[regular|13ZV2F8JAR]] and singular ones, and it constrains the possible values of cardinal exponentiation.

## Explanations

### L1 human

To reach the top of $\omega$, the natural numbers, you need infinitely many steps $0, 1, 2, \ldots$, so its cofinality is $\omega$.
The much bigger cardinal $\aleph_\omega$ can also be reached in $\omega$ steps, through $\aleph_0, \aleph_1, \aleph_2, \ldots$, so its cofinality is also $\omega$.
In contrast, $\aleph_1$ cannot be reached by any countable sequence of countable ordinals, since a countable union of countable sets is countable.
So the cofinality of $\aleph_1$ is $\aleph_1$ itself.
Cofinality measures how hard it is to climb to the top.
</document>

<document source="https://tamnd.github.io/math2/node/BB0TSMW93P.md" tag="BB0TSMW93P" kind="concept">
# Flag manifold

## Summary

A flag manifold is a quotient $G/P$ of a connected complex semisimple Lie group $G$ by a parabolic subgroup $P$.
The basic example is the space of complete flags $V_1 \subset V_2 \subset \cdots \subset V_{n-1}$ of subspaces of $\C^n$ with $\dim V_i = i$.
Projective spaces and Grassmannians are flag manifolds too.
Flag manifolds are compact complex manifolds on which $G$ acts, and they carry the geometry behind the Borel–Weil theorem and Schubert calculus.

## Explanations

### L1 human

A flag in $\C^3$ is a line through the origin together with a plane that contains it, like a flagpole and a flag.
The set of all such flags is a smooth, compact space of complex dimension $3$.
Invertible matrices move flags around, and any flag can be moved to any other.
So the space of flags is a group divided by the subgroup that fixes one flag, the upper triangular matrices.
Other flag manifolds keep only some of the subspaces, as the projective plane keeps only the line.
</document>

<document source="https://tamnd.github.io/math2/node/BB0X043XD1.md" tag="BB0X043XD1" kind="definition">
# Néron–Severi group

## Definition

Let $X$ be a smooth [[projective variety|A4H4CZ2V35]] over an [[algebraically closed field|02SJG1HC32]] $k$, with [[Picard group|3Z9DCWYXQZ]] $\operatorname{Pic}(X)$ of isomorphism classes of [[invertible sheaves|7JCF2TE3V5]].
Consider a connected variety $T$ over $k$, points $t_0, t_1 \in T(k)$ and an invertible sheaf $M$ on $X \times T$.
An invertible sheaf $L$ on $X$ is *algebraically equivalent to $0$* if such data exist with:

1. $M|_{X \times \set{t_0}} \cong \mathcal{O}_X$;
2. $M|_{X \times \set{t_1}} \cong L$.

Let $\operatorname{Pic}^0(X) \subset \operatorname{Pic}(X)$ be the subgroup of classes algebraically equivalent to $0$.
The *Néron–Severi group* of $X$ is
$$\operatorname{NS}(X) = \operatorname{Pic}(X) / \operatorname{Pic}^0(X).$$

## Remarks

The theorem of the base says that $\operatorname{NS}(X)$ is finitely generated; its rank $\rho(X)$ is the *Picard number*.
Example: for $X = \mathbb{P}^n$, $\operatorname{Pic}(X) \cong \Z$, generated by $\mathcal{O}(1)$, and $\operatorname{Pic}^0(X) = 0$, so $\operatorname{NS}(\mathbb{P}^n) \cong \Z$.
For an elliptic curve $E$, $\operatorname{Pic}(E) \cong \Z \oplus E(k)$, with $\operatorname{Pic}^0(E) \cong E(k)$ the classes of degree $0$, so $\operatorname{NS}(E) \cong \Z$ through the degree.
Non-example: $\operatorname{Pic}(X)$ itself is usually not a candidate for this role: for an elliptic curve over $\C$, $\operatorname{Pic}(E)$ contains $E(\C)$, which is not finitely generated.
Over $\C$, the first Chern class identifies $\operatorname{NS}(X)$ with the image of $\operatorname{Pic}(X) \to H^2(X(\C), \Z)$, which is the group of integral classes of Hodge type $(1, 1)$ by the Lefschetz theorem on $(1, 1)$-classes.
</document>

<document source="https://tamnd.github.io/math2/node/BB15T3J891.md" tag="BB15T3J891" kind="concept">
# Differentiation operator p

## Summary

In [[operational calculus|8050R3Z6HA]], $p$ denotes the operator $d/dt$ that sends a function of time to its [[derivative|0V1EJCT8G6]].
It is treated as an algebraic symbol: one forms polynomials and fractions in $p$, and $1/p$ is integration from $0$.
Because $p$ undoes integration but integration does not quite undo $p$, initial values appear in the rules, as in $p^{-1} p f = f - f(0)$.

## Explanations

### L1 human

Writing $p$ for "take the derivative" lets one write $y'' + 3y' + 2y$ as $(p^2 + 3p + 2) y$, and even factor it as $(p + 1)(p + 2) y$.
Its partner $1/p$ means "integrate from $0$".
Differentiating an integral gives back the function, but integrating a derivative from $0$ loses the starting value.
For example, for $f(t) = t + 5$, integrating $f' = 1$ from $0$ gives $t$, not $t + 5$.
</document>

<document source="https://tamnd.github.io/math2/node/BB28P2ED5H.md" tag="BB28P2ED5H" kind="statement">
# Hyperbolic Dehn surgery theorem

## Statement

Let $M$ be an orientable [[hyperbolic 3-manifold|GHVW1Z35BS]] of finite volume that is the interior of a compact $3$-manifold $\bar{M}$ whose boundary consists of tori $T_1, \ldots, T_n$.
A *slope* on $T_i$ is an isotopy class of essential simple closed curves on $T_i$.
For slopes $s_1, \ldots, s_n$, let $M(s_1, \ldots, s_n)$ be the closed manifold obtained by [[Dehn filling|ZA405B0Z1S]].
It is made by gluing a solid torus to each $T_i$ so that a meridian is glued to $s_i$.
Then for each $i$ there is a finite set $E_i$ of slopes on $T_i$ such that $M(s_1, \ldots, s_n)$ is hyperbolic whenever $s_i \notin E_i$ for every $i$.

## Remarks

The theorem is due to Thurston.
The complete structure on $M$ lies in a space of incomplete hyperbolic structures, parametrised near it by the shapes of ideal tetrahedra.
Nearby incomplete structures whose metric completion is a closed manifold give the filled manifolds.
Thurston also showed that the volumes of the filled manifolds are smaller than the volume of $M$ and tend to it as all slopes go to infinity.
For the complement of the [[figure-eight knot|C01XBGAHC6]], exactly ten slopes give a non-hyperbolic filling: $\infty$, $0$, $\pm 1$, $\pm 2$, $\pm 3$ and $\pm 4$.
The finite set cannot be empty in general: the filling along the meridian of a knot complement gives back $S^3$, which is not hyperbolic.

## Explanations

### L1 human

Start from a hyperbolic space with some tube-shaped ends, and close each end by gluing in a solid doughnut.
There are infinitely many ways to glue each doughnut.
Thurston's theorem says that all of them, except for finitely many on each end, give a closed hyperbolic manifold.

### L2 human

The theorem produces infinitely many closed hyperbolic $3$-manifolds from each cusped one, for example from knot complements.
Since every closed orientable $3$-manifold is obtained by surgery on a link in $S^3$, it suggests that hyperbolic manifolds are generic.
The exceptional slopes are well studied: Lackenby and Meyerhoff proved that a one-cusped manifold has at most ten of them, so the figure-eight knot complement is extremal.
</document>

<document source="https://tamnd.github.io/math2/node/BB4FJ1FFGM.md" tag="BB4FJ1FFGM" kind="conjecture">
# Artin's conjecture on primitive roots

## Statement

Let $a$ be an integer that is not a perfect square and is not $-1$.
Then there are infinitely many [[prime numbers|01826KMG2H]] $p$ such that $a$ is a [[primitive root modulo|0F33KX9Q6F]] $p$.

## Remarks

The conjecture is open for every single value of $a$.
The full conjecture also predicts a positive density for this set of primes; for many $a$ the predicted density is Artin's constant, about $0.3739558$.
Christopher Hooley proved the conjecture in 1967 under the assumption of the [[generalized Riemann hypothesis|DNKA4WBKDS]].
Roger Heath-Brown proved in 1986 that at most two primes $a$ fail the conjecture; so at least one of $2$, $3$, $5$ is a primitive root modulo infinitely many primes.
The two hypotheses are needed: if $a = b^2$ is a square and $p \nmid a$, then $a^{(p-1)/2} \equiv b^{p-1} \equiv 1 \pmod p$, so $a$ is never a primitive root modulo an odd prime $p$.
For $a = -1$, the [[multiplicative order|02RRHJ7R98]] of $-1$ is at most $2$, so it is a primitive root only modulo $2$ and $3$.
For example, $2$ is a primitive root modulo $3$, $5$, $11$, $13$, $19$ and $29$.

## Explanations

### L1 human

A primitive root modulo $p$ is a number whose powers run through all nonzero remainders modulo $p$.
Artin guessed that a fixed number such as $2$ is a primitive root for infinitely many primes, unless there is an obvious reason against it.
Squares are such an obvious reason, and so is $-1$.
The guess is still unproved, even for $a = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/BB7S4NTZ6V.md" tag="BB7S4NTZ6V" kind="concept">
# Bergman metric

## Summary

The Bergman metric of a bounded domain $\Omega \subseteq \C^n$ is the Kähler metric whose potential is the logarithm of the [[Bergman kernel|8RWDHW0CXC]] on the diagonal.
It is defined from the Hilbert space of square-integrable holomorphic functions, the [[Bergman space|2D619WTPYC]].
Hence every biholomorphic map between bounded domains is an isometry for it.
Stefan Bergman introduced it, and on the unit disc it is the hyperbolic metric up to a constant factor.
It is a basic tool for the study of biholomorphic maps, for example in Fefferman's theorem on their boundary extension.

## Explanations

### L1 human

A bounded domain has a natural Hilbert space of holomorphic functions, those whose squared modulus has finite integral.
From this space one builds a kernel $K(z, w)$, and the logarithm of $K(z, z)$ serves as a potential for a Riemannian metric, the Bergman metric.
Because the construction uses only holomorphic functions and the integral, any biholomorphic map carries the metric of one domain to that of the other.
For the unit disc the Bergman metric is $\frac{2 \abs{dz}^2}{(1 - \abs{z}^2)^2}$, a multiple of the hyperbolic metric.
</document>

<document source="https://tamnd.github.io/math2/node/BBF9SCQRM9.md" tag="BBF9SCQRM9" kind="definition">
# Semitopological group

## Definition

A *semitopological group* is a [[group|G5S1M9]] $G$ together with a [[topology|0CQHFPGJ45]] on $G$ such that the multiplication $G \times G \to G$ is separately continuous.
That is, for each $a \in G$ the left translation $x \mapsto ax$ and the right translation $x \mapsto xa$ are [[continuous|0VGFHTM75Y]] maps $G \to G$.

## Remarks

Example: $(\Z, +)$ with the cofinite topology.
Each translation $x \mapsto x + a$ is a bijection, so it maps cofinite sets to cofinite sets and is continuous.
This group is not a [[topological group|92700QAB5Y]]: every nonempty basic open set $U \times V$ of $\Z \times \Z$ contains a pair $(x, -x)$, so the preimage of $\Z \setminus \set{0}$ under addition is not open.
Non-example: the multiplicative group $\R \setminus \set{0}$ with the topology whose open sets are $\emptyset$, the whole set, and the sets $(a, \infty) \setminus \set{0}$ for $a \in \R$.
The left translation $x \mapsto (-1) x$ is not continuous: the preimage of the open set $(1, \infty)$ is $(-\infty, -1)$, which is not open.
Every topological group is a semitopological group, since joint continuity of multiplication implies separate continuity.
In a semitopological group each translation is a homeomorphism, with inverse the translation by $a^{-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/BBHM1A36M7.md" tag="BBHM1A36M7" kind="statement">
# Dynkin's π-λ theorem

## Statement

Let $X$ be a set, let $\mathcal{P}$ be a [[π-system|4W9NM803BB]] on $X$, and let $\mathcal{D}$ be a [[Dynkin system|V5M2CKB591]] on $X$ with $\mathcal{P} \subseteq \mathcal{D}$.
Let $\sigma(\mathcal{P})$ be the [[σ-algebra|0SV8K0TDGY]] generated by $\mathcal{P}$.
Then $\sigma(\mathcal{P}) \subseteq \mathcal{D}$.

## Remarks

Let $d(\mathcal{P})$ be the smallest Dynkin system that contains $\mathcal{P}$.
For a fixed $A \in \mathcal{P}$, the sets $B$ with $A \cap B \in d(\mathcal{P})$ form a Dynkin system that contains $\mathcal{P}$, so it contains $d(\mathcal{P})$.
Repeating the argument with $A \in d(\mathcal{P})$ shows that $d(\mathcal{P})$ is a π-system.
A Dynkin system that is a π-system is a σ-algebra, so $\sigma(\mathcal{P}) = d(\mathcal{P}) \subseteq \mathcal{D}$.

The main use is uniqueness of measures.
Let $\mu$ and $\nu$ be [[probability measures|059TJN0NZ1]] on $(X, \sigma(\mathcal{P}))$ that agree on $\mathcal{P}$.
The sets on which $\mu$ and $\nu$ agree form a Dynkin system that contains $\mathcal{P}$, so $\mu = \nu$ on $\sigma(\mathcal{P})$.
For example, a probability measure on the Borel sets of $\R$ is fixed by its values on the rays $(-\infty, a]$.

The hypothesis that $\mathcal{P}$ is a π-system is needed.
On $X = \set{1, 2, 3, 4}$, let $\mathcal{P} = \set{\set{1, 2}, \set{2, 3}}$ and let $\mathcal{D}$ be the family of subsets with an even number of elements.
Then $\mathcal{D}$ is a Dynkin system that contains $\mathcal{P}$, but $\set{2} = \set{1, 2} \cap \set{2, 3}$ lies in $\sigma(\mathcal{P})$ and not in $\mathcal{D}$.

## Explanations

### L1 human

To show that a property holds for every set of a σ-algebra, it is often enough to check it on a simple generating family.
The theorem says this works when the simple family is closed under intersections and the good sets form a Dynkin system.
Dynkin systems are easy to check, since they only ask for complements and disjoint unions.

### L2 human

The π-λ theorem is the workhorse behind uniqueness statements in measure theory and probability.
It shows that a distribution function determines a probability measure on $\R$, and that a measure on a product is fixed by its values on rectangles.
It also gives the independence of σ-algebras generated by independent π-systems.
The monotone class theorem plays the same role with algebras of sets in place of π-systems.
</document>

<document source="https://tamnd.github.io/math2/node/BBR7QAD87Y.md" tag="BBR7QAD87Y" kind="definition">
# Expander graph

## Definition

Let $d \ge 3$ be an integer and let $\varepsilon > 0$.
For a finite graph $G$ with at least $2$ vertices, let $h(G)$ be its [[Cheeger constant|Y7S6W12N64]].
It is the minimum of $\abs{\partial S} / \abs{S}$ over the sets $S$ of vertices with $1 \le \abs{S} \le \abs{V(G)}/2$.
Here $\partial S$ is the set of edges with exactly one end in $S$.
A finite graph $G$ is an $\varepsilon$-expander if $h(G) \ge \varepsilon$.
A family of expander graphs of degree $d$ is a sequence $(G_n)_{n \ge 1}$ of finite $d$-[[regular graphs|0KA4S8R39M]] such that:

1. The number of vertices of $G_n$ tends to infinity as $n \to \infty$.
2. There is $\varepsilon > 0$ such that $h(G_n) \ge \varepsilon$ for every $n$.

## Remarks

The complete graph $K_4$ is a $2$-expander: a set $S$ of $1$ or $2$ vertices has $\abs{S}(4 - \abs{S})$ edges leaving it, so the ratio is $4 - \abs{S} \ge 2$.
The cycles $C_n$ do not form a family of expanders: they are $2$-regular, and an arc $S$ of $\floor{n/2}$ consecutive vertices has $\abs{\partial S} = 2$, so $h(C_n) \le 2/\floor{n/2} \to 0$, and clause 2 fails.
Families of expanders of every degree $d \ge 3$ exist.
For example, for each $d \ge 3$ there is $\varepsilon > 0$ such that a random $d$-regular graph on $n$ vertices is an $\varepsilon$-expander with probability tending to $1$ as $n \to \infty$.
An equivalent spectral form uses the second largest eigenvalue $\lambda_2$ of the adjacency matrix.
The discrete Cheeger inequalities say $\frac{d - \lambda_2}{2} \le h(G) \le \sqrt{2d(d - \lambda_2)}$ for a $d$-regular graph $G$.
So clause 2 holds if and only if the spectral gap $d - \lambda_2(G_n)$ is bounded away from $0$.
</document>

<document source="https://tamnd.github.io/math2/node/BBRSEM0C4H.md" tag="BBRSEM0C4H" kind="definition">
# Rank of an abelian group

## Definition

Let $A$ be an [[abelian group|082GHNPJ53]], written additively.
A subset $X \subseteq A$ is *linearly independent* if for all distinct $x_1, \ldots, x_k \in X$ and integers $n_1, \ldots, n_k$,
$$n_1 x_1 + \cdots + n_k x_k = 0 \implies n_1 = \cdots = n_k = 0.$$
The *rank* of $A$ is the [[cardinality|0EBACQFAZW]] of a maximal linearly independent subset of $A$.
It does not depend on the choice of the maximal subset.

## Remarks

Example: $A = \Z^2 \oplus \Z/3\Z$ has rank $2$.
The set $\set{(1, 0, 0), (0, 1, 0)}$ is independent, and it is maximal: for any $y \in A$, $3y$ lies in $\Z^2 \oplus 0$, so $3y = m(1,0,0) + n(0,1,0)$ gives a relation with nonzero coefficient on $y$.

Non-example of independence: in $\Z/6\Z$, the set $\set{1}$ is not independent, since $6 \cdot 1 = 0$ with coefficient $6 \neq 0$.
In fact a group in which every element has finite order, such as the [[torsion subgroup|0DBWXQRXQJ]] of any abelian group, has rank $0$.

Equivalently, the rank of $A$ is the [[dimension|03GSZ8RTZJ]] of the $\Q$-vector space $A \otimes_\Z \Q$.
Rank is additive: if $0 \to A \to B \to C \to 0$ is exact, then $\rank B = \rank A + \rank C$.
The [[free abelian group|0WJX8MGG37]] $\Z^n$ has rank $n$, and $\Q$ has rank $1$.
</document>

<document source="https://tamnd.github.io/math2/node/BBYNPDWR5R.md" tag="BBYNPDWR5R" kind="example">
# Wang tile

## Example

A Wang tile is a unit square whose four edges carry colours from a finite set.
Fix a finite set $W$ of Wang tiles.
A tiling by $W$ is a map $\tau \colon \Z^2 \to W$ such that adjacent edges match.
That is, for each $(i, j)$, the right edge of $\tau(i, j)$ has the colour of the left edge of $\tau(i+1, j)$.
Also the top edge of $\tau(i, j)$ has the colour of the bottom edge of $\tau(i, j+1)$.
Tiles may not be rotated or reflected.

The set of all tilings by $W$ is a two-dimensional [[subshift of finite type|624XDXX8FN]] in $W^{\Z^2}$.
Indeed, its forbidden patterns are the pairs of adjacent tiles whose common edge has two different colours.
There are finitely many of them.

Check of a small case: let $W$ consist of one tile whose four edges all have the colour $a$.
All matching conditions hold, so the constant map is a tiling, and it is periodic.
Now let $W$ consist of one tile with left edge $a$ and right edge $b \ne a$.
Two such tiles can never sit side by side, so $W$ admits no tiling.

## Remarks

The domino problem asks for an algorithm that decides, given $W$, whether $W$ tiles the plane.
Berger proved that no such algorithm exists, by simulating a [[Turing machine|W6ESDYBF8Z]] with Wang tiles and reducing to the [[halting problem|Z7TN2B35B0]].
A key step is the existence of an aperiodic set of Wang tiles: a set that tiles the plane, but only in non-periodic ways.
Jeandel and Rao found an aperiodic set of $11$ Wang tiles and showed that no set with fewer tiles is aperiodic.

## Explanations

### L1 human

Wang tiles are square dominoes with coloured edges, and neighbouring edges must have the same colour.
Some sets of such tiles cover the plane in a repeating pattern, some cannot cover it at all.
Surprisingly, some sets cover the plane but never in a repeating way.
Because of this, no computer program can decide for every set of tiles whether it covers the plane.
</document>

<document source="https://tamnd.github.io/math2/node/BBZP1R872N.md" tag="BBZP1R872N" kind="concept">
# Dynamical system

## Summary

A dynamical system is a space of states together with a rule that moves each state forward in time.
Time can be discrete, as in the repeated application of one map, or continuous, as in the solutions of an autonomous differential equation.
The rule must be consistent: waiting for time $s$ and then for time $t$ is the same as waiting for time $s + t$.
Dynamical systems model planets, populations, chemical reactions and many other processes, and their study asks what happens in the long run.

## Explanations

### L1 human

Think of a state as a snapshot of a system, such as the position and speed of a pendulum.
The dynamical system tells where the snapshot will be after any amount of time.
The set of all snapshots, the phase space, is often pictured as a landscape on which each point moves along a path.
A simple example is the map $x \mapsto x/2$ on the real line, with time counted in whole steps.
After $n$ steps the state $x$ has become $x / 2^n$, so every state drifts to $0$.
</document>

<document source="https://tamnd.github.io/math2/node/BCETMS8K3X.md" tag="BCETMS8K3X" kind="statement">
# Arthur–Selberg trace formula

## Statement

Let $F$ be a number field with [[adele ring|SEH68BA1QQ]] $\mathbb A$, and let $G$ be a connected reductive group over $F$.
Let $G(\mathbb A)^1$ be the common kernel of the maps $x \mapsto \abs{\chi(x)}$ for the rational characters $\chi$ of $G$, and assume that $G(F) \backslash G(\mathbb A)^1$ is compact.
Let $R$ be the right regular representation of $G(\mathbb A)^1$ on $L^2(G(F) \backslash G(\mathbb A)^1)$.
For a smooth compactly supported function $f$ on $G(\mathbb A)^1$, let $R(f) = \int f(y) R(y) \, dy$, with a fixed [[Haar measure|VY0HXD3GC4]].
Then $R(f)$ is a [[trace class operator|Z2CP530WDE]].
Its trace is given by
$$\sum_{\pi} m(\pi) \operatorname{tr} \pi(f) = \sum_{\gamma} \operatorname{vol}\big(G_\gamma(F) \backslash G_\gamma(\mathbb A)^1\big) \int_{G_\gamma(\mathbb A) \backslash G(\mathbb A)^1} f(x^{-1} \gamma x) \, dx.$$
Here $\pi$ runs over the irreducible unitary representations of $G(\mathbb A)^1$, and $m(\pi)$ is the multiplicity of $\pi$ in $R$.
The index $\gamma$ runs over representatives of the [[conjugacy classes|0DEBXAJ5SA]] of $G(F)$, and $G_\gamma$ is the centralizer of $\gamma$.
Both sides equal $\operatorname{tr} R(f)$.

## Remarks

This compact-quotient case is the direct generalisation of the [[Selberg trace formula|BDAWKPBNQP]].
The proof writes $R(f)$ as an integral operator with kernel $K(x, y) = \sum_{\gamma \in G(F)} f(x^{-1} \gamma y)$ and integrates the kernel over the diagonal.
Grouping the terms by conjugacy class gives the geometric side; the discrete decomposition of $L^2$ gives the spectral side.
For general $G$, such as $\mathrm{GL}_n$ with $n \ge 2$, the quotient is not compact and $R(f)$ is not trace class.
Arthur's trace formula then replaces both sides by truncated distributions $J_{\mathrm{geom}}(f) = J_{\mathrm{spec}}(f)$, which contain extra terms from the continuous spectrum and from non-semisimple conjugacy classes.
The spectral side encodes [[automorphic forms|3C73HND6XA]], the geometric side encodes orbital integrals, and comparing the formulas for two groups is the main tool for proving cases of Langlands functoriality.

## Explanations

### L1 human

The formula computes one number, the trace of an operator on functions on an adelic quotient, in two ways.
One way lists the automorphic representations with their multiplicities.
The other way lists the conjugacy classes of rational elements of the group, weighted by volumes and integrals.
Equating them turns questions about automorphic forms into questions about conjugacy classes, which are often easier.

### L2 human

The Selberg trace formula is the case of a cocompact Fuchsian group, seen through the real place only.
Jacquet and Langlands used a trace formula comparison to relate automorphic forms on $\mathrm{GL}_2$ and on a quaternion algebra.
The stabilization of the trace formula, completed with the fundamental lemma, allowed Arthur to classify the automorphic representations of classical groups.
</document>

<document source="https://tamnd.github.io/math2/node/BCG5KM83AP.md" tag="BCG5KM83AP" kind="statement">
# Jordan–Hölder theorem for modular lattices

## Statement

Let $L$ be a [[modular lattice|QX7WZ92QAK]] and let $a \le b$ in $L$.
Write $x \lessdot y$ when $y$ [[covers|4HHDPZ4BDY]] $x$.
For $u \le v$ in $L$, write $[u, v] = \set{z \in L : u \le z \le v}$.
Two such intervals are *transposed* if they have the form $[x \wedge y, x]$ and $[y, x \vee y]$ for some $x, y \in L$, in either order.
Two intervals are *projective* if they are joined by a finite sequence of intervals in which each consecutive pair is transposed.

Let
$$a = x_0 \lessdot x_1 \lessdot \cdots \lessdot x_m = b \quad\text{and}\quad a = y_0 \lessdot y_1 \lessdot \cdots \lessdot y_n = b$$
be two finite [[maximal chains|3AX549FFFM]] from $a$ to $b$.
Then $m = n$, and there is a permutation $\sigma$ of $\set{1, \dots, m}$ such that $[x_{i-1}, x_i]$ is projective to $[y_{\sigma(i)-1}, y_{\sigma(i)}]$ for each $i$.

## Remarks

The proof is by induction on $m$.
If $x_1 = y_1$, apply induction to the interval $[x_1, b]$.
Otherwise $x_1$ and $y_1$ both cover $a = x_1 \wedge y_1$, and modularity makes the transposed intervals $[y_1, x_1 \vee y_1]$ and $[a, x_1]$ isomorphic, so $x_1 \vee y_1$ covers both $x_1$ and $y_1$.
Choosing a maximal chain from $x_1 \vee y_1$ to $b$ and applying induction in $[x_1, b]$ and in $[y_1, b]$ gives the result.

Example: the lattice of submodules of a module $M$ is modular.
For submodules $X$ and $Y$, the transposed intervals give the isomorphism $X / (X \cap Y) \cong (X + Y) / Y$, so projective intervals of length one give isomorphic simple quotients.
So the theorem yields the [[Jordan–Hölder theorem for modules|M1ABX18FC4]]: two [[composition series|0XV2A49X6Q]] of a module have the same length and isomorphic factors up to order.

Without modularity the equal-length conclusion can fail.
In the [[pentagon lattice N5|DHC8TZXGYS]], the maximal chains $0 < a < c < 1$ and $0 < b < 1$ have lengths $3$ and $2$.

## Explanations

### L1 human

In a modular lattice, any two ways of climbing from $a$ to $b$ by smallest possible steps use the same number of steps.
Moreover, the steps of one climb match the steps of the other, up to order, in a precise sense.
This is the lattice form of the familiar fact that composition series of a module have the same length and the same factors.

### L2 human

The theorem shows that modular lattices satisfy the [[Jordan–Dedekind chain condition|RW7YC3HG0S]] on every interval of finite length, so such intervals have a well-defined length.
For submodule lattices it recovers the Jordan–Hölder theorem for modules, with projectivity of intervals giving isomorphism of factors.
The [[Jordan–Hölder theorem|0E088ZP533]] for groups is a close relative, with subnormal series in place of chains of submodules.
The same equal-length conclusion holds in [[semimodular lattices|C1NET23GG0]], which is why matroids and geometric lattices have a rank function.
</document>

<document source="https://tamnd.github.io/math2/node/BCGBQ3HK26.md" tag="BCGBQ3HK26" kind="concept">
# Iterative functional equation

## Summary

An iterative functional equation is a [[functional equation|SRAN4EKWS0]] in one variable in which the unknown function is composed with a given map or with itself.
A typical form is $\varphi(f(x)) = g(x, \varphi(x))$, where $f$ and $g$ are given and $\varphi$ is unknown.
Such equations describe how a quantity changes under one step of an [[iterated map|AYVTR2ES5B]].
The Abel equation, the Schröder equation and the equation $\varphi(\varphi(x)) = f(x)$ for a functional square root are the main examples.

## Explanations

### L1 human

Iteration means applying the same map again and again.
An iterative functional equation asks for a function that interacts in a fixed way with one step of this process.
For example, a function $\varphi$ with $\varphi(2x) = \varphi(x) + 1$ counts how many doublings one has made, and $\varphi(x) = \log_2 x$ works on the positive numbers.
Such a function lets one ask what half a step of the process should be.
</document>

<document source="https://tamnd.github.io/math2/node/BCJD3SR975.md" tag="BCJD3SR975" kind="definition">
# H-space

## Definition

An *H-space* is a [[pointed space|T9DG8DDE75]] $(X, e)$ together with a continuous map $\mu \colon X \times X \to X$ with these properties.

1. $\mu(e, e) = e$.
2. The map $X \to X$, $x \mapsto \mu(e, x)$, is [[homotopic|F2F143KQ2M]] to $\id_X$ through pointed maps.
3. The map $X \to X$, $x \mapsto \mu(x, e)$, is homotopic to $\id_X$ through pointed maps.

## Remarks

Some authors drop the condition that the homotopies fix $e$; for CW complexes the two definitions give the same class of spaces.
An H-space is *homotopy associative* if $\mu \circ (\mu \times \id) \simeq \mu \circ (\id \times \mu)$, and *homotopy commutative* if $\mu \simeq \mu \circ \tau$, where $\tau$ swaps the factors.

Example: every topological group $G$ with $e$ its identity and $\mu$ its multiplication is an H-space, since $\mu(e, x) = \mu(x, e) = x$ on the nose.

Example: the [[loop space|2JFF5F3EGF]] $\Omega Y$ with concatenation of loops is an H-space; the constant loop is a unit up to the standard reparametrization homotopy.

Non-example: $S^2$ admits no such $\mu$.
If it did, let $x \in H^2(S^2; \Z)$ be a generator; the unit condition forces $\mu^*(x) = x \times 1 + 1 \times x$ in $H^2(S^2 \times S^2; \Z)$.
Then $0 = \mu^*(x^2) = (x \times 1 + 1 \times x)^2 = 2 (x \times x)$ for the [[cup product|JX47MS1R1H]], which is false since $x \times x$ generates $H^4(S^2 \times S^2; \Z) \cong \Z$.
</document>

<document source="https://tamnd.github.io/math2/node/BCJHNHKQFG.md" tag="BCJHNHKQFG" kind="concept">
# Densely defined operator

## Summary

A densely defined operator is an unbounded operator whose domain is a dense subspace of the space.
Density of the domain is the condition under which the adjoint of an operator on a Hilbert space is well defined.
Differential operators on $L^2$ spaces, defined on smooth functions, are the standard examples.

## Explanations

### L1 human

An unbounded operator often cannot act on every vector, but it is good if it acts on vectors that come arbitrarily close to any vector.
That is what a dense domain gives.
For example, differentiation acts on polynomials, and every continuous function on $[0, 1]$ is a uniform limit of polynomials.
A dense domain also makes the operator determined by enough data to define its adjoint.
</document>

<document source="https://tamnd.github.io/math2/node/BD1CNVV5P0.md" tag="BD1CNVV5P0" kind="statement">
# Varadhan's lemma

## Statement

Let $\mathcal{X}$ be a [[topological space|0CQHFPGJ45]], and let $(\mu_n)$ be a sequence of [[probability measures|059TJN0NZ1]] on $\mathcal{X}$.
Assume that $(\mu_n)$ satisfies the [[large deviation principle|MNPWBBNCAD]] with a good [[rate function|140N2PQSTE]] $I$.
Let $\varphi \colon \mathcal{X} \to \R$ be a [[continuous|0X3WGX95FE]] function that is bounded above.
Then
$$\lim_{n \to \infty} \frac{1}{n} \log \int_{\mathcal{X}} e^{n \varphi(x)} \, \mu_n(dx) = \sup_{x \in \mathcal{X}} \bigl(\varphi(x) - I(x)\bigr).$$

## Remarks

The lemma is an infinite-dimensional form of Laplace's method: the integral is dominated by the neighbourhood of the point where $\varphi - I$ is largest.
For the lower bound, take $x$ with $I(x) < \infty$ and a small open neighbourhood $G$ of $x$ on which $\varphi > \varphi(x) - \delta$; the large deviation lower bound gives $\mu_n(G) \ge e^{-n(I(x) + \delta)}$ for large $n$.
For the upper bound, cover a compact sublevel set of $I$ by finitely many small closed neighbourhoods.
Use the large deviation upper bound on each, and the bound on $\varphi$ outside.

The boundedness of $\varphi$ from above can be replaced by the tail condition
$$\lim_{M \to \infty} \limsup_{n \to \infty} \frac{1}{n} \log \int_{\set{\varphi \ge M}} e^{n \varphi} \, d\mu_n = -\infty.$$
Some such condition is needed, since otherwise rare but huge values of $e^{n\varphi}$ can dominate the integral.

Example: let $\mu_n$ be the normal law with mean $0$ and variance $1/n$, with $I(x) = x^2/2$, and let $\varphi(x) = -\abs{x - 1}$.
The lemma gives the limit $\sup_x (-\abs{x - 1} - x^2/2) = -1/2$, attained at $x = 1$.

## Explanations

### L1 human

Suppose a random quantity is very unlikely to be far from typical, with a known exponential cost $I$.
If you reward each outcome $x$ by an exponential weight $e^{n \varphi(x)}$, then the average reward grows or decays at the rate $\max(\varphi - I)$.
The winner is the outcome with the best balance between reward and probability cost.

### L2 human

Varadhan's lemma is the main tool to compute free energies and other exponential averages once a large deviation principle is known.
In statistical mechanics, the limit is the pressure and the maximisers of $\varphi - I$ are the equilibrium states.
Its converse, Bryc's theorem, recovers a large deviation principle from the limits of such exponential integrals.
</document>

<document source="https://tamnd.github.io/math2/node/BD756KR8FM.md" tag="BD756KR8FM" kind="concept">
# Lambda calculus

## Summary

The lambda calculus is a formal system for describing functions by abstraction and application, with computation done by substituting arguments into function bodies.
In its untyped form it computes exactly the same functions on natural numbers as [[Turing machines|758EYCN27B]].
It was introduced by Alonzo Church and is the basis of functional programming languages and of type theory.

## Explanations

### L1 human

In the lambda calculus, $\lambda x.\, M$ means "the function that sends $x$ to $M$".
To apply a function to an argument, one replaces the variable by the argument.
For example, $(\lambda x.\, x)\, y$ becomes $y$: the identity function applied to $y$ gives $y$.
Nothing else is built in, yet numbers, truth values and recursion can all be encoded as functions.
</document>

<document source="https://tamnd.github.io/math2/node/BDA5TTB1F0.md" tag="BDA5TTB1F0" kind="statement">
# Azuma's inequality

## Statement

Let $(X_k)_{k = 0}^n$ be a [[martingale|0D0GBNDKDH]] with respect to a filtration $(\mathcal{F}_k)_{k = 0}^n$.
Assume that there are constants $c_1, \dots, c_n > 0$ with $\abs{X_k - X_{k-1}} \le c_k$ almost surely for $k = 1, \dots, n$.
Then for every $t > 0$,
$$P(X_n - X_0 \ge t) \le \exp\left(-\frac{t^2}{2 \sum_{k=1}^n c_k^2}\right),$$
and hence $P(\abs{X_n - X_0} \ge t) \le 2 \exp\left(-\frac{t^2}{2 \sum_{k=1}^n c_k^2}\right)$.

## Remarks

Proof idea: let $D_k = X_k - X_{k-1}$. Since $E[D_k \mid \mathcal{F}_{k-1}] = 0$ and $D_k \in [-c_k, c_k]$, Hoeffding's lemma applied to the [[conditional expectation|0TTDGTEFMN]] gives $E[e^{\lambda D_k} \mid \mathcal{F}_{k-1}] \le e^{\lambda^2 c_k^2 / 2}$.
By induction $E[e^{\lambda (X_n - X_0)}] \le e^{\lambda^2 \sum_k c_k^2 / 2}$, and Markov's inequality with $\lambda = t / \sum_k c_k^2$ gives the bound.
For independent centred $D_k$ the result is a form of [[Hoeffding's inequality|0WPDNXGSSN]], and it is also called the Azuma–Hoeffding inequality.
Example: for a simple random walk $X_n = D_1 + \dots + D_n$ with independent fair $\pm 1$ steps, $P(X_n \ge t) \le e^{-t^2 / (2n)}$.

## Explanations

### L1 human

A martingale is a fair game: on average each step adds nothing.
Suppose each step can change the fortune by at most a fixed amount.
Then the total after $n$ steps is very unlikely to be far from the start.
When each step is at most $1$, the chance of being more than $t$ away drops like $e^{-t^2 / (2n)}$.

### L2 human

Azuma's inequality is applied to [[Doob martingales|Z57CG6W9S6]] $X_k = E[f \mid \mathcal{F}_k]$ of a function $f$ of independent variables, revealed one at a time.
This gives concentration of quantities such as the chromatic number of a random graph, and it leads to [[McDiarmid's inequality|8APFT1RPM4]].
</document>

<document source="https://tamnd.github.io/math2/node/BDAWKPBNQP.md" tag="BDAWKPBNQP" kind="statement">
# Selberg trace formula

## Statement

Let $\Gamma \subset \mathrm{PSL}_2(\R)$ be a torsion-free [[Fuchsian group|VNJWSCVP6V]] such that $X = \Gamma \backslash \mathbb H$ is compact, where $\mathbb H$ is the [[upper half-plane|X6XGGH5FPW]] with the hyperbolic metric $(dx^2 + dy^2)/y^2$.
Let $0 = \lambda_0 < \lambda_1 \le \lambda_2 \le \cdots$ be the [[eigenvalues|0BQAA2BK3Q]] of the [[Laplace–Beltrami operator|F4NWZNTRMD]] $\Delta = -y^2 (\partial_x^2 + \partial_y^2)$ on $X$, repeated with multiplicity, and write $\lambda_j = \tfrac14 + r_j^2$ with $r_j \in \C$.
Let $h$ be an even function, holomorphic on a strip $\abs{\operatorname{Im} r} \le \tfrac12 + \varepsilon$ with $\varepsilon > 0$, such that $\abs{h(r)} \le C (1 + \abs{r})^{-2-\varepsilon}$ there.
Let $g(u) = \frac{1}{2\pi} \int_{-\infty}^{\infty} h(r) e^{-iru} \, dr$.
Then
$$\sum_{j=0}^{\infty} h(r_j) = \frac{\operatorname{area}(X)}{4\pi} \int_{-\infty}^{\infty} r \, h(r) \tanh(\pi r) \, dr + \sum_{\gamma} \sum_{k=1}^{\infty} \frac{\ell(\gamma)}{2 \sinh(k \ell(\gamma)/2)} \, g(k \ell(\gamma)),$$
where $\gamma$ runs over the primitive oriented closed [[geodesics|NSYMQMHPDN]] of $X$ and $\ell(\gamma)$ is the length of $\gamma$.
All sums and integrals converge absolutely.

## Remarks

The closed geodesics of $X$ correspond to the nontrivial [[conjugacy classes|0DEBXAJ5SA]] of $\Gamma$, and $k \ell(\gamma)$ is the length of the $k$-th power.
The proof computes the trace of an integral operator on $L^2(X)$ in two ways.
The kernel $\sum_{\sigma \in \Gamma} k(z, \sigma w)$ is built from a point-pair invariant $k$ whose spectral transform is $h$.
Expanding in eigenfunctions gives the left side; integrating the kernel over the diagonal and grouping $\sigma$ by conjugacy classes gives the right side, with the identity giving the area term.
Example: $h(r) = e^{-t(r^2 + 1/4)}$ with $t > 0$ gives the trace of the heat kernel $\sum_j e^{-t \lambda_j}$, and the area term behaves like $\operatorname{area}(X)/(4\pi t)$ as $t \to 0$.
This yields Weyl's law: the number of $\lambda_j \le T$ is asymptotic to $\operatorname{area}(X) \, T / (4\pi)$.
For non-compact $X$ of finite area, such as $\mathrm{SL}_2(\Z) \backslash \mathbb H$, the formula needs extra terms from the cusps and from the continuous spectrum.

## Explanations

### L1 human

On a compact hyperbolic surface there are two natural lists of numbers.
One list is the frequencies at which the surface can vibrate, the eigenvalues of the Laplacian.
The other is the lengths of the closed geodesics, the loops that a light ray could travel forever.
The Selberg trace formula is an exact identity that links the two lists.

### L2 human

The formula is a non-abelian analogue of the Poisson summation formula, which links a lattice in $\R$ with its dual lattice.
It gives the prime geodesic theorem, the analytic properties of the Selberg zeta function, and Weyl's law for $X$.
Applied to congruence subgroups with Hecke operators inserted, it links Maass forms with arithmetic, and it is the prototype of the Arthur–Selberg trace formula.
</document>

<document source="https://tamnd.github.io/math2/node/BDB06N91FN.md" tag="BDB06N91FN" kind="concept">
# Tautological line bundle

## Summary

Let $\mathbb{P}(V)$ be the projective space of lines in a vector space $V$.
The tautological line bundle on $\mathbb{P}(V)$ is the line bundle whose fibre over a point is the line that the point represents.
Its sheaf of sections is $\mathcal{O}(-1)$, the dual of Serre's twisting sheaf.
It is universal: a line bundle generated by finitely many global sections is pulled back from its dual $\mathcal{O}(1)$ by a map to projective space.

## Explanations

### L1 human

A point of projective space is a line through the origin of a vector space.
The tautological line bundle attaches to that point the line itself, so it is defined by a tautology.
On the real projective line, the result is a Möbius band, which shows that the bundle is twisted.
Over the complex numbers, the tautological bundle on $\mathbb{P}^n$ has no nonzero global sections for $n \ge 1$.
</document>

<document source="https://tamnd.github.io/math2/node/BDBDJSP6DF.md" tag="BDBDJSP6DF" kind="statement">
# Integration operator as convolution with the unit step

## Statement

Let $f$ be a locally integrable function on $[0, \infty)$.
Let $h$ be the constant function $h(t) = 1$ on $[0, \infty)$, the restriction of the [[Heaviside step function|09C4JP7Y0M]].
Write $f * g$ for the [[convolution on the half-line|FCKX8KFVGN]], $(f * g)(t) = \int_0^t f(\tau) \, g(t - \tau) \, d\tau$.
Then:

1. $(h * f)(t) = \int_0^t f(\tau) \, d\tau$ for all $t \ge 0$.
2. For every integer $n \ge 1$, the $n$-fold convolution power of $h$ is $h^{*n}(t) = \dfrac{t^{n-1}}{(n-1)!}$.
3. For every integer $n \ge 1$, $(h^{*n} * f)(t) = \displaystyle \int_0^t \frac{(t - \tau)^{n-1}}{(n-1)!} f(\tau) \, d\tau$, and this is the $n$-fold iterated integral of $f$ from $0$ to $t$.

## Remarks

Clause 1 is the definition of convolution with $h(t - \tau) = 1$ for $0 \le \tau \le t$.
Clause 2 follows by induction, since $\int_0^t \tau^{n-1}/(n-1)! \, d\tau = t^n / n!$.
Clause 3 follows from clauses 1 and 2 and the associativity of convolution; it is the Cauchy formula for repeated integration.
For example, with $f(t) = \cos t$, one gets $(h * f)(t) = \sin t$ and $(h * h * f)(t) = 1 - \cos t$.
In the [[Mikusiński operational calculus|TD5XK5AHNV]] this result lets one treat integration as multiplication by the element $h$ of a commutative ring.

## Explanations

### L1 human

Integrating a function from $0$ to $t$ is the same as convolving it with the function that is constantly $1$.
So the operation "integrate" becomes multiplication by a fixed object in a ring whose product is convolution.
Integrating twice is multiplication by $h * h$, which is the function $t$.

### L2 human

This is the starting point of Mikusiński's calculus.
The convolution ring of functions on $[0, \infty)$ has no zero divisors by the Titchmarsh convolution theorem, so it has a field of fractions.
In that field $h$ has an inverse $s$, which plays the role of differentiation.
Under the Laplace transform, $h$ corresponds to $1/s$, in line with the rule that the transform of $\int_0^t f$ is $F(s)/s$.
</document>

<document source="https://tamnd.github.io/math2/node/BDEX7HET71.md" tag="BDEX7HET71" kind="concept">
# Interior algebra

## Summary

An interior algebra is a [[Boolean algebra|0XADJKM699]] with one more operation that behaves like the [[interior|0RZH6W98HK]] operation of a [[topological space|0CQHFPGJ45]].
It is the algebraic form of topology, in the same way as a Boolean algebra is the algebraic form of set operations.
Interior algebras are the algebraic models of the [[modal logic|C3RZY20ETK]] S4, where the interior operation interprets "necessarily".
The dual notion, with a closure operation instead, is called a closure algebra.

## Explanations

### L1 human

Take a topological space, such as the real line.
For any set of points you can take its interior, the points that have some room around them inside the set.
The interior of the interval $[0, 1]$ is $(0, 1)$.
Taking the interior twice gives nothing new, and the interior of an intersection is the intersection of the interiors.
An interior algebra keeps only these rules and forgets the points.
</document>

<document source="https://tamnd.github.io/math2/node/BDFXWM8XK1.md" tag="BDFXWM8XK1" kind="definition">
# Fréchet manifold

## Definition

Let $F$ and $G$ be real [[Fréchet spaces|3GAA614673]], and let $U \subseteq F$ be open.
A map $f \colon U \to G$ is of class $C^1$ if it is continuous, the [[Gateaux derivative|EAZF9NJW12]] $Df(x) h = \lim_{t \to 0} (f(x + t h) - f(x)) / t$ exists for all $x \in U$ and $h \in F$, and $Df \colon U \times F \to G$ is continuous.
It is of class $C^{k+1}$ if it is $C^1$ and $Df$ is $C^k$, and *smooth* if it is $C^k$ for all $k$.

Let $M$ be a [[Hausdorff space|0MXB4CKHFW]].
A *smooth Fréchet atlas* on $M$ is a family of charts $(U_i, \varphi_i)_{i \in I}$ such that:

1. Each $U_i$ is open in $M$ and the sets $U_i$ cover $M$.
2. Each $\varphi_i$ is a [[homeomorphism|0QT00PB5G2]] from $U_i$ onto an [[open subset|02BH3MCBQB]] of a Fréchet space.
3. Each transition map $\varphi_j \circ \varphi_i^{-1} \colon \varphi_i(U_i \cap U_j) \to \varphi_j(U_i \cap U_j)$ is smooth.

A *Fréchet manifold* is a space $M$ with an equivalence class of smooth Fréchet atlases, two atlases being equivalent when their union is an atlas.

## Remarks

Every Banach space is a Fréchet space, and every smooth [[Banach manifold|9K15AMGBV9]] is a Fréchet manifold.

Example: the Fréchet space $C^\infty(S^1)$ of smooth functions on the circle, with the single chart $\id$, is a Fréchet manifold.
More generally, for a compact manifold $K$ and a manifold $N$, the space $C^\infty(K, N)$ of smooth maps is a Fréchet manifold.
The diffeomorphisms of $K$ form a Fréchet Lie group.

Non-example: take charts with values in the space of compactly supported smooth functions on $\R$ with the sup norm.
That normed space is not complete, so it is not a Fréchet space and clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/BDHADVMJ8G.md" tag="BDHADVMJ8G" kind="statement">
# Final value theorem

## Statement

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$.
Assume that the [[limit|0TXTSEMHR5]] $f(\infty) = \lim_{t \to \infty} f(t)$ exists and is finite.
Then the Laplace integral $F(s)$ of $f$ converges for every real $s > 0$, and
$$\lim_{s \to 0^+} s F(s) = f(\infty),$$
where $s$ tends to $0$ through positive real values.

## Remarks

Since $f$ has a finite limit, it is bounded for large $t$, so $F(s)$ converges for $s > 0$.
Since $s \int_0^\infty e^{-s t} \, dt = 1$, the difference $s F(s) - f(\infty)$ equals $s \int_0^\infty e^{-s t} (f(t) - f(\infty)) \, dt$.
As $s \to 0^+$ the weight $s e^{-s t}$ spreads its mass over large $t$, where $f(t) - f(\infty)$ is small, and the part over a fixed interval $[0, T]$ tends to $0$.
Example: for $f(t) = 1 - e^{-t}$, $s F(s) = 1 - s/(s + 1) \to 1 = f(\infty)$.
The hypothesis that $f(\infty)$ exists is needed: for $f(t) = \sin t$, $s F(s) = s / (s^2 + 1) \to 0$, but $\sin t$ has no limit.

## Explanations

### L1 human

The behaviour of the transform for small $s$ tells the long-run value of the function, if that value exists.
One can read the steady state from $F$ without inverting the transform.

### L2 human

In control theory one uses it to find the steady-state error of a stable system.
For a rational $F$, the limit $f(\infty)$ exists when all poles of $s F(s)$ lie in the open left half-plane; then the theorem applies.
If $s F(s)$ has a pole on the imaginary axis or to its right, the formula can give a wrong answer, as the example $\sin t$ shows.
The converse direction, from $F$ to $f$, needs extra hypotheses and belongs to the [[Tauberian theorems|NV54MWKT6T]].
Its companion is the [[initial value theorem|KZPVHDKV9E]].
</document>

<document source="https://tamnd.github.io/math2/node/BDJ1AX4XBB.md" tag="BDJ1AX4XBB" kind="concept">
# Multiply transitive group action

## Summary

A group action is $k$-transitive when one group element can move any $k$ distinct points onto any other $k$ distinct points, in order.
For $k = 1$ this is a [[transitive action|02GK6HZYBT]], and larger $k$ means a stronger kind of symmetry.
The symmetric group $S_n$ is $n$-transitive and the alternating group $A_n$ is $(n-2)$-transitive on $n$ points, and a $2$-transitive group is always [[primitive|Y10S0751KG]].
Apart from symmetric and alternating groups, finite multiply transitive groups are rare: the [[Mathieu groups|T1KR4KPX8P]] are famous examples.

## Explanations

### L1 human

A group acting on a set is $2$-transitive if, for any two ordered pairs of different points, some element sends the first pair to the second.
Think of a set of seats and a group of allowed ways to reseat people.
If the group is $2$-transitive, any two people can be moved into any two chosen seats.
All permutations of $n$ points form an $n$-transitive group.
The rotations and reflections of a square act transitively on its four corners, but not $2$-transitively: two neighbouring corners cannot be moved onto two opposite corners.
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# Energy drift

## Summary

Energy drift is the gradual, systematic change of the total energy of a [[Hamiltonian system|0EDKGA2FAK]] in a numerical simulation.
The exact solution keeps the energy constant.
It comes from the accumulation of the small errors of the integrator over many steps.
Its size and growth are a basic test of the quality of a long simulation, for example in molecular dynamics.
Its absence is one of the main benefits of [[symplectic integrators|B4C25SXFRE]].

## Explanations

### L1 human

A frictionless pendulum keeps its energy forever.
A computer simulation of it makes a small error at each step, and with many methods these errors push the energy steadily up or down.
After a long run, the simulated pendulum swings much higher or lower than it should: this is energy drift.
With the explicit Euler method on the oscillator $q' = p$, $p' = -q$, the energy is multiplied by $1 + h^2$ at every step, so it grows without bound.
Good methods for long runs let the energy wobble a little around its true value, without a trend.
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<document source="https://tamnd.github.io/math2/node/BDZYBXBFDE.md" tag="BDZYBXBFDE" kind="concept">
# Topological entropy

## Summary

Topological entropy is a nonnegative number, possibly infinite, that measures the exponential growth rate of the number of distinguishable orbits of a continuous map.
It is an invariant of topological conjugacy, so it can show that two systems are not conjugate.
Positive entropy is a common sign of complicated, chaotic behaviour.
Through the [[variational principle|HHR8HP3R6G]] it equals the supremum of the measure-theoretic entropies of the invariant measures.

## Explanations

### L1 human

Suppose you can only see the state of a system up to a fixed small precision.
Count how many different orbit segments of length $n$ you can tell apart.
If this number grows like $e^{hn}$, the topological entropy is $h$.
For a rotation, nearby points stay close forever, so the count does not grow and the entropy is $0$.
For the doubling map of the circle, the count doubles at each step, and the entropy is $\log 2$.
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# Gaussian free field

## Definition

Let $G = (V, E)$ be a finite connected [[graph|00C0BR7D7H]], let $B \subseteq V$ be a nonempty set of boundary vertices, and let $U = V \setminus B$ be nonempty.
For $f \colon V \to \R$, let $\Delta f(x) = \sum_{y : \set{x, y} \in E} (f(y) - f(x))$ be the graph [[Laplacian|0E0B6RA8K0]], and let $\Delta_U$ be its restriction to functions that vanish on $B$, viewed as a $U \times U$ matrix.
The *discrete Gaussian free field* on $G$ with zero boundary values on $B$ is a random function $h \colon V \to \R$ with $h = 0$ on $B$.
Its values $(h(x))_{x \in U}$ have the following density on $\R^U$:
$$\frac{1}{Z} \exp\Big( -\frac12 \sum_{\set{x, y} \in E} \big(h(x) - h(y)\big)^2 \Big),$$
where $Z$ is the normalising constant.

## Remarks

The exponent equals $-\frac12 \langle h, -\Delta_U h \rangle$, and $-\Delta_U$ is positive definite because $B$ is nonempty and $G$ is connected.
So $h$ restricted to $U$ is a centred [[Gaussian process|22WEM6RTW0]] with [[covariance|0DJ8W3HP2D]] matrix $(-\Delta_U)^{-1}$.
This matrix is the Green function of the continuous-time random walk that crosses each edge at rate $1$ and is killed on $B$.
Example: let $V = \set{a, x, b}$ with edges $\set{a, x}$ and $\set{x, b}$, and $B = \set{a, b}$.
The exponent is $-\frac12 (h(x)^2 + h(x)^2) = -h(x)^2$, so $h(x)$ is normal with mean $0$ and variance $1/2$, and indeed $-\Delta_U = (2)$ has inverse $1/2$.
Non-example: with $B$ empty, the exponent is unchanged when a constant is added to $h$, so no normalising constant $Z$ exists.
The continuum Gaussian free field on a domain $D \subseteq \R^d$ is the centred Gaussian random generalised function whose covariance is the Dirichlet Green function of $-\Delta$ on $D$.
It is not a function when $d \ge 2$.
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# Parry measure

## Definition

Let $A$ be a $k \times k$ matrix with entries in $\set{0, 1}$ that is [[irreducible|ZEZPYJC06B]].
Let $\Sigma_A = \set{x \in \set{1, \ldots, k}^{\Z} : A_{x_i x_{i+1}} = 1 \text{ for all } i}$ be its vertex shift, a [[subshift of finite type|M7RPDE39FM]], with shift map $\sigma$.
By the [[Perron–Frobenius theorem|62NS7PV1M9]], $A$ has a positive eigenvalue $\lambda$ of largest modulus, with eigenvectors $u, v$ with positive entries such that $u^T A = \lambda u^T$ and $A v = \lambda v$.
Normalise them so that $\sum_i u_i v_i = 1$.
Put $p_i = u_i v_i$ and $P_{ij} = A_{ij} v_j / (\lambda v_i)$.
The *Parry measure* is the [[probability measure|0VGMEBVW1M]] $\mu$ on $\Sigma_A$ such that, for all $n \in \Z$, $m \ge 0$ and states $i_0, \ldots, i_m$,
$$\mu\set{x : x_{n} = i_0, \ldots, x_{n+m} = i_m} = p_{i_0} P_{i_0 i_1} \cdots P_{i_{m-1} i_m}.$$

## Remarks

The matrix $P$ is stochastic and $p^T P = p^T$, so these values are consistent and $\mu$ is a $\sigma$-[[invariant measure|B232EZ1FMB]].
Its entropy is $\log \lambda$, which is the topological entropy of $\sigma$ on $\Sigma_A$, and it is the only invariant probability measure with this entropy.
Example: the golden mean shift has $A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ and $\lambda = \varphi = (1 + \sqrt5)/2$.
Here $u = v = (\varphi, 1)$ up to scaling, since $\varphi + 1 = \varphi^2$.
After normalisation, $p_1 = \varphi^2/(\varphi^2 + 1)$ and $p_2 = 1/(\varphi^2 + 1)$, and $P_{11} = 1/\varphi$, $P_{12} = 1/\varphi^2$, $P_{21} = 1$, $P_{22} = 0$.
Each row of $P$ sums to $1$, because $1/\varphi + 1/\varphi^2 = 1$.
Non-example: on the same shift, the Markov measure with $P_{11} = P_{12} = 1/2$ is invariant for its stationary vector but is not the Parry measure, and its entropy is smaller than $\log \varphi$.
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# Conjugacy of maximal tori

## Statement

Let $G$ be a connected [[compact Lie group|9TFC6ADC1C]] and let $T$ be a [[maximal torus|3JVGZVVTB0]] of $G$.

1. For every maximal torus $T'$ of $G$ there is $g \in G$ with $T' = gTg^{-1}$.
2. Every element of $G$ lies in some maximal torus; equivalently, $G = \bigcup_{g \in G} gTg^{-1}$.

## Remarks

One proof uses the Lefschetz fixed point theorem on the manifold $G/T$.
For $x \in G$, the map $gT \mapsto xgT$ of $G/T$ is homotopic to the identity, since $G$ is connected.
So its Lefschetz number is the Euler characteristic of $G/T$, which equals the order of the Weyl group and is nonzero.
A fixed point $gT$ means $xgT = gT$, that is, $g^{-1}xg \in T$; this gives part 2, and part 1 follows by taking for $x$ a generator of a dense cyclic subgroup of $T'$.
Example: for $G = U(n)$ and $T$ the diagonal unitary matrices, part 2 is the spectral theorem: every unitary matrix is $g D g^{-1}$ with $g$ unitary and $D$ diagonal.
Connectedness is needed for part 2: in $O(2)$, every torus lies in the identity component $SO(2)$, so a reflection lies in no maximal torus.

## Explanations

### L1 human

In a compact connected Lie group there is essentially only one maximal torus: all of them are rotated copies of each other.
Moreover these copies cover the whole group.
For unitary matrices this is the familiar fact that every unitary matrix becomes diagonal in a suitable orthonormal basis.

### L2 human

Part 2 implies that a continuous class function on $G$ is determined by its restriction to $T$, and that restriction is invariant under the [[Weyl group|EZGTEK3HQN]].
This is why characters of representations are studied on $T$, and why the [[Weyl integration formula|QMH530N27M]] reduces integrals of class functions to integrals over $T$.
It also implies that the exponential map of a compact connected Lie group is surjective, since it is surjective on each torus.
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# Arithmetic dynamics

## Summary

Arithmetic dynamics studies the number-theoretic properties of points under iteration of polynomial and rational maps, and more generally of self-maps of algebraic varieties.
Typical questions ask which rational or algebraic points are [[periodic|T75CZ5H47N]] or preperiodic.
Others ask how the [[height|F1FF505PWH]] of a point grows along its orbit, and how orbits behave modulo primes or over $p$-adic fields.
It combines complex dynamics with Diophantine geometry.
Many of its conjectures are dynamical analogues of results on elliptic curves and abelian varieties, where torsion points play the role of preperiodic points.

## Explanations

### L1 human

Take a simple rule like $x \mapsto x^2 - 1$ and apply it again and again to a fraction.
Some starting values, such as $0$, run in a cycle: $0 \mapsto -1 \mapsto 0$.
Most starting values, such as $2$, produce numbers with more and more digits: $2, 3, 8, 63, \ldots$.
Arithmetic dynamics asks which starting values give cycles, how fast the digits grow, and what such patterns look like over other number systems.
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# Yang–Baxter equation

## Definition

Let $V$ be a [[vector space|05EGHEQZAW]] over a field $k$ and let $R \colon V \otimes V \to V \otimes V$ be a linear map, where $\otimes$ is the [[tensor product|0BF2DE2SYK]] over $k$.
Define linear maps of $V \otimes V \otimes V$ by $R_{12} = R \otimes \id_V$ and $R_{23} = \id_V \otimes R$.
Let $R_{13}$ act as $R$ on the first and third factors and as the identity on the second.
If $\tau_{23}$ swaps the second and third factors, then $R_{13} = \tau_{23} R_{12} \tau_{23}$.
The map $R$ satisfies the *(quantum) Yang–Baxter equation* if
$$R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}.$$

## Remarks

Let $\tau \colon V \otimes V \to V \otimes V$ be the flip $u \otimes v \mapsto v \otimes u$ and $c = \tau \circ R$.
Then $R$ satisfies the Yang–Baxter equation if and only if $c$ satisfies the *braid relation* $c_{12} c_{23} c_{12} = c_{23} c_{12} c_{23}$, with $c_{12} = c \otimes \id_V$ and $c_{23} = \id_V \otimes c$.
Example: $R = \tau$.
On $u \otimes v \otimes w$, the left side sends it through $u \otimes w \otimes v$ and $v \otimes w \otimes u$ to $w \otimes v \otimes u$.
The right side sends it through $v \otimes u \otimes w$ and $w \otimes u \otimes v$ to $w \otimes v \otimes u$, so the two sides agree.
Example and non-example: let $R = A \otimes B$ for linear maps $A, B \colon V \to V$.
Then $R_{12} R_{13} R_{23} = A^2 \otimes BA \otimes B^2$ and $R_{23} R_{13} R_{12} = A^2 \otimes AB \otimes B^2$.
If $AB = BA$ the equation holds.
If $A, B$ are invertible and $AB \ne BA$, it fails, because $A^2 \otimes (BA - AB) \otimes B^2 \ne 0$.
Versions with a spectral parameter, $R(u)$ depending on $u \in \C$, are also called Yang–Baxter equations.
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# Weitzenböck identity

## Statement

Let $(M, g)$ be an oriented [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$, with [[Levi-Civita connection|CAZYGE1T4E]] $\nabla$.
On [[differential forms|0E3G0GRPZ6]] of degree $k$, let $d$ be the [[exterior derivative|0ZHSTAG8ZY]].
Let $\delta = (-1)^{n(k+1)+1} \star d \star$ be the codifferential, where $\star$ is the [[Hodge star|1Y4WX4XQWJ]].
Let $\Delta_H = d\delta + \delta d$ be the Hodge Laplacian.
Let $\nabla^* \nabla = -\sum_{i=1}^n \bigl(\nabla_{e_i} \nabla_{e_i} - \nabla_{\nabla_{e_i} e_i}\bigr)$ be the rough Laplacian, for a local orthonormal frame $e_1, \dots, e_n$.
Then there is a bundle endomorphism $\mathcal{R}_k$ of $k$-forms such that
$$\Delta_H \omega = \nabla^* \nabla \omega + \mathcal{R}_k(\omega)$$
for every $k$-form $\omega$.
For $k = 1$, the term is the [[Ricci curvature|4GTNM0WHDH]]: $\mathcal{R}_1(\omega)(X) = \operatorname{Ric}(\omega^\sharp, X)$, where $\omega^\sharp$ is the vector field dual to $\omega$.
At each point, $\mathcal{R}_k$ is given by a linear expression in the Riemann curvature tensor.

## Remarks

The idea of the proof: express $d$ and $\delta$ through $\nabla$, namely $d\omega = \sum_i e^i \wedge \nabla_{e_i} \omega$ and $\delta\omega = -\sum_i \iota_{e_i} \nabla_{e_i} \omega$.
Expanding $d\delta + \delta d$ gives the second-order part $\nabla^* \nabla$ plus commutators $[\nabla_{e_i}, \nabla_{e_j}]$, which are curvature terms.

Example: for $\omega = du$ the identity, paired with $du$, gives [[Bochner's formula|DF7KYMAGTC]] for the function $u$.

Bochner's vanishing theorem follows: let $M$ be compact.
If $\omega$ is a harmonic 1-form, integrating $\ip{\Delta_H \omega}{\omega} = 0$ gives $\int_M \abs{\nabla \omega}^2 + \operatorname{Ric}(\omega^\sharp, \omega^\sharp) = 0$.
So if $\operatorname{Ric} \ge 0$, every harmonic 1-form is parallel and the first Betti number is at most $n$; if $\operatorname{Ric} > 0$, the first Betti number is $0$.

## Explanations

### L1 human

There are two natural Laplacians on differential forms: one built from the exterior derivative, and one built from the covariant derivative.
The identity says they differ only by a curvature term with no derivatives.
For 1-forms that term is the Ricci curvature.

### L2 human

Weitzenböck identities are the engine of the Bochner technique.
Combined with the Hodge theorem, which identifies cohomology with harmonic forms, a sign of the curvature term forces harmonic forms to vanish or to be parallel.
The same pattern holds for other Dirac-type operators; for the spin Dirac operator the curvature term is a quarter of the scalar curvature (the Lichnerowicz formula).
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# Whitney sum

## Summary

The Whitney sum $E \oplus E'$ of two vector bundles over the same base has as fiber over each point the direct sum of the two fibers.
It is the basic operation on vector bundles, and with it the isomorphism classes of vector bundles over a space form a commutative monoid.
[[Characteristic classes|7HACJC0JE2]] behave multiplicatively under Whitney sum, which is the content of the Whitney product formula.
Over a compact Hausdorff space, topological K-theory is the group completion of this monoid.

## Explanations

### L1 human

Over each point of a base space, a vector bundle provides a vector space.
Given two bundles over the same base, place the two vector spaces side by side at each point to form their direct sum.
The result is a new bundle whose rank is the sum of the two ranks.
For the sphere, the tangent planes together with the normal lines give, at each point, the whole of $\R^3$.
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# Uniformly convex space

## Summary

A normed space is uniformly convex when midpoints of two unit vectors that are far apart lie uniformly deep inside the unit ball.
It is a quantitative form of [[strict convexity|GDTKW9KPH8]] of the unit ball.
Hilbert spaces and $L^p$ spaces for $1 < p < \infty$ are uniformly convex, and the [[Milman–Pettis theorem|AJ5RG9XQYZ]] says that every uniformly convex Banach space is reflexive.

## Explanations

### L1 human

Picture the unit ball as a round ball.
If two points on or in the ball are a fixed distance apart, the midpoint of the segment between them is noticeably inside the ball, not near its edge.
Uniform convexity asks that this happen with a margin that depends only on the distance between the points.
A square ball in the plane fails this: the midpoint of two corners on the same side lies on the boundary.
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# Artin–Schreier theory

## Definition

Let $p$ be a prime and let $K$ be a [[field|0QA8CHH8MG]] of [[characteristic|1ZB13N78XW]] $p$.
The *Artin–Schreier map* is the additive homomorphism $\wp \colon K \to K$, $\wp(x) = x^p - x$.
For $a \in K$, the *Artin–Schreier polynomial* of $a$ is $x^p - x - a$, and an *Artin–Schreier extension* of $K$ is a field $K(\alpha)$ with $\alpha^p - \alpha = a$ for some $a \in K$.
*Artin–Schreier theory* is the description of the [[cyclic extensions|VFE5B71E7Y]] of degree $p$ of $K$ by these polynomials:

1. if $a \in \wp(K)$, then $x^p - x - a$ splits into linear factors over $K$;
2. if $a \notin \wp(K)$, then $x^p - x - a$ is irreducible over $K$, and $K(\alpha)/K$ is a [[Galois extension|0DS2F7GK05]] that is cyclic of degree $p$;
3. every cyclic extension of degree $p$ of $K$ inside a [[separable closure|JJ91A1V8NQ]] $K^{s}$ arises as in clause 2;
4. the subgroups of order $p$ of $K/\wp(K)$ correspond bijectively to the cyclic extensions of degree $p$ of $K$ inside $K^{s}$.

In clause 4, the subgroup generated by the class of $a \notin \wp(K)$ corresponds to $K(\alpha) \subseteq K^{s}$, where $\alpha^p - \alpha = a$.

## Remarks

Clauses 1 and 2 rest on one fact: if $\alpha$ is a root, the roots are $\alpha + i$ for $i \in \F_p$, since $(\alpha + i)^p - (\alpha + i) = \alpha^p - \alpha$.
Example: over $K = \F_2$, the map $\wp(x) = x^2 - x$ sends $0$ and $1$ to $0$, so $a = 1 \notin \wp(K)$.
Then $x^2 + x + 1$ is irreducible over $\F_2$, and $\F_2(\alpha) = \F_4$ is cyclic of degree $2$ over $\F_2$.
Non-example: $x^p - a$ is not an Artin–Schreier polynomial, and over $K = \F_p(t)$ with $a = t$ it gives a purely inseparable extension, which is not Galois.
More generally, the finite subgroups of $K/\wp(K)$ correspond to the finite abelian extensions of $K$ of exponent $p$.
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# Kernel density estimation

## Definition

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with an unknown [[probability density function|04YG690N5V]] $f$.
Let $K \colon \R \to \R$ be an integrable function with $\int_{\R} K(u) \, du = 1$, called the kernel, and let $h > 0$, called the bandwidth.
The *kernel density estimator* of $f$ is
$$\hat f_h(x) = \frac{1}{n h} \sum_{i=1}^{n} K\Big(\frac{x - X_i}{h}\Big), \quad x \in \R.$$

## Remarks

The substitution $u = (x - X_i)/h$ gives $\int_{\R} \hat f_h(x) \, dx = 1$; if moreover $K \ge 0$, then $\hat f_h$ is a probability density.
Common kernels are the Gaussian kernel $K(u) = e^{-u^2/2}/\sqrt{2\pi}$ and the Epanechnikov kernel $K(u) = \frac{3}{4}(1 - u^2)$ for $\abs{u} \le 1$, $0$ otherwise.
Example: with $K = \frac{1}{2} \mathbf{1}_{[-1, 1]}$, $\hat f_h(x)$ is the fraction of data in $[x - h, x + h]$ divided by $2h$.
Non-example: $K = \mathbf{1}_{[-1, 1]}$ has integral $2$, so the resulting $\hat f_h$ integrates to $2$ and is not a density estimate.
If $f$ is twice continuously differentiable and $K$ is a bounded symmetric density with finite variance, the mean integrated squared error is smallest for $h$ of order $n^{-1/5}$.
In $\R^d$, one uses $\frac{1}{n h^d} \sum_i K((x - X_i)/h)$ with a kernel on $\R^d$.
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# Eta invariant

## Definition

Let $Y$ be a closed smooth manifold of dimension $n$ with a smooth positive density, and let $E$ be a Hermitian vector bundle over $Y$.
Let $A$ be a formally [[self-adjoint|0NEGN5CRSC]] [[elliptic differential operator|MT9P2X85V8]] of order $m \ge 1$ on sections of $E$.
Its spectrum is a discrete set of real [[eigenvalues|0BQAA2BK3Q]] $\lambda$ of finite multiplicity.
The *eta function* of $A$ is
$$\eta_A(s) = \sum_{\lambda \ne 0} \operatorname{sign}(\lambda) \, \abs{\lambda}^{-s},$$
summed over the nonzero eigenvalues counted with multiplicity.
It converges for $\operatorname{Re} s > n / m$, and it has a meromorphic [[continuation|0M3YQJ81NB]] to $\C$ that is holomorphic at $s = 0$.
The *eta invariant* of $A$ is $\eta_A(0)$.

## Remarks

The regularity at $s = 0$ was proved by Atiyah, Patodi and Singer, with the general case due to Gilkey.
Some authors call $\xi_A = \tfrac12(\eta_A(0) + \dim \ker A)$ the reduced eta invariant.

Example: on $S^1 = \R / 2\pi\Z$ let $A = -i \, d/d\theta + a$ with $0 < a < 1$.
The eigenfunctions are $e^{ik\theta}$, $k \in \Z$, with eigenvalues $k + a$.
So $\eta_A(s) = \sum_{k \ge 0} (k + a)^{-s} - \sum_{k \ge 1} (k - a)^{-s} = \zeta(s, a) - \zeta(s, 1 - a)$, with $\zeta(s, a)$ the [[Hurwitz zeta function|G4SW7HJG88]].
Since $\zeta(0, a) = \tfrac12 - a$, the eta invariant is $\eta_A(0) = 1 - 2a$.
For $a = 0$ the nonzero spectrum $\Z \setminus \{0\}$ is symmetric and $\eta_A(0) = 0$.

Non-example: $B = d/d\theta$ on $S^1$ is elliptic but not self-adjoint; its eigenvalues $ik$ are imaginary, so $\operatorname{sign}(\lambda)$ has no meaning and $B$ has no eta invariant.
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# Class field theory

## Summary

Class field theory describes the abelian extensions of a number field $K$ in terms of arithmetic data of $K$ itself.
The finite abelian extensions of $K$ correspond to subgroups of generalised ideal class groups.
Equivalently, they correspond to open subgroups of finite index of the [[idele class group|9GHD9MXNYV]].
The [[Artin reciprocity law|7938EKMMVR]] gives an isomorphism between the Galois group of such an extension and the corresponding quotient, through [[Frobenius elements|H5145REFMW]].
The [[Kronecker–Weber theorem|AW00HPFB54]] and the [[Hilbert class field|WWCJ7CR2R3]] are special cases, and [[local class field theory|SHEN158T3M]] treats the completions.

## Explanations

### L1 human

Over $\Q$, whether a prime $p$ splits in $\Q(i)$ depends only on $p$ modulo $4$.
Class field theory says this is typical.
In any extension of a number field with commutative Galois group, the way a prime splits depends only on a congruence condition on the prime.
Conversely, every such congruence condition comes from an abelian extension.
So the abelian extensions of a field are classified by data that can be read off inside the field.
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# Pólya's recurrence theorem

## Statement

Let $d \ge 1$ and let $(S_n)_{n \ge 0}$ be the simple [[random walk|04VZE3FZXF]] on $\Z^d$ with $S_0 = 0$: its steps are independent and uniform on the $2d$ vectors $\pm e_1, \ldots, \pm e_d$.

1. If $d = 1$ or $d = 2$, then $P(S_n = 0 \text{ for some } n \ge 1) = 1$.
2. If $d \ge 3$, then $P(S_n = 0 \text{ for some } n \ge 1) < 1$.

## Remarks

The walk is a [[Markov chain|0G48PYM96Q]], and the origin is [[recurrent|PBZTBWZ4GS]] if and only if $\sum_{n} P(S_{2n} = 0) = \infty$.
For $d = 1$, $P(S_{2n} = 0) = \binom{2n}{n} 4^{-n}$, which behaves like $(\pi n)^{-1/2}$.
For $d = 2$, $P(S_{2n} = 0) = \left(\binom{2n}{n} 4^{-n}\right)^2$, which behaves like $(\pi n)^{-1}$, so the sum still diverges.
For general $d$, $P(S_{2n} = 0)$ is of order $n^{-d/2}$, and the sum converges for $d \ge 3$.
In dimension $3$, the probability of return is about $0.34$.
In dimensions $1$ and $2$ the walk returns infinitely often, and in dimension $d \ge 3$ it satisfies $\abs{S_n} \to \infty$ almost surely.

## Explanations

### L1 human

A walker on a grid takes unit steps in random directions.
On a line or in the plane, she is sure to come back home eventually.
In space of three or more dimensions, there is a fair chance that she wanders off and never returns.

### L2 human

The proof reduces recurrence to the divergence of the expected number of returns, which local limit estimates for $P(S_{2n} = 0)$ settle.
Another proof views the lattice as an electrical network: the walk is recurrent if and only if the effective resistance from the origin to infinity is infinite.
The same dichotomy between $d \le 2$ and $d \ge 3$ appears for Brownian motion, which is neighbourhood recurrent in the plane and transient in $\R^3$.
</document>

<document source="https://tamnd.github.io/math2/node/BETJYRCRJB.md" tag="BETJYRCRJB" kind="definition">
# Vertex of a polytope

## Definition

Let $P \subset \R^d$ be a [[convex polytope|300K6CB1KH]].
A *vertex* of $P$ is a [[face|KY7SB854VH]] of $P$ of dimension $0$, identified with the single point it contains.
Equivalently, a point $v \in P$ is a vertex if there is a vector $c \in \R^d$ such that $c \cdot v > c \cdot x$ for every $x \in P$ with $x \ne v$.

## Remarks

Example: in the square $P = [0,1]^2$, the point $v = (1,1)$ is a vertex.
Take $c = (1,1)$: then $c \cdot v = 2$, and $c \cdot x = x_1 + x_2 < 2$ for every other point $x$ of $P$.

Non-example: the point $(1, \tfrac12)$ of the same square is not a vertex.
For any $c$, the points $(1, \tfrac12 \pm t)$ for small $t > 0$ lie in $P$, and $c \cdot x$ is at least $c \cdot (1,\tfrac12)$ at one of them, so no $c$ singles it out.

The vertices of $P$ are exactly its [[extreme points|3MSGFJM2NX]].
If $V$ is the set of vertices, then $P$ is the [[convex hull|0FQSDC2EQ2]] of $V$, and every finite set whose convex hull is $P$ contains $V$.
</document>

<document source="https://tamnd.github.io/math2/node/BEW48TA9WZ.md" tag="BEW48TA9WZ" kind="concept">
# Brauer group

## Summary

The Brauer group of a field $K$ is the group of classes of [[central simple algebras|5HRKJBAY2V]] over $K$ up to matrix algebras, see the [[definition|EX34J8PP61]].
Its product is given by the tensor product.
Each class contains exactly one central [[division ring|0EA5BTA0CX]] up to isomorphism, so the group classifies the division algebras with center $K$ and finite dimension.
It is isomorphic to the Galois cohomology group $H^2(K, (K^s)^\times)$, see [[Galois cohomology|9MPCZ3EAJM]].
Its computation for local and global fields is a main part of class field theory.

## Explanations

### L1 human

Over the real numbers there are exactly two finite-dimensional division algebras with center $\R$: $\R$ itself and the quaternions.
The Brauer group packs this kind of information into a group.
Each division algebra with center $K$ gives an element, and the product of two elements comes from the tensor product of the algebras.
For $\R$ the group has two elements, and the quaternions square to the trivial class, because $\mathbb{H} \otimes_\R \mathbb{H}$ is the algebra of $4 \times 4$ real matrices.
</document>

<document source="https://tamnd.github.io/math2/node/BF7T7VE57V.md" tag="BF7T7VE57V" kind="statement">
# Hardy–Littlewood maximal inequality

## Statement

Let $n \ge 1$ and let $\abs{E}$ be the [[Lebesgue measure|2Z2QVWZ262]] of a set $E \subseteq \R^n$.
For a locally integrable function $f$ on $\R^n$, let $Mf$ be its [[Hardy–Littlewood maximal function|VX1A56P5KQ]].

1. For every $f \in L^1(\R^n)$ and every $\lambda > 0$,
$$\abs{\set{x \in \R^n : Mf(x) > \lambda}} \le \frac{3^n}{\lambda} \norm{f}_{L^1}.$$
2. For every $p$ with $1 < p \le \infty$ there is a constant $C = C(n, p)$ such that $\norm{Mf}_{L^p} \le C \norm{f}_{L^p}$ for every $f \in L^p(\R^n)$.

Here $L^p(\R^n)$ is the [[Lp space|SASP14MA1Z]] of Lebesgue measure.

## Remarks

Part 1 says that $M$ is of [[weak type|9534R6VMY0]] $(1, 1)$.
Its proof uses the [[Vitali covering lemma|FCA0J1X07K]].
Each point of a compact subset $K$ of $\set{Mf > \lambda}$ is the centre of a ball $B$ with $\int_B \abs{f} > \lambda \abs{B}$.
The covering lemma selects finitely many disjoint such balls $B_1, \dots, B_m$ whose dilates $3B_k$ cover $K$, so $\abs{K} \le 3^n \sum_k \abs{B_k} \le 3^n \lambda^{-1} \norm{f}_1$.
For $p = \infty$, part 2 is the trivial bound $Mf \le \norm{f}_\infty$, and for $1 < p < \infty$ it follows from part 1 by the [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]].

The constant in part 2 grows like $1/(p - 1)$ as $p \to 1$, and part 2 fails for $p = 1$.
For example, $f = \mathbf{1}_{[0, 1]}$ on $\R$ has $Mf(x) = 1/(2x)$ for $x > 1$, which is not integrable.
In fact $Mf$ is integrable only when $f = 0$ almost everywhere.

## Explanations

### L1 human

The maximal function records, at each point, the largest average of $\abs{f}$ over balls centred there.
One might fear that taking the largest of infinitely many averages makes the function huge.
The theorem says that this does not happen.
For $p > 1$, the $L^p$ norm of $Mf$ is at most a constant times that of $f$.
For $p = 1$, a weaker bound holds on the size of the sets where $Mf$ is large.

### L2 human

The inequality is the main tool for proving that averages converge almost everywhere.
Combined with density of continuous functions it gives the Lebesgue differentiation theorem: the averages of $f$ over balls shrinking to $x$ tend to $f(x)$ for almost every $x$.
Many operators in harmonic analysis, such as convolutions with radially decreasing integrable kernels, are bounded pointwise by a multiple of $Mf$, so they inherit its $L^p$ bounds.
The weighted version of the inequality leads to the Muckenhoupt $A_p$ condition.
</document>

<document source="https://tamnd.github.io/math2/node/BF9XR5SVPV.md" tag="BF9XR5SVPV" kind="concept">
# Type and cotype

## Summary

Type and cotype measure how far a [[Banach space|06GZ956657]] is from a Hilbert space.
They compare random sign sums $\sum_i \pm x_i$ with the sizes of the $x_i$.
A space of type $p$ makes such sums small, and a space of cotype $q$ makes them large.
A theorem of Kwapień says that a Banach space has type $2$ and cotype $2$ exactly when it is isomorphic to a [[Hilbert space|HM2HHE3J51]].
These notions organise much of the local theory of Banach spaces, around results such as [[Dvoretzky's theorem|NRPQP49N9F]].

## Explanations

### L1 human

Take some vectors and add them with random signs.
In the plane with the usual length, the average squared length of the result is the sum of the squared lengths, by the theorem of Pythagoras.
In other spaces the random sum can be longer or shorter than this.
Type and cotype record the best inequalities between the two sides.
For example, in $\ell^1$ the unit vectors $e_1, \dots, e_n$ give a random sum of norm exactly $n$, much larger than the Pythagorean value $\sqrt{n}$.
</document>

<document source="https://tamnd.github.io/math2/node/BFBNQ36QS8.md" tag="BFBNQ36QS8" kind="concept">
# Cartesian closed category

## Summary

A Cartesian closed category is a [[category|A2DE66937D]] with finite [[products|S5744QZV4N]] in which every pair of objects has an [[exponential object|RHWD8DY41R]].
In it, morphisms $X \times Y \to Z$ match morphisms $X \to Z^Y$, as in currying of functions.
Sets, small categories, presheaf categories and every [[topos|6837RS7N82]] are Cartesian closed.
Cartesian closed categories are the models of the simply typed [[lambda calculus|BD756KR8FM]].

## Explanations

### L1 human

In the world of sets you can form pairs, and you can form the set of all functions between two sets.
A Cartesian closed category is any category where both of these constructions make sense and fit together as they do for sets.
The key rule is that a function of two inputs is the same as a function of one input that returns a function.
The category of all small categories is an example: the functors between two categories form a category again.
</document>

<document source="https://tamnd.github.io/math2/node/BFD7T0YY42.md" tag="BFD7T0YY42" kind="concept">
# Radical axis

## Summary

The radical axis of two non-concentric [[circles|060W5CBZP6]] is the set of points that have the same [[power|02E9VT1RVQ]] with respect to both circles.
It is a straight line perpendicular to the line through the two centres.
When the circles meet in two points, it is the line through those points.
Radical axes give short proofs of many concurrency and collinearity results.

## Explanations

### L1 human

From a point outside a circle, draw a tangent segment to the circle.
The radical axis of two circles is where these tangent segments to the two circles have equal length.
If two circles cross, the radical axis is the line through their two crossing points.
For example, the unit circles centred at $(0, 0)$ and $(2, 0)$ touch at $(1, 0)$, and their radical axis is the vertical line $x = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/BFFA03GE3V.md" tag="BFFA03GE3V" kind="concept">
# Projective module

## Summary

A projective module is a module $P$ such that every linear map from $P$ lifts along every surjective linear map.
Equivalently, $P$ is a direct summand of a [[free module|0DZ9DKAKZR]].
Projective modules keep the most useful property of free modules, lifting of maps, but need not have a basis.
They are the building blocks of projective resolutions in homological algebra, and finitely generated projective modules correspond to [[vector bundles|P4DZM975BV]] in geometry.

## Explanations

### L1 human

Free modules are the easiest modules: you can send a basis anywhere you like.
A projective module is a piece cut off from a free module, so it shares this freedom when you map out of it.
Every free module is projective.
Over the ring $\Z/6\Z \cong \Z/2\Z \times \Z/3\Z$, the module $\Z/2\Z$ is a piece of the free module $\Z/6\Z$, so it is projective, but it is too small to be free.
</document>

<document source="https://tamnd.github.io/math2/node/BFP0ZST7V3.md" tag="BFP0ZST7V3" kind="conjecture">
# Jacobson's conjecture

## Statement

Let $R$ be a [[ring|0894J980RK]] with unit that is [[Noetherian|0MVH4Y1CRZ]] on both sides, that is, left Noetherian and right Noetherian.
Let $J$ be the [[Jacobson radical|7HWZ06SRMF]] of $R$.
Then
$$\bigcap_{n \ge 1} J^n = 0.$$

## Remarks

The conjecture is open in general.
For commutative Noetherian rings it holds, by the Krull intersection theorem.
The one-sided version, for rings that are only right Noetherian, is false; I. N. Herstein gave a counterexample.
The conjecture is known for several classes of two-sided Noetherian rings, among them the fully bounded Noetherian rings.

## Explanations

### L1 human

The Jacobson radical $J$ collects the elements of a ring that are "small" in a precise sense.
Its powers $J, J^2, J^3, \ldots$ shrink, and the conjecture says that in a Noetherian ring nothing survives in all of them.
For commutative rings this is a classical theorem of Krull, and the question is whether the same holds without commutativity.
</document>

<document source="https://tamnd.github.io/math2/node/BFQBHBN5F5.md" tag="BFQBHBN5F5" kind="concept">
# Empirical likelihood

## Summary

Empirical likelihood is a method of inference that uses a [[likelihood|0VW9E4QXR6]] ratio without assuming a parametric family.
It assigns unknown weights to the observed data points and maximises the product of the weights under a constraint given by the parameter value.
It then compares the result with the unconstrained maximum.
Art Owen developed it in the late 1980s.
Like a parametric likelihood ratio, it yields tests and confidence regions with chi-squared calibration, and its confidence regions take their shape from the data.

## Explanations

### L1 human

A parametric likelihood asks how probable the data are under a formula for the distribution.
Empirical likelihood avoids the formula: it only allows distributions that put weight on the observed points.
To judge a proposed mean $\mu$, it tilts the weights as little as possible so that the weighted average of the data equals $\mu$.
The more the weights must be tilted, the less plausible $\mu$ is.
If $\mu$ is the sample mean, no tilting is needed and the ratio is $1$.
</document>

<document source="https://tamnd.github.io/math2/node/BFV3MKCBHZ.md" tag="BFV3MKCBHZ" kind="definition">
# Schlegel diagram

## Definition

Let $P \subset \R^d$ be a [[convex polytope|300K6CB1KH]] of dimension $d \ge 2$, and let $F$ be a [[facet|DPDXN2FJC4]] of $P$.
For each facet $G$, let $H_G$ be the hyperplane spanned by $G$.
Choose a point $p \in \R^d$ such that:

1. $p$ and $P$ lie on opposite sides of $H_F$, strictly;
2. for every facet $G \ne F$, $p$ lies strictly on the same side of $H_G$ as the interior of $P$.

For $x \in P$, let $\pi(x)$ be the point where the segment from $p$ to $x$ meets $H_F$.
The *Schlegel diagram* of $P$ based at $F$ is the collection of the sets $\pi(G')$.
Here $G'$ runs over the [[faces|KY7SB854VH]] of $P$ other than $P$ and $F$.

## Remarks

Each $\pi(G')$ is a convex polytope contained in $F$, of the same dimension as $G'$.
These polytopes cover $F$, two of them meet in a common face, and their inclusions are those of the faces of $P$.
So the diagram is a polytopal subdivision of $F$ that records the whole face lattice of $P$.
Points $p$ as in the definition exist; any point beyond $F$ close enough to an interior point of $F$ works.

Example: let $P = [0,1]^3$, $F$ the facet $x_3 = 1$, and $p = (\tfrac12, \tfrac12, 1 + \epsilon)$ with $\epsilon > 0$ small.
Then $p$ is above $x_3 = 1$, and on the inner side of the four side planes and of $x_3 = 0$.
The bottom square projects to a smaller square in the middle of $F$, and the four side squares project to trapezoids between the two squares.

Non-example: in the same example, $p = (5, \tfrac12, 2)$ satisfies condition 1 but fails condition 2 for the facet $x_1 = 1$, since $p$ has $x_1 > 1$.
Then the images are not even inside $F$: the segment from $p$ to the vertex $(1, 0, 0)$ meets the plane $x_3 = 1$ at $(3, \tfrac14, 1) \notin F$.
</document>

<document source="https://tamnd.github.io/math2/node/BFWTGP7F92.md" tag="BFWTGP7F92" kind="definition">
# Nevanlinna theory of difference equations

## Definition

Let $c \in \C \setminus \set{0}$ and let $n \ge 1$.
Let $F$ be a polynomial or a rational function in its last $n + 1$ arguments, with coefficients that are meromorphic functions of $z$.
A *complex difference equation* for an unknown [[meromorphic function|0ZZ7ZYDR8W]] $f$ on $\C$ is an equation
$$F\big(z, f(z), f(z + c), \dots, f(z + nc)\big) = 0.$$
For such a solution $f$, write $T(r, f)$ for its [[Nevanlinna characteristic|YT6Z0GWS0W]] and $m(r, f)$ for its proximity function.
A quantity $S(r, f)$ is *small* with respect to $f$ if $S(r, f) = o(T(r, f))$ as $r \to \infty$ outside a set of $r$ of finite logarithmic measure.
*Nevanlinna theory of difference equations* is the study of the meromorphic solutions of such difference equations through $T$, $m$ and the counting function $N$.
Its basic tools are estimates of the type $m\big(r, f(z + c)/f(z)\big) = S(r, f)$.

## Remarks

Example: $f(z) = e^{z}$ solves $f(z + 1) = e f(z)$.
Here $T(r, f) = r/\pi$ and $f(z + 1)/f(z) = e$ is constant, so $m\big(r, f(z+1)/f(z)\big) = 1$, which is small compared with $T(r, f)$.

Non-example: for $f(z) = e^{e^z}$, which has infinite order, the quotient $f(z + 1)/f(z) = e^{(e - 1)e^z}$ has proximity function comparable to $T(r, f)$, so the small-term estimate fails; this is why the difference lemma requires finite order.

The basic estimate holds for every meromorphic $f$ of finite order and every $c \in \C$, outside an exceptional set of finite logarithmic measure.
</document>

<document source="https://tamnd.github.io/math2/node/BG3EH18KRE.md" tag="BG3EH18KRE" kind="statement">
# Dehn–Lickorish theorem

## Statement

Let $g \ge 1$ and let $S_g$ be a closed orientable [[surface|VGJJVBZ1MQ]] of [[genus|NJ787GZS1N]] $g$.
Let $\operatorname{Mod}(S_g)$ be its [[mapping class group|JV5NYG913V]], the group of isotopy classes of orientation-preserving homeomorphisms of $S_g$.
Then $\operatorname{Mod}(S_g)$ is generated by finitely many [[Dehn twists|W890NRHGS4]] about non-separating simple closed curves.
More precisely, Dehn twists about $3g - 1$ suitable curves suffice.

## Remarks

Dehn proved that finitely many twists generate the group, and Lickorish later gave the explicit set of $3g - 1$ curves.
Humphries reduced the number to $2g + 1$, which is the smallest possible for $g \ge 2$.
Idea of the proof: argue by induction on the genus with the curve complex or a connected graph of non-separating curves.
Two non-separating curves can be joined by a chain of curves, each meeting the next in one point.
Twists along such chains move any non-separating curve to a fixed one.
This reduces the problem to the stabiliser of one curve, which is related to the mapping class group of a surface of smaller complexity.

The theorem at work: for $g = 1$ the mapping class group of the torus is $SL(2, \Z)$, acting on $H_1(T^2; \Z) = \Z^2$.
The twists about the two standard curves act by $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}$, and these two matrices generate $SL(2, \Z)$.
For $g = 0$ the mapping class group of the sphere is trivial.

## Explanations

### L1 human

The ways to deform a closed surface onto itself, counted up to continuous wiggling, form a group.
The theorem says that every such deformation is a composition of a few basic moves, twists along loops.
For a surface with $g$ holes, $3g - 1$ fixed loops are enough.

### L2 human

Finite generation by twists is the starting point of most work on mapping class groups.
It is the starting point for finite presentations of the group, and it lets one compute its abelianization, which is trivial for $g \ge 3$.
It is the key step in the proof of the [[Lickorish–Wallace theorem|1YP4RF1QXV]].
There a Heegaard gluing map is written as a product of twists, and each twist is realised by surgery on a curve.
It is also used to show that the action on homology maps $\operatorname{Mod}(S_g)$ onto the symplectic group $Sp(2g, \Z)$.
</document>

<document source="https://tamnd.github.io/math2/node/BG3XQF6V58.md" tag="BG3XQF6V58" kind="statement">
# Bellows conjecture

## Statement

Let $P_t$, for $t \in [0, 1]$, be a continuous flex of a closed orientable polyhedral surface in $\R^3$.
That is, each face of $P_t$ stays congruent to the same face of $P_0$, and the combinatorics stay the same.
Then the oriented volume enclosed by $P_t$ does not depend on $t$.

## Remarks

The statement was proved in 1997 by Connelly, Sabitov and Walz, so it is now a theorem.
The proof rests on a result of Sabitov: the volume of a polyhedron is a root of a monic polynomial whose coefficients are polynomials in the squared edge lengths.
Along a flex the edge lengths are fixed, so the volume takes only finitely many values.
Since it varies continuously in $t$, it is constant.

Example: every [[flexible polyhedron|ZGD3TQX0K6]], such as Steffen's polyhedron, keeps its volume while it moves.
So such a polyhedron cannot work as a bellows that pumps air by changing its volume.

## Explanations

### L1 human

Some polyhedra made of rigid faces with hinged edges can still wobble.
One might hope to use such a shape as a bellows, squeezing air in and out.
The theorem says this is impossible: the volume inside stays the same during the whole motion.

### L2 human

The result complements [[Cauchy's rigidity theorem|CJ7KCDDF31]].
Cauchy rules out any flex of a convex polyhedron, while the bellows theorem allows flexes of nonconvex surfaces but fixes their volume.
Its algebraic proof treats the volume as an algebraic function of the edge lengths, a three-dimensional relative of [[Heron's formula|0FRBZ2X4HJ]].
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<document source="https://tamnd.github.io/math2/node/BG5A4CB6NB.md" tag="BG5A4CB6NB" kind="concept">
# Benz plane

## Summary

Benz plane is the common name for three kinds of circle geometry: Möbius planes, Laguerre planes and Minkowski planes.
Each kind is an incidence structure of points and circles in which three suitable points lie on exactly one circle and circles can touch.
The classical examples are the plane sections of a sphere, of a circular cylinder and of a hyperboloid of one sheet.
The name honours Walter Benz, who developed their common theory.

## Explanations

### L1 human

Cut a sphere, a cylinder or a hyperboloid with planes and call the curves you get circles.
In each case three points that are not in some special position lie on exactly one circle.
The three resulting geometries share many features, and the word Benz plane names all three at once.
They differ in their parallel points: none on a sphere, one family of lines on a cylinder, two families on a hyperboloid.
</document>

<document source="https://tamnd.github.io/math2/node/BG86NXRJ5F.md" tag="BG86NXRJ5F" kind="statement">
# Kaplansky's theorem on projective modules

## Statement

1. Let $R$ be a [[ring|0894J980RK]] with identity. Every [[projective|CRE6VEKT32]] left $R$-module is a direct sum of countably generated projective left $R$-modules.
2. Let $R$ be a local ring, that is, a ring with identity whose quotient $R/J(R)$ by the [[Jacobson radical|XA0DCNH7WY]] is a [[division ring|0WZXK05ME7]]. Every projective left $R$-module is [[free|02E1T12PP4]].

## Remarks

Part 1 is proved by a transfinite construction that builds a projective module up from countably generated direct summands.
For part 2 one reduces to countably generated modules by part 1.
A finitely generated projective module over a local ring is free by Nakayama's lemma.
In the countably generated case, an inductive argument splits off free summands one element at a time.
The point of part 2 is that no finiteness hypothesis is needed.
Example: every projective module over the [[local ring|MZ02BKPHJA]] $\Z_{(p)}$ is free.
The local hypothesis is necessary.
Over $\Z/6\Z \cong \Z/2\Z \times \Z/3\Z$, the module $\Z/2\Z$ is projective, since it is a direct summand of $\Z/6\Z$.
It is not free, since it has $2$ elements while a nonzero free module has at least $6$.

## Explanations

### L1 human

Projective modules are pieces of free modules.
Over a local ring, a ring with a single "point", such pieces are always free themselves, even when they are very large.
The first part says that every projective module can be cut into pieces that are each generated by countably many elements.

### L2 human

For finitely generated modules, part 2 is a standard consequence of Nakayama's lemma; Kaplansky's contribution is the infinitely generated case.
Part 1 is a general reduction used in many other questions about large [[projective modules|BFFA03GE3V]].
Over [[semiperfect rings|ZRC2S3ECFK]] a related result holds: every finitely generated projective module is a direct sum of modules $Re$ with $e$ an idempotent such that $eRe$ is local.
</document>

<document source="https://tamnd.github.io/math2/node/BG8Q0BYSQ3.md" tag="BG8Q0BYSQ3" kind="statement">
# Fagin's theorem

## Statement

Let $\sigma$ be a finite relational signature.
Let $\mathcal K$ be a class of finite $\sigma$-[[structures|983GZBQYT4]] that is closed under isomorphism.
Then the following are equivalent.

1. The class $\mathcal K$ is in NP: some nondeterministic [[Turing machine|758EYCN27B]] decides, in time polynomial in the size of a finite $\sigma$-structure $A$, whether $A \in \mathcal K$.
2. There is an existential [[second-order|G6X7SZSNC5]] [[sentence|DKFGQTRYZE]] $\exists R_1 \cdots \exists R_k \, \varphi$ such that a finite $\sigma$-structure belongs to $\mathcal K$ if and only if it satisfies this sentence.

Here $R_1, \ldots, R_k$ are relation variables, and $\varphi$ is a first-order sentence in the signature $\sigma \cup \set{R_1, \ldots, R_k}$.

## Remarks

From 2 to 1: a machine guesses the relations $R_1, \ldots, R_k$ on the finite universe, which takes polynomially many bits.
It then checks the first-order sentence $\varphi$, which takes polynomial time for a fixed $\varphi$.
From 1 to 2: the sentence guesses a linear order of the universe and relations that encode an accepting computation of the machine.
The computation of length $n^c$ is coded with tuples of length $c$, and $\varphi$ checks that each step follows the transition rules.

Example: the class of finite graphs that admit a proper colouring with three colours is in NP.
It is defined by the sentence $\exists R \, \exists G \, \exists B \, \varphi$.
Here $\varphi$ says that each vertex lies in $R$, $G$ or $B$, and that no edge joins two vertices in the same set.

## Explanations

### L1 human

The theorem matches a kind of computing problem with a kind of logical description.
A problem is in NP when a proposed solution can be checked quickly.
Fagin's theorem says these are exactly the properties of finite structures of the form "there exist some relations such that a first-order condition holds".
The relations play the role of the solution that is guessed.

### L2 human

Fagin's theorem started descriptive complexity, which characterises complexity classes by logics without any reference to machines.
It shows that NP is closed under complement if and only if existential second-order logic has the same power as universal second-order logic on finite structures.
Other results of this kind characterise P on ordered finite structures by first-order logic with a least fixed point operator.
The theorem belongs to [[finite model theory|4XRVJE0KAN]], where classical tools such as compactness are not available.
</document>

<document source="https://tamnd.github.io/math2/node/BGH41KPWQ6.md" tag="BGH41KPWQ6" kind="definition">
# Canonical bundle

## Definition

Let $k$ be a field and let $X$ be a scheme that is [[smooth|EXKGG9X6J0]] over $k$ and of pure dimension $n$.
Then the [[sheaf of differentials|6VC6HQTFSA]] $\Omega_{X/k}$ is locally free of rank $n$.
The *canonical bundle* of $X$ is the [[invertible sheaf|7JCF2TE3V5]]
$$\omega_X = \bigwedge^n \Omega_{X/k},$$
the top [[exterior power|0MDMZ965VE]] of $\Omega_{X/k}$.
A *canonical divisor* is a divisor $K_X$ with $\mathcal{O}_X(K_X) \cong \omega_X$.

## Remarks

Locally, if $x_1, \dots, x_n$ are local coordinates (an étale chart), then $\omega_X$ is free with basis $dx_1 \wedge \dots \wedge dx_n$.

Example: for $X = \mathbb{P}^n_k$, the Euler sequence $0 \to \Omega_X \to \mathcal{O}(-1)^{\oplus (n+1)} \to \mathcal{O}_X \to 0$ gives, by taking determinants, $\omega_X \cong \mathcal{O}(-n-1)$.
For $n = 1$ this is the degree $-2$ bundle on $\mathbb{P}^1_k$.

Non-example: for the cuspidal curve $y^2 = x^3$, $\Omega_{X/k}$ is not locally free at the cusp, so the definition does not apply.
For singular varieties one uses instead the dualizing sheaf, which agrees with $\omega_X$ on smooth projective varieties.
</document>

<document source="https://tamnd.github.io/math2/node/BGRWCAT4SZ.md" tag="BGRWCAT4SZ" kind="definition">
# Lopatinskii–Shapiro condition

## Definition

Let $\Omega \subseteq \R^n$ be an open set with smooth boundary $\partial \Omega$, with $n \ge 2$, and let $m \ge 1$.
Let $A$ be a linear differential operator of order $2m$ with smooth coefficients on $\overline{\Omega}$, with [[principal symbol|F31T650DKY]] $a(x, \xi)$.
For $x \in \partial \Omega$, let $\nu(x)$ be the inner unit normal.
Assume that $A$ is properly [[elliptic|XY8GEQSWPR]].
That is, let $x \in \partial \Omega$ and let $\xi' \in \R^n$ be nonzero and tangent to $\partial \Omega$ at $x$.
Then the polynomial $\tau \mapsto a(x, \xi' + \tau \nu(x))$ has exactly $m$ roots with positive imaginary part.
The roots are counted with multiplicity.
Let $B_1, \ldots, B_m$ be linear differential operators on $\partial \Omega$ with smooth coefficients, of orders $m_j < 2m$ and principal symbols $b_j(x, \xi)$.
Write $D_t = -i \, d/dt$.
The operators $B_1, \ldots, B_m$ satisfy the *Lopatinskii–Shapiro condition* with respect to $A$ if the following holds.
Let $x \in \partial \Omega$ and let $\xi'$ be a nonzero tangent vector at $x$.
The only bounded solution $v \colon [0, \infty) \to \C$ of
$$a\big(x, \xi' + \nu(x) D_t\big) v(t) = 0 \quad (t > 0), \qquad b_j\big(x, \xi' + \nu(x) D_t\big) v(0) = 0 \quad (j = 1, \ldots, m)$$
is $v = 0$.

## Remarks

Equivalently, the polynomials $\tau \mapsto b_j(x, \xi' + \tau \nu)$ are linearly independent modulo $\prod_k (\tau - \tau_k^+)$, where $\tau_k^+$ are the $m$ roots of $a(x, \xi' + \tau \nu)$ with positive imaginary part.
Example: $A = -\Delta$ with the Dirichlet condition $B_1 u = u$ on the half-space $\set{x_n > 0}$.
Here $a(\xi) = \abs{\xi}^2$, $\nu = e_n$, and the ordinary differential equation is $-v'' + \abs{\xi'}^2 v = 0$, whose bounded solutions are $v(t) = c \, e^{-\abs{\xi'} t}$; the condition $v(0) = 0$ forces $c = 0$.
The Neumann condition $\partial_\nu u$ also passes: $v'(0) = -c \abs{\xi'} = 0$ forces $c = 0$.
Non-example: in $\R^3$, take $A = -\Delta$ on $\set{x_3 > 0}$ with $B_1 u = \partial_1 u$ on the boundary.
For $\xi' = (0, 1, 0)$, the function $v(t) = e^{-t}$ is a bounded nonzero solution, since the symbol of $B_1$ is $i \xi_1 = 0$, so the condition fails.
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<document source="https://tamnd.github.io/math2/node/BGXEC1SVQV.md" tag="BGXEC1SVQV" kind="definition">
# Regular chain

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $S = k[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]] with the variables ordered $x_1 < x_2 < \cdots < x_n$.
Let $p \in S \setminus k$.
Its *main variable* $\operatorname{mvar}(p)$ is the largest variable that occurs in $p$.
Its *initial* $\operatorname{init}(p)$ is the [[leading coefficient|0V5GYWR8R8]] of $p$ as a polynomial in $\operatorname{mvar}(p)$.
A *triangular set* is a finite set $T$ of elements of $S \setminus k$ with pairwise distinct main variables.
For a triangular set $T$ with product of initials $h_T$, the *saturated ideal* is the [[ideal quotient|16J9DM8V57]] $\operatorname{sat}(T) = (T) : h_T^\infty = \set{ f \in S : h_T^m f \in (T) \text{ for some } m \ge 0 }$.
A triangular set $T$ is a *regular chain* if either $T = \emptyset$, or $T = T' \cup \set{t}$ where:

1. $\operatorname{mvar}(t)$ is larger than the main variable of every element of $T'$.
2. $T'$ is a regular chain.
3. $\operatorname{init}(t)$ is not a [[zero divisor|0NNRDW10RZ]] in the [[quotient ring|0076HWHGHQ]] $S/\operatorname{sat}(T')$.

## Remarks

For $T = \emptyset$, the saturated ideal is $(0)$, so a set $\set{t}$ with one element is always a regular chain.

Example: in $k[x, y]$ with $x < y$, let $T = \set{x^2 - 1, \; xy - 1}$.
Here $T' = \set{x^2 - 1}$ has initial $1$, so $\operatorname{sat}(T') = (x^2 - 1)$.
The initial of $t = xy - 1$ is $x$, and $x$ is a unit in $k[x, y]/(x^2 - 1)$, because $x \cdot x = 1$ there; so it is not a zero divisor, and $T$ is a regular chain.

Non-example: $T = \set{x^2 - x, \; xy - 1}$ fails clause 3.
Here $\operatorname{sat}(\set{x^2 - x}) = (x^2 - x)$, and $x \cdot (x - 1) = 0$ in $k[x, y]/(x^2 - x)$ with $x - 1 \ne 0$, so the initial $x$ is a zero divisor.
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<document source="https://tamnd.github.io/math2/node/BH32RQBX6A.md" tag="BH32RQBX6A" kind="concept">
# Deck transformation group

## Summary

The deck transformation group of a [[covering map|3XZRRE165N]] $p \colon \tilde X \to X$ is the group of homeomorphisms of $\tilde X$ that commute with $p$.
These maps permute the points of each fibre, and they measure the symmetry of the covering.
For the [[universal cover|Y1YPE6PM52]] of a reasonable space, the deck transformation group is isomorphic to the [[fundamental group|S1XK3HXHFX]] of the space.

## Explanations

### L1 human

A deck transformation shuffles the sheets of a covering without moving anything in the base.
For the helix $\R$ over the circle, shifting the whole helix up by one full turn is such a symmetry.
The shifts by an integer number of turns are all of them, so the deck transformation group is $\Z$.
This matches the fact that a loop can wind around the circle any integer number of times.
</document>

<document source="https://tamnd.github.io/math2/node/BH3NNAYJT5.md" tag="BH3NNAYJT5" kind="statement">
# Prime-factor FFT algorithm

## Statement

For $M \ge 1$ let $\omega_M = e^{-2\pi i/M}$.
Let $N = N_1 N_2$ where $N_1, N_2 \ge 1$ are coprime integers.
Let $x \in \C^N$ and let $X$ be its [[discrete Fourier transform|TST0GH66A5]], $X_k = \sum_{n=0}^{N-1} x_n \omega_N^{nk}$.
For $0 \le n_1 < N_1$ and $0 \le n_2 < N_2$ let $n(n_1, n_2) = (N_2 n_1 + N_1 n_2) \bmod N$.
For $0 \le k_1 < N_1$ and $0 \le k_2 < N_2$ let $k(k_1, k_2)$ be the unique $k \in \set{0, \ldots, N-1}$ with $k \equiv k_1 \pmod{N_1}$ and $k \equiv k_2 \pmod{N_2}$.
Then both index maps are bijections onto $\set{0, \ldots, N-1}$, and
$$X_{k(k_1, k_2)} = \sum_{n_1=0}^{N_1-1} \sum_{n_2=0}^{N_2-1} x_{n(n_1, n_2)}\, \omega_{N_1}^{n_1 k_1}\, \omega_{N_2}^{n_2 k_2}.$$
So a DFT of length $N$ is a two-dimensional DFT of size $N_1 \times N_2$, with no twiddle factors.

## Remarks

The output map is a bijection by the [[Chinese remainder theorem|0FNN9B4W64]].
The input map is a bijection because $n(n_1, n_2) \equiv N_2 n_1 \pmod{N_1}$ and $N_2$ is invertible modulo $N_1$, and similarly modulo $N_2$.
For the formula, $\omega_N^{N_2 n_1 k} = \omega_{N_1}^{n_1 k} = \omega_{N_1}^{n_1 k_1}$ since $k \equiv k_1 \pmod{N_1}$, and in the same way $\omega_N^{N_1 n_2 k} = \omega_{N_2}^{n_2 k_2}$.
Example: $N = 6 = 2 \cdot 3$; the input order $n(n_1, n_2) = (3n_1 + 2n_2) \bmod 6$ lists $0, 2, 4$ for $n_1 = 0$ and $3, 5, 1$ for $n_1 = 1$.
Coprimality is necessary: for $N_1 = N_2 = 2$ the input map gives $2n_1 + 2n_2 \bmod 4$, which takes only the values $0$ and $2$.
For non-coprime factors one uses the [[Cooley–Tukey algorithm|9N8D9E5TB2]], which pays with twiddle factors.

## Explanations

### L1 human

Suppose the length of a Fourier transform is a product of two coprime numbers, such as $6 = 2 \cdot 3$ or $15 = 3 \cdot 5$.
Then the input and output can be reordered so that the transform becomes a grid of small transforms.
The reordering is the Chinese remainder theorem in action.
Unlike the Cooley–Tukey split, no extra multiplications are needed between the two stages.

### L2 human

The algorithm is due to Good and to Thomas, and is often called the Good–Thomas algorithm.
It is used for lengths with several coprime factors, with each small length done by a hand-optimised module, for example Winograd's.
The price is the more complicated index arithmetic.
A prime-power length such as $8$ or $9$ cannot be split this way, and a Cooley–Tukey or another FFT is used for such factors.
</document>

<document source="https://tamnd.github.io/math2/node/BH57D2FBNR.md" tag="BH57D2FBNR" kind="concept">
# Several complex variables

## Summary

A function of several complex variables is a function $f(z_1, \dots, z_n)$ whose arguments are $n$ complex numbers, that is, a function on a subset of $\C^n$.
The theory of several complex variables studies such functions, mainly the [[holomorphic|507C34W0ZT]] ones, and the domains and spaces on which they live.
Many facts of one complex variable carry over, such as power series expansions and the [[Cauchy integral formula on a polydisc|FG5448NZ18]].
Other facts change deeply when $n \ge 2$.
Zeros are never isolated, some domains force holomorphic functions to extend to larger domains, and the [[unit ball|PDQ9KW9FAP]] and the [[polydisc|DJBWP7D3M8]] are not biholomorphic.

## Explanations

### L1 human

In one complex variable a function takes one complex number $z$ and returns a complex number.
In several complex variables it takes a list $(z_1, \dots, z_n)$ of complex numbers.
A picture: $\C^n$ is like $\R^{2n}$, with each coordinate $z_j = x_j + i y_j$ giving two real directions.
For example, $f(z_1, z_2) = z_1 z_2 + z_2^2$ is a function of two complex variables.
It is a polynomial in each variable, and such functions are the simplest examples of the theory.
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<document source="https://tamnd.github.io/math2/node/BH5TK7DC2X.md" tag="BH5TK7DC2X" kind="definition">
# Caputo fractional derivative

## Definition

Let $a < b$ be real numbers and let $\alpha > 0$ be a real number that is not an integer.
Let $n = \ceil{\alpha}$.
Let $f \colon [a, b] \to \R$ have $n - 1$ continuous derivatives such that $f^{(n-1)}$ is absolutely continuous, so that $f^{(n)}$ exists almost everywhere and is integrable.
The *Caputo fractional derivative* of order $\alpha$ of $f$ is
$${}^{C}D_a^\alpha f(x) = \frac{1}{\Gamma(n - \alpha)} \int_a^x (x - t)^{n - \alpha - 1} f^{(n)}(t) \, dt, \qquad a < x \le b,$$
that is, the [[Riemann–Liouville integral|JTKCZ44NN3]] $I_a^{n - \alpha}$ applied to the [[higher derivative|04TKTF8KDK]] $f^{(n)}$.
Here $\Gamma$ is the [[gamma function|0H5SYBSED9]].
For an integer $\alpha = n$, one sets ${}^{C}D_a^n f = f^{(n)}$.

## Remarks

Example: let $a = 0$, $\alpha = \tfrac{1}{2}$, so $n = 1$, and $f(x) = x$.
Then $f' = 1$ and ${}^{C}D_0^{1/2} f(x) = \frac{1}{\Gamma(1/2)} \int_0^x (x - t)^{-1/2} \, dt = \frac{2\sqrt{x}}{\sqrt{\pi}}$.
For a constant $f = c$, $f^{(n)} = 0$ and the Caputo derivative is $0$.
Non-example: the Riemann–Liouville derivative $\frac{d}{dx} I_0^{1/2} 1 = \frac{1}{\sqrt{\pi x}}$ is not $0$, so it does not agree with the Caputo derivative on constants.
In general, under the hypotheses above, the Caputo derivative equals the Riemann–Liouville derivative of $f(x) - \sum_{k=0}^{n-1} \frac{f^{(k)}(a)}{k!} (x - a)^k$.

## Explanations

### L1 human

The Caputo derivative takes an ordinary derivative of whole order first, and then smooths with a fractional integral.
This keeps the familiar rule that constants have derivative zero.
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# Symmetric spectrum

## Definition

Let $S^1 = \Delta[1] / \partial \Delta[1]$ be the simplicial circle and $S^p = S^1 \wedge \cdots \wedge S^1$ ($p$ factors), with the [[symmetric group|0RJMNEN3BJ]] $\Sigma_p$ acting by permuting the factors.
A *symmetric spectrum* $X$ consists of:

1. for each $n \ge 0$, a pointed [[simplicial set|WW81ARB8K9]] $X_n$ with a base-point-preserving left action of $\Sigma_n$;
2. for each $n \ge 0$, a pointed map $\sigma_n \colon S^1 \wedge X_n \to X_{n+1}$;
3. such that for all $n, p \ge 0$ the iterated structure map $S^p \wedge X_n \to X_{n+p}$ is $\Sigma_p \times \Sigma_n$-equivariant, where $\Sigma_p \times \Sigma_n$ acts on $X_{n+p}$ through the inclusion $\Sigma_p \times \Sigma_n \subseteq \Sigma_{p+n}$.

A map $f \colon X \to Y$ of symmetric spectra is a sequence of $\Sigma_n$-equivariant pointed maps $f_n \colon X_n \to Y_n$ that commute with the structure maps.

## Remarks

Forgetting the actions gives a sequential [[spectrum|1GHF4EKBXS]], after geometric realization.
The same definition with [[pointed|T9DG8DDE75]] topological spaces in place of simplicial sets gives symmetric spectra of spaces.

Example: the *symmetric sphere spectrum* $\mathbb{S}$ has $\mathbb{S}_n = S^n$ with the permutation action of $\Sigma_n$.
Its structure maps are the identifications $S^1 \wedge S^n = S^{n+1}$.
The equivariance in clause 3 holds because both actions permute the same circle factors.

Non-example: on the sequence $X_n = S^n$ with the same structure maps, let every $\Sigma_n$ act as the identity.
Clauses 1 and 2 hold, but clause 3 fails for $p = 2$ and $n = 0$.
The map $S^2 \wedge S^0 \to X_2$ is the identity of $S^2$.
It does not carry the swap of the two circle factors, which is not the identity of $S^2$, to the identity action on $X_2$.

Symmetric spectra form a closed symmetric monoidal category under a smash product with unit $\mathbb{S}$.
They carry a stable model structure whose homotopy category is equivalent to the [[stable homotopy category|26GTNMAF7B]]; this is the theorem of Hovey, Shipley and Smith.
Monoids and commutative monoids in symmetric spectra model $A_\infty$ and $E_\infty$ ring spectra.
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<document source="https://tamnd.github.io/math2/node/BH7YPXCYDK.md" tag="BH7YPXCYDK" kind="concept">
# Percolation theory

## Summary

Percolation theory studies the connected clusters of a random subgraph of a large graph, where each edge or vertex is kept independently with probability $p$.
Its main question is whether an infinite cluster appears.
On many graphs there is a sharp [[percolation threshold|MP8VSZK19A]]: below it all clusters are finite, above it an infinite cluster exists.
It is a basic model of a phase transition and of flow through a porous medium.

## Explanations

### L1 human

Take a huge square grid of pipes and block each pipe at random, keeping it open with probability $p$.
Pour water at one point and ask whether it can flow arbitrarily far.
If $p$ is small, the water stays in a small region; if $p$ is close to $1$, it can reach infinity with positive probability.
The change happens suddenly at a critical value of $p$, which for this grid is $1/2$, by [[Kesten's theorem|B8Y0ZRQP17]].
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<document source="https://tamnd.github.io/math2/node/BH8R1M1KVF.md" tag="BH8R1M1KVF" kind="statement">
# Upper bound theorem

## Statement

Let $n > d \ge 2$, and let $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d$ with $n$ [[vertices|R7V3S9V891]].
Let $C(n, d)$ be the [[cyclic polytope|M6YBMXKJYD]] of dimension $d$ with $n$ vertices.
Then for every $i$ with $0 \le i \le d - 1$,
$$f_i(P) \le f_i(C(n, d)),$$
where $f_i$ is the number of $i$-dimensional faces, the $i$-th entry of the [[f-vector|872BAKDQQK]].

## Remarks

The theorem was proved by Peter McMullen in 1970, after Theodore Motzkin had conjectured it.
Proof idea: a small perturbation of the vertices makes $P$ simplicial without lowering any $f_i$, so one may assume $P$ simplicial.
Then one passes to the h-vector of the boundary, uses the [[Dehn–Sommerville equations|4WKHF5BCTA]] $h_i = h_{d-i}$, and proves $h_i \le \binom{n - d + i - 1}{i}$ for $i \le d/2$ with a shelling of the boundary.
Cyclic polytopes attain all these bounds, because they are neighborly.

The number of facets of $C(n, d)$ is
$$f_{d-1}(C(n,d)) = \frac{n}{n - m} \binom{n - m}{m} \text{ if } d = 2m, \qquad 2 \binom{n - m - 1}{m} \text{ if } d = 2m + 1.$$
For $d = 3$ this gives $2n - 4$, the number of triangles of any simplicial $3$-polytope with $n$ vertices.
For $d = 4$ and $n = 8$ it gives $8 \cdot 15 / 6 = 20$ facets.

By [[polar polytope|6DXF1T506B]] duality, the theorem also bounds the number of vertices of a $d$-polytope with $n$ facets.

## Explanations

### L1 human

Among all polytopes with a given dimension and a given number of corners, which one has the most faces?
The upper bound theorem says: the cyclic polytope, and it wins in every dimension of face at once.
So there is a single shape that is the champion for edges, for two-dimensional faces, and so on.

### L2 human

In the dual form, a polytope with $n$ facets in dimension $d$ has at most about $n^{\floor{d/2}}$ vertices.
This bounds the number of vertices that the simplex method or a vertex enumeration algorithm may meet, and the size of convex hulls in computational geometry.
The theorem was extended to simplicial spheres by Richard Stanley using commutative algebra.
Its lower-bound partner, for simplicial polytopes, is the lower bound theorem, and both are contained in the [[g-theorem|4CZMWQVC30]].
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# Cyclotomic field

## Definition

Let $n \ge 1$ be an integer and let $\zeta_n = e^{2\pi i/n} \in \C$, a [[primitive $n$-th root of unity|0EHM5ZP5QH]].
The *$n$-th cyclotomic field* is the subfield $\Q(\zeta_n)$ of $\C$ generated by $\zeta_n$ over $\Q$.
A *cyclotomic field* is a field of this form.

## Remarks

The minimal polynomial of $\zeta_n$ over $\Q$ is the [[cyclotomic polynomial|08KEYFAH66]] $\Phi_n$, which is irreducible over $\Q$ by the [[irreducibility of cyclotomic polynomials|0SK1VHZ5R2]].
So $[\Q(\zeta_n) : \Q] = \varphi(n)$, where $\varphi$ is [[Euler's totient function|0DAMHCT17R]], and $\Q(\zeta_n)$ is an [[algebraic number field|MWBDFS2KJ5]].
Example: for $n = 4$, $\zeta_4 = i$ and $\Phi_4 = x^2 + 1$, so $\Q(\zeta_4) = \Q(i)$ has degree $\varphi(4) = 2$.
Non-example: $\Q(\sqrt[3]{2})$ is not a cyclotomic field: every cyclotomic field is a Galois extension of $\Q$, but $\Q(\sqrt[3]{2}) \subset \R$ does not contain the non-real roots of $x^3 - 2$.
The Galois group of $\Q(\zeta_n)/\Q$ is isomorphic to $(\Z/n\Z)^{\times}$, with $a$ acting by $\zeta_n \mapsto \zeta_n^a$.
The ring of integers of $\Q(\zeta_n)$ is $\Z[\zeta_n]$.
For odd $n$, $\Q(\zeta_{2n}) = \Q(\zeta_n)$, because $-\zeta_n$ is a primitive $2n$-th root of unity.
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# Group von Neumann algebra

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] with a left [[Haar measure|VY0HXD3GC4]] $\mu$.
For $g \in G$, let $\lambda(g)$ be the unitary operator on $L^2(G, \mu)$ given by $(\lambda(g)\xi)(x) = \xi(g^{-1}x)$, the left regular representation.
The *group von Neumann algebra* of $G$ is
$$L(G) = \set{\lambda(g) : g \in G}'',$$
the double [[commutant|1VC7EJ1ZBZ]] of the set of operators $\lambda(g)$ in the bounded operators on $L^2(G)$.

## Remarks

By the von Neumann bicommutant theorem, $L(G)$ is the closure in the [[strong operator topology|KHJ2G2R5G7]] of the linear span of the operators $\lambda(g)$, and it is a [[von Neumann algebra|PSFWQVZ0FS]].

Example: $G = \Z$.
Fourier series identify $\ell^2(\Z)$ with $L^2(\mathbb{T})$, and $\lambda(n)$ with multiplication by $z^n$.
The commutant of these multiplications is the algebra of multiplications by functions in $L^\infty(\mathbb{T})$, which is its own commutant, so $L(\Z) \cong L^\infty(\mathbb{T})$.

Non-example: the reduced group C\*-algebra $C^*_r(\Z) \cong C(\mathbb{T})$, acting by multiplication on $L^2(\mathbb{T})$, is not a von Neumann algebra.
Its double commutant is $L^\infty(\mathbb{T})$, which is strictly larger than $C(\mathbb{T})$.

For a discrete group $G$, the vector $\delta_e$ gives a faithful normal trace $\tau(a) = \ip{a\delta_e}{\delta_e}$ on $L(G)$.
If $G$ is a nontrivial discrete group and every conjugacy class of $G$ other than $\set{e}$ is infinite, then $L(G)$ is a factor of type $\mathrm{II}_1$.
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# Multijet transversality theorem

## Statement

Let $X$ and $Y$ be [[smooth manifolds|F45VQ47GPV]], and let $s \ge 1$ and $k \ge 0$ be integers.
Let $X^{(s)} = \set{(x_1, \dots, x_s) \in X^s : x_i \ne x_j \text{ for } i \ne j}$.
Let $J^k(X, Y)$ be the [[bundle of $k$-jets|HC3VBBCJQP]] of smooth maps from $X$ to $Y$, with source map $\alpha \colon J^k(X, Y) \to X$.
The $s$-fold *multijet bundle* is
$$J^k_s(X, Y) = \set{(\sigma_1, \dots, \sigma_s) \in J^k(X, Y)^s : (\alpha(\sigma_1), \dots, \alpha(\sigma_s)) \in X^{(s)}}.$$
For a smooth map $f \colon X \to Y$, its multijet extension is $j^k_s f \colon X^{(s)} \to J^k_s(X, Y)$, $(x_1, \dots, x_s) \mapsto (j^k f(x_1), \dots, j^k f(x_s))$, where $j^k f(x)$ is the [[$k$-jet|1QE0Z0756D]] of $f$ at $x$.
Let $W$ be a [[submanifold|VKFW27WXEZ]] of $J^k_s(X, Y)$, and give $C^\infty(X, Y)$ the [[Whitney $C^\infty$ topology|7PN9MD1PQB]].
Then the set of $f \in C^\infty(X, Y)$ such that $j^k_s f$ is [[transverse|A2PDZ8GA9C]] to $W$ is a [[residual set|GRH8MCVN3N]] in $C^\infty(X, Y)$.

## Remarks

For $s = 1$ this is the [[Thom transversality theorem|KWNRS3RA09]].
The proof follows the same pattern: one perturbs $f$ near the $s$ distinct points $x_1, \dots, x_s$ independently.
The perturbations are polynomial maps of degree at most $k$, multiplied by bump functions with disjoint supports.
The family of perturbed multijets is a submersion onto the fibres, so the parametric transversality theorem applies, and a countable covering argument gives a residual set.
The points must be distinct, since the jets of $f$ at the same point cannot be perturbed independently.

Example: let $s = 2$, $k = 0$, and let $W = \set{((x_1, y_1), (x_2, y_2)) : y_1 = y_2}$, which has codimension $\dim Y$ in $J^0_2(X, Y) = X^{(2)} \times Y \times Y$.
If $\dim Y > 2 \dim X = \dim X^{(2)}$, transversality to $W$ means that $j^0_2 f$ misses $W$, that is, $f$ is injective.
So injective maps are generic when $\dim Y > 2 \dim X$.
Combined with the genericity of immersions, this gives the easy form of the Whitney embedding theorem for compact $X$.

## Explanations

### L1 human

Thom's theorem says that a generic map behaves well at each single point.
The multijet version says that a generic map also behaves well at several points at once.
For example, two different points are not sent to the same place in a degenerate way.

This controls self-intersections of maps, not only their local shape.

### L2 human

The theorem is the main tool for global properties of generic maps.
It shows that generic immersions have only transverse self-intersections, that is, normal crossings, and it is a basic tool in the theory of stable maps.
</document>

<document source="https://tamnd.github.io/math2/node/BJ314H17HC.md" tag="BJ314H17HC" kind="concept">
# Outer measure

## Summary

An outer measure assigns a size in $[0, \infty]$ to every subset of a set, not only to the sets of a $\sigma$-algebra.
It is monotone and countably subadditive, but in general not additive.
Through [[Carathéodory's criterion|JDGQ135YK2]] every outer measure produces a $\sigma$-algebra on which it is a complete [[measure|PR11BZQQPG]].
This is the standard way to build Lebesgue measure and Hausdorff measures.

## Explanations

### L1 human

To measure an odd-shaped set, cover it by simple pieces whose size is known, and take the smallest total size of such a cover.
This over-estimate is the outer measure.
It can be applied to any set at all, but for very wild sets it is not additive.
The union of two disjoint sets can have an outer measure smaller than the sum of their outer measures.
For example, Lebesgue outer measure covers a subset of $\R$ by countably many intervals and takes the infimum of the total length.
Every countable set gets outer measure $0$, because the $n$-th point can be covered by an interval of length $\varepsilon / 2^n$.
</document>

<document source="https://tamnd.github.io/math2/node/BJ33Y5H49M.md" tag="BJ33Y5H49M" kind="definition">
# Copula

## Definition

Let $d \ge 2$.
A *$d$-dimensional copula* is a function $C \colon [0, 1]^d \to [0, 1]$ such that there is a random vector $U = (U_1, \ldots, U_d)$ with:

1. each $U_i$ [[uniformly distributed|0AREFQ9Z65]] on $[0, 1]$, and
2. $C(u_1, \ldots, u_d) = P(U_1 \le u_1, \ldots, U_d \le u_d)$ for all $(u_1, \ldots, u_d) \in [0, 1]^d$.

So a copula is the restriction to $[0, 1]^d$ of the joint [[cumulative distribution function|0H8H7ZTHW2]] of a random vector with uniform [[marginals|0MXT6PRZZ6]].

## Remarks

Example: if $U_1, \ldots, U_d$ are independent and uniform on $[0, 1]$, then $C(u) = u_1 u_2 \cdots u_d$, the independence copula.
Example: if $U_1 = U_2 = U$ with $U$ uniform, then $C(u_1, u_2) = P(U \le \min(u_1, u_2)) = \min(u_1, u_2)$.
Non-example: $H(u_1, u_2) = u_1^2 u_2^2$ on $[0, 1]^2$ is the joint distribution function of two independent variables with distribution function $t \mapsto t^2$; its marginals are not uniform, since $H(u_1, 1) = u_1^2$, so it is not a copula.
Equivalently, $C$ is a copula when it is grounded ($C(u) = 0$ if some $u_i = 0$), satisfies $C(1, \ldots, 1, u_i, 1, \ldots, 1) = u_i$, and is $d$-increasing (it gives non-negative mass to every box).
Every copula satisfies the Fréchet–Hoeffding bounds $\max(u_1 + \cdots + u_d - d + 1, 0) \le C(u) \le \min(u_1, \ldots, u_d)$.
</document>

<document source="https://tamnd.github.io/math2/node/BJE8P9TH88.md" tag="BJE8P9TH88" kind="statement">
# Divide-and-conquer eigenvalue algorithm

## Statement

Let $T \in \R^{n \times n}$ be a symmetric [[tridiagonal matrix|DARX1KZRHZ]] with diagonal entries $a_1, \dots, a_n$ and off-diagonal entries $b_1, \dots, b_{n-1}$, and let $1 \le k < n$ with $\beta = b_k$.

1. (Splitting.) Let $T_1$ be the leading $k \times k$ block of $T$ with $a_k$ replaced by $a_k - \beta$, and $T_2$ the trailing $(n-k) \times (n-k)$ block with $a_{k+1}$ replaced by $a_{k+1} - \beta$. With $u = e_k + e_{k+1}$,
$$T = \begin{pmatrix} T_1 & 0 \\ 0 & T_2 \end{pmatrix} + \beta \, u u^T.$$
If $T_i = Q_i D_i Q_i^T$ with $Q_i$ [[orthogonal|0W02QESPJQ]] and $D_i$ [[diagonal|044JNC7ZAA]], then $T = Q (D + \beta z z^T) Q^T$ with $Q = \mathrm{diag}(Q_1, Q_2)$, $D = \mathrm{diag}(D_1, D_2)$ and $z = Q^T u$.
2. (Secular equation.) Let $D = \mathrm{diag}(d_1, \dots, d_n)$ with $d_1 < d_2 < \cdots < d_n$, let $z \in \R^n$ have all entries nonzero, and let $\beta > 0$. Then the [[eigenvalues|0BQAA2BK3Q]] of $D + \beta z z^T$ are the $n$ roots of
$$f(\lambda) = 1 + \beta \sum_{i=1}^{n} \frac{z_i^2}{d_i - \lambda},$$
they satisfy $d_1 < \lambda_1 < d_2 < \lambda_2 < \cdots < d_n < \lambda_n$, and $(D - \lambda_j I)^{-1} z$ is an [[eigenvector|0N6960W0BV]] for $\lambda_j$.

## Remarks

Part 1 is a direct check of entries: $u u^T$ has ones exactly in positions $(k, k)$, $(k, k+1)$, $(k+1, k)$, $(k+1, k+1)$.
For part 2, if $(D + \beta z z^T) x = \lambda x$ and $\lambda$ is not a $d_i$, then $x = -\beta (z^T x)(D - \lambda I)^{-1} z$ with $z^T x \ne 0$, and taking the inner product with $z$ gives $f(\lambda) = 0$.
Since $f'(\lambda) = \beta \sum_i z_i^2/(d_i - \lambda)^2 > 0$ and $f$ goes from $-\infty$ to $+\infty$ between consecutive poles, and from $-\infty$ to $1$ after $d_n$, there is exactly one root in each of these $n$ intervals.
For $\beta < 0$ the interlacing is reversed: $\lambda_1 < d_1 < \lambda_2 < \cdots < \lambda_n < d_n$.
If some $z_i = 0$ or some $d_i$ coincide, then some $d_i$ is already an eigenvalue and the problem is reduced in size first; this step is called deflation.

The algorithm applies the splitting recursively, solves the small problems directly, and merges with the secular equation.
Each root is found by a safeguarded Newton-type iteration on $f$ in its interval.
Computing the eigenvectors in a numerically orthogonal way needs a careful recomputation of $z$, due to Gu and Eisenstat.

## Explanations

### L1 human

To find the eigenvalues of a big symmetric tridiagonal matrix, cut it into two halves at one entry.
Solve the two halves, possibly by cutting them again, and then glue the answers.
Gluing changes the matrix by a simple rank-one piece.
The new eigenvalues are the roots of one scalar equation, with exactly one root between each pair of neighboring old eigenvalues.

### L2 human

Symmetric eigenvalue problems are first reduced to tridiagonal form by orthogonal transformations, so this algorithm handles the core step.
It is among the fastest methods when all eigenvalues and eigenvectors are wanted, because deflation is frequent in practice and the merges use matrix multiplication.
The interlacing in part 2 is a rank-one analogue of the [[Cauchy interlacing theorem|8QQPVNR80F]].
The method was proposed by Cuppen in 1981.
</document>

<document source="https://tamnd.github.io/math2/node/BJMEXPKRYM.md" tag="BJMEXPKRYM" kind="definition">
# Perfect ring

## Definition

Let $R$ be a [[ring|0894J980RK]] with identity and $J = J(R)$ its [[Jacobson radical|XA0DCNH7WY]].
A subset $S \subseteq R$ is *left T-nilpotent* if for every sequence $a_1, a_2, a_3, \ldots$ of elements of $S$ there is $n \ge 1$ with $a_1 a_2 \cdots a_n = 0$.
$R$ is *left perfect* if the following hold.

1. $R$ is [[semilocal|YH75M7XAM1]], that is, $R/J$ is a semisimple ring.
2. $J$ is left T-nilpotent.

*Right perfect* rings are defined with *right T-nilpotent*, which asks for $a_n \cdots a_2 a_1 = 0$ instead.

## Remarks

Example: a left Artinian ring is left and right perfect: $R/J$ is semisimple, and $J^n = 0$ for some $n$, so every product of $n$ elements of $J$ is $0$.
Non-example: $\Z_{(p)}$ is semilocal, but $J = p\Z_{(p)}$ is not left T-nilpotent, since the constant sequence $p, p, p, \ldots$ has all products $p^n \ne 0$.
By [[Bass's theorem P|D2M8BSW720]], $R$ is left perfect if and only if every left $R$-module has a projective cover.
This is also equivalent to the descending chain condition on principal right ideals.
Many authors take one of these equivalent conditions as the definition.
</document>

<document source="https://tamnd.github.io/math2/node/BJQTEPEZ9G.md" tag="BJQTEPEZ9G" kind="concept">
# Asymptotic expansion

## Summary

An asymptotic expansion of a function $f$ near a point is a formal series $\sum a_n \varphi_n$ in a scale of functions that shrink one after another.
Every partial sum must approximate $f$ with an error smaller than its last term.
The series need not converge.
Asymptotic expansions describe special functions, integrals and solutions of differential equations for large or small parameters.
They are closely tied to the theory of [[divergent series|0NBRP6093V]] and to [[Borel summation|8DJPSYSHGJ]].

## Explanations

### L1 human

Think of an approximation that gets better in stages: first a rough main term, then a correction, then a smaller correction.
An asymptotic expansion lists all these stages at once.
Each stage is guaranteed to help when the variable is close enough to the limit point, even if adding infinitely many stages does not converge.
For example, as $x \to \infty$, the function $\frac{1}{x - 1}$ has the expansion $\frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + \cdots$.
The [[Stirling series|K29QYWB6A5]] for $\ln \Gamma(z)$ is an expansion of this kind that diverges for every $z$ and is still very accurate.
</document>

<document source="https://tamnd.github.io/math2/node/BJS1CGTKQ5.md" tag="BJS1CGTKQ5" kind="concept">
# Colimit

## Summary

The colimit of a [[diagram|HJK75HDWFE]] in a [[category|A2DE66937D]] is a universal cocone under it.
It maps to every other cocone in exactly one compatible way.
It is the notion dual to a [[limit|CHS5AFJGWE]], and it covers coproducts, coequalizers, pushouts, direct limits and initial objects.
Colimits glue objects together, while limits cut out compatible families.

## Explanations

### L1 human

The colimit of a diagram is the most economical object that receives a map from every part of the diagram in a consistent way.
In the category of sets, put all the sets of the diagram side by side.
Then glue each element to its images under the arrows of the diagram.
For two sets with no arrows between them, nothing is glued, and the colimit is their disjoint union.
For an increasing chain of sets $X_0 \subseteq X_1 \subseteq \cdots$, the colimit is the union of the chain.
</document>

<document source="https://tamnd.github.io/math2/node/BJVRFFJTV6.md" tag="BJVRFFJTV6" kind="concept">
# Vietoris topology

## Summary

The Vietoris topology is a topology on a set of subsets of a topological space, usually the closed subsets or the compact subsets.
A set is near a set $A$ when it lies inside a given open neighbourhood of $A$.
It must also meet some given open sets that $A$ meets.
It turns a family of subsets into a space, called a hyperspace.
On the nonempty compact subsets of a metric space, it is the topology of the Hausdorff metric.

## Explanations

### L1 human

Sometimes the objects you want to move around are sets, not points.
The Vietoris topology says when two sets are close.
A set $B$ is close to $A$ when $B$ stays inside a thin open region around $A$, and $B$ reaches every part of $A$ you chose to watch.
For example, in the plane, a fine grid of points inside a disc is close to the disc in this sense.
</document>

<document source="https://tamnd.github.io/math2/node/BJVZXPB297.md" tag="BJVZXPB297" kind="concept">
# Floquet theory

## Summary

Floquet theory studies linear systems $x' = A(t)\, x$ whose coefficient matrix $A(t)$ is periodic in time.
Its main result is that every [[fundamental matrix|0H1GN6P9R5]] is a periodic matrix times a matrix exponential $e^{tB}$.
So a periodic change of variables turns the system into one with constant coefficients.
The behaviour of solutions, such as stability, is read off from the eigenvalues of the monodromy matrix, the [[Floquet multipliers|FPS7YFGX32]].
It applies to the [[Hill equation|YE7H8DGDPY]], to the stability of periodic orbits and to waves in periodic media.
See the [[definition|GW8SBF37K1]].

## Explanations

### L1 human

A child on a swing who pumps the legs periodically changes the equation of motion periodically in time.
Floquet theory answers whether such a periodically forced linear system has growing or bounded motion.
The key is to look at the system once per period, like taking a photo every period.
The map from one photo to the next is a fixed matrix, the monodromy matrix, and its eigenvalues decide everything.
</document>

<document source="https://tamnd.github.io/math2/node/BJX6QS0ANT.md" tag="BJX6QS0ANT" kind="concept">
# Euler class

## Summary

The Euler class $e(E) \in H^n(B; \Z)$ is a [[characteristic class|7HACJC0JE2]] of oriented real vector bundles $E \to B$ of rank $n$.
It is the first obstruction to a nowhere-zero section: if $E$ has a section that never vanishes, then $e(E) = 0$.
For the tangent bundle of a closed oriented manifold, its value on the [[fundamental class|ZEA0PEM3B2]] is the [[Euler characteristic|RNDDWDAFJ5]].
It appears in the [[Gysin sequence|QJSPWM9F9P]] and is defined from the Thom class of the [[Thom isomorphism|N685RXHDZX]].

## Explanations

### L1 human

The Euler class counts, with signs, the zeros that a section of a bundle is forced to have.
For the tangent bundle of a surface, a section is a field of arrows, and the count is the Euler characteristic of the surface.
On the sphere the count is $2$, so every field of arrows on the sphere has a zero somewhere; this is the hairy ball theorem.
On the torus the count is $0$, and indeed the torus carries a field of arrows with no zeros.
</document>

<document source="https://tamnd.github.io/math2/node/BK4DDS24XC.md" tag="BK4DDS24XC" kind="concept">
# Kleene algebra

## Summary

A Kleene algebra is an [[idempotent semiring|JCSAC9Y1CV]] with a star operation $a^*$ that behaves like "any number of repetitions of $a$".
The model is the algebra of formal languages, with union, concatenation and the Kleene star.
Kleene algebras give an equational theory of regular expressions, and they are used to reason about programs and automata.
Binary relations on a set, with union, composition and reflexive transitive closure, also form a Kleene algebra.

## Explanations

### L1 human

A regular expression such as $(ab)^*c$ describes strings: any number of copies of $ab$, followed by $c$.
Its building blocks are choice ($+$), sequencing (writing one thing after another) and repetition ($^*$).
A Kleene algebra keeps the laws that these three operations obey, such as $a^* = 1 + a a^*$, where $1$ is the empty string.
With these laws one can prove that two regular expressions describe the same strings without listing the strings.
</document>

<document source="https://tamnd.github.io/math2/node/BK9WBMF56G.md" tag="BK9WBMF56G" kind="concept">
# Tate–Shafarevich group

## Summary

Let $E$ be an [[elliptic curve|427N6S6QQW]] over a [[number field|MWBDFS2KJ5]] $K$.
Its Tate–Shafarevich group $\text{Ш}(E/K)$ measures the failure of the [[Hasse principle|EMAR4X3W8D]] for twisted forms of $E$.
Its elements are classes of genus one curves that become isomorphic to $E$ over an algebraic closure.
These curves have points over every completion of $K$, and the trivial class is that of curves with a point over $K$.
It is conjectured to be finite, and its order enters the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].

## Explanations

### L1 human

To find rational points on a curve, one first checks whether the curve has points over the real numbers and over every $p$-adic field.
For some curves of genus one, all these local tests succeed, and yet there is no rational point.
The Tate–Shafarevich group collects exactly these deceptive curves, attached to a given elliptic curve.
A classical example is Selmer's curve $3x^3 + 4y^3 + 5z^3 = 0$: it has points over $\R$ and every $\Q_p$, but no rational point other than $(0,0,0)$.
When the group is trivial, local information is enough to decide such questions for that elliptic curve.
</document>

<document source="https://tamnd.github.io/math2/node/BKKAA5JKT0.md" tag="BKKAA5JKT0" kind="concept">
# Spencer cohomology

## Summary

Spencer cohomology is a family of cohomology groups attached to the symbol of a system of linear or nonlinear [[partial differential equations|GC5BDZ6QBC]].
It is computed from the Spencer complex, an algebraic analogue of the de Rham complex built from polynomials and [[exterior forms|0MDMZ965VE]].
It measures the obstructions to solving a system order by order in formal power series.
It also gives an algebraic criterion for involutivity in the sense of Cartan.
The name also refers to a related sheaf cohomology, introduced by D. C. Spencer.
It is the cohomology of a manifold with coefficients in the sheaf of solutions of a linear differential operator.

## Explanations

### L1 human

When one tries to solve a system of partial differential equations by power series, one finds the coefficients degree by degree.
At each degree some equations may contradict each other or may force new hidden equations.
Spencer cohomology is the bookkeeping device that records these problems in the language of linear algebra.
When the relevant groups vanish, the order-by-order construction never gets stuck.
</document>

<document source="https://tamnd.github.io/math2/node/BKM0TQSCD9.md" tag="BKM0TQSCD9" kind="concept">
# Multiplier ideal

## Summary

The multiplier ideal of a [[plurisubharmonic function|FF2P7ER3QT]] $\varphi$ is the sheaf of holomorphic germs $f$ for which $\abs{f}^2 e^{-2\varphi}$ is locally integrable.
It measures the singularities of $\varphi$: the more singular $\varphi$ is, the smaller the ideal.
Nadel proved that it is a [[coherent analytic sheaf|DSJET02WPR]].
The Nadel vanishing theorem, a consequence of [[Hörmander's L2 estimates|TPSH4ETK67]], makes it a basic tool in complex and algebraic geometry.
In algebraic geometry, the multiplier ideal of an ideal sheaf with a rational exponent is defined through a log resolution.
It agrees with the analytic definition.

## Explanations

### L1 human

Take a function $\varphi$ that tends to $-\infty$ at some points, such as $\varphi = c \log\abs{z}$ near $0$ in $\C$.
Then $e^{-2\varphi} = \abs{z}^{-2c}$ blows up at $0$, and may fail to be integrable.
Multiplying by a holomorphic $f$ that vanishes at $0$ can tame the blow-up.
The multiplier ideal is the set of such "multipliers" $f$ that make $\abs{f}^2 e^{-2\varphi}$ integrable.
For $\varphi = c \log\abs{z}$ with $c > 0$, these are the germs that vanish to order at least $\floor{c}$ at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/BKXYBS57T9.md" tag="BKXYBS57T9" kind="concept">
# Dynamic equation on time scales

## Summary

A dynamic equation on a time scale is an equation that involves the delta derivative of an unknown function.
The function lives on a time scale $\mathbb{T}$, a nonempty closed subset of $\R$.
It is the basic object of [[time-scale calculus|30G1M35TK1]].
On $\mathbb{T} = \R$ it is an [[ordinary differential equation|0NT64JV96A]], and on $\mathbb{T} = \Z$ it is a [[recurrence relation|0FYX44GP92]], so one theory covers both cases at once.

## Explanations

### L1 human

Some quantities change continuously, others change in steps, and some do both.
A time scale is the set of times at which the quantity lives, for example all reals, all integers, or a union of intervals.
A dynamic equation tells how the quantity changes from each time to the next one, using a derivative where time flows and a difference quotient across each gap.
For example, the equation $x^\Delta = x$ with $x(0) = 1$ has the solution $e^t$ on $\R$ but $2^t$ on $\Z$, since on $\Z$ it says $x(t + 1) - x(t) = x(t)$.
</document>

<document source="https://tamnd.github.io/math2/node/BKZNC76RC8.md" tag="BKZNC76RC8" kind="concept">
# Spherical function

## Summary

A spherical function for a [[Gelfand pair|W38ZDXDVYC]] $(G, K)$ is a continuous function on $G$, invariant under $K$ on both sides.
It satisfies a product formula with an average over $K$.
Spherical functions are exactly the multiplicative functionals on the commutative algebra of $K$-bi-invariant functions.
They give the spherical transform, the analogue of the Fourier transform on the homogeneous space $G/K$.
Positive-definite spherical functions correspond to the irreducible unitary representations of $G$ with a nonzero $K$-fixed vector.

## Explanations

### L1 human

On the sphere, a zonal function depends only on the angle from the north pole.
Among such functions, the Legendre polynomials $P_\ell(\cos \theta)$ are special: averaging them over rotations behaves like multiplying exponentials.
These are the spherical functions of the pair of the rotation group and the rotations around one axis.
On the real line, with the trivial subgroup, the spherical functions are the exponentials $e^{s x}$.
They are the building blocks for analysis on spaces with a lot of symmetry.
</document>

<document source="https://tamnd.github.io/math2/node/BM3TWJ0D4P.md" tag="BM3TWJ0D4P" kind="statement">
# Langlands decomposition

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center, with an [[Iwasawa decomposition|4YDR9Q36DS]] $G = KAN$.
Let $\mathfrak{a}$ be the Lie algebra of $A$, and let $\Sigma^+$ be the positive restricted roots used to define $N$.
Let $\Pi \subseteq \Sigma^+$ be the simple ones.
Fix a subset $F \subseteq \Pi$ and let $\Sigma_F$ be the set of restricted roots in the linear span of $F$.
Put $\mathfrak{a}_F = \set{H \in \mathfrak{a} : \alpha(H) = 0 \text{ for all } \alpha \in F}$ and $\mathfrak{n}_F = \bigoplus_{\lambda \in \Sigma^+ \setminus \Sigma_F} \mathfrak{g}_\lambda$.
Let $A_F = \exp \mathfrak{a}_F$, let $N_F = \exp \mathfrak{n}_F$, and let $L_F = Z_G(\mathfrak{a}_F)$ be the centralizer of $\mathfrak{a}_F$ under the adjoint action.
Let $M_F$ be the set of $x \in L_F$ with $\abs{\chi(x)} = 1$ for every continuous homomorphism $\chi \colon L_F \to \R^\times$.
Then $P_F = L_F N_F$ is a [[parabolic subgroup|Z2FEBW5MQM]] of $G$, and:

1. $M_F$ and $A_F$ are closed subgroups that commute with each other, $M_F \cap A_F = \set{e}$, and $L_F = M_F A_F$;
2. $N_F$ is a closed, simply connected, [[nilpotent|DK87YN458D]] normal subgroup of $P_F$;
3. the map $M_F \times A_F \times N_F \to P_F$, $(m, a, n) \mapsto man$, is a diffeomorphism.

The factorization $P_F = M_F A_F N_F$ is the *Langlands decomposition* of $P_F$.

## Remarks

Every parabolic subgroup that contains $MAN$ equals $P_F$ for exactly one $F$, and every parabolic subgroup is conjugate to such a $P_F$.
The group $L_F = M_F A_F$ is a Levi factor of $P_F$; it is reductive, and $A_F$ is a vector group, isomorphic to $\R^k$ with $k = \abs{\Pi \setminus F}$.

For $F = \emptyset$, the group $M_\emptyset$ is $M = Z_K(\mathfrak{a})$ and $P_\emptyset = MAN$ is the minimal parabolic subgroup.
For $F = \Pi$, one gets $\mathfrak{a}_F = 0$, $N_F = \set{e}$ and $P_F = M_F = G$, since a semisimple group has no nontrivial continuous homomorphism to $\R_{>0}$.

Example: in $\mathrm{SL}_n(\R)$, a subset $F$ corresponds to block sizes $n_1 + \cdots + n_r = n$.
Then $P_F$ is the group of block upper triangular matrices, and $N_F$ is the group of block unipotent ones.
The group $A_F$ consists of the matrices $\operatorname{diag}(t_1 I_{n_1}, \dots, t_r I_{n_r})$ with $t_i > 0$ and determinant $1$.
The group $M_F$ consists of the block diagonal matrices of determinant $1$ whose blocks have determinant $\pm 1$.

## Explanations

### L1 human

A parabolic subgroup splits into three pieces in a unique way.
One piece $M$ is a smaller reductive group, similar in kind to $G$.
One piece $A$ is a group of scalings, a copy of $\R^k$.
One piece $N$ is a group of shears, and it is normal.
For block upper triangular matrices, $M$ and $A$ together give the diagonal blocks, and $N$ gives the part above the blocks.

### L2 human

The decomposition is the basis of parabolic induction.
A representation $\sigma$ of $M_F$ and a character $\nu$ of $A_F$ give a representation $\sigma \otimes \nu \otimes 1$ of $P_F$, which one induces to $G$.
The principal series come from $F = \emptyset$, and Langlands used general $F$ to classify the irreducible admissible representations of $G$.
The groups $M_F$ are again reductive groups, which makes induction on the size of the group possible.
</document>

<document source="https://tamnd.github.io/math2/node/BM4VXG92DT.md" tag="BM4VXG92DT" kind="example">
# The cosets of 3Z in the integers

## Example

Let $G = (\mathbb{Z}, +)$ be the [[integers|0C4JS9RF5M]] with addition, and let $H = 3\mathbb{Z}$ be the multiples of $3$.
The [[cosets|05FD317V2W]] of $H$ are $3\mathbb{Z}$, $1 + 3\mathbb{Z}$ and $2 + 3\mathbb{Z}$.
Each integer is in exactly one of them, given by its remainder after division by $3$.
The group is abelian, so each left coset is also a right coset.

## Remarks

The three cosets are the elements of the group [[$\mathbb{Z}/3\mathbb{Z}$|03VFZ2WETT]].
</document>

<document source="https://tamnd.github.io/math2/node/BM703GV34G.md" tag="BM703GV34G" kind="concept">
# Multivariate interpolation

## Summary

Multivariate interpolation is the construction of a function of several variables that takes given values at given points of $\R^d$ with $d \ge 2$.
In one variable, polynomials of degree at most $n$ always interpolate uniquely at $n + 1$ distinct points.
In several variables, this depends on the position of the points, see [[unisolvent point set|5276MDSDQV]].
For gridded data one uses tensor products of one-dimensional methods, such as bilinear and bicubic interpolation.
For scattered data one uses methods adapted to the points, such as [[radial basis function interpolation|XZ33P3HBB9]], [[inverse distance weighting|QHN67MSH8S]], [[moving least squares|T0JZN6WP5X]] and [[kriging|VP9S3PNJQ0]].

## Explanations

### L1 human

Imagine temperature readings taken at weather stations scattered over a country.
Multivariate interpolation fills in a temperature map that agrees with the readings at the stations.
On a regular grid one can interpolate first along rows and then along columns.
For example, bilinear interpolation on the unit square with corner values $f_{00}, f_{10}, f_{01}, f_{11}$ gives $f_{00}(1 - x)(1 - y) + f_{10} x (1 - y) + f_{01} (1 - x) y + f_{11} x y$.
For scattered stations other methods are needed.
The [[definition|C4B5FB6A34]] states the problem.
</document>

<document source="https://tamnd.github.io/math2/node/BMCCQ8JXNE.md" tag="BMCCQ8JXNE" kind="concept">
# Prolongation of a section to a jet bundle

## Summary

Let $s$ be a section of a fiber bundle.
Its $k$-th prolongation is the section $j^k s$ of the [[jet bundle|HC3VBBCJQP]] that sends each point $p$ to the [[$k$-jet|1QE0Z0756D]] of $s$ at $p$.
It lifts a section to a section that also records all derivatives up to order $k$.
A section of the jet bundle that arises in this way is called holonomic.
Prolongations let one write a differential equation as the condition that $j^k s$ takes values in a given subset of the jet bundle.

## Explanations

### L1 human

Take a function $u = s(x)$ of one variable.
Its first prolongation is the curve $x \mapsto (x, s(x), s'(x))$ in the space with coordinates $(x, u, u_x)$.
For $s(x) = x^2$ this curve is $x \mapsto (x, x^2, 2x)$.
Not every curve $x \mapsto (x, u(x), p(x))$ in that space is a prolongation: the third coordinate must be the derivative of the second.
That is the whole point of prolongation: it ties the slope coordinate to the actual slope.
</document>

<document source="https://tamnd.github.io/math2/node/BMD35Z5SGC.md" tag="BMD35Z5SGC" kind="proof">
# Proof that the order of an element divides the order of the group

## Proof

1. Let $G$ be a finite group and let $g \in G$.
2. The [[cyclic subgroup|0YT586TKXF]] $\langle g \rangle$ is a subgroup of $G$.
3. [[The order of $\langle g \rangle$ is the order of $g$|031M8HX7DA]].
4. By [[Lagrange's theorem|3C53TD]], $|\langle g \rangle|$ divides $|G|$.
5. Thus the order of $g$ divides $|G|$.

## Idea

The element generates a subgroup whose order is the order of the element. Lagrange's theorem then gives the result.
</document>

<document source="https://tamnd.github.io/math2/node/BMD7KQQA9C.md" tag="BMD7KQQA9C" kind="definition">
# Regular singular point

## Definition

Let $p_1, \ldots, p_n$ be meromorphic functions near $z_0 \in \C$, and consider the [[linear equation|0QQRP17Z7S]]
$$y^{(n)} + p_1(z)\, y^{(n-1)} + \cdots + p_n(z)\, y = 0.$$
The point $z_0$ is a *regular singular point* of the equation if:

1. $z_0$ is a [[singular point|QK2J038VB4]] of the equation, and
2. for each $k = 1, \ldots, n$, the function $(z - z_0)^k\, p_k(z)$ is [[analytic|081WKEDK6C]] at $z_0$.

The point $\infty$ is a regular singular point if $w = 0$ is one for the equation obtained by the substitution $z = 1/w$.

## Remarks

For $n = 2$, written $y'' + p(z) y' + q(z) y = 0$, condition 2 says that $(z - z_0) p(z)$ and $(z - z_0)^2 q(z)$ are analytic at $z_0$.
For $n = 2$, the point $\infty$ is ordinary or regular singular if and only if $z\, p(z)$ and $z^2 q(z)$ are analytic at $\infty$.

Example: the [[Cauchy–Euler equation|03F7N3915D]] $z^2 y'' + a z y' + b y = 0$ with $(a, b) \ne (0, 0)$ has $p = \frac{a}{z}$ and $q = \frac{b}{z^2}$.
So $0$ is singular, while $z p = a$ and $z^2 q = b$ are constant; hence $0$ is a regular singular point.

Non-example: for $z^2 y'' + y' = 0$, one has $p = \frac{1}{z^2}$, and $z p = \frac{1}{z}$ is not analytic at $0$.
So $0$ is singular but not regular singular; it is an irregular singular point.

Fuchs proved a characterisation by growth.
A singular point is regular singular if and only if every solution grows at most like a power of $\abs{z - z_0}^{-1}$ as $z \to z_0$ in each sector with vertex $z_0$.
</document>

<document source="https://tamnd.github.io/math2/node/BMDG4JQPEH.md" tag="BMDG4JQPEH" kind="object">
# Algebra of bounded operators on a Hilbert space

## Description

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] with inner product $\ip{\cdot}{\cdot}$.
Let $B(H)$ be the set of bounded linear operators $T \colon H \to H$, with pointwise addition and scalar multiplication, composition as product, and the [[operator norm|06EKW1AKXF]]
$$\norm{T} = \sup \set{\norm{T x} : x \in H, \norm{x} \le 1}.$$
The involution is the [[adjoint|6HSJBJ8TTQ]] $T \mapsto T^*$, defined by $\ip{T x}{y} = \ip{x}{T^* y}$ for all $x, y \in H$.
With these operations $B(H)$ is a C*-algebra with unit the identity operator.

Check: $B(H)$ is complete in the operator norm because $H$ is complete, and $\norm{S T} \le \norm{S} \norm{T}$, so it is a Banach algebra.
The adjoint is conjugate linear, satisfies $(S T)^* = T^* S^*$ and $T^{**} = T$, and $\norm{T^*} = \norm{T}$.
For the C* identity, $\norm{T x}^2 = \ip{T^* T x}{x} \le \norm{T^* T} \norm{x}^2$ gives $\norm{T}^2 \le \norm{T^* T}$, and $\norm{T^* T} \le \norm{T^*} \norm{T} = \norm{T}^2$ gives the reverse inequality.

## Remarks

If $H = \C^n$, then $B(H)$ is the algebra $M_n(\C)$ of $n \times n$ complex matrices, and $T^*$ is the conjugate transpose.
By the Gelfand–Naimark theorem every C*-algebra is isometrically $*$-isomorphic to a norm-closed $*$-subalgebra of some $B(H)$.
The algebra $B(H)$ is also a von Neumann algebra: it is closed in the weak operator topology, and its commutant consists of the scalar multiples of the identity.
If $H$ is infinite-dimensional, the compact operators form a proper closed two-sided ideal of $B(H)$, and when $H$ is separable it is the only one other than $0$ and $B(H)$.

## Explanations

### L1 human

The bounded operators on a Hilbert space are the infinite-dimensional version of square matrices.
They can be added, multiplied by composition, and they have an adjoint, the analogue of the conjugate transpose.
The size of an operator is the largest factor by which it stretches a vector.
Most of operator theory studies single elements, subalgebras or subspaces of this one algebra.
</document>

<document source="https://tamnd.github.io/math2/node/BMGE1ZAFQM.md" tag="BMGE1ZAFQM" kind="definition">
# Galois theory

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]], and let $\Aut(L/K)$ be the group of field automorphisms of $L$ that fix every element of $K$.
*Galois theory* is the study of field extensions $L/K$ through the groups $\Aut(L/K)$ and the [[Galois connection between subgroups and subfields|R3625YKQC0]].
Its basic objects are:

1. the [[Galois extensions|0DS2F7GK05]], that is, the algebraic extensions that are normal and separable;
2. their [[Galois groups|0XVR72M51M]] $\Gal(L/K) = \Aut(L/K)$;
3. the maps $H \mapsto L^H$, from subgroups of $\Gal(L/K)$ to their [[fixed fields|0KMJ6919WS]], and $E \mapsto \Gal(L/E)$, from [[intermediate fields|02NBKVQFRQ]] to subgroups.

## Remarks

Example: for $L = \Q(i)$ over $\Q$, the group $\Gal(L/\Q)$ has two elements, the identity and complex conjugation, and the fixed field of the whole group is $\Q$.
Non-example of a Galois extension: $\Q(\sqrt[3]{2})/\Q$ is separable but not normal, since $x^3 - 2$ has only one root in it.
Its automorphism group is trivial, so the base field $\Q$ is not the fixed field of any subgroup.
The Galois group of a separable polynomial $f \in K[x]$ is the Galois group of a splitting field of $f$ over $K$.
</document>

<document source="https://tamnd.github.io/math2/node/BMKATJ802A.md" tag="BMKATJ802A" kind="concept">
# Spectrum of an operator

## Summary

The spectrum of a bounded operator $T$ on a complex Banach space is the set of complex numbers $\lambda$ for which $T - \lambda I$ is not invertible.
It generalises the set of eigenvalues of a square matrix, the [[spectrum of a matrix|0XPX5NC67E]].
In infinite dimensions it can contain numbers that are not eigenvalues.
It is always a nonempty compact subset of the plane when the space is not zero.

## Explanations

### L1 human

For a matrix, $\lambda$ is an eigenvalue exactly when $A - \lambda I$ cannot be inverted.
The spectrum keeps this second description and uses it for operators in infinite dimensions.
There, $T - \lambda I$ can fail to be invertible even when it is injective, because it may miss some vectors.
For the right shift on square-summable sequences, there are no eigenvalues, yet the spectrum is the whole closed unit disc.
</document>

<document source="https://tamnd.github.io/math2/node/BMRX3X7095.md" tag="BMRX3X7095" kind="definition">
# Weakly singular kernel

## Definition

Let $m \ge 1$ and let $\Omega \subseteq \R^m$ be a closed bounded set.
Write $\abs{x - s}$ for the Euclidean distance of $x, s \in \Omega$.
A *weakly singular kernel* on $\Omega$ is a [[kernel|GSMBXT7FPK]] $K$ defined for $x, s \in \Omega$ with $x \ne s$ such that:

1. $K$ is a [[continuous function|0X3WGX95FE]] on $\set{(x, s) \in \Omega \times \Omega : x \ne s}$.
2. There are constants $M > 0$ and $\alpha \in [0, m)$ such that $\abs{K(x, s)} \le M \abs{x - s}^{-\alpha}$ for all $x \ne s$.

## Remarks

Example: $K(x, s) = \abs{x - s}^{-1/2}$ on $\Omega = [0, 1]$, with $m = 1$, $M = 1$ and $\alpha = 1/2 < 1$.
It is continuous off the diagonal.

Non-example: $K(x, s) = 1/(x - s)$ on $[0, 1]$.
It satisfies the bound only with $\alpha = 1$, which is not less than $m = 1$; the integral $\int_0^1 \abs{x - s}^{-1} \, ds$ diverges.

Since $\alpha < m$, the function $s \mapsto \abs{x - s}^{-\alpha}$ is integrable over $\Omega$, so $\int_\Omega K(x, s) \varphi(s) \, ds$ exists for bounded continuous $\varphi$.
Assume also that $\Omega$ is Jordan measurable, for example a closed interval or a closed ball.
Then the integral operator of a weakly singular kernel maps continuous functions on $\Omega$ to continuous functions and is compact for the maximum norm.
The kernel $\log \abs{x - s}$ is weakly singular on a bounded set: it satisfies the bound for every $\alpha \in (0, m)$.
</document>

<document source="https://tamnd.github.io/math2/node/BMXDMHRXBG.md" tag="BMXDMHRXBG" kind="statement">
# Nyquist–Shannon sampling theorem

## Statement

Let $B > 0$.
Let $f, g \in L^2(\R)$ be continuous [[band-limited functions|P3T104G564]] with bandwidth $B$, that is, their [[Fourier transforms|DPVM34ADZV]] vanish almost everywhere outside $[-B, B]$.
If $f(k / (2B)) = g(k / (2B))$ for every $k \in \Z$, then $f = g$.

## Remarks

The difference $h = f - g$ is band-limited and vanishes at all points $k/(2B)$.
On $[-B, B]$, $\hat h$ has the Fourier coefficients $\frac{1}{2B} \int_{-B}^{B} \hat h(\xi) e^{-2\pi i k \xi / (2B)}\, d\xi = \frac{1}{2B} h(-\frac{k}{2B})$ by Fourier inversion, and they are all zero.
So $\hat h = 0$ in [[$L^2$|SASP14MA1Z]], and $h = 0$.
The [[Whittaker–Shannon interpolation formula|Z4ZFP6FNK2]] gives $f$ explicitly from its samples.

The rate $2B$ samples per unit length is called the Nyquist rate.
It cannot be lowered: if the spacing is $h > 1/(2B)$, then $f(x) = \sin(\pi x / h)\, \operatorname{sinc}(2\delta x)^2$ with $\delta = \frac12 (B - \frac{1}{2h}) > 0$ is a nonzero function, band-limited with bandwidth $B$, in $L^2(\R)$, that vanishes at every point $kh$.
Here $\operatorname{sinc}(t) = \sin(\pi t)/(\pi t)$, and $\operatorname{sinc}(2\delta x)^2$ has transform supported in $[-2\delta, 2\delta]$.

The theorem is named after H. Nyquist and C. E. Shannon; E. T. Whittaker and V. A. Kotelnikov found equivalent results.

## Explanations

### L1 human

If a signal has no frequencies above $B$ cycles per second, then sampling it $2B$ times per second loses nothing: the samples determine the signal.
Sampling more slowly can confuse a high frequency with a lower one, which is called aliasing.
This is why audio is sampled at a little more than twice the highest audible frequency.

### L2 human

The proof is a duality between the line and the circle.
The samples of $f$ are the Fourier coefficients of $\hat f$, viewed as a periodic function on an interval of length $2B$.
Equivalently, multiplying $f$ by a Dirac comb of spacing $1/(2B)$ periodizes $\hat f$ with period $2B$, and the copies do not overlap exactly when the band fits.
Landau's density theorem extends the necessity of the Nyquist rate to irregular sampling sets.
</document>

<document source="https://tamnd.github.io/math2/node/BMXZSXETDW.md" tag="BMXZSXETDW" kind="statement">
# Hankel inversion theorem

## Statement

Let $\nu \ge -\frac12$, and let $J_\nu$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $\nu$.
Let $f \colon (0, \infty) \to \C$ satisfy $\int_0^\infty \abs{f(r)}\, r^{1/2}\, dr < \infty$, and let $F_\nu(k) = \int_0^\infty f(r) J_\nu(kr)\, r\, dr$ be its [[Hankel transform|2P9YWDGKC4]] of order $\nu$.
Let $r > 0$, and assume that $f$ is of bounded variation on a neighbourhood of $r$.
Then
$$\lim_{K \to \infty} \int_0^K F_\nu(k)\, J_\nu(kr)\, k\, dk = \frac{f(r^+) + f(r^-)}{2}.$$
In particular, the limit is $f(r)$ at every point $r$ where $f$ is continuous.

## Remarks

The proof follows the pattern of the Fourier inversion theorem for Fourier integrals.
After exchanging the integrals, the inner integral $\int_0^K J_\nu(kr) J_\nu(kt)\, k\, dk$ is computed in closed form from Bessel's equation, and it acts as an approximate identity concentrated at $t = r$ as $K \to \infty$.
A Riemann–Lebesgue argument disposes of the parts away from $r$, and the bounded variation near $r$ controls the part near $r$, exactly as in Dirichlet's theorem for Fourier series.
There is also an $L^2$ version: the Hankel transform of order $\nu$ extends to a unitary operator of $L^2((0, \infty), r\, dr)$ that is its own inverse.

Example: for $\nu = 0$ and $f(r) = e^{-r^2/2}$ one has $F_0(k) = e^{-k^2/2}$, and transforming again gives back $e^{-r^2/2}$.
For the indicator function of $[0, 1]$ and $\nu = 0$, $F_0(k) = J_1(k)/k$, and at $r = 1$ the inversion integral converges to the mean value $\frac12$ of the jump.

## Explanations

### L1 human

The Hankel transform breaks a radial function into circular waves.
This theorem says that adding the waves back together recovers the function.
At a jump, the reconstruction lands exactly halfway between the two values, as for Fourier series.

### L2 human

The theorem is the radial form of the Fourier inversion theorem, and for $\nu = \frac{n}{2} - 1$ it can be derived from it for radial functions on $\R^n$.
It justifies the solution of boundary value problems in cylindrical coordinates: transform in $r$, solve an ordinary differential equation in the remaining variable, and transform back.
The hypotheses parallel those of Dirichlet's pointwise convergence theorem for Fourier series.
</document>

<document source="https://tamnd.github.io/math2/node/BMY55EW3RF.md" tag="BMY55EW3RF" kind="concept">
# Submersion

## Summary

A submersion is a smooth map between [[smooth manifolds|F45VQ47GPV]] whose [[differential|6P9PXQDSEN]] is surjective at every point.
Locally it looks like the projection of $\R^m$ onto $\R^n$.
Its fibres are smooth submanifolds, which makes submersions the basic way to cut a manifold into smooth pieces.
Projections of products and of fibre bundles are typical examples.

## Explanations

### L1 human

A submersion is a map that is locally a projection, like the shadow map $(x, y, z) \mapsto (x, y)$ that forgets the height.
Near each point it can move the point in every direction of the target.
Because of this, the set of points that land on a given value is a smooth piece of space, with no pinches.
For example, $f(x, y) = x^2 + y^2$ is a submersion from $\R^2 \setminus \set{0}$ to $(0, \infty)$, and its fibres are the circles around the origin.
</document>

<document source="https://tamnd.github.io/math2/node/BND6A96S3D.md" tag="BND6A96S3D" kind="definition">
# Heyting algebra

## Definition

A *Heyting algebra* is a structure $(H, \wedge, \vee, \to, 0, 1)$ such that:

1. $(H, \wedge, \vee, 0, 1)$ is a bounded lattice, with [[partial order|0QV2JS9RJ3]] $a \le b$ if and only if $a \wedge b = a$, least element $0$ and greatest element $1$.
2. $\to$ is a binary operation on $H$.
3. For all $a, b, c \in H$, $c \le (a \to b)$ if and only if $c \wedge a \le b$.

The *pseudocomplement* of $a$ is $\neg a = a \to 0$.

## Remarks

Clause 3 says that $a \to b$ is the largest element $c$ with $c \wedge a \le b$, so $\to$ is determined by the lattice.

Example: let $X$ be a [[topological space|0CQHFPGJ45]] and let $H$ be the set of its [[open sets|02BH3MCBQB]], with $\wedge = \cap$, $\vee = \cup$, $0 = \emptyset$, $1 = X$, and $U \to V$ the [[interior|0RZH6W98HK]] of $(X \setminus U) \cup V$.
For an open set $W$, the inclusion $W \subseteq (X \setminus U) \cup V$ holds if and only if $W \cap U \subseteq V$.
Since $W$ is open, the first inclusion holds if and only if $W$ lies in the interior; this is clause 3.

Non-example: the five-element lattice $M_3$ has a least element, a greatest element and three pairwise incomparable elements between them.
It is a bounded lattice but carries no operation $\to$ satisfying clause 3.
Indeed, every Heyting algebra is a [[distributive lattice|0HM0HQCK6N]], and $M_3$ is not distributive.

A [[Boolean algebra|0XADJKM699]] is a Heyting algebra with $a \to b = \neg a \vee b$, and a Heyting algebra is Boolean if and only if $a \vee \neg a = 1$ for all $a$.
Every finite distributive lattice is a Heyting algebra.
</document>

<document source="https://tamnd.github.io/math2/node/BNDRN0G9BR.md" tag="BNDRN0G9BR" kind="object">
# E8 manifold

## Description

The *$E_8$ manifold* is the closed, simply connected, topological [[4-manifold|8H58Q6YJCH]] whose [[intersection form|WK8GKRKAVD]] is the form of the [[$E_8$ lattice|2CXH65EKY7]].
This form is positive definite, even and unimodular, of rank $8$ and [[signature|07EZA80NEH]] $8$.
Its existence and its uniqueness up to [[homeomorphism|0QT00PB5G2]] follow from [[Freedman's classification|HSD6FKCYRC]], since the form is even.
One construction plumbs together eight disk bundles over the $2$-sphere with Euler number $2$, following the pattern of the $E_8$ Dynkin diagram.
The boundary of this plumbing is the [[Poincaré homology sphere|NHERCK7C43]].
Freedman showed that this homology sphere bounds a contractible topological $4$-manifold, and gluing it to the plumbing along the common boundary gives the $E_8$ manifold.

## Remarks

The $E_8$ manifold admits no [[smooth structure|R2TQ82PCNB]].
Indeed its intersection form is even, so a smooth structure would give a closed smooth spin $4$-manifold of signature $8$, against [[Rokhlin's theorem|12BXVH5W5H]], which demands signature divisible by $16$.
It also follows from [[Donaldson's theorem|34B7K77P31]], since the $E_8$ form is definite but not diagonalizable over $\Z$.
Its Kirby–Siebenmann invariant is $1$, equal to the signature divided by $8$ modulo $2$.
Andrew Casson showed that it admits no triangulation at all.

## Explanations

### L1 human

The $E_8$ manifold is a four-dimensional space that can be built by gluing topological pieces but that has no smooth version.
Its fingerprint is the $E_8$ pattern, a famous $8 \times 8$ table of numbers that records how surfaces inside it cross.
It was one of the first proofs that topological and smooth manifolds really differ in dimension four.
</document>

<document source="https://tamnd.github.io/math2/node/BNEXZF75QC.md" tag="BNEXZF75QC" kind="concept">
# Smoothing spline

## Summary

A smoothing spline is the function that best balances closeness to noisy data against roughness, measured by the integral of the squared second derivative.
The balance is set by a smoothing parameter $\lambda > 0$.
The minimizer turns out to be a natural [[cubic spline|V42H0R3EKX]] with knots at the data points.
Smoothing splines are a basic tool of nonparametric regression and data smoothing.

## Explanations

### L1 human

Imagine noisy measurements scattered around an unknown smooth curve.
Passing a curve through every point copies the noise, while a straight line may miss the shape.
A smoothing spline pays a price for missing the points and a price for bending, and takes the curve with the smallest total price.
A small $\lambda$ gives a wiggly curve close to the data, and a large $\lambda$ gives a curve close to the [[least squares|0BYTQ4X1HC]] line.
</document>

<document source="https://tamnd.github.io/math2/node/BNQVS83GZV.md" tag="BNQVS83GZV" kind="concept">
# Hilbert–Samuel function

## Summary

Let $M$ be a finitely generated module over a Noetherian [[local ring|MZ02BKPHJA]], and let $I$ be an ideal primary to the maximal ideal.
The Hilbert–Samuel function of $M$ is $n \mapsto \ell(M/I^n M)$, where $\ell$ is the [[length|CZSCNZH8VY]].
For large $n$ it agrees with a polynomial, the Hilbert–Samuel polynomial.
If $M \neq 0$, the degree of this polynomial is the dimension of $M$, and its leading coefficient gives the [[multiplicity|JX20MNDDXT]].

## Explanations

### L1 human

Look at a ring near a point and ask how much information is visible up to order $n$.
The Hilbert–Samuel function counts this, as a length, for each $n$.
For the plane at the origin, the polynomials up to degree $n - 1$ are spanned by $n(n+1)/2$ monomials, so the function is $n(n+1)/2$.
Its growth like $n^2$ reveals that the plane has dimension $2$.
The [[definition|22J8E7AB8V]] makes this precise.
</document>

<document source="https://tamnd.github.io/math2/node/BNR4H5GCTP.md" tag="BNR4H5GCTP" kind="concept">
# Braid group

## Summary

The braid group $B_n$ is the [[group|G5S1M9]] of braids with $n$ strands.
Two braids are equal when one can be deformed into the other, and the product places one braid after the other.
Artin gave its presentation with generators $\sigma_1, \ldots, \sigma_{n-1}$, the crossings of neighbouring strands, so $B_n$ is an [[Artin group|SAWBDBHE41]].
It is also the [[fundamental group|S1XK3HXHFX]] of the space of sets of $n$ distinct points in the plane.
It is also the [[mapping class group|JV5NYG913V]] of a disk with $n$ marked points.
Braid groups link algebra with knot theory: by [[Alexander's theorem|PRWP1JF486]], every link is the closure of a braid.

## Explanations

### L1 human

Fix $n$ pegs on a top bar and $n$ pegs on a bottom bar, and join them with $n$ strings that always go downward.
Two such braids are the same if one can be moved into the other without cutting the strings or moving the pegs.
To multiply two braids, hang one below the other.
The basic braid $\sigma_i$ crosses string $i$ over string $i + 1$, and its inverse crosses it back under.
For example, with three strands, $\sigma_1 \sigma_2 \sigma_1$ and $\sigma_2 \sigma_1 \sigma_2$ are the same braid: both reverse the order of the three strings.
</document>

<document source="https://tamnd.github.io/math2/node/BNSC4P3PHS.md" tag="BNSC4P3PHS" kind="statement">
# Serre's vanishing theorem on affine schemes

## Statement

Let $X$ be an [[affine scheme|B3V3XJ2W8G]] and let $\mathcal{F}$ be a [[quasi-coherent sheaf|KPKND0K7W6]] on $X$.
Then the [[sheaf cohomology|PQ6Q143ARJ]] groups vanish in positive degrees:
$$H^i(X, \mathcal{F}) = 0 \quad \text{for all } i > 0.$$

## Remarks

The idea of the proof for $X = \operatorname{Spec} A$ with $A$ Noetherian: if $I$ is an injective $A$-module, the associated sheaf $\tilde{I}$ is flasque.
Resolve $\mathcal{F} = \tilde{M}$ by $\tilde{I}^\bullet$ for an injective resolution $M \to I^\bullet$; this is a flasque resolution, and taking global sections gives back the exact complex $I^\bullet$.
In general one uses that the Čech complex of a quasi-coherent sheaf for a cover by standard open sets $D(f_i)$ is exact in positive degrees.

Example: on a separated scheme $X$ with an affine open cover, the theorem shows that the cover is acyclic for quasi-coherent sheaves, so Čech cohomology of this cover computes $H^i(X, \mathcal{F})$.

The affine hypothesis is necessary: on $X = \mathbb{P}^1_k$, which is not affine, $H^1(X, \mathcal{O}(-2)) \cong k \ne 0$.
Conversely, Serre's criterion says that a Noetherian scheme $X$ is affine if $H^1(X, \mathcal{F}) = 0$ for every quasi-coherent sheaf $\mathcal{F}$, and it suffices to ask this for coherent ideal sheaves.

## Explanations

### L1 human

On an affine scheme, every quasi-coherent sheaf comes from a module, and global sections of a short exact sequence of such sheaves stay exact.
So there is never an obstruction to passing from local to global data, and all higher cohomology vanishes.
Affine schemes play for quasi-coherent sheaves the role that contractible open sets play in topology.

### L2 human

The theorem is the base of every computation of coherent cohomology: one covers a separated scheme by affine open sets and computes the Čech complex.
It gives the long exact sequence argument that a surjection of quasi-coherent sheaves on an affine scheme is surjective on global sections.
With Serre's converse criterion, it characterizes affine schemes cohomologically, in the same way as Cartan's Theorem B for Stein manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/BNVRR925X8.md" tag="BNVRR925X8" kind="concept">
# Lattice homomorphism

## Summary

A lattice homomorphism is a map between two [[lattices|WH4SFAACAS]] that preserves [[join and meet|J2T75ATRV5]].
Every lattice homomorphism is [[monotone|0A7KV2WB7A]], but a monotone map need not preserve joins or meets.
Lattice homomorphisms are the morphisms of the category of lattices; their kernels are the lattice congruences, and the bijective ones are the lattice isomorphisms.

## Explanations

### L1 human

A lattice homomorphism carries one lattice into another and keeps best common bounds.
The best common upper bound of two elements goes to the best common upper bound of their images, and the same for lower bounds.
For example, fix a subset $A$ of a set $X$ and send each subset $S$ of $X$ to $S \cap A$.
Then $(S \cup T) \cap A = (S \cap A) \cup (T \cap A)$ and $(S \cap T) \cap A = (S \cap A) \cap (T \cap A)$.
So this map from the subsets of $X$ to the subsets of $A$ is a lattice homomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/BNYW1SNP00.md" tag="BNYW1SNP00" kind="definition">
# Hyperbolic 3-manifold

## Definition

A *hyperbolic $3$-manifold* is a connected [[3-manifold|R20TB2KEQQ]] $M$ with a [[Riemannian metric|7CBT4SM83Y]] $g$ such that:

1. $(M, g)$ is complete as a metric space, and
2. every [[sectional curvature|1X42JAA349]] of $g$ is equal to $-1$.

## Remarks

Example: hyperbolic $3$-space $\mathbb{H}^3 = \set{(x, y, t) \in \R^3 : t > 0}$ with the metric $(dx^2 + dy^2 + dt^2)/t^2$ is complete and has constant sectional curvature $-1$.
More generally, if $\Gamma$ is a discrete group of isometries of $\mathbb{H}^3$ acting freely, then $\mathbb{H}^3 / \Gamma$ with the quotient metric is a hyperbolic $3$-manifold.

Non-example: the open subset $\mathbb{H}^3 \setminus \set{p}$, for a point $p$, has curvature $-1$, but it is not complete, since a geodesic heading to $p$ stops after finite length.

By the Killing–Hopf theorem, every hyperbolic $3$-manifold is isometric to some $\mathbb{H}^3 / \Gamma$ as above.
It is orientable exactly when $\Gamma$ lies in the group of orientation-preserving isometries, which is isomorphic to $PSL(2, \C)$.
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<document source="https://tamnd.github.io/math2/node/BP0ZYHGD0S.md" tag="BP0ZYHGD0S" kind="statement">
# Oka's normalization theorem

## Statement

Let $X$ be a [[reduced complex space|831N2Q4GCB]] and let $N \subseteq X$ be the set of points of $X$ that are not normal.
Then $N$ is a [[nowhere dense|0VQYCCQVJ5]] [[analytic subset|GB7M582M5Y]] of $X$, and there are a [[normal complex space|PN55VEF0J7]] $\tilde{X}$ and a holomorphic map $\nu \colon \tilde{X} \to X$ such that:

1. $\nu$ is finite, that is, closed with finite fibres, and surjective;
2. $\nu^{-1}(N)$ is nowhere dense in $\tilde{X}$;
3. $\nu$ restricts to a biholomorphic map $\tilde{X} \setminus \nu^{-1}(N) \to X \setminus N$.

The pair $(\tilde{X}, \nu)$ is unique up to a unique isomorphism over $X$; it is the [[normalization|M2FAMD9HZN]] of $X$.

## Remarks

The key step, due to Oka, is that the sheaf $\hat{\mathcal{O}}_X$ of weakly holomorphic functions is a [[coherent analytic sheaf|DSJET02WPR]] of $\mathcal{O}_X$-modules.
A weakly holomorphic function is one that is holomorphic on the regular points and locally bounded.
Then $\tilde{X}$ is the analytic spectrum of the finite $\mathcal{O}_X$-algebra $\hat{\mathcal{O}}_X$.

Example: for the cusp $X = \set{z_1^2 = z_2^3} \subseteq \C^2$, the map $\nu \colon \C \to X$, $t \mapsto (t^3, t^2)$, is the normalization.
It is bijective and finite, $N = \set{0}$, and off $t = 0$ its inverse $(z_1, z_2) \mapsto z_1 / z_2$ is holomorphic.
For the cross $X = \set{z_1 z_2 = 0}$, the normalization is the disjoint union of two copies of $\C$, mapped onto the two axes; here $\nu$ is not injective over $0$.

## Explanations

### L1 human

A singular space can be improved by separating sheets that cross and by opening up pinches, without changing it elsewhere.
The theorem says that this can always be done for complex spaces, and in exactly one natural way.
The result is a normal space that maps onto the original one, with finite fibres.

### L2 human

Normalization reduces many questions about reduced spaces to normal spaces, where the Riemann extension theorem and local irreducibility are available.
For curves the normalization is smooth, so it is a resolution of singularities.
It is the analytic counterpart of the normalization of an integral scheme by integral closure of its rings.
</document>

<document source="https://tamnd.github.io/math2/node/BP331JP8SF.md" tag="BP331JP8SF" kind="definition">
# Følner condition

## Definition

Let $G$ be a [[group|G5S1M9]], and for a subset $F \subseteq G$ and $g \in G$ let $gF = \set{gx : x \in F}$.
The group $G$ satisfies the *Følner condition* if the following holds.
For every finite set $S \subseteq G$ and every $\varepsilon > 0$, there is a finite nonempty set $F \subseteq G$ with
$$\abs{sF \,\triangle\, F} < \varepsilon \abs{F} \quad \text{for all } s \in S,$$
where $\triangle$ is the symmetric difference.
For a countable group, a *Følner sequence* is a sequence $(F_n)$ of finite nonempty subsets with $\abs{gF_n \triangle F_n} / \abs{F_n} \to 0$ for every $g \in G$.

## Remarks

Example: $G = \Z$ with $F_n = \set{0, 1, \ldots, n-1}$.
For $k \in \Z$ with $\abs{k} \le n$, the sets $k + F_n$ and $F_n$ differ in $2\abs{k}$ points, so $\abs{(k + F_n) \triangle F_n} / n = 2\abs{k}/n \to 0$.
So $(F_n)$ is a Følner sequence.

Non-example: the free group $F_2$ on two generators fails the condition.
Every finite nonempty $F \subseteq F_2$ has some generator $s$ with $\abs{sF \triangle F}$ at least a fixed positive fraction of $\abs{F}$.
This reflects that $F_2$ is not [[amenable|CEAT9BESKK]].

Følner's theorem says that a discrete group satisfies the Følner condition exactly when it is amenable.
For a [[locally compact group|AHV05ZBCYW]], one uses compact sets $F$ of positive [[Haar measure|VY0HXD3GC4]] and compact sets $S$ instead.
</document>

<document source="https://tamnd.github.io/math2/node/BP3WR8NG1W.md" tag="BP3WR8NG1W" kind="concept">
# Frame operator

## Summary

The frame operator of a frame $(f_i)$ in a [[Hilbert space|HM2HHE3J51]] sends a vector $f$ to $\sum_i \ip{f}{f_i} f_i$.
It is bounded, [[self-adjoint|0NEGN5CRSC]], [[positive|77020CY5ZW]] and [[invertible|CPAD5W1CGW]], and its spectrum lies between the frame bounds.
Its inverse gives the canonical dual frame, and so a formula that rebuilds every vector from its frame coefficients.

## Explanations

### L1 human

A frame lets one measure a vector by taking inner products with many fixed vectors.
The frame operator first measures, then adds up the fixed vectors weighted by the measurements.
For an orthonormal basis this gives back the vector itself.
For a general frame the result is distorted, but in a way that can be undone, because the operator is invertible.
For example, the three Mercedes-Benz star vectors in the plane give the frame operator $\tfrac32 I$.
</document>

<document source="https://tamnd.github.io/math2/node/BP8S3Z5TW9.md" tag="BP8S3Z5TW9" kind="definition">
# Convergence in measure

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]], and let $f_1, f_2, \ldots$ and $f$ be [[measurable functions|YYNCJSR0PJ]] from $X$ to $\R$.
The sequence $(f_n)$ *converges in measure* to $f$ if for every $\varepsilon > 0$
$$\lim_{n \to \infty} \mu\big(\set{x \in X : \abs{f_n(x) - f(x)} \ge \varepsilon}\big) = 0.$$

## Remarks

Example: on $[0, 1]$ with Lebesgue measure, write $n = 2^k + j$ with $0 \le j < 2^k$ and let $f_n = \mathbf{1}_{[j 2^{-k}, (j+1) 2^{-k}]}$.
For $0 < \varepsilon \le 1$, the set where $\abs{f_n} \ge \varepsilon$ has measure $2^{-k}$, which tends to $0$, so $f_n \to 0$ in measure.
But each $x \in [0, 1]$ has $f_n(x) = 1$ for infinitely many $n$, so $f_n(x)$ converges for no $x$.
Non-example: on $\R$ with Lebesgue measure, $f_n = \mathbf{1}_{[n, n+1]}$ tends to $0$ at every point, but the set where $\abs{f_n} \ge 1/2$ has measure $1$ for all $n$.
A local variant asks the limit to be $0$ only for the restriction to each set of finite measure.
Convergence in $L^p$ for some $1 \le p < \infty$ implies convergence in measure, by Chebyshev's inequality.
</document>

<document source="https://tamnd.github.io/math2/node/BPH9SJ30Y0.md" tag="BPH9SJ30Y0" kind="concept">
# Enriched category

## Summary

A category enriched over a [[monoidal category|M3MSQ3NKAE]] $\mathcal{V}$ has, instead of a set of morphisms between two objects, an object of $\mathcal{V}$.
Composition and identities are morphisms of $\mathcal{V}$ that satisfy associativity and unit laws.
Ordinary locally small [[categories|A2DE66937D]] are the case $\mathcal{V} = \mathbf{Set}$; [[preadditive categories|SC6FD5F4J0]] are the case of abelian groups; strict 2-categories are the case of small categories.
Lawvere observed that [[metric spaces|0VKD1TP7F3]] are categories enriched over the non-negative reals.

## Explanations

### L1 human

In many categories the morphisms between two objects carry extra structure.
Linear maps between vector spaces can be added, continuous maps between spaces form a space, and so on.
An enriched category builds this structure into the definition: the "hom-set" is replaced by a hom-object of some fixed kind.
A surprising example: a metric space is a category whose objects are points and whose "hom-object" from $x$ to $y$ is the number $d(x, y)$.
Composition becomes the triangle inequality.
</document>

<document source="https://tamnd.github.io/math2/node/BPKZV9W7AT.md" tag="BPKZV9W7AT" kind="definition">
# Matrix pencil

## Definition

Let $\mathbb{K}$ be a [[field|04C5ZBG1WT]], let $m, n \ge 1$, and let $A$ and $B$ be $m \times n$ [[matrices|0VD9ZSWWJG]] over $\mathbb{K}$.
The *matrix pencil* of $A$ and $B$ is the matrix $A - \lambda B$ with entries in the polynomial ring $\mathbb{K}[\lambda]$.
When $m = n$, the pencil is *regular* if the polynomial $\det(A - \lambda B)$ is not the zero polynomial, and *singular* otherwise.
For a regular pencil, an *eigenvalue* is a root $\lambda_0$ of $\det(A - \lambda B)$, and an *eigenvector* for $\lambda_0$ is a vector $v \ne 0$ with $Av = \lambda_0 Bv$.

## Remarks

Some authors write $\lambda B - A$ or $A + \lambda B$; this changes only signs.
When $B$ is singular, a regular pencil is often said to have an eigenvalue $\infty$, since $\mu A - B$ is singular at $\mu = 0$.
Example: $A = \begin{pmatrix} 2 & 0 \\ 0 & 6 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$ give $\det(A - \lambda B) = (2 - \lambda)(6 - 2\lambda)$, so the pencil is regular with eigenvalues $2$ and $3$.
Non-example: $A = B = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ give $\det(A - \lambda B) = (1 - \lambda) \cdot 0 = 0$, so the pencil is singular.
If $B$ is invertible, the eigenvalues of the pencil are the ordinary [[eigenvalues|0BQAA2BK3Q]] of $B^{-1}A$.
</document>

<document source="https://tamnd.github.io/math2/node/BPV58EPYCQ.md" tag="BPV58EPYCQ" kind="concept">
# Galois insertion

## Summary

A Galois insertion is a [[Galois connection|MBVJ4JQXC5]] $(f, g)$ in which the lower adjoint undoes the upper adjoint: $f(g(y)) = y$ for every $y$.
Then $g$ embeds $Q$ into $P$ as the set of closed elements of the [[closure operator|17FVAN6HXX]] $g \circ f$.
Galois insertions describe how a structure such as "subgroups" sits inside a larger ordered set such as "all subsets".

## Explanations

### L1 human

In a Galois insertion one ordered set sits inside the other, and each element of the big set has a best approximation from above in the small set.
For a group, the subgroups sit inside the collection of all subsets.
Each subset $S$ has a smallest subgroup containing it, the subgroup generated by $S$.
Going from a subgroup to its underlying set and back gives the same subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/BPXY5A22BB.md" tag="BPXY5A22BB" kind="definition">
# Interior operator

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
An *interior operator* on $P$ is a map $i \colon P \to P$ such that, for all $x, y \in P$:

1. $i(x) \le x$ (deflationary);
2. if $x \le y$, then $i(x) \le i(y)$ (monotone);
3. $i(i(x)) = i(x)$ (idempotent).

An element $x$ with $i(x) = x$ is *open*.

## Remarks

Let $X$ be a topological space and $P$ the set of subsets of $X$ ordered by inclusion.
The map sending $A$ to its [[interior|0C38S0KMJK]] is an interior operator: the interior lies in $A$, grows with $A$, and is open, so taking it twice changes nothing.
On $\R$, the map $i(x) = x - 1$ is deflationary and monotone but not idempotent, since $i(i(0)) = -2 \ne -1 = i(0)$.
A map $i$ is an interior operator on $P$ if and only if it is a [[closure operator|17FVAN6HXX]] on the [[dual|BS1RD4F5MD]] of $P$.
On a power set, $A \mapsto X \setminus c(X \setminus A)$ is an interior operator for each closure operator $c$.
Other names are kernel operator and dual closure operator.

## Explanations

### L1 human

An interior operator moves each element down to the nearest open element.
Open elements stay where they are.
</document>

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# Pol–Inv Galois connection

## Statement

Let $A$ be a finite nonempty set.
For a set $F$ of finitary operations on $A$, let $\operatorname{Inv}(F)$ be the set of finitary relations on $A$ preserved by every member of $F$.
For a set $\Gamma$ of finitary relations on $A$, let $\operatorname{Pol}(\Gamma)$ be the set of operations that are [[polymorphisms|A0K9EZRM7E]] of all members of $\Gamma$.
Then:

1. $\operatorname{Pol}(\operatorname{Inv}(F))$ is the [[clone|PTKR703G6A]] generated by $F$.
2. $\operatorname{Inv}(\operatorname{Pol}(\Gamma))$ is the [[relational clone|MC25MTN2N7]] generated by $\Gamma$.
3. Consequently, $C \mapsto \operatorname{Inv}(C)$ is an inclusion-reversing bijection from the clones on $A$ to the relational clones on $A$, with inverse $\Gamma \mapsto \operatorname{Pol}(\Gamma)$.

## Remarks

The maps $\operatorname{Pol}$ and $\operatorname{Inv}$ come from the relation "$f$ preserves $R$" between operations and relations, so they always form a [[Galois connection|Q5JHEPRYQ6]]; the theorem identifies its closed sets.
For part 1, take an $n$-ary $f \in \operatorname{Pol}(\operatorname{Inv}(F))$ and the relation $R$ of arity $\abs{A}^n$ formed by the columns of all $n$-ary operations in the clone generated by $F$.
This $R$ is preserved by $F$ and contains the columns of the $n$ projections.
So $f$ applied to these columns gives the column of $f$, which lies in $R$, and $f$ is in the clone.
Part 2 is proved in a similar way, with a relation built from $\operatorname{Pol}(\Gamma)$ and a primitive positive definition of it from $\Gamma$.
The result is due to Geiger and, independently, to Bodnarčuk, Kalužnin, Kotov and Romov.
For example, on $\set{0, 1}$ the clone of monotone operations is $\operatorname{Pol}(\set{\le})$.
Finiteness of $A$ is needed: on an infinite set, the closed sets are clones that are locally closed, and some clones are not of the form $\operatorname{Pol}(\Gamma)$.

## Explanations

### L1 human

A family of operations can be described in two ways: by listing operations that generate it, or by listing the relations that all of them respect.
On a finite set, the theorem says that the second description is always possible and exact.
Every clone is the set of all operations that respect some family of relations.
Dually, a family of relations closed under the natural constructions is the set of all relations respected by some operations.

### L2 human

The theorem turns questions about clones into questions about relations, which are often easier to handle.
Maximal clones, for example, are of the form $\operatorname{Pol}(\set{\rho})$ for a single relation $\rho$, and [[Rosenberg's classification|GF1RK91VED]] lists those relations.
In constraint satisfaction, adding a primitive positive definable relation to a template does not change the complexity.
So the complexity of a finite template depends only on its polymorphism clone.
The same reasoning underlies the [[Baker–Pixley theorem|97SRYBF41G]], which bounds the arity of the relations needed for a clone with a near-unanimity operation.
</document>

<document source="https://tamnd.github.io/math2/node/BQ6CYHQW02.md" tag="BQ6CYHQW02" kind="concept">
# Regularity structures

## Summary

A regularity structure is an algebraic framework, introduced by Martin Hairer, for describing functions and distributions locally by generalized Taylor expansions.
The building blocks are abstract symbols with assigned degrees of regularity, together with a group that says how to re-expand them around another point.
The theory gives a solution theory for singular [[stochastic partial differential equations|PRCQG82QTK]], whose nonlinear terms are not defined classically.
Examples include the [[Kardar–Parisi–Zhang equation|RQ196EHXKC]] and the dynamical $\Phi^4_3$ model.

## Explanations

### L1 human

A smooth function is described near a point by its Taylor polynomial, built from the monomials $1, x, x^2, \ldots$.
Solutions of rough random equations have no Taylor polynomial, because they are not differentiable.
The idea is to add new building blocks, made from the noise itself, and to expand the solution in terms of them.
Each block gets a degree, which may be negative or fractional, saying how it behaves at small scales.
A rule for moving the base point, like re-expanding $(x - a)^2$ around another point $b$, completes the structure.
</document>

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# Dahlquist barriers

## Statement

Consider a [[linear $k$-step method|BX4D85A79P]] $\sum_{i=0}^k \alpha_i y_{j+i} = h \sum_{i=0}^k \beta_i f_{j+i}$ with $\alpha_k = 1$ and characteristic polynomials $\rho(\zeta) = \sum_i \alpha_i \zeta^i$, $\sigma(\zeta) = \sum_i \beta_i \zeta^i$, where $\rho$ and $\sigma$ have no common root.
The method has *order* $p$ if $\rho(e^z) - z \sigma(e^z) = O(z^{p+1})$ as $z \to 0$.
It is *zero-stable* if every root of $\rho$ has modulus at most $1$ and the roots of modulus $1$ are simple.
It is *A-stable* if, for every complex $\lambda$ with $\operatorname{Re} \lambda < 0$ and every $h > 0$, every sequence produced by the method for $y' = \lambda y$ tends to $0$.

1. First Dahlquist barrier: a zero-stable linear $k$-step method has order $p \le k + 2$ if $k$ is even, and $p \le k + 1$ if $k$ is odd.
If it is moreover explicit, then $p \le k$.
2. Second Dahlquist barrier: an A-stable linear multistep method has order $p \le 2$, and no explicit linear multistep method is A-stable.

## Remarks

For the first barrier, the substitution $\zeta = \frac{1 + w}{1 - w}$ maps the left half plane $\operatorname{Re} w < 0$ onto the open unit disk.
It turns the order conditions into conditions on the power series of a rational function of $w$.
Zero-stability forces sign conditions on the coefficients that are incompatible with a higher order.
For the second barrier, A-stability means that $\rho(\zeta) - z \sigma(\zeta)$ has all its roots in the open unit disk whenever $\operatorname{Re} z < 0$.
Hence $\operatorname{Re} \frac{\rho(\zeta)}{\sigma(\zeta)} \ge 0$ for $\abs{\zeta} > 1$, and this positivity is incompatible with order greater than $2$.
Example: Simpson's two-step method $y_{j+2} - y_j = \frac{h}{3} (f_{j+2} + 4 f_{j+1} + f_j)$ has order $4 = k + 2$ and is zero-stable, so the first barrier is sharp for $k = 2$.
The trapezoidal rule $y_{j+1} - y_j = \frac{h}{2} (f_{j+1} + f_j)$ has order $2$ and is A-stable, so the second barrier is sharp; among A-stable methods of order $2$ it has the smallest error constant.
The [[backward differentiation formulas|TDA94C2JYT]] of order $3$ to $6$ are zero-stable but not A-stable, as the second barrier requires.

## Explanations

### L1 human

One might hope to make a multistep method more accurate just by using more old values and choosing the weights cleverly.
The first barrier says that this hope fails beyond a fixed limit: past that order, the method becomes unstable and useless.
The second barrier says that methods that are safe for every decaying problem, however stiff, cannot be more than second-order accurate.
So accuracy and stability pull against each other, and the barriers say exactly where.

### L2 human

A $k$-step method can reach order $2k$, but such methods are not zero-stable for $k \ge 3$.
The first barrier explains why practical codes use Adams methods instead, of order $k$ or $k + 1$ for $k$ steps.
The second barrier explains why stiff solvers use [[BDF methods|TDA94C2JYT]].
Above order $2$ these methods give up full A-stability, but they keep stability in a large sector of the left half plane.
It also motivates implicit Runge–Kutta methods, which escape the barrier because they are not linear multistep methods: the Gauss–Legendre methods are A-stable of every even order.
Both barriers were proved by Germund Dahlquist.
</document>

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# Generator

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
An object $G$ of $\mathcal{C}$ is a *generator* (or *separator*) if it has the following property.
For all [[morphisms|859F0TFVA3]] $f, g \colon X \to Y$ with $f \neq g$, there is a morphism $h \colon G \to X$ with $f \circ h \neq g \circ h$.
More generally, a set $\mathcal{G}$ of objects is a *generating family* if for all $f \neq g \colon X \to Y$ there are $G \in \mathcal{G}$ and $h \colon G \to X$ with $f \circ h \neq g \circ h$.

## Remarks

When $\mathcal{C}$ is locally small, $G$ is a generator exactly when the [[hom functor|VJXS8AK67E]] $\Hom(G, -) \colon \mathcal{C} \to \mathbf{Set}$ is [[faithful|GRG8WEBSH5]].
The dual notion is a cogenerator (or coseparator): an object $C$ such that $f \neq g \colon X \to Y$ implies $k \circ f \neq k \circ g$ for some $k \colon Y \to C$.

Example: in the category of [[abelian groups|082GHNPJ53]], $\Z$ is a generator.
If $f \neq g \colon A \to B$, pick $a \in A$ with $f(a) \neq g(a)$ and let $h \colon \Z \to A$ be $n \mapsto na$; then $(f \circ h)(1) = f(a) \neq g(a) = (g \circ h)(1)$.

Non-example: in the same category, the trivial group $0$ is not a generator.
The identity and the zero map of $\Z/2\Z$ differ, but the only morphism $h \colon 0 \to \Z/2\Z$ is zero, and both composites with $h$ are zero.
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# Embedding of structures

## Definition

Let $\sigma$ be a signature and let $\mathbf{A}$ and $\mathbf{B}$ be $\sigma$-[[structures|983GZBQYT4]] with domains $A$ and $B$.
An *embedding* of $\mathbf{A}$ into $\mathbf{B}$ is a map $h \colon A \to B$ such that:

1. $h$ is a [[homomorphism|KBXSZC6BV1]] from $\mathbf{A}$ to $\mathbf{B}$;
2. $h$ is [[injective|09NWETVTNY]];
3. for each relation symbol $R \in \sigma$ of arity $n$ and all $a_1, \dots, a_n \in A$, if $(h(a_1), \dots, h(a_n)) \in R^{\mathbf{B}}$ then $(a_1, \dots, a_n) \in R^{\mathbf{A}}$.

## Remarks

A map $h$ is an embedding if and only if its image is the domain of a [[substructure|676H7EGTKJ]] $\mathbf{C}$ of $\mathbf{B}$ and $h$ is an [[isomorphism|PZMGQMDN25]] from $\mathbf{A}$ to $\mathbf{C}$.
For a signature without relation symbols, clause 3 is empty, and an embedding is an injective homomorphism.

Example: for the signature $\set{+, \cdot, 0, 1}$, the inclusion $\Z \to \Q$ is an embedding, because it is injective and respects sums, products, $0$ and $1$.

Non-example: view graphs as structures with one binary relation.
Let $A$ be the graph with vertices $1, 2$ and no edge, and $B$ the graph with vertices $1, 2$ and one edge between them.
The identity map is an injective homomorphism from $A$ to $B$ but not an embedding, because clause 3 fails for the pair $(1, 2)$.
</document>

<document source="https://tamnd.github.io/math2/node/BQJ9C1CD5X.md" tag="BQJ9C1CD5X" kind="statement">
# Mean value theorem for divided differences

## Statement

Let $n \ge 1$, let $I \subseteq \R$ be an open interval, and let $f \colon I \to \R$ be $n$ times differentiable.
Let $x_0, \dots, x_n \in I$ be pairwise distinct, and let $m$ and $M$ be the smallest and largest of them.
Then there is $\xi \in (m, M)$ such that
$$f[x_0, \dots, x_n] = \frac{f^{(n)}(\xi)}{n!},$$
where $f[x_0, \dots, x_n]$ is the [[divided difference|N5ZHV4HWDC]].

## Remarks

Let $p$ be the [[interpolation polynomial|SS1YMS2NPC]] of $f$ at the nodes.
Then $g = f - p$ has $n + 1$ distinct zeros in $[m, M]$.
Applying [[Rolle's theorem|0194DY35BG]] $n$ times gives a point $\xi \in (m, M)$ with $g^{(n)}(\xi) = 0$.
Since $p^{(n)}$ is the constant $n! \, f[x_0, \dots, x_n]$, the claim follows.

For $n = 1$ this is the [[mean value theorem|0BARQY52N5]].
Example: for $f(x) = x^3$ and the nodes $0, 1, 2$, one has $f[0, 1] = 1$, $f[1, 2] = 7$ and $f[0, 1, 2] = 3$.
Since $f''(\xi)/2 = 3\xi$, the theorem holds with $\xi = 1$.

## Explanations

### L1 human

The ordinary mean value theorem says that the slope of a chord equals the slope of the tangent somewhere in between.
This theorem says the same thing one level up: a higher divided difference equals a scaled higher derivative at some point between the nodes.

### L2 human

It turns statements about derivatives into statements about divided differences and back.
For example, if $f^{(n)} \ge 0$ on $I$, then every $n$-th divided difference of $f$ is nonnegative.
Combined with the [[Newton form|9W77FPTM9E]] of the interpolation polynomial, it gives the [[interpolation error formula|HASA2XSG66]].
</document>

<document source="https://tamnd.github.io/math2/node/BQJ9ZX9P3V.md" tag="BQJ9ZX9P3V" kind="concept">
# Ramification group

## Summary

The ramification groups of a finite Galois extension of local fields form a decreasing chain of normal subgroups $G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots$ of its [[Galois group|0XTY86BZKY]].
The group $G_0$ is the inertia group, and $G_1$ is the wild inertia group.
The chain measures how deeply the extension is [[ramified|H5RC67SD24]].
It controls the different and the conductor of the extension.

## Explanations

### L1 human

An automorphism of a field extension moves elements of the ring of integers.
Some automorphisms move every integer only a little, in the sense of the valuation.
The group $G_i$ collects the automorphisms that move every integer by an amount divisible by the $(i+1)$-st power of a uniformizer.
The larger $i$ is, the closer to the identity these automorphisms are.
For the extension $\Q_2(i) / \Q_2$, complex conjugation lies in $G_0$ and $G_1$, but not in $G_2$.
</document>

<document source="https://tamnd.github.io/math2/node/BQPQAY1FDJ.md" tag="BQPQAY1FDJ" kind="concept">
# Newton–Cotes formulas

## Summary

The Newton–Cotes formulas are the [[quadrature rules|ADSVMCY77A]] obtained by integrating the interpolating polynomial of the integrand at equally spaced nodes.
The closed formulas include the endpoints; the trapezoidal rule and Simpson's rule are the first two.
They are simple and exact for polynomials of the interpolation degree.
High order formulas have negative weights and are unstable, so in practice low order formulas are applied on many small subintervals.

## Explanations

### L1 human

Sample the function at evenly spaced points, draw the polynomial curve through the samples, and take the area under that curve as the estimate.
With two points the curve is a line and the estimate is a trapezoid.
With three points the curve is a parabola, and the result is Simpson's rule with weights proportional to $1, 4, 1$.
Using many points at once does not work well, because polynomials through many equally spaced points can swing wildly.
</document>

<document source="https://tamnd.github.io/math2/node/BQQ0FPFZ67.md" tag="BQQ0FPFZ67" kind="definition">
# Term

## Definition

Let $\sigma$ be a [[signature|WDVVEYD6AT]] and let $V$ be a set of variables, disjoint from the symbols of $\sigma$.
The set of *terms* over $\sigma$ and $V$ is the smallest set of finite strings of symbols such that:

1. every variable $x \in V$ is a term;
2. every constant symbol $c$ of $\sigma$ is a term;
3. if $f$ is a function symbol of $\sigma$ of arity $n \ge 1$ and $t_1, \dots, t_n$ are terms, then $f(t_1, \dots, t_n)$ is a term.

A term with no variables is a *closed term*, or a *ground term*.

## Remarks

In the signature of rings, $\cdot(x, +(y, 1))$ is a term, usually written $x \cdot (y + 1)$.
It is a term by clause 1 for $x$ and $y$, clause 2 for $1$, and clause 3 twice.
In the signature of ordered sets, $x < y$ is not a term: $<$ is a relation symbol, not a function symbol, so clause 3 does not apply.
Every term has a unique reading, so one can define functions on terms by recursion on this construction.
In Mathlib terms are `FirstOrder.Language.Term`.
</document>

<document source="https://tamnd.github.io/math2/node/BQQZ0ZCGQF.md" tag="BQQZ0ZCGQF" kind="concept">
# Complex dynamics

## Summary

Complex dynamics studies what happens when a [[holomorphic function|0NAHXR1FRH]] is applied to its own output again and again.
The main examples are [[rational functions|0XDK9CGYKJ]] on the [[Riemann sphere|0J78KBPFZB]], polynomials on the plane and [[entire functions|0G4F0XB9X2]].
The space splits into the [[Fatou set|C0N0XGC6GK]], where the behaviour is stable, and the [[Julia set|5M4S7P04GR]], where it is chaotic.
The subject joins complex analysis, topology and geometry, and it produced famous fractals such as the [[Mandelbrot set|QSWSVYRCS1]].

## Explanations

### L1 human

Pick a complex number $z$ and a rule, such as $z \mapsto z^2 + c$, and apply the rule over and over.
The list of points you get is the orbit of $z$.
Complex dynamics asks how this orbit behaves and how the answer changes when $z$ or the rule changes a little.
For $z \mapsto z^2$, points inside the unit circle shrink to $0$, points outside run off to infinity, and points on the circle stay on it.
The circle is where the two kinds of behaviour meet, and in other examples this border becomes a fractal.
</document>

<document source="https://tamnd.github.io/math2/node/BQTM3PBQC9.md" tag="BQTM3PBQC9" kind="definition">
# Algebra of sets

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]].
An *algebra of sets* on $X$ is a subset $\mathcal{A}$ of the [[power set|0BC7KX51Z8]] of $X$ such that:

1. $X \in \mathcal{A}$.
2. If $A \in \mathcal{A}$, then its [[complement|0B1FSS1VZW]] $X \setminus A$ is in $\mathcal{A}$.
3. If $A, B \in \mathcal{A}$, then the [[union|07KVB4JSC7]] $A \cup B$ is in $\mathcal{A}$.

## Remarks

On $X = \N$, the family $\mathcal{A}$ of the subsets $A$ such that $A$ is finite or $\N \setminus A$ is finite is an algebra.
Clause 1 holds since $\N \setminus \N = \varnothing$ is finite.
Clause 2 holds since the condition is symmetric in $A$ and $\N \setminus A$.
Clause 3 holds since a union of two finite sets is finite, and a union that contains a set with finite complement has finite complement.
This algebra is not a [[σ-algebra|04WSW8T3SY]]: the set of even numbers is a countable union of singletons, but neither it nor its complement is finite.
The family of all finite subsets of $\N$ is not an algebra, since clause 1 fails: $\N$ is not finite.
An algebra contains $\varnothing$ and is closed under finite intersections and differences, by De Morgan's laws.
So an algebra of sets is exactly a [[ring of sets|KWE1DBMN9B]] that contains $X$.
The name *field of sets* is also common.
</document>

<document source="https://tamnd.github.io/math2/node/BQXB8H365P.md" tag="BQXB8H365P" kind="definition">
# Fano variety

## Definition

Let $k$ be a field and let $X$ be a smooth [[projective variety|A4H4CZ2V35]] over $k$ with [[canonical bundle|W05VYY4HBR]] $\omega_X$.
The variety $X$ is a *Fano variety* if the anticanonical bundle $\omega_X^{-1}$ is [[ample|RRK8P6TXG2]].

## Remarks

In divisor notation, $X$ is Fano when $-K_X$ is ample.
Example: on $\mathbb{P}^n$ one has $\omega = \mathcal{O}(-n-1)$, so $\omega^{-1} = \mathcal{O}(n+1)$ is ample and $\mathbb{P}^n$ is Fano.
More generally, a smooth hypersurface $X \subset \mathbb{P}^n$ of degree $d \ge 1$, with $n \ge 2$, has $\omega_X = \mathcal{O}_X(d - n - 1)$ by the adjunction formula, so it is Fano exactly when $d \le n$.
Non-example: a smooth plane cubic curve has $\omega = \mathcal{O}(3 - 3) = \mathcal{O}$, and the trivial bundle on a curve is not ample.
Many authors also allow mild singularities, such as terminal or log terminal ones, and then ask that $-K_X$ be an ample $\Q$-Cartier divisor.
Over $\C$, smooth Fano varieties are simply connected and rationally connected, and in each dimension they form finitely many deformation families.
</document>

<document source="https://tamnd.github.io/math2/node/BQXRD4GVKZ.md" tag="BQXRD4GVKZ" kind="example">
# Blancmange curve

## Example

For $x \in \R$, let $s(x)$ be the distance from $x$ to the nearest integer, so $0 \le s(x) \le 1/2$.
The *Takagi function* is
$$T(x) = \sum_{n=0}^{\infty} \frac{s(2^n x)}{2^n}.$$
Its graph is the blancmange curve.

Each term is continuous and bounded by $2^{-n-1}$.
By the [[Weierstrass M-test|0NX5NWAFC9]], the series converges uniformly on $\R$, so $T$ is [[continuous|0P27W9RW0G]].
The function $T$ has period $1$ and satisfies $T(0) = 0$ and $T(1/2) = 1/2$.
At $x = 1/3$, each $2^n/3$ has distance $1/3$ to the nearest integer, so $T(1/3) = \sum_{n \ge 0} \frac{1}{3 \cdot 2^n} = \frac 23$.
The maximum value of $T$ is $2/3$.

## Remarks

The function $T$ is [[differentiable|0A61B60B5R]] at no point of $\R$.
The idea of the proof: the partial sum $T_N = \sum_{n < N} s(2^n x)/2^n$ is linear on each interval $[k 2^{-N}, (k+1) 2^{-N}]$.
Its slope there is an integer that changes by $\pm 1$ from $T_N$ to $T_{N+1}$.
So the slopes of $T$ on small dyadic intervals around a point cannot converge.
It is a simpler companion of the [[Weierstrass function|01A0MK1730]], built from tent functions instead of cosines.

## Explanations

### L1 human

Start with a tent of height $1/2$ over each unit interval.
Add tents half as tall and half as wide, then tents a quarter as tall and a quarter as wide, and so on forever.
The sum is a continuous curve that looks like a pudding with ripples at every scale.
Because it has corners at every scale, it has no tangent line anywhere.
It is a concrete continuous nowhere differentiable function.
</document>

<document source="https://tamnd.github.io/math2/node/BR3HD0RRA4.md" tag="BR3HD0RRA4" kind="statement">
# Minimal deviation property of Chebyshev polynomials

## Statement

Let $n \ge 1$ and let $T_n$ be the [[Chebyshev polynomial|M6EAR1JHPP]] of the first kind of degree $n$.
Let $p$ be a [[monic|0C3QBCR46M]] real polynomial of [[degree|0DPF6WRZW6]] $n$.
Then
$$\max_{-1 \le x \le 1} \abs{p(x)} \ge 2^{1-n},$$
and equality holds if and only if $p = 2^{1-n} T_n$.

## Remarks

Proof idea: the polynomial $\tilde T_n = 2^{1-n} T_n$ is monic and equals $(-1)^k 2^{1-n}$ at the $n + 1$ points $\cos(k\pi/n)$, $k = 0, \dots, n$.
If $\max \abs{p} < 2^{1-n}$, then $\tilde T_n - p$ has degree at most $n - 1$ and alternates in sign at these $n + 1$ points.
So it has $n$ zeros, hence it is $0$, a contradiction.
The equality case follows from a refinement of the same counting, or from the uniqueness of best uniform approximation.

Example: for $n = 2$, $x^2$ has maximum $1$ on $[-1, 1]$, while $\tilde T_2 = x^2 - 1/2$ has maximum $1/2$, the smallest possible value.

Equivalently, $x^n - \tilde T_n$ is the best uniform approximation of $x^n$ on $[-1, 1]$ by polynomials of degree at most $n - 1$, with error $2^{1-n}$.
On an interval $[a, b]$, an affine change of variable gives the minimum $2 \left(\frac{b - a}{4}\right)^n$.

## Explanations

### L1 human

A monic polynomial has leading term $x^n$, which grows fast, and lower terms can partly cancel it.
The theorem says how small such a polynomial can stay on $[-1, 1]$: never below $2^{1-n}$ everywhere.
The scaled Chebyshev polynomial is the one that reaches this bound, because it wiggles evenly between $\pm 2^{1-n}$.

### L2 human

This is the first and model case of Chebyshev's theory of best uniform approximation.
It explains why [[Chebyshev nodes|HYK8J56PSH]] are good for interpolation: the node polynomial $\prod (x - x_k)$ appears in the interpolation error, and the zeros of $T_n$ make its maximum as small as possible.
It also gives the minimax error of $x^n$ and lower bounds for polynomial approximation of smooth functions.
</document>

<document source="https://tamnd.github.io/math2/node/BR6KJEJD1C.md" tag="BR6KJEJD1C" kind="definition">
# Hölder space

## Definition

Let $U \subseteq \R^n$ be open, let $k \ge 0$ be an integer, and let $0 < \alpha \le 1$.
For a function $v \colon U \to \R$, its Hölder seminorm of exponent $\alpha$ is
$$[v]_{\alpha, U} = \sup_{x, y \in U, \; x \ne y} \frac{\abs{v(x) - v(y)}}{\abs{x - y}^{\alpha}}.$$
The *Hölder space* $C^{k,\alpha}(\overline{U})$ is the set of functions $u \colon U \to \R$ such that:

1. $u$ is $k$ times continuously differentiable on $U$;
2. every [[partial derivative|09EGC6F8PT]] $\partial^\beta u$ with $\abs{\beta} \le k$ is bounded on $U$;
3. every partial derivative $\partial^\beta u$ with $\abs{\beta} = k$ satisfies $[\partial^\beta u]_{\alpha, U} < \infty$.

Its norm is
$$\norm{u}_{C^{k,\alpha}(\overline{U})} = \sum_{\abs{\beta} \le k} \sup_{U} \abs{\partial^\beta u} + \sum_{\abs{\beta} = k} [\partial^\beta u]_{\alpha, U}.$$

## Remarks

With this norm, $C^{k,\alpha}(\overline{U})$ is a [[Banach space|06GZ956657]].
Condition 3 says that the derivatives of order $k$ are [[Hölder continuous|012PBA0FMD]] with exponent $\alpha$ on $U$, so they extend continuously to the closure $\overline{U}$.
The space $C^{0,1}(\overline{U})$ is the space of bounded Lipschitz functions.
On a bounded set $U$, $C^{k,\beta}(\overline{U}) \subseteq C^{k,\alpha}(\overline{U})$ for $0 < \alpha < \beta \le 1$.
The local space $C^{k,\alpha}(U)$ is the set of $u$ with $u \in C^{k,\alpha}(\overline{V})$ for every open $V$ with compact closure in $U$.

Example: $u(x) = \sqrt{\abs{x}}$ lies in $C^{0,1/2}([-1, 1])$.
It is continuous and bounded by $1$, and $\abs{\sqrt{\abs{x}} - \sqrt{\abs{y}}} \le \sqrt{\abs{\abs{x} - \abs{y}}} \le \abs{x - y}^{1/2}$.

Non-example: the same $u$ is not in $C^{0,1}([-1, 1])$, so it fails condition 3 for $\alpha = 1$.
Indeed $\abs{u(x) - u(0)} / \abs{x - 0} = \abs{x}^{-1/2}$ is unbounded as $x \to 0$.
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<document source="https://tamnd.github.io/math2/node/BRGGGZVQDS.md" tag="BRGGGZVQDS" kind="concept">
# Breakdown point

## Summary

The breakdown point of an estimator is the smallest fraction of the data that can ruin the estimate.
Ruin means that replacing this fraction by arbitrary values can carry the estimate arbitrarily far.
It measures the global robustness of an estimator against gross outliers.
The sample mean has breakdown point $1/n$, which tends to $0$.
The median has breakdown point close to $1/2$, the largest possible value for equivariant estimators of location.

## Explanations

### L1 human

Ask how many data points an adversary must corrupt to push your estimate as far away as they like.
For the average, one corrupted point is enough: making it huge makes the average huge.
For the median, the adversary needs to corrupt about half of the points.
The breakdown point is this number of points divided by the sample size.
</document>

<document source="https://tamnd.github.io/math2/node/BRHQQTZB9K.md" tag="BRHQQTZB9K" kind="definition">
# Extraspecial group

## Definition

Let $p$ be a prime and let $P$ be a finite [[$p$-group|06QYRTX9KQ]].
Let $Z(P)$ be the [[centre|0DZ93493VN]] of $P$.
The group $P$ is *extraspecial* when:

1. $Z(P)$ has order $p$;
2. the [[quotient group|0KYX2N69J0]] $P/Z(P)$ is a nontrivial [[elementary abelian|DZSZ8ZS554]] $p$-group.

## Remarks

In an extraspecial group, the centre, the [[commutator subgroup|0ASGSTXSBW]] and the [[Frattini subgroup|KXJG5VV340]] coincide.
The order of an extraspecial group is $p^{1+2n}$ for some $n \ge 1$, and for each $p$ and $n$ there are exactly two extraspecial groups of that order up to isomorphism.
Example: the [[quaternion group|0NRJGW864D]] $Q_8$ is extraspecial with $p = 2$.
Its centre is $\{\pm 1\}$, of order $2$, and $Q_8/\{\pm 1\}$ has order $4$ with every element of order at most $2$, so it is the Klein four-group, which is elementary abelian.
Non-example: the cyclic group $\Z/p^2\Z$ is not extraspecial, because its centre is the whole group, of order $p^2$, so clause 1 fails.

## Explanations

### L1 human

An extraspecial group is a $p$-group with a centre of order $p$ such that the group becomes elementary abelian, and not trivial, once the centre is factored out.
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<document source="https://tamnd.github.io/math2/node/BRKFX8BQ60.md" tag="BRKFX8BQ60" kind="statement">
# Torelli theorem

## Statement

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]].
Let $C$ and $C'$ be smooth projective connected [[curves|XJM2MT4S3Y]] over $k$.
Let $(J(C), \Theta)$ and $(J(C'), \Theta')$ be their [[Jacobian varieties|ZQ84M9D9TC]] with the principal polarizations given by their theta divisors.
Suppose that an isomorphism of [[abelian varieties|R7W03TXRS9]] $J(C) \to J(C')$ carries the polarization of $\Theta$ to that of $\Theta'$.
Then $C$ and $C'$ are isomorphic.

## Remarks

One proof, due to Andreotti, recovers $C$ from the geometry of $\Theta$.
Let $C$ be a curve that is not hyperelliptic.
The Gauss map sends a smooth point of $\Theta$ to its tangent hyperplane, and its branch locus is the dual of the canonical model of $C$.
Projective duality then gives back $C$.
Over $\C$, let $M_g$ be the moduli of curves of genus $g$ and $A_g$ the moduli of principally polarized abelian varieties of dimension $g$.
The theorem says that the Torelli map $M_g \to A_g$ is injective on points.
For example, two elliptic curves are isomorphic exactly when their Jacobians, which are the curves themselves, are isomorphic.
The polarization cannot be dropped in general: there are non-isomorphic curves whose Jacobians are isomorphic as unpolarized abelian varieties.

## Explanations

### L1 human

Every curve has a Jacobian, an abelian variety built from the curve, together with a special divisor $\Theta$ on it.
The theorem says that the Jacobian with this divisor remembers the curve completely.
So no information is lost when one passes from a curve to its polarized Jacobian.

### L2 human

The theorem reduces questions about curves to questions about principally polarized abelian varieties, which are described by period matrices.
Over $\C$, the period matrix of $C$ in the Siegel upper half space determines $C$.
The image of $M_g$ in $A_g$ is the Jacobian locus, and describing it is the Schottky problem.
There are Torelli theorems for other varieties, such as the global Torelli theorem for K3 surfaces.
In them the curve is replaced by the variety and the Jacobian by a Hodge structure.
</document>

<document source="https://tamnd.github.io/math2/node/BRKPE99MFC.md" tag="BRKPE99MFC" kind="concept">
# Zimmer program

## Summary

The Zimmer program is the study of smooth actions of [[lattices|BTYF5Y5PFY]] in higher-rank [[semisimple Lie groups|XJS07AJ50Y]] on compact manifolds.
Its guiding idea is that such actions are very restricted.
In low dimension they should be finite, and in general they should come from algebraic constructions.
It extends the [[Margulis superrigidity theorem|DSKV65NHZZ]], which describes linear representations of these lattices, to actions by diffeomorphisms.
[[Zimmer's conjecture|JW71J3AKY4]] is its best known test case.

## Explanations

### L1 human

A group such as $\mathrm{SL}_3(\Z)$ is huge and rigid.
The Zimmer program asks in which ways such a group can act by smooth symmetries on a compact space like a circle, a sphere or a torus.
The expected answer is that it can hardly act at all on small spaces, and that on larger spaces its actions come from known algebraic models.
For example, $\mathrm{SL}_3(\Z)$ acts on the torus $\R^3/\Z^3$ through its linear action on $\R^3$, and actions of this kind are expected to be typical.
</document>

<document source="https://tamnd.github.io/math2/node/BRXKAGYEFQ.md" tag="BRXKAGYEFQ" kind="definition">
# Linear canonical transformation

## Definition

Let $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ be a real matrix with $ad - bc = 1$, an element of the [[special linear group|0V9PMN0RBH]] $SL(2, \R)$, and assume $b \ne 0$.
Fix a square root $\sqrt{1/(ib)}$ of the complex number $1/(ib)$.
The *linear canonical transformation* with matrix $M$ sends an integrable function $f \colon \R \to \C$ to the function
$$L_M f(u) = \sqrt{\frac{1}{i b}}\; e^{i\pi \frac{d}{b} u^2} \int_\R e^{-2\pi i \frac{u t}{b}}\, e^{i\pi \frac{a}{b} t^2} f(t)\, dt, \qquad u \in \R.$$

## Remarks

The map $L_M$ is the composite of a multiplication by the chirp $e^{i\pi a t^2 / b}$, a [[Fourier transform|DPVM34ADZV]] rescaled by $1/b$, a multiplication by $e^{i\pi d u^2 / b}$, and the factor $\sqrt{1/(ib)}$ of modulus $\abs{b}^{-1/2}$.
The rescaled Fourier transform multiplies the $L^2$ norm by $\abs{b}^{1/2}$, the factor divides it by the same amount, and the chirps do not change it.
So $L_M$ preserves the $L^2$ norm, and $L_M$ extends to a [[unitary operator|MM0EJ2GDE4]] of $L^2(\R)$, the [[$L^2$ space|SASP14MA1Z]].
For $b = 0$ the transform is defined by $L_M f(u) = \sqrt{d}\, e^{i\pi c d u^2} f(du)$, a scaling followed by a chirp multiplication.
For two matrices one has $L_{M_1} L_{M_2} = \pm L_{M_1 M_2}$; the sign ambiguity comes from the choice of square roots.

Example: $M = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, which has determinant $0 \cdot 0 - 1 \cdot (-1) = 1$.
Here $a = d = 0$ and $b = 1$, so $L_M f(u) = \sqrt{1/i} \int_\R e^{-2\pi i u t} f(t)\, dt$.
With $\sqrt{1/i} = e^{-i\pi/4}$, this is $e^{-i\pi/4} \hat f(u)$, the Fourier transform up to a constant phase.
The matrices $\begin{pmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{pmatrix}$ give the fractional Fourier transforms, in the normalization with the kernel $e^{-2\pi i u t}$, up to a constant phase.

Non-example: $M = \begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}$ has determinant $2$, so it is not in $SL(2, \R)$ and has no linear canonical transformation.
The formula above does not use $c$, and with $a = 1$, $b = 1$, $d = 2$ it gives the transform of $\begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$, the only matrix of determinant $1$ with these entries $a$, $b$, $d$.
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<document source="https://tamnd.github.io/math2/node/BRXX5Y929C.md" tag="BRXX5Y929C" kind="definition">
# Embedding dimension

## Definition

Let $(R, \mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] with residue field $k = R/\mathfrak m$.
The *embedding dimension* of $R$ is the [[dimension|03GSZ8RTZJ]] of its [[cotangent space|Z1CGB483J6]]:
$$\operatorname{edim} R = \dim_k \mathfrak m / \mathfrak m^2.$$

## Remarks

Since $\mathfrak m$ is finitely generated, [[Nakayama's lemma|15JHBYR6KH]] shows that $\operatorname{edim} R$ is the minimal number of generators of the ideal $\mathfrak m$.
The [[dimension theorem for local rings|KW3HY77V6D]] gives $\dim R \le \operatorname{edim} R$, where $\dim R$ is the [[Krull dimension|MVGBS3J1P7]].
When equality holds, $R$ is a [[regular local ring|63XXTF6B5M]].

Example: for a field $k$, the ring $k[\![x_1, \ldots, x_n]\!]$ has $\mathfrak m = (x_1, \ldots, x_n)$, and the classes of the $x_i$ form a basis of $\mathfrak m/\mathfrak m^2$, so $\operatorname{edim} = n$.
For the cusp $R = k[x, y]_{(x,y)}/(y^2 - x^3)$, one has $\operatorname{edim} R = 2$ but $\dim R = 1$.

Non-example: without the Noetherian hypothesis, $\dim_k \mathfrak m / \mathfrak m^2$ need not count generators of $\mathfrak m$.
In a valuation ring with value group $\Q$, every element of positive value $q$ is a product of two elements of value $q/2$.
So $\mathfrak m = \mathfrak m^2$ and $\mathfrak m / \mathfrak m^2 = 0$, although $\mathfrak m \ne 0$; here $\mathfrak m$ is not finitely generated.
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<document source="https://tamnd.github.io/math2/node/BS1RD4F5MD.md" tag="BS1RD4F5MD" kind="definition">
# Duality in order theory

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
The *dual* of $P$ is the poset $P^{\mathrm{op}} = (P, \ge)$ on the same set, where $x \ge y$ means $y \le x$.
The *dual* of a statement about partial orders is the statement obtained by replacing $\le$ with $\ge$ throughout.
Then each defined notion is replaced by its dual notion, such as [[upper bound|0H5J02P8S9]] by lower bound and [[greatest element|0Z7HDDXX3X]] by [[least element|0T2NTJCV83]].

## Remarks

The relation $\ge$ is again a partial order, since reflexivity, antisymmetry and transitivity are symmetric in the two sides.
For example, the dual of $(\N, \le)$ is $\N$ with the reverse order, in which $0$ is the greatest element and there is no least element.
The dual of the dual of $P$ is $P$ itself.
Duality principle: if a statement holds in every poset, then its dual statement holds in every poset.
The reason is that the dual statement for $P$ is the original statement for $P^{\mathrm{op}}$.
The dual of $P$ is also written $P^{\partial}$ or $P^{*}$.

## Explanations

### L1 human

The dual order reads the same order backwards.
A theorem about orders, read backwards, is again a theorem.
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<document source="https://tamnd.github.io/math2/node/BS24BGYFJJ.md" tag="BS24BGYFJJ" kind="definition">
# Strong Cesàro summability

## Definition

Let $\sum_{k \ge 0} a_k$ be a [[series|0JH4MN4K1C]] of real or complex numbers with [[partial sums|0NB643K60Q]] $s_k = a_0 + \dots + a_k$.
Let $q > 0$ be a real number and let $s$ be a number.
The series is *strongly Cesàro summable* with exponent $q$ to $s$ if
$$\lim_{n \to \infty} \frac{1}{n + 1} \sum_{k = 0}^{n} \abs{s_k - s}^q = 0.$$

## Remarks

Example: a convergent series with sum $s$ is strongly Cesàro summable to $s$ for every $q > 0$.
Indeed $\abs{s_k - s}^q \to 0$, and the arithmetic means of a sequence tending to $0$ also tend to $0$.

Non-example: Grandi's series $1 - 1 + 1 - \cdots$ is [[Cesàro summable|N8ZGG7EW6J]] to $1/2$, but it is not strongly Cesàro summable with any exponent.
Its partial sums alternate between $1$ and $0$.
For any $s$, one of $\abs{1 - s}$ and $\abs{s}$ is at least $1/2$.
So at least half of the terms $\abs{s_k - s}^q$ are at least $2^{-q}$, and the means do not tend to $0$.

For $q \ge 1$, the power mean inequality gives
$$\abs{\frac{1}{n+1} \sum_{k=0}^{n} (s_k - s)} \le \left( \frac{1}{n+1} \sum_{k=0}^{n} \abs{s_k - s}^q \right)^{1/q},$$
so strong Cesàro summability to $s$ implies Cesàro summability to $s$.
</document>

<document source="https://tamnd.github.io/math2/node/BS2H0G0T90.md" tag="BS2H0G0T90" kind="definition">
# Random walk on a group

## Definition

Let $G$ be a countable [[group|G5S1M9]] with identity $e$, and let $\mu$ be a [[probability measure|059TJN0NZ1]] on $G$.
Let $X_1, X_2, \dots$ be independent [[random variables|0PNRKMRC55]] with values in $G$, each with distribution $\mu$.
The *random walk* on $G$ with step distribution $\mu$ is the sequence $(Z_n)_{n \ge 0}$ with
$$Z_0 = e, \qquad Z_n = X_1 X_2 \cdots X_n \quad (n \ge 1).$$

## Remarks

The sequence $(Z_n)$ is a [[Markov chain|0G48PYM96Q]] on $G$ with transition probabilities $p(g, h) = \mu(g^{-1} h)$.
These satisfy $p(kg, kh) = p(g, h)$ for all $k \in G$.
The distribution of $Z_n$ is the $n$-fold convolution $\mu^{*n}$.

Example: on $\Z^d$, written additively, let $\mu$ give mass $\frac{1}{2d}$ to each of the $2d$ vectors $\pm e_1, \dots, \pm e_d$.
Then $Z_n = X_1 + \cdots + X_n$ is the simple random walk on the lattice $\Z^d$, which is the walk on the [[Cayley graph|0GYQK9NDJF]] for the standard generators.

Non-example: on $\Z$, let $X_1$ be $\pm 1$ with probability $\frac{1}{2}$ each, and let $X_n = X_1$ for all $n$.
Each $X_n$ has the same distribution, but they are not independent, so $Z_n = n X_1$ is not a random walk on $\Z$ in this sense.
</document>

<document source="https://tamnd.github.io/math2/node/BS7NSJFFW3.md" tag="BS7NSJFFW3" kind="definition">
# Bring radical

## Definition

Let $a \in \R$.
The *Bring radical* of $a$, written $\operatorname{BR}(a)$, is the unique real [[root|00EHX6FWC0]] of the polynomial $x^5 + x + a$.

## Remarks

The definition makes sense: $g(x) = x^5 + x + a$ is strictly increasing on $\R$, since $g'(x) = 5x^4 + 1 > 0$, and it tends to $\pm\infty$ as $x \to \pm\infty$, so it has exactly one real root.
Example: $\operatorname{BR}(-2) = 1$, since $1^5 + 1 - 2 = 0$; and $\operatorname{BR}(0) = 0$.
Non-example: for the polynomial $x^5 - x + a$, the derivative $5x^4 - 1$ changes sign, and for $a = 0$ there are three real roots $-1, 0, 1$, so no "unique real root" is defined in the same way.
The function $\operatorname{BR}$ is odd, continuous and strictly decreasing on $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/BSA6GFT8SQ.md" tag="BSA6GFT8SQ" kind="statement">
# Von Neumann bicommutant theorem

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] with [[algebra of bounded operators|BMDG4JQPEH]] $B(H)$.
Let $M$ be a subalgebra of $B(H)$ that contains the identity $I$ and is closed under the [[adjoint|6HSJBJ8TTQ]].
Let $M''$ be the [[bicommutant|1VC7EJ1ZBZ]] of $M$.
Then the following are equivalent:

1. $M = M''$;
2. $M$ is closed in the [[weak operator topology|4XM1VN1C1N]];
3. $M$ is closed in the [[strong operator topology|KHJ2G2R5G7]].

In particular, $M''$ is the closure of $M$ in each of these two topologies.

## Remarks

Every commutant is weakly closed, so 1 implies 2, and 2 implies 3 because the strong topology is finer.
For 3 implies 1, let $T \in M''$ and $x \in H$.
The [[orthogonal projection|0B8KZRT28M]] $P$ onto the closure of $Mx$ commutes with $M$ because $M$ is closed under adjoints, so $T$ commutes with $P$ and $Tx \in \overline{Mx}$, using $x = Ix \in Mx$.
Applying this to the diagonal action of $M$ on $H^n$ handles $n$ vectors at once, which shows that $T$ is a strong limit of elements of $M$.

The identity is needed: the algebra $M = \set{0}$ is closed under adjoints and weakly closed, but $M' = B(H)$ and $M'' = \C I \ne M$.
More generally, without the identity the theorem holds in the form $M'' = \overline{M}$ only when $M$ acts nondegenerately.

For example, on $\C^n$ the diagonal matrices $D$ satisfy $D' = D$, so $D'' = D' = D$, as the theorem predicts for a finite-dimensional, hence closed, algebra.

## Explanations

### L1 human

A von Neumann algebra can be described in two very different ways.
One is analytic: it is closed under weak limits of operators.
The other is algebraic: it consists of all operators that commute with everything that commutes with it.
The theorem says these two descriptions agree, as long as the algebra contains the identity and is closed under adjoints.

### L2 human

The theorem lets one produce von Neumann algebras for free.
The commutant of any self-adjoint set $S$ of operators is one, and so is the bicommutant $S''$, the von Neumann algebra generated by $S$.
It turns questions about topological closure into algebraic questions about commuting operators.
This is how one shows, for example, that the spectral projections of a self-adjoint element of $M$ lie in $M$.
Together with the Kaplansky density theorem, it is the basic tool for approximating elements of $M''$ by elements of $M$.
</document>

<document source="https://tamnd.github.io/math2/node/BSCYBP6AS6.md" tag="BSCYBP6AS6" kind="concept">
# Nonlinear integral equation

## Summary

A nonlinear integral equation is an [[integral equation|R3SG1SN7JQ]] that is not a [[linear integral equation|AYZCKG34QK]].
Typical forms are the Urysohn equation $\varphi(x) = f(x) + \int_a^b H(x, s, \varphi(s)) \, ds$ and the Hammerstein equation $\varphi(x) = f(x) + \int_a^b K(x, s) g(s, \varphi(s)) \, ds$.
They arise from nonlinear differential equations, such as initial value problems for $y' = F(x, y)$.
The main tools are fixed point theorems, such as the [[Banach fixed point theorem|0VK6VEH345]].

## Explanations

### L1 human

In a nonlinear equation the unknown function appears in a curved way, for example squared or inside a sine.
Then solutions cannot be added, and there may be no solution, one solution or several.
For example, $\varphi(x) = 1 + \int_0^x \varphi(s)^2 \, ds$ is the integral form of $y' = y^2$, $y(0) = 1$.
Its solution $\varphi(x) = 1/(1 - x)$ exists only for $x < 1$.
</document>

<document source="https://tamnd.github.io/math2/node/BSCZN8Y1AN.md" tag="BSCZN8Y1AN" kind="example">
# Arnold's cat map

## Example

Let $\mathbb{T}^2 = \R^2/\Z^2$ be the [[torus|03NM7DZXXC]] and let $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$.
Arnold's cat map is $f(x, y) = (2x + y, \; x + y) \bmod 1$.
It is well defined because $A$ maps $\Z^2$ to $\Z^2$, and it is a diffeomorphism because $\det A = 1$, so $A^{-1}$ also has integer entries.

Check that it is an [[Anosov diffeomorphism|RCQH8P6DPQ]].
The derivative of $f$ is $A$ at every point.
The characteristic polynomial is $t^2 - 3t + 1$, so the [[eigenvalues|0ZND7NQ970]] are $\lambda_\pm = (3 \pm \sqrt5)/2$, with $\lambda_+ \approx 2.618$ and $\lambda_- = 1/\lambda_+ \approx 0.382$.
Let $E^u$ and $E^s$ be the eigenlines for $\lambda_+$ and $\lambda_-$; since $A$ is symmetric they are orthogonal.
Then $\norm{A^n v} = \lambda_-^n \norm{v}$ on $E^s$ and $\norm{A^{-n} v} = \lambda_+^{-n}\norm{v}$ on $E^u$, which is the required hyperbolicity with $C = 1$.

## Remarks

The map preserves [[Lebesgue measure|2Z2QVWZ262]], since $\abs{\det A} = 1$, and it is mixing for this measure.
Its periodic points are exactly the points with rational coordinates, so they are dense.
The number of fixed points of $f^n$ is $\abs{\det(A^n - I)} = \lambda_+^n + \lambda_+^{-n} - 2$; for $n = 1$ this is $1$, the origin.
Its topological entropy is $\log \lambda_+$.

## Explanations

### L1 human

Draw a picture on a square whose opposite sides are glued, so that it is really a torus.
Shear and stretch the picture by the matrix, then cut the pieces that stick out and shift them back into the square.
After a few steps the picture looks like random noise, because nearby points are pulled apart by a factor of about $2.6$ each time.
Yet the map is exactly invertible, and every point with rational coordinates returns to where it started.
</document>

<document source="https://tamnd.github.io/math2/node/BSDCDJR6RQ.md" tag="BSDCDJR6RQ" kind="concept">
# Bring radical

## Summary

The Bring radical of a real number $a$ is the unique real root of $x^5 + x + a$.
It plays for quintic equations the role that $n$-th roots play for the equations $x^n = a$.
Using only radicals, every quintic equation can be reduced to the Bring–Jerrard form $x^5 + px + q = 0$.
With the Bring radical extended to complex arguments, the roots of every quintic can then be written with radicals and Bring radicals.

## Explanations

### L1 human

The [[Abel–Ruffini theorem|163A2WQQJX]] says that ordinary roots are not enough to solve all quintic equations.
The Bring radical is a new kind of root that fills this gap.
Just as $\sqrt[3]{a}$ is the real number whose cube is $a$, the Bring radical of $a$ is the real number $x$ with $x^5 + x = -a$.
For example, the Bring radical of $-2$ is $1$, since $1^5 + 1 - 2 = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/BSJ8X2XRN9.md" tag="BSJ8X2XRN9" kind="concept">
# Fundamental solution of the Laplacian

## Summary

The fundamental solution of the Laplacian is a radial function $\Phi$ on $\R^n \setminus \set{0}$ with $-\Delta \Phi = \delta$, the Dirac delta at the origin.
It equals $-\frac{1}{2\pi} \log \abs{x}$ in the plane and $\frac{1}{n(n-2)\omega_n} \abs{x}^{2-n}$ in dimension $n \ge 3$, where $\omega_n$ is the volume of the unit ball.
It is the [[fundamental solution|K2NK9A8M5F]] of the Laplace operator that is used throughout potential theory, up to sign.
Convolving it with a source $f$ solves Poisson's equation, and it is the building block of Green's functions and of the Newtonian and logarithmic potentials.

## Explanations

### L1 human

Think of a single unit point charge at the origin of space.
The electrostatic potential it creates is proportional to one over the distance to the origin, and this is the fundamental solution in three dimensions.
In the plane the potential of a point source grows like the logarithm of the distance instead.
Away from the origin the function is harmonic; all of its "source" sits at the origin.
For example, in $\R^3$ it is $\Phi(x) = \frac{1}{4\pi \abs{x}}$.
</document>

<document source="https://tamnd.github.io/math2/node/BSRJN5M0R4.md" tag="BSRJN5M0R4" kind="definition">
# Abel equation

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a given map.
The *Abel equation* for $f$ is the [[iterative functional equation|BCGBQ3HK26]]
$$\alpha(f(x)) = \alpha(x) + 1 \quad \text{for all } x \in X,$$
for an unknown [[function|0EPAT51DXQ]] $\alpha \colon X \to \R$ or $\alpha \colon X \to \C$.
A solution is called an [[Abel function|2QBWT7T02Y]] of $f$.

## Remarks

Example: for $f(x) = x^2$ on $X = (1, \infty)$, $\alpha(x) = \log_2 \ln x$ is a solution, since $\log_2 \ln(x^2) = \log_2(2 \ln x) = \log_2 \ln x + 1$.
Non-example: for $f(x) = 2x$ on $(0, \infty)$, $\alpha(x) = x$ is not a solution, since $\alpha(f(2)) = 4$ while $\alpha(2) + 1 = 3$.
If $\alpha$ is a real-valued solution and $p \colon \R \to \R$ satisfies $p(t + 1) = p(t)$, then $\alpha + p \circ \alpha$ is again a solution, so solutions are far from unique without extra conditions.
If $f$ has a fixed point $x_0$, no solution is defined at $x_0$, since $\alpha(x_0) = \alpha(x_0) + 1$ is impossible; so the equation is usually posed on a domain without fixed points.
</document>

<document source="https://tamnd.github.io/math2/node/BT4QFQ9MJH.md" tag="BT4QFQ9MJH" kind="definition">
# Tilting theory

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $k$.
A finitely generated left $A$-module $T$ is a *tilting module* if:

1. the [[projective dimension|MR9QCH1PNA]] of $T$ is at most $1$;
2. $\operatorname{Ext}^1_A(T, T) = 0$, where $\operatorname{Ext}$ is the [[Ext functor|4P7GBWJ4RR]];
3. the number of isomorphism classes of [[indecomposable|E2P29GCD9T]] direct summands of $T$ equals the number of isomorphism classes of [[simple|G00C22AEDH]] $A$-modules.

*Tilting theory* studies the relation between $A$ and the [[endomorphism ring|AWSBZWWMG9]] $B = \End_A(T)$ of a tilting module $T$, through the functors $\Hom_A(T, \cdot)$ and $\operatorname{Ext}^1_A(T, \cdot)$.

## Remarks

Example: $T = A$ is a tilting module, since it is projective and its indecomposable summands are the indecomposable projectives, one for each simple module.
Example: let $A$ be the path algebra of $1 \to 2$, with $S_2 = P_2$, $P_1$ and $S_1$ as indecomposables, and let $T = P_1 \oplus S_1$.
The projective resolution $0 \to P_2 \to P_1 \to S_1 \to 0$ gives clause 1.
Clause 2 holds since $P_1$ is projective and injective and $\operatorname{Ext}^1_A(S_1, S_1) = 0$, and clause 3 holds with $2$ summands.
Non-example: $T = S_1 \oplus S_2$ fails clause 2, since the non-split sequence $0 \to S_2 \to P_1 \to S_1 \to 0$ gives $\operatorname{Ext}^1_A(S_1, S_2) \ne 0$.
Happel proved that for a tilting module $T$ the bounded derived categories of $\operatorname{mod} A$ and $\operatorname{mod} B$ are equivalent as triangulated categories.
</document>

<document source="https://tamnd.github.io/math2/node/BT5B00W898.md" tag="BT5B00W898" kind="definition">
# Spherical geometry

## Definition

Let $S^2 = \set{x \in \R^3 : \abs{x} = 1}$ be the unit [[sphere|T62369RFH6]], and let $x \cdot y$ be the [[dot product|0S52TNNTH9]] of $\R^3$.
*Spherical geometry* is the geometry on $S^2$ with these data:

1. The points are the points of $S^2$.
2. The lines are the *great circles*, the sets $S^2 \cap P$ where $P$ is a [[plane|0V3XAMF3C1]] through the origin.
3. The distance is $d(x, y) = \arccos(x \cdot y) \in [0, \pi]$, the length of a shortest arc of great circle from $x$ to $y$.
4. The angle between two great circles at a common point is the angle between their planes.

## Remarks

Example: the equator $\set{x_3 = 0} \cap S^2$ is a line, since the plane $x_3 = 0$ passes through the origin.
The points $N = (0, 0, 1)$, $A = (1, 0, 0)$, $B = (0, 1, 0)$ are pairwise at distance $\arccos 0 = \pi/2$, and the three planes $x_1 = 0$, $x_2 = 0$, $x_3 = 0$ are pairwise perpendicular, so the triangle $NAB$ has three right angles.
Non-example: the circle of latitude $\set{x_3 = 1/2} \cap S^2$ is not a line, because the plane $x_3 = 1/2$ does not pass through the origin.
Two distinct great circles always meet in a pair of antipodal points, so there are no parallels.
Two points that are not antipodal lie on exactly one great circle, but antipodal points lie on infinitely many.
On a sphere of radius $R$ the same definitions apply with distance $R \arccos(x \cdot y / R^2)$; the geometry has constant curvature $1/R^2$.
</document>

<document source="https://tamnd.github.io/math2/node/BT5V3TSNY6.md" tag="BT5V3TSNY6" kind="concept">
# Place (number field)

## Summary

A place of an [[algebraic number field|MWBDFS2KJ5]] $K$ is a class of equivalent nontrivial absolute values on $K$.
The finite places correspond to the nonzero prime ideals of the ring of integers.
The infinite places correspond to the real embeddings and the pairs of complex conjugate embeddings of $K$.
Each place gives a completion of $K$, which is a [[local field|J85FZEQG38]] or $\R$ or $\C$.
The places together are the index set of the [[adele ring|SEH68BA1QQ]].
For $K = \Q$, [[Ostrowski's theorem|GCQ23XT6VD]] says that the places are the primes $p$ and one infinite place.

## Explanations

### L1 human

A place is a way of measuring the size of the numbers of a field.
On $\Q$ there is the usual size, and for each prime $p$ there is the [[p-adic size|WGN82TCEF2]].
For the $p$-adic size, a number is small when it is divisible by a high power of $p$.
For example, $\abs{75}_5 = 1/25$, because $75 = 3 \cdot 5^2$.
Number theorists treat all these ways of measuring on an equal footing, and the places are their list.
</document>

<document source="https://tamnd.github.io/math2/node/BT6T702C28.md" tag="BT6T702C28" kind="concept">
# p-adic number

## Summary

For a [[prime number|01826KMG2H]] $p$, the $p$-adic numbers form the field $\Q_p$, the completion of $\Q$ for the [[p-adic absolute value|WGN82TCEF2]].
In $\Q_p$, a number is small when it is divisible by a high power of $p$.
The field $\Q_p$ is a [[local field|J85FZEQG38]] of characteristic $0$.
Questions about congruences modulo all powers of $p$ become questions about equations in $\Q_p$.

## Explanations

### L1 human

A real number can be written with infinitely many digits after the decimal point.
A $p$-adic number is written in base $p$ with infinitely many digits to the left instead, and only finitely many to the right.
Two numbers are close when their difference is divisible by a large power of $p$.
For example, in $\Q_5$ the powers $5, 25, 125, \ldots$ get smaller and smaller and tend to $0$.
The sum $4 + 4 \cdot 5 + 4 \cdot 5^2 + \cdots$ converges in $\Q_5$, and its value is $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/BT7D4VQ6Y0.md" tag="BT7D4VQ6Y0" kind="definition">
# Initial algebra semantics

## Definition

Let $\Sigma$ be a signature of function symbols and let $K$ be a class of $\Sigma$-algebras.
An algebra $I \in K$ is *initial in $K$* if for every $A \in K$ there is exactly one [[homomorphism|KBXSZC6BV1]] $I \to A$.
The [[term algebra|3YYKGWJJND]] $T_\Sigma$ of ground terms is initial in the class of all $\Sigma$-algebras.
In *initial algebra semantics*, the meaning of a ground term $t$ in a $\Sigma$-algebra $A$ is $[\![t]\!]_A = h_A(t)$, where $h_A \colon T_\Sigma \to A$ is the unique homomorphism.
Let $E$ be a set of [[equations|PKD440G2XW]].
The meaning of the specification $(\Sigma, E)$ is the initial algebra of the class of $\Sigma$-algebras that satisfy $E$.
It is determined up to isomorphism.

## Remarks

Since $h_A$ is a homomorphism, $[\![f(t_1, \dots, t_n)]\!]_A = f^A([\![t_1]\!]_A, \dots, [\![t_n]\!]_A)$, so meanings are compositional.
The initial algebra of the algebras satisfying $E$ exists: it is the quotient of $T_\Sigma$ by the least congruence that contains all ground instances of the equations of $E$.

Example: let $\Sigma = \set{0, s, +}$ with arities $0, 1, 2$, and let $E$ consist of $x + 0 = x$ and $x + s(y) = s(x + y)$.
The natural numbers $\N$ with zero, successor and addition satisfy $E$.
Every ground term equals some $s^n(0)$ by $E$, and distinct numerals $s^m(0) \ne s^n(0)$ are separated in $\N$, so $\N$ is initial among the models of $E$.

Non-example: $\N$ is not initial in the class of all $\Sigma$-algebras.
In $T_\Sigma$ the terms $s(0)$ and $s(0) + 0$ are different, but in $\N$ they have the same value $1$.
A homomorphism $\N \to T_\Sigma$ would send $1$ to both terms, so there is none.

The general notion of an [[initial algebra|R76EC56V9R]] for a functor contains this one: $T_\Sigma$ is the initial algebra of the polynomial functor of $\Sigma$.
</document>

<document source="https://tamnd.github.io/math2/node/BTCXG67H69.md" tag="BTCXG67H69" kind="definition">
# Wythoff construction

## Definition

Let $W$ be a finite group of isometries of $\R^n$ generated by [[reflections|06BRDF3F0E]] in hyperplanes through the origin, a finite [[Coxeter group|G6FND7X69J]].
Assume that no nonzero vector is fixed by all of $W$.
Let $C$ be a fundamental chamber of $W$: the closure of one connected component of the complement of all the reflecting hyperplanes of $W$.
Then $C$ is a simplicial cone bounded by $n$ hyperplanes $H_1, \dots, H_n$, and the reflections $s_1, \dots, s_n$ in them generate $W$.
Choose a point $v \in C$ with $v \ne 0$.
The *Wythoff construction* produces the polytope
$$P(v) = \operatorname{conv}(W v),$$
the [[convex hull|0FQSDC2EQ2]] of the [[orbit|0YQPZB5Q1G]] $W v = \set{w(v) : w \in W}$.

## Remarks

All points of $W v$ lie on the sphere of radius $\norm{v}$, so each of them is a vertex of $P(v)$, and $W$ acts transitively on the vertices of $P(v)$.
The polytope depends, up to similarity, mainly on the set of mirrors $H_i$ that do not contain $v$; in a Coxeter diagram one marks the corresponding nodes with rings.
If $v$ does not lie on the mirror $H_i$, then $v$ and $s_i(v)$ are joined by an edge of length twice the distance from $v$ to $H_i$.
When $v$ has the same distance to every mirror that does not contain it, all edges of $P(v)$ have the same length, and $P(v)$ is a [[uniform polytope|BZXHGNGQAX]].
Let $W$ be the symmetry group of a regular polytope.
If $v$ lies on all mirrors except the one at an end of the Coxeter diagram, then $P(v)$ is that regular polytope or its dual, up to similarity.

Example: let $W$ be the symmetry group of the square, of order $8$, generated by the reflections in the lines $y = 0$ and $y = x$, with chamber $C = \set{(x, y) : 0 \le y \le x}$.
For $v = (1, 0)$, on the mirror $y = 0$, the orbit is $\set{(\pm1, 0), (0, \pm1)}$, and $P(v)$ is a square.
For $v = (1, 1)$, on the mirror $y = x$, the orbit is $\set{(\pm1, \pm1)}$, again a square.
For $v = (1, \tan(\pi/8))$, which is at equal distance from both mirrors, the orbit has $8$ points spaced by angles $\pi/4$, and $P(v)$ is a regular octagon.

Non-example: the choice $v = 0$ is excluded; its orbit is the single point $0$, and the construction gives no polytope of dimension $n$.
</document>

<document source="https://tamnd.github.io/math2/node/BTDZHD9HG3.md" tag="BTDZHD9HG3" kind="example">
# Projective special linear group

## Example

Let $F$ be a field and let $n \ge 2$ be an integer.
The *projective special linear group* is the [[quotient group|0KYX2N69J0]]
$$PSL_n(F) = SL_n(F) / Z,$$
where $SL_n(F)$ is the [[special linear group|0V9PMN0RBH]] and $Z$ is its [[centre|0DZ93493VN]].
The centre $Z$ consists of the scalar matrices $\lambda I_n$ with $\lambda^n = 1$.
For a [[finite field|0ZXZ82FTX2]] $\F_q$ with $q$ elements, write $PSL_n(q)$.

Take $n = 2$ and $q = 7$.
The order of $SL_2(7)$ is $q(q^2 - 1) = 7 \cdot 48 = 336$.
The scalars $\lambda I_2$ with $\lambda^2 = 1$ are $\pm I_2$, so $\abs{Z} = 2$.
Hence $PSL_2(7)$ has order $168$.
It is a [[simple group|0YMYMNQ0QZ]], and it is the second smallest non-abelian simple group, after the alternating group $A_5$ of order $60$.

## Remarks

The group $PSL_n(q)$ has order
$$\frac{1}{\gcd(n, q - 1)} \, q^{n(n-1)/2} \prod_{i=2}^{n} (q^i - 1).$$
It is simple for all $n \ge 2$ and all prime powers $q$, except $PSL_2(2) \cong S_3$ and $PSL_2(3) \cong A_4$.
These groups form one of the infinite families of [[groups of Lie type|NPEBABQ49M]] in the [[classification of finite simple groups|06J6P03MG7]].
The group $PSL_n(F)$ acts faithfully on the projective space of lines in $F^n$, which explains the name.

## Explanations

### L1 human

Start with the $n \times n$ matrices of determinant $1$ over a field.
Then declare two matrices equal when one is a scalar multiple of the other.
What is left is the projective special linear group.
Over finite fields these groups give an endless supply of simple groups, the building blocks of all finite groups.
The smallest interesting one, $PSL_2(7)$, has $168$ elements and is the symmetry group of the Fano plane.
</document>

<document source="https://tamnd.github.io/math2/node/BTH3GSAWY3.md" tag="BTH3GSAWY3" kind="statement">
# Soul theorem

## Statement

Let $(M, g)$ be a connected complete noncompact [[Riemannian manifold|7CBT4SM83Y]] with nonnegative [[sectional curvature|1X42JAA349]].
Then $M$ contains a compact, totally convex, totally geodesic embedded submanifold $S$ without boundary, called a *soul* of $M$.
Moreover, $M$ is [[diffeomorphic|0KZZ70ZHXP]] to the total space of the normal [[vector bundle|P4DZM975BV]] of $S$ in $M$.

## Remarks

The theorem is due to Cheeger and Gromoll (1972).
The idea of the proof: fix a point $p$ and remove from $M$ the open balls $B(\gamma(t), t)$ for every ray $\gamma$ from $p$ and every $t > 0$.
The remaining set is compact, and it is totally convex by the Toponogov comparison theorem.
Shrinking this set step by step through levels of the distance to its boundary ends at a totally convex set without boundary, the soul.
A deformation along the gradient-like flow of the distance to $S$ gives the diffeomorphism with the normal bundle.

Example: the flat cylinder $S^1 \times \R$ has the soul $S^1 \times \set{0}$, and it is the normal bundle of this circle.
The paraboloid $z = x^2 + y^2$ has positive curvature, its soul is a single point, and it is diffeomorphic to $\R^2$.

A soul is not unique as a subset, but any two souls are isometric.
Perelman proved in 1994 the soul conjecture of Cheeger and Gromoll.
It says: if moreover the sectional curvature is positive at some point, then the soul is a point and $M$ is diffeomorphic to $\R^n$.

## Explanations

### L1 human

A complete, infinite space that is nowhere negatively curved has a compact core, the soul.
The whole space is this core with directions going off to infinity attached, like a cylinder around its central circle.
So the topology of the space is that of the soul.

### L2 human

The soul theorem reduces the topology of open manifolds with $K \ge 0$ to that of compact ones: $M$ is homotopy equivalent to its soul.
It is a main example of how Toponogov's comparison theorem is used to build convex sets.
</document>

<document source="https://tamnd.github.io/math2/node/BTJR5NZPE3.md" tag="BTJR5NZPE3" kind="definition">
# Landau's constants

## Definition

Let $\mathbb{D}$ be the [[unit disc|8G5ABXRK85]], and let $\mathcal{F}$ be the set of [[holomorphic|0DG0Z72CQD]] functions $f \colon \mathbb{D} \to \C$ with $f'(0) = 1$.
For $f \in \mathcal{F}$:

1. $L_f$ is the supremum of the radii $r$ such that $f(\mathbb{D})$ contains an open disc of radius $r$.
2. $B_f$ is the supremum of the radii $r$ of the open discs $f(\Omega)$, where $\Omega \subseteq \mathbb{D}$ is open and $f$ is [[injective|0PVG1CRAJN]] on $\Omega$.

*Landau's constants* are the three numbers
$$B = \inf_{f \in \mathcal{F}} B_f, \qquad L = \inf_{f \in \mathcal{F}} L_f, \qquad A = \inf \set{L_f : f \in \mathcal{F},\ f \text{ injective}}.$$
The number $B$ is the *Bloch constant*, $L$ is the *Landau constant*, and $A$ is the *univalent Landau constant*.

## Remarks

Since a disc covered injectively is a disc contained in the image, $B_f \le L_f$ for all $f$, so $B \le L$.
The infimum $A$ is taken over a smaller set of functions than $L$, so $L \le A$.
Example: for $f(z) = z$, the image is $\mathbb{D}$, so $L_f = B_f = 1$.
For an injective $f \in \mathcal{F}$, the [[Koebe quarter theorem|WX0HMW4GT7]] gives a disc of radius $1/4$ in the image, so $A \ge 1/4$.
Non-example: $f(z) = 2z$ is not in $\mathcal{F}$, because $f'(0) = 2$; without the normalisation the infimum would be $0$, as $f(z) = \varepsilon z$ shows.
[[Bloch's theorem|KW31KDX7TB]] is the statement $B > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/BTM6WJ0A52.md" tag="BTM6WJ0A52" kind="statement">
# Implicit function theorem in Banach spaces

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $X$, $Y$ and $Z$ be [[Banach spaces|06GZ956657]] over $\mathbb{K}$.
Let $\Omega \subseteq X \times Y$ be open, and let $F \colon \Omega \to Z$ be of class $C^1$ in the sense of the [[Fréchet derivative|06EH97V2A7]].
Let $D_x F$ and $D_y F$ be the partial derivatives of $F$, with values in the continuous linear maps $X \to Z$ and $Y \to Z$.
Let $(a, b) \in \Omega$ with $F(a, b) = 0$, and suppose that $D_y F(a, b) \colon Y \to Z$ is bijective.
Then there are open sets $U \subseteq X$ and $V \subseteq Y$ with $a \in U$, $b \in V$ and $U \times V \subseteq \Omega$, and a unique map $g \colon U \to V$, such that the following holds.
For $(x, y) \in U \times V$, $F(x, y) = 0$ if and only if $y = g(x)$.
After shrinking $U$ if needed, $g$ is of class $C^1$, $D_y F(x, g(x))$ is bijective for each $x \in U$, and
$$Dg(x) = -D_y F(x, g(x))^{-1}\, D_x F(x, g(x)).$$

## Remarks

Apply the [[inverse function theorem in Banach spaces|M218MWZ62X]] to $\Phi(x, y) = (x, F(x, y))$ from $\Omega$ to $X \times Z$.
Its derivative at $(a, b)$ is bijective because $D_y F(a, b)$ is, and the local inverse of $\Phi$ has the form $(x, z) \mapsto (x, h(x, z))$; then $g(x) = h(x, 0)$.
The formula for $Dg$ comes from differentiating $F(x, g(x)) = 0$ with the chain rule.

Example: let $h \in C([0,1])$ and $F(\lambda, u) = u - h - \lambda \sin \circ u$ on $\R \times C([0,1])$.
Then $F(0, h) = 0$ and $D_u F(0, h) = \id$.
So for $\lambda$ near $0$, the equation $u = h + \lambda \sin(u)$ has exactly one solution $u = g(\lambda)$ near $h$, and $g$ is $C^1$.

For finite-dimensional $X$, $Y$ and $Z$, the theorem is the [[implicit function theorem|0BXBADR94C]] of several-variable calculus.

## Explanations

### L1 human

The theorem says when an equation $F(x, y) = 0$ can be solved for $y$ as a function of $x$ near a known solution.
The test is linear: the derivative in $y$ must be invertible.
It works when $x$ and $y$ are functions themselves, as long as the spaces are complete normed spaces.

### L2 human

The theorem is the basic tool of perturbation theory.
A solution of a problem at one parameter value persists for nearby parameters when the linearized problem is uniquely solvable.
The solution then depends smoothly on the parameter.
Bifurcation theory studies exactly the points where $D_y F$ fails to be invertible.
It is equivalent to the [[inverse function theorem in Banach spaces|M218MWZ62X]], which is the special case $F(x, y) = f(y) - x$.
</document>

<document source="https://tamnd.github.io/math2/node/BTPQHQNJVY.md" tag="BTPQHQNJVY" kind="statement">
# Inversion of the Weierstrass transform

## Statement

Let $f \in L^2(\R)$, an element of the [[$L^2$ space|SASP14MA1Z]], and let $F = Wf$ be its [[Weierstrass transform|5HYB6DDZH8]].
Let $\hat g(\xi) = \int_\R g(x)\, e^{-i x \xi}\, dx$ denote the [[Fourier transform|DPVM34ADZV]], extended to $L^2(\R)$.
Then $F \in L^2(\R)$, $\hat F(\xi) = e^{-\xi^2} \hat f(\xi)$ for almost every $\xi$, and $f$ is recovered from $F$ as the function in $L^2(\R)$ whose Fourier transform is
$$\hat f(\xi) = e^{\xi^2}\, \hat F(\xi).$$
In particular $W$ is injective on $L^2(\R)$.

## Remarks

The kernel $k(u) = \frac{1}{\sqrt{4\pi}} e^{-u^2/4}$ is integrable with $\hat k(\xi) = e^{-\xi^2}$, and $F = k * f$.
The convolution theorem on $L^1 * L^2$ gives $\hat F = \hat k \hat f$, and the [[Plancherel theorem|HPZHRZ0502]] makes the Fourier transform a bijection of $L^2(\R)$ up to the factor $2\pi$, which yields the formula.
Formally $W = e^{D^2}$ with $D = d/dx$, and the inverse is $e^{-D^2}$; for example, $W[e^{ax}] = e^{a^2} e^{ax}$ and $e^{-D^2} e^{ax} = e^{-a^2} e^{ax}$.
The inversion is unstable: the factor $e^{\xi^2}$ grows very fast, so a small change of $F$ at high frequencies can change $f$ a lot.
Not every function in $L^2$ is a Weierstrass transform: $F$ must satisfy $e^{\xi^2} \hat F \in L^2$, and every such $F$ extends to an entire function.

## Explanations

### L1 human

The Weierstrass transform blurs a function as heat flow blurs a temperature profile.
This theorem says the blur loses no information: the original profile can be computed back from the blurred one.
In practice the computation is very sensitive to small errors, as running heat flow backwards in time always is.

### L2 human

The inversion is the uniqueness theorem for the backward heat equation at a fixed time, in the class of square-integrable data.
On the Fourier side it is division by the Gaussian $e^{-\xi^2}$, which never vanishes; this is why the transform is injective.
The same reasoning applies to any convolution with an integrable kernel whose Fourier transform has no zeros.
The operator notation $e^{D^2}$ also explains how $W$ acts on polynomials: it maps the Hermite polynomial $H_n(x/2)$ to $x^n$, since $e^{-D^2} x^n = H_n(x/2)$ for the physicists' Hermite polynomials $H_n$.
</document>

<document source="https://tamnd.github.io/math2/node/BTRNZKN1BP.md" tag="BTRNZKN1BP" kind="definition">
# Totally positive matrix

## Definition

Let $A$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] with real entries.
The matrix $A$ is *totally positive* when every [[minor|0BT9BYWVRY]] of $A$ is positive.
That is, for all $1 \le k \le \min(m, n)$, all rows $i_1 < \cdots < i_k$ and all columns $j_1 < \cdots < j_k$, the $k \times k$ submatrix on these rows and columns has positive determinant.

## Remarks

Example: $\begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$ is totally positive: its $1 \times 1$ minors $1, 1, 1, 2$ are positive, and its only $2 \times 2$ minor is $2 - 1 = 1 > 0$.
Non-example: $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$ is not totally positive: all entries are positive, but the determinant is $1 - 4 = -3$.
The matrix is *totally nonnegative* when every minor is nonnegative.
Some authors use "totally positive" for this weaker condition and "strictly totally positive" for the condition above.
The [[Vandermonde matrix|Q3JWBZ7EFY]] of real numbers $0 < x_1 < \cdots < x_n$ is totally positive.
</document>

<document source="https://tamnd.github.io/math2/node/BTX4G1FDF6.md" tag="BTX4G1FDF6" kind="definition">
# First fundamental form

## Definition

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]] and let $p \in S$.
Let $T_p S \subseteq \R^3$ be the [[tangent plane|0BJDQ2HBH1]] of $S$ at $p$, viewed as a $2$-dimensional linear subspace.
The *first fundamental form* of $S$ at $p$ is the [[quadratic form|0HZNE5QZDJ]] $I_p \colon T_p S \to \R$, $I_p(w) = \ip{w}{w}$, where $\ip{\cdot}{\cdot}$ is the [[dot product|05HQYM0R4G]] of $\R^3$.
In a local parametrisation $x(u, v)$ of $S$, its coefficients are
$$E = \ip{x_u}{x_u}, \qquad F = \ip{x_u}{x_v}, \qquad G = \ip{x_v}{x_v}.$$
For $w = a x_u + b x_v$, this gives $I_p(w) = E a^2 + 2F ab + G b^2$.

## Remarks

One writes $I = E \, du^2 + 2F \, du \, dv + G \, dv^2$.
The length of a curve $t \mapsto x(u(t), v(t))$, $t \in [a, b]$, is $\int_a^b \sqrt{E u'^2 + 2F u' v' + G v'^2} \, dt$, and the area of $x(R)$ is $\iint_R \sqrt{EG - F^2} \, du \, dv$.
Example: for the unit sphere with $x(\theta, \varphi) = (\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta)$, one gets $E = 1$, $F = 0$ and $G = \sin^2 \theta$.
For $0 < \theta < \pi$ the form is positive definite, as it must be, since $EG - F^2 = \sin^2 \theta > 0$.
Non-example: at $\theta = 0$ the same formulas give $G = 0$, a degenerate form; the map is not a local parametrisation there, because $x_\varphi = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/BTYF5Y5PFY.md" tag="BTYF5Y5PFY" kind="concept">
# Lattice in a Lie group

## Summary

A lattice in a [[Lie group|7JBRGCVJRB]] $G$ is a [[discrete subgroup|5M7D1BSHVZ]] $\Gamma$ such that the quotient $G/\Gamma$ has finite volume for a $G$-invariant measure.
It is the case of Lie groups of a [[lattice in a locally compact group|NS2057MFWP]].
The lattices in $\R^n$ are the classical [[lattices|WN9V7WSJKF]] of the geometry of numbers, and $\mathrm{SL}_n(\Z)$ is a lattice in $\mathrm{SL}_n(\R)$.
Lattices link Lie groups with number theory, through [[arithmetic groups|4XWG6S7R4Z]], and with geometry, as fundamental groups of finite-volume locally symmetric spaces.

## Explanations

### L1 human

A lattice is a discrete set of symmetries of a continuous group that is large enough to fill the group up to a piece of finite size.
The integer points $\Z^2$ in the plane are the model: they are spread out, yet one unit square of the plane already contains a copy of every point modulo $\Z^2$.
In a non-commutative group such as $\mathrm{SL}_2(\R)$, the integer matrices $\mathrm{SL}_2(\Z)$ play the same part.
Here the leftover piece is not compact, but it still has finite volume.
</document>

<document source="https://tamnd.github.io/math2/node/BV08DGBD7C.md" tag="BV08DGBD7C" kind="example">
# Reduction modulo n is a homomorphism

## Example

Let $n \ge 1$. Define $\varphi : \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ by $\varphi(u) = u \bmod n$.
Then $\varphi(u + v) = \varphi(u) + \varphi(v)$, so $\varphi$ is a group homomorphism.
The map is onto, and its [[kernel|0Z424HK8CQ]] is $n\mathbb{Z}$, the integers that $n$ divides.
By the [[first isomorphism theorem|0CXXJNG5B6]], $\mathbb{Z}/n\mathbb{Z}$ as a quotient of $\mathbb{Z}$ is isomorphic to the image, the [[integers modulo $n$|03VFZ2WETT]].
</document>

<document source="https://tamnd.github.io/math2/node/BV0NXXGQZB.md" tag="BV0NXXGQZB" kind="definition">
# Biorthogonal wavelet

## Definition

Let $\psi$ and $\tilde\psi$ be functions in the [[Lp space|SASP14MA1Z]] $L^2(\R)$.
For $j, k \in \Z$ put $\psi_{j,k}(x) = 2^{j/2}\psi(2^j x - k)$ and $\tilde\psi_{j,k}(x) = 2^{j/2}\tilde\psi(2^j x - k)$.
The pair $(\psi, \tilde\psi)$ is a *biorthogonal wavelet* pair if:

1. the family $\set{\psi_{j,k} : j, k \in \Z}$ is a [[Riesz basis|ZYDYD7FB1Z]] of $L^2(\R)$;
2. the family $\set{\tilde\psi_{j,k} : j, k \in \Z}$ is a Riesz basis of $L^2(\R)$;
3. $\ip{\psi_{j,k}}{\tilde\psi_{j',k'}} = 1$ if $(j,k) = (j',k')$ and $0$ otherwise.

The function $\tilde\psi$ is called the dual wavelet of $\psi$.

## Remarks

Under these conditions every $f \in L^2(\R)$ has the expansions
$$f = \sum_{j,k \in \Z} \ip{f}{\tilde\psi_{j,k}} \, \psi_{j,k} = \sum_{j,k \in \Z} \ip{f}{\psi_{j,k}} \, \tilde\psi_{j,k},$$
with convergence in $L^2(\R)$.

Example: if $\psi$ is an orthonormal [[wavelet|4PZ35C0GV3]], such as the Haar wavelet, then $(\psi, \psi)$ is a biorthogonal pair.
Each family is an orthonormal basis, hence a Riesz basis, and clause 3 is orthonormality.

Non-example: take $\psi$ the Haar wavelet and $\tilde\psi = 2\psi$.
Both families are Riesz bases, but $\ip{\psi_{0,0}}{\tilde\psi_{0,0}} = 2$, so clause 3 fails.

In practice biorthogonal pairs come from two multiresolution analyses with dual scaling functions $\varphi$ and $\tilde\varphi$, and the transform is computed with two pairs of finite filters.
The Cohen–Daubechies–Feauveau family contains the 5/3 and 9/7 biorthogonal wavelets used in JPEG 2000.

## Explanations

### L1 human

Two families of wavelets are used: one to measure a signal and one to rebuild it.
They must be dual, which means each measuring function gives $1$ on its own partner and $0$ on every other rebuilding function.
</document>

<document source="https://tamnd.github.io/math2/node/BV2GCXPDDJ.md" tag="BV2GCXPDDJ" kind="definition">
# Discriminator variety

## Definition

Let $\mathcal{K}$ be a class of algebras of the same signature.
Assume that there is a ternary [[term|N9M2N27END]] $s(x, y, z)$ whose term operation on every $A \in \mathcal{K}$ is the [[ternary discriminator|72PG6XCQ7D]] on $A$.
The [[variety|AJNZJD8442]] $V(\mathcal{K})$ generated by $\mathcal{K}$ is then called a *discriminator variety*, with discriminator term $s$.
A variety is a discriminator variety if it is of the form $V(\mathcal{K})$ for such a class $\mathcal{K}$.

## Remarks

Example: the variety of Boolean algebras is generated by the two-element Boolean algebra $\mathbf{2}$.
On $\mathbf{2}$, the term $s(x, y, z) = (x \wedge (x \oplus y)) \vee (z \wedge \neg(x \oplus y))$, with $x \oplus y = (x \wedge \neg y) \vee (\neg x \wedge y)$, is the discriminator.
Check: if $x = y$ then $x \oplus y = 0$ and $s = z$; if $x \neq y$ then $x \oplus y = 1$ and $s = x$.
Non-example: the variety of lattices is not a discriminator variety.
A discriminator term $s$ satisfies $s(x, y, y) \approx x$ and $s(x, x, y) \approx y$ on $\mathcal{K}$, hence on $V(\mathcal{K})$.
So a discriminator variety is congruence-permutable, while the three-element chain is a lattice with non-permuting congruences.
The identities $s(x, y, y) \approx x \approx s(x, y, x)$ and $s(x, x, y) \approx y$ show that $s$ is a Pixley term, so every discriminator variety is an [[arithmetical variety|GW78A05TNA]].
The term $s$ need not act as the discriminator on every member of $V(\mathcal{K})$; for Boolean algebras it does so only on $\mathbf{2}$ and the trivial algebra.
</document>

<document source="https://tamnd.github.io/math2/node/BV6HGA075P.md" tag="BV6HGA075P" kind="definition">
# Value function

## Definition

Let $U \subseteq \R^m$ be a nonempty [[compact set|0R6EWFKRCF]] and $T > 0$.
Let $f \colon [0, T] \times \R^n \times U \to \R^n$ be continuous and Lipschitz in $x$ uniformly in $(t, u)$.
Let $L \colon [0, T] \times \R^n \times U \to \R$ and $g \colon \R^n \to \R$ be continuous.
For $(t, x) \in [0, T] \times \R^n$ and a measurable control $u \colon [t, T] \to U$, let $y$ be the trajectory of the [[control system|ZPDTVZQN5K]] $y' = f(s, y, u)$ with $y(t) = x$, and set
$$J(t, x; u) = \int_t^T L(s, y(s), u(s)) \, ds + g(y(T)).$$
The *value function* of this [[optimal control|T3V6ZNAMDN]] problem is the function $V \colon [0, T] \times \R^n \to \R$ given by
$$V(t, x) = \inf \set{ J(t, x; u) : u \colon [t, T] \to U \text{ measurable} }.$$

## Remarks

Under these hypotheses the trajectories from $(t, x)$ stay in a bounded set, so $V$ is finite.
By definition $V(T, x) = g(x)$, and a control $u$ is optimal from $(t, x)$ exactly when $J(t, x; u) = V(t, x)$.
Example: $n = m = 1$, $U = [-1, 1]$, $f(t, x, u) = u$, $L = 0$ and $g(x) = \abs{x}$.
From $(t, x)$ one can move toward $0$ at speed at most $1$, so $V(t, x) = \max(\abs{x} - (T - t), 0)$.
This $V$ is Lipschitz but not differentiable where $\abs{x} = T - t$ and $t < T$.
Non-example: the cost $J(t, x; u_0)$ of one fixed control $u_0$ is in general not the value function.
For $u_0 = 0$ in the example it is $\abs{x}$, which is larger than $V(t, x)$ when $x \ne 0$ and $t < T$.
For maximization problems the infimum is replaced by a supremum.
For infinite-horizon problems with discounting, $V$ depends only on the initial state.
</document>

<document source="https://tamnd.github.io/math2/node/BVA6XT55CS.md" tag="BVA6XT55CS" kind="statement">
# Euler product for multiplicative functions

## Statement

Let $f$ be a [[multiplicative function|0AP80TT660]] from the positive integers to $\C$ such that $\sum_{n=1}^{\infty} f(n)$ converges [[absolutely|0W3K3QPEWT]].
Then for each [[prime|0SN0MAEKYX]] $p$ the series $\sum_{k=0}^{\infty} f(p^k)$ converges absolutely, and
$$\sum_{n=1}^{\infty} f(n) = \lim_{x \to \infty} \prod_{p \le x} \sum_{k=0}^{\infty} f(p^k),$$
where the [[product|RZ0Z2BTMT4]] runs over the primes $p \le x$.
If $f$ is completely multiplicative, then $\abs{f(p)} < 1$ for each prime $p$, and the factor for $p$ equals $1/(1 - f(p))$.

## Remarks

For finite $x$, multiplying out the finitely many absolutely convergent factors gives $\sum f(n)$ over all $n$ whose prime factors are at most $x$.
Here one uses that $f$ is multiplicative and the [[fundamental theorem of arithmetic|084HH1NDCB]].
Every missing $n$ is larger than $x$, so the difference from the full sum is at most $\sum_{n > x} \abs{f(n)}$, which tends to $0$.
With $f(n) = n^{-s}$ and $\operatorname{Re}(s) > 1$ this is the [[Euler product formula|0C3PN4075R]] for the zeta function.
With $f(n) = \mu(n) n^{-s}$, where $\mu$ is the [[Möbius function|0C8QENHYZW]], it gives $\sum \mu(n) n^{-s} = \prod_p (1 - p^{-s})$.

## Explanations

### L1 human

An Euler product writes a sum over all positive integers as a product over the primes.
It works because each integer is built from primes in exactly one way, and a multiplicative function respects this building.
It is the basic bridge between sums over integers and the primes.

### L2 human

Euler products are the starting point of analytic number theory.
The zeta function, Dirichlet $L$-functions and many other Dirichlet series have them in their half-plane of absolute convergence.
From the product one reads off facts about primes and zeros.
For example, the zeta function has no zeros where its Euler product converges, since a convergent product of nonzero factors is nonzero.
</document>

<document source="https://tamnd.github.io/math2/node/BVFGX5JH7X.md" tag="BVFGX5JH7X" kind="example">
# Wiener filter equation

## Example

Let $x(t)$ and $s(t)$, $t \in \R$, be real, jointly wide-sense [[stationary processes|DT0QXJE705]] with mean $0$.
Write $R_{xx}(\tau) = E[x(u + \tau)\, x(u)]$ and $R_{sx}(\tau) = E[s(u + \tau)\, x(u)]$; they do not depend on $u$.
The task is to estimate $s(t)$ by a causal linear filter of the observations, $\hat s(t) = \int_0^\infty h(\tau)\, x(t - \tau)\, d\tau$, with the smallest mean square error $E\bigl[(s(t) - \hat s(t))^2\bigr]$.
The optimal impulse response $h$ satisfies the *Wiener filter equation*
$$R_{sx}(\sigma) = \int_0^\infty h(\tau)\, R_{xx}(\sigma - \tau)\, d\tau, \quad \sigma \ge 0.$$
This is a [[Wiener–Hopf equation|QYY4NMDY5H]] of the first kind with kernel $R_{xx}$.

Check of the equation.
The error $e(t) = s(t) - \hat s(t)$ of the best filter is orthogonal to every observation $x(t - \sigma)$ with $\sigma \ge 0$.
Otherwise a small change of $h$ would lower the error.
So $E[s(t)\, x(t - \sigma)] = \int_0^\infty h(\tau)\, E[x(t - \tau)\, x(t - \sigma)]\, d\tau$.
The left side is $R_{sx}(\sigma)$ and the expectation under the integral is $R_{xx}(\sigma - \tau)$, which gives the equation.
The condition holds only for $\sigma \ge 0$, because the filter may use only the past.

## Remarks

Without the causality condition, the equation holds for all $\sigma \in \R$, and the Fourier transform solves it: $H = S_{sx} / S_{xx}$, with $S$ the spectral densities given by the [[Wiener–Khinchin theorem|3ZXN26PW4G]].
With causality, the [[Wiener–Hopf method|H4NXEY6QTZ]] applies.
One factors $S_{xx} = S^+ S^-$ into factors without zeros in the two half-planes.
The causal filter is then $H = \bigl[S_{sx}/S^-\bigr]_+ / S^+$, where $[\cdot]_+$ keeps the part that comes from a function supported on $[0, \infty)$.
This step is called spectral factorization.

## Explanations

### L1 human

One wants to clean up a noisy signal as it arrives, using only what has been seen so far.
The best linear way to do it makes the error uncorrelated with all past observations.
Writing this down gives an integral equation on a half-line, because only the past counts.
That half-line is what makes it a Wiener–Hopf equation instead of a plain convolution.
</document>

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# Central difference

## Summary

The central difference of a function $f$ with step $h$ is $\delta_h f(x) = f(x + \tfrac{h}{2}) - f(x - \tfrac{h}{2})$.
It is symmetric around the point $x$.
For a smooth function, the quotient $\delta_h f(x) / h$ approximates $f'(x)$ with an error of order $h^2$, better than the order $h$ of the one-sided quotients.

## Explanations

### L1 human

To estimate the slope of a curve at a point, you can look a little to the right and a little to the left of it, and compare.
The central difference does exactly this, with the point in the middle.
For $f(x) = x^2$ and step $h$ the result is $(x + h/2)^2 - (x - h/2)^2 = 2hx$.
Dividing by $h$ gives $2x$, which is the exact derivative, while a one-sided difference gives $2x + h$.
</document>

<document source="https://tamnd.github.io/math2/node/BVJ2ERSQBM.md" tag="BVJ2ERSQBM" kind="definition">
# Bloch–Wigner function

## Definition

Let $\operatorname{Li}_2$ be the [[dilogarithm|G0X0NWSDB7]], taken as its principal branch on $\C \setminus [1, \infty)$.
For $z \in \C \setminus [1, \infty)$, let $\arg(1 - z) \in (-\pi, \pi)$ be the principal argument of $1 - z$, as for the [[complex logarithm|02QGGD5G87]].
The *Bloch–Wigner function* is
$$D(z) = \operatorname{Im}\big(\operatorname{Li}_2(z)\big) + \arg(1 - z) \, \log \abs{z}, \qquad z \in \C \setminus \big(\set{0} \cup [1, \infty)\big),$$
extended to all of $\C$ by continuity, with $D(0) = D(1) = 0$.

## Remarks

The function is real-analytic on $\C \setminus \set{0, 1}$ and continuous on $\C$.
Across $(1, \infty)$ the term $\operatorname{Im} \operatorname{Li}_2$ jumps by $2\pi \log x$ at the point $x$, and the term $\arg(1 - z) \log \abs{z}$ jumps by $-2\pi \log x$, so the extension is well defined.

Example: for $z = e^{i\theta}$ one has $\log \abs{z} = 0$, so $D(e^{i\theta}) = \operatorname{Im} \operatorname{Li}_2(e^{i\theta}) = \operatorname{Cl}_2(\theta)$, the [[Clausen function|AVS6G54711]].
In particular $D(e^{i\pi/3}) = \operatorname{Cl}_2(\pi/3) \approx 1.01494$.

Non-example: the function $\operatorname{Im} \operatorname{Li}_2(z)$ alone does not extend continuously across $(1, \infty)$, because of the jump $2\pi \log x$ described above.

Symmetries: $D(\bar z) = -D(z)$, and
$$D(z) = D\left(1 - \tfrac1z\right) = D\left(\tfrac{1}{1-z}\right) = -D\left(\tfrac1z\right) = -D(1 - z) = -D\left(\tfrac{z}{z-1}\right).$$
For $\operatorname{Im} z > 0$, $D(z)$ is the hyperbolic volume of the ideal tetrahedron with vertices $0, 1, \infty, z$.
</document>

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# Exposed point

## Definition

Let $n \ge 1$ and let $C \subseteq \R^n$.
A point $x \in C$ is an *exposed point* of $C$ if there is a [[linear form|07YQ1C78G0]] $f \colon \R^n \to \R$ such that $f(x) > f(y)$ for all $y \in C$ with $y \ne x$.

## Remarks

Equivalently, there is a [[supporting hyperplane|K7N565NKCV]] $H$ of $C$ with $H \cap C = \set{x}$.
In a real topological vector space the same definition is used with continuous linear functionals.

Every exposed point is an [[extreme point|3MSGFJM2NX]].
Suppose $x = (1 - t) y + t z$ with $y, z \in C$, $y \ne z$ and $0 < t < 1$.
Then $x \ne y$ and $x \ne z$, so $f(y) < f(x)$ and $f(z) < f(x)$, and hence $f(x) = (1 - t) f(y) + t f(z) < f(x)$, a contradiction.

Example: in $[0, 1]^2$, the corner $(1, 1)$ is exposed by $f(x_1, x_2) = x_1 + x_2$, since $f \le 2$ on the square with equality only at $(1, 1)$.

Non-example: let $C$ be the stadium, the convex hull of the discs of radius $1$ centred at $(-1, 0)$ and $(1, 0)$.
The point $(1, 1)$ is extreme, but the only supporting line of $C$ through it is $y = 1$, which meets $C$ in the whole segment from $(-1, 1)$ to $(1, 1)$.
So no linear form is maximised on $C$ only at $(1, 1)$.
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<document source="https://tamnd.github.io/math2/node/BVR56STQ32.md" tag="BVR56STQ32" kind="definition">
# Difference algebra

## Definition

A *difference ring* is a pair $(R, \sigma)$, where $R$ is a commutative [[ring|0B1261B5ED]] with identity and $\sigma \colon R \to R$ is a [[ring homomorphism|0VTH8KPCG2]], that is:

1. $\sigma(a + b) = \sigma(a) + \sigma(b)$ for all $a, b \in R$;
2. $\sigma(a b) = \sigma(a) \sigma(b)$ for all $a, b \in R$;
3. $\sigma(1) = 1$.

A *difference field* is a difference ring $(R, \sigma)$ in which $R$ is a [[field|04C5ZBG1WT]].
A difference ring is *inversive* if $\sigma$ is an automorphism.
*Difference algebra* is the study of difference rings, difference fields, and the difference equations and ideals over them.

## Remarks

Some authors require $\sigma$ to be an automorphism in the definition of a difference ring or field.
A *difference polynomial* over $(R, \sigma)$ is a polynomial in the variables $y, \sigma y, \sigma^2 y, \dots$, and a solution is an element $a$ of a difference ring extension at which it vanishes.

Example: $(\C(x), \sigma)$ with $\sigma(f)(x) = f(x + 1)$ is an inversive difference field.
The shift is additive and multiplicative, it fixes $1$, and its inverse is $f(x) \mapsto f(x - 1)$.

Example: a field $K$ of characteristic $p > 0$ with the [[Frobenius map|0T3WVTD675]] $\sigma(a) = a^p$ is a difference field.

Non-example: $(\Z, \sigma)$ with $\sigma(n) = 2n$ is not a difference ring.
It satisfies clause 1, but $\sigma(1) = 2 \ne 1$, so clause 3 fails, and $\sigma(1 \cdot 1) = 2 \ne 4 = \sigma(1)\sigma(1)$, so clause 2 fails too.
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# Manifold of smooth mappings

## Summary

Let $M$ be a compact manifold and $N$ a manifold.
The set $C^\infty(M, N)$ of all smooth maps is itself an infinite-dimensional manifold, modelled on [[Fréchet spaces|3GAA614673]] of smooth sections.
The charts come from the exponential map of a Riemannian metric on $N$.
A map near $f$ is written as $x \mapsto \exp_{f(x)}(s(x))$ for a small vector field $s$ along $f$.
Loop spaces, [[diffeomorphism groups|WTCNKST134]] and spaces of embeddings are open subsets or submanifolds of such manifolds.
Maps of class $C^k$, or of Sobolev class $H^s$ with $s > \dim M / 2$, form Banach or [[Hilbert manifolds|R8DMMW0E9M]] of maps, which are better suited to analysis.

## Explanations

### L1 human

A map from a circle to a sphere is a closed curve on the sphere.
All such curves together form a huge space, with one point for each curve.
Near a given curve, any nearby curve is obtained by pushing each point a little along a tangent arrow.
So nearby curves correspond to small fields of tangent arrows along the given curve.

These fields form a vector space, and this gives coordinates on the space of curves, making it a manifold of infinite dimension.
</document>

<document source="https://tamnd.github.io/math2/node/BW21VJW403.md" tag="BW21VJW403" kind="definition">
# Lattice in a locally compact group

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] and let $\Gamma$ be a [[discrete subgroup|5M7D1BSHVZ]] of $G$.
Let $G/\Gamma$ be the space of left cosets $g\Gamma$ with the quotient topology, on which $G$ acts by $h \cdot g\Gamma = hg\Gamma$.
The subgroup $\Gamma$ is a *lattice* in $G$ if there is a nonzero [[Radon measure|HZNEQQBCW4]] $\nu$ on $G/\Gamma$ such that:

1. $\nu$ is $G$-invariant: $\nu(hE) = \nu(E)$ for every $h \in G$ and every Borel set $E \subseteq G/\Gamma$;
2. $\nu$ is finite: $\nu(G/\Gamma) < \infty$.

A lattice $\Gamma$ is *uniform*, or *cocompact*, if $G/\Gamma$ is compact.

## Remarks

Example: $\Z^n$ in $(\R^n, +)$.
The quotient $\R^n/\Z^n$ is the torus, and the image of Lebesgue measure on $[0, 1)^n$ is a translation-invariant measure on it of total mass $1$.
Since the torus is compact, $\Z^n$ is a uniform lattice.

Non-example: $\Gamma = \Z \times \set{0}$ in $\R^2$ is discrete, but fails clause 2.
The quotient is the cylinder $(\R/\Z) \times \R$.
Every invariant Radon measure on it is a multiple of the product of the [[Haar measures|VY0HXD3GC4]] of the two factors.
The nonzero ones have infinite total mass.

If $G$ has a lattice, then $G$ is [[unimodular|CXG6JZGMCK]].
A measurable [[fundamental domain|KNJA2NEZ9F]] for the action of $\Gamma$ on $G$ by right multiplication has finite Haar measure exactly when $\Gamma$ is a lattice.
The group $SL_2(\Z)$ is a lattice in $SL_2(\R)$ that is not uniform.
</document>

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# Ultraweak topology

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.
For sequences $x = (x_n)$ and $y = (y_n)$ in $H$ with $\sum_n \|x_n\|^2 < \infty$ and $\sum_n \|y_n\|^2 < \infty$, let
$$\omega_{x,y}(T) = \sum_{n} \langle T x_n, y_n \rangle, \quad T \in B(H).$$
The *ultraweak topology*, or *$\sigma$-weak topology*, on $B(H)$ is the [[topology generated by the seminorms|ZVFXQ7S6FY]] $T \mapsto \abs{\omega_{x,y}(T)}$ for all such pairs of sequences.

## Remarks

The series defining $\omega_{x,y}(T)$ converges absolutely, by the Cauchy–Schwarz inequality.
Equivalently, the ultraweak topology is the weak-$*$ topology of the duality $B(H) \cong \mathcal{L}^1(H)^*$, where $\mathcal{L}^1(H)$ is the space of [[trace class operators|Z2CP530WDE]] and $T$ acts by $\rho \mapsto \operatorname{tr}(T \rho)$.

Example: for a trace class operator $\rho$, the functional $T \mapsto \operatorname{tr}(T\rho)$ is ultraweakly continuous.
Indeed, writing $\rho = \sum_n s_n \langle \cdot, e_n \rangle f_n$ in Schmidt form with $\sum_n s_n < \infty$ gives $\operatorname{tr}(T\rho) = \omega_{x,y}(T)$ with $x_n = \sqrt{s_n} f_n$ and $y_n = \sqrt{s_n} e_n$.

Non-example: let $H$ be infinite-dimensional and let $\varphi$ be a state on $B(H)$ that vanishes on all compact operators.
Such states exist because the quotient algebra by the compact operators has states.
Then $\varphi$ is norm-continuous but not ultraweakly continuous: every ultraweakly continuous functional has the form $T \mapsto \operatorname{tr}(T\rho)$, vanishing on all rank-one operators forces $\rho = 0$, and yet $\varphi(I) = 1$.

Every [[weak operator topology|4XM1VN1C1N]] open set is ultraweakly open, and the two topologies agree on every norm-bounded subset of $B(H)$.
They differ on all of $B(H)$ when $H$ is infinite-dimensional.
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# Automatic differentiation

## Summary

Automatic differentiation computes exact values of [[derivatives|0V1EJCT8G6]] of a function given as a computer program.
It applies the [[chain rule|0RXX2PT1MQ]] to each elementary operation of the program.
So its only errors are rounding errors, unlike [[numerical differentiation|NVH3F8FTYS]].
In forward mode it propagates directional derivatives along with the values; in reverse mode it propagates sensitivities backwards from the outputs.
Reverse mode computes the gradient of a scalar function at a small constant multiple of the cost of the function.
The backpropagation algorithm of machine learning is a special case of reverse mode.

## Explanations

### L1 human

A program computes a complicated function by a long chain of simple steps: additions, multiplications, sines, exponentials.
The derivative of each simple step is known, and the chain rule says how to combine them.
Automatic differentiation lets the computer carry along the derivative next to each intermediate value.
For example, for $f(x) = x \sin x$ at $x = 1$, it keeps the pairs $(1, 1)$ for $x$ and $(\sin 1, \cos 1)$ for $\sin x$, and the product rule gives the pair $(\sin 1, \sin 1 + \cos 1)$, whose second entry is $f'(1)$.
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<document source="https://tamnd.github.io/math2/node/BW7NGTGXAY.md" tag="BW7NGTGXAY" kind="definition">
# Melnikov method

## Definition

Let $H \colon \R^2 \to \R$ be smooth and let $f = (\partial H / \partial y, -\partial H / \partial x)$, so that $u' = f(u)$ is a planar [[Hamiltonian system|0EDKGA2FAK]].
Let $p$ be a [[hyperbolic|0C5PREPZWG]] saddle of this system, and let $q_0(t)$ be a solution whose orbit is a [[homoclinic orbit|6DBSQKZK4W]] of $p$.
Let $g \colon \R^2 \times \R \to \R^2$ be smooth and [[periodic|0ATF6Q0633]] of period $T > 0$ in its last variable.
Consider the perturbed system
$$u' = f(u) + \varepsilon\, g(u, t), \qquad \varepsilon \text{ small}.$$
For $a, b \in \R^2$ put $a \wedge b = a_1 b_2 - a_2 b_1$.
The *Melnikov function* of the perturbation along $q_0$ is
$$M(t_0) = \int_{-\infty}^{\infty} f(q_0(t)) \wedge g\big(q_0(t), t + t_0\big)\, dt, \qquad t_0 \in \R.$$
The *Melnikov method* is the study of the perturbed system through the zeros of $M$.

## Remarks

The integral converges because $f(q_0(t)) \to f(p) = 0$ exponentially fast as $t \to \pm\infty$.
The function $M$ is $T$-periodic in $t_0$.
For small $\varepsilon$, the perturbed system has a hyperbolic periodic orbit near $p$.
The distance between its stable and unstable manifolds, measured along the normal to the unperturbed loop at $q_0(0)$, is $\varepsilon M(t_0) / \norm{f(q_0(0))} + O(\varepsilon^2)$, with a suitable sign convention.
The basic theorem says: if $M$ has a simple zero, then for small $\varepsilon \ne 0$ these manifolds intersect transversally; if $M$ has no zero, they do not intersect.

Example: for $H = \frac{1}{2} y^2 - \frac{1}{2} x^2 + \frac{1}{4} x^4$ and $g = (0, -\delta y + \gamma \cos(\omega t))$, the system is the damped, forced Duffing equation.
Here $q_0(t) = (\sqrt{2} \operatorname{sech} t, -\sqrt{2} \operatorname{sech} t \tanh t)$, and a direct computation gives $M(t_0) = -\frac{4}{3} \delta + c\, \gamma \sin(\omega t_0)$ with a constant $c$ that depends only on $\omega$.

Non-example: if $g(u, t) = (0, t)$, then $g$ is not periodic in $t$.
The perturbed system is then not periodic in time, there is no Poincaré map of period $T$, and the theorem above does not apply.
</document>

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# Watson's lemma

## Statement

Let $0 < T \le \infty$, let $\lambda > -1$, and let $f \colon (0, T) \to \C$ be measurable.
Assume that:

1. $f$ has the [[asymptotic expansion|B53THC8F79]] $f(t) \sim \sum_{n \ge 0} a_n t^{n + \lambda}$ as $t \to 0^+$, with complex $a_n$.
2. There is a real number $x_0$ such that $\int_0^T e^{-x_0 t} \abs{f(t)} \, dt < \infty$.

Then the integral $F(x) = \int_0^T e^{-xt} f(t) \, dt$ converges absolutely for every $x \ge x_0$, and
$$F(x) \sim \sum_{n \ge 0} a_n \frac{\Gamma(n + \lambda + 1)}{x^{n + \lambda + 1}} \quad \text{as } x \to +\infty,$$
where $\Gamma$ is the [[gamma function|0V3FZMVB4K]].

## Remarks

Idea of the proof: for $x$ large, the factor $e^{-xt}$ makes the integral over $[\delta, T)$ exponentially small in $x$, for any fixed $\delta > 0$.
On $(0, \delta)$, replace $f$ by its first $N$ terms plus an error bounded by $C t^{N + \lambda}$.
Each term gives $\int_0^\infty e^{-xt} t^{n + \lambda} \, dt = \Gamma(n + \lambda + 1) x^{-n-\lambda-1}$, up to an exponentially small change, and the error contributes $O(x^{-N-\lambda-1})$.
Example: $\int_0^\infty \frac{e^{-xt}}{1 + t} \, dt \sim \sum_{n \ge 0} (-1)^n \frac{n!}{x^{n+1}}$, a divergent series obtained from $\frac{1}{1+t} = \sum (-1)^n t^n$ near $0$.
The condition $\lambda > -1$ is needed: if $\lambda \le -1$ and $a_0 \ne 0$, then $\abs{f(t)}$ is of order $t^{\lambda}$ near $0$ and the integral diverges at $t = 0$.

## Explanations

### L1 human

For a large parameter $x$, the weight $e^{-xt}$ is tiny except for $t$ very close to $0$.
So the integral only sees the behaviour of $f$ near $0$.
Watson's lemma makes this precise: expand $f$ in powers of $t$ near $0$, integrate term by term, and the result is the correct asymptotic expansion of the integral.

### L2 human

The lemma gives the large-$x$ behaviour of the [[Laplace transform|05XT5AVJBK]] $F(x)$ of $f$ from the small-$t$ behaviour of $f$.
Together with a change of variables it yields [[Laplace's method|9G7K8DBT9Y]] and the method of steepest descent for integrals $\int e^{x \phi(t)} g(t) \, dt$.
The resulting series often diverge, and [[Borel summation|8DJPSYSHGJ]] can be seen as the inverse process that recovers $F$ from its series.
</document>

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# Connected sum

## Definition

Let $n \ge 1$ and let $M$ and $N$ be [[connected|0V032VWTHD]] [[topological manifolds|QQQ9B1DK4P]] of dimension $n$.
Let $D^n$ be the [[closed unit ball|0Q0H13G8ZB]] of $\R^n$, with interior $B^n$ and boundary sphere $S^{n-1}$.
Let $2D^n$ be the closed ball of radius $2$.
Let $f \colon 2D^n \to M$ and $g \colon 2D^n \to N$ be [[topological embeddings|00ZG9Y9TR4]] onto subsets of the interiors.
The *connected sum* $M \# N$ is the [[quotient space|0NQ1H2X495]] of the disjoint union
$$\bigl(M \setminus f(B^n)\bigr) \sqcup \bigl(N \setminus g(B^n)\bigr)$$
by the relation that identifies $f(x)$ with $g(x)$ for every $x \in S^{n-1}$.
If $M$ and $N$ are [[oriented|702SNFQG0W]], one takes $f$ preserving and $g$ reversing orientation.
Then $M \# N$ gets an orientation that extends those of both pieces.

## Remarks

For two tori $T^2 \# T^2$ is the closed orientable surface of genus $2$.
Each torus minus an open disk is a torus with one boundary circle, and gluing the two circles gives a connected surface with two handles.

The disjoint union $M \sqcup N$ is not a connected sum: no balls are removed and no spheres are glued, and the result is not connected.

The result is a connected topological $n$-manifold.
Up to homeomorphism, it does not depend on the choice of the embeddings when $M$ and $N$ are oriented and the orientation rule above is followed.
Without orientations the choice can matter: $\mathbb{CP}^2 \# \mathbb{CP}^2$ and $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$ are not homotopy equivalent, since their intersection forms differ.
The sphere is a unit: $M \# S^n$ is homeomorphic to $M$.
The real projective plane gives $\mathbb{RP}^2 \# \mathbb{RP}^2$ homeomorphic to the [[Klein bottle|6J4C2P3ZHC]].
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# Locally convex space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a [[topological vector space|T32W553W95]] over $\mathbb{K}$.
$E$ is *locally convex* if every [[neighbourhood|0ZCY0Z80B6]] of $0$ in $E$ contains a [[convex|02AS45T3H9]] neighbourhood of $0$.

## Remarks

Equivalently, the topology of $E$ is the [[topology generated by a family of seminorms|ZVFXQ7S6FY]], namely the family of all continuous seminorms on $E$.
Many authors also require $E$ to be Hausdorff.
Example: every normed space is locally convex, since the open balls $\set{x : \norm{x} < r}$ are convex and form a neighbourhood base at $0$.
Convexity of a ball: if $\norm{x}, \norm{y} < r$ and $0 \le t \le 1$, then $\norm{t x + (1 - t) y} \le t \norm{x} + (1 - t) \norm{y} < r$.
Non-example: for $0 < p < 1$, the space $L^p([0, 1])$ with the metric $d(f, g) = \int_0^1 \abs{f - g}^p$ is a topological vector space.
But its only convex open set containing $0$ is the whole space, so it is not locally convex.
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# Monadic functor

## Definition

Let $U \colon \mathcal{D} \to \mathcal{C}$ be a functor with a [[left adjoint|TZHF6ZM92S]] $F$ and counit $\varepsilon \colon F U \Rightarrow \mathrm{Id}_{\mathcal{D}}$.
Let $T = U F$ be the [[monad of the adjunction|DF5N0PK3ZS]], and let $\mathcal{C}^T$ be its [[Eilenberg–Moore category|FEPRJR3MND]].
The *comparison functor* $K \colon \mathcal{D} \to \mathcal{C}^T$ sends an object $d$ to the algebra $(U(d), U(\varepsilon_d))$ and a morphism $f$ to $U(f)$.
The functor $U$ is *monadic* if $K$ is an [[equivalence of categories|K7A04ZBD92]].

## Remarks

A functor without a left adjoint is never monadic.
Some authors call $U$ strictly monadic when $K$ is an isomorphism of categories.
The comparison functor always satisfies $U^T \circ K = U$, where $U^T$ is the forgetful functor of $\mathcal{C}^T$.

Example: the forgetful functor from groups to sets is monadic.
Its left adjoint is the free group functor.
An algebra of the monad $T$ is a set $A$ with a map from the free group on $A$ to $A$ satisfying the algebra laws.
Such a map comes from a unique group structure on $A$, and algebra morphisms are group homomorphisms, so $K$ is an isomorphism.

Non-example: the forgetful functor $U$ from topological spaces to sets has the discrete space functor as left adjoint.
The monad $U F$ is the identity monad, whose algebras are just sets, so $\mathcal{C}^T$ is the category of sets.
The comparison functor sends a space to its underlying set, and it is not full, since not every function between spaces is continuous.
So $K$ is not an equivalence and $U$ is not monadic.
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# Centerpoint theorem

## Statement

Let $d \ge 1$ and let $X$ be a finite set of $n$ points in $\R^d$.
Then there is a point $c \in \R^d$ such that every closed half-space of $\R^d$ that contains $c$ contains at least $\dfrac{n}{d+1}$ points of $X$.
Such a point $c$ is called a *centerpoint* of $X$.

## Remarks

Consider all closed half-spaces that contain more than $\frac{d}{d+1} n$ points of $X$, and the [[convex hulls|0FQSDC2EQ2]] of the points of $X$ in each of them.
Each such half-space misses fewer than $\frac{n}{d+1}$ points of $X$, so $d + 1$ of them together miss fewer than $n$ points, and some point of $X$ lies in all $d + 1$ hulls.
By [[Helly's theorem|F1W3QRBNGJ]] all these hulls have a common point $c$.
Suppose a closed half-space containing $c$ had fewer than $\frac{n}{d+1}$ points.
Its open complement would contain more than $\frac{d}{d+1} n$ points, and a slightly smaller closed half-space inside it would give a hull that misses $c$, a contradiction.
In dimension $1$ a median of $X$ is a centerpoint, with the bound $n/2$.
The bound $n/(d+1)$ cannot be improved in general.
Take $d + 1$ equal small clusters of points near the vertices of a simplex: for any point $c$, some closed half-space through $c$ meets only one cluster.

## Explanations

### L1 human

A centerpoint is a point that sits "deep" inside a cloud of points.
However one cuts space in two halves by a flat wall through it, each half keeps a fair share of the points.
In the plane that fair share is at least one third of the points.
On a line it is the median, which has half the points on each side.

### L2 human

The centerpoint theorem is a standard consequence of [[Helly's theorem|F1W3QRBNGJ]].
It generalizes the median to higher dimensions and gives a notion of depth for data.
It is used in divide-and-conquer algorithms and in the proof of separator theorems for planar graphs and meshes.
The [[ham sandwich theorem|GDXX60SNG5]] is a different statement about dividing point sets by hyperplanes.
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# Excessive function

## Definition

Let $(E, \mathcal{E})$ be a measurable space.
Let $(P_t)_{t \ge 0}$ be a sub-Markov transition semigroup on $E$.
That is, each $P_t$ is a kernel as for a [[Markov kernel|2AVNDP8BJP]], except that the total mass satisfies $P_t(x, E) \le 1$, $P_0$ is the identity, and $P_s P_t = P_{s+t}$.
For a measurable $f \colon E \to [0, \infty]$, write $P_t f(x) = \int_E f(y) \, P_t(x, dy)$.
A measurable function $f \colon E \to [0, \infty]$ is *excessive* for $(P_t)$ if:

1. $P_t f \le f$ for every $t \ge 0$.
2. $P_t f(x) \to f(x)$ as $t \downarrow 0$, for every $x \in E$.

## Remarks

Under clause 1, $t \mapsto P_t f$ is decreasing, since $P_{t+s} f = P_t (P_s f) \le P_t f$; so the limit in clause 2 exists and is at most $f$.
A function with clause 1 alone is called supermedian; for $\alpha > 0$, the $\alpha$-excessive functions are those that are excessive for the semigroup $e^{-\alpha t} P_t$.
Example: let $(P_t)$ be the semigroup of [[Brownian motion|22735FGJV6]] in $\R^3$ and $f(x) = 1/\abs{x}$, with $f(0) = \infty$.
Then $f$ is a nonnegative [[superharmonic function|X4V2S56YQQ]], and averaging it against the Gaussian kernel gives $P_t f \le f$ with $P_t f \uparrow f$ as $t \downarrow 0$, so $f$ is excessive.
Non-example: for the same semigroup, let $f$ be the indicator of the single point $0$.
Then $P_t f = 0 \le f$, so clause 1 holds, but $P_t f(0) = 0$ does not tend to $f(0) = 1$, so clause 2 fails.
For Brownian motion in $\R^d$, the excessive functions are exactly the nonnegative superharmonic functions on $\R^d$, together with the constant $+\infty$.
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<document source="https://tamnd.github.io/math2/node/BWQ8PVA9Q3.md" tag="BWQ8PVA9Q3" kind="statement">
# Aberth method

## Statement

Let $p$ be a polynomial of degree $n \ge 2$ with [[complex|09FTAET5Z4]] coefficients, whose [[roots|019MA1VTR2]] $r_1, \dots, r_n$ are pairwise distinct.
The *Aberth method*, also called the Aberth–Ehrlich method, updates $n$ approximations $z_1, \dots, z_n$ simultaneously by $z_i \mapsto z_i - w_i$, where
$$w_i = \frac{\dfrac{p(z_i)}{p'(z_i)}}{1 - \dfrac{p(z_i)}{p'(z_i)} \displaystyle\sum_{j \ne i} \frac{1}{z_i - z_j}}, \qquad i = 1, \dots, n.$$
Then there is $\delta > 0$ such that, if each starting value satisfies $\abs{z_i - r_i} < \delta$, the iterates are defined and converge to $r_1, \dots, r_n$, with cubic [[convergence|YWZQC5E54B]].

## Remarks

The step is [[Newton's method|07EXVNF9VP]] applied to the rational function
$$F_i(x) = \frac{p(x)}{\prod_{j \ne i} (x - z_j)},$$
which has the same root $r_i$ as $p$ but whose poles push $z_i$ away from the other approximations.
Indeed $F_i'/F_i = p'/p - \sum_{j \ne i} 1/(x - z_j)$, and $x - F_i(x)/F_i'(x)$ at $x = z_i$ is exactly $z_i - w_i$.
This repulsion tends to keep two approximations from converging to the same root.
At multiple roots the convergence is only linear.

## Explanations

### L1 human

The Aberth method moves guesses for all the roots of a polynomial at the same time.
Each guess takes a Newton step, but it is also repelled by the other guesses, like electric charges of the same sign.
So the guesses spread out, and each one settles on its own root.

### L2 human

It improves the Durand–Kerner method, which converges only quadratically near simple roots.
Each step costs $O(n^2)$ operations, and the steps can be done in parallel or in Gauss–Seidel fashion, using new values as soon as they are known.
It is the core of the MPSolve software for computing polynomial roots to high precision.
</document>

<document source="https://tamnd.github.io/math2/node/BWS042NC2P.md" tag="BWS042NC2P" kind="concept">
# Decreasing rearrangement

## Summary

The decreasing rearrangement of a measurable function $f$ is the non-increasing, right-continuous function $f^*$ on $[0, \infty)$ that has the same distribution function as $\abs{f}$.
It sorts the values of $\abs{f}$ from the largest to the smallest.
Every norm that depends only on the distribution function, such as the $L^p$ norms, takes the same value on $f$ and on $f^*$.
It is the basic tool of Lorentz spaces and of rearrangement-invariant function spaces.

## Explanations

### L1 human

Imagine the values of a function as piles of sand placed along a line.
The decreasing rearrangement moves the piles so that the tallest come first and the heights go down from left to right.
No sand is added or removed, so the amount of land above each height stays the same.
For example, on $[0, 1]$ the function $f(x) = x$ has decreasing rearrangement $f^*(t) = 1 - t$ for $0 \le t < 1$ and $f^*(t) = 0$ for $t \ge 1$.
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<document source="https://tamnd.github.io/math2/node/BWSN2JAMTK.md" tag="BWSN2JAMTK" kind="concept">
# Covering density

## Summary

The covering density of a covering of space by bodies measures, on average, how many times each point is covered.
It is at least $1$, and the covering problem asks for the smallest possible value, the covering density of the body.
It is the counterpart of the [[packing density|4P5PSWADJD]], with overlaps allowed and gaps forbidden.
For discs in the plane the optimum is $2\pi/\sqrt{27} \approx 1.2092$, given by the hexagonal lattice arrangement.

## Explanations

### L1 human

Suppose you must light a large square with lamps that each light a disc, and every spot must be lit.
The discs must overlap somewhere, and the covering density measures how much light is wasted on average.
For equal discs, the best is to place the centres in a triangular grid, so that each disc is cut to a regular hexagon by its neighbours.
Even then about $21\%$ of the light is spent on overlaps.
</document>

<document source="https://tamnd.github.io/math2/node/BWT2AAEK9R.md" tag="BWT2AAEK9R" kind="definition">
# q-difference equation

## Definition

Let $q \in \C$ with $q \ne 0$ and $q \ne 1$, let $n \ge 1$, and let $U \subseteq \C$ be a set with $qU \subseteq U$.
Let $F$ be a given [[function|0EPAT51DXQ]] of $n + 2$ complex variables.
A *$q$-difference equation of order $n$* for an unknown function $y \colon U \to \C$ is a condition
$$F\big(x, y(x), y(qx), \dots, y(q^n x)\big) = 0 \quad \text{for all } x \in U.$$
It is *linear* if it has the form
$$a_n(x) y(q^n x) + \cdots + a_1(x) y(qx) + a_0(x) y(x) = b(x)$$
with given functions $a_0, \dots, a_n, b$ on $U$.

## Remarks

Example: let $0 < \abs{q} < 1$ and $e_q(x) = \sum_{k \ge 0} x^k / [k]_q!$ for $\abs{x} < 1/\abs{1 - q}$, where $[k]_q = (1 - q^k)/(1 - q)$ and $[k]_q! = [1]_q \cdots [k]_q$.
Since the [[q-derivative|PMJT3FZE0Y]] satisfies $D_q x^k = [k]_q x^{k-1}$, one gets $D_q e_q = e_q$.
Written out, this is the first-order linear $q$-difference equation $e_q(qx) = \big(1 + (q - 1)x\big) e_q(x)$.

Non-example: $y(x + 1) = x\, y(x)$, satisfied by the gamma function, relates $y$ at $x$ and at the translate $x + 1$.
It involves no dilate $q^j x$, so it is an ordinary difference equation and not a $q$-difference equation.

The $q$-derivative $D_q y(x) = \big(y(qx) - y(x)\big)/\big((q - 1)x\big)$ lets one rewrite any $q$-difference equation in terms of $D_q, D_q^2, \dots$, as a $q$-analog of a differential equation.
Many authors assume $\abs{q} \ne 1$, or at least that $q$ is not a root of unity.
</document>

<document source="https://tamnd.github.io/math2/node/BWWZB1F0HY.md" tag="BWWZB1F0HY" kind="statement">
# Hardy's Tauberian theorem for Cesàro means

## Statement

Let $\sum_{n \ge 0} a_n$ be a series of complex numbers.
Assume that it is [[Cesàro summable|N8ZGG7EW6J]] to $s$.
Assume also that $a_n = O(1/n)$ in [[big O notation|0RYTGBJAGN]], that is, there is a constant $K$ with $n \abs{a_n} \le K$ for all $n \ge 1$.
Then $\sum_{n \ge 0} a_n$ [[converges|0CVVKY7GY2]] and its sum is $s$.

## Remarks

Let $s_n$ be the partial sums and $\sigma_n$ their Cesàro means.
For $m > n$ one has the identity
$$s_n = \frac{(m+1)\sigma_m - (n+1)\sigma_n}{m - n} - \frac{1}{m - n} \sum_{k=n+1}^{m} (s_k - s_n).$$
The first term tends to $s$ when $m \approx (1 + \varepsilon) n$ and $n \to \infty$.
The second term is at most $\max_{n < k \le m} \abs{s_k - s_n} \le K \frac{m - n}{n} \approx K\varepsilon$, so $\limsup \abs{s_n - s} \le K\varepsilon$ for every $\varepsilon > 0$.

The Tauberian condition cannot be dropped.
The series $1 - 1 + 1 - \cdots$ is Cesàro summable to $\frac12$ and diverges; here $n \abs{a_n} = n$ is unbounded.
For real $a_n$, the one-sided condition $n a_n \ge -K$ is already enough.

## Explanations

### L1 human

Averaging the partial sums can give a value to a series that diverges.
Hardy's theorem says this cannot happen when the terms shrink at least like $1/n$.
Then a Cesàro sum is an honest sum.

### L2 human

This is one of the first Tauberian theorems in which the condition $n a_n \to 0$ of [[Tauber's theorem|8SK3GQ4S0M]] is relaxed to $n a_n = O(1)$.
The analogous result for Abel summation is [[Littlewood's Tauberian theorem|Z35TD4ZZRD]], which is harder, because Abel summation is stronger than Cesàro summation.
Hardy's theorem is used to show that a Fourier series of a function of bounded variation converges wherever its Cesàro means do, since its coefficients are $O(1/n)$.
</document>

<document source="https://tamnd.github.io/math2/node/BWYFKCWRW5.md" tag="BWYFKCWRW5" kind="definition">
# Upper set

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $U$ be a [[subset|0FE8Q5S42Y]] of $P$.
The set $U$ is an *upper set* when, for all $x, y \in P$, if $x \in U$ and $x \le y$, then $y \in U$.
Dually, a subset $L$ of $P$ is a *lower set* when, for all $x, y \in P$, if $y \in L$ and $x \le y$, then $x \in L$.

## Remarks

In $(\N, \le)$, the set $\set{n \in \N : n \ge 5}$ is an upper set: if $n \ge 5$ and $n \le m$, then $m \ge 5$.
The set of even numbers is not an upper set of $(\N, \le)$, since $2$ is in it, $2 \le 3$, and $3$ is not.
A subset is an upper set if and only if its complement is a lower set.
Unions and intersections of any family of upper sets are upper sets.
Other names are up-set and upward closed set for upper sets, and down-set, decreasing set and order ideal for lower sets.
The smallest lower set containing a subset $S$ is its [[down-closure|TGW5NTV3AA]].

## Explanations

### L1 human

An upper set never stops partway up: whatever lies above a member is also a member.
A lower set is the same thing turned upside down.
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# Wiman–Valiron theory

## Summary

Wiman–Valiron theory studies a transcendental [[entire function|0419794R0T]] through the largest term of its [[Taylor series|0225QPWRRE]] on each circle $\abs{z} = r$.
The index of this maximal term, the central index $\nu(r)$, grows to infinity with $r$.
Near points where $\abs{f}$ attains its maximum on the circle, $f$ behaves like the single monomial $a_{\nu(r)} z^{\nu(r)}$, outside a small exceptional set of radii.
A main application is to the growth of entire solutions of linear differential equations with polynomial coefficients.

## Explanations

### L1 human

An entire function is an infinite sum of terms $a_n z^n$.
On a large circle, one term is usually much bigger than the others, and its index is the central index.
Near the point where the function is largest on the circle, the function looks like this one term.
In particular, each derivative multiplies the function by about $\nu(r)/z$, as for a monomial.
For $e^z$ the central index on the circle of radius $r$ is about $r$, and indeed $(e^z)' = e^z \approx (r/z) e^z$ near $z = r$.
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# Infinitely divisible distribution

## Summary

A probability distribution is infinitely divisible if, for every $n$, it is the law of a sum of $n$ [[independent and identically distributed|00MRN0FARV]] random variables.
The [[normal|05AQE0QR0D]], [[Poisson|0XB9GQTN9G]], [[gamma|0B80PAQD3C]] and [[Cauchy|0GPG24Q3GR]] laws are infinitely divisible, while no nondegenerate bounded law is.
Infinitely divisible laws are exactly the laws at time $1$ of [[Lévy processes|8NT5CFG2ZR]], and they are described by the [[Lévy–Khintchine representation|Q82B4H4Z4J]].
They are also the possible limits of sums of many small independent terms.

## Explanations

### L1 human

A law is infinitely divisible if it can be split into any number of equal independent parts.
Suppose the number of calls to a call centre in one hour is Poisson with mean $60$.
It is the sum of the calls in $60$ independent minutes, each Poisson with mean $1$.
It is also the sum over $3600$ independent seconds, each Poisson with mean $1/60$.
This works for every number of parts, so the Poisson law is infinitely divisible.
A fair coin with values $0$ and $1$ cannot be split even into two equal independent parts.
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# Linear multistep method

## Summary

A linear multistep method solves an [[initial value problem|0FXDW992DS]] $y' = f(t, y)$ numerically.
It computes the next approximation from a fixed number of earlier approximations and the values of $f$ there.
It needs only one new evaluation of $f$ per step, which makes it cheap compared with [[Runge–Kutta methods|7DBPJKYC6C]] of the same order.
The Adams methods and the [[backward differentiation formulas|TDA94C2JYT]] are the main families.
Its convergence is governed by consistency and zero-stability, and its possible order is limited by the [[Dahlquist barriers|BQBZ7QMJE2]].

## Explanations

### L1 human

When you walk along a curve and have already recorded several points, you can use all of them to guess where the curve goes next.
A linear multistep method does this: it combines the last few positions and slopes with fixed weights.
For example, the two-step Adams–Bashforth method steps forward by $h \left( \frac32 f_{k+1} - \frac12 f_k \right)$, an extrapolation of the last two slopes.
The first few values must be supplied by another method, such as a Runge–Kutta method.
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<document source="https://tamnd.github.io/math2/node/BX558DTV2D.md" tag="BX558DTV2D" kind="statement">
# Gross–Zagier theorem

## Statement

Let $E$ be an [[elliptic curve|427N6S6QQW]] over $\Q$ of conductor $N$, with modular parametrisation $\varphi \colon X_0(N) \to E$ given by the [[modularity theorem|CS1NVJBZMH]].
Let $K$ be an imaginary [[quadratic field|7SV4WA92SM]] of discriminant $D$ such that every prime dividing $N$ splits in $K$.
Let $x$ be a [[Heegner point|2230ZKE4W9]] for $K$; it is defined over the [[Hilbert class field|WWCJ7CR2R3]] $H$ of $K$.
Let $y_K \in E(K)$ be the trace of $\varphi(x)$ from $H$ to $K$.
Let $L(E/K, s) = L(E, s) \, L(E^D, s)$, where $E^D$ is the quadratic twist of $E$ by $K$, and let $\hat{h}$ be the Néron–Tate canonical [[height|F1FF505PWH]] on $E(K)$.
Then $L(E/K, 1) = 0$, and there is a real constant $c > 0$, depending on $E$ and $K$ and given explicitly by periods and other invariants, such that
$$L'(E/K, 1) = c \cdot \hat{h}(y_K).$$

## Remarks

In particular, $y_K$ has infinite order if and only if $L'(E/K, 1) \ne 0$, since $\hat{h}$ vanishes exactly on torsion points.
The Heegner hypothesis forces the sign of the functional equation of $L(E/K, s)$ to be $-1$, which gives $L(E/K, 1) = 0$.
The proof computes both sides.
The derivative is computed by the Rankin–Selberg method, and the height by local intersection theory of Heegner points on $X_0(N)$.
Gross and Zagier then compare the two as Fourier coefficients of modular forms.
Gross and Zagier proved it in 1986, with some additional conditions on $D$ that later work removed.
Kolyvagin's Euler system argument adds the following consequence.
If $L(E, s)$ has a zero of order at most $1$ at $s = 1$, then the rank of $E(\Q)$ equals that order and the [[Tate–Shafarevich group|BK9WBMF56G]] of $E$ is finite.

## Explanations

### L1 human

The theorem gives an exact formula linking two very different numbers.
One is the slope of the $L$-function of $E$ at the central point, an analytic quantity.
The other is the height of a specific point built from complex multiplication, which measures how complicated the point's coordinates are.
So a nonzero slope guarantees an explicit point of infinite order.

### L2 human

This is the strongest known evidence for the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]], covering analytic rank $0$ and $1$.
Gross and Zagier also used it, with Goldfeld's earlier work, to give an effective lower bound for class numbers of imaginary quadratic fields.
Generalisations to Shimura curves and higher-dimensional settings are an active area.
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# Semialgebraic set

## Summary

A semialgebraic set is a subset of $\R^n$ described by finitely many polynomial equations and strict inequalities combined with "and", "or" and "not".
By the [[Tarski–Seidenberg theorem|3W5JGDZAN0]], the projection of a semialgebraic set is again semialgebraic.
Semialgebraic sets have finitely many connected components and admit triangulations.
They are the natural objects of [[real algebraic geometry|PPWJ8NFH7G]] and the model example of an [[o-minimal structure|DTWHAW1N0V]].

## Explanations

### L1 human

The closed disc $x^2 + y^2 \le 1$ is not the zero set of polynomials, but it is described by one polynomial inequality.
Semialgebraic sets are all shapes obtained in this way from finitely many polynomial equations and inequalities.
A half-plane, a triangle and the region between two parabolas are examples.
The graph of the sine function is not, because it crosses the axis infinitely often, while a polynomial description allows only finitely many such crossings.
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# Absolutely convergent Fourier series

## Definition

Let $f \colon \R \to \C$ be a $2\pi$-periodic function that is [[integrable|X7GTF3D98F]] on $[-\pi, \pi]$.
For $n \in \Z$ let $\hat f(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-int} \, dt$ be its [[Fourier coefficients|5NN2YZZR3V]].
The [[Fourier series|G12KXDHPPS]] of $f$ is *absolutely convergent* if
$$\sum_{n \in \Z} \abs{\hat f(n)} < \infty.$$

## Remarks

If the series converges absolutely, then $\sum_n \hat f(n) e^{inx}$ converges uniformly on $\R$ to a continuous function $g$.
The function $g$ has the same Fourier coefficients as $f$, so $g = f$ almost everywhere.
One often says simply that $f$ has an absolutely convergent Fourier series.

Example: let $f(x) = \abs{x}$ for $-\pi \le x \le \pi$, extended with period $2\pi$.
Integration by parts gives $\hat f(0) = \pi/2$ and $\hat f(n) = \frac{(-1)^n - 1}{\pi n^2}$ for $n \ne 0$.
So $\abs{\hat f(n)} \le 2/(\pi n^2)$, and the sum $\sum_n \abs{\hat f(n)}$ is finite.

Non-example: the square wave equal to $1$ on $(0, \pi)$ and $-1$ on $(-\pi, 0)$ has $\abs{\hat f(n)} = \frac{2}{\pi \abs{n}}$ for odd $n$.
The sum of these over odd $n$ diverges like the harmonic series.
This also follows from the remark above: the square wave is not equal almost everywhere to a continuous function.
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# Dirac bundle

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]], and let $\mathrm{Cl}(M)$ be its Clifford bundle, whose fibre at $x$ is the [[Clifford algebra|A38B1QFK20]] of $(T_x M, g_x)$ with $v \cdot v = -g(v, v)$.
The [[Levi-Civita connection|CAZYGE1T4E]] induces a connection on $\mathrm{Cl}(M)$.
A *Dirac bundle* over $M$ is a real or complex vector bundle $S \to M$ with these data and properties.

1. $S$ is a bundle of left [[Clifford modules|31Q0RDPZ00]] over $\mathrm{Cl}(M)$.
That is, a smooth bundle map $\mathrm{Cl}(M) \otimes S \to S$, $\varphi \otimes \psi \mapsto \varphi \cdot \psi$, makes each fibre $S_x$ a module over $\mathrm{Cl}(T_x M)$.
2. $S$ has a Riemannian or Hermitian metric $\ip{\cdot}{\cdot}$ such that $\ip{e \cdot \psi_1}{e \cdot \psi_2} = \ip{\psi_1}{\psi_2}$ for every unit vector $e \in T_x M$ and all $\psi_1, \psi_2 \in S_x$.
3. $S$ has a [[connection|5ZYRRVE8BW]] $\nabla$ that preserves this metric.
4. $\nabla_X(\varphi \cdot \psi) = (\nabla_X \varphi) \cdot \psi + \varphi \cdot \nabla_X \psi$ for every vector field $X$, every section $\varphi$ of $\mathrm{Cl}(M)$ and every section $\psi$ of $S$.

The *Dirac operator* of $S$ is $D \psi = \sum_{i=1}^n e_i \cdot \nabla_{e_i} \psi$ for a local orthonormal frame $e_1, \ldots, e_n$.

## Remarks

Clause 2 is equivalent to the skew-adjointness of Clifford multiplication by each tangent vector.
Clauses 2 to 4 make $D$ formally self-adjoint, and its square satisfies $D^2 = \nabla^* \nabla + \mathfrak{R}$ for an explicit curvature endomorphism $\mathfrak{R}$.
The [[spinor bundle|PYYQQKAZ0C]] of a spin manifold, with the spin connection, is a Dirac bundle, and its Dirac operator is the Atiyah–Singer operator.

Example: let $S = \Lambda^* T^*M$ with the metric and connection induced by $g$, and let $v \cdot \omega = v^\flat \wedge \omega - \iota_v \omega$.
Then $v \cdot v \cdot \omega = -(v^\flat \wedge \iota_v \omega + \iota_v(v^\flat \wedge \omega)) = -g(v, v) \, \omega$, so clause 1 holds, and clauses 2 to 4 hold because the Levi-Civita connection is metric and commutes with wedge and contraction.
Its Dirac operator is $\sum_i (e_i^\flat \wedge \nabla_{e_i} - \iota_{e_i} \nabla_{e_i}) = d + \delta$, whose square is the [[Hodge Laplacian|PWG3GKZG80]].

Non-example: on $M = \R^n$ with $n \ge 2$, take the same bundle and Clifford multiplication, but the connection $\nabla'_X = \nabla_X + dx_1(X) \, A$, where $A$ is Clifford multiplication by $e_1 = \partial_1$.
Since $A$ is skew-adjoint, $\nabla'$ still preserves the metric, so clause 3 holds.
Clause 4 fails: for $X = e_1$ and $\varphi = e_2$, it would require $A(e_2 \cdot \psi) = e_2 \cdot A\psi$, but $e_1 e_2 = -e_2 e_1 \ne e_2 e_1$.

## Explanations

### L1 human

A Dirac bundle is a bundle of Clifford modules with a metric and a connection that are compatible with Clifford multiplication, so that it carries a well-behaved Dirac operator.
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# Mehler–Heine formula

## Statement

Let $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$ and let $J_0$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $0$.
Then
$$\lim_{n \to \infty} P_n\left( \cos \frac{z}{n} \right) = J_0(z),$$
uniformly for $z$ in each bounded subset of $\C$.

## Remarks

Proof idea: write $P_n(\cos \theta) = {}_2F_1\left( -n, n + 1; 1; \sin^2 \frac{\theta}{2} \right) = \sum_{k=0}^{n} \frac{(-n)_k (n + 1)_k}{(k!)^2} \sin^{2k} \frac{\theta}{2}$.
With $\theta = z/n$ and $k$ fixed, $(-n)_k (n + 1)_k \sim (-1)^k n^{2k}$ and $\sin^{2k} \frac{z}{2n} \sim \left( \frac{z}{2n} \right)^{2k}$.
So the $k$-th term tends to $\frac{(-1)^k}{(k!)^2} \left( \frac{z}{2} \right)^{2k}$, the $k$-th term of the series of $J_0(z)$, and a domination argument allows the termwise limit.
The same holds for [[Jacobi polynomials|TRD7DSF2KR]]: $\lim_{n \to \infty} n^{-\alpha} P_n^{(\alpha, \beta)}\left( \cos \frac{z}{n} \right) = \left( \frac{z}{2} \right)^{-\alpha} J_\alpha(z)$.
A consequence via Hurwitz's theorem: if $\theta_{n,k}$ is the $k$-th smallest zero of $\theta \mapsto P_n(\cos \theta)$ in $(0, \pi)$, then $n \theta_{n,k} \to j_{0,k}$, the $k$-th positive zero of $J_0$.

## Explanations

### L1 human

Near the end point $x = 1$, a Legendre polynomial of high degree oscillates faster and faster.
If one zooms in at the right scale, with the angle shrinking like $1/n$, the picture settles down to a fixed wave.
This limiting wave is the Bessel function $J_0$.

### L2 human

The formula describes the behaviour of orthogonal polynomials at the edge of their interval, where the oscillatory asymptotics in the interior break down.
In the interior, the classical asymptotic formulas give cosine waves with slowly varying amplitude; near $x = \pm 1$ Bessel functions take their place.
It explains why the zeros of $P_n$ closest to $1$, measured in the angle $\theta$, are about $j_{0,k}/n$.
Its versions for Jacobi and Laguerre polynomials, and their analogues for random matrices, describe the "hard edge" behaviour of eigenvalues.
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# Bernoulli polynomials

## Description

The Bernoulli polynomials are the polynomials $B_n(x) \in \Q[x]$, $n \ge 0$, given by the generating function
$$\frac{t \, e^{x t}}{e^t - 1} = \sum_{n=0}^{\infty} B_n(x) \frac{t^n}{n!}.$$
Equivalently, $B_n(x) = \sum_{k=0}^{n} \binom{n}{k} B_k \, x^{n-k}$, where $B_k$ are the [[Bernoulli numbers|0KC00K7KY4]] with $B_1 = -\frac{1}{2}$.

## Remarks

The first values are $B_0(x) = 1$, $B_1(x) = x - \frac{1}{2}$, $B_2(x) = x^2 - x + \frac{1}{6}$ and $B_3(x) = x^3 - \frac{3}{2} x^2 + \frac{1}{2} x$.
Each $B_n$ is monic of degree $n$, and $B_n(0) = B_n$.
They satisfy $B_n'(x) = n B_{n-1}(x)$ and $B_n(x + 1) - B_n(x) = n x^{n-1}$ for $n \ge 1$.
Summing the second identity gives the sums of powers: for $p \ge 0$ and $m \ge 1$,
$$\sum_{k=0}^{m-1} k^p = \frac{B_{p+1}(m) - B_{p+1}(0)}{p + 1}.$$
Check with $p = 1$, $m = 4$: $\frac{B_2(4) - B_2(0)}{2} = \frac{12}{2} = 6 = 0 + 1 + 2 + 3$.
They also satisfy $B_n(1 - x) = (-1)^n B_n(x)$.

## Explanations

### L1 human

The Bernoulli polynomials are a family of polynomials that turn sums of powers into polynomial formulas.
Their values at $0$ are the Bernoulli numbers.
The difference $B_n(x + 1) - B_n(x)$ is the simple monomial $n x^{n-1}$, which is what makes the sums telescope.
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# Regular summability method

## Definition

Let $M \colon D_M \to \mathbb{K}$ be a [[summability method|X80DZEXQDQ]], where $\mathbb{K}$ is $\R$ or $\C$.
The method $M$ is *regular* if every [[convergent sequence|0AZ3R52593]] $s$ belongs to $D_M$ and
$$M(s) = \lim_{n \to \infty} s_n.$$
Equivalently, every convergent series is $M$-summable, and its $M$-sum is its ordinary sum.

## Remarks

Example: Cesàro summation is regular, because the [[Cesàro means of a convergent sequence converge to the same limit|0RDRPBMNAC]].
Non-example: the method with $D_M$ the convergent sequences and $M(s) = 2 \lim s_n$ is a summability method, but it is not regular, since it sends the constant sequence $1$ to $2$.
A method that sums every convergent sequence, but not necessarily to its ordinary limit, is called *conservative*.

## Explanations

### L1 human

A regular method extends the ordinary limit: it never changes the value of a convergent sequence.
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# Sub-mean value property

## Statement

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be open, and let $u \colon \Omega \to [-\infty, \infty)$ be [[upper semicontinuous|RN4G8SGP6P]].
For $a \in \R^n$ and $r > 0$ let $\sigma_{a,r}$ be the normalised surface measure on the sphere $S(a, r)$, and let $\lambda$ be Lebesgue measure.
Assume that each $a \in \Omega$ has some $\rho(a) > 0$ such that $u(a) \le \int u \, d\sigma_{a,r}$ for all $0 < r < \rho(a)$, that is, $u$ is [[subharmonic|Z1MHMGFNFW]] in the local sense.
Then for every $a \in \Omega$ and every $r > 0$ with $\overline{B(a, r)} \subseteq \Omega$:

1. $u(a) \le \displaystyle\int_{S(a, r)} u \, d\sigma_{a,r}$;
2. $u(a) \le \dfrac{1}{\lambda(B(a, r))} \displaystyle\int_{B(a, r)} u \, d\lambda$.

## Remarks

Proof idea: since $u$ is upper semicontinuous, there are continuous functions $\varphi_k$ on $S(a, r)$ that decrease to $u$ there.
Let $h_k$ be the harmonic function in $B(a, r)$ with boundary values $\varphi_k$, given by the [[Poisson integral formula|PCZBE7EX01]].
The function $u - h_k$ again satisfies the local sub-mean inequality, so the [[maximum principle for subharmonic functions|9XJ5FBQD7M]] gives $u \le h_k$ in the ball.
Hence $u(a) \le h_k(a) = \int \varphi_k \, d\sigma_{a,r}$, and monotone convergence gives part 1.
Part 2 follows by integrating part 1 over the radii from $0$ to $r$.

So a locally defined subharmonic function satisfies the inequality on every ball whose closure lies in $\Omega$, not only on small ones.
Example: $u(x) = \abs{x}^2$ on $\R^n$ has sphere mean $\abs{a}^2 + r^2$ and ball mean $\abs{a}^2 + \frac{n}{n+2} r^2$, both at least $\abs{a}^2$.
For a harmonic function both inequalities become equalities, which is the [[mean value property|0HYPR64JM2]].
The ball $\overline{B(a, r)}$ must lie in $\Omega$: for $u(x) = -\log\abs{x}$ on $\R^2 \setminus \set{0}$, which is harmonic there, the circle mean about $a = (1, 0)$ with $r = 2$ is $-\log 2 < 0 = u(a)$.

## Explanations

### L1 human

A subharmonic function at a point is never larger than its average over a sphere or a ball around that point.
The definition only asks this for tiny spheres, and this result says it then holds for every sphere that fits in the domain.

### L2 human

The global inequality is what makes subharmonic functions useful: it gives the maximum principle, local integrability, and comparison with harmonic functions on balls.
The averages over spheres of radius $r$ are increasing in $r$ and tend to $u(a)$ as $r \to 0$.
For $C^2$ functions the condition is equivalent to $\Delta u \ge 0$.
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# Lorentzian manifold

## Summary

A Lorentzian manifold is a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] whose metric has exactly one negative direction at each point.
It is the mathematical model of spacetime in general relativity.
Its tangent vectors split into timelike, null and spacelike vectors, which gives each tangent space a light cone.

## Explanations

### L1 human

In a Lorentzian manifold one direction plays the role of time and the others play the role of space.
The metric measures time with the opposite sign from space.
At each point the vectors of squared length zero form a double cone, the light cone, along which light travels.
Minkowski space $\R^4$ with metric $-dt^2 + dx^2 + dy^2 + dz^2$ is the flat example.
The vector $(1, 0, 0, 0)$ has squared length $-1$ and points in a time direction, while $(1, 1, 0, 0)$ lies on the light cone.
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# Erdős–Kac theorem

## Statement

For an integer $n \ge 1$, let $\omega(n)$ be the [[number of distinct prime factors|08RQ62VAVA]] of $n$, and let $\log$ be the [[natural logarithm|04DXAFN2ES]].
For all real numbers $a < b$,
$$\lim_{x \to \infty} \frac{1}{x}\, \#\set{3 \le n \le x : a \le \frac{\omega(n) - \log\log n}{\sqrt{\log\log n}} \le b} = \frac{1}{\sqrt{2\pi}} \int_{a}^{b} e^{-t^2/2}\, dt.$$

## Remarks

In words, $\omega(n)$ has an approximately [[normal distribution|05AQE0QR0D]] with mean and variance $\log\log n$.
A heuristic proof: $\omega(n) = \sum_{p \le n} \mathbf{1}_{p \mid n}$, and for a random $n$ the events $p \mid n$ have probability $1/p$ and are nearly independent, so the central limit theorem applies with variance $\sum_{p \le x} 1/p \approx \log\log x$.
A rigorous proof restricts to small primes, shows near independence with a sieve, and compares moments with those of the normal law.
The earlier theorem of Hardy and Ramanujan says that $\omega(n)$ has normal order $\log\log n$: for every $\varepsilon > 0$, almost all $n$ satisfy $\abs{\omega(n) - \log\log n} < \varepsilon \log\log n$.
Since $\log\log n$ grows very slowly, the limit is reached slowly; for $n$ near $10^{9}$, $\log\log n \approx 3$.

## Explanations

### L1 human

Pick a large number at random and count its different prime factors.
The theorem says the count behaves like a bell curve: its average is about $\log\log n$, and the spread is about $\sqrt{\log\log n}$.
So most numbers have roughly $\log\log n$ prime factors, which is a very small number even for huge $n$.

### L2 human

The Erdős–Kac theorem is the founding result of probabilistic number theory.
It treats the indicators of divisibility by different primes as almost independent random variables, and turns this into a [[central limit theorem|0EPZC4Z1FC]] for the additive function $\omega$.
The same holds for $\Omega(n)$, the number of prime factors counted with multiplicity, and more generally for many additive functions.
It sharpens the Hardy–Ramanujan normal order result, which gives only the center and not the shape of the distribution.
</document>

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# Positive current

## Summary

A positive current is a [[current|360H7MD9RD]] of bidegree $(p, p)$ on a [[complex manifold|JSGWAN2RJV]] with a positivity property.
Its wedge product with products of the forms $i \alpha \wedge \bar\alpha$, for $(1, 0)$-forms $\alpha$, is a nonnegative measure.
Pierre Lelong introduced closed positive currents, and they are now a central tool of complex geometry.
The two basic examples are the current of integration over a complex analytic subvariety and the current $dd^c u$ of a [[plurisubharmonic function|FF2P7ER3QT]] $u$.
The coefficients of a positive current are complex measures, and closed positive currents carry [[Lelong numbers|WVB60RKC3Z]] that measure their singularities.

## Explanations

### L1 human

A current is a generalised differential form whose coefficients may be measures or even distributions.
In complex geometry there is a natural notion of positivity for forms of type $(p, p)$, which extends the idea of a nonnegative function or of a positive semidefinite Hermitian matrix.
A positive current is a current that is positive in this sense.
The simplest example is integration over a complex curve $C$ in $\C^2$: it sends a $2$-form to its integral over $C$, and complex curves always carry a positive orientation.
Another example is $i \partial\bar\partial \abs{z}^2$, the standard Kähler form of $\C^n$.
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# Wiener–Hopf factorization

## Statement

Let $k \in L^1(\R)$ and write $\hat g(\xi) = \int_{\R} g(x)\, e^{i\xi x}\, dx$ for the [[Fourier transform|DPVM34ADZV]].
Assume that $1 - \hat k(\xi) \ne 0$ for all $\xi \in \R$.
Since $\hat k(\xi) \to 0$ as $\abs{\xi} \to \infty$, the curve $\xi \mapsto 1 - \hat k(\xi)$, $\xi \in [-\infty, \infty]$, is closed; let $\nu$ be its [[winding number|0X7SBS8E0F]] about $0$.
Let $r(\xi) = (\xi - i)/(\xi + i)$.

1. There are $k_+, k_- \in L^1(\R)$, with $k_+ = 0$ on $(-\infty, 0)$ and $k_- = 0$ on $(0, \infty)$, and
$$1 - \hat k(\xi) = \bigl(1 - \hat k_-(\xi)\bigr)\, r(\xi)^{\nu}\, \bigl(1 - \hat k_+(\xi)\bigr), \quad \xi \in \R.$$
Here $1 - \hat k_+$ has no zeros in the closed upper half-plane and $1 - \hat k_-$ has no zeros in the closed lower half-plane.
2. Let $1 \le p < \infty$ and $W \varphi(x) = \varphi(x) - \int_0^\infty k(x - t) \varphi(t)\, dt$ on $L^p(0, \infty)$. Then $W$ is a [[Fredholm operator|8HWGKWD6ZM]] of [[index|2V9Q4KHY1T]] $-\nu$. If $\nu = 0$ it is invertible. If $\nu > 0$ it is injective and its range has codimension $\nu$. If $\nu < 0$ it is surjective and its kernel has dimension $-\nu$.

## Remarks

The functions $1 - \hat k_\pm$ extend [[holomorphically|0NAHXR1FRH]] to the corresponding half-planes, because $k_\pm$ vanish on a half-line.
When $\nu = 0$, a continuous branch of $\log\bigl(1 - \hat k\bigr)$ on $\R$ tends to $0$ at infinity; splitting it into an upper and a lower part and exponentiating gives the two factors.
The difficult point is that the split pieces again come from $L^1$ functions, which uses [[Wiener's lemma|NC768TWZ3D]] on absolutely convergent Fourier integrals.
For $\nu \ne 0$ one first divides by $r^\nu$, whose winding number is $\nu$.

Example: $k(x) = 2 e^{-x}$ for $x > 0$ and $k = 0$ otherwise gives $1 - \hat k(\xi) = 1 - \frac{2}{1 - i\xi} = \frac{\xi - i}{\xi + i} = r(\xi)$.
So $\nu = 1$, $k_\pm = 0$, and $W$ is injective with range of codimension $1$.

The hypothesis $1 - \hat k \ne 0$ on $\R$ is needed: for $k(x) = \frac{1}{2} e^{-\abs{x}}$ the symbol $\frac{\xi^2}{1 + \xi^2}$ vanishes at $0$, and then $W$ is not Fredholm on $L^p(0, \infty)$.

## Explanations

### L1 human

A Wiener–Hopf equation is a convolution equation cut off at $0$.
Its Fourier symbol can be written as a product of a part that "looks only forward" and a part that "looks only backward".
A simple extra factor counts how many times the symbol winds around $0$.
This winding number decides everything: if it is $0$ the equation has exactly one solution for each right side.

### L2 human

The theorem is the continuous analogue of the theory of Toeplitz operators, where the symbol lives on the unit circle instead of the real line.
It turns the Wiener–Hopf method into a complete solvability theory for kernels in $L^1(\R)$.
The sign convention of the Fourier transform matters.
With the kernel $e^{-i\xi x}$ the half-planes swap roles and the winding number changes sign, so the index becomes $+\nu$.
For systems of equations the symbol is a matrix function, and a single winding number is replaced by several partial indices, which is the matrix Wiener–Hopf factorization.
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# Dominated convergence theorem

## Statement

Let $(X, \Sigma, \mu)$ be a measure space.
Let $f_1, f_2, \dots$ and $f$ be real or complex measurable functions on $X$ with $f_n \to f$ [[almost everywhere|RARTMA0ZEC]].
Suppose there is an [[integrable function|X7GTF3D98F]] $g \colon X \to [0, \infty]$ with $\abs{f_n} \le g$ almost everywhere for every $n$.
Then $f$ and each $f_n$ are integrable, and
$$\lim_{n \to \infty} \int_X \abs{f_n - f} \, d\mu = 0, \qquad \lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu,$$
where the integrals are [[Lebesgue integrals|DWJF7ABPR1]].

## Remarks

Proof idea: $\abs{f} \le g$ almost everywhere, so $f$ is integrable.
The functions $2g - \abs{f_n - f}$ are non-negative almost everywhere and tend to $2g$.
[[Fatou's lemma|S95TR54TCG]] gives $\int 2g \le \int 2g - \limsup_n \int \abs{f_n - f}$, so $\limsup_n \int \abs{f_n - f} = 0$.
The second limit follows from $\abs{\int f_n - \int f} \le \int \abs{f_n - f}$.

Example: $\int_0^1 x^n \, dx \to 0$, since $x^n \to 0$ for $x \in [0, 1)$ and $\abs{x^n} \le 1$, which is integrable on $[0, 1]$.

The dominating function is needed.
For $f_n = n \mathbf{1}_{(0, 1/n)}$ on $\R$, $f_n \to 0$ everywhere but $\int f_n = 1$.
Here $\sup_n f_n$ behaves like $1/x$ near $0$ and is not integrable.

## Explanations

### L1 human

Suppose a sequence of functions converges at almost every point, and all of them stay below one fixed function of finite integral.
Then their integrals converge to the integral of the limit.
The fixed bound stops mass from escaping to infinity or piling up in a spike.

### L2 human

This is the most used convergence theorem of analysis.
It justifies differentiating under the integral sign, exchanging sums and integrals, and passing to the limit in many approximation arguments.
For finite measure spaces, a constant bound is enough, which gives the bounded convergence theorem.
The Vitali convergence theorem replaces domination by the weaker uniform integrability and gives a necessary and sufficient condition.
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# Grönwall area theorem

## Statement

Let $g$ be an injective meromorphic function on $\set{z \in \C : \abs{z} > 1}$ with [[Laurent expansion|0X3XD2ATB0]]
$$g(z) = z + b_0 + \sum_{n=1}^{\infty} b_n z^{-n}, \qquad \abs{z} > 1.$$
Then
$$\sum_{n=1}^{\infty} n \abs{b_n}^2 \le 1.$$

## Remarks

For $r > 1$, the image of the circle $\abs{z} = r$ is a Jordan curve, and the bounded region $E_r$ it encloses is the complement of $g(\set{\abs{z} > r})$.
Green's formula gives the area of $E_r$ as $\frac{1}{2i}\oint \bar{w}\, dw$ over this curve, which works out to $\pi\bigl(r^2 - \sum_{n \ge 1} n \abs{b_n}^2 r^{-2n}\bigr)$.
This area is nonnegative, and letting $r \to 1$ gives the inequality.
Equality holds exactly when the complement of $g(\set{\abs{z} > 1})$ has area $0$.
Example: $g(z) = z + 1/z$ maps $\set{\abs{z} > 1}$ onto $\C \setminus [-2, 2]$, and $b_1 = 1$ gives equality.
In particular $\abs{b_1} \le 1$.
Applied to $g(z) = 1/\sqrt{f(1/z^2)}$ for $f$ in the class $S$ of normalised [[univalent functions|R91E83SDBD]], it gives Bieberbach's inequality $\abs{a_2} \le 2$.

## Explanations

### L1 human

The function $g$ maps the outside of the unit disc onto the outside of some compact set in the plane.
That compact set has a nonnegative area, and the area can be written with the coefficients of $g$.
So these coefficients cannot be too large, which is the inequality.

### L2 human

The area theorem is the starting point of the coefficient theory of univalent functions.
Through the substitution $f \mapsto 1/\sqrt{f(1/z^2)}$, it gives $\abs{a_2} \le 2$ for $f \in S$, hence the [[Koebe quarter theorem|WX0HMW4GT7]] and the distortion theorems.
The Grunsky inequalities on the [[Grunsky matrix|XKRR37N916]] refine it to a statement about all the coefficients of $\log\frac{g(z) - g(\zeta)}{z - \zeta}$.
The equality case shows that the bound $\abs{b_1} \le 1$ is attained exactly by maps onto the complement of a segment.
</document>

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# Poincaré's functional equation

## Definition

Let $U \subseteq \C$ be an open set containing $0$, and let $f \colon U \to \C$ be [[holomorphic|0NAHXR1FRH]] with $f(0) = 0$.
Let $\lambda = f'(0)$.
*Poincaré's functional equation* for $f$ is the [[functional equation|SRAN4EKWS0]]
$$\Psi(\lambda z) = f(\Psi(z)) \quad \text{for all } z \text{ near } 0.$$
The unknown is a holomorphic function $\Psi$ on a neighbourhood of $0$ with $\Psi(0) = 0$ and $\Psi'(0) = 1$.

## Remarks

Example: $f(w) = w^2 + 2w$ has $f(0) = 0$ and $\lambda = f'(0) = 2$.
The function $\Psi(z) = e^z - 1$ satisfies $\Psi(0) = 0$, $\Psi'(0) = 1$ and $f(\Psi(z)) = (e^z - 1)^2 + 2(e^z - 1) = e^{2z} - 1 = \Psi(2z)$.

Non-example: for the same $f$, the function $\Psi(z) = z$ fails, since $f(z) = z^2 + 2z \ne 2z$ for $z \ne 0$.

If $\Phi$ is a solution of [[Schröder's equation|VTJV7JVJEA]] $\Phi \circ f = \lambda \Phi$ with $\Phi(0) = 0$ and $\Phi'(0) = 1$, then its local inverse $\Psi = \Phi^{-1}$ solves Poincaré's equation.
So for $\abs{\lambda} > 1$ a unique solution exists, by the Koenigs linearization theorem applied to $f$.
Assume moreover that $f$ is [[entire|0G4F0XB9X2]].
Then the formula $\Psi(z) = f^n(\Psi(z / \lambda^n))$, with $n$ so large that $z / \lambda^n$ lies in the domain of $\Psi$, extends $\Psi$ to an entire function.
This function is the Poincaré function of $f$ at $0$.
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# Cayley–Bacharach theorem

## Statement

Let $k$ be an algebraically closed field, and let $\mathbb{P}^2$ be the [[projective plane|ZT9A6JK8QF]] over $k$.
Let $C_1$ and $C_2$ be [[plane curves|EWS4WFTA7W]] of degrees $d$ and $e$, defined by [[homogeneous polynomials|2C3PAQ7H50]] with no common factor.
Assume that $C_1$ and $C_2$ meet in exactly $de$ distinct points.
Then every plane curve of degree $d + e - 3$ that passes through all but one of these $de$ points also passes through the remaining point.

## Remarks

The most quoted case is $d = e = 3$: if two plane cubics meet in nine distinct points, then every cubic through eight of them passes through the ninth.
For this case, the cubics through the eight points form a vector space of dimension $10 - 8 = 2$, because the eight points impose independent conditions on cubics.
The defining polynomials $F_1, F_2$ of $C_1, C_2$ are independent and lie in this space, so every cubic through the eight points is $\lambda F_1 + \mu F_2$ and vanishes at the ninth point.
The general case is usually proved with duality on the zero-dimensional complete intersection $C_1 \cap C_2$.
By [[Bézout's theorem|KWNVF194D8]], two curves of degrees $d$ and $e$ without common component meet in $de$ points counted with multiplicity, so the hypothesis says that all intersections are transverse.
The cubic case gives a proof of the associativity of the group law on an elliptic curve, and of Pascal's theorem on six points of a conic.

## Explanations

### L1 human

Two curves in the plane cross in a predictable number of points, for two cubics nine.
The theorem says that these points are not independent of each other.
A curve of the right degree that goes through all of them but one is forced to go through the last one as well.
For cubics, eight of the nine points already decide the ninth.

### L2 human

The theorem says that the intersection points fail to impose independent conditions on curves of degree $d + e - 3$.
This is the degree that appears in the adjunction formula for the canonical divisor of a plane curve.
It is the source of many classical incidence theorems, such as Pappus's and Pascal's theorems, which come from degenerate cubics made of lines and conics.
In modern language it is a consequence of duality for the complete intersection $C_1 \cap C_2$.
</document>

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# Sperner property

## Definition

Let $P$ be a finite [[graded poset|1C7TKGTB2F]] with rank function $\rho$, and for each integer $i \ge 0$ let $P_i = \set{x \in P : \rho(x) = i}$ be the $i$-th rank level.
The poset $P$ has the *Sperner property* when every [[antichain|03HF498P1J]] $A \subseteq P$ satisfies $\abs{A} \le \max_i \abs{P_i}$.

## Remarks

Since each $P_i$ is an antichain, the condition says that the [[width|RA5SQ908G1]] of $P$ equals $\max_i \abs{P_i}$.

Example: the lattice of subsets of $\set{1, \ldots, n}$, ranked by size, has the Sperner property.
Check: by Sperner's theorem, an antichain has at most $\binom{n}{\floor{n/2}}$ members, which is the size of the middle level.

Non-example: let $P$ have elements $a_1, a_2, b$ of rank $0$ and $c, d_1, d_2$ of rank $1$, with the relations $a_1 < c$, $a_2 < c$, $b < d_1$, $b < d_2$ and no others.
Each level has $3$ elements, but $\set{a_1, a_2, d_1, d_2}$ is an antichain with $4$ elements.

A poset is strongly Sperner when, for every $k$, the largest union of $k$ antichains is no larger than the largest union of $k$ rank levels.
</document>

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# Biproduct

## Summary

A biproduct of two objects is a single object that is at the same time their [[product|S5744QZV4N]] and their [[coproduct|4FJ0S71P58]], with compatible projections and inclusions.
It generalizes the direct sum $A \oplus B$ of abelian groups or modules, which has both universal properties.
Additive categories are the preadditive categories with a zero object in which all binary biproducts exist.
In a preadditive category, a finite product is automatically a biproduct.

## Explanations

### L1 human

The direct sum $\R^2 = \R \oplus \R$ can be seen in two ways.
It is built from two copies of $\R$ by taking pairs, with the two coordinate projections.
It also receives the two copies of $\R$ as the axes, so that a pair of maps out of the two copies combines into one map out of $\R^2$.
A biproduct is an object with both these features, where projecting after including the same copy gives the identity and after including the other copy gives zero.
</document>

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# Mordell's theorem

## Statement

Let $E$ be an [[elliptic curve|427N6S6QQW]] over the field $\Q$ of [[rational numbers|0NCJ5WXN6Z]].
Then the group $E(\Q)$ of rational points of $E$, with the [[chord-and-tangent law|P7V1BQJPSS]], is a [[finitely generated|082DWWXMRJ]] [[abelian group|082GHNPJ53]].

## Remarks

By the structure theorem for finitely generated abelian groups, $E(\Q) \cong \Z^r \oplus T$, where $T$ is the finite torsion subgroup and $r \ge 0$ is the rank of $E$.
The proof has two steps.
The weak Mordell theorem says that $E(\Q)/2E(\Q)$ is finite; it is proved with a descent map into a finite group built from square classes of rationals.
Then a height function $h$ on $E(\Q)$, with finitely many points of bounded height and $h(2P) \approx 4h(P)$, lets one show by descent that finitely many points together with $2E(\Q)$ generate everything.
Example: for $y^2 = x^3 - x$ one has $E(\Q) = \set{O, (0,0), (1,0), (-1,0)}$, of rank $0$.
Weil extended the theorem to abelian varieties over number fields; the general result is the Mordell–Weil theorem.

## Explanations

### L1 human

An elliptic curve over the rationals can have infinitely many rational points.
Mordell's theorem says that they are never a shapeless infinite mess: finitely many of them generate all others by the chord-and-tangent construction.

### L2 human

The theorem gives the two basic invariants of an elliptic curve over $\Q$.
The torsion subgroup is described by [[Mazur's torsion theorem|RQS6YHAR8N]], and the rank is the subject of the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].
The descent step uses [[height functions|F1FF505PWH]], and the obstruction to making descent effective is measured by the [[Selmer group|36YNSTESMQ]] and the [[Tate–Shafarevich group|BK9WBMF56G]].
No algorithm is known that provably computes the rank in all cases.
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# Particle filter

## Summary

A particle filter is a Monte Carlo method that tracks the hidden state of a dynamical system from noisy observations.
It represents the conditional distribution of the current state given the observations so far by a cloud of weighted samples, called particles.
At each time step the particles are moved by the system dynamics, reweighted by how well they explain the new observation, and resampled.
It works for nonlinear and non-Gaussian state-space models, such as [[hidden Markov models|HJVM390354]] with continuous states, where exact filtering formulas are not available.

## Explanations

### L1 human

Think of tracking a robot that you cannot see directly, using only a noisy sensor.
You scatter many guesses of where the robot might be.
When the robot moves, you move every guess in the same way, with some random jitter.
When a sensor reading arrives, guesses that agree with it get more weight, and guesses that disagree get less.
Then you copy the heavy guesses and drop the light ones, and the cloud of guesses follows the robot.
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# Larson–Sweedler theorem

## Statement

Let $K$ be a field and let $H$ be a finite-dimensional [[Hopf algebra|7XFYP0N84H]] over $K$.
Then:

1. The space of left [[integrals|RJA5026RGX]] in $H$ is one-dimensional over $K$.
2. $H$ is a [[Frobenius algebra|8EVP1K2CZH]].

## Remarks

The proof applies the fundamental theorem of [[Hopf modules|N8X7XJBKT0]] to the dual space $H^*$, which is a Hopf module over $H$.
This gives an isomorphism $H^* \cong (H^*)^{\mathrm{co}H} \otimes H$, and the coinvariants of $H^*$ are the left integrals on $H$.
Comparing dimensions shows that this space is one-dimensional, and the isomorphism $H^* \cong H$ of $H$-modules is the Frobenius property; applying this to $H^*$ gives the statement about integrals in $H$.
Example: for a finite group $G$, the left integrals in $KG$ are the multiples of $\sum_{g \in G} g$.
Indeed, if $\Lambda = \sum_g a_g g$ satisfies $h \Lambda = \Lambda$ for all $h \in G$, then all $a_g$ are equal.
Without finite dimension the conclusion fails: in $K[x]$ with $x$ primitive the only integral is $0$.
The same work shows that the antipode of a finite-dimensional Hopf algebra is bijective.

## Explanations

### L1 human

An integral is a Hopf algebra version of the sum of all elements of a finite group.
The theorem says that, in a finite-dimensional Hopf algebra, such an element always exists and is unique up to a scalar.
It also says that the algebra is Frobenius, a strong symmetry property: as a module over itself it looks like its own dual.

### L2 human

The theorem is the starting point for the structure theory of finite-dimensional Hopf algebras.
Together with it, Larson and Sweedler proved the Hopf algebra form of Maschke's theorem: $H$ is semisimple if and only if $\varepsilon(\Lambda) \ne 0$ for a nonzero left integral $\Lambda$.
For $H = KG$ this gives $\varepsilon(\sum_g g) = \abs{G}$, and so recovers the classical condition that the characteristic of $K$ does not divide $\abs{G}$.
Integrals also define the distinguished group-like elements of $H$ and $H^*$, which appear in Radford's formula for $S^4$.
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# Rado's theorem

## Statement

Let $A$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] with rational entries, with columns $c_1, \dots, c_n \in \Q^m$.
The system $Ax = 0$ is called partition regular if the following holds for every finite colouring of the positive integers.
There is a solution $x = (x_1, \dots, x_n)$ in positive integers whose entries all have the same colour.
The matrix $A$ satisfies the columns condition if the index set $\set{1, \dots, n}$ has a partition into nonempty sets $I_1, \dots, I_k$ with these properties, where $s_j = \sum_{i \in I_j} c_i$:

1. $s_1 = 0$.
2. For each $j$ with $2 \le j \le k$, the vector $s_j$ is a linear combination with rational coefficients of the columns $c_i$ with $i \in I_1 \cup \dots \cup I_{j-1}$.

Then the system $Ax = 0$ is partition regular if and only if $A$ satisfies the columns condition.

## Remarks

For a single equation $a_1 x_1 + \dots + a_n x_n = 0$ with nonzero integer coefficients, the columns condition says that some nonempty set of the coefficients $a_i$ has sum $0$.
So $x + y - z = 0$ is partition regular, because $1 + (-1) = 0$; this is [[Schur's theorem|V6XJQXZ7E8]].
The equation $x + y - 3z = 0$ is not partition regular, because no nonempty subset of $\set{1, 1, -3}$ sums to $0$.
The system $x_i - x_0 - i y = 0$ for $1 \le i \le k - 1$, in the unknowns $x_0, \dots, x_{k-1}, y$, satisfies the columns condition with $I_1 = \set{x_0, \dots, x_{k-1}}$ and $I_2 = \set{y}$.
Its monochromatic solutions are arithmetic progressions $x_0, x_0 + y, \dots, x_0 + (k-1)y$ with $y \ge 1$, so [[van der Waerden's theorem|V96FRZ2ME1]] is a special case.
The proof of sufficiency builds the monochromatic solution from many uses of van der Waerden's theorem.
The proof of necessity colours each integer by its last nonzero digit in base $p$, for a suitable prime $p$.

## Explanations

### L1 human

Colour every positive integer with one of finitely many colours.
Some systems of linear equations always have a solution inside a single colour, whatever the colouring is.
Rado's theorem tells exactly which systems have this property, with a simple test on the columns of the matrix.
For one equation the test is easy: some of the coefficients must add up to zero.

### L2 human

Rado's theorem is a central result of [[Ramsey theory|G5TWXV565S]] on the integers.
It contains Schur's theorem and van der Waerden's theorem as special cases.
The columns condition is checked by linear algebra over $\Q$, so partition regularity is decidable.
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# Uniform polytope

## Summary

A uniform polytope is a polytope whose symmetries can carry any vertex to any other vertex and whose facets are themselves uniform.
In dimension two the uniform polytopes are the regular polygons.
Every [[regular polytope|F60PD912D5]] is uniform, but uniform polytopes are far more numerous.
In dimension three the convex ones are the Platonic solids, the thirteen Archimedean solids, and the infinite families of prisms and antiprisms.
Most uniform polytopes arise from the [[Wythoff construction|J4EDDYCXMR]].

## Explanations

### L1 human

A uniform polytope is a solid with regular polygon faces in which every corner looks exactly like every other corner.
The faces need not all be the same: the cuboctahedron has squares and triangles, arranged the same way around each corner.
A triangular prism with square sides is uniform too.
A square pyramid is not, since its tip looks different from its base corners.
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# Scaling function and refinement equation

## Definition

Let $(V_j)_{j \in \Z}$ be a [[multiresolution analysis|1GHQMBR7DE]] of $L^2(\R)$.
A *scaling function* of it is a function $\varphi \in V_0$ whose integer translates $(\varphi(\cdot - k))_{k \in \Z}$ form an orthonormal basis of $V_0$.
The functions $\sqrt2 \, \varphi(2 \cdot - k)$, $k \in \Z$, form an orthonormal basis of $V_1$, and $\varphi \in V_1$.
Hence there is a unique sequence $(h_k)_{k \in \Z} \in \ell^2(\Z)$ with
$$\varphi(x) = \sqrt2 \sum_{k \in \Z} h_k \, \varphi(2x - k),$$
with convergence in $L^2(\R)$ (see [[Lp space|SASP14MA1Z]]).
This identity is the *refinement equation* (or two-scale equation) of $\varphi$, and $(h_k)$ is its *low-pass filter*; explicitly $h_k = \ip{\varphi}{\sqrt2 \, \varphi(2 \cdot - k)}$.

## Remarks

Orthonormality of the translates of $\varphi$ gives $\sum_k h_k \overline{h_{k + 2m}} = \delta_{m,0}$ for all $m \in \Z$.
If $\varphi$ is integrable with $\int \varphi \ne 0$, integrating the refinement equation gives $\sum_k h_k = \sqrt2$.
The function $\psi(x) = \sqrt2 \sum_k (-1)^k \overline{h_{1-k}} \, \varphi(2x - k)$ is then an orthonormal wavelet.

Example: in the Haar multiresolution analysis, $\varphi = 1_{[0, 1)}$ and $\varphi(x) = \varphi(2x) + \varphi(2x - 1)$.
So $h_0 = h_1 = 1/\sqrt2$ and $h_k = 0$ otherwise.
The formula for $\psi$ gives $\psi(x) = \varphi(2x) - \varphi(2x - 1) = 1_{[0, 1/2)}(x) - 1_{[1/2, 1)}(x)$, the Haar wavelet.

Non-example: $\varphi = 1_{[0, 2)}/\sqrt2$ satisfies the refinement equation $\varphi(x) = \varphi(2x) + \varphi(2x - 2)$, but its integer translates overlap, so $\ip{\varphi}{\varphi(\cdot - 1)} = \tfrac12 \ne 0$ and it is not the scaling function of a multiresolution analysis in the sense above.
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# Figure-eight knot

## Example

The *figure-eight knot* is the [[knot|QYEAZTYGPT]] obtained by closing up the figure-eight knot of rope.
Its standard diagram has $4$ crossings that alternate over and under along the curve.
One parametrization of it is
$$t \mapsto \bigl((2 + \cos 2t) \cos 3t, \ (2 + \cos 2t) \sin 3t, \ \sin 4t\bigr), \qquad t \in [0, 2\pi).$$
Its [[crossing number|SWJZ4QRF9W]] is $4$.
Indeed, the only knots with crossing number less than $4$ are the unknot and the two trefoils, and the figure-eight knot is neither.

## Remarks

Its Alexander polynomial is $-t^{-1} + 3 - t$, which differs from that of the unknot, $1$, and from that of the trefoil, $t^{-1} - 1 + t$.
It is not [[tricolorable|KNAGK0W65G]], because its determinant is $5$, not divisible by $3$.
It is equivalent to its mirror image, unlike the trefoil.
It is a [[hyperbolic knot|K47G2Q24DT]]: its complement carries a complete hyperbolic metric of finite volume, made of two regular ideal tetrahedra.

## Explanations

### L1 human

The figure-eight knot is the next simplest knot after the trefoil, with four crossings in its simplest picture.
Its name comes from the figure-eight knot that climbers and sailors tie, with the ends joined.
Unlike the trefoil, it can be deformed into its own mirror image.
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# Vinogradov's theorem

## Statement

There is an integer $N_0$ such that every odd integer $N > N_0$ is a sum of three [[prime numbers|01826KMG2H]].

## Remarks

Vinogradov proved the theorem in 1937 by the [[Hardy–Littlewood circle method|ZQFBEX62D0]].
The proof gives an asymptotic formula: for every $A > 0$,
$$\sum_{p_1 + p_2 + p_3 = N} \log p_1 \log p_2 \log p_3 = \tfrac12 \mathfrak{S}(N) N^2 + O\!\left(\frac{N^2}{(\log N)^{A}}\right),$$
where
$$\mathfrak{S}(N) = \prod_{p \mid N} \left(1 - \frac{1}{(p-1)^2}\right) \prod_{p \nmid N} \left(1 + \frac{1}{(p-1)^3}\right).$$
For odd $N$ the singular series satisfies $\mathfrak{S}(N) > 0.6$, so the main term dominates for large $N$; for even $N$ the factor at $p = 2$ is $0$.
The major arcs are handled by the [[Siegel–Walfisz theorem|XEPW251A97]], and the new step was Vinogradov's bound for the exponential sum $\sum_{p \le N} \log p \; e(p\alpha)$ on the minor arcs.
The first proof gave no explicit $N_0$, because the Siegel–Walfisz constants are ineffective; later proofs made $N_0$ explicit.
In 2013 Harald Helfgott proved that every odd integer greater than $5$ is a sum of three primes.

## Explanations

### L1 human

The weak Goldbach conjecture says that every odd number greater than $5$ is a sum of three primes, as in $21 = 3 + 7 + 11$.
Vinogradov proved this for all odd numbers beyond some huge bound.
His method also counts, approximately, how many ways there are to do it.

### L2 human

The theorem is the model application of the circle method to primes.
The case of three primes works because the minor arcs can be bounded with one factor estimated pointwise and the other two in mean square, using Parseval's identity.
For two primes, which is [[Goldbach's conjecture|063JNBSMJ0]], this argument fails: the mean square alone is as large as the expected main term.
The strongest results towards Goldbach's conjecture by other means are sieve results such as [[Chen's theorem|WE0RQTR68H]].
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# Cobordism ring

## Summary

The cobordism ring is the graded ring whose elements in degree $n$ are [[cobordism|T8BVGPPSZP]] classes of closed smooth $n$-manifolds.
Its addition is disjoint union and its multiplication is cartesian product.
There are versions for unoriented, oriented, complex and other structured manifolds.
Thom computed the unoriented ring and showed that cobordism groups are homotopy groups of Thom spectra.
This brought homotopy theory into the classification of manifolds.

## Explanations

### L1 human

Two closed manifolds are cobordant if together they form the boundary of a compact manifold one dimension higher.
For example, two circles are cobordant, since they bound a cylinder.
Cobordism classes can be added by putting manifolds side by side and multiplied by taking products.
The result is a ring that records which manifolds can be turned into one another by such a filling.
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# Sine and cosine transforms

## Summary

The sine and cosine transforms are the real forms of the [[Fourier transform|DPVM34ADZV]] for functions on the half-line $[0, \infty)$.
The cosine transform integrates $f$ against $\cos(2\pi x \xi)$, and the sine transform integrates it against $\sin(2\pi x \xi)$.
They are the Fourier transforms of the even and of the odd extension of $f$, up to constant factors.
They are used to solve boundary value problems on a half-line, where the boundary condition picks the even or the odd extension.
See the [[definition|Q4HR7AF2R9]].

## Explanations

### L1 human

A signal that starts at time $0$ can be mirrored to negative times in two ways: as an even copy or as an odd copy.
The mirrored signal has a Fourier transform made only of cosines in the even case and only of sines in the odd case.
So one real transform suffices instead of a complex one.
For $f(x) = e^{-x}$ on $[0, \infty)$, the cosine transform is $1/(1 + 4\pi^2 \xi^2)$.
These transforms are the continuous analogues of the [[sine and cosine series|Y6EPBG6S8G]] of a function on an interval.
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# q-exponential

## Definition

Let $q \in \C$ with $0 < \abs{q} < 1$, and let $[n]_q! = \prod_{k=1}^{n} \frac{1 - q^k}{1 - q}$ be the $q$-factorial, a [[q-analog|GFSDMJHHVA]] of $n!$.
The *$q$-exponential* is the [[power series|0MZB06ST2B]]
$$e_q(z) = \sum_{n=0}^{\infty} \frac{z^n}{[n]_q!},$$
for $z \in \C$ with $\abs{z} < \frac{1}{\abs{1 - q}}$.
The second $q$-exponential is
$$E_q(z) = \sum_{n=0}^{\infty} \frac{q^{\binom{n}{2}} z^n}{[n]_q!}, \qquad z \in \C.$$

## Remarks

Since $\frac{z^n}{[n]_q!} = \frac{((1 - q) z)^n}{(q; q)_n}$, the [[q-binomial theorem|VVT27RB46B]] gives the products $e_q(z) = \frac{1}{((1 - q) z; q)_\infty}$ and $E_q(z) = (-(1 - q) z; q)_\infty$, with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].
The ratio test gives the radius of convergence $\frac{1}{\abs{1-q}}$ of $e_q$, since $[n+1]_q \to \frac{1}{1 - q}$.
With $D_q f(z) = \frac{f(qz) - f(z)}{(q - 1) z}$, one has $D_q z^n = [n]_q z^{n-1}$, hence $D_q e_q(z) = e_q(z)$.

Example: the products show $e_q(z) E_q(-z) = \frac{((1-q)z; q)_\infty}{((1-q)z; q)_\infty} = 1$.
Check on the first terms: $e_q(z) E_q(-z) = (1 + z + \cdots)(1 - z + \cdots) = 1 + 0 \cdot z + \cdots$.

Non-example: $e_q(x) e_q(y) \ne e_q(x + y)$ in general.
The coefficient of $xy$ is $1$ on the left, and on the right it is $\frac{2}{[2]_q} = \frac{2}{1 + q}$, which differs from $1$ for $q \ne 1$.

Some authors use $\sum_n z^n / (q; q)_n$ instead, which is $e_q\left(\frac{z}{1-q}\right)$.
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# Error of best approximation

## Definition

Let $X$ be a [[normed vector space|03J5RAKB30]], let $A \subseteq X$ be nonempty, and let $x \in X$.
The *error of best approximation* of $x$ from $A$ is the [[infimum|04KFTA34C5]]
$$E(x, A) = \inf_{a \in A} \norm{x - a}.$$
In particular, for a continuous $f \colon [a, b] \to \R$ with the supremum norm and the set $\mathcal{P}_n$ of real [[polynomial functions|1G0B73WPX8]] of degree at most $n$,
$$E_n(f) = \inf_{p \in \mathcal{P}_n} \max_{a \le x \le b} \abs{f(x) - p(x)}.$$

## Remarks

Example: for $f(x) = x^2$ on $[0, 1]$ one has $E_1(f) = 1/8$.
The line $p(x) = x - 1/8$ gives $\abs{x^2 - x + 1/8} \le 1/8$ on $[0, 1]$, and no line does better, because the error of $p$ takes the values $1/8, -1/8, 1/8$ at $0, 1/2, 1$.

Non-example: the error of a particular approximation is in general larger.
The line $x$ has error $\max_{[0, 1]} \abs{x^2 - x} = 1/4$, which is not $E_1(f)$.

$E(x, A) = 0$ if and only if $x$ lies in the closure of $A$.
The infimum equals $\norm{x - a^*}$ when $a^*$ is a [[best approximation|7N4988HN3P]].
Since $\mathcal{P}_0 \subseteq \mathcal{P}_1 \subseteq \cdots$, the sequence $E_n(f)$ is nonincreasing.
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# Meijer G-function transforms

## Summary

A Meijer G-function transform, or G-transform, is an [[integral transform|YCK709QB06]] on $(0, \infty)$ whose kernel is a [[Meijer G-function|19JNNN5KV7]] of the product $xy$.
Many classical transforms are special cases, among them the [[Laplace transform|0V49JEZW06]], the [[Stieltjes transform|RG1HFAYD98]] and transforms with Bessel function kernels.
The [[Mellin transform|N5PVGRJE1T]] of a G-function is a ratio of products of gamma functions.
So the whole family can be studied at once through the Mellin transform.

## Explanations

### L1 human

The Meijer G-function is one very flexible special function that contains the exponential, the Bessel functions, the hypergeometric functions and many others.
Using it as the kernel of a transform gives one formula that covers a whole zoo of transforms.
For example, one choice of parameters makes the kernel $e^{-xy}$, and the G-transform becomes the Laplace transform.
Results proved once for G-transforms, such as inversion and composition rules, then hold for all these transforms.
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# Polynomial identity ring

## Summary

A polynomial identity ring, or PI ring, is a [[ring|0B1261B5ED]] in which some nonzero noncommutative polynomial vanishes for all choices of elements.
Commutative rings are PI rings, because $xy - yx$ vanishes on them, and so are matrix rings over commutative rings by the [[Amitsur–Levitzki theorem|YDH60KPTWT]].
PI rings form the class of noncommutative rings that are closest to commutative rings, and many results of commutative algebra extend to them.
Their theory is built on [[free algebras|2P3Y2FRCSZ]], [[central polynomials|667YZTHE96]] and [[generic matrices|GJGEVMZ828]].

## Explanations

### L1 human

In a commutative ring the rule $xy = yx$ holds for all elements.
A PI ring is a ring where some other, more complicated rule of this kind holds for all elements.
For $2 \times 2$ matrices, $xy$ and $yx$ differ, but the rule $(xy - yx)^2 z = z (xy - yx)^2$ always holds.
Such a rule is called a polynomial identity, and its presence forces the ring to behave much like a commutative ring or a ring of matrices.
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# Zero object

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
An object $0$ of $\mathcal{C}$ is a *zero object* if it is both an [[initial object and a terminal object|EK85VSSRMR]].
So for every object $X$ there is exactly one [[morphism|859F0TFVA3]] $0 \to X$ and exactly one morphism $X \to 0$.

## Remarks

A zero object is unique up to unique isomorphism, because an initial object is.
A category with a zero object is called pointed.

Example: in the category of groups, the [[trivial group|0BNPC9850Y]] $\{e\}$ is a zero object.
The only homomorphism $\{e\} \to G$ sends $e$ to the identity of $G$, and the only map $G \to \{e\}$ is the constant map, which is a homomorphism.
The same check shows that the zero module is a zero object in the category of modules over a ring.
It also shows that a one-point set is a zero object in the category of pointed sets.

Non-example: in the category of sets, a one-point set $\{*\}$ is terminal but not initial, since there are two functions $\{*\} \to \{0, 1\}$.
The empty set is initial but not terminal, since there is no function from a non-empty set to it.
So the category of sets has no zero object.
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# Stable ∞-category

## Definition

Let $\mathcal{C}$ be an [[(∞,1)-category|YFY1FWA1MY]] with a [[zero object|AQET9YHH6F]] $0$.
A *triangle* in $\mathcal{C}$ is a commutative square
$$\begin{array}{ccc} X & \xrightarrow{f} & Y \\ \downarrow & & \downarrow g \\ 0 & \longrightarrow & Z \end{array}$$
whose lower left corner is $0$.
It is a fiber sequence if the square is a [[pullback|Q05K04QJ71]], and a cofiber sequence if the square is a [[pushout|Q699KKMVJD]].
The category $\mathcal{C}$ is *stable* if:

1. $\mathcal{C}$ has a zero object.
2. Each morphism $g \colon Y \to Z$ has a fiber, that is, a pullback of $Y \to Z \leftarrow 0$, and each morphism $f \colon X \to Y$ has a cofiber, that is, a pushout of $0 \leftarrow X \to Y$.
3. A triangle in $\mathcal{C}$ is a fiber sequence if and only if it is a cofiber sequence.

Pullbacks and pushouts are meant in the ∞-categorical sense, that is, as homotopy pullbacks and homotopy pushouts.

## Remarks

Equivalently, $\mathcal{C}$ is stable if it has a zero object and all finite limits and finite colimits, and its pullback squares are exactly its pushout squares.

Example: the ∞-category of [[spectra|1GHF4EKBXS]] is stable.
The zero spectrum is a zero object, and spectra have all finite limits and colimits, so clauses 1 and 2 hold.
Clause 3 holds because the suspension functor on spectra is an equivalence.
In a pointed ∞-category with finite limits and colimits, this forces fiber and cofiber sequences to agree.

Non-example: the ∞-category of pointed spaces has a zero object, the one-point space, and has fibers and cofibers, but clause 3 fails.
The cofiber of $S^1 \to *$ is $S^2$, while the fiber of $* \to S^2$ is the loop space $\Omega S^2$.
This is not equivalent to $S^1$, because $\pi_2(\Omega S^2) \cong \pi_3(S^2) \cong \Z$ while $\pi_2(S^1) = 0$.

The suspension $\Sigma X$ is the cofiber of $X \to 0$, and the loop object $\Omega X$ is the fiber of $0 \to X$.
In a stable ∞-category, $\Sigma$ and $\Omega$ are inverse equivalences.
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# Fatou set

## Summary

The Fatou set of a [[rational function|0XDK9CGYKJ]] is the open set of points near which its iterates behave in a stable, predictable way.
Precisely, it is where the iterates form a [[normal family|JTVWRXXAKM]].
Its complement is the [[Julia set|5M4S7P04GR]].
The map permutes the connected components of the Fatou set.
The possible types of periodic components are completely classified.

## Explanations

### L1 human

Iterate a rational map and look at a small disc of starting points.
If all points of the disc move together and do the same thing in the long run, the disc is in the Fatou set.
For $z \mapsto z^2$ the Fatou set has two pieces.
One is the open unit disc, whose points are pulled to $0$.
The other is the outside of the unit circle, whose points run off to infinity.
The Fatou set is the calm region, and the Julia set is its chaotic border.
</document>

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# Brahmagupta's formula

## Statement

Let $ABCD$ be a convex [[cyclic quadrilateral|0RCP929Y1M]] in the Euclidean plane with side lengths $a$, $b$, $c$, $d$.
Let $s = \frac{1}{2}(a + b + c + d)$ be its semiperimeter.
Then the area $K$ of $ABCD$ is
$$K = \sqrt{(s - a)(s - b)(s - c)(s - d)}.$$

## Remarks

The diagonal $AC$ splits $ABCD$ into two triangles, and the opposite angles at $B$ and $D$ add up to $\pi$ because $ABCD$ is cyclic.
Computing $AC^2$ by the law of cosines in both triangles and adding the two areas gives the formula after factoring.
Example: a $3 \times 4$ rectangle is cyclic, $s = 7$, and $\sqrt{4 \cdot 3 \cdot 4 \cdot 3} = 12$.
The hypothesis that $ABCD$ is cyclic is needed.
A rhombus with side $1$ and an angle $\theta \ne \pi/2$ has area $\sin \theta < 1$, but the formula would give $1$.
Letting $d = 0$ turns the formula into [[Heron's formula|0FRBZ2X4HJ]] for a triangle.

## Explanations

### L1 human

For a quadrilateral whose corners all lie on one circle, the area depends only on the four side lengths.
Brahmagupta's formula computes it directly from them.
It looks just like Heron's formula for triangles, with one more factor.

### L2 human

For a general quadrilateral the four sides do not fix the area, since the shape can flex.
Among all quadrilaterals with given sides, the cyclic one has the largest area, and Brahmagupta's formula gives that maximum.
Bretschneider's formula extends it to all convex quadrilaterals by an extra term that involves the sum of two opposite angles.
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<document source="https://tamnd.github.io/math2/node/C0RDHFD00Y.md" tag="C0RDHFD00Y" kind="definition">
# Moment map

## Definition

Let $(M, \omega)$ be a [[symplectic manifold|CAN490EPRT]] and let $G$ be a [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{g}$ and dual space $\mathfrak{g}^*$.
Let $G$ [[act|0WCY42EV9V]] smoothly on $M$ by symplectomorphisms.
For $\xi \in \mathfrak{g}$, let $\xi_M$ be the vector field $\xi_M(x) = \frac{d}{dt}\big|_{t=0} \exp(t\xi) \cdot x$.
A *moment map* for the action is a smooth map $\mu \colon M \to \mathfrak{g}^*$ such that:

1. For every $\xi \in \mathfrak{g}$, the function $\mu^\xi(x) = \langle \mu(x), \xi \rangle$ satisfies $\iota_{\xi_M} \omega = d\mu^\xi$, so $\xi_M$ is the [[Hamiltonian vector field|C5EH3B4BEP]] of $\mu^\xi$.
2. $\mu$ is equivariant: $\mu(g \cdot x) = \mathrm{Ad}^*_g \mu(x)$ for all $g \in G$ and $x \in M$, where $\mathrm{Ad}^*$ is the coadjoint action.

An action with a moment map is called a *Hamiltonian action*.

## Remarks

Example: let the circle $S^1 = \R / 2\pi\Z$ act on $\R^2$ by rotations, with $\omega = dx \wedge dy$.
Identify $\mathfrak{g}$ and $\mathfrak{g}^*$ with $\R$; the generator is $\xi_M = -y\,\partial_x + x\,\partial_y$.
Let $\mu(x, y) = -\frac{1}{2}(x^2 + y^2)$.
Then $\iota_{\xi_M} \omega = -y\,dy - x\,dx = d\mu$, and clause 2 holds because $G$ is abelian and $\mu$ is rotation invariant.

Non-example: the translations of the torus $T^2 = \R^2 / \Z^2$ in the $x$ direction preserve $\omega = dx \wedge dy$, but there is no moment map.
Clause 1 would need $\iota_{\partial_x} \omega = dy$ to be the differential of a function on $T^2$, and $dy$ is closed but not exact there.

For a connected group, clause 2 can be replaced by an infinitesimal condition.
It says that $\xi \mapsto \mu^\xi$ is a Lie algebra homomorphism into functions with the [[Poisson bracket|2S0MXDT4YH]], up to the sign convention.
When $G$ is abelian, a moment map can be changed by adding any constant in $\mathfrak{g}^*$.
Many authors use the opposite sign convention in clause 1.
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# Shanks transformation

## Summary

The Shanks transformation is a nonlinear [[series acceleration|B1S2XJPGYD]] method.
Its first-order form takes three consecutive partial sums and returns the limit they would have if the error were exactly geometric.
This form coincides with [[Aitken's delta-squared process|WF08BMN6Q6]].
Higher orders $e_k$ use $2k + 1$ terms and model the error as a sum of $k$ geometric sequences.
In practice it is computed with [[Wynn's epsilon algorithm|6X0C4HNMAZ]], and applied to the partial sums of a power series it produces [[Padé approximants|1F16G4V64X]].

## Explanations

### L1 human

Many slowly converging series have partial sums that overshoot and undershoot their limit, like a damped oscillation.
The Shanks transformation fits a simple model to a few consecutive partial sums and reads off the limit of the model.
For the Leibniz series $1 - \frac13 + \frac15 - \cdots = \frac{\pi}{4}$, the first three partial sums $1, \frac23, \frac{13}{15}$ already give $\frac{19}{24} \approx 0.792$, close to $\frac{\pi}{4} \approx 0.785$.
Applying the transformation again to the new sequence improves the values further.
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# Stieltjes inversion formula

## Statement

Let $\mu$ be a finite positive [[Borel measure|DYR1QFNGNJ]] on $\R$, and let $S_\mu(z) = \int_\R \frac{d\mu(t)}{t - z}$ be its [[Stieltjes transform|RG1HFAYD98]] for $z \in \C \setminus \R$.
Then for all real numbers $a < b$,
$$\lim_{\varepsilon \to 0^+} \frac{1}{\pi} \int_a^b \operatorname{Im} S_\mu(x + i\varepsilon)\, dx = \mu\big((a, b)\big) + \tfrac12 \mu(\set{a}) + \tfrac12 \mu(\set{b}).$$
In particular, $S_\mu$ determines $\mu$.

## Remarks

For $\varepsilon > 0$ one has $\frac{1}{\pi} \operatorname{Im} S_\mu(x + i\varepsilon) = \int_\R P_\varepsilon(x - t)\, d\mu(t)$ with the [[Poisson kernel|TCYSWCTYYM]] $P_\varepsilon(u) = \frac{1}{\pi} \frac{\varepsilon}{u^2 + \varepsilon^2}$ of the upper half-plane.
By Fubini, the left side before the limit is $\int_\R \frac{1}{\pi}\big(\arctan\frac{b - t}{\varepsilon} - \arctan\frac{a - t}{\varepsilon}\big)\, d\mu(t)$.
The integrand is bounded by $1$ and tends to $1$ for $a < t < b$, to $\frac12$ for $t = a$ or $t = b$, and to $0$ otherwise, so dominated convergence gives the formula.
If $\mu$ has a continuous density $\rho$ near $x$, then $\rho(x) = \lim_{\varepsilon \to 0^+} \frac{1}{\pi} \operatorname{Im} S_\mu(x + i\varepsilon)$.

Example: for $\mu = \delta_0$, $\frac{1}{\pi} \operatorname{Im} S_\mu(x + i\varepsilon) = \frac{1}{\pi} \frac{\varepsilon}{x^2 + \varepsilon^2}$, which concentrates at $0$ with total mass $1$.

## Explanations

### L1 human

The Stieltjes transform loses no information: the measure can be recovered from it.
Push the variable $z$ down to the real axis from above and look at the imaginary part.
In the limit it becomes the density of the measure, up to the factor $\pi$.

### L2 human

The formula is the reason why the Stieltjes transform is the standard tool to identify a measure.
In random matrix theory one computes the limit of the Stieltjes transforms of empirical eigenvalue distributions, and the inversion formula then gives the limiting law, such as the semicircle law.
In the moment problem and in the spectral theory of self-adjoint operators the same formula recovers a spectral measure from its resolvent.
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# Associated Legendre polynomials

## Definition

Let $\ell$ and $m$ be integers with $0 \le m \le \ell$, and let $P_\ell$ be the [[Legendre polynomial|QFYGJVA996]] of degree $\ell$.
The *associated Legendre polynomial* $P_\ell^m$ is the function on $[-1, 1]$ given by
$$P_\ell^m(x) = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_\ell(x).$$
For $0 < m \le \ell$ one also sets
$$P_\ell^{-m}(x) = (-1)^m \frac{(\ell - m)!}{(\ell + m)!} P_\ell^m(x).$$

## Remarks

The factor $(-1)^m$ is the Condon–Shortley phase; some authors leave it out.
Examples: $P_\ell^0 = P_\ell$, $P_1^1(x) = -(1 - x^2)^{1/2}$, $P_2^1(x) = -3x (1 - x^2)^{1/2}$ and $P_2^2(x) = 3(1 - x^2)$.
Check for $P_2^2$: $P_2(x) = (3x^2 - 1)/2$ has second derivative $3$, so $P_2^2(x) = (1 - x^2) \cdot 3$.
Each $P_\ell^m$ solves the associated Legendre equation $(1 - x^2) y'' - 2x y' + \left( \ell(\ell + 1) - \frac{m^2}{1 - x^2} \right) y = 0$ on $(-1, 1)$.
For fixed $m$ they are orthogonal: $\int_{-1}^{1} P_k^m(x) P_\ell^m(x) \, dx = \frac{2 (\ell + m)!}{(2\ell + 1)(\ell - m)!} \delta_{k\ell}$.
Non-example: the case $m > \ell$ is excluded, because then the $m$-th derivative of $P_\ell$ is zero and the formula gives only the zero function.
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# Pro-p group

## Summary

For a prime $p$, a pro-$p$ group is a [[profinite group|KA0YDB6QXE]] that is an inverse limit of finite [[p-groups|06QYRTX9KQ]].
The $p$-adic integers are the basic example.
Pro-$p$ groups occur as Sylow subgroups of profinite groups and as Galois groups of maximal $p$-extensions of fields.
Every $p$-adic analytic group has an open subgroup that is a pro-$p$ group.
Much of the theory of finite $p$-groups carries over to them.

## Explanations

### L1 human

Fix a prime number $p$, such as $2$ or $3$.
A finite $p$-group is a finite group whose number of elements is a power of $p$.
A pro-$p$ group is built from a compatible tower of such finite groups, much as the $p$-adic integers are built from the integers modulo $p$, $p^2$, $p^3$, and so on.
Each finite quotient of a pro-$p$ group by an open normal subgroup has order a power of $p$.
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# Proof-theoretic ordinal

## Definition

Let $T$ be a recursively axiomatized theory that contains enough arithmetic to speak about [[computable|KZSB5E471K]] relations on $\N$.
For a computable binary relation $\prec$ on $\N$, let $\mathrm{WF}(\prec)$ be the statement in the language of $T$ that $\prec$ is a [[well-order|0F1D6PK9RZ]].
For first-order theories, $\mathrm{WF}(\prec)$ is replaced by the scheme of [[transfinite induction|0ZHQMQ2C1N]] along $\prec$ for all formulas.
The *proof-theoretic ordinal* of $T$ is written $\abs{T}$.
It is the supremum of the order types of all computable well-orders $\prec$ on $\N$ such that $T$ proves $\mathrm{WF}(\prec)$.

## Remarks

Example: for [[Peano arithmetic|0QFT7Z76N5]], $\abs{\mathrm{PA}} = \varepsilon_0$, the ordinal [[epsilon nought|KT50SMH3EC]].
Gentzen showed that PA proves transfinite induction along a standard notation system for each ordinal below $\varepsilon_0$, and not along the one for $\varepsilon_0$ itself.
Non-example: the least uncountable [[ordinal|0RDJK7EQYY]] $\omega_1$ is not the proof-theoretic ordinal of any such theory, because a computable well-order on $\N$ has countable order type.
Suppose that every computable order that $T$ proves well-founded is in fact well-founded.
Then $\abs{T}$ is below the Church–Kleene ordinal $\omega_1^{\mathrm{CK}}$, the least ordinal that is not the order type of a computable well-order.
Several inequivalent definitions are in use, for example with primitive recursive in place of computable orders, or with $\Pi^0_2$ consequences in place of well-foundedness.
For the usual theories they give the same ordinal.
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# Elliptic boundary value problem

## Summary

An elliptic boundary value problem asks for a solution of an elliptic equation in a domain, subject to conditions on the boundary of the domain.
The model case is the Dirichlet problem for the [[Laplacian|0E0B6RA8K0]]: $\Delta u = f$ inside and $u = g$ on the boundary.
In general, the operator must be elliptic and the boundary conditions must satisfy the [[Lopatinskii–Shapiro condition|48MEJEMEXM]].
Such problems have a priori estimates, regularity up to the boundary and a Fredholm solution theory.

## Explanations

### L1 human

Think of the steady temperature in a metal plate whose edge is held at given temperatures.
Inside, heat flow balances out, which gives an elliptic equation; on the edge, the temperature is prescribed.
Together these determine the temperature everywhere.
Elliptic boundary value problems are the general form of this kind of question, for equilibrium states in physics and geometry.
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# Covariant derivative

## Definition

Let $M$ be a smooth manifold, let $E \to M$ be a smooth vector bundle, and let $\nabla$ be a [[connection|5ZYRRVE8BW]] on $E$.

1. Let $p \in M$, $v \in T_pM$, and let $s$ be a smooth [[section|WWX7H5W1RP]] of $E$. The *covariant derivative* of $s$ in the direction $v$ is $\nabla_v s = (\nabla_X s)(p)$, where $X$ is any smooth vector field with $X_p = v$.
2. Let $\gamma : I \to M$ be a smooth curve on an interval $I$. The *covariant derivative along $\gamma$* is the unique map $D_t$ from smooth sections of $E$ along $\gamma$ to themselves with three properties. It is $\R$-linear. It satisfies $D_t (f V) = f' V + f D_t V$ for every smooth $f : I \to \R$. And $D_t V(t) = \nabla_{\gamma'(t)} s$ whenever $V = s \circ \gamma$ for a section $s$ of $E$.

## Remarks

Clause 1 is well defined because $\nabla_X s$ is $C^\infty(M)$-linear in $X$, so its value at $p$ depends only on $X_p$.
The existence and uniqueness in clause 2 follow by writing $V(t) = \sum_i V^i(t) e_i(\gamma(t))$ in a local frame: then $D_t V = \sum_i \big(\dot V^i + \sum_j \omega^i_j(\gamma'(t)) V^j\big) e_i$, where $\nabla e_j = \sum_i \omega^i_j \otimes e_i$.
Example: for the flat connection on $\R^n$ and $E = T\R^n$, $D_t V = \dot V$ is the ordinary derivative of the components; so $D_t \gamma' = \ddot \gamma$ is the acceleration.
Non-example: on the round sphere $S^2 \subset \R^3$ with its Levi-Civita connection, $D_t \gamma'$ is the tangential part of $\ddot \gamma$, not $\ddot\gamma$ itself.
Along a great circle $\ddot \gamma$ is normal to the sphere, so $D_t \gamma' = 0$ although $\ddot \gamma \ne 0$.
A connection on $TM$ extends uniquely to covariant derivatives of all [[tensor fields|WX302T93TW]] that obey the product rule for $\otimes$, commute with contractions and act on functions by $\nabla_X f = X f$.
In index notation $\nabla_i Y^k = \partial_i Y^k + \Gamma^k_{ij} Y^j$, often written $Y^k{}_{;i}$.
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# Semimodular lattice

## Summary

A semimodular lattice is a [[lattice|WH4SFAACAS]] in which joining with an element preserves the covering relation: if $a$ covers $a \wedge b$, then $a \vee b$ covers $b$.
It is a one-sided weakening of the [[modular law|QX7WZ92QAK]], and every modular lattice is semimodular.
Lattices of flats of [[matroids|PX31ZNYS3Z]] and lattices of partitions of a set are semimodular without being modular in general.
In lattices of finite length, semimodularity is what makes a well-behaved rank function exist.

## Explanations

### L1 human

Think of the elements of a lattice as subspaces and of a cover as a step up by one dimension.
Semimodularity says: if $a$ is one step above $a \wedge b$, then adding $a$ to $b$ moves $b$ up by at most one step, and in fact by exactly one.
So "one more generator" never gives more than "one more dimension".
In the lattice of partitions of a set, merging two blocks is one step up, and this rule holds there too.
</document>

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# Complete intersection ring

## Definition

Let $(R, \mathfrak{m})$ be a commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] and let $\hat{R}$ be its $\mathfrak{m}$-adic [[completion|KBN13NPYJQ]].
Then $R$ is a *complete intersection ring* if the following holds.
There are a [[complete|G9Q61B8A7C]] [[regular local ring|63XXTF6B5M]] $S$ and an [[$S$-regular sequence|69R5R2FCS5]] $f_1, \ldots, f_r$ such that
$$\hat{R} \cong S/(f_1, \ldots, f_r).$$

## Remarks

The completion is used because a Noetherian local ring need not be a quotient of a regular local ring, while its completion always is, by the [[Cohen structure theorem|N3XH42KVV3]].
If $R = S/(f_1, \ldots, f_r)$ with $S$ a regular local ring and $f_1, \ldots, f_r$ an $S$-regular sequence, then $R$ is a complete intersection.
Example: $K[x,y]$ localized at $(x, y)$ and divided by $(xy)$ is a complete intersection, since $xy \ne 0$ is a non-zero-divisor in the domain $K[x,y]_{(x,y)}$.
Example: $K[x]/(x^n)$ is a complete intersection, with $S = K[\![x]\!]$ and $f_1 = x^n$.
Non-example: $K[x,y]/(x^2, xy, y^2)$ is not a complete intersection, because it is not [[Gorenstein|T6HCX959PJ]], while every complete intersection is Gorenstein.
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# Bredon cohomology

## Definition

Let $G$ be a discrete group.
The *orbit category* $\mathcal{O}_G$ has as objects the left $G$-sets $G/H$ for subgroups $H \le G$, and as morphisms the $G$-equivariant maps.
A *coefficient system* is a contravariant [[functor|28NQ0KRHD3]] $M \colon \mathcal{O}_G \to \mathrm{Ab}$ to [[abelian groups|082GHNPJ53]].

Let $X$ be a $G$-CW complex, that is, a [[CW complex|5W95QFHXPR]] with a cellular [[action|0WCY42EV9V]] of $G$.
It is required that each element of $G$ that maps a cell to itself fixes that cell pointwise.
For each subgroup $H$, the fixed set $X^H$ is a subcomplex.
For $n \ge 0$, let $\underline{C}_n(X)$ be the contravariant functor $\mathcal{O}_G \to \mathrm{Ab}$ given by $G/H \mapsto C_n(X^H)$, the [[cellular chains|W4DG4WCQ5W]] of $X^H$.
A $G$-map $G/H \to G/K$ with $H \mapsto gK$ acts by $C_n(X^K) \to C_n(X^H)$, $c \mapsto g c$.
The cellular boundary maps make $\underline{C}_\bullet(X)$ a chain complex of functors.
The *Bredon cochain complex* is
$$C^n_G(X; M) = \operatorname{Nat}(\underline{C}_n(X), M),$$
the group of [[natural transformations|KYCPCC57FT]], with coboundary induced by the boundary of $\underline{C}_\bullet(X)$.
The *Bredon cohomology* $H^n_G(X; M)$ is the $n$-th cohomology of $C^\bullet_G(X; M)$.

## Remarks

Example: let $X$ be a point with the trivial action.
Then $X^H$ is a point for every $H$, so $\underline{C}_0(X)$ is the constant functor $\Z$ and $\underline{C}_n(X) = 0$ for $n > 0$.
A natural transformation from the constant functor $\Z$ to $M$ is fixed by its value at $G/G$, the terminal object of $\mathcal{O}_G$.
So $H^0_G(\mathrm{pt}; M) \cong M(G/G)$ and $H^n_G(\mathrm{pt}; M) = 0$ for $n > 0$.

If $G$ acts freely on $X$ and $M$ is the constant functor with value $A$, then $X^H$ is empty for $H \neq 1$ and $H^n_G(X; M) \cong H^n(X/G; A)$.

Non-example: a family of groups $M(G/H)$ with maps going in the covariant direction is not a coefficient system for this cohomology.
Covariant functors on $\mathcal{O}_G$ are the coefficients of Bredon homology instead.
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# Helgason support theorem

## Statement

Let $n \ge 2$ and let $f \colon \R^n \to \C$ be continuous and rapidly decreasing, that is, $\abs{x}^k f(x)$ is bounded on $\R^n$ for every integer $k \ge 0$.
Let $K \subseteq \R^n$ be a compact convex set.
Assume that the [[Radon transform|VJNZJ352PC]] of $f$ vanishes on every hyperplane that does not meet $K$.
That is, $R f(\theta, s) = 0$ whenever the hyperplane $x \cdot \theta = s$ is disjoint from $K$.
Then $f(x) = 0$ for every $x \notin K$.

## Remarks

The decay hypothesis cannot be dropped.
In $\R^2 = \C$, let $k \ge 2$ be an integer and let $f$ be continuous with $f(z) = z^{-k}$ for $\abs{z} \ge 1$.
Take a line $z = z_0 + t w$, $t \in \R$, that misses the closed unit disk.
The function $t \mapsto (z_0 + tw)^{-k}$ has its only pole off the real axis, and it decays like $\abs{t}^{-k}$.
Closing the contour in the half-plane without the pole gives integral $0$.
So $R f$ vanishes on all lines missing the unit disk, while $f \ne 0$ outside it; here $\abs{x}^k f(x)$ is bounded but $\abs{x}^{k+1} f(x)$ is not.

The convexity of $K$ is also needed.
The hyperplanes that miss a compact set are those that miss its convex hull, so the data can only locate the support up to the convex hull.

## Explanations

### L1 human

Suppose every X-ray that misses a certain convex region sees nothing.
The theorem says that, under a mild decay condition, there is indeed nothing outside that region.
It is not obvious: a ray that misses the region could in principle pass through material whose contributions cancel.

### L2 human

The theorem gives the exterior problem of tomography its uniqueness: a body is determined outside a convex obstacle by the lines that avoid the obstacle.
It is a sharper form of injectivity of the [[Radon transform|VJNZJ352PC]], local in the set of hyperplanes.
The example with $z^{-k}$ shows that rapid decrease is the right hypothesis.
The theorem is due to Helgason and has analogues for Radon transforms on symmetric spaces.
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# Hilbert's sixteenth problem

## Problem

The problem has two parts.
The real points of a nonsingular real plane curve form finitely many closed curves in the real projective plane.
Those that bound a disc are called ovals.

1. For each $n$, determine the possible relative positions of the ovals of a nonsingular real [[plane algebraic curve|EWS4WFTA7W]] of degree $n$. Ask the same for real algebraic surfaces in space.
2. Consider planar vector fields $\dot{x} = P(x, y)$, $\dot{y} = Q(x, y)$ with $P, Q$ real polynomials of degree at most $n$. For each $n$, bound the number $H(n)$ of their [[limit cycles|0AAQ73WRFB]], and describe their possible positions.

## Remarks

Hilbert included the problem in his list of 1900.
For part 1, Harnack's curve theorem says that a real plane curve of degree $n$ has at most $(n-1)(n-2)/2 + 1$ connected components, and this bound is attained.
Gudkov completed the classification of the arrangements for degree $6$, and the classification is known up to degree $7$ and open beyond.
For part 2, Écalle and Ilyashenko proved independently that each single polynomial vector field has finitely many limit cycles.
Whether $H(n)$ is finite is unknown for every $n \ge 2$, and quadratic vector fields with $4$ limit cycles are known.

## Explanations

### L1 human

A real polynomial curve in the plane can consist of several closed loops, some nested inside others.
The first part asks which nesting patterns can occur for a given degree.
The second part asks how many isolated closed orbits a polynomial system of differential equations in the plane can have.
Both questions are easy to state and remain largely open after more than a century.
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# Blaschke product

## Definition

Let $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ be the [[unit disc|8G5ABXRK85]].
Let $m \ge 0$ be an integer.
Let $(a_n)$ be a finite or infinite sequence of points of $\mathbb{D} \setminus \set{0}$, with repetitions allowed, such that
$$\sum_n (1 - \abs{a_n}) < \infty.$$
The *Blaschke product* with zeros $0$ (of order $m$) and $(a_n)$ is the function $B \colon \mathbb{D} \to \C$ given by
$$B(z) = z^m \prod_n \frac{\abs{a_n}}{a_n} \, \frac{a_n - z}{1 - \overline{a_n} z}.$$
It is a *finite Blaschke product* when the sequence $(a_n)$ is finite.

## Remarks

Under the condition above, the product converges uniformly on compact subsets of $\mathbb{D}$.
So $B$ is [[holomorphic|0DG0Z72CQD]] with $\abs{B} \le 1$, and its zeros are exactly $0$ and the $a_n$, with multiplicity.
Moreover the radial limit of $B$ has modulus $1$ at almost every point of the circle, so $B$ is an [[inner function|NN8XH5RJYB]].
The factor $\abs{a_n}/a_n$ makes the $n$-th factor positive at $z = 0$; some authors also allow a constant factor of modulus $1$.

Example: for $a \in \mathbb{D} \setminus \set{0}$, the single factor $B(z) = \frac{\abs{a}}{a} \frac{a - z}{1 - \bar a z}$ is a finite Blaschke product.
For $\abs{z} = 1$ one has $\abs{1 - \bar a z} = \abs{\bar z - \bar a} = \abs{a - z}$, so $\abs{B(z)} = 1$ on the circle.

Non-example: the points $a_n = 1 - 1/n$ for $n \ge 2$ fail the summability condition, since $\sum 1/n = \infty$.
For them the partial products tend to $0$ at every point of $\mathbb{D}$, so no Blaschke product has these zeros.
In fact no bounded holomorphic function other than $0$ vanishes at all of them.
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<document source="https://tamnd.github.io/math2/node/C210P5AK7X.md" tag="C210P5AK7X" kind="problem">
# Schottky problem

## Problem

Let $g \ge 1$, and let $\mathcal{A}_g$ be the moduli space of principally polarized complex [[abelian varieties|R7W03TXRS9]] of dimension $g$.
The [[Jacobian variety|ZQ84M9D9TC]] of a smooth projective complex curve of genus $g$, with its theta divisor, is a point of $\mathcal{A}_g$.
Let $\mathcal{J}_g \subseteq \mathcal{A}_g$ be the set of these points.
Characterize the points of $\mathcal{J}_g$, or of its closure, among all points of $\mathcal{A}_g$.

## Remarks

By the [[Torelli theorem|BRKFX8BQ60]], a curve is determined by its Jacobian with its polarization, so $\mathcal{J}_g$ has the dimension $3g - 3$ of the moduli of curves when $g \ge 2$.
The space $\mathcal{A}_g$ has dimension $g(g + 1)/2$.
For $g \le 3$ the closure of $\mathcal{J}_g$ is all of $\mathcal{A}_g$, and for $g \ge 4$ it is a proper closed subset.
For $g = 4$, Schottky found a modular form, built from [[theta functions|2F5CYQJVAF]], that vanishes on $\mathcal{J}_4$.
Igusa showed that its zero locus is exactly the closure of $\mathcal{J}_4$.
In 1986 Shiota proved a conjecture of Novikov on this problem.
It says that an indecomposable principally polarized abelian variety is a Jacobian if and only if its theta function solves the KP equation.
Other characterizations, for example by trisecants of the Kummer variety, are also known.

## Explanations

### L1 human

Every curve has an abelian variety attached to it, its Jacobian.
There are many more abelian varieties than curves once the dimension is at least $4$.
The Schottky problem asks how to recognize, from its equations or its theta function, which abelian varieties come from curves.
</document>

<document source="https://tamnd.github.io/math2/node/C21ECQ1H84.md" tag="C21ECQ1H84" kind="concept">
# Mittag-Leffler star

## Summary

The Mittag-Leffler star of a power series is the set of points that can be reached from its centre by [[analytic continuation|0M3YQJ81NB]] along straight segments.
It is an open set that is star-shaped with respect to the centre, and it contains the disc of convergence.
Mittag-Leffler showed that the function is given on its whole star by a series of polynomials.
The star is a natural domain for representing a function by summation methods beyond the disc of convergence.

## Explanations

### L1 human

Start at the centre of a power series and walk outwards along a ray.
Continue the function along the ray as long as possible, and stop at the first singular point.
The points reached on all rays form the star.
For $1/(1 - z)$ with centre $0$, every ray except the positive real axis can be followed forever, and the positive real axis is blocked at $1$.
So the star is the plane with the half-line $[1, \infty)$ removed.
</document>

<document source="https://tamnd.github.io/math2/node/C22331MBY7.md" tag="C22331MBY7" kind="definition">
# Summability field of a matrix method

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $\omega$ be the space of all sequences in $\mathbb{K}$.
Let $A = (a_{nk})_{n, k \ge 0}$ be an infinite matrix with entries in $\mathbb{K}$, viewed as a [[matrix summability method|M6311DWJ1E]].
For $x = (x_k)_{k \ge 0} \in \omega$, write $(Ax)_n = \sum_{k \ge 0} a_{nk} x_k$ when this series converges.
The *summability field* (or *convergence domain*) of $A$ is
$$c_A = \set{x \in \omega : (Ax)_n \text{ converges for every } n, \text{ and } \lim_{n \to \infty} (Ax)_n \text{ exists}}.$$

## Remarks

The set $c_A$ is a [[sequence space|XYTN19EPF2]], since both conditions are preserved by sums and scalar multiples.

Example: for the identity matrix, $(Ax)_n = x_n$, so $c_A$ is the space $c$ of [[convergent sequences|0W4CN9E2W5]].

Example: for the Cesàro matrix, $a_{nk} = 1/(n+1)$ for $k \le n$ and $a_{nk} = 0$ for $k > n$, the sequence $x = (1, 0, 1, 0, \dots)$ lies in $c_A$.
Indeed $(Ax)_n$ is the proportion of ones among $x_0, \dots, x_n$, which tends to $1/2$.

Non-example: for the same matrix, $x_k = (-1)^k k$ is not in $c_A$.
The sum $x_0 + \dots + x_n$ equals $n/2$ for even $n$ and $-(n+1)/2$ for odd $n$, so $(Ax)_n$ comes close to $1/2$ and to $-1/2$ infinitely often.

A matrix $A$ is [[regular|29C7C41A52]] when $c \subseteq c_A$ and $\lim_n (Ax)_n = \lim_k x_k$ for every $x \in c$.
With a suitable topology, $c_A$ is an [[FK-space|71DMQHW6JZ]].
</document>

<document source="https://tamnd.github.io/math2/node/C22WGRP3T8.md" tag="C22WGRP3T8" kind="concept">
# Strongly regular graph

## Summary

A strongly regular graph is a [[regular graph|0Q70YFB6ZM]] with a second kind of regularity.
The number of common neighbours of two distinct vertices depends only on whether they are adjacent.
Its parameters are written $(v, k, \lambda, \mu)$.
A connected strongly regular graph that is not complete has exactly three distinct adjacency eigenvalues.
These graphs are a meeting point of graph theory, design theory and group theory.
The node [[strongly regular graph|J50MA9C0EH]] gives the precise definition.

## Explanations

### L1 human

In a strongly regular graph every vertex has the same number of neighbours.
Moreover, any two friends have the same number of common friends, and any two strangers have the same number of common friends.
The pentagon $C_5$ is an example: two adjacent vertices have no common neighbour, and two non-adjacent vertices have exactly one.
The [[Petersen graph|0NR95CTFCZ]] is a larger example.
</document>

<document source="https://tamnd.github.io/math2/node/C24D9MWRV3.md" tag="C24D9MWRV3" kind="definition">
# Parametrix

## Definition

Let $M$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary with a smooth positive density $\mu$.
Let $E$ and $F$ be smooth [[vector bundles|P4DZM975BV]] over $M$.
For a smooth vector bundle $V$ over $M$, let $\Hom(V, V)$ also denote the bundle over $M \times M$ with fibre $\Hom(V_y, V_x)$ at $(x, y)$.
A linear operator $R \colon \Gamma(V) \to \Gamma(V)$ is *smoothing* if there is a smooth section $K$ of this bundle such that
$$(R u)(x) = \int_M K(x, y) \, u(y) \, d\mu(y) \quad \text{for all } u \in \Gamma(V).$$
Let $P \colon \Gamma(E) \to \Gamma(F)$ be a [[differential operator|CGBX1N9V8H]].
A *parametrix* for $P$ is a continuous linear operator $Q \colon \Gamma(F) \to \Gamma(E)$ such that $QP - \id$ and $PQ - \id$ are smoothing.

## Remarks

An operator $Q$ with only $QP - \id$ smoothing is a left parametrix, and one with only $PQ - \id$ smoothing is a right parametrix.
Two parametrices of the same operator differ by a smoothing operator.
For constant-coefficient operators on $\R^n$, a parametrix corresponds to a distribution $e$ with $P e = \delta + \omega$ for a smooth function $\omega$, an approximate [[fundamental solution|K2NK9A8M5F]].
An elliptic operator of order $m$ has a parametrix that is a [[pseudo-differential operator|FF27N0BWTP]] of order $-m$.

Example: on the circle $S^1 = \R / 2\pi\Z$ with $d\mu = dx$, let $P = -d^2/dx^2$ on functions, and define
$$Q\Bigl(\sum_{n \in \Z} c_n e^{inx}\Bigr) = \sum_{n \ne 0} \frac{c_n}{n^2} \, e^{inx}.$$
Then $PQu = QPu = u - c_0$, where $c_0 = \frac{1}{2\pi} \int_0^{2\pi} u(y) \, dy$.
So $QP - \id$ and $PQ - \id$ are both the operator with the constant smooth kernel $K(x, y) = -\frac{1}{2\pi}$, and $Q$ is a parametrix.

Non-example: the zero operator is not a parametrix for $P = -d^2/dx^2$.
Here $PQ - \id = -\id$, and the identity is not smoothing.
An operator with a continuous kernel extends to a compact operator on $L^2(S^1)$, while the identity of an infinite-dimensional Hilbert space is not compact.

## Explanations

### L1 human

A parametrix inverts an operator up to an error that has a smooth kernel, an error that makes every input smooth.
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<document source="https://tamnd.github.io/math2/node/C28YT5DQFF.md" tag="C28YT5DQFF" kind="definition">
# Tauberian condition

## Definition

Let $M$ be a [[summability method|8X25Q4XYMN]] that assigns a value to some [[series|0JH4MN4K1C]] $\sum a_n$ of real or complex numbers.
Let $P$ be a property of sequences $(a_n)_{n \ge 0}$.
The property $P$ is a *Tauberian condition* for $M$ if the following holds.
Every series $\sum a_n$ whose terms have property $P$ and which is $M$-summable to $s$ [[converges|0CVVKY7GY2]] with sum $s$.

## Remarks

Example: for Abel summation, the property $n a_n \to 0$ is a Tauberian condition.
This is [[Tauber's theorem|8SK3GQ4S0M]].
Stronger theorems allow weaker conditions; for example, $n a_n = O(1)$ is a Tauberian condition both for Cesàro summation and for Abel summation.

Non-example: the property $a_n \to 0$ is not a Tauberian condition for Cesàro summation.
Let the partial sums start at $-1$.
For $k = 1, 2, 3, \dots$ in turn, let them run up to $1$ in $k$ steps of size $2/k$ and then back down to $-1$ in $k$ such steps.
The terms are $\pm 2/k$, so $a_n \to 0$.
Over each full up-and-down run the partial sums add up to $0$.
The incomplete last run contributes at most $2k + 2$ in absolute value, while $n$ is at least of order $k^2$.
So the arithmetic means of the partial sums tend to $0$, but the partial sums reach $1$ and $-1$ infinitely often and do not converge.

A theorem whose hypothesis is a Tauberian condition is called a Tauberian theorem.
It is a partial converse of an Abelian theorem such as [[Abel's theorem|0S4EMDQS2F]].
</document>

<document source="https://tamnd.github.io/math2/node/C2ATM03215.md" tag="C2ATM03215" kind="definition">
# Random group

## Definition

Fix an integer $m \ge 2$ and a real number $d$ with $0 \le d \le 1$, the *density*.
Let $F_m$ be the [[free group|0KTW26C80J]] on $a_1, \dots, a_m$.
For each integer $\ell \ge 1$, let $C_\ell$ be the [[finite set|0BBTH3YSZA]] of cyclically reduced words of length $\ell$ in $F_m$.
Let $R$ be a subset of $C_\ell$ with $\floor{(2m - 1)^{d \ell}}$ elements, chosen uniformly at random among all such subsets.
A *random group at density $d$ and length $\ell$* is the group
$$G = \langle a_1, \dots, a_m \mid R \rangle.$$
A property holds *for random groups at density $d$* if the probability that $G$ has it tends to $1$ as $\ell \to \infty$.

## Remarks

This is Gromov's density model.
The number of cyclically reduced words of length $\ell$ grows like $(2m - 1)^\ell$, so the density $d$ measures the number of relators on a logarithmic scale.
Gromov's theorem: for $d < \frac{1}{2}$, random groups at density $d$ are infinite, torsion-free and [[hyperbolic|JWJ8YPR7EB]]; for $d > \frac{1}{2}$, random groups at density $d$ are trivial or of order $2$.

Example: at density $d = 0$, the set $R$ has exactly one element, so the model produces a random one-relator group.

Non-example: a single fixed group, such as $\Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle$, is not "random" in itself.
The definition describes a sequence of probability distributions on presentations, and only properties that hold with probability tending to $1$ are meaningful.

Other models fix the number of relators and let their length grow (the few-relator model), or use relators of length $3$ (the triangular model).
</document>

<document source="https://tamnd.github.io/math2/node/C2CK6KSSTV.md" tag="C2CK6KSSTV" kind="concept">
# Intrinsic volume

## Summary

The intrinsic volumes $V_0(K), \dots, V_n(K)$ of a convex body $K$ in $\R^n$ are rescaled [[quermassintegrals|H1K36H9ZWB]].
The scaling makes them independent of the dimension of the ambient space.
$V_n$ is the volume, $2 V_{n-1}$ is the surface area, and $V_0$ is the Euler characteristic, equal to $1$ for every nonempty convex body.
By [[Hadwiger's theorem|7V2X68AMEY]] they span all continuous valuations on convex bodies that are invariant under rigid motions.

## Explanations

### L1 human

Intrinsic volumes are the natural sizes of a convex shape in each dimension.
For a box with sides $a, b, c$ they are $abc$ (volume), $ab + bc + ca$ (half the surface area), $a + b + c$ (a total length), and $1$.
If the box is flattened into a rectangle with sides $a, b$, one gets $0$, $ab$, $a + b$ and $1$, exactly the intrinsic volumes of the rectangle in the plane.
This independence of the surrounding space is what "intrinsic" means.
</document>

<document source="https://tamnd.github.io/math2/node/C2DAKWAZ28.md" tag="C2DAKWAZ28" kind="definition">
# Wedge sum

## Definition

Let $(X_i, x_i)_{i \in I}$ be a family of [[pointed spaces|T9DG8DDE75]].
The *wedge sum* of the family is the space
$$\bigvee_{i \in I} X_i = \Bigl( \bigsqcup_{i \in I} X_i \Bigr) / \set{x_i : i \in I}.$$
It is the quotient of the [[disjoint union|ZXJKSB30WF]] that collapses the set of base points to one point.
It carries the [[quotient topology|0NQ1H2X495]].
The image of the base points is the base point of the wedge sum.

## Remarks

For two spaces one writes $X \vee Y$.
Each $X_i$ embeds in the wedge sum, and a pointed map out of $\bigvee_i X_i$ is the same as a family of pointed maps out of the $X_i$.
Example: $S^1 \vee S^1$ is homeomorphic to the figure eight $\set{z \in \C : \abs{z - 1} = 1 \text{ or } \abs{z + 1} = 1}$, where the two circles meet only at $0$.
Non-example: the disjoint union $S^1 \sqcup S^1$ without collapsing the base points is not a wedge sum; it is disconnected, while the wedge of connected spaces is connected.
The wedge sum depends on the choice of base points in general, but not up to homotopy for well-behaved spaces such as connected CW complexes.
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# Borel–Ritt theorem

## Statement

Let $0 < r \le \infty$ and $\alpha < \beta$ be real numbers, and let
$$S = \set{z \in \C : 0 < \abs{z} < r, \ \alpha < \arg z < \beta}$$
be an open sector with vertex $0$, of any finite opening $\beta - \alpha$.
Let $\sum_{n \ge 0} a_n z^n$ be a [[formal power series|072W67KFYK]] with complex coefficients.
Then there is a [[holomorphic function|0DG0Z72CQD]] $f$ on $S$ with the following property.
On every closed subsector $\set{z : 0 < \abs{z} \le \rho, \ \alpha' \le \arg z \le \beta'}$ with $\rho < r$ and $\alpha < \alpha' < \beta' < \beta$,
$$f(z) \sim \sum_{n \ge 0} a_n z^n \quad \text{as } z \to 0,$$
in the sense of [[asymptotic expansions|B53THC8F79]] in powers of $z$.

## Remarks

Idea of the proof: after a rotation, one picks $\gamma > 0$ so small that $\operatorname{Re}(z^{-\gamma}) > 0$ on $S$ for a branch of $z^{-\gamma}$.
Then one sets $f(z) = \sum_{n} a_n z^n \left(1 - e^{-b_n z^{-\gamma}}\right)$ with positive numbers $b_n$ that tend to $0$ fast.
Each factor $1 - e^{-b_n z^{-\gamma}}$ is bounded on $S$ and at most $b_n \abs{z}^{-\gamma}$ in absolute value, which makes the series converge and keeps the tail small near $0$.
The function $f$ is never unique: adding $e^{-z^{-\gamma}}$ for a suitable $\gamma$ does not change the expansion.
The real analogue, with smooth functions on an interval, is [[Borel's lemma|KZHNA19FSS]].
If the opening of $S$ is larger than $2\pi$, $S$ is understood on the Riemann surface of the logarithm, and the theorem still holds.

## Explanations

### L1 human

Any formal power series, even one that diverges everywhere, is the asymptotic expansion of some holomorphic function on a sector.
So asymptotic expansions on sectors carry no growth condition on their coefficients.
This is the complex counterpart of the fact that every sequence of numbers is the sequence of derivatives at $0$ of a smooth function.

### L2 human

The theorem shows that the map from functions with an asymptotic expansion on a sector to formal power series is onto.
Its kernel consists of functions that are flat at $0$ in the sector, such as $e^{-1/z}$ on $\abs{\arg z} < \pi/2$.
To recover uniqueness one needs stronger conditions, such as Gevrey estimates on a sector of opening larger than $\pi/k$.
This is the starting point of $k$-summability and of [[Borel summation|8DJPSYSHGJ]] for divergent solutions of differential equations.
</document>

<document source="https://tamnd.github.io/math2/node/C2H26BYWHE.md" tag="C2H26BYWHE" kind="concept">
# Nilpotent ideal

## Summary

A nilpotent ideal of a [[ring|0B1261B5ED]] is a one-sided or two-sided ideal $I$ such that some power $I^n$ is zero.
This means that every product of $n$ elements of $I$ vanishes, a uniform version of being a [[nil ideal|NZ1G9WRWFB]].
Every nilpotent ideal is nil, but a nil ideal need not be nilpotent.
A ring is called semiprime when it has no nonzero nilpotent ideals.
The Jacobson radical of a left Artinian ring is a nilpotent ideal.

## Explanations

### L1 human

In a nil ideal each element dies after enough self-multiplications.
In a nilpotent ideal a single number $n$ works for all of them at once: any product of $n$ elements of the ideal, equal or not, is zero.
In the ring of upper triangular $2 \times 2$ matrices, the matrices with zero diagonal form a nilpotent ideal.
The product of any two of them is zero.
</document>

<document source="https://tamnd.github.io/math2/node/C2R5Q2Q717.md" tag="C2R5Q2Q717" kind="definition">
# Dimension of an analytic set

## Definition

Let $X$ be a complex manifold and let $A$ be an [[analytic subset|VMDTGTMN9P]] of $X$.
Let $\operatorname{reg} A$ be the set of [[regular points|JPP8N6MZGP]] of $A$.
At a regular point $z$, the set $A$ is near $z$ a complex submanifold, and $\dim_z A$ is its complex dimension at $z$.
For an arbitrary point $a \in A$, the *dimension of $A$ at $a$* is
$$\dim_a A = \limsup_{z \to a,\ z \in \operatorname{reg} A} \dim_z A.$$
The *dimension of $A$* is $\dim A = \sup_{a \in A} \dim_a A$, with $\dim \emptyset = -\infty$.
The set $A$ has *pure dimension* $k$ if $\dim_a A = k$ for every $a \in A$.

## Remarks

The definition makes sense because $\operatorname{reg} A$ is dense in $A$; at a regular point $a$ it gives back the manifold dimension.
Example: let $A = \set{z_1 z_3 = 0,\ z_2 z_3 = 0}$ in $\C^3$, the union of the plane $P = \set{z_3 = 0}$ and the line $L = \set{z_1 = z_2 = 0}$.
The regular points are the points of $P \setminus \set{0}$, of dimension $2$, and the points of $L \setminus \set{0}$, of dimension $1$.
So $\dim_0 A = 2$, $\dim_a A = 1$ for $a \in L \setminus \set{0}$, and $\dim A = 2$.
Non-example of pure dimension: this set $A$ does not have pure dimension, since its local dimension takes the values $1$ and $2$.
It also shows that the dimension is not the number of variables minus the number of equations, which would be $1$ here.
There are equivalent forms of $\dim_a A$ when $X = \C^n$.
It is the Krull dimension of $\mathcal{O}_{n,a}/\mathcal{I}(A_a)$.
It is the largest of the integers $d$ given by the local parametrization theorem for the irreducible components of the germ $A_a$.
It is the smallest $k$ such that some complex affine subspace of codimension $k$ through $a$ meets $A$ in the isolated point $a$.
</document>

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# Cohomology operation

## Definition

Let $G$ and $H$ be [[abelian groups|082GHNPJ53]] and let $n, m \ge 0$ be integers.
Write $H^n(X; G)$ for [[singular cohomology|6N7BC70CX0]] with coefficients in $G$.
A *cohomology operation* of type $(G, n; H, m)$ is a family of functions $\theta_X \colon H^n(X; G) \to H^m(X; H)$, one for each topological space $X$.
The family must be natural.
That is, $f^* \circ \theta_X = \theta_Y \circ f^*$ for every [[continuous map|0X3WGX95FE]] $f \colon Y \to X$.
Here $f^*$ denotes the induced maps in cohomology.

## Remarks

In the language of categories, a cohomology operation is a natural transformation between the functors $H^n(-; G)$ and $H^m(-; H)$ to sets.
The functions $\theta_X$ need not be homomorphisms.
Operations of type $(G, n; H, m)$ correspond bijectively to elements of $H^m(K(G, n); H)$, through $\theta \mapsto \theta(\iota_n)$, where $\iota_n \in H^n(K(G, n); G)$ is the fundamental class of the Eilenberg–MacLane space.

Example: the cup square $\theta_X(x) = x \smile x$ is an operation of type $(\Z, n; \Z, 2n)$.
The check: $f^*(x \smile x) = f^* x \smile f^* x$, because $f^*$ is a ring homomorphism.
It is not additive in general: $(x + y)^2 = x^2 + x y + y x + y^2$.

Non-example: on $H^0(-; \Z)$, let $\theta_X(x) = x$ if $X$ is path-connected and $\theta_X(x) = 0$ otherwise.
Take $X = \set{a, b}$ discrete, $Y$ a point, $f$ the map to $a$, and $x = (1, 1) \in H^0(X; \Z) \cong \Z^2$.
Then $f^* \theta_X(x) = 0$ but $\theta_Y(f^* x) = 1$, so the naturality condition fails.
</document>

<document source="https://tamnd.github.io/math2/node/C2V5P3G2FS.md" tag="C2V5P3G2FS" kind="example">
# Kuramoto model

## Example

Let $N \ge 2$, let $K \ge 0$ be a coupling strength, and let $\omega_1, \ldots, \omega_N \in \R$ be natural frequencies.
The *Kuramoto model* is the [[autonomous system|0673D85QKZ]] for phases $\theta_1, \ldots, \theta_N \in \R / 2\pi\Z$:
$$\theta_i' = \omega_i + \frac{K}{N} \sum_{j=1}^{N} \sin(\theta_j - \theta_i), \qquad i = 1, \ldots, N.$$
Each oscillator turns at its own speed and is pulled towards the phases of the others.

For $N = 2$ the phase difference $\psi = \theta_2 - \theta_1$ satisfies
$$\psi' = (\omega_2 - \omega_1) - K \sin \psi.$$
If $K > 0$ and $K \ge \abs{\omega_2 - \omega_1}$, this equation has an equilibrium $\psi^*$ with $\sin \psi^* = (\omega_2 - \omega_1)/K$.
Then the two oscillators lock their phases at the difference $\psi^*$ and turn at the common speed $(\omega_1 + \omega_2)/2$.
If $K < \abs{\omega_2 - \omega_1}$, then $\psi' \ne 0$ always, and the phases drift apart.

## Remarks

The degree of synchrony is measured by the order parameter $r e^{i\phi} = \frac{1}{N} \sum_j e^{i \theta_j}$, with $0 \le r \le 1$.
With it the equations become $\theta_i' = \omega_i + K r \sin(\phi - \theta_i)$, so each oscillator couples only to the mean field.
In the limit $N \to \infty$ with frequencies drawn from a symmetric unimodal density $g$, a partially synchronized state with $r > 0$ appears when $K$ exceeds $K_c = 2 / (\pi g(0))$.
For the Lorentzian density $g(\omega) = \gamma / (\pi(\gamma^2 + \omega^2))$ this gives $K_c = 2\gamma$, and the stationary order parameter is $r = \sqrt{1 - K_c/K}$ for $K > K_c$.

Kuramoto introduced the model in 1975.
It is the basic model of [[synchronization|YP1PMW0HE3]] of oscillators in physics, chemistry and biology.

## Explanations

### L1 human

Picture many runners on a circular track, each with a favourite speed, and each slightly attracted to the others.
If the attraction is weak, they spread out and run independently.
If it is strong enough, a group forms that runs together, and the group grows as the attraction increases.
The Kuramoto model is the simplest equation that shows this sudden onset of collective rhythm, seen in fireflies flashing together and in pacemaker cells of the heart.
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<document source="https://tamnd.github.io/math2/node/C31PYYPRGC.md" tag="C31PYYPRGC" kind="concept">
# Faithfully flat module

## Summary

A faithfully flat module is a [[flat module|73RYYEVDKA]] $M$ such that tensoring with $M$ kills no nonzero module.
Then a sequence of modules is exact if and only if it stays exact after tensoring with $M$.
A ring map $R \to S$ is faithfully flat when $S$ is faithfully flat as an $R$-module; such maps are the setting of faithfully flat descent.
Examples are nonzero free modules, polynomial rings $R[x]$ over $R$, and the completion of a Noetherian local ring.

## Explanations

### L1 human

A flat module never destroys injectivity, but it can still make a whole module vanish.
For example, $\Q$ is flat over $\Z$, yet $\Z/2\Z \otimes_\Z \Q = 0$.
A faithfully flat module never does this: it reflects all the information.
So a statement can be checked after a faithfully flat change of rings, where it may be easier to prove.
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<document source="https://tamnd.github.io/math2/node/C338GKB4VJ.md" tag="C338GKB4VJ" kind="concept">
# Association scheme

## Summary

An association scheme is a partition of the ordered pairs of a finite set into relations with a strong regularity property.
The number of paths of two given types between two points depends only on the relation between the points.
The adjacency matrices of the relations span a matrix algebra, the Bose–Mesner algebra, which makes linear algebra available for combinatorial questions.
Association schemes came from the design of statistical experiments and are now basic in coding theory, design theory and the study of [[strongly regular graphs|C22WGRP3T8]].

## Explanations

### L1 human

Think of a finite set of points where every pair of points stands in exactly one of a few possible relations, such as "equal", "near" or "far".
The scheme is very regular.
Take two points in a given relation; the number of third points near the first and far from the second depends only on that relation.
The words of length $n$ over an alphabet, with "differ in exactly $i$ places" as the $i$-th relation, form such a scheme, called the Hamming scheme.
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<document source="https://tamnd.github.io/math2/node/C34RZMKKA4.md" tag="C34RZMKKA4" kind="statement">
# Kazhdan–Margulis theorem

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] without compact factors.
Then there is an open [[neighbourhood|0ZCY0Z80B6]] $U$ of the identity $e$ in $G$ with the following property.
For every [[discrete subgroup|5M7D1BSHVZ]] $\Gamma$ of $G$ there is $g \in G$ such that $g \Gamma g^{-1} \cap U = \set{e}$.

## Remarks

The proof starts from a [[Zassenhaus neighbourhood|FZ798EZ6M0]]: the elements of $\Gamma$ close to $e$ generate a nilpotent group.
One then conjugates by suitable elements of $G$ that expand this nilpotent part, which is possible because $G$ has no compact factors.

Example: $G = \mathrm{SL}_2(\R)$ and $\Gamma = \mathrm{SL}_2(\Z)$.
Conjugating the element $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ by $\operatorname{diag}(t, t^{-1})$ gives $\begin{pmatrix} 1 & t^2 \\ 0 & 1 \end{pmatrix}$, which is close to $e$ for small $t$.
So some conjugates of $\Gamma$ come close to $e$, and the theorem only asserts that some conjugate stays away.

The hypothesis on compact factors is necessary.
In the compact semisimple group $\mathrm{SO}(3)$, let $\Gamma_n$ be the finite cyclic group of rotations about one axis by multiples of $2\pi/n$.
Every conjugate of $\Gamma_n$ contains a rotation by the angle $2\pi/n$.
So for a given $U$, every conjugate of $\Gamma_n$ meets $U \setminus \set{e}$ once $n$ is large.

## Explanations

### L1 human

A discrete subgroup has no elements arbitrarily close to the identity, but it might still have some fairly close ones.
The theorem says that, after moving the subgroup by a conjugation, one can keep all its elements outside a fixed neighbourhood of the identity.
The size of this neighbourhood depends only on the big group, not on the subgroup.

### L2 human

The main consequence concerns [[lattices|BTYF5Y5PFY]].
The volume of $G/\Gamma$ is at least a constant $c(G) > 0$ that does not depend on the lattice $\Gamma$.
Indeed a small ball around $e$ injects into $G/(g\Gamma g^{-1})$, and $G/\Gamma$ and $G/(g\Gamma g^{-1})$ have the same volume.
For quotients of symmetric spaces, this gives a uniform lower bound for the volume of locally symmetric orbifolds of a given type.
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<document source="https://tamnd.github.io/math2/node/C37R11WREF.md" tag="C37R11WREF" kind="definition">
# Trace class operator

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]], and let $T$ be a bounded operator on $H$ with [[adjoint|6HSJBJ8TTQ]] $T^*$.
Let $\abs{T} = (T^* T)^{1/2}$ be the [[square root|3JY5HNFRMF]] of the [[positive operator|77020CY5ZW]] $T^* T$.
The operator $T$ is *trace class* if for some [[orthonormal basis|0S9GGTR0EX]] $(e_i)_{i \in I}$ of $H$,
$$\norm{T}_1 = \sum_{i \in I} \ip{\abs{T} e_i}{e_i} < \infty.$$
The number $\norm{T}_1$ does not depend on the orthonormal basis; it is the *trace norm* of $T$.

## Remarks

If $T$ is trace class, then for every orthonormal basis $(e_i)$ the sum $\operatorname{tr} T = \sum_i \ip{T e_i}{e_i}$ converges absolutely, and its value does not depend on the basis.
On $\C^n$, this is the usual [[trace|0XC1RDBY7A]] of a matrix.
Equivalently, $T$ is trace class if and only if $T$ is compact and its [[singular values|QSJKPKPM7H]] satisfy $\sum_n s_n(T) < \infty$, so the trace class is the [[Schatten class|153N5DW55Z]] $S_1$.
Example: on $\ell^2$, let $T e_n = e_n / n^2$; then $\abs{T} = T$ and $\norm{T}_1 = \sum 1/n^2 < \infty$, so $T$ is trace class with $\operatorname{tr} T = \pi^2 / 6$.
Non-example: $D e_n = e_n / n$ gives $\sum_n \ip{\abs{D} e_n}{e_n} = \sum 1/n = \infty$, so $D$ is compact but not trace class.
The trace class operators form a two-sided ideal of the bounded operators, and $\operatorname{tr}(AT) = \operatorname{tr}(TA)$ for $T$ trace class and $A$ bounded.
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# Indeterminate moment problem

## Definition

Let $K$ be a closed subset of $\R$.
Let $\mu$ be a positive Borel [[measure|PR11BZQQPG]] on $\R$ with support in $K$, such that every [[moment|01XAHPADBA]] $m_n = \int x^n \, d\mu(x)$, $n \ge 0$, is finite.
The [[moment problem|CS7TWHY7JM]] for $\mu$ on $K$ is *indeterminate* if there is a positive Borel measure $\nu \ne \mu$ with the following properties.
The support of $\nu$ lies in $K$, and $\int x^n \, d\nu(x) = m_n$ for all $n \ge 0$.
Equivalently, it is not [[determinate|H0GY2J14CA]].

## Remarks

Example: the log-normal density $f(x) = (x \sqrt{2\pi})^{-1} e^{-(\ln x)^2/2}$ on $(0, \infty)$.
The substitution $x = e^{u}$ shows $\int_0^\infty x^n f(x) \sin(2\pi \ln x) \, dx = e^{n^2/2} \int_{\R} \frac{e^{-v^2/2}}{\sqrt{2\pi}} \sin(2\pi v) \, dv = 0$, with $v = u - n$, since the last integrand is odd.
So $\nu$ with density $f(x)(1 + \sin(2\pi \ln x))$ is a different positive measure on $[0, \infty)$ with the same moments.
Non-example: the uniform distribution on $[0, 1]$ is not indeterminate, since a measure on a compact set is determined by its moments.
Every measure with finitely many atoms is determinate, so an indeterminate measure has infinite support; its support must also be unbounded.
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# Vitali convergence theorem

## Statement

Let $(X, \Sigma, \mu)$ be a [[finite measure|63Z1Q4FS23]] space.
Let $f_1, f_2, \dots$ be real or complex [[integrable functions|X7GTF3D98F]] on $X$.
Let $f$ be a measurable function with $f_n \to f$ [[almost everywhere|RARTMA0ZEC]].
Then the following are equivalent.

1. The family $\set{f_n : n \ge 1}$ is [[uniformly integrable|VRVVKMXDGV]].
2. The function $f$ is integrable and $\lim_{n \to \infty} \int_X \abs{f_n - f} \, d\mu = 0$.

## Remarks

Proof idea for 1 implies 2: uniform integrability bounds $\int \abs{f_n}$, so $f$ is integrable by [[Fatou's lemma|S95TR54TCG]].
Split $\int \abs{f_n - f}$ into the part on a set where $f_n \to f$ uniformly, given by Egorov's theorem, and the part on a small set.
Uniform integrability makes the second part small, uniformly in $n$.
For 2 implies 1, a sequence converging in $L^1$ is uniformly integrable, because finitely many integrable functions are, and the tail is close to $f$.

Example: if $\sup_n \int \abs{f_n}^2 \, d\mu < \infty$ and $f_n \to f$ almost everywhere, then the family is uniformly integrable and $f_n \to f$ in $L^1$.

The finiteness of $\mu$ is needed in this form.
On $\R$ with Lebesgue measure, $f_n = \frac{1}{n} \mathbf{1}_{[0, n]}$ satisfies the condition of uniform integrability, since $\abs{f_n} \le 1$, and tends to $0$ everywhere, but $\int \abs{f_n} = 1$ for all $n$.
The theorem also holds with convergence in measure in place of convergence almost everywhere.

## Explanations

### L1 human

Pointwise convergence alone does not make integrals converge, because mass can escape into thin tall spikes.
On a space of finite total measure, uniform integrability is exactly the condition that forbids such spikes.
So it is the precise condition for convergence of the integrals.

### L2 human

The theorem refines the dominated convergence theorem: a dominated sequence is uniformly integrable, but not conversely.
It is the form of the convergence theorem used in probability, where one shows uniform integrability with a moment bound such as $\sup_n \mathbb{E}[\abs{Y_n}^p] < \infty$ for some $p > 1$.
There are versions for $L^p$ and for infinite measure spaces, where a condition of tightness is added.
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# Landau–Kolmogorov inequality

## Statement

Let $n \ge 2$ and $1 \le k < n$ be integers.
There is a constant $C(n, k)$, depending only on $n$ and $k$, with the following property.
Let $f \colon \R \to \R$ be $n$ times differentiable, with $f$ and its $n$-th [[derivative|084NGSTWM7]] $f^{(n)}$ bounded on $\R$.
Then $f^{(k)}$ is bounded on $\R$ and
$$\norm{f^{(k)}}_\infty \le C(n, k) \, \norm{f}_\infty^{1 - k/n} \, \norm{f^{(n)}}_\infty^{k/n},$$
where $\norm{g}_\infty = \sup_{x \in \R} \abs{g(x)}$ is the [[supremum|0BTMQ04556]] of $\abs{g}$.

## Remarks

Proof idea for $n = 2$ and $k = 1$: let $M_0 = \norm{f}_\infty$ and $M_2 = \norm{f''}_\infty$.
By [[Taylor's theorem|03QHBVW4TT]], $f(x \pm h) = f(x) \pm h f'(x) + \frac{h^2}{2} f''(\xi_\pm)$ for some $\xi_\pm$.
Subtracting gives $\abs{f'(x)} \le \frac{M_0}{h} + \frac{h M_2}{2}$ for every $h > 0$, and the choice $h = \sqrt{2 M_0 / M_2}$ gives $\norm{f'}_\infty^2 \le 2 M_0 M_2$.
The constant $2$ here is the best possible on $\R$.
On the half-line $[0, \infty)$ the same kind of bound holds with the best constant $4$ in place of $2$.

The exponents are forced by scaling: replacing $f(x)$ by $f(\lambda x)$ multiplies $\norm{f^{(j)}}_\infty$ by $\lambda^j$.

## Explanations

### L1 human

If a function stays small and its $n$-th derivative also stays small, then every derivative in between must stay small too.
A function cannot wiggle fast without either growing large or having a large high-order derivative.
The inequality gives the precise trade-off.

### L2 human

These are interpolation inequalities: the middle derivative is controlled by a weighted geometric mean of the two ends.
They are a model for the Gagliardo–Nirenberg inequalities for Sobolev norms.
The best constants $C(n, k)$ on $\R$ are known for all $n$ and $k$, and they are attained by functions related to Euler splines.
For $n = 2$ and $k = 1$ the best constant is $\sqrt{2}$.
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# Weak derivative

## Summary

A weak derivative of a locally integrable function is a function that obeys the integration by parts formula against all smooth compactly supported test functions.
It agrees with the usual [[partial derivative|09EGC6F8PT]] for continuously differentiable functions, but it also exists for many functions with corners.
Weak derivatives are used to define [[Sobolev spaces|M4CEZ5K9HP]] and weak solutions of partial differential equations.

## Explanations

### L1 human

For a smooth function $u$ and a test function $\varphi$ that vanishes near the ends of an interval, integration by parts gives $\int u \varphi' = -\int u' \varphi$.
The weak derivative turns this around: $v$ is called the derivative of $u$ if $\int u \varphi' = -\int v \varphi$ for every test function $\varphi$.
The function $\abs{x}$ has a corner at $0$ and no classical derivative there.
Its weak derivative is the sign function, which is $-1$ for $x < 0$ and $1$ for $x > 0$.
The value at the single point $0$ does not matter, since weak derivatives are only defined up to sets of measure zero.
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<document source="https://tamnd.github.io/math2/node/C3JHS6W73G.md" tag="C3JHS6W73G" kind="definition">
# Natural transformation

## Definition

Let $F, G \colon \mathcal{C} \to \mathcal{D}$ be [[functors|28NQ0KRHD3]].
A *natural transformation* $\alpha \colon F \Rightarrow G$ is a family of [[morphisms|859F0TFVA3]] $\alpha_A \colon F(A) \to G(A)$ of $\mathcal{D}$, one for each object $A$ of $\mathcal{C}$.
The family must satisfy
$$G(f) \circ \alpha_A = \alpha_B \circ F(f)$$
for every morphism $f \colon A \to B$ of $\mathcal{C}$.
The morphism $\alpha_A$ is the *component* of $\alpha$ at $A$.
If every component is an isomorphism, $\alpha$ is a *natural isomorphism*, and $F$ and $G$ are *naturally isomorphic*, written $F \cong G$.

## Remarks

The condition says that the naturality square with sides $F(f)$, $\alpha_B$, $\alpha_A$ and $G(f)$ is a [[commutative diagram|WV898P4X16]].
Example: on [[vector spaces|000AYPA9AJ]] over a field $K$, let $\mathrm{ev}_V \colon V \to V^{**}$ send $v$ to the map $\varphi \mapsto \varphi(v)$, where $V^*$ is the [[dual space|07JBSZE0V4]].
For a linear map $f \colon V \to W$ and $\psi \in W^*$, both $f^{**}(\mathrm{ev}_V(v))$ and $\mathrm{ev}_W(f(v))$ send $\psi$ to $\psi(f(v))$.
So $\mathrm{ev}$ is a natural transformation from the identity functor to the double dual functor.
On finite-dimensional spaces it is a natural isomorphism, by the [[canonical isomorphism with the double dual|0ZJM9D7NPS]].
Non-example: on the category of nonempty sets, choose for each set $X$ a point $x_X \in X$ and let $\alpha_X \colon X \to X$ be the constant map with value $x_X$.
This family is not a natural transformation from the identity functor to itself.
For a two-element set $Y$ and the constant map $f \colon X \to Y$ whose value is not $x_Y$, the condition would require $f(x_X) = x_Y$.
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# Modal logic

## Summary

Modal logic extends [[propositional logic|09K9MWD2QQ]] with operators $\square$ and $\lozenge$, read for example as "necessarily" and "possibly".
Different readings of $\square$, such as "it is known that", "it will always be true" or "it is provable that", give different modal logics.
Its usual semantics uses [[Kripke models|NGHWZ9Y7ZV]], which are sets of possible worlds with an accessibility relation.

## Explanations

### L1 human

Ordinary logic says whether a statement is true.
Modal logic can also say whether it must be true, or could be true.
The formula $\square p$ says "$p$ is necessary", and $\lozenge p$ says "$p$ is possible".
For example, $\square p \to p$ says "what is necessary is true", which holds for necessity but fails if $\square$ means "it is believed that".
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# Infinitesimal stability

## Summary

A smooth map is infinitesimally stable if every infinitesimal deformation of it can be undone.
The undoing uses infinitesimal changes of coordinates in the source and in the target.
It is the linearised form of [[stability|1991WETZF9]]: an equation for vector fields replaces a condition on diffeomorphisms.
Mather's theorem says that for proper maps the two notions agree.
This turns stability into a problem of linear algebra over rings of smooth functions.

## Explanations

### L1 human

Push a smooth map a tiny bit in some direction at each point; that push is a vector field along the map.
The map is infinitesimally stable if every such push can be produced in two steps.
One slides the points of the source along a vector field and the points of the target along another vector field.
A submersion, such as the projection of the plane onto a line, is infinitesimally stable, since any push can be produced by sliding the source alone.
A constant map from a circle to the line is not: sliding the target moves all points by the same amount, and sliding the source does nothing.
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# Stable theory

## Definition

Let $T$ be a [[complete theory|655K8Y7VS8]] in a countable signature, with infinite models, and let $\lambda$ be an infinite [[cardinal|0EBACQFAZW]].
The theory $T$ is *$\lambda$-stable* if the following holds for every [[model|MVCH6N9SNW]] $M$ of $T$ and every set $A$ of elements of $M$ with $\abs{A} \le \lambda$.
The set $S_1^M(A)$ of complete $1$-[[types|9V71Q3SD1G]] over $A$ has [[cardinality|0G0ASKNA6Z]] at most $\lambda$.
The theory $T$ is *stable* if it is $\lambda$-stable for some infinite cardinal $\lambda$.

## Remarks

An $\aleph_0$-stable theory is called $\omega$-stable.
Shelah proved that $T$ is stable if and only if no formula $\varphi(\bar{x}, \bar{y})$ has the order property.
The order property means that some model of $T$ has tuples $\bar{a}_i, \bar{b}_i$ for $i \in \N$ with $\varphi(\bar{a}_i, \bar{b}_j)$ true exactly when $i < j$.

Example: the theory $\mathrm{ACF}_0$ of [[algebraically closed fields|MT8TDVPBXB]] of characteristic $0$ is $\omega$-stable.
For a countable set $A$ in a model $K$, let $k$ be the subfield generated by $A$.
By quantifier elimination, a complete $1$-type over $A$ is determined by the ideal of polynomials in $k[x]$ that vanish at $x$.
This ideal is $0$ or is generated by an irreducible polynomial, and $k[x]$ has countably many polynomials, so there are countably many types.

Non-example: the theory of [[dense linear orders without endpoints|2W4S2H4VYA]] is not stable.
The formula $x < y$ has the order property: in $(\Q, <)$ take $a_i = b_i = i$, so $a_i < b_j$ exactly when $i < j$.
For $\lambda = \aleph_0$ one sees the failure directly, with $A = \Q$ inside the model $\R$.
Each irrational number $r$ realizes its own complete type over $\Q$, determined by the cut $\set{q \in \Q : q < r}$.
So $S_1^{\R}(\Q)$ has size at least $2^{\aleph_0}$.
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# Matched Z-transform method

## Summary

The matched Z-transform method turns a continuous-time filter with a rational transfer function into a discrete-time filter.
It sends each [[pole|066E7PJCYE]] and each zero $a$ in the $s$-plane to the point $e^{aT}$ in the $z$-plane, where $T$ is the sampling period.
Then it fixes the overall gain.
It is also called the pole–zero mapping method.
It keeps the poles exactly where sampling puts them, and it is simple to apply, but it can distort the frequency response.

## Explanations

### L1 human

A filter for continuous signals is described by its poles and zeros, points of the complex plane.
To build a digital version, one moves each of these points with the map $a \mapsto e^{aT}$, which is how sampling moves exponentials.
Then one scales the result so that, for example, constant signals pass with the same gain as before.
For $H(s) = 1/(s + 1)$ and $T = 0.1$, one gets $H(z) = k/(1 - e^{-0.1} z^{-1})$ with $k = 1 - e^{-0.1}$ for equal gain at $s = 0$, $z = 1$.
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# Time-scale calculus

## Definition

A *time scale* is a nonempty [[closed subset|0ZCK8DQYN1]] $\mathbb{T}$ of $\R$.
The forward jump operator $\sigma \colon \mathbb{T} \to \mathbb{T}$ is $\sigma(t) = \inf \set{s \in \mathbb{T} : s > t}$, with $\sigma(t) = t$ when $t$ is the largest element of $\mathbb{T}$.
The graininess is $\mu(t) = \sigma(t) - t$.
Let $\mathbb{T}^\kappa$ be $\mathbb{T}$ without its largest element $m$ if $m$ exists and $\sup \set{s \in \mathbb{T} : s < m} < m$; otherwise $\mathbb{T}^\kappa = \mathbb{T}$.
Let $f \colon \mathbb{T} \to \R$ and $t \in \mathbb{T}^\kappa$.
The function $f$ is *delta differentiable* at $t$, with *delta derivative* $f^\Delta(t) \in \R$, if for every $\varepsilon > 0$ there is $\delta > 0$ such that
$$\abs{f(\sigma(t)) - f(s) - f^\Delta(t) \, (\sigma(t) - s)} \le \varepsilon \abs{\sigma(t) - s} \quad \text{for all } s \in \mathbb{T} \text{ with } \abs{t - s} < \delta.$$
*Time-scale calculus* is the study of delta derivatives, of the matching integrals, and of dynamic equations built from them.

## Remarks

The delta derivative is unique when it exists.
Example: for $\mathbb{T} = \R$, $\sigma(t) = t$ and the condition says that $f$ is differentiable at $t$ with $f^\Delta(t) = f'(t)$, the ordinary [[derivative|0V1EJCT8G6]].
Example: for $\mathbb{T} = \Z$, $\sigma(t) = t + 1$, and with $\delta = \frac{1}{2}$ the only admissible $s$ is $t$, so the condition holds exactly for $f^\Delta(t) = f(t + 1) - f(t)$.
More generally, if $\mu(t) > 0$ and $f$ is continuous at $t$, then $f^\Delta(t) = \frac{f(\sigma(t)) - f(t)}{\mu(t)}$, a [[difference quotient|AZBDB5F4M4]].
Non-example: on $\mathbb{T} = \R$, the function $f(t) = \abs{t}$ is not delta differentiable at $0$, because it has no ordinary derivative there.
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# Multivariate interpolation

## Definition

Let $d \ge 2$, let $\Omega \subseteq \R^d$, let $x_1, \dots, x_N \in \Omega$ be pairwise distinct points of [[Euclidean space|0XPR1E85W0]], and let $y_1, \dots, y_N \in \R$.
Let $V$ be a set of real [[functions of several variables|0JPCHTXWG0]] on $\Omega$.
A *multivariate interpolant* of the data $(x_i, y_i)$ from $V$ is a function $s \in V$ such that $s(x_i) = y_i$ for $i = 1, \dots, N$.
*Multivariate interpolation* is the problem of finding such an $s$, usually from a space $V$ chosen so that it exists and is unique.

## Remarks

For $d = 1$ and $V$ the polynomials of degree at most $N - 1$ this is [[polynomial interpolation|SS1YMS2NPC]].
For $d \ge 2$ and $V$ a fixed space of [[polynomials in several variables|07GNX8GCYP]] with $\dim V = N$, the interpolant is unique exactly when the points are a [[unisolvent point set|5276MDSDQV]] for $V$.

Example: $d = 2$, $V = \set{a + bx + cy}$, points $(0, 0)$, $(1, 0)$, $(0, 1)$ with values $1, 3, 4$.
Then $s(x, y) = 1 + 2x + 3y$ is an interpolant: $s(0, 0) = 1$, $s(1, 0) = 3$, $s(0, 1) = 4$.

Non-example: with the same $V$ and the collinear points $(0, 0)$, $(1, 1)$, $(2, 2)$ with values $0, 1, 3$, there is no interpolant.
Indeed $s(t, t) = a + (b + c) t$ is affine in $t$, but the values $0, 1, 3$ at $t = 0, 1, 2$ do not lie on a line.
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# Observability

## Summary

A [[control system|ZPDTVZQN5K]] with an output is observable if the initial state can be recovered from the known input and the measured output over time.
Observability matters because in practice only some quantities can be measured, while a feedback law usually needs the full state.
For linear systems it is dual to [[controllability|A9WFM77Z4P]]: the pair $(A, C)$ is observable exactly when $(A^T, C^T)$ is controllable.

## Explanations

### L1 human

Suppose you can only see the position of a moving car, not its speed.
By watching the position for a while, you can still find the speed, since the speed is the rate of change of the position.
So this system is observable even though one quantity is hidden.
If you could only see the speed, you would never know where the car started, and the system would not be observable.
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# Blowing up

## Summary

Blowing up is a construction that replaces a closed subvariety of a variety by the set of normal directions to it.
The result maps birationally onto the original variety and is an isomorphism away from the centre.
The preimage of the centre, the exceptional divisor, is a hypersurface.
Blowing up is the basic tool of [[birational geometry|51YVV85GWB]] and of [[resolution of singularities|MB52NDA6WN]].

## Explanations

### L1 human

To blow up the origin of the plane, keep every other point, but replace the origin by a whole projective line, one point for each line through the origin.
Lines through the origin that used to meet there are now separated, because they arrive at different points of the new line.
A picture is a spiral staircase over the plane: above the origin sits a full circle of directions, and above other points one point.
For example, the curve $y^2 = x^2(x + 1)$ crosses itself at the origin, and after blowing up its two branches pass through different points and no longer meet.
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# Canonical height

## Summary

Let $\phi$ be a morphism of degree $d \ge 2$ of projective space over a number field.
Its canonical height is the [[height function|F1FF505PWH]] adapted to $\phi$: it multiplies by exactly $d$ under $\phi$.
It differs from the naive Weil height by a bounded amount, and it vanishes exactly at the preperiodic points of $\phi$.
For the multiplication maps on an elliptic curve the same construction gives the Néron–Tate height, a quadratic form on the group of rational points.
Canonical heights are a basic tool of [[arithmetic dynamics|BE4X0QG8XW]].

## Explanations

### L1 human

The height of a fraction measures how many digits it needs.
Applying a map like $z \mapsto z^2 + 1$ to a fraction roughly doubles the number of digits, but not exactly.
The canonical height corrects this: it is a slightly adjusted size for which applying the map doubles the size exactly.
Points of canonical height zero are those whose orbit never grows, which are exactly the points that eventually cycle.
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<document source="https://tamnd.github.io/math2/node/C57T562NKB.md" tag="C57T562NKB" kind="concept">
# Differentiable curve

## Summary

A differentiable curve is a [[parametrised curve|0G7WM8DX4P]] whose coordinate functions have continuous derivatives up to some order $k \ge 1$.
It is the basic object of the differential geometry of curves.
Tangent vectors, arc length, curvature and torsion are all defined for differentiable curves of suitable order.

## Explanations

### L1 human

Think of a point that moves smoothly through space, with a velocity at each moment.
A differentiable curve records the position of that point as a function of time.
Being differentiable means that the velocity exists and changes continuously.
For example, $t \mapsto (\cos t, \sin t, t)$ is a differentiable curve, the helix, which winds around a cylinder.
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<document source="https://tamnd.github.io/math2/node/C5971PGKPH.md" tag="C5971PGKPH" kind="concept">
# Classical ring of quotients

## Summary

The classical ring of quotients of a ring $R$ is obtained by inverting all regular elements of $R$.
These are the elements that are not [[zero divisors|0NNRDW10RZ]] on either side.
It is the [[Ore localization|SG7AQ9T8HD]] of $R$ at the set of regular elements, when that localization exists.
For an integral domain it is the [[field of fractions|0F2PTGX1A2]], and for a commutative ring it is the [[total ring of fractions|K5Z67HR9ET]].
[[Goldie's theorem|MDS8655N0Z]] says exactly when this ring is semisimple Artinian.

## Explanations

### L1 human

The rationals are built from the integers by allowing division by every nonzero integer.
For a general ring one cannot divide by an element that kills some other nonzero element, so one only tries to divide by the regular elements.
The classical ring of quotients is the ring of fractions whose denominators are all the regular elements, if it exists.
For a ring of $2 \times 2$ matrices over $\Z$, it is the ring of $2 \times 2$ matrices over $\Q$, since an integer matrix with nonzero determinant becomes invertible over $\Q$.
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<document source="https://tamnd.github.io/math2/node/C5ANXPS4YN.md" tag="C5ANXPS4YN" kind="definition">
# Cone

## Definition

Let $J$ and $\mathcal{C}$ be [[categories|A2DE66937D]] and let $D \colon J \to \mathcal{C}$ be a [[diagram|HJK75HDWFE]].
A *cone* over $D$ is an object $N$ of $\mathcal{C}$, the *apex*, with a [[morphism|859F0TFVA3]] $\psi_j \colon N \to D(j)$ for each object $j$ of $J$.
The morphisms must satisfy $D(u) \circ \psi_j = \psi_k$ for every morphism $u \colon j \to k$ of $J$.
A *cocone* under $D$ is an object $N$ together with morphisms $\psi_j \colon D(j) \to N$ such that $\psi_k \circ D(u) = \psi_j$ for every $u \colon j \to k$.
A morphism of cones $(N, \psi) \to (N', \psi')$ over $D$ is a morphism $h \colon N \to N'$ with $\psi'_j \circ h = \psi_j$ for all $j$.

## Remarks

A cone with apex $N$ is the same as a [[natural transformation|KYCPCC57FT]] $\Delta N \Rightarrow D$, where $\Delta N \colon J \to \mathcal{C}$ is the constant functor with value $N$; the condition $D(u) \circ \psi_j = \psi_k$ is its naturality square.
Cones over $D$ and their morphisms form a category, written $\mathrm{Cone}(D)$.

Example: let $D$ be the diagram $f, g \colon X \rightrightarrows Y$ of two parallel functions between sets.
Let $N = \set{x \in X : f(x) = g(x)}$, let $\psi_X$ be the inclusion and let $\psi_Y = f \circ \psi_X$.
The conditions $f \circ \psi_X = \psi_Y$ and $g \circ \psi_X = \psi_Y$ hold because $f$ and $g$ agree on $N$, so this is a cone.

Non-example: in the same diagram take $N = X$, $\psi_X = \id_X$ and $\psi_Y = f$.
If $f \neq g$, then $g \circ \psi_X = g \neq \psi_Y$, so the condition fails for the morphism $g$ and this is not a cone.
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# Ordinal analysis

## Summary

Ordinal analysis is the part of [[proof theory|N4EDJNMTEF]] that assigns to a formal theory its [[proof-theoretic ordinal|C72RBH0CDR]].
It measures the strength of a theory by the transfinite inductions that the theory can prove.
It started with Gentzen's consistency proof for [[Peano arithmetic|0QFT7Z76N5]], which gives the ordinal $\varepsilon_0$.
An ordinal analysis usually also yields a consistency proof and a description of the provably total functions of the theory.

## Explanations

### L1 human

Theories of arithmetic and of sets differ in strength, and ordinal analysis gives a ruler to measure them.
The marks on the ruler are countable ordinals, written in a finite notation system.
To place a theory, one shows that it proves induction up to every mark below some ordinal, and that it cannot reach that ordinal.
For Peano arithmetic the answer is $\varepsilon_0$.
Stronger theories, such as parts of second-order arithmetic, get much larger ordinals.
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# Hamiltonian vector field

## Summary

On a [[symplectic manifold|CAN490EPRT]], every smooth function $H$ determines a [[vector field|0299VCW1TH]] $X_H$, its Hamiltonian vector field.
The symplectic form turns the differential $dH$ into a tangent vector at each point.
The flow of $X_H$ is the motion of a [[Hamiltonian system|0EDKGA2FAK]] with energy $H$, so this notion carries classical mechanics onto curved phase spaces.

## Explanations

### L1 human

Think of $H$ as the energy of a mechanical system, written as a function of position and momentum.
The gradient of $H$ points to where the energy grows fastest.
The Hamiltonian vector field is the gradient turned by a quarter turn, so it points along the level sets of $H$ instead of across them.
For $H(q, p) = \frac{1}{2}(q^2 + p^2)$ on the plane, the field is $X_H = p\,\partial_q - q\,\partial_p$.
Its flow turns the plane around the origin and keeps the energy constant, which is the motion of a harmonic oscillator.
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<document source="https://tamnd.github.io/math2/node/C5FG6ARWVV.md" tag="C5FG6ARWVV" kind="concept">
# Pseudorandom number generator

## Summary

A pseudorandom number generator is a deterministic algorithm that produces a sequence of numbers meant to look like independent uniform random numbers.
It has a finite internal state, updated by a fixed map at each step.
An output function reads a number from the state.
Because the state space is finite, the sequence is eventually periodic, and the same seed always reproduces the same sequence.
Such generators supply the randomness of almost all Monte Carlo computations and simulations.

## Explanations

### L1 human

A computer cannot toss coins, but it can run a rule that scrambles a number again and again.
If the rule is good, the outputs pass statistical tests of randomness, even though each one is completely determined by the starting value, the seed.
A toy example is $s_{i+1} = 5 s_i + 1 \bmod 8$ with output $s_i/8$: from $s_0 = 0$ it gives $0, 1/8, 6/8, 7/8, 4/8, 5/8, 2/8, 3/8$ and then repeats.
Real generators work the same way with huge state spaces, so that the period is far longer than any computation.
Being reproducible is a feature: a simulation can be rerun exactly from its seed.
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# Adaptive mesh refinement

## Definition

Let $\mathcal{T}_0$ be an initial mesh of a domain $\Omega$, and fix a discretization such as the [[finite element method|98GZ9DVHTA]].
Let an error indicator assign a number $\eta_K(\mathcal{T}) \ge 0$ to each element $K$ of each mesh $\mathcal{T}$.
*Adaptive mesh refinement* builds meshes $\mathcal{T}_0, \mathcal{T}_1, \ldots$ by repeating four steps for $\ell = 0, 1, \ldots$:

1. Solve: compute the discrete solution $u_\ell$ on $\mathcal{T}_\ell$.
2. Estimate: compute $\eta_K = \eta_K(\mathcal{T}_\ell)$ for all $K \in \mathcal{T}_\ell$ from $u_\ell$ and the data.
3. Mark: choose a set $\mathcal{M}_\ell \subseteq \mathcal{T}_\ell$ of elements with large indicators.
4. Refine: subdivide at least the elements of $\mathcal{M}_\ell$, plus the neighbours needed to keep the mesh conforming, to get $\mathcal{T}_{\ell + 1}$.

The loop stops when $\big(\sum_K \eta_K^2\big)^{1/2}$ is below a tolerance.

## Remarks

A common marking rule is the Dörfler criterion with a parameter $0 < \theta \le 1$.
It takes a set $\mathcal{M}_\ell$ of minimal size with $\sum_{K \in \mathcal{M}_\ell} \eta_K^2 \ge \theta \sum_{K \in \mathcal{T}_\ell} \eta_K^2$.
For the Poisson equation, a standard residual indicator combines $h_K^2 \norm{f + \Delta u_\ell}_{L^2(K)}^2$ with the jumps of the normal derivative of $u_\ell$ across the faces of $K$.

Example: on the L-shaped domain $(-1, 1)^2 \setminus [0, 1) \times (-1, 0]$, the solution of $-\Delta u = 0$ with boundary values $r^{2/3} \sin(2\varphi/3)$ is $u = r^{2/3} \sin(2\varphi/3)$ in polar coordinates $(r, \varphi)$ with $0 < \varphi < 3\pi/2$.
Its gradient blows up at the corner $0$, the indicators are largest there, and the meshes become graded towards $0$.

Non-example: uniform refinement subdivides every element of $\mathcal{T}_\ell$ in each step.
Its refinement does not depend on any indicator, so step 3 is missing.
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# Projective determinacy

## Definition

*Projective determinacy* (PD) is the following statement in the language of set theory.
For every projective set $X \subseteq \N^{\N}$, that is, every set in some level $\boldsymbol{\Sigma}^1_n$ of the [[projective hierarchy|8DZTNVTGY2]], the game $G(\N, X)$ is [[determined|QXECDPAYTN]].

## Remarks

Example: every Borel set $X$ of the [[Baire space|P2WS16HKD7]] is projective, and its game is determined in ZFC by the Borel determinacy theorem.
So the Borel instances of PD are theorems of ZFC.

Non-example: the axiom of choice gives a set $X \subseteq \N^{\N}$ whose game is not determined.
PD does not cover this set, and under PD such a set is not projective.

For each $n$, the statement that all $\boldsymbol{\Sigma}^1_n$ games are determined is a separate hypothesis, and PD is the conjunction of all of them.
The statement that all analytic games, the case $n = 1$, are determined follows from the existence of a measurable cardinal.
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# Tychonoff space

## Summary

A Tychonoff space is a [[T1 space|0NW6CH38YR]] with many continuous real functions.
Every point can be separated by such a function from every closed set that does not contain it.
Such spaces are exactly the subspaces of products of copies of $[0, 1]$, up to homeomorphism.
They are also exactly the spaces that embed in some compact Hausdorff space.
Metric spaces and compact Hausdorff spaces are Tychonoff.

## Explanations

### L1 human

In a Tychonoff space, there are enough continuous real functions to describe the topology.
Given a point and a closed set away from it, there is a function that is $0$ at the point and $1$ on the set.
You can think of such a function as a continuous hill that rises from the point to the set.
In the plane, for a point $p$ and a closed set $F$ with $p \notin F$, the function $\min(1, d(x, p)/d(p, F))$ does the job.
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# Covering density

## Definition

Let $n \ge 1$, let $\operatorname{vol}$ be the [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$, and let $B_R$ be the [[closed ball|0Q0H13G8ZB]] of radius $R$ centred at $0$.
A *covering* of $\R^n$ is a countable family $\mathcal C = (K_i)_{i \in I}$ of compact sets whose union is $\R^n$.
It is required that each bounded set meets only finitely many $K_i$.
The *lower density* of $\mathcal C$ is
$$\underline{d}(\mathcal C) = \liminf_{R \to \infty} \frac{\sum_{i \in I} \operatorname{vol}(K_i \cap B_R)}{\operatorname{vol}(B_R)},$$
with the [[limit inferior|0H4R3RESGV]]; when the limit exists it is the *density* of $\mathcal C$.
Let $K$ be a compact set with nonempty interior.
The *covering density* of $K$ is $\vartheta(K) = \inf \underline{d}(\mathcal C)$, over all coverings $\mathcal C$ of $\R^n$ by congruent copies of $K$.

## Remarks

The ratio is the same as for the [[packing density|4P5PSWADJD]]; packings forbid overlaps, coverings forbid gaps.
Since the sets cover $B_R$, the ratio is at least $1$, so $\vartheta(K) \ge 1$, with equality when $K$ tiles space, as a cube does.
Example: unit discs centred at the points of the triangular lattice spanned by $(\sqrt 3, 0)$ and $(\sqrt 3/2, 3/2)$.
Each disc contains the regular hexagon of circumradius $1$ around its centre, and these hexagons tile the plane, so the discs cover.
The density is $\pi$ divided by the hexagon area $3\sqrt 3/2$, that is $2\pi/\sqrt{27}$.
Non-example: unit discs centred at $2\Z^2$ do not form a covering, since the point $(1, 1)$ is at distance $\sqrt 2 > 1$ from every centre.
For a lattice covering $K + \Lambda$, the density exists and equals $\operatorname{vol}(K)/\det \Lambda$.
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# Milstein method

## Statement

Let $W$ be a one-dimensional [[Wiener process|22735FGJV6]], and consider the scalar Itô [[stochastic differential equation|B219G9SHJA]]
$$dX_t = a(X_t)\, dt + b(X_t)\, dW_t, \qquad X_0 = x_0 \in \R, \qquad 0 \le t \le T.$$
Assume that $a$ and $b$ are twice continuously differentiable with bounded first and second derivatives, and that $b b'$ is globally Lipschitz continuous.
Let $h = T/N$, $t_n = n h$ and $\Delta W_n = W_{t_{n+1}} - W_{t_n}$.
The Milstein method sets $Y_0 = x_0$ and
$$Y_{n+1} = Y_n + a(Y_n)\, h + b(Y_n)\, \Delta W_n + \tfrac12\, b(Y_n)\, b'(Y_n) \left( (\Delta W_n)^2 - h \right).$$
Then the method has strong order $1$: there is a constant $C$, independent of $N$, such that
$$\max_{0 \le n \le N} \left( \mathbb{E} \abs{X_{t_n} - Y_n}^2 \right)^{1/2} \le C h.$$

## Remarks

The scheme is the Itô–Taylor expansion of the solution over one step, cut after the terms of order $h$ in mean square.
Applying [[Itô's lemma|2J1RVQB0CM]] to $b(X_t)$ shows that the leading neglected term of the [[Euler–Maruyama method|22T0NVC56P]] is $b b' \int_{t_n}^{t_{n+1}} (W_s - W_{t_n})\, dW_s$, and this double integral equals $\frac12((\Delta W_n)^2 - h)$.
The Euler–Maruyama method omits this term and has strong order only $1/2$ when $b' \ne 0$; when $b$ is constant, both methods coincide.
Example: for geometric Brownian motion $dX = \mu X\, dt + \sigma X\, dW$, the method reads $Y_{n+1} = Y_n \left( 1 + \mu h + \sigma \Delta W_n + \frac{\sigma^2}{2} ((\Delta W_n)^2 - h) \right)$.
For systems driven by several Wiener processes, the analogous scheme needs the iterated integrals $\int\!\!\int dW^i\, dW^j$, which are not functions of the increments unless a commutativity condition holds.

## Explanations

### L1 human

Simulating a random differential equation step by step, the simplest method treats the noise strength as frozen over each step.
When the noise strength depends on the state, this ignores how the noise changes itself during the step.
Milstein's method adds one correction term for this effect, built from the square of the random increment.
With it, halving the step size halves the pathwise error, instead of reducing it only by a factor $\sqrt{2}$.

### L2 human

Strong order measures the error of individual paths and matters for multilevel Monte Carlo and for pathwise simulation.
Weak order measures the error of expectations, and there Euler–Maruyama and Milstein both have order $1$.
The extra term costs one derivative of $b$; derivative-free variants replace $b'$ by a difference quotient, as in stochastic Runge–Kutta methods.
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# Localization of a Noetherian ring is Noetherian

## Statement

Let $R$ be a [[Noetherian|0670T59HGE]] commutative ring and let $S \subseteq R$ be a [[multiplicative set|AN9SKEZ3VY]].
Then the [[localization|4C1AE84N5N]] $S^{-1}R$ is a Noetherian ring.

## Remarks

Let $\lambda \colon R \to S^{-1}R$ be the canonical map.
Each ideal $J$ of $S^{-1}R$ is the [[extension|5AFH94P4S3]] of its contraction: $J = S^{-1}\lambda^{-1}(J)$.
Indeed, if $a/s \in J$, then $a/1 = s \cdot (a/s) \in J$, so $a \in \lambda^{-1}(J)$.
Hence a strictly ascending chain of ideals in $S^{-1}R$ gives a strictly ascending chain of their contractions in $R$, which cannot be infinite.
The same argument shows that $S^{-1}M$ is a Noetherian $S^{-1}R$-module when $M$ is a Noetherian $R$-module.

Example: $\Z_{(p)}$, $\Z[1/n]$, $\Q$ and the local rings $k[x_1, \ldots, x_n]_{\mathfrak p}$ of a polynomial ring over a field $k$ are Noetherian.

The converse fails: Noetherianity is not a [[local property|3S58E70Z7Z]].
An infinite product $R = \prod_{i \in \N} k$ of copies of a field $k$ is not Noetherian, but each localization $R_{\mathfrak p}$ at a prime ideal is a field.

## Explanations

### L1 human

A ring is Noetherian when it has no infinite strictly increasing chain of ideals.
Allowing new denominators never creates such a chain, because every ideal of the new ring comes from an ideal of the old ring.
So the good finiteness property of the old ring passes to the new one.

### L2 human

This fact is used all the time with $S = R \setminus \mathfrak p$: the local rings $R_{\mathfrak p}$ of a Noetherian ring are Noetherian local rings.
Thus the theory of Noetherian local rings, such as [[Krull dimension|MVGBS3J1P7]], the [[embedding dimension|DP3KSCRKFP]] and [[regular local rings|63XXTF6B5M]], applies to every prime of a Noetherian ring.
Together with [[Hilbert's basis theorem|0YKA9J3XEE]] and passage to quotients, it shows that the coordinate rings of affine varieties and all their localizations are Noetherian.
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# Latent variable model

## Summary

A latent variable model is a statistical model that links observed variables to variables that are never observed.
The hidden variables explain the dependence between the observed ones.
Typical examples are [[factor analysis|4QQQP1Q886]], [[mixture models|C98DKQSTNC]] and hidden Markov models.
The likelihood of the observed data is obtained by integrating out the hidden variables, often with the [[expectation–maximization algorithm|11N1AXSXYK]].

## Explanations

### L1 human

Intelligence, wealth or the type of a customer cannot be measured directly, but they leave traces in things that can be measured.
A latent variable model treats such a quantity as a hidden variable and describes how it shapes the measured data.
For example, the answers of a student to many test questions are linked because they all depend on one hidden ability.
Once the ability is known, the answers are often modelled as independent of each other.
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# Algebraic logic

## Definition

Algebraic logic is the part of mathematical logic that studies a logic through classes of algebras attached to it.
Its main objects are the following.

1. A logic, given by a formal language and a relation of consequence or provability between formulas.
2. Its algebras, often obtained as quotients of the algebra of formulas by provable equivalence, as in the [[Lindenbaum–Tarski algebra|QC9M3Q8B8Q]].
3. A translation between the two sides, under which theorems of the logic correspond to equations valid in the class of algebras.

## Remarks

Example: the completeness of [[classical propositional logic|09K9MWD2QQ]] with respect to [[Boolean algebras|0XADJKM699]] is a result of algebraic logic.
A formula is provable exactly when it is identically $1$ in every Boolean algebra, and already in the two-element Boolean algebra.

Another example: [[intuitionistic logic|Q11N8X99FK]] corresponds in the same way to the class of [[Heyting algebras|NG943XNQWH]].

Non-example: a truth-table check of a single formula is a computation in one algebra.
It is not by itself a study of the correspondence between a logic and a class of algebras.

Abstract algebraic logic studies in general which logics admit such a correspondence.
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# Potential operator of a Markov process

## Definition

Let $(E, \mathcal{E})$ be a measurable space and let $(P_t)_{t \ge 0}$ be a sub-Markov transition semigroup on $E$.
So the $P_t$ are kernels as for [[Markov kernels|2AVNDP8BJP]] but with $P_t(x, E) \le 1$, and $P_s P_t = P_{s+t}$.
Assume that $(t, x) \mapsto P_t(x, A)$ is measurable on $[0, \infty) \times E$ for each $A \in \mathcal{E}$.
The *potential operator* (or potential kernel) of $(P_t)$ is the kernel $U$ given by
$$U(x, A) = \int_0^\infty P_t(x, A) \, dt \in [0, \infty], \qquad x \in E, \ A \in \mathcal{E}.$$
For measurable $f \colon E \to [0, \infty]$ it acts by $U f(x) = \int_0^\infty P_t f(x) \, dt$.

## Remarks

If $(X_t)$ is a Markov process with semigroup $(P_t)$ and law $P^x$ when started at $x$, then $U f(x) = E^x \int_0^\infty f(X_t) \, dt$.
So $U(x, A)$ is the expected total time that the process spends in $A$.
For each measurable $f \ge 0$, the function $U f$ is [[excessive|46436KE7NT]].
Example: for [[Brownian motion|22735FGJV6]] in $\R^3$, the Gaussian integral $\int_0^\infty (2\pi t)^{-3/2} e^{-r^2/2t} \, dt = 1/(2\pi r)$ gives
$$U(x, A) = \int_A \frac{dy}{2\pi \abs{x - y}}.$$
Non-example of finiteness: for Brownian motion in $\R^2$, $U(x, A) = \infty$ for every nonempty open $A$, so $U$ is not a useful kernel there; the process is recurrent.
$U$ is the limit as $\alpha \downarrow 0$ of the [[resolvent|J5YNNKSB9P]] $U_\alpha = \int_0^\infty e^{-\alpha t} P_t \, dt$.
When $L$ is the [[infinitesimal generator|VPA41PWRR1]] and $U f$ is finite and in the domain of $L$, one has $-L U f = f$.
In discrete time the analogue is $U = \sum_{n \ge 0} P^n$, the Green kernel of a Markov chain.
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# Scott continuity

## Summary

A map between partially ordered sets is Scott-continuous if it preserves suprema of directed subsets.
Scott-continuous maps are exactly the maps that are continuous for the [[Scott topologies|ZR8A03FV67]] on both sides.
In [[domain theory|Q2P5D74Q1V]] they model computable functions: the output on a limit of approximations is the limit of the outputs.
A Scott-continuous self-map of a pointed dcpo has a [[least fixed point|DBD64TH52D]], which gives meaning to recursive definitions.

## Explanations

### L1 human

Feed a program better and better approximations of its input.
If the program is well behaved, its outputs also improve, and the output for the full input is the limit of these outputs.
Scott continuity is this property stated for ordered sets: the map respects order and respects limits of directed families.
For example, on the subsets of $\N$ ordered by inclusion, the map $A \mapsto A \cup \set{0}$ is Scott-continuous, since it commutes with unions of directed families.
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# M3–N5 characterization of distributivity

## Statement

Let $L$ be a [[lattice|WH4SFAACAS]].
Let $M_3$ be the [[diamond lattice|JQ6FJH2TP8]] and $N_5$ the [[pentagon lattice|DHC8TZXGYS]].
Then $L$ is a [[distributive lattice|0HM0HQCK6N]] if and only if $L$ has no [[sublattice|PV90EW00XK]] isomorphic to $M_3$ and no sublattice isomorphic to $N_5$.

## Remarks

If $L$ is distributive, so is every sublattice, and $M_3$ and $N_5$ are not distributive.
Conversely, if $L$ has no copy of $N_5$, then $L$ is [[modular|QX7WZ92QAK]] by [[Dedekind's characterization|ANRXAG227H]].
In a modular lattice in which $a \wedge (b \vee c) \ne (a \wedge b) \vee (a \wedge c)$, put $d = (a \wedge b) \vee (b \wedge c) \vee (c \wedge a)$ and $e = (a \vee b) \wedge (b \vee c) \wedge (c \vee a)$.
Then $d < e$, and the elements $d$, $e$, $(a \wedge e) \vee d$, $(b \wedge e) \vee d$, $(c \wedge e) \vee d$ form a sublattice isomorphic to $M_3$.

Example: the lattice of subspaces of $\R^2$ is not distributive, because the zero subspace, three distinct lines and $\R^2$ form a copy of $M_3$.
The lattice of divisors of $12$ under divisibility, with $\gcd$ as meet and $\lcm$ as join, is distributive, so it contains neither shape.

## Explanations

### L1 human

The distributive law is the familiar rule that "and" distributes over "or".
The theorem says that a lattice breaks this rule only if one of two small five-element lattices, the diamond or the pentagon, sits inside it.
So one can test distributivity by looking for these two shapes.

### L2 human

This theorem is the standard way to show that a lattice is not distributive: one exhibits a diamond or a pentagon.
It refines [[Dedekind's characterization|ANRXAG227H]] of modular lattices, which forbids only the pentagon.
For a finite distributive lattice, the structure is then described completely by [[Birkhoff's representation theorem|6RDM3069KC]].
The proof constructs the forbidden sublattice explicitly from any triple that violates distributivity.
</document>

<document source="https://tamnd.github.io/math2/node/C6J40QF82R.md" tag="C6J40QF82R" kind="concept">
# Simplicial homology

## Summary

Simplicial homology assigns to a [[simplicial complex|V8GZ0W1ZZ5]] a sequence of abelian groups $H_n(K)$ that count its $n$-dimensional holes.
For a finite complex it is computed by finite linear algebra over $\Z$.
Chains are formal sums of oriented simplices, and the boundary operator sends a simplex to the signed sum of its faces.
It agrees with the [[singular homology|M0Q5CFJ7ZH]] of the underlying space, so it is a topological invariant even though its definition uses a triangulation.

## Explanations

### L1 human

A cycle is a combination of simplices with no boundary, like a loop of edges.
A boundary is a cycle that is filled in by higher simplices, like the edge loop around a triangle that is present in the complex.
The $n$-th homology group counts cycles that are not boundaries.
For the hollow triangle, made of three edges, the loop around it is a cycle that is not filled, so $H_1 \cong \Z$.
If the triangle is filled in, the loop becomes a boundary and $H_1 = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/C6QQ3HM5S4.md" tag="C6QQ3HM5S4" kind="definition">
# Tropical variety

## Definition

Let $K$ be a field with a [[valuation|B85T5VKR3M]] $\operatorname{val} \colon K^\times \to \R$, and let $K[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$ be the ring of Laurent polynomials.
For $f = \sum_{u \in \Z^n} c_u x^u$ with finitely many nonzero $c_u$, the *tropicalization* of $f$ is the function $\operatorname{trop}(f) \colon \R^n \to \R$,
$$\operatorname{trop}(f)(w) = \min_{u : c_u \neq 0} \big( \operatorname{val}(c_u) + u \cdot w \big).$$
The *tropical hypersurface* of $f \neq 0$ is the set $\operatorname{Trop}(f)$ of all $w \in \R^n$ at which this minimum is attained for at least two exponents $u$.
For an [[ideal|096Z9MP2ME]] $I$ of $K[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$, the *tropical variety* of $I$ is
$$\operatorname{Trop}(I) = \bigcap_{f \in I,\ f \neq 0} \operatorname{Trop}(f).$$

## Remarks

Fundamental theorem: let $K$ be [[algebraically closed|02SJG1HC32]] with $\operatorname{val}$ nontrivial.
Then $\operatorname{Trop}(I)$ is the closure in $\R^n$ of the points $(\operatorname{val}(z_1), \dots, \operatorname{val}(z_n))$, for $z$ a zero of $I$ in the [[torus|66FQFAJNS4]] $(K^\times)^n$.
A typical such field is the field of [[Puiseux series|SENTYNSKSF]] over $\C$, with the valuation given by the lowest exponent.
If moreover $I$ is prime and its zero set has dimension $d$, then $\operatorname{Trop}(I)$ is the support of a pure $d$-dimensional polyhedral complex.

Example: let $f = x + y + 1$ with all coefficients of valuation $0$.
Then $\operatorname{trop}(f)(w_1, w_2) = \min(w_1, w_2, 0)$.
The minimum is attained twice exactly on the three rays $\set{w_1 = w_2 \le 0}$, $\set{w_1 = 0 \le w_2}$ and $\set{w_2 = 0 \le w_1}$, which form the tropical line.
Non-example: the point $(1, 1)$ is not on it, since there $\min(1, 1, 0) = 0$ is attained only by the constant term.

The intersection must run over all of $I$: the intersection of $\operatorname{Trop}(f)$ over a set of generators of $I$ can be strictly larger than $\operatorname{Trop}(I)$.
A finite generating set for which equality holds is a tropical basis.
</document>

<document source="https://tamnd.github.io/math2/node/C72RBH0CDR.md" tag="C72RBH0CDR" kind="concept">
# Proof-theoretic ordinal

## Summary

The proof-theoretic ordinal of a formal theory measures how much [[transfinite induction|0ZHQMQ2C1N]] the theory can prove.
It is the supremum of the order types of the computable well-orderings that the theory proves to be well-founded.
It is a countable [[ordinal|0YSCSX5YZC]], and it gives a scale on which theories can be compared by strength.
For [[Peano arithmetic|0QFT7Z76N5]] it is [[epsilon nought|KT50SMH3EC]].

## Explanations

### L1 human

Some orderings of the natural numbers are well-orderings: every decreasing sequence in them stops.
A theory can prove this for some such orderings and not for others.
The longest orderings that the theory still trusts give an ordinal, its proof-theoretic ordinal.
A stronger theory trusts longer orderings, so it gets a larger ordinal.
For example, Peano arithmetic trusts each ordering shorter than $\varepsilon_0$, but not the ordering of length $\varepsilon_0$ itself.
</document>

<document source="https://tamnd.github.io/math2/node/C73TGW752D.md" tag="C73TGW752D" kind="concept">
# Sine and cosine addition equations

## Summary

The sine and cosine addition equations are the system of [[functional equations|SRAN4EKWS0]] $S(x + y) = S(x) C(y) + C(x) S(y)$ and $C(x + y) = C(x) C(y) - S(x) S(y)$ for two unknown functions $S, C \colon \R \to \R$.
They turn the angle addition formulas of [[sine and cosine|08WFZP6P89]] into equations.
They are equivalent to Cauchy's exponential equation for the complex-valued function $C + iS$.
So they characterize sine and cosine, up to a scaling of the variable and an exponential factor, among continuous functions.

## Explanations

### L1 human

Rotating by an angle $x$ and then by an angle $y$ is the same as rotating by $x + y$.
Writing this in coordinates gives the addition formulas for sine and cosine.
The equations ask which pairs of functions obey these formulas.
Besides $\sin$ and $\cos$, the pair $S(x) = \sin(2x)$, $C(x) = \cos(2x)$ also works, which corresponds to rotating twice as fast.
</document>

<document source="https://tamnd.github.io/math2/node/C7E5EMV89Z.md" tag="C7E5EMV89Z" kind="definition">
# Sheaf cohomology

## Definition

Let $(X, \mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]] and let $\mathrm{Ab}(X)$ be the category of [[sheaves|TB6TM6R4MD]] of [[abelian groups|082GHNPJ53]] on $X$.
The global sections functor $\Gamma(X, -) \colon \mathrm{Ab}(X) \to \mathrm{Ab}$ is left exact, and $\mathrm{Ab}(X)$ has enough injective objects.
For a sheaf $\mathcal{F}$ in $\mathrm{Ab}(X)$, the *$i$-th cohomology group* $H^i(X, \mathcal{F})$ is the value at $\mathcal{F}$ of the $i$-th right [[derived functor|NPMC3K6J2F]] of $\Gamma(X, -)$.
Concretely, for an [[injective resolution|ZW3W6FX4PW]] $0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \to \cdots$ in $\mathrm{Ab}(X)$,
$$H^i(X, \mathcal{F}) = H^i\big(\Gamma(X, \mathcal{I}^0) \to \Gamma(X, \mathcal{I}^1) \to \cdots\big).$$
For a [[sheaf of $\mathcal{O}_X$-modules|S33B2H3MYD]] $\mathcal{F}$, the groups $H^i(X, \mathcal{F})$ are taken of its underlying sheaf of abelian groups; they are modules over $\Gamma(X, \mathcal{O}_X)$.

## Remarks

The groups $H^i(X, \mathcal{F})$ for an $\mathcal{O}_X$-module $\mathcal{F}$ agree with the derived functors of $\Gamma$ computed in the category of $\mathcal{O}_X$-modules, since injective $\mathcal{O}_X$-modules are flasque and flasque sheaves are acyclic.
This is the same construction as [[sheaf cohomology in topology|KB76923A1M]], applied to the topological space of a scheme.

Example: always $H^0(X, \mathcal{F}) = \mathcal{F}(X)$.
For $X = \mathbb{P}^n_k$ and $\mathcal{F} = \mathcal{O}(d)$, $H^0$ is the space of homogeneous polynomials of degree $d$, $H^n \cong k$ when $d = -n-1$, and $H^i = 0$ for $0 < i < n$.

Non-example: a resolution of $\mathcal{F}$ by sheaves that are not acyclic, for example by arbitrary locally free sheaves, does not in general compute $H^i(X, \mathcal{F})$.
In practice one uses flasque resolutions or, on separated schemes with quasi-coherent sheaves, the Čech complex of an affine cover.
</document>

<document source="https://tamnd.github.io/math2/node/C7E69V030F.md" tag="C7E69V030F" kind="concept">
# Nonlinear regression

## Summary

Nonlinear regression fits a model in which the mean response is a nonlinear function of the unknown parameters.
Examples are an exponential decay or a growth curve.
The parameters are usually estimated by nonlinear [[least squares|0BYTQ4X1HC]], which has no closed formula and is solved by iterative methods such as Gauss–Newton or Levenberg–Marquardt.
Inference relies on a linear approximation of the model near the estimate, so it is approximate and asymptotic.
It is the natural tool when a scientific theory dictates the form of the mean function.

## Explanations

### L1 human

In [[linear regression|41FVC4P5JB]] the unknowns enter as coefficients of known terms, and one formula gives the fit.
In a model like $y = \theta_1 e^{\theta_2 x}$, the unknown $\theta_2$ sits inside the exponential, and no such formula exists.
One starts from a guess, approximates the model by a linear one near that guess, solves that, and repeats.
A polynomial $y = \theta_0 + \theta_1 x + \theta_2 x^2$ looks curved but is still linear regression, because it is linear in the parameters.
</document>

<document source="https://tamnd.github.io/math2/node/C7MZ3SFGYF.md" tag="C7MZ3SFGYF" kind="concept">
# Equivariant cohomology

## Summary

Equivariant cohomology is a cohomology theory for spaces with an action of a topological group $G$.
In the Borel construction, one replaces the space $X$ by the homotopy quotient $EG \times_G X$ and takes its ordinary cohomology.
For a free action of a compact Lie group this is the cohomology of the orbit space.
For the action on a point it is the cohomology of the [[classifying space|WSZ40H5E4T]] $BG$.
It is a main tool for localization theorems, such as the Atiyah–Bott formula, and for the study of fixed points.

## Explanations

### L1 human

When a group acts on a space, the plain quotient often loses information, for example when points have large stabilisers.
The Borel construction first lets the group act freely by pairing each point with a point of a big contractible space on which the group acts freely.
Then it takes the quotient, which is now well behaved, and computes its cohomology.
For the circle group acting on a single point, the result is a polynomial ring in one variable of degree $2$.
Yet the plain quotient is just a point.
</document>

<document source="https://tamnd.github.io/math2/node/C7Q2MVD933.md" tag="C7Q2MVD933" kind="definition">
# Ideal class group

## Definition

Let $R$ be a [[Dedekind domain|7NPY2CN5KB]] with field of fractions $K$.
Let $I(R)$ be the set of nonzero [[fractional ideals|YB4JTB4WJN]] of $R$; under the product of fractional ideals it is an [[abelian group|082GHNPJ53]] with identity $R$.
Let $P(R) = \set{xR : x \in K^{\times}}$ be its subgroup of principal fractional ideals.
The *ideal class group* of $R$ is the [[quotient group|0KYX2N69J0]]
$$\operatorname{Cl}(R) = I(R) / P(R).$$
For a [[number field|MWBDFS2KJ5]] $K$ with ring of integers $\mathcal{O}_K$, the class group of $K$ is $\operatorname{Cl}(K) = \operatorname{Cl}(\mathcal{O}_K)$.

## Remarks

Example: $R = \Z$ is a principal ideal domain, so every fractional ideal is $x\Z$ with $x \in \Q^{\times}$, $I(\Z) = P(\Z)$, and $\operatorname{Cl}(\Z)$ is trivial.
Example: for $R = \Z[\sqrt{-5}]$, let $\mathfrak{p} = (2, 1 + \sqrt{-5})$.
Then $\mathfrak{p}^2$ is generated by $4$, $2 + 2\sqrt{-5}$ and $(1 + \sqrt{-5})^2 = -4 + 2\sqrt{-5}$; all are multiples of $2$ and the ideal contains $4$ and $(2 + 2\sqrt{-5}) - (-4 + 2\sqrt{-5}) = 6$, hence $2$, so $\mathfrak{p}^2 = (2)$.
The ideal $\mathfrak{p}$ is not principal, since $a^2 + 5b^2 = 2$ has no integer solution; so its class has order $2$, and in fact $\operatorname{Cl}(\Z[\sqrt{-5}]) \cong \Z/2\Z$.
Non-example: the nonzero integral ideals of $R$ alone do not form a group under multiplication.
A proper ideal $\mathfrak{a}$ has no integral ideal $\mathfrak{b}$ with $\mathfrak{a}\mathfrak{b} = R$, since $\mathfrak{a}\mathfrak{b} \subseteq \mathfrak{a}$.
This is why fractional ideals are used.
Equivalently, $\operatorname{Cl}(R)$ is the set of nonzero ideals of $R$ modulo the relation $\mathfrak{a} \sim \mathfrak{b}$ if $x\mathfrak{a} = y\mathfrak{b}$ for some nonzero $x, y \in R$.
The class group is trivial if and only if $R$ is a principal ideal domain.
</document>

<document source="https://tamnd.github.io/math2/node/C7WPR8W54K.md" tag="C7WPR8W54K" kind="example">
# Boy's surface

## Example

Boy's surface is an [[immersion|6RW68JGBHC]] of the [[real projective plane|MWDB7RACWQ]] $\R P^2$ in $\R^3$.
That is, it is a smooth map $f \colon \R P^2 \to \R^3$ whose differential is injective at every point.
So the image has a tangent plane everywhere and no pinch points.
The map is not injective: its self-intersection is a closed curve made of three loops, all passing through one triple point.
The picture has a threefold rotational symmetry.
Werner Boy found it in 1901.

## Remarks

The real projective plane is a closed non-orientable [[surface|VGJJVBZ1MQ]], so it has no embedding in $\R^3$; Boy's surface shows that it does have an immersion.
Other classical models of $\R P^2$ in $\R^3$, such as the cross-cap and the Roman surface, have singular points where the differential is not injective, so they are not immersions.
Any generic immersion of a closed surface with odd Euler characteristic in $\R^3$ has an odd number of triple points, and $\chi(\R P^2) = 1$, so one triple point is the least possible.

## Explanations

### L1 human

The projective plane cannot sit in ordinary space without crossing itself.
Boy's surface is a way to place it in space so that it crosses itself only along a curve, with no sharp points or creases.
It looks like a rounded shape with three lobes, whose walls pass through each other along three loops meeting at a single point.
</document>

<document source="https://tamnd.github.io/math2/node/C7X657TF19.md" tag="C7X657TF19" kind="statement">
# Akizuki–Hopkins–Levitzki theorem

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]].
Then $R$ is [[Artinian|ZV04WWYE2Y]] if and only if $R$ is [[Noetherian|0MVH4Y1CRZ]] and every [[prime ideal|06HC6HK0Z8]] of $R$ is [[maximal|09J6KGT65G]].

## Remarks

The condition that every prime ideal is maximal says that $R$ has [[Krull dimension|MVGBS3J1P7]] $0$ when $R$ is not the zero ring.
If $R$ is Artinian, each prime $P$ gives an Artinian domain $R/P$, which is a field, so $P$ is maximal.
Then $R$ has finitely many maximal ideals $\mathfrak{m}_1, \dots, \mathfrak{m}_n$, and $(\mathfrak{m}_1 \cdots \mathfrak{m}_n)^k = 0$ for some $k$.
The chain $R \supseteq \mathfrak{m}_1 \supseteq \mathfrak{m}_1 \mathfrak{m}_2 \supseteq \cdots \supseteq 0$ has factors that are vector spaces over fields $R/\mathfrak{m}_i$, and for vector spaces the two chain conditions agree, so $R$ has finite [[length|CZSCNZH8VY]] and is Noetherian.
For example, $\Z/12\Z$ is Artinian and Noetherian, while $\Z$ is Noetherian but has the non-maximal prime $(0)$, so it is not Artinian.
For noncommutative rings, the Hopkins–Levitzki theorem says that every left Artinian ring is left Noetherian, while the converse needs extra hypotheses.

## Explanations

### L1 human

For commutative rings, the decreasing chain condition is stronger than the increasing one.
The theorem says exactly how much stronger: Artinian means Noetherian plus "no prime ideal sits strictly inside another".

### L2 human

The theorem shows that Artinian rings are the Noetherian rings of dimension $0$, and it is often used together with the [[structure theorem for Artinian rings|HE7MP4HP6Q]].
A typical use: if $R$ is a Noetherian local ring and $I$ is an ideal whose radical is the maximal ideal, then $R/I$ is Artinian, so it has finite length.
This fact is the start of dimension theory and of the [[Krull–Akizuki theorem|FSXA3CF365]].
</document>

<document source="https://tamnd.github.io/math2/node/C7YWCR87QJ.md" tag="C7YWCR87QJ" kind="concept">
# Paraconsistent logic

## Summary

A paraconsistent logic is a logic in which a contradiction does not imply every statement.
In classical and [[intuitionistic logic|Q11N8X99FK]], the principle of explosion derives any $\psi$ from $\varphi$ and $\neg\varphi$; paraconsistent logics reject it.
They allow reasoning with inconsistent information, such as a database with conflicting entries, without collapse.

## Explanations

### L1 human

In ordinary logic, one contradiction ruins everything: from "$p$" and "not $p$" you can prove anything at all.
A paraconsistent logic blocks this, so a contradiction stays local.
Suppose a database says both "the train leaves at 9" and "the train does not leave at 9".
It should not let you conclude that the moon is made of cheese.
</document>

<document source="https://tamnd.github.io/math2/node/C81JPZ51NA.md" tag="C81JPZ51NA" kind="concept">
# Laman graph

## Summary

A Laman graph is a [[graph|00C0BR7D7H]] with $2n - 3$ edges on $n$ vertices in which no set of $k \ge 2$ vertices spans more than $2k - 3$ edges.
These are exactly the graphs that are minimally rigid in the plane for generic positions of the vertices.
So they reduce a question of [[rigidity of frameworks|V5YS9BZX0N]] in the plane to a count of edges, which can be checked by efficient algorithms.

## Explanations

### L1 human

A point in the plane has two degrees of freedom, and a rigid shape in the plane can still slide and turn in three ways.
So $n$ points need $2n - 3$ bars to be held rigid, and no part may waste bars on a piece that is already rigid.
A triangle has $3$ vertices and $3 = 2 \cdot 3 - 3$ edges, and it is a Laman graph.
The complete graph on $4$ vertices has $6 > 5$ edges, so it has one bar too many.
</document>

<document source="https://tamnd.github.io/math2/node/C81RAXD6DV.md" tag="C81RAXD6DV" kind="concept">
# Tutte polynomial

## Summary

The Tutte polynomial $T(x, y)$ is a polynomial in two variables attached to a graph or, more generally, to a [[matroid|PX31ZNYS3Z]].
It is the most general invariant that satisfies the deletion–contraction recurrence, and many counting invariants are specializations of it.
Among them are the number of [[spanning trees|0S4ZEZM7KE]], the [[chromatic polynomial|0B621RBVS9]], the flow polynomial and the reliability polynomial.
The partition function of the Potts model in statistical physics is another specialization.

## Explanations

### L1 human

Many questions about a graph can be answered by removing one edge or squeezing one edge to a point and combining the answers for the two smaller graphs.
The Tutte polynomial records all such answers at once.
Plugging in particular numbers gives particular counts: for example, $T(1, 1)$ is the number of spanning trees of a connected graph.
For a triangle, $T(x, y) = x^2 + x + y$, and $T(1, 1) = 3$ is its number of spanning trees.
</document>

<document source="https://tamnd.github.io/math2/node/C82AXZK2BS.md" tag="C82AXZK2BS" kind="statement">
# Hochster's theorem on spectral spaces

## Statement

A [[topological space|0CQHFPGJ45]] $X$ is *spectral* if:

1. $X$ is quasi-compact;
2. the quasi-compact open subsets of $X$ form a basis of the topology and are closed under finite intersections;
3. $X$ is sober: every irreducible [[closed subset|0ZCK8DQYN1]] $Z$ of $X$ has exactly one point $z$ whose closure is $Z$.

Here a closed set is irreducible if it is nonempty and not the union of two closed proper subsets.

Hochster's theorem: a topological space $X$ is spectral if and only if it is homeomorphic to $\operatorname{Spec} R$ for some [[commutative ring|0310REJJY8]] $R$.
Here $\operatorname{Spec} R$ is the [[spectrum|FB45BHHA2J]] with the [[Zariski topology|8SQK7RTB63]].

## Remarks

One direction is a direct check.
$\operatorname{Spec} R$ is [[quasi-compact|J4YW23MD6N]], the [[basic open sets|N6JX9BCKQP]] $D(f) \cong \operatorname{Spec} R_f$ are quasi-compact, and a quasi-compact open set is a finite union of them, so finite intersections stay quasi-compact by $D(f) \cap D(g) = D(fg)$.
An irreducible closed set is $V(\mathfrak p)$ for a unique prime $\mathfrak p$, which is its generic point.
The other direction, building a ring from a spectral space, is the hard part of the theorem.

Example: a finite $T_0$ space is spectral, so it is the spectrum of some ring.
Non-example: the real line with its usual topology is not spectral; it is not quasi-compact.
A Hausdorff spectral space is a Stone space, and every Stone space is the spectrum of a Boolean ring.
An infinite Stone space is never the spectrum of a Noetherian ring.

## Explanations

### L1 human

The theorem says exactly which spaces arise as spaces of prime ideals.
There are three conditions, all of them about finiteness and about "generic points".
Any space with these properties, however strange, comes from some ring.

### L2 human

The conditions make the class of spectra purely topological, with no reference to rings.
To show that a space is homeomorphic to a spectrum, one may check the three conditions instead of finding a ring.
Spectral spaces also appear in logic and lattice theory, as the spaces dual to bounded distributive lattices.
</document>

<document source="https://tamnd.github.io/math2/node/C82WDEQ3DC.md" tag="C82WDEQ3DC" kind="statement">
# Asymptotic behavior of solutions of renewal equations

## Statement

Let $F$ be a [[cumulative distribution function|0H8H7ZTHW2]] with $F(0) = 0$, so that $F$ is the law of a positive random variable.
Assume that $F$ is not arithmetic, that is, $F$ is not concentrated on a set $\set{0, h, 2h, \dots}$ for any $h > 0$.
Let $\mu = \int_{[0, \infty)} t \, dF(t)$ be its mean, with $0 < \mu \le \infty$.
Let $g \colon [0, \infty) \to \R$ be directly Riemann integrable.
This means that, for the partition of $[0, \infty)$ into intervals of length $h$, the upper and lower Riemann sums of $g$ are finite.
It also means that they tend to the same limit as $h \to 0$.
Consider the renewal equation
$$u(t) = g(t) + \int_{[0, t]} u(t - s) \, dF(s), \qquad t \ge 0,$$
where the integral is a [[Lebesgue–Stieltjes integral|FK3QBZAZH7]].
Then:

1. The equation has exactly one solution $u$ that is bounded on bounded intervals, namely $u(t) = \int_{[0, t]} g(t - s) \, dU(s)$. Here $U = \sum_{n \ge 0} F^{*n}$, where $F^{*n}$ is the $n$-fold convolution power of $F$ and $F^{*0}$ is the unit mass at $0$.
2. As $t \to \infty$,
$$u(t) \to \frac{1}{\mu} \int_0^\infty g(s) \, ds,$$
where $1/\mu$ is read as $0$ when $\mu = \infty$.

## Remarks

Part 1 follows by iterating the equation: the series of convolution powers converges because $F^{*n}(t) \to 0$ fast for each fixed $t$.
Part 2 is the key renewal theorem.
Its proof first treats $g$ equal to the indicator function of $[0, h)$, which is Blackwell's renewal theorem.
It then approximates a directly Riemann integrable $g$ from above and below by step functions.
A nonnegative, nonincreasing, Lebesgue integrable function on $[0, \infty)$ is directly Riemann integrable.

Example: for $g = 1 - F$, the constant $u = 1$ solves the equation, since $1 - F(t) + F(t) = 1$.
The limit in part 2 is $\frac{1}{\mu} \int_0^\infty (1 - F(s)) \, ds = \mu / \mu = 1$, as it must be.

The hypothesis that $F$ is not arithmetic is needed.
If $F$ is the unit mass at $1$, then $U$ puts mass $1$ on each integer $n \ge 0$.
For $g$ the indicator function of $[0, 1/2)$, the solution is $u(t) = 1$ when the fractional part of $t$ is less than $1/2$ and $u(t) = 0$ otherwise, so $u(t)$ has no limit.

## Explanations

### L1 human

A renewal equation describes a quantity that is fed by a source $g$ and by its own past, where the past is weighted by a waiting time law $F$.
The theorem says that after a long time the quantity settles down to a constant.
The constant is the total size of the source divided by the mean waiting time $\mu$.
For example, the expected number of light bulb replacements in a time window of length $h$ tends to $h / \mu$.

### L2 human

In [[renewal theory|45V0W838BQ]], $U(t)$ is the expected number of renewals in $[0, t]$, counting the start.
Many quantities of a renewal process, such as the law of the residual waiting time at time $t$, satisfy a renewal equation with an explicit $g$.
The theorem then gives their limits as $t \to \infty$.
The same method describes the long-time behaviour of age-structured populations through the [[renewal equation|EWVVH3EKMR]] and the [[Euler–Lotka equation|E0YVMBPK7A]], after a change of variable that makes $F$ a probability distribution.
From the point of view of [[Volterra integral equations|DND8CBEVXT]], it is a statement about the [[Volterra equation of convolution type|SNFYTC1TTA]] whose kernel is a probability measure.
</document>

<document source="https://tamnd.github.io/math2/node/C831AN3RFM.md" tag="C831AN3RFM" kind="concept">
# Link of a singularity

## Summary

The link of a singular point of a complex analytic set is its intersection with a small sphere centred at the point, see the [[definition|MHGZEW6RBM]].
Near the point, the set is homeomorphic to the cone over its link, so the link carries the local topology of the singularity.
For an [[isolated hypersurface singularity|VTGYEES3VS]] in $\C^{n+1}$ the link is a smooth closed manifold of dimension $2n - 1$.
For plane curves it is a [[link|1A5WRCNTM3]] in the knot theory sense, a disjoint union of circles in $S^3$.

## Explanations

### L1 human

To see what a singular point looks like, cut the space with a tiny sphere around it and look at what is left on the sphere.
The space near the point is then the cone over this cut, made of rays from the point to the cut.
For the curve $x^2 = y^3$ in $\C^2$, the cut is a knotted circle in the $3$-sphere, the trefoil knot.
For two lines crossing at a point, it is two circles that link once, the Hopf link.
</document>

<document source="https://tamnd.github.io/math2/node/C83KPT47R0.md" tag="C83KPT47R0" kind="concept">
# Gröbner basis

## Summary

Fix a [[monomial order|E874NARWPV]] on a polynomial ring.
A Gröbner basis of an ideal is a finite generating set whose leading terms generate the ideal of all leading terms of the ideal.
With a Gröbner basis, the [[multivariate division algorithm|T8EN3CAEQF]] has a unique remainder.
So one can decide whether a polynomial lies in the ideal, and one can compute in the quotient ring.
Every ideal has a Gröbner basis, and [[Buchberger's algorithm|4RH1D40WG1]] computes one.
Gröbner bases are the basic tool of computational commutative algebra and algebraic geometry, including [[elimination theory|9PA8CVSWPD]].

## Explanations

### L1 human

For polynomials in one variable, an ideal is generated by one polynomial, and division by it decides membership.
In several variables, a set of generators may be badly chosen: a polynomial in the ideal can have a leading term that none of the generators' leading terms divides.
A Gröbner basis is a set of generators without this defect.
For example, $\set{x + y, x}$ generates the ideal $(x, y)$.
But $y = x - (x + y)$ is in the ideal, and for the order with $x > y$ its leading term $y$ is not divisible by $x$.
So this set is not a Gröbner basis, while the set $\set{x, y}$ is.
</document>

<document source="https://tamnd.github.io/math2/node/C846XS3Y0T.md" tag="C846XS3Y0T" kind="statement">
# KAK decomposition

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center and Lie algebra $\mathfrak{g}$.
Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be a Cartan decomposition, as in the [[Cartan decomposition of a Lie group|A5JB9JQKR0]].
Let $K$ be the connected subgroup with Lie algebra $\mathfrak{k}$.
Let $\mathfrak{a} \subseteq \mathfrak{p}$ be a maximal abelian subspace, and let $A = \exp \mathfrak{a}$.
Let $N_K(\mathfrak{a})$ and $Z_K(\mathfrak{a})$ be the normalizer and the centralizer of $\mathfrak{a}$ in $K$ under the adjoint action.
Let $W = N_K(\mathfrak{a}) / Z_K(\mathfrak{a})$.
Then $G = KAK$: each $g \in G$ can be written $g = k_1 \exp(H) k_2$ with $k_1, k_2 \in K$ and $H \in \mathfrak{a}$.
In such a product, $H$ is unique up to the action of $W$ on $\mathfrak{a}$.

## Remarks

The idea of the proof: by the Cartan decomposition, $g = k \exp X$ with $X \in \mathfrak{p}$.
Each element of $\mathfrak{p}$ is conjugate under $\mathrm{Ad}(K)$ to an element of $\mathfrak{a}$, so $X = \mathrm{Ad}(k_2^{-1}) H$ and $\exp X = k_2^{-1} \exp(H) k_2$.
Uniqueness comes from the fact that two elements of $\mathfrak{a}$ conjugate under $\mathrm{Ad}(K)$ are conjugate under the Weyl group $W$.

Example: $G = \mathrm{SL}_n(\R)$, $K = \mathrm{SO}(n)$, $A$ the positive diagonal matrices of determinant $1$.
The theorem is the singular value decomposition $g = k_1 \operatorname{diag}(\sigma_1, \dots, \sigma_n) k_2$.
Here $W$ is the symmetric group $S_n$, and the singular values are unique up to order.

The factors $k_1$ and $k_2$ are not unique: for $g = e$ one may take $k_1 = k$, $H = 0$, $k_2 = k^{-1}$ for any $k \in K$.

## Explanations

### L1 human

Every element of a semisimple Lie group is a rotation, then a stretching along fixed axes, then another rotation.
For matrices this is the singular value decomposition.
The amounts of stretching are well defined up to the order of the axes.

### L2 human

The decomposition is the group form of the fact that $K$ acts on the symmetric space $G/K$ and each orbit meets the flat $A \cdot o$.
It gives polar coordinates on $G$.
In these coordinates the Haar measure has a density in $H$, a product of powers of $\abs{\sinh \lambda(H)}$ over the positive restricted roots $\lambda$.
This is the starting point of harmonic analysis on $G/K$ and of the study of spherical functions.
</document>

<document source="https://tamnd.github.io/math2/node/C84CHVTZ4J.md" tag="C84CHVTZ4J" kind="definition">
# Injective resolution

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $X$ be an object of $\mathcal{A}$.
An *injective resolution* of $X$ is a sequence of objects and morphisms
$$0 \to X \xrightarrow{\eta} I^0 \xrightarrow{d^0} I^1 \xrightarrow{d^1} I^2 \to \cdots$$
such that:

1. every $I^n$ is an [[injective object|5QRFTBF4PG]];
2. the sequence is [[exact|JB7ZBFBA9C]].

## Remarks

Example: over $\Z$, the sequence $0 \to \Z \to \Q \to \Q/\Z \to 0$ is an injective resolution of $\Z$.
It is exact, and $\Q$ and $\Q/\Z$ are divisible groups, which are [[injective modules|85H7PRHDPZ]] over $\Z$.
Non-example: $0 \to \Z \xrightarrow{\cdot 2} \Z \to \Z/2\Z \to 0$ is exact, but $\Z$ is not an injective $\Z$-module, since it is not divisible, so clause 1 fails.
One writes the resolution as the cochain complex $I^\bullet$ with the map $\eta$; then $H^0(I^\bullet) \cong X$ and $H^n(I^\bullet) = 0$ for $n > 0$.
If $\mathcal{A}$ has enough injectives, every object has an injective resolution; this holds for modules over any ring, since every module embeds into an injective module.
</document>

<document source="https://tamnd.github.io/math2/node/C87X3ACS76.md" tag="C87X3ACS76" kind="definition">
# Ramanujan summation

## Definition

Let $f$ be a [[smooth function|0BE3QDH73A]] on an open interval that contains $[0, \infty)$, and let $B_{2k}$ be the [[Bernoulli numbers|0RAS24E1R5]], with $B_2 = \frac16$.
Assume that the series $\sum_{k \ge 1} \frac{B_{2k}}{(2k)!} f^{(2k-1)}(0)$ converges.
The *Ramanujan sum* of the series $\sum_{n \ge 1} f(n)$ is
$$\sum_{n \ge 1}^{\mathcal{R}} f(n) = -\frac{f(0)}{2} - \sum_{k \ge 1} \frac{B_{2k}}{(2k)!} f^{(2k-1)}(0).$$

## Remarks

If $f$ is a polynomial, only finitely many derivatives are nonzero and the series in the definition is a finite sum.

Example: for $f(x) = x$, one has $f(0) = 0$, $f'(0) = 1$ and all higher derivatives vanish.
So the Ramanujan sum of $1 + 2 + 3 + \cdots$ is $-\frac{B_2}{2} = -\frac{1}{12}$.
For $f(x) = 1$, the Ramanujan sum of $1 + 1 + 1 + \cdots$ is $-\frac12$.

Not a regular method: for $f(x) = e^{-x}$ the series $\sum_{n \ge 1} e^{-n}$ converges to $\frac{1}{e - 1}$.
From $\frac{t}{e^t - 1} = 1 - \frac{t}{2} + \sum_{k \ge 1} \frac{B_{2k}}{(2k)!} t^{2k}$ at $t = 1$ one gets $\sum_{k \ge 1} \frac{B_{2k}}{(2k)!} = \frac{1}{e-1} - \frac12$.
Since $f^{(2k-1)}(0) = -1$, the Ramanujan sum is $-\frac12 + \frac{1}{e-1} - \frac12 = \frac{1}{e-1} - 1$.
It differs from the ordinary sum by $\int_0^\infty e^{-x} \, dx = 1$.

Not stable: for $g(x) = f(x + 1)$ with $f = 1$, the Ramanujan sum of $\sum g(n)$ is $-\frac12$, not $-\frac12 - f(1) = -\frac32$.
</document>

<document source="https://tamnd.github.io/math2/node/C8E0TWY3WT.md" tag="C8E0TWY3WT" kind="concept">
# Liouville's equation

## Summary

Liouville's equation is the [[semilinear|RCNY89A4NY]] elliptic equation $\Delta u + K e^{2u} = 0$ for a function $u$ of two real variables, with $K$ a constant.
It says that the metric $e^{2u}(dx^2 + dy^2)$ has constant [[Gaussian curvature|SQVQ5DQT6W]] $K$.
Joseph Liouville found its general local solution in terms of holomorphic functions.
Equations of this exponential type also appear in models of self-gravitating gases and of vortices.

## Explanations

### L1 human

Take a flat piece of the plane and rescale lengths by a factor $e^{u}$ that changes from point to point.
The rescaled surface can look curved, like a piece of a sphere, and its curvature depends on how $u$ varies.
Liouville's equation asks for the rescalings that give the same curvature everywhere.
For example, $u = \log\big(2/(1 + x^2 + y^2)\big)$ turns the plane into the round sphere of radius one with a point removed, which has curvature $1$.
</document>

<document source="https://tamnd.github.io/math2/node/C8HD62AB6Q.md" tag="C8HD62AB6Q" kind="concept">
# Monodromy

## Summary

Monodromy describes how a multivalued object changes when it is continued around a loop.
Take a linear differential equation with singular points, and continue a basis of solutions around one of them.
The result is a new basis, related to the old one by an invertible matrix.
These matrices form a representation of the fundamental group of the plane minus the singular points, the monodromy representation.
It is a global invariant of the equation, and it is the subject of [[Hilbert's twenty-first problem|7RHW0KA5RY]].
See the [[definition|SK224Y1HNG]] and the related [[monodromy theorem|54QMKJF64A]].

## Explanations

### L1 human

The function $\sqrt{z}$ changes sign when one goes once around $0$.
The function $\log z$ gains $2 \pi i$.
These are the simplest cases of monodromy: walking around a singular point and coming back changes the value.
For a differential equation, the solutions are mixed among themselves in the same way, and a matrix records the mixing.
</document>

<document source="https://tamnd.github.io/math2/node/C8N3ER51WT.md" tag="C8N3ER51WT" kind="definition">
# Borel–Moore homology

## Definition

Let $X$ be a [[locally compact|0FE2SB8WVJ]] [[Hausdorff space|0S1FJ8VEX9]] and let $A$ be an [[abelian group|082GHNPJ53]].
A *locally finite singular $i$-chain* on $X$ is a formal sum $c = \sum_\sigma a_\sigma \sigma$, possibly infinite, over singular $i$-simplices $\sigma \colon \Delta^i \to X$ with $a_\sigma \in A$.
It is required that every point of $X$ has a neighbourhood that meets the image of only finitely many $\sigma$ with $a_\sigma \neq 0$.
These chains form a group $C^{\mathrm{lf}}_i(X; A)$, and the usual boundary formula of [[singular homology|M0Q5CFJ7ZH]] defines $\partial \colon C^{\mathrm{lf}}_i(X; A) \to C^{\mathrm{lf}}_{i-1}(X; A)$ with $\partial \circ \partial = 0$.
The *Borel–Moore homology* of $X$ is the homology of this [[chain complex|WQ82C0M3QB]]:
$$H^{\mathrm{BM}}_i(X; A) = H_i(C^{\mathrm{lf}}_\bullet(X; A)).$$

## Remarks

The boundary of a locally finite chain is locally finite, because each face of $\sigma$ has image inside the image of $\sigma$.
If $X$ is compact, a locally finite chain is a finite chain, so $H^{\mathrm{BM}}_i(X; A) = H_i(X; A)$.
Some authors define the theory with sheaves.
If $X = Y \setminus Z$ for a finite CW complex $Y$ and a subcomplex $Z$, then $H^{\mathrm{BM}}_i(X; A) \cong H_i(Y, Z; A)$.

Example: on $X = \R$, the chain $c = \sum_{k \in \Z} \sigma_k$, where $\sigma_k$ is the affine path from $k$ to $k+1$, is locally finite, and $\partial c = \sum_k (\set{k+1} - \set{k}) = 0$.
Its class generates $H^{\mathrm{BM}}_1(\R; \Z) \cong \Z$, while $H^{\mathrm{BM}}_0(\R; \Z) = 0$.

Non-example: the infinite sum $\sum_{k \ge 1} \tau_k$, where $\tau_k$ is the affine path from $0$ to $1/k$ in $\R$, is not locally finite: every neighbourhood of $0$ meets all the $\tau_k$.

For an oriented $n$-manifold $M$, Poincaré duality takes the form $H^{\mathrm{BM}}_i(M; \Z) \cong H^{n-i}(M; \Z)$.
</document>

<document source="https://tamnd.github.io/math2/node/C8S7XX0VZM.md" tag="C8S7XX0VZM" kind="statement">
# Bairstow's method

## Statement

Let $p$ be a polynomial of degree $n \ge 3$ with real coefficients, and let $u, v$ be real numbers.
By [[polynomial long division|D6D0SR1FK3]] by $B(x) = x^2 + ux + v$, write
$$p(x) = B(x) \, q(x) + c x + d, \qquad q(x) = B(x) \, s(x) + g x + h,$$
where $q$ and $s$ are polynomials and $c, d, g, h$ are real numbers that depend on $(u, v)$.
Then $c$ and $d$ are polynomial functions of $(u, v)$ with partial derivatives
$$\frac{\partial c}{\partial u} = ug - h, \quad \frac{\partial c}{\partial v} = -g, \quad \frac{\partial d}{\partial u} = vg, \quad \frac{\partial d}{\partial v} = -h.$$
The step of *Bairstow's method* is
$$\begin{pmatrix} u \\ v \end{pmatrix} \mapsto \begin{pmatrix} u \\ v \end{pmatrix} - \frac{1}{vg^2 + h(h - ug)} \begin{pmatrix} -h & g \\ -vg & ug - h \end{pmatrix} \begin{pmatrix} c \\ d \end{pmatrix}.$$
It is one step of [[Newton's method|07EXVNF9VP]] for the system $c(u, v) = 0$, $d(u, v) = 0$.
The solutions of this system are the pairs $(u, v)$ for which $x^2 + ux + v$ divides $p$.

## Remarks

Differentiate $p = Bq + cx + d$ with respect to $v$: $0 = q + B \, \partial_v q + (\partial_v c) x + \partial_v d$.
So $q$ leaves the remainder $-(\partial_v c) x - \partial_v d$ on division by $B$, which gives $\partial_v c = -g$ and $\partial_v d = -h$.
Differentiating with respect to $u$ gives that $x q(x)$ leaves the remainder $-(\partial_u c) x - \partial_u d$; since $x(gx + h) \equiv (h - ug) x - vg$ modulo $B$, this gives the other two derivatives.
The $2 \times 2$ matrix in the step is the inverse of the [[Jacobian matrix|047NDTEPCV]] of $(c, d)$, times its determinant $vg^2 + h(h - ug)$.
So, when this Jacobian is invertible at a quadratic factor, the method converges quadratically from nearby starting values.
A quadratic factor gives two [[roots|019MA1VTR2]] of $p$ by the quadratic formula, which may be a pair of complex conjugate roots, all in real arithmetic.

## Explanations

### L1 human

A real polynomial can have complex roots, which come in conjugate pairs.
Each pair is the pair of roots of a real quadratic factor $x^2 + ux + v$.
Bairstow's method searches for such a factor directly, by adjusting $u$ and $v$ until the division of $p$ by $x^2 + ux + v$ leaves no remainder.

### L2 human

All the work is two synthetic divisions per step, so it is cheap.
After a factor is found, one divides it out and repeats on the quotient of degree $n - 2$.
Like Newton's method, it needs good starting values and converges only linearly at multiple factors.
</document>

<document source="https://tamnd.github.io/math2/node/C8SEGT1AZS.md" tag="C8SEGT1AZS" kind="example">
# Weak order of permutations

## Example

Let $n \ge 1$ and let $S_n$ be the [[symmetric group|0515A6D2Y1]] of [[permutations|05RWFKVTJ7]] of $\set{1, \ldots, n}$, written in one-line notation $w = w(1) w(2) \cdots w(n)$.
The inversion set $\mathrm{Inv}(w)$ is the set of pairs of values $(a, b)$ with $a < b$ such that $b$ stands left of $a$ in $w$.
The *weak order* on $S_n$ is defined by $u \le w$ when $\mathrm{Inv}(u) \subseteq \mathrm{Inv}(w)$.
It is a partial order, and it is a [[lattice|WH4SFAACAS]].
The covering relations are the moves that swap two adjacent entries $a < b$ standing as $\ldots a b \ldots$.
Such a move adds the single pair $(a, b)$ to the inversion set.
The least element is the identity $12 \cdots n$, and the greatest element is the reversal $n \cdots 21$.

Check for $n = 3$.
The inversion sets are $\mathrm{Inv}(123) = \emptyset$, $\mathrm{Inv}(213) = \set{(1,2)}$, $\mathrm{Inv}(132) = \set{(2,3)}$, $\mathrm{Inv}(231) = \set{(1,2), (1,3)}$, $\mathrm{Inv}(312) = \set{(1,3), (2,3)}$, and $\mathrm{Inv}(321)$ has all three pairs.
So the weak order is the two chains $123 < 213 < 231 < 321$ and $123 < 132 < 312 < 321$.
The elements $213$ and $312$ are incomparable, since $(1,2) \notin \mathrm{Inv}(312)$.

## Remarks

The weak order is graded by the number of inversions.
Its Hasse diagram for $S_3$ is a hexagon.
The Bruhat order has more relations: there $213 < 312$.

## Explanations

### L1 human

Start with the numbers in order, $1 2 \cdots n$, and swap neighbours that are in the right order, one swap at a time.
Each swap creates one new out-of-order pair.
A permutation is below another when the pairs it has out of order are among those of the other.
For three numbers you get a hexagon, from $123$ at the bottom to $321$ at the top.
</document>

<document source="https://tamnd.github.io/math2/node/C8TTRMVVS7.md" tag="C8TTRMVVS7" kind="problem">
# Kadison–Singer problem

## Problem

Let $H = \ell^2(\N)$ be the [[Hilbert space|HM2HHE3J51]] of square-summable sequences, and let $B(H)$ be the C*-algebra of bounded operators on $H$.
Let $D \subseteq B(H)$ be the C\*-subalgebra of diagonal operators with respect to the standard basis, which is isomorphic to $\ell^\infty(\N)$.
A pure [[state|F7MB2WREBJ]] on a C\*-algebra is a state that is not a proper convex combination of two other states.
Does every pure state on $D$ extend to a unique state on $B(H)$?

## Remarks

Every state on $D$ extends to some state on $B(H)$ by the Hahn–Banach theorem, so the question is about uniqueness.
Kadison and Singer posed the question in their 1959 paper "Extensions of pure states".
Over the following decades it was shown to be equivalent to many problems in other fields, among them Anderson's paving conjecture, the Feichtinger conjecture on frames, and Weaver's discrepancy conjecture.
Adam Marcus, Daniel Spielman and Nikhil Srivastava proved Weaver's conjecture, and so solved the problem with a positive answer; their proof was posted in 2013 and published in 2015.
Their method uses interlacing families of polynomials.

## Explanations

### L1 human

In quantum mechanics, a state assigns expected values to observables.
The question asks: if one knows the expected values of all diagonal observables in a very sharp way, does that determine the expected values of all observables?
The answer turned out to be yes.
The solution came from an unexpected direction, through a result about sums of random rank-one matrices and the roots of polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/C8VE2SEVEB.md" tag="C8VE2SEVEB" kind="definition">
# Archimedean equivalence classes

## Definition

Let $(G, +, \le)$ be a [[linearly ordered group|1P8087YGYQ]] that is an [[abelian group|071DDFVG4J]].
For $a \in G$, let $\abs{a} = \max(a, -a)$.
Two elements $a, b \in G$ are *Archimedean equivalent*, written $a \sim b$, if there are integers $m, n \ge 1$ with
$$\abs{a} \le n \abs{b} \quad \text{and} \quad \abs{b} \le m \abs{a}.$$
This is an equivalence relation on $G$.
Its classes are the *Archimedean classes* of $G$, and the class of $0$ is $\set{0}$.
For nonzero $a, b$ with $a \not\sim b$, one says that $a$ is *infinitely larger* than $b$ if $n \abs{b} < \abs{a}$ for all $n \ge 1$.

## Remarks

Exactly one of $a \sim b$, "$a$ is infinitely larger than $b$" and "$b$ is infinitely larger than $a$" holds, so the classes of nonzero elements are totally ordered by size.
Example: in $\Z^2$ with the lexicographic order, $(0, 2) \sim (0, 3)$, since $\abs{(0, 2)} \le 1 \cdot (0, 3)$ and $(0, 3) \le 2 \cdot (0, 2)$.
Non-example: $(1, 0) \not\sim (0, 1)$, since $n \cdot (0, 1) = (0, n) < (1, 0)$ for all $n$, so no multiple of $(0, 1)$ bounds $(1, 0)$.
So $\Z^2$ with this order has exactly two classes of nonzero elements, those with first coordinate $0$ and the others.
The group $G$ is Archimedean if and only if all its nonzero elements are equivalent.
Conventions on the direction of the order of classes differ between texts.
</document>

<document source="https://tamnd.github.io/math2/node/C8W7J0S73V.md" tag="C8W7J0S73V" kind="definition">
# Local time

## Definition

Let $(B_t)_{t \ge 0}$ be a standard one-dimensional [[Brownian motion|22735FGJV6]] on a [[probability space|0NFPJVJJHX]] $(\Omega, \mathcal{F}, P)$.
The *local time* of $B$ is a random field $L = (L(t, x))_{t \ge 0, \, x \in \R}$ such that, almost surely:

1. the map $(t, x) \mapsto L(t, x)$ is continuous;
2. for every $t \ge 0$ and every bounded Borel function $f \colon \R \to \R$,
$$\int_0^t f(B_s) \, ds = \int_{\R} f(x) \, L(t, x) \, dx.$$

Such a field exists, and it is unique up to indistinguishability.
For each $t$ and $x$ it satisfies, almost surely,
$$L(t, x) = \lim_{\varepsilon \downarrow 0} \frac{1}{2\varepsilon} \int_0^t \mathbf{1}_{\set{\abs{B_s - x} < \varepsilon}} \, ds.$$

## Remarks

Example: $L(0, x) = 0$ for all $x$, and $E[L(t, 0)] = \int_0^t (2\pi s)^{-1/2} \, ds = \sqrt{2t/\pi}$, because the density of $B_s$ at $0$ is $(2\pi s)^{-1/2}$.
Non-example: the field $\int_0^t \mathbf{1}_{\set{B_s = x}} \, ds$ is almost surely $0$ for every $x$, so it fails clause 2 with $f = 1$ and $t > 0$.
For fixed $x$, the map $t \mapsto L(t, x)$ is nondecreasing and increases only on the set of times $t$ with $B_t = x$.
The notations $L_t^x$ and $\ell_t(x)$ are also common.
With this normalisation, $L(t, 0)$ has the same law as $\max_{s \le t} B_s$.
</document>

<document source="https://tamnd.github.io/math2/node/C8XY22X5NP.md" tag="C8XY22X5NP" kind="statement">
# Müntz–Szász theorem

## Statement

Let $0 < \lambda_1 < \lambda_2 < \cdots$ be real numbers with $\lambda_n \to \infty$.
Let $C[0, 1]$ be the space of [[continuous functions|0X3WGX95FE]] $[0, 1] \to \R$ with the supremum norm.
Then the linear span of the functions $1, x^{\lambda_1}, x^{\lambda_2}, \dots$ is [[dense|0BSJSX7ME2]] in $C[0, 1]$ if and only if
$$\sum_{n=1}^{\infty} \frac{1}{\lambda_n} = \infty.$$

## Remarks

The idea of the proof: one first works in $L^2[0, 1]$.
There the distance from $x^m$ to the span of finitely many powers $x^{\lambda_k}$ can be computed exactly with Gram determinants, which here are Cauchy determinants.
The distance is a product of factors $\abs{m - \lambda_k} / (m + \lambda_k + 1)$ times a constant, and this product tends to $0$ exactly when $\sum 1/\lambda_k$ diverges.
A further argument passes from $L^2$ to [[uniform approximation|XAARVNKXF1]] of the monomials $x^m$, and then the Weierstrass theorem finishes the proof.

Example: the exponents $\lambda_n = n$ give the ordinary polynomials, and $\sum 1/n = \infty$, which recovers the Weierstrass approximation theorem on $[0, 1]$.
The primes also work, since $\sum 1/p = \infty$, so combinations of $1$ and $x^p$ for primes $p$ are dense.
The squares do not: $\sum 1/n^2 < \infty$, so the combinations of $1, x, x^4, x^9, \dots$ are not dense in $C[0, 1]$.

The constant function $1$ is needed in the span: every $x^{\lambda_n}$ vanishes at $0$, so without $1$ no function with $f(0) \ne 0$ could be approximated.

## Explanations

### L1 human

Polynomials can approximate every continuous function on $[0, 1]$.
The theorem asks what happens if you may use only some of the powers of $x$.
The answer is a single counting test: you need enough powers, in the sense that the sum of the reciprocals of the exponents is infinite.

### L2 human

The theorem sharpens the Weierstrass theorem and shows how much freedom there is in choosing building blocks.
It is a classic application of complex analysis and Hilbert space methods to approximation; a common proof uses the zeros of bounded analytic functions in a half-plane.
When the series converges, the proof shows more: no power $x^\lambda$ with $\lambda > 0$ outside the set $\set{\lambda_n}$ lies in the closure of the span.
</document>

<document source="https://tamnd.github.io/math2/node/C8ZPFCX1SJ.md" tag="C8ZPFCX1SJ" kind="definition">
# Free Boolean algebra

## Definition

Let $S$ be a set.
Let $F$ be a [[Boolean algebra|0XADJKM699]] and let $\iota \colon S \to F$ be a map.
The pair $(F, \iota)$ is a *free Boolean algebra* on $S$ if the following holds.
For every Boolean algebra $B$ and every map $g \colon S \to B$, there is exactly one homomorphism $\bar g \colon F \to B$ with $\bar g \circ \iota = g$.

## Remarks

It exists for every $S$ and is unique up to a unique isomorphism compatible with $\iota$.
For finite $S = \set{1, \ldots, n}$, one model is the algebra of all [[Boolean functions|0X05GS525E]] $\set{0, 1}^n \to \set{0, 1}$, with $\iota(i)$ the $i$-th coordinate function; it has $2^{2^n}$ elements and $2^n$ atoms.
For any $S$, one model is the algebra of clopen subsets of the product space $\set{0, 1}^S$.
The free Boolean algebra on a countably infinite set is atomless.

Example: for $S = \set{x}$, the algebra $\set{0, x, \neg x, 1}$ with $x$ and $\neg x$ incomparable is free.
A map $g$ extends by $\neg x \mapsto \neg g(x)$, $0 \mapsto 0$, $1 \mapsto 1$, and this extension is forced.

Non-example: the power set of $\set{1, 2}$ with $\iota(x) = \set{1}$ and $\iota(y) = \set{2}$ is generated by the image of $\iota$ but is not free on $\set{x, y}$.
For the map $g(x) = g(y) = 1$ into the two-element algebra, a homomorphism $\bar g$ would send $\set{1} \cap \set{2} = \emptyset$ to $1 \wedge 1 = 1$, but a homomorphism sends $\emptyset$ to $0$.
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<document source="https://tamnd.github.io/math2/node/C95QJ7YH5T.md" tag="C95QJ7YH5T" kind="concept">
# Polar set

## Summary

A polar set in $\R^n$, $n \ge 2$, is a set on which some subharmonic function, not identically $-\infty$, takes the value $-\infty$.
Polar sets are the negligible sets of [[potential theory|9PM8Q2E30Q]].
A Borel set is polar exactly when it has [[capacity|4YZVFGJVSZ]] zero, Newtonian capacity for $n \ge 3$ and [[logarithmic capacity|DBQXRTC0VN]] for $n = 2$.
Such sets are removable for bounded harmonic functions, and Brownian motion almost surely never hits them.

## Explanations

### L1 human

A [[subharmonic function|Z1MHMGFNFW]] may take the value $-\infty$ at some points, like $\log \abs{z}$ at $0$ in the plane.
A polar set is a set that is small enough to be sent to $-\infty$ by one subharmonic function at once.
Points and countable sets are polar.
A line segment in the plane is not polar, even though it has area zero.
So polar sets are much smaller than sets of measure zero.
</document>

<document source="https://tamnd.github.io/math2/node/C96N6EB236.md" tag="C96N6EB236" kind="definition">
# Wiener algebra

## Definition

Let $\mathbb{T} = \R / 2\pi\Z$, and for an integrable function $f$ on $\mathbb{T}$ let $\hat f(n)$, $n \in \Z$, be its [[Fourier coefficients|5NN2YZZR3V]].
The *Wiener algebra* is the set
$$A(\mathbb{T}) = \Big\{ f \colon \mathbb{T} \to \C \text{ continuous} : \sum_{n \in \Z} \abs{\hat f(n)} < \infty \Big\},$$
of [[continuous|0X3WGX95FE]] functions with an [[absolutely convergent Fourier series|HHBGB5R5S2]], with pointwise addition and multiplication and the norm
$$\norm{f}_A = \sum_{n \in \Z} \abs{\hat f(n)}.$$

## Remarks

The map $f \mapsto (\hat f(n))_{n \in \Z}$ is an isometric isomorphism of $A(\mathbb{T})$ onto $\ell^1(\Z)$.
It turns pointwise products into convolutions: $\widehat{fg}(n) = \sum_{k \in \Z} \hat f(k) \hat g(n - k)$.
So $\norm{fg}_A \le \norm{f}_A \norm{g}_A$, and $A(\mathbb{T})$ is a commutative unital [[Banach algebra|9KYWN94QDN]].
Every $f \in A(\mathbb{T})$ satisfies $\sup_x \abs{f(x)} \le \norm{f}_A$.

Example: $f(x) = 2 + \cos x$ lies in $A(\mathbb{T})$ with $\hat f(0) = 2$, $\hat f(\pm 1) = 1/2$, and $\norm{f}_A = 3$.

Non-example: $f(x) = \sum_{n \ge 2} \frac{\sin nx}{n \log n}$.
Its coefficients decrease to $0$ and $n \cdot \frac{1}{n \log n} \to 0$, so by the Chaundy–Jolliffe theorem the series converges uniformly and $f$ is continuous.
But $\abs{\hat f(\pm n)} = \frac{1}{2 n \log n}$ and $\sum_{n \ge 2} \frac{1}{n \log n} = \infty$, so $f \notin A(\mathbb{T})$.

The characters of $A(\mathbb{T})$ are the evaluations $f \mapsto f(x)$ at points $x \in \mathbb{T}$; this is the key to [[Wiener's lemma|NC768TWZ3D]].
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<document source="https://tamnd.github.io/math2/node/C98DKQSTNC.md" tag="C98DKQSTNC" kind="concept">
# Mixture model

## Summary

A mixture model describes a population made of several subpopulations, each with its own distribution.
The density of an observation is a weighted average of the component densities, with weights equal to the subpopulation proportions.
It is a [[latent variable model|C63RVH96TF]] in which the hidden variable is the label of the subpopulation.
Mixture models are usually fitted by the [[expectation–maximization algorithm|11N1AXSXYK]] and are a basis of model-based [[cluster analysis|D9Y3A9RH6J]].

## Explanations

### L1 human

The heights of adults form a mixture of the heights of women and the heights of men.
Each group alone gives a bell-shaped curve, and the population curve is a weighted sum of the two bells.
If the two groups are far apart, the population curve has two humps.
A mixture model starts from such a sum and tries to recover the groups, their proportions and their shapes from the data.
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<document source="https://tamnd.github.io/math2/node/C98T2T5FJS.md" tag="C98T2T5FJS" kind="concept">
# Normal number

## Summary

A real number is normal in a base $b$ when every block of $k$ digits appears in its base-$b$ expansion with limiting frequency $b^{-k}$.
A number is absolutely normal when it is normal in every base $b \ge 2$.
Borel showed that almost every real number, in the sense of Lebesgue measure, is absolutely normal.
Yet for familiar constants such as $\sqrt{2}$, $\pi$ and $e$, normality is not known in any base.

## Explanations

### L1 human

Think of the digits of a number as the results of rolling a fair ten-sided die.
A normal number is one whose digits look random in this sense.
Each digit appears one tenth of the time, each pair of digits one hundredth of the time, and so on.
The number $0.123456789101112131415\ldots$, made by writing the positive integers one after the other, is normal in base $10$.
A [[rational number|0NCJ5WXN6Z]] is never normal, because its digits repeat with a fixed period.
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<document source="https://tamnd.github.io/math2/node/C9DT4Y63CY.md" tag="C9DT4Y63CY" kind="statement">
# Cheeger–Müller theorem

## Statement

Let $M$ be a closed oriented Riemannian manifold, and let $\rho$ be an acyclic orthogonal or unitary representation of $\pi_1(M)$.
Then the [[analytic torsion|MZT9D34B0A]] of $(M, \rho)$ equals the Reidemeister torsion of $(M, \rho)$, computed from any smooth triangulation of $M$.
Here both torsions are normalized with the same convention.

## Remarks

Ray and Singer conjectured the equality, and Cheeger and Müller proved it independently.
Cheeger's proof compares the two sides under surgery; Müller's approximates the de Rham complex by the combinatorial cochain complex of finer and finer triangulations.
Müller extended it to unimodular representations, and Bismut and Zhang gave a proof by Witten deformation that covers all flat bundles, with an explicit correction term.
Example: on the circle with the character $\rho(1) = e^{2\pi i a}$, $0 < a < 1$, the Reidemeister torsion is $\abs{1 - e^{2\pi i a}}$, and the analytic torsion is $2\sin(\pi a)$, the same number.
For lens spaces the theorem gives an analytic way to compute the Reidemeister torsions that distinguish non-homeomorphic homotopy equivalent lens spaces.

## Explanations

### L1 human

Reidemeister torsion is computed by cutting a manifold into simplices and doing linear algebra.
Analytic torsion is computed from the eigenvalues of Laplacians, with no triangulation at all.
The theorem says that these two very different recipes give the same number.

### L2 human

The theorem identifies a spectral invariant with a combinatorial one, in the spirit of index theory but for a secondary invariant.
It motivated the study of torsion in families (the Bismut–Lott torsion forms) and the use of analytic torsion to study torsion in the cohomology of arithmetic groups.
</document>

<document source="https://tamnd.github.io/math2/node/C9EDBYCRAY.md" tag="C9EDBYCRAY" kind="definition">
# Detailed balance

## Definition

Let $P = (p_{ij})_{i, j \in S}$ be the [[transition matrix|0F462493EA]] of a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable set $S$.
A probability distribution $\pi = (\pi_i)_{i \in S}$ on $S$ satisfies *detailed balance* for $P$ if
$$\pi_i \, p_{ij} = \pi_j \, p_{ji} \quad \text{for all } i, j \in S.$$
The chain is then called *reversible* with respect to $\pi$.

## Remarks

Detailed balance implies that $\pi$ is [[stationary|0KRH5157WP]]: $\sum_i \pi_i p_{ij} = \sum_i \pi_j p_{ji} = \pi_j$.
If $X_0$ has law $\pi$, then $(X_0, \ldots, X_n)$ and $(X_n, \ldots, X_0)$ have the same law.
Example: on a finite connected [[graph|00C0BR7D7H]] with edge set $E$, let $p_{ij} = 1/\deg(i)$ when $j$ is a neighbour of $i$.
With $\pi_i = \deg(i) / (2\abs{E})$, both sides equal $1/(2\abs{E})$ for neighbours, and both vanish otherwise.
Non-example: on $S = \set{1, 2, 3}$ let the chain move deterministically $1 \to 2 \to 3 \to 1$.
The uniform law is stationary, but $\pi_1 p_{12} = 1/3$ while $\pi_2 p_{21} = 0$, so detailed balance fails.
Kolmogorov's criterion says that an irreducible chain with a stationary distribution is reversible if and only if $p_{i_1 i_2} p_{i_2 i_3} \cdots p_{i_n i_1} = p_{i_1 i_n} p_{i_n i_{n-1}} \cdots p_{i_2 i_1}$ for every finite cycle of states.
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# Sullivan's no wandering domains theorem

## Statement

Let $f \colon \hat{\C} \to \hat{\C}$ be a [[rational function|0XDK9CGYKJ]] of degree at least $2$ on the [[Riemann sphere|0J78KBPFZB]].
Let $U$ be a connected component of the [[Fatou set|C0N0XGC6GK]] of $f$.
Then $U$ is eventually periodic: there are integers $m \ge 0$ and $p \ge 1$ with $f^{m+p}(U) = f^m(U)$.

## Remarks

A Fatou component that is not eventually periodic is called a wandering domain, so the theorem says that rational maps have none.
The proof, due to Dennis Sullivan, uses [[quasiconformal mappings|SB8KG46NHP]].
A wandering domain would support an infinite-dimensional family of quasiconformal deformations of $f$.
Each deformation is again a rational map of the same degree, and these maps form a finite-dimensional space, a contradiction.
The hypothesis that $f$ is rational is needed.
The transcendental [[entire function|0G4F0XB9X2]] $g(z) = z + \sin z + 2\pi$ has wandering domains.
Indeed $h(z) = z + \sin z$ has superattracting fixed points at the odd multiples of $\pi$, and $g = h + 2\pi$ moves each of their basins to the next one.

## Explanations

### L1 human

The Fatou set of a rational map breaks into pieces, and the map moves these pieces around.
The theorem says that no piece can wander forever through new pieces.
After finitely many steps every piece falls into a cycle of pieces that repeats.
This reduces the study of the Fatou set to its periodic pieces, whose types are known.

### L2 human

Together with the classification of periodic Fatou components, the theorem gives a complete picture of the Fatou set of a rational map.
Each component eventually lands on a cycle of attracting basins, parabolic basins, Siegel discs or Herman rings.
The method of proof, quasiconformal deformation, became a basic tool of the subject.
For transcendental entire functions wandering domains exist, and they remain an active topic of research.
</document>

<document source="https://tamnd.github.io/math2/node/C9GDH7264N.md" tag="C9GDH7264N" kind="definition">
# Strict total order

## Definition

Let $A$ be a set.
A *strict total order* on $A$ is a [[strict partial order|SWB1ZXFVKY]] $<$ on $A$ such that, for all $a, b \in A$ with $a \ne b$, $a < b$ or $b < a$.

## Remarks

The relation $<$ on $\R$ is a strict total order: it is irreflexive, transitive, and for distinct reals $x$ and $y$, $x < y$ or $y < x$.
The relation $\subsetneq$ on the [[power set|0BC7KX51Z8]] of $\set{1, 2}$ is a strict partial order but not a strict total order.
The sets $\set{1}$ and $\set{2}$ are distinct, and neither is a proper subset of the other.
The condition gives trichotomy: for all $a, b \in A$, exactly one of $a < b$, $a = b$, $b < a$ holds.
The strict total orders on $A$ correspond one to one with the [[total orders|0316AYH1SC]] on $A$, by $a \le b$ if and only if $a < b$ or $a = b$.
Other names are strict linear order and strict simple order.

## Explanations

### L1 human

A strict total order ranks all elements in one line with no ties.
Given two different elements, it tells which one comes first.
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# Quasi-isomorphism

## Summary

A quasi-isomorphism is a [[chain map|HTK9WQZJ1G]] that induces an isomorphism on [[homology|2J4JJJYP22]] in every degree.
Every chain homotopy equivalence is a quasi-isomorphism, but not conversely.
The [[derived category|5ST7NVJFSM]] is obtained by formally inverting the quasi-isomorphisms, so that complexes with "the same homology in a coherent way" become isomorphic.

## Explanations

### L1 human

Homology is the information one usually wants from a chain complex.
A quasi-isomorphism is a map between two complexes that matches their homology exactly, even if the complexes themselves look very different.
For example, the complex $0 \to \Z \xrightarrow{2} \Z \to 0$ and the single group $\Z/2\Z$ both have homology $\Z/2\Z$ in degree $0$ and nothing else.
The reduction map from the first to the second is a quasi-isomorphism.
This is how a resolution replaces a module by a nicer complex with the same homology.
</document>

<document source="https://tamnd.github.io/math2/node/C9JJD2AVJR.md" tag="C9JJD2AVJR" kind="definition">
# Pfaffian

## Definition

Let $K$ be a [[commutative ring|0310REJJY8]], let $n \ge 1$, and let $A = (a_{ij})$ be a $2n \times 2n$ matrix over $K$.
Assume that $A$ is alternating: $a_{ji} = -a_{ij}$ and $a_{ii} = 0$ for all $i, j$.
Let $\Pi$ be the set of [[perfect matchings|0NHK9MFFW9]] of $\set{1, \dots, 2n}$, that is, partitions into $n$ pairs.
Write $M \in \Pi$ as $\set{\set{i_1, j_1}, \dots, \set{i_n, j_n}}$ with $i_k < j_k$, and let $\pi_M$ be the permutation $1 \mapsto i_1, 2 \mapsto j_1, \dots, 2n-1 \mapsto i_n, 2n \mapsto j_n$.
The *Pfaffian* of $A$ is
$$\operatorname{pf}(A) = \sum_{M \in \Pi} \sgn(\pi_M) \prod_{k=1}^{n} a_{i_k j_k},$$
where $\sgn$ is the [[sign of a permutation|02DKV69ZH0]].

## Remarks

The term of $M$ does not depend on the order in which the pairs are listed, because exchanging two pairs is an even permutation.
For an alternating matrix of odd size, the Pfaffian is set to $0$.

Example: for $n = 2$ there are three matchings, and
$$\operatorname{pf}(A) = a_{12} a_{34} - a_{13} a_{24} + a_{14} a_{23}.$$
For the matching $\set{\set{1,3}, \set{2,4}}$, the permutation $\pi_M$ is $1 \mapsto 1, 2 \mapsto 3, 3 \mapsto 2, 4 \mapsto 4$, a transposition, so its sign is $-1$.

Non-example: the symmetric matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is not alternating, because $a_{21} \ne -a_{12}$ unless $2 = 0$ in $K$, so it has no Pfaffian in this sense.

The main facts are $\operatorname{pf}(A)^2 = \det A$ and $\operatorname{pf}(B A B^T) = \det(B) \operatorname{pf}(A)$ for every $2n \times 2n$ matrix $B$, where $\det$ is the [[determinant|00KDZRW4C0]].
Alternating matrices over a field are the same as [[skew-symmetric matrices|0PBY4E9GQF]] when the characteristic is not $2$.

## Explanations

### L1 human

The Pfaffian adds up, with signs, one product for each way to split the indices into pairs.
It plays for skew-symmetric matrices the role that the determinant plays for all square matrices.
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# Universally closed morphism

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].
The morphism $f$ is *universally closed* if the following holds for every morphism of schemes $Y' \to Y$.
The base change
$$f' \colon X \times_Y Y' \to Y'$$
is a [[closed map|00FSJTXPJM]] of the underlying topological spaces.
Here $X \times_Y Y'$ is the [[fiber product|1DZMT4S9FB]] and $f'$ is the second projection.

## Remarks

A closed immersion $Z \to Y$ is universally closed.
Its base change along any $Y' \to Y$ is again a closed immersion, and a closed immersion is a homeomorphism onto a closed subset, so it is a closed map.
For a field $k$, the morphism $\mathbb{A}^1_k \to \operatorname{Spec} k$ is closed but not universally closed.
Its base change along $\mathbb{A}^1_k \to \operatorname{Spec} k$ is the projection $\mathbb{A}^2_k \to \mathbb{A}^1_k$, which sends the closed set $V(xy - 1)$ onto $\mathbb{A}^1_k \setminus \set{0}$, a set that is not closed.
Universally closed morphisms are stable under composition and base change.
</document>

<document source="https://tamnd.github.io/math2/node/C9VJ8DTZWC.md" tag="C9VJ8DTZWC" kind="definition">
# M-matrix

## Definition

Let $n \ge 1$ and let $A = (a_{ij})$ be a real $n \times n$ [[matrix|0JXZKX526R]].
The matrix $A$ is a *Z-matrix* if $a_{ij} \le 0$ for all $i \ne j$.
The matrix $A$ is an *M-matrix* if it can be written as $A = sI - B$, where:

1. $I$ is the $n \times n$ [[identity matrix|06DW9BAWB1]] and $s$ is a real number.
2. $B$ is a [[nonnegative matrix|JPQCCFCY8X]].
3. $s \ge \rho(B)$, where $\rho(B)$ is the [[spectral radius|9V7YQA85PC]] of $B$.

An M-matrix is *nonsingular* if moreover $s > \rho(B)$.

## Remarks

Every M-matrix is a Z-matrix.
For a Z-matrix the choice of $s$ does not matter: whenever $s \ge \max_i a_{ii}$, the number $s - \rho(sI - A)$ is the same.
Example: $A = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} = 2I - B$ with $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.
The eigenvalues of $B$ are $\pm 1$, so $\rho(B) = 1 < 2$, and $A$ is a nonsingular M-matrix.
Non-example: $A = \begin{pmatrix} 1 & -2 \\ -2 & 1 \end{pmatrix} = I - B$ with $B = \begin{pmatrix} 0 & 2 \\ 2 & 0 \end{pmatrix}$.
Here $\rho(B) = 2 > 1$, so clause 3 fails; indeed $A$ has the negative eigenvalue $-1$.
For a Z-matrix $A$, each of the following conditions is equivalent to $A$ being a nonsingular M-matrix.

1. $A$ is invertible and $A^{-1}$ is nonnegative.
2. Every [[eigenvalue|0BQAA2BK3Q]] of $A$ has positive real part.
3. All leading principal minors of $A$ are positive.
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# Monster group

## Description

The Monster group $\mathbb{M}$ is the largest of the $26$ [[sporadic groups|PZ49Z4CBFZ]].
It is a finite [[simple group|0YMYMNQ0QZ]] of [[order|0B85S2V2DR]]
$$\abs{\mathbb{M}} = 2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71,$$
which is $808017424794512875886459904961710757005754368000000000$, about $8 \cdot 10^{53}$.

## Remarks

The smallest nontrivial [[irreducible representation|FX3NEFYFFN]] of $\mathbb{M}$ over $\C$ has dimension $196883$.
Robert Griess constructed $\mathbb{M}$ as the automorphism group of a commutative non-associative algebra of dimension $196884$, the [[Griess algebra|ZMPYZ8KD9N]].
Its existence had been predicted earlier by Bernd Fischer and by Griess.
The [[$j$-invariant|FEC2XH822R]] has the expansion $j(\tau) = q^{-1} + 744 + 196884 q + \cdots$ with $q = e^{2 \pi i \tau}$, and $196884 = 196883 + 1$.
This coincidence led to the [[monstrous moonshine|E0Y7Y75T6C]] conjecture of Conway and Norton, proved by Richard Borcherds.
Twenty of the $26$ sporadic groups are subquotients of $\mathbb{M}$, that is, quotients of subgroups of $\mathbb{M}$; the other six are the [[pariah groups|ZYZ5FYG7QC]].

## Explanations

### L1 human

The Monster is the largest of the exceptional finite simple groups, with about $8 \cdot 10^{53}$ elements.
Its smallest faithful action by complex matrices needs matrices of size $196883 \times 196883$.
It contains most of the other exceptional simple groups, and it is linked in surprising ways to number theory through monstrous moonshine.
</document>

<document source="https://tamnd.github.io/math2/node/C9WE6180A7.md" tag="C9WE6180A7" kind="definition">
# Amoeba

## Definition

Let $n \ge 1$ and let $\operatorname{Log} \colon (\C^\times)^n \to \R^n$ be the map
$$\operatorname{Log}(z_1, \dots, z_n) = (\log \abs{z_1}, \dots, \log \abs{z_n}).$$
Let $f$ be a nonzero Laurent polynomial in $z_1, \dots, z_n$ with [[complex|09FTAET5Z4]] coefficients, and let $V_f = \set{z \in (\C^\times)^n : f(z) = 0}$.
The *amoeba* of $f$ is the set $\mathcal{A}_f = \operatorname{Log}(V_f) \subseteq \R^n$.

## Remarks

The amoeba is a closed subset of $\R^n$, and each [[connected component|0JG9W3KA6B]] of its complement is convex.
Let $f$ be nonconstant.
The number of these components is at least the number of vertices of the [[Newton polytope|CYTTX8KHC8]] of $f$, and at most its number of lattice points.

Example: let $f = 1 + z + w$.
The point $(0, 0)$ lies in $\mathcal{A}_f$: the point $z = e^{2\pi i/3}$, $w = e^{-2\pi i/3}$ has $\abs{z} = \abs{w} = 1$ and $1 + z + w = 0$.

Non-example: the point $(1, 0)$ does not lie in $\mathcal{A}_f$.
It would need $\abs{z} = e$, $\abs{w} = 1$ and $z = -1 - w$, but $\abs{1 + w} \le 2 < e$.

In one variable, $f(z) = z - 2$ has $V_f = \set{2}$ and $\mathcal{A}_f = \set{\log 2}$.
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# Radó's theorem on triangulation of surfaces

## Statement

Let $S$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $2$, that is, a Hausdorff, second-countable space locally homeomorphic to $\R^2$.
Then $S$ has a [[triangulation|YH170WBQ6M]].

## Remarks

Tibor Radó proved the theorem in 1925.
One proof covers $S$ by countably many closed disks coming from charts, and modifies their boundary circles so that, on each compact part, they meet in finitely many points.
The pieces cut out by these circles are bounded by Jordan curves, so the [[Schoenflies theorem|3VX06XRVZG]] shows that they are closed disks, which can then be divided into triangles.
For example, the torus is triangulable, with as few as $7$ vertices.
The theorem is the reason why the classification of closed surfaces, first proved for triangulated surfaces, applies to all closed topological surfaces.
The triangulation is also unique up to piecewise linear homeomorphism, so every surface has a unique piecewise linear structure.
A related result, also called Radó's theorem, says that every connected Riemann surface is second-countable; its proof uses the same kind of construction.

## Explanations

### L1 human

Every surface, however it is given, can be cut into triangles glued edge to edge.
This lets one count vertices, edges and triangles, and so compute the Euler characteristic of any closed surface.

### L2 human

In a course on surfaces, theorems are often proved first for triangulated surfaces, using the combinatorics of the triangles.
Radó's theorem removes this hypothesis.
The analogous statement holds in dimension $3$ by Moise's theorem, but fails in dimension $4$, where some topological manifolds have no triangulation.
</document>

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# Lawvere–Tierney topology

## Definition

Let $\mathcal{E}$ be a [[topos|6837RS7N82]] with [[subobject classifier|NKP5TPQ79X]] $\top \colon 1 \to \Omega$.
Let $\wedge \colon \Omega \times \Omega \to \Omega$ be the characteristic morphism of the subobject $(\top, \top) \colon 1 \to \Omega \times \Omega$.
A *Lawvere–Tierney topology* on $\mathcal{E}$ is a morphism $j \colon \Omega \to \Omega$ such that:

1. $j \circ \top = \top$;
2. $j \circ j = j$;
3. $j \circ \wedge = \wedge \circ (j \times j)$.

## Remarks

Example: the identity $\id_\Omega$ satisfies all three equations.
Example: the double negation $\neg\neg \colon \Omega \to \Omega$ is a Lawvere–Tierney topology in every topos.
Non-example: in $\mathbf{Set}$, with $\Omega = \set{0, 1}$ and $\top = 1$, the negation $\neg$ sends $1$ to $0$, so $\neg \circ \top \ne \top$ and clause 1 fails.
An object $F$ is a $j$-sheaf when, for every monomorphism $m \colon A \to B$ whose characteristic morphism satisfies $j \circ \chi_m = \top \circ {!}$, where ${!} \colon B \to 1$ is the unique morphism, every morphism $A \to F$ extends uniquely along $m$.
The $j$-sheaves form a topos $\mathrm{Sh}_j(\mathcal{E})$, and the inclusion has a left adjoint that preserves finite limits.
Another name for a Lawvere–Tierney topology is a local operator.
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# Laplace transform on a convex cone

## Definition

Let $\Gamma \subseteq \R^n$ be a closed [[convex cone|6VE4KA12VD]] with nonempty interior, and let $\ip{\cdot}{\cdot}$ be the standard inner product on $\R^n$.
Let $\Gamma^* = \set{y \in \R^n : \ip{y}{x} \ge 0 \text{ for all } x \in \Gamma}$ be its [[dual cone|QFY1JR1TK4]], and let $\Omega$ be the interior of $\Gamma^*$.
Let $f \colon \Gamma \to \C$ be locally integrable.
The *Laplace transform* of $f$ on $\Gamma$ is the function
$$(\mathcal{L}_\Gamma f)(z) = \int_\Gamma e^{-\ip{z}{x}} f(x)\, dx, \qquad z = y + i\eta \in \C^n,$$
defined at those $z$ where the integral converges absolutely; here $\ip{z}{x} = \sum_j z_j x_j$.

## Remarks

If $y \in \Omega$, there is $c > 0$ with $\ip{y}{x} \ge c \abs{x}$ for all $x \in \Gamma$, by compactness of $\Gamma \cap \set{\abs{x} = 1}$.
So if $\abs{f(x)} \le C (1 + \abs{x})^N$, the transform converges absolutely on the tube $\Omega + i \R^n$ and is holomorphic there.

Example: $\Gamma = [0, \infty)^n$ is self-dual, and $\mathcal{L}_\Gamma$ is the [[multidimensional Laplace transform|QT3M720TCV]].
For $f = 1$ one gets $\mathcal{L}_\Gamma f(z) = 1/(z_1 \cdots z_n)$ on $\operatorname{Re} z_j > 0$.

Example: let $\Gamma$ be the cone of positive semidefinite real symmetric $m \times m$ matrices, with $\ip{y}{x} = \tr(yx)$.
Let $dx$ be the Lebesgue measure in the entries $x_{ij}$, $i \le j$.
The cone is self-dual, and $\Omega$ is the cone of [[positive definite matrices|0KX66CZMBB]].
For real $s > (m - 1)/2$ and $y \in \Omega$,
$$\int_\Omega e^{-\tr(yx)} (\det x)^{s - (m+1)/2}\, dx = \Gamma_m(s)\, (\det y)^{-s}, \qquad \Gamma_m(s) = \pi^{m(m-1)/4} \prod_{j=0}^{m-1} \Gamma\left(s - \frac{j}{2}\right).$$

Non-example: let $\Gamma$ contain a line $\R v$.
Then $\Gamma^*$ lies in the hyperplane $v^\perp$ and $\Omega$ is empty.
Since $\Gamma + \R v = \Gamma$, Fubini's theorem along the direction $v$ shows that even $f = 1$ has a divergent transform at every $z$.
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<document source="https://tamnd.github.io/math2/node/C9Z22J9SQE.md" tag="C9Z22J9SQE" kind="definition">
# Witt vectors

## Definition

Let $p$ be a prime number and let $A$ be a [[commutative ring|0310REJJY8]].
For $n \ge 0$, the $n$-th ghost polynomial is
$$w_n(X_0, \ldots, X_n) = \sum_{i=0}^{n} p^i X_i^{p^{n-i}} = X_0^{p^n} + p X_1^{p^{n-1}} + \cdots + p^n X_n.$$
There are unique sequences of [[polynomials|0SDC934R52]] $S_n, P_n \in \Z[X_0, \ldots, X_n, Y_0, \ldots, Y_n]$ such that for every $n \ge 0$:
$$w_n(S_0, \ldots, S_n) = w_n(X) + w_n(Y), \qquad w_n(P_0, \ldots, P_n) = w_n(X) \, w_n(Y).$$
The ring of *$p$-typical Witt vectors* $W(A)$ is the set $A^{\N}$ of sequences $a = (a_0, a_1, a_2, \ldots)$ with entries in $A$, with the operations
$$(a + b)_n = S_n(a_0, \ldots, a_n, b_0, \ldots, b_n), \qquad (ab)_n = P_n(a_0, \ldots, a_n, b_0, \ldots, b_n).$$
With these operations $W(A)$ is a commutative ring with zero $(0, 0, \ldots)$ and identity $(1, 0, 0, \ldots)$.

## Remarks

The first sum polynomials are $S_0 = X_0 + Y_0$ and $S_1 = X_1 + Y_1 + \frac{X_0^p + Y_0^p - (X_0 + Y_0)^p}{p}$, which has integer coefficients.
The ghost map $a \mapsto (w_0(a), w_1(a), \ldots)$ is a [[ring homomorphism|0VTH8KPCG2]] $W(A) \to A^{\N}$, and it is an isomorphism when $p$ is a unit in $A$.
A ring homomorphism $f \colon A \to B$ induces $W(f) \colon W(A) \to W(B)$ by applying $f$ to each entry.

Example: $W(\F_p)$ is isomorphic to the ring $\Z_p$ of [[p-adic integers|V6TJAFZ4MD]].
For $p = 2$, compute $(1, 0, \ldots) + (1, 0, \ldots)$ in $W(\Z)$ through ghost components: $w_0 = 1 + 1 = 2$ and $w_1 = 1 + 1 = 2$.
So $a_0 = 2$ and $a_0^2 + 2 a_1 = 2$, which gives $a_1 = -1$.
Reducing modulo $2$, in $W(\F_2)$ the sum is $(0, 1, \ldots)$, which corresponds to $2 \in \Z_2$.

Non-example: the componentwise addition on $\F_2^{\N}$ gives $(1, 0, \ldots) + (1, 0, \ldots) = (0, 0, \ldots)$.
This is not the Witt sum, because it does not respect the ghost components $w_1$ over $\Z$.

The Teichmüller representative of $x \in A$ is $[x] = (x, 0, 0, \ldots)$, and it satisfies $[xy] = [x][y]$.
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<document source="https://tamnd.github.io/math2/node/CA1J41NWGS.md" tag="CA1J41NWGS" kind="definition">
# Schur complement

## Definition

Let $F$ be a field and let $p, q \ge 1$.
Let $M$ be a $(p + q) \times (p + q)$ [[block matrix|0ASDFEQS6P]] over $F$,
$$M = \begin{pmatrix} A & B \\ C & D \end{pmatrix},$$
where $A$ is $p \times p$, $B$ is $p \times q$, $C$ is $q \times p$ and $D$ is $q \times q$.
If $A$ is [[invertible|0EB9N0S33C]], the *Schur complement of $A$ in $M$* is the $q \times q$ matrix
$$M/A = D - C A^{-1} B.$$
If $D$ is invertible, the *Schur complement of $D$ in $M$* is the $p \times p$ matrix
$$M/D = A - B D^{-1} C.$$

## Remarks

Example: let $M = \begin{pmatrix} 2 & 1 \\ 4 & 5 \end{pmatrix}$ with $p = q = 1$ and $A = (2)$.
Then $M/A = 5 - 4 \cdot \tfrac{1}{2} \cdot 1 = 3$, and indeed $\det M = 10 - 4 = 6 = 2 \cdot 3$.
Non-example: for $M = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $A = (0)$, the block $A$ is not invertible, so $M/A$ is not defined, although $M$ itself is invertible.
The definition comes from the factorisation
$$M = \begin{pmatrix} I_p & 0 \\ C A^{-1} & I_q \end{pmatrix} \begin{pmatrix} A & 0 \\ 0 & M/A \end{pmatrix} \begin{pmatrix} I_p & A^{-1} B \\ 0 & I_q \end{pmatrix},$$
which gives $\det M = \det A \cdot \det(M/A)$; see [[determinant via the Schur complement|0KA7A4ENVJ]].
In particular, when $A$ is invertible, $M$ is invertible if and only if $M/A$ is invertible.
The same formulas hold over any commutative ring when $A$ is invertible.
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<document source="https://tamnd.github.io/math2/node/CA38T5ZH14.md" tag="CA38T5ZH14" kind="concept">
# Ultraweak topology

## Summary

The ultraweak topology, also called the $\sigma$-weak topology, is a topology on the bounded operators $B(H)$ on a Hilbert space $H$.
It is the weak-$*$ topology that comes from viewing $B(H)$ as the dual of the space of [[trace class operators|Z2CP530WDE]].
It is finer than the [[weak operator topology|4XM1VN1C1N]] but agrees with it on bounded sets.
The linear functionals that are continuous for it are exactly the normal functionals, which makes it the intrinsic topology of a [[von Neumann algebra|PSFWQVZ0FS]].

## Explanations

### L1 human

The weak operator topology tests an operator $T$ against finitely many pairs of vectors at a time.
The ultraweak topology allows infinitely many pairs at once, as long as the total size of the vectors is finite.
This is the same as testing $T$ against density matrices $\rho$ through the number $\operatorname{tr}(T \rho)$, the expected value of $T$ in the state $\rho$ in quantum mechanics.
On the unit ball nothing changes, but globally the topology has more open sets.
Its advantage is that it depends only on the algebra and not on the Hilbert space on which it acts.
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<document source="https://tamnd.github.io/math2/node/CA49F5KQGB.md" tag="CA49F5KQGB" kind="definition">
# Logical connective

## Definition

Let $n \ge 0$ be an integer, and write $\mathsf{T}$ and $\mathsf{F}$ for the two truth values.
An $n$-ary *logical connective* of classical [[propositional logic|09K9MWD2QQ]] is a symbol $c$ with a truth function $f_c \colon \set{\mathsf{T}, \mathsf{F}}^n \to \set{\mathsf{T}, \mathsf{F}}$.
For formulas $\varphi_1, \dots, \varphi_n$, the expression $c(\varphi_1, \dots, \varphi_n)$ is again a formula.
Under a truth assignment, its value is $f_c$ applied to the values of $\varphi_1, \dots, \varphi_n$.

## Remarks

Conjunction $\land$ is a binary connective: $f_\land(a, b) = \mathsf{T}$ exactly when $a = b = \mathsf{T}$.
So $p \land q$ is true only when [[propositions|0ZGC7ZESYN]] $p$ and $q$ are both true.
The modal operator "it is necessary that" is not a connective of classical propositional logic.
It has no truth function: the truth of $p$ does not determine whether $p$ is necessary.
Binary connectives are written in infix form, such as $\varphi \to \psi$.
The set $\set{\neg, \land}$ is functionally complete: every truth function is expressed by a formula built with these two.
The single connective NAND, with $f(a, b) = \mathsf{F}$ exactly when $a = b = \mathsf{T}$, is also functionally complete.
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<document source="https://tamnd.github.io/math2/node/CA49GFN9ZA.md" tag="CA49GFN9ZA" kind="definition">
# Complete elliptic integral of the second kind

## Definition

Let $k$ be a real number with $0 \le k \le 1$.
The *complete elliptic integral of the second kind* is
$$E(k) = \int_0^{\pi/2} \sqrt{1 - k^2 \sin^2\theta} \, d\theta = \int_0^1 \frac{\sqrt{1 - k^2 t^2}}{\sqrt{1 - t^2}} \, dt.$$
It is the value at $\varphi = \pi/2$ of the incomplete [[elliptic integral|NXMCYGCB59]] $E(\varphi, k)$ of the second kind.
With $k' = \sqrt{1 - k^2}$ one writes $E'(k) = E(k')$.

## Remarks

Example: $E(0) = \pi/2$, and $E(1) = \int_0^{\pi/2} \cos\theta \, d\theta = 1$.
The ellipse $x = a\sin\theta$, $y = b\cos\theta$ with $0 < b \le a$ has perimeter $4aE(e)$, where $e = \sqrt{1 - b^2/a^2}$ is its eccentricity, because its arc length element is $a\sqrt{1 - e^2\sin^2\theta}\,d\theta$.
Non-example: for real $k > 1$ the integrand $\sqrt{1 - k^2\sin^2\theta}$ is not real for $\theta$ near $\pi/2$, so the formula does not define a real number there.
Series: $E(k) = \frac{\pi}{2}\left(1 - \sum_{n=1}^{\infty} \left(\frac{(2n)!}{2^{2n}(n!)^2}\right)^2 \frac{k^{2n}}{2n - 1}\right)$.
Some authors write $E(m)$ with the parameter $m = k^2$.
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# Coupling from the past

## Definition

Let $S$ be a finite set and let $P$ be the [[transition matrix|0F462493EA]] of an irreducible and aperiodic [[Markov chain|033TJBZDG0]] on $S$.
Let $\pi$ be its [[stationary distribution|0KRH5157WP]].
Let $\phi \colon S \times [0, 1] \to S$ be an *update function*: if $U$ is [[uniform|0AREFQ9Z65]] on $[0, 1]$, then $P(\phi(x, U) = y) = P(x, y)$ for all $x, y \in S$.
Let $U_{-1}, U_{-2}, \ldots$ be independent and uniform on $[0, 1]$.
For $t \ge 1$ define the map $F_{-t} \colon S \to S$ by
$$F_{-t} = \phi_{-1} \circ \phi_{-2} \circ \cdots \circ \phi_{-t}, \quad \phi_{-k}(x) = \phi(x, U_{-k}).$$
*Coupling from the past* is the algorithm that computes $F_{-t}$ for $t = 1, 2, 4, 8, \ldots$, always with the same $U_{-k}$.
It stops at the first such $t$ for which $F_{-t}$ is constant, and it outputs the common value $F_{-t}(x)$.

## Remarks

If $\tau = \min\set{t : F_{-t} \text{ is constant}}$ is finite almost surely, the output has law exactly $\pi$.
Indeed, $F_{-t} = F_{-\tau} \circ (\phi_{-\tau-1} \circ \cdots \circ \phi_{-t})$ has the same constant value for all $t \ge \tau$, while $F_{-t}(x)$ has law $P^t(x, \cdot)$, which tends to $\pi$.
Example: take a chain that jumps to a fresh sample of $\pi$ at each step, with an update $\phi(x, u)$ that does not depend on $x$.
Then $F_{-1}$ is constant, and the output $\phi(x, U_{-1})$ has law $\pi$.
Non-example: running the [[coupled|HTR7J0W66V]] copies forward from time $0$ and outputting their value at the first meeting time is biased.
Take $S = \set{0, 1}$, $\phi(0, u) = 0$ for $u < 1/2$, $\phi(0, u) = 1$ otherwise, and $\phi(1, u) = 0$.
Then the forward copies always meet at $0$, while $\pi = (2/3, 1/3)$.
If $S$ has a partial order with a top and a bottom element and $\phi(\cdot, u)$ is monotone, it suffices to follow the top and the bottom states.
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# Arithmetic group

## Definition

Let $G$ be a [[linear algebraic group|GA81VVQHEP]] defined over $\Q$, together with a faithful representation $G \hookrightarrow \mathrm{GL}_n$ defined over $\Q$.
Let $G(\Z) = G(\Q) \cap \mathrm{GL}_n(\Z)$.
A [[subgroup|0X1JXT47GA]] $\Gamma$ of $G(\Q)$ is an *arithmetic subgroup* of $G$ if it is commensurable with $G(\Z)$.
This means that $\Gamma \cap G(\Z)$ has finite [[index|0DTXBQPA2S]] both in $\Gamma$ and in $G(\Z)$.
An *arithmetic group* is a group that is an arithmetic subgroup of some such $G$.

## Remarks

The class of arithmetic subgroups does not depend on the choice of the faithful representation.

Example: the [[special linear group|0V9PMN0RBH]] $\mathrm{SL}_n$ in its standard representation has $G(\Z) = \mathrm{SL}_n(\Z)$, which is arithmetic.
For an integer $N \ge 1$, the [[congruence subgroup|1H6R856P3C]] $\Gamma(N)$, the kernel of reduction modulo $N$ from $\mathrm{SL}_n(\Z)$ to $\mathrm{SL}_n(\Z/N\Z)$, is arithmetic too.
Indeed, $\Gamma(N) \cap \mathrm{SL}_n(\Z) = \Gamma(N)$, and its index in $\mathrm{SL}_n(\Z)$ is at most the order of the finite group $\mathrm{SL}_n(\Z/N\Z)$.

Non-example: $\Gamma = \mathrm{SL}_2(\Z[1/2])$ inside $\mathrm{SL}_2(\Q)$.
Here $\Gamma \cap \mathrm{SL}_2(\Z) = \mathrm{SL}_2(\Z)$.
The matrices $d_k = \mathrm{diag}(2^k, 2^{-k})$, $k \ge 0$, lie in $\Gamma$, and $d_k^{-1} d_m \notin \mathrm{SL}_2(\Z)$ for $k \ne m$.
So $\mathrm{SL}_2(\Z)$ has infinite index in $\Gamma$, and $\Gamma$ is not arithmetic in $\mathrm{SL}_2$.

Over a number field $K$ with ring of integers $\mathcal{O}_K$, one uses $G(\mathcal{O}_K)$ in place of $G(\Z)$; restriction of scalars reduces this case to the one over $\Q$.
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<document source="https://tamnd.github.io/math2/node/CAB1ARHH31.md" tag="CAB1ARHH31" kind="definition">
# Finite volume method

## Definition

Let $\Omega \subset \R^d$ be a bounded polygonal domain and $F \colon \R \to \R^d$ a $C^1$ flux.
Consider the [[conservation law|8RA7AMWSAF]] $u_t + \nabla \cdot F(u) = 0$, with $\nabla \cdot$ the [[divergence|0MWV6ZTK6S]].
Partition $\Omega$ into polygonal control volumes $K_i$, and write $e_{ij}$ for the common face of $K_i$ and $K_j$.
Let $n_{ij}$ be the unit normal of $e_{ij}$ pointing from $K_i$ to $K_j$.
A numerical flux is a function $H(a, b, n)$ with two properties:

1. consistency: $H(a, a, n) = F(a) \cdot n$ for all $a$ and $n$;
2. conservation: $H(a, b, n) = -H(b, a, -n)$.

Let $u_i(t)$ approximate the average of $u$ over $K_i$.
The *finite volume method* evolves these numbers by
$$\abs{K_i} \frac{d u_i}{dt} = -\sum_{j} \abs{e_{ij}} \, H(u_i, u_j, n_{ij}),$$
where the sum runs over the neighbours $K_j$ of $K_i$, and $\abs{\cdot}$ denotes volume or area.
Boundary faces use fluxes built from the boundary conditions.

## Remarks

The scheme is the exact balance law $\frac{d}{dt} \int_{K_i} u = -\int_{\partial K_i} F(u) \cdot n$, with the face integrals replaced by numerical fluxes.
By clause 2, the flux from $K_i$ to $K_j$ is the negative of the flux from $K_j$ to $K_i$.
So $\sum_i \abs{K_i} u_i$ changes only through the boundary of $\Omega$.

Example: for $u_t + a u_x = 0$ with $a > 0$ on cells of width $\Delta x$, take the upwind flux $H(u_i, u_{i+1}, +1) = a u_i$.
This gives $u_i' = -a (u_i - u_{i-1}) / \Delta x$.
Consistency holds since $H(v, v, +1) = a v = F(v)$.

Non-example: the scheme $u_i' = -u_i (u_{i+1} - u_{i-1}) / (2 \Delta x)$ for Burgers' equation $u_t + (u^2/2)_x = 0$ is not in flux form.
It has no numerical flux, so clause 2 has no meaning for it.
It does not conserve $\sum_i u_i$ and can produce shocks with wrong speeds.
</document>

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# Peetre K-functional

## Summary

The Peetre K-functional $K(t, x)$ measures how well an element $x$ of a sum $X_0 + X_1$ of two Banach spaces splits into pieces in $X_0$ and $X_1$.
The size of the second piece carries the weight $t$.
It is the basic tool of the real method of [[interpolation spaces|VX1R6FN6PS]].
In approximation theory it links smoothness to approximation, since for suitable pairs of spaces it is equivalent to a [[modulus of smoothness|CRPZ1RVNKQ]].

## Explanations

### L1 human

Think of $X_0$ as rough functions measured in a weak norm and $X_1$ as smooth functions measured in a strong norm.
To compute $K(t, f)$ you write $f$ as a smooth part plus a rough remainder and pay the size of the remainder plus $t$ times the size of the smooth part.
For small $t$ smoothness is cheap, so you can take a smooth part very close to $f$.
How fast $K(t, f)$ goes to $0$ as $t \to 0$ tells you how well $f$ can be approached by smooth functions.
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# Steinitz's theorem on algebraically closed fields

## Statement

Let $K$ and $L$ be [[algebraically closed fields|02SJG1HC32]], and let $P_K$ and $P_L$ be their prime subfields.
Then $K$ and $L$ are isomorphic if and only if they have the same [[characteristic|1ZB13N78XW]] and $\operatorname{trdeg}_{P_K} K = \operatorname{trdeg}_{P_L} L$, where $\operatorname{trdeg}$ is the [[transcendence degree|D7TR0YWR9H]].

## Remarks

If the characteristics agree, then $P_K \cong P_L$, and a bijection between [[transcendence bases|E45PA267S4]] $B$ of $K$ and $B'$ of $L$ gives an isomorphism $P_K(B) \cong P_L(B')$.
Since $K$ and $L$ are [[algebraic closures|0MCXX29HWY]] of these fields, the isomorphism extends to $K \cong L$ by [[uniqueness of the algebraic closure|01VQ7C4FTB]].
The converse holds because an isomorphism preserves the characteristic and maps a transcendence basis to a transcendence basis.
For an uncountable algebraically closed field, the transcendence degree over the prime field equals the [[cardinality|0JTGRV4WT4]] of the field.
So two uncountable algebraically closed fields of the same characteristic and the same cardinality are isomorphic; for example, $\C$ is isomorphic to an algebraic closure of $\Q(x_i : i \in \R)$.

## Explanations

### L1 human

An algebraically closed field is determined by two numbers: its characteristic and how many free variables it has.
Everything else follows by adding all roots of all polynomials.
For instance, $\C$ and the algebraic closure of the field of $p$-adic numbers are isomorphic as fields, although their topologies are very different.

### L2 human

The theorem shows that the algebraic closure $\overline{\Q}$ and $\C$ are not isomorphic: one has transcendence degree $0$, the other has transcendence degree the continuum.
It is the field-theoretic source of the fact that the theory of algebraically closed fields of a fixed characteristic is categorical in each uncountable cardinality.
Its proof uses the existence of transcendence bases and the uniqueness of algebraic closures, both of which rest on Zorn's lemma.
</document>

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# Nonharmonic Fourier series

## Definition

Let $A > 0$ and let $\Lambda = (\lambda_n)_{n \in \Z}$ be a sequence of distinct complex numbers.
For $n \in \Z$ let $e_n(t) = e^{i \lambda_n t}$, a function in the [[Lebesgue space|SASP14MA1Z]] $L^2(-A, A)$.
A *nonharmonic Fourier series* with frequencies $\Lambda$ is a series
$$\sum_{n \in \Z} c_n \, e^{i \lambda_n t}, \quad t \in (-A, A),$$
with complex coefficients $c_n$, considered with convergence in $L^2(-A, A)$.

## Remarks

When $A = \pi$ and $\lambda_n = n$, this is an ordinary [[Fourier series|G12KXDHPPS]], and the $e_n / \sqrt{2\pi}$ form an orthonormal basis.
In general the $e_n$ are not orthogonal, and the theory asks which sequences $\Lambda$ make $(e_n)$ complete, a [[Riesz basis|ZYDYD7FB1Z]] or a frame of $L^2(-A, A)$.
When $(e_n)$ is a Riesz basis, every $f \in L^2(-A, A)$ has a unique expansion $f = \sum c_n e_n$, and $\sum \abs{c_n}^2$ is comparable to $\norm{f}^2_{L^2}$.

Example: let $A = \pi$ and $\lambda_n = n + \delta_n$ with real $\delta_n$ and $\sup_n \abs{\delta_n} < 1/4$.
By Kadec's $1/4$ theorem, $(e^{i\lambda_n t})$ is a Riesz basis of $L^2(-\pi, \pi)$, so every function in $L^2(-\pi, \pi)$ is the sum of a unique nonharmonic Fourier series with these frequencies.
The constant $1/4$ cannot be replaced by a larger number.

Non-example: let $A = \pi$ and $\lambda_n = 2n$.
The functions $e^{2int}$ all have period $\pi$, so they cannot represent $f = 1_{(0, \pi)}$, which is not $\pi$-periodic on $(-\pi, \pi)$.
So the closed span of $(e_n)$ is not all of $L^2(-\pi, \pi)$ and the system is not complete.
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# Irreducible ideal

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
An [[ideal|0MYSA6MYNA]] $I$ of $R$ is *irreducible* if these two conditions hold:

1. $I \ne R$.
2. For all ideals $J$ and $K$ of $R$ with $I = J \cap K$, one has $I = J$ or $I = K$.

## Remarks

Example: every prime ideal $P$ is irreducible; if $P = J \cap K$ with $a \in J \setminus P$ and $b \in K \setminus P$, then $ab \in J \cap K = P$, a contradiction.
Non-example: $(6)$ in $\Z$ fails clause 2, since $(6) = (2) \cap (3)$ with both terms strictly larger.
In a Noetherian ring, every irreducible ideal is [[primary|V026AAV8E6]].
The converse fails: in $k[x, y]$ over a field $k$, the ideal $(x^2, xy, y^2)$ is primary, but it equals $(x^2, y) \cap (x, y^2)$.
Some authors call an ideal with clause 2 alone meet-irreducible, and then allow $I = R$.

## Explanations

### L1 human

An irreducible ideal is a proper ideal that is not the intersection of two strictly larger ideals.
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# Copula

## Summary

A copula is a joint [[cumulative distribution function|0H8H7ZTHW2]] on the unit cube whose one-dimensional [[marginals|0MXT6PRZZ6]] are uniform on $[0, 1]$.
By [[Sklar's theorem|ZD7V7R27R9]], every joint distribution splits into its marginal distributions and a copula that carries the dependence between the coordinates.
Copulas let one model the marginals and the dependence separately.
They are used in finance, insurance and hydrology.

## Explanations

### L1 human

Two quantities, like the rainfall in two nearby towns, each have their own distribution, and they also influence each other.
A copula describes only the way they influence each other, after each one is rescaled to a number between $0$ and $1$ by its own distribution.
The independence copula $C(u, v) = u v$ says there is no influence at all.
The copula $C(u, v) = \min(u, v)$ says that both quantities always move in perfect step.
</document>

<document source="https://tamnd.github.io/math2/node/CAN490EPRT.md" tag="CAN490EPRT" kind="concept">
# Symplectic manifold

## Summary

A symplectic manifold is a smooth manifold with a closed nondegenerate $2$-form, its symplectic form.
It is the geometric setting of Hamiltonian mechanics: phase spaces of mechanical systems are symplectic manifolds.
By Darboux's theorem all symplectic manifolds of the same dimension look alike near a point, so the subject is global and close to topology.
Each tangent space is a [[symplectic vector space|PQ7JJY0VPH]], so the dimension is even.

## Explanations

### L1 human

A symplectic form measures oriented area on each little piece of surface in the manifold, rather than lengths.
In the plane with coordinates $(q, p)$, the form $dq \wedge dp$ assigns to a small parallelogram its signed area.
Mechanics uses such a plane: $q$ is the position of a particle on a line and $p$ its momentum.
For several particles one takes many such pairs, and the symplectic form adds up the areas in each pair of coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/CAPQBQDRA7.md" tag="CAPQBQDRA7" kind="definition">
# Graded ring

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]].
A *grading* of $R$ is a family $(R_n)_{n \ge 0}$ of subgroups of the [[abelian group|082GHNPJ53]] $(R, +)$ such that:

1. $R = \bigoplus_{n \ge 0} R_n$ as abelian groups, that is, every $f \in R$ is uniquely a finite sum $f = \sum_n f_n$ with $f_n \in R_n$;
2. $R_m R_n \subseteq R_{m+n}$ for all $m, n \ge 0$.

A *graded ring* is a ring with a grading.
The elements of $R_n$ are the homogeneous elements of degree $n$, and $f_n$ is the homogeneous component of $f$ of degree $n$.

## Remarks

Example: $R = k[x_1, \dots, x_r]$, with $k$ a field and $R_n$ the span of the monomials of degree $n$; the product of two monomials of degrees $m$ and $n$ has degree $m + n$.
Non-example: in $R = k[x]/(x^2 - 1)$ take $R_0 = k$, $R_1 = k \bar x$ and $R_n = 0$ for $n \ge 2$; clause 1 holds, but $\bar x \cdot \bar x = 1$ is a nonzero element of $R_0$, not of $R_2 = 0$, so clause 2 fails.
It follows from the definition that $1 \in R_0$ and that $R_0$ is a [[subring|0GNCRPYXYV]].
One also grades by $\Z$ or by another monoid in the same way; the ring $R$ is then $\Z$-graded, and so on.
</document>

<document source="https://tamnd.github.io/math2/node/CAPWE7KWRY.md" tag="CAPWE7KWRY" kind="definition">
# Gegenbauer polynomials

## Definition

Let $\lambda > -\frac12$ be a real number with $\lambda \ne 0$.
The *Gegenbauer polynomials* $C_n^{(\lambda)}$, $n \ge 0$, are the polynomials defined by the [[generating function|0PHAC833JF]]
$$\frac{1}{(1 - 2xt + t^2)^{\lambda}} = \sum_{n=0}^{\infty} C_n^{(\lambda)}(x) \, t^n,$$
valid for $x \in [-1, 1]$ and $\abs{t} < 1$.

## Remarks

Equivalently, $C_0^{(\lambda)} = 1$, $C_1^{(\lambda)} = 2\lambda x$ and $(n + 1) C_{n+1}^{(\lambda)} = 2(n + \lambda) x \, C_n^{(\lambda)} - (n + 2\lambda - 1) C_{n-1}^{(\lambda)}$ for $n \ge 1$.
Example: $C_2^{(\lambda)}(x) = 2\lambda(\lambda + 1) x^2 - \lambda$.
Check: with $u = 2xt - t^2$, the expansion $(1 - u)^{-\lambda} = 1 + \lambda u + \frac{\lambda(\lambda + 1)}{2} u^2 + \cdots$ has $t^2$-coefficient $-\lambda + 2\lambda(\lambda + 1) x^2$.
They are orthogonal: $\int_{-1}^{1} C_m^{(\lambda)}(x) C_n^{(\lambda)}(x) (1 - x^2)^{\lambda - 1/2} \, dx = 0$ for $m \ne n$.
Special cases: $C_n^{(1/2)}$ is the [[Legendre polynomial|QFYGJVA996]] $P_n$, and $C_n^{(1)}$ is the [[Chebyshev polynomial of the second kind|7CATGEN6KH]] $U_n$.
Non-example: $\lambda = 0$ is excluded, because the generating function is then the constant $1$ and gives $C_n^{(0)} = 0$ for all $n \ge 1$.
</document>

<document source="https://tamnd.github.io/math2/node/CAWK9DA1PH.md" tag="CAWK9DA1PH" kind="definition">
# Cohen–Macaulay ring

## Definition

Let $(R, \mathfrak{m})$ be a commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]].
Then $R$ is a *Cohen–Macaulay local ring* if $\operatorname{depth} R = \dim R$, where $\operatorname{depth}$ is the [[depth|M2RADBFW99]] with respect to $\mathfrak{m}$ and $\dim$ is the [[Krull dimension|MVGBS3J1P7]].
A commutative Noetherian ring $R$ is a *Cohen–Macaulay ring* if the [[localization|X6JD4AG973]] $R_{\mathfrak{p}}$ is a Cohen–Macaulay local ring for every [[prime ideal|00MHHZ9SNE]] $\mathfrak{p}$ of $R$.

## Remarks

For a Noetherian local ring one always has $\operatorname{depth} R \le \dim R$, so the condition is that the depth is as large as possible.
It suffices to test the localizations at maximal ideals, since localizations of Cohen–Macaulay local rings at primes are Cohen–Macaulay.
Example: every Noetherian local ring of dimension $0$ is Cohen–Macaulay, since then $\operatorname{depth} R = \dim R = 0$.
Example: $K[x_1, \ldots, x_n]$ localized at $(x_1, \ldots, x_n)$ is Cohen–Macaulay, since $x_1, \ldots, x_n$ is a regular sequence of length $n = \dim R$.
Non-example: $R = K[x,y]/(x^2, xy)$ localized at $(x, y)$ has dimension $1$, because its primes are $(x)$ and $(x, y)$, but depth $0$, because $x$ kills the maximal ideal.
For a local ring, Cohen–Macaulay is equivalent to: some (equivalently every) system of parameters is a regular sequence.
A finitely generated module $M \ne 0$ over a Noetherian local ring is a *Cohen–Macaulay module* if $\operatorname{depth} M = \dim M$.
</document>

<document source="https://tamnd.github.io/math2/node/CAYQ1S3RM8.md" tag="CAYQ1S3RM8" kind="definition">
# Method of matched asymptotic expansions

## Definition

Let $y_\varepsilon$ solve a [[singularly perturbed|G6J21TGRMM]] problem on an interval $[a, b]$ that has a [[boundary layer|YCFPD0GE5G]] at $a$ of thickness $\delta(\varepsilon) \to 0$.
The *method of matched asymptotic expansions* builds an approximation in four steps.

1. Outer expansion: solve the reduced equations order by order in $\varepsilon$, with the boundary conditions away from $a$, to get $y^{\mathrm{out}}(x)$.
2. Inner expansion: set $X = (x - a) / \delta(\varepsilon)$ and rewrite the problem in $X$.
   Solve it order by order with the boundary conditions at $a$, to get $Y^{\mathrm{in}}(X)$, which may contain free constants.
3. Matching: choose the free constants so that the two expansions agree in the overlap.
   At leading order this is Prandtl's rule $\lim_{X \to \infty} Y^{\mathrm{in}}(X) = \lim_{x \to a^+} y^{\mathrm{out}}(x)$.
   This common value is the *common part*.
4. Composite expansion: $y^{\mathrm{comp}}(x) = y^{\mathrm{out}}(x) + Y^{\mathrm{in}}((x - a)/\delta(\varepsilon)) - (\text{common part})$.

Each expansion is an [[asymptotic expansion|Q26QN6GYZX]] valid in its own region, and the composite one is meant to be valid uniformly on $[a, b]$.

## Remarks

Example: $\varepsilon y'' + y' + y = 0$ on $[0, 1]$, $y(0) = 0$, $y(1) = 1$.
Outer: $y' + y = 0$ with $y(1) = 1$ gives $y^{\mathrm{out}} = e^{1 - x}$.
Inner: with $X = x / \varepsilon$, the leading equation is $Y'' + Y' = 0$ with $Y(0) = 0$, so $Y^{\mathrm{in}} = A (1 - e^{-X})$.
Matching gives $A = y^{\mathrm{out}}(0) = e$, so the common part is $e$.
Composite: $y^{\mathrm{comp}} = e^{1 - x} - e^{1 - x/\varepsilon}$; it satisfies $y(0) = 0$ exactly and $y(1) = 1 - e^{1 - 1/\varepsilon}$, which is exponentially close to $1$.

Non-example: putting the outer solution and the inner solution side by side without step 3 fails, since the constant $A$ stays undetermined.
The layer must also be placed correctly: a trial layer at $x = 1$ leads to inner solutions that grow like $e^{X}$ away from the boundary and cannot match.

## Explanations

### L1 human

Solve the easy problem far from the layer, solve the zoomed-in problem in the layer, and glue the two by making them agree in between.
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<document source="https://tamnd.github.io/math2/node/CAZYGE1T4E.md" tag="CAZYGE1T4E" kind="concept">
# Levi-Civita connection

## Summary

The Levi-Civita connection of a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] is the [[affine connection|JX65KSVHSJ]] that has no [[torsion|JZNQAN00ZJ]] and preserves the metric.
By the [[fundamental theorem of Riemannian geometry|78JRW8KTH8]] there is exactly one such connection.
It is the connection used to define geodesics, parallel transport and curvature of a Riemannian manifold.

## Explanations

### L1 human

On a curved surface there is no obvious way to compare vectors at different points, so differentiating a vector field is not obvious.
A connection is a rule that does this, and there are many such rules.
The Levi-Civita connection is the one rule that respects lengths and angles and does not twist.
On a surface in $\R^3$ it is what one gets by differentiating in $\R^3$ and then projecting onto the tangent plane.
For $\R^n$ with the usual metric it is ordinary differentiation of the components.
</document>

<document source="https://tamnd.github.io/math2/node/CB4Y87RC4V.md" tag="CB4Y87RC4V" kind="definition">
# Rigid analytic space

## Definition

Let $K$ be a field complete for a nontrivial [[non-archimedean absolute value|1H5F1BP6XS]] $\abs{\cdot}$.
The *Tate algebra* in $n$ variables is
$$T_n = K\langle x_1, \dots, x_n \rangle = \set{\textstyle\sum_{\nu \in \N^n} a_\nu x^\nu : a_\nu \in K,\ \abs{a_\nu} \to 0 \text{ as } \abs{\nu} \to \infty},$$
the algebra of power series that converge on the closed unit polydisc.
An *affinoid algebra* is a $K$-algebra $A$ isomorphic to a quotient $T_n / I$ for some $n$ and some ideal $I$.
The *affinoid space* $\operatorname{Sp} A$ is the set of maximal ideals of $A$ with a Grothendieck topology and a structure [[sheaf|TB6TM6R4MD]] $\mathcal{O}$.
In this topology the admissible opens are the finite unions of affinoid subdomains.
The admissible coverings are the finite coverings by such sets.
A *rigid analytic space* over $K$ is a set $X$ with a Grothendieck topology and a sheaf of $K$-algebras, as a [[ringed space|KNDXDE7K8M]].
It must have an admissible covering by open subsets each isomorphic, with these structures, to an affinoid space.

## Remarks

That $\mathcal{O}$ is a sheaf for finite affinoid coverings is Tate's acyclicity theorem.
The general admissible opens and coverings are defined from these by a standard extension of the topology.

Example: $\operatorname{Sp} T_1$ is the closed unit disc.
Its points are the maximal ideals of $T_1$, which correspond to elements $x$ of an algebraic closure of $K$ with $\abs{x} \le 1$, up to conjugation over $K$.

Non-example of an admissible covering: for $K = \Q_p$, the closed disc $\operatorname{Sp} T_1$ is the disjoint union of the open disc $\set{\abs{x} < 1}$ and the circle $\set{\abs{x} = 1}$.
This covering is not admissible.
Every affinoid subdomain of the open disc lies in a disc $\set{\abs{x} \le r}$ with $r < 1$.
So no finite family of affinoid subdomains of the two pieces covers the points with $r < \abs{x} < 1$.
This is what keeps the closed disc connected in rigid geometry.

The analytification of a variety of finite type over $K$ is a rigid analytic space, and a rigid GAGA theorem compares coherent sheaves on both sides for proper varieties.
</document>

<document source="https://tamnd.github.io/math2/node/CB54N55JD9.md" tag="CB54N55JD9" kind="statement">
# Baily–Borel compactification

## Statement

Let $\mathbf{G}$ be a connected semisimple [[algebraic group|GH81SYAPPN]] over $\Q$.
Suppose that the symmetric space $D$ of $\mathbf{G}(\R)$ is Hermitian, so that $D$ is a [[bounded symmetric domain|Y7Q2SNKG0C]].
Let $\Gamma \subset \mathbf{G}(\Q)$ be an [[arithmetic subgroup|4XWG6S7R4Z]].
Then the quotient $\Gamma \backslash D$ has a compactification $(\Gamma \backslash D)^*$ with the following properties:

1. $(\Gamma \backslash D)^*$ is a normal [[projective variety|J7NBAJMQND]] over $\C$;
2. $\Gamma \backslash D$ is a Zariski-open dense subset of $(\Gamma \backslash D)^*$, so $\Gamma \backslash D$ is a normal quasi-projective variety;
3. the boundary $(\Gamma \backslash D)^* \setminus \Gamma \backslash D$ is a finite union of quotients of lower-dimensional bounded symmetric domains by arithmetic groups.

## Remarks

The theorem was proved by Walter Baily and Armand Borel in 1966.
As a topological space the compactification is Satake's: one adds the rational boundary components of $D$ and divides by $\Gamma$.
The projective embedding is given by automorphic forms on $D$ of sufficiently high weight, through Poincaré–Eisenstein series; $(\Gamma\backslash D)^*$ is the Proj of the graded ring of automorphic forms for $\Gamma$.

Example: for $D$ the upper half-plane and $\Gamma = SL_2(\Z)$, the boundary is one cusp, and $(\Gamma \backslash D)^* = \Gamma \backslash D \cup \set{\infty}$ is the projective line, with coordinate the $j$-invariant.
For $D$ the [[Siegel upper half-space|5PTYE97PF6]] $\mathbb{H}_g$ and $\Gamma = Sp(2g, \Z)$, the boundary strata are the quotients $Sp(2h, \Z) \backslash \mathbb{H}_h$ with $0 \le h < g$.

## Explanations

### L1 human

Quotients of bounded symmetric domains by arithmetic groups, such as the moduli spaces of elliptic curves or of abelian varieties, are not compact.
The theorem adds finitely many smaller pieces of the same kind at infinity and obtains a compact space.
This compact space is a projective algebraic variety, so the original quotient is an algebraic variety too.

### L2 human

The algebraicity of $\Gamma \backslash D$ is what makes it possible to study these spaces, such as Shimura varieties, with algebraic geometry and to define them over number fields.
The Baily–Borel compactification is minimal and usually very singular along the boundary.
The toroidal compactifications of Ash, Mumford, Rapoport and Tai can be smooth, and each of them maps onto the Baily–Borel compactification.
Borel's extension theorem says that holomorphic maps from a punctured polydisc into $\Gamma\backslash D$, with $\Gamma$ torsion-free, extend across the punctures to $(\Gamma\backslash D)^*$.
</document>

<document source="https://tamnd.github.io/math2/node/CB5G212Q03.md" tag="CB5G212Q03" kind="concept">
# Hilbert–Schmidt integral operator

## Summary

A Hilbert–Schmidt integral operator is an operator $f \mapsto \int K(\cdot, y) f(y) \, dy$ on a space $L^2$ whose kernel $K$ is square-integrable in both variables together.
Such operators are bounded by the $L^2$ norm of the kernel and are [[compact|M2NEP1K5AE]].
They form the [[Schatten class|153N5DW55Z]] of exponent $2$ on $L^2$, and they are the natural setting for the $L^2$ theory of integral equations.

## Explanations

### L1 human

The kernel $K(x, y)$ is like an infinite matrix, with rows indexed by $x$ and columns indexed by $y$.
The Hilbert–Schmidt condition asks that the sum of the squares of all entries be finite, here as a double integral.
This makes the operator behave much like a matrix.
For example, $K(x, y) = \min(x, y)$ on $[0, 1]$ is bounded, so its square has a finite integral, and its operator is Hilbert–Schmidt.
</document>

<document source="https://tamnd.github.io/math2/node/CB945JVQ0X.md" tag="CB945JVQ0X" kind="definition">
# q-derivative

## Definition

Let $q$ be a real number with $q \ne 0$ and $q \ne 1$.
Let $A \subseteq \R$ be a set with $qx \in A$ for every $x \in A$, and let $f \colon A \to \R$.
For $x \in A$ with $x \ne 0$, the *$q$-derivative* of $f$ at $x$ is
$$D_q f(x) = \frac{f(qx) - f(x)}{(q - 1) x}.$$
If $0 \in A$, one sets $D_q f(0) = f'(0)$ when the ordinary [[derivative|0V1EJCT8G6]] $f'(0)$ exists.

## Remarks

It is a [[difference quotient|AZBDB5F4M4]] of $f$ between $x$ and $qx$.
Example: for an integer $n \ge 1$ and $f(x) = x^n$ on $\R$, one gets $D_q x^n = \frac{q^n - 1}{q - 1} x^{n-1} = [n]_q \, x^{n-1}$, where $[n]_q = 1 + q + \cdots + q^{n-1}$ is the $q$-analog of $n$.
As $q \to 1$, $[n]_q \to n$, and more generally $D_q f(x) \to f'(x)$ at each $x \ne 0$ where $f$ is differentiable.
Non-example: with $q = 2$ and $f$ defined only on $A = [1, 2]$, the point $qx$ lies outside $A$ for $x > 1$, so $D_2 f(x)$ is not defined there.
The operator is linear and satisfies the product rule $D_q (fg)(x) = f(qx) \, D_q g(x) + g(x) \, D_q f(x)$.
</document>

<document source="https://tamnd.github.io/math2/node/CB9WJCN9BM.md" tag="CB9WJCN9BM" kind="concept">
# Krull domain

## Summary

A Krull domain is an [[integral domain|09S3PFBEPM]] that is the intersection of a family of discrete valuation rings of its fraction field.
The family must satisfy a finiteness condition: each nonzero element has nonzero valuation for only finitely many of them.
[[Unique factorisation domains|06YMXJMXYS]], Dedekind domains and Noetherian [[integrally closed domains|H20P600HQS]] are Krull domains.
Krull domains are the natural setting for the [[divisor class group|WN8PK6RQZR]], which measures their failure to be factorial.

## Explanations

### L1 human

In $\Z$, a rational number is an integer exactly when no prime appears in its denominator.
For each prime $p$, the exponent of $p$ in a fraction is a "valuation", and $\Z$ is the set of fractions where all these exponents are $\ge 0$.
A Krull domain is a ring that can be described in the same way by a family of such exponent functions.
Each nonzero element must have nonzero exponent for only finitely many of them.
This gives a theory of divisibility by "prime divisors" even when elements do not factor uniquely.
</document>

<document source="https://tamnd.github.io/math2/node/CBDM280TDF.md" tag="CBDM280TDF" kind="definition">
# Exact sequence

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]].
Consider a finite or infinite sequence of [[$R$-modules|0NZ8JA25WK]] and $R$-[[linear maps|0NRYHJPQG2]]
$$\cdots \to M_{i-1} \xrightarrow{f_{i-1}} M_i \xrightarrow{f_i} M_{i+1} \to \cdots.$$
The sequence is *exact at* $M_i$ if $\im f_{i-1} = \ker f_i$, where $\im$ is the [[image|0S21GNW9ZB]] and $\ker$ the [[kernel|0W88CEGQP1]].
The sequence is *exact* if it is exact at every module that has both an incoming and an outgoing map.
A *short exact sequence* is an exact sequence of the form $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$.

## Remarks

A sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is exact exactly when $f$ is injective, $g$ is surjective, and $\im f = \ker g$; then $C \cong B / f(A)$.
Exactness of $0 \to A \xrightarrow{f} B$ means that $f$ is injective, and exactness of $B \xrightarrow{g} C \to 0$ means that $g$ is surjective.
Exactness at $M_i$ implies $f_i \circ f_{i-1} = 0$, but not conversely.

Example: $0 \to \Z \xrightarrow{2} \Z \xrightarrow{\pi} \Z/2\Z \to 0$, with $\pi$ the reduction, is a short exact sequence of $\Z$-modules.
Multiplication by $2$ is injective, $\pi$ is onto, and $\im(2) = 2\Z = \ker \pi$.

Non-example: the sequence $\Z \xrightarrow{2} \Z \xrightarrow{\rho} \Z/4\Z$, with $\rho$ the reduction modulo $4$, is not exact at the middle term.
The image of the first map is $2\Z$, but $\ker \rho = 4\Z \ne 2\Z$.
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<document source="https://tamnd.github.io/math2/node/CBNYG6QV03.md" tag="CBNYG6QV03" kind="definition">
# Robust statistics

## Definition

*Robust statistics* is the study of statistical procedures that remain reliable when the true distribution of the data is only close to the assumed model.
The basic precise notion is Hampel's qualitative robustness.
Let $X_1, X_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with distribution $F$, and let $T_n = T_n(X_1, \dots, X_n)$, $n \ge 1$, be a sequence of [[estimators|0TTNRP5PFQ]].
Write $\mathcal{L}_F(T_n)$ for the law of $T_n$ when the $X_i$ have distribution $F$, and let $\pi$ be the [[Lévy–Prokhorov metric|TASMEPWNA6]] on probability distributions.
The sequence $(T_n)$ is *qualitatively robust* at $F$ if for every $\varepsilon > 0$ there is $\delta > 0$ such that for every distribution $G$ and every $n$,
$$\pi(F, G) < \delta \implies \pi\big(\mathcal{L}_F(T_n), \mathcal{L}_G(T_n)\big) < \varepsilon.$$

## Remarks

Qualitative robustness asks that the maps $G \mapsto \mathcal{L}_G(T_n)$ be continuous at $F$, uniformly in $n$.
Closeness in $\pi$ allows both small rounding of all data and a small fraction of arbitrary outliers.
The quantitative measures are the [[breakdown point|BRGGGZVQDS]] and the [[influence function|EB2QJQNE5Z]].

Example: if $F$ has a unique median $m$, the sample median is qualitatively robust at $F$.
A small fraction of outliers can shift it only to nearby order statistics, so its law changes little.

Non-example: the sample mean is not qualitatively robust at any $F$.
Let $0 < \eta < \delta$ and $G = (1 - \eta) F + \eta \, \delta_x$, with $\delta_x$ the point mass at $x$; then $\pi(F, G) \le \eta < \delta$.
For large $n$, with high probability about $n\eta$ observations equal $x$, and the mean moves by about $\eta x$, which is arbitrarily large for large $x$.
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<document source="https://tamnd.github.io/math2/node/CBRTFS0RVC.md" tag="CBRTFS0RVC" kind="statement">
# Entropic uncertainty principle

## Statement

Let $f \in L^2(\R)$ with $\norm{f}_2 = 1$, and let $\hat f$ be its Fourier transform on $L^2(\R)$, normalized so that $\hat f(\xi) = \int_\R f(x) e^{-2\pi i x \xi}\, dx$ for integrable $f$.
For a probability density $p$ on $\R$, let $H(p) = -\int_\R p(x) \log p(x)\, dx$ be its differential entropy, with the natural logarithm.
Suppose that the integrals defining $H(\abs{f}^2)$ and $H(\abs{\hat f}^2)$ converge absolutely.
Then
$$H(\abs{f}^2) + H(\abs{\hat f}^2) \ge \log \frac{e}{2}.$$
Equality holds when $f$ is a Gaussian $f(x) = (2a)^{1/4} e^{-\pi a x^2}$ with $a > 0$.

## Remarks

By the [[Plancherel theorem|HPZHRZ0502]], $\abs{\hat f}^2$ is also a probability density, so both entropies make sense.
The proof differentiates the [[Babenko–Beckner inequality|FFM7Y61KRT]] $\norm{\hat f}_q \le A_p \norm{f}_p$ with respect to $p$ at $p = 2$.
At $p = 2$ both sides are equal to $1$, so the derivative of the difference has a sign, and that sign is the entropy inequality.
I. I. Hirschman proved a weaker constant and conjectured this one; W. Beckner's sharp constant in the Hausdorff–Young inequality gave the conjecture.

On $\R^n$ the bound is $n \log \frac{e}{2}$.

The inequality implies the Heisenberg inequality.
For a density $p$ with variance $\sigma^2$, the entropy satisfies $H(p) \le \frac12 \log(2\pi e \sigma^2)$, with equality for normal densities.
Inserting this for $\abs{f}^2$ and $\abs{\hat f}^2$ gives $\sigma_x^2 \sigma_\xi^2 \ge 1/(16\pi^2)$.

## Explanations

### L1 human

Entropy measures how spread out a probability distribution is.
The theorem says that the spread of $\abs{f}^2$ and the spread of $\abs{\hat f}^2$, measured by entropy, cannot both be small.
Their sum is at least the number $\log(e/2)$, and Gaussians reach this bound.

### L2 human

Entropy is a finer measure of spread than variance: a density can have infinite variance and still finite entropy.
So this form of the uncertainty principle is stronger than the Heisenberg inequality, which it implies.
It is the model for entropic uncertainty relations in quantum information, where the Fourier transform is replaced by a pair of measurements in two bases.
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<document source="https://tamnd.github.io/math2/node/CBT8F8S8JT.md" tag="CBT8F8S8JT" kind="problem">
# Euler's elastica

## Problem

Let $\ell > 0$.
Consider plane curves $\gamma \colon [0, \ell] \to \R^2$ of class $C^2$, parametrised by arc length.
Prescribe the end points $\gamma(0)$, $\gamma(\ell)$ and the unit tangents $\gamma'(0)$, $\gamma'(\ell)$.
Among such curves, find those that make the bending energy
$$E(\gamma) = \frac{1}{2} \int_0^\ell \kappa(s)^2 \, ds$$
stationary, where $\kappa$ is the signed [[curvature|7JQS6VKFEZ]] of $\gamma$.
The solutions are the *elasticae*, the shapes of a thin elastic rod held at its ends.

## Remarks

Write $\gamma'(s) = (\cos \theta(s), \sin \theta(s))$, so that $\kappa = \theta'$.
Let $(X, Y) = \gamma(\ell) - \gamma(0)$.
The end points give the constraints $\int_0^\ell \cos \theta \, ds = X$ and $\int_0^\ell \sin \theta \, ds = Y$.
So this is an [[isoperimetric problem|6FY0KHMK5A]] with two constraints.
With multipliers, the Euler–Lagrange equation becomes
$$\theta'' + c \sin(\theta - \theta_0) = 0$$
for constants $c$ and $\theta_0$, the equation of a [[pendulum|0SHPXM2HWH]].
So the solutions are expressed through elliptic integrals and [[elliptic functions|FF9ZPCZ59K]].
Euler classified the possible shapes of the elasticae in 1744.

## Explanations

### L1 human

Bend a thin flexible ruler or a steel wire and hold its two ends.
It takes the shape that stores the least bending energy, which is measured by how sharply it curves along its length.
The angle of the wire along its length obeys the same equation as a swinging pendulum, and this gives a whole family of shapes, from gentle arcs to loops.
</document>

<document source="https://tamnd.github.io/math2/node/CBXJVT7M6H.md" tag="CBXJVT7M6H" kind="definition">
# Handlebody

## Definition

Let $g \ge 0$ be an integer.
A *handlebody of genus $g$* is a compact, connected, orientable $3$-dimensional smooth [[manifold with boundary|7DS6B2BFK8]] $V$ with the following property.
It has a [[handle decomposition|9V11N8VVTP]] with one $0$-handle, $g$ one-handles, and no other handles.

## Remarks

The [[orientability|702SNFQG0W]] condition fixes the way each $1$-handle is attached, and then a handlebody of genus $g$ is unique up to diffeomorphism.
Its boundary $\partial V$ is the closed orientable surface of [[genus|NJ787GZS1N]] $g$, and $V$ is homotopy equivalent to a wedge of $g$ circles.
Equivalently, for $g \ge 1$, $V$ is the boundary connected sum of $g$ solid tori $S^1 \times D^2$.
Also, $V$ is a closed regular neighbourhood of a connected finite graph in $\R^3$ with Euler characteristic $1 - g$.
Example: the solid torus $S^1 \times D^2$ is a handlebody of genus $1$.
It is the $3$-ball $D^3$ with a $1$-handle $D^1 \times D^2$ attached along two disjoint discs of $\partial D^3$, in an orientation-compatible way.
Non-example: the solid Klein bottle is obtained from $D^3$ by attaching one $1$-handle so that the result is non-orientable.
It has the right handles but is not orientable, so it is not a handlebody.
In higher-dimensional topology, *handlebody* often means any manifold with a handle decomposition, with no restriction on dimension or on the indices of the handles.
</document>

<document source="https://tamnd.github.io/math2/node/CBYB39FTFG.md" tag="CBYB39FTFG" kind="problem">
# Ulam's stability problem

## Problem

Let $G_1$ be a group and let $G_2$ be a group with a metric $d$.
Given $\varepsilon > 0$, does there exist $\delta > 0$ with the following property?
If a map $h \colon G_1 \to G_2$ satisfies $d(h(xy), h(x) h(y)) < \delta$ for all $x, y \in G_1$, then there is a [[group homomorphism|0JYTG5K9Y4]] $H \colon G_1 \to G_2$ with $d(h(x), H(x)) < \varepsilon$ for all $x \in G_1$.

## Remarks

The question asks when an approximate homomorphism lies close to an exact one.
Ulam posed it in a talk at the University of Wisconsin in 1940.
D. H. Hyers answered it in 1941 for additive maps between Banach spaces, with $\delta = \varepsilon$.
This answer is [[Hyers' theorem|VFRJ0P2P15]].
The answer depends on the groups: for some pairs of groups the property fails, so the problem is a family of questions rather than one.
Questions of this kind for other functional equations form the theory of [[Hyers–Ulam stability|TPYXQTYND6]].

## Explanations

### L1 human

Suppose a map respects the group law only up to a small error.
Ulam asked whether such a map must be close to a map that respects the law exactly.
For additive maps between Banach spaces the answer is yes, and this started a large field.
</document>

<document source="https://tamnd.github.io/math2/node/CBZ20SS8RQ.md" tag="CBZ20SS8RQ" kind="statement">
# Riesz decomposition theorem

## Statement

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]], and let $u$ be [[subharmonic|Z1MHMGFNFW]] on $\Omega$ and not identically $-\infty$ on any connected component.
Let $\mu$ be the [[Riesz measure|PR8KWEA3MH]] of $u$, so $\mu = \Delta u$ in the sense of distributions.
Let
$$E(x) = \frac{1}{2\pi} \log\abs{x} \ (n = 2), \qquad E(x) = -\frac{\abs{x}^{2-n}}{(n-2)\,\omega_n} \ (n \ge 3),$$
where $\omega_n$ is the area of the unit sphere in $\R^n$, so that $\Delta E = \delta_0$.
Let $U$ be an open set whose closure is compact and contained in $\Omega$.
Then there is a [[harmonic function|0DEB0JMN0W]] $h$ on $U$ such that
$$u(x) = \int_U E(x - y)\, d\mu(y) + h(x) \qquad \text{for all } x \in U.$$

## Remarks

Proof idea: the measure $\mu$ restricted to $U$ is finite, so $p(x) = \int_U E(x - y)\, d\mu(y)$ is subharmonic on $\R^n$ and $\Delta p = \mu$ on $U$ in the sense of distributions.
Then $\Delta(u - p) = 0$ on $U$ as a distribution, and Weyl's lemma makes $u - p$ equal, almost everywhere, to a harmonic function $h$.
Both $u$ and $p + h$ are subharmonic, and two subharmonic functions that agree almost everywhere agree everywhere, since each is the limit of its ball averages.

Example: for $u(z) = \log\abs{z}$ on $\C$, $\mu = 2\pi\delta_0$, so $\int E(z - w)\, d\mu(w) = \log\abs{z}$ and $h = 0$.
Let $f$ be a polynomial with leading coefficient $c$ and zeros $a_1, \dots, a_k$ listed with multiplicity.
Then $\log\abs{f(z)} = \sum_j \log\abs{z - a_j} + \log\abs{c}$ on all of $\C$, which has the same form, with the constant $h = \log\abs{c}$ as harmonic part.

A global form holds on a domain $\Omega$ with a Green function $G$, normalised by $-\Delta G(\cdot, y) = \delta_y$.
Let $u$ be a [[superharmonic|X4V2S56YQQ]] function on $\Omega$ that has a subharmonic minorant.
Then $u$ is the sum of the Green potential of the measure $-\Delta u$ and the greatest harmonic minorant of $u$.

## Explanations

### L1 human

Every subharmonic function is, locally, the potential of some charge distribution plus a harmonic function.
The charge distribution is its Riesz measure, and the harmonic part is the "smooth background".
So subharmonic functions are exactly the negatives of potentials of positive charges, up to harmonic corrections.

### L2 human

The theorem reduces many questions about subharmonic functions to questions about measures and their potentials.
For $\log\abs{f}$ with $f$ holomorphic it is the analytic form of factoring $f$ through its zeros, as in Jensen's formula and the Hadamard factorisation of entire functions.
In the global form on Green domains, it identifies the positive superharmonic functions with zero greatest harmonic minorant, the potentials, with Green potentials of measures.
</document>

<document source="https://tamnd.github.io/math2/node/CC666JT89X.md" tag="CC666JT89X" kind="definition">
# Lyapunov–Schmidt reduction

## Definition

Let $X$, $Y$ and $\Lambda$ be [[Banach spaces|06GZ956657]], let $U \subseteq X \times \Lambda$ be an open neighbourhood of $(0, 0)$, and let $F \colon U \to Y$ be [[continuously differentiable|0VW90BBSJ4]] with $F(0, 0) = 0$.
Let $L = D_x F(0, 0) \colon X \to Y$, and assume that $L$ is a Fredholm operator:

1. the [[kernel|0S23E6YE83]] $N = \ker L$ has finite dimension;
2. the range $R = L(X)$ is closed and has a complement of finite dimension in $Y$.

Choose closed subspaces $X_1$ and $Y_1$ with $X = N \oplus X_1$ and $Y = R \oplus Y_1$, and let $Q \colon Y \to R$ be the continuous [[projection|033DA1WE8Y]] onto $R$ along $Y_1$.
The restriction of $QL$ to $X_1$ is an isomorphism onto $R$.
Write $x = v + w$ with $v \in N$ and $w \in X_1$.
By the [[implicit function theorem|0BXBADR94C]], the equation $Q F(v + w, \lambda) = 0$ has a unique solution $w = W(v, \lambda)$ near $0$.
Here $(v, \lambda)$ is near $(0, 0)$, and $W(0, 0) = 0$.
The *Lyapunov–Schmidt reduction* of $F = 0$ at $(0, 0)$ is the *bifurcation equation*
$$(I - Q)\, F\big(v + W(v, \lambda), \lambda\big) = 0,$$
for $v \in N$ and $\lambda$ near $0$.
Its values lie in $Y_1$.

## Remarks

Near $(0, 0)$, the solutions of $F(x, \lambda) = 0$ are exactly the points $x = v + W(v, \lambda)$ with $(v, \lambda)$ a solution of the bifurcation equation.
Since $N$ and $Y_1$ are finite-dimensional, this is a system of $\dim Y_1$ equations in $\dim N$ unknowns.

Example: $X = Y = \R^2$, $\Lambda = \R$, $F(x, y, \lambda) = (\lambda x - x y,\ y - x^2)$.
Here $L = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$, so $N$ is the $x$-axis and $R$ is the $y$-axis; take $X_1 = R$ and $Y_1 = N$.
The equation $Q F = 0$ is $y - x^2 = 0$, so $W(x, \lambda) = x^2$.
The bifurcation equation is $\lambda x - x^3 = 0$, which has a pitchfork bifurcation at $\lambda = 0$.

Non-example: on the sequence space $\ell^2$, the operator $L(x_1, x_2, \ldots) = (x_1, x_2/2, x_3/3, \ldots)$ has trivial kernel, but its range is dense and not closed, so clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/CC8KNGMZYS.md" tag="CC8KNGMZYS" kind="definition">
# Lebesgue–Stieltjes integration

## Definition

Let $g \colon \R \to \R$ be a [[non-decreasing|0WFF6AZPBP]], right-continuous function.
Let $\mu_g$ be its [[Lebesgue–Stieltjes measure|QV75XJBFCS]], the Borel measure with $\mu_g((a, b]) = g(b) - g(a)$ for all $a < b$.
Let $f \colon \R \to \R$ be a Borel measurable function that is [[integrable|X7GTF3D98F]] with respect to $\mu_g$.
The *Lebesgue–Stieltjes integral* of $f$ with respect to $g$ is the [[Lebesgue integral|DWJF7ABPR1]]
$$\int_\R f \, dg = \int_\R f \, d\mu_g.$$

## Remarks

Example: let $g = \mathbf{1}_{[0, \infty)}$, the Heaviside step function with $g(0) = 1$, which is non-decreasing and right-continuous.
Then $\mu_g$ is the Dirac measure at $0$, so $\int f \, dg = f(0)$ for every Borel function $f$.
For $g(x) = x$ the measure $\mu_g$ is Lebesgue measure, and the integral is the usual Lebesgue integral.

Non-example: $g(x) = -x$ is not non-decreasing, so the definition does not apply directly.
The formula $\mu_g((a, b]) = g(b) - g(a)$ would give negative mass.

For $g$ of bounded variation and right-continuous, one writes $g = g_1 - g_2$ with $g_1, g_2$ non-decreasing and right-continuous, and sets $\int f \, dg = \int f \, dg_1 - \int f \, dg_2$.
On $[a, b]$ one writes $\int_{(a, b]} f \, dg$, and the endpoints matter when $g$ jumps there.
If $f$ is continuous on $[a, b]$, the integral over $(a, b]$ equals the Riemann–Stieltjes integral $\int_a^b f \, dg$.
If $g$ is absolutely continuous on bounded intervals, then $\int f \, dg = \int f(x) g'(x) \, dx$.
</document>

<document source="https://tamnd.github.io/math2/node/CCB8VNQCV0.md" tag="CCB8VNQCV0" kind="statement">
# Witt's theorem

## Statement

Let $F$ be a field whose [[characteristic|0F6ES3BRYP]] is not $2$.
Let $V$ be a [[finite-dimensional vector space|0ZHWKQMVX5]] over $F$ with a [[nondegenerate|01NHFZF2SK]] [[symmetric bilinear form|0AXZNE5XNH]] $B$.
Let $U$ and $W$ be [[linear subspaces|00YC1D652K]] of $V$, and let $\sigma \colon U \to W$ be a linear bijection with $B(\sigma(x), \sigma(y)) = B(x, y)$ for all $x, y \in U$.
Then there is a linear bijection $\tau \colon V \to V$ with $B(\tau(x), \tau(y)) = B(x, y)$ for all $x, y \in V$ and $\tau(x) = \sigma(x)$ for all $x \in U$.

## Remarks

The usual proof is by induction on $\dim U$, and it builds $\tau$ as a product of reflections in non-isotropic vectors; dividing by $2$ is needed to write these reflections.
A consequence is Witt cancellation: if $V_1 \perp V_2$ and $V_1 \perp V_3$ are isometric and $V_1, V_2, V_3$ are nondegenerate, then $V_2$ and $V_3$ are isometric.
Another consequence: all maximal totally isotropic subspaces of $V$ have the same dimension, the Witt index.
For example, in $\R^n$ with the dot product, any isometric map between two subspaces extends to an orthogonal map of $\R^n$.

The hypothesis on the characteristic is necessary for symmetric forms.
Over $\F_2$, take $V = \F_2^3$ with the dot product, $u = (1,1,1)$ and $e_1 = (1,0,0)$; both have $B(x,x) = 1$, so $e_1 \mapsto u$ is an isometry between lines.
An extension to $V$ would map the orthogonal complement of $e_1$ onto that of $u$.
But the complement of $e_1$ contains $(0,1,0)$ with $B(x,x) = 1$, while every $x$ in the complement of $u$ has $B(x,x) = x_1 + x_2 + x_3 = 0$.
The theorem holds in every characteristic for nondegenerate [[alternating forms|0DGDS5XY84]] and for quadratic forms with nondegenerate polar form.

## Explanations

### L1 human

A symmetric bilinear form measures something like lengths and angles, possibly with signs.
Witt's theorem says that if two configurations of vectors look the same for the form, then a symmetry of the whole space carries one onto the other.
So the symmetries of the form can carry any configuration onto any other configuration that the form cannot tell apart.

### L2 human

Witt's theorem is the foundation of the algebraic theory of [[quadratic forms|0RBQS1JJCR]].
Cancellation reduces the classification of forms to anisotropic forms plus a number of hyperbolic planes, the Witt decomposition.
This decomposition leads to the [[Witt group|KFVY0RPGF2]] of a field.
For real forms, cancellation gives a short proof of Sylvester's law of inertia.
</document>

<document source="https://tamnd.github.io/math2/node/CCEKM4241P.md" tag="CCEKM4241P" kind="statement">
# Kakutani's solution of the Dirichlet problem

## Statement

Let $d \ge 2$ and let $D \subset \R^d$ be a bounded [[domain|0Q4CBD3HX0]].
For $x \in \R^d$, let $(B_t)_{t \ge 0}$ be a $d$-dimensional [[Brownian motion|22735FGJV6]] started at $x$, with law $P^x$ and expectation $E^x$.
Let $\tau = \inf\set{t > 0 : B_t \notin D}$ be the exit time of $D$.
Let $\varphi \colon \partial D \to \R$ be continuous, and define
$$u(x) = E^x\big[\varphi(B_\tau)\big], \qquad x \in D.$$

1. $\tau < \infty$ almost surely, and $u$ is [[harmonic|0DEB0JMN0W]] on $D$.
2. If $\zeta \in \partial D$ is a [[regular boundary point|B7948M365K]], then $u(x) \to \varphi(\zeta)$ as $x \to \zeta$ with $x \in D$.
3. If every point of $\partial D$ is regular, then $u$ extended by $\varphi$ is the unique solution of the [[Dirichlet problem|8D56GQ4AG8]] with data $\varphi$.

## Remarks

Proof idea: $\tau$ is a [[stopping time|WND87FBAKC]], and $\tau < \infty$ because $D$ is bounded.
Take a ball $\overline{B}(x, r) \subset D$.
The strong Markov property at the exit time of the ball and rotation invariance show that $u(x)$ is the average of $u$ over the sphere $\partial B(x, r)$.
A bounded measurable function with this [[mean value property|0HYPR64JM2]] is harmonic.
At a regular point $\zeta$, a path started near $\zeta$ leaves $D$ quickly and close to $\zeta$, with high probability, which gives clause 2.
Example: for the unit disc and $x = 0$, the exit point $B_\tau$ is uniform on the circle, so $u(0)$ is the average of $\varphi$, as the mean value property requires.
Regularity is needed: on the punctured disc $D = \set{0 < \abs{x} < 1}$ in $\R^2$, Brownian motion never hits $0$, so $u$ ignores $\varphi(0)$.
With $\varphi(0) = 1$ and $\varphi = 0$ on the unit circle one gets $u \equiv 0$, which does not tend to $1$ at $0$.

## Explanations

### L1 human

To solve the Dirichlet problem at a point, start a random walk there and let it wander until it hits the boundary.
Read the boundary value where it hits, and average over all possible paths.
The result is the harmonic function with the given boundary values.

### L2 human

The theorem identifies classical potential theory with the theory of Brownian motion.
The probabilistic regularity criterion, $P^\zeta(\tau = 0) = 1$, agrees with the analytic notion of a regular point given by barriers or Wiener's criterion.
The formula also gives Monte Carlo methods for harmonic functions, such as the walk-on-spheres method.
The distribution of $B_\tau$ under $P^x$ is the harmonic measure of $D$ at $x$.
</document>

<document source="https://tamnd.github.io/math2/node/CCNCGN3P59.md" tag="CCNCGN3P59" kind="counterexample">
# Burali-Forti paradox

## Counterexample

Use von Neumann [[ordinals|0YSCSX5YZC]]: an ordinal is a transitive set that is [[well-ordered|01CGSJ0GNQ]] by the membership relation $\in$.
Suppose that the collection $\Omega$ of all ordinals is a [[set|0BSJ7RQ9BW]].
Every element of an ordinal is an ordinal, so $\Omega$ is transitive.
The ordinals are well-ordered by $\in$, so $\Omega$ is a transitive set well-ordered by $\in$.
Hence $\Omega$ is an ordinal, and so $\Omega \in \Omega$.
This contradicts the fact that no ordinal is an element of itself.
So the collection of all ordinals is not a set.

## Remarks

In [[naive set theory|XGAMQBSE1G]] every property defines a set, so the argument there is a paradox.
In [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] it is a theorem: the ordinals form a proper [[class|Y48FA69563]], not a set.
In the original form, the order type of all ordinals would be an ordinal greater than every ordinal, including itself.
Cesare Burali-Forti published the argument in 1897.

## Explanations

### L1 human

Each ordinal counts the length of a well-ordered list.
If all ordinals formed one set, that set would itself be a well-ordered list, so it would have a length.
That length would be an ordinal bigger than all ordinals, which is absurd.
The way out is that "all ordinals" is too big to be a set.
</document>

<document source="https://tamnd.github.io/math2/node/CCNKA8C711.md" tag="CCNKA8C711" kind="concept">
# Path homotopy

## Summary

A path homotopy is a deformation of one path into another with both end points held fixed.
It is a [[homotopy relative|ADDS6HE5SV]] to the end points $\set{0, 1}$ of the interval $[0, 1]$.
Classes of paths up to path homotopy form the [[fundamental group|S1XK3HXHFX]] and the [[fundamental groupoid|ZBHGXJF24E]].

## Explanations

### L1 human

Think of a path as a string from a point $p$ to a point $q$, with both ends nailed down.
A path homotopy moves the string around without moving the nails and without cutting it.
In the plane, any two paths from $p$ to $q$ are path homotopic.
In the plane without the origin, a path that passes above the origin and one that passes below it are not.
</document>

<document source="https://tamnd.github.io/math2/node/CCPS738AH8.md" tag="CCPS738AH8" kind="statement">
# Haar's uniqueness theorem

## Statement

Let $K$ be a [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]] with at least $n$ points.
Let $C(K)$ be the space of continuous functions $K \to \R$ with the supremum norm.
Let $V \subseteq C(K)$ be a linear subspace of dimension $n \ge 1$.
Then the following are equivalent.

1. Every $f \in C(K)$ has exactly one [[best approximation|7N4988HN3P]] from $V$ in the supremum norm.
2. $V$ is a Haar subspace: every basis of $V$ is a [[Chebyshev system|NY5AK617TR]], that is, every nonzero $v \in V$ has at most $n - 1$ zeros in $K$.

## Remarks

The idea of the proof: best approximations always exist, since $V$ is finite-dimensional.
If $V$ is Haar and $v_1, v_2$ are both best, then so is $(v_1 + v_2)/2$.
Its error attains the norm only at points where $v_1 - v_2$ vanishes.
For a Haar subspace the error of a best approximation attains its norm at $n + 1$ or more points, too many zeros for $v_1 - v_2 \ne 0$.
Conversely, if a nonzero $v \in V$ vanishes at $n$ points, one builds a continuous $f$ that has $0$ and all small multiples of a suitable element of $V$ as best approximations.

Example: the polynomials of degree at most $n - 1$ on $[a, b]$ form a Haar subspace, so best [[uniform approximation|XAARVNKXF1]] of a continuous function by polynomials of a fixed degree is unique.

Non-example: on $[-1, 1]$, the span of $1$ and $x^2$ is not Haar, since $x^2 - 1$ has two zeros.
Correspondingly $f(x) = x$ has many best approximations from it.
For every $g = a + c x^2$, the error $x - g$ takes the values $1 - a - c$ and $-1 - a - c$ at $\pm 1$, which differ by $2$, so $\norm{x - g}_\infty \ge 1$.
For $\abs{c} \le 1/2$, the error $h(x) = x - c(x^2 - 1)$ has $h' = 1 - 2cx \ge 0$ on $[-1, 1]$ and $h(\pm 1) = \pm 1$, so $\norm{h}_\infty = 1$.
Hence every $c(x^2 - 1)$ with $\abs{c} \le 1/2$ is a best approximation of $x$.

## Explanations

### L1 human

In the uniform norm, best approximations are not always unique.
Haar's theorem says exactly when they are: when no nonzero function of the approximating space has too many zeros.
Polynomials of a fixed degree pass this test, which is why the best uniform polynomial approximation is unique.

### L2 human

The theorem explains the special role of polynomials and of Chebyshev systems in uniform approximation.
For Haar subspaces the best approximation is characterized by alternation of the error at $n + 1$ points, which generalizes the equioscillation theorem.
The Mairhuber–Curtis theorem shows that Haar subspaces of dimension at least $2$ do not exist on open sets in several variables.
So best uniform approximation by multivariate polynomials of a fixed degree is not always unique.
</document>

<document source="https://tamnd.github.io/math2/node/CCQ0QVJY4R.md" tag="CCQ0QVJY4R" kind="definition">
# Facet

## Definition

Let $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d \ge 1$, where the dimension of a set is the dimension of its [[affine span|0YMB1676R1]].
A *facet* of $P$ is a [[face|KY7SB854VH]] $F$ of $P$ with $\dim F = d - 1$.

## Remarks

Example: the cube $[0,1]^3$ has dimension $3$.
The square $\set{x \in [0,1]^3 : x_3 = 1}$ is the intersection of the cube with the supporting plane $x_3 = 1$, and its affine span is a plane, so it is a facet.
The cube has six facets, one for each equation $x_i = 0$ or $x_i = 1$.

Non-example: the edge $\set{(t, 0, 0) : 0 \le t \le 1}$ of the same cube is a face, but its dimension is $1 \ne 2$, so it is not a facet.

Equivalently, when $P \subset \R^d$ is full-dimensional, $F$ is a facet if and only if $F = P \cap H$ for a supporting hyperplane $H$ such that $F$ spans $H$ affinely.
A full-dimensional polytope has an irredundant description by linear inequalities, unique up to positive scaling of each inequality.
Its inequalities correspond one to one to its facets.
The faces of dimension $0$ and $1$ are the vertices and edges; for $d = 3$ the facets are the usual two-dimensional faces.
</document>

<document source="https://tamnd.github.io/math2/node/CCR7BAQ5N5.md" tag="CCR7BAQ5N5" kind="concept">
# Oscillator representation

## Summary

The oscillator representation is a unitary representation of the [[metaplectic group|8B0JM2YDE2]], the double cover of the real symplectic group, on $L^2(\R^n)$.
It exists because symplectic linear maps preserve the canonical commutation relations, and the [[Stone–von Neumann theorem|Z877XDBMQ3]] then gives an intertwining operator for each of them.
It is the real case of the Weil representation, and it underlies the theory of theta functions and the [[theta correspondence|ZBGT50BW0Z]].
Its name comes from the quantum harmonic oscillator, whose Hamiltonian generates the action of a compact one-parameter subgroup.

## Explanations

### L1 human

Linear changes of the coordinates $(x, \xi)$ of position and momentum that keep the symplectic area form are the symplectic maps.
Quantum mechanics replaces $(x, \xi)$ by operators on wave functions $\psi(x)$.
Each symplectic change of coordinates then corresponds to a unitary operator on wave functions, defined up to a sign.
For example, a quarter turn in the $(x, \xi)$ plane corresponds, up to a constant, to the Fourier transform, which swaps position and momentum.
These operators together form the oscillator representation.
</document>

<document source="https://tamnd.github.io/math2/node/CCT8SQKHW4.md" tag="CCT8SQKHW4" kind="object">
# Levi-Civita field

## Description

A function $a \colon \Q \to \R$ has *left-finite support* if for every $q \in \Q$ there are only finitely many $r < q$ with $a(r) \ne 0$.
The Levi-Civita field $\mathcal{R}$ is the set of such functions, written as formal sums $a = \sum_{q \in \Q} a(q)\, \varepsilon^q$.
Addition is termwise, and the product is $(ab)(q) = \sum_{r + s = q} a(r) b(s)$, a finite sum because both supports are left-finite.
For $a \ne 0$, let $q_0$ be the least $q$ with $a(q) \ne 0$; then $a > 0$ means $a(q_0) > 0$.
With these operations and this order, $\mathcal{R}$ is an [[ordered field|010JCV9J4E]] that contains $\R$ as the constant series.

## Remarks

Check that $\varepsilon = 1 \cdot \varepsilon^1$ is a positive infinitesimal: $\varepsilon > 0$ since its leading coefficient is $1$, and for each integer $n \ge 1$ the element $1/n - \varepsilon$ has leading term $\frac{1}{n} \varepsilon^0$, so it is positive.
Hence $\mathcal{R}$ is not [[Archimedean|0HQARMKEV5]], and $\varepsilon^{-1}$ is larger than every real number.
Since exponents are rational, $\varepsilon$ has roots of every order, for example $(\varepsilon^{1/2})^2 = \varepsilon$.
The field $\mathcal{R}$ is [[real closed|778KFSM09P]], and it is Cauchy complete for the topology of its order.
Adjoining a square root of $-1$ gives an algebraically closed field.

## Explanations

### L1 human

The Levi-Civita field adds to the real numbers an infinitely small positive number $\varepsilon$ and its rational powers.
Its elements are series in $\varepsilon$ such as $3 + 2\varepsilon^{1/2} - \varepsilon^{5}$, with at most finitely many terms below each power.
A number is positive when its biggest term, the one with the lowest power of $\varepsilon$, is positive.
Such fields are used to compute with infinitesimals exactly, for example in automatic differentiation.
</document>

<document source="https://tamnd.github.io/math2/node/CCZA91NE5X.md" tag="CCZA91NE5X" kind="concept">
# Open segment

## Summary

The open segment between two points is the [[line segment|VQ4Q2V45RP]] between them with the two endpoints removed, when the points are distinct.
Open segments appear in the definitions of [[extreme points|3MSGFJM2NX]], extreme sets and faces.
These notions ask what happens when a point of a set lies strictly between two other points.
They also define strict convexity of sets.

## Explanations

### L1 human

Draw the segment from $x$ to $y$ and erase its two ends.
What is left is the open segment: the points strictly between $x$ and $y$.
In coordinates these are the points $(1 - t) x + t y$ with $0 < t < 1$.
For example, the open segment from $0$ to $1$ on the real line is the open interval $(0, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/CD2XB4Y1SF.md" tag="CD2XB4Y1SF" kind="concept">
# Treewidth

## Summary

Treewidth is a number that measures how close a graph is to a [[tree|0MJ0JS72DZ]].
It is defined through tree decompositions, which arrange overlapping sets of vertices along a tree.
Forests have treewidth at most $1$, and the complete graph $K_n$ has treewidth $n - 1$.
Treewidth is central in the theory of [[graph minors|VYQ32ZBPEE]].
Many problems that are hard in general can be solved efficiently on graphs of bounded treewidth.

## Explanations

### L1 human

A tree is easy to handle because removing any vertex that is not a leaf splits it into separate pieces.
A graph of small treewidth has the same property in a fuzzy way.
It can be cut into pieces by removing small sets of vertices, again and again, in a tree-like pattern.
For example, a cycle is not a tree, but it has treewidth $2$, so it is almost as simple as a tree.
A large square grid has large treewidth, because no small set of vertices splits it into small pieces.
</document>

<document source="https://tamnd.github.io/math2/node/CD35MSBNE7.md" tag="CD35MSBNE7" kind="statement">
# Frequency shift theorem

## Statement

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$, and let $F$ be its [[Laplace transform|05XT5AVJBK]].
Let $a$ be a complex number, and let $s$ be a complex number such that the Laplace integral of $f$ converges at $s - a$.
Then the Laplace integral of $t \mapsto e^{a t} f(t)$ converges at $s$, and
$$\mathcal{L}\{e^{a t} f(t)\}(s) = F(s - a).$$

## Remarks

The proof is one line: $e^{a t} f(t) e^{-s t} = f(t) e^{-(s - a) t}$, so the two integrals are the same integral.
In particular, the half-plane of convergence moves to the right by $\operatorname{Re} a$.
Example: from $\mathcal{L}\{\sin \omega t\}(s) = \omega / (s^2 + \omega^2)$ one gets $\mathcal{L}\{e^{a t} \sin \omega t\}(s) = \omega / ((s - a)^2 + \omega^2)$.
Likewise $\mathcal{L}\{t^n e^{a t}\}(s) = n! / (s - a)^{n + 1}$ for $\operatorname{Re} s > \operatorname{Re} a$.

## Explanations

### L1 human

Multiplying a signal by the [[exponential|0J0FCWV32D]] $e^{a t}$ just slides its transform sideways by $a$.
So a damped oscillation has the same transform as a plain oscillation, moved by the damping rate.

### L2 human

This rule is also called the first shifting theorem or the $s$-shift rule.
It is the main tool for inverting terms such as $1/((s + 2)^2 + 9)$: complete the square, then read off $\tfrac{1}{3} e^{-2t} \sin 3t$.
It is the mirror image of the [[time shift theorem|M4VJHPZJMQ]], where a shift in $t$ becomes a factor $e^{-a s}$.
</document>

<document source="https://tamnd.github.io/math2/node/CD6CAQ6XDN.md" tag="CD6CAQ6XDN" kind="statement">
# Raabe's test

## Statement

Let $(a_n)$ be a sequence of positive real numbers, and put
$$R_n = n \left( \frac{a_n}{a_{n+1}} - 1 \right).$$

1. If there are $r > 1$ and $N$ with $R_n \ge r$ for all $n \ge N$, then $\sum a_n$ [[converges|0CVVKY7GY2]].
2. If there is $N$ with $R_n \le 1$ for all $n \ge N$, then $\sum a_n$ [[diverges|0NBRP6093V]].

In particular, if $R_n \to R$, the series converges when $R > 1$ and diverges when $R < 1$.

## Remarks

This is [[Kummer's test|9JQEWK5405]] with $c_n = n$.
Then $\rho_n = n a_n / a_{n+1} - (n + 1) = R_n - 1$, and $\sum 1/n$ diverges.
For $R = 1$ the limit form gives no answer, and [[Bertrand's test|6T3E3AXZT7]] can then help.
Example: let $a_1 = 1/2$ and $a_{n+1} = a_n (2n+1)/(2n+2)$, so $a_n = \frac{1 \cdot 3 \cdots (2n-1)}{2 \cdot 4 \cdots (2n)}$.
The [[ratio test|0QNY6ZBXPN]] fails, since $a_{n+1}/a_n \to 1$.
Here $R_n = n/(2n+1) \le 1$, so $\sum a_n$ diverges.
For $a_n = 1/n^p$ one gets $R_n \to p$, which recovers the [[p-series test|0ZNSSK8ZK1]] for $p \ne 1$.

## Explanations

### L1 human

Raabe's test is a sharper version of the ratio test.
It is used when the ratio of consecutive terms tends to $1$.
It measures how fast the ratio approaches $1$: a gap like $p/n$ with $p > 1$ means convergence.

### L2 human

The ratio test compares a series with a geometric series.
Raabe's test compares it with the $p$-series $\sum 1/n^p$ instead, through Kummer's test with weights $c_n = n$.
It settles many series whose terms are quotients of products, such as binomial and hypergeometric coefficients.
[[Gauss's test|A7MJXJERNY]] is a convenient form when $a_n / a_{n+1}$ has an expansion in powers of $1/n$.
</document>

<document source="https://tamnd.github.io/math2/node/CD6SMMZSW8.md" tag="CD6SMMZSW8" kind="concept">
# Holomorphically convex hull

## Summary

Let $K$ be a compact subset of an open set $D \subseteq \C^n$.
Its holomorphically convex hull is the set of points of $D$ where no [[holomorphic function|507C34W0ZT]] on $D$ beats, in absolute value, its maximum on $K$.
It is the analogue of the convex hull, with holomorphic functions in place of linear functions.
Whether these hulls stay away from the boundary of $D$ decides whether $D$ is a domain of holomorphy.

## Explanations

### L1 human

The closed [[convex hull|0FQSDC2EQ2]] of a compact set consists of the points that every closed half-space containing the set also contains.
The holomorphic hull replaces half-spaces by the regions $\abs{f} \le c$ for holomorphic functions $f$.
For the unit circle in $\C$, the hull is the closed unit disc.
By the maximum principle, no holomorphic function on $\C$ can be larger inside the disc than on the circle.
If the origin is removed from the domain, the hull of the circle is just the circle, because the function $1/z$ is larger inside.
So the hull depends on the domain $D$ as well as on $K$.
</document>

<document source="https://tamnd.github.io/math2/node/CD7JZ43JBA.md" tag="CD7JZ43JBA" kind="definition">
# Clifford analysis

## Definition

Let $n \ge 1$ and let $\mathbb{R}_{0,n}$ be the real [[Clifford algebra|A38B1QFK20]] generated by $e_1, \ldots, e_n$ with the relations $e_j^2 = -1$ and $e_j e_k = -e_k e_j$ for $j \ne k$.
The *Dirac operator* on $\R^n$ is $D = \sum_{j=1}^{n} e_j \dfrac{\partial}{\partial x_j}$, acting on differentiable functions from open subsets of $\R^n$ to $\mathbb{R}_{0,n}$.
*Clifford analysis* is the study of functions from open subsets of $\R^n$ to $\mathbb{R}_{0,n}$ by means of $D$ and related [[Dirac operators|3DXGJTEQZH]].
Its main objects are the [[monogenic functions|646PNPT1R6]], the solutions of $Df = 0$.

## Remarks

The anticommutation relations give $D^2 = -\Delta$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] of $\R^n$: the mixed terms $e_j e_k \partial_j \partial_k + e_k e_j \partial_k \partial_j$ cancel.
So every monogenic function of class $C^2$ has harmonic components.

Example: for $n = 1$, $\mathbb{R}_{0,1} = \R + \R e_1$ with $e_1^2 = -1$ is a copy of $\C$, and $D = e_1 \partial_x$ only sees one real variable.
To recover complex analysis one uses instead the operator $\partial_{x_0} + e_1 \partial_{x_1}$ on $\R^2$, which is twice the [[Wirtinger derivative|WMXDVF36ZT]] $\partial_{\bar z}$; many authors work with this operator $\partial_{x_0} + \sum_j e_j \partial_{x_j}$ on $\R^{n+1}$.

Non-example: harmonic analysis of real-valued functions alone is not Clifford analysis, since it uses only $\Delta$ and not its first-order factor $D$.
</document>

<document source="https://tamnd.github.io/math2/node/CD9Y7YP4EJ.md" tag="CD9Y7YP4EJ" kind="statement">
# Esakia duality

## Statement

Let $f \colon X \to Y$ be a continuous order-preserving map between [[Esakia spaces|996BWM6ZQ7]].
Call $f$ an *Esakia morphism* if for all $x \in X$ and $y \in Y$ with $f(x) \le y$ there is $z \in X$ with $x \le z$ and $f(z) = y$.
Let $\mathbf{HA}$ be the category of [[Heyting algebras|NG943XNQWH]] with homomorphisms that preserve $0$, $1$, $\wedge$, $\vee$ and $\to$.
Let $\mathbf{Esa}$ be the category of Esakia spaces with Esakia morphisms.
Then $\mathbf{HA}$ is dually equivalent to $\mathbf{Esa}$, that is, $\mathbf{HA}$ is [[equivalent|K7A04ZBD92]] to the opposite category of $\mathbf{Esa}$.

1. A Heyting algebra $H$ goes to the set $X_H$ of prime filters of $H$, ordered by inclusion.
Its topology is generated by the sets $\set{P : a \in P}$ and $\set{P : a \notin P}$ for $a \in H$.
2. An Esakia space $X$ goes to the Heyting algebra of its clopen up-sets, with $U \to V = X \setminus {\downarrow}(U \setminus V)$.
3. The map $a \mapsto \set{P \in X_H : a \in P}$ is an isomorphism from $H$ onto the clopen up-sets of $X_H$.

## Remarks

The theorem refines [[Priestley duality|SA22SEC75K]] between bounded distributive lattices and Priestley spaces.
A Heyting algebra is in particular a bounded distributive lattice, and the extra structure of the implication matches exactly the condition that ${\downarrow} C$ is clopen for clopen $C$.
A lattice homomorphism between Heyting algebras preserves $\to$ exactly when its dual map satisfies the lifting condition above.
For a finite Heyting algebra the topology is discrete, and the theorem says that finite Heyting algebras are the algebras of up-sets of finite posets.

## Explanations

### L1 human

Every Heyting algebra can be rebuilt from a space of points with an order and a topology.
Algebra questions about Heyting algebras then turn into geometric questions about these spaces, and back.
Since Heyting algebras model intuitionistic logic, the theorem links the algebraic and the relational pictures of that logic.

### L2 human

The points of the dual space are the prime filters, which play the role of possible worlds in [[Kripke semantics|NGHWZ9Y7ZV]].
The duality turns subalgebras into images of Esakia morphisms and homomorphic images into closed up-sets, which makes it a working tool in the study of intermediate logics.
It is one member of the family of Stone-type dualities, next to [[Stone duality|B9T33D5MJK]] for Boolean algebras and [[Priestley duality|SA22SEC75K]] for distributive lattices.
</document>

<document source="https://tamnd.github.io/math2/node/CDGBK86D23.md" tag="CDGBK86D23" kind="definition">
# De Morgan algebra

## Definition

A *De Morgan algebra* is a structure $(A, \vee, \wedge, 0, 1, \sim)$ such that:

1. $(A, \vee, \wedge, 0, 1)$ is a [[bounded lattice|72V1Z82M5F]] that is a [[distributive lattice|0HM0HQCK6N]].
2. ${\sim} \colon A \to A$ satisfies ${\sim}{\sim} a = a$ for every $a \in A$.
3. ${\sim}(a \vee b) = {\sim} a \wedge {\sim} b$ for all $a, b \in A$.

## Remarks

From clauses 2 and 3 one gets ${\sim}(a \wedge b) = {\sim} a \vee {\sim} b$, ${\sim} 0 = 1$ and ${\sim} 1 = 0$.
So ${\sim}$ is an order-reversing bijection of $A$ onto itself.

Example: the interval $[0, 1]$ with $a \vee b = \max(a, b)$, $a \wedge b = \min(a, b)$ and ${\sim} a = 1 - a$.
A chain is a distributive lattice, $1 - (1 - a) = a$, and $1 - \max(a, b) = \min(1 - a, 1 - b)$.
It is not a Boolean algebra, since $\tfrac12 \wedge {\sim} \tfrac12 = \tfrac12 \ne 0$.

Non-example: the chain $0 < m < 1$ with ${\sim}$ the [[pseudocomplement|WMR73J0S26]], so ${\sim} 0 = 1$, ${\sim} m = 0$, ${\sim} 1 = 0$.
Clause 2 fails, since ${\sim}{\sim} m = 1 \ne m$.
</document>

<document source="https://tamnd.github.io/math2/node/CDJNQ3SQTG.md" tag="CDJNQ3SQTG" kind="definition">
# Lipschitz domain

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]].
The set $\Omega$ is a *Lipschitz domain* if for every point $p$ of its [[boundary|05D5EGMFYV]] $\partial\Omega$ there are:

1. a radius $r > 0$;
2. a rigid motion of $\R^n$ giving coordinates $x = (x', x_n)$ with $x' \in \R^{n-1}$ and $x_n \in \R$;
3. a [[Lipschitz|0AW1E6YTJY]] function $\varphi \colon \R^{n-1} \to \R$,

such that, in these coordinates,
$$\Omega \cap B(p, r) = \set{x \in B(p, r) : x_n > \varphi(x')},$$
where $B(p, r)$ is the [[open ball|0DC3MPR7C9]] of radius $r$ about $p$.

## Remarks

It follows that $\partial\Omega \cap B(p, r) = \set{x \in B(p, r) : x_n = \varphi(x')}$.
Many authors also require $\Omega$ to be bounded, and sometimes connected.
If $\Omega$ is bounded, finitely many such charts cover $\partial\Omega$, so one can use a common radius and a common Lipschitz constant.
These numbers are called the Lipschitz character of $\Omega$.
Example: the square $(0, 1)^2$.
Near an edge point the square is the region above a horizontal line.
Near the corner $(0, 0)$, in coordinates rotated by $45$ degrees, it is the region $\set{x_2 > \abs{x_1}}$ near the corner, and $\abs{x_1}$ is Lipschitz with constant $1$.
Non-example: the slit disc $\set{\abs{z} < 1} \setminus [0, 1)$ in $\R^2$.
Near the point $1/2$ of the slit, the set lies on both sides of the slit, so it cannot be the region on one side of a graph.
Every bounded open set with $C^1$ boundary is a Lipschitz domain.
</document>

<document source="https://tamnd.github.io/math2/node/CDM40MT3HZ.md" tag="CDM40MT3HZ" kind="statement">
# Hölder's theorem on Archimedean groups

## Statement

Let $(G, \le)$ be a [[linearly ordered group|1P8087YGYQ]] with identity $e$.
Suppose that $G$ is *Archimedean*: for all $a, b \in G$ with $a > e$, there is an integer $n \ge 1$ with $b < a^n$.
Then there is an injective [[group homomorphism|0HV9Y7WG5R]] $\varphi \colon G \to (\R, +)$ such that $a \le b$ if and only if $\varphi(a) \le \varphi(b)$.
In particular, $G$ is abelian.

## Remarks

If $G = \set{e}$ there is nothing to prove.
Otherwise one fixes an element $u > e$ and sends $g$ to the real number $\sup \set{m/n : n \ge 1, \ u^m \le g^n}$.
The Archimedean property makes this set bounded above and nonempty, and an argument with [[Dedekind cuts|04QNZZ4HHY]] shows that the map is an [[injective|0PVG1CRAJN]] order-preserving homomorphism.
Commutativity of $G$ then follows from commutativity of $\R$.
Example: the subgroup $\Z + \Z\sqrt{2}$ of $\R$ is Archimedean and is already a subgroup of [[the reals|08J2M1JYAG]].
The Archimedean hypothesis is needed: $\Z^2$ with the lexicographic order is linearly ordered, but $n \cdot (0, 1) < (1, 0)$ for all $n$.
No order-preserving injective homomorphism to $\R$ exists for it, since the image of $(0, 1)$ would be positive with all its multiples below the image of $(1, 0)$.

## Explanations

### L1 human

A linearly ordered group is Archimedean if no element is infinitely large compared with another positive element.
Hölder's theorem says that such a group is just a group of real numbers in disguise.
So the real numbers contain every way of measuring with a single scale and no infinitely large or small quantities.

### L2 human

The theorem is the base case for the structure theory of linearly ordered abelian groups.
A general linearly ordered abelian group splits into Archimedean classes, and the quotient attached to each class is Archimedean, so Hölder's theorem embeds it into $\R$.
The [[Hahn embedding theorem|FVPXXFNVQY]] puts these real pieces together in a lexicographic product.
The theorem also explains why the value group of a rank one valuation can be taken inside $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/CDMXAH4A7G.md" tag="CDMXAH4A7G" kind="statement">
# Dormand–Prince method

## Statement

Consider the explicit seven-stage [[Butcher tableau|RH8BD7PW5F]] with two weight rows
$$\begin{array}{c|ccccccc}
0 \\
\frac15 & \frac15 \\
\frac{3}{10} & \frac{3}{40} & \frac{9}{40} \\
\frac45 & \frac{44}{45} & -\frac{56}{15} & \frac{32}{9} \\
\frac89 & \frac{19372}{6561} & -\frac{25360}{2187} & \frac{64448}{6561} & -\frac{212}{729} \\
1 & \frac{9017}{3168} & -\frac{355}{33} & \frac{46732}{5247} & \frac{49}{176} & -\frac{5103}{18656} \\
1 & \frac{35}{384} & 0 & \frac{500}{1113} & \frac{125}{192} & -\frac{2187}{6784} & \frac{11}{84} \\
\hline
b^{T} & \frac{35}{384} & 0 & \frac{500}{1113} & \frac{125}{192} & -\frac{2187}{6784} & \frac{11}{84} & 0 \\
\hat b^{T} & \frac{5179}{57600} & 0 & \frac{7571}{16695} & \frac{393}{640} & -\frac{92097}{339200} & \frac{187}{2100} & \frac{1}{40}
\end{array}$$
Then:

1. The [[Runge–Kutta method|7DBPJKYC6C]] with weights $b$ has order $5$, and the one with weights $\hat b$ has order $4$.
2. The last row of the matrix equals $b^{T}$ and $c_7 = 1$.
   So the seventh stage of a step equals $f(t_{n+1}, y_{n+1})$, which is the first stage of the next step.
3. The difference $y_{n+1} - \hat y_{n+1}$ of the two results equals the [[local truncation error|T220HF0562]] of the fourth-order step up to a term $O(h^6)$, for $f$ smooth enough.

## Remarks

Clause 1 is checked against Butcher's order conditions on rooted trees; for example $\sum b_i = \sum \hat b_i = 1$ and $\sum b_i c_i^4 = \frac15$, while $\sum \hat b_i c_i^4 \ne \frac15$.
Clause 2 is the "first same as last" property: an accepted step needs only six new evaluations of $f$.
The method propagates the fifth-order value $y_{n+1}$, a practice called local extrapolation, and uses the error estimate of clause 3 for [[adaptive step size|6MREWBZR7R]] control.
John Dormand and Peter Prince published the pair in 1980, choosing the free coefficients to make the error constants of the fifth-order formula small.

## Explanations

### L1 human

The Dormand–Prince method is a Runge–Kutta method that computes two answers per step, one of order five and one of order four.
Their difference tells the solver how large the error is, so it can adjust the step size.
It reuses its last slope evaluation as the first of the next step, which saves work.

### L2 human

This pair, often called DOPRI5 or RK45, is the default non-stiff solver in many software packages, for example the ode45 function of MATLAB.
Compared with the [[Runge–Kutta–Fehlberg method|6D73FFD9DC]], it advances with the more accurate formula and has smaller error coefficients.
It also admits a dense output, a polynomial interpolant that gives accurate values between the grid points.
For stiff problems it needs very small steps, and implicit methods are preferred.
</document>

<document source="https://tamnd.github.io/math2/node/CDS1VWB7A3.md" tag="CDS1VWB7A3" kind="concept">
# Morphism of varieties

## Summary

A morphism of varieties is a map between algebraic varieties that is given locally by polynomials, or by quotients of polynomials with nonvanishing denominators.
Morphisms are the maps of the category of varieties, and an isomorphism is a morphism with an inverse morphism.
Between affine varieties, morphisms correspond exactly to homomorphisms of the coordinate rings in the opposite direction.

## Explanations

### L1 human

A morphism is a map between solution sets of polynomial equations that is itself written with polynomials.
The map $t \mapsto (t, t^2)$ from the line onto the parabola $y = x^2$ is a morphism, and so is its inverse $(x, y) \mapsto x$.
So the line and the parabola are isomorphic varieties.
The map $t \mapsto (t^2, t^3)$ onto the cusp $y^2 = x^3$ is a morphism and a bijection, but its inverse is not a morphism, so these two are not isomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/CDVA7EJSSD.md" tag="CDVA7EJSSD" kind="concept">
# Ambiguity function

## Summary

The ambiguity function of a signal $f$ measures how much $f$ correlates with copies of itself shifted in time and in frequency.
In radar it describes how well a transmitted pulse lets one tell targets apart by range (time delay) and by speed (Doppler frequency shift).
It is the [[short-time Fourier transform|3DHNP5NEWH]] of $f$ with window $f$, up to a phase factor.
It is related to the [[Wigner distribution|FCHW5QGG5V]] by a two-dimensional Fourier transform.

## Explanations

### L1 human

A radar sends a pulse and listens for its echo.
The echo comes back delayed, because the target is far, and shifted in frequency, because the target moves.
The ambiguity function tells, for each delay and frequency shift, how much the shifted echo still looks like the original pulse.
A good pulse has an ambiguity function with one sharp peak at zero, so that different targets cannot be confused.
</document>

<document source="https://tamnd.github.io/math2/node/CDVRMVED2V.md" tag="CDVRMVED2V" kind="concept">
# Latin square

## Summary

A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols.
Each symbol occurs exactly once in each row and exactly once in each column.
Latin squares are the multiplication tables of quasigroups, and the Cayley tables of finite groups are examples.
They are used in the design of experiments and in coding theory, and orthogonal families of them are tied to finite projective planes.
The node [[Latin square|44HTG620GB]] gives the precise definition.

## Explanations

### L1 human

A completed Sudoku grid is a Latin square of order $9$ with extra conditions on the boxes.
Without the boxes, the rule is only: no symbol repeats in a row or in a column.
For order $3$, the rows $1\,2\,3$, then $2\,3\,1$, then $3\,1\,2$ form a Latin square.
Each new row is the previous row shifted by one place.
</document>

<document source="https://tamnd.github.io/math2/node/CDVYCKZ2XZ.md" tag="CDVYCKZ2XZ" kind="definition">
# Hidden Markov model

## Definition

Let $S$ and $O$ be finite sets, the *hidden states* and the *observations*.
Let $\lambda$ be a probability distribution on $S$, and let $A = (a_{ij})_{i, j \in S}$ be a [[transition matrix|0F462493EA]].
Let $E = (e_i(o))_{i \in S, o \in O}$ be a matrix with $e_i(o) \ge 0$ and $\sum_{o \in O} e_i(o) = 1$ for each $i$.
A *hidden Markov model* with these parameters is a pair of processes $(X_n, Y_n)_{n \ge 0}$ with values in $S \times O$ such that:

1. $(X_n)$ is a [[Markov chain|033TJBZDG0]] with initial law $\lambda$ and transition matrix $A$;
2. given $(X_n)_{n \ge 0}$, the variables $Y_0, Y_1, \ldots$ are [[conditionally independent|0THJ7416DX]];
3. $P(Y_n = o \mid (X_k)_{k \ge 0}) = e_{X_n}(o)$ for all $n \ge 0$ and $o \in O$.

Only $(Y_n)$ is observed.

## Remarks

The joint law is $P(X_0 = i_0, Y_0 = o_0, \ldots, X_n = i_n, Y_n = o_n) = \lambda_{i_0} e_{i_0}(o_0) \prod_{k=1}^{n} a_{i_{k-1} i_k} e_{i_k}(o_k)$.
Example: let $S = \set{F, L}$ for a fair and a loaded die, with $a_{FL} = a_{LF} = 0.05$, and let $O = \set{1, \ldots, 6}$.
Take $e_F(o) = 1/6$ for all $o$, $e_L(6) = 1/2$ and $e_L(o) = 1/10$ for $o \le 5$; each row sums to $1$, so clauses 1 to 3 define a model.
Non-example: if $Y_n$ is allowed to depend on $Y_{n-1}$ as well as on $X_n$, clause 3 fails.
The observed process $(Y_n)$ alone is in general not a [[Markov chain|0G48PYM96Q]].
The forward algorithm computes the likelihood $P(Y_0 = o_0, \ldots, Y_n = o_n)$ in time of order $n \abs{S}^2$, and the Viterbi algorithm finds a most likely hidden path.
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# Approximate identity

## Summary

An approximate identity in a [[Banach algebra|9KYWN94QDN]] is a [[net|4DVJQJV0DG]] of elements that behaves more and more like a unit.
Multiplying by it changes each element less and less.
Many important algebras have no unit, such as the convolution algebra $L^1(\R)$ and the algebra of continuous functions vanishing at infinity.
In these algebras an approximate identity often replaces the unit in proofs.
Every C*-algebra has one, and a bounded approximate identity is the hypothesis of the Cohen factorization theorem.

## Explanations

### L1 human

Think of a sequence of bumps that equal $1$ on larger and larger intervals.
Multiplying a function that dies out at infinity by such a bump changes it only far away, where it is already small.
So the bumps act almost like the constant function $1$, which itself is not in the algebra.
In $L^1(\R)$ with convolution, taller and thinner spikes of area $1$ play the same role: convolving with them averages over tiny windows and changes a function very little.
</document>

<document source="https://tamnd.github.io/math2/node/CE0V9686M8.md" tag="CE0V9686M8" kind="definition">
# Haar measure

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] and let $\mathcal{B}$ be the $\sigma$-algebra of [[Borel sets|0WQXY32Q8D]] of $G$.
A *left Haar measure* on $G$ is a measure $\mu$ on $\mathcal{B}$ such that:

1. $\mu$ is a [[Radon measure|HZNEQQBCW4]]: it is finite on compact sets, outer regular on Borel sets and inner regular on open sets;
2. $\mu$ is not the zero measure;
3. $\mu$ is left-invariant: $\mu(gA) = \mu(A)$ for every $g \in G$ and every $A \in \mathcal{B}$, where $gA = \set{ga : a \in A}$.

A *right Haar measure* is defined in the same way with $\mu(Ag) = \mu(A)$ in clause 3.

## Remarks

Example: Lebesgue measure $\lambda$ on $(\R^n, +)$.
It is finite on compact sets, regular and nonzero, and $\lambda(x + A) = \lambda(A)$ for every $x \in \R^n$ and Borel set $A$.

Non-example: the Dirac measure $\delta_0$ on $(\R, +)$ is a nonzero Radon measure but fails clause 3, since $\delta_0(\set{0}) = 1$ and $\delta_0(1 + \set{0}) = \delta_0(\set{1}) = 0$.

In integral form, left invariance says $\int_G f(gx) \, d\mu(x) = \int_G f(x) \, d\mu(x)$ for every $g \in G$ and every compactly supported continuous $f$.
If $\mu$ is a left Haar measure, then $A \mapsto \mu(A^{-1})$ is a right Haar measure.
A left Haar measure gives positive measure to every nonempty open set.
</document>

<document source="https://tamnd.github.io/math2/node/CE8GX6ADZS.md" tag="CE8GX6ADZS" kind="concept">
# Betweenness

## Summary

Betweenness is the relation "the point $y$ lies between the points $x$ and $z$".
In an [[affine space|09XNAPTJQ7]] over an [[ordered field|0P3VCBDEE2]] it means that $y$ lies on the segment from $x$ to $z$.
In axiomatic geometry it is a primitive relation, and the axioms of order, such as [[Pasch's axiom|KJT98G16Y4]], describe it.
Segments, rays, half-planes and the inside of a triangle are all defined from it.

## Explanations

### L1 human

Walk along a straight road from $x$ to $z$.
Every point you pass on the way is between $x$ and $z$.
On the number line, $2$ is between $1$ and $5$, and $7$ is not.
This simple idea of order along a line is enough to define segments and sides of a line.
</document>

<document source="https://tamnd.github.io/math2/node/CEAT9BESKK.md" tag="CEAT9BESKK" kind="concept">
# Amenable group

## Summary

An amenable group is a group with a finitely additive, translation-invariant way to measure the size of all of its subsets, with total size $1$.
Finite groups, abelian groups and solvable groups are amenable; groups containing a non-abelian [[free group|0KTW26C80J]] are not.
Amenability has many equivalent forms, such as the [[Følner condition|82R5XGCART]].
It governs the existence of paradoxical decompositions such as the one in the [[Banach–Tarski paradox|FH250K068Z]].

## Explanations

### L1 human

Try to assign to every subset of a group the "fraction of the group" it fills.
Disjoint pieces should add up, and moving a set by a group element should not change its fraction.
For a finite group this is easy: take the number of elements divided by the order of the group.
For the integers one can do it too, with a kind of limiting density, though not by an explicit formula.
For the free group on two generators it is impossible: the group can be cut into pieces that, after moving, make two full copies of the group.
Amenable groups are those where such a fair notion of size exists.
</document>

<document source="https://tamnd.github.io/math2/node/CEC3DESCFT.md" tag="CEC3DESCFT" kind="concept">
# Meixner polynomials

## Summary

The Meixner polynomials $M_n(x; \beta, c)$ are the orthogonal polynomials of the negative binomial (Pascal) distribution on the nonnegative integers.
They are a discrete analogue of the [[Laguerre polynomials|WAMRHWW82R]], which arise from them in a limit.
They sit in the [[Askey scheme|RR2S05C09B]] at the level of the ${}_2F_1$ series, together with the Jacobi and Krawtchouk polynomials.

## Explanations

### L1 human

Repeat a trial with success probability $c$ until a fixed number $\beta$ of failures, and count the successes.
Such counts follow a negative binomial distribution, with weights proportional to $\frac{(\beta)_x}{x!} c^x$ on $x = 0, 1, 2, \dots$.
The Meixner polynomials are the polynomials in $x$ that are perpendicular to each other for these weights.
They play for this distribution the role that the Laguerre polynomials play for the gamma distribution.
</document>

<document source="https://tamnd.github.io/math2/node/CECPTP66EK.md" tag="CECPTP66EK" kind="definition">
# Algebraic variety

## Definition

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]].
An *algebraic variety* over $k$ (in the classical sense) is one of the following, with its Zariski topology:

1. an [[affine variety|JERVQT4C12]], that is, an irreducible algebraic set in $k^n$;
2. a quasi-affine variety, that is, a nonempty open subset of an affine variety;
3. a [[projective variety|A4H4CZ2V35]], that is, an irreducible projective algebraic set in $\mathbb{P}^n$;
4. a [[quasi-projective variety|EQSHJPE7H8]], that is, a nonempty open subset of a projective variety.

## Remarks

Each of the four kinds is a quasi-projective variety, so the definition says that a variety is a quasi-projective variety.
Every variety is [[irreducible|SY2MWC434Z]] and its topology is Noetherian.

Example: the affine line $k$, the projective plane $\mathbb{P}^2$, and the punctured plane $k^2 \setminus \set{(0, 0)}$ are varieties.

Non-example: $V(xy) \subseteq k^2$ is not a variety in this sense, since it is reducible.

Other conventions exist.
Some authors allow reducible varieties.
In the language of schemes, a variety over any field $k$ is often an integral separated scheme of finite type over $k$.
This includes abstract varieties that are not quasi-projective.
</document>

<document source="https://tamnd.github.io/math2/node/CED4Q4TFCD.md" tag="CED4Q4TFCD" kind="concept">
# Boolean product

## Summary

A Boolean product is a [[subdirect product|88WN3AD2VQ]] of [[algebras|QJGPT2W0FT]] indexed by the points of a [[Stone space|ZV38HA5PAQ]].
Its elements behave like continuous functions on that space.
The sets where two elements agree are clopen, and elements can be cut and pasted along clopen sets.
Boolean products give sheaf-like representations of algebras.
They are a central tool in the study of discriminator varieties.

## Explanations

### L1 human

Think of an element of a Boolean product as a function on a totally disconnected space, taking its value at each point in a factor algebra.
Two such functions agree on a set that is both open and closed.
Given two elements and a clopen set, you can take the first element on the clopen set and the second one elsewhere, and the result is again an element.
Every Boolean algebra is a Boolean product of copies of the two-element Boolean algebra, over its Stone space.
</document>

<document source="https://tamnd.github.io/math2/node/CEGF8W64VQ.md" tag="CEGF8W64VQ" kind="concept">
# Cauchy's exponential equation

## Summary

Cauchy's exponential equation is the [[functional equation|SRAN4EKWS0]] $f(x + y) = f(x) f(y)$ for an unknown function $f \colon \R \to \R$.
It asks which functions turn sums into products.
Its continuous solutions are the zero function and the [[exponential functions|0J0FCWV32D]] $x \mapsto a^x$ with $a > 0$.
It is one of the four basic equations of Cauchy, together with the additive, logarithmic and power equations.

## Explanations

### L1 human

Growth by a fixed factor in equal steps is the picture behind this equation.
If a quantity is multiplied by $f(x)$ over a time $x$ and by $f(y)$ over a further time $y$, it is multiplied by $f(x) f(y)$ over the time $x + y$.
For example, $f(x) = 2^x$ satisfies $2^{x + y} = 2^x \cdot 2^y$.
The equation says that this rule alone almost determines the function.
</document>

<document source="https://tamnd.github.io/math2/node/CEHHBBD6NT.md" tag="CEHHBBD6NT" kind="concept">
# Translation surface

## Summary

A translation surface is a closed surface with a flat metric whose charts change by translations, except at finitely many cone points.
Equivalently, it is a compact [[Riemann surface|5K0N8QMV6Q]] with a nonzero holomorphic 1-form, an [[abelian differential|P014C13K5H]].
It can also be given by finitely many polygons in the plane with sides glued in pairs by translations.
Straight-line flows on translation surfaces generalize linear flows on the torus and are closely tied to [[interval exchange transformations|19JFF3GY9F]] and to billiards in rational polygons.
The space of translation surfaces carries an action of $SL(2, \R)$, whose study links dynamics with [[Teichmüller theory|ZZGC6BZNJ5]].

## Explanations

### L1 human

Take a square of paper and glue opposite sides: you get a torus that is flat everywhere, on which straight lines make sense.
Do the same with a regular octagon, gluing each side to the opposite one, and you get a surface with two holes.
It is flat everywhere except at one point, where all eight corners meet and the angles add up to three full turns instead of one.
Such surfaces, made of polygons glued by translations, are translation surfaces.
</document>

<document source="https://tamnd.github.io/math2/node/CEK4HKMPB4.md" tag="CEK4HKMPB4" kind="definition">
# Formally smooth map

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $A$ be a commutative [[$R$-algebra|DPBJ04JMH8]].
Then $A$ is *formally smooth* over $R$, and the structure map $R \to A$ is a *formally smooth map*, if the following holds.
Let $C$ be a commutative $R$-algebra and let $J \subseteq C$ be an [[ideal|096Z9MP2ME]] with $J^2 = 0$.
Let $\pi \colon C \to C/J$ be the [[quotient map|0076HWHGHQ]], and let $u \colon A \to C/J$ be an $R$-algebra [[homomorphism|0VTH8KPCG2]].
The condition is that for every such $C$, $J$ and $u$ there is an $R$-algebra homomorphism $v \colon A \to C$ with $\pi \circ v = u$.

## Remarks

The lift $v$ need not be unique.
If the lift always exists and is unique, $A$ is *formally étale* over $R$; if it is always unique, $A$ is *formally unramified* over $R$.

Example: let $A = R[x_i : i \in I]$ be a [[polynomial ring|0SDC934R52]] in any set of variables.
Given $u \colon A \to C/J$, choose for each $i$ an element $c_i \in C$ with $\pi(c_i) = u(x_i)$, and let $v$ be the $R$-algebra map with $v(x_i) = c_i$.
Then $\pi \circ v$ and $u$ agree on the variables, so they are equal.

Non-example: let $k$ be a field and $A = k[x]/(x^2)$.
Take $C = k[x]/(x^3)$ and $J = (x^2)/(x^3)$, so $J^2 = 0$ and $C/J = A$, and let $u$ be the identity of $A$.
A lift $v$ would send $x$ to $x + a x^2$ for some $a \in k$, and then $v(x)^2 = x^2 \ne 0$ in $C$, while $v(x^2) = v(0) = 0$.
So no lift exists, and $A$ is not formally smooth over $k$.
</document>

<document source="https://tamnd.github.io/math2/node/CEK4SC20WK.md" tag="CEK4SC20WK" kind="concept">
# Deformation retract

## Summary

A subspace $A$ of a space $X$ is a deformation retract when $X$ can be deformed continuously onto $A$.
At the end of the deformation, the points of $A$ are back where they started.
Then the inclusion of $A$ in $X$ is a [[homotopy equivalence|XZG6MTSWCN]], so $A$ and $X$ have the same homotopy invariants.
Deformation retractions are the main tool for replacing a space by a simpler one of the same [[homotopy type|SK0ZJGWXDP]].

## Explanations

### L1 human

Picture a thick object melting slowly onto a thin skeleton inside it, with the skeleton itself held in place.
At the end of the melting, everything lies on the skeleton.
For example, the plane without the origin melts onto the unit circle if each point moves along its ray towards the circle.
So the punctured plane and the circle have the same holes.
</document>

<document source="https://tamnd.github.io/math2/node/CEZTVHC1KS.md" tag="CEZTVHC1KS" kind="concept">
# Tietze transformations

## Summary

Tietze transformations are four elementary moves on a [[group presentation|0JGEQXQ4AC]] that do not change the group up to isomorphism.
They add or remove a redundant relation, or add or remove a redundant generator.
Two finite presentations define isomorphic groups if and only if one can pass from one to the other by finitely many such moves.
They are the basic tool to simplify presentations and to recognise when two presentations give the same group.

## Explanations

### L1 human

A presentation describes a group by letters and rules, and many different presentations describe the same group.
Tietze moves are the safe edits.
You may add a rule that already follows from the others, or delete such a rule.
You may introduce a new letter as an abbreviation for a word, or remove a letter that is only an abbreviation.
For example, $\langle a, b \mid b = a^2 \rangle$ becomes $\langle a \mid \ \rangle$ by removing the abbreviation $b$, so it is the infinite cyclic group.
</document>

<document source="https://tamnd.github.io/math2/node/CFAG3VRXEA.md" tag="CFAG3VRXEA" kind="concept">
# Group homology

## Summary

Group homology attaches to a [[group|G5S1M9]] $G$ and a $G$-module $M$ a sequence of abelian groups $H_n(G, M)$, for $n \ge 0$.
It is defined with the [[Tor functors|DDQRERAQCV]] over the [[group ring|4NMTSX0XDS]] $\Z[G]$, and it is dual to [[group cohomology|YXJXA0Q52R]].
In low degrees it has concrete meaning: $H_1(G, \Z)$ is the abelianization of $G$, and $H_2(G, \Z)$ is given by [[Hopf's formula|S31M1ZYJRK]].
Topologically, $H_n(G, \Z)$ is the [[singular homology|M0Q5CFJ7ZH]] of a [[classifying space|WA796ZCNRX]] of $G$.

## Explanations

### L1 human

Group homology is a list of abelian groups, one for each degree, that record algebraic information about a group.
The first homology group records what is left of the group when every two elements are forced to commute.
For the integers $\Z$, the first homology is $\Z$ and all higher ones vanish, matching the circle, whose loops are counted by $\Z$.
For the group $\Z^2$, the second homology is $\Z$, matching the single two-dimensional hole of the torus.
</document>

<document source="https://tamnd.github.io/math2/node/CFASPSF7BB.md" tag="CFASPSF7BB" kind="concept">
# Composition operator

## Summary

A composition operator acts on a space of functions by precomposing with a fixed map: $C_\varphi f = f \circ \varphi$.
It is always linear.
The questions are when it maps the space into itself, when it is bounded or compact, and how its properties reflect the symbol $\varphi$.
On the [[Hardy space|PNNNFJVF6W]] of the disc, every holomorphic self-map $\varphi$ of the disc gives a bounded composition operator, by Littlewood's subordination principle.
In ergodic theory the same construction is called the Koopman operator.

## Explanations

### L1 human

A composition operator changes a function by first moving the point with a fixed map $\varphi$ and then applying the function.
If $\varphi(z) = z/2$, then $C_\varphi$ turns $f(z)$ into $f(z/2)$, a squeezed copy of $f$.
On power series $\sum a_n z^n$ this multiplies the $n$-th coefficient by $2^{-n}$.
Properties of the operator, such as its norm or its spectrum, then become questions about how $\varphi$ moves points around.
</document>

<document source="https://tamnd.github.io/math2/node/CFB41AZX87.md" tag="CFB41AZX87" kind="statement">
# Reduction to diagonal form by transvections

## Statement

Let $F$ be a [[field|04C5ZBG1WT]], let $n \ge 1$ and let $M$ be an $n \times n$ matrix over $F$.
Then there are finitely many [[transvection matrices|0DTCBCKEJP]] $L_1, \dots, L_r$ and $R_1, \dots, R_s$ and a [[diagonal matrix|044JNC7ZAA]] $D$ over $F$ such that
$$M = L_1 \cdots L_r \, D \, R_1 \cdots R_s.$$

## Remarks

Multiplying by a transvection on the left adds a multiple of one row to another row; on the right it adds a multiple of one column to another column.
The proof is an induction on $n$.
If the last row or the last column of $M$ has a nonzero entry, one row or column addition puts a nonzero entry at position $(n, n)$.
Further additions of multiples of row $n$ and column $n$ then clear the rest of the last row and the last column.
The upper left block of size $n - 1$ is treated by induction.
Example: $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ becomes $\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}$ after adding column $2$ to column $1$, and then $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ after subtracting column $1$ from column $2$.
Each transvection has [[determinant|00KDZRW4C0]] $1$, so $\det M = \det D$.
A refinement of the argument shows that an invertible $M$ is a product of transvections and the matrix $\operatorname{diag}(1, \dots, 1, \det M)$.
Hence the transvections generate the [[special linear group|0V9PMN0RBH]] $\mathrm{SL}_n(F)$.

## Explanations

### L1 human

Using only the move "add a multiple of one row (or column) to another", every square matrix over a field can be made diagonal.
No swaps and no scalings are needed.
These moves do not change the determinant, so the diagonal matrix keeps the determinant of the original matrix.

### L2 human

This is a sharper form of Gaussian elimination: [[elementary matrices|01REXW3WWE]] of the row-addition type already suffice, as long as both row and column operations are allowed.
The theorem is the main step in the proof that $\mathrm{SL}_n(F)$ is generated by transvections.
It is also a first step towards the Smith normal form, where the field is replaced by a principal ideal domain and more care is needed.
</document>

<document source="https://tamnd.github.io/math2/node/CFD7BPM51W.md" tag="CFD7BPM51W" kind="concept">
# Harmonic measure

## Summary

The harmonic measure of a domain, seen from a point $z$, is a probability measure on the boundary of the domain.
It is the measure that represents the value at $z$ of the solution of the [[Dirichlet problem|8D56GQ4AG8]] in terms of the boundary data.
Equivalently, it is the distribution of the first exit point of a Brownian motion started at $z$.
It is invariant under conformal maps, and it is a central tool for estimating harmonic and holomorphic functions near the boundary.

## Explanations

### L1 human

Release a random walker at a point $z$ inside a region and wait until it first hits the boundary.
The harmonic measure of a piece of the boundary is the probability that the walker hits that piece first.
Parts of the boundary that are near $z$ and in the open get a large measure, while parts hidden deep inside narrow fjords get a small one.
For the centre of a disc, every arc of the circle gets a measure proportional to its length.
</document>

<document source="https://tamnd.github.io/math2/node/CFDZJ55N95.md" tag="CFDZJ55N95" kind="statement">
# Gelfand–Raikov theorem

## Statement

Let $G$ be a [[locally compact group|AHV05ZBCYW]] and let $g \in G$ with $g \ne e$.
Then there is an [[irreducible|FX3NEFYFFN]] [[unitary representation|74BPYJNZTM]] $\pi$ of $G$ on a Hilbert space $H$ such that $\pi(g) \ne \id_H$.

## Remarks

In short, the irreducible unitary representations of a locally compact group separate its points.
The proof works with continuous [[positive-definite functions|3PRZ02APFA]] $\varphi$ with $\varphi(e) \le 1$.
These form a convex set, compact in the weak-$*$ topology of $L^\infty(G)$.
Its nonzero extreme points are exactly the functions $x \mapsto \ip{\pi(x)v}{v}$ with $\pi$ irreducible and $v$ a unit vector.
The left regular representation supplies a positive-definite function that is not constant on the pair $\set{e, g}$, and the Krein–Milman theorem then yields an extreme point with the same property.
Example: for $G = \R$, the characters $x \mapsto e^{ixy}$, $y \in \R$, are one-dimensional irreducible unitary representations, and for $g \ne 0$ one can choose $y$ with $e^{igy} \ne 1$.
For a compact group the representations can be chosen finite-dimensional, which is part of the Peter–Weyl theorem.

## Explanations

### L1 human

A unitary representation lets a group act by rotations on a Hilbert space, and irreducible ones are the building blocks that cannot be split further.
The theorem says that there are enough building blocks to see every element: each element other than the identity acts nontrivially in at least one of them.
So no information about the group is lost when one studies it through its irreducible unitary representations.

### L2 human

The theorem justifies the program of harmonic analysis on locally compact groups, which aims to decompose functions and representations into irreducible pieces.
It is the starting point for the study of the unitary dual and of the group C\*-algebra, since it shows that the irreducible representations already detect the group.
For abelian groups, it says that characters separate points, which is a step towards Pontryagin duality.
Irreducible representations may need to be infinite-dimensional: for a connected simple Lie group such as $SL_2(\R)$, every finite-dimensional unitary representation is trivial.
</document>

<document source="https://tamnd.github.io/math2/node/CFHNV77ZMD.md" tag="CFHNV77ZMD" kind="object">
# Path space of a manifold

## Description

Let $M$ be a smooth manifold of dimension $n$ and fix points $p, q \in M$.
The *path space* $\Omega_{p,q} M$ is the set of paths $\gamma \colon [0, 1] \to M$ of [[Sobolev class|M4CEZ5K9HP]] $H^1$ with $\gamma(0) = p$ and $\gamma(1) = q$.
Here $H^1$ means that in local coordinates the path is absolutely continuous with square-integrable derivative.

## Remarks

The path space is a Hilbert manifold.
Choose a complete [[Riemannian metric|7CBT4SM83Y]] on $M$.
For a smooth path $\gamma$ from $p$ to $q$, let $T_\gamma$ be the Hilbert space of $H^1$ vector fields $\xi$ along $\gamma$ with $\xi(0) = \xi(1) = 0$.
Small $\xi \in T_\gamma$ give paths $t \mapsto \exp_{\gamma(t)}(\xi(t))$ from $p$ to $q$, and these maps are charts with smooth chart changes.
Every $H^1$ path is close to a smooth one, so these charts cover $\Omega_{p,q} M$, and each model space $T_\gamma$ is isomorphic to $H^1_0([0, 1]; \R^n)$.

On this Hilbert manifold the energy $E(\gamma) = \frac{1}{2} \int_0^1 \abs{\gamma'(t)}^2 \, dt$ is a smooth function whose critical points are the [[geodesics|NSYMQMHPDN]] from $p$ to $q$.
The inclusion of piecewise smooth or smooth paths into $\Omega_{p,q} M$ is a homotopy equivalence.
For connected $M$ the space is homotopy equivalent to the based [[loop space|2JFF5F3EGF]] of $M$.


## Explanations

### L1 human

The path space collects all routes from $p$ to $q$ in $M$, each route being one point of the space.
Using paths with square-integrable speed, rather than smooth ones, makes the space a Hilbert manifold, which is the right setting for calculus of variations.
The shortest and, more generally, the locally straightest routes are the geodesics, and they appear as the critical points of the energy on this space.
This is the starting point of Morse theory of geodesics.
</document>

<document source="https://tamnd.github.io/math2/node/CFQXJ4FHR4.md" tag="CFQXJ4FHR4" kind="definition">
# Finite-dimensional distributions

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]], and let $(S, \mathcal{S})$ be a measurable space.
Let $X = (X_t)_{t \in T}$ be a [[stochastic process|0QH5ENKZM1]] with values in $S$, indexed by a set $T$.
Let $\mathcal{S}^{\otimes k}$ be the product [[sigma-algebra|0SV8K0TDGY]] on $S^k$.
For $k \ge 1$ and distinct $t_1, \dots, t_k \in T$, the *finite-dimensional distribution* of $X$ at $(t_1, \dots, t_k)$ is the measure $\mu_{t_1, \dots, t_k}$ on $(S^k, \mathcal{S}^{\otimes k})$ given by
$$\mu_{t_1, \dots, t_k}(B) = P\bigl((X_{t_1}, \dots, X_{t_k}) \in B\bigr) \quad \text{for } B \in \mathcal{S}^{\otimes k}.$$
So $\mu_{t_1, \dots, t_k}$ is the [[law|01EQPZ6RYZ]] of the random vector $(X_{t_1}, \dots, X_{t_k})$.
The *finite-dimensional distributions* of $X$ are the family of all these measures.

## Remarks

The family is consistent.
Permuting the times permutes the coordinates of the measure, and $\mu_{t_1, \dots, t_k}(B \times S) = \mu_{t_1, \dots, t_{k-1}}(B)$.
Conversely, for $S$ a Polish space, the [[Kolmogorov extension theorem|WR4X7MMTH2]] gives a process with any prescribed consistent family.

Example: for the standard [[Wiener process|22735FGJV6]] $W$ on $T = [0, \infty)$, the law of $(W_{t_1}, \dots, W_{t_k})$ is the [[multivariate normal distribution|0XN3K7RSZY]] with mean $0$ and covariance matrix $(\min(t_i, t_j))_{i, j}$.

Two processes with the same finite-dimensional distributions can have different paths.
Let $U$ be uniform on $[0, 1]$, let $X_t = 0$ and let $Y_t = \mathbf{1}_{\set{t = U}}$ for $t \in [0, 1]$.
For any fixed times, $P(Y_{t_i} = 0 \text{ for all } i) = 1$, so $X$ and $Y$ have the same finite-dimensional distributions.
But every path of $X$ is continuous, and no path of $Y$ is continuous.
</document>

<document source="https://tamnd.github.io/math2/node/CFRN0G7GDB.md" tag="CFRN0G7GDB" kind="statement">
# Continuous mapping theorem

## Statement

Let $(S, d)$ and $(S', d')$ be [[metric spaces|0VKD1TP7F3]] with their Borel $\sigma$-algebras.
Let $X, X_1, X_2, \dots$ be random elements of $S$.
Let $g \colon S \to S'$ be a [[measurable map|YYNCJSR0PJ]], and let $D_g$ be the set of points of $S$ at which $g$ is not [[continuous|0X3WGX95FE]].
Assume that $P(X \in D_g) = 0$.

1. If $X_n \to X$ [[in distribution|0SRYJGXKR4]], then $g(X_n) \to g(X)$ in distribution.
2. If the $X_n$ and $X$ are defined on one probability space and $X_n \to X$ [[in probability|0QA1HBD9Z4]], then $g(X_n) \to g(X)$ in probability.
3. If the $X_n$ and $X$ are defined on one probability space and $X_n \to X$ [[almost surely|0AAXWGT5TD]], then $g(X_n) \to g(X)$ almost surely.

## Remarks

Part 3 is direct: on the event where $X_n \to X$ and $X \notin D_g$, the continuity of $g$ at $X$ gives $g(X_n) \to g(X)$, and this event has probability $1$.
Part 1 follows from the [[portmanteau theorem|4SSJ589G6T]]: for a closed set $F \subseteq S'$, the closure of $g^{-1}(F)$ lies in $g^{-1}(F) \cup D_g$.
Part 2 follows from part 3 applied along subsequences, since convergence in probability holds exactly when every subsequence has a further subsequence that converges almost surely.

Example: if $X_n \to Z$ in distribution with $Z$ standard normal, then $X_n^2$ converges in distribution to a chi-squared variable with one degree of freedom.

The hypothesis $P(X \in D_g) = 0$ cannot be dropped.
Take $X_n = 1/n$, $X = 0$ and $g = \mathbf{1}_{(0, \infty)}$.
Then $X_n \to X$ surely, but $g(X_n) = 1$ for all $n$ while $g(X) = 0$.
Here $D_g = \set{0}$ and $P(X \in D_g) = 1$.

## Explanations

### L1 human

If random quantities settle down to a limit, then any continuous function of them settles down to the same function of the limit.
This holds for each of the three usual kinds of convergence.
The function may even jump at some points, as long as the limit almost never lands on such a point.

### L2 human

The theorem is the main tool to move limit results from one statistic to another.
With [[Slutsky's theorem|GCV10B5HES]] and the [[central limit theorem|0EPZC4Z1FC]], it gives the limit laws of sums, products, quotients and squares of asymptotically normal statistics.
It also underlies the delta method, where $g$ is a differentiable function.
For random paths, applied with $g$ the supremum map, it turns [[Donsker's theorem|818832PMSV]] into a limit law for the maximum of a random walk.
</document>

<document source="https://tamnd.github.io/math2/node/CFWMB0S5NK.md" tag="CFWMB0S5NK" kind="definition">
# Trigonometric functions

## Definition

Let $\sin, \cos \colon \R \to \R$ be the [[sine and cosine|0WB57D9488]] functions, given by their power series.
Let $Z_{\cos} = \set{x \in \R : \cos x = 0}$ and $Z_{\sin} = \set{x \in \R : \sin x = 0}$.
The *trigonometric functions* are $\sin$, $\cos$, and the four functions
$$\tan x = \frac{\sin x}{\cos x}, \quad \sec x = \frac{1}{\cos x} \quad (x \in \R \setminus Z_{\cos}),$$
$$\cot x = \frac{\cos x}{\sin x}, \quad \csc x = \frac{1}{\sin x} \quad (x \in \R \setminus Z_{\sin}).$$
They are called tangent, secant, cotangent and cosecant.

## Remarks

Here $Z_{\cos} = \set{\pi/2 + k\pi : k \in \Z}$ and $Z_{\sin} = \set{k\pi : k \in \Z}$.

Example: $x = \pi/3$ lies in neither set, and $\sin(\pi/3) = \sqrt3/2$, $\cos(\pi/3) = 1/2$, so $\tan(\pi/3) = \sqrt3$, $\sec(\pi/3) = 2$, $\cot(\pi/3) = 1/\sqrt3$ and $\csc(\pi/3) = 2/\sqrt3$.

Non-example of a value: $\tan(\pi/2)$ is not defined, because $\cos(\pi/2) = 0$, so $\pi/2 \in Z_{\cos}$; the tangent tends to $+\infty$ as $x \to \pi/2$ from the left.

The functions $\sin$, $\cos$, $\sec$ and $\csc$ have period $2\pi$, and $\tan$ and $\cot$ have period $\pi$.
The [[tangent function|06T0HS14PP]] has its own node.
</document>

<document source="https://tamnd.github.io/math2/node/CFZ1J559MK.md" tag="CFZ1J559MK" kind="concept">
# Concentration of measure

## Summary

Concentration of measure is a phenomenon of many high-dimensional [[metric spaces|0VKD1TP7F3]] with a [[probability measure|059TJN0NZ1]].
On such a space, almost all of the space lies close to every set of measure at least $1/2$.
Equivalently, every [[Lipschitz|0AW1E6YTJY]] function is nearly constant, close to its median, outside a set of very small measure.
The sphere $S^{n-1}$ and the [[Gaussian measure|P0DV2TCH1N]] on $\R^n$ are the model examples.
Milman put the phenomenon forward in the local theory of Banach spaces.
It is now a basic tool in asymptotic convex geometry, probability, statistics and computer science.

## Explanations

### L1 human

On a sphere in very high dimension, almost all of the area lies near the equator, whichever equator one picks.
So if a set covers half the sphere, then a thin band around it already covers almost everything.
In the same way, a function that does not change too fast takes nearly the same value at almost every point.
For example, the average of many independent coin flips is very close to $1/2$ with overwhelming probability, because it depends only a little on each flip.
</document>

<document source="https://tamnd.github.io/math2/node/CG0BWPC9ZY.md" tag="CG0BWPC9ZY" kind="definition">
# Tensor product of algebras

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $A$, $B$ be [[$R$-algebras|DPBJ04JMH8]].
The *tensor product* $A \otimes_R B$ is the [[tensor product of $R$-modules|F0HQXQ0HCD]] with the multiplication determined by
$$(a \otimes b)(a' \otimes b') = aa' \otimes bb' \quad \text{for } a, a' \in A,\ b, b' \in B,$$
extended bilinearly to finite sums.
With this multiplication and identity $1 \otimes 1$, it is an $R$-algebra, with structure map $r \mapsto r \otimes 1 = 1 \otimes r$.

## Remarks

The multiplication is well defined because the formula is $R$-bilinear in each pair of arguments.
The maps $a \mapsto a \otimes 1$ and $b \mapsto 1 \otimes b$ are $R$-algebra [[homomorphisms|0VTH8KPCG2]], and their images commute.
If $A$ and $B$ are commutative, then $A \otimes_R B$ is commutative, and it has a universal property.
For every commutative $R$-algebra $C$ and $R$-algebra maps $f \colon A \to C$, $g \colon B \to C$, there is exactly one $R$-algebra map $h \colon A \otimes_R B \to C$ with $h(a \otimes b) = f(a) g(b)$.

Example: $R[x] \otimes_R R[y] \cong R[x, y]$, with $x^i \otimes y^j \mapsto x^i y^j$; both sides are free $R$-modules with these bases, and the map respects products.
Another example is $\C \otimes_\R \C \cong \C[t]/(t^2 + 1) \cong \C \times \C$, by the Chinese remainder theorem, since $t^2 + 1 = (t - i)(t + i)$ over $\C$.

Non-example: the direct product $A \times B$ with componentwise operations is in general not $A \otimes_R B$.
For $A = B = \Q[x]$ over $R = \Q$, the product $\Q[x] \times \Q[x]$ has zero divisors, such as $(1, 0)(0, 1) = 0$, while $\Q[x] \otimes_\Q \Q[x] \cong \Q[x, y]$ is an integral domain.
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<document source="https://tamnd.github.io/math2/node/CG17WSS64W.md" tag="CG17WSS64W" kind="statement">
# Jacobi's four-square theorem

## Statement

Let $n \ge 1$ be an integer, and let $r_4(n)$ be the number of quadruples $(a, b, c, d)$ of [[integers|0C4JS9RF5M]] with $a^2 + b^2 + c^2 + d^2 = n$.
Then
$$r_4(n) = 8 \sum_{\substack{d \mid n \\ 4 \nmid d}} d.$$
Let $\sigma$ be the [[sum of divisors|06TK51263Q]], and let $m$ be the largest odd divisor of $n$.
Equivalently, $r_4(n) = 8\sigma(n)$ if $n$ is odd, and $r_4(n) = 24\sigma(m)$ if $n$ is even.

## Remarks

One proof uses the series $\sum_{n \in \Z} q^{n^2}$ with $q = e^{2 \pi i \tau}$, which is $\theta(2\tau)$ for the theta constant $\theta$ of the [[theta function|2F5CYQJVAF]].
Its fourth power is $\sum_{n \ge 0} r_4(n) q^n$.
The function $\theta(2\tau)^4$ is a [[modular form|GWKN5VG78A]] of weight $2$ for the [[congruence subgroup|1H6R856P3C]] $\Gamma_0(4)$.
That space has dimension $2$ and is spanned by Eisenstein series, and comparing two coefficients gives the formula.
Elementary proofs also exist.
Example: $n = 4$ has divisors $1, 2$ not divisible by $4$, so $r_4(4) = 8 \cdot 3 = 24$.
Check: the solutions are the $8$ quadruples with one entry $\pm 2$ and three zeros, and the $16$ quadruples with all entries $\pm 1$.
Since the sum always contains $d = 1$, $r_4(n) \ge 8 > 0$, which gives [[Lagrange's four-square theorem|0KYYSMEVHH]].

## Explanations

### L1 human

Every positive integer is a sum of four squares, and this theorem says exactly in how many ways.
The count is $8$ times the sum of the divisors of $n$ that are not multiples of $4$.
Order and signs count: $1 = 1^2 + 0^2 + 0^2 + 0^2$ gives $8$ representations, one for each position and sign.
It turns an existence result into an exact formula.

### L2 human

The theorem is a model case of how modular forms solve counting problems.
A generating function of a counting problem is shown to be modular, and the finite dimension of the space of modular forms identifies it.
For two squares the same method gives $r_2(n) = 4 \sum_{d \mid n,\, d \text{ odd}} \chi(d)$, where $\chi(d) = (-1)^{(d-1)/2}$.
For larger numbers of squares, cusp forms can appear, and the exact formula then has a cusp form term besides the divisor sums.
</document>

<document source="https://tamnd.github.io/math2/node/CG7BFJ04YE.md" tag="CG7BFJ04YE" kind="statement">
# Mellin transform under a power substitution

## Statement

Let $f \colon (0, \infty) \to \C$ be measurable, let $a \in \R$ with $a \ne 0$, and let $g(x) = f(x^{a})$ for $x > 0$.
Let $s \in \C$.
Then $g$ is [[Mellin convergent|SGZZ4Y0D36]] at $s$ if and only if $f$ is Mellin convergent at $s/a$.
In that case
$$\mathcal{M}g(s) = \frac{1}{\abs{a}} \, \mathcal{M}f\left(\frac{s}{a}\right),$$
where $\mathcal{M}$ is the [[Mellin transform|N5PVGRJE1T]].

## Remarks

For $a > 0$, substitute $u = x^{a}$, so $x = u^{1/a}$ and $dx = \frac{1}{a} u^{1/a - 1} \, du$:
$$\int_0^\infty x^{s-1} f(x^{a}) \, dx = \frac{1}{a} \int_0^\infty u^{(s-1)/a} \, u^{1/a - 1} f(u) \, du = \frac{1}{a} \int_0^\infty u^{s/a - 1} f(u) \, du.$$
For $a < 0$ the substitution reverses the orientation of $(0, \infty)$, which turns the factor $1/a$ into $1/\abs{a}$.
The case $a = -1$ is the rule for the inversion $x \mapsto 1/x$.
Example: $f(x) = e^{-x}$ and $a = 2$ give $\int_0^\infty x^{s-1} e^{-x^2} \, dx = \frac{1}{2} \Gamma(s/2)$ for $\operatorname{Re} s > 0$; at $s = 1$ this is $\int_0^\infty e^{-x^2} \, dx = \frac{1}{2}\Gamma(1/2) = \frac{\sqrt{\pi}}{2}$.
The hypothesis $a \ne 0$ is needed: for $a = 0$, $g$ is the constant $f(1)$, which is Mellin convergent at no $s$ unless $f(1) = 0$.

## Explanations

### L1 human

Replacing $x$ by a power $x^{a}$ rescales the Mellin variable: $s$ becomes $s/a$, and a factor $1/\abs{a}$ appears.
It is the logarithmic version of the rule that stretching a function by a factor rescales its Fourier transform.

### L2 human

This rule computes many transforms from a few, for instance Gaussian integrals from the gamma function.
It is used in the Riemann proof of the functional equation, where the theta function $\sum_n e^{-\pi n^2 x}$ meets $\Gamma(s/2)$ rather than $\Gamma(s)$.
Combined with the rules for dilations and for multiplication by powers, it describes the Mellin transform of $x^{b} f(c x^{a})$ for all $a \ne 0$, $b \in \C$ and $c > 0$.
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<document source="https://tamnd.github.io/math2/node/CG91WNMG2T.md" tag="CG91WNMG2T" kind="concept">
# Graph embedding

## Summary

An embedding of a graph in a surface is a drawing of the graph on the surface in which edges meet only at shared ends.
Embeddings in the plane or sphere are the subject of planar graph theory, and embeddings in other surfaces lead to the genus of a graph.
Many results, such as [[Euler's formula|0W1C598HMS]] and the notion of a [[dual graph|1QHTD1DS5J]], are statements about embedded graphs.

## Explanations

### L1 human

To embed a graph is to draw it on a surface with no two edges crossing.
Some graphs, like $K_5$, cannot be drawn on a sheet of paper without crossings, but they can be drawn on the surface of a doughnut.
The genus of a graph measures how many handles a surface needs before the graph fits on it.
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<document source="https://tamnd.github.io/math2/node/CG9EH0TN27.md" tag="CG9EH0TN27" kind="concept">
# Profinite completion

## Summary

The profinite completion of a [[group|G5S1M9]] $G$ is the inverse limit of all its finite quotient groups.
It is a [[profinite group|KA0YDB6QXE]] that records every finite quotient of $G$ at once, together with a natural homomorphism from $G$.
This homomorphism is injective exactly when $G$ is [[residually finite|HEE922EHQ8]].
The profinite completion of $\Z$ is the group $\hat\Z$ of profinite integers, which is the product of the $p$-adic integers over all primes $p$.

## Explanations

### L1 human

A group can be studied through its finite images, such as the integers modulo $n$ for the group of integers.
The profinite completion collects all of these finite pictures into one object.
An element of it is a compatible choice of an image in every finite quotient.
For the integers, it is a choice of a residue modulo every $n$, consistent under reduction, and such choices form the group $\hat\Z$.
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<document source="https://tamnd.github.io/math2/node/CG9JEATW3Q.md" tag="CG9JEATW3Q" kind="concept">
# Schwartz space

## Summary

The Schwartz space $\mathcal{S}(\R^n)$ is the space of smooth functions on $\R^n$ that, together with all their derivatives, decay faster than any power of $1/\abs{x}$.
The [[Fourier transform|DPVM34ADZV]] maps it bijectively onto itself, which makes it the natural home of Fourier analysis on $\R^n$.
Its dual space is the space of [[tempered distributions|AES2T5JPFP]].
See the [[definition|GE020MY43M]].

## Explanations

### L1 human

A Schwartz function is smooth and dies off very fast at infinity, and so do all its derivatives.
The bell curve $e^{-x^2}$ is the basic example: it decays faster than $1/x^{100}$ or any other power.
The function $1/(1 + x^2)$ is smooth but decays only like $1/x^2$, so it is not a Schwartz function.
The point of the space is that differentiation, multiplication by polynomials and the Fourier transform never lead out of it.
</document>

<document source="https://tamnd.github.io/math2/node/CGBX1N9V8H.md" tag="CGBX1N9V8H" kind="concept">
# Differential operator between vector bundles

## Summary

A differential operator between [[vector bundles|P4DZM975BV]] over a manifold is a linear map from [[sections|WWX7H5W1RP]] of one bundle to sections of the other.
In local coordinates and trivialisations, it is a matrix of partial derivatives with smooth coefficients.
It is the coordinate-free form of a [[linear system of partial differential equations|6DZETZ5T9G]] on a manifold.
The exterior derivative, the Laplacian on forms, Dirac operators and connections are all examples.
Global analysis studies these operators on compact manifolds, where their properties are tied to the topology of the manifold.

## Explanations

### L1 human

On $\R^n$, a linear differential operator takes a function and returns a combination of its partial derivatives, with coefficients depending on the point.
On a manifold, the functions are often replaced by sections of a vector bundle, such as vector fields or differential forms.
A differential operator between bundles does the same job: in each small coordinate patch it is a matrix of ordinary partial derivative operators.
For example, the exterior derivative turns $1$-forms into $2$-forms, and in coordinates it takes $f \, dx + g \, dy$ to $(\partial_x g - \partial_y f) \, dx \wedge dy$.
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<document source="https://tamnd.github.io/math2/node/CGDN1X32NQ.md" tag="CGDN1X32NQ" kind="statement">
# Hadamard's inequality

## Statement

Let $A$ be an $n \times n$ complex matrix with columns $v_1, \dots, v_n \in \C^n$.
Let $\norm{v}$ denote the Euclidean [[norm|0V2WP9B1MF]] of $v \in \C^n$.
Then
$$\abs{\det A} \le \prod_{i=1}^{n} \norm{v_i},$$
where $\det$ is the [[determinant|00KDZRW4C0]].
Equality holds if and only if some $v_i$ is $0$ or the vectors $v_1, \dots, v_n$ are pairwise [[orthogonal|0Y2X7P2WB7]].

## Remarks

If the columns are linearly dependent, the left side is $0$.
Otherwise the [[Gram–Schmidt process|0VT6P0GNQR]] gives $A = QR$ with $Q$ unitary and $R$ upper triangular with diagonal entries $r_{ii}$.
Then $\abs{\det A} = \prod_i \abs{r_{ii}}$, and $\abs{r_{ii}}$ is the distance from $v_i$ to the span of $v_1, \dots, v_{i-1}$, so $\abs{r_{ii}} \le \norm{v_i}$.

Example: for $A = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$, the columns are orthogonal with norm $\sqrt{2}$, and indeed $\abs{\det A} = 2 = \sqrt{2} \cdot \sqrt{2}$.
For $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the bound is $1 \cdot \sqrt{2}$, while $\det A = 1$.

An equivalent form: if $P$ is an $n \times n$ [[positive semidefinite|01MMVW8CKC]] matrix, then $\det P \le \prod_i p_{ii}$.
Apply it to $P = A^* A$, whose diagonal entries are $\norm{v_i}^2$.
If all entries of $A$ satisfy $\abs{a_{ij}} \le 1$, the inequality gives $\abs{\det A} \le n^{n/2}$, which is the starting point of [[Hadamard's maximal determinant problem|GCH1Z9YFNG]].

## Explanations

### L1 human

The absolute value of a determinant is the volume of the box spanned by the columns.
A box with given edge lengths has the largest volume when its edges are at right angles.
So the volume is at most the product of the edge lengths, with equality for a rectangular box.

### L2 human

The inequality bounds the size of a determinant without computing it.
It is used to bound the size of integer determinants in algorithms, and it gives the bound $n^{n/2}$ for matrices with entries of absolute value at most $1$.
The positive semidefinite form is a special case of Fischer's inequality for block matrices.
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# Geometric quantization

## Definition

Let $(M, \omega)$ be a [[symplectic manifold|CAN490EPRT]] of dimension $2n$ and let $\hbar > 0$.
Assume the integrality condition: the [[de Rham class|GX74C07T0G]] of $\omega / (2\pi\hbar)$ lies in the image of $H^2(M; \Z) \to H^2(M; \R)$.
Geometric quantization consists of the following data and steps.

1. (Prequantization) A *prequantum line bundle* is a Hermitian complex line bundle $L \to M$, a [[vector bundle|P4DZM975BV]] of rank one, with a connection $\nabla$. The connection must preserve the Hermitian metric and have curvature $-\frac{i}{\hbar}\omega$.
The *prequantum Hilbert space* is the space of sections $s$ of $L$ with $\int_M \abs{s}^2 \, \omega^n / n! < \infty$.
To a smooth function $f$ it assigns the operator $\hat{f} = -i\hbar \nabla_{X_f} + f$ on sections, where $X_f$ is the [[Hamiltonian vector field|C5EH3B4BEP]] of $f$.
2. (Polarization) A *polarization* is a subbundle $P \subset TM \otimes \C$ of complex rank $n$ that is closed under the Lie bracket of vector fields. The complex-bilinear extension of $\omega$ must vanish on $P$.
3. (Quantization) The *quantum Hilbert space* is the [[Hilbert space|HM2HHE3J51]] of square-integrable sections $s$ that are polarized: $\nabla_X s = 0$ for every section $X$ of $P$.

## Remarks

The integrality condition holds exactly when a prequantum line bundle exists.
With consistent conventions, $f \mapsto \hat{f}$ sends the [[Poisson bracket|2S0MXDT4YH]] to the commutator up to the factor $\pm i\hbar$, which is Dirac's quantization condition.
Only functions whose flows preserve $P$ act directly on the polarized sections; in practice one also adds half-forms, the metaplectic correction, to obtain the correct spectra.
Sign and factor conventions differ from author to author.

Example: take $M = \C^n$ with the standard form and the Kähler polarization spanned by the vectors $\partial / \partial \bar{z}_j$.
Then the quantum Hilbert space is the Segal–Bargmann space of holomorphic functions that are square-integrable for a Gaussian weight.
For the sphere $S^2 = \mathbb{CP}^1$ with total area $2\pi\hbar k$, $k \ge 1$ an integer, the prequantum bundle is the line bundle of degree $k$.
The polarized sections are its holomorphic sections, a space of dimension $k + 1$.

Non-example: for the sphere with total area $\pi\hbar$, the integrality condition fails, since $\int_{S^2} \omega / (2\pi\hbar) = \frac{1}{2}$ is not an integer, so no prequantum line bundle exists.
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<document source="https://tamnd.github.io/math2/node/CGH8C8EA9N.md" tag="CGH8C8EA9N" kind="concept">
# Elementary embedding

## Summary

An elementary embedding is a map between two [[structures|983GZBQYT4]] of the same signature that preserves and reflects the truth of every first-order formula.
Its image is an [[elementary substructure|8WZF9SBP39]] of the target, isomorphic to the source.
Elementary embeddings are the natural maps of model theory, and in set theory they are used to define large cardinals.

## Explanations

### L1 human

An ordinary embedding copies a structure into another one while keeping the operations and relations.
An elementary embedding keeps much more: every first-order statement about elements holds before the map exactly when it holds after.
The inclusion of $(\Q, <)$ into $(\R, <)$ is elementary.
The map $n \mapsto n + 1$ from $(\N, <)$ to $(\N, <)$ is an embedding, but not an elementary one: $0$ has no element below it, while its image $1$ has $0$ below it.
</document>

<document source="https://tamnd.github.io/math2/node/CGY299QMC3.md" tag="CGY299QMC3" kind="concept">
# Completely positive map

## Summary

A completely positive map between C*-algebras is a linear map that stays positive after tensoring with the identity on every matrix algebra $M_n(\C)$.
These maps are the natural morphisms of operator algebras when one keeps only the order structure.
The [[Stinespring dilation theorem|X8K0Z7484R]] describes them all as compressions of $*$-homomorphisms.
In quantum information theory, completely positive trace-preserving maps are the quantum channels.

## Explanations

### L1 human

A positive map sends [[positive elements|NA8DF5X3M4]] to positive elements.
Complete positivity asks more: if the map is applied entrywise to matrices whose entries are in the algebra, positive block matrices must still go to positive block matrices.
Physically, an operation on one system must remain a valid operation when the system is part of a larger one.
The transpose of $2 \times 2$ matrices is positive but not completely positive, which shows that the extra condition is real.
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<document source="https://tamnd.github.io/math2/node/CH2JRZNX7V.md" tag="CH2JRZNX7V" kind="concept">
# Hypergraph

## Summary

A hypergraph is a generalisation of a [[graph|00C0BR7D7H]] in which an edge may join any number of vertices.
Formally it is a set of vertices together with a family of subsets of it, the hyperedges.
Hypergraphs are the natural language for set systems, [[block designs|4JCSS2Z1BA]] and many problems of extremal combinatorics.
The node [[hypergraph|2KM46K20DZ]] gives the precise definition.

## Explanations

### L1 human

In a graph, an edge links two people, like a friendship.
In a hypergraph, an edge can be a whole club with any number of members.
For example, the students of a school are the vertices and each class is a hyperedge.
When every hyperedge has exactly two members, a hypergraph is just a graph.
</document>

<document source="https://tamnd.github.io/math2/node/CH6655HZXS.md" tag="CH6655HZXS" kind="statement">
# Existence of an eigenvalue for a nonzero symmetric kernel

## Statement

Let $a < b$ and let $K \in L^2([a, b] \times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]] that is [[Hermitian|2ZB3NFZ94T]], that is, $K(y, x) = \overline{K(x, y)}$ for almost all $x, y$.
Assume that $K$ is not $0$ almost everywhere.
Let $T_K$ be its [[Hilbert–Schmidt integral operator|CB5G212Q03]] on $L^2[a, b]$.
Then $T_K$ has an [[eigenfunction|4NSACSWRJZ]] with a real eigenvalue $\mu$ such that $\abs{\mu} = \norm{T_K} > 0$.
In particular $K$ has at least one [[characteristic value|DC7HMHVREP]], namely $1/\mu$.

## Remarks

Since $T_K$ is self-adjoint, $\norm{T_K} = \sup_{\norm{\varphi} = 1} \abs{\ip{T_K \varphi}{\varphi}}$, and $\norm{T_K} > 0$ because a kernel whose operator is $0$ is $0$ almost everywhere.
Take unit vectors $\varphi_k$ with $\ip{T_K \varphi_k}{\varphi_k} \to \mu$, where $\mu = \pm \norm{T_K}$.
Then $\norm{T_K \varphi_k - \mu \varphi_k}^2 \le 2\mu^2 - 2\mu \ip{T_K \varphi_k}{\varphi_k} \to 0$.
Since $T_K$ is [[compact|M2NEP1K5AE]], a subsequence of $T_K \varphi_k$ converges, so $\varphi_k$ converges to some unit vector $\varphi$, and $T_K \varphi = \mu \varphi$.

Example: for $K(x, y) = \min(x, y)$ on $[0, 1]$, the largest eigenvalue is $4 / \pi^2$, with eigenfunction $\sin(\pi x / 2)$, and $\norm{T_K} = 4 / \pi^2$.

Symmetry is necessary.
The Volterra kernel, equal to $1$ for $y \le x$ and $0$ otherwise, is square-integrable and not zero.
But its operator $\varphi \mapsto \int_a^x \varphi$ has no eigenfunction at all, so this kernel has no characteristic value.
In finite dimension, the statement is the fact that a nonzero Hermitian matrix has an eigenvalue whose absolute value is its operator norm.

## Explanations

### L1 human

A symmetric kernel that is not zero always has at least one natural mode: a function that the integral operator only stretches.
The stretching factor of the best mode equals the largest amount by which the operator can stretch any function.
Without symmetry this can fail completely.

### L2 human

This is the first step of the Hilbert–Schmidt theory.
Applying it again on the orthogonal complement of the eigenfunctions already found produces the whole sequence of eigenvalues and eigenfunctions, and leads to the [[Hilbert–Schmidt expansion theorem|3CXAZ1G5RJ]].
The variational form $\abs{\mu} = \max \abs{\ip{T_K \varphi}{\varphi}}$ is the starting point of the min-max characterization of the further characteristic values.
</document>

<document source="https://tamnd.github.io/math2/node/CH73TVRT27.md" tag="CH73TVRT27" kind="conjecture">
# Brennan conjecture

## Statement

Let $\Omega \subsetneq \C$ be a [[simply connected domain|0QW64ARSHB]], and let $\varphi \colon \Omega \to \mathbb{D}$ be a [[conformal map|0YYT36R4YD]] onto the unit disc $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$.
Then for every real $p$ with $4/3 < p < 4$,
$$\int_\Omega \abs{\varphi'(z)}^p \, dA(z) < \infty,$$
where $dA$ is area measure.

## Remarks

Such a map $\varphi$ exists by the [[Riemann mapping theorem|02WHMEG5ZR]].
For $p = 2$ the integral is the area of $\mathbb{D}$, which is $\pi$.
The range is sharp at both ends.
Take the Koebe domain $\Omega = \C \setminus (-\infty, -1/4]$ and the inverse $\varphi$ of the [[Koebe function|EZQFWKF03Q]].
Then $\abs{\varphi'(z)}$ behaves like $\abs{z}^{-3/2}$ at infinity and like $\abs{z + 1/4}^{-1/2}$ near the tip of the slit.
So the integral diverges for $p \le 4/3$ and for $p \ge 4$.
The conjecture is known for $4/3 < p < 3 + \delta$ with some $\delta > 0$; Brennan proved this range.
The full conjecture is open.

## Explanations

### L1 human

The derivative of a Riemann map measures how much the map stretches the domain near each point.
Near a sharp inward corner or the tip of a slit, this stretching blows up.
The conjecture says that it never blows up too badly: a certain power of it always has finite total integral.
The slit plane is the worst case and shows that the claimed range of powers cannot be enlarged.
</document>

<document source="https://tamnd.github.io/math2/node/CH9B4RZPMW.md" tag="CH9B4RZPMW" kind="definition">
# Least fixed point

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]] and let $f \colon P \to P$ be a map.
A *fixed point* of $f$ is an element $x \in P$ with $f(x) = x$.
A *least fixed point* of $f$ is a fixed point $x$ of $f$ such that $x \le y$ for every fixed point $y$ of $f$.
In other words, it is the [[least element|0AANKKNHS4]] of the set $\set{x \in P : f(x) = x}$.

## Remarks

A least fixed point is unique when it exists, by antisymmetry of $\le$.
It is written $\operatorname{lfp}(f)$ or $\mu f$.
The *greatest fixed point* is defined dually and written $\operatorname{gfp}(f)$ or $\nu f$.
Example: for $f \colon \R \to \R$, $f(x) = x^2$, the fixed points are the solutions of $x^2 = x$, namely $0$ and $1$.
Since $0 \le 1$, the least fixed point is $0$.
Non-example: for $g \colon \R \to \R$, $g(x) = x$, every real number is a fixed point, and $\R$ has no least element, so $g$ has no least fixed point.
The map $x \mapsto x + 1$ on $\R$ has no fixed point at all.
</document>

<document source="https://tamnd.github.io/math2/node/CHEQ235HW9.md" tag="CHEQ235HW9" kind="concept">
# Stability of a difference equation

## Summary

An [[equilibrium|D62PEKHRV5]] of a difference equation is stable if solutions that start close to it stay close to it at all later times.
It is asymptotically stable if, in addition, nearby solutions converge to it.
These notions are the discrete-time versions of [[Lyapunov stability|0F2SYCKDT2]] for differential equations.
They decide which equilibria can be observed in practice, for example which population levels persist in a model.

## Explanations

### L1 human

Put a ball at the bottom of a bowl and nudge it: it stays near the bottom, so the bottom is stable.
Balance it on top of a hill and nudge it: it rolls away, so the top is unstable.
For the rule $x_{n+1} = x_n / 2$, the equilibrium $0$ is asymptotically stable: every solution halves at each step and tends to $0$.
For $x_{n+1} = 2 x_n$, the equilibrium $0$ is unstable: every nonzero start doubles away.
</document>

<document source="https://tamnd.github.io/math2/node/CHESR1W1CY.md" tag="CHESR1W1CY" kind="concept">
# Inverse Mellin transform

## Summary

The inverse Mellin transform turns a function $\varphi$ on a vertical line $\operatorname{Re} s = c$ into the function $x \mapsto \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} x^{-s} \varphi(s) \, ds$ on $(0, \infty)$.
By the [[Mellin inversion theorem|XJWC13NY9E]], it undoes the [[Mellin transform|N5PVGRJE1T]] under suitable hypotheses.
In logarithmic variables it is an [[inverse Fourier transform|JB8QMT8DCH]], and when $\varphi$ is meromorphic it can often be computed with the [[residue theorem|032HKYS329]].

## Explanations

### L1 human

The Mellin transform breaks a function on $(0, \infty)$ into the powers $x^{-s}$, with $s$ on a vertical line.
The inverse Mellin transform adds these powers back together, with the weights $\varphi(s)$.
For example, with $\varphi = \Gamma$ and any $c > 0$, the inverse transform gives back $e^{-x}$.
Moving the line of integration to the left and collecting residues of $\Gamma$ at $0, -1, -2, \dots$ then gives the power series of $e^{-x}$.
</document>

<document source="https://tamnd.github.io/math2/node/CHHD44WDHJ.md" tag="CHHD44WDHJ" kind="concept">
# Robbins–Monro algorithm

## Summary

The Robbins–Monro algorithm is the first [[stochastic approximation|ZPCPW4W1Y9]] method, introduced by Herbert Robbins and Sutton Monro in 1951.
It finds the point where an increasing function, observed only through noisy measurements, takes a given value.
It updates the current guess by a step proportional to the observed error, with step sizes that shrink at a suitable rate.

## Explanations

### L1 human

Suppose the dose of a drug sets the average response, but each patient responds a bit differently.
To find the dose whose average response equals a target, give a dose and observe the response.
Then lower the dose if the response was too high, or raise it if it was too low.
The corrections shrink like $1/n$, so random errors cancel out over time.
With target $\alpha$, dose $\theta_n$ and observed response $N_n$, the update is $\theta_{n+1} = \theta_n - \frac{c}{n} (N_n - \alpha)$ for a constant $c > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/CHJ4TAH0QX.md" tag="CHJ4TAH0QX" kind="statement">
# Lichnerowicz vanishing theorem

## Statement

Let $M$ be a closed manifold of dimension $n$ with a [[spin structure|RSPZ9TCFQ8]].
Suppose that $M$ carries a [[Riemannian metric|7CBT4SM83Y]] whose [[scalar curvature|ZHX2VYQDZ3]] is positive at every point.

1. The [[Dirac operator|QMRAVX14BJ]] of $M$ for this metric has trivial kernel: every harmonic spinor is zero.
2. If $n = 4k$, then the $\hat{A}$-genus of $M$ vanishes: $\hat{A}(M) = 0$.

## Remarks

Let $\psi$ be a spinor with $D\psi = 0$.
Integrating the [[Lichnerowicz formula|4WCJ49Z90S]] $D^2 = \nabla^* \nabla + \frac{R}{4}$ against $\psi$ gives $0 = \int_M \abs{\nabla \psi}^2 + \frac{1}{4} \int_M R \abs{\psi}^2$.
Both terms are nonnegative and $R > 0$, so $\psi = 0$.
By the Atiyah–Singer index theorem, $\hat{A}(M)$ is the [[index of the positive part of the Dirac operator|JR2M58RCW3]], $\dim \ker D^+ - \dim \ker D^-$, which is then $0$.
The argument works as well when $R \ge 0$ everywhere and $R > 0$ at some point.

Example: a K3 surface is a closed spin $4$-manifold with signature $-16$, so $\hat{A} = -\frac{\sigma}{8} = 2 \ne 0$.
Hence a K3 surface has no metric of positive scalar curvature.

The spin hypothesis is necessary.
The complex projective plane $\C P^2$ with the Fubini–Study metric has positive scalar curvature, but $\hat{A}(\C P^2) = -\frac{1}{8}$, computed from its Pontryagin number $p_1 = 3$ by $\hat{A} = -\frac{p_1}{24}$.
This is consistent because $\C P^2$ is not spin.

## Explanations

### L1 human

Positive scalar curvature is incompatible with certain topological invariants of spin manifolds.
The theorem shows that if the $\hat{A}$-genus of a spin manifold is not zero, no metric on it can have positive scalar curvature everywhere.
The proof is short: positive curvature forces the Dirac operator to have no kernel, and the index theorem turns this into a statement about topology.

### L2 human

This is the first and basic obstruction to metrics of positive scalar curvature.
Hitchin refined it to a $KO$-theoretic invariant.
Gromov and Lawson extended the method to Dirac operators twisted by bundles, which shows for example that tori admit no metric of positive scalar curvature.
For simply connected manifolds of dimension at least $5$, the refined invariant decides the question for spin manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/CHP0QWHFWM.md" tag="CHP0QWHFWM" kind="definition">
# Path homotopy

## Definition

Let $X$ be a topological space and let $I = [0, 1]$ be the unit [[interval|0Q1D85B8G6]].
A *path* in $X$ is a continuous map $\gamma \colon I \to X$, from $\gamma(0)$ to $\gamma(1)$.
Let $\gamma_0, \gamma_1 \colon I \to X$ be paths with $\gamma_0(0) = \gamma_1(0) = x_0$ and $\gamma_0(1) = \gamma_1(1) = x_1$.
A *path homotopy* from $\gamma_0$ to $\gamma_1$ is a continuous map $H \colon I \times I \to X$ such that:

1. $H(s, 0) = \gamma_0(s)$ and $H(s, 1) = \gamma_1(s)$ for all $s \in I$;
2. $H(0, t) = x_0$ and $H(1, t) = x_1$ for all $t \in I$.

The paths are *path homotopic*, written $\gamma_0 \simeq_p \gamma_1$, if a path homotopy exists.

## Remarks

A path homotopy is the same as a [[homotopy relative|ADDS6HE5SV]] to $\set{0, 1}$.
Path homotopy is an equivalence relation on paths from $x_0$ to $x_1$.
Example: in a convex set $C \subseteq \R^n$, the map $H(s, t) = (1 - t)\gamma_0(s) + t \gamma_1(s)$ is a path homotopy, since $H(0, t) = x_0$ and $H(1, t) = x_1$.
Non-example: in $S^1$, the loop $s \mapsto e^{2\pi i s}$ and the constant loop at $1$ are not path homotopic; the [[fundamental group of the circle|6SQXPE0JPN]] tells them apart.
</document>

<document source="https://tamnd.github.io/math2/node/CHS5AFJGWE.md" tag="CHS5AFJGWE" kind="concept">
# Limit

## Summary

The limit of a [[diagram|HJK75HDWFE]] in a [[category|A2DE66937D]] is a universal [[cone|7TPBQPPN97]] over it.
Every other cone factors through it in exactly one way.
Limits unify many constructions, such as products, equalizers, pullbacks, inverse limits and terminal objects.
A limit is unique up to unique isomorphism when it exists, and in the category of sets every small diagram has one.

## Explanations

### L1 human

The limit of a diagram is the best object that maps to every part of the diagram in a consistent way.
"Best" means that any other object with consistent maps to the diagram maps to the limit in exactly one way that matches.
In the category of sets, the limit is the set of all consistent choices of one element in each set of the diagram.
For two sets with no arrows between them, a consistent choice is a pair $(x, y)$, so the limit is the Cartesian product.
</document>

<document source="https://tamnd.github.io/math2/node/CHT5JYM6QW.md" tag="CHT5JYM6QW" kind="definition">
# p-adic number

## Definition

Let $p$ be a [[prime number|01826KMG2H]] and let $\abs{\cdot}_p$ be the [[p-adic absolute value|WGN82TCEF2]] on the [[rational numbers|0NCJ5WXN6Z]] $\Q$.
The field $\Q_p$ of *$p$-adic numbers* is the [[completion|06RW4RKWH7]] of $\Q$ for the metric $d(x, y) = \abs{x - y}_p$.
A $p$-adic number is an element of $\Q_p$.
The field operations and the absolute value $\abs{\cdot}_p$ extend from $\Q$ to $\Q_p$ by continuity.

## Remarks

Every $x \in \Q_p$ with $x \neq 0$ has a unique expansion $x = \sum_{n \ge m} a_n p^n$ with $m \in \Z$, digits $a_n \in \set{0, 1, \ldots, p - 1}$ and $a_m \neq 0$.
Then $\abs{x}_p = p^{-m}$.
Example: $-1 = \sum_{n \ge 0} (p - 1) p^n$ in $\Q_p$.
Check: the partial sum up to $p^{N-1}$ is $p^N - 1$, and $\abs{(p^N - 1) - (-1)}_p = p^{-N}$ tends to $0$.
Non-example: the formal sum $\sum_{n \ge 0} p^{-n}$ is not a $p$-adic number, because its terms have $\abs{p^{-n}}_p = p^n$ and do not tend to $0$.
The elements with $\abs{x}_p \le 1$ are the [[p-adic integers|V6TJAFZ4MD]] $\Z_p$.
By [[Ostrowski's theorem|GCQ23XT6VD]], the completions of $\Q$ are $\R$ and the fields $\Q_p$.
</document>

<document source="https://tamnd.github.io/math2/node/CJ57DT86S8.md" tag="CJ57DT86S8" kind="definition">
# Line graph

## Definition

Let $G = (V, E)$ be a [[graph|0JD5C82QMB]].
The line graph of $G$ is the graph $L(G)$ whose vertex set is the [[edge set|0WNGQBPPQX]] $E$ of $G$.
Two distinct edges $e, f \in E$ are adjacent in $L(G)$ if and only if they have a common endpoint.

## Remarks

If $e = uv$ is an edge of $G$, its degree in $L(G)$ is $\deg u + \deg v - 2$.
Example: the [[cycle graph|0QQ0F12962]] $C_n$ with $n \ge 3$ has $L(C_n) \cong C_n$, since each edge meets exactly the two edges next to it around the cycle.
Also $L(K_{1,3}) \cong K_3$: the three edges of the star share the centre, so they are pairwise adjacent.
Non-example: the star $K_{1,3}$ is itself not the line graph of any graph.
Indeed, suppose a vertex $e$ of $L(H)$ had three pairwise non-adjacent neighbours.
These would be three edges meeting $e$, so two of them would meet $e$ at the same endpoint, and these two would be adjacent.
Whitney showed that two connected graphs with isomorphic line graphs are isomorphic, except for the pair $K_3$ and $K_{1,3}$.
</document>

<document source="https://tamnd.github.io/math2/node/CJ5EC820S7.md" tag="CJ5EC820S7" kind="statement">
# Arnoldi iteration

## Statement

Let $A$ be a complex $n \times n$ matrix and let $b \in \C^n$ be nonzero.
The *Arnoldi iteration* sets $q_1 = b / \norm{b}$ and, for $j = 1, 2, \dots$, computes
$$h_{ij} = q_i^* A q_j \ (1 \le i \le j), \quad w_j = A q_j - \sum_{i=1}^{j} h_{ij} q_i, \quad h_{j+1, j} = \norm{w_j}, \quad q_{j+1} = w_j / h_{j+1, j}.$$
Here $\norm{\cdot}$ is the Euclidean norm.
Suppose $h_{j+1, j} \ne 0$ for $j = 1, \dots, m$.
Let $Q_m$ be the $n \times m$ matrix with columns $q_1, \dots, q_m$.
Let $\tilde{H}_m$ be the $(m+1) \times m$ matrix of the $h_{ij}$, with $h_{ij} = 0$ for $i > j + 1$, and let $H_m$ be its first $m$ rows.
Then:

1. $q_1, \dots, q_{m+1}$ is an orthonormal basis of the [[Krylov subspace|QB45PCGQPV]] $\mathcal{K}_{m+1}(A, b)$;
2. $A Q_m = Q_{m+1} \tilde{H}_m$, and $H_m = Q_m^* A Q_m$ is an upper [[Hessenberg matrix|YCEEWJTX56]];
3. if $h_{m+1, m} = 0$ instead, the iteration stops at step $m$, $\mathcal{K}_m(A, b)$ is invariant under $A$, and every eigenvalue of $H_m$ is an eigenvalue of $A$.

## Remarks

The iteration is the [[Gram–Schmidt process|0VT6P0GNQR]], in its modified form in practice, applied to the vectors $b, A q_1, A q_2, \dots$.
Since $A q_j$ lies in $\mathcal{K}_{j+1}(A, b)$, clause 1 follows by induction, and clause 2 is the matrix form of the defining recurrence.
Clause 3 holds because $A Q_m = Q_m H_m$ when $h_{m+1, m} = 0$.
The [[eigenvalues|0BQAA2BK3Q]] of $H_m$ are the Ritz values; the extreme ones often approximate eigenvalues of $A$ well for $m$ much smaller than $n$.
Example: if $b$ is an eigenvector of $A$ for $\lambda$, then $w_1 = 0$, the iteration stops at $m = 1$, and $H_1 = (\lambda)$.
If $A$ is [[Hermitian|0P17NGQQMQ]], then $H_m$ is Hermitian and Hessenberg, hence [[tridiagonal|DARX1KZRHZ]].
Then the recurrence has only three terms, and it becomes the [[Lanczos algorithm|7RRB4PCRJB]].

## Explanations

### L1 human

The Arnoldi iteration builds an orthonormal basis of the space spanned by $b, A b, A^2 b, \dots$, one vector at a time.
In this basis, the action of $A$ looks like a small matrix that is almost triangular.
Studying this small matrix reveals information about the large one, such as its extreme eigenvalues.

### L2 human

Arnoldi is the engine of GMRES for nonsymmetric linear systems and of the implicitly restarted Arnoldi method for eigenvalues.
Its cost grows with $m$: step $j$ needs one product with $A$ and $j$ inner products, and all $q_j$ must be stored.
This is why practical codes restart after a fixed number of steps.
The relation $A Q_m = Q_{m+1} \tilde{H}_m$ reduces large problems on $\mathcal{K}_m$ to small problems with $\tilde{H}_m$.
</document>

<document source="https://tamnd.github.io/math2/node/CJ7KCDDF31.md" tag="CJ7KCDDF31" kind="statement">
# Cauchy's rigidity theorem

## Statement

Let $P$ and $P'$ be convex [[polyhedra|0325SGSJAR]] in $\R^3$.
Suppose there is a bijection between their faces that preserves the incidences of faces, edges and vertices.
Suppose also that each face of $P$ is congruent to the corresponding face of $P'$.
Then there is an [[isometry|02HPP5VCZH]] of $\R^3$ that maps $P$ onto $P'$.

## Remarks

The proof compares the dihedral angles at corresponding edges and marks each edge $+$, $-$ or $0$ according to the change.
Cauchy's arm lemma shows that around each vertex with a marked edge, the signs change at least four times.
A count of sign changes with Euler's formula $V - E + F = 2$ then shows that no edge can be marked, so all dihedral angles agree.
Cauchy's original proof of the arm lemma had a gap, which was later repaired.

Convexity is needed.
Take a convex polyhedron with a vertex whose neighbours all lie in one plane.
Reflect the small pyramid at that vertex inward through that plane.
The new nonconvex polyhedron has the same combinatorics and congruent faces, but it is not congruent to the old one.
Flexible polyhedra show that even a single nonconvex polyhedron can deform continuously with rigid faces.

## Explanations

### L1 human

Make a convex solid from rigid cardboard faces joined by hinges at the edges.
Cauchy's theorem says the solid cannot wobble: the faces alone fix its whole shape.
In particular, a convex polyhedron is determined by its net of faces and the way they are glued.

### L2 human

The theorem is a statement of global rigidity for convex polyhedral surfaces among all convex realizations.
It implies that a convex polyhedron cannot be deformed continuously with rigid faces.
Dehn later proved the stronger infinitesimal rigidity of convex polyhedra.
For nonconvex polyhedra rigidity can fail, as Connelly's flexible polyhedron shows, but the bellows conjecture says the volume is still preserved.
</document>

<document source="https://tamnd.github.io/math2/node/CJ9MSY7CRM.md" tag="CJ9MSY7CRM" kind="concept">
# Coulomb wave function

## Summary

Coulomb wave functions are solutions of the Coulomb wave equation $u'' + \left( 1 - \frac{2\eta}{\rho} - \frac{L(L + 1)}{\rho^2} \right) u = 0$.
This equation is the radial part of the [[Schrödinger equation|EPZR6H2WJN]] for a charged particle in the field of a point charge.
The regular solution $F_L(\eta, \rho)$ vanishes at $\rho = 0$, and the irregular solution $G_L(\eta, \rho)$ does not.
Both are [[confluent hypergeometric functions|6AZ4ZBN1M8]] in disguise.

## Explanations

### L1 human

Without the charge, that is for $\eta = 0$, the equation describes a free wave, and for $L = 0$ its solutions are $\sin \rho$ and $\cos \rho$.
The term $2\eta / \rho$ adds the electric attraction or repulsion, and $L(L + 1)/\rho^2$ adds the effect of angular momentum.
Far from the origin the solutions still look like sine waves, but with a phase that drifts slowly like $\eta \log 2\rho$.
The regular function $F_L$ generalises $\sin \rho$, and the irregular function $G_L$ generalises $\cos \rho$.
</document>

<document source="https://tamnd.github.io/math2/node/CJCPD4DZ65.md" tag="CJCPD4DZ65" kind="statement">
# Positivity of Hankel matrices of moments

## Statement

Let $\mu$ be a positive Borel [[measure|PR11BZQQPG]] on $\R$ such that every [[moment|01XAHPADBA]] $m_k = \int x^k \, d\mu(x)$, $k \ge 0$, is finite.
For $n \ge 0$, let $H_n = (m_{j+k})_{j,k=0}^{n}$ be the $(n+1) \times (n+1)$ [[Hankel matrix|PQ1D8FQB3E]] of the moments.
Then:

1. $H_n$ is [[positive semidefinite|01MMVW8CKC]] for every $n \ge 0$.
2. If the [[support|FQTXV6BQZG]] of $\mu$ is an infinite set, then $H_n$ is [[positive definite|0KX66CZMBB]] for every $n \ge 0$, so $\det H_n > 0$.

## Remarks

For real $c = (c_0, \ldots, c_n)$, let $p(x) = \sum_{j=0}^{n} c_j x^j$.
Then $c^{T} H_n c = \sum_{j,k} c_j c_k \int x^{j+k} \, d\mu = \int p(x)^2 \, d\mu(x) \ge 0$, which is clause 1.
If $c \ne 0$, then $p$ is a nonzero polynomial with at most $n$ real roots.
If the support is infinite, then $p^2 > 0$ on a set of positive measure, so the integral is positive, which is clause 2.
Example: for Lebesgue measure on $[0, 1]$, $m_k = 1/(k+1)$, and $H_n$ is the Hilbert matrix $(1/(j + k + 1))$, which is positive definite.
For $\mu = \delta_0$, the support is finite, $H_n$ has $m_0 = 1$ in one corner and zeros elsewhere, and it is only semidefinite for $n \ge 1$.
The converse of clause 1 is Hamburger's theorem, part of the [[Hamburger moment problem|TBNCFXTMMW]].
It says that a real sequence with all $H_n$ positive semidefinite is the moment sequence of a positive measure on $\R$.

## Explanations

### L1 human

From the moments of a mass distribution, build square tables whose entry in row $j$ and column $k$ is the moment of order $j + k$.
These tables always behave like sums of squares: they never give a negative value.
The reason is that each value is the average of the square of a polynomial.

### L2 human

This is the necessary condition in the Hamburger moment problem, and Hamburger showed that it is also sufficient.
For the Stieltjes problem on $[0, \infty)$ one adds the shifted Hankel matrices $(m_{j+k+1})$, whose positivity reflects $x \ge 0$.
The positive definite case gives an inner product $\ip{p}{q} = \int p q \, d\mu$ on polynomials, and Gram–Schmidt then yields the orthogonal polynomials of $\mu$.
</document>

<document source="https://tamnd.github.io/math2/node/CJDS1BGDVH.md" tag="CJDS1BGDVH" kind="definition">
# Lyapunov function for a difference equation

## Definition

Let $D \subseteq \R^k$, let $f \colon D \to D$ be [[continuous|0X3WGX95FE]], and consider the difference equation $x_{n+1} = f(x_n)$.
Let $x^* \in D$ be an [[equilibrium|D62PEKHRV5]], so $f(x^*) = x^*$.
Let $G \subseteq D$ be a [[neighbourhood|0ZCY0Z80B6]] of $x^*$ relative to $D$.
A *Lyapunov function* for the equation on $G$ is a continuous function $V \colon G \to \R$ such that:

1. $V(x^*) = 0$.
2. $V(x) > 0$ for every $x \in G$ with $x \ne x^*$.
3. $\Delta V(x) = V(f(x)) - V(x) \le 0$ for every $x \in G$ with $f(x) \in G$.

The function $V$ is a *strict* Lyapunov function if moreover $\Delta V(x) < 0$ for every $x \ne x^*$ in $G$ with $f(x) \in G$.

## Remarks

Example: for $f(x) = x / (1 + x^2)$ on $\R$ with $x^* = 0$, take $V(x) = x^2$.
Then $V(0) = 0$, $V(x) > 0$ for $x \ne 0$, and $\Delta V(x) = x^2 \left( (1 + x^2)^{-2} - 1 \right) < 0$ for $x \ne 0$, so $V$ is a strict Lyapunov function.

Non-example: for $f(x) = 2x$ and $V(x) = x^2$, clauses 1 and 2 hold but $\Delta V(x) = 3x^2 > 0$ for $x \ne 0$, so clause 3 fails.

The quantity $\Delta V$ is the forward difference of $V$ along solutions, the analogue of the derivative $\dot V$ along solutions of a differential equation.
If a Lyapunov function exists, the equilibrium is stable; if a strict one exists, it is asymptotically stable.
For a linear map $f(x) = Ax$ one often uses a quadratic form $V(x) = x^T P x$ with $P$ positive definite and $A^T P A - P$ negative definite.
</document>

<document source="https://tamnd.github.io/math2/node/CJFEPPP9KA.md" tag="CJFEPPP9KA" kind="concept">
# Mahlo cardinal

## Summary

A Mahlo cardinal is an [[inaccessible cardinal|NW4DREQ5DQ]] $\kappa$ such that the inaccessible cardinals below $\kappa$ form a [[stationary set|YE6WZR46RQ]] in $\kappa$.
So a Mahlo cardinal has very many inaccessible cardinals below it, and it is the $\kappa$-th inaccessible cardinal.
The notion is named after Paul Mahlo.
Mahlo cardinals are stronger than inaccessible cardinals, and weaker than [[weakly compact cardinals|1RHTVMH34G]].

## Explanations

### L1 human

An inaccessible cardinal is a size of infinity that cannot be reached from below.
A Mahlo cardinal is an inaccessible cardinal that has inaccessible cardinals below it in a strong sense: any closed unbounded set of smaller ordinals contains one of them.
The first inaccessible cardinal is not Mahlo, since there is no inaccessible cardinal below it at all.
So Mahlo cardinals are a step up in the hierarchy of large cardinals.
</document>

<document source="https://tamnd.github.io/math2/node/CJGKW80KBR.md" tag="CJGKW80KBR" kind="concept">
# Krawtchouk polynomials

## Summary

The Krawtchouk (or Kravchuk) polynomials $K_n(x; p, N)$ are the orthogonal polynomials of the [[binomial distribution|0WCWA1KT3M]] on $\set{0, 1, \dots, N}$.
They are a finite family: there is one polynomial for each degree $n = 0, \dots, N$.
In coding theory they describe the Hamming scheme, and they enter the MacWilliams identities and Delsarte's linear programming bound.

## Explanations

### L1 human

Toss a coin $N$ times, with probability $p$ of heads, and count the heads $x$.
The count follows the binomial distribution on the numbers $0$ to $N$.
The Krawtchouk polynomials are the polynomials in $x$ that are perpendicular to each other for this distribution.
The first nonconstant one is $1 - \frac{x}{Np}$, which averages to zero because the mean number of heads is $Np$.
</document>

<document source="https://tamnd.github.io/math2/node/CJK2HS2508.md" tag="CJK2HS2508" kind="statement">
# Courant–Fischer min-max theorem

## Statement

Let $A$ be an $n \times n$ [[Hermitian matrix|0P17NGQQMQ]] with [[eigenvalues|0BQAA2BK3Q]] $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n$, listed with multiplicity.
For $x \in \C^n$ with $x \ne 0$, let $R_A(x) = \dfrac{x^* A x}{x^* x}$ be the [[Rayleigh quotient|06WHP8SE98]].
Then, for each $k = 1, \dots, n$,
$$\lambda_k = \max_{\dim U = k} \; \min_{x \in U \setminus \set{0}} R_A(x) = \min_{\dim U = n-k+1} \; \max_{x \in U \setminus \set{0}} R_A(x),$$
where $U$ runs over the [[linear subspaces|00YC1D652K]] of $\C^n$ of the given [[dimension|03GSZ8RTZJ]].

## Remarks

Let $u_1, \dots, u_n$ be an orthonormal basis of eigenvectors with $A u_i = \lambda_i u_i$.
On the span of $u_1, \dots, u_k$, the Rayleigh quotient is at least $\lambda_k$, so the max-min is at least $\lambda_k$.
Every subspace $U$ of dimension $k$ meets the span of $u_k, \dots, u_n$ in a nonzero vector, by counting dimensions, and there $R_A \le \lambda_k$.
So the max-min is exactly $\lambda_k$; the min-max form is proved in the same way.

For $k = 1$ it gives $\lambda_1 = \max_{x \ne 0} R_A(x)$, and for $k = n$ it gives $\lambda_n = \min_{x \ne 0} R_A(x)$.
For $A = \operatorname{diag}(3, 1)$, $R_A(x) = (3x_1^2 + x_2^2)/(x_1^2 + x_2^2)$ for real $x$, which takes all values in $[1, 3]$.

The Hermitian hypothesis is needed: for a non-Hermitian matrix the eigenvalues need not be real, and $R_A$ need not be real either.

## Explanations

### L1 human

The eigenvalues of a symmetric matrix can be found by an optimisation, without solving any polynomial equation.
The largest eigenvalue is the largest value of the Rayleigh quotient.
The second one is the best value one can guarantee on every plane, and so on.

### L2 human

The theorem turns questions about eigenvalues into questions about subspaces.
It gives the [[Cauchy interlacing theorem|8QQPVNR80F]] and [[Weyl's inequality|RACSNQXDQV]] in a few lines.
The real symmetric case is the same, with $\R^n$ and $x^T A x$.
A version holds for compact self-adjoint operators on Hilbert spaces.
</document>

<document source="https://tamnd.github.io/math2/node/CJQ8D9FTQN.md" tag="CJQ8D9FTQN" kind="statement">
# Bias–variance decomposition

## Statement

Let $x$ be a fixed point and let $Y = f(x) + \varepsilon$, where $f(x)$ is a real number and $\varepsilon$ is a random variable with $E(\varepsilon) = 0$ and $\operatorname{Var}(\varepsilon) = \sigma^2$.
Let $\hat{f}(x)$ be a random variable with $E(\hat{f}(x)^2) < \infty$ that is [[independent|0E9502NT92]] of $\varepsilon$.
For example, $\hat{f}(x)$ is a prediction built from a training sample independent of the new observation.
Then
$$E\big[(Y - \hat{f}(x))^2\big] = \big(E[\hat{f}(x)] - f(x)\big)^2 + \operatorname{Var}\big(\hat{f}(x)\big) + \sigma^2.$$

## Remarks

Proof: write $Y - \hat{f} = \varepsilon + (f - E\hat{f}) + (E\hat{f} - \hat{f})$.
Expand the square and take [[expectations|0DCT0QZVQ1]]: the three cross terms vanish, because $E\varepsilon = 0$, $E(E\hat{f} - \hat{f}) = 0$, and $\varepsilon$ is independent of $\hat{f}$.
The remaining terms are $\sigma^2$, the squared bias and the [[variance|007QED3E0Y]] of $\hat{f}$.

The case without noise is the decomposition of the [[mean squared error|0V4VSCBXP3]] of an estimator $\hat{\theta}$ of a parameter $\theta$:
$$E\big[(\hat{\theta} - \theta)^2\big] = \operatorname{Bias}(\hat{\theta})^2 + \operatorname{Var}(\hat{\theta}),$$
with the [[bias|01BD2TFFP4]] $\operatorname{Bias}(\hat{\theta}) = E(\hat{\theta}) - \theta$.

Example: for i.i.d. $X_1, \dots, X_n$ with variance $\sigma^2$, the estimator $\hat{\sigma}^2 = \frac{1}{n}\sum_i (X_i - \bar{X})^2$ has bias $-\sigma^2/n$.
For normal data its variance is $2(n-1)\sigma^4/n^2$, so its mean squared error is $(2n - 1)\sigma^4/n^2$, which is smaller than the mean squared error $2\sigma^4/(n-1)$ of the unbiased sample variance.
A small bias can thus buy a larger reduction of variance.

The irreducible term $\sigma^2$ does not depend on $\hat{f}$; no predictor can do better than it on average.

## Explanations

### L1 human

The average squared error of a prediction has three parts.
The bias is how far the prediction is off on average.
The variance is how much the prediction jumps around when the training data change.
The noise is the part of the outcome that nobody can predict.
Simple models tend to have large bias and small variance, flexible models the opposite.

### L2 human

The decomposition is the basis of the bias–variance tradeoff in model selection: adding complexity lowers the bias but raises the variance, and the best expected error lies in between.
Shrinkage methods such as ridge regression accept bias to reduce variance.
In point estimation it explains why an unbiased estimator need not have the smallest mean squared error.
Note that the decomposition holds at each fixed point $x$; averaging over $x$ gives the decomposition of the expected prediction error.
</document>

<document source="https://tamnd.github.io/math2/node/CK08JW32WB.md" tag="CK08JW32WB" kind="concept">
# Generator

## Summary

A generator, also called a separator, of a [[category|A2DE66937D]] is an object $G$ with the following property.
Its morphisms into other objects are enough to tell any two different parallel morphisms apart.
Equivalently, the [[hom functor|VJXS8AK67E]] $\Hom(G, -)$ is faithful.
Generators and the dual cogenerators appear as hypotheses in the special adjoint functor theorem, and generators are part of the definition of a Grothendieck category.

## Explanations

### L1 human

Two functions $f, g \colon X \to Y$ are different exactly when they differ at some point $x$ of $X$.
In the category of sets, a point of $X$ is the same as a map from the one-point set $\{*\}$ to $X$.
So $\{*\}$ can "probe" every difference between maps: it is a generator.
In the category of abelian groups, the group $\Z$ plays the same role, since a homomorphism $\Z \to A$ is the same as an element of $A$.
A generator is an object whose probes see everything.
</document>

<document source="https://tamnd.github.io/math2/node/CK1DCPC3D3.md" tag="CK1DCPC3D3" kind="concept">
# Hermitian manifold

## Summary

A Hermitian manifold is a complex manifold with a Riemannian metric that is compatible with multiplication by $i$ on tangent vectors.
It is the complex analogue of a Riemannian manifold, and every complex manifold admits such a metric.
Each Hermitian metric has a fundamental $2$-form, and when this form is closed the manifold is a Kähler manifold.
Kähler manifolds are the setting of Hodge theory and of much of complex algebraic geometry.

## Explanations

### L1 human

On a complex manifold one can turn each tangent vector by a quarter turn, using multiplication by $i$.
A Hermitian metric measures lengths and angles so that this quarter turn never changes a length or an angle.
The plane $\C$ with the usual distance is the simplest example: rotating an arrow by $90$ degrees keeps its length.
Pairing the metric with the quarter turn gives a $2$-form, which measures oriented areas.
</document>

<document source="https://tamnd.github.io/math2/node/CK4QHTVKN4.md" tag="CK4QHTVKN4" kind="definition">
# Space of functions of bounded variation

## Definition

Let $a < b$ be real numbers.
The *space of functions of bounded variation* $BV([a, b])$ is the set of functions $f \colon [a, b] \to \R$ of [[bounded variation|A8251BHCMP]], with pointwise operations.
Its norm is
$$\norm{f}_{BV} = \abs{f(a)} + V_a^b(f),$$
where $V_a^b(f)$ is the [[total variation|8QA5DEWNYD]] of $f$ on $[a, b]$.

## Remarks

This is a [[normed space|03J5RAKB30]].
The variation is subadditive and $V_a^b(t f) = \abs{t} V_a^b(f)$, which gives the triangle inequality and homogeneity.
If $\norm{f}_{BV} = 0$, then $V_a^b(f) = 0$, so $f$ is constant, and $f(a) = 0$ gives $f = 0$.
The space is complete, so it is a Banach space.

Example: $f(x) = x$ on $[0, 1]$.
It is increasing, so $V_0^1(f) = f(1) - f(0) = 1$ and $\norm{f}_{BV} = 0 + 1 = 1$.

Non-example: $f(x) = x \sin(1/x)$ for $0 < x \le 1$, with $f(0) = 0$.
It is continuous on $[0, 1]$, but its total variation is infinite, so it is not in $BV([0, 1])$.

An equivalent norm is $\sup_{x} \abs{f(x)} + V_a^b(f)$.
</document>

<document source="https://tamnd.github.io/math2/node/CK80HQNHEK.md" tag="CK80HQNHEK" kind="definition">
# q-analog

## Definition

Let $q$ be an indeterminate, or a complex number close to $1$.
A *q-analog* of an expression $E$ is an expression $E_q$, depending on $q$, such that $\lim_{q \to 1} E_q = E$.
A q-analog of an identity or a theorem is an identity or a theorem about q-analogs that reduces to the original one as $q \to 1$.
For an integer $n \ge 0$, the *q-integer* is
$$[n]_q = \frac{1 - q^n}{1 - q} = 1 + q + \cdots + q^{n-1},$$
and the *q-factorial* is $[n]_q! = [1]_q [2]_q \cdots [n]_q$, with $[0]_q! = 1$.

## Remarks

Example: $[n]_q$ is a q-analog of $n$, since $1 + q + \cdots + q^{n-1} \to n$ as $q \to 1$.
Likewise $[n]_q!$ is a q-analog of $n!$.
It is the generating function of the number of inversions over the [[permutations|0BQMWCNS8S]] of $\set{1, \ldots, n}$: $[n]_q! = \sum_{\pi} q^{\operatorname{inv}(\pi)}$.
For $n = 3$, the inversion numbers are $0, 1, 1, 2, 2, 3$, and $[3]_q! = (1)(1 + q)(1 + q + q^2) = 1 + 2q + 2q^2 + q^3$.

Non-example: $q^n$ is not a q-analog of $n$, since $q^n \to 1$ as $q \to 1$, and $1 \ne n$ for $n \ne 1$.

The definition by a limit alone does not fix a unique q-analog.
In practice a q-analog is chosen because it keeps the structure of the original, such as a recurrence or a combinatorial meaning.
Main examples are the [[Gaussian binomial coefficients|YS09Z1YNEK]], the [[q-Pochhammer symbol|Y1X8ZZQ7VM]] and the [[q-binomial theorem|VVT27RB46B]].
</document>

<document source="https://tamnd.github.io/math2/node/CK8ZQRN183.md" tag="CK8ZQRN183" kind="concept">
# Heyting arithmetic

## Summary

Heyting arithmetic, HA, is the theory of the natural numbers with the axioms of [[Peano arithmetic|0QFT7Z76N5]] and the rules of [[intuitionistic logic|Q11N8X99FK]].
It is the standard formal system for constructive arithmetic.
Every existence theorem of HA comes with a way to compute a witness.
It proves the same formulas as Peano arithmetic up to a double-negation translation, so the two theories are equally consistent.

## Explanations

### L1 human

Heyting arithmetic keeps the usual axioms of the natural numbers, including induction.
It only removes one rule of logic: the law that every statement is either true or false.
So it can prove "$A$ or $B$" only when it can tell which of the two holds.
For example, it proves that every number is even or odd, because one can compute which.
It cannot prove the statement "every program halts or does not halt", since no algorithm decides which.
</document>

<document source="https://tamnd.github.io/math2/node/CKF9KDXBP7.md" tag="CKF9KDXBP7" kind="concept">
# Mann–Whitney U test

## Summary

The Mann–Whitney U test, also called the Wilcoxon rank-sum test, is a [[hypothesis test|087P03NEND]] that compares two independent samples.
Its statistic counts the pairs, one value from each sample, in which the first value is larger.
Under the null hypothesis that both samples come from the same continuous distribution, the law of this count does not depend on that distribution.
It is the rank-based counterpart of the two-sample $t$-test, and it does not need normality.

## Explanations

### L1 human

Put the two samples together and ask, for every pair made of one value from each sample, which is larger.
If the two samples come from the same population, the first sample should win about half of these duels.
If it wins nearly all of them, its values tend to be larger.
For example, with samples $(3, 5)$ and $(1, 4)$, the first sample wins $3$ of the $4$ duels.
</document>

<document source="https://tamnd.github.io/math2/node/CKGDR9P9PA.md" tag="CKGDR9P9PA" kind="concept">
# Yang–Mills functional

## Summary

The Yang–Mills functional assigns to a [[connection|NJ21PN5ZGJ]] on a [[principal bundle|8E37E74XVP]] over a compact Riemannian manifold the integral of the squared norm of its [[curvature|ZAJFW1S078]].
It does not change under gauge transformations.
Its critical points are the solutions of the [[Yang–Mills equations|JJWETW5AT8]], and its absolute minima include the flat connections.
In dimension four its minima in a given bundle are often the instantons, which led to deep results on four-manifolds.

## Explanations

### L1 human

A connection tells how to compare the fibres of a bundle at nearby points, and its curvature measures how much this comparison twists around small loops.
The Yang–Mills functional adds up the square of this twisting over the whole manifold, like the total energy of a field.
For the circle group the functional is the energy of an electromagnetic field, half the integral of the squared field strength.
Connections that make it as small as possible, or that balance it to first order, are the natural fields of gauge theory.
</document>

<document source="https://tamnd.github.io/math2/node/CKGN4ZH6P2.md" tag="CKGN4ZH6P2" kind="definition">
# Strict Henselization

## Definition

Let $(R, \mathfrak{m}, k)$ be a [[local ring|MZ02BKPHJA]] with [[residue field|NJ3APQ8BHH]] $k$, and let $k^{s}$ be a [[separable closure|JJ91A1V8NQ]] of $k$.
A local ring is *strictly Henselian* if it is a [[Henselian local ring|RNXPV3KC5J]] and its residue field is [[separably closed|8B57B7EGK4]].
A *strict Henselization* of $R$ is a local ring $R^{sh}$ with a [[local homomorphism|P8YTMGEN9R]] $i \colon R \to R^{sh}$.
The residue field of $R^{sh}$ is identified with $k^{s}$, and the following hold:

1. $R^{sh}$ is strictly Henselian.
2. The map of residue fields induced by $i$ is the inclusion $k \subseteq k^{s}$.
3. Let $S$ be a strictly Henselian local ring with residue field $\kappa_S$, let $f \colon R \to S$ be a local homomorphism, and let $\sigma \colon k^{s} \to \kappa_S$ be a $k$-embedding.
Then there is exactly one local homomorphism $g \colon R^{sh} \to S$ with $g \circ i = f$ that induces $\sigma$ on residue fields.

## Remarks

A strict Henselization exists and is unique up to a unique isomorphism once $k^{s}$ is fixed.
It is a direct limit of localizations of [[étale|FTRW0679E2]] $R$-algebras, and the map $i$ is [[faithfully flat|C31PYYPRGC]].
The map $i$ factors through the [[Henselization|917PGHVZPM]] of $R$.

Example: for a field $k$, the field $k^{s}$ with the inclusion $k \to k^{s}$ is a strict Henselization.
A field is a Henselian local ring with maximal ideal $0$, its residue field is itself, and $k^{s}$ is separably closed.

Non-example: the ring $\Z_p$ of $p$-adic integers with the identity map fails clause 1.
It is Henselian, but its residue field $\F_p$ is not separably closed.
</document>

<document source="https://tamnd.github.io/math2/node/CKQRZVBGTS.md" tag="CKQRZVBGTS" kind="object">
# Kepler–Poinsot polyhedron

## Description

A *Kepler–Poinsot polyhedron* is a regular star [[polyhedron|0325SGSJAR]].
It is non-convex, its faces are congruent regular polygons or regular star polygons, and the same number of faces meet at each vertex.
Its symmetries act transitively on its flags.
There are exactly four, given here with their [[Schläfli symbols|MAE6BF4DFK]], where $5/2$ denotes the pentagram:

1. the small stellated dodecahedron $\set{5/2, 5}$;
2. the great dodecahedron $\set{5, 5/2}$;
3. the great stellated dodecahedron $\set{5/2, 3}$;
4. the great icosahedron $\set{3, 5/2}$.

## Remarks

Their vertices, edges and faces number $(V, E, F) = (12, 30, 12)$, $(12, 30, 12)$, $(20, 30, 12)$ and $(12, 30, 20)$ in this order.
So $V - E + F$ equals $-6$ for the first two and $2$ for the last two.
The first two do not satisfy Euler's formula $V - E + F = 2$, since their pentagram or crossing pentagon faces do not fit together into a surface shaped like a sphere.
All four share the icosahedral symmetry group of the icosahedron and dodecahedron.
They are regular in the flag sense of a [[regular polytope|F60PD912D5]], but not convex.
Kepler described the two stellated dodecahedra, Poinsot the other two, and Cauchy showed that there are no other regular star polyhedra.

## Explanations

### L1 human

Extend the faces of a dodecahedron or an icosahedron until they meet again, and spiky star-shaped solids appear.
Four of these are as regular as the Platonic solids: every face, edge and corner looks like every other.
Their faces pass through each other, which is why they are not convex.
</document>

<document source="https://tamnd.github.io/math2/node/CKTDR0VE8D.md" tag="CKTDR0VE8D" kind="concept">
# Solution set condition

## Summary

The solution set condition on a [[functor|28NQ0KRHD3]] $G \colon \mathcal{D} \to \mathcal{C}$ asks the following.
For each object $c$, a mere set of morphisms $c \to G d$ is enough to factor all of them.
It is a smallness condition.
Together with completeness and preservation of limits, it gives the existence of a left [[adjoint|TZHF6ZM92S]] in Freyd's adjoint functor theorem.

## Explanations

### L1 human

To build a free group on a set $X$, one would like to look at all maps from $X$ into all groups at once.
There are too many groups for that to be a set.
But every such map lands in the subgroup generated by its image, and that subgroup is not much bigger than $X$.
So a set of small groups is enough to see all maps from $X$.
The solution set condition asks for such a set in general.
</document>

<document source="https://tamnd.github.io/math2/node/CM3KYNZZJ6.md" tag="CM3KYNZZJ6" kind="definition">
# Hyponormal operator

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $T \colon H \to H$ be a bounded linear operator with [[adjoint|6HSJBJ8TTQ]] $T^*$.
The operator $T$ is *hyponormal* when $T^*T - TT^*$ is a [[positive operator|77020CY5ZW]], that is,
$$\ip{(T^*T - TT^*)x}{x} \ge 0 \quad \text{for all } x \in H.$$

## Remarks

Example: for the unilateral shift $S(x_0, x_1, \ldots) = (0, x_0, x_1, \ldots)$ on $\ell^2(\N)$, one has $S^*S = I$ and $SS^*$ is the projection onto the vectors with $x_0 = 0$.
So $S^*S - SS^*$ is the projection onto the first coordinate, which is positive; $S$ is hyponormal.

Non-example: the backward shift $S^*$ gives $(S^*)^* S^* - S^*(S^*)^* = SS^* - S^*S$, the negative of that projection, and $\ip{(SS^* - S^*S)e_0}{e_0} = -1 < 0$ for $e_0 = (1, 0, 0, \ldots)$.

Equivalent form: $T$ is hyponormal if and only if $\norm{T^*x} \le \norm{Tx}$ for all $x \in H$, since $\ip{(T^*T - TT^*)x}{x} = \norm{Tx}^2 - \norm{T^*x}^2$.
The operator $T$ is normal exactly when both $T$ and $T^*$ are hyponormal.
</document>

<document source="https://tamnd.github.io/math2/node/CM6XF84MNY.md" tag="CM6XF84MNY" kind="definition">
# Loeb measure

## Definition

Work in a nonstandard extension of the real numbers that is countably saturated.
Let $\Omega$ be an internal set and let $\mathcal{A}$ be an internal algebra of subsets of $\Omega$.
Let $\mu \colon \mathcal{A} \to {}^*[0, \infty)$ be an internal [[finitely additive measure|KFWQ24D9M6]] such that $\mu(\Omega)$ is a finite hyperreal number.
For $A \in \mathcal{A}$ let $\operatorname{st}\mu(A)$ be the standard part of $\mu(A)$, that is, the unique real number infinitely close to $\mu(A)$.
The *Loeb measure* $L_\mu$ of $\mu$ is the unique [[measure|PR11BZQQPG]] on the σ-algebra $\sigma(\mathcal{A})$ generated by $\mathcal{A}$ such that $L_\mu(A) = \operatorname{st}\mu(A)$ for all $A \in \mathcal{A}$.
The [[measure space|P15VQS5DJ9]] $(\Omega, \sigma(\mathcal{A}), L_\mu)$ is the *Loeb space* of $\mu$.

## Remarks

The map $\operatorname{st}\mu$ is finitely additive on $\mathcal{A}$.
By countable saturation, if a countable union of sets of $\mathcal{A}$ lies in $\mathcal{A}$, then it is already a finite union, so $\operatorname{st}\mu$ is countably additive on $\mathcal{A}$.
Hence [[Carathéodory's extension theorem|Q8EXG7EKYW]] gives the extension, and it is unique because $\operatorname{st}\mu(\Omega)$ is finite.
Many authors call the [[completion|KJNSKV75VQ]] of this measure the Loeb measure.

Example: let $N$ be an infinite hypernatural number, let $\Omega = \set{1, \ldots, N}$, and let $\mathcal{A}$ be the internal power set of $\Omega$.
Let $\mu(A) = \abs{A}/N$, where $\abs{A}$ is the internal number of elements of $A$.
Then $\mu(\Omega) = 1$ is finite, and $L_\mu$ is a probability measure.
The standard part map $k \mapsto \operatorname{st}(k/N)$ pushes $L_\mu$ forward to Lebesgue measure on $[0, 1]$.

Non-example: on the same $\Omega$, the internal counting measure $\mu(A) = \abs{A}$ has $\mu(\Omega) = N$, which is infinite.
It fails the finiteness clause, and the construction above does not apply directly.
For such $\mu$ one can still define a Loeb measure, through an outer measure, but it need not be the unique extension.
</document>

<document source="https://tamnd.github.io/math2/node/CM7D3ES438.md" tag="CM7D3ES438" kind="definition">
# Keller–Segel model

## Definition

Let $\Omega \subseteq \R^n$ be open and let $\chi > 0$, $\tau \ge 0$, $\alpha \ge 0$ be constants.
The *Keller–Segel model* is the system for a cell density $u(x, t) \ge 0$ and a chemical concentration $v(x, t)$:
$$\partial_t u = \Delta u - \chi\, \nabla \cdot (u \nabla v), \qquad \tau\, \partial_t v = \Delta v - \alpha v + u.$$
Here $\Delta$ is the [[Laplacian|0E0B6RA8K0]], $\nabla$ the [[gradient|0ABK9V9Y64]] and $\nabla \cdot$ the [[divergence|0MWV6ZTK6S]] in $x$.
On a bounded $\Omega$ one usually imposes no-flux conditions $\partial_\nu u = \partial_\nu v = 0$ on $\partial\Omega$.
For $\tau > 0$ the system is [[parabolic|1CCTH8X2FT]]; the case $\tau = 0$ is the parabolic-elliptic Keller–Segel model.

## Remarks

Example: on a bounded $\Omega$ with $\alpha > 0$, the constants $u \equiv \alpha c$, $v \equiv c$ with $c > 0$ form a stationary solution, since all derivatives vanish and $-\alpha c + \alpha c = 0$.
Non-example: with $\chi = 0$ the cells do not respond to the chemical, so the system decouples into a heat equation for $u$; it is excluded by $\chi > 0$.
The total mass $\int_\Omega u\, dx$ is conserved under the no-flux conditions.
Consider the parabolic-elliptic model on $\R^2$ with $\tau = \alpha = 0$ and $\chi = 1$.
Solutions with mass less than $8\pi$ and suitable integrability of the initial data exist for all time.
Solutions with mass greater than $8\pi$ and finite second moment blow up in finite time.
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<document source="https://tamnd.github.io/math2/node/CM9BDNEA10.md" tag="CM9BDNEA10" kind="statement">
# Makarov's theorem on harmonic measure

## Statement

Let $\Omega \subseteq \C$ be a bounded domain whose boundary $\partial\Omega$ is a Jordan curve.
Let $z \in \Omega$, and let $\omega$ be the [[harmonic measure|CFD7BPM51W]] of $\Omega$ at $z$.
Then:

1. $\omega(E) = 0$ for every Borel set $E \subseteq \partial\Omega$ of [[Hausdorff dimension|SSNFTHZ6YJ]] less than $1$.
2. There is a Borel set $F \subseteq \partial\Omega$ of Hausdorff dimension $1$ with $\omega(\partial\Omega \setminus F) = 0$.

## Remarks

In short, the harmonic measure of a [[simply connected|0QW64ARSHB]] planar domain has dimension exactly $1$, whatever the dimension of the boundary.
Let $f$ be a conformal map from the unit disc onto $\Omega$ with $f(0) = z$.
By Carathéodory's theorem $f$ extends to the circle, and $\omega$ is the image of normalised arc length under this extension.
Idea of the proof: the growth of $\log f'$, a Bloch function, is controlled by a law of the iterated logarithm.
This shows that $f$ distorts the size of small arcs at almost every point only by subpower factors, which gives both parts.
Example: for the domain bounded by the Koch snowflake, the boundary has dimension $\log 4 / \log 3 > 1$, yet $\omega$ is carried by a subset of dimension $1$.
Simple connectivity matters for part 1: let $K$ be the middle-thirds Cantor set, of dimension $\log 2 / \log 3 < 1$ and of positive logarithmic capacity.
The harmonic measure of the complement of $K$ in the Riemann sphere, at $\infty$, is carried by $K$ and so by a set of dimension less than $1$.
The theorem was proved by N. G. Makarov in 1985.

## Explanations

### L1 human

Release a random walker inside a planar region bounded by a single closed curve, and record where it first hits the curve.
Even if the curve is wildly wiggly and has dimension larger than $1$, the hitting places concentrate on a part of dimension exactly $1$.
The walker almost never reaches the deep fjords of the curve.

### L2 human

Makarov's theorem started the study of the dimension of harmonic measure.
For general planar domains, Jones and Wolff showed that harmonic measure is always carried by a set of dimension at most $1$.
In higher dimensions the picture is different.
Bourgain proved a bound strictly below the dimension of the space, and Wolff built domains in $\R^3$ whose harmonic measure has dimension larger than $2$.
The proof links harmonic measure with the boundary behaviour of conformal maps and with probabilistic limit theorems.
</document>

<document source="https://tamnd.github.io/math2/node/CMAX3KF9H0.md" tag="CMAX3KF9H0" kind="concept">
# Akaike information criterion

## Summary

The Akaike information criterion (AIC) scores a fitted statistical model by its maximized [[likelihood|0VW9E4QXR6]], with a penalty for the number of parameters.
Among candidate models, one prefers the model with the smallest AIC.
It rests on an estimate of the expected [[Kullback–Leibler divergence|PQBW43X5RE]] between the true law and the fitted model, and it aims at good prediction.

## Explanations

### L1 human

A model with more parameters always fits the data at hand at least as well, but it may only be fitting noise.
AIC balances the fit, measured by the likelihood, against the complexity, measured by the number of parameters.
Each extra parameter costs $2$ units, so it must raise twice the log-likelihood by more than $2$ to be worth adding.
For example, a cubic regression is preferred to a straight line only if its better fit pays for its two extra coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/CMC2C1D53W.md" tag="CMC2C1D53W" kind="concept">
# Woodin cardinal

## Summary

A Woodin cardinal is a [[large cardinal|X1FG7PSCTY]] defined by [[elementary embeddings|CGH8C8EA9N]] that reach high into the universe.
It is named after W. Hugh Woodin.
Woodin cardinals are the large cardinals behind [[determinacy|QXECDPAYTN]]: Martin and Steel proved that infinitely many Woodin cardinals imply [[projective determinacy|2GY3131ZPS]].
A Woodin cardinal is Mahlo, but it need not be weakly compact or measurable.

## Explanations

### L1 human

A Woodin cardinal is a size of infinity below which there are many cardinals that are the starting point of embeddings of the universe into a copy of itself.
These embeddings must keep a large part of the universe intact, and how large is controlled by any given function.
Woodin cardinals are the large cardinals one needs to show that many infinite games on the reals are determined.
So they connect the theory of large cardinals with the study of sets of real numbers.
</document>

<document source="https://tamnd.github.io/math2/node/CMGKQNMY77.md" tag="CMGKQNMY77" kind="statement">
# Shapley–Folkman lemma

## Statement

Let $d \ge 1$ and $N \ge 1$, and let $S_1, \dots, S_N$ be subsets of $\R^d$.
Let $S_1 + \cdots + S_N$ be their [[Minkowski sum|SXS6D87PHE]] and let $\operatorname{conv}$ denote the [[convex hull|0FQSDC2EQ2]].
Then every point $x \in \operatorname{conv}(S_1 + \cdots + S_N)$ can be written as $x = x_1 + \cdots + x_N$ with $x_i \in \operatorname{conv}(S_i)$ for each $i$, and with $x_i \in S_i$ for all but at most $d$ indices $i$.

## Remarks

First, $\operatorname{conv}(S_1 + \cdots + S_N) = \operatorname{conv}(S_1) + \cdots + \operatorname{conv}(S_N)$, so $x = \sum_i \sum_j \lambda_{ij} y_{ij}$ with $y_{ij} \in S_i$, $\lambda_{ij} \ge 0$ and $\sum_j \lambda_{ij} = 1$ for each $i$.
Then the point $(x, 1, \dots, 1) \in \R^{d+N}$ is a nonnegative combination of the vectors $(y_{ij}, e_i)$, where $e_i$ is the $i$-th unit vector of $\R^N$.
By the conic form of [[Carathéodory's theorem|0201Q4P2B5]], at most $d + N$ of these vectors are needed.
Each index $i$ needs at least one of them, so at most $d$ indices use two or more, and the other indices use a single point of $S_i$.
Example: $d = 1$ and $S_i = \set{0, 1}$ for all $i$.
The point $x = 2.5$ of $\operatorname{conv}(S_1 + S_2 + S_3) = [0, 3]$ is $1 + 1 + 0.5$, and only one summand is outside its $S_i$.

## Explanations

### L1 human

Adding many sets together makes the sum look almost convex, even if the sets themselves are far from convex.
The lemma makes this precise: a point of the convex hull of the sum can be reached by taking actual points of all but $d$ of the sets.
The number $d$ is the dimension, and it does not grow with the number of sets.

### L2 human

A quantitative consequence, the Shapley–Folkman theorem, bounds the distance between $\operatorname{conv}(S_1 + \cdots + S_N)$ and $S_1 + \cdots + S_N$ in terms of $d$ and the sizes of the $S_i$, independently of $N$.
In mathematical economics this explains why equilibria nearly exist in large markets with nonconvex preferences.
In optimization it bounds the duality gap of separable nonconvex problems with many terms.
</document>

<document source="https://tamnd.github.io/math2/node/CMK92XDD0W.md" tag="CMK92XDD0W" kind="definition">
# Willmore flow

## Definition

Let $\Sigma$ be a closed oriented surface and let $T > 0$.
Let $f \colon \Sigma \times [0, T) \to \R^3$ be a smooth family of immersions, and write $f_t = f(\cdot, t)$.
For each $t$, let $\nu$ be the unit normal of $f_t$.
Let $H = \tfrac{1}{2}(\kappa_1 + \kappa_2)$ be the [[mean curvature|PA78NFEPCF]] of $f_t$ with respect to $\nu$, so that the mean curvature vector is $2H\nu$.
Let $K$ be the [[Gaussian curvature|SQVQ5DQT6W]] of $f_t$.
Let $\Delta$ be the [[Laplace–Beltrami operator|F4NWZNTRMD]] of the metric induced by $f_t$.
The family is a solution of the *Willmore flow* if
$$\frac{\partial f}{\partial t} = -\left(\Delta H + 2H\left(H^2 - K\right)\right) \nu.$$

## Remarks

The right side is minus the $L^2$ gradient of the [[Willmore energy|R7WFJGJBV4]] $W(f_t) = \int_\Sigma H^2 \, dA$, so $W(f_t)$ does not increase.
Example: the constant family of a round sphere of radius $r$ is a solution.
Indeed $H = \pm 1/r$ is constant, so $\Delta H = 0$, and $H^2 - K = 1/r^2 - 1/r^2 = 0$, so the right side vanishes.
Non-example: the spheres of radius $1 + t$ do not form a solution, since the right side vanishes for each of them but the surfaces move.
A surface for which $\Delta H + 2H(H^2 - K) = 0$ is a Willmore surface; these are exactly the stationary solutions.
The equation is of fourth order and parabolic, so the maximum principle fails and convexity need not be preserved.
</document>

<document source="https://tamnd.github.io/math2/node/CMMDQ13YEE.md" tag="CMMDQ13YEE" kind="definition">
# Harmonic analysis

## Definition

Let $X$ be a space with a group of symmetries, such as the circle $\mathbb{T} = \R / 2\pi\Z$ or the space $\R^n$ with translations.
*Harmonic analysis* is the branch of analysis that studies functions on such $X$ through the following questions:

1. how a function decomposes into basic oscillations, such as the exponentials $e^{inx}$ on $\mathbb{T}$ or $e^{2\pi i x \cdot \xi}$ on $\R^n$;
2. in which sense, and in which function spaces, the function is recovered from this decomposition;
3. how operators that commute with translations, such as convolutions and differential operators with constant coefficients, act on the decomposition, and which norm estimates they satisfy.

## Remarks

Example: the theory of [[Fourier series|G12KXDHPPS]] answers these questions on $\mathbb{T}$, and the [[Fourier transform|DPVM34ADZV]] answers them on $\R^n$.
Non-example: the study of the integer solutions of a single polynomial equation by elementary congruences decomposes no function into oscillations, so it is not harmonic analysis.
The part over $\R^n$ and $\mathbb{T}^n$ is often called Euclidean or classical harmonic analysis.
The part over locally compact groups is called abstract harmonic analysis, or [[harmonic analysis on groups|5E6KB3V8R0]].
</document>

<document source="https://tamnd.github.io/math2/node/CMXHRNWY35.md" tag="CMXHRNWY35" kind="concept">
# Backward error analysis

## Summary

Backward error analysis studies a computed answer by asking for which nearby input it is the exact answer.
The size of that input change is the backward error.
The error of the answer, the forward error, is then bounded by about the [[condition number|AB3S6MHJ8X]] times the backward error.
This splits the analysis into a property of the algorithm (small backward error, that is, [[numerical stability|1AKB4XHMTH]]) and a property of the problem (its conditioning).
It was developed and made systematic by James H. Wilkinson in his work on numerical linear algebra.

## Explanations

### L1 human

Instead of asking "how wrong is my answer?", backward error analysis asks "for which slightly different question is my answer exactly right?".
If the question had to change only by an amount comparable to the rounding errors, the algorithm did as well as one can hope.
For example, a computed root $\hat r$ of a polynomial may be an exact root of a polynomial whose coefficients differ from the given ones only in the last digit.
Whether the answer is then close to the true root depends on how sensitive the problem is, not on the algorithm.
</document>

<document source="https://tamnd.github.io/math2/node/CMXS9P2F13.md" tag="CMXS9P2F13" kind="statement">
# Inversion of the Hilbert transform

## Statement

Let $1 < p < \infty$ and let $f \in L^p(\R)$, an element of the [[$L^p$ space|SASP14MA1Z]].
Let $H$ denote the [[Hilbert transform|NSCEBXM7ZM]].
Then $Hf \in L^p(\R)$ and
$$H(Hf) = -f$$
almost everywhere.
So $H$ is a bijection of $L^p(\R)$ onto itself, with inverse $H^{-1} = -H$.

## Remarks

For $f \in L^2(\R)$ the [[Fourier transform|DPVM34ADZV]] of $Hf$ is $-i \sgn(\xi)\, \hat f(\xi)$.
Applying this twice multiplies $\hat f$ by $(-i \sgn \xi)^2 = -1$ for $\xi \ne 0$, which gives $H^2 f = -f$ on $L^2$.
For general $p$, [[M. Riesz's theorem|PJFH51Z48M]] says that $H$ is bounded on $L^p$, so $H^2$ is bounded too, and the identity extends from the dense subspace $L^2 \cap L^p$ by continuity.
The hypothesis $p > 1$ is needed: for $f \in L^1(\R)$ the function $Hf$ need not lie in $L^1$.
For example, the indicator function of $[0, 1]$ has $Hf(x) = \frac{1}{\pi} \log \abs{\frac{x}{x - 1}}$, which behaves like $\frac{1}{\pi x}$ as $\abs{x} \to \infty$ and is not integrable.

Example: for $f(x) = \frac{1}{1 + x^2}$ one has $Hf(x) = \frac{x}{1 + x^2}$ and then $H\big[\frac{x}{1 + x^2}\big] = -\frac{1}{1 + x^2}$.

## Explanations

### L1 human

Doing the Hilbert transform twice gives back the original function with a minus sign.
So the transform loses no information, and its inverse is the same operation with a minus sign.
On waves, it is a quarter-period shift, and two quarter-period shifts turn a wave upside down.

### L2 human

The inversion formula says that $H$ is, up to sign, its own inverse; on $L^2$ it is a unitary operator with $H^* = -H$.
It explains the symmetry between the two halves of a Hilbert transform pair.
An example is the pair of real and imaginary parts of the boundary values of a function in the [[Hardy space|PNNNFJVF6W]] of the upper half-plane.
Together with the boundedness on $L^p$ for $1 < p < \infty$, it is the starting point of the theory of singular integral operators.
</document>

<document source="https://tamnd.github.io/math2/node/CMYVK9CEKZ.md" tag="CMYVK9CEKZ" kind="concept">
# Specialization preorder

## Summary

The specialization preorder of a topological space orders its points by closure: $x \le y$ when $x$ lies in the closure of $\set{y}$.
It is a partial order exactly when the space is [[T0|07SG12TQH3]], and it is the equality relation exactly when the space is T1.
In algebraic geometry it records which points of a scheme specialize to which.
In domain theory it recovers the order of a poset from its Scott topology, and in general it links topological spaces with ordered sets.

## Explanations

### L1 human

In a nice space like the plane, every point is closed and the specialization preorder says nothing.
In spaces that are not T1 some points sit "inside the closure" of others, and the preorder records this.
In the Sierpiński space with points $0$ and $1$ and open sets $\emptyset$, $\set{1}$ and $\set{0, 1}$, the closure of $\set{1}$ is $\set{0, 1}$.
So $0 \le 1$: the point $1$ is a "general" point and $0$ is a "special" one.
</document>

<document source="https://tamnd.github.io/math2/node/CMZS6KA033.md" tag="CMZS6KA033" kind="statement">
# Weil's converse to Haar's theorem

## Statement

Let $G$ be a group with a [[standard Borel|ABEFWF4JHH]] structure $\mathcal{B}$ such that the map $G \times G \to G$, $(x, y) \mapsto x y^{-1}$, is Borel.
Let $\mu$ be a nonzero [[σ-finite measure|WEBBT18D4X]] on $\mathcal{B}$ that is left [[quasi-invariant|FCNQP0A60D]]: for every $g \in G$ and $A \in \mathcal{B}$, $\mu(gA) = 0$ if and only if $\mu(A) = 0$.
Then there is exactly one topology on $G$ with the following two properties.

1. With it, $G$ is a Hausdorff [[locally compact group|AHV05ZBCYW]] that is [[second countable|03PPNCM9GG]].
2. Its Borel $\sigma$-algebra is $\mathcal{B}$.

For this topology, $\mu$ has the same null sets as a left [[Haar measure|VY0HXD3GC4]] of $G$.

## Remarks

[[Haar's theorem|KB792QTG4F]] goes from a topology to an invariant measure; this result goes back from an invariant measure to a topology.
Weil's original version starts from a group with a left-invariant measure that separates points in a suitable sense.
It defines a topology from the measure, called the Weil topology.
The completion of the group in this topology is locally compact, and the given measure is essentially a Haar measure on it.
The form stated here, for standard Borel groups and quasi-invariant measures, is due to Mackey.
The idea is that the sets $\set{g : \mu(gA \,\triangle\, A) < \varepsilon}$, for $A$ of finite positive measure and $\varepsilon > 0$, generate the neighbourhoods of the identity: their finite intersections form a base.
Example: a Polish group that is not locally compact, such as the additive group of an infinite-dimensional separable Hilbert space, carries no nonzero σ-finite left quasi-invariant Borel measure.
This explains why there is no analogue of Lebesgue measure on such a space.

## Explanations

### L1 human

Haar's theorem says that every locally compact group has a way to measure size that does not change under translation.
This theorem says the converse.
If a reasonable group has a translation-invariant way to measure size, at least up to sets of size zero, then the group is locally compact.
So invariant measures and local compactness go together.

### L2 human

The theorem is used to show that a group has no Haar-like measure: one checks that it cannot be locally compact.
A standard case is an infinite-dimensional Banach space, which is a Polish group under addition but is not locally compact.
In representation theory, the result lets one recover the topology of a locally compact group from its Borel structure and its measure class.
It connects to [[quasi-invariant measures|FCNQP0A60D]]: on a second countable locally compact group, every σ-finite left quasi-invariant measure has the same null sets as Haar measure.
</document>

<document source="https://tamnd.github.io/math2/node/CN7NP6R1AQ.md" tag="CN7NP6R1AQ" kind="statement">
# Boundary integral equation for the Neumann problem

## Statement

Let $m \in \set{2, 3}$, and let $D \subset \R^m$ be a bounded domain whose boundary $\partial D$ is a connected $C^2$ curve or surface with outward unit normal $\nu$.
Let $\Phi$ be the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] with $-\Delta \Phi = \delta_0$.
For a continuous density $\psi$ on $\partial D$ put
$$(K'\psi)(x) = \int_{\partial D} \frac{\partial \Phi(x - y)}{\partial \nu(x)} \, \psi(y) \, ds(y), \quad x \in \partial D.$$
Let $g \colon \partial D \to \R$ be continuous.
Then:

1. The [[integral equation of the second kind|8KJCN49V64]] $\psi + 2K'\psi = 2g$ has a continuous solution $\psi$ if and only if $\int_{\partial D} g \, ds = 0$.
2. For such a solution $\psi$, the [[single layer potential|9T6ZEV4125]] $u(x) = \int_{\partial D} \Phi(x - y) \psi(y) \, ds(y)$ is [[harmonic|0DEB0JMN0W]] in $D$, and its normal derivative from inside equals $g$ on $\partial D$.

So the interior Neumann problem $\Delta u = 0$ in $D$, $\partial_\nu u = g$ on $\partial D$, is solvable exactly when $\int_{\partial D} g \, ds = 0$, and its solutions differ by constants.

## Remarks

Part 2 follows from the [[jump relations for layer potentials|J58YA8QNNX]]: the normal derivative of the single layer potential from inside is $K'\psi + \frac12 \psi$.
For part 1, $K'$ is compact on $C(\partial D)$ and is the adjoint of the double layer operator $K$ with respect to the pairing $\int_{\partial D} \varphi \psi \, ds$.
The null space of $I + 2K$ consists of the constants, since $K1 = -\frac12$ by Gauss's formula for the double layer potential of a constant density.
By the [[Fredholm alternative|4HH8C9V8JQ]], $\psi + 2K'\psi = 2g$ is solvable if and only if $2g$ is orthogonal to the constants.

Example: on the unit disc in $\R^2$, $\frac{\partial \Phi(x - y)}{\partial \nu(x)} = -\frac{1}{4\pi}$ for $x \ne y$ on the unit circle, so $K'\psi = -\frac{1}{4\pi} \int \psi \, ds$.
For $g(\theta) = \cos\theta$ the equation $\psi + 2K'\psi = 2g$ has the solution $\psi = 2\cos\theta$, since $\int_0^{2\pi} \cos\theta \, d\theta = 0$; the resulting single layer potential is $u(x) = x_1$ in the disc, whose normal derivative on the circle is $\cos\theta$.
For $g = 1$ the condition $\int g \, ds = 0$ fails, and indeed no harmonic $u$ in $D$ has $\partial_\nu u = 1$, because $\int_{\partial D} \partial_\nu u \, ds = \int_D \Delta u \, dx = 0$.

## Explanations

### L1 human

Here the flux of a harmonic function through the boundary is prescribed instead of its values.
Since nothing is created inside the region, the total flux must be zero, and that is the only obstruction.
One looks for the solution as the potential of a layer of charges on the boundary, whose density solves an integral equation on the boundary.

### L2 human

This is the companion of the [[boundary integral equation for the Dirichlet problem|ZYDVFT6JR7]], which uses the double layer potential.
The two boundary operators $K$ and $K'$ are adjoint to each other, and the solvability condition comes from the Fredholm alternative applied to this pair.
The same equations, with the signs changed, treat the exterior problems.
Numerically, both equations are discretized by boundary element or Nyström methods, which reduce the dimension of the problem by one.
</document>

<document source="https://tamnd.github.io/math2/node/CN91KKNSQK.md" tag="CN91KKNSQK" kind="definition">
# Iterative method

## Definition

Let $X$ be a [[normed vector space|03J5RAKB30]], let $D \subseteq X$, and let $x^* \in X$ be a solution of a given problem.
An *iterative method* is a sequence of maps $\Phi_k \colon D^{k+1} \to D$ for $k \ge 0$.
From an initial guess $x_0 \in D$, it produces the iterates $x_{k+1} = \Phi_k(x_0, \dots, x_k)$.
The method is *convergent* for $x_0$ if the iterates [[converge|090AA928Y7]] to $x^*$.
It is *stationary* if there is one map $\Phi \colon D \to D$ with $\Phi_k(x_0, \dots, x_k) = \Phi(x_k)$ for all $k$.

## Remarks

A stationary method is a [[fixed-point iteration|K6ZF91051B]] for $\Phi$.
For a [[linear system|0GT4WWGWH2]] $A x = b$ with $A$ invertible, a *stationary linear method* has the form $x_{k+1} = G x_k + c$.
It is *consistent* if $x^* = A^{-1} b$ satisfies $x^* = G x^* + c$.
Then the error $e_k = x_k - x^*$ satisfies $e_{k+1} = G e_k$.
So the method converges for every $x_0$ if and only if the [[spectral radius|9V7YQA85PC]] of $G$ is less than $1$.
Example: $x_{k+1} = x_k / 2 + 1$ on $\R$ is stationary and consistent for $x / 2 = 1$, with $x^* = 2$.
Its error halves at each step, so it converges for every $x_0$.
Non-example: $x_{k+1} = 2 x_k - 2$ has the fixed point $2$, but the error doubles at each step.
It is not convergent for any $x_0 \ne 2$.
Krylov subspace methods are nonstationary: each iterate depends on all earlier ones, or on several of them.
</document>

<document source="https://tamnd.github.io/math2/node/CNADH7BENS.md" tag="CNADH7BENS" kind="counterexample">
# Hartogs triangle

## Counterexample

The *Hartogs triangle* is the domain
$$H = \set{(z, w) \in \C^2 : \abs{z} < \abs{w} < 1}.$$
It is a bounded [[pseudoconvex|AHYZMSBBNH]] domain whose closure $\overline{H}$ has no basis of pseudoconvex neighbourhoods.

Check that $H$ is pseudoconvex.
On $H$ the function $g(z, w) = z/w$ is holomorphic with $\abs{g} < 1$.
Let
$$\varphi(z, w) = -\log\big(1 - \abs{z/w}^2\big) - \log\big(1 - \abs{w}^2\big) - \log \abs{w}.$$
The first two terms are [[plurisubharmonic|FF2P7ER3QT]], since $-\log(1 - \abs{h}^2) = \sum_{k \ge 1} \abs{h^k}^2 / k$ for a holomorphic $h$ with $\abs{h} < 1$, and the last term is pluriharmonic.
The boundary of $H$ consists of the points with $\abs{z} = \abs{w} \le 1$ and the points with $\abs{w} = 1$, $\abs{z} \le 1$.
All three terms are nonnegative on $H$.
Near the first part the first term tends to $+\infty$, except at the origin, where the last term does.
Near the second part the second term tends to $+\infty$.
So $\varphi$ is a continuous [[plurisubharmonic exhaustion function|FGSKR6ZS3Q]], and $H$ is pseudoconvex, hence a [[domain of holomorphy|4487CSC0MB]] by [[Oka's theorem|6BJB149MDK]].

Check the neighbourhoods.
Let $U$ be an open set containing $\overline{H} = \set{\abs{z} \le \abs{w} \le 1}$.
Since $\overline{H}$ is compact, $U$ contains, for some $\varepsilon > 0$, the set $\set{\abs{z} < \varepsilon, \abs{w} < 1 + \varepsilon}$ and the set $\set{\abs{z} < 1 + \varepsilon, 1 - \varepsilon < \abs{w} < 1 + \varepsilon}$.
The union of these two sets is a [[Hartogs figure|GHJC53RFBT]].
Let $f$ be holomorphic on $U$.
The Laurent series of $w \mapsto f(z, w)$ on the annulus has no negative powers when $\abs{z} < \varepsilon$.
Its coefficients are holomorphic in $z$, so by the identity theorem it has none when $\abs{z} < 1$.
So $f$ extends holomorphically to the bidisc $\set{\abs{z} < 1, \abs{w} < 1 + \varepsilon}$.
Hence any domain of holomorphy that contains $\overline{H}$ contains the whole unit bidisc, and the pseudoconvex neighbourhoods of $\overline{H}$ cannot shrink to $\overline{H}$.

## Remarks

The boundary of $H$ is not a Lipschitz graph near the origin, and this singular point is the source of the pathology.
For bounded pseudoconvex domains with smooth boundary, the closure can still fail to have a pseudoconvex neighbourhood basis.
The examples are much harder, such as the worm domains of Diederich and Fornæss.

## Explanations

### L1 human

The Hartogs triangle is the set of pairs $(z, w)$ of complex numbers with $\abs{z} < \abs{w} < 1$.
If one draws it with axes $\abs{z}$ and $\abs{w}$, it is a triangle.
It is a perfectly good domain for holomorphic functions, a domain of holomorphy.
But every domain of holomorphy that contains its closure must contain the whole unit bidisc, because holomorphic functions near the closure always extend that far.
So the closure of a domain of holomorphy need not be approximable from outside by domains of holomorphy.
</document>

<document source="https://tamnd.github.io/math2/node/CNCXNWKC6R.md" tag="CNCXNWKC6R" kind="concept">
# Sieve

## Summary

A sieve on an object $c$ of a [[category|A2DE66937D]] is a collection of morphisms with codomain $c$ that is closed under composition on the right.
Equivalently, it is a subfunctor of the [[representable functor|KCCHH87H6E]] $\Hom(-, c)$.
Sieves are the building blocks of [[Grothendieck topologies|GDRFSJ8E2B]]: a Grothendieck topology chooses which sieves count as coverings.
They play the role of open covers in a topological space.

## Explanations

### L1 human

Think of $c$ as an open set and of morphisms $d \to c$ as smaller open sets inside it.
A sieve on $c$ is a family of such smaller sets that is closed under shrinking.
If $V$ is in the family and $W \subseteq V$ is open, then $W$ is in the family too.
For example, for an open cover $\set{U_i}$ of $U$, all open sets contained in some $U_i$ form a sieve on $U$.
Working with sieves rather than covers removes the choice of a particular list of sets.
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# Legendre duplication formula

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\C$.
For every $z \in \C$ with $2z \notin \set{0, -1, -2, \ldots}$,
$$\Gamma(z) \, \Gamma\!\left(z + \tfrac12\right) = 2^{1 - 2z} \sqrt{\pi} \; \Gamma(2z),$$
where $2^{1-2z} = e^{(1-2z)\ln 2}$.

## Remarks

One proof applies the [[Bohr–Mollerup theorem|QFF87AXWN2]] to $f(x) = 2^{x-1}\Gamma(\tfrac{x}{2})\Gamma(\tfrac{x+1}{2})/\sqrt{\pi}$ for $x > 0$: it satisfies $f(1) = 1$, $f(x+1) = x f(x)$, and $\ln f$ is convex, so $f = \Gamma$; then put $x = 2z$.
Another proof uses the [[beta function|QHNWVSC6AQ]]: the substitution $t \mapsto 4t(1-t)$ gives $B(z, z) = 2^{1-2z} B(z, \tfrac12)$.
The identity theorem extends the formula from $(0, \infty)$ to all allowed complex $z$.
Check at $z = 1$: the left side is $\Gamma(1)\Gamma(\tfrac32) = \tfrac{\sqrt{\pi}}{2}$, and the right side is $2^{-1}\sqrt{\pi}\,\Gamma(2) = \tfrac{\sqrt{\pi}}{2}$.
It is the case $m = 2$ of the [[Gauss multiplication formula|Y4759V8CY6]].
In Mathlib the statement is `Complex.Gamma_mul_Gamma_add_half`.

## Explanations

### L1 human

The duplication formula expresses $\Gamma$ at a doubled argument through two values of $\Gamma$ at half that size, half a unit apart.
For whole numbers it is a statement about factorials: $(2n)! = \frac{4^n}{\sqrt{\pi}}\, n! \, \Gamma(n + \tfrac12)$.

### L2 human

The formula is how one computes $\Gamma$ at half-integers, giving $\Gamma(n + \tfrac12) = \frac{(2n)!}{4^n n!}\sqrt{\pi}$.
It appears in the functional equation of the Riemann zeta function, where it converts between the symmetric form with $\Gamma(s/2)$ and the form with $\Gamma(s)$ and a cosine.
It also gives the volume of the unit ball in odd dimensions in closed form.
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# Characteristic class

## Definition

Let $G$ be a [[topological group|92700QAB5Y]], let $R$ be an [[abelian group|082GHNPJ53]] and let $k \ge 0$ be an integer.
Write $H^k(X; R)$ for [[singular cohomology|6N7BC70CX0]] with coefficients in $R$.
A *characteristic class* of degree $k$ for principal $G$-bundles is a rule $c$ that assigns to each [[principal $G$-bundle|8E37E74XVP]] $P \to X$ a class $c(P) \in H^k(X; R)$, such that:

1. If $P \to X$ and $P' \to X$ are isomorphic, then $c(P) = c(P')$.
2. For every continuous map $f \colon Y \to X$, $c(f^* P) = f^* c(P)$, where $f^* P$ is the pullback bundle and $f^*$ on the right is the induced map in cohomology.

## Remarks

The same definition applies to real or complex vector bundles of a fixed rank, which correspond to principal $GL_n(\R)$- or $GL_n(\C)$-bundles.
Over CW complexes, characteristic classes of degree $k$ correspond bijectively to elements of $H^k(BG; R)$: the class $u$ gives $c(P) = f_P^* u$, where $f_P \colon X \to BG$ classifies $P$.

Example: for real line bundles, the first Stiefel–Whitney class $w_1 \in H^1(X; \Z/2\Z)$ is a characteristic class.
For the Möbius line bundle $M$ over $S^1$, $w_1(M)$ is the generator of $H^1(S^1; \Z/2\Z)$, while $w_1$ of the trivial bundle is $0$.

Non-example: for line bundles over spaces $X$, the rule $c(P) = 1 \in H^0(X; \Z)$ if $X$ is compact and $c(P) = 0$ otherwise fails clause 2.
For the inclusion $f \colon [0, 1] \to \R$ and the trivial bundle $P$ over $\R$, $c(f^* P) = 1$ but $f^* c(P) = 0$.

Clause 2 implies that a trivial bundle $X \times G$ has $c(X \times G) = 0$ when $k > 0$, since it is pulled back from a point.
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# Category of posets

## Definition

The *category of posets* $\mathbf{Pos}$ is the [[category|A2DE66937D]] given by the following data.

1. Its objects are the [[partially ordered sets|011C29Q1S1]].
2. A morphism from $(P, \le_P)$ to $(Q, \le_Q)$ is a [[monotone function|0WFF6AZPBP]] $f \colon P \to Q$, that is, $x \le_P y$ implies $f(x) \le_Q f(y)$.
3. Composition is the usual [[composition of functions|07HF9YCWPP]], and the identity morphism of $P$ is the identity map of $P$.

## Remarks

The data form a category.
The identity map is monotone, and if $f$ and $g$ are monotone, then $x \le y$ gives $f(x) \le f(y)$ and then $g(f(x)) \le g(f(y))$.
Associativity and the identity laws hold because they hold for functions.
Posets with order-reversing maps do not form a category in the same way: the composite of two order-reversing maps preserves order.
A morphism of $\mathbf{Pos}$ is an isomorphism exactly when it is an [[order isomorphism|0Y74KN99A5]].
A monotone bijection need not be one: the identity from a two-element antichain to a two-element chain is a monotone bijection whose inverse is not monotone.

## Explanations

### L1 human

The category of posets is the world of ordered sets together with the maps that respect order.
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# Turing degree

## Definition

Let $\le_T$ be [[Turing reducibility|PA4PYQ1PZ0]] on subsets of $\N$, and let $A \equiv_T B$ mean $A \le_T B$ and $B \le_T A$.
The relation $\equiv_T$ is an equivalence relation on the power set of $\N$.
The *Turing degree* of a set $A \subseteq \N$ is its equivalence class
$$\deg(A) = \set{B \subseteq \N : B \equiv_T A}.$$
The degrees are partially ordered by $\deg(A) \le \deg(B)$ if and only if $A \le_T B$; this does not depend on the choice of representatives.

## Remarks

Example: the set $E$ of even numbers and the set $\set{0}$ have the same degree.
Both are [[computable sets|PPGD5KW8JC]], and every computable set is computable from every set, so $E \le_T \set{0}$ and $\set{0} \le_T E$.
This common degree is written $\mathbf{0}$; it is the least degree.

Non-example: the [[halting problem|Z7TN2B35B0]] $K$ does not belong to $\mathbf{0}$, because $K \not\le_T \emptyset$.
Its degree is written $\mathbf{0}'$.

Every degree is a countable set, since there are only countably many oracle machines.
Hence there are uncountably many degrees.
Any two degrees $\deg(A)$ and $\deg(B)$ have a least upper bound, the degree of $A \oplus B = \set{2n : n \in A} \cup \set{2n + 1 : n \in B}$.
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# External ray

## Summary

An external ray is a curve that comes in from infinity towards the filled Julia set of a polynomial, or towards the [[Mandelbrot set|QSWSVYRCS1]].
It is the image of a radial line under the inverse of the Böttcher coordinate, and it is labelled by an angle.
The polynomial maps each ray to another ray, multiplying its angle by the degree.
Rays and the points where they land turn the topology of [[Julia sets|5M4S7P04GR]] into combinatorics of angles.

## Explanations

### L1 human

Outside the filled Julia set of a polynomial with connected Julia set, there is a coordinate in which everything looks like the outside of the unit disc.
In that picture, draw the straight rays that leave the disc at each angle.
Pulled back to the original plane, these become curves that spiral and bend towards the Julia set.
For $z \mapsto z^2$, the rays are just the straight half-lines from the unit circle to infinity.
Squaring doubles the angle of each ray, and this simple rule organises the whole dynamics.
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# Transfer operator

## Summary

The transfer operator of a map $f$ with weight $g$ acts on functions $\varphi$.
The value of the new function at $x$ is the sum of $g \varphi$ over the preimages of $x$.
It moves densities forward under the map, while the [[Koopman operator|37TRG5DBQC]] moves observables, and the two are dual to each other.
Invariant densities are fixed points of the transfer operator, and its spectrum controls the speed of mixing and decay of correlations.
It is also called the Ruelle operator or, for the weight given by the Jacobian, the Perron–Frobenius operator.

## Explanations

### L1 human

Spread some dust on an interval with a density $\varphi$, then move every grain by the map $f$.
The new density at a point $x$ collects the dust from all points that land on $x$, weighted by how much the map stretches near them.
This rule is linear in $\varphi$, and the transfer operator is that linear rule.
For the doubling map $x \mapsto 2x \bmod 1$, each $x$ has the two preimages $x/2$ and $(x+1)/2$, and the map stretches by $2$.
So the new density is the average of $\varphi$ at these two points.
The constant density is unchanged, which says that the doubling map preserves length.
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# Finite element method

## Definition

Let $V$ be a real [[Hilbert space|HM2HHE3J51]] of functions on a bounded polygonal domain $\Omega \subset \R^d$, for example the [[Sobolev space|M4CEZ5K9HP]] $H^1_0(\Omega)$.
Let $a \colon V \times V \to \R$ be a bounded [[bilinear form|0SQZMXB6GP]] and $\ell \colon V \to \R$ a bounded linear functional.
The [[weak form|SJZX81A783]] of the problem is: find $u \in V$ with $a(u, v) = \ell(v)$ for all $v \in V$.
The *finite element method* consists of:

1. A mesh $\mathcal{T}_h$: a partition of $\overline{\Omega}$ into finitely many closed simplices of diameter at most $h$, meeting face to face.
2. A finite-dimensional space $V_h \subseteq V$ of functions that are polynomials of degree at most $k$ on each element of $\mathcal{T}_h$, with a locally supported basis.
3. The discrete problem: find $u_h \in V_h$ with $a(u_h, v_h) = \ell(v_h)$ for all $v_h \in V_h$.

## Remarks

In a basis $\varphi_1, \ldots, \varphi_N$ of $V_h$, the discrete problem is the linear system $A c = b$ with $A_{ij} = a(\varphi_j, \varphi_i)$ and $b_i = \ell(\varphi_i)$, and $u_h = \sum_j c_j \varphi_j$.
If $a$ is coercive, $a(v, v) \ge \alpha \norm{v}^2$, and $\abs{a(u, v)} \le M \norm{u} \norm{v}$, then $u_h$ exists, is unique, and Céa's lemma gives $\norm{u - u_h} \le (M / \alpha) \min_{v_h \in V_h} \norm{u - v_h}$.
Example: $-u'' = f$ on $(0, 1)$ with $u(0) = u(1) = 0$, $V = H^1_0(0, 1)$, $a(u, v) = \int_0^1 u' v'$, nodes $x_i = ih$ with $h = 1/(N+1)$, and hat functions $\varphi_i$ with $\varphi_i(x_j) = \delta_{ij}$.
Then $\int \varphi_i'^2 = 2/h$ and $\int \varphi_i' \varphi_{i \pm 1}' = -1/h$, so $A = h^{-1} \operatorname{tridiag}(-1, 2, -1)$.
Non-example: the Galerkin method with $V_N$ spanned by $\sin(\pi x), \ldots, \sin(N \pi x)$ satisfies clause 3 but not clause 2.
These functions are not piecewise polynomials on a mesh, so this is a spectral method.
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# Hamilton–Jacobi–Bellman equation

## Definition

Let $U \subseteq \R^m$ be a nonempty set, let $T > 0$, and let $f \colon \R^n \times U \to \R^n$, $L \colon \R^n \times U \to \R$ and $g \colon \R^n \to \R$ be continuous.
Consider the [[control system|ZPDTVZQN5K]] $x' = f(x, u)$ with running cost $L$ and terminal cost $g$.
The *Hamilton–Jacobi–Bellman equation* of this problem is the partial differential equation
$$\frac{\partial V}{\partial t}(t, x) + \min_{u \in U} \left\{ \nabla_x V(t, x) \cdot f(x, u) + L(x, u) \right\} = 0, \qquad (t, x) \in (0, T) \times \R^n,$$
with terminal condition $V(T, x) = g(x)$, for an unknown function $V \colon [0, T] \times \R^n \to \R$.
When the minimum is not attained, it is replaced by the infimum.

## Remarks

Its intended solution is the [[value function|K6G0DRQ3CP]] $V(t, x) = \inf_u \left( \int_t^T L(x(s), u(s)) \, ds + g(x(T)) \right)$, where the infimum is over admissible controls on $[t, T]$ and $x$ is the trajectory with $x(t) = x$.
If the value function is continuously differentiable, it satisfies the equation.
Under standard hypotheses, such as $U$ compact and $f$, $L$, $g$ bounded and Lipschitz, it is the unique bounded uniformly continuous [[viscosity solution|H1ERJ3T75T]].
Example: for $x' = u$, $U = \R$, $L = u^2 / 2$ and $g(x) = x^2 / 2$, the minimum over $u$ is $-\frac{1}{2} (\partial_x V)^2$, and $V(t, x) = \dfrac{x^2}{2 (1 + T - t)}$ solves the equation.
Indeed $\partial_t V = \dfrac{x^2}{2 (1 + T - t)^2}$ and $\frac{1}{2} (\partial_x V)^2 = \dfrac{x^2}{2 (1 + T - t)^2}$, and $V(T, x) = x^2 / 2$.
Non-example: $W(t, x) = x^2 / 2$ satisfies the terminal condition but not the equation, since $\partial_t W - \frac{1}{2} (\partial_x W)^2 = -x^2 / 2 \ne 0$ for $x \ne 0$.
For infinite-horizon problems with discount rate $\rho > 0$ the equation becomes stationary: $\rho V(x) = \min_{u \in U} \{ \nabla V(x) \cdot f(x, u) + L(x, u) \}$.
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# Numerical continuation

## Definition

Let $H \colon \R^{n+1} \to \R^n$ be a [[continuously differentiable|0HHHDCXZV4]] map, and let $u_0 \in \R^{n+1}$ satisfy $H(u_0) = 0$.
A point $u$ with $H(u) = 0$ is *regular* if the [[Jacobian matrix|047NDTEPCV]] $H'(u)$, of size $n \times (n+1)$, has rank $n$.
Near a regular point, the zero set $H^{-1}(0)$ is a smooth curve, by the [[implicit function theorem|0BXBADR94C]].
A *numerical continuation method* for $H$ from $u_0$ is an algorithm that produces points $u_1, u_2, \dots$ such that:

1. each $u_k$ approximately satisfies $H(u_k) = 0$, within a prescribed tolerance;
2. each $u_{k+1}$ is obtained from $u_k$ (and possibly earlier points) by a step of controlled length along the curve;
3. the points follow the connected component of $H^{-1}(0)$ through $u_0$ in one direction.

## Remarks

The usual scheme is predictor–corrector.
The predictor moves from $u_k$ to $v = u_k + h t_k$ for a step length $h > 0$.
Here the unit tangent $t_k$ is a unit vector in the kernel of $H'(u_k)$, oriented in the direction of travel.
The corrector applies [[Newton's method|07EXVNF9VP]] to the $n + 1$ equations $H(u) = 0$ and $t_k \cdot (u - v) = 0$; this is *pseudo-arclength continuation*.
Example: for $H(x, \lambda) = x^2 + \lambda^2 - 1$ on $\R^2$, every zero is regular, and pseudo-arclength continuation from $(1, 0)$ traces the whole unit circle.
Non-example: *natural parameter continuation* fixes $\lambda$ at successive values and solves for $x$; on the same circle it fails at $(0, \pm 1)$, where $\partial H / \partial x = 0$, because the curve turns back in $\lambda$.
Such points are called turning points or folds.

## Explanations

### L1 human

Continuation follows a curve of solutions in small steps.
Each step guesses ahead along the tangent and then corrects the guess back onto the curve.
Measuring progress by arc length, rather than by the parameter, lets the method go around bends where the parameter turns back.
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# Lévy process

## Definition

Let $d \ge 1$.
A *Lévy process* in $\R^d$ is a [[stochastic process|0QH5ENKZM1]] $(X_t)_{t \ge 0}$ with values in $\R^d$ such that:

1. $X_0 = 0$ almost surely;
2. for all $0 \le t_0 < t_1 < \cdots < t_n$, the increments $X_{t_1} - X_{t_0}, \ldots, X_{t_n} - X_{t_{n-1}}$ are [[independent|0GQ938GDE0]];
3. for all $0 \le s \le t$, the increment $X_t - X_s$ has the same [[distribution|0DVDCMA2DK]] as $X_{t-s}$;
4. for every $t \ge 0$ and $\varepsilon > 0$, $\lim_{h \to 0} P(\abs{X_{t+h} - X_t} > \varepsilon) = 0$.

## Remarks

Example: the Poisson process $N$ of rate $\lambda$ satisfies the four clauses; $N_t - N_s$ is Poisson with mean $\lambda(t - s)$, and $P(N_{t+h} \ne N_t) = 1 - e^{-\lambda \abs{h}} \to 0$.
Non-example: the deterministic process $X_t = t^2$ satisfies clauses 1, 2 and 4, but $X_{s+1} - X_s = 2s + 1$ depends on $s$, so clause 3 fails.
Every Lévy process has a modification whose paths are right-continuous with left limits, and one usually works with it.
There is a function $\psi$, the characteristic exponent, with $\mathbb{E}[e^{i \langle u, X_t \rangle}] = e^{t \psi(u)}$ for all $u \in \R^d$ and $t \ge 0$.
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# Score

## Summary

The score is the gradient of the log-likelihood with respect to the parameter.
It measures how sensitive the log-likelihood of the observed data is to small changes of the parameter.
Under regularity conditions its expectation at the true parameter is zero, and its covariance is the Fisher information.
Setting the score equal to zero gives the likelihood equations, whose solutions are candidates for the maximum likelihood estimate.

## Explanations

### L1 human

Imagine the log-likelihood as a hill over the parameter axis.
The score is the slope of the hill at a given parameter value.
At the top of the hill, the maximum likelihood estimate, the slope is zero.
For one observation $x$ from a normal distribution with mean $\mu$ and variance $1$, the score is $x - \mu$: it is positive when the observation suggests a larger mean.
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# Torsor

## Definition

Let $G$ be a group.
A *$G$-torsor* is a nonempty set $X$ with a [[group action|0WCY42EV9V]] of $G$ such that for all $x, y \in X$ there is exactly one $g \in G$ with $g \cdot x = y$.

Let $k$ be a field and let $G$ be an [[algebraic group|GH81SYAPPN]] over $k$.
A *$G$-torsor* over $k$ is a nonempty variety $X$ over $k$ with an [[action|RW85C34GQW]] $\sigma$ of $G$ such that the morphism
$$G \times_k X \to X \times_k X, \quad (g, x) \mapsto (\sigma(g, x), x)$$
is an isomorphism.

## Remarks

In the set case, the condition says that the action is free and transitive.
In the variety case, $X$ becomes isomorphic to $G$, with $G$ acting by left translation, after a finite extension of $k$.
The torsor is *trivial* if $X \cong G$ over $k$, and this holds if and only if $X$ has a $k$-point.

Example: let $V$ be a vector space and let $A$ be an affine space with translation space $V$.
For any two points $a, b \in A$ there is exactly one vector $v$ with $a + v = b$, so $A$ is a $V$-torsor.

Example over a field: let $G = \mu_2 = \set{\pm 1}$ over $\R$ act on $X = \set{x^2 = -1}$ by $x \mapsto \pm x$.
Over $\C$, $X$ has the two points $\pm i$, and $G$ swaps them freely.
So $X$ is a $\mu_2$-torsor, and it is not trivial, since $X(\R)$ is empty.

Non-example: $\mathbb{G}_m$ acting on $\mathbb{A}^1$ by multiplication is not a torsor.
The point $0$ is fixed by every $t$, so the action is neither free nor transitive.
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# Abstract Wiener space

## Summary

An abstract Wiener space is a [[Hilbert space|HM2HHE3J51]] sitting densely inside a larger [[Banach space|06GZ956657]].
The standard Gaussian [[cylinder set measure|G95YDNFXQN]] of the Hilbert space must become a true [[Gaussian measure|P0DV2TCH1N]] on the Banach space.
The notion was introduced by Leonard Gross.
It is the general frame for Wiener measure, Malliavin calculus and Gaussian free fields.

## Explanations

### L1 human

There is no standard Gaussian measure on an infinite-dimensional Hilbert space: a typical point would have infinite length.
The fix is to enlarge the space, with a weaker norm, until the typical points fit.
The classical example is Brownian motion.
The Hilbert space is the space of paths with square-integrable derivative, and the larger space is the space of all continuous paths, where Brownian paths actually live.
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# Invertible operator

## Summary

A bounded operator between normed spaces is invertible when it has a bounded inverse.
Invertibility is the basic notion behind the spectrum: a number $\lambda$ lies in the spectrum of $T$ exactly when $T - \lambda I$ is not invertible.
In infinite dimensions, an injective operator need not be surjective, and a bijective operator needs a theorem to have a bounded inverse.

## Explanations

### L1 human

An operator is invertible when it can be undone by another bounded operator.
Think of a change of coordinates that can be reversed without blowing up small errors.
For example, the map $x \mapsto 2x$ on any normed space is invertible, with inverse $x \mapsto \tfrac12 x$.
The right shift on sequences is not invertible: it is injective but misses every sequence with a nonzero first entry.
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# Bedford–Taylor theory

## Definition

Let $\Omega \subseteq \C^n$ be open, and let $d^c = i(\bar\partial - \partial)$.
Let $u$ be a locally bounded [[plurisubharmonic function|FF2P7ER3QT]] on $\Omega$, and let $T$ be a closed [[positive current|BYJHP6A906]] of bidegree $(p, p)$ on $\Omega$, with $p \le n - 1$.
The coefficients of $T$ are measures and $u$ is locally bounded and Borel, so $uT$ is a well defined [[current|360H7MD9RD]].
The *Bedford–Taylor product* is the current
$$dd^c u \wedge T = dd^c (u T),$$
where $dd^c$ is taken in the sense of currents.
For locally bounded plurisubharmonic functions $u_1, \dots, u_k$ on $\Omega$, with $k \le n$, the product is defined by induction:
$$dd^c u_1 \wedge dd^c u_2 \wedge \cdots \wedge dd^c u_k = dd^c \left( u_1 \, dd^c u_2 \wedge \cdots \wedge dd^c u_k \right).$$
In particular $(dd^c u)^n$ is the *Monge–Ampère measure* of $u$.

## Remarks

The main facts of the theory are these.
The product $dd^c u \wedge T$ is again a closed positive current, which makes the induction possible.
When the $u_j$ are of class $C^2$ the product agrees with the wedge product of smooth forms, so $(dd^c u)^n$ agrees with the [[complex Monge–Ampère operator|MXRC891TQS]].
If $u_j^{(m)}$ decreases to $u_j$ as $m \to \infty$, for each $j$, then the products of the $u_j^{(m)}$ converge weakly to the product of the $u_j$.
The product is symmetric in $u_1, \dots, u_k$.
Example: for $n = 1$ and $T = 1$, $dd^c u = \Delta u$ is the Riesz measure of the subharmonic function $u$, which is defined for every subharmonic $u$.
Non-example: the definition does not apply to $u = \log\abs{z_1}$ on $\C^2$, which is not locally bounded.
For unbounded plurisubharmonic functions in general, $(dd^c u)^n$ cannot be defined as a measure with good continuity properties.
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# Translation equation

## Summary

The translation equation is the [[functional equation|SRAN4EKWS0]] $F(F(x, s), t) = F(x, s + t)$ for an unknown map $F \colon X \times \R \to X$.
Its solutions with $F(x, 0) = x$ are the flows on the set $X$, that is, actions of the additive group $\R$ on $X$.
So it is the basic equation of continuous-time [[dynamical systems|BBZP1R872N]] and of iteration with non-integer exponents.
With the real line replaced by a group, it becomes the equation of a [[group action|0WCY42EV9V]].

## Explanations

### L1 human

Think of $F(x, t)$ as the position at time $t$ of a particle that starts at $x$.
Moving for time $s$ and then for time $t$ should give the same position as moving for time $s + t$.
For example, $F(x, t) = x + t$ moves every point at unit speed along the line.
The map $F(x, t) = x e^t$ is another solution, in which points move away from $0$ exponentially fast.
</document>

<document source="https://tamnd.github.io/math2/node/CPEC2NW2KF.md" tag="CPEC2NW2KF" kind="concept">
# Phase space

## Summary

The phase space of a system of differential equations is the set of all its possible states.
Each solution traces a curve in it, its [[trajectory|0BVT6VSB3S]], and the collection of these curves is the [[phase portrait|0T3GK0XCBC]].
Thinking in the phase space turns questions about formulas for solutions into questions about the geometry of curves.
See the [[definition|RY2RAH3802]].

## Explanations

### L1 human

A state is everything one needs to know now to predict the future.
For a moving particle this is its position together with its velocity, not the position alone.
So the phase space of a particle on a line is a plane, with position on one axis and velocity on the other.
For the [[harmonic oscillator|0YG0ZJPTWG]] $x'' = -x$, the motions are circles around the origin in this plane.
The [[phase plane|05VDGH129F]] and the [[phase line|081QSEAGEY]] are the phase spaces of dimension two and one.
</document>

<document source="https://tamnd.github.io/math2/node/CPF9Q1GR10.md" tag="CPF9Q1GR10" kind="example">
# Electric field integral equation

## Example

Let $S$ be the smooth closed surface of a perfectly conducting body in $\R^3$, with outward unit normal $n$.
Let a time-harmonic incident field $E^{\mathrm{inc}}$ with time factor $e^{-i\omega t}$ illuminate the body in a medium with permittivity $\varepsilon$ and permeability $\mu$, and let $k = \omega \sqrt{\varepsilon \mu}$.
Let $G(x, y) = \frac{e^{ik\abs{x - y}}}{4\pi \abs{x - y}}$.
The unknown is the tangential surface current density $J$ on $S$.
The electric field integral equation is
$$n(x) \times E^{\mathrm{inc}}(x) = -i\omega\mu \, n(x) \times \left[\int_S G(x, y) J(y) \, dS(y) + \frac{1}{k^2} \nabla_x \int_S G(x, y) \, (\nabla_S \cdot J)(y) \, dS(y)\right], \quad x \in S.$$
The unknown $J$ appears only under the integrals, so it is an [[integral equation of the first kind|ZDS77Q0F06]].

Check: the current $J$ radiates the scattered field $E^{\mathrm{s}} = i\omega A - \nabla \Phi$.
Here $A = \mu \int_S G J \, dS$ is the vector potential and $\Phi = \frac{1}{\varepsilon} \int_S G \sigma \, dS$ the scalar potential.
The charge density is $\sigma = \frac{1}{i\omega} \nabla_S \cdot J$ by the continuity equation.
Since $\frac{1}{i\omega\varepsilon} = \frac{-i\omega\mu}{k^2}$, this gives $E^{\mathrm{s}} = i\omega\mu \left[\int_S G J \, dS + \frac{1}{k^2} \nabla \int_S G \, \nabla_S \cdot J \, dS\right]$.
On a perfect conductor the tangential total field vanishes, $n \times (E^{\mathrm{inc}} + E^{\mathrm{s}}) = 0$ on $S$, which is the equation above.

## Remarks

The equation is uniquely solvable except at the countably many wavenumbers $k$ at which the interior of $S$ has a nonzero cavity resonance field.
Near these wavenumbers numerical solutions become unstable; combining it with the magnetic field integral equation removes the problem.
Discretized with the method of moments, it is a standard tool for computing radar scattering and antenna fields.

## Explanations

### L1 human

When a radio wave hits a metal object, it drives electric currents on the surface, and these currents send out the reflected wave.
On metal the total electric field along the surface must be zero.
This condition gives an equation for the unknown surface currents, in which each current element contributes through an integral over the whole surface.
</document>

<document source="https://tamnd.github.io/math2/node/CPFSBBHXT5.md" tag="CPFSBBHXT5" kind="concept">
# Statistical decision theory

## Summary

Statistical decision theory treats statistical problems as decisions under uncertainty.
A statistician observes data from a model with an unknown parameter and chooses an action by a decision rule.
The statistician then pays a [[loss|3BTG6VVCGZ]] that depends on the action and the true parameter.
Rules are compared through their [[risk functions|N26TSVRH2S]], which gives a common frame for estimation, testing and prediction.

## Explanations

### L1 human

Estimating a number, testing a hypothesis or predicting a value are all ways of making a decision from data.
Decision theory asks the user to say how costly each kind of mistake is, and then looks for the rule with the smallest average cost.
Usually no rule is best for every possible truth at once, so one needs a criterion.
Common criteria are to protect against the worst case (minimax), to average over a prior (Bayes), or at least to discard rules that are beaten everywhere (admissibility).
</document>

<document source="https://tamnd.github.io/math2/node/CPGEJWQS8A.md" tag="CPGEJWQS8A" kind="definition">
# Mixture model

## Definition

Let $K \ge 1$ be an integer and let $\set{f_\theta : \theta \in \Theta}$ be a parametric family of [[probability densities|0ZY9DX8FBF]] on $\R^p$.
A *finite mixture model* with $K$ components is the [[statistical model|K3AMMSY8DD]] of the densities
$$f(x) = \sum_{k=1}^K \pi_k f_{\theta_k}(x), \qquad x \in \R^p,$$
where $\theta_1, \ldots, \theta_K \in \Theta$ and the *mixing weights* satisfy $\pi_k \ge 0$ and $\pi_1 + \cdots + \pi_K = 1$.
The densities $f_{\theta_k}$ are the *components*.

## Remarks

Equivalently, let $Z$ be a random label with $P(Z = k) = \pi_k$, and let $X$ have density $f_{\theta_k}$ given $Z = k$.
Then $X$ has density $f$, so a mixture model is a [[latent variable model|C63RVH96TF]] with hidden label $Z$.
Example: $f(x) = \tfrac{1}{2} \varphi(x) + \tfrac{1}{2} \varphi(x - 3)$, where $\varphi$ is the standard normal density, is a two-component Gaussian mixture with $\pi_1 = \pi_2 = 1/2$.
Non-example: $\tfrac{1}{2} \varphi(x) + \tfrac{1}{4} \varphi(x - 3)$ has weights adding up to $3/4$, so it is not a mixture and not even a probability density.
The parameters are identifiable at best up to a permutation of the components, since relabelling the components does not change $f$.
The same definition with discrete distributions replacing densities gives mixtures of discrete [[distributions|0DVDCMA2DK]].
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<document source="https://tamnd.github.io/math2/node/CPN6SD3BTV.md" tag="CPN6SD3BTV" kind="concept">
# Round-off error

## Summary

Round-off error, or rounding error, is the difference between the exact result of a computation and the result computed with [[rounded|5NPVX29Y0C]] numbers.
In [[floating-point arithmetic|RTKRSC3KPH]] each single rounding has relative error at most the unit roundoff, about $10^{-16}$ in double precision.
Many roundings can add up, and a subtraction of nearly equal numbers can magnify them, as in [[catastrophic cancellation|TJPNYW7J8E]].
It is distinct from [[truncation error|4AKCQERM89]], which comes from the method even in exact arithmetic.

## Explanations

### L1 human

A computer keeps only a fixed number of digits, so it must round almost every result.
Each rounding error is tiny, but a long computation does millions of operations.
Sometimes the errors cancel each other, and sometimes they pile up.
A classic example: adding $0.1$ ten times in double precision does not give exactly $1$, because $0.1$ is rounded when stored and again after each addition.
</document>

<document source="https://tamnd.github.io/math2/node/CPR1B0WEWA.md" tag="CPR1B0WEWA" kind="definition">
# Plurisubharmonic function

## Definition

Let $\Omega \subseteq \C^n$ be an open set.
A function $u \colon \Omega \to [-\infty, \infty)$ is *plurisubharmonic* if:

1. $u$ is [[upper semicontinuous|RN4G8SGP6P]];
2. for every $a \in \Omega$ and $b \in \C^n$ such that the set $\set{a + \lambda b : \lambda \in \C, \abs{\lambda} \le 1}$ lies in $\Omega$,
$$u(a) \le \frac{1}{2\pi} \int_0^{2\pi} u(a + e^{i\theta} b) \, d\theta .$$

## Remarks

Clause 2 says that for all $a, b$ the function $\lambda \mapsto u(a + \lambda b)$ is [[subharmonic|Z1MHMGFNFW]] or identically $-\infty$ on each component of its domain.
Many authors also require that $u$ is not identically $-\infty$ on any connected component of $\Omega$.
A function $u$ of class $C^2$ is plurisubharmonic if and only if
$$\sum_{j,k=1}^n \frac{\partial^2 u}{\partial z_j \partial \bar z_k}(z) \, w_j \bar w_k \ge 0 \quad \text{for all } z \in \Omega,\ w \in \C^n .$$
Example: $u(z) = \abs{z_1}^2 + \cdots + \abs{z_n}^2$ has complex Hessian equal to the identity matrix, which is positive definite, so $u$ is plurisubharmonic.
Non-example: $u(z_1, z_2) = \abs{z_1}^2 - \abs{z_2}^2$ is smooth, but on the line $\lambda \mapsto (0, \lambda)$ it is $-\abs{\lambda}^2$, which is not subharmonic, so clause 2 fails at $a = 0$, $b = (0, 1)$.
Plurisubharmonic functions are subharmonic as functions on $\R^{2n}$, but the converse fails, as the non-example shows.
The abbreviation "psh" is common.
</document>

<document source="https://tamnd.github.io/math2/node/CPZKZ2AV0G.md" tag="CPZKZ2AV0G" kind="statement">
# Gleason's theorem on projective compact spaces

## Statement

A [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]] $P$ is *projective* if it satisfies the following lifting property.
For all compact Hausdorff spaces $Y$ and $Z$, every continuous surjection $f \colon Y \to Z$ and every [[continuous map|0VGFHTM75Y]] $g \colon P \to Z$, there is a continuous map $h \colon P \to Y$ with $f \circ h = g$.

Let $P$ be a compact Hausdorff space.
Then $P$ is projective if and only if $P$ is [[extremally disconnected|370PJ5BF70]].

## Remarks

For one direction, let $D$ be the set $P$ with the discrete topology.
The identity $D \to P$ extends to a surjection $\beta D \to P$ from the [[Stone–Čech compactification|D9MWP5C85W]].
If $P$ is projective, it is a retract of $\beta D$, which is extremally disconnected, and retracts inherit the property.
The other direction uses a closed subset of the pullback that is minimal among those mapping onto $P$, found by Zorn's lemma.

Example at work: $\beta \N$ is extremally disconnected, so it is projective.

## Explanations

### L1 human

Projective compact Hausdorff spaces are those from which maps can always be lifted along surjections.
Gleason's theorem says that, among compact Hausdorff spaces, these are exactly the spaces where closures of open sets are open.
So a categorical property is matched with a concrete topological one.

### L2 human

Every compact Hausdorff space $X$ is a continuous image of a projective one, namely $\beta D$ for $D$ the set $X$ with the discrete topology.
In fact $X$ has a projective cover, often called its Gleason cover.
Through Stone duality, extremally disconnected compact Hausdorff spaces correspond to complete [[Boolean algebras|0XADJKM699]].
</document>

<document source="https://tamnd.github.io/math2/node/CQ3616BH5C.md" tag="CQ3616BH5C" kind="definition">
# Morita equivalence

## Definition

Let $R$ and $S$ be [[rings|0B1261B5ED]] with identity.
Let $R\text{-Mod}$ be the category of left $R$-[[modules|0NZ8JA25WK]] and $R$-linear maps, and likewise $S\text{-Mod}$.
The rings $R$ and $S$ are *Morita equivalent* if there is an [[equivalence of categories|K7A04ZBD92]] $F \colon R\text{-Mod} \to S\text{-Mod}$.

## Remarks

Example: for a field $k$, the rings $k$ and $M_2(k)$ are Morita equivalent.
The functor sends a vector space $V$ to $V^2 = k^2 \otimes_k V$, the column vectors with entries in $V$, on which $M_2(k)$ acts by matrix multiplication.
A quasi-inverse sends a module $N$ to $e_{11} N$, where $e_{11}$ is the matrix unit.
Non-example: $k$ and $k \times k$ are not Morita equivalent.
In $k\text{-Mod}$ every nonzero module has a nonzero map to every other nonzero module, while the simple $k \times k$-modules $k \times 0$ and $0 \times k$ have no nonzero map between them.
An equivalence of categories preserves this property.
An equivalence between module categories is automatically additive and is naturally isomorphic to $P \otimes_R -$ for an $S$-$R$-bimodule $P$.
Morita's theorem: $R$ and $S$ are Morita equivalent if and only if $S \cong \End_R(P)^{\mathrm{op}}$ for some [[progenerator|ZQ9ABBKF1N]] $P$ of $R\text{-Mod}$, where $\End_R(P)$ acts on $P$ from the left.
The definition with right modules gives the same relation.
Isomorphic rings are Morita equivalent, and for commutative rings the converse holds.
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<document source="https://tamnd.github.io/math2/node/CQ4C1WKMF6.md" tag="CQ4C1WKMF6" kind="definition">
# Combinatorial design

## Definition

Let $t, v, k, \lambda$ be positive integers with $t \le k \le v$.
Let $X$ be a [[finite set|070R982G4B]], whose elements are called points.
Let $\mathcal{B}$ be a finite family of subsets of $X$, called blocks, with repetitions allowed.
The pair $(X, \mathcal{B})$ is a *$t$-$(v, k, \lambda)$ design* if:

1. $\abs{X} = v$.
2. Every block has exactly $k$ points.
3. Every subset of $X$ with $t$ points is contained in exactly $\lambda$ blocks.

A *combinatorial design* in the usual sense is a $t$-design of this kind.
More broadly, the term covers any finite set with a family of subsets that satisfies a prescribed balance condition of this sort.

## Remarks

A $2$-$(v, k, \lambda)$ design is a balanced incomplete [[block design|4JCSS2Z1BA]], and a $t$-$(v, k, 1)$ design is a [[Steiner system|7BPSC4T4M3]] $S(t, k, v)$.
Counting gives the number of blocks: $\abs{\mathcal{B}} = \lambda \binom{v}{t} / \binom{k}{t}$.

Example: $X = \set{1, \dots, 7}$ with blocks $\set{1,2,4}$, $\set{2,3,5}$, $\set{3,4,6}$, $\set{4,5,7}$, $\set{5,6,1}$, $\set{6,7,2}$, $\set{7,1,3}$ is a $2$-$(7, 3, 1)$ design, the Fano plane.
The blocks are the translates of $\set{1, 2, 4}$ modulo $7$, whose pairwise differences $\pm 1, \pm 2, \pm 3$ cover each nonzero residue exactly once, so every pair of points lies in exactly one block.

Non-example: $X = \set{1, 2, 3, 4}$ with blocks $\set{1, 2}$ and $\set{3, 4}$ is not a $2$-$(4, 2, 1)$ design, because the pair $\set{1, 3}$ lies in no block.
</document>

<document source="https://tamnd.github.io/math2/node/CQ7E0V6T2G.md" tag="CQ7E0V6T2G" kind="statement">
# Converse of the mean value property

## Statement

Let $U \subseteq \R^n$ be an open set and let $u \colon U \to \R$ be [[continuous|0X3WGX95FE]].
Assume that for every $x \in U$ and every $r > 0$ with closed ball $\overline{B}(x, r) \subseteq U$,
$$u(x) = \frac{1}{\abs{\partial B(x, r)}} \int_{\partial B(x, r)} u \, dS,$$
where $\abs{\partial B(x, r)}$ is the area of the sphere $\partial B(x, r)$.
Then $u$ is [[harmonic|0DEB0JMN0W]] on $U$; in particular $u$ is infinitely differentiable.

## Remarks

Proof idea: let $\eta_\varepsilon$ be a smooth radial mollifier supported in $B(0, \varepsilon)$ with integral $1$.
Polar coordinates and the mean value property on each sphere give $u * \eta_\varepsilon = u$ at the points at distance more than $\varepsilon$ from the complement of $U$.
So $u$ is smooth there.
For a smooth function, the average over $\partial B(x, r)$ minus $u(x)$ equals $\frac{r^2}{2n} \Delta u(x) + o(r^2)$; since it is $0$, the [[Laplacian|0E0B6RA8K0]] vanishes.
Together with the [[mean value property|0HYPR64JM2]] of harmonic functions, this shows that, among continuous functions, harmonicity is the same as the mean value property on spheres.
The same holds with averages over balls in place of spheres.
For the version with ball averages, local integrability of $u$ is enough, because a function that equals its ball averages is automatically continuous.
Example: in $\R^2$, $u(x, y) = x y$ is continuous and its average over each circle about $(a, b)$ is $a b$, so $u$ is harmonic, as one checks directly from $\Delta(xy) = 0$.

## Explanations

### L1 human

Harmonic functions have the property that the value at a point is the average of the values on any sphere around it.
This theorem says the converse: any continuous function with this averaging property is harmonic.
So "harmonic" can be defined either by the Laplace equation or by the averaging property.

### L2 human

The averaging definition needs no derivatives, which makes it the natural starting point for harmonic functions on general spaces and for subharmonic functions.
It is also the key step in proving that a locally uniform limit of harmonic functions is harmonic: the mean value property passes to uniform limits, and continuity is preserved.
A stronger result, Koebe's theorem, only asks for the mean value property on all small enough spheres about each point.
</document>

<document source="https://tamnd.github.io/math2/node/CQACXG10JZ.md" tag="CQACXG10JZ" kind="concept">
# 2-category

## Summary

A (strict) 2-category has objects, morphisms between objects, and 2-morphisms between parallel morphisms.
The 2-morphisms compose in two directions, vertically and horizontally, and the two compositions satisfy the interchange law.
The main example is the 2-category of small [[categories|A2DE66937D]], [[functors|28NQ0KRHD3]] and [[natural transformations|KYCPCC57FT]].
Formally, a 2-category is a category [[enriched|BPH9SJ30Y0]] over the category of small categories.

## Explanations

### L1 human

In a 2-category there are not only arrows between objects, but also arrows between arrows.
Picture objects as points, morphisms as paths between points, and 2-morphisms as surfaces filling the space between two paths with the same ends.
Surfaces can be glued along a common path (vertical composition) or side by side at a common point (horizontal composition).
Categories, functors and natural transformations behave exactly like this: a natural transformation goes from one functor to another.
</document>

<document source="https://tamnd.github.io/math2/node/CQE378ZVY5.md" tag="CQE378ZVY5" kind="example">
# Newton fractal

## Example

Let $p(z) = z^3 - 1$, with roots $1$, $\omega = e^{2\pi i/3}$ and $\omega^2$.
[[Newton's method|07EXVNF9VP]] for $p$ iterates the [[rational function|0XDK9CGYKJ]]
$$N(z) = z - \frac{p(z)}{p'(z)} = \frac{2z^3 + 1}{3z^2}.$$
The *Newton fractal* of $p$ is the [[Julia set|5M4S7P04GR]] $J(N)$.
It is usually drawn with each point of the [[Fatou set|C0N0XGC6GK]] coloured by the root its orbit tends to.

Check of the main features:

1. Each root $r$ is a superattracting fixed point: $N(r) = r$ since $p(r) = 0$, and $N'(z) = p(z) p''(z) / p'(z)^2$ vanishes at $r$.
2. So each root has an open basin of attraction, contained in the Fatou set.
3. The map $N$ commutes with $z \mapsto \omega z$, so the picture has threefold rotational symmetry.
4. The Julia set is the common boundary of the three basins: every point of $J(N)$ lies in the closure of each basin.

## Remarks

Newton's method can therefore fail in a complicated way: arbitrarily close to the boundary of one basin there are starting points that converge to each of the other roots.
For a quadratic polynomial with two distinct roots, the Newton map is conjugate to $z \mapsto z^2$.
Then the Julia set is the perpendicular bisector of the two roots, a result going back to Cayley.

## Explanations

### L1 human

Newton's method guesses a root of a polynomial and improves the guess again and again.
Colour each starting point in the plane by the root that the guesses settle on.
For $z^3 - 1$ there are three colours, and the borders between them form an endlessly detailed fractal.
Near any border point all three colours appear, so a tiny change of the starting guess can change which root is found.
</document>

<document source="https://tamnd.github.io/math2/node/CQJ1FPKKHT.md" tag="CQJ1FPKKHT" kind="statement">
# Direct summand conjecture

## Statement

Let $R$ be a [[regular local ring|63XXTF6B5M]] and let $R \subseteq S$ be an extension of commutative rings such that $S$ is a [[finitely generated|05QY1VEBYD]] $R$-module.
Then the inclusion $R \to S$ splits as a map of $R$-modules: there is an $R$-linear map $\rho \colon S \to R$ with $\rho(1) = 1$.

## Remarks

Suppose that $R$ contains the rational numbers and $S$ is a domain, and let $F$ and $L$ be the fraction fields of $R$ and $S$.
Then $\rho = \frac{1}{[L : F]} \operatorname{Tr}_{L/F}$ is a splitting.
It maps $S$ into $R$ because $R$ is regular, hence integrally closed.
Hochster proved the statement when $R$ contains a field, and André proved it in mixed characteristic, using perfectoid methods.
It is one of [[Hochster's homological conjectures|1TNS604ZST]], and it kept the name "conjecture" after its proof.
Regularity cannot be dropped: for $R = K[\![t^2, t^3]\!] \subseteq S = K[\![t]\!]$, a splitting $\rho$ would give $t^3 = \rho(t^3) = t^2 \rho(t)$, so $\rho(t) = t \notin R$, a contradiction.

## Explanations

### L1 human

If a nice ring sits inside a finite extension ring, one can project the big ring back onto the small one in a linear way.
For rings containing the rationals this is easy, by averaging with a trace.
For rings like the $p$-adic integers it was a famous open problem for decades.

### L2 human

The statement is equivalent to the monomial conjecture, and it implies several other homological conjectures.
A ring that is a direct summand of a ring is often forced to inherit good properties; for example, a direct summand of a normal domain is normal.
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<document source="https://tamnd.github.io/math2/node/CQJ2E2ARRF.md" tag="CQJ2E2ARRF" kind="concept">
# Characteristic variety

## Summary

The characteristic variety of a linear differential operator is the set of points and nonzero covectors where its [[principal symbol|F31T650DKY]] vanishes.
A hypersurface whose normal never lies in this set is called noncharacteristic, and the Cauchy problem with data on such a hypersurface is well behaved.
In algebraic analysis the same idea defines the characteristic variety of a D-module.
See the [[definition|JMJEVS7S1V]].

## Explanations

### L1 human

The characteristic variety lists the directions in which an operator fails to control the highest derivatives.
Along such directions, data on a surface do not determine how the solution leaves the surface.
For the [[wave equation|5EHD2XBQDG]] $\partial_t^2 u - \partial_x^2 u$, the symbol $\tau^2 - \xi^2$ vanishes when $\tau = \pm \xi$.
The corresponding curves $x \pm t = \text{const}$ are the light rays along which waves travel.
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<document source="https://tamnd.github.io/math2/node/CQKTEBK13V.md" tag="CQKTEBK13V" kind="definition">
# Closed immersion

## Definition

Let $Z$ and $X$ be [[schemes|5A35TYZD02]].
A [[morphism of schemes|JXRKH6P1Z4]] $(f, f^\#) \colon Z \to X$ is a *closed immersion* if:

1. $f$ is a [[homeomorphism|0QT00PB5G2]] of $Z$ onto a [[closed subset|0ZCK8DQYN1]] of $X$;
2. the [[sheaf|TB6TM6R4MD]] map $f^\# \colon \mathcal{O}_X \to f_* \mathcal{O}_Z$ is surjective, that is, surjective on every stalk.

## Remarks

Example: for a ring $A$ and an ideal $I$, the morphism $\operatorname{Spec} A/I \to \operatorname{Spec} A$ induced by $A \to A/I$ is a closed immersion.
Its image is the closed set $V(I)$, and on the stalk at $\mathfrak{p} \supseteq I$ the map is $A_{\mathfrak p} \to (A/I)_{\mathfrak p}$, which is onto.
Every closed immersion into $\operatorname{Spec} A$ is isomorphic to one of these.
Non-example: let $k$ be a field and let $f \colon \operatorname{Spec} k[t] \to \operatorname{Spec} k[t^2, t^3]$ come from the inclusion $k[t^2, t^3] \subseteq k[t]$.
Then $f$ is a homeomorphism onto the whole space, so clause 1 holds.
Clause 2 fails at the origin: $t$ does not lie in the image of the stalk map.
Non-example: the open immersion $\operatorname{Spec} k[x, x^{-1}] \to \operatorname{Spec} k[x]$ fails clause 1, since its image $D(x)$ is not closed.
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<document source="https://tamnd.github.io/math2/node/CQM84EW10G.md" tag="CQM84EW10G" kind="concept">
# Ultrafilter on a Boolean algebra

## Summary

An ultrafilter on a [[Boolean algebra|0XADJKM699]] $B$ is a proper filter of $B$ that is maximal among proper filters.
Equivalently, it is a filter that contains exactly one of $a$ and $\neg a$ for every $a \in B$.
Ultrafilters are the same as homomorphisms from $B$ onto the two-element Boolean algebra, and they are the points of the [[Stone space|ZV38HA5PAQ]] of $B$.
For the power set of a set $X$ one gets the [[ultrafilters|J64Y3JC0RE]] on $X$.

## Explanations

### L1 human

An ultrafilter picks out the elements of a Boolean algebra that count as "true" in one consistent way of deciding everything.
It decides every element: either $a$ is true or $\neg a$ is, never both.
If $a$ and $b$ are true, so is "$a$ and $b$", and anything above a true element is true.
In the algebra of subsets of a finite set, every ultrafilter consists of the subsets that contain one fixed point.
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<document source="https://tamnd.github.io/math2/node/CQMK4YPE1J.md" tag="CQMK4YPE1J" kind="concept">
# Mollifier

## Summary

A mollifier is a smooth, nonnegative [[bump function|HS65DW0HXH]] with integral $1$, supported in the unit ball.
Rescaling it gives a family $\varphi_\varepsilon$ that concentrates at the origin as $\varepsilon \to 0$.
[[Convolution|2Z29JN3VES]] with $\varphi_\varepsilon$ turns a rough function into a [[smooth function|0V33BNKZ9S]] that is close to it.
Mollifiers are the standard tool to approximate functions, prove density results and regularize distributions.

## Explanations

### L1 human

Think of a mollifier as a small smooth blob of total mass $1$.
Convolving a function with the blob replaces each value by a weighted average of the nearby values.
This blurs away corners and jumps, and the result is infinitely differentiable.
If the blob is made narrower, the averages are taken over smaller regions, and the blurred function gets closer to the original one.
A standard blob on the line is $c \, e^{-1/(1 - x^2)}$ for $\abs{x} < 1$ and $0$ elsewhere.
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<document source="https://tamnd.github.io/math2/node/CQSZTPMSEP.md" tag="CQSZTPMSEP" kind="definition">
# Center of a ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
The *center* of $R$ is the set
$$Z(R) = \set{z \in R : zr = rz \text{ for all } r \in R}.$$

## Remarks

$Z(R)$ is a commutative [[subring|0859D58WPD]] of $R$, and $R$ is commutative if and only if $Z(R) = R$.
Example: for a field $K$ and $n \ge 1$, the center of $M_n(K)$ is the set of scalar matrices $cI_n$ with $c \in K$.
The check: a matrix $Z$ that commutes with every matrix unit $E_{ij}$ satisfies $Z E_{ij} = E_{ij} Z$; comparing entries shows that $Z$ is diagonal with equal diagonal entries.
Non-example: for $n \ge 2$, the matrix $E_{11}$ is not in the center of $M_n(K)$, since $E_{11} E_{12} = E_{12}$ but $E_{12} E_{11} = 0$.
Some authors write $C(R)$ for the center.
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<document source="https://tamnd.github.io/math2/node/CR14D9YQTY.md" tag="CR14D9YQTY" kind="concept">
# Ricci flow

## Summary

The Ricci flow is the evolution equation $\partial_t g = -2 \operatorname{Ric}(g)$ for a family of [[Riemannian metrics|Q61VFHCH2J]] on a manifold.
It was introduced by Richard Hamilton in 1982.
It is a nonlinear, weakly parabolic system that behaves like a heat equation for the metric, smoothing out irregularities of the curvature.
Grigori Perelman used the Ricci flow with surgery to prove the Poincaré conjecture and Thurston's geometrization conjecture.

## Explanations

### L1 human

Think of a lumpy shape that you let relax, with each region changing its size according to how curved it is.
Under the Ricci flow, regions of positive [[Ricci curvature|4GTNM0WHDH]] shrink and regions of negative Ricci curvature expand.
A round sphere stays round and shrinks to a point in finite time.
The hope, made true in many cases, is that a manifold flows toward a simple standard geometry, which then reveals its shape.
</document>

<document source="https://tamnd.github.io/math2/node/CRE6VEKT32.md" tag="CRE6VEKT32" kind="definition">
# Projective module

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $P$ be an [[$R$-module|0NZ8JA25WK]].
The module $P$ is *projective* if it satisfies the following lifting property.
Let $M$, $N$ be $R$-modules, $g \colon M \to N$ a surjective $R$-[[linear map|0NRYHJPQG2]], and $f \colon P \to N$ an $R$-linear map.
Then there is an $R$-linear map $h \colon P \to M$ with $g \circ h = f$.

## Remarks

The following are equivalent for an $R$-module $P$:

1. $P$ is projective.
2. Every surjective linear map $M \to P$ has a right inverse, that is, it splits.
3. $P$ is a direct summand of a [[free module|0DZ9DKAKZR]]: there is a module $Q$ with $P \oplus Q$ free.
4. The functor $\Hom_R(P, -)$ takes [[exact sequences|JB7ZBFBA9C]] to exact sequences.

The same definition works over any ring, commutative or not.

Example: every free module $F$ with basis $(e_i)$ is projective.
Given $g$ and $f$, choose $m_i \in M$ with $g(m_i) = f(e_i)$ and let $h(e_i) = m_i$; then $g \circ h$ and $f$ agree on a basis, so they are equal.
Over $R = \Z/6\Z$, the module $\Z/2\Z$ is projective but not free, since $R \cong \Z/2\Z \oplus \Z/3\Z$ as $R$-modules.

Non-example: over $R = \Z$, the module $\Z/2\Z$ is not projective.
Take $g \colon \Z \to \Z/2\Z$ the reduction and $f$ the identity of $\Z/2\Z$.
A lift $h \colon \Z/2\Z \to \Z$ must be zero, because $2h(1) = h(0) = 0$ and $\Z$ has no nonzero element of order $2$; then $g \circ h = 0 \ne f$.
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<document source="https://tamnd.github.io/math2/node/CRF6HCM0Q5.md" tag="CRF6HCM0Q5" kind="definition">
# Kernel approximation in reproducing kernel Hilbert spaces

## Definition

Let $X$ be a set and let $K \colon X \times X \to \R$ be a symmetric positive definite kernel.
Let $H_K$ be its [[reproducing kernel Hilbert space|4CK2GSRAGD]], so that $f(x) = \ip{f}{K(\cdot, x)}_{H_K}$ for all $f \in H_K$ and $x \in X$.
Let $x_1, \dots, x_n \in X$ be distinct points and let $y_1, \dots, y_n \in \R$.
The *kernel matrix* is the $n \times n$ [[matrix|0VD9ZSWWJG]] $\mathbf{K} = (K(x_i, x_j))_{i,j}$.
If $\mathbf{K}$ is invertible, the *kernel interpolant* of the data is
$$s = \sum_{j=1}^{n} c_j K(\cdot, x_j), \qquad \mathbf{K} c = y ,$$
which satisfies $s(x_i) = y_i$ for $i = 1, \dots, n$.
For $\lambda > 0$, the *regularized kernel approximant* is $s_\lambda = \sum_{j} c_j K(\cdot, x_j)$ with $(\mathbf{K} + \lambda I) c = y$.
*Kernel approximation* in $H_K$ is the approximation of a function $f$ by $s$ or $s_\lambda$ built from the data $y_i = f(x_i)$.

## Remarks

The interpolant $s$ is the unique element of minimal norm in $H_K$ among those $g$ with $g(x_i) = y_i$ for all $i$.
If $f \in H_K$ and $y_i = f(x_i)$, then $s$ is the orthogonal projection of $f$ onto the span of $K(\cdot, x_1), \dots, K(\cdot, x_n)$.
The regularized approximant $s_\lambda$ minimizes $\sum_{i} (g(x_i) - y_i)^2 + \lambda \norm{g}_{H_K}^2$ over $g \in H_K$.
For $f \in H_K$, $\abs{f(x) - s(x)} \le P(x) \norm{f}_{H_K}$, where the power function is $P(x)^2 = K(x, x) - k(x)^{T} \mathbf{K}^{-1} k(x)$ with $k(x) = (K(x, x_j))_j$.
Example: $K(x, y) = e^{-(x - y)^2}$ on $X = \R$, points $x_1 = 0$, $x_2 = 1$, data $y = (1, 1)$.
Here $\mathbf{K} = \begin{pmatrix} 1 & e^{-1} \\ e^{-1} & 1 \end{pmatrix}$ is invertible, and $c_1 = c_2 = 1/(1 + e^{-1})$ solves $\mathbf{K} c = y$.
Non-example: the kernel $K(x, y) = xy$ on $\R$ is only positive semidefinite; for the points $1$ and $2$, $\mathbf{K} = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$ is singular, and the data $y = (1, 0)$ cannot be interpolated by any $c\,x$.
</document>

<document source="https://tamnd.github.io/math2/node/CRFGFCNG3Q.md" tag="CRFGFCNG3Q" kind="definition">
# Group scheme

## Definition

Let $S$ be a [[scheme|5A35TYZD02]], and let $\pi \colon G \to S$ be a scheme over $S$.
Products below are [[fiber products|1DZMT4S9FB]] over $S$, and all maps are [[morphisms|JXRKH6P1Z4]] over $S$.
A *group scheme* over $S$ is such a $G$ with morphisms $m \colon G \times_S G \to G$, $e \colon S \to G$ and $i \colon G \to G$ such that:

1. $m \circ (m \times \id_G) = m \circ (\id_G \times m)$ as morphisms $G \times_S G \times_S G \to G$.
2. $m \circ (e \circ \pi, \id_G) = \id_G = m \circ (\id_G, e \circ \pi)$.
3. $m \circ (i, \id_G) = e \circ \pi = m \circ (\id_G, i)$.

## Remarks

Equivalently, for every $S$-scheme $T$, the set $G(T)$ of $S$-morphisms $T \to G$ carries a [[group|G5S1M9]] structure, and the maps $G(T') \to G(T)$ induced by $S$-morphisms $T \to T'$ are group homomorphisms.
This is the [[functor of points|SWWW9X4A3N]] form of the definition.
For example, the multiplicative group $\mathbb{G}_{m,S} = \operatorname{Spec} \Z[t, t^{-1}] \times_{\operatorname{Spec} \Z} S$ is a group scheme over $S$.
For an affine $S$-scheme $T = \operatorname{Spec} R$, its points are $\mathbb{G}_m(T) = R^\times$, the units of $R$ under multiplication.
The affine line $\mathbb{A}^1_S$ with multiplication $(a, b) \mapsto ab$ is not a group scheme.
Clause 3 fails, since the point $0 \in R$ has no multiplicative inverse in a nonzero ring $R$.
</document>

<document source="https://tamnd.github.io/math2/node/CRFQEGJDA9.md" tag="CRFQEGJDA9" kind="concept">
# Fuzzy measure

## Summary

A fuzzy measure, also called a capacity or a monotone measure, is a set function that vanishes on the empty set and is monotone.
It drops the additivity of a [[measure|PR11BZQQPG]] and keeps only the rule that bigger sets get bigger values.
Fuzzy measures model the worth of coalitions in cooperative games and the importance of groups of criteria in decision making.
One integrates against them with the [[Choquet integral|36TXCRZ6GC]] or the Sugeno integral.

## Explanations

### L1 human

Suppose several criteria judge a product, and a group of criteria gets a weight.
Two criteria that measure almost the same thing should not count twice, so the weight of the pair may be less than the sum of the two weights.
Two criteria that support each other may get more than the sum.
A fuzzy measure allows both effects and asks only that adding a criterion never lowers the weight.
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<document source="https://tamnd.github.io/math2/node/CRGZZKQ7Z9.md" tag="CRGZZKQ7Z9" kind="concept">
# Minimal set

## Summary

A minimal set of a continuous map is a nonempty closed [[invariant set|WWZTW11664]] that contains no smaller nonempty closed invariant set.
Equivalently, the forward orbit of each of its points is dense in it.
Minimal sets are the smallest building blocks of [[topological dynamics|5X6E97EMFR]].
On a nonempty compact space every continuous map has at least one minimal set, by Zorn's lemma.

## Explanations

### L1 human

A minimal set is a piece of the space that the motion cannot split further.
Start anywhere in it, and the orbit comes as close as you like to every point of it.
A periodic orbit is a minimal set: it is finite, closed and invariant, and any invariant subset contains a whole orbit.
For an irrational rotation of the circle, the whole circle is a minimal set, because every orbit is dense.
</document>

<document source="https://tamnd.github.io/math2/node/CRJSW18XVM.md" tag="CRJSW18XVM" kind="definition">
# Smoothness classes

## Definition

Let $I \subseteq \R$ be an open interval and let $f \colon I \to \R$.

1. The function $f$ is of *class* $C^0$ on $I$ if it is [[continuous|0P27W9RW0G]] on $I$.
2. Let $k \ge 1$ be an integer. The function $f$ is of *class* $C^k$ on $I$ if the [[higher derivatives|084NGSTWM7]] $f', \ldots, f^{(k)}$ exist on $I$. Moreover $f^{(k)}$ must be continuous on $I$.
3. The function $f$ is of *class* $C^\infty$ on $I$ if it is of class $C^k$ for every $k \ge 0$.

The set of functions of class $C^k$ on $I$ is written $C^k(I)$.

## Remarks

The classes are nested: $C^\infty(I) \subseteq \cdots \subseteq C^2(I) \subseteq C^1(I) \subseteq C^0(I)$.
The class $C^1$ is the class of [[continuously differentiable functions|06SKCHDQT5]], and $C^\infty$ is the class of [[smooth functions|0BE3QDH73A]].
The class $C^\omega$ of real [[analytic functions|0JD5ZNW0F4]] lies inside $C^\infty$.
For functions of several variables, $C^k$ is defined by continuous partial derivatives up to order $k$; see [[the characterisation by derivatives|0TCBZMEYGN]].

Example: $f(x) = x \abs{x}$ on $\R$ is of class $C^1$.
Its derivative is $f'(x) = 2 \abs{x}$, which is continuous.
It is not of class $C^2$, because $2\abs{x}$ is not differentiable at $0$.
More generally, $x^k \abs{x}$ is of class $C^k$ but not of class $C^{k+1}$.

Non-example: $g(x) = x^2 \sin(1/x)$ for $x \ne 0$, with $g(0) = 0$, is differentiable on $\R$ but not of class $C^1$.
Its derivative is $g'(x) = 2x \sin(1/x) - \cos(1/x)$ for $x \ne 0$ and $g'(0) = 0$, and $g'$ has no limit at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/CRPZ1RVNKQ.md" tag="CRPZ1RVNKQ" kind="concept">
# Modulus of smoothness

## Summary

The modulus of smoothness of order $k$ of a function measures the size of its $k$-th finite differences with small step.
For $k = 1$ in the uniform norm it is the [[modulus of continuity|AXCJBSQZWF]].
Higher orders detect smoothness beyond Lipschitz, and they give the sharp form of the direct and inverse theorems of approximation by polynomials.

## Explanations

### L1 human

The first difference $f(x + h) - f(x)$ is small when $f$ changes little over a step $h$.
The second difference $f(x + 2h) - 2 f(x + h) + f(x)$ is small when $f$ is nearly a straight line over two steps; it vanishes for every linear function.
The modulus of smoothness records the largest such difference over all steps up to $t$.
For $f(x) = x$ the first-order modulus is $t$ and the second-order modulus is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/CRZH89G0J1.md" tag="CRZH89G0J1" kind="definition">
# Hahn polynomials

## Definition

Let $N \ge 0$ be an integer, let $\alpha, \beta > -1$ be real numbers, and let $n$ be an integer with $0 \le n \le N$.
Write $(a)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]].
The *Hahn polynomial* of degree $n$ is
$$Q_n(x; \alpha, \beta, N) = \sum_{k=0}^{n} \frac{(-n)_k \, (n + \alpha + \beta + 1)_k \, (-x)_k}{(\alpha + 1)_k \, (-N)_k \, k!},$$
that is, the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_3F_2\left( -n, n + \alpha + \beta + 1, -x; \alpha + 1, -N; 1 \right)$.

## Remarks

They are orthogonal on $\set{0, 1, \dots, N}$:
$$\sum_{x=0}^{N} \binom{\alpha + x}{x} \binom{\beta + N - x}{N - x} Q_m(x) Q_n(x) = 0 \quad (m \ne n),$$
where the [[binomial coefficients|0GBZM0MT0F]] are $\binom{a + j}{j} = \frac{(a + 1)_j}{j!}$.
Example: $Q_1(x) = 1 - \frac{(\alpha + \beta + 2) x}{(\alpha + 1) N}$.
Check: the weight is proportional to a beta-binomial distribution with mean $\frac{N(\alpha + 1)}{\alpha + \beta + 2}$, so $Q_1$ has weighted mean $0$.
Non-example: $n = N + 1$ is not allowed, since $(-N)_k = 0$ for $k = N + 1$ and the defining sum has a zero denominator.
The value at $0$ is $Q_n(0) = 1$.
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<document source="https://tamnd.github.io/math2/node/CS10HH24VT.md" tag="CS10HH24VT" kind="definition">
# Gross–Pitaevskii equation

## Definition

Let $d \ge 1$, let $\hbar, m > 0$ and $g \in \R$ be constants, and let $V \colon \R^d \to \R$ be a given function, the external potential.
The *Gross–Pitaevskii equation* is the equation
$$i \hbar \, \partial_t \psi = - \frac{\hbar^2}{2m} \Delta \psi + V(x) \psi + g \abs{\psi}^2 \psi$$
for an unknown function $\psi \colon \R \times \R^d \to \C$, $(t, x) \mapsto \psi(t, x)$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] in $x$.
The constant $g$ measures the strength of the interaction, repulsive for $g > 0$ and attractive for $g < 0$.

## Remarks

For $V = 0$ it is, after scaling, the cubic [[nonlinear Schrödinger equation|GEJHRXC176]].
A standing wave $\psi(t, x) = e^{-i\mu t/\hbar} \phi(x)$ solves it exactly when $\phi$ solves the stationary equation $\mu \phi = -\tfrac{\hbar^2}{2m} \Delta \phi + V \phi + g \abs{\phi}^2 \phi$, and $\mu$ is the chemical potential.

Example: for $V = 0$ and $n > 0$, the constant state $\phi = \sqrt{n}$ solves the stationary equation with $\mu = g n$, since $\Delta \phi = 0$ and $g \abs{\phi}^2 \phi = g n \phi$.

Non-example: if $V$ is not constant, then no nonzero constant $\phi$ solves the stationary equation, because $\mu \phi = (V(x) + g \abs{\phi}^2) \phi$ would force $V$ to be constant.

The ground state minimises the Gross–Pitaevskii energy $\int \big( \tfrac{\hbar^2}{2m} \abs{\nabla \phi}^2 + V \abs{\phi}^2 + \tfrac{g}{2} \abs{\phi}^4 \big) dx$ under the constraint $\int \abs{\phi}^2 \, dx = N$, the number of particles.
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# Modularity theorem

## Statement

Let $E$ be an [[elliptic curve|427N6S6QQW]] over $\Q$ with conductor $N$, a positive integer whose prime factors are the primes where $E$ has bad reduction.
For each prime $p \nmid N$, let $\tilde{E}$ be the reduction modulo $p$ of a minimal Weierstrass model of $E$.
It is a curve over the [[finite field|0ZXZ82FTX2]] $\F_p$; let $a_p = p + 1 - \# \tilde{E}(\F_p)$.
Then there is a [[cusp form|A50Y012RBP]] $f(\tau) = \sum_{n \ge 1} b_n q^n$, with $q = e^{2\pi i \tau}$, of weight $2$ for the [[congruence subgroup|1H6R856P3C]] $\Gamma_0(N)$, such that:

1. $f$ is a normalised newform: $b_1 = 1$ and $f$ is an eigenform of all [[Hecke operators|S4SV3QZ8B1]] that does not come from a lower level;
2. $b_n \in \Z$ for all $n$;
3. $b_p = a_p$ for every prime $p \nmid N$.

## Remarks

Equivalently, the $L$-function of $E$ equals the $L$-function of $f$, or there is a nonconstant morphism $X_0(N) \to E$ defined over $\Q$ from the modular curve $X_0(N)$.
Wiles, with a key step completed jointly with Taylor, proved the theorem for semistable curves in 1995; Breuil, Conrad, Diamond and Taylor completed the general case in 2001.
The proof shows that the $\ell$-adic [[Galois representation|JQEYEQJY2G]] of $E$ is modular.
It lifts the modularity of the reduction modulo $3$, known by the Langlands–Tunnell theorem, with a modularity lifting theorem.
A switch to the prime $5$ handles the cases where the reduction modulo $3$ is not irreducible.
Example: $y^2 + y = x^3 - x^2$ has conductor $11$, and it corresponds to $f = q \prod_{n \ge 1} (1 - q^n)^2 (1 - q^{11n})^2$.
The semistable case, combined with the earlier work of Frey, Serre and Ribet, implies Fermat's Last Theorem.

## Explanations

### L1 human

Counting the points of an elliptic curve modulo each prime produces a list of numbers.
The theorem says that this list always appears as the coefficients of a very symmetric function, a modular form.
It was known as the Taniyama–Shimura–Weil conjecture before its proof.

### L2 human

Modularity gives the analytic continuation and functional equation of $L(E, s)$, which are needed even to state the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].
The parametrisation $X_0(N) \to E$ lets one construct [[Heegner points|2230ZKE4W9]] on $E$, the input of the [[Gross–Zagier theorem|BX558DTV2D]].
It is a case of the reciprocity conjectures of the [[Langlands program|J9T2M084W5]].
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# Moment problem

## Problem

Let $K$ be a closed subset of $\R$ and let $(m_n)_{n \ge 0}$ be a [[sequence|0G9EKM0E15]] of real numbers.

1. Existence: decide whether there is a positive Borel [[measure|PR11BZQQPG]] $\mu$ on $\R$ with support in $K$ and the following property. Every $x^n$ is $\mu$-integrable and $m_n = \int x^n \, d\mu(x)$ for all $n \ge 0$.
2. Uniqueness: when such a measure exists, decide whether it is the only one.

The numbers $m_n$ are then the [[moments|01XAHPADBA]] of $\mu$.

## Remarks

The classical cases are named after the set $K$.
For $K = \R$ it is the [[Hamburger moment problem|TBNCFXTMMW]], for $K = [0, \infty)$ the [[Stieltjes moment problem|Y9KH8P9JC0]], and for $K = [0, 1]$ the [[Hausdorff moment problem|G7VG61YD4J]].
For these three cases the existence question is solved by positivity conditions on Hankel matrices built from the moments, or on differences of the sequence for $[0, 1]$.
For compact $K$ the measure is unique when it exists, since polynomials are dense in the continuous functions on $K$.
For unbounded $K$ uniqueness can fail; the log-normal distribution is the standard example.
Many variants are studied: finitely many moments (the truncated problem), measures on $\R^d$, and the trigonometric problem on the circle.

## Explanations

### L1 human

The moments of a mass distribution are its total mass, its average position, its average squared position, and so on.
The moment problem runs the other way: given a list of numbers, is there a distribution with exactly these moments, and is there only one?
The answer depends on where the mass may sit, on a bounded interval, a half-line or the whole line.
It matters in probability, where one often knows the moments of a random variable and wants to identify its law.
</document>

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# Prenex normal form

## Summary

A formula is in prenex normal form when all its quantifiers stand at the front, followed by a part without quantifiers.
In classical first-order logic every formula is equivalent to a formula in prenex normal form.
The string of quantifiers in front measures the logical complexity of the formula, as in the [[arithmetical hierarchy|7CVJCQKY41]].
Prenex form is also the first step towards [[Skolem normal form|GCPQGMJYEV]].

## Explanations

### L1 human

The formula $\forall x \, \exists y \, (x < y \land y < x + 2)$ has all its quantifiers at the start, so it is in prenex form.
The formula $(\forall x \, P(x)) \to Q(c)$ is not, because the quantifier sits inside the implication.
It can be rewritten as $\exists x \, (P(x) \to Q(c))$, which means the same thing and is in prenex form.
Pulling quantifiers to the front like this is always possible in classical logic.
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# Dougall's formula

## Statement

Let $m$ be a nonnegative integer and let $a, b, c, d, e$ be complex numbers with
$$1 + 2a = b + c + d + e - m.$$
Assume that none of the lower parameters $\frac{a}{2}, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a + m$ lies in $\set{0, -1, \dots, 1 - m}$, and that $(1 + a - b - c - d)_m \ne 0$.
Write $(x)_k = x (x + 1) \cdots (x + k - 1)$ for the [[Pochhammer symbol|YZHM3HSM0F]].
Then the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_7F_6$ at $1$ is
$${}_7F_6\left( \begin{matrix} a,\ 1 + \frac{a}{2},\ b,\ c,\ d,\ e,\ -m \\ \frac{a}{2},\ 1 + a - b,\ 1 + a - c,\ 1 + a - d,\ 1 + a - e,\ 1 + a + m \end{matrix} ; 1 \right) = \frac{(1 + a)_m\, (1 + a - b - c)_m\, (1 + a - b - d)_m\, (1 + a - c - d)_m}{(1 + a - b)_m\, (1 + a - c)_m\, (1 + a - d)_m\, (1 + a - b - c - d)_m}.$$

## Remarks

The series is *very well-poised*: each upper parameter plus the matching lower parameter gives $1 + a$, and the pair $1 + \frac{a}{2}, \frac{a}{2}$ is present.
The condition $1 + 2a = b + c + d + e - m$ says that the series is $2$-balanced: the lower parameters sum to $2$ more than the upper ones.
One proof applies [[Whipple's transformation|QNGGCTXK9S]], which turns the ${}_7F_6$ into a multiple of ${}_4F_3(-m, d, e, 1 + a - b - c; 1 + a - b, 1 + a - c, d + e - a - m; 1)$.
Under the condition $1 + 2a = b + c + d + e - m$ one has $d + e - a - m = 1 + a - b - c$, so this ${}_4F_3$ reduces to a balanced ${}_3F_2$.
The [[Pfaff–Saalschütz theorem|9X6KG770GV]] then sums it, and the product on the right comes out.
For $m = 0$ both sides equal $1$.
Letting $m \to \infty$ term by term gives the summation of the well-poised ${}_5F_4$ with upper parameters $a, 1 + \frac{a}{2}, b, c, d$, which in turn contains part 3 of [[Dixon's identity|QC4TP5D6HE]].

## Explanations

### L1 human

Dougall's formula evaluates a finite sum of ratios of rising factorials with seven parameters on top and six on the bottom.
The parameters must follow a rigid pattern, and one linear relation must hold among them.
When they do, the whole sum is a single product, which is rare for sums of this size.

### L2 human

Dougall's formula sits near the top of the classical summation formulas for hypergeometric series at $1$.
Specialising parameters gives many other summations, which is why it is a hub in tables of identities.
Its $q$-analogue is Jackson's summation for a very well-poised ${}_8\phi_7$.
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# Serre subcategory

## Summary

A Serre subcategory of an [[abelian category|Y9CZ3MTNF1]] is a nonempty [[full subcategory|V3BFGKBJF7]] closed under subobjects, quotient objects and extensions.
These are the subcategories that can be "divided out".
For each Serre subcategory there is a quotient abelian category in which its objects become zero.
The objects sent to zero by an [[exact functor|S1G61TDW0K]] always form a Serre subcategory.

## Explanations

### L1 human

Think of a Serre subcategory as a class of objects that one is willing to ignore.
For this to be consistent, the class must contain every piece of its objects and everything glued together from two of its objects.
The torsion abelian groups form such a class: a subgroup or quotient of a torsion group is torsion, and so is a group built from two torsion groups.
Dividing out the torsion groups leaves a category in which a map of abelian groups counts as an isomorphism when it becomes one after tensoring with $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/CSETD54YDT.md" tag="CSETD54YDT" kind="statement">
# Kaczmarz method

## Statement

Let $A$ be a real $m \times n$ matrix with nonzero rows $a_1^T, \dots, a_m^T$, and let $b \in \R^m$ be such that the [[linear system|0GT4WWGWH2]] $A x = b$ has a solution.
For an index $i$, the *Kaczmarz step* maps $x \in \R^n$ to
$$x + \frac{b_i - \ip{a_i}{x}}{\norm{a_i}^2} \, a_i,$$
the [[orthogonal projection|0B8KZRT28M]] of $x$ onto the hyperplane $\set{y : \ip{a_i}{y} = b_i}$.
Start from any $x_0 \in \R^n$.

1. (Cyclic version) Let $x_{k+1}$ be the Kaczmarz step from $x_k$ with $i = (k \bmod m) + 1$. Then $x_k$ converges to a solution of $A x = b$. If $x_0 = 0$, the limit is the solution of least Euclidean norm.
2. (Randomized version, Strohmer and Vershynin) Suppose $A$ has [[rank|0ZC9YTXZRY]] $n$, and let $x^*$ be the solution. Let $x_{k+1}$ be the Kaczmarz step from $x_k$ with $i$ drawn independently, equal to $j$ with probability $\norm{a_j}^2 / \norm{A}_F^2$. Then $\mathbb{E} \norm{x_k - x^*}^2 \le \left( 1 - \kappa^{-2} \right)^k \norm{x_0 - x^*}^2$, where $\kappa = \norm{A}_F \norm{A^{+}}_2$.

Here $\norm{\cdot}$ is the Euclidean norm, $\norm{A}_F$ is the Frobenius norm and $A^{+}$ is the pseudoinverse.

## Remarks

Each step is a projection onto a convex set that contains every solution.
So for every solution $z$, the distance $\norm{x_k - z}$ never increases, which drives the convergence proof.
Each step adds a multiple of a row $a_i$, so $x_k - x_0$ stays in the row space of $A$.
The solution of least norm is the only solution in the row space, which gives the claim for $x_0 = 0$.
Example: for the system $x_1 = 1$, $x_2 = 1$ in $\R^2$, two cyclic steps from $0$ give $(1, 0)$ and then the solution $(1, 1)$.
For orthogonal rows the method finishes after one sweep; for nearly parallel rows it zigzags and is slow.
The number $\kappa$ is a scaled [[condition number|AB3S6MHJ8X]], and it measures this effect in the randomized bound.

## Explanations

### L1 human

Each equation of a linear system describes a flat set, a line in the plane or a plane in space.
The Kaczmarz method moves the current point straight onto one of these sets, then onto the next, and so on.
The point gets closer to the common intersection at each move.
This simple idea is the algebraic reconstruction technique of computed tomography.

### L2 human

The Kaczmarz method touches one row at a time, so it suits huge systems whose rows are generated on the fly.
It is a row-action [[iterative method|2RPE77ACER]], related to the Gauss–Seidel method applied to $A A^T y = b$ with $x = A^T y$.
Choosing rows at random with probability proportional to their squared norms gives a clean linear rate in expectation.
This randomized version links the method to stochastic gradient descent for least squares.
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<document source="https://tamnd.github.io/math2/node/CSK0KBMARJ.md" tag="CSK0KBMARJ" kind="definition">
# Simplex

## Definition

Let $n \ge 0$ and $N \ge n$ be integers.
Let $v_0, \ldots, v_n$ be [[affinely independent|JWK49HMHYT]] points of $\R^N$.
The *$n$-simplex* with vertices $v_0, \ldots, v_n$ is their [[convex hull|0FQSDC2EQ2]]
$$\Delta = \set{t_0 v_0 + \cdots + t_n v_n : t_0, \ldots, t_n \ge 0, \ t_0 + \cdots + t_n = 1}.$$
The number $n$ is its dimension.
The convex hull of a nonempty subset of the vertices is a *face* of $\Delta$.

## Remarks

The numbers $t_0, \ldots, t_n$ are the barycentric coordinates of a point of $\Delta$, and they are unique by affine independence.

Example: $v_0 = (0, 0)$, $v_1 = (1, 0)$ and $v_2 = (0, 1)$ are affinely independent, since $v_1 - v_0$ and $v_2 - v_0$ are linearly independent.
Their $2$-simplex is the triangle $\set{(x, y) : x, y \ge 0, \ x + y \le 1}$.

Non-example: the points $(0, 0)$, $(1, 0)$ and $(2, 0)$ are not affinely independent.
Their convex hull is the segment from $(0, 0)$ to $(2, 0)$, a $1$-simplex but not a $2$-simplex.

The *standard $n$-simplex* is the simplex in $\R^{n+1}$ with the standard basis vectors as vertices.
Some texts, and Mathlib, use the word simplex for the family of vertices rather than for their convex hull.
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<document source="https://tamnd.github.io/math2/node/CSMB49BDPT.md" tag="CSMB49BDPT" kind="example">
# Projective geometry over a finite field

## Example

Let $q$ be a prime power, let $\F_q$ be the [[finite field|0ZXZ82FTX2]] with $q$ elements, and let $n \ge 1$.
The projective space $PG(n, q)$ is the [[projective space|ZT9A6JK8QF]] of $V = \F_q^{n+1}$.
Its points are the $1$-dimensional subspaces of $V$, and its lines are the $2$-dimensional subspaces.
In general its $k$-dimensional subspaces are the $(k+1)$-dimensional subspaces of $V$, with incidence given by inclusion.

Count of points: $V$ has $q^{n+1} - 1$ nonzero vectors, and each $1$-dimensional subspace contains $q - 1$ of them, so $PG(n, q)$ has
$$\frac{q^{n+1} - 1}{q - 1} = q^n + q^{n-1} + \cdots + q + 1$$
points.
Each line has $(q^2 - 1)/(q - 1) = q + 1$ points.

Check for $PG(3, 2)$: it has $15$ points and each line has $3$ points.
Two distinct points span a unique line, so the number of lines is $\binom{15}{2} / \binom{3}{2} = 105 / 3 = 35$.
Its planes are the $3$-dimensional subspaces of $\F_2^4$, and there are $15$ of them, one for each nonzero linear form up to scaling.

For $n = 2$, the plane $PG(2, q)$ is a [[finite projective plane|2HWB7P1DZN]] of order $q$, and $PG(2, 2)$ is the Fano plane.

## Explanations

### L1 human

Take the usual recipe for projective space, lines through the origin of a vector space, and use a finite field in place of the real numbers.
The result is a projective space with finitely many points, one of the basic objects of [[finite geometry|MCDZP31ZF3]].
Every axiom of projective geometry still holds, and all counts become explicit formulas in $q$.
For $q = 2$ and $n = 2$ one gets the seven points and seven lines of the Fano plane.
</document>

<document source="https://tamnd.github.io/math2/node/CSMF0WVH67.md" tag="CSMF0WVH67" kind="definition">
# Deficiency indices

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]], and let $A$ be a [[symmetric operator|H971WGMV6C]] on $H$ with dense domain and [[adjoint|86B7HF79PQ]] $A^*$.
The *deficiency subspaces* of $A$ are the closed subspaces
$$K_+ = \ker(A^* - iI), \qquad K_- = \ker(A^* + iI)$$
of $H$, where $\ker$ denotes the [[kernel|0S23E6YE83]].
The *deficiency indices* of $A$ are the [[Hilbert dimensions|40SV748R64]] $n_+ = \dim K_+$ and $n_- = \dim K_-$, which are cardinal numbers, possibly infinite.

## Remarks

Equivalently, $K_\pm$ is the orthogonal complement of the [[image|0WSFKKT6FF]] of $A \pm iI$.
Some authors exchange the names of $K_+$ and $K_-$; the pair $(n_+, n_-)$ is then reversed.
The operator $A$ is [[essentially self-adjoint|DT49GKR9AS]] if and only if $n_+ = n_- = 0$.
Example: on $L^2([0,1])$, let $A = -i \, d/dx$ on the smooth functions with compact support in $(0,1)$.
Then $A^*$ is $-i \, d/dx$ on the Sobolev space $H^1([0,1])$, and $A^* u = \pm i u$ means $u' = \mp u$.
So $K_+$ is spanned by $e^{-x}$ and $K_-$ by $e^{x}$, and $(n_+, n_-) = (1, 1)$.
Example with unequal indices: on $L^2([0, \infty))$, the same construction gives $K_+$ spanned by $e^{-x}$, while $e^{x}$ is not square integrable, so $(n_+, n_-) = (1, 0)$.
Non-example: the deficiency indices are not defined for an operator that is not densely defined, because then the adjoint is not a well-defined operator.
</document>

<document source="https://tamnd.github.io/math2/node/CSRSDP9PXJ.md" tag="CSRSDP9PXJ" kind="example">
# Beverton–Holt model

## Example

Let $R_0 > 1$ and $M > 0$ be constants, and let $K = (R_0 - 1) M$.
The *Beverton–Holt model* is the recurrence
$$n_{t+1} = \frac{R_0\, n_t}{1 + n_t / M} \qquad (t \ge 0)$$
for a population size $n_t \ge 0$.
It is a [[Riccati difference equation|D4Q8VM7826]] with $a = R_0$, $b = 0$, $c = 1/M$, $d = 1$, so $c \ne 0$ and $ad - bc = R_0 \ne 0$.
For $n_0 > 0$, the solution is
$$n_t = \frac{K\, n_0}{n_0 + (K - n_0)\, R_0^{-t}}.$$
Check: $z_t = 1/n_t$ satisfies the linear recurrence $z_{t+1} = z_t / R_0 + 1/(R_0 M)$, whose solution is $z_t = R_0^{-t} z_0 + (1 - R_0^{-t})/K$; inverting gives the formula.
Since $R_0^{-t} \to 0$, every solution with $n_0 > 0$ tends to the [[equilibrium|D62PEKHRV5]] $K$, the carrying capacity.

## Explanations

### L1 human

This is a discrete-time model of a population with limited resources, used in fisheries.
When the population is small, it grows by the factor $R_0$ each generation.
As it grows, crowding reduces the growth, and the population levels off at the carrying capacity $K$.
Unlike many nonlinear models, it has an exact formula for its solution, because taking reciprocals makes it linear.
</document>

<document source="https://tamnd.github.io/math2/node/CSXT0BG2CM.md" tag="CSXT0BG2CM" kind="concept">
# Incidence geometry

## Summary

Incidence geometry is the part of geometry that studies points, lines and the relation "lies on", and nothing else.
Its objects are [[incidence structures|FGKNAFNHA3]] that satisfy axioms such as "two points lie on exactly one line".
[[Affine planes|EC7XT2ZM5S]], [[projective planes|DANFNN8C4Y]] and finite geometries are its main examples.
It asks which configurations the axioms force.
It also asks which algebra, such as a field or a division ring, can be rebuilt from the points and lines alone.

## Explanations

### L1 human

Ordinary geometry talks about lengths and angles.
Incidence geometry throws these away and keeps only the question of which points lie on which lines.
Surprisingly much survives: theorems such as those of Desargues and Pappus only speak about points being on lines.
It also allows small finite worlds, such as a geometry with $7$ points and $7$ lines, that behave like the plane in this respect.
</document>

<document source="https://tamnd.github.io/math2/node/CSYZGA5YKS.md" tag="CSYZGA5YKS" kind="statement">
# Uniform convergence of absolutely summable Fourier series

## Statement

Let $f \colon \R \to \C$ be a [[continuous|0X3WGX95FE]] $2\pi$-periodic function whose [[Fourier coefficients|5NN2YZZR3V]] satisfy
$$\sum_{n \in \Z} \abs{\hat f(n)} < \infty.$$
Then the partial sums $S_N f(x) = \sum_{n=-N}^{N} \hat f(n) e^{inx}$ of its [[Fourier series|G12KXDHPPS]] converge to $f$ [[uniformly|08VZ57PT26]] on $\R$.

## Remarks

Since $\abs{\hat f(n) e^{inx}} = \abs{\hat f(n)}$, the [[Weierstrass M-test|0NX5NWAFC9]] shows that $S_N f$ converges uniformly to a continuous $2\pi$-periodic function $g$.
Uniform convergence allows integration term by term, so $\hat g(n) = \hat f(n)$ for all $n$.
Then $f - g$ is continuous with all Fourier coefficients zero, and its Fejér means, which converge uniformly to $f - g$ by [[Fejér's theorem|TPJZCSXSVN]], are all zero; hence $f = g$.
Example: for $f(x) = \abs{x}$ on $[-\pi, \pi]$, extended periodically, $\hat f(0) = \frac{\pi}{2}$ and $\hat f(n) = \frac{(-1)^n - 1}{\pi n^2}$ for $n \ne 0$, which is summable.
Without continuity the conclusion fails: if $f$ is only integrable with summable coefficients, the series converges uniformly to a continuous function that equals $f$ only almost everywhere.
Every $f \in C^1$ periodic satisfies the hypothesis, by the Cauchy–Schwarz inequality and Parseval's identity applied to $f'$.

## Explanations

### L1 human

If the sizes of the Fourier coefficients add up to a finite number, the Fourier series converges as nicely as possible: uniformly, to the function itself.
The finite sum of sizes bounds the error of all the tails at once.

### L2 human

This is the simplest sufficient condition for uniform convergence, and it is easy to check from decay of the coefficients, for instance $\abs{\hat f(n)} \le C \abs{n}^{-1-\delta}$ with $\delta > 0$.
The functions with summable Fourier coefficients form the [[Wiener algebra|5H8XESYCHQ]] $A(\mathbb{T})$.
Bernstein's theorem shows that Hölder continuity of order $\alpha > \frac{1}{2}$ is enough to be in $A(\mathbb{T})$.
The condition is far from necessary: there are continuous functions whose Fourier series converge uniformly but not absolutely.
</document>

<document source="https://tamnd.github.io/math2/node/CT3PNBCYMX.md" tag="CT3PNBCYMX" kind="concept">
# Multivariate approximation

## Summary

Multivariate approximation is the part of [[approximation theory|YPJSH79SHQ]] that approximates functions of several variables.
Its tools include [[polynomials in several variables|07GNX8GCYP]], tensor product methods, [[radial basis functions|XZ33P3HBB9]], box splines, [[sparse grids|84BR7X6HET]] and neural networks.
Many one-dimensional facts fail or change form here: for instance, interpolation by polynomials at arbitrary distinct points is no longer always possible.
A central question is how the error depends on the dimension, which leads to the [[curse of dimensionality|YS424FTY4E]].

## Explanations

### L1 human

A function of one variable is a curve, and a function of two variables is a surface; in many applications there are dozens or thousands of variables.
Multivariate approximation asks how to replace such a function by a simpler one, such as a polynomial, that is close to it everywhere.
The number of simple building blocks grows quickly with the number of variables.
For example, polynomials of degree at most $2$ in one variable form a space of dimension $3$, but in $10$ variables they form a space of dimension $\binom{12}{2} = 66$.
</document>

<document source="https://tamnd.github.io/math2/node/CT9GYCJV4V.md" tag="CT9GYCJV4V" kind="definition">
# Derived functor

## Definition

Let $\mathcal{A}$ and $\mathcal{B}$ be [[abelian categories|Y9CZ3MTNF1]], and assume that every object of $\mathcal{A}$ has an [[injective resolution|ZW3W6FX4PW]].
Let $F \colon \mathcal{A} \to \mathcal{B}$ be an [[additive functor|KAN07YC9T2]] that is left [[exact|S1G61TDW0K]].
For an object $X$ of $\mathcal{A}$, choose an injective resolution $X \to I^\bullet$.
For $n \ge 0$, the $n$-th *right derived functor* of $F$ at $X$ is the [[cohomology|2J4JJJYP22]]
$$R^n F(X) = H^n(F(I^\bullet)).$$
Dually, if every object has a [[projective resolution|HK2AA1YSQ2]] $P_\bullet \to X$ and $F$ is additive and right exact, the *left derived functors* are $L_n F(X) = H_n(F(P_\bullet))$.

## Remarks

Up to a canonical isomorphism, $R^n F(X)$ does not depend on the resolution, since any two injective resolutions are chain homotopy equivalent; the same argument makes $R^n F$ a functor.
Left exactness gives $R^0 F \cong F$, and a short exact sequence in $\mathcal{A}$ gives a long exact sequence of the $R^n F$.
Example: let $F = \Hom_\Z(\Z/2\Z, -)$ on abelian groups and $X = \Z$, with the injective resolution $0 \to \Z \to \Q \to \Q/\Z \to 0$.
Then $F(\Q) = 0$ and $F(\Q/\Z) = \set{0, 1/2} \cong \Z/2\Z$, so $R^0 F(\Z) = 0$ and $R^1 F(\Z) \cong \Z/2\Z$, which is $\operatorname{Ext}^1_\Z(\Z/2\Z, \Z)$.
Non-example: the complex $0 \to \Z \xrightarrow{\id} \Z \to 0$ is exact, but $\Z$ is not injective, and using it in place of $I^\bullet$ would give $0$ in degree $1$ instead of $\Z/2\Z$.
In general $R^n \Hom_R(M, -) = \operatorname{Ext}^n_R(M, -)$ and $L_n(M \otimes_R -) = \operatorname{Tor}_n^R(M, -)$.
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<document source="https://tamnd.github.io/math2/node/CTBNKE0B03.md" tag="CTBNKE0B03" kind="definition">
# p-adic Lie group

## Definition

Let $p$ be a prime and let $\Q_p$ be the field of [[p-adic numbers|BT6T702C28]].
Let $U \subseteq \Q_p^n$ be open.
A map $f \colon U \to \Q_p^m$ is *analytic* if each point of $U$ has an open neighbourhood on which each coordinate of $f$ is a convergent [[power series|0MZB06ST2B]].
A *$p$-adic analytic manifold* of dimension $n$ is a Hausdorff space $M$ with an atlas of charts.
The charts are homeomorphisms from open sets of $M$ onto open sets of $\Q_p^n$, and their transition maps are analytic.
A *$p$-adic Lie group* is a group $G$ with the structure of a $p$-adic analytic manifold such that:

1. the multiplication $G \times G \to G$, $(x, y) \mapsto xy$, is analytic;
2. the inversion $G \to G$, $x \mapsto x^{-1}$, is analytic.

## Remarks

Example: the group $\Z_p$ of $p$-adic integers under addition is an open subset of $\Q_p$, so the identity map is a chart.
Addition $(x, y) \mapsto x + y$ and negation $x \mapsto -x$ are polynomials, hence analytic.
So $\Z_p$ is a $p$-adic Lie group of dimension $1$.
The group $GL_n(\Q_p)$ is open in the space $\Q_p^{n^2}$ of matrices.
Its product and inverse are rational in the entries with nonvanishing denominators, so it is a $p$-adic Lie group of dimension $n^2$.

Non-example: the additive group $\R$ is not a $p$-adic Lie group with its usual topology.
Every open subset of $\Q_p^n$ is totally disconnected, so every $p$-adic analytic manifold is totally disconnected.
But $\R$ is connected and has more than one point.

Such groups are also called $p$-adic analytic groups.
Every $p$-adic Lie group is a totally disconnected locally compact Hausdorff group.
</document>

<document source="https://tamnd.github.io/math2/node/CTBYV71CEJ.md" tag="CTBYV71CEJ" kind="definition">
# Affine scheme

## Definition

Let $A$ be a [[commutative ring|0310REJJY8]] with identity, and let $X = \operatorname{Spec} A$ be its [[spectrum|FB45BHHA2J]] with the [[Zariski topology|8SQK7RTB63]].
For $f \in A$, write $D(f) = \set{\mathfrak{p} \in X : f \notin \mathfrak{p}}$; these sets form a basis of open sets of $X$.
Let $A_f$ be the [[localization|4C1AE84N5N]] of $A$ at the powers of $f$.
The *structure sheaf* $\mathcal{O}_X$ is the unique [[sheaf|TB6TM6R4MD]] of rings on $X$ with $\mathcal{O}_X(D(f)) = A_f$ for every $f \in A$.
Its restriction maps $A_f \to A_g$ for $D(g) \subseteq D(f)$ are the natural maps.
Its stalk at $\mathfrak{p}$ is the [[localization at the prime ideal|X6JD4AG973]] $A_{\mathfrak p}$, which is a local ring.

An *affine scheme* is a [[locally ringed space|B7FPT8JTXR]] isomorphic to $(\operatorname{Spec} A, \mathcal{O}_{\operatorname{Spec} A})$ for some commutative ring $A$.

## Remarks

Example: $\operatorname{Spec} \Z$ is an affine scheme; its points are $(0)$ and $(p)$ for the primes $p$, and $\mathcal{O}(D(n)) = \Z[1/n]$ for $n \ne 0$.
Example: the affine $n$-space over a ring $R$ is $\mathbb{A}^n_R = \operatorname{Spec} R[x_1, \ldots, x_n]$.
Non-example: for a field $k$, the open subset $U = \mathbb{A}^2_k \setminus \set{(x, y)}$ of the plane, with the restricted sheaf, is a locally ringed space but not an affine scheme.
Its ring of global functions is $k[x, y]$, so if it were affine it would be $\operatorname{Spec} k[x, y]$, which also contains the origin $(x, y)$.
The ring is recovered as $A \cong \mathcal{O}_X(X)$, the global sections.
</document>

<document source="https://tamnd.github.io/math2/node/CTCSAWBSB8.md" tag="CTCSAWBSB8" kind="definition">
# Topological conjugacy

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], and let $f \colon X \to X$ and $g \colon Y \to Y$ be [[continuous maps|0X3WGX95FE]].
A *topological conjugacy* from $f$ to $g$ is a [[homeomorphism|0QT00PB5G2]] $h \colon X \to Y$ such that
$$h \circ f = g \circ h,$$
where $\circ$ is [[composition|0THBK9V825]].
The maps $f$ and $g$ are *topologically conjugate* if such an $h$ exists.

For flows $\varphi$ on $X$ and $\psi$ on $Y$, a topological conjugacy is a homeomorphism $h \colon X \to Y$ with $h \circ \varphi_t = \psi_t \circ h$ for all $t \in \R$.

## Remarks

The relation $h \circ f = g \circ h$ gives $h \circ f^n = g^n \circ h$ for all $n \ge 0$, so $h$ maps orbits of $f$ onto orbits of $g$.
In particular $h$ maps fixed points to fixed points and periodic points of minimal period $p$ to periodic points of minimal period $p$.
Topological conjugacy is an equivalence relation on continuous self-maps.

Example: the tent map $T(x) = 1 - \abs{2x - 1}$ and the logistic map $f(x) = 4x(1 - x)$ on $[0, 1]$ are conjugate through $h(x) = \sin^2(\pi x / 2)$.
The map $h$ is an increasing homeomorphism of $[0, 1]$.
One has $f(h(x)) = 4 \sin^2(\pi x/2) \cos^2(\pi x/2) = \sin^2(\pi x)$.
Also $h(T(x)) = \sin^2(\pi x)$ for $x \le 1/2$, and $h(T(x)) = \sin^2(\pi - \pi x) = \sin^2(\pi x)$ for $x \ge 1/2$.

Non-example: $f(x) = x/2$ and $g(x) = -x/2$ on $\R$ are not conjugate.
A conjugacy $h$ would map the only fixed point $0$ to $0$, so it would map $(0, \infty)$ onto $(0, \infty)$ or onto $(-\infty, 0)$.
Since $f$ maps $(0, \infty)$ into itself while $g$ maps each half-line into the other one, $h \circ f = g \circ h$ fails.

For flows, the weaker notion of topological equivalence only asks that $h$ map orbits onto orbits and preserve their direction, not the time parametrisation.
</document>

<document source="https://tamnd.github.io/math2/node/CTDG4MZC8D.md" tag="CTDG4MZC8D" kind="statement">
# Wald's equation

## Statement

Let $X_1, X_2, \dots$ be [[independent and identically distributed|00MRN0FARV]] real random variables with $E\abs{X_1} < \infty$.
Let $S_n = X_1 + \cdots + X_n$.
Let $N$ be a [[stopping time|WND87FBAKC]] with values in $\set{1, 2, \dots}$ with respect to the filtration $\mathcal{F}_n = \sigma(X_1, \dots, X_n)$, and assume $E[N] < \infty$.
Then
$$E[S_N] = E[N] \, E[X_1],$$
where $E$ is the [[expectation|0WQNRCNY3M]].

## Remarks

Write $S_N = \sum_{i \ge 1} X_i \mathbf{1}_{\set{N \ge i}}$.
The event $\set{N \ge i}$ is the complement of $\set{N \le i - 1}$, so it depends only on $X_1, \dots, X_{i-1}$ and is independent of $X_i$.
Hence $E[X_i \mathbf{1}_{\set{N \ge i}}] = E[X_1] \, P(N \ge i)$, and summing over $i$ gives $E[X_1] \sum_i P(N \ge i) = E[X_1] \, E[N]$.
The same computation with $\abs{X_i}$ shows that the series converges absolutely, which justifies exchanging sum and expectation.

The hypothesis $E[N] < \infty$ is necessary.
For the simple symmetric [[random walk|04VZE3FZXF]], with $P(X_i = 1) = P(X_i = -1) = 1/2$, let $N$ be the first $n$ with $S_n = 1$.
Then $N$ is finite with probability one and $S_N = 1$, so $E[S_N] = 1$, while $E[X_1] = 0$; here $E[N] = \infty$.

## Explanations

### L1 human

A gambler plays a game many times and decides when to stop using only what has happened so far.
Wald's equation says that the expected total winnings equal the expected number of games times the expected winnings per game.
So no stopping strategy turns a fair game into a profitable one, provided the expected number of games is finite.

### L2 human

Wald's equation is a key tool in [[sequential analysis|9CWYNB9003]], where it gives the expected sample size of the sequential probability ratio test from the drift of the log-likelihood ratio.
It is also used in renewal theory and queueing theory.
It can also be derived from the [[optional stopping theorem|DC9HZT2HVJ]] applied to the martingale $S_n - n E[X_1]$.
</document>

<document source="https://tamnd.github.io/math2/node/CTGN6CK1ZS.md" tag="CTGN6CK1ZS" kind="concept">
# Detailed balance

## Summary

Detailed balance is a property of a distribution $\pi$ and a [[Markov chain|0G48PYM96Q]].
It says that, in equilibrium, the probability flow from each state $i$ to each state $j$ equals the flow from $j$ back to $i$.
Detailed balance implies that $\pi$ is a [[stationary distribution|0KRH5157WP]], and a chain with such a $\pi$ is called reversible.
It is the usual design principle of Markov chain Monte Carlo algorithms.

## Explanations

### L1 human

Picture many walkers spread over the states according to $\pi$, all moving at once by the rules of the chain.
Stationarity only asks that the total number in each state stay the same.
Detailed balance asks more: for every pair of states, as many walkers go from $i$ to $j$ as from $j$ to $i$.
A film of the chain in equilibrium then looks the same when played backwards.
A random walk on a graph that moves to a uniformly chosen neighbour has this property, with $\pi$ proportional to the degrees.
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# Equalizer

## Summary

The equalizer of two parallel morphisms $f, g \colon X \to Y$ in a [[category|A2DE66937D]] is a universal morphism $e \colon E \to X$ with $f \circ e = g \circ e$.
It is the [[limit|CHS5AFJGWE]] of the diagram formed by $f$ and $g$.
In sets it is the subset where $f$ and $g$ agree, and in abelian groups it is the kernel of $f - g$.
Every equalizer is a monomorphism, and products together with equalizers build all limits.

## Explanations

### L1 human

Given two functions $f$ and $g$ from $X$ to $Y$, ask where they give the same answer.
The answer is the set $E = \set{x \in X : f(x) = g(x)}$, which sits inside $X$.
For example, for $f(x) = x^2$ and $g(x) = x$ on the real numbers, the equalizer is $\{0, 1\}$, the set of solutions of $x^2 = x$.
The categorical equalizer describes this "solution set" through maps only: anything mapping into $X$ on which $f$ and $g$ agree must map into $E$.
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# Pexider equation

## Summary

The Pexider equation is the [[functional equation|SRAN4EKWS0]] $f(x + y) = g(x) + h(y)$ with three unknown functions $f, g, h \colon \R \to \R$.
It generalizes Cauchy's additive equation, which is the case $f = g = h$.
Its solutions are $f = A + a + b$, $g = A + a$ and $h = A + b$, where $A$ is an [[additive function|KF719SZD5T]] and $a, b$ are constants.
Equations of this kind, with several unknown functions in the place of one, are called Pexiderized forms of the classical equations.

## Explanations

### L1 human

The equation asks for three functions that fit together: the first one at a sum is the second at one term plus the third at the other.
Allowing three different functions gives only a little more freedom than Cauchy's equation.
For example, $f(x) = x + 3$, $g(x) = x + 1$ and $h(x) = x + 2$ solve it.
Every solution differs from an additive function only by constant shifts.
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# Cartan connection

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] and $H$ a closed [[subgroup|NM8XCPS1Q3]] of $G$, with Lie algebras $\mathfrak{h} \subset \mathfrak{g}$.
Let $M$ be a smooth manifold with $\dim M = \dim G - \dim H$.
A *Cartan connection on $M$ modelled on $(G, H)$* is a [[principal $H$-bundle|8E37E74XVP]] $\pi : P \to M$ together with a smooth $\mathfrak{g}$-valued $1$-form $\omega$ on $P$ such that:

1. $\omega_u : T_uP \to \mathfrak{g}$ is a linear isomorphism for every $u \in P$.
2. $R_h^* \omega = \mathrm{Ad}(h^{-1}) \, \omega$ for every $h \in H$, where $\mathrm{Ad}$ is the [[adjoint representation|P6ZP0TE07M]] of $G$ restricted to $H$.
3. $\omega(X^\#) = X$ for every $X \in \mathfrak{h}$, where $X^\#$ is the fundamental vector field of $X$ on $P$.

## Remarks

The *curvature* of $\omega$ is $\Omega = d\omega + \frac12 [\omega \wedge \omega]$.
Example: the model itself, $P = G \to G/H$ with the left [[Maurer–Cartan form|P1JVA39H5E]] $\omega = \theta$ of $G$.
Clause 1 holds because $\theta_g = dL_{g^{-1}}$ is an isomorphism $T_gG \to \mathfrak{g}$, and clauses 2 and 3 are standard properties of $\theta$.
Its curvature vanishes by the Maurer–Cartan equation, and a Cartan connection is locally isomorphic to this model exactly when $\Omega = 0$.
Non-example: a principal connection form on $P$ has values in $\mathfrak{h}$, so it fails clause 1 unless $\mathfrak{h} = \mathfrak{g}$ and $M$ is a point.
For $G$ the Euclidean group $\R^n \rtimes \mathrm{O}(n)$ and $H = \mathrm{O}(n)$, a torsion-free Cartan connection is the same as a Riemannian metric.
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# Euclidean tilings by convex regular polygons

## Example

Consider [[tilings|GKJK293DXS]] of the plane whose tiles are convex [[regular polygons|01KJJ9A400]].
The tiling is edge-to-edge if two tiles that meet share either one full edge or one vertex.
The interior angle of a regular $n$-gon is $(n-2)\pi/n$.
A tiling by copies of a single regular $n$-gon is called *regular*.
At each vertex of such a tiling some number $k \ge 3$ of tiles meet, so $k(n-2)\pi/n = 2\pi$, which gives $(n-2)(k-2) = 4$.
The only solutions are $(n, k) = (3, 6), (4, 4), (6, 3)$.
So there are exactly three regular tilings.
They use equilateral triangles, six at each vertex; squares, four at each vertex; and regular hexagons, three at each vertex.
In particular, regular pentagons do not tile the plane edge-to-edge, since $3 \cdot 3\pi/5 < 2\pi < 4 \cdot 3\pi/5$.

## Remarks

An edge-to-edge tiling by regular polygons is *semiregular*, or Archimedean, if it uses more than one kind of polygon and its symmetries map any vertex to any other vertex.
There are exactly eight semiregular tilings.
One of them has at each vertex a triangle, a square, a hexagon and a square, in this cyclic order; the angles add up to $\pi/3 + \pi/2 + 2\pi/3 + \pi/2 = 2\pi$.
The three regular and the eight semiregular tilings are all [[periodic|WFRGSBFZGV]].

## Explanations

### L1 human

Which regular shapes can cover a floor with no gaps, all meeting edge to edge?
The corners meeting at a point must add up to a full turn of $360^\circ$.
Triangles have $60^\circ$ corners, squares $90^\circ$ and hexagons $120^\circ$, and these divide $360^\circ$ evenly.
No other regular polygon works on its own, which is why floors use these three shapes so often.
Mixing several regular shapes gives eight more patterns in which every corner looks the same.
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# Exact category

## Definition

Let $\mathcal{E}$ be an [[additive category|B2VP96NJSK]].
A *kernel–cokernel pair* is a pair of composable morphisms $A \xrightarrow{i} B \xrightarrow{p} C$ such that $i$ is a [[kernel|S417YNEJ6M]] of $p$ and $p$ is a [[cokernel|H8W38933NC]] of $i$.
An *exact structure* on $\mathcal{E}$ is a class $\mathcal{S}$ of kernel–cokernel pairs, closed under isomorphism of pairs, whose members are called *conflations*.
In a conflation $(i, p)$, the morphism $i$ is called an *inflation* and $p$ a *deflation*.
The class $\mathcal{S}$ must satisfy:

1. For every object $A$, the identity $1_A$ is an inflation and a deflation.
2. A composite of two inflations is an inflation, and a composite of two deflations is a deflation.
3. For every inflation $i \colon A \to B$ and every morphism $f \colon A \to A'$, the [[pushout|Q699KKMVJD]] of $i$ along $f$ exists and is an inflation $A' \to B'$.
4. For every deflation $p \colon B \to C$ and every morphism $g \colon C' \to C$, the [[pullback|Q05K04QJ71]] of $p$ along $g$ exists and is a deflation $B' \to C'$.

An *exact category* is an additive category together with an exact structure.

## Remarks

This is Quillen's notion, in the streamlined form of Keller.
Inflations are also called admissible monomorphisms, and deflations admissible epimorphisms.

Example: every [[abelian category|Y9CZ3MTNF1]] with the class of all short exact sequences is an exact category.
Every morphism has a kernel and a cokernel, pushouts and pullbacks exist, and pushouts of monomorphisms are monomorphisms while pullbacks of epimorphisms are epimorphisms.
Example: every additive category with the class of split exact sequences $A \to A \oplus C \to C$ is an exact category.

Non-example: in the category of abelian groups, the class of short exact sequences whose three terms are finite groups fails clause 1.
The identity of $\Z$ is not an inflation, because the only kernel–cokernel pair $\Z \xrightarrow{1} \Z \to 0$ that contains it has an infinite term.
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# Natural density

## Definition

Let $A$ be a set of positive integers.
For each integer $n \ge 1$, let $A(n)$ be the number of elements $a \in A$ with $a \le n$.
If the [[limit|05M5SWSZZD]]
$$d(A) = \lim_{n \to \infty} \frac{A(n)}{n}$$
exists, then $d(A)$ is the *natural density* (or *asymptotic density*) of $A$.
The *upper density* $\overline{d}(A)$ and the *lower density* $\underline{d}(A)$ are the limit superior and the [[limit inferior|033RZ32T4Q]] of $A(n)/n$; they always exist.

## Remarks

The natural density exists if and only if $\overline{d}(A) = \underline{d}(A)$, and then all three numbers are equal.

Example: for the multiples of a positive integer $m$, $A(n) = \floor{n/m}$, and $n/m - 1 < A(n) \le n/m$; so $d(A) = 1/m$.

Non-example: let $A$ be the set of positive integers whose first decimal digit is $1$.
For $n = 2 \cdot 10^k - 1$ one has $A(n) = 1 + 10 + \dots + 10^k = (10^{k+1} - 1)/9$, so $A(n)/n \to 5/9$.
For $n = 10^{k+1} - 1$ the count $A(n)$ is the same, so $A(n)/n \to 1/9$.
Hence the limit does not exist; here $\overline{d}(A) = 5/9$ and $\underline{d}(A) = 1/9$.

If $d(A)$ and $d(B)$ exist and $A \cap B$ is empty, then $d(A \cup B) = d(A) + d(B)$.
Natural density is not the same as [[Schnirelmann density|QHH8RY9XCZ]], which is an infimum over all $n$ and is $0$ for every set not containing $1$.
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# Mackey imprimitivity theorem

## Statement

Let $G$ be a second countable [[locally compact group|AHV05ZBCYW]], and let $H$ be a closed subgroup of $G$.
Let $\pi$ be a [[unitary representation|74BPYJNZTM]] of $G$ on a [[Hilbert space|HM2HHE3J51]] $\mathcal{H}$.
A *system of imprimitivity* for $\pi$ based on $G/H$ is a [[projection-valued measure|ATGT25E5FF]] $P$ on the Borel sets of $G/H$, acting on $\mathcal{H}$.
It must satisfy
$$\pi(g) P(E) \pi(g)^{-1} = P(gE)$$
for all $g \in G$ and all Borel sets $E \subseteq G/H$.
Every [[induced representation|6Q7AHAZ6Q4]] $\operatorname{Ind}_H^G \sigma$ has a canonical system of imprimitivity, where $P(E)$ multiplies a function by the indicator of $E$.
Let $(\pi, P)$ be any system of imprimitivity based on $G/H$.
Then there is a unitary representation $\sigma$ of $H$ such that $(\pi, P)$ is unitarily equivalent to the canonical system of $\operatorname{Ind}_H^G \sigma$.
The representation $\sigma$ is unique up to unitary equivalence.

## Remarks

In operator-algebra terms, a system of imprimitivity is a covariant representation of the action of $G$ on $C_0(G/H)$.
Such pairs correspond to representations of the crossed product $C_0(G/H) \rtimes G$.
This crossed product is Morita equivalent to the group $C^*$-algebra of $H$, and this equivalence gives one modern proof.

Example: take $G = \R$ and $H = \set{0}$, so $G/H = \R$.
A system of imprimitivity is a unitary group $\pi(t)$ together with a self-adjoint operator $Q$, through the spectral measure $P$ of $Q$.
The covariance condition is the Weyl form of the canonical commutation relation.
The theorem says that every such pair is a multiple of translations and multiplication by $x$ on $L^2(\R)$.
This is the [[Stone–von Neumann theorem|Z877XDBMQ3]].

The theorem also gives a criterion for being induced: $\pi$ is induced from $H$ exactly when it admits a system of imprimitivity based on $G/H$.

## Explanations

### L1 human

Some representations come with a way of cutting the space into pieces labelled by points of $G/H$, and $G$ permutes the pieces.
The theorem says that this extra structure is exactly what induced representations have.
So to recognise an induced representation, one only has to find such a compatible way of cutting the space.
In quantum mechanics, the pieces are the positions of a particle, and the group moves the positions.

### L2 human

The imprimitivity theorem is the engine of the [[Mackey machine|12CEWPMSG4]].
If $N$ is a closed abelian normal subgroup, the spectral measure of $\pi|_N$ lives on the dual $\widehat{N}$, and $G$ permutes it.
When $\pi$ is irreducible and the action is regular, this measure sits on one orbit $G/G_\chi$, and the theorem shows that $\pi$ is induced from the stabiliser $G_\chi$.
Wigner's classification of the particles through representations of the Poincaré group is an instance.
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# Free monoid

## Summary

The free monoid on a set $A$ is the [[monoid|0N6R902FG1]] $A^*$ of all finite words over the alphabet $A$.
Its operation is concatenation, and its identity is the empty word.
It is free in the sense that any map from $A$ to a monoid extends in exactly one way to a monoid homomorphism from $A^*$.
Free monoids are the setting of formal languages and automata, and they are the monoid analogue of [[free groups|0KTW26C80J]].

## Explanations

### L1 human

Take an alphabet, such as the letters $a$ and $b$.
The free monoid consists of all strings you can write with them, including the empty string: $a$, $ba$, $abba$, and so on.
To multiply two strings, write one after the other, so $ab$ times $ba$ is $abba$.
No two different strings are ever identified, which is what "free" means.
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# Arithmetical hierarchy

## Definition

Work in the first-order language of arithmetic with symbols $0$, $S$, $+$, $\cdot$ and $<$, interpreted in the standard model $\N$ of the [[natural numbers|0MWKJ44V5A]].
A bounded quantifier has the form $\exists x < t$ or $\forall x < t$, where $t$ is a term not containing $x$.
The classes of [[formulas|HYJ10AF6RT]] $\Sigma_n$ and $\Pi_n$ are defined by induction on $n$.

1. $\Sigma_0 = \Pi_0$ is the class of formulas in which every quantifier is bounded.
2. A formula is $\Sigma_{n+1}$ if it has the form $\exists x_1 \cdots \exists x_k \, \psi$ with $\psi$ in $\Pi_n$.
3. A formula is $\Pi_{n+1}$ if it has the form $\forall x_1 \cdots \forall x_k \, \psi$ with $\psi$ in $\Sigma_n$.

A set $A \subseteq \N^m$ is $\Sigma^0_n$ if $A = \set{\bar a : \N \models \varphi(\bar a)}$ for some $\Sigma_n$ formula $\varphi$, and $\Pi^0_n$ if the same holds with a $\Pi_n$ formula.
A set is $\Delta^0_n$ if it is both $\Sigma^0_n$ and $\Pi^0_n$.
A set is *arithmetical* if it is $\Sigma^0_n$ for some $n$.

## Remarks

Example: the halting set $K = \set{e : \exists c \, T(e, e, c)}$ is $\Sigma^0_1$, where $T$ is [[Kleene's T predicate|S9DK13K5PS]].
Every primitive recursive relation is $\Delta^0_1$, so $T$ is defined by a $\Sigma_1$ formula, and so is $K$.
The set $\set{e : \forall x \, \exists c \, T(e, x, c)}$ of programs computing total functions is $\Pi^0_2$.

Non-example: the set of Gödel numbers of sentences true in $\N$ is not arithmetical, by [[Tarski's undefinability theorem|YRV6SYK2BY]].

A set is $\Sigma^0_n$ if and only if its complement is $\Pi^0_n$.
The inclusions $\Sigma^0_n \cup \Pi^0_n \subseteq \Delta^0_{n+1}$ hold, and each is strict.
The classes $\Sigma^0_1$ and $\Delta^0_1$ are the computably enumerable and the computable sets.
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# Random ordinary differential equation

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]].
Let $\eta = (\eta_t)_{t \ge 0}$ be a [[stochastic process|0QH5ENKZM1]] with values in $\R^m$ whose sample paths $t \mapsto \eta_t(\omega)$ are continuous.
Let $f \colon \R^n \times \R^m \to \R^n$ be continuous.
The *random ordinary differential equation* driven by $\eta$ with right side $f$ is the family of [[ordinary differential equations|0NT64JV96A]]
$$\frac{dx}{dt} = f(x, \eta_t(\omega)), \qquad \omega \in \Omega.$$
A *solution* is a process $(X_t)_{t \ge 0}$ such that for $P$-almost every $\omega$ the path $t \mapsto X_t(\omega)$ is continuously differentiable and solves the equation for that $\omega$.

## Remarks

For fixed $\omega$ the right side $(t, x) \mapsto f(x, \eta_t(\omega))$ is continuous, so the usual existence theory for ordinary differential equations applies path by path.
Example: with $n = m = 1$ and $f(x, y) = -x + y$, the process $X_t = e^{-t} X_0 + \int_0^t e^{-(t - s)} \eta_s \, ds$ is a solution; differentiating gives $X_t' = -X_t + \eta_t$.
Non-example: let $W$ be a [[Wiener process|22735FGJV6]].
The equation $dX_t = X_t \, dW_t$ is not a random ordinary differential equation as written, since the noise enters through $dW_t$ and the paths of $W$ are not differentiable.
A [[stochastic differential equation|B219G9SHJA]] with additive noise, $dX_t = a(X_t) \, dt + dW_t$, becomes a random ordinary differential equation after the change of variables $Z_t = X_t - W_t$: then $Z_t' = a(Z_t + W_t)$.
Some authors allow $f$ to depend on $t$ and $\eta$ to have only measurable paths; then solutions are taken in the sense of Carathéodory.
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# Local existence of harmonic conjugates

## Statement

Let $U \subseteq \C$ be an open set and let $u \colon U \to \R$ be a [[harmonic function|0DEB0JMN0W]].
Let $D = \set{z \in \C : \abs{z - a} < r}$ be an open disc with $D \subseteq U$.
Then there is a [[holomorphic function|0NAHXR1FRH]] $f \colon D \to \C$ with $\operatorname{Re} f = u$ on $D$.
Its imaginary part $v = \operatorname{Im} f$ is a [[harmonic conjugate|050F2FMTKF]] of $u$ on $D$, and it is unique up to an additive real constant.

## Remarks

Proof idea: write $z = x + iy$ and set $g = u_x - i u_y$.
The equation $u_{xx} + u_{yy} = 0$ and the symmetry $u_{xy} = u_{yx}$ are exactly the [[Cauchy–Riemann equations|0T71SSYZ39]] for $g$, so $g$ is holomorphic on $U$.
A holomorphic function on a disc [[has a primitive|02AC7JR6G6]] $F$, and $F' = g$ gives $\nabla(\operatorname{Re} F) = \nabla u$.
So $\operatorname{Re} F - u$ is a real constant $c$ on $D$, and $f = F - c$ works.
The same proof works on every [[simply connected domain|0QW64ARSHB]] in place of the disc.
The result is local: on $U = \C \setminus \set{0}$ the function $u(z) = \log\abs{z}$ is harmonic.
On each disc that avoids $0$ it has the conjugate $v = \arg z$ for a continuous branch of the argument.
It has no harmonic conjugate on all of $U$, because no continuous argument exists on a circle around $0$.

## Explanations

### L1 human

A harmonic function in the plane looks like the real part of a holomorphic function.
This theorem says that this is always true, at least on small discs.
So questions about harmonic functions in two variables can often be turned into questions about holomorphic functions, where many tools are available.

### L2 human

The theorem lets one transfer results from complex analysis to planar harmonic functions.
For example, the mean value property and the maximum principle for $u$ follow from those for a local holomorphic $f$ with $\operatorname{Re} f = u$.
Globally, a harmonic conjugate exists exactly when the period $\oint (-u_y \, dx + u_x \, dy)$ vanishes on each closed curve in $U$.
These periods are the obstruction on multiply connected domains, as the example $\log\abs{z}$ shows.
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# Brenier's theorem

## Statement

Let $\mu$ and $\nu$ be Borel probability measures on $\R^n$ with finite second moments, $\int \abs{x}^2 \, d\mu < \infty$ and $\int \abs{y}^2 \, d\nu < \infty$.
Suppose that $\mu$ is [[absolutely continuous|P0P21KDV6E]] with respect to Lebesgue measure.
Then:

1. There is a [[convex function|07Q7CNM5WF]] $\varphi \colon \R^n \to \R \cup \set{+\infty}$, differentiable at $\mu$-almost every point, such that the [[pushforward|3N66TM3TSQ]] of $\mu$ by its [[gradient|0ABK9V9Y64]] is $\nu$: $(\nabla \varphi)_\# \mu = \nu$.
2. The [[coupling|HTR7J0W66V]] $\pi = (\id, \nabla \varphi)_\# \mu$ is the unique coupling of $\mu$ and $\nu$ that minimizes $\int \abs{x - y}^2 \, d\pi(x, y)$.
3. The map $\nabla \varphi$ is unique: if $\psi$ is another convex function with $(\nabla \psi)_\# \mu = \nu$, then $\nabla \psi = \nabla \varphi$ $\mu$-almost everywhere.

## Remarks

Yann Brenier proved the theorem in work on the polar factorization of vector fields, published in 1987 and 1991.
The proof uses Kantorovich duality: since $\abs{x - y}^2 = \abs{x}^2 + \abs{y}^2 - 2 \ip{x}{y}$, an optimal plan maximizes $\int \ip{x}{y} \, d\pi$, and its support lies in the subdifferential of a convex function $\varphi$.
A convex function is differentiable outside a Lebesgue null set.
So absolute continuity of $\mu$ makes the subdifferential single-valued $\mu$-almost everywhere, and the plan is carried by the graph of $\nabla \varphi$.
Example: if $\nu$ is the image of $\mu$ under $x \mapsto A x + b$ with $A$ symmetric positive definite, then $\varphi(x) = \frac12 \ip{Ax}{x} + \ip{b}{x}$ works, and the affine map is optimal.
Absolute continuity cannot be dropped: if $\mu = \delta_0$ and $\nu$ is not a Dirac mass, no map pushes $\mu$ to $\nu$.
When $\mu$ and $\nu$ have densities $f$ and $g$, the potential $\varphi$ solves the [[Monge–Ampère equation|14AKAC0S0T]] $g(\nabla \varphi) \det D^2 \varphi = f$ in a weak sense.

## Explanations

### L1 human

To move one distribution of mass in space onto another at the least total squared distance, there is a best plan, and it moves each point to a single destination.
Brenier's theorem says that this best map is the gradient of a convex function, the higher-dimensional version of an increasing function.
In one dimension it is the map that sends the left part of one distribution to the left part of the other, in order.

### L2 human

The theorem solves the [[Monge problem|K8EBASR5VG]] for the quadratic cost.
It gives the polar factorization: a nondegenerate map $u$ pushing $\mu$ to $\nu$ factors as $u = \nabla \varphi \circ s$, where $\varphi$ is convex and $s$ preserves $\mu$.
It is the basis of the Wasserstein geometry of probability measures, of displacement convexity, and of regularity theory for the Monge–Ampère equation through Caffarelli's work.
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# Kelvin problem

## Problem

Among all partitions of $\R^3$ into cells of equal volume, find one whose interfaces have the least [[surface area|0S29Y4ZW0Y]] per cell.

To make the question precise, fix the cell volume to $1$.
Then ask for the least value of the quantity
$$\limsup_{R \to \infty} \frac{\mathcal{H}^2(\Sigma \cap B_R)}{\abs{B_R}}.$$
Here $\Sigma$ is the union of the cell boundaries, $\mathcal{H}^2$ is area, and $B_R$ is the ball of radius $R$ about the origin.

## Remarks

Kelvin proposed as the answer a partition into congruent cells shaped like truncated octahedra, arranged as in the bitruncated cubic [[honeycomb|HTJ03RFKFH]].
His cells are not exactly polyhedra: their hexagonal faces are slightly curved, so that they satisfy [[Plateau's laws|381RB634WS]].
In 1993 Denis Weaire and Robert Phelan found a partition into two kinds of cells of equal volume with about $0.3\%$ less area than Kelvin's.
So Kelvin's proposal is not optimal.
The problem is open: no partition is known to be optimal, and it is not known whether the Weaire–Phelan structure is.
The two-dimensional analogue is solved: by the [[honeycomb theorem|3JST9B967R]] of Thomas Hales, the regular hexagonal tiling is optimal in the plane.

## Explanations

### L1 human

How should one divide space into bubbles of the same size, so that the soap film needed is as small as possible?
Kelvin suggested a pattern of identical fourteen-faced cells.
A century later, a better foam was found with two kinds of cells, and nobody yet knows the best one.
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# Inverse spectral problem

## Problem

Let $q$ be a real potential on an interval, and consider the [[Schrödinger operator|89AA6XJY5K]] $-y'' + q y$ with given boundary conditions.
Determine which spectral data of this operator determine $q$ uniquely, and reconstruct $q$ from such data.

## Remarks

The direct problem computes the eigenvalues from $q$; the inverse problem goes the other way.
Ambartsumian showed in 1929 that for continuous $q$ on $[0, \pi]$ with the conditions $y'(0) = y'(\pi) = 0$, if the eigenvalues are exactly $0, 1, 4, 9, \ldots$, then $q = 0$.
In general one spectrum does not determine $q$: for example, $q(x)$ and $q(\pi - x)$ on $[0, \pi]$ with Dirichlet conditions at both ends have the same eigenvalues.
Göran Borg proved that two spectra, for two different boundary conditions at one end and the same condition at the other, determine $q$.
On a half-line, the spectral measure, or equivalently the Weyl–Titchmarsh function, determines $q$; the [[Gelfand–Levitan|R4MRNP9EQT]] method reconstructs it.

## Explanations

### L1 human

If you hear all the notes a string can play, can you tell how its thickness varies along its length?
This is the inverse spectral problem for a string.
The answer is no from the notes alone, because a string and its mirror image sound the same, but yes with a little more information.
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<document source="https://tamnd.github.io/math2/node/CW1TJDGCKA.md" tag="CW1TJDGCKA" kind="definition">
# Circumsphere

## Definition

Let $n \ge 1$ and let $\Delta$ be an $n$-[[simplex|YX1DJK2QAG]] in $\R^N$ with vertices $v_0, \ldots, v_n$.
Let $S$ be the [[affine span|0FFAE5P6R4]] of the vertices.
There is exactly one point $o \in S$ with $\norm{o - v_0} = \norm{o - v_1} = \cdots = \norm{o - v_n}$.
This point is the *circumcentre* of $\Delta$, and this common distance $r$ is its *circumradius*.
The *circumsphere* of $\Delta$ is the [[sphere|P9CM4GQ2T0]] $\set{x \in S : \norm{x - o} = r}$ in $S$.

## Remarks

The point $o$ exists and is unique because the conditions $\norm{o - v_i}^2 = \norm{o - v_0}^2$ for $i = 1, \ldots, n$ are $n$ affine equations in $S$.
Their linear parts $v_i - v_0$ are linearly independent, by affine independence of the vertices.

Example: the triangle with vertices $(0, 0)$, $(2, 0)$ and $(0, 2)$ has circumcentre $(1, 1)$ and circumradius $\sqrt{2}$.
Each vertex is at distance $\sqrt{1 + 1} = \sqrt{2}$ from $(1, 1)$.
For $n = 2$ this is the [[circumcircle|03VZJE6C8C]].

Non-example: for four points $(0, 0)$, $(2, 0)$, $(0, 2)$ and $(3, 3)$ in the plane, there is no circle through all of them.
The only circle through the first three has centre $(1, 1)$, and $(3, 3)$ is at distance $2\sqrt{2}$ from it.
The definition needs the vertices of a simplex, which are affinely independent.

Without the condition $o \in S$, the centre would not be unique when $n < N$: every point of the affine subspace through $o$ orthogonal to $S$ is equidistant from the vertices.
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<document source="https://tamnd.github.io/math2/node/CW5CEF84T2.md" tag="CW5CEF84T2" kind="definition">
# Sigma-finite measure

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
The [[measure|PR11BZQQPG]] $\mu$ is *σ-finite* if there are sets $A_1, A_2, \ldots$ in $\Sigma$ such that $X = \bigcup_{n=1}^{\infty} A_n$ and $\mu(A_n) < \infty$ for every $n$.

## Remarks

Lebesgue measure on $\R$ is σ-finite: take $A_n = [-n, n]$, of measure $2n$.
Counting measure on the power set of $\R$ is not σ-finite.
A set of finite counting measure is finite, a [[countable union|01TQB88H4C]] of finite sets is countable, and $\R$ is uncountable.
Every [[finite measure|63Z1Q4FS23]] is σ-finite, with $A_n = X$.
The sets $A_n$ can be chosen increasing, or pairwise disjoint, by the usual rearrangements.
</document>

<document source="https://tamnd.github.io/math2/node/CWDF4KGHS3.md" tag="CWDF4KGHS3" kind="concept">
# Principal series representation

## Summary

A principal series representation of a real reductive group is induced from a minimal parabolic subgroup.
The inducing data are a character of its split part and an irreducible representation of its compact part.
For $\mathrm{SL}_2(\R)$, it acts on homogeneous functions on the plane, or equivalently on functions on the circle.
The unitary principal series supply the continuous part of the Plancherel decomposition of $L^2(G)$.
By a theorem of Casselman, every irreducible admissible representation embeds into some principal series, which makes these representations a universal source.

## Explanations

### L1 human

Pick a large subgroup $P$ of $G$, such as the upper triangular matrices in $\mathrm{SL}_2(\R)$.
A character of $P$ is a simple one-dimensional rule.
Inducing spreads this rule over the whole group: the space consists of functions on $G$ that transform under $P$ by the chosen character.
The resulting family depends on a complex parameter $\nu$, and almost all its members are irreducible.
</document>

<document source="https://tamnd.github.io/math2/node/CWKNRTVERF.md" tag="CWKNRTVERF" kind="definition">
# Karoubi envelope

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
An endomorphism $e \colon A \to A$ is *idempotent* if $e \circ e = e$.
The *Karoubi envelope* $\mathrm{Kar}(\mathcal{C})$ is the category with these parts.

1. Its objects are pairs $(A, e)$ with $A$ an object of $\mathcal{C}$ and $e$ an idempotent endomorphism of $A$.
2. A morphism $f \colon (A, e) \to (B, e')$ is a morphism $f \colon A \to B$ of $\mathcal{C}$ with $e' \circ f \circ e = f$.
3. Composition is that of $\mathcal{C}$, and the identity of $(A, e)$ is $e$.

## Remarks

The condition in clause 2 is equivalent to $f \circ e = f$ and $e' \circ f = f$.
The functor $\mathcal{C} \to \mathrm{Kar}(\mathcal{C})$ with $A \mapsto (A, \id_A)$ is fully faithful.
Every idempotent of $\mathrm{Kar}(\mathcal{C})$ splits.
For example, an idempotent $e$ of $(A, \id_A)$ is the composite of the [[retraction|DXJH3DGXBC]] $e \colon (A, \id_A) \to (A, e)$ and the section $e \colon (A, e) \to (A, \id_A)$.
Other names are idempotent completion and Cauchy completion.

Example: let $\mathcal{C}$ have one object $*$ and morphisms $1$ and $e$, with $e \circ e = e$.
Then $\mathrm{Kar}(\mathcal{C})$ has the objects $(*, 1)$ and $(*, e)$.
The morphisms $(*, e) \to (*, e)$ are the $f$ with $e f e = f$, and only $f = e$ qualifies, so $(*, e)$ has only its identity $e$.

Non-example: with the same $\mathcal{C}$, the morphism $1 \colon (*, e) \to (*, e)$ is not a morphism of $\mathrm{Kar}(\mathcal{C})$.
Indeed $e \circ 1 \circ e = e \ne 1$, so clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/CWP26Y1TCZ.md" tag="CWP26Y1TCZ" kind="definition">
# Double series

## Definition

Let $(a_{m,n})_{m, n \in \N}$ be a [[double sequence|WK5V372TG3]] of [[complex numbers|09FTAET5Z4]].
The double series $\sum_{m, n = 0}^{\infty} a_{m,n}$ is the double sequence of rectangular partial sums
$$S_{M,N} = \sum_{m=0}^{M} \sum_{n=0}^{N} a_{m,n}.$$
The double series converges to $s \in \C$ if $(S_{M,N})$ converges to $s$ in the sense of [[Pringsheim|QYTGCW1GTK]].
That is, for every $\varepsilon > 0$ there is $K$ with $\abs{S_{M,N} - s} < \varepsilon$ for all $M, N \ge K$.

## Remarks

Example: for $a_{m,n} = 2^{-m} 3^{-n}$ one has $S_{M,N} = \left(\sum_{m \le M} 2^{-m}\right)\left(\sum_{n \le N} 3^{-n}\right)$.
The two factors tend to $2$ and $\tfrac32$, so the double series converges to $3$.
Non-example: for $a_{m,n} = 1$ when $m = n$ and $0$ otherwise, $S_{M,N} = \min(M, N) + 1$, which is unbounded, so the double series diverges.
A convergent double series can have unbounded terms.
With $a_{0,n} = n$, $a_{1,n} = -n$ and $a_{m,n} = 0$ for $m \ge 2$, one has $S_{M,N} = 0$ for all $M \ge 1$.
So it converges to $0$, although its rows do not form convergent series.

## Explanations

### L1 human

A double series is the sum of all entries of an infinite table, taken as the limit of sums over larger and larger rectangles.
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<document source="https://tamnd.github.io/math2/node/CWQV9PTCB3.md" tag="CWQV9PTCB3" kind="definition">
# Model

## Definition

Let $L$ be a signature and let $T$ be an $L$-[[theory|92SWXST1ZM]].
An $L$-[[structure|983GZBQYT4]] $\mathcal{M}$ is a model of $T$ when $\mathcal{M} \models \varphi$ for every [[sentence|DKFGQTRYZE]] $\varphi$ in $T$.
Here $\mathcal{M} \models \varphi$ means that $\varphi$ is true in $\mathcal{M}$ under the usual inductive definition of truth.
The notation is $\mathcal{M} \models T$.

## Remarks

Example: let $T$ be the theory of groups in the signature $\set{\cdot, e, {}^{-1}}$.
The structure $(\Z, +, 0, -)$, with $\cdot$ read as $+$, $e$ as $0$ and ${}^{-1}$ as $-$, is a model of $T$.
Indeed addition of integers is associative, $0$ is neutral and $-x$ is an inverse of $x$.

Non-example: the structure $(\N, +, 0, \id)$, with ${}^{-1}$ read as the identity map, is not a model of $T$.
The sentence $\forall x \, (x \cdot x^{-1} = e)$ fails at $x = 1$, because $1 + 1 \ne 0$.

A theory is satisfiable when it has a model.
The class of all models of $T$ is often written $\mathrm{Mod}(T)$.

## Explanations

### L1 human

To check that a structure is a model of a theory, you check each sentence of the theory in the structure.
Every group is a model of the group axioms, and every model of the group axioms is a group.
</document>

<document source="https://tamnd.github.io/math2/node/CX07CWPCDX.md" tag="CX07CWPCDX" kind="concept">
# Partial linear space

## Summary

A partial linear space is an [[incidence structure|FGKNAFNHA3]] of points and lines with two rules.
Two distinct points lie on at most one line, and every line has at least two points.
It is the common ground of most point-line geometries: projective and affine planes, [[generalized polygons|PCT2HY9AAZ]] and [[polar spaces|Z7T0V2VAWJ]] are all partial linear spaces.
When any two points lie on exactly one line, it is called a linear space.

## Explanations

### L1 human

In ordinary geometry, two points determine a line.
A partial linear space keeps only half of this rule: two points determine at most one line, and some pairs of points may lie on no common line.
For example, take the nine points of a $3 \times 3$ grid and use only the three rows and three columns as lines.
Two points in the same row or column share exactly one line, and two points in different rows and columns share none.
</document>

<document source="https://tamnd.github.io/math2/node/CX4M8K61AB.md" tag="CX4M8K61AB" kind="concept">
# Fiber bundle

## Summary

A fiber bundle is a space that looks locally like a product of a base space and a fixed fiber, but may be twisted globally.
It comes with a projection to the base whose preimages, the fibers, are all copies of the same space.
Vector bundles, covering spaces, principal bundles and the Hopf fibration are all fiber bundles.
They describe families of spaces parametrized by a base, and they are central in topology, geometry and physics.

## Explanations

### L1 human

Take a strip of paper: it is a product of a long segment and a short one.
Give it a half twist before gluing the ends and you get a Möbius band.
Over each small piece of the central circle the band still looks like a product of an arc and a segment, but as a whole it is twisted.
The Möbius band is a fiber bundle over the circle with fiber a segment, and the plain cylinder is the untwisted one.
</document>

<document source="https://tamnd.github.io/math2/node/CX7BDQ47ZK.md" tag="CX7BDQ47ZK" kind="concept">
# Grothendieck's Galois theory

## Summary

Grothendieck's Galois theory recasts [[Galois theory|WNQDKQYZ2R]] as an equivalence of categories.
For a field $k$, it matches finite étale $k$-algebras with finite sets on which the [[absolute Galois group|PGKWBJDTKX]] of $k$ acts continuously.
The same pattern, with Galois categories and fibre functors, defines the étale fundamental group of a scheme.
It thus puts Galois groups of fields and fundamental groups of spaces in one frame.

## Explanations

### L1 human

Classical Galois theory matches intermediate fields with subgroups of a Galois group.
Grothendieck's view looks instead at all finite separable algebras at once, such as products of fields.
Each such algebra is replaced by the finite set of its embeddings into a separable closure, and the Galois group permutes this set.
For $k = \R$, the algebra $\C$ gives a set of two points swapped by complex conjugation, while $\R \times \R$ gives two points that stay fixed.
The picture is the same as for covering spaces, where the fundamental group permutes the points of a fibre.
</document>

<document source="https://tamnd.github.io/math2/node/CXES7915R7.md" tag="CXES7915R7" kind="statement">
# Non-vanishing of the zeta function on the line Re s = 1

## Statement

Let $\zeta$ be the [[Riemann zeta function|1XWBY4N0VT]].
For every complex number $s \ne 1$ with $\operatorname{Re} s \ge 1$, one has $\zeta(s) \ne 0$.
In particular $\zeta(1 + it) \ne 0$ for every real $t \ne 0$.

## Remarks

For $\operatorname{Re} s > 1$ this follows from the [[Euler product formula|0C3PN4075R]], since a convergent product of nonzero factors is nonzero.
For the line $\operatorname{Re} s = 1$, the classical argument uses the inequality $3 + 4\cos\theta + \cos 2\theta = 2(1 + \cos\theta)^2 \ge 0$.
With the Euler product it gives $\abs{\zeta(\sigma)^3 \, \zeta(\sigma + it)^4 \, \zeta(\sigma + 2it)} \ge 1$ for $\sigma > 1$.
If $\zeta(1 + it) = 0$ with $t \ne 0$, then as $\sigma \to 1^+$ the left side behaves like $(\sigma - 1)^{-3} (\sigma - 1)^{4} \to 0$, because $\zeta$ has a simple pole at $1$ and $\zeta(1 + 2it)$ is finite; this is a contradiction.

## Explanations

### L1 human

The zeros of the zeta function control how regularly the primes are spread.
This theorem says that no zero lies on the line $\operatorname{Re} s = 1$, the edge of the region where the series converges.
It is the key analytic input in the proof of the prime number theorem.

### L2 human

The [[prime number theorem|0EHBY5Y84N]] $\pi(x) \sim x / \log x$ follows from this non-vanishing by a Tauberian theorem, such as the Wiener–Ikehara theorem, applied to $-\zeta'(s)/\zeta(s) = \sum_n \Lambda(n) n^{-s}$.
Hadamard and de la Vallée Poussin proved the prime number theorem this way in 1896.
Quantitative zero-free regions, such as $\sigma > 1 - c/\log(\abs{t} + 2)$, give the prime number theorem with an error term.
The same method shows $L(1 + it, \chi) \ne 0$ for [[Dirichlet L-functions|9D2Z543EJQ]].
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<document source="https://tamnd.github.io/math2/node/CXG6JZGMCK.md" tag="CXG6JZGMCK" kind="concept">
# Unimodular group

## Summary

A unimodular group is a [[locally compact group|AHV05ZBCYW]] whose left [[Haar measure|VY0HXD3GC4]] is also right-invariant, that is, whose [[modular function|63TX60RTRQ]] is identically $1$.
Abelian groups, compact groups, discrete groups, semisimple Lie groups and connected nilpotent Lie groups are unimodular.
Many results of harmonic analysis, such as the existence of lattices or the simplest form of trace formulas, require unimodularity.

## Explanations

### L1 human

In some groups, measuring sizes in a way that ignores left shifts automatically ignores right shifts as well.
These groups are called unimodular.
Commutative groups are the obvious example, because left and right shifts agree.
The group of affine maps $x \mapsto ax + b$ with $a > 0$ is the standard group that is not unimodular: its left and right Haar measures are $da \, db / a^2$ and $da \, db / a$.
</document>

<document source="https://tamnd.github.io/math2/node/CXV1969XGB.md" tag="CXV1969XGB" kind="definition">
# Topological ring

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $\mathcal{T}$ be a [[topology|0CNQXVBHCX]] on the set $R$.
Give $R \times R$ the [[product topology|0Z57EC6YX3]].
The pair $(R, \mathcal{T})$ is a *topological ring* if the following maps are [[continuous|0SY9ECQ19Q]]:

1. addition $R \times R \to R$, $(a, b) \mapsto a + b$;
2. negation $R \to R$, $a \mapsto -a$;
3. multiplication $R \times R \to R$, $(a, b) \mapsto ab$.

## Remarks

Clauses 1 and 2 say that $(R, +)$ is a topological group.
Example: $\R$ with the usual topology; continuity of $(a, b) \mapsto ab$ follows from $ab - a'b' = a(b - b') + (a - a')b'$.
Example: any ring with the discrete topology, since every map from a discrete space is continuous.
Example: for an ideal $I$ of a commutative ring $R$, the $I$-adic topology, with the ideals $I^n$ as a base of neighbourhoods of $0$.
Non-example: $\R$ with the lower limit topology, whose base is the intervals $[a, b)$.
Negation is not continuous: the preimage of the open set $[0, 1)$ is $(-1, 0]$, which is not open there.
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<document source="https://tamnd.github.io/math2/node/CXXM6ZYXDX.md" tag="CXXM6ZYXDX" kind="definition">
# Homoclinic orbit

## Definition

Let $U$ be an open subset of $\R^n$, let $f \colon U \to \R^n$ be locally Lipschitz, and consider the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.
Let $x_0 \in U$ be an [[equilibrium point|0G4X36AE0X]].
A solution $\gamma \colon \R \to U$ is homoclinic to $x_0$ if it is not constant and
$$\lim_{t \to -\infty} \gamma(t) = x_0 = \lim_{t \to +\infty} \gamma(t).$$
A homoclinic orbit is the [[trajectory|07TZ2Q4C1M]] $\gamma(\R)$ of such a solution.

## Remarks

A homoclinic orbit lies in the intersection of the stable and the unstable set of $x_0$, and it does not contain $x_0$.
Example: for $x' = y$, $y' = x - x^2$, the function $\gamma(t) = (x(t), x'(t))$ with $x(t) = \frac{3}{2} \operatorname{sech}^2(t/2)$ is a solution.
Indeed $x'' = \frac{3}{2} s - \frac{9}{4} s^2 = x - x^2$ with $s = \operatorname{sech}^2(t/2)$, and $x(t) \to 0$, $x'(t) \to 0$ as $t \to \pm\infty$, so $\gamma$ is homoclinic to the origin.
Its trajectory is the part of the curve $y^2 = x^2 - \frac{2}{3} x^3$ with $x > 0$.
Non-example: for the pendulum $x' = y$, $y' = -\sin x$, the solution with $x(t) \to -\pi$ as $t \to -\infty$ and $x(t) \to \pi$ as $t \to +\infty$ joins two different equilibria in $\R^2$.
So it is a heteroclinic orbit, not a homoclinic one.
A periodic orbit is also not homoclinic, since it does not tend to an equilibrium.

## Explanations

### L1 human

A homoclinic orbit is a path that leaves an equilibrium and comes back to the same one, taking infinite time in both directions.
In the plane it looks like a loop pinned at a saddle point.
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<document source="https://tamnd.github.io/math2/node/CXZZJ1QV5K.md" tag="CXZZJ1QV5K" kind="definition">
# Alexandrov topology

## Definition

Let $X$ be a set and let $\tau$ be a [[topology|0CQHFPGJ45]] on $X$.
The topology $\tau$ is an *Alexandrov topology* if for every family $(U_i)_{i \in I}$ of [[open sets|02BH3MCBQB]] of $\tau$, the intersection $\bigcap_{i \in I} U_i$ is open.
A topological space whose topology is an Alexandrov topology is called an *Alexandrov space* or an Alexandrov-discrete space.

## Remarks

Equivalently, every point $x \in X$ has a smallest open set $U_x$ containing it, namely the intersection of all open sets that contain $x$.

For a [[preorder|0VABYWJWH9]] $\le$ on $X$, the family of all [[upper sets|ETCE287E2R]] of $(X, \le)$ is an Alexandrov topology.
Conversely, every Alexandrov topology arises in this way from exactly one preorder, its [[specialization preorder|CMYVK9CEKZ]].

Example: every topology on a finite set is an Alexandrov topology, because then every family of open sets is finite.

Non-example: the usual topology of $\R$ is not an Alexandrov topology.
The open sets $(-1/n, 1/n)$ for $n \ge 1$ have intersection $\set{0}$, which is not open.

The name "Alexandrov space" is also used in metric geometry for spaces with curvature bounded below; that notion is unrelated.
</document>

<document source="https://tamnd.github.io/math2/node/CY1TXHF6KQ.md" tag="CY1TXHF6KQ" kind="concept">
# Definable set

## Summary

A definable set in a [[structure|983GZBQYT4]] is a set of tuples of elements that is cut out by a first-order formula, possibly with parameters.
Definable sets are the main objects of study in modern model theory.
Their behaviour, such as in o-minimal or stable structures, describes how complicated a structure is.

## Explanations

### L1 human

A definable set is a set that you can describe by a first-order formula in the language of the structure.
In the field of real numbers, the set of nonnegative numbers is definable by the formula "$x$ is a square", that is, $\exists y \, (y \cdot y = x)$.
The unit circle is definable by $x \cdot x + y \cdot y = 1$.
In the ordered set $(\R, <)$ without parameters, no single number can be defined, because a shift $x \mapsto x + c$ preserves the order.
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<document source="https://tamnd.github.io/math2/node/CYAWQF7AJZ.md" tag="CYAWQF7AJZ" kind="definition">
# Stein space

## Definition

Let $(X, \mathcal{O}_X)$ be a [[complex analytic space|HXSZA44J5J]] whose underlying topological space is [[Hausdorff|0S1FJ8VEX9]] and [[second countable|03PPNCM9GG]].
Let $\mathcal{O}(X)$ be the algebra of global holomorphic functions on $X$, and let $f(x) \in \C$ denote the value of $f \in \mathcal{O}(X)$ at $x \in X$.
For a [[compact set|0R6EWFKRCF]] $K \subseteq X$, its [[holomorphically convex hull|CD6SMMZSW8]] is $\widehat{K} = \set{x \in X : \abs{f(x)} \le \sup_K \abs{f} \text{ for all } f \in \mathcal{O}(X)}$.
The space $X$ is a *Stein space* if:

1. $X$ is holomorphically convex: $\widehat{K}$ is compact for every compact $K \subseteq X$.
2. $X$ is holomorphically separable: for all $x \ne y$ in $X$ there is $f \in \mathcal{O}(X)$ with $f(x) \ne f(y)$.

## Remarks

A Stein space whose underlying space is a complex manifold is a Stein manifold.
Example: every closed complex subspace $X$ of $\C^N$ is Stein.
The coordinate functions restricted to $X$ separate points, and $\widehat{K}$ is a closed subset of $X$ contained in a closed ball of $\C^N$, hence compact.
Non-example: the complex projective line $\mathbb{P}^1$ is a complex space whose global holomorphic functions are constant, so clause 2 fails.
An equivalent form: $X$ is Stein if and only if $H^q(X, \mathcal{F}) = 0$ for every coherent analytic sheaf $\mathcal{F}$ on $X$ and every $q \ge 1$.
The direction from Stein to vanishing is Cartan's theorem B.
</document>

<document source="https://tamnd.github.io/math2/node/CYCJ46W6P6.md" tag="CYCJ46W6P6" kind="definition">
# Transcendence basis

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
A subset $B \subseteq L$ is a *transcendence basis* of $L/K$ if:

1. $B$ is [[algebraically independent|KV4PT6KW10]] over $K$;
2. $L$ is an [[algebraic extension|09EYMKMSB9]] of the subfield $K(B)$ generated by $K$ and $B$.

## Remarks

Equivalently, $B$ is a maximal algebraically independent subset of $L$.
Example: in the [[field of rational functions|4V81611P9N]] $K(x_1, \dots, x_n)$, the set $\set{x_1, \dots, x_n}$ is a transcendence basis over $K$, and so is $\set{x_1^2, x_2, \dots, x_n}$.
Non-example: $\set{x_1}$ is not a transcendence basis of $K(x_1, x_2)/K$, because $x_2$ is not algebraic over $K(x_1)$, so clause 2 fails.
For an algebraic extension, the empty set is the unique transcendence basis.
</document>

<document source="https://tamnd.github.io/math2/node/CYCTQ0CR3K.md" tag="CYCTQ0CR3K" kind="concept">
# Clean ring

## Summary

A clean ring is a ring in which every element is the sum of an [[idempotent|0N5NJZKKN3]] and a [[unit|0XVEZDE9J8]].
The notion was introduced by W. K. Nicholson in his study of exchange rings, and every clean ring is an exchange ring.
Local rings, matrix rings over local rings and [[unit-regular rings|H4K920SVYT]] are clean.
Cleanness gives a supply of idempotents and units that helps in decomposing modules.

## Explanations

### L1 human

In a field every element is either $0$ or invertible, and $0 = 1 + (-1)$ is an idempotent plus a unit.
A clean ring generalizes this: every element can be split as "a projection plus something invertible".
In the integers this fails: the idempotents are $0$ and $1$ and the units are $\pm 1$, so the only sums are $-1, 0, 1, 2$, and $3$ is not among them.
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<document source="https://tamnd.github.io/math2/node/CYEMGH98WN.md" tag="CYEMGH98WN" kind="concept">
# Pointless topology

## Summary

Pointless topology studies spaces through their lattices of open sets instead of through their points.
Its basic objects are [[frames|306PWGH7BF]], complete lattices in which finite meets distribute over arbitrary joins, and [[locales|G2QNFT210W]], which are frames viewed as generalized spaces.
Every topological space gives a locale, its frame of open sets, and for [[sober spaces|4N4DNDR22Z]] no information is lost.
Some theorems that need the axiom of choice for spaces, such as the product theorem for compact spaces, hold for locales without it.

## Explanations

### L1 human

In ordinary topology a space is a set of points together with a choice of open sets.
Pointless topology forgets the points and keeps only the open sets, with their order by inclusion, their unions and their finite intersections.
The open sets of the real line, for example, form such a structure, and the points can be recovered from it.
Working without points makes some constructions simpler and more constructive.
It also allows "spaces" that have open sets but no points at all.
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# Grassmannian

## Description

Let $F$ be a field and let $0 \le r \le n$ be integers.
The *Grassmannian* $\operatorname{Gr}(r, n)$ is the set of $r$-dimensional [[linear subspaces|00YC1D652K]] of $F^n$.

The Plücker embedding makes it a subset of a [[projective space|ZT9A6JK8QF]].
Let $\bigwedge^r F^n$ be the $r$-th exterior power, a part of the [[exterior algebra|0MDMZ965VE]], of dimension $\binom{n}{r}$.
A subspace $W$ with basis $w_1, \dots, w_r$ is sent to
$$W \mapsto [w_1 \wedge \cdots \wedge w_r] \in \mathbb{P}\left(\bigwedge\nolimits^r F^n\right) \cong \mathbb{P}^{\binom{n}{r} - 1}.$$
A change of basis multiplies $w_1 \wedge \cdots \wedge w_r$ by the determinant of the change of basis matrix, so the point is well defined.
The map is injective, since $W = \set{v \in F^n : v \wedge w_1 \wedge \cdots \wedge w_r = 0}$.
The coordinates of the image are the Plücker coordinates $p_I$, the $r \times r$ minors of the $r \times n$ matrix with rows $w_1, \dots, w_r$.

## Remarks

When $F$ is algebraically closed, the image is the zero set of quadratic equations, the Plücker relations, and it is a [[projective variety|A4H4CZ2V35]] of [[dimension|36X2WTHYCB]] $r(n - r)$.

For $r = 1$, $\operatorname{Gr}(1, n) = \mathbb{P}^{n-1}$.
For $r = 2$ and $n = 4$, the Grassmannian of lines in $\mathbb{P}^3$ is the quadric hypersurface
$$p_{12} p_{34} - p_{13} p_{24} + p_{14} p_{23} = 0$$
in $\mathbb{P}^5$, of dimension $4 = 2 \cdot (4 - 2)$.

Over $\R$ or $\C$, Grassmannians are also compact manifolds, and infinite Grassmannians are [[classifying spaces for vector bundles|RF8WD9TSE7]].

## Explanations

### L1 human

The Grassmannian is a space whose points are planes, or more generally subspaces of a fixed dimension, inside a bigger space.
The lines through the origin of $F^n$ form projective space, so the Grassmannian generalises projective space.
To give it coordinates, take a basis of the subspace, write it as the rows of a matrix, and record all the maximal minors of that matrix.
Changing the basis multiplies all minors by the same number, so the minors give a well-defined point of projective space.
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# Pólya's theorem on integer-valued entire functions

## Statement

Let $f$ be an [[entire function|0419794R0T]] such that $f(n) \in \Z$ for every integer $n \ge 0$.
Let $M(r)$ be the maximum of $\abs{f(z)}$ on the circle $\abs{z} = r$.
If
$$\limsup_{r \to \infty} \frac{\log M(r)}{r} < \log 2,$$
then $f$ is a polynomial.

## Remarks

The idea of the proof uses the Newton series $f(z) = \sum_{k \ge 0} \Delta^k f(0) \binom{z}{k}$, where $\Delta g(z) = g(z+1) - g(z)$ and $\binom{z}{k}$ is the [[binomial coefficient|092GEF38YD]] polynomial.
The differences $\Delta^k f(0) = \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} f(j)$ are integers.
A contour integral estimate shows that the growth bound forces $\Delta^k f(0) \to 0$, so $\Delta^k f(0) = 0$ for all large $k$, and the series is a polynomial.
The constant $\log 2$ is sharp: $f(z) = 2^z = e^{z \log 2}$ takes integer values at the nonnegative integers, is not a polynomial, and has $\log M(r) = r \log 2$.

## Explanations

### L1 human

Polynomials such as $z(z-1)/2$ take integer values at all nonnegative integers.
So does $2^z$, which grows exponentially.
Pólya's theorem says that any entire function with integer values at $0, 1, 2, \dots$ that grows more slowly than $2^z$ must be a polynomial.
So the growth of $2^z$ is the smallest possible for such a function that is not a polynomial.

### L2 human

The condition says that $f$ is of [[exponential type|9P8D8YAHQS]] less than $\log 2$.
The polynomials that the theorem produces are integer-valued polynomials, which are exactly the integer combinations of the polynomials $\binom{z}{k}$.
The theorem started a series of arithmetic results on entire functions.
In these results, integrality conditions on values combined with growth bounds force algebraic behaviour.
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# Newton polytope

## Summary

The Newton polytope of a polynomial in several variables is the [[convex hull|0FQSDC2EQ2]] of the exponent vectors of its monomials with nonzero coefficient.
It records which monomials can dominate, much as the [[Newton polygon|H7QNCE2QZJ]] of a polynomial over a valued field does in one variable.
The Newton polytope of a product is the Minkowski sum of the Newton polytopes of the factors.
Mixed volumes of Newton polytopes count the solutions of generic polynomial systems by the [[Bernstein–Kushnirenko theorem|VP32769PVR]].
Newton polytopes link polynomials with [[toric varieties|YWB981MEF2]] and [[tropical geometry|7YTC3BHK60]].

## Explanations

### L1 human

Write a polynomial in $x$ and $y$ and mark, for each term $c\,x^a y^b$ with $c \ne 0$, the point $(a, b)$ in the plane.
The smallest convex region containing all the marked points is the Newton polytope.
For $1 + x + y + xy$ it is the unit square, and for $1 + x^2 + y^3$ it is a triangle.
Its shape predicts much about the polynomial, for example how many common zeros two generic polynomials with these shapes have.
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# Gauss–Newton algorithm

## Statement

Let $D \subseteq \R^n$ be open, let $m \ge n$, and let $r \colon D \to \R^m$ be continuously differentiable with [[Jacobian matrix|047NDTEPCV]] $J(x)$.
The aim is to minimize $S(x) = \tfrac12 \norm{r(x)}_2^2$.
The *Gauss–Newton algorithm* starts from $x_0 \in D$ and sets
$$x_{k+1} = x_k - \bigl( J(x_k)^T J(x_k) \bigr)^{-1} J(x_k)^T r(x_k),$$
as long as $J(x_k)$ has full column rank.
Suppose that $x^* \in D$ satisfies $r(x^*) = 0$, that $J(x^*)$ has full column rank, and that $J$ is [[Lipschitz continuous|0QNVQ4VD9T]] near $x^*$.
Then there are $\delta > 0$ and $C > 0$ such that, for every $x_0$ with $\norm{x_0 - x^*} < \delta$, the iterates are defined, converge to $x^*$, and satisfy
$$\norm{x_{k+1} - x^*} \le C \norm{x_k - x^*}^2.$$

## Remarks

The step $p_k = x_{k+1} - x_k$ is the [[least squares|0BYTQ4X1HC]] solution of the linearized problem $\min_p \norm{r(x_k) + J(x_k) p}_2$.
In practice one computes it by a QR factorization of $J(x_k)$ rather than by forming $J^T J$.
The Hessian of $S$ is $J^T J + \sum_i r_i \nabla^2 r_i$.
Gauss–Newton drops the second term, so near a zero-residual solution it behaves like [[Newton's method|07EXVNF9VP]], which gives the quadratic rate.

If $r(x^*) \ne 0$, the dropped term need not vanish, and the convergence is then in general only linear.
For large residuals or strongly nonlinear $r_i$, the iteration may diverge even from good starting points.
A line search or the [[Levenberg–Marquardt algorithm|TH54SPVN1N]] makes the method robust.
Example: fitting $y = e^{c t}$ to the data $(0, 1)$, $(1, e)$ has zero residual at $c = 1$, so Gauss–Newton converges quadratically from any $c_0$ near $1$.

## Explanations

### L1 human

Many fitting problems ask for parameters that make a nonlinear model match data as closely as possible in the least squares sense.
Gauss–Newton replaces the model by its linear approximation at the current guess, solves the resulting linear least squares problem, and repeats.
When the model can fit the data exactly, it converges very fast near the answer.

### L2 human

The method needs only first derivatives of the residuals, unlike Newton's method for $S$, which needs second derivatives.
It is the basic method of [[nonlinear regression|C7E69V030F]] and of many calibration and inverse problems.
The rank condition on $J(x^*)$ is needed: without it the linearized problems have no unique solution, and the convergence theory fails.
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# Finite difference

## Definition

Let $f \colon \R \to \R$ be a [[function|0XHDTE5YMZ]], and let $a, b \in \R$ with $a \ne b$.
The *finite difference* of $f$ with offsets $a$ and $b$ is the function
$$x \mapsto f(x + b) - f(x + a).$$
For a step $h > 0$, the three standard cases are:

1. the forward difference $\Delta_h f(x) = f(x + h) - f(x)$, with $a = 0$ and $b = h$;
2. the backward difference $\nabla_h f(x) = f(x) - f(x - h)$, with $a = -h$ and $b = 0$;
3. the central difference $\delta_h f(x) = f(x + \tfrac{h}{2}) - f(x - \tfrac{h}{2})$, with $a = -\tfrac{h}{2}$ and $b = \tfrac{h}{2}$.

## Remarks

Example: for $f(x) = x^3$ and $h = 1$, the forward difference is $(x+1)^3 - x^3 = 3x^2 + 3x + 1$.
Non-example: $f(x + h) + f(x)$ is not a finite difference, since it is a sum of values and not a difference.
Dividing by $b - a$ gives the difference quotient $\frac{f(x + b) - f(x + a)}{b - a}$.
The same formulas make sense for functions from an abelian group to an abelian group, and for sequences, where the step is $1$.
See the [[forward difference operator|8E0EBA8X07]], the [[backward difference operator|MK5JGVDPS9]] and the [[central difference|BVGYDP6YKF]].
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# Berezin integral

## Summary

The Berezin integral is the notion of integration over anticommuting variables, the odd coordinates of a [[supermanifold|Z0SB9HQBAJ]].
Over one odd variable $\theta$ it sends $a + b\theta$ to $b$; over several it picks out the coefficient of the product of all odd variables.
It is translation invariant and kills derivatives, like the usual integral, but it changes under linear substitutions by the determinant instead of its inverse.
It is the basic tool of fermionic path integrals in quantum field theory, where Gaussian integrals give [[Pfaffians|NH5HWBPWS9]] and determinants.

## Explanations

### L1 human

A function of anticommuting variables is a finite polynomial, because each variable squares to zero.
The Berezin integral simply reads off the coefficient of the top term, the product of all variables.
This sounds strange, but it is the only rule that keeps the familiar properties of integration, such as the integral of a derivative being zero.
For instance, with two odd variables, $\int e^{a \theta_1 \theta_2} = \int (1 + a\theta_1\theta_2) = a$.
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# Join-irreducible element

## Definition

Let $(L, \vee, \wedge)$ be a [[lattice|WH4SFAACAS]].
An element $a \in L$ is *join-irreducible* if:

1. $a$ is not a [[least element|0AANKKNHS4]] of $L$;
2. for all $b, c \in L$, if $a = b \vee c$, then $a = b$ or $a = c$.

An element is *meet-irreducible* if it satisfies the dual conditions, with the greatest element and the [[meet|J2T75ATRV5]].

## Remarks

Clause 2 says that $a$ is not the join of two elements strictly below it.
Example: in the power set of a set $X$, each singleton $\set{x}$ is join-irreducible.
It is not $\emptyset$, and if $\set{x} = B \cup C$ then $B$ or $C$ contains $x$, so it equals $\set{x}$.
Non-example: $\set{1, 2} = \set{1} \cup \set{2}$ with $\set{1} \ne \set{1, 2}$ and $\set{2} \ne \set{1, 2}$, so clause 2 fails.
In a chain, every element other than the least element is join-irreducible, since $b \vee c = \max(b, c)$.
In a finite lattice, every element is the join of the join-irreducible elements below it.
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# Hemicontinuity

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]].
Let $\Gamma$ be a [[multivalued function|03CPD1HGF3]] from $X$ to $Y$, that is, a map that sends each $x \in X$ to a nonempty subset $\Gamma(x) \subseteq Y$.
Let $x_0 \in X$.

1. $\Gamma$ is *upper hemicontinuous* at $x_0$ if the following holds for every [[open|02BH3MCBQB]] set $V \subseteq Y$ with $\Gamma(x_0) \subseteq V$.
   There is a [[neighbourhood|0ZCY0Z80B6]] $U$ of $x_0$ with $\Gamma(x) \subseteq V$ for all $x \in U$.
2. $\Gamma$ is *lower hemicontinuous* at $x_0$ if the following holds for every open set $V \subseteq Y$ with $\Gamma(x_0) \cap V \ne \emptyset$.
   There is a neighbourhood $U$ of $x_0$ with $\Gamma(x) \cap V \ne \emptyset$ for all $x \in U$.

$\Gamma$ is upper (lower) hemicontinuous if it is so at every point of $X$, and *continuous* if it is both.

## Remarks

Some authors say upper and lower semicontinuous instead of hemicontinuous.
If $\Gamma(x) = \set{f(x)}$ for a function $f \colon X \to Y$, then each of the two conditions is equivalent to the [[continuity|0VGFHTM75Y]] of $f$ at $x_0$.

Example: let $\Gamma \colon \R \to \R$ with $\Gamma(0) = [0, 1]$ and $\Gamma(x) = \set{0}$ for $x \ne 0$.
It is upper hemicontinuous at $0$: an open $V \supseteq [0, 1]$ contains $0$, so $\Gamma(x) \subseteq V$ for all $x$.

Non-example: the same $\Gamma$ is not lower hemicontinuous at $0$.
The open set $V = (1/2, 2)$ meets $\Gamma(0)$, but $\Gamma(x) \cap V = \emptyset$ for every $x \ne 0$.
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# Bessel function of the first kind

## Summary

The Bessel function of the first kind $J_\nu(z)$ is the solution of [[Bessel's equation|EAV97QH9DF]] $z^2 y'' + z y' + (z^2 - \nu^2) y = 0$ that behaves like $\frac{1}{\Gamma(\nu + 1)} \left( \frac z2 \right)^\nu$ near $z = 0$.
This description assumes that $\nu$ is not a negative integer.
It is given by a power series that converges for every $z$.
Bessel functions describe waves and heat flow in cylinders and discs, such as the vibration modes of a drum.
They appear throughout analysis, number theory and probability.

## Explanations

### L1 human

A Bessel function of the first kind looks like a cosine wave whose height slowly shrinks, roughly like $1 / \sqrt{x}$.
For example $J_0(x)$ starts at $1$ for $x = 0$ and then oscillates around $0$ with decreasing amplitude.
Its zeros give the frequencies at which a circular drum can vibrate.
For half-integer order the function is elementary: $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$ for $x > 0$.
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<document source="https://tamnd.github.io/math2/node/CZCWFPNS4N.md" tag="CZCWFPNS4N" kind="example">
# Special unitary group SU(n)

## Example

Let $n \ge 1$.
The *special unitary group* is
$$SU(n) = \set{U \in M_n(\C) : U^* U = I_n, \ \det U = 1},$$
where $U^*$ is the conjugate transpose of $U$ and $I_n$ is the identity matrix.
It is the kernel of the [[determinant|00KDZRW4C0]] $\det \colon U(n) \to U(1)$ on the [[unitary group|0R9R0Y1HDK]], so it is a normal subgroup of $U(n)$.

Check that it is a [[compact Lie group|9TFC6ADC1C]].
The two conditions are polynomial equations in the real and imaginary parts of the entries, so $SU(n)$ is closed in $GL_n(\C)$.
By the closed subgroup theorem it is a [[Lie group|7JBRGCVJRB]].
Each column of a unitary matrix is a unit vector, so $SU(n)$ is bounded in $M_n(\C)$, hence compact.

## Remarks

Its Lie algebra $\mathfrak{su}(n)$ consists of the skew-Hermitian $n \times n$ matrices with trace $0$, so its real dimension is $n^2 - 1$.
$SU(n)$ is connected and simply connected.
Its centre is the set of scalar matrices $\omega I_n$ with $\omega^n = 1$, a cyclic group of order $n$.
For $n \ge 2$ its Lie algebra is simple, of type $A_{n-1}$.
In Mathlib this group is `Matrix.specialUnitaryGroup`.

## Explanations

### L1 human

The special unitary group collects the complex matrices that keep lengths and also have determinant $1$.
The unitary group is a little larger: its determinants are all complex numbers of absolute value $1$.
Fixing the determinant to $1$ removes one real degree of freedom, a single overall phase.
For $n = 1$ the group is trivial, and for $n = 2$ it is a copy of the $3$-sphere.
The groups $SU(2)$ and $SU(3)$ are central in particle physics.
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# Birkhoff's subdirect representation theorem

## Statement

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with at least two elements.
Then $\mathbf{A}$ is isomorphic to a [[subdirect product|88WN3AD2VQ]] of a family of [[subdirectly irreducible algebras|DASF939439]].
Moreover, each factor of the family can be chosen to be a [[quotient|N5KE7FH6CX]] of $\mathbf{A}$.

## Remarks

Proof idea.
For each pair $a \ne b$ in $A$, [[Zorn's lemma|04RXTQA5ZA]] gives a [[congruence|DCSXXT4SER]] $\theta_{a,b}$ that is maximal among the congruences not containing $(a, b)$.
Every congruence of $\mathbf{A}/\theta_{a,b}$ other than equality comes from a strictly larger congruence, which contains $(a, b)$.
So $\mathbf{A}/\theta_{a,b}$ is subdirectly irreducible, with monolith generated by the classes of $a$ and $b$.
The intersection of all $\theta_{a,b}$ is equality, so $a \mapsto (a/\theta_{a,b})$ is a subdirect embedding.

Example: the ring $\Z$ is a subdirect product of the fields $\Z/p\Z$, $p$ prime, which are simple and hence subdirectly irreducible.
Another example: the only subdirectly irreducible distributive lattice is the two-element chain, so every distributive lattice with at least two elements is a subdirect product of copies of it.

A subdirect product may need infinitely many factors even for a countable algebra, as for $\Z$.
For a finite algebra, finitely many factors suffice, and Zorn's lemma is not needed.

## Explanations

### L1 human

Every algebra can be taken apart into pieces that cannot be taken apart further.
The pieces are quotients of the algebra, and the algebra sits inside their product, with nothing lost.
It is like recovering an integer from its remainders modulo many numbers.

### L2 human

The theorem reduces many questions about a variety to its subdirectly irreducible members.
A variety is generated by its subdirectly irreducible algebras, so two varieties with the same subdirectly irreducible members are equal.
For example, it shows that every Boolean algebra is a subdirect product of copies of the two-element Boolean algebra.
This leads to notions such as [[residually finite algebras|X9072A41NK]], which are subdirect products of finite algebras.
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# Totally disconnected group

## Definition

A *totally disconnected group* is a [[topological group|92700QAB5Y]] $G$ whose underlying topological space is [[totally disconnected|0RJSBR9GW1]].
That is, the only connected subsets of $G$ are the empty set and the one-point sets.

## Remarks

Example: the [[p-adic integers|V6TJAFZ4MD]] $\Z_p$ under addition.
Each subgroup $p^n \Z_p$ is open and closed, and two distinct elements $x, y$ lie in different cosets of $p^n \Z_p$ for $n$ large, so no connected set contains both.
Non-example: $(\R, +)$ with the usual topology is connected, so it is not totally disconnected.
Equivalently, $G$ is totally disconnected if and only if its identity component is $\set{e}$, because each connected component is a coset of the identity component.
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# Conditionally complete lattice

## Summary

A conditionally complete lattice is a lattice in which every nonempty bounded-above subset has a supremum and every nonempty bounded-below subset has an infimum.
It is the order-theoretic form of the completeness of the [[real numbers|08J2M1JYAG]].
Unlike a [[complete lattice|D7A0ZGZT5S]], it need not have a least or a greatest element.

## Explanations

### L1 human

The real numbers are not a complete lattice, because the set of all integers has no upper bound.
But every nonempty set of reals that does have an upper bound has a least one.
A conditionally complete lattice is an ordered set with this weaker property, together with the matching property for lower bounds.
The integers are another example: a nonempty set of integers that is bounded above has a largest element, which is its supremum.
The rational numbers are not an example, since the rationals below $\sqrt{2}$ have no least rational upper bound.
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# Kodaira–Nakano vanishing theorem

## Statement

Let $X$ be a compact [[complex manifold|JSGWAN2RJV]] of dimension $n$, and let $L$ be a [[positive line bundle|J45KWFHENP]] on $X$.
For $p \ge 0$ let $\Omega_X^p$ be the sheaf of holomorphic $p$-forms on $X$.
Then
$$H^q(X, \Omega_X^p \otimes L) = 0 \quad \text{whenever } p + q > n,$$
where $H^q$ is [[sheaf cohomology|PQ6Q143ARJ]].

## Remarks

The theorem is also called the Akizuki–Nakano vanishing theorem.
A compact manifold with a positive line bundle is Kähler, with Kähler form $\omega = i\Theta_h$ given by the curvature of a positive metric $h$.
By Dolbeault's theorem, a class in $H^q(X, \Omega^p_X \otimes L)$ is represented by an $L$-valued harmonic [[$(p,q)$-form|DC81TCR98Y]] $u$.
The Bochner–Kodaira–Nakano identity gives $\langle [i\Theta_h, \Lambda] u, u \rangle \le 0$ for such $u$, and with $\omega = i\Theta_h$ the operator $[i\Theta_h, \Lambda]$ is multiplication by $p + q - n$ on $(p,q)$-forms.
So $(p + q - n)\norm{u}^2 \le 0$, which forces $u = 0$ when $p + q > n$.

The case $p = n$ is the [[Kodaira vanishing theorem|NY77DQ2AX8]]: $\Omega^n_X = K_X$ and $H^q(X, K_X \otimes L) = 0$ for $q \ge 1$.
By Serre duality the theorem is equivalent to $H^q(X, \Omega^p_X \otimes L^{-1}) = 0$ for $p + q < n$.

Positivity of $L$ cannot be dropped.
For $X = \mathbb{P}^1$, $n = 1$, $p = q = 1$ and $L$ trivial, one has $H^1(\mathbb{P}^1, \Omega^1) \cong \C \ne 0$, though $p + q = 2 > 1$.

## Explanations

### L1 human

The theorem says that many cohomology groups of a positive line bundle, twisted by holomorphic forms, are zero.
Zero cohomology means that local problems have global solutions, so it is a strong and useful conclusion.
The hypothesis is a positivity of curvature, the analytic form of ampleness.

### L2 human

The proof is a model of the Bochner technique: a curvature identity shows that harmonic representatives must vanish.
It extends Kodaira vanishing, which is the case $p = n$, to all bundles of holomorphic forms.
It is used to prove the Lefschetz hyperplane theorem for cohomology and to control cohomology of $\Omega^p$ on projective manifolds.
For vector bundles the analogous result needs Nakano positivity, not just Griffiths positivity.
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# Length of a module

## Summary

The length of a module is the largest number of strict steps in a chain of its submodules.
It measures the size of a module, as dimension measures the size of a vector space.
A module has finite length exactly when it is both a [[Noetherian module|AK1ZRCXR32]] and an [[Artinian module|YQNZTQGP1X]].
Length is additive along short exact sequences, which makes it a basic tool in dimension theory and multiplicity theory.

## Explanations

### L1 human

Climb from the zero submodule to the whole module, one strict step at a time, and count the steps.
The length is the largest number of steps that you can take.
For a vector space this is its dimension.
For $\Z/12\Z$ as a $\Z$-module the length is $3$, as in $0 \subset 6\Z/12\Z \subset 3\Z/12\Z \subset \Z/12\Z$, matching the three prime factors of $12 = 2 \cdot 2 \cdot 3$.
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# Endomorphism monoid

## Definition

Let $\mathbf{A}$ be a [[structure|983GZBQYT4]] with domain $A$.
An *endomorphism* of $\mathbf{A}$ is a [[homomorphism|KBXSZC6BV1]] from $\mathbf{A}$ to $\mathbf{A}$.
Let $E$ be the set of endomorphisms of $\mathbf{A}$, let $\circ$ be [[composition|0THBK9V825]] and let $\id_A$ be the identity map of $A$.
The *endomorphism monoid* of $\mathbf{A}$ is the [[monoid|0N6R902FG1]] $\End(\mathbf{A}) = (E, \circ, \id_A)$.

## Remarks

The composite of two homomorphisms is a homomorphism and $\id_A$ is a homomorphism, so $E$ is closed under $\circ$ and contains $\id_A$; composition is associative.

Example: let $\mathbf{A} = (\Z, +, 0)$.
Each endomorphism $h$ satisfies $h(x) = x\,h(1)$ for all $x \in \Z$, by induction on $x \ge 0$ and $h(-x) = -h(x)$.
So $h \mapsto h(1)$ is an isomorphism from $\End(\mathbf{A})$ to the monoid $(\Z, \cdot, 1)$, since $(g \circ h)(1) = g(1)\,h(1)$.

Non-example: the map $x \mapsto x + 1$ lies in the monoid of all maps $\Z \to \Z$ but not in $\End(\Z, +, 0)$, because it sends $0$ to $1$.

The invertible elements of $\End(\mathbf{A})$ are the automorphisms of $\mathbf{A}$.
</document>

<document source="https://tamnd.github.io/math2/node/CZWDNA1HA7.md" tag="CZWDNA1HA7" kind="definition">
# Algebra over a monad

## Definition

Let $(T, \eta, \mu)$ be a [[monad|FF6XXA6D1A]] on a category $\mathcal{C}$.
A *$T$-algebra* is a pair $(A, a)$ of an object $A$ of $\mathcal{C}$ and a morphism $a \colon T(A) \to A$ with these two properties.

1. Unit law: $a \circ \eta_A = \id_A$.
2. Associativity law: $a \circ T(a) = a \circ \mu_A$ as morphisms $T(T(A)) \to A$.

A *morphism of $T$-algebras* $f \colon (A, a) \to (B, b)$ is a morphism $f \colon A \to B$ of $\mathcal{C}$ with $f \circ a = b \circ T(f)$.

## Remarks

Example: let $T$ be the list monad on sets, so $T(X)$ is the [[free monoid|CV013KHVJG]] of finite words in $X$.
For a [[monoid|0N6R902FG1]] $M$, let $a$ send a word $(m_1, \dots, m_n)$ to the product $m_1 m_2 \cdots m_n$ in $M$, with the empty word sent to the identity.
The unit law holds since a one-letter word $m$ evaluates to $m$.
The associativity law holds since multiplying the blocks of a word of words and then multiplying the results gives the product of the concatenated word.
Every $T$-algebra arises in this way, so $T$-algebras are the same as monoids.

Non-example: for the same monad, let $a \colon T(\N) \to \N$ send a word to its length.
Then $a(\eta_{\N}(5)) = 1 \ne 5$, so the unit law fails.

For every object $X$, the pair $(T(X), \mu_X)$ is a $T$-algebra, the free $T$-algebra on $X$.
</document>

<document source="https://tamnd.github.io/math2/node/CZYDQ96X2T.md" tag="CZYDQ96X2T" kind="statement">
# Filtered colimits commute with finite limits

## Statement

Let $I$ be a small [[filtered category|975PMY4FG8]] and let $J$ be a finite category.
Let $F \colon I \times J \to \mathbf{Set}$ be a functor on the [[product category|AFNE9C4NH6]] with values in the [[category of sets|WP0YB7VC59]].
Then the canonical map
$$\operatorname{colim}_{i \in I} \lim_{j \in J} F(i, j) \longrightarrow \lim_{j \in J} \operatorname{colim}_{i \in I} F(i, j)$$
is a bijection.

## Remarks

The proof uses the explicit form of a filtered [[colimit|BJS1CGTKQ5]] of sets.
It is the disjoint union of the sets, where two elements are identified when they become equal at some later stage.
A finite [[limit|CHS5AFJGWE]] involves only finitely many elements and finitely many equations.
Because $I$ is filtered, all of them can be moved to one common stage, which gives both injectivity and surjectivity.

Both hypotheses matter.
For $J$ infinite, take $I = \N$, $J = \N$ seen as a discrete category, and $F(n, j)$ a one-point set when $j < n$ and empty otherwise.
Then each $\prod_j F(n, j)$ is empty, while each $\operatorname{colim}_n F(n, j)$ is a point.
So the left side is empty and the right side is a point.
For $I$ not filtered, coproducts do not commute with products: $(A \sqcup B) \times (A \sqcup B)$ is larger than $(A \times A) \sqcup (B \times B)$ when $A$ and $B$ are not empty.

## Explanations

### L1 human

A filtered colimit of sets is a kind of union of a growing family.
A finite limit is a set of solutions of finitely many equations.
The theorem says that solving the equations and then taking the union gives the same as taking the union and then solving.
This works because any finite amount of data already lives at some stage of the growing family.

### L2 human

The theorem explains why filtered colimits are so well behaved in algebra.
For example, a filtered colimit of groups or modules can be computed on underlying sets, because the group operations are maps out of finite products, which are finite limits.
It also shows that filtered colimits of exact sequences of modules are exact.
The same result holds in any locally finitely presentable category, and finitely presentable objects are defined by a closely related property of the hom functor.
</document>

<document source="https://tamnd.github.io/math2/node/D05NAAC3XK.md" tag="D05NAAC3XK" kind="proof">
# Proof that a homomorphism is injective iff its kernel is trivial

## Proof

1. Let $\varphi : G \to H$ be a group homomorphism with [[kernel|0Z424HK8CQ]] $K$.
2. Suppose that $\varphi$ is injective, and let $k \in K$. [[$\varphi(e) = e$|08FKQD7NJ3]], so $\varphi(k) = \varphi(e)$. Thus $k = e$, and $K = \{e\}$.
3. Now suppose that $K = \{e\}$, and let $a, b \in G$ with $\varphi(a) = \varphi(b)$.
4. Then $\varphi(a^{-1}b) = \varphi(a)^{-1}\varphi(b) = e$, so $a^{-1}b \in K$.
5. Thus $a^{-1}b = e$, so $a = b$, and $\varphi$ is injective.

## Idea

Two elements have the same image exactly when their quotient is in the kernel.
</document>

<document source="https://tamnd.github.io/math2/node/D07MBM0S3V.md" tag="D07MBM0S3V" kind="statement">
# Conley's fundamental theorem of dynamical systems

## Statement

Let $(X, d)$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]] and let $\varphi \colon \R \times X \to X$, $(t, x) \mapsto \varphi_t(x)$, be a continuous [[flow|BBZP1R872N]].
Let $\mathcal{R}(\varphi)$ be its [[chain recurrent set|6MX5R482KJ]].
Call $x, y \in \mathcal{R}(\varphi)$ chain equivalent if for all $\varepsilon > 0$ and $T > 0$ there are $(\varepsilon, T)$-chains from $x$ to $y$ and from $y$ to $x$.
The classes of this relation are the chain components.
Then there is a continuous function $V \colon X \to \R$, a complete [[Lyapunov function|01REYCJ371]] for $\varphi$, such that:

1. $V(\varphi_t(x)) < V(x)$ for every $x \notin \mathcal{R}(\varphi)$ and every $t > 0$.
2. For $x, y \in \mathcal{R}(\varphi)$, $V(x) = V(y)$ if and only if $x$ and $y$ lie in the same chain component.
3. The set $V(\mathcal{R}(\varphi))$ is a compact [[nowhere dense|0VQYCCQVJ5]] subset of $\R$.

## Remarks

The proof uses attractor–repeller pairs $(A, A^\ast)$.
For each such pair, Conley builds a continuous function equal to $0$ on $A$ and to $1$ on $A^\ast$ that strictly decreases along orbits between them.
A compact metric space carries only countably many attractors $A_1, A_2, \ldots$, and $V = \sum_n 3^{-n} V_n$ combines the corresponding functions.
The key fact is that $\mathcal{R}(\varphi)$ is the intersection of the sets $A \cup A^\ast$ over all attractor–repeller pairs.
Example: let $f$ be a Morse function on a compact Riemannian manifold with pairwise distinct critical values.
For the flow of $-\nabla f$, the chain recurrent set is the set of critical points, and $V = f$ satisfies the three conditions.
Compactness is needed for the countability of attractors and for the existence of the attractor–repeller decomposition used in the proof.

## Explanations

### L1 human

The theorem splits every continuous flow on a compact metric space into two parts.
One part is the chain recurrent set, where the motion can come back to itself up to tiny errors.
On the rest, the motion runs steadily downhill for some height function, as water flows down a landscape.
So every flow on a compact space looks like a gradient flow once the recurrent pieces are collapsed to points.

### L2 human

The function $V$ orders the chain components like levels of a landscape and shows that orbits outside $\mathcal{R}(\varphi)$ go from higher components to lower ones.
This is why the result is sometimes called the fundamental theorem of dynamical systems: the chain components are the indecomposable pieces, and between them there is only gradient-like motion.
The statement is the starting point for Morse decompositions and for the Conley index.
Versions for continuous maps of compact metric spaces, with $\varepsilon$-chains in place of $(\varepsilon, T)$-chains, also hold.
</document>

<document source="https://tamnd.github.io/math2/node/D08SKTH2T2.md" tag="D08SKTH2T2" kind="example">
# Almost Mathieu operator

## Example

Let $\lambda, \alpha, \theta$ be real numbers.
The *almost Mathieu operator* $H_{\lambda,\alpha,\theta}$ acts on the Hilbert space $\ell^2(\Z)$ of square-summable sequences $u = (u(n))_{n \in \Z}$ by
$$(H_{\lambda,\alpha,\theta}u)(n) = u(n + 1) + u(n - 1) + 2\lambda\cos\big(2\pi(n\alpha + \theta)\big)u(n).$$
It is a bounded [[self-adjoint operator|0NEGN5CRSC]].
Check: each of the three terms has norm at most $1$, $1$ and $2\abs{\lambda}$, so $\norm{H_{\lambda,\alpha,\theta}} \le 2 + 2\abs{\lambda}$.
The shifts $u(n) \mapsto u(n + 1)$ and $u(n) \mapsto u(n - 1)$ are adjoint to each other, so their sum is self-adjoint, and multiplication by a real sequence is self-adjoint.

## Remarks

The operator is a discrete [[Schrödinger operator|89AA6XJY5K]] with a quasi-periodic potential when $\alpha$ is irrational.
For irrational $\alpha$, its [[spectrum|BMKATJ802A]] does not depend on $\theta$.
Aubry duality relates the operators with coupling $\lambda$ and $1/\lambda$, and $\lambda = 1$ is the critical coupling.
For irrational $\alpha$, the Lebesgue measure of the spectrum is $\abs{4 - 4\abs{\lambda}}$; in particular it is zero at $\lambda = 1$.
The [[ten martini problem|3X4ZQ41PXG]] asked whether the spectrum is a Cantor set for all irrational $\alpha$ and all $\lambda \ne 0$; Avila and Jitomirskaya proved that it is.
Plotting the spectrum against $\alpha$ for $\lambda = 1$ gives Hofstadter's butterfly.

## Explanations

### L1 human

The operator models an electron hopping along a chain of atoms, one site to the next, with an energy at each site that varies like a cosine.
When $\alpha$ is irrational, the pattern of site energies never repeats exactly, but it comes close to repeating again and again.
This in-between situation, neither periodic nor random, gives a spectrum of remarkable fractal structure.
The operator is a test case for the spectral theory of quasi-periodic systems and also models electrons in a crystal under a magnetic field.
</document>

<document source="https://tamnd.github.io/math2/node/D0A22MEB1F.md" tag="D0A22MEB1F" kind="example">
# Tent map

## Example

For a parameter $\mu \in [0, 2]$, the *tent map* is
$$T_\mu \colon [0, 1] \to [0, 1], \qquad T_\mu(x) = \mu \min(x, 1 - x).$$
Its graph is a tent: it rises with slope $\mu$ on $[0, 1/2]$ and falls with slope $-\mu$ on $[1/2, 1]$.
Its top value is $\mu/2 \le 1$, so it maps $[0, 1]$ into itself and is a [[discrete-time dynamical system|5AA9AJEARJ]].

Check that [[Lebesgue measure|2Z2QVWZ262]] $\lambda$ is [[invariant|B232EZ1FMB]] for $\mu = 2$.
For $0 \le a < b \le 1$, the preimage $T_2^{-1}([a, b])$ is $[a/2, b/2] \cup [1 - b/2, 1 - a/2]$, two intervals of length $(b - a)/2$ each.
So $\lambda(T_2^{-1}[a, b]) = b - a = \lambda([a, b])$, and this extends from intervals to all Borel sets.

## Remarks

For $\mu = 2$ the map is ergodic for Lebesgue measure, has [[topological entropy|BDZYBXBFDE]] $\log 2$, and has [[Lyapunov exponent|NJ1FJSNK6A]] $\log 2$ at every point whose orbit avoids $1/2$, since $\abs{T_2'} = 2$.
For $0 \le \mu < 1$ every orbit tends to the fixed point $0$.
The map $T_2$ is conjugate to the logistic map $x \mapsto 4x(1 - x)$.

## Explanations

### L1 human

The tent map stretches the unit interval to twice its length and folds it back onto itself, like kneading dough.
Each step doubles small distances between nearby points, so tiny errors grow fast and long-term prediction fails.
Yet the stretching and folding is so even that a random starting point spends equal time in intervals of equal length.
</document>

<document source="https://tamnd.github.io/math2/node/D0DAFG8QK6.md" tag="D0DAFG8QK6" kind="concept">
# Relative CW complex

## Summary

A relative CW complex is a pair $(X, A)$ of spaces in which $X$ is built from $A$ by attaching cells of increasing dimension.
When $A$ is empty, it is the same as a [[CW complex|5W95QFHXPR]].
The inclusion $A \to X$ of a relative CW complex has the homotopy extension property, so it is a cofibration.

## Explanations

### L1 human

Start with any space $A$, not necessarily a CW complex.
Glue points to it, then intervals along their ends, then disks along their boundary circles, and so on.
The result $X$ is a relative CW complex over $A$.
For example, the closed disk $D^2$ is obtained from its boundary circle $S^1$ by gluing one $2$-cell, so $(D^2, S^1)$ is a relative CW complex.
</document>

<document source="https://tamnd.github.io/math2/node/D0Q6GE7FG4.md" tag="D0Q6GE7FG4" kind="definition">
# Solvmanifold

## Definition

Let $G$ be a connected [[solvable Lie group|9CNSVFQ801]] and let $H$ be a closed subgroup of $G$.
The [[homogeneous space|85TWR6E75P]] $G/H$, with its manifold structure as a quotient of [[Lie groups|7JBRGCVJRB]], is a *solvmanifold*.
Equivalently, a solvmanifold is a manifold with a smooth transitive action of a connected solvable Lie group.

## Remarks

Example: $G$ the group of maps $x \mapsto ax + b$ of $\R$ with $a > 0$ and $b \in \R$.
It is solvable, since its commutator subgroup lies in the abelian group of translations.
It acts transitively on $\R$, and the stabilizer of $0$ is the closed subgroup $H = \set{x \mapsto ax}$.
So $\R \cong G/H$ is a solvmanifold.

Non-example: the presentation $S^2 = \mathrm{SO}(3)/\mathrm{SO}(2)$ does not show that $S^2$ is a solvmanifold, because $\mathrm{SO}(3)$ is not solvable: its Lie algebra $\mathfrak{so}(3)$ is simple.

Many authors call only the compact quotients $G/\Gamma$ by a discrete subgroup $\Gamma$ solvmanifolds.
Each nilmanifold is a solvmanifold, since nilpotent groups are solvable.
</document>

<document source="https://tamnd.github.io/math2/node/D0R8455C26.md" tag="D0R8455C26" kind="concept">
# Set-valued analysis

## Summary

Set-valued analysis is the study of maps $F \colon X \rightrightarrows Y$ that assign to each point $x$ a subset $F(x)$ of $Y$, possibly empty or with many points.
Such maps are the [[multivalued functions|03CPD1HGF3]] of analysis, studied with tools of topology, convexity and differentiation.
Its main notions are the graph and inverse of a set-valued map, its [[hemicontinuity|XNMKX7H4FD]], limits of sets, selections and derivatives of set-valued maps.
Set-valued maps arise as solution sets of parametrized problems, as subdifferentials of nonsmooth functions, and as right-hand sides of [[differential inclusions|9ZSMF6NYKM]] in control theory.
Typical results are the [[Kakutani fixed-point theorem|7V3HWGR32Z]] and the [[Michael selection theorem|3RJ1K36A9A]].

## Explanations

### L1 human

An ordinary function gives one answer for each input.
Many questions in optimization and control have several answers, or none, for a given input.
Examples are the set of minimizers of a problem, or the set of velocities a controlled system can choose.
Set-valued analysis treats the whole set of answers as the value of a map.
For example, the map $F(a) = \set{x \in \R : x^2 = a}$ gives two points for $a > 0$, one point for $a = 0$ and the empty set for $a < 0$.
The subject asks how such sets move when the input changes.
It also asks whether one can pick a single continuous answer from them, and how to differentiate them.
</document>

<document source="https://tamnd.github.io/math2/node/D0TQ9F2ZDW.md" tag="D0TQ9F2ZDW" kind="concept">
# Fresnel integrals

## Summary

The Fresnel integrals $S(x) = \int_0^x \sin(t^2)\, dt$ and $C(x) = \int_0^x \cos(t^2)\, dt$ are integrals of [[sine and cosine|08WFZP6P89]] of a square.
They arise in the optics of diffraction near an edge, studied by Augustin-Jean Fresnel.
The plane curve $(C(t), S(t))$ is the Euler spiral, whose curvature grows linearly with arc length.
Both integrals tend to $\sqrt{\pi/8}$ as $x \to \infty$.

## Explanations

### L1 human

The function $\sin(t^2)$ oscillates faster and faster as $t$ grows.
Its area from $0$ to $x$ therefore wobbles less and less and settles down to a limit.
If one draws the point $(C(t), S(t))$ as $t$ increases, one gets a spiral that winds tighter and tighter around a point.
Road and railway designers use this spiral to pass smoothly from a straight line into a curve.
</document>

<document source="https://tamnd.github.io/math2/node/D0X6EBNMW6.md" tag="D0X6EBNMW6" kind="definition">
# Chromatic homotopy theory

## Definition

Fix a [[prime number|01826KMG2H]] $p$ and work with $p$-local [[spectra|1GHF4EKBXS]] in the [[stable homotopy category|26GTNMAF7B]].
For a spectrum $E$, a spectrum $X$ is *$E$-acyclic* if $E \wedge X \simeq 0$, and *$E$-local* if every map from an $E$-acyclic spectrum to $X$ is null.
The *Bousfield localization* $X \to L_E X$ is the map to an $E$-local spectrum whose fibre is $E$-acyclic; it exists and is unique up to equivalence.
For $n \ge 1$, let $E(n)$ be the Johnson–Wilson spectrum, with $\pi_* E(n) = \Z_{(p)}[v_1, \ldots, v_n, v_n^{-1}]$, a quotient of [[Brown–Peterson cohomology|S8S72AA9C8]] with $v_n$ inverted; let $E(0) = H\Q$.
Write $L_n = L_{E(n)}$.
Since $E(n-1)$-local spectra are $E(n)$-local, there are natural maps $L_n X \to L_{n-1} X$.
The *chromatic tower* of $X$ is
$$X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X = X \wedge H\Q.$$
Let $K(n)$ be the $n$-th Morava K-theory, with $\pi_* K(n) = \mathbb{F}_p[v_n, v_n^{-1}]$ and $\abs{v_n} = 2(p^n - 1)$.
*Chromatic homotopy theory* is the study of spectra through their chromatic towers, their *monochromatic layers* $M_n X = \operatorname{fib}(L_n X \to L_{n-1} X)$, and their localizations $L_{K(n)} X$.

## Remarks

Example: for the $p$-local sphere spectrum $S$, the layer $L_0 S = H\Q$ records the rational stable homotopy, which is $\Q$ in degree $0$ only.
The layer $L_1 S$ is computed from K-theory and contains the image of the J-homomorphism.

Non-example: the Eilenberg–MacLane spectrum $H\mathbb{F}_p$ is not $E(n)$-local for any $n$.
Indeed $E(n) \wedge H\mathbb{F}_p \simeq 0$, so $H\mathbb{F}_p$ is a nonzero $E(n)$-acyclic spectrum, and its identity map is not null.
So all terms of its chromatic tower vanish.

By the chromatic convergence theorem of Hopkins and Ravenel, a finite $p$-local spectrum is the homotopy limit of its chromatic tower.
The example of $H\mathbb{F}_p$ shows that this fails without finiteness.
</document>

<document source="https://tamnd.github.io/math2/node/D0ZHVD797D.md" tag="D0ZHVD797D" kind="concept">
# Localization of a space

## Summary

Localizing a space at a set of primes $P$ is the topological analogue of [[localizing|4C1AE84N5N]] the integers at $P$.
For a simply connected space $X$ it produces a map $X \to X_P$.
This map tensors every [[homotopy group|P976WYEFMB]] and every homology group with $\Z_{(P)}$.
This lets one study a space one prime at a time.
The empty set of primes gives rationalization, the starting point of [[rational homotopy theory|8G9QPMBT7R]].

## Explanations

### L1 human

Homotopy groups of spaces are often full of torsion at many primes, which makes them hard to compute all at once.
Localization keeps only the information at chosen primes and makes all other primes invertible.
For example, the rationalization of an odd-dimensional sphere $S^n$ is the Eilenberg–MacLane space $K(\Q, n)$.
Rationally, an odd sphere has just one homotopy group, $\Q$ in degree $n$, and all the torsion disappears.
</document>

<document source="https://tamnd.github.io/math2/node/D0ZXXEYNK8.md" tag="D0ZXXEYNK8" kind="definition">
# Euler–Maruyama method

## Definition

Let $T > 0$, let $x_0 \in \R$, and let $a, b \colon [0, T] \times \R \to \R$ be functions.
Consider the stochastic differential equation $dX_t = a(t, X_t) \, dt + b(t, X_t) \, dB_t$ with $X_0 = x_0$, where $B$ is a [[Brownian motion|22735FGJV6]].
Let $N \ge 1$, let $\Delta t = T / N$, and let $t_n = n \, \Delta t$ for $0 \le n \le N$.
The *Euler–Maruyama approximation* is the sequence $(Y_n)_{0 \le n \le N}$ with $Y_0 = x_0$ and
$$Y_{n+1} = Y_n + a(t_n, Y_n) \, \Delta t + b(t_n, Y_n) \, \Delta B_n, \quad \Delta B_n = B_{t_{n+1}} - B_{t_n}.$$

## Remarks

The increments $\Delta B_n$ are independent and [[normal|05AQE0QR0D]] with mean $0$ and variance $\Delta t$.
In a simulation one takes $\Delta B_n = \sqrt{\Delta t} \, Z_n$ with independent standard normal $Z_n$.
Example: for [[geometric Brownian motion|D48DR46B97]], $a(t, x) = \mu x$ and $b(t, x) = \sigma x$, so $Y_{n+1} = Y_n (1 + \mu \Delta t + \sigma \Delta B_n)$.
Non-example: the update $Y_{n+1} = Y_n + a(t_n, Y_n) \Delta t + b(t_n, Y_n) \Delta t \, Z_n$ uses noise of size $\Delta t$ instead of $\sqrt{\Delta t}$, so its noise vanishes in the limit.
If $b = 0$, the scheme is the [[Euler method|0JARTXX4XP]] for the ordinary equation $x' = a(t, x)$.
Assume that $a$ and $b$ are Lipschitz in $x$ uniformly in $t$, grow at most linearly, and are Hölder continuous of exponent $1/2$ in $t$.
Then $E\abs{X_T - Y_N} \le C \sqrt{\Delta t}$ for a constant $C$.
</document>

<document source="https://tamnd.github.io/math2/node/D115PG3T7K.md" tag="D115PG3T7K" kind="statement">
# Mahler's theorem

## Statement

Let $p$ be a prime, let $\Z_p$ be the ring of [[$p$-adic integers|GBN9F8FAQ3]], and let $\Q_p$ be the field of [[$p$-adic numbers|CHT5JYM6QW]].
For $k \in \N$ and $x \in \Z_p$ let $\binom{x}{k} = \frac{x(x-1)\cdots(x-k+1)}{k!}$.
Let $f \colon \Z_p \to \Q_p$ be a [[continuous function|0P27W9RW0G]], and let $a_k = \Delta^k f(0)$ be its [[$k$-th forward difference|E38801KVJ1]] at $0$ with step $1$.
Then $a_k \to 0$ in $\Q_p$, and
$$f(x) = \sum_{k=0}^{\infty} a_k \binom{x}{k} \quad \text{for all } x \in \Z_p,$$
where the series [[converges uniformly|08FPABDQ4C]] on $\Z_p$.
Conversely, for every sequence $(a_k)$ in $\Q_p$ with $a_k \to 0$, this series defines a continuous function $f \colon \Z_p \to \Q_p$ with $\Delta^k f(0) = a_k$ for all $k$.

## Remarks

The binomial polynomials $\binom{x}{k}$ take integer values on $\N$, which is dense in $\Z_p$, so they map $\Z_p$ into $\Z_p$; hence $\abs{\binom{x}{k}}_p \le 1$ and the series converges uniformly when $a_k \to 0$.
On $\N$ the expansion is [[Newton's forward difference formula|DDHCMKHD2T]], and both sides are continuous, so the main point is that $a_k \to 0$.
This is proved by showing that $\Delta^{p^m}$ is small on continuous functions for large $m$, using the expansion of $\Delta^{p^m}$ in shifts and the divisibility of binomial coefficients by $p$.
Moreover, $\sup_{x \in \Z_p} \abs{f(x)}_p = \max_k \abs{a_k}_p$, so the functions $\binom{x}{k}$ form an orthonormal basis of the Banach space of continuous functions $\Z_p \to \Q_p$.
Example: for $u \in \Z_p$ with $\abs{u - 1}_p < 1$, the function $x \mapsto u^x$ is continuous, and its expansion is $\sum_k (u - 1)^k \binom{x}{k}$, with $a_k = (u-1)^k \to 0$.

## Explanations

### L1 human

Every continuous function on the $p$-adic integers is given by a Newton series that converges everywhere.
The coefficients are the forward differences at $0$, which can be read off from the values $f(0), f(1), f(2), \dots$.
Over the real numbers nothing like this holds, since a continuous function on $\R$ is far from being fixed by its values at the natural numbers.

### L2 human

The theorem identifies the space of continuous functions $\Z_p \to \Q_p$ with the space of null sequences, isometrically for the sup norm.
It is a basic tool in $p$-adic analysis, for example to construct $p$-adic interpolations of functions defined on the natural numbers, and to describe $p$-adic measures by their moments.
It is the $p$-adic counterpart of the [[Newton series|Y2HXSEA2EE]] expansion, where in the real case convergence is a delicate question.
</document>

<document source="https://tamnd.github.io/math2/node/D13K9GJT0B.md" tag="D13K9GJT0B" kind="concept">
# Girth

## Summary

The girth of a graph is the length of its shortest [[cycle|0Z05VCBQ7Y]], and it is infinite for a graph without cycles.
Large girth means that the graph looks like a tree near each vertex.
Graphs with both large girth and large chromatic number exist, a classical result of the probabilistic method.
The node [[girth|WZ4JGTG6ZD]] gives the precise definition.

## Explanations

### L1 human

Walk around the smallest loop you can find in a graph and count its edges: that number is the girth.
A triangle gives girth $3$, and a graph whose smallest loops are squares has girth $4$, like a grid of streets.
The [[Petersen graph|0NR95CTFCZ]] has girth $5$: it has pentagons but no triangles and no squares.
</document>

<document source="https://tamnd.github.io/math2/node/D14XVCMGX9.md" tag="D14XVCMGX9" kind="definition">
# Extremally disconnected space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *extremally disconnected* if for every [[open set|02BH3MCBQB]] $U \subseteq X$, the [[closure|0QEAJB91GZ]] $\overline{U}$ is open in $X$.

## Remarks

Example: a discrete space is extremally disconnected.
Every subset is open, so in particular every closure is open.

Non-example: $\R$ with the usual topology is not extremally disconnected.
The set $(0, 1)$ is open, and its closure $[0, 1]$ is not open.

An equivalent condition: for all disjoint open sets $U, V \subseteq X$, the closures $\overline{U}$ and $\overline{V}$ are disjoint.
Some authors also require $X$ to be Hausdorff.
An extremally disconnected Hausdorff space is [[totally disconnected|0RJSBR9GW1]].
</document>

<document source="https://tamnd.github.io/math2/node/D17XX5N9VF.md" tag="D17XX5N9VF" kind="concept">
# 6-j symbol

## Summary

The Wigner 6-j symbol $\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix}$ is a number attached to six spins.
It compares the two ways to couple three irreducible representations of [[SU(2)|E5P8AG6CK7]]: first the first two, or first the last two.
It is a sum of products of four [[3-j symbols|YJRJVZ998D]], and it is closely related to the Racah W-coefficient and to the [[Racah polynomials|P6JYNDX5DR]].
6-j symbols are used in atomic and nuclear physics, and in the construction of invariants of 3-manifolds.

## Explanations

### L1 human

Three spins can be added in two orders: $(j_1 + j_2) + j_3$ or $j_1 + (j_2 + j_3)$.
Both orders give a basis of the same space, and the change of basis between them is given by 6-j symbols.
So a 6-j symbol is the analogue, for representations, of the associativity of addition.
Its six entries can be drawn on the edges of a tetrahedron, and the symbol has the symmetries of the tetrahedron.
</document>

<document source="https://tamnd.github.io/math2/node/D186VNHFZJ.md" tag="D186VNHFZJ" kind="definition">
# Newton polygon

## Definition

Let $K$ be a field with a [[valuation|SVY882HKSJ]] $v \colon K \to \R \cup \set{\infty}$.
Let $f = a_0 + a_1 X + \cdots + a_n X^n \in K[X]$ with $n \ge 1$ and $a_0 a_n \ne 0$.
Let $S = \set{(i, v(a_i)) : 0 \le i \le n,\ a_i \ne 0} \subseteq \R^2$.
The *Newton polygon* of $f$ is the lower boundary of the [[convex hull|0TCHAS04WK]] of $S$.
It is the graph of the largest convex function $\varphi \colon [0, n] \to \R$ with $\varphi(i) \le v(a_i)$ for every $(i, v(a_i)) \in S$.

It is a broken line from $(0, v(a_0))$ to $(n, v(a_n))$ whose vertices lie in $S$ and whose slopes increase from left to right.

## Remarks

Main property: let $L$ be a splitting field of $f$ over $K$ and $w$ an extension of $v$ to $L$.
If the Newton polygon has a segment of slope $s$ and horizontal length $\ell$, then exactly $\ell$ roots of $f$ in $L$ have $w$-value $-s$, counted with multiplicity.
Example: over $\Q_p$, take $f = X^2 + X + p$.
The points are $(0, 1)$, $(1, 0)$, $(2, 0)$, and the lower boundary has a segment of slope $-1$ from $(0, 1)$ to $(1, 0)$ and a segment of slope $0$ from $(1, 0)$ to $(2, 0)$.
So one root has valuation $1$ and one has valuation $0$; this agrees with the product of the roots being $p$ and their sum being $-1$.
Non-example: for $f = 1 + pX + X^2$ the points are $(0, 0)$, $(1, 1)$, $(2, 0)$.
The broken line through all three points is not the Newton polygon, because it is not the graph of a convex function.
The point $(1, 1)$ lies above the segment from $(0, 0)$ to $(2, 0)$, and this segment is the whole Newton polygon.
So both roots have valuation $0$.
</document>

<document source="https://tamnd.github.io/math2/node/D1EBAAFEWV.md" tag="D1EBAAFEWV" kind="concept">
# Hilbert's invariant integral

## Summary

Hilbert's invariant integral is a line integral attached to a [[field of extremals|K4DFT3H9BJ]] of a variational problem.
Its value depends only on the end points of the path, not on the path inside the field.
Along curves of the field it equals the functional itself.
Comparing it with the functional on an arbitrary curve gives the Weierstrass formula, which expresses $J(y) - J(y_0)$ as an integral of the [[Weierstrass excess function|R8TWRPSPNV]].

## Explanations

### L1 human

Inside a field of extremals, each point comes with a preferred slope, the slope of the field curve through it.
Hilbert built from these slopes a line integral that does not care which path one takes between two points.
On a field curve it gives exactly the value of the functional.
So to compare a competitor curve with the field curve, one compares the functional on the competitor with this path-independent integral.
The difference is an integral of the excess function, and if the excess is nonnegative, the field curve wins.
</document>

<document source="https://tamnd.github.io/math2/node/D1FM4ECHKD.md" tag="D1FM4ECHKD" kind="definition">
# Interval arithmetic

## Definition

Let $\mathbb{IR}$ be the set of closed bounded [[intervals|0Q1D85B8G6]] $[a, b] = \set{x \in \R : a \le x \le b}$ with real $a \le b$.
For $X, Y \in \mathbb{IR}$ and an operation $\circ \in \set{+, -, \times, \div}$, set
$$X \circ Y = \set{x \circ y : x \in X, \ y \in Y},$$
where for $\div$ one requires $0 \notin Y$.
*Interval arithmetic* is $\mathbb{IR}$ with these four operations.
For $X = [a, b]$ and $Y = [c, d]$ they are given by:

1. $X + Y = [a + c, \ b + d]$;
2. $X - Y = [a - d, \ b - c]$;
3. $X \times Y = [\min(ac, ad, bc, bd), \ \max(ac, ad, bc, bd)]$;
4. $X \div Y = X \times [1/d, \ 1/c]$ when $0 \notin Y$.

## Remarks

Each set $X \circ Y$ is again a closed bounded interval, because it is the image of the compact connected set $X \times Y$ under a continuous map.
Example: $[1, 2] \times [-1, 3]$ has the products $-1, 3, -2, 6$ of the end points, so it is $[-2, 6]$; indeed $x y$ takes every value between $-2$ and $6$ for $x \in [1, 2]$ and $y \in [-1, 3]$.
Non-example: $[1, 2] \div [-1, 1]$ is not defined, because $0 \in [-1, 1]$ and the set $\set{x / y : y \ne 0}$ is unbounded.
Addition and multiplication are associative and commutative, but $X - X \ne [0, 0]$ unless $X$ has one point, and only the subdistributive law $X \times (Y + Z) \subseteq X \times Y + X \times Z$ holds in general.
For example, $[-1, 1] \times ([1, 1] + [-1, -1]) = [0, 0]$, while $[-1, 1] \times [1, 1] + [-1, 1] \times [-1, -1] = [-2, 2]$.
</document>

<document source="https://tamnd.github.io/math2/node/D1GQNNRDDN.md" tag="D1GQNNRDDN" kind="statement">
# LYM inequality

## Statement

Let $n \ge 1$ be an integer and let $\mathcal{F}$ be a [[family|0C21Q694AG]] of subsets of $[n] = \set{1, \ldots, n}$.
Suppose that $\mathcal{F}$ is an [[antichain|03HF498P1J]] in the [[power set|00E0YZCERH]] ordered by inclusion: no member of $\mathcal{F}$ is a proper subset of another.
Then
$$\sum_{A \in \mathcal{F}} \frac{1}{\binom{n}{|A|}} \le 1.$$

## Remarks

A maximal chain $\emptyset \subset \set{a_1} \subset \set{a_1, a_2} \subset \cdots \subset [n]$ corresponds to an ordering $a_1, \ldots, a_n$ of $[n]$, so there are $n!$ of them.
A set $A$ with $|A| = k$ lies on exactly $k! (n - k)!$ maximal chains.
A chain meets an antichain at most once, so $\sum_{A \in \mathcal{F}} |A|! (n - |A|)! \le n!$, and division by $n!$ gives the inequality.

Since $\binom{n}{k} \le \binom{n}{\floor{n/2}}$ for all $k$, the inequality gives $|\mathcal{F}| \le \binom{n}{\floor{n/2}}$, which is [[Sperner's theorem|0J87V9K1GE]].
Equality holds for the family of all $k$-subsets of $[n]$, for any $k$.
Without the antichain hypothesis the sum can exceed $1$: for $\mathcal{F} = \set{\emptyset, [n]}$ it equals $2$.

## Explanations

### L1 human

Each set of size $k$ gets the weight $1/\binom{n}{k}$, so a full layer of sets of the same size has total weight $1$.
The inequality says that a family in which no set contains another can never have total weight more than one full layer.
So sets of rare sizes are expensive, and sets of the middle size are cheap.

### L2 human

The LYM inequality is the strong form of [[Sperner's theorem|0J87V9K1GE]], and its chain-counting proof is a model double-counting argument.
It extends to posets with a suitable chain structure, called LYM posets.
It is also used, through the same idea of averaging over orderings, in Katona's proof of the [[Erdős–Ko–Rado theorem|FJ92Q6VTJH]].
The [[Kruskal–Katona theorem|G6WNYB0WMY]] gives the exact minimum size of the shadow of a family of sets of one size.
</document>

<document source="https://tamnd.github.io/math2/node/D1MYE6WVZ9.md" tag="D1MYE6WVZ9" kind="definition">
# Derived category of coherent sheaves

## Definition

Let $X$ be a [[Noetherian scheme|ND6XEY68P4]], and let $\operatorname{Coh}(X)$ be the abelian category of [[coherent sheaves|56RXEJ8Z6S]] on $X$.
A [[chain complex|WQ82C0M3QB]] $F^\bullet$ of coherent sheaves is *bounded* if $F^i = 0$ for all but finitely many $i$.
Consider the category of bounded complexes in $\operatorname{Coh}(X)$ with chain maps.
The *bounded derived category of coherent sheaves* of $X$ is the category $D^b(X) = D^b(\operatorname{Coh}(X))$ obtained from it by formally inverting all [[quasi-isomorphisms|C9JEC2WQKV]].
It is the bounded [[derived category|5ST7NVJFSM]] of $\operatorname{Coh}(X)$.

## Remarks

$D^b(X)$ is a [[triangulated category|T2V7GJJ2EN]] with shift $F^\bullet \mapsto F^\bullet[1]$.
It is equivalent to the full subcategory of the derived category of all $\mathcal{O}_X$-modules whose objects are the complexes with coherent cohomology sheaves, nonzero in finitely many degrees.
Example: for $X = \operatorname{Spec} k$ with $k$ a field, coherent sheaves are finite-dimensional vector spaces, and every object of $D^b(X)$ is isomorphic to $\bigoplus_i H^i(F^\bullet)[-i]$, its cohomology placed in each degree.
Non-example: on $X = \mathbb{A}^1_k = \operatorname{Spec} k[x]$, the complex $k[x] \xrightarrow{x} k[x]$ in degrees $-1, 0$ maps to $k[x]/(x)$ in degree $0$ by a quasi-isomorphism.
In the homotopy category of complexes, before quasi-isomorphisms are inverted, these two complexes are not isomorphic, since there is no nonzero map of modules $k[x]/(x) \to k[x]$.
In $D^b(X)$ they are isomorphic.
Theorems on $D^b$: Beilinson showed that $D^b(\mathbb{P}^n)$ is generated by $\mathcal{O}, \mathcal{O}(1), \ldots, \mathcal{O}(n)$, and Mukai showed that an abelian variety and its dual have equivalent derived categories.
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<document source="https://tamnd.github.io/math2/node/D1PAT0K446.md" tag="D1PAT0K446" kind="concept">
# Littlewood–Paley decomposition

## Summary

The Littlewood–Paley decomposition writes a function $f$ on $\R^n$ as a sum $f = \sum_{j \in \Z} \Delta_j f$ of pieces whose [[Fourier transforms|DPVM34ADZV]] are supported in dyadic annuli $\set{2^{j-1} < \abs{\xi} < 2^{j+1}}$.
The pieces are obtained by smooth cut-offs in frequency, so each $\Delta_j f$ is a convolution of $f$ with a rescaled Schwartz function.
It is the basic construction of [[Littlewood–Paley theory|AA29VBWBFH]], and it is used to define Besov and Triebel–Lizorkin spaces and paraproducts.

## Explanations

### L1 human

The decomposition separates a function into frequency bands, each one octave wide.
The piece $\Delta_j f$ contains the oscillations of $f$ at wavelengths about $2^{-j}$, while coarser and finer details are filtered out.
Smooth filters are used instead of sharp cut-offs, so that each piece is still nicely localized in space.
For instance, $\cos(2\pi \cdot 5x)$ on $\R$ has frequencies $\pm 5$, and only the pieces with $j = 2$ and $j = 3$ can be nonzero, since $2^{j-1} < 5 < 2^{j+1}$ holds only for these $j$.
</document>

<document source="https://tamnd.github.io/math2/node/D1QA6N9EWJ.md" tag="D1QA6N9EWJ" kind="definition">
# Difference set

## Definition

Let $G$ be a finite [[group|G5S1M9]] of order $v$ with identity $e$, and let $k$ and $\lambda$ be positive integers.
A $(v, k, \lambda)$ difference set in $G$ is a [[subset|0BJR9QR08E]] $D \subseteq G$ with $k$ elements with the following property.
Each element $g \ne e$ of $G$ can be written as $g = d_1 d_2^{-1}$ with $d_1, d_2 \in D$ in exactly $\lambda$ ways.

## Remarks

When $G$ is abelian and written additively, the condition is about differences $d_1 - d_2$.
Counting the $k(k - 1)$ pairs $(d_1, d_2)$ with $d_1 \ne d_2$ gives $k(k-1) = \lambda (v - 1)$.
Example: $D = \set{1, 2, 4}$ in $\Z/7\Z$.
The six differences are $2 - 1 = 1$, $1 - 2 = 6$, $4 - 1 = 3$, $1 - 4 = 4$, $4 - 2 = 2$, $2 - 4 = 5$, so each nonzero residue occurs once, and $D$ is a $(7, 3, 1)$ difference set.
Non-example: $D = \set{0, 1, 2}$ in $\Z/7\Z$ is not a difference set, since $1 = 1 - 0 = 2 - 1$ occurs twice while $3$ does not occur.
If $D$ is a $(v, k, \lambda)$ difference set with $k < v$, the translates $Dg = \set{dg : d \in D}$ for $g \in G$ are the blocks of a symmetric $2$-$(v, k, \lambda)$ [[block design|TGZATGBHTH]].
</document>

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# Noncommutative ring

## Summary

A noncommutative ring is a [[ring|0B1261B5ED]] in which the order of the factors in a product can matter.
Matrix rings, quaternions, endomorphism rings and group rings of nonabelian groups are the main sources of examples.
Many tools of commutative algebra split into left and right versions in this setting.
The study of these rings is the subject of noncommutative ring theory.

## Explanations

### L1 human

In ordinary arithmetic $ab = ba$, but many natural products do not behave this way.
Doing one rotation of space and then another is in general not the same as doing them in the other order.
Square matrices model this: with
$$A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},$$
one gets $AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $BA = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$.
So the ring of $2 \times 2$ real matrices is a noncommutative ring.
</document>

<document source="https://tamnd.github.io/math2/node/D1SYF8F8RQ.md" tag="D1SYF8F8RQ" kind="definition">
# Lorentzian manifold

## Definition

Let $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] of dimension $n \ge 2$.
It is a *Lorentzian manifold* if, at each point $p \in M$, the form $g_p$ has [[signature|07EZA80NEH]] $(n - 1, 1)$.
That is, in a suitable basis of $T_pM$ the matrix of $g_p$ is $\operatorname{diag}(-1, 1, \ldots, 1)$.

## Remarks

Example: Minkowski space is $\R^n$ with coordinates $(t, x_1, \ldots, x_{n-1})$ and $g = -dt^2 + dx_1^2 + \cdots + dx_{n-1}^2$.
Its matrix is $\operatorname{diag}(-1, 1, \ldots, 1)$ at every point, with one negative and $n - 1$ positive entries.

Non-example: $\R^4$ with $g = -dt^2 - dx^2 + dy^2 + dz^2$ is pseudo-Riemannian, but its signature has two negative entries, so it is not Lorentzian.

Many physics texts use the opposite convention, signature $(1, n - 1)$, that is $\operatorname{diag}(1, -1, \ldots, -1)$; the two conventions differ by replacing $g$ with $-g$.
A nonzero tangent vector $v$ is called timelike if $g(v, v) < 0$, null if $g(v, v) = 0$, and spacelike if $g(v, v) > 0$ (in the convention used here).
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<document source="https://tamnd.github.io/math2/node/D1T39KHPDN.md" tag="D1T39KHPDN" kind="problem">
# Suslin's problem

## Problem

Let $(L, <)$ be a nonempty [[total order|0316AYH1SC]] with these properties:

1. $L$ has no least and no greatest element.
2. $L$ is dense: between any two elements there is a third.
3. $L$ is complete: each nonempty subset of $L$ that is bounded above has a least upper bound.
4. Each family of pairwise disjoint nonempty open intervals of $L$ is countable.

Must $L$ be order isomorphic to the real line $\R$?

## Remarks

The real line is characterised by clauses 1 to 3 together with separability: $L$ has a countable subset that meets each nonempty open interval.
So the question asks whether the countable chain condition, clause 4, can replace [[separability|0GKJ78948V]].
A counterexample is called a Suslin line, and Suslin's hypothesis says that no Suslin line exists.
The problem is independent of ZFC, if ZFC is consistent.
Jensen showed that the [[diamond principle|HFQSSB1Z23]], which holds in the constructible universe, implies that a Suslin line exists.
Solovay and Tennenbaum showed that Suslin's hypothesis is consistent, and it follows from [[Martin's axiom|Z8XMCCZFA4]] together with the failure of the continuum hypothesis.
A Suslin line exists if and only if a Suslin tree exists, that is, an [[Aronszajn tree|KKT4JYJEJJ]] in which each antichain is countable.

## Explanations

### L1 human

The real numbers form a line with no ends and no gaps, and a countable set, the rationals, is spread all through it.
Suslin asked whether a weaker condition can replace the countable dense set: that one can only fit countably many disjoint intervals into the line.
The answer turned out to be that the usual axioms of mathematics cannot decide the question.
In some models of set theory there are strange lines, Suslin lines, that pass this test but are not the real line.
</document>

<document source="https://tamnd.github.io/math2/node/D1TK8AHMCM.md" tag="D1TK8AHMCM" kind="definition">
# Moments and orthogonal polynomials

## Definition

Let $(m_n)_{n \ge 0}$ be a sequence of real numbers.
For $n \ge 0$ let $\Delta_n = \det (m_{i+j})_{i,j=0}^{n}$ be the [[determinant|00KDZRW4C0]] of its $(n+1) \times (n+1)$ [[Hankel matrix|PQ1D8FQB3E]], and put $\Delta_{-1} = 1$.
Assume that $\Delta_n > 0$ for all $n \ge 0$.
The *moment functional* of $(m_n)$ is the [[linear form|07YQ1C78G0]] $L$ on the real polynomials $\R[x]$ with $L(x^n) = m_n$ for all $n \ge 0$.
The *monic orthogonal polynomials* of $(m_n)$ are the polynomials $P_0 = 1$ and, for $n \ge 1$,
$$P_n(x) = \frac{1}{\Delta_{n-1}} \det \begin{pmatrix} m_0 & m_1 & \cdots & m_n \\ m_1 & m_2 & \cdots & m_{n+1} \\ \vdots & \vdots & & \vdots \\ m_{n-1} & m_n & \cdots & m_{2n-1} \\ 1 & x & \cdots & x^n \end{pmatrix}.$$
Each $P_n$ is monic of degree $n$, and $L(P_m P_n) = 0$ for $m \ne n$.

## Remarks

Orthogonality holds because, for $k < n$, applying $L$ to $x^k P_n$ replaces the last row by $(m_k, \ldots, m_{n+k})$, which equals row $k$, so the determinant vanishes.
The leading coefficient is $1$ since the cofactor of $x^n$ is $\Delta_{n-1}$.
Also $L(P_n^2) = \Delta_n / \Delta_{n-1} > 0$.
Let $(m_n)$ be the moment sequence of a positive Borel measure $\mu$ on $\R$ with infinite support and all moments finite.
Then $\Delta_n > 0$ for all $n$, and the $P_n$ are the monic [[orthogonal polynomials|A18HPFPC4D]] of $\mu$ in $L^2(\mu)$.

Example: for Lebesgue measure on $[-1, 1]$ one has $m_0 = 2$, $m_1 = 0$, $m_2 = \tfrac23$ and $\Delta_0 = 2$, $\Delta_1 = \tfrac43$.
The formula gives $P_1(x) = \tfrac12 \det \begin{pmatrix} 2 & 0 \\ 1 & x \end{pmatrix} = x$ and
$P_2(x) = \tfrac34 \left( \tfrac43 x^2 - \tfrac49 \right) = x^2 - \tfrac13$, the monic Legendre polynomials.

Non-example: the moments of the Dirac measure at $0$ are $m_0 = 1$ and $m_n = 0$ for $n \ge 1$.
Then $\Delta_1 = \det \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = 0$, so the condition $\Delta_n > 0$ fails.
Indeed $L(x^2) = 0$, and no monic polynomial of degree $1$ is orthogonal to $1$ with nonzero norm.

## Explanations

### L1 human

All the information needed to build orthogonal polynomials sits in the moments.
The determinant formula turns a list of moments directly into the polynomials, one degree at a time.
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# Order of a projective plane

## Definition

Let $\Pi$ be a [[projective plane|DANFNN8C4Y]] with a finite set of points.
The *order* of $\Pi$ is the integer $n$ such that one line of $\Pi$, and then every line, contains exactly $n + 1$ points.

## Remarks

The definition does not depend on the line.
Given two distinct lines $\ell$ and $m$, choose a point $O$ on neither; the map $P \mapsto OP \cap m$ is a bijection from the points of $\ell$ to the points of $m$.
For a point $P$ and a line $\ell$ not through $P$, the map $Q \mapsto PQ$ is a bijection from the points of $\ell$ to the lines through $P$.
So every point lies on exactly $n + 1$ lines.
Since a projective plane has four points with no three on a line, every line has at least three points, so $n \ge 2$.

Example: in the [[Fano plane|QQ8GW1ASZM]] each line has $3$ points, so its order is $2$.
In $PG(2, q)$, a line is a $2$-dimensional subspace of $\F_q^3$, which contains $(q^2 - 1)/(q - 1) = q + 1$ one-dimensional subspaces; so the order is $q$.

Non-example: the real projective plane has infinitely many points on each line, so it has no order in this sense.

In Mathlib the order of a finite projective plane is `Configuration.ProjectivePlane.order`, defined through the number of lines through a point, which gives the same number.
</document>

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# Topological transitivity

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
The map $f$ is *topologically transitive* if for every pair of nonempty [[open sets|02BH3MCBQB]] $U, V \subseteq X$ there is an integer $n \ge 1$ with $f^n(U) \cap V \neq \emptyset$.

## Remarks

An equivalent form is $U \cap f^{-n}(V) \neq \emptyset$, since both say that some point of $U$ is sent into $V$ by $f^n$.
Let $X$ be a compact metric space without isolated points.
Then $f$ is topologically transitive if and only if some forward orbit $\set{f^n(x) : n \ge 0}$ is [[dense|0BSJSX7ME2]] in $X$.
In that case the points with dense forward orbit form a dense $G_\delta$ set.
Example: let $f(x) = x + \alpha \bmod 1$ on the circle $\R / \Z$ with $\alpha$ irrational.
Every forward orbit is dense, so for nonempty open $U, V$ and a point $x \in U$, some $f^n(x)$ with $n \ge 1$ lies in $V$.
Non-example: the identity map of $[0, 1]$, with $U = (0, 1/3)$ and $V = (2/3, 1)$, has $f^n(U) \cap V = U \cap V = \emptyset$ for all $n$.

## Explanations

### L1 human

A transitive map moves some point of each open region into each other open region, at some later time.
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# Maximum principle

## Statement

Let $U \subset \R^n$ be a bounded open set.
Let
$$L u = \sum_{i,j=1}^{n} a_{ij}(x) \, \partial_i \partial_j u + \sum_{i=1}^{n} b_i(x) \, \partial_i u$$
with continuous coefficients on the closure $\overline{U}$, where $a_{ij} = a_{ji}$.
Assume that $L$ is uniformly [[elliptic|XY8GEQSWPR]]: there is $\theta > 0$ with $\sum_{i,j} a_{ij}(x) \xi_i \xi_j \ge \theta \abs{\xi}^2$ for all $x \in U$ and $\xi \in \R^n$.
Let $u$ be twice continuously differentiable on $U$ and continuous on $\overline{U}$, with $L u \ge 0$ in $U$.

1. (Weak maximum principle.) $\max_{\overline{U}} u = \max_{\partial U} u$.
2. (Strong maximum principle.) If moreover $U$ is connected and $u$ attains its maximum over $\overline{U}$ at a point of $U$, then $u$ is constant on $U$.

## Remarks

Proof idea for part 1: suppose that $L u > 0$ and $u$ had an interior maximum at $x_0$.
Then $\nabla u(x_0) = 0$ and the Hessian of $u$ at $x_0$ would be negative semidefinite, which forces $L u(x_0) \le 0$.
For $L u \ge 0$, apply this to $u + \varepsilon e^{\lambda x_1}$ with $\lambda$ large, and let $\varepsilon \to 0$.
Part 2 follows from the Hopf lemma on the boundary normal derivative at a maximum point.
Replacing $u$ by $-u$ gives the corresponding minimum principles for $L u \le 0$.

Example: every solution of [[Laplace's equation|S8RG6N1PRT]] on a bounded open set takes its maximum and its minimum on the boundary.
So a solution of the Dirichlet problem with boundary values between $0$ and $1$ lies between $0$ and $1$, and the Dirichlet problem has at most one solution.

The hypotheses are needed.
Boundedness of $U$: on the half-plane $\set{x_1 > 0}$, the harmonic function $u = x_1$ vanishes on the boundary but is positive inside.
The absence of a zeroth-order term with the wrong sign: on $(0, \pi)$, $u = \sin x$ satisfies $u'' + u = 0$, vanishes at both ends and equals $1$ at $\pi/2$.
With a term $c(x) u$ and $c \le 0$, part 1 still holds in the form $\max_{\overline{U}} u \le \max_{\partial U} u^+$, where $u^+ = \max(u, 0)$.

## Explanations

### L1 human

The maximum principle says that a solution of an elliptic equation cannot have a peak inside the region: its largest value is on the boundary.
For a temperature in equilibrium, the hottest point is on the edge, since a hot spot inside would spread its heat to the cooler surroundings.
If the largest value is reached inside a connected region, the solution must be constant.

### L2 human

The maximum principle is the main tool for uniqueness, comparison and bounds for second-order elliptic equations, without any integral estimates.
It gives uniqueness for the Dirichlet problem, and comparison: if $L u \ge L v$ in $U$ and $u \le v$ on $\partial U$, then $u \le v$ in $U$.
It also underlies Perron's method for the Dirichlet problem, Harnack inequalities and the theory of viscosity solutions.
There are parabolic versions for the heat equation, where the maximum is attained on the bottom or sides of a space-time cylinder.
</document>

<document source="https://tamnd.github.io/math2/node/D2KC8NEAPS.md" tag="D2KC8NEAPS" kind="example">
# Liénard equation

## Example

Let $f \colon \R \to \R$ be a continuous [[even function|0FP31SX5EA]] and let $g \colon \R \to \R$ be a continuous odd function.
The Liénard equation is the second-order [[ordinary differential equation|0NT64JV96A]]
$$x'' + f(x)\, x' + g(x) = 0.$$
Let $F(x) = \int_0^x f(s)\, ds$.
With $y = x' + F(x)$, the equation becomes the planar [[autonomous system|0673D85QKZ]]
$$x' = y - F(x), \qquad y' = -g(x).$$
Check: from $x' = y - F(x)$ follows $x'' = y' - f(x) x' = -g(x) - f(x) x'$, which is the Liénard equation.

## Remarks

The term $f(x) x'$ is a damping that depends on the position.
Where $f(x) < 0$ it pumps energy into the motion, and where $f(x) > 0$ it removes energy.
The term $g(x)$ is a restoring force.
For $f = 0$ and $g(x) = x$, the equation is the [[harmonic oscillator|0YG0ZJPTWG]].
For $f(x) = -\mu (1 - x^2)$ with $\mu > 0$ and $g(x) = x$, it is the van der Pol equation $x'' - \mu (1 - x^2) x' + x = 0$.
Liénard's theorem gives conditions on $f$ and $g$ under which the system has exactly one limit cycle.

## Explanations

### L1 human

The Liénard equation describes a particle on a line pulled back toward the origin by the force $-g(x)$, with a friction coefficient $f(x)$ that depends on where the particle is.
If the friction is negative near the origin and positive far away, small motions grow and large motions shrink.
The motion then settles on one oscillation of a definite size.
</document>

<document source="https://tamnd.github.io/math2/node/D2M4VXJ7PA.md" tag="D2M4VXJ7PA" kind="concept">
# Nuclear C*-algebra

## Summary

A nuclear C\*-algebra is a C*-algebra whose algebraic tensor product with any other C\*-algebra has only one C\*-norm.
Equivalently, the identity map of the algebra can be approximated by [[completely positive maps|CGY299QMC3]] that factor through matrix algebras.
Nuclearity is a finiteness or amenability property: for C\*-algebras it is equivalent to amenability as a Banach algebra.
It is a standing hypothesis in the classification program for simple C\*-algebras.

## Explanations

### L1 human

To combine two C\*-algebras into one, one forms their tensor product and must choose a norm on it.
In general there can be many reasonable choices, which give different completions.
Nuclear algebras are the well-behaved ones for which there is no choice to make.
All commutative C\*-algebras, all finite-dimensional ones and all AF algebras are nuclear.
</document>

<document source="https://tamnd.github.io/math2/node/D2M8BSW720.md" tag="D2M8BSW720" kind="statement">
# Bass's theorem P

## Statement

Let $R$ be a [[ring|0894J980RK]] with identity and $J = J(R)$ its [[Jacobson radical|XA0DCNH7WY]].
The following are equivalent.

1. $R/J$ is a semisimple ring and $J$ is left T-nilpotent, that is, $R$ is [[left perfect|BJMEXPKRYM]].
2. Every left $R$-module has a [[projective cover|WGYXR97NPK]].
3. $R$ satisfies the descending chain condition on principal right ideals.
4. Every flat left $R$-module is [[projective|CRE6VEKT32]].

## Remarks

The link between 1 and 2 goes through Nakayama-type lemmas.
Left T-nilpotence of $J$ is exactly what makes $JM \ne M$ for every nonzero left module $M$, finitely generated or not.
Then a projective cover of $M$ can be built by lifting a projective cover of the semisimple module $M/JM$.
Condition 3 is notable for its side switch: a condition on left modules is equivalent to a chain condition on right ideals.
Example: left Artinian rings satisfy all four conditions, so over them every left module has a projective cover and flat left modules are projective.
Over $\Z$, which is not perfect, the flat module $\Q$ is not projective, and the module $\Z/2\Z$ has no projective cover.

## Explanations

### L1 human

The theorem gives four very different descriptions of the same class of rings.
One is about the structure of the ring, one about covers of modules, one about chains of ideals, and one about flat modules.
That they all agree is a sign that perfect rings are a natural class.

### L2 human

The theorem is the main characterization of [[perfect rings|49Y9Q949PK]] and appears in the study of projective covers and flat modules.
Its finitely generated analogue characterizes [[semiperfect rings|ZRC2S3ECFK]]: every finitely generated left module has a projective cover exactly when $R$ is semiperfect.
Because [[semiprimary rings|XVP517QRH5]] have nilpotent radical, they satisfy condition 1 on both sides.
</document>

<document source="https://tamnd.github.io/math2/node/D2R7BNFZNS.md" tag="D2R7BNFZNS" kind="definition">
# Ore extension

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity and let $\sigma \colon R \to R$ be a [[ring endomorphism|0VTH8KPCG2]] with $\sigma(1) = 1$.
A *$\sigma$-derivation* of $R$ is an additive map $\delta \colon R \to R$ with $\delta(ab) = \sigma(a)\delta(b) + \delta(a) b$ for all $a, b \in R$.
Given such $\sigma$ and $\delta$, the *Ore extension* $R[x; \sigma, \delta]$ is the ring $S$ with these properties:

1. $S$ contains $R$ as a subring and contains an element $x$.
2. As a left $R$-[[module|0NZ8JA25WK]], $S$ is [[free|0DZ9DKAKZR]] with basis $1, x, x^2, \dots$.
3. $x a = \sigma(a) x + \delta(a)$ for all $a \in R$.

## Remarks

For every endomorphism $\sigma$ and $\sigma$-derivation $\delta$, such a ring exists and is unique up to an isomorphism fixing $R$ and $x$.
Its elements are written $a_0 + a_1 x + \cdots + a_n x^n$, and rule 3 together with associativity determines all products.
Example: let $k$ be a field, $R = k[y]$, $\sigma = \id$ and $\delta = d/dy$.
The map $\delta$ satisfies $\delta(ab) = a \delta(b) + \delta(a) b$, the Leibniz rule, so it is an $\id$-derivation, and rule 3 reads $x y = y x + 1$; this ring is the first [[Weyl algebra|98QE7KY55W]].
Non-example: take $R = k[y]$, $\sigma = \id$ and $\delta = \id$.
Then $\delta(1 \cdot 1) = 1$ but $1 \cdot \delta(1) + \delta(1) \cdot 1 = 2$, and $1 \ne 2$ in every field, so $\delta$ is not an $\id$-derivation.
No ring as above exists, since rule 3 would give $x = x \cdot 1 = 1 \cdot x + \delta(1) = x + 1$.
When $\delta = 0$ one writes $R[x; \sigma]$, the skew polynomial ring; when $\sigma = \id$ one writes $R[x; \delta]$, a differential polynomial ring.
If $R$ is a [[domain|0Q4CBD3HX0]] and $\sigma$ is injective, then $R[x; \sigma, \delta]$ is a domain.
If $R$ is left Noetherian and $\sigma$ is an automorphism, then $R[x; \sigma, \delta]$ is left Noetherian, a form of the Hilbert basis theorem.
</document>

<document source="https://tamnd.github.io/math2/node/D31S3GR51T.md" tag="D31S3GR51T" kind="concept">
# Semiring

## Summary

A semiring is a set with an addition and a multiplication that obey the axioms of a [[ring|0B1261B5ED]], except that negatives need not exist.
The natural numbers with the usual operations are the first example.
Every ring is a semiring, but semirings also include the Boolean semiring and the tropical semirings.
Semirings appear in combinatorics, in tropical geometry and in computer science, for example in shortest path problems and formal languages.

## Explanations

### L1 human

A semiring is a place where you can add and multiply as usual, but you may not be able to subtract.
The counting numbers $0, 1, 2, \dots$ are the model: $3 - 5$ is not a counting number.
Another example uses "minimum" as addition and ordinary $+$ as multiplication on the real numbers together with $\infty$.
There, the "sum" of $3$ and $5$ is $3$, and their "product" is $8$.
This strange arithmetic is exactly what you need to compute shortest paths in a road network.
</document>

<document source="https://tamnd.github.io/math2/node/D33JK3AXRZ.md" tag="D33JK3AXRZ" kind="example">
# Bernoulli scheme

## Example

Let $A = \set{1, \dots, k}$ be a finite alphabet and let $p = (p_1, \dots, p_k)$ be a probability vector, with $p_i \ge 0$ and $p_1 + \cdots + p_k = 1$.
Let $X = A^{\Z}$ be the set of two-sided sequences $x = (x_n)_{n \in \Z}$, with the σ-algebra generated by the cylinder sets
$$[a_m, \dots, a_n]_m = \set{x \in X : x_i = a_i \text{ for } m \le i \le n}.$$
Let $\mu$ be the [[infinite product measure|2HWVFHM13R]] of copies of $p$, so $\mu([a_m, \dots, a_n]_m) = p_{a_m} p_{a_{m+1}} \cdots p_{a_n}$.
The shift $\sigma \colon X \to X$, $(\sigma x)_n = x_{n+1}$, is the *Bernoulli scheme* or *Bernoulli shift* $B(p_1, \dots, p_k)$.

Check that $\sigma$ is a [[measure-preserving transformation|556WBDAHNP]]: $x \in \sigma^{-1}[a_m, \dots, a_n]_m$ means $x_{i+1} = a_i$ for $m \le i \le n$.
So $\sigma^{-1}[a_m, \dots, a_n]_m = [a_m, \dots, a_n]_{m+1}$, which has the same measure $p_{a_m} \cdots p_{a_n}$.
Cylinder sets form a π-system generating the σ-algebra, so $\mu(\sigma^{-1} E) = \mu(E)$ for every measurable $E$.

The coordinate functions $x \mapsto x_n$ form an i.i.d. sequence with law $p$, a two-sided version of the [[Bernoulli process|0CQQG3DEW9]] when $k = 2$.
The shift is invertible, [[mixing|S3RJQ9CX2P]], and a [[Kolmogorov automorphism|YRHJA7C7HS]].
Its [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] is $-\sum_{i=1}^{k} p_i \log p_i$, computed with the generating partition into the sets $\set{x : x_0 = i}$.
The one-sided version uses $A^{\N}$ instead of $A^{\Z}$ and gives a non-invertible measure-preserving map.

## Explanations

### L1 human

A Bernoulli scheme is the model of rolling the same loaded die once every second, forever into the past and into the future.
A point of the space is a full record of all the rolls, and the shift moves the clock forward by one second.
Because the rolls are independent and identically distributed, moving the clock does not change the probabilities, so the shift preserves the measure.
Bernoulli schemes are the most random measure-preserving systems and serve as the yardstick for the others.
</document>

<document source="https://tamnd.github.io/math2/node/D33V8HE0KY.md" tag="D33V8HE0KY" kind="concept">
# Opposite structure

## Summary

The opposite of a structure with a [[binary operation|0M418FE7BB]] $\ast$ has the same elements and the reversed operation $a \ast^{\mathrm{op}} b = b \ast a$.
The opposite of a [[group|G5S1M9]] is a group, and the opposite of a [[monoid|0N6R902FG1]] is a monoid.
The [[opposite ring|M8Q24XB0AF]] reverses the multiplication and keeps the addition.
Opposite structures turn left notions into right notions, for example left modules over a ring into right modules over its opposite ring.
A structure equals its opposite exactly when its operation is [[commutative|099E72KTF8]].

## Explanations

### L1 human

The opposite structure reads every product backwards.
If $a$ then $b$ gives $a \ast b$, in the opposite structure $a$ then $b$ gives $b \ast a$.
For the composition of moves of a puzzle, the opposite means doing the moves in the other order.
For addition of numbers, nothing changes, because $a + b = b + a$.
</document>

<document source="https://tamnd.github.io/math2/node/D34QHZVPES.md" tag="D34QHZVPES" kind="statement">
# Adams–Novikov spectral sequence

## Statement

Let $MU$ be the [[complex cobordism|5CZHFNY9F3]] spectrum, with coefficient ring $MU_* = \pi_* MU$ and $MU_* MU = \pi_*(MU \wedge MU)$.
Then $(MU_*, MU_* MU)$ is a Hopf algebroid, and for every [[spectrum|1GHF4EKBXS]] $X$ the group $MU_* X$ is a comodule over it.
Let $X$ be a connective spectrum.
Then there is a [[spectral sequence|GG8AYV847V]] $(E_r^{s,t}, d_r)_{r \ge 2}$.
Its differentials are $d_r \colon E_r^{s,t} \to E_r^{s+r, t+r-1}$, and its second page is
$$E_2^{s,t} = \operatorname{Ext}^{s,t}_{MU_* MU}(MU_*, MU_* X),$$
the [[Ext groups|4P7GBWJ4RR]] in the category of $MU_* MU$-comodules, and the spectral sequence converges to $\pi_{t-s}(X)$.

## Remarks

The construction is the same as for the [[Adams spectral sequence|DD4D0EKX74]], with the mod $p$ Eilenberg–MacLane spectrum replaced by $MU$.
Since $\pi_0 MU = \Z$ and $X$ is connective, the $MU$-based tower recovers $X$ itself, which gives convergence to $\pi_*(X)$ with no completion.
After localizing at a prime $p$, one may replace $MU$ by the [[Brown–Peterson spectrum|S8S72AA9C8]] $BP$, with $E_2^{s,t} = \operatorname{Ext}^{s,t}_{BP_* BP}(BP_*, BP_* X)$, which is smaller and easier to compute.

Example: for $X = S$, the [[sphere spectrum|14CGGHA85T]], the groups $MU_*$ and $MU_* MU$ are concentrated in even degrees, so $E_2^{s,t} = 0$ for odd $t$.
Also $E_2^{0,t}$ is $\Z$ for $t = 0$ and $0$ otherwise.

## Explanations

### L1 human

The Adams–Novikov spectral sequence is a machine for computing stable homotopy groups, like the Adams spectral sequence.
It replaces ordinary mod $p$ cohomology by complex cobordism, a richer cohomology theory built from complex manifolds.
The starting page is harder to compute, but far fewer corrections are needed afterwards.

### L2 human

Through Quillen's theorem, $MU_* MU$-comodules are related to formal group laws, so the $E_2$ term is a cohomology of the moduli of formal groups.
This links the stable homotopy groups of spheres to the height filtration of formal groups and to chromatic homotopy theory.
At odd primes the first differential in the $BP$ version that can be nonzero is $d_{2p-1}$, because the $E_2$ term is concentrated in internal degrees divisible by $2(p-1)$.
</document>

<document source="https://tamnd.github.io/math2/node/D3649AJ30F.md" tag="D3649AJ30F" kind="concept">
# Whitehead tower

## Summary

The Whitehead tower of a space $X$ is a sequence of spaces $\cdots \to X\langle 2 \rangle \to X\langle 1 \rangle \to X$.
The stage $X\langle n \rangle$ is $n$-connected and has the same homotopy groups as $X$ above degree $n$.
It is dual to the [[Postnikov tower|EKGEZGHYBW]]: instead of discarding the high homotopy groups, it discards the low ones.
Its first stage is the [[universal cover|Y1YPE6PM52]], and each further stage is a fibration over the previous one with fibre an [[Eilenberg–MacLane space|AJHEYJE5XY]].
It is used to compute homotopy groups with spectral sequences and to define string structures through the stage $BO\langle 8 \rangle$.

## Explanations

### L1 human

The Whitehead tower peels off the homotopy groups of a space from the bottom.
The first step replaces the space by its universal cover, which removes the fundamental group.
The next step removes the second homotopy group, and so on.
Each stage keeps all the higher homotopy groups unchanged.
For example, the $3$-connected cover of the $3$-sphere has the same homotopy groups as $S^3$ in degrees $4$ and above, but none below.
</document>

<document source="https://tamnd.github.io/math2/node/D38YWGXRKJ.md" tag="D38YWGXRKJ" kind="definition">
# Space of bounded continuous functions

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $\mathbb{K}$ be $\R$ or $\C$.
The *space of bounded continuous functions* $C_b(X)$ is the set of [[continuous functions|0X3WGX95FE]] $f \colon X \to \mathbb{K}$ such that $\sup_{x \in X} \abs{f(x)} < \infty$.
It is a vector space under pointwise addition and scalar multiplication.
It is a [[normed space|03J5RAKB30]] with the norm
$$\norm{f}_\infty = \sup_{x \in X} \abs{f(x)},$$
with $\norm{f}_\infty = 0$ when $X$ is empty.

## Remarks

Example: $f(x) = \sin x$ on $X = \R$ is continuous, and $\abs{\sin x} \le 1$ with equality at $x = \pi/2$, so $f \in C_b(\R)$ with $\norm{f}_\infty = 1$.

Non-example: $g(x) = x$ on $\R$ is continuous, but $\sup_x \abs{g(x)} = \infty$, so $g \notin C_b(\R)$.

The [[supremum|0BTMQ04556]] is finite by the definition, and the norm axioms follow from those of $\abs{\cdot}$.
The space $C_b(X)$ is complete, so it is a Banach space.
If $X$ is compact, $C_b(X)$ is the space $C(X)$ of all continuous functions on $X$.
If $X$ is a set with the discrete topology, $C_b(X)$ is the space $\ell^\infty(X)$ of bounded functions.
</document>

<document source="https://tamnd.github.io/math2/node/D38ZP0YX0D.md" tag="D38ZP0YX0D" kind="definition">
# Quasi-measure-preserving map

## Definition

Let $(X, \mathcal{A}, \mu)$ and $(Y, \mathcal{B}, \nu)$ be [[measure spaces|P15VQS5DJ9]].
A map $T \colon X \to Y$ is *quasi-measure-preserving* if two conditions hold.

1. $T$ is [[measurable|YYNCJSR0PJ]]: $T^{-1}(B) \in \mathcal{A}$ for every $B \in \mathcal{B}$.
2. If $B \in \mathcal{B}$ and $\nu(B) = 0$, then $\mu(T^{-1}(B)) = 0$.

## Remarks

Condition 2 says that the pushforward $T_* \mu$ is [[absolutely continuous|P0P21KDV6E]] with respect to $\nu$, that is, preimages of [[null sets|VQQM4Y550T]] are null sets.
Every [[measure-preserving|556WBDAHNP]] map satisfies it.
Example: $T(x) = 2x$ on $\R$ with Lebesgue measure $\lambda$.
For a Borel set $B$ one has $\lambda(T^{-1}(B)) = \lambda(B)/2$, so null sets pull back to null sets, but $T$ is not measure-preserving.
Non-example: the constant map $T(x) = 0$ from $[0,1]$ to $\R$, both with Lebesgue measure, is measurable.
But $\lambda(\set{0}) = 0$ while $\lambda(T^{-1}(\set{0})) = \lambda([0,1]) = 1$, so condition 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/D39NBQH3H6.md" tag="D39NBQH3H6" kind="statement">
# Euler's formula for polytopes

## Statement

Let $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d \ge 1$, with [[f-vector|872BAKDQQK]] $(f_0, \dots, f_{d-1})$.
Then
$$\sum_{i=0}^{d-1} (-1)^i f_i = f_0 - f_1 + f_2 - \cdots + (-1)^{d-1} f_{d-1} = 1 - (-1)^d.$$

## Remarks

With $f_{-1} = 1$ for the empty face and $f_d = 1$ for $P$, the relation reads $\sum_{i=-1}^{d} (-1)^i f_i = 0$.
One proof is topological: the boundary of $P$ is homeomorphic to the sphere $S^{d-1}$, and its faces form a cell decomposition.
So the alternating sum is the [[Euler characteristic|RNDDWDAFJ5]] $1 + (-1)^{d-1}$ of the sphere.
A combinatorial proof builds the boundary facet by facet in a shelling order and checks that the alternating sum changes correctly at each step.

For $d = 3$ the formula is [[Euler's polyhedron formula|09HT3P35YP]] $f_0 - f_1 + f_2 = 2$.
For $d = 4$ the alternating sum is $0$; the $4$-cube has $16 - 32 + 24 - 8 = 0$.
For a polygon with $n$ vertices, $n - n = 0 = 1 - (-1)^2$.

It is the only linear equation satisfied by the f-vectors of all $d$-polytopes, up to scaling.
For simplicial polytopes there are further linear relations, the [[Dehn–Sommerville equations|4WKHF5BCTA]].

## Explanations

### L1 human

Count the faces of each dimension of a convex polytope and add them with alternating signs.
The answer does not depend on the polytope, only on its dimension: it is $2$ in odd dimensions and $0$ in even dimensions.
In three dimensions this is the classical rule vertices minus edges plus faces equals $2$.

### L2 human

The relation $\sum (-1)^i f_i = 1 - (-1)^d$ is the Euler–Poincaré relation.
It is the first constraint any candidate f-vector must meet, and in dimension $3$ together with the inequalities $f_2 \le 2 f_0 - 4$ and $f_0 \le 2 f_2 - 4$ it characterises the f-vectors of $3$-polytopes.
Its proof by shellings was historically delicate: early arguments assumed that the boundary of a polytope can be shelled, which was proved only later by Bruggesser and Mani.
The same alternating sum, for general cell complexes, is the Euler characteristic of algebraic topology.
</document>

<document source="https://tamnd.github.io/math2/node/D3A9V8DKZV.md" tag="D3A9V8DKZV" kind="example">
# Capacity of a disc and a segment

## Example

Let $D_r$ be the closed disc of radius $r > 0$ and let $S_L$ be a closed segment of length $L > 0$ in $\C$.
Their [[logarithmic capacities|DBQXRTC0VN]] are
$$\operatorname{cap}(D_r) = r, \qquad \operatorname{cap}(S_L) = L/4.$$

Check for the disc, with the [[Chebyshev constant|DMNN7E2BYA]].
Translate the disc so that its centre is $0$; capacity does not change under translations.
For a monic polynomial $p(z) = z^n + a_{n-1} z^{n-1} + \cdots + a_0$, Parseval's identity on the circle $\abs{z} = r$ gives
$$\frac{1}{2\pi} \int_0^{2\pi} \abs{p(r e^{i\theta})}^2 \, d\theta = r^{2n} + \sum_{k < n} \abs{a_k}^2 r^{2k} \ge r^{2n}.$$
So $\max_{D_r} \abs{p} \ge r^n$, with equality for $p = z^n$.
Hence the Chebyshev constant is $r$, and by the [[equality of capacity and Chebyshev constant|V0SH1DY9FJ]] the capacity is $r$.

Check for the segment.
Move it by a rotation and a translation to $[-L/2, L/2]$, which does not change capacity.
For $[-1, 1]$, the monic [[Chebyshev polynomial|M6EAR1JHPP]] $2^{1-n} T_n$ has maximum $2^{1-n}$ on the interval, and no monic polynomial of degree $n$ has a smaller maximum there.
So the Chebyshev constant of $[-1, 1]$ is $\lim_n (2^{1-n})^{1/n} = 1/2$.
Capacity scales linearly, so the segment of length $L$ has capacity $(L/2) \cdot (1/2) = L/4$.

The two results show that capacity is not a measure of area.
The segment has area zero but positive capacity $L/4$.
The disc of diameter $L$ has capacity $L/2$.

## Explanations

### L1 human

A disc of radius $r$ has capacity $r$, the same as its boundary circle, because the charge in equilibrium sits on the edge anyway.
A segment of length $L$ has capacity $L/4$.
So a thin segment can hold charge, unlike a single point, but it holds less than the disc whose diameter it is.
The computation uses the fact that capacity equals the Chebyshev constant, which is easy to find for these two shapes.
</document>

<document source="https://tamnd.github.io/math2/node/D3AC1N7PNX.md" tag="D3AC1N7PNX" kind="definition">
# Homomorphism

## Definition

Let $\sigma$ be a [[signature|WDVVEYD6AT]] and let $\mathbf{A}$ and $\mathbf{B}$ be $\sigma$-[[structures|983GZBQYT4]] with domains $A$ and $B$.
A *homomorphism* from $\mathbf{A}$ to $\mathbf{B}$ is a [[function|0EPAT51DXQ]] $h \colon A \to B$ such that:

1. for each function symbol $f \in \sigma$ of arity $n$ and all $a_1, \dots, a_n \in A$, $h(f^{\mathbf{A}}(a_1, \dots, a_n)) = f^{\mathbf{B}}(h(a_1), \dots, h(a_n))$;
2. for each constant symbol $c \in \sigma$, $h(c^{\mathbf{A}}) = c^{\mathbf{B}}$;
3. for each relation symbol $R \in \sigma$ of arity $n$ and all $a_1, \dots, a_n \in A$, if $(a_1, \dots, a_n) \in R^{\mathbf{A}}$ then $(h(a_1), \dots, h(a_n)) \in R^{\mathbf{B}}$.

## Remarks

Example: for the signature $\set{+, 0}$, the map $h \colon \Z \to \Z/n\Z$ that sends an integer to its residue class is a homomorphism.
Indeed, the class of $a + b$ is the sum of the classes of $a$ and $b$, and the class of $0$ is $0$.

Non-example: for the same signature, the map $x \mapsto 2x + 1$ from $\Z$ to $\Z$ fails clause 2, since $0 \mapsto 1$; it also fails clause 1, since $h(0 + 0) = 1 \ne 2 = h(0) + h(0)$.

For a signature with only function and constant symbols, clauses 1 and 2 are the whole condition; this is the notion of homomorphism of [[universal algebra|W82R6N0GM9]].
Whether a map is a homomorphism depends on the signature.
In the signature of groups $\set{\cdot, {}^{-1}, e}$ the inverse is preserved by definition, while for $\set{\cdot}$ alone it follows from the group axioms.
</document>

<document source="https://tamnd.github.io/math2/node/D3AVHVYQW9.md" tag="D3AVHVYQW9" kind="statement">
# Alexandrov–Fenchel inequality

## Statement

Let $n \ge 2$ and let $K_1, K_2, \dots, K_n$ be nonempty compact convex subsets of $\R^n$ ([[convex bodies|437FSG3419]]).
Let $V$ denote the [[mixed volume|QB3Q5FBJ08]].
Then
$$V(K_1, K_2, K_3, \dots, K_n)^2 \ge V(K_1, K_1, K_3, \dots, K_n)\, V(K_2, K_2, K_3, \dots, K_n).$$

## Remarks

The inequality was proved by Alexandrov, and independently by Fenchel, in the 1930s.
Alexandrov's proofs treat polytopes or smooth bodies first, by induction on the dimension, and then pass to general bodies by approximation.
Khovanskii and Teissier found a proof through the Hodge index theorem applied to toric varieties.

Example: for $n = 2$ the inequality reads $V(K, L)^2 \ge \operatorname{area}(K) \operatorname{area}(L)$.
With $L$ the unit disk, $V(K, L) = \frac12 \operatorname{per}(K)$, and one gets $\operatorname{per}(K)^2 \ge 4 \pi \operatorname{area}(K)$, the planar [[isoperimetric inequality|KBX8RBZ626]].

## Explanations

### L1 human

Mixed volumes measure how several convex shapes interact.
The Alexandrov–Fenchel inequality says that a mixed volume of two different shapes is at least the geometric mean of the "pure" mixed volumes of each shape with itself.
It is a far-reaching relative of the Cauchy–Schwarz inequality and of the isoperimetric inequality.

### L2 human

Applied with the remaining arguments chosen among $K_1$ and $K_2$, it shows that the sequence $V(K_1[i], K_2[n - i])$, for $i = 0, \dots, n$, is log-concave in $i$.
It implies [[Minkowski's first inequality|9M8XJX4DQ7]] and the [[Brunn–Minkowski theorem|9B02RPN7XD]] for convex bodies.
Stanley used it to prove that the numbers of linear extensions of a poset with a fixed element at a given position form a log-concave sequence.
Its equality cases are understood only in special situations, and their full description is a hard problem.
</document>

<document source="https://tamnd.github.io/math2/node/D3BN7MBKTR.md" tag="D3BN7MBKTR" kind="concept">
# Path space fibration

## Summary

Let $(X, x_0)$ be a [[pointed space|T9DG8DDE75]] and $PX$ the space of paths that start at $x_0$.
The path space fibration is the map $PX \to X$ that sends each path to its end point.
It is a fibration, its total space $PX$ is contractible, and its fiber over $x_0$ is the [[loop space|2JFF5F3EGF]] $\Omega X$.
From its [[long exact sequence of homotopy groups|E0RV5D1GQ0]] one gets $\pi_n(\Omega X) \cong \pi_{n+1}(X)$.
It plays the role of a universal bundle for loops and is the basic input of the Serre spectral sequence computations of loop spaces.

## Explanations

### L1 human

Fix a home point $x_0$ in $X$.
A point of the path space is a walk in $X$ that starts at home, and the map records where the walk ends.
All walks can be shrunk back to the walk that stays at home, so the path space has no shape of its own.
The walks that end at home are the loops, so the loops form the fiber over $x_0$.
</document>

<document source="https://tamnd.github.io/math2/node/D3FHYZ4P1R.md" tag="D3FHYZ4P1R" kind="definition">
# Feedback linearization

## Definition

Let $\Omega \subseteq \R^n$ be open, and let $f, g \colon \Omega \to \R^n$ be smooth [[vector fields|0299VCW1TH]].
Consider the single-input control-affine [[control system|ZPDTVZQN5K]] $x' = f(x) + g(x) u$ with $u \in \R$, and let $x_0 \in \Omega$.
The system is *locally feedback linearizable* at $x_0$ if there are the following data.
An open neighbourhood $W \subseteq \Omega$ of $x_0$, and a smooth [[diffeomorphism|0KZZ70ZHXP]] $\Phi$ from $W$ onto an open subset of $\R^n$.
Smooth functions $\alpha, \beta \colon W \to \R$ with $\beta(x) \ne 0$ for all $x \in W$.
These data must satisfy two conditions:

1. under the feedback $u = \alpha(x) + \beta(x) v$, the new state $z = \Phi(x)$ satisfies $z' = A z + B v$ for a real $n \times n$ matrix $A$ and a vector $B \in \R^n$;
2. the linear system $z' = A z + B v$ is [[controllable|A9WFM77Z4P]].

*Feedback linearization* is the construction of such $\Phi$, $\alpha$ and $\beta$.

## Remarks

Example: the pendulum $x_1' = x_2$, $x_2' = -\sin x_1 + u$ has $f(x) = (x_2, -\sin x_1)$ and $g(x) = (0, 1)$; with $\Phi = \id$, $\alpha(x) = \sin x_1$ and $\beta = 1$ one gets $z_1' = z_2$, $z_2' = v$, which is controllable.
Non-example: with $g \equiv 0$ the input has no effect, so clause 2 fails for every choice of $\Phi$, $\alpha$ and $\beta$.
Write $\mathrm{ad}_f g = [f, g]$ for the [[Lie bracket|68FGQEBD6J]].
The system is locally feedback linearizable at $x_0$ if and only if two conditions hold.
The vectors $g, \mathrm{ad}_f g, \ldots, \mathrm{ad}_f^{n-1} g$ are linearly independent at $x_0$.
The distribution spanned by $g, \mathrm{ad}_f g, \ldots, \mathrm{ad}_f^{n-2} g$ is involutive near $x_0$.
The related input–output linearization only makes the map from $v$ to a chosen output linear, and may leave internal dynamics.
</document>

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# Closure of an operator

## Definition

Let $X$ and $Y$ be normed spaces, and let $T$ be a [[closable operator|3TWMBFQJJ8]] from $X$ to $Y$.
The *closure* of $T$ is the operator $\overline{T}$ whose [[graph|3RCAY97W1E]] is the [[closure|0QEAJB91GZ]] of the graph of $T$:
$$\Gamma(\overline{T}) = \overline{\Gamma(T)} \subseteq X \times Y.$$

## Remarks

Explicitly, $x \in D(\overline{T})$ if and only if there is a sequence $(x_n)$ in $D(T)$ with $x_n \to x$ such that $(T x_n)$ converges, and then $\overline{T} x = \lim T x_n$.
The operator $\overline{T}$ is [[closed|VM7B38PA87]], it extends $T$, and it is contained in every closed extension of $T$.

Example: on $C[0, 1]$ with the supremum norm, let $T$ be differentiation on polynomials.
Every $f \in C^1[0, 1]$ is a limit of polynomials $p_n$ with $p_n' \to f'$ uniformly: take polynomials $q_n \to f'$ uniformly and set $p_n(x) = f(0) + \int_0^x q_n$.
Combined with the closedness of differentiation on $C^1[0, 1]$, this gives $D(\overline{T}) = C^1[0, 1]$ and $\overline{T} f = f'$.

Non-example: an operator that is not closable has no closure, because the closure of its graph contains a pair $(0, y)$ with $y \ne 0$ and is not a graph.

A closed operator is its own closure.
A subspace $D \subseteq D(\overline{T})$ with $\overline{T|_D} = \overline{T}$ is called a *core* of $\overline{T}$.
</document>

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# Similarity transformation

## Summary

A similarity transformation of Euclidean space is a bijection that multiplies all distances by the same positive factor, its ratio.
Similarities are the compositions of [[isometries|02HPP5VCZH]] and [[homotheties|0ZG1PNEP35]], and they preserve angles, shapes and ratios of lengths.
Two figures are similar when a similarity maps one onto the other.
The theory of similar figures, such as similar triangles, rests on this notion.

## Explanations

### L1 human

A similarity is a change of scale followed by a rigid motion.
Photographs enlarged or reduced, maps drawn at a scale, and models of buildings all use similarities.
The map that doubles every coordinate, $(x, y) \mapsto (2x, 2y)$, is a similarity of ratio $2$: every distance doubles.
Angles do not change, so a triangle is mapped to a triangle with the same angles.
</document>

<document source="https://tamnd.github.io/math2/node/D3HEJTVY7G.md" tag="D3HEJTVY7G" kind="concept">
# Complex multiplication

## Summary

An [[elliptic curve|427N6S6QQW]] has complex multiplication when it has more [[endomorphisms|0J7H0YBFDJ]] than the multiplication-by-$n$ maps.
In characteristic zero its endomorphism ring is then an order in an imaginary [[quadratic field|7SV4WA92SM]].
The theory of complex multiplication describes the [[j-invariants|FEC2XH822R]] and torsion points of such curves.
These values generate abelian extensions of imaginary quadratic fields, such as the [[Hilbert class field|WWCJ7CR2R3]].

## Explanations

### L1 human

On every elliptic curve one can add a point to itself $n$ times, which gives an endomorphism "multiplication by $n$".
Some special curves have extra symmetries.
For example, on $y^2 = x^3 - x$ the map $(x, y) \mapsto (-x, iy)$ sends the curve to itself and respects addition, and doing it twice gives $P \mapsto -P$.
So this map behaves like multiplication by the complex number $i$, hence the name.
These special curves are rare, but they have a particularly rich arithmetic.
</document>

<document source="https://tamnd.github.io/math2/node/D3JWE193EM.md" tag="D3JWE193EM" kind="definition">
# Kelvin functions

## Definition

Let $\nu$ be real and let $x > 0$.
Let $J_\nu$ be the [[Bessel function of the first kind|CZCMRDWDYW]] and $K_\nu$ the [[modified Bessel function|2ET068BZZH]] of the second kind, with principal branches.
The *Kelvin functions* of order $\nu$ are the real functions $\operatorname{ber}_\nu$, $\operatorname{bei}_\nu$, $\operatorname{ker}_\nu$, $\operatorname{kei}_\nu$ on $(0, \infty)$ defined by
$$\operatorname{ber}_\nu(x) + i \operatorname{bei}_\nu(x) = J_\nu\left( x e^{3\pi i/4} \right),$$
$$\operatorname{ker}_\nu(x) + i \operatorname{kei}_\nu(x) = e^{-\nu\pi i/2}\, K_\nu\left( x e^{\pi i/4} \right).$$

## Remarks

The functions $J_\nu(x e^{3\pi i/4})$ and $K_\nu(x e^{\pi i/4})$ solve $x^2 w'' + x w' - (i x^2 + \nu^2) w = 0$, since $\left( x e^{3\pi i/4} \right)^2 = -i x^2$.
Example: for $\nu = 0$, $J_0(z) = \sum_m \frac{(-1)^m (z/2)^{2m}}{(m!)^2}$ and $(-1)^m e^{3\pi i m/2} = i^m$, so the real part keeps the terms with $m = 2k$ and gives $\operatorname{ber}_0(x) = \sum_{k \ge 0} \frac{(-1)^k (x/2)^{4k}}{((2k)!)^2}$.
The odd terms give $\operatorname{bei}_0(x) = \sum_{k \ge 0} \frac{(-1)^k (x/2)^{4k+2}}{((2k+1)!)^2}$.
A non-example: $J_0(x)$ on the real axis is not $\operatorname{ber}_0(x)$; its series has the term $-\frac{x^2}{4}$, which $\operatorname{ber}_0$ lacks, because the argument is not rotated to the ray $\arg z = \frac{3\pi}{4}$.
When $\nu = 0$ the index is usually dropped: $\operatorname{ber} = \operatorname{ber}_0$, and so on.
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# Homotopy extension property

## Summary

Let $A$ be a subspace of a space $X$.
The pair $(X, A)$ has the homotopy extension property when each [[homotopy|9YA6S6HD9V]] on $A$ extends to all of $X$, once its start on $X$ is given.
Pairs with this property let one move maps on a subspace without losing control of the whole space.
It holds for CW pairs and is the defining property of [[cofibrations|W5WS5EY8ZM]].

## Explanations

### L1 human

Suppose you have a map on a big space $X$ and you start to wiggle it on a small part $A$.
The homotopy extension property says that you can wiggle the map on the rest of $X$ in a way that matches.
For example, a disk with its boundary circle has this property: any deformation of a map on the circle spreads into the disk.
Badly attached subspaces, such as a limit point of a sequence of isolated points, can fail it.
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# Tychonoff plank

## Counterexample

Let $\omega$ be the first infinite [[ordinal|0YSCSX5YZC]] and $\omega_1$ the first uncountable ordinal.
Give $[0, \omega_1]$ and $[0, \omega]$ the [[order topology|RCWHDMWP1T]], and let $T = [0, \omega_1] \times [0, \omega]$ with the [[product topology|0GBG8B49AA]].
This is the Tychonoff plank.
The deleted Tychonoff plank is the [[subspace|0WS9P12FQR]] $T_\infty = T \setminus \set{(\omega_1, \omega)}$.
The plank $T$ is [[compact|0TVJZYB6SH]] and [[Hausdorff|0S1FJ8VEX9]], hence [[normal|0NSR9V8TED]], but the open subspace $T_\infty$ is not normal.

## Remarks

The sets $A = \set{\omega_1} \times [0, \omega)$ and $B = [0, \omega_1) \times \set{\omega}$ are disjoint and closed in $T_\infty$.
Let $U$ be an open set of $T_\infty$ containing $A$.
For each $n < \omega$ there is $\alpha_n < \omega_1$ with $(\alpha_n, \omega_1] \times \set{n} \subseteq U$.
The supremum $\alpha$ of the countably many $\alpha_n$ is still countable, so $\alpha + 1 < \omega_1$ and $(\alpha + 1, n) \in U$ for every $n$.
Any open set $V$ containing $(\alpha + 1, \omega) \in B$ contains $(\alpha + 1, n)$ for all large $n$, so $V$ meets $U$.
Hence $A$ and $B$ cannot be separated, and $T_\infty$ is not normal.
Since $T_\infty$ is a subspace of the compact Hausdorff space $T$, it is a [[Tychonoff space|C5N2J2K6Z3]]; so an open subspace of a normal space need not be normal.

## Explanations

### L1 human

The plank is a product of two well-ordered "lines", one countably long and one uncountably long, each with a top point.
It is compact, so it behaves well.
Removing the single top corner point leaves two edges that run towards the missing corner.
Any open set around one edge must swallow a whole column near the corner, and so it always touches the other edge.
</document>

<document source="https://tamnd.github.io/math2/node/D44KAF8PNK.md" tag="D44KAF8PNK" kind="problem">
# Second Cousin problem

## Problem

Let $X$ be a connected [[complex manifold|JSGWAN2RJV]] and let $(U_i)_{i \in I}$ be an open cover of $X$.
For each $i$, let $f_i$ be a [[meromorphic function|W6FTSF57T1]] on $U_i$ that is not identically zero on any component of $U_i$.
Assume that $f_i / f_j$ is [[holomorphic|507C34W0ZT]] and nowhere zero on $U_i \cap U_j$ for all $i, j$.
Find a meromorphic function $f$ on $X$ such that $f / f_i$ is holomorphic and nowhere zero on $U_i$ for every $i$.

## Remarks

The data prescribe the zeros and poles of $f$ with multiplicities, that is, a divisor on $X$.
The problem asks whether this divisor is the divisor of a global meromorphic function.
The quotients $g_{ij} = f_i / f_j$ form a Čech $1$-cocycle with values in the sheaf $\mathcal{O}^*$ of nowhere-zero holomorphic functions, and they define a [[holomorphic line bundle|QAC7BYK9KX]].
The problem is solvable for these data if and only if this line bundle is holomorphically trivial.
So if $H^1(X, \mathcal{O}^*) = 0$, every second Cousin problem on $X$ is solvable.
On a Stein manifold, Cartan's theorem B and the exponential sequence $0 \to \Z \to \mathcal{O} \to \mathcal{O}^* \to 0$ give an isomorphism $H^1(X, \mathcal{O}^*) \cong H^2(X, \Z)$.
Hence on a Stein manifold the problem is solvable if and only if the line bundle is topologically trivial, a result of Oka.
In particular it is always solvable on a Stein manifold with $H^2(X, \Z) = 0$, for example on $\C^n$ or on a convex domain.
For a domain in $\C$ the problem is solved by the Weierstrass factorization theorem.
There are domains of holomorphy on which some second Cousin problem has no solution.
An example is $X = \set{(z, w) \in \C^2 : (\log \abs{z})^2 + (\log \abs{w})^2 < 1}$, a logarithmically convex Reinhardt domain that retracts onto the torus $\abs{z} = \abs{w} = 1$, so $H^2(X, \Z) \cong \Z$.
A holomorphic line bundle on $X$ with nonzero Chern class has a meromorphic section that is not identically zero, and its local expressions are second Cousin data without solution.

## Explanations

### L1 human

In one variable, Weierstrass showed how to build a holomorphic function with prescribed zeros and multiplicities.
The second Cousin problem asks the same in several variables, where zeros form hypersurfaces rather than isolated points.
Given local functions that define the wanted zeros and poles near each point, one wants one global function that defines them everywhere.
Unlike the first Cousin problem, this one can fail even on very nice domains, for a topological reason.
Oka showed that on Stein manifolds topology is the only obstruction.
</document>

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# Frölicher–Nijenhuis bracket

## Summary

The Frölicher–Nijenhuis bracket is a graded Lie bracket on differential forms with values in the tangent bundle of a [[smooth manifold|F45VQ47GPV]].
On vector fields, which are the forms of degree $0$, it is the usual [[Lie bracket of vector fields|68FGQEBD6J]].
It is defined through the derivations of the algebra of [[differential forms|0E3G0GRPZ6]] that commute with the [[exterior derivative|0ZHSTAG8ZY]].
It expresses integrability conditions, such as those of almost complex structures and of connections, as the vanishing of a bracket.

## Explanations

### L1 human

The Lie bracket of two vector fields measures how far their flows fail to commute.
Vector-valued forms are more general objects, such as a field of linear maps on tangent spaces.
The Frölicher–Nijenhuis bracket extends the Lie bracket to all of them in a consistent way.
For example, for an almost complex structure $J$, the bracket of $J$ with itself measures whether $J$ comes from complex coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/D48DR46B97.md" tag="D48DR46B97" kind="concept">
# Geometric Brownian motion

## Summary

Geometric Brownian motion is a positive process whose logarithm is a Brownian motion with drift.
It solves the stochastic differential equation $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$.
It is the model of stock prices in the Black–Scholes theory of option pricing.

## Explanations

### L1 human

In geometric Brownian motion, the random changes are proportional to the current value, so it grows or shrinks by percentages rather than by fixed amounts.
This is why it never becomes negative, which suits prices of shares.
Over a time $t$, the value is multiplied by a [[log-normal|4W40NHWPQT]] random factor.
For example, with drift $\mu = 0.05$ and volatility $\sigma$ the expected value grows like $e^{0.05 t}$, while the median value grows only like $e^{(0.05 - \sigma^2/2) t}$.
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# Structural stability

## Summary

A dynamical system is structurally stable if every small enough perturbation of it has the same qualitative behaviour.
More precisely, the [[phase portrait|0T3GK0XCBC]] of the perturbed system is the image of the original one under a homeomorphism that maps trajectories to trajectories.
Structural stability is about the stability of the whole system under changes of the equations, not of one solution under changes of the initial state.
Models of the real world are only known approximately, so structurally stable models are the ones whose qualitative predictions can be trusted.
See the [[definition|KQX43WB57R]].

## Explanations

### L1 human

Imagine drawing all the trajectories of a system as a picture of arrows and curves.
If a tiny change in the equations only bends and stretches this picture, without changing its pattern, the system is structurally stable.
On a circle, the flow $\theta' = \sin \theta$ has one repelling and one attracting rest point, and small changes keep this pattern.
The flow $\theta' = 1 - \cos \theta$ has a single rest point that attracts from one side and repels on the other.
Adding a small positive constant removes this rest point, so this flow is not structurally stable.
</document>

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# Locally connected space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *locally connected* if the following holds.
For every point $x \in X$ and every [[open set|02BH3MCBQB]] $U$ containing $x$, there is a [[connected|0SPE8WNWFV]] open set $V$ with $x \in V \subseteq U$.

## Remarks

Example: $\R$ is locally connected.
For $x \in U$ with $U$ open, some interval $(x - \varepsilon, x + \varepsilon)$ lies in $U$, and intervals are connected.

Non-example: $\Q$ with the subspace topology from $\R$ is not locally connected.
Its only nonempty connected subsets are single points, and single points are not open in $\Q$.

Equivalently, $X$ is locally connected if and only if every [[connected component|0JG9W3KA6B]] of every open subset of $X$ is open in $X$.
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# Riccati difference equation

## Summary

The Riccati difference equation is the first-order [[rational difference equation|T8YDEASD1H]] $x_{n+1} = (a x_n + b)/(c x_n + d)$ with $c \ne 0$.
It is the iteration of a [[Möbius transformation|0SG74WBFA2]].
A substitution turns it into a linear recurrence of order two, so it can be solved in closed form.
It is the discrete analogue of the [[Riccati equation|0VT2QK6X6Q]] for differential equations, which is linearized by a similar substitution.

## Explanations

### L1 human

Each new term is a fraction with $x_n$ in the top and in the bottom.
Such a rule looks nonlinear, but it hides a linear rule.
If you write $c x_n + d$ as a ratio $y_{n+1}/y_n$, the numbers $y_n$ obey an ordinary two-step linear recurrence.
So the behaviour of the fraction rule is decided by the roots of a quadratic equation.
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<document source="https://tamnd.github.io/math2/node/D4TYZN7BRH.md" tag="D4TYZN7BRH" kind="definition">
# Socle of a module

## Definition

Let $R$ be a ring and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.
The *socle* of $M$, written $\operatorname{soc}(M)$, is the sum of all [[simple submodules|G00C22AEDH]] of $M$.
If $M$ has no simple [[submodule|05WDSN597C]], then $\operatorname{soc}(M) = 0$.

## Remarks

Example: let $M = \Z/12\Z$ as a $\Z$-module.
A submodule is simple exactly when it has prime order, so the simple submodules are $6\Z/12\Z$ and $4\Z/12\Z$, of orders $2$ and $3$.
Their sum is $2\Z/12\Z$, so $\operatorname{soc}(\Z/12\Z) = 2\Z/12\Z$.

Non-example of a nonzero socle: the $\Z$-module $\Z$ has socle $0$.
Each nonzero submodule $n\Z$ contains the proper nonzero submodule $2n\Z$, so no submodule of $\Z$ is simple.

The socle is a semisimple module, and it contains every semisimple submodule of $M$.
It also equals the intersection of all essential submodules of $M$.
The socle of the left module ${}_R R$ is a two-sided ideal of $R$.
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# Polyharmonic spline

## Definition

Let $d \ge 1$ and $m \ge 1$ be integers with $2m > d$.
Define $\varphi \colon [0, \infty) \to \R$ by
$$\varphi(r) = \begin{cases} r^{2m-d} \log r & \text{if } d \text{ is even}, \\ r^{2m-d} & \text{if } d \text{ is odd}, \end{cases}$$
with $\varphi(0) = 0$.
Let $\Pi_{m-1}$ be the space of polynomials on $\R^d$ of total degree at most $m - 1$.
A *polyharmonic spline* of order $m$ with distinct centres $x_1, \dots, x_N \in \R^d$ is a function
$$s(x) = \sum_{i=1}^{N} w_i\, \varphi(\norm{x - x_i}) + p(x),$$
where $p \in \Pi_{m-1}$ and the real weights satisfy $\sum_{i=1}^{N} w_i\, q(x_i) = 0$ for every $q \in \Pi_{m-1}$.

## Remarks

Suppose the centres are unisolvent for $\Pi_{m-1}$, that is, the only $q \in \Pi_{m-1}$ vanishing at all $x_i$ is $0$.
Then for any data $y_1, \dots, y_N$ there is exactly one polyharmonic spline with $s(x_i) = y_i$.
This is [[radial basis function interpolation|XZ33P3HBB9]] with the [[radial basis function|ZBSXSTQM2J]] $\varphi$.
Up to a constant factor, $\varphi(\norm{x})$ is a [[fundamental solution|K2NK9A8M5F]] of $\Delta^m$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]], so $s$ satisfies $\Delta^m s = 0$ away from the centres.
The interpolant minimizes the seminorm $\sum_{\abs{\alpha} = m} \frac{m!}{\alpha!} \int_{\R^d} (\partial^\alpha f)^2$ among suitable functions with $f(x_i) = y_i$.
Example: $d = 1$, $m = 2$ gives $\varphi(r) = r^3$ and $s(x) = \sum w_i \abs{x - x_i}^3 + a + bx$ with $\sum w_i = \sum w_i x_i = 0$; this is a natural cubic spline.
Non-example: in the same setting, $s(x) = \abs{x}^3$ (one centre $0$, $w_1 = 1$) violates the side condition $\sum w_i = 0$, and indeed it is not linear outside the centres.
For $d = 2$, $m = 2$ one gets $\varphi(r) = r^2 \log r$, the thin plate spline.
</document>

<document source="https://tamnd.github.io/math2/node/D4WF2BJE2M.md" tag="D4WF2BJE2M" kind="example">
# Iwasawa manifold

## Example

Let $G$ be the complex [[Heisenberg group|ZZYCV5EX9J]] of matrices
$$\begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{pmatrix}, \qquad x, y, z \in \C,$$
a complex Lie group isomorphic to $\C^3$ as a complex manifold.
Let $\Gamma \subset G$ be the subgroup of matrices with $x, y, z$ in the [[Gaussian integers|0K5W69AC0J]] $\Z[i]$.
The *Iwasawa manifold* is the quotient $X = \Gamma \backslash G$ of $G$ by the left action of $\Gamma$.
It is a compact [[complex manifold|JSGWAN2RJV]] of dimension $3$: $\Gamma$ is a discrete cocompact subgroup, a [[lattice|BTYF5Y5PFY]] of $G$, and acts freely by holomorphic maps.

Check that $X$ is not [[Kähler|X60JG9AEQJ]].
Left multiplication by the matrix with entries $(a, b, c)$ sends $(x, y, z)$ to $(x + a, \, y + b, \, z + a y + c)$.
So the holomorphic $1$-forms $dx$, $dy$ and $\theta = dz - x\, dy$ are invariant: $d(z + ay + c) - (x + a)\, d(y + b) = dz - x\, dy$.
They descend to holomorphic $1$-forms on $X$.
But $d\theta = -dx \wedge dy \ne 0$, so $\theta$ is not [[closed|0ZV14XRNF0]].
On a compact Kähler manifold every holomorphic form is closed, so $X$ is not Kähler.

## Remarks

$X$ is a [[nilmanifold|43NRBE72C6]] and a fibre bundle over the complex torus $\C^2 / \Z[i]^2$, via $(x, y, z) \mapsto (x, y)$, with fibre the elliptic curve $\C / \Z[i]$.
Its holomorphic tangent bundle is trivial, spanned by the left-invariant vector fields dual to $dx, dy, \theta$, so $X$ is a compact complex parallelizable manifold.
Its first Betti number is $4$, an even number, so non-Kählerness is not visible from $b_1$ here.

## Explanations

### L1 human

The Iwasawa manifold is made from $3 \times 3$ upper triangular complex matrices with ones on the diagonal.
Two matrices count as the same when they differ by one with Gaussian integer entries.
It is a compact complex manifold of dimension $3$ with three independent holomorphic vector fields at each point.
It has a holomorphic $1$-form whose derivative is not zero, which cannot happen on a compact Kähler manifold.
So it is a basic example of a compact complex manifold that is not Kähler, even though its Betti numbers look normal.
</document>

<document source="https://tamnd.github.io/math2/node/D59HBKKRP9.md" tag="D59HBKKRP9" kind="concept">
# Spline interpolation

## Summary

Spline interpolation passes a [[spline|84TJ8KQ926]] through given data points, using the data nodes as knots.
Unlike [[polynomial interpolation|SS1YMS2NPC]] of high degree, it keeps the degree of each piece low, which avoids [[Runge's phenomenon|ZMT8JF21R6]] and gives small errors for smooth data.
The most common choice is the [[cubic spline|V42H0R3EKX]], for which two extra conditions at the ends of the interval make the interpolant unique.
The coefficients solve a [[tridiagonal|DARX1KZRHZ]] linear system, so the cost grows only linearly with the number of points.

## Explanations

### L1 human

Connecting data points by straight segments gives a curve with corners.
Spline interpolation uses curved pieces instead, typically cubics, and chooses them so that neighbouring pieces meet with the same slope and the same bending.
The result looks like a smooth curve drawn through the points by hand.
Unlike one polynomial through all the points, it does not swing wildly between the points when there are many of them.
The [[definition|RGT7WEZG8X]] states the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/D5BDZZW9D4.md" tag="D5BDZZW9D4" kind="definition">
# Puiseux series

## Definition

Let $k$ be a [[field|0QA8CHH8MG]] and let $X$ be a variable.
Let $m$ be a positive integer, $N$ an integer, and $a_n \in k$ for all $n \ge N$.
A *Puiseux series* in $X$ over $k$ is a formal sum of the form
$$f = \sum_{n \ge N} a_n X^{n/m}.$$
Equivalently, $f$ is an element of the [[field of formal Laurent series|YZWYF7DXTW]] $k((X^{1/m}))$ in the variable $X^{1/m}$, for some $m \ge 1$.

For $m \mid m'$, the field $k((X^{1/m}))$ is identified with a subfield of $k((X^{1/m'}))$ through $X^{1/m} = (X^{1/m'})^{m'/m}$.
The field of Puiseux series is the union $\bigcup_{m \ge 1} k((X^{1/m}))$ with these identifications.

## Remarks

Example: $f = X^{1/2}$ is a Puiseux series with $m = 2$, and it is a root of $Y^2 - X$ in the field of Puiseux series, since $(X^{1/2})^2 = X$ in $k((X^{1/2}))$.
Non-example: the formal sum $\sum_{n \ge 1} X^{1 - 1/n}$ has the exponents $0, 1/2, 2/3, 3/4, \ldots$, whose denominators are unbounded, so there is no single $m$ and it is not a Puiseux series.
The union is a field because any two Puiseux series lie in a common field $k((X^{1/m}))$.
The order $v(f)$ of the lowest exponent with nonzero coefficient is a valuation on the field of Puiseux series with value group $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/D5CKZCK3Z2.md" tag="D5CKZCK3Z2" kind="definition">
# Uniform algebra

## Definition

Let $X$ be a [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]], and let $C(X)$ be the algebra of [[continuous|0X3WGX95FE]] functions $X \to \C$ with the sup norm $\norm{f}_\infty = \max_{x \in X} \abs{f(x)}$.
A *uniform algebra* on $X$ is a [[subalgebra|7CHG6SD365]] $A \subseteq C(X)$ such that:

1. $A$ is closed in $C(X)$ for the sup norm, that is, under [[uniform convergence|08VZ57PT26]];
2. $A$ contains the constant functions;
3. $A$ separates the points of $X$: for $x \ne y$ in $X$ there is $f \in A$ with $f(x) \ne f(y)$.

## Remarks

A uniform algebra is a commutative unital Banach algebra with $\norm{f^2} = \norm{f}^2$ for all $f$.
Conversely, a commutative unital complex Banach algebra with $\norm{a^2} = \norm{a}^2$ for all $a$ is isometrically isomorphic, through the Gelfand transform, to a uniform algebra on its character space.

Example: the disc algebra $A(\mathbb{D})$ on the closed unit disc $X = \overline{\mathbb{D}}$, the continuous functions on $X$ that are holomorphic in the open disc.
It is a subalgebra; it is closed because [[uniform limits of holomorphic functions are holomorphic|0431RN0R1N]]; it contains the constants; and $f(z) = z$ separates points.

Non-example: on $X = [0, 1]$, let $A = \set{f \in C([0,1]) : f(0) = f(1)}$.
It is a closed subalgebra containing the constants, but no $f \in A$ distinguishes $0$ from $1$, so clause 3 fails.

By the [[Stone–Weierstrass theorem|1P8D3MK79H]], a uniform algebra on $X$ that is closed under complex conjugation equals $C(X)$.
</document>

<document source="https://tamnd.github.io/math2/node/D5HZKE8E1A.md" tag="D5HZKE8E1A" kind="concept">
# Oscillatory integral

## Summary

An oscillatory integral is an integral of the form $\int e^{i\lambda\phi(x)} \psi(x) \, dx$.
Here $\phi$ is a real phase function, $\psi$ is an amplitude and $\lambda$ is a large real parameter.
The main question is how fast it decays as $\lambda \to \infty$, which depends on the critical points of $\phi$.
Fourier transforms of measures on curved surfaces, Bessel functions and solutions of dispersive equations are all oscillatory integrals.
Their estimates are basic tools in harmonic analysis, number theory and PDE.

## Explanations

### L1 human

When $\lambda$ is large, the factor $e^{i\lambda\phi(x)}$ spins around the unit circle very fast as $x$ moves.
Its positive and negative parts nearly cancel, so the integral is small.
The cancellation is weakest where $\phi$ is flat, at points where its derivative vanishes, and these points decide the size of the integral.
For example, $\int_{-\infty}^{\infty} e^{i\lambda x^2} e^{-x^2} \, dx = \sqrt{\pi/(1 - i\lambda)}$, which is of size $\lambda^{-1/2}$ for large $\lambda$, because the phase $x^2$ is flat at $x = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/D5J6CP5V2Z.md" tag="D5J6CP5V2Z" kind="concept">
# Holomorphically convex domain

## Summary

An open set $D \subseteq \C^n$ is holomorphically convex if the [[holomorphically convex hull|CD6SMMZSW8]] of every compact subset of $D$ is again compact.
This is the holomorphic analogue of convexity, where hulls are taken with respect to holomorphic functions on $D$ instead of linear functions.
For domains in $\C^n$, holomorphic convexity is equivalent to being a domain of holomorphy.
On complex manifolds it is one of the conditions defining Stein manifolds.

## Explanations

### L1 human

Take any compact piece $K$ of the domain and form its hull: all points where holomorphic functions cannot beat their maximum over $K$.
In a holomorphically convex domain these hulls never creep out to the boundary.
Every convex open set in $\C^n$ has this property.
The punctured ball $\set{0 < \abs{z} < 1}$ in $\C^2$ does not: the hull of a small sphere around the origin contains points arbitrarily close to the missing origin.
</document>

<document source="https://tamnd.github.io/math2/node/D5M7TRD3MY.md" tag="D5M7TRD3MY" kind="statement">
# Lax equivalence theorem

## Statement

Let $X$ be a [[Banach space|06GZ956657]] and $T > 0$.
Let $A$ be a linear operator in $X$ such that the [[initial value problem|075VBY4K4X]] $u'(t) = A u(t)$, $u(0) = u_0$, is [[well-posed|W09B9NZ9B5]].
This means there are bounded linear solution operators $E(t)$, $t \ge 0$, with $\sup_{0 \le t \le T} \norm{E(t)} < \infty$.
Moreover $u(t) = E(t) u_0$ is a genuine solution for $u_0$ in a dense subspace $D \subseteq X$.
Let $C(k)$, $0 < k \le k_0$, be bounded linear operators on $X$ defining the one-step [[finite difference scheme|X7HYEBA5SA]] $u^{n+1} = C(k) u^n$ with time step $k$.
Assume the scheme is consistent: for every $u_0 \in D$,
$$\lim_{k \to 0} \sup_{0 \le t \le T} \frac{\norm{C(k) u(t) - u(t + k)}}{k} = 0, \qquad u(t) = E(t) u_0.$$
Then the following are equivalent:

1. The scheme is stable: there are constants $K$ and $k_1 > 0$ with $\norm{C(k)^n} \le K$ whenever $0 < k \le k_1$ and $0 \le nk \le T$, where $\norm{\cdot}$ is the [[operator norm|06EKW1AKXF]].
2. The scheme is convergent: for every $u_0 \in X$, every $t \in [0, T]$, and all sequences $k_j \to 0$ and integers $n_j$ with $n_j k_j \to t$, one has $C(k_j)^{n_j} u_0 \to E(t) u_0$ in $X$.

## Remarks

Stability implies convergence by a telescoping sum: $C(k)^n u_0 - E(nk) u_0 = \sum_{m=0}^{n-1} C(k)^{n-1-m} \left( C(k) E(mk) - E((m+1)k) \right) u_0$.
Each factor $C(k)^{n-1-m}$ is bounded by $K$, and the $n$ consistency errors are each $o(k)$.
Density of $D$ and the uniform bounds then cover all $u_0$.
Convergence implies stability by the uniform boundedness principle, since the orbits $C(k)^n u_0$ stay bounded for each $u_0$.
Example: for $u_t = u_{xx}$ on $\R$ in the maximum norm, the explicit scheme $u_j^{n+1} = u_j^n + r(u_{j+1}^n - 2u_j^n + u_{j-1}^n)$ with fixed $r = k/h^2 \le \tfrac12$ satisfies $\norm{C(k)} \le 1$, so it converges.
With $r > \tfrac12$ the mode $u_j^0 = (-1)^j$ is multiplied by $1 - 4r < -1$ at each step, so the scheme is unstable and does not converge.
The theorem is for linear problems; for nonlinear schemes stability and consistency do not in general characterize convergence.

## Explanations

### L1 human

A numerical method for a time-dependent equation is consistent if each single step makes only a small error.
It is stable if errors do not blow up when many steps are taken.
The theorem says that, for linear problems that are well-posed, a consistent method gives the right answer as the steps shrink exactly when it is stable.
So the hard question "does the method converge?" reduces to the easier question "is it stable?".

### L2 human

In practice consistency is checked by Taylor expansion, and stability by a norm bound on the step operator $C(k)$, often through von Neumann's Fourier analysis in $L^2$.
For explicit schemes for hyperbolic equations, stability leads to the [[Courant–Friedrichs–Lewy condition|FQVVNYV6N8]] on $k/h$.
The result is due to Lax and Richtmyer, in their 1956 paper on the stability of linear finite difference equations.
The same structure, "consistency plus stability gives convergence", reappears for boundary value problems and for finite element methods, where Céa's lemma plays the role of stability.
</document>

<document source="https://tamnd.github.io/math2/node/D5MMVPJ8M1.md" tag="D5MMVPJ8M1" kind="definition">
# Endomorphism ring

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $M$ be a left [[$R$-module|09VDS2W1E8]].
An *endomorphism* of $M$ is a map $f \colon M \to M$ with $f(x + y) = f(x) + f(y)$ and $f(rx) = r f(x)$ for all $x, y \in M$ and $r \in R$.
The *endomorphism ring* $\End_R(M)$ is the set of endomorphisms of $M$ with the operations
$$(f + g)(x) = f(x) + g(x), \qquad (fg)(x) = f(g(x)).$$

## Remarks

$\End_R(M)$ is a ring with zero map as zero and the identity map $\id_M$ as unit element.
For an abelian group $A$, viewed as a $\Z$-module, one writes $\End(A) = \End_\Z(A)$.
Example: $\End_\Z(\Z/n\Z) \cong \Z/n\Z$.
The check: an endomorphism $f$ is determined by $f(1) = a$, since $f(k) = ka$, and every $a$ gives one; composition corresponds to multiplication of the values $a$.
Non-example: the map $x \mapsto x + 1$ on $\Z$ is not in $\End_\Z(\Z)$, since it does not preserve addition: it sends $0$ to $1$.
For a field $K$, choosing a basis gives $\End_K(K^n) \cong M_n(K)$.
For the ring $R$ as a left module over itself, $\End_R(R) \cong R^{\mathrm{op}}$, the [[opposite ring|M8Q24XB0AF]], through $f \mapsto f(1)$.
</document>

<document source="https://tamnd.github.io/math2/node/D5PN3G41D6.md" tag="D5PN3G41D6" kind="concept">
# k-means clustering

## Summary

$k$-means clustering splits points of $\R^d$ into $k$ groups.
It makes the sum of squared distances from each point to the mean of its group as small as possible.
It is one of the most widely used methods of [[cluster analysis|D9Y3A9RH6J]].
Finding the exact optimum is computationally hard in general, so it is approximated by Lloyd's algorithm, which alternates between assigning points and updating means.
The method works best for compact, roughly round clusters of similar size.

## Explanations

### L1 human

Choose $k$ starting centres.
Give each point to its nearest centre, then move each centre to the average of the points it received.
Repeat until nothing changes.
For the points $0, 1, 10, 11$, $k = 2$ and the starting centres $0$ and $1$, the method ends with the groups $\set{0, 1}$ and $\set{10, 11}$, with centres $0.5$ and $10.5$.
</document>

<document source="https://tamnd.github.io/math2/node/D5VM0MC3EB.md" tag="D5VM0MC3EB" kind="statement">
# De Giorgi's structure theorem

## Statement

Let $n \ge 2$ and let $E \subseteq \R^n$ be a set of locally finite [[perimeter|Y68BHXZJBV]], that is, a local [[Caccioppoli set|W6R7QNEAF1]].
Let $\mu = \abs{D\chi_E}$ be the total variation measure of the distributional gradient of its indicator function $\chi_E$, a [[Radon measure|HZNEQQBCW4]].
Write $D\chi_E = -\nu_E \, \mu$ with a Borel map $\nu_E \colon \R^n \to \R^n$ such that $\abs{\nu_E} = 1$ $\mu$-almost everywhere.
For $x \in \R^n$ and $r > 0$ let $B_r(x)$ be the open ball of radius $r$ about $x$.
Consider the limit
$$\lim_{r \to 0} \frac{D\chi_E(B_r(x))}{\mu(B_r(x))}.$$
The *reduced boundary* $\partial^* E$ is the set of points $x$ in the support of $\mu$ at which this limit exists and has norm $1$.
Let $\mathcal{H}^{n-1}$ be the $(n-1)$-dimensional [[Hausdorff measure|Q1TGZK8RN6]].
Then:

1. The set $\partial^* E$ is countably $(n-1)$-[[rectifiable|7N0GFMW05R]]: up to an $\mathcal{H}^{n-1}$-null set, it is contained in a countable union of $C^1$ hypersurfaces.
2. $\mu = \mathcal{H}^{n-1} \llcorner \partial^* E$, that is, $\mu(A) = \mathcal{H}^{n-1}(A \cap \partial^* E)$ for every Borel set $A$.
3. For every $x \in \partial^* E$, the rescaled sets $(E - x)/r$ converge in $L^1_{\mathrm{loc}}(\R^n)$, as $r \to 0$, to the half-space $\set{y : y \cdot \nu_E(x) < 0}$.
4. For every $\varphi \in C^1_c(\R^n; \R^n)$, the generalized Gauss–Green formula holds:
$$\int_E \operatorname{div} \varphi \, dx = \int_{\partial^* E} \varphi \cdot \nu_E \, d\mathcal{H}^{n-1}.$$

## Remarks

The proof starts from the Besicovitch differentiation theorem, which shows that $\mu$-almost every point lies in $\partial^* E$.
At a point of $\partial^* E$, the isoperimetric inequality gives density estimates comparing $\mu(B_r(x))$ with $r^{n-1}$.
With the compactness of sets of finite perimeter, a blow-up argument then shows that the rescaled sets converge to a half-space.
The half-space limit identifies $\mu$ with $\mathcal{H}^{n-1}$ on $\partial^* E$.
Rectifiability then follows from the existence of approximate tangent planes at all points of $\partial^* E$.
For a bounded open set with $C^1$ boundary, $\partial^* E = \partial E$, $\nu_E$ is the outer unit normal, and item 4 is the classical divergence theorem.
For the square $[0, 1]^2$, the reduced boundary is the boundary minus the four corners; the corners form an $\mathcal{H}^1$-null set.
The topological boundary can be much larger than $\partial^* E$, even of positive Lebesgue measure, so the reduced boundary is the right notion of boundary here.

## Explanations

### L1 human

A set of finite perimeter can have a very rough boundary.
The theorem says that the part of the boundary that matters is still made of pieces of smooth surfaces, with a well-defined outward direction at almost every point.
Zooming in at such a point, the set looks more and more like half of space.
With this boundary, the usual formula relating the flux through the boundary to the integral of the divergence still holds.

### L2 human

The theorem is due to Ennio De Giorgi, and it is the foundation of the theory of sets of finite perimeter.
It turns a set known only through the measure $D\chi_E$ into a geometric object with a measure-theoretic normal.
Federer later showed that $\partial^* E$ coincides, up to an $\mathcal{H}^{n-1}$-null set, with the essential boundary, the set of points where $E$ has density neither $0$ nor $1$.
These results are the starting point of the regularity theory of perimeter minimizers and of the study of free discontinuity problems.
</document>

<document source="https://tamnd.github.io/math2/node/D5VS0YFTEZ.md" tag="D5VS0YFTEZ" kind="definition">
# Simson line

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane and let $\Gamma$ be its [[circumcircle|03VZJE6C8C]].
Let $P$ be a point of $\Gamma$.
Let $D$, $E$ and $F$ be the feet of the perpendiculars from $P$ to the [[lines|0MP9BCF6DP]] $BC$, $CA$ and $AB$.
The points $D$, $E$ and $F$ are [[collinear|0AFNNTRTJE]], and the *Simson line* of $P$ with respect to $ABC$ is the line that contains them.

## Remarks

The collinearity in the definition is the Simson–Wallace theorem.
Conversely, if the three feet from a point $P$ are collinear, then $P$ lies on $\Gamma$.

Example: let $P = A'$ be the point of $\Gamma$ opposite to $A$.
Since $AA'$ is a diameter, the angles $\angle ABA'$ and $\angle ACA'$ are right angles, so the feet on $AB$ and $CA$ are $B$ and $C$.
The foot on $BC$ lies on the line $BC$, so the Simson line of $A'$ is the line $BC$.

Non-example: let $P$ be the [[circumcentre|0F98FZK14A]] $O$, which is not on $\Gamma$.
The feet of the perpendiculars from $O$ are the [[midpoints|0S706JBZBG]] of the three sides.
These form the medial triangle, so they are not collinear, and no Simson line is defined.
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<document source="https://tamnd.github.io/math2/node/D5Z60BRP18.md" tag="D5Z60BRP18" kind="concept">
# Horn

## Summary

A horn is the union of all the codimension-one faces of a standard simplex except one.
In the theory of [[simplicial sets|WW81ARB8K9]], horns are the shapes against which one tests extension properties.
A Kan complex is a simplicial set in which every horn can be filled.
A quasi-category is one in which every inner horn can be filled.

## Explanations

### L1 human

Take a triangle and keep only two of its three edges.
The two edges form a corner shaped like the letter V, which is the horn $\Lambda^2_k$, where $k$ is the vertex where the two edges meet.
Filling the horn means finding the third edge and the triangle itself.
In a space, a horn of paths can always be filled.
Two paths that meet at a point compose to a third path, and a triangle witnesses the composition.
</document>

<document source="https://tamnd.github.io/math2/node/D62PEKHRV5.md" tag="D62PEKHRV5" kind="concept">
# Equilibrium of a difference equation

## Summary

An equilibrium of an autonomous difference equation $x_{n+1} = f(x_n)$ is a point $x^*$ with $f(x^*) = x^*$, that is, a [[fixed point|R6QVS6P21N]] of $f$.
Starting there, the solution stays there forever, so equilibria are the constant solutions.
They are the discrete-time analogue of the [[equilibrium points|0N8NGKDZDZ]] of differential equations.
The first question about a nonlinear difference equation is usually where its equilibria are and whether nearby solutions approach them.

## Explanations

### L1 human

A difference equation tells you how a state changes from one time step to the next.
An equilibrium is a state that the rule leaves unchanged.
On a graph of $f$, the equilibria are where the graph crosses the line $y = x$.
For the population rule $x_{n+1} = 2 x_n (1 - x_n)$, the equilibria are $0$ and $1/2$: a population of $1/2$ stays at $1/2$.
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<document source="https://tamnd.github.io/math2/node/D637VGM7GV.md" tag="D637VGM7GV" kind="definition">
# Quasi-algebraically closed field

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
The field $K$ is *quasi-algebraically closed*, or a *$C_1$ field*, if the following holds.
For all integers $n > d \ge 1$, every [[homogeneous polynomial|SD3M01318E]] $f \in K[x_1, \ldots, x_n]$ of degree $d$ has a zero $(a_1, \ldots, a_n) \in K^n$ with $(a_1, \ldots, a_n) \ne (0, \ldots, 0)$.

## Remarks

Every [[algebraically closed field|02SJG1HC32]] is $C_1$.
Indeed, let $g(t) = f(t, 1, 0, \ldots, 0)$.
If $g$ is not constant, it has a root $t_0$, and $(t_0, 1, 0, \ldots, 0)$ is a nontrivial zero of $f$.
If $g$ is constant, the coefficient of $x_1^d$ in $f$ is $0$, so $(1, 0, \ldots, 0)$ is a nontrivial zero of $f$.

Example: a [[finite field|08RTDHX1SK]] $\F_q$ is $C_1$.
Check: by the [[Chevalley–Warning theorem|08895Q5W96]], the number of zeros in $\F_q^n$ of a polynomial of degree $d < n$ is divisible by the characteristic.
The zero vector is one zero, so there is another.

Non-example: $\R$ is not $C_1$.
The form $x^2 + y^2 + z^2$ has degree $2$ in $3$ variables, but its only real zero is $(0, 0, 0)$.

More generally, $K$ is a $C_i$ field if every homogeneous polynomial of degree $d \ge 1$ in $n > d^i$ variables has a nontrivial zero; $C_1$ is the case $i = 1$.
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<document source="https://tamnd.github.io/math2/node/D6BM80QYNE.md" tag="D6BM80QYNE" kind="concept">
# Cyclic algebra

## Summary

A cyclic algebra is a [[crossed product algebra|G21MZQCME9]] for a [[cyclic extension|VFE5B71E7Y]] of fields.
It is described by a generator $\sigma$ of the Galois group and one nonzero element $a$ of the base field.
Every cyclic algebra of degree $2$ is a quaternion algebra, and over local and global fields every central simple algebra is cyclic.
Whether a cyclic algebra splits is decided by the [[field norm|KRNFEZWPY0]]: it splits exactly when $a$ is a norm.

## Explanations

### L1 human

Take a field extension $L/K$ whose symmetries form a cyclic group of order $n$ generated by $\sigma$.
Add one new symbol $u$ to $L$ with two rules.
Moving a number $x$ of $L$ past $u$ turns it into $\sigma(x)$, and $u^n$ equals a fixed number $a$ of $K$.
The result is the cyclic algebra.
For $L = \C$, $K = \R$, $\sigma$ complex conjugation and $a = -1$, this gives Hamilton's quaternions with $u = j$.
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# Polynomial long division

## Summary

Polynomial long division is the algorithm that divides a polynomial $f$ by a polynomial $g$ with invertible leading coefficient.
It produces the quotient $q$ and the remainder $r$ with $f = q g + r$ and $\deg r < \deg g$.
These are the quotient and remainder of the [[division algorithm for polynomials|0MES3KS0ZC]].
It is the basic step of the [[Euclidean algorithm|023VY493K0]] for polynomials and of many computations in algebra.

## Explanations

### L1 human

It works like long division of integers, with degrees in place of digits.
At each step you divide the leading term of what is left by the leading term of $g$.
You write this term in the quotient, subtract its product with $g$, and go on.
For example, dividing $x^2 + 1$ by $x - 1$ gives the quotient $x + 1$ and the remainder $2$, since $x^2 + 1 = (x + 1)(x - 1) + 2$.
The [[definition|FTAK2F0TR3]] gives the steps.
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<document source="https://tamnd.github.io/math2/node/D6DF1ARPVD.md" tag="D6DF1ARPVD" kind="statement">
# Yoneda lemma

## Statement

Let $\mathcal{C}$ be a locally small [[category|A2DE66937D]] and let $A$ be an object of $\mathcal{C}$.
Let $\mathbf{Set}$ be the category of sets and let $\mathcal{C}^{\mathrm{op}}$ be the [[opposite category|DAMK8XE1FR]].
Let $F \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ be a [[presheaf|AND2WHZC8P]] on $\mathcal{C}$.
Let $\mathrm{Nat}(\Hom(-, A), F)$ be the class of [[natural transformations|KYCPCC57FT]] from the [[hom functor|VJXS8AK67E]] $\Hom(-, A)$ to $F$.
Then the map
$$\mathrm{Nat}(\Hom(-, A), F) \to F(A), \qquad \alpha \mapsto \alpha_A(\id_A)$$
is a bijection.
Its inverse sends $x \in F(A)$ to the natural transformation $\alpha$ with $\alpha_X(f) = F(f)(x)$ for every object $X$ and every $f \colon X \to A$.
The bijection is natural in both $A$ and $F$.

## Remarks

Proof idea: let $\alpha$ be natural and $x = \alpha_A(\id_A)$.
For $f \colon X \to A$, naturality applied to $f$ and the element $\id_A$ gives $\alpha_X(\id_A \circ f) = F(f)(\alpha_A(\id_A))$, that is, $\alpha_X(f) = F(f)(x)$.
So $\alpha$ is determined by $x$, and a direct check shows that every $x$ defines a natural transformation in this way.

Example: take $F = \Hom(-, B)$ for an object $B$.
The lemma says that natural transformations $\Hom(-, A) \to \Hom(-, B)$ correspond to elements of $\Hom(A, B)$, that is, to morphisms $A \to B$; each one is post-composition with that morphism.

The covariant form is the same: for a functor $G \colon \mathcal{C} \to \mathbf{Set}$, natural transformations $\Hom(A, -) \to G$ correspond to elements of $G(A)$.
Local smallness is needed for $\Hom(-, A)$ to take values in $\mathbf{Set}$.

## Explanations

### L1 human

The Yoneda lemma says that an object is known completely by the arrows that point into it.
A natural way of turning arrows into $A$ into elements of some set-valued structure $F$ is decided by one choice: where the identity arrow of $A$ goes.
Everything else is forced by naturality.

### L2 human

The lemma is the reason that the Yoneda embedding $A \mapsto \Hom(-, A)$ is fully faithful, so every category sits inside a category of presheaves without loss of information.
It implies that a representing object of a representable functor is unique up to unique isomorphism.
This is why objects defined by a universal property are unique up to unique isomorphism.
In practice it is used to prove that two objects are isomorphic by showing that their hom functors are naturally isomorphic.
Applied to a group seen as a one-object category, it gives Cayley's theorem, that every group embeds in a group of permutations of its own elements.
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<document source="https://tamnd.github.io/math2/node/D6FMEQYA76.md" tag="D6FMEQYA76" kind="statement">
# Prokhorov's theorem

## Statement

Let $X$ be a [[metric space|0VKD1TP7F3]] and let $\mathcal{M}$ be a set of Borel [[probability measures|059TJN0NZ1]] on $X$.
Call $\mathcal{M}$ relatively compact if every sequence in $\mathcal{M}$ has a subsequence that [[converges weakly|MBW9JCGZC1]] to some Borel probability measure on $X$.

1. If $\mathcal{M}$ is [[tight|MNMJC8YVBS]], then $\mathcal{M}$ is relatively compact.
2. If $X$ is [[separable|0Z2P41SVDP]] and [[complete|0BY52JW6NK]], and $\mathcal{M}$ is relatively compact, then $\mathcal{M}$ is tight.

## Remarks

Tight means that for every $\varepsilon > 0$ there is a [[compact set|0R6EWFKRCF]] $K$ with $\mu(K) > 1 - \varepsilon$ for all $\mu \in \mathcal{M}$.
For a compact space $X$, part 1 follows from the Riesz representation theorem and the weak-* compactness of the unit ball of the dual of $C(X)$.
The general case reduces to compact sets by tightness.
For $X = \R$, part 1 is close to Helly's selection theorem for distribution functions.

Example: the normal distributions $N(m, \sigma^2)$ with $\abs{m} \le 1$ and $0 < \sigma \le 1$ form a tight family, using $K = [-M, M]$ with $M$ large.
So every sequence of them has a weakly convergent subsequence.
The family of Dirac measures $\delta_n$, $n \in \N$, on $\R$ is not tight, and no subsequence converges weakly.
The mass escapes to infinity.

## Explanations

### L1 human

A family of probability measures is tight when no mass runs away to infinity.
The theorem says that tightness is exactly what one needs to extract convergent subsequences.
It plays the role of the Bolzano–Weierstrass theorem for measures.

### L2 human

In probability, one proves convergence in distribution in two steps.
First one shows tightness, which gives limit points by part 1.
Then one shows that all limit points are the same measure, for example through characteristic functions or finite-dimensional distributions.
This scheme proves Donsker's invariance principle for random walks.
On a Polish space, the set of probability measures with the weak topology is itself a Polish space, metrized by the Lévy–Prokhorov metric.
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# Lelong number

## Definition

Let $\Omega \subseteq \C^n$ be open, let $u$ be a [[plurisubharmonic function|FF2P7ER3QT]] on $\Omega$ and let $x \in \Omega$.
The *Lelong number* of $u$ at $x$ is
$$\nu(u, x) = \lim_{r \to 0^+} \frac{\sup_{\abs{z - x} < r} u(z)}{\log r}.$$
Here $\abs{\cdot}$ is the Euclidean norm on $\C^n$.

## Remarks

The limit exists and lies in $[0, +\infty)$, because $\sup_{\abs{z - x} < r} u(z)$ is a convex increasing function of $\log r$.
An equivalent form is $\nu(u, x) = \liminf_{z \to x} u(z) / \log \abs{z - x}$.

Example: let $f$ be holomorphic on $\Omega$ with order of vanishing $m$ at $x$, and $u = \log \abs{f}$.
Near $x$ one has $c_1 r^m \le \sup_{\abs{z - x} < r} \abs{f(z)} \le c_2 r^m$ for constants $c_1, c_2 > 0$, so $\nu(u, x) = m$.
In particular $\nu(\log \abs{z_1}, 0) = 1$.

If $u(x) > -\infty$, then $\nu(u, x) = 0$.
The converse fails.
On the unit ball of $\C^n$ let $u(z) = -\sqrt{-\log \abs{z}}$, a plurisubharmonic function because $t \mapsto -\sqrt{-t}$ is convex and increasing on $(-\infty, 0)$.
Then $u(0) = -\infty$, but $\sup_{\abs{z} < r} u(z) / \log r = 1 / \sqrt{-\log r} \to 0$, so $\nu(u, 0) = 0$.

Let $T$ be a closed [[positive current|BYJHP6A906]] of bidimension $(p, p)$.
Its Lelong number $\nu(T, x)$ is the limit as $r \to 0$ of the trace mass of $T$ on the ball $B(x, r)$, divided by $\pi^p r^{2p} / p!$.
The divisor $\pi^p r^{2p} / p!$ is the volume of a ball of radius $r$ in $\C^p$.
With a suitable normalisation of $dd^c$, one has $\nu(u, x) = \nu(dd^c u, x)$.
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# Post's inversion formula

## Statement

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$.
Assume that the Laplace integral of $f$ converges at some real number $s_0$, and let $F$ be the [[Laplace transform|05XT5AVJBK]] of $f$.
Then for every $t > 0$ at which $f$ is continuous,
$$f(t) = \lim_{k \to \infty} \frac{(-1)^k}{k!} \left(\frac{k}{t}\right)^{k+1} F^{(k)}\!\left(\frac{k}{t}\right),$$
where $F^{(k)}$ is the $k$-th derivative of $F$ on the real half-line $s > s_0$.

## Remarks

By [[differentiation in the transform variable|VEPJN1NK3C]], $\frac{(-1)^k}{k!} \left(\frac{k}{t}\right)^{k+1} F^{(k)}\!\left(\frac{k}{t}\right) = \int_0^\infty f(u) \, \frac{1}{k!} \left(\frac{k}{t}\right)^{k+1} u^k e^{-k u / t} \, du$.
The kernel in this integral is a probability density in $u$ (a gamma density) with mean $t (k+1)/k$ and variance $t^2 (k+1)/k^2$.
As $k \to \infty$ it concentrates at $u = t$, so the integral tends to $f(t)$ at points of continuity.
Example: for $f(t) = e^{-t}$, $F(s) = 1/(s + 1)$ and $\frac{(-1)^k}{k!} F^{(k)}(s) = (s+1)^{-k-1}$, so the expression is $\left(\frac{k/t}{k/t + 1}\right)^{k+1} = \left(1 + \frac{t}{k}\right)^{-k-1} \to e^{-t}$.

## Explanations

### L1 human

This formula recovers a function from its Laplace transform using only values of the transform and its derivatives at real points.
No complex integration is needed.
The price is that one needs derivatives of very high order.

### L2 human

The formula matters in theory, for example in the theory of semigroups, where it gives the exponential formula for $T(t)$.
Numerically it is poor: high derivatives amplify small errors, which reflects the [[ill-posedness of real Laplace inversion|TRTN840ZB0]].
The method of proof, a kernel that concentrates at one point, is the same as in many approximation theorems.
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# Extension and contraction of ideals

## Definition

Let $A$ and $B$ be [[commutative rings|0GYGHKX834]] and let $f \colon A \to B$ be a [[ring homomorphism|0GGE0Z0JQ1]].

1. For an [[ideal|0MYSA6MYNA]] $I$ of $A$, the *extension* of $I$ is the [[ideal generated|030TZ0DQWT]] by $f(I)$ in $B$:
$$I^e = f(I) B = \set{\textstyle\sum_{k=1}^{n} b_k f(a_k) : n \ge 0,\ a_k \in I,\ b_k \in B}.$$
2. For an ideal $J$ of $B$, the *contraction* of $J$ is the preimage $J^c = f^{-1}(J)$, an ideal of $A$.

## Remarks

The relations $I \subseteq I^{ec}$, $J^{ce} \subseteq J$, $I^{e} = I^{ece}$ and $J^{c} = J^{cec}$ hold.
If $J$ is a [[prime ideal|06HC6HK0Z8]] of $B$, then $J^c$ is a prime ideal of $A$, because $A/J^c$ embeds into the integral domain $B/J$.
Example: for the inclusion $\Z \to \Z[i]$, $(2 + i)^c = 5\Z$, since $(2 + i) \cap \Z$ contains $5 = (2+i)(2-i)$ and is a proper ideal containing $5\Z$.
Non-example of the analogous property for extension: $2\Z$ is prime in $\Z$, but $(2\Z)^e = 2\Z[i]$ is not prime, since $(1 + i)^2 = 2i$ lies in it while $1 + i$ does not.
The image $f(I)$ itself need not be an ideal of $B$; for $\Z \subseteq \Q$, the image of $2\Z$ is not an ideal of $\Q$, and its extension is all of $\Q$.
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# Abelian and Tauberian theorems

## Summary

Abelian and Tauberian theorems are two opposite kinds of results about [[summability methods|8X25Q4XYMN]].
An Abelian theorem says that a method gives the usual value to every convergent series.
More generally, it says that summability by one method implies summability by another, with the same value.
[[Abel's theorem|0S4EMDQS2F]] is the model.
A Tauberian theorem is a partial converse.
It says that a summable series converges, provided its terms satisfy an extra [[Tauberian condition|6ZDP261K7R]].
[[Tauber's theorem|8SK3GQ4S0M]] is the model.
Tauberian theorems for integral transforms, such as the [[Wiener–Ikehara theorem|KQ57DC3AKM]], give short proofs of the [[prime number theorem|0EHBY5Y84N]].

## Explanations

### L1 human

Summation methods are ways to give a value to a series that may diverge.
An Abelian theorem checks that a method does no harm: a series that already has a sum keeps the same sum.
A Tauberian theorem goes the other way: if a method gives a value and the terms are small enough, the series really converges to that value.
For example, if $\sum a_n x^n \to s$ as $x \to 1^-$ and $n a_n \to 0$, then $\sum a_n = s$.
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<document source="https://tamnd.github.io/math2/node/D75XZC18PP.md" tag="D75XZC18PP" kind="statement">
# Beck's theorem

## Statement

There are constants $c_1 > 0$ and $c_2 > 0$ with the following property.
Let $P$ be a finite set of $n$ points in the [[Euclidean plane|0XW7V9ZZ05]].
Say that a [[line|0MP9BCF6DP]] is *determined* by $P$ if it contains at least two points of $P$.
Then at least one of the following holds.

1. Some line contains at least $c_1 n$ points of $P$.
2. $P$ determines at least $c_2 n^2$ distinct lines.

## Remarks

The proof counts pairs of points of $P$ according to the number of points on the line through them.
The [[Szemerédi–Trotter theorem|NQBWFSC1E0]] bounds the number of lines containing between $k$ and $2k$ points of $P$ by $O(n^2/k^3 + n/k)$.
Sum this bound over dyadic ranges of $k$.
Unless some line is very rich, a constant fraction of the $\binom{n}{2}$ pairs then lie on lines with few points.
Those lines are then of order $n^2$ in number.
Example: if $P$ consists of $n - 1$ [[collinear|0AFNNTRTJE]] points and one point off their line, clause 1 holds, while $P$ determines only $n$ lines.
If $P$ is the set of vertices of a regular $n$-gon, no line contains more than $2$ points, so clause 2 must hold; indeed all $\binom{n}{2}$ lines are distinct.

## Explanations

### L1 human

Take $n$ points in the plane.
Either many of them lie on one line, or the points determine a huge number of lines, about as many as there are pairs of points.
There is no middle ground where the points are spread out but determine few lines.

### L2 human

József Beck proved the theorem in 1983, and it is a standard application of the Szemerédi–Trotter incidence bound.
A consequence is a weak form of Dirac's conjecture: some point of $P$ lies on at least $c n$ of the lines determined by $P$, unless most points are collinear.
It belongs to the circle of results around the [[Sylvester–Gallai theorem|T3GGJN70P2]] and the orchard-planting problem, which count lines with few points.
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# Complete lattice

## Summary

A complete lattice is a partially ordered set in which every subset, finite or infinite, has a supremum and an infimum.
It is the setting of many fixed-point theorems, such as the [[Knaster–Tarski theorem|RHXP3HA64M]].
Power sets, lattices of subgroups and lattices of closed sets of a topological space are complete lattices.
Every partially ordered set embeds in a complete lattice, its [[Dedekind–MacNeille completion|DKH4933HMQ]].

## Explanations

### L1 human

In a lattice, any two elements have a least upper bound and a greatest lower bound.
A complete lattice asks the same for every collection of elements, even an infinite one.
The subsets of a set, ordered by inclusion, form a complete lattice: the least upper bound of any family is its union and the greatest lower bound is its intersection.
The real line is not complete in this sense, since the set of all integers has no upper bound, but the closed interval $[0, 1]$ is.
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# Eigendecomposition

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $F$.
Suppose that $A$ has $n$ [[linearly independent|0CRMGPAH83]] [[eigenvectors|0N6960W0BV]] $q_1, \dots, q_n$ in $F^n$, with $A q_i = \lambda_i q_i$.
Let $Q$ be the matrix with columns $q_1, \dots, q_n$, and let $\Lambda$ be the [[diagonal matrix|044JNC7ZAA]] with diagonal entries $\lambda_1, \dots, \lambda_n$.
Then $Q$ is [[invertible|0EB9N0S33C]] and
$$A = Q \Lambda Q^{-1}.$$
Conversely, let $A = Q \Lambda Q^{-1}$ with $Q$ invertible and $\Lambda$ diagonal.
Then the columns of $Q$ are linearly independent eigenvectors of $A$.
The diagonal entries of $\Lambda$ are the corresponding [[eigenvalues|0BQAA2BK3Q]].

## Remarks

The columns of $AQ$ are the vectors $A q_i = \lambda_i q_i$, and these are also the columns of $Q \Lambda$.
Thus $AQ = Q\Lambda$, and $Q$ is invertible because its columns are independent.
For example, $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ has the eigenvector $(1, 0)^T$ for $2$ and $(1, 1)^T$ for $3$.
Thus $A = Q \Lambda Q^{-1}$ with $Q = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $\Lambda = \operatorname{diag}(2, 3)$.
The hypothesis is necessary: $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ has only one independent eigenvector, and it has no eigendecomposition.
When $A$ is real symmetric or complex [[normal|0MR0BYF0K2]], $Q$ can be chosen orthogonal or unitary, by the [[spectral theorem|06J4PXRG22]].

## Explanations

### L1 human

A matrix with enough eigenvectors acts, in the basis of those eigenvectors, by stretching each axis by its eigenvalue.
The eigendecomposition writes this as: change of basis, stretch, change back.
Not every matrix has one; exactly the [[diagonalisable|07SQCNV233]] matrices do.

### L2 human

The decomposition makes powers and functions of $A$ easy: $A^k = Q \Lambda^k Q^{-1}$, and the same holds for the exponential.
It is the matrix form of the criterion that [[a matrix is diagonalisable iff it has a basis of eigenvectors|056SW1E0AC]].
Distinct eigenvalues give enough eigenvectors, by [[this statement|0HNMFG8F8K]].
When eigenvectors are missing, the [[Jordan normal form|085HQ5G5TE]] or the [[Schur decomposition|V1JG4C7R7B]] takes its place.
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# Puppe sequence

## Statement

Let $f \colon (X, x_0) \to (Y, y_0)$ be a map of [[pointed spaces|T9DG8DDE75]], and let $I = [0, 1]$.
Let $CX = (X \times I) / (X \times \set{1} \cup \set{x_0} \times I)$ be the reduced cone on $X$.
Let $\Sigma X = CX / (X \times \set{0})$ be the reduced [[suspension|Q7RFNJH76Q]] of $X$.
Let $C_f = (Y \sqcup CX) / \sim$, with $[(x, 0)] \sim f(x)$ for $x \in X$, be the reduced mapping cone of $f$.
Let $i \colon Y \to C_f$ be the inclusion, and let $q \colon C_f \to \Sigma X$ be the map that collapses $Y$ to the base point.
Then there is a sequence of pointed maps
$$X \xrightarrow{f} Y \xrightarrow{i} C_f \xrightarrow{q} \Sigma X \xrightarrow{\Sigma f} \Sigma Y \xrightarrow{\Sigma i} \Sigma C_f \xrightarrow{\Sigma q} \Sigma^2 X \xrightarrow{\Sigma^2 f} \cdots$$
with the following property.
For every pointed space $Z$, the induced sequence of sets of pointed homotopy classes
$$\cdots \to [\Sigma Y, Z]_* \to [\Sigma X, Z]_* \to [C_f, Z]_* \to [Y, Z]_* \to [X, Z]_*$$
is exact.
Exactness at a term means this: the image of the incoming map is the set of classes sent to the class of the constant map.

## Remarks

The key step is that the mapping cone of $i \colon Y \to C_f$ is homotopy equivalent to $\Sigma X$, by collapsing the cone on $Y$, which is a contractible subspace attached along a [[cofibration|W5WS5EY8ZM]].
Exactness at $[Y, Z]_*$ is direct.
A map $g \colon Y \to Z$ extends over $C_f$ exactly when $g \circ f$ is [[null-homotopic|SMEV5BSQ5S]] by a pointed homotopy, since such a null-homotopy is a map on $CX$.
Repeating the step gives the whole sequence, and up to sign the maps after $q$ are suspensions of the earlier ones.
The terms $[\Sigma^n X, Z]_*$ are groups for $n \ge 1$ and abelian groups for $n \ge 2$, and there the [[exact sequence|JB7ZBFBA9C]] is one of groups.
When $f$ is the inclusion of a subcomplex $X$ of a CW complex $Y$, the space $C_f$ is homotopy equivalent to the quotient $Y / X$.
The dual construction with loop spaces and homotopy fibres gives the fibre sequence $\cdots \to \Omega Y \to F \to X \to Y$.

## Explanations

### L1 human

The Puppe sequence turns a single map $f \colon X \to Y$ into an infinite chain of spaces and maps.
Each new space is built by coning off the previous map, and after three steps the chain repeats with suspended spaces.
Mapping this chain into any space $Z$ gives an exact sequence, so information about maps out of $X$ and $Y$ controls maps out of the cone $C_f$.

### L2 human

The Puppe sequence is the source of long exact sequences in homotopy theory.
Applied to the inclusion of a subcomplex $A \subseteq X$ and to a representing space $Z$, such as an Eilenberg–MacLane space, it yields the long exact sequence of a pair in cohomology.
Its [[Eckmann–Hilton dual|1EK3355WMC]] is the fibre sequence, which yields the long exact sequence of homotopy groups of a fibration.
It is a basic example of a triangulated structure, which in the stable homotopy category becomes the distinguished triangles $X \to Y \to C_f \to \Sigma X$.
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# Countable additivity

## Summary

A set function is countably additive, or σ-additive, if its value on a countable disjoint union is the sum of its values on the pieces.
Countable additivity is the defining property of measures.
It is stronger than finite additivity, and it is what makes limit theorems such as monotone convergence work.

## Explanations

### L1 human

Countable additivity says that sizes add up even when a set is cut into infinitely many pieces, as long as there are only countably many.
For example, $[0, 1)$ is the disjoint union of $[1 - 2^{-n+1}, 1 - 2^{-n})$ for $n \ge 1$, of lengths $1/2, 1/4, 1/8, \ldots$.
These lengths add up to $1$, the length of $[0, 1)$.
Without this rule, one could not pass to limits when computing sizes.
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# Fusion category

## Summary

A fusion category is a semisimple [[rigid monoidal category|A10WW8GEKR]] over a field, with finite-dimensional spaces of morphisms.
It has finitely many simple objects up to isomorphism, and its unit object is simple.
Its structure is recorded by fusion rules: how the tensor product of two simple objects splits into simple objects.
The finite-dimensional representations of a finite group over $\C$ form the basic example.
Fusion categories appear in subfactor theory, in conformal field theory and in topological phases of matter.

## Explanations

### L1 human

A fusion category is a finite "multiplication table" for building blocks that can be combined and decomposed.
There are finitely many basic objects, every object is a sum of basic ones, and combining two basic objects gives a sum of basic objects again.
For example, the group $\Z/2$ has two irreducible complex representations, the trivial one $1$ and the sign $s$.
Their tensor products are $1 \otimes 1 = 1$, $1 \otimes s = s$ and $s \otimes s = 1$.
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# Fourier analysis

## Definition

*Fourier analysis* is the part of [[harmonic analysis|7QNRC9SJVV]] that studies the decomposition of functions into [[complex exponentials|07GJV36896]] and the reconstruction of functions from these components.
It covers the following transforms:

1. the [[Fourier series|G12KXDHPPS]] of a $2\pi$-periodic function, with components $e^{inx}$ for $n \in \Z$;
2. the [[Fourier transform|DPVM34ADZV]] of a function on $\R^n$, with components $e^{2\pi i x \cdot \xi}$ for $\xi \in \R^n$;
3. the discrete Fourier transform of a function on $\Z / N\Z$, with components $e^{2\pi i jk/N}$ for $k \in \Z / N\Z$.

## Remarks

Example: the $2\pi$-periodic function $f(x) = \sin^3 x$ satisfies $f(x) = \frac{3}{4} \sin x - \frac{1}{4} \sin 3x$, by the identity $\sin 3x = 3 \sin x - 4 \sin^3 x$.
So its Fourier decomposition has only the frequencies $\pm 1$ and $\pm 3$.
Non-example: writing a function as a power series $\sum a_n x^n$ is an expansion into monomials, not into exponentials, so it is not Fourier analysis.
The word analysis refers to the decomposition step, and the word synthesis to the reconstruction step.
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# Transcendence degree

## Summary

The transcendence degree of a [[field extension|0EQWVHF3W5]] $L/K$ is the cardinality of a [[transcendence basis|E45PA267S4]].
It counts the algebraically independent elements needed so that the rest is algebraic.
It does not depend on the choice of basis.
It is the field-theoretic notion of dimension: the function field of an irreducible variety has transcendence degree equal to the dimension of the variety.

## Explanations

### L1 human

The transcendence degree counts how many free variables a field extension has.
The field $\C(x, y)$ has transcendence degree $2$ over $\C$, like the plane.
The field of rational functions on a curve has transcendence degree $1$, like a line.
An extension has transcendence degree $0$ exactly when it is algebraic.
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# Topological field

## Summary

A topological field is a [[field|04C5ZBG1WT]] with a [[topology|0CQHFPGJ45]] in which addition, multiplication, negation and inversion are continuous, see the [[definition|H4FS63VY0Y]].
It is a field that is also a [[topological ring|TSF8RZDD5B]] and in which taking inverses is continuous.
The real numbers, the complex numbers, the [[p-adic numbers|BT6T702C28]] and every field with an absolute value are topological fields.
The notion is the common setting for analysis over fields other than $\R$ and $\C$.

## Explanations

### L1 human

In $\R$, small changes of $x$ and $y$ give small changes of $x + y$, $xy$ and, when $x \ne 0$, of $1/x$.
A topological field is any field with a notion of "closeness" for which the same is true.
The $p$-adic numbers are an example where two numbers are close when their difference is divisible by a high power of $p$.
The rule that $1/x$ moves little when $x$ moves little, away from $0$, is the extra demand compared with a topological ring.
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<document source="https://tamnd.github.io/math2/node/D83NMTHT7N.md" tag="D83NMTHT7N" kind="concept">
# Projective connection

## Summary

A projective connection is the structure a manifold carries when it knows which curves are its straight lines.
It does not know how these lines are parametrized.
In Cartan's language it is a [[Cartan connection|QPR7DCHXPG]] modelled on real projective space, viewed as a homogeneous space of the projective linear group.
Classically it is described by a class of torsion-free affine connections that have the same geodesics up to reparametrization.
Projective connections appear in projective differential geometry and in the study of second-order ordinary differential equations.

## Explanations

### L1 human

In projective geometry there are lines but no distances and no preferred speed along a line.
A projective connection gives a curved space the same kind of structure: a family of curves that play the role of straight lines.
Two different ways of measuring acceleration can produce exactly the same paths with different speeds; they define the same projective connection.
The flat example is projective space itself, or ordinary space with its usual straight lines.
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<document source="https://tamnd.github.io/math2/node/D85VZWS9QK.md" tag="D85VZWS9QK" kind="statement">
# Hilbert's Nullstellensatz

## Statement

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]], let $n \ge 1$, and let $I$ be an [[ideal|096Z9MP2ME]] of the [[polynomial ring|0SDC934R52]] $k[x_1, \ldots, x_n]$.
Let $V(I) = \set{a \in k^n : f(a) = 0 \text{ for all } f \in I}$ be the [[zero set|VHFMQQYSTN]] of $I$.
Let $I(V(I)) = \set{f \in k[x_1, \ldots, x_n] : f(a) = 0 \text{ for all } a \in V(I)}$ be the [[ideal of polynomials vanishing on it|HH6VEVM6Z0]].
Then
$$I(V(I)) = \sqrt{I},$$
where $\sqrt{I} = \set{f : f^m \in I \text{ for some } m \ge 1}$ is the [[radical|437YRCNYRJ]] of $I$.

## Remarks

The inclusion $\sqrt{I} \subseteq I(V(I))$ is clear: if $f^m$ vanishes at a point, so does $f$.
For the other inclusion, let $f$ vanish on $V(I)$ and add a new variable $y$ (the Rabinowitsch trick).
The ideal generated by $I$ and $1 - y f$ in $k[x_1, \ldots, x_n, y]$ has no zeros, so it is the unit ideal by the [[weak Nullstellensatz|ZWS0YWGF17]].
Writing $1$ as a combination and substituting $y = 1/f$, then clearing denominators, gives $f^m \in I$ for some $m$.

Example: in $\C[x]$, let $I = (x^2)$; then $V(I) = \set{0}$ and $I(V(I)) = (x) = \sqrt{(x^2)}$.

The field must be algebraically closed: over $\R$, let $I = (x^2 + 1)$ in $\R[x]$.
Then $V(I)$ is empty, so $I(V(I)) = \R[x]$, but $\sqrt{I} = I$ is a proper ideal, since $I$ is prime.

## Explanations

### L1 human

A polynomial vanishes at all common zeros of some polynomials exactly when one of its powers is a combination of them.
So, over $\C$, systems of polynomial equations and ideals of polynomials carry the same information, up to taking radicals.

### L2 human

The theorem gives an inclusion-reversing bijection between radical ideals of $k[x_1, \ldots, x_n]$ and algebraic subsets of $k^n$.
Prime ideals correspond to irreducible algebraic sets and maximal ideals to points.
It is the bridge between commutative algebra and classical algebraic geometry.
In ring-theoretic form it says that $k[x_1, \ldots, x_n]$ is a [[Jacobson ring|7EB64ZBXAT]]: every radical ideal is an intersection of maximal ideals.
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<document source="https://tamnd.github.io/math2/node/D86X15A8WS.md" tag="D86X15A8WS" kind="concept">
# Singular point of an algebraic variety

## Summary

A point of an algebraic variety is singular when the tangent space there is larger than the dimension of the variety.
Otherwise it is nonsingular, or smooth.
For a hypersurface $f = 0$ with $f$ irreducible, the singular points are those where $f$ and all its partial derivatives vanish.
Singular points are where the variety fails to look like a manifold, such as cusps and self-crossings.
They form a proper closed subset of every variety.

## Explanations

### L1 human

Most points of a curve look like a small piece of a line when you zoom in.
A singular point is one that does not: the curve crosses itself there, or has a sharp cusp.
The curve $y^2 = x^2 (x + 1)$ crosses itself at the origin, and the curve $y^2 = x^3$ has a cusp there.
At those points the partial derivatives of the equation both vanish, and there is no well-defined tangent line.
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<document source="https://tamnd.github.io/math2/node/D8A4VEQR98.md" tag="D8A4VEQR98" kind="statement">
# Ulm's theorem

## Statement

Let $p$ be a [[prime|01826KMG2H]].
An abelian $p$-group is an [[abelian group|082GHNPJ53]] in which every element has order a power of $p$.
It is *reduced* if its only [[divisible|ZXVBEPMTCN]] subgroup is $\set{0}$.
For an abelian $p$-group $A$, define subgroups $p^\alpha A$ for each [[ordinal|0YSCSX5YZC]] $\alpha$ by $p^0 A = A$, $p^{\alpha+1} A = p(p^\alpha A)$, and $p^\lambda A = \bigcap_{\alpha < \lambda} p^\alpha A$ for limit ordinals $\lambda$.
Let $P_\alpha = \set{x \in p^\alpha A : px = 0}$, a vector space over the [[finite field|0ZXZ82FTX2]] $\F_p$.
The *$\alpha$-th Ulm invariant* of $A$ is $u_\alpha(A) = \dim_{\F_p} (P_\alpha / P_{\alpha+1})$.

Let $A$ and $B$ be [[countable|01TQB88H4C]] reduced abelian $p$-groups.
Then $A \cong B$ if and only if $u_\alpha(A) = u_\alpha(B)$ for every ordinal $\alpha$.

## Remarks

Isomorphic groups have the same Ulm invariants, since an isomorphism carries each $P_\alpha$ onto the corresponding subgroup.
For the converse one builds an isomorphism step by step, by a back-and-forth argument over the countably many elements.
At each step one keeps a partial isomorphism between finite subgroups that preserves the heights of elements.
The equality of the Ulm invariants is what makes each extension step possible.

Example: for $A = \Z/p^k\Z$ with $k \ge 1$, $P_n$ is cyclic of order $p$ for $n < k$ and zero for $n \ge k$, so $u_{k-1}(A) = 1$ and all other invariants vanish.
For a finite abelian $p$-group, $u_n$ counts the cyclic summands of order $p^{n+1}$, and the theorem reduces to the classification of finite abelian $p$-groups.

Countability is necessary.
Let $B = \bigoplus_{n \ge 1} \Z/p^n\Z$, and let $\bar B$ be the torsion part of $\prod_{n \ge 1} \Z/p^n\Z$.
Both are reduced, both have $u_n = 1$ for every finite $n$ and $u_\alpha = 0$ for $\alpha \ge \omega$, but $B$ is countable and $\bar B$ is uncountable, so they are not isomorphic.

## Explanations

### L1 human

A countable abelian group in which every element has order a power of $p$ can be very complicated, with elements divisible by $p$ many times, infinitely many times, and beyond.
Ulm attached to such a group a list of numbers, indexed by ordinals, that measure how many independent elements of order $p$ sit at each level of divisibility.
The theorem says that, for countable groups with no divisible part, this list determines the group completely.

### L2 human

Ulm's theorem extends the classification of finite abelian $p$-groups by cyclic decompositions to all countable reduced $p$-groups, where no cyclic decomposition need exist.
A general countable abelian $p$-group is the direct sum of a [[divisible|ZXVBEPMTCN]] group and a reduced group.
The divisible part is a direct sum of [[Prüfer groups|F0XAMM2PPS]], determined by the number of summands.
So all countable abelian $p$-groups are classified.
The uncountable case has no such classification, as the example above shows.
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# Subcategory

## Summary

A subcategory of a [[category|A2DE66937D]] $\mathcal{C}$ is a choice of some objects and some [[morphisms|859F0TFVA3]] of $\mathcal{C}$.
The choice must be closed under composition and contain the identities of its objects.
It is a category in its own right, and its inclusion into $\mathcal{C}$ is a [[faithful functor|GRG8WEBSH5]].
Important special cases are the [[full subcategories|V3BFGKBJF7]], which keep all morphisms between the chosen objects, and the wide subcategories, which keep all objects.
The [[category of abelian groups|082GHNPJ53]] inside the category of groups is a standard example.

## Explanations

### L1 human

Start from a category and keep only part of it: some of the objects, and some of the arrows between them.
To still have a category, the identity arrow of each kept object must be kept, and a chain of kept arrows must compose to a kept arrow.
For example, among all sets and functions, keep all sets but only the injective functions.
Identities are injective and composites of injective functions are injective, so this is a subcategory.
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# Commutative diagram

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
A *diagram* in $\mathcal{C}$ is a directed graph with each vertex labelled by an object and each edge labelled by a [[morphism|859F0TFVA3]] of $\mathcal{C}$.
An edge from a vertex labelled $A$ to a vertex labelled $B$ must be labelled by a morphism $A \to B$.
The composite of a directed path of edges $e_1, \dots, e_n$ with $n \ge 1$ and labels $f_1, \dots, f_n$ is $f_n \circ \dots \circ f_1$.
The diagram *commutes* if any two directed paths of length at least $1$ with the same start and the same end have the same composite.

## Remarks

Example: the square with vertices $A, B, C, D$ and edges $f \colon A \to B$, $g \colon B \to D$, $h \colon A \to C$, $k \colon C \to D$ commutes exactly when $g \circ f = k \circ h$.
For instance, in the category of sets with $A = B = C = D = \R$, the maps $f = k = (x \mapsto 2x)$ and $g = h = (x \mapsto 3x)$ give a commuting square, since both composites are $x \mapsto 6x$.
Non-example: with $f = k = (x \mapsto x + 1)$ and $g = h = (x \mapsto 2x)$, the square does not commute.
Indeed $g(f(x)) = 2x + 2$ while $k(h(x)) = 2x + 1$.
In practice one only asks for commutativity of the paths that the text names.
For a diagram with a loop $f$, the convention above would force $f = f \circ f$.
More formally, a diagram of shape $J$ is a functor from a category $J$ to $\mathcal{C}$.
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<document source="https://tamnd.github.io/math2/node/D8GA2R3QT5.md" tag="D8GA2R3QT5" kind="conjecture">
# Standard conjectures on algebraic cycles

## Statement

Let $k$ be an algebraically closed field, and let $\ell$ be a prime different from the characteristic of $k$.
Let $X$ be a smooth irreducible [[projective variety|A4H4CZ2V35]] of dimension $n$ over $k$.
Write $H^i(X) = H^i(X, \Q_\ell)$ for [[ℓ-adic cohomology|E1DVHN06CR]], with the [[cup product|JX47MS1R1H]].
Let $\mathrm{cl} \colon Z^j(X) \to H^{2j}(X)$ be the cycle class map on [[algebraic cycles|9VFWPZEVGN]] of codimension $j$, and let $A^j(X) \subset H^{2j}(X)$ be the $\Q$-span of its image.
A class in $H^\bullet(X)$ is *algebraic* if it lies in the span of the $A^j(X)$.
Let $L \colon H^i(X) \to H^{i+2}(X)$ be the cup product with the class of a hyperplane section.
By the $\ell$-adic form of the [[hard Lefschetz theorem|816AHR3DRT]], due to Deligne, $L^{n-i} \colon H^i(X) \to H^{2n-i}(X)$ is an isomorphism for $i \le n$.
The standard conjectures are the following.

1. Lefschetz type (conjecture B): the operator $\Lambda$ on $H^\bullet(X)$ that inverts $L$ in the Lefschetz decomposition is induced by an algebraic class in $H^\bullet(X \times X)$.
2. Künneth type (conjecture C): the projectors $\pi_i$ of $H^\bullet(X)$ onto $H^i(X)$ are induced by algebraic classes in $H^{2n}(X \times X)$.
3. Conjecture D: for $Z \in Z^j(X) \otimes \Q$, one has $\mathrm{cl}(Z) = 0$ if and only if $\deg(Z \cdot W) = 0$ for every $W \in Z^{n-j}(X)$; that is, homological and numerical equivalence agree.
4. Hodge type (conjecture I): for $2j \le n$, the pairing $(x, y) \mapsto (-1)^j \deg\big(L^{n-2j} x \cdot y\big)$ is positive definite on the primitive part $\set{x \in A^j(X) : L^{n-2j+1} x = 0}$.

## Remarks

Grothendieck stated the conjectures in his 1968 lecture at the Bombay colloquium on algebraic geometry, published in 1969.
They remain open in general.
The standard conjectures imply the Riemann hypothesis part of the [[Weil conjectures|81TXP32VDM]], which Deligne later proved by other means.
In characteristic $0$, conjecture I follows from the Hodge–Riemann bilinear relations of Hodge theory.
Conjecture B is known for curves, surfaces and abelian varieties.
Conjecture D is known for cycles of codimension $1$.
If the conjectures hold, the category of [[motives|K1AZ4YF2RE]] for numerical equivalence has the properties Grothendieck wanted for a universal cohomology.

## Explanations

### L1 human

Many results about varieties concern the classes in cohomology that come from subvarieties, the algebraic classes.
The standard conjectures predict that some natural operations on cohomology, such as the projection onto one degree, are themselves given by subvarieties of $X \times X$.
They also predict that a class from subvarieties is zero in cohomology as soon as it meets every other such class with total multiplicity zero.
Grothendieck saw them as the key to a theory of motives and to a natural proof of the Weil conjectures.
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# Topology of pointwise convergence

## Definition

Let $X$ be a set, let $Y$ be a [[topological space|0CQHFPGJ45]], and let $Y^X$ be the set of all maps from $X$ to $Y$.
For $x \in X$ and an [[open|02BH3MCBQB]] set $U \subseteq Y$, put
$$W(x, U) = \set{f \in Y^X : f(x) \in U}.$$
The *topology of pointwise convergence* on $Y^X$ is the topology generated by the [[subbasis|09S8DV34Y0]] of all sets $W(x, U)$.
On a subset $F \subseteq Y^X$, such as the set of continuous maps when $X$ is a space, it is the [[subspace topology|0WS9P12FQR]].

## Remarks

Viewing $Y^X$ as the product $\prod_{x \in X} Y$, this topology is exactly the [[product topology|0GBG8B49AA]].
A sequence $(f_n)$ converges to $f$ in it if and only if $f_n(x) \to f(x)$ in $Y$ for every $x \in X$, which is [[pointwise convergence|0GFGK40582]].
The same holds for nets.

Example: on $\R^{[0, 1]}$, let $f_n(x) = x^n$, and let $f(x) = 0$ for $x < 1$ and $f(1) = 1$.
For each fixed $x$, the numbers $x^n$ tend to $f(x)$, so $f_n \to f$ in this topology.

Non-example: the discrete topology on $\R^{[0, 1]}$ is not the topology of pointwise convergence.
In the discrete topology, the sequence $f_n$ above does not converge, because $\set{f}$ is open and contains no $f_n$.

For an uncountable set $X$ and $Y = \R$, the space $\R^X$ with this topology is not first-countable, so it is not metrizable.
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# Boundary integral method for the Dirichlet problem

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a bounded open set whose [[boundary|05D5EGMFYV]] $\partial\Omega$ is a $C^2$ hypersurface.
Let $f \colon \partial\Omega \to \R$ be continuous, and consider the interior [[Dirichlet problem|8D56GQ4AG8]] with data $f$.
For a continuous density $\varphi$ on $\partial\Omega$, let $D\varphi$ be its [[double layer potential|EG8Z3FWV9A]], and let $K\varphi(x) = \int_{\partial\Omega} \frac{\partial \Phi(x - y)}{\partial \nu(y)} \varphi(y)\, d\sigma(y)$ for $x \in \partial\Omega$, as in the [[jump relations|J58YA8QNNX]].
Here $\Phi$ is the fundamental solution of the Laplacian with $-\Delta\Phi = \delta_0$, $\sigma$ is surface measure on $\partial\Omega$ and $\nu$ is the outward unit normal.
The *boundary integral method* (double layer method) for this problem consists of the following steps.

1. Seek the solution in the form $u = D\varphi$ on $\Omega$, with $\varphi$ unknown.
2. Solve the boundary integral equation
$$-\tfrac12 \varphi(x) + K\varphi(x) = f(x), \qquad x \in \partial\Omega,$$
for a continuous $\varphi$.
3. Take $u = D\varphi$ on $\Omega$ and $u = f$ on $\partial\Omega$.

## Remarks

By the jump relations, the interior boundary values of $D\varphi$ are $K\varphi - \tfrac12\varphi$, so step 2 is exactly the condition that $u$ is continuous on $\overline{\Omega}$ with $u = f$ on $\partial\Omega$.
Since $\partial\Omega$ is $C^2$, the kernel of $K$ is weakly singular and $K$ is a [[compact operator|M2NEP1K5AE]] on $C(\partial\Omega)$.
So the equation is a Fredholm equation of the second kind.
By the [[Fredholm alternative|4HH8C9V8JQ]], it is solvable for every $f$ once the homogeneous equation has only the solution $\varphi = 0$.
This holds when $\R^n \setminus \overline{\Omega}$ is connected.

Example: let $\Omega$ be the unit ball in $\R^3$ and $f \equiv 1$.
Gauss's integral gives $K1 = -\tfrac12$ on the sphere, so $\varphi \equiv -1$ solves $-\tfrac12\varphi + K\varphi = \tfrac12 + \tfrac12 = 1$.
Then $u = D(-1) = 1$ in the ball, the correct solution.

Non-example: let $\Omega$ be the shell $\set{x \in \R^3 : 1 < \abs{x} < 2}$, whose complement has the bounded component $\set{\abs{x} \le 1}$.
Let $\varphi = 1$ on the inner sphere and $\varphi = 0$ on the outer sphere.
The outward normal of $\Omega$ on the inner sphere points toward the origin, so $D\varphi = 1$ in the inner ball and $D\varphi = 0$ outside it, in particular on $\Omega$.
Thus $\varphi \ne 0$ solves the homogeneous equation, and the equation of step 2 is not uniquely solvable.
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# Specht module

## Definition

Let $n \ge 1$, let $F$ be a field, and let $\lambda$ be a [[partition|0JV30XN73Z]] of $n$ with [[Young diagram|096AHMB3P4]] $D_\lambda$.
A [[Young tableau|Q6TH9PH2R7]] $t$ of shape $\lambda$ is a bijective filling of the boxes of $D_\lambda$ with $1, \ldots, n$.
The [[symmetric group|0515A6D2Y1]] $S_n$ acts on tableaux by acting on the entries.
Two tableaux are row-equivalent if corresponding rows have the same sets of entries.
The class $\set{t}$ of $t$ under this relation is a tabloid.
Let $M^\lambda$ be the vector space over $F$ with basis the tabloids of shape $\lambda$, with the induced action of $S_n$.
Let $C_t \le S_n$ be the column [[subgroup|0X1JXT47GA]] of $t$, the permutations that keep each column of $t$ as a set.
The polytabloid of $t$ is
$$e_t = \sum_{\sigma \in C_t} \sgn(\sigma) \, \sigma \set{t} \in M^\lambda,$$
where $\sgn$ is the [[sign|02DKV69ZH0]].
The *Specht module* $S^\lambda$ is the subspace of $M^\lambda$ spanned by the $e_t$ for all tableaux $t$ of shape $\lambda$.

## Remarks

Since $\pi e_t = e_{\pi t}$ for $\pi \in S_n$, the space $S^\lambda$ is a representation of $S_n$, that is, an $F[S_n]$-module.
Example: for $\lambda = (n)$, one row, there is only one tabloid and $C_t$ is trivial, so $S^{(n)}$ is the trivial representation of degree $1$.
For $\lambda = (1^n)$, one column, each row has a single box, so there are $n!$ tabloids, $C_t = S_n$ and $\pi e_t = \sgn(\pi) e_t$, so $S^{(1^n)}$ is the sign representation.
Non-example: the space $M^\lambda$ itself is not in general a Specht module; for $\lambda = (n-1, 1)$ it is the permutation representation on $n$ points, of dimension $n$, while $S^{(n-1,1)}$ has dimension $n - 1$.
The polytabloids $e_t$ for standard tableaux $t$, increasing along rows and down columns, form a basis of $S^\lambda$.
If $F$ has characteristic $0$, the modules $S^\lambda$ are irreducible, pairwise non-isomorphic, and every irreducible representation of $S_n$ over $F$ is isomorphic to one of them.
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# Positive solutions of integral equations

## Summary

A positive solution of an [[integral equation|R3SG1SN7JQ]] is a solution that is nonnegative and not identically zero, or in a stronger sense strictly positive.
In models of densities, concentrations, populations or probabilities only such solutions have a meaning, so existence of a positive solution is often the real question.
The main tools are order methods: the cone of nonnegative functions, [[positive operators|77020CY5ZW]] that map it into itself, and monotone iteration.
Fixed point and eigenvalue theorems on cones, such as the [[Krein–Rutman theorem|VADXDK3TKM]], complete the toolbox.
For linear equations with nonnegative kernels, positivity is closely tied to the spectral radius of the integral operator.

## Explanations

### L1 human

If the unknown is a density of particles, a negative value makes no sense, so one wants solutions that are never negative.
Integral operators with nonnegative kernels send nonnegative functions to nonnegative functions, and this helps.
But it is not enough: $\varphi(x) = 1 + 2\int_0^1 \varphi(t) \, dt$ on $[0, 1]$ has the unique solution $\varphi = -1$, although all its data are positive.
The trouble is that the integral term is too strong; with the factor $\frac12$ instead of $2$, the solution is $\varphi = 2$.
</document>

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# Domain of convergence of a power series

## Statement

Let $\sum_{\alpha \in \N^n} c_\alpha z^\alpha$ be a [[power series in several variables|R3QTCPTJK7]] with centre $0$.
Let $S$ be the set of points of $\C^n$ where it converges, and let $D$ be the interior of $S$, its *domain of convergence*.
Suppose that $D$ is nonempty.
Then:

1. $D$ is a complete [[Reinhardt domain|WA0SE56FVH]].
2. $D$ is [[logarithmically convex|EHWH6EF4QY]].
3. The series converges absolutely and uniformly on every compact subset of $D$, and its sum is a [[holomorphic function|507C34W0ZT]] on $D$.

## Remarks

If $z \in D$, there is a point $w \in S$ with $\abs{w_j} > \abs{z_j}$ for all $j$.
Then [[Abel's lemma|6YT8ZM6B0F]] gives convergence on the polydisc $\set{\abs{w'_j} < \abs{w_j}}$, which contains $z$ and every $w'$ with $\abs{w'_j} \le \abs{z_j}$.
This gives the first and third parts.
For the second part, suppose the terms are bounded by $M$ at $z$ and at $w$, and let $0 \le t \le 1$.
If $\abs{u_j} = \abs{z_j}^{1-t} \abs{w_j}^{t}$ for all $j$, then $\abs{c_\alpha u^\alpha} = \abs{c_\alpha z^\alpha}^{1-t} \abs{c_\alpha w^\alpha}^{t} \le M$.
So the set of logarithms of absolute values is convex.
Example: the series $\sum_{k \ge 0} (z_1 z_2)^k$ has domain of convergence $\set{\abs{z_1 z_2} < 1}$, whose image under $z \mapsto (\log\abs{z_1}, \log\abs{z_2})$ is the half-plane $x_1 + x_2 < 0$.
This domain is unbounded and is not a polydisc, which never happens for $n = 1$, where the domain of convergence is a disc or the whole plane.
Conversely, every logarithmically convex complete Reinhardt domain with centre $0$ is the domain of convergence of some power series.

## Explanations

### L1 human

In one variable, a power series converges on a disc, and the disc is fixed by one number, the radius of convergence.
In several variables, the region of convergence can have many shapes.
The theorem says which shapes can occur.
They are symmetric under rotating each coordinate, they are "filled in" towards the origin, and they become convex after taking logarithms of absolute values.

### L2 human

This result explains why Reinhardt domains and logarithmic convexity appear so early in the theory.
It is a first example of the convexity conditions that characterise domains of holomorphy.
For instance, a holomorphic function on a complete Reinhardt domain $D$ with centre $0$ has a power series at $0$ that converges on all of $D$.
Hence it converges on the smallest logarithmically convex complete Reinhardt domain containing $D$.
So the holomorphic functions on a complete Reinhardt domain that is not logarithmically convex, such as a union of two crossed polydiscs, all extend to a larger domain.
</document>

<document source="https://tamnd.github.io/math2/node/D8VX6XQ3TT.md" tag="D8VX6XQ3TT" kind="concept">
# Heteroscedasticity

## Summary

A collection of random variables is heteroscedastic when their variances are not all equal; it is homoscedastic when they are.
In regression the term refers to errors whose variance changes with the regressors, for example noise that grows with the size of the response.
Under heteroscedasticity [[ordinary least squares|YM708KHV2W]] remains unbiased but is no longer the best linear unbiased estimator.
Also, the usual formula for its standard errors is wrong.
Remedies include [[weighted least squares|H5CGZHBR58]] and heteroscedasticity-consistent standard errors, and tests such as the Breusch–Pagan and White tests detect it.
The spelling heteroskedasticity is also common.

## Explanations

### L1 human

Plot household spending on food against income.
Poor households spend similar amounts, but rich households differ a lot, so the cloud of points fans out as income grows.
That fanning out is heteroscedasticity: the scatter around the trend is not the same everywhere.
It matters because a method that treats all points as equally precise will be overconfident about some conclusions.
</document>

<document source="https://tamnd.github.io/math2/node/D8ZD10YF3Y.md" tag="D8ZD10YF3Y" kind="conjecture">
# Rota's basis conjecture

## Statement

Let $V$ be a [[vector space|000AYPA9AJ]] of dimension $n \ge 1$ over a field.
Let $B_1, \dots, B_n$ be [[bases|0KCQ9Y3QKA]] of $V$, regarded as $n$ disjoint labelled families of $n$ vectors.
Then the $n^2$ vectors can be arranged in an $n \times n$ grid with the following properties.
For each $i$, the $i$-th row consists of the vectors of $B_i$.
Each column is a basis of $V$.

## Remarks

The conjecture is due to Gian-Carlo Rota, and it is open.
It has a natural form for [[matroids|PX31ZNYS3Z]]: given $n$ [[bases|QTSQR7130G]] of a matroid of rank $n$, the elements can be arranged in such a grid with each column a basis.
For $n = 2$ the statement is easy to check by hand: with $B_1 = \set{a, b}$ and $B_2 = \set{c, d}$, at least one of the pairings $\set{a, c}, \set{b, d}$ or $\set{a, d}, \set{b, c}$ gives two bases.
Huang and Rota related the vector space form to the Alon–Tarsi conjecture on Latin squares, which gives the conjecture for some even $n$ over fields of characteristic zero.
Asymptotic versions are known that produce many disjoint column bases, but not $n$ of them in general.

## Explanations

### L1 human

Start with $n$ bases of an $n$-dimensional space, each written as a row of $n$ vectors.
The question is whether one can reorder each row so that every column also becomes a basis.
It looks like a puzzle in the style of Latin squares, and it is still unsolved in general.
</document>

<document source="https://tamnd.github.io/math2/node/D94626DTTW.md" tag="D94626DTTW" kind="definition">
# Nonlinear approximation

## Definition

Let $X$ be a [[normed space|03J5RAKB30]] and let $\Sigma_0 \subseteq \Sigma_1 \subseteq \Sigma_2 \subseteq \cdots$ be subsets of $X$, which in general are not linear subspaces.
Usually there is a constant $c$ with $\Sigma_n + \Sigma_n \subseteq \Sigma_{cn}$.
*Nonlinear approximation* is the approximation of elements $f \in X$ by elements of $\Sigma_n$, measured by the [[error of best approximation|YDN0T4MEX0]]
$$\sigma_n(f) = \inf_{s \in \Sigma_n} \norm{f - s}_X.$$
The main example is $n$-term approximation from a dictionary $\mathcal{D} \subseteq X$, with $\Sigma_n = \set{\sum_{k=1}^{n} c_k g_k : c_k \in \R,\ g_k \in \mathcal{D}}$.

## Remarks

Example: let $X$ be a separable Hilbert space with [[orthonormal basis|0S9GGTR0EX]] $(e_j)$, and let $\Sigma_n$ be the vectors with at most $n$ nonzero coordinates.
The set $\Sigma_1$ is not a subspace: $e_1, e_2 \in \Sigma_1$ but $e_1 + e_2 \notin \Sigma_1$.
For $f = \sum_j c_j e_j$, the best element of $\Sigma_n$ keeps the $n$ coefficients of largest absolute value, so $\sigma_n(f)^2$ is the sum of $\abs{c_j}^2$ over all other $j$.

Non-example: $\Sigma_n$ the polynomials of degree at most $n$ in $C[0, 1]$ is a linear subspace, so approximation from it is linear approximation.
</document>

<document source="https://tamnd.github.io/math2/node/D95SQEJJSK.md" tag="D95SQEJJSK" kind="conjecture">
# Hopf conjecture

## Statement

Let $M$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] without boundary, of even dimension $2d$.
Let $\chi(M)$ be its [[Euler characteristic|RNDDWDAFJ5]].

1. If $M$ has positive [[sectional curvature|1X42JAA349]], then $\chi(M) > 0$.
2. If $M$ has negative sectional curvature, then $(-1)^d \chi(M) > 0$.

## Remarks

Both parts are open in dimension $6$ and higher.
In dimension $2$ both parts follow from the [[Gauss–Bonnet theorem|ESGEHKPD60]], since the integral of the Gaussian curvature equals $2\pi \chi(M)$.
In dimension $4$ both parts follow from the [[Chern–Gauss–Bonnet theorem|GG6NYPKMHX]], because in that dimension the integrand is positive when the sectional curvature is either everywhere positive or everywhere negative.
For the positive case in dimension $4$ there is also a topological argument: by [[Synge's theorem|Z8JJ04X1JX]] an orientable such $M$ is simply connected, so $b_1 = b_3 = 0$ and $\chi(M) = 2 + b_2 > 0$.
In dimensions $2d \ge 6$ the Chern–Gauss–Bonnet integrand can be negative at a point even when the sectional curvature is positive, so this route fails.
The positive case is known for manifolds whose curvature is pinched enough, since then the [[sphere theorem|7GRSF20H15]] shows that $M$ is covered by a sphere.
Weaker forms with non-strict inequalities are also attributed to Hopf: nonnegative curvature should give $\chi(M) \ge 0$, and nonpositive curvature should give $(-1)^d \chi(M) \ge 0$.
A different question of Hopf, often also called the Hopf conjecture, asks whether $S^2 \times S^2$ carries a metric of positive sectional curvature; this is also open.

## Explanations

### L1 human

Curvature is a local quantity, measured near each point.
The Euler characteristic is a global number: for a surface cut into polygons it is vertices minus edges plus faces.
Hopf asked whether the sign of the curvature everywhere forces the sign of this global count.
For surfaces the answer is yes.
A positively curved closed surface is a sphere or a projective plane, and both have positive Euler characteristic.
A negatively curved closed surface always has negative Euler characteristic.
The conjecture asks the same in every even dimension, and in general it is proved only up to dimension $4$.
</document>

<document source="https://tamnd.github.io/math2/node/D96K8S8R61.md" tag="D96K8S8R61" kind="concept">
# Axiom schema of replacement

## Summary

The axiom schema of replacement is a list of axioms of [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], one for each formula.
It says that the image of a [[set|0BSJ7RQ9BW]] under a definable [[function|0EPAT51DXQ]] is again a set.
Fraenkel and Skolem proposed it in 1922 as an addition to Zermelo's axioms.
It is needed to build large sets such as $\aleph_\omega$ and to define functions on [[ordinals|0YSCSX5YZC]] by transfinite recursion.

## Explanations

### L1 human

Suppose you have a set, and a rule that assigns to each element of it a unique object.
Replacement says that the objects you get form a set too.
You replace each element by its image, and the result is still a set.
For example, from the set of natural numbers $\N$ and the rule $n \mapsto \mathcal{P}^n(\N)$, the $n$-fold power set, one gets the set $\set{\N, \mathcal{P}(\N), \mathcal{P}(\mathcal{P}(\N)), \dots}$.
Without replacement, this set cannot be proved to exist.
</document>

<document source="https://tamnd.github.io/math2/node/D99FZQ2SQT.md" tag="D99FZQ2SQT" kind="statement">
# Gauss–Legendre Runge–Kutta methods

## Statement

Let $s \ge 1$ and let $c_1 < \cdots < c_s$ be the zeros in $(0, 1)$ of the [[shifted Legendre polynomial|9V3QA0KTW8]] of degree $s$.
Let $\ell_1, \ldots, \ell_s$ be the [[Lagrange basis polynomials|Y7KSGE97GK]] of these nodes.
The *$s$-stage Gauss–Legendre method* is the implicit [[Runge–Kutta method|7DBPJKYC6C]] with
$$a_{ij} = \int_0^{c_i} \ell_j(\theta) \, d\theta, \qquad b_j = \int_0^1 \ell_j(\theta) \, d\theta, \qquad i, j = 1, \ldots, s.$$
Then:

1. It has order $2s$, and no Runge–Kutta method with $s$ stages has order greater than $2s$.
2. Its stability function $R(z)$, defined by $y_{n+1} = R(h\lambda) y_n$ for $y' = \lambda y$, is the $(s, s)$ [[Padé approximant|1F16G4V64X]] of $e^z$.
   Moreover $\abs{R(z)} \le 1$ whenever $\operatorname{Re} z \le 0$, so the method is A-stable.
3. Its coefficients satisfy $b_i a_{ij} + b_j a_{ji} = b_i b_j$ for all $i, j$, so the method is a [[symplectic integrator|B4C25SXFRE]] and preserves every quadratic first integral of the equation.

## Remarks

The method is the collocation method at the nodes $c_i$: it finds the polynomial $u$ of degree $s$ with $u(t_n) = y_n$ and $u'(t_n + c_i h) = f(t_n + c_i h, u(t_n + c_i h))$ for each $i$, and sets $y_{n+1} = u(t_n + h)$.
The weights $b_j$ and nodes $c_j$ form the Gauss–Legendre quadrature rule on $[0, 1]$, which is exact up to degree $2s - 1$; collocation theory turns this into order $2s$.
Example: $s = 1$ gives $c_1 = \frac12$, $a_{11} = \frac12$, $b_1 = 1$, the implicit midpoint method $y_{n+1} = y_n + h f\big(t_n + \frac h2, \frac{y_n + y_{n+1}}{2}\big)$, with $R(z) = \frac{1 + z/2}{1 - z/2}$; clause 3 reads $\frac12 + \frac12 = 1$.
Example: $s = 2$ gives $c = \frac12 \mp \frac{\sqrt3}{6}$, $b = (\frac12, \frac12)$, $a_{11} = a_{22} = \frac14$, $a_{12} = \frac14 - \frac{\sqrt3}{6}$, $a_{21} = \frac14 + \frac{\sqrt3}{6}$, a method of order $4$.
The price of these properties is that each step needs the solution of a nonlinear system of dimension $sd$ for a system of dimension $d$.

## Explanations

### L1 human

Gauss–Legendre methods are the most accurate Runge–Kutta methods for a given number of stages: with $s$ stages they reach order $2s$.
They place their internal evaluation points at the same special positions as Gaussian quadrature.
They are implicit, so each step requires solving equations, but they stay stable for every step size on decaying problems and they conserve energy-like quantities very well over long times.

### L2 human

The family links three topics: Gaussian quadrature, which fixes the nodes; collocation, which gives the order; and geometric integration, through the symplecticity condition.
For Hamiltonian systems they are standard high-order symplectic methods, used for long-time simulations in celestial mechanics and molecular dynamics.
Their stability function satisfies $\abs{R(iy)} = 1$ on the imaginary axis and $\abs{R(z)} \to 1$ as $z \to -\infty$, so unlike [[backward Euler|M7XKVNATFA]] they do not damp very stiff components.
</document>

<document source="https://tamnd.github.io/math2/node/D9AWNH60RM.md" tag="D9AWNH60RM" kind="definition">
# Computable function

## Definition

Let $k \ge 1$ and let $f \colon \N^k \rightharpoonup \N$ be a partial function, where $\N$ is the set of [[natural numbers|0MWKJ44V5A]].
Fix an encoding of tuples of natural numbers as words on the input alphabet, for example in unary separated by a marker.
The function $f$ is *computable*, or *partial computable*, if there is a [[Turing machine|758EYCN27B]] $M$ with the following property for every $\bar x \in \N^k$.

1. If $f(\bar x)$ is defined, then $M$ halts on input $\bar x$ with the code of $f(\bar x)$ on its tape.
2. If $f(\bar x)$ is undefined, then $M$ does not halt on input $\bar x$.

A *total computable function* is a computable function defined at every point of $\N^k$.

## Remarks

Example: addition $(x, y) \mapsto x + y$ is computable.
With unary codes $1^x 0 1^y$, a machine that replaces the $0$ by $1$ and erases the last $1$ halts with $1^{x+y}$ on its tape.

Non-example: the function $h(e) = 1$ if the machine with index $e$ halts on input $e$, and $h(e) = 0$ otherwise, is total but not computable.
No machine satisfies clause 1 for $h$; see the [[halting problem|Z7TN2B35B0]].

A partial function is computable if and only if it is partial [[general recursive|HBK5QHC1GC]].
The definition does not depend on the choice of a reasonable encoding or a variant of Turing machines.
</document>

<document source="https://tamnd.github.io/math2/node/D9B4HE4B4Q.md" tag="D9B4HE4B4Q" kind="statement">
# Kolmogorov's inequality

## Statement

Let $X_1, \dots, X_n$ be [[independent|0GQ938GDE0]] real [[random variables|0PNRKMRC55]] with $E[X_k] = 0$ and $E[X_k^2] < \infty$ for each $k$.
Let $S_k = X_1 + \cdots + X_k$ for $1 \le k \le n$.
Then for every $\lambda > 0$,
$$P\left(\max_{1 \le k \le n} \abs{S_k} \ge \lambda\right) \le \frac{\operatorname{Var}(S_n)}{\lambda^2} = \frac{1}{\lambda^2} \sum_{k=1}^{n} E[X_k^2].$$

## Remarks

Let $\tau$ be the first $k$ with $\abs{S_k} \ge \lambda$.
The event $\set{\tau = k}$ depends only on $X_1, \dots, X_k$.
By independence the cross term $E[S_k (S_n - S_k) \mathbf{1}_{\tau = k}]$ vanishes, so $E[S_n^2 \mathbf{1}_{\tau = k}] \ge E[S_k^2 \mathbf{1}_{\tau = k}] \ge \lambda^2 P(\tau = k)$.
Summing over $k$ gives $\lambda^2 P(\tau \le n) \le E[S_n^2]$.

The bound has the same right side as [[Chebyshev's inequality|02HZ1BK3MW]] for the single variable $S_n$, but it controls the whole path $S_1, \dots, S_n$.
It is a special case of [[Doob's martingale inequality|4G53E7BRXF]], since $(S_k)$ is a [[martingale|0D0GBNDKDH]].

Example: let the $X_k$ be independent signs, equal to $1$ or $-1$ with probability $1/2$ each.
Then the probability that $\abs{S_k} \ge \lambda$ for some $k \le n$ is at most $n / \lambda^2$.
With $\lambda = 10 \sqrt{n}$ this is at most $1/100$.

## Explanations

### L1 human

A sum of independent centred random steps rarely wanders far from zero.
Chebyshev's inequality says this for the final position.
Kolmogorov's inequality says that the same bound holds for the farthest point reached along the way.

### L2 human

The inequality is the key step in the proof that a series $\sum X_k$ of independent centred variables with $\sum \operatorname{Var}(X_k) < \infty$ converges almost surely.
That result leads to [[Kolmogorov's three-series theorem|TTTFQ693RJ]] and to the [[strong law of large numbers|0DFWHQ839T]] for independent variables with suitable finite variances.
</document>

<document source="https://tamnd.github.io/math2/node/D9B92SQ2GF.md" tag="D9B92SQ2GF" kind="statement">
# Ahlfors–Bers theorem

## Statement

Let $\mu \colon \C \to \C$ be measurable with $\norm{\mu}_\infty < 1$.
Let $f^\mu \colon \C \to \C$ be the unique [[quasiconformal|SB8KG46NHP]] homeomorphism that fixes $0$ and $1$ and solves the [[Beltrami equation|9J7GFJ1GPF]] $f_{\bar z} = \mu f_z$ almost everywhere.
Its existence and uniqueness is the [[measurable Riemann mapping theorem|TZHD2WKNSD]].

Let $U \subset \C$ be open.
Let $t \mapsto \mu_t$ be a map from $U$ to the open unit ball of $L^\infty(\C)$ that is holomorphic as a map into the Banach space $L^\infty(\C)$.
Then for each $z \in \C$, the function $t \mapsto f^{\mu_t}(z)$ is [[holomorphic|0NAHXR1FRH]] on $U$.

## Remarks

For $\mu$ with compact support, the proof writes the derivative $\partial_{\bar z}$ of the principal solution as a Neumann series in the [[Beurling transform|TX32GDQ330]] $T$: $\mu + \mu T \mu + \mu T(\mu T \mu) + \cdots$.
The series converges in $L^p$ for some $p > 2$, because $T$ is an isometry of $L^2$ and its norm on $L^p$ tends to $1$ as $p \to 2$.
Each term depends holomorphically on $\mu$, and the convergence is locally uniform.
The same argument gives real-analytic dependence when $\mu_t$ depends real-analytically on a real parameter.

Example: for $\mu_t = t \chi$ with $\chi$ the indicator of a disc and $\abs{t} < 1$, the normalized solutions $f^{\mu_t}$ form a holomorphic family of quasiconformal maps through $f^{0} = \id$.

The condition $\norm{\mu_t}_\infty < 1$ is needed: when $\norm{\mu}_\infty = 1$ the Beltrami equation is degenerate, and a homeomorphic solution need not exist.

## Explanations

### L1 human

A Beltrami coefficient $\mu$ prescribes how a map should distort infinitesimal circles into ellipses.
The measurable Riemann mapping theorem gives exactly one normalized map with this distortion.
The Ahlfors–Bers theorem adds that if the prescribed distortion moves holomorphically with a complex parameter, then so does the map.

### L2 human

This holomorphic dependence is what makes [[Teichmüller space|ZZGC6BZNJ5]] a complex manifold: Beltrami coefficients give coordinates, and the solution maps vary holomorphically with them.
It is also the tool behind holomorphic motions and quasiconformal surgery in complex dynamics, where one deforms a rational map by moving an invariant Beltrami coefficient in a holomorphic family.
</document>

<document source="https://tamnd.github.io/math2/node/D9C581V5HG.md" tag="D9C581V5HG" kind="statement">
# Blaschke selection theorem

## Statement

Let $n \ge 1$ and let $(K_m)_{m \ge 1}$ be a sequence of nonempty [[compact|0R6EWFKRCF]] [[convex sets|02AS45T3H9]] in $\R^n$, all contained in one [[bounded set|09QSZVVM26]].
Then $(K_m)$ has a [[subsequence|0F1PR74BRX]] that converges, in the [[Hausdorff distance|F79WYYC07C]], to a nonempty compact convex set $K \subseteq \R^n$.

## Remarks

The nonempty compact subsets of a compact metric space form a compact metric space under the Hausdorff distance.
All $K_m$ lie in a closed ball, which is compact, so a subsequence converges to some nonempty compact set $K$.
A Hausdorff limit of convex sets is convex, which gives the theorem.
The limit can be smaller in dimension than the $K_m$: the rectangles $K_m = [0, 1] \times [0, \frac1m]$ converge to the segment $[0,1] \times \set{0}$.
Boundedness is needed: the segments $[0, m] \times \set{0}$ have no convergent subsequence, since their Hausdorff distances to any fixed compact set tend to infinity.

## Explanations

### L1 human

Take infinitely many convex shapes that all fit in one box.
Then one can pick some of them, one after another, that settle down to a limiting convex shape.
This is the Bolzano–Weierstrass theorem for convex shapes instead of numbers.

### L2 human

The theorem makes the space of convex bodies with the Hausdorff metric locally compact.
It is used to prove the existence of extremal bodies.
A continuous functional, such as volume or [[mixed volume|QB3Q5FBJ08]], attains its maximum and minimum over a bounded closed family of convex sets.
Existence proofs for the [[John ellipsoid|WNHXE7FCG4]] and for solutions of isoperimetric-type problems use it in this way.
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<document source="https://tamnd.github.io/math2/node/D9H2AGW198.md" tag="D9H2AGW198" kind="definition">
# Laplace–Stieltjes transform

## Definition

Let $g \colon [0, \infty) \to \C$ be a function of [[bounded variation|F34S6P61SW]] on each interval $[0, T]$ with $T > 0$.
The *Laplace–Stieltjes transform* of $g$ is the function
$$\mathcal{L}^*\{g\}(s) = \int_0^\infty e^{-s t} \, dg(t) = \lim_{T \to \infty} \int_0^T e^{-s t} \, dg(t),$$
defined for each complex $s$ for which the limit exists.
Each integral $\int_0^T e^{-s t} \, dg(t)$ is a [[Riemann–Stieltjes integral|8NMH42GJRK]].

## Remarks

Example: let $a > 0$ and $g(t) = 0$ for $t < a$, $g(t) = 1$ for $t \ge a$.
Then $\int_0^T e^{-s t} \, dg(t) = e^{-a s}$ for every $T > a$, so $\mathcal{L}^*\{g\}(s) = e^{-a s}$ for every complex $s$.
If $g(t) = \int_0^t f(\tau) \, d\tau$ with $f$ continuous, then $dg = f(t) \, dt$ and $\mathcal{L}^*\{g\}$ is the [[Laplace transform|05XT5AVJBK]] of $f$.
Non-example: $g(t) = e^{t^2}$ is of bounded variation on each $[0, T]$, but $\int_0^T e^{-s t} \, dg(t) = \int_0^T 2 t e^{t^2 - s t} \, dt$ diverges as $T \to \infty$ for every $s$, so no transform exists at any point.
In probability one often uses the Lebesgue–Stieltjes form $\int_{[0, \infty)} e^{-s t} \, dG(t)$ for a distribution function $G$, which counts a possible jump at $t = 0$.
With $G(t) = 0$ for $t < 0$ it equals $\mathbb{E}[e^{-s X}]$ for $X \ge 0$ with distribution function $G$.
If $g(0) = 0$ and $\abs{g(t)} \le M e^{a t}$ for all $t \ge 0$, integration by parts gives $\mathcal{L}^*\{g\}(s) = s \, \mathcal{L}\{g\}(s)$ for $\operatorname{Re} s > a$.
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# Stone–Čech compactification

## Summary

The Stone–Čech compactification $\beta X$ of a [[Tychonoff space|C5N2J2K6Z3]] $X$ is the largest Hausdorff [[compactification|Q6XNC8JAZD]] of $X$.
It is characterised by a universal property.
Every continuous map from $X$ to a compact Hausdorff space extends uniquely to $\beta X$.
It links topology with algebra and set theory, for example through the description of $\beta \N$ by [[ultrafilters|J64Y3JC0RE]].

## Explanations

### L1 human

A compactification adds ideal points to make a space compact.
The Stone–Čech compactification adds as many points as possible, in a precise sense.
Every compact Hausdorff compactification of $X$ is a quotient of $\beta X$.
For the natural numbers, the new points of $\beta \N$ correspond to the free ultrafilters on $\N$.
</document>

<document source="https://tamnd.github.io/math2/node/D9PDDPC61K.md" tag="D9PDDPC61K" kind="definition">
# Standard Borel space

## Definition

A *Polish space* is a [[topological space|0CQHFPGJ45]] $(X, \tau)$ that is [[separable|0Z2P41SVDP]] and has the following property.
There is a metric $d$ on $X$ that induces $\tau$ such that $(X, d)$ is a [[complete metric space|0BY52JW6NK]].
Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]].
It is a *standard Borel space* if $X$ has a topology $\tau$ with these two properties.
The space $(X, \tau)$ is a Polish space, and $\Sigma$ is the [[Borel σ-algebra|0WQXY32Q8D]] of $\tau$.

## Remarks

The real line $\R$ with its Borel σ-algebra is a standard Borel space, since the usual metric on $\R$ is complete and $\Q$ is a countable dense subset.
A countable set with its power set is a standard Borel space, using the discrete metric.
Let $\Sigma$ be the family of subsets $A \subseteq \R$ such that $A$ or $\R \setminus A$ is countable.
Then $(\R, \Sigma)$ is a measurable space but not a standard Borel space.
Indeed, the Borel σ-algebra of a Polish space is generated by countably many sets.
These are the open balls with rational radius around the points of a countable dense set.
The σ-algebra $\Sigma$ is not generated by countably many sets.
Equivalently, a standard Borel space is a measurable space isomorphic to a Polish space with its Borel σ-algebra.
Kuratowski's theorem says that a standard Borel space is countable or has the cardinality of $\R$, and that two standard Borel spaces of the same cardinality are isomorphic.
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<document source="https://tamnd.github.io/math2/node/D9RY4667K6.md" tag="D9RY4667K6" kind="concept">
# Directed set

## Summary

A directed set is a nonempty [[preordered set|0VABYWJWH9]] in which every two elements have a common upper bound.
Directed sets are the index sets of [[nets|4DVJQJV0DG]], which generalize sequences in topology, and of direct limits in algebra.
They also underlie directed-complete partial orders in domain theory.

## Explanations

### L1 human

In a directed set, any two elements can be "caught up with" by a third that is above both.
So the set has a sense of going further and further in one direction, even if it is not a line.
The natural numbers are directed, and so are the finite subsets of a set ordered by inclusion: $A \cup B$ is above both $A$ and $B$.
A set with two incomparable elements and nothing else is not directed.
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<document source="https://tamnd.github.io/math2/node/D9RZQGM0QP.md" tag="D9RZQGM0QP" kind="definition">
# Jackson kernel

## Definition

Let $n \ge 1$ be an integer.
The *Jackson kernel* of order $n$ is the function
$$J_n(t) = \lambda_n \left(\frac{\sin(nt/2)}{\sin(t/2)}\right)^4, \qquad t \in \R,$$
with the value $\lambda_n n^4$ at the multiples of $2\pi$.
The constant $\lambda_n > 0$ is chosen so that $\frac{1}{2\pi} \int_{-\pi}^{\pi} J_n(t) \, dt = 1$.

## Remarks

Since $\left(\frac{\sin(nt/2)}{\sin(t/2)}\right)^2 = n F_{n-1}(t)$, where $F_{n-1}$ is the [[Fejér kernel|DCCA2T9GKH]], $J_n$ is a constant times $F_{n-1}^2$.
So $J_n$ is an even, nonnegative [[trigonometric polynomial|J3NJQ5FZJ5]] of degree $2n - 2$.
Its key property is $\frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{t}^k J_n(t) \, dt \le \frac{C}{n^k}$ for $k = 1, 2$, with $C$ independent of $n$.

Example: for $n = 1$, the quotient is $1$, so $J_1 = 1$ and $\lambda_1 = 1$.

Non-example: the Fejér kernel $F_n$ is nonnegative with mean $1$, but it fails the key property for $k = 1$.
Its first moment $\frac{1}{2\pi} \int \abs{t} F_n(t) \, dt$ is of order $\frac{\log n}{n}$, not $\frac{1}{n}$.

Texts differ in the indexing; some call $J_m$ with $m \approx n/2$ the kernel of degree $n$.
The convolution $f \mapsto f * J_n$ is the Jackson operator.
</document>

<document source="https://tamnd.github.io/math2/node/D9TW46P19Y.md" tag="D9TW46P19Y" kind="concept">
# Bifurcation theory

## Summary

Bifurcation theory studies how the qualitative behaviour of a family of [[dynamical systems|BBZP1R872N]] changes as parameters vary.
A bifurcation happens at a parameter value where the [[phase portrait|0T3GK0XCBC]] changes its topological type, for example when [[equilibria|0N8NGKDZDZ]] appear, disappear or change stability.
The theory classifies the typical bifurcations, such as the [[saddle-node|MAP5J6HK0T]] and [[Hopf|F5H8GW1XYS]] bifurcations, and gives normal forms for them.
It explains sudden transitions in models from physics, biology and engineering.

## Explanations

### L1 human

Many systems change smoothly when you turn a knob, but at certain settings their behaviour changes in kind.
A pendulum with friction comes to rest; push it with a motor and past some strength it starts to swing forever.
Bifurcation theory finds these critical settings and describes what happens there.
A simple example is $\dot x = \mu - x^2$: for $\mu < 0$ there is no resting state, and for $\mu > 0$ there are two, which are born together at $\mu = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/D9W79NKYZJ.md" tag="D9W79NKYZJ" kind="statement">
# Quantum ergodicity

## Statement

Let $M$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] without boundary, with volume measure $dv$ and total volume $V$.
Assume that the [[geodesic flow|VF43FTDA1H]] on the unit cotangent bundle $S^*M$ is [[ergodic|AZRPHS478B]] for the Liouville measure.
Let $(\varphi_j)_{j \ge 0}$ be an orthonormal basis of $L^2(M)$ of real eigenfunctions of the [[Laplace–Beltrami operator|F4NWZNTRMD]], with eigenvalues $\lambda_0 \le \lambda_1 \le \lambda_2 \le \cdots$.
Then there is a set $S \subseteq \N$ of density one, that is, with $\abs{S \cap \set{0, \ldots, N}} / (N + 1) \to 1$ as $N \to \infty$, such that for every continuous function $a$ on $M$
$$\lim_{j \to \infty, \ j \in S} \int_M a \, \varphi_j^2 \, dv = \frac{1}{V} \int_M a \, dv.$$

## Remarks

The full theorem is stated in phase space.
Let $A$ be a pseudodifferential operator of order $0$ with principal symbol $\sigma_A$, and let $\bar{\sigma}_A$ be the average of $\sigma_A$ over $S^*M$ for the normalised Liouville measure.
Then $\ip{A \varphi_j}{\varphi_j}$ tends to $\bar{\sigma}_A$ as $j \to \infty$ in $S$.
The version above is the case of multiplication operators.
The proof combines Egorov's theorem, which relates the quantum evolution to the geodesic flow, with the local [[Weyl law|2JK9YMZV2B]] and the mean ergodic theorem.
These give that the variance $\frac{1}{N} \sum_{j < N} \abs{\ip{A \varphi_j}{\varphi_j} - \bar{\sigma}_A}^2$ over the first $N$ eigenfunctions tends to $0$ as $N \to \infty$, which yields a density-one subsequence.
The result is due to Shnirelman, Zelditch and Colin de Verdière.
The ergodicity hypothesis is needed: on the round sphere $S^2$ the geodesic flow is not ergodic, and there are orthonormal bases of eigenfunctions for which the conclusion fails.
For example, the normalised real parts of $(x_1 + i x_2)^k$ restricted to $S^2$ are eigenfunctions whose mass concentrates on the equator $x_3 = 0$ as $k \to \infty$.
Whether the whole sequence converges, with no exceptional subsequence, is the [[quantum unique ergodicity conjecture|F3EB7WZFAC]].

## Explanations

### L1 human

Eigenfunctions of the Laplacian are the standing waves of a manifold.
If a particle sliding freely along geodesics explores the whole manifold evenly, then almost all high-frequency standing waves also spread their mass evenly.
This is the meaning of quantum ergodicity: classical chaos forces the quantum waves to equidistribute.
The words "almost all" matter, since the theorem allows a thin set of exceptions.

### L2 human

The theorem is the basic bridge between the dynamics of the geodesic flow and the spectral theory of the Laplacian.
It is a statement about averages, proved by a variance estimate, and so it cannot rule out a sparse subsequence of eigenfunctions that concentrates.
Negatively curved compact manifolds satisfy the hypothesis, since their geodesic flows are ergodic.
Possible limits of the measures $\varphi_j^2 \, dv$ along subsequences are called quantum limits, and much later work studies which measures can occur.
</document>

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# Cluster analysis

## Summary

Cluster analysis groups a set of objects into clusters.
Objects in the same cluster should be more similar to each other than to objects in other clusters.
Unlike classification, it uses no labels known in advance.
Main families of methods are centroid methods such as [[k-means clustering|D5PN3G41D6]], hierarchical clustering, density-based methods and model-based clustering.
There is no single best notion of a cluster, and different methods can give different answers.

## Explanations

### L1 human

Throw a handful of coins on a table: you see them lying in a few heaps.
Cluster analysis tries to find such heaps in data, for instance groups of customers who buy similar things.
Nobody tells the method which group each object belongs to; it must discover the groups from the data alone.
A simple rule is to put points that are close to each other in the same group.
</document>

<document source="https://tamnd.github.io/math2/node/D9YHA46YWM.md" tag="D9YHA46YWM" kind="concept">
# Kleisli category

## Summary

The Kleisli category of a [[monad|FF6XXA6D1A]] $T$ on a category $\mathcal{C}$ has the same objects as $\mathcal{C}$.
A morphism from $X$ to $Y$ in it is a morphism $X \to T Y$ of $\mathcal{C}$.
Composition uses the multiplication of the monad to flatten $T T$ to $T$.
It is equivalent to the full subcategory of free algebras in the [[Eilenberg–Moore category|FEPRJR3MND]].
Among all adjunctions that induce the monad, it gives the initial one.
In computer science, Kleisli morphisms model programs with side effects, such as partiality, nondeterminism or state.

## Explanations

### L1 human

For the powerset monad, a morphism from $X$ to $Y$ in the Kleisli category is a map that sends each element of $X$ to a set of elements of $Y$.
Such a map is the same as a relation between $X$ and $Y$.
Composing two of them means following all possible paths, which is the usual composition of relations.
So the Kleisli category of the powerset monad is the category of sets and relations.
</document>

<document source="https://tamnd.github.io/math2/node/DA03KQ311J.md" tag="DA03KQ311J" kind="definition">
# Schur polynomial

## Definition

Let $n \ge 1$ and let $\lambda = (\lambda_1, \ldots, \lambda_n)$ be an [[integer partition|0JV30XN73Z]] with at most $n$ nonzero parts, padded with zeros, so $\lambda_1 \ge \cdots \ge \lambda_n \ge 0$.
The *Schur polynomial* of $\lambda$ in the variables $x_1, \ldots, x_n$ is
$$s_\lambda(x_1, \ldots, x_n) = \frac{\det\big(x_i^{\lambda_j + n - j}\big)_{1 \le i, j \le n}}{\det\big(x_i^{n - j}\big)_{1 \le i, j \le n}}.$$
Here $\det$ is the [[determinant|0C3CFHDN6R]], and the denominator is the [[Vandermonde determinant|0P84B554AS]] $\prod_{i < j} (x_i - x_j)$.

## Remarks

The numerator is an alternating polynomial, so it is divisible by the Vandermonde determinant, and the quotient is a [[symmetric polynomial|0VHSHB9B6J]] in $\Z[x_1, \ldots, x_n]$.
It is homogeneous of degree $\abs{\lambda} = \lambda_1 + \cdots + \lambda_n$.

Example: for $n = 2$ and $\lambda = (2, 0)$, the exponents in the numerator are $\lambda_1 + 1 = 3$ and $\lambda_2 + 0 = 0$.
So $s_{(2)} = \frac{x_1^3 - x_2^3}{x_1 - x_2} = x_1^2 + x_1 x_2 + x_2^2$.
For $\lambda = (1, 1)$, the same computation gives $s_{(1,1)} = \frac{x_1^2 x_2 - x_1 x_2^2}{x_1 - x_2} = x_1 x_2$.

Non-example: the sequence $(1, 2)$ is not weakly decreasing, so it is not a partition, and $s_{(1,2)}$ is not a Schur polynomial.
The formula would give a numerator with two equal columns, hence $0$.

Combinatorial form: $s_\lambda(x_1, \ldots, x_n) = \sum_T x^T$, a sum over the semistandard [[Young tableaux|Q6TH9PH2R7]] $T$ of shape $\lambda$ with entries in $\set{1, \ldots, n}$.
Here $x^T = \prod_i x_i^{t_i}$, where $t_i$ is the number of entries of $T$ equal to $i$.
Special cases are $s_{(k)} = h_k$, the complete homogeneous symmetric polynomial, and $s_{(1, \ldots, 1)} = e_k$, the [[elementary symmetric polynomial|0X5QVCF06N]] for $k$ ones.
Letting $n \to \infty$ gives the Schur functions in the ring of [[symmetric functions|PAJEHWXW6M]].
</document>

<document source="https://tamnd.github.io/math2/node/DA3CQJ5MJ6.md" tag="DA3CQJ5MJ6" kind="concept">
# Lagrange basis polynomials

## Summary

The Lagrange basis polynomials of distinct nodes $x_0, \dots, x_n$ are $n + 1$ polynomials of degree $n$.
Each one equals $1$ at its own node and $0$ at the other nodes.
They form a basis of the polynomials of degree at most $n$.
In this basis the coordinates of a polynomial are its values at the nodes.
So the [[Lagrange polynomial|Y7KSGE97GK]] of given data is simply the sum of the basis polynomials weighted by the data values.

## Explanations

### L1 human

Think of each basis polynomial as a switch attached to one node: it is on, with value $1$, at its node and off, with value $0$, at all the others.
To get a polynomial with prescribed values, turn on each switch with the strength you want and add them up.
For the nodes $0$ and $1$ the two basis polynomials are $1 - x$ and $x$.
Then the line with value $a$ at $0$ and $b$ at $1$ is $a(1 - x) + bx$.
The [[definition|ZCVB4ZQ0SA]] gives the general formula.
</document>

<document source="https://tamnd.github.io/math2/node/DA515YH1J0.md" tag="DA515YH1J0" kind="definition">
# Residually finite group

## Definition

Let $G$ be a [[group|G5S1M9]].
$G$ is *residually finite* if for every $g \in G$ with $g \ne 1$ there is a [[normal subgroup|0CB0DX5BVV]] $N$ of $G$ of finite [[index|0DTXBQPA2S]] with $g \notin N$.

## Remarks

Equivalently, for every $g \ne 1$ there are a finite group $Q$ and a [[homomorphism|0JYTG5K9Y4]] $\varphi \colon G \to Q$ with $\varphi(g) \ne 1$.
Equivalently, the intersection of all normal subgroups of finite index of $G$ is trivial.

Example: $\Z$ is residually finite.
For $n \ne 0$, the subgroup $N = (\abs{n} + 1)\Z$ is normal of index $\abs{n} + 1$, and $n \notin N$ since $0 < \abs{n} < \abs{n} + 1$.

Non-example: the additive group $\Q$ is not residually finite.
Each quotient of $\Q$ is divisible, and a finite divisible abelian group is trivial, since multiplication by its order is surjective and also zero.
So $\Q$ has no proper normal subgroup of finite index, and the condition fails for the element $1 \in \Q$, which is not the identity element $0$.
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<document source="https://tamnd.github.io/math2/node/DA619M38C3.md" tag="DA619M38C3" kind="concept">
# Quantale

## Summary

A quantale is a [[complete lattice|D7A0ZGZT5S]] with an associative multiplication that distributes over arbitrary joins on both sides.
Lattices of ideals of a ring, frames of open sets, and power sets of monoids are quantales.
Quantales generalise frames and locales to a non-commutative or non-idempotent setting.
They appear in noncommutative topology and linear logic.
The multiplication of a quantale always has left and right residuals, which links quantales with [[residuated lattices|E337A6WVH3]].

## Explanations

### L1 human

Take the ideals of a commutative ring.
You can join any family of ideals by taking the ideal they generate, and you can multiply two ideals.
Multiplying an ideal by the join of a family gives the join of the products.
A quantale is any complete lattice with a multiplication with this property.
</document>

<document source="https://tamnd.github.io/math2/node/DA8M8VZZHC.md" tag="DA8M8VZZHC" kind="statement">
# Hadamard gap theorem

## Statement

Let $(n_k)_{k \ge 1}$ be a strictly increasing sequence of positive integers, and assume there is $\lambda > 1$ with $n_{k+1} / n_k \ge \lambda$ for all $k$.
Let $(a_k)$ be complex numbers such that the [[power series|0VARVQ9ZW1]] $f(z) = \sum_{k \ge 1} a_k z^{n_k}$ has [[radius of convergence|0660QX1VPR]] $1$.
Then the unit circle is a [[natural boundary|WER1M071EK]] of $f$.

## Remarks

Proof idea: suppose that $f$ extends holomorphically across an arc of the circle, and rotate so that the arc contains $1$.
A standard proof composes $f$ with the polynomial $w \mapsto (w^m + w^{m+1})/2$, for an integer $m$ chosen from $\lambda$.
This map sends the closed unit disc into the unit disc except at $1$, so the composed series would converge on a disc of radius greater than $1$.
Because of the gaps, the expansion of $f$ composed with this map in powers of $w$ splits into disjoint blocks, one for each term $a_k z^{n_k}$.
So the series of $f$ would converge at some real point greater than $1$, a contradiction.
Example: $\sum_{n \ge 0} z^{2^n}$ and $\sum_{n \ge 1} z^{n!}$ have the unit circle as natural boundary.
In the second series, $n_{k+1}/n_k = k + 1 \ge 2$.
The gap condition cannot just be dropped: $\sum_{n \ge 0} z^n = \frac{1}{1 - z}$ has radius $1$ but extends across every point except $1$.
The [[Fabry gap theorem|GE0HX2KMMZ]] reaches the same conclusion under the weaker condition $n_k / k \to \infty$.

## Explanations

### L1 human

A power series with only very few terms, spaced out so that the exponents grow at least geometrically, behaves wildly at every point of its circle of convergence.
It cannot be extended past any arc of the circle.
So sparse power series give simple explicit functions whose domain cannot be enlarged at all.

### L2 human

The theorem is the classical source of functions with a natural boundary, called lacunary functions.
It shows that the obstruction to [[analytic continuation|0M3YQJ81NB]] can be spread over the whole circle of convergence.
It is a special case of the [[Fabry gap theorem|GE0HX2KMMZ]], and its proof is closely related to the phenomenon of overconvergence studied by Ostrowski.
</document>

<document source="https://tamnd.github.io/math2/node/DAEHVRJVSD.md" tag="DAEHVRJVSD" kind="statement">
# Transversality conditions for variable endpoints

## Statement

Let $L \colon \R \times \R \times \R \to \R$, written $L(x, z, p)$, be of class $C^2$, with [[partial derivative|09EGC6F8PT]] $L_p$ in $p$.
Let $\psi \colon \R \to \R$ be of class $C^1$, and fix $a, A \in \R$.
Consider pairs $(y, b)$ with $b > a$, $y \in C^2([a, b])$, $y(a) = A$ and $y(b) = \psi(b)$, and set $J(y, b) = \int_a^b L(x, y, y') \, dx$.
Suppose that $(y_0, b_0)$ minimises $J$ among all such pairs.
Then:

1. $y_0$ solves the [[Euler–Lagrange equation|PS0VV149ET]] $\frac{d}{dx} L_p(x, y_0, y_0') = L_z(x, y_0, y_0')$ on $[a, b_0]$;
2. at the free end, the *transversality condition*
$$\Big[ L(x, y_0, y_0') + \big(\psi'(x) - y_0'(x)\big) L_p(x, y_0, y_0') \Big]_{x = b_0} = 0$$
holds.

## Remarks

The proof computes the [[first variation|5B8NFYB2XQ]] of $J$ along a family $(y_t, b_t)$ of competitors with $y_t(b_t) = \psi(b_t)$.
Variations that fix both ends give the Euler–Lagrange equation.
Moving the end along the curve $y = \psi(x)$ gives a boundary term, $L \, \delta b + L_p (\delta y - y' \delta b)$ at $b_0$, with $\delta y = \psi' \delta b$; it must vanish, which is the transversality condition.

Special cases.
If instead $b$ is fixed and $y(b)$ is free, the end moves on the vertical line $x = b$, and the same computation gives the [[natural boundary condition|QTRPNFDFJ8]] $L_p = 0$ at $b$.
If $L(x, z, p) = f(x, z) \sqrt{1 + p^2}$ with $f(b_0, y_0(b_0)) \ne 0$, the condition simplifies to $\psi'(b_0) \, y_0'(b_0) = -1$: the extremal meets the curve $y = \psi(x)$ at a right angle.

Example: for $f = 1$, $J$ is arc length, and the condition says that the shortest segment from the point $(a, A)$ to a curve meets the curve orthogonally.

## Explanations

### L1 human

When the end of a curve may slide along a given line or curve, the best curve must meet that curve in a special way.
For the shortest path from a point to a curve, the special way is a right angle.
For other functionals, the angle is fixed by a formula involving the integrand.

### L2 human

The transversality condition replaces the missing boundary value.
The Euler–Lagrange equation has order two and one end is fixed, so one more condition fixes the extremal; then $y(b_0) = \psi(b_0)$ fixes the unknown $b_0$.
In optimal control, the [[Pontryagin maximum principle|7RPV2T29CF]] has transversality conditions of the same kind on the costate at a free final state or final time.
</document>

<document source="https://tamnd.github.io/math2/node/DAK6EJK20R.md" tag="DAK6EJK20R" kind="statement">
# Fast marching method

## Statement

Let $h > 0$ and let $G$ be a finite set of $N$ nodes $x_{ij} = (ih, jh)$ of a square grid in $\R^2$.
Let $F_{ij} > 0$ be a speed at each node, and let $S \subset G$ be a nonempty set of source nodes with given values $T_{ij} \ge 0$.
For $T$ on $G$ write $a = \min(T_{i-1,j}, T_{i+1,j})$ and $b = \min(T_{i,j-1}, T_{i,j+1})$, with missing neighbours ignored.
The upwind discretization of the [[eikonal equation|HGMX1QGTAZ]] $\abs{\nabla T} = 1/F$ at a node $x_{ij} \notin S$ is
$$\big( \max(T_{ij} - a, 0) \big)^2 + \big( \max(T_{ij} - b, 0) \big)^2 = \frac{h^2}{F_{ij}^2}.$$
The *fast marching method* first marks the nodes of $S$ as accepted.
It then repeats two steps until all nodes are accepted.
First, it computes a tentative value at each non-accepted neighbour of an accepted node.
This value solves the equation above, with only accepted neighbours used in $a$ and $b$.
Second, it accepts the node with the smallest tentative value.
Assume the grid graph on $G$ is connected.
Then:

1. every node is accepted exactly once, and the method stops;
2. the nodes not in $S$ are accepted in nondecreasing order of their values;
3. the final values satisfy the upwind equation at every node not in $S$;
4. with the tentative values kept in a binary [[heap|V389A4WGH9]], the method uses $O(N \log N)$ operations.

## Remarks

The upwind equation has the causality property: its solution $T_{ij}$ is larger than every neighbour value that enters it with a nonzero term.
Hence a newly computed tentative value is never smaller than the last accepted value, which gives claim 2, and accepting the smallest tentative value never needs to be undone.
The source values are arbitrary, so claim 2 is only about the nodes outside $S$.
This is the same argument as for Dijkstra's algorithm, which the method resembles; Dijkstra's algorithm on the grid graph would measure distance along grid edges only.
Each node is inserted and updated a bounded number of times, and each heap operation costs $O(\log N)$, which gives claim 4.

Example: with $F \equiv 1$ and one source with $T = 0$ at the origin, the computed $T$ approximates the Euclidean distance to the origin.
At the node $(h, h)$, both neighbours $(h, 0)$ and $(0, h)$ have value $h$, and the equation gives $T = h + h/\sqrt{2} \approx 1.71 h$, near the true value $\sqrt{2} h \approx 1.41 h$.
The error is of first order in $h$.

## Explanations

### L1 human

Imagine a fire that starts on a set of points and spreads at a given speed, which may vary from place to place.
The fast marching method computes the arrival time of the fire at every grid point.
It sweeps outward from the start like a front, always fixing the point that the fire reaches next.
Each point is fixed once, so the whole computation is nearly as fast as visiting every point once.

### L2 human

The eikonal equation is a stationary Hamilton–Jacobi equation, and its relevant solution is the [[viscosity solution|H1ERJ3T75T]].
The upwind scheme above is monotone and consistent, and as $h \to 0$ its solutions approximate the viscosity solution.
The fast marching method solves this nonlinear system in one pass, instead of by repeated sweeps or iterations.
It is used to build signed distance functions for the [[level-set method|7KWR08FJET]], to compute shortest paths around obstacles, and in seismic travel time computations.
</document>

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# Opposite category

## Summary

The opposite category $\mathcal{C}^{\mathrm{op}}$ of a [[category|A2DE66937D]] $\mathcal{C}$ has the same objects and the same morphisms, with every arrow reversed.
It turns [[monomorphisms|Y2NP1JFR86]] into [[epimorphisms|RVXDSY04KG]], products into coproducts, and limits into colimits.
It is the basis of [[duality in category theory|Z5EETTFWAN]].
It also lets one write a [[contravariant functor|RECR96V97B]] on $\mathcal{C}$ as an ordinary functor on $\mathcal{C}^{\mathrm{op}}$.

## Explanations

### L1 human

Take a drawing of a category and reverse the direction of every arrow.
The result is again a category, because chains of arrows can still be composed, in the reverse order.
For a [[preorder|0VABYWJWH9]] seen as a category, with an arrow $a \to b$ when $a \le b$, the opposite category is the reversed order $\ge$.
For the numbers $1 \le 2 \le 3$, the opposite category has arrows $3 \to 2 \to 1$.
</document>

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# Projective plane

## Summary

A projective plane is an [[incidence structure|FGKNAFNHA3]] in which two points lie on exactly one line and two lines meet in exactly one point.
It is the plane of perspective drawing, where parallel lines meet at a point at infinity.
The real projective plane and the planes $\mathbb{P}^2(k)$ over fields are examples, and there are also planes that come from no field.
Finite projective planes are central objects of combinatorics.

## Explanations

### L1 human

In a perspective drawing, railway tracks meet at a point on the horizon.
A projective plane makes this a rule: any two lines meet, and the horizon becomes a line like the others.
Through two points there is still exactly one line.
The smallest projective plane, the Fano plane, has $7$ points and $7$ lines, with $3$ points on each line and $3$ lines through each point.
</document>

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# Thiele's interpolation formula

## Statement

Let $n \ge 1$, let $x_1, \dots, x_{n+1} \in \R$ be pairwise distinct, and let $f$ be a real [[function|0EPAT51DXQ]] defined at these points.
The *reciprocal differences* $\rho_k$ of $f$ are defined by $\rho_0(x_i) = f(x_i)$,
$$\rho_1(x_i, x_{i+1}) = \frac{x_i - x_{i+1}}{f(x_i) - f(x_{i+1})},$$
and, for $k \ge 2$,
$$\rho_k(x_i, \dots, x_{i+k}) = \frac{x_i - x_{i+k}}{\rho_{k-1}(x_i, \dots, x_{i+k-1}) - \rho_{k-1}(x_{i+1}, \dots, x_{i+k})} + \rho_{k-2}(x_{i+1}, \dots, x_{i+k-1}).$$
Set $a_0 = f(x_1)$, $a_1 = \rho_1(x_1, x_2)$ and $a_k = \rho_k(x_1, \dots, x_{k+1}) - \rho_{k-2}(x_1, \dots, x_{k-1})$ for $2 \le k \le n$.
Assume that no denominator above is zero and that the finite [[continued fraction|04P2RJ4WPS]]
$$R(x) = a_0 + \cfrac{x - x_1}{a_1 + \cfrac{x - x_2}{a_2 + \cfrac{x - x_3}{\ddots + \cfrac{x - x_n}{a_n}}}}$$
can be evaluated at each node without division by zero.
Then $R$ is a [[rational function|0XDK9CGYKJ]] and $R(x_i) = f(x_i)$ for $i = 1, \dots, n + 1$.

## Remarks

Reciprocal differences play for rational interpolation the role that [[divided differences|N5ZHV4HWDC]] play for polynomial interpolation.
The proof is by induction on $n$: the tail of the continued fraction starting at $a_1$ interpolates the reciprocal data $(x - x_1)/(f(x) - f(x_1))$ at $x_2, \dots, x_{n+1}$.

Example: $n = 1$ gives $R(x) = f(x_1) + (x - x_1)\dfrac{f(x_2) - f(x_1)}{x_2 - x_1}$, the line through the two data points.

The non-degeneracy hypothesis is needed.
The formula can break down even when a simple interpolant exists.
For instance, for equal values $f(x_1) = f(x_2)$, interpolated by a constant, the first reciprocal difference already divides by $0$.

## Explanations

### L1 human

Polynomials are bad at imitating functions with poles or with horizontal asymptotes.
Thiele's formula fits a ratio of polynomials instead, written as a continued fraction.
Its coefficients come from a table of reciprocal differences, built much like a table of divided differences.

### L2 human

Thiele's formula is the rational analogue of the [[Newton polynomial|9W77FPTM9E]]: each new node adds one more level to the continued fraction and leaves the earlier levels unchanged.
Its truncations give rational interpolants whose numerator and denominator degrees grow alternately.
It is closely related to [[Padé approximants|1F16G4V64X]], which match derivatives at one point instead of values at many points.
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# Topological quantum field theory

## Summary

A topological quantum field theory assigns a vector space to each closed manifold of a fixed dimension $n$.
It assigns a linear map to each $(n+1)$-manifold whose boundary joins two such manifolds.
Gluing manifolds must correspond to composing linear maps, and disjoint union must correspond to the tensor product.
A closed manifold of the top dimension then receives a number, which is an invariant of the manifold.
Atiyah stated these axioms, drawing on work of Witten and Segal in physics.
A main example is [[Chern–Simons theory|F28G886PWD]], which produces [[quantum invariants of 3-manifolds|YGPR4VR3GP]] and of knots.

## Explanations

### L1 human

Think of a closed surface as a moment in time and a solid shape between two surfaces as a history from one moment to the next.
A topological quantum field theory gives each moment a space of possible states and each history a rule that turns old states into new ones.
Doing two histories one after the other must give the same rule as composing the two rules.
The rules only see the shape up to smooth deformation, not lengths or angles.
In the simplest case, with points as moments and curves as histories, a theory picks a vector space $V$, and the circle receives the number $\dim V$.
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<document source="https://tamnd.github.io/math2/node/DAR3QJ5H7Y.md" tag="DAR3QJ5H7Y" kind="statement">
# Monte Carlo integration

## Statement

Let $d \ge 1$, let $\Omega \subseteq \R^d$ be a measurable set with [[Lebesgue measure|2Z2QVWZ262]] $V$, $0 < V < \infty$, and let $f \colon \Omega \to \R$ be integrable, with $I = \int_\Omega f(x)\, dx$.
Let $X_1, X_2, \ldots$ be [[independent and identically distributed|00MRN0FARV]] random points with the [[uniform distribution|0AREFQ9Z65]] on $\Omega$, and let
$$Q_n = \frac{V}{n} \sum_{i=1}^{n} f(X_i).$$
Then:

1. $\mathbb E Q_n = I$ for every $n \ge 1$;
2. $Q_n \to I$ almost surely as $n \to \infty$;
3. if moreover $\int_\Omega f^2 < \infty$, then $\operatorname{Var} Q_n = V^2 \sigma^2 / n$, where $\sigma^2 = \operatorname{Var} f(X_1) = \frac{1}{V}\int_\Omega f^2 - \frac{I^2}{V^2}$;
4. under the hypothesis of clause 3 and $\sigma > 0$, $\sqrt{n}\,(Q_n - I) / (V \sigma)$ converges in distribution to the standard normal distribution.

## Remarks

The density of $X_1$ is $1/V$ on $\Omega$, so $\mathbb E[V f(X_1)] = \int_\Omega f$; this gives clause 1.
Clause 2 is the [[strong law of large numbers|0DFWHQ839T]] for the variables $V f(X_i)$, and clause 4 is the [[central limit theorem|0EPZC4Z1FC]].
Clause 3 follows from the [[variance of a sum of independent random variables|0V0778DCW9]].
So the root-mean-square error is $V\sigma / \sqrt{n}$; the rate $n^{-1/2}$ does not depend on the dimension $d$ or on the smoothness of $f$.
Example: for $f(x) = x^2$ on $\Omega = [0, 1]$ one has $I = 1/3$ and $\sigma^2 = 1/5 - 1/9 = 4/45$, so with $n = 10^4$ the standard error is about $0.003$.
In practice $\sigma$ is unknown and is replaced by the sample standard deviation, which gives approximate confidence intervals.
Without square-integrability clause 3 fails, and the convergence in clause 2 may be much slower than $n^{-1/2}$.

## Explanations

### L1 human

To find the integral of a function over a region, pick many random points in the region, average the function values, and multiply by the volume.
On average this gives exactly the right answer, and the scatter of the answer shrinks like one over the square root of the number of points.
The remarkable feature is that this rate is the same in dimension $1$ and in dimension $1000$.
Grid-based rules, by contrast, need exponentially many points as the dimension grows.

### L2 human

This is the basic [[Monte Carlo method|HWP11G3HTN]] applied to $Y = V f(X)$ with $X$ uniform.
In low dimension and for smooth $f$, classical quadrature rules converge much faster than $n^{-1/2}$, so Monte Carlo is mainly useful when $d$ is large or $f$ is rough.
Variance reduction techniques, such as importance sampling, control variates and stratification, keep the rate $n^{-1/2}$ but reduce the constant $\sigma$.
Quasi-Monte Carlo methods replace the random points by deterministic low-discrepancy points.
For functions of bounded variation in the sense of Hardy and Krause, they reach a rate close to $n^{-1}$.
Uniform sampling on $\Omega$ is often replaced by sampling from another density, which is the idea of importance sampling.
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# Tridiagonal matrix

## Summary

A tridiagonal matrix is a square matrix whose nonzero entries lie only on the main diagonal and on the two diagonals next to it.
It is a [[band matrix|Z58FRZCF8P]] of bandwidth one on each side, and it is both upper and lower [[Hessenberg|YCEEWJTX56]].
Linear systems with a tridiagonal matrix can be solved with a number of operations proportional to the size, not to its cube.
Such matrices arise from finite difference schemes, from three-term recurrences and from the reduction of symmetric matrices for eigenvalue computations.

## Explanations

### L1 human

Picture a square matrix where only a narrow strip of three diagonals is filled and everything else is zero.
For example, $\begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}$ is tridiagonal.
This one comes from approximating a second derivative at evenly spaced points: each point only talks to its two neighbours.
Because each unknown meets only its neighbours, elimination proceeds row by row without creating new nonzero entries.
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# Subdirectly irreducible algebra

## Summary

A subdirectly irreducible algebra is a nontrivial [[algebra|QJGPT2W0FT]] that cannot be split as a [[subdirect product|88WN3AD2VQ]] in a nontrivial way.
In every subdirect representation of it, one of the projections is already an isomorphism.
Equivalently, among its [[congruences|DCSXXT4SER]] other than equality there is a smallest one, called the monolith.
By [[Birkhoff's subdirect representation theorem|CZF5696TPY]], these algebras are the building blocks of all algebras, and a variety is determined by its subdirectly irreducible members.

## Explanations

### L1 human

Any nontrivial gluing of a subdirectly irreducible algebra, however small, must glue a certain fixed pair of elements.
So the algebra cannot be recovered from several smaller pictures of it, since each picture loses that pair.
Every [[simple algebra|9CJCXNYNK8]] is subdirectly irreducible.
The group $\Z/4\Z$ is subdirectly irreducible but not simple: every nontrivial gluing identifies $0$ and $2$.
The group $\Z/6\Z$ is not, because it is the product of $\Z/2\Z$ and $\Z/3\Z$.
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# Solving linear difference equations by the Z-transform

## Summary

The Z-transform method solves a [[linear recurrence with constant coefficients|0NDVHBVBXB]] with given initial values.
One applies the [[unilateral Z-transform|EPG6JPVJE7]] to the equation, solves the resulting algebraic equation for the transform of the unknown, and inverts.
The initial values enter by themselves through the [[shift property|JWH2KYCGBQ]].
It is the discrete analogue of the Laplace transform method for linear differential equations.

## Explanations

### L1 human

A recurrence such as $y_{n+2} = y_{n+1} + y_n$ ties each term to earlier ones.
After the Z-transform, each shift becomes multiplication by $z$, so the recurrence becomes an equation that one solves with ordinary algebra.
The answer is a rational function of $z$.
Splitting it into simple fractions and reading each one back as a geometric sequence gives a closed formula for $y_n$.
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# Relative homology

## Definition

Let $X$ be a topological space and let $A \subseteq X$ be a subspace with the [[subspace topology|0WS9P12FQR]].
Let $C_\bullet(X)$ be the singular [[chain complex|WQ82C0M3QB]] of $X$.
Every singular simplex in $A$ is a singular simplex in $X$, so $C_\bullet(A)$ is a subcomplex of $C_\bullet(X)$.
The *relative chain complex* is the quotient $C_n(X, A) = C_n(X) / C_n(A)$, with the boundary map induced by the boundary of $C_\bullet(X)$.
The *relative [[homology|2J4JJJYP22]]* $H_n(X, A)$ is the homology of $C_\bullet(X, A)$ in degree $n$.

## Remarks

A class in $H_n(X, A)$ is represented by a *relative cycle*, a chain $c \in C_n(X)$ with $\partial c \in C_{n-1}(A)$.
Two relative cycles give the same class when their difference is $\partial b + a$ with $b \in C_{n+1}(X)$ and $a \in C_n(A)$.
For $A = \emptyset$, the group $H_n(X, \emptyset)$ is the [[singular homology|M0Q5CFJ7ZH]] $H_n(X)$.
A map of pairs $f \colon (X, A) \to (Y, B)$, that is, a continuous map with $f(A) \subseteq B$, induces maps $f_* \colon H_n(X, A) \to H_n(Y, B)$.

Example: for $n \ge 1$, $H_n(D^n, S^{n-1}) \cong \Z$ and the other relative groups vanish.
For $n = 1$, the path $\sigma(t) = 2t - 1$ in $D^1 = [-1, 1]$ is a relative cycle, since $\partial \sigma = (1) - (-1)$ lies in $C_0(S^0)$.

Non-example: in $X = [-1, 1]$ with $A = \set{-1}$, the same path is not a relative cycle, because its boundary contains the point $1 \notin A$.
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# Nelson–Aalen estimator

## Definition

Let $(\tilde t_1, \delta_1), \dots, (\tilde t_n, \delta_n)$ be right-[[censored|MY8ZZP84NP]] observations of a lifetime, with $\delta_i = 1$ for an event and $\delta_i = 0$ for a censored time.
Let $t_{(1)} < \dots < t_{(m)}$ be the distinct event times, $d_j$ the number of events at $t_{(j)}$, and $n_j = \#\set{i : \tilde t_i \ge t_{(j)}}$ the number at risk just before $t_{(j)}$.
The *Nelson–Aalen estimator* of the cumulative [[hazard function|Y959NKQ3XW]] $H$ is
$$\hat H(t) = \sum_{j \,:\, t_{(j)} \le t} \frac{d_j}{n_j}, \qquad t \ge 0,$$
with the empty sum equal to $0$.

## Remarks

In counting process notation, with $N(t)$ the number of observed events up to $t$ and $Y(t)$ the number at risk at $t$, the [[estimator|0SSS6ERQ0S]] is $\hat H(t) = \int_0^t \frac{dN(s)}{Y(s)}$, with $0/0 = 0$.
A common variance estimator is $\sum_{j : t_{(j)} \le t} d_j / n_j^2$.
The survival estimate $e^{-\hat H(t)}$ is at least the Kaplan–Meier estimate, since $e^{-x} \ge 1 - x$ for all $x$.

Example: for the observations $2, 3^+, 5, 7$, with $3^+$ censored, $(n_j, d_j)$ is $(4, 1)$ at $2$, $(2, 1)$ at $5$ and $(1, 1)$ at $7$.
So $\hat H = 0$ on $[0, 2)$, $1/4$ on $[2, 5)$, $3/4$ on $[5, 7)$ and $7/4$ from $7$ on.

Non-example: the cumulative sum $\sum_{t_{(j)} \le t} d_j / n$, which divides by the initial sample size $n$, is not the Nelson–Aalen estimator.
It ignores that subjects leave the risk set, and it underestimates the cumulative hazard.
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<document source="https://tamnd.github.io/math2/node/DBD64TH52D.md" tag="DBD64TH52D" kind="concept">
# Least fixed point

## Summary

The least fixed point of a map $f$ from a [[partially ordered set|0QV2JS9RJ3]] to itself is a point $x$ with $f(x) = x$.
It must lie below every other point with this property.
When it exists it is unique.
Least fixed points give meaning to recursive and inductive definitions: the set defined inductively by some rules is the least set closed under the rules.
The [[Knaster–Tarski theorem|RHXP3HA64M]] and the [[Kleene fixed-point theorem|DDAHG7QRY8]] guarantee least fixed points of monotone maps under completeness hypotheses.

## Explanations

### L1 human

A fixed point of a map is a point that the map leaves where it is.
Among all fixed points, the least one is the one that lies below all the others.
For the map $x \mapsto x^2$ on the real numbers, the fixed points are $0$ and $1$, and the least fixed point is $0$.
The set of even natural numbers is the least set that contains $0$ and contains $n + 2$ whenever it contains $n$.
It is the least fixed point of the map $S \mapsto \set{0} \cup \set{n + 2 : n \in S}$ on subsets of $\N$.
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# Time series

## Definition

Let $T$ be a set of time points, usually $T = \Z$, $T = \set{1, \dots, n}$ or an interval of $\R$.
A *time series* is a family of observations $(x_t)_{t \in T}$ indexed by time, together with the order of $T$.
As a statistical model, a time series is a realization of a [[stochastic process|0QH5ENKZM1]] $(X_t)_{t \in T}$.
The same name is often used for the process itself.
The series is in *discrete time* when $T$ is a subset of $\Z$, and in *continuous time* when $T$ is an interval of $\R$.

## Remarks

Example: the yearly maximum temperatures $x_{1990}, \dots, x_{2020}$ at one weather station form a discrete-time series with $T = \set{1990, \dots, 2020}$.
Non-example: the heights of $30$ pupils, listed in alphabetical order of their names, are not a time series.
The index is not time, and its order carries no meaning.
A discrete-time series is a finite or infinite [[sequence|0G9EKM0E15]] of real numbers or vectors; when its values are vectors it is called multivariate.
Many methods assume that the underlying process is stationary, possibly after removing a trend or a seasonal component.
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<document source="https://tamnd.github.io/math2/node/DBEGF38V9H.md" tag="DBEGF38V9H" kind="statement">
# Karamata's Tauberian theorem

## Statement

Let $U \colon \R \to \R$ be a nondecreasing, right-continuous [[monotone function|0WFF6AZPBP]] with $U(x) = 0$ for $x < 0$.
Assume that its Laplace–Stieltjes transform
$$\widehat{U}(s) = \int_{[0, \infty)} e^{-sx} \, dU(x),$$
a [[Riemann–Stieltjes integral|8NMH42GJRK]] over $[0, \infty)$ that includes a possible jump of $U$ at $0$, is finite for every $s > 0$.
Let $\rho \ge 0$ and $c > 0$ be real numbers, let $L$ be a [[slowly varying function|EHJ7D5CRZS]], and let $\Gamma$ be the [[gamma function|0V3FZMVB4K]].
Then the following are equivalent:

1. $\widehat{U}(s) \sim c \, s^{-\rho} L(1/s)$ as $s \to 0^+$;
2. $U(x) \sim \dfrac{c}{\Gamma(1 + \rho)} \, x^{\rho} L(x)$ as $x \to \infty$.

Here $\sim$ denotes [[asymptotic equivalence|0QR92XBV6H]].

## Remarks

The direction from 2 to 1 is an Abelian statement and follows by integration by parts and standard estimates for slowly varying functions.
The direction from 1 to 2 is the Tauberian part, where monotonicity of $U$ is essential.
Karamata's proof shows that $\int_0^\infty g(e^{-sx}) \, dU(x)$ has the expected behaviour for polynomials $g$, then for the indicator function of $[e^{-1}, 1]$ by one-sided polynomial approximation, which recovers $U(1/s)$.

Example: let $a_n \ge 0$ and $U(x) = \sum_{n \le x} a_n$.
Then $\widehat{U}(s) = \sum_{n \ge 0} a_n e^{-sn}$, and the case $\rho = 1$, $c = 1$, $L = 1$ is the [[Hardy–Littlewood Tauberian theorem|N42JDBT0VA]] with $x = e^{-s}$.

Monotonicity cannot be dropped in the Tauberian direction: the counterexample to the Hardy–Littlewood theorem with $a_n = 1 + (-1)^n n$ also violates statement 2 here.

## Explanations

### L1 human

The Laplace transform turns a growing function into a function of $s$ that blows up as $s \to 0$.
Karamata's theorem says that, for a nondecreasing function, a power-law growth (up to a slowly varying factor) of one side is exactly matched by a power-law blow-up of the other.
So one can read off the growth of $U$ from its transform near $0$, and back.

### L2 human

The theorem is the standard form of Tauberian theorems for Laplace–Stieltjes transforms, and it is central in the theory of regular variation.
In probability it relates the tail of a distribution or the growth of a renewal function to the behaviour of its Laplace transform near $0$.
Its proof by polynomial approximation, through the [[Weierstrass approximation theorem|0JQE130DHW]], gave a short route to the Hardy–Littlewood theorem.
</document>

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# Chaos theory

## Summary

Chaos theory studies deterministic [[dynamical systems|BBZP1R872N]] whose long-time behaviour looks random and cannot be predicted in practice.
The key feature is sensitive dependence on initial conditions, the [[butterfly effect|G0ZT1SDQGS]]: tiny differences in the starting state grow until they dominate.
Typical tools are [[Lyapunov exponents|NJ1FJSNK6A]], [[strange attractors|9WKCZGG906]], symbolic dynamics and invariant measures, and a standard mathematical definition of a chaotic map is [[Devaney chaos|21B8AHPD4P]].
Classical examples are the [[Lorenz system|WYRC41CQN9]] in meteorology and the [[logistic map|QN9X9BNKTX]] in population dynamics.

## Explanations

### L1 human

A chaotic system follows exact rules, with no chance involved, and yet its future is practically unpredictable.
The reason is that any small error in the present, such as rounding a measurement, is magnified again and again.
The doubling map $x \mapsto 2x \bmod 1$ shows this well: it shifts the binary digits of $x$ one place to the left.
Two starting values that agree in their first $50$ binary digits give orbits that look unrelated after about $50$ steps.
Weather is the most famous example, which is why forecasts become unreliable after a week or two.
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<document source="https://tamnd.github.io/math2/node/DBM13YSJAM.md" tag="DBM13YSJAM" kind="statement">
# Differentiation operator as inverse of integration

## Statement

Let $\mathcal{M}$ be the [[Mikusiński field of operators|82YAXZMMQ1]], the field of fractions of the convolution ring of locally integrable functions on $[0, \infty)$.
Let $h$ be the constant function $1$ on $[0, \infty)$, and let $s = 1/h$ be its inverse in $\mathcal{M}$.
For a complex number $c$, write $c$ also for the numerical operator $c \cdot 1$ of $\mathcal{M}$, which corresponds to the function $c\,h$ divided by $h$.
Let $f$ be continuously differentiable on $[0, \infty)$.
Then
$$s f = f' + f(0)$$
in $\mathcal{M}$.
More generally, if $f$ is $n$ times continuously differentiable on $[0, \infty)$, then
$$s^n f = f^{(n)} + \sum_{k=0}^{n-1} f^{(k)}(0) \, s^{n-1-k}.$$

## Remarks

By the [[fundamental theorem of calculus|0NHGY0N0JB]], $\int_0^t f'(\tau) \, d\tau = f(t) - f(0)$.
By the [[integration operator result|BDBDJSP6DF]] this reads $h f' = f - f(0) h$ in the convolution ring.
Multiplying both sides by $s$ gives $f' = s f - f(0)$, which is the first formula.
The general formula follows by induction on $n$.
For example, with $f(t) = e^{a t}$, one gets $s f = a f + 1$, so $f = 1/(s - a)$ in $\mathcal{M}$.
The operator $s$ is not a function: it is a quotient of functions, and $s f$ equals the function $f'$ only when $f(0) = 0$.

## Explanations

### L1 human

Integration from $0$ is multiplication by the constant function $1$.
Its inverse in the field of operators is an operator $s$ that acts like differentiation.
It differs from plain differentiation by a term that remembers the starting value $f(0)$.

### L2 human

This formula has the same shape as the rule $\mathcal{L}\{f'\}(s) = s F(s) - f(0)$ for the Laplace transform.
Mikusiński's calculus gets it with no condition on the growth of $f$ and no convergence of integrals.
A linear differential equation with constant coefficients and initial values becomes an algebraic equation in $s$ in the field $\mathcal{M}$.
One then solves for the unknown and reads off a function, often after a partial fraction expansion in $s$.
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# Logarithmic capacity

## Summary

The logarithmic capacity of a compact set $K$ in the plane measures how large $K$ is from the point of view of [[potential theory|9PM8Q2E30Q]].
It is defined from the least possible logarithmic energy of a unit charge distributed on $K$.
A disc of radius $r$ has capacity $r$, and a line segment of length $L$ has capacity $L/4$.
Sets of capacity zero, the polar sets, are negligible for many questions on harmonic and holomorphic functions.
Examples are removable singularities and the solvability of the [[Dirichlet problem|8D56GQ4AG8]].

## Explanations

### L1 human

Spread one unit of electric charge on a conducting plate shaped like $K$, in a two-dimensional world where the potential of a point charge is logarithmic.
The charge arranges itself to make the energy as small as possible.
The smaller that minimal energy, the larger the capacity.
A finite set of points cannot hold any charge with finite energy, so it has capacity $0$, while a disc of radius $r$ has capacity $r$.
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<document source="https://tamnd.github.io/math2/node/DBVE636BXN.md" tag="DBVE636BXN" kind="definition">
# Mixed finite element method

## Definition

Let $V$ and $Q$ be real [[Hilbert spaces|HM2HHE3J51]] of functions on a bounded polygonal domain $\Omega \subset \R^d$.
Let $a \colon V \times V \to \R$ and $b \colon V \times Q \to \R$ be bounded [[bilinear forms|0SQZMXB6GP]], and let $f \in V'$ and $g \in Q'$.
The *mixed formulation* is the problem: find $(u, p) \in V \times Q$ with
$$a(u, v) + b(v, p) = f(v) \ \text{ for all } v \in V, \qquad b(u, q) = g(q) \ \text{ for all } q \in Q.$$
A *mixed finite element method* for it consists of:

1. a mesh $\mathcal{T}_h$ of $\Omega$;
2. two [[finite element|98GZ9DVHTA]] spaces $V_h \subseteq V$ and $Q_h \subseteq Q$ of piecewise polynomial functions on $\mathcal{T}_h$, chosen separately;
3. the [[Galerkin|Y5B0PEBKE8]] problem: find $(u_h, p_h) \in V_h \times Q_h$ with the two equations above for all $v \in V_h$ and $q \in Q_h$.

## Remarks

Example: the mixed Poisson problem.
For $-\Delta u = s$ with $s \in L^2(\Omega)$ in $\Omega$ with $u = 0$ on $\partial \Omega$, set $\sigma = \nabla u$.
Take $V = H(\operatorname{div}, \Omega)$, the vector fields in $L^2$ whose [[divergence|0MWV6ZTK6S]] is in $L^2$, and $Q = L^2(\Omega)$.
Multiplying $\sigma = \nabla u$ by $\tau \in V$ and integrating by parts with $u = 0$ on the boundary gives $\int_\Omega \sigma \cdot \tau + \int_\Omega u \, \nabla \cdot \tau = 0$, and the equation gives $\int_\Omega (\nabla \cdot \sigma) q = -\int_\Omega s q$.
This has the form above with $a(\sigma, \tau) = \int_\Omega \sigma \cdot \tau$, $b(\tau, q) = \int_\Omega q \, \nabla \cdot \tau$, $f = 0$ and $g(q) = -\int_\Omega s q$.
A stable choice is the lowest-order Raviart–Thomas space for $V_h$ with piecewise constants for $Q_h$.

Non-example: the standard method seeks $u_h$ alone, in continuous piecewise linear functions in the [[Sobolev space|M4CEZ5K9HP]] $H^1_0(\Omega)$.
It has one unknown field and no form $b$, so it is not a mixed method.

The matrix of the discrete problem has the block form $\begin{pmatrix} A & B^T \\ B & 0 \end{pmatrix}$, which is symmetric when $a$ is but never positive definite.
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# Ternary discriminator

## Definition

Let $A$ be a nonempty set.
The *ternary discriminator* on $A$ is the following [[ternary operation|WC47MFR6GW]] $t \colon A^3 \to A$.
$$t(x, y, z) = \begin{cases} z & \text{if } x = y, \\ x & \text{if } x \neq y. \end{cases}$$
A term $s(x, y, z)$ is a *discriminator term* for an algebra $A$ if its term operation on $A$ is the ternary discriminator on $A$.

## Remarks

Example: on a field $F$ with $q$ elements, the polynomial $s(x, y, z) = x + (z - x)\bigl(1 - (x - y)^{q - 1}\bigr)$ gives the discriminator.
Check: if $x = y$ then $(x - y)^{q - 1} = 0$ and $s = z$; if $x \neq y$ then $(x - y)^{q - 1} = 1$ and $s = x$.
Non-example: on $\set{0, 1}$ the lattice term $(x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$ is not the discriminator, since at $(0, 1, 1)$ it gives $1$ while the discriminator gives $0$.
The discriminator satisfies $t(x, y, y) = x$, $t(x, x, y) = y$ and $t(x, y, x) = x$, so it is a [[Pixley term|5C91SFVTY7]] operation.
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# Graph of groups

## Summary

A graph of groups is a connected graph with a group attached to each vertex and each edge.
Each edge group comes with injective homomorphisms into the groups of the end vertices of the edge.
Its fundamental group glues the vertex groups along the edge groups.
It is the central object of [[Bass–Serre theory|6M731X66RD]]: groups acting on trees are exactly fundamental groups of graphs of groups.
The simplest cases give the [[amalgamated free product|M848AHBRCM]] and the [[HNN extension|YEYERRY37K]].

## Explanations

### L1 human

Think of each vertex group as a building block, and each edge as an instruction to glue two blocks along a common piece, the edge group.
With two vertices and one edge one gets an amalgam $A *_C B$: two groups glued along a shared subgroup.
With one vertex and a loop one glues a group to itself, which gives an HNN extension.
If all the groups are trivial, the fundamental group is just the fundamental group of the graph, a free group.
</document>

<document source="https://tamnd.github.io/math2/node/DC4JPVPXEJ.md" tag="DC4JPVPXEJ" kind="definition">
# Obstruction theory

## Definition

Obstruction theory is built on the obstruction cocycle, defined as follows.
Let $(X, A)$ be a [[relative CW complex|D0DAFG8QK6]], and let $X_k$ be the union of $A$ with the cells of $X$ of dimension at most $k$.
Let $n \ge 1$ and let $Y$ be a path-connected space whose fundamental group acts as the identity on the [[homotopy group|P976WYEFMB]] $\pi_n(Y)$.
Then free homotopy classes of maps $S^n \to Y$ correspond to elements of $\pi_n(Y)$.
Let $g \colon X_n \to Y$ be a map.
Fix an orientation of each $(n+1)$-cell $e$ of $X \setminus A$, with attaching map $\varphi_e \colon S^n \to X_n$.
The *obstruction cocycle* of $g$ is the cellular cochain
$$c(g) \in C^{n+1}(X, A; \pi_n(Y)), \qquad c(g)(e) = [g \circ \varphi_e].$$
Its class $[c(g)]$ lies in $H^{n+1}(X, A; \pi_n(Y))$, which is isomorphic to the cellular cohomology and to [[singular cohomology|6N7BC70CX0]] of the pair.

## Remarks

The basic facts of obstruction theory are these.

1. The cochain $c(g)$ is a cocycle.
2. The map $g$ extends to $X_{n+1}$ if and only if $c(g) = 0$.
3. The class $[c(g)]$ is zero if and only if the restriction of $g$ to $X_{n-1}$ extends to a map $X_{n+1} \to Y$.

Fact 2 holds because a map $S^n \to Y$ extends over $D^{n+1}$ exactly when it is null-homotopic.
Fact 3 is proved with the difference cochain of two maps that agree on $X_{n-1}$.

Example: let $(X, A) = (D^{n+1}, S^n)$, where $X \setminus A$ is a single $(n+1)$-cell, so $X_n = S^n$.
Let $n \ge 1$, $Y = S^n$, and let $g$ be the identity of $S^n$.
Then $c(g)$ sends the top cell to the generator of $\pi_n(S^n) \cong \Z$, so $g$ does not extend; this is the no-retraction theorem.

Non-example: if $\pi_1(Y)$ acts nontrivially on $\pi_n(Y)$, the value $[g \circ \varphi_e]$ is only defined up to this action, so $c(g)$ is not defined with constant coefficients; one then uses local coefficients.
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<document source="https://tamnd.github.io/math2/node/DC60FYCSPM.md" tag="DC60FYCSPM" kind="concept">
# Second-order arithmetic

## Summary

Second-order arithmetic, often written $\mathsf{Z}_2$, is a theory with two kinds of variables.
One kind ranges over natural numbers, and the other over sets of natural numbers.
Its axioms are basic facts about arithmetic, induction for sets, and comprehension, which says that every definable collection of numbers is a set.
A large part of ordinary mathematics, including real analysis, can be developed in it by coding real numbers and other objects as sets of numbers.
Its subsystems are the setting of [[reverse mathematics|7SQS69T17R]].

## Explanations

### L1 human

First-order arithmetic talks only about numbers, such as $3$ or $17$.
Second-order arithmetic can also talk about sets of numbers, such as the set of all primes.
A real number can be coded as a set of numbers, for example by a sequence of rational approximations, so the theory can talk about real numbers too.
This makes it strong enough for most of calculus and analysis, while it stays a theory about numbers at heart.
</document>

<document source="https://tamnd.github.io/math2/node/DC7HMHVREP.md" tag="DC7HMHVREP" kind="concept">
# Characteristic value of a kernel

## Summary

A characteristic value of a kernel $K$ is a number $\lambda$ for which the homogeneous integral equation $\varphi(x) = \lambda \int K(x, y) \varphi(y) \, dy$ has a nonzero solution.
It is the reciprocal of a nonzero [[eigenvalue|0BQAA2BK3Q]] of the integral operator with kernel $K$.
Characteristic values are the exceptional parameters of the [[Fredholm equation of the second kind|DEYYNVCV9J]].
Away from them, the equation has exactly one solution for each right-hand side.

## Explanations

### L1 human

An integral equation of the second kind contains a parameter $\lambda$ in front of the integral.
For most values of $\lambda$ the equation is well behaved, but for some special values the homogeneous equation has a nonzero solution, and uniqueness fails.
These special values are the characteristic values.
For $K(x, y) = xy$ on $[0, 1]$, the only one is $\lambda = 3$, with solution $\varphi(x) = x$, because $\int_0^1 x y \cdot y \, dy = x/3$.
</document>

<document source="https://tamnd.github.io/math2/node/DC81TCR98Y.md" tag="DC81TCR98Y" kind="concept">
# (p,q)-form

## Summary

A $(p, q)$-form on an open subset of $\C^n$, or on a complex manifold, is a complex-valued [[differential form|0E3G0GRPZ6]] of a special type.
Each of its terms contains $p$ of the differentials $dz_j$ and $q$ of the conjugate differentials $d\bar z_j$.
Every complex differential form of degree $k$ splits in a unique way as a sum of $(p, q)$-forms with $p + q = k$.
This splitting does not depend on the choice of holomorphic coordinates.
It is the setting of the [[$\bar\partial$ operator|M1VKCW3RFW]] and of Dolbeault cohomology.

## Explanations

### L1 human

In $\C$, the differential $dx$ of the real part and $dy$ of the imaginary part can be traded for $dz = dx + i\,dy$ and $d\bar z = dx - i\,dy$.
A $(p, q)$-form is a form whose terms each contain exactly $p$ factors of the kind $dz_j$ and $q$ of the kind $d\bar z_j$.
For example, $dz_1 \wedge d\bar z_2$ in $\C^2$ is a $(1, 1)$-form.
The form $dx_1 = \tfrac12 (dz_1 + d\bar z_1)$ is a sum of a $(1, 0)$-form and a $(0, 1)$-form, so it has no single type.
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<document source="https://tamnd.github.io/math2/node/DC9HZT2HVJ.md" tag="DC9HZT2HVJ" kind="statement">
# Optional stopping theorem

## Statement

Let $(X_n)_{n \ge 0}$ be a [[martingale|0CDVK9RQBH]] with respect to a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_n)_{n \ge 0}$, and let $\tau$ be a [[stopping time|WND87FBAKC]] for $(\mathcal{F}_n)$.
Assume that one of the following holds:

1. There is a constant $N$ with $\tau \le N$ almost surely.
2. $E[\tau] < \infty$, and there is a constant $c$ with $E\bigl[\abs{X_{n+1} - X_n} \mid \mathcal{F}_n\bigr] \le c$ almost surely on the event $\{\tau > n\}$, for all $n \ge 0$.
3. $\tau < \infty$ almost surely, and there is a constant $c$ with $\abs{X_{n \wedge \tau}} \le c$ almost surely for all $n \ge 0$.

Then $X_\tau$ is integrable and $E[X_\tau] = E[X_0]$.
If $(X_n)$ is only a [[submartingale|EQ055QMACW]], the same hypotheses give $E[X_\tau] \ge E[X_0]$.

## Remarks

The [[stopped process|NG35MGYDZR]] $(X_{n \wedge \tau})_{n \ge 0}$ is again a martingale, so $E[X_{n \wedge \tau}] = E[X_0]$ for every $n$.
Each hypothesis lets one pass to the limit $n \to \infty$.
In case 1 take $n = N$.
In cases 2 and 3 use dominated convergence, with the bound $\abs{X_0} + \sum_{k < \tau} \abs{X_{k+1} - X_k}$ in case 2 and the constant $c$ in case 3.
Example: let $S_n$ be the simple symmetric random walk from $0$, let $a, b$ be positive integers, and let $\tau$ be the first time $S_n \in \{-a, b\}$.
Case 3 applies, so $0 = E[S_\tau] = b\,P(S_\tau = b) - a\,(1 - P(S_\tau = b))$, which gives $P(S_\tau = b) = a/(a + b)$.
Some hypothesis is needed: if $\tau$ is the first time the same walk hits $1$, then $\tau < \infty$ almost surely but $E[S_\tau] = 1 \ne 0 = E[S_0]$; here $E[\tau] = \infty$ and the stopped walk is unbounded below.

## Explanations

### L1 human

In a fair game, no rule for deciding when to quit can give you an expected profit.
This is true as long as the rule only uses the past, and the game cannot drag on in an uncontrolled way.
The theorem makes "uncontrolled" precise with three alternative conditions.
Without such conditions, the strategy "play until you are one unit ahead" would beat a fair game, but it may take an unbounded time and unbounded credit.

### L2 human

The theorem is the main tool for computing hitting probabilities and expected hitting times of random walks and Brownian motion.
For example, applying it to the martingale $S_n^2 - n$ in the gambler's ruin gives $E[\tau] = ab$.
It gives a short proof of [[Doob's martingale inequality|4G53E7BRXF]].
In continuous time the same result holds for right-continuous martingales.
For uniformly integrable martingales it holds for all stopping times, with $X_\infty$ the limit from [[Doob's martingale convergence theorems|MRXBC4ZNRR]].
</document>

<document source="https://tamnd.github.io/math2/node/DCCA2T9GKH.md" tag="DCCA2T9GKH" kind="concept">
# Fejér kernel

## Summary

The Fejér kernel $F_N$ is the average of the first $N$ Dirichlet kernels, the trigonometric polynomials that produce the partial sums of a [[Fourier series|G12KXDHPPS]].
Convolving a periodic function with $F_N$ gives the $N$-th [[Cesàro mean|08R2V9GZF7]] of its Fourier partial sums.
Unlike the Dirichlet kernel, the Fejér kernel is nonnegative, has total mass $1$ and concentrates near $0$.
These three properties are the heart of [[Fejér's theorem|TPJZCSXSVN]].

## Explanations

### L1 human

A Fourier partial sum is an average of the function against a weight, the Dirichlet kernel, which takes negative values and oscillates strongly.
Averaging these weights over the first $N$ steps gives the Fejér kernel, which is never negative.
Its graph is a tall narrow bump at $0$ with small ripples elsewhere, and the bump gets narrower as $N$ grows.
So convolving with it computes a local average of the function, which approaches the value of the function at each point of continuity.
</document>

<document source="https://tamnd.github.io/math2/node/DCH91F9YDK.md" tag="DCH91F9YDK" kind="definition">
# de Casteljau's algorithm

## Definition

Let $n \ge 1$, let $P_0, \dots, P_n$ be points of $\R^d$ (or real numbers), and let $t \in \R$.
*De Casteljau's algorithm* computes points $P_i^{(j)}$ for $0 \le j \le n$ and $0 \le i \le n - j$ by

1. $P_i^{(0)} = P_i$ for $i = 0, \dots, n$;
2. $P_i^{(j)} = (1 - t)\, P_i^{(j-1)} + t\, P_{i+1}^{(j-1)}$ for $j = 1, \dots, n$ and $i = 0, \dots, n - j$;

and returns $P_0^{(n)}$.

## Remarks

The output equals $B(t) = \sum_{k=0}^{n} \binom{n}{k} (1-t)^{n-k} t^k P_k$, the point of the [[Bézier curve|NEXBTGYYRG]] with these control points; by induction on $j$, $P_i^{(j)} = \sum_{k=0}^{j} \binom{j}{k} (1-t)^{j-k} t^k P_{i+k}$.
For $t \in [0, 1]$ each step is a [[convex combination|21SXYJYC59]].
Example: $n = 2$, $P_0 = (0, 0)$, $P_1 = (1, 2)$, $P_2 = (2, 0)$, $t = 1/2$: then $P_0^{(1)} = (1/2, 1)$, $P_1^{(1)} = (3/2, 1)$ and $P_0^{(2)} = (1, 1)$, which is $B(1/2)$.
Non-example: replacing step 2 by the average $\tfrac{1}{2}(P_i^{(j-1)} + P_{i+1}^{(j-1)})$ for all $t$ returns $B(1/2)$ no matter what $t$ is, so it does not compute $B(t)$ for $t \ne 1/2$.
The points $P_0^{(0)}, P_0^{(1)}, \dots, P_0^{(n)}$ and $P_0^{(n)}, P_1^{(n-1)}, \dots, P_n^{(0)}$ are the control points of the curve restricted to $[0, t]$ and to $[t, 1]$, reparametrized to $[0, 1]$.
The algorithm costs $O(n^2)$ operations, more than Horner's scheme, but it is more stable.
</document>

<document source="https://tamnd.github.io/math2/node/DCJTJXN3SR.md" tag="DCJTJXN3SR" kind="definition">
# Favard constant

## Definition

For an integer $r \ge 0$, the *Favard constant* $K_r$ is the number
$$K_r = \frac{4}{\pi} \sum_{k=0}^{\infty} \left(\frac{(-1)^k}{2k + 1}\right)^{r + 1}.$$

## Remarks

The series converges for every $r \ge 0$: for $r = 0$ it is an alternating series, and for $r \ge 1$ it converges absolutely.

Example: $K_0 = \frac{4}{\pi} \left(1 - \frac{1}{3} + \frac{1}{5} - \cdots\right) = \frac{4}{\pi} \cdot \frac{\pi}{4} = 1$.
Likewise $K_1 = \frac{4}{\pi} \sum_{k \ge 0} \frac{1}{(2k+1)^2} = \frac{4}{\pi} \cdot \frac{\pi^2}{8} = \frac{\pi}{2}$.
Further values are $K_2 = \pi^2/8$ and $K_3 = \pi^3/24$.

Non-example: $\frac{4}{\pi} \sum_{k \ge 0} \frac{1}{2k + 1}$ is not a Favard constant; it lacks the signs of the case $r = 0$ and the series diverges.

The even-indexed constants increase and the odd-indexed constants decrease, and both tend to $4/\pi$.
So $1 = K_0 < K_2 < K_4 < \cdots < \frac{4}{\pi} < \cdots < K_3 < K_1 = \frac{\pi}{2}$.
</document>

<document source="https://tamnd.github.io/math2/node/DCJW3WZ7ZJ.md" tag="DCJW3WZ7ZJ" kind="example">
# Ricker wavelet

## Example

Fix a width $\sigma > 0$.
The *Ricker wavelet*, also called the Mexican hat wavelet, is
$$\psi(t) = \frac{2}{\sqrt{3\sigma} \, \pi^{1/4}} \left(1 - \frac{t^2}{\sigma^2}\right) e^{-t^2/(2\sigma^2)}, \qquad t \in \R.$$
Up to a negative constant factor, it is the second derivative of the Gaussian $e^{-t^2/(2\sigma^2)}$.

Check of the zero mean, with $\sigma = 1$.
Since $\int_{\R} e^{-t^2/2} \, dt = \sqrt{2\pi}$ and $\int_{\R} t^2 e^{-t^2/2} \, dt = \sqrt{2\pi}$, the integral of $(1 - t^2) e^{-t^2/2}$ is $0$.
Check of the norm, with $\sigma = 1$.
The integral of $(1 - t^2)^2 e^{-t^2}$ is $\sqrt{\pi}\,(1 - 1 + 3/4) = 3\sqrt{\pi}/4$, and multiplying by $4/(3\sqrt{\pi})$ gives $\norm{\psi}_2^2 = 1$.
The general case follows by the substitution $t = \sigma s$.

## Remarks

The Fourier transform of $\psi$ is a constant times $\omega^2 e^{-\sigma^2\omega^2/2}$.
It vanishes to second order at $\omega = 0$, so $\psi$ satisfies the [[admissibility condition|2BBB8NY5ZA]] and has two [[vanishing moments|7QERXM0C52]]: $\int \psi = 0$ and $\int t \psi(t) \, dt = 0$, while $\int t^2 \psi(t) \, dt \ne 0$.
It does not generate an orthonormal basis; it is used in the [[continuous wavelet transform|57FR21NDGV]].
In two variables, the analogous function is the negative Laplacian of a Gaussian, which detects blobs and edges in images.
The Ricker wavelet is also a standard model of a seismic source pulse.

## Explanations

### L1 human

Draw a bell curve, push its sides down below zero, and keep the middle as a bump: that is the Mexican hat shape.
The positive bump and the two negative dips balance, so the total area is zero.
Sliding this shape along a signal gives a strong response where the signal has a peak of matching width.
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<document source="https://tamnd.github.io/math2/node/DCK63WTY5E.md" tag="DCK63WTY5E" kind="statement">
# Marchenko–Pastur law

## Statement

Let $(Y_{ij})_{i, j \ge 1}$ be [[independent and identically distributed|00MRN0FARV]] real random variables with mean $0$ and variance $1$.
Let $p = p(n)$ be positive integers with $p / n \to y \in (0, \infty)$ as $n \to \infty$.
Let $X_n$ be the $p \times n$ matrix $(Y_{ij})_{i \le p, j \le n}$, let $S_n = \frac1n X_n X_n^{\mathsf{T}}$, and let $\mu_n = \frac1p \sum_{k=1}^p \delta_{\lambda_k}$, where $\lambda_1, \ldots, \lambda_p$ are the [[eigenvalues|0BQAA2BK3Q]] of $S_n$.
Let $a = (1 - \sqrt y)^2$ and $b = (1 + \sqrt y)^2$.
Then, almost surely, $\mu_n$ [[converges weakly|0SRYJGXKR4]] to the Marchenko–Pastur distribution with ratio $y$, the probability measure
$$\nu_y(dx) = \frac{1}{2\pi x y} \sqrt{(b - x)(x - a)} \, \mathbf{1}_{[a, b]}(x) \, dx + \Big(1 - \frac1y\Big)^{+} \delta_0(dx),$$
where $t^{+} = \max(t, 0)$.

## Remarks

The proof can use the moment method or the Stieltjes transform.
The Stieltjes transform $m(z) = \int \frac{\nu_y(dx)}{x - z}$ of the limit solves the quadratic equation $y z m^2 + (z + y - 1) m + 1 = 0$, and inverting it gives the density.
When $y > 1$ the matrix $S_n$ has rank at most $n < p$, which explains the atom of mass $1 - 1/y$ at $0$.
The limit has mean $1$ and second moment $1 + y$.
Example: for $y = 1$ the support is $[0, 4]$ and the density is $\frac{1}{2\pi} \sqrt{(4 - x)/x}$.

## Explanations

### L1 human

Take $n$ independent samples of a vector of $p$ independent standard measurements and form the sample covariance matrix.
If $p$ is small compared with $n$, its eigenvalues are all close to $1$, the true value.
If $p$ and $n$ are both large and comparable, the eigenvalues spread over an interval around $1$ with a definite shape, the Marchenko–Pastur law.

### L2 human

It is the analogue of the [[Wigner semicircle law|3QTT95DMFH]] for sample covariance matrices.
In high-dimensional statistics it shows that the sample covariance matrix is a poor estimate of the true covariance when $p/n$ is not small.
Eigenvalues of data that fall outside $[a, b]$ are read as signals that are not pure noise.
</document>

<document source="https://tamnd.github.io/math2/node/DCK64GFPD7.md" tag="DCK64GFPD7" kind="concept">
# f-vector and h-vector

## Summary

The f-vector of a [[simplicial complex|V8GZ0W1ZZ5]] or of a polytope lists the number of its faces of each dimension.
The h-vector is an invertible linear transform of the f-vector that often has simpler properties.
It is symmetric for simplicial spheres and nonnegative for many important complexes.
The [[Dehn–Sommerville equations|4WKHF5BCTA]] and the [[g-theorem|4CZMWQVC30]] are stated in terms of the h-vector.

## Explanations

### L1 human

Count the vertices, edges, triangles, and so on of a shape built from simplices, and you get its f-vector.
For the surface of an octahedron the counts are $6$ vertices, $12$ edges and $8$ triangles.
Rewriting these numbers as an h-vector gives $(1, 3, 3, 1)$, which reads the same forwards and backwards.
This hidden symmetry is one reason to use the h-vector.
</document>

<document source="https://tamnd.github.io/math2/node/DCMFQ01QGS.md" tag="DCMFQ01QGS" kind="concept">
# Newtonian potential

## Summary

The Newtonian potential of a mass distribution $\mu$ in $\R^n$, $n \ge 3$, is the function $U^\mu(x) = \int \abs{x - y}^{2-n} \, d\mu(y)$.
It is the [[potential of the measure|JHKSF3XFY2]] $\mu$ with respect to the [[Newtonian kernel|6TJTWQ5HCR]].
In $\R^3$ it is the gravitational potential of Newton's theory, up to the gravitational constant and a sign.
It is harmonic off the support of $\mu$.
In the sense of distributions it solves Poisson's equation $\Delta U^\mu = -(n-2)\sigma_n \mu$, where $\sigma_n$ is the area of the unit sphere.

## Explanations

### L1 human

Newton's law says that a point mass $m$ at $y$ creates the potential $m / \abs{x - y}$ at the point $x$ of space.
A planet, a star or a cloud of dust is a sum of many small masses.
Its Newtonian potential is the sum, or integral, of all these contributions.
For a uniform ball of mass $M$, the potential outside the ball is the same as that of a point mass $M$ at the centre.
</document>

<document source="https://tamnd.github.io/math2/node/DCMP1A0138.md" tag="DCMP1A0138" kind="concept">
# Duality in projective geometry

## Summary

In a [[projective plane|DANFNN8C4Y]], points and lines play symmetric roles: two points lie on one line, and two lines meet in one point.
Duality is the exchange of the words "point" and "line", together with "lies on" and "passes through".
The principle of duality says that the dual of a theorem valid in all projective planes is again a theorem valid in all projective planes.
In a projective space of dimension $n$, duality exchanges subspaces of dimension $d$ and $n - 1 - d$.

## Explanations

### L1 human

Take any true statement about points and lines of the projective plane and swap the two words.
"Two points determine a line" becomes "two lines determine a point", which is also true in the projective plane.
So every theorem comes with a free twin.
For example, [[Pascal's theorem|94AKWRHND0]] about six points on a conic has [[Brianchon's theorem|3THE3Q96KK]] about six lines tangent to a conic as its dual twin.
</document>

<document source="https://tamnd.github.io/math2/node/DCPFVVZZ54.md" tag="DCPFVVZZ54" kind="statement">
# Schauder fixed-point theorem

## Statement

Let $X$ be a real [[normed vector space|03J5RAKB30]] and let $K \subseteq X$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].
Let $f \colon K \to K$ be a [[continuous map|0X3WGX95FE]].
Then $f$ has a fixed point: there is $x \in K$ with $f(x) = x$.

## Remarks

The proof reduces to the [[Brouwer fixed-point theorem|N7AAJ2MW5T]].
For $\varepsilon > 0$, compactness gives a finite $\varepsilon$-net $\set{y_1, \ldots, y_m}$ of $K$.
The Schauder projection maps $K$ continuously into the convex hull of the $y_i$ and moves each point by less than $\varepsilon$.
Brouwer's theorem gives a fixed point of the projected map on this finite-dimensional convex set, and it is an $\varepsilon$-fixed point of $f$.
A limit as $\varepsilon \to 0$ gives a fixed point of $f$.

An equivalent form is often used in analysis.
Let $C$ be a nonempty closed convex subset of a Banach space and let $f \colon C \to C$ be continuous with $f(C)$ contained in a compact set; then $f$ has a fixed point.

Example: Peano's existence theorem for $y' = F(t, y)$, $y(0) = y_0$ with continuous $F$ follows.
The integral operator $y \mapsto y_0 + \int_0^t F(s, y(s)) \, ds$ maps a closed convex set of continuous functions into a compact set, by the Arzelà–Ascoli theorem.
Its fixed point is a solution.

Compactness cannot be dropped: on the closed unit ball of $\ell^2$, Kakutani's map $x \mapsto \big(\sqrt{1 - \norm{x}^2}, x_1, x_2, \ldots\big)$ is continuous with no fixed point.

## Explanations

### L1 human

Brouwer's theorem says that a continuous map of a disc or ball into itself fixes some point.
Schauder's theorem extends this to infinite-dimensional spaces, such as spaces of functions, provided the set is compact and convex.
It turns many existence problems for differential and integral equations into the search for a fixed point.

### L2 human

The theorem is the main tool for nonlinear problems without uniqueness, where the Banach fixed point theorem does not apply.
Compactness usually comes from an Arzelà–Ascoli argument or a compact Sobolev embedding.
Tychonoff extended it to compact convex subsets of locally convex spaces, and [[Schaefer's theorem|7H3YSSDEBK]] replaces the invariant set by an a priori bound.
For set-valued maps the analogue is the [[Kakutani fixed-point theorem|7V3HWGR32Z]] and its infinite-dimensional versions.
</document>

<document source="https://tamnd.github.io/math2/node/DCSXXT4SER.md" tag="DCSXXT4SER" kind="concept">
# Congruence relation

## Summary

A congruence relation on an [[algebra|QJGPT2W0FT]] is an [[equivalence relation|0G8VEZ3WTK]] that is compatible with all the operations.
Replacing elements by congruent ones never changes the class of the result.
Congruences are exactly the relations that allow a [[quotient algebra|N5KE7FH6CX]], and they are exactly the kernels of homomorphisms.
In groups they correspond to normal subgroups, and in rings to two-sided ideals.

## Explanations

### L1 human

A congruence is a way of declaring some elements "the same" so that the operations still make sense on the classes.
The model is [[congruence modulo n|0WAY5WX8NY]] on the integers: if $a \equiv a'$ and $b \equiv b' \pmod n$, then $a + b \equiv a' + b'$ and $ab \equiv a'b'$.
So one can add and multiply remainders, which gives the ring $\Z/n\Z$.
Not every equivalence relation works: grouping integers by sign is not compatible with addition, since $1 \sim 2$ but $1 + (-1) = 0$ and $2 + (-1) = 1$ have different signs.
</document>

<document source="https://tamnd.github.io/math2/node/DCXJGS6E6D.md" tag="DCXJGS6E6D" kind="statement">
# FKG inequality

## Statement

Let $L$ be a finite [[distributive lattice|0HM0HQCK6N]] with join $\vee$ and meet $\wedge$.
Let $\mu \colon L \to [0, \infty)$ be log-supermodular, that is, $\mu(x \vee y) \, \mu(x \wedge y) \ge \mu(x) \, \mu(y)$ for all $x, y \in L$.
Let $f, g \colon L \to \R$ be [[increasing|0A7KV2WB7A]]: $x \le y$ implies $f(x) \le f(y)$ and $g(x) \le g(y)$.
Then
$$\Big( \sum_{x \in L} f(x) g(x) \mu(x) \Big) \Big( \sum_{x \in L} \mu(x) \Big) \ge \Big( \sum_{x \in L} f(x) \mu(x) \Big) \Big( \sum_{x \in L} g(x) \mu(x) \Big).$$

## Remarks

When $\mu$ is a probability measure, the inequality reads $\mathbb{E}[f g] \ge \mathbb{E}[f] \, \mathbb{E}[g]$: increasing functions are positively correlated.
It follows from the [[Ahlswede–Daykin inequality|4085B8A528]].
Adding a constant to $f$ or to $g$ does not change the difference of the two sides, so one may assume $f, g \ge 0$.
Then the four functions $f \mu$, $g \mu$, $f g \mu$ and $\mu$ satisfy $f(x) \mu(x) \, g(y) \mu(y) \le f(x \vee y) g(x \vee y) \mu(x \vee y) \, \mu(x \wedge y)$, by monotonicity and log-supermodularity, and Ahlswede–Daykin with both sets equal to $L$ gives the result.

Example: a product measure on the lattice of subsets of $\set{1, \ldots, n}$ satisfies the hypothesis with equality, and the theorem becomes the [[Harris inequality|W1YH3YD3S4]].
If $f$ is increasing and $g$ is decreasing, the inequality reverses.
Log-supermodularity cannot be dropped.
On $\set{0, 1}^2$, the measure giving mass $1/2$ to each of $(1, 0)$ and $(0, 1)$ is not log-supermodular.
For it, the increasing indicators of $\set{x_1 = 1}$ and $\set{x_2 = 1}$ are negatively correlated.

## Explanations

### L1 human

Imagine random configurations in which "more" of one thing tends to come with "more" of another.
The FKG inequality makes this precise: if the weights favour combining configurations, then any two increasing quantities are positively correlated.
For instance, in a random graph, being connected and containing a triangle tend to happen together.

### L2 human

The inequality was proved by Fortuin, Kasteleyn and Ginibre for models of statistical mechanics such as the Ising model with ferromagnetic interactions, whose Gibbs measures are log-supermodular.
In combinatorics and probability it is used to compare probabilities of monotone events in random graphs and percolation.
The Harris inequality is its product-measure case, and the Holley inequality is a two-measure generalization.
The XYZ inequality for linear extensions of posets is proved with it.
</document>

<document source="https://tamnd.github.io/math2/node/DD4D0EKX74.md" tag="DD4D0EKX74" kind="statement">
# Adams spectral sequence

## Statement

Let $p$ be a prime, let $\mathcal{A}$ be the mod $p$ [[Steenrod algebra|JWTTXSGD0E]], and write $H^*(-)$ for mod $p$ cohomology of [[spectra|1GHF4EKBXS]].
Let $X$ be a finite CW spectrum, and let $Y$ be a bounded-below spectrum whose homotopy groups are finitely generated in each degree.
For $n \in \Z$, let $[X, Y]_n$ be the group of maps $\Sigma^n X \to Y$ in the [[stable homotopy category|26GTNMAF7B]].
Then there is a [[spectral sequence|GG8AYV847V]] $(E_r^{s,t}, d_r)_{r \ge 2}$ with $s \ge 0$.
Its differentials are $d_r \colon E_r^{s,t} \to E_r^{s+r, t+r-1}$.
Its second page is
$$E_2^{s,t} = \operatorname{Ext}_{\mathcal{A}}^{s,t}\big(H^*(Y), H^*(X)\big).$$
Here $\operatorname{Ext}^{s,t}$ is the [[Ext group|4P7GBWJ4RR]] of $\mathcal{A}$-modules in homological degree $s$ and internal degree $t$.
The spectral sequence converges strongly to $[X, Y]_{t-s} \otimes \Z_p$, where $\Z_p$ is the ring of $p$-adic integers.

## Remarks

The construction resolves $Y$ by a tower of spectra built from mod $p$ Eilenberg–MacLane spectra, the Adams resolution, and applies $[X, -]_*$.
The filtration degree $s$ counts how many maps in the tower a class passes through.

The main case is $X = Y = S$, the [[sphere spectrum|14CGGHA85T]].
Then $E_2^{s,t} = \operatorname{Ext}_{\mathcal{A}}^{s,t}(\F_p, \F_p)$ and the abutment is the $p$-primary part of the [[stable homotopy groups of spheres|HGZAYPQBXW]] $\pi_{t-s}^s$ for $t - s > 0$.
For $p = 2$, the group $\operatorname{Ext}^{1,t}$ has a basis of classes $h_j$ with $t = 2^j$, one for each $j \ge 0$, coming from the indecomposable squares $Sq^{2^j}$.
The classes $h_0, h_1, h_2, h_3$ detect $2$, $\eta$, $\nu$ and $\sigma$, while $d_2(h_j) = h_0 h_{j-1}^2 \neq 0$ for $j \ge 4$; this is Adams' theorem on Hopf invariant one.

Without the finiteness hypotheses the spectral sequence still exists, but it may converge only to the homotopy of a $p$-completion of $Y$, or not converge at all.

## Explanations

### L1 human

Stable homotopy groups of spheres are very hard to compute directly.
The Adams spectral sequence starts from a purely algebraic object, built from the Steenrod operations on mod $p$ cohomology.
After a series of corrections, it gives the $p$-primary part of these groups.

### L2 human

The $E_2$ term can be computed by machine with minimal resolutions over the Steenrod algebra.
In a range of degrees the differentials are known, and this gives the stable stems far beyond what other methods reach.
Its algebraic structure, such as Massey products and the May spectral sequence for $E_2$, controls much of the computation.
The Adams–Novikov spectral sequence replaces mod $p$ cohomology by complex cobordism and is better suited for periodic phenomena.
</document>

<document source="https://tamnd.github.io/math2/node/DD66THZBDX.md" tag="DD66THZBDX" kind="statement">
# Finite speed of propagation

## Statement

Let $n \ge 1$ and let $u \in C^2(\R^n \times [0, \infty))$ solve the [[wave equation|5EHD2XBQDG]] $u_{tt} = \Delta u$ on $\R^n \times (0, \infty)$.
Let $x_0 \in \R^n$ and $t_0 > 0$.
Suppose that $u(x, 0) = u_t(x, 0) = 0$ for all $x$ with $\abs{x - x_0} \le t_0$.
Then $u(x, t) = 0$ for all $(x, t)$ in the cone $\set{(x, t) : 0 \le t \le t_0, \ \abs{x - x_0} \le t_0 - t}$.

## Remarks

The proof uses the [[energy method|G8J0YQRPA3]] on shrinking balls.
Let $e(t) = \frac{1}{2}\int_{B(x_0, t_0 - t)} \big(u_t^2 + \abs{\nabla u}^2\big)\, dx$ for $0 \le t \le t_0$.
Differentiating, integrating by parts and using $\abs{u_t\, \partial_\nu u} \le \frac{1}{2}(u_t^2 + \abs{\nabla u}^2)$ on the boundary sphere gives $e'(t) \le 0$.
Since $e(0) = 0$, it follows that $e \equiv 0$, so $u_t$ and $\nabla u$ vanish in the cone and $u$ equals its initial value $0$ there.
For the equation $u_{tt} = c^2 \Delta u$, the same holds with the cone $\abs{x - x_0} \le c(t_0 - t)$.
Example: in one dimension, d'Alembert's formula shows directly that $u(x, t)$ depends only on the data on $[x - t, x + t]$.
By linearity, two solutions whose data agree on $B(x_0, t_0)$ agree in the cone, so a change of the data travels with speed at most $1$.
The [[heat equation|VP2HN86VAG]] has no such property: a nonnegative nonzero initial temperature with compact support gives a positive temperature everywhere at every positive time.

## Explanations

### L1 human

Waves need time to travel.
If nothing is happening in a ball of radius $t_0$ at the start, then the centre stays undisturbed until time $t_0$, because no disturbance from outside can arrive sooner.
The undisturbed region is a cone in space-time that shrinks at the wave speed.
This is the mathematical form of the fact that sound or light has a finite speed.

### L2 human

Finite speed of propagation is the characteristic feature of [[hyperbolic equations|92C3S03AQZ]], in contrast with parabolic and elliptic ones.
It gives the domain of dependence of a point, the backward cone, and the domain of influence of a region of data, the forward cone.
The energy proof works for general second-order hyperbolic equations with variable coefficients, with the cone replaced by a region bounded by characteristic surfaces.
It also proves uniqueness for the Cauchy problem and allows one to solve the problem locally and patch the local solutions together.
</document>

<document source="https://tamnd.github.io/math2/node/DD8SAAED09.md" tag="DD8SAAED09" kind="statement">
# Bailey's lemma

## Statement

Let $q, a, \rho_1, \rho_2 \in \C \setminus \set{0}$ be such that no denominator below vanishes, and write $(c_1, \dots, c_k; q)_n = (c_1; q)_n \cdots (c_k; q)_n$ with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].
A pair of sequences $(\alpha_n)_{n \ge 0}$, $(\beta_n)_{n \ge 0}$ is a *Bailey pair relative to $a$* if
$$\beta_n = \sum_{r=0}^{n} \frac{\alpha_r}{(q; q)_{n-r}\, (aq; q)_{n+r}} \quad \text{for all } n \ge 0.$$
Put $c = \frac{aq}{\rho_1 \rho_2}$.
If $(\alpha_n, \beta_n)$ is a Bailey pair relative to $a$, then so is $(\alpha_n', \beta_n')$, where
$$\alpha_n' = \frac{(\rho_1, \rho_2; q)_n\, c^n}{(aq/\rho_1, aq/\rho_2; q)_n}\, \alpha_n,
\qquad
\beta_n' = \sum_{j=0}^{n} \frac{(\rho_1, \rho_2; q)_j\, (c; q)_{n-j}\, c^j}{(q; q)_{n-j}\, (aq/\rho_1, aq/\rho_2; q)_n}\, \beta_j.$$

## Remarks

Proof idea: insert the definition of $\beta_j$ into $\beta_n'$ and exchange the sums.
The inner sum over $j$ is a terminating balanced ${}_3\phi_2$ series, which the $q$-Pfaff–Saalschütz summation evaluates in closed form.
Example: the *unit Bailey pair* relative to $a$ is $\beta_n = \delta_{n0}$ with $\alpha_0 = 1$ and $\alpha_n = \frac{(1 - a q^{2n})(a; q)_n (-1)^n q^{\binom{n}{2}}}{(1 - a)(q; q)_n}$ for $n \ge 1$; applying the lemma to it produces new pairs.
Letting $\rho_1, \rho_2 \to \infty$ gives $c^n (\rho_1, \rho_2; q)_n \to a^n q^{n^2}$, and with $n \to \infty$ one gets the identity $\sum_{n \ge 0} a^n q^{n^2} \beta_n = \frac{1}{(aq; q)_\infty} \sum_{n \ge 0} a^n q^{n^2} \alpha_n$ for suitable pairs.
This limit, applied to a suitable pair with $a = 1$ or $a = q$ and combined with the [[Jacobi triple product|JFP2DGMNXT]], gives the [[Rogers–Ramanujan identities|M24H4GJPPW]].

## Explanations

### L1 human

A Bailey pair is two sequences linked by a fixed triangular sum.
Bailey's lemma is a machine: put one Bailey pair in, and a new Bailey pair comes out, with two free parameters.
Each pair gives an identity between a sum and, often, a product, so repeating the machine produces infinitely many identities.

### L2 human

Bailey proved the lemma in the 1940s, and Andrews later noticed that it can be iterated, which gives the Bailey chain.
The Bailey chain explains the Rogers–Ramanujan identities and their many generalizations, such as the Andrews–Gordon identities, in a uniform way.
The proof only needs the $q$-Pfaff–Saalschütz sum, the $q$-analogue of the [[Pfaff–Saalschütz theorem|9X6KG770GV]].
</document>

<document source="https://tamnd.github.io/math2/node/DDAHG7QRY8.md" tag="DDAHG7QRY8" kind="statement">
# Kleene fixed-point theorem

## Statement

Let $(P, \le)$ be an [[omega-complete partial order|T4BCQHR68Y]]: every increasing sequence $x_0 \le x_1 \le x_2 \le \cdots$ in $P$ has a [[supremum|0R3H5PANRQ]] in $P$.
Assume that $P$ has a [[least element|0AANKKNHS4]] $\bot$.
Let $f \colon P \to P$ be [[continuous|C6GAHS1Z2P]] in this sense: $f$ is monotone, and $f\left(\sup_n x_n\right) = \sup_n f(x_n)$ for every increasing sequence $(x_n)$.
Then $f$ has a [[least fixed point|DBD64TH52D]], and
$$\operatorname{lfp}(f) = \sup_{n \ge 0} f^n(\bot),$$
where $f^0(\bot) = \bot$ and $f^{n+1}(\bot) = f(f^n(\bot))$.

## Remarks

Proof idea: $\bot \le f(\bot)$, and monotonicity gives $f^n(\bot) \le f^{n+1}(\bot)$ by induction, so the iterates form an increasing sequence with supremum $x$.
Continuity gives $f(x) = \sup_n f^{n+1}(\bot) = x$.
If $y$ is any fixed point, then $\bot \le y$ and induction gives $f^n(\bot) \le f^n(y) = y$, so $x \le y$.
The same statement holds for a directed-complete partial order with least element and a Scott-continuous $f$.
Example: on the power set of $\N$, the map $f(S) = \set{0} \cup \set{n + 2 : n \in S}$ preserves unions of increasing sequences.
Its iterates from $\emptyset$ are $\set{0}$, $\set{0, 2}$, $\set{0, 2, 4}$, and so on, and their union, the even numbers, is the least fixed point.
Continuity cannot be dropped from the formula: on the chain $\N \cup \set{\omega, \omega + 1}$, the monotone map with $f(n) = n + 1$, $f(\omega) = \omega + 1$ and $f(\omega + 1) = \omega + 1$ has least fixed point $\omega + 1$, while $\sup_n f^n(0) = \omega$.

## Explanations

### L1 human

Start at the bottom and apply $f$ again and again.
The values climb up step by step, and their limit is the least fixed point of $f$.
This gives a way to compute least fixed points, not only to know that they exist.
It is how the meaning of a recursive program is defined: run the recursion $0, 1, 2, \ldots$ levels deep and take the limit.

### L2 human

The theorem is the main tool of denotational semantics, where recursive definitions are interpreted as least fixed points of continuous maps on domains.
Compared with the [[Knaster–Tarski theorem|RHXP3HA64M]], it needs less of the order, only suprema of increasing sequences, but more of the map, namely continuity, and it gives an explicit construction.
For a monotone map that is not continuous, the iteration has to continue transfinitely, through $f^\omega(\bot) = \sup_n f^n(\bot)$, $f^{\omega + 1}(\bot)$, and so on, until it stabilises.
In Datalog and in dataflow analysis on finite lattices, the iteration stops after finitely many steps.
</document>

<document source="https://tamnd.github.io/math2/node/DDH18HWKZ9.md" tag="DDH18HWKZ9" kind="statement">
# Earnshaw's theorem

## Statement

Let $U \subseteq \R^n$ be an open set, $n \ge 2$, and let $\varphi \colon U \to \R$ be a [[harmonic function|0DEB0JMN0W]].
Then $\varphi$ has no strict local minimum and no strict local maximum in $U$.
Consequently, consider a charged particle with charge $q$ that moves under the force $F = -q \nabla \varphi$.
No [[equilibrium point|0N8NGKDZDZ]] in $U$ is a strict local minimum of its potential energy $q\varphi$.
In this sense the particle has no stable equilibrium in $U$.

## Remarks

Proof: suppose that $\varphi(x) > \varphi(a)$ for all $x \ne a$ in a ball $B(a, \rho) \subseteq U$.
Take $0 < r < \rho$.
By the [[mean value property|0HYPR64JM2]], $\varphi(a)$ is the average of $\varphi$ over the sphere of radius $r$ about $a$.
But every value on that sphere is larger than $\varphi(a)$, a contradiction.
A strict local maximum is excluded in the same way, or by applying the argument to $-\varphi$.
In physics, $\varphi$ is the potential of charges that lie outside $U$; it is harmonic in the charge-free region $U$.
Example: in $\R^3$, with $e_1 = (1, 0, 0)$ and $e_2 = (0, 1, 0)$, let $\varphi(x) = 1/\abs{x - e_1} + 1/\abs{x + e_1}$ be the potential of two unit charges at $\pm e_1$.
The origin is an equilibrium point by symmetry.
Along the $x_1$-axis, $\varphi(t e_1) = 2/(1 - t^2)$ has a minimum at $t = 0$, but along the $x_2$-axis, $\varphi(t e_2) = 2/\sqrt{1 + t^2}$ has a maximum, so the origin is a saddle point.
The theorem is about potentials that are harmonic; diamagnetic materials, feedback control and time-varying fields are classical ways around it in physics.

## Explanations

### L1 human

You cannot hold a charged particle at rest in a stable way using fixed charges alone.
At any point where the forces balance, there is a direction in which a small push makes the particle run away.
Mathematically, a harmonic function cannot have a bowl-shaped minimum, because its value at a point is the average of its values around the point.

### L2 human

The theorem is a direct consequence of the mean value property, and it is a weak form of the strong maximum and minimum principles.
It applies to every inverse-square force law, so it also rules out stable static configurations of point masses under Newtonian gravity.
At an equilibrium of a harmonic potential, the Hessian has trace $0$, so it cannot be positive definite.
This is another way to see that an equilibrium cannot be a nondegenerate minimum.
</document>

<document source="https://tamnd.github.io/math2/node/DDHCMKHD2T.md" tag="DDHCMKHD2T" kind="statement">
# Newton's forward difference formula

## Statement

Let $M$ be an additive commutative monoid, let $G$ be an additive abelian group, let $h \in M$, and let $f \colon M \to G$ be a function.
Let $\Delta_h^k f$ be the [[$k$-th forward difference|E38801KVJ1]] of $f$ with step $h$.
Then for every $n \in \N$ and every $x \in M$,
$$f(x + nh) = \sum_{k=0}^{n} \binom{n}{k} \Delta_h^k f(x),$$
where $\binom{n}{k}$ is the [[binomial coefficient|092GEF38YD]].

## Remarks

The [[shift operator|91QMF4R1CY]] satisfies $E_h = I + \Delta_h$, and $I$ commutes with $\Delta_h$.
So the [[binomial theorem|0EHWPWGHB2]] gives $E_h^n = \sum_k \binom{n}{k} \Delta_h^k$, which is the formula applied at $x$.
Example: for $f(x) = x^2$, $h = 1$ and $x = 0$, the differences at $0$ are $f(0) = 0$, $\Delta f(0) = 1$, $\Delta^2 f(0) = 2$, so $f(n) = n + 2\binom{n}{2} = n + n(n-1) = n^2$.
Let $f \colon \R \to \R$ and $h \ne 0$, and replace the top entry $n$ of $\binom{n}{k}$ by a real number $s$.
This gives the Newton forward interpolation polynomial
$$p(x_0 + sh) = \sum_{k=0}^{n} \binom{s}{k} \Delta_h^k f(x_0), \qquad \binom{s}{k} = \frac{s(s-1)\cdots(s-k+1)}{k!}.$$
It is the unique polynomial of degree at most $n$ that agrees with $f$ at $x_0, x_0 + h, \dots, x_0 + nh$.

## Explanations

### L1 human

If you know the value of a function at one point and all its differences there, you can rebuild its values further along.
This is like Taylor's formula, with differences in place of derivatives and binomial coefficients in place of $s^k / k!$.
For a polynomial only finitely many differences are nonzero, so the formula recovers it from a finite table.

### L2 human

The formula and the [[expansion of iterated differences in shifts|24D1EGHP4G]] are inverse to each other: they express the binomial transform and its inverse.
In numerical analysis it gives the interpolation polynomial at equally spaced nodes, a special case of [[polynomial interpolation|SS1YMS2NPC]].
Letting $n \to \infty$ with real $s$ leads to the [[Newton series|Y2HXSEA2EE]] of $f$.
Over the $p$-adic integers, the same expansion with infinitely many terms is [[Mahler's theorem|D115PG3T7K]].
</document>

<document source="https://tamnd.github.io/math2/node/DDQ2WBQ9GB.md" tag="DDQ2WBQ9GB" kind="concept">
# Weyl's unitary trick

## Summary

Weyl's unitary trick proves facts about finite-dimensional representations of complex semisimple Lie groups and Lie algebras.
It does so by passing to a compact real form.
Representations of the compact form can be averaged over the group, and this gives results such as complete reducibility.
The trick then carries the results back, because the complex Lie algebra and its compact real form have the same finite-dimensional complex representations.

## Explanations

### L1 human

The group $SL(2, \C)$ is not compact, so one cannot average over it.
But it contains the compact group $SU(2)$, and the Lie algebra of $SL(2, \C)$ is the complexification of the Lie algebra of $SU(2)$.
A subspace is invariant under $SL(2,\C)$ exactly when it is invariant under $SU(2)$, at least for holomorphic representations.
So one can prove that representations of $SL(2,\C)$ split into irreducible pieces by averaging over $SU(2)$, where the proof is easy.
</document>

<document source="https://tamnd.github.io/math2/node/DDQJGJGQGC.md" tag="DDQJGJGQGC" kind="statement">
# Cartan's theorem on continuous homomorphisms

## Statement

Let $G$ and $H$ be real [[Lie groups|7JBRGCVJRB]].
Let $\varphi \colon G \to H$ be a [[continuous group homomorphism|9J5PQMG3HJ]].
Then $\varphi$ is smooth; in fact it is real-analytic.

## Remarks

The graph $\Gamma = \set{(g, \varphi(g)) : g \in G}$ is a subgroup of $G \times H$, and it is closed since $\varphi$ is continuous.
By the [[closed subgroup theorem|3NJGY8QSTE]], $\Gamma$ is an embedded Lie subgroup.
The projection $\Gamma \to G$ is a bijective smooth homomorphism; such a map has constant rank, so it is a diffeomorphism, and $\varphi$ is its inverse followed by the smooth projection $\Gamma \to H$.
Example: a continuous homomorphism $\varphi \colon \R \to \R$ satisfies $\varphi(x + y) = \varphi(x) + \varphi(y)$ and is therefore $\varphi(x) = cx$, which is smooth.
Continuity is necessary: using a basis of $\R$ as a vector space over $\Q$, there are additive maps $\R \to \R$ that are not continuous, hence not smooth.

## Explanations

### L1 human

For a map between Lie groups that respects the group laws, mere continuity already forces smoothness.
So the smooth structure of a Lie group adds nothing beyond its topology and group law.

### L2 human

A direct consequence is that a topological group carries at most one smooth structure making it a Lie group.
Indeed, the identity map between two such structures is a continuous homomorphism, so it is a diffeomorphism.
This is why one can speak of "the" Lie group structure of a group such as $SO(n)$.
It also means continuous representations of Lie groups on finite-dimensional spaces are automatically smooth, so their differentials give representations of the Lie algebra.
The question of which topological groups admit a Lie group structure at all is Hilbert's fifth problem.
</document>

<document source="https://tamnd.github.io/math2/node/DDQRERAQCV.md" tag="DDQRERAQCV" kind="concept">
# Tor functor

## Summary

The Tor functors $\operatorname{Tor}_n^R(M, N)$ are the left derived functors of the [[tensor product|F0HQXQ0HCD]] $M \otimes_R N$.
They measure how far tensoring is from being exact.
The vanishing of $\operatorname{Tor}_1^R(M, -)$ characterizes the [[flat modules|73RYYEVDKA]], and Tor groups over $\Z$ appear in the universal coefficient and Künneth theorems of topology.

## Explanations

### L1 human

Tensoring with a module can turn an injective map into one that is not injective.
The Tor groups record what is lost.
To compute them, replace $M$ by a [[projective resolution|HK2AA1YSQ2]], tensor with $N$, and take homology.
For example, $\operatorname{Tor}_1^\Z(\Z/2\Z, \Z/2\Z) \cong \Z/2\Z$, because multiplication by $2$ is injective on $\Z$ but becomes the zero map on $\Z/2\Z$.
The name comes from torsion: $\operatorname{Tor}_1^\Z(\Z/n\Z, A)$ is the set of elements of the abelian group $A$ killed by $n$.
</document>

<document source="https://tamnd.github.io/math2/node/DDV6E39VSX.md" tag="DDV6E39VSX" kind="statement">
# Beurling's projection theorem

## Statement

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]] and let $E \subseteq \overline{\mathbb{D}}$ be a closed set with $0 \notin E$.
Let $E^* = \set{\abs{z} : z \in E} \subseteq (0, 1]$ be the circular projection of $E$ onto the positive radius.
For a closed set $A \subseteq \overline{\mathbb{D}}$ with $0 \notin A$, let $\Omega_A$ be the component of $\mathbb{D} \setminus A$ that contains $0$.
Let $\omega(0, A)$ be the [[harmonic measure|CFD7BPM51W]] of $A \cap \partial\Omega_A$ for $\Omega_A$ at the point $0$.
Then
$$\omega(0, E) \ge \omega(0, E^*).$$

## Remarks

The projection keeps, for each radius $r$, whether $E$ meets the circle $\abs{z} = r$, and squeezes the part of $E$ on that circle to a single point on the positive axis.
So among sets that meet the same circles, a radial slit is the hardest to hit from the centre.
Projecting onto any other radius gives the same value of $\omega(0, E^*)$, by rotation invariance.
Proofs use symmetrisation methods.
Example: if $E$ is a connected set that meets both circles $\abs{z} = r$ and $\abs{z} = 1$, then $E^* = [r, 1]$.
So $\omega(0, E)$ is at least the harmonic measure of the slit $[r, 1]$ in $\mathbb{D} \setminus [r, 1]$, which can be computed by a conformal map.
This gives Beurling's estimate that a Brownian path from $0$ hits $E$ before leaving the disc with probability bounded below in terms of $r$ only.

## Explanations

### L1 human

Place an obstacle inside a round room and release a random walker at the centre.
Compare obstacles that reach the same distances from the centre.
The theorem says that the obstacle squeezed onto a single radius is the one the walker is least likely to hit before it reaches the wall.
Spreading the obstacle around the circles can only make it easier to hit.

### L2 human

Beurling's projection theorem is a basic comparison tool for harmonic measure in the plane.
It gives lower bounds for the harmonic measure of connected sets.
These lead to estimates on the boundary behaviour of conformal maps and to Hölder bounds for harmonic measure of connected boundaries.
It is closely related to the Milloux problem, which asks for the same kind of extremal set.
</document>

<document source="https://tamnd.github.io/math2/node/DDY3Q3PJ6W.md" tag="DDY3Q3PJ6W" kind="example">
# Real closed field

## Example

A *real closed field* is an [[ordered field|0P3VCBDEE2]] $(R, <)$ with two properties.
Every positive element has a square root in $R$.
Every polynomial of odd degree with coefficients in $R$ has a root in $R$.
The [[real numbers|08J2M1JYAG]] form a real closed field.
A positive real number has a real square root.
A real polynomial of odd degree takes values of both signs, so it has a root by the intermediate value theorem.
The real algebraic numbers also form a real closed field, and it is countable.

The [[rational numbers|0NCJ5WXN6Z]] do not form a real closed field: $2 > 0$ has no rational square root.

## Remarks

The theory $\mathrm{RCF}$ of real closed fields in the signature $\set{+, -, \cdot, 0, 1, <}$ has [[quantifier elimination|FSJ9NGRRPA]].
So it is model complete, it is [[complete|655K8Y7VS8]], and it is decidable.
Every real closed field therefore satisfies the same first-order sentences as $\R$.
In the signature of rings without $<$, the theory is still model complete, but it no longer has quantifier elimination, since the order $x \ge 0$ is defined by $\exists y \, (x = y^2)$.

A field $R$ is real closed, for some ordering, exactly when $-1$ is not a sum of squares in $R$ and $R(\sqrt{-1})$ is algebraically closed.
The ordering is then unique: the positive elements are the nonzero squares.

## Explanations

### L1 human

Real closed fields are the fields that behave like the real numbers for all algebraic purposes.
Every positive number has a square root and odd-degree polynomials always have a root.
Take any statement written with $+$, $\cdot$, $<$ and quantifiers over the elements of the field.
Model theory shows that it is true in a real closed field exactly when it is true in $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/DE1Z7R9T05.md" tag="DE1Z7R9T05" kind="statement">
# Tameness theorem

## Statement

Let $M$ be a [[hyperbolic 3-manifold|GHVW1Z35BS]] whose [[fundamental group|S1XK3HXHFX]] is finitely generated.
Then $M$ is homeomorphic to the interior of a compact $3$-manifold.

## Remarks

The statement was conjectured by Marden and is also called the Marden conjecture.
It was proved in 2004 by Agol and, independently, by Calegari and Gabai.
Both proofs approximate the ends of $M$ by surfaces of controlled geometry, building on work of Canary and others on tame ends.
For example, if $M = \mathbb{H}^3 / \Gamma$ with $\Gamma$ a quasi-Fuchsian group isomorphic to the fundamental group of a closed surface $S$, then $M$ is homeomorphic to $S \times \R$, the interior of $S \times [0, 1]$.
The hypothesis that $\pi_1(M)$ is finitely generated is needed: there are hyperbolic $3$-manifolds with infinitely generated fundamental group that are not the interior of any compact manifold.
Among its consequences is the Ahlfors measure conjecture: the limit set of a finitely generated Kleinian group is either the whole sphere or has Lebesgue measure zero.

## Explanations

### L1 human

A hyperbolic $3$-dimensional space may be infinite, with ends that stretch off to infinity in complicated ways.
The theorem says that, when the fundamental group is finitely generated, these ends are as simple as possible topologically.
The whole space is a compact manifold with its boundary removed.

### L2 human

Tameness is the topological input for the classification of hyperbolic $3$-manifolds with finitely generated fundamental group.
Together with the ending lamination theorem and the density theorem, it gives a complete picture of the deformation space of Kleinian groups.
It also implies that such a manifold has finitely many ends, each homeomorphic to a surface times a half-line.
</document>

<document source="https://tamnd.github.io/math2/node/DE29ZC399Z.md" tag="DE29ZC399Z" kind="statement">
# Segal conjecture

## Statement

Let $G$ be a finite group, let $BG$ be its [[classifying space|7XRBR7AV3C]], and let $BG_+$ be $BG$ with a disjoint basepoint added.
Let $A(G)$ be the Burnside ring of $G$.
Let $I \subseteq A(G)$ be its augmentation ideal, the kernel of the ring map $A(G) \to \Z$ that sends a finite $G$-set to its size.
Let $\pi^0_s$ denote stable cohomotopy in degree $0$.
Then there is a natural isomorphism
$$\pi^0_s(BG_+) \cong A(G)^\wedge_I,$$
where $A(G)^\wedge_I = \varprojlim_k A(G)/I^k$ is the $I$-adic completion.

## Remarks

The map comes from the natural ring map $A(G) \to \pi^0_s(BG_+)$, which sends a finite $G$-set $S$ to the transfer of the covering $EG \times_G S \to BG$.
The theorem says that this map induces an isomorphism after $I$-adic completion.
Lin proved the case $G = \Z/2$ and Gunawardena the case $G = \Z/p$ for $p$ odd.
Carlsson proved the conjecture for all finite groups in 1984, after a reduction to elementary abelian $p$-groups.
The analogous statement for complex representation rings and K-theory, $K^0(BG) \cong R(G)^\wedge_I$, is the Atiyah–Segal completion theorem, which motivated the conjecture.

At work: for the trivial group, $A(G) = \Z$ and $I = 0$, and the statement is $\pi^0_s(S^0) \cong \Z$, the degree of stable self-maps of spheres.

## Explanations

### L1 human

The Burnside ring of a finite group is built from the finite sets on which the group acts.
Stable cohomotopy is the most basic generalized cohomology theory, built from maps to spheres.
The Segal conjecture says that the stable cohomotopy of the classifying space of $G$ is the Burnside ring, after a completion.
So pure algebra about $G$-sets computes a hard homotopy-theoretic invariant.

### L2 human

The theorem has a stronger equivariant form: the map from the fixed points of the $G$-equivariant sphere spectrum to its homotopy fixed points becomes an equivalence after completion at $I$.
It gives a description of the stable maps between classifying spaces of finite groups, and it is a model for later completion and descent theorems in equivariant homotopy theory.
It is parallel to the [[Sullivan conjecture|9EA7KJ50AE]], which compares fixed points and homotopy fixed points for spaces instead of spectra.
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<document source="https://tamnd.github.io/math2/node/DE2KEQ85ZT.md" tag="DE2KEQ85ZT" kind="concept">
# Inverse limit of algebras

## Summary

The inverse limit of an inverse system of algebras is the algebra of all compatible families of elements.
Such a family has one element in each algebra of the system.
It is a subalgebra of the [[direct product|WJ73XYBNE4]] of the algebras.
It is the limit of the system in the category of algebras of the given signature.
Inverse limits build large algebras, such as the $p$-adic integers and profinite groups, out of a coherent family of smaller ones.

## Explanations

### L1 human

Picture a tower of algebras, each one mapping to the one below it by a [[homomorphism|KBXSZC6BV1]].
A point of the inverse limit is a choice of one element on every floor, such that each choice maps to the choice on the floor below.
For example, take the rings $\Z/2\Z$, $\Z/4\Z$, $\Z/8\Z$, and so on, with reduction maps between them.
A compatible family such as $(1, 3, 7, 15, \ldots)$ is an element of the inverse limit, which is the ring of $2$-adic integers.
</document>

<document source="https://tamnd.github.io/math2/node/DE42T92C4R.md" tag="DE42T92C4R" kind="definition">
# Hilbert sum

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $(H_i)_{i \in I}$ be a family of [[Hilbert spaces|HM2HHE3J51]] over $\mathbb{K}$.
The *Hilbert sum* $\widehat{\bigoplus}_{i \in I} H_i$ is the set of families $x = (x_i)_{i \in I}$ with $x_i \in H_i$ for each $i$ and $\sum_{i \in I} \norm{x_i}^2 < \infty$.
Its vector operations are componentwise, and its inner product is
$$\ip{x}{y} = \sum_{i \in I} \ip{x_i}{y_i}.$$
With this inner product it is a Hilbert space.

A Hilbert space $H$ is the *internal Hilbert sum* of a family $(V_i)_{i \in I}$ of closed subspaces when these two conditions hold:

1. $V_i \perp V_j$ for all $i \neq j$.
2. The linear span of $\bigcup_{i \in I} V_i$ is dense in $H$.

## Remarks

The series defining $\ip{x}{y}$ converges absolutely, because $\abs{\ip{x_i}{y_i}} \le \tfrac12 (\norm{x_i}^2 + \norm{y_i}^2)$.
If $H$ is the internal Hilbert sum of $(V_i)$, then $x \mapsto (P_i x)_{i}$, where $P_i$ is the orthogonal projection onto $V_i$, is a unitary map from $H$ onto $\widehat{\bigoplus} V_i$.

Example: if every $H_i$ is $\mathbb{K}$, the Hilbert sum is $\ell^2(I)$.
For each $i$, let $e_i$ be the family with $1$ at $i$ and $0$ elsewhere; the lines $\mathbb{K} e_i$ are pairwise orthogonal and their span, the finitely supported families, is dense.

Non-example: for infinite $I$ and all $H_i \neq 0$, the [[external direct sum|0S7W6EG32V]] $\bigoplus_{i} H_i$ of finitely supported families is not complete.
So it is not a Hilbert space for the same inner product.
The Hilbert sum is its completion.

For a closed subspace $V$ of $H$, the space $H$ is the internal Hilbert sum of $V$ and its [[orthogonal complement|0XJXTBCB2T]] $V^\perp$.
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<document source="https://tamnd.github.io/math2/node/DED008EJVH.md" tag="DED008EJVH" kind="concept">
# Möbius plane

## Summary

A Möbius plane is an incidence structure of points and circles that behaves like the circles on a sphere.
Any three points lie on exactly one circle, and circles can touch.
The classical example is the plane with one point at infinity, where circles and lines through infinity are the circles.
Möbius planes are one of the three kinds of [[Benz plane|BG5A4CB6NB]], and they give the axiomatic setting of [[inversive geometry|HFBW3SXQVM]].

## Explanations

### L1 human

On a sphere, any three points lie on exactly one circle.
Two circles can meet in two points, in no point, or touch in one point.
A Möbius plane keeps exactly these facts as axioms and forgets the rest.
Stereographic projection turns the sphere into the plane plus one point at infinity, and circles through that point become straight lines.
</document>

<document source="https://tamnd.github.io/math2/node/DED6JFAV1K.md" tag="DED6JFAV1K" kind="concept">
# Complete statistic

## Summary

A [[statistic|0Q5CG2NEQ8]] is complete when no non-trivial function of it has expectation zero under every parameter value.
Completeness means that the family of laws of the statistic is rich enough to separate functions of it.
It is the key hypothesis of the [[Lehmann–Scheffé theorem|QQZPWDWS29]] and of [[Basu's theorem|6GATRFMJYY]].
By Lehmann–Scheffé, an unbiased estimator that is a function of a complete [[sufficient statistic|WP4TGXJQJG]] is the unique best one.

## Explanations

### L1 human

Suppose a function of the statistic averages to zero whatever the parameter is.
For a complete statistic, this can only happen if the function is zero.
So two unbiased estimators that are both functions of a complete statistic must agree.
For example, the number of heads in $n$ tosses of a coin with any chance $p$ is complete, because a polynomial in $p$ that vanishes for all $p$ is zero.
</document>

<document source="https://tamnd.github.io/math2/node/DEHZAQ1N92.md" tag="DEHZAQ1N92" kind="definition">
# c-cyclical monotonicity

## Definition

Let $X$ and $Y$ be sets and let $c \colon X \times Y \to (-\infty, \infty]$ be a function, the cost.
A set $\Gamma \subseteq X \times Y$ is *$c$-cyclically monotone* if for every integer $N \ge 1$ and all pairs $(x_1, y_1), \dots, (x_N, y_N) \in \Gamma$,
$$\sum_{i=1}^{N} c(x_i, y_i) \le \sum_{i=1}^{N} c(x_i, y_{i+1}),$$
where $y_{N+1} = y_1$.
Let $X$ and $Y$ be topological spaces, and let $\pi$ be a Borel measure on $X \times Y$, such as a [[coupling|HTR7J0W66V]].
Then $\pi$ is *$c$-cyclically monotone* if it is concentrated on a $c$-cyclically monotone Borel set.

## Remarks

Example: let $X = Y = \R$, $c(x, y) = (x - y)^2$ and $\Gamma = \set{(x, x) : x \in \R}$, the diagonal.
The left side of the inequality is $0$ and the right side is a sum of squares, so the inequality holds and $\Gamma$ is $c$-cyclically monotone.

Non-example: with the same cost, $\Gamma = \set{(0, 1), (1, 0)}$ fails for $N = 2$: the left side is $1 + 1 = 2$ and the right side is $c(0, 0) + c(1, 1) = 0$.

Equivalently, the inequality may be required for every permutation $\sigma$ of $\set{1, \dots, N}$, with $y_{\sigma(i)}$ in place of $y_{i+1}$, since every permutation is a product of disjoint cycles.
For $X = Y = \R^n$ and $c(x, y) = \frac12 \abs{x - y}^2$, a set is $c$-cyclically monotone exactly when it is cyclically monotone in the sense of Rockafellar, that is, $\sum_i \ip{y_i}{x_{i+1} - x_i} \le 0$ with $x_{N+1} = x_1$.
</document>

<document source="https://tamnd.github.io/math2/node/DEK2SCEDJ2.md" tag="DEK2SCEDJ2" kind="definition">
# Hilbert space adjoint

## Definition

Let $H$ and $K$ be [[Hilbert spaces|HM2HHE3J51]] over $\C$, with inner products $\ip{\cdot}{\cdot}$ linear in the first argument.
Let $T \colon H \to K$ be a [[continuous linear map|03TM135G0N]].
The *adjoint* of $T$ is the unique map $T^* \colon K \to H$ such that
$$\ip{Tx}{y}_K = \ip{x}{T^* y}_H \qquad \text{for all } x \in H \text{ and } y \in K.$$

## Remarks

For fixed $y$, the map $x \mapsto \ip{Tx}{y}$ is a bounded linear functional on $H$, so the [[Riesz representation theorem|TTC40N49M6]] gives a unique vector $T^* y$ representing it.
The map $T^*$ is linear and bounded, with [[operator norm|06EKW1AKXF]] $\norm{T^*} = \norm{T}$ and $\norm{T^* T} = \norm{T}^2$.
It satisfies $(S + T)^* = S^* + T^*$, $(aT)^* = \bar a T^*$, $(ST)^* = T^* S^*$ and $T^{**} = T$.
For finite-dimensional spaces it is the [[adjoint of a linear map|0R9NMSM1ZP]].

Example: let $S(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$ be the right shift on $\ell^2$, and $L(y_1, y_2, \ldots) = (y_2, y_3, \ldots)$ the left shift.
Then $\ip{Sx}{y} = \sum_{n \ge 2} x_{n-1} \overline{y_n} = \sum_{m \ge 1} x_m \overline{y_{m+1}} = \ip{x}{Ly}$, so $S^* = L$.

Non-example: $S$ itself does not satisfy the defining identity in place of $S^*$.
With unit vectors $e_n$, $\ip{S e_1}{e_2} = \ip{e_2}{e_2} = 1$, while $\ip{e_1}{S e_2} = \ip{e_1}{e_3} = 0$.
So $S^* \ne S$, and $S$ is not self-adjoint.

Over $\R$, the same definition applies, and then $(aT)^* = a T^*$.
</document>

<document source="https://tamnd.github.io/math2/node/DERJPQ04BQ.md" tag="DERJPQ04BQ" kind="concept">
# Nevanlinna theory in several variables

## Summary

Nevanlinna theory in several variables is the value distribution theory of holomorphic maps whose source or target has dimension greater than one.
Its model case is a holomorphic curve $f \colon \C \to \mathbb{P}^n$, studied against hyperplanes, and more generally holomorphic maps from $\C^m$ to projective varieties, studied against divisors.
It extends the one-variable [[Nevanlinna theory|1EHH63VNAD]], with a characteristic function, counting functions and proximity functions, and first and second main theorems.
It is closely tied to [[Kobayashi hyperbolicity|HRPAX8V7HA]] and, through an analogy found by Vojta, to Diophantine approximation.

## Explanations

### L1 human

One-variable Nevanlinna theory measures how often a meromorphic function on the plane hits each value.
In several variables the questions are the same, for maps into higher-dimensional spaces such as projective space.
For a curve in $\mathbb{P}^n$, the role of the values is played by hyperplanes, and one asks how often the curve meets each hyperplane.
Cartan's second main theorem says, roughly, that a curve not contained in any hyperplane cannot avoid too many hyperplanes in general position.
For $n = 1$ it gives Picard's theorem: a nonconstant meromorphic function on $\C$ omits at most two values.
</document>

<document source="https://tamnd.github.io/math2/node/DEV24TSWQZ.md" tag="DEV24TSWQZ" kind="definition">
# von Neumann regular ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity, not necessarily commutative.
$R$ is *von Neumann regular* if for every $a \in R$ there is $x \in R$ with $a x a = a$.

## Remarks

If $a x a = a$, then $e = a x$ is an [[idempotent|0N5NJZKKN3]] with $e R = a R$; so $R$ is von Neumann regular if and only if every principal right ideal is generated by an idempotent.
A commutative ring is von Neumann regular if and only if it is [[reduced|SZHY975FFX]] of [[Krull dimension|MVGBS3J1P7]] $0$, and if and only if every $R$-module is [[flat|73RYYEVDKA]].
Every [[semisimple ring|NY25HXA478]], for instance the ring of $n \times n$ [[matrices|0VD9ZSWWJG]] over a [[division ring|0EA5BTA0CX]], is von Neumann regular.

Example: a [[Boolean ring|0RCV1ZFP8G]] $R$, in which $a^2 = a$ for all $a$, is von Neumann regular with $x = a$: $a a a = a^2 = a$.
Example: a [[field|04C5ZBG1WT]] is von Neumann regular, with $x = a^{-1}$ for $a \ne 0$ and $x = 0$ for $a = 0$; so is any [[product|0QM2G72DXA]] of fields, componentwise.

Non-example: $\Z$ is not von Neumann regular.
For $a = 2$, the condition $2 x 2 = 2$ means $4x = 1$, which has no solution in $\Z$.
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<document source="https://tamnd.github.io/math2/node/DEYYNVCV9J.md" tag="DEYYNVCV9J" kind="concept">
# Fredholm equation of the second kind

## Summary

A Fredholm equation of the second kind is an integral equation $\varphi(x) - \lambda \int_a^b K(x, y) \varphi(y) \, dy = f(x)$ with fixed limits $a$ and $b$.
The unknown $\varphi$ appears both outside and under the integral.
It is the [[Fredholm integral equation|526TCVMSHP]] form of an [[integral equation of the second kind|8KJCN49V64]].
Boundary value problems for differential equations lead to it, and its theory is very complete.
For all but a discrete set of values of the parameter $\lambda$, it has exactly one solution for every right-hand side.

## Explanations

### L1 human

The unknown function equals a known function plus a weighted average of its own values.
So the equation describes a balance, such as the temperature at a point being fed by the temperature at all other points.
When $\lambda$ is small, the averaged part is only a small correction, and one can solve by repeated substitution.
For example, on $[0, 1]$ the equation $\varphi(x) - \int_0^1 x y \, \varphi(y) \, dy = x$ has the solution $\varphi(x) = 3x/2$.
</document>

<document source="https://tamnd.github.io/math2/node/DF5N0PK3ZS.md" tag="DF5N0PK3ZS" kind="statement">
# Monad from an adjunction

## Statement

Let $F \colon \mathcal{C} \to \mathcal{D}$ be [[left adjoint|TZHF6ZM92S]] to $G \colon \mathcal{D} \to \mathcal{C}$.
Let $\eta \colon \mathrm{Id}_{\mathcal{C}} \Rightarrow G F$ be the [[unit and $\varepsilon \colon F G \Rightarrow \mathrm{Id}_{\mathcal{D}}$ the counit|MYGWT305AB]] of the adjunction.
Then $(G F, \eta, G \varepsilon F)$ is a [[monad|FF6XXA6D1A]] on $\mathcal{C}$.
Here $G \varepsilon F$ is the natural transformation $G F G F \Rightarrow G F$ with components $G(\varepsilon_{F(X)})$.

## Remarks

The unit laws follow from the triangle identities $\varepsilon F \circ F \eta = \id_F$ and $G \varepsilon \circ \eta G = \id_G$.
Applying $G$ to the first gives $G\varepsilon F \circ G F \eta = \id_{G F}$, and whiskering the second by $F$ gives $G \varepsilon F \circ \eta G F = \id_{G F}$.
Associativity reduces to $\varepsilon \circ F G \varepsilon = \varepsilon \circ \varepsilon F G$, which is the naturality of $\varepsilon$.

Example: the free monoid functor from sets to monoids is left adjoint to the forgetful functor.
The monad $G F$ sends a set $X$ to the set of finite words in $X$, and its multiplication concatenates a word of words.
This is the list monad.

Dually, $(F G, \varepsilon, F \eta G)$ is a [[comonad|97NPWM3S6E]] on $\mathcal{D}$.
Conversely, every monad arises from some adjunction, for example from its Eilenberg–Moore category or its Kleisli category.
Different adjunctions can give the same monad.

## Explanations

### L1 human

When a free construction and a forgetful functor are adjoint, going free and then forgetting is a monad.
Its unit puts an element into the free object as a generator.
Its multiplication evaluates a free expression in free expressions as one free expression.
This is the main source of monads in practice.

### L2 human

This result is half of the link between adjunctions and monads.
The other half says that every monad comes from an adjunction, and that among these there is an initial one, the Kleisli adjunction, and a terminal one, the Eilenberg–Moore adjunction.
A right adjoint is called monadic when the comparison functor to the Eilenberg–Moore category is an equivalence.
Beck's monadicity theorem tells when this happens.
For example, the free group adjunction gives a monad on sets whose algebras are exactly the groups.
</document>

<document source="https://tamnd.github.io/math2/node/DF69BF6841.md" tag="DF69BF6841" kind="definition">
# Permutation pattern

## Definition

Let $k \le n$ be positive integers.
Write a [[permutation|0BQMWCNS8S]] $\pi$ of $\set{1, \ldots, n}$ in one-line notation $\pi(1) \pi(2) \cdots \pi(n)$, and let $\sigma$ be a permutation of $\set{1, \ldots, k}$.
The permutation $\pi$ *contains* $\sigma$ as a *pattern* if there are indices $i_1 < i_2 < \cdots < i_k$ such that, for all $a, b \in \set{1, \ldots, k}$,
$$\pi(i_a) < \pi(i_b) \iff \sigma(a) < \sigma(b).$$
Otherwise $\pi$ *avoids* $\sigma$.
The set of permutations of $\set{1, \ldots, n}$ that avoid $\sigma$ is written $\operatorname{Av}_n(\sigma)$.

## Remarks

Example: $\pi = 3\,1\,4\,2$ contains $\sigma = 312$.
Take the indices $1, 2, 4$, with values $3, 1, 2$; the first is the largest and the second is the smallest, as in $312$.

Non-example: $\pi = 3\,1\,4\,2$ avoids $123$.
Its increasing [[subsequences|0R8J3ZQMJS]] are $3\,4$, $1\,4$ and $1\,2$ and single entries, and none has length $3$.

For every $\sigma$ of length $3$, the number $\abs{\operatorname{Av}_n(\sigma)}$ is the [[Catalan number|0YWZJEABB2]] $C_n$.
Knuth showed that the permutations that can be sorted by a single stack are exactly those that avoid $231$.
The [[Stanley–Wilf conjecture|KWJS2QYJ1K]] gives an exponential bound on $\abs{\operatorname{Av}_n(\sigma)}$ for every $\sigma$.
</document>

<document source="https://tamnd.github.io/math2/node/DF7KYMAGTC.md" tag="DF7KYMAGTC" kind="statement">
# Bochner's formula

## Statement

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $u \colon M \to \R$ be a smooth function.
Write $\nabla u$ for the gradient of $u$ and $\nabla^2 u$ for its Hessian with respect to the [[Levi-Civita connection|CAZYGE1T4E]].
Write $\Delta u = \tr \nabla^2 u$ for the [[Laplace–Beltrami operator|F4NWZNTRMD]] and $\operatorname{Ric}$ for the [[Ricci curvature|4GTNM0WHDH]].
Then
$$\frac{1}{2} \Delta \abs{\nabla u}^2 = \abs{\nabla^2 u}^2 + \ip{\nabla u}{\nabla (\Delta u)} + \operatorname{Ric}(\nabla u, \nabla u).$$

## Remarks

The idea of the proof: compute at a point in normal coordinates.
The left side is $\sum_{i,j} (\nabla_i \nabla_j u)^2 + \sum_{i,j} \nabla_j u \, \nabla_i \nabla_i \nabla_j u$.
Commuting the covariant derivatives in the last term gives $\nabla_j \nabla_i \nabla_i u$ plus a curvature term, and that curvature term is exactly $\operatorname{Ric}(\nabla u, \nabla u)$.

Example: in $\R^n$ the Ricci term is $0$.
For $u(x) = \abs{x}^2 / 2$ one has $\nabla u = x$, $\nabla^2 u = \id$ and $\Delta u = n$, so the left side is $\frac{1}{2} \Delta \abs{x}^2 = n$ and the right side is $n + 0 + 0$.

A standard application: let $M$ be compact of dimension $n$ with $\operatorname{Ric} \ge (n-1) k \, g$ for some $k > 0$.
Integrating the formula for an eigenfunction shows that every nonzero eigenvalue $\lambda$ of $-\Delta$ satisfies $\lambda \ge n k$.
This is Lichnerowicz's eigenvalue estimate.

## Explanations

### L1 human

The formula compares how a function bends with how the space bends.
It says that the Laplacian of the squared size of the gradient splits into three parts.
They are the size of the second derivatives, a term with the Laplacian of the function, and a Ricci curvature term.
When the Ricci curvature is nonnegative, the squared length of the gradient of a harmonic function is subharmonic, and this gives strong control on the function.

### L2 human

Bochner's formula is the main identity behind the "Bochner technique": a curvature sign forces certain objects to be parallel or to vanish.
It is used in the Cheng–Yau gradient estimate, in the Laplacian comparison theorem, in the proof of the Cheeger–Gromoll splitting theorem and in eigenvalue estimates.
The same computation for 1-forms is the case of the Weitzenböck identity that leads to Bochner's vanishing theorem for the first Betti number.
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# Morin singularities

## Summary

The Morin singularities are the stable [[singularities|1JG3HFWX3T]] of smooth maps whose differential drops rank by exactly one.
For maps between manifolds of the same dimension $n$ they form one sequence, with one member for each order $k$ with $1 \le k \le n$.
It begins with the fold, the cusp and the swallowtail.
Morin gave their normal forms, and they are exactly the generic corank one singularities.
Their [[Thom–Boardman symbol|A9TYDPD5D7]] is $(1, 1, \ldots, 1, 0)$ with $k$ ones.

## Explanations

### L1 human

The simplest way a map can degenerate is to crush exactly one direction.
Among such degenerations, the fold of a sheet of paper is the mildest.
The next is the cusp, where a fold line ends.
In more dimensions there are further steps, such as the swallowtail, each one rarer than the last.
The Morin singularities are this whole ladder, written in simple coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/DFCJCRTTBR.md" tag="DFCJCRTTBR" kind="definition">
# Mixing

## Definition

Let $(X, \mathcal{B}, \mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \colon X \to X$ be a [[measure-preserving transformation|556WBDAHNP]].
The map $T$ is *mixing*, or *strongly mixing*, if for all $A, B \in \mathcal{B}$,
$$\lim_{n \to \infty} \mu(T^{-n} A \cap B) = \mu(A) \, \mu(B).$$

## Remarks

Example: let $T(x) = 2x \bmod 1$ on $[0,1)$ with Lebesgue measure $\mu$.
Let $A$ and $B$ be dyadic intervals of length $2^{-k}$, that is, of the form $[j 2^{-k}, (j+1) 2^{-k})$.
For $n \ge k$, the set $T^{-n} A$ is a union of $2^n$ intervals of length $2^{-n-k}$, one in each interval $[i 2^{-n}, (i+1) 2^{-n})$.
Exactly $2^{n-k}$ of them lie in $B$, so $\mu(T^{-n} A \cap B) = 2^{-2k} = \mu(A) \mu(B)$.
Dyadic intervals generate the Borel sets, and an approximation argument gives the limit for all $A$ and $B$.

Non-example: let $T(x) = x + \alpha \bmod 1$ with $\alpha$ irrational, and let $A = B = [0, 1/4)$.
The numbers $n \alpha \bmod 1$ come arbitrarily close to $0$ for infinitely many $n$, and for those $n$ the measure $\mu(T^{-n} A \cap A)$ is close to $1/4$.
So it does not tend to $1/16$, and the rotation is not mixing, although it is ergodic.

Taking $A = B$ with $T^{-1} A = A$ shows that a mixing map is ergodic: then $\mu(A) = \mu(A)^2$, so $\mu(A) \in \set{0, 1}$.
In terms of functions, $T$ is mixing if and only if $\int (f \circ T^n) \, g \, d\mu \to \int f \, d\mu \int g \, d\mu$ for all $f, g \in L^2(\mu)$.
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# Coherent ring

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
The ring $R$ is *coherent* if every [[finitely generated ideal|030TZ0DQWT]] of $R$ is [[finitely presented|3EH4C3M5GV]] as an $R$-[[module|09VDS2W1E8]].

## Remarks

Equivalently, for every $a_1, \dots, a_n \in R$, the module of relations $\set{(r_1, \dots, r_n) \in R^n : r_1 a_1 + \cdots + r_n a_n = 0}$ is finitely generated.
Example: every [[Noetherian ring|0MVH4Y1CRZ]] is coherent, because the module of relations is a submodule of $R^n$, which is a Noetherian module.
Non-example: let $R = k[x, y_1, y_2, \dots]/(x y_1, x y_2, \dots)$ over a field $k$.
The relations of the single element $x$ form the ideal $(y_1, y_2, \dots)$ of $R$, which is not finitely generated.
So the ideal $(x)$ is finitely generated but not finitely presented.
For a noncommutative ring one defines left coherent rings: every finitely generated left ideal is finitely presented.

## Explanations

### L1 human

A ring is coherent when every finite list of elements has a finite list of basic relations, from which all relations follow.
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<document source="https://tamnd.github.io/math2/node/DFM7NJTX3J.md" tag="DFM7NJTX3J" kind="definition">
# Minimal surface

## Definition

Let $S \subset \R^3$ be a [[regular surface|6080X3FAD4]] with [[mean curvature|PA78NFEPCF]] $H \colon S \to \R$.
The surface $S$ is *minimal* if $H(p) = 0$ for every point $p \in S$.

## Remarks

The [[catenoid|6NN1HP54GB]] $x(u, v) = (\cosh v \cos u, \cosh v \sin u, v)$ is minimal.
Its principal curvatures are $\pm 1/\cosh^2 v$, so $H = 0$.
The sphere of radius $r$ is not minimal: both principal curvatures equal $1/r$ for the inward normal, so $H = 1/r \neq 0$.

Equivalent forms:

1. $S$ is minimal if and only if, for every compact region $R \subset S$ and every smooth normal variation of $R$ fixed on $\partial R$, the first variation of area is zero.
2. If $x$ is a conformal parametrization of $S$, then $S$ is minimal if and only if each coordinate function of $x$ is harmonic.
3. The graph of a smooth function $u(x, y)$ is minimal if and only if $(1 + u_y^2) u_{xx} - 2 u_x u_y u_{xy} + (1 + u_x^2) u_{yy} = 0$, the [[minimal surface equation|3GV2E2YTZX]].

A minimal surface need not minimize area among all surfaces with the same boundary; it is only a critical point.

## Explanations

### L1 human

A minimal surface is a surface whose mean curvature vanishes everywhere.
At every point the bending in one principal direction cancels the bending in the other.
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<document source="https://tamnd.github.io/math2/node/DFNM3BGMAY.md" tag="DFNM3BGMAY" kind="statement">
# Mal'cev's characterization of quasivarieties

## Statement

Let $\sigma$ be an algebraic signature and let $K$ be a class of $\sigma$-algebras.
Then $K$ is a [[quasivariety|WRXEKT4NQS]] if and only if $K$ is closed under the following four constructions:

1. [[isomorphic|PZMGQMDN25]] copies;
2. [[subalgebras|7CHG6SD365]];
3. [[direct products|WJ73XYBNE4]] of arbitrary families, where the product of the empty family is the one-element algebra;
4. ultraproducts.

## Remarks

The forward direction is a check: a [[quasi-identity|94GVY3APPQ]] is preserved by subalgebras and by products, coordinate by coordinate, and by ultraproducts by Łoś's theorem.
For the converse, let $A$ satisfy every quasi-identity that holds in $K$, and let $a \ne b$ in $A$.
For each finite set $\Phi$ of equations true in $A$ among names of its elements, the quasi-identity "$\bigwedge \Phi \rightarrow a \approx b$" fails in $A$, so it fails in some $B_\Phi \in K$.
An ultraproduct of the algebras $B_\Phi$ over a suitable ultrafilter gives a member of $K$ with a homomorphism from $A$ that separates $a$ and $b$.
The product of these maps over all pairs $a \ne b$ embeds $A$ into a product of members of $K$, so $A \in K$.

The same argument shows that the quasivariety generated by any class $K$ is $\mathbf{ISPP_U}(K)$, the class of algebras isomorphic to subalgebras of products of ultraproducts of members of $K$.
Compare [[Birkhoff's HSP theorem|5083DHSK82]], where closure under homomorphic images is added and identities replace quasi-identities.

Closure under ultraproducts cannot be dropped.
Residually finite groups are closed under the first three constructions, but not under ultraproducts, and they do not form a quasivariety.

## Explanations

### L1 human

Varieties are the classes of algebras that you cannot leave by taking subalgebras, products or quotients.
Quasivarieties are described in the same way, with two changes.
Quotients are no longer allowed, and a new construction, the ultraproduct, is added.
So a property defined by conditional equations is exactly a property that survives these four constructions.

### L2 human

The theorem lets one show that a class is a quasivariety without finding its axioms, and that a class is not a quasivariety by exhibiting a failure of closure.
In the theory of finite algebras, the quasivariety generated by one finite algebra $M$ is $\mathbf{ISP}(M)$, because ultraproducts of copies of a finite algebra are isomorphic to it.
This class is the starting point of the theory of [[natural dualities|W5VG6HS8KW]].
In abstract algebraic logic, the algebraic counterparts of many deductive systems are quasivarieties, and the theorem is the tool to recognise them.
</document>

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# Critical point theory

## Definition

Let $M$ be a $C^1$ [[Banach manifold|9K15AMGBV9]], for example a finite-dimensional smooth manifold.
Let $f \colon M \to \R$ be a $C^1$ function, with differential $df_p$ at $p \in M$ in the sense of the [[Fréchet derivative|06EH97V2A7]].

1. A [[critical point|0YTCMFB4HC]] of $f$ is a point $p \in M$ with $df_p = 0$.
2. A *critical value* of $f$ is a number $c = f(p)$ with $p$ a critical point; other real numbers are *regular values*.
3. For $c \in \R$ the *sublevel set* is $M^c = \set{p \in M : f(p) \le c}$.

*Critical point theory* studies the critical points and critical values of such $f$ through the topology of $M$ and of the sublevel sets $M^c$.
A central topic is the change of $M^c$ as $c$ passes a critical value.

## Remarks

The basic mechanism is a deformation along a gradient flow.
If $[a, b]$ contains no critical value and a compactness condition holds, it shows that $M^a$ is a deformation retract of $M^b$.
So the topology of the sublevel sets changes only at critical values.

Example: let $M = S^1 \subset \R^2$ and $f(x, y) = y$.
The critical points are $(0, 1)$ and $(0, -1)$, with critical values $1$ and $-1$.
The sublevel set $M^c$ is empty for $c < -1$, a point for $c = -1$, an arc for $-1 < c < 1$, and the whole circle for $c \ge 1$.
So its topology changes exactly at the two critical values.

Non-example: for $f(x) = e^x$ on $\R$, every real number is a regular value and $f$ has no critical point.
The sublevel set $M^c$ is empty for $c \le 0$ and is the half-line $(-\infty, \log c]$ for $c > 0$.
So its topology changes at $c = 0$, although $0$ is not a critical value, and $f$ has no minimum.
This is possible because the compactness condition, the [[Palais–Smale condition|4724KRNHAT]], fails.
Along $x_n = -n$ the values $f(x_n)$ are bounded and $f'(x_n) \to 0$, but $(x_n)$ has no convergent subsequence.

The two classical frameworks are Morse theory, through [[Morse functions|KPQ96640ZJ]], and [[Lusternik–Schnirelmann theory|QARH15D2PN]].
</document>

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# Mal'cev's theorem on congruence permutability

## Statement

Let $\mathcal{V}$ be a variety of algebras.
Then $\mathcal{V}$ is a [[congruence-permutable variety|1B3N9T68YV]] if and only if $\mathcal{V}$ has a [[Mal'cev term|Y203PPTS61]].
That is a ternary term $p$ such that $\mathcal{V}$ satisfies $p(x, y, y) = x$ and $p(x, x, y) = y$.

## Remarks

Suppose $p$ is a Mal'cev term, $A \in \mathcal{V}$, and $a \mathrel{\alpha} b \mathrel{\beta} c$ for [[congruences|DCSXXT4SER]] $\alpha$ and $\beta$ of $A$.
Put $d = p(a, b, c)$.
Then $d \mathrel{\beta} p(a, b, b) = a$ and $d \mathrel{\alpha} p(b, b, c) = c$, so $a \mathrel{\beta} d \mathrel{\alpha} c$, and the two compositions are equal.
Conversely, work in the [[free algebra|GTS4DRRMZR]] $F$ of $\mathcal{V}$ on $x, y, z$, with $\alpha$ generated by $(x, y)$ and $\beta$ generated by $(y, z)$.
Since $x \mathrel{\alpha} y \mathrel{\beta} z$, permutability gives $w = p(x, y, z) \in F$ with $x \mathrel{\beta} w \mathrel{\alpha} z$, and identifying $z$ with $y$, or $y$ with $x$, turns these relations into the two identities.

Example: groups have the Mal'cev term $x y^{-1} z$, so the congruences of every group permute; this is the familiar fact that $MN = NM$ for normal subgroups.
Lattices have no Mal'cev term, since the three-element chain has two congruences that do not permute.

## Explanations

### L1 human

The theorem says that a property of all congruences of all algebras in a variety is the same as the existence of one formula with two simple equations.
To show that a variety has permuting congruences, it is enough to find this formula.

### L2 human

This is the first Mal'cev condition, and it is the model for later ones, such as Jónsson's characterisation of congruence distributivity and Day's characterisation of congruence modularity.
The proof pattern is general: an implication in the free algebra on few generators gives terms, and terms give the property in every algebra.
Quasigroups, with $p(x, y, z) = (x / (y \backslash y)) \cdot (y \backslash z)$, also have a Mal'cev term, so their congruences permute.
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# Integral of a non-negative function

## Summary

The integral of a non-negative measurable function on a measure space is the supremum of the integrals of the simple functions that lie below it.
It always exists, as a number in $[0, \infty]$.
It is the first step of the Lebesgue integral, and the monotone convergence theorem and Fatou's lemma are statements about it.

## Explanations

### L1 human

For a function that takes finitely many values, the integral is easy: add up each value times the measure of the set where it is taken.
A general non-negative function is approached from below by such staircase functions.
The integral is the best value that these staircases reach.
For example, the function $f(x) = x$ on $[0, 1]$ has staircases below it with integrals as close to $1/2$ as one wants, and its integral is $1/2$.
The value $\infty$ is allowed, as for the constant function $1$ on the whole real line.
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<document source="https://tamnd.github.io/math2/node/DG7DZ37217.md" tag="DG7DZ37217" kind="definition">
# Oscillation theory

## Definition

Let $t_0 \in \R$ and let $q \colon [t_0, \infty) \to \R$ be continuous.
Consider the [[linear differential equation|0QQRP17Z7S]]
$$y'' + q(t) y = 0 \quad \text{on } [t_0, \infty).$$
A real [[solution|07VX34QXAV]] $y$ that is not identically zero is oscillatory if it has arbitrarily large zeros.
That is, for each $T \ge t_0$ there is $t > T$ with $y(t) = 0$.
Otherwise $y$ is nonoscillatory.
The equation is oscillatory if it has an oscillatory solution, and nonoscillatory otherwise.
Oscillation theory is the study of the zeros of solutions of such equations and their generalisations.
It looks in particular for criteria for oscillation in terms of the coefficients.

## Remarks

By the Sturm separation theorem, if one nonzero solution is oscillatory then every nonzero solution is oscillatory.
So the equation is oscillatory exactly when all its nonzero solutions are.
Example: $y'' + y = 0$ is oscillatory, since $y = \sin t$ vanishes at $t = k\pi$ for every integer $k$.
Non-example: for $y'' = 0$ every nonzero solution $y = A t + B$ has at most one zero, so the equation is nonoscillatory.
The Euler equation $y'' + \frac{c}{t^2} y = 0$ on $[1, \infty)$ is oscillatory exactly when $c > \frac{1}{4}$.
A classical criterion says that the equation is oscillatory if $\int_{t_0}^\infty q(t) \, dt = +\infty$.

## Explanations

### L1 human

A solution oscillates if its graph keeps crossing the axis forever.
Oscillation theory looks for conditions on the equation that force, or forbid, this behaviour.
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<document source="https://tamnd.github.io/math2/node/DG8FBQ8FPY.md" tag="DG8FBQ8FPY" kind="definition">
# Complete measure

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
The measure $\mu$, or the measure space, is *complete* if for every [[null set|VQQM4Y550T]] $N \in \Sigma$ and every subset $S \subseteq N$, one has $S \in \Sigma$.

## Remarks

Counting measure on the power set of a set is complete, since every subset is measurable.
Let $X = \set{1, 2, 3}$, let $\Sigma = \set{\varnothing, \set{1, 2}, \set{3}, X}$, and let $\mu(A) = 1$ if $3 \in A$ and $\mu(A) = 0$ otherwise.
Then $\set{1, 2}$ is a null set, but its subset $\set{1}$ is not in $\Sigma$, so $\mu$ is not complete.
For a complete measure, a subset of a null set is again a null set, by monotonicity.
Lebesgue measure on the σ-algebra of Lebesgue measurable sets is complete.
Its restriction to the Borel sets is not complete: the Cantor set is a Borel null set with more subsets than there are Borel sets.
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# Gibbs measure

## Summary

A Gibbs measure is a probability measure in which the weight of a configuration is proportional to $e^{-\text{energy}}$.
It comes from statistical mechanics, where it describes a system in thermal equilibrium.
In dynamics, the thermodynamic formalism uses Gibbs measures on [[shift spaces|A1XE59ZHNM]].
There the measure of a cylinder of length $n$ is comparable to the exponential of a Birkhoff sum of a potential.
For a Hölder potential on a mixing [[subshift of finite type|624XDXX8FN]] the Gibbs measure is unique and is the equilibrium state of the potential.

## Explanations

### L1 human

Imagine a long row of coins, and give each possible row an energy.
A Gibbs measure picks rows at random so that low-energy rows are more likely, with odds given by the exponential of minus the energy.
In dynamics the "row" is the sequence of regions an orbit visits, and the energy is the sum of a function along the orbit.
On the full shift on two symbols, when the function is constant, every block of $n$ symbols gets the same weight, and one gets fair coin tossing.
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<document source="https://tamnd.github.io/math2/node/DGJEC8MTMQ.md" tag="DGJEC8MTMQ" kind="definition">
# Conformal radius

## Definition

Let $\Omega \subsetneq \C$ be a [[simply connected domain|0QW64ARSHB]] and let $a \in \Omega$.
Let $\mathbb{D}$ be the unit disc.
By the [[Riemann mapping theorem|02WHMEG5ZR]] there is a unique [[conformal map|0YYT36R4YD]] $f \colon \mathbb{D} \to \Omega$ onto $\Omega$ with $f(0) = a$ and $f'(0) > 0$.
The *conformal radius* of $\Omega$ at $a$ is the number $\operatorname{rad}(a, \Omega) = f'(0)$.

## Remarks

Equivalently, $\operatorname{rad}(a, \Omega) = 1/\abs{\varphi'(a)}$ for any conformal map $\varphi$ of $\Omega$ onto $\mathbb{D}$ with $\varphi(a) = 0$.
Example: let $\Omega = \mathbb{D}$ and $a \in \mathbb{D}$.
The map $f(w) = (w + a)/(1 + \bar a w)$ sends $\mathbb{D}$ onto $\mathbb{D}$ with $f(0) = a$ and $f'(0) = 1 - \abs{a}^2 > 0$, so $\operatorname{rad}(a, \mathbb{D}) = 1 - \abs{a}^2$.
Non-example: $\Omega = \C$ is simply connected, but no conformal map of $\mathbb{D}$ onto $\C$ exists, so the definition does not apply.
With $d = \operatorname{dist}(a, \partial\Omega)$ one has $d \le \operatorname{rad}(a, \Omega) \le 4d$.
The lower bound follows from the [[Schwarz lemma|0MB48XDBY3]] applied to $f^{-1}$ on the disc of centre $a$ and radius $d$; the upper bound is the [[Koebe quarter theorem|WX0HMW4GT7]].
The [[Green's function|M36SD56MWQ]] of $\Omega$ satisfies $g(z, a) = -\log\abs{z - a} + \log \operatorname{rad}(a, \Omega) + o(1)$ as $z \to a$.
Under a conformal map $F$ of $\Omega$ onto $\Omega'$, $\operatorname{rad}(F(a), \Omega') = \abs{F'(a)} \operatorname{rad}(a, \Omega)$.
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<document source="https://tamnd.github.io/math2/node/DGVNNYKBHC.md" tag="DGVNNYKBHC" kind="concept">
# Porous medium equation

## Summary

The porous medium equation is the nonlinear diffusion equation $u_t = \Delta(u^m)$ with $m > 1$.
It models the flow of a gas through a porous rock, among other things.
The diffusion coefficient $m u^{m-1}$ vanishes where $u = 0$, so the equation is degenerate there.
As a consequence, disturbances travel with finite speed and solutions can have compact support with a moving free boundary.

## Explanations

### L1 human

In the ordinary [[heat equation|VP2HN86VAG]], heat spreads at once to the whole space, however little of it there is.
In the porous medium equation the spreading rate grows with the density, so where there is almost nothing, almost nothing moves.
A blob of gas therefore keeps a sharp edge that advances at finite speed.
The explicit Barenblatt solution shows this: it is a bump of compact support whose radius grows like a power of $t$.
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<document source="https://tamnd.github.io/math2/node/DGWYF0B6NT.md" tag="DGWYF0B6NT" kind="concept">
# Disk algebra

## Summary

The disk algebra $A(\mathbb{D})$ is the algebra of continuous functions on the closed unit disc that are holomorphic inside it.
With the supremum norm it is a commutative [[Banach algebra|9KYWN94QDN]], and the polynomials are dense in it.
It is the basic example of a uniform algebra and a natural setting for boundary problems on the [[unit disc|8G5ABXRK85]].

## Explanations

### L1 human

Take a function on the closed disc that is continuous up to the edge and holomorphic in the interior.
Such functions can be added and multiplied, and uniform limits of them have the same properties.
Polynomials are examples, and every element is a uniform limit of polynomials.
The function $\sum_{n \ge 1} z^n / n^2$ is in the disk algebra, since the series converges uniformly on the closed disc.
The function $1/(1 - z)$ is holomorphic in the open disc but does not belong, because it is unbounded near $1$.
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# Lawvere theory

## Definition

Let $\aleph_0$ be the [[category|A2DE66937D]] whose objects are the natural numbers $n \ge 0$.
Each $n$ is viewed as the set $\set{1, \ldots, n}$, and the morphisms are all functions between these sets.
A *Lawvere theory* is a small category $\mathcal{T}$ with finite [[products|S5744QZV4N]], together with a [[functor|28NQ0KRHD3]] $I \colon \aleph_0^{\mathrm{op}} \to \mathcal{T}$ that is bijective on objects and preserves finite products.
A *model* of $\mathcal{T}$ is a functor $\mathcal{T} \to \mathbf{Set}$ that preserves finite products.

## Remarks

In $\aleph_0^{\mathrm{op}}$, the object $n$ is the product of $n$ copies of $1$, with projections given by the inclusions $\set{1} \to \set{1, \ldots, n}$.
So in $\mathcal{T}$ every object is a power $1^n$ of the object $1$.
A model $M$ is determined by the set $M(1)$ and the maps $M(1)^n \to M(1)$ given by the morphisms $n \to 1$.

Example: for a [[variety|AJNZJD8442]] $\mathcal{V}$ with free algebras $F(n)$ on $n$ generators, let $\mathcal{T}_{\mathcal{V}}(n, m)$ be the set of homomorphisms $F(m) \to F(n)$, composed in the opposite order.
This is a Lawvere theory.
Here $\mathcal{T}_{\mathcal{V}}(n, 1) \cong F(n)$ is the set of $n$-ary terms modulo the identities of $\mathcal{V}$.
The object $n$ is the product of $n$ copies of $1$ because $F(n)$ is the coproduct of $n$ copies of $F(1)$ in $\mathcal{V}$.
Its category of models is equivalent to $\mathcal{V}$.

Non-example: the category $\aleph_0$ itself, not its opposite, has finite products, given by cartesian products, so the product of $n$ and $m$ is $n m$.
It is not a Lawvere theory, since in a Lawvere theory every object is a power of one object $x$.
Here $2$ and $3$ are not both powers of the same natural number.
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<document source="https://tamnd.github.io/math2/node/DH30WTA26T.md" tag="DH30WTA26T" kind="concept">
# Term evaluation

## Summary

Term evaluation computes the value of a [[term|6D3WA7N2W4]] in an algebra once each variable has been given a value.
The value is defined by recursion on the term.
Variables take their assigned values, and a function symbol applied to subterms is computed with the matching operation of the algebra.
It is the bridge between syntax and algebra: identities, term operations and the free algebras of a variety are all defined through it.

## Explanations

### L1 human

A term such as $x \cdot (y + 1)$ is only a formula on paper.
To evaluate it in the ring of integers, choose numbers for the variables, say $x = 3$ and $y = 4$.
Then compute from the inside out: $y + 1 = 5$, and $x \cdot 5 = 15$.
The same term evaluated in the ring $\Z/7\Z$ with the same choices gives $1$, so the value depends on the algebra as well as on the assignment.
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<document source="https://tamnd.github.io/math2/node/DH3PYQJXH0.md" tag="DH3PYQJXH0" kind="statement">
# Plancherel theorem for groups

## Statement

Let $G$ be a second countable, [[unimodular|CXG6JZGMCK]] [[locally compact group|AHV05ZBCYW]] of type I, with a fixed [[Haar measure|VY0HXD3GC4]] $dg$.
Let $\widehat{G}$ be its [[unitary dual|95Q8WZMN0W]], the set of equivalence classes of irreducible [[unitary representations|74BPYJNZTM]] of $G$, with its Borel structure.
For $f \in L^1(G)$ and a unitary representation $\pi$ of $G$ on a [[Hilbert space|HM2HHE3J51]] $\mathcal{H}_\pi$, let $\pi(f) = \int_G f(g) \pi(g) \, dg$, a bounded operator on $\mathcal{H}_\pi$.
Then there is a unique positive Borel measure $\mu$ on $\widehat{G}$, the *Plancherel measure*, with the following property.
For every $f \in L^1(G) \cap L^2(G)$, the operator $\pi(f)$ is Hilbert–Schmidt for $\mu$-almost every $\pi$, and
$$\int_G \abs{f(g)}^2 \, dg = \int_{\widehat{G}} \norm{\pi(f)}_{\mathrm{HS}}^2 \, d\mu(\pi).$$

## Remarks

The idea of the proof is to decompose the two-sided regular representation of $G \times G$ on $L^2(G)$.
For a type I group, the left regular representation is a direct integral of irreducible representations over $\widehat{G}$.
The Plancherel measure is the measure in this direct integral, normalised by the choice of $dg$.
The identity then extends by continuity to a unitary isomorphism from $L^2(G)$ onto a direct integral of the spaces of Hilbert–Schmidt operators on the $\mathcal{H}_\pi$.

For $G = \R$ with Lebesgue measure, $\widehat{G}$ is the set of characters $x \mapsto e^{i \xi x}$ with $\xi \in \R$, each $\pi(f)$ is multiplication by a value of the Fourier transform, and $\mu = d\xi / 2\pi$.
This is the classical Plancherel theorem.
For a [[compact group|EVHCE5VR6Y]] with Haar measure of total mass $1$, the dual $\widehat{G}$ is discrete, $\mu(\set{\pi}) = \dim \mathcal{H}_\pi$, and the formula is the Parseval identity of the [[Peter–Weyl theorem|6VC9QE39XH]].

The type I hypothesis cannot simply be dropped.
For groups that are not of type I, the regular representation can decompose in essentially different ways, and the formula needs factor representations.
Without unimodularity, the operators $\pi(f)$ must be corrected by unbounded positive operators before the formula holds.

## Explanations

### L1 human

For functions on the line, the Plancherel theorem says that the energy of a function equals the energy of its Fourier transform.
For a general group, the role of the frequencies is played by the irreducible representations of the group.
The theorem gives a measure on these representations, the Plancherel measure, that makes the energy identity true.
It tells which representations really occur when one breaks $L^2(G)$ into pieces.

### L2 human

Many groups met in practice are of type I: abelian groups, compact groups, connected semisimple Lie groups and connected nilpotent Lie groups.
For abelian groups the Plancherel measure is the Haar measure on the Pontryagin dual, suitably normalised.
For a semisimple Lie group, Harish-Chandra computed the Plancherel measure explicitly; the discrete series representations are exactly the atoms of $\mu$.
For a [[Gelfand pair|W38ZDXDVYC]] $(G, K)$ and functions that are bi-invariant under $K$, the theorem becomes the Plancherel theorem for [[spherical functions|BKZNC76RC8]].
</document>

<document source="https://tamnd.github.io/math2/node/DH4P0W4V3J.md" tag="DH4P0W4V3J" kind="definition">
# Moving frame

## Definition

Let $M$ be a smooth manifold of dimension $n$, and let $U \subset M$ be open.
A *moving frame*, or local frame, on $U$ is an ordered $n$-tuple $(E_1, \dots, E_n)$ of smooth vector fields on $U$.
The condition is that $(E_1(p), \dots, E_n(p))$ is a basis of $T_pM$ for every $p \in U$.
The *dual coframe* is the $n$-tuple of $1$-forms $(\theta^1, \dots, \theta^n)$ on $U$ with $\theta^i(E_j) = \delta^i_j$.

## Remarks

A moving frame on $U$ is the same as a smooth section over $U$ of the [[frame bundle|F4HWTRNXAP]] of $M$.
Example: on $\R^2 \setminus \set{0}$ with polar coordinates, $E_1 = \partial_r$ and $E_2 = \frac1r \partial_\theta$ form a moving frame; at each point they are orthonormal for the Euclidean metric, so they form a basis.
Non-example: on $\R^2$, the pair $(\partial_x, x \, \partial_y)$ is not a moving frame, because at the points with $x = 0$ the second field vanishes.
Given an affine connection $\nabla$, the connection forms $\omega^i_j$ of the frame are defined by $\nabla E_j = \sum_i \omega^i_j \otimes E_i$, and Cartan's structure equations express torsion and curvature through $d\theta^i$ and $d\omega^i_j$.
A moving frame on all of $M$ exists only when the [[tangent bundle|WY6K3P4YCG]] is trivial; for example $S^2$ has none.
</document>

<document source="https://tamnd.github.io/math2/node/DHC8TZXGYS.md" tag="DHC8TZXGYS" kind="example">
# Pentagon lattice N5

## Example

The *pentagon lattice* $N_5$ has five elements $0, a, b, c, 1$.
Its order is generated by $0 < a < c < 1$ and $0 < b < 1$, and $b$ is incomparable to $a$ and to $c$.
Every pair of elements has a least upper bound and a greatest lower bound, so $N_5$ is a [[lattice|WH4SFAACAS]].
For instance $a \vee b = c \vee b = 1$ and $a \wedge b = c \wedge b = 0$.

$N_5$ is not a [[modular lattice|QX7WZ92QAK]].
Take $a \le c$ and $x = b$.
Then $a \vee (b \wedge c) = a \vee 0 = a$, while $(a \vee b) \wedge c = 1 \wedge c = c \ne a$.

## Remarks

By [[Dedekind's characterization|ANRXAG227H]], a lattice is modular exactly when it has no sublattice isomorphic to $N_5$.
Since every distributive lattice is modular, $N_5$ is also not distributive.
The Hasse diagram of $N_5$ is a pentagon, which gives the lattice its name.

## Explanations

### L1 human

Draw a pentagon: a bottom, a top, a short side with one middle point $b$ and a long side with two points $a$ below $c$.
Going up from $a$ through $b$ reaches the top, and coming back down to the long side lands at $c$, not at $a$.
This mismatch is exactly the failure of the modular law.
It is the smallest lattice where this happens.
</document>

<document source="https://tamnd.github.io/math2/node/DHC96EYCN6.md" tag="DHC96EYCN6" kind="statement">
# Curry–Schoenberg theorem

## Statement

Let $k \ge 1$ be an integer, the order, so that polynomial pieces have degree at most $k - 1$.
Let $\xi_1 < \xi_2 < \cdots < \xi_{l+1}$ be breakpoints, and for $i = 2, \dots, l$ let $\nu_i$ be an integer with $0 \le \nu_i \le k$.
Let $S$ be the space of functions $s$ on $[\xi_1, \xi_{l+1})$ such that:

1. on each interval $[\xi_i, \xi_{i+1})$, $s$ agrees with a polynomial of degree at most $k - 1$;
2. at each interior breakpoint $\xi_i$, the two adjacent polynomial pieces have equal derivatives of orders $0, 1, \dots, \nu_i - 1$ at $\xi_i$.

Let $n = k + \sum_{i=2}^{l} (k - \nu_i)$.
Let $t_1 \le t_2 \le \cdots \le t_{n+k}$ be a knot sequence with $t_k = \xi_1$, $t_{n+1} = \xi_{l+1}$, $t_1 \le \dots \le t_k$, $t_{n+1} \le \dots \le t_{n+k}$, and in which each interior breakpoint $\xi_i$ occurs exactly $k - \nu_i$ times among $t_{k+1}, \dots, t_n$.
Then the [[B-splines|62AFX32JGT]] $B_{1,k}, \dots, B_{n,k}$ of order $k$ for the knots $t$, restricted to $[\xi_1, \xi_{l+1})$, form a [[basis|0KCQ9Y3QKA]] of $S$.
In particular, $\dim S = n$.

## Remarks

Each $B_{j,k}$ is a piecewise polynomial of degree at most $k - 1$, and a knot of multiplicity $m$ costs it exactly $m$ orders of smoothness, so each $B_{j,k}$ lies in $S$.
The dimension of $S$ is $n$ by counting: $l$ pieces give $kl$ coefficients, and the conditions at $\xi_i$ remove $\nu_i$ of them, which are independent.
Linear independence of the B-splines can be proved with the dual functionals of de Boor and Fix, and $n$ independent elements of an $n$-dimensional space form a basis.
Example: for $k = 4$ and $\nu_i = 3$ at every interior breakpoint, $S$ is the space of cubic [[splines|84TJ8KQ926]] of class $C^2$, and $n = l + 3$.
The end knots $t_1, \dots, t_k$ and $t_{n+1}, \dots, t_{n+k}$ can be chosen freely in the stated ranges; a common choice is to make them all equal to the endpoints.
The intervals are half-open because B-splines are right-continuous.
With the common choice of end knots, all of them vanish at $\xi_{l+1}$, and one extends a spline there by its left limit.

## Explanations

### L1 human

A spline is built from polynomial pieces glued together with some smoothness at the joints.
The theorem says that the B-splines are exactly the right building blocks: every such spline is a unique sum of multiples of them.
The number of B-splines needed equals the number of free parameters of the spline space.
The only thing to choose is how often each joint is repeated as a knot, and that encodes how smooth the spline is there.

### L2 human

The theorem turns the abstract space of piecewise polynomials into a concrete space with a well-conditioned basis of small support.
It is why spline software stores a spline as a knot vector and a list of B-spline coefficients.
Interpolation and least squares fitting with splines become banded linear systems in this basis.
Raising the multiplicity of a knot by one lowers the smoothness there by one, and multiplicity $k$ allows a jump.
</document>

<document source="https://tamnd.github.io/math2/node/DHCHJNHD02.md" tag="DHCHJNHD02" kind="statement">
# Cartier–Milnor–Moore theorem

## Statement

Let $K$ be a field of [[characteristic|0F6ES3BRYP]] $0$.
Let $H$ be a [[Hopf algebra|7XFYP0N84H]] over $K$ that is cocommutative and connected, which means that $K \cdot 1$ is the only simple subcoalgebra of $H$.
Let $P(H) = \set{x \in H : \Delta(x) = x \otimes 1 + 1 \otimes x}$ be the space of primitive elements, a Lie algebra under the commutator $[x, y] = xy - yx$.
Then the inclusion $P(H) \to H$ extends to an isomorphism of Hopf algebras $U(P(H)) \to H$, where $U(P(H))$ is the [[universal enveloping algebra|SCZ8PRVY00]] of $P(H)$.

## Remarks

The map $U(P(H)) \to H$ exists by the universal property of $U$, and it respects the coproducts because both are generated by primitive elements.
Injectivity uses the Poincaré–Birkhoff–Witt theorem and the fact that in characteristic $0$ the primitive elements of $U(\mathfrak{g})$ are exactly $\mathfrak{g}$.
Surjectivity is proved by induction along the coradical filtration of $H$.
Example: $H = K[x]$ with $x$ primitive; then $P(H) = Kx$, an abelian Lie algebra of dimension $1$, and $U(Kx) = K[x]$.
Characteristic $0$ is needed: if $K$ has characteristic $p > 0$, then $x^p$ is also primitive in $K[x]$, since $\Delta(x)^p = x^p \otimes 1 + 1 \otimes x^p$.
Then $U(P(K[x]))$ is a polynomial ring in the independent variables $x, x^p, x^{p^2}, \dots$, and the map to $K[x]$ is not injective.
Connectedness is needed: for a group $G \ne 1$, $KG$ is cocommutative but $P(KG) = 0$, so $U(P(KG)) = K \ne KG$.
Milnor and Moore proved the graded version.
A connected graded cocommutative Hopf algebra over a field of characteristic $0$ is the universal enveloping algebra of its graded Lie algebra of primitives.

## Explanations

### L1 human

Lie algebras and their universal enveloping algebras give many Hopf algebras.
The theorem says that, over a field of characteristic $0$, such as $\Q$ or $\C$, every cocommutative Hopf algebra that is connected arises in this way.
So such a Hopf algebra carries no information beyond the Lie algebra of its primitive elements.

### L2 human

The theorem is one half of the classification of cocommutative Hopf algebras in characteristic $0$.
Over an algebraically closed field, the Cartier–Gabriel–Kostant theorem adds group algebras: $H$ is the smash product $U(P(H)) \# KG(H)$, where $G(H)$ is the group of group-like elements.
In algebraic topology, let $X$ be a simply connected space with loop space $\Omega X$.
The graded version identifies the rational homology of $\Omega X$ with the universal enveloping algebra of the Lie algebra $\pi_*(\Omega X) \otimes \Q$.
It is also the reason why the Hopf algebras of renormalization in physics are dual to universal enveloping algebras.
</document>

<document source="https://tamnd.github.io/math2/node/DHHR5PZZ7E.md" tag="DHHR5PZZ7E" kind="statement">
# Existence of Leray weak solutions

## Statement

Let $\nu > 0$ and let $u_0 \in L^2(\R^3; \R^3)$ satisfy $\nabla \cdot u_0 = 0$ in the sense of distributions.
Then there exists a vector field $u \in L^\infty\big(0, \infty; L^2(\R^3; \R^3)\big)$ with $\nabla u \in L^2\big(\R^3 \times (0, \infty)\big)$ such that:

1. $u$ is a [[weak solution|SJZX81A783]] of the [[Navier–Stokes equations|18KXZHVF4M]] with no external force and initial value $u_0$.
That is, $\nabla \cdot u(\cdot, t) = 0$ for almost every $t$.
Also, let $\varphi \colon \R^3 \times [0, \infty) \to \R^3$ be any smooth compactly supported test field with $\nabla \cdot \varphi = 0$.
Then
$$\int_0^\infty \!\! \int_{\R^3} \Big( u \cdot \partial_t \varphi + \sum_{i,j=1}^3 u_i u_j \, \partial_j \varphi_i - \nu \nabla u : \nabla \varphi \Big) dx \, dt + \int_{\R^3} u_0 \cdot \varphi(x, 0) \, dx = 0.$$
2. $u$ satisfies the energy inequality: for every $t > 0$,
$$\norm{u(\cdot, t)}_{L^2}^2 + 2 \nu \int_0^t \norm{\nabla u(\cdot, s)}_{L^2}^2 \, ds \le \norm{u_0}_{L^2}^2.$$

Here $\nabla u : \nabla \varphi = \sum_{i,j} \partial_j u_i \, \partial_j \varphi_i$.
A solution with these properties is called a *Leray weak solution*, or a Leray–Hopf weak solution.

## Remarks

Jean Leray proved this theorem in 1934, and Eberhard Hopf later proved the analogue on bounded domains.
The proof regularises the nonlinear term, solves the regularised problem, and uses the energy identity to bound the approximate solutions uniformly in the [[Sobolev space|M4CEZ5K9HP]] norm.
A compactness argument then gives a subsequence whose limit solves the equations in the weak sense, and the energy inequality survives the limit by lower semicontinuity of the norm.
The pressure does not appear, because $\int \nabla p \cdot \varphi \, dx = 0$ for divergence-free $\varphi$.
In two space dimensions the Leray weak solution is unique and smooth for positive times.
In three dimensions its uniqueness and smoothness are open, and they are the heart of the Navier–Stokes existence and smoothness problem.

## Explanations

### L1 human

For the equations of fluid flow in three dimensions, nobody knows how to build smooth solutions that last forever.
Leray found a weaker kind of solution that does always exist: it may be rough, but it obeys the equations on average against smooth test fields.
It also never gains kinetic energy, as a real viscous fluid should not.

### L2 human

The theorem is the starting point of the mathematical theory of the Navier–Stokes equations.
The energy inequality gives the only a priori bound that is known for large data, and the class of solutions it defines is the natural one for studying regularity.
A weak solution that is smooth agrees with the classical solution, and a classical solution is unique among Leray weak solutions while it exists.
Results such as the partial regularity theorem of Caffarelli, Kohn and Nirenberg concern this class of solutions.
Buckmaster and Vicol showed that weak solutions without the energy inequality and with less regularity are not unique.
</document>

<document source="https://tamnd.github.io/math2/node/DHPXDFEEMF.md" tag="DHPXDFEEMF" kind="example">
# Pólya urn model

## Example

Let $a \ge 1$ and $b \ge 1$ be integers.
An urn holds $a$ red balls and $b$ black balls.
At each step one draws a ball uniformly at random and notes its colour.
Then one puts it back together with one more ball of the same colour.
Let $R_n$ and $B_n$ be the numbers of red and black balls after $n$ steps, so $R_n + B_n = a + b + n$.
Then $(R_n, B_n)$ is a [[Markov chain|0G48PYM96Q]], since the next draw depends only on the current contents.
From $(r, k)$ it moves to $(r + 1, k)$ with probability $r/(r + k)$ and to $(r, k + 1)$ otherwise.

The fraction $M_n = R_n / (a + b + n)$ of red balls is a [[martingale|0D0GBNDKDH]]:
$$\mathbb{E}[M_{n+1} \mid R_n, B_n] = \frac{R_n}{R_n + B_n} \cdot \frac{R_n + 1}{R_n + B_n + 1} + \frac{B_n}{R_n + B_n} \cdot \frac{R_n}{R_n + B_n + 1} = \frac{R_n}{R_n + B_n} = M_n.$$
It takes values in $[0, 1]$, so it converges almost surely, and its limit has the [[beta distribution|0EW824Z7G2]] with parameters $a$ and $b$.

## Remarks

For $a = b = 1$ the limit is uniform on $[0, 1]$, and $R_n$ is uniform on $\set{1, 2, \ldots, n + 1}$ for each $n$.
The sequence of colours drawn is [[exchangeable|4MR4YJS1ZD]]: the probability of a sequence depends only on how many red and black draws it contains.
By [[de Finetti's theorem|W19129PVKQ]], given the limit fraction $M$, the draws are independent with probability $M$ of red.
The urn shows that a random process can have a random limit: the rich get richer.

## Explanations

### L1 human

Each draw of a colour makes that colour more likely next time.
Early luck has a lasting effect, so the proportion of red settles down, but to a value that is itself random.
Starting with one red and one black ball, every final proportion between $0$ and $1$ is equally likely.
</document>

<document source="https://tamnd.github.io/math2/node/DHRDTHFBFA.md" tag="DHRDTHFBFA" kind="definition">
# Five-point stencil

## Definition

Let $h > 0$ be a grid spacing.

1. In one dimension, the *five-point stencil* at $x \in \R$ is the set of points $x - 2h, x - h, x, x + h, x + 2h$.
The associated formulas for a function $f$ are
$$f'(x) \approx \frac{-f(x + 2h) + 8 f(x + h) - 8 f(x - h) + f(x - 2h)}{12 h},$$
$$f''(x) \approx \frac{-f(x + 2h) + 16 f(x + h) - 30 f(x) + 16 f(x - h) - f(x - 2h)}{12 h^2}.$$
2. In two dimensions, the *five-point stencil* at $(x, y) \in \R^2$ is the set of points $(x, y)$, $(x \pm h, y)$, $(x, y \pm h)$.
The associated *five-point Laplacian* of a function $u$ is
$$\Delta_h u(x, y) = \frac{u(x + h, y) + u(x - h, y) + u(x, y + h) + u(x, y - h) - 4 u(x, y)}{h^2}.$$

## Remarks

The one-dimensional weights are the [[finite difference coefficients|752Z7DRT7P]] for the stencil $-2, -1, 0, 1, 2$.
For $f \in C^5$, the first formula has error $-\frac{h^4}{30} f^{(5)}(\xi)$ for some $\xi$ in $(x - 2h, x + 2h)$, so it has order $4$; the second has order $4$ for $f \in C^6$.
Check of the second formula on $f(s) = s^2$ at $x = 0$: $\frac{-4h^2 + 16h^2 + 16h^2 - 4h^2}{12h^2} = 2 = f''(0)$.
For $u \in C^4$, each one-dimensional second difference has error $\frac{h^2}{12} \partial^4 u$ at an intermediate point, so $\abs{\Delta_h u - \Delta u} \le \frac{h^2}{12} \big( \max \abs{u_{xxxx}} + \max \abs{u_{yyyy}} \big)$ and the five-point Laplacian approximates the [[Laplacian|0E0B6RA8K0]] to second order.
Non-example: the nine-point and the diagonal five-point stencil $(x \pm h, y \pm h)$ also approximate $\Delta u$, the latter as $\frac{1}{2h^2}\big(\sum u(x \pm h, y \pm h) - 4u(x,y)\big)$, but it uses different points and is not the five-point stencil defined here.

## Explanations

### L1 human

In one dimension, the five-point stencil uses two points on each side of $x$ and gives very accurate derivative estimates.
In two dimensions, it uses the four neighbours of a grid point and estimates the Laplacian, which measures how a value differs from the average of its surroundings.
</document>

<document source="https://tamnd.github.io/math2/node/DHXXSN12JC.md" tag="DHXXSN12JC" kind="concept">
# Subnormal number

## Summary

A subnormal number, also called a denormal number, is a nonzero [[floating-point|RTKRSC3KPH]] number smaller in absolute value than the smallest normalized floating-point number.
It has the minimal exponent and a leading significand digit $0$, so it has fewer significant digits.
Subnormal numbers fill the gap between $0$ and the smallest normalized number with evenly spaced values.
This *gradual underflow* ensures that $x - y = 0$ in floating-point arithmetic only when $x = y$.

## Explanations

### L1 human

Normal floating-point numbers get closer and closer together as they approach $0$, but the smallest one still leaves a relatively big jump down to $0$.
Subnormal numbers are extra points placed evenly in that jump.
They give up some precision so that very small results do not suddenly become $0$.
In double precision the smallest normal number is $2^{-1022}$, and the subnormal numbers go down to $2^{-1074}$.
</document>

<document source="https://tamnd.github.io/math2/node/DHYS1W3Y4N.md" tag="DHYS1W3Y4N" kind="definition">
# Group algebra of a locally compact group

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] with a left [[Haar measure|VY0HXD3GC4]] $\mu$ and [[modular function|63TX60RTRQ]] $\Delta$.
The *group algebra* of $G$ is the space [[$L^1(G) = L^1(G, \mu)$|SASP14MA1Z]] of integrable functions, with the following operations.

1. The product is [[convolution|2Z29JN3VES]]: $(f * h)(x) = \int_G f(y) \, h(y^{-1}x) \, d\mu(y)$.
2. The involution is $f^*(x) = \Delta(x^{-1}) \, \overline{f(x^{-1})}$.
3. The norm is $\norm{f}_1 = \int_G \abs{f} \, d\mu$.

## Remarks

With these operations $L^1(G)$ is a [[Banach algebra|9KYWN94QDN]] with an isometric involution, that is, a Banach $*$-algebra: $\norm{f * h}_1 \le \norm{f}_1 \norm{h}_1$, $(f * h)^* = h^* * f^*$ and $\norm{f^*}_1 = \norm{f}_1$.

Example: $G = \Z$ with counting measure.
Then $L^1(\Z) = \ell^1(\Z)$, $\Delta \equiv 1$, the product is $(f * h)(n) = \sum_m f(m) h(n - m)$ and $f^*(n) = \overline{f(-n)}$.
The sequence $\delta_0$ is a unit: $(\delta_0 * h)(n) = h(n)$.
Here $\ell^1(\Z)$ contains the [[group algebra|MNP2HBWB5Q]] $\C[\Z]$ of finitely supported sequences as a dense subalgebra.

Non-example of a unit: $L^1(\R)$ has no unit.
If $e * f = f$ for all $f \in L^1(\R)$, then taking Fourier transforms gives $\hat e = 1$ wherever some $\hat f$ is nonzero, so $\hat e \equiv 1$, which contradicts the Riemann–Lebesgue lemma $\hat e(\xi) \to 0$.
In general $L^1(G)$ has a unit exactly when $G$ is discrete, but it always has a bounded approximate unit.

$L^1(G)$ is commutative if and only if $G$ is abelian.
</document>

<document source="https://tamnd.github.io/math2/node/DHYSPSYSCB.md" tag="DHYSPSYSCB" kind="statement">
# Yau–Tian–Donaldson conjecture

## Statement

Let $X$ be a Fano manifold, that is a compact complex manifold whose anticanonical line bundle $-K_X$ is ample.
Then $X$ admits a [[Kähler–Einstein metric|6ETEBA8827]] if and only if the pair $(X, -K_X)$ is K-polystable.

Here K-polystable means the following.
A test configuration of $(X, -K_X)$ is a $\C^*$-equivariant degeneration of $X$ with a line bundle extending a power of $-K_X$.
For every normal test configuration, the Donaldson–Futaki invariant is nonnegative.
Moreover, it vanishes only for normal test configurations that are products of $X$ with $\C$ twisted by a $\C^*$-action on $X$.

## Remarks

This form of the conjecture, for Fano manifolds, was proved by Xiuxiong Chen, Simon Donaldson and Song Sun.
The direction "Kähler–Einstein implies K-polystable" is the easier one.
The converse uses the continuity method with metrics having cone singularities along a divisor, and [[Gromov–Hausdorff limits|GT56RTG9X5]] of these metrics, which are shown to be algebraic varieties.

Example: the blow-up of $\C P^2$ at one point is Fano but has no Kähler–Einstein metric.
Its automorphism group is not reductive, which rules out a Kähler–Einstein metric by Matsushima's theorem, and a test configuration coming from a $\C^*$-action has negative Donaldson–Futaki invariant.
By contrast $\C P^n$ with the Fubini–Study metric is Kähler–Einstein and K-polystable.

The general conjecture asks for a constant scalar curvature Kähler metric in $c_1(L)$ on a polarized manifold $(X, L)$, under a suitable notion of K-stability.
That general form is open, and the right stability notion for it is still discussed.

## Explanations

### L1 human

For manifolds with positive first Chern class, a Kähler–Einstein metric may or may not exist.
The theorem says that existence is equivalent to an algebraic condition called K-polystability.
This condition looks at all ways the manifold can degenerate and checks a sign on each of them.

### L2 human

The negative and zero cases of the Kähler–Einstein problem follow from the [[Calabi conjecture|WJKA5YBTJ7]] methods with no obstruction.
In the Fano case there are obstructions, such as the Futaki invariant and Matsushima's theorem, and K-stability collects them all.
The statement links a nonlinear partial differential equation to geometric invariant theory, in the spirit of the Hitchin–Kobayashi correspondence for vector bundles.
After the proof, much work has gone into deciding K-stability of explicit Fano manifolds by algebraic means.
</document>

<document source="https://tamnd.github.io/math2/node/DJ6F3DZX42.md" tag="DJ6F3DZX42" kind="definition">
# Measurable space

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]].
A *measurable space* is a pair $(X, \Sigma)$, where $\Sigma$ is a [[σ-algebra|04WSW8T3SY]] on $X$.
The elements of $\Sigma$ are the *measurable sets* of $(X, \Sigma)$.

## Remarks

The real line with its [[Borel σ-algebra|0BVHZXWKCD]] $\mathcal{B}(\R)$ is a measurable space, because $\mathcal{B}(\R)$ is a σ-algebra by its definition.
Any set $X$ with its power set is a measurable space, and so is $X$ with $\set{\varnothing, X}$.
For $X = \set{1, 2}$, the pair $(X, \set{\varnothing, \set{1}, X})$ is not a measurable space.
The family is not closed under complements, since $X \setminus \set{1} = \set{2}$ is missing.
A map $f \colon X \to Y$ between measurable spaces $(X, \Sigma)$ and $(Y, T)$ is called measurable when $f^{-1}(B) \in \Sigma$ for every $B \in T$.
Some authors call a measurable space a *Borel space*.
</document>

<document source="https://tamnd.github.io/math2/node/DJB8MAV5BH.md" tag="DJB8MAV5BH" kind="problem">
# Linear-quadratic-Gaussian control

## Problem

Let $A$, $B$, $C$ be real matrices of sizes $n \times n$, $n \times m$ and $k \times n$, and let $T > 0$.
Let $Q, F$ be symmetric [[positive semidefinite|01MMVW8CKC]] $n \times n$ matrices, and let $R$ be a symmetric [[positive definite|0KX66CZMBB]] $m \times m$ matrix.
Let $W$ and $V$ be independent [[Wiener processes|22735FGJV6]] in $\R^n$ and $\R^k$ with covariance matrices $\Xi t$ and $\Theta t$.
Here $\Xi$ is positive semidefinite and $\Theta$ is positive definite.
Let the initial state $x(0)$ be Gaussian with mean $m_0$ and covariance $\Sigma_0$, independent of $W$ and $V$.
The state $x$ and the observation $y$ solve the [[stochastic differential equations|B219G9SHJA]]
$$dx = (A x + B u) \, dt + dW, \qquad dy = C x \, dt + dV, \qquad y(0) = 0.$$
A control is admissible if $u(t)$ depends only on the observations $y(s)$, $s \le t$, that is, $u$ is adapted to the [[filtration|HNK1Y62G7V]] generated by $y$.
The *linear-quadratic-Gaussian control problem* is to minimize the [[expected value|0WQNRCNY3M]]
$$J(u) = \mathbb{E}\Big[ \int_0^T \big( x^{T} Q x + u^{T} R u \big) \, dt + x(T)^{T} F x(T) \Big]$$
over all admissible controls $u$.

## Remarks

The problem is solved, and its solution is the separation principle.
The optimal control is $u = -R^{-1} B^{T} P(t) \hat{x}(t)$, where $P$ solves the Riccati equation of the [[linear-quadratic regulator|HR720TJZZR]] and $\hat{x}(t)$ is the Kalman–Bucy filter estimate of $x(t)$ from the observations.
Without noise and with full observation of the state, the problem reduces to the linear-quadratic regulator.
The infinite-horizon version uses the average cost per unit time and the algebraic Riccati equations of the regulator and of the filter.
LQG controllers can have poor robustness margins, which was one motivation for robust control methods.

## Explanations

### L1 human

The linear-quadratic-Gaussian problem is the linear-quadratic regulator in a noisy world.
The system is pushed by random disturbances, and one cannot see its state, only noisy measurements of some of its parts.
One still wants to keep the state near zero with little control effort, on average.
The answer splits into two simple parts: estimate the state as well as possible, then act as if the estimate were the true state.
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<document source="https://tamnd.github.io/math2/node/DJBWP7D3M8.md" tag="DJBWP7D3M8" kind="concept">
# Polydisc

## Summary

A polydisc in $\C^n$ is a product of $n$ open discs of the complex plane, one in each coordinate.
It is the basic domain of [[several complex variables|BH57D2FBNR]], playing the role that the disc plays in one variable.
The [[Cauchy integral formula on a polydisc|FG5448NZ18]] follows from the one-variable formula applied in each coordinate, and it leads to power series expansions.
Its [[distinguished boundary|A6HN207Y2T]] is a torus, much smaller than its topological boundary.

## Explanations

### L1 human

A polydisc is what one gets by letting each coordinate $z_j$ range over its own disc, independently of the others.
In $\C^2$, the unit polydisc is the set of pairs $(z_1, z_2)$ with $\abs{z_1} < 1$ and $\abs{z_2} < 1$.
It is a complex analogue of a box, or of a square in the plane.
The point $(0.9, 0.9)$ lies in the unit polydisc of $\C^2$, but not in the [[unit ball|PDQ9KW9FAP]], since $0.81 + 0.81 > 1$.
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<document source="https://tamnd.github.io/math2/node/DJBY0QQA80.md" tag="DJBY0QQA80" kind="definition">
# Integral operator

## Definition

Let $(X, \mu)$ and $(Y, \nu)$ be [[measure spaces|56MJSVTHG0]] whose measures are [[sigma-finite|CW5CEF84T2]].
Let $K \colon X \times Y \to \C$ be a measurable function, the *kernel*.
The *integral operator* with kernel $K$ is the [[linear map|0RJF668P1T]] $T$ given by
$$(Tf)(x) = \int_Y K(x, y)\, f(y)\, d\nu(y).$$
Its domain is the set of measurable functions $f \colon Y \to \C$ for which the [[Lebesgue integral|XVEWR93EMV]] $\int_Y \abs{K(x, y) f(y)}\, d\nu(y)$ is finite for $\mu$-almost every $x \in X$.

## Remarks

Example: on $X = Y = [0, 1]$ with Lebesgue measure, the kernel $K(x, y) = 1$ for $y < x$ and $K(x, y) = 0$ for $y \ge x$ gives the Volterra operator $(Vf)(x) = \int_0^x f(y)\, dy$.
It is defined for every integrable $f$, because $\int_0^1 \abs{K(x, y) f(y)}\, dy \le \int_0^1 \abs{f}$.
Non-example: the identity map of $L^2([0,1])$ is linear but is not an integral operator with a function kernel.
Such a kernel would give $\int_A K(x, y)\, dy = 1$ for almost every $x \in A$ and $0$ for almost every $x \notin A$, for every interval $A$.
This would force the measure $K(x, y)\, dy$ to be a point mass at $x$, which is impossible.
If $K$ is in $L^2(X \times Y)$, the operator maps the [[Lebesgue space|7KGF78FGEF]] $L^2(Y)$ to $L^2(X)$ and is bounded with norm at most $\norm{K}_{L^2}$.
Such operators are called Hilbert–Schmidt integral operators, and they are compact.
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<document source="https://tamnd.github.io/math2/node/DJEK11CXXJ.md" tag="DJEK11CXXJ" kind="definition">
# Quasi-primal algebra

## Definition

Let $A$ be an algebra whose universe is a nonempty [[finite set|070R982G4B]].
The algebra $A$ is *quasi-primal* if the [[ternary discriminator|72PG6XCQ7D]] $t$ on $A$ is a [[term operation|N9M2N27END]] of $A$.
Here $t(x, y, z) = z$ if $x = y$, and $t(x, y, z) = x$ if $x \ne y$.

## Remarks

Example: every finite field $F$ with $q$ elements, as a ring, is quasi-primal.
Check: the term $x + (z - x)\bigl(1 - (x - y)^{q - 1}\bigr)$ gives $z$ when $x = y$, and gives $x$ otherwise since then $(x - y)^{q - 1} = 1$.
Non-example: the two-element lattice $(\set{0, 1}, \wedge, \vee)$ is not quasi-primal.
Its term operations are monotone, but the discriminator is not, since $t(0, 0, 1) = 1$ while $t(0, 1, 1) = 0$.
A finite algebra is quasi-primal if and only if it generates an arithmetical variety and every nontrivial subalgebra of it is simple.
This characterization is due to Alden Pixley.
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<document source="https://tamnd.github.io/math2/node/DJFQWTQG9V.md" tag="DJFQWTQG9V" kind="concept">
# McKenzie–Hobby tame congruence theory

## Summary

Tame congruence theory is a structure theory for finite algebras and [[locally finite varieties|8QPW65PZVW]], developed by David Hobby and Ralph McKenzie.
It studies each covering pair $\alpha \prec \beta$ in the [[congruence lattice|REE07CSXHR]] of a finite algebra through the polynomial operations restricted to small subsets, the minimal sets.
On these subsets the algebra behaves in one of five ways, which gives each covering pair a type: unary, affine, Boolean, lattice or semilattice.
The types that occur in a locally finite variety control its congruence lattices and its Mal'cev conditions.
For example, a locally finite variety omits the unary type exactly when it has a [[Taylor term|QEAMRNRJJD]].

## Explanations

### L1 human

A finite algebra can be very complicated, but tame congruence theory looks at it one small step at a time.
Each step goes from one congruence to the next larger one, and the theory finds a small piece of the algebra where this step is visible.
On that piece, the algebra looks like one of five simple kinds of structure: a set with permutations, a vector space, a Boolean algebra, a lattice, or a semilattice.
Knowing which kinds appear tells a great deal about the whole algebra and the variety it generates.
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<document source="https://tamnd.github.io/math2/node/DJFYTQJ3SH.md" tag="DJFYTQJ3SH" kind="concept">
# Spearman's rank correlation coefficient

## Summary

Spearman's rank correlation coefficient $\rho_s$ is the Pearson [[correlation coefficient|0DNN9B6Q2F]] of the ranks of two variables.
It measures how well the relation between the variables is described by a monotone function, not only by a straight line.
It lies between $-1$ and $1$, and it equals $1$ exactly when one variable is a strictly increasing function of the other on the sample.
It is unchanged by any strictly increasing change of scale of either variable.

## Explanations

### L1 human

Replace each value of $x$ by its place in the sorted list of $x$ values, and do the same for $y$.
Then compute the ordinary correlation of these places.
If larger $x$ always goes with larger $y$, the places agree and the coefficient is $1$, even when the relation is curved.
For example, the points $(1, 1), (2, 4), (3, 9)$ lie on a parabola, not a line, yet their Spearman coefficient is $1$.
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<document source="https://tamnd.github.io/math2/node/DJHVKR63H5.md" tag="DJHVKR63H5" kind="definition">
# Unitary operator

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $U \colon H \to H$ be a bounded linear operator with [[adjoint|6HSJBJ8TTQ]] $U^*$.
The operator $U$ is *unitary* when
$$U^* U = U U^* = I,$$
where $I$ is the identity operator of $H$.

## Remarks

Example: on $\ell^2(\Z)$ the bilateral shift $(Ux)_n = x_{n-1}$ has adjoint $(U^*x)_n = x_{n+1}$, and both $U^*U$ and $UU^*$ are the identity, so $U$ is unitary.

Non-example: on $\ell^2(\N)$ the unilateral shift $S(x_0, x_1, \ldots) = (0, x_0, x_1, \ldots)$ satisfies $S^*S = I$, but $SS^*$ sends $(1, 0, 0, \ldots)$ to $0$, so $SS^* \ne I$ and $S$ is not unitary.
It is an isometry that is not surjective.

Equivalent forms: $U$ is unitary if and only if $U$ is surjective and $\ip{Ux}{Uy} = \ip{x}{y}$ for all $x, y$, and if and only if $U$ is a surjective isometry.
In finite dimensions $U^*U = I$ already implies $UU^* = I$, but in infinite dimensions both conditions are needed, as the unilateral shift shows.
The spectrum of a unitary operator lies on the unit circle.
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<document source="https://tamnd.github.io/math2/node/DJM5H15WV5.md" tag="DJM5H15WV5" kind="definition">
# Singular simplicial set

## Definition

For $n \ge 0$ let $[n] = \set{0, 1, \dots, n}$, and let
$$\abs{\Delta^n} = \set{(t_0, \dots, t_n) \in \R^{n+1} : t_i \ge 0, \ t_0 + \cdots + t_n = 1}$$
be the standard topological $n$-simplex, with vertices $e_0, \dots, e_n$.
An order-preserving map $\theta \colon [m] \to [n]$ induces the affine map $\theta_* \colon \abs{\Delta^m} \to \abs{\Delta^n}$ with $\theta_*(e_i) = e_{\theta(i)}$.
Let $X$ be a [[topological space|0CNQXVBHCX]].
The *singular [[simplicial set|WW81ARB8K9]]* $\operatorname{Sing} X$ is the simplicial set with:

1. $(\operatorname{Sing} X)_n$ the set of [[continuous maps|0SY9ECQ19Q]] $\sigma \colon \abs{\Delta^n} \to X$, called singular $n$-simplices.
2. For $\theta \colon [m] \to [n]$ order-preserving, the map $\theta^* \colon (\operatorname{Sing} X)_n \to (\operatorname{Sing} X)_m$ given by $\theta^*(\sigma) = \sigma \circ \theta_*$.

## Remarks

The identities $(\theta \circ \psi)_* = \theta_* \circ \psi_*$ and $\id_* = \id$ show that $\operatorname{Sing} X$ is a functor $\Delta^{\mathrm{op}} \to \mathrm{Set}$, that is, a simplicial set.
A continuous map $f \colon X \to Y$ induces $\operatorname{Sing} f$ by $\sigma \mapsto f \circ \sigma$, so $\operatorname{Sing}$ is a functor from spaces to simplicial sets.

Example: if $X$ is a single point, each $(\operatorname{Sing} X)_n$ has exactly one element, so $\operatorname{Sing} X$ is the terminal simplicial set $\Delta^0$.

Non-example: the simplicial set whose $n$-simplices are all functions $\abs{\Delta^n} \to X$, continuous or not, is a simplicial set, but it is not $\operatorname{Sing} X$ for $X = [0, 1]$.
It contains discontinuous functions $\abs{\Delta^1} \to [0, 1]$, which clause 1 excludes.

$\operatorname{Sing} X$ is always a Kan complex.
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<document source="https://tamnd.github.io/math2/node/DJMXZX0G9G.md" tag="DJMXZX0G9G" kind="definition">
# Oracle machine

## Definition

An *oracle machine* is a [[Turing machine|758EYCN27B]] with these extra parts:

1. a second tape, the *oracle tape*, on which the machine can read and write;
2. three special states: a *query state* $q_?$, a *yes state* $q_{\mathrm{yes}}$ and a *no state* $q_{\mathrm{no}}$.

Let $O \subseteq \N$ be a set, the *oracle*.
The machine $M$ *with oracle* $O$, written $M^O$, runs like a Turing machine, except at the query state.
Suppose $M^O$ enters $q_?$ while the oracle tape holds the binary code of a number $n$.
Then its next state is $q_{\mathrm{yes}}$ if $n \in O$ and $q_{\mathrm{no}}$ if $n \notin O$.
A partial function $f \colon \N \to \N$ is *computable relative to* $O$ if there is an oracle machine $M$ with this property.
For every $x$, the run of $M^O$ on input $x$ halts with output $f(x)$ when $f(x)$ is defined, and does not halt otherwise.

## Remarks

Example: let $M$ copy its input $x$ to the oracle tape, enter $q_?$, and then output $1$ from $q_{\mathrm{no}}$ and $0$ from $q_{\mathrm{yes}}$.
For each oracle $O$, the machine $M^O$ halts on every input and computes the characteristic function of $\N \setminus O$.

Non-example: a device that may query two sets $O_1$ and $O_2$ is not an oracle machine, because an oracle machine queries one fixed set.
It can be simulated by an oracle machine with the single oracle $\set{2n : n \in O_1} \cup \set{2n + 1 : n \in O_2}$.

A machine that never enters $q_?$ is an ordinary Turing machine, so every computable function is computable relative to every oracle.
Oracle machines can be listed effectively as $M_0, M_1, \ldots$, and $\Phi_e^O$ often denotes the partial function computed by $M_e^O$.
</document>

<document source="https://tamnd.github.io/math2/node/DJXX3X6TEQ.md" tag="DJXX3X6TEQ" kind="definition">
# Affine arithmetic

## Definition

An *affine form* is an expression
$$\hat x = x_0 + \sum_{i=1}^{k} x_i \varepsilon_i,$$
with real coefficients $x_0, \ldots, x_k$ and *noise symbols* $\varepsilon_1, \ldots, \varepsilon_k$.
The noise symbols stand for unknown reals in $[-1, 1]$, shared by all affine forms of a computation.
Its *range* is the [[interval|0Q1D85B8G6]] $[x_0 - r, x_0 + r]$ with radius $r = \sum_{i=1}^{k} \abs{x_i}$.
*Affine arithmetic* computes with affine forms by the following rules.

1. Affine operations are exact: $\hat x \pm \hat y = (x_0 \pm y_0) + \sum (x_i \pm y_i) \varepsilon_i$ and $c \hat x + d = (c x_0 + d) + \sum c x_i \varepsilon_i$ for reals $c, d$.
2. A nonlinear operation $g$ is replaced by an affine form $\hat z = z_0 + \sum z_i \varepsilon_i + z_{k+1} \varepsilon_{k+1}$ with a new noise symbol $\varepsilon_{k+1}$. It is chosen so that $g(x, y, \ldots) \in \hat z$ for all values of $\varepsilon_1, \ldots, \varepsilon_k$ in $[-1, 1]$.
3. For multiplication one may take $\hat x \hat y = x_0 y_0 + \sum_{i=1}^{k} (x_0 y_i + y_0 x_i) \varepsilon_i + r_x r_y \, \varepsilon_{k+1}$, where $r_x, r_y$ are the radii of $\hat x, \hat y$.

## Remarks

Example: $\hat x = 2 + \varepsilon_1$ represents $x \in [1, 3]$, and $\hat x - \hat x = 0$ by rule 1, while [[interval arithmetic|QPKKTE1ZY0]] gives $[1, 3] - [1, 3] = [-2, 2]$.
Example: $\hat x \cdot \hat x = 4 + 4\varepsilon_1 + \varepsilon_2$ by rule 3, with range $[-1, 9]$, which contains the true range $[1, 9]$ of $x^2$.
Non-example: replacing $x^2$ by its tangent $4 + 4\varepsilon_1$ alone, without a new noise term, violates rule 2: at $\varepsilon_1 = 1$ it gives $8$, but $x^2 = 9$.
In a computer the coefficients are rounded outward, and the rounding errors are also absorbed into a new noise term.
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# Discrete differential geometry

## Summary

Discrete differential geometry builds discrete counterparts of notions of differential geometry, such as curvature, geodesics and conformal maps.
It works with polyhedral surfaces, meshes and other finite structures.
A good discrete notion keeps the key theorems of the smooth theory exactly, not only in the limit.
The angle defect at a vertex of a [[polyhedron|0325SGSJAR]] plays the role of [[Gaussian curvature|SQVQ5DQT6W]], and [[circle packings|S360JBEYKD]] serve as discrete conformal maps.
The field supplies the mathematics behind computer graphics, geometry processing and architectural design.

## Explanations

### L1 human

Smooth surfaces are approximated on computers by surfaces made of flat triangles.
Discrete differential geometry asks how to define curvature and related ideas directly on such triangle meshes.
At a vertex of a polyhedral surface, the curvature is measured by how much the angles around the vertex fall short of a full turn $2\pi$.
At a corner of a cube three right angles meet, so the defect is $2\pi - 3 \cdot \frac{\pi}{2} = \frac{\pi}{2}$.
The eight corners give a total of $4\pi$, exactly as the Gauss–Bonnet theorem predicts for a sphere; this is [[Descartes's theorem|PT67C5ANSD]].
</document>

<document source="https://tamnd.github.io/math2/node/DK3492C42N.md" tag="DK3492C42N" kind="definition">
# Doob's h-transform

## Definition

Let $(E, \mathcal{E})$ be a measurable space and let $(P_t)_{t \ge 0}$ be a sub-Markov transition semigroup on $E$.
So the $P_t$ are kernels as for [[Markov kernels|2AVNDP8BJP]] but with $P_t(x, E) \le 1$, and $P_s P_t = P_{s+t}$.
Let $h \colon E \to (0, \infty)$ be a measurable function that is [[excessive|46436KE7NT]] for $(P_t)$.
The *h-transform* of $(P_t)$ is the family of kernels
$$P^h_t(x, dy) = \frac{h(y)}{h(x)} \, P_t(x, dy), \qquad t \ge 0, \ x \in E.$$
A Markov process with transition semigroup $(P^h_t)$ is called an *h-process*.

## Remarks

The kernels $P^h_t$ form a sub-Markov semigroup.
Indeed, $P^h_t(x, E) = P_t h(x)/h(x) \le 1$ since $h$ is excessive, and
$$\int P^h_s(x, dy) P^h_t(y, dz) = \frac{h(z)}{h(x)} \int P_s(x, dy) P_t(y, dz) = \frac{h(z)}{h(x)} P_{s+t}(x, dz).$$
If $P_t h = h$ for all $t$, then each $P^h_t$ is a Markov kernel and the h-process has infinite lifetime.
Example: let $(P_t)$ be [[Brownian motion|22735FGJV6]] on $(0, \infty)$ killed at $0$, and $h(x) = x$.
Then $P_t h(x) = E^x[B_t ; t < \tau_0] = x$ by optional stopping, so $h$ is invariant, and the h-process is the three-dimensional Bessel process, Brownian motion conditioned never to hit $0$.
Non-example: for Brownian motion on $\R$ and $h(x) = 1 + x^2$, one has $P_t h(x) = 1 + x^2 + t > h(x)$, so $h$ is not excessive and $P^h_t(x, \R) > 1$.
If $L$ is the [[infinitesimal generator|VPA41PWRR1]] of $(P_t)$, the generator of the h-process acts by $L^h f = h^{-1} L(h f)$ on suitable $f$.
For $L = \frac12 \Delta$ and harmonic $h$, this is $L^h f = \frac12 \Delta f + \nabla(\log h) \cdot \nabla f$, a Brownian motion with drift $\nabla \log h$.
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<document source="https://tamnd.github.io/math2/node/DK3MT3HYX5.md" tag="DK3MT3HYX5" kind="concept">
# Absolute neighborhood retract

## Summary

An absolute neighborhood retract, or ANR, is a [[metrizable space|091GJ3FYD7]] with the following property.
Whenever it sits as a closed subset in a metrizable space, it is a retract of some neighbourhood of itself.
ANRs include manifolds, polyhedra, open subsets of Banach spaces and convex subsets of normed spaces.
They are the spaces with good local behaviour for homotopy theory.
They are locally contractible, and compact ANRs have the homotopy type of finite CW complexes.

## Explanations

### L1 human

A retraction pushes a big space onto a subspace while keeping each point of the subspace fixed.
Place an ANR as a closed set inside a larger metric space.
Then some thin open layer around it can be pushed onto it by a retraction.
The interval $[0, 1]$ is an ANR, and so is the circle.
A set with infinitely many points piling up, like $\set{0} \cup \set{1/n : n \ge 1}$, is not an ANR.
A thin layer around it in $\R$ contains a whole interval around $0$, and this interval cannot be pushed continuously onto separate points.
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<document source="https://tamnd.github.io/math2/node/DK3Q4RPRZJ.md" tag="DK3Q4RPRZJ" kind="concept">
# Nilpotent operator

## Summary

A nilpotent operator is a bounded linear operator $T$ on a Banach space with $T^n = 0$ for some positive integer $n$.
It is the operator version of a [[nilpotent endomorphism|063D5Z2C70]] and of a [[nilpotent matrix|07BSPZPSMF]].
On a nonzero space its spectrum is $\set{0}$.
Operators with spectrum $\set{0}$ are called quasinilpotent; every nilpotent operator is quasinilpotent, but the Volterra operator is quasinilpotent without being nilpotent.

## Explanations

### L1 human

A nilpotent operator dies after a fixed number of applications.
On pairs of sequences, the map $(x, y) \mapsto (0, x)$ is an example: applied twice it gives $(0, 0)$.
In infinite dimensions one must be careful: an operator can shrink everything faster and faster under powers without ever reaching $0$, and such an operator is not nilpotent.
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# Nilpotent Lie group

## Summary

A nilpotent Lie group is a connected [[Lie group|7JBRGCVJRB]] that is a [[nilpotent group|0Q7MMQHC1P]].
So its lower central series reaches the trivial group after finitely many steps.
Equivalently, its Lie algebra is a nilpotent Lie algebra.
Every nilpotent Lie group is a [[solvable Lie group|9CNSVFQ801]], but not conversely.
For a connected and simply connected nilpotent Lie group, the exponential map from the Lie algebra is a diffeomorphism.
So such a group can be studied almost entirely through its Lie algebra.

## Explanations

### L1 human

A nilpotent Lie group is close to being commutative.
The commutators of its elements may be nontrivial, but commutators with commutators, repeated a fixed number of times, are always trivial.
The basic example is the Heisenberg group of upper triangular $3 \times 3$ matrices with $1$ on the diagonal.
There, any two elements commute up to a central element, and central elements commute with everything.
Groups of this kind model the position and momentum operators of quantum mechanics.
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<document source="https://tamnd.github.io/math2/node/DKFGQTRYZE.md" tag="DKFGQTRYZE" kind="concept">
# Sentence

## Summary

A sentence is a [[formula|HYJ10AF6RT]] with no free variables.
Every variable in it is bound by a quantifier, so it makes a complete claim.
In a given [[structure|983GZBQYT4]] a sentence is simply true or false, with no values to choose.
Axioms of theories and theorems of mathematics are sentences.

## Explanations

### L1 human

The formula $x > 0$ is neither true nor false until you know $x$.
The sentence $\forall x \, (x \cdot x \ge 0)$ is a full claim: it is true in the real numbers.
A sentence is a formula in which every variable is tied to a "for all" or a "there exists".
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<document source="https://tamnd.github.io/math2/node/DKH4933HMQ.md" tag="DKH4933HMQ" kind="concept">
# Dedekind–MacNeille completion

## Summary

The Dedekind–MacNeille completion of a partially ordered set is the smallest [[complete lattice|D7A0ZGZT5S]] that contains it as an ordered subset.
It is built from cuts, pairs of sets of lower and upper bounds.
The real numbers are built in the same way from [[Dedekind cuts|0G3RRM41X9]] of the rationals.
Every existing supremum and infimum of the original poset is kept in the completion.

## Explanations

### L1 human

The rational numbers have gaps, such as the place where $\sqrt{2}$ should be.
Dedekind filled these gaps by describing each missing number by the rationals below it.
MacNeille did the same for any ordered set: each new element is a set of elements that is exactly the set of lower bounds of its own upper bounds.
For the rationals, the completion is the real line with $-\infty$ and $+\infty$ added.
For a finite lattice nothing new is added, because it is already complete.
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<document source="https://tamnd.github.io/math2/node/DKN15NVKSB.md" tag="DKN15NVKSB" kind="concept">
# Cahn–Hilliard equation

## Summary

The Cahn–Hilliard equation is the fourth-order parabolic equation $c_t = \Delta(c^3 - c - \gamma \Delta c)$.
It describes how a mixture of two substances separates into regions rich in one or the other.
Unlike the [[Allen–Cahn equation|7S813PEV69]], it conserves the total amount $\int c\,dx$.
It is the gradient flow of the Ginzburg–Landau energy in the $H^{-1}$ metric.

## Explanations

### L1 human

Shake oil and vinegar together, then let the mixture rest: droplets form and slowly grow.
The Cahn–Hilliard equation models this.
The concentration $c$ is near $+1$ in one substance and near $-1$ in the other.
Matter is only moved around, never created, so the average of $c$ stays fixed while the pattern coarsens.
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# Shooting method

## Definition

Let $a < b$, let $\alpha, \beta \in \R$, and let $f \colon [a, b] \times \R^2 \to \R$ be continuous and locally Lipschitz in its last two arguments.
Consider the [[boundary value problem|TM8VQJWGHZ]]
$$y'' = f(x, y, y'), \qquad y(a) = \alpha, \qquad y(b) = \beta.$$
For $s \in \R$, let $y(\cdot\,; s)$ be the solution of the [[initial value problem|0FXDW992DS]] $y'' = f(x, y, y')$, $y(a) = \alpha$, $y'(a) = s$.
Let $S$ be the set of $s$ for which this solution exists on all of $[a, b]$, and define $F \colon S \to \R$ by
$$F(s) = y(b; s) - \beta.$$
The shooting method is the procedure that finds a root $s^*$ of $F$ and returns $y(\cdot\,; s^*)$.
The root is found, for example, by bisection, by the secant method or by [[Newton's method|07EXVNF9VP]].

## Remarks

Each root $s^*$ of $F$ gives a solution $y(\cdot\,; s^*)$ of the boundary value problem, and each solution arises this way from $s^* = y'(a)$.
Example: for $y'' = 0$, $y(0) = 0$, $y(1) = 3$, one gets $y(x; s) = s x$ and $F(s) = s - 3$, so $s^* = 3$ and $y = 3x$.
Non-example of success: for $y'' = -y$, $y(0) = 0$, $y(\pi) = 1$, one gets $y(x; s) = s \sin x$ and $F(s) = -1$ for every $s$.
Then $F$ has no root, and indeed the boundary value problem has no solution.
When the solutions of the initial value problems grow fast, small errors in $s$ blow up.
Then one splits $[a, b]$ into pieces and shoots on each piece; this is called multiple shooting.

## Explanations

### L1 human

The method guesses the slope at the left end, solves the equation from there, and looks at where the solution ends up on the right.
The miss distance is a function of the guess.
Making the miss zero is a problem of finding a root of one function of one variable.
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# Generalized polygon

## Definition

Let $\mathcal{S} = (P, L, I)$ be an [[incidence structure|FGKNAFNHA3]] with point set $P$, line set $L$ and incidence relation $I \subseteq P \times L$.
Its *incidence graph* is the [[bipartite graph|0ZVQKTRKVQ]] with vertex set $P \cup L$, with an edge between $p \in P$ and $\ell \in L$ when $(p, \ell) \in I$.
Let $n \ge 2$ be an integer.
The structure $\mathcal{S}$ is a *generalized $n$-gon* if its incidence graph is connected, has diameter $n$, and has [[girth|D13K9GJT0B]] $2n$.
Here the diameter is the largest distance between two vertices of the [[graph|00C0BR7D7H]].
A generalized $n$-gon is *thick* if every point lies on at least three lines and every line has at least three points.

## Remarks

Example: the [[Fano plane|QQ8GW1ASZM]] is a generalized $3$-gon.
Two points lie on a common line, and two lines meet in a point, so in both cases the distance is $2$.
A point and a line not through it are at distance $3$, so the diameter is $3$.
A cycle of length $4$ would give two points on two common lines, which is impossible.
A triangle of the plane gives a cycle of length $6$, so the girth is $6$.
It is thick, since each line has $3$ points and each point is on $3$ lines.

Non-example: the affine plane of order $2$, with $4$ points and the $6$ lines of $2$ points each.
Its girth is $6$, but two parallel lines have no common point, so they are at distance $4$; the diameter is $4 \ne 3$.

For $n \ge 3$, any two points lie on at most one line, so a generalized $n$-gon is a partial linear space.
The generalized $3$-gons with at least three points on each line and three lines on each point are exactly the [[projective planes|DANFNN8C4Y]], and the generalized $4$-gons are the generalized quadrangles.
A finite thick generalized $n$-gon has an order $(s, t)$: every line has $s + 1$ points and every point is on $t + 1$ lines.
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# p-Laplacian

## Definition

Let $n \ge 1$, let $\Omega \subseteq \R^n$ be an open set, and let $1 < p < \infty$.
For a function $u \colon \Omega \to \R$ whose [[gradient|0ABK9V9Y64]] $\nabla u$ is regular enough, the *$p$-Laplacian* of $u$ is
$$\Delta_p u = \operatorname{div} \big( \abs{\nabla u}^{p-2} \nabla u \big),$$
where $\operatorname{div}$ is the [[divergence|0MWV6ZTK6S]].
Let $u$ be in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}_{\mathrm{loc}}(\Omega)$.
The equation $\Delta_p u = 0$ holds in the [[weak sense|SJZX81A783]] if
$$\int_\Omega \abs{\nabla u}^{p-2} \, \nabla u \cdot \nabla \varphi \, dx = 0$$
for every smooth function $\varphi$ with compact support in $\Omega$.
Such a $u$ is called *$p$-harmonic*.

## Remarks

For $p = 2$, $\Delta_2 = \Delta$ is the [[Laplacian|0E0B6RA8K0]].
Example: on $\R^n \setminus \set{0}$ with $p \ne n$, the function $u(x) = \abs{x}^{(p-n)/(p-1)}$ is $p$-harmonic.
Indeed, for $u = r^\beta$ with $r = \abs{x}$, the field $\abs{\nabla u}^{p-2} \nabla u$ is radial with size $\abs{\beta}^{p-1} r^{(\beta - 1)(p-1)}$, and its divergence vanishes when $r^{n-1} r^{(\beta-1)(p-1)}$ is constant, that is, when $\beta = (p - n)/(p - 1)$.
Non-example: for $p = 3$, the function $u(x) = x_1^2$ is not $3$-harmonic, since $\abs{\nabla u} \nabla u = (4 x_1 \abs{x_1}, 0, \ldots, 0)$ has divergence $8 \abs{x_1} \ne 0$ for $x_1 \ne 0$.
The weak form is the Euler–Lagrange equation of $\int_\Omega \abs{\nabla u}^p \, dx$.
Some authors call $-\Delta_p$ the $p$-Laplacian.
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# Meijer G-function

## Definition

Let $m, n, p, q$ be integers with $0 \le m \le q$ and $0 \le n \le p$.
Let $a_1, \dots, a_p$ and $b_1, \dots, b_q$ be complex numbers such that $a_k - b_j$ is not a positive integer for $1 \le k \le n$ and $1 \le j \le m$.
Let $\Gamma$ be the [[gamma function|0H5SYBSED9]], and let $z \ne 0$ be complex with a fixed branch of $\log z$.
The *Meijer G-function* is the [[Barnes integral|1PDGBGEHM2]]
$$G^{m, n}_{p, q}\left( \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \,\middle|\, z \right) = \frac{1}{2\pi i} \int_L \frac{\prod_{j = 1}^{m} \Gamma(b_j - s) \prod_{k = 1}^{n} \Gamma(1 - a_k + s)}{\prod_{j = m + 1}^{q} \Gamma(1 - b_j + s) \prod_{k = n + 1}^{p} \Gamma(a_k - s)}\, z^{s}\, ds.$$
The path $L$ separates the poles $s = b_j + \ell$ of the factors $\Gamma(b_j - s)$ from the poles $s = a_k - 1 - \ell$ of the factors $\Gamma(1 - a_k + s)$, where $\ell \in \N$.
Call these the $b$-poles and the $a$-poles.
The path $L$ is one of the following:

1. a path from $-i\infty$ to $+i\infty$, where the integral converges;
2. a loop from $+\infty$ to $+\infty$ around all $b$-poles once in the negative direction, when $q \ge 1$ and either $p < q$, or $p = q$ and $\abs{z} < 1$;
3. a loop from $-\infty$ to $-\infty$ around all $a$-poles once in the positive direction, when $p \ge 1$ and either $p > q$, or $p = q$ and $\abs{z} > 1$.

## Remarks

The condition on $a_k - b_j$ makes the two families of poles disjoint, so a separating path exists.
When more than one choice of $L$ is possible, they give the same value.
Example: $G^{1, 0}_{0, 1}(- ; 0 \mid z) = e^{-z}$, where $-$ marks the empty list of parameters $a_k$.
Check: the integrand is $\Gamma(-s) z^s$, and with the loop of type 2 the residues at $s = \ell$ give $\sum_{\ell \ge 0} \frac{(-1)^\ell}{\ell!} z^\ell$.
Non-example: $m = 2$ with $q = 1$ is not allowed, since $m \le q$ fails; there are not enough parameters $b_j$ for two factors $\Gamma(b_j - s)$.
Every [[generalized hypergeometric function|VDBAW9PN88]] is a G-function: if no $a_k$ is a nonpositive integer, no $b_j$ is a nonpositive integer, $p \le q + 1$, and $\abs{z} < 1$ when $p = q + 1$, then
$${}_pF_q(a_1, \dots, a_p; b_1, \dots, b_q; z) = \frac{\prod_{j} \Gamma(b_j)}{\prod_{k} \Gamma(a_k)}\, G^{1, p}_{p, q + 1}\left( \begin{matrix} 1 - a_1, \dots, 1 - a_p \\ 0, 1 - b_1, \dots, 1 - b_q \end{matrix} \,\middle|\, -z \right).$$
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# Clifford analysis

## Summary

Clifford analysis extends complex analysis to $\R^n$ by letting functions take values in a [[Clifford algebra|A38B1QFK20]].
Its central objects are the [[monogenic functions|646PNPT1R6]], the functions annihilated by a [[Dirac operator|3DXGJTEQZH]].
The square of the Dirac operator is the Laplacian up to sign.
So the theory refines harmonic analysis, as holomorphic functions refine harmonic functions in the plane.
It has a Cauchy integral formula, Taylor and Laurent series, and links to singular integrals, spin geometry and representation theory.
It grew out of [[quaternionic analysis|SAT7ST5WP7]], which is closely related to its low-dimensional cases.

## Explanations

### L1 human

In the plane, holomorphic functions are the solutions of the Cauchy–Riemann equations, a first-order system whose "square" is the Laplace equation.
In higher dimensions there is no such system with complex numbers alone.
A Clifford algebra supplies anticommuting units $e_1, \ldots, e_n$ with $e_j^2 = -1$, and with them the operator $\sum_j e_j \partial_{x_j}$ squares to minus the Laplacian.
Functions killed by this operator behave much like holomorphic functions: they have an integral formula, power series, and a maximum principle.
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# Open subgroups are closed

## Statement

Let $G$ be a [[topological group|92700QAB5Y]] and let $H$ be an [[open subgroup|QYCFH68RD1]] of $G$.
Then $H$ is a [[closed set|0ZCK8DQYN1]] in $G$.

## Remarks

The [[left cosets|03DEH43F4H]] $gH$ are open, because left translation by $g$ is a homeomorphism.
The complement of $H$ is the union of the cosets $gH$ with $g \notin H$, so it is open, and $H$ is closed.
Conversely, a closed subgroup of finite [[index|0DTXBQPA2S]] is open, since its complement is a finite union of closed cosets.
Example: in the $p$-adic integers $\Z_p$, each subgroup $p^n \Z_p$ is open, hence also closed.
The converse of the statement fails: $\Z$ is a closed subgroup of $\R$ that is not open.

## Explanations

### L1 human

If a subgroup of a topological group is open, it is automatically closed as well.
The reason is that the group splits into copies of the subgroup, all shifted, and each copy is open.
The part outside the subgroup is a union of these open copies, so it is open too.

### L2 human

A first consequence: a connected topological group has no open subgroup except itself, since an open subgroup is a nonempty set that is both open and closed.
For instance, a connected group is generated by any neighbourhood of the identity, because the subgroup generated by an open set is open.
In profinite groups the open subgroups are exactly the closed subgroups of finite index, which links the topology with finite quotients.
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# Hadamard matrix

## Definition

Let $n$ be a positive integer.
A Hadamard matrix of order $n$ is an $n \times n$ [[matrix|0N1MY1T0W9]] $H$ with entries in $\set{1, -1}$ such that $H H^T = n I_n$.
Here $H^T$ is the [[transpose|0W809K6AW0]] of $H$ and $I_n$ is the [[identity matrix|06DW9BAWB1]] of size $n$.

## Remarks

The condition says that any two distinct rows have dot product $0$, so they agree in exactly $n/2$ places.
Since $H^{-1} = H^T / n$, also $H^T H = n I_n$, so the columns are pairwise orthogonal as well.
Example: $H_2 = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ satisfies $H_2 H_2^T = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = 2 I_2$.
The [[Kronecker product|0JK6M18QZH]] of two Hadamard matrices is a Hadamard matrix, so the Kronecker powers of $H_2$ give Hadamard matrices of every order $2^k$.
Non-example: no $3 \times 3$ matrix with entries $\pm 1$ is Hadamard, since the dot product of two rows is a sum of three terms $\pm 1$, which is odd and so not $0$.
A Hadamard matrix of order $n$ can exist only if $n = 1$, $n = 2$ or $n$ is a multiple of $4$; the [[Hadamard conjecture|6X9S10E3M2]] asks whether all multiples of $4$ occur.
For such $H$, $\abs{\det H} = n^{n/2}$.
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# Breakdown point

## Definition

Let $T$ be an [[estimator|0TTNRP5PFQ]] that maps each sample $x = (x_1, \dots, x_n)$ of points in $\R^p$ to a value $T(x) \in \R^k$.
For $0 \le m \le n$, let $\mathcal{Y}_m(x)$ be the set of samples obtained from $x$ by replacing at most $m$ points by arbitrary points of $\R^p$.
The *finite-sample breakdown point* of $T$ at $x$ is
$$\varepsilon^*_n(T, x) = \min\Big\{ \frac{m}{n} \;:\; \sup_{y \in \mathcal{Y}_m(x)} \norm{T(y) - T(x)} = \infty \Big\}.$$
The *asymptotic breakdown point* is the limit of $\varepsilon^*_n$ as $n \to \infty$, when it exists.

## Remarks

Example: the sample median of real data has $\varepsilon^*_n = \floor{(n+1)/2}/n$.
For $n = 2k + 1$, changing $k$ points leaves an original point at the middle position, between original values; changing $k + 1$ points to a huge value $M$ makes the median $M$.
So the asymptotic breakdown point of the [[median|08W2CW8PKW]] is $1/2$.

Non-example of robustness: the [[sample mean|0QNRBHWV35]] has $\varepsilon^*_n = 1/n$, since replacing one point $x_1$ by $x_1 + nM$ moves the mean by $M$.
Its asymptotic breakdown point is $0$.

The $k$-trimmed mean, which drops the $k$ smallest and $k$ largest values, has $\varepsilon^*_n = (k+1)/n$.
For estimators of location that are translation equivariant, the asymptotic breakdown point is at most $1/2$.
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# Delta method

## Statement

Let $\theta \in \R$ and $\sigma^2 > 0$.
Let $(T_n)$ be a sequence of real random variables such that
$$\sqrt{n}\,(T_n - \theta) \xrightarrow{d} \mathcal{N}(0, \sigma^2),$$
where $\xrightarrow{d}$ is [[convergence in distribution|0JDAQHWA9N]] and $\mathcal{N}(0, \sigma^2)$ is the [[normal distribution|07RXBVTVZB]].
Let $g \colon \R \to \R$ be [[differentiable|0A61B60B5R]] at $\theta$.
Then
$$\sqrt{n}\,\big(g(T_n) - g(\theta)\big) \xrightarrow{d} \mathcal{N}\big(0, g'(\theta)^2 \sigma^2\big).$$

## Remarks

Idea of proof: by differentiability, $g(T_n) - g(\theta) = g'(\theta)(T_n - \theta) + r(T_n)(T_n - \theta)$ with $r(t) \to 0$ as $t \to \theta$.
Since $T_n \to \theta$ in probability, $r(T_n) \to 0$ in probability, and $\sqrt{n}(T_n - \theta)$ is bounded in probability.
So the remainder times $\sqrt{n}$ tends to $0$ in probability, and Slutsky's lemma gives the limit $g'(\theta)\,\mathcal{N}(0, \sigma^2)$.

Multivariate form: if $\sqrt{n}(T_n - \theta) \xrightarrow{d} \mathcal{N}_k(0, \Sigma)$ in $\R^k$, with a [[multivariate normal|0G558GJQMW]] limit, and $g \colon \R^k \to \R^m$ is [[differentiable|0135XZZQ8M]] at $\theta$ with [[Jacobian matrix|047NDTEPCV]] $J$, then $\sqrt{n}(g(T_n) - g(\theta)) \xrightarrow{d} \mathcal{N}_m(0, J \Sigma J^T)$.

Example: for i.i.d. [[Poisson|0VEFSSKX18]] variables with mean $\lambda > 0$, the [[central limit theorem|0EPZC4Z1FC]] gives $\sqrt{n}(\bar{X}_n - \lambda) \xrightarrow{d} \mathcal{N}(0, \lambda)$.
With $g(t) = 2\sqrt{\abs{t}}$, $g'(\lambda) = 1/\sqrt{\lambda}$, so $\sqrt{n}\big(2\sqrt{\bar{X}_n} - 2\sqrt{\lambda}\big) \xrightarrow{d} \mathcal{N}(0, 1)$.
The limit variance no longer depends on $\lambda$: the square root is a variance-stabilizing transformation for Poisson data.

If $g'(\theta) = 0$, the limit is the point mass at $0$, and one needs a second-order expansion: if $g''(\theta) \ne 0$ exists, then $n(g(T_n) - g(\theta)) \xrightarrow{d} \frac{1}{2} g''(\theta) \sigma^2 \chi^2_1$.

## Explanations

### L1 human

If an estimate is approximately normal around the truth, then a smooth function of the estimate is also approximately normal around the function of the truth.
Near the true value, a smooth function looks like a straight line with slope $g'(\theta)$, and a straight line stretches the spread by that slope.
So the standard deviation is multiplied by $\abs{g'(\theta)}$.

### L2 human

The delta method turns the central limit theorem for averages into limit laws for smooth functions of averages: ratios, logarithms, correlation coefficients, odds ratios.
It gives approximate standard errors and confidence intervals for such quantities.
It is also the step that passes from the asymptotic normality of an estimator $\hat{\theta}$ to that of $g(\hat{\theta})$, for example for maximum likelihood estimators.
Variance-stabilizing transformations are chosen with it: one picks $g$ so that $g'(\theta)^2 \sigma^2(\theta)$ does not depend on $\theta$.
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# Chebyshev constant

## Summary

The Chebyshev constant of a compact set $K$ in the plane measures how small a monic polynomial can be on $K$.
For each degree $n$ one takes the least maximum of $\abs{p}$ on $K$ over [[monic polynomials|0C3QBCR46M]] $p$ of degree $n$.
Then one takes its $n$-th root and lets $n$ tend to infinity.
The extremal polynomials generalize the classical [[Chebyshev polynomials|M6EAR1JHPP]] of an interval.
A theorem of Fekete and Szegő says that the Chebyshev constant equals the [[logarithmic capacity|DBQXRTC0VN]] of $K$.

## Explanations

### L1 human

A monic polynomial of degree $n$ has leading coefficient $1$, so it cannot be small everywhere: far from the origin it is about $\abs{z}^n$.
The question is how small it can stay on a given set $K$.
On the unit circle the best choice is $z^n$, with maximum $1$, so the Chebyshev constant is $1$.
On the interval $[-1, 1]$ the best choice is the Chebyshev polynomial divided by $2^{n-1}$, with maximum $2^{1-n}$, so the Chebyshev constant is $1/2$.
A small set lets monic polynomials stay small, and so has a small Chebyshev constant.
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# Random dynamical system

## Summary

A random dynamical system is a [[dynamical system|BBZP1R872N]] whose rule of evolution is chosen at random at each time.
The randomness comes from a noise that is itself a measure-preserving dynamical system.
The evolution is a cocycle over the noise.
The state after time $s + t$ is obtained by first evolving for time $s$ and then evolving for time $t$ with the shifted noise.
Stochastic differential equations and random iterations of maps give random dynamical systems.
The framework allows notions of deterministic dynamics, such as attractors, invariant measures and [[Lyapunov exponents|NJ1FJSNK6A]], to be used for systems with noise.

## Explanations

### L1 human

Imagine moving a point by one of two maps, and choosing which map by tossing a coin each time.
Each sequence of coin tosses gives one sequence of maps, and the point follows the composition of these maps.
A random dynamical system keeps track of all coin sequences at once, together with their probabilities.
The rule that ties it together is simple.
Running for $s$ steps and then $t$ steps is the same as running for $s + t$ steps, as long as the coin tosses are read in the right order.
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# Pointed cone

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]].
A *pointed cone* in $V$ is a [[convex cone|6VE4KA12VD]] $C \subseteq V$ with $0 \in C$.

## Remarks

Equivalently, $C$ is nonempty and $\alpha x + \beta y \in C$ for all $x, y \in C$ and all real $\alpha, \beta \ge 0$.
In this form a pointed cone is a submodule of $V$ over the semiring of nonnegative reals.

Example: the closed upper half-plane $H = \set{(x, y) \in \R^2 : y \ge 0}$.
It is closed under sums and positive multiples, and it contains $(0, 0)$.
It contains the whole $x$-axis, so it is pointed in this sense but not salient.

Non-example: the open quadrant $\set{(x, y) : x > 0, y > 0}$ is a convex cone, but it does not contain $(0, 0)$.

Warning on terminology: many texts on convex analysis say pointed for a convex cone $C$ with $C \cap (-C) \subseteq \set{0}$, that is, a cone that contains no line.
That property is called salient here.
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# Linear algebra method

## Summary

The linear algebra method bounds the size of a combinatorial family by attaching to each member a vector, or a polynomial, in a [[vector space|000AYPA9AJ]].
One then proves that these vectors are [[linearly independent|0CRMGPAH83]].
The size of the family is then at most the [[dimension|03GSZ8RTZJ]] of the space.
Classic results proved this way include [[Fisher's inequality|M9XF82GGW8]] and the Oddtown theorem.
Its polynomial versions, such as the [[combinatorial Nullstellensatz|9ZW7BCJMDT]] and the slice rank argument for the [[cap set problem|1SF8Y9M8YB]], form the polynomial method in combinatorics.

## Explanations

### L1 human

To show that a family cannot be large, turn each of its members into a vector and show that no vector is a combination of the others.
Then there can be no more vectors than the dimension of the space.
A standard example is the Oddtown theorem.
Let $\mathcal{F}$ be a family of subsets of $\set{1, \ldots, n}$ in which every set has odd size and any two distinct sets meet in an even number of elements.
Over the [[finite field|0ZXZ82FTX2]] $\F_2$, the indicator vectors $v_A$ satisfy $v_A \cdot v_A = 1$ and $v_A \cdot v_B = 0$ for $A \ne B$, so they are linearly independent, and $|\mathcal{F}| \le n$.
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# The local ring of holomorphic germs is Noetherian and factorial

## Statement

Let $n \ge 0$ and let $\mathcal{O}_n = \C\{z_1, \dots, z_n\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]], which is isomorphic to the ring of [[germs of holomorphic functions|7PJ8KVZZCQ]] at $0 \in \C^n$.
Then:

1. $\mathcal{O}_n$ is a [[local ring|MZ02BKPHJA]] with maximal ideal $\mathfrak{m} = (z_1, \dots, z_n)$.
2. $\mathcal{O}_n$ is a [[Noetherian ring|0670T59HGE]].
3. $\mathcal{O}_n$ is a [[unique factorization domain|0W4X9G6263]].

## Remarks

Part 1 holds because a series with nonzero constant term has a convergent inverse.
Parts 2 and 3 are proved together by induction on $n$, with the Weierstrass preparation and division theorems as the main tools.
For a nonzero ideal $I$, after a linear change of coordinates $I$ contains an element $f$ regular of order $d$ in $z_n$.
Division by $f$ shows that $\mathcal{O}_n/(f)$ is a finitely generated module over $\mathcal{O}_{n-1}$, which is Noetherian by induction, so $I/(f)$ and hence $I$ is finitely generated.
For factoriality, preparation reduces factorization in $\mathcal{O}_n$ to factorization of Weierstrass polynomials in $\mathcal{O}_{n-1}[z_n]$, which is a unique factorization domain by Gauss's lemma, since $\mathcal{O}_{n-1}$ is one by induction.
Example: in $\mathcal{O}_2$, the germ $z_1^2 - z_2^3$ is irreducible, while $z_2^2 - z_1^2(1 + z_1)$ factors as $\big(z_2 - z_1\sqrt{1 + z_1}\big)\big(z_2 + z_1\sqrt{1 + z_1}\big)$, where $\sqrt{1 + z_1}$ is the convergent binomial series.
The ring of holomorphic functions on an open set, rather than germs, is not Noetherian.
On $\C$, the entire functions vanishing at all but finitely many of the points $1, 2, 3, \dots$ form an ideal that is not finitely generated.

## Explanations

### L1 human

Near a point, holomorphic functions behave algebraically like polynomials.
Every ideal of germs is generated by finitely many germs, and every germ factors into irreducible germs in an essentially unique way.
This is what makes it possible to study the local shape of zero sets of holomorphic functions with algebra.

### L2 human

The theorem is the starting point of local analytic geometry.
Noetherianity gives that every germ of an analytic set is a finite union of irreducible germs, and that descending chains of analytic germs stabilize.
Factoriality shows that the germ of a hypersurface $\set{f = 0}$ decomposes according to the irreducible factors of $f$.
The ring $\mathcal{O}_n$ is moreover regular of dimension $n$, and Oka's coherence theorem extends its Noetherian property to the sheaf of holomorphic functions.
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# Pontryagin class

## Summary

The Pontryagin classes $p_i(E) \in H^{4i}(B; \Z)$ are [[characteristic classes|7HACJC0JE2]] of real vector bundles $E \to B$ with integer coefficients.
They are defined from the [[Chern classes|PAD37GKDPD]] of the complexification $E \otimes \C$.
Pontryagin classes of the tangent bundle of a manifold give the Pontryagin numbers, which are invariants of oriented cobordism.
They enter the Hirzebruch signature theorem and Milnor's construction of exotic spheres.

## Explanations

### L1 human

A real vector bundle can be turned into a complex one by allowing complex scalars in each fiber.
Complex bundles have Chern classes, which are integer cohomology classes.
The Pontryagin classes are the Chern classes of this complexified bundle in degrees divisible by four, with a sign.
For the complex projective plane, the first Pontryagin class is three times the square of the generator, and this three is the reason its signature is one.
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# Transformation semigroup

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]].
Let $T_X$ be the set of all [[maps|0XHDTE5YMZ]] $X \to X$, with the product $fg = f \circ g$ given by [[composition|0THBK9V825]].
A *transformation semigroup* on $X$ is a nonempty subset $S \subseteq T_X$ such that $f \circ g \in S$ for all $f, g \in S$.
It is often written as the pair $(X, S)$.

## Remarks

Composition of maps is associative, so a transformation semigroup is a [[semigroup|0XMEM406JM]].
The whole set $T_X$ is the full transformation [[monoid|0N6R902FG1]]; for $X$ with $n$ elements it has $n^n$ elements.
Its invertible elements form the [[symmetric group|0515A6D2Y1]] on $X$.
Every semigroup $S$ is isomorphic to a transformation semigroup on the set $S^1$, by sending $a$ to the map $x \mapsto ax$.

Example: the constant maps $c_a \colon x \mapsto a$, for $a \in X$.
For all $a, b$, $c_a \circ c_b = c_a$, so they form a transformation semigroup.

Non-example: on $X = \set{1, 2}$, let $s$ swap $1$ and $2$, and let $c_1$ be the constant map to $1$.
The set $\set{\id, s, c_1}$ is not closed under composition, because $s \circ c_1 = c_2$ is not in it.
</document>

<document source="https://tamnd.github.io/math2/node/DN5G4NAF95.md" tag="DN5G4NAF95" kind="definition">
# Computable set

## Definition

Let $A \subseteq \N$ be a [[subset|0BJR9QR08E]] of the natural numbers.
The characteristic function of $A$ is $\chi_A \colon \N \to \set{0, 1}$, with $\chi_A(n) = 1$ if $n \in A$ and $\chi_A(n) = 0$ if $n \notin A$.
The set $A$ is *computable*, or *recursive*, or *decidable*, if $\chi_A$ is a total [[computable function|KZSB5E471K]].
The same definition applies to subsets of $\N^k$.

## Remarks

Example: the set of prime numbers is computable.
A machine can test $n \ge 2$ for divisors $d$ with $2 \le d < n$, which takes finitely many steps, and so computes $\chi_A$.

Non-example: the set $K = \set{e : \text{program } e \text{ halts on input } e}$ is not computable, by the undecidability of the [[halting problem|Z7TN2B35B0]].
Its characteristic function is total but not computable.

A set $A$ is computable if and only if both $A$ and $\N \setminus A$ are [[computably enumerable|TN9G24VABN]].
The computable sets are closed under union, intersection and complement.
</document>

<document source="https://tamnd.github.io/math2/node/DN7VSSDYPQ.md" tag="DN7VSSDYPQ" kind="concept">
# Median algebra

## Summary

A median algebra is a set with a ternary operation $\langle x, y, z \rangle$ that behaves like the median of three numbers.
Every [[distributive lattice|0HM0HQCK6N]] is a median algebra with $\langle x, y, z \rangle = (x \wedge y) \vee (y \wedge z) \vee (z \wedge x)$.
The vertex sets of median graphs, such as trees and hypercubes, are median algebras.
There $\langle x, y, z \rangle$ is the unique vertex lying on shortest paths between each pair of the three vertices.
Median algebras give a common language for distributive lattices, median graphs and some structures of geometric group theory.

## Explanations

### L1 human

Given three real numbers, the median is the one in the middle.
It does not depend on the order of the three numbers, and if two of them are equal, the median is that repeated value.
A median algebra keeps these rules and one more, and forgets everything else about numbers.
In a tree, the median of three vertices is the point where the three paths between them meet.
</document>

<document source="https://tamnd.github.io/math2/node/DN98H15VZF.md" tag="DN98H15VZF" kind="statement">
# Nehari's theorem on univalence

## Statement

Let $\mathbb{D}$ be the [[unit disc|8G5ABXRK85]], and let $f \colon \mathbb{D} \to \C$ be [[holomorphic|0DG0Z72CQD]] with $f'(z) \ne 0$ for all $z \in \mathbb{D}$.
Let $S(f)$ be its [[Schwarzian derivative|6ZPKWE0VAZ]].
If
$$\abs{S(f)(z)} \le \frac{2}{(1 - \abs{z}^2)^2} \qquad \text{for all } z \in \mathbb{D},$$
then $f$ is [[univalent|R91E83SDBD]] on $\mathbb{D}$.

## Remarks

Locally, $f = u_1/u_2$ for two independent solutions of the linear equation $u'' + \frac{1}{2} S(f)\, u = 0$.
If $f(a) = f(b)$ with $a \ne b$, then some solution of this equation vanishes at both $a$ and $b$.
A disc automorphism moves $a$ and $b$ onto a diameter.
The bound on $S(f)$ then allows a comparison of Sturm type with the equation $u'' + (1 - x^2)^{-2} u = 0$ on $(-1, 1)$.
That equation has the solution $\sqrt{1 - x^2}$ without zeros, which leads to a contradiction.
The constant $2$ cannot be replaced by any larger constant.
In the other direction, the Kraus–Nehari theorem says that every univalent $f$ on $\mathbb{D}$ satisfies $\abs{S(f)(z)} \le 6/(1 - \abs{z}^2)^2$, with equality at $z = 0$ for the [[Koebe function|EZQFWKF03Q]].
Example: $f(z) = e^{cz}$ has $S(f) = -c^2/2$, so for $\abs{c} \le 2$ the bound holds, since $(1 - \abs{z}^2)^{-2} \ge 1$; such $f$ is univalent on $\mathbb{D}$, as two points of $\mathbb{D}$ cannot differ by $2\pi i k / c$ with $k \ne 0$.

## Explanations

### L1 human

The Schwarzian derivative measures how far a map is from being a Möbius map, and Möbius maps are injective.
Nehari's theorem says that if the Schwarzian derivative is small enough, measured against the size of the disc near the boundary, then the map is injective.
It turns a global property, injectivity, into a pointwise bound that can be checked.

### L2 human

The weight $(1 - \abs{z}^2)^{-2}$ is the square of the density of the hyperbolic metric, so the condition is invariant under automorphisms of the disc.
Together with the necessary bound $6$, it shows that the Schwarzian derivatives of univalent functions all have bounded hyperbolic size, and that all small enough ones come from univalent functions.
This point of view leads to the Bers embedding of the universal Teichmüller space.
It also leads to the theorem of Ahlfors and Weill that a slightly stronger bound gives a quasiconformal extension of $f$ to the plane.
</document>

<document source="https://tamnd.github.io/math2/node/DNBS22EV21.md" tag="DNBS22EV21" kind="definition">
# Dedekind–MacNeille completion

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
For $A \subseteq P$, let $A^u$ be the set of [[upper bounds|0S60DVMVPN]] of $A$ in $P$ and $A^l$ the set of lower bounds of $A$ in $P$.
The *Dedekind–MacNeille completion* of $P$ is the set
$$\operatorname{DM}(P) = \set{A \subseteq P : (A^u)^l = A},$$
ordered by inclusion, together with the map $\iota \colon P \to \operatorname{DM}(P)$, $\iota(x) = \set{y \in P : y \le x}$.

## Remarks

$\operatorname{DM}(P)$ is a [[complete lattice|D7A0ZGZT5S]], whose meets are intersections, and $\iota$ is an order embedding: $x \le y$ if and only if $\iota(x) \subseteq \iota(y)$.
The image of $\iota$ is join-dense and meet-dense: each element of $\operatorname{DM}(P)$ is a supremum and an infimum of elements of $\iota(P)$.
These properties characterize $\operatorname{DM}(P)$ up to isomorphism, and it embeds in every complete lattice that contains $P$ as an ordered subset.
Example: for $P = \Q$, the sets $A = (A^u)^l$ are $\varnothing$, $\Q$, and the sets $\set{q \in \Q : q \le r}$ for $r \in \R$.
So $\operatorname{DM}(\Q)$ is the [[extended real line|0GSZ7B0CGR]], and its proper nonempty elements are closely related to [[Dedekind cuts|0G3RRM41X9]].
Non-example: in $\Q$, the set $A = \set{q \in \Q : q < 0}$ is not in $\operatorname{DM}(\Q)$, because $A^u$ is the set of nonnegative rationals and $(A^u)^l = \set{q : q \le 0} \ne A$.
</document>

<document source="https://tamnd.github.io/math2/node/DND8CBEVXT.md" tag="DND8CBEVXT" kind="concept">
# Volterra integral equation

## Summary

A Volterra integral equation is an integral equation in which the integral runs from a fixed point up to the variable $t$.
The typical form of the second kind is $x(t) = f(t) + \int_a^t K(t, s) x(s) \, ds$.
Every [[initial value problem|0FXDW992DS]] for an ordinary differential equation can be written as a Volterra equation of the second kind, in general a nonlinear one.
Under mild conditions, the linear equation of the second kind has a unique solution, given by a convergent series of iterated kernels.

## Explanations

### L1 human

In a Volterra equation, the value at time $t$ is built from a known part and a weighted sum over all earlier times.
So the solution depends only on the past, as in a process that unfolds in time.
The equation $x(t) = 1 + \int_0^t x(s) \, ds$ is the integral form of $x' = x$, $x(0) = 1$.
Its solution is $x(t) = e^t$.
</document>

<document source="https://tamnd.github.io/math2/node/DNDKBVHS3H.md" tag="DNDKBVHS3H" kind="concept">
# Point process

## Summary

A point process is a random collection of points in a space, such as the line or the plane.
Each bounded region contains only finitely many of the points.
It is described as a random counting measure: for each region it gives the random number of points inside.
The [[Poisson point process|0MTNR28K60]] is the basic example.
Point processes model arrival times, positions of trees or stars, and eigenvalues of random matrices.

## Explanations

### L1 human

Throw a handful of sand on a floor and look at where the grains land.
The positions form a random pattern of points.
A point process is a model of such a pattern.
One describes it by counting: how many points fall in this square, how many in that disc.
The times at which customers enter a shop give a point process on the time line.
</document>

<document source="https://tamnd.github.io/math2/node/DNFR3YP0XF.md" tag="DNFR3YP0XF" kind="concept">
# Localization of a category

## Summary

Localizing a [[category|A2DE66937D]] at a class of morphisms means adding formal inverses so that these morphisms become isomorphisms, in the most economical way.
The result is a new category with a functor from the old one, characterized by a universal property.
Homotopy categories and derived categories are localizations: one inverts weak equivalences or quasi-isomorphisms.
The construction generalizes the localization of a commutative ring, where chosen elements are made invertible.

## Explanations

### L1 human

Sometimes two objects should count as the same because a chosen map connects them, even though the map has no inverse.
Localization forces such maps to have inverses by allowing formal backward steps in chains of maps.
A morphism of the new category is a zigzag of old morphisms, some of which point the wrong way and are chosen ones.
For example, in the category with two objects $0$ and $1$ and one arrow $0 \to 1$, inverting the arrow makes $0$ and $1$ isomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/DNKA4WBKDS.md" tag="DNKA4WBKDS" kind="conjecture">
# Generalized Riemann hypothesis

## Statement

Let $\chi$ be a [[Dirichlet character|7XB5VJKDPG]] and let $L(s, \chi)$ be its [[Dirichlet L-function|9D2Z543EJQ]], continued to all of $\C$.
Then every [[zero|00GC8ZQP4Q]] $s$ of $L(s, \chi)$ with $0 < \operatorname{Re} s < 1$ satisfies $\operatorname{Re} s = 1/2$.

## Remarks

The conjecture is open.
For the principal character modulo $1$, $L(s, \chi)$ is the Riemann zeta function, so the conjecture contains the [[Riemann hypothesis|VCGF9YR36X]].
The extended Riemann hypothesis makes the same claim for the [[Dedekind zeta function|KXS5DQDFYD]] of every [[number field|MWBDFS2KJ5]].
The conjecture implies that there is no [[Siegel zero|5VBZBXSFDQ]].
It also implies $\psi(x; q, a) = x/\varphi(q) + O(x^{1/2} (\log x)^2)$ for all $q \le x$ and all $a$ coprime to $q$, with an absolute implied constant.
Under the conjecture, the Miller test becomes a deterministic primality test: it suffices to test the bases $a < 2(\ln n)^2$, a bound of Bach.
The [[Bombieri–Vinogradov theorem|B3JMY54GJF]] proves an average form of these consequences without any hypothesis.

## Explanations

### L1 human

The Riemann hypothesis says where the zeros of the Riemann zeta function lie: all the nontrivial ones are on one vertical line.
The generalized Riemann hypothesis says the same thing for a whole family of similar functions, the Dirichlet L-functions, one for each way of sorting numbers by residue classes.
If it is true, primes are spread over residue classes as evenly as one could hope, with a small and explicit error.
Many results in number theory and in the analysis of algorithms are first proved under this hypothesis.
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<document source="https://tamnd.github.io/math2/node/DNKXHMGQY1.md" tag="DNKXHMGQY1" kind="concept">
# M-matrix

## Summary

An M-matrix is a real square matrix of the form $sI - B$, where $B$ is a [[nonnegative matrix|JPQCCFCY8X]].
The number $s$ must be at least the [[spectral radius|9V7YQA85PC]] of $B$.
Since $B$ has no negative entry, the entries of an M-matrix off the diagonal are nonpositive.
Such matrices appear when a linear system is discretised, in economic input-output models and in Markov chains.
A nonsingular M-matrix has an inverse with all entries nonnegative, so it turns nonnegative data into nonnegative solutions.

## Explanations

### L1 human

Think of a matrix with positive numbers on the diagonal and negative or zero numbers elsewhere.
It is an M-matrix when the diagonal is strong enough to dominate the rest in a precise sense.
The matrix $\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$, which comes from the second difference of a function, is an example.
Its inverse $\frac{1}{3}\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ has only nonnegative entries.
</document>

<document source="https://tamnd.github.io/math2/node/DNTDCKZBKK.md" tag="DNTDCKZBKK" kind="concept">
# Monadic functor

## Summary

A functor $U \colon \mathcal{D} \to \mathcal{C}$ is monadic when it has a left adjoint and $\mathcal{D}$ is, up to equivalence, the category of algebras of the resulting [[monad|FF6XXA6D1A]].
So the objects of $\mathcal{D}$ are exactly objects of $\mathcal{C}$ with algebraic structure described by the monad.
The forgetful functors from groups, rings, modules and compact Hausdorff spaces to sets are monadic.
Beck's monadicity theorem characterizes monadic functors.

## Explanations

### L1 human

A forgetful functor forgets structure, for example it sends a group to its underlying set.
It is monadic when the forgotten structure can be fully recovered as a way of evaluating formal expressions.
Groups are monadic over sets: a group is a set with a consistent way to evaluate words in its elements.
Topological spaces are not monadic over sets, since a topology is not an algebraic structure of this kind.
</document>

<document source="https://tamnd.github.io/math2/node/DNVGM3YT8Q.md" tag="DNVGM3YT8Q" kind="definition">
# Logarithmic kernel

## Definition

Identify $\R^2$ with $\C$.
The *logarithmic kernel* is the function $k \colon \C \times \C \to (-\infty, +\infty]$ given by
$$k(z, w) = \log \frac{1}{\abs{z - w}} \quad (z \ne w), \qquad k(z, z) = +\infty.$$

## Remarks

Example: $k(2, 0) = -\log 2 < 0$ and $k(1/2, 0) = \log 2 > 0$.
For fixed $w$, the function $z \mapsto k(z, w)$ is [[harmonic|0DEB0JMN0W]] on $\C \setminus \set{w}$.
Indeed, on every disc that does not contain $w$ it is the real part of a holomorphic branch of $-\log(z - w)$.
Non-example: in $\R^3$ the function $\log(1/\abs{x - y})$ is not harmonic off the pole, since for $r = \abs{x - y}$ one has $\Delta \log(1/r) = -1/r^2$; the three-dimensional analogue is the Newtonian kernel.
The kernel is symmetric and lower semicontinuous, and $k(z, w) = 2\pi \, \Phi(z - w)$, where $\Phi$ is the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] in the plane.
So $\Delta_z k(z, w) = -2\pi \delta_w$ in the sense of distributions.
</document>

<document source="https://tamnd.github.io/math2/node/DP39F5KQ2W.md" tag="DP39F5KQ2W" kind="example">
# Lucas sequence

## Example

Fix integers $P$ and $Q$.
The *Lucas sequences* $U_n(P, Q)$ and $V_n(P, Q)$ are the [[solutions|06MJC8YDYR]] of the recurrence
$$x_{n+2} = P\, x_{n+1} - Q\, x_n \qquad (n \ge 0)$$
with initial values $U_0 = 0$, $U_1 = 1$ and $V_0 = 2$, $V_1 = P$.
The [[characteristic polynomial|0YA7X86K05]] is $X^2 - P X + Q$.
If its discriminant $D = P^2 - 4Q$ is not $0$, let $a, b$ be its two distinct complex roots; then
$$U_n = \frac{a^n - b^n}{a - b}, \qquad V_n = a^n + b^n.$$
Check: both formulas solve the recurrence, since $a^{n+2} = P a^{n+1} - Q a^n$ and likewise for $b$, and they give $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = a + b = P$.

Special cases:

1. $P = 1$, $Q = -1$: $U_n$ are the [[Fibonacci numbers|0GA9K57A39]] $0, 1, 1, 2, 3, 5, \dots$ and $V_n$ are the Lucas numbers $2, 1, 3, 4, 7, 11, \dots$.
2. $P = 2$, $Q = -1$: $U_n$ are the [[Pell numbers|JY4ZN779YG]] $0, 1, 2, 5, 12, 29, \dots$.
3. $P = 3$, $Q = 2$: $U_n = 2^n - 1$, the Mersenne numbers $0, 1, 3, 7, 15, \dots$, since the roots are $2$ and $1$.

## Explanations

### L1 human

A Lucas sequence is a two-parameter family that contains the Fibonacci numbers and many of their relatives.
Each term is $P$ times the previous term minus $Q$ times the one before.
Changing $P$ and $Q$ gives the Fibonacci numbers, the Pell numbers, or the numbers $2^n - 1$.
Because they share one formula, facts proved once for Lucas sequences hold for all of these at the same time.
They are used in primality tests, for example the [[Lucas–Lehmer test|061BTYA0RN]] for [[Mersenne primes|0F3Z22M2GT]].
</document>

<document source="https://tamnd.github.io/math2/node/DP3KSCRKFP.md" tag="DP3KSCRKFP" kind="concept">
# Embedding dimension

## Summary

The embedding dimension of a Noetherian [[local ring|MZ02BKPHJA]] $(R, \mathfrak m, k)$ is the dimension of its [[cotangent space|Z1CGB483J6]] $\mathfrak m / \mathfrak m^2$ over $k$.
By [[Nakayama's lemma|15JHBYR6KH]] it equals the minimal number of generators of $\mathfrak m$.
It is always at least the [[Krull dimension|MVGBS3J1P7]] of $R$, and equality defines a [[regular local ring|63XXTF6B5M]].

## Explanations

### L1 human

The embedding dimension counts how many coordinates you need to describe a space near a point.
A smooth curve needs one coordinate, but the cusp $y^2 = x^3$ at the origin needs two, even though it is still a curve.
The difference between the embedding dimension and the actual dimension measures how singular the point is.
The [[definition|BRXX5Y929C]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/DP4WWQBJJF.md" tag="DP4WWQBJJF" kind="concept">
# Rotation number

## Summary

The rotation number of an orientation-preserving [[homeomorphism|0QT00PB5G2]] of the [[circle|060W5CBZP6]] is the average amount by which the map turns points around the circle, per step.
For the rotation by an angle $\alpha$ it is $\alpha$.
It is a topological invariant: homeomorphisms conjugate by an orientation-preserving homeomorphism have the same rotation number.
Poincaré introduced it, and showed that it decides much of the dynamics: rational rotation number means periodic orbits, irrational rotation number means none.
It is the basic invariant of circle dynamics and of quasi-periodic motion.

## Explanations

### L1 human

Follow a point as a map moves it around a circle many times, and count the total number of turns made.
Divide by the number of steps; in the long run this ratio settles to a number, the rotation number.
For a clock hand that moves a quarter turn each tick, the rotation number is $1/4$.
A map can move different points by different amounts, but the long-run average turn is the same for every starting point.
</document>

<document source="https://tamnd.github.io/math2/node/DPBJ04JMH8.md" tag="DPBJ04JMH8" kind="concept">
# Commutative algebra over a ring

## Summary

A commutative algebra over a commutative ring $R$ is a [[commutative ring|0GYGHKX834]] $A$ together with a [[ring homomorphism|0GGE0Z0JQ1]] $R \to A$.
The homomorphism lets one multiply elements of $A$ by elements of $R$, so $A$ is at the same time a ring and an $R$-module.
Most rings in commutative algebra and algebraic geometry come with such a structure, for example polynomial rings over $R$ and their quotients.
The notion lets one compare rings over a fixed base and speak of $R$-algebra homomorphisms.

## Explanations

### L1 human

An $R$-algebra is a ring whose elements can be scaled by the elements of a base ring $R$.
Think of the polynomial ring $\R[x]$: one can add and multiply polynomials, and one can also multiply a polynomial by a real number.
The map that sends a real number $c$ to the constant polynomial $c$ is the structure map $\R \to \R[x]$.
Every commutative ring is an algebra over $\Z$ in exactly one way, since there is exactly one ring homomorphism $\Z \to A$.
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<document source="https://tamnd.github.io/math2/node/DPDXN2FJC4.md" tag="DPDXN2FJC4" kind="concept">
# Facet

## Summary

A facet of a [[convex polytope|300K6CB1KH]] is a [[face|KY7SB854VH]] of the largest possible dimension below that of the polytope itself.
A polytope of dimension $d$ has facets of dimension $d - 1$.
The facets determine the polytope: a full-dimensional polytope is the intersection of the closed half-spaces bounded by the hyperplanes of its facets.
Counting facets and studying their shapes is a central theme of [[polytope combinatorics|879SBTVE7V]].

## Explanations

### L1 human

The facets of a solid are the flat pieces of its surface that are "one dimension down".
For a cube in space they are the six square faces.
For a polygon in the plane they are its edges.
For a four-dimensional cube they are eight ordinary cubes.
Each facet sits on a wall, and the polytope is what lies on the inner side of all the walls.
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<document source="https://tamnd.github.io/math2/node/DPE8EDD9W7.md" tag="DPE8EDD9W7" kind="definition">
# Dual graph

## Definition

Let $G$ be a [[planar graph|029JQEK15B]] together with a fixed [[embedding|CG91WNMG2T]] in the plane, called a plane graph.
The faces of $G$ are the connected components of the complement of the drawing in the plane.
The dual $G^*$ is the [[multigraph|01M5T0KDG7]], possibly with loops, defined as follows.

1. The vertices of $G^*$ are the faces of $G$.
2. For each edge $e$ of $G$ there is one edge $e^*$ of $G^*$.
3. The ends of $e^*$ are the faces on the two sides of $e$; if the same face lies on both sides, $e^*$ is a loop.

## Remarks

Example: draw $K_4$ in the plane as a triangle with a central vertex joined to the corners.
This drawing has $4$ faces, and every two of them share exactly one edge, so its dual is again $K_4$.
Non-example: $K_5$ has no dual in this sense, since it is not planar and so it has no plane embedding, which the setting requires.
An edge $e$ is a [[bridge|0D81KWH9ZM]] exactly when $e^*$ is a loop.
If $G$ is connected, then $(G^*)^*$ is isomorphic to $G$.
A set of edges is a cycle of $G$ exactly when the dual edges form a minimal cut of $G^*$.
Different embeddings of the same planar graph can have non-isomorphic duals.
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<document source="https://tamnd.github.io/math2/node/DPFMXS71D1.md" tag="DPFMXS71D1" kind="definition">
# Absolute Galois group

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $K^{s}$ be a [[separable closure|JJ91A1V8NQ]] of $K$.
The *absolute Galois group* of $K$ is the [[Galois group|0XVR72M51M]]
$$G_K = \Gal(K^{s}/K)$$
of field automorphisms of $K^{s}$ that fix $K$ pointwise, with the [[Krull topology|F3GTF5NT4M]].

## Remarks

It does not depend on the choice of $K^{s}$ up to isomorphism, because two separable closures are isomorphic over $K$.
The induced isomorphism of Galois groups is unique only up to an inner automorphism.
If $K$ is perfect, for example of characteristic $0$, then $K^{s}$ is an algebraic closure of $K$.
Example: $G_\R = \Gal(\C/\R)$ has order $2$, generated by complex conjugation, since $\C$ is algebraically closed and $[\C : \R] = 2$.
Example: for a finite field $\F_q$, the group $G_{\F_q}$ is isomorphic to the profinite completion $\hat{\Z}$ of $\Z$, topologically generated by $x \mapsto x^q$.
Non-example: $\Gal(\Q(i)/\Q)$ is not the absolute Galois group of $\Q$, since $\Q(i)$ is not separably closed; $x^2 - 2$ has no root in it.
If $K$ is separably closed, then $G_K$ is trivial.
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<document source="https://tamnd.github.io/math2/node/DPR9DJRPAF.md" tag="DPR9DJRPAF" kind="definition">
# Stiffness matrix

## Definition

Let $V$ be a real [[Hilbert space|HM2HHE3J51]], let $a \colon V \times V \to \R$ be a bounded [[bilinear form|0SQZMXB6GP]], and let $\ell \colon V \to \R$ be a bounded linear functional.
Let $V_h \subseteq V$ be a finite-dimensional subspace with a basis $\varphi_1, \ldots, \varphi_N$, as in the [[Galerkin method|Y5B0PEBKE8]] or the [[finite element method|98GZ9DVHTA]].
The *stiffness matrix* is the $N \times N$ matrix $K$ with entries
$$K_{ij} = a(\varphi_j, \varphi_i), \qquad 1 \le i, j \le N.$$
The vector $F$ with $F_i = \ell(\varphi_i)$ is the *load vector*.

## Remarks

The function $u_h = \sum_j c_j \varphi_j$ satisfies $a(u_h, v) = \ell(v)$ for all $v \in V_h$ exactly when $K c = F$.
If $a$ is symmetric, then $K$ is symmetric.
If $a$ is coercive, $a(v, v) \ge \alpha \norm{v}^2$ with $\alpha > 0$, then $c^T K c = a(v, v) > 0$ for $v = \sum_j c_j \varphi_j \ne 0$, so $K$ is positive definite in the symmetric case and invertible in general.
If $\varphi_i$ and $\varphi_j$ have disjoint supports and $a$ is an integral of local terms, then $K_{ij} = 0$.

Example: for $-u'' = f$ on $(0, 1)$ with $u(0) = u(1) = 0$, take $a(u, v) = \int_0^1 u' v' \, dx$, nodes $x_i = ih$ with $h = 1/(N + 1)$, and hat functions $\varphi_i$ with $\varphi_i(x_j) = \delta_{ij}$.
On each of its two intervals $\varphi_i'$ equals $\pm 1/h$, so $K_{ii} = 2/h$, $K_{i, i \pm 1} = -1/h$, and all other entries are $0$.
So $K = h^{-1} \operatorname{tridiag}(-1, 2, -1)$.

Non-example: the matrix $M_{ij} = \int_0^1 \varphi_j \varphi_i \, dx$ is built from the $L^2$ inner product, not from the form $a$ of the problem.
It is the mass matrix, which has $M_{ii} = 2h/3$ and $M_{i, i \pm 1} = h/6$ for the hat functions above.
</document>

<document source="https://tamnd.github.io/math2/node/DPS8QZBVB0.md" tag="DPS8QZBVB0" kind="concept">
# Relaxation of a variational problem

## Summary

The relaxation of a variational problem replaces a functional $J$ that has no minimiser by its lower semicontinuous envelope $\bar J$.
This is the largest lower semicontinuous functional below $J$.
The relaxed problem has the same infimum as the original one.
Every limit of a minimising sequence of the original problem, in the topology used, is a minimiser of the relaxed problem.
For integral functionals, relaxation typically replaces the integrand by its convex or quasiconvex envelope in the gradient variable.

## Explanations

### L1 human

Sometimes a minimisation problem has no solution because minimising sequences oscillate faster and faster, like a zigzag with ever finer teeth.
The limit of such a sequence is a smooth function, but the original energy undervalues what the zigzags achieved.
Relaxation fixes the energy so that it assigns to each function the best value that nearby functions can approach.
Then the limit becomes a true minimiser of the corrected energy.
For the double-well cost $(\xi^2 - 1)^2$, relaxation replaces it by $0$ on $[-1, 1]$, because zigzags of slopes $\pm 1$ can produce any average slope in between at no cost.
</document>

<document source="https://tamnd.github.io/math2/node/DPVM34ADZV.md" tag="DPVM34ADZV" kind="concept">
# Fourier transform

## Summary

The Fourier transform writes a function on $\R^n$ as a superposition of plane waves $e^{2\pi i x \cdot \xi}$ and records the amplitude of each frequency $\xi$.
It turns derivatives into multiplication by polynomials in $\xi$, and convolution into ordinary multiplication.
So it changes linear partial differential equations with constant coefficients into algebraic or ordinary differential equations.
It extends to $L^2$ functions and to [[tempered distributions|AES2T5JPFP]].
See the [[definition|6V59T5KJBX]].

## Explanations

### L1 human

A sound is a mix of pure tones of different pitches.
The Fourier transform takes the sound as a function of time and tells how much of each pitch it contains.
On the frequency side, taking a derivative just multiplies each pitch by a factor proportional to its frequency.
This is why the transform solves equations such as the heat equation so easily.
The bell curve $e^{-\pi x^2}$ is special: its transform is the same bell curve.
</document>

<document source="https://tamnd.github.io/math2/node/DPW8KNHE6P.md" tag="DPW8KNHE6P" kind="concept">
# Wald test

## Summary

The Wald test is a [[hypothesis test|087P03NEND]] that measures how far an unrestricted estimate of a parameter lies from its null value.
The distance is scaled by the estimated standard error, usually from the [[Fisher information|G1Z3WSTE8Q]] at the [[maximum likelihood estimate|0FREPFKNS7]].
It needs only the unrestricted fit, and it is the test behind the usual "estimate divided by standard error" output of regression software.

## Explanations

### L1 human

Estimate the parameter, then ask how many standard errors the estimate is from the value claimed by the null hypothesis.
If it is more than about two standard errors away, reject at level about $5\%$.
For a coin with $60$ heads in $100$ tosses, the estimate is $0.6$ with standard error $\sqrt{0.6 \cdot 0.4/100} \approx 0.049$.
So the estimate is about $2.04$ standard errors from $0.5$, and the Wald test rejects a fair coin at level $5\%$.
</document>

<document source="https://tamnd.github.io/math2/node/DPZ76D18DD.md" tag="DPZ76D18DD" kind="definition">
# Stable homotopy category

## Definition

Let $\mathbf{Sp}$ be the category of [[spectra|1GHF4EKBXS]] and maps of spectra.
A map $f \colon E \to F$ of spectra is a *stable weak equivalence* if $\pi_k(f) \colon \pi_k(E) \to \pi_k(F)$ is an isomorphism of [[homotopy groups|P976WYEFMB]] for every $k \in \Z$.
The *stable [[homotopy category|48QQ7T98Z9]]* $\mathrm{Ho}(\mathbf{Sp})$ is the [[localization|DNFR3YP0XF]] of $\mathbf{Sp}$ at the stable weak equivalences.
It has the same objects as $\mathbf{Sp}$.
It is the universal category under $\mathbf{Sp}$ in which every stable weak equivalence becomes an isomorphism.

## Remarks

The stable weak equivalences are the weak equivalences of a model structure on spectra.
So the localization exists with small morphism sets, and it is the homotopy category of that model structure.

Basic properties:

1. Every morphism set $[E, F]$ is an abelian group, and composition is bilinear.
2. The suspension $\Sigma$ is an equivalence of categories, with inverse the loop functor $\Omega$.
3. With $\Sigma$ as shift and cofibre sequences as distinguished triangles, $\mathrm{Ho}(\mathbf{Sp})$ is a [[triangulated category|T2V7GJJ2EN]].
4. The smash product makes it a symmetric monoidal category, with the sphere spectrum as unit.

Example: for finite pointed CW complexes $X$ and $Y$, $[\Sigma^\infty X, \Sigma^\infty Y] \cong \operatorname{colim}_k [\Sigma^k X, \Sigma^k Y]_*$, the group of stable maps.

Non-example: in the homotopy category of pointed CW complexes, suspension is not an equivalence.
The suspension map $\pi_3(S^2) = [S^3, S^2]_* \to [S^4, S^3]_* = \pi_4(S^3)$ goes from $\Z$ onto $\Z/2$, so it is not injective, and $\Sigma$ is not fully faithful there.

The first construction of a stable homotopy category with a well-behaved smash product is due to Boardman.
Modern models, such as symmetric spectra and orthogonal spectra, have homotopy categories equivalent to it.
</document>

<document source="https://tamnd.github.io/math2/node/DQ0TSVK7ZT.md" tag="DQ0TSVK7ZT" kind="definition">
# Local solvability

## Definition

Let $\Omega \subseteq \R^n$ be open and let $P$ be a [[linear differential operator|6DZETZ5T9G]] on $\Omega$ with [[smooth|0V33BNKZ9S]] complex coefficients.
A distribution on an open set $U$ is a continuous linear functional on the space of [[test functions|HS65DW0HXH]] on $U$.
The operator $P$ is *locally solvable at* $x_0 \in \Omega$ if there is an open neighbourhood $U \subseteq \Omega$ of $x_0$ with the following property.
For every smooth function $f$ on $\Omega$ there is a distribution $u$ on $U$ with $P u = f$ in $U$, in the sense of distributions.
It is *locally solvable in* $\Omega$ if it is locally solvable at every point of $\Omega$.

## Remarks

Example: $P = \partial_1$ on $\R^n$ is locally solvable at every point: $u(x) = \int_0^{x_1} f(s, x_2, \dots, x_n) \, ds$ is smooth and satisfies $\partial_1 u = f$.
More generally, every nonzero operator with constant coefficients is locally solvable, by the [[Malgrange–Ehrenpreis theorem|3173Y813V7]].
Non-example: Lewy's operator $L = \partial_x + i \partial_y - 2 i (x + i y) \partial_t$ on $\R^3$ with variables $(x, y, t)$.
There is a smooth function $f$ on $\R^3$ such that $L u = f$ has no distribution solution in any open subset of $\R^3$, so $L$ is locally solvable at no point.
Some texts state the definition with slightly different classes of right-hand sides or of solutions.

## Explanations

### L1 human

An operator is locally solvable at a point when, for every smooth right-hand side, the equation can be solved on a fixed small neighbourhood of that point.
</document>

<document source="https://tamnd.github.io/math2/node/DQ14ERMRTZ.md" tag="DQ14ERMRTZ" kind="definition">
# Majority term

## Definition

Let $\mathcal{V}$ be a [[variety of algebras|AJNZJD8442]].
A *majority term* for $\mathcal{V}$ is a ternary [[term|N9M2N27END]] $m(x, y, z)$ such that $\mathcal{V}$ satisfies the [[identities|PKD440G2XW]]
$$m(x, x, y) \approx m(x, y, x) \approx m(y, x, x) \approx x.$$
Equivalently, a majority term is a [[near-unanimity term|4CRH8NNFEF]] of arity $3$.

## Remarks

Example: on $\set{0, 1}$ the Boolean majority function $m(x, y, z) = (x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$ is a majority term of the two-element Boolean algebra.
Check: if two arguments equal $a$, then one of the three meets equals $a$ and the other two are at most $a$, so the join is $a$.
Non-example: the term $p(x, y, z) = x - y + z$ of abelian groups satisfies $p(x, x, y) \approx y$, not $x$, so it fails the first identity.
In fact no variety of groups other than the trivial variety has a majority term, since such a variety is not congruence-distributive.
</document>

<document source="https://tamnd.github.io/math2/node/DQ4E55T3BW.md" tag="DQ4E55T3BW" kind="concept">
# Clopen set

## Summary

A clopen set in a [[topological space|0CQHFPGJ45]] is a subset that is both [[open|02BH3MCBQB]] and [[closed|0ZCK8DQYN1]].
The empty set and the whole space are always clopen.
A space is [[connected|0V032VWTHD]] exactly when it has no other clopen sets, so clopen sets measure how a space splits into pieces.

## Explanations

### L1 human

Usually "open" and "closed" look like opposite properties, but a set can have both.
Picture a space made of two separate islands: each island is open, and its complement, the other island, is open too, so each island is also closed.
For example, in the space $[0, 1] \cup [2, 3]$ with the usual distance, the piece $[0, 1]$ is clopen.
In the real line $\R$ itself, only $\emptyset$ and $\R$ are clopen, because $\R$ is in one piece.
</document>

<document source="https://tamnd.github.io/math2/node/DQ564JNFYE.md" tag="DQ564JNFYE" kind="definition">
# Function field

## Definition

Let $k$ be a [[field|0QA8CHH8MG]] and let $n \ge 0$ be an integer.
A *function field in $n$ variables over $k$* is a [[field extension|0WQ1A8SHTZ]] $K/k$ with two properties.
First, $K = k(a_1, \dots, a_m)$ for finitely many $a_1, \dots, a_m \in K$.
Second, the [[transcendence degree|D7TR0YWR9H]] of $K/k$ is $n$.

## Remarks

Many authors also ask that $k$ be algebraically closed in $K$, and some ask $n \ge 1$.
Example: the [[field of rational functions|4V81611P9N]] $k(x_1, \dots, x_n)$ is a function field in $n$ variables.
Example: if $k$ has characteristic not $2$, the field $k(x)[y]/(y^2 - x^3 - 1)$ is a function field in one variable, with [[transcendence basis|E45PA267S4]] $\set{x}$; it is the function field of the curve $y^2 = x^3 + 1$.
Non-example: the field $k(x_1, x_2, x_3, \dots)$ in infinitely many variables is not finitely generated over $k$.
For an irreducible variety $V$ over $k$, the field $k(V)$ of rational functions on $V$ is a function field in $\dim V$ variables over $k$.
</document>

<document source="https://tamnd.github.io/math2/node/DQA60Z3FD7.md" tag="DQA60Z3FD7" kind="example">
# Veronese embedding

## Example

Let $k$ be an algebraically closed field, and let $n \ge 1$ and $d \ge 1$ be integers.
There are $\binom{n + d}{d}$ monomials of degree $d$ in $x_0, \dots, x_n$; let $N = \binom{n + d}{d} - 1$.
The *Veronese embedding* of degree $d$ is the map of [[projective spaces|ZT9A6JK8QF]]
$$\nu_d \colon \mathbb{P}^n \to \mathbb{P}^N, \qquad [x_0 : \dots : x_n] \mapsto [x^{\alpha}]_{\abs{\alpha} = d},$$
which lists all monomials $x^\alpha = x_0^{\alpha_0} \cdots x_n^{\alpha_n}$ with $\alpha_0 + \dots + \alpha_n = d$.

Check that it is a [[morphism|CDS1VWB7A3]]: the monomials are [[homogeneous polynomials|2C3PAQ7H50]] of the same degree $d$.
So scaling $x$ by $\lambda$ scales all of them by $\lambda^d$.
They never vanish together, since $x_i^d$ is among them and some $x_i \ne 0$.

Check that it is injective: if $x_i \ne 0$, then the monomials $x_i^{d-1} x_j$ for $j = 0, \dots, n$ are the coordinates of $x_i^{d-1} \cdot (x_0, \dots, x_n)$, which recover $[x]$.
In fact $\nu_d$ is an isomorphism onto its image, which is a [[projective variety|A4H4CZ2V35]] of dimension $n$.

For $n = 1$ and $d = 2$, $\nu_2([s : t]) = [s^2 : st : t^2]$, and the image is the conic $z_0 z_2 = z_1^2$ in $\mathbb{P}^2$.
For $n = 1$ and $d = 3$, the image of $[s : t] \mapsto [s^3 : s^2 t : s t^2 : t^3]$ is the twisted cubic curve in $\mathbb{P}^3$.
For $n = 2$ and $d = 2$, the image is the Veronese surface in $\mathbb{P}^5$.

Under $\nu_d$, a hypersurface of degree $d$ in $\mathbb{P}^n$ becomes the intersection of the image with a hyperplane of $\mathbb{P}^N$.

## Explanations

### L1 human

The Veronese map replaces a point by the list of all products of its coordinates of a fixed degree.
For the projective line and degree two, the point $[s : t]$ goes to $[s^2 : st : t^2]$, and the image is a conic.
The trick turns curved equations of degree $d$ into linear equations in the new coordinates.
This lets one study hypersurfaces of degree $d$ with the tools of linear algebra on a bigger space.
</document>

<document source="https://tamnd.github.io/math2/node/DQB1BVQJWD.md" tag="DQB1BVQJWD" kind="definition">
# Tangent bundle

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $C^\infty(M)$ be the algebra of [[smooth functions|0V33BNKZ9S]] $M \to \R$.
For $p \in M$, a *tangent vector* at $p$ is a linear map $v \colon C^\infty(M) \to \R$ such that $v(fg) = f(p)\, v(g) + g(p)\, v(f)$ for all $f, g \in C^\infty(M)$; that is, a derivation at $p$.
The tangent vectors at $p$ form a real vector space $T_pM$, the *tangent space* at $p$.
The *tangent bundle* of $M$ is the disjoint union
$$TM = \bigsqcup_{p \in M} T_pM = \set{(p, v) : p \in M, \ v \in T_pM},$$
with the projection $\pi \colon TM \to M$, $\pi(p, v) = p$.
Let $(U, (x^1, \ldots, x^n))$ be a chart of $M$.
$TM$ has a unique smooth structure in which each such chart gives a chart on $\pi^{-1}(U)$, namely
$$(p, v) \mapsto \big(x^1(p), \ldots, x^n(p), v^1, \ldots, v^n\big), \qquad v = \sum_{i=1}^n v^i \left.\frac{\partial}{\partial x^i}\right|_p.$$

## Remarks

Each $T_pM$ has dimension $n$, with basis $\partial/\partial x^1|_p, \ldots, \partial/\partial x^n|_p$ for any chart around $p$, so $TM$ is a smooth manifold of dimension $2n$.
For $M = \R^n$ a single chart covers everything, and $T\R^n = \R^n \times \R^n$.
For the circle $S^1$, the vector field $\partial/\partial\theta$ is nowhere zero, and $TS^1$ is diffeomorphic to $S^1 \times \R$.

The tangent bundle of $S^2$ is not isomorphic, as a vector bundle, to the product $S^2 \times \R^2$.
By the hairy ball theorem every tangent vector field on $S^2$ vanishes somewhere, while $S^2 \times \R^2$ has nowhere zero sections.

Other common definitions of $T_pM$ use equivalence classes of curves through $p$, or vectors in coordinates that change by the Jacobian matrix; they give isomorphic spaces.

## Explanations

### L1 human

The tangent bundle is the set of all pairs made of a point of the manifold and a tangent vector at that point.
</document>

<document source="https://tamnd.github.io/math2/node/DQCD11X1VM.md" tag="DQCD11X1VM" kind="concept">
# Shape operator

## Summary

The shape operator of a surface in $\R^3$ at a point is a linear map of the tangent plane to itself.
It tells how the unit normal turns as one moves on the surface.
It is minus the differential of the Gauss map, and it is also called the Weingarten map.
Its eigenvalues are the principal curvatures, its determinant is the Gaussian curvature, and half its trace is the mean curvature.

## Explanations

### L1 human

Imagine a tiny arrow standing straight out of a surface at each point, like a hair.
Walk a little in some direction, and watch how the hair tilts.
The shape operator turns your step into the change of tilt of the hair.
On a flat floor the hair never tilts, and on a ball it tilts in the direction you walk, at the same rate in every direction.
</document>

<document source="https://tamnd.github.io/math2/node/DQCW86BXEP.md" tag="DQCW86BXEP" kind="definition">
# Subnormal subgroup

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
The subgroup $H$ is *subnormal* in $G$ when there are subgroups $H_0, H_1, \dots, H_n$ of $G$ with these properties.
First, $H_0 = H$ and $H_n = G$.
Second, $H_{i-1}$ is a [[normal subgroup|0CB0DX5BVV]] of $H_i$ for $1 \le i \le n$.
In symbols, $H = H_0 \trianglelefteq H_1 \trianglelefteq \cdots \trianglelefteq H_n = G$.

## Remarks

One writes $H \trianglelefteq\trianglelefteq G$.
The least such $n$ is the defect of $H$ in $G$; the subgroups of defect at most $1$ are the normal subgroups.
Example: in the alternating group $A_4$, let $V = \{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}$ and $H = \{e, (1\,2)(3\,4)\}$.
Then $H \trianglelefteq V$ because $V$ is abelian, and $V \trianglelefteq A_4$, so $H$ is subnormal of defect $2$; it is not normal, since conjugation by $(1\,2\,3)$ sends $(1\,2)(3\,4)$ to $(2\,3)(1\,4)$.
Non-example: in the symmetric group $S_3$, the subgroup $H = \{e, (1\,2)\}$ is not subnormal.
The only subgroups of $S_3$ that contain $H$ are $H$ and $S_3$.
So a chain from $H$ to $S_3$ must contain a step $H \trianglelefteq S_3$, but $H$ is not normal in $S_3$.
In a [[nilpotent group|0Q7MMQHC1P]] every subgroup is subnormal.

## Explanations

### L1 human

A subgroup is subnormal when you can climb from it to the whole group by a ladder in which each rung is normal in the next one.
</document>

<document source="https://tamnd.github.io/math2/node/DQEHD56649.md" tag="DQEHD56649" kind="statement">
# Euler's sine product

## Statement

For every $z \in \C$,
$$\sin(\pi z) = \pi z \prod_{n=1}^{\infty} \left(1 - \frac{z^2}{n^2}\right),$$
where $\sin$ is the [[complex sine|0A7YDKRNR6]].
The product converges absolutely for every $z$, and [[uniformly|08FPABDQ4C]] on each compact subset of $\C$.

## Remarks

The convergence follows from $\sum_n \abs{z}^2 / n^2 < \infty$.
Both sides are [[entire functions|0419794R0T]] with simple zeros exactly at the integers.
By the [[Weierstrass factorization theorem|11VC51YHQJ]], their quotient is $e^{g(z)}$ for an entire function $g$.
One shows that $g$ is constant, for example by comparing logarithmic derivatives with the partial fraction expansion $\pi \cot(\pi z) = \frac{1}{z} + \sum_{n \ge 1} \frac{2z}{z^2 - n^2}$; the limit at $z = 0$ then gives $e^{g} = 1$.
Comparing the coefficients of $z^3$ on both sides gives $\sum_{n \ge 1} 1/n^2 = \pi^2/6$, the [[Basel problem|0DMY75D6FV]].
At $z = 1/2$ it gives the [[Wallis product|0XCYA4YERS]] $\frac{2}{\pi} = \prod_{n \ge 1} \left(1 - \frac{1}{4n^2}\right)$.
In Mathlib it is `Complex.tendsto_euler_sin_prod`.

## Explanations

### L1 human

A polynomial is determined, up to a constant, by its roots.
The sine function behaves like a polynomial of infinite degree with roots at all the integers.
Euler's formula writes $\sin(\pi z)$ as the product of the factors $(1 - z/n)(1 + z/n)$ over all its roots, times $\pi z$.

### L2 human

The formula is the standard example of a [[Weierstrass product|ACSF58HYDH]].
Here no exponential correction factors are needed, because the factors $1 - z/n$ and $1 + z/n$ are paired.
Taking the logarithmic derivative gives the partial fraction expansion of $\pi\cot(\pi z)$.
Combined with the product for $1/\Gamma$, it yields the [[reflection formula|WF7KQAQTS2]] for the gamma function.
</document>

<document source="https://tamnd.github.io/math2/node/DQEW4MSD5H.md" tag="DQEW4MSD5H" kind="definition">
# Hausdorff distance

## Definition

Let $(M, d)$ be a [[metric space|0VKD1TP7F3]].
For $x \in M$ and a nonempty subset $S \subseteq M$, let $d(x, S) = \inf_{s \in S} d(x, s)$, an [[infimum|0QF1MZ9FRG]].
The *Hausdorff distance* between nonempty subsets $A, B \subseteq M$ is
$$d_H(A, B) = \max\Big\{ \sup_{a \in A} d(a, B), \ \sup_{b \in B} d(b, A) \Big\} \in [0, +\infty],$$
where $\sup$ is the [[supremum|0R3H5PANRQ]].

## Remarks

Equivalently, $d_H(A, B)$ is the infimum of the $\varepsilon \ge 0$ with $A \subseteq B_\varepsilon$ and $B \subseteq A_\varepsilon$, where $S_\varepsilon = \set{x \in M : d(x, S) \le \varepsilon}$.
In $\R^n$ with the Euclidean distance and closed unit ball $D$, this reads $d_H(A, B) = \inf\set{\varepsilon \ge 0 : A \subseteq B + \varepsilon D, \ B \subseteq A + \varepsilon D}$ for nonempty compact $A, B$, with [[Minkowski sums|SXS6D87PHE]].

Example: $A = [0, 1]$ and $B = [0, 2]$ in $\R$.
Every point of $A$ lies in $B$, so $\sup_{a \in A} d(a, B) = 0$; the points of $B$ have $d(b, A) = \max(b - 1, 0)$, with supremum $1$ at $b = 2$.
So $d_H(A, B) = 1$.

Non-example of a metric: on all nonempty subsets, $d_H$ is not a metric.
For $A = (0, 1)$ and $B = [0, 1]$ in $\R$ one has $d_H(A, B) = 0$ although $A \neq B$, and $d_H(\set{0}, [0, \infty)) = +\infty$.
On the nonempty closed bounded subsets of $M$, and in particular on the nonempty compact subsets, $d_H$ is a metric.
If $M$ is complete, then so is this space of closed bounded subsets; if $M$ is compact, then so is the space of nonempty compact subsets.
</document>

<document source="https://tamnd.github.io/math2/node/DQGJMP9WHC.md" tag="DQGJMP9WHC" kind="concept">
# Contiguity of measures

## Summary

Contiguity is an asymptotic form of absolute continuity for sequences of probability measures.
A sequence $(Q_n)$ is contiguous with respect to $(P_n)$ if every sequence of events whose $P_n$-probability tends to zero also has $Q_n$-probability tending to zero.
It was introduced by Lucien Le Cam.
It lets one transfer convergence in probability from a null hypothesis to nearby alternatives.
This is the basis of asymptotic power calculations and of local asymptotic theory.

## Explanations

### L1 human

Two sequences of probability models are contiguous if, as the sample grows, events that become impossible under one also become impossible under the other.
The models may differ, but never so much that the data could tell them apart with certainty.
For $n$ normal observations, a mean of $0$ and a mean of $1/\sqrt{n}$ stay contiguous, because the shift is of the size of the statistical noise.
A mean of $0$ and a mean of $1$ are not contiguous: with many observations one can tell them apart almost surely.
</document>

<document source="https://tamnd.github.io/math2/node/DQP63REKEK.md" tag="DQP63REKEK" kind="statement">
# Monodromy theorem for singularities

## Statement

Let $n \ge 1$ and let $f$ be a holomorphic function near $0$ in $\C^{n+1}$ with an [[isolated singularity|VTGYEES3VS]] at $0$.
Let $F$ be the Milnor fibre of the [[Milnor fibration|KTFG1S2SMG]] of $f$ at $0$.
Going once around the base circle gives a diffeomorphism $h \colon F \to F$, defined up to isotopy, and its action $T = h_* \colon H_n(F; \C) \to H_n(F; \C)$ is the *[[monodromy|C8HD62AB6Q]]* of $f$.
Then:

1. every [[eigenvalue|0BQAA2BK3Q]] of $T$ is a [[root of unity|0YV2CKP2AV]];
2. every [[Jordan block|07PH4RX0X5]] of $T$ has size at most $n + 1$.

Equivalently, there is an integer $N \ge 1$ such that $(T^N - \id)^{n+1} = 0$: the monodromy is quasi-unipotent.

## Remarks

One proof uses an embedded resolution of $f^{-1}(0)$, in which the pull-back of $f$ is locally a monomial $u \, x_1^{N_1} \cdots x_k^{N_k}$ with $u$ a unit.
Locally the monodromy then has finite order, and one finds that $T^N$ is unipotent for $N$ the least common multiple of the multiplicities $N_i$.
The nilpotency index of $T^N - \id$ is bounded by the number of divisors that can meet at a point, which is at most $n + 1$.
Other proofs go through the regularity of the Gauss–Manin connection or through Hodge theory.

Example: for $f = x^p + y^q$ with $p, q \ge 2$, the monodromy on $H_1(F; \C)$ is diagonalizable.
Its eigenvalues are the products $\zeta \eta$, where $\zeta \ne 1$ is a $p$-th root of unity and $\eta \ne 1$ is a $q$-th root of unity.
So $T^N = \id$ with $N = \lcm(p, q)$.

## Explanations

### L1 human

Going once around the singular value, the Milnor fibre comes back to itself, but its cycles may be moved around.
The theorem says that this motion is almost periodic: after enough turns, it differs from doing nothing only by a nilpotent part.
In particular, no eigenvalue can be a number like $2$ or $e^{i\sqrt{2}\pi}$; they are all roots of unity.

### L2 human

The monodromy theorem is the local version of a result for families of algebraic varieties over a punctured disc, proved in several ways by Grothendieck, Landman, Clemens and others.
For singularities, the eigenvalues of $T$ are linked to the roots of the [[Bernstein–Sato polynomial|7M5P2A0BZ6]] of $f$.
By results of Malgrange and Kashiwara, the numbers $e^{2\pi i \alpha}$ for the roots $\alpha$ are the eigenvalues of the monodromy at points of $f^{-1}(0)$.
The size of the Jordan blocks is related to the weight filtration of the limit mixed Hodge structure on the cohomology of the Milnor fibre.
</document>

<document source="https://tamnd.github.io/math2/node/DQRXS8PJ15.md" tag="DQRXS8PJ15" kind="definition">
# Heaviside operational calculus

## Definition

Let $H$ be the [[Heaviside step function|09C4JP7Y0M]], and consider functions of $t$ that vanish for $t < 0$.
The *Heaviside operational calculus* is the formal method that works with the following conventions.

1. The [[differentiation operator|BB15T3J891]] $d/dt$ is written $p$, and $p^{-1}$ means integration from $0$: $(p^{-1} f)(t) = \int_0^t f(\tau) \, d\tau$.
2. For each integer $n \ge 0$, $p^{-n} H = \frac{t^n}{n!} H$.
3. Consider a system governed by a [[linear differential equation|0QQRP17Z7S]] with constant coefficients, at rest before $t = 0$. Replace $d/dt$ by $p$ and solve the equation algebraically for $x$. The response $x$ to the input $H$ is then written $x = \varphi(p) H$ with a rational function $\varphi$.
4. $\varphi(p) H$ is evaluated by expanding $\varphi(p)$ in powers of $p^{-1}$ and applying rule 2 term by term, or by the expansion theorem.

## Remarks

In modern terms, $\varphi(p) H$ is the function whose Laplace transform is $\varphi(s)/s$, and rule 2 says that $\mathcal{L}(t^n/n!) = 1/s^{n+1}$.
For rational $\varphi$ that is bounded at infinity, the series in rule 4 converges for all $t$ and the method is correct.
Heaviside also applied the rules to non-rational expressions such as $\sqrt{p}$, where they need a separate justification.

Example: $x' - a x = a H$, with $x = 0$ for $t < 0$.
Rule 3 gives $(p - a) x = a H$, so $x = \frac{a}{p - a} H = \frac{a/p}{1 - a/p} H = \sum_{n \ge 1} a^n p^{-n} H = \sum_{n \ge 1} \frac{a^n t^n}{n!} = e^{at} - 1$ for $t \ge 0$.
Check: $x' - ax = a e^{at} - a e^{at} + a = a$ and $x(0) = 0$.

Non-example: the rules do not apply to an equation with variable coefficients such as $x' - t x = H$.
There $p$ does not commute with multiplication by $t$, so the equation cannot be solved for $x$ as $\varphi(p) H$.
</document>

<document source="https://tamnd.github.io/math2/node/DQWX2Y8EA5.md" tag="DQWX2Y8EA5" kind="definition">
# Trace form

## Definition

Let $L/K$ be a [[finite extension|0NRYPM9Y27]] and let $\mathrm{Tr}_{L/K}$ be the [[field trace|73XV4RP5PT]].
The *trace form* of $L/K$ is the map
$$B \colon L \times L \to K, \qquad B(x, y) = \mathrm{Tr}_{L/K}(xy).$$

## Remarks

Since $\mathrm{Tr}_{L/K}$ is $K$-linear and multiplication is commutative and distributive, $B$ is a [[symmetric bilinear form|0744CN4MDA]] on the $K$-vector space $L$.
It is [[nondegenerate|0KAN330PAH]] if and only if $L/K$ is separable.
Its quadratic form is $x \mapsto \mathrm{Tr}_{L/K}(x^2)$.

Example: $L = \Q(i)$ over $K = \Q$.
Check: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$, so $B(a + bi, c + di) = 2(ac - bd)$; in the basis $1, i$ the matrix of $B$ is $\begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix}$, which is invertible.

Non-example of nondegeneracy: let $K = \F_p(t)$ and $L = K(s)$ with $s^p = t$.
The trace of every element of $L$ is $0$, so the trace form of $L/K$ is identically zero and degenerate, as expected for an inseparable extension.
</document>

<document source="https://tamnd.github.io/math2/node/DR9BXFWMQB.md" tag="DR9BXFWMQB" kind="statement">
# Supporting hyperplane theorem

## Statement

Let $n \ge 1$, let $S \subseteq \R^n$ be a [[convex set|02AS45T3H9]], and let $x_0$ be a point of the [[boundary|05D5EGMFYV]] of $S$.
Then there is a [[supporting hyperplane|K7N565NKCV]] of $S$ through $x_0$.
That is, there is a vector $a \in \R^n$ with $a \ne 0$ such that $a \cdot x \le a \cdot x_0$ for all $x \in S$, where $a \cdot x$ is the [[dot product|0S52TNNTH9]].

## Remarks

Proof idea: since $x_0$ is a boundary point of the convex set $S$, it is not an interior point of the closure $\overline{S}$, so there are points $y_k \notin \overline{S}$ with $y_k \to x_0$.
For each $k$, the nearest point of $\overline{S}$ to $y_k$ gives a unit vector $a_k$ with $a_k \cdot x \le a_k \cdot y_k$ for all $x \in S$.
A subsequence of the unit vectors $a_k$ converges to a unit vector $a$, and passing to the limit gives $a \cdot x \le a \cdot x_0$ for all $x \in S$.

Example: for the square $[0, 1]^2$ and the corner $x_0 = (1, 1)$, every $a = (a_1, a_2) \ne 0$ with $a_1, a_2 \ge 0$ works, so the supporting line at a corner is not unique.

Convexity is needed: for the annulus $\set{x \in \R^2 : 1 \le \abs{x} \le 2}$, the inner boundary point $(1, 0)$ has no supporting line, since every line through it has points of the annulus on both sides.
A partial converse holds: a closed set with nonempty interior that has a supporting hyperplane at each of its boundary points is convex.

## Explanations

### L1 human

Take a convex shape and any point on its edge.
The theorem says you can always lay a flat wall against the shape at that point, with the whole shape on one side.
For a smooth shape the wall is the tangent plane; at a corner there may be many possible walls.
This fails for shapes with dents, where the wall would have to cut through the shape.

### L2 human

The theorem is the finite-dimensional, boundary-point form of the separation theorems; the general version is the [[Hahn–Banach separation theorem|6GWMGFGFJJ]].
It is used to prove that a compact convex set in $\R^n$ is the convex hull of its extreme points.
That proof passes to the face cut out by a supporting hyperplane and inducts on dimension.
It gives the existence of subgradients of a convex function at interior points of its domain, by applying it to the epigraph.
In optimisation it is the geometric root of Lagrange multipliers and of the Karush–Kuhn–Tucker conditions.
</document>

<document source="https://tamnd.github.io/math2/node/DRFKZKE229.md" tag="DRFKZKE229" kind="definition">
# Regular surface

## Definition

Let $S \subseteq \R^3$ be a subset.
The set $S$ is a *regular surface* if each $p \in S$ has the following property.
There are an [[open set|01ZQQDCERP]] $V \subseteq \R^3$ containing $p$, an open set $U \subseteq \R^2$, and a map $x \colon U \to V \cap S$ such that:

1. The map $x$ is [[smooth|0BE3QDH73A]] as a map from $U$ to $\R^3$.
2. The map $x$ is a [[homeomorphism|05GZ11PWRY]] from $U$ onto $V \cap S$, with the subspace topology on $V \cap S$.
3. For each $q \in U$, the [[Jacobian matrix|0YF0VXKXGB]] of $x$ at $q$ has rank $2$, that is, $x_u(q)$ and $x_v(q)$ are linearly independent.

Such a map $x$ is a *local parametrisation* of $S$ at $p$.

## Remarks

Example: the unit sphere is a regular surface.
At points with $z > 0$, the map $x(u, v) = (u, v, \sqrt{1 - u^2 - v^2})$ on the open unit disc satisfies the three clauses, and five similar maps cover the rest.
More generally, if $f \colon \R^3 \to \R$ is smooth and $\nabla f \ne 0$ on $f^{-1}(c)$, then the level set $f^{-1}(c)$, when nonempty, is a regular surface.
Non-example: the double cone $x^2 + y^2 = z^2$ fails clause 2 at the origin.
Indeed, removing the origin from a small neighbourhood of it in the cone leaves two pieces, while a punctured disc is connected.
Each local parametrisation is a regular [[parametrised surface|05788J51WZ]] in the sense of a map, but a regular surface is a set, covered by such maps.
</document>

<document source="https://tamnd.github.io/math2/node/DRGS5TT784.md" tag="DRGS5TT784" kind="definition">
# Extremal graph theory

## Definition

Let $H$ be a finite simple [[graph|00C0BR7D7H]] with at least one edge, and let $n \ge 1$ be an integer.
A graph $G$ is *$H$-free* if no [[subgraph|0WBFFC8XZ9]] of $G$ is [[isomorphic|0F7QKGJS6B]] to $H$.
The *extremal number* $\mathrm{ex}(n, H)$ is the largest number of edges of an $H$-free graph with $n$ vertices.
An $H$-free graph with $n$ vertices and $\mathrm{ex}(n, H)$ edges is an *extremal graph* for $H$.
Extremal graph theory is the study of $\mathrm{ex}(n, H)$, of the extremal graphs, and of similar maximum and minimum problems for graphs.

## Remarks

Example: $\mathrm{ex}(4, K_3) = 4$.
The $4$-cycle has $4$ edges and no triangle.
A graph on $4$ vertices with $5$ edges is $K_4$ minus one edge, and it contains a triangle.

Non-example: $K_4$ is not counted for $\mathrm{ex}(4, K_3)$, because it is not $K_3$-free.

The maximum exists, since the empty graph on $n$ vertices is $H$-free and the number of edges is at most $\binom{n}{2}$.
For complete graphs, [[Turán's theorem|0JG1KZSPFZ]] says that $\mathrm{ex}(n, K_{r+1})$ is the number of edges of the [[Turán graph|0VTCABADBE]] $T(n, r)$.
</document>

<document source="https://tamnd.github.io/math2/node/DRJPG78875.md" tag="DRJPG78875" kind="definition">
# Integral closure

## Definition

Let $B$ be a commutative ring and let $A$ be a [[subring|0GNCRPYXYV]] of $B$.
The *integral closure of $A$ in $B$* is the set
$$\overline{A} = \set{b \in B : b \text{ is integral over } A}$$
of the elements of $B$ that are [[integral|1EHN4KPSFS]] over $A$.
The ring $A$ is *integrally closed in $B$* if $\overline{A} = A$.
For an [[integral domain|0QVD7WBFGN]] $A$, the integral closure of $A$ in its [[field of fractions|0F2PTGX1A2]] is the *normalization* of $A$.

## Remarks

The set $\overline{A}$ is a subring of $B$ that contains $A$, and it is integrally closed in $B$ by [[transitivity of integrality|ZXZMQM818D]].

Example: the integral closure of $\Z$ in $\Q(i)$ is $\Z[i]$.
If $a + bi$ with $a, b \in \Q$ is integral, then so is its conjugate, so $2a$ and $a^2 + b^2$ are rational and integral over $\Z$, hence integers, and this forces $a, b \in \Z$.

Non-example: $\Z[\sqrt{5}]$ is not integrally closed in $\Q(\sqrt{5})$.
The element $(1 + \sqrt{5})/2$ is a root of $x^2 - x - 1$, so it is integral over $\Z$, but it is not in $\Z[\sqrt{5}]$.
The integral closure of $\Z$ in $\Q(\sqrt{5})$ is $\Z[(1 + \sqrt{5})/2]$.
</document>

<document source="https://tamnd.github.io/math2/node/DRNH767W84.md" tag="DRNH767W84" kind="definition">
# Donaldson theory

## Definition

Let $M$ be a closed, oriented, [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]] with a [[Riemannian metric|Q61VFHCH2J]] $g$, and let $\ast$ be the Hodge star of $g$ on $2$-forms.
Let $P \to M$ be a [[principal bundle|8E37E74XVP]] with structure group [[$SU(2)$|E5P8AG6CK7]] and second [[Chern class|PAD37GKDPD]] $c_2(P) = k$.
For a [[connection|NJ21PN5ZGJ]] $A$ on $P$ with curvature $F_A$, write $F_A = F_A^+ + F_A^-$ with $\ast F_A^{\pm} = \pm F_A^{\pm}$.

1. The connection $A$ is *anti-self-dual* if $F_A^+ = 0$.
2. The gauge group $\mathcal{G}$ of bundle automorphisms of $P$ acts on anti-self-dual connections.
3. The *moduli space* $\mathcal{M}_k(g)$ is the set of anti-self-dual connections on $P$ modulo $\mathcal{G}$.

*Donaldson theory* is the study of the spaces $\mathcal{M}_k(g)$ and of the invariants of $M$ obtained from them.
The main such invariants are the Donaldson polynomial invariants, defined by integrating cohomology classes over $\mathcal{M}_k(g)$.

## Remarks

Let $b_1(M)$ be the first Betti number and $b_2^+(M)$ the number of positive eigenvalues of the intersection form.
For a generic metric, the irreducible part of $\mathcal{M}_k(g)$ is a smooth manifold of dimension $8k - 3(1 - b_1(M) + b_2^+(M))$.
For example, for $k = 1$ and $M$ simply connected with negative definite form, the dimension is $8 - 3 = 5$; this is the moduli space used in the proof of [[Donaldson's theorem|34B7K77P31]].
A connection with $F_A^- = 0$ but $F_A^+ \ne 0$ is self-dual, not anti-self-dual, and so does not lie in $\mathcal{M}_k(g)$; reversing the orientation of $M$ exchanges the two notions.
Anti-self-dual connections are the absolute minima of the Yang–Mills functional in their bundle.
The polynomial invariants are defined when $b_2^+(M) \ge 2$ and do not depend on the metric.
</document>

<document source="https://tamnd.github.io/math2/node/DRT1E6WY6K.md" tag="DRT1E6WY6K" kind="statement">
# Sokhotski–Plemelj formulae

## Statement

Let $\Gamma$ be a smooth simple closed curve in $\C$, oriented counterclockwise, with interior domain $D^+$ and exterior domain $D^-$.
Let $\varphi \colon \Gamma \to \C$ be [[Hölder continuous|012PBA0FMD]], and let $\Phi(z) = \frac{1}{2\pi i} \int_\Gamma \frac{\varphi(\tau)}{\tau - z} \, d\tau$ for $z \notin \Gamma$ be its [[Cauchy-type integral|MGBKF52TQE]].
Let $t \in \Gamma$.
Then the limit $\Phi^+(t)$ of $\Phi(z)$ as $z \to t$ with $z \in D^+$ exists, and so does the limit $\Phi^-(t)$ as $z \to t$ with $z \in D^-$.
They are
$$\Phi^{\pm}(t) = \pm \frac12 \varphi(t) + \frac{1}{2\pi i} \, \mathrm{p.v.} \int_\Gamma \frac{\varphi(\tau)}{\tau - t} \, d\tau,$$
where $\mathrm{p.v.}$ is the [[Cauchy principal value|0SYR10HD4R]].
Equivalently,
$$\Phi^+(t) - \Phi^-(t) = \varphi(t), \qquad \Phi^+(t) + \Phi^-(t) = \frac{1}{\pi i} \, \mathrm{p.v.} \int_\Gamma \frac{\varphi(\tau)}{\tau - t} \, d\tau.$$

## Remarks

Proof idea: write $\int_\Gamma \frac{\varphi(\tau)}{\tau - z} \, d\tau = \int_\Gamma \frac{\varphi(\tau) - \varphi(t)}{\tau - z} \, d\tau + \varphi(t) \int_\Gamma \frac{d\tau}{\tau - z}$.
The last integral is $2\pi i$ for $z \in D^+$ and $0$ for $z \in D^-$.
The Hölder condition makes the first integral continuous up to $\Gamma$ from both sides, with value $\mathrm{p.v.} \int_\Gamma \frac{\varphi(\tau) - \varphi(t)}{\tau - t} \, d\tau$ at $t$.
Finally, $\mathrm{p.v.} \int_\Gamma \frac{d\tau}{\tau - t} = \pi i$, half of $2\pi i$, because the excised small arc around $t$ leaves half a turn.

Example: on the unit circle with $\varphi(\tau) = \tau^{-1}$, one has $\Phi = 0$ on $D^+$ and $\Phi(z) = -1/z$ on $D^-$.
So $\Phi^+(t) - \Phi^-(t) = t^{-1} = \varphi(t)$, and the principal value integral equals $\pi i (\Phi^+(t) + \Phi^-(t)) = -\pi i \, t^{-1}$.

Continuity of $\varphi$ alone is not enough: there are continuous densities whose principal value integral fails to exist at some points.
The formulae also hold on a smooth open arc, at every point other than the endpoints.
There $\Phi^+$ is the limit from the left of the arc and $\Phi^-$ the limit from the right.
For the real line they reduce to the [[Sokhotski–Plemelj theorem|RNB6QX0GCA]].

## Explanations

### L1 human

A Cauchy-type integral defines two holomorphic functions, one inside the curve and one outside.
At the curve, the inside one and the outside one have limits.
Their difference is the density $\varphi$, and their average is a principal value integral over the curve.
So the integral jumps by exactly $\varphi$ when one crosses the curve.

### L2 human

The formulae are the bridge between boundary value problems for holomorphic functions and singular integral equations.
The Riemann–Hilbert problem $\Phi^+ = G \Phi^- + g$ on $\Gamma$ is solved by writing the unknowns as Cauchy-type integrals.
Conversely, substituting $\Phi^\pm$ into the formulae turns the [[singular integral equation|E30EA6ETZ4]] $a \varphi + b S \varphi = f$, with $S \varphi = \frac{1}{\pi i} \mathrm{p.v.} \int_\Gamma \frac{\varphi(\tau)}{\tau - t} d\tau$, into the boundary condition $(a + b) \Phi^+ - (a - b) \Phi^- = f$.
This reduction is the basis of the theory of the [[index of a singular integral operator|GW3B4K1WA5]].
</document>

<document source="https://tamnd.github.io/math2/node/DRY1HXP1W9.md" tag="DRY1HXP1W9" kind="statement">
# Hopf's theorem on cohomology of compact Lie groups

## Statement

Let $G$ be a connected [[compact Lie group|9TFC6ADC1C]] and let $k$ be a field of characteristic $0$.
Then the [[cohomology|HKZABKRPN6]] $H^*(G; k)$ is isomorphic, as a graded $k$-algebra, to an [[exterior algebra|0MDMZ965VE]] $\Lambda_k(x_1, \dots, x_r)$ on finitely many generators $x_i$ of odd degrees $d_1, \dots, d_r$.

## Remarks

Hopf's proof uses only the [[Hopf algebra|WJXK0V9NE9]] structure given by the multiplication of $G$, together with finite-dimensionality.
Over a field of characteristic $0$, a connected graded-commutative Hopf algebra is free as a graded-commutative algebra.
That is, it is a polynomial algebra on generators of even degree tensored with an exterior algebra on generators of odd degree.
A polynomial generator of even degree would have infinitely many nonzero powers, which is impossible in finite dimension.
So all generators have odd degree.
The same argument works for any connected H-space with finite-dimensional rational cohomology.

Further facts: the number $r$ of generators equals the rank of $G$, the dimension of a maximal torus, and $d_1 + \cdots + d_r = \dim G$, since the product $x_1 \cdots x_r$ spans the top cohomology.

Example: for $G = \mathrm{SU}(n)$, the degrees are $3, 5, \dots, 2n - 1$, with $r = n - 1$ and $3 + 5 + \cdots + (2n - 1) = n^2 - 1 = \dim \mathrm{SU}(n)$.
For $G = \mathrm{SU}(2) \cong S^3$, this is $H^*(S^3; \Q) = \Lambda(x_3)$.

The characteristic $0$ hypothesis is needed.
The group $\mathrm{SO}(3)$ is homeomorphic to $\R P^3$, and $H^*(\R P^3; \F_2) = \F_2[x]/(x^4)$ with $\abs{x} = 1$, which is not an exterior algebra since $x^2 \ne 0$.

## Explanations

### L1 human

The holes of a compact connected Lie group have a very simple pattern.
There are a few basic holes, all of odd dimension, and every other hole is a combination of distinct basic holes.
For the group $\mathrm{SU}(2)$, a $3$-sphere, there is just one basic hole of dimension $3$.

### L2 human

The theorem explains why the rational cohomology of a compact Lie group looks like that of a product of odd spheres.
For example $H^*(\mathrm{U}(n); \Q)$ is that of $S^1 \times S^3 \times \cdots \times S^{2n-1}$.
The degrees $d_i$ are $2m_i + 1$, where the $m_i$ are the exponents of the Weyl group, and the $m_i + 1$ are the degrees of its basic invariant polynomials.
With integer or mod $p$ coefficients the cohomology can have torsion, and its study is much harder.
</document>

<document source="https://tamnd.github.io/math2/node/DRYYK2WX0M.md" tag="DRYYK2WX0M" kind="concept">
# Complexification of a Lie group

## Summary

The complexification of a real [[Lie group|7JBRGCVJRB]] $G$ is a complex Lie group $G_\C$ with a homomorphism $G \to G_\C$.
Every continuous homomorphism of $G$ to a complex Lie group factors through it, in a unique holomorphic way.
It is the group analogue of passing from a real Lie algebra $\mathfrak{g}$ to $\mathfrak{g} \otimes_\R \C$.
For example $\mathrm{SL}_n(\C)$ is the complexification of both $\mathrm{SL}_n(\R)$ and $\mathrm{SU}(n)$.
It links real groups, compact groups and complex groups, as in Weyl's unitary trick.

## Explanations

### L1 human

A real Lie group is described by real parameters, such as angles or real matrix entries.
Its complexification lets these parameters become complex numbers.
For the circle of angles $e^{i\theta}$, letting $\theta$ be complex gives all nonzero complex numbers $\C^\times$.
So $\C^\times$ is the complexification of the circle group.
Any continuous homomorphism from the circle to a complex Lie group extends in exactly one way to a holomorphic homomorphism on $\C^\times$.
</document>

<document source="https://tamnd.github.io/math2/node/DS2AS9TSZP.md" tag="DS2AS9TSZP" kind="conjecture">
# Hilbert–Smith conjecture

## Statement

Let $M$ be a connected [[topological manifold|QQQ9B1DK4P]] and let $G$ be a [[locally compact group|AHV05ZBCYW]].
Suppose that $G$ acts on $M$ continuously and [[faithfully|0P7G9NGG2G]].
That is, the map $G \times M \to M$ is continuous, and only the identity of $G$ fixes every point of $M$.
Then $G$ is a [[Lie group|7JBRGCVJRB]].

## Remarks

The conjecture is a form of [[Hilbert's fifth problem|NR70GHZ37D]], which Hilbert stated for transformation groups; it is named after David Hilbert and Paul A. Smith.
It is known to be equivalent to the claim that, for every prime $p$, the group $\Z_p$ of $p$-adic integers cannot act continuously and faithfully on a connected manifold.
The conjecture is proved for manifolds of dimension at most $3$; the case of dimension $3$ is due to John Pardon.
It is also proved for actions by Lipschitz maps and for actions by diffeomorphisms.
In general it is open.

## Explanations

### L1 human

A group can act on a space by moving its points around, as rotations move the points of a sphere.
The conjecture says that if a group acts on a manifold in a continuous and faithful way, the group must be a smooth Lie group.
In other words, exotic groups such as the $p$-adic integers should not act continuously on a manifold unless some element other than the identity leaves every point in place.
</document>

<document source="https://tamnd.github.io/math2/node/DS3W88FESM.md" tag="DS3W88FESM" kind="definition">
# Diffeology

## Definition

Let $X$ be a set.
A *parametrization* of $X$ is a map $p \colon U \to X$, where $U$ is an open subset of $\R^n$ for some $n \ge 0$.
A *diffeology* on $X$ is a set $\mathcal{D}$ of parametrizations of $X$, called *plots*, with these properties.

1. Covering: every constant parametrization $U \to X$ is a plot.
2. Locality: if a parametrization $p \colon U \to X$ is such that each $u \in U$ has an open neighbourhood $V \subseteq U$ with $p|_V \in \mathcal{D}$, then $p \in \mathcal{D}$.
3. Smooth compatibility: let $p \colon U \to X$ be a plot and let $V$ be an open subset of some $\R^m$. For every [[smooth map|0V33BNKZ9S]] $f \colon V \to U$, the map $p \circ f$ is a plot.

A *diffeological space* is a set with a diffeology.
A map $F \colon X \to Y$ between diffeological spaces is *smooth* if $F \circ p$ is a plot of $Y$ for every plot $p$ of $X$.

## Remarks

Example: let $M$ be a [[smooth manifold|F45VQ47GPV]] and let the plots be all smooth maps $p \colon U \to M$.
Constant maps are smooth, smoothness is a local property, and composites of smooth maps are smooth, so the three clauses hold.
A map between manifolds is smooth in the usual sense exactly when it is smooth for these diffeologies.

Non-example: on $X = \R$, let the parametrizations be the real-[[analytic|081WKEDK6C]] maps $U \to \R$.
Clauses 1 and 2 hold, but clause 3 fails: the identity of $\R$ is analytic, while its composite with the smooth non-analytic map $f(t) = e^{-1/t^2}$ for $t \ne 0$, $f(0) = 0$, is not analytic.

Every subset and every quotient of a diffeological space carries a natural diffeology.
Diffeological spaces with smooth maps form a [[cartesian closed category|BFBNQ36QS8]], so the set of smooth maps between two diffeological spaces is again a diffeological space.
</document>

<document source="https://tamnd.github.io/math2/node/DS6TC7FSDQ.md" tag="DS6TC7FSDQ" kind="definition">
# Kan extension

## Definition

Let $K \colon \mathcal{A} \to \mathcal{B}$ and $F \colon \mathcal{A} \to \mathcal{C}$ be [[functors|28NQ0KRHD3]].
A *left Kan extension* of $F$ along $K$ is a functor $L \colon \mathcal{B} \to \mathcal{C}$ together with a [[natural transformation|KYCPCC57FT]] $\eta \colon F \Rightarrow L \circ K$ with this property.
For every functor $G \colon \mathcal{B} \to \mathcal{C}$ and every natural transformation $\alpha \colon F \Rightarrow G \circ K$, there is a unique natural transformation $\beta \colon L \Rightarrow G$ such that $\alpha = (\beta K) \circ \eta$.
Here $\beta K$ is the natural transformation with components $\beta_{K(a)}$ for $a$ in $\mathcal{A}$.

A *right Kan extension* of $F$ along $K$ is a functor $R \colon \mathcal{B} \to \mathcal{C}$ together with a natural transformation $\varepsilon \colon R \circ K \Rightarrow F$ with this property.
For every functor $G \colon \mathcal{B} \to \mathcal{C}$ and every natural transformation $\alpha \colon G \circ K \Rightarrow F$, there is a unique natural transformation $\beta \colon G \Rightarrow R$ such that $\alpha = \varepsilon \circ (\beta K)$.

## Remarks

The usual notation is $\mathrm{Lan}_K F$ for the left and $\mathrm{Ran}_K F$ for the right Kan extension.
Both are unique up to a unique natural isomorphism when they exist.
Suppose these extensions exist for all $F$.
Then, between the [[functor categories|1X0WKFQX8K]], $\mathrm{Lan}_K$ is left adjoint and $\mathrm{Ran}_K$ is right adjoint to the restriction $G \mapsto G \circ K$ from $[\mathcal{B}, \mathcal{C}]$ to $[\mathcal{A}, \mathcal{C}]$.

Example: let $\mathcal{B} = \mathbf{1}$ be the category with one object and one morphism, and $K$ the unique functor.
A functor $G \colon \mathbf{1} \to \mathcal{C}$ is an object $c$, and $G \circ K$ is the constant functor at $c$.
A natural transformation $F \Rightarrow G \circ K$ is then a cocone from $F$ to $c$.
So the clause of the definition says exactly that $\eta$ is a universal cocone, and $\mathrm{Lan}_K F$ is the colimit of $F$.

Non-example: let $\mathcal{A}$ be the empty category, $\mathcal{B} = \mathbf{1}$ and $\mathcal{C}$ the category of sets.
Then $F$ and $\eta$ are empty, and a left Kan extension is an initial object of $\mathcal{C}$, namely the empty set.
The constant functor $L$ at a one-point set is not a left Kan extension.
For $G$ the constant functor at the empty set there is no natural transformation $L \Rightarrow G$, so the existence part of the clause fails.
</document>

<document source="https://tamnd.github.io/math2/node/DSE72WQ5A2.md" tag="DSE72WQ5A2" kind="concept">
# Near-field

## Summary

A near-field is an algebraic structure like a [[division ring|0EA5BTA0CX]], except that only one of the two distributive laws is required.
It is a [[near-ring|ARAEEM8CVF]] with identity in which every nonzero element has a multiplicative inverse.
Near-fields were first constructed by Dickson, and they coordinatize certain [[projective planes|DANFNN8C4Y]] that are not Desarguesian.

## Explanations

### L1 human

In a field one can add, subtract, multiply and divide, and multiplication distributes over addition from both sides.
A near-field keeps everything except one side of distributivity: $(a + b)c = ac + bc$ always holds, but $a(b + c) = ab + ac$ may fail.
Every field and every division ring is a near-field.
The smallest near-field that is not a division ring has $9$ elements.
It is built from the [[finite field|0ZXZ82FTX2]] with $9$ elements by twisting its multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/DSF8C6QPAG.md" tag="DSF8C6QPAG" kind="conjecture">
# Kaplansky's conjectures

## Statement

Let $K$ be a field and let $G$ be a [[torsion-free group|PEMNN32A43]].
That is, every element of $G$ other than the identity has infinite order.
Let $K[G]$ be the [[group ring|4NMTSX0XDS]] of $G$ over $K$.

1. (Zero divisor conjecture) $K[G]$ has no [[zero divisors|0NNRDW10RZ]] other than $0$; see [[Kaplansky's zero divisor conjecture|E1NE8V7EX8]].
2. (Idempotent conjecture) The only [[idempotents|0N5NJZKKN3]] of $K[G]$ are $0$ and $1$.
3. (Unit conjecture) Every [[unit|0XVEZDE9J8]] of $K[G]$ has the form $c g$ with $c \in K$ nonzero and $g \in G$.

## Remarks

The conjectures are linked: the unit conjecture implies the zero divisor conjecture, and the zero divisor conjecture implies the idempotent conjecture, since an idempotent $e$ satisfies $e(1 - e) = 0$.
The hypothesis that $G$ is torsion-free is needed: if $g \ne 1$ has finite order $n$, then $(1 - g)(1 + g + \cdots + g^{n-1}) = 1 - g^n = 0$.
The unit conjecture goes back to the 1940 thesis of Graham Higman.
In 2021 Giles Gardam disproved the unit conjecture over the field with two elements, with a nontrivial unit in the group ring of the fundamental group of the Hantzsche–Wendt manifold.
All three conjectures hold for groups with the unique product property, and the zero divisor conjecture holds for torsion-free elementary amenable groups.
The zero divisor and idempotent conjectures are open in general.
Kaplansky also made conjectures in other fields, on Banach algebras and on quadratic forms; they are separate questions.

## Explanations

### L1 human

The group ring of a group mixes the group with a field, and elements of finite order in the group always create zero divisors and idempotents in it.
Kaplansky asked whether these are the only sources: if the group has no elements of finite order, is the group ring as well behaved as a ring of polynomials?
Three precise forms of the question concern zero divisors, idempotents and units.
The one about units turned out to be false, and the other two are still open.
</document>

<document source="https://tamnd.github.io/math2/node/DSJET02WPR.md" tag="DSJET02WPR" kind="concept">
# Coherent analytic sheaf

## Summary

A coherent analytic sheaf on a [[complex analytic space|HXSZA44J5J]] is a [[sheaf of modules|S33B2H3MYD]] over the structure sheaf that is locally finitely generated.
Moreover, the relations among finitely many local sections must again be locally finitely generated.
Coherent sheaves are the right class of sheaves for complex analytic geometry.
They include the structure sheaf, ideal sheaves of analytic subsets and sheaves of sections of holomorphic vector bundles.
Kernels, cokernels and images of morphisms between coherent sheaves are coherent.
They are the analytic counterpart of [[coherent sheaves|56RXEJ8Z6S]] on schemes.

## Explanations

### L1 human

A coherent sheaf attaches to each small region of a space a module of functions or sections.
Locally everything is described by finitely many generators and finitely many relations.
The model is a system of linear equations with holomorphic coefficients: its solutions form a coherent sheaf.
For example, on $\C^2$ the germs $(g_1, g_2)$ with $g_1 z_1 + g_2 z_2 = 0$ form a coherent sheaf, generated by the single relation $(z_2, -z_1)$.
</document>

<document source="https://tamnd.github.io/math2/node/DSK9GN63GQ.md" tag="DSK9GN63GQ" kind="statement">
# Lazard's theorem on p-adic analytic groups

## Statement

Let $p$ be a prime.
For a [[pro-p group|C13D6W3MM9]] $U$ and $k \ge 1$, let $\overline{U^k}$ be the closed subgroup generated by all $k$-th powers.
A finitely generated pro-$p$ group $U$ is *powerful* if $U / \overline{U^p}$ is abelian for odd $p$, or $U / \overline{U^4}$ is abelian for $p = 2$.
It is *uniform* if it is finitely generated, powerful and torsion-free.
Let $G$ be a topological group.
Then $G$ is a [[p-adic Lie group|3AKM6EPV2F]] if and only if $G$ has an [[open subgroup|QYCFH68RD1]] that is a uniform pro-$p$ group.

## Remarks

In one direction, a $p$-adic Lie group has small open subgroups on which the logarithm and exponential maps converge.
A small enough such subgroup is uniform.
In the other direction, fix a minimal topological generating set $u_1, \dots, u_d$ of a uniform group $U$.
Every element of $U$ is a unique product $u_1^{\lambda_1} \cdots u_d^{\lambda_d}$ with $\lambda_i \in \Z_p$.
In these coordinates the group law is analytic.
Example: for an odd prime $p$, the congruence subgroup $1 + p M_n(\Z_p)$ of $GL_n(\Z_p)$ is open and uniform, which shows again that $GL_n(\Z_p)$ is a $p$-adic Lie group.
The result was proved by Michel Lazard in his work on $p$-adic analytic groups.
For a [[profinite group|KA0YDB6QXE]], it turns a geometric condition, being a $p$-adic manifold, into a purely group-theoretic one.

## Explanations

### L1 human

A $p$-adic Lie group is a group built from the $p$-adic numbers in a smooth way.
Lazard's theorem says that one can recognize these groups by algebra alone.
A group is of this kind exactly when it has an open subgroup of a special algebraic type, called a uniform pro-$p$ group.
Such a group is a limit of finite groups of $p$-power order.

### L2 human

The theorem is the basis of the group-theoretic treatment of $p$-adic analytic groups, as in the book of Dixon, du Sautoy, Mann and Segal.
Among compact groups, it shows that the compact $p$-adic Lie groups are the profinite groups that contain an open uniform pro-$p$ subgroup.
The dimension of $G$ as a manifold equals the minimal number of topological generators of any uniform open subgroup.
The Lie algebra of $G$ can be built from a uniform subgroup $U$ by turning $U$ into a $\Z_p$-lattice with a Lie bracket.
This is a close analogue of the real Lie group–Lie algebra correspondence.
</document>

<document source="https://tamnd.github.io/math2/node/DSKV65NHZZ.md" tag="DSKV65NHZZ" kind="statement">
# Margulis superrigidity theorem

## Statement

Let $G$ be a connected semisimple [[algebraic group|GH81SYAPPN]] defined over $\R$ whose real rank is at least $2$.
Assume that the identity component of $G(\R)$ has no compact factors.
Let $\Gamma$ be an irreducible [[lattice|BTYF5Y5PFY]] in $G(\R)$.
Let $H$ be a connected simple algebraic group over $\R$ with trivial center.
Let $\rho \colon \Gamma \to H(\R)$ be a [[group homomorphism|0JYTG5K9Y4]] such that $\rho(\Gamma)$ is Zariski dense in $H$ and its closure in $H(\R)$ is not compact.
Then there is a continuous homomorphism $\varphi \colon G(\R)^\circ \to H(\R)$, where $G(\R)^\circ$ is the identity component.
The map $\varphi$ agrees with $\rho$ on a subgroup of finite index in $\Gamma$.
In other words, $\rho$ virtually extends to $G(\R)^\circ$.

## Remarks

The proof builds a $\Gamma$-equivariant measurable map from the boundary $G(\R)/P$, with $P$ a minimal parabolic subgroup, to a projective variety of $H$.
This uses the amenability of $P$ and the ergodicity of the action of $\Gamma$ on $G(\R)/P$.
The higher rank is then used to show that this map is rational, and a rational boundary map gives the virtual extension.

Example: take $\Gamma = \mathrm{SL}_3(\Z)$ in $G = \mathrm{SL}_3$.
Every homomorphism of $\Gamma$ to such an $H(\R)$ with Zariski dense and unbounded image agrees, on a subgroup of finite index, with a continuous homomorphism of $\mathrm{SL}_3(\R)$.

The rank condition is necessary.
The group $\mathrm{SL}_2(\R)$ has real rank $1$, and $\mathrm{SL}_2(\Z)$ contains a [[free group|0KTW26C80J]] $F_2$ of finite index, which is a lattice in $\mathrm{SL}_2(\R)$.
A subgroup of index $2$ in $F_2$ is a free lattice $F_3$, and it maps onto $F_2$ by killing one free generator $x$.
Composing with the inclusion of $F_2$ into $\mathrm{SL}_2(\R)$ and then the projection to $\mathrm{PSL}_2(\R)$ gives a homomorphism $\rho$ with Zariski dense, unbounded image.
Suppose a continuous $\varphi \colon \mathrm{SL}_2(\R) \to \mathrm{PSL}_2(\R)$ agreed with $\rho$ on a subgroup $\Lambda$ of finite index in $F_3$.
Then $\varphi$ would be nontrivial, so its kernel, a proper normal subgroup of $\mathrm{SL}_2(\R)$, would lie in $\set{\pm I}$.
But some power $x^k$ with $k \ge 1$ lies in $\Lambda$ and in the kernel of $\rho$, and $x^k \ne \pm I$ because a free group has no torsion, so no such $\varphi$ exists.

## Explanations

### L1 human

Superrigidity says that certain discrete groups of matrices can only be represented in the ways the surrounding continuous group allows.
Any reasonable homomorphism of such a discrete group into another matrix group agrees, after passing to a subgroup of finite index, with a continuous homomorphism of the big group.
This holds for lattices in groups like $\mathrm{SL}_n(\R)$ with $n \ge 3$, but not for $\mathrm{SL}_2(\R)$.

### L2 human

Superrigidity is the main step in the proof of the [[Margulis arithmeticity theorem|FEJK846N5X]]: every irreducible lattice in a group of real rank at least $2$ is an [[arithmetic group|4XWG6S7R4Z]].
It strengthens [[Mostow rigidity|4BEWKR0EG7]], which concerns isomorphisms between lattices, to arbitrary homomorphisms.
The rank one groups $\mathrm{Sp}(n, 1)$ and $F_{4(-20)}$ also satisfy superrigidity, by work of Corlette and of Gromov and Schoen.
</document>

<document source="https://tamnd.github.io/math2/node/DSQ17TNW1C.md" tag="DSQ17TNW1C" kind="concept">
# Automorphism

## Summary

An automorphism of a [[structure|983GZBQYT4]] is an [[isomorphism|PZMGQMDN25]] from the structure to itself.
It is a symmetry of the structure: a permutation of its elements that keeps all operations and relations.
The automorphisms of a structure form a group under composition, the [[automorphism group|0K1SCPBNKJ]].
Complex conjugation, as an automorphism of the field $\C$, and the [[Frobenius automorphism of a finite field|0CGCPX8Z70]] are classical examples.

## Explanations

### L1 human

An automorphism is a way to shuffle the elements of a structure so that nobody can tell the difference afterwards.
For the integers with addition, the map $x \mapsto -x$ is an automorphism: it turns sums into sums and $0$ into $0$.
For a square drawn as a graph with four corners and four sides, each rotation and reflection of the square gives an automorphism.
The identity map is always an automorphism.
</document>

<document source="https://tamnd.github.io/math2/node/DSTF6X8DRJ.md" tag="DSTF6X8DRJ" kind="concept">
# Admissible decision rule

## Summary

A decision rule is admissible if no other rule beats it in the following sense.
The other rule is at least as good for every value of the parameter and strictly better for some value.
Admissibility is a minimal requirement of rationality in [[statistical decision theory|CPFSBBHXT5]]: an inadmissible rule can be replaced by one that never does worse.
[[Stein's example|7SC9E65XDN]] shows that even very natural estimators can fail to be admissible.

## Explanations

### L1 human

Compare two rules by their average loss for each possible truth.
If one rule is never worse than another and sometimes better, the worse one should never be used: it is inadmissible.
An admissible rule is one that cannot be beaten in this complete way.
Being admissible does not mean being good.
The rule that always guesses $0$, ignoring the data, is admissible for estimating a normal mean, because nothing beats it when the truth is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/DT0QXJE705.md" tag="DT0QXJE705" kind="concept">
# Stationary process

## Summary

A stationary process is a [[stochastic process|0QH5ENKZM1]] whose statistical behaviour does not change under shifts of time.
All its [[finite-dimensional distributions|Y0CN7V3B7E]] are invariant under a common time shift.
A weaker notion, weak or wide-sense stationarity, asks only that the mean be constant and the covariance depend only on the time lag.
Stationary processes are the basic models of time series analysis and of ergodic theory, and their covariance has a spectral representation by the [[Wiener–Khinchin theorem|3ZXN26PW4G]].

## Explanations

### L1 human

A process is stationary if it looks the same whenever you start watching it.
Daily noise in a measuring device is often modelled this way, while a growing population is not.
For example, a sequence of independent rolls of a fair die is stationary: any stretch of five rolls has the same joint law as any other stretch of five rolls.
A random walk is not stationary, since its spread grows with time.
</document>

<document source="https://tamnd.github.io/math2/node/DT45YNKWSQ.md" tag="DT45YNKWSQ" kind="definition">
# Direct collocation

## Definition

Consider a [[trajectory optimization|58895A7T4H]] problem on a fixed interval $[t_0, t_f]$: minimise $\phi(x(t_f)) + \int_{t_0}^{t_f} L(x, u, t) \, dt$ subject to $x' = f(x, u, t)$, path constraints $c(x, u, t) \le 0$ and boundary conditions $b(x(t_0), x(t_f)) = 0$.
Choose a mesh $t_0 = \tau_0 < \tau_1 < \cdots < \tau_N = t_f$ and an integer $s \ge 1$.
On each interval $[\tau_k, \tau_{k+1}]$ choose $s$ collocation points $\tau_k \le \tau_{k,1} < \cdots < \tau_{k,s} \le \tau_{k+1}$.
The *direct collocation* method replaces the problem by the following finite nonlinear program.

1. The unknowns are the states $x_k$ at the mesh points and the controls at the collocation points. On each interval the state is a polynomial $p_k$ of degree at most $s$ with $p_k(\tau_k) = x_k$.
2. The dynamics are imposed only at the collocation points: $p_k'(\tau_{k,j}) = f(p_k(\tau_{k,j}), u(\tau_{k,j}), \tau_{k,j})$ for all $k$ and $j$, and the pieces join continuously, $p_k(\tau_{k+1}) = x_{k+1}$.
3. The path constraints hold at the collocation points and the boundary conditions at $x_0$ and $x_N$. The integral of $L$ becomes a quadrature sum.

The solution of this program is an approximate optimal trajectory.

## Remarks

Each choice of collocation points corresponds to an implicit [[Runge–Kutta method|7DBPJKYC6C]]; the state on each interval is a [[polynomial interpolant|SS1YMS2NPC]] of its values.
With $s = 2$ and the endpoints $\tau_k, \tau_{k+1}$ as collocation points, clause 2 becomes the trapezoidal rule $x_{k+1} - x_k = \frac{h_k}{2} \left( f(x_k, u_k, \tau_k) + f(x_{k+1}, u_{k+1}, \tau_{k+1}) \right)$ with $h_k = \tau_{k+1} - \tau_k$.
Hermite–Simpson collocation uses the endpoints and the midpoint with a cubic state.
The nonlinear program is large but sparse, and it is usually solved by sequential quadratic programming or by interior-point methods.

Example: minimise $\int_0^1 \frac{1}{2} u^2 \, dt$ with $x' = u$, $x(0) = 0$, $x(1) = 1$, using trapezoidal collocation on a uniform mesh of step $h = 1/N$.
The program is: minimise $\sum_{k=0}^{N-1} \frac{h}{4} (u_k^2 + u_{k+1}^2)$ subject to $x_{k+1} - x_k = \frac{h}{2} (u_k + u_{k+1})$, $x_0 = 0$, $x_N = 1$.
The choice $u_k = 1$, $x_k = k h$ satisfies all constraints.
It also satisfies the optimality conditions, with all multipliers of the dynamics constraints equal to one constant ($1$ or $-1$, by sign convention).
It reproduces the exact solution $u = 1$, $x(t) = t$ at the grid points.

Non-example: direct single shooting integrates the dynamics forward from a guessed control and optimises only over the control values.
It is not collocation, because the states are not unknowns of the program.

## Explanations

### L1 human

The trajectory is replaced by grid values, the equations of motion are checked only at a few points per step, and a large finite optimisation problem is solved.
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<document source="https://tamnd.github.io/math2/node/DT49GKR9AS.md" tag="DT49GKR9AS" kind="concept">
# Essentially self-adjoint operator

## Summary

An essentially self-adjoint operator is a symmetric operator on a Hilbert space whose closure is self-adjoint.
Such an operator has exactly one self-adjoint extension, so it determines a unique quantum observable and a unique spectral theory.
In practice, differential operators are first defined on small spaces of smooth functions, and essential self-adjointness is what makes this choice harmless.

## Explanations

### L1 human

Unbounded operators, such as derivatives, can only be defined on part of a Hilbert space, and the choice of this part matters.
A symmetric operator on a small domain may admit many self-adjoint extensions, which correspond to different boundary conditions and different physics.
An essentially self-adjoint operator leaves no choice: there is exactly one way to extend it to a self-adjoint operator.
For example, $-i \, d/dx$ on smooth compactly supported functions on the whole line $\R$ is essentially self-adjoint, because the line has no boundary where a condition could be imposed.
</document>

<document source="https://tamnd.github.io/math2/node/DT5PH9QVQ7.md" tag="DT5PH9QVQ7" kind="statement">
# Compactness via ultrafilters

## Statement

Let $X$ be a [[topological space|0CQHFPGJ45]].
A [[filter|15PF73BKY6]] $\mathcal F$ on $X$ [[converges|70M1VAPFN0]] to $x \in X$ when every [[neighbourhood|0ZCY0Z80B6]] of $x$ belongs to $\mathcal F$.
Then $X$ is [[compact|0TVJZYB6SH]] if and only if every [[ultrafilter|J64Y3JC0RE]] on $X$ converges to at least one point of $X$.

## Remarks

Suppose $X$ is compact and an ultrafilter $\mathcal U$ converges to no point.
Then each $x$ has an open neighbourhood $V_x \notin \mathcal U$, so $X \setminus V_x \in \mathcal U$.
Finitely many $V_x$ cover $X$, and the intersection of their complements is empty but lies in $\mathcal U$, a contradiction.
Conversely, if an open cover has no finite subcover, the complements of its members generate a proper filter; an ultrafilter that contains it cannot converge to any point.
This direction uses the ultrafilter lemma, a weak form of the axiom of choice.

The theorem gives a short proof of [[Tychonoff's theorem|0JTRSJKCFB]]: an ultrafilter on a product converges as soon as all its images under the projections converge.
If $X$ is also [[Hausdorff|0S1FJ8VEX9]], each ultrafilter has exactly one limit.
For example, on $X = [0, 1]$, the ultrafilter of all sets that contain a fixed point $p$ converges to $p$.

## Explanations

### L1 human

An ultrafilter on a space is a way of choosing, for every subset, whether it is large or small, in a consistent way.
The theorem says that a space is compact exactly when every such choice points to some place in the space.
It is the filter version of the fact that, in a compact metric space, every sequence has a convergent subsequence.

### L2 human

For general spaces, sequences are not enough to describe compactness, but ultrafilters are.
The criterion turns proofs about open covers into proofs about limits.
Its best-known use is the proof of Tychonoff's theorem, where convergence in a product is checked coordinate by coordinate.
It also leads to the Stone–Čech compactification of a discrete space as the space of ultrafilters on it.
</document>

<document source="https://tamnd.github.io/math2/node/DT60HNP41N.md" tag="DT60HNP41N" kind="definition">
# Pfaffian system

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.
A *Pfaffian system* on $M$ is an [[exterior differential system|901CCKQHVY]] $\mathcal{I}$ generated, as a differential ideal, by [[one-forms|02Z5B2HA3F]] $\theta^1, \dots, \theta^s$.
So $\mathcal{I}$ is the algebraic ideal generated by $\theta^1, \dots, \theta^s$ and $d\theta^1, \dots, d\theta^s$.
One usually asks that $\theta^1, \dots, \theta^s$ are linearly independent at each point.
Then $\mathcal{D} = \set{ v \in TM : \theta^a(v) = 0 \text{ for } a = 1, \dots, s }$ is a distribution of rank $n - s$, and an immersed submanifold is an integral manifold of $\mathcal{I}$ exactly when it is tangent to $\mathcal{D}$.

## Remarks

The system is *completely integrable* if $d\theta^a \wedge \theta^1 \wedge \dots \wedge \theta^s = 0$ for $a = 1, \dots, s$, that is, if each $d\theta^a$ lies in the algebraic ideal generated by the $\theta^b$.
By the [[Frobenius theorem|5Q4BHBFXJY]], this holds exactly when every point lies on an integral manifold of dimension $n - s$.

Example: on $\R^3$, the system generated by $\theta = dz$ has $d\theta = 0$, so it is completely integrable.
Its integral surfaces are the planes $z = c$.

Example: on $\R^3$, let $\theta = dz - y\, dx$.
Then $d\theta = dx \wedge dy$ and $\theta \wedge d\theta = dz \wedge dx \wedge dy \ne 0$, so the system is not completely integrable.
It has integral curves, such as $t \mapsto (t, c, ct)$, but no integral surfaces.

Non-example: the exterior differential system on $\R^3$ generated by the [[two-form|0E3G0GRPZ6]] $dx \wedge dy$ is not a Pfaffian system.
It contains no nonzero one-form, so it is not generated by one-forms.
</document>

<document source="https://tamnd.github.io/math2/node/DT6WEXJANH.md" tag="DT6WEXJANH" kind="definition">
# Spherical harmonics

## Definition

Let $d \ge 2$ and $\ell \ge 0$ be integers, and let $S^{d-1} = \set{x \in \R^d : \norm{x} = 1}$ be the unit [[sphere|T62369RFH6]].
A *spherical harmonic of degree $\ell$* is a function $Y \colon S^{d-1} \to \C$ such that $Y = p|_{S^{d-1}}$ for some polynomial $p$ on $\R^d$ that is:

1. a [[homogeneous polynomial|2C3PAQ7H50]] of degree $\ell$, and
2. [[harmonic|0DEB0JMN0W]], that is, $\Delta p = \sum_{j=1}^{d} \frac{\partial^2 p}{\partial x_j^2} = 0$.

The spherical harmonics of degree $\ell$ form a complex vector space $\mathcal{H}_\ell(S^{d-1})$.

## Remarks

Example: on $S^2$, $Y(x, y, z) = x^2 - y^2$ is a spherical harmonic of degree $2$, since $\Delta(x^2 - y^2) = 2 - 2 = 0$.
Non-example: $x^2$ is homogeneous of degree $2$ but $\Delta x^2 = 2 \ne 0$.
On $S^2$ it equals $\left( x^2 - \frac13 (x^2 + y^2 + z^2) \right) + \frac13$, a sum of harmonics of degrees $2$ and $0$, so it is not a spherical harmonic of degree $2$.
For $d \ge 3$, the dimension of $\mathcal{H}_\ell(S^{d-1})$ is $\binom{\ell + d - 1}{d - 1} - \binom{\ell + d - 3}{d - 1}$, which is $2\ell + 1$ for $d = 3$; for $d = 2$ it is $2$ when $\ell \ge 1$.
Each $Y \in \mathcal{H}_\ell$ satisfies $\Delta_{S^{d-1}} Y = -\ell(\ell + d - 2) Y$ for the Laplace–Beltrami operator of the sphere.
The spaces $\mathcal{H}_\ell$ are pairwise orthogonal in $L^2(S^{d-1})$, and their sum is dense.
For $d = 3$, in spherical coordinates, a standard orthonormal basis is $Y_\ell^m(\theta, \varphi) = \sqrt{\frac{2\ell + 1}{4\pi} \frac{(\ell - m)!}{(\ell + m)!}} \, P_\ell^m(\cos \theta) \, e^{im\varphi}$ for $-\ell \le m \le \ell$, where $P_\ell^m$ are the associated Legendre polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/DTNDR45FTH.md" tag="DTNDR45FTH" kind="definition">
# Topos

## Definition

An *elementary topos* is a [[category|A2DE66937D]] $\mathcal{E}$ such that:

1. $\mathcal{E}$ has all finite limits (it is [[finitely complete|PYSBNWCHCF]]);
2. $\mathcal{E}$ is [[Cartesian closed|BFBNQ36QS8]]: for all objects $A$ and $B$ there is an [[exponential object|RHWD8DY41R]] $B^A$;
3. $\mathcal{E}$ has a [[subobject classifier|NKP5TPQ79X]] $\top \colon 1 \to \Omega$.

## Remarks

An equivalent definition asks for finite limits and power objects: for each object $A$, an object $PA$ with a natural bijection between subobjects of $B \times A$ and morphisms $B \to PA$; then $PA = \Omega^A$.
Every topos also has finite colimits, and its subobjects of any object form a Heyting algebra.
Example: $\mathbf{Set}$ is a topos: it has finite limits, the exponential $B^A$ is the set of functions $A \to B$, and $\set{0, 1}$ classifies subsets.
Example: for a group $G$, the category of $G$-sets is a topos with $\Omega = \set{0, 1}$ and trivial action.
Non-example: the category of real vector spaces fails clause 3.
Its terminal object is $0$, so the pullback of $\top$ along $\chi \colon X \to \Omega$ is $\ker \chi$.
If $\chi \colon \R \to \Omega$ classifies the zero subspace, then $\chi$ is injective, so $2\chi \ne \chi$, yet $\ker(2\chi) = 0$ too, and uniqueness fails.
Non-example: the category $\mathbf{Top}$ of topological spaces has finite limits but is not Cartesian closed, so it fails clause 2.
</document>

<document source="https://tamnd.github.io/math2/node/DTVFW3QMNW.md" tag="DTVFW3QMNW" kind="problem">
# Moser's worm problem

## Problem

Call a region $R$ in the plane a *worm cover* if it has the following property.
For every [[plane curve|EWS4WFTA7W]] $\gamma$ of [[length|0TH78S7GBZ]] $1$, some [[isometry|02HPP5VCZH]] $g$ of the plane satisfies $g(\gamma) \subseteq R$.
Find the smallest possible area of a worm cover.

## Remarks

The problem is open; most work concerns convex worm covers.
A disc of radius $1/2$ is such a region, so the answer is at most $\pi / 4$.
Indeed, let $m$ be the point of $\gamma$ at arc length $1/2$ from an end.
Every point of $\gamma$ is joined to $m$ by an arc of length at most $1/2$, so it lies within distance $1/2$ of $m$.
Much smaller convex worm covers are known.

## Explanations

### L1 human

Picture a worm of length one that can curl up in any way.
What is the smallest blanket that can always be placed over the worm, whatever its shape?
A round blanket of diameter one works, but smaller shapes also work, and the best one is not known.
</document>

<document source="https://tamnd.github.io/math2/node/DTWDS6KAKN.md" tag="DTWDS6KAKN" kind="definition">
# Left ideal

## Definition

Let $R$ be a [[ring|0894J980RK]].
A *left ideal* of $R$ is a subset $I \subseteq R$ such that:

1. $I$ is a subgroup of the additive [[group|G5S1M9]] $(R, +)$.
2. $ra \in I$ for all $r \in R$ and $a \in I$.

## Remarks

A *right ideal* is defined in the same way with $ar \in I$ in clause 2.
A [[two-sided ideal|0DDA6CY7GJ]] is a subset that is both a left and a right ideal.
Example: let $K$ be a field and let $I \subseteq M_2(K)$ be the set of matrices whose second column is zero.
It is an additive subgroup, and for $A \in M_2(K)$ and $X \in I$ the second column of $AX$ is $A$ times the second column of $X$, which is zero.
So $I$ is a left ideal.
Non-example: the set $J$ of matrices whose second row is zero is an additive subgroup, but $E_{21} E_{11} = E_{21} \notin J$, where $E_{ij}$ is the matrix with $1$ in position $(i,j)$ and $0$ elsewhere.
So clause 2 fails for $J$; it is a right ideal instead.
The left ideals of $R$ are the submodules of $R$ as a left module over itself.
For each $a \in R$, the set $Ra = \set{ra : r \in R}$ is the left ideal generated by $a$.
</document>

<document source="https://tamnd.github.io/math2/node/DTWHAW1N0V.md" tag="DTWHAW1N0V" kind="concept">
# O-minimal theory

## Summary

An o-minimal structure is a [[totally ordered|0FRXBVVR03]] structure in which every definable subset of the line is a finite union of points and open [[intervals|0Q1D85B8G6]].
An o-minimal theory is a theory all of whose models are o-minimal.
This condition on sets in one variable forces tame behaviour in all dimensions.
Definable sets admit cell decompositions and have a good dimension theory.
The ordered field of real numbers is the model case.
O-minimality is the basis of "tame geometry", used in real algebraic geometry and in number theory.

## Explanations

### L1 human

In an o-minimal structure, a formula about one number can only describe a set made of finitely many points and intervals.
You can never describe an infinite discrete set such as the integers, or a set with infinitely many gaps.
For the real numbers with addition, multiplication and order, the inequality $x^2 - 2 > 0$ describes $(-\infty, -\sqrt 2) \cup (\sqrt 2, \infty)$, a union of two intervals.
Adding the sine function would break this, because the zeros of sine form an infinite discrete set.
</document>

<document source="https://tamnd.github.io/math2/node/DV00798WNX.md" tag="DV00798WNX" kind="definition">
# Initial value problem for a recurrence

## Definition

Let $X$ be a set, let $k \ge 1$ be an integer, and let $f \colon \N \times X^k \to X$ be a function.
Let $c_0, \dots, c_{k-1} \in X$.
The *initial value problem* for the [[recurrence relation|07TZV87Y0T]] of [[order|SNRN446BHC]] $k$ given by $f$, with initial values $c_0, \dots, c_{k-1}$, is the system
$$u_{n+k} = f(n, u_n, \dots, u_{n+k-1}) \quad (n \ge 0), \qquad u_0 = c_0, \ \dots, \ u_{k-1} = c_{k-1}.$$
A *solution* is a sequence $(u_n)_{n \ge 0}$ in $X$ that satisfies all these equations.

## Remarks

Every such problem has exactly one solution.
The terms $u_k, u_{k+1}, \dots$ are computed one after the other from the recurrence, and induction on $n$ shows that no other choice is possible.
Example: $u_{n+2} = u_{n+1} + u_n$ with $u_0 = 0$ and $u_1 = 1$ has the solution $0, 1, 1, 2, 3, 5, \dots$, the Fibonacci numbers.
Non-example: giving only $u_0 = 0$ for this order $2$ recurrence is not an initial value problem in this sense.
Both $0, 0, 0, \dots$ and $0, 1, 1, 2, \dots$ satisfy the recurrence with $u_0 = 0$, so one initial value is too few.
Initial conditions may also be placed at another starting index $n_0$ instead of $0$.
</document>

<document source="https://tamnd.github.io/math2/node/DV3PWY8NVW.md" tag="DV3PWY8NVW" kind="definition">
# Presheaf

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]], let $\mathcal{C}^{\mathrm{op}}$ be its [[opposite category|DAMK8XE1FR]], and let $\mathbf{Set}$ be the category of sets and functions.
A *presheaf* on $\mathcal{C}$ is a [[functor|28NQ0KRHD3]] $F \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$.
So $F$ assigns a set $F(X)$ to each object $X$ and a function $F(f) \colon F(Y) \to F(X)$ to each morphism $f \colon X \to Y$, with $F(\id_X) = \id_{F(X)}$ and $F(g \circ f) = F(f) \circ F(g)$.
A morphism of presheaves is a [[natural transformation|KYCPCC57FT]], and the presheaves on $\mathcal{C}$ form the category $[\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]$, often written $\widehat{\mathcal{C}}$.

## Remarks

Example: let $T$ be a [[topological space|0CQHFPGJ45]] and let $\mathcal{O}(T)$ be the category whose objects are the [[open sets|02BH3MCBQB]] of $T$, with one morphism $V \to U$ when $V \subseteq U$.
Let $F(U)$ be the set of continuous functions $U \to \R$ and let $F(V \to U)$ be restriction $s \mapsto s|_V$.
Restriction to $U$ is the identity and restricting to $V$ then to $W$ equals restricting to $W$, so $F$ is a presheaf.

Non-example: on $\mathcal{O}(T)$, let $G(U)$ be the set of all functions $U \to \R$ and send $V \subseteq U$ to extension by zero $G(V) \to G(U)$.
This is a functor $\mathcal{O}(T) \to \mathbf{Set}$, but its maps go from $V$ to $U$, so it is covariant and not a presheaf on $\mathcal{O}(T)$.

Every hom functor $\Hom(-, A)$ is a presheaf, and a presheaf on $\mathcal{O}(T)$ is a presheaf on $T$ in the sense of sheaf theory.
</document>

<document source="https://tamnd.github.io/math2/node/DV8GH6HJJ8.md" tag="DV8GH6HJJ8" kind="definition">
# Countably compact space

## Definition

A [[topological space|0CQHFPGJ45]] $X$ is *countably compact* when every [[countable|01TQB88H4C]] [[open cover|049PXYHW1T]] of $X$ has a finite subcover.

## Remarks

The following are equivalent for any topological space $X$:

1. $X$ is countably compact;
2. every sequence in $X$ has a cluster point, that is, a point $x$ such that each neighbourhood of $x$ contains $x_n$ for infinitely many $n$;
3. every countable family of closed sets with the finite intersection property has nonempty intersection.

Every [[compact|0TVJZYB6SH]] space and every [[sequentially compact|0FRMJH3CY7]] space is countably compact.
Every countably compact space is [[limit point compact|E8XKSQY76M]], and the converse holds for [[T1 spaces|0NW6CH38YR]].
A countably compact space that is also [[Lindelöf|SW0SN1NEJE]] is compact.

Example: $[0, 1]$ is countably compact, since it is compact by the Heine–Borel theorem.
Example: the space $[0, \omega_1)$ of countable ordinals with the order topology is countably compact but not compact.

Non-example: $\R$ is not countably compact.
The countable open cover by the intervals $(-n, n)$ for $n \ge 1$ has no finite subcover.
</document>

<document source="https://tamnd.github.io/math2/node/DV92N2YWZJ.md" tag="DV92N2YWZJ" kind="definition">
# Quasilinear PDE

## Definition

Let $\Omega$ be an open set in $\R^n$ and let $m \ge 1$.
A [[partial differential equation|GC5BDZ6QBC]] of [[order|E7JDFDPM1X]] $m$ in $\Omega$ is *quasilinear* if it has the following form.
$$\sum_{\abs{\alpha} = m} a_\alpha\left(D^{m-1} u, \dots, D u, u, x\right) \partial^\alpha u + a_0\left(D^{m-1} u, \dots, D u, u, x\right) = 0.$$
Here the functions $a_\alpha$ and $a_0$ are given.
Here $\alpha$ runs over [[multi-indices|0PJXBR6QA3]], and $D^j u$ denotes the collection of all partial derivatives of order $j$.
So the equation is affine in the derivatives of order $m$, and its coefficients depend at most on the derivatives of order less than $m$.

## Remarks

Example: the minimal surface equation $\operatorname{div}\left(\nabla u / \sqrt{1 + \abs{\nabla u}^2}\right) = 0$ in two variables expands to $(1 + u_y^2) u_{xx} - 2 u_x u_y u_{xy} + (1 + u_x^2) u_{yy} = 0$.
The coefficients of $u_{xx}$, $u_{xy}$, $u_{yy}$ depend only on first derivatives, so it is quasilinear of order $2$.
Non-example: the Monge–Ampère equation $u_{xx} u_{yy} - u_{xy}^2 = f(x, y)$ is not quasilinear, because it contains products of second derivatives; it is [[fully nonlinear|2D3N1ZJWBW]].
Every [[semilinear PDE|RCNY89A4NY]] is quasilinear.
For first-order equations, quasilinear means $\sum_{i} a_i(u, x) \partial_i u + a_0(u, x) = 0$.

## Explanations

### L1 human

Quasilinear means linear in the highest derivatives, once the solution and its lower derivatives are frozen.
</document>

<document source="https://tamnd.github.io/math2/node/DVA5DBQYBA.md" tag="DVA5DBQYBA" kind="definition">
# Atomistic lattice

## Definition

Let $L$ be a [[lattice|WH4SFAACAS]] with a [[least element|0AANKKNHS4]] $0$.
$L$ is *atomistic* if every $b \in L$ is the [[join|J2T75ATRV5]] in $L$ of the set $\set{a \in L : a \text{ is an atom and } a \le b}$.

## Remarks

The least element $0$ is the join of the empty set of atoms.
An atomistic lattice is [[atomic|5SE8EMSDNY]]: if $b \ne 0$ had no atom below it, $b$ would be the join of the empty set, that is $b = 0$.
Example: the power set of a set $X$ is atomistic, since every subset $B$ is the union of the singletons $\set{x}$ with $x \in B$.
Non-example: the chain $0 < 1 < 2$ is atomic, with the single atom $1$, but not atomistic.
The join of the atoms below $2$ is $1 \ne 2$.
</document>

<document source="https://tamnd.github.io/math2/node/DVA7AZ0HDJ.md" tag="DVA7AZ0HDJ" kind="definition">
# Optimal control of partial differential equations

## Definition

Let $Y$, $Z$ and $V$ be real [[Banach spaces|06GZ956657]], called the state space, the target space of the equation, and the control space.
Let $e \colon Y \times V \to Z$ be a map that encodes a [[partial differential equation|GC5BDZ6QBC]] with its boundary conditions.
Thus $e(y, u) = 0$ means that $y$ solves the equation with control $u$.
Let $V_{\mathrm{ad}} \subseteq V$ be a nonempty set of admissible controls, and let $J \colon Y \times V \to \R$ be a cost functional.
An *optimal control problem for a partial differential equation* is the problem
$$\text{minimize } J(y, u) \quad \text{subject to } e(y, u) = 0, \; u \in V_{\mathrm{ad}}.$$
A pair $(y^*, u^*)$ with $e(y^*, u^*) = 0$ and $u^* \in V_{\mathrm{ad}}$ is *optimal* if $J(y^*, u^*) \le J(y, u)$ for all such pairs $(y, u)$.

## Remarks

Example: let $\Omega \subseteq \R^d$ be a bounded open set, let $y_d \in L^2(\Omega)$, $\alpha > 0$ and $a < b$.
Minimize $\frac12 \norm{y - y_d}_{L^2(\Omega)}^2 + \frac{\alpha}{2} \norm{u}_{L^2(\Omega)}^2$ over $u \in L^2(\Omega)$ with $a \le u \le b$ almost everywhere, where $y \in H^1_0(\Omega)$ is the weak solution of $-\Delta y = u$ in $\Omega$, $y = 0$ on $\partial \Omega$.
This fits the definition with $Y = H^1_0(\Omega)$, $V = L^2(\Omega)$, $Z = H^{-1}(\Omega)$ and $e(y, u) = -\Delta y - u$.
Here the problem has a unique solution, because the control-to-state map is linear and bounded.
So the reduced cost is strictly [[convex|02AS45T3H9]], continuous and coercive on the closed convex set $V_{\mathrm{ad}}$.
Non-example: if the state equation is an ordinary differential equation in $\R^n$, the problem is an ordinary [[optimal control|T3V6ZNAMDN]] problem, not one for a partial differential equation.
When $e(y, u) = 0$ has a unique solution $y = S(u)$ for each $u$, the problem reduces to minimizing $f(u) = J(S(u), u)$ over $V_{\mathrm{ad}}$.
</document>

<document source="https://tamnd.github.io/math2/node/DVAWCZWT3B.md" tag="DVAWCZWT3B" kind="definition">
# Dirichlet-to-Neumann operator

## Definition

Let $\Omega \subset \R^n$, $n \ge 2$, be a bounded connected open set with smooth boundary $\partial\Omega$ and outward unit normal $\nu$.
Let $H^{1/2}(\partial\Omega)$ and $H^{-1/2}(\partial\Omega)$ be the boundary [[Sobolev spaces|M4CEZ5K9HP]] of order $\tfrac12$ and $-\tfrac12$.
For $f \in H^{1/2}(\partial\Omega)$, let $u_f \in H^1(\Omega)$ be the weak solution of the [[Dirichlet problem|8D56GQ4AG8]]
$$\Delta u = 0 \text{ in } \Omega, \qquad u = f \text{ on } \partial\Omega.$$
The boundary condition is taken in the sense of the [[trace operator|Z2T5R88QDB]].
The *Dirichlet-to-Neumann operator* is the map
$$\Lambda \colon H^{1/2}(\partial\Omega) \to H^{-1/2}(\partial\Omega), \qquad \Lambda f = \partial_\nu u_f|_{\partial\Omega}.$$
More generally, let $\gamma$ be measurable on $\Omega$ with $c \le \gamma \le c^{-1}$ for some $c > 0$.
The operator $\Lambda_\gamma$ uses the equation $\nabla \cdot (\gamma \nabla u) = 0$ and the weakly defined [[Neumann data|GS3KT2H645]] $\gamma \, \partial_\nu u|_{\partial\Omega}$.

## Remarks

Example: let $\Omega$ be the unit disk and $f(\theta) = e^{ik\theta}$ with $k \in \Z$.
Then $u_f = r^{\abs{k}} e^{ik\theta}$ is harmonic, and $\partial_r u_f = \abs{k}$ at $r = 1$, so $\Lambda f = \abs{k} f$.
Thus on the circle $\Lambda = (-\partial_\theta^2)^{1/2}$.
Non-example: replacing $\Delta u = 0$ by $\Delta u + \lambda u = 0$, where $\lambda$ is a Dirichlet eigenvalue of $-\Delta$ on $\Omega$, does not define a map, because the boundary value problem has no unique solution.
In general $\Lambda$ is self-adjoint and nonnegative on $L^2(\partial\Omega)$, with kernel the constants, and it is a first-order elliptic [[pseudo-differential operator|FF27N0BWTP]] with principal symbol $\abs{\xi}$.
It is also called the Poincaré–Steklov operator, and the weak form is $\int_{\partial\Omega} (\Lambda f) \, g = \int_\Omega \nabla u_f \cdot \nabla u_g$.
</document>

<document source="https://tamnd.github.io/math2/node/DVB79BK06Y.md" tag="DVB79BK06Y" kind="object">
# Raviart–Thomas elements

## Description

Let $T \subset \R^d$ be a simplex, a triangle when $d = 2$, and let $k \ge 0$ be an integer.
Let $P_k(T)$ be the [[polynomials|07GNX8GCYP]] of degree at most $k$ on $T$, and $\tilde P_k(T)$ the homogeneous ones of degree $k$.
The *Raviart–Thomas space* of index $k$ on $T$ is
$$RT_k(T) = P_k(T)^d + x \, \tilde P_k(T),$$
the vector fields $p(x) + x \, q(x)$ with $p \in P_k(T)^d$ and $q \in \tilde P_k(T)$.
The degrees of freedom are the moments of the normal component $v \cdot n$ on each face against $P_k$ of that face.
For $k \ge 1$ they also include the moments of $v$ against $P_{k-1}(T)^d$.
On a mesh, the global space is formed by the piecewise $RT_k$ fields whose normal moments agree on each interior face.

## Remarks

Lowest order on a triangle: $RT_0(T) = \set{x \mapsto a + b x : a \in \R^2, \ b \in \R}$, of dimension $3$.
On an edge $e$ with unit normal $n$, the points satisfy $x \cdot n = c$ for a constant $c$.
So $(a + b x) \cdot n = a \cdot n + b c$ is constant on $e$, and it is fixed by the flux $\int_e v \cdot n$.
The three fluxes determine $v$, which gives the three degrees of freedom.

The dimension of $RT_k(T)$ on a triangle is $(k + 1)(k + 3)$.
Matching the normal moments makes the normal component continuous across faces.
So the global space lies in $H(\operatorname{div})$, the $L^2$ fields with [[divergence|0MWV6ZTK6S]] in $L^2$.
The divergence maps $RT_k(T)$ onto $P_k(T)$.
Paired with discontinuous piecewise polynomials of degree $k$, these spaces give a stable [[mixed finite element method|87VMQ892VW]] for the Poisson equation.
They were introduced by Pierre-Arnaud Raviart and Jean-Marie Thomas.

## Explanations

### L1 human

Raviart–Thomas elements are building blocks for approximating flows, such as the flux of heat or of water through soil.
Instead of storing values at the corners, the lowest-order element stores how much flows through each edge of a triangle.
Neighbouring triangles share these edge fluxes, so whatever leaves one triangle enters the next.
This makes the computed flow conserve mass exactly, element by element.
</document>

<document source="https://tamnd.github.io/math2/node/DVGFRRYK10.md" tag="DVGFRRYK10" kind="concept">
# Curve-shortening flow

## Summary

The curve-shortening flow moves each point of a plane curve along its normal vector.
The speed is the [[curvature|7JQS6VKFEZ]] of the curve at that point.
It is the one-dimensional case of [[mean curvature flow|2YYV01C7QQ]] and the steepest descent flow for the length of the curve.
A heat-equation-like smoothing effect makes curves rounder over time, and embedded closed curves shrink to round points by the [[Gage–Hamilton–Grayson theorem|FVGW1381TZ]].

## Explanations

### L1 human

Imagine a closed loop of string in the plane in which every bit moves inward at a speed equal to how sharply the string bends there.
Sharp corners move fast and flat parts move slowly, so the loop smooths out.
A circle of radius $r_0$ stays a circle, and its radius $r(t)$ satisfies $r' = -1/r$, so $r(t) = \sqrt{r_0^2 - 2t}$ and the circle disappears at time $r_0^2 / 2$.
The flow decreases the length as fast as possible, which is the reason for its name.
</document>

<document source="https://tamnd.github.io/math2/node/DVGQB92HPJ.md" tag="DVGQB92HPJ" kind="statement">
# Stone's theorem on one-parameter unitary groups

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]].
Let $(U(t))_{t \in \R}$ be a family of [[unitary operators|MM0EJ2GDE4]] on $H$ such that:

1. $U(0) = I$ and $U(s + t) = U(s)U(t)$ for all $s, t \in \R$;
2. $t \mapsto U(t)x$ is continuous from $\R$ to $H$ for every $x \in H$.

Then there is a unique [[self-adjoint operator|0NEGN5CRSC]] $A$ on $H$ with $U(t) = e^{itA}$ for all $t \in \R$.
Its domain is the set of $x$ for which $\lim_{t \to 0} (U(t)x - x)/t$ exists, and $Ax = \lim_{t \to 0} (U(t)x - x)/(it)$.
Conversely, for every self-adjoint operator $A$ on $H$, the operators $e^{itA}$, defined by the [[spectral theorem|18KPYRRPZY]], satisfy clauses 1 and 2.

## Remarks

The converse is a direct consequence of the spectral theorem and dominated convergence.
For the main direction, $A$ is defined by the limit in the statement.
Vectors $\int \varphi(t) U(t)x \, dt$ with smooth compactly supported $\varphi$ lie in its domain, so $A$ is densely defined.
The group law shows that $A$ is symmetric, and the vectors $\int_0^\infty e^{-t} U(\pm t)x \, dt$ show that $A \pm iI$ are onto, which makes $A$ self-adjoint.

Example: on $L^2(\R)$, the translations $(U(t)f)(s) = f(s + t)$ form such a group.
Its generator is $A = -i \, d/ds$ with domain the Sobolev space $H^1(\R)$, the momentum operator of quantum mechanics.

Strong continuity in clause 2 is needed only in a weak form: on a separable Hilbert space, weak measurability of $t \mapsto \ip{U(t)x}{y}$ already implies it, by a theorem of von Neumann.

## Explanations

### L1 human

A one-parameter unitary group is a way of moving the vectors of a Hilbert space continuously in time without changing lengths.
Stone's theorem says that every such motion has the form $e^{itA}$ for one self-adjoint operator $A$.
In quantum mechanics, the motion is the time evolution of a state and $A$ is the energy, the Hamiltonian.
So the theorem explains why observables and symmetries correspond to self-adjoint operators.

### L2 human

The theorem is the unbounded counterpart of the fact that a continuous group of unitary matrices is $e^{itA}$ for a Hermitian matrix $A$.
It gives a bijection between self-adjoint operators and strongly continuous unitary groups.
It is the special case of semigroup theory where the semigroup extends to negative times and each operator is unitary; compare the [[Lumer–Phillips theorem|Z2N8MPD8DH]].
It is the starting point of the representation theory of $\R$, and with the SNAG theorem, of locally compact abelian groups.
</document>

<document source="https://tamnd.github.io/math2/node/DVJ17KVWXP.md" tag="DVJ17KVWXP" kind="concept">
# Morrey quasiconvexity

## Summary

A function $f$ on $m \times n$ matrices is quasiconvex in the sense of Morrey if perturbations with zero boundary values never help.
That is, the average of $f(\xi + \nabla \varphi)$ is never below $f(\xi)$.
Under suitable growth conditions on $f$, it is exactly the condition under which $\int_\Omega f(\nabla u) \, dx$ is weakly lower semicontinuous on Sobolev spaces of vector-valued maps.
It lies between polyconvexity and [[rank-one convexity|GK4P8G0PB2]].
It coincides with convexity when $m = 1$ or $n = 1$.
It is a different notion from the [[quasiconvex functions|S4RMH7JQB8]] of optimisation, which are defined by convex sublevel sets.

## Explanations

### L1 human

Take a linear map with constant gradient $\xi$ on a region, and wiggle it inside while keeping its boundary values.
Quasiconvexity says that the wiggled map always has at least as much energy as the linear one.
So, on average, an affine map is the cheapest way to achieve given affine boundary values.
Convex functions have this property by Jensen's inequality, because the average of $\nabla \varphi$ over the region is zero.
The condition is hard to check in general, because it is not local in $\xi$: it involves all test maps $\varphi$.
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<document source="https://tamnd.github.io/math2/node/DVJM7NBSF2.md" tag="DVJM7NBSF2" kind="concept">
# Bessel potential space

## Summary

A Bessel potential space $H^{s,p}(\R^n)$ consists of the functions or tempered distributions whose "derivative of order $s$" lies in $L^p$.
The derivative of order $s$ is defined on the Fourier side by multiplication with $(1 + \abs{\xi}^2)^{s/2}$, so $s$ may be any real number.
These spaces extend the [[Sobolev spaces|M4CEZ5K9HP]] $W^{k,p}$ to fractional and negative orders, and for $p = 2$ they are the [[fractional Sobolev spaces|N6NX1VT837]] $H^s$.
They are a standard scale of spaces for elliptic equations and for interpolation.

## Explanations

### L1 human

Differentiating a function multiplies its Fourier transform by the frequency.
So a function is smooth when its Fourier transform still has small size after multiplication by a power of the frequency.
The Bessel potential space of order $s$ asks exactly this, with the power $(1 + \abs{\xi}^2)^{s/2}$ and with size measured in $L^p$.
For $s = 0$ the space is just $L^p$.
For $s = 1$ and $1 < p < \infty$ it is the space of $L^p$ functions with first derivatives in $L^p$.
</document>

<document source="https://tamnd.github.io/math2/node/DVSTWGVCSZ.md" tag="DVSTWGVCSZ" kind="concept">
# Coequalizer

## Summary

The coequalizer of two parallel morphisms $f, g \colon X \to Y$ in a [[category|A2DE66937D]] is a universal morphism $q \colon Y \to Q$ with $q \circ f = q \circ g$.
It is the [[colimit|BJS1CGTKQ5]] of the diagram formed by $f$ and $g$, and it is dual to the [[equalizer|CTJ5AD8ST5]].
In sets it is the quotient of $Y$ that identifies $f(x)$ with $g(x)$ for each $x$.
In abelian groups it is the cokernel of $f - g$.

## Explanations

### L1 human

Given two functions $f$ and $g$ from $X$ to $Y$, glue $f(x)$ to $g(x)$ in $Y$ for every $x$, and keep gluing until the identifications are consistent.
The result is a quotient set $Q$ of $Y$, and $q$ sends each element of $Y$ to its glued class.
For example, let $X = \{*\}$, $Y = [0, 1]$, $f(*) = 0$ and $g(*) = 1$.
The coequalizer glues the two ends of the interval, and with the quotient topology it gives a circle.
</document>

<document source="https://tamnd.github.io/math2/node/DVYA6GD8RN.md" tag="DVYA6GD8RN" kind="problem">
# Plateau's problem

## Problem

Let $\Gamma$ be a Jordan curve in $\R^3$, that is, a subset homeomorphic to a circle.
Is there a surface of least area among all surfaces with boundary $\Gamma$?

## Remarks

The problem is named after Joseph Plateau, who studied soap films spanning wire frames by experiment.
Jesse Douglas and Tibor Radó solved it independently for surfaces of the type of a disc, around 1930.
Their result applies to a rectifiable $\Gamma$ and gives a continuous map $X$ from the closed unit disc $\overline{D}$ to $\R^3$.
The map sends the boundary circle homeomorphically onto $\Gamma$.
It is [[harmonic|0DEB0JMN0W]] in the open disc and [[conformal|0YYT36R4YD]] away from isolated branch points.
It has least area among all such disc-type surfaces spanning $\Gamma$.
The image of $X$ is then a [[minimal surface|41X0YW2R9N]], possibly with branch points.
Douglas received one of the first two Fields Medals in 1936, in part for this work.
Later, geometric measure theory treated the problem in higher dimensions and without a fixed topological type, with solutions such as integral currents.
The least-area surface need not be unique, and it need not be a disc: some curves are spanned by surfaces of other types with smaller area.

## Explanations

### L1 human

Bend a wire into a closed loop, dip it in soapy water, and a film appears that spans the loop with as little area as it can.
Plateau's problem asks whether such a least-area surface always exists, as a mathematical object, for every loop.
The answer is yes for disc-shaped surfaces, and the proof was one of the great successes of analysis in the twentieth century.
</document>

<document source="https://tamnd.github.io/math2/node/DW5G0SQ2G5.md" tag="DW5G0SQ2G5" kind="definition">
# Dirichlet form

## Definition

Let $(X, \mathcal{A}, m)$ be a $\sigma$-finite [[measure|7V9TSDJFZR]] space.
Let $H = L^2(X, m)$ be the real [[Hilbert space|HM2HHE3J51]] of square-integrable functions (an [[Lp space|SASP14MA1Z]]), with inner product $(\cdot, \cdot)$.
Let $\mathcal{F}$ be a dense linear subspace of $H$ and let $\mathcal{E} \colon \mathcal{F} \times \mathcal{F} \to \R$ be a [[bilinear form|0SQZMXB6GP]].
The pair $(\mathcal{E}, \mathcal{F})$ is a *Dirichlet form* on $L^2(X, m)$ if:

1. $\mathcal{E}$ is symmetric: $\mathcal{E}(u, v) = \mathcal{E}(v, u)$ for all $u, v \in \mathcal{F}$.
2. $\mathcal{E}$ is nonnegative: $\mathcal{E}(u, u) \ge 0$ for all $u \in \mathcal{F}$.
3. $\mathcal{E}$ is closed: $\mathcal{F}$ is complete for the norm $\norm{u}_1 = \left(\mathcal{E}(u, u) + (u, u)\right)^{1/2}$.
4. $\mathcal{E}$ is Markovian: for every $u \in \mathcal{F}$, the function $v = \min(\max(u, 0), 1)$ lies in $\mathcal{F}$ and $\mathcal{E}(v, v) \le \mathcal{E}(u, u)$.

## Remarks

Example: on $L^2(\R^n)$ with Lebesgue measure, take $\mathcal{F}$ to be the [[Sobolev space|M4CEZ5K9HP]] $W^{1,2}(\R^n)$ and $\mathcal{E}(u, v) = \int_{\R^n} \nabla u \cdot \nabla v \, dx$.
It is symmetric and nonnegative, and $\norm{\cdot}_1$ is the Sobolev norm, which is complete.
The truncation $v = \min(\max(u, 0), 1)$ of $u \in W^{1,2}$ lies in $W^{1,2}$, with $\nabla v = \nabla u$ almost everywhere on $\set{0 < u < 1}$ and $\nabla v = 0$ almost everywhere elsewhere, so $\mathcal{E}(v, v) \le \mathcal{E}(u, u)$.

Non-example: on $L^2(\R)$ take $\mathcal{F} = W^{2,2}(\R)$ and $\mathcal{E}(u, v) = \int u'' v'' \, dx$.
Clauses 1 to 3 hold, but clause 4 fails.
If $u \in W^{2,2}(\R)$ is smooth and crosses the level $1$ with nonzero slope, its truncation has a corner there, so it is not in $W^{2,2}(\R)$.

A closed, symmetric, nonnegative, densely defined form corresponds to a nonnegative [[self-adjoint operator|0NEGN5CRSC]] $A$ with $\mathcal{E}(u, v) = (A^{1/2} u, A^{1/2} v)$ and $\mathcal{F}$ the domain of $A^{1/2}$.
Clause 4 holds if and only if the semigroup $T_t = e^{-tA}$ is Markovian: $0 \le f \le 1$ almost everywhere implies $0 \le T_t f \le 1$ almost everywhere, for every $t > 0$.
Let $X$ be a locally compact separable metric space and $m$ a Radon measure of full support.
The form is then *regular* if $\mathcal{F} \cap C_c(X)$ is dense in $\mathcal{F}$ for $\norm{\cdot}_1$ and dense in $C_c(X)$ for the uniform norm.
</document>

<document source="https://tamnd.github.io/math2/node/DW5SBZXK6R.md" tag="DW5SBZXK6R" kind="concept">
# Loeb measure

## Summary

A Loeb measure is a genuine countably additive [[measure|PR11BZQQPG]] built from a finitely additive measure inside a nonstandard model of analysis.
One takes the standard part of the nonstandard measure and extends it to a σ-algebra.
Introduced by Peter Loeb, it lets hyperfinite objects, such as a finite set with an infinite number of points, serve as ordinary measure spaces.
It is the main tool of nonstandard measure theory and gives nonstandard constructions of Lebesgue measure and of Brownian motion.

## Explanations

### L1 human

In nonstandard analysis there are "finite" sets whose number of elements $N$ is an infinitely large hypernatural number.
On such a set one can give each point the infinitesimal weight $1/N$, as for a fair die with $N$ faces.
The resulting sizes are hyperreal numbers, so one rounds each of them to the nearest real number.
Loeb showed that the rounded sizes extend to a true probability measure.
For example, mapping the points $k/N$ for $k = 1, \ldots, N$ to their standard parts in $[0, 1]$ turns this uniform measure into Lebesgue measure on $[0, 1]$.
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<document source="https://tamnd.github.io/math2/node/DW7FG506QJ.md" tag="DW7FG506QJ" kind="concept">
# Validated numerics

## Summary

Validated numerics, also called verified or rigorous computation, is numerical computation whose output comes with a mathematical guarantee.
Instead of a single approximate number, it returns a set, usually an interval or a box, that provably contains the exact answer.
All [[round-off errors|CPN6SD3BTV]] and [[truncation errors|4AKCQERM89]] are bounded, typically with [[interval arithmetic|QPKKTE1ZY0]] or [[affine arithmetic|YPE8E9CVXP]] and directed rounding.
Its results can serve as steps of a proof; an example is Warwick Tucker's computer-assisted proof that the Lorenz attractor exists.

## Explanations

### L1 human

An ordinary numerical program says "the answer is about $1.41421$" and hopes the digits are right.
A validated program says "the answer lies between $1.414213$ and $1.414214$", and this claim is certain, rounding errors included.
It does this by computing with intervals and by rounding lower bounds down and upper bounds up.
Such enclosures can turn a computer calculation into part of a mathematical proof.
</document>

<document source="https://tamnd.github.io/math2/node/DWFPDF8XTB.md" tag="DWFPDF8XTB" kind="statement">
# Riesz–Markov–Kakutani representation theorem

## Statement

Let $X$ be a [[locally compact|0FE2SB8WVJ]] [[Hausdorff space|0S1FJ8VEX9]].
Let $C_c(X)$ be the real vector space of [[continuous functions|0X3WGX95FE]] $f \colon X \to \R$ with compact support.
Let $\Lambda \colon C_c(X) \to \R$ be a [[linear form|07YQ1C78G0]] that is positive, that is, $\Lambda f \ge 0$ whenever $f \ge 0$.
Then there is a unique measure $\mu$ on the Borel sets of $X$ with these four properties:

1. $\Lambda f = \int_X f \, d\mu$ for every $f \in C_c(X)$, where the integral is the [[Lebesgue integral|DWJF7ABPR1]].
2. $\mu(K) < \infty$ for every compact set $K \subseteq X$.
3. $\mu(E) = \inf \set{\mu(U) : U \supseteq E, \ U \text{ open}}$ for every Borel set $E$.
4. $\mu(E) = \sup \set{\mu(K) : K \subseteq E, \ K \text{ compact}}$ for every open set $E$ and for every Borel set $E$ with $\mu(E) < \infty$.

## Remarks

Clauses 2 to 4 say that $\mu$ is a [[Radon measure|HZNEQQBCW4]].
The proof sets $\mu(U) = \sup \set{\Lambda f : 0 \le f \le 1, \ \operatorname{supp} f \subseteq U}$ for open $U$, extends it to an outer measure, and checks that the Borel sets are measurable.
Urysohn's lemma for locally compact Hausdorff spaces supplies the functions $f$ that the construction needs.

Example: on $X = \R$, the Riemann integral $\Lambda f = \int_{-\infty}^{\infty} f(x) \, dx$ is positive, and the measure it gives is Lebesgue measure.
The evaluation $\Lambda f = f(0)$ gives the Dirac measure at $0$.

Let $X$ be compact Hausdorff and give $C(X)$ the sup norm.
A related form identifies the dual of $C(X)$ with the space of regular signed Borel measures on $X$, normed by total variation.

## Explanations

### L1 human

A positive linear rule that assigns a number to each continuous function always comes from integration against a measure.
So measures and positive linear functionals on continuous functions are two descriptions of the same thing.
This is often used to build measures: one writes down the functional first.

### L2 human

The theorem is the bridge between functional analysis and measure theory.
It identifies the dual of $C(X)$ for compact $X$ with a space of measures, and so gives the weak-* topology on measures that underlies weak convergence of probability measures.
Haar measure on a locally compact group is often built by constructing an invariant positive functional and then applying this theorem.
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<document source="https://tamnd.github.io/math2/node/DWJF7ABPR1.md" tag="DWJF7ABPR1" kind="concept">
# Lebesgue integral

## Summary

The Lebesgue integral assigns a number to each integrable function on a measure space.
It is built from simple functions, then non-negative functions, then general functions split into positive and negative parts.
It extends the Riemann integral to many more functions and to any measure space.
Its great strength is the convergence theorems, which let one exchange limits and integrals under mild conditions.

## Explanations

### L1 human

The Riemann integral cuts the horizontal axis into small pieces.
The Lebesgue integral cuts the vertical axis instead: for each range of values it asks how large the set of points with those values is.
This handles functions that jump around wildly.
For example, the indicator function of the rationals on $[0, 1]$ has no Riemann integral.
Its Lebesgue integral is $0$, because the rationals form a set of measure zero.
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<document source="https://tamnd.github.io/math2/node/DWK5N5SKTK.md" tag="DWK5N5SKTK" kind="concept">
# Poisson's equation

## Summary

Poisson's equation is the partial differential equation $\Delta u = f$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $f$ is a given function.
With $f = 0$ it is [[Laplace's equation|S8RG6N1PRT]], so it is the inhomogeneous version of the equation of harmonic functions.
It describes a potential created by a source: the gravitational potential of a mass density, or the electrostatic potential of a charge density.
On the whole space a solution is given by convolving $f$ with the [[fundamental solution of the Laplacian|BSJ8X2XRN9]].
On bounded domains it is solved with Green's functions.

## Explanations

### L1 human

Laplace's equation describes a steady state with no sources inside, such as the temperature in a plate with no heater.
Poisson's equation allows sources: the right-hand side $f$ says how much heat is produced, or how much charge sits, at each point.
For example, $u = (x^2 + y^2)/4$ solves $\Delta u = 1$ in the plane, a constant source everywhere.
Two solutions differ by a harmonic function, so on a bounded domain the boundary values pin down a single solution.
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<document source="https://tamnd.github.io/math2/node/DWMQ3V205V.md" tag="DWMQ3V205V" kind="definition">
# Universal covering group

## Definition

Let $G$ be a [[connected|0V032VWTHD]] [[Lie group|7JBRGCVJRB]].
A *universal covering group* of $G$ is a Lie group $\tilde G$ together with a map $p \colon \tilde G \to G$ such that:

1. $\tilde G$ is connected and simply connected, that is, its [[fundamental group|S1XK3HXHFX]] is trivial;
2. $p$ is a homomorphism of Lie groups;
3. $p$ is a [[covering map|3XZRRE165N]].

## Remarks

Existence: let $p \colon \tilde G \to G$ be the [[universal cover|Y1YPE6PM52]] of the manifold $G$, and choose $\tilde e$ over the identity $e$.
There is a unique Lie group structure on $\tilde G$ with identity $\tilde e$ for which $p$ is a homomorphism.
The universal covering group is unique up to isomorphism over $G$.
The kernel of $p$ is a discrete central subgroup of $\tilde G$, isomorphic to $\pi_1(G)$.

Example: $p \colon \R \to S^1$, $p(t) = e^{2\pi i t}$, is a homomorphism from $(\R, +)$, a covering map, and $\R$ is simply connected; the kernel is $\Z = \pi_1(S^1)$.

Example: identify $SU(2)$ with the unit quaternions and $\R^3$ with the purely imaginary quaternions.
The map sending $q$ to the rotation $v \mapsto q v q^{-1}$ is a universal covering group $SU(2) \to SO(3)$ with kernel $\set{\pm 1}$.

Non-example: the identity map $SO(3) \to SO(3)$ is a homomorphism and a covering map, but $SO(3)$ is not simply connected, since $\pi_1(SO(3)) \cong \Z/2$.
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<document source="https://tamnd.github.io/math2/node/DWN0XEQHHK.md" tag="DWN0XEQHHK" kind="concept">
# Harmonic space

## Summary

A harmonic space is a locally compact space with a sheaf of continuous functions, called harmonic, that obey axioms modelled on classical [[harmonic functions|0DEB0JMN0W]].
The axioms ask that the [[Dirichlet problem|8D56GQ4AG8]] be uniquely solvable on a base of small open sets.
They also ask that suitable increasing limits of harmonic functions be harmonic.
Harmonic spaces are the main setting of [[axiomatic potential theory|YS7RF1PE42]].
Euclidean space with the Laplace equation is a harmonic space, and so is space-time with the heat equation.

## Explanations

### L1 human

In the plane, a harmonic function on a disc is determined by its values on the boundary circle.
A harmonic space is any space where some chosen functions behave in this way on enough small sets.
The chosen functions must also be stable under natural operations, such as gluing local pieces and taking increasing limits.
This is enough to build much of potential theory: superharmonic functions, potentials and the solution of boundary value problems.
The solutions of the heat equation, which describe temperature changing in time, form a harmonic space too.
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<document source="https://tamnd.github.io/math2/node/DWV8BQP6GA.md" tag="DWV8BQP6GA" kind="definition">
# Perfectly normal space

## Definition

A [[topological space|0CQHFPGJ45]] $X$ is *perfectly normal* when the following two conditions hold.

1. $X$ is a [[normal space|0NSR9V8TED]].
2. Every [[closed set|0ZCK8DQYN1]] of $X$ is a [[G-delta set|V1TBNP8R3S]], that is, a countable intersection of [[open sets|02BH3MCBQB]].

## Remarks

Equivalently, for all disjoint closed sets $A, B \subseteq X$ there is a [[continuous function|0X3WGX95FE]] $f \colon X \to [0, 1]$ with $f^{-1}(0) = A$ and $f^{-1}(1) = B$.
Another equivalent form: every closed set is $f^{-1}(0)$ for some continuous $f \colon X \to \R$.
A perfectly normal space is [[completely normal|9HBDQ2W9CF]].
A perfectly normal $T_1$ space is called a $T_6$ space.

Example: every metric space $(X, d)$ is perfectly normal.
It is normal, and a closed set $A$ equals $\bigcap_{n \ge 1} \set{x : d(x, A) < 1/n}$, an intersection of countably many open sets.

Non-example: the space $[0, \omega_1]$ of countable ordinals together with $\omega_1$, with the order topology, is compact Hausdorff and so normal.
The closed set $\set{\omega_1}$ is not a G-delta set: any countably many neighbourhoods $(\alpha_n, \omega_1]$ contain $(\sup_n \alpha_n, \omega_1]$, which has more than one point.
So clause 2 fails.
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<document source="https://tamnd.github.io/math2/node/DWYCKAKXZD.md" tag="DWYCKAKXZD" kind="concept">
# Many-valued logic

## Summary

A many-valued logic is a propositional logic with more than two truth values.
Its connectives are given by tables over the set of truth values, and some values are designated as "true enough".
Examples are the three-valued logics of Łukasiewicz and Kleene, and the infinite-valued Łukasiewicz logic behind fuzzy logic and [[MV-algebras|8T4M7RN9AC]].

## Explanations

### L1 human

Classical logic has two truth values, true and false.
A many-valued logic allows others, such as "unknown" or degrees of truth between $0$ and $1$.
In Łukasiewicz's three-valued logic, the values are $0$, $\tfrac12$ and $1$, and "not $x$" has the value $1 - x$.
There, "$p$ or not $p$", with "or" as maximum, has value $\tfrac12$ when $p$ has value $\tfrac12$, so it is not a law.
</document>

<document source="https://tamnd.github.io/math2/node/DWYPS58A7Q.md" tag="DWYPS58A7Q" kind="definition">
# Affine connection

## Definition

Let $M$ be a smooth manifold with [[tangent bundle|WY6K3P4YCG]] $TM$, and let $\mathfrak{X}(M)$ be the space of smooth vector fields on $M$.
An *affine connection* on $M$ is a [[connection|5ZYRRVE8BW]] on the vector bundle $TM$.
That is, it is a map $\nabla : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M)$, $(X, Y) \mapsto \nabla_X Y$, that is $C^\infty(M)$-linear in $X$, $\R$-linear in $Y$, and satisfies $\nabla_X (fY) = (Xf) Y + f \nabla_X Y$ for every smooth function $f$.

## Remarks

Example: on $\R^n$ with $Y = \sum_j Y^j \partial_j$, the formula $\nabla_X Y = \sum_j (X Y^j) \, \partial_j$ defines an affine connection, the flat connection; the three conditions follow from the linearity and the product rule of directional derivatives.
Non-example: $(X, Y) \mapsto [X, Y]$ is not an affine connection, since $[fX, Y] = f [X, Y] - (Yf) X$ is not $C^\infty(M)$-linear in $X$.
In a chart, $\nabla$ is given by its Christoffel symbols $\Gamma^k_{ij}$, defined by $\nabla_{\partial_i} \partial_j = \Gamma^k_{ij} \partial_k$.
Every smooth manifold has affine connections: glue the flat connections of charts with a partition of unity.
Two important tensors measure an affine connection: its torsion $\nabla_X Y - \nabla_Y X - [X, Y]$, built with the [[Lie bracket|68FGQEBD6J]], and its curvature.
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<document source="https://tamnd.github.io/math2/node/DWZA03W93P.md" tag="DWZA03W93P" kind="definition">
# C-infinity ring

## Definition

A *$C^\infty$-ring* is a set $A$ together with, for each $n \ge 0$ and each [[smooth function|0V33BNKZ9S]] $f \colon \R^n \to \R$, a map $\Phi_f \colon A^n \to A$, such that both conditions hold.

1. For each $n$ and each coordinate projection $\pi_i \colon \R^n \to \R$, $(x_1, \dots, x_n) \mapsto x_i$, one has $\Phi_{\pi_i}(a_1, \dots, a_n) = a_i$.
2. For smooth $f \colon \R^n \to \R$ and $g_1, \dots, g_n \colon \R^m \to \R$, and $h = f \circ (g_1, \dots, g_n)$, one has
$$\Phi_h(a) = \Phi_f\big(\Phi_{g_1}(a), \dots, \Phi_{g_n}(a)\big) \quad \text{for all } a \in A^m.$$

A morphism of $C^\infty$-rings is a map $\phi \colon A \to B$ with $\phi(\Phi_f(a_1, \dots, a_n)) = \Phi_f(\phi(a_1), \dots, \phi(a_n))$ for all $f$ and $a_i$.

## Remarks

Equivalently, a $C^\infty$-ring is a model of the [[Lawvere theory|NTJ94H2CNS]] whose $n$-ary operations are the smooth maps $\R^n \to \R$.
In other words, it is a product-preserving functor to sets from the category of the spaces $\R^n$ with smooth maps.
Using $f(x, y) = x + y$, $f(x, y) = xy$ and the constant functions, every $C^\infty$-ring is a [[commutative ring|0310REJJY8]] and an $\R$-algebra.

Example: for a [[smooth manifold|F45VQ47GPV]] $M$, let $A = C^\infty(M)$ and $\Phi_f(a_1, \dots, a_n) = f \circ (a_1, \dots, a_n)$.
Clause 1 holds since $\pi_i \circ (a_1, \dots, a_n) = a_i$, and clause 2 is associativity of composition.
The ring of [[germs of smooth functions|7SWSRGDT4A]] at a point is a $C^\infty$-ring in the same way.
The ring $\R[\varepsilon]/(\varepsilon^2)$ is a $C^\infty$-ring with $\Phi_f(a + b\varepsilon) = f(a) + f'(a) b \varepsilon$ for $n = 1$, and the analogous formula with the gradient for larger $n$; clause 2 is the chain rule.

Non-example: on $A = \R$, the rule $\Phi_f(a_1, \dots, a_n) = f(0, \dots, 0)$ fails clause 1, since $\Phi_{\pi_1}(a) = 0 \ne a$ for $a \ne 0$.
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<document source="https://tamnd.github.io/math2/node/DWZD2EKC1Q.md" tag="DWZD2EKC1Q" kind="example">
# Laplace transform of the error function

## Example

Let $\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-u^2}\, du$ be the [[error function|1EG3DWPNZN]].
The [[Laplace transform|05XT5AVJBK]] of $f(t) = \operatorname{erf}(\sqrt{t})$ is
$$\mathcal{L}\{\operatorname{erf}(\sqrt{t})\}(s) = \frac{1}{s \sqrt{s + 1}} \quad (s > 0).$$
Check: $f(0) = 0$ and, for $t > 0$, $f'(t) = \frac{2}{\sqrt{\pi}} e^{-t} \cdot \frac{1}{2\sqrt{t}} = \frac{e^{-t}}{\sqrt{\pi t}}$.
By the [[transform of a power|RDFYHS2Q55]] with $\alpha = -1/2$ and the shift $s \mapsto s + 1$ caused by the factor $e^{-t}$,
$$\mathcal{L}\{f'\}(s) = \frac{1}{\sqrt{\pi}} \cdot \frac{\Gamma(1/2)}{\sqrt{s + 1}} = \frac{1}{\sqrt{s + 1}},$$
using $\Gamma(1/2) = \sqrt{\pi}$ for the [[gamma function|0V3FZMVB4K]].
Since $f(t) = \int_0^t f'(u)\, du$, its transform is that of $f'$ divided by $s$, which gives $1/(s\sqrt{s + 1})$.
The formula extends to $\operatorname{Re} s > 0$ with the principal square root.

## Explanations

### L1 human

The error function measures the area under the bell curve, and it appears in heat flow and probability.
Its transform, with argument $\sqrt{t}$, is a simple algebraic expression in $s$.
This is why Laplace transforms solve heat conduction problems: the hard function erf on the time side becomes a square root on the transform side.
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<document source="https://tamnd.github.io/math2/node/DWZYGTXKNS.md" tag="DWZYGTXKNS" kind="definition">
# Minimum description length

## Definition

Let $D$ be data from a finite or countable set, and let $\mathcal{H}$ be a countable set of hypotheses.
Each hypothesis $H \in \mathcal{H}$ is a [[probability distribution|0QB913WM5E]] $P_H$ on the possible data, as in a [[statistical model|K3AMMSY8DD]].
Fix a prefix code for $\mathcal{H}$, with length $L(H)$ in bits for the codeword of $H$.
Let $L(D \mid H) = -\log_2 P_H(D)$ be the code length of $D$ with the code built from $P_H$.
The *two-part minimum description length* principle selects a hypothesis $H \in \mathcal{H}$ that minimizes
$$L(H) + L(D \mid H).$$

## Remarks

The length $-\log_2 P_H(D)$ is the length, up to rounding, of a Shannon code for $D$ built from $P_H$.
Example: for $n$ coin tosses with $h$ heads, take $\mathcal{H} = \set{\text{fair coin}, \text{coin with } p = h/n}$ with $L(H) = 1$ bit each.
The fair coin costs $1 + n$ bits; the other costs $1 + n\, \mathrm{H}(h/n)$ bits, where $\mathrm{H}$ is the binary entropy, so this toy rule picks it whenever $h \ne n/2$.
This is too generous, because the second hypothesis depends on the data.
A real two-part code must also encode $h$, which costs about $\log_2 (n + 1)$ bits, and then MDL picks the biased coin only when $h/n$ is far enough from $1/2$.
Non-example: minimizing $L(D \mid H)$ alone is maximum likelihood, not MDL, and it ignores the cost of complex hypotheses.
Refined versions of MDL replace the two-part code by a single universal code, such as the normalized maximum likelihood code.
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<document source="https://tamnd.github.io/math2/node/DX2JE47SXH.md" tag="DX2JE47SXH" kind="definition">
# Surface of general type

## Definition

Let $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field.
$X$ is a *surface of general type* if its [[Kodaira dimension|ZW00Q0SAGR]] is $2$.

## Remarks

Equivalently, with $K_X$ the [[canonical divisor|W05VYY4HBR]], there is a constant $c > 0$ with $h^0(X, nK_X) \ge c n^2$ for all large divisible $n$; that is, $K_X$ is big.
Being of general type is a birational property.
A minimal surface of general type satisfies $K_X^2 \ge 1$ and $\chi(\mathcal{O}_X) \ge 1$.
Example: a smooth surface $X \subset \mathbb{P}^3$ of degree $d \ge 5$ has $K_X = (d - 4) H$ by adjunction, with $H$ a plane section.
Since $d - 4 \ge 1$, $K_X$ is ample and $h^0(nK_X)$ grows like $n^2$, so $X$ is of general type.
Non-example: a smooth quartic surface in $\mathbb{P}^3$ has $K_X = 0$, so $h^0(nK_X) = 1$ for all $n \ge 1$ and its Kodaira dimension is $0$.
Example: the product $C_1 \times C_2$ of two curves of genus at least $2$ is of general type.
</document>

<document source="https://tamnd.github.io/math2/node/DX3DJY91QH.md" tag="DX3DJY91QH" kind="definition">
# Degree of a continuous map

## Definition

Let $n \ge 1$ and let $f \colon S^n \to S^n$ be a [[continuous map|0VGFHTM75Y]] of the [[$n$-sphere|T62369RFH6]].
The induced homomorphism $f_\ast \colon H_n(S^n; \Z) \to H_n(S^n; \Z)$ on [[singular homology|M0Q5CFJ7ZH]] is an endomorphism of a group isomorphic to $\Z$.
So $f_\ast(a) = d \, a$ for a unique integer $d$ and all $a$.
This integer is the *degree* of $f$, written $\deg f$.

More generally, let $M$ and $N$ be closed connected [[oriented|913WEQERBE]] [[topological $n$-manifolds|QQQ9B1DK4P]] with [[fundamental classes|ZEA0PEM3B2]] $[M]$ and $[N]$, and let $f \colon M \to N$ be continuous.
The degree of $f$ is the integer $\deg f$ with $f_\ast [M] = (\deg f) [N]$ in $H_n(N; \Z)$.

## Remarks

Homotopic maps have the same degree, and $\deg (g \circ f) = \deg g \cdot \deg f$.

Example: on $S^1 \subset \C$, the map $f(z) = z^k$ has degree $k$.
For a reflection $r$ of $S^n$ in a hyperplane, $\deg r = -1$; the antipodal map $x \mapsto -x$ is a composite of $n + 1$ reflections, so its degree is $(-1)^{n+1}$.

Non-example: a map $S^m \to S^n$ with $m \neq n$ has no degree in this sense, since the source and target manifolds must have the same dimension.

If $f$ is smooth and $y$ is a regular value, then $\deg f = \sum_{x \in f^{-1}(y)} \sgn \det Df(x)$, where the sign is computed in oriented charts.
For $n = 1$, the degree of a map $S^1 \to S^1$ equals its [[winding number|0NQSV2EWDR]] about $0$.
</document>

<document source="https://tamnd.github.io/math2/node/DX6X1D21FE.md" tag="DX6X1D21FE" kind="example">
# The Klein four-group is abelian but not cyclic

## Example

The [[Klein four-group|0PMRQJVTME]] is $V = \{e, a, b, c\}$, where $a^2 = b^2 = c^2 = e$ and $ab = ba = c$.
The group is abelian.
Each element that is not $e$ has [[order|0JPYSD5E0B]] $2$. Thus each element generates a subgroup with at most $2$ elements.
No element generates all $4$ elements, so $V$ is not [[cyclic|0B85Q41RP6]].

## Remarks

Thus the converse of the result that cyclic groups are abelian is false.
Up to isomorphism, $V$ and $\mathbb{Z}/4\mathbb{Z}$ are the only groups of order $4$.
</document>

<document source="https://tamnd.github.io/math2/node/DX8RV2XG6H.md" tag="DX8RV2XG6H" kind="statement">
# Darcy's law

## Statement

Consider the slow flow of a fluid of viscosity $\mu > 0$ through a porous medium of permeability $k > 0$ that fills a region $\Omega \subseteq \R^3$.
Let $p \colon \Omega \to \R$ be the pressure of the fluid, and neglect gravity.
Let $q \colon \Omega \to \R^3$ be the volumetric flux, the [[vector field|0299VCW1TH]] of the volume of fluid that crosses a unit area per unit time.
Then
$$q = - \frac{k}{\mu} \nabla p.$$
Here $\nabla p$ is the [[gradient|0ABK9V9Y64]] of the pressure.

## Remarks

Darcy's law is an empirical law, found by Henry Darcy from experiments on water flowing through sand.
It can also be derived from the equations of [[Stokes flow|13CB7V2YM4]] by homogenization in a periodic porous structure.
It is valid when the flow is slow, that is, when the Reynolds number of the flow in the pores is small.

For an incompressible fluid without sources, the [[divergence|0MWV6ZTK6S]] of the flux is zero.
With constant $k$ and $\mu$ this gives $\Delta p = 0$, so the pressure is a harmonic function, and boundary values of $p$ determine the flow.
With gravity the law reads $q = -\tfrac{k}{\mu} (\nabla p - \rho g)$, where $\rho$ is the density and $g$ the gravitational acceleration vector.

## Explanations

### L1 human

Water seeps through sand or soil from high pressure to low pressure.
Darcy's law says that the flow rate is proportional to the drop in pressure and to how easily the medium lets fluid through.
It is inversely proportional to the viscosity, that is, to how thick and sticky the fluid is.
It is the analogue for porous media of Ohm's law for electric current and Fourier's law for heat.

### L2 human

Darcy's law turns problems of groundwater, oil reservoirs and filtration into elliptic and parabolic partial differential equations for the pressure.
Combined with conservation of mass for a compressible fluid or a gas, it leads to diffusion equations such as the porous medium equation.
When the permeability varies in space, the pressure solves the divergence-form equation $\nabla \cdot (k \nabla p) = 0$, a standard model of an elliptic equation with variable coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/DXCTK6M7ZV.md" tag="DXCTK6M7ZV" kind="definition">
# CAT(0) group

## Definition

Let $X$ be a [[CAT(0) space|1FAF2VNA4G]], that is, a geodesic metric space with the following property.
For every geodesic triangle $\Delta$, let $\bar\Delta$ be a triangle in $\R^2$ with the same side lengths.
Then $d(p, q) \le \norm{\bar p - \bar q}$ for all points $p, q$ of $\Delta$ and their comparison points $\bar p, \bar q$ in $\bar\Delta$.
Let a group $G$ [[act|0WCY42EV9V]] on $X$ by [[isometries|02HPP5VCZH]].

1. The action is *proper* if each $x \in X$ has a radius $r > 0$ such that $\set{g \in G : g B(x, r) \cap B(x, r) \neq \emptyset}$ is finite.
2. The action is *cocompact* if there is a compact set $K \subseteq X$ with $GK = X$.

The group $G$ is a *CAT(0) group* if it admits a proper and cocompact action by isometries on some CAT(0) space.

## Remarks

Example: $\Z^n$ acts on $\R^n$ by translations.
The Euclidean space $\R^n$ is CAT(0), since its triangles are exactly as thin as their comparison triangles.
The action is proper, because a ball of radius $1/3$ meets only its own translate, and cocompact, with $K = [0, 1]^n$.

Example: a free group of finite rank acts properly and cocompactly on its Cayley graph for a free basis, which is a tree, and trees are CAT(0).

Non-example: the integer Heisenberg group is not a CAT(0) group.
In a CAT(0) group every solvable subgroup is virtually abelian, and the Heisenberg group is nilpotent but not virtually abelian.

Such an action is also called geometric.
Every CAT(0) group is finitely presented, and its [[Dehn function|Q4HGMRA9JQ]] is at most quadratic.
</document>

<document source="https://tamnd.github.io/math2/node/DXJH3DGXBC.md" tag="DXJH3DGXBC" kind="concept">
# Section and retraction

## Summary

When two [[morphisms|859F0TFVA3]] $f \colon A \to B$ and $r \colon B \to A$ satisfy $r \circ f = 1_A$, the morphism $f$ is a section of $r$ and $r$ is a retraction of $f$.
A morphism with a retraction is a split [[monomorphism|Y2NP1JFR86]], and a morphism with a section is a split [[epimorphism|RVXDSY04KG]].
These are one-sided forms of invertibility, and every functor preserves them.
In linear algebra and homological algebra, split monomorphisms and split epimorphisms describe direct summands.

## Explanations

### L1 human

A section is a morphism that can be undone from one side, though perhaps not from the other.
Think of $f$ as placing $A$ inside $B$ and of $r$ as folding $B$ back onto $A$ so that each placed point returns to where it came from.
For example, put $f \colon \R \to \R^2$, $f(x) = (x, 0)$, and $r \colon \R^2 \to \R$, $r(x, y) = x$.
Then $r(f(x)) = x$ for every $x$, so $f$ is a section of $r$ and $r$ is a retraction of $f$.
The other composite $f(r(x, y)) = (x, 0)$ is not the identity, so neither map is an isomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/DXJXVFV79H.md" tag="DXJXVFV79H" kind="concept">
# Expander graph

## Summary

An expander graph is a sparse graph in which every set of at most half of the vertices has many edges leaving it.
One usually means a family of [[regular graphs|0KA4S8R39M]] of fixed degree whose [[Cheeger constant|Y7S6W12N64]] stays above a fixed positive bound.
Expanders combine few edges with strong connectivity, and they are used in network design, error-correcting codes, derandomization and group theory.

## Explanations

### L1 human

Picture a network of computers in which each machine has only three cables.
In a good expander, any group of up to half the machines is joined to the rest by many cables, in proportion to the size of the group.
So the network has no bottleneck, although it is cheap to build.
A long cycle is the opposite: half of the cycle is cut off from the other half by just $2$ edges.
</document>

<document source="https://tamnd.github.io/math2/node/DXWBWMM5H7.md" tag="DXWBWMM5H7" kind="definition">
# Pixley term

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
A *Pixley term* for $\mathcal{V}$ is a ternary term $p(x, y, z)$ such that $\mathcal{V}$ satisfies the [[identities|PKD440G2XW]]
$$p(x, y, y) = x, \qquad p(x, x, y) = y, \qquad p(x, y, x) = x.$$

## Remarks

The first two identities say that $p$ is a [[Mal'cev term|Y203PPTS61]].
The term $m(x, y, z) = p(x, p(x, y, z), z)$ is then a [[majority term|9879ZMW5AA]]: $m(x, x, z) = p(x, z, z) = x$, $m(x, y, x) = p(x, x, x) = x$ and $m(x, z, z) = p(x, x, z) = z$.
Pixley's theorem says that $\mathcal{V}$ has a Pixley term if and only if $\mathcal{V}$ is an [[arithmetical variety|GW78A05TNA]].
Example: for Boolean algebras, $p(x, y, z) = (x \wedge \neg y) \vee (x \wedge z) \vee (\neg y \wedge z)$ is a Pixley term.
Indeed $p(x, y, y) = (x \wedge \neg y) \vee (x \wedge y) \vee 0 = x$, $p(x, x, y) = 0 \vee (x \wedge y) \vee (\neg x \wedge y) = y$, and $p(x, y, x) = (x \wedge \neg y) \vee x \vee (\neg y \wedge x) = x$.
Non-example: for groups, the Mal'cev term $x y^{-1} z$ is not a Pixley term.
In the group $\Z$, the third identity would read $x - y + x = x$, which fails for $x \ne y$.

## Explanations

### L1 human

A Pixley term is a three-input formula that returns $x$ on the inputs $x, y, y$ and $x, y, x$, and returns $y$ on the inputs $x, x, y$.
</document>

<document source="https://tamnd.github.io/math2/node/DXX7GNY3HE.md" tag="DXX7GNY3HE" kind="concept">
# von Neumann regular ring

## Summary

A von Neumann regular ring is a ring in which every element $a$ has a weak inverse $x$, with $a x a = a$.
Fields, division rings, matrix rings over division rings, products of fields and Boolean rings are examples.
John von Neumann introduced these rings in his work on continuous geometry.
In the commutative case they are exactly the reduced rings of Krull dimension $0$, and also the rings over which every module is flat.
The name is unrelated to regular local rings.

## Explanations

### L1 human

In a field every nonzero element has an inverse.
A von Neumann regular ring asks for less: each element $a$ has a partner $x$ that undoes $a$ "on its image", in the sense that $a x a = a$.
In the product $\Q \times \Q$ of two fields, the element $(2, 0)$ has no inverse, but $x = (\frac12, 0)$ satisfies $a x a = a$.
In the integers, $2 x 2 = 2$ would need $4x = 1$, which has no solution, so $\Z$ is not von Neumann regular.
</document>

<document source="https://tamnd.github.io/math2/node/DXZP6V23Z7.md" tag="DXZP6V23Z7" kind="concept">
# Hypergeometric function of a matrix argument

## Summary

A hypergeometric function of a matrix argument ${}_pF_q^{(\alpha)}(a; b; X)$ is a generalisation of the [[generalized hypergeometric function|VDBAW9PN88]] in which the variable is a matrix $X$.
The series runs over [[integer partitions|0JV30XN73Z]].
The powers $z^k$ are replaced by Jack polynomials in the eigenvalues of $X$, and the rising factorials by generalized Pochhammer symbols.
These functions appear in multivariate statistics and random matrix theory, where densities of eigenvalues of Wishart and related matrices are written with them.

## Explanations

### L1 human

For a number $z$, the series ${}_pF_q$ is a sum over $k$ of rational coefficients times $z^k / k!$.
For a matrix, the role of $z^k$ is played by a family of symmetric polynomials of degree $k$ in the eigenvalues, one for each way of splitting $k$ into parts.
These polynomials add up to $(\operatorname{tr} X)^k$, so the matrix series keeps track of finer information than the trace alone.
The simplest case is ${}_0F_0(X) = e^{\operatorname{tr} X}$, which comes out because the polynomials of degree $k$ sum to $(\operatorname{tr} X)^k$.
The parameter $\alpha$ selects the setting: $\alpha = 2$ for real symmetric matrices and $\alpha = 1$ for complex Hermitian matrices.
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<document source="https://tamnd.github.io/math2/node/DY8QQ5GTTX.md" tag="DY8QQ5GTTX" kind="concept">
# Kobayashi metric

## Summary

The Kobayashi metric, or Kobayashi pseudodistance, of a complex manifold $X$ is a pseudodistance on $X$ built from holomorphic discs.
It is the largest pseudodistance for which every [[holomorphic map|3MQTWKBBBT]] from the unit disc, with its Poincaré distance, is distance-decreasing.
It is built from chains of holomorphic discs, and Shoshichi Kobayashi introduced it in 1967.
Every holomorphic map between complex manifolds is distance-decreasing for the Kobayashi pseudodistances, so it is a biholomorphic invariant.
It vanishes identically on $\C^n$ and is a true distance on bounded domains.
The manifolds on which it is a distance are the [[Kobayashi hyperbolic manifolds|HRPAX8V7HA]].

## Explanations

### L1 human

The unit disc carries a natural distance, the hyperbolic distance of the [[Poincaré disk model|85MM2EA0TK]], which holomorphic self-maps of the disc never increase.
To measure the distance between two points of a complex manifold, connect them by a chain of holomorphic discs and add up the hyperbolic lengths of the pieces.
The Kobayashi distance is the smallest possible total.
In $\C$ one can join two points by a huge disc, in which they are very close in the hyperbolic sense, so the Kobayashi distance of $\C$ is $0$.
In the unit disc itself, the Kobayashi distance is the hyperbolic distance.
</document>

<document source="https://tamnd.github.io/math2/node/DYAWGXE2GV.md" tag="DYAWGXE2GV" kind="example">
# James space

## Example

Let $J$ be the set of real sequences $x = (x_n)_{n \ge 1}$ with $x_n \to 0$ and $\norm{x}_J < \infty$, where
$$\norm{x}_J = \sup \Big( \sum_{i=1}^{k-1} (x_{p_i} - x_{p_{i+1}})^2 \Big)^{1/2}.$$
The supremum runs over all $k \ge 2$ and all integers $p_1 < p_2 < \cdots < p_k$.
Then $\norm{\cdot}_J$ is a norm on $J$, and $J$ is a [[Banach space|06GZ956657]], the James space.

Check that $x_n = 1/\sqrt{n}$ lies in $J$.
For $p_1 < \cdots < p_k$, the differences $a_i = x_{p_i} - x_{p_{i+1}}$ are positive.
Then $\sum_i a_i^2 \le (\sum_i a_i)^2 = (x_{p_1} - x_{p_k})^2 \le 1$.
So $\norm{x}_J \le 1$, although $x$ is not in $\ell^2$, since $\sum_n 1/n$ diverges.

## Remarks

James built this space in 1951.
The canonical image of $J$ in its [[bidual|QKR3D65TTH]] $J^{**}$ has codimension $1$, so $J$ is not [[reflexive|NN8RWPC8WG]].
Yet $J$ is isomorphic to $J^{**}$.
So a Banach space can be isomorphic to its bidual without being reflexive.
The norm vanishes only at $0$ because $x_n \to 0$: the difference $x_p - x_q$ with $q$ large is close to $x_p$.

## Explanations

### L1 human

The norm of James space measures the total squared jumps of a sequence along any chosen subsequence.
A sequence that slides down to $0$ without wiggling has a small norm, even if it decreases slowly.
The space is famous as a counterexample: it is as big as its double dual, but not in the canonical way.
</document>

<document source="https://tamnd.github.io/math2/node/DYFQ2EQ7J6.md" tag="DYFQ2EQ7J6" kind="definition">
# Hyperbolic knot

## Definition

Let $K \subset S^3$ be a tame [[knot|QYEAZTYGPT]].
The knot $K$ is *hyperbolic* if its complement $S^3 \setminus K$ admits a complete Riemannian metric of constant sectional curvature $-1$ with finite volume.

## Remarks

Equivalently, $S^3 \setminus K$ is a [[hyperbolic 3-manifold|GHVW1Z35BS]] of finite volume, the quotient $\mathbb{H}^3 / \Gamma$ by a discrete torsion-free group $\Gamma$ of isometries.
By Mostow–Prasad rigidity the metric is unique up to isometry, so the hyperbolic volume is a [[knot invariant|GZQ5ZE3X6G]].
Example: the [[figure-eight knot|C01XBGAHC6]] complement is obtained by gluing two regular ideal hyperbolic tetrahedra, so the knot is hyperbolic with volume $2.0298832\ldots$.
Non-example: the complement of a [[torus knot|NS60VV786W]] is Seifert fibered and has no hyperbolic metric.
The complement of the [[unknot|1TQ06R4Q66]] is an open solid torus, whose fundamental group $\Z$ cannot be a lattice in the isometry group of $\mathbb{H}^3$; so the unknot is not hyperbolic.
By Thurston, a nontrivial knot is hyperbolic if and only if it is neither a torus knot nor a [[satellite knot|A1E8TDEVJ9]].
</document>

<document source="https://tamnd.github.io/math2/node/DYFY16SFHZ.md" tag="DYFY16SFHZ" kind="concept">
# Kendall rank correlation coefficient

## Summary

The Kendall rank correlation coefficient $\tau$ measures the agreement between the orderings of two variables.
It compares every pair of observations.
It counts the pairs ordered the same way by both variables, minus the pairs ordered in opposite ways, and divides by the number of pairs.
It lies between $-1$ and $1$ and depends only on the ranks of the data, like [[Spearman's coefficient|DJFYTQJ3SH]].

## Explanations

### L1 human

Take two judges who rank the same contestants.
For each pair of contestants, check whether the two judges put them in the same order.
Kendall's $\tau$ is the share of pairs on which they agree minus the share on which they disagree.
If they agree on every pair, $\tau = 1$; if they disagree on every pair, $\tau = -1$.
With three contestants ranked $1, 2, 3$ and $1, 3, 2$, the judges agree on two of three pairs, so $\tau = 2/3 - 1/3 = 1/3$.
</document>

<document source="https://tamnd.github.io/math2/node/DYM1EFAC59.md" tag="DYM1EFAC59" kind="definition">
# Priestley space

## Definition

Let $X$ be a set with a [[topology|0CQHFPGJ45]] $\tau$ and a [[partial order|0QV2JS9RJ3]] $\le$.
The triple $(X, \tau, \le)$ is a *Priestley space* if:

1. $(X, \tau)$ is [[compact|0TVJZYB6SH]];
2. for all $x, y \in X$ with $x \not\le y$, some $U \subseteq X$ has $x \in U$ and $y \notin U$, is open and closed in $\tau$, and is an [[upper set|ETCE287E2R]].

Clause 2 is called the Priestley separation axiom.

## Remarks

A Priestley space is Hausdorff and totally disconnected, so its underlying space is a [[Stone space|ZV38HA5PAQ]].
Indeed, for $x \ne y$, say $x \not\le y$, the sets $U$ and $X \setminus U$ from clause 2 separate $x$ and $y$.

Example: a finite poset with the discrete topology.
Every subset is clopen, and for $x \not\le y$ the upper set ${\uparrow} x = \set{z : x \le z}$ contains $x$ but not $y$.

Non-example: the interval $[0, 1]$ with its usual topology and usual order.
It is compact, but its only clopen sets are $\emptyset$ and $[0, 1]$, so no clopen set separates $1 \not\le 0$, and clause 2 fails.

The morphisms of Priestley spaces are the continuous order-preserving maps.
</document>

<document source="https://tamnd.github.io/math2/node/DYR1QFNGNJ.md" tag="DYR1QFNGNJ" kind="concept">
# Borel measure

## Summary

A Borel measure on a topological space is a [[measure|PR11BZQQPG]] defined on the [[Borel sets|0WQXY32Q8D]], the sets generated by the open sets.
It is the natural kind of measure on spaces such as $\R^n$, manifolds and topological groups, where open sets and continuous functions are given.
Every continuous real function is measurable for a Borel measure.
Some authors also require a Borel measure to be finite on compact sets.

## Explanations

### L1 human

On a space with a notion of open sets, the Borel sets are all the sets one can build from open sets by complements and countable unions.
A Borel measure assigns a size to each of these sets.
Length on the real line, area in the plane and the point mass at a point are Borel measures.
They give sizes to intervals, open discs, closed sets, countable sets and much more.
</document>

<document source="https://tamnd.github.io/math2/node/DYXEBVDC6W.md" tag="DYXEBVDC6W" kind="statement">
# Levi–Malcev decomposition

## Statement

Let $G$ be a connected and simply connected real [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{g}$.
Let $\mathfrak{r}$ be the radical of $\mathfrak{g}$, its largest solvable ideal, and let $R$ be the connected Lie subgroup of $G$ with Lie algebra $\mathfrak{r}$.
Then the following hold.

1. The subgroup $R$ is closed, normal, simply connected and [[solvable|9CNSVFQ801]].
2. There is a closed, simply connected, [[semisimple|XJS07AJ50Y]] Lie subgroup $S$ of $G$ such that $G = R \rtimes S$, that is, $G = RS$ and $R \cap S = \set{e}$.
3. If $S'$ is a second subgroup as in 2, then $S' = x S x^{-1}$ for some $x \in R$.

A subgroup $S$ as in 2 is a *Levi subgroup* of $G$.

## Remarks

The proof starts from the Levi decomposition of the Lie algebra, $\mathfrak{g} = \mathfrak{r} \rtimes \mathfrak{s}$ with $\mathfrak{s}$ a semisimple subalgebra.
The quotient $G/R$ is simply connected with Lie algebra $\mathfrak{s}$, so the connected subgroup $S$ with Lie algebra $\mathfrak{s}$ maps isomorphically onto $G/R$.
Thus $S$ is the image of a continuous section of $G \to G/R$, which gives 2.
Part 3 comes from Malcev's theorem that two Levi subalgebras are conjugate by $\exp(\operatorname{ad} z)$ for some $z$ in $\mathfrak{r}$; then $x = \exp z$ works.

Example: the group $G$ of affine maps $v \mapsto Av + b$ of $\R^2$ with $A \in \mathrm{SL}_2(\R)$ has radical the translations $R \cong \R^2$ and Levi subgroup $\mathrm{SL}_2(\R)$.
This $G$ is not simply connected; its [[universal cover|Y1YPE6PM52]] is $\R^2 \rtimes \widetilde{\mathrm{SL}}_2(\R)$.

Simple connectedness is needed for 2.
For a general connected Lie group, a Levi subgroup $S$ still exists and $G = RS$, but $R \cap S$ need not be trivial, and $S$ need not be closed.

## Explanations

### L1 human

Every connected Lie group is built from two kinds of pieces.
One piece is solvable, close to a group of upper triangular matrices.
The other piece is semisimple, like a group of rotations or $\mathrm{SL}_n$.
The theorem says that, for simply connected groups, the group is exactly a twisted product of one piece of each kind.

### L2 human

The decomposition reduces many questions about Lie groups to two separate theories.
Solvable groups are studied by induction on dimension, using Lie's theorem and the structure of nilpotent groups.
Semisimple groups are studied with root systems, the Cartan and Iwasawa decompositions, and Weyl groups.
The Lie algebra version is due to Levi, and the conjugacy statement to Malcev.
</document>

<document source="https://tamnd.github.io/math2/node/DZ0B0JD11D.md" tag="DZ0B0JD11D" kind="definition">
# Minimal variety

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
The variety $\mathcal{V}$ is *minimal* if both of these hold:

1. $\mathcal{V}$ contains an algebra with more than one element.
2. Every [[subvariety|4YSCK1DF26]] of $\mathcal{V}$ is either $\mathcal{V}$ itself or the trivial variety of one-element algebras.

## Remarks

Example: the variety of Boolean algebras is minimal.
Every nontrivial Boolean algebra contains the two-element Boolean algebra $\mathbf{2}$ as a subalgebra, and $\mathbf{2}$ generates the variety of all Boolean algebras.
So a subvariety with a nontrivial member contains $\mathbf{2}$ and hence every Boolean algebra.
Non-example: the variety of all groups is not minimal.
Its subvariety of abelian groups contains $\Z/2\Z$, so it is not trivial, and it is a proper subvariety.
In the [[lattice of subvarieties|GJAYCVYZYF]] of a variety, the minimal varieties are exactly the atoms, the elements that cover the trivial variety.

## Explanations

### L1 human

A minimal variety cannot be cut down further by identities without becoming trivial.
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<document source="https://tamnd.github.io/math2/node/DZ297MMGV8.md" tag="DZ297MMGV8" kind="concept">
# Locally finite family

## Summary

A family of subsets of a [[topological space|0CQHFPGJ45]] is locally finite if each point has a [[neighbourhood|0ZCY0Z80B6]] meeting only finitely many members.
The family may be infinite, but near each point it looks finite.
Local finiteness lets one treat infinite unions and sums as if they were finite.
For example, the closure of the union of a locally finite family is the union of the closures.
It is the key notion in the definition of [[paracompact spaces|8R29G6NG9Q]] and of partitions of unity.

## Explanations

### L1 human

Think of tiles on an infinite floor: there are infinitely many tiles, but around any spot you stand on, only a few tiles are close by.
The intervals $[n, n + 1]$ for integers $n$ form such a family on the real line.
The intervals $(0, 1/n)$ for $n \ge 1$ do not, because every small interval around $0$ meets all of them.
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<document source="https://tamnd.github.io/math2/node/DZ47JVKEF5.md" tag="DZ47JVKEF5" kind="concept">
# Integral equation of the third kind

## Summary

An integral equation of the third kind is a [[linear integral equation|AYZCKG34QK]] $a_0(x) \varphi(x) = f(x) + \lambda \int K(x, s) \varphi(s) \, ds$ whose coefficient $a_0$ vanishes at some points but not everywhere.
It sits between the [[first kind|ZDS77Q0F06]], where $a_0 = 0$, and the [[second kind|8KJCN49V64]], where $a_0 = 1$.
Near the zeros of $a_0$ the equation behaves like one of the first kind, so its solutions may be singular there.
Such equations occur in transport theory and in some problems of mathematical physics.

## Explanations

### L1 human

The unknown appears outside the integral with a weight $a_0(x)$.
Where the weight is not zero one can divide by it and get an ordinary equation of the second kind.
At the points where the weight is zero this division fails, and the solution can blow up or need not exist.
For example, $x \varphi(x) = f(x) + \int_{-1}^{1} \varphi(s) \, ds$ on $[-1, 1]$ is of the third kind, since $x$ vanishes at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/DZB378CNCK.md" tag="DZB378CNCK" kind="concept">
# Inverse limit

## Summary

An inverse limit, or projective limit, is the [[limit|CHS5AFJGWE]] of a system of objects indexed by a [[directed set|D9RY4667K6]].
Its maps go from later objects to earlier ones.
It collects the compatible families of elements, one in each object of the system.
The [[p-adic integers|V6TJAFZ4MD]] and the [[profinite groups|KA0YDB6QXE]] are inverse limits of finite objects.
The dual notion is the [[direct limit|X9HEQ29Q49]].

## Explanations

### L1 human

Think of a sequence of ever finer pictures of the same thing, each one able to forget detail and give the previous one.
The inverse limit is the object whose points are whole towers of compatible pictures.
For example, take the rings $\Z/p\Z$, $\Z/p^2\Z$, $\Z/p^3\Z$, and so on, with reduction maps between them.
A point of the inverse limit is a sequence of residues $a_n$ modulo $p^n$, with $a_{n+1}$ reducing to $a_n$.
These sequences form the ring of $p$-adic integers.
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<document source="https://tamnd.github.io/math2/node/DZD0Q4H4DS.md" tag="DZD0Q4H4DS" kind="definition">
# Newton series

## Definition

Let $f \colon \C \to \C$ be a function and let $a \in \C$.
For $k \in \N$ let $\Delta^k f(a)$ be the [[$k$-th forward difference|E38801KVJ1]] of $f$ at $a$ with step $1$.
For $s \in \C$ let $\binom{s}{k} = \frac{s(s-1)\cdots(s-k+1)}{k!}$, with $\binom{s}{0} = 1$.
The *Newton series* of $f$ at $a$ is the [[series|0JH4MN4K1C]]
$$\sum_{k=0}^{\infty} \binom{x - a}{k} \Delta^k f(a),$$
viewed as a series of functions of $x \in \C$.

## Remarks

Example: for $f(x) = 2^x$ and $a = 0$, $\Delta^k f(0) = 1$ for all $k$, so the Newton series is $\sum_k \binom{x}{k}$.
This is the binomial series of $(1 + z)^x$ at $z = 1$, which converges to $2^x$ for real $x > -1$.
Non-example: $\sum_k \frac{x^k}{k!}$ is the Taylor series of $e^x$ at $0$, not its Newton series.
The Newton series of $e^x$ at $0$ has coefficients $\Delta^k e^x |_{x=0} = (e - 1)^k$, and since $e - 1 > 1$ it diverges at every real $x$ that is not a natural number.
At $x = a + n$ with $n \in \N$, only the terms with $k \le n$ are nonzero, and the sum is $f(a + n)$ by [[Newton's forward difference formula|DDHCMKHD2T]].
</document>

<document source="https://tamnd.github.io/math2/node/DZEQY398HQ.md" tag="DZEQY398HQ" kind="definition">
# Schubert variety

## Definition

Let $k$ be a field, let $1 \le d < n$, and let $\mathrm{Gr}(d, n)$ be the [[Grassmannian|CYPA882RB9]] of $d$-dimensional subspaces of $k^n$.
Fix a full flag $F_1 \subset F_2 \subset \cdots \subset F_n = k^n$ with $\dim F_j = j$.
Let $\lambda = (\lambda_1, \ldots, \lambda_d)$ be a partition with $n - d \ge \lambda_1 \ge \cdots \ge \lambda_d \ge 0$.
The *Schubert variety* $X_\lambda$ is
$$X_\lambda = \set{\Lambda \in \mathrm{Gr}(d, n) : \dim(\Lambda \cap F_{n - d + i - \lambda_i}) \ge i \text{ for } i = 1, \ldots, d}.$$

## Remarks

$X_\lambda$ is a closed irreducible subvariety of codimension $\abs{\lambda} = \lambda_1 + \cdots + \lambda_d$.

Example: $\lambda = 0$ gives all of $\mathrm{Gr}(d, n)$.
Indeed $\dim(\Lambda \cap F_{n - d + i}) \ge d + (n - d + i) - n = i$ holds for every $\Lambda$.
For $d = 2$, $n = 4$ and $\lambda = (1, 0)$, the conditions reduce to $\dim(\Lambda \cap F_2) \ge 1$.
In projective terms these are the lines in $\mathbb{P}^3$ that meet a fixed line, a hypersurface in $\mathrm{Gr}(2, 4)$.

Non-example: the set of $\Lambda \in \mathrm{Gr}(2, 4)$ with $\Lambda \cap F_2 = 0$ is open and dense, and it is not closed.
So it is not a Schubert variety; it is the open Schubert cell.

General form: let $G$ be a connected reductive group with a [[Borel subgroup|5JSYA1JNCC]] $B$ and a parabolic subgroup $P \supseteq B$.
A Schubert variety in the [[flag variety|NHRWCQQKTH]] $G / P$ is the Zariski closure of a $B$-orbit.
</document>

<document source="https://tamnd.github.io/math2/node/DZF985Y137.md" tag="DZF985Y137" kind="definition">
# Unitary representation

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] and let $H$ be a complex [[Hilbert space|HM2HHE3J51]].
Let $U(H)$ be the group of [[unitary operators|MM0EJ2GDE4]] on $H$.
A *unitary representation* of $G$ on $H$ is a map $\pi \colon G \to U(H)$ such that:

1. $\pi$ is a group homomorphism: $\pi(gh) = \pi(g)\pi(h)$ for all $g, h \in G$;
2. $\pi$ is strongly continuous: for every $v \in H$, the map $G \to H$, $g \mapsto \pi(g)v$, is continuous.

## Remarks

Clause 2 says that $\pi$ is continuous for the [[strong operator topology|KHJ2G2R5G7]] on $U(H)$.
It is equivalent to weak continuity, the continuity of $g \mapsto \ip{\pi(g)u}{v}$ for all $u, v \in H$.

Example: $G = (\R, +)$ on $H = L^2(\R)$ by $(\pi(t)f)(x) = f(x - t)$.
Translation preserves $\int \abs{f}^2$ and has inverse $\pi(-t)$, so each $\pi(t)$ is unitary, and $\pi(s + t) = \pi(s)\pi(t)$.
For continuity, $\norm{\pi(t)f - f}_2 \to 0$ as $t \to 0$ holds for compactly supported continuous $f$ by uniform continuity, and then for all $f \in L^2(\R)$ since these are dense and each $\pi(t)$ has norm $1$.

Non-example: let $\ell^2(\R)$ be the Hilbert space of square-summable functions on the set $\R$, with orthonormal basis $(\delta_s)_{s \in \R}$, and let $\pi(t)\delta_s = \delta_{s + t}$.
Each $\pi(t)$ is unitary and $\pi$ is a homomorphism, but clause 2 fails for the usual topology of $\R$.
Indeed $\norm{\pi(t)\delta_0 - \delta_0} = \sqrt{2}$ for every $t \ne 0$, so $\pi(t)\delta_0$ does not tend to $\delta_0$ as $t \to 0$.

Two unitary representations $\pi_1$ on $H_1$ and $\pi_2$ on $H_2$ are *unitarily equivalent* if there is a unitary $T \colon H_1 \to H_2$ with $T\pi_1(g) = \pi_2(g)T$ for all $g \in G$.
A unitary representation is *irreducible* if $H \ne 0$ and the only closed subspaces invariant under all $\pi(g)$ are $0$ and $H$.
</document>

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# Birkhoff–Frink theorem

## Statement

Let $L$ be a [[complete lattice|D7A0ZGZT5S]].
The following are equivalent.

1. $L$ is an [[algebraic lattice|3J0BTXSCVN]].
2. $L$ is isomorphic to the lattice of subalgebras of some algebra $A$, a set with a family of finitary operations, ordered by inclusion.
3. $L$ is isomorphic to the lattice of closed sets of an algebraic [[closure operator|2HJGRABSPZ]] $c$ on some set $X$.
   Algebraic means that $c(S) = \bigcup \set{c(F) : F \subseteq S \text{ finite}}$ for every $S \subseteq X$.

## Remarks

From 2 to 3: the subalgebra generated by a set is an algebraic closure operator, since each operation has finitely many arguments.
From 3 to 1: in the lattice of closed sets, the [[compact elements|9F7TYES4KE]] are the closures of finite sets, and every closed set is the union of these.
From 1 to 2: one equips the set $K$ of compact elements of $L$ with operations whose subalgebras are exactly the ideals of the join-semilattice $K$.
The map $a \mapsto \set{k \in K : k \le a}$ identifies $L$ with these ideals.
For example, the lattice of subgroups of a group and the lattice of ideals of a ring are algebraic.
The lattice of closed subsets of $[0, 1]$ is complete but not algebraic, so it is not a subalgebra lattice.

## Explanations

### L1 human

The subgroups of a group, the subspaces of a vector space and the ideals of a ring all form complete lattices of a special kind, called algebraic.
The theorem says that the lattices of "sub-things" of algebraic structures are exactly the algebraic lattices.
Every algebraic lattice is the lattice of substructures of some suitable algebra.

### L2 human

The theorem explains why compactness is the right abstraction of "finitely generated": the compact elements of a subalgebra lattice are the finitely generated subalgebras.
It is the first of two representation results: the [[Grätzer–Schmidt theorem|2BZ2ZXH03B]] shows that the same lattices are also exactly the congruence lattices of algebras.
Together with the description of algebraic lattices as ideal lattices of join-semilattices with $0$, it gives several equivalent pictures of the same class.
</document>

<document source="https://tamnd.github.io/math2/node/DZKBF576W5.md" tag="DZKBF576W5" kind="problem">
# Merton's portfolio problem

## Problem

Consider a market with a riskless asset with interest rate $r$ and a stock.
The stock price follows the [[geometric Brownian motion|D48DR46B97]] $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$, where $\mu > r$, $\sigma > 0$ and $W$ is a Wiener process.
An investor with initial wealth $x > 0$ chooses, at each time $t$, the fraction $\pi_t$ of wealth held in the stock and a consumption rate $c_t \ge 0$.
Both are adapted to the information generated by $W$.
The wealth then evolves by
$$dX_t = \left( r X_t + \pi_t (\mu - r) X_t - c_t \right) dt + \pi_t \sigma X_t \, dW_t, \qquad X_0 = x.$$
Let $\rho > 0$ be a discount rate and let $U(c) = \dfrac{c^{1 - \gamma}}{1 - \gamma}$ with $\gamma > 0$ and $\gamma \ne 1$.
The problem is to find the strategies $(\pi, c)$ with $X_t \ge 0$ for all $t$ that maximise the expected discounted utility of consumption
$$E \int_0^\infty e^{-\rho t} U(c_t) \, dt.$$

## Remarks

The problem is solved, and it is a standard example of [[stochastic optimal control|B24T6DQCDR]].
Write $\theta = (\mu - r) / \sigma$ and assume $\nu = \dfrac{\rho - (1 - \gamma)\left( r + \theta^2 / (2 \gamma) \right)}{\gamma} > 0$.
The value function is $V(x) = \nu^{-\gamma} U(x)$.
The optimal strategy holds the constant fraction $\pi^* = \dfrac{\mu - r}{\gamma \sigma^2}$ in the stock and consumes at the rate $c_t = \nu X_t$.
These follow from the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] $\rho V = \max_{\pi, c} \left\{ U(c) + V'(x) \left( r x + \pi (\mu - r) x - c \right) + \frac{1}{2} \sigma^2 \pi^2 x^2 V''(x) \right\}$ with the guess $V = A U$ for a constant $A > 0$.
Merton also treated a finite horizon with utility of final wealth, and the logarithmic utility $U = \ln$, which is the limit case $\gamma = 1$ and gives $\pi^* = (\mu - r) / \sigma^2$ and $c_t = \rho X_t$.

## Explanations

### L1 human

How much of your savings should you keep in risky stocks, and how fast should you spend?
Merton's answer, for this model, is to keep a fixed share of your wealth in stocks and to spend a fixed share of it per unit time.
The share in stocks grows with the excess return $\mu - r$ and falls with the risk $\sigma^2$ and with your aversion to risk $\gamma$.
</document>

<document source="https://tamnd.github.io/math2/node/DZQ4EW6907.md" tag="DZQ4EW6907" kind="definition">
# Hermite interpolation

## Definition

Let $m \ge 0$, let $x_0, \dots, x_m \in \R$ be pairwise distinct, and let $k_0, \dots, k_m \ge 1$ be integers with $N = k_0 + \cdots + k_m$.
For each $i$ and each $0 \le j \le k_i - 1$ let a real number $y_i^{(j)}$ be given.
The *Hermite interpolation polynomial* of these data is the polynomial $p$ with real coefficients such that:

1. $p = 0$ or $\deg p \le N - 1$;
2. $p^{(j)}(x_i) = y_i^{(j)}$ for all $0 \le i \le m$ and $0 \le j \le k_i - 1$, where $p^{(j)}$ is the $j$-th [[derivative|0V1EJCT8G6]].

## Remarks

Such a $p$ exists and is unique.
Uniqueness: the difference $q$ of two solutions is $0$ or has [[degree|0P4JRMSWNH]] at most $N - 1$.
It has a root of [[multiplicity|047SX3TD69]] at least $k_i$ at each $x_i$, so at least $N$ roots counted with multiplicity, and hence $q = 0$.
Existence follows because the conditions form a square linear system whose only solution for zero data is $0$.
If all $k_i = 1$ this is [[polynomial interpolation|SS1YMS2NPC]]; if $m = 0$ the solution is the Taylor polynomial of order $k_0 - 1$ at $x_0$.

Example: $m = 1$, nodes $0$ and $1$, $k_0 = k_1 = 2$, data $p(0) = 0$, $p'(0) = 0$, $p(1) = 1$, $p'(1) = 0$.
Then $N = 4$ and $p(x) = 3x^2 - 2x^3$: indeed $p(0) = 0$, $p'(x) = 6x - 6x^2$ gives $p'(0) = p'(1) = 0$, and $p(1) = 1$.

Non-example: $3x^2 - 2x^3 + x^2 (x - 1)^2$ satisfies clause 2 for the same data, since the added term has double roots at $0$ and $1$, but its degree is $4 > 3$, so clause 1 fails.

The conditions must cover consecutive derivative orders $0, 1, \dots, k_i - 1$ at each node; dropping this gives [[Birkhoff interpolation|FT1YKVBFC9]], where uniqueness can fail.
</document>

<document source="https://tamnd.github.io/math2/node/DZSZ8ZS554.md" tag="DZSZ8ZS554" kind="concept">
# Elementary abelian group

## Summary

An elementary abelian $p$-group, for a prime $p$, is an [[abelian group|082GHNPJ53]] in which every element other than the identity has order $p$.
Such a group is the same thing as a [[vector space|000AYPA9AJ]] over the field $\F_p$.
So a finite one is isomorphic to $(\Z/p\Z)^n$ for some $n$.
They appear everywhere in finite group theory.
For example, minimal normal subgroups of finite solvable groups are elementary abelian, and so is the quotient of a finite [[p-group|06QYRTX9KQ]] by its Frattini subgroup.

## Explanations

### L1 human

Take $n$ switches, each of which can be on or off, and combine two settings by flipping the switches that are on in the second one.
Doing the same move twice brings you back to the start, so every element has order $2$.
This is the elementary abelian group $(\Z/2\Z)^n$ with $2^n$ elements.
For $n = 2$ it is the Klein four-group.
Replacing on/off by the $p$ residues modulo a prime $p$ gives $(\Z/p\Z)^n$.
</document>

<document source="https://tamnd.github.io/math2/node/DZW58R6T2Q.md" tag="DZW58R6T2Q" kind="concept">
# Picard group of a ring

## Summary

The [[Picard group|3Z9DCWYXQZ]] $\operatorname{Pic}(R)$ of a [[commutative ring|0310REJJY8]] $R$ collects the isomorphism classes of rank one [[projective modules|BFFA03GE3V]] over $R$.
Its group law is the [[tensor product|F0HQXQ0HCD]].
It measures how far such modules are from being free.
For an [[integral domain|0QVD7WBFGN]] it is the group of [[invertible ideals|E190TAV2S6]] modulo principal ones, and for a [[Dedekind domain|7NPY2CN5KB]] it is the [[ideal class group|WQ46BD3ARY]].
It is the Picard group of the affine scheme $\operatorname{Spec} R$.

## Explanations

### L1 human

A rank one projective module looks like a copy of the ring near every point, but it may be twisted globally.
The Picard group collects all these twisted copies, and tensoring two of them gives a third one.
The ring itself is the untwisted copy and plays the role of the identity.
Over $\Z[\sqrt{-5}]$, the ideal $(2, 1 + \sqrt{-5})$ is such a twisted copy, and the Picard group has two elements.
</document>

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# Coordinate ring

## Definition

Let $k$ be a field, let $n \ge 0$, and let $X \subseteq k^n$ be an [[algebraic set|VHFMQQYSTN]] with [[ideal|HH6VEVM6Z0]] $I(X)$.
The *coordinate ring* of $X$ is the [[quotient ring|0076HWHGHQ]]
$$k[X] = k[x_1, \dots, x_n] / I(X)$$
of the [[polynomial ring|0SDC934R52]].
It is also written $A(X)$ or $\Gamma(X)$.

## Remarks

Two polynomials have the same class in $k[X]$ if and only if they define the same function on $X$.
So $k[X]$ is the ring of [[polynomial functions|1G0B73WPX8]] $X \to k$.
It is a finitely generated $k$-algebra, generated by the classes of $x_1, \dots, x_n$.
It is a [[reduced ring|SZHY975FFX]], since $I(X)$ is a radical ideal.

Example: for the parabola $X = V(y - x^2)$ over an infinite field, $k[X] = k[x, y]/(y - x^2) \cong k[x]$, through $x \mapsto x$ and $y \mapsto x^2$.

Non-example: the ring $k[x]/(x^2)$ is not the coordinate ring of any algebraic set, because it has the nonzero nilpotent element $x$.

$X$ is irreducible if and only if $k[X]$ is an [[integral domain|0QVD7WBFGN]]; when $k$ is algebraically closed, this says that $X$ is an [[affine variety|JERVQT4C12]].
If $k$ is algebraically closed, [[Hilbert's Nullstellensatz|D85VZWS9QK]] shows that the points of $X$ correspond to the [[maximal ideals|0PV74TR5FP]] of $k[X]$.
The point $a$ corresponds to the ideal of functions that vanish at $a$.
</document>

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# Fenichel's theorem

## Statement

Let $r \ge 1$ be an integer, and let $f \colon \R^n \times \R^m \times \R \to \R^n$ and $g \colon \R^n \times \R^m \times \R \to \R^m$ be of class $C^{r+1}$.
Consider the [[singularly perturbed|G6J21TGRMM]] slow-fast system of [[ordinary differential equations|0NT64JV96A]]
$$\dot x = f(x, y, \varepsilon), \quad \varepsilon \dot y = g(x, y, \varepsilon).$$
Let $K \subseteq \R^n$ be a compact set that is the closure of an open set with smooth boundary.
Let $h_0 \colon K \to \R^m$ be of class $C^{r+1}$ with $g(x, h_0(x), 0) = 0$ for all $x \in K$.
So $M_0 = \set{(x, h_0(x)) : x \in K}$ is part of the *critical manifold* $\set{g = 0}$.
Assume that $M_0$ is *normally hyperbolic*: for each $x \in K$, no [[eigenvalue|0BQAA2BK3Q]] of $\partial_y g(x, h_0(x), 0)$ has real part $0$.
Then there is $\varepsilon_1 > 0$ such that for $0 < \varepsilon \le \varepsilon_1$ there is a $C^r$ function $h_\varepsilon \colon K \to \R^m$ with these properties.

1. The *slow manifold* $M_\varepsilon = \set{(x, h_\varepsilon(x)) : x \in K}$ is locally [[invariant|WWZTW11664]].
   A solution that starts on $M_\varepsilon$ stays on it as long as $x$ remains in the interior of $K$.
2. $h_\varepsilon$ depends in a $C^r$ way on $\varepsilon$, and $h_\varepsilon = h_0 + O(\varepsilon)$ uniformly on $K$.
3. Suppose that along $M_0$, the matrix $\partial_y g$ has $k$ eigenvalues with negative and $m - k$ with positive real part.
   Then $M_\varepsilon$ has local stable and unstable manifolds of dimensions $n + k$ and $n + m - k$.
   They are $O(\varepsilon)$-close to the stable and unstable manifolds of $M_0$ for the fast system at $\varepsilon = 0$.

## Remarks

At $\varepsilon = 0$, every point of $M_0$ is an equilibrium of the fast system $y' = g(x, y, 0)$, where the prime is the derivative in $\tau = t / \varepsilon$.
Normal hyperbolicity means that the contraction or expansion transverse to $M_0$ dominates the dynamics along $M_0$.
The proof applies the persistence theory of normally hyperbolic invariant manifolds, developed by Fenichel, to the system in the fast time with $\varepsilon$ added as a variable.

Example: for $\dot x = -x$, $\varepsilon \dot y = -y + x^2$, the critical manifold is $y = x^2$, normally attracting since $\partial_y g = -1$.
One checks that $y = h_\varepsilon(x) = x^2 / (1 - 2\varepsilon)$ is invariant for $\varepsilon < 1/2$: along it $\varepsilon \dot y = -2\varepsilon x^2 / (1 - 2\varepsilon) = -y + x^2$.
So $h_\varepsilon = x^2 + O(\varepsilon)$, as the theorem predicts.

Normal hyperbolicity is needed: at fold points, where an eigenvalue of $\partial_y g$ crosses $0$, the slow manifold can fail to continue, as in relaxation oscillations and canard phenomena.
The slow manifold is in general not unique, but any two choices are exponentially close in $1/\varepsilon$.

## Explanations

### L1 human

When a system has fast and slow parts, the fast part quickly settles near a surface of rest points.
Fenichel's theorem says that, for a small but positive speed ratio, there is a true surface nearby that the motion follows exactly.
On that surface, the system can be studied with fewer variables.

### L2 human

The theorem is the foundation of geometric singular perturbation theory.
It complements [[Tikhonov's theorem|8X8Y61MGHW]], which gives convergence on finite time intervals.
On the slow manifold, the flow is a regular perturbation of the reduced flow $\dot x = f(x, h_0(x), 0)$.
Combined with the stable and unstable fibrations, it is used to construct travelling waves and periodic orbits of relaxation type.
</document>

<document source="https://tamnd.github.io/math2/node/E09VVB7BJQ.md" tag="E09VVB7BJQ" kind="definition">
# Hall subgroup

## Definition

Let $G$ be a finite [[group|G5S1M9]].
A *Hall subgroup* of $G$ is a [[subgroup|0X1JXT47GA]] $H$ whose [[order|0B85S2V2DR]] and [[index|0DTXBQPA2S]] are [[coprime|0VZ48N083F]]: $\gcd(\abs{H}, [G : H]) = 1$.

Let $\pi$ be a set of [[prime numbers|01826KMG2H]].
A *$\pi$-number* is a positive integer whose prime factors all lie in $\pi$.
A *Hall $\pi$-subgroup* of $G$ is a subgroup $H$ such that $\abs{H}$ is a $\pi$-number and $[G : H]$ has no prime factor in $\pi$.

## Remarks

A Hall subgroup $H$ is a Hall $\pi$-subgroup for $\pi$ the set of primes dividing $\abs{H}$.
For $\pi = \set{p}$, the Hall $\pi$-subgroups are the [[Sylow $p$-subgroups|0053CQE9Y2]].

Example: $A_4 \le A_5$.
Here $\abs{A_4} = 12$ and $[A_5 : A_4] = 5$, and $\gcd(12, 5) = 1$, so $A_4$ is a Hall $\set{2, 3}$-subgroup of $A_5$.

Non-example: the subgroup $\set{0, 2}$ of $\Z/4\Z$.
Its order is $2$ and its index is $2$, and $\gcd(2, 2) = 2 \ne 1$.

Hall $\pi$-subgroups need not exist: $A_5$ has no subgroup of order $15$ or $20$, so it has no Hall $\set{3, 5}$-subgroup and no Hall $\set{2, 5}$-subgroup.
In solvable groups they always exist, by [[Hall's theorem|8Z8PE69SJS]].
A normal Hall subgroup always has a complement, by the [[Schur–Zassenhaus theorem|64K1A69C80]].
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# Dowker space

## Definition

Let $X$ be a topological space.
$X$ is *countably paracompact* if every countable open cover of $X$ has an open refinement that is a [[locally finite family|DZ297MMGV8]].
$X$ is a *Dowker space* if:

1. $X$ is [[Hausdorff|0S1FJ8VEX9]],
2. $X$ is [[normal|0NSR9V8TED]], and
3. $X$ is not countably paracompact.

## Remarks

By a theorem of Dowker, a normal space $X$ is countably paracompact if and only if the product $X \times [0, 1]$ is normal.
So a Dowker space is the same as a normal Hausdorff space $X$ with $X \times [0, 1]$ not normal.
Another equivalent condition for a normal space to be countably paracompact: for every decreasing sequence $F_1 \supseteq F_2 \supseteq \cdots$ of closed sets with empty intersection there are open sets $U_n \supseteq F_n$ with empty intersection.
An example is the Dowker space constructed by Rudin in ZFC; the check of its properties is long.
A non-example is the real line $\R$: it is metrizable, hence [[paracompact|8R29G6NG9Q]], hence countably paracompact, so clause 3 fails.
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<document source="https://tamnd.github.io/math2/node/E0NXWECXTB.md" tag="E0NXWECXTB" kind="definition">
# Chern–Simons theory

## Definition

Let $G = SU(n)$, a compact [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{su}(n)$, and let $M$ be a closed oriented [[3-manifold|R20TB2KEQQ]].
Every [[principal $G$-bundle|8E37E74XVP]] over $M$ is trivial, so a [[connection|NJ21PN5ZGJ]] on it is given by a $\mathfrak{su}(n)$-valued [[differential $1$-form|0E3G0GRPZ6]] $A$ on $M$.
The *Chern–Simons functional* is
$$CS(A) = \frac{1}{8\pi^2} \int_M \tr\left(A \wedge dA + \tfrac{2}{3} A \wedge A \wedge A\right).$$
For an integer $k$, called the level, *Chern–Simons theory* is the gauge theory on $M$ with gauge group $G$ and action $S_k(A) = 2\pi k \, CS(A)$.
Its quantum partition function is the formal path integral
$$Z_k(M) = \int e^{i S_k(A)} \, \mathcal{D}A$$
over connections modulo gauge transformations.

## Remarks

Example: the trivial connection $A = 0$ has $CS(0) = 0$.
The first variation of $CS$ is $\delta CS = \frac{1}{4\pi^2} \int_M \tr(\delta A \wedge F_A)$, where $F_A = dA + A \wedge A$ is the curvature, so the critical points of $CS$ are exactly the flat connections.

$CS$ is not invariant under gauge transformations: a gauge transformation $g \colon M \to G$ changes $CS(A)$ by an integer, the winding number of $g$ up to sign.
So $CS$ is not a function on connections modulo gauge, but $e^{i S_k}$ is, because $k$ is an integer; this is why the level must be an integer.

The path integral is not a rigorous mathematical definition.
Its predicted values are realised rigorously by the Witten–Reshetikhin–Turaev invariants.
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# Ternary operation

## Definition

Let $A$ be a set and let $A^3$ be the [[Cartesian product|09CEDJYBKP]] $A \times A \times A$.
A *ternary operation* on $A$ is a [[finitary operation|EJ8YJVM9BW]] of arity $3$, that is, a [[function|0EPAT51DXQ]] $f \colon A^3 \to A$.

## Remarks

Example: on $\R$, the map $m(x, y, z)$ that returns the middle value of $x, y, z$ is a ternary operation.
It is defined for every triple and its value is one of $x, y, z$, so it lies in $\R$.
It satisfies $m(x, x, y) = m(x, y, x) = m(y, x, x) = x$.

Non-example: the map $(x, y, z) \mapsto x / (y - z)$ is not a ternary operation on $\R$, because it is not defined when $y = z$.

A ternary operation is usually written in prefix form $f(x, y, z)$, since there is no standard infix notation for three arguments.
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# Del Pezzo surface

## Definition

Let $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field, with [[canonical divisor|W05VYY4HBR]] $K_X$.
$X$ is a *del Pezzo surface* if the anticanonical divisor $-K_X$ is [[ample|RRK8P6TXG2]].
Its *degree* is the self-intersection number $d = K_X^2$.

## Remarks

The degree satisfies $1 \le d \le 9$.
Classification: $X$ is either $\mathbb{P}^1 \times \mathbb{P}^1$, of degree $8$, or the [[blow-up|C4KGYAFPN7]] of $\mathbb{P}^2$ at $9 - d$ points in general position.
Here general position means three conditions.
No three of the points lie on a line, and no six lie on a conic.
No eight lie on a cubic that is singular at one of them.
Example: $\mathbb{P}^2$ has $K = -3L$ for a line $L$, so $-K = 3L$ is ample and $\mathbb{P}^2$ is a del Pezzo surface of degree $9$.
Example: a smooth cubic surface $X \subset \mathbb{P}^3$ has $K_X = -H$ by adjunction, where $H$ is a plane section, so $-K_X$ is ample and $d = H^2 = 3$.
Non-example: blow up $\mathbb{P}^2$ at three points on a line $L$, and let $\tilde L$ be the strict transform of $L$.
Then $\tilde L^2 = 1 - 3 = -2$, and adjunction gives $K \cdot \tilde L = -2 - \tilde L^2 = 0$.
So $-K \cdot \tilde L = 0$, and $-K$ is not ample.
Some authors allow mild singularities, such as ADE singularities, and call the smooth ones nonsingular del Pezzo surfaces.
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# Long exact sequence of homotopy groups

## Statement

Let $p \colon E \to B$ be a [[Serre fibration|PB9JXQ2KC3]], and let $B$ be [[path-connected|0WB6DH499T]].
Let $b_0 \in B$, let $F = p^{-1}(b_0)$ be the fibre, and let $x_0 \in F$.
Then there is an [[exact sequence|JB7ZBFBA9C]]
$$\cdots \to \pi_n(F, x_0) \xrightarrow{i_*} \pi_n(E, x_0) \xrightarrow{p_*} \pi_n(B, b_0) \xrightarrow{\partial} \pi_{n-1}(F, x_0) \to \cdots \to \pi_0(F, x_0) \to \pi_0(E, x_0) \to 0.$$
Here $\pi_n$ are the [[homotopy groups|P976WYEFMB]], $i \colon F \to E$ is the inclusion, and $\partial$ is a homomorphism for $n \ge 2$.
At the terms $\pi_0$ and at $\pi_1(B, b_0)$, exactness is meant as pointed sets: the kernel of a map is the preimage of the base point.

## Remarks

The key step is that $p_* \colon \pi_n(E, F, x_0) \to \pi_n(B, b_0)$ is an isomorphism for $n \ge 1$, proved by lifting maps of disks with the homotopy lifting property.
The sequence then comes from the long exact sequence of the pair $(E, F)$ in [[relative homotopy groups|ZHHMDZWA7F]].
The map $\partial$ lifts a based map of a disk into $B$ to $E$ and restricts it to the boundary sphere, which lands in $F$.
For the Hopf fibration $S^1 \to S^3 \to S^2$, the sequence gives $\pi_2(S^2) \cong \pi_1(S^1) \cong \Z$ and $\pi_n(S^3) \cong \pi_n(S^2)$ for $n \ge 3$.
For the covering $\R \to S^1$ the fibre $\Z$ is discrete, so $\pi_n(S^1) \cong \pi_n(\R) = 0$ for $n \ge 2$.

## Explanations

### L1 human

A fibration is a space $E$ built over a base $B$ with copies of a fibre $F$ over each point.
The sequence says that the homotopy groups of the three spaces are tied together tightly.
If two of them are known, the third is often determined.

### L2 human

This is the most used tool for computing homotopy groups.
It gives $\pi_3(S^2) \cong \Z$ from the Hopf fibration, and computes homotopy groups of Lie groups and homogeneous spaces from fibrations such as $SO(n) \to SO(n+1) \to S^n$.
For the path space fibration $\Omega B \to PB \to B$ with contractible total space, it gives $\pi_n(\Omega B) \cong \pi_{n+1}(B)$.
The analogous sequence of a pair $A \subseteq X$ has the form $\cdots \to \pi_n(A) \to \pi_n(X) \to \pi_n(X, A) \to \pi_{n-1}(A) \to \cdots$.
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# Arithmetic–geometric mean and elliptic integrals

## Statement

Let $a, b > 0$.
Define sequences by $a_0 = a$, $b_0 = b$ and
$$a_{n+1} = \frac{a_n + b_n}{2}, \qquad b_{n+1} = \sqrt{a_n b_n}.$$
Then:

1. Both sequences [[converge|05M5SWSZZD]] to the same limit $M(a, b) > 0$, the *arithmetic–geometric mean* of $a$ and $b$.
2. $$\int_0^{\pi/2} \frac{d\theta}{\sqrt{a^2\cos^2\theta + b^2\sin^2\theta}} = \frac{\pi}{2 M(a, b)}.$$
3. For $0 \le k < 1$ and $k' = \sqrt{1 - k^2}$, the [[complete elliptic integral of the first kind|RM54CPPE12]] satisfies $K(k) = \dfrac{\pi}{2 M(1, k')}$.

## Remarks

Proof idea: by the [[inequality of arithmetic and geometric means|0DRF0F3XC9]], $b_n \le b_{n+1} \le a_{n+1} \le a_n$ for $n \ge 1$, so both sequences are monotone and bounded.
Since $b_{n+1} \ge b_n$, one gets $a_{n+1} - b_{n+1} \le a_{n+1} - b_n = (a_n - b_n)/2$, so the difference tends to $0$ and the two limits agree.
For part 2, call the integral $I(a, b)$; a change of variables found by Gauss shows that $I(a, b) = I(a_1, b_1)$.
Hence $I(a, b) = I(a_n, b_n)$ for all $n$, and letting $n \to \infty$ gives $I(M, M) = \pi / (2M)$.
Part 3 is part 2 with $a = 1$ and $b = k'$, since $\cos^2\theta + k'^2\sin^2\theta = 1 - k^2\sin^2\theta$.

Example: $M(1, \sqrt2) \approx 1.19814$, and $1/M(1, \sqrt 2)$ is Gauss's constant, equal to $\varpi/\pi$ where $\varpi$ is the [[lemniscate constant|V5A2S8QYQQ]].

## Explanations

### L1 human

Start with two positive numbers, and replace them by their average and their geometric mean, again and again.
The two numbers squeeze together very fast, and their common limit is the arithmetic–geometric mean.
The surprise is that this simple process computes an elliptic integral, which has no elementary formula.

### L2 human

The convergence is quadratic: once $a_n$ and $b_n$ are close, the number of correct digits roughly doubles at each step, because $a_{n+1} - b_{n+1} = (\sqrt{a_n} - \sqrt{b_n})^2/2$.
This makes the formula $K(k) = \pi/(2M(1, k'))$ the standard way to evaluate $K$ numerically.
Its invariance step $I(a, b) = I(a_1, b_1)$ is a form of [[Landen's transformation|A0V7ZAEQBW]].
Combined with [[Legendre's relation|8GSKDNZ4TK]] it gives the Gauss–Legendre algorithm for $\pi$.
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<document source="https://tamnd.github.io/math2/node/E0VZTNPJXA.md" tag="E0VZTNPJXA" kind="concept">
# Discriminant of a number field

## Summary

The discriminant of an [[algebraic number field|MWBDFS2KJ5]] $K$ is a nonzero integer $d_K$ that measures the size of the ring of integers of $K$.
View the ring of integers as a lattice in a real vector space.
Up to sign and a power of $2$, $d_K$ is the square of the volume of a fundamental domain of this lattice.
A prime number ramifies in $K$ exactly when it divides $d_K$.
The discriminant appears in the [[Minkowski bound|1TQ2F9CFVE]], the [[class number formula|55XX5A7XZV]] and the [[Dedekind zeta function|KXS5DQDFYD]].

## Explanations

### L1 human

Take a basis of the ring of integers of $K$ and place it inside the complex numbers in all possible ways.
The discriminant is the squared determinant of the square table of numbers that this gives.
For the Gaussian integers $\Z[i]$, with basis $1, i$, the table is $\begin{pmatrix} 1 & i \\ 1 & -i \end{pmatrix}$ and the discriminant is $(-2i)^2 = -4$.
The only prime dividing $-4$ is $2$, and $2 = -i(1+i)^2$ is the only prime that becomes a square in $\Z[i]$, up to a unit.
A large discriminant means that the field is arithmetically complicated.
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<document source="https://tamnd.github.io/math2/node/E0WKQ3XBAC.md" tag="E0WKQ3XBAC" kind="concept">
# Canonical correlation

## Summary

Canonical correlation analysis studies the linear relations between two groups of variables.
It finds a linear combination of each group such that the two combinations have the largest possible [[correlation|0DNN9B6Q2F]].
Further pairs of combinations are found under the condition that they are uncorrelated with the earlier ones.
The resulting correlations are the canonical correlations.

## Explanations

### L1 human

Suppose that for each student there are several test scores and several measures of study habits.
Canonical correlation asks for the mix of scores and the mix of habits that move together most closely.
The correlation between these two best mixes is the first canonical correlation.
With only one variable in each group, it is just the absolute value of the ordinary correlation.
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# Monstrous moonshine

## Statement

Let $\mathbb{M}$ be the [[Monster group|C9VY20TAPT]].
Let $J(\tau) = j(\tau) - 744 = q^{-1} + \sum_{n \ge 1} c(n) q^n$ with $q = e^{2\pi i \tau}$, where $j$ is the [[j-invariant|FEC2XH822R]].
There is a graded complex [[representation|WYGRXCG2RN]] $V^\natural = \bigoplus_{n \ge -1} V_n$ of $\mathbb{M}$, with each $V_n$ finite-dimensional, such that:

1. $\dim V_{-1} = 1$, $V_0 = 0$, and $\dim V_n = c(n)$ for all $n \ge 1$.
2. For each $g \in \mathbb{M}$, the McKay–Thompson series $T_g(\tau) = \sum_{n \ge -1} \tr(g \mid V_n)\, q^n$ is the normalised Hauptmodul of a genus zero subgroup $\Gamma_g$ of $\mathrm{SL}_2(\R)$.
That is, $T_g$ is invariant under $\Gamma_g$ and $T_g = q^{-1} + O(q)$.
Also, $T_g$ induces an isomorphism from the compactification of $\Gamma_g \backslash \mathbb{H}$ onto the Riemann sphere.
3. Each $\Gamma_g$ contains the [[congruence subgroup|1H6R856P3C]] $\Gamma_0(N)$ for some $N \ge 1$.

## Remarks

The representation $V^\natural$ is the moonshine module, constructed by Frenkel, Lepowsky and Meurman as a vertex operator algebra with an action of $\mathbb{M}$.
The statement was conjectured by Conway and Norton, after McKay observed that $196884 = 1 + 196883$, where $196883$ is the smallest dimension of a nontrivial irreducible representation of $\mathbb{M}$.
Borcherds proved it in 1992.
He constructed the monster Lie algebra from $V^\natural$.
Its denominator identity shows that the series $T_g$ satisfy the replication formulas, which determine them from finitely many coefficients.
These coefficients were then checked against the known Hauptmoduln.
For $g = 1$, the series $T_1$ is $J$ itself, the Hauptmodul of $\mathrm{SL}_2(\Z)$.

## Explanations

### L1 human

The Monster is the largest sporadic finite simple group, an object from pure algebra.
The $j$-function is a classical function from complex analysis and number theory.
Moonshine says that the coefficients of $j$ are dimensions of natural pieces of one representation of the Monster.
Even more, twisting by each element of the Monster gives another function of the same special kind.

### L2 human

The coefficient identity $196884 = 1 + 196883$ is the first visible trace of the theorem, at $n = 1$.
The proof goes through vertex operator algebras and generalized Kac–Moody algebras, which link finite groups, modular functions and string theory.
Analogous phenomena, such as umbral moonshine, were found later for other groups.
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<document source="https://tamnd.github.io/math2/node/E0YVMBPK7A.md" tag="E0YVMBPK7A" kind="example">
# Euler–Lotka equation

## Example

Consider an age-structured population, and let $a \ge 0$ denote age.
Let $l(a)$ be the probability of surviving to age $a$, and $m(a)$ the birth rate of individuals of age $a$.
Assume $l m$ is nonnegative, integrable on $[0, \infty)$ and not almost everywhere zero.
The birth rate $B(t)$ at time $t \ge 0$ satisfies a [[renewal equation|EWVVH3EKMR]]
$$B(t) = G(t) + \int_0^t B(t - a) \, l(a) \, m(a) \, da,$$
where $G(t)$ collects the births from individuals already alive at time $0$.
This is a [[Volterra equation of convolution type|SNFYTC1TTA]] with kernel $k = l m$.

Looking for exponential solutions $B(t) = Q e^{rt}$ of the equation without the term $G$, with the integral taken over all ages $a \ge 0$, gives the *Euler–Lotka equation*
$$\int_0^\infty e^{-r a} \, l(a) \, m(a) \, da = 1.$$
Its left side is the [[Laplace transform|0V49JEZW06]] $\hat k(r)$ of the kernel, so $r$ is a root of $1 - \hat k(r) = 0$, the denominator of the Laplace transform method.

Check on an instance: let $l(a) m(a) = b \, e^{-\mu a}$ with constants $b > 0$ and $\mu \ge 0$.
For $r > -\mu$ the left side is $\int_0^\infty b \, e^{-(r + \mu) a} \, da = \frac{b}{r + \mu}$.
It equals $1$ exactly when $r = b - \mu$, the birth rate minus the death rate.

## Remarks

The left side is strictly decreasing in real $r$, where it converges, so the Euler–Lotka equation has at most one real root.
This real root is the intrinsic rate of increase of the population.
The net reproduction number is $R_0 = \int_0^\infty l(a) m(a) \, da$, the value of the left side at $r = 0$.
If $R_0 > 1$, the left side tends to $0$ as $r \to \infty$, so there is a real root and it is positive.
If $R_0 \le 1$ and a real root exists, it is not positive.

## Explanations

### L1 human

Each newborn will, over its life, produce offspring at various ages.
If the population grows like $e^{rt}$, then births long ago count less, by the factor $e^{-ra}$, compared with births today.
The Euler–Lotka equation says that, with this discount, each newborn exactly replaces itself.
Solving it gives the long-run growth rate $r$ of the population.
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# Complexification of a Lie group

## Definition

Let $G$ be a real [[Lie group|7JBRGCVJRB]].
A *complexification* (or universal complexification) of $G$ is a pair $(G_\C, \gamma)$ with two properties.

1. $G_\C$ is a complex Lie group and $\gamma \colon G \to G_\C$ is a continuous homomorphism.
2. For each complex Lie group $H$ and each continuous homomorphism $\varphi \colon G \to H$, there is exactly one holomorphic homomorphism $\psi \colon G_\C \to H$ with $\varphi = \psi \circ \gamma$.

## Remarks

By the universal property, a complexification is unique up to a unique isomorphism compatible with $\gamma$; it always exists.

Example: $G = \R$ with $G_\C = \C$ and $\gamma$ the inclusion.
A continuous homomorphism $\varphi \colon \R \to H$ is $\varphi(t) = \exp(tX)$ for some $X$ in the Lie algebra of $H$.
Then $\psi(z) = \exp(zX)$ is a holomorphic homomorphism extending $\varphi$.
It is the only one, since two holomorphic maps on $\C$ that agree on $\R$ are equal.

Non-example: $G = \R$ with $G_\C = \C^\times$ and $\gamma(t) = e^t$ fails clause 2.
Take $H = \C$ and $\varphi(t) = t$.
For a holomorphic homomorphism $\psi \colon \C^\times \to \C$, the map $z \mapsto \psi(e^z)$ is a holomorphic homomorphism $\C \to \C$, so $\psi(e^z) = cz$ for a constant $c$, and $c \cdot 2\pi i = \psi(1) = 0$ gives $\psi = 0$, so $\psi \circ \gamma \ne \varphi$.

The map $\gamma$ need not be injective.
For the universal cover $\widetilde{\mathrm{SL}}_2(\R)$, the complexification is $\mathrm{SL}_2(\C)$, and $\gamma$ is the covering map onto $\mathrm{SL}_2(\R)$, with infinite kernel.
For a compact Lie group $G$, the map $\gamma$ is injective, and $G$ is a maximal compact subgroup of $G_\C$.
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<document source="https://tamnd.github.io/math2/node/E14W2V09HM.md" tag="E14W2V09HM" kind="concept">
# Residue calculus for Laplace inversion

## Summary

Residue calculus for Laplace inversion evaluates the [[Bromwich integral|5KNAXNHRY6]] with the [[residue theorem|032HKYS329]].
One closes the vertical line of integration with a large arc to the left.
When $F$ decays on such arcs, the [[inverse Laplace transform|07WBTX1MHA]] is the sum of the [[residues|0PHN07KVQT]] of $e^{s t} F(s)$ at the poles of $F$.
It turns inversion into a finite or infinite sum of explicit terms, one for each pole.

## Explanations

### L1 human

Each pole of the transform $F$ produces one piece of the original function of time.
A simple pole at $s = a$ produces a multiple of $e^{a t}$; a double pole produces terms like $t e^{a t}$.
For $F(s) = 1/(s^2 + 1)$, the poles $\pm i$ give $\frac{e^{i t}}{2 i} - \frac{e^{-i t}}{2 i} = \sin t$.
The method needs $F$ to be small on large arcs in the left half-plane; this fails for factors such as $e^{-s}$, which encode a time delay.
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# Admissible representation

## Definition

Let $G$ be a [[real reductive Lie group|42ZCE14AD2]] and let $K$ be a [[maximal compact subgroup|T2EB8R9HEV]] of $G$.
Let $\pi$ be a [[representation|WYGRXCG2RN]] of $G$ on a [[Hilbert space|HM2HHE3J51]] $V$ such that the map $G \times V \to V$, $(g, v) \mapsto \pi(g) v$, is continuous.
The representation $\pi$ is *admissible* if:

1. the restriction $\pi|_K$ is unitary;
2. for each [[irreducible representation|FX3NEFYFFN]] $\tau$ of $K$, the multiplicity of $\tau$ in $\pi|_K$ is finite.

## Remarks

The multiplicity of $\tau$ is the dimension of $\Hom_K(V_\tau, V)$, the space of continuous $K$-equivariant linear maps from the space of $\tau$ to $V$.

Example: let $G = \mathrm{SL}_2(\R)$ and $K = \mathrm{SO}(2)$, and let $G$ act on functions on $\R^2 \setminus \set{0}$ that satisfy $f(tx) = \abs{t}^{-1-\nu} f(x)$ for $t \ne 0$, with $\nu \in i\R$, by $(\pi(g) f)(x) = f(g^{-1} x)$.
Such an $f$ is determined by its values on the unit circle, and the norm is the $L^2$ norm there.
The characters of $K$ are $e^{i\theta} \mapsto e^{in\theta}$ with $n \in \Z$.
Each one with $n$ even occurs exactly once, spanned by the function $e^{in\theta}$ on the circle, and the odd ones do not occur, so $\pi$ is admissible.

Non-example: the right regular representation of $G = \mathrm{SL}_2(\R)$ on $L^2(G)$ is not admissible.
Its vectors fixed by $K$ are the functions invariant under right translation by $K$, which form the infinite-dimensional space $L^2(G/K)$.
So clause 2 fails for the trivial representation of $K$.

Over a non-archimedean local field $F$, a representation of $G(F)$ on a complex vector space $V$ is admissible under two conditions.
Every vector is fixed by some compact open subgroup, and the space $V^{K'}$ is finite-dimensional for each compact open subgroup $K'$.
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# Heat potential

## Definition

Let $n \ge 1$, and let $\Phi \colon \R^n \times \R \to [0, \infty)$ be the fundamental solution of the [[heat equation|VP2HN86VAG]] $\partial_t u = \Delta u$:
$$\Phi(x, t) = (4\pi t)^{-n/2} \exp\!\left(-\frac{\abs{x}^2}{4t}\right) \text{ for } t > 0, \qquad \Phi(x, t) = 0 \text{ for } t \le 0.$$
Let $\mu$ be a positive [[Radon measure|HZNEQQBCW4]] on $\R^n \times \R$.
The *heat potential* of $\mu$ is the function $\Phi\mu \colon \R^n \times \R \to [0, \infty]$ given by
$$\Phi\mu(x, t) = \int_{\R^n \times \R} \Phi(x - y, t - s) \, d\mu(y, s).$$
When $d\mu = f \, dy \, ds$ for a function $f \ge 0$, the heat potential $\Phi\mu$ is also called the volume potential of $f$.

## Remarks

Only the part of $\mu$ with $s < t$ contributes to $\Phi\mu(x, t)$, since $\Phi$ vanishes for nonpositive times.
Example: for the unit point mass $\mu = \delta_{(0, 0)}$, the heat potential is $\Phi$ itself, the [[heat kernel|KD404MF042]] started at time $0$; it solves the heat equation away from $(0, 0)$.
Degenerate case: for Lebesgue measure on $\R^n \times \R$, $\Phi\mu(x, t) = \int_{-\infty}^{t} \int_{\R^n} \Phi(x - y, t - s) \, dy \, ds = \int_{-\infty}^t 1 \, ds = \infty$ at every point, so finiteness of a heat potential needs growth conditions on $\mu$.
For a bounded function $f$ on $\R^n \times (0, T)$ that is locally Hölder continuous, the function $u(x, t) = \int_0^t \int_{\R^n} \Phi(x - y, t - s) f(y, s) \, dy \, ds$ solves $\partial_t u - \Delta u = f$ with $u(\cdot, 0) = 0$.
A measure carried by the lateral boundary of a space-time cylinder gives a single layer heat potential.
Using a normal derivative of $\Phi$ instead gives a double layer heat potential.
Both are used to solve boundary value problems for the heat equation.
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# Invertible ideal

## Summary

An invertible ideal of an [[integral domain|0QVD7WBFGN]] $R$ is a fractional ideal $I$ that has an inverse $J$ with $IJ = R$.
Fractional ideals are $R$-submodules of the [[field of fractions|0F2PTGX1A2]] that become ordinary [[ideals|096Z9MP2ME]] after clearing one denominator.
The invertible fractional ideals form a group under multiplication, and its quotient by the principal ones is the [[Picard group|DZW58R6T2Q]] of $R$.
A [[Dedekind domain|7NPY2CN5KB]] is exactly a domain in which every nonzero fractional ideal is invertible.

## Explanations

### L1 human

Ideals can be multiplied, but in general they cannot be divided.
An invertible ideal is one that can be divided by: some fractional ideal multiplies it back to the whole ring.
Every nonzero principal ideal $(a)$ is invertible, with inverse $(1/a)$.
In $\Z[\sqrt{-5}]$, the ideal $P = (2, 1 + \sqrt{-5})$ is not principal, but $P^2 = (2)$, so $\frac{1}{2} P$ is an inverse of $P$.
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<document source="https://tamnd.github.io/math2/node/E192Q8BQWJ.md" tag="E192Q8BQWJ" kind="example">
# Fold catastrophe

## Example

The *fold catastrophe* is the one-parameter family of functions
$$V_a(x) = x^3 + a x, \qquad x, a \in \R.$$
Its [[critical points|0YTCMFB4HC]] solve $V_a'(x) = 3x^2 + a = 0$.

1. For $a < 0$ there are two critical points $x = \pm \sqrt{-a/3}$; since $V_a''(x) = 6x$, the point $-\sqrt{-a/3}$ is a local maximum and $\sqrt{-a/3}$ is a local minimum.
2. For $a = 0$ there is one critical point, $x = 0$, and it is degenerate, since $V_0''(0) = 0$.
3. For $a > 0$ there are no critical points.

So as $a$ increases through $0$, the minimum and the maximum approach each other, merge at $x = 0$ and disappear.
The bifurcation set, the set of parameters where a degenerate critical point occurs, is the single point $a = 0$.
The family is the [[universal unfolding|KSAFWZVB8K]] of the germ $x^3$.
Indeed, the derivative of $V_a$ in $a$ is $x$, and $1$ and $x$ span $\mathcal{E}_1$ modulo the ideal $(x^2)$.

## Explanations

### L1 human

Picture a ball resting in a dip of a curve that has a dip next to a hump.
As the parameter $a$ grows, the dip and the hump become shallower and slide toward each other.
At $a = 0$ they merge into a flat inflection, and after that the dip is gone, so the ball suddenly rolls away.
This sudden loss of a resting state is the fold catastrophe, and in dynamics it appears as the [[saddle-node bifurcation|MAP5J6HK0T]].
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<document source="https://tamnd.github.io/math2/node/E1B3J3XB1T.md" tag="E1B3J3XB1T" kind="definition">
# Recurrent point

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
A point $x \in X$ is *recurrent* for $f$ if there is a strictly increasing sequence of positive integers $n_1 < n_2 < \cdots$ such that $f^{n_k}(x) \to x$ as $k \to \infty$.

## Remarks

Equivalent form: $x$ is recurrent if and only if $x \in \omega(x)$, where $\omega(x)$ is the $\omega$-[[limit set|YF4XH2BEG5]] of $x$, the set of limits of sequences $f^{n_k}(x)$ with $n_k \to \infty$.
The same condition $x \in \omega(x)$ defines recurrence on a general topological space.
Every point on a [[periodic orbit|0AA97XA70K]] is recurrent: if $f^p(x) = x$, take $n_k = kp$.
Every point of a [[minimal set|CRGZZKQ7Z9]] is recurrent.
Example: for $f(x) = x + \alpha \bmod 1$ on the circle with $\alpha$ irrational, Dirichlet's approximation theorem gives infinitely many $n$ with $n\alpha$ within $1/n$ of an integer, so $f^{n}(x) \to x$ along these $n$.
Non-example: for $f(x) = x/2$ on $\R$, the point $x = 1$ has $f^n(1) = 2^{-n} \to 0$, so no subsequence tends to $1$.

## Explanations

### L1 human

A recurrent point is a point that its own orbit approaches again and again.
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<document source="https://tamnd.github.io/math2/node/E1CA8QFEHB.md" tag="E1CA8QFEHB" kind="concept">
# Graded manifold

## Summary

A graded manifold is a smooth manifold whose sheaf of functions is enlarged by anticommuting odd coordinates.
Locally its functions are smooth functions of $n$ ordinary variables with values in a Grassmann algebra on $m$ generators.
It is the algebraic form, due to Kostant and to Berezin and Leites, of a [[supermanifold|Z0SB9HQBAJ]].
Graded manifolds give a geometric home to fermionic fields and to the odd variables of the BRST and BV formalisms.

## Explanations

### L1 human

On an ordinary manifold a function of two variables $x, y$ is a number at each point.
On a graded manifold one adds symbols $\theta_1, \dots, \theta_m$ that anticommute, so $\theta_i \theta_j = -\theta_j \theta_i$ and $\theta_i^2 = 0$.
A function is then a finite sum such as $f(x) + g(x) \theta_1 + h(x) \theta_1 \theta_2$ with smooth coefficients.
The odd symbols carry no points of their own: setting all of them to zero gives back the ordinary manifold.
For example, the sections of the exterior algebra bundle of the cotangent bundle, that is the differential forms on a manifold, form a graded manifold.
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<document source="https://tamnd.github.io/math2/node/E1DVHN06CR.md" tag="E1DVHN06CR" kind="concept">
# ℓ-adic cohomology

## Summary

Let $X$ be a variety over a separably closed field, and let $\ell$ be a prime different from the characteristic.
The $\ell$-adic cohomology of $X$ is the inverse limit of the [[étale cohomology|7YX76G0838]] groups of $X$ with coefficients $\Z/\ell^n\Z$.
It gives modules over the [[$\ell$-adic integers|GBN9F8FAQ3]], and after tensoring, vector spaces over $\Q_\ell$, a field of characteristic $0$.
These groups behave like the singular cohomology of a complex variety, with Poincaré duality, a Künneth formula and a Lefschetz trace formula.
They are the cohomology used in the proof of the [[Weil conjectures|81TXP32VDM]], and the Galois action on them gives [[Galois representations|JQEYEQJY2G]].

## Explanations

### L1 human

Étale cohomology with finite coefficients $\Z/n\Z$ works well, but one wants coefficients in a field of characteristic $0$, like $\Q$ in topology.
Coefficients $\Q$ directly give the wrong answer, so one takes the coefficients $\Z/\ell, \Z/\ell^2, \Z/\ell^3, \ldots$ and passes to the limit.
For an elliptic curve over an algebraically closed field, the first $\ell$-adic cohomology is a $2$-dimensional space over $\Q_\ell$.
It is dual to the space built from the [[Tate module|4CSMQFTGD8]], made of the points of order $\ell^n$ of the curve.
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# s-cobordism theorem

## Statement

Let $n \ge 5$.
Let $W$ be a compact connected smooth $(n+1)$-manifold whose boundary is a disjoint union $\partial W = M_0 \sqcup M_1$ of closed $n$-manifolds.
Assume that both inclusions $M_i \hookrightarrow W$ are homotopy equivalences.
Let $\pi = \pi_1(M_0)$ be the [[fundamental group|S1XK3HXHFX]], let $\operatorname{Wh}(\pi)$ be its Whitehead group, and let $\tau(W, M_0) \in \operatorname{Wh}(\pi)$ be the Whitehead torsion of the inclusion $M_0 \hookrightarrow W$.

1. $W$ is diffeomorphic to $M_0 \times [0, 1]$ relative to $M_0$ if and only if $\tau(W, M_0) = 0$.
2. For every closed connected $n$-manifold $M_0$ with fundamental group $\pi$ and every $\tau \in \operatorname{Wh}(\pi)$, there is such an h-cobordism $W$ with $\tau(W, M_0) = \tau$.
3. Two such h-cobordisms on $M_0$ are diffeomorphic relative to $M_0$ if and only if they have the same torsion.

## Remarks

The theorem is due independently to Barden, Mazur and Stallings.
The same statements hold for PL manifolds, and for topological manifolds by work of Kirby and Siebenmann.
Idea of the proof: as for the [[h-cobordism theorem|56YF8W9NXP]], take a handle decomposition and cancel handles.
Now the handle chain complex is a free based complex over $\Z[\pi]$, and the obstruction to cancelling all handles is exactly its torsion in $\operatorname{Wh}(\pi)$.

The theorem at work: $\operatorname{Wh}(1) = 0$, which gives back the h-cobordism theorem, and $\operatorname{Wh}(\Z^k) = 0$ by the computation for free abelian groups.
So every h-cobordism of dimension at least $6$ on a torus is a product.
For $\pi = \Z/5$ the Whitehead group is infinite cyclic, so a closed $5$-manifold with this fundamental group carries h-cobordisms that are not products.

## Explanations

### L1 human

The h-cobordism theorem needs the manifolds to have no nontrivial loops.
The s-cobordism theorem drops this hypothesis and says exactly what can go wrong.
There is one algebraic invariant, the Whitehead torsion, and the bigger manifold is a cylinder exactly when this invariant vanishes.

### L2 human

The theorem explains why surgery theory works with simple homotopy equivalences: an h-cobordism with zero torsion is an s-cobordism, and s-cobordant manifolds of dimension at least $5$ are diffeomorphic.
It is one input of the [[surgery exact sequence|Q7C480HQ9J]], where the structure set is formed up to this relation.
Whitehead torsion is computed from algebraic K-theory, since $\operatorname{Wh}(\pi)$ is a quotient of $K_1(\Z[\pi])$.
Classical examples come from lens spaces, whose distinction up to simple homotopy uses Reidemeister torsion.
</document>

<document source="https://tamnd.github.io/math2/node/E1NE8V7EX8.md" tag="E1NE8V7EX8" kind="conjecture">
# Kaplansky's zero divisor conjecture

## Statement

Let $K$ be a field and let $G$ be a [[torsion-free group|PEMNN32A43]].
Then the [[group ring|4NMTSX0XDS]] $K[G]$ has no [[zero divisors|0NNRDW10RZ]] other than $0$: if $a, b \in K[G]$ and $ab = 0$, then $a = 0$ or $b = 0$.
In other words, $K[G]$ is a [[domain|0Q4CBD3HX0]].

## Remarks

The conjecture is open.
Torsion-freeness is necessary: if $g \in G$ has finite order $n > 1$, then $(1 - g)(1 + g + \cdots + g^{n-1}) = 1 - g^n = 0$ with both factors nonzero.
It holds when $G$ has the unique product property: for nonempty finite sets $A, B \subseteq G$, some product $ab$ with $a \in A$, $b \in B$ arises in only one way.
For such $G$, take $A$ and $B$ to be the supports of $a$ and $b$.
The coefficient of that unique product in $ab$ is a product of two nonzero field elements, so $ab \ne 0$.
Left-orderable groups, and in particular torsion-free abelian groups and free groups, have this property.
For $G = \Z$, the group ring $K[\Z]$ is the Laurent polynomial ring $K[x, x^{-1}]$, which is a domain.
The conjecture is also known for torsion-free elementary amenable groups, whose group rings are Ore domains.
In characteristic zero it follows from the strong Atiyah conjecture for $G$.

## Explanations

### L1 human

In the ring of polynomials, a product of two nonzero polynomials is never zero, because the top degree terms multiply to a nonzero term.
The conjecture says the same happens for any group without elements of finite order.
For groups that can be ordered there is a "top term" and the polynomial argument works.
For general torsion-free groups no such argument is known.
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# Involute

## Summary

An involute of a curve is the path traced by the end of a taut string that is unwound from the curve.
At each point it meets the tangent line of the original curve at a right angle.
Involutes and [[evolutes|Y3ZNV24V2C]] are inverse operations: the evolute of an involute is the original curve.
The involute of a circle is the profile of the teeth of most gears.

## Explanations

### L1 human

Wrap a string tightly around a spool and hold its free end.
Pull the string away so it stays tight, and unwind it.
The free end draws a spiral-like curve: this is the involute of the circle.
Different starting lengths of string give different involutes, all parallel to each other.
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<document source="https://tamnd.github.io/math2/node/E1SQWQ2PAV.md" tag="E1SQWQ2PAV" kind="concept">
# Kloosterman sum

## Summary

A Kloosterman sum is an [[exponential sum|97PBMQCPPM]] over the invertible residues $x$ modulo $m$ with phase $(ax + b\bar{x})/m$.
Here $\bar{x}$ is the inverse of $x$ modulo $m$.
It is a real number, and for a prime modulus $p$ not dividing $ab$ Weil's bound gives $\abs{K(a, b; p)} \le 2\sqrt{p}$.
Kloosterman sums appear in the Fourier coefficients of Poincaré series, in the circle method for quadratic forms, and in the spectral theory of automorphic forms.

## Explanations

### L1 human

Pair each invertible residue $x$ modulo $m$ with its inverse $\bar{x}$, and turn $ax + b\bar{x}$ into a point on the unit circle.
The Kloosterman sum adds these points.
For a prime modulus $p$ there are $p - 1$ terms, but the inverse map scrambles them so much that the sum has size at most $2\sqrt{p}$.
For $m = 5$ and $a = b = 1$, the sum equals $2 + 2\cos(4\pi/5) \approx 0.38$.
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<document source="https://tamnd.github.io/math2/node/E1V22QQHYJ.md" tag="E1V22QQHYJ" kind="concept">
# Semianalytic set

## Summary

A semianalytic set is a subset of a [[real analytic manifold|Z7H98E1NHP]] with a simple local description.
Near each point it is given by finitely many equations and strict inequalities between [[real analytic functions|081WKEDK6C]].
It is the analytic analogue of a [[semialgebraic set|BX5WSK8ZCT]], with polynomials replaced by real analytic functions.
Semianalytic sets are locally finite unions of connected real analytic submanifolds, and they can be triangulated.
Since images of semianalytic sets under proper analytic maps need not be semianalytic, one also studies the larger class of [[subanalytic sets|K1HTQF26MA]].

## Explanations

### L1 human

Take some real analytic functions, such as polynomials, $\sin$ or $\exp$, and form sets by equations $f = 0$ and inequalities $g > 0$, combined with "and", "or" and "not".
A semianalytic set is a set that can be described this way near every point.
For example, the open half disc $\set{(x, y) : y > 0,\ x^2 + y^2 < 1}$ is semianalytic in $\R^2$.
The set $\set{0} \cup \set{1/n : n \ge 1}$ is not, because near $0$ it consists of infinitely many isolated points.
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# Subdivision surface

## Definition

A *control mesh* is a polygon mesh $M$ in $\R^3$.
It is a finite set of vertices with positions in $\R^3$, together with edges and faces that form a combinatorial surface.
A *subdivision scheme* is a rule that maps each control mesh $M_k$ to a refined control mesh $M_{k+1}$ such that:

1. the connectivity of $M_{k+1}$ is obtained from that of $M_k$ by a fixed combinatorial refinement, such as splitting every face;
2. the position of each vertex of $M_{k+1}$ is a finite affine combination $\sum_v w_v P_v$, with $\sum_v w_v = 1$, of positions $P_v$ of vertices of $M_k$ near it;
3. the weights $w_v$ depend only on the local connectivity of $M_k$, not on the positions.

The *subdivision surface* of $M_0$ is the limit of the meshes $M_0, M_1, M_2, \dots$, when this limit exists as a continuous surface.

## Remarks

Example: the curve analogue, Chaikin's scheme, maps a polygon with vertices $P_i$ to the polygon with vertices $\tfrac34 P_i + \tfrac14 P_{i+1}$ and $\tfrac14 P_i + \tfrac34 P_{i+1}$.
These are affine combinations with weights summing to $1$ that depend only on adjacency, so clauses 2 and 3 hold.
The limit curve is the uniform quadratic [[B-spline|62AFX32JGT]] with control points $P_i$.
The surface schemes of Doo–Sabin, [[Catmull–Clark|PF5Z57DDG2]] and Loop are subdivision schemes in this sense; on regular parts of the mesh their limits are biquadratic, bicubic and box-spline surfaces.
Non-example: the rule $P \mapsto 2P$ applied to all vertices has weights summing to $2$, so clause 2 fails.
The meshes grow without bound, and the result depends on the choice of origin.
The condition $\sum_v w_v = 1$ makes the scheme commute with affine maps of space, so the limit surface does not depend on the coordinate system.
Near extraordinary vertices, those with a nonstandard number of neighbors, the smoothness of the limit is studied through the eigenvalues of the local subdivision matrix.

## Explanations

### L1 human

A subdivision scheme is a recipe that turns a coarse mesh into a finer one by averaging nearby points with fixed weights.
Applying it over and over gives a sequence of meshes that converges to a smooth surface, the subdivision surface.
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# Closing lemma

## Statement

Let $M$ be a compact smooth manifold and let $f \colon M \to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]].
Let $p$ be a [[non-wandering point|EY2W34JZA4]] of $f$.
Give the space of $C^1$ diffeomorphisms of $M$ the $C^1$ topology, and let $\mathcal{U}$ be a neighbourhood of $f$ in it.
Then there is $g \in \mathcal{U}$ such that $p$ is a [[periodic point|T75CZ5H47N]] of $g$.

## Remarks

The result is due to Pugh, and it is often called Pugh's closing lemma.
Since $p$ is non-wandering, some orbit starting near $p$ returns near $p$.
The difficulty is to close this orbit by a perturbation that is small in $C^1$.
A naive push changes the derivative by about the size of the gap divided by the size of the region where one pushes.
Pugh's argument spreads the push over many returns of the orbit, so that each step is small in $C^1$.
A consequence is the general density theorem: for a $C^1$-generic diffeomorphism, the periodic points are dense in the non-wandering set.
The analogous statement with the $C^r$ topology for $r \ge 2$ is open in general; it is the tenth of Smale's problems.

## Explanations

### L1 human

A non-wandering point is a point near which some orbits come back again and again.
The closing lemma says that by changing the system only very slightly, one can make the point lie on a closed, periodic orbit.
So recurrence can always be turned into periodicity at small cost.

### L2 human

The closing lemma is the basic perturbation tool of $C^1$ generic dynamics.
Together with the Kupka–Smale theorem it shows that a $C^1$-generic diffeomorphism has dense hyperbolic periodic points in its non-wandering set.
It is a key ingredient in Mañé's proof of the stability conjecture.
Related tools are the connecting lemma of Hayashi and Franks' lemma, which also rely on $C^1$-small perturbations.
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# Toric variety

## Definition

Let $k$ be a field, and let $T \cong \mathbb{G}_m^n$ be a split [[algebraic torus|66FQFAJNS4]] over $k$.
A *toric variety* with torus $T$ is a [[variety|HV6YBS0YDH]] $X$ over $k$ with an open embedding $T \hookrightarrow X$ such that:

1. $X$ is [[normal|EK261G1XP9]];
2. the image of $T$ is dense in $X$;
3. the action of $T$ on itself by multiplication extends to an [[action|RW85C34GQW]] $T \times X \to X$.

## Remarks

Some authors drop clause 1 and speak of normal toric varieties when it holds.
Normal toric varieties correspond to fans: finite collections of strongly convex rational polyhedral cones in $\R^n$, closed under faces, any two of which meet in a common face.

Example: $\mathbb{P}^n$ with the torus $T = \set{[x_0 : \cdots : x_n] : x_0 x_1 \cdots x_n \ne 0} \cong \mathbb{G}_m^n$.
Here $T$ is open and dense, $\mathbb{P}^n$ is smooth and hence normal, and $(t_1, \ldots, t_n) \cdot [x_0 : x_1 : \cdots : x_n] = [x_0 : t_1 x_1 : \cdots : t_n x_n]$ extends the multiplication.

Non-example: the cuspidal curve $C = \set{y^2 = x^3}$ in $\mathbb{A}^2$ contains the dense open torus $\set{(t^2, t^3) : t \ne 0}$.
The action $s \cdot (x, y) = (s^2 x, s^3 y)$ extends the multiplication, so clauses 2 and 3 hold.
But $C$ is not normal at the origin, so it fails clause 1.
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# Proximal operator

## Summary

The proximal operator of a convex function $f$ on a Hilbert space sends a point $v$ to a nearby point where $f$ is small.
The new point minimizes $f(x) + \tfrac12 \norm{x - v}^2$, which balances the two aims.
It generalizes the projection onto a closed convex set, which is the proximal operator of the indicator function of the set.
Proximal operators are firmly nonexpansive, and they are the building blocks of proximal gradient methods and other splitting algorithms in convex optimization.

## Explanations

### L1 human

Imagine standing at a point $v$ and wanting to move to a place where $f$ is small, but each step away costs you.
The proximal point is the best compromise: it minimizes $f(x)$ plus half the squared distance from $x$ to $v$.
If $f$ is $0$ on a convex region and $+\infty$ outside, the compromise is simply the nearest point of the region.
For $f(x) = \abs{x}$ on $\R$, the proximal operator moves $v$ toward $0$ by $1$ and stops at $0$: it sends $3$ to $2$ and $0.4$ to $0$.
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# Weak equivalence

## Summary

In abstract homotopy theory, the weak equivalences of a category are the morphisms that one wants to treat as if they were isomorphisms.
They form one of the three classes of a model category, and the homotopy category is obtained by inverting them.
The motivating examples are the [[weak homotopy equivalences|4Q3WY1RX91]] of topological spaces and the [[quasi-isomorphisms|C9JEC2WQKV]] of chain complexes.
A weak equivalence need not have an inverse morphism, which is why the word weak is used.

## Explanations

### L1 human

Two spaces can have the same holes in every dimension without there being a nice map back and forth between them.
A weak equivalence is a map that sees no difference between its source and target from the chosen point of view, for example in all homotopy groups.
Homotopy theory then declares such a map to be as good as an isomorphism.
For chain complexes, a map that induces isomorphisms on all homology groups is a weak equivalence, even when it is far from invertible.
An example is the map from the complex $\Z \xrightarrow{2} \Z$, placed in degrees one and zero, to the complex with $\Z/2$ in degree zero and $0$ elsewhere.
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# Self-avoiding walk

## Summary

A self-avoiding walk is a path on a lattice that never visits the same point twice.
Counting such walks and describing a typical long one are hard problems, much harder than for [[random walks|04VZE3FZXF]].
The growth rate of their number is the connective constant of the lattice.
Self-avoiding walks are a model of long polymer molecules, which cannot occupy the same place twice.

## Explanations

### L1 human

Draw a path on squared paper along the grid lines, one unit step at a time, and never come back to a corner you have visited.
That path is a self-avoiding walk.
On the square grid there are $4$ walks of one step, $12$ of two steps and $36$ of three steps.
The number grows roughly like $\mu^n$ for some number $\mu$ between $2$ and $3$.
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# Craig interpolation theorem

## Statement

Let $\varphi$ and $\psi$ be first-order [[sentences|DKFGQTRYZE]] in a language with equality.
Assume that $\varphi \models \psi$, in the sense of [[logical consequence|1HK0VYWTV7]].
Then there is a first-order sentence $\theta$ with $\varphi \models \theta$ and $\theta \models \psi$.
Moreover, every relation, function and constant symbol of $\theta$ occurs in both $\varphi$ and $\psi$.

## Remarks

The sentence $\theta$ is called an interpolant.
One proof uses a cut-free [[sequent calculus|W72CA3R9MP]] and builds $\theta$ by induction on a cut-free proof of $\varphi \to \psi$.
Another proof is model-theoretic and uses the [[compactness theorem|1NR3HAWG9D]].
Example: $P(c) \land Q(c) \models P(c) \lor R(c)$, and $\theta = P(c)$ is an interpolant in the common symbols $P$ and $c$.
When $\varphi$ and $\psi$ share no symbols, $\theta$ is a sentence of pure equality, such as $\forall x \, (x = x)$ or its negation.
The propositional version holds too, with $\theta$ built from the common variables and the constants $\top$ and $\bot$.
William Craig proved the theorem in 1957.

## Explanations

### L1 human

Suppose one statement implies another.
The theorem says that there is a middle statement, using only the vocabulary the two have in common, that carries the implication.
So the reason for the implication can always be expressed in the shared language.

### L2 human

Craig interpolation is a basic structural property of first-order logic.
It implies the [[Beth definability theorem|RJBDSA2ETP]]: an implicitly definable relation is explicitly definable.
It is closely related to Robinson's joint consistency theorem.
Many other logics fail to have it, so it is also used to compare logics.
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# Regular function

## Definition

Let $k$ be an algebraically closed field.

1. Let $X$ be a quasi-affine variety, that is, a nonempty open subset of an [[affine variety|JERVQT4C12]] in $k^n$. A function $f \colon X \to k$ is *regular at* $p \in X$ if the following holds. There are an [[open|02BH3MCBQB]] neighbourhood $U \subseteq X$ of $p$ and polynomials $g, h \in k[x_1, \dots, x_n]$ such that $h$ has no zero on $U$ and $f = g/h$ on $U$.
2. Let $X$ be a [[quasi-projective variety|EQSHJPE7H8]] in $\mathbb{P}^n$. A function $f \colon X \to k$ is *regular at* $p \in X$ if the following holds. There are an open neighbourhood $U \subseteq X$ of $p$ and [[homogeneous polynomials|2C3PAQ7H50]] $g, h \in k[x_0, \dots, x_n]$ of the same degree. The polynomial $h$ has no zero on $U$, and $f = g/h$ on $U$.

In both cases, $f$ is *regular* on $X$ if it is regular at every point of $X$.

## Remarks

In case 2, $g/h$ is a well-defined function on $U$, because $g$ and $h$ have the same degree $d$ and so $g(\lambda a)/h(\lambda a) = g(a)/h(a)$.
A regular function is [[continuous|0VGFHTM75Y]] when $k$ is identified with $\mathbb{A}^1$ with its Zariski topology.

Example: on $X = k \setminus \set{0}$, the function $f(x) = 1/x$ is regular, with $g = 1$ and $h = x$ on $U = X$.

Non-example: the function on $k$ given by $f(0) = 1$ and $f(x) = 0$ for $x \ne 0$ is not regular at $0$.
Near $0$ it would equal $g/h$ on a nonempty open set, so $g$ would vanish on a cofinite set and hence be $0$, which contradicts $f(0) = 1$.

On an affine variety $X$, the ring of regular functions on all of $X$ equals the [[coordinate ring|G9WB27VWP7]] $k[X]$; this uses Hilbert's Nullstellensatz.
On a [[projective variety|A4H4CZ2V35]], every regular function on the whole variety is constant.
</document>

<document source="https://tamnd.github.io/math2/node/E2K0MM2MQQ.md" tag="E2K0MM2MQQ" kind="counterexample">
# Line with two origins

## Counterexample

Let $Y = \R \times \set{0, 1}$, the disjoint union of two copies of the real line.
Identify $(x, 0)$ with $(x, 1)$ for every $x \ne 0$, and give the set $X$ of classes the quotient topology.
The two classes $0_a$ of $(0, 0)$ and $0_b$ of $(0, 1)$ are the two origins.
The space $X$ is locally homeomorphic to $\R$, [[second-countable|03PPNCM9GG]] and $T_1$, but it is not [[Hausdorff|0S1FJ8VEX9]].

## Remarks

Removing $0_b$ leaves an open set homeomorphic to $\R$, and the same holds for $0_a$; these two open sets cover $X$.
Every [[neighbourhood|0ZCY0Z80B6]] of $0_a$ contains the image of $(-\varepsilon, 0) \cup (0, \varepsilon)$ for some $\varepsilon > 0$, and so does every neighbourhood of $0_b$.
So any neighbourhood of $0_a$ meets any neighbourhood of $0_b$, and the two origins cannot be separated.
For the same reason the sequence of classes of $1/n$ [[converges|0W4CN9E2W5]] to both $0_a$ and $0_b$, so limits of sequences are not unique.
This is why the usual definition of a [[topological manifold|QQQ9B1DK4P]] requires the Hausdorff condition: without it, this space would count as a $1$-manifold.

## Explanations

### L1 human

Take the real line and double its zero point, keeping everything else the same.
Near each point, the new space still looks like a piece of the line.
But the two zeros are glued so tightly to the same nearby points that no pair of open sets can keep them apart.
A sequence that goes to zero now goes to both zeros at once.
</document>

<document source="https://tamnd.github.io/math2/node/E2P29GCD9T.md" tag="E2P29GCD9T" kind="concept">
# Indecomposable module

## Summary

An indecomposable module is a nonzero [[module|0NZ8JA25WK]] that cannot be written as a direct sum of two nonzero submodules.
Every [[simple module|G00C22AEDH]] is indecomposable, but an indecomposable module may have many submodules.
By the Krull–Schmidt theorem, a module of finite length is a direct sum of indecomposable modules in an essentially unique way.
So classifying modules of finite length over a ring comes down to classifying its indecomposable modules.

## Explanations

### L1 human

An indecomposable module cannot be split into two independent nonzero parts.
It may still have smaller pieces inside, but none of them can be split off cleanly.
The group $\Z/4\Z$ is an example: it contains $\set{0, 2}$, but no subgroup is a complement to it.
The group $\Z/6\Z$ is not indecomposable, since it is $\Z/2\Z \oplus \Z/3\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/E2ZFQ76N7F.md" tag="E2ZFQ76N7F" kind="statement">
# Rosenbrock methods

## Statement

Let $f \colon \R^d \to \R^d$ be continuously differentiable, and consider the autonomous [[initial value problem|0FXDW992DS]] $y' = f(y)$, $y(t_0) = y_0$.
Let $J = f'(y_n)$ be the [[Jacobian matrix|047NDTEPCV]] of $f$ at the current value $y_n$.
An $s$-stage Rosenbrock method with real coefficients $\alpha_{ij}$, $\gamma_{ij}$, $b_i$ and $\gamma_{ii} = \gamma > 0$ computes, for $i = 1, \ldots, s$,
$$(I - h \gamma J)\, k_i = h f\Big(y_n + \sum_{j<i} \alpha_{ij} k_j\Big) + h J \sum_{j<i} \gamma_{ij} k_j,$$
and sets $y_{n+1} = y_n + \sum_{i=1}^s b_i k_i$.

1. Each step requires only the solution of $s$ linear systems, all with the same matrix $I - h\gamma J$, and no nonlinear equation.
2. Applied to the test equation $y' = \lambda y$ with $\lambda \in \C$, the method gives $y_{n+1} = R(h\lambda)\, y_n$, where $R$ is a rational function whose denominator is $(1 - \gamma z)^s$.
3. For $s = 1$, $\gamma = 1$ and $b_1 = 1$ (the linearly implicit Euler method), $R(z) = 1/(1 - z)$; the method has order $1$, and it is A-stable and L-stable.

## Remarks

Clause 1 holds because the right-hand side of the $i$-th equation involves only $k_1, \ldots, k_{i-1}$, so the stages are computed one after the other.
For clause 2, with $f(y) = \lambda y$ and $J = \lambda$, each $k_i$ is $y_n$ times a rational function of $z = h\lambda$ with denominator $(1 - \gamma z)^i$.
For clause 3, $(1 - h\lambda) k_1 = h\lambda y_n$ gives $y_{n+1} = y_n / (1 - h\lambda)$, the same as the [[backward Euler method|M7XKVNATFA]]: $\abs{R(z)} \le 1$ when $\operatorname{Re} z \le 0$, and $R(z) \to 0$ as $\abs{z} \to \infty$.
The order conditions of Rosenbrock methods use the exact Jacobian; methods that allow an approximate $J$ are called W-methods.

## Explanations

### L1 human

[[Stiff equations|RR5FCG92PN]] need implicit methods, but implicit methods need the solution of a nonlinear equation at every step.
A Rosenbrock method replaces this nonlinear equation by a few linear ones, built from the derivative of $f$.
In effect, it does one Newton step per stage and builds the error control into the coefficients.
The simplest one has the same behaviour as backward Euler on linear problems.

### L2 human

Rosenbrock methods are linearly implicit [[Runge–Kutta methods|7DBPJKYC6C]]: the coefficients are chosen so that the order conditions hold when $J$ is the exact Jacobian.
One LU factorisation of $I - h\gamma J$ per step serves all stages, which is why all diagonal coefficients are equal.
They are popular for moderately stiff problems of medium size, for example in chemical kinetics, and embedded pairs give error estimates for step size control.
</document>

<document source="https://tamnd.github.io/math2/node/E30EA6ETZ4.md" tag="E30EA6ETZ4" kind="concept">
# Singular integral equation

## Summary

A singular integral equation is an [[integral equation|R3SG1SN7JQ]] whose kernel has a strong singularity.
The integral does not converge absolutely, and it is taken as a [[Cauchy principal value|0SYR10HD4R]].
The main case is the one-dimensional equation on a curve with the Cauchy kernel $1/(\tau - t)$.
Its integral operator is not compact, so the Fredholm theory does not apply directly.
Instead, the theory of Noether, Muskhelishvili and others gives an index that measures the failure of the Fredholm alternative.
Such equations arise in elasticity, aerodynamics, fracture mechanics and boundary value problems for holomorphic functions.

## Explanations

### L1 human

In an ordinary integral equation, the kernel stays bounded or blows up mildly, so the integral makes sense as it is.
In a singular integral equation, the kernel blows up like $1/(\tau - t)$ at the point $t$ of evaluation, which is too strong.
The integral only makes sense by cutting out a small symmetric piece around $t$ and letting it shrink, so that the positive and negative parts cancel.
For example, on the real line, $\frac{1}{\pi} \, \mathrm{p.v.} \int \frac{\varphi(\tau)}{t - \tau} \, d\tau = g(t)$ asks for the function whose [[Hilbert transform|NSCEBXM7ZM]] is $g$.
</document>

<document source="https://tamnd.github.io/math2/node/E337A6WVH3.md" tag="E337A6WVH3" kind="concept">
# Residuated lattice

## Summary

A residuated lattice is a [[lattice|WH4SFAACAS]] with a [[monoid|0N6R902FG1]] multiplication that has left and right division operations, called residuals.
The residuals are characterised by an adjunction with the multiplication and the order.
[[Heyting algebras|NG943XNQWH]], [[Boolean algebras|0XADJKM699]], lattice-ordered groups and the lattices of ideals of a ring are examples.
Residuated lattices are the algebraic semantics of substructural logics, where the multiplication is a conjunction and the residuals are implications.

## Explanations

### L1 human

In the positive real numbers, $xy \le z$ holds exactly when $y \le z/x$.
So division is the best answer to the question "how large can $y$ be so that $x$ times $y$ stays below $z$?".
A residuated lattice asks for such a best answer in an abstract ordered setting, from the left and from the right.
In logic, "$x$ and $y$ imply $z$" holds exactly when "$y$ implies ($x$ implies $z$)", and this is the same rule.
</document>

<document source="https://tamnd.github.io/math2/node/E33H1J8S53.md" tag="E33H1J8S53" kind="statement">
# Special adjoint functor theorem

## Statement

Let $\mathcal{D}$ be a [[complete category|FWGPKXCPB7]] whose hom-sets are sets.
Assume that $\mathcal{D}$ is [[well-powered|XVG7FPG227]] and has a cogenerating set.
Let $\mathcal{C}$ be a category whose hom-sets are sets, and let $G \colon \mathcal{D} \to \mathcal{C}$ be a functor.
Then $G$ has a [[left adjoint|TZHF6ZM92S]] if and only if $G$ [[preserves limits|H4MY8JN8W4]] of small diagrams.

## Remarks

Here a set $Q$ of objects is cogenerating when, for any two different morphisms $f, g \colon a \to b$, some $q \colon b \to x$ with $x \in Q$ has $q f \neq q g$.
This is the dual of a [[generator|CK08JW32WB]].
The proof constructs an initial object of each comma category $c \downarrow G$.
Let $P$ be the product of copies of $x$, one for each $x \in Q$ and each morphism $c \to G x$.
Then one intersects all subobjects $S$ of $P$ such that the induced map $c \to G P$ factors through $G S$.
Well-poweredness makes this intersection a small limit.

Example: the category of compact Hausdorff spaces is complete and well-powered, and the unit interval $[0, 1]$ is a cogenerator.
The inclusion into topological spaces preserves limits.
So it has a left adjoint, the [[Stone–Čech compactification|D9MWP5C85W]].

## Explanations

### L1 human

This theorem is another test for a left adjoint.
Instead of a size condition on the functor, it asks for size conditions on the category where the functor starts.
Then preserving limits is all that is needed.
Its most famous use is to show that every space has a Stone–Čech compactification.

### L2 human

Compared with Freyd's general theorem, the solution set condition is replaced by well-poweredness and a cogenerating set in $\mathcal{D}$.
These conditions concern only $\mathcal{D}$, so they are checked once for all functors out of $\mathcal{D}$.
The dual theorem gives right adjoints of colimit-preserving functors out of a cocomplete, co-well-powered category with a generating set.
For instance, every colimit-preserving functor from a category of presheaves on a small category has a right adjoint.
</document>

<document source="https://tamnd.github.io/math2/node/E38801KVJ1.md" tag="E38801KVJ1" kind="definition">
# Higher-order differences

## Definition

Let $G$ and $H$ be additive abelian groups, let $h \in G$, and let $f \colon G \to H$ be a function.
Let $\Delta_h$ be the [[forward difference operator|8E0EBA8X07]] with step $h$.
The *forward differences of order $n$* of $f$, for $n \in \N$, are defined by induction:
$$\Delta_h^0 f = f, \qquad \Delta_h^{n+1} f = \Delta_h\left(\Delta_h^n f\right).$$
The backward differences $\nabla_h^n f$ of order $n$ are defined in the same way from the [[backward difference operator|MK5JGVDPS9]] $\nabla_h$.

## Remarks

Example: for $f(x) = x^2$ on $\R$, $\Delta_h f(x) = 2hx + h^2$ and $\Delta_h^2 f(x) = 2h(x + h) + h^2 - 2hx - h^2 = 2h^2$.
Non-example: $\left(\Delta_h f\right)^2$, the square of the first difference, is not $\Delta_h^2 f$; for $f(x) = x$ and $h = 1$ it is $1$, while $\Delta^2 f = 0$.
In terms of values, $\Delta_h^2 f(x) = f(x + 2h) - 2f(x + h) + f(x)$.
The two kinds are related by $\nabla_h^n f(x) = \Delta_h^n f(x - nh)$.
</document>

<document source="https://tamnd.github.io/math2/node/E38H8QVJK9.md" tag="E38H8QVJK9" kind="definition">
# Berezin integral

## Definition

Let $n \ge 1$ and let $\Lambda_n = \Lambda[\theta_1, \dots, \theta_n]$ be the real [[exterior algebra|0MDMZ965VE]] on anticommuting generators $\theta_1, \dots, \theta_n$, so that $\theta_i \theta_j = -\theta_j \theta_i$.
Every $f \in \Lambda_n$ has a unique expansion $f = \sum_{I} a_I \theta_I$, where $I = \set{i_1 < \dots < i_k}$ runs over subsets of $\set{1, \dots, n}$, $\theta_I = \theta_{i_1} \cdots \theta_{i_k}$ and $a_I \in \R$.
The *Berezin integral* is the linear map $\Lambda_n \to \R$,
$$\int f \, d^n\theta = a_{\set{1, \dots, n}},$$
the coefficient of $\theta_1 \theta_2 \cdots \theta_n$.
The same definition applies with coefficients in any commutative ring, for instance $a_I \in C^\infty(U)$.

## Remarks

The order of the symbols $d\theta_i$ in the notation varies between authors and changes the result by a sign.
For $n = 1$: $\int (a + b\theta)\, d\theta = b$.

The integral kills derivatives: for each $i$, $\int \frac{\partial f}{\partial \theta_i}\, d^n\theta = 0$, where $\partial / \partial\theta_i$ is the odd derivation with $\partial \theta_j / \partial \theta_i = \delta_{ij}$, since a derivative has no top term.
Every monomial other than $\theta_1 \cdots \theta_n$ is, up to sign, such a derivative, so this property fixes the functional up to a constant factor.
Under a linear substitution $\theta = A\eta$ with $A$ an invertible real $n \times n$ matrix, $\theta_1 \cdots \theta_n = \det(A)\, \eta_1 \cdots \eta_n$, so $\int f(A\eta)\, d^n\eta = \det(A) \int f(\theta)\, d^n\theta$; for ordinary integrals the factor would be $\abs{\det A}^{-1}$.

Example: for $n = 2$ and $a \in \R$, $e^{a\theta_1\theta_2} = 1 + a\theta_1\theta_2$, so $\int e^{a\theta_1\theta_2}\, d^2\theta = a$.
This is the [[Pfaffian|NH5HWBPWS9]] of the antisymmetric matrix with entries $A_{12} = a$, $A_{21} = -a$, and in general the Gaussian integral $\int \exp(\frac{1}{2} \sum_{i,j} A_{ij}\theta_i\theta_j)\, d^n\theta$ equals $\operatorname{Pf}(A)$ up to a sign fixed by the conventions.

Non-example: the map $f \mapsto a_\emptyset$, which takes the constant term, is linear but is not the Berezin integral.
For $n = 1$ it gives $1$ on $\partial(\theta)/\partial\theta = 1$, so it does not kill derivatives.

On a [[supermanifold|Z0SB9HQBAJ]] of dimension $p|q$, consider a compactly supported function on a superdomain.
Its integral is the Berezin integral in the odd variables followed by the ordinary integral in the even ones.
</document>

<document source="https://tamnd.github.io/math2/node/E3C4F0RKJY.md" tag="E3C4F0RKJY" kind="concept">
# L-estimator

## Summary

An L-estimator is an [[estimator|0SSS6ERQ0S]] that is a linear combination of the [[order statistics|58NHGZDSRE]] of the sample.
The sample median, the trimmed means, the midrange and the interquartile range are all L-estimators.
Small or zero weights on the extreme order statistics give estimators that resist outliers.
This makes L-estimators a basic family of [[robust statistics|J7410PG2DR]].

## Explanations

### L1 human

Sort the data from smallest to largest.
An L-estimator multiplies each sorted value by a fixed weight and adds the results.
The mean gives every value the same weight, while the median puts all the weight on the middle value.
A trimmed mean drops a few values at each end and averages the rest, so a single wild value cannot pull it far.
</document>

<document source="https://tamnd.github.io/math2/node/E3CPZNJZNG.md" tag="E3CPZNJZNG" kind="statement">
# Completeness of the real numbers

## Statement

Let $F$ be an [[Archimedean ordered field|0HQARMKEV5]].
Then the following five conditions on $F$ are equivalent.

1. Each nonempty subset of $F$ that is bounded above has a supremum in $F$ (the [[least-upper-bound property|0AXWET80YW]]).
2. Each [[Cauchy sequence|0QZMTN5AXX]] in $F$ converges in $F$.
3. For each sequence of closed bounded intervals $[a_1, b_1] \supseteq [a_2, b_2] \supseteq \cdots$ in $F$, some point of $F$ lies in all of them (the [[nested intervals property|MF56GK7QY2]]).
4. Each bounded [[monotone sequence|0KN22AAW5R]] in $F$ converges in $F$.
5. Each bounded sequence in $F$ has a subsequence that converges in $F$.

The field $\R$ of [[real numbers|08J2M1JYAG]] satisfies all five conditions.

## Remarks

The standard proof goes in a cycle.
From (1), a bounded increasing sequence converges to the supremum of its terms, which gives (4).
From (4), each sequence has a monotone subsequence, which gives (5).
A Cauchy sequence is bounded, and if a subsequence converges then the whole sequence converges, so (5) gives (2).
From (2), repeated bisection between a point of a set and an upper bound gives a Cauchy sequence.
Its limit is the supremum, and the Archimedean property makes the interval lengths tend to $0$.
For (3), the left endpoints of nested closed intervals have a supremum under (1), and this supremum lies in every interval.
Conversely, (3) gives (1) by the same bisection, with the Archimedean property again making the lengths tend to $0$.
The field $\Q$ satisfies none of the conditions: the decimal approximations $1, 1.4, 1.41, 1.414, \ldots$ of $\sqrt 2$ form a Cauchy sequence with no limit in $\Q$.
The Archimedean hypothesis matters for (2): there are non-Archimedean ordered fields in which each Cauchy sequence converges, but which lack the least-upper-bound property.
Conditions (1), (4) and (5) each imply the Archimedean property by themselves.
Up to isomorphism, $\R$ is the only [[complete ordered field|0A9E6X0S3E]].

## Explanations

### L1 human

The real line has no gaps, and there are several ways to say this.
A bounded set has a least upper bound; a sequence whose terms crowd together has a limit; shrinking closed intervals always share a point.
These sound different, but each one implies the others.
The rational numbers fail all of them, because $\sqrt 2$ is missing.

### L2 human

In a course, one of the conditions is taken as an axiom or proved from a construction of $\R$, and the others follow as theorems.
Dedekind cuts give the least-upper-bound property directly, while Cauchy sequences give condition (2) directly.
The [[Bolzano–Weierstrass theorem|0KDJD249DT]] is condition (5), and the [[monotone convergence theorem|0FM6N4R0VB]] is condition (4).
The intermediate value theorem and the extreme value theorem both rest on completeness, and both fail over $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/E3ECET66DQ.md" tag="E3ECET66DQ" kind="definition">
# Hyperbolic group

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] that is geodesic: any two points are joined by a geodesic segment.
A geodesic segment is an isometric image of a closed interval of $\R$.
A geodesic triangle in $X$ consists of three points and three geodesic segments joining them in pairs.
For $\delta \ge 0$, a geodesic triangle is *$\delta$-slim* if each of its sides lies in the closed $\delta$-neighbourhood of the union of the other two sides.
The space $X$ is *$\delta$-hyperbolic* if every geodesic triangle in $X$ is $\delta$-slim.

Let $G$ be a [[finitely generated group|082DWWXMRJ]] and let $S$ be a finite [[generating set|0RP7VWQ0YN]].
Give the [[Cayley graph|0GYQK9NDJF]] $\Gamma(G, S)$ the path metric in which each edge has length $1$.
The group $G$ is *hyperbolic* if $\Gamma(G, S)$ is $\delta$-hyperbolic for some $\delta \ge 0$.

## Remarks

The definition does not depend on $S$: Cayley graphs for different finite generating sets are [[quasi-isometric|6FKYSH729T]], and among geodesic spaces hyperbolicity is invariant under quasi-isometry, though $\delta$ may change.

Example: the [[free group|0KTW26C80J]] $F_2$ with a free basis $S$.
Its Cayley graph is a tree, in which every geodesic triangle is a tripod, so each side lies in the union of the other two and the graph is $0$-hyperbolic.

Example: every finite group is hyperbolic, with $\delta$ equal to the diameter of its Cayley graph.

Non-example: $\Z^2$ is not hyperbolic.
Its Cayley graph is quasi-isometric to the Euclidean plane.
In the plane, take the triangle with vertices $(0, 0)$, $(n, 0)$ and $(0, n)$.
The point $(n/2, n/2)$ of its hypotenuse is at distance $n/2$ from the other two sides, which is unbounded in $n$.

Equivalent definitions use the Gromov product, or ask that the group be finitely presented with a linear [[Dehn function|Q4HGMRA9JQ]].
Such groups are also called word-hyperbolic or Gromov hyperbolic groups.
</document>

<document source="https://tamnd.github.io/math2/node/E3EYXM7GJN.md" tag="E3EYXM7GJN" kind="concept">
# Lie pseudogroup

## Summary

A Lie pseudogroup is a [[pseudogroup|3017J7P6JD]] of local diffeomorphisms of a manifold that is cut out by a system of partial differential equations.
Membership of a local diffeomorphism is decided by its [[jets|1QE0Z0756D]] up to a fixed order at each point.
Isometries, conformal maps, symplectomorphisms, holomorphic maps and volume-preserving maps all form Lie pseudogroups.
Lie pseudogroups were introduced by Lie and studied by Cartan as the infinite-dimensional analogues of Lie groups.

## Explanations

### L1 human

Many geometric structures are defined by the maps that preserve them.
Often those maps are the solutions of differential equations: a map of the plane preserves area when its Jacobian determinant equals $1$.
The set of all local solutions of such an equation forms a pseudogroup, and this is a Lie pseudogroup.
Unlike a Lie group, it can be infinite-dimensional, as in the area-preserving example, where there are as many solutions as there are functions of two variables.
</document>

<document source="https://tamnd.github.io/math2/node/E3GQVC04BC.md" tag="E3GQVC04BC" kind="concept">
# Fréchet mean

## Summary

The Fréchet mean generalises the average to data in a [[metric space|0VKD1TP7F3]]: it is a point that minimises the expected squared distance to the data.
In Euclidean space it is the usual mean.
It also makes sense for points on a sphere, on a [[Riemannian manifold|7CBT4SM83Y]], or in spaces of trees, shapes or covariance matrices.
In these spaces adding and dividing points is not defined.
It need not exist or be unique, and the conditions for uniqueness involve the curvature of the space.
It is named after Maurice Fréchet; local minimisers on Riemannian manifolds are also called Karcher means.

## Explanations

### L1 human

To average two cities on the globe, you cannot average their coordinates in 3D space, since the result lies inside the Earth.
Instead, look for the place on the surface that minimises the sum of squared distances, measured along the surface, to the two cities.
That place is the midpoint of the shortest route between them.
The same recipe works for averaging directions, rotations or shapes.
For two exactly opposite points on a sphere, every point of the great circle halfway between them is a Fréchet mean, so the mean is not unique.
</document>

<document source="https://tamnd.github.io/math2/node/E3RQ0J3ACZ.md" tag="E3RQ0J3ACZ" kind="statement">
# Wigner's classification

## Statement

Let $\tilde P = \R^{1,3} \rtimes \mathrm{SL}_2(\C)$ be the universal cover of the identity component of the [[Poincaré group|8ZQ9TA4Q2Q]].
Here $\R^{1,3}$ is Minkowski space with the form $p \cdot p = p_0^2 - p_1^2 - p_2^2 - p_3^2$, and [[SL(2,C)|J58ED1GV6N]] acts on it through the covering map onto the restricted Lorentz group.
The group $\mathrm{SL}_2(\C)$ also acts on the momentum space $\widehat{\R^{1,3}} \cong \R^{1,3}$ of characters $x \mapsto e^{i p \cdot x}$ of the translations.
For a momentum $p$, let $H_p \subseteq \mathrm{SL}_2(\C)$ be its stabilizer, the little group.

1. The irreducible [[unitary representations|74BPYJNZTM]] of $\tilde P$, up to equivalence, are in bijection with certain pairs $(\mathcal{O}, \sigma)$. Here $\mathcal{O}$ is an [[orbit|0YQPZB5Q1G]] of $\mathrm{SL}_2(\C)$ in momentum space. And $\sigma$ is an [[irreducible|FX3NEFYFFN]] unitary representation of $H_p$ for a chosen $p \in \mathcal{O}$, up to equivalence.
2. The representation attached to $(\mathcal{O}, \sigma)$ is the [[induced representation|6Q7AHAZ6Q4]] $\mathrm{Ind}_{\R^{1,3} \rtimes H_p}^{\tilde P} (e^{i p \cdot x} \sigma)$.
3. For the orbit $p \cdot p = m^2$ with $m > 0$ and $p_0 > 0$, the little group is isomorphic to [[SU(2)|E5P8AG6CK7]], and $\sigma$ is labelled by a spin $s \in \set{0, \tfrac12, 1, \tfrac32, \dots}$.
4. For the orbit $p \cdot p = 0$ with $p_0 > 0$, the little group is isomorphic to a double cover of the Euclidean group of the plane. The representations $\sigma$ that are trivial on its translation part are characters labelled by a helicity $h \in \tfrac12 \Z$; the others form the continuous spin representations.

## Remarks

The translations form an abelian normal subgroup, so the [[Mackey machine|12CEWPMSG4]] applies.
The orbits of $\mathrm{SL}_2(\C)$ on momentum space are the hyperboloids $p \cdot p = m^2$, the two halves of the light cone, the hyperboloids $p \cdot p < 0$ of one sheet, and the point $0$.
These orbits are locally closed, which is the condition under which Mackey's theory gives a complete classification by orbits and little groups.
The remaining orbits ($p \cdot p < 0$, $p_0 < 0$, and $p = 0$) give representations that are not used for physical particles.
For example, an electron is described by the orbit with mass $m > 0$ and spin $s = \tfrac12$, and a photon by the light cone orbit with helicities $\pm 1$.
The [[semidirect product|0H3WZXCZNF]] structure is essential: without the normal abelian subgroup of translations there is no momentum space to start from.

## Explanations

### L1 human

Symmetry of spacetime under shifts, rotations and boosts forms the Poincaré group.
Wigner found all the elementary ways in which this group can act on the states of a quantum system.
Each such way is labelled by a mass and by a spin or a helicity.
This is why physicists say that a particle "is" an irreducible representation of the Poincaré group.

### L2 human

The method is the general one for semidirect products $A \rtimes H$ with $A$ abelian.
First choose a character of $A$, here a momentum $p$, then a representation of its stabilizer in $H$, and induce.
For massive particles the stabilizer of $(m, 0, 0, 0)$ is $\mathrm{SU}(2)$, which explains why spin takes half-integer values.
The same method classifies the representations of the Euclidean motion groups and of the Heisenberg-type semidirect products.
</document>

<document source="https://tamnd.github.io/math2/node/E3SRYCE5VN.md" tag="E3SRYCE5VN" kind="concept">
# Ramanujan summation

## Summary

Ramanujan summation assigns a value to a series $\sum_{n \ge 1} f(n)$ by taking the constant term in the [[Euler–Maclaurin formula|9P839VB8S7]] for its partial sums.
The value is expressed through $f(0)$, the odd derivatives of $f$ at $0$ and the [[Bernoulli numbers|0KC00K7KY4]].
It gives $1 + 2 + 3 + \cdots = -\frac{1}{12}$ and $1 + 1 + 1 + \cdots = -\frac12$, the values $\zeta(-1)$ and $\zeta(0)$ of the [[Riemann zeta function|1XWBY4N0VT]].
Unlike most [[summability methods|8X25Q4XYMN]], it is neither regular nor stable under shifting the index.

## Explanations

### L1 human

The partial sums $f(1) + \cdots + f(N)$ of a series often behave like an integral of $f$ plus correction terms plus a constant.
Ramanujan proposed to call that constant the "sum" of the series, even when the series diverges.
For $f(n) = n$, the partial sums are $\frac{N^2}{2} + \frac{N}{2}$ exactly, and the constant obtained by his rule is $-\frac{1}{12}$.
This number does not mean that the positive integers add up to a negative number; it is a different kind of value, attached to the function $f$.
</document>

<document source="https://tamnd.github.io/math2/node/E3TEYV0D89.md" tag="E3TEYV0D89" kind="definition">
# Noncentral chi-squared distribution

## Definition

Let $k$ be a positive integer and let $\mu_1, \dots, \mu_k$ be real numbers.
Let $X_1, \dots, X_k$ be [[independent|0E9502NT92]] random variables with $X_i \sim \mathcal{N}(\mu_i, 1)$, the [[normal distribution|07RXBVTVZB]] with mean $\mu_i$ and variance $1$.
The distribution of
$$Q = \sum_{i=1}^k X_i^2$$
depends only on $k$ and on $\lambda = \sum_{i=1}^k \mu_i^2$.
It is the *noncentral chi-squared distribution* with $k$ degrees of freedom and noncentrality parameter $\lambda$, written $\chi'^2_k(\lambda)$.

## Remarks

For $\lambda = 0$ it is the [[chi-squared distribution|0RFMMBHFTV]] with $k$ degrees of freedom.
Its mean is $k + \lambda$ and its variance is $2(k + 2\lambda)$.
Its density is a [[Poisson|0VEFSSKX18]] mixture of central chi-squared densities:
$$f(x) = \sum_{j=0}^{\infty} e^{-\lambda/2} \frac{(\lambda/2)^j}{j!} f_{k + 2j}(x), \quad x > 0,$$
where $f_m$ is the chi-squared density with $m$ degrees of freedom.

Example: if $X \sim \mathcal{N}(\mu, 1)$, then $X^2 \sim \chi'^2_1(\mu^2)$, and $E(X^2) = 1 + \mu^2$ agrees with the mean $k + \lambda$.

Non-example: the variances must be $1$.
If $X \sim \mathcal{N}(0, 4)$, then $X^2$ has mean $4$ and variance $32$.
A $\chi'^2_1(\lambda)$ variable with mean $4$ has $\lambda = 3$ and variance $2(1 + 6) = 14$, so $X^2$ has no noncentral chi-squared distribution with one degree of freedom.
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<document source="https://tamnd.github.io/math2/node/E459H2B2EM.md" tag="E459H2B2EM" kind="statement">
# Furstenberg's multiple recurrence theorem

## Statement

Let $(X, \mathcal{B}, \mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \colon X \to X$ be a [[measure-preserving transformation|556WBDAHNP]].
Let $A \in \mathcal{B}$ with $\mu(A) > 0$, and let $k \ge 1$ be an integer.
Then
$$\liminf_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} \mu\big(A \cap T^{-n}A \cap T^{-2n}A \cap \cdots \cap T^{-kn}A\big) > 0.$$
In particular, there is an integer $n \ge 1$ with $\mu(A \cap T^{-n}A \cap \cdots \cap T^{-kn}A) > 0$.

## Remarks

For $k = 1$ the weaker conclusion is the [[Poincaré recurrence theorem|TVYJPBHD0S]], and the average form follows from the mean ergodic theorem.
For general $k$ the proof goes through a structure theorem.
One first reduces to ergodic systems.
Every ergodic system is built from a trivial system by a possibly transfinite tower of compact extensions, followed possibly by one weakly mixing extension.
The property holds for compact extensions, where orbits come back almost periodically.
It also holds for weakly mixing extensions, where the sets $T^{-in}A$ behave as if independent.
Then it passes up the tower and to limits.
Example: for an irrational rotation and an arc $A$ of length $\ell$, take $n$ with $n\alpha$ within $\ell/(2k)$ of an integer.
Then $A \cap T^{-n}A \cap \cdots \cap T^{-kn}A$ contains an arc of length at least $\ell/2$.
Example: for a Bernoulli scheme and a cylinder set $A$, independence gives $\mu(A \cap T^{-n}A \cap \cdots \cap T^{-kn}A) = \mu(A)^{k+1}$ once $n$ exceeds the length of the cylinder.
The hypothesis $\mu(X) < \infty$ cannot be dropped: the translation $x \mapsto x + 1$ on $\R$ with Lebesgue measure and $A = [0, 1)$ gives $A \cap T^{-n}A = \emptyset$ for every $n \ge 1$.

## Explanations

### L1 human

In a system that preserves a finite volume, a set of positive volume does not only come back to itself.
It comes back at evenly spaced times $n, 2n, \dots, kn$ simultaneously, for some $n$ and for any length $k$ you ask for.
This is a dynamical form of finding long arithmetic progressions.

### L2 human

Furstenberg proved the theorem in 1977 to give an ergodic-theoretic proof of [[Szemerédi's theorem|B23B0QEMN5]].
The Furstenberg correspondence principle turns a set of integers of positive upper density into a set $A$ of positive measure in a measure-preserving system.
It does so in such a way that a positive value of $\mu(A \cap T^{-n}A \cap \cdots \cap T^{-kn}A)$ yields an arithmetic progression $a, a+n, \dots, a+kn$ in the original set.
The method grew into ergodic Ramsey theory, with extensions to several commuting transformations, which gives multidimensional Szemerédi theorems, and to polynomial times.
</document>

<document source="https://tamnd.github.io/math2/node/E45PA267S4.md" tag="E45PA267S4" kind="concept">
# Transcendence basis

## Summary

A transcendence basis of a [[field extension|0EQWVHF3W5]] $L/K$ is a set $B$ of [[algebraically independent|KV4PT6KW10]] elements such that $L$ is [[algebraic|0PTBM7K0R2]] over $K(B)$.
It plays the role that a [[basis|0KCQ9Y3QKA]] plays for a vector space, with polynomial relations in place of linear ones.
[[Every extension has one|W635VBXEXG]], and all of them have the same cardinality, the [[transcendence degree|D7TR0YWR9H]].

## Explanations

### L1 human

A transcendence basis is a largest set of free variables inside a field extension.
Everything else in the big field is tied to these variables by polynomial equations.
In $\Q(x, \sqrt{x^3 + 1})$, the single element $x$ is a transcendence basis over $\Q$, since $y = \sqrt{x^3 + 1}$ satisfies $y^2 = x^3 + 1$.
</document>

<document source="https://tamnd.github.io/math2/node/E4F316AR19.md" tag="E4F316AR19" kind="definition">
# Blowing up

## Definition

Let $X$ be a Noetherian [[scheme|5A35TYZD02]] and let $Z \subseteq X$ be a closed subscheme with [[ideal sheaf|N3PGYJJQ7W]] $\mathcal{I} \subseteq \mathcal{O}_X$.
The *blow-up* of $X$ along $Z$ is the scheme
$$\operatorname{Bl}_Z X = \operatorname{Proj}_X \bigoplus_{d \ge 0} \mathcal{I}^d,$$
given by the relative [[Proj construction|A39XQHCVNW]], with $\mathcal{I}^0 = \mathcal{O}_X$, together with its projection $\pi \colon \operatorname{Bl}_Z X \to X$.
The closed subscheme $E = \pi^{-1}(Z)$ is the *exceptional divisor*, and $Z$ is the *centre* of the blow-up.

## Remarks

The ideal sheaf $\pi^{-1}\mathcal{I} \cdot \mathcal{O}_{\operatorname{Bl}_Z X}$ of $E$ is invertible, so $E$ is an effective [[Cartier divisor|FHV05ZK99F]].
The morphism $\pi$ is projective, and it is an isomorphism over $X \setminus Z$.
It has a universal property: every morphism $f \colon Y \to X$ such that $f^{-1}\mathcal{I} \cdot \mathcal{O}_Y$ is invertible factors uniquely through $\pi$.

Example: the blow-up of $\mathbb{A}^2 = \operatorname{Spec} k[x, y]$ at the origin, with $\mathcal{I} = (x, y)$, is the closed subvariety
$$\set{((x, y), [u : v]) \in \mathbb{A}^2 \times \mathbb{P}^1 : x v = y u}.$$
Over a point $(x, y) \ne (0, 0)$ the condition forces $[u : v] = [x : y]$, so $\pi$ is bijective there.
Over the origin every $[u : v]$ is allowed, so $E \cong \mathbb{P}^1$, the [[projective line|ZT9A6JK8QF]] of directions through the origin.

Non-example: the projection $\mathbb{A}^2 \times \mathbb{P}^1 \to \mathbb{A}^2$ is not the blow-up at the origin.
It fails to be an isomorphism over $\mathbb{A}^2 \setminus \set{0}$, since every fibre is a whole $\mathbb{P}^1$.

Blowing up an effective Cartier divisor changes nothing: $\pi$ is then an isomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/E4FMVDQ2GR.md" tag="E4FMVDQ2GR" kind="statement">
# Muller's method

## Statement

Let $f$ be a complex function that is holomorphic near a simple zero $r$, so $f(r) = 0$ and $f'(r) \ne 0$.
Given three distinct points $x_{k-2}, x_{k-1}, x_k$, write $f[x_i, x_j] = \dfrac{f(x_i) - f(x_j)}{x_i - x_j}$ and $f[x_k, x_{k-1}, x_{k-2}] = \dfrac{f[x_k, x_{k-1}] - f[x_{k-1}, x_{k-2}]}{x_k - x_{k-2}}$, and put
$$w = f[x_k, x_{k-1}] + f[x_k, x_{k-2}] - f[x_{k-1}, x_{k-2}].$$
*Muller's method* defines
$$x_{k+1} = x_k - \frac{2 f(x_k)}{w \pm \sqrt{w^2 - 4 f(x_k) \, f[x_k, x_{k-1}, x_{k-2}]}},$$
where the sign is chosen so that the denominator has the larger absolute value.
Then there is $\delta > 0$ with the following property.
For all starting points within $\delta$ of $r$ for which the iteration is defined, the iterates [[converge|YWZQC5E54B]] to $r$ with order $q \approx 1.839$.
Here $q$ is the real root of $q^3 = q^2 + q + 1$.

## Remarks

The point $x_{k+1}$ is the zero, closest to $x_k$, of the [[interpolating polynomial|SS1YMS2NPC]] of degree at most $2$ through $(x_{k-2}, f(x_{k-2}))$, $(x_{k-1}, f(x_{k-1}))$ and $(x_k, f(x_k))$.
The formula is the quadratic formula written in the stable form that avoids cancellation.
The errors $e_k = x_k - r$ satisfy $e_{k+1} \approx C e_k e_{k-1} e_{k-2}$, which gives the stated order.
Because of the square root, the iterates can leave the real line, so the method finds [[complex|09FTAET5Z4]] zeros even from real starting points.
This makes it useful for zeros of real polynomials that are not real.
With a line through two points instead of a parabola through three, the same idea gives the [[secant method|9ZC177PVPC]].

## Explanations

### L1 human

The secant method draws a line through two points of the graph and takes its crossing with the axis.
Muller's method draws a parabola through three points and takes its crossing nearest to the last point.
A parabola can miss the real axis, and then the method moves into the complex numbers, which lets it find complex roots.

### L2 human

The method needs no derivatives and one new function value per step.
Its order is about $1.84$, between the secant method and Newton's method.
It is a common choice for finding all roots of a polynomial one at a time, combined with deflation.
</document>

<document source="https://tamnd.github.io/math2/node/E4NJER1RFQ.md" tag="E4NJER1RFQ" kind="concept">
# Kirby–Siebenmann class

## Summary

The Kirby–Siebenmann class of a topological manifold $M$ is a cohomology class $\operatorname{ks}(M) \in H^4(M; \Z/2)$.
It is the obstruction to putting a piecewise linear structure on $M \times \R$, and, when $\dim M \ge 5$, on $M$ itself.
It shows that topological manifolds and piecewise linear manifolds really differ.
In dimension $4$, a manifold whose class is nonzero, such as the $E_8$ manifold, has no smooth structure.

## Explanations

### L1 human

Some topological manifolds can be cut into simplices in a way that makes them piecewise linear, and some cannot.
Kirby and Siebenmann found a single test for this, in the form of one cohomology class with coefficients in $\Z/2$.
If the class is nonzero, no piecewise linear or smooth structure exists.
In dimensions five and higher, a zero class means that a piecewise linear structure does exist.
</document>

<document source="https://tamnd.github.io/math2/node/E4P30YHWEQ.md" tag="E4P30YHWEQ" kind="definition">
# Differential equations in Banach spaces

## Definition

Let $X$ be a [[Banach space|06GZ956657]] over $\R$ or $\C$ with norm $\norm{\cdot}$, and let $J \subseteq \R$ be an interval.
A function $u \colon J \to X$ is *differentiable* at $t \in J$ if the limit $u'(t) = \lim_{h \to 0} \frac{u(t + h) - u(t)}{h}$ exists in the norm of $X$.
Let $D \subseteq X$ and $F \colon J \times D \to X$.
A *differential equation in $X$* is the equation $u'(t) = F(t, u(t))$.
A *(classical) solution* on $J$ is a continuously differentiable $u \colon J \to X$ with $u(t) \in D$ and $u'(t) = F(t, u(t))$ for all $t \in J$.
In the main linear case, $F(t, u) = A u + f(t)$, where $A \colon D(A) \to X$ is a linear operator, $D = D(A)$ and $f \colon J \to X$.
The [[initial value problem|0FXDW992DS]] $u' = A u$, $u(0) = u_0$ is the *abstract Cauchy problem*.

## Remarks

Example: if $A$ is a bounded operator on $X$, then $u(t) = e^{tA} u_0 = \sum_{k \ge 0} \frac{t^k}{k!} A^k u_0$ solves the abstract Cauchy problem on $\R$.
The series converges in [[operator norm|06EKW1AKXF]] since $\norm{A^k} \le \norm{A}^k$, and term-by-term differentiation gives $u' = A u$.
For $X = \R^n$ this is the [[matrix exponential|020M04FN1H]] solution of a linear system of [[ordinary differential equations|0NT64JV96A]].

Non-example: take $X = C([0, 1])$ with the supremum norm and $u(t) = (s \mapsto \abs{s - t})$ for $t \in [0, 1]$.
For each fixed $s$, the number $u(t)(s)$ has a derivative in $t$ for $t \neq s$, equal to $-\sgn(s - t)$.
But the difference quotients $\frac{u(t + h) - u(t)}{h}$ do not converge in the supremum norm: their pointwise limit $s \mapsto -\sgn(s - t)$ is not continuous for $0 < t < 1$.
So $u$ is not differentiable as a map into $X$, and it is not a classical solution of any differential equation in $X$.

When $A$ is closed but unbounded, the Cauchy problem is well posed exactly when $A$ generates a strongly continuous semigroup $T(t)$, and then $u(t) = T(t) u_0$.

## Explanations

### L1 human

The unknown is a moving point in a Banach space, and its velocity is computed in the norm of that space.
Bounded operators behave like matrices; unbounded ones, such as derivatives in space, need more care.
</document>

<document source="https://tamnd.github.io/math2/node/E4TEZEDHY8.md" tag="E4TEZEDHY8" kind="definition">
# Order topology

## Definition

Let $(X, <)$ be a [[totally ordered set|0FRXBVVR03]].
The *order topology* on $X$ is the [[topology|0CQHFPGJ45]] on $X$ with [[subbasis|09S8DV34Y0]] the open rays
$$(a, \to) = \set{x \in X : a < x} \quad\text{and}\quad (\gets, b) = \set{x \in X : x < b}, \qquad a, b \in X.$$

## Remarks

When $X$ has more than one point, the open intervals $(a, b)$ together with the open rays form a basis of the order topology.

Example: on $\R$ with its usual order, the open rays generate the open intervals, so the order topology is the usual topology of $\R$.

Non-example: let $Y = \set{-1} \cup \set{1/n : n \ge 1}$ as a subset of $\R$.
With the subspace topology from $\R$, the point $-1$ is isolated.
In the order topology of $Y$, every open set containing $-1$ contains a ray $(\gets, 1/n)$, which contains $1/m$ for all $m > n$.
So the subspace topology of $Y$ is not its order topology, although the order of $Y$ is the restriction of the order of $\R$.

Some authors also use the name for the topology generated by the same rays on an arbitrary partially ordered set.
</document>

<document source="https://tamnd.github.io/math2/node/E5D4BD4193.md" tag="E5D4BD4193" kind="statement">
# Characterizations of semisimple modules

## Statement

Let $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].
The following are equivalent:

1. $M$ is [[semisimple|N3MRHDKDGC]], that is, a direct sum of [[simple|G00C22AEDH]] submodules;
2. $M$ is a sum of simple submodules;
3. every [[submodule|0RGZ10DFPG]] $N$ of $M$ is a direct summand: there is a submodule $N'$ with $M = N \oplus N'$.

## Remarks

The step from 2 to 1, and to 3, uses [[Zorn's lemma|04RXTQA5ZA]].
Given $M = \sum_{i \in I} S_i$ and a submodule $N$, choose a set $J \subseteq I$ maximal such that the sum $N + \sum_{j \in J} S_j$ is direct.
Each $S_i$ meets this sum, and being simple it lies inside it, so the sum is all of $M$; taking $N = 0$ gives 1.
For 3 implies 2, one shows that every nonzero submodule contains a simple submodule, and that the sum of all simple submodules has a complement which must be zero.
Consequences: submodules and quotient modules of semisimple modules are semisimple.
Example: in a vector space every subspace has a complementary subspace, matching condition 3.

## Explanations

### L1 human

There are three ways to say that a module is built cleanly from simple pieces.
It is a direct sum of simple pieces; or it is merely covered by simple pieces, possibly overlapping; or every piece of it can be split off.
The theorem says these all mean the same thing.

### L2 human

Condition 3 is usually the easiest to check, and it is how Maschke's theorem proves semisimplicity of group representations.
Condition 2 shows quickly that quotients of semisimple modules are semisimple.
The equivalence is the starting point of the theory of semisimple rings, where every module satisfies these conditions.
</document>

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# Maximal order

## Summary

A maximal order is an [[order|1ZBX4TKVZ9]] in an algebra that is not contained in any larger order.
In an [[algebraic number field|MWBDFS2KJ5]] the ring of integers is the unique maximal order, so maximal orders generalize rings of integers to noncommutative algebras.
In noncommutative algebras maximal orders exist under mild hypotheses but are usually not unique.
For example, the conjugates of $M_2(\Z)$ by elements of $GL_2(\Q)$ are maximal orders in $M_2(\Q)$.
Maximal orders in [[quaternion algebras|ZWXW0T3EAC]] are central in the arithmetic of quaternions and of Shimura curves.

## Explanations

### L1 human

An order is a ring of "integers" inside an algebra, but an algebra can have many orders, some inside others.
For example $\Z[\sqrt{-3}]$ is an order in $\Q(\sqrt{-3})$, but it sits inside the bigger order $\Z[(1+\sqrt{-3})/2]$.
A maximal order is one that cannot be enlarged.
For number fields there is exactly one, the ring of all algebraic integers of the field.
For matrix algebras and quaternion algebras there can be several, and studying how they relate is part of the theory.
</document>

<document source="https://tamnd.github.io/math2/node/E5G2SSTBP3.md" tag="E5G2SSTBP3" kind="definition">
# Schouten–Nijenhuis bracket

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]].
For $p \ge 0$, let $\mathfrak{X}^p(M)$ be the space of $p$-vector fields, the smooth sections of the [[exterior power|0MDMZ965VE]] $\Lambda^p TM$.
So $\mathfrak{X}^0(M) = C^\infty(M)$, and $\mathfrak{X}^1(M)$ is the space of [[vector fields|0299VCW1TH]].
The *Schouten–Nijenhuis bracket* is the unique family of $\R$-bilinear maps $[\cdot, \cdot] \colon \mathfrak{X}^p(M) \times \mathfrak{X}^q(M) \to \mathfrak{X}^{p+q-1}(M)$ with these properties, for $P \in \mathfrak{X}^p(M)$, $Q \in \mathfrak{X}^q(M)$ and $R \in \mathfrak{X}^r(M)$.

1. For functions $f, g$: $[f, g] = 0$.
2. For a vector field $X$: $[X, P] = \mathcal{L}_X P$, the [[Lie derivative|PJWK0PKN6Q]]; in particular $[X, f] = X(f)$ and $[X, Y]$ is the [[Lie bracket|68FGQEBD6J]].
3. Graded antisymmetry: $[P, Q] = -(-1)^{(p-1)(q-1)} [Q, P]$.
4. Graded Leibniz rule: $[P, Q \wedge R] = [P, Q] \wedge R + (-1)^{(p-1)q}\, Q \wedge [P, R]$.

## Remarks

For vector fields $X_1, \dots, X_p$ and $Y_1, \dots, Y_q$ with $p, q \ge 1$, the bracket is
$$[X_1 \wedge \dots \wedge X_p,\ Y_1 \wedge \dots \wedge Y_q] = \sum_{i=1}^{p} \sum_{j=1}^{q} (-1)^{i+j} [X_i, Y_j] \wedge X_1 \wedge \dots \widehat{X_i} \dots \wedge X_p \wedge Y_1 \wedge \dots \widehat{Y_j} \dots \wedge Y_q,$$
where a hat marks an omitted factor.
The bracket also satisfies a graded Jacobi identity, so $\mathfrak{X}^{\bullet + 1}(M)$, with degrees shifted by one, is a graded Lie algebra.

A bivector field $\pi \in \mathfrak{X}^2(M)$ defines the bracket $\{f, g\} = \pi(df, dg)$ on functions.
This bracket satisfies the Jacobi identity, so that $\pi$ is a [[Poisson structure|A2KQ49Y4NF]], exactly when $[\pi, \pi] = 0$.

Example: on $\R^2$, let $\pi = \partial_x \wedge \partial_y$.
All brackets of the coordinate vector fields vanish, so the formula gives $[\pi, \pi] = 0$, and $\{f, g\} = f_x g_y - f_y g_x$ is the standard Poisson bracket.

Non-example: the wedge product $(P, Q) \mapsto P \wedge Q$ is bilinear but is not this bracket.
It fails property 2: for a vector field $X$ and a function $f$, it gives $X \wedge f = fX$, while property 2 requires the function $X(f)$.
</document>

<document source="https://tamnd.github.io/math2/node/E5H2GFPK42.md" tag="E5H2GFPK42" kind="statement">
# Hájek–Le Cam convolution theorem

## Statement

Let $\Theta \subseteq \R^k$ be open and let the models $\set{P_{n,\theta} : \theta \in \Theta}$ be [[locally asymptotically normal|6DMQ97EBVG]] at $\theta$, with nonsingular information matrix $I_\theta$.
Let $T_n$ be [[estimators|0TTNRP5PFQ]] of $\theta$ that are *regular* at $\theta$.
This means that there is a probability measure $L_\theta$ on $\R^k$ such that, for every $h \in \R^k$,
$$\sqrt{n}\,\Big(T_n - \big(\theta + \tfrac{h}{\sqrt{n}}\big)\Big) \xrightarrow{d} L_\theta \quad \text{under } P_{n, \theta + h/\sqrt{n}},$$
where $\xrightarrow{d}$ is [[convergence in distribution|0JDAQHWA9N]].
Then there is a probability measure $M_\theta$ on $\R^k$ such that
$$L_\theta = \mathcal{N}_k\big(0, I_\theta^{-1}\big) * M_\theta,$$
the [[convolution|07CHJ5YX28]] of the [[multivariate normal distribution|0G558GJQMW]] $\mathcal{N}_k(0, I_\theta^{-1})$ with $M_\theta$.

## Remarks

Idea of proof: by LAN, the pair $\big(\sqrt{n}(T_n - \theta), \Delta_{n,\theta}\big)$ has a joint limit along subsequences, and Le Cam's third lemma gives its limit under the local alternatives.
Regularity says that this limit does not change with $h$; comparing characteristic functions then shows that the limit of $\sqrt{n}(T_n - \theta) - I_\theta^{-1}\Delta_{n,\theta}$ is independent of the limit of $I_\theta^{-1}\Delta_{n,\theta}$, which is $\mathcal{N}_k(0, I_\theta^{-1})$.

Consequence: if $L_\theta$ has covariance $\Sigma$, then $\Sigma - I_\theta^{-1}$ is positive semidefinite, so $I_\theta^{-1}$ is the best possible asymptotic covariance for regular estimators.
An estimator with $L_\theta = \mathcal{N}_k(0, I_\theta^{-1})$, such as the maximum likelihood estimator in smooth models, is called asymptotically efficient.

The regularity hypothesis is necessary: there exist superefficient estimators that beat $I_\theta^{-1}$ at single points, but they are not regular there.

## Explanations

### L1 human

Any well-behaved estimator, in large samples, behaves like the best possible estimator plus some independent extra noise.
The best possible error is normal with covariance equal to the inverse Fisher information.
So no well-behaved estimator can beat that, and the best ones add no extra noise at all.

### L2 human

The theorem is the asymptotic counterpart of the Cramér–Rao bound, but it needs no unbiasedness and it describes the whole limit law, not just its variance.
"Regular" rules out estimators that are tuned to one parameter value, which could otherwise do better there.
Together with the local asymptotic minimax theorem, it makes precise in what sense maximum likelihood estimators are asymptotically optimal in LAN models.
</document>

<document source="https://tamnd.github.io/math2/node/E5NNY7XM44.md" tag="E5NNY7XM44" kind="counterexample">
# Wente torus

## Counterexample

There is an immersed torus in $\R^3$ whose mean curvature is a nonzero constant.
Such a torus, constructed by Wente in 1986, is called a Wente torus.
It is a compact [[constant-mean-curvature surface|RPBKZ7E9VE]] that is not a round sphere.

## Remarks

Hopf asked whether every compact immersed surface of constant mean curvature in $\R^3$ must be a round sphere.
Hopf proved this for immersed spheres, and Alexandrov proved it for embedded surfaces of any genus.
The Wente torus shows that without embeddedness the answer is no in genus one.
The torus has self-intersections, as it must by Alexandrov's theorem.
Wente's construction reduces the problem to doubly periodic solutions of a sinh-Gordon equation for the conformal factor, together with conditions that make the surface close up.

## Explanations

### L1 human

A soap bubble that does not touch itself is always a round sphere.
For a long time people suspected that the same holds even if the surface may pass through itself.
Wente found a doughnut-shaped surface that passes through itself and still has the same mean curvature everywhere, which settled the question.
</document>

<document source="https://tamnd.github.io/math2/node/E5P8AG6CK7.md" tag="E5P8AG6CK7" kind="example">
# SU(2)

## Example

The group $SU(2)$ is the [[special unitary group|CZCWFPNS4N]] for $n = 2$.
Its elements are exactly the matrices
$$U = \begin{pmatrix} a & -\bar{b} \\ b & \bar{a} \end{pmatrix}, \qquad a, b \in \C, \quad \abs{a}^2 + \abs{b}^2 = 1.$$
Check: for such a matrix, $U^* U = \begin{pmatrix} \abs{a}^2 + \abs{b}^2 & 0 \\ 0 & \abs{a}^2 + \abs{b}^2 \end{pmatrix} = I_2$ and $\det U = \abs{a}^2 + \abs{b}^2 = 1$.
Conversely, a unitary matrix with determinant $1$ has its second column determined by its first column in this way.

So the map $U \mapsto (a, b)$ identifies $SU(2)$ with the unit sphere $S^3 \subset \C^2 = \R^4$.
It is a [[compact Lie group|9TFC6ADC1C]] of dimension $3$.

## Remarks

Under the usual identification of $\C^2$ with the [[quaternions|0GE4DVQKR8]], $SU(2)$ becomes isomorphic to the group of unit quaternions.
Since $S^3$ is simply connected, $SU(2)$ is simply connected.
$SU(2)$ acts on the trace-zero Hermitian $2 \times 2$ matrices, a real vector space of dimension $3$, by $X \mapsto U X U^*$.
This action preserves $-\det X$, which is a positive definite quadratic form on that space.
It gives a surjective homomorphism $SU(2) \to SO(3)$ onto the [[rotation group|263GGJFA21]] with kernel $\set{\pm I_2}$.
So $SU(2)$ is the universal cover of $SO(3)$, and it is also the spin group $\mathrm{Spin}(3)$.
Its Lie algebra $\mathfrak{su}(2)$ has basis $i\sigma_1, i\sigma_2, i\sigma_3$, where $\sigma_1, \sigma_2, \sigma_3$ are the Pauli matrices.

## Explanations

### L1 human

$SU(2)$ is one of the smallest non-commutative compact connected Lie groups.
As a space it is the $3$-sphere, the set of points at distance $1$ from the origin in four dimensions.
Each element gives a rotation of ordinary space, and exactly two elements, $U$ and $-U$, give the same rotation.
Physicists use $SU(2)$ to describe the spin of particles such as the electron.
</document>

<document source="https://tamnd.github.io/math2/node/E6F61940F9.md" tag="E6F61940F9" kind="definition">
# Planar ternary ring

## Definition

A *planar ternary ring* is a set $R$ with two distinct elements $0$ and $1$ and a [[ternary operation|WC47MFR6GW]] $T \colon R^3 \to R$ such that:

1. $T(a, 0, c) = T(0, b, c) = c$ for all $a, b, c \in R$;
2. $T(1, a, 0) = T(a, 1, 0) = a$ for all $a \in R$;
3. for all $a, b, c, d \in R$ with $a \ne c$, there is a unique $x \in R$ with $T(x, a, b) = T(x, c, d)$;
4. for all $a, b, c \in R$, there is a unique $x \in R$ with $T(a, b, x) = c$;
5. for all $a, b, c, d \in R$ with $a \ne c$, there is a unique pair $(x, y) \in R^2$ with $T(a, x, y) = b$ and $T(c, x, y) = d$.

## Remarks

The intended reading is $T(x, m, b) = x m + b$.
Clause 3 says that two lines of different slopes meet in exactly one point.
Clause 4 says that there is one line of given slope through a given point.
Clause 5 says that two points with different first coordinates lie on exactly one non-vertical line.

Example: a field $k$ with $T(a, b, c) = a b + c$ is a planar ternary ring.
Check: clauses 1 and 2 are $a \cdot 0 + c = 0 \cdot b + c = c$ and $1 \cdot a = a \cdot 1 = a$.
Clause 3 is $x(a - c) = d - b$, solved by $x = (d - b)(a - c)^{-1}$, and clause 4 is solved by $x = c - a b$.
Clause 5 is a $2 \times 2$ linear system with determinant $a - c \ne 0$.

Non-example: $\Z$ with $T(a, b, c) = a b + c$ fails clause 3.
For $a = 2$, $b = 0$, $c = 0$, $d = 1$ the equation $2x = 1$ has no solution in $\Z$.

A planar ternary ring has an addition $a + b = T(a, 1, b)$ and a multiplication $a \cdot b = T(a, b, 0)$.
It is *linear* when $T(a, b, c) = a \cdot b + c$ for all $a, b, c$.
Choosing four points in general position in a projective plane gives a planar ternary ring that coordinatizes it, and every planar ternary ring arises this way.
</document>

<document source="https://tamnd.github.io/math2/node/E6GYGZDBS7.md" tag="E6GYGZDBS7" kind="statement">
# Menger's theorem

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]], and let $u$ and $v$ be distinct vertices of $G$.
Two $u$–$v$ [[paths|0JDE56HWQ8]] are internally disjoint if they have no vertex in common other than $u$ and $v$.
A set $S$ of vertices or edges separates $u$ from $v$ if every $u$–$v$ path meets $S$.

1. Suppose that $u$ and $v$ are not adjacent. Consider the sets of vertices, other than $u$ and $v$, that separate $u$ from $v$. The least size of such a set equals the greatest number of pairwise internally disjoint $u$–$v$ paths.
2. The least size of a set of edges that separates $u$ from $v$ equals the greatest number of pairwise edge-disjoint $u$–$v$ paths.

## Remarks

In both parts, the inequality "at least" is easy: each separating set must contain a vertex or edge from each of the disjoint paths.
The other inequality can be proved by induction on the number of edges.
It also follows from the [[max-flow min-cut theorem|4HMR4KQ01T]], with capacity $1$ on each edge or each vertex and integral flows.
In part 1 the hypothesis that $u$ and $v$ are not adjacent is needed: if $uv$ is an edge, no set of other vertices separates $u$ from $v$.
For example, in the cycle $C_6$ two opposite vertices are joined by two internally disjoint paths, and removing one inner vertex from each of these paths separates them.
A global form, due to Whitney, follows.
A graph with more than $k$ vertices is [[$k$-vertex-connected|7FZB7YYBXA]] if and only if every two distinct vertices are joined by $k$ internally disjoint paths.

## Explanations

### L1 human

Think of a road map and two towns.
Count the fewest roads you must close to cut one town off from the other.
Count the most routes between the two towns that share no road.
Menger's theorem says that these two numbers are always equal.

### L2 human

Menger's theorem is the basic min-max theorem of graph connectivity.
It turns the definition of $k$-connectivity, which is about removing vertices, into a statement about disjoint paths, which is often easier to use.
It is a special case of max-flow min-cut, and it is equivalent in strength to [[Kőnig's theorem|0JP0ZF57PN]] and [[Hall's marriage theorem|08WCB90WND]] on bipartite matchings.
</document>

<document source="https://tamnd.github.io/math2/node/E6H1CTT439.md" tag="E6H1CTT439" kind="concept">
# Lawvere–Tierney topology

## Summary

A Lawvere–Tierney topology on a [[topos|6837RS7N82]] is a morphism $j \colon \Omega \to \Omega$ from the [[subobject classifier|NKP5TPQ79X]] to itself.
It behaves like a modality "it is locally true that".
It is the elementary counterpart of a [[Grothendieck topology|GDRFSJ8E2B]]: on a presheaf topos, Lawvere–Tierney topologies correspond to Grothendieck topologies on the base category.
Each such $j$ determines a full subcategory of $j$-sheaves, which is again a topos.
The double negation topology $\neg\neg$ is a standard example and is used in forcing arguments.

## Explanations

### L1 human

In a topos, truth values may be partial, like "true on this part of a space".
A Lawvere–Tierney topology is a rule that makes some statements count as true once they hold "locally enough".
Applying the rule twice gives nothing new, true statements stay true, and the rule respects "and".
For example, take presheaves on the open sets of a space, with the topology given by open covers.
Then a property counts as true on $U$ as soon as it holds on the pieces of a cover of $U$.
</document>

<document source="https://tamnd.github.io/math2/node/E6HDE1DFXD.md" tag="E6HDE1DFXD" kind="concept">
# Energy of a map between Riemannian manifolds

## Summary

The energy of a smooth map between [[Riemannian manifolds|7CBT4SM83Y]] is half the integral, over the domain, of the squared norm of its differential.
It generalizes the Dirichlet energy of a function and the [[energy of a path|Q12AB1MJ64]].
Its critical points are the [[harmonic maps|KZJYG9DS3M]], and its gradient is minus the [[tension field|W8F8S8DX5X]].
It measures how much a map stretches the domain on average.

## Explanations

### L1 human

Picture the domain as a rubber sheet that the map stretches and lays onto the target.
The energy density at a point is the total squared stretching in all directions there, and the energy adds it up over the sheet.
A map that is constant does not stretch at all and has energy $0$.
For example, the map $x \mapsto 2x$ from the interval $[0, 1]$ to the line stretches everything by $2$, so its energy is $\frac{1}{2} \int_0^1 4 \, dx = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/E6MB6B3YQ0.md" tag="E6MB6B3YQ0" kind="definition">
# Realizability

## Definition

Let $\{e\}$ denote the partial [[computable function|KZSB5E471K]] with code $e \in \N$, and write $\{e\}(n)\downarrow$ when it is defined at $n$.
Fix a computable pairing of natural numbers with projections $(e)_0$ and $(e)_1$.
*Number realizability* is the relation "$e$ realizes $A$", written $e \Vdash A$, between a number $e$ and a closed formula $A$ of [[Heyting arithmetic|RXHJPVMPJH]].
It is defined by induction on $A$.

1. $e \Vdash (s = t)$ if $s = t$ is true. No number realizes $0 = 1$.
2. $e \Vdash A \wedge B$ if $(e)_0 \Vdash A$ and $(e)_1 \Vdash B$.
3. $e \Vdash A \vee B$ if either $(e)_0 = 0$ and $(e)_1 \Vdash A$, or $(e)_0 \ne 0$ and $(e)_1 \Vdash B$.
4. $e \Vdash A \to B$ if for every $n$ with $n \Vdash A$, $\{e\}(n)\downarrow$ and $\{e\}(n) \Vdash B$.
5. $e \Vdash \exists x\, A(x)$ if $(e)_1 \Vdash A(\overline{(e)_0})$, where $\overline{k}$ is the numeral of $k$.
6. $e \Vdash \forall x\, A(x)$ if for every $n$, $\{e\}(n)\downarrow$ and $\{e\}(n) \Vdash A(\overline{n})$.

A closed formula is *realizable* if some number realizes it.
Negation $\neg A$ is read as $A \to (0 = 1)$.

## Remarks

Soundness theorem (Kleene): if Heyting arithmetic proves a closed formula $A$, then $A$ is realizable.
Example: $\forall x\, \exists y\, (y = x + x)$ is realized by a code $e$ of a total function with $\{e\}(n) = \langle 2n, 0 \rangle$.
Indeed, $\{e\}(n)$ is defined, its first part is $2n$, and its second part realizes the true equation $2n = n + n$ by clause 1.
Non-example: let $H(x)$ say that the program $x$ halts on input $x$.
A realizer of $\forall x\, (H(x) \vee \neg H(x))$ would compute, from $x$, a bit telling whether $H(x)$ holds, and so solve the halting problem.
So this instance of the law of excluded middle is not realizable, and hence not provable in Heyting arithmetic.
Many variants exist, such as realizability by functions, by terms of a typed calculus, and in toposes.
</document>

<document source="https://tamnd.github.io/math2/node/E6MGJDRXJ0.md" tag="E6MGJDRXJ0" kind="definition">
# Infinite product measure

## Definition

Let $I$ be a nonempty index set and, for each $i \in I$, let $(X_i, \Sigma_i, \mu_i)$ be a [[probability space|0NFPJVJJHX]].
Let $X = \prod_{i \in I} X_i$ with the [[product σ-algebra|6HVCTJGB61]] $\Sigma = \bigotimes_{i \in I} \Sigma_i$.
The *product measure* $\bigotimes_{i \in I} \mu_i$ is the unique [[probability measure|059TJN0NZ1]] $\mu$ on $(X, \Sigma)$ with
$$\mu\Big( \prod_{i \in J} A_i \times \prod_{i \in I \setminus J} X_i \Big) = \prod_{i \in J} \mu_i(A_i)$$
for every finite $J \subseteq I$ and all $A_i \in \Sigma_i$, $i \in J$.

## Remarks

Such a measure exists for every family of probability spaces, without topological assumptions, and it is unique because the cylinder sets form a π-system that generates $\Sigma$.
For countable $I$, existence also follows from the [[Ionescu-Tulcea theorem|94ASD187AQ]] with kernels that do not depend on the past.

Example: let $X_n = \set{0, 1}$ and $\mu_n(\set{0}) = \mu_n(\set{1}) = \tfrac12$ for $n = 1, 2, 3, \ldots$
The product measure on the space of sequences $(x_1, x_2, \ldots)$ with $x_n \in \set{0, 1}$ is the law of a sequence of fair coin tosses.
The set of sequences starting with $0, 1$ is a cylinder set with $J = \set{1, 2}$ and measure $\tfrac12 \cdot \tfrac12 = \tfrac14$.

Non-example: on the same space, the Dirac measure at the constant sequence $(0, 0, 0, \ldots)$ is a probability measure.
It gives the cylinder set $\set{x : x_1 = 0}$ measure $1$ instead of $\tfrac12$, so it is not the product of the fair coin measures.

Under the product measure, the coordinate maps $\pi_i$ are independent random variables, and $\pi_i$ has law $\mu_i$.
</document>

<document source="https://tamnd.github.io/math2/node/E7063PCV9P.md" tag="E7063PCV9P" kind="definition">
# Green's function of a Markov process

## Definition

Let $(E, \mathcal{E})$ be a measurable space with a σ-finite [[measure|PR11BZQQPG]] $m$.
Let $(P_t)$ be a sub-Markov transition semigroup on $E$ with [[potential kernel|595AZJAZNA]] $U$.
A *Green's function* (or potential density) of $(P_t)$ with respect to $m$ is a measurable function $u \colon E \times E \to [0, \infty]$ such that
$$U(x, A) = \int_A u(x, y) \, m(dy) \qquad \text{for every } x \in E \text{ and } A \in \mathcal{E}.$$

## Remarks

For each $x$ the function $u(x, \cdot)$ is determined only $m$-almost everywhere.
One usually fixes a version for which $x \mapsto u(x, y)$ is [[excessive|46436KE7NT]] for each $y$; under standard duality hypotheses such a version exists and is unique.
Example: for [[Brownian motion|22735FGJV6]] in $\R^3$ with Lebesgue measure, $u(x, y) = 1/(2\pi\abs{x - y})$, because this is the density of $U(x, \cdot)$.
For Brownian motion killed on leaving a planar domain $D$, the Green's function is $u = g_D/\pi$.
Here $g_D$ is the classical [[Green's function|M36SD56MWQ]] with $\Delta g_D(\cdot, y) = -2\pi\delta_y$, and the factor comes from the generator $\frac12 \Delta$.
For a [[Markov chain|0G48PYM96Q]] on a countable set with counting measure, $u(x, y) = \sum_{n \ge 0} P^n(x, y)$ is the expected number of visits to $y$ starting from $x$.
Non-example: Brownian motion in $\R^2$ on the whole plane has no Green's function with finite values, since $U(x, A) = \infty$ for every nonempty open $A$.
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<document source="https://tamnd.github.io/math2/node/E70DQGM8X5.md" tag="E70DQGM8X5" kind="concept">
# Kernel pair

## Summary

The kernel pair of a morphism $f \colon X \to Y$ is the [[pullback|Q05K04QJ71]] of $f$ along itself.
It records which pairs of elements $f$ sends to the same place.
So it is the categorical form of the equivalence relation induced by a map.
A morphism $f \colon X \to Y$ is a [[monomorphism|Y2NP1JFR86]] exactly when $X$ with two identity maps is a kernel pair of $f$.
In a [[regular category|2SM3JRJXA4]], a regular epimorphism is the coequalizer of its kernel pair.

## Explanations

### L1 human

A function can send different inputs to the same output.
The kernel pair collects all pairs of inputs with the same output.
For the map from the integers to the remainders modulo $3$, it is the set of pairs $(a, b)$ with $a - b$ divisible by $3$.
For a group homomorphism, it carries the same information as the kernel.
</document>

<document source="https://tamnd.github.io/math2/node/E76951YS23.md" tag="E76951YS23" kind="definition">
# Bessel function of the second kind

## Definition

Let $J_\nu$ denote the [[Bessel function of the first kind|CZCMRDWDYW]] of order $\nu$, taken on $\C \setminus (-\infty, 0]$ with the principal branch of $z^\nu$.
For $\nu \in \C \setminus \Z$, the *Bessel function of the second kind* of order $\nu$ is
$$Y_\nu(z) = \frac{J_\nu(z) \cos(\nu \pi) - J_{-\nu}(z)}{\sin(\nu \pi)}.$$
For an integer $n$, it is the limit
$$Y_n(z) = \lim_{\nu \to n} Y_\nu(z).$$

## Remarks

The function $Y_\nu$ solves Bessel's equation $z^2 y'' + z y' + (z^2 - \nu^2) y = 0$, because it is a linear combination of solutions for non-integer $\nu$ and a locally uniform limit of solutions for integer $\nu$.
Its Wronskian with $J_\nu$ is $W(J_\nu, Y_\nu)(z) = \frac{2}{\pi z}$, which never vanishes, so $J_\nu$ and $Y_\nu$ are independent for every $\nu$, integers included.
This is the point of the definition: for an integer $n$ one has $J_{-n} = (-1)^n J_n$, so $J_n$ and $J_{-n}$ do not form a basis, while $J_n$ and $Y_n$ do.
For $\nu = \frac{1}{2}$ the formula gives $Y_{1/2}(x) = -J_{-1/2}(x) = -\sqrt{\frac{2}{\pi x}} \cos x$ for $x > 0$.
A non-example: the function $J_{-\nu}$ alone is not $Y_\nu$; it is a second solution only when $\nu \notin \Z$.
The notation $N_\nu$ is also used for $Y_\nu$.
</document>

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# Total variation of a measure

## Definition

Let $(X, \Sigma)$ be a measurable space and let $\nu$ be a [[signed measure|VG2P7R91Z7]] or a [[complex measure|NY6TYHRGMS]] on it.
The *total variation* of $\nu$ is the map $\abs{\nu} \colon \Sigma \to [0, \infty]$ given by
$$\abs{\nu}(E) = \sup \set{ \sum_{i=1}^{n} \abs{\nu(E_i)} : n \ge 1, \ E_1, \ldots, E_n \in \Sigma \text{ pairwise disjoint}, \ E = E_1 \cup \cdots \cup E_n },$$
where the [[supremum|0BTMQ04556]] is over all finite partitions of $E$ into measurable sets.
The *total variation norm* of $\nu$ is $\norm{\nu} = \abs{\nu}(X)$.

## Remarks

The map $\abs{\nu}$ is a measure, and $\abs{\nu(E)} \le \abs{\nu}(E)$ for all $E \in \Sigma$.
For a complex measure, $\abs{\nu}$ is a finite measure.

Example: let $\nu = \delta_0 - \delta_1$ on the subsets of $\R$, where $\delta_a$ is the Dirac measure at $a$.
For $E = \set{0, 1}$, one has $\nu(E) = 0$, but the partition $\set{0}, \set{1}$ gives $\abs{1} + \abs{-1} = 2$.
No partition gives more, so $\abs{\nu}(E) = 2$ and $\norm{\nu} = 2$.

Non-example: the map $E \mapsto \abs{\nu(E)}$ is not the total variation in general.
For the same $\nu$ it gives $0$ on $\set{0, 1}$, and it is not even additive: $\abs{\nu(\set{0})} + \abs{\nu(\set{1})} = 2 \ne 0$.

For a signed measure with Jordan decomposition $\nu = \nu^+ - \nu^-$, one has $\abs{\nu} = \nu^+ + \nu^-$.
If $\nu(E) = \int_E f \, d\mu$ with $f \in L^1(\mu)$, then $\abs{\nu}(E) = \int_E \abs{f} \, d\mu$.
</document>

<document source="https://tamnd.github.io/math2/node/E77T9G4DV7.md" tag="E77T9G4DV7" kind="definition">
# Littlewood's three principles

## Definition

Let $\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\R$, and let "nearly" mean "outside a set of arbitrarily small measure".
*Littlewood's three principles* are the following three heuristic statements:

1. Every Lebesgue measurable set of finite measure is nearly a finite union of intervals.
2. Every [[measurable function|YYNCJSR0PJ]] is nearly [[continuous|0X3WGX95FE]].
3. Every pointwise convergent sequence of measurable functions is nearly [[uniformly convergent|08VZ57PT26]].

## Remarks

The precise forms on a set $E \subseteq \R$ with $\lambda(E) < \infty$ are as follows.
For principle 1: for every $\varepsilon > 0$ there is a finite union $U$ of intervals with $\lambda(E \mathbin{\triangle} U) < \varepsilon$.
For principle 2: this is [[Lusin's theorem|GZMAH9W2SK]], which gives a closed $F \subseteq E$ with $\lambda(E \setminus F) < \varepsilon$ on which $f$ is continuous.
For principle 3: this is [[Egorov's theorem|762Y2NYA7W]], which gives uniform convergence outside a set of measure less than $\varepsilon$.
Example: for $f = \mathbf{1}_{\Q}$ on $[0, 1]$, take $F$ a closed set of irrationals with $\lambda(F) > 1 - \varepsilon$; then $f|_F = 0$ is continuous.
Without finite measure, principle 3 fails: on $\R$, the functions $\mathbf{1}_{[n, \infty)}$ tend to $0$ pointwise, but not uniformly outside any set of finite measure.

## Explanations

### L1 human

The three principles say that measure theory is close to ordinary analysis, as long as one is allowed to ignore a small set.
Sets are almost intervals, functions are almost continuous, and convergence is almost uniform.
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<document source="https://tamnd.github.io/math2/node/E7890PHMGP.md" tag="E7890PHMGP" kind="concept">
# Functional square root

## Summary

A functional square root of a map $f$ is a map $g$ with $g(g(x)) = f(x)$ for all $x$.
It is the [[iterative root|27C6J3YNZY]] of order $2$, and it is often called a half-iterate of $f$.
It answers the question of what it means to apply $f$ half a time.
Functional square roots of the exponential function and of other simple maps have a long history in iteration theory.

## Explanations

### L1 human

Squaring a number has an inverse question: which number squared gives the result?
For maps, the analogue of squaring is applying the map twice.
A functional square root of $f$ is a map that, done twice, has the same effect as $f$ done once.
For $f(x) = x + 2$ one root is $g(x) = x + 1$; for $f(x) = 4x$ two roots are $2x$ and $-2x$.
</document>

<document source="https://tamnd.github.io/math2/node/E7A5DHFT01.md" tag="E7A5DHFT01" kind="concept">
# Zariski tangent space

## Summary

The Zariski tangent space of a variety at a point is the linear space of directions in which the defining equations vanish to first order.
For a variety in $k^n$ it is the kernel of the Jacobian matrix of the defining polynomials at the point.
Intrinsically it is the dual of $\mathfrak m / \mathfrak m^2$, where $\mathfrak m$ is the maximal ideal of the point.
Its dimension is at least the dimension of the variety, with equality exactly at the nonsingular points.

## Explanations

### L1 human

Near a point, replace each defining equation by its best linear approximation, its differential at the point.
The common solutions of these linear equations form the tangent space.
For the circle $x^2 + y^2 = 1$ at $(1, 0)$, the differential is $2x \, dx + 2y \, dy = 2 \, dx$, so the tangent space is the vertical line $dx = 0$.
For the cusp $y^2 = x^3$ at the origin, all the differentials vanish, so the tangent space is the whole plane, larger than the curve's dimension $1$.
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<document source="https://tamnd.github.io/math2/node/E7B78X780X.md" tag="E7B78X780X" kind="statement">
# Glivenko–Cantelli theorem

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|00MRN0FARV]] real random variables with [[cumulative distribution function|0H8H7ZTHW2]] $F$.
For $n \ge 1$ and $x \in \R$, let
$$F_n(x) = \frac{1}{n} \#\set{k \le n : X_k \le x}$$
be the [[empirical distribution function|A05WR6TEK5]].
Then, [[almost surely|0ST21PGDAP]],
$$\sup_{x \in \R} \abs{F_n(x) - F(x)} \to 0 \quad \text{as } n \to \infty.$$

## Remarks

For each fixed $x$, the [[strong law of large numbers|0DFWHQ839T]] applied to the indicators $\mathbf{1}_{\set{X_k \le x}}$ gives $F_n(x) \to F(x)$ almost surely.
The same holds for the left limits $F_n(x-) \to F(x-)$.
To make the convergence [[uniform|08VZ57PT26]], choose finitely many points $x_0 < x_1 < \cdots < x_m$ such that $F$ grows by at most $\varepsilon$ between consecutive points, apart from jumps at the points themselves.
Since $F_n$ and $F$ are non-decreasing, the uniform error is at most $\varepsilon$ plus the error at these finitely many points.

Example: if the $X_k$ are uniform on $[0, 1]$, then the step function $F_n$ lies within any fixed distance of the line $F(x) = x$ on $[0, 1]$, for all large $n$, almost surely.

The rate is given by the Dvoretzky–Kiefer–Wolfowitz inequality, and the fluctuations of order $n^{-1/2}$ are described by the [[empirical process|XKFG1676CG]] and [[Donsker's theorem|818832PMSV]].

## Explanations

### L1 human

Take a large sample from an unknown distribution and draw the step function that records which fraction of the sample lies below each value.
This step function gets close to the true distribution function everywhere at once as the sample grows.
So a large sample reveals the whole distribution, not only its mean.

### L2 human

The theorem is often called the fundamental theorem of statistics, since it justifies estimating a distribution by its empirical version.
It is the first uniform law of large numbers.
Empirical process theory generalises it to classes of sets or functions, called Glivenko–Cantelli classes, on which the empirical averages converge uniformly; the Vapnik–Chervonenkis condition is a sufficient criterion.
</document>

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# Complexification of a real analytic manifold

## Statement

Let $M$ be a [[real analytic manifold|Z7H98E1NHP]] of dimension $n$ whose topology has a countable basis.
Then there is a [[complex manifold|JSGWAN2RJV]] $X$ of complex dimension $n$ that contains $M$ as a closed subset, with the following property.
Each point $p \in M$ has a holomorphic chart $\varphi \colon U \to \C^n$ of $X$ around $p$ with $\varphi(U \cap M) = \varphi(U) \cap \R^n$.
Moreover $\varphi|_{U \cap M}$ is a real analytic chart of $M$.
Such an $X$ is a complexification of $M$.
It is unique as a germ, in the following sense.
Let $X$ and $X'$ be two complexifications of $M$.
Then there are open neighbourhoods $V \subseteq X$ and $V' \subseteq X'$ of $M$ and a biholomorphic map $V \to V'$ that is the identity on $M$.

## Remarks

The existence is due to Whitney and Bruhat.
Locally, a real analytic chart extends to a holomorphic chart by replacing real variables by complex ones in the convergent power series of the transition maps.
The transition maps of $M$ then extend holomorphically to small complex neighbourhoods.
One glues these local pieces along a neighbourhood of $M$ that is chosen thin enough for the extensions to agree.
Uniqueness follows from the identity theorem: a holomorphic map is determined by its values on a totally real submanifold of maximal dimension.

Examples: $\R^n$ has the complexification $\C^n$.
The circle $S^1 = \set{x^2 + y^2 = 1} \subseteq \R^2$ has the complexification $\set{(z, w) \in \C^2 : z^2 + w^2 = 1}$, which is biholomorphic to $\C \setminus \set{0}$ by $(z, w) \mapsto z + i w$.

By a theorem of Grauert, the complexification $X$ can be chosen to be a Stein manifold.

## Explanations

### L1 human

A real analytic manifold is described by convergent power series in real variables.
Feeding complex numbers into these series gives a complex manifold of the same dimension that contains the real one, as $\C^n$ contains $\R^n$.
Near the real manifold this complex manifold is unique.

### L2 human

Complexification moves problems about real analytic objects to complex analysis, where stronger tools are available.
A real analytic function on $M$ extends to a holomorphic function on a neighbourhood of $M$ in $X$.
With a Stein complexification, Cartan's theorems give many such functions, which leads to Grauert's analytic embedding of $M$ in some $\R^N$.
</document>

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# Roth's theorem on arithmetic progressions

## Statement

Let $A$ be a set of positive integers with
$$\limsup_{N \to \infty} \frac{\abs{A \cap \set{1, \dots, N}}}{N} > 0,$$
where $\limsup$ is the [[limit superior|0T3RZQXDYD]].
Then $A$ contains three terms $a, a + d, a + 2d$ of an arithmetic progression with $d \ge 1$.

## Remarks

Let $r_3(N)$ be the largest size of a subset of $\set{1, \dots, N}$ with no three-term arithmetic progression.
The theorem is equivalent to $r_3(N) / N \to 0$ as $N \to \infty$.
Roth's proof uses Fourier analysis and a density increment.
A set without three-term progressions has a large Fourier coefficient, so it is denser on some long sub-progression.
This step cannot repeat forever, because the density cannot exceed $1$.
Another proof goes through the [[triangle removal lemma|XMS1QRAQZC]].
Behrend constructed sets showing $r_3(N) \ge N \exp(-c\sqrt{\log N})$ for a constant $c > 0$, so $r_3(N)$ is not bounded by $N^{1 - \varepsilon}$ for any $\varepsilon > 0$.
Kelley and Meka proved in 2023 an upper bound of the form $r_3(N) \le N \exp(-c (\log N)^{\beta})$ with constants $c, \beta > 0$.
The analogue in $\F_3^n$ is the [[cap set problem|1SF8Y9M8YB]].

## Explanations

### L1 human

Take any set of whole numbers that fills a fixed positive fraction of the numbers.
Then it contains three numbers equally spaced, like $5, 9, 13$.
The odd numbers, for example, contain $1, 3, 5$.

### L2 human

Roth's theorem is the case $k = 3$ of [[Szemerédi's theorem|B23B0QEMN5]], and it was proved first.
Its Fourier-analytic density-increment argument is a model for many proofs in [[additive combinatorics|3NM362A2JG]].
The gap between Behrend's lower bound and the best upper bounds for $r_3(N)$ is still being studied.
</document>

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# Order of a PDE

## Summary

The order of a [[partial differential equation|GC5BDZ6QBC]] is the highest order of a partial derivative of the unknown function that occurs in it.
It is the first number used to sort PDEs, and it controls how many initial or boundary conditions a well-posed problem needs.
See the [[definition|YFFEKD37RP]].

## Explanations

### L1 human

Count how many times the unknown is differentiated in each term, and take the largest count.
The [[transport equation|PVD7VZJZ9B]] $\partial_t u + \partial_x u = 0$ has order $1$.
The [[wave equation|5EHD2XBQDG]] $\partial_t^2 u = \partial_x^2 u$ has order $2$.
A mixed derivative such as $\partial_x \partial_y u$ also counts as order $2$.
</document>

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# Projective tensor product

## Definition

Let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\mathbb{K} = \R$ or $\C$, and let $X \otimes Y$ be their algebraic [[tensor product|0BF2DE2SYK]].
For $u \in X \otimes Y$, let
$$\pi(u) = \inf \Big\{ \sum_{i=1}^n \norm{x_i} \, \norm{y_i} : u = \sum_{i=1}^n x_i \otimes y_i \Big\}.$$
The infimum runs over all finite representations of $u$.
Then $\pi$ is a norm on $X \otimes Y$, the *projective norm*.
The *projective tensor product* $X \widehat{\otimes}_\pi Y$ is the completion of $X \otimes Y$ for $\pi$.

## Remarks

The norm is a cross norm: $\pi(x \otimes y) = \norm{x} \, \norm{y}$.
The bound $\le$ comes from the representation with one term.
For $\ge$, take functionals $f \in X^*$ and $g \in Y^*$ of norm $1$ with $f(x) = \norm{x}$ and $g(y) = \norm{y}$, given by the Hahn–Banach theorem.
For any representation, $\norm{x} \, \norm{y} = \abs{\sum_i f(x_i) g(y_i)} \le \sum_i \norm{x_i} \, \norm{y_i}$.

The infimum is needed: $u = 0$ can be written as $x \otimes y + x \otimes (-y)$, with cost $2 \norm{x} \, \norm{y}$.
A single representation does not give a well-defined number.

Universal property: bounded bilinear maps $B \colon X \times Y \to Z$ into a Banach space $Z$ correspond to bounded linear maps $X \widehat{\otimes}_\pi Y \to Z$ with the same norm.
In particular, the dual of $X \widehat{\otimes}_\pi Y$ is the space of bounded linear maps from $X$ to $Y^*$.
For example, $\ell^1 \widehat{\otimes}_\pi \ell^1$ is isometric to $\ell^1(\N \times \N)$.
</document>

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# Faber–Krahn inequality

## Statement

Let $n \ge 1$ and let $\Omega \subseteq \R^n$ be a nonempty bounded open set.
Let $\Omega^*$ be the open ball in $\R^n$ with the same [[Lebesgue measure|2Z2QVWZ262]] as $\Omega$.
Let $\lambda_1(\Omega)$ be the first [[Dirichlet eigenvalue|4VD8XXQ4P3]] of $\Omega$.
Then
$$\lambda_1(\Omega) \ge \lambda_1(\Omega^*).$$
Equality holds if and only if $\Omega$ is a ball up to a set of zero capacity.

## Remarks

Proof idea: let $u \ge 0$ be a first eigenfunction of $\Omega$ and let $u^*$ be its [[symmetric decreasing rearrangement|2GRPCP5XK9]], which lies in $H^1_0(\Omega^*)$.
Rearrangement keeps $\int u^2$ and does not increase $\int \abs{\nabla u}^2$, by the Pólya–Szegő inequality.
So the Rayleigh quotient of $u^*$ on $\Omega^*$ is at most $\lambda_1(\Omega)$, which bounds $\lambda_1(\Omega^*)$.

Example: in the plane, a square of side $\sqrt{\pi}$ has area $\pi$, the same as the unit disc.
Its first eigenvalue is $2\pi^2 / \pi = 2\pi \approx 6.28$.
The unit disc has $\lambda_1 = j_{0,1}^2 \approx 5.78$, where $j_{0,1}$ is the first zero of the Bessel function $J_0$.

The volume constraint is needed: dilating a ball by a factor $s$ divides $\lambda_1$ by $s^2$, so without fixing the volume there is no lower bound.

## Explanations

### L1 human

Among all drums of the same area, the round drum has the lowest fundamental tone.
This was conjectured by Lord Rayleigh and proved by Faber and Krahn.
It is a spectral analogue of the [[isoperimetric inequality|KBX8RBZ626]], which says the disc has the shortest boundary for a given area.

### L2 human

The inequality is the prototype of shape optimisation for eigenvalues.
Its proof combines the variational characterisation of $\lambda_1$ with symmetrisation, and the same method gives the Pólya–Szegő and Saint-Venant inequalities.
Related results include the Szegő–Weinberger inequality, where the ball maximises the first nonzero Neumann eigenvalue, and the Hong–Krahn–Szegő inequality for $\lambda_2$, where two equal disjoint balls are optimal.
</document>

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# Ramification

## Definition

Let $L/K$ be a finite extension of [[algebraic number fields|MWBDFS2KJ5]], with rings of integers $\mathcal{O}_K \subseteq \mathcal{O}_L$.
Let $\mathfrak{p}$ be a nonzero [[prime ideal|00MHHZ9SNE]] of $\mathcal{O}_K$, and let $\mathfrak{P}$ be a prime ideal of $\mathcal{O}_L$ with $\mathfrak{P} \cap \mathcal{O}_K = \mathfrak{p}$.
The *ramification index* $e(\mathfrak{P} \mid \mathfrak{p})$ is the exponent of $\mathfrak{P}$ in the factorisation of $\mathfrak{p}\mathcal{O}_L$ into prime ideals of the [[Dedekind domain|7NPY2CN5KB]] $\mathcal{O}_L$.
The prime $\mathfrak{P}$ is *ramified* over $\mathfrak{p}$ if $e(\mathfrak{P} \mid \mathfrak{p}) > 1$.
The prime $\mathfrak{p}$ is *ramified in $L$* if some prime of $\mathcal{O}_L$ over $\mathfrak{p}$ is ramified, and *unramified in $L$* otherwise.

## Remarks

For number fields the residue field extensions are finite extensions of finite fields, hence separable, so no separability condition is needed.
A prime number $p$ ramifies in a number field $L$ if and only if $p$ divides the [[discriminant|E0VZTNPJXA]] $d_L$; in particular only finitely many primes ramify.
See [[splitting of prime ideals|B7KNZCBFYT]] for the full description of the factorisation.

Example: in $L = \Q(\sqrt{-5})$, the ideal $2\mathcal{O}_L$ equals $\mathfrak{P}^2$ with $\mathfrak{P} = (2, 1 + \sqrt{-5})$.
So $e(\mathfrak{P} \mid 2) = 2$ and $2$ is ramified, in line with $2 \mid d_L = -20$.

Non-example: $3$ is unramified in $\Q(\sqrt{-5})$.
Indeed $x^2 + 5 \equiv (x - 1)(x + 1) \pmod 3$ has distinct roots, so $3\mathcal{O}_L = (3, \sqrt{-5} - 1)(3, \sqrt{-5} + 1)$ with two distinct primes, each of exponent $1$; and $3 \nmid 20$.
</document>

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# Characterizations of discrete valuation rings

## Statement

Let $R$ be a [[Noetherian|0670T59HGE]] [[local|MZ02BKPHJA]] [[integral domain|0QVD7WBFGN]] of [[Krull dimension|MVGBS3J1P7]] $1$, with maximal ideal $\mathfrak{m}$ and [[residue field|NJ3APQ8BHH]] $k = R/\mathfrak{m}$.
The following conditions are equivalent:

1. $R$ is a [[discrete valuation ring|MGM3CNJW9V]].
2. $R$ is [[integrally closed|H20P600HQS]] in its field of fractions.
3. $\mathfrak{m}$ is a [[principal ideal|0SVYSP8HCG]].
4. $\dim_k \mathfrak{m}/\mathfrak{m}^2 = 1$.
5. Every nonzero ideal of $R$ is a power $\mathfrak{m}^n$ with $n \ge 0$.
6. There is $x \in R$ such that every nonzero ideal of $R$ is $(x^n)$ for some $n \ge 0$.

## Remarks

The key step is that 2 implies 3.
Take $a \in \mathfrak{m}$ nonzero; since $\mathfrak{m}$ is the only nonzero prime, some power $\mathfrak{m}^n$ lies in $(a)$; with $n$ minimal one can pick $b \in \mathfrak{m}^{n-1}$ with $b \notin (a)$.
For $y = a/b$, the element $y^{-1}$ is not in $R$, so by integral closedness it is not integral, and then $y^{-1}\mathfrak{m} \not\subseteq \mathfrak{m}$; this forces $y^{-1}\mathfrak{m} = R$, so $\mathfrak{m} = (y)$.
The other implications use Nakayama's lemma and the fact that $\bigcap_n \mathfrak{m}^n = 0$.

Condition 4 says that $R$ is a [[regular local ring|63XXTF6B5M]] of dimension $1$.
Example: the local ring of $k[x, y]/(y - x^2)$ at the origin is a discrete valuation ring, with $\mathfrak{m} = (x)$, since $y = x^2$ there.
The local ring of $k[x, y]/(y^2 - x^3)$ at the origin is not: its maximal ideal $(x, y)$ needs two generators, and $t = y/x$ satisfies $t^2 = x$, so it is not integrally closed.

The hypotheses matter.
Without dimension $1$, the ring $k[\![x, y]\!]$ is a Noetherian integrally closed local domain, but not a discrete valuation ring.
Without the Noetherian condition, a valuation ring with value group $\Q$ is an integrally closed local domain of dimension $1$, but its maximal ideal is not principal.

## Explanations

### L1 human

Discrete valuation rings can be recognised in several ways.
Among the "one-dimensional" local rings, they are exactly the ones without singularity.
"Without singularity" can mean that no fraction is missing which should be there.
It can also mean that the maximal ideal has a single generator, or that all ideals are powers of one ideal.
The theorem says that all these meanings agree.

### L2 human

This result is the local step in the theory of [[Dedekind domains|7NPY2CN5KB]].
A Noetherian domain of dimension $1$ is a Dedekind domain exactly when its localizations at nonzero primes are discrete valuation rings.
In geometry, it says that a point on a curve is nonsingular exactly when the local ring there is integrally closed, which is why normalization resolves the singularities of curves.
Condition 4 links the theory with [[regular local rings|63XXTF6B5M]], which generalise discrete valuation rings to higher dimension.
</document>

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# Local time

## Summary

The local time $L(t, x)$ of a [[Brownian motion|22735FGJV6]] measures how much time the path spends near the level $x$ up to time $t$.
It is the density of the occupation measure of the path with respect to Lebesgue measure.
Local time is the correction term in [[Tanaka's formula|GEM0ERJ5ZB]], which extends [[Itô's lemma|2J1RVQB0CM]] to the function $\abs{x}$.

## Explanations

### L1 human

A Brownian path hits each level it visits infinitely often, but the set of times it sits exactly at a level has length zero.
So the plain time spent at $x$ is always zero, and it says nothing.
Local time instead counts the time spent in a thin band $(x - \varepsilon, x + \varepsilon)$ and divides by the width $2\varepsilon$.
As $\varepsilon$ shrinks, this ratio has a finite positive limit at levels the path has crossed.
For a standard Brownian motion started at $0$, the expected local time at $0$ up to time $t$ is $\sqrt{2t/\pi}$.
</document>

<document source="https://tamnd.github.io/math2/node/E820CTMJVM.md" tag="E820CTMJVM" kind="statement">
# Convex functions of subharmonic functions

## Statement

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]].

1. Let $u$ be [[subharmonic|Z1MHMGFNFW]] on $\Omega$ and let $\varphi \colon \R \to \R$ be [[convex|07Q7CNM5WF]] and increasing.
Put $\varphi(-\infty) = \lim_{t \to -\infty} \varphi(t) \in [-\infty, \infty)$.
Then $\varphi \circ u$ is subharmonic on $\Omega$.
2. Let $h$ be [[harmonic|0DEB0JMN0W]] on $\Omega$ and let $\varphi \colon \R \to \R$ be convex.
Then $\varphi \circ h$ is subharmonic on $\Omega$.

## Remarks

Proof idea: a convex function on $\R$ is continuous, and an increasing continuous function preserves upper semicontinuity.
For the sub-mean inequality on a sphere with normalised measure $\sigma$, monotonicity and [[Jensen's inequality|0RWMRTPC87]] give
$$\varphi(u(a)) \le \varphi\Big(\int u \, d\sigma\Big) \le \int \varphi(u) \, d\sigma.$$
In part 2 the first step is an equality, by the mean value property of $h$, so monotonicity of $\varphi$ is not needed.
If $\int u \, d\sigma = -\infty$, then $u(a) = -\infty$ and $\varphi(u(a)) = \inf \varphi$, which is below every value of $\varphi \circ u$.

Examples: if $u$ is subharmonic, so are $e^{u}$ and $\max(u, 0)$; if $h$ is harmonic, so are $\abs{h}$ and $h^2$.
Monotonicity is needed in part 1: $u(x) = \abs{x}^2$ is subharmonic and $\varphi(t) = -t$ is convex, but $\varphi \circ u = -\abs{x}^2$ is not subharmonic.

## Explanations

### L1 human

Subharmonic functions sit below their averages.
Applying an increasing convex function keeps them below their averages, because a convex function of an average is at most the average of the convex function.
So for example $e^u$ is subharmonic whenever $u$ is.

### L2 human

This is the multi-dimensional analogue of the fact that an increasing convex function of a convex function is convex.
Combined with the subharmonicity of $\log\abs{f}$ for holomorphic $f$, it shows that $\abs{f}^p$ is subharmonic for all $p > 0$, which is the starting point of the theory of Hardy spaces.
It also shows that $\log u$ need not be subharmonic when $u$ is, since $\log$ is concave; functions with $\log u$ subharmonic form the smaller class of log-subharmonic functions.
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# Monomial order

## Summary

A monomial order is a total order on the monomials of a polynomial ring that is compatible with multiplication and has no infinite descending chains.
It singles out a leading term in every nonzero polynomial, which is what makes the [[multivariate division algorithm|T8EN3CAEQF]] and [[Gröbner bases|C83KPT47R0]] work.
Common examples are the lexicographic order, the graded lexicographic order and the graded reverse lexicographic order.

## Explanations

### L1 human

For polynomials in one variable, terms are ordered by degree, and the leading term is the one of highest degree.
With several variables, there is no single natural choice, so one fixes a rule that compares any two monomials, such as $x^2 y$ and $x y^3$.
The rule must respect multiplication: if $u$ comes before $v$, then $wu$ comes before $wv$.
It must also make $1$ the smallest monomial, so that repeatedly cancelling leading terms always stops.
In the lexicographic order with $x > y$, one compares the powers of $x$ first, so $x^2 y > x y^3$.
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<document source="https://tamnd.github.io/math2/node/E8DAD7052E.md" tag="E8DAD7052E" kind="concept">
# Birth–death process

## Summary

A birth–death process is a [[continuous-time Markov chain|TWQFE0FZZS]] on the integers $0, 1, 2, \ldots$ that moves one step up or one step down at each jump.
The state is read as the size of a population: a step up is a birth and a step down is a death.
When it has a stationary distribution, that distribution has an explicit product formula.
It is the basic model for queues, population growth and chemical reactions.

## Explanations

### L1 human

Think of the number of people waiting in a shop.
A new customer arrives, so the number goes up by one, or a customer leaves, so it goes down by one.
Two things never happen at the same instant.
The speed of arrivals and of departures may depend on how many people are already there.
For example, the [[M/M/1 queue|GTFRPF5J2K]] has arrivals at a constant rate and departures at a constant rate whenever the shop is not empty.
</document>

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# Immersed boundary method

## Definition

Let $\Omega \subset \R^d$ be a fluid domain with density $\rho > 0$ and viscosity $\mu > 0$.
Let a structure be a curve or surface $X(s, t) \in \Omega$, with parameter $s$ in a set $S$.
Let $F(s, t)$ be the elastic force density of the structure, a given function of the configuration $X(\cdot, t)$.
The *immersed boundary formulation* couples the incompressible [[Navier–Stokes equations|18KXZHVF4M]]
$$\rho (u_t + u \cdot \nabla u) = -\nabla p + \mu \Delta u + f, \qquad \nabla \cdot u = 0,$$
with two interaction equations, written with the [[Dirac delta function|0KPHENZ5S7]] $\delta$:
$$f(x, t) = \int_S F(s, t) \, \delta(x - X(s, t)) \, ds, \qquad \frac{\partial X}{\partial t}(s, t) = \int_\Omega u(x, t) \, \delta(x - X(s, t)) \, dx.$$
The *immersed boundary method* discretizes it in three parts:

1. the fluid equations on a fixed Cartesian grid of width $h$, by [[finite differences|X7HYEBA5SA]] or another grid method;
2. the structure by a finite set of points $X_k$;
3. $\delta$ by a smooth discrete delta function $\delta_h$, supported on a few grid cells.

## Remarks

The second interaction equation says that the structure moves with the local fluid velocity, which is the no-slip condition.
The same $\delta_h$ spreads forces and interpolates velocities, so spreading is the adjoint of interpolation.
Hence the power given by the structure equals the power received by the fluid.
A standard $\delta_h$ in $d$ dimensions is a product $\prod_{i=1}^{d} h^{-1} \phi(x_i / h)$ of one-dimensional kernels $\phi$ supported in $[-2, 2]$.

Example: a closed elastic membrane $X(s, t)$, $s \in [0, 2\pi)$, in a fluid at rest, with tension force $F = \kappa \, \partial_s^2 X$ and $\kappa > 0$.
If the initial membrane is an ellipse, the forces drive the fluid, and the membrane relaxes towards a circle of the same enclosed area.

Non-example: a method that meshes the fluid region outside the structure with a body-fitted mesh that moves with the structure fails clause 1.
Its grid is not fixed, so it is not an immersed boundary method.
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# Prime ideal in a noncommutative ring

## Summary

A prime ideal of a noncommutative ring is a proper two-sided ideal $P$ with the following property.
A product $AB$ of two-sided ideals lies in $P$ only when $A$ or $B$ does.
For commutative rings this is the usual [[prime ideal|00MHHZ9SNE]].
Ideals are used instead of elements because, for $n \ge 2$, the matrix ring $M_n(k)$ over a field has no proper ideal with the element-wise property.
Prime ideals define the [[prime radical|WF117EJ2HF]] and are the kernels of maps onto [[prime rings|Z9TJSZZYG1]].

## Explanations

### L1 human

For integers, a prime $p$ divides $ab$ only if it divides $a$ or $b$.
For ideals of a noncommutative ring, this is asked for products of whole ideals, not of single elements.
In the ring $M_2(\Q)$ the zero ideal is prime, because the only ideals are $0$ and the whole ring.
But it is not prime element-wise, since $e_{11} e_{22} = 0$ with $e_{11} \ne 0$ and $e_{22} \ne 0$.
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# Holomorphy on Banach spaces

## Summary

A map between complex Banach spaces is holomorphic if it has a complex-linear Fréchet derivative at every point of its domain.
This extends holomorphic functions of one and several complex variables to infinite dimension.
Holomorphic maps are exactly the complex analytic maps, and they are also the continuous maps that are holomorphic on each complex line.
They appear in operator theory, in spectral theory and in the study of moduli spaces.

## Explanations

### L1 human

A holomorphic function of one complex variable is one that has a complex derivative at each point.
For a map between spaces of vectors, the derivative becomes a complex-linear map that approximates the map near the point.
A good test is to restrict the map to complex lines $z \mapsto a + z h$, where it must be an ordinary holomorphic function.
For example, $x \mapsto x^2$ in an algebra of complex matrices is holomorphic.
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<document source="https://tamnd.github.io/math2/node/E8QJJVC1KV.md" tag="E8QJJVC1KV" kind="statement">
# Gauss–Kronrod quadrature

## Statement

Let $n \ge 1$ and let $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$.
Its zeros $x_1, \ldots, x_n$ are the nodes of the $n$-point [[Gauss–Legendre rule|RGGXDRCJVE]] on $[-1, 1]$.
Let $E_{n+1}$ be the Stieltjes polynomial: the unique monic polynomial of degree $n + 1$ with
$$\int_{-1}^{1} P_n(x) E_{n+1}(x) x^k \, dx = 0 \quad \text{for } k = 0, 1, \ldots, n.$$
Then:

1. the zeros $y_1, \ldots, y_{n+1}$ of $E_{n+1}$ are real, simple, lie in $(-1, 1)$, and interlace with $x_1, \ldots, x_n$;
2. the interpolatory rule $K_{2n+1}(f) = \sum_{i=1}^{n} a_i f(x_i) + \sum_{j=1}^{n+1} b_j f(y_j)$ on these $2n + 1$ nodes satisfies $K_{2n+1}(p) = \int_{-1}^{1} p(x) \, dx$ for every polynomial $p$ of degree at most $3n + 1$.

## Remarks

Clause 2 follows from clause 1 by division.
Write $p = q \, P_n E_{n+1} + r$ with $\deg q \le n$ and $\deg r \le 2n$.
The term $q P_n E_{n+1}$ vanishes at all $2n+1$ nodes and has integral $0$ by the defining property of $E_{n+1}$.
The remainder $r$ is integrated exactly, because an interpolatory rule with $2n + 1$ nodes is exact up to degree $2n$.
Clause 1 is harder; it was proved by Szegő for the Legendre weight, and it can fail for other weight functions.
Example: for $n = 1$, $P_1 = x$ and the conditions give $E_2(x) = x^2 - \tfrac35$, so the Kronrod rule uses the nodes $0, \pm\sqrt{3/5}$ and is the $3$-point Gauss–Legendre rule, exact up to degree $5$.
In practice the difference $\abs{K_{2n+1}(f) - G_n(f)}$, where $G_n$ is the Gauss rule, serves as an error estimate that costs only $n + 1$ new function values.

## Explanations

### L1 human

A Gauss rule samples a function at a few clever points and gives a very accurate integral, but it does not tell how accurate.
The Kronrod extension adds new points between the old ones, keeps the old samples, and gives a second, more accurate value.
Comparing the two values tells how large the error is.

### L2 human

The $n$-point Gauss rule has [[degree of exactness|ZYT1JNT1Y0]] $2n - 1$, and a fresh $(2n+1)$-point Gauss rule would have degree $4n + 1$ but it would reuse almost none of the old nodes.
The Kronrod rule gives up some degree, $3n + 1$ instead of $4n + 1$, in exchange for reusing all $n$ old nodes.
Pairs such as $G_7$–$K_{15}$ and $G_{10}$–$K_{21}$ are the basis of adaptive integrators, for example in the QUADPACK library.
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# Heat kernel on a Riemannian manifold

## Definition

Let $(M, g)$ be a closed [[Riemannian manifold|7CBT4SM83Y]] with volume measure $dv$, and let $\Delta = -\operatorname{div} \operatorname{grad}$ be its [[Laplace–Beltrami operator|F4NWZNTRMD]], with the sign that makes it nonnegative.
The *heat kernel* of $M$ is the function $K \colon (0, \infty) \times M \times M \to \R$ such that:

1. $K$ is continuous, and it is smooth in $t$ and $x$;
2. $\partial_t K(t, x, y) + \Delta_x K(t, x, y) = 0$ for all $t > 0$ and $x, y \in M$;
3. for every continuous $f \colon M \to \R$ and every $x \in M$, $\int_M K(t, x, y) f(y) \, dv(y) \to f(x)$ as $t \to 0^+$.

Such a function exists and is unique.

## Remarks

If $\phi_0, \phi_1, \ldots$ is an orthonormal basis of $L^2(M)$ of eigenfunctions with $\Delta \phi_j = \lambda_j \phi_j$, then
$$K(t, x, y) = \sum_{j \ge 0} e^{-t \lambda_j} \phi_j(x) \phi_j(y),$$
the series converging uniformly for $t \ge t_0 > 0$.
So $K(t, \cdot, \cdot)$ is the integral kernel of the operator $e^{-t\Delta}$, and $K$ is smooth, symmetric in $x$ and $y$, and positive.
The same definition, with $\Delta$ replaced by a Laplace-type operator on a vector bundle, gives heat kernels of operators such as the Hodge Laplacian.
On a complete noncompact manifold one takes the minimal positive fundamental solution; for $\R^n$ it is the Gaussian [[heat kernel|KD404MF042]].

Example: on $S^1 = \R / 2\pi\Z$ with $\Delta = -d^2/d\theta^2$, the eigenvalues are $k^2$ with eigenfunctions $e^{ik\theta}$, and
$$K(t, \theta, \theta') = \frac{1}{2\pi} \sum_{k \in \Z} e^{-k^2 t} e^{ik(\theta - \theta')}.$$
Each term solves the heat equation.
For fixed $t$, $K(t, \theta, \cdot)$ is the sum over $m \in \Z$ of the Gaussian kernel of $\R$ translated by $2\pi m$.
It is a positive kernel of total mass $1$ that concentrates at $\theta$ as $t \to 0^+$, which gives clause 3.

Non-example: on $S^1$ the constant $K \equiv 1/(2\pi)$ satisfies clauses 1 and 2, but $\int K f \to \frac{1}{2\pi}\int f$, not $f(x)$, so clause 3 fails.
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# Fourier monomial on the circle

## Definition

Let $\mathbb{T} = \R / 2\pi\Z$ be the circle, and write $[x]$ for the class of $x \in \R$ in $\mathbb{T}$.
For $n \in \Z$, the *Fourier monomial* of index $n$ is the function $e_n \colon \mathbb{T} \to \C$ given by
$$e_n([x]) = e^{inx},$$
where $e^{inx}$ is the [[complex exponential|07GJV36896]].

## Remarks

The function is well defined: if $x' = x + 2\pi k$ with $k \in \Z$, then $e^{inx'} = e^{inx} e^{2\pi i nk} = e^{inx}$.
It is continuous, takes values of absolute value $1$, and satisfies $e_n e_m = e_{n+m}$ and $e_n(s + t) = e_n(s) e_n(t)$.
So each $e_n$ is a continuous homomorphism from $\mathbb{T}$ to the [[circle group|04EAFVFTF7]], and every continuous homomorphism of this kind is one of the $e_n$.
For the inner product $\langle f, g \rangle = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \overline{g(x)} \, dx$, one has $\langle e_n, e_m \rangle = 1$ if $n = m$ and $0$ otherwise.
Example: $e_3([x]) = e^{3ix} = \cos 3x + i \sin 3x$.
Non-example: $x \mapsto e^{ix/2}$ is not a function on $\mathbb{T}$, since $x = 0$ and $x = 2\pi$ give the same point of $\mathbb{T}$ but the values $1$ and $-1$.
On the circle $\R / T\Z$ of length $T > 0$, the Fourier monomials are $x \mapsto e^{2\pi i n x / T}$.
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# Injectivity radius

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and $p \in M$.
For $r > 0$, let $B_r(0) = \set{v \in T_pM : g_p(v, v) < r^2}$.
Let $A_p$ be the set of $r > 0$ such that:

1. $B_r(0)$ lies in the domain of the [[exponential map|Y7Z0AGY1A7]] $\exp_p$;
2. $\exp_p$ restricted to $B_r(0)$ is a [[diffeomorphism|0KZZ70ZHXP]] onto an open subset of $M$.

The *injectivity radius* of $M$ at $p$ is $\operatorname{inj}(p) = \sup A_p$.
The injectivity radius of $M$ is the [[infimum|0QF1MZ9FRG]] $\operatorname{inj}(M) = \inf_{p \in M} \operatorname{inj}(p)$.

## Remarks

Example: on the unit sphere $S^n$, for any $p$, $\exp_p$ is a diffeomorphism from $B_\pi(0)$ onto $S^n$ minus the antipode $-p$.
It maps the whole sphere $\set{\abs{v} = \pi}$ to $-p$, so no larger ball works, and $\operatorname{inj}(p) = \pi$.

Non-example: for the same $p$ on $S^n$, the radius $r = 2\pi$ does not belong to $A_p$, because $\exp_p$ is not injective on $B_{2\pi}(0)$.

For $\R^n$ with the Euclidean metric, $\operatorname{inj}(p) = \infty$ for every $p$.
The injectivity radius is always positive at each point; if $M$ is compact, then $\operatorname{inj}(M) > 0$.
If $M$ is complete, $\operatorname{inj}(p)$ is the distance from $p$ to its [[cut locus|X5WKGC5Q6T]].
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# Limit point compact space

## Summary

A limit point compact space is a topological space in which every infinite subset has a [[limit point|0E3X7TXS67]].
It is the form of compactness that appears in the Bolzano–Weierstrass theorem.
Every [[countably compact|S3N3CHZV89]] space is limit point compact, and the two notions agree for T1 spaces.
For metric spaces, it is equivalent to compactness.

## Explanations

### L1 human

If you put infinitely many points in a limit point compact space, they must pile up somewhere.
There is a point whose every neighbourhood contains points of the set other than itself.
For example, in $[0, 1]$ any infinite set of points accumulates somewhere, by the Bolzano–Weierstrass theorem.
In $\R$ this fails: the integers are infinite but do not accumulate anywhere.
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# Homotopy principle

## Definition

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], let $r \ge 1$, and let $J^r(M, N)$ be the [[jet bundle|HC3VBBCJQP]] of $r$-jets of smooth maps $M \to N$, with projection $J^r(M, N) \to M$.
A *differential relation* of order $r$ is a subset $\mathcal{R} \subseteq J^r(M, N)$.
A *formal solution* of $\mathcal{R}$ is a continuous section $F \colon M \to J^r(M, N)$ with $F(M) \subseteq \mathcal{R}$.
A formal solution is *holonomic* if $F = j^r f$ for a smooth map $f \colon M \to N$, where $j^r f$ is the $r$-jet extension of $f$.
In that case $f$ is a *genuine solution* of $\mathcal{R}$.
The relation $\mathcal{R}$ satisfies the *h-principle* if every formal solution is [[homotopic|9YA6S6HD9V]] to a holonomic one.
The homotopy must run through formal solutions of $\mathcal{R}$.

## Remarks

Every genuine solution gives a formal solution, so the h-principle says that the only obstruction to solving $\mathcal{R}$ is the homotopy-theoretic obstruction to finding a formal solution.
Stronger versions ask that the inclusion of holonomic solutions into formal solutions be a weak homotopy equivalence; the parametric h-principle is of this kind.
Example: let $\dim M < \dim N$ and let $\mathcal{R} \subseteq J^1(M, N)$ be the set of $1$-jets whose linear part is injective.
Genuine solutions are immersions, formal solutions are fibrewise injective bundle maps $TM \to TN$, and the Smale–Hirsch theorem shows that $\mathcal{R}$ satisfies the h-principle.
Non-example: for $M = S^1$, $N = \R$ and the analogous relation of injective $1$-jets, the section $F(x) = (0, \text{the linear map } v \mapsto v)$, using the standard trivialisation of $TS^1$, is a formal solution.
But there is no genuine solution, because a smooth function on the compact circle has a maximum, where its derivative is zero; so the h-principle fails.
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# Logarithmically convex function

## Definition

Let $I$ be an [[interval|0TK8Y3J5RW]] of $\R$ and let $f \colon I \to (0, \infty)$ be a function.
The function $f$ is *logarithmically convex*, or *log-convex*, if the function $\ln \circ f \colon I \to \R$ is [[convex|0VV0HSW1ZS]], where $\ln$ is the [[natural logarithm|0044CPHJBV]].

## Remarks

Equivalently, $f(t x + (1 - t) y) \le f(x)^{t} f(y)^{1 - t}$ for all $x, y \in I$ and $t \in [0, 1]$.
By the [[AM–GM inequality|0DRF0F3XC9]], $f(x)^t f(y)^{1-t} \le t f(x) + (1 - t) f(y)$, so a log-convex function is convex.

Example: $f(x) = e^{x^2}$ on $\R$ is log-convex, because $\ln f(x) = x^2$ and $x^2$ is convex.
The [[gamma function|0V3FZMVB4K]] on $(0, \infty)$ is also log-convex.

Non-example: $f(x) = x^2$ on $(0, \infty)$ is convex, but $\ln f(x) = 2 \ln x$ is strictly concave, so $f$ is not log-convex.

Some authors allow $f$ to take the value $0$, with the convention $\ln 0 = -\infty$.
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# Standard part function

## Definition

Let ${}^*\R$ be a field of [[hyperreal numbers|421QMTFG9T]] containing the [[real numbers|08J2M1JYAG]].
A hyperreal $x$ is *finite* if $\abs{x} < r$ for some real $r$.
For each finite $x$ there is exactly one real number $r$ such that $x - r$ is infinitesimal.
This $r$ is the *standard part* of $x$, written $\operatorname{st}(x)$.
The *standard part function* is the map $\operatorname{st}$ from the set of finite hyperreals to $\R$.

## Remarks

Existence: $\operatorname{st}(x)$ is the [[supremum|0R3H5PANRQ]] of the set $\set{q \in \R : q < x}$, which is nonempty and bounded above when $x$ is finite.
Uniqueness: if $x - r$ and $x - r'$ are infinitesimal for reals $r$ and $r'$, then $r - r'$ is a real infinitesimal, so $r = r'$.

Example: for a positive infinitesimal $\varepsilon$, $\operatorname{st}(3 + \varepsilon) = 3$, since $(3 + \varepsilon) - 3 = \varepsilon$ is infinitesimal.

Non-example: $H = 1/\varepsilon$ is not finite, and for every real $r$ the difference $H - r$ is larger than $1$, so it is not infinitesimal and $H$ has no standard part.

The finite hyperreals form a subring of ${}^*\R$, and $\operatorname{st}$ is a [[ring homomorphism|0VTH8KPCG2]] onto $\R$ whose kernel is the set of infinitesimals.
It is also monotone: $x \le y$ implies $\operatorname{st}(x) \le \operatorname{st}(y)$.
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# Periodic solutions of integral equations

## Definition

Let $T > 0$.
Let $f \colon \R \to \R$ be continuous and $T$-periodic.
Let $C$ be a continuous function on $\set{(t, s) \in \R^2 : s \le t}$ with $C(t + T, s + T) = C(t, s)$ for all $s \le t$.
Let $g \colon \R \times \R \to \R$ be continuous and $T$-periodic in its first variable.
Consider the [[integral equation|R3SG1SN7JQ]]
$$x(t) = f(t) + \int_{-\infty}^t C(t, s) \, g(s, x(s)) \, ds, \qquad t \in \R.$$
A *$T$-periodic solution* of this equation is a function $x \colon \R \to \R$ such that:

1. $x$ is continuous;
2. $x(t + T) = x(t)$ for all $t \in \R$, that is, $x$ is a $T$-[[periodic function|0ATF6Q0633]];
3. for every $t \in \R$ the integral converges absolutely and the equation holds.

## Remarks

The same notion applies to systems, where $x$ takes values in $\R^n$, and to equations of Fredholm type over one period, such as $x(t) = f(t) + \int_0^T k(t - s) \, x(s) \, ds$ with a $T$-periodic kernel $k$.

Example: $T = 2\pi$, $C(t, s) = e^{-(t - s)}$, $g(s, x) = x$ and $f(t) = \frac{1}{2}(\sin t + \cos t)$.
The function $x(t) = \sin t$ is continuous and $2\pi$-periodic.
The integral $\int_{-\infty}^t e^{-(t - s)} \sin s \, ds$ converges absolutely and equals $\frac{1}{2}(\sin t - \cos t)$, as the imaginary part of $\int_{-\infty}^t e^{-(t - s)} e^{i s} \, ds = e^{i t} / (1 + i)$.
Adding $f(t)$ gives $\sin t$, so all three clauses hold.

Non-example: periodic data do not force periodic solutions.
The equation $x(t) = 1 + \int_0^t x(s) \, ds$ on $[0, \infty)$ has constant data, but its only continuous solution is $x(t) = e^t$, which fails clause 2 for every $T$.

## Explanations

### L1 human

A periodic solution is a solution that comes back to the same value after each period and is valid for all times, past and future.
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# Lyapunov function for a difference equation

## Summary

A Lyapunov function for a difference equation $x_{n+1} = f(x_n)$ is a function that is positive away from an [[equilibrium|D62PEKHRV5]] and does not increase along solutions.
It is the discrete-time version of the [[Lyapunov function|01REYCJ371]] of a differential equation.
Its existence proves [[stability|CHEQ235HW9]] of the equilibrium without solving the equation, and a strict decrease proves asymptotic stability.
This is the main method for nonlinear difference equations whose linearization gives no answer.

## Explanations

### L1 human

Think of the function as an energy or a height above the bottom of a bowl.
At each step of the recursion the energy goes down or stays the same, so the state cannot climb away from the bottom.
For $x_{n+1} = x_n / 2$ the square $V(x) = x^2$ works: it changes to $x^2/4$ at each step, which is smaller.
So every solution slides down to $0$.
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# Convolution ring has no zero divisors

## Statement

Let $\mathcal{C}$ be the set of continuous functions $[0, \infty) \to \C$, with pointwise addition and the [[convolution on the half-line|FCKX8KFVGN]]
$$(f * g)(t) = \int_0^t f(\tau) \, g(t - \tau) \, d\tau$$
as multiplication.
Then $\mathcal{C}$ is a commutative [[rng|MBK7NFE74S]], that is, a ring that need not have a unit.
It has no [[zero divisors|0NNRDW10RZ]]: if $f, g \in \mathcal{C}$ and $f * g = 0$, then $f = 0$ or $g = 0$.

## Remarks

Commutativity and associativity follow from the corresponding facts on the line, and distributivity is linearity of the integral.
For the absence of zero divisors, suppose $f * g = 0$ and $f(t_0) \ne 0$.
By continuity $f$ is nonzero near $t_0$, so $f$ does not vanish almost everywhere on $(0, \lambda)$ for any $\lambda > t_0$.
For each $T > t_0$, the [[Titchmarsh convolution theorem|WGY75H4VWW]] then gives $g = 0$ almost everywhere on $(0, T - t_0)$; letting $T \to \infty$ and using continuity gives $g = 0$.

$\mathcal{C}$ has no unit element.
If $e * f = f$ for all $f$, then $e * 1 = 1$, but $(e * 1)(0) = 0 \ne 1$.

Example: $1 * 1 = t$ and $t * t = t^3/6$, and in general no product of nonzero polynomials in $\mathcal{C}$ is $0$, in line with the formula $t^m * t^n = \frac{m!\,n!}{(m+n+1)!} t^{m+n+1}$.

## Explanations

### L1 human

In ordinary arithmetic, a product is zero only when one factor is zero.
The same rule holds when continuous functions on $[0, \infty)$ are "multiplied" by convolution.
This is far from obvious, because convolution mixes values of the two functions in a complicated way.

### L2 human

A commutative rng without zero divisors has a field of fractions, built from pairs $f/g$ as for the rational numbers.
For $\mathcal{C}$ this is the [[Mikusiński field of operators|82YAXZMMQ1]], the setting of the Mikusiński operational calculus.
The same conclusion holds for locally integrable functions, with equality almost everywhere.
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# Butterfly theorem

## Statement

Let $PQ$ be a chord of a [[circle|060W5CBZP6]] in the Euclidean plane and let $M$ be its [[midpoint|0S706JBZBG]].
Let $AB$ and $CD$ be two further chords through $M$.
Suppose that the segment $AD$ meets $PQ$ at $X$ and the segment $BC$ meets $PQ$ at $Y$.
Then $M$ is the midpoint of $XY$.

## Remarks

One proof drops perpendiculars from $X$ and $Y$ to the chords and uses similar triangles together with the power of the point $M$, $MA \cdot MB = MC \cdot MD = MP \cdot MQ$.
A projective proof uses the involution that the pencil of conics through $A, B, C, D$ induces on the line $PQ$.
Example: let $CD$ be the mirror image of $AB$ in the diameter through $M$ perpendicular to $PQ$, with $C$ the image of $A$.
Then the figure is symmetric and $MX = MY$ is clear.
The theorem stays true when the circle is replaced by any conic.

## Explanations

### L1 human

Take a chord of a circle and its midpoint.
Through the midpoint draw two more chords, and join their ends crosswise to make a shape like the wings of a butterfly.
The wings cut the first chord at two points that are exactly equally far from the midpoint.
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# Differentiability of the Mellin transform

## Statement

Let $f \colon (0, \infty) \to \C$ be integrable on every compact subinterval of $(0, \infty)$.
Let $a, b \in \R$ with $b < a$, and assume that $f(x) = O(x^{-a})$ as $x \to \infty$ and $f(x) = O(x^{-b})$ as $x \to 0^+$, in [[big O notation|0TGV80E6A4]].
Let $S = \set{s \in \C : b < \operatorname{Re} s < a}$.
Then:

1. For every $s \in S$, the function $x \mapsto (\log x) f(x)$ is [[Mellin convergent|SGZZ4Y0D36]] at $s$.
2. The [[Mellin transform|N5PVGRJE1T]] $\mathcal{M}f$ is complex differentiable at every $s \in S$, with
$$(\mathcal{M}f)'(s) = \int_0^\infty x^{s-1} (\log x) f(x) \, dx = \mathcal{M}[(\log x) f(x)](s).$$

In particular $\mathcal{M}f$ is [[holomorphic|0NAHXR1FRH]] on the open strip $S$.

## Remarks

The derivative of $x^{s-1}$ in $s$ is $x^{s-1} \log x$.
Fix $s \in S$ and $\varepsilon > 0$ with $b < \operatorname{Re} s - 2\varepsilon$ and $\operatorname{Re} s + 2\varepsilon < a$.
For $\abs{z - s} \le \varepsilon$, the bound $\abs{\log x} \le \varepsilon^{-1} (x^{\varepsilon} + x^{-\varepsilon})$ dominates $\abs{x^{z-1} (\log x) f(x)}$ by a function that is integrable by the estimates of the fundamental strip.
So [[differentiation under the integral sign|0AR06W1E1B]] applies.
Iterating gives $(\mathcal{M}f)^{(n)}(s) = \mathcal{M}[(\log x)^n f(x)](s)$ for all $n \ge 0$.
Example: for $f(x) = e^{-x}$ the statement gives $\Gamma'(s) = \int_0^\infty x^{s-1} (\log x) e^{-x} \, dx$ for $\operatorname{Re} s > 0$, and at $s = 1$ this is $\Gamma'(1) = -\gamma$, with $\gamma$ the Euler–Mascheroni constant.
The open strip is needed: for $f(x) = 1/(1 + x)$, $\mathcal{M}f(s) = \pi / \sin(\pi s)$ on $0 < \operatorname{Re} s < 1$, and this function has poles at $s = 0$ and $s = 1$, the edges of the strip.

## Explanations

### L1 human

Under the usual bounds on $f$ at $0$ and at $\infty$, the Mellin transform is a smooth function of $s$ in the complex sense.
Its derivative is the Mellin transform of $f$ multiplied by $\log x$.

### L2 human

Holomorphy on the strip is what makes Mellin methods work: one can continue $\mathcal{M}f$ analytically beyond the strip, shift lines of integration, and pick up residues.
For instance the gamma function is first defined on $\operatorname{Re} s > 0$ as a Mellin transform, holomorphic there by this result, and then continued meromorphically to $\C$.
The same holds for the completed zeta function $\pi^{-s/2} \Gamma(s/2) \zeta(s)$, which is a Mellin transform of a theta series.
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# Time series

## Summary

A time series is a sequence of observations recorded in time order, such as daily temperatures or monthly unemployment rates.
Time series analysis models such data as one realization of a [[stochastic process|0QH5ENKZM1]], in which nearby observations are usually dependent.
The goals are to describe this dependence, to estimate models, and to forecast future values.

## Explanations

### L1 human

Most statistical methods assume that the observations are independent, like separate draws from a hat.
A time series is different: today's value says a lot about tomorrow's value.
Plotting the data against time shows trends, seasonal cycles and random wiggles.
For example, the monthly sales $x_1, \dots, x_{36}$ of a shop over three years form a time series, and a peak every December is a seasonal pattern.
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# Nontangential limit

## Definition

Let $n \ge 2$, let $B = \set{x \in \R^n : \abs{x} < 1}$ and let $S = \partial B$.
For $\zeta \in S$ and $\alpha > 1$, the *nontangential approach region* at $\zeta$ is
$$\Gamma_\alpha(\zeta) = \set{x \in B : \abs{x - \zeta} < \alpha (1 - \abs{x})}.$$
Let $u \colon B \to \C$ and $L \in \C$.
The function $u$ has *nontangential limit* $L$ at $\zeta$ if, for every $\alpha > 1$, $u(x) \to L$ as $x \to \zeta$ with $x \in \Gamma_\alpha(\zeta)$.

## Remarks

The limit along $\Gamma_\alpha(\zeta)$ is a [[limit of a function of several variables|0J8Y1NCY0K]] restricted to that set; $\zeta$ is a limit point of each $\Gamma_\alpha(\zeta)$.
For the [[unit disc|8G5ABXRK85]] the regions $\Gamma_\alpha(\zeta)$ are the Stolz angles, and the same definition works for the upper half-space with cones $\set{(x, t) : \abs{x - x_0} < \alpha t}$.
A nontangential limit implies the radial limit $\lim_{r \to 1} u(r\zeta) = L$.
Example: let $f(z) = \exp\left(-\frac{1 + z}{1 - z}\right)$ on the unit disc and $\zeta = 1$.
Since $\operatorname{Re} \frac{1 + z}{1 - z} = \frac{1 - \abs{z}^2}{\abs{1 - z}^2}$, on $\Gamma_\alpha(1)$ this is at least $\frac{1}{\alpha^2 (1 - \abs{z})}$, which tends to $+\infty$.
So $f$ has nontangential limit $0$ at $1$.
On the circle $\abs{z - 1/2} = 1/2$ the same real part equals $1$, so $\abs{f} = e^{-1}$ there, and $f$ has no unrestricted limit at $1$.
Non-example: the Poisson kernel $u(z) = \frac{1 - \abs{z}^2}{\abs{1 - z}^2}$ has $u(r) = \frac{1 + r}{1 - r} \to +\infty$, so it has no finite nontangential limit at $1$.
By [[Fatou's theorem|Q2N917JQ6A]], the Poisson integral of an integrable function has a nontangential limit at almost every boundary point.
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<document source="https://tamnd.github.io/math2/node/EAEFNBF94A.md" tag="EAEFNBF94A" kind="definition">
# Degree of a central simple algebra

## Definition

Let $K$ be a field and let $A$ be a [[central simple algebra|5HRKJBAY2V]] over $K$.
The dimension $\dim_K A$ is a perfect square $n^2$ with $n \ge 1$.
The *degree* of $A$ is the integer $n = \sqrt{\dim_K A}$, written $\deg A$.

## Remarks

Let $\bar{K}$ be an [[algebraic closure|0X6AB5P3G1]] of $K$.
The [[tensor product|1735VGRT2P]] $A \otimes_K \bar{K}$ is central simple over $\bar{K}$, hence isomorphic to $M_n(\bar{K})$ for some $n$ by Wedderburn's theorem.
So $\dim_K A = \dim_{\bar{K}} M_n(\bar{K}) = n^2$ is a square.

Example: $\deg M_n(K) = n$, and the quaternion algebra $\mathbb{H}$ over $\R$ has degree $2$, since $\dim_\R \mathbb{H} = 4$.

Non-example: $\C$ has dimension $2$ over $\R$, which is not a square, so it has no degree as an algebra over $\R$; it is not central over $\R$.

If $A \cong M_m(D)$ with $D$ a division algebra, then $\deg A = m \deg D$.
Degree is multiplicative: $\deg(A \otimes_K B) = \deg A \cdot \deg B$.
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<document source="https://tamnd.github.io/math2/node/EANPWS9G8N.md" tag="EANPWS9G8N" kind="definition">
# Modality of a singularity

## Definition

Let $f \colon (\C^n, 0) \to (\C, 0)$ be a [[germ|VZXF7RNFKS]] of a [[holomorphic function|0NAHXR1FRH]] with an isolated [[critical point|0YTCMFB4HC]] at $0$.
Let $k$ be so large that $f$ is $k$-determined.
Let $J^k$ be the space of [[$k$-jets|1QE0Z0756D]] at $0$ of germs vanishing at $0$.
The group of $k$-jets of germs of biholomorphic maps $(\C^n, 0) \to (\C^n, 0)$ acts on $J^k$ by composition on the right.
The *modality* of $f$ is the least integer $m \ge 0$ with this property.
Some neighbourhood of $j^k f$ in $J^k$ is covered by finitely many families of orbits, each depending on at most $m$ parameters.
A germ of modality $0$ is called *simple*, of modality $1$ *unimodal*, and of modality $2$ *bimodal*.

## Remarks

Informally, a family of orbits depending on $m$ parameters is the union of the orbits of the points of an $m$-dimensional smooth submanifold of $J^k$.
The modality does not depend on the choice of $k$, as long as $f$ is $k$-determined.
Example: $f = x^3$ on $\C$ is simple.
Any small perturbation of the jet of $x^3$ in $J^3$ is $a_1 x + a_2 x^2 + a_3 x^3$ with $a_3 \ne 0$.
If $a_1 \ne 0$ it is right equivalent to $x$, a regular point.
If $a_1 = 0 \ne a_2$ it is right equivalent to $x^2$, a Morse point.
If $a_1 = a_2 = 0$ it is right equivalent to $x^3$.
Non-example: $f = x^4 + y^4$ on $\C^2$ is not simple.
For the germs $x^4 + y^4 + a x^2 y^2$ with $a^2 \ne 4$, the zero set is four distinct lines through $0$.
Their cross-ratio, taken up to the order of the lines, depends on $a$ and is invariant under right equivalence.
So infinitely many orbits meet every neighbourhood of $f$, and in fact the modality of $f$ is $1$.
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<document source="https://tamnd.github.io/math2/node/EAP9N3VEBV.md" tag="EAP9N3VEBV" kind="concept">
# Generalized convolution of an integral transform

## Summary

A convolution of an [[integral transform|YCK709QB06]] $T$ is a bilinear product $\circledast$ of functions that $T$ turns into the pointwise product: $T(f \circledast g) = Tf \cdot Tg$.
The usual [[convolution on the line|ERP5737VZ1]] plays this role for the Fourier transform.
The [[convolution on the half-line|FCKX8KFVGN]] plays it for the Laplace transform, and the [[Mellin convolution|A2PMJG7J68]] for the Mellin transform.
Finding such a product for a given transform gives it a convolution theorem, which is what makes it useful for solving integral and differential equations.

## Explanations

### L1 human

Many transforms are useful because they turn a hard operation into multiplication.
For each transform one can ask which operation on the original functions becomes multiplication on the transform side.
That operation is the convolution of the transform.
For the Fourier transform it is the familiar convolution; for other transforms it is a different integral, adapted to the kernel of the transform.
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<document source="https://tamnd.github.io/math2/node/EAPHAQVVD5.md" tag="EAPHAQVVD5" kind="concept">
# Profunctor

## Summary

A profunctor, also called a distributor or bimodule, from a category $\mathcal{C}$ to a category $\mathcal{D}$ is a [[functor|28NQ0KRHD3]] $\mathcal{D}^{\mathrm{op}} \times \mathcal{C} \to \mathbf{Set}$.
It generalizes functors: every functor gives a profunctor.
It also generalizes relations: a relation between sets is a profunctor between discrete categories with values in truth values.
Profunctors compose by a [[coend|W9QTFV0Z8D]], and categories, profunctors and natural transformations form a bicategory.
They are the categorical analogue of bimodules between rings.

## Explanations

### L1 human

A function from $X$ to $Y$ assigns to each $x$ one $y$, while a relation only says which pairs $(x, y)$ are related.
A profunctor does the same for categories, and it says how $x$ and $y$ are related by a set of "witnesses" $P(d, c)$.
The witnesses can be moved along morphisms in both categories, in opposite directions.
For example, the hom-sets $\Hom(d, c)$ of a category form a profunctor from the category to itself: a morphism is a witness that $d$ maps to $c$.
</document>

<document source="https://tamnd.github.io/math2/node/EAQDSQD4DW.md" tag="EAQDSQD4DW" kind="concept">
# Lerch zeta function

## Summary

The Lerch zeta function is the series $\sum_{n \ge 0} e^{2\pi i \lambda n} (n + \alpha)^{-s}$, a generalized [[Dirichlet series|8MWTEF6462]] with a phase factor and a shift.
In the closely related form $\Phi(z, s, a) = \sum_{n \ge 0} z^n (n + a)^{-s}$ it is called the Lerch transcendent.
It contains the [[Hurwitz zeta function|G4SW7HJG88]], the [[Riemann zeta function|1XWBY4N0VT]], and the [[polylogarithm|96CHDZKGQ0]] as special cases.
Dirichlet $L$-functions are finite linear combinations of its values.
So identities proved for it hold for all of them at once.

## Explanations

### L1 human

The Riemann zeta function adds up $1/n^s$.
The Hurwitz zeta function shifts the integers to $n + \alpha$, and the polylogarithm weights the terms by powers $z^n$.
The Lerch zeta function does both: it adds up $z^n / (n + \alpha)^s$.
For instance, with $z = -1$, $\alpha = 1$ and $s = 1$, the sum is $1 - \frac12 + \frac13 - \cdots = \log 2$.
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<document source="https://tamnd.github.io/math2/node/EASHTX8EG5.md" tag="EASHTX8EG5" kind="concept">
# H-space

## Summary

An H-space is a pointed space with a continuous multiplication for which the basepoint is a two-sided unit up to homotopy.
Topological groups and [[loop spaces|2JFF5F3EGF]] are the main examples.
The multiplication gives the cohomology of an H-space extra structure, such as a [[Hopf algebra structure|WJXK0V9NE9]].
This strongly restricts which spaces can be H-spaces.
By a theorem of Adams, the only spheres that are H-spaces are $S^0$, $S^1$, $S^3$ and $S^7$.

## Explanations

### L1 human

A topological group has a multiplication with an exact identity element.
An H-space keeps only part of this: a multiplication with an identity that works up to deformation, and no inverses or associativity are required.
The circle $S^1$, seen as the unit complex numbers, is an H-space.
The sphere $S^7$, seen as the unit octonions, is an H-space although the octonions are not associative, so $S^7$ is not a group.
</document>

<document source="https://tamnd.github.io/math2/node/EAV97QH9DF.md" tag="EAV97QH9DF" kind="example">
# Bessel's equation

## Example

Let $\nu \in \C$. *Bessel's equation* of order $\nu$ is
$$z^2 y'' + z y' + (z^2 - \nu^2)\, y = 0.$$
In normalised form, $p(z) = \frac{1}{z}$ and $q(z) = 1 - \frac{\nu^2}{z^2}$.
Then $z p(z) = 1$ and $z^2 q(z) = z^2 - \nu^2$ are analytic at $0$, so $0$ is a [[regular singular point|H96HXG6CA0]].
The indicial equation is $r (r - 1) + r - \nu^2 = r^2 - \nu^2 = 0$, with exponents $r = \pm \nu$.
If $\nu$ is not a negative integer, the [[Frobenius method|W8ZWXC2HQS]] with $r = \nu$ gives the recurrence $\left( (\nu + k)^2 - \nu^2 \right) a_k = - a_{k-2}$, that is, $k (k + 2 \nu)\, a_k = - a_{k-2}$, with $a_1 = 0$.
With $a_0 = \frac{1}{2^\nu \Gamma(\nu + 1)}$ it gives the Bessel function of the first kind
$$J_\nu(z) = \sum_{m \ge 0} \frac{(-1)^m}{m!\, \Gamma(m + \nu + 1)} \left( \frac{z}{2} \right)^{2m + \nu},$$
where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
The series converges for all $z$, since the coefficients of the equation are analytic on $\C \setminus \set{0}$.
At $\infty$, the function $z^2 q(z) = z^2 - \nu^2$ is not bounded, so $\infty$ is an irregular singular point.

## Remarks

If $\nu \notin \Z$, then $J_\nu$ and $J_{-\nu}$ are linearly independent solutions.
If $\nu$ is an integer, then $J_{-\nu} = (-1)^\nu J_\nu$, and a second solution contains a logarithm, as Fuchs' theorem predicts.
For $\nu = \frac{1}{2}$, one gets $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}}\, \sin x$ for $x > 0$.
The equation arises from separating variables in the Laplace or wave equation in cylindrical coordinates.

## Explanations

### L1 human

Bessel's equation describes waves on a round drum and heat flow in a cylinder.
Its solutions near $0$ behave like the power $z^{\nu}$, and the Frobenius method finds them as series.
Far from $0$ they oscillate like a sine wave whose size decays like $\frac{1}{\sqrt{x}}$.
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<document source="https://tamnd.github.io/math2/node/EAX2SYK8C5.md" tag="EAX2SYK8C5" kind="definition">
# Homotopy category

## Definition

Let $\mathcal{M}$ be a [[model category|SDTZVZ6B4Q]] with class of [[weak equivalences|E2870TEDG7]] $W$.
The *homotopy category* of $\mathcal{M}$ is the [[localization|DNFR3YP0XF]]
$$\gamma \colon \mathcal{M} \to \mathrm{Ho}(\mathcal{M}) = \mathcal{M}[W^{-1}].$$

## Remarks

Quillen's fundamental theorem says that this localization exists with small hom-sets.
If $X$ is cofibrant and $Y$ is fibrant (see [[fibration and cofibration|HSV1J7BJXX]]), then $\mathrm{Ho}(\mathcal{M})(\gamma X, \gamma Y)$ is the set of homotopy classes of morphisms $X \to Y$.
Every object is isomorphic in $\mathrm{Ho}(\mathcal{M})$ to one that is both fibrant and cofibrant, by the factorization axiom.

Example: for topological spaces with the Quillen model structure, $W$ consists of the [[weak homotopy equivalences|4Q3WY1RX91]].
Every space is fibrant, and the [[CW complexes|5W95QFHXPR]] are cofibrant.
So $\mathrm{Ho}(\mathrm{Top})$ is equivalent to the category of CW complexes with [[homotopy|9YA6S6HD9V]] classes of continuous maps as morphisms.
Non-example: the naive homotopy category, with all spaces as objects and homotopy classes of maps as morphisms, is a different category.
In it, the map from the Warsaw circle to a point is not invertible, although it is a weak homotopy equivalence.

The name is also used for the homotopy category of chain complexes, whose morphisms are chain homotopy classes of chain maps.
That category is related to, but different from, the derived category.
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<document source="https://tamnd.github.io/math2/node/EAZF9NJW12.md" tag="EAZF9NJW12" kind="concept">
# Gateaux derivative

## Summary

The Gateaux derivative of a map between vector spaces at a point measures the rate of change of the map along each direction.
It generalizes the directional derivative of calculus to normed and locally convex spaces of any dimension.
It is weaker than the Fréchet derivative: a map can have a Gateaux derivative at a point without being continuous there.
It is the derivative used in the calculus of variations, where it gives the first variation of a functional.

## Explanations

### L1 human

To differentiate a map at a point $x$, move along a straight line $x + t h$ and look at the rate of change at $t = 0$.
The Gateaux derivative asks only for this rate along every line through $x$, one line at a time.
For $f(x) = \norm{x}^2$ on a real Hilbert space, the rate along $h$ is $2\ip{x}{h}$.
Since nothing is asked about how the lines fit together, the notion is weaker than the usual derivative.
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<document source="https://tamnd.github.io/math2/node/EAZMKENA4N.md" tag="EAZMKENA4N" kind="concept">
# Fox H-function

## Summary

The Fox H-function is a generalisation of the [[Meijer G-function|19JNNN5KV7]].
In its [[Barnes integral|1PDGBGEHM2]] the gamma factors may have the form $\Gamma(b + B s)$ with a positive scale $B$, not only $\Gamma(b + s)$.
When all scales equal $1$, it is a Meijer G-function.
The extra freedom covers functions such as the Mittag-Leffler and Fox–Wright functions.
H-functions appear in fractional calculus and as densities in probability, for instance of products and quotients of independent random variables.

## Explanations

### L1 human

A Meijer G-function is described by a Mellin transform that is a ratio of gamma values at points $b + s$.
Fox allowed points $b + B s$ with any positive stretch $B$.
This small change brings in functions whose series have factorials of non-integer multiples of the index, such as $\sum_k z^k / \Gamma(\alpha k + 1)$.
A basic example is $H^{1, 0}_{0, 1}\left[ z \,\middle|\, (0, 1) \right] = e^{-z}$, which comes from the Mellin transform $\Gamma(s)$ of $e^{-z}$.
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<document source="https://tamnd.github.io/math2/node/EB2QJQNE5Z.md" tag="EB2QJQNE5Z" kind="concept">
# Influence function

## Summary

The influence function of a statistical functional measures the effect on the functional of adding an infinitesimal amount of contamination at a point $x$.
It is a directional derivative of the functional in the direction of a point mass, a [[Gateaux derivative|EAZF9NJW12]].
A bounded influence function means that no single observation can have a large effect, and the supremum of its absolute value is the gross-error sensitivity.
Under regularity conditions it also gives the asymptotic variance of the estimator, so it links robustness and efficiency.

## Explanations

### L1 human

Imagine a huge data set, and add one more point at position $x$.
The estimate moves a little; the influence function records how much, per unit of added weight, as a function of $x$.
For the mean, the effect grows without limit as $x$ moves away, so a far outlier has a large effect.
For the median, the effect is the same for every point above the median, however far away, so outliers have limited effect.
</document>

<document source="https://tamnd.github.io/math2/node/EB4N9RQR51.md" tag="EB4N9RQR51" kind="concept">
# Basic hypergeometric series

## Summary

A basic hypergeometric series, or $q$-hypergeometric series, is a power series $\sum_n c_n z^n$ in which the ratio $c_{n+1}/c_n$ is a rational function of $q^n$.
It is written ${}_r\phi_s(a_1, \dots, a_r; b_1, \dots, b_s; q, z)$ and is built from [[q-Pochhammer symbols|Y1X8ZZQ7VM]].
It is the [[q-analog|GFSDMJHHVA]] of the [[generalized hypergeometric series|VDBAW9PN88]], to which it tends as $q \to 1$ after a change of parameters.
Basic hypergeometric series contain the partition generating functions, the theta products and the $q$-orthogonal polynomials of the Askey scheme.

## Explanations

### L1 human

In an ordinary hypergeometric series, the ratio of two consecutive coefficients is a rational function of the index $n$.
In a basic hypergeometric series, the index $n$ appears only through the power $q^n$ of a fixed number $q$.
The simplest example is $\sum_{n \ge 0} \frac{z^n}{(1-q)(1-q^2)\cdots(1-q^n)}$, which equals $\prod_{k \ge 0} \frac{1}{1 - z q^k}$ for $\abs{q} < 1$ and $\abs{z} < 1$.
With $z = q$, it counts the partitions of each number, by the [[generating function for partitions|0578PA53VA]].
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<document source="https://tamnd.github.io/math2/node/EBASBW5TYS.md" tag="EBASBW5TYS" kind="statement">
# Gordon–Luecke theorem

## Statement

Let $K$ and $K'$ be tame [[knots|QYEAZTYGPT]] in $S^3$.
Suppose that there is a [[homeomorphism|0QT00PB5G2]] $S^3 \setminus K \to S^3 \setminus K'$.
Then there is a homeomorphism $h \colon S^3 \to S^3$ with $h(K) = K'$.

## Remarks

The proof reduces the theorem to a statement about [[Dehn surgery|ZA405B0Z1S]].
A homeomorphism of complements that does not carry meridian to meridian would give a nontrivial surgery on $K'$ that yields $S^3$.
Gordon and Luecke showed that a nontrivial Dehn surgery on a nontrivial knot never gives $S^3$, using a careful study of the graphs formed by intersecting planar surfaces.
The homeomorphism $h$ may reverse orientation, so a knot and its mirror image have homeomorphic complements and count as equivalent here.
The analogue for links fails: twisting a link along an unknotted component changes the link but not the homeomorphism type of its complement.
The [[knot group|AQVHM3FP0D]] alone does not determine the knot: the square knot and the granny knot have isomorphic groups but are different knots.

## Explanations

### L1 human

Remove a knot from space and keep the space that is left.
The theorem says that this leftover space remembers the knot completely.
If two knots leave behind the same space, up to stretching, then the knots themselves are the same, up to mirror image.
So questions about knots can be turned into questions about $3$-manifolds.

### L2 human

The theorem justifies studying a knot through its complement, for instance through the hyperbolic structure of the complement of a [[hyperbolic knot|K47G2Q24DT]].
Combined with Mostow–Prasad rigidity, it shows that a hyperbolic knot is determined by the isometry type of its complement.
It is close to the Property P question: both are about which surgeries on a knot can give simple manifolds.
For links, and for knots in other $3$-manifolds, the complement can lose information.
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<document source="https://tamnd.github.io/math2/node/EBD8TWFNS0.md" tag="EBD8TWFNS0" kind="concept">
# Wolff potential

## Summary

The Wolff potential of a measure is a nonlinear potential that measures how much mass the measure puts near a point at all scales.
At each scale a power $1/(p - 1)$ is applied before the scales are summed.
For $p = 2$ it reduces to a [[Riesz potential|1V2N8MFSHC]], up to a constant factor.
Hedberg and Wolff showed that it controls the energy of measures in nonlinear [[capacity|4YZVFGJVSZ]] theory.
Kilpeläinen and Malý showed that it gives pointwise bounds for $p$-superharmonic functions in terms of their Riesz measure, a basic tool of [[nonlinear potential theory|M923VKQVZM]].

## Explanations

### L1 human

A potential adds up the influence of a mass distribution on a point.
The Wolff potential looks at balls of every radius around the point and asks how much mass each ball holds, compared with what its size allows.
Each of these ratios is raised to the power $1/(p - 1)$ and the results are summed over all scales.
For a single point mass at the origin, the Wolff potential at $x$ is a negative power of $\abs{x}$, like the fundamental solution of the $p$-Laplace equation.
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<document source="https://tamnd.github.io/math2/node/EBEF6PEK4N.md" tag="EBEF6PEK4N" kind="concept">
# Chow group

## Summary

The Chow groups of a variety $X$ are the groups of [[algebraic cycles|9VFWPZEVGN]] on $X$ modulo [[rational equivalence|K76N026JFK]], one for each dimension.
They are the algebraic analogue of homology groups.
For a smooth variety, the intersection product turns their sum into a graded ring, the Chow ring, which is the setting of [[intersection theory|4HV4WTAX9P]].
In codimension one they recover the divisor class group and the [[Picard group|3Z9DCWYXQZ]] of a smooth variety.

## Explanations

### L1 human

A Chow group counts subvarieties of a given dimension, where two of them count as the same when one can be deformed into the other through a rational family.
On the projective plane, every curve of degree $d$ is equivalent to $d$ times a line, so the group of curves is $\Z$, generated by the class of a line.
On an elliptic curve, the group of points is much bigger: it records the degree and also a point of the curve itself.
The Chow groups are thus simple for rational varieties and can be enormous in general.
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<document source="https://tamnd.github.io/math2/node/EBEVWSRD2E.md" tag="EBEVWSRD2E" kind="statement">
# Plünnecke–Ruzsa inequality

## Statement

Let $G$ be an [[abelian group|082GHNPJ53]], and let $A$ and $B$ be finite nonempty subsets of $G$.
Let $K \ge 1$ be a real number.
Suppose that the [[sumset|XNBN31HQWB]] satisfies $\abs{A + B} \le K \abs{A}$.
Then for all integers $m, n \ge 0$ with $m + n \ge 1$,
$$\abs{mB - nB} \le K^{m + n} \abs{A}.$$
Here $mB - nB$ is the set of all $b_1 + \dots + b_m - b'_1 - \dots - b'_n$ with all $b_i, b'_j \in B$.

## Remarks

Petridis gave a short proof.
Choose a nonempty $X \subseteq A$ that minimizes the ratio $\abs{X + B} / \abs{X}$, and call the minimum $K'$, so $K' \le K$.
He showed that $\abs{X + B + C} \le K' \abs{X + C}$ for every finite set $C$, and induction gives $\abs{X + mB} \le K^m \abs{X}$.
The [[Ruzsa triangle inequality|R7E5A4W6FJ]] then handles the differences.
With $B = A$, a set with $\abs{A + A} \le K\abs{A}$ satisfies $\abs{mA - nA} \le K^{m+n}\abs{A}$, so small doubling controls all iterated sums.
For $A = \set{1, \dots, N}$ one has $\abs{A + A} = 2N - 1 \le 2\abs{A}$, and indeed $\abs{mA - nA} \le (m + n)N \le 2^{m+n} N$.

## Explanations

### L1 human

If adding a set $B$ to a set $A$ makes it only a little larger, then adding $B$ many times, or subtracting it, also cannot make it much larger.
The growth is at most a fixed power of the first growth factor.
So a set with few sums is controlled at every later step.

### L2 human

The Plünnecke–Ruzsa inequality is a basic tool in [[additive combinatorics|3NM362A2JG]].
It is used in the proof of [[Freiman's theorem|KEPFYE6WYZ]].
It is often combined with the [[Balog–Szemerédi–Gowers theorem|W5BZZ59CVM]].
Plünnecke's original proof used a graph-theoretic argument with layered commutative graphs, and Ruzsa turned it into the form above.
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# Mutually singular measures

## Summary

Two [[measures|PR11BZQQPG]] on the same space are mutually singular when they live on disjoint parts of the space.
One of them gives zero mass to a set, and the other gives zero mass to its complement.
Mutual singularity is the opposite extreme of [[absolute continuity of measures|P0P21KDV6E]].
The [[Lebesgue decomposition theorem|74VMD07DRX]] splits a σ-finite measure, relative to a σ-finite reference measure, into an absolutely continuous part and a singular part.

## Explanations

### L1 human

Picture two piles of sand on the same floor that never touch.
Each pile sits on its own region, and the two regions do not overlap.
On the real line, Lebesgue measure spreads mass evenly, while the point mass at $0$ puts all its mass on the single point $0$.
Lebesgue measure gives the set $\{0\}$ length zero, and the point mass gives the rest of the line mass zero.
So these two measures are mutually singular.
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<document source="https://tamnd.github.io/math2/node/EBHKKRTRFS.md" tag="EBHKKRTRFS" kind="example">
# Wilkie's identity

## Example

Write $A = 1 + x$, $B = 1 + x + x^2$, $C = 1 + x^3$ and $D = 1 + x^2 + x^4$, where $x^2$ means $x \cdot x$ and so on.
*Wilkie's identity* is
$$(A^y + B^y)^x \cdot (C^x + D^x)^y = (A^x + B^x)^y \cdot (C^y + D^y)^x.$$
It is an [[identity|PKD440G2XW]] in the signature $1, +, \cdot$, exponentiation.

Check that it holds for all positive integers $x, y$.
Let $Q = 1 - x + x^2$, a positive integer for $x \ge 1$.
Then $C = A \cdot Q$ and $D = B \cdot Q$.
So $C^x + D^x = Q^x (A^x + B^x)$ and $C^y + D^y = Q^y (A^y + B^y)$.
Both sides become $(A^y + B^y)^x \cdot (A^x + B^x)^y \cdot Q^{xy}$.

Alex Wilkie showed that this identity cannot be derived from the eleven high school identities, which answers [[Tarski's high school algebra problem|RCRD8CAVB4]] negatively.
The check above uses the term $1 - x + x^2$, which needs subtraction and so is not a term of the signature.

## Explanations

### L1 human

Both sides of Wilkie's identity are built only from $1$, addition, multiplication and powers, and they are always equal for positive whole numbers.
The quick proof factors $1 + x^3$ and $1 + x^2 + x^4$ with the help of $1 - x + x^2$, which uses a minus sign.
School rules have no minus sign, and Wilkie proved that they cannot prove the identity in any other way.
</document>

<document source="https://tamnd.github.io/math2/node/EBHS26MHZT.md" tag="EBHS26MHZT" kind="definition">
# Successive minima

## Definition

Let $n \ge 1$ and let $\Lambda \subset \R^n$ be a [[lattice|WN9V7WSJKF]].
Let $K \subset \R^n$ be a [[convex|02AS45T3H9]] [[compact|0R6EWFKRCF]] set with nonempty interior that is symmetric about the origin, that is, $-x \in K$ for all $x \in K$.
For $\lambda > 0$ write $\lambda K = \set{\lambda x : x \in K}$.
Let $k \in \set{1, \dots, n}$.
The *$k$-th successive minimum* $\lambda_k(K, \Lambda)$ of $\Lambda$ for $K$ is the infimum of all $\lambda > 0$ such that $\lambda K \cap \Lambda$ contains $k$ [[linearly independent|0CRMGPAH83]] vectors.

## Remarks

The infimum is attained, and $0 < \lambda_1 \le \lambda_2 \le \dots \le \lambda_n < \infty$.
Example: for $\Lambda = \Z^2$ and $K = [-1, 1] \times [-2, 2]$, one has $\lambda K = [-\lambda, \lambda] \times [-2\lambda, 2\lambda]$.
The vector $(0, 1)$ lies in $\lambda K$ exactly when $\lambda \ge 1/2$, and no nonzero integer vector lies in $\lambda K$ for $\lambda < 1/2$, so $\lambda_1 = 1/2$.
A vector independent of $(0, 1)$ has first coordinate of absolute value at least $1$, so it needs $\lambda \ge 1$, and $(1, 0)$ enters at $\lambda = 1$; so $\lambda_2 = 1$.
Non-example: the strip $K = \R \times [-1, 1]$ is convex and symmetric but not compact.
Then $(1, 0) \in \lambda K$ for every $\lambda > 0$, so the infimum defining $\lambda_1$ is $0$ and the minima lose their meaning.
When $K$ is the closed unit ball, $\lambda_k$ is the smallest $r$ such that $\Lambda$ has $k$ linearly independent vectors of length at most $r$.
</document>

<document source="https://tamnd.github.io/math2/node/EBMCFQT0JN.md" tag="EBMCFQT0JN" kind="concept">
# Fréchet manifold

## Summary

A Fréchet manifold is a space that looks locally like an open subset of a Fréchet space, with smooth changes of coordinates.
It is the natural setting for spaces of smooth maps, such as the smooth loops in a manifold or the diffeomorphisms of a compact manifold.
Calculus on Fréchet manifolds is harder than on Banach manifolds, because the inverse function theorem fails in general Fréchet spaces.
The Nash–Moser theorem gives a replacement under extra hypotheses.

## Explanations

### L1 human

Spaces of smooth functions are not Banach spaces, because no single norm controls all derivatives at once.
They are Fréchet spaces, controlled by a sequence of norms.
A Fréchet manifold is built from patches that look like pieces of such spaces.
For example, the space of all smooth loops in a sphere is a Fréchet manifold.
</document>

<document source="https://tamnd.github.io/math2/node/EBNHD7FQ23.md" tag="EBNHD7FQ23" kind="concept">
# Repair radii and villainy

## Summary

A repair radius measures how far a colouring of a graph $G$ with a given profile of class sizes can be from a proper one.

The radius $R_G(m)$ is the largest Hamming [[distance from a point to a set|07TXYYVYZR]], over colourings with profile $m$, to the set of [[proper colourings|0874138YP6]] with profile $m$.

The weak radius $R^w_G(m)$ lets the nearest proper colouring have any profile.

The villainy $B(G)$ and the weak villainy $B_w(G)$ are the maxima of these radii over feasible profiles, and both are zero for the empty graph.
</document>

<document source="https://tamnd.github.io/math2/node/EBV9M9C435.md" tag="EBV9M9C435" kind="concept">
# Magnus expansion

## Summary

The Magnus expansion writes the solution of a linear system $Y' = A(t)\, Y$, $Y(0) = I$, as one [[matrix exponential|020M04FN1H]] $Y(t) = e^{\Omega(t)}$.
The exponent $\Omega(t)$ is a series of iterated integrals of commutators of $A$ at different times.
When the matrices $A(t)$ commute with each other, only the first term $\int_0^t A$ remains.
Truncations of the series keep structure, for example they stay in a matrix group, which makes them useful in numerical methods and quantum mechanics.
See the [[definition|TD9T02HKPD]].

## Explanations

### L1 human

For a scalar equation $y' = a(t)\, y$ the solution is $y = e^{\int a}$.
For matrices this formula is wrong in general, because matrices at different times need not commute.
The Magnus expansion corrects the exponent by extra terms built from commutators $[A(t_1), A(t_2)] = A(t_1) A(t_2) - A(t_2) A(t_1)$.
These terms measure how much the matrices fail to commute.
</document>

<document source="https://tamnd.github.io/math2/node/EBXRM4PFRC.md" tag="EBXRM4PFRC" kind="problem">
# Fermi–Pasta–Ulam–Tsingou problem

## Problem

Let $N \ge 2$ and $\alpha \in \R$.
Consider $N$ particles on a line with displacements $q_1, \ldots, q_N$ and momenta $p_1, \ldots, p_N$, with fixed ends $q_0 = q_{N+1} = 0$, and the [[Hamiltonian system|0EDKGA2FAK]] with Hamiltonian
$$H = \sum_{k=1}^{N} \frac{p_k^2}{2} + \sum_{k=0}^{N} \left( \frac{1}{2} (q_{k+1} - q_k)^2 + \frac{\alpha}{3} (q_{k+1} - q_k)^3 \right).$$
For $\alpha = 0$ the chain is a set of $N$ independent normal modes, each with a conserved energy.
Start with all energy in the lowest normal mode and $\alpha \ne 0$ small.
Explain the long-time behaviour of the energies of the normal modes.
In particular, explain why the energy does not spread evenly over all modes.
Instead it stays in a few low modes and returns almost completely to the initial mode after some time.

## Remarks

Fermi, Pasta, Ulam and Tsingou expected the nonlinear term to share the energy evenly between the modes, as statistical mechanics predicts for an ergodic system.
Their computer simulation on the MANIAC computer at Los Alamos showed the opposite: near recurrences to the initial state.
In 1965 Zabusky and Kruskal studied a continuum limit of the chain, which is the [[Korteweg–De Vries equation|KTT75QC72Z]], and discovered [[solitons|63D4AXYCT4]] in it.
Other explanations use the closeness of the chain to an [[integrable system|N0N7CMRCMH]], namely the [[Toda lattice|ADBEEPPKRR]], and KAM-type and Nekhoroshev-type stability.
It is now understood that for small energies the approach to equipartition, if any, happens only on very long time scales.
A full quantitative description is still a subject of research.

## Explanations

### L1 human

Take a chain of masses joined by springs that are slightly nonlinear, and pluck it in its simplest way of vibrating.
Physicists expected the energy to spread to all the ways of vibrating, like heat spreading out.
One of the first computer experiments showed that it does not: the energy moves to a few nearby ways of vibrating and then comes back almost entirely.
Explaining this surprise led to the discovery of solitons and to a large part of modern nonlinear science.
</document>

<document source="https://tamnd.github.io/math2/node/EC6C1MR0C1.md" tag="EC6C1MR0C1" kind="definition">
# Extensive category

## Definition

Let $\mathcal{C}$ be a category with finite [[coproducts|4FJ0S71P58]].
For objects $A$ and $B$ consider the functor
$$+ \colon \mathcal{C}/A \times \mathcal{C}/B \to \mathcal{C}/(A + B)$$
that sends a pair $(f \colon X \to A, \, g \colon Y \to B)$ to $f + g \colon X + Y \to A + B$.
Here $\mathcal{C}/A$ is the [[slice category|1TC7W7BQT3]] over $A$ and the domain is a [[product category|AFNE9C4NH6]].
The category $\mathcal{C}$ is *extensive*, or finitarily extensive, if this functor is an [[equivalence of categories|K7A04ZBD92]] for all objects $A$ and $B$.

## Remarks

An equivalent form: $\mathcal{C}$ has [[pullbacks|Q05K04QJ71]] along the coproduct injections, and the following holds.
For every commutative diagram whose bottom row is $A \to A + B \leftarrow B$ and whose top row is $X_1 \to X \leftarrow X_2$, the top row is a coproduct if and only if both squares are pullbacks.
In an extensive category coproducts are disjoint, and the initial object is strict.

Example: the category of sets is extensive.
A map $h \colon X \to A + B$ is, up to isomorphism, the sum of its restrictions $h^{-1}(A) \to A$ and $h^{-1}(B) \to B$, and this decomposition is unique.

Non-example: the category of real vector spaces, where the coproduct is the direct sum.
Let $\Delta \colon \R \to \R \oplus \R$ be the diagonal $x \mapsto (x, x)$.
The pullbacks of $\Delta$ along both injections are the zero space, since $(x, x) = (a, 0)$ forces $x = 0$.
But $0 \oplus 0 = 0$ is not $\R$, so in the equivalent form the squares are pullbacks while the top row is not a coproduct.
</document>

<document source="https://tamnd.github.io/math2/node/EC7XT2ZM5S.md" tag="EC7XT2ZM5S" kind="concept">
# Affine plane

## Summary

An affine plane is an [[incidence structure|FGKNAFNHA3]] of points and lines in which two points lie on exactly one line and the parallel axiom holds.
It abstracts the incidence properties of the ordinary Euclidean plane.
Every plane $k^2$ over a field is one, but there are also affine planes that come from no field.
Adding a line at infinity turns an affine plane into a [[projective plane|DANFNN8C4Y]].

## Explanations

### L1 human

Forget distances and angles in the plane, and keep only points, lines, and which point is on which line.
What remains are two rules: through two points there is one line, and through a point off a line there is exactly one parallel.
Any system of points and lines that obeys these rules, and is not too small, is an affine plane.
The smallest one has four points and six lines: every pair of points is a line, and the six lines fall into three pairs of parallels.
</document>

<document source="https://tamnd.github.io/math2/node/ECB3W5EVK3.md" tag="ECB3W5EVK3" kind="definition">
# Quantale

## Definition

Let $Q$ be a [[complete lattice|D7A0ZGZT5S]], with join $\bigvee$ of arbitrary subsets.
Let $\cdot$ be a binary operation on $Q$ such that $(Q, \cdot)$ is a [[semigroup|0V3CFG1KK9]].
The structure $(Q, \cdot)$ is a *quantale* if for every $a \in Q$ and every family $(b_i)_{i \in I}$ of elements of $Q$,
$$a \cdot \bigvee_{i \in I} b_i = \bigvee_{i \in I} (a \cdot b_i) \quad \text{and} \quad \Big(\bigvee_{i \in I} b_i\Big) \cdot a = \bigvee_{i \in I} (b_i \cdot a).$$
The index set $I$ may be empty.

## Remarks

Example: the [[ideals|0MYSA6MYNA]] of a [[commutative ring|0GYGHKX834]] $R$, ordered by inclusion, with the product of ideals, form a quantale.
The join of a family of ideals is their sum, and $J \sum_i K_i = \sum_i J K_i$ because the product of ideals distributes over sums.
Example: a complete lattice in which $\wedge$ distributes over arbitrary joins is a quantale with $a \cdot b = a \wedge b$; these are the frames.
Non-example: on the two-element lattice $\set{0, 1}$, let $a \cdot b = 1$ for all $a, b$.
This is associative, but for the empty family $a \cdot \bigvee \emptyset = a \cdot 0 = 1 \ne 0 = \bigvee \emptyset$, so distributivity fails.
The case $I = \emptyset$ says that $a \cdot 0 = 0 \cdot a = 0$, where $0$ is the least element.
A quantale is *unital* if the semigroup has an identity element, and *commutative* if the multiplication is commutative.
</document>

<document source="https://tamnd.github.io/math2/node/ECDJ8DD9RV.md" tag="ECDJ8DD9RV" kind="definition">
# Radon–Nikodym derivative

## Definition

Let $(X, \Sigma)$ be a measurable space and let $\mu$ and $\nu$ be [[σ-finite measures|WEBBT18D4X]] on $\Sigma$ with $\nu$ [[absolutely continuous|P0P21KDV6E]] with respect to $\mu$.
By the [[Radon–Nikodym theorem|1ZSZS5G1Q4]] there is a measurable $f \colon X \to [0, \infty)$ with $\nu(A) = \int_A f \, d\mu$ for all $A \in \Sigma$, and $f$ is unique [[$\mu$-almost everywhere|RARTMA0ZEC]].
The *Radon–Nikodym derivative* of $\nu$ with respect to $\mu$ is this function $f$, determined up to $\mu$-null sets.
It is written $\dfrac{d\nu}{d\mu}$.

## Remarks

Example: let $\mu$ be Lebesgue measure on $\R$ and $\nu(A) = \int_A 2x \, \mathbf{1}_{[0,1]}(x) \, dx$.
Then $\nu$ is a [[measure with density|Y98C5T956Q]], and $\frac{d\nu}{d\mu}(x) = 2x$ for $x \in [0,1]$ and $0$ elsewhere, up to a null set.

Non-example: the Dirac measure $\delta_0$ has no Radon–Nikodym derivative with respect to Lebesgue measure $\lambda$.
Indeed $\lambda(\{0\}) = 0$ but $\delta_0(\{0\}) = 1$, so $\delta_0$ is not absolutely continuous with respect to $\lambda$.

For every measurable $g \colon X \to [0, \infty]$,
$$\int_X g \, d\nu = \int_X g \, \frac{d\nu}{d\mu} \, d\mu.$$
If $\nu \ll \mu \ll \rho$ with all three σ-finite, then the chain rule $\frac{d\nu}{d\rho} = \frac{d\nu}{d\mu} \cdot \frac{d\mu}{d\rho}$ holds $\rho$-almost everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/ECDKZ8G998.md" tag="ECDKZ8G998" kind="definition">
# Concrete category

## Definition

A *concrete category* is a pair $(\mathcal{C}, U)$ where $\mathcal{C}$ is a [[category|A2DE66937D]] and $U \colon \mathcal{C} \to \mathbf{Set}$ is a [[faithful|GRG8WEBSH5]] [[functor|28NQ0KRHD3]] to the category of sets and functions.
The functor $U$ is the *forgetful functor*, and $U(A)$ is the *underlying set* of $A$.
A category $\mathcal{C}$ is *concretizable* if such a faithful functor exists.

## Remarks

Example: the category of [[groups|G5S1M9]] with the functor sending a group to its underlying set is concrete.
Two [[homomorphisms|0JYTG5K9Y4]] $G \to H$ with the same underlying function are equal, so $U$ is faithful.
Non-example: the pair (groups, $P$), where $P$ sends every group to a one-point set, is not a concrete category.
The homomorphisms $x \mapsto 0$ and $x \mapsto x$ of $\Z$ are different but have the same image under $P$, so $P$ is not faithful.
Every small category $\mathcal{C}$ is concretizable.
Send $A$ to the set of all morphisms with codomain $A$, and $f \colon A \to B$ to the map $g \mapsto f \circ g$.
This functor is faithful, since $f$ is the image of $1_A$.
</document>

<document source="https://tamnd.github.io/math2/node/ECHTH3P9RE.md" tag="ECHTH3P9RE" kind="example">
# Ginzburg–Landau functional

## Example

Let $\Omega \subseteq \R^2$ be a bounded open set and let $\varepsilon > 0$.
The simplified *Ginzburg–Landau functional* acts on maps $u \colon \Omega \to \C$ in the [[Sobolev space|M4CEZ5K9HP]] $H^1(\Omega; \C)$ by
$$E_\varepsilon(u) = \int_\Omega \Big( \frac12 \abs{\nabla u}^2 + \frac{1}{4 \varepsilon^2} \big(1 - \abs{u}^2\big)^2 \Big) \, dx.$$
The first term penalizes variation of $u$, and the second term pushes $\abs{u}$ towards $1$.

Check of the zero set: $E_\varepsilon(u) = 0$ holds exactly when $\nabla u = 0$ and $\abs{u} = 1$ almost everywhere.
When $\Omega$ is connected, this means that $u$ is a constant of modulus $1$.

Euler–Lagrange equation: the derivative of $E_\varepsilon$ at $u$ in a direction $v$ is
$$\int_\Omega \Big( \operatorname{Re}\big(\nabla u \cdot \overline{\nabla v}\big) - \frac{1}{\varepsilon^2} \big(1 - \abs{u}^2\big) \operatorname{Re}\big(u \, \overline{v}\big) \Big) \, dx.$$
So critical points solve $-\Delta u = \varepsilon^{-2} u \big(1 - \abs{u}^2\big)$ weakly, an equation of [[Ginzburg–Landau type|713MRSZ9XA]] without magnetic field.

## Remarks

Let $\Omega$ be smooth and simply connected.
Impose a boundary condition $u = g$ on $\partial \Omega$, where $g \colon \partial \Omega \to S^1$ is smooth with winding number $d \ne 0$.
No map in $H^1$ with these boundary values has $\abs{u} = 1$ everywhere, so minimizers must have zeros, called vortices.
The minimal energy grows like $\pi \abs{d} \log(1/\varepsilon)$ as $\varepsilon \to 0$.
The study of the limit $\varepsilon \to 0$, by direct estimates and by [[Γ-convergence|NW9MM370A5]] of $E_\varepsilon / \log(1/\varepsilon)$, locates the vortices.
The full functional of Ginzburg–Landau theory also contains a magnetic vector potential.

## Explanations

### L1 human

The functional models a superconductor or a superfluid by a complex number $u$ at each point of a plane domain.
The material prefers $\abs{u} = 1$, and it prefers $u$ not to change.
If the boundary values wind around the circle, both wishes cannot be met, and $u$ must vanish at a few points.
Near these points, the vortices, $u$ turns around once like the hands of a clock.
As $\varepsilon$ gets small, the vortices shrink to points and their positions are what the limit theory describes.
</document>

<document source="https://tamnd.github.io/math2/node/ECJG25SAJR.md" tag="ECJG25SAJR" kind="statement">
# Hartogs's theorem on separate holomorphicity

## Statement

Let $D \subseteq \C^n$ be an open set and let $f \colon D \to \C$ be a function.
Suppose that $f$ is holomorphic in each variable separately.
That is, for each $j$ and each choice of the other coordinates, the function $z_j \mapsto f(z_1, \dots, z_n)$ is [[holomorphic|0NAHXR1FRH]] where it is defined.
Then $f$ is a [[holomorphic function|507C34W0ZT]] on $D$; in particular, $f$ is continuous.

## Remarks

If $f$ is also assumed continuous, the result is easier: it is [[Osgood's lemma|NBFNXA2WFM]].
The same easy argument works if $f$ is only assumed locally bounded.
In that case the iterated Cauchy formula on small polydiscs gives the power series of $f$.
Hartogs's proof first finds, by a Baire category argument, small polydiscs on which $f$ is bounded.
It then uses [[Hartogs's lemma for subharmonic functions|ZENZT8596D]] on the power series in one variable to get holomorphy at every point.
No such result holds for real variables.
The function $g(x, y) = xy/(x^2 + y^2)$ with $g(0, 0) = 0$ is real analytic in each variable separately, yet it is not continuous at the origin.
Example: $f(z_1, z_2) = e^{z_1 z_2}$ is entire in $z_1$ for each fixed $z_2$ and in $z_2$ for each fixed $z_1$, so it is holomorphic on $\C^2$.

## Explanations

### L1 human

To check that a function of several complex variables is holomorphic, it is enough to check it one variable at a time.
No continuity needs to be assumed; it comes for free.
This is special to complex variables: the same statement fails for real smooth or analytic functions.

### L2 human

The theorem shows that the definition of holomorphy in several variables is very robust.
Complex differentiability, holomorphy in each variable, local power series expansion, and the Cauchy–Riemann equations for $C^1$ functions all give the same class of functions.
In practice one often uses the easier Osgood version, since continuity is usually known.
The full theorem is a striking early result of the theory, and its proof introduced methods that reappear in the study of plurisubharmonic functions.
</document>

<document source="https://tamnd.github.io/math2/node/ECS1GDJ734.md" tag="ECS1GDJ734" kind="conjecture">
# Hilbert–Pólya conjecture

## Statement

There is a complex [[Hilbert space|HM2HHE3J51]] $H$ with a self-adjoint operator $T$ on $H$ that has the following property.
Here self-adjoint means $T = T^*$, where $T^*$ is the [[adjoint|86B7HF79PQ]] of $T$.
A complex number $\rho$ with $0 < \operatorname{Re} \rho < 1$ is a zero of the [[Riemann zeta function|1XWBY4N0VT]] if and only if $\rho = \tfrac{1}{2} + i t$ for an [[eigenvalue|0BQAA2BK3Q]] $t$ of $T$.

## Remarks

The eigenvalues of a self-adjoint operator are real, so the conjecture implies the [[Riemann hypothesis|VCGF9YR36X]]: every nontrivial zero $\rho = \tfrac{1}{2} + it$ then has real part $\tfrac{1}{2}$.
The conjecture is open.
As stated above, it is in fact equivalent to the Riemann hypothesis, since under that hypothesis a diagonal operator with the required eigenvalues can be written down directly.
So the real content of the idea lies in finding a natural operator, for instance one coming from geometry or physics, whose spectrum is given by the zeros.
Evidence for the approach comes from random matrix theory.
Montgomery's pair correlation conjecture of 1973 and Odlyzko's numerical computations indicate that the imaginary parts of the zeros are spaced like the eigenvalues of large random Hermitian matrices.
The idea is attributed to Hilbert and Pólya.
It is known through a 1982 letter of Pólya to Odlyzko, in which Pólya recalled suggesting it in Göttingen around 1912 to 1914.

## Explanations

### L1 human

The Riemann hypothesis says that all the interesting zeros of the zeta function lie on one vertical line in the complex plane.
Symmetric matrices, and more generally self-adjoint operators, always have real eigenvalues.
The Hilbert–Pólya idea is to find an operator whose eigenvalues are the heights of the zeros on that line.
Reality of the eigenvalues would then put all the zeros on the line.
No such natural operator is known.
</document>

<document source="https://tamnd.github.io/math2/node/ECSW5HZGE3.md" tag="ECSW5HZGE3" kind="definition">
# Luzin N property

## Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function.
Write $\lambda$ for [[Lebesgue measure|2Z2QVWZ262]] on $\R$.
The function $f$ has the *Luzin N property* if for every set $E \subseteq [a, b]$ with $\lambda(E) = 0$, the image $f(E)$ satisfies $\lambda(f(E)) = 0$.

## Remarks

Example: a [[Lipschitz function|0QNVQ4VD9T]] $f$ with constant $L$ has the property.
Let $\lambda(E) = 0$ and $\varepsilon > 0$, and cover $E$ by intervals $I_k$ of total length less than $\varepsilon$.
Each set $f(E \cap I_k)$ lies in an interval of length at most $L \cdot \abs{I_k}$.
So $f(E)$ is covered by intervals of total length less than $L \varepsilon$.

Non-example: the [[Cantor function|NPBYYK63FE]] maps the Cantor set $C$, which has measure $0$, onto $[0, 1]$, which has measure $1$.

The image $f(E)$ need not be measurable for a general $f$, so the condition is read with Lebesgue outer measure.
A continuous function on $[a, b]$ has the N property if and only if it maps Lebesgue measurable sets to Lebesgue measurable sets.
</document>

<document source="https://tamnd.github.io/math2/node/ECT6RNF865.md" tag="ECT6RNF865" kind="concept">
# Shooting method

## Summary

The shooting method solves a [[boundary value problem|TM8VQJWGHZ]] by solving a family of [[initial value problems|0FXDW992DS]] instead.
One guesses the missing initial data, solves forward, and adjusts the guess until the condition at the far end holds.
This turns the boundary value problem into a root-finding problem for a function of the missing initial data.
It is used both as a numerical method and as a tool in existence proofs.

## Explanations

### L1 human

Think of firing a cannon at a target on a hill.
You know where the cannon stands but not the right angle.
You fire, see whether the ball lands too short or too far, and correct the angle.
For $y'' = -y$, $y(0) = 0$, $y(\pi/2) = 1$, one shoots with $y'(0) = s$ and gets $y = s \sin x$.
The ball hits the target $y(\pi/2) = 1$ exactly when $s = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/ED0SZS1VED.md" tag="ED0SZS1VED" kind="concept">
# Barrier

## Summary

A barrier at a boundary point $\xi$ of a bounded open set is a [[superharmonic function|X4V2S56YQQ]] that vanishes at $\xi$.
It is positive at every other point of the closure.
Barriers are the local test for whether the [[Perron method|AEZ7PPCNQS]] gives the right boundary value at $\xi$.
A boundary point is regular for the [[Dirichlet problem|8D56GQ4AG8]] if and only if it has a barrier.
Simple geometric conditions, such as an exterior ball or an exterior cone at $\xi$, produce explicit barriers.

## Explanations

### L1 human

A barrier is a superharmonic function that dips down to zero exactly at the chosen boundary point.
It acts like a funnel that forces the Perron solution to approach the boundary value at that point.
In $\R^3$, suppose a closed ball with centre $y$ and radius $R$ meets the closure of the set only at $\xi$.
Then the function $\frac{1}{R} - \frac{1}{\abs{x - y}}$ is a barrier at $\xi$.
</document>

<document source="https://tamnd.github.io/math2/node/ED3DAZVQZ4.md" tag="ED3DAZVQZ4" kind="example">
# Simons cone

## Example

Write points of $\R^8$ as $(x, y)$ with $x, y \in \R^4$.
The *Simons cone* is the hypersurface
$$C = \set{(x, y) \in \R^4 \times \R^4 : \abs{x} = \abs{y}}.$$
It is a [[cone|7TPBQPPN97]]: if $(x, y) \in C$ and $t > 0$, then $(tx, ty) \in C$.
It is smooth away from the origin, and the origin is its only singular point.

Check that $C \setminus \set{0}$ is a [[minimal surface|41X0YW2R9N]], that is, has [[mean curvature|PA78NFEPCF]] zero.
The cone is the zero set of $u(x, y) = \abs{x}^2 - \abs{y}^2$, and $\nabla u = 2(x, -y)$ vanishes only at the origin.
The unit normal is $\nu = (x, -y)/r$ with $r = \sqrt{\abs{x}^2 + \abs{y}^2}$, and the mean curvature is, up to a constant factor, $\operatorname{div} \nu$.
Since $\operatorname{div}(x, -y) = 4 - 4 = 0$ and $(x, -y) \cdot \nabla (1/r) = -(\abs{x}^2 - \abs{y}^2)/r^3$, one gets $\operatorname{div} \nu = -(\abs{x}^2 - \abs{y}^2)/r^3$, which is $0$ on $C$.

## Remarks

The intersection of $C$ with the unit sphere $S^7$ is the product $S^3(1/\sqrt{2}) \times S^3(1/\sqrt{2})$ of two $3$-spheres of radius $1/\sqrt{2}$, a minimal hypersurface of $S^7$.
James Simons showed in 1968 that $C$ is stable, that is, the second variation of area is nonnegative.
Bombieri, De Giorgi and Giusti proved in 1969 that $C$ is in fact area-minimizing: the region $\set{\abs{x} < \abs{y}}$ minimizes perimeter among all sets that agree with it outside a ball.
So area-minimizing hypersurfaces in $\R^8$ can have isolated singular points, and the dimension bound for the singular set of such hypersurfaces is sharp.
The analogous cones $\set{\abs{x} = \abs{y}}$ with $x, y \in \R^k$ for $k \le 3$ are not area-minimizing.
The cone is the reason why the answer to [[Bernstein's problem|NP948D6FR0]] changes at dimension $8$.

## Explanations

### L1 human

Take two copies of four-dimensional space and look at the points where the distance from the origin in the first copy equals the distance in the second.
This set is a cone in eight dimensions, with its tip at the origin.
It balances its own surface tension perfectly away from the tip, like a soap film.
Surprisingly, it is even the least-area surface with its boundary, so least-area surfaces in eight dimensions can have a sharp point.
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# Bohr almost periodic function

## Summary

A Bohr almost periodic function is a continuous function on the real line whose $\varepsilon$-almost periods are spread evenly over the line, for every $\varepsilon > 0$.
This was Harald Bohr's original definition of an [[almost periodic function|GZBSV8DYGS]].
These functions are exactly the uniform limits of trigonometric polynomials with arbitrary real frequencies, by [[Bohr's approximation theorem|GWCY1C1KQV]].
They have a mean value and a Fourier series, and they form a closed subalgebra of the bounded continuous functions.

## Explanations

### L1 human

A periodic function with period $p$ repeats exactly after every multiple of $p$.
A Bohr almost periodic function repeats only approximately, up to an error $\varepsilon$.
The shifts that achieve this are not equally spaced, but they never leave large gaps: every interval of some length $L$ contains one.
For a continuous periodic function one can take the multiples of the period, with $L$ equal to the period.
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# Heegaard splitting

## Summary

A Heegaard splitting cuts a closed orientable [[3-manifold|R20TB2KEQQ]] along a closed surface into two [[handlebodies|AP1KY4EHMA]] of the same genus.
Every closed orientable $3$-manifold has one, so each such manifold is described by a surface and a gluing map between the boundaries of two handlebodies.
The smallest possible genus is the Heegaard genus, an invariant of the manifold.

## Explanations

### L1 human

A handlebody is a solid ball with some solid handles attached, like the inside of a pretzel.
Take two such solids with the same number of handles and glue their surfaces together.
The result is a closed $3$-dimensional space, and every closed orientable one arises in this way.
For example, the $3$-sphere is two solid balls glued along their boundary spheres, and also two solid doughnuts glued along their boundary tori.
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# Galois category

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $F \colon \mathcal{C} \to \mathbf{FinSet}$ be a [[functor|28NQ0KRHD3]] to the category of finite sets.
Let $q \colon Y \to X$ be a morphism of $\mathcal{C}$ such that the [[pullback|Q05K04QJ71]] $Y \times_X Y$ exists.
The morphism $q$ is a *strict epimorphism* when it is the [[coequalizer|DVSTWGVCSZ]] of the two projections $Y \times_X Y \to Y$, its [[kernel pair|E70DQGM8X5]].
The pair $(\mathcal{C}, F)$ is a *Galois category*, and $F$ is a *fibre functor*, when these axioms hold:

1. $\mathcal{C}$ has a [[terminal object|EK85VSSRMR]] and all fibre products.
2. $\mathcal{C}$ has finite [[coproducts|4FJ0S71P58]], in particular an initial object, and for each object $X$ and each finite subgroup $G$ of $\Aut(X)$ the quotient $X/G$ exists.
3. Each morphism $u \colon Y \to X$ factors as $u = u' \circ u''$ with $u''$ a strict epimorphism and $u'$ a [[monomorphism|Y2NP1JFR86]].
Moreover, each monomorphism $Y \to X$ is an isomorphism of $Y$ onto a direct summand of $X$, that is, onto a coproduct component.
4. $F$ sends a terminal object to a terminal object and preserves fibre products.
5. $F$ preserves finite coproducts, sends strict epimorphisms to surjections, and preserves quotients by finite groups of automorphisms.
6. If $u$ is a morphism of $\mathcal{C}$ and $F(u)$ is a bijection, then $u$ is an isomorphism.

## Remarks

Here $X/G$ means the coequalizer of the family of maps $g \colon X \to X$, $g \in G$.
Example: for a [[profinite group|KA0YDB6QXE]] $\pi$, the category of finite sets with a continuous action of $\pi$, with $F$ the functor that forgets the action, satisfies all six axioms.
Main theorem: for every Galois category $(\mathcal{C}, F)$, the group $\pi = \Aut(F)$ is profinite and $F$ induces an equivalence of $\mathcal{C}$ with finite continuous $\pi$-sets.
Non-example: the category of all finite sets with $F$ the functor that doubles each set, $F(X) = X \sqcup X$, fails axiom 4, since it sends the one-point set to a two-point set.
The axioms follow Grothendieck's SGA 1; for a field $k$, finite étale $k$-algebras with arrows reversed form a Galois category whose group is the absolute Galois group of $k$.
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# Stratonovich integral

## Summary

The Stratonovich integral $\int Y \circ dX$ is a stochastic integral.
It evaluates the integrand at the midpoint of each time step, not at the left end.
It differs from the [[Itô integral|JA95NH206E]] by half the [[quadratic covariation|X4DHMZD5JA]] of $Y$ and $X$.
With this integral the ordinary chain rule of calculus holds, which makes it natural in physics and in stochastic differential geometry.

## Explanations

### L1 human

To integrate against a Brownian path, one sums values of the integrand times small increments of the path.
The Itô integral takes the value at the start of each step, while the Stratonovich integral takes the average of the start and end values.
For smooth paths both choices give the same limit, but for Brownian paths they do not.
For example, $\int_0^t B_s \circ dB_s = \tfrac12 B_t^2$ when $B_0 = 0$, just as in ordinary calculus.
The Itô version instead gives $\tfrac12 (B_t^2 - t)$.
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# Bifurcation in a family of maps

## Summary

A bifurcation in a family of maps $f_\mu$ is a parameter value $\mu_0$ at which the qualitative behaviour of the iterates of $f_\mu$ changes.
Typical signs are fixed points or periodic orbits that appear, disappear or change their stability.
It is the setting of [[bifurcation theory|D9TW46P19Y]] for [[discrete-time dynamical systems|5AA9AJEARJ]], that is, for difference equations $x_{n+1} = f_\mu(x_n)$.
For one-dimensional maps the basic local types are the saddle-node, the transcritical, the pitchfork and the [[period-doubling bifurcation|NQDTTZ59X8]].

## Explanations

### L1 human

Picture a recursion with a knob, such as the growth rate in a population model.
For most knob settings, turning it a little changes nothing important: the same number of resting states, each attracting or repelling as before.
At a few special settings something new happens, for example a resting state loses its pull and a two-step cycle appears.
Those settings are the bifurcations.
In the logistic map $x \mapsto r x (1 - x)$, the value $r = 3$ is one: below it solutions settle to one value, just above it they alternate between two values.
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# Prevariety

## Summary

A prevariety is a class of algebras of one signature that is closed under isomorphic copies, [[subalgebras|7CHG6SD365]] and [[direct products|WJ73XYBNE4]].
Every [[quasivariety|WRXEKT4NQS]] and every [[variety|AJNZJD8442]] is a prevariety, but not conversely.
The prevariety generated by a class $K$ consists of the algebras that embed into a product of members of $K$.
Every prevariety is reflective: each algebra has a best approximation inside the class, as in the [[reflection onto a variety|6EWEMZ6FX1]].

## Explanations

### L1 human

Think of a class of algebras that you can only enlarge by two moves: taking a part (a subalgebra) and putting several members side by side (a product).
If the class is stable under both moves, it is a prevariety.
For example, residually finite groups form a prevariety: a subgroup or a product of residually finite groups is again residually finite.
Fields do not, because the product of two fields has zero divisors.
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# Wave front of a Legendrian map

## Definition

Let $(E, \xi)$ be a [[contact manifold|HBQ8BZ7XB5]] of dimension $2n + 1$ and let $\rho \colon E \to B$ be a smooth fibre bundle over a [[smooth manifold|F45VQ47GPV]] $B$ of dimension $n + 1$.
The bundle $\rho$ is a *Legendrian fibration* if each fibre is a [[Legendrian submanifold|Q2DMSTWZYZ]] of $E$.
Let $\iota \colon L \to E$ be a Legendrian [[immersion|6RW68JGBHC]], that is, $\dim L = n$ and $d\iota$ maps each tangent space of $L$ into $\xi$.
The composition $\rho \circ \iota \colon L \to B$ is a *Legendrian map*.
Its *wave front* (or *front*) is the image $\rho(\iota(L)) \subset B$.

## Remarks

The model case is the [[contact structure on the jet space|GCS702QT54]] $J^1(Q) = T^*Q \times \R$ of a manifold $Q$ of dimension $n$, with coordinates $(q, p, z)$, contact form $dz - \sum_i p_i \, dq_i$, and $\rho(q, p, z) = (q, z) \in Q \times \R$.
The fibres $\set{(q, z)} \times T_q^* Q$ are Legendrian, since $dq = 0$ and $dz = 0$ along them.

Example: $Q = \R$ and $\iota(t) = (t^2, t, \tfrac23 t^3)$.
Along $\iota$, $dz = 2t^2 \, dt$ and $p \, dq = t \cdot 2t \, dt = 2t^2 \, dt$, so $\iota^*(dz - p\,dq) = 0$ and $\iota$ is a Legendrian embedding.
Its front is the curve $(t^2, \tfrac23 t^3)$ in the $(q, z)$-plane, a semicubical parabola with a cusp at the origin, although $\iota$ itself is smooth with nonzero velocity.

Non-example: the curve $\gamma(t) = (t, 0, t)$ in $J^1(\R)$ is not Legendrian, since $\gamma^*(dz - p\,dq) = dt \ne 0$; so its projection $(t, t)$ is not a front in this sense.

For the $1$-jet lift $q \mapsto (q, df(q), f(q))$ of a smooth function $f$, the front is the graph of $f$.
Where the front is a smooth hypersurface not tangent to the $z$-direction, the Legendrian submanifold is recovered from it by $p = \partial z / \partial q$.
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# Bedford–Taylor theory

## Summary

Bedford–Taylor theory defines the [[complex Monge–Ampère operator|MXRC891TQS]] $(dd^c u)^n$ for plurisubharmonic functions $u$ that are only locally bounded, not differentiable.
The products $dd^c u_1 \wedge \cdots \wedge dd^c u_k$ are defined inductively as closed [[positive currents|BYJHP6A906]], and they are continuous along decreasing sequences of plurisubharmonic functions.
Eric Bedford and B. A. Taylor developed the theory in work published in 1976 and 1982.
It gives the Monge–Ampère capacity and solves the Dirichlet problem for the complex Monge–Ampère equation with continuous data on strongly pseudoconvex domains.
It is the starting point of modern pluripotential theory.

## Explanations

### L1 human

For a smooth function the Monge–Ampère operator is a determinant of second derivatives, but many important plurisubharmonic functions are not smooth.
For example, $\max(\log\abs{z_1}, \log\abs{z_2}, 0)$ is continuous and plurisubharmonic on $\C^2$, but it has corners.
Bedford and Taylor showed how to give a meaning to the Monge–Ampère measure of such functions by using derivatives in the sense of distributions, one factor at a time.
For the function above, the Monge–Ampère measure turns out to be a constant multiple of the uniform measure on the torus $\abs{z_1} = \abs{z_2} = 1$.
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# Galois cohomology

## Definition

Let $K$ be a field, $K^s$ a [[separable closure|JJ91A1V8NQ]] of $K$, and $G_K = \Gal(K^s/K)$ the [[absolute Galois group|PGKWBJDTKX]] with its Krull topology, a [[profinite group|KA0YDB6QXE]].
A *discrete $G_K$-module* is an [[abelian group|071DDFVG4J]] $M$ with a [[group action|0YSE802YCQ]] of $G_K$ by group automorphisms.
The stabilizer of each $m \in M$ must be an open subgroup of $G_K$.

For $n \ge 0$, let $C^n(G_K, M)$ be the group of continuous maps $G_K^n \to M$, where $M$ has the discrete topology and $C^0(G_K, M) = M$.
Define $d \colon C^n(G_K, M) \to C^{n+1}(G_K, M)$ by
$$(d\varphi)(g_1, \ldots, g_{n+1}) = g_1 \varphi(g_2, \ldots, g_{n+1}) + \sum_{i=1}^{n} (-1)^i \varphi(g_1, \ldots, g_i g_{i+1}, \ldots, g_{n+1}) + (-1)^{n+1} \varphi(g_1, \ldots, g_n).$$
Then $d \circ d = 0$.
The *Galois cohomology groups* of $K$ with coefficients in $M$ are
$$H^n(K, M) = H^n(G_K, M) = \ker\big(d \colon C^n \to C^{n+1}\big) / \im\big(d \colon C^{n-1} \to C^n\big),$$
with $C^{-1} = 0$.

## Remarks

The group $H^0(K, M)$ is the group $M^{G_K}$ of fixed elements.
The group $H^n(K, M)$ is also the direct limit, over finite Galois extensions $L/K$ inside $K^s$, of the [[group cohomology|YXJXA0Q52R]] groups $H^n(\Gal(L/K), M^{\Gal(K^s/L)})$.
Example: $M = (K^s)^\times$ is a discrete $G_K$-module, since each element of $K^s$ lies in a finite extension of $K$, and its stabilizer contains the open subgroup fixing that extension.
For it, $H^0(K, (K^s)^\times) = K^\times$, $H^1(K, (K^s)^\times) = 0$ by [[Hilbert's Theorem 90|YFF8B5A0QG]], and $H^2(K, (K^s)^\times)$ is the [[Brauer group|BEW48TA9WZ]] of $K$.
For an integer $m$ not divisible by the characteristic of $K$, with $\mu_m$ the group of $m$-th roots of unity in $K^s$, [[Kummer theory|XZQENYV15Y]] gives $H^1(K, \mu_m) \cong K^\times / (K^\times)^m$.
Non-example: let $\ell$ be a prime and $\Z_\ell(1)$ the inverse limit of the groups $\mu_{\ell^n}$ in $\overline{\Q}$, with $G_\Q$ acting on each $\mu_{\ell^n}$.
With the discrete topology, $\Z_\ell(1)$ is not a discrete $G_\Q$-module.
The stabilizer of a generator is the subgroup fixing all $\ell$-power roots of unity.
It is closed but not open, because the field of all $\ell$-power roots of unity has infinite degree over $\Q$.
For such modules one uses continuous cochains for the $\ell$-adic topology instead.
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# Dirichlet boundary condition

## Summary

A Dirichlet boundary condition prescribes the values of the unknown function on the boundary of the domain.
For an ordinary differential equation on $[a, b]$, it fixes $y(a)$ and $y(b)$.
For a partial differential equation on a domain $\Omega$, it fixes $u$ on the boundary $\partial \Omega$.
It is the most common type of condition in a [[boundary value problem|TM8VQJWGHZ]], next to the Neumann and Robin conditions.
See the [[definition|G3XFT0MWKH]].

## Explanations

### L1 human

A Dirichlet condition says what the solution must be at the edges.
A string with both ends nailed at height $0$ satisfies $y(a) = y(b) = 0$.
A metal plate whose edge is held at a fixed temperature gives a Dirichlet condition for the heat equation.
For $y'' = 0$ on $[0, 1]$ with $y(0) = 1$ and $y(1) = 3$, the solution is the straight line $y = 1 + 2x$.
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# Differential Galois theory

## Summary

Differential Galois theory studies linear differential equations over a [[differential field|GA5PTQGC6G]], see the [[definition|RW8DQQYJMR]].
It uses the group of symmetries of their solutions, as [[Galois theory|WNQDKQYZ2R]] does for polynomial equations.
The solutions generate a [[Picard–Vessiot extension|TB53Y5NMJD]], and the differential Galois group is the group of its automorphisms that commute with the derivation.
This group is a linear algebraic group over the constants, and there is a Galois correspondence between its closed subgroups and the intermediate differential fields.
The theory explains why some equations, and some integrals such as $\int e^{-x^2} \, dx$, cannot be solved in elementary terms, in line with [[Liouville's theorem|8VRTPECBPZ]].

## Explanations

### L1 human

Classical Galois theory explains why the quintic equation has no formula in radicals, by looking at the symmetries of its roots.
Differential Galois theory does the same for linear differential equations such as $y'' + x y = 0$.
The symmetries now form a group of matrices that can be continuous, such as all rotations or all scalings.
For $y' = y$ over the rational functions, the solutions are $c e^x$, and the symmetries are the scalings $e^x \mapsto c e^x$ with $c \ne 0$.
Whether an equation can be solved by integrals, exponentials and algebraic functions depends on whether this group is solvable in a suitable sense.
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# Costa's minimal surface

## Example

Costa's surface is a complete [[minimal surface|41X0YW2R9N]] in $\R^3$ that is embedded and has finite total curvature.
Topologically it is a torus with three points removed.
Near the three removed points the surface has three ends: two look like the ends of a [[catenoid|6NN1HP54GB]] and one looks like a plane.

It is built with the [[Weierstrass–Enneper parameterization|YWV6J23AP6]].
The domain is the square torus $\C / (\Z + i\Z)$ minus three points, and the data are given by the Weierstrass $\wp$ function of this lattice.
One checks that the real periods of the integrals vanish, so the surface is well defined, and minimality then follows from the parameterization.
The hard part, proved later, is that the surface has no self-intersections.

## Remarks

Costa described the surface in 1982, and Hoffman and Meeks proved that it is embedded.
Before it, the only known complete embedded minimal surfaces of finite topology were the plane, the catenoid and the helicoid.
Its total curvature is $-12\pi$.

## Explanations

### L1 human

Costa's surface looks like a catenoid with a flat sheet slicing through its waist, where the sheet and the tube are joined by two tunnels instead of crossing.
It has a hole like a doughnut and goes off to infinity in three directions.
Its discovery showed that minimal surfaces without self-crossings are much more varied than had been thought.
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# Equivalence of varieties

## Definition

Let $\mathcal{V}$ and $\mathcal{W}$ be [[varieties|AJNZJD8442]] of algebras.
Let $\rho$ be an [[interpretation|32PDB3WQ4D]] of $\mathcal{V}$ in $\mathcal{W}$, which sends each $A \in \mathcal{W}$ to an algebra $A^{\rho} \in \mathcal{V}$ on the same set.
Let $\theta$ be an interpretation of $\mathcal{W}$ in $\mathcal{V}$, which sends each $B \in \mathcal{V}$ to an algebra $B^{\theta} \in \mathcal{W}$ on the same set.
The pair $(\rho, \theta)$ is an *equivalence* if $(A^{\rho})^{\theta} = A$ for all $A \in \mathcal{W}$ and $(B^{\theta})^{\rho} = B$ for all $B \in \mathcal{V}$.
The varieties $\mathcal{V}$ and $\mathcal{W}$ are *equivalent*, or *term equivalent*, if such a pair exists.

## Remarks

Example: Boolean algebras and Boolean rings are equivalent.
One translation is $x + y \mapsto (x \wedge \neg y) \vee (\neg x \wedge y)$, $x \cdot y \mapsto x \wedge y$, with $0$ and $1$ kept.
The other is $x \vee y \mapsto x + y + xy$, $x \wedge y \mapsto xy$, $\neg x \mapsto 1 + x$.
Going there and back recovers each operation, for example $x + y + xy$ becomes the join again in every Boolean algebra.
Non-example: the variety of Boolean algebras is not equivalent to the variety of distributive lattices with signature $\vee, \wedge$.
There is not even an interpretation of Boolean algebras in distributive lattices, because every lattice term contains a variable, so no term can stand for the constant $0$.
Equivalent varieties have, on each set, the same subalgebras, the same congruences and the same term operations.

## Explanations

### L1 human

Two varieties are equivalent when they describe the same structures with different but interchangeable basic operations.
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# Cohomological dimension of a group

## Summary

The cohomological dimension of a [[group|G5S1M9]] $G$ is the largest degree in which [[group cohomology|YXJXA0Q52R]] of $G$ can be nonzero.
It equals the shortest length of a projective resolution of $\Z$ over the [[group ring|4NMTSX0XDS]] $\Z[G]$.
Free groups are exactly the groups of dimension at most $1$, by the [[Stallings–Swan theorem|B41GE4M134]], and $\Z^n$ has dimension $n$.
A group of finite cohomological dimension is [[torsion-free|PEMNN32A43]].

## Explanations

### L1 human

The cohomological dimension says how many dimensions a group needs to be modelled as a shape.
The integers $\Z$ need one dimension: they are the loops on a circle.
The group $\Z^2$ needs two, like a torus.
A nontrivial finite cyclic group cannot be modelled in any finite dimension, and its dimension is infinite.
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# Factorization of polynomials

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be a nonzero polynomial.
A factorization of $f$ into irreducibles is an expression
$$f = c \, p_1^{e_1} \cdots p_k^{e_k}$$
such that:

1. $c \in K$ is nonzero;
2. $p_1, \dots, p_k$ are pairwise distinct [[monic|0BDX4MBKT4]] [[irreducible polynomials|00X2A9G9W3]] in $K[x]$, with $k \ge 0$;
3. $e_1, \dots, e_k$ are positive integers.

## Remarks

Every nonzero $f$ has such a factorization, and it is unique up to the order of the factors, since $K[x]$ is a [[unique factorisation domain|06YMXJMXYS]].
The constant $c$ is the leading coefficient of $f$.
Example: over $\Q$, $2x^3 - 2x = 2 \, x (x - 1)(x + 1)$; each factor is monic of degree $1$, hence irreducible.
Non-example: $x^4 - 1 = (x^2 - 1)(x^2 + 1)$ is a factorization but not into irreducibles, since $x^2 - 1 = (x - 1)(x + 1)$ violates clause 2.
Over a unique factorisation domain $R$ such as $\Z$, a nonzero $f \in R[x]$ is its content times a primitive polynomial.
The primitive part factors into primitive irreducibles, by [[polynomial ring over a UFD is a UFD|0D6KH8Y59N]].
Over finite fields, see [[factorization over finite fields|XNE7F0BAGE]].

## Explanations

### L1 human

The definition fixes the shape of the answer: a constant times powers of different monic irreducible polynomials.
The exponents record repeated factors.
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# Lebesgue spine

## Counterexample

In $\R^3$, write points as $x = (x_1, x_2, x_3)$ and consider the closed thin spike
$$K = \set{x \in \R^3 : 0 < x_1 \le 1, \; x_2^2 + x_3^2 \le e^{-2/x_1}} \cup \set{0}.$$
Its cross-section at distance $x_1$ from the origin is a disc of radius $e^{-1/x_1}$.
Let $\Omega = B \setminus K$, where $B$ is the [[open ball|0YE5868HZB]] of radius $2$ centred at $0$.
The origin $0$ is a boundary point of $\Omega$, the tip of the spike, and it is not a [[regular boundary point|B7948M365K]].
Check with [[Wiener's criterion|STMT1K1865]], using the part $E_k$ of the complement in the shell $2^{-k-1} \le \abs{x} \le 2^{-k}$.
For large $k$, $E_k$ lies in a thin cylinder of length about $2^{-k-1}$ and radius at most $e^{-2^k}$.
A cylinder of length $\ell$ and radius $r \le \ell/2$ has Newtonian [[capacity|4YZVFGJVSZ]] at most a constant times $\ell / \log(\ell / r)$.
So $\operatorname{cap}(E_k)$ is at most a constant times $2^{-k} / 2^k = 4^{-k}$, and $\sum_k 2^k \operatorname{cap}(E_k)$ converges.
Hence $0$ is irregular.

## Remarks

The example is due to Lebesgue.
The spike is connected and has interior points, so a boundary point can be irregular even when the complement of $\Omega$ near it is a connected set of positive volume.
Suppose the spike is less sharp, for example with radius $x_1^m$ for some $m > 1$.
Then $E_k$ contains a cylinder whose capacity is at least a constant times $2^{-k}/k$.
The terms of Wiener's series are then at least a constant times $1/k$, the series diverges, and the tip is regular.
In the plane no such example exists: a boundary point that lies in a connected component of the complement with more than one point is always regular.

## Explanations

### L1 human

Push a very sharp needle into a ball from the inside, so that the needle tip is a point of the boundary.
If the needle gets thin fast enough near its tip, the solution of the Dirichlet problem does not feel the tip.
Then a prescribed boundary value at the tip can be ignored by the solution.
</document>

<document source="https://tamnd.github.io/math2/node/EFFRJFW67Y.md" tag="EFFRJFW67Y" kind="concept">
# Foundations of mathematics

## Summary

The foundations of mathematics study the basic framework in which all of mathematics is built.
They ask which language mathematics is written in, which axioms it starts from, and what counts as a [[proof|0B94TKFQJX]].
Today the usual answer is [[predicate logic|086QKPF0A6]] together with the axioms of [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].
The field also studies the limits of this framework, such as the [[incompleteness theorems|X9WYNYE29C]].

## Explanations

### L1 human

Every theorem rests on earlier theorems, and these rest on definitions and axioms.
The foundations of mathematics look at the bottom of this tower.
For example, the natural numbers can be built from sets: $0$ is the empty set, $1$ is $\set{0}$, $2$ is $\set{0, 1}$, and so on.
Then facts about numbers become facts about sets, and only the axioms of set theory are needed.
Around 1900, paradoxes in naive set theory made this question urgent, and [[logicism|5PGSQYB5Y1]], [[formalism|46JCQ6KDA4]] and [[intuitionism|XQHF63BBYF]] gave different answers.
</document>

<document source="https://tamnd.github.io/math2/node/EFJ77PFF88.md" tag="EFJ77PFF88" kind="definition">
# Cofibration

## Definition

Let $A$ and $X$ be topological spaces and let $I = [0, 1]$.
Let $i \colon A \to X$ be a [[continuous map|0VGFHTM75Y]].
The map $i$ is a *cofibration* if the following holds for every topological space $Y$.
For every continuous map $f \colon X \to Y$ and every [[homotopy|9YA6S6HD9V]] $h \colon A \times I \to Y$ with $h(a, 0) = f(i(a))$ for all $a \in A$, there is a homotopy $H \colon X \times I \to Y$ with:

1. $H(x, 0) = f(x)$ for all $x \in X$;
2. $H(i(a), t) = h(a, t)$ for all $a \in A$ and $t \in I$.

## Remarks

For the inclusion of a subspace $A \subseteq X$, this says that the pair $(X, A)$ has the [[homotopy extension property|D42X0F1EG1]].
Every cofibration is a homeomorphism onto its image, and if $X$ is Hausdorff the image is closed in $X$.
Example: the inclusion $S^{n-1} \to D^n$ is a cofibration, and so is the inclusion of a subcomplex of a CW complex.
Non-example: the inclusion $(0, 1) \to [0, 1]$ is not a cofibration, since $[0, 1]$ is Hausdorff and $(0, 1)$ is not closed in it.
Cofibrations are stable under pushout and under composition.
</document>

<document source="https://tamnd.github.io/math2/node/EFY92E7GC7.md" tag="EFY92E7GC7" kind="definition">
# John ellipsoid

## Definition

Let $n \ge 1$ and let $B$ be the closed unit ball of $\R^n$.
An *ellipsoid* in $\R^n$ is a set $E = c + A B = \set{c + Ax : \abs{x} \le 1}$, where $c \in \R^n$ and $A$ is an invertible real $n \times n$ [[matrix|0VD9ZSWWJG]].
Its volume is $\abs{\det A}\, \kappa_n$, where $\kappa_n$ is the [[Lebesgue measure|2Z2QVWZ262]] of $B$.
Let $K \subseteq \R^n$ be a compact [[convex body|437FSG3419]] with nonempty interior.
The *John ellipsoid* of $K$ is the ellipsoid of maximal volume among all ellipsoids $E$ with $E \subseteq K$.

## Remarks

The [[Löwner–John theorem|TM5SYH4XHY]] shows that this ellipsoid exists and is unique.
The ellipsoid of minimal volume containing $K$ is the *Löwner ellipsoid* (or Löwner–John ellipsoid) of $K$.

Example: $K = [-1, 1]^2$.
Its John ellipsoid is the unit disk $B$.
The problem is invariant under the symmetries of the square, so by uniqueness the maximal ellipse is invariant under them too.
An ellipse invariant under rotation by $90°$ about the origin is a disk centered at $0$, and the largest such disk in $K$ is $B$.

Non-example: in the rectangle $K = [-2, 2] \times [-1, 1]$ the unit disk lies in $K$ but is not the John ellipsoid.
The ellipse $\set{(x, y) : x^2/4 + y^2 \le 1}$ also lies in $K$ and has area $2\pi > \pi$.

The condition that $K$ has nonempty interior is needed: otherwise every ellipsoid in $K$ would be degenerate, and no ellipsoid in the above sense lies in $K$.
</document>

<document source="https://tamnd.github.io/math2/node/EFYGR426DF.md" tag="EFYGR426DF" kind="statement">
# Differences of polynomials vanish beyond the degree

## Statement

Let $R$ be a commutative ring, let $h \in R$, and let $p \in R[x]$ be a [[polynomial|0Q9EZS90GY]].
Let $n \in \N$ with $\deg p < n$, where the [[degree|0DPF6WRZW6]] of the zero polynomial is $-\infty$.
Then the [[$n$-th forward difference|E38801KVJ1]] of the polynomial function $x \mapsto p(x)$ on $R$ with step $h$ is zero:
$$\Delta_h^n p(x) = 0 \quad \text{for all } x \in R.$$

## Remarks

The key step is that $\Delta_h$ lowers the degree: for $m \ge 1$, $(x + h)^m - x^m = \sum_{j<m} \binom{m}{j} h^{m-j} x^j$ has degree less than $m$, and $\Delta_h$ sends constants to $0$.
Induction on $n$ then gives the claim.
The same computation shows more: if $p$ has degree $d$ and leading coefficient $a$, then $\Delta_h^d p$ is the constant $d!\, a\, h^d$.
Example: for $p(x) = x^3$ and $h = 1$, the third differences of $0, 1, 8, 27, 64$ are all $6 = 3!$, and the fourth differences are $0$.
Over $\R$ there is a converse: a function $f \colon \Z \to \R$ with $\Delta^n f = 0$ agrees on $\Z$ with a polynomial of degree less than $n$.

## Explanations

### L1 human

Write the values of a polynomial at equally spaced points and take differences again and again.
For a polynomial of degree $d$, the $d$-th row is constant and every later row is zero.
This is the discrete form of the fact that the $(d+1)$-th derivative of a polynomial of degree $d$ is zero.

### L2 human

The fact gives a practical test: if a table of values has constant $d$-th differences, the data fit a polynomial of degree $d$.
With the [[expansion of iterated differences in shifts|24D1EGHP4G]] it yields identities such as $\sum_{k=0}^{n} (-1)^k \binom{n}{k} k^m = 0$ for $0 \le m < n$.
Combined with [[Newton's forward difference formula|DDHCMKHD2T]], it shows that a polynomial of degree $d$ is determined by its values at $d + 1$ equally spaced points.
It is then given by a finite [[Newton series|Y2HXSEA2EE]].
</document>

<document source="https://tamnd.github.io/math2/node/EG1N82CV0R.md" tag="EG1N82CV0R" kind="definition">
# Boolean algebras canonically defined

## Definition

Let $\mathbf{2} = (\set{0, 1}, \wedge, \vee, \neg, 0, 1)$ be the [[two-element Boolean algebra|0WG43AC0EV]].
A *term* is an expression built from variables with the binary symbols $\wedge, \vee$, the unary symbol $\neg$ and the constants $0, 1$.
An *equation* is a pair of terms, written $s = t$.
Let $A$ be a set with operations $\wedge, \vee, \neg$ and constants $0, 1$.
An equation *holds* in $A$ if both sides take the same value for every assignment of elements of $A$ to the variables.
In the canonical definition, a *Boolean algebra* is such a set $A$ in which every equation that holds in $\mathbf{2}$ also holds.

## Remarks

This class coincides with the class given by the usual axioms of a [[Boolean algebra|06GKBTAA5E]].
One direction is clear, since the usual axioms hold in $\mathbf{2}$; the other uses the fact that every equation true in $\mathbf{2}$ follows from those axioms.
In the language of universal algebra, the Boolean algebras form the variety generated by $\mathbf{2}$.

Example: the power set of a set $X$, with intersection, union and complement.
A subset $S$ of $X$ is the same as a function $X \to \set{0, 1}$, and the operations act point by point.
So an equation true in $\mathbf{2}$ holds at each point of $X$, and hence in the power set.

Non-example: the chain $0 < m < 1$ with $\wedge = \min$, $\vee = \max$, $\neg 0 = 1$, $\neg m = 0$ and $\neg 1 = 0$.
The equation $x \vee \neg x = 1$ holds in $\mathbf{2}$, but for $x = m$ it gives $m \vee 0 = m \ne 1$.
</document>

<document source="https://tamnd.github.io/math2/node/EG41JQS3PC.md" tag="EG41JQS3PC" kind="definition">
# WKB approximation

## Definition

Let $I$ be an open interval, let $Q \colon I \to \R$ be smooth with $Q(x) \neq 0$ for all $x \in I$, and let $\varepsilon > 0$ be a small parameter.
Consider the linear [[ordinary differential equation|0NT64JV96A]]
$$\varepsilon^2 y''(x) = Q(x) \, y(x), \quad x \in I.$$
The *WKB ansatz* is $y = \exp\bigl(\frac{1}{\varepsilon} \sum_{n \ge 0} \varepsilon^n S_n(x)\bigr)$.
Inserting it and collecting powers of $\varepsilon$ gives $S_0'^2 = Q$ at order $\varepsilon^0$ and $2 S_0' S_1' + S_0'' = 0$ at order $\varepsilon^1$.
The *WKB approximation*, or *Liouville–Green approximation*, keeps these two orders.
It gives the two functions
$$y_\pm(x) = Q(x)^{-1/4} \exp\Bigl(\pm \frac{1}{\varepsilon} \int_{x_0}^{x} \sqrt{Q(s)} \, ds\Bigr),$$
where $x_0 \in I$ and the roots are taken with a fixed branch.
On compact subintervals of $I$, the equation has solutions $y = y_\pm \, (1 + O(\varepsilon))$ as $\varepsilon \to 0$.

## Remarks

Example: for constant $Q = q > 0$, the formula gives $q^{-1/4} e^{\pm \sqrt{q} (x - x_0) / \varepsilon}$, which solve the equation exactly.
Check of the first order: with $S_0' = \sqrt{Q}$, the second relation gives $S_1' = -S_0'' / (2 S_0') = -Q' / (4 Q)$, so $e^{S_1} = Q^{-1/4}$.

Non-example: the hypothesis $Q \neq 0$ fails for the Airy-type equation $\varepsilon^2 y'' = x y$ on an interval containing $0$.
The point $x = 0$ is a *turning point*, where $Q^{-1/4}$ blows up; there the WKB formulas are not valid, and solutions are matched through Airy functions.

The series $\sum \varepsilon^n S_n$ is in general only an [[asymptotic expansion|Q26QN6GYZX]], not a convergent series.
When $Q < 0$, one writes $Q = -k^2$, and the two solutions become $k^{-1/2} e^{\pm i \int k / \varepsilon}$, which oscillate.

## Explanations

### L1 human

The method guesses that the solution is an exponential with a slowly changing exponent.
The first equation fixes the phase, and the second fixes how the amplitude changes.
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<document source="https://tamnd.github.io/math2/node/EG4AJVEFT2.md" tag="EG4AJVEFT2" kind="concept">
# Integral current

## Summary

An integral current is a [[current|360H7MD9RD]] that behaves like an oriented surface with integer multiplicities.
It is given by integration over a [[rectifiable set|7N0GFMW05R]] with an orientation and a positive integer multiplicity.
Both it and its boundary have finite mass.
Integral currents include all compact oriented smooth submanifolds with boundary.
By the [[Federer–Fleming compactness theorem|TPXYARH912]], they are closed under the limits that matter for area minimisation.
They are the setting of the Plateau problem in all dimensions and codimensions.

## Explanations

### L1 human

An integral current is a generalised surface that can have corners and singularities, and parts counted twice or three times, but never a fractional number of times.
You integrate a differential form over it as over an ordinary oriented surface, counting each part with its multiplicity.
For example, the unit disc in the plane, traversed once, is an integral $2$-current with boundary the unit circle.
The same disc counted twice is also one, but the disc counted half a time is not.
</document>

<document source="https://tamnd.github.io/math2/node/EG4VCBXDKZ.md" tag="EG4VCBXDKZ" kind="concept">
# Kruskal–Wallis test

## Summary

The Kruskal–Wallis test is a [[hypothesis test|087P03NEND]] of whether several independent samples come from the same distribution.
It replaces the data by their ranks in the pooled sample and measures how much the mean ranks of the groups differ.
It is the rank-based counterpart of one-way [[analysis of variance|M9AQ5HKGP4]], and for two groups it is equivalent to the [[Mann–Whitney U test|CKF9KDXBP7]].

## Explanations

### L1 human

Suppose three fertilisers are tried on separate plots and the yields are recorded.
Line up all the yields together and number them from smallest to largest.
If the fertilisers make no difference, each group should get a fair mix of low and high numbers, and the average number in each group should be near the middle.
The test statistic grows when some groups get mostly high numbers and others mostly low ones.
</document>

<document source="https://tamnd.github.io/math2/node/EG8Z3FWV9A.md" tag="EG8Z3FWV9A" kind="concept">
# Double layer potential

## Summary

A double layer potential is the potential of a layer of dipoles spread over a surface, with dipoles pointing along the normal.
It is the integral of the normal derivative of the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] against a density on the boundary of a domain.
It is [[harmonic|0DEB0JMN0W]] off the surface, and its value jumps across the surface by exactly the density, as the [[jump relations|J58YA8QNNX]] say.
This jump makes it the standard tool for turning the [[Dirichlet problem|8D56GQ4AG8]] into a Fredholm integral equation, see the [[boundary integral method|QQ62R5ZAXJ]].

## Explanations

### L1 human

Put a thin sheet of tiny magnets on a surface, each with its north pole just outside and its south pole just inside.
The potential they create is a double layer potential.
When you cross the sheet, the potential jumps up by the strength of the magnets at that place.
For the unit sphere in space with constant strength $1$, the potential is $-1$ inside the sphere and $0$ outside.
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<document source="https://tamnd.github.io/math2/node/EGAJPZ71VW.md" tag="EGAJPZ71VW" kind="concept">
# Quasiregular map

## Summary

A quasiregular map is a map between domains of $\R^n$ whose infinitesimal distortion is bounded.
It sends small balls to sets that are round up to a fixed factor.
Unlike a [[quasiconformal map|SB8KG46NHP]], it need not be injective, so it generalizes holomorphic functions rather than conformal maps.
In the plane, quasiregular maps are holomorphic functions composed with quasiconformal homeomorphisms.
In dimension $n \ge 3$ they give a rich substitute for holomorphic functions, since conformal maps there are only Möbius maps.

## Explanations

### L1 human

A holomorphic function stretches the plane near each point by the same amount in all directions.
A quasiregular map may stretch more in one direction than in another, but the ratio is bounded by a fixed constant at all points.
It may fold and cover some points several times, like $z \mapsto z^2$.
Results of Reshetnyak show that such maps, when not constant, still keep key features of holomorphic functions: they are open and their preimages of points are discrete.
</document>

<document source="https://tamnd.github.io/math2/node/EGBAZMS66W.md" tag="EGBAZMS66W" kind="statement">
# Maximum principle for potentials

## Statement

Let $n \ge 3$ and let $\mu$ be a positive [[Radon measure|HZNEQQBCW4]] on $\R^n$ with compact [[support|FQTXV6BQZG]] $S$.
Let
$$U^\mu(x) = \int_{\R^n} \abs{x - y}^{2-n} \, d\mu(y)$$
be its [[Newtonian potential|DCMFQ01QGS]], with values in $[0, \infty]$.
Let $M$ be a real number with $U^\mu(x) \le M$ for every $x \in S$.
Then $U^\mu(x) \le M$ for every $x \in \R^n$.

## Remarks

Proof idea: first assume that $U^\mu$ restricted to $S$ is continuous.
The [[continuity principle|GPJC1NY5RE]] then makes $U^\mu$ continuous on $\R^n$.
On the open set $\R^n \setminus S$ the potential is harmonic, it is at most $M$ on the boundary, which lies in $S$, and it tends to $0$ at infinity.
The maximum principle for harmonic functions gives $U^\mu \le M$ there.
The general case follows by restricting $\mu$ to compact subsets of $S$ on which $U^\mu$ is continuous, which exist by Lusin's theorem, and letting these subsets increase.

Example: let $\mu$ be the uniform measure of total mass $m$ on the unit sphere in $\R^3$.
By the shell theorem, $U^\mu(x) = m$ for $\abs{x} \le 1$ and $U^\mu(x) = m / \abs{x}$ for $\abs{x} \ge 1$.
So $U^\mu \le m$ on the support, and indeed $U^\mu \le m$ everywhere.

The positivity of $\mu$ is needed.
Let $\sigma$ be the uniform probability measure on the unit sphere of $\R^3$ and let $\mu = -\sigma$.
Then $U^\mu = -1$ on the support, but $U^\mu(x) = -1/\abs{x} > -1$ for $\abs{x} > 1$.
For logarithmic potentials in the plane the same statement holds: if $\int \log(1/\abs{z - w}) \, d\mu(w) \le M$ on the support of $\mu$, then it is at most $M$ on all of $\C$.

## Explanations

### L1 human

The potential of a positive charge is largest on the charge itself.
If the potential stays below a level $M$ wherever there is charge, it stays below $M$ everywhere in space.
This is the potential-theory version of the fact that a harmonic function has no interior maximum.

### L2 human

The principle is due to Maria and to Frostman.
It is the key step that turns an inequality on the support of a measure into an inequality everywhere.
It is used in the proof of Frostman's theorem to show that the equilibrium potential never exceeds its value on the conductor.
For Riesz kernels $\abs{x - y}^{\alpha - n}$ with $0 < \alpha < n$ and $\alpha > 2$, only a weaker version holds, with $M$ replaced by a constant multiple of $M$.
</document>

<document source="https://tamnd.github.io/math2/node/EGCDMWYKQB.md" tag="EGCDMWYKQB" kind="concept">
# Gaussian free field

## Summary

The Gaussian free field is a random function whose law is Gaussian, with covariance given by the Green function of the [[Laplacian|0E0B6RA8K0]].
In the continuum it is a random generalised function rather than a function.
It is the natural Gaussian analogue of a [[harmonic function|0DEB0JMN0W]] with random fluctuations.
In two dimensions the continuum field is conformally invariant and log-correlated.
It appears in statistical physics, in random surfaces and in the theory of the Schramm–Loewner evolution.

## Explanations

### L1 human

Think of a random landscape over a grid whose height is pinned to $0$ at the boundary.
Each height is pulled towards the average height of its neighbours, and randomness pushes it away.
Landscapes with large differences between neighbours are unlikely, so the surface is rough but not wild.
The discrete Gaussian free field is this random landscape, and its fine-mesh limit is the continuum free field.
</document>

<document source="https://tamnd.github.io/math2/node/EGKVP4KK6V.md" tag="EGKVP4KK6V" kind="definition">
# Abelian scheme

## Definition

Let $S$ be a scheme.
An *abelian scheme* over $S$ is a [[group scheme|TJ04VQP4NB]] $\pi \colon A \to S$ such that:

1. $\pi$ is [[smooth|EXKGG9X6J0]];
2. $\pi$ is [[proper|9537RKBZPP]];
3. every fibre of $\pi$ is geometrically connected.

## Remarks

For each point $s \in S$, the fibre $A_s$ is an [[abelian variety|R7W03TXRS9]] over the residue field $\kappa(s)$.
An abelian scheme over $\operatorname{Spec} k$, for a field $k$, is the same as an abelian variety over $k$.
The group law of an abelian scheme is commutative.

Example: let $S = \operatorname{Spec} \Z[1/2]$ and let $E \subseteq \mathbb{P}^2_S$ be the closure of $y^2 = x^3 - x$, with unit section the point at infinity.
The discriminant of this Weierstrass equation is $64 = 2^6$, a unit on $S$.
So every fibre is a smooth curve of genus $1$, and $E$ is an abelian scheme of relative dimension $1$.

Non-example: the same equation over $\operatorname{Spec} \Z$ fails clause 1.
Modulo $2$, the curve $y^2 = x^3 + x$ is singular at the point $(1, 0)$, where both partial derivatives $2y$ and $3x^2 + 1$ vanish.
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<document source="https://tamnd.github.io/math2/node/EGPP69XA4F.md" tag="EGPP69XA4F" kind="statement">
# Finite determinacy theorem

## Statement

Let $\mathcal{E}_n$ be the ring of [[germs|7SWSRGDT4A]] at $0$ of smooth functions $\R^n \to \R$.
Let $\mathfrak{m} \subset \mathcal{E}_n$ be its maximal [[ideal|096Z9MP2ME]], the germs that vanish at $0$.
For $f \in \mathcal{E}_n$, let $J(f)$ be the Jacobian ideal, generated by $\partial f / \partial x_1, \ldots, \partial f / \partial x_n$.
A germ $g$ is *right equivalent* to $f$ if $g = f \circ \varphi$ for a germ of [[diffeomorphism|0KZZ70ZHXP]] $\varphi \colon (\R^n, 0) \to (\R^n, 0)$.
Call $f$ *$k$-determined* if every $g \in \mathcal{E}_n$ with the same [[$k$-jet|1QE0Z0756D]] at $0$ as $f$ is right equivalent to $f$.

1. If $\mathfrak{m}^{k+1} \subseteq \mathfrak{m}^2 J(f)$, then $f$ is $k$-determined.
2. If $f$ is $k$-determined, then $\mathfrak{m}^{k+1} \subseteq \mathfrak{m} J(f)$.
3. Hence $f$ is $k$-determined for some $k$ if and only if $\mathfrak{m}^r \subseteq J(f)$ for some $r$.

## Remarks

The proof of the first part uses the homotopy method: join $f$ to $g$ by $f_t = f + t(g - f)$ and look for diffeomorphisms $\varphi_t$ with $f_t \circ \varphi_t = f$.
Differentiating in $t$ leads to an equation $g - f = -\sum_i \xi_i \, \partial f_t / \partial x_i$ for a time-dependent vector field $\xi$ vanishing at $0$, which can be solved by Nakayama's lemma because $g - f \in \mathfrak{m}^{k+1}$.
Example: if $f$ has a nondegenerate [[critical point|0YTCMFB4HC]] at $0$, then $J(f) = \mathfrak{m}$, so $\mathfrak{m}^3 = \mathfrak{m}^2 J(f)$ and $f$ is $2$-determined.
With the normal form of a quadratic form this gives the [[Morse lemma|ERSD8D4HWX]].
For $f = x^3$ on $\R$, $J(f) = (x^2)$ and $\mathfrak{m}^2 J(f) = (x^4) = \mathfrak{m}^4$, so $x^3$ is $3$-determined.
For $f = x^2 y$ on $\R^2$, the ideal $J(f) = (xy, x^2)$ contains no power of $y$, so $f$ is not finitely determined.
Mather and Tougeron proved such criteria; similar results hold for other equivalence relations and for holomorphic germs.

## Explanations

### L1 human

A smooth function near a point has a Taylor expansion with infinitely many terms.
Finite determinacy says that, for most interesting critical points, only finitely many terms matter.
After the first $k$ terms, any change of the higher terms can be absorbed by a change of coordinates.
For example, near a nondegenerate minimum, the quadratic terms already decide the shape completely.

### L2 human

The theorem is what makes the classification of singularities a finite problem.
A $k$-determined germ is fixed by a polynomial, its $k$-jet, so the classification takes place in a finite-dimensional jet space under the action of a finite-dimensional group.
The condition $\mathfrak{m}^r \subseteq J(f)$ says that the local algebra $\mathcal{E}_n / J(f)$ is finite-dimensional; for holomorphic germs its dimension is the Milnor number.
This underlies [[Arnold's classification of simple critical points|M5437FP73G]] and the theory of [[versal unfoldings|KSAFWZVB8K]].
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<document source="https://tamnd.github.io/math2/node/EGQ3CJDH0D.md" tag="EGQ3CJDH0D" kind="statement">
# Lebedev quadrature

## Statement

Let $S^2 \subset \R^3$ be the unit [[sphere|P9CM4GQ2T0]] with its surface measure $\sigma$.
Let $\mathcal{O}_h$ be the [[group|G5S1M9]] of the $48$ orthogonal maps of $\R^3$ that map the cube $[-1, 1]^3$ onto itself.
Consider a rule $Q(f) = \sum_{x \in X} w_x f(x)$ on a finite set $X \subset S^2$ that is invariant: $g(X) = X$ and $w_{g(x)} = w_x$ for all $g \in \mathcal{O}_h$ and $x \in X$.
Let $p \ge 0$.
Then two conditions are equivalent.
The first is that $Q(f) = \int_{S^2} f \, d\sigma$ for every polynomial $f$ on $\R^3$ of degree at most $p$.
The second is that this holds for every such polynomial with $f \circ g = f$ for all $g \in \mathcal{O}_h$.

A *Lebedev quadrature* rule is an invariant rule of this kind with nodes on orbits of $\mathcal{O}_h$.
It is chosen to be exact for all polynomials of degree at most $p$.

## Remarks

Proof idea: for any $f$ let $\bar f = \frac{1}{48} \sum_{g \in \mathcal{O}_h} f \circ g$.
Both $Q$ and the integral are invariant under $\mathcal{O}_h$, so $Q(f) = Q(\bar f)$ and $\int f \, d\sigma = \int \bar f \, d\sigma$, and $\bar f$ is an invariant polynomial of the same degree bound.
This reduction is an instance of a theorem of Sobolev on invariant cubature.
It cuts the number of equations from about $(p+1)^2$, the dimension of the [[spherical harmonics|3F32Y21HP4]] of degree at most $p$, to the number of invariant ones.
Example: the $6$-point rule with nodes $\pm e_1, \pm e_2, \pm e_3$ and weights $\frac{4\pi}{6}$ is invariant.
On $S^2$ the invariant polynomials of degree at most $3$ are constants, because odd ones change sign under $x \mapsto -x$ and the only invariant quadratic is $x_1^2 + x_2^2 + x_3^2 = 1$.
Since the weights add up to $4\pi = \sigma(S^2)$, the rule is exact up to degree $3$; it fails for $x_1^4$, where it gives $\frac{4\pi}{3}$ instead of $\frac{4\pi}{5}$.

## Explanations

### L1 human

Lebedev quadrature approximates an average over the surface of a sphere by a weighted sum over a few well-placed points.
The points are arranged with the symmetry of a cube, which makes the rule fair in all directions.
Thanks to that symmetry, only a few conditions need to be checked to make the rule exact for all polynomials up to a given degree.

### L2 human

Vyacheslav Lebedev computed such rules with many points and high degree, and they are standard in quantum chemistry, where integrals over directions appear in density functional theory.
Compared with product rules in spherical coordinates, they use fewer points for the same degree and do not crowd points near the poles.
The underlying principle, reducing exactness to the invariant functions of a symmetry group, is used for cubature on many other domains.
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<document source="https://tamnd.github.io/math2/node/EGQ4GHH127.md" tag="EGQ4GHH127" kind="definition">
# Quasi-invariant measure

## Definition

Let $(X, \Sigma)$ be a measurable space and let $\mu$ be a [[measure|PR11BZQQPG]] on $\Sigma$.
Let $G$ be a group that acts on $X$ by [[group action|0WCY42EV9V]], such that each map $T_g \colon X \to X$, $x \mapsto g \cdot x$, is measurable.
The measure $\mu$ is *quasi-invariant* under $G$ if for every $g \in G$ and every $A \in \Sigma$,
$$\mu(A) = 0 \iff \mu(T_g^{-1}(A)) = 0.$$

## Remarks

Equivalently, for each $g$ the [[pushforward|3N66TM3TSQ]] $(T_g)_* \mu$ and $\mu$ are each [[absolutely continuous|P0P21KDV6E]] with respect to the other.
Every invariant measure is quasi-invariant.

Example: let $G = \R$ act on $X = \R$ by translations, and let $\mu$ be the standard [[Gaussian measure|P0DV2TCH1N]].
Its density is positive everywhere, so $\mu(A) = 0$ exactly when $A$ has Lebesgue measure $0$.
Lebesgue [[null sets|VQQM4Y550T]] are preserved by translations, so $\mu$ is quasi-invariant.

Non-example: under the same action, the [[Dirac measure|0ACRGYNHSR]] $\delta_0$ is not quasi-invariant.
For $g = 1$ and $A = \set{1}$, one has $\delta_0(A) = 0$ but $\delta_0(T_1^{-1}(A)) = \delta_0(\set{0}) = 1$.

By the [[Cameron–Martin theorem|WAE4ZGJXJ7]], Wiener measure is quasi-invariant under translations by the Cameron–Martin space, but not under all translations of the path space.

## Explanations

### L1 human

Moving a set by the group never turns a negligible set into a non-negligible one.
The measure can change in size, but its null sets stay the same.
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<document source="https://tamnd.github.io/math2/node/EGRH1KN03C.md" tag="EGRH1KN03C" kind="definition">
# Harmonic measure

## Definition

Let $\Omega \subset \C$ be a bounded [[domain|097KNJFJ7V]] with [[boundary|0JJ9J1XZ8G]] $\partial\Omega$.
Assume that the [[Dirichlet problem|8D56GQ4AG8]] on $\Omega$ is solvable for continuous data.
That is, for every continuous $\varphi \colon \partial\Omega \to \R$ there is a continuous $u_\varphi \colon \overline{\Omega} \to \R$ that is [[harmonic|0W5NJQ8MKY]] on $\Omega$ and equals $\varphi$ on $\partial\Omega$.
By the maximum principle, $u_\varphi$ is unique.
Let $z \in \Omega$.
The *harmonic measure* of $\Omega$ at $z$ is the unique Borel probability measure $\omega_z = \omega(z, \cdot, \Omega)$ on $\partial\Omega$ such that
$$u_\varphi(z) = \int_{\partial\Omega} \varphi \, d\omega_z \qquad \text{for every continuous } \varphi \colon \partial\Omega \to \R.$$
For a [[Borel set|0BVHZXWKCD]] $E \subseteq \partial\Omega$, the number $\omega(z, E, \Omega) = \omega_z(E)$ is the harmonic measure of $E$ at $z$.

## Remarks

The measure exists by the Riesz representation theorem.
Indeed, by the maximum principle, $\varphi \mapsto u_\varphi(z)$ is a positive linear functional on the continuous functions on $\partial\Omega$, and it sends $1$ to $1$.
For fixed $E$, the function $z \mapsto \omega(z, E, \Omega)$ is harmonic on $\Omega$ with values in $[0, 1]$.
Example: for the unit disc $\mathbb{D}$, the [[Poisson kernel|TCYSWCTYYM]] gives $d\omega_z(e^{i\theta}) = \frac{1 - \abs{z}^2}{2\pi \abs{e^{i\theta} - z}^2} \, d\theta$.
At $z = 0$ this is $d\theta / 2\pi$, so the harmonic measure of an arc is its length divided by $2\pi$.
Non-example: the punctured disc $\mathbb{D} \setminus \set{0}$ does not satisfy the hypothesis.
The data $1$ at $0$ and $0$ on the unit circle have no harmonic solution that is continuous up to the boundary.
With the more general Perron solution, the point $0$ gets harmonic measure $0$.
If $f \colon \Omega \to \Omega'$ is a conformal map that extends to a homeomorphism of the closures, then $\omega(f(z), f(E), \Omega') = \omega(z, E, \Omega)$.
Probabilistically, $\omega(z, E, \Omega)$ is the probability that a Brownian motion started at $z$ first leaves $\Omega$ through $E$.
</document>

<document source="https://tamnd.github.io/math2/node/EGWG9RKDBQ.md" tag="EGWG9RKDBQ" kind="definition">
# Inner measure

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
Let $\mathcal{P}(X)$ be the [[power set|00E0YZCERH]] of $X$.
The *inner measure* induced by $\mu$ is the map $\mu_* \colon \mathcal{P}(X) \to [0, \infty]$ given by
$$\mu_*(A) = \sup \set{ \mu(B) : B \in \Sigma, \ B \subseteq A },$$
the [[supremum|0R3H5PANRQ]] of the measures of the measurable subsets of $A$.

## Remarks

The outer measure induced by $\mu$ is $\mu^*(A) = \inf \set{ \mu(B) : B \in \Sigma, \ A \subseteq B }$, and $\mu_*(A) \le \mu^*(A)$ for all $A$.
If $A \in \Sigma$, then $\mu_*(A) = \mu(A) = \mu^*(A)$.
For Lebesgue measure $\lambda$ on $\R^n$, the supremum may be taken over compact subsets of $A$.
For Lebesgue measure and a set $A$ with $\lambda^*(A) < \infty$, the set $A$ is Lebesgue measurable if and only if $\lambda_*(A) = \lambda^*(A)$.
Example: for $A = \Q \cap [0, 1]$ and Lebesgue measure, $A$ is measurable with $\lambda_*(A) = \lambda^*(A) = 0$.
Non-example of measurability: a Vitali set $V \subseteq [0, 1]$ has $\lambda_*(V) = 0 < \lambda^*(V)$.
Unlike an outer measure, an inner measure is superadditive: $\mu_*(A \cup B) \ge \mu_*(A) + \mu_*(B)$ for disjoint $A$ and $B$.
Some authors define inner measures abstractly by such axioms, without a given measure.
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<document source="https://tamnd.github.io/math2/node/EH0CZ19RKP.md" tag="EH0CZ19RKP" kind="concept">
# Spectrum of an element

## Summary

The spectrum of an element $a$ of a unital algebra is the set of scalars $\lambda$ for which $a - \lambda 1$ has no inverse.
For square matrices it is the set of eigenvalues, and for bounded operators it can be larger than the set of eigenvalues.
In a complex Banach algebra the spectrum is a nonempty compact set.
It is the starting point of spectral theory and of the functional calculus.

## Explanations

### L1 human

For a matrix $a$, the number $\lambda$ is an eigenvalue exactly when $a - \lambda$ cannot be inverted.
The spectrum keeps this second description and uses it in any algebra where one can multiply and invert.
For a continuous function $f$ on $[0,1]$, the function $f - \lambda$ has an inverse exactly when it never vanishes.
So the spectrum of $f$ is the set of values of $f$.
</document>

<document source="https://tamnd.github.io/math2/node/EHA5MF9VKE.md" tag="EHA5MF9VKE" kind="statement">
# Markov brothers' inequality

## Statement

Let $n \ge 1$ and let $P$ be a real [[polynomial function|MAM9H3SY3Y]] of [[degree|0DPF6WRZW6]] at most $n$.
Write $\norm{Q} = \max_{-1 \le x \le 1} \abs{Q(x)}$ for a function $Q$ continuous on $[-1, 1]$.

1. $\norm{P'} \le n^2 \norm{P}$.
2. More generally, for $1 \le k \le n$, the $k$-th [[derivative|084NGSTWM7]] satisfies
$$\norm{P^{(k)}} \le \frac{n^2 (n^2 - 1^2)(n^2 - 2^2) \cdots (n^2 - (k-1)^2)}{1 \cdot 3 \cdot 5 \cdots (2k - 1)} \, \norm{P}.$$

## Remarks

Both bounds are attained by the [[Chebyshev polynomial|M6EAR1JHPP]] $T_n$, defined by $T_n(\cos \theta) = \cos(n \theta)$.
The constant in the second bound equals $T_n^{(k)}(1)$; for $k = 1$ it is $n^2$.

Example: $P(x) = T_2(x) = 2x^2 - 1$ has $\norm{P} = 1$ and $P'(x) = 4x$, so $\norm{P'} = 4 = 2^2 \norm{P}$.

In the interior, [[Bernstein's inequality for polynomials|F0YGC89S9R]] gives the stronger pointwise bound $\abs{P'(x)} \le \frac{n}{\sqrt{1 - x^2}} \norm{P}$ for $-1 < x < 1$.
The factor $n^2$ in the first bound is caused by the endpoints $\pm 1$, where this pointwise bound degenerates.

## Explanations

### L1 human

A polynomial of degree $n$ that stays between $-1$ and $1$ on the interval $[-1, 1]$ cannot be too steep there.
Its slope is at most $n^2$ in absolute value.
The Chebyshev polynomials, which oscillate as much as possible, are the steepest ones.

### L2 human

Markov-type inequalities bound the derivative of a polynomial by its size, with a constant depending only on the degree.
They are used in approximation theory to prove inverse theorems: if a function is well approximated by polynomials, then it is smooth.
The first inequality is due to A. A. Markov and the general one to his brother V. A. Markov.
</document>

<document source="https://tamnd.github.io/math2/node/EHASX3TBRQ.md" tag="EHASX3TBRQ" kind="concept">
# Abelian algebra

## Summary

An abelian algebra is an algebra in which the [[commutator|6DSWPHSWQW]] of the full congruence with itself is the trivial congruence.
Equivalently, its term operations satisfy the term condition with equality.
For groups this notion gives exactly the abelian groups, and modules over a ring are abelian algebras.
Abelian algebras play the role of abelian groups in the general theory of solvable and nilpotent algebras.

## Explanations

### L1 human

In an abelian group, changing one input of a formula has an effect that does not depend on the other inputs.
An abelian algebra is an algebra with the same property for every formula built from its operations.
Modules and vector spaces are abelian in this sense.
Nonabelian groups and lattices with more than one element are not.
</document>

<document source="https://tamnd.github.io/math2/node/EHBQ295J3Q.md" tag="EHBQ295J3Q" kind="concept">
# Lipschitz class

## Summary

The Lipschitz class $\mathrm{Lip}\,\alpha$, for $0 < \alpha \le 1$, is the set of functions on an interval that satisfy $\abs{f(x) - f(y)} \le M \abs{x - y}^\alpha$ for some constant $M$.
For $\alpha = 1$ these are the [[Lipschitz continuous|0QNVQ4VD9T]] functions, and for $\alpha < 1$ the [[Hölder continuous|012PBA0FMD]] functions of exponent $\alpha$.
In approximation theory these classes are the standard scale of smoothness.
For $0 < \alpha < 1$, a continuous $2\pi$-periodic function is in $\mathrm{Lip}\,\alpha$ exactly when its best trigonometric approximation of degree $n$ has error of order $n^{-\alpha}$.

## Explanations

### L1 human

The exponent $\alpha$ says how fast the function may change over short distances.
With $\alpha = 1$ the change is at most proportional to the distance, like a graph with bounded slope.
With smaller $\alpha$ the function may be steeper near some points: $\sqrt{x}$ on $[0, 1]$ is in $\mathrm{Lip}\,\tfrac12$ but not in $\mathrm{Lip}\,1$, because its slope blows up at $0$.
Smaller $\alpha$ means a larger class of rougher functions.
</document>

<document source="https://tamnd.github.io/math2/node/EHDXZV05JR.md" tag="EHDXZV05JR" kind="definition">
# Cohomology ring

## Definition

Let $X$ be a topological space and let $R$ be a [[commutative ring|0310REJJY8]] with $1$.
The *cohomology ring* of $X$ with coefficients in $R$ is the direct sum of the [[singular cohomology|6N7BC70CX0]] groups
$$H^\bullet(X; R) = \bigoplus_{k \ge 0} H^k(X; R).$$
Its multiplication extends the [[cup product|JX47MS1R1H]] $H^k \times H^l \to H^{k+l}$ bilinearly to all of $H^\bullet(X; R)$.

## Remarks

It is an associative [[graded ring|8KB4SQGC0C]] with unit, and an $R$-algebra.
It is graded commutative: $ab = (-1)^{kl} ba$ for $a \in H^k$ and $b \in H^l$.
A continuous map $f \colon X \to Y$ induces a ring homomorphism $f^* \colon H^\bullet(Y; R) \to H^\bullet(X; R)$.

Example: $H^\bullet(S^n; \Z) \cong \Z[x]/(x^2)$ with $x$ of degree $n \ge 1$, since $x^2$ would lie in $H^{2n}(S^n) = 0$.
For the [[complex projective space|7GJ4STCNNM]] $\C P^n$, $H^\bullet(\C P^n; \Z) \cong \Z[x]/(x^{n+1})$ with $x$ of degree $2$.

Non-example: [[singular homology|M0Q5CFJ7ZH]] $H_\bullet(X)$ is a graded group of the same shape, but in general it carries no natural product of this kind.
The cup product needs the diagonal map $X \to X \times X$, which goes the wrong way for homology.
</document>

<document source="https://tamnd.github.io/math2/node/EHE26PR17K.md" tag="EHE26PR17K" kind="statement">
# Hermite normal form

## Statement

Let $A$ be an $m \times n$ matrix with integer entries.
Say that an $m \times n$ integer matrix $H = (h_{ij})$ is in *Hermite normal form* if:

1. the nonzero rows of $H$ come before the zero rows;
2. the first nonzero entry of each nonzero row, its pivot, is positive and lies strictly to the right of the pivot of the row above;
3. the entries below each pivot are $0$;
4. each entry $h$ above a pivot $p$ in the same column satisfies $0 \le h < p$.

Then there is a [[unimodular|R6RBQHW191]] $m \times m$ integer matrix $U$, that is, one with $\det U = \pm 1$, such that $UA$ is in Hermite normal form.
The matrix $H = UA$ is unique.

## Remarks

The proof is integer Gaussian elimination.
The Euclidean algorithm on a column, by unimodular row operations, leaves the gcd of the entries in the pivot position and zeros below it.
Then the entries above each pivot are reduced modulo the pivot.
Uniqueness holds because the rows of $H$ are a special basis of the lattice spanned by the rows of $A$, and this basis is determined by the lattice.

Example: for $A = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix}$, subtract $3$ times row $1$ from row $2$ to get $\begin{pmatrix} 2 & 4 \\ 0 & -4 \end{pmatrix}$.
Negate row $2$ and subtract it from row $1$ to get $H = \begin{pmatrix} 2 & 0 \\ 0 & 4 \end{pmatrix}$, which satisfies all four conditions.

Without condition 4 the form is not unique: $\begin{pmatrix} 2 & 4 \\ 0 & 4 \end{pmatrix}$ is also a [[row echelon form|0F27PBRWD7]] of $A$ reachable by unimodular operations.
Some authors use the column version, $AU = H$ with $U$ unimodular of size $n \times n$ and $H$ lower triangular.

## Explanations

### L1 human

The Hermite normal form is the integer analogue of the reduced row echelon form.
Only operations that can be undone over the integers are allowed: swapping rows, negating a row, and adding an integer multiple of one row to another.
The result is a staircase-shaped matrix with a fixed choice of the entries above the pivots, and it is unique.

### L2 human

Two integer matrices have the same Hermite normal form exactly when their rows span the same lattice in $\Z^n$.
So the form solves the membership problem for lattices and gives a canonical basis.
It is used to solve linear Diophantine systems, to compute with ideals in number fields, and in integer programming.
Unlike the [[Smith normal form|MV28Z5N9XX]], it uses only row operations, and it keeps the row space.
</document>

<document source="https://tamnd.github.io/math2/node/EHEZM8TXHJ.md" tag="EHEZM8TXHJ" kind="definition">
# Continuous dual space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a [[topological vector space|T32W553W95]] over $\mathbb{K}$.
The *continuous dual space* of $E$ is the set $E'$ of all [[continuous linear maps|03TM135G0N]] $f \colon E \to \mathbb{K}$, that is, of all continuous [[linear forms|07YQ1C78G0]] on $E$.
With pointwise addition and scalar multiplication, $E'$ is a linear subspace of the algebraic [[dual space|07JBSZE0V4]] of $E$.

## Remarks

The notations $E^*$ and $E'$ are both common; some texts reserve $E^*$ for the algebraic dual.
A linear form $f$ is continuous if and only if it is bounded on some neighbourhood of $0$.
For a normed space $E$, $f$ is continuous if and only if $\norm{f} = \sup_{\norm{x} \le 1} \abs{f(x)} < \infty$, and $E'$ with this norm is a Banach space.
Example: on $\ell^2$, for fixed $y \in \ell^2$, the form $f(x) = \sum_n x_n y_n$ is in $(\ell^2)'$, since $\abs{f(x)} \le \norm{y} \norm{x}$ by the Cauchy–Schwarz inequality.
Non-example: on the space $c_{00}$ of finitely supported sequences with the sup norm, $f(x) = \sum_n x_n$ is linear but not continuous, since $x = e_1 + \cdots + e_k$ has norm $1$ and $f(x) = k$.
For a Hausdorff locally convex space $E$, the Hahn–Banach theorem gives that $E'$ separates the points of $E$; for $L^p([0, 1])$ with $0 < p < 1$, by contrast, $E' = \set{0}$.
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<document source="https://tamnd.github.io/math2/node/EHFQK1WXKS.md" tag="EHFQK1WXKS" kind="concept">
# Rational approximation

## Summary

Rational approximation is the approximation of functions by [[rational functions|0XDK9CGYKJ]] $p/q$ with prescribed bounds on the degrees of $p$ and $q$.
Rational functions can imitate poles, steep fronts, and singular points much better than polynomials of the same number of parameters.
A striking case is [[Newman's theorem|JEQVZXHE6A]]: $\abs{x}$ on $[-1, 1]$ has rational approximation error of order $e^{-c\sqrt{n}}$, while polynomials only achieve order $1/n$.
[[Padé approximants|1F16G4V64X]] and best uniform rational approximations are the main constructions.

## Explanations

### L1 human

A polynomial is a sum of powers and can only grow without bound far away; it struggles to follow a function with a sharp corner or a near blow-up.
A quotient of two polynomials has more freedom: its denominator can be small where the function is large.
Rational approximation asks how close such quotients of given degrees can come to a given function.
For example, $1/(1 + 25x^2)$ on $[-1, 1]$ is itself a rational function of degree $2$, but polynomials need high degree to approximate it well.
</document>

<document source="https://tamnd.github.io/math2/node/EHGECREYB4.md" tag="EHGECREYB4" kind="concept">
# Laplace transform method for convolution equations

## Summary

The Laplace transform method solves a [[Volterra equation of convolution type|SNFYTC1TTA]] on $[0, \infty)$ by turning it into an algebraic equation.
The [[convolution theorem for the Laplace transform|0JNMJEDD93]] changes the convolution $k * x$ into the product $\hat k \, \hat x$ of the transforms.
One solves for $\hat x$ and then inverts the transform.
It is the standard way to find explicit solutions, and also to study the long-time behaviour of solutions.

## Explanations

### L1 human

A convolution equation mixes the unknown with its own past, which makes it hard to solve directly.
After a [[Laplace transform|0V49JEZW06]], the mixing becomes a plain multiplication.
For example, $x(t) = 1 + \int_0^t x(s) \, ds$ becomes $\hat x(p) = \frac1p + \frac1p \hat x(p)$.
So $\hat x(p) = \frac{1}{p - 1}$, which is the transform of $e^t$, and indeed $x(t) = e^t$.
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<document source="https://tamnd.github.io/math2/node/EHGRPCTM3J.md" tag="EHGRPCTM3J" kind="definition">
# Haagerup property

## Definition

Let $G$ be a second countable [[locally compact group|AHV05ZBCYW]].
A [[unitary representation|74BPYJNZTM]] $\pi$ of $G$ on a [[Hilbert space|HM2HHE3J51]] $\mathcal{H}$ is *mixing* if $\ip{\pi(g)\xi}{\eta} \to 0$ as $g \to \infty$, for all $\xi, \eta \in \mathcal{H}$.
Here $g \to \infty$ means that $g$ leaves every [[compact|0TVJZYB6SH]] subset of $G$.
The representation $\pi$ *almost has invariant vectors* if the following holds.
For every compact $Q \subseteq G$ and $\varepsilon > 0$, there is a unit vector $\xi$ with $\norm{\pi(g)\xi - \xi} < \varepsilon$ for all $g \in Q$.
The group $G$ has the *Haagerup property* if it has a mixing unitary representation that almost has invariant vectors.

## Remarks

Example: $G = \Z$ and the regular representation on $\ell^2(\Z)$.
For finitely supported $\xi$ and $\eta$, the coefficient $\ip{\pi(n)\xi}{\eta}$ is $0$ once $\abs{n}$ is large, and by density the representation is mixing.
The unit vectors $N^{-1/2} \mathbf{1}_{\set{0, \ldots, N-1}}$ are almost invariant for large $N$.

Non-example: $\mathrm{SL}_3(\Z)$ does not have the Haagerup property.
It has [[property (T)|GTFZ277047]], so a representation that almost has invariant vectors has a nonzero invariant vector $\xi$.
Then $\ip{\pi(g)\xi}{\xi} = \norm{\xi}^2$ for all $g$, which does not tend to $0$ on the infinite group.

Equivalently, $G$ has a proper continuous conditionally negative definite function $\psi \colon G \to [0, \infty)$.
Another equivalent form: $G$ has a continuous affine isometric action on a Hilbert space that is metrically proper.
Amenable groups, free groups and $\mathrm{SL}_2(\R)$ have the Haagerup property.
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<document source="https://tamnd.github.io/math2/node/EHJ3R589ZX.md" tag="EHJ3R589ZX" kind="concept">
# Factorial experiment

## Summary

A factorial experiment studies several factors at once, each at a few levels, and runs every combination of levels.
It estimates the main effect of each factor and the interactions between factors.
An interaction shows whether the effect of one factor depends on the level of another.
Each run contributes to the estimate of every main effect, so a factorial experiment is more efficient than changing one factor at a time.
The $2^k$ design, with $k$ factors at two levels each, is the basic case, and [[fractional factorial designs|PBK2XY4M0A]] run only a chosen part of it.

## Explanations

### L1 human

To bake a better cake, one might vary the oven temperature (low or high) and the baking time (short or long).
A factorial experiment bakes all four combinations, instead of changing one thing at a time.
Every cake then says something about both temperature and time.
It also shows interactions: maybe a long time helps at low temperature but burns the cake at high temperature.
</document>

<document source="https://tamnd.github.io/math2/node/EHJ7D5CRZS.md" tag="EHJ7D5CRZS" kind="concept">
# Slowly varying function

## Summary

A slowly varying function is a positive function $L$ on $(0, \infty)$ such that $L(ax)/L(x) \to 1$ as $x \to \infty$ for every $a > 0$.
Such functions grow or decay more slowly than any positive power of $x$, like the logarithm.
They appear as correction factors in [[Karamata's Tauberian theorem|DBEGF38V9H]], in probability theory and in the study of asymptotic behaviour.

## Explanations

### L1 human

A slowly varying function hardly notices if you double or triple its argument, once the argument is large.
The logarithm is the standard example: $\log(2x) = \log x + \log 2$, and the extra $\log 2$ is tiny compared with $\log x$ for large $x$.
So $\log(2x) / \log x$ gets closer and closer to $1$.
A power such as $x^2$ is not slowly varying, because doubling $x$ multiplies $x^2$ by $4$.
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<document source="https://tamnd.github.io/math2/node/EHW4EAWJBF.md" tag="EHW4EAWJBF" kind="definition">
# Multiresolution analysis

## Definition

A *multiresolution analysis* of $L^2(\R)$ (see [[Lp space|SASP14MA1Z]]) is a sequence $(V_j)_{j \in \Z}$ of closed linear subspaces of $L^2(\R)$ with these properties.

1. Nesting: $V_j \subseteq V_{j+1}$ for all $j \in \Z$.
2. Scaling: for all $j \in \Z$ and $f \in L^2(\R)$, $f \in V_j$ if and only if the function $x \mapsto f(2x)$ lies in $V_{j+1}$.
3. Separation and density: $\bigcap_{j \in \Z} V_j = \set{0}$, and $\bigcup_{j \in \Z} V_j$ is [[dense|0BSJSX7ME2]] in $L^2(\R)$.
4. Scaling function: there is $\varphi \in V_0$ such that the translates $(\varphi(\cdot - k))_{k \in \Z}$ form an [[orthonormal basis|0S9GGTR0EX]] of $V_0$.

## Remarks

Some authors only ask in clause 4 that the translates of $\varphi$ form a Riesz basis of $V_0$; an orthonormalization step then yields a function as in clause 4.
Clauses 1 to 3 imply that $f \in V_0$ if and only if $f(2^j \cdot) \in V_j$.
Let $W_j$ be the orthogonal complement of $V_j$ in $V_{j+1}$.
Then $L^2(\R)$ is the orthogonal direct sum of all $W_j$.
There is an orthonormal [[wavelet|4PZ35C0GV3]] $\psi$ whose integer translates form an orthonormal basis of $W_0$.

Example: let $V_j$ be the set of $f \in L^2(\R)$ that are constant on each interval $[k 2^{-j}, (k+1) 2^{-j})$, $k \in \Z$.
Nesting and scaling are clear, and step functions on dyadic intervals are dense in $L^2(\R)$.
A function in every $V_j$ is constant on $[0, 2^{-j})$ and on $[-2^{-j}, 0)$ for all $j \le 0$, hence constant on $[0, \infty)$ and on $(-\infty, 0)$, hence $0$ since it is square-integrable.
With $\varphi = 1_{[0, 1)}$, clause 4 holds; this is the Haar multiresolution analysis.

Non-example: $V_j = L^2(\R)$ for all $j$ satisfies nesting, scaling and density, but $\bigcap_j V_j = L^2(\R) \ne \set{0}$, so separation fails.
</document>

<document source="https://tamnd.github.io/math2/node/EHWH6EF4QY.md" tag="EHWH6EF4QY" kind="concept">
# Logarithmically convex domain

## Summary

A logarithmically convex domain is a [[Reinhardt domain|WA0SE56FVH]] whose image under the map $z \mapsto (\log\abs{z_1}, \dots, \log\abs{z_n})$ is a convex subset of $\R^n$.
The [[domain of convergence of a power series|D8S8JWE45R]] always has this property.
For complete Reinhardt domains, logarithmic convexity is exactly the condition that the domain is a domain of holomorphy.
The notion is close to that of a [[logarithmically convex function|8FRWATNNMX]]: a function is logarithmically convex when its logarithm is convex.

## Explanations

### L1 human

A Reinhardt domain can be drawn by the absolute values of its points.
Take logarithms of these absolute values: the picture then lives in $\R^n$.
The domain is logarithmically convex when this new picture is convex, that is, contains the segment between any two of its points.
For example, $\set{\abs{z_1 z_2} < 1}$ becomes the half-plane $\set{x_1 + x_2 < 0}$, which is convex.
A union of two thin polydiscs crossing like a plus sign becomes an L-shape, which is not convex.
</document>

<document source="https://tamnd.github.io/math2/node/EHXHC4ZNPA.md" tag="EHXHC4ZNPA" kind="definition">
# Covering relation

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]], and write $x < y$ for $x \le y$ and $x \ne y$.
For $x, y \in P$, the element $y$ *covers* $x$, written $x \lessdot y$, when both conditions hold.

1. $x < y$.
2. There is no $z \in P$ with $x < z < y$.

The covering relation of $P$ is the set of pairs $(x, y)$ with $x \lessdot y$.

## Remarks

Order the positive integers by divisibility.
Then $2 \lessdot 6$: the only divisors of $6$ that $2$ divides are $2$ and $6$, so nothing lies strictly between.
In general $m \lessdot n$ exactly when $m \mid n$ and $n / m$ is prime.
The pair $2 < 8$ is not a cover, since $2 < 4 < 8$, so the second condition fails.
In a finite poset, $x < y$ holds if and only if there is a finite sequence $x = z_0 \lessdot z_1 \lessdot \cdots \lessdot z_k = y$ with $k \ge 1$.
So the covering relation determines a finite partial order.
This fails for infinite posets: in $\Q$ the covering relation is empty.

## Explanations

### L1 human

A cover is a single step up in the order, with no room for anything in between.
The lines of a Hasse diagram are exactly the covers.
</document>

<document source="https://tamnd.github.io/math2/node/EHY450Q5CW.md" tag="EHY450Q5CW" kind="concept">
# Conjugate prior

## Summary

A conjugate prior for a likelihood is a [[prior distribution|00NBBSXRQS]] from a family that the update by Bayes' theorem maps into itself.
The [[posterior distribution|0KJ7ZS8AJA]] then lies in the same family, and only its parameters change.
This gives closed-form posteriors, which made conjugate priors central in [[Bayesian inference|0JWRJX2RMP]] before simulation methods became common.

## Explanations

### L1 human

A Bayesian update multiplies the prior by the likelihood.
For a well-chosen family of priors the product has the same shape as the prior, with new parameters.
For a coin with unknown chance $p$, a beta prior $\mathrm{Beta}(a, b)$ plus $k$ heads and $m$ tails gives the posterior $\mathrm{Beta}(a + k, b + m)$.
So the prior acts like $a$ earlier heads and $b$ earlier tails, and the data just add to these counts.
</document>

<document source="https://tamnd.github.io/math2/node/EHY4PYSMGZ.md" tag="EHY4PYSMGZ" kind="concept">
# Growth of algebras

## Summary

The growth of a finitely generated [[associative algebra|3CTW1P4HBP]] over a field measures how fast a dimension increases with $n$.
This is the dimension of the span of all products of at most $n$ generators.
Up to a natural equivalence, this rate does not depend on the chosen generators.
Algebras then split into algebras of polynomial growth, of exponential growth and of intermediate growth.
The notion parallels the [[growth rate of a group|4CMRM3975W]], and its numerical summary is the [[Gelfand–Kirillov dimension|MDVSY2NMD2]].

## Explanations

### L1 human

Pick a few generators of an algebra and count how many linearly independent products of length at most $n$ they give.
For the polynomial ring $k[x]$ this number is $n + 1$, which grows like a line.
For $k[x, y]$ it is about $n^2/2$, which grows like an area.
For the free algebra on two generators, where $xy \ne yx$, it is $2^{n+1} - 1$, which grows exponentially.
The growth of an algebra is the shape of this counting function, ignoring constant factors in $n$.
</document>

<document source="https://tamnd.github.io/math2/node/EJ0N7F9TYQ.md" tag="EJ0N7F9TYQ" kind="definition">
# Morrey quasiconvexity

## Definition

Let $m, n \ge 1$ and let $f \colon \R^{m \times n} \to \R$ be continuous.
The function $f$ is *quasiconvex* (in the sense of Morrey) if
$$f(\xi) \le \frac{1}{\abs{D}} \int_D f\big(\xi + \nabla \varphi(x)\big) \, dx$$
for every bounded open set $D \subseteq \R^n$, every $\xi \in \R^{m \times n}$ and every $\varphi \in W^{1,\infty}_0(D; \R^m)$.
Here $W^{1,\infty}_0(D; \R^m)$ is the [[Sobolev space|M4CEZ5K9HP]] of Lipschitz maps that vanish on $\partial D$.
The number $\abs{D}$ is the Lebesgue measure of $D$.

## Remarks

If the inequality holds for one bounded open set $D$ with $\abs{\partial D} = 0$, it holds for all of them; often $D$ is the unit cube.

Example: every [[convex|07GR3TNW4H]] $f$ is quasiconvex.
Since $\varphi = 0$ on $\partial D$, the average of $\nabla \varphi$ over $D$ is $0$, and Jensen's inequality gives the inequality.

Non-example: $f(\xi) = -\abs{\xi}^2$ is not quasiconvex.
For $\varphi \ne 0$, the cross term $2 \int_D \xi : \nabla \varphi \, dx$ vanishes, so the average of $f(\xi + \nabla \varphi)$ is $-\abs{\xi}^2 - \frac{1}{\abs{D}} \int_D \abs{\nabla \varphi}^2 \, dx < f(\xi)$.

Every quasiconvex function is [[rank-one convex|GK4P8G0PB2]].
For $m = 1$ or $n = 1$, quasiconvexity is ordinary convexity.
For $m \ge 3$ and $n \ge 2$, Šverák gave a rank-one convex function that is not quasiconvex; the case $m = 2$ is open.
</document>

<document source="https://tamnd.github.io/math2/node/EJ3W3836VZ.md" tag="EJ3W3836VZ" kind="problem">
# Behrens–Fisher problem

## Problem

Let $X_1, \dots, X_m$ be i.i.d. [[normal|07RXBVTVZB]] $\mathcal{N}(\mu_1, \sigma_1^2)$ and let $Y_1, \dots, Y_n$ be i.i.d. $\mathcal{N}(\mu_2, \sigma_2^2)$, independent of the $X_i$.
All four parameters are unknown, and $\sigma_1^2 = \sigma_2^2$ is not assumed.
The problem asks for a [[test|087P03NEND]] of $H_0 \colon \mu_1 = \mu_2$ whose type I error probability is exactly $\alpha$ for all values of $\sigma_1^2$ and $\sigma_2^2$.
Equivalently, it asks for a [[confidence interval|0K4NMMJBG2]] for $\mu_1 - \mu_2$ with exact coverage $1 - \alpha$ for all values of the variances.

## Remarks

The natural statistic is $(\bar X - \bar Y)/\sqrt{S_1^2/m + S_2^2/n}$, where $S_1^2$ and $S_2^2$ are the sample variances.
Its law under $H_0$ depends on the unknown ratio $\sigma_1^2/\sigma_2^2$, which is the source of the difficulty.
If the ratio $\sigma_1^2/\sigma_2^2$ is known, a two-sample [[Student's t-test|0HYP606RYF]] solves the problem exactly.
Welch's test compares the statistic with a t-distribution whose degrees of freedom are estimated from the data; its level is close to $\alpha$ but not exactly $\alpha$ in general.
Fisher proposed a fiducial solution, and its level as a frequentist test is not exactly $\alpha$ either.

## Explanations

### L1 human

Two groups are measured, and one wants to know whether their true means differ.
When both groups have the same spread, the classical t-test gives an exact answer.
When the spreads may differ and are unknown, the exact answer becomes surprisingly hard, and this is the Behrens–Fisher problem.
In practice, people use Welch's approximate test, which works well for most sample sizes.
</document>

<document source="https://tamnd.github.io/math2/node/EJ556BJNFM.md" tag="EJ556BJNFM" kind="statement">
# Perron's formula

## Statement

Let $a \colon \set{1, 2, 3, \dots} \to \C$ and let $F(s) = \sum_{n \ge 1} a(n) n^{-s}$ be its [[Dirichlet series|8MWTEF6462]].
Let $\sigma_a$ be the abscissa of [[absolute convergence|0YBDSJARWN]] of $F$, assume $\sigma_a < \infty$, and let $c > \max(0, \sigma_a)$.
For $x > 0$, let $A^*(x) = \sum_{n < x} a(n) + \frac{1}{2} a(x)$, where the last term is present only when $x$ is an integer.
Then for every $x > 0$,
$$A^*(x) = \lim_{T \to \infty} \frac{1}{2\pi i} \int_{c - iT}^{c + iT} F(s) \, \frac{x^{s}}{s} \, ds.$$

## Remarks

The proof rests on the discontinuous integral.
Fix $c > 0$, and for $y > 0$ let $I(y) = \lim_{T \to \infty} \frac{1}{2\pi i} \int_{c - iT}^{c + iT} \frac{y^{s}}{s} \, ds$.
Then $I(y) = 1$ for $y > 1$, $I(1) = 1/2$, and $I(y) = 0$ for $0 < y < 1$.
For $y > 1$ one closes the path to the left and picks up the residue $1$ at $s = 0$; for $y < 1$ one closes it to the right, where there is no pole.
One applies this with $y = x/n$ to each term $a(n) n^{-s}$ and exchanges sum and limit.
The exchange uses the explicit error bound of the discontinuous integral and the absolute convergence of $F$ on $\operatorname{Re} s = c$.
The limit cannot be replaced by an absolutely convergent integral in general.
For $F = 1$, that is $a(1) = 1$ and $a(n) = 0$ otherwise, the integrand $x^{s}/s$ has modulus of order $1/\abs{s}$ on the line.
Example: take $a = \Lambda$, the [[von Mangoldt function|0NK8JTE1NN]], so that $F(s) = -\zeta'(s)/\zeta(s)$, and take $c > 1$.
Then the formula expresses the Chebyshev function $\psi(x)$ as a contour integral.
Shifting the contour to the left and collecting the residue $x$ at $s = 1$ is the route to the [[prime number theorem|0EHBY5Y84N]].

## Explanations

### L1 human

Perron's formula recovers the partial sums of the coefficients of a Dirichlet series from the values of the series on a vertical line.
It works because the integral of $x^{s}/s$ along a vertical line acts as a switch: it is $1$ when $x > 1$ and $0$ when $x < 1$.
So each term $a(n)$ is counted exactly when $n < x$.

### L2 human

The formula is an inverse Mellin transform: by the Mellin representation of Dirichlet series, $F(s)/s$ is the Mellin transform at $-s$ of the summatory function.
In practice one uses a truncated version with an explicit error term in $T$ and $x$, then moves the line of integration to the left of $\operatorname{Re} s = 1$.
The poles of $F(s) x^{s}/s$ then give the main terms, and bounds for $F$ in the critical strip control the error.
This is how the prime number theorem and its explicit formulas are derived from properties of the zeta function.
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# Finitary operation

## Summary

A finitary operation on a set $A$ takes $n$ elements of $A$ and returns an element of $A$, for a fixed natural number $n$.
The number $n$ is its [[arity|MSXX2P86CY]].
[[Binary operations|0M418FE7BB]] such as addition are the most common case, but [[unary|Q2W26ZY10B]], [[nullary|9HPF7TE18C]] and [[ternary|WC47MFR6GW]] operations also occur.
[[Algebraic structures|QJGPT2W0FT]] are sets equipped with finitary operations.

## Explanations

### L1 human

An operation is a rule that turns some elements of a set into one element of the same set.
"Finitary" means that the number of inputs is a fixed natural number.
On the integers, $x \mapsto -x$ uses one input, $(x, y) \mapsto x + y$ uses two, and $(x, y, z) \mapsto x - y + z$ uses three.
The result must stay inside the set: subtraction is an operation on the integers but not on the natural numbers.
Operations with infinitely many inputs are called [[infinitary|WFVC137TYB]].
</document>

<document source="https://tamnd.github.io/math2/node/EJD3C6QW9T.md" tag="EJD3C6QW9T" kind="definition">
# Stein manifold

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of complex dimension $n$ that is [[second countable|03PPNCM9GG]], and let $\mathcal{O}(X)$ be the algebra of [[holomorphic functions|507C34W0ZT]] on $X$.
For a [[compact set|0R6EWFKRCF]] $K \subseteq X$, its [[holomorphically convex hull|CD6SMMZSW8]] is
$$\widehat{K} = \set{x \in X : \abs{f(x)} \le \sup_{K} \abs{f} \text{ for all } f \in \mathcal{O}(X)}.$$
The manifold $X$ is a *Stein manifold* if:

1. $X$ is holomorphically convex: $\widehat{K}$ is compact for every compact $K \subseteq X$.
2. $X$ is holomorphically separable: for all $x \ne y$ in $X$ there is $f \in \mathcal{O}(X)$ with $f(x) \ne f(y)$.

## Remarks

Some authors add a third condition: at every point $x$ there are $f_1, \dots, f_n \in \mathcal{O}(X)$ that form local coordinates near $x$.
This condition follows from the first two.
Example: $X = \C^n$ is Stein.
The coordinate functions separate points.
The hull $\widehat{K}$ is closed, and $\abs{z_j} \le \max_K \abs{z_j}$ on $\widehat{K}$ for each $j$.
So $\widehat{K}$ lies in a compact polydisc, and it is compact.
Non-example: a compact connected complex manifold $X$ of positive dimension is not Stein.
Every $f \in \mathcal{O}(X)$ is constant, by the maximum principle, so clause 2 fails.
A domain in $\C^n$ is a Stein manifold if and only if it is a domain of holomorphy.
</document>

<document source="https://tamnd.github.io/math2/node/EJE15QHXZN.md" tag="EJE15QHXZN" kind="definition">
# Dirichlet kernel

## Definition

Let $N \ge 0$ be an integer.
The *Dirichlet kernel* of order $N$ is the [[trigonometric polynomial|J3NJQ5FZJ5]] $D_N \colon \R \to \R$ given by
$$D_N(x) = \sum_{n=-N}^{N} e^{inx} = 1 + 2 \sum_{n=1}^{N} \cos nx.$$

## Remarks

Summing the geometric progression gives $D_N(x) = \frac{\sin((N + \frac{1}{2}) x)}{\sin(x/2)}$ for $x \notin 2\pi\Z$, and $D_N(x) = 2N + 1$ for $x \in 2\pi\Z$.
It is even, $2\pi$-periodic, and $\frac{1}{2\pi} \int_{-\pi}^{\pi} D_N(x) \, dx = 1$, since only the term $n = 0$ contributes.
For $f$ integrable and $2\pi$-periodic, the partial sums of its [[Fourier series|G12KXDHPPS]] are the [[convolutions|2Z29JN3VES]]
$$S_N f(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x - t) D_N(t) \, dt.$$
Example: $D_1(x) = 1 + 2 \cos x$, and indeed $\frac{\sin(3x/2)}{\sin(x/2)} = 3 - 4 \sin^2(x/2) = 1 + 2 \cos x$.
Non-example: $\sum_{n=0}^{N} e^{inx}$ is not a Dirichlet kernel, since its sum is not symmetric in $n$; it is not real-valued for $N \ge 1$.
Some authors use the factor $\frac{1}{2\pi}$ in $D_N$ itself, or the period $1$ with $D_N(x) = \sum_{\abs{n} \le N} e^{2\pi i n x}$.
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<document source="https://tamnd.github.io/math2/node/EJJYCPWHPA.md" tag="EJJYCPWHPA" kind="statement">
# Eckmann–Hilton argument

## Statement

Let $X$ be a set with two [[binary operations|0M418FE7BB]] $\circ$ and $\otimes$.
Assume that $\circ$ has an [[identity element|07WYQPB8E3]] $1_\circ$, that $\otimes$ has an identity element $1_\otimes$, and that the interchange law
$$(a \otimes b) \circ (c \otimes d) = (a \circ c) \otimes (b \circ d)$$
holds for all $a, b, c, d \in X$.
Then $1_\circ = 1_\otimes$, the two operations are equal, and this common operation is commutative and associative.

## Remarks

Put $a = d = 1_\circ$ and $b = c = 1_\otimes$ in the interchange law: the left side is $1_\circ \circ 1_\circ = 1_\circ$ and the right side is $1_\otimes \otimes 1_\otimes = 1_\otimes$, so the units agree; write $1$ for both.
Then $a \circ d = (a \otimes 1) \circ (1 \otimes d) = (a \circ 1) \otimes (1 \circ d) = a \otimes d$, so the operations agree.
Also $b \circ c = (1 \otimes b) \circ (c \otimes 1) = (1 \circ c) \otimes (b \circ 1) = c \otimes b = c \circ b$, which is commutativity.
Associativity follows in the same way from $(a \otimes b) \circ (1 \otimes c) = (a \circ 1) \otimes (b \circ c)$.
The units are needed: on a set with at least two elements, the operations $a \circ b = a$ and $a \otimes b = b$ satisfy the interchange law, but they differ.
The main application: for $n \ge 2$, the homotopy group $\pi_n(X, x_0)$ carries two compatible products, concatenation in two different coordinates, so it is abelian.

## Explanations

### L1 human

Suppose a set has two ways of combining elements, each with a neutral element, and the two ways commute with each other in the sense of the interchange law.
Then they are secretly one and the same operation, and this operation is commutative.
A picture: arrange four elements in a square grid.
Combine them first by rows and then by columns, or first by columns and then by rows; the interchange law says the results agree.

### L2 human

The argument explains why the higher homotopy groups $\pi_n$ for $n \ge 2$ are abelian, while $\pi_1$ need not be.
In dimension $n \ge 2$ there are two independent directions in which to concatenate maps of cubes.
For the same reason, the fundamental group of an H-space, such as a topological group, is abelian.
There, pointwise multiplication of loops satisfies the interchange law with concatenation.
In category theory it shows that a monoid object in the category of monoids is a commutative monoid.
It also shows that the endomorphisms of the unit object in a monoidal category form a commutative monoid.
</document>

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# Stieltjes transform

## Definition

Let $\mu$ be a finite positive [[Borel measure|DYR1QFNGNJ]] on $\R$.
The *Stieltjes transform* of $\mu$ is the function $S_\mu \colon \C \setminus \R \to \C$ given by
$$S_\mu(z) = \int_\R \frac{d\mu(t)}{t - z}.$$
If $\mu$ is concentrated on $[0, \infty)$, the same formula defines $S_\mu$ on $\C \setminus [0, \infty)$.

## Remarks

The integral converges absolutely, since $\abs{t - z} \ge \abs{\operatorname{Im} z} > 0$ for real $t$.
The function $S_\mu$ is [[holomorphic|0NAHXR1FRH]], satisfies $S_\mu(\bar z) = \overline{S_\mu(z)}$, and has $\operatorname{Im} S_\mu(z) > 0$ when $\operatorname{Im} z > 0$ and $\mu \ne 0$, because $\operatorname{Im} \frac{1}{t - z} = \frac{\operatorname{Im} z}{\abs{t - z}^2}$.
Many authors use the opposite sign, $\int \frac{d\mu(t)}{z - t}$, or call $S_\mu$ the Cauchy transform of $\mu$.
For $d\mu = f(t)\, dt$ on $[0, \infty)$ and $x > 0$, $S_\mu(-x) = \int_0^\infty \frac{f(t)}{x + t}\, dt$, the classical Stieltjes transform of the function $f$.

Example: for the unit point mass $\mu = \delta_0$, $S_\mu(z) = \frac{1}{0 - z} = -\frac{1}{z}$.
For the uniform probability measure on $[0, 1]$, $S_\mu(z) = \int_0^1 \frac{dt}{t - z} = \log\frac{1 - z}{-z}$ with the principal branch of the logarithm; the argument $1 - 1/z$ is a nonpositive real number only for $z \in (0, 1]$.

Non-example: Lebesgue measure on $\R$ is not finite, and $\int_\R \frac{dt}{\abs{t - z}}$ diverges for every $z$, so the formula does not define a transform.
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# FK-space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $X$ be a [[sequence space|XYTN19EPF2]] over $\mathbb{K}$.
An *FK-space* is such an $X$ together with a topology $\tau$ on $X$ such that:

1. $(X, \tau)$ is a [[Fréchet space|3GAA614673]], that is, a complete metrizable locally convex topological vector space.
2. For each $n$, the coordinate map $X \to \mathbb{K}$, $x \mapsto x_n$, is [[continuous|0X3WGX95FE]].

## Remarks

Example: the space $\omega$ of all sequences, with the [[product topology|0GBG8B49AA]] of $\mathbb{K}^{\N}$, is an FK-space.
This topology is given by the seminorms $p_n(x) = \abs{x_n}$, so it is locally convex and makes each coordinate continuous.
It is metrizable by $d(x, y) = \sum_n 2^{-n} \min(1, \abs{x_n - y_n})$, and it is complete because a Cauchy sequence converges in each coordinate.

Example: each [[space ℓ^p|08P6T2WH6R]] with $1 \le p \le \infty$ and its usual norm is an FK-space, since $\abs{x_n} \le \norm{x}_p$.

Non-example: the space of sequences with finitely many nonzero terms, with the norm $\sup_n \abs{x_n}$, is not an FK-space under this norm, because it is not complete.
The sequences $(1, 1/2, \dots, 1/k, 0, 0, \dots)$ form a Cauchy sequence whose coordinatewise limit $(1/n)$ is not in the space.

An FK-space whose topology is given by a complete norm is a BK-space.
By the closed graph theorem, a sequence space has at most one FK topology, and every inclusion map between FK-spaces is continuous.
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# Homotopy group with coefficients

## Definition

Let $k \ge 2$ and $n \ge 2$ be integers.
Let $P^n(k) = S^{n-1} \cup_k e^n$ be the space obtained by attaching an $n$-cell to $S^{n-1}$ along a map of degree $k$; it is a [[Moore space|34SN01NWPS]] $M(\Z/k, n - 1)$.
Let $X$ be a [[pointed space|T9DG8DDE75]].
The *$n$-th homotopy group of $X$ with coefficients in $\Z/k$* is the set of pointed [[homotopy classes|F2F143KQ2M]] of pointed maps
$$\pi_n(X; \Z/k) = [P^n(k), X]_*.$$

## Remarks

For $n \ge 3$ the space $P^n(k)$ is the suspension of $P^{n-1}(k)$, so $\pi_n(X; \Z/k)$ is a group.
For $n \ge 4$ it is a double suspension, so the group is abelian.
For $n = 2$ the set $\pi_2(X; \Z/k)$ is in general only a pointed set.

For $n \ge 3$ the cofibre sequence $S^{n-1} \xrightarrow{k} S^{n-1} \to P^n(k)$ gives a short exact sequence
$$0 \to \pi_n(X) \otimes \Z/k \to \pi_n(X; \Z/k) \to \set{x \in \pi_{n-1}(X) : kx = 0} \to 0,$$
where $\pi_n(X)$ is the ordinary [[homotopy group|P976WYEFMB]].

Example: for $X = S^n$ with $n \ge 3$, $\pi_n(S^n) \otimes \Z/k \cong \Z/k$ and $\pi_{n-1}(S^n) = 0$, so $\pi_n(S^n; \Z/k) \cong \Z/k$.

Non-example: maps from $S^n$ to $X$ do not give $\pi_n(X; \Z/k)$, since $S^n$ is not a Moore space $M(\Z/k, n-1)$; its homology in degree $n - 1$ is $0$ for $n \ge 2$, not $\Z/k$.
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# Indecomposable continuum

## Definition

Let $X$ be a [[continuum|NJQ196M1WR]].
The continuum $X$ is *decomposable* if there are subcontinua $A$ and $B$ of $X$ with $A \neq X$, $B \neq X$ and $X = A \cup B$.
The continuum $X$ is *indecomposable* if it is not decomposable.
It is *hereditarily indecomposable* if every subcontinuum of $X$ is indecomposable.

## Remarks

The interval $[0, 1]$ is decomposable, since $[0, 1] = [0, 1/2] \cup [1/2, 1]$ with both pieces proper subcontinua.
A one-point continuum is indecomposable in a trivial way, because its only subcontinuum is itself.
The Knaster bucket handle and the [[pseudo-arc|VGCTDCGS6V]] are nondegenerate indecomposable continua.
A continuum is indecomposable if and only if every proper subcontinuum has empty [[interior|0RZH6W98HK]].

## Explanations

### L1 human

An indecomposable continuum is a compact space in one piece that cannot be written as the union of two smaller compact pieces in one piece.
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# Jacobson density theorem

## Statement

Let $R$ be a [[ring|0894J980RK]] with identity and let $U$ be a [[simple|G00C22AEDH]] left $R$-module.
Let $D = \End_R(U)$ be its [[endomorphism ring|AWSBZWWMG9]], which is a [[division ring|0WZXK05ME7]], and view $U$ as a vector space over $D$.
Let $u_1, \ldots, u_n \in U$ be linearly independent over $D$, and let $v_1, \ldots, v_n \in U$ be arbitrary.
Then there is $r \in R$ with $r u_i = v_i$ for $i = 1, \ldots, n$.

## Remarks

$D$ is a division ring by [[Schur's lemma|TRMYSF91EV]].
The proof is by induction on $n$; the key step shows that if $W \subseteq U$ is a finite-dimensional $D$-subspace and $u \notin W$, then some $r \in R$ kills $W$ but not $u$.
Example: let $R = M_n(k)$ for a field $k$ and $U = k^n$, so that $D = k$.
The theorem says that any $n$ linearly independent vectors can be sent to any $n$ vectors by a matrix.
If $U$ is finite-dimensional over $D$, the map $R \to \End_D(U)$ is surjective; when $R$ is a finite-dimensional algebra over an algebraically closed field this gives Burnside's theorem.

## Explanations

### L1 human

A ring acting irreducibly on a module acts as richly as possible.
On any finite set of independent vectors, it can do whatever a linear map can do.
So a primitive ring looks like a ring of linear maps that is "dense" in the full ring of linear maps.

### L2 human

The theorem is the main tool for the structure of [[primitive rings|8F6EHDVJ62]]: a left primitive ring is isomorphic to a dense subring of $\End_D(U)$.
If in addition $R$ is left Artinian, the subring is all of $\End_D(U)$ with $U$ finite-dimensional, so $R \cong M_n(D)$.
This is the route to [[Wedderburn's theorem on simple Artinian rings|YJM07BT0KK]] and to the [[Artin–Wedderburn theorem|R1F271180N]].
</document>

<document source="https://tamnd.github.io/math2/node/EK261G1XP9.md" tag="EK261G1XP9" kind="concept">
# Normal scheme

## Summary

A normal scheme is a [[scheme|5A35TYZD02]] whose local rings are [[integrally closed domains|H20P600HQS]].
Normality is a mild regularity condition.
Over an algebraically closed field, a normal curve is smooth, and a normal variety is smooth outside a closed set of codimension at least $2$.
Every integral scheme has a normal model, its normalization.

## Explanations

### L1 human

Normal schemes have no singularities of the simplest kinds, such as a curve crossing itself or a cusp.
The test is algebraic: a fraction that satisfies a monic polynomial equation over the local ring must already lie in the ring.
On the cusp $y^2 = x^3$, the fraction $t = y/x$ satisfies $t^2 = x$ but is not a polynomial function on the cusp, so the cusp is not normal.
The plane and $\operatorname{Spec} \Z$ are normal.
</document>

<document source="https://tamnd.github.io/math2/node/EK58S2872F.md" tag="EK58S2872F" kind="statement">
# Beurling–Deny formula

## Statement

Let $X$ be a [[locally compact|0FE2SB8WVJ]] separable metric space and let $m$ be a positive [[Radon measure|HZNEQQBCW4]] on $X$ with full support.
Let $(\mathcal{E}, \mathcal{F})$ be a regular [[Dirichlet form|MH20Q2N1AH]] on $L^2(X, m)$, and let $d = \set{(x, x) : x \in X}$ be the diagonal of $X \times X$.
Then there are
1. a symmetric form $\mathcal{E}^{(c)}$ on $\mathcal{F} \cap C_c(X)$ with the strong local property: $\mathcal{E}^{(c)}(u, v) = 0$ whenever $v$ is constant on a neighbourhood of the support of $u$;
2. a symmetric positive Radon measure $J$ on $X \times X \setminus d$, the jumping measure;
3. a positive Radon measure $k$ on $X$, the killing measure;

such that for all $u, v \in \mathcal{F} \cap C_c(X)$
$$\mathcal{E}(u, v) = \mathcal{E}^{(c)}(u, v) + \iint_{X \times X \setminus d} \big(u(x) - u(y)\big)\big(v(x) - v(y)\big) \, J(dx, dy) + \int_X u(x) v(x) \, k(dx).$$
The form $\mathcal{E}^{(c)}$ and the measures $J$ and $k$ are uniquely determined by $\mathcal{E}$.

## Remarks

The proof starts from the Markov property, which gives $\mathcal{E}(u, v) \le 0$ for nonnegative $u, v \in \mathcal{F} \cap C_c(X)$ with disjoint supports.
So $-\mathcal{E}(u, v)$ is a positive functional of such pairs, and a Riesz representation argument turns it into the measure $J$.
The killing measure comes from the behaviour of $\mathcal{E}$ on functions equal to $1$ near a compact set, and $\mathcal{E}^{(c)}$ is what remains.
Some texts put a factor $2$ or $\tfrac12$ in front of the jump term, which only rescales $J$.
For the form $\int_{\R^n} \nabla u \cdot \nabla v \, dx$ on $W^{1,2}(\R^n)$, the form is all local: $J = 0$ and $k = 0$.
For the fractional form $\iint (u(x) - u(y))(v(x) - v(y)) \abs{x - y}^{-n - \alpha} \, dx \, dy$ with $0 < \alpha < 2$, only the jump part is present.
For a Hunt process attached to $\mathcal{E}$, the three parts describe its continuous motion, its jumps and its killing.

## Explanations

### L1 human

A Dirichlet form measures the energy of a function, and this formula splits that energy into three kinds.
One kind comes from smooth local variation, as with the gradient of a function.
One kind comes from differences between values at distant points, which correspond to jumps of a random particle.
The last kind comes from the function's size alone, which corresponds to the particle being killed.

### L2 human

The formula is the analytic counterpart of the Lévy–Khintchine decomposition of a Lévy process into a diffusion part, a jump part and a killing rate.
A regular Dirichlet form is local, meaning that $J = 0$, exactly when its associated Hunt process has continuous paths up to its lifetime.
In practice one reads off $J$ and $k$ from a given form to understand the process, or one builds forms from chosen $J$ and $k$ to construct jump processes.
</document>

<document source="https://tamnd.github.io/math2/node/EK85VSSRMR.md" tag="EK85VSSRMR" kind="concept">
# Initial and terminal objects

## Summary

An initial object of a [[category|A2DE66937D]] is an object with exactly one morphism to every object.
A terminal object is an object with exactly one morphism from every object.
They are the simplest universal properties: the empty set and the one-point set play these roles among sets.
Each is unique up to a unique isomorphism when it exists, and many constructions are initial or terminal objects of suitable auxiliary categories.

## Explanations

### L1 human

An initial object is a starting point: from it there is one and only one arrow to anything.
A terminal object is an end point: from anything there is one and only one arrow to it.
Among sets, the [[empty set|0ECVK59Y03]] is initial, because there is exactly one function from it to any set, the empty function.
A one-point set is terminal, because every set has exactly one function to it, sending everything to the single point.
Among groups, the trivial group is both initial and terminal.
</document>

<document source="https://tamnd.github.io/math2/node/EKDSDN89FR.md" tag="EKDSDN89FR" kind="definition">
# Polynomial interpolation

## Definition

Let $K$ be a [[field|0QA8CHH8MG]], let $n \ge 0$, let $x_0, \dots, x_n \in K$ be pairwise distinct, and let $y_0, \dots, y_n \in K$.
An interpolation polynomial of the data $(x_i, y_i)$ is a polynomial $p \in K[x]$ such that:

1. $p = 0$ or $\deg p \le n$;
2. $p(x_i) = y_i$ for $i = 0, \dots, n$.

## Remarks

There is exactly one interpolation polynomial.
Uniqueness: the difference of two of them has degree at most $n$ and $n + 1$ roots, so it is $0$ by the [[bound on the number of roots|0KJVQV3E0F]].
Existence: the [[Lagrange formula|237KR6FRCK]] gives one.
Equivalently, the coefficients solve a linear system whose matrix has the nonzero [[Vandermonde determinant|0P84B554AS]].
Example: for $(0, 1)$, $(1, 3)$, $(2, 7)$ the interpolation polynomial is $x^2 + x + 1$.
Non-example: $x^2 + x + 1 + x(x - 1)(x - 2)$ takes the same values at $0, 1, 2$, but it has degree $3 > 2$, so clause 1 fails.

## Explanations

### L1 human

The degree bound is what makes the answer unique.
Without it, you can add any multiple of $(x - x_0) \cdots (x - x_n)$ and still pass through the points.
</document>

<document source="https://tamnd.github.io/math2/node/EKEPQC0T50.md" tag="EKEPQC0T50" kind="definition">
# Radical axis

## Definition

Let $\Gamma_1$ and $\Gamma_2$ be [[circles|060W5CBZP6]] in the Euclidean plane with distinct centres $c_1 \ne c_2$ and radii $r_1, r_2$.
For a point $P$, its [[power|02E9VT1RVQ]] with respect to $\Gamma_i$ is $\pi_i(P) = \norm{P - c_i}^2 - r_i^2$.
The *radical axis* of $\Gamma_1$ and $\Gamma_2$ is the set
$$\set{P : \pi_1(P) = \pi_2(P)}.$$

## Remarks

Expanding the two powers, the condition becomes
$$2 \ip{P}{c_2 - c_1} = \norm{c_2}^2 - \norm{c_1}^2 - r_2^2 + r_1^2.$$
Since $c_2 - c_1 \ne 0$, this is a [[line|0MP9BCF6DP]] perpendicular to $c_1 c_2$.
For three circles with non-collinear centres, the three radical axes meet in one point, the radical centre.

Example: let $\Gamma_1$ be $x^2 + y^2 = 1$ and $\Gamma_2$ be $(x - 2)^2 + y^2 = 1$.
Then $\pi_1 = x^2 + y^2 - 1$ and $\pi_2 = x^2 - 4x + 4 + y^2 - 1$, and they are equal exactly when $x = 1$.

Non-example: concentric circles fail the condition $c_1 \ne c_2$.
For $c_1 = c_2$ and $r_1 \ne r_2$, the powers differ by the constant $r_1^2 - r_2^2 \ne 0$, so no point has equal powers.
</document>

<document source="https://tamnd.github.io/math2/node/EKGEZGHYBW.md" tag="EKGEZGHYBW" kind="concept">
# Postnikov tower

## Summary

A Postnikov tower of a path-connected space $X$ is a sequence of spaces $X_1 \leftarrow X_2 \leftarrow X_3 \leftarrow \cdots$ with maps from $X$.
The stage $X_n$ has the homotopy groups of $X$ up to degree $n$ and none above degree $n$.
Each stage maps to the previous one by a fibration with fibre an [[Eilenberg–MacLane space|AJHEYJE5XY]].
So the tower builds $X$ from Eilenberg–MacLane spaces, one homotopy group at a time.
The extra data needed to rebuild $X$ are cohomology classes called $k$-invariants.

## Explanations

### L1 human

Think of the homotopy groups of a space as layers.
The $n$-th stage of the Postnikov tower keeps the first $n$ layers and throws away all the layers above.
Going up the tower adds one layer at a time, and in the limit one recovers the space up to weak homotopy equivalence.
For example, for the circle every stage is the circle itself, since the circle has only one layer.
</document>

<document source="https://tamnd.github.io/math2/node/EKJBDD07TY.md" tag="EKJBDD07TY" kind="concept">
# Closed monoidal category

## Summary

A closed monoidal category is a [[monoidal category|M3MSQ3NKAE]] in which morphisms between two objects form again an object, the internal hom.
The internal hom $[B, C]$ is characterised by a currying bijection between morphisms $A \otimes B \to C$ and morphisms $A \to [B, C]$.
[[Cartesian closed categories|BFBNQ36QS8]] are the case where $\otimes$ is the product.
Vector spaces, abelian groups and modules over a commutative ring, with the tensor product, are closed monoidal.

## Explanations

### L1 human

In many categories, the set of maps from $B$ to $C$ carries the same kind of structure as $B$ and $C$.
For example, the linear maps between two vector spaces form a vector space.
A closed monoidal category is one where this happens in a way that matches the tensor product.
A linear map out of $A \otimes B$ is the same thing as a linear map from $A$ into the space of linear maps from $B$ to $C$.
</document>

<document source="https://tamnd.github.io/math2/node/EKXQFD7EMF.md" tag="EKXQFD7EMF" kind="concept">
# Busemann G-space

## Summary

A Busemann G-space is a [[metric space|0VKD1TP7F3]] in which shortest paths exist and extend a little past their ends in exactly one way.
These axioms, due to Herbert Busemann, capture the behaviour of [[geodesics|NSYMQMHPDN]] with no smoothness assumed.
Complete [[Riemannian manifolds|7CBT4SM83Y]] and many [[Finsler manifolds|5R6QZ388BB]] are examples, so G-spaces give a common synthetic frame for their geometry.

## Explanations

### L1 human

In a G-space you can always walk from one point to another along a shortest route.
You can also keep walking straight a bit past the end, and there is only one way to do so.
The round sphere has this property: a great circle arc continues as a unique great circle.
The plane with the taxicab distance fails it, because a shortest route can be continued straight on or bent at a corner with the same length.
</document>

<document source="https://tamnd.github.io/math2/node/EKZB41Y82N.md" tag="EKZB41Y82N" kind="concept">
# Neyman construction

## Summary

The Neyman construction is a general method to build a [[confidence interval|0S6T9KQNC0]] for a parameter.
It starts from a family of acceptance regions, one for each value of the parameter.
The confidence set collects all parameter values whose acceptance region contains the observed data.
It is the inversion of a family of [[hypothesis tests|087P03NEND]], and it gives the required coverage without any pivot.

## Explanations

### L1 human

For each possible value of the parameter, draw a band of data values that are likely, with probability $1 - \alpha$, under that value.
After the experiment, keep every parameter value whose band contains what you saw.
Picture a plot with the parameter on one axis and the data on the other.
The bands form a belt, and the confidence interval is the slice of the belt at the observed data.
For a coin with unknown head probability $p$, the bands are ranges of head counts, and the slice gives an interval for $p$ with at least the required coverage.
</document>

<document source="https://tamnd.github.io/math2/node/EM7FHAFD3N.md" tag="EM7FHAFD3N" kind="concept">
# Bimodule

## Summary

A bimodule over two rings $R$ and $S$ is a left [[module|0NZ8JA25WK]] over $R$ that is also a right module over $S$.
The two actions must commute.
Bimodules are the natural objects between two rings: tensoring with an $(R, S)$-bimodule turns right $R$-modules into right $S$-modules.
The two-sided ideals of a ring are exactly its sub-bimodules.
Bimodules underlie Morita theory and Hochschild cohomology.

## Explanations

### L1 human

A bimodule is acted on from both sides, by one ring on the left and by another ring on the right.
The only condition linking the two actions is that it does not matter which side acts first.
Rectangular $m \times n$ matrices are the model: square $m \times m$ matrices multiply them on the left, square $n \times n$ matrices multiply them on the right, and $(AX)B = A(XB)$.
Every ring is a bimodule over itself, with multiplication on both sides.
</document>

<document source="https://tamnd.github.io/math2/node/EM8MFGFAGN.md" tag="EM8MFGFAGN" kind="concept">
# Berlekamp's algorithm

## Summary

Berlekamp's algorithm factors square-free polynomials over a [[finite field|0ZXZ82FTX2]] by linear algebra.
It finds the polynomials $g$ of degree less than $\deg f$ with $g^q \equiv g$ modulo $f$.
They form a vector space whose dimension is the number of irreducible factors of $f$.
Greatest common divisors of $f$ with $g - s$ for constants $s$ then split $f$.
It is deterministic, and it was the main method of [[factorization over finite fields|XNE7F0BAGE]] until the randomised Cantor–Zassenhaus algorithm of 1981.

## Explanations

### L1 human

The algorithm turns factoring into solving a system of linear equations over the finite field.
Each solution $g$ of the system acts like a test that sorts the irreducible factors of $f$ into groups.
For example, over $\F_5$ the polynomial $x^2 + 1$ splits as $(x - 2)(x - 3)$, and the algorithm finds this by computing $\gcd(x^2 + 1, x - s)$ for $s \in \F_5$.
The [[definition|HJW1KSQZWA]] gives the steps.
</document>

<document source="https://tamnd.github.io/math2/node/EM8Q0636YT.md" tag="EM8Q0636YT" kind="definition">
# Schatten class

## Definition

Let $H$ and $K$ be complex [[Hilbert spaces|HM2HHE3J51]], and let $1 \le p < \infty$.
A [[compact operator|M2NEP1K5AE]] $T \colon H \to K$ with [[singular values|QSJKPKPM7H]] $s_1(T) \ge s_2(T) \ge \cdots$ belongs to the *Schatten $p$-class* $S_p(H, K)$ if
$$\norm{T}_p = \Big( \sum_{n=1}^{\infty} s_n(T)^p \Big)^{1/p} < \infty.$$
The number $\norm{T}_p$ is the Schatten $p$-norm of $T$.

## Remarks

Example: on $\ell^2$, the diagonal operator $T e_n = e_n / n$ has $s_n(T) = 1/n$, so $T \in S_p$ for every $p > 1$.
Non-example: the same operator is not in $S_1$, because $\sum 1/n$ diverges.
$S_1$ is the trace class, and $S_2$ is the class of Hilbert–Schmidt operators, with $\norm{T}_2^2 = \sum_{i} \norm{T e_i}^2$ for any orthonormal basis $(e_i)$ of $H$.
For $p \le q$, $S_p \subseteq S_q$ and $\norm{T}_q \le \norm{T}_p$.
Each $S_p(H) = S_p(H, H)$ is a two-sided ideal of the algebra of bounded operators on $H$, and a Banach space under $\norm{\cdot}_p$.
The finite-rank operators are dense in each $S_p$, and the operator norm satisfies $\norm{T} \le \norm{T}_p$.
Some authors also allow $0 < p < 1$, where $\norm{\cdot}_p$ is only a quasi-norm.
</document>

<document source="https://tamnd.github.io/math2/node/EMAR4X3W8D.md" tag="EMAR4X3W8D" kind="concept">
# Hasse principle

## Summary

The Hasse principle, or local-global principle, concerns polynomial equations with rational coefficients.
It says that such an equation should have a rational solution when it has a real solution and a [[p-adic|BT6T702C28]] solution for each prime $p$.
Local solvability is easy to test, so when the principle holds it turns a hard question into finitely many checks.
The [[Hasse–Minkowski theorem|R8HCWA981S]] shows that it holds for quadratic forms.
It fails for some cubic equations, and the failures are measured by objects such as the [[Tate–Shafarevich group|BK9WBMF56G]].

## Explanations

### L1 human

A solution in rational numbers gives a solution modulo every power of every prime, and a real solution.
These "local" solutions are necessary conditions, and they can be checked by finite computations.
The Hasse principle asks whether the necessary conditions are also enough.
For $x^2 + y^2 = 3z^2$ there is no solution modulo $9$ with $x$, $y$, $z$ not all divisible by $3$, so there is no nonzero rational solution either.
For quadratic equations, the local conditions are always enough; for the cubic $3x^3 + 4y^3 + 5z^3 = 0$ they are not.
</document>

<document source="https://tamnd.github.io/math2/node/EMCTDD3HZ1.md" tag="EMCTDD3HZ1" kind="concept">
# Rank of an abelian group

## Summary

The rank of an [[abelian group|082GHNPJ53]] is the largest number of its elements that are independent over the integers.
It equals the [[dimension|03GSZ8RTZJ]] of the rational vector space obtained by tensoring the group with $\Q$.
It plays for abelian groups the role that dimension plays for vector spaces, but it ignores torsion: finite groups have rank $0$.
For a finitely generated abelian group, the rank is the number of copies of $\Z$ in its decomposition.

## Explanations

### L1 human

The rank counts how many independent directions an abelian group has, the way $\Z^n$ has $n$ coordinate directions.
Elements of finite order do not count as directions, so $\Z \oplus \Z/2\Z$ has rank $1$.
The group $\Q$ also has rank $1$: any two nonzero rationals $a, b$ satisfy a relation $ma - nb = 0$ with integers $m, n$ not both zero.
So $\Q$, though not finitely generated, has only one direction.
</document>

<document source="https://tamnd.github.io/math2/node/EMDVD4E83K.md" tag="EMDVD4E83K" kind="statement">
# Parabolic maximum principle

## Statement

Let $\Omega \subseteq \R^n$ be a bounded open set and let $T > 0$.
Let $\Omega_T = \Omega \times (0, T]$, and let $\Gamma_T = \overline{\Omega_T} \setminus \Omega_T$ be its parabolic boundary, which is the bottom $\overline{\Omega} \times \set{0}$ together with the sides $\partial \Omega \times [0, T]$.
Let $u \colon \overline{\Omega_T} \to \R$ be continuous, with $\partial_t u$ and the second partial derivatives in $x$ continuous on $\Omega_T$, and assume that
$$\partial_t u - \Delta u \le 0 \quad \text{in } \Omega_T.$$
Then
$$\max_{\overline{\Omega_T}} u = \max_{\Gamma_T} u.$$

## Remarks

First assume the strict inequality $\partial_t u - \Delta u < 0$.
At a maximum point $(x_0, t_0)$ in $\Omega_T$ one would have $\Delta u(x_0, t_0) \le 0$ and $\partial_t u(x_0, t_0) \ge 0$, the latter also when $t_0 = T$, which is a contradiction.
In general apply this to $v = u + \varepsilon \abs{x}^2$, for which $\partial_t v - \Delta v \le -2n\varepsilon < 0$, and let $\varepsilon \to 0$.
Applied to $u$ and $-u$, it shows that a solution of the [[heat equation|VP2HN86VAG]] on $\Omega_T$ is bounded by its values on the parabolic boundary.
So the initial-boundary value problem has at most one solution.
The strong form says more: if $\Omega$ is connected and the maximum is reached at a point $(x_0, t_0)$ of $\Omega_T$, then $u$ is constant on $\overline{\Omega} \times [0, t_0]$.
Boundedness of $\Omega$ is needed.
On $\R \times (0, T]$, Tychonoff's example is a nonzero solution of the heat equation with zero initial values, so its maximum is not controlled by the initial values.
Without the side boundary, uniqueness only holds under growth conditions, such as $\abs{u(x, t)} \le A e^{a \abs{x}^2}$.

## Explanations

### L1 human

In a body with no heat sources, the hottest moment and place must lie at the start or on the edge.
Heat cannot pile up in the interior by itself; it can only spread out and even out.
This gives a simple way to compare and bound temperatures without solving the equation.

### L2 human

The principle is the main tool for uniqueness and comparison results for parabolic equations.
It extends to operators $\partial_t u - \sum_{i,j} a_{ij} \partial_i \partial_j u - \sum_i b_i \partial_i u$ with $(a_{ij})$ positive definite.
It is the time-dependent analogue of the maximum principle for harmonic and subharmonic functions.
Its strong form follows from a mean value property on heat balls or from a parabolic Harnack inequality.
</document>

<document source="https://tamnd.github.io/math2/node/EMGBZ46GCD.md" tag="EMGBZ46GCD" kind="statement">
# Cohen's independence of CH

## Statement

If [[ZF|0HE5WDQ9FJ]] is consistent, then ZFC together with the negation of the [[continuum hypothesis|0MN2MER9WN]] is consistent.
Together with [[Gödel's theorem|Q9176JRJ5M]], it follows that if ZFC is consistent, then the continuum hypothesis is neither provable nor refutable in ZFC.

## Remarks

One starts from a countable transitive model $M$ of ZFC and uses [[forcing|F2DTWW9HP7]] with the set $P$ of finite partial functions from $\omega_2^M \times \omega$ to $\set{0, 1}$, ordered by reverse inclusion.
A [[generic filter|WEDP73Y7GK]] $G$ gives a function from $\omega_2^M \times \omega$ to $\set{0, 1}$, and so $\aleph_2^M$ different new reals.
Every antichain of $P$ is countable, and this chain condition implies that $M$ and $M[G]$ have the same cardinals.
So $M[G] \models 2^{\aleph_0} \ge \aleph_2$.
The existence of a countable transitive model of ZFC does not follow from the consistency of ZFC.
The standard way around this uses countable transitive models of large finite fragments of ZFC, or Boolean-valued models.

## Explanations

### L1 human

The continuum hypothesis asks whether there is a size of infinity between the natural numbers and the real numbers.
Gödel showed that the usual axioms cannot disprove it, and Cohen showed that they cannot prove it either.
Cohen took a universe of sets and added many new real numbers to it, so many that the continuum became larger than the next infinite size.
So the usual axioms of set theory leave the question open.

### L2 human

The proof is the first use of forcing, and its pattern is used in almost all independence proofs.
The first key step is to choose a partial order whose generic filter adds the desired objects.
The second is to check that the forcing does not collapse cardinals, here by the countable chain condition.
Changing $\omega_2$ to other cardinals gives many possible values of $2^{\aleph_0}$; Easton's theorem describes the possible values of $2^\kappa$ for all regular $\kappa$.
Cohen also used forcing to show that the axiom of choice is independent of ZF.
</document>

<document source="https://tamnd.github.io/math2/node/EMKFC3R1WM.md" tag="EMKFC3R1WM" kind="definition">
# Shallow water equations

## Definition

Let $g > 0$ be a constant, the gravitational acceleration.
The one-dimensional *shallow water equations* over a flat bottom are the system
$$\partial_t h + \partial_x (h u) = 0, \qquad \partial_t (h u) + \partial_x \big( h u^2 + \tfrac{1}{2} g h^2 \big) = 0,$$
for unknown functions $h, u \colon \R \times [0, \infty) \to \R$ with $h > 0$.
Here $h$ is the depth of the fluid and $u$ its horizontal velocity.
The symbols $\partial_t$ and $\partial_x$ denote [[partial derivatives|09EGC6F8PT]].

## Remarks

The system has the form $\partial_t U + \partial_x \Phi(U) = 0$ for $U = (h, hu)$, so it is a system of [[hyperbolic conservation laws|XN081NCPF5]].
The eigenvalues of the Jacobian matrix of $\Phi$ are $u \pm \sqrt{g h}$, which are real and distinct when $h > 0$.

Example: for constants $H > 0$ and $U_0 \in \R$, the uniform flow $h = H$, $u = U_0$ is a solution, since all derivatives vanish.

Non-example: $h(x, t) = 1 + x^2$ and $u = 0$ do not form a solution, because $\partial_x(\tfrac{1}{2} g h^2) = g h \, \partial_x h = 2 g x (1 + x^2)$ is not identically $0$.

In two dimensions, with velocity $(u, v)$, the equations are $\partial_t h + \nabla \cdot (h u, h v) = 0$ together with two momentum equations of the same form.
Variants add a varying bottom, friction or the Coriolis force.
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# Total curvature

## Definition

Let $\gamma \colon [a, b] \to \R^n$ be a [[regular|06JMR2XWXW]] curve of class $C^2$ with [[curvature|7JQS6VKFEZ]] $\kappa$.
The *total curvature* of $\gamma$ is
$$\int_a^b \kappa(t) \, \norm{\gamma'(t)} \, dt,$$
that is, the integral of $\kappa$ with respect to arc length.
For a plane curve with signed curvature $k$, the *total signed curvature* is $\int_a^b k(t) \, \norm{\gamma'(t)} \, dt$.

## Remarks

Example: the circle $\gamma(t) = (r \cos t, r \sin t)$, $t \in [0, 2\pi]$, has $\kappa = 1/r$ and $\norm{\gamma'} = r$, so its total curvature is $2\pi$.
Non-example: the boundary of a square is a [[closed curve|0D032NSZKY]] that is not of class $C^2$, so the definition does not apply.
For convex polygons one uses instead the sum of the exterior angles, which is again $2\pi$.
The total curvature does not change under a regular reparametrisation or a rigid motion.
For a closed regular plane curve, the total signed curvature equals $2\pi$ times the turning number of the curve.
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<document source="https://tamnd.github.io/math2/node/EMVECB93EQ.md" tag="EMVECB93EQ" kind="statement">
# Cubic reciprocity

## Statement

Let $\omega = e^{2\pi i/3}$ be a primitive cube [[root of unity|0YV2CKP2AV]], and let $\Z[\omega] = \set{a + b\omega : a, b \in \Z}$ be the ring of Eisenstein integers, with norm $N(a + b\omega) = a^2 - ab + b^2$.
Let $\pi$ be a prime element of $\Z[\omega]$ with $N(\pi) \ne 3$.
For $\alpha \in \Z[\omega]$ not divisible by $\pi$, the cubic residue character $\left(\frac{\alpha}{\pi}\right)_3$ is the unique element of $\set{1, \omega, \omega^2}$ with
$$\alpha^{(N(\pi) - 1)/3} \equiv \left(\frac{\alpha}{\pi}\right)_3 \pmod{\pi}.$$
A prime $\pi$ is called primary if $\pi \equiv 2 \pmod 3$ in $\Z[\omega]$.
Let $\pi_1$ and $\pi_2$ be primary primes of $\Z[\omega]$ with $N(\pi_1) \ne 3$, $N(\pi_2) \ne 3$ and $N(\pi_1) \ne N(\pi_2)$.
Then
$$\left(\frac{\pi_1}{\pi_2}\right)_3 = \left(\frac{\pi_2}{\pi_1}\right)_3.$$

## Remarks

The standard proof uses cubic [[Gauss sums|EQ6A3ZN48K]] and [[Jacobi sums|2767J23XHN]].
For a prime $\pi$ of norm $p \equiv 1 \pmod 3$, the Jacobi sum of the cubic character is a primary element of norm $p$.
The cube of the Gauss sum equals $p$ times this Jacobi sum.
Comparing this identity modulo the other prime gives the law.
Each prime of $\Z[\omega]$ not dividing $3$ has exactly one primary associate, so the hypothesis "primary" only fixes the choice among the six associates.
A classical consequence: for a prime $p \equiv 1 \pmod 3$, the integer $2$ is a cube modulo $p$ if and only if $p = x^2 + 27y^2$ for some integers $x, y$.
For example, $31 = 2^2 + 27 \cdot 1^2$, and indeed $4^3 = 64 \equiv 2 \pmod{31}$.
The prime $7$ is not of this form, and the cubes modulo $7$ are only $0$, $1$ and $6$.

## Explanations

### L1 human

Cubic reciprocity is the analogue of [[quadratic reciprocity|085MJS6907]] for cubes instead of squares.
It relates the question "is $\pi_1$ a cube modulo $\pi_2$?" to the question "is $\pi_2$ a cube modulo $\pi_1$?".
To state it cleanly one must work in the Eisenstein integers, which contain the cube roots of unity.
There the answer to both questions is the same.

### L2 human

The law is usually completed by supplementary laws for the units and for the prime $1 - \omega$ above $3$.
Together they let one compute any cubic residue character by repeated swapping and reduction, just as with the Legendre symbol.
The proof through Gauss and Jacobi sums is the model for the higher reciprocity laws, and the general case is part of class field theory through the [[Artin reciprocity law|7938EKMMVR]].
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# Projective geometry

## Definition

A *projective geometry*, or projective space in the axiomatic sense, is an [[incidence structure|FGKNAFNHA3]] whose lines are sets of points, incidence being membership, such that:

1. Any two distinct points lie on exactly one line.
2. (Veblen's axiom) Let $a, b, c, d$ be distinct points; if the lines $ab$ and $cd$ meet, then the lines $ac$ and $bd$ meet.
3. Every line contains at least three points.

The subject *projective geometry* is the study of these spaces and of the properties preserved by their isomorphisms.

## Remarks

Example: let $V$ be a vector space of dimension at least $3$ over a field $k$.
The points of the [[projective space|ZT9A6JK8QF]] $\mathbb{P}(V)$, with the lines $\mathbb{P}(W)$ for $\dim W = 2$, form a projective geometry.
Check: two distinct points span a unique $2$-dimensional $W$.
If $ab$ and $cd$ meet, then $a, b, c, d$ lie in a subspace $U$ of dimension at most $3$.
The planes for $ac$ and $bd$ lie in $U$, so they meet in a nonzero vector.
A $2$-dimensional $W$ has $\abs{k} + 1 \ge 3$ points.

Non-example: the real [[affine plane|EC7XT2ZM5S]] $\R^2$ satisfies clauses 1 and 3 but fails Veblen's axiom.
Take $a = (0,0)$, $b = (1,0)$, $c = (0,1)$ and $d = (1,2)$.
The lines $ab$ (given by $y = 0$) and $cd$ (given by $y = x + 1$) meet at $(-1, 0)$, but the lines $ac$ (given by $x = 0$) and $bd$ (given by $x = 1$) are parallel.

A projective geometry of dimension $2$ is the same as a [[projective plane|DANFNN8C4Y]].
By the [[Veblen–Young theorem|NPNNXWMVCC]], every projective geometry of dimension at least $3$ is some $\mathbb{P}(V)$ over a division ring.
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<document source="https://tamnd.github.io/math2/node/EMXK9T0X1Q.md" tag="EMXK9T0X1Q" kind="definition">
# Hamilton–Jacobi theory of the calculus of variations

## Definition

Let $I \subseteq \R$ be an open interval and let $L \colon I \times \R^n \times \R^n \to \R$, $(t, q, v) \mapsto L(t, q, v)$, be of class $C^2$.
Assume that for each $(t, q)$ the map $v \mapsto \partial L / \partial v (t, q, v)$ is a diffeomorphism of $\R^n$ and $\partial^2 L / \partial v^2$ is positive definite.
The variational problem is to find extremals of $\int L(t, q, q') \, dt$, the solutions of the [[Euler–Lagrange equation|PS0VV149ET]].
Its Hamiltonian is $H(t, q, p) = p \cdot v - L(t, q, v)$, where $v$ is the unique vector with $p = \partial L / \partial v(t, q, v)$.
The *Hamilton–Jacobi equation* of the problem is the [[partial differential equation|5TRXQKP7DY]]
$$\frac{\partial S}{\partial t} + H\Big(t, q, \frac{\partial S}{\partial q}\Big) = 0$$
for a function $S$ of class $C^2$ on an open set $G \subseteq I \times \R^n$.
*Hamilton–Jacobi theory* is the study of the variational problem through the solutions $S$ of this equation and the curves $q$ in $G$ with
$$q'(t) = \frac{\partial H}{\partial p}\Big(t, q(t), \frac{\partial S}{\partial q}(t, q(t))\Big).$$

## Remarks

The basic fact is that each such curve is an extremal, with momentum $p(t) = \partial S / \partial q(t, q(t))$ solving Hamilton's canonical equations.
These curves fill $G$ and form a [[field of extremals|K4DFT3H9BJ]], and $S$ is the action along them.
Example: $L = \frac12 \abs{v}^2$, so $H = \frac12 \abs{p}^2$, on $G = \set{(t, q) : t > t_0}$.
The function $S(t, q) = \abs{q - a}^2 / (2(t - t_0))$ solves the equation, since $S_t = -\abs{q - a}^2 / (2 (t - t_0)^2)$ and $\frac12 \abs{\nabla_q S}^2 = \abs{q - a}^2 / (2 (t - t_0)^2)$.
The curves solve $q' = (q - a)/(t - t_0)$, so they are the straight lines through $a$ at time $t_0$, which are the extremals of $L$.
Non-example: for $L = \frac12 \abs{v}^2$ the function $S(t, q) = \abs{q}^2$ does not solve the equation, since $S_t + \frac12 \abs{\nabla S}^2 = 2 \abs{q}^2 \ne 0$ for $q \ne 0$.
Jacobi's theorem completes the theory: from a family $S(t, q, a)$ of solutions with $\det(\partial^2 S / \partial q \, \partial a) \ne 0$, the equations $\partial S / \partial a = b$ define all solutions of Hamilton's equations near a point.
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# Incomplete LU factorization

## Definition

Let $A$ be a real $n \times n$ [[sparse matrix|JGACZYEEY7]].
Let $P \subseteq \set{1, \dots, n}^2$ be a set of positions that contains every diagonal position $(i, i)$.
An *incomplete LU factorization* of $A$ with pattern $P$ is a pair $(L, U)$ such that:

1. $L$ is unit [[lower triangular|GTG8MJGDKC]] and $U$ is upper triangular;
2. $l_{ij} = 0$ and $u_{ij} = 0$ for every position $(i, j) \notin P$;
3. $(L U)_{ij} = a_{ij}$ for every $(i, j) \in P$.

When $P$ is the set of positions with $a_{ij} \ne 0$ together with the diagonal, it is called *ILU(0)*.

## Remarks

If $P$ is the set of all positions and $A$ has an [[LU decomposition|0RKG5YW66Y]] with unit lower triangular $L$, the two factorizations coincide.
Example: if $A$ is tridiagonal and has an LU decomposition, its exact factors are bidiagonal.
They vanish outside the pattern of $A$, so they form the ILU(0) factorization.
Non-example: let $A$ be the $2 \times 2$ identity, let $P$ be all positions, and let $L$ have rows $(1, 0)$ and $(1, 1)$, with $U = I$.
Clauses 1 and 2 hold, but clause 3 fails, since $(L U)_{21} = 1 \ne 0 = a_{21}$.
The factors are computed by Gaussian elimination in which each update outside $P$ is discarded.
For a general matrix a zero pivot can occur, and then the factorization does not exist.
It exists for every pattern when $A$ is an [[M-matrix|DNKXHMGQY1]].
For symmetric positive definite $A$ the symmetric analogue is the [[incomplete Cholesky factorization|QYYCFQ8E5D]].
Variants use levels of fill, written ILU($k$), or thresholds, written ILUT.
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# Petersson inner product

## Summary

The Petersson inner product is an [[inner product|0NB0AWGJBW]] on the space $S_k$ of [[cusp forms|A50Y012RBP]] of weight $k$.
It integrates $f \overline{g}\, y^k$ against the hyperbolic area over a fundamental domain of the [[modular group|VEKV1EM77B]].
The [[Hecke operators|S4SV3QZ8B1]] are self-adjoint for it, so $S_k$ has a basis of common eigenforms.
The [[Eisenstein series|EYREVY31BN]] are orthogonal to the cusp forms in a suitable sense.

## Explanations

### L1 human

To speak of angles and lengths in a space of functions, one needs an inner product.
For cusp forms the natural one multiplies $f$ by the conjugate of $g$, adds the weight factor $y^k$, and integrates over one copy of the quotient of the upper half-plane.
The weight factor makes the integrand invariant, so the choice of copy does not matter.
With this inner product, the Hecke operators behave like symmetric matrices, and so they can be diagonalized together.
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<document source="https://tamnd.github.io/math2/node/EN4A9MM7RP.md" tag="EN4A9MM7RP" kind="definition">
# Anosov flow

## Definition

Let $M$ be a [[compact|0QPHW703PV]] Riemannian manifold with norm $\norm{\cdot}$ on each tangent space.
Let $X$ be a $C^1$ [[vector field|0N67NBKQ5T]] on $M$ with no zeros, and let $\varphi_t$ be its [[flow|02903SSA48]].
The flow $\varphi_t$ is an *Anosov flow* if the following hold.

1. For each $x \in M$ there is a splitting $T_x M = E^s_x \oplus E^0_x \oplus E^u_x$, where $E^0_x$ is the line spanned by $X(x)$.
2. The splitting is invariant: $D\varphi_t(E^s_x) = E^s_{\varphi_t(x)}$ and $D\varphi_t(E^u_x) = E^u_{\varphi_t(x)}$ for all $t \in \R$.
3. There are constants $C > 0$ and $\lambda > 0$ such that, for all $x \in M$ and $t \ge 0$, $\norm{D\varphi_t v} \le C e^{-\lambda t}\norm{v}$ for $v \in E^s_x$ and $\norm{D\varphi_{-t} v} \le C e^{-\lambda t}\norm{v}$ for $v \in E^u_x$.

## Remarks

Example: the [[geodesic flow|VF43FTDA1H]] on the unit tangent bundle of a compact surface of constant curvature $-1$ is an Anosov flow.
In the model $\Gamma \backslash \mathrm{PSL}_2(\R)$, the flow is right multiplication by $a_t = \operatorname{diag}(e^{t/2}, e^{-t/2})$, and $E^s$, $E^u$ are tangent to the orbits of the two [[horocycle flows|2DC88BMQVX]].
Conjugating $\begin{pmatrix} 1 & s \\ 0 & 1\end{pmatrix}$ by $a_t$ multiplies $s$ by $e^{t}$ or by $e^{-t}$, which gives clause 3 with $\lambda = 1$.
Another example is the suspension flow of an [[Anosov diffeomorphism|X1P48KQFXQ]] with constant roof function.
Non-example: the linear flow $\varphi_t(x) = x + t v$ on the torus $\R^n/\Z^n$ has derivative the identity, so no direction is contracted and clause 3 fails.
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# Universal property

## Summary

A universal property describes an object with a morphism by a factorisation rule.
Every other morphism of the same type factors through it in exactly one way.
Products, quotients, free groups, tensor products, limits and colimits are all defined by universal properties.
An object with a universal property is unique up to a unique isomorphism, so the property pins it down without choosing a construction.

## Explanations

### L1 human

Instead of saying what an object is made of, a universal property says how it relates to everything else.
For example, the [[free group|0KTW26C80J]] on a set $S$ is a group that contains $S$.
Any function from $S$ to any group extends to exactly one group homomorphism on it.
Any two objects with the same universal property are matched by a unique isomorphism, so it does not matter how the object was built.
This is why mathematicians often say "the" product or "the" free group.
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<document source="https://tamnd.github.io/math2/node/ENA9XD5B77.md" tag="ENA9XD5B77" kind="definition">
# Hamiltonian of an optimal control problem

## Definition

Let $U \subseteq \R^m$ be a nonempty set and $I \subseteq \R$ an interval.
Consider an [[optimal control|T3V6ZNAMDN]] problem with [[control system|ZPDTVZQN5K]] $x' = f(t, x, u)$ and running cost $\int L(t, x, u) \, dt$, where $f \colon I \times \R^n \times U \to \R^n$ and $L \colon I \times \R^n \times U \to \R$.
The *Hamiltonian* of the problem is the function $H \colon I \times \R^n \times U \times \R^n \times \R \to \R$ given by
$$H(t, x, u, p, \lambda_0) = p \cdot f(t, x, u) - \lambda_0 L(t, x, u).$$
The vector $p \in \R^n$ is the costate, and the number $\lambda_0 \ge 0$ is the cost multiplier.
In the normal case $\lambda_0 = 1$ one writes $H(t, x, u, p) = p \cdot f(t, x, u) - L(t, x, u)$.

## Remarks

Example: for $x' = u$ in $\R$ with running cost $\int \frac12 (x^2 + u^2) \, dt$, the normal Hamiltonian is $H = p u - \frac12 (x^2 + u^2)$.
For fixed $(x, p)$ it is a concave function of $u$ with maximum at $u = p$.
The state equation is recovered as $x' = \partial H / \partial p$.
Many texts use the opposite sign convention $H = L + p \cdot f$, under which optimal controls minimize $H$; this gives the minimum principle.
The two conventions differ by the change $p \mapsto -p$ and an overall sign.
A non-example is the Lagrangian $L(t, x, u)$ alone: it does not involve the costate, so it does not carry the dynamics.
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# Strictly singular operator

## Definition

Let $X$ and $Y$ be [[Banach spaces|06GZ956657]], and let $T \colon X \to Y$ be a [[bounded linear map|03TM135G0N]].
The map $T$ is *strictly singular* if for every infinite-dimensional closed subspace $M$ of $X$,
$$\inf \set{\norm{Tx} : x \in M, \ \norm{x} = 1} = 0.$$
Equivalently, no restriction $T|_M$ to an infinite-dimensional closed subspace $M$ is an isomorphism onto its image.

## Remarks

Example: every [[compact operator|M2NEP1K5AE]] is strictly singular.
Indeed, suppose that $T|_M$ had a bounded inverse on $T(M)$.
Then the closed unit ball of $M$ would be the image of a relatively compact set under a bounded map, so it would be compact, and $M$ would be finite-dimensional.
For $1 \le p < q < \infty$, the inclusion $\ell^p \to \ell^q$ is strictly singular but not compact.
Non-example: the identity map of an infinite-dimensional Banach space $X$ is not strictly singular, since it is an isomorphism on $M = X$.
On Hilbert spaces, the strictly singular operators are exactly the compact operators.
For a Banach space $X$, the strictly singular operators from $X$ to $X$ form a closed two-sided ideal of the algebra of bounded operators.
If $F$ is a [[Fredholm operator|8HWGKWD6ZM]] and $S$ is strictly singular, then $F + S$ is Fredholm with the same index.
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# Bornological space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a [[locally convex space|B6GY2KJMWW]] over $\mathbb{K}$.
A subset $A \subseteq E$ *absorbs* a subset $B \subseteq E$ if there is $r > 0$ such that $B \subseteq cA$ for every $c \in \mathbb{K}$ with $\abs{c} \ge r$.
$E$ is *bornological* if every [[convex|02AS45T3H9]] [[balanced|2TDSHJJYS4]] subset of $E$ that absorbs every [[bounded set|3BAJ8ZTVPX]] of $E$ is a [[neighbourhood|0ZCY0Z80B6]] of $0$.

## Remarks

Equivalently, $E$ is bornological if and only if every linear map from $E$ to a locally convex space that maps bounded sets to bounded sets is continuous.
Example: every normed space $E$ is bornological.
If a convex balanced set $A$ absorbs the closed unit ball $B$, then $B \subseteq c A$ for some $c > 0$, so $A$ contains the ball of radius $1 / c$.
More generally, every metrizable locally convex space is bornological.
Non-example: let $E$ be an infinite-dimensional Banach space with its weak topology $\sigma(E, E')$.
Its weakly bounded sets are norm bounded, by the uniform boundedness principle, so the closed unit ball absorbs all of them.
But the ball is not a weak neighbourhood of $0$, since every weak neighbourhood of $0$ contains a nonzero linear subspace.
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# Arithmetic genus

## Definition

Let $k$ be a field and let $X$ be a [[projective|A4H4CZ2V35]] scheme over $k$ of dimension $n$.
Let
$$\chi(\mathcal{O}_X) = \sum_{i \ge 0} (-1)^i \dim_k H^i(X, \mathcal{O}_X)$$
be the [[Euler characteristic|RNDDWDAFJ5]] of its structure sheaf, computed with [[sheaf cohomology|PQ6Q143ARJ]].
The *arithmetic genus* of $X$ is
$$p_a(X) = (-1)^n \left( \chi(\mathcal{O}_X) - 1 \right).$$

## Remarks

The sum is finite because $X$ is projective, so each $H^i(X, \mathcal{O}_X)$ is finite-dimensional and vanishes for $i > n$.
Equivalently, $p_a(X) = (-1)^n (P_X(0) - 1)$, where $P_X$ is the Hilbert polynomial of $X$ for a chosen embedding.
For a curve $C$ with $H^0(C, \mathcal{O}_C) = k$, for example a connected reduced projective curve over an algebraically closed field, $p_a(C) = \dim_k H^1(C, \mathcal{O}_C)$.
Example: $H^1(\mathbb{P}^1, \mathcal{O}) = 0$, so $p_a(\mathbb{P}^1) = 0$.
Example: a plane cubic curve, smooth or nodal, has $p_a = 1$; this follows from the exact sequence $0 \to \mathcal{O}_{\mathbb{P}^2}(-3) \to \mathcal{O}_{\mathbb{P}^2} \to \mathcal{O}_C \to 0$.
Non-example: the affine line $\mathbb{A}^1$ is not projective, and $H^0(\mathbb{A}^1, \mathcal{O}) = k[t]$ is infinite-dimensional, so $\chi$ is not defined.
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# Bochner integral

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $E$ be a [[Banach space|0BDTTM24DP]] over $\R$ or $\C$.
An *integrable simple function* is a function $s = \sum_{i=1}^{n} \mathbf{1}_{A_i} e_i$ with $e_i \in E$ and pairwise disjoint $A_i \in \Sigma$ with $\mu(A_i) < \infty$.
Its integral is $\int_X s \, d\mu = \sum_{i=1}^{n} \mu(A_i) \, e_i$.
A function $f \colon X \to E$ is *Bochner integrable* if both of the following hold.

1. The function $f$ is [[strongly measurable|JDM2M1FRY6]].
2. There are integrable simple functions $s_1, s_2, \dots$ with $\lim_{n \to \infty} \int_X \norm{f - s_n} \, d\mu = 0$.

The integrals in clause 2 are [[integrals of non-negative functions|DG76G5Z6XW]].
The *Bochner integral* of $f$ is then $\int_X f \, d\mu = \lim_{n \to \infty} \int_X s_n \, d\mu$, a limit in $E$.

## Remarks

The limit exists, since $\norm{\int s_n - \int s_m} \le \int \norm{s_n - s_m}$, so the integrals form a Cauchy sequence in $E$.
It does not depend on the choice of the $s_n$.

Example: let $X = [0, 1]$ with Lebesgue measure and let $f \colon X \to E$ be continuous.
The step functions $s_n$ equal to $f(k/n)$ on $[k/n, (k+1)/n)$ for $0 \le k < n$ and to $f(1)$ at $1$ converge to $f$ uniformly, so $\int \norm{f - s_n} \to 0$.
So $f$ is Bochner integrable, and its integral is the limit of the Riemann sums $\frac{1}{n} \sum_{k=0}^{n-1} f(k/n)$.

Non-example: take $X = \R$ with Lebesgue measure, $E = \R$ and $f = 1$.
It is strongly measurable, but every integrable simple function vanishes outside a set of finite measure, so $\int \norm{f - s} = \infty$ for each of them.

Bochner's theorem: a strongly measurable function $f$ is Bochner integrable if and only if $\int_X \norm{f} \, d\mu < \infty$.
For $E = \R$ or $E = \C$ the Bochner integral is the Lebesgue integral.
For a bounded linear map $T \colon E \to F$ one has $T \int f \, d\mu = \int T f \, d\mu$.
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# Siegel's theorem on meromorphic function fields

## Statement

Let $X$ be a compact, connected [[complex manifold|JSGWAN2RJV]] of dimension $n$, and let $\mathcal{M}(X)$ be its field of [[meromorphic functions|W6FTSF57T1]].
Then the [[transcendence degree|D7TR0YWR9H]] of $\mathcal{M}(X)$ over $\C$ is at most $n$.
Equivalently, any $n + 1$ meromorphic functions on $X$ satisfy a nonzero polynomial relation with complex coefficients.

## Remarks

Proof idea: let $f_1, \ldots, f_{n+1}$ be meromorphic on $X$ and consider all polynomials of degree at most $k$ in them.
These form a space of dimension of order $k^{n+1}$.
Cover $X$ by finitely many polydiscs and use a Schwarz lemma.
This shows that such a polynomial expression vanishing to high order at finitely many chosen points vanishes identically.
The number of conditions needed grows only like $k^n$.
For large $k$ some nonzero polynomial in the $f_i$ is therefore identically zero.

Example: on a compact Riemann surface ($n = 1$), any two meromorphic functions are algebraically dependent; for $\hat\C$, $f$ and $g$ are both rational functions of $z$.

Compactness is necessary: on the non-compact manifold $\C$, the meromorphic functions $z$ and $e^z$ are algebraically independent, so the transcendence degree is at least $2 > 1$.

In terms of the [[algebraic dimension|F0GF0JEZTG]], the theorem says $a(X) \le \dim X$.
The field $\mathcal{M}(X)$ is moreover finitely generated over $\C$, so it is the function field of an algebraic variety of dimension $a(X)$.

## Explanations

### L1 human

On a compact complex manifold of dimension $n$ there cannot be more than $n$ independent meromorphic functions.
Any $n + 1$ of them are tied together by a polynomial equation, just as $x$ and $y$ are tied by $x^2 + y^2 = 1$ on a circle.
This mirrors algebraic geometry, where an $n$-dimensional variety has a function field of transcendence degree $n$.

### L2 human

The theorem makes the [[algebraic dimension|F0GF0JEZTG]] a number between $0$ and $\dim X$, and the case of equality defines [[Moishezon manifolds|8ZW9P9JRQD]].
It shows that compact complex manifolds behave like algebraic varieties as far as their meromorphic functions go, even when they are far from algebraic.
The proof uses only local estimates and compactness, in the spirit of the Schwarz lemma.
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<document source="https://tamnd.github.io/math2/node/ENYXKW76AA.md" tag="ENYXKW76AA" kind="concept">
# Ruelle zeta function

## Summary

The Ruelle zeta function of a map is a weighted count of its [[periodic points|T75CZ5H47N]].
Each fixed point of $f^n$ is counted with the product of a weight function along its orbit.
With the weight $1$ it is the [[Artin–Mazur zeta function|2A5Z21CE4P]].
It is a basic tool of the thermodynamic formalism, where its poles and zeros are related to the eigenvalues of [[transfer operators|CNZV00JQRC]].
Ruelle also defined a version for flows, with a product over closed orbits.
For geodesic flows of hyperbolic surfaces, this version is close to the Selberg zeta function.

## Explanations

### L1 human

Instead of just counting periodic points, one can give each periodic orbit a weight, for example a probability or a stretching factor.
The Ruelle zeta function packs these weighted counts for all periods into one function.
Its singularities then tell how fast the weighted counts grow, which in turn gives the pressure and the rate of mixing of the system.
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<document source="https://tamnd.github.io/math2/node/ENZ88T9WWQ.md" tag="ENZ88T9WWQ" kind="definition">
# Discrete Painlevé equations

## Definition

Let $R(n, x, y)$ be a [[rational function|0XDK9CGYKJ]] of $x$ and $y$ whose coefficients depend on $n \in \Z$.
Consider the second-order difference equation
$$x_{n+1} = R(n, x_n, x_{n-1}), \qquad n \in \Z.$$
It is a *discrete Painlevé equation* if the following hold.

1. the map $(x_{n-1}, x_n) \mapsto (x_n, x_{n+1})$ is birational for each $n$;
2. the equation is integrable, for example in the sense that it is the compatibility condition of a Lax pair of linear problems;
3. the equation has one of the [[Painlevé equations|AZ17JCMRKB]] as a continuum limit.

In Sakai's geometric classification, these equations arise as translations of affine Weyl groups acting on families of rational surfaces.
The dependence on $n$ may also enter through $q^n$ for a fixed $q$, which gives $q$-discrete Painlevé equations.

## Remarks

Example: the equation called d-$\mathrm{P_I}$,
$$x_{n+1} + x_n + x_{n-1} = \frac{\alpha n + \beta}{x_n} + \gamma,$$
with constants $\alpha \ne 0$, $\beta$, $\gamma$, is a discrete Painlevé equation; under a suitable scaling it tends to $\mathrm{P_I}$, $y'' = 6y^2 + t$.

Non-example: the logistic recurrence $x_{n+1} = r x_n (1 - x_n)$ is a rational recurrence, but its map $x \mapsto r x (1 - x)$ is two-to-one, so it is not birational and clause 1 fails.

Clause 2 has several versions in the literature, so the definition is less rigid than most.
A practical test is singularity confinement: a singularity met at one step disappears after finitely many steps.
Another standard example is d-$\mathrm{P_{II}}$, $x_{n+1} + x_{n-1} = \dfrac{(\alpha n + \beta) x_n + a}{1 - x_n^2}$.
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<document source="https://tamnd.github.io/math2/node/ENZF7KF1TC.md" tag="ENZF7KF1TC" kind="definition">
# Bézier curve

## Definition

Let $n \ge 1$ and let $P_0, P_1, \dots, P_n$ be points of $\R^d$, the control points.
The *Bézier curve* of degree $n$ with these control points is the map $B \colon [0, 1] \to \R^d$,
$$B(t) = \sum_{k=0}^{n} \binom{n}{k} (1 - t)^{n-k}\, t^k\, P_k.$$

## Remarks

The coefficients $\binom{n}{k}(1-t)^{n-k} t^k$ are the [[Bernstein basis polynomials|0C4VWBFY73]] of degree $n$; they are nonnegative on $[0, 1]$ and sum to $1$.
So $B(t)$ lies in the [[convex hull|0FQSDC2EQ2]] of the control points, $B(0) = P_0$, $B(1) = P_n$, $B'(0) = n(P_1 - P_0)$ and $B'(1) = n(P_n - P_{n-1})$.
Example: $n = 2$ with $P_0 = (0, 0)$, $P_1 = (1, 2)$, $P_2 = (2, 0)$ gives $B(t) = (2t, 4t - 4t^2)$, an arc of the parabola $y = 2x - x^2$.
Non-example: no arc of a circle of positive length is a Bézier curve.
Indeed, a polynomial map $t \mapsto (x(t), y(t))$ with $x^2 + y^2 = 1$ must be constant.
If $x$ and $y$ had top degree $m \ge 1$, the coefficient of $t^{2m}$ in $x^2 + y^2$ would be a positive sum of squares.
A rational Bézier curve adds positive weights, $B(t) = \sum_k w_k b_{k,n}(t) P_k / \sum_k w_k b_{k,n}(t)$, and does represent circular arcs.
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<document source="https://tamnd.github.io/math2/node/EP681WKN00.md" tag="EP681WKN00" kind="definition">
# Jacobi amplitude

## Definition

Let $0 \le k < 1$.
For real $\varphi$ let $F(\varphi, k) = \int_0^\varphi d\theta / \sqrt{1 - k^2\sin^2\theta}$ be the incomplete [[elliptic integral|NXMCYGCB59]] of the first kind.
The map $\varphi \mapsto F(\varphi, k)$ is a strictly increasing bijection from $\R$ onto $\R$.
The *Jacobi amplitude* $\operatorname{am}(\cdot, k) \colon \R \to \R$ is its inverse function: $\varphi = \operatorname{am}(u, k)$ exactly when $u = F(\varphi, k)$.

## Remarks

The map is strictly increasing because its derivative $1/\sqrt{1 - k^2\sin^2\varphi}$ lies between $1$ and $1/k'$, where $k' = \sqrt{1 - k^2}$; the lower bound $1$ makes it unbounded in both directions.
So $\operatorname{am}(\cdot, k)$ is smooth, odd, and $\frac{d}{du}\operatorname{am}(u, k) = \sqrt{1 - k^2\sin^2\operatorname{am}(u, k)}$.
Since the integrand has period $\pi$, $F(\varphi + \pi, k) = F(\varphi, k) + 2K$, with $K = K(k)$ the [[complete elliptic integral of the first kind|RM54CPPE12]]; hence $\operatorname{am}(u + 2K, k) = \operatorname{am}(u, k) + \pi$.
Example: for $k = 0$, $F(\varphi, 0) = \varphi$, so $\operatorname{am}(u, 0) = u$; and $\operatorname{am}(K, k) = \pi/2$ for every $k$.
Non-example: for $k = 1$ the integral $F(\varphi, 1) = \int_0^\varphi d\theta/\cos\theta$ only makes sense for $\abs{\varphi} < \pi/2$, so the definition above does not apply.
The inverse there is the Gudermannian function $\operatorname{am}(u, 1) = 2\arctan(e^u) - \pi/2$, a bijection from $\R$ onto $(-\pi/2, \pi/2)$.
The Jacobi elliptic functions are $\operatorname{sn} u = \sin\operatorname{am} u$ and $\operatorname{cn} u = \cos\operatorname{am} u$.
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# Exponential family

## Summary

An exponential family is a parametric family of probability distributions whose densities have the form $h(x) \exp\big(\sum_i \eta_i(\theta) T_i(x) - A(\theta)\big)$.
Most standard distributions belong to one: normal, binomial, Poisson, gamma, beta and many more.
The statistics $T_i$ are sufficient, and the structure makes estimation, testing and Bayesian updating explicit.

## Explanations

### L1 human

In an exponential family the parameter and the data meet only through a few products $\eta_i(\theta) T_i(x)$ in an exponent.
So the data can be compressed to the values of $T_1, \dots, T_k$ without losing any information about the parameter.
For example, the Poisson probabilities $e^{-\lambda} \lambda^x / x!$ can be written as $\frac{1}{x!} \exp(x \log \lambda - \lambda)$, with $T(x) = x$ and $\eta = \log \lambda$.
For a sample of Poisson counts, only the total count matters for $\lambda$.
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<document source="https://tamnd.github.io/math2/node/EPAAKD80H9.md" tag="EPAAKD80H9" kind="concept">
# Homotopy type theory

## Summary

Homotopy type theory reads [[intuitionistic type theory|ESVYD9CZ8C]] in terms of homotopy theory: types are spaces, terms are points, and terms of identity types are paths.
It adds Voevodsky's univalence axiom, which says that equivalent types are equal, and higher inductive types such as the circle.
These additions give a new foundation of mathematics, called univalent foundations.
The book "Homotopy Type Theory: Univalent Foundations of Mathematics", written in 2013 at the Institute for Advanced Study, set out the theory.

## Explanations

### L1 human

In ordinary [[type theory|Q48KGV33AH]], a proof that $a = b$ is just a certificate.
In homotopy type theory, it is a path from $a$ to $b$ in a space, and two such paths need not be the same.
A proof that two proofs of $a = b$ are equal is then a deformation of one path into the other.
The univalence axiom says that two types with a one-to-one correspondence between them are equal, so isomorphic structures can always be swapped.
This matches how mathematicians already treat isomorphic objects.
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<document source="https://tamnd.github.io/math2/node/EPAZPBPFF9.md" tag="EPAZPBPFF9" kind="definition">
# Simplicial set

## Definition

Let $\Delta$ be the [[simplex category|FRFYHJGAN3]].
A *simplicial set* is a functor $X \colon \Delta^{\mathrm{op}} \to \mathbf{Set}$, that is, a [[presheaf|AND2WHZC8P]] on $\Delta$.
The elements of $X_n = X([n])$ are the $n$-simplices of $X$.
A map of simplicial sets is a natural transformation.
Thus a simplicial set is a [[simplicial object|WK0A32TZHB]] in the category of sets.

## Remarks

Concretely, a simplicial set is a family of sets $X_n$ with face maps $d_i \colon X_n \to X_{n-1}$ and degeneracy maps $s_i \colon X_n \to X_{n+1}$, $0 \le i \le n$, that satisfy the simplicial identities.
An $n$-simplex is degenerate when it lies in the image of some $s_i$.
Example: the standard $n$-simplex $\Delta^n = \Hom_{\Delta}(-, [n])$ is a representable functor, so it is a simplicial set; its nondegenerate $m$-simplices are the injective order-preserving maps $[m] \to [n]$.
Non-example: a family of sets $X_n$ with only face maps and no degeneracy maps is not a simplicial set; such data form a semi-simplicial set.
The category of simplicial sets is often written $\mathbf{sSet}$.
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<document source="https://tamnd.github.io/math2/node/EPBND51CR5.md" tag="EPBND51CR5" kind="concept">
# Total variation of a measure

## Summary

Let $\nu$ be a [[signed measure|VG2P7R91Z7]] or a [[complex measure|NY6TYHRGMS]].
Its total variation $\abs{\nu}$ is the smallest positive measure with $\abs{\nu(E)} \le \abs{\nu}(E)$ for every set $E$.
It is computed as the supremum of $\sum \abs{\nu(E_i)}$ over partitions of a set into measurable pieces.
For a signed measure it equals $\nu^+ + \nu^-$ from the Jordan decomposition.
The number $\abs{\nu}(X)$ is the total variation norm, which makes the finite signed or complex measures a Banach space.
It extends the [[total variation|8QA5DEWNYD]] of a function to measures.

## Explanations

### L1 human

A signed measure can make a set look small by cancelling positive and negative parts.
The total variation removes this cancellation: it counts all the positive mass and all the negative mass.
For example, on $[-1, 1]$ the signed measure $\nu(E) = \int_E x \, dx$ gives the whole interval measure $0$, but its total variation gives it $\int_{-1}^{1} \abs{x} \, dx = 1$.
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<document source="https://tamnd.github.io/math2/node/EPG6JPVJE7.md" tag="EPG6JPVJE7" kind="concept">
# Unilateral Z-transform

## Summary

The unilateral Z-transform of a sequence $(x_n)$ uses only its terms with $n \ge 0$: $X(z) = \sum_{n \ge 0} x_n z^{-n}$.
It is the form of the [[Z-transform|5FY0JQGHYZ]] used for initial value problems.
A forward shift of the sequence brings in its first terms, so initial values enter the transformed equation by themselves.
It plays for difference equations the part that the one-sided Laplace transform plays for differential equations.

## Explanations

### L1 human

Many processes start at a time zero, and only the values from then on matter.
The unilateral Z-transform ignores everything before time zero.
For $x_n = 2^n$ it gives $\sum_{n \ge 0} 2^n z^{-n} = z/(z - 2)$, valid when $\abs{z} > 2$.
Its set of convergence is always the outside of a circle centred at $0$, possibly together with some points of that circle.
It can also be empty or the whole plane without $0$.
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<document source="https://tamnd.github.io/math2/node/EPM2HKGMZS.md" tag="EPM2HKGMZS" kind="concept">
# Complete intersection ring

## Summary

A complete intersection ring is a local ring that is, after completion, a [[regular local ring|63XXTF6B5M]] divided by an ideal generated by a [[regular sequence|69R5R2FCS5]].
It is the algebraic version of a space cut out by exactly as many equations as its codimension.
Every complete intersection ring is [[Gorenstein|T6HCX959PJ]], and hence [[Cohen–Macaulay|N1QJQEGBFC]].

## Explanations

### L1 human

In three-dimensional space, a surface is usually cut out by one equation and a curve by two.
When a space of codimension $r$ is cut out by exactly $r$ equations, it is a complete intersection.
The plane curve $xy = 0$, the union of two axes, is a complete intersection: one equation in the plane.
The three coordinate axes in space form a curve that needs three equations, $xy = yz = zx = 0$, so they are not a complete intersection.
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<document source="https://tamnd.github.io/math2/node/EPN6C0X93V.md" tag="EPN6C0X93V" kind="definition">
# Symplectic capacity

## Definition

Fix $n \ge 1$.
Let $B^{2n}(1)$ be the open unit ball and $Z^{2n}(1) = \set{(x, y) \in \R^{2n} : x_1^2 + y_1^2 < 1}$ the open unit cylinder, both with the standard symplectic form $\omega_0$.
A *symplectic capacity* on [[symplectic manifolds|CAN490EPRT]] of dimension $2n$ is a function $c$ assigning to each such manifold $(M, \omega)$ a value $c(M, \omega) \in [0, \infty]$ such that:

1. (Monotonicity) If there is a symplectic embedding $(M, \omega) \to (N, \tau)$, then $c(M, \omega) \le c(N, \tau)$.
2. (Conformality) $c(M, \lambda \omega) = \abs{\lambda} \, c(M, \omega)$ for every real $\lambda \ne 0$.
3. (Normalization) $c(B^{2n}(1), \omega_0) = c(Z^{2n}(1), \omega_0) = \pi$.

## Remarks

Example: the *Gromov width* $w(M, \omega) = \sup \set{\pi r^2 : B^{2n}(r) \text{ embeds symplectically in } (M, \omega)}$.
Monotonicity holds because an embedding into $M$ composed with $M \to N$ is an embedding into $N$.
Conformality holds because scaling by $\sqrt{\abs{\lambda}}$ (and swapping $x$ and $y$ if $\lambda < 0$) maps $(B(r), \lambda\omega_0)$ to $(B(\sqrt{\abs{\lambda}}\, r), \omega_0)$ symplectically.
The value $\pi$ on the ball is clear, and the value $\pi$ on the cylinder is exactly [[Gromov's non-squeezing theorem|RDX55HQ3XT]].

Non-example: for $n \ge 2$, the function $c(M, \omega) = \left( \int_M \omega^n / n! \right)^{1/n}$, suitably rescaled, satisfies monotonicity and conformality, but fails normalization, because the cylinder has infinite volume.

Some authors drop the normalization and only ask for nontriviality, $0 < c(B^{2n}(1))$ and $c(Z^{2n}(1)) < \infty$.
Other examples are the Hofer–Zehnder capacity and the Ekeland–Hofer capacities.
The Gromov width is the smallest normalized capacity.
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<document source="https://tamnd.github.io/math2/node/EPPR3DAFP6.md" tag="EPPR3DAFP6" kind="concept">
# Noncommutative Gröbner basis

## Summary

A noncommutative Gröbner basis, or Gröbner–Shirshov basis, generates a two-sided ideal of a [[free algebra|2P3Y2FRCSZ]] $k\langle X \rangle$.
Its leading words control the leading words of all elements of the ideal.
It extends the commutative [[Gröbner basis|C83KPT47R0]] from polynomial rings to free algebras, with words in place of monomials.
The words that avoid all leading words give a linear basis of the quotient algebra.
So a Gröbner basis solves the normal form problem for an algebra given by generators and relations.
Unlike the commutative case, a finitely generated ideal may have no finite Gröbner basis, and the completion procedure need not terminate.

## Explanations

### L1 human

An algebra given by generators and relations, such as $yx = xy$, raises the question of when two expressions are equal.
The idea is to orient each relation as a rewriting rule from a larger word to a smaller one, such as $yx \to xy$.
If the rules are complete, every expression rewrites to a unique normal form, and two expressions are equal exactly when their normal forms agree.
A Gröbner basis is such a complete set of rules.
With the single rule $yx \to xy$ the normal forms are the words $x^a y^b$, which recovers the polynomial ring $k[x, y]$.
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<document source="https://tamnd.github.io/math2/node/EPS9JT55MV.md" tag="EPS9JT55MV" kind="concept">
# Restriction of a measure

## Summary

The restriction of a [[measure|PR11BZQQPG]] to a measurable set $A$ keeps the mass that lies inside $A$ and forgets the rest.
It is again a measure on the same space.
Restriction lets one integrate over a subset, since the integral of $f$ over $A$ is the integral of $f$ for the restricted measure.

## Explanations

### L1 human

Picture a measure as paint spread over a space.
Restricting it to a region $A$ wipes away all the paint outside $A$.
A set $B$ then gets the amount of paint that lies in both $B$ and $A$.
For example, restrict length on the real line to $A = [0, 1]$.
Then the interval $[0, 5]$ gets the value $1$, and the interval $[2, 3]$ gets the value $0$.
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<document source="https://tamnd.github.io/math2/node/EPWB2JEXPE.md" tag="EPWB2JEXPE" kind="problem">
# Unit distance problem

## Problem

Let $n \ge 2$, and let $\norm{p - q}$ be the [[Euclidean distance|0NPPGHQNXQ]] in the [[Euclidean plane|0XW7V9ZZ05]].
Let $u(n)$ be the largest number of pairs $\set{p, q}$ with $\norm{p - q} = 1$ in a set of $n$ points of the plane.
Determine the order of growth of $u(n)$ as $n \to \infty$.

## Remarks

Paul Erdős posed the problem in 1946.
In the same paper he showed $u(n) = O(n^{3/2})$, and, using a suitably scaled square grid, $u(n) \ge n^{1 + c/\log \log n}$ for a constant $c > 0$; here $O$ is [[big O notation|0RYTGBJAGN]].
Joel Spencer, Endre Szemerédi and William Trotter proved $u(n) = O(n^{4/3})$ in 1984.
A short proof of this bound by László Székely uses the [[crossing number inequality|34B1PW5EE2]].
The exact order of growth is still not known.
The pairs at distance $1$ form the edges of a unit distance graph on the $n$ points.

## Explanations

### L1 human

Place $n$ points in the plane and count the pairs that are exactly one unit apart.
How large can this count be?
A square grid of the right size does a little better than $n$, and nobody has done much better.
The best upper bound is about $n^{4/3}$, and closing the gap is a famous open question of Erdős.
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# Mountain pass theorem

## Statement

Let $X$ be a real [[Banach space|06GZ956657]] and let $I \colon X \to \R$ be a continuously [[Fréchet differentiable|06EH97V2A7]] functional that satisfies the [[Palais–Smale condition|4724KRNHAT]].
Assume that:

1. $I(0) = 0$;
2. there are $r > 0$ and $a > 0$ such that $I(u) \ge a$ for every $u \in X$ with $\norm{u} = r$;
3. there is $e \in X$ with $\norm{e} > r$ and $I(e) \le 0$.

Let $\Gamma$ be the set of continuous paths $\gamma \colon [0, 1] \to X$ with $\gamma(0) = 0$ and $\gamma(1) = e$, and let
$$c = \inf_{\gamma \in \Gamma} \max_{t \in [0, 1]} I(\gamma(t)).$$
Then $c \ge a$, and $c$ is a critical value of $I$: there is $u \in X$ with $I(u) = c$ and $I'(u) = 0$.
In particular $u$ is a [[critical point|0YTCMFB4HC]] of $I$ different from $0$ and from $e$.

## Remarks

Every path from $0$ to $e$ crosses the sphere $\norm{u} = r$, so its maximum is at least $a$, which gives $c \ge a$.
Suppose $c$ were not a critical value.
Then a deformation lemma, which uses the Palais–Smale condition, would push the sublevel set $\set{I \le c + \varepsilon}$ into $\set{I \le c - \varepsilon}$ while fixing $0$ and $e$.
This would produce a path with maximum below $c$.
The Palais–Smale condition cannot be dropped, even in $\R^2$.
The function $f(x, y) = x^2 + (1 - x)^3 y^2$ has a strict local minimum at the origin with value $0$, so it is at least some $a > 0$ on a small circle around it.
It also takes negative values, for example at $(2, 3)$, but its only critical point is the origin.
Indeed, $\partial_y f = 2(1 - x)^3 y$ vanishes only for $x = 1$ or $y = 0$; at $x = 1$ one has $\partial_x f = 2$, and for $y = 0$ one needs $x = 0$.
Antonio Ambrosetti and Paul Rabinowitz proved the theorem.
A typical application gives a nonzero weak solution of $-\Delta u = \abs{u}^{p-1} u$ with zero boundary values on a bounded domain of $\R^n$, for $1 < p < (n+2)/(n-2)$ when $n \ge 3$.

## Explanations

### L1 human

Picture a valley surrounded by a ring of mountains, with lower ground somewhere beyond the ring.
Any route from the valley to that lower ground has to climb, and the best route goes through the lowest mountain pass.
The theorem says that this pass is a point where the ground is flat, that is, a critical point.
For functionals on infinite-dimensional spaces one needs a compactness condition, the Palais–Smale condition, to make the argument work.

### L2 human

The theorem is a basic tool of critical point theory and of variational methods for nonlinear elliptic equations.
It finds solutions that are neither minima nor maxima of the energy, but saddle points.
For semilinear problems, $0$ is often a trivial solution and a local minimum, and the theorem gives a second, nontrivial solution.
It is the starting point of minimax methods, such as the linking theorems and Lusternik–Schnirelmann theory.
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# Schrödinger equation

## Summary

The Schrödinger equation $i\hbar\, \partial_t \psi = -\frac{\hbar^2}{2m} \Delta \psi + V \psi$ describes how the wave function $\psi$ of a quantum particle of mass $m$ in a potential $V$ evolves in time.
It is a linear dispersive partial differential equation with complex-valued solutions.
Its right-hand side is a [[Schrödinger operator|89AA6XJY5K]] applied to $\psi$.
The solutions are given by a unitary group of operators, so the total probability $\int \abs{\psi}^2 \, dx$ is conserved.
It is the basic equation of non-relativistic quantum mechanics and a model problem for dispersive equations.

## Explanations

### L1 human

In quantum mechanics a particle is described by a complex wave, and $\abs{\psi(x, t)}^2$ is the probability density of finding it at $x$ at time $t$.
The Schrödinger equation says how this wave moves: the kinetic energy term spreads it out, and the potential bends it.
A free particle is a superposition of plane waves $e^{i(k \cdot x - \abs{k}^2 t)}$, in units where $\hbar = 1$ and $2m = 1$.
Waves with larger $k$ move faster, so a localised wave packet spreads out in time, which is the typical behaviour of dispersive equations.
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# Submartingale

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]], let $T \subseteq [0, \infty)$ be an index set, and let $(\mathcal{F}_t)_{t \in T}$ be a [[filtration|HNK1Y62G7V]].
A *submartingale* with respect to $(\mathcal{F}_t)$ is a family $(X_t)_{t \in T}$ of real random variables such that:

1. $(X_t)$ is [[adapted|YRNFE016FF]]: $X_t$ is $\mathcal{F}_t$-measurable for each $t \in T$.
2. $E[\abs{X_t}] < \infty$ for each $t \in T$.
3. $E[X_t \mid \mathcal{F}_s] \ge X_s$ almost surely for all $s \le t$ in $T$, where $E[\cdot \mid \mathcal{F}_s]$ is the [[conditional expectation|0YAMJRK04E]].

## Remarks

Let $S_n = \xi_1 + \dots + \xi_n$, where the $\xi_i$ are independent with $P(\xi_i = 1) = P(\xi_i = -1) = 1/2$, and let $\mathcal{F}_n = \sigma(\xi_1, \dots, \xi_n)$.
Then $X_n = S_n^2$ is a submartingale: it is adapted, $\abs{X_n} \le n^2$, and $E[S_{n+1}^2 \mid \mathcal{F}_n] = S_n^2 + 2 S_n E[\xi_{n+1}] + E[\xi_{n+1}^2] = S_n^2 + 1 \ge S_n^2$.
The process $-S_n^2$ is adapted and integrable but fails clause 3, since $E[-S_{n+1}^2 \mid \mathcal{F}_n] = -S_n^2 - 1$.
In discrete time $T = \{0, 1, 2, \dots\}$, clause 3 is equivalent to $E[X_{n+1} \mid \mathcal{F}_n] \ge X_n$ for each $n$, by the tower property.
A process is a [[martingale|0CDVK9RQBH]] if and only if it is both a submartingale and a [[supermartingale|KYW1PV0SX5]].
If $(M_t)$ is a martingale and $\varphi$ is convex with $E[\abs{\varphi(M_t)}] < \infty$ for all $t$, then $(\varphi(M_t))$ is a submartingale by the conditional form of [[Jensen's inequality|0RWMRTPC87]].
Taking expectations in clause 3 shows that $t \mapsto E[X_t]$ is non-decreasing.
</document>

<document source="https://tamnd.github.io/math2/node/EQ23YX9CFG.md" tag="EQ23YX9CFG" kind="definition">
# Superharmonic potential

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a connected [[open set|02BH3MCBQB]].
A *potential* on $\Omega$ is a function $u$ on $\Omega$ such that:

1. $u$ is [[superharmonic|X4V2S56YQQ]] on $\Omega$.
2. $u \ge 0$ on $\Omega$.
3. The greatest harmonic minorant of $u$ is $0$; equivalently, every [[harmonic function|0DEB0JMN0W]] $h$ on $\Omega$ with $h \le u$ satisfies $h \le 0$.

## Remarks

Since $u \ge 0$, the function $0$ is a harmonic minorant, so the greatest harmonic minorant exists; it is $-h_0$, where $h_0$ is the [[least harmonic majorant|G0H97JZRS9]] of $-u$.
Let $\Omega$ have a Green function $G$.
Then the [[Riesz decomposition theorem|CBZ20SS8RQ]] shows that the potentials are exactly the functions $x \mapsto \int G(x, y)\, d\mu(y)$.
Here $\mu$ runs over the positive measures for which this function is not identically $+\infty$.
Every nonnegative superharmonic function is uniquely the sum of a potential and a nonnegative harmonic function.

Example: for $n \ge 3$ let $u(x) = \abs{x}^{2-n}$ on $\Omega = \R^n$, with $u(0) = +\infty$.
It is superharmonic and positive.
Let $h$ be harmonic with $h \le u$, and fix $a$.
By the [[mean value property|0HYPR64JM2]], $h(a)$ is the mean of $h$ over the sphere $S(a, R)$, which is at most the mean of $u$ there, namely $\max(\abs{a}, R)^{2-n}$.
Letting $R \to \infty$ gives $h(a) \le 0$, so $u$ is a potential.

Non-example: on a bounded domain the constant $u = 1$ is superharmonic and nonnegative, but $h = 1$ is a harmonic minorant with $h > 0$, so clause 3 fails.
</document>

<document source="https://tamnd.github.io/math2/node/EQ4QB1QW60.md" tag="EQ4QB1QW60" kind="concept">
# Dawson function

## Summary

The Dawson function is $F(x) = e^{-x^2} \int_0^x e^{t^2}\, dt$.
It is a scaled form of the imaginary error function: $F(x) = \frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi} x$, where $\operatorname{erfi} x = -i \operatorname{erf}(ix)$ and $\operatorname{erf}$ is the [[error function|7Y56QCZSZJ]].
It is the imaginary part of the [[Faddeeva function|3GHGFTMFQR]] on the real line, up to the factor $\frac{2}{\sqrt{\pi}}$.
It appears in plasma physics, in spectroscopy and in the Hilbert transform of the Gaussian.

## Explanations

### L1 human

The integral of $e^{t^2}$ grows extremely fast, and the factor $e^{-x^2}$ tames this growth.
What is left is a small odd function: it rises from $0$, reaches a maximum of about $0.54$ near $x \approx 0.92$, and then decays like $\frac{1}{2x}$.
So the Dawson function is a bounded smooth bump on the positive side with its mirror image on the negative side.
</document>

<document source="https://tamnd.github.io/math2/node/EQ5JSYGWG2.md" tag="EQ5JSYGWG2" kind="example">
# Helicoid

## Example

Fix $c > 0$.
For $u \in \R$ and $v \in \R$, let
$$x(u, v) = (v \cos u, \; v \sin u, \; c u).$$
Its image, the helicoid, is swept out by a horizontal line that turns about the $z$-axis at a constant rate while it moves up.
It is a [[minimal surface|41X0YW2R9N]].

Check.
The partial derivatives are $x_u = (-v \sin u, v \cos u, c)$ and $x_v = (\cos u, \sin u, 0)$.
So $E = v^2 + c^2$, $F = 0$ and $G = 1$.
The cross product is $x_u \times x_v = (-c \sin u, c \cos u, -v)$, with length $W = \sqrt{v^2 + c^2}$, and $N = (x_u \times x_v)/W$.
The second derivatives are $x_{uu} = (-v \cos u, -v \sin u, 0)$, $x_{uv} = (-\sin u, \cos u, 0)$ and $x_{vv} = 0$.
So the coefficients of the [[second fundamental form|9KHQR16P1B]] are $e = 0$, $f = c/W$ and $g = 0$.
The [[mean curvature|PA78NFEPCF]] is $H = \dfrac{eG - 2fF + gE}{2(EG - F^2)} = 0$.

## Remarks

The Gaussian curvature is $K = -f^2/(EG) = -c^2/(v^2 + c^2)^2$.
The helicoid is a ruled surface, and together with the plane it is the only ruled minimal surface in $\R^3$.
It is locally isometric to the [[catenoid|6NN1HP54GB]].

## Explanations

### L1 human

Picture a spiral staircase whose steps are thin lines that extend to infinity on both sides of the central pole.
That surface is a helicoid.
The points at a fixed distance from the pole trace out helices, and a soap film can take this shape.
</document>

<document source="https://tamnd.github.io/math2/node/EQ6A3ZN48K.md" tag="EQ6A3ZN48K" kind="concept">
# Gauss sum

## Summary

A Gauss sum is a finite sum $\sum_a \chi(a)\psi(a)$ over a [[finite field|0ZXZ82FTX2]], where $\chi$ is a multiplicative character and $\psi$ an additive character.
It mixes the multiplicative and the additive structure of the field.
Its absolute value is the square root of the field size when both characters are nontrivial.
Gauss sums are a basic tool in proofs of reciprocity laws and in the study of [[character sums|VTYJ9419C7]] and [[Dirichlet L-functions|9D2Z543EJQ]].

## Explanations

### L1 human

Take an odd prime $p$ and the complex number $\zeta = e^{2\pi i/p}$, a [[root of unity|0YV2CKP2AV]].
Weight each power $\zeta^a$, for $a = 1, \dots, p - 1$, by $+1$ or $-1$, according to whether $a$ is a square modulo $p$ or not, and add.
For $p = 3$ this gives $\zeta - \zeta^2 = i\sqrt{3}$.
The surprise is that the sum always has absolute value exactly $\sqrt{p}$, much smaller than the $p - 1$ terms suggest.
This square root is where a lot of number theory hides.
</document>

<document source="https://tamnd.github.io/math2/node/EQ7R998DYJ.md" tag="EQ7R998DYJ" kind="definition">
# Congruence subgroup

## Definition

Let $N \ge 1$ be an integer.
The *principal congruence subgroup of level $N$* is the [[kernel|0W88CEGQP1]] of the reduction map $\mathrm{SL}_2(\Z) \to \mathrm{SL}_2(\Z / N\Z)$:
$$\Gamma(N) = \set{\gamma \in \mathrm{SL}_2(\Z) : \gamma \equiv I \pmod N}.$$
A *congruence subgroup* is a [[subgroup|0X1JXT47GA]] $\Gamma$ of the [[modular group|VEKV1EM77B]] $\mathrm{SL}_2(\Z)$ such that $\Gamma(N) \subseteq \Gamma$ for some $N \ge 1$.

## Remarks

Example: $\Gamma_0(N) = \set{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\Z) : c \equiv 0 \pmod N}$ is a congruence subgroup, since each matrix of $\Gamma(N)$ has $c \equiv 0 \pmod N$.
Another example is $\Gamma_1(N)$, defined by $c \equiv 0$ and $a \equiv d \equiv 1 \pmod N$.
Since $\mathrm{SL}_2(\Z / N\Z)$ is finite, $\Gamma(N)$ has finite [[index|0DTXBQPA2S]], and so does every congruence subgroup.
Non-example: the subgroup $\set{\begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix} : n \in \Z}$ has infinite index, so it contains no $\Gamma(N)$.
For a prime $p$, the index of $\Gamma_0(p)$ in $\mathrm{SL}_2(\Z)$ is $p + 1$.
Not every subgroup of finite index in $\mathrm{SL}_2(\Z)$ is a congruence subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/EQB39X4HGB.md" tag="EQB39X4HGB" kind="conjecture">
# Slice-ribbon conjecture

## Statement

A tame [[knot|QYEAZTYGPT]] $K \subset S^3$ is a *ribbon knot* if it bounds an immersed disk $D$ in $S^3$ whose only self-intersections are ribbon singularities.
A ribbon singularity is an arc of double points with two preimages in $D$.
One preimage is an arc interior to $D$, and the other is an arc meeting $\partial D$ in its two endpoints.
The conjecture states that every smoothly [[slice knot|P68G28E4AD]] is a ribbon knot.

## Remarks

The converse holds: pushing the interior of a ribbon disk into $B^4$ removes the singularities, so every ribbon knot is slice.
Equivalently, $K$ is ribbon if and only if it bounds a smoothly embedded disk in $B^4$ on which the radial distance function has no local maxima.
Lisca proved the conjecture for two-bridge knots.
The conjecture is open in general.

## Explanations

### L1 human

A ribbon knot bounds a disk in ordinary space that passes through itself only in a mild way.
The disk looks like a ribbon threaded through a slit in itself.
Such a disk can be pushed into four dimensions so that it no longer meets itself, so ribbon knots are slice.
The conjecture says that this is the only way a knot can be slice.
It would turn a four-dimensional question into a three-dimensional one.
</document>

<document source="https://tamnd.github.io/math2/node/EQBC3P901V.md" tag="EQBC3P901V" kind="statement">
# Heisenberg inequality for the Fourier transform

## Statement

Let $f \in L^2(\R)$, and let $\hat f$ be its Fourier transform on $L^2(\R)$, normalized so that $\hat f(\xi) = \int_\R f(x) e^{-2\pi i x \xi}\, dx$ for integrable $f$.
Then for all real numbers $a$ and $b$,
$$\left( \int_\R (x - a)^2 \abs{f(x)}^2\, dx \right) \left( \int_\R (\xi - b)^2 \abs{\hat f(\xi)}^2\, d\xi \right) \ge \frac{\norm{f}_2^4}{16 \pi^2}.$$
Equality holds if and only if $f(x) = C e^{2\pi i b x} e^{-c (x - a)^2}$ for some $C \in \C$ and $c > 0$.

## Remarks

For a Schwartz function $f$ and $a = b = 0$, integrate by parts: $\int \abs{f}^2\, dx = -\int x\, \frac{d}{dx}\abs{f(x)}^2\, dx = -2 \operatorname{Re} \int x f(x) \overline{f'(x)}\, dx$.
The [[Cauchy–Schwarz inequality|0RFWW56Y22]] bounds this by $2 \norm{x f}_2 \norm{f'}_2$.
By the [[differentiation rule|HHNR31W3WD]] and the [[Plancherel theorem|HPZHRZ0502]], $\norm{f'}_2 = 2\pi \norm{\xi \hat f}_2$.
Squaring gives the inequality; translation in $x$ and modulation by $e^{2\pi i b x}$ give general $a$ and $b$, and a density argument gives all $f$ in [[$L^2(\R)$|SASP14MA1Z]].
Equality in Cauchy–Schwarz forces $f' = -2c x f$, whose solutions are Gaussians.

If one of the two integrals is infinite and $f \ne 0$, then the other is positive, so the left side is $+\infty$ and the inequality holds.

Example: for $f(x) = e^{-\pi x^2}$, one has $\hat f = f$, $\norm{f}_2^2 = 2^{-1/2}$ and $\int x^2 e^{-2\pi x^2}\, dx = \frac{1}{4\pi} 2^{-1/2}$, so the left side equals $\frac{1}{16\pi^2} \cdot \frac12 = \frac{\norm{f}_2^4}{16\pi^2}$.

## Explanations

### L1 human

If $\abs{f}^2$ is a probability density, its spread around a point $a$ and the spread of $\abs{\hat f}^2$ around a point $b$ cannot both be small.
Their product is at least $1/(16\pi^2)$.
In quantum mechanics, position and momentum are related by this Fourier transform, and the inequality becomes Heisenberg's uncertainty principle.

### L2 human

The proof is a commutator argument: the operators of multiplication by $x$ and of differentiation do not commute, and Cauchy–Schwarz turns this into a lower bound.
The same scheme gives the Robertson uncertainty relation for any two self-adjoint operators.
The inequality also follows from the entropic uncertainty principle, since among densities with a given variance the Gaussian has the largest entropy.
In $\R^n$ the analogous bound is $\norm{\abs{x} f}_2 \norm{\abs{\xi} \hat f}_2 \ge \frac{n}{4\pi} \norm{f}_2^2$.
</document>

<document source="https://tamnd.github.io/math2/node/EQDDVAKSVE.md" tag="EQDDVAKSVE" kind="definition">
# Manifold with corners

## Definition

Let $n \ge 1$ be an integer and let $\overline{\R}^n_+ = \set{x \in \R^n : x_1 \ge 0, \ldots, x_n \ge 0}$ be the closed orthant, with the subspace topology.
Let $A \subseteq \R^n$ be any subset.
A map $f \colon A \to \R^n$ is *smooth* if each $a \in A$ has an open neighbourhood $W$ in $\R^n$ with a [[smooth|0V33BNKZ9S]] map $g \colon W \to \R^n$ equal to $f$ on $A \cap W$.
Let $M$ be a [[topological space|0CQHFPGJ45]].
A *chart with corners* on $M$ is a [[homeomorphism|0QT00PB5G2]] $\varphi$ from an open subset $U \subseteq M$ onto an open subset of $\overline{\R}^n_+$.
A *smooth atlas with corners* is a family of charts with corners whose domains cover $M$.
Its transition maps $\varphi_j \circ \varphi_i^{-1}$ must be smooth in the above sense.
A *smooth manifold with corners of dimension $n$* is a space $M$ such that:

1. $M$ is [[Hausdorff|0S1FJ8VEX9]].
2. $M$ is [[second-countable|03PPNCM9GG]].
3. $M$ carries a maximal smooth atlas with corners of dimension $n$.

## Remarks

The square $[0, 1]^2$ is a manifold with corners.
Near the vertex $(0, 0)$ the identity is a chart onto an open subset of $\overline{\R}^2_+$.
Near $(1, 1)$ the chart is $(x, y) \mapsto (1 - x, 1 - y)$, and near the other points one uses similar affine maps.
All transition maps are affine, hence smooth, and the charts generate a maximal atlas.

The solid square pyramid in $\R^3$ is not a manifold with corners for this structure, because clause 3 fails at its apex.
The tangent cone of the pyramid at the apex has four faces.
The derivative of a chart at the apex would map it linearly onto the cone $\overline{\R}^3_+$, which has only three faces.

A chart with corners is in particular a chart in the sense of an [[atlas|J7P32JQTVE]] modelled on $\overline{\R}^n_+$.
As a topological space, every manifold with corners is a [[manifold with boundary|7DS6B2BFK8]], because $\overline{\R}^n_+$ is homeomorphic to the half-space $\set{x_n \ge 0}$.
In Mathlib, manifolds with corners are modelled through a `ModelWithCorners`.
</document>

<document source="https://tamnd.github.io/math2/node/EQDEXBWGJH.md" tag="EQDEXBWGJH" kind="concept">
# Abelian extension

## Summary

An abelian extension is a [[Galois extension|0DS2F7GK05]] whose [[Galois group|0XVR72M51M]] is abelian.
By the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], every intermediate field of a finite abelian extension is again Galois over the base.
[[Cyclic extensions|VFE5B71E7Y]] and [[cyclotomic extensions|VD6VC3ZMMY]] of $\Q$ are abelian.
Class field theory describes the abelian extensions of number fields and local fields.

## Explanations

### L1 human

An extension is abelian when its symmetries commute with each other.
Then the order in which two symmetries are applied does not matter.
For example, $\Q(\sqrt{2}, \sqrt{3})$ over $\Q$ has four symmetries, each changing the signs of $\sqrt{2}$ and of $\sqrt{3}$ independently.
Changing the two signs in either order gives the same result, so the extension is abelian.
</document>

<document source="https://tamnd.github.io/math2/node/EQG2K8ZFM4.md" tag="EQG2K8ZFM4" kind="statement">
# Choi's theorem on completely positive maps

## Statement

Let $n, m \ge 1$, and let $\Phi \colon M_n(\C) \to M_m(\C)$ be a linear map between [[matrix rings|0FHVPRAE60]].
Let $E_{ij} \in M_n(\C)$ be the matrix units, and let
$$C_\Phi = \sum_{i,j=1}^{n} E_{ij} \otimes \Phi(E_{ij}) \in M_n(\C) \otimes M_m(\C) \cong M_{nm}(\C)$$
be the Choi matrix of $\Phi$, where $\otimes$ is the [[Kronecker product|0N7ENQRCXN]].
The following are equivalent:

1. $\Phi$ is [[completely positive|CGY299QMC3]];
2. $\Phi$ is $n$-positive;
3. $C_\Phi$ is [[positive semidefinite|01MMVW8CKC]];
4. there are $m \times n$ complex matrices $V_1, \ldots, V_r$, with $r \le nm$, such that $\Phi(a) = \sum_{k=1}^{r} V_k a V_k^*$ for all $a \in M_n(\C)$.

## Remarks

Proof idea: $1 \Rightarrow 2$ is clear, and $2 \Rightarrow 3$ holds because $C_\Phi = \Phi_n(x)$ for the positive block matrix $x = [E_{ij}]_{i,j}$.
For $3 \Rightarrow 4$, write $C_\Phi = \sum_k v_k v_k^*$ with at most $nm$ vectors $v_k \in \C^{nm}$, using the spectral decomposition, and reshape each $v_k$ into an $m \times n$ matrix $V_k$.
For $4 \Rightarrow 1$, each map $a \mapsto V a V^*$ is completely positive, and sums of completely positive maps are completely positive.

Example: for the transpose $\Phi(a) = a^T$ on $M_2(\C)$, the Choi matrix is the swap matrix on $\C^2 \otimes \C^2$, which has eigenvalue $-1$.
So the transpose is not completely positive, although it is positive.
The matrices $V_k$ in clause 4 are called Kraus operators.

## Explanations

### L1 human

Complete positivity is a condition about infinitely many matrix sizes, so it looks hard to check.
Choi's theorem reduces it to a single test: build one $nm \times nm$ matrix from $\Phi$ and check that it has no negative eigenvalue.
It also gives a concrete formula for every completely positive map between matrix algebras.

### L2 human

The theorem is the finite-dimensional form of the [[Stinespring dilation theorem|X8K0Z7484R]].
In quantum information, it gives the Kraus representation of quantum channels; the channel is trace-preserving exactly when $\sum_k V_k^* V_k = I$.
The correspondence $\Phi \mapsto C_\Phi$ between maps and matrices is the Choi–Jamiołkowski isomorphism, and condition 2 shows that $n$-positivity already implies complete positivity for maps defined on $M_n(\C)$.
</document>

<document source="https://tamnd.github.io/math2/node/EQHPSKC5SY.md" tag="EQHPSKC5SY" kind="statement">
# Cartan's uniqueness theorem

## Statement

Let $D \subseteq \C^n$ be a [[bounded|09QSZVVM26]] connected open set and let $p \in D$.
Let $f \colon D \to D$ be a [[holomorphic map|3MQTWKBBBT]] with $f(p) = p$.
Suppose that the complex [[Jacobian matrix|047NDTEPCV]] of $f$ at $p$ is the identity matrix.
Then $f(z) = z$ for all $z \in D$.

## Remarks

The proof uses iteration.
Take $p = 0$ and suppose that $f(z) = z + P_k(z) + \cdots$, where $P_k$ is the first nonzero homogeneous term of degree $k \ge 2$ in the Taylor expansion of $f - \id$.
The $m$-th iterate of $f$ then has the expansion $z + m P_k(z) + \cdots$.
All iterates map $D$ into the bounded set $D$, so the Cauchy estimates bound their Taylor coefficients independently of $m$, which forces $P_k = 0$.

A consequence: if $g, h$ are [[biholomorphisms|FX92PP3HZR]] of $D$ onto a domain $D'$ with $g(p) = h(p)$ and the same Jacobian matrix at $p$, then $g = h$.
So an automorphism of a bounded domain is determined by its value and its derivative at one point.

Boundedness is necessary.
On $\C^2$ the map $f(z_1, z_2) = (z_1, z_2 + z_1^2)$ fixes $0$ and has the identity as Jacobian matrix at $0$, but it is not the identity map.

## Explanations

### L1 human

A holomorphic self-map of a bounded domain that fixes a point and looks like the identity to first order at that point must be the identity everywhere.
It is a rigidity result: first-order information pins down the whole map.
In one variable it is a cousin of the [[Schwarz lemma|0MB48XDBY3]].

### L2 human

The theorem is the basic tool for studying the automorphism group of a bounded domain.
It shows that the map sending an automorphism fixing $p$ to its Jacobian matrix at $p$ is injective.
With it one proves that automorphisms of a bounded circular domain fixing the origin are linear maps.
It is also a step toward showing that the automorphism group of a bounded domain is a Lie group.
</document>

<document source="https://tamnd.github.io/math2/node/EQSFAA5D1M.md" tag="EQSFAA5D1M" kind="definition">
# Integral functional with a Lagrangian density

## Definition

Let $n, m \ge 1$, let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]], and let $\overline{\Omega}$ be its closure.
Let $L \colon \overline{\Omega} \times \R^m \times \R^{m \times n} \to \R$, written $L(x, z, p)$, be a continuous function, called the *Lagrangian density*.
For a [[continuously differentiable|0VW90BBSJ4]] map $u \in C^1(\overline{\Omega}; \R^m)$ with Jacobian matrix $Du$, set
$$J(u) = \int_\Omega L\big(x, u(x), Du(x)\big) \, dx.$$
The [[functional|1EQM0DJ4NC]] $J \colon C^1(\overline{\Omega}; \R^m) \to \R$ is the *integral functional* with Lagrangian density $L$.
The integrand is continuous and bounded on $\Omega$, so the [[Lebesgue integral|DWJF7ABPR1]] is finite.

## Remarks

In one variable, with $n = m = 1$ and $\Omega = (a, b)$, the functional reads $J(y) = \int_a^b L(x, y(x), y'(x)) \, dx$.
Example: $L(x, z, p) = \frac{1}{2} \abs{p}^2$ with $m = 1$ gives the Dirichlet energy $J(u) = \frac{1}{2} \int_\Omega \abs{\nabla u}^2 \, dx$.
It is an integral functional because this $L$ is continuous on all of $\overline{\Omega} \times \R \times \R^n$.
Non-example: on $\Omega = (0, 1)$, the expression $L(x, z, p) = 1/x$ is not defined at $x = 0$ and has no continuous extension to $[0, 1] \times \R \times \R$, and indeed $\int_0^1 dx / x = \infty$ for every $y$.
The same name is used for Lagrangians with higher derivatives, as in the [[Euler–Poisson equation|XBNAJ1RRET]], and for $u$ in [[Sobolev spaces|M4CEZ5K9HP]] under growth conditions on $L$.
</document>

<document source="https://tamnd.github.io/math2/node/EQSHJPE7H8.md" tag="EQSHJPE7H8" kind="concept">
# Quasi-projective variety

## Summary

A quasi-projective variety is an open subset of a projective variety, with the induced Zariski topology.
The class contains both the affine varieties and the projective varieties, and it is closed under taking open and closed subvarieties.
In classical algebraic geometry, the quasi-projective varieties are exactly the varieties one works with.

## Explanations

### L1 human

Start with a projective variety and remove a closed piece of it, defined by some equations.
What is left is a quasi-projective variety.
Removing the line at infinity from the projective plane leaves the affine plane, so affine varieties are quasi-projective.
Removing a single point from the affine plane leaves a quasi-projective variety that is neither affine nor projective.
</document>

<document source="https://tamnd.github.io/math2/node/EQVVDS0ZQT.md" tag="EQVVDS0ZQT" kind="concept">
# Mapping cylinder

## Summary

The mapping cylinder of a continuous map $f \colon X \to Y$ is the space obtained from the cylinder $X \times [0, 1]$ by gluing its end $X \times \set{1}$ to $Y$ along $f$.
It contains $Y$ as a [[deformation retract|CEK4SC20WK]] and contains $X$ as the other end.
So every map is, up to [[homotopy equivalence|XZG6MTSWCN]], the inclusion of a subspace.
The construction is used to build mapping cones, cofibrations and long exact sequences.

## Explanations

### L1 human

Take a tube shaped like $X$, and glue one end of the tube onto $Y$ by following $f$.
The other end of the tube is a copy of $X$ that sits freely in the new space.
Sliding along the tube pushes everything onto $Y$, so the new space has the same shape as $Y$.
For example, if $Y$ is a single point, the tube closes up into a cone on $X$.
</document>

<document source="https://tamnd.github.io/math2/node/EQYDBQHY5B.md" tag="EQYDBQHY5B" kind="definition">
# Morphism of schemes

## Definition

Let $X$ and $Y$ be [[schemes|5A35TYZD02]].
A *morphism of schemes* $X \to Y$ is a morphism of [[locally ringed spaces|B7FPT8JTXR]], that is, a pair $(f, f^\#)$ such that:

1. $f \colon X \to Y$ is a [[continuous map|0VGFHTM75Y]].
2. $f^\# \colon \mathcal{O}_Y \to f_* \mathcal{O}_X$ is a morphism of sheaves of rings, so for each open $V \subseteq Y$ it gives a ring map $\mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}(V))$, compatible with restrictions.
3. For every $x \in X$, the induced map of stalks $f^\#_x \colon \mathcal{O}_{Y,f(x)} \to \mathcal{O}_{X,x}$ is a [[local homomorphism|P8YTMGEN9R]].

The composite of $(f, f^\#) \colon X \to Y$ and $(g, g^\#) \colon Y \to Z$ is $(g \circ f, g_*(f^\#) \circ g^\#)$.

## Remarks

Example: a ring homomorphism $\varphi \colon A \to B$ gives a morphism $\operatorname{Spec} B \to \operatorname{Spec} A$ with $f(\mathfrak{q}) = \varphi^{-1}(\mathfrak{q})$.
On stalks it is the map $A_{\varphi^{-1}(\mathfrak q)} \to B_{\mathfrak q}$, which sends the maximal ideal into $\mathfrak{q} B_{\mathfrak q}$, so it is local.
Non-example: let $R$ be a discrete valuation ring with fraction field $K$.
Let $f$ send the single point of $\operatorname{Spec} K$ to the closed point of $\operatorname{Spec} R$, and let $f^\#$ come from the inclusion $R \to K$.
This is a morphism of ringed spaces, but it fails clause 3.
The stalk map $R \to K$ is not local, since it sends the maximal ideal of $R$ into the units of $K$.
</document>

<document source="https://tamnd.github.io/math2/node/ERFZ6S8X0X.md" tag="ERFZ6S8X0X" kind="concept">
# Associated prime

## Summary

An associated prime of a module $M$ over a commutative ring is a prime ideal that is the [[annihilator|HB4XDKJTW8]] of a single element of $M$.
For a finitely generated module over a [[Noetherian ring|0670T59HGE]], there are only finitely many, and their union is the set of zero divisors on $M$.
When $R$ is Noetherian, the associated primes of $R/I$ are the primes that appear in a [[primary decomposition|ZD34THCXQ0]] of the ideal $I$.

## Explanations

### L1 human

Take one element $m$ of a module and collect all ring elements that kill it.
Sometimes this set is a prime ideal; such primes are the associated primes.
For $\Z/12\Z$, the element $6$ is killed exactly by the even numbers, and the element $4$ exactly by the multiples of $3$.
So $(2)$ and $(3)$ are the associated primes of $\Z/12\Z$, the primes dividing $12$.
</document>

<document source="https://tamnd.github.io/math2/node/ERHGXV009H.md" tag="ERHGXV009H" kind="definition">
# Total variation

## Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function.
The *total variation* of $f$ on $[a, b]$ is
$$V_a^b(f) = \sup_P \sum_{i=1}^{n} \abs{f(x_i) - f(x_{i-1})},$$
where the [[supremum|0BTMQ04556]] runs over all [[partitions|069P20KYQT]] $P$ of $[a, b]$ given by $a = x_0 < x_1 < \cdots < x_n = b$.
The value $V_a^b(f)$ lies in $[0, \infty]$.

## Remarks

Example: for $f(x) = \sin x$ on $[0, 2\pi]$, the total variation is $4$.
The function is monotone on $[0, \pi/2]$, $[\pi/2, 3\pi/2]$ and $[3\pi/2, 2\pi]$, and on each of these pieces the variation is the absolute change $1$, $2$ and $1$.
Total variation is additive over adjacent intervals: $V_a^c(f) = V_a^b(f) + V_b^c(f)$ for $a < b < c$.

Non-example of finite total variation: the indicator function of the rationals on $[0, 1]$.
A partition alternating between rational and irrational points with $n$ subintervals gives the sum $n$, so the total variation is $\infty$.

If $f$ is continuously differentiable on $[a, b]$, then $V_a^b(f) = \int_a^b \abs{f'(x)}\, dx$.
</document>

<document source="https://tamnd.github.io/math2/node/ERP5737VZ1.md" tag="ERP5737VZ1" kind="concept">
# Convolution on the line

## Summary

The convolution of two functions $f$ and $g$ on the real line is the function $(f * g)(x) = \int_\R f(y) g(x - y) \, dy$.
It is the case $G = \R$ of the general [[convolution|2Z29JN3VES]] on a group.
It models the output of a linear time-invariant system and the density of a sum of independent random variables.
It also describes the smoothing of a function by a kernel.
The [[Fourier transform|DPVM34ADZV]] turns it into a pointwise product.

## Explanations

### L1 human

To compute $(f * g)(x)$, flip $g$, slide it so that it sits at $x$, multiply by $f$, and take the area under the product.
Doing this for every $x$ gives a new function, usually smoother than either of the two.
For example, the indicator function of $[0, 1]$ convolved with itself is a tent of height $1$ over $[0, 2]$.
Convolving a signal with a narrow bump of area $1$ replaces each value by a local weighted average.
</document>

<document source="https://tamnd.github.io/math2/node/ERSD8D4HWX.md" tag="ERSD8D4HWX" kind="statement">
# Morse lemma

## Statement

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $f \colon M \to \R$ be smooth.
Let $p$ be a nondegenerate [[critical point|0YTCMFB4HC]] of $f$ of index $\lambda$, in the sense of [[Morse functions|KPQ96640ZJ]].
Then there is a chart $(U, (x_1, \dots, x_n))$ of $M$ with $p \in U$ and $x_i(p) = 0$ for all $i$, such that on $U$
$$f = f(p) - x_1^2 - \cdots - x_\lambda^2 + x_{\lambda+1}^2 + \cdots + x_n^2.$$

## Remarks

The idea of the proof is this.
Work in a chart centred at $p$ with $f(p) = 0$; by Taylor's formula with integral remainder, $f(x) = \sum_{i,j} x_i x_j h_{ij}(x)$ with $h_{ij}$ smooth, symmetric in $i, j$, and $h_{ij}(0) = \frac{1}{2} \partial^2 f / \partial x_i \partial x_j(0)$.
Since the matrix $(h_{ij}(0))$ is invertible, one can complete squares one variable at a time, as in the diagonalisation of a quadratic form, with smooth coefficients near $0$.
Sylvester's law of inertia shows that the number of minus signs is the index of the [[Hessian|0693Y2METT]].
Example: $f(x, y) = x^2 - y^2 + y^3$ has a nondegenerate critical point of index $1$ at the origin, and near it the change $u = x$, $v = y \sqrt{1 - y}$ gives $f = u^2 - v^2$.
Nondegeneracy is needed: $f(x) = x^3$ cannot be written as $\pm x_1^2$ in any chart near $0$, since it changes sign at $0$.

## Explanations

### L1 human

Near a clean peak, pit or saddle, a function looks exactly like a sum of squares with some minus signs, once one chooses the right curved coordinates.
The number of minus signs counts the independent downhill directions.
So the local picture at a nondegenerate critical point is completely determined by one integer.

### L2 human

The Morse lemma is the local input of Morse theory.
It shows that nondegenerate critical points are isolated, and it describes how the sublevel set $\set{f \le c}$ changes when $c$ passes a critical value: a handle of index $\lambda$ is attached.
This leads to handle decompositions, to the Morse inequalities, and to Morse homology.
It is the simplest case of the classification of singularities of smooth functions.
</document>

<document source="https://tamnd.github.io/math2/node/ERTAM0SFVB.md" tag="ERTAM0SFVB" kind="concept">
# Normal function

## Summary

A normal function is a function from [[ordinals|0YSCSX5YZC]] to ordinals that is strictly increasing and continuous at limit ordinals.
The [[precise definition|PXTF18X433]] asks that the value at a limit be the supremum of the earlier values.
Each normal function has arbitrarily large fixed points, and this fixed-point lemma is the engine behind the [[Veblen function|JQFAE9S2F4]] and many ordinal notations.

## Explanations

### L1 human

A normal function climbs through the ordinals without jumps at limit points.
At a limit stage, its value is exactly the place that the earlier values were approaching.
For example, $\alpha \mapsto \omega + \alpha$ is normal: at $\omega$ its value is $\omega + \omega$, the limit of $\omega + 1, \omega + 2, \ldots$.
The function $\alpha \mapsto \alpha + 1$ is not normal, because at $\omega$ it jumps to $\omega + 1$ while the earlier values approach only $\omega$.
Because a normal function never skips a limit, it must eventually catch its own argument, and it has fixed points.
</document>

<document source="https://tamnd.github.io/math2/node/ERTRZPKA1Q.md" tag="ERTRZPKA1Q" kind="definition">
# Lattice homomorphism

## Definition

Let $(L, \vee, \wedge)$ and $(M, \vee, \wedge)$ be [[lattices|WH4SFAACAS]].
A *lattice homomorphism* is a map $f \colon L \to M$ such that, for all $x, y \in L$:

1. $f(x \vee y) = f(x) \vee f(y)$;
2. $f(x \wedge y) = f(x) \wedge f(y)$.

If $L$ and $M$ are bounded lattices, a *bounded lattice homomorphism* also satisfies $f(0) = 0$ and $f(1) = 1$.

## Remarks

A lattice homomorphism is [[monotone|0A7KV2WB7A]]: if $x \le y$ then $x \vee y = y$, so $f(x) \vee f(y) = f(y)$, that is, $f(x) \le f(y)$.
A bijective lattice homomorphism is a lattice isomorphism, and its inverse is again a lattice homomorphism.
A bijection between lattices is a lattice isomorphism exactly when it and its inverse are monotone.
Example: every monotone map between two chains is a lattice homomorphism, since join and meet in a chain are $\max$ and $\min$, and monotone maps preserve both.
Non-example: let $N = \set{\emptyset, \set{1}, \set{2}, \set{1, 2, 3}}$, a lattice under $\subseteq$ in which $\set{1} \vee \set{2} = \set{1, 2, 3}$.
The inclusion map of $N$ into the power set of $\set{1, 2, 3}$ is monotone and preserves meets.
It fails clause 1, because the join of $\set{1}$ and $\set{2}$ in the power set is $\set{1, 2} \ne \set{1, 2, 3}$.
</document>

<document source="https://tamnd.github.io/math2/node/ERVV6RWNNX.md" tag="ERVV6RWNNX" kind="definition">
# Model complete theory

## Definition

Let $T$ be a [[theory|92SWXST1ZM]] in a first-order signature $L$.
The theory $T$ is *model complete* if the following holds for all [[models|MVCH6N9SNW]] $M$ and $N$ of $T$.
If $M$ is a [[substructure|676H7EGTKJ]] of $N$, then $M$ is an [[elementary substructure|8WZF9SBP39]] of $N$.

## Remarks

Equivalently, every embedding between models of $T$ is elementary.
By Robinson's test, $T$ is model complete if and only if every $L$-formula is equivalent modulo $T$ to an existential formula $\exists \bar{y} \, \chi(\bar{x}, \bar{y})$ with $\chi$ quantifier-free.
A theory with [[quantifier elimination|7FDDXV3JK7]] is model complete.

Example: the theory of dense linear orders without endpoints is model complete, since it has quantifier elimination.
Let $M \subseteq N$ be two such orders and let $\bar{a}$ be a tuple of $M$.
A formula $\varphi(\bar{a})$ is equivalent to a quantifier-free formula.
That formula only compares elements of $\bar{a}$, so it has the same truth value in $M$ and in $N$.

Non-example: the theory of dense linear orders with a least and a greatest element is not model complete.
The orders $[0, 1] \cap \Q$ and $[0, 2] \cap \Q$ are models, and the first is a substructure of the second.
The element $1$ satisfies $\forall y \, (y \le x)$ in $[0, 1] \cap \Q$ but not in $[0, 2] \cap \Q$.

The theory of real closed fields in the signature $\set{+, -, \cdot, 0, 1}$ is model complete but does not have quantifier elimination.
</document>

<document source="https://tamnd.github.io/math2/node/ERXF29KNPC.md" tag="ERXF29KNPC" kind="definition">
# Hodges–Lehmann estimator

## Definition

Let $x_1, \dots, x_n$ be real numbers.
The *Walsh averages* are the $n(n+1)/2$ numbers $(x_i + x_j)/2$ for $1 \le i \le j \le n$.
The *one-sample Hodges–Lehmann estimator* is the [[median|08W2CW8PKW]] of the Walsh averages:
$$\hat\theta = \operatorname{med}\Big\{ \frac{x_i + x_j}{2} : 1 \le i \le j \le n \Big\}.$$
For two samples $x_1, \dots, x_m$ and $y_1, \dots, y_n$, the *two-sample Hodges–Lehmann estimator* of the shift is $\hat\Delta = \operatorname{med}\set{y_j - x_i : 1 \le i \le m, \ 1 \le j \le n}$.

## Remarks

It is an [[estimator|0TTNRP5PFQ]] of the centre of symmetry $\theta$ in the model of the [[Wilcoxon signed-rank test|38SETDT2HN]].
The estimate $\hat\theta$ is the value of $t$ for which the signed-rank statistic of $x_1 - t, \dots, x_n - t$ is closest to its null mean $n(n+1)/4$.
Its asymptotic breakdown point is $1 - 1/\sqrt{2} \approx 0.29$.

Example: data $(1, 2, 9)$.
The Walsh averages are $1, 2, 9$ (for $i = j$) and $1.5, 5, 5.5$ (for $i < j$); sorted, $1, 1.5, 2, 5, 5.5, 9$.
Their median is $(2 + 5)/2 = 3.5$.

Non-example: the median of the averages $(x_i + x_j)/2$ over $i < j$ only, without the terms $i = j$, is a different estimator; for $(1, 2, 9)$ it gives $5$.
Both versions appear in the literature, so the convention should be stated.
</document>

<document source="https://tamnd.github.io/math2/node/ES08X651A3.md" tag="ES08X651A3" kind="definition">
# Irreducible analytic germ

## Definition

Let $a \in \C^n$ and let $A_a$ be a [[germ of an analytic set|A2WHCVP6JN]] at $a$.
The germ $A_a$ is *irreducible* if:

1. $A_a$ is not the empty germ, and
2. whenever $A_a = B_a \cup C_a$ for germs of analytic sets $B_a, C_a$ at $a$, one has $A_a = B_a$ or $A_a = C_a$.

## Remarks

Equivalently, the ideal $\mathcal{I}(A_a)$ of [[holomorphic germs|2TGPZ8VW2D]] vanishing on $A_a$ is a [[prime ideal|00MHHZ9SNE]] of $\mathcal{O}_{n,a}$.
Example: the germ at $0$ of the cusp $A = \set{z_1^2 = z_2^3}$ in $\C^2$ is irreducible.
The germ $f = z_1^2 - z_2^3$ is irreducible in $\mathcal{O}_{2,0}$, since it is a Weierstrass polynomial of degree $2$ in $z_1$.
A factorization would give a germ $h$ with $h(z_2)^2 = z_2^3$, which is impossible by comparing orders of vanishing.
Since $\mathcal{O}_{2,0}$ is a unique factorization domain, the ideal $(f)$ is prime, hence radical, and the [[Rückert Nullstellensatz|3VRZ80X45P]] gives $\mathcal{I}(A_0) = (f)$, a prime ideal.
Non-example: the germ at $0$ of $\set{z_2^2 = z_1^2(1 + z_1)}$ is reducible.
With $s = \sqrt{1 + z_1}$, the convergent binomial series, it is the union of the germs of the two smooth curves $\set{z_2 = z_1 s}$ and $\set{z_2 = -z_1 s}$, and neither contains the other.
Every germ of an analytic set has a decomposition $A_a = A_{1,a} \cup \cdots \cup A_{k,a}$ into irreducible germs, unique up to order when no $A_{i,a}$ is contained in another.
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<document source="https://tamnd.github.io/math2/node/ES34ZH23RS.md" tag="ES34ZH23RS" kind="concept">
# Order theory

## Summary

Order theory is the branch of mathematics that studies orders, that is, binary relations that compare elements, such as [[partial orders|011C29Q1S1]], [[total orders|0316AYH1SC]] and [[preorders|0RXS8R6K37]].
Its basic objects are partially ordered sets and the [[monotone functions|0WFF6AZPBP]] between them.
It includes the theory of [[lattices|WH4SFAACAS]], [[Galois connections|MBVJ4JQXC5]] and [[closure operators|17FVAN6HXX]].
Its ideas appear in algebra, topology, logic and computer science, wherever objects can be compared by size, inclusion or strength.

## Explanations

### L1 human

Many collections in mathematics come with a natural way to say that one element is "below" another.
Numbers are compared by size, sets by inclusion, and whole numbers by divisibility.
Order theory studies what all these comparisons have in common.
A picture of a finite order is a [[Hasse diagram|00EYM7MDKS]]: dots for the elements and upward lines for the steps between them.
For example, the divisors $1, 2, 3, 6$ of $6$ form a diamond: $1$ at the bottom, $2$ and $3$ in the middle, $6$ at the top.
</document>

<document source="https://tamnd.github.io/math2/node/ES4MC30ZES.md" tag="ES4MC30ZES" kind="concept">
# Missing data mechanisms

## Summary

A missing data mechanism is the probability law that decides which entries of a data set are not observed.
The standard classification has three classes: missing completely at random (MCAR), missing at random (MAR) and missing not at random (MNAR).
The class decides which analyses stay valid.
Under MAR, if the mechanism has its own separate parameters, likelihood and Bayesian inference can ignore it.
Under MNAR the mechanism must be modelled.

## Explanations

### L1 human

Data are often incomplete: people skip survey questions, sensors fail, patients leave a study.
The key question is whether the fact that a value is missing is linked to the values themselves.
If a lab machine breaks at random times, the missing values are MCAR, and the rest of the data is still a fair sample.
If older people skip an income question more often and age is recorded, the data are MAR.
If people with high incomes skip the income question because their income is high, the data are MNAR, and a naive average of the answers is too low.
</document>

<document source="https://tamnd.github.io/math2/node/ES6P7WHJG9.md" tag="ES6P7WHJG9" kind="example">
# Trivial variety

## Example

Fix a signature $\sigma$.
The *trivial variety* of signature $\sigma$ is the class of all algebras of signature $\sigma$ that satisfy the [[identity|PKD440G2XW]] $x = y$.
An algebra satisfies $x = y$ exactly when any two of its elements are equal, so the trivial variety consists of the one-element algebras.
It is a [[variety|AJNZJD8442]] because it is defined by an identity.
The check by closure is also direct.
A subalgebra or a homomorphic image of a one-element algebra has one element.
A direct product of one-element algebras has one element, and the empty product is a one-element algebra too.

## Remarks

Every variety of signature $\sigma$ contains all one-element algebras of signature $\sigma$, since a one-element algebra satisfies every identity.
So the trivial variety is the least element of the [[lattice of subvarieties|GJAYCVYZYF]] of any variety.
Its equational theory is the largest one: it contains every identity of signature $\sigma$.

## Explanations

### L1 human

The trivial variety is the smallest possible variety: it only has algebras with one element.
In such an algebra, every operation has only one possible value, so there is nothing to compute.
Every other variety contains it.
</document>

<document source="https://tamnd.github.io/math2/node/ES9XR9CQ3D.md" tag="ES9XR9CQ3D" kind="concept">
# Relatively complemented lattice

## Summary

A relatively complemented lattice is a [[lattice|WH4SFAACAS]] in which every interval $[c, d]$ is a [[complemented lattice|ZT47BK9Y6K]].
So inside every interval each element has a complement relative to the ends of the interval.
A bounded relatively complemented lattice is complemented, and a complemented [[modular lattice|QX7WZ92QAK]] is relatively complemented.
Boolean algebras and lattices of subspaces of a vector space are examples.

## Explanations

### L1 human

Being complemented is a statement about the whole lattice, from its bottom to its top.
Relative complements ask the same thing for every smaller window $[c, d]$ of the lattice.
For subspaces $C \subseteq A \subseteq D$ of a vector space, there is a subspace $B$ between $C$ and $D$ with $A \cap B = C$ and $A + B = D$.
So subspace lattices are complemented not only globally but in every window.
</document>

<document source="https://tamnd.github.io/math2/node/ES9YATDWA3.md" tag="ES9YATDWA3" kind="concept">
# Algebra over an operad

## Summary

An operad lists abstract operations with several inputs and one output, together with rules for plugging operations into each other.
An algebra over an operad is an object on which these abstract operations act as actual operations, compatibly with plugging, units and permutations of inputs.
Monoids, commutative monoids, Lie algebras and loop spaces are all algebras over suitable operads.
The notion lets one treat many algebraic structures, and their homotopy-coherent versions, in one framework.

## Explanations

### L1 human

Think of an operad as a menu of recipes, each taking $n$ ingredients and producing one dish.
An algebra over the operad is a kitchen that can actually cook every recipe on the menu with its own ingredients.
Cooking a recipe whose ingredients are dishes of other recipes must give the same result as cooking the combined recipe.
Take the operad with exactly one $n$-ary recipe for every $n$.
An algebra over it is a set with an associative and commutative multiplication with a unit, that is, a commutative [[monoid|0N6R902FG1]].
Loop spaces are algebras over the [[little cubes operad|FWT0003QS1]], which is the origin of the theory.
</document>

<document source="https://tamnd.github.io/math2/node/ESBGJHPEDT.md" tag="ESBGJHPEDT" kind="definition">
# Nakano positivity

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $n$ and let $E \to X$ be a [[holomorphic vector bundle|TKA958Z4KZ]] of rank $r$ with a [[Hermitian metric|Q6X6W2N29G]] $h$.
At a point $x$, write the curvature $\Theta$ of the [[Chern connection|7RC9CN2Q49]] in local coordinates and a frame orthonormal at $x$ as
$$i \Theta(x) = i \sum_{j,k=1}^{n} \sum_{\lambda,\mu=1}^{r} c_{jk\lambda\mu}\, dz_j \wedge d\bar z_k \otimes e_\lambda^* \otimes e_\mu.$$
The bundle $(E, h)$ is *Nakano positive* if at every point $x \in X$
$$\sum_{j,k,\lambda,\mu} c_{jk\lambda\mu}\, \tau_{j\lambda} \bar\tau_{k\mu} > 0$$
for every nonzero complex $n \times r$ matrix $\tau = (\tau_{j\lambda})$.
Such a matrix is the same as a tensor $\tau \in T_x X \otimes E_x$.

## Remarks

Taking $\tau_{j\lambda} = \xi_j v_\lambda$ recovers the condition of [[Griffiths positivity|FVCDGMNSAN]], so Nakano positive bundles are Griffiths positive.
For $r = 1$ or $n = 1$ every tensor $\tau$ is of this form, and the two notions agree.

Example: $E = L^{\oplus r}$, where $L$ is a positive line bundle with curvature $i \sum a_{jk} dz_j \wedge d\bar z_k$, with the direct sum metric.
Then $c_{jk\lambda\mu} = a_{jk} \delta_{\lambda\mu}$, and the sum is $\sum_\lambda \sum_{j,k} a_{jk} \tau_{j\lambda} \bar\tau_{k\lambda}$.
Each inner sum is $\ge 0$ because $(a_{jk})$ is positive definite, and some inner sum is $> 0$ because $\tau \ne 0$.

Non-example: a trivial bundle $X \times \C^r$ with the standard metric has $\Theta = 0$, so the sum is $0$ for every $\tau$.

Nakano's vanishing theorem: if $X$ is compact Kähler and $E$ is Nakano positive, then $H^q(X, K_X \otimes E) = 0$ for $q \ge 1$, where $K_X$ is the canonical bundle.
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<document source="https://tamnd.github.io/math2/node/ESGEHKPD60.md" tag="ESGEHKPD60" kind="statement">
# Gauss–Bonnet theorem

## Statement

Let $S \subset \R^3$ be an [[oriented|0TD904MG06]] [[regular surface|6080X3FAD4]], and let $R \subset S$ be a compact region.
Assume that the boundary $\partial R$ is a finite union of simple closed, piecewise regular curves $C_1, \ldots, C_m$, positively oriented.
Let $\theta_1, \ldots, \theta_p$ be the external angles at the vertices of these curves.
Let $K$ be the [[Gaussian curvature|SQVQ5DQT6W]] of $S$, and let $k_g$ be the [[geodesic curvature|550P8PFMVC]] of the curves $C_i$.
Let $\chi(R)$ be the [[Euler characteristic|RNDDWDAFJ5]] of $R$.
Then
$$\int_R K \, dA + \sum_{i=1}^{m} \int_{C_i} k_g \, ds + \sum_{l=1}^{p} \theta_l = 2\pi \chi(R).$$
In particular, if $S$ is compact, then
$$\int_S K \, dA = 2\pi \chi(S).$$

## Remarks

The proof starts from the local case of a small simple region in one coordinate patch.
There the formula follows from Green's theorem, applied to the turning of a tangent vector along the boundary.
One then takes a [[triangulation|YH170WBQ6M]] of $R$ fine enough that each triangle lies in one patch, and adds the local formulas.
Interior edges are traversed twice in opposite directions, so their geodesic curvature terms cancel.
The angle terms then add up to $2\pi(V - E + F) = 2\pi\chi(R)$.

For the round sphere of radius $r$, one has $K = 1/r^2$ and area $4\pi r^2$, so the total curvature is $4\pi = 2\pi \cdot 2$, and indeed $\chi(S^2) = 2$.
For any torus in $\R^3$, the total curvature is $0$, since $\chi = 0$.
Compactness is needed for the second formula: the plane has $K = 0$ but $\chi = 1$.

The theorem extends to compact two-dimensional Riemannian manifolds, and the [[Chern–Gauss–Bonnet theorem|GG6NYPKMHX]] extends it to all even dimensions.

## Explanations

### L1 human

Gaussian curvature measures how a surface bends at each point.
The theorem says that if you add up all the curvature over a closed surface, you get $2\pi$ times a whole number that depends only on the topological shape.
So a sphere, however you dent it, always has total curvature $4\pi$.
A local, metric quantity is tied to a global, topological one.

### L2 human

The formula is the standard bridge between the local geometry and the topology of surfaces.
It shows that a compact surface with $K > 0$ everywhere must have $\chi > 0$, so it is a sphere when orientable.
Applied to a geodesic triangle, where $k_g = 0$, it gives $\int_T K \, dA = \alpha + \beta + \gamma - \pi$ for the interior angles $\alpha, \beta, \gamma$.
This explains why angle sums exceed $\pi$ on a sphere and fall short of $\pi$ on surfaces of negative curvature.
It is also the first case of the theory of characteristic classes, where the Euler class is represented by a curvature form.
</document>

<document source="https://tamnd.github.io/math2/node/ESPDZT91QP.md" tag="ESPDZT91QP" kind="example">
# Young's lattice

## Example

Young's lattice $\mathbb{Y}$ is the set of all [[integer partitions|0JV30XN73Z]] $\lambda = (\lambda_1 \ge \lambda_2 \ge \cdots)$, including the empty partition.
It is ordered by $\lambda \le \mu$ when $\lambda_i \le \mu_i$ for all $i$, that is, when the [[Young diagram|096AHMB3P4]] of $\lambda$ fits inside that of $\mu$.

It is a [[distributive lattice|0HM0HQCK6N]].
The meet and join are $(\lambda \wedge \mu)_i = \min(\lambda_i, \mu_i)$ and $(\lambda \vee \mu)_i = \max(\lambda_i, \mu_i)$, the intersection and union of the diagrams.
Both are again weakly decreasing sequences, and $\min$ and $\max$ distribute over each other.

It is a [[graded poset|1C7TKGTB2F]] with rank function $\rho(\lambda) = \abs{\lambda} = \lambda_1 + \lambda_2 + \cdots$.
Indeed, $\mu$ covers $\lambda$ exactly when the diagram of $\mu$ is that of $\lambda$ with one more box.

## Remarks

The saturated chains from $\varnothing$ to $\lambda$ add one box at a time, so they correspond to the standard [[Young tableaux|Q6TH9PH2R7]] of shape $\lambda$.
For $\lambda = (2, 1)$ there are two of them: $\varnothing, (1), (2), (2, 1)$ and $\varnothing, (1), (1, 1), (2, 1)$.
The partitions of $n$ form the rank $n$ level, and their number is the partition number $p(n)$.

## Explanations

### L1 human

Draw each partition as a staircase of boxes in the top-left corner of a grid.
One staircase is below another if it fits inside it.
Going up one step in the lattice means adding one box, so climbing from nothing to a shape is a way of building that shape box by box.
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<document source="https://tamnd.github.io/math2/node/ESPG0BJERZ.md" tag="ESPG0BJERZ" kind="concept">
# Regular scheme

## Summary

A regular scheme is a locally Noetherian [[scheme|5A35TYZD02]] whose local rings are all [[regular local rings|63XXTF6B5M]].
It is the algebraic notion of a space without singularities that works over any base, including $\Z$.
Over a perfect field, a scheme of finite type is regular exactly when it is smooth.

## Explanations

### L1 human

At a smooth point of a surface, two coordinates are enough to describe everything nearby.
At a singular point, such as the tip of a cone, more are needed than the dimension.
Regularity says exactly this, in algebra: near each point, the number of local generators equals the dimension.
The plane is regular, and so is $\operatorname{Spec} \Z$, but the cusp $y^2 = x^3$ is not regular at the origin.
</document>

<document source="https://tamnd.github.io/math2/node/ESQCJ06KZD.md" tag="ESQCJ06KZD" kind="definition">
# Kruskal–Wallis test

## Definition

Let $k \ge 2$, and for $g = 1, \dots, k$ let $X_{g1}, \dots, X_{g n_g}$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with a continuous distribution function $F_g$, the $k$ samples being independent.
Let $N = n_1 + \dots + n_k$.
Rank all $N$ observations together from $1$ to $N$, let $R_g$ be the sum of the ranks in sample $g$, and let $\bar R_g = R_g / n_g$.
The *Kruskal–Wallis statistic* is
$$H = \frac{12}{N(N+1)} \sum_{g=1}^{k} n_g \Big( \bar R_g - \frac{N + 1}{2} \Big)^2 = \frac{12}{N(N+1)} \sum_{g=1}^{k} \frac{R_g^2}{n_g} - 3(N + 1).$$
The *Kruskal–Wallis test* of $H_0 \colon F_1 = \dots = F_k$ rejects $H_0$ when $H$ is large.

## Remarks

Under $H_0$ the law of $H$ depends only on $n_1, \dots, n_k$.
As all $n_g \to \infty$, $H$ [[converges in distribution|0JDAQHWA9N]] to the [[chi-squared distribution|0RFMMBHFTV]] with $k - 1$ degrees of freedom, which gives approximate critical values.
The number $(N+1)/2$ is the mean of the ranks $1, \dots, N$.

Example: $k = 2$, samples $(1, 2)$ and $(3, 4)$, so $N = 4$, $R_1 = 3$, $R_2 = 7$.
Then $H = \frac{12}{20}\big(\frac{9}{2} + \frac{49}{2}\big) - 15 = 17.4 - 15 = 2.4$.
The first form gives the same: $\frac{12}{20}\big(2(1.5 - 2.5)^2 + 2(3.5 - 2.5)^2\big) = 0.6 \cdot 4 = 2.4$.

Non-example: repeated measurements on the same subjects under $k$ conditions are not independent samples.
The clause on independence fails; the Friedman test is used for such blocked data.
</document>

<document source="https://tamnd.github.io/math2/node/ESRDNKF6WJ.md" tag="ESRDNKF6WJ" kind="definition">
# Smash product

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $H$ be a [[Hopf algebra|7XFYP0N84H]] over $k$ with comultiplication $\Delta$ and counit $\varepsilon$.
Write $\Delta(h) = \sum h_{(1)} \otimes h_{(2)}$ in [[Sweedler notation|NC26CE0Y8B]].
A *left $H$-module algebra* is an [[associative algebra|3CTW1P4HBP]] $A$ over $k$ with identity that is also a left $H$-[[module|0NZ8JA25WK]], written $h \cdot a$, such that:

1. $h \cdot (ab) = \sum (h_{(1)} \cdot a)(h_{(2)} \cdot b)$ for all $h \in H$ and $a, b \in A$.
2. $h \cdot 1_A = \varepsilon(h) 1_A$ for all $h \in H$.

The *smash product* $A \# H$ is the vector space $A \otimes_k H$, with $a \otimes h$ written $a \# h$, and with the multiplication
$$(a \# h)(b \# g) = \sum a \, (h_{(1)} \cdot b) \# h_{(2)} g.$$

## Remarks

With this product $A \# H$ is an associative algebra with identity $1 \# 1$, and $a \mapsto a \# 1$ and $h \mapsto 1 \# h$ embed $A$ and $H$ as subalgebras.
Example: let $G$ be a group acting on $A$ by algebra automorphisms and let $H = kG$, with $\Delta(g) = g \otimes g$ and $\varepsilon(g) = 1$.
Then clause 1 reads $g \cdot (ab) = (g \cdot a)(g \cdot b)$, clause 2 reads $g \cdot 1 = 1$, and the product is $(a \# g)(b \# h) = a (g \cdot b) \# gh$, so $A \# kG$ is the [[skew group ring|PCKK4SFZEK]] $A * G$.
Example: if $H = U(\mathfrak g)$ for a Lie algebra $\mathfrak g$, with $\Delta(x) = x \otimes 1 + 1 \otimes x$ for $x \in \mathfrak g$, clause 1 says that each $x$ acts by a derivation of $A$.
Non-example: let $G = \set{e, g}$ have order $2$, $A = k[x]$, and let $g$ act linearly by $g \cdot 1 = 1$ and $g \cdot x^n = -x^n$ for $n \ge 1$.
This is a $kG$-module with clause 2, but $g \cdot (x \cdot x) = -x^2$ while $(g \cdot x)(g \cdot x) = x^2$, so clause 1 fails when the characteristic of $k$ is not $2$.
</document>

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# Intuitionistic type theory

## Summary

Intuitionistic type theory is the dependent [[type theory|Q48KGV33AH]] introduced by Per Martin-Löf from the 1970s on, also called Martin-Löf type theory.
It is a foundation of constructive mathematics based on [[intuitionistic logic|Q11N8X99FK]].
Through the [[Curry–Howard correspondence|A77FER1Q5H]], its types are also propositions, and its terms are both proofs and programs.
It is the basis of proof assistants such as Agda, and it is close to the systems behind Coq and Lean.

## Explanations

### L1 human

In this theory, a statement is a type, and a proof of the statement is an element of that type.
Types can depend on numbers or other data, as in "lists of length $n$".
A proof of "for every $n$ there is $m$ with $P(n, m)$" is a function that sends each $n$ to a pair: a number $m$ and a proof of $P(n, m)$.
So every proof of existence is also a program that computes the witness.
Checking a proof is the same as checking that a program has the right type.
</document>

<document source="https://tamnd.github.io/math2/node/ESWC2PXNZX.md" tag="ESWC2PXNZX" kind="definition">
# Hypergeometric function

## Definition

Let $a, b, c \in \C$ with $c \notin \set{0, -1, -2, \ldots}$.
The *hypergeometric function* ${}_2F_1(a, b; c; \cdot)$ is the unique [[holomorphic function|0NAHXR1FRH]] on $\C \setminus [1, \infty)$ that agrees on the disc $\abs{z} < 1$ with the sum of the [[hypergeometric series|428M453CR2]]
$$\sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \, \frac{z^n}{n!}.$$

## Remarks

Existence of this [[analytic continuation|0M3YQJ81NB]] follows from Euler's integral representation when $\operatorname{Re} c > \operatorname{Re} b > 0$.
In general it follows from the hypergeometric differential equation, whose only finite singular points are $0$ and $1$.
Uniqueness holds because $\C \setminus [1, \infty)$ is connected.
This is the principal branch; continuation across the cut gives other branches of a multivalued function.

Example: ${}_2F_1(a, b; b; z) = (1 - z)^{-a}$ on the whole cut plane, with the principal power, since both sides are holomorphic there and agree for $\abs{z} < 1$.

Non-example: ${}_2F_1(1, 1; 2; z) = -\log(1 - z)/z$ has no holomorphic extension to a neighbourhood of $z = 1$ or to any annulus around $1$: going once around $z = 1$ changes $\log(1 - z)$ by $2\pi i$.
So the cut, or a choice of branch, is needed.

Gauss's notation $F(a, b; c; z)$ is common, and the normalized function ${}_2F_1(a, b; c; z)/\Gamma(c)$ is entire in $c$.
</document>

<document source="https://tamnd.github.io/math2/node/ESWE8Y3JD4.md" tag="ESWE8Y3JD4" kind="statement">
# Verlet integration

## Statement

Let $a \colon \R^d \to \R^d$ be twice continuously differentiable, and consider the equation $x'' = a(x)$ with $x(0) = x_0$ and $x'(0) = v_0$.
Fix a step size $h > 0$.
The Störmer–Verlet method sets $x_1 = x_0 + h v_0 + \frac{h^2}{2} a(x_0)$ and
$$x_{n+1} = 2 x_n - x_{n-1} + h^2 a(x_n), \qquad n \ge 1.$$

1. The positions $x_n$ are the same as the positions computed by the [[leapfrog method|3HGJCQBWVY]] (velocity Verlet) from $(x_0, v_0)$ with step size $h$.
2. On a bounded interval $[0, T]$, the error $\max_{nh \le T} \abs{x_n - x(nh)}$ is $O(h^2)$ as $h \to 0$.

## Remarks

The two-step formula comes from adding the Taylor expansions of $x(t + h)$ and $x(t - h)$: the odd powers cancel, and $x(t+h) - 2x(t) + x(t-h) = h^2 x''(t) + O(h^4)$.
So the local error is $O(h^4)$.
In a two-step recurrence of this kind, a local error made at one step is carried forward with a factor of size at most $O(n)$.
Summing over $n \le T/h$ steps gives a global error $O(h^2)$.
For clause 1, the leapfrog positions satisfy $x_{n+1} - x_n = h v_{n+1/2}$ and $v_{n+1/2} = v_{n-1/2} + h a(x_n)$, so $x_{n+1} - x_n = x_n - x_{n-1} + h^2 a(x_n)$, and both start from the same $x_1$.
Velocities do not appear in the two-step form; they can be recovered by $v_n \approx (x_{n+1} - x_{n-1}) / (2h)$, with error $O(h^2)$.
Example: for $x'' = 0$ the recurrence gives $x_n = x_0 + n h v_0$, the exact solution.

## Explanations

### L1 human

Verlet integration moves a particle using its last two positions and the current force.
The new position is the current one, plus the last displacement, plus a correction from the acceleration.
It is cheap, accurate to second order, and does not let energy drift away over long runs.
This is why it is the workhorse of molecular dynamics.

### L2 human

The two-step form needs only positions, which halves the storage.
Its drawbacks are that velocities are only implicit and that round-off is larger, because a small term $h^2 a$ is added to large positions.
The velocity Verlet form fixes both points and is the same method.
Its good long-time behaviour comes from it being symmetric and symplectic.
</document>

<document source="https://tamnd.github.io/math2/node/ESX3ERNP4G.md" tag="ESX3ERNP4G" kind="definition">
# Geometric invariant theory

## Definition

Let $k$ be an algebraically closed field and let $G$ be a [[reductive group|2XS84PWE3S]] over $k$.
Let $G$ [[act|RW85C34GQW]] on a [[projective variety|A4H4CZ2V35]] $X$.
Let $L$ be an [[ample line bundle|RRK8P6TXG2]] on $X$ with a $G$-linearization, that is, a lift of the action of $G$ to $L$.
Write $R = \bigoplus_{m \ge 0} H^0(X, L^{m})$ and let $R^G$ be its [[ring of invariants|6VTR5EBNYC]].

1. A point $x \in X$ is *semistable* if there are $m \ge 1$ and an invariant section $s \in H^0(X, L^m)^G$ with $s(x) \ne 0$.
2. A semistable point $x$ is *stable* if its orbit $G x$ is closed in the semistable locus $X^{ss}$ and its stabilizer $G_x$ is finite.
3. A point that is not semistable is *unstable*.

The *GIT quotient* is the projective variety
$$X /\!\!/_L\, G = \operatorname{Proj} R^G,$$
built with the [[Proj construction|A39XQHCVNW]], together with the morphism $\pi \colon X^{ss} \to X /\!\!/_L\, G$ induced by the inclusion $R^G \subseteq R$.

## Remarks

The ring $R^G$ is finitely generated because $G$ is reductive, so the quotient is a projective variety.
The map $\pi$ is a good categorical quotient: two semistable points have the same image exactly when the closures of their orbits meet in $X^{ss}$.
On the stable locus $X^s$, which is open, $\pi$ restricts to a geometric quotient $X^s \to X^s / G$ whose fibres are single orbits.

Example: let $G = \mathbb{G}_m$ act on $X = \mathbb{P}^1$ by $t \cdot [x : y] = [t x : t^{-1} y]$, linearized on $L = \mathcal{O}(1)$ by the weights $1$ and $-1$.
A monomial $x^a y^b$ of degree $m = a + b$ is invariant exactly when $a = b$.
So the invariant sections are the multiples of powers of $xy$, and the semistable points are those with $xy \ne 0$.
They form a single orbit, which is closed in $X^{ss}$, and the stabilizer of $[1 : 1]$ is $\set{\pm 1}$, so these points are stable and the quotient is a point.

Non-example: the point $[1 : 0]$ is unstable, since every invariant section of positive degree is a multiple of $xy$ and vanishes there.

For an affine variety $X = \operatorname{Spec} A$ with the trivial linearization, the quotient is $X /\!\!/ G = \operatorname{Spec} A^G$.
</document>

<document source="https://tamnd.github.io/math2/node/ESYKDNZF8E.md" tag="ESYKDNZF8E" kind="concept">
# Stone algebra

## Summary

A Stone algebra is a bounded [[distributive lattice|0HM0HQCK6N]] in which every element $a$ has a [[pseudocomplement|WMR73J0S26]] $a^*$ and $a^* \vee a^{**} = 1$ holds.
The identity $a^* \vee a^{**} = 1$ is called the Stone identity.
Every [[Boolean algebra|0XADJKM699]] is a Stone algebra, and so is every bounded chain.
Stone algebras lie between Boolean algebras and general pseudocomplemented distributive lattices.

## Explanations

### L1 human

In a Boolean algebra, every element and its complement together fill everything.
A pseudocomplement is a weaker kind of complement, for which this can fail.
A Stone algebra asks for a little less: the pseudocomplement of $a$ and the pseudocomplement of that together fill everything.
The three-element chain $0 < m < 1$ is an example: $m^* = 0$ and $m^{**} = 1$, so $m^* \vee m^{**} = 1$, even though $m \vee m^* = m$.
</document>

<document source="https://tamnd.github.io/math2/node/ET1HPME8VV.md" tag="ET1HPME8VV" kind="definition">
# Hilbert system

## Definition

Consider formulas of [[propositional logic|09K9MWD2QQ]] built from variables with [[negation|029JBBF693]] $\neg$ and [[implication|0NMG7W7Q11]] $\to$.
The standard *Hilbert system* for this language has the following axioms, for all formulas $\varphi$, $\psi$, $\chi$:

1. $\varphi \to (\psi \to \varphi)$;
2. $(\varphi \to (\psi \to \chi)) \to ((\varphi \to \psi) \to (\varphi \to \chi))$;
3. $(\neg \varphi \to \neg \psi) \to (\psi \to \varphi)$.

Its only rule of inference is modus ponens: from $\varphi$ and $\varphi \to \psi$, infer $\psi$.
Let $\Gamma$ be a set of formulas.
A *derivation* of $\varphi$ from $\Gamma$ is a finite sequence of formulas that ends with $\varphi$.
Each formula in it is an axiom, a member of $\Gamma$, or follows from two earlier formulas by modus ponens.
One writes $\Gamma \vdash \varphi$ when such a derivation exists.

## Remarks

Here is a derivation of $\varphi \to \varphi$ from no hypotheses.

1. $(\varphi \to ((\varphi \to \varphi) \to \varphi)) \to ((\varphi \to (\varphi \to \varphi)) \to (\varphi \to \varphi))$, axiom 2.
2. $\varphi \to ((\varphi \to \varphi) \to \varphi)$, axiom 1.
3. $(\varphi \to (\varphi \to \varphi)) \to (\varphi \to \varphi)$, modus ponens from lines 2 and 1.
4. $\varphi \to (\varphi \to \varphi)$, axiom 1.
5. $\varphi \to \varphi$, modus ponens from lines 4 and 3.

The sequence $\varphi, \; \varphi \to \psi$ is not a derivation of $\varphi \to \psi$ from no hypotheses: its first line is neither an axiom nor obtained by a rule.
More generally, a *Hilbert system* is any proof system of this shape: axiom schemas, few rules, and proofs as lists.
For [[predicate logic|086QKPF0A6]] one adds axioms for the quantifiers and the rule of generalization.
</document>

<document source="https://tamnd.github.io/math2/node/ET5G9CT0RC.md" tag="ET5G9CT0RC" kind="definition">
# Polytope

## Definition

Let $n \ge 1$.
A *polytope* in $\R^n$ is a set $P = P_1 \cup \cdots \cup P_k$ that is the union of finitely many [[convex polytopes|300K6CB1KH]] $P_1, \ldots, P_k$ in $\R^n$.

## Remarks

Example: the L-shaped region $[0, 2] \times [0, 1] \cup [0, 1] \times [0, 2]$ is the union of two rectangles, which are convex polytopes, so it is a polytope.
It is not convex: it contains $(2, 1)$ and $(1, 2)$, but not their midpoint $(3/2, 3/2)$.
Non-example: the closed unit disc in $\R^2$ is not a polytope.
The boundary of a finite union of convex polygons lies in finitely many lines.
Each line meets the unit circle in at most two points, so this boundary cannot contain the whole circle.
Conventions differ between authors.
In convex geometry and in polyhedral combinatorics "polytope" usually means convex polytope, and some authors require a polytope to be a union of simplices that meet in common faces.
</document>

<document source="https://tamnd.github.io/math2/node/ET71R6CSAZ.md" tag="ET71R6CSAZ" kind="definition">
# Amenable Banach algebra

## Definition

Let $A$ be a complex [[Banach algebra|9KYWN94QDN]].
A *Banach $A$-bimodule* is a Banach space $X$ with bounded bilinear left and right actions $A \times X \to X$ and $X \times A \to X$ that make $X$ an $A$-bimodule.
The [[dual space|4A82S5FVRT]] $X^*$ is then a Banach $A$-bimodule with $(a \cdot f)(x) = f(x \cdot a)$ and $(f \cdot a)(x) = f(a \cdot x)$ for $a \in A$, $f \in X^*$, $x \in X$.
A bounded linear map $D \colon A \to X^*$ is a *[[derivation|3J89REV9WQ]]* if $D(ab) = a \cdot D(b) + D(a) \cdot b$ for all $a, b \in A$.
It is *inner* if there is $f \in X^*$ with $D(a) = a \cdot f - f \cdot a$ for all $a \in A$.
The algebra $A$ is *amenable* if, for every Banach $A$-bimodule $X$, every bounded derivation $A \to X^*$ is inner.

## Remarks

In cohomological terms, $A$ is amenable if and only if $H^1(A, X^*) = 0$ for every Banach $A$-bimodule $X$.

Example: $A = \C$.
Let $D \colon \C \to X^*$ be a derivation and put $u = D(1)$.
From $D(1) = D(1 \cdot 1) = 1 \cdot u + u \cdot 1$ one gets $1 \cdot u \cdot 1 = 0$.
Put $f = 1 \cdot u - u \cdot 1$.
Then $1 \cdot f - f \cdot 1 = 1 \cdot u - 2 (1 \cdot u \cdot 1) + u \cdot 1 = u = D(1)$, and by linearity $D(\lambda) = \lambda \cdot f - f \cdot \lambda$ for all $\lambda \in \C$.
So every bounded derivation is inner.

Non-example: let $\mathbb{F}_2$ be the free group on two generators, which is not an [[amenable group|CEAT9BESKK]].
By Johnson's theorem, $L^1(G)$ is amenable if and only if the locally compact group $G$ is amenable.
So $\ell^1(\mathbb{F}_2)$ is not amenable: some bounded derivation into a dual bimodule is not inner.

A C*-algebra is amenable if and only if it is [[nuclear|D2M4VXJ7PA]].
</document>

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# Loop-erased random walk

## Summary

A loop-erased random walk is the simple path obtained from a [[random walk|04VZE3FZXF]] by erasing its loops in the order in which they are created.
It is a random self-avoiding path, but its law differs from the uniform [[self-avoiding walk|E2BVEB570S]].
Through Wilson's algorithm it is the building block of the uniform [[spanning tree|0S4ZEZM7KE]] of a graph.
In the plane its scaling limit is the Schramm–Loewner evolution with parameter $2$.

## Explanations

### L1 human

Let a random walker wander until it reaches a target, and record its path.
Each time the walker comes back to a place it has already visited, cut out the loop it has just made.
What remains is a path from start to target that never crosses itself.
For example, on the integer line the path $0, 1, 2, 1, 2, 3$ becomes $0, 1, 2, 3$ after the loop $2, 1, 2$ is erased.
</document>

<document source="https://tamnd.github.io/math2/node/ET9QA03J3F.md" tag="ET9QA03J3F" kind="definition">
# Pontryagin dual

## Definition

Let $G$ be a [[locally compact abelian group|ZR09S9R3AW]], and let $\mathbb{T}$ be the [[circle group|04EAFVFTF7]].
The *Pontryagin dual* of $G$ is the set $\widehat{G}$ of all continuous [[characters|H1WNVN5V7F]] $\chi \colon G \to \mathbb{T}$, with:

1. the group operation $(\chi\psi)(x) = \chi(x)\psi(x)$;
2. the [[compact-open topology|RG15F0TWPW]], that is, the topology of uniform convergence on compact subsets of $G$.

## Remarks

Example: for $G = \Z$ with the discrete topology, a character $\chi$ is fixed by $z = \chi(1) \in \mathbb{T}$, since $\chi(n) = z^n$, and each $z \in \mathbb{T}$ gives a character.
Uniform convergence on finite sets is pointwise convergence, so $\widehat{\Z} \cong \mathbb{T}$ as topological groups.
Non-example: the group of all characters of $\R$, continuous or not, is not the Pontryagin dual of $\R$.
There are discontinuous homomorphisms $\R \to \mathbb{T}$, built from a Hamel basis, and the dual excludes them.
With the compact-open topology, $\widehat{G}$ is again a locally compact abelian group.
If $G$ is compact, then $\widehat{G}$ is discrete, and if $G$ is discrete, then $\widehat{G}$ is compact.
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<document source="https://tamnd.github.io/math2/node/ETCE287E2R.md" tag="ETCE287E2R" kind="concept">
# Upper set

## Summary

An upper set of a [[partially ordered set|011C29Q1S1]] is a subset that contains everything above each of its elements.
The dual notion is a lower set, which contains everything below each of its elements.
Upper and lower sets are used to define order filters, the Alexandrov topology, and the lattice of down-sets of a poset.

## Explanations

### L1 human

An upper set is closed upwards: once an element is in, every larger element is in too.
In the real numbers, the interval $[3, \infty)$ is an upper set, and so is $(3, \infty)$.
In the subsets of $\set{1, 2, 3}$ ordered by inclusion, the family of all subsets containing $1$ is an upper set.
The set $\set{3}$ alone is not an upper set of $\R$, because $4 > 3$ is missing.
</document>

<document source="https://tamnd.github.io/math2/node/ETCVR65W3M.md" tag="ETCVR65W3M" kind="definition">
# Paraconsistent logic

## Definition

Let $\vdash$ be a [[consequence relation|1HK0VYWTV7]] between sets of formulas and formulas of a language with a [[negation|029JBBF693]] $\neg$.
The relation $\vdash$ is explosive when $\set{\varphi, \neg\varphi} \vdash \psi$ for all formulas $\varphi$ and $\psi$.
A logic is paraconsistent when its consequence relation is not explosive.
That is, there are formulas $\varphi$ and $\psi$ with $\set{\varphi, \neg\varphi} \nvdash \psi$.

## Remarks

Example: Priest's logic of paradox $\mathrm{LP}$ has the truth values $t$, $b$ ("both") and $f$, ordered $f < b < t$.
Negation swaps $t$ and $f$ and fixes $b$; conjunction is the minimum and disjunction the maximum.
The designated values are $t$ and $b$, and $\Gamma \vdash \psi$ means that every valuation giving all of $\Gamma$ designated values gives $\psi$ one too.
The valuation $p \mapsto b$, $q \mapsto f$ makes $p$ and $\neg p$ designated and $q$ not, so $\set{p, \neg p} \nvdash q$.

Non-example: classical [[propositional logic|09K9MWD2QQ]] is explosive.
No valuation makes both $\varphi$ and $\neg\varphi$ true, so every $\psi$ follows from them.
Intuitionistic logic is explosive too, through the axiom $\bot \to \psi$.

## Explanations

### L1 human

A paraconsistent logic is any logic in which some contradiction fails to prove some statement.
Such logics weaken negation, or the notion of consequence, so that a contradiction can be true in a model without making everything true.
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<document source="https://tamnd.github.io/math2/node/ETDW4AK1W1.md" tag="ETDW4AK1W1" kind="definition">
# Lindelöf space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is a *Lindelöf space* if every [[open cover|049PXYHW1T]] $\mathcal{U}$ of $X$ has a countable subcover.
That is, there is a countable family $\mathcal{V} \subseteq \mathcal{U}$ with $\bigcup \mathcal{V} = X$.

## Remarks

Example: $\R$ with the usual topology is Lindelöf.
It is [[second-countable|03PPNCM9GG]], with the countable base of open intervals with rational end points.
For an open cover $\mathcal{U}$, pick for each basic interval contained in some member of $\mathcal{U}$ one such member; these countably many members cover $\R$.

Non-example: an uncountable set $X$ with the discrete topology is not Lindelöf.
The cover $\set{\set{x} : x \in X}$ is an open cover, and every subcover must contain all of its uncountably many members.

The [[Sorgenfrey line|Y4T3GVK8GV]] is Lindelöf, but its square, the Sorgenfrey plane, is not.
So a product of two Lindelöf spaces need not be Lindelöf.
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<document source="https://tamnd.github.io/math2/node/ETH7FW1KKA.md" tag="ETH7FW1KKA" kind="definition">
# Resolutivity of boundary functions

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]] with [[boundary|05D5EGMFYV]] $\partial\Omega$, and let $f \colon \partial\Omega \to [-\infty, +\infty]$ be a function.

1. The *upper class* $\mathcal{U}_f$ is the set of functions $u$ on $\Omega$ with these properties.
On each connected component of $\Omega$, $u$ is [[superharmonic|X4V2S56YQQ]] or identically $+\infty$.
The function $u$ is bounded below, and $\liminf_{x \to \zeta,\, x \in \Omega} u(x) \ge f(\zeta)$ for every $\zeta \in \partial\Omega$.
2. The *lower class* $\mathcal{L}_f$ is the set of functions $v$ on $\Omega$ with these properties.
On each connected component of $\Omega$, $v$ is [[subharmonic|Z1MHMGFNFW]] or identically $-\infty$.
The function $v$ is bounded above, and $\limsup_{x \to \zeta,\, x \in \Omega} v(x) \le f(\zeta)$ for every $\zeta \in \partial\Omega$.
3. The *upper solution* is $\overline{H}_f(x) = \inf\set{u(x) : u \in \mathcal{U}_f}$ and the *lower solution* is $\underline{H}_f(x) = \sup\set{v(x) : v \in \mathcal{L}_f}$, for $x \in \Omega$.

The function $f$ is *resolutive* if $\overline{H}_f = \underline{H}_f$ on $\Omega$ and this common function is [[harmonic|0DEB0JMN0W]] on $\Omega$, in particular finite.
The common function is then written $H_f$.

## Remarks

The inequality $\underline{H}_f \le \overline{H}_f$ always holds, by the maximum principle applied to $v - u$ for $v \in \mathcal{L}_f$ and $u \in \mathcal{U}_f$.
Wiener's theorem says that every continuous $f$ is resolutive, so $H_f$ is a generalised solution of the [[Dirichlet problem|8D56GQ4AG8]].
For a bounded domain, Brelot showed that $f$ is resolutive if and only if $f$ is integrable with respect to [[harmonic measure|CFD7BPM51W]].
Then $H_f(x)$ is the integral of $f$ against the harmonic measure for $x$.

Example: let $\Omega$ be the unit disc in $\R^2$ and $f$ continuous on the unit circle.
The Poisson integral $P[f]$ is harmonic in $\Omega$ and tends to $f(\zeta)$ at each boundary point $\zeta$.
So $P[f]$ lies in both $\mathcal{U}_f$ and $\mathcal{L}_f$, which gives $\overline{H}_f \le P[f] \le \underline{H}_f$.
With $\underline{H}_f \le \overline{H}_f$ this forces $\overline{H}_f = \underline{H}_f = P[f]$, which is harmonic, so $f$ is resolutive.

Non-example: let $f \equiv +\infty$ on $\partial\Omega$.
The only member of $\mathcal{U}_f$ is the constant $+\infty$, so $\overline{H}_f \equiv +\infty$.
Every constant $c$ lies in $\mathcal{L}_f$, so $\underline{H}_f \equiv +\infty$ too.
The two solutions agree but are not harmonic, so $f$ is not resolutive.
</document>

<document source="https://tamnd.github.io/math2/node/ETJ4WRXC41.md" tag="ETJ4WRXC41" kind="definition">
# Brauer character

## Definition

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a prime, and let $k$ be an [[algebraically closed field|0PF6TM8TWT]] of characteristic $p$.
An element $g \in G$ is $p$-regular if its [[order|0WE2NHZJ7B]] is not divisible by $p$; let $G_{p'}$ be the set of $p$-regular elements.
Let $\mu_{p'}(k)$ and $\mu_{p'}(\C)$ be the groups of [[roots of unity|0YV2CKP2AV]] of order prime to $p$ in $k$ and in the [[complex numbers|09FTAET5Z4]].
Fix a group isomorphism $\phi \colon \mu_{p'}(k) \to \mu_{p'}(\C)$.
Let $\rho \colon G \to GL(V)$ be a [[representation|WYGRXCG2RN]] of $G$ on a vector space $V$ over $k$ of finite dimension $d$.
For $g \in G_{p'}$, the map $\rho(g)$ has [[eigenvalues|0BQAA2BK3Q]] $\lambda_1, \ldots, \lambda_d \in \mu_{p'}(k)$, listed with multiplicity.
The *Brauer character* of $\rho$ is the function $\beta_\rho \colon G_{p'} \to \C$ given by
$$\beta_\rho(g) = \phi(\lambda_1) + \cdots + \phi(\lambda_d).$$

## Remarks

The eigenvalues lie in $\mu_{p'}(k)$ because $\rho(g)^m = \id$ with $m$ the order of $g$, prime to $p$.
Such an isomorphism $\phi$ exists; it is usually fixed through a $p$-modular system, and other choices change $\beta_\rho$ by a Galois automorphism.
Example: for the trivial representation of degree $d$, all eigenvalues are $1$, so $\beta(g) = d$ for every $g \in G_{p'}$; for $d = p$ this is $p$, while the trace in $k$ is $0$.
Non-example: let $G = \Z/p$ and let $V = k^2$, with the generator acting by $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.
The generator is not $p$-regular, so the Brauer character has no value there.
It is defined only at the identity, with value $2$.
Brauer characters are constant on conjugacy classes, additive on short exact sequences, and $\beta_\rho(1) = \dim V$.
The irreducible Brauer characters are linearly independent, and their number is the number of conjugacy classes of $p$-regular elements.
</document>

<document source="https://tamnd.github.io/math2/node/ETMAPNQY7T.md" tag="ETMAPNQY7T" kind="statement">
# Szemerédi regularity lemma

## Statement

Let $G$ be a finite simple [[graph|00C0BR7D7H]] with vertex set $V$.
For disjoint nonempty sets $X, Y \subseteq V$, let $e(X, Y)$ be the number of edges with one end in $X$ and one in $Y$, and let $d(X, Y) = e(X, Y) / (|X| |Y|)$.
Let $\varepsilon > 0$.
A pair $(A, B)$ of disjoint nonempty sets is *$\varepsilon$-regular* if $\abs{d(X, Y) - d(A, B)} \le \varepsilon$ whenever $X \subseteq A$, $Y \subseteq B$, $|X| \ge \varepsilon |A|$ and $|Y| \ge \varepsilon |B|$.
A [[partition|08QMKTWF9T]] $V = V_1 \cup \cdots \cup V_k$ is *equitable* if any two parts differ in size by at most $1$.

For every $\varepsilon > 0$ and every integer $m \ge 1$ there is an integer $M = M(\varepsilon, m)$ with the following property.
Let $G$ have at least $m$ vertices.
Then $G$ has an equitable partition $V_1, \ldots, V_k$ of its vertex set with $m \le k \le M$.
In this partition, all but at most $\varepsilon k^2$ of the pairs $(V_i, V_j)$ with $i < j$ are $\varepsilon$-regular.

## Remarks

The proof uses an energy, the mean square of the densities $d(V_i, V_j)$, which lies between $0$ and $1$.
If the current partition has too many irregular pairs, the sets $X, Y$ that witness irregularity are used to refine it.
Then the energy increases by at least a fixed amount depending on $\varepsilon$.
So the process stops after a bounded number of steps.
Each step multiplies the number of parts by an exponential, so $M$ is a tower of exponentials of height polynomial in $1/\varepsilon$.
Gowers showed that bounds of tower type are necessary.

The lemma comes with a counting lemma: if parts $V_1, V_2, V_3$ are pairwise $\varepsilon$-regular with densities at least $\delta$ and $\varepsilon$ is small compared with $\delta$, then they span many triangles.
Together they prove the [[triangle removal lemma|XMS1QRAQZC]] and the [[Erdős–Stone theorem|XZF9BFJFXB]].

## Explanations

### L1 human

Every large graph, however messy, can be cut into a bounded number of pieces of equal size so that between most pairs of pieces the edges look random.
"Look random" means that the edge density is about the same between any two large subsets of the two pieces.
The number of pieces depends only on the accuracy you ask for, not on the size of the graph.

### L2 human

The lemma replaces a huge graph by a small weighted "reduced graph" on the parts, with the densities as weights.
Questions about copies of a fixed small graph can then be answered in the reduced graph through the counting lemma.
This is the route to the [[triangle removal lemma|XMS1QRAQZC]], to [[Roth's theorem|E7H8NWXESF]], and to the [[Erdős–Stone theorem|XZF9BFJFXB]].
Szemerédi first proved a version for bipartite graphs as a step towards [[Szemerédi's theorem|B23B0QEMN5]] on arithmetic progressions.
The limit object of this point of view is the [[graphon|QMJ5PNSF9X]].
</document>

<document source="https://tamnd.github.io/math2/node/ETNDTECY97.md" tag="ETNDTECY97" kind="statement">
# Schur orthogonality for compact groups

## Statement

Let $G$ be a [[compact group|EVHCE5VR6Y]] with normalised [[Haar measure|VY0HXD3GC4]] $dg$, and for $\alpha, \beta \in L^2(G)$ write $\ip{\alpha}{\beta} = \int_G \alpha(g)\overline{\beta(g)} \, dg$.
Let $\pi$ and $\sigma$ be finite-dimensional continuous [[irreducible|FX3NEFYFFN]] [[unitary representations|74BPYJNZTM]] of $G$, of dimension $d_\pi$ and $d_\sigma$.
Let $\pi_{ij}$ and $\sigma_{kl}$ be their matrix coefficients in orthonormal bases.

1. If $\pi$ and $\sigma$ are not isomorphic, then $\ip{\pi_{ij}}{\sigma_{kl}} = 0$ for all indices.
2. For all indices $i, j, k, l$, one has $\ip{\pi_{ij}}{\pi_{kl}} = \delta_{ik}\delta_{jl}/d_\pi$.
3. The [[characters|RMWMYBY8TC]] $\chi_\pi = \tr \pi$ and $\chi_\sigma = \tr \sigma$ satisfy $\ip{\chi_\pi}{\chi_\sigma} = 1$ if $\pi \cong \sigma$ and $\ip{\chi_\pi}{\chi_\sigma} = 0$ otherwise.

## Remarks

For a linear map $A$ between the two spaces, the average $\int_G \sigma(g) A \pi(g)^{-1} \, dg$ intertwines $\pi$ and $\sigma$.
By Schur's lemma it is $0$ when they are not isomorphic and a scalar when $\pi = \sigma$.
Taking $A$ to be a matrix unit gives parts 1 and 2, the scalar being found by taking traces; summing over $i = j$ and $k = l$ gives part 3.
Example: for $G = \mathbb{T}$ with the characters $\chi_n(z) = z^n$, part 3 reads $\frac{1}{2\pi}\int_0^{2\pi} e^{in\theta} e^{-im\theta} \, d\theta = \delta_{nm}$.
For a finite group with Haar measure the counting measure divided by $\abs{G}$, the statement is the [[Schur orthogonality relations|AY0B20QC04]] of finite groups.
A consequence: a finite-dimensional continuous representation with character $\chi$ is irreducible if and only if $\ip{\chi}{\chi} = 1$.

## Explanations

### L1 human

The matrix entries of different irreducible representations of a compact group are perpendicular to each other as functions on the group.
The characters of irreducible representations are perpendicular unit vectors.
So one can find how often an irreducible piece occurs in a representation by an integral, just like taking a coordinate.

### L2 human

These relations give the orthonormality half of the [[Peter–Weyl theorem|6VC9QE39XH]]; the completeness half is the harder part.
For compact connected Lie groups, the [[Weyl integration formula|QMH530N27M]] turns the character relation into an integral over a maximal torus, which is the route to the [[Weyl character formula|230F5FXPMQ]].
The relations apply to every finite-dimensional continuous representation after a change of basis, since each one is unitary for a suitable inner product by [[complete reducibility|N6JC6KNXJJ]].
</document>

<document source="https://tamnd.github.io/math2/node/ETQ87X5CWS.md" tag="ETQ87X5CWS" kind="definition">
# Jacobi sum

## Definition

Let $F$ be a [[finite field|0ZXZ82FTX2]] with $q$ elements.
Let $\chi, \lambda \colon F^\times \to \C^\times$ be multiplicative characters, that is, [[group homomorphisms|0JYTG5K9Y4]], each extended to $F$ by the value $0$ at $0$.
The *Jacobi sum* of $\chi$ and $\lambda$ is
$$J(\chi, \lambda) = \sum_{a \in F} \chi(a) \lambda(1 - a).$$

## Remarks

Some books set the trivial character equal to $1$ at $0$; this changes $J$ only when $\chi$ or $\lambda$ is trivial.
If $\chi$, $\lambda$ and $\chi\lambda$ are nontrivial, then $J(\chi, \lambda) = g(\chi)g(\lambda)/g(\chi\lambda)$, where $g$ is the [[Gauss sum|EQ6A3ZN48K]] with a fixed nontrivial additive character, and so $\abs{J(\chi, \lambda)} = \sqrt{q}$.

Example: let $q = 5$ and $\chi = \lambda$ the [[Legendre symbol|0JTC045NNZ]] modulo $5$.
The terms with $a = 0$ and $a = 1$ vanish.
For $a = 2, 3, 4$ the products $\chi(a)\chi(1-a)$ are $(-1)(1) = -1$, $(-1)(-1) = 1$ and $(1)(-1) = -1$.
So $J(\chi, \chi) = -1$; here $\chi^2$ is trivial and the general rule gives $J(\chi, \chi^{-1}) = -\chi(-1) = -1$.

Non-example: the sum $\sum_{a} \chi(a) \lambda(a)$ is not a Jacobi sum, since the second argument is $a$ and not $1 - a$.
By orthogonality it equals $q - 1$ if $\chi\lambda$ is trivial and $0$ otherwise.
</document>

<document source="https://tamnd.github.io/math2/node/ETRF2FHAEE.md" tag="ETRF2FHAEE" kind="object">
# Gromov–Hausdorff space

## Description

Let $Z$ be a [[metric space|0VKD1TP7F3]] and let $A, B \subseteq Z$ be nonempty.
The *Hausdorff distance* $d_H(A, B)$ is the infimum of the numbers $r \ge 0$ with the following property.
Each point of $A$ lies within distance $r$ of $B$, and each point of $B$ lies within distance $r$ of $A$.

Let $X$ and $Y$ be nonempty [[compact|0TVJZYB6SH]] metric spaces.
Their *Gromov–Hausdorff distance* is
$$d_{GH}(X, Y) = \inf d_H(f(X), g(Y)),$$
where the infimum runs over all metric spaces $Z$ and all [[isometric embeddings|02HPP5VCZH]] $f \colon X \to Z$ and $g \colon Y \to Z$.

The *Gromov–Hausdorff space* is the set of isometry classes of nonempty compact metric spaces, with the distance $d_{GH}$.

## Remarks

The function $d_{GH}$ is a metric on isometry classes.
In particular, $d_{GH}(X, Y) = 0$ for compact $X$ and $Y$ only when $X$ and $Y$ are isometric.
The Gromov–Hausdorff space is a [[complete metric space|0BY52JW6NK]] and is [[separable|0Z2P41SVDP]].
Finite metric spaces with rational distances form a countable dense subset.

For a one-point space $P$ and a compact metric space $X$, one has $d_{GH}(P, X) = \operatorname{diam}(X) / 2$.

## Explanations

### L1 human

The Gromov–Hausdorff space is a space whose points are themselves shapes, namely compact metric spaces.
Two shapes are close when they can be placed in a common space so that each is near the other.
For example, a fine finite grid of points is close to the square it sits in.
This lets one speak of a sequence of spaces converging to a limit space.
</document>

<document source="https://tamnd.github.io/math2/node/ETSXVDDCYK.md" tag="ETSXVDDCYK" kind="definition">
# Canonical module

## Definition

Let $(R, \mathfrak{m}, k)$ be a [[Cohen–Macaulay|N1QJQEGBFC]] [[local ring|MZ02BKPHJA]] of dimension $d$ with [[residue field|NJ3APQ8BHH]] $k$.
A *canonical module* of $R$ is a [[finitely generated|05QY1VEBYD]] $R$-module $\omega_R$ such that:

1. $\omega_R$ is a maximal Cohen–Macaulay module, that is, $\operatorname{depth} \omega_R = d$, where $\operatorname{depth}$ is the [[depth|M2RADBFW99]];
2. $\omega_R$ has finite injective dimension;
3. $\dim_k \operatorname{Ext}^d_R(k, \omega_R) = 1$, where $\operatorname{Ext}$ is the [[Ext functor|4P7GBWJ4RR]].

## Remarks

A canonical module is unique up to isomorphism when it exists.
It exists if and only if $R$ is a quotient of a Gorenstein local ring; in particular every complete Cohen–Macaulay local ring has one.
Example: $R$ is [[Gorenstein|T6HCX959PJ]] if and only if $R$ itself is a canonical module; for instance $\omega_R = R$ for $R = K[\![x_1, \ldots, x_n]\!]$.
Example: for an Artinian local ring $R$, the canonical module is the [[injective hull|QCJN05GN64]] $E_R(k)$ of the residue field.
Non-example: for $R = K[x,y]/(x^2, xy, y^2)$ the module $R$ is not a canonical module, because $\operatorname{Hom}_R(k, R)$ is the socle $Kx \oplus Ky$, of dimension $2$, so clause 3 fails.
</document>

<document source="https://tamnd.github.io/math2/node/EV25PMXVQT.md" tag="EV25PMXVQT" kind="definition">
# Hankel transform

## Definition

Let $\nu \ge -\frac12$, and let $J_\nu$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $\nu$.
Let $f \colon (0, \infty) \to \C$ be measurable with $\int_0^\infty \abs{f(r)}\, r^{1/2}\, dr < \infty$.
The *Hankel transform of order $\nu$* of $f$ is the function on $(0, \infty)$ given by the [[Lebesgue integral|DWJF7ABPR1]]
$$F_\nu(k) = \int_0^\infty f(r)\, J_\nu(kr)\, r\, dr.$$

## Remarks

The integral converges absolutely, because $\sqrt{z}\, \abs{J_\nu(z)}$ is bounded on $(0, \infty)$ for $\nu \ge -\frac12$, so $\abs{J_\nu(kr)}\, r \le C k^{-1/2} r^{1/2}$.
The inverse transform has the same form: under suitable conditions $f(r) = \int_0^\infty F_\nu(k)\, J_\nu(kr)\, k\, dk$.
Other normalizations use the kernel $J_\nu(kr) \sqrt{kr}$ with $dr$ instead of $r\, dr$.
Through the [[Fourier transform|DPVM34ADZV]] in the plane, the order-$0$ transform describes radial functions: if $g(x) = f(\abs{x})$ on $\R^2$, then $\int_{\R^2} g(x) e^{-i x \cdot \xi}\, dx = 2\pi F_0(\abs{\xi})$.

Example: $\nu = 0$ and $f(r) = e^{-r^2/2}$.
The plane Gaussian $e^{-\abs{x}^2/2}$ has Fourier transform $2\pi e^{-\abs{\xi}^2/2}$ with this kernel, so $F_0(k) = e^{-k^2/2}$.

Non-example: $f(r) = 1$ fails the integrability condition, and $\int_0^R J_0(kr)\, r\, dr = \frac{R}{k} J_1(kR)$ has no limit as $R \to \infty$, since $J_1(kR)$ decays only like $R^{-1/2}$.
</document>

<document source="https://tamnd.github.io/math2/node/EV6ZDQ0NDY.md" tag="EV6ZDQ0NDY" kind="concept">
# Exponential integral

## Summary

The exponential integrals are the functions $E_1(x) = \int_x^\infty \frac{e^{-t}}{t}\, dt$ for $x > 0$ and $\operatorname{Ei}(x)$, the principal value of $\int_{-\infty}^x \frac{e^t}{t}\, dt$ for $x \ne 0$.
They are related by $E_1(x) = -\operatorname{Ei}(-x)$ for $x > 0$, and $E_1(x)$ is the [[upper incomplete gamma function|YB7ZV0HG7K]] $\Gamma(0, x)$.
The [[logarithmic integral|VP98WSTZWT]] is $\operatorname{li}(x) = \operatorname{Ei}(\ln x)$.
They occur in heat transfer, in groundwater flow and in radiative transfer.

## Explanations

### L1 human

The function $e^t / t$ has no antiderivative made from elementary functions.
The exponential integral gives a name to its antiderivative, fixed by a choice of starting point.
The function $E_1(x)$ is the area under $e^{-t}/t$ to the right of $x$; it is large near $0$ and very small for large $x$.
Near $0$ it behaves like $-\ln x$, because $e^{-t}/t$ is close to $1/t$ there.
</document>

<document source="https://tamnd.github.io/math2/node/EV7HZCHQ5V.md" tag="EV7HZCHQ5V" kind="definition">
# Mixed model

## Definition

A *linear mixed model* is a statistical model for a random vector $y \in \R^n$ of the form
$$y = X\beta + Z u + \varepsilon,$$
where:

1. $X$ is a known real $n \times p$ matrix and $\beta \in \R^p$ is an unknown vector of *fixed effects*;
2. $Z$ is a known real $n \times q$ matrix and $u$ is a random vector of *random effects* with $u \sim N_q(0, G)$;
3. $\varepsilon \sim N_n(0, R)$ is a random error vector, independent of $u$;
4. the [[covariance matrices|0V7G6WMFK0]] $G$ and $R$ are [[positive definite|0KX66CZMBB]] and depend on unknown variance parameters.

Here $N_k(m, S)$ is the [[multivariate normal distribution|0XN3K7RSZY]].

## Remarks

The model implies $y \sim N_n(X\beta, \, Z G Z^T + R)$.
Without the term $Z u$ it is a [[general linear model|A06SGFCSCE]] with one response.

Example: the random intercept model.
For groups $i = 1, \dots, a$ and members $j = 1, \dots, m$, let $y_{ij} = \beta_0 + b_i + \varepsilon_{ij}$ with independent $b_i \sim N(0, \tau^2)$ and $\varepsilon_{ij} \sim N(0, \sigma^2)$.
This fits the definition with $p = 1$, $X$ the column of ones, $q = a$, $Z$ the matrix of group indicators, $G = \tau^2 I_a$ and $R = \sigma^2 I_{am}$.
Two members of the same group have covariance $\tau^2$ and correlation $\tau^2 / (\tau^2 + \sigma^2)$, the intraclass correlation.

Non-example: if the group shifts $b_i$ are treated as unknown constants rather than random variables, clause 2 fails.
The model is then a fixed effects model, a general linear model with the group indicators among the columns of $X$.

Given the variance parameters, $\beta$ is estimated by generalized least squares and $u$ is predicted by its best linear unbiased predictor.
The variance parameters are usually estimated by restricted maximum likelihood.
</document>

<document source="https://tamnd.github.io/math2/node/EVB1XHQTJR.md" tag="EVB1XHQTJR" kind="definition">
# Local coefficient system

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] with [[fundamental groupoid|ZBHGXJF24E]] $\Pi_1(X)$.
A *local coefficient system* (or *local system*) of [[abelian groups|082GHNPJ53]] on $X$ is a functor $L \colon \Pi_1(X) \to \mathrm{Ab}$.
In detail, it consists of:

1. an abelian group $L_x$ for each point $x \in X$;
2. an isomorphism $L_\gamma \colon L_x \to L_y$ for each path-homotopy class $\gamma$ of paths from $x$ to $y$;

such that $L_{\gamma \cdot \gamma'} = L_{\gamma'} \circ L_\gamma$ for composable classes and $L_\gamma = \id$ for the class of a constant path.

## Remarks

Let $X$ be path-connected and $x_0 \in X$.
Then a local system is determined up to isomorphism by the group $L_{x_0}$ with its action of the [[fundamental group|S1XK3HXHFX]] $\pi_1(X, x_0)$.
This is the same as a [[module|0NZ8JA25WK]] over the group ring $\Z[\pi_1(X, x_0)]$.
If $X$ is locally path-connected and semilocally simply connected, local systems are the same as locally constant sheaves of abelian groups on $X$.

Example: on the circle $S^1$, let $L_x = \Z$ for every $x$, and let transport along a path that winds $k$ times multiply by $(-1)^k$.
This respects composition, since $(-1)^{k + l} = (-1)^k (-1)^l$, so it is a local system.
It is not isomorphic to the constant system $\Z$, since the generator of $\pi_1(S^1)$ acts by $-1$ rather than $1$.

Non-example: on $S^1$, assigning to every path the same isomorphism $n \mapsto -n$ fails the clause on constant paths, which must act by the identity.

The orientation system of an $n$-manifold $M$, with $L_x = H_n(M, M \setminus \set{x}; \Z)$, is the main example; it is constant exactly when $M$ is orientable.
</document>

<document source="https://tamnd.github.io/math2/node/EVF180QJXY.md" tag="EVF180QJXY" kind="concept">
# C0-semigroup

## Summary

A $C_0$-semigroup, or strongly continuous semigroup, is a family $T(t)$, $t \ge 0$, of bounded linear operators on a [[Banach space|06GZ956657]] that behaves like $e^{tA}$.
It satisfies $T(0) = I$ and $T(t + s) = T(t) T(s)$, and $T(t) x \to x$ as $t \to 0$ for each vector $x$.
Its generator $A$ is in general an unbounded operator, and $u(t) = T(t) u_0$ solves $u' = A u$.
$C_0$-semigroups give the solutions of linear evolution equations such as the heat equation, transport equations and Markov processes.

## Explanations

### L1 human

Let $T(t)$ be the map that takes a state now to the state after time $t$.
Running for $s$ and then for $t$ is the same as running for $t + s$, which is the semigroup law.
For the transport equation $u_t = u_s$ on the line, $T(t)$ shifts a profile: $(T(t) f)(s) = f(s + t)$.
Strong continuity says that a short shift barely changes the profile, measured in the chosen norm.
For a matrix $A$, $T(t) = e^{tA}$ is the simplest example.
</document>

<document source="https://tamnd.github.io/math2/node/EVHCE5VR6Y.md" tag="EVHCE5VR6Y" kind="concept">
# Compact group

## Summary

A compact group is a topological group whose topology is compact and Hausdorff.
Compact groups carry a Haar measure of total mass $1$, so one can average over the group as over a finite group.
So their representation theory looks much like that of finite groups.
Every finite-dimensional representation splits into irreducible ones, and the Peter–Weyl theorem decomposes $L^2(G)$.

## Explanations

### L1 human

A compact group is a group that is also a "closed and bounded" space, with no way to run off to infinity.
The circle of rotations of the plane is the simplest infinite example, and the group of rotations of space is another.
Every finite group is a compact group with the discrete topology.
The key feature is that one can take the average of a function over all elements of the group, just as one averages over a finite set.
</document>

<document source="https://tamnd.github.io/math2/node/EVPS59M8X4.md" tag="EVPS59M8X4" kind="definition">
# Separably closed field

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
The field $K$ is *separably closed* if every [[separable polynomial|03J8BS7FMV]] $f \in K[x]$ of degree at least $1$ has a root in $K$.

## Remarks

The following are equivalent for a field $K$:

1. $K$ is separably closed;
2. every separable polynomial in $K[x]$ splits into linear factors over $K$;
3. $K$ has no [[separable extension|0N019F6RE9]] of degree greater than $1$.

Every [[algebraically closed field|02SJG1HC32]] is separably closed.
A field is algebraically closed if and only if it is separably closed and perfect.

Example: the separable closure $K^{\mathrm{sep}}$ of $K = \F_p(t)$ is separably closed but not algebraically closed.
Check: the roots of a separable polynomial over $K^{\mathrm{sep}}$ are separable over $K^{\mathrm{sep}}$, hence over $K$, so they lie in $K^{\mathrm{sep}}$.
But the root $t^{1/p}$ of $x^p - t$ is not separable over $K$.

Non-example: a finite field $\F_q$ is not separably closed.
The polynomial $x^q - x - 1$ is separable, since its derivative is $-1$, and it has no root in $\F_q$, since $a^q - a - 1 = -1$ for every $a \in \F_q$.
</document>

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# Ring of convergent power series

## Summary

The ring of convergent power series $\C\{z_1, \dots, z_n\}$ consists of the formal power series in $n$ variables that converge on some neighbourhood of the origin.
It is a subring of the ring of [[multivariate formal power series|3E8ZG3NJYZ]] $\C[\![z_1, \dots, z_n]\!]$.
It is isomorphic to the ring of [[germs of holomorphic functions|7PJ8KVZZCQ]] at $0 \in \C^n$, and it is the basic algebraic object of local analytic geometry.
It is a Noetherian local unique factorization domain, as is its formal completion.

## Explanations

### L1 human

A power series in several variables is an infinite sum of monomials $z_1^{k_1} \cdots z_n^{k_n}$ with coefficients.
Some such series converge near the origin, such as $\sum_{k} (z_1 z_2)^k$, and some converge only at the origin, such as $\sum_k k!\, z_1^k$.
The convergent ones can be added and multiplied, and they form a ring.
This ring encodes all holomorphic functions near the origin, through their Taylor series.
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# Fourier integral operator

## Summary

A Fourier integral operator is an operator of the form $u \mapsto \int e^{i \varphi(x, \xi)} a(x, \xi) \hat{u}(\xi) \, d\xi / (2\pi)^n$.
Here the phase $\varphi$ is homogeneous of degree one in $\xi$ and $a$ is a symbol.
The phase $x \cdot \xi$ gives the [[pseudo-differential operators|FF27N0BWTP]], and other phases move singularities along a canonical transformation of the cotangent bundle.
Solution operators of strictly hyperbolic equations, such as the wave equation, are Fourier integral operators.
The general theory was developed by Lars Hörmander.

## Explanations

### L1 human

A pseudo-differential operator can smooth or sharpen a function, but it does not move its singularities, such as jumps or kinks.
The solution of the wave equation does move them: a sharp wave front travels at the speed of light.
Fourier integral operators are the class of operators that can move singularities in a controlled way.
For example, the operator $e^{it\sqrt{-\Delta}}$, which multiplies the frequency $\xi$ by $e^{it\abs{\xi}}$, moves each singularity a distance $t$ in a direction set by its frequency.
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<document source="https://tamnd.github.io/math2/node/EVXCN06FW3.md" tag="EVXCN06FW3" kind="concept">
# Crystalline cohomology

## Summary

Crystalline cohomology is a cohomology theory for varieties over a perfect field $k$ of characteristic $p > 0$.
Its values are modules over the ring $W(k)$ of [[Witt vectors|6N3X2QX4W9]].
Grothendieck proposed it and Berthelot developed it, to obtain a $p$-adic cohomology in characteristic $p$, where [[ℓ-adic cohomology|E1DVHN06CR]] with $\ell = p$ fails.
When the variety lifts to $W(k)$, it is the [[algebraic de Rham cohomology|13F5JZPDD0]] of the lift.
So it is a de Rham theory that does not depend on the choice of a lift.
It carries a Frobenius action and is a main object of $p$-adic Hodge theory.

## Explanations

### L1 human

De Rham cohomology over a field of characteristic $p$ gives vector spaces over that field, and it loses information, because differentiating $t^p$ gives $0$.
One would like to lift the variety to the $p$-adic integers, or more generally to $W(k)$, and compute de Rham cohomology there.
Lifts do not always exist and are not unique.
Crystalline cohomology computes what the answer would be from the variety alone, using all its thickenings at once.
For the projective line over $\F_p$ it gives $\Z_p$ in degrees $0$ and $2$, as the de Rham cohomology of the projective line over $\Z_p$ does.
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# Weak operator topology

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] with [[inner product|0NB0AWGJBW]] $\langle \cdot, \cdot \rangle$, and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.
The *weak operator topology* on $B(H)$ is the coarsest topology such that, for all $x, y \in H$, the map $B(H) \to \C$, $T \mapsto \langle T x, y \rangle$, is continuous.

## Remarks

Equivalently, it is the [[topology generated by the seminorms|ZVFXQ7S6FY]] $T \mapsto \abs{\langle T x, y \rangle}$ for $x, y \in H$.
A net $(T_i)$ converges to $T$ in this topology if and only if $\langle T_i x, y \rangle \to \langle T x, y \rangle$ for all $x, y \in H$.

Example: on $H = \ell^2(\N)$, let $S$ be the right shift, $S(x_0, x_1, \dots) = (0, x_0, x_1, \dots)$.
Then $\langle S^n x, y \rangle = \sum_{k \ge 0} x_k \overline{y_{k+n}}$, and by the Cauchy–Schwarz inequality its modulus is at most $\|x\| \, (\sum_{k \ge n} \abs{y_k}^2)^{1/2}$, which tends to $0$.
So $S^n \to 0$ in the weak operator topology, while $\|S^n\| = 1$ for all $n$.

Non-example: let $\lambda \in \C$ with $\abs{\lambda} = 1$ and $\lambda \ne 1$, and let $T_n = \lambda^n I$.
For $x = y \ne 0$, the numbers $\langle T_n x, x \rangle = \lambda^n \|x\|^2$ do not converge, so $(T_n)$ has no limit in this topology.

The weak operator topology is coarser than the strong operator topology, which is coarser than the norm topology.
The closed unit ball of $B(H)$ is compact in the weak operator topology.
Multiplication is separately continuous but not jointly continuous when $H$ is infinite-dimensional, and $T \mapsto T^*$ is continuous.
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# Eilenberg swindle

## Definition

Let $R$ be a ring, and let $P$ and $Q$ be left $R$-[[modules|0NZ8JA25WK]] such that $P \oplus Q \cong F$ for a [[free module|0DZ9DKAKZR]] $F$.
Let $F^{(\infty)} = F \oplus F \oplus \cdots$ be the direct sum of countably many copies of $F$.
The *Eilenberg swindle* is the chain of isomorphisms
$$P \oplus F^{(\infty)} \cong P \oplus (Q \oplus P) \oplus (Q \oplus P) \oplus \cdots \cong (P \oplus Q) \oplus (P \oplus Q) \oplus \cdots \cong F^{(\infty)},$$
obtained by regrouping the summands of a countable direct sum.
Its conclusion is that $P \oplus F^{(\infty)} \cong F^{(\infty)}$, where $F^{(\infty)}$ is free.

## Remarks

Example: every [[projective module|BFFA03GE3V]] $P$ is a direct summand of a free module $F$, so the swindle applies to it.
Take $R = \Z/6\Z$ and $P = 3\Z/6\Z$, $Q = 2\Z/6\Z$; then $P \oplus Q \cong R$, and the swindle gives $P \oplus R^{(\infty)} \cong R^{(\infty)}$, although $P$ itself is not free.

Non-example: the regrouping needs infinitely many summands.
With finitely many summands the argument breaks.
Indeed, for $R$ commutative and $P \ne 0$, the isomorphism $P \oplus R^n \cong R^n$ fails.
To see this, compare dimensions over the residue field at a prime where $P$ is nonzero.

For this reason the Grothendieck group $K_0(R)$ uses only finitely generated projective modules: with countably generated ones every class would vanish.
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# Entropy solution

## Definition

Let $n \ge 1$, let $f \colon \R \to \R^n$ be [[continuously differentiable|0VW90BBSJ4]], and let $g \colon \R^n \to \R$ be bounded and measurable.
Consider the scalar conservation law
$$u_t + \operatorname{div} f(u) = 0 \text{ in } \R^n \times (0, \infty), \qquad u(x, 0) = g(x).$$
A bounded measurable function $u \colon \R^n \times (0, \infty) \to \R$ is an *entropy solution* in the sense of Kruzhkov if the following holds.
For every real number $k$ and every nonnegative test function $\varphi \in C_c^\infty(\R^n \times \R)$,
$$\int_0^\infty \!\! \int_{\R^n} \Big( \abs{u - k} \, \varphi_t + \sgn(u - k) \big( f(u) - f(k) \big) \cdot \nabla_x \varphi \Big) \, dx \, dt + \int_{\R^n} \abs{g - k} \, \varphi(x, 0) \, dx \ge 0.$$

## Remarks

Take $k$ larger than $\sup \abs{u}$ and $\sup \abs{g}$, and then smaller than $-\sup \abs{u}$ and $-\sup \abs{g}$.
This gives two opposite inequalities whose combination says that $u$ is a [[weak solution|SJZX81A783]] of the initial value problem.
So every entropy solution is a weak solution.
Kruzhkov's theorem states that for each bounded measurable $g$ there is exactly one entropy solution, and that it depends continuously on $g$ in $L^1_{\mathrm{loc}}$.
Let $u$ be piecewise smooth in one space dimension, with $f$ strictly convex.
Then the condition is equivalent to the Rankine–Hugoniot jump condition together with the Lax condition $u_- > u_+$ across each shock.
Here $u_-$ and $u_+$ are the values to the left and to the right of the shock.

Example: for Burgers' equation, $n = 1$ and $f(u) = u^2/2$, with $g = 1$ for $x < 0$ and $g = 0$ for $x > 0$, the shock $u = 1$ for $x < t/2$ and $u = 0$ for $x > t/2$ is the entropy solution.
Its speed $1/2$ is $(f(1) - f(0))/(1 - 0)$, as Rankine–Hugoniot requires, and $u_- = 1 > 0 = u_+$.

Non-example: with $g = 0$ for $x < 0$ and $g = 1$ for $x > 0$, the shock $u = 0$ for $x < t/2$ and $u = 1$ for $x > t/2$ is a weak solution but not an entropy solution, because $u_- = 0 < 1 = u_+$.
The entropy solution is the rarefaction fan $u = 0$ for $x < 0$, $u = x/t$ for $0 \le x \le t$, and $u = 1$ for $x > t$.
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# Symbolic dynamics

## Summary

Symbolic dynamics studies systems whose states are infinite sequences of symbols from a finite alphabet.
Their time evolution is the shift of a sequence by one place.
These systems are the [[shift spaces|A1XE59ZHNM]].
It is also a method: one codes an orbit of a general system by the pieces of a finite partition that it visits.
Through such codings, questions about smooth or hyperbolic systems become combinatorial questions about sequences, as for [[subshifts of finite type|624XDXX8FN]].

## Explanations

### L1 human

Divide the space of a system into a few labelled regions, say $0$ and $1$.
Instead of following a point exactly, write down the label of the region it visits at each time.
The orbit becomes an infinite word such as $0110100\ldots$, and moving forward in time just deletes the first letter.
For the doubling map $x \mapsto 2x \bmod 1$, with the regions $[0, 1/2)$ and $[1/2, 1)$, the word of $x$ is its binary expansion.
Symbolic dynamics studies the sets of words that arise this way.
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# Duistermaat–Heckman formula

## Statement

Let $(M, \omega)$ be a compact [[symplectic manifold|CAN490EPRT]] of dimension $2n$.
Let a torus $T$ act on $M$ with a [[moment map|ZARM34FZ6Z]] $\mu \colon M \to \mathfrak{t}^*$, and assume that the fixed points of $T$ are isolated.
At a fixed point $p$, identify $(T_p M, \omega_p)$ with $\C^n$ by an $\omega_p$-compatible complex structure.
Let $w_{1,p}, \ldots, w_{n,p} \in \mathfrak{t}^*$ be the weights of the action of $T$ on it, with signs fixed by the same convention as the sign of $\mu$.
Then for every $X \in \mathfrak{t}$ with $\langle w_{j,p}, X \rangle \ne 0$ for all $j$ and $p$,
$$\int_M e^{\langle \mu, X \rangle} \frac{\omega^n}{n!} = (2\pi)^n \sum_{p \in M^T} \frac{e^{\langle \mu(p), X \rangle}}{\prod_{j=1}^n \langle w_{j,p}, X \rangle}.$$

## Remarks

If $X$ is replaced by $iX$, the integral on the left becomes an oscillatory integral.
The right side is then exactly its stationary phase approximation, so this approximation has no error.
An equivalent form concerns the [[pushforward measure|3N66TM3TSQ]] $\mu_*(\omega^n / n!)$, the Duistermaat–Heckman measure.
It has a piecewise polynomial density with respect to [[Lebesgue measure|2Z2QVWZ262]] on the affine span of $\mu(M)$.
This form holds without the assumption of isolated fixed points.
A modern proof uses equivariant cohomology and the Atiyah–Bott–Berline–Vergne localization formula.

Example: the circle acts on the unit sphere $S^2$ by rotation about the vertical axis, $\omega$ is the area form of total area $4\pi$, and $\mu$ is the height $z$.
By Archimedes, the area between heights $a < b$ is $2\pi(b - a)$, so the pushforward measure is $2\pi$ times Lebesgue measure on $[-1, 1]$.
Then $\int_{S^2} e^{tz} \omega = 2\pi (e^t - e^{-t}) / t$, which matches the two fixed points $z = 1$ and $z = -1$ with weights $1$ and $-1$.

## Explanations

### L1 human

When a torus symmetry acts on a compact symplectic manifold, certain integrals over the whole manifold can be computed from the few points that the symmetry fixes.
The formula replaces an integral by a finite sum.
It also says that the symplectic volume is spread over the moment polytope by a piecewise polynomial density.

### L2 human

On $S^2$ the density is constant, which is Archimedes' theorem that slices of a sphere of equal height have equal area.
The formula is an early instance of localization in equivariant cohomology, and it inspired many results on exact integration, including path integral heuristics in physics.
Together with the [[Atiyah–Guillemin–Sternberg convexity theorem|5CTHC6612N]] it gives a detailed picture of the image of a moment map.
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# Plane curve

## Summary

A plane algebraic curve is a curve in the affine or projective plane defined by one polynomial equation.
Its degree is the degree of that polynomial.
By [[Bézout's theorem|KWNVF194D8]], it is the number of points in which a general line meets the curve.
Lines, conics and cubics are the plane curves of degree $1$, $2$ and $3$.
More generally, in geometry a plane curve is any curve that lies in a plane.

## Explanations

### L1 human

A plane curve is the set of points $(x, y)$ satisfying one polynomial equation $f(x, y) = 0$.
The circle $x^2 + y^2 = 1$ has degree $2$, and the curve $y^2 = x^3 - x$ has degree $3$.
Adding points at infinity gives a projective plane curve, defined by a homogeneous polynomial in three variables.
Many properties of the curve, such as its [[genus|6W8TDV9VVN]] when it is smooth, can be read off from the degree alone.
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# Macdonald polynomial

## Definition

Let $\Lambda$ be the ring of [[symmetric functions|PAJEHWXW6M]] in variables $x = (x_1, x_2, \dots)$ with coefficients in the field $\Q(q, t)$.
For a [[partition|0JV30XN73Z]] $\lambda$, let $m_\lambda$ be the monomial symmetric function and let $p_\lambda = p_{\lambda_1} p_{\lambda_2} \cdots$ be the product of [[power sums|0ZNXZB4HNB]].
Let $z_\lambda = \prod_{i \ge 1} i^{m_i} m_i!$, where $m_i$ is the number of parts of $\lambda$ equal to $i$.
Define a bilinear form on $\Lambda$ by
$$\ip{p_\lambda}{p_\mu}_{q,t} = \delta_{\lambda\mu} \, z_\lambda \prod_{i} \frac{1 - q^{\lambda_i}}{1 - t^{\lambda_i}}.$$
For partitions $\lambda, \mu$ of the same number, write $\mu \le \lambda$ (dominance order) if $\mu_1 + \dots + \mu_k \le \lambda_1 + \dots + \lambda_k$ for all $k$.
The *Macdonald polynomials* are the unique family $(P_\lambda(x; q, t))_\lambda$ in $\Lambda$ such that:

1. $P_\lambda = m_\lambda + \sum_{\mu < \lambda} c_{\lambda\mu} m_\mu$ for some $c_{\lambda\mu} \in \Q(q, t)$.
2. $\ip{P_\lambda}{P_\mu}_{q,t} = 0$ for $\lambda \ne \mu$.

## Remarks

The existence and uniqueness of this family is a theorem of Macdonald; it is not automatic, because dominance is only a partial order.

Example: for $\lambda = (2)$, $P_{(2)} = m_{(2)} + \frac{(1+q)(1-t)}{1-qt} m_{(1,1)}$.
Check: $m_{(2)} = p_2$ and $m_{(1,1)} = \tfrac12 (p_1^2 - p_2)$, and $z_{(2)} = z_{(1,1)} = 2$.
Then $\ip{p_2}{m_{(1,1)}}_{q,t} = -\frac{1-q^2}{1-t^2}$ and $\ip{m_{(1,1)}}{m_{(1,1)}}_{q,t} = \frac{(1-q)(1-qt)}{(1-t)(1-t^2)}$, so $\ip{P_{(2)}}{P_{(1,1)}}_{q,t} = 0$ with $P_{(1,1)} = m_{(1,1)}$.

Non-example: the Schur function $s_{(2)} = m_{(2)} + m_{(1,1)}$ satisfies clause 1 but not clause 2 when $q \ne t$.
Indeed $\ip{s_{(2)}}{m_{(1,1)}}_{q,t} = \frac{(1-q)(t-q)}{(1-t)(1-t^2)} \ne 0$.

At $q = t$ the form is the Hall inner product and $P_\lambda = s_\lambda$.
At $q = 0$, $P_\lambda$ is the Hall–Littlewood polynomial $P_\lambda(x; t)$.
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# Transversality

## Definition

Let $X$ and $Y$ be [[smooth manifolds|F45VQ47GPV]], let $Z \subseteq Y$ be an [[embedded submanifold|VKFW27WXEZ]], and let $f \colon X \to Y$ be a smooth map.
The map $f$ is *transverse* to $Z$, written $f \pitchfork Z$, if for every $x \in X$ with $f(x) \in Z$,
$$df_x(T_xX) + T_{f(x)}Z = T_{f(x)}Y,$$
where $df_x$ is the [[differential|6P9PXQDSEN]] of $f$ at $x$.
Two embedded submanifolds $A, B \subseteq Y$ are *transverse*, written $A \pitchfork B$, if $T_pA + T_pB = T_pY$ for every $p \in A \cap B$; this is transversality of the inclusion $A \to Y$ to $B$.

## Remarks

The condition is empty where $f(X)$ misses $Z$, so a map whose image avoids $Z$ is transverse to $Z$.
If $Z$ is a point $y$, then $f \pitchfork \set{y}$ means that $y$ is a regular value of $f$.
If $f \pitchfork Z$, then $f^{-1}(Z)$ is an embedded submanifold of $X$ with $\dim X - \dim f^{-1}(Z) = \dim Y - \dim Z$.
Example: in $\R^2$ the $x$-axis and the line $y = x$ are transverse at the origin, since their directions $(1, 0)$ and $(1, 1)$ span $\R^2$.
Non-example: the $x$-axis and the parabola $y = x^2$ meet at the origin, where both tangent lines equal the $x$-axis, which does not span $\R^2$; so they are not transverse.
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# Renewal equation

## Example

Let $F$ be the distribution function of a nonnegative random variable with $F(0) = 0$, and let $g$ be a locally bounded function on $[0, \infty)$.
The *renewal equation* for an unknown locally bounded function $u$ on $[0, \infty)$ is
$$u(t) = g(t) + \int_0^t u(t - s) \, dF(s), \quad t \ge 0.$$
When $F$ has a density $k = F'$, the integral is $\int_0^t k(t - s) \, u(s) \, ds$, and the equation is a [[Volterra equation of convolution type|SNFYTC1TTA]] of the second kind.

In [[renewal theory|45V0W838BQ]], let $X_1, X_2, \dots$ be independent waiting times with distribution function $F$, and let $N(t)$ be the number of $n \ge 1$ with $X_1 + \cdots + X_n \le t$.
Conditioning on the first waiting time $X_1$ shows that the renewal function $m(t) = \mathbb{E}[N(t)]$ solves the renewal equation with $g = F$:
$$m(t) = F(t) + \int_0^t m(t - s) \, dF(s).$$

Check on the exponential case: let $F(t) = 1 - e^{-\lambda t}$ with $\lambda > 0$, so $dF(s) = \lambda e^{-\lambda s} \, ds$.
Then $N$ is a Poisson process of rate $\lambda$, and $m(t) = \lambda t$.
Indeed $\int_0^t \lambda (t - s) \lambda e^{-\lambda s} \, ds = \lambda t - 1 + e^{-\lambda t}$, and adding $F(t) = 1 - e^{-\lambda t}$ gives $\lambda t$.

## Remarks

The renewal equation has a unique locally bounded solution, $u = g + m * g$, that is, $u(t) = g(t) + \int_0^t g(t - s) \, dm(s)$.
Let $\varphi_F(p) = \int_{[0, \infty)} e^{-ps} \, dF(s)$.
Taking Laplace transforms gives $\hat u = \hat g / (1 - \varphi_F)$, as in the [[Laplace transform method|EHGECREYB4]].
For the exponential case, $\hat F(p) = \frac{\lambda}{p(p + \lambda)}$ and $\varphi_F(p) = \frac{\lambda}{p + \lambda}$, so $\hat m(p) = \frac{\lambda}{p(p + \lambda)} \cdot \frac{p + \lambda}{p} = \frac{\lambda}{p^2}$, the transform of $\lambda t$.

## Explanations

### L1 human

Think of a light bulb that is replaced each time it fails.
The expected number of replacements by time $t$ is the chance of a first failure by $t$, plus what happens after a first failure at time $s$.
After that failure, the same problem starts again with $t - s$ time left.
This "restart" argument gives the renewal equation, an integral equation in which the unknown appears with a time shift.
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# Brown–McCoy radical

## Summary

The Brown–McCoy radical of a ring with identity is the intersection of its maximal two-sided ideals.
It is the two-sided analogue of the [[Jacobson radical|XA0DCNH7WY]], which uses maximal left ideals instead.
It contains the Jacobson radical, and for a [[commutative ring|0310REJJY8]] the two radicals coincide.
It is a radical in the sense of the [[Kurosh–Amitsur radical theory|A72J7ZVNR5]].

## Explanations

### L1 human

The quotients of a ring by its maximal two-sided ideals are the simple rings it maps onto.
The Brown–McCoy radical is what all these maps kill.
For the integers, the maximal ideals are $p\Z$ for primes $p$, and their intersection is $0$.
Consider the ring of all linear maps of a countably infinite-dimensional vector space.
Its Brown–McCoy radical is the ideal of maps of finite rank, although its Jacobson radical is $0$.
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# Kodaira–Spencer map

## Summary

The Kodaira–Spencer map of a [[deformation|VKQSM3BCE4]] measures how fast the complex structure of the fibre changes.
It sends a tangent vector of the base at $t$ to a class in $H^1(X_t, \Theta_{X_t})$.
Here $\Theta_{X_t}$ is the sheaf of holomorphic vector fields on the fibre $X_t$.
It vanishes identically for a trivial family, and it is the basic tool to decide whether a family is complete or effectively parametrized.
It is the first-order part of the deformation theory of compact complex manifolds.

## Explanations

### L1 human

A family of complex manifolds is glued, locally over the base, from pieces of a product.
Moving in the base changes the gluing maps, and the derivative of this change is a holomorphic vector field on each overlap.
These vector fields form a Čech cocycle, and its class in $H^1(X_t, \Theta_{X_t})$ does not depend on the choices.
If the class is zero, the complex structure does not change to first order in that direction.
For the family of elliptic curves $\C/(\Z + \tau\Z)$, the map is nonzero at every $\tau$.
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# Brauer group

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
A [[central simple algebra|5HRKJBAY2V]] over $K$ is a finite-dimensional associative $K$-algebra $A \ne 0$ whose center is $K$ and whose only two-sided ideals are $0$ and $A$.
For a $K$-algebra $A$, let $M_m(A)$ be the [[matrix ring|0FHVPRAE60]] of $m \times m$ matrices over $A$.
Two central simple algebras $A$ and $B$ over $K$ are *Brauer equivalent* if $M_m(A) \cong M_n(B)$ as $K$-algebras for some integers $m, n \ge 1$.
The *Brauer group* $\mathrm{Br}(K)$ is the set of Brauer equivalence classes $[A]$, with the operation
$$[A] \cdot [B] = [A \otimes_K B],$$
the [[tensor product|F0HQXQ0HCD]] over $K$.

## Remarks

The operation is well defined, and $\mathrm{Br}(K)$ is an [[abelian group|071DDFVG4J]] with identity $[K]$ and inverse $[A]^{-1} = [A^{\mathrm{op}}]$, the opposite algebra; this rests on $A \otimes_K A^{\mathrm{op}} \cong M_d(K)$ with $d = \dim_K A$.
By Wedderburn's theorem each central simple algebra is $M_n(D)$ for a central division algebra $D$ unique up to isomorphism, so each class contains exactly one such $D$.
Example: $\mathrm{Br}(\R)$ has two elements, $[\R]$ and $[\mathbb{H}]$, by the [[Frobenius theorem|QDJ7PQE9SW]].
Example: $\mathrm{Br}(K) = 0$ for $K$ algebraically closed and for $K$ finite.
Non-example: the $K$-algebra $K \times K$ has no class in $\mathrm{Br}(K)$, since its center is $K \times K$, not $K$, and $K \times \set{0}$ is a two-sided ideal.
There is an isomorphism $\mathrm{Br}(K) \cong H^2(K, (K^s)^\times)$ with [[Galois cohomology|9MPCZ3EAJM]], where $K^s$ is a separable closure of $K$.
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<document source="https://tamnd.github.io/math2/node/EX382NQTGP.md" tag="EX382NQTGP" kind="concept">
# Enumerative combinatorics

## Summary

Enumerative combinatorics is the part of [[combinatorics|0JKW7W7XFR]] that counts finite structures, see the [[definition|NWGARTY61Y]].
It asks for the number of objects of size $n$ in a family, as a function of $n$.
Its answers take the form of closed formulas, [[recurrence relations|0FYX44GP92]], [[generating functions|0PHAC833JF]] or asymptotic estimates.
Its tools range from bijections and the inclusion-exclusion principle to complex analysis in [[analytic combinatorics|YJK71RKVEB]].

## Explanations

### L1 human

Enumerative combinatorics answers questions that begin with "how many".
How many ways are there to climb $n$ stairs taking one or two steps at a time?
The answer is a Fibonacci number, and a recurrence finds it: the last step is a single or a double step.
The aim is a formula or a fast method that works for every $n$ at once, not a list of cases.
</document>

<document source="https://tamnd.github.io/math2/node/EX6YSYN7B4.md" tag="EX6YSYN7B4" kind="statement">
# Lefschetz hyperplane theorem

## Statement

Let $X$ be a complex [[projective variety|A4H4CZ2V35]] of dimension $n$.
Let $D \subseteq X$ be an effective [[ample|RRK8P6TXG2]] [[Cartier divisor|FHV05ZK99F]], for example a hyperplane section, such that $X \setminus D$ is smooth.
Then, for [[singular cohomology|6N7BC70CX0]] with integer coefficients, the restriction map
$$H^i(X, \Z) \to H^i(D, \Z)$$
is an isomorphism for $i \le n - 2$ and injective for $i = n - 1$.

## Remarks

The idea of the proof: $X \setminus D$ is a smooth affine variety of dimension $n$.
By the Andreotti–Frankel theorem it has the homotopy type of a CW complex of real dimension at most $n$.
Lefschetz duality then gives $H^i(X, D; \Z) \cong H_{2n - i}(X \setminus D; \Z) = 0$ for $i < n$, and the long exact sequence of the pair gives the claim.
The homotopy version says that $\pi_i(D) \to \pi_i(X)$ is an isomorphism for $i \le n - 2$ and surjective for $i = n - 1$.

Example: a smooth hypersurface $X \subseteq \mathbb{P}^{n+1}_\C$ of dimension $n$ is an ample divisor in $\mathbb{P}^{n+1}$.
So $H^i(X, \Z) \cong H^i(\mathbb{P}^{n+1}, \Z)$ for $i \le n - 1$, which is $\Z$ for even $i$ and $0$ for odd $i$; in particular $X$ is simply connected when $n \ge 2$.

The bound $n - 1$ is sharp.
Let $X = \mathbb{P}^1 \times \mathbb{P}^1$, of dimension $n = 2$, embedded in $\mathbb{P}^3$ as a quadric by the Segre embedding.
A hyperplane section is a smooth conic $D \cong \mathbb{P}^1$, and $H^2(X, \Z) \cong \Z^2$ while $H^2(D, \Z) \cong \Z$.
So the map in degree $n = 2$ is not injective.

## Explanations

### L1 human

Cut a projective variety by a hyperplane.
The theorem says that the slice has the same cohomology as the whole variety in all low degrees, below the middle dimension of the slice.
So the topology of a variety is largely determined by the topology of its hyperplane sections, and new topology can only appear in the middle degree.

### L2 human

The theorem is the basis for induction on dimension in the topology of algebraic varieties, through Lefschetz pencils.
Together with Poincaré duality it shows that all the interesting cohomology of a smooth complete intersection sits in the middle degree.
It is complemented by the hard Lefschetz theorem, which describes the middle and upper degrees, and it has analogues in étale cohomology used in the proof of the Weil conjectures.
</document>

<document source="https://tamnd.github.io/math2/node/EX7N9D1YCD.md" tag="EX7N9D1YCD" kind="concept">
# Krull dimension of a poset

## Summary

The Krull dimension of a [[partially ordered set|0QV2JS9RJ3]] is the supremum of the lengths of its finite [[chains|0S8FE7ZMFM]].
Here a chain with $n + 1$ elements has length $n$.
It generalizes the [[Krull dimension of a commutative ring|MVGBS3J1P7]], which is the Krull dimension of the set of prime ideals ordered by inclusion.
For a nonempty finite poset it is one less than the [[height|MPR6MH4YSQ]] counted in elements.

## Explanations

### L1 human

Count the number of upward steps in the longest climb through the poset.
A single point has dimension $0$, a two-element chain has dimension $1$, and the subsets of a set with $n$ elements have dimension $n$.
The name comes from algebra, where a point, a curve and a surface give chains of prime ideals with $0$, $1$ and $2$ steps.
</document>

<document source="https://tamnd.github.io/math2/node/EX9NCDFYC7.md" tag="EX9NCDFYC7" kind="concept">
# Lattice of clones

## Summary

The lattice of clones on a set $A$ is the set of all [[clones|PTKR703G6A]] on $A$, ordered by inclusion.
It is a complete lattice: the meet of a family of clones is its intersection, and the join is the clone generated by the union.
Its least element is the clone of [[projections|769B35T6WG]] and its greatest element is the clone of all finitary operations.
On a two-element set it is countable, and [[Post's lattice|MA20X7JCET]] describes it completely.
On a finite set with at least three elements it has the cardinality of the continuum.

## Explanations

### L1 human

Each clone is a toolbox of operations, and a bigger clone can build more.
Ordering the toolboxes by inclusion gives the lattice of clones.
At the bottom sits the clone that can only copy inputs; at the top sits the clone that can build every operation.
On $\set{0, 1}$ there are countably many clones, while on $\set{0, 1, 2}$ there are already uncountably many.
</document>

<document source="https://tamnd.github.io/math2/node/EX9YYRMAJA.md" tag="EX9YYRMAJA" kind="example">
# Radiative transfer equation

## Example

Let $D \subseteq \R^3$ be a region filled with a medium, and let $S^2$ be the unit sphere of directions.
The unknown is the specific intensity $I(x, \Omega)$ of radiation at a point $x \in D$ travelling in the direction $\Omega \in S^2$, at one fixed frequency.
Let $\kappa(x) \ge 0$ and $\sigma(x) \ge 0$ be the absorption and scattering coefficients, $B(x)$ the emission source, and $p(\Omega' \cdot \Omega) \ge 0$ the phase function, normalized by $\frac{1}{4\pi} \int_{S^2} p(\Omega' \cdot \Omega) \, d\Omega' = 1$.
The stationary radiative transfer equation is
$$\Omega \cdot \nabla_x I(x, \Omega) + \big(\kappa(x) + \sigma(x)\big) I(x, \Omega) = \kappa(x) B(x) + \frac{\sigma(x)}{4\pi} \int_{S^2} p(\Omega' \cdot \Omega) \, I(x, \Omega') \, d\Omega'.$$
It contains a derivative of $I$ along the direction $\Omega$ and an integral of $I$ over all directions.
So it is an [[integro-partial differential equation|TH4NGC4016]].

Check: if $\kappa$, $\sigma$ and $B$ are constant, the constant intensity $I = B$ is a solution.
The left side is $(\kappa + \sigma) B$, and by the normalization of $p$ the right side is $\kappa B + \sigma B$.

## Remarks

The left side describes the loss of intensity along a ray by absorption and by scattering out of the direction $\Omega$.
The right side describes thermal emission and scattering into $\Omega$ from all other directions.
Along each ray $x + s\Omega$ the equation is a linear ordinary differential equation, and integrating it turns the problem into an integral equation for $I$.
For a plane-parallel atmosphere with isotropic scattering this leads to equations such as the [[Milne problem|KKT4Y0TBRZ]].

## Explanations

### L1 human

Light travelling through fog, a star or a furnace is partly absorbed, partly scattered into new directions, and new light is emitted.
The equation keeps track of how much light goes in each direction at each point.
The scattering term collects light from all directions, which is why an integral over directions appears next to the derivative along the ray.
</document>

<document source="https://tamnd.github.io/math2/node/EXD1197NPH.md" tag="EXD1197NPH" kind="definition">
# Geometric morphism

## Definition

Let $\mathcal{E}$ and $\mathcal{F}$ be [[toposes|6837RS7N82]].
A *geometric morphism* $f \colon \mathcal{E} \to \mathcal{F}$ is a pair of functors $f_* \colon \mathcal{E} \to \mathcal{F}$ and $f^* \colon \mathcal{F} \to \mathcal{E}$ such that:

1. $f^*$ is [[left adjoint|TZHF6ZM92S]] to $f_*$;
2. $f^*$ [[preserves finite limits|H4MY8JN8W4]].

The functor $f_*$ is the *direct image* and $f^*$ is the *inverse image* of $f$.

## Remarks

Example: let $g \colon X \to Y$ be a continuous map of topological spaces.
The direct image $g_* F(V) = F(g^{-1}(V))$ and the inverse image $g^*$ of sheaves form a geometric morphism $\mathrm{Sh}(X) \to \mathrm{Sh}(Y)$.
The inverse image preserves finite limits, because stalks of $g^*G$ are stalks of $G$ and finite limits of sheaves are computed stalkwise.
Non-example: on $\mathbf{Set}$, take $f^* = - \times A$ and $f_* = (-)^A$ for a set $A$ with two elements.
Then $f^* \dashv f_*$, but $f^*$ sends the terminal set $1$ to $A$, which is not terminal, so clause 2 fails.
For a topos $\mathcal{E}$, a *point* of $\mathcal{E}$ is a geometric morphism $\mathbf{Set} \to \mathcal{E}$.
Every Grothendieck topos has, up to isomorphism, exactly one geometric morphism to $\mathbf{Set}$.
</document>

<document source="https://tamnd.github.io/math2/node/EXHJHA1TBK.md" tag="EXHJHA1TBK" kind="definition">
# Quillen equivalence

## Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be [[model categories|SDTZVZ6B4Q]], and let $F \colon \mathcal{C} \rightleftarrows \mathcal{D} \colon G$ be a [[Quillen adjunction|SY38F954FE]], with $F$ [[left adjoint|TZHF6ZM92S]] to $G$.
For $X$ in $\mathcal{C}$ and $Y$ in $\mathcal{D}$, write $g^\flat \colon X \to GY$ for the adjoint of a map $g \colon FX \to Y$.
The adjunction is a *Quillen equivalence* if the following holds for every cofibrant object $X$ of $\mathcal{C}$ and every fibrant object $Y$ of $\mathcal{D}$.
A map $g \colon FX \to Y$ is a weak equivalence in $\mathcal{D}$ if and only if $g^\flat$ is a weak equivalence in $\mathcal{C}$.

## Remarks

Equivalently, the derived adjunction $\mathbf{L}F \colon \operatorname{Ho}(\mathcal{C}) \rightleftarrows \operatorname{Ho}(\mathcal{D}) \colon \mathbf{R}G$ is an [[equivalence of categories|K7A04ZBD92]].

Example: geometric realization $\abs{\cdot}$ from simplicial sets to topological spaces is left adjoint to the singular complex functor $\operatorname{Sing}$.
It is a Quillen equivalence between the Kan–Quillen model structure and the Quillen model structure.
Every simplicial set is cofibrant and every space is fibrant.
The check reduces to two facts: the unit $K \to \operatorname{Sing}\abs{K}$ is a weak equivalence, and the counit $\abs{\operatorname{Sing} Y} \to Y$ is a weak homotopy equivalence.

Non-example: give the category of sets the model structure whose weak equivalences are the bijections and in which every map is a cofibration and a fibration.
The functor $\pi_0$ from simplicial sets to sets is left adjoint to the functor that sends a set to the discrete simplicial set on it.
This is a Quillen adjunction.
It is not a Quillen equivalence.
Take the simplicial circle $K = \Delta^1 / \partial \Delta^1$; the identity map of the one-point set $\pi_0 K$ is a bijection.
Its adjoint $K \to \Delta^0$ is not a weak equivalence, since $\abs{K}$ is a circle.
</document>

<document source="https://tamnd.github.io/math2/node/EXJAXVXW51.md" tag="EXJAXVXW51" kind="definition">
# Zariski topology

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $X$ be the set of [[prime ideals|00MHHZ9SNE]] of $R$.
For a subset $E \subseteq R$, let
$$V(E) = \set{\mathfrak p \in X : E \subseteq \mathfrak p}.$$
The *Zariski topology* on $X$ is the [[topology|0CQHFPGJ45]] whose [[closed sets|0ZCK8DQYN1]] are exactly the sets $V(E)$ for $E \subseteq R$.

## Remarks

These sets are the closed sets of a topology:

1. $V(\set{0}) = X$ and $V(\set{1}) = \emptyset$;
2. $\bigcap_i V(E_i) = V\left(\bigcup_i E_i\right)$ for any family of subsets $E_i$;
3. $V(I) \cup V(J) = V(IJ)$ for ideals $I, J$, because a prime ideal contains $IJ$ only if it contains $I$ or $J$.

Also $V(E) = V(I)$ for the ideal $I$ generated by $E$, and $V(I) = V(\sqrt I)$.
The closure of a point $\mathfrak p$ is $V(\mathfrak p)$, so $\set{\mathfrak p}$ is closed exactly when $\mathfrak p$ is a [[maximal ideal|0PV74TR5FP]].

Example: in $\operatorname{Spec} \Z$, the closed sets are $\operatorname{Spec} \Z$ itself and the finite sets of nonzero prime ideals.
Indeed $V(n\Z)$ for $n \ne 0$ is the finite set of $p\Z$ with $p \mid n$.

Non-example: $\operatorname{Spec} \Z$ with this topology is not [[Hausdorff|0S1FJ8VEX9]].
The point $(0)$ is not closed: its closure is $V(0)$, the whole space.
Any nonempty open set contains $(0)$, so no two nonempty open sets are disjoint.

The open sets $D(f) = X \setminus V(\set{f})$ are the [[basic open sets|N6JX9BCKQP]] and form a basis of the topology.
</document>

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# Smooth morphism

## Summary

A smooth morphism of schemes is a flat morphism, locally of finite presentation, whose fibres are smooth schemes over their residue fields.
It is the algebraic analogue of a submersion of differentiable manifolds.
A variety over a field is smooth exactly when its structure morphism is smooth, and smooth morphisms are stable under composition and base change.

## Explanations

### L1 human

A smooth morphism is a family of smooth spaces that fit together without jumps.
Near each point, it looks like the projection of a product of the target with an affine space, up to an étale change of coordinates.
The projection $\mathbb{A}^2 \to \mathbb{A}^1$ is smooth: each fibre is a line.
The map from the curve $y^2 = x^2(x + 1)$ to a point is not smooth, because the curve has a node at the origin.
</document>

<document source="https://tamnd.github.io/math2/node/EXPS8MH5S2.md" tag="EXPS8MH5S2" kind="definition">
# De Rham cohomology of varieties

## Definition

Let $k$ be a field and let $X$ be a [[scheme|5A35TYZD02]] that is [[smooth|EXKGG9X6J0]] over $k$.
For $p \ge 0$ let $\Omega^p_{X/k} = \bigwedge^p \Omega^1_{X/k}$, where $\Omega^1_{X/k}$ is the [[sheaf of differentials|6VC6HQTFSA]], built from [[Kähler differentials|RG6WFFW7K3]].
The exterior derivative $d \colon \Omega^p_{X/k} \to \Omega^{p+1}_{X/k}$ is the unique $k$-linear map with $d(f) = df$ on functions, $d \circ d = 0$, and $d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^p \alpha \wedge d\beta$ for $\alpha$ of degree $p$.
The complex $\Omega^\bullet_{X/k} = (\Omega^0_{X/k} \to \Omega^1_{X/k} \to \Omega^2_{X/k} \to \cdots)$ is the *algebraic de Rham complex*.
The *algebraic de Rham cohomology* of $X$ is the hypercohomology $H^i_{\mathrm{dR}}(X/k) = \mathbb{H}^i(X, \Omega^\bullet_{X/k})$.
This is the $i$-th [[sheaf cohomology|PQ6Q143ARJ]] of the complex $\Omega^\bullet_{X/k}$, computed from an injective resolution of the complex.

## Remarks

If $X = \operatorname{Spec} A$ is affine, the sheaves $\Omega^p_{X/k}$ are quasi-coherent and have no higher cohomology.
Then $H^i_{\mathrm{dR}}(X/k)$ is the $i$-th cohomology of the complex of $A$-modules $\Omega^\bullet_{A/k}$.
Example: let $k$ have characteristic $0$ and $A = k[t, t^{-1}]$.
Then $\Omega^1_{A/k} = A \, dt$ and $d(t^n) = n t^{n-1} dt$.
The kernel of $d$ on $A$ is $k$, and the image of $d$ contains $t^m \, dt$ for every $m \ne -1$.
So $H^0_{\mathrm{dR}} = k$ and $H^1_{\mathrm{dR}}$ is spanned by the class of $dt/t$.
Non-example of good behaviour: if $k$ has characteristic $p > 0$ and $A = k[t]$, then $d(t^p) = 0$.
So $H^0_{\mathrm{dR}}(\mathbb{A}^1_k/k) = k[t^p]$ is infinite-dimensional, and the theory does not count connected components as in characteristic $0$.
Grothendieck's comparison theorem: if $X$ is smooth over $\C$, then $H^i_{\mathrm{dR}}(X/\C) \cong H^i(X(\C), \C)$ for all $i$.
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<document source="https://tamnd.github.io/math2/node/EXZA8ZA8P6.md" tag="EXZA8ZA8P6" kind="statement">
# Lyusternik–Fet theorem

## Statement

Let $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]] without boundary, of dimension at least $1$.
Then $M$ has a [[closed geodesic|4MCCP8GG4J]].

## Remarks

If the fundamental group of $M$ is nontrivial, a loop of least length in a nontrivial free homotopy class exists by compactness, and it is a closed geodesic.
If $M$ is simply connected, then $M$ is not contractible, because a closed $n$-manifold has nonzero homology in degree $n$ with coefficients in $\Z/2$.
By the Hurewicz theorem, some homotopy group $\pi_k(M)$ with $k \ge 2$ is nonzero.
Then a minimax over $(k-1)$-parameter families of loops in the [[free loop space|SGGXZ3TBSA]] gives a critical value of the energy.
The Palais–Smale condition for the energy makes the minimax work.
The nontriviality of the class makes the critical value positive, so the critical point is not a constant loop.

Example: on a round sphere every great circle is a closed geodesic.
On an ellipsoid in $\R^3$ the theorem gives at least one, and the intersections with the three coordinate planes are in fact closed geodesics.
Compactness is needed: in $\R^n$ with the Euclidean metric no geodesic is closed.

## Explanations

### L1 human

On any closed curved space, there is a loop that runs straight ahead and closes up on itself.
If the space has a hole that a loop can wind around, one can tighten such a loop as much as possible.
If not, the space still has a higher-dimensional hole.
One sweeps it by a family of loops and looks for the loop at the highest point of the best sweep.

### L2 human

The theorem was the first general existence result for closed geodesics, and its proof is the model for minimax methods on loop spaces.
It extends Birkhoff's earlier result for spheres, which uses the [[Birkhoff min-max method|1R5X9HW6D3]].
Questions about the number of geometrically distinct closed geodesics are much harder; see the [[Gromoll–Meyer theorem|XC8BRYJN93]] and the [[theorem of Bangert and Franks|X93AEE88PX]].
</document>

<document source="https://tamnd.github.io/math2/node/EXZHFNRMV8.md" tag="EXZHFNRMV8" kind="definition">
# Non-uniform discrete Fourier transform

## Definition

Let $N \ge 1$ and $K \ge 1$ be integers.
Let $p_0, \ldots, p_{N-1}$ be real numbers, the sample points, and let $f_0, \ldots, f_{K-1}$ be real numbers, the frequencies.
The *non-uniform discrete Fourier transform* of a vector $x = (x_0, \ldots, x_{N-1}) \in \C^N$ is the vector $X \in \C^K$ given by
$$X_k = \sum_{n=0}^{N-1} x_n\, e^{-2\pi i\, p_n f_k}, \qquad k = 0, \ldots, K-1.$$
It is of *type I* if $f_k = k$ for all $k$, with arbitrary $p_n$.
It is of *type II* if $p_n = n/N$ for all $n$, with arbitrary $f_k$.
It is of *type III* if both families are arbitrary.
The numbering of types I and II varies between authors; some, including the cited source, swap the two names.

## Remarks

Example: with $K = N$, $p_n = n/N$ and $f_k = k$, the sum is $\sum_n x_n e^{-2\pi i nk/N}$, the ordinary [[discrete Fourier transform|TST0GH66A5]]; it is both of type I and of type II.
Non-example: the sum $\sum_{n \in \Z} x_n e^{-2\pi i n f}$ over all integers is not a non-uniform DFT, because the definition requires finitely many sample points.
The map $x \mapsto X$ is linear, with the $K \times N$ matrix $A_{kn} = e^{-2\pi i p_n f_k}$.
Unlike the DFT matrix, $A$ is in general not a multiple of a unitary matrix, and it can be badly conditioned or singular, for example when two sample points coincide.
The adjoint transform, with $e^{+2\pi i p_n f_k}$, is used in place of an inverse.
Some authors use multidimensional points $p_n, f_k \in \R^d$ with the dot product $p_n \cdot f_k$ in the exponent.

## Explanations

### L1 human

The definition keeps the formula of the discrete Fourier transform but lets the positions $p_n$ and frequencies $f_k$ be any real numbers.
The three types say which of the two sides is still on a regular grid.
Computing it directly costs about $NK$ operations.
Fast algorithms, called non-uniform FFTs, approximate it to a chosen accuracy with far fewer operations when $N$ and $K$ are large.
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# Tensor product of Hilbert spaces

## Summary

The tensor product of two Hilbert spaces is the completion of their algebraic tensor product.
Its inner product multiplies the inner products of the factors.
It is again a Hilbert space, and products of orthonormal basis vectors form an orthonormal basis of it.
In quantum mechanics it describes a system made of two subsystems.
In analysis, $L^2(X) \mathbin{\widehat{\otimes}} L^2(Y)$ is $L^2(X \times Y)$.

## Explanations

### L1 human

If one Hilbert space describes functions of $x$ and another describes functions of $y$, the tensor product describes functions of the pair $(x, y)$.
The simplest elements are products $f(x) g(y)$, and general elements are limits of sums of such products.
For example, $\C^2 \otimes \C^3$ is $\C^6$, with the six products $e_i \otimes f_j$ of basis vectors as an orthonormal basis.
</document>

<document source="https://tamnd.github.io/math2/node/EY18HD35RG.md" tag="EY18HD35RG" kind="definition">
# Charlier polynomials

## Definition

Let $a > 0$ be a real number and let $n \ge 0$ be an integer.
Write $x^{\underline{k}} = x (x - 1) \cdots (x - k + 1)$ for the falling factorial, with $x^{\underline{0}} = 1$.
The *Charlier polynomial* of degree $n$ with parameter $a$ is
$$C_n(x; a) = \sum_{k=0}^{n} \binom{n}{k} \left( -\frac{1}{a} \right)^k x^{\underline{k}},$$
where $\binom{n}{k}$ is the [[binomial coefficient|0GBZM0MT0F]].

## Remarks

In hypergeometric notation, $C_n(x; a) = {}_2F_0\left( -n, -x; ; -\frac{1}{a} \right)$.
They are orthogonal for the [[Poisson distribution|0XB9GQTN9G]]: $\sum_{x=0}^{\infty} \frac{a^x}{x!} C_m(x; a) C_n(x; a) = a^{-n} e^a n! \, \delta_{mn}$.
Example: $C_1(x; a) = 1 - \frac{x}{a}$.
Check: $\sum_{x \ge 0} \frac{a^x}{x!} \left( 1 - \frac{x}{a} \right) = e^a - \frac{1}{a} \cdot a e^a = 0$, and $\sum_{x \ge 0} \frac{a^x}{x!} \left( 1 - \frac{x}{a} \right)^2 = e^a \left( 1 - 2 + \frac{a + a^2}{a^2} \right) = \frac{e^a}{a}$.
They are symmetric in degree and variable: $C_n(x; a) = C_x(n; a)$ for nonnegative integers $n$ and $x$.
Non-example: the monomials $1$ and $x$ are not orthogonal for this weight, since $\sum_{x \ge 0} \frac{a^x}{x!} x = a e^a \ne 0$.
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# Non-wandering set

## Summary

The non-wandering set of a continuous map consists of the points $x$ whose every neighbourhood meets one of its own later images.
It is the complement of the set of wandering points from the notion of [[wandering set|VNW5FHCQ2M]].
It is closed and invariant, and it contains every [[periodic orbit|0AA97XA70K]], every [[recurrent point|4SQHHRPKMV]] and every $\omega$-[[limit set|YF4XH2BEG5]].
In smooth dynamics, hyperbolicity conditions such as Smale's Axiom A are imposed on the non-wandering set.

## Explanations

### L1 human

Some points of a system are only passing through: a small region around them drifts away and never overlaps itself again.
The non-wandering set is what is left when all such passing points are removed.
It is where the motion keeps coming back to, so it holds all the long-term behaviour.
For a pendulum with friction, only the rest positions are non-wandering; every other state slowly drifts toward rest.
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<document source="https://tamnd.github.io/math2/node/EY362KR0KB.md" tag="EY362KR0KB" kind="concept">
# Differential-algebraic system of equations

## Summary

A differential-algebraic system of equations, or DAE, is a system $F(t, x, x') = 0$ that cannot be solved for the derivative $x'$.
Typically some equations contain derivatives and others are pure constraints between the unknowns.
Such systems arise in electrical circuits, in constrained mechanics and in chemical engineering.
See the [[definition|FD7H5AM1R7]].

## Explanations

### L1 human

An ordinary differential equation tells how every unknown changes in time.
In a DAE some unknowns have no equation for their rate of change; they are tied to the others by algebraic rules instead.
A pendulum written in Cartesian coordinates is an example: the position moves by Newton's law, but it must also stay at a fixed distance from the pivot.
Because of the constraints, the starting values cannot be chosen freely.
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# Convolution integral equation

## Summary

A convolution integral equation is an [[integral equation|R3SG1SN7JQ]] whose kernel depends only on the difference $x - t$ of its two variables.
The integral in it is then a [[convolution|2Z29JN3VES]] of the kernel with the unknown function.
Because integral transforms turn convolutions into products, such equations can often be solved explicitly with the [[Fourier transform|DPVM34ADZV]] or the [[Laplace transform|0V49JEZW06]].
They model systems that are invariant under translation, such as linear filters, renewal processes and homogeneous media.

## Explanations

### L1 human

Think of a signal that passes through a blurring device that treats every position in the same way.
The output at a point is a weighted average of the input around that point, with weights that depend only on the distance.
Asking for the input that produced a given output is a convolution integral equation.
For example, $\varphi(x) - \frac{1}{2} \int_{-\infty}^{\infty} e^{-\abs{x - t}} \varphi(t) \, dt = f(x)$ asks for $\varphi$ when $f$ is known.
Since the weight $e^{-\abs{x - t}}$ depends only on $x - t$, the equation is of convolution type.
</document>

<document source="https://tamnd.github.io/math2/node/EYA944RVSY.md" tag="EYA944RVSY" kind="definition">
# Born series

## Definition

Let $X$ be a [[Banach space|06GZ956657]] of functions on $\R^n$, and let $\phi \in X$ be an incoming wave.
Let $V$ be multiplication by a potential.
Let $G_0$ be the outgoing free resolvent, the convolution with the outgoing Green's function of $\Delta + k^2$.
Assume that $T = G_0 V$ is a bounded linear operator on $X$.
Consider the scattering equation $\psi = \phi + T \psi$, an [[integral equation of the second kind|8KJCN49V64]].
The *Born series* of $\psi$ is the formal series
$$\psi = \sum_{n = 0}^{\infty} T^n \phi = \phi + G_0 V \phi + G_0 V G_0 V \phi + \cdots.$$
Its partial sum $\phi + G_0 V \phi$ is the *(first) Born approximation*.

## Remarks

If the [[operator norm|06EKW1AKXF]] satisfies $\norm{T} < 1$, the series converges in $X$ and its sum is the unique solution of $\psi = \phi + T \psi$, by the [[Neumann series|8QJ91709V2]].
Replacing $V$ by $g V$ with a coupling constant $g$, this holds for all small enough $\abs{g}$.

Example: in one dimension, take the point interaction $V = g \delta_0$ with $g \in \R$; then the outgoing Green's function is $\frac{e^{ik\abs{x}}}{2ik}$, and $G_0 V \psi(x) = \frac{g}{2ik} e^{ik\abs{x}} \psi(0)$.
For $\phi(x) = e^{ikx}$ each term is a multiple of $e^{ik\abs{x}}$, and the series sums to $\psi(x) = e^{ikx} + \frac{g/(2ik)}{1 - g/(2ik)} e^{ik\abs{x}}$ when $\abs{g} < 2k$.
At $x = 0$ this gives $\psi(0) = \frac{1}{1 - g/(2ik)}$, which solves $\psi(0) = 1 + \frac{g}{2ik} \psi(0)$, as it must.

Non-example: for $\abs{g} > 2k$ in the same model the geometric series diverges, although the equation still has the solution given by the closed formula.
So the Born series may fail to converge even when the scattering problem is well posed.
</document>

<document source="https://tamnd.github.io/math2/node/EYDGR9D2Y9.md" tag="EYDGR9D2Y9" kind="definition">
# Euler–Boole summation

## Definition

The *Euler polynomials* $E_k(x)$, for $k \ge 0$, are defined by the generating function
$$\frac{2 e^{xt}}{e^t + 1} = \sum_{k \ge 0} E_k(x) \frac{t^k}{k!}.$$
The *periodic Euler functions* $\widetilde{E}_k$ are the functions on $\R$ with $\widetilde{E}_k(x) = E_k(x)$ for $0 \le x < 1$ and $\widetilde{E}_k(x + 1) = -\widetilde{E}_k(x)$ for all $x$.

Let $m \ge 1$ and $n \ge 1$ be integers, let $0 \le h \le 1$, and let $f$ be a [[smooth function|0BE3QDH73A]] on an open interval that contains $[0, n]$.
*Euler–Boole summation* is the identity
$$\sum_{j=0}^{n-1} (-1)^j f(j + h) = \frac12 \sum_{k=0}^{m-1} \frac{E_k(h)}{k!} \left( (-1)^{n-1} f^{(k)}(n) + f^{(k)}(0) \right) + \frac{1}{2 (m-1)!} \int_0^n f^{(m)}(x) \, \widetilde{E}_{m-1}(h - x) \, dx,$$
and the finite sum over $k$ is the Euler–Boole expansion of the alternating sum to order $m$.

## Remarks

The first Euler polynomials are $E_0(x) = 1$ and $E_1(x) = x - \frac12$.
They satisfy $E_k(x + 1) + E_k(x) = 2x^k$, which is the source of the formula.

Example: take $f(x) = x$, $h = 0$, $n = 2$ and $m = 2$.
The left side is $0 - 1 = -1$.
The term $k = 0$ gives $\frac12 (-f(2) + f(0)) = -1$, the term $k = 1$ gives $\frac12 \cdot (-\frac12) \cdot (-f'(2) + f'(0)) = 0$, and the integral vanishes because $f'' = 0$.
Both sides equal $-1$.

Non-example: the identity is for alternating sums.
For the plain sum $\sum_{j=0}^{n-1} f(j)$ the right side is wrong in general, and the Euler–Maclaurin formula applies instead.
</document>

<document source="https://tamnd.github.io/math2/node/EYF0H8AP91.md" tag="EYF0H8AP91" kind="statement">
# Comparison principle

## Statement

Let $\Omega \subseteq \R^n$ be a bounded domain, that is, a bounded connected open set.
Let
$$L u = \sum_{i,j=1}^n a^{ij}(x) \, \partial_i \partial_j u + \sum_{i=1}^n b^i(x) \, \partial_i u + c(x) \, u$$
be a uniformly [[elliptic|B4TTNHSTRX]] operator in $\Omega$ with bounded coefficients, $a^{ij} = a^{ji}$, and $c \le 0$ in $\Omega$.
Let $u, v \in C^2(\Omega) \cap C^0(\overline{\Omega})$ satisfy $L u \ge L v$ in $\Omega$ and $u \le v$ on $\partial\Omega$.
Then $u \le v$ in $\Omega$.

## Remarks

Proof idea: the function $w = u - v$ satisfies $L w \ge 0$ in $\Omega$ and $w \le 0$ on $\partial\Omega$.
The weak [[maximum principle|D2JF19HF91]] for operators with $c \le 0$ gives $\sup_\Omega w \le \sup_{\partial\Omega} w^+ = 0$.

Example: if $\Delta u \ge 0$ in the unit ball $B$ and $u \le 1$ on $\partial B$, comparing with $v = 1$ gives $u \le 1$ in $B$.
If $-\Delta u = 1$ in $B \subseteq \R^n$ with $u = 0$ on $\partial B$, comparison with $0$ and with $(1 - \abs{x}^2)/(2n)$ shows $0 \le u \le (1 - \abs{x}^2)/(2n)$; in fact $u$ equals this function.

The sign condition on $c$ is needed: for $L u = u'' + u$ on $(0, \pi)$, the functions $u = \sin x$ and $v = 0$ satisfy $L u = L v = 0$ and $u = v$ at the endpoints, but $u > v$ inside.

Uniqueness for the Dirichlet problem $L u = f$ in $\Omega$, $u = g$ on $\partial\Omega$, follows by applying the theorem twice.

## Explanations

### L1 human

If two solutions of the same kind of equation are ordered on the boundary, they stay ordered inside.
So to bound an unknown solution, it is enough to find an explicit function that dominates it on the boundary and satisfies the right inequality inside.
Such functions are called barriers, or super- and subsolutions.

### L2 human

The comparison principle is the main use of the maximum principle.
It gives uniqueness for boundary value problems, a priori bounds by explicit barriers, and monotone iteration schemes between a subsolution and a supersolution.
Versions exist for parabolic equations, through the [[parabolic maximum principle|EMDVD4E83K]], and for fully nonlinear equations, where comparison for [[viscosity solutions|H1ERJ3T75T]] is the key uniqueness tool.
It is the PDE analogue of the [[comparison theorem|7XTBR347P5]] for ordinary differential equations.
</document>

<document source="https://tamnd.github.io/math2/node/EYG946BY5G.md" tag="EYG946BY5G" kind="statement">
# Steinhaus theorem

## Statement

Let $n \ge 1$ and let $A \subseteq \R^n$ be a Lebesgue measurable set with $\lambda(A) > 0$, where $\lambda$ is [[Lebesgue measure|2Z2QVWZ262]].
Then the difference set
$$A - A = \set{a - b : a, b \in A}$$
contains an open ball centred at $0$.

## Remarks

One proof uses inner regularity to find a compact set $K \subseteq A$ with $0 < \lambda(K) < \infty$.
The function $f(x) = \lambda(K \cap (K + x))$ is continuous and $f(0) = \lambda(K) > 0$, so $f(x) > 0$ for $x$ near $0$.
If $f(x) > 0$, then some $k \in K$ also lies in $K + x$, so $x = k - k'$ with $k, k' \in K$, and $x \in A - A$.
Example: if $A \subseteq \R$ is a measurable set of positive measure, then every real number $t$ with $\abs{t}$ small enough is a difference of two points of $A$.
The hypothesis $\lambda(A) > 0$ is needed: $A = \Z$ has measure zero, and $\Z - \Z = \Z$ contains no interval around $0$.
The same statement holds in every locally compact group for a measurable set of positive [[Haar measure|VY0HXD3GC4]], with $A A^{-1}$ in place of $A - A$.

## Explanations

### L1 human

Take any set of real numbers that has positive length, however scattered it is.
Then all small numbers can be written as a difference of two of its points.
So a set of positive measure, even with no interval inside it, produces a whole interval of differences.

### L2 human

The theorem is a key step in automatic continuity results: for instance, a Lebesgue measurable additive function $f \colon \R \to \R$ is continuous, hence linear.
It also shows that a measurable proper subgroup of $\R$ has measure zero, since a subgroup containing an interval around $0$ is all of $\R$.
Its analogue for Baire category is the Pettis lemma.
</document>

<document source="https://tamnd.github.io/math2/node/EYMHCPTZ8T.md" tag="EYMHCPTZ8T" kind="definition">
# Dehn invariant

## Definition

Let $P$ be a [[polyhedron|0325SGSJAR]] in $\R^3$.
For each edge $e$ of $P$, let $\ell(e)$ be its length and let $\theta(e)$ be the [[dihedral angle|05HN08ZSRJ]] of $P$ at $e$.
The *Dehn invariant* of $P$ is the element
$$D(P) = \sum_{e} \ell(e) \otimes \theta(e)$$
of the [[tensor product|F0HQXQ0HCD]] $\R \otimes_\Z (\R / \pi\Z)$ of [[abelian groups|082GHNPJ53]].
The sum runs over all edges of $P$.

## Remarks

Example: for the cube of side $a$, every dihedral angle is $\pi/2$.
So each edge gives $a \otimes \pi/2 = (a/2) \otimes \pi = 0$, and $D(\text{cube}) = 0$.
A polyhedron with nonzero invariant: the regular tetrahedron of side $a$ has dihedral angle $\arccos(1/3)$, which is not a rational multiple of $\pi$.
Its invariant $6a \otimes \arccos(1/3)$ is not $0$, since $\R \otimes_\Z (\R/\pi\Z)$ is isomorphic to the $\Q$-vector space $\R \otimes_\Q (\R/\Q\pi)$.
The invariant is additive: if $P$ is cut into polyhedra $P_1, \dots, P_k$, then $D(P) = D(P_1) + \cdots + D(P_k)$.
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<document source="https://tamnd.github.io/math2/node/EYMQ82Z8XJ.md" tag="EYMQ82Z8XJ" kind="concept">
# Positive harmonic function

## Summary

A positive harmonic function is a [[harmonic function|0DEB0JMN0W]] that takes only positive values on its domain.
Positivity is a strong constraint: on a connected open set, the values of such a function at two points control each other, by [[Harnack's inequality|YNH832D50W]].
On the unit disc and on balls, positive harmonic functions are exactly the Poisson integrals of nonzero positive finite Borel measures on the boundary sphere.
They are the starting point of the theory of the Martin boundary.

## Explanations

### L1 human

Think of a steady temperature distribution that is above zero everywhere in a room.
Such a temperature cannot be very large at one point and very small at a nearby point.
For example, on the half-plane $y > 0$ the function $u(x, y) = y$ is harmonic and positive.
On the unit disc, the [[Poisson kernel|TCYSWCTYYM]] $(1 - \abs{z}^2)/\abs{1 - z}^2$ is another example.
</document>

<document source="https://tamnd.github.io/math2/node/EYREVY31BN.md" tag="EYREVY31BN" kind="concept">
# Eisenstein series

## Summary

The Eisenstein series $G_k$ of even weight $k \ge 4$ is the sum of $(m\tau + n)^{-k}$ over all pairs of integers $(m, n) \neq (0, 0)$.
It is the simplest [[modular form|GWKN5VG78A]] of weight $k$ for the [[modular group|VEKV1EM77B]] that is not a [[cusp form|A50Y012RBP]].
Its $q$-expansion involves the [[Riemann zeta function|1XWBY4N0VT]] and the divisor sums $\sigma_{k-1}(n)$.
The series $E_4$ and $E_6$ generate the ring of all modular forms for $\mathrm{SL}_2(\Z)$.

## Explanations

### L1 human

Take the lattice of points $m\tau + n$ in the complex plane and add up the $k$-th powers of their reciprocals.
Changing the basis of the lattice by a matrix of $\mathrm{SL}_2(\Z)$ only reorders the sum, and this gives the modular symmetry.
The result has a neat expansion: $E_4 = 1 + 240 \sum_{n \ge 1} \sigma_3(n) q^n$, where $\sigma_3(n)$ is the sum of the cubes of the divisors of $n$.
Comparing such expansions proves surprising identities between divisor sums.
</document>

<document source="https://tamnd.github.io/math2/node/EYXG11V76E.md" tag="EYXG11V76E" kind="statement">
# Cauchy formula for repeated integration

## Statement

Let $a < b$ be real numbers, let $f \colon [a, b] \to \R$ be [[continuous|0X3WGX95FE]], and let $n \ge 1$ be an integer.
Define $F_0 = f$ and $F_k(x) = \int_a^x F_{k-1}(t) \, dt$ for $k \ge 1$ and $x \in [a, b]$.
Then for all $x \in [a, b]$,
$$F_n(x) = \frac{1}{(n - 1)!} \int_a^x (x - t)^{n - 1} f(t) \, dt.$$

## Remarks

The proof is by induction on $n$.
The case $n = 1$ is the definition.
For the step, note that both sides vanish at $x = a$.
Differentiating the right side under the integral sign gives the same formula with $n - 1$ in place of $n$.
The [[fundamental theorem of calculus|0NHGY0N0JB]] then gives equality.
Equivalently, one writes $F_n$ as an [[iterated integral|0WZCHE1ZF9]] over a simplex and integrates out all variables but the last one.
Example: for $f = 1$, $a = 0$ and $n = 2$, the right side is $\int_0^x (x - t) \, dt = x^2 / 2$, which is the twice repeated integral of $1$.

## Explanations

### L1 human

Integrating a function $n$ times in a row seems to need $n$ nested integrals.
Cauchy's formula collapses them into one integral with the weight $(x - t)^{n-1} / (n - 1)!$.

### L2 human

The formula is used in the proof of [[Taylor's theorem|03QHBVW4TT]] with the integral remainder.
Replacing $(n - 1)!$ by $\Gamma(n)$ and letting $n$ be any positive real number gives the [[Riemann–Liouville integral|JTKCZ44NN3]], the starting point of [[fractional calculus|T73XK15SA2]].
</document>

<document source="https://tamnd.github.io/math2/node/EYXNKYMJJH.md" tag="EYXNKYMJJH" kind="statement">
# Lidskii's trace formula for integral operators

## Statement

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ be continuous, and let $T$ be the integral operator $(T x)(s) = \int_a^b K(s, t) \, x(t) \, dt$ on the [[Lp space|SASP14MA1Z]] $L^2(a, b)$.
Assume that $T$ is a [[trace class operator|Z2CP530WDE]].
Let $(\lambda_n)$ be the nonzero eigenvalues of $T$, each repeated according to its algebraic multiplicity.
Then $\sum_n \abs{\lambda_n} < \infty$ and
$$\sum_n \lambda_n = \int_a^b K(s, s) \, ds.$$

## Remarks

The proof has two steps.
First, for a trace class integral operator with continuous kernel, the operator trace equals the [[trace of the kernel|3JVT83K8A1]] $\int_a^b K(s, s) \, ds$.
Second, [[Lidskii's theorem|QD23PJR6RP]] says that the operator trace equals the sum of the eigenvalues, counted with algebraic multiplicity.

Example: on $[0, 1]$ let $K(s, t) = \min(s, t)$.
The [[eigenfunctions|4NSACSWRJZ]] are $\sin((n - \tfrac12)\pi s)$ with eigenvalues $\lambda_n = 1/((n - \tfrac12)^2 \pi^2)$, $n \ge 1$.
Since $K$ is a positive kernel and $\sum_n \lambda_n < \infty$, the operator is trace class.
The formula reads $\sum_{n \ge 1} \frac{4}{(2n - 1)^2 \pi^2} = \int_0^1 s \, ds = \frac12$, which is the identity $\sum_{n \ge 1} (2n-1)^{-2} = \pi^2/8$.

The trace class hypothesis cannot be dropped.
There are continuous kernels whose operators are not trace class, and for them the sum of the eigenvalues need not converge absolutely.
Take the Volterra kernel $K(s, t) = 1$ for $t \le s$ and $K(s, t) = 0$ otherwise.
Its operator has no nonzero eigenvalue, while the diagonal integral is $b - a$.
There is no contradiction: this kernel is not continuous, and the operator is not trace class.

## Explanations

### L1 human

For a matrix, the sum of the eigenvalues equals the sum of the diagonal entries.
For an integral operator, the diagonal entries become the values $K(s, s)$, and the sum becomes an integral.
The formula says that, for good enough kernels, the sum of all eigenvalues equals this integral.

### L2 human

For a continuous Hermitian kernel of positive type, the formula follows from [[Mercer's theorem|PKGADWXPYH]] by integrating the expansion $K(s, s) = \sum_n \lambda_n \abs{\varphi_n(s)}^2$.
Lidskii's theorem extends it to kernels that are not self-adjoint, where the eigenvalues may be complex and the eigenfunctions need not form a basis.
The formula is the first case of the trace identities $\sum_n \lambda_n^m = \int_a^b K_m(s, s) \, ds$ for the iterated kernels $K_m$, which hold for every continuous kernel when $m \ge 2$, since then $T^m$ is trace class.
</document>

<document source="https://tamnd.github.io/math2/node/EZ21NHHAKG.md" tag="EZ21NHHAKG" kind="statement">
# Varieties as categories of algebras over a monad

## Statement

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras of a finitary signature, regarded as a category with homomorphisms.
Let $U \colon \mathcal{V} \to \mathbf{Set}$ be the [[forgetful functor|WV3BPYCHVG]] and let $F \colon \mathbf{Set} \to \mathcal{V}$ be the [[free functor|N353JF8RBK]], its left adjoint.
Let $\mathbb{T} = (U F, \eta, \mu)$ be the [[monad|FF6XXA6D1A]] on $\mathbf{Set}$ [[induced by this adjunction|DF5N0PK3ZS]], and let $\mathbf{Set}^{\mathbb{T}}$ be its [[Eilenberg–Moore category|FEPRJR3MND]] of [[$\mathbb{T}$-algebras|QG1ZXRC8EH]].
Then the comparison functor $K \colon \mathcal{V} \to \mathbf{Set}^{\mathbb{T}}$, which sends an algebra $A$ to the $\mathbb{T}$-algebra $(U A, U \varepsilon_A)$ where $\varepsilon_A \colon F U A \to A$ is the counit, is an isomorphism of categories.

## Remarks

The underlying set of $U F(X)$ is the set of terms in the variables $X$ modulo the identities of $\mathcal{V}$.
The map $U \varepsilon_A \colon U F U A \to U A$ evaluates such a term at elements of $A$.
Conversely, a $\mathbb{T}$-algebra $h \colon U F(X) \to X$ defines each basic operation $f$ of arity $n$ by $f(x_1, \ldots, x_n) = h(f(x_1, \ldots, x_n))$.
On the right, $h$ is applied to the term $f(x_1, \ldots, x_n)$.
The unit and associativity laws of $h$ force the identities of $\mathcal{V}$ to hold.
Another proof applies Beck's monadicity theorem: $U$ creates coequalizers of $U$-split pairs.

Example: for the variety of monoids, $U F(X)$ is the set $X^*$ of words.
A $\mathbb{T}$-algebra structure $X^* \to X$ is the same as a monoid structure on $X$: it sends a word to the product of its letters.

The monads that arise in this way are finitary, and every finitary monad on $\mathbf{Set}$ arises from a variety, possibly with infinitely many operations of each finite arity.
Monads that are not finitary describe infinitary structures; for instance, the algebras of the ultrafilter monad on $\mathbf{Set}$ are the compact Hausdorff spaces.

## Explanations

### L1 human

A variety, such as groups or rings, is usually described by operations and equations.
The theorem gives another description: only the free algebras and the way to evaluate formal expressions are needed.
An algebra of the variety is then a set with a rule that turns every formal expression in its elements into an element, consistently.

### L2 human

The result says that the forgetful functor of a variety is monadic, in the strict sense that the comparison functor is an isomorphism, not only an equivalence.
It explains why constructions on varieties, such as limits computed on underlying sets and the existence of colimits, follow from general monad theory.
Together with [[Lawvere theories|NTJ94H2CNS]], it gives three equivalent views of a variety: by a presentation, by a Lawvere theory, and by a finitary monad on $\mathbf{Set}$.
The [[Kleisli category|D9YHA46YWM]] of the same monad is equivalent to the full subcategory of free algebras.
</document>

<document source="https://tamnd.github.io/math2/node/EZ24HDESPB.md" tag="EZ24HDESPB" kind="definition">
# Attractor

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \colon X \to X$ be a [[continuous map|0X3WGX95FE]], with [[iterates|AYVTR2ES5B]] $f^n$.
For $x \in X$ and $A \subseteq X$ nonempty, let $d(x, A) = \inf_{a \in A} d(x, a)$.
A nonempty [[compact|0TVJZYB6SH]] set $A \subseteq X$ is an *attractor* of $f$ if:

1. $A$ is invariant: $f(A) = A$;
2. $A$ attracts an open neighbourhood: there is an [[open set|02BH3MCBQB]] $U \supseteq A$ such that $d(f^n(x), A) \to 0$ as $n \to \infty$ for every $x \in U$;
3. $A$ is minimal: no nonempty compact proper subset of $A$ satisfies clauses 1 and 2.

The *basin of attraction* of $A$ is the set of all $x \in X$ with $d(f^n(x), A) \to 0$.
For a flow $(\varphi_t)$ the definition is the same, with $\varphi_t(A) = A$ for all $t$ and $t \to \infty$ in place of $n \to \infty$.

## Remarks

A set that satisfies clauses 1 and 2 is often called an attracting set.
The literature uses several other versions.
Some replace clause 3 by the existence of a dense orbit in $A$.
Some ask for a neighbourhood $U$ with $f(\overline{U}) \subseteq U$ and $A = \bigcap_{n \ge 0} f^n(U)$.
Milnor's version replaces the open neighbourhood in clause 2 by a basin of positive measure.

Example: $f(x) = x/2$ on $\R$ and $A = \set{0}$.
Clause 1 holds since $f(0) = 0$; clause 2 holds with $U = \R$, since $d(f^n(x), A) = \abs{x}/2^n \to 0$; clause 3 holds since $A$ has no nonempty proper subset.

Non-example: let $f$ be the time-$1$ map of the flow of $u' = u - u^3$ on $\R$, and let $A = [-1, 1]$.
Every solution tends to $-1$, $0$ or $1$, so $A$ is invariant and attracts all of $\R$.
But the proper subset $\set{1}$ also satisfies clauses 1 and 2, with $U = (0, \infty)$, so clause 3 fails for $A$.
The attractors of this map are $\set{-1}$ and $\set{1}$.
</document>

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# Multidimensional scaling

## Summary

Multidimensional scaling places objects as points in a low-dimensional space so that their distances match given dissimilarities as well as possible.
Its input is only a table of pairwise dissimilarities, not coordinates.
Classical multidimensional scaling does this exactly when the dissimilarities are [[Euclidean distances|0NPPGHQNXQ]], using the [[eigenvectors|0N6960W0BV]] of a centred matrix.
It is used to visualise similarity data in two or three dimensions.

## Explanations

### L1 human

Given only the road distances between cities, one can try to redraw the map.
Multidimensional scaling does exactly this: it finds points in the plane whose distances are close to the given ones.
The map comes out correct up to shifting, turning and mirroring, since these do not change distances.
For three points with distances $1$, $1$ and $2$, it puts them on a line at $-1$, $0$ and $1$.
</document>

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# Optimal control

## Definition

Let $x' = f(x, u)$ be a [[control system|SBQVGQPXT1]] on $\R^n$ with control set $U \subseteq \R^m$.
Fix a time horizon $T > 0$, an initial state $x_0 \in \R^n$ and a closed target set $S \subseteq \R^n$.
Let $L \colon \R^n \times U \to \R$ and $\varphi \colon \R^n \to \R$ be [[continuous functions|0X3WGX95FE]], the running cost and the terminal cost.
For an [[admissible control|8M31QKB0GW]] $u$ for $(x_0, T, S)$ with trajectory $x_u$, the *cost* is
$$J(u) = \varphi\big(x_u(T)\big) + \int_0^T L\big(x_u(t), u(t)\big) \, dt.$$
An admissible control $u^*$ is an *optimal control* if $J(u^*) \le J(u)$ for every admissible control $u$ for $(x_0, T, S)$.
The pair $(u^*, x_{u^*})$ is then an optimal pair, and $x_{u^*}$ is an optimal trajectory.

## Remarks

The integral exists, because $x_u$ is continuous and $u$ is essentially bounded, so the integrand is a bounded measurable function.
A cost with both terms is in Bolza form; $L = 0$ gives the Mayer form and $\varphi = 0$ the Lagrange form.
An optimal control need not exist, and when it exists it need not be unique.

Example: $x' = u$ on $\R$ with $U = [-1, 1]$, $x_0 = 0$, $T = 1$, $S = \R$, $L = 0$ and $\varphi(x) = -x$.
Every admissible $u$ gives $x_u(1) = \int_0^1 u(t) \, dt \le 1$, so $J(u) \ge -1$.
The control $u^* = 1$ gives $J(u^*) = -1$, so it is optimal.

Non-example: the control $u = 0$ in the same problem is admissible but not optimal, since $J(0) = 0 > -1 = J(u^*)$.
</document>

<document source="https://tamnd.github.io/math2/node/EZ8NCHN7SJ.md" tag="EZ8NCHN7SJ" kind="definition">
# Skolem normal form

## Definition

Let $\sigma$ be a [[signature|WDVVEYD6AT]].
A [[sentence|DKFGQTRYZE]] over $\sigma$ is in *Skolem normal form* if it has the form
$$\forall x_1 \, \forall x_2 \cdots \forall x_n \, \psi,$$
where $n \ge 0$ and $\psi$ is a formula with no quantifiers.
So it is a sentence in [[prenex normal form|CS82V969DE]] whose prefix has only universal quantifiers.

## Remarks

Let $\varphi = \forall x_1 \cdots \forall x_k \, \exists y \, \chi$ be in prenex normal form.
Its Skolemization step replaces $\varphi$ by $\forall x_1 \cdots \forall x_k \, \chi[f(x_1, \dots, x_k)/y]$.
Here $f$ is a new function symbol of arity $k$; for $k = 0$ it is a new constant symbol.
Repeating the step gives a sentence in Skolem normal form over a larger signature.
It is satisfiable if and only if $\varphi$ is satisfiable.
For example, $\forall x \, \exists y \, R(x, y)$ becomes $\forall x \, R(x, f(x))$, which is in Skolem normal form.
The sentence $\forall x \, \exists y \, R(x, y)$ itself is not in Skolem normal form, since its prefix contains $\exists$.
The sentence $\forall x \, R(x, f(x))$ implies $\forall x \, \exists y \, R(x, y)$, but not conversely, so the two are not equivalent.
</document>

<document source="https://tamnd.github.io/math2/node/EZGTEK3HQN.md" tag="EZGTEK3HQN" kind="concept">
# Weyl group of a compact Lie group

## Summary

Let $G$ be a compact connected Lie group with maximal torus $T$.
Its Weyl group is the finite group $N_G(T)/T$, where $N_G(T)$ is the normaliser of $T$.
It acts on $T$ by conjugation and records how elements of $T$ can be conjugate in $G$.
It is isomorphic to the Weyl group of the root system of $G$, which is generated by reflections.
It appears in the Weyl integration and character formulas.

## Explanations

### L1 human

The maximal torus is a large commutative piece of the group, but the group can still move this piece onto itself in a few ways.
The Weyl group lists these ways, up to the trivial moves inside the torus.
For the unitary group $U(n)$ with the diagonal torus, the Weyl group is the symmetric group $S_n$: one may permute the diagonal entries by conjugating with permutation matrices.
So two diagonal unitary matrices are conjugate in $U(n)$ exactly when their entries agree up to order.
</document>

<document source="https://tamnd.github.io/math2/node/EZQFWKF03Q.md" tag="EZQFWKF03Q" kind="example">
# Koebe function

## Example

The *Koebe function* is
$$k(z) = \frac{z}{(1 - z)^2} = \sum_{n=1}^{\infty} n z^n, \qquad \abs{z} < 1.$$
It is a [[univalent function|R91E83SDBD]] on the unit disc $\mathbb{D}$ in the class $S$, and $k(\mathbb{D}) = \C \setminus (-\infty, -1/4]$.

Check: one has
$$k(z) = \frac{1}{4}\left( \left(\frac{1 + z}{1 - z}\right)^2 - 1 \right).$$
The Möbius map $z \mapsto (1+z)/(1-z)$ maps $\mathbb{D}$ bijectively onto the right half-plane $\set{\operatorname{Re} w > 0}$.
Squaring maps the right half-plane bijectively onto $\C \setminus (-\infty, 0]$.
Finally $w \mapsto (w - 1)/4$ maps $\C \setminus (-\infty, 0]$ onto $\C \setminus (-\infty, -1/4]$.
A composition of injective holomorphic maps is injective and holomorphic, and $k(0) = 0$, $k'(0) = 1$.

## Remarks

The point $-1/4$ is not in the image, so the radius $1/4$ in the [[Koebe quarter theorem|WX0HMW4GT7]] cannot be improved.
The coefficients are $a_n = n$, so the bound $\abs{a_n} \le n$ of the [[Bieberbach conjecture|ZMRFSYVWQT]] is attained for every $n$.
The rotations $e^{-i\theta} k(e^{i\theta} z)$ with $\theta \in \R$ are also in $S$ and are the extremal functions of many problems on $S$.

## Explanations

### L1 human

The Koebe function maps the unit disc onto the whole plane with one slit cut out along the negative real axis, from $-1/4$ to $-\infty$.
It is the extreme case of the class of normalised univalent functions: it stretches the disc as much as possible.
Its boundary slit comes as close as $1/4$ to the origin, and its power series coefficients $1, 2, 3, \ldots$ are as large as allowed.
</document>

<document source="https://tamnd.github.io/math2/node/EZSWDV6ZZQ.md" tag="EZSWDV6ZZQ" kind="definition">
# Noncommutative Gröbner basis

## Definition

Let $k$ be a [[field|0QA8CHH8MG]], let $X$ be a finite set, and let $X^*$ be the [[free monoid|CV013KHVJG]] of words in $X$.
Let $k\langle X \rangle$ be the [[free algebra|2P3Y2FRCSZ]] on $X$, whose basis is $X^*$.
Fix an *admissible order* $<$ on $X^*$: a well-order such that $u < v$ implies $p u q < p v q$ for all words $p, q$.
For a nonzero $f \in k\langle X \rangle$, its *leading word* $\operatorname{LW}(f)$ is the largest word that occurs in $f$ with a nonzero coefficient.
Let $I$ be a [[two-sided ideal|0DDA6CY7GJ]] of $k\langle X \rangle$.
A subset $G \subseteq I$ is a *(noncommutative) Gröbner basis* of $I$ if for every nonzero $f \in I$ there are $g \in G$ and words $p, q$ with
$$\operatorname{LW}(f) = p \operatorname{LW}(g)\, q.$$

## Remarks

A Gröbner basis $G$ of $I$ generates $I$ as a two-sided ideal.
The words that contain no $\operatorname{LW}(g)$, $g \in G$, as a subword are the *normal words*, and their images form a $k$-basis of $k\langle X \rangle / I$.
Example: $X = \set{x, y}$ with the degree-lexicographic order, $x < y$, and $I$ generated by $g = yx - xy$, so $\operatorname{LW}(g) = yx$.
The word $yx$ has no overlap with itself, so by the diamond lemma $\set{g}$ is a Gröbner basis; the normal words are $x^a y^b$, and $k\langle x, y \rangle / I \cong k[x, y]$.
Non-example: with the same order, let $I$ be generated by $g = xx - y$, so $\operatorname{LW}(g) = xx$.
The element $gx - xg = xy - yx$ lies in $I$, and its leading word is $yx$, which does not contain $xx$.
So $\set{g}$ is not a Gröbner basis of $I$.
</document>

<document source="https://tamnd.github.io/math2/node/EZVN87C88J.md" tag="EZVN87C88J" kind="concept">
# Field homomorphism

## Summary

A field homomorphism is a map between two fields that respects addition, multiplication and the unit element.
It is a [[ring homomorphism|0VTH8KPCG2]] whose source and target are fields.
Every field homomorphism is injective, so it identifies the first field with a subfield of the second.
This is why field theory speaks of embeddings and of [[field extensions|0EQWVHF3W5]].

## Explanations

### L1 human

A field homomorphism copies one field into another without breaking any computation.
The inclusion of the rational numbers into the real numbers is one.
Complex conjugation $a + bi \mapsto a - bi$ is a field homomorphism from $\C$ to itself.
Such a map can never send a nonzero number to $0$, so it never merges two elements.
The [[definition|JBZMRZY4YF]] lists the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/EZXPY4A19C.md" tag="EZXPY4A19C" kind="concept">
# Abel summation

## Summary

Abel summation is the [[summability method|8X25Q4XYMN]] that assigns to a series $\sum a_n$ the limit of the [[power series|0VP56B4EMX]] $\sum a_n x^n$ as $x \to 1$ from below.
By [[Abel's theorem|0S4EMDQS2F]], it gives the usual sum to every convergent series.
It sums every series that [[Cesàro summation|N8ZGG7EW6J]] sums, to the same value, and more.
Its partial converses, such as [[Tauber's theorem|8SK3GQ4S0M]] and [[Littlewood's Tauberian theorem|Z35TD4ZZRD]], started the theory of Tauberian theorems.

## Explanations

### L1 human

Damp the $n$-th term of the series by the factor $x^n$, with $x$ slightly below $1$.
The damped series often converges even when the original does not.
Then let $x$ grow to $1$ and see whether the damped sums approach a limit.
For $1 - 1 + 1 - \cdots$, the damped sum is $1 - x + x^2 - \cdots = \frac{1}{1+x}$, which tends to $1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/EZYHSE1E8P.md" tag="EZYHSE1E8P" kind="concept">
# Steenrod reduced power

## Summary

For an odd prime $p$, the Steenrod reduced powers $P^i$ are [[cohomology operations|K8T15QHN3W]] on mod $p$ [[singular cohomology|6N7BC70CX0]] that raise degree by $2i(p-1)$.
They play for odd primes the role that the Steenrod squares play for the prime $2$.
Together with the Bockstein homomorphism they generate all stable mod $p$ cohomology operations.
They detect maps that ordinary cohomology groups and [[cup products|JX47MS1R1H]] cannot see.

## Explanations

### L1 human

Cohomology with coefficients in $\F_p$ already has a product, so a class $x$ has a $p$-th power $x^p$.
The reduced power $P^i$ is a natural way to produce new classes from $x$ that interpolate between $x$ itself and $x^p$.
If $x$ has degree $2i$, then $P^i x = x^p$, and if the degree of $x$ is less than $2i$, then $P^i x = 0$.
For example, on $\C P^\infty$ with generator $x$ of degree $2$, one has $P^1 x = x^p$.
Because these operations commute with every map of spaces, a space where they act differently from another space cannot be homotopy equivalent to it.
</document>

<document source="https://tamnd.github.io/math2/node/F0047A6GDG.md" tag="F0047A6GDG" kind="definition">
# Viscosity solution

## Definition

Let $U \subseteq \R^n$ be open, and let $F \colon U \times \R \times \R^n \to \R$ be continuous.
Consider the first-order equation $F(x, u, Du) = 0$ in $U$, where $Du$ is the gradient of $u$.
Let $u \colon U \to \R$ be continuous.

1. $u$ is a *viscosity subsolution* if the following holds for every [[continuously differentiable function|0VW90BBSJ4]] $\varphi$ on $U$. At every point $x_0 \in U$ at which $u - \varphi$ has a local maximum, $F(x_0, u(x_0), D\varphi(x_0)) \le 0$.
2. $u$ is a *viscosity supersolution* if, for every continuously differentiable $\varphi$ on $U$ and every point $x_0 \in U$ at which $u - \varphi$ has a local minimum, $F(x_0, u(x_0), D\varphi(x_0)) \ge 0$.
3. $u$ is a *viscosity solution* if it is both a viscosity subsolution and a viscosity supersolution.

## Remarks

If $u$ is continuously differentiable and $u - \varphi$ has a local extremum at $x_0$, then $Du(x_0) = D\varphi(x_0)$.
So a continuously differentiable $u$ is a viscosity solution exactly when it solves the equation pointwise.
One gets the same notion with smooth test functions $\varphi$ instead of continuously differentiable ones.
The definition depends on the sign of $F$: the equations $F = 0$ and $-F = 0$ have different viscosity solutions in general.
For second-order equations $F(x, u, Du, D^2 u) = 0$, the Hessian $D^2 \varphi(x_0)$ enters as well, and $F$ is required to be degenerate elliptic, which means non-increasing in the matrix variable for this sign convention.

Example: on $U = (-1, 1)$, the function $u(x) = 1 - \abs{x}$ is a viscosity solution of $\abs{u'} - 1 = 0$.
Away from $0$, $u$ is smooth with $\abs{u'} = 1$.
At $0$, if $u - \varphi$ has a local maximum then $-\abs{x} \le \varphi(x) - \varphi(0)$ near $0$, so $\abs{\varphi'(0)} \le 1$ and the subsolution condition holds.
No continuously differentiable $\varphi$ makes $u - \varphi$ have a local minimum at $0$, because that would need $\varphi(x) - \varphi(0) \le -\abs{x}$ near $0$, so the supersolution condition is empty there.

Non-example: $w(x) = \abs{x} - 1$ is not a viscosity solution of the same equation, although $\abs{w'} = 1$ wherever $w'$ exists.
With $\varphi = 0$, the function $w - \varphi$ has a local minimum at $0$, and $\abs{\varphi'(0)} - 1 = -1 < 0$, so the supersolution condition fails.
</document>

<document source="https://tamnd.github.io/math2/node/F01D990B61.md" tag="F01D990B61" kind="definition">
# Simple homotopy equivalence

## Definition

Let $X$ be a finite [[CW complex|5W95QFHXPR]] and let $n \ge 1$.
Let $D^n$ be the [[closed unit ball|0Q0H13G8ZB]] of $\R^n$.
Its boundary sphere is the union of the closed lower hemisphere $D^{n-1}_-$ and the closed upper hemisphere $D^{n-1}_+$.
The two hemispheres meet in $S^{n-2}$.
Let $g \colon D^{n-1}_- \to X$ be a continuous map with $g(S^{n-2}) \subseteq X^{(n-2)}$ and $g(D^{n-1}_-) \subseteq X^{(n-1)}$, where $X^{(k)}$ is the [[$k$-skeleton|VBM8ZYTT2X]].
Let $Y = X \cup_g D^n$ be the space obtained by gluing $D^n$ to $X$ along $g$.
It is a CW complex with two new cells: the interior of $D^{n-1}_+$ and the interior of $D^n$.

1. The inclusion $X \hookrightarrow Y$ is an *elementary expansion*, and $Y$ is said to *collapse elementarily* onto $X$.
2. A *formal deformation* from $X$ to $Z$ is a finite sequence $X = X_0, X_1, \ldots, X_m = Z$ of finite CW complexes. Each step is an elementary expansion or an elementary collapse.
3. A cellular map $f \colon X \to Z$ is a *simple homotopy equivalence* if it is homotopic to the composite along a formal deformation from $X$ to $Z$. In this composite, an expansion contributes its inclusion and a collapse contributes a deformation retraction.

## Remarks

Let $X$ be a point and $n = 1$: gluing $D^1 = [-1, 1]$ to $X$ along its lower end point gives $Y = [-1, 1]$, with cells $\set{-1}$, $\set{1}$ and $(-1, 1)$.
So the inclusion of a point into an interval is an elementary expansion, and the constant map from the interval to the point is a simple homotopy equivalence.

The lens spaces $L(7, 1)$ and $L(7, 2)$ are [[homotopy equivalent|XZG6MTSWCN]], but no homotopy equivalence between them is simple.
Their Reidemeister torsions differ, and Reidemeister torsion is invariant under formal deformations.

Every simple homotopy equivalence is a homotopy equivalence.
Let $f \colon X \to Z$ be a homotopy equivalence of finite CW complexes.
Then $f$ is simple if and only if its Whitehead torsion $\tau(f)$ vanishes in the Whitehead group of the [[fundamental group|S1XK3HXHFX]] of $Z$.
In particular, if $Z$ is simply connected, every homotopy equivalence $X \to Z$ is simple, since the Whitehead group of the trivial group is zero.
By a theorem of Thomas Chapman, every homeomorphism between finite CW complexes is a simple homotopy equivalence.
</document>

<document source="https://tamnd.github.io/math2/node/F02PJ2ZK9X.md" tag="F02PJ2ZK9X" kind="concept">
# Sperner property

## Summary

A finite [[graded poset|1C7TKGTB2F]] has the Sperner property when no [[antichain|03HF498P1J]] is larger than its largest rank level.
Each rank level is an antichain, so the property says that the [[width|RA5SQ908G1]] of the poset equals the size of a largest level.
[[Sperner's theorem|0J87V9K1GE]] is the statement that the lattice of subsets of a finite set has this property.

## Explanations

### L1 human

Sort the elements of a graded poset into levels by rank.
Elements on one level are never comparable, so each level is an antichain.
The poset is Sperner when you cannot beat the biggest level by picking incomparable elements from several levels at once.
For the subsets of $\set{1, 2, 3, 4}$, the biggest level has the $6$ two-element subsets, and no family of $7$ subsets avoids all inclusions.
</document>

<document source="https://tamnd.github.io/math2/node/F02XA3MYX8.md" tag="F02XA3MYX8" kind="statement">
# Steiner formula

## Statement

Let $n \ge 1$, let $K$ be a nonempty compact convex subset of $\R^n$ (a [[convex body|437FSG3419]]), and let $B$ be the [[closed unit ball|0Q0H13G8ZB]] of $\R^n$.
Then for every $t \ge 0$,
$$\operatorname{vol}(K + tB) = \sum_{j=0}^{n} \binom{n}{j} V(K[n - j], B[j]) \, t^j.$$
Here $K + tB$ is the [[Minkowski sum|SXS6D87PHE]] and $V$ is the [[mixed volume|QB3Q5FBJ08]].
The notation $K[n-j], B[j]$ means $K$ repeated $n - j$ times and $B$ repeated $j$ times.
In particular $\operatorname{vol}(K + tB)$ is a polynomial in $t$ of degree at most $n$.

## Remarks

The formula is the defining expansion of mixed volumes for $K$ and $B$, with $\lambda_1 = 1$ and $\lambda_2 = t$.
One groups the terms by the number of copies of $B$.
The coefficients $W_j(K) = V(K[n-j], B[j])$ are the [[quermassintegrals|H1K36H9ZWB]].
Rewritten with the [[intrinsic volumes|C2CK6KSSTV]] $V_j$, it reads $\operatorname{vol}(K + tB) = \sum_{j=0}^n \kappa_{n-j} V_j(K)\, t^{n-j}$, where $\kappa_m$ is the volume of the unit ball of $\R^m$.

Example: for a cube $K = [0, a]^3$ in $\R^3$,
$$\operatorname{vol}(K + tB) = a^3 + 6 a^2 t + 3 \pi a t^2 + \tfrac{4}{3} \pi t^3.$$
The terms come from the cube, six slabs over the faces, twelve quarter cylinders along the edges, and eight eighths of a ball at the corners.

Convexity is needed for a polynomial: for $K = \set{0, 1} \subseteq \R$ the length of $K + tB$ is $4t$ for $t \le \frac12$ and $1 + 2t$ for $t \ge \frac12$.

## Explanations

### L1 human

Thicken a convex shape by a margin $t$ on every side.
The volume of the thickened shape is a polynomial in $t$.
In the plane it is area plus perimeter times $t$ plus $\pi t^2$.
The coefficients are the basic size measurements of the shape: volume, surface area, and so on down to a constant.

### L2 human

The Steiner formula is the origin of quermassintegrals and intrinsic volumes.
It explains why surface area is a derivative of volume: $\frac{d}{dt} \operatorname{vol}(K + tB)$ at $t = 0$ is $n V(K[n-1], B)$, the surface area of $K$.
Combined with the [[Brunn–Minkowski theorem|9B02RPN7XD]] it yields the [[isoperimetric inequality|KBX8RBZ626]].
Its local versions lead to curvature measures, and a version for sets of positive reach is due to Federer.
</document>

<document source="https://tamnd.github.io/math2/node/F03A4MKA5V.md" tag="F03A4MKA5V" kind="conjecture">
# Voronoi's conjecture on parallelohedra

## Statement

Let $d \ge 1$ and let $P \subset \R^d$ be a [[parallelohedron|MR2JWBCZQZ]].
Then there are a [[lattice|WN9V7WSJKF]] $\Lambda \subset \R^d$ and an affine bijection $A \colon \R^d \to \R^d$ such that $A(P)$ is the [[Voronoi cell|V1R1R88GV6]] of $\Lambda$.

## Remarks

The converse holds: the Voronoi cell of every lattice is a parallelohedron.
Voronoi proved the conjecture for primitive parallelohedra, those whose tiling has exactly $d + 1$ tiles at each vertex.
Delone proved it in dimensions $d \le 4$.
Erdahl proved it for parallelohedra that are zonotopes.
The general case is open.

## Explanations

### L1 human

A parallelohedron is a convex tile that fills space when you only slide copies of it.
Every lattice gives such a tile: the region of points nearer to one lattice point than to all others.
Voronoi guessed that this is the only source of such tiles, once you allow stretching and shearing.
</document>

<document source="https://tamnd.github.io/math2/node/F05PSBZP8Z.md" tag="F05PSBZP8Z" kind="definition">
# Heteroclinic orbit

## Definition

Let $U \subseteq \R^n$ be open, let $f \colon U \to \R^n$ be continuously differentiable, and let $\varphi$ be the [[flow|02903SSA48]] of the system $x' = f(x)$.
Let $p$ and $q$ be two distinct [[equilibrium points|0G4X36AE0X]] of the system.
A *heteroclinic orbit* from $p$ to $q$ is the [[trajectory|0BVT6VSB3S]] $\set{\varphi_t(x) : t \in \R}$ of a point $x \in U$ such that:

1. the solution through $x$ exists for all $t \in \R$;
2. $\varphi_t(x) \to p$ as $t \to -\infty$;
3. $\varphi_t(x) \to q$ as $t \to +\infty$.

## Remarks

Example: for $x' = 1 - x^2$ on $\R$, the equilibria are $-1$ and $1$.
The solution with $x(0) = 0$ is $x(t) = \tanh t$, since $(\tanh t)' = 1 - \tanh^2 t$.
It exists for all $t$, tends to $-1$ as $t \to -\infty$ and tends to $1$ as $t \to +\infty$.
So $(-1, 1)$ is a heteroclinic orbit from $-1$ to $1$.

Non-example: for the same equation, the solution with $x(0) = 2$ is $x(t) = \coth(t + c)$, where $c > 0$ and $\coth c = 2$.
It tends to $1$ as $t \to +\infty$, but it blows up at $t = -c$, so clause 1 fails.

A heteroclinic orbit from $p$ to $q$ lies in the unstable set of $p$ and in the stable set of $q$.
If $p = q$ in the same construction, the orbit is called a homoclinic orbit.
Some authors also allow $p$ and $q$ to be periodic orbits instead of equilibria.
</document>

<document source="https://tamnd.github.io/math2/node/F0ABT2ZFQF.md" tag="F0ABT2ZFQF" kind="definition">
# Projection operation

## Definition

Let $A$ be a set and let $1 \le k \le n$ be integers.
The $k$-th $n$-ary *projection* on $A$ is the [[operation|EJ8YJVM9BW]] $\pi^n_k \colon A^n \to A$ on the [[Cartesian power|09CEDJYBKP]] $A^n$ given by
$$\pi^n_k(a_1, \dots, a_n) = a_k.$$

## Remarks

Example: on $A = \Z$, the operation $\pi^2_1(a, b) = a$ is a projection; it returns the first of its two arguments for every input.
Non-example: on $\Z$, the operation $(a, b) \mapsto \max(a, b)$ is not a projection, since $\max(1, 0) = 1 \ne 0$ rules out $\pi^2_2$ and $\max(0, 1) = 1 \ne 0$ rules out $\pi^2_1$.
The unary projection $\pi^1_1$ is the identity map of $A$.
In any algebra with universe $A$, the term $x_k$, read as a term in the variables $x_1, \dots, x_n$, has the term operation $\pi^n_k$.
</document>

<document source="https://tamnd.github.io/math2/node/F0B6BRJA5Z.md" tag="F0B6BRJA5Z" kind="problem">
# Whitehead problem

## Problem

Let $A$ be an [[abelian group|082GHNPJ53]].
Then $A$ is a *Whitehead group* if the following holds for every abelian group $B$.
For every surjective [[homomorphism|0JYTG5K9Y4]] $f \colon B \to A$ with kernel isomorphic to $\Z$, there is a homomorphism $g \colon A \to B$ with $f \circ g = \id_A$.
Equivalently, $\mathrm{Ext}^1(A, \Z) = 0$, in terms of the [[Ext functor|4P7GBWJ4RR]].
The Whitehead problem asks: is every Whitehead group a [[free abelian group|0WJX8MGG37]]?

## Remarks

Every free abelian group is a Whitehead group, because a surjection onto a free abelian group always splits: send each basis element to any preimage.
Every countable Whitehead group is free, so the question concerns uncountable groups.
Shelah showed in 1974 that the problem is independent of [[ZFC|0HE5WDQ9FJ]].
If the [[axiom of constructibility|4QKRPRCSXH]] $V = L$ holds, then every Whitehead group is free.
If [[Martin's axiom|V8KGT8MRFE]] holds together with the negation of the continuum hypothesis, then there is a Whitehead group that is not free.

## Explanations

### L1 human

An abelian group is free if it has a basis, like $\Z^n$.
Free groups have a lifting property: any map onto them from a bigger group, with kernel $\Z$, can be reversed.
J. H. C. Whitehead asked whether this lifting property characterises the free groups.
The surprising answer is that the usual axioms of set theory cannot decide it: in some models of set theory the answer is yes, in others it is no.
</document>

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# Fenchel's theorem

## Statement

Let $\gamma \colon [a, b] \to \R^3$ be a [[regular|06JMR2XWXW]] [[closed curve|0D032NSZKY]] of class $C^2$ whose derivatives up to order $2$ also agree at $a$ and $b$.
Then the [[total curvature|QP9DY5K8QW]] of $\gamma$ satisfies
$$\int_a^b \kappa(t) \, \norm{\gamma'(t)} \, dt \ge 2\pi.$$
Equality holds if and only if $\gamma$ is a simple closed curve that lies in a plane and bounds a [[convex|0K06X9QXDX]] region of that plane.

## Remarks

The unit tangent $T = \gamma' / \norm{\gamma'}$ is a closed curve on the unit sphere, and its length equals the total curvature.
For each unit vector $v$, the integral of $\ip{T}{v}$ over a period is $\ip{\gamma(b) - \gamma(a)}{v} = 0$.
So $T$ is not contained in any open hemisphere, and a closed curve on the sphere with this property has length at least $2\pi$.
Example: a circle has total curvature exactly $2\pi$, and an ellipse too, since both are plane convex curves.
A figure-eight curve in the plane has total curvature greater than $2\pi$, since it is not simple.
The theorem holds in the same form in $\R^n$ for each $n \ge 2$.
Werner Fenchel proved it in 1929.

## Explanations

### L1 human

Any closed loop in space must turn by at least one full turn in all.
Only flat convex loops, such as a circle or an ellipse, manage with exactly one full turn.
Every other loop wastes some turning.

### L2 human

The proof turns a statement about curves in space into a statement about curves on the sphere, through the tangent indicatrix $T$.
This trick, the Gauss map of a curve, is the one-dimensional model of the Gauss map of a surface.
The [[Fáry–Milnor theorem|B1Z1C8AHVD]] sharpens Fenchel's bound for knotted curves, from $2\pi$ to more than $4\pi$.
</document>

<document source="https://tamnd.github.io/math2/node/F0EXKKKEFW.md" tag="F0EXKKKEFW" kind="statement">
# Straszewicz's theorem

## Statement

Let $n \ge 1$ and let $K \subseteq \R^n$ be a [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].
Then every [[extreme point|3MSGFJM2NX]] of $K$ is a limit of [[exposed points|VN4M72SQW5]] of $K$.
That is, the set of extreme points of $K$ lies in the [[closure|0QEAJB91GZ]] of the set of exposed points of $K$.

## Remarks

Proof idea: for a point $z \in \R^n$, a point $p$ of $K$ farthest from $z$ is exposed.
Indeed, the sphere around $z$ through $p$ encloses $K$, and its tangent hyperplane at $p$ meets $K$ only in $p$.
One then shows that every extreme point can be approached by such farthest points, by choosing $z$ suitably far away in suitable directions.

Combined with the [[Minkowski–Carathéodory theorem|NT4SKSYKG9]], it follows that $K$ is the [[convex hull|0FQSDC2EQ2]] of the closure of its exposed points.

Example: let $K$ be the stadium, the convex hull of the discs of radius $1$ centred at $(-1, 0)$ and $(1, 0)$.
The four points $(\pm 1, \pm 1)$ are extreme but not exposed.
Each of them is a limit of points on the round ends, and those points are exposed.

## Explanations

### L1 human

Extreme points are corners in a broad sense, and exposed points are corners that a flat wall can touch alone.
Some corners are not exposed, such as the points where the straight sides of a stadium meet its round ends.
The theorem says these exceptions are rare: each extreme point is a limit of exposed points.

### L2 human

The theorem lets one replace extreme points by exposed points in many arguments, at the cost of a closure.
Exposed points are often easier to handle, since each comes with a linear function maximised exactly there.
It is used in the study of faces of convex bodies and in convex analysis, where exposed points correspond to directions in which the support function is differentiable.
</document>

<document source="https://tamnd.github.io/math2/node/F0G92Q55DM.md" tag="F0G92Q55DM" kind="concept">
# Stability condition

## Summary

A stability condition on a [[triangulated category|T2V7GJJ2EN]] is a pair made of a central charge and a slicing.
The central charge is a homomorphism from the Grothendieck group to $\C$.
The slicing assigns to each real number a class of semistable objects of that phase.
The notion was introduced by Tom Bridgeland to give a uniform meaning to stability of sheaves and of quiver representations.
The stability conditions on a fixed category form a space with a natural topology.
The study of this space links algebraic geometry with string theory and mirror symmetry.

## Explanations

### L1 human

A stability condition gives each object of a category a complex number, its central charge, which records a kind of mass and an angle.
The angle, divided by $\pi$, is called the phase.
The semistable objects are the basic building blocks, and every object splits in a unique way into layers of semistable pieces whose phases decrease.
For a smooth projective curve, the charge $Z(E) = -\deg E + i \operatorname{rk} E$ of a coherent sheaf $E$ recovers the classical slope stability of vector bundles.
Each stability condition has a heart, which is the heart of a bounded [[t-structure|2GD9V7SA0E]], so it refines the choice of a t-structure.
</document>

<document source="https://tamnd.github.io/math2/node/F0GBTYKN31.md" tag="F0GBTYKN31" kind="definition">
# Midpoint convex function

## Definition

Let $I \subseteq \R$ be an [[interval|0Q1D85B8G6]].
A [[function|0EPAT51DXQ]] $f \colon I \to \R$ is *midpoint convex*, or *Jensen-convex*, if
$$f\left(\frac{x + y}{2}\right) \le \frac{f(x) + f(y)}{2} \quad \text{for all } x, y \in I.$$

## Remarks

Example: $f(x) = x^2$ on $\R$ is midpoint convex.
Check: $\frac{x^2 + y^2}{2} - \left(\frac{x + y}{2}\right)^2 = \frac{(x - y)^2}{4} \ge 0$.

Non-example: $f(x) = -x^2$ on $\R$ is not midpoint convex, since $f(0) = 0 > -1 = \frac{f(-1) + f(1)}{2}$.

Every [[convex function|07Q7CNM5WF]] is midpoint convex: take $t = 1/2$ in the definition of convexity.
The converse is false in general.
A [[discontinuous additive function|FFTNS66XG6]] $a \colon \R \to \R$ satisfies $a\big(\frac{x+y}{2}\big) = \frac{a(x) + a(y)}{2}$, so it is midpoint convex.
But it is not convex, since convex functions on open intervals are continuous.
A midpoint convex function that is continuous is convex.
</document>

<document source="https://tamnd.github.io/math2/node/F0GF0JEZTG.md" tag="F0GF0JEZTG" kind="concept">
# Algebraic dimension

## Summary

The algebraic dimension $a(X)$ of a compact connected [[complex manifold|JSGWAN2RJV]] $X$ is the [[transcendence degree|D7TR0YWR9H]] over $\C$ of its field of meromorphic functions.
It measures how many independent meromorphic functions $X$ carries.
By [[Siegel's theorem|ENXCWPTG92]], $0 \le a(X) \le \dim X$.
The extreme case $a(X) = \dim X$ defines [[Moishezon manifolds|8ZW9P9JRQD]], which are close to projective varieties, while $a(X) = 0$ occurs for generic complex tori.

## Explanations

### L1 human

On a compact complex manifold the holomorphic functions are all constant, but meromorphic functions, which may have poles, can be interesting.
The algebraic dimension counts how many of them are independent, in the sense that no polynomial relation links them.
On complex projective space of dimension $n$ there are $n$ independent ones, such as $z_1/z_0, \ldots, z_n/z_0$.
On some complex tori there are none besides the constants.
So the algebraic dimension tells how far a compact complex manifold is from being algebraic.
</document>

<document source="https://tamnd.github.io/math2/node/F0HQXQ0HCD.md" tag="F0HQXQ0HCD" kind="concept">
# Tensor product of modules

## Summary

The tensor product $M \otimes_R N$ of two modules over a [[commutative ring|0310REJJY8]] $R$ is the module that turns bilinear maps on $M \times N$ into linear maps.
It is generated by the symbols $m \otimes n$, which are additive in each variable and let scalars move between the two factors.
It generalises the [[tensor product of vector spaces|0BF2DE2SYK]], but over a ring it can be much smaller than expected, even zero.
Tensor products are used for change of rings, for [[flatness|73RYYEVDKA]], and for the [[tensor-hom adjunction|AVXARAXHKY]].

## Explanations

### L1 human

The tensor product combines two modules into one, in which you can multiply an element of the first by an element of the second.
The rules are those of a product: $(m + m') \otimes n = m \otimes n + m' \otimes n$, and a scalar can be moved from one side to the other.
Sometimes everything cancels.
For example, $\Z/2\Z \otimes_\Z \Z/3\Z = 0$: since $3 \cdot 1 = 1$ in $\Z/2\Z$, one has $a \otimes b = 3a \otimes b = a \otimes 3b = a \otimes 0 = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/F0JQ1TGKB0.md" tag="F0JQ1TGKB0" kind="concept">
# Laguerre plane

## Summary

A Laguerre plane is an incidence structure of points and circles with a parallelism of points.
It is modelled on the plane sections of a circular cylinder.
Three pairwise non-parallel points lie on exactly one circle, and each circle meets each parallel class in exactly one point.
It is one of the three kinds of [[Benz plane|BG5A4CB6NB]].

## Explanations

### L1 human

Take a round cylinder and cut it with planes that are not parallel to its axis; the cuts are ellipses, called circles here.
Two points on the same vertical line of the cylinder are called parallel, and no circle passes through both.
Through three points with no two parallel there is exactly one such circle.
A flat picture of the same geometry uses parabolas $y = ax^2 + bx + c$ in the plane, with vertical lines as parallel classes.
</document>

<document source="https://tamnd.github.io/math2/node/F0M6BY5BH7.md" tag="F0M6BY5BH7" kind="definition">
# Exact couple

## Definition

An *exact couple* consists of two [[abelian groups|071DDFVG4J]] $D$ and $E$ and three homomorphisms
$$i \colon D \to D, \qquad j \colon D \to E, \qquad k \colon E \to D,$$
such that the triangle is [[exact|CBDM280TDF]] at each vertex:

1. $\ker j = \im i$;
2. $\ker k = \im j$;
3. $\ker i = \im k$.

## Remarks

The map $d = j \circ k \colon E \to E$ satisfies $d \circ d = j (k j) k = 0$, by clause 2.
The *derived couple* has $D' = i(D)$ and $E' = \ker d / \im d$, with $i'$ the restriction of $i$, $j'(i(x)) = [j(x)]$ and $k'([e]) = k(e)$; it is again an exact couple.
Iterating gives groups $E^r$ with differentials $d^r$ such that $E^{r+1}$ is the homology of $(E^r, d^r)$, that is, a [[spectral sequence|GG8AYV847V]].
In practice $D$ and $E$ are bigraded, and the bidegrees of $i, j, k$ fix the bidegree of each $d^r$.

Example: let $C$ be a chain complex of free abelian groups and $p$ a prime.
The short exact sequence $0 \to C \xrightarrow{p} C \to C / pC \to 0$ gives a long exact sequence in homology.
It is an exact couple with $D = H_*(C)$, $E = H_*(C/pC)$, $i$ multiplication by $p$, $j$ reduction mod $p$ and $k$ the connecting map.

Non-example: take $D = E = \Z$ with $i = 0$, $j = \id$ and $k = 0$.
Clause 1 holds, since $\ker j = 0 = \im i$, and clause 2 holds, since $\ker k = \Z = \im j$.
Clause 3 fails, since $\ker i = \Z$ while $\im k = 0$.

Exact couples were introduced by William S. Massey.
</document>

<document source="https://tamnd.github.io/math2/node/F0QRCDYBXK.md" tag="F0QRCDYBXK" kind="definition">
# Pluripolar set

## Definition

Let $E \subseteq \C^n$.
The set $E$ is *pluripolar* if each point $a \in E$ has an open neighbourhood $U$ with the following property.
There is a [[plurisubharmonic function|FF2P7ER3QT]] $u$ on $U$, not identically $-\infty$ on any component of $U$, such that
$$E \cap U \subseteq \set{z \in U : u(z) = -\infty}.$$
The set $E$ is *complete pluripolar* in an open set $\Omega$ if $E = \set{z \in \Omega : u(z) = -\infty}$ for some function $u$ of the following kind.
The function $u$ is plurisubharmonic on $\Omega$ and not identically $-\infty$ on any component of $\Omega$.

## Remarks

Some authors call the sets of the first definition locally pluripolar, and call $E$ globally pluripolar if one function $u$ plurisubharmonic on all of $\C^n$, not identically $-\infty$, has $E \subseteq \set{u = -\infty}$.
By [[Josefson's theorem|V6XSDT5C21]] the two notions agree.
Example: $E = \set{z \in \C^2 : z_1 = 0}$ is pluripolar; take $U = \C^2$ and $u(z) = \log\abs{z_1}$, which is plurisubharmonic and equals $-\infty$ exactly on $E$.
Non-example: $\R^n \subset \C^n$ is not pluripolar.
For $n = 1$: if $u$ is subharmonic near a point of $\R$ and $u = -\infty$ on an interval, then $u \equiv -\infty$ near that point, since intervals are not [[polar|C95QJ7YH5T]].
Applying this on complex lines parallel to the coordinate axes, one coordinate at a time, gives the general case.
For $n = 1$, pluripolar sets are exactly polar sets.
</document>

<document source="https://tamnd.github.io/math2/node/F0RVAF11K9.md" tag="F0RVAF11K9" kind="concept">
# Fekete points

## Summary

Let $K$ be a compact set in the plane.
Fekete points of $K$ are $n$ points of $K$ that make the product of their mutual distances as large as possible.
They realize the $n$-th diameter in the definition of the [[transfinite diameter|VSGTSQ7RY8]].
Let $K$ have positive capacity.
As $n$ grows, the uniform distribution on the Fekete points tends to the [[equilibrium measure|464NN2K4FZ]] of $K$.
They are used as interpolation nodes and as discrete models of an electric charge in equilibrium.

## Explanations

### L1 human

Put $n$ equal point charges on a conductor $K$ and let them push each other apart with the logarithmic law.
The positions where they come to rest with the least energy are the Fekete points.
On a circle they are $n$ equally spaced points.
On an interval they crowd near the two ends, just as the equilibrium charge of an interval is densest at its ends.
</document>

<document source="https://tamnd.github.io/math2/node/F0XAMM2PPS.md" tag="F0XAMM2PPS" kind="example">
# Prüfer group

## Example

Let $p$ be a [[prime|01826KMG2H]].
The *Prüfer $p$-group* is
$$\Z(p^\infty) = \Z[1/p] / \Z = \set{m/p^k + \Z : m \in \Z, \ k \ge 0},$$
a subgroup of $\Q/\Z$.
Equivalently, it is the group of all complex [[roots of unity|0S61N75H45]] of order a power of $p$, a subgroup of the [[circle group|04EAFVFTF7]].

It is a [[divisible group|ZXVBEPMTCN]].
Let $x = m/p^k + \Z$ and $n \ge 1$, and write $n = p^j r$ with $p \nmid r$.
Choose an integer $s$ with $rs \equiv 1 \pmod{p^{k+j}}$, and put $y = sm/p^{k+j} + \Z$.
Then $ny = rs \, m / p^k + \Z$, and $rs \, m/p^k - m/p^k = (rs - 1) m / p^k$ is an integer, because $p^{k+j}$ divides $rs - 1$; so $ny = x$.

## Remarks

The Prüfer group is the union of the chain of cyclic groups $\Z/p\Z \subset \Z/p^2\Z \subset \cdots$, where $\Z/p^k\Z$ is identified with $\set{m/p^k + \Z}$.
Its proper subgroups are exactly these finite cyclic groups, so it is not finitely generated.
It is a [[p-group|06QYRTX9KQ]] in the sense that every element has order a power of $p$.
Together with $\Q$, the Prüfer groups are the building blocks of all divisible groups.
Its Pontryagin dual is the group of [[p-adic integers|V6TJAFZ4MD]].

## Explanations

### L1 human

Take the circle and mark the points that you reach by rotating by $1/p$ of a turn, by $1/p^2$ of a turn, and so on.
Add all their combinations.
This set of rotations is the Prüfer group.
Every rotation in it can be split into $n$ equal smaller rotations that still lie in it.
It is infinite, but each of its proper subgroups is finite.
</document>

<document source="https://tamnd.github.io/math2/node/F0YGC89S9R.md" tag="F0YGC89S9R" kind="statement">
# Bernstein's inequality for polynomials

## Statement

Let $n \ge 0$ and let $P$ be a polynomial with [[complex|09FTAET5Z4]] coefficients of [[degree|0DPF6WRZW6]] at most $n$.
Then
$$\max_{\abs{z} = 1} \abs{P'(z)} \le n \max_{\abs{z} = 1} \abs{P(z)}.$$
Equality holds if and only if $P(z) = a z^n$ for some $a \in \C$.

## Remarks

Proof idea: let $M = \max_{\abs{z} = 1} \abs{P(z)} > 0$ and let $\lambda \in \C$ with $\abs{\lambda} > 1$.
On $\abs{z} = 1$ one has $\abs{\lambda M z^n} > \abs{P(z)}$, so by Rouché's theorem all $n$ roots of $P(z) - \lambda M z^n$ lie in $\abs{z} < 1$.
By the [[Gauss–Lucas theorem|A56086N8XQ]], so do the roots of $P'(z) - \lambda n M z^{n-1}$.
If $\abs{P'(z_0)} > n M$ for some $\abs{z_0} = 1$, then $\lambda = P'(z_0) / (n M z_0^{n-1})$ has $\abs{\lambda} > 1$ and makes $z_0$ such a root, a contradiction.

The inequality has an equivalent trigonometric form: a real or complex trigonometric polynomial $T$ of degree at most $n$ satisfies $\max \abs{T'} \le n \max \abs{T}$.
With $x = \cos \theta$, the trigonometric form gives, for a real polynomial $P$ of degree at most $n$ and $-1 < x < 1$, the bound $\abs{P'(x)} \le \frac{n}{\sqrt{1 - x^2}} \max_{[-1, 1]} \abs{P}$.

Example: $P(z) = z^n$ gives $\abs{P'(z)} = n$ on the unit circle, so equality holds.

## Explanations

### L1 human

A polynomial of degree $n$ that is at most $1$ in size on the unit circle has derivative at most $n$ there.
The pure power $z^n$, which turns around the circle $n$ times as fast, is the extreme case.

### L2 human

Bernstein's inequality is the basic tool for inverse theorems in approximation theory: good approximation by polynomials of low degree forces smoothness.
Its form on $[-1, 1]$ is sharp in the interior, while near the endpoints the [[Markov brothers' inequality|EHA5MF9VKE]] gives the correct bound $n^2$.
The proof through the Gauss–Lucas theorem shows how geometric facts about roots give analytic estimates.
</document>

<document source="https://tamnd.github.io/math2/node/F12A5QKW8Z.md" tag="F12A5QKW8Z" kind="concept">
# Walsh function

## Summary

The Walsh functions are functions on $[0, 1)$ that take only the values $1$ and $-1$.
They are the finite products of [[Rademacher functions|51GRFQ4WY6]], and together they form a complete orthonormal system of $L^2([0, 1))$.
They are a digital analogue of the trigonometric system, and they lead to the [[Walsh–Hadamard transform|K784BA2WHV]].

## Explanations

### L1 human

Think of a square wave that jumps between $+1$ and $-1$.
The Rademacher functions are square waves that flip sign faster and faster, one for each binary digit.
A Walsh function multiplies some of these square waves together, chosen by the binary digits of an index $n$.
For example, $w_3 = r_1 r_2$ equals $1$ on $[0, \tfrac14)$, $-1$ on $[\tfrac14, \tfrac34)$, and $1$ on $[\tfrac34, 1)$.
Every square-integrable function on $[0, 1)$ can be expanded in Walsh functions, much as in a Fourier series.
</document>

<document source="https://tamnd.github.io/math2/node/F12V24NF4R.md" tag="F12V24NF4R" kind="definition">
# Von Neumann algebra

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$, with the [[adjoint|6HSJBJ8TTQ]] $T \mapsto T^*$.
A *von Neumann algebra* on $H$ is a subset $M \subseteq B(H)$ such that:

1. $M$ is a subalgebra of $B(H)$ that contains the identity operator $I$;
2. $T^* \in M$ for every $T \in M$;
3. $M$ is closed in the [[weak operator topology|4XM1VN1C1N]].

## Remarks

Example: let $(X, \mu)$ be a $\sigma$-finite measure space and $H = L^2(X, \mu)$.
For $f \in L^\infty(X, \mu)$ let $M_f$ be multiplication by $f$.
The set $\set{M_f : f \in L^\infty}$ contains $I = M_1$, is closed under products and under $M_f^* = M_{\bar f}$, and one checks that it equals its own commutant, so it is weakly closed.
Other examples are $B(H)$ itself and the matrix algebras $M_n(\C)$ acting on $\C^n$.

Non-example: on $H = L^2[0,1]$, the multiplication operators $M_f$ with $f$ continuous satisfy clauses 1 and 2 and form a C*-algebra.
But clause 3 fails.
The indicator function of a subinterval is a pointwise limit of uniformly bounded continuous functions.
By dominated convergence, its multiplication operator is a weak limit of elements of the algebra, but it is not continuous.
The weak closure of this algebra is the whole multiplication algebra of $L^\infty[0,1]$.

By the von Neumann bicommutant theorem, clause 3 can be replaced by closedness in the strong operator topology, or by the condition $M = M''$, where $M'$ is the commutant.
Every von Neumann algebra is a C*-algebra.
By Sakai's theorem, an abstract C*-algebra is isomorphic to a von Neumann algebra exactly when it is a dual Banach space.
Such algebras are called W*-algebras.
</document>

<document source="https://tamnd.github.io/math2/node/F18MBATVVP.md" tag="F18MBATVVP" kind="concept">
# Nonlinear Volterra equation

## Summary

A nonlinear Volterra equation is an integral equation $\varphi(t) = f(t) + \int_a^t K(t, s, \varphi(s))\, ds$.
The upper limit of integration is the variable itself, and the unknown may enter the integrand nonlinearly.
It extends the linear [[Volterra integral equation|DND8CBEVXT]] and contains every initial value problem $y' = F(t, y)$, $y(a) = y_0$, in integral form.
When $K$ is [[Lipschitz|0AW1E6YTJY]] in its last variable, successive approximations give a unique solution on the whole interval.
When the Lipschitz bound holds only on bounded sets, a solution may exist only on a short interval.

## Explanations

### L1 human

The value of the unknown at time $t$ depends on its whole past, through a rule that may be nonlinear.
Because only the past matters, the solution can be built forward step by step.
For example, $\varphi(t) = 1 + \int_0^t \varphi(s)^2\, ds$ is solved by $\varphi(t) = 1/(1 - t)$, which blows up at $t = 1$.
So, unlike the linear case, a solution need not last for all time.
</document>

<document source="https://tamnd.github.io/math2/node/F19BQTN832.md" tag="F19BQTN832" kind="definition">
# Whitney stratification

## Definition

Let $X$ be a closed subset of $\R^N$.
Let $S$ and $S'$ be disjoint smooth [[submanifolds|VKFW27WXEZ]] of $\R^N$ and let $y \in S'$.
The pair $(S, S')$ satisfies *Whitney's condition (b)* at $y$ if the following holds.
Take any sequences $x_k \in S$ and $y_k \in S'$ that both converge to $y$, with $x_k \ne y_k$.
If the [[tangent spaces|WY6K3P4YCG]] $T_{x_k} S$ converge to a subspace $\tau$ and the lines through $x_k$ and $y_k$ converge to a line $\ell$, then $\ell \subseteq \tau$.
The pair satisfies *condition (a)* at $y$ if the following holds.
For every sequence $x_k \in S$ converging to $y$ such that $T_{x_k} S$ converges to a subspace $\tau$, one has $T_y S' \subseteq \tau$.

A *Whitney stratification* of $X$ is a locally finite partition of $X$ into connected smooth submanifolds of $\R^N$, called *strata*, such that:

1. each stratum is locally closed in $X$;
2. if a stratum $S'$ meets the closure $\overline{S}$ of a stratum $S \ne S'$, then $S' \subseteq \overline{S} \setminus S$ (the frontier condition);
3. every such pair $(S, S')$ satisfies condition (b) at every point of $S'$.

## Remarks

Condition (b) implies condition (a).
The same definition applies to subsets of $\C^N = \R^{2N}$, and for complex analytic sets one asks the strata to be complex submanifolds.

Example: the cone $X = \set{x^2 + y^2 = z^2} \subset \R^3$ with three strata: $\set{0}$ and the two connected components of $X \setminus \set{0}$, where $z > 0$ and $z < 0$.
The frontier condition holds, and condition (b) holds at $0$.
Indeed, the cone is a union of lines through $0$, so the line from $0$ to $x_k$ lies in the tangent plane $T_{x_k} X$, hence also in the limit plane.

Non-example: the complex Whitney umbrella $X = \set{x^2 = z y^2} \subset \C^3$ with the two strata $S'$, the $z$-axis, and $S = X \setminus S'$.
The surface $S$ is smooth, since the gradient $(2x, -2zy, -y^2)$ of $x^2 - z y^2$ is nonzero off the $z$-axis.
Every point $(0, 0, c)$ is a limit of points $(\sqrt{c}\, t, t, c)$ of $S$ as $t \to 0$, so the frontier condition holds.
For small real $s > 0$, the point $x_s = (s^{3/2}, s^{1/2}, s^2)$ lies on $S$ and tends to $0$.
There the normal vector is $(2 s^{3/2}, -2 s^{5/2}, -s)$, which, divided by $s$, tends to $(0, 0, -1)$.
So $T_{x_s} S$ tends to the complex plane $z = 0$, which does not contain $T_0 S'$, the $z$-axis.
Condition (a) fails at $0$, so condition (b) fails too.
The origin must be made a stratum of its own: the strata $S$, the $z$-axis minus $0$, and $\set{0}$ form a Whitney stratification.
</document>

<document source="https://tamnd.github.io/math2/node/F1CCCF4V83.md" tag="F1CCCF4V83" kind="statement">
# Rokhlin lemma

## Statement

Let $(X, \mathcal{B}, \mu)$ be a [[standard probability space|QDDRDW9PXS]] and let $T \colon X \to X$ be an invertible [[measure-preserving transformation|556WBDAHNP]] with measurable inverse.
Assume that $T$ is aperiodic: for every $n \ge 1$, the set $\set{x \in X : T^n x = x}$ has measure $0$.
Then for every integer $n \ge 1$ and every $\varepsilon > 0$ there is a set $B \in \mathcal{B}$ such that:

1. the sets $B, TB, T^2B, \dots, T^{n-1}B$ are pairwise disjoint, and
2. $\mu(B \cup TB \cup \cdots \cup T^{n-1}B) > 1 - \varepsilon$.

## Remarks

The sets $B, TB, \dots, T^{n-1}B$ form a Rokhlin tower of height $n$ with base $B$, and $T$ moves each level to the next one.
The proof for ergodic $T$ starts from a set $F$ of small positive measure and slices the space by the first return time to $F$.
Each column of height at least $n$ is cut into blocks of height $n$, and the leftover has measure at most about $n \mu(F)$.
Example at work: a tower gives a periodic approximation of $T$.
Define $P = T$ on $B \cup \cdots \cup T^{n-2}B$, let $P = T^{-(n-1)}$ on $T^{n-1}B$, and let $P$ be the identity elsewhere.
Then $P^n$ is the identity, $P$ preserves $\mu$, and $P$ agrees with $T$ outside a set of measure less than $\mu(B) + \varepsilon \le 1/n + \varepsilon$.
The aperiodicity hypothesis is necessary: if $T^2 = \mathrm{id}$ and $n = 3$, then $T^2 B = B$, so condition 1 forces $\mu(B) = 0$ and condition 2 fails.
The error $\varepsilon$ cannot be removed in general.
If a tower of height $n \ge 2$ covers $X$ up to a null set, then $T^n B = B$ up to a null set, and $f = \sum_{j=0}^{n-1} e^{2\pi i j/n} \mathbf{1}_{T^j B}$ satisfies $f \circ T = e^{2\pi i/n} f$.
So $f$ is a nonconstant eigenfunction, which is impossible when $T$ is weakly mixing, for example a Bernoulli scheme.

## Explanations

### L1 human

Any measure-preserving map without periodic behaviour can be pictured, up to a tiny leftover, as a tall stack of $n$ floors.
The map moves each floor up to the next one, and only at the top floor does something complicated happen.
The height $n$ can be as large as you like and the leftover as small as you like.

### L2 human

The lemma is the main tool for building approximations and codings in ergodic theory.
It shows that every aperiodic transformation can be approximated in the weak topology by periodic ones.
It is used in the proofs of Sinai's factor theorem and of Ornstein's isomorphism theorem.
It extends to free actions of $\Z^d$ and of countable amenable groups, where the role of intervals of times is played by Følner sets.
</document>

<document source="https://tamnd.github.io/math2/node/F1CGWAW74E.md" tag="F1CGWAW74E" kind="definition">
# Stalk

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]], let $\mathcal F$ be a presheaf or a [[sheaf|TB6TM6R4MD]] of sets on $X$, and let $x \in X$.
Consider pairs $(U, s)$ where $U$ is an [[open set|02BH3MCBQB]] containing $x$ and $s \in \mathcal F(U)$.
Two pairs $(U, s)$ and $(V, t)$ are equivalent if there is an open set $W$ with $x \in W \subseteq U \cap V$ and $s|_W = t|_W$.
The *stalk* of $\mathcal F$ at $x$ is the set $\mathcal F_x$ of equivalence classes.
The class of $(U, s)$ is the *germ* of $s$ at $x$, written $s_x$.

## Remarks

The relation is an [[equivalence relation|0G8VEZ3WTK]]: for transitivity, intersect the two open sets $W$.
In the language of categories, $\mathcal F_x$ is the direct limit of the sets $\mathcal F(U)$ over the open neighbourhoods $U$ of $x$, ordered by reverse inclusion.
If $\mathcal F$ is a sheaf of abelian groups or of rings, then each stalk is an abelian group or a ring, with $s_x + t_x = (s|_W + t|_W)_x$ for $W$ a common neighbourhood.

Example: for the sheaf of continuous real functions on $\R$, the stalk at $0$ is the ring of [[germs|VZXF7RNFKS]] of continuous functions at $0$.
It is a [[local ring|MZ02BKPHJA]]: the germs with $f(0) \ne 0$ are units, since $1/f$ is continuous near $0$, and the other germs form the unique maximal ideal.

Example: for the sheaf of holomorphic functions on $\C$, the stalk at $0$ is the ring of power series $\sum a_n z^n$ with positive radius of convergence.

Non-example: a germ is not a function value.
The functions $0$ and $x \mapsto \max(x, 0)$ have the same value at $0$, but they have different germs at $0$, since they differ on every interval around $0$.

For a sheaf, a section $s \in \mathcal F(U)$ is determined by its germs $s_x$ for $x \in U$, by the locality condition.
</document>

<document source="https://tamnd.github.io/math2/node/F1EHT2CPHC.md" tag="F1EHT2CPHC" kind="object">
# Feigenbaum constants

## Description

The Feigenbaum constants are two real numbers that describe the cascade of [[period-doubling bifurcations|NQDTTZ59X8]] of a one-parameter family of unimodal maps with a quadratic maximum.
Let $f_r$ be such a family, for example the [[logistic map|QN9X9BNKTX]] $f_r(x) = r x (1 - x)$.
Let $r_n$ be the parameter at which the attracting orbit of period $2^{n-1}$ loses stability and an attracting orbit of period $2^n$ appears.
For the logistic map $r_1 = 3$ and $r_2 = 1 + \sqrt 6 \approx 3.4495$.
The first Feigenbaum constant is
$$\delta = \lim_{n \to \infty} \frac{r_n - r_{n-1}}{r_{n+1} - r_n} = 4.669201609\ldots.$$
Let $c$ be the critical point, and let $R_n$ be the parameter, between $r_n$ and $r_{n+1}$, at which $c$ is periodic of least period $2^n$.
Let $d_n = f_{R_n}^{2^{n-1}}(c) - c$, the signed distance from $c$ to the point of its orbit halfway around the cycle.
The second Feigenbaum constant is
$$\alpha = \lim_{n \to \infty} \abs{\frac{d_n}{d_{n+1}}} = 2.502907875\ldots.$$

## Remarks

Check for the logistic map: the period-$2$ orbit is stable while $\abs{(f_r^2)'} < 1$ on it, and solving gives $r_2 = 1 + \sqrt 6$; the parameters $r_n$ accumulate at $r_\infty \approx 3.5699$.
The ratios $d_n / d_{n+1}$ alternate in sign, which is why $\alpha$ is often given as $-2.5029\ldots$.
The constants are universal: they take the same values for every smooth family of unimodal maps with a nondegenerate quadratic maximum that has such a cascade.
Feigenbaum explained this by a renormalization operator acting on maps of the interval, with a fixed point whose unstable eigenvalue is $\delta$ and whose scaling factor is $\alpha$.
Lanford gave a computer-assisted proof of the existence of this fixed point in 1982.
It is not known whether $\delta$ or $\alpha$ is irrational.

## Explanations

### L1 human

When a parameter is turned up, many systems double their period again and again: a cycle of $2$, then $4$, $8$, and so on, until chaos.
The gaps between these doublings shrink, each about $4.669$ times smaller than the one before.
The same number appears for dripping taps, electronic circuits and many simple formulas, which made it a surprise when Feigenbaum found it in the 1970s.
</document>

<document source="https://tamnd.github.io/math2/node/F1FF505PWH.md" tag="F1FF505PWH" kind="concept">
# Height function

## Summary

A height function measures the arithmetic complexity of a rational or algebraic point, roughly the number of digits needed to write down its coordinates.
The basic example is the Weil height on projective space; composing it with a morphism gives heights on any projective variety.
The key property, due to Northcott, is that there are only finitely many points of bounded height and bounded degree.
Heights drive finiteness proofs such as [[Mordell's theorem|BZHGKN6Q3K]] and appear in the [[Gross–Zagier theorem|BX558DTV2D]] and the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].

## Explanations

### L1 human

The rational numbers $1/2$ and $1000001/2000001$ are close in size, but the second is much more complicated.
A height captures this: the height of a fraction $a/b$ in lowest terms is $\max(\abs{a}, \abs{b})$, so the first has height $2$ and the second has height $2000001$.
There are only finitely many fractions of height at most a given bound, so heights let one turn "infinitely many points" into "points of ever larger complexity".
For points with several coordinates, one takes the largest coordinate after clearing denominators.
On an elliptic curve, a refined height behaves like a quadratic form under the group law, which is the engine of the descent proofs.
</document>

<document source="https://tamnd.github.io/math2/node/F1M51B4MDG.md" tag="F1M51B4MDG" kind="concept">
# Equational theory

## Summary

The equational theory of a class of algebras is the set of all [[identities|PKD440G2XW]] that hold in every member of the class.
A set of identities is called an equational theory when it arises in this way.
Equational theories correspond one to one to varieties of algebras, with larger theories giving smaller varieties.

## Explanations

### L1 human

Collect every law that holds in all groups: associativity, $x \cdot e \approx x$, $(x y)^{-1} \approx y^{-1} x^{-1}$, and infinitely many more.
This collection is the equational theory of groups.
It does not contain $x y \approx y x$, because some groups are not commutative.
The equational theory of abelian groups is bigger: it contains $x y \approx y x$ and everything that follows from it.
</document>

<document source="https://tamnd.github.io/math2/node/F1MCB9P7KJ.md" tag="F1MCB9P7KJ" kind="definition">
# Integrable system

## Definition

Let $(M, \omega)$ be a [[symplectic manifold|CAN490EPRT]] of dimension $2n$, and let $H \colon M \to \R$ be a smooth function.
Let $\{F, G\}$ denote the [[Poisson bracket|2S0MXDT4YH]] of smooth functions $F, G$ on $M$.
The [[Hamiltonian system|0EDKGA2FAK]] with Hamiltonian $H$ is *integrable*, or *Liouville integrable*, if there are smooth functions $F_1 = H, F_2, \ldots, F_n \colon M \to \R$ such that:

1. $\{F_i, F_j\} = 0$ for all $i, j \in \set{1, \ldots, n}$;
2. the differentials $dF_1, \ldots, dF_n$ are linearly independent at each point of an open dense subset of $M$.

## Remarks

Clause 1 with $i = 1$ says that each $F_j$ is a [[first integral|0262JP0NV9]] of the system, since $\frac{d}{dt} F_j = \{F_j, H\}$ along solutions.
Functions that satisfy clause 1 are said to be in involution.

Example: on $\R^4$ with coordinates $(q_1, q_2, p_1, p_2)$, let $H = \frac{1}{2}(p_1^2 + q_1^2) + \frac{1}{2}(p_2^2 + q_2^2)$ and $F_2 = \frac{1}{2}(p_2^2 + q_2^2)$.
Then $\{H, F_2\} = \{\frac{1}{2}(p_2^2 + q_2^2), \frac{1}{2}(p_2^2 + q_2^2)\} = 0$, since the first summand of $H$ does not depend on $q_2, p_2$.
The differentials $dH$ and $dF_2$ are independent wherever $(q_1, p_1) \ne (0, 0)$ and $(q_2, p_2) \ne (0, 0)$, an open dense set.

Non-example: the same $H$ with $F_2 = H$ satisfies clause 1 but not clause 2, since $dF_1 = dF_2$ everywhere.

Every Hamiltonian system with one degree of freedom ($n = 1$) is integrable as soon as $dH \ne 0$ on an open dense set.
The [[Liouville–Arnold theorem|PNQCW85HTJ]] describes the motion of an integrable system near a compact regular level set.
</document>

<document source="https://tamnd.github.io/math2/node/F1P5SZ1XPH.md" tag="F1P5SZ1XPH" kind="definition">
# Constructivism

## Definition

*Constructivism* is the position in the philosophy of mathematics that consists of these two principles.

1. A mathematical object exists only if there is a method to construct it.
2. A proof of "there exists $x$ such that $P(x)$" must give a method to construct such an $x$. It must also prove $P(x)$ for this $x$.

*Constructive mathematics* is mathematics carried out under these principles.
Its logic is intuitionistic logic: classical logic without the general [[law of excluded middle|0RYYK8PG4S]] $P \vee \neg P$.

## Remarks

Example: the proof that for each natural number $n$ there is a prime $p > n$, by taking a prime factor of $n! + 1$, is constructive.
It gives a method: factor $n! + 1$ and take any prime factor $p$; then $p > n$, because no number from $2$ to $n$ divides $n! + 1$.
Non-example: the classical proof that there are irrational numbers $a, b$ with $a^b$ rational is not constructive.
It argues that either $\sqrt{2}^{\sqrt{2}}$ is rational, or it is not and then $(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = 2$; it does not say which case holds.
It uses the law of excluded middle, which fails principle 2.
In constructive mathematics, a [[proof by contradiction|0J0XBKG64S]] proves negated statements $\neg P$, but in general does not prove $P$ from $\neg \neg P$; see the [[double negation law|0NFAGWH84K]].
</document>

<document source="https://tamnd.github.io/math2/node/F1QWA1PEQQ.md" tag="F1QWA1PEQQ" kind="statement">
# Tridiagonal matrix algorithm

## Statement

Let $n \ge 2$ and consider the [[linear system|0GT4WWGWH2]]
$$a_i x_{i-1} + b_i x_i + c_i x_{i+1} = d_i, \qquad i = 1, \dots, n,$$
with real or complex coefficients, $a_1 = 0$ and $c_n = 0$, whose matrix $A$ is [[tridiagonal|DARX1KZRHZ]].
Assume that $A$ is strictly [[diagonally dominant|89P5GTKW87]] by rows: $\abs{b_i} > \abs{a_i} + \abs{c_i}$ for all $i$.
The *tridiagonal matrix algorithm* (or *Thomas algorithm*) computes
$$c'_1 = \frac{c_1}{b_1}, \quad d'_1 = \frac{d_1}{b_1}, \qquad c'_i = \frac{c_i}{b_i - a_i c'_{i-1}}, \quad d'_i = \frac{d_i - a_i d'_{i-1}}{b_i - a_i c'_{i-1}} \quad (i = 2, \dots, n),$$
and then
$$x_n = d'_n, \qquad x_i = d'_i - c'_i x_{i+1} \quad (i = n - 1, \dots, 1).$$
Then every denominator $b_i - a_i c'_{i-1}$ is nonzero, $\abs{c'_i} < 1$ for all $i$, and $(x_1, \dots, x_n)$ is the unique solution of the system.
The algorithm uses $O(n)$ arithmetic operations.

## Remarks

The proof is an induction on $i$.
If $\abs{c'_{i-1}} < 1$, then $\abs{b_i - a_i c'_{i-1}} \ge \abs{b_i} - \abs{a_i} > \abs{c_i} \ge 0$, so the denominator is nonzero and $\abs{c'_i} < 1$.
The forward sweep is [[Gaussian elimination|0JVJ2WCXZH]] without pivoting, which keeps the tridiagonal shape, and it computes the [[LU decomposition|0RKG5YW66Y]] of $A$ with bidiagonal factors; the second sweep is [[back substitution|0ATFVE08MW]].
For diagonally dominant or symmetric positive definite matrices, the algorithm is also numerically stable.
Without such a hypothesis a denominator can vanish even when $A$ is invertible: for $b_1 = 0$, $c_1 = 1$, $a_2 = 1$, $b_2 = 0$, the matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is invertible but the first step divides by $0$.
Example: the system $2x_1 - x_2 = 1$, $-x_1 + 2x_2 = 1$ is diagonally dominant by rows; the algorithm gives $c'_1 = -1/2$, $d'_1 = 1/2$, $d'_2 = (1 + 1/2)/(2 - 1/2) = 1$, so $x_2 = 1$ and $x_1 = 1/2 + 1/2 = 1$.

## Explanations

### L1 human

Many problems, such as heat flow along a rod or spline fitting, give systems where each equation involves only an unknown and its two neighbours.
The Thomas algorithm solves such a system in two sweeps: down the list, eliminating one unknown from each equation, and back up, reading off the unknowns one by one.
The work grows only in proportion to the number of unknowns, instead of its cube.

### L2 human

The algorithm is Gaussian elimination specialized to tridiagonal matrices, with no fill-in and no pivoting.
Strict diagonal dominance is a simple condition that guarantees it never divides by zero.
The finite difference matrix of $-u'' = f$, with rows $(-1, 2, -1)$, is only weakly diagonally dominant, but it is symmetric positive definite, and the algorithm works for it too.
For tridiagonal systems that are not diagonally dominant, one uses Gaussian elimination with partial pivoting on the band instead.
</document>

<document source="https://tamnd.github.io/math2/node/F1W3QRBNGJ.md" tag="F1W3QRBNGJ" kind="statement">
# Helly's theorem

## Statement

Let $d \ge 1$ and let $C_1, \dots, C_m$ be [[convex sets|02AS45T3H9]] in $\R^d$, with $m \ge d + 1$.
Suppose that every $d + 1$ of the sets have a point in common.
Then all $m$ sets have a point in common: $C_1 \cap \cdots \cap C_m \ne \emptyset$.

## Remarks

Proof idea: induct on $m$, starting from $m = d + 1$.
For $m \ge d + 2$, the induction hypothesis gives, for each $j$, a point $x_j$ in all the sets except possibly $C_j$.
[[Radon's theorem|888R24ZGNY]] applied to $x_1, \dots, x_m$ gives a partition into $A$ and $B$ with a common point $p$ of their convex hulls.
Each $C_i$ contains all of $A$ or all of $B$, so it contains $p$ by convexity.

Example: if every three of finitely many convex sets in the plane meet, then all of them meet.

The number $d + 1$ cannot be lowered: the three lines that contain the sides of a triangle in $\R^2$ meet in pairs but have no common point.
For an infinite family the statement can fail: the half-lines $[k, \infty)$, $k = 1, 2, \dots$, in $\R$ meet in pairs but have empty intersection.
The infinite version holds when all the sets are [[compact|0R6EWFKRCF]] and convex.

## Explanations

### L1 human

To check that many convex sets share a point, it is enough to check small groups.
In the plane, groups of three suffice; in space, groups of four.
Without convexity this fails.
Cut a circle in the plane into four disjoint arcs, and for each arc let $S_i$ be the circle with that arc removed.
Any three of the sets $S_i$ share the fourth arc, but no point lies in all four.

### L2 human

Helly's theorem is the model of a Helly-type theorem: a global property follows from the same property for all small subfamilies.
It has many consequences, such as the centerpoint theorem and Jung's theorem on the smallest ball containing a set.
It is used in Kirchberger's theorem and has a fractional version and a colourful version.
In optimisation it explains why an infeasible system of convex constraints in $\R^d$ has an infeasible subsystem with at most $d + 1$ constraints.
</document>

<document source="https://tamnd.github.io/math2/node/F1WYS82NNB.md" tag="F1WYS82NNB" kind="concept">
# Left-orderable group

## Summary

A left-orderable group is a [[group|G5S1M9]] with a total order that is preserved by multiplication on the left.
Every [[orderable group|9BM25E20JQ]] is left-orderable, but the converse fails.
[[Braid groups|BNR4H5GCTP]] are left-orderable, a result of Dehornoy.
Left orders link group theory with dynamics, because countable left-orderable groups are the countable groups that act faithfully on the real line by order-preserving homeomorphisms.

## Explanations

### L1 human

Imagine lining up the elements of a group so that multiplying everything on the left by a fixed element keeps the line in order.
Multiplying on the right may scramble it.
This is weaker than asking both sides to respect the order.
For countable groups, this is the same as acting on the real line by moving points around while keeping their order.
</document>

<document source="https://tamnd.github.io/math2/node/F20VTSH2FM.md" tag="F20VTSH2FM" kind="concept">
# Pseudospectrum

## Summary

The $\varepsilon$-pseudospectrum of a square matrix is a set of complex numbers.
It consists of the numbers that become [[eigenvalues|0BQAA2BK3Q]] after some perturbation of the matrix of norm less than $\varepsilon$.
Equivalently, it is the region where the norm of the [[resolvent|J5YNNKSB9P]] $(zI - A)^{-1}$ exceeds $1/\varepsilon$.
For non-normal matrices it can be much larger than the set of eigenvalues.
It then explains transient growth, slow convergence of iterative methods and sensitivity of computed eigenvalues, which the eigenvalues alone do not show.

## Explanations

### L1 human

Eigenvalues of some matrices are very fragile: a tiny change of the entries moves them far away.
The pseudospectrum draws, for a given tolerance $\varepsilon$, all the places the eigenvalues could move to under a change of size $\varepsilon$.
For a symmetric matrix this is just a disc of radius $\varepsilon$ around each eigenvalue.
For the matrix $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, whose only eigenvalue is $0$, it contains a disc of radius $\sqrt{\varepsilon}$, which is far larger than $\varepsilon$ when $\varepsilon$ is small.
</document>

<document source="https://tamnd.github.io/math2/node/F28G886PWD.md" tag="F28G886PWD" kind="concept">
# Chern–Simons theory

## Summary

Chern–Simons theory is a gauge theory on $3$-manifolds whose action is the Chern–Simons functional of a connection.
The functional does not use a metric, so the quantum theory is a [[topological quantum field theory|DAP92B80J2]].
Witten showed in 1989 that its expectation values of Wilson loops give the [[Jones polynomial|4SPC17FPXE]].
Its partition functions give new invariants of [[3-manifolds|R20TB2KEQQ]], the [[quantum invariants|YGPR4VR3GP]].
Its classical solutions are the flat connections, which link it with representations of fundamental groups and with the [[Casson invariant|33GY6P5A4T]].

## Explanations

### L1 human

Chern–Simons theory is a physical theory of fields on a $3$-dimensional space that does not care about lengths or angles, only about shape.
Because of that, the numbers it produces are invariants of the space and of knots inside it.
Its most famous output is the Jones polynomial of a knot, which was first found by quite different methods.
It is a meeting point of knot theory, topology and quantum physics.
</document>

<document source="https://tamnd.github.io/math2/node/F2C174HHSE.md" tag="F2C174HHSE" kind="concept">
# Semiorder

## Summary

A semiorder is a strict partial order in which one element is below another when it is smaller by more than a fixed threshold.
Semiorders model preferences in which small differences cannot be noticed.
They are the [[interval orders|VSNQV29KFJ]] that can be represented by intervals of the same length.

## Explanations

### L1 human

A person may not taste the difference between coffee with $10$ grains of sugar and coffee with $11$ grains.
Yet the same person prefers $10$ grains to $500$ grains.
So "not noticeably different" is not transitive, while "noticeably less sweet" is still an order.
On the real numbers, the relation $x \prec y$ when $x + 1 < y$ is a semiorder of this kind.
</document>

<document source="https://tamnd.github.io/math2/node/F2CMPBHQ02.md" tag="F2CMPBHQ02" kind="concept">
# Acyclic object

## Summary

For a left exact functor $F$, an object $X$ is $F$-acyclic if all the higher right [[derived functors|NPMC3K6J2F]] $R^n F(X)$ with $n \ge 1$ vanish.
Injective objects are acyclic for every such $F$, but many more objects are acyclic for a given $F$.
Derived functors can be computed with resolutions by $F$-acyclic objects, which are often much easier to find than injective resolutions.

## Explanations

### L1 human

An acyclic object is one on which a functor behaves as if it were exact.
To compute derived functors one needs a resolution by objects of this kind, and injective objects are the universal choice.
But they are large and hard to handle, so one uses objects adapted to the given functor.
For sheaf cohomology, flasque sheaves, whose sections over open sets always extend, are acyclic for the global sections functor.
For the tensor product, flat modules are acyclic, and this is why Tor can be computed with flat resolutions.
</document>

<document source="https://tamnd.github.io/math2/node/F2DTWW9HP7.md" tag="F2DTWW9HP7" kind="concept">
# Forcing

## Summary

Forcing is a method to build new models of set theory by adding a new set to a given model.
One starts from a model $M$ of ZFC and a [[partial order|0QV2JS9RJ3]] in $M$, whose elements are approximations of the new set.
A [[generic filter|WEDP73Y7GK]] $G$ over $M$ collects compatible approximations, and the extension $M[G]$ is the smallest transitive model of ZFC that contains $M$ and $G$.
Paul Cohen introduced forcing to prove that the [[continuum hypothesis|0MN2MER9WN]] cannot be proved in ZFC, and it is now the main tool for independence proofs.

## Explanations

### L1 human

Forcing adds a new object to a universe of sets, such as a new real number that was not there before.
The new object is described by its finite pieces of information, called conditions.
For a new real, a condition says what finitely many of its binary digits are.
A generic choice of conditions avoids every pattern that the old universe can describe, so the real it defines is truly new.
Facts about the new universe are decided by single conditions, and this is how one controls what is true there.
</document>

<document source="https://tamnd.github.io/math2/node/F2E3317P78.md" tag="F2E3317P78" kind="concept">
# Harish-Chandra character

## Summary

The Harish-Chandra character extends the [[character of a representation|RMWMYBY8TC]] to infinite-dimensional representations of reductive groups.
The operators $\pi(g)$ have no trace, but their averages $\int f(g) \pi(g) \, dg$ against smooth compactly supported functions $f$ do.
The map $f \mapsto \operatorname{tr} \pi(f)$ is a distribution on the group, the character of $\pi$.
It determines an irreducible admissible representation up to infinitesimal equivalence, and it is the basic tool of harmonic analysis on reductive groups.

## Explanations

### L1 human

For a finite group, the character of a representation records the trace of each group element and tells the representation apart from all others.
For a group like $\mathrm{SL}_2(\R)$ acting on an infinite-dimensional space, the trace of a single element is usually infinite.
The trick is to smear the group element out with a smooth bump function before taking the trace.
The result is no longer a function on the group but a "generalised function", and it still remembers the representation.
</document>

<document source="https://tamnd.github.io/math2/node/F2F143KQ2M.md" tag="F2F143KQ2M" kind="concept">
# Homotopic maps

## Summary

Two continuous maps $f, g \colon X \to Y$ are homotopic, written $f \simeq g$, when there is a [[homotopy|9YA6S6HD9V]] from $f$ to $g$.
Being homotopic is an equivalence relation on the set of continuous maps from $X$ to $Y$.
Its equivalence classes are the homotopy classes, and the set of them is written $[X, Y]$.
Algebraic topology studies spaces through such sets of homotopy classes.

## Explanations

### L1 human

Two maps are homotopic when one can be slid continuously into the other inside $Y$.
Homotopy forgets the fine details of a map and keeps only its shape up to deformation.
For example, every two maps into $\R^n$ are homotopic, by moving each point along a straight segment.
Two loops in the circle that wind around a different number of times are not homotopic.
</document>

<document source="https://tamnd.github.io/math2/node/F2HHDD85EM.md" tag="F2HHDD85EM" kind="statement">
# Hartman–Wintner theorem

## Statement

Let $t_0 \ge 0$ and let $q \colon [t_0, \infty) \to \R$ be continuous.
Consider the [[linear differential equation|0QQRP17Z7S]]
$$y'' + q(t) y = 0 \quad \text{on } [t_0, \infty).$$
For $t > t_0$ put
$$A(t) = \frac{1}{t} \int_{t_0}^{t} \int_{t_0}^{s} q(u) \, du \, ds.$$
Assume that the equation is nonoscillatory in the sense of [[oscillation theory|ZH2S50NSZ6]], and that $\liminf_{t \to \infty} A(t) > -\infty$, where $\liminf$ is the [[limit inferior|033RZ32T4Q]].
Then $\lim_{t \to \infty} A(t)$ exists and is a finite real number.

## Remarks

The proof goes through the [[Riccati equation|0VT2QK6X6Q]].
A nonoscillatory equation has a solution $y > 0$ on some interval $[T, \infty)$, and $w = y'/y$ satisfies $w' + w^2 + q = 0$ there.
Integrating gives $w(t) = w(T) - \int_T^t q - \int_T^t w^2$.
Averaging this identity and using the lower bound on $A$, one shows that $\int_T^\infty w^2 < \infty$, and then the average $A(t)$ converges.

Read as a test for oscillation, the theorem says: if $\liminf A(t) > -\infty$ and either $\liminf A(t) < \limsup A(t)$ or $A(t) \to \infty$, then the equation is oscillatory.
The case $A(t) \to \infty$ is Wintner's oscillation criterion.

Example: for $q = 1$ on $[0, \infty)$ one gets $A(t) = t/2 \to \infty$, so the equation is oscillatory; indeed $\sin t$ is a solution.
For $q(t) = 1/(4t^2)$ on $[1, \infty)$ the equation has the positive solution $\sqrt{t}$, so it is nonoscillatory.
Here $A(t) = (t - 1 - \ln t)/(4t)$, which tends to the finite limit $1/4$, as the theorem predicts.

## Explanations

### L1 human

The equation $y'' + q y = 0$ describes a spring whose stiffness $q(t)$ changes with time.
The solutions either keep crossing zero forever or eventually stay on one side.
The theorem looks at a double average of the stiffness.
If the solutions stop crossing zero and this average is bounded below, then the average must settle to a finite value.
So an average that grows without bound, or that keeps swinging, forces endless oscillation.

### L2 human

The Sturm comparison theorem compares $q$ with a constant or an Euler coefficient pointwise.
The Hartman–Wintner theorem uses only averages of $q$, so it also applies when $q$ changes sign.
It improves Wintner's criterion, which asks that $A(t) \to \infty$.
The key tool is the Riccati substitution $w = y'/y$, which turns a nonoscillatory linear equation into a first-order nonlinear one with a solution on a half-line.
Later criteria of Kamenev and Philos replace the weight $1/t$ by other weights in the same Riccati argument.
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<document source="https://tamnd.github.io/math2/node/F2JX0R86WQ.md" tag="F2JX0R86WQ" kind="definition">
# Curve of constant width

## Definition

Let $K \subseteq \R^2$ be a [[convex body|437FSG3419]] with [[support function|1H92CYA4ZC]] $h_K$, and let $w > 0$.
For a unit vector $u$, the *width* of $K$ in the direction $u$ is $w_K(u) = h_K(u) + h_K(-u)$.
It is the distance between the two [[supporting lines|K7N565NKCV]] of $K$ orthogonal to $u$.
The body $K$ has *constant width* $w$ if $w_K(u) = w$ for every unit vector $u$.
A *curve of constant width* $w$ is the [[boundary|05D5EGMFYV]] of a plane convex body of constant width $w$.

## Remarks

Example: the circle of radius $r$ centred at $c$ bounds the disc with $h(u) = c \cdot u + r$, so $w(u) = 2r$ for all $u$, and the circle has constant width $2r$.
Non-example: the square $[0, 1]^2$ has width $1$ in the direction $(1, 0)$ and width $\sqrt{2}$ in the direction $(1, 1)/\sqrt{2}$, so its boundary is not a curve of constant width.
A body of constant width $w$ has diameter $w$.
The same definition in $\R^n$ gives bodies of constant width.
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# Delta functor

## Definition

Let $\mathcal{A}$ and $\mathcal{B}$ be [[abelian categories|Y9CZ3MTNF1]].
A *cohomological $\delta$-functor* from $\mathcal{A}$ to $\mathcal{B}$ consists of [[additive functors|KAN07YC9T2]] $T^n \colon \mathcal{A} \to \mathcal{B}$ for $n \ge 0$ and of connecting morphisms.
For every [[short exact sequence|KJKNJMGKZZ]] $0 \to A' \to A \to A'' \to 0$ in $\mathcal{A}$ and every $n \ge 0$, there is a morphism $\delta^n \colon T^n(A'') \to T^{n+1}(A')$.
These data must satisfy:

1. for every short exact sequence, the sequence
$$0 \to T^0(A') \to T^0(A) \to T^0(A'') \xrightarrow{\delta^0} T^1(A') \to T^1(A) \to T^1(A'') \xrightarrow{\delta^1} T^2(A') \to \cdots$$
is [[exact|JB7ZBFBA9C]];
2. the maps $\delta^n$ are natural. Let a morphism from $0 \to A' \to A \to A'' \to 0$ to $0 \to B' \to B \to B'' \to 0$ have components $f' \colon A' \to B'$ and $f'' \colon A'' \to B''$. Then $\delta^n \circ T^n(f'') = T^{n+1}(f') \circ \delta^n$.

A $\delta$-functor $(T^n)$ is *universal* if the following holds for every $\delta$-functor $(S^n)$ and every [[natural transformation|KYCPCC57FT]] $T^0 \to S^0$.
There is a unique family of natural transformations $T^n \to S^n$ extending it and commuting with the maps $\delta^n$.

## Remarks

Example: for an $R$-module $M$, the functors $T^n = \operatorname{Ext}^n_R(M, -)$ with their connecting maps form a $\delta$-functor, since Ext has the long exact sequence in the second variable, and it is natural.
Non-example: on abelian groups, put $T^0 = \Hom_\Z(\Z/2\Z, -)$ and $T^n = 0$ for $n \ge 1$.
For $0 \to \Z \xrightarrow{\cdot 2} \Z \to \Z/2\Z \to 0$ the sequence in clause 1 starts $0 \to 0 \to 0 \to \Z/2\Z \to 0$, which is not exact, so clause 1 fails.
A theorem of Grothendieck says that a $\delta$-functor is universal when each $T^n$ with $n \ge 1$ is effaceable, meaning that every object $A$ embeds by a monomorphism $u$ with $T^n(u) = 0$.
Hence the right derived functors of a left exact functor form a universal $\delta$-functor.
Homological $\delta$-functors $(T_n)$ are defined in the same way with the arrows $\delta_n \colon T_n(A'') \to T_{n-1}(A')$.
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# Principal symbol

## Summary

Take a linear differential operator of order $m$, keep only its terms of order $m$, and replace each derivative $\partial_j$ by a variable $\xi_j$.
The resulting polynomial in $\xi$ is the principal symbol.
It captures the behaviour of the operator at high frequencies.
Ellipticity, hyperbolicity and characteristic directions are all defined through the principal symbol.
See the [[definition|MX6D07DDAZ]].

## Explanations

### L1 human

Apply a differential operator to a fast oscillation $e^{i \lambda x \cdot \xi}$ with large $\lambda$.
The highest derivatives bring the largest powers of $\lambda$, so they dominate the result.
The principal symbol records exactly this dominant part.
For the [[Laplacian|0E0B6RA8K0]] $\partial_x^2 + \partial_y^2$ the principal symbol is $\xi_1^2 + \xi_2^2$, with the sign convention that replaces $\partial_j$ by $\xi_j$.
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# Bounded variation

## Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function.
For a [[partition|069P20KYQT]] $P$ given by $a = x_0 < x_1 < \cdots < x_n = b$, put
$$V(f, P) = \sum_{i=1}^{n} \abs{f(x_i) - f(x_{i-1})}.$$
The [[total variation|8QA5DEWNYD]] of $f$ on $[a, b]$ is $V_a^b(f) = \sup_P V(f, P)$, where the supremum runs over all partitions of $[a, b]$.
The function $f$ has *bounded variation* on $[a, b]$ if $V_a^b(f) < \infty$.

## Remarks

Example: a nondecreasing function $f$ on $[a, b]$ has bounded variation.
For every partition the sum telescopes: $V(f, P) = \sum_i (f(x_i) - f(x_{i-1})) = f(b) - f(a)$, so $V_a^b(f) = f(b) - f(a)$.

Non-example: let $f(0) = 0$ and $f(x) = x \sin(1/x)$ for $0 < x \le 1$.
The function $f$ is continuous, but it does not have bounded variation.
At the points $x_k = 1/((k + \frac12)\pi)$ the values are $\pm x_k$ with alternating signs, so a partition through $x_1, \dots, x_N$ gives $V(f, P) \ge \sum_{k=1}^{N} x_k$, and this sum diverges like the harmonic series.

The set of functions of bounded variation on $[a, b]$ is often written $BV([a, b])$.
Every function of bounded variation is bounded, since $\abs{f(x)} \le \abs{f(a)} + V_a^b(f)$.
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# Center manifold

## Definition

Let $U$ be an open subset of $\R^n$ and let $f \colon U \to \R^n$ be of class $C^k$ with $k \ge 1$.
Let $\varphi$ be the flow of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.
Let $x_0 \in U$ be an [[equilibrium point|0G4X36AE0X]].
Let $E^c$ be the sum of the generalised eigenspaces of the [[Jacobian matrix|047NDTEPCV]] $Df(x_0)$ for eigenvalues with real part $0$.
A local center manifold of $x_0$ is a set $W^c$ with these properties:

1. $W^c$ is an embedded $C^1$ submanifold of an open neighbourhood $V \subseteq U$ of $x_0$, it contains $x_0$, and its tangent space at $x_0$ is $E^c$.
2. $W^c$ is locally [[invariant|WWZTW11664]]: if $x \in W^c$ and $\varphi_s(x) \in V$ for all $s$ between $0$ and $t$, then $\varphi_t(x) \in W^c$.

## Remarks

The center manifold theorem states that a local center manifold exists and can be chosen of class $C^k$, but it is not unique in general.
Example: for $x' = x^2$, $y' = -y$ at the origin, $E^c$ is the $x$-axis, and the $x$-axis is a center manifold.
For each $c \in \R$, the curve $y = c e^{1/x}$ for $x < 0$ together with $y = 0$ for $x \ge 0$ is also a center manifold.
On it $y' = (-c e^{1/x} / x^2) \cdot x^2 = -y$, and the curve is smooth and tangent to the $x$-axis at $0$.
Non-example: the $y$-axis is invariant and passes through the origin, but its tangent space is the stable direction, not $E^c$, so clause 1 fails.
When $Df(x_0)$ has no eigenvalue with positive real part, the stability of $x_0$ is decided by the reduced equation on a center manifold.

## Explanations

### L1 human

A center manifold is an invariant curved surface through the equilibrium that is tangent to the neutral directions of the linearisation.
On it the motion is slow and decided by nonlinear terms, while off it solutions are attracted or repelled exponentially.
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# Atomic domain

## Summary

An atomic domain is an [[integral domain|09S3PFBEPM]] in which every nonzero nonunit is a finite product of [[irreducible elements|052SDEPZH7]].
It asks for the existence of factorizations, but not for their uniqueness.
Every Noetherian domain is atomic, and more generally every domain with the ascending chain condition on principal ideals.
A [[unique factorisation domain|06YMXJMXYS]] is exactly an atomic domain in which every irreducible element is [[prime|0MNHXBSX3M]].

## Explanations

### L1 human

The irreducible elements are the "atoms" of multiplication: they cannot be split into two nonunit factors.
In an atomic domain, every nonzero nonunit can be broken down into atoms in at least one way.
In $\Z[\sqrt{-5}]$ this works, but the breakdown is not unique: $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$.
In the ring of all algebraic integers it fails completely: every nonzero nonunit $\alpha$ is $\sqrt{\alpha} \cdot \sqrt{\alpha}$, so there are no atoms at all.
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<document source="https://tamnd.github.io/math2/node/F39N9NBFPZ.md" tag="F39N9NBFPZ" kind="concept">
# Lattice polytope

## Summary

A lattice polytope, also called an integral polytope, is a [[convex polytope|300K6CB1KH]] whose vertices all have integer coordinates.
Counting the integer points in such a polytope and in its dilates leads to the [[Ehrhart polynomial|9DEG0WRDHR]].
Lattice polytopes link convex geometry with combinatorics, with integer programming and, through their [[normal fans|XJ83PTS3H0]], with [[toric varieties|YWB981MEF2]].

## Explanations

### L1 human

Draw a square grid of points and join some of the grid points to make a convex shape with straight sides.
If every corner of the shape sits on a grid point, the shape is a lattice polygon.
The square with corners $(0,0)$, $(2,0)$, $(2,2)$, $(0,2)$ is one; it contains $9$ grid points.
Lattice polytopes are the same idea in any dimension.
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<document source="https://tamnd.github.io/math2/node/F3DXBJZAME.md" tag="F3DXBJZAME" kind="definition">
# Lamé equations of elasticity

## Definition

Let $\Omega \subseteq \R^3$ be an open set, and let $\lambda, \mu$ be real constants, the Lamé parameters, with $\mu > 0$ and $3\lambda + 2\mu > 0$.
Let $f \colon \Omega \to \R^3$ be a given body force.
The *Lamé equations* of static linear elasticity are the system
$$\mu \Delta u + (\lambda + \mu) \nabla (\nabla \cdot u) + f = 0 \quad \text{in } \Omega$$
for an unknown [[vector field|0299VCW1TH]] $u \colon \Omega \to \R^3$, the displacement.
Here $\Delta$ is the [[Laplacian|0E0B6RA8K0]] acting on each component, $\nabla \cdot u$ is the [[divergence|0MWV6ZTK6S]] and $\nabla$ is the [[gradient|0ABK9V9Y64]].
For a body of constant density $\rho > 0$, the *dynamic Lamé equations* are $\rho \, \partial_t^2 u = \mu \Delta u + (\lambda + \mu) \nabla (\nabla \cdot u) + f$.

## Remarks

The conditions $\mu > 0$ and $3\lambda + 2\mu > 0$ make the elastic energy positive definite on symmetric strains.
They imply $\lambda + 2\mu > 0$, so the system is strongly elliptic.

Example: for $a, b \in \R^3$ and $f = 0$, the rigid displacement $u(x) = a + b \times x$ is a solution.
Each component of $u$ is affine, so $\Delta u = 0$, and $\nabla \cdot u = 0$.

Non-example: for $f = 0$, $u(x) = (x_1^2, 0, 0)$ is not a solution.
Here $\Delta u = (2, 0, 0)$ and $\nabla(\nabla \cdot u) = (2, 0, 0)$, so the left side is $(2(\lambda + 2\mu), 0, 0) \ne 0$.

The equations come from the balance of forces $\nabla \cdot \sigma + f = 0$ with Hooke's law $\sigma = \lambda (\nabla \cdot u) I + \mu (\nabla u + \nabla u^T)$ for the stress $\sigma$.
In the dynamic equations, pressure waves travel at speed $\sqrt{(\lambda + 2\mu)/\rho}$ and shear waves at speed $\sqrt{\mu / \rho}$.
</document>

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# Quantum unique ergodicity conjecture

## Statement

Let $M$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] without boundary and with negative [[sectional curvature|1X42JAA349]], with volume measure $dv$ and total volume $V$.
Let $(\varphi_j)$ be any sequence of $L^2$-normalised eigenfunctions of the [[Laplace–Beltrami operator|F4NWZNTRMD]] with eigenvalues $\lambda_j \to \infty$.
The conjecture says that for every continuous function $a$ on $M$
$$\lim_{j \to \infty} \int_M a \, \abs{\varphi_j}^2 \, dv = \frac{1}{V} \int_M a \, dv.$$
In its full form it says that the microlocal lifts of the measures $\abs{\varphi_j}^2 \, dv$ converge to the normalised Liouville measure on the unit cotangent bundle.

## Remarks

[[Quantum ergodicity|D9W79NKYZJ]] gives this convergence along a subsequence of density one; the conjecture asks that there be no exceptional subsequence at all.
The conjecture is open in general.
Lindenstrauss proved the analogue for compact arithmetic hyperbolic surfaces and joint eigenfunctions of the Laplacian and the Hecke operators.
Together with work of Soundararajan that rules out escape of mass, this also covers the modular surface.
Anantharaman proved that on compact manifolds with Anosov geodesic flow every quantum limit has positive entropy, so eigenfunctions cannot concentrate entirely on a closed geodesic.
For some ergodic billiards the analogue fails: Hassell proved that for almost every stadium-shaped billiard there are exceptional sequences of eigenfunctions.

## Explanations

### L1 human

On a negatively curved surface, a freely moving particle explores every region evenly.
The conjecture says that every high-frequency standing wave on such a surface also spreads its mass evenly, with no exceptions at all.
Quantum ergodicity already proves this for almost all waves; the conjecture removes the word "almost".
</document>

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# Krull topology

## Summary

The Krull topology is the natural topology on the [[Galois group|0XVR72M51M]] of a Galois extension, possibly of infinite degree.
A basis of neighbourhoods of the identity is given by the subgroups that fix a finite subextension.
With it, the Galois group is a [[profinite group|KA0YDB6QXE]], and [[infinite Galois theory|RVR5QFFD77]] matches intermediate fields with closed subgroups.

## Explanations

### L1 human

Two symmetries of a large field are declared close when they agree on a big finite piece of the field.
The more elements they agree on, the closer they are.
For a finite extension, every subgroup is open and the topology carries no new information.
For the algebraic closure of $\Q$, the topology is what makes the group of symmetries behave like a limit of finite groups.
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# Stable distribution

## Summary

A stable distribution is a law on $\R$ whose type is preserved under independent sums.
A sum of independent copies has the same law up to a change of scale and location.
Each stable law has an index $\alpha \in (0, 2]$; the [[normal distribution|05AQE0QR0D]] is the case $\alpha = 2$ and the [[Cauchy distribution|0GPG24Q3GR]] is a case with $\alpha = 1$.
Stable laws are exactly the possible nondegenerate limits of suitably normalised sums of i.i.d. random variables, which extends the [[central limit theorem|0EPZC4Z1FC]] to variables of infinite variance.
Every stable law is [[infinitely divisible|BX3V3NYM1Y]].

## Explanations

### L1 human

Add two independent normal variables and you get a normal variable again, only wider; stable laws are the laws with this property.
For an index $\alpha < 2$ the tails are heavy: the chance that $\abs{X}$ exceeds $x$ falls like $x^{-\alpha}$, and the variance is infinite.
For example, the average of $n$ independent standard Cauchy variables is again standard Cauchy, so averaging does not reduce the spread at all.
Stable laws appear as limits whenever many independent heavy-tailed effects add up.
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<document source="https://tamnd.github.io/math2/node/F3STQVV22B.md" tag="F3STQVV22B" kind="definition">
# Intermittency

## Definition

Let $(\phi_\mu)$ be a one-parameter family of dynamical systems, flows or maps, with $\mu \in \R$.
Assume that for $\mu$ on one side of $\mu_0$ the system has a stable [[periodic orbit|0AA97XA70K]], and that it disappears or loses stability at $\mu_0$.
The system shows *intermittency* for $\mu$ just past $\mu_0$ if typical orbits alternate between two kinds of phases.

1. Laminar phases: long stretches of time during which the orbit stays close to the former periodic orbit.
2. Bursts: short irregular stretches, after which the orbit is reinjected near the former periodic orbit.

The *type* is I when the bifurcation at $\mu_0$ is a [[saddle-node bifurcation|MAP5J6HK0T]].
It is II for a subcritical [[Hopf|F5H8GW1XYS]] or Neimark–Sacker bifurcation, and III for a subcritical [[period-doubling bifurcation|NQDTTZ59X8]].

## Remarks

Example: the [[logistic map|QN9X9BNKTX]] $x \mapsto r x (1 - x)$ near $r_c = 1 + 2\sqrt{2}$.
At $r_c$ a stable and an unstable orbit of period $3$ are born in a saddle-node bifurcation of the third iterate.
For $r$ slightly below $r_c$, orbits pass slowly through the narrow gaps left between the graph of $f^3$ and the diagonal, so they look nearly periodic of period $3$.
Between these passages they move chaotically: this is type I intermittency.
For type I the mean length of the laminar phases grows like $(r_c - r)^{-1/2}$.
Non-example: just after a supercritical period-doubling bifurcation the new stable orbit of double period attracts nearby orbits, so there are no bursts.
Intermittency is a route to chaos studied in [[bifurcation theory|D9TW46P19Y]].
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# Synthetic differential geometry

## Summary

Synthetic differential geometry develops differential calculus and differential geometry with genuine infinitesimals, inside a [[topos|6837RS7N82]] instead of the category of sets.
Its basic object is a number line $R$ whose set $D$ of elements of square zero is not reduced to $0$.
By the Kock–Lawvere axiom, every function from $D$ to $R$ is affine.
Tangent vectors become maps from the infinitesimal set $D = \set{d : d^2 = 0}$ into a space, and derivatives are defined without limits.
Models built from [[C-infinity rings|4EJED0YZBB]] show that the theory is consistent and contains ordinary smooth manifolds.

## Explanations

### L1 human

Physicists and early analysts often computed with tiny quantities whose square is zero.
Synthetic differential geometry makes this rigorous.
In it, $f(x + d) = f(x) + d\, f'(x)$ holds exactly for every infinitesimal $d$, and this equation defines $f'(x)$.
The price is a different logic: one cannot assume that every number is either zero or not zero.
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<document source="https://tamnd.github.io/math2/node/F3Z30M606H.md" tag="F3Z30M606H" kind="statement">
# Kleene's recursion theorem

## Statement

Fix a standard [[Gödel numbering|MFJXPAF0R0]] of programs, and let $\varphi_e$ be the partial [[computable function|KZSB5E471K]] of one argument computed by program $e$.
Let $f \colon \N \to \N$ be a total computable function.
Then there is $e \in \N$ such that $\varphi_e = \varphi_{f(e)}$ as partial functions.

## Remarks

By the [[smn theorem|YWX4FCFPAP]], there is a total computable $h$ such that $\varphi_{h(u)} = \varphi_{\varphi_u(u)}$ whenever $\varphi_u(u)$ is defined.
The composite $f \circ h$ is total computable, so it equals $\varphi_v$ for some $v$.
Put $e = h(v)$.
Then $\varphi_e = \varphi_{\varphi_v(v)} = \varphi_{f(h(v))} = \varphi_{f(e)}$.

An equivalent form: for every partial computable $\psi(e, x)$ there is $e$ with $\varphi_e(x) \simeq \psi(e, x)$ for all $x$.
Here $\simeq$ means that both sides are undefined, or both are defined and equal.
Example: with $\psi(e, x) = e$ one gets a program $e$ that outputs its own number on every input, a "quine".

The function $f$ need not have a fixed point $e = f(e)$; the theorem only gives a fixed point of the programs' behaviour.
For instance $f(e) = e + 1$ has no fixed point in $\N$, but some $e$ and $e + 1$ compute the same function.

## Explanations

### L1 human

The theorem says that a program may refer to its own code.
Any mechanical transformation of programs leaves the behaviour of at least one program unchanged.
A classic consequence is that there are programs that print their own source code.

### L2 human

The recursion theorem justifies definitions in which a program calls itself by index, which is how general recursion is handled inside computability theory.
Its proof is a diagonal argument and is closely related to the diagonal lemma used in [[Gödel's incompleteness theorems|X9WYNYE29C]].
It gives a short proof of [[Rice's theorem|AKDR5YBBE2]] and of the undecidability of the [[halting problem|Z7TN2B35B0]].
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# Monad

## Definition

Let $\mathcal{C}$ be a category.
A *monad* on $\mathcal{C}$ is a triple $(T, \eta, \mu)$ made of these data.

1. $T \colon \mathcal{C} \to \mathcal{C}$ is a [[functor|28NQ0KRHD3]].
2. $\eta \colon \mathrm{Id}_{\mathcal{C}} \Rightarrow T$ is a [[natural transformation|KYCPCC57FT]], the unit.
3. $\mu \colon T \circ T \Rightarrow T$ is a natural transformation, the multiplication.

The data must satisfy two laws.
The associativity law is $\mu \circ T\mu = \mu \circ \mu T$ as natural transformations $T T T \Rightarrow T$.
The unit laws are $\mu \circ T\eta = \id_T$ and $\mu \circ \eta T = \id_T$.
Here $(T\mu)_X = T(\mu_X)$ and $(\mu T)_X = \mu_{T(X)}$, and similarly for $\eta$.

## Remarks

Example: the power set monad on the category of sets.
Here $T(X)$ is the power set of $X$, and $T(f)$ sends a subset $A$ to its image $f(A)$.
The unit is $\eta_X(x) = \set{x}$ and the multiplication is $\mu_X(\mathcal{S}) = \bigcup \mathcal{S}$.
Both sides of the associativity law send a set of sets of subsets to the union of all the subsets that occur in it.
The unit laws hold because $\bigcup \set{A} = A$ and $\bigcup \set{\set{a} : a \in A} = A$.

Non-example: on the category of sets let $T(X) = X \times X$, $\eta_X(x) = (x, x)$ and $\mu_X((a, b), (c, d)) = (a, c)$.
Then $(\mu \circ \eta T)_X$ sends $(x, y)$ to $\mu_X((x, y), (x, y)) = (x, x)$.
This differs from $(x, y)$ when $x \ne y$, so a unit law fails.
With $\mu_X((a, b), (c, d)) = (a, d)$ instead, one gets a monad.

A monad on $\mathcal{C}$ is the same as a monoid object in the monoidal category of endofunctors of $\mathcal{C}$, with composition as the product.
The dual notion is a comonad.
</document>

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# Quasi-ergodicity

## Summary

A quasi-ergodic map is a [[quasi-measure-preserving map|F608XX51KA]] of a measure space to itself whose invariant sets are all trivial: null or of null complement.
It extends the notion of an [[ergodic transformation|AZRPHS478B]] to maps that need not preserve the measure, but only send null sets back to null sets.
This is the natural setting of nonsingular ergodic theory, where measures are only known up to equivalence.
A quasi-ergodic map cannot be split into two parts of positive measure that the map keeps apart.

## Explanations

### L1 human

Ergodicity says that a measure-preserving map cannot be split: it has no invariant pieces except null sets and sets of full measure.
Quasi-ergodicity asks for the same indivisibility, but drops the request that the map keeps the size of sets.
The map may stretch and shrink sets, as long as it never pulls a set of size zero back to a set of positive size.
For example, an irrational rotation of the circle is ergodic for length.
So it is quasi-ergodic for any measure with a positive density, even when it does not preserve that measure.
</document>

<document source="https://tamnd.github.io/math2/node/F456KQGPFV.md" tag="F456KQGPFV" kind="statement">
# Einstein problem

## Statement

There is a single closed topological disk $T$ in the plane $\R^2$ with the following two properties.

1. The plane has a [[tiling|GKJK293DXS]] in which every tile is congruent to $T$.
2. No tiling of the plane in which every tile is congruent to $T$ is [[periodic|WFRGSBFZGV]].

In other words, the one-element set $\set{T}$ is an [[aperiodic set of prototiles|Z3FCJZF6WK]], and $T$ is called an *aperiodic monotile* or *einstein*.

## Remarks

The question whether one tile can be aperiodic, with no matching rules, is the einstein problem; the name plays on the German "ein Stein", one stone.
David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss answered it in 2023 with the "hat", a polygon made of eight kites with angles $60^\circ, 90^\circ, 120^\circ, 90^\circ$.
Tilings by the hat use both the hat and its mirror image.
The same authors later found the "spectre", which tiles the plane only non-periodically even when only rotations and translations are allowed.
The proof groups the tiles of any hat tiling into larger clusters that again form a tiling of the same kind.
This hierarchical structure at all scales rules out periods.

## Explanations

### L1 human

For a long time people asked whether one single shape could cover the whole plane, but only in patterns that never repeat.
Two shapes were known to do it, such as the Penrose tiles.
In 2023 an amateur, David Smith, found a single shape, called the hat, that does it.
Any tiling by hats is forced to be non-repeating.

### L2 human

Aperiodic sets of prototiles had shrunk from Berger's huge set of tiles in 1966 to the two Penrose tiles in the 1970s.
The einstein problem asked whether the number can be $1$.
The hat solves it if reflections are allowed, and the spectre solves it even with rotations and translations only.
The key technique, as for Penrose tilings, is a substitution: tilings are built from clusters that copy the structure on a larger scale.
</document>

<document source="https://tamnd.github.io/math2/node/F45VQ47GPV.md" tag="F45VQ47GPV" kind="concept">
# Smooth manifold

## Summary

A smooth manifold is a space that looks like a piece of $\R^n$ near each point, glued together so that calculus makes sense on it.
The gluing maps between overlapping coordinate charts are required to be infinitely differentiable.
This lets one speak of smooth functions, tangent vectors, vector fields and differential forms on curved spaces such as spheres and tori.
Smooth manifolds are the basic objects of differential geometry and differential topology.

## Explanations

### L1 human

The surface of the Earth is curved, but a small region of it can be drawn on a flat map.
An atlas is a collection of such maps that together cover the whole Earth, with overlaps.
A smooth manifold is a space with an atlas in which passing from one map to an overlapping one is a smooth change of coordinates.
The circle is a one-dimensional example: near each point it looks like a piece of a line, and the angle coordinate works as a chart away from one point.
</document>

<document source="https://tamnd.github.io/math2/node/F46GGH7RJ5.md" tag="F46GGH7RJ5" kind="example">
# Sturm–Liouville problem as an integral equation

## Example

Consider the simplest [[Sturm–Liouville problem|7WV6B2502K]]: find $\lambda$ and a nonzero $u \in C^2[0, 1]$ with
$$-u''(x) = \lambda u(x), \quad u(0) = u(1) = 0.$$
The [[Green's function|SH6JY9RN2E]] of $-u''$ with these boundary conditions is
$$G(x, s) = \begin{cases} s(1 - x), & 0 \le s \le x \le 1, \\ x(1 - s), & 0 \le x \le s \le 1. \end{cases}$$
For continuous $f$, the unique solution of $-u'' = f$, $u(0) = u(1) = 0$ is $u(x) = \int_0^1 G(x, s) f(s) \, ds$.
Taking $f = \lambda u$, the boundary value problem becomes the homogeneous Fredholm equation of the second kind
$$u(x) = \lambda \int_0^1 G(x, s) \, u(s) \, ds.$$
Conversely, a continuous solution of this integral equation is $C^2$ and solves the boundary value problem.

Check: the kernel $G$ is continuous and [[symmetric|ADTYF8FG6C]].
For $u_n(x) = \sin(n \pi x)$, the function $v = u_n / (n^2 \pi^2)$ satisfies $-v'' = u_n$ and $v(0) = v(1) = 0$.
So $\int_0^1 G(x, s) \sin(n \pi s) \, ds = \sin(n \pi x) / (n^2 \pi^2)$, and the [[characteristic values|DC7HMHVREP]] of $G$ are $\lambda_n = n^2 \pi^2$, $n \ge 1$.
These are exactly the eigenvalues of the boundary value problem.

## Remarks

Since $G$ is symmetric, the Hilbert–Schmidt theorem gives the expansion of every function $\int_0^1 G(x, s) f(s) \, ds$ in the sine series $\sum_n c_n \sin(n \pi x)$.
The [[trace|3JVT83K8A1]] of $G$ is $\int_0^1 x(1 - x) \, dx = 1/6$.
It equals $\sum_{n \ge 1} 1/\lambda_n = \sum_{n \ge 1} 1/(n^2 \pi^2)$, which is Euler's identity $\sum_{n \ge 1} 1/n^2 = \pi^2/6$.

## Explanations

### L1 human

A differential equation with boundary conditions can be turned around.
Instead of asking which $u$ has $-u''$ proportional to $u$, one asks which $u$ is reproduced, up to a factor, by integrating against $G$.
The Green's function $G$ is a tent-shaped kernel that undoes $-d^2/dx^2$.
The two problems have the same solutions, and the integral form brings the tools of compact symmetric operators: real eigenvalues, orthogonal eigenfunctions and expansions.
</document>

<document source="https://tamnd.github.io/math2/node/F4BVTGH58G.md" tag="F4BVTGH58G" kind="definition">
# Domain theory

## Definition

Let $P$ be a [[directed-complete partial order|7TSXP95QN6]] and let $\ll$ be its [[way-below relation|RDM7J5J9HQ]].
The dcpo $P$ is a *continuous domain* if for every $x \in P$ the set $\set{y \in P : y \ll x}$ is [[directed|D9RY4667K6]] with supremum $x$.
It is an *algebraic domain* if for every $x \in P$ the set of compact elements below $x$ is directed with supremum $x$.
*Domain theory* is the study of continuous domains and related classes such as [[Scott domains|1SAAZG46DG]].
It studies them together with the [[Scott-continuous maps|C6GAHS1Z2P]] between them and their [[Scott topologies|ZR8A03FV67]].

## Remarks

Example: the interval $[0, 1]$ is a continuous domain that is not algebraic; $\set{y : y \ll x} = [0, x)$ for $x > 0$, with supremum $x$, while the only compact element is $0$.
Example: the partial functions from $\N$ to $\N$, ordered by extension, form an algebraic domain whose compact elements are the finite partial functions.
Non-example: the lattice of closed subsets of $[0, 1]$, ordered by inclusion, is a dcpo, but only $\varnothing$ is way below $[0, 1]$, so it is not a continuous domain.
A central tool is the [[least fixed point|DBD64TH52D]]: a Scott-continuous map $f$ on a pointed dcpo has the least fixed point $\bigvee_n f^n(\bot)$.
Such fixed points give the meaning of recursive programs and of terms of the [[lambda calculus|BD756KR8FM]].
</document>

<document source="https://tamnd.github.io/math2/node/F4HWTRNXAP.md" tag="F4HWTRNXAP" kind="concept">
# Frame bundle

## Summary

The frame bundle of a manifold is the space of all ordered bases of all tangent spaces.
A point of it is a point of the manifold together with a choice of coordinate axes for the tangent space there.
The general linear group acts on it by changing the basis, which makes it a principal bundle.
It is the natural home of connections and of G-structures, such as Riemannian metrics and orientations.

## Explanations

### L1 human

At each point of a surface you can draw two arrows tangent to it that are not parallel; they form a frame.
The frame bundle collects all possible frames at all points.
Two frames at the same point differ by an invertible $2 \times 2$ matrix.
If you only allow perpendicular arrows of unit length, you get the smaller orthonormal frame bundle.
</document>

<document source="https://tamnd.github.io/math2/node/F4J5BZ4HWD.md" tag="F4J5BZ4HWD" kind="definition">
# Minkowski addition

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]] and let $A, B \subseteq V$.
The *Minkowski sum* of $A$ and $B$ is the set
$$A + B = \set{a + b : a \in A, \ b \in B}.$$
For a real number $\lambda$, the *dilate* of $A$ is $\lambda A = \set{\lambda a : a \in A}$.
The operation $(A, B) \mapsto A + B$ is *Minkowski addition*.

## Remarks

Example: in $\R^2$, let $A = [0,1] \times \set{0}$ and $B = \set{0} \times [0, 1]$.
A point $(s, 0) + (0, t)$ with $s, t \in [0, 1]$ is $(s, t)$, so $A + B$ is the unit square $[0,1]^2$.

Minkowski addition is associative and commutative, with neutral element $\set{0}$, and $A + \emptyset = \emptyset$.
If $A$ and $B$ are convex, then $A + B$ is convex; if they are compact, then $A + B$ is compact.

Non-example of a familiar rule: $A + A = 2A$ can fail.
For $A = \set{0, 1} \subseteq \R$ one has $A + A = \set{0, 1, 2}$ but $2A = \set{0, 2}$.
For convex $A$ and $\lambda, \mu \ge 0$ one has $\lambda A + \mu A = (\lambda + \mu) A$.
In terms of [[support functions|1H92CYA4ZC]], $h_{A + B} = h_A + h_B$ for nonempty compact convex $A, B \subseteq \R^n$.
For polytopes, see the [[Minkowski sum of polytopes|7GK5417BJF]].
</document>

<document source="https://tamnd.github.io/math2/node/F4NWZNTRMD.md" tag="F4NWZNTRMD" kind="concept">
# Laplace–Beltrami operator

## Summary

The Laplace–Beltrami operator is the [[Laplacian|0E0B6RA8K0]] of a [[Riemannian manifold|7CBT4SM83Y]]: the divergence of the gradient of a function, both taken with respect to the metric.
It is a second-order elliptic operator that depends only on the metric, not on coordinates.
Its eigenvalues and eigenfunctions form the subject of spectral geometry, and it governs heat flow, waves and harmonic functions on manifolds.

## Explanations

### L1 human

On flat space, the Laplacian of a function compares its value at a point with its average on small spheres around that point.
The Laplace–Beltrami operator does the same thing on a curved space, measuring distances with the curved metric.
On the unit sphere, the function giving the height above the equator is an eigenfunction: its Laplacian is $-2$ times itself.
The vibrations of a curved membrane and the way heat spreads on a curved surface are both described by this operator.
</document>

<document source="https://tamnd.github.io/math2/node/F4RH4MZQDG.md" tag="F4RH4MZQDG" kind="concept">
# Banach bundle

## Summary

A Banach bundle is a [[vector bundle|P4DZM975BV]] whose fibres are Banach spaces, possibly infinite-dimensional, over a [[Banach manifold|9K15AMGBV9]].
The transition maps between local trivialisations must vary smoothly in the operator norm.
Tangent bundles of Banach manifolds are Banach bundles, and so are the bundles of Sobolev sections that appear in nonlinear analysis on manifolds.
Many constructions of moduli spaces present the moduli space as the zero set of a section of a Banach bundle.

## Explanations

### L1 human

A vector bundle attaches a vector space to each point of a base, such as the tangent plane at each point of a surface.
In a Banach bundle these vector spaces may be infinite-dimensional, for example spaces of functions.
Locally the bundle is a product of a piece of the base and one fixed Banach space.
The way these local products are glued must change smoothly from point to point, measured in the operator norm.
</document>

<document source="https://tamnd.github.io/math2/node/F4VMHVWK41.md" tag="F4VMHVWK41" kind="statement">
# Cauchy–Kovalevskaya theorem for ODEs

## Statement

Let $z_0 \in \C$, $w_0 \in \C^n$, and let $F$ be a function from a neighbourhood of $(z_0, w_0)$ in $\C \times \C^n$ to $\C^n$ that is [[holomorphic|0NAHXR1FRH]] in all its variables.
Then there is $r > 0$ such that the [[initial value problem|0FXDW992DS]]
$$w'(z) = F(z, w(z)), \qquad w(z_0) = w_0$$
has a holomorphic solution $w$ on the disc $\abs{z - z_0} < r$.
Any two holomorphic solutions on a disc around $z_0$ agree on the smaller disc.

## Remarks

Writing $w(z) = \sum_{k \ge 0} a_k (z - z_0)^k$ and differentiating the equation repeatedly, each $a_k$ is a polynomial with nonnegative coefficients in $a_0 = w_0$ and the Taylor coefficients of $F$.
So the Taylor series of any holomorphic solution is determined by the equation, which gives uniqueness.
Cauchy's method of majorants proves convergence.
One replaces $F$ by a function with larger Taylor coefficients in absolute value, for which the equation can be solved explicitly.
The solution of this new equation dominates the formal [[power series|0MZB06ST2B]].
The real version holds as well: if $F$ is real [[analytic|081WKEDK6C]], the solution is real analytic.

Example: $w' = w^2$, $w(0) = 1$ has the holomorphic solution $w(z) = \frac{1}{1 - z} = \sum_{k \ge 0} z^k$ on $\abs{z} < 1$.
The radius $r$ cannot be taken arbitrarily large, since this solution has a pole at $z = 1$.

Analyticity of $F$ is needed for an analytic solution: for a smooth but not analytic function $g$ on $\R$, the solution $w(x) = \int_0^x g$ of $w' = g(x)$, $w(0) = 0$, is not analytic.

## Explanations

### L1 human

If the right side of a differential equation is given by convergent power series, then so is the solution.
One can compute the Taylor coefficients of the solution one by one from the equation.
The theorem says that the resulting series really converges near the starting point.

### L2 human

This is the case of ordinary differential equations in the general Cauchy–Kovalevskaya theorem for partial differential equations.
It is the starting point of the theory of differential equations in the complex domain.
Unlike the Picard–Lindelöf theorem, it gives analyticity of the solution, but it needs analytic data.
For linear equations, the solution even extends along every path in the domain where the coefficients are holomorphic, which leads to [[monodromy|C8HD62AB6Q]].
</document>

<document source="https://tamnd.github.io/math2/node/F4WDYPQ9VP.md" tag="F4WDYPQ9VP" kind="statement">
# McKean–Singer formula

## Statement

Let $M$ be a closed smooth manifold with a smooth positive density, and let $E$, $F$ be Hermitian vector bundles over $M$.
Let $P \colon C^\infty(M; E) \to C^\infty(M; F)$ be an [[elliptic differential operator|MT9P2X85V8]] of positive order with formal adjoint $P^*$.
Then for every $t > 0$ the operators $e^{-tP^*P}$ and $e^{-tPP^*}$ are [[trace class|Z2CP530WDE]], and
$$\operatorname{ind} P = \operatorname{tr} e^{-tP^*P} - \operatorname{tr} e^{-tPP^*},$$
where $\operatorname{ind} P$ is the [[analytic index|44GBFZ16SG]].

## Remarks

Idea of the proof: $P^*P$ and $PP^*$ have discrete spectra, with $\ker P^*P = \ker P$ and $\ker PP^* = \ker P^*$.
For $\lambda > 0$, $P$ maps the $\lambda$-eigenspace of $P^*P$ isomorphically onto the $\lambda$-eigenspace of $PP^*$, with inverse $\lambda^{-1} P^*$.
So all terms with $\lambda > 0$ cancel in the difference of the traces, and only $\dim \ker P - \dim \ker P^*$ remains.
The left side does not depend on $t$, so one may let $t \to 0^+$.
Then each trace is an integral of the [[heat kernel|327NP6S7H5]] on the diagonal, which has an asymptotic expansion in powers of $t$ with local coefficients.
Hence the index is the integral of a local density built from the coefficients of $P$, and the singular terms of the expansion cancel.
Example: for $P = d + d^*$ from even to odd forms, the formula reads $\chi(M) = \sum_k (-1)^k \operatorname{tr} e^{-t \Delta_k}$, where $\Delta_k$ is the Hodge Laplacian on $k$-forms.

## Explanations

### L1 human

The index counts solutions of $Pu = 0$ minus solutions of $P^*v = 0$.
The heat operators $e^{-tP^*P}$ and $e^{-tPP^*}$ weight every eigenvalue by $e^{-t\lambda}$.
All nonzero eigenvalues appear in pairs, one on each side, so they cancel, and only the zero modes, which give the index, survive.

### L2 human

The formula is the starting point of the heat equation proof of the index theorem.
For small $t$ the heat traces are given by the [[Minakshisundaram–Pleijel expansion|45P2SEGGR1]], so the index is the integral of the constant term of a local expansion.
The local index theorem identifies this constant term with a characteristic form, which proves the Atiyah–Singer theorem for Dirac operators.
</document>

<document source="https://tamnd.github.io/math2/node/F4WRD396PB.md" tag="F4WRD396PB" kind="concept">
# Reduced product

## Summary

A reduced product of [[algebras|QJGPT2W0FT]] is a quotient of their [[direct product|WJ73XYBNE4]].
It identifies two elements when they agree on a "large" set of indices.
A [[filter|15PF73BKY6]] on the index set says which sets count as large.
When the filter is an [[ultrafilter|J64Y3JC0RE]], the reduced product is an [[ultraproduct|AX9Q86GB8M]].
Reduced products preserve all identities and, more generally, all Horn sentences, which makes them useful in universal algebra and model theory.

## Explanations

### L1 human

Elements of a product are lists with one entry from each factor.
In a reduced product, two lists count as equal if they agree almost everywhere, where "almost everywhere" is fixed in advance by a filter.
For example, with the filter of cofinite subsets of $\N$, two sequences are equal when they differ in only finitely many places.
The operations are still computed entry by entry.
</document>

<document source="https://tamnd.github.io/math2/node/F4ZYFEKKZW.md" tag="F4ZYFEKKZW" kind="definition">
# Compact Lie group

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]].
The group $G$ is a *compact Lie group* if its underlying topological space is [[compact|0TVJZYB6SH]].

## Remarks

Example: the special unitary group $\mathrm{SU}(2)$ is the set of complex matrices $\begin{pmatrix} a & -\bar b \\ b & \bar a \end{pmatrix}$ with $\abs{a}^2 + \abs{b}^2 = 1$.
It is a closed subgroup of $\mathrm{GL}_2(\C)$, so it is a Lie group.
As a space it is the unit sphere $S^3$ in $\C^2 \cong \R^4$, which is closed and bounded, hence compact.

Non-example: the additive group $\R$ is a Lie group, but it is not compact, since the open cover by the intervals $(-n, n)$ has no finite subcover.

A compact group need not be a Lie group.
The group $\Z_p$ of $p$-adic integers is compact, but each neighbourhood of $0$ contains the nontrivial subgroup $p^k \Z_p$ for $k$ large, so it is not a Lie group.

Each compact Lie group is isomorphic to a closed subgroup of some unitary group $\mathrm{U}(n)$.
</document>

<document source="https://tamnd.github.io/math2/node/F52B2JMEHY.md" tag="F52B2JMEHY" kind="statement">
# Fitting's lemma

## Statement

Let $R$ be a ring, let $M$ be a left [[$R$-module|09VDS2W1E8]] of [[finite length|V6JBKT168T]], and let $f \in \End_R(M)$ be an element of its [[endomorphism ring|AWSBZWWMG9]].
Then there is an integer $n \ge 1$ such that:

1. $M = \im f^n \oplus \ker f^n$, a direct sum of [[submodules|0RGZ10DFPG]];
2. the restriction of $f$ to $\im f^n$ is an automorphism of $\im f^n$;
3. the restriction of $f$ to $\ker f^n$ is nilpotent.

In particular, if $M$ is [[indecomposable|E2P29GCD9T]], then every endomorphism of $M$ is either an automorphism or nilpotent.

## Remarks

A module of finite length is both Noetherian and Artinian.
So the chain $\ker f \subseteq \ker f^2 \subseteq \cdots$ and the chain $\im f \supseteq \im f^2 \supseteq \cdots$ both become constant, say from step $n$ on.
If $x \in \im f^n \cap \ker f^n$, then $x = f^n(y)$ with $f^{2n}(y) = 0$, so $y \in \ker f^{2n} = \ker f^n$ and $x = 0$.
For $x \in M$, pick $z$ with $f^n(x) = f^{2n}(z)$; then $x = f^n(z) + (x - f^n(z))$ splits $x$ into the two parts.
For an indecomposable $M$ the corollary shows that the non-invertible endomorphisms form an ideal, so $\End_R(M)$ is a local ring.
Finite length is needed: multiplication by $2$ on the indecomposable $\Z$-module $\Z$ is neither an automorphism nor nilpotent.

## Explanations

### L1 human

Repeat a self-map of a module of finite length many times.
Eventually the module splits into two parts: on one part the map keeps shuffling elements bijectively, and on the other it pushes everything to zero.
If the module cannot be split at all, only one of these behaviours can occur.

### L2 human

Fitting's lemma is the module version of the splitting of a linear map of a finite-dimensional space into an invertible part and a nilpotent part.
Its main use is the corollary that the endomorphism ring of an indecomposable module of finite length is local.
This locality is exactly what the proof of the [[Krull–Schmidt theorem|94M0BPV1W4]] needs.
</document>

<document source="https://tamnd.github.io/math2/node/F52PFPZ4H9.md" tag="F52PFPZ4H9" kind="definition">
# Stochastic difference equation

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space, let $k, l \ge 1$, and let $(\xi_n)_{n \ge 1}$ be a sequence of [[random variables|0PNRKMRC55]] with values in $\R^l$.
For each $n \ge 0$ let $f_n \colon \R^k \times \R^l \to \R^k$ be a Borel [[measurable function|YYNCJSR0PJ]].
A *stochastic difference equation* is a recursion
$$X_{n+1} = f_n(X_n, \xi_{n+1}), \qquad n \ge 0,$$
together with an initial random variable $X_0$ with values in $\R^k$.
Its *solution* is the [[stochastic process|0QH5ENKZM1]] $(X_n)_{n \ge 0}$ defined recursively by these equations.

## Remarks

Example: the autoregressive equation of order $1$, $X_{n+1} = a X_n + \varepsilon_{n+1}$ with $a \in \R$ and independent identically distributed $\varepsilon_n$, is a stochastic difference equation with $f_n(x, e) = ax + e$.
Its solution is $X_n = a^n X_0 + \sum_{j=1}^{n} a^{n-j} \varepsilon_j$.

Non-example: $X_{n+1} = a X_n$ with a constant $X_0$ involves no random input, so it is a deterministic difference equation whose solution is a single sequence.

Higher-order equations, such as the autoregressive model of order $p$, $X_{n+1} = \sum_{i=1}^{p} a_i X_{n+1-i} + \varepsilon_{n+1}$, become first order on $\R^p$ by stacking $p$ consecutive values.
If $X_0, \xi_1, \xi_2, \dots$ are independent, the solution is a Markov chain.
It is time-homogeneous when $f_n$ does not depend on $n$ and the $\xi_n$ are identically distributed.
</document>

<document source="https://tamnd.github.io/math2/node/F55M92H5JY.md" tag="F55M92H5JY" kind="definition">
# Subanalytic set

## Definition

Let $n \ge 1$ and let $X \subseteq \R^n$.
The set $X$ is *subanalytic* if each point $a \in \R^n$ has an open neighbourhood $U$ with the following property.
There are an integer $m \ge 0$ and a bounded [[semianalytic set|E1V22QQHYJ]] $A \subseteq \R^{n+m}$ such that $X \cap U = \pi(A)$.
Here $\pi \colon \R^{n+m} \to \R^n$ is the [[projection|033DA1WE8Y]] onto the first $n$ coordinates.

## Remarks

Every semianalytic set $X \subseteq \R^n$ is subanalytic.
Indeed, for a point $a$ take an open ball $U$ around $a$, take $m = 0$ and $A = X \cap U$.
This set $A$ is semianalytic and bounded, and $\pi$ is the identity, so $X \cap U = \pi(A)$.

The set $X = \set{1/k : k = 1, 2, 3, \dots} \subseteq \R$ is not subanalytic.
A subanalytic set meets each small enough neighbourhood of a point in finitely many connected components.
But every neighbourhood of $0$ contains infinitely many points of $X$, each of them a connected component of the intersection.

The same definition works in a real analytic manifold $M$.
One replaces $\R^m$ by another real analytic manifold $N$ and asks for a relatively compact semianalytic subset $A$ of $M \times N$.
</document>

<document source="https://tamnd.github.io/math2/node/F59TS79V34.md" tag="F59TS79V34" kind="conjecture">
# Dixmier conjecture

## Statement

Let $k$ be a field of characteristic $0$ and let $A_1(k)$ be the first [[Weyl algebra|98QE7KY55W]] over $k$.
Then every endomorphism of $A_1(k)$ as a $k$-algebra is an automorphism.

## Remarks

Dixmier asked the question in a 1968 paper on Weyl algebras.
The same question for the $n$-th Weyl algebra $A_n(k)$ is called the generalized Dixmier conjecture.
An endomorphism of $A_1(k)$ is injective, because $A_1(k)$ is a simple ring, so the question is whether it is also surjective.
Tsuchimoto (2005) and Belov-Kanel and Kontsevich (2007) proved that the generalized Dixmier conjecture for all $n$ is equivalent to the [[Jacobian conjecture|B5JFXEQQD8]] for all $n$.
The case of $A_1(k)$ is open.

## Explanations

### L1 human

The first Weyl algebra is the algebra of polynomial differential operators in one variable.
A map from this algebra to itself that respects sums and products is always one-to-one.
Dixmier asked whether it must also be onto.
The question turned out to be closely tied to the Jacobian conjecture about polynomial maps.
</document>

<document source="https://tamnd.github.io/math2/node/F5EAWFMC8A.md" tag="F5EAWFMC8A" kind="concept">
# Gross–Pitaevskii equation

## Summary

The Gross–Pitaevskii equation describes the ground state and the dynamics of a Bose–Einstein condensate.
It is a [[nonlinear Schrödinger equation|GEJHRXC176]] with a cubic term, which models the interaction between the particles, and an external trapping potential.
Its unknown is a single complex wave function for the whole condensate.
It can be derived from the many-body quantum mechanics of bosons in a suitable limit of many weakly interacting particles.

## Explanations

### L1 human

At very low temperatures a gas of bosonic atoms can collapse into one shared quantum state, a Bose–Einstein condensate.
Then a single wave function describes all the atoms at once.
That wave function feels the trap that holds the gas and also the repulsion or attraction between the atoms, which depends on how dense the gas is.
The Gross–Pitaevskii equation is the Schrödinger equation with this density-dependent extra term.
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# Hopf bifurcation

## Summary

A Hopf bifurcation is the birth of a [[periodic orbit|0AA97XA70K]] from an [[equilibrium point|0N8NGKDZDZ]] as a parameter crosses a critical value.
It happens when a pair of complex conjugate eigenvalues of the linearisation crosses the imaginary axis.
It is the main way in which a steady state of a system turns into a small oscillation.
See the [[definition|1JPGWFMAGK]].

## Explanations

### L1 human

Think of a system resting at an equilibrium, where small disturbances spiral back to rest.
As a parameter changes, the spiral becomes slower, and at a critical value it turns outward.
Then the motion no longer returns to rest but settles on a small loop, a [[limit cycle|0AAQ73WRFB]].
In polar coordinates, the model $r' = \mu r - r^3$, $\theta' = 1$ shows this.
For $\mu \le 0$ every solution spirals into the origin, and for $\mu > 0$ solutions approach the circle $r = \sqrt{\mu}$.
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# Dini test

## Statement

Let $f \colon \R \to \C$ be a [[periodic function|0ATF6Q0633]] of period $2\pi$ that is integrable on $[-\pi, \pi]$, let $x \in \R$ and let $s \in \C$.
Suppose that
$$\int_0^{\pi} \frac{\abs{f(x + t) + f(x - t) - 2s}}{t} \, dt < \infty.$$
Then the partial sums $S_N f(x)$ of the [[Fourier series|G12KXDHPPS]] of $f$ converge to $s$ as $N \to \infty$.

## Remarks

Since the [[Dirichlet kernel|3JY4BVE4DW]] is even with mean $1$, one has $S_N f(x) - s = \frac{1}{2\pi} \int_0^{\pi} \big(f(x + t) + f(x - t) - 2s\big) D_N(t) \, dt$.
Write $D_N(t) = \frac{\sin((N + \frac{1}{2}) t)}{\sin(t/2)}$; the hypothesis says that $t \mapsto \frac{f(x + t) + f(x - t) - 2s}{\sin(t/2)}$ is integrable on $[0, \pi]$, because $\sin(t/2) \ge t/\pi$ there.
The [[Riemann–Lebesgue lemma|WVKDC011KP]] then sends the integral to $0$.
Example: if $f$ is differentiable at $x$, or only satisfies $\abs{f(x + t) - f(x)} \le C \abs{t}^\alpha$ for some $\alpha > 0$ and small $t$, then the hypothesis holds with $s = f(x)$, so the Fourier series converges to $f(x)$.
Continuity of $f$ at $x$ alone does not suffice, since there are continuous functions whose Fourier series diverge at a point.

## Explanations

### L1 human

The Dini test checks convergence of a Fourier series at one point by looking only at how fast the function approaches a value near that point.
If the symmetric difference $f(x + t) + f(x - t) - 2s$ goes to zero fast enough that dividing by $t$ still leaves something integrable, the series converges to $s$.
A small power of $t$ is enough, so all differentiable and all Hölder continuous functions pass.

### L2 human

The test is local: it depends only on $f$ near $x$, in line with the Riemann localization principle.
At a jump with one-sided Hölder conditions, it applies with $s = \frac{1}{2}(f(x^+) + f(x^-))$.
It is independent of the Dirichlet–Jordan test: some functions of bounded variation fail the Dini condition, and some functions satisfying it at every point are not of bounded variation.
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# Holomorphic line bundle

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]].
A *holomorphic line bundle* on $X$ is a [[holomorphic vector bundle|TKA958Z4KZ]] $\pi \colon L \to X$ of rank $1$.

## Remarks

Equivalently, $L$ is given by an open cover $(U_i)$ of $X$ and nowhere vanishing holomorphic functions $g_{ij} \colon U_i \cap U_j \to \C^*$ with $g_{ij} g_{jk} = g_{ik}$ on $U_i \cap U_j \cap U_k$.

Example: the [[tautological line bundle|BDB06N91FN]] $\mathcal{O}(-1)$ on $\mathbb{P}^n$ is $\set{(\ell, v) \in \mathbb{P}^n \times \C^{n+1} : v \in \ell}$ with the projection to $\mathbb{P}^n$.
On $U_i = \set{z_i \ne 0}$ the point $(\ell, v)$ goes to $(\ell, v_i)$.
The fibre coordinates satisfy $v_i = (z_i / z_j) v_j$, and $z_i / z_j$ is holomorphic and nonzero on $U_i \cap U_j$.
Its dual is $\mathcal{O}(1)$, and $\mathcal{O}(k)$ is the $k$-th tensor power of $\mathcal{O}(1)$.

Non-example: the holomorphic tangent bundle of $\mathbb{P}^2$ is a holomorphic vector bundle of rank $2$, so it is not a line bundle.
Its top exterior power is a line bundle, the anticanonical bundle $\mathcal{O}(3)$.

The tensor product of two line bundles with transition functions $g_{ij}$ and $h_{ij}$ has transition functions $g_{ij} h_{ij}$, and the dual of the first has $g_{ij}^{-1}$.
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# Hörmander–Mikhlin multiplier theorem

## Statement

Let $n \ge 1$ and let $k = \floor{n/2} + 1$.
Let $m \colon \R^n \setminus \set{0} \to \C$ be a function of class $C^k$.
Assume that there is a constant $A$ such that
$$\abs{\partial^\alpha m(\xi)} \le A \abs{\xi}^{-\abs{\alpha}} \quad \text{for all } \xi \ne 0$$
and all multi-indices $\alpha$ with $\abs{\alpha} \le k$.
Then $m$ is an [[$L^p$ multiplier|RAGR38Q40B]] for every $1 < p < \infty$.
Let $\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$.
There is a constant $C$, depending only on $n$ and $p$, with
$$\norm{(m \hat f)^\vee}_{L^p(\R^n)} \le C A \norm{f}_{L^p(\R^n)}$$
for every Schwartz function $f$.
Moreover, the operator $f \mapsto (m \hat f)^\vee$ is of weak type $(1, 1)$.

## Remarks

The condition with $\alpha = 0$ says that $m$ is bounded, so the operator is bounded on $L^2$ by the Plancherel theorem.
The derivative conditions make the inverse Fourier transform $K$ of $m$ a kernel that satisfies the Hörmander condition $\int_{\abs{x} > 2\abs{y}} \abs{K(x - y) - K(x)} \, dx \le CA$.
Then the [[Calderón–Zygmund theorem|AE1HJ8D8GV]] gives the weak type $(1, 1)$ bound, interpolation gives $1 < p \le 2$, and duality gives $2 \le p < \infty$.
Hörmander weakened the pointwise derivative bounds to bounds on the $L^2$ averages of the derivatives over dyadic shells.

Example: in $\R^n$, $m(\xi) = \xi_j / \abs{\xi}$ is smooth away from $0$ and homogeneous of degree $0$.
So $\partial^\alpha m$ is homogeneous of degree $-\abs{\alpha}$ and bounded on the unit sphere, which gives the hypothesis.
Hence the [[Riesz transforms|FRNDFJ9WQ5]] are bounded on $L^p(\R^n)$ for $1 < p < \infty$.

Some smoothness is needed.
The indicator of a ball is bounded but not smooth.
By [[Fefferman's ball multiplier theorem|4A5MJY4XPV]], it is not an $L^p$ multiplier for $p \ne 2$ when $n \ge 2$.

## Explanations

### L1 human

Take a multiplier that is smooth away from the origin, whose derivatives shrink at the right rate as the frequency grows.
Such a multiplier keeps $L^p$ size under control for every $p$ strictly between $1$ and $\infty$.
The typical case is a function that depends only on the direction of $\xi$, or that changes slowly on each dyadic scale.
One only needs to check about half as many derivatives as there are dimensions.

### L2 human

This is the most used multiplier criterion in PDE.
It shows at once that the Riesz transforms, the operators $\partial_i \partial_j \Delta^{-1}$, and the imaginary powers $(-\Delta)^{it}$ are bounded on $L^p(\R^n)$ for $1 < p < \infty$.
It sits between the [[Marcinkiewicz multiplier theorem|375ZGE0V2W]], which uses dyadic rectangles, and the general Calderón–Zygmund theory, which supplies its proof.
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# Asymptotics of Bessel functions

## Statement

Let $\nu$ be a real number, and let $J_\nu$ and $Y_\nu$ be the [[Bessel functions of the first kind|CZCMRDWDYW]] and of the [[second kind|GRSNDJG363]].
Let $\Gamma$ be the [[gamma function|0H5SYBSED9]], and let $\sim$ denote [[asymptotic equivalence|0QR92XBV6H]].

1. (Large argument.) As $x \to +\infty$,
$$J_\nu(x) = \sqrt{\frac{2}{\pi x}} \left( \cos\left( x - \frac{\nu\pi}{2} - \frac{\pi}{4} \right) + O\left( \frac{1}{x} \right) \right),$$
$$Y_\nu(x) = \sqrt{\frac{2}{\pi x}} \left( \sin\left( x - \frac{\nu\pi}{2} - \frac{\pi}{4} \right) + O\left( \frac{1}{x} \right) \right).$$
2. (Small argument.) As $x \to 0^+$, if $\nu$ is not a negative integer then $J_\nu(x) \sim \frac{1}{\Gamma(\nu + 1)} \left( \frac{x}{2} \right)^\nu$; if $\nu > 0$ then $Y_\nu(x) \sim -\frac{\Gamma(\nu)}{\pi} \left( \frac{2}{x} \right)^\nu$; and $Y_0(x) \sim \frac{2}{\pi} \ln x$.

## Remarks

The small-argument forms are the leading terms of the power series of $J_\nu$ and of the series of $Y_\nu$ obtained from its definition.
The large-argument forms come from an integral representation of the Hankel functions $H^{(1)}_\nu = J_\nu + iY_\nu$ and the stationary-phase method.
They are the first terms of full asymptotic expansions in powers of $\frac{1}{x}$.
Example: for $\nu = \frac{1}{2}$, $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$ exactly, and indeed $\cos\left( x - \frac{\pi}{2} \right) = \sin x$, with no error term.
The large-argument form holds for fixed $\nu$; it fails when $\nu$ grows with $x$, for instance when $\nu$ is close to $x$, where the transition is described by Airy functions.

## Explanations

### L1 human

Far from the origin a Bessel function looks like a cosine wave whose amplitude shrinks like $\frac{1}{\sqrt{x}}$.
The order $\nu$ only shifts the phase of the wave.
Near the origin $J_\nu$ behaves like the power $x^\nu$, while $Y_\nu$ blows up like $x^{-\nu}$ or like $\ln x$.

### L2 human

The large-argument form explains why the positive zeros of $J_\nu$ are eventually spaced by about $\pi$.
The amplitude $\frac{1}{\sqrt{x}}$ reflects conservation of energy for cylindrical waves, whose wavefronts have length proportional to $x$.
The small-argument forms decide which solution is allowed in a boundary value problem on a disc: regularity at the centre excludes $Y_\nu$.
For modified Bessel functions the analogues are $I_\nu(x) \sim \frac{e^x}{\sqrt{2\pi x}}$ and $K_\nu(x) \sim \sqrt{\frac{\pi}{2x}}\, e^{-x}$.
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# Quasi-measure-preserving map

## Summary

A quasi-measure-preserving map is a measurable map between measure spaces under which the preimage of every null set is a null set.
Equivalently, the pushforward of the first measure is [[absolutely continuous|P0P21KDV6E]] with respect to the second one.
Such maps respect "almost everywhere" statements, so they act on equivalence classes of measurable functions.
Every [[measure-preserving transformation|556WBDAHNP]] is quasi-measure-preserving.

## Explanations

### L1 human

A measure-preserving map keeps the size of every set.
A quasi-measure-preserving map is allowed to change sizes, but it may not turn something of size zero into something of positive size when one pulls back.
For example, the map $x \mapsto 2x$ on the real line doubles lengths, so it does not preserve length.
Still, the preimage of a set of length zero has length zero, so the map is quasi-measure-preserving.
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# Regular polytope

## Summary

A regular polytope is a polytope with the largest possible symmetry: its symmetries can carry any flag of faces to any other flag.
In the plane the regular polytopes are the [[regular polygons|01KJJ9A400]], and in space the convex ones are the five [[Platonic solids|017E5PRS1Q]].
Each regular polytope is described by its [[Schläfli symbol|MAE6BF4DFK]], and their full list is given by the [[classification of regular polytopes|XYCD53PB5K]].
Regular polytopes are the model objects of the theory of polytopes and of finite reflection groups.

## Explanations

### L1 human

A regular polytope looks the same from every corner, every edge, every face, and so on.
More precisely, pick a corner, an edge at that corner, and a face at that edge; any such choice can be moved to any other by a symmetry.
The cube is regular: any corner, edge and face that fit together can be rotated or reflected to any other such triple.
A box with sides of three different lengths is not regular, because a short edge cannot be moved onto a long one.
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# Dynamic programming in continuous time

## Definition

Consider a finite-horizon [[optimal control|T3V6ZNAMDN]] problem on $[0, T]$ with dynamics $y' = f(s, y, u)$, controls with values in a set $U$, running cost $L$ and terminal cost $g$.
Let $V$ be its [[value function|K6G0DRQ3CP]].
The *dynamic programming principle* for the problem is the identity, for all $0 \le t \le s \le T$ and $x \in \R^n$,
$$V(t, x) = \inf_{u} \Big( \int_t^s L(r, y(r), u(r)) \, dr + V(s, y(s)) \Big).$$
Here the infimum is over measurable controls $u \colon [t, s] \to U$, and $y$ is the trajectory with $y(t) = x$.
*Dynamic programming in continuous time* is the method that derives from this principle a partial differential equation for $V$.
That equation is the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] $V_t + \min_{u \in U} \big( L(t, x, u) + \nabla_x V \cdot f(t, x, u) \big) = 0$ with $V(T, x) = g(x)$.
One then obtains optimal controls by choosing, at each $(t, x)$, a minimizer $u$ of this expression.

## Remarks

The principle holds under the standing hypotheses of the value function: $U$ compact, $f$ continuous and Lipschitz in the state, and $L$, $g$ continuous.
Its proof splits a control on $[t, T]$ into its parts on $[t, s]$ and $[s, T]$.
Example: for $y' = u$, $U = [-1, 1]$, $L = 0$ and $g(x) = \abs{x}$, the value function is $V(t, x) = \max(\abs{x} - (T - t), 0)$.
Where $\abs{x} > T - t$ it equals $\abs{x} - (T - t)$, so $V_t = 1$ and $\min_{\abs{u} \le 1} V_x u = -1$, and the equation holds.
The minimizer $u = -\sgn(x)$ is the optimal feedback.
Non-example: a greedy rule that minimizes only the running cost $L$ at each instant, ignoring $V$, does not follow the principle and is in general not optimal.
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# Information geometry

## Summary

Information geometry studies families of [[probability distributions|0DVDCMA2DK]] as manifolds, using the tools of differential geometry.
The central structure is the Fisher information metric, a [[Riemannian metric|Q61VFHCH2J]] on the parameter space, together with a family of [[affine connections|JX65KSVHSJ]] called the $\alpha$-connections.
On finite sample spaces, Chentsov's theorem shows that the Fisher metric is, up to a constant factor, the only metric invariant under sufficient statistics.
The field was developed largely by Shun-ichi Amari and has applications in statistics, machine learning and information theory.

## Explanations

### L1 human

A statistical model is a family of distributions described by a few parameters, like the mean and standard deviation of a normal distribution.
Each choice of parameters is a point, so the model is a space of points.
Information geometry measures the distance between two nearby points by how easily data can tell the two distributions apart.
For normal distributions, two distributions with means $0$ and $1$ are far apart when the standard deviation is small and close when it is large.
With this distance the family of normal distributions becomes a copy of the hyperbolic plane.
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# Siegel disc

## Definition

Let $f \colon \hat{\C} \to \hat{\C}$ be a [[rational function|0XDK9CGYKJ]] of degree at least $2$.
A *Siegel disc* of $f$ is a component $U$ of the [[Fatou set|C0N0XGC6GK]] of $f$ with the following properties.

1. There is $p \ge 1$ with $f^p(U) = U$.
2. There is a [[conformal map|0YYT36R4YD]] $\varphi \colon U \to \mathbb{D}$ onto the [[unit disc|8G5ABXRK85]] $\mathbb{D}$.
3. There is an [[irrational number|0DZXAXEJQ9]] $\theta$ with $\varphi(f^p(z)) = e^{2\pi i \theta} \varphi(z)$ for all $z \in U$.

## Remarks

The point $z_0 = \varphi^{-1}(0)$ is a [[periodic point|T75CZ5H47N]] of $f$, and its multiplier is $(f^p)'(z_0) = e^{2\pi i \theta}$.
So the periodic point is neutral and irrationally indifferent, and the map is linearizable near it.
Example: Siegel proved in 1942 that $f(z) = e^{2\pi i \theta} z + z^2$ is linearizable at $0$ when $\theta$ is Diophantine, for instance the golden mean $(\sqrt{5} - 1)/2$.
The component of the Fatou set containing $0$ is then a Siegel disc.
Non-example: the unit disc for $f(z) = z^2$ fails clause 3, since $f$ is not injective there, while a rotation is.
The analogous domain with an annulus in place of the disc is a Herman ring.
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# Pólya's conjecture on Laplacian eigenvalues

## Statement

Let $d \ge 1$ and let $\Omega \subseteq \R^d$ be a nonempty bounded open set with [[Lebesgue measure|2Z2QVWZ262]] $\abs{\Omega}$.
Let $\omega_d$ be the volume of the unit ball of $\R^d$.
Let $0 < \lambda_1 \le \lambda_2 \le \cdots$ be the [[Dirichlet eigenvalues|4VD8XXQ4P3]] of $-\Delta$ on $\Omega$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
If $\Omega$ is also connected with Lipschitz boundary, let $0 = \mu_1 < \mu_2 \le \mu_3 \le \cdots$ be its eigenvalues with the [[Neumann boundary condition|GS3KT2H645]].
All are repeated according to multiplicity.
Let $C_d = \dfrac{4 \pi^2}{\omega_d^{2/d}}$.
Then, for every $k \ge 1$,
$$\lambda_k \ge C_d \left( \frac{k}{\abs{\Omega}} \right)^{2/d} \qquad \text{and} \qquad \mu_{k+1} \le C_d \left( \frac{k}{\abs{\Omega}} \right)^{2/d}.$$

## Remarks

The constant is that of [[Weyl's law|2JK9YMZV2B]], $\lambda_k \sim C_d (k / \abs{\Omega})^{2/d}$ as $k \to \infty$, so the conjecture says that the Weyl term is a one-sided bound for every $k$, not only asymptotically.
For $d = 2$ the bounds read $\lambda_k \ge 4 \pi k / \abs{\Omega}$ and $\mu_{k+1} \le 4 \pi k / \abs{\Omega}$.
The conjecture is open in general.
Pólya proved the Dirichlet bound for plane domains that tile the plane.
Li and Yau proved the weaker bound $\lambda_k \ge \frac{d}{d + 2} C_d (k / \abs{\Omega})^{2/d}$ for all bounded domains, now called the Berezin–Li–Yau inequality.
Filonov, Levitin, Polterovich and Sher proved both bounds for the disk.
For $k = 1$ the Dirichlet bound follows from the [[Faber–Krahn inequality|E7TY2TPXX7]].

## Explanations

### L1 human

The eigenvalues of the Laplacian on a region are the squared pitches of a drum of that shape.
Weyl's law says how the $k$-th pitch grows with $k$, on average.
Pólya guessed that, with the Dirichlet condition, every pitch lies above the Weyl prediction, and with the Neumann condition every pitch lies below it.
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# Log-rank test

## Summary

The log-rank test is a nonparametric [[hypothesis test|087P03NEND]] that compares the survival distributions of two or more groups from right-[[censored|MY8ZZP84NP]] data.
At each event time it compares the observed number of events in a group with an expected number.
The expected number is computed as if all groups had the same [[hazard function|Y959NKQ3XW]].
Summing these differences and standardising gives a statistic that is approximately standard normal under the null hypothesis.
It is the standard test for comparing [[Kaplan–Meier|B7AJ8RWZRC]] curves, and it is most efficient when the hazards of the groups are proportional.

## Explanations

### L1 human

Two groups of patients get different treatments, and we record when each patient has a relapse.
Each time someone relapses, we look at how many patients of each group were still at risk.
From this we compute how many of these relapses we would expect in group 1 if the treatments were equally good.
The test adds up "observed minus expected" over all relapse times.
A large total, compared with its natural spread, suggests that the treatments really differ.
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# Sine-Gordon equation

## Definition

The *sine-Gordon equation* is the [[partial differential equation|GC5BDZ6QBC]]
$$u_{tt} - u_{xx} + \sin u = 0$$
for an unknown real function $u(x, t)$ of $x \in \R$ and $t \in \R$, where subscripts denote [[partial derivatives|09EGC6F8PT]].

## Remarks

Example: the static kink $u(x,t) = 4 \arctan(e^x)$ is a solution.
Put $\theta = \arctan(e^x)$; then $\sin 2\theta = \mathrm{sech}\, x$ and $\cos 2\theta = -\tanh x$.
So $\sin u = \sin 4\theta = -2\, \mathrm{sech}\, x \tanh x$.
Also $u_x = 2\, \mathrm{sech}\, x$, so $u_{xx} = -2\, \mathrm{sech}\, x \tanh x = \sin u$, and $u_{tt} = 0$.
For $\abs{v} < 1$, the moving kink $4 \arctan\!\big(\exp\big((x - vt)/\sqrt{1 - v^2}\big)\big)$ is also a solution, by Lorentz invariance.

Non-example: $u(x,t) = x$ is not a solution, because $u_{tt} - u_{xx} + \sin u = \sin x$ is not identically $0$.

In the coordinates $\xi = (x+t)/2$ and $\eta = (x-t)/2$ the equation becomes $u_{\xi\eta} = \sin u$.
This is the form in which it arises for surfaces of Gaussian curvature $-1$.
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# Bloch space

## Summary

The Bloch space is the space of [[holomorphic functions|0NAHXR1FRH]] $f$ on the unit disc for which $(1 - \abs{z}^2)\abs{f'(z)}$ stays bounded.
With a natural norm it is a [[Banach space|06GZ956657]], and the seminorm $\sup_z (1 - \abs{z}^2)\abs{f'(z)}$ does not change under composition with disc automorphisms.
It contains all bounded holomorphic functions and also some unbounded ones such as $\log(1 - z)$.
It is closely tied to [[Bloch's theorem|KW31KDX7TB]], to the space of [[bounded mean oscillation|QJYTXFT8Y4]], and to the duality theory of [[Bergman spaces|2D619WTPYC]].

## Explanations

### L1 human

A function in the Bloch space may grow near the edge of the disc, but its derivative may grow at most like $1/(1 - \abs{z})$.
The factor $1 - \abs{z}^2$ is the natural scale of the disc in hyperbolic geometry, so the condition says that $f$ changes by a bounded amount over each hyperbolic step.
For example, $\log(1 - z)$ is unbounded near $z = 1$, yet it belongs to the Bloch space.
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# Filter limit

## Summary

A filter limit is the limit of a function $f \colon S \to X$ along a [[filter|15PF73BKY6]] on $S$, where $X$ is a [[topological space|0CQHFPGJ45]].
The function tends to $x$ along the filter when the preimage of every [[neighbourhood|0ZCY0Z80B6]] of $x$ belongs to the filter.
One notion covers limits of sequences, limits of functions at a point or at infinity, one-sided limits and sums of families, by choosing the filter.
It is the same as the [[convergence|70M1VAPFN0]] of the image filter $f(\mathcal{F})$ to $x$.

## Explanations

### L1 human

A filter on $S$ describes a way of "going to the limit" in $S$, by saying which sets are eventually entered.
For sequences, the sets that are eventually entered are the sets that contain all $n$ from some point on.
A function has limit $x$ along the filter if, for every small region around $x$, the points where the function lands in that region form an eventually entered set.
With the filter of tails of $\N$, this is exactly the usual limit of a sequence.
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# Adic space

## Summary

An adic space is a geometric space, introduced by Roland Huber, whose points are continuous valuations on topological rings, see the [[definition|5AXZG4PXTE]].
Adic spaces give one framework for non-Archimedean geometry that contains the [[rigid analytic spaces|YAH6MZVG7K]] over a complete non-Archimedean field and also locally Noetherian formal schemes.
Their points include the classical points and extra points of higher rank, which make the topology work well.
[[Perfectoid spaces|Q9Y4CJE1X5]] are adic spaces of a special kind.

## Explanations

### L1 human

In ordinary geometry, a point of a space is a way to evaluate functions.
In an adic space, a point is a way to measure the size of functions, given by a valuation.
The model case is the $p$-adic closed unit disc, built from power series with $p$-adic coefficients tending to $0$, which converge on the disc.
Besides the usual points, it has points that measure functions by their largest size on a smaller disc, and some infinitesimal refinements of these.
With these extra points, coverings behave as they should, so functions defined piece by piece can be glued.
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# Cauchy's exponential equation

## Definition

*Cauchy's exponential equation* is the [[functional equation|SRAN4EKWS0]]
$$f(x + y) = f(x)\, f(y) \quad \text{for all } x, y \in \R,$$
for an unknown [[function|0EPAT51DXQ]] $f \colon \R \to \R$.
A solution is a function $f$ that satisfies this identity for all real $x$ and $y$.

## Remarks

Example: $f(x) = e^{3x}$ is a solution, since $e^{3(x + y)} = e^{3x} e^{3y}$.
Non-example: $f(x) = 1 + x$ is not a solution, since $f(1 + 1) = 3$ while $f(1) f(1) = 4$.
Every solution satisfies $f(x) = f(x/2)^2 \ge 0$.
If $f(x_0) = 0$ for one $x_0$, then $f(x) = f(x - x_0) f(x_0) = 0$ for all $x$.
Otherwise $f > 0$ and $\ln f$ is an [[additive function|KF719SZD5T]], so the solutions are $f = 0$ and $f = e^{A}$ with $A$ additive.
The continuous solutions are $f = 0$ and $f(x) = a^x$ with $a > 0$.
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# Positive-definite function

## Summary

A positive-definite function on $\R^n$ is a complex function $f$ such that every matrix $\big(f(x_i - x_j)\big)_{i,j}$ built from finitely many points is positive semidefinite.
It is the case $G = \R^n$ of a [[positive-definite function on a group|3PRZ02APFA]].
By [[Bochner's theorem|ZRHAG4NXHC]], the continuous positive-definite functions on $\R^n$ are exactly the Fourier transforms of finite positive measures.
They appear as characteristic functions in probability, as covariance functions of stationary processes, and as kernels in interpolation and machine learning.

## Explanations

### L1 human

Pick some points on the line and form the table of values $f(x_i - x_j)$.
For a positive-definite function, this table always behaves like a table of inner products, so it never produces a negative "squared length".
The cosine is the simplest example, and so is the Gaussian $e^{-x^2}$.
Being positive-definite forces $f(0)$ to be the largest value of $\abs{f}$.
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# Hausdorff distance

## Summary

The Hausdorff distance measures how far two subsets of a [[metric space|0VKD1TP7F3]] are from each other.
It is the smallest $\varepsilon$ such that each set lies in the $\varepsilon$-neighbourhood of the other.
On the nonempty compact subsets of a metric space it is a metric.
With it, the compact convex sets of $\R^n$ form the basic space of convex geometry.

## Explanations

### L1 human

Two shapes are close in the Hausdorff distance if every point of each shape is near some point of the other shape.
Picture thickening one shape by a margin $\varepsilon$ until it swallows the other, and doing the same the other way round.
The smallest margin that works both ways is the Hausdorff distance.
For the intervals $[0, 1]$ and $[0, 2]$ on the line, the distance is $1$: the point $2$ is at distance $1$ from $[0,1]$.
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<document source="https://tamnd.github.io/math2/node/F7A335JYKF.md" tag="F7A335JYKF" kind="definition">
# Whittaker transform

## Definition

Fix complex parameters $\kappa$ and $\mu$, and let $W_{\kappa, \mu}$ be the [[Whittaker function|7EFDMJSVDP]] on $(0, \infty)$.
Let $k(z) = e^{-z/2} z^{-\kappa} W_{\kappa, \mu}(z)$ for $z > 0$.
Let $f \colon (0, \infty) \to \C$ be such that the integral below converges for $y$ in a set $Y \subseteq (0, \infty)$.
The *Whittaker transform* of $f$, in this normalization, is the function
$$g(y) = \int_0^\infty k(xy)\, f(x)\, dx, \qquad y \in Y.$$

## Remarks

Other authors put other powers of $xy$ and other exponential factors in front of $W_{\kappa, \mu}(xy)$; the name covers this whole family of kernels.

Example: for $\kappa = \mu + 1/2$ one has $W_{\mu + 1/2, \mu}(z) = e^{-z/2} z^{\mu + 1/2}$.
Then $k(z) = e^{-z/2} z^{-\mu - 1/2} e^{-z/2} z^{\mu + 1/2} = e^{-z}$, and the transform is the [[Laplace transform|0V49JEZW06]] $g(y) = \int_0^\infty e^{-xy} f(x)\, dx$.

Non-example: $W_{\kappa, \mu}(z)$ behaves like $e^{-z/2} z^{\kappa}$ as $z \to \infty$, so $k(xy)$ behaves like $e^{-xy}$ for large $x$.
Hence $f(x) = e^{x^2}$ gives a divergent integral for every $y > 0$ and has no Whittaker transform.
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# Dodgson condensation

## Definition

Let $F$ be a [[field|04C5ZBG1WT]], let $n \ge 2$, and let $A$ be an $n \times n$ matrix over $F$.
Let $A^{(-1)}$ be the $(n+1) \times (n+1)$ matrix with all entries $1$, and let $A^{(0)} = A$.
For $k = 1, \dots, n-1$, let $A^{(k)}$ be the $(n-k) \times (n-k)$ matrix with entries
$$a^{(k)}_{ij} = \frac{a^{(k-1)}_{ij}\, a^{(k-1)}_{i+1,j+1} - a^{(k-1)}_{i,j+1}\, a^{(k-1)}_{i+1,j}}{a^{(k-2)}_{i+1,j+1}}, \qquad 1 \le i, j \le n-k.$$
This is defined when every divisor $a^{(k-2)}_{i+1,j+1}$ is nonzero.
*Dodgson condensation* is the computation of the sequence $A^{(0)}, A^{(1)}, \dots, A^{(n-1)}$.
When it is defined, the single entry of $A^{(n-1)}$ equals the [[determinant|00KDZRW4C0]] of $A$.

## Remarks

The divisors in step $k$ are the interior entries of $A^{(k-2)}$, that is, the entries not in its first or last row or column.
For $k = 1$ all divisors are $1$, so $A^{(1)}$ is the matrix of connected $2 \times 2$ [[minors|0BT9BYWVRY]] of $A$.

Example: for $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{pmatrix}$, the first step gives $A^{(1)} = \begin{pmatrix} -3 & -3 \\ -3 & 2 \end{pmatrix}$.
The only divisor in the second step is the central entry $5$ of $A$, so $A^{(2)} = \big((-3)(2) - (-3)(-3)\big)/5 = -3$.
Expansion along the first row also gives $\det A = 2 + 4 - 9 = -3$.

Non-example: for $A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$, the central entry is $0$, so $A^{(2)}$ is not defined, although $\det A = -1$.
In such a case one can first permute rows or columns, or use a different order of elimination.
The correctness of the method follows from the Desnanot–Jacobi identity, a form of the case $k = n - 2$ of [[Sylvester's determinant identity|7SSKBQYFW1]].

## Explanations

### L1 human

The method shrinks a matrix by one row and one column at each step, using only $2 \times 2$ determinants and divisions.
It stops when one number is left, and that number is the determinant.
It fails when it must divide by zero.
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<document source="https://tamnd.github.io/math2/node/F7EKTEESQ5.md" tag="F7EKTEESQ5" kind="statement">
# Beltrami identity

## Statement

Let $a < b$, and let $L \colon \R \times \R \to \R$, written $L(z, p)$, be a function of class $C^2$ that does not depend on the independent variable $x$.
Write $L_z$ and $L_p$ for its [[partial derivatives|09EGC6F8PT]].
Let $y \in C^2([a, b])$ solve the [[Euler–Lagrange equation|PS0VV149ET]]
$$\frac{d}{dx} L_p\big(y(x), y'(x)\big) = L_z\big(y(x), y'(x)\big) \quad \text{on } [a, b].$$
Then there is a constant $C$ with
$$L\big(y(x), y'(x)\big) - y'(x) \, L_p\big(y(x), y'(x)\big) = C \quad \text{for all } x \in [a, b].$$

## Remarks

The proof is one differentiation.
The derivative of $L(y, y') - y' L_p(y, y')$ is $L_z y' + L_p y'' - y'' L_p - y' \frac{d}{dx} L_p = y' \big(L_z - \frac{d}{dx} L_p\big)$, which is $0$ by the Euler–Lagrange equation.

Example: for the [[minimal surface of revolution|AEG6QWEFRG]] one has $L(z, p) = z \sqrt{1 + p^2}$.
Then $L - p L_p = z / \sqrt{1 + p^2}$, so $y / \sqrt{1 + y'^2} = C$.
For $C > 0$ the solutions are the catenaries $y(x) = C \cosh\big((x - d)/C\big)$.

A partial converse holds: if the quantity is constant and $y'(x) \ne 0$, then the Euler–Lagrange equation holds at $x$.
So constant solutions $y' = 0$ of the identity must be checked separately.

## Explanations

### L1 human

When the integrand of a variational problem does not depend on position along the axis, one quantity stays constant along every solution.
This replaces a second-order equation by a first-order one, which is often easy to solve.
It is how the brachistochrone and the hanging chain are solved by hand.

### L2 human

In mechanics, with $x$ as time and $L$ the Lagrangian, the quantity $y' L_p - L$ is the energy, the Hamiltonian.
The identity is then conservation of energy for a time-independent Lagrangian, a special case of [[Noether's theorem|XHPDM5T2JG]] for time translations.
It is used in the [[brachistochrone problem|ZWQ5JBR6XF]] and the [[catenary problem|8REBFQD6E0]].
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<document source="https://tamnd.github.io/math2/node/F7FYTD7Z56.md" tag="F7FYTD7Z56" kind="definition">
# Lagrange resolvent

## Definition

Let $n \ge 1$ be an integer and let $K$ be a field that contains a [[primitive $n$-th root of unity|0J817RTVG6]] $\zeta$.
Let $L/K$ be a [[cyclic extension|VFE5B71E7Y]] of degree $n$, and let $\sigma$ be a generator of its [[Galois group|0XVR72M51M]].
For $\alpha \in L$, the *Lagrange resolvent* of $\alpha$ with respect to $\zeta$ is
$$(\zeta, \alpha) = \sum_{i=0}^{n-1} \zeta^i \sigma^i(\alpha) = \alpha + \zeta \sigma(\alpha) + \cdots + \zeta^{n-1} \sigma^{n-1}(\alpha).$$

## Remarks

Since $\sigma$ fixes $\zeta$ and $\sigma^n = \id$, one has $\sigma((\zeta, \alpha)) = \sum_{i} \zeta^i \sigma^{i+1}(\alpha) = \zeta^{-1} (\zeta, \alpha)$.
Hence $(\zeta, \alpha)^n$ is fixed by $\sigma$ and lies in $K$.
By the [[independence of characters|N43G50YHG2]] applied to the automorphisms $\id, \sigma, \ldots, \sigma^{n-1}$, some $\alpha$ has $(\zeta, \alpha) \ne 0$.
For such $\alpha$, the automorphism $\sigma^j$ multiplies $(\zeta, \alpha)$ by $\zeta^{-j}$, which equals $1$ only when $n$ divides $j$.
So only the identity of the Galois group fixes $(\zeta, \alpha)$, hence $L = K((\zeta, \alpha))$ is generated by an $n$-th root of an element of $K$.

Example: let $K = \Q$, $n = 2$, $\zeta = -1$, $L = \Q(\sqrt{2})$ and $\sigma(\sqrt{2}) = -\sqrt{2}$.
For $\alpha = \sqrt{2}$, the resolvent is $\sqrt{2} - \sigma(\sqrt{2}) = 2\sqrt{2}$, and its square $8$ lies in $\Q$.

Non-example: with $\zeta = 1$ the same sum is $\alpha + \sigma(\alpha) + \cdots + \sigma^{n-1}(\alpha)$, the trace of $\alpha$.
This is not a Lagrange resolvent for $n \ge 2$, because $1$ is not a primitive $n$-th root of unity, and it lies in $K$ instead of generating $L$.

Classically, for a polynomial with roots $x_1, \ldots, x_n$, the Lagrange resolvent is the expression $x_1 + \zeta x_2 + \cdots + \zeta^{n-1} x_n$.
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# Minkowski–Bouligand dimension

## Definition

Let $A$ be a nonempty [[bounded set|09QSZVVM26]] in $\R^n$.
For $\varepsilon > 0$, let $N(A, \varepsilon)$ be the smallest number of sets of [[diameter|0KW3PN547K]] at most $\varepsilon$ that cover $A$.
The *upper* and *lower box-counting dimensions* of $A$ are
$$\overline{\dim}_B A = \limsup_{\varepsilon \to 0^+} \frac{\log N(A, \varepsilon)}{\log (1/\varepsilon)}, \qquad \underline{\dim}_B A = \liminf_{\varepsilon \to 0^+} \frac{\log N(A, \varepsilon)}{\log (1/\varepsilon)}.$$
If they are equal, the common value $\dim_B A$ is the *Minkowski–Bouligand dimension*, or box-counting dimension, of $A$.

## Remarks

One gets the same values if $N(A, \varepsilon)$ counts the cubes of an $\varepsilon$-mesh grid that meet $A$, or the largest number of disjoint balls of radius $\varepsilon$ centred in $A$.
For bounded $A$, $\overline{\dim}_B A = \inf \set{ s : \mathcal{M}^{*s}(A) = 0 }$, where $\mathcal{M}^{*s}$ is the upper [[Minkowski content|HS2NTCMZ4T]].
Always $\dim_H A \le \underline{\dim}_B A \le \overline{\dim}_B A$, where $\dim_H$ is the [[Hausdorff dimension|SSNFTHZ6YJ]].

Example: let $C$ be the middle-thirds [[Cantor set|0BXNDZFQME]].
It is covered by $2^k$ intervals of length $3^{-k}$, and each set of diameter at most $3^{-k}$ meets at most two of them.
So $2^{k-1} \le N(C, 3^{-k}) \le 2^k$, and the ratio tends to $\log 2 / \log 3$, so $\dim_B C = \log 2 / \log 3$.

The box dimension does not ignore countable sets.
The set $\Q \cap [0, 1]$ is dense, so $N(\Q \cap [0, 1], \varepsilon) = N([0, 1], \varepsilon)$, and its box dimension is $1$.
Its Hausdorff dimension is $0$.

## Explanations

### L1 human

Count how many small sets of size $\varepsilon$ are needed to cover the set.
If the count grows like $\varepsilon^{-d}$ as $\varepsilon$ shrinks, the box dimension is $d$.
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# Adaptive step size

## Definition

Let a [[one-step method|558XADG64M]] for $y' = f(t, y)$ come with an error estimator.
From $t_n$, $y_n$ and a trial step $h$ it computes $y_{n+1}$ and a number $\mathrm{err}_n \ge 0$.
This number estimates the norm of the [[local truncation error|T220HF0562]] of the step, which behaves like $C h^{q+1}$ for some integer $q \ge 1$.
Let $\mathrm{tol} > 0$ be a tolerance, $0 < \rho < 1$ a safety factor, and $0 < \alpha_{\min} < 1 < \alpha_{\max}$ bounds.
*Adaptive step size control* runs the method with steps chosen as follows.

1. If $\mathrm{err}_n \le \mathrm{tol}$, the step is accepted: $t_{n+1} = t_n + h$, and $y_{n+1}$ is kept.
2. If $\mathrm{err}_n > \mathrm{tol}$, the step is rejected and repeated from $t_n$.
3. In both cases the next trial step is
$$h_{\mathrm{new}} = h \cdot \min\Big(\alpha_{\max}, \max\Big(\alpha_{\min}, \; \rho \, \Big(\frac{\mathrm{tol}}{\mathrm{err}_n}\Big)^{1/(q+1)}\Big)\Big).$$

## Remarks

The formula of clause 3 comes from $\mathrm{err} \approx C h^{q+1}$: the step $h'$ with $C h'^{q+1} = \mathrm{tol}$ is $h (\mathrm{tol}/\mathrm{err})^{1/(q+1)}$, and $\rho$, $\alpha_{\min}$, $\alpha_{\max}$ protect against a poor estimate.
Example: with an embedded pair of orders $4$ and $5$ such as the [[Runge–Kutta–Fehlberg method|6D73FFD9DC]], $\mathrm{err}_n = \norm{\hat y_{n+1} - y_{n+1}}$ and $q = 4$; if $\mathrm{err}_n = 32\,\mathrm{tol}$ the step is rejected and, with $\rho = 0.9$, retried with $h_{\mathrm{new}} = 0.9 \cdot 32^{-1/5} h = 0.45 h$.
Other estimators use step doubling: compare one step of size $h$ with two steps of size $h/2$, as in [[Richardson extrapolation|WMTZG71K2Y]].
Non-example: a schedule fixed in advance, such as $h_n = 2^{-n}$, varies the step but is not adaptive, since it ignores the error estimate.
The tolerance controls the local error per step; the global error is usually of a similar size but is not guaranteed to be below $\mathrm{tol}$.

## Explanations

### L1 human

After each step, the solver estimates how much error it just made.
If the error is too big, it throws the step away and tries again with a smaller one.
If it is small, it accepts the step and tries a larger one next time.
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# Maximal compact subgroup

## Definition

Let $G$ be a [[topological group|92700QAB5Y]].
A subgroup $K$ of $G$ is a *maximal compact subgroup* if two conditions hold.

1. The subgroup $K$ is [[compact|0TVJZYB6SH]] in the subspace topology.
2. If $L$ is a compact subgroup of $G$ with $K \subseteq L$, then $L = K$.

## Remarks

Example: $K = \mathrm{O}(n)$ in $G = \mathrm{GL}_n(\R)$.
It is closed and bounded in the space of matrices, so it is compact.
Let $L \supseteq \mathrm{O}(n)$ be a compact subgroup.
Averaging the standard inner product over $L$ with the Haar measure gives an inner product $\ip{\cdot}{\cdot}'$ that $L$ preserves.
The group $\mathrm{O}(n)$ preserves it too, and the only $\mathrm{O}(n)$-invariant inner products are the positive multiples of the standard one.
So $L$ preserves the standard inner product, and $L \subseteq \mathrm{O}(n)$.

Non-example: $\mathrm{SO}(n)$ in $\mathrm{GL}_n(\R)$ is compact, but it fails clause 2, since $\mathrm{SO}(n) \subsetneq \mathrm{O}(n)$.

A maximal compact subgroup need not exist.
In the countable direct sum $\bigoplus_{i \in \N} \Z/2$ with the discrete topology, the compact subgroups are the finite ones, and each finite subgroup lies in a larger finite subgroup.
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# Cauchy–Davenport theorem

## Statement

Let $p$ be a [[prime number|01826KMG2H]], and let $A$ and $B$ be nonempty subsets of the [[integers modulo $p$|03VFZ2WETT]], $\Z/p\Z$.
Then the [[sumset|XNBN31HQWB]] $A + B$ satisfies
$$\abs{A + B} \ge \min(p, \ \abs{A} + \abs{B} - 1).$$

## Remarks

One proof is by induction on $\abs{B}$.
It replaces $(A, B)$ by $(A \cup (B + e), \ B \cap (A - e))$ for a suitable $e$, which keeps $\abs{A} + \abs{B}$ and does not enlarge the sumset.
Another proof uses the [[combinatorial Nullstellensatz|9ZW7BCJMDT]] with a polynomial that vanishes on a too small set $C \supseteq A + B$.
The bound is sharp: for $A = \set{0, 1, \dots, k-1}$ and $B = \set{0, 1, \dots, l-1}$ with $k + l - 1 \le p$, the sumset is $\set{0, \dots, k + l - 2}$.
The hypothesis that $p$ is prime is needed.
In $\Z/4\Z$, take $A = B = \set{0, 2}$; then $A + B = \set{0, 2}$ has $2$ elements, while $\min(4, 3) = 3$.

## Explanations

### L1 human

Work with numbers modulo a prime $p$, on a clock with $p$ hours.
Add every element of a set $A$ to every element of a set $B$.
You get at least $\abs{A} + \abs{B} - 1$ different results, unless you already get all $p$ of them.
So sums modulo a prime cannot bunch up too much.

### L2 human

The Cauchy–Davenport theorem is the analogue for $\Z/p\Z$ of the easy bound $\abs{A + B} \ge \abs{A} + \abs{B} - 1$ for sets of integers.
Over composite moduli, subgroups give smaller sumsets, and Kneser's theorem describes this.
The Dias da Silva–Hamidoune theorem bounds the set of sums of two distinct elements of $A$ by $\min(p, 2\abs{A} - 3)$, as conjectured by Erdős and Heilbronn.
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# State on a C*-algebra

## Summary

A state on a C*-algebra is a positive linear functional of norm $1$.
On a unital C\*-algebra this means a linear functional that takes nonnegative values on [[positive elements|NA8DF5X3M4]] and the value $1$ at the unit.
States play the role of probability measures.
For a compact Hausdorff space $X$, the states on $C(X)$ are the integrals against regular Borel probability measures on $X$.
In quantum mechanics, the states of a system with observable algebra $A$ are the states on $A$.

## Explanations

### L1 human

A state assigns to each observable an expected value.
For a quantum system described by a unit vector $\xi$, the state assigns to an operator $T$ the number $\ip{T \xi}{\xi}$.
For a classical system on a space $X$, a state assigns to a function its average with respect to a probability distribution on $X$.
The states that cannot be written as mixtures of two other states are called pure states.
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# Join and meet

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]] and let $x, y \in P$.
The *join* of $x$ and $y$ is the [[supremum|0R3H5PANRQ]] of $\set{x, y}$ in $P$, when it exists.
So it is an element $j \in P$ with $x \le j$ and $y \le j$, such that $j \le z$ for every $z \in P$ with $x \le z$ and $y \le z$.
It is written $x \vee y$.
The *meet* of $x$ and $y$ is the [[infimum|0QF1MZ9FRG]] of $\set{x, y}$ in $P$, when it exists.
It is written $x \wedge y$.
More generally, the join $\bigvee S$ and the meet $\bigwedge S$ of a subset $S \subseteq P$ are its supremum and its infimum.

## Remarks

Example: in the power set of a set $X$, ordered by $\subseteq$, the join of $A$ and $B$ is $A \cup B$.
Indeed $A \cup B$ contains $A$ and $B$, and every set that contains $A$ and $B$ contains $A \cup B$.
Dually, the meet is $A \cap B$.
Non-example: in the poset $\set{a, b, c, d}$ with $a < c$, $a < d$, $b < c$, $b < d$ and no other strict relations, the elements $a$ and $b$ have no join.
They have two upper bounds, $c$ and $d$, and neither is below the other, so there is no least upper bound.
When they exist, joins and meets are unique, and $x \le y$ holds exactly when $x \vee y = y$, and exactly when $x \wedge y = x$.
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<document source="https://tamnd.github.io/math2/node/F7ZGDKRMB4.md" tag="F7ZGDKRMB4" kind="definition">
# Epsilon-net of a set system

## Definition

Let $X$ be a finite nonempty set, let $\mathcal{R}$ be a [[family of subsets|0C21Q694AG]] of $X$, called ranges, and let $0 < \varepsilon \le 1$.
A set $N \subseteq X$ is an *$\varepsilon$-net* of $(X, \mathcal{R})$ if every range $R \in \mathcal{R}$ with $\abs{R} \ge \varepsilon \abs{X}$ satisfies $R \cap N \ne \emptyset$.

## Remarks

Example: let $X = \set{1, 2, \dots, 100}$, let $\mathcal{R}$ be the sets of the form $\set{a, a+1, \dots, b}$, and let $\varepsilon = 1/10$.
The set $N = \set{10, 20, \dots, 100}$ is an $\varepsilon$-net.
Indeed, a range with at least $10$ elements consists of at least $10$ consecutive integers, so it contains a multiple of $10$.

Non-example: for the same $X$ and $\mathcal{R}$, the set $N' = \set{20, 40, \dots, 100}$ is not a $1/10$-net.
The range $\set{1, \dots, 19}$ has $19 \ge 10$ elements but misses $N'$.

The same notion is used for a probability measure $\mu$ on a set $X$: then $N$ is an $\varepsilon$-net if it meets every range $R$ with $\mu(R) \ge \varepsilon$.
The case above is $\mu$ equal to the uniform measure on a finite $X$.
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# Mathematical formulation of quantum mechanics

## Definition

In the mathematical formulation of quantum mechanics, a physical system is described by the following data and rules.

1. States: the system has a complex separable [[Hilbert space|HM2HHE3J51]] $H$. A pure state is a unit vector $\psi \in H$. The vectors $\psi$ and $e^{i\alpha}\psi$ with $\alpha \in \R$ give the same state.
2. Observables: each measurable quantity is a [[self-adjoint operator|0NEGN5CRSC]] $A$ on $H$, possibly [[unbounded|KRJX5RA0PW]], with [[projection-valued measure|ATGT25E5FF]] $P^A$ from the spectral theorem, so that $A = \int_\R \lambda \, dP^A(\lambda)$.
3. Measurement (Born rule): measuring $A$ in the state $\psi$ gives a result in a Borel set $E \subseteq \R$ with probability $\langle P^A(E)\psi, \psi \rangle$. So the possible results lie in the spectrum of $A$. For $\psi$ in the domain of $A$, the expected value is $\langle A\psi, \psi \rangle$.
4. Dynamics: there is a self-adjoint operator $\hat H$, the Hamiltonian. A state evolves by $\psi(t) = e^{-it\hat H/\hbar}\psi(0)$, where $\hbar$ is the reduced Planck constant. The operators $e^{-it\hat H/\hbar}$ form a strongly continuous one-parameter group of [[unitary operators|MM0EJ2GDE4]].
5. Composite systems: the Hilbert space of a system made of two parts with spaces $H_1$ and $H_2$ is the [[tensor product|EY0X2NKA3S]] $H_1 \otimes H_2$.

Mixed states are density operators: positive [[trace class operators|Z2CP530WDE]] $\rho$ with $\operatorname{tr}\rho = 1$; the probability in clause 3 is then $\operatorname{tr}(\rho P^A(E))$.

## Remarks

Example: a spin-$\tfrac12$ system has $H = \C^2$ and spin observable $\sigma_z = \operatorname{diag}(1, -1)$, whose spectral projections are $P(\set{1}) = \operatorname{diag}(1, 0)$ and $P(\set{-1}) = \operatorname{diag}(0, 1)$.
For $\psi = (1, 1)/\sqrt 2$, the two results each have probability $\tfrac12$, and the expected value is $\langle \sigma_z \psi, \psi \rangle = 0$.

Non-example: the matrix $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ is not self-adjoint, so clause 2 does not make it an observable.
It has no spectral decomposition into orthogonal projections: its only eigenvalue is $0$, but $N \ne 0$.

When $\psi$ is a unit vector, the pure state $\psi$ corresponds to the density operator $\rho = \langle \cdot, \psi \rangle \psi$, the projection onto $\C\psi$.
By Stone's theorem, every strongly continuous one-parameter unitary group has the form of clause 4 for a unique self-adjoint generator.
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<document source="https://tamnd.github.io/math2/node/F8RSNQBMBE.md" tag="F8RSNQBMBE" kind="concept">
# Fractional Laplacian

## Summary

The fractional Laplacian $(-\Delta)^s$, for $0 < s < 1$, is a power of the operator $-\Delta$ built from the [[Laplacian|0E0B6RA8K0]].
It acts on the [[Fourier transform|DPVM34ADZV]] side as multiplication by $\abs{\xi}^{2s}$.
Unlike the Laplacian, it is nonlocal: $(-\Delta)^s u(x)$ depends on the values of $u$ everywhere.
Its negative $-(-\Delta)^s$ is the generator of the rotationally symmetric $2s$-stable Lévy processes.
It appears in models of anomalous diffusion, in the thin obstacle problem and in [[fractional calculus|4V4AGE7XP4]].

## Explanations

### L1 human

The Laplacian compares the value of a function at a point with its average over tiny spheres around the point.
The fractional Laplacian compares it with values at all other points, with weights that decay like a power of the distance.
So a change far away still has a small effect here, which models particles that sometimes make long jumps.
For a wave $\cos(k x)$ on the line, $-\Delta$ multiplies it by $k^2$, and $(-\Delta)^s$ multiplies it by $\abs{k}^{2s}$.
For $s = \tfrac12$ one gets a "square root" of $-\Delta$, which on $\R^n$ is the [[Dirichlet-to-Neumann operator|W783KDGT8E]] of the upper half-space $\R^n \times (0, \infty)$.
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<document source="https://tamnd.github.io/math2/node/F8T3MJX3VJ.md" tag="F8T3MJX3VJ" kind="definition">
# Autoregressive–moving-average model

## Definition

Let $(\varepsilon_t)_{t \in \Z}$ be a white noise with variance $\sigma^2 > 0$.
Let $p, q \ge 0$, and let $\phi(z) = 1 - \phi_1 z - \cdots - \phi_p z^p$ and $\theta(z) = 1 + \theta_1 z + \cdots + \theta_q z^q$ be real polynomials with no common [[root|019MA1VTR2]].
An *ARMA($p$, $q$) process* is a weakly [[stationary process|DT0QXJE705]] $(X_t)_{t \in \Z}$ with mean $\mu$ such that
$$X_t - \mu - \sum_{i=1}^{p} \phi_i (X_{t-i} - \mu) = \varepsilon_t + \sum_{j=1}^{q} \theta_j \varepsilon_{t-j} \qquad \text{for all } t \in \Z.$$
With the backshift operator $B X_t = X_{t-1}$, this reads $\phi(B)(X_t - \mu) = \theta(B) \varepsilon_t$.

## Remarks

The case $q = 0$ is the [[autoregressive model|2JC96FMGA1]] and the case $p = 0$ is the [[moving-average model|PEB9DN0AHA]].
A stationary solution exists, and is unique, exactly when $\phi(z) \ne 0$ for $\abs{z} = 1$.
It is causal, a function of present and past shocks only, when $\phi$ has no root with $\abs{z} \le 1$, and invertible when $\theta$ has no root with $\abs{z} \le 1$.
The condition of no common roots excludes redundant parameters.

Example: $X_t = 0.5 X_{t-1} + \varepsilon_t + 0.3 \varepsilon_{t-1}$ has $\phi(z) = 1 - 0.5 z$ with root $2$ and $\theta(z) = 1 + 0.3 z$ with root $-10/3$.
Both roots lie outside the unit disk and they differ, so this is a causal and invertible ARMA($1$, $1$) process.

Non-example: $X_t - 0.5 X_{t-1} = \varepsilon_t - 0.5 \varepsilon_{t-1}$ fails the condition on common roots, since both polynomials equal $1 - 0.5 z$.
Its stationary solution is just $X_t = \varepsilon_t$, a white noise, so the two parameters are not identifiable.
</document>

<document source="https://tamnd.github.io/math2/node/F8XD8CDB0H.md" tag="F8XD8CDB0H" kind="statement">
# Borel–Carathéodory theorem

## Statement

Let $R > 0$ and let $f$ be [[holomorphic|0DG0Z72CQD]] on an open set containing the closed disc $\set{z : \abs{z} \le R}$.
Let $A(R) = \max_{\abs{z} = R} \operatorname{Re} f(z)$.
Then for each $r$ with $0 < r < R$,
$$\max_{\abs{z} = r} \abs{f(z)} \le \frac{2r}{R - r} A(R) + \frac{R + r}{R - r} \abs{f(0)}.$$

## Remarks

Proof idea: first let $f(0) = 0$ and $f$ not constant, so that $A(R) > 0$ by the mean value property of $\operatorname{Re} f$.
Then $g = f / (2A(R) - f)$ maps the disc of radius $R$ into the unit disc with $g(0) = 0$.
The [[Schwarz lemma|0MB48XDBY3]] gives $\abs{g(z)} \le \abs{z}/R$, and solving for $f$ gives $\abs{f(z)} \le \frac{2r A(R)}{R - r}$ on $\abs{z} = r$.
In general, apply this to $f - f(0)$, whose real part is at most $A(R) + \abs{f(0)}$ on $\abs{z} = R$.
Example: a real constant $f = c \le 0$ gives equality.
The left side is $\abs{c}$, and the right side is $\frac{2rc + (R + r)\abs{c}}{R - r} = \abs{c}$.

## Explanations

### L1 human

Usually one needs a bound on $\abs{f}$ to control $\abs{f}$.
This theorem says that an upper bound on the real part alone is enough, on a slightly smaller disc.
The price is a factor that blows up as the smaller radius approaches the larger one.

### L2 human

The theorem is a standard tool for entire functions and in analytic number theory.
A typical use is to bound $\log f$ when $f$ has no zeros in a disc.
There $\operatorname{Re} \log f = \log \abs{f}$ is easy to bound from above.
The theorem then bounds $\abs{\log f}$ and, through Cauchy's estimates, its derivatives.
This is a step in the proof of the [[Hadamard factorization theorem|KYSR7NHNFQ]] and in zero-free region arguments for the Riemann zeta function.
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<document source="https://tamnd.github.io/math2/node/F92JJR2NZE.md" tag="F92JJR2NZE" kind="definition">
# Hierarchical matrix

## Definition

Let $I = \set{1, \dots, n}$ be an index set.
A *cluster tree* on $I$ is a rooted [[tree|062D9KXJB1]] whose nodes are nonempty subsets of $I$.
Its root is $I$, and the children of each node partition that node.
Let $\mathcal{P}$ be a partition of $I \times I$ into blocks $\tau \times \sigma$ with $\tau$ and $\sigma$ nodes of the cluster tree.
Let $\mathcal{P} = \mathcal{P}^{+} \cup \mathcal{P}^{-}$ be a split into *admissible* blocks $\mathcal{P}^{+}$ and the remaining blocks $\mathcal{P}^{-}$.
Let $k \ge 0$ be an integer.
Let $M$ be a real [[matrix|0VD9ZSWWJG]] with rows and columns indexed by $I$.
It is a *hierarchical matrix* ($\mathcal{H}$-matrix) of local rank $k$ for $\mathcal{P}$ if each admissible block has small rank.
That is, $M|_{\tau \times \sigma}$ has [[rank|0ZC9YTXZRY]] at most $k$ for every $\tau \times \sigma \in \mathcal{P}^{+}$.

## Remarks

Each admissible block is stored as a product $X Y^T$ with $X$ of size $\abs{\tau} \times k$ and $Y$ of size $\abs{\sigma} \times k$.
The blocks in $\mathcal{P}^{-}$ are stored in full; the partition is built so that they are small.
In applications each index $i$ carries a point $x_i \in \R^d$, and $\Omega_\tau$ is the set of points of a cluster $\tau$.
The standard admissibility condition is $\min(\operatorname{diam} \Omega_\tau, \operatorname{diam} \Omega_\sigma) \le \eta \operatorname{dist}(\Omega_\tau, \Omega_\sigma)$ for a fixed $\eta > 0$.
With this condition, the matrices of asymptotically smooth kernels such as $1 / \abs{x - y}$ are well approximated by $\mathcal{H}$-matrices of small rank.
Example: let $n = 2m$, split $I$ into halves $\tau_1$ and $\tau_2$, and call the two off-diagonal blocks admissible.
A block diagonal matrix has zero off-diagonal blocks, of rank $0$, so it is an $\mathcal{H}$-matrix of every local rank $k$.
Non-example: the exchange matrix, with ones on the antidiagonal, has off-diagonal blocks of rank $m$.
So it is not an $\mathcal{H}$-matrix of local rank $k < m$ for this partition.
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<document source="https://tamnd.github.io/math2/node/F9410EB0N1.md" tag="F9410EB0N1" kind="problem">
# Problem of Apollonius

## Problem

Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ be three [[circles|060W5CBZP6]] in the Euclidean plane.
Construct, with [[straightedge and compass|0DAECMZVRT]], every circle that is tangent to all three of them.

## Remarks

The problem is solved: straightedge and compass constructions of all solutions are known.
Apollonius posed it in his work *Tangencies*, which is lost, so his own solution is not known in detail.
François Viète gave a straightedge and compass solution.
A solution circle may touch each $\Gamma_i$ from outside or from inside, which gives $2^3 = 8$ cases.
So, outside special positions with infinitely many solutions, there are at most eight solution circles, and some positions give fewer or none.
Inversion in a circle and the algebra of three quadratic equations both lead to solutions.
Variants replace some of the circles by points or by lines.

## Explanations

### L1 human

You are given three circles drawn on a sheet of paper.
The task is to draw a fourth circle that just touches each of the three.
There are usually several answers, up to eight, depending on which of the given circles the new circle wraps around.
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<document source="https://tamnd.github.io/math2/node/F961Q88FHK.md" tag="F961Q88FHK" kind="concept">
# Darboux integral

## Summary

The Darboux integral defines the integral of a bounded function on a closed interval by squeezing it between [[upper and lower Darboux sums|0QRQVH0CRW]].
A function is Darboux integrable when its [[upper and lower Darboux integrals|09FQT057T7]] agree, and the integral is the common value.
This gives exactly the same integrable functions and the same values as the [[Riemann integral|0QM4TXT3DR]], by [[a classical theorem|021XJAQVDC]].
It is often the most convenient way to prove that a given function is integrable.

## Explanations

### L1 human

Cut the interval into pieces and draw on each piece a rectangle as high as the largest value of the function there, and another as high as the smallest value.
The tall rectangles cover the area under the graph from above, and the short ones fill it from below.
If refining the cuts can bring the two totals as close as you like, the area is the number caught between them.
For $f(x) = x$ on $[0, 1]$ with $n$ equal pieces, the two totals are $\frac{n+1}{2n}$ and $\frac{n-1}{2n}$, and both tend to $\frac12$.
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<document source="https://tamnd.github.io/math2/node/F99RM1T02C.md" tag="F99RM1T02C" kind="statement">
# Finiteness of the class number

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]].
Then the [[ideal class group|WQ46BD3ARY]] $\operatorname{Cl}(K)$ is finite.

## Remarks

The standard proof uses the [[Minkowski bound|1TQ2F9CFVE]].
Let $n = [K : \Q]$, let $2r_2$ be the number of non-real embeddings of $K$ into $\C$, and let $d_K$ be the [[discriminant|E0VZTNPJXA]] of $K$.
By [[Minkowski's theorem|0K387S2PKQ]] on lattice points, every ideal class contains an integral ideal $\mathfrak{a}$ with [[norm|M54ADM0CVF]]
$$N(\mathfrak{a}) \le \frac{n!}{n^n} \left(\frac{4}{\pi}\right)^{r_2} \sqrt{\abs{d_K}}.$$
There are only finitely many ideals of $\mathcal{O}_K$ of bounded norm, so there are finitely many classes.
Example: for $K = \Q(\sqrt{-5})$, $n = 2$, $r_2 = 1$ and $d_K = -20$, so the bound is $\tfrac12 \cdot \tfrac4\pi \cdot \sqrt{20} \approx 2.85$; the ideals of norm at most $2$ are $\mathcal{O}_K$ and $(2, 1 + \sqrt{-5})$, which gives $h_K \le 2$.
The hypothesis that $K$ is a number field is needed: for a general [[Dedekind domain|7NPY2CN5KB]] the class group can be infinite.

## Explanations

### L1 human

In a number ring, unique factorisation can fail, and the class group measures by how much.
This theorem says that the failure is always finite: there are only finitely many kinds of ideals up to multiplication by numbers.
It turns the question of unique factorisation in a given field into a finite computation.

### L2 human

The theorem is one of the two basic finiteness results of algebraic number theory, the other being [[Dirichlet's unit theorem|1D99CSKX6N]].
Both come from the geometry of numbers: $\mathcal{O}_K$ embeds as a lattice in $\R^n$, and Minkowski's convex body theorem produces small elements.
The explicit bound makes class groups computable; for example it shows at once that $\Q(i)$, $\Q(\sqrt{-2})$ and $\Q(\sqrt{-3})$ have class number $1$.
The size of $h_K$ is described by the [[class number formula|55XX5A7XZV]], which links it to the residue of the [[Dedekind zeta function|KXS5DQDFYD]] at $s = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/F9FTASTFSX.md" tag="F9FTASTFSX" kind="definition">
# Homotopy continuation for polynomial systems

## Definition

Let $f = (f_1, \dots, f_n)$ be a system of [[polynomials|07GNX8GCYP]] in $n$ variables with [[complex|09FTAET5Z4]] coefficients, and let $d_i \ge 1$ be the degree of $f_i$.
Let $g = (g_1, \dots, g_n)$ be a *start system* with known solutions.
An example is the total-degree start system $g_i(x) = x_i^{d_i} - 1$, whose $d_1 d_2 \cdots d_n$ solutions are the tuples of roots of unity.
Fix a complex number $\gamma$ and define the [[homotopy|9YA6S6HD9V]]
$$H(x, t) = (1 - t) \, \gamma \, g(x) + t \, f(x), \qquad x \in \C^n, \ t \in [0, 1].$$
*Homotopy continuation* for $f$ is the following procedure.
For each solution $x_0$ of $g$, it follows the path $x(t)$ with $H(x(t), t) = 0$ and $x(0) = x_0$ by [[numerical continuation|5V3GJ84CPQ]], for $t$ from $0$ towards $1$.
It returns the limits of $x(t)$ as $t \to 1$.

## Remarks

With the total-degree start system and a generic choice of $\gamma$, the paths are smooth and do not meet for $t \in [0, 1)$.
Every isolated solution of $f$ is then the limit of at least one path, and the number of paths is the Bézout number $d_1 \cdots d_n$, the bound of [[Bézout's theorem|KWNVF194D8]].
Choosing $\gamma$ at random is called the gamma trick; with probability one it avoids the values of $\gamma$ for which paths collide or blow up before $t = 1$.
Example: for $f(x) = x^2 - 4$ in one variable, $g(x) = x^2 - 1$ and $\gamma = 1$, one gets $H(x, t) = x^2 - 1 - 3t$, and the two paths $x(t) = \pm\sqrt{1 + 3t}$ start at $\pm 1$ and end at $\pm 2$.
Non-example: with $\gamma = -1$ and the same $f$ and $g$, one gets $H(x, t) = (2t - 1) x^2 + 1 - 5t$, whose leading coefficient vanishes at $t = 1/2$, so both paths go to infinity there and the procedure fails.
Some paths may diverge or end at singular or non-isolated solutions; practical codes detect and handle these endgames.

## Explanations

### L1 human

Start with equations whose solutions you know, and blend them gradually into the equations you want to solve.
Each known solution moves continuously during the blend, and you track it with small steps.
Where the tracks end, you read off the solutions you were looking for.
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# Subdirect product

## Definition

Let $(\mathbf{A}_i)_{i \in I}$ be a family of [[algebras|QJGPT2W0FT]] of the same signature, and let $\pi_j$ denote the projection of the [[direct product|WJ73XYBNE4]] $\prod_{i \in I} \mathbf{A}_i$ onto $\mathbf{A}_j$.
A *subdirect product* of the family is a [[subalgebra|7CHG6SD365]] $\mathbf{B}$ of $\prod_{i \in I} \mathbf{A}_i$ such that $\pi_j$ restricted to $B$ is [[surjective|01RVCMSN5N]] onto $A_j$ for every $j \in I$.
An embedding $\alpha \colon \mathbf{A} \to \prod_{i \in I} \mathbf{A}_i$ is *subdirect* when $\alpha(\mathbf{A})$ is a subdirect product.

## Remarks

Let $(\theta_i)_{i \in I}$ be a family of congruences on an algebra $\mathbf{A}$.
The map $a \mapsto (a/\theta_i)_{i \in I}$ is a subdirect embedding of $\mathbf{A}$ into $\prod_{i \in I} \mathbf{A}/\theta_i$ if and only if $\bigcap_{i \in I} \theta_i = \Delta_A$.

Example: the map $n \mapsto (n \bmod p)_p$ is a subdirect embedding of the ring $\Z$ into $\prod_{p \text{ prime}} \Z/p\Z$.
It is a homomorphism, it is injective because a nonzero integer has only finitely many prime divisors, and each coordinate map $\Z \to \Z/p\Z$ is onto.

Non-example: the subgroup $\set{(x, 0) : x \in \Z/2\Z}$ of $\Z/2\Z \times \Z/2\Z$ is not a subdirect product.
Its projection onto the second factor is $\set{0}$, not all of $\Z/2\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/F9Q17JRV5K.md" tag="F9Q17JRV5K" kind="concept">
# Shearlet

## Summary

Shearlets are a multiscale system of functions on $\R^2$ obtained from one generating function by parabolic dilations, shears and translations.
Like [[curvelets|VEVPDJ8S7B]], they resolve directions and represent images with smooth edges sparsely.
Unlike curvelets, they change direction by shearing instead of rotating, and shears with integer parameters preserve the integer lattice.
This makes the passage from the continuous theory to digital algorithms on pixel grids direct.

## Explanations

### L1 human

A shear tilts a picture sideways, as when a deck of cards is pushed so that it leans.
A shearlet is a small thin wave packet that is stretched more in one direction than the other and then tilted by a shear.
Different shears point the packet in different directions, and different scales make it finer.
Since shears with integer amounts move grid points to grid points, the system fits naturally on a grid of pixels.
</document>

<document source="https://tamnd.github.io/math2/node/F9RQZ6EPS7.md" tag="F9RQZ6EPS7" kind="definition">
# Yoneda embedding

## Definition

Let $\mathcal{C}$ be a locally small [[category|A2DE66937D]], and let $[\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]$ be the [[functor category|1X0WKFQX8K]] of [[presheaves|AND2WHZC8P]] on $\mathcal{C}$, with [[natural transformations|KYCPCC57FT]] as morphisms.
The *Yoneda embedding* is the [[functor|28NQ0KRHD3]] $y \colon \mathcal{C} \to [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]$ given by:

1. on objects, $y(A) = \Hom(-, A)$, the contravariant [[hom functor|VJXS8AK67E]] of $A$;
2. on morphisms, for $f \colon A \to B$, $y(f) \colon \Hom(-, A) \to \Hom(-, B)$ is the natural transformation with components $y(f)_X(g) = f \circ g$ for $g \colon X \to A$.

## Remarks

The components $y(f)_X$ are natural in $X$ because $f \circ (g \circ h) = (f \circ g) \circ h$.
By the [[Yoneda lemma|D6DF1ARPVD]] with $F = \Hom(-, B)$, the map $f \mapsto y(f)$ is a bijection $\Hom(A, B) \to \mathrm{Nat}(y(A), y(B))$, so $y$ is [[full and faithful|GRG8WEBSH5]].
Since $y$ is fully faithful, $y(A) \cong y(B)$ implies $A \cong B$.
The dual functor $A \mapsto \Hom(A, -)$ embeds $\mathcal{C}^{\mathrm{op}}$ into $[\mathcal{C}, \mathbf{Set}]$.

Example: let $P$ be a preorder seen as a category, with one morphism $a \to b$ when $a \le b$.
Then $y(a)$ is the presheaf with $y(a)(x)$ a one-point set when $x \le a$ and empty otherwise, so $y$ sends $a$ to the down-set $\set{x : x \le a}$.
The map $a \mapsto \set{x : x \le a}$ is the classical embedding of a preorder into its down-sets ordered by inclusion.

Non-example: the assignment $A \mapsto \Hom(A, -)$ is not a functor $\mathcal{C} \to [\mathcal{C}, \mathbf{Set}]$; a morphism $f \colon A \to B$ induces $\Hom(B, -) \to \Hom(A, -)$, which goes the wrong way, so clause 2 fails for this assignment.
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<document source="https://tamnd.github.io/math2/node/F9ZCCKATES.md" tag="F9ZCCKATES" kind="definition">
# Arithmetic dynamics

## Definition

Let $K$ be a field, let $N \ge 1$, and let $\phi \colon \mathbb{P}^N \to \mathbb{P}^N$ be a morphism of [[projective space|ZT9A6JK8QF]] defined over $K$.
For $N = 1$ such a morphism is given by a [[rational function|0XDK9CGYKJ]] $\phi \in K(z)$.
Write $\phi^n$ for the $n$-fold composition of $\phi$ with itself and $\phi^0 = \id$.
For a point $P \in \mathbb{P}^N(K)$:

1. the *forward orbit* of $P$ is $\mathcal{O}_\phi(P) = \set{\phi^n(P) : n \ge 0}$;
2. $P$ is *periodic* if $\phi^n(P) = P$ for some $n \ge 1$;
3. $P$ is *preperiodic* if $\phi^m(P)$ is periodic for some $m \ge 0$, equivalently if $\mathcal{O}_\phi(P)$ is finite.

*Arithmetic dynamics* is the study of these orbits when $K$ is a field of arithmetic interest.
Examples of such fields are an [[algebraic number field|MWBDFS2KJ5]], a $p$-adic field or a finite field.

## Remarks

The set of preperiodic points of $\phi$ in $\mathbb{P}^N(K)$ is written $\operatorname{PrePer}(\phi, K)$.

Example: let $K = \Q$ and $\phi(z) = z^2 - 1$.
Then $0 \mapsto -1 \mapsto 0$, so $0$ and $-1$ are [[periodic|T75CZ5H47N]] of period $2$.
Also $1 \mapsto 0$, so $1$ is preperiodic but not periodic, since $\phi^n(1) \in \set{0, -1}$ for all $n \ge 1$.

Non-example: for the same $\phi$, the point $2$ is not preperiodic.
Its orbit is $2, 3, 8, 63, \ldots$, and $\phi(z) > z \ge 2$ for $z \ge 2$, so the orbit is strictly increasing and infinite.

When $K$ is a number field and $\phi$ has degree at least $2$, Northcott's theorem on [[heights|F1FF505PWH]] implies that $\operatorname{PrePer}(\phi, K)$ is finite.
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# Weak mixing

## Definition

Let $(X, \mathcal{B}, \mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \colon X \to X$ be a [[measure-preserving transformation|556WBDAHNP]].
The map $T$ is *weakly mixing* if for all $A, B \in \mathcal{B}$,
$$\lim_{N \to \infty} \frac{1}{N} \sum_{n=0}^{N-1} \abs{\mu(T^{-n} A \cap B) - \mu(A) \, \mu(B)} = 0.$$

## Remarks

Example: every [[mixing|S3RJQ9CX2P]] map is weakly mixing, since the averages of a sequence that tends to $0$ also tend to $0$.
So the doubling map $x \mapsto 2x \bmod 1$ with Lebesgue measure is weakly mixing.

Non-example: let $T(x) = x + \alpha \bmod 1$ with $\alpha$ irrational, and let $f(x) = e^{2\pi i x}$.
Then $f \circ T = e^{2\pi i \alpha} f$, so $f$ is a nonconstant eigenfunction of the [[Koopman operator|37TRG5DBQC]].
By form 2 below, $T$ is not weakly mixing, although it is ergodic.

Equivalent forms, for $T$ as above:

1. The product map $T \times T$ is ergodic for the product measure $\mu \times \mu$.
2. Every $f \in L^2(\mu)$ with $f \circ T = \lambda f$ for some $\lambda \in \C$ is constant almost everywhere.
3. For all $A, B \in \mathcal{B}$ there is a set $J \subseteq \N$ of density zero such that $\mu(T^{-n} A \cap B) \to \mu(A)\mu(B)$ as $n \to \infty$ with $n \notin J$.

There are weakly mixing maps that are not mixing; Chacon's transformation is a classical example.
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<document source="https://tamnd.github.io/math2/node/FA37X4AFFB.md" tag="FA37X4AFFB" kind="statement">
# Classification of Fatou components

## Statement

Let $f \colon \hat{\C} \to \hat{\C}$ be a [[rational function|0XDK9CGYKJ]] of degree at least $2$.
Let $U$ be a component of the [[Fatou set|C0N0XGC6GK]] of $f$ with $f^p(U) = U$ for some $p \ge 1$, and let $g = f^p$.
Then exactly one of the following holds.

1. Attracting case: $U$ contains a fixed point $z_0$ of $g$ with $\abs{g'(z_0)} < 1$, and $g^n \to z_0$ locally uniformly on $U$.
2. Parabolic case: $\partial U$ contains a fixed point $z_0$ of $g$ with $g'(z_0) = 1$, and $g^n \to z_0$ locally uniformly on $U$.
3. Siegel case: $U$ is a [[Siegel disc|KNDYA9T8ZK]].
4. Herman case: $U$ is a [[Herman ring|HZX6Q514X8]].

## Remarks

In the first two cases $z_0$ is a [[periodic point|T75CZ5H47N]] of $f$, and the superattracting case $g'(z_0) = 0$ is included in the first case.
In the last two cases $g$ restricted to $U$ is conformally conjugate to an irrational rotation of a disc or of a round annulus.
The proof uses the theory of normal families: limits of subsequences of $g^n$ on $U$ are either constant, giving the first two cases, or not, giving the last two.
For $f(z) = z^2$, the unit disc is an attracting component with $z_0 = 0$.
For $f(z) = z + z^2$, the fixed point $0$ has multiplier $1$ and lies on the boundary of a parabolic component.
By [[Sullivan's theorem|C9FDR4BGMG]] every Fatou component is eventually periodic, so these four types describe the long-run behaviour on the whole Fatou set.

## Explanations

### L1 human

A periodic piece of the Fatou set can behave in only four ways.
Its points can be sucked into a fixed point inside it, or creep towards a fixed point on its edge.
Or the piece can be a disc or a ring on which the map simply turns, like a wheel by an irrational angle.
Nothing else can happen.

### L2 human

The theorem is the starting point for describing the Fatou set of any rational map.
Attracting and parabolic cycles each attract at least one critical point.
So a map of degree $d$ has at most $2d - 2$ such cycles, the number of its critical points counted with multiplicity.
Siegel discs and Herman rings are rotation domains; their boundaries lie in the closure of the critical orbits.
Polynomials have no Herman rings, so for them only three cases occur.
</document>

<document source="https://tamnd.github.io/math2/node/FA5V5KRWDF.md" tag="FA5V5KRWDF" kind="statement">
# Compactness of the spectrum

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $A$ be a [[Banach algebra|9KYWN94QDN]] over $\mathbb{K}$ with unit $1$, so that $\norm{xy} \le \norm{x} \norm{y}$ for all $x, y \in A$.
Let $a \in A$.
Then the [[spectrum|EH0CZ19RKP]] $\sigma(a)$ is a [[compact subset|0R6EWFKRCF]] of $\mathbb{K}$, and $\abs{\lambda} \le \norm{a}$ for every $\lambda \in \sigma(a)$.

## Remarks

If $\abs{\lambda} > \norm{a}$, then $a - \lambda 1 = -\lambda (1 - a/\lambda)$, and the Neumann series $\sum_{n \ge 0} (a/\lambda)^n$ converges to an inverse of $1 - a/\lambda$.
So $\sigma(a)$ lies in the closed disc of radius $\norm{a}$.
If $u$ is invertible and $\norm{b} < 1/\norm{u^{-1}}$, then $u - b = u(1 - u^{-1} b)$ is invertible by the same series.
Hence the invertible elements form an open set, the [[resolvent set|X0243CZD9D]] is open, and $\sigma(a)$ is closed and bounded.

Completeness is necessary.
In the [[normed algebra|4DJ4V5VBSH]] of complex polynomials with the sup norm on $[0,1]$, only nonzero constants are invertible.
So the element $t$ has spectrum $\C$, which is not bounded.

For a complex Banach algebra the spectrum is also nonempty.
For instance, for $a \in M_n(\C)$ with the operator norm, the theorem says that every eigenvalue has modulus at most $\norm{a}$.

## Explanations

### L1 human

The numbers $\lambda$ for which $a - \lambda$ is not invertible form a closed and bounded set, and they are never larger than the size of $a$.
The reason is that $1 - x$ is invertible as soon as $x$ is small, by the geometric series $1 + x + x^2 + \cdots$.

### L2 human

This is the first basic fact of spectral theory in Banach algebras.
Together with the holomorphy of the resolvent and [[Liouville's theorem|0EHKVRF4J6]], it gives the [[nonemptiness of the spectrum|QWQC5SV0SD]] in the complex case.
It is refined by [[Gelfand's spectral radius formula|T4RFYT8HDN]], $\max \set{\abs{\lambda} : \lambda \in \sigma(a)} = \lim_n \norm{a^n}^{1/n}$, which can be smaller than $\norm{a}$.
The compactness allows the continuous functional calculus, which needs $\sigma(a)$ to be a compact space.
</document>

<document source="https://tamnd.github.io/math2/node/FA8QE1SGP4.md" tag="FA8QE1SGP4" kind="concept">
# Principal curvature

## Summary

The principal curvatures of a surface at a point are the largest and the smallest bending of the surface over all tangent directions.
They are the eigenvalues of the [[shape operator|DQCD11X1VM]], and the corresponding eigenvectors are the principal directions.
Their product is the Gaussian curvature and their average is the mean curvature.

## Explanations

### L1 human

Stand on a saddle-shaped surface and look in different directions.
In one direction the surface curves up the most, in another it curves down the most.
These two extreme amounts of bending are the principal curvatures.
When they differ, the two directions are at right angles.
On a cylinder of radius $r$, one principal curvature is $1/r$, around the cylinder, and the other is $0$, along it.
</document>

<document source="https://tamnd.github.io/math2/node/FABJSZ9D3S.md" tag="FABJSZ9D3S" kind="problem">
# Dido's problem

## Problem

Let $\ell > 0$.
Consider [[plane curves|EWS4WFTA7W]] of length $\ell$ that start and end on a fixed straight line and otherwise lie on one side of it.
Which of them encloses, together with the line, the largest area?

In coordinates, the line is the $x$-axis and the curve is a piecewise $C^1$ map $t \mapsto (x(t), y(t))$, $0 \le t \le 1$, with $y(0) = y(1) = 0$ and $y \ge 0$.
When the curve has no self-crossings, the enclosed area is $\abs{\int_0^1 y(t) \, x'(t) \, dt}$.

## Remarks

The answer is a semicircle of length $\ell$, with its diameter on the line; it encloses the area $\ell^2 / (2\pi)$.
Reflecting a curve in the line turns the problem into the classical isoperimetric problem for closed curves of length $2\ell$.
So the answer follows from the [[isoperimetric inequality|KBX8RBZ626]] in the plane.
Written with such a parametrised curve, it is an [[isoperimetric problem of the calculus of variations|6FY0KHMK5A]].
Its Euler–Lagrange equation with a multiplier says that the curve has constant curvature, so it is an arc of a circle.
The name comes from the legend of the founding of Carthage.
Dido was granted as much land as an ox hide could enclose; she cut the hide into a thin strip and laid it out against the coast.

## Explanations

### L1 human

You have a rope of fixed length and a straight coastline.
You want to fence off the largest piece of land, using the coast as one side.
The best shape is a half disc, with the rope as the round edge.
</document>

<document source="https://tamnd.github.io/math2/node/FAGMBTRN7Z.md" tag="FAGMBTRN7Z" kind="definition">
# Bidual and canonical embedding

## Definition

Let $X$ be a [[normed vector space|03J5RAKB30]] over $\mathbb{K} = \R$ or $\C$, and let $X^*$ be its [[continuous dual space|4A82S5FVRT]] with the operator norm.
The *bidual* of $X$ is the Banach space $X^{**} = (X^*)^*$.
The *canonical embedding* is the map $J \colon X \to X^{**}$ given by
$$(Jx)(f) = f(x) \quad \text{for } x \in X \text{ and } f \in X^*.$$

## Remarks

For each $x$, the map $Jx$ is linear on $X^*$ and $\abs{(Jx)(f)} \le \norm{f}\,\norm{x}$, so $Jx \in X^{**}$ with $\norm{Jx} \le \norm{x}$.
The [[Hahn–Banach theorem|QB5AE3VYZZ]] gives $f$ with $\norm{f} = 1$ and $f(x) = \norm{x}$, so in fact $J$ is a [[linear isometry|07SJXTWVQW]].
Example: for $X = \mathbb{K}^n$ with any norm, $J$ is an isometric isomorphism onto $X^{**}$, because $\dim X^{**} = \dim X$; compare [[the double dual in linear algebra|0ZJM9D7NPS]].
Example: for $X = c_0$ one has $X^* = \ell^1$ and $X^{**} = \ell^\infty$, and $J$ is the inclusion of $c_0$ in $\ell^\infty$, which is not onto.
Non-example: the algebraic double dual, the space of all linear forms on $X^*$, is in general larger than $X^{**}$, because it does not require continuity.
The image $J(X)$ is closed in $X^{**}$ exactly when $X$ is complete; in general the closure of $J(X)$ is a [[completion|2W7R5JHF0C]] of $X$.
</document>

<document source="https://tamnd.github.io/math2/node/FAHERE12SH.md" tag="FAHERE12SH" kind="definition">
# f-ring

## Definition

A *lattice-ordered ring* is a [[partially ordered ring|A0NP6HQNPX]] $(R, \le)$ whose order is a [[lattice|WH4SFAACAS]], with meet $\wedge$ and join $\vee$.
A lattice-ordered ring $R$ is an *f-ring* if for all $a, b, c \in R$,
$$a \wedge b = 0 \text{ and } c \ge 0 \implies ca \wedge b = 0 \text{ and } ac \wedge b = 0.$$

## Remarks

Example: the ring $C(X)$ of [[continuous real functions|0P27W9RW0G]] on a topological space $X$, with the pointwise order, is an f-ring.
If $\min(a, b) = 0$ pointwise and $c \ge 0$, then at each point where $b > 0$ one has $a = 0$, so $ca = 0$ there, and $\min(ca, b) = 0$.
Non-example: the ring of real $2 \times 2$ matrices with the entrywise order is lattice-ordered but is not an f-ring.
Let $E_{ij}$ be the matrix unit; then $E_{11} \wedge E_{21} = 0$ and $E_{21} \ge 0$, but $E_{21} E_{11} = E_{21}$ and $E_{21} \wedge E_{21} = E_{21} \ne 0$.
Every totally ordered ring is an f-ring, and in an f-ring every square is nonnegative.
</document>

<document source="https://tamnd.github.io/math2/node/FAMZ5Z5JK4.md" tag="FAMZ5Z5JK4" kind="definition">
# Besov space

## Definition

Let $n \ge 1$.
Let $\psi_0$ be a function in the [[Schwartz space|CG9JEATW3Q]] $\mathcal{S}(\R^n)$ with $\psi_0(\xi) = 1$ for $\abs{\xi} \le 1$ and $\psi_0(\xi) = 0$ for $\abs{\xi} \ge 2$.
For $j \ge 1$ let $\psi_j(\xi) = \psi_0(2^{-j} \xi) - \psi_0(2^{-j+1} \xi)$, so that $\sum_{j \ge 0} \psi_j = 1$ and $\psi_j$ is supported in $2^{j-1} \le \abs{\xi} \le 2^{j+1}$.
For a [[tempered distribution|AES2T5JPFP]] $f$, let $\Delta_j f = \mathcal{F}^{-1}(\psi_j \, \mathcal{F} f)$, where $\mathcal{F}$ is the [[Fourier transform|DPVM34ADZV]]; each $\Delta_j f$ is a smooth function.
Let $s \in \R$ and $0 < p, q \le \infty$.
The *Besov space* $B^s_{p,q}(\R^n)$ is the set of tempered distributions $f$ with
$$\norm{f}_{B^s_{p,q}} = \Big( \sum_{j=0}^{\infty} 2^{jsq} \norm{\Delta_j f}_{L^p(\R^n)}^q \Big)^{1/q} < \infty,$$
with the usual change to $\sup_{j \ge 0} 2^{js} \norm{\Delta_j f}_{L^p}$ when $q = \infty$.

## Remarks

A different choice of $\psi_0$ gives the same space with an equivalent quasi-norm.
For $p, q \ge 1$ the space is a Banach space; otherwise it is a complete quasi-normed space.
Here $L^p(\R^n)$ is the [[Lp space|SASP14MA1Z]], with the same formula when $0 < p < 1$.

Example: $B^s_{2,2}(\R^n)$ is the space $H^s(\R^n)$ of tempered distributions with $(1 + \abs{\xi}^2)^{s/2} \hat f \in L^2$.
Indeed, by Plancherel, $\norm{\Delta_j f}_{L^2}^2 = \int \psi_j^2 \abs{\hat f}^2 \, d\xi$ up to a constant factor, and $2^{2js}$ is comparable with $(1 + \abs{\xi}^2)^s$ on the support of $\psi_j$.

Non-example: the Dirac delta $\delta$ on $\R^n$ is not in $B^0_{1,q}(\R^n)$ for $q < \infty$.
Here $\Delta_j \delta = \mathcal{F}^{-1} \psi_j$, and for $j \ge 1$ this function is $x \mapsto 2^{(j-1)n} (\mathcal{F}^{-1} \psi_1)(2^{j-1} x)$.
So $\norm{\Delta_j \delta}_{L^1}$ is the same nonzero number for all $j \ge 1$, and the sum defining the norm diverges.
The same computation shows that $\delta \in B^0_{1,\infty}(\R^n)$.

For $s > 0$ not an integer, $B^s_{\infty,\infty}(\R^n)$ is the Hölder space $C^{k,\alpha}(\R^n)$ with $k = \floor{s}$ and $\alpha = s - k$.
</document>

<document source="https://tamnd.github.io/math2/node/FANW5G7YZC.md" tag="FANW5G7YZC" kind="definition">
# Square-integrable kernel

## Definition

Let $a < b$.
A *square-integrable kernel* on $[a, b]$ is a [[kernel|GSMBXT7FPK]] $K \colon [a, b] \times [a, b] \to \C$ that is Lebesgue [[measurable|YYNCJSR0PJ]] and satisfies
$$\norm{K}_{2} = \left( \int_a^b \int_a^b \abs{K(x, s)}^2 \, ds \, dx \right)^{1/2} < \infty,$$
where the integral is the [[Lebesgue integral|DWJF7ABPR1]].

## Remarks

Example: $K(x, s) = \abs{x - s}^{-1/4}$ for $x \ne s$ on $[0, 1]$.
Here $\abs{K}^2 = \abs{x - s}^{-1/2}$, and $\int_0^1 \abs{x - s}^{-1/2} \, ds \le 2 \int_0^{1} t^{-1/2} \, dt = 4$ for each $x$, so $\norm{K}_2^2 \le 4$.

Non-example: $K(x, s) = \abs{x - s}^{-1/2}$ on $[0, 1]$.
For $0 < x < 1$, $\int_0^1 \abs{x - s}^{-1} \, ds = \infty$, so the double integral of $\abs{K}^2$ is infinite.

In other words, $K$ lies in the [[Lp space|SASP14MA1Z]] $L^2([a, b] \times [a, b])$.
Every continuous kernel on $[a, b]$ is square-integrable.
For $\varphi \in L^2(a, b)$, the function $(K \varphi)(x) = \int_a^b K(x, s) \varphi(s) \, ds$ lies in $L^2(a, b)$ and $\norm{K \varphi}_2 \le \norm{K}_2 \norm{\varphi}_2$, by the Cauchy–Schwarz inequality.
The same definition is used on a measurable subset of $\R^m$.
</document>

<document source="https://tamnd.github.io/math2/node/FAQ8ZJ4M0M.md" tag="FAQ8ZJ4M0M" kind="definition">
# Stone algebra

## Definition

A *Stone algebra* is a [[bounded lattice|72V1Z82M5F]] $(L, \vee, \wedge, 0, 1)$ such that:

1. $L$ is a [[distributive lattice|0HM0HQCK6N]].
2. Every $a \in L$ has a [[pseudocomplement|WMR73J0S26]] $a^*$.
3. $a^* \vee a^{**} = 1$ for every $a \in L$.

## Remarks

Example: every bounded chain $C$ with $0 \ne 1$ is a Stone algebra.
Its pseudocomplement is $0^* = 1$ and $a^* = 0$ for $a \ne 0$.
If $a = 0$ then $a^* \vee a^{**} = 1 \vee 0 = 1$, and if $a \ne 0$ then $a^* \vee a^{**} = 0 \vee 1 = 1$.

Non-example: the lattice of open sets of $\R$, with $U^* = \operatorname{int}(\R \setminus U)$, satisfies clauses 1 and 2 but not 3.
For $U = (0, \infty)$, one has $U^* = (-\infty, 0)$ and $U^{**} = (0, \infty)$, so $U^* \cup U^{**} = \R \setminus \set{0} \ne \R$.

Every Boolean algebra is a Stone algebra, with $a^* = \neg a$ and $a^{**} = a$.
</document>

<document source="https://tamnd.github.io/math2/node/FAQ9CH2Z76.md" tag="FAQ9CH2Z76" kind="statement">
# Hamilton's principle

## Statement

Consider $N$ particles in $\R^3$ with masses $m_1, \dots, m_N > 0$ and positions $r = (r_1, \dots, r_N) \in \R^{3N}$.
Let $V \colon \R \times \R^{3N} \to \R$ be a potential of class $C^2$.
Let $T(\dot r) = \sum_{i=1}^N \frac12 m_i \abs{\dot r_i}^2$ be the kinetic energy and $L = T - V$ the Lagrangian.
Let $t_0 < t_1$ and let $r \colon [t_0, t_1] \to \R^{3N}$ be a $C^2$ path.
Consider Newton's equations
$$m_i \ddot r_i(t) = -\nabla_{r_i} V\big(t, r(t)\big), \qquad i = 1, \dots, N, \ t \in [t_0, t_1].$$
Let $S[r] = \int_{t_0}^{t_1} \big( T(\dot r) - V(t, r) \big) \, dt$ be the action.
Then $r$ satisfies Newton's equations if and only if $r$ is a stationary point of $S$ among $C^2$ paths with the same endpoints.

## Remarks

This is the case of mechanics of the [[stationary-action principle|Q104C67P26]].
For the proof, the [[first variation|5B8NFYB2XQ]] of $S$ in a direction $h$ with $h(t_0) = h(t_1) = 0$ is $\int_{t_0}^{t_1} \sum_i \left( m_i \dot r_i \cdot \dot h_i - \nabla_{r_i} V \cdot h_i \right) dt$.
An integration by parts turns it into $-\int_{t_0}^{t_1} \sum_i \left( m_i \ddot r_i + \nabla_{r_i} V \right) \cdot h_i \, dt$, which vanishes for all such $h$ exactly when Newton's equations hold, by the fundamental lemma of the calculus of variations.
So Newton's equations are the [[Euler–Lagrange equations|PS0VV149ET]] of $L = T - V$.
Example: a body falling near the Earth's surface, with $V(r) = m g z$, gives $\ddot z = -g$ and $\ddot x = \ddot y = 0$.
The endpoint times are fixed; Maupertuis's principle instead fixes the energy and lets the time vary.
For a system with ideal holonomic constraints, the same equivalence holds in generalized coordinates on the constraint manifold, and the constraint forces drop out.
William Rowan Hamilton formulated the principle in 1834 and 1835.

## Explanations

### L1 human

Hamilton's principle restates Newton's laws as an optimization rule.
Take the kinetic energy minus the potential energy, and add it up over the time of the motion.
The true motion is the path for which this total does not change, to first order, when the path is slightly bent with the endpoints held fixed.

### L2 human

The principle makes mechanics independent of coordinates: $T - V$ can be written in polar, spherical or any other coordinates, and the Euler–Lagrange equations in those coordinates are the equations of motion.
It is the starting point of Lagrangian and Hamiltonian mechanics, and through the Legendre transform it leads to Hamilton's canonical equations.
It also connects mechanics with the calculus of variations, so that tools such as the Jacobi condition decide when the action is in fact a minimum.
</document>

<document source="https://tamnd.github.io/math2/node/FAYWXJ68SB.md" tag="FAYWXJ68SB" kind="concept">
# Fourier analysis

## Summary

Fourier analysis is the study of how functions are represented by sums or integrals of sines, cosines and complex exponentials.
Its main tools are the [[Fourier series|G12KXDHPPS]] of periodic functions, the [[Fourier transform|DPVM34ADZV]] on $\R^n$ and the [[discrete Fourier transform|TST0GH66A5]] on $\Z / N\Z$.
It is the classical core of [[harmonic analysis|7QNRC9SJVV]].
It turns differentiation into multiplication and [[convolution|2Z29JN3VES]] into pointwise product, which makes it central for differential equations and signal processing.

## Explanations

### L1 human

Fourier analysis splits a signal into its frequencies, much as a prism splits white light into colours.
A periodic signal is a sum of waves whose frequencies are whole multiples of one base frequency.
A non-periodic signal needs all frequencies, and the sum becomes an integral.
For example, the function $\cos^2 x$ equals $\frac{1}{2} + \frac{1}{2} \cos 2x$, so it contains only the frequencies $0$ and $2$.
Joseph Fourier used such decompositions to solve the [[heat equation|VP2HN86VAG]].
</document>

<document source="https://tamnd.github.io/math2/node/FB45BHHA2J.md" tag="FB45BHHA2J" kind="concept">
# Spectrum of a ring

## Summary

The spectrum $\operatorname{Spec} R$ of a [[commutative ring|0310REJJY8]] $R$ is the set of its [[prime ideals|00MHHZ9SNE]], with the [[Zariski topology|8SQK7RTB63]].
It turns a ring into a geometric space, on which the elements of $R$ behave like functions.
A ring homomorphism $R \to A$ gives a continuous map $\operatorname{Spec} A \to \operatorname{Spec} R$ in the opposite direction.
The spectrum is the underlying space of an affine scheme, the basic object of modern algebraic geometry.

## Explanations

### L1 human

For the integers, the points of $\operatorname{Spec} \Z$ are the ideals $(2), (3), (5), \ldots$, one for each prime number, and one extra point $(0)$.
An integer $n$ "vanishes" at the point $(p)$ when $p$ divides $n$, so a nonzero integer $n$ behaves like a function with finitely many zeros.
For the polynomial ring $\C[x]$, the maximal ideals $(x - a)$ correspond to the complex numbers $a$, so the spectrum contains a copy of the complex line.
The [[definition|QJAXM2SF1Z]] gives the set and its topology.
</document>

<document source="https://tamnd.github.io/math2/node/FB692YN0BZ.md" tag="FB692YN0BZ" kind="statement">
# Bayes risk is at most minimax risk

## Statement

Let $\set{P_\theta : \theta \in \Theta}$ be a statistical model with a loss function, with [[risk function|N26TSVRH2S]] $R(\theta, \delta)$.
Let $\pi$ be a prior on $\Theta$ such that $\theta \mapsto R(\theta, \delta)$ is measurable for every rule $\delta$.
Then the [[Bayes risk|VEBDNCJJTN]] of $\pi$ is at most the [[minimax risk|8EG1M1WZ9C]]:
$$r(\pi) = \inf_\delta \int_\Theta R(\theta, \delta) \, \pi(d\theta) \le \inf_\delta \sup_{\theta \in \Theta} R(\theta, \delta).$$

## Remarks

For each fixed rule $\delta$, an average of $R(\cdot, \delta)$ is at most its supremum, so $\int R(\theta, \delta) \, \pi(d\theta) \le \sup_\theta R(\theta, \delta)$.
Taking the infimum over $\delta$ on both sides gives the claim.
Hence also $\sup_\pi r(\pi) \le R^*$, where $R^*$ is the minimax risk.

This inequality is the standard way to prove lower bounds for the minimax risk.
If a rule $\delta^*$ has $\sup_\theta R(\theta, \delta^*) = r(\pi)$ for some prior $\pi$, then $\delta^*$ is minimax.
For instance, a Bayes rule with constant risk is minimax, as for the binomial proportion with the beta prior of parameters $\sqrt{n}/2$ and $\sqrt{n}/2$.

## Explanations

### L1 human

The worst case is always at least as bad as any average.
So the best achievable average risk, for any choice of prior, is never more than the best achievable worst-case risk.

### L2 human

To show that no estimator can do better than some value in the worst case, it is enough to find a prior whose Bayes risk equals that value.
This is the Bayesian route to minimax lower bounds, often used with a sequence of priors whose Bayes risks increase to the candidate value.
For a finite parameter set with randomized rules allowed, minimax theorems show that the two sides are equal when the supremum over priors is taken.
</document>

<document source="https://tamnd.github.io/math2/node/FB6J5WAT5V.md" tag="FB6J5WAT5V" kind="statement">
# Cubic formula

## Statement

Let $a, b, c, d \in \C$ with $a \ne 0$.
Put
$$p = \frac{3ac - b^2}{3a^2}, \qquad q = \frac{2b^3 - 9abc + 27a^2 d}{27a^3}.$$
Then $x$ is a [[root|00EHX6FWC0]] of $ax^3 + bx^2 + cx + d$ if and only if $t = x + \frac{b}{3a}$ is a root of $t^3 + pt + q$.
If $p = q = 0$, the only root of $t^3 + pt + q$ is $t = 0$.
Otherwise, let $\delta$ be a square root of $\frac{q^2}{4} + \frac{p^3}{27}$, chosen so that $w = -\frac{q}{2} + \delta \ne 0$.
Let $u$ be a cube root of $w$ and let $\omega = e^{2\pi i/3}$.
Then the three roots of $t^3 + pt + q$, counted with multiplicity, are
$$t_k = \omega^k u - \frac{p}{3\omega^k u}, \qquad k = 0, 1, 2.$$

## Remarks

A choice of $\delta$ with $w \ne 0$ exists: if both choices gave $w = 0$, then $\delta = 0$ and $q = 0$, hence $p = 0$.
For the proof, put $v = -p/(3u)$, so that $uv = -p/3$ and $(u + v)^3 = u^3 + v^3 - p(u + v)$.
The numbers $u^3 = w$ and $v^3 = -p^3/(27w)$ are the two roots of $z^2 + qz - p^3/27$, so $u^3 + v^3 = -q$ and $u + v$ is a root.
The same holds for $\omega^k u$ in place of $u$, and expanding $\prod_k (t - t_k)$ gives $t^3 + pt + q$.
Example: for $t^3 - 6t - 9$, the number $\frac{q^2}{4} + \frac{p^3}{27} = \frac{81}{4} - 8 = \frac{49}{4}$; with $\delta = \frac{7}{2}$, $w = 8$, $u = 2$ and $t_0 = 2 + 1 = 3$.
When the cubic has real coefficients and three distinct real roots, $\delta$ is not real, and the formula passes through nonreal numbers.
By the [[casus irreducibilis|MJ864PBRRJ]], this detour cannot be avoided when the cubic is irreducible over $\Q$.

## Explanations

### L1 human

The cubic formula gives the roots of any equation of degree $3$ using arithmetic, square roots and cube roots.
First a shift of $x$ removes the $x^2$ term.
Then the root is split as a sum $u + v$ of two cube roots, and $u^3$ and $v^3$ solve a quadratic equation.

### L2 human

The formula shows that every cubic is [[solvable by radicals|J7YS0DA3HQ]], as [[Galois' theorem|VHCWCFA3MX]] predicts, since the Galois group lies in $S_3$, which is solvable.
The quantity under the square root is $-\frac{1}{108}$ times the discriminant of $t^3 + pt + q$, which is $-4p^3 - 27q^2$.
The same method of shifting and resolving into a lower-degree equation leads to the [[quartic formula|4CDFMKKZ01]].
</document>

<document source="https://tamnd.github.io/math2/node/FB79N7AHZA.md" tag="FB79N7AHZA" kind="example">
# Integers modulo n form a cyclic group

## Example

Let $n \ge 1$. The [[integers modulo $n$|03VFZ2WETT]], $\mathbb{Z}/n\mathbb{Z} = \{0, 1, \dots, n-1\}$, form a group under addition modulo $n$.
Each element $k$ is the sum $1 + 1 + \dots + 1$ of $k$ copies of $1$. Thus $1$ is a generator, and the group is [[cyclic|0B85Q41RP6]] of order $n$.
Addition modulo $n$ is commutative, so the group is abelian. This agrees with the result that [[cyclic groups are abelian|04RG02S6BA]].
</document>

<document source="https://tamnd.github.io/math2/node/FBERT0KQXN.md" tag="FBERT0KQXN" kind="statement">
# Whitney decomposition

## Statement

Let $n \ge 1$ and let $\Omega \subseteq \R^n$ be an open set whose complement $F = \R^n \setminus \Omega$ is nonempty.
For a set $E$ write $\operatorname{diam}(E)$ for its diameter and $\operatorname{dist}(E, F)$ for its distance to $F$.
Then there is a countable family $(Q_k)$ of closed cubes with sides parallel to the axes such that:

1. $\Omega = \bigcup_k Q_k$;
2. the interiors of the cubes $Q_k$ are pairwise disjoint;
3. for every $k$,
$$\operatorname{diam}(Q_k) \le \operatorname{dist}(Q_k, F) \le 4 \operatorname{diam}(Q_k).$$

## Remarks

The cubes can be chosen dyadic, that is, of the form $2^{-j}([0, 1]^n + m)$ with $j \in \Z$ and $m \in \Z^n$.
Proof idea: for each $j$, take the dyadic cubes of side $2^{-j}$ that meet the layer $\set{x \in \Omega : 2^{-j+1}\sqrt{n} < \operatorname{dist}(x, F) \le 2^{-j+2}\sqrt{n}}$.
These cubes satisfy clause 3 by the triangle inequality, and they cover $\Omega$.
Two dyadic cubes are either nested or have disjoint interiors, so keeping only the maximal cubes in the collection gives clause 2.

Clause 3 implies that neighbouring cubes have comparable sizes: if $Q_k$ and $Q_l$ touch, then $\operatorname{diam}(Q_l) \le \operatorname{dist}(Q_l, F) \le \operatorname{dist}(Q_k, F) + \operatorname{diam}(Q_k) \le 5 \operatorname{diam}(Q_k)$.
It follows that, for a fixed small enlargement factor such as $1 + \frac{1}{4}$, each point of $\Omega$ lies in at most $C(n)$ of the enlarged cubes.

Example: for $\Omega = (0, \infty) \subseteq \R$ and $F = (-\infty, 0]$, the intervals $[2^{j}, 2^{j+1}]$ for $j \in \Z$ cover $\Omega$ with disjoint interiors.
Their distance to $F$ equals their length, so clause 3 holds.

## Explanations

### L1 human

An open set can be tiled by cubes whose size is proportional to their distance from the boundary.
Near the boundary the cubes are tiny, far from it they are large, like a mosaic that gets finer and finer as it approaches the edge.
The proportionality makes the tiling well adapted to the shape of the set.

### L2 human

Whitney introduced this decomposition to extend smooth functions from a closed set $F$ to all of $\R^n$, in the Whitney extension theorem.
In harmonic analysis it is applied to the open set $\set{Mf > \lambda}$ in the Calderón–Zygmund decomposition, in the atomic decomposition of Hardy spaces, and in proofs of good-$\lambda$ inequalities.
A partition of unity subordinate to the slightly enlarged cubes has derivatives bounded by $C \operatorname{dist}(x, F)^{-k}$, which is what such arguments need.
</document>

<document source="https://tamnd.github.io/math2/node/FBFFS10WGS.md" tag="FBFFS10WGS" kind="definition">
# Chern–Weil homomorphism

## Definition

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $\mathbb{K}$ be $\R$ or $\C$.
For $k \ge 0$, let $I^k(G)$ be the space of symmetric $k$-linear maps $f : \mathfrak{g}^k \to \mathbb{K}$ that are invariant under the [[adjoint action|P6ZP0TE07M]]: $f(\mathrm{Ad}_g X_1, \dots, \mathrm{Ad}_g X_k) = f(X_1, \dots, X_k)$ for all $g \in G$.
Let $\pi : P \to M$ be a principal $G$-bundle with a [[connection form|NJ21PN5ZGJ]] $\omega$ and [[curvature form|ZAJFW1S078]] $\Omega$.
For $f \in I^k(G)$, the $2k$-form $f(\Omega, \dots, \Omega)$ on $P$ (formed with the wedge product) equals $\pi^* \alpha_f$ for a unique closed $2k$-form $\alpha_f$ on $M$.
The *Chern–Weil homomorphism* is the map
$$I^*(G) = \bigoplus_{k \ge 0} I^k(G) \to H^{\mathrm{even}}_{\mathrm{dR}}(M; \mathbb{K}), \quad f \mapsto [\alpha_f],$$
into [[de Rham cohomology|GX74C07T0G]], which does not depend on the choice of $\omega$.

## Remarks

The form $f(\Omega, \dots, \Omega)$ descends to $M$ because $\Omega$ vanishes on vertical vectors and $f$ is $\mathrm{Ad}$-invariant; it is closed by the Bianchi identity $d\Omega = [\Omega \wedge \omega]$ and invariance.
For two connections $\omega_0, \omega_1$, the forms $\alpha_f$ differ by the exterior derivative of a transgression form built from the path $\omega_t = (1-t)\omega_0 + t\omega_1$; so the class is independent of $\omega$.
With the product of symmetric multilinear forms, $I^*(G)$ is a commutative algebra and the map is an algebra homomorphism.
Example: for $G = \mathrm{U}(1)$, $\mathfrak{g} = i\R$ and $f(X) = \frac{i}{2\pi} X$, the class $[\alpha_f]$ is the first Chern class of the associated line bundle.
For $G = \mathrm{U}(n)$, $\det\big(I + \frac{i}{2\pi} \Omega\big)$ gives the total Chern class, and its images are the [[characteristic classes|7HACJC0JE2]] with real coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/FBGXAXAEJ1.md" tag="FBGXAXAEJ1" kind="definition">
# Complete category

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
$\mathcal{C}$ is *complete* if every [[diagram|HJK75HDWFE]] $F \colon J \to \mathcal{C}$ with $J$ a small category has a [[limit|CHS5AFJGWE]] in $\mathcal{C}$.
Here a category $J$ is small when its objects and its morphisms form sets.

## Remarks

Dually, $\mathcal{C}$ is *cocomplete* if every small diagram has a colimit.
A category is complete if and only if it has all small products and all equalizers.

Example: the category of sets is complete.
The limit of $F \colon J \to \mathbf{Set}$ is the set of families $(x_j)_{j \in J}$ with $x_j \in F(j)$ and $F(u)(x_j) = x_k$ for every morphism $u \colon j \to k$ of $J$.

Non-example: the category of finite sets is not complete.
Suppose $P$ were a product of countably many copies of $\set{0, 1}$ there.
Maps from a one-point set to $P$ would match the elements of $P$ with all sequences in $\set{0, 1}$, which form an uncountable set.

A small category that is complete is a preorder, by a theorem of Freyd.
So completeness is a property of large categories.
</document>

<document source="https://tamnd.github.io/math2/node/FBH6KG484Z.md" tag="FBH6KG484Z" kind="statement">
# Carleson's theorem

## Statement

Let $f \colon \R \to \C$ be a $2\pi$-periodic function that is square integrable on $[-\pi, \pi]$, that is, $f \in L^2(\mathbb{T})$ (see [[Lp space|SASP14MA1Z]]).
For $k \in \Z$ let $\hat f(k) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-ikt} \, dt$ be its [[Fourier coefficients|5NN2YZZR3V]].
For $N \ge 0$ let $S_N f(x) = \sum_{k=-N}^{N} \hat f(k) e^{ikx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]].
Then $S_N f(x) \to f(x)$ as $N \to \infty$ for [[almost every|RARTMA0ZEC]] $x \in \R$.

## Remarks

The proof reduces the theorem to an estimate for the maximal operator $Cf(x) = \sup_N \abs{S_N f(x)}$, namely that it is of weak type $(2, 2)$.
Convergence holds at every point for trigonometric polynomials, which are dense in $L^2(\mathbb{T})$, and the maximal estimate passes it to all of $L^2(\mathbb{T})$.
Carleson proved the maximal estimate in 1966 by a delicate decomposition of the function in time and frequency.
Later proofs were given by Fefferman and by Lacey and Thiele.

The theorem settles a conjecture of Lusin.
Since continuous periodic functions are square integrable, the Fourier series of a continuous function converges almost everywhere, although it can [[diverge at a point|FPDWWD12FE]].
The [[Kahane–Katznelson theorem|42DDDZHP7F]] shows that "almost everywhere" cannot be improved, even for continuous functions.
Square integrability cannot be weakened to integrability: [[Kolmogorov's example|25V2MHPHZ7]] is an integrable function whose Fourier series diverges everywhere.

## Explanations

### L1 human

The Fourier series of a periodic function is an infinite sum of waves meant to rebuild the function.
For a long time it was unknown whether this sum gives back the function at most points, even for continuous functions.
Carleson's theorem says yes for every square-integrable function: the sum converges to the right value except on a set of length zero.

### L2 human

Convergence in the $L^2$ norm follows from Parseval's identity and is easy.
Pointwise convergence is much harder because it needs control of all partial sums at once, through the Carleson maximal operator.
[[Hunt's extension|NAX51M6025]] proves the same for $L^p$ with $1 < p < \infty$.
The time-frequency methods behind the proof became a central tool of modern harmonic analysis.
</document>

<document source="https://tamnd.github.io/math2/node/FBJ4JCP5SR.md" tag="FBJ4JCP5SR" kind="definition">
# Godunov's scheme

## Definition

Let $u_t + F(u)_x = 0$ be a [[hyperbolic system of conservation laws|XN081NCPF5]] with $F \colon \R^m \to \R^m$ smooth.
Fix $\Delta x, \Delta t > 0$, put $\lambda = \Delta t / \Delta x$, and let the cells be $C_j = (x_{j-1/2}, x_{j+1/2})$ with $x_{j+1/2} = (j + \tfrac12) \Delta x$.
Let $\Lambda$ bound the absolute values of the eigenvalues of $DF$ on the relevant states, and assume $\Lambda \lambda \le 1/2$.
Given cell values $u_j^n$, *Godunov's scheme* computes $u_j^{n+1}$ as follows.

1. Let $\tilde u$ be the function equal to $u_j^n$ on $C_j$ for each $j$.
2. Let $v$ be the entropy solution on $t_n \le t \le t_{n+1}$ with $v(\cdot, t_n) = \tilde u$, made of the solutions of the [[Riemann problems|GJWKX22VVE]] at the points $x_{j+1/2}$.
3. Set $u_j^{n+1} = \frac{1}{\Delta x} \int_{C_j} v(x, t_{n+1})\, dx$.

## Remarks

Let $u^*(u_l, u_r)$ be the value at $x/t = 0$ of the Riemann solution with states $u_l, u_r$.
Integrating the equation over $C_j \times (t_n, t_{n+1})$ gives the conservation form
$$u_j^{n+1} = u_j^n - \lambda \left( F(u^*(u_j^n, u_{j+1}^n)) - F(u^*(u_{j-1}^n, u_j^n)) \right).$$
In this form the scheme only needs $\Lambda \lambda \le 1$.
For a scalar law the numerical flux is $\min_{u_l \le w \le u_r} F(w)$ if $u_l \le u_r$, and $\max_{u_r \le w \le u_l} F(w)$ if $u_l > u_r$.

Example: for $u_t + a u_x = 0$ with $a > 0$, the Riemann solution at $x/t = 0$ is $u_l$.
So the flux at $x_{j+1/2}$ is $a u_j^n$, and the scheme is the upwind scheme $u_j^{n+1} = u_j^n - a \lambda (u_j^n - u_{j-1}^n)$.

Non-example: the downwind scheme $u_j^{n+1} = u_j^n - a\lambda(u_{j+1}^n - u_j^n)$ uses the flux $a u_{j+1}^n$ instead.
This is not the Riemann solution value, and the scheme is unstable for every $\lambda > 0$ when $a > 0$.

## Explanations

### L1 human

The scheme thinks of the solution as constant in each cell.
It moves the solution forward exactly for one short step, then takes averages again.
Only the jumps at the cell boundaries matter, and each of them is a Riemann problem.
</document>

<document source="https://tamnd.github.io/math2/node/FBTWQEMTPS.md" tag="FBTWQEMTPS" kind="definition">
# Contact structure on a jet bundle

## Definition

Let $\pi \colon E \to M$ be a smooth fiber bundle, let $k \ge 1$, and let $J^k\pi$ be its $k$-th [[jet bundle|HC3VBBCJQP]].
A [[one-form|02Z5B2HA3F]] $\theta$ on an open set of $J^k\pi$ is a *contact form* if
$$(j^k s)^* \theta = 0$$
for every local section $s$ of $\pi$.
Here $j^k s$ is the [[prolongation|BMCCQ8JXNE]] of $s$, and the pullback is taken where it is defined.
The *contact distribution*, or Cartan distribution, of $J^k\pi$ is the subbundle $\mathcal{C} \subseteq T J^k\pi$ of vectors on which every contact form vanishes.
The contact forms, or equivalently the distribution $\mathcal{C}$, form the *contact structure* of $J^k\pi$.

## Remarks

In adapted coordinates $(x^i, u^\alpha, u^\alpha_I)$, the contact forms are the combinations with smooth coefficients of
$$\theta^\alpha_I = du^\alpha_I - \sum_{i} u^\alpha_{I + i}\, dx^i, \qquad 0 \le \abs{I} \le k - 1,$$
where $u^\alpha_\emptyset = u^\alpha$ and $I + i$ is the multi-index $I$ with $i$ added once.

A local section $\sigma$ of $J^k\pi \to M$ is holonomic, that is, of the form $j^k s$, exactly when $\sigma^*\theta = 0$ for every contact form $\theta$.
For $k = 1$, $M = \R$ and one fiber coordinate this is a direct check: if $\sigma(x) = (x, s(x), q(x))$ then $\sigma^*(du - q\, dx) = (s'(x) - q(x))\, dx$, which vanishes exactly when $q = s'$.

Example: for $\pi \colon \R \times \R \to \R$, $J^1\pi = \R^3$ with coordinates $(x, u, q)$ and $\theta = du - q\, dx$.
Then $d\theta = dx \wedge dq$ and $\theta \wedge d\theta = du \wedge dx \wedge dq \ne 0$.
So $\theta$ is a contact form in the sense of [[contact geometry|HBQ8BZ7XB5]], and $J^1\pi$ is a contact manifold.

Non-example: on the same space the [[form|0E3G0GRPZ6]] $du$ is not a contact form.
For $s(x) = x$ one gets $(j^1 s)^* du = dx \ne 0$.

For $k = 1$ and one fiber coordinate, $J^1\pi$ is a contact manifold for every $\dim M$, with contact form $du - \sum_i u_i\, dx^i$.
For $k > 1$ or more than one fiber coordinate, the contact distribution has codimension greater than $1$.
So it is not a contact structure in the sense of contact geometry, though the name is the same.
</document>

<document source="https://tamnd.github.io/math2/node/FBVEG8HVTP.md" tag="FBVEG8HVTP" kind="example">
# Seifert–Weber space

## Example

Take a solid regular dodecahedron.
Glue each face to the opposite face after a turn of $3/10$ of a full turn, that is, by the angle $3\pi/5$.
The result is the Seifert–Weber space $W$, a closed orientable [[3-manifold|R20TB2KEQQ]].

Under these gluings, the $30$ edges of the dodecahedron fall into $6$ classes of $5$ edges, and all $20$ vertices become one point.
The cell structure has $1$ vertex, $6$ edges, $6$ faces and $1$ cell.
So its Euler characteristic is $1 - 6 + 6 - 1 = 0$, as it must be for a closed $3$-manifold.
In hyperbolic space $\mathbb{H}^3$ there is a regular dodecahedron with dihedral angles $2\pi/5$.
With this dodecahedron the $5$ angles around each edge class add up to $2\pi$, and the gluing maps are isometries.
So $W$ is a closed [[hyperbolic 3-manifold|GHVW1Z35BS]].

## Remarks

Its first homology group is $(\Z/5)^3$.
Gluing opposite faces with a turn of $1/10$ instead gives the [[Poincaré homology sphere|NHERCK7C43]], and a turn of $1/2$ gives the [[real projective space|1DDYB6VVSR]] $\mathbb{RP}^3$.
Seifert and Weber described the space in 1933; it was one of the first known closed hyperbolic $3$-manifolds.

## Explanations

### L1 human

The Seifert–Weber space is made from a dodecahedron, the solid with twelve pentagonal faces, by gluing opposite faces with a twist.
In ordinary space the corners do not fit together, but in hyperbolic space a large enough dodecahedron has smaller angles, and one size makes them exactly right.
So a hyperbolic creature living inside would see a perfect tiling of its space by dodecahedra.
</document>

<document source="https://tamnd.github.io/math2/node/FC34XP2DQV.md" tag="FC34XP2DQV" kind="definition">
# Polynomial operation

## Definition

Let $A$ be an algebra and let $n \ge 1$.
An operation $p \colon A^n \to A$ is an $n$-ary *polynomial operation* of $A$ if the following holds.
There are $m \ge 0$, an $(n + m)$-ary [[term operation|N9M2N27END]] $t^A$ of $A$, and elements $c_1, \ldots, c_m \in A$ such that
$$p(a_1, \ldots, a_n) = t^A(a_1, \ldots, a_n, c_1, \ldots, c_m)$$
for all $a_1, \ldots, a_n \in A$.

## Remarks

Equivalently, the polynomial operations of $A$ are the term operations of the algebra $A_A$ obtained from $A$ by adding a constant symbol for each element of $A$.
Every term operation is a polynomial operation, with $m = 0$.
An equivalence relation on $A$ is a [[congruence|DCSXXT4SER]] exactly when it is preserved by every unary polynomial operation of $A$.
For a commutative ring $R$, the polynomial operations of $R$ are the [[polynomial functions|1G0B73WPX8]] $R^n \to R$ given by polynomials with coefficients in $R$.

Example: in the group $(\Z, +, -, 0)$, the map $p(a) = 2a + 5$ is a polynomial operation.
Take $t = (x_1 + x_1) + x_2$ and $c_1 = 5$; then $t^A(a, 5) = 2a + 5$.

Non-example: on $(\Z, +, -, 0)$ the map $a \mapsto a^2$ is not a polynomial operation.
By induction on terms, every unary polynomial operation of this group has the form $a \mapsto k a + c$ with $k, c \in \Z$.
The map $a \mapsto a^2$ is not of this form, since its values $0, 1, 4$ at $0, 1, 2$ are not in arithmetic progression.
</document>

<document source="https://tamnd.github.io/math2/node/FC78F2E87E.md" tag="FC78F2E87E" kind="statement">
# Gabriel's theorem

## Statement

Let $k$ be an algebraically closed field and let $Q$ be a finite connected [[quiver|RE9YFFS3TS]].
For $x \in \Z^{Q_0}$ let $q(x) = \sum_{i \in Q_0} x_i^2 - \sum_{a \in Q_1} x_{s(a)} x_{t(a)}$.

1. $Q$ has only finitely many isomorphism classes of finite-dimensional [[indecomposable|E2P29GCD9T]] [[representations|TXVWFWP6VQ]] over $k$ if and only if $Q$ is a [[Dynkin quiver|92RQG73SC5]].
2. If $Q$ is a Dynkin quiver, then the dimension vector gives a bijection from the isomorphism classes of indecomposable representations to the set $\Phi_+ = \set{x \in \N^{Q_0} : x \ne 0, \ q(x) = 1}$.
The set $\Phi_+$ is the set of positive roots of the root system of the underlying Dynkin diagram.

## Remarks

Bernstein, Gelfand and Ponomarev gave a proof with [[reflection functors|MG4RNMA8AR]].
A reflection functor at a sink or source changes the orientation at one vertex, and acts on dimension vectors by the simple reflection of the root system.
Applying them repeatedly brings every indecomposable representation to a simple one, whose dimension vector is a simple root.
Example: for $1 \to 2$, of type $A_2$, the positive roots are $(1, 0)$, $(0, 1)$ and $(1, 1)$, matching the indecomposables $k \to 0$, $0 \to k$ and $k \xrightarrow{\id} k$.
The Dynkin hypothesis is necessary: for the Kronecker quiver, with two arrows $a, b \colon 1 \to 2$, the representations $V_1 = V_2 = k$, $V_a = \id$, $V_b = \lambda \id$ are indecomposable and pairwise non-isomorphic for $\lambda \in k$.
The proof by reflection functors works over every field.

## Explanations

### L1 human

The theorem answers the question: for which quivers can all the representations be listed?
The answer is exactly the quivers whose shape is one of the Dynkin diagrams $A_n$, $D_n$, $E_6$, $E_7$, $E_8$.
For those quivers, the building blocks are in one-to-one correspondence with the positive roots of a root system, so they can be counted.
For type $A_n$ there are $n(n+1)/2$ building blocks, whatever the directions of the arrows.

### L2 human

The quadratic form $q$ is the Tits form of the quiver; it does not depend on the orientation, and it is positive definite exactly for the Dynkin graphs.
This links representation theory of quivers with the classification of simple Lie algebras, where the same diagrams appear.
[[Kac's theorem|59QJC3ZXA1]] extends the description of dimension vectors to all quivers without loops, through the roots of Kac–Moody algebras.
For path algebras of Dynkin quivers, the Auslander–Reiten quiver can be built from the projective modules by the Auslander–Reiten translation.
</document>

<document source="https://tamnd.github.io/math2/node/FC7VB4WSG0.md" tag="FC7VB4WSG0" kind="concept">
# Reflective subcategory

## Summary

A reflective subcategory is a [[full subcategory|V3BFGKBJF7]] whose inclusion functor has a left [[adjoint|TZHF6ZM92S]], called the reflector.
The reflector sends each object to its best approximation inside the subcategory.
Abelianization, [[sheafification|2HYK3KBQNA]] and the [[Stone–Čech compactification|D9MWP5C85W]] are reflectors.
Reflective subcategories inherit limits from the ambient category, and colimits through the reflector.

## Explanations

### L1 human

Abelian groups form a part of the world of all groups.
Every group $H$ has a best abelian approximation, the quotient $H/[H, H]$ by its commutator subgroup.
Every homomorphism from $H$ to an abelian group passes through this quotient in exactly one way.
A reflective subcategory is a part of a category where every object has such a best approximation.
</document>

<document source="https://tamnd.github.io/math2/node/FC8C0V4TFZ.md" tag="FC8C0V4TFZ" kind="concept">
# Accessible category

## Summary

An accessible category is a large category that is generated, under highly [[filtered colimits|975PMY4FG8]], by a small set of small objects.
Here small objects are the $\lambda$-presentable ones, those whose hom functor preserves $\lambda$-filtered colimits for a [[regular cardinal|13ZV2F8JAR]] $\lambda$.
Accessible categories are exactly the categories of models of sketches, and they include the categories of models of many first-order theories.
The cocomplete ones are the locally presentable categories.

## Explanations

### L1 human

Many large categories are determined by a small part of them.
Every set is the union of its finite subsets, and every vector space is the union of its finite-dimensional subspaces.
An accessible category is one where every object is glued in this way from a fixed small stock of small objects.
This lets one handle a huge category with set-sized data.
</document>

<document source="https://tamnd.github.io/math2/node/FC9NRV5GSP.md" tag="FC9NRV5GSP" kind="definition">
# Sampling distribution

## Definition

Let $\mathcal{P} = \set{P_\theta : \theta \in \Theta}$ be a [[statistical model|JZ6R7J36K5]] on a measurable space $(S, \mathcal{S})$, and let $X$ be a random element of $S$.
Let $T = g(X)$ be a [[statistic|036756NFD6]] with values in $\R^k$.
The *sampling distribution* of $T$ under $\theta$ is the [[probability distribution|0QB913WM5E]] of $T$ when $X$ has law $P_\theta$:
$$P_\theta(T \in B) = P_\theta\left(g^{-1}(B)\right)$$
for every Borel set $B \subseteq \R^k$.

## Remarks

Example: let $X_1, \dots, X_n$ be i.i.d. $\mathcal{N}(\mu, \sigma^2)$.
The sample mean $\bar{X}_n$ is a linear combination of independent normal variables, so its sampling distribution is $\mathcal{N}(\mu, \sigma^2/n)$.
Non-example: the histogram of one observed data set is not the sampling distribution of anything; it is a single realization, while a sampling distribution ranges over all possible data.
The standard deviation of the sampling distribution of an estimator is its standard error.
In general the sampling distribution depends on $\theta$; a statistic whose sampling distribution is the same for all $\theta$ is [[ancillary|VBW2A4H6TD]].
</document>

<document source="https://tamnd.github.io/math2/node/FCA0J1X07K.md" tag="FCA0J1X07K" kind="statement">
# Vitali covering lemma

## Statement

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]].
For a [[closed ball|05DSV40EB9]] $B = \bar B(x, r)$ with $r > 0$ and $c > 0$, let $cB = \bar B(x, cr)$.

1. Finite version: for closed balls $B_1, \dots, B_k$ of positive radius in $X$, there is a subfamily $B_{j_1}, \dots, B_{j_m}$ of pairwise disjoint balls with $B_1 \cup \dots \cup B_k \subseteq 3B_{j_1} \cup \dots \cup 3B_{j_m}$.
2. Infinite version: let $\mathcal{F}$ be a family of closed balls of positive radius in $X$ with bounded radii. There is a subfamily $\mathcal{G} \subseteq \mathcal{F}$ of pairwise disjoint balls with this property: each $B \in \mathcal{F}$ meets some $B' \in \mathcal{G}$ of radius at least half that of $B$. In particular $\bigcup_{B \in \mathcal{F}} B \subseteq \bigcup_{B' \in \mathcal{G}} 5B'$.

If $X$ is [[separable|0Z2P41SVDP]], for example $X = \R^n$, the subfamily $\mathcal{G}$ in part 2 is countable.

## Remarks

The finite version is greedy: pick the largest ball, discard every ball that meets it, and repeat.
A discarded ball meets a chosen ball of at least its radius, so it lies in the triple of that ball.
The infinite version sorts the balls by radius into dyadic classes $R/2^{k+1} < r \le R/2^k$ and chooses a maximal disjoint subfamily in each class in turn, using Zorn's lemma.
If $B$ meets $B'$ and the radius of $B$ is at most twice that of $B'$, then $B \subseteq 5B'$.

Example: in $\R^n$ the lemma gives $\lambda\big(\bigcup_i B_i\big) \le 3^n \sum_m \lambda(B_{j_m})$, which is the key step in the weak type estimate for the Hardy–Littlewood maximal function.

The bound on the radii is needed in part 2.
The balls $\bar B(0, k)$ for $k = 1, 2, \dots$ in $\R$ pairwise meet, so a disjoint subfamily has one ball, and no fixed enlargement of it covers $\R$.

## Explanations

### L1 human

Given a bunch of overlapping balls, one can keep a few of them that do not overlap at all.
After blowing the kept balls up by a fixed factor, they cover everything the original balls covered.
So the total size of a union of balls is controlled by the size of a disjoint subfamily.

### L2 human

The lemma turns overlapping covers into disjoint ones at the price of a constant factor.
It gives the weak $(1,1)$ bound for the maximal function and so the Lebesgue differentiation theorem.
The Vitali covering theorem built on it says that a fine cover of a set by balls contains a disjoint subfamily that covers almost all of the set.
</document>

<document source="https://tamnd.github.io/math2/node/FCAA02SN0E.md" tag="FCAA02SN0E" kind="statement">
# Nikodym convergence theorem

## Statement

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]] and let $(\mu_n)_{n \ge 1}$ be a sequence of [[complex measures|NY6TYHRGMS]] on $\Sigma$.
Assume that for every $E \in \Sigma$ the limit $\mu(E) = \lim_{n \to \infty} \mu_n(E)$ exists in $\C$.
Then:

1. The $\mu_n$ are uniformly countably additive: for every sequence $E_1 \supseteq E_2 \supseteq \cdots$ in $\Sigma$ with $\bigcap_k E_k = \emptyset$, one has $\sup_n \abs{\mu_n(E_k)} \to 0$ as $k \to \infty$.
2. The set function $\mu$ is [[countably additive|D7KB9HFCGZ]], so it is a complex measure on $\Sigma$.

## Remarks

One proof reduces to the Vitali–Hahn–Saks theorem.
Put $\lambda = \sum_n 2^{-n} \abs{\mu_n} / (1 + \abs{\mu_n}(X))$, a finite measure.
Each $\mu_n$ is absolutely continuous with respect to $\lambda$, and the Vitali–Hahn–Saks theorem gives the conclusion.

Example: let $f_n$ be integrable functions on a measure space with $\int_E f_n$ convergent for every measurable $E$.
Then the limit $E \mapsto \lim_n \int_E f_n$ is again countably additive.

## Explanations

### L1 human

If a sequence of measures has a limit on each measurable set, the limit is a measure too.
This is surprising, because pointwise limits usually lose properties such as countable additivity.

### L2 human

The theorem is often stated together with the Nikodym boundedness theorem, which says that a family of complex measures bounded on each set is uniformly bounded.
Both are measure-theoretic forms of the uniform boundedness principle.
They are basic tools in the study of weak convergence in spaces of measures and in $L^1$.
</document>

<document source="https://tamnd.github.io/math2/node/FCGHXDGVV9.md" tag="FCGHXDGVV9" kind="concept">
# Henstock–Kurzweil integral

## Summary

The Henstock–Kurzweil integral, also called the gauge integral, is an integral of Riemann type.
In its sums the allowed size of a subinterval may vary from point to point.
This one change makes it integrate every derivative, so that $\int_a^b F' = F(b) - F(a)$ holds whenever $F$ is differentiable everywhere on $[a, b]$.
It integrates every [[Lebesgue integrable|DWJF7ABPR1]] function with the same value, and it gives the same integral as the [[Perron integral|M077AXJACA]].
Unlike the Lebesgue integral, it also integrates some functions whose absolute value is not integrable.

## Explanations

### L1 human

In a Riemann sum all pieces must be short, with one bound on their length.
In the Henstock–Kurzweil sum each point $t$ comes with its own allowed size $\delta(t)$, and a piece tagged by $t$ must fit within distance $\delta(t)$ of $t$.
Near a point where the function misbehaves, one demands very short pieces; elsewhere longer ones are fine.
For example, the derivative of $x^2 \sin(1/x^2)$ is not Lebesgue integrable on $[0, 1]$, but it is Henstock–Kurzweil integrable with integral $\sin 1$.
</document>

<document source="https://tamnd.github.io/math2/node/FCHW5QGG5V.md" tag="FCHW5QGG5V" kind="concept">
# Wigner distribution function

## Summary

The Wigner distribution of a function $f$ on $\R$ is a real-valued function on the time–frequency plane built from $f$ and its complex conjugate.
Its integral over frequency gives $\abs{f}^2$, and its integral over time gives $\abs{\hat f}^2$, so it behaves like a joint density of time and frequency.
It can take negative values, so it is not a true probability density.
Eugene Wigner introduced it in 1932 in quantum mechanics, as a quasi-probability distribution on phase space.
It is also a basic tool of [[time–frequency analysis|GQFPVXEWK1]].

## Explanations

### L1 human

One would like a picture that tells how much of a signal sits at each time and each frequency.
The Wigner distribution compares the signal at times a little before and a little after each moment, and takes a Fourier transform of that comparison.
Summing it over all frequencies gives back the energy at each time, and summing over all times gives the energy at each frequency.
For a Gaussian pulse it is a Gaussian bump in the plane.
For a sum of two pulses it shows an extra wavy pattern between them, which can be negative.
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<document source="https://tamnd.github.io/math2/node/FCKX8KFVGN.md" tag="FCKX8KFVGN" kind="concept">
# Convolution on the half-line

## Summary

The convolution on the half-line combines two functions $f, g$ on $[0, \infty)$ into $(f * g)(t) = \int_0^t f(\tau) g(t - \tau) \, d\tau$.
It is the [[convolution on the line|ERP5737VZ1]] of the two functions extended by zero to negative times.
The [[Laplace transform|0V49JEZW06]] turns it into a pointwise product.
It is also the product of the [[Mikusiński operational calculus|TD5XK5AHNV]].

## Explanations

### L1 human

Think of $t$ as time and of $f$ and $g$ as signals that start at time $0$.
The value $(f * g)(t)$ adds up the effects at time $t$ of all the inputs $f(\tau)$ from earlier times $\tau$, each weighted by $g$ at the elapsed time $t - \tau$.
Only the past matters, so the integral runs from $0$ to $t$.
For example, convolving the constant $1$ with itself gives $\int_0^t 1 \, d\tau = t$.
</document>

<document source="https://tamnd.github.io/math2/node/FCM1T5EXFV.md" tag="FCM1T5EXFV" kind="concept">
# Primary obstruction

## Summary

The primary obstruction is the first obstruction that appears when one tries to extend a map into an $(n-1)$-connected space over a CW complex.
It is a single cohomology class in degree $n + 1$ with coefficients in $\pi_n$ of the target, and it does not depend on any choice.
Many characteristic classes, such as the Euler class of an oriented sphere bundle, are primary obstructions.

## Explanations

### L1 human

If the target space $Y$ has no holes in dimensions below $n$, a map can always be extended over cells of dimension at most $n$.
The first real difficulty comes with the cells of dimension $n + 1$.
The primary obstruction is the cohomology class that measures this difficulty.
Try to build a nonvanishing vector field on a closed oriented surface.
It can always be built over the vertices and edges, and the primary obstruction lives on the top cells.
There it equals the Euler characteristic of the surface times a generator.
</document>

<document source="https://tamnd.github.io/math2/node/FCNQP0A60D.md" tag="FCNQP0A60D" kind="concept">
# Quasi-invariant measure

## Summary

A [[measure|PR11BZQQPG]] is quasi-invariant under a [[group action|0WCY42EV9V]] if moving by any group element keeps the same [[null sets|VQQM4Y550T]].
The measure itself may change, but only by a density.
This weaker form of invariance is what one finds on homogeneous spaces without invariant measures and on infinite-dimensional spaces.

## Explanations

### L1 human

An invariant measure gives the same size to a set and to each of its moved copies.
A quasi-invariant measure only promises that a set of size zero stays of size zero when moved.
For example, the bell-curve measure on the real line is not invariant under shifts, since it puts more mass near $0$.
But a shifted set has zero bell-curve measure exactly when it has zero length, so the measure is quasi-invariant under shifts.
</document>

<document source="https://tamnd.github.io/math2/node/FCSBB1HAHH.md" tag="FCSBB1HAHH" kind="definition">
# Support function

## Definition

Let $n \ge 1$ and let $A \subseteq \R^n$ be nonempty.
The *support function* of $A$ is the map $h_A \colon \R^n \to \R \cup \set{+\infty}$ given by
$$h_A(x) = \sup_{a \in A} \, x \cdot a,$$
where $x \cdot a$ is the [[dot product|0S52TNNTH9]] and $\sup$ is the [[supremum|0R3H5PANRQ]].

## Remarks

If $A$ is bounded, then $h_A$ takes only real values.

Example: for $A = [0, 1]^2$, $h_A(x_1, x_2) = \max(x_1, 0) + \max(x_2, 0)$.
Indeed $x \cdot a = x_1 a_1 + x_2 a_2$, and each term is largest at $a_i = 1$ when $x_i > 0$ and at $a_i = 0$ when $x_i \le 0$.

Non-example: the function $f(x) = \abs{x}^2$ is not the support function of any nonempty set.
Every support function satisfies $h_A(t x) = t\, h_A(x)$ for $t > 0$, but $f(2x) = 4 f(x) \ne 2 f(x)$ for $x \ne 0$.

Each support function is positively homogeneous and subadditive, $h_A(x + y) \le h_A(x) + h_A(y)$, hence convex.
One has $h_A = h_B$ when $B$ is the closure of the convex hull of $A$, and a nonempty compact convex set is determined by its support function.
</document>

<document source="https://tamnd.github.io/math2/node/FCTRS2SS16.md" tag="FCTRS2SS16" kind="example">
# Nested radical

## Example

The infinite nested radical
$$\sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}$$
is defined as the limit of the sequence $x_1 = 1$, $x_{n+1} = \sqrt{1 + x_n}$.
This is an [[infinite composition|B88PFYXWBK]] of the map $f(x) = \sqrt{1 + x}$, since $x_n = f^{\circ n}(0)$.
Its value is the [[golden ratio|0SC8T7548D]] $\varphi = \frac{1 + \sqrt5}{2}$.

Check: the sequence is increasing, since $x_2 = \sqrt2 > x_1$ and $f$ is increasing.
It is bounded above by $2$, since $x_n < 2$ gives $x_{n+1} < \sqrt3 < 2$.
By the [[monotone convergence theorem|0FM6N4R0VB]] it has a limit $L \ge 1$, and continuity of $f$ gives $L = \sqrt{1 + L}$.
So $L^2 - L - 1 = 0$, and the positive root is $L = \varphi$.

The same method gives $\sqrt{2 + \sqrt{2 + \cdots}} = 2$, from $L = \sqrt{2 + L}$.

## Explanations

### L1 human

An infinite nested radical is a square root inside a square root, again and again.
You give it a meaning by stopping after $n$ roots and letting $n$ grow.
The partial values $1, 1.414, 1.554, 1.598, \dots$ climb towards $1.618\ldots$, the golden ratio.
</document>

<document source="https://tamnd.github.io/math2/node/FD3T25A1DX.md" tag="FD3T25A1DX" kind="definition">
# Local property

## Definition

Let $P$ be a property of modules over commutative rings.
The property $P$ is *local* if, for every [[commutative ring|0310REJJY8]] $R$ and every $R$-[[module|0NZ8JA25WK]] $M$, the following are equivalent:

1. $M$ has $P$ as an $R$-module;
2. for every [[prime ideal|00MHHZ9SNE]] $\mathfrak p$ of $R$, the [[localized module|68E0F8N94E]] $M_{\mathfrak p}$ has $P$ as an $R_{\mathfrak p}$-module.

A property of commutative rings is *local* in the same way, with $R$ in place of $M$ and the [[localizations|X6JD4AG973]] $R_{\mathfrak p}$ in place of $M_{\mathfrak p}$.
A property of module homomorphisms $\varphi \colon M \to N$ is local in the same way, with the localized maps $\varphi_{\mathfrak p} \colon M_{\mathfrak p} \to N_{\mathfrak p}$.

## Remarks

Example: the property "$M = 0$" is local.
If $M \ne 0$, pick $m \ne 0$ in $M$; its [[annihilator|HB4XDKJTW8]] is a proper ideal, so it lies in a [[maximal ideal|0PV74TR5FP]] $\mathfrak m$.
Then $m/1 \ne 0$ in $M_{\mathfrak m}$, since $um \ne 0$ for all $u \notin \mathfrak m$.
The converse direction is clear.
It follows that injectivity and surjectivity of a module map, exactness of a sequence, and [[flatness|73RYYEVDKA]] are local.

In each case it is enough to check the maximal ideals instead of all prime ideals.

Non-example: being an [[integral domain|0QVD7WBFGN]] is not a local property of rings.
For a field $k$, the ring $R = k \times k$ has the two prime ideals $k \times 0$ and $0 \times k$, and both localizations are isomorphic to $k$, a domain.
But $(1, 0) \cdot (0, 1) = 0$ in $R$, so $R$ is not a domain.
Being [[Noetherian|0670T59HGE]] is also not local; see [[localization of a Noetherian ring is Noetherian|C618HAK3X6]].
</document>

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# Differential-algebraic system of equations

## Definition

Let $I \subseteq \R$ be an open interval, let $\Omega \subseteq \R^n \times \R^n$ be an open set, and let $F : I \times \Omega \to \R^n$ be continuously differentiable.
Write $F = F(t, x, p)$ with $t \in I$ and $x, p \in \R^n$.
The system
$$F(t, x(t), x'(t)) = 0$$
is a *differential-algebraic system of equations* if the [[Jacobian matrix|047NDTEPCV]] $\partial F / \partial p$ of size $n \times n$ is singular at every point of $I \times \Omega$.
A solution is a differentiable map $x$ on a subinterval of $I$ with $(x(t), x'(t)) \in \Omega$ and $F(t, x(t), x'(t)) = 0$ for each $t$.

## Remarks

Example: for $x = (u, w)$, take the system $u' = -u + w$ and $0 = u - w + \sin t$.
Here $F(t, x, p) = (p_1 + u - w,\ u - w + \sin t)$ and $\partial F / \partial p$ has rows $(1, 0)$ and $(0, 0)$, so it is singular everywhere.
The constraint gives $w = u + \sin t$, hence $u' = \sin t$, so $u = C - \cos t$ and $w = C - \cos t + \sin t$.
Initial values must satisfy the constraint: $w(0) = u(0)$.
Non-example: if $\partial F / \partial p$ is invertible at a point, the [[implicit function theorem|0BXBADR94C]] solves for $x'$ near that point, and the system is locally an [[ordinary differential equation|0P97V20RW7]] $x' = f(t, x)$.
The form $u' = f(t, u, w)$, $0 = g(t, u, w)$ is called semi-explicit.
The differentiation index is the least number of differentiations of the system needed to determine $x'$ as a continuous function of $(t, x)$; the example above has index $1$.
</document>

<document source="https://tamnd.github.io/math2/node/FD8VEKWZNR.md" tag="FD8VEKWZNR" kind="definition">
# Dirichlet space

## Definition

Let $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ be the [[unit disc|8G5ABXRK85]] and let $dA$ be area measure.
The *Dirichlet space* $\mathcal{D}$ is the set of [[holomorphic functions|0NAHXR1FRH]] $f \colon \mathbb{D} \to \C$ such that
$$\int_{\mathbb{D}} \abs{f'(z)}^2 \, dA(z) < \infty.$$
Its norm is given by $\norm{f}_{\mathcal{D}}^2 = \abs{f(0)}^2 + \frac{1}{\pi}\int_{\mathbb{D}} \abs{f'(z)}^2 \, dA(z)$.

## Remarks

For $f(z) = \sum_{n \ge 0} a_n z^n$, a [[power series|0D6KXBHKC0]] on $\mathbb{D}$, one has $\int_{\mathbb{D}} \abs{z^{n-1}}^2 dA = \pi/n$, hence $\int_{\mathbb{D}} \abs{f'}^2 \, dA = \pi\sum_{n \ge 1} n \abs{a_n}^2$ and $\norm{f}_{\mathcal{D}}^2 = \abs{a_0}^2 + \sum_{n \ge 1} n\abs{a_n}^2$.
This norm comes from an [[inner product|0QVNCJXRY3]], and $\mathcal{D}$ is a Hilbert space.
Example: $f(z) = z$ has $\int_{\mathbb{D}} \abs{f'}^2 \, dA = \pi$, the area of $f(\mathbb{D}) = \mathbb{D}$.
Non-example: $f(z) = \log(1 - z) = -\sum_{n \ge 1} z^n/n$ gives $\sum_{n \ge 1} n \cdot \frac{1}{n^2} = \sum 1/n = \infty$, so $f \notin \mathcal{D}$, although $f$ is in the Hardy space $H^2$.
If $f$ is injective, $\int_{\mathbb{D}} \abs{f'}^2 \, dA$ is the area of $f(\mathbb{D})$, since $\abs{f'}^2$ is the Jacobian determinant of $f$.
Some authors omit the term $\abs{f(0)}^2$ or replace it by $\norm{f}_{H^2}^2$; these give equivalent norms.
</document>

<document source="https://tamnd.github.io/math2/node/FDARANFQ0B.md" tag="FDARANFQ0B" kind="definition">
# Biholomorphism

## Definition

Let $U$ and $V$ be open subsets of $\C^n$.
A *biholomorphism* from $U$ to $V$ is a [[holomorphic map|3MQTWKBBBT]] $F \colon U \to V$ such that:

1. $F$ is a bijection from $U$ onto $V$.
2. The [[inverse map|0J2WAR83WP]] $F^{-1} \colon V \to U$ is holomorphic.

The sets $U$ and $V$ are *biholomorphic*, or biholomorphically equivalent, if there is a biholomorphism from $U$ to $V$.

## Remarks

Example: $F(z_1, z_2) = (2 z_1, \ z_2 + z_1^2)$ is a biholomorphism of $\C^2$, since it is polynomial, bijective, and its inverse $(w_1, w_2) \mapsto (w_1/2, \ w_2 - w_1^2/4)$ is polynomial.
Non-example: $F(z) = z^3$ from $\C$ to $\C$ is holomorphic and onto, but not injective, since $F(1) = F(e^{2\pi i/3})$; so clause 1 fails.
In fact clause 2 follows from clause 1: an injective holomorphic map from an open set $U \subseteq \C^n$ to $\C^n$ has open image and a holomorphic inverse.
This differs from the real case, where $x \mapsto x^3$ is a smooth bijection of $\R$ whose inverse is not differentiable at $0$.
The biholomorphisms of $U$ onto itself form a group, the holomorphic automorphism group $\Aut(U)$.
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# Noether theorems for singular integral equations

## Statement

Let $\Gamma$ be a smooth simple closed curve in $\C$, oriented counterclockwise.
Let $a$ and $b$ be [[Hölder continuous|012PBA0FMD]] functions on $\Gamma$ with $a(t)^2 - b(t)^2 \ne 0$ for all $t \in \Gamma$, and let $k$ be Hölder continuous on $\Gamma \times \Gamma$.
Let $K$ be the operator of the [[singular integral equation|E30EA6ETZ4]]
$$K \varphi(t) = a(t) \, \varphi(t) + \frac{b(t)}{\pi i} \, \mathrm{p.v.} \int_\Gamma \frac{\varphi(\tau)}{\tau - t} \, d\tau + \int_\Gamma k(t, \tau) \, \varphi(\tau) \, d\tau,$$
and let $K'$ be its adjoint operator
$$K' \psi(t) = a(t) \, \psi(t) - \frac{1}{\pi i} \, \mathrm{p.v.} \int_\Gamma \frac{b(\tau) \, \psi(\tau)}{\tau - t} \, d\tau + \int_\Gamma k(\tau, t) \, \psi(\tau) \, d\tau.$$
All functions below are Hölder continuous on $\Gamma$.
Let $\kappa$ be the [[index|GW3B4K1WA5]] of $K$.
Then:

1. The homogeneous equations $K \varphi = 0$ and $K' \psi = 0$ each have a finite number of linearly independent solutions; call these numbers $\ell$ and $\ell'$.
2. For Hölder continuous $f$, the equation $K \varphi = f$ has a solution if and only if $\int_\Gamma f(t) \, \psi(t) \, dt = 0$ for every solution $\psi$ of $K' \psi = 0$.
3. $\ell - \ell' = \kappa$.

## Remarks

The adjoint is taken for the bilinear pairing $\int_\Gamma u(t) \, v(t) \, dt$; exchanging the order of integration in $\int_\Gamma (K \varphi) \psi \, dt$ gives $\int_\Gamma \varphi \, (K' \psi) \, dt$.
Proof idea: the dominant operator $K^0 = a + b S$, with $S \varphi = \frac{1}{\pi i} \mathrm{p.v.} \int_\Gamma \frac{\varphi(\tau)}{\tau - t} \, d\tau$, is solved explicitly through the [[Sokhotski–Plemelj formulae|DRT1E6WY6K]] and a Riemann–Hilbert problem, which shows that $\ell - \ell' = \kappa$ for $K^0$.
One then builds a regularizer $M$ with $M K = I + T$ for a compact operator $T$, and the Fredholm theory of $I + T$ transfers the results to $K$.
The term with $k$ is compact, so it changes neither the finiteness nor the index.

Example: on the unit circle, take $a(t) = (t + 1)/2$, $b(t) = (t - 1)/2$ and $k = 0$, so $\kappa = -1$.
With $P = \frac12 (I + S)$ and $Q = \frac12 (I - S)$, one has $K = a + bS = t P + Q$.
On $\varphi = \sum_n c_n t^n$, $P$ keeps the terms with $n \ge 0$ and $Q$ the terms with $n < 0$, so $K \varphi = \sum_{n \ge 0} c_n t^{n+1} + \sum_{n < 0} c_n t^n$.
So $K \varphi = 0$ forces $\varphi = 0$, and $K \varphi = f$ is solvable only if the constant Fourier coefficient of $f$ vanishes: $\ell = 0$, $\ell' = 1$, and $\ell - \ell' = -1 = \kappa$.

The normal type condition $a^2 - b^2 \ne 0$ is needed: without it, the homogeneous equation may have infinitely many independent solutions or the range may fail to be closed.
When $\kappa = 0$, the three statements give the classical Fredholm alternative, but in general $\ell \ne \ell'$, unlike for Fredholm equations of the second kind.

## Explanations

### L1 human

A singular integral equation of normal type behaves almost like a system of linear equations.
There are finitely many independent solutions of the homogeneous equation and finitely many conditions for the general equation to be solvable.
The difference of these two numbers is the index, and it can be computed by counting how many times a function made from the coefficients winds around $0$.

### L2 human

These are the singular analogues of Fredholm's theorems.
They were proved by Fritz Noether, and the singular integral operators of normal type are classical examples of Fredholm operators with nonzero [[Fredholm index|2V9Q4KHY1T]].
The index formula $\ell - \ell' = \kappa$ is an early case of an index theorem, which expresses an analytic quantity through a topological one, here a winding number.
In the Hölder setting the theorems are presented in Muskhelishvili's book on singular integral equations; the same results hold for operators on $L^p(\Gamma)$, $1 < p < \infty$, with continuous coefficients.
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# Iterated kernel

## Definition

Let $a < b$ and let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]].
The *iterated kernels* of $K$ are the functions $K_n \colon [a, b] \times [a, b] \to \C$, for $n \ge 1$, defined by $K_1 = K$ and
$$K_{n}(x, s) = \int_a^b K(x, t) \, K_{n-1}(t, s) \, dt \quad \text{for } n \ge 2.$$
The function $K_n$ is the $n$-th iterated kernel of $K$.

## Remarks

Example: $K(x, s) = x s$ on $[0, 1]$.
Then $K_2(x, s) = \int_0^1 x t^2 s \, dt = x s / 3$, and by induction $K_n(x, s) = x s / 3^{n-1}$.

Non-example: the pointwise power $K(x, s)^2 = x^2 s^2$ is not the second iterated kernel $x s / 3$; iteration composes the integral operators, it does not multiply kernel values.

Each $K_n$ is continuous, and $K_n$ is the kernel of the $n$-th power of the integral operator of $K$.
More generally $K_{n + m}(x, s) = \int_a^b K_n(x, t) K_m(t, s) \, dt$ for $n, m \ge 1$.
If $\abs{K} \le M$, then $\abs{K_n(x, s)} \le M^n (b - a)^{n - 1}$, by induction on $n$.
The same definition is used for square-integrable kernels, with equality almost everywhere.
For a Volterra kernel, which vanishes for $s > x$, the integral runs over $s \le t \le x$.
</document>

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# Kan complex

## Summary

A Kan complex is a [[simplicial set|WW81ARB8K9]] in which every horn can be filled to a full simplex.
Kan complexes behave like topological spaces.
Their simplicial homotopy groups are well defined and agree with those of their geometric realization.
The singular simplicial set of every space is a Kan complex.
Kan complexes are the fibrant objects of the Kan–Quillen model structure on simplicial sets.
They are a standard model of $\infty$-groupoids.

## Explanations

### L1 human

A simplicial set is built from points, edges, triangles and higher simplices glued along faces.
A horn is a simplex with one face and its interior removed, like two edges of a triangle that meet at a corner.
In a Kan complex every such horn is part of a full simplex, so two edges that meet can always be completed to a triangle.
This lets one compose and invert paths, as in a space.
The singular simplicial set of a space, whose $n$-simplices are all continuous maps from the standard $n$-simplex, is the basic example.
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# Locally connected space

## Summary

A locally connected space is a [[topological space|0CQHFPGJ45]] in which every point has arbitrarily small connected open [[neighbourhoods|0ZCY0Z80B6]].
Equivalently, the [[connected components|0JG9W3KA6B]] of every open subset are open.
Local connectedness and [[connectedness|0V032VWTHD]] are independent: neither implies the other.
It is a hypothesis in many results of continuum theory, such as the [[Hahn–Mazurkiewicz theorem|Q5QFYKD004]].

## Explanations

### L1 human

A space is locally connected if, zooming in near any point, you can always find a small open piece around it that is all in one piece.
Euclidean space $\R^n$ is locally connected, since small open balls are connected.
Two disjoint open intervals form a locally connected space that is not connected.
The [[comb space|A2Q4RXD162]] is connected but not locally connected.
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<document source="https://tamnd.github.io/math2/node/FDWY67RFRT.md" tag="FDWY67RFRT" kind="definition">
# Dynamical billiards

## Definition

Let $Q \subset \R^2$ be a [[compact|0TVJZYB6SH]] connected set whose [[boundary|05D5EGMFYV]] $\partial Q$ is a finite union of smooth curves, called the table.
A *billiard trajectory* in $Q$ is a path $t \mapsto q(t)$ in $Q$ with the following properties:

1. Between two times at which $q(t) \in \partial Q$, the path is a straight segment run at unit speed.
2. At a time when $q(t)$ is a smooth point of $\partial Q$, the velocity after the collision is the mirror image of the velocity before it. The component tangent to $\partial Q$ is kept and the normal component changes sign.

The *billiard flow* is the flow on the unit tangent vectors $(q, v)$, $q \in Q$, $\abs{v} = 1$, that moves $(q, v)$ along its trajectory.
The *billiard map* acts on the pairs $(s, \theta)$.
Here $s$ is the arc-length parameter of a point of $\partial Q$, and $\theta \in (0, \pi)$ is the angle between an inward unit vector and the boundary.
The map sends $(s, \theta)$ to the next collision point and its outgoing angle.
These two [[dynamical systems|BBZP1R872N]] are called *dynamical billiards*.

## Remarks

Example: let $Q$ be the unit disc.
A chord leaving the circle at angle $\theta$ to the tangent subtends the arc $2\theta$ and meets the circle again at the same angle $\theta$.
So the billiard map is $(s, \theta) \mapsto (s + 2\theta \bmod 2\pi, \theta)$, and the angle is a conserved quantity.

Non-example: a trajectory that hits a corner of $\partial Q$, where the boundary has no tangent, is not continued by clause 2.
Such trajectories form a set of measure zero and are left out.

The billiard flow preserves the Liouville measure $dq \, dv$.
The billiard map preserves the [[invariant measure|B232EZ1FMB]] $\sin \theta \, ds \, d\theta$.
The same definitions make sense for tables in $\R^n$ and on Riemannian manifolds, such as the flat torus.
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<document source="https://tamnd.github.io/math2/node/FE1VTGF349.md" tag="FE1VTGF349" kind="definition">
# Kontsevich integral

## Definition

Identify $\R^3$ with $\C \times \R$, with complex coordinate $z$ and height $t$.
Let $K$ be an oriented [[knot|QYEAZTYGPT]] in $\C \times \R$ such that the height $t$ restricted to $K$ is a Morse function.
Let $t_{\min}$ and $t_{\max}$ be the minimum and the maximum of $t$ on $K$.
Let $\mathcal{A}$ be the $\C$-vector space of chord diagrams on the circle modulo the four-term relation and the one-term relation.
The one-term relation kills every diagram with an isolated chord.
Let $\widehat{\mathcal{A}}$ be the completion of $\mathcal{A}$ by degree.
For $t_{\min} < t_1 < \cdots < t_m < t_{\max}$, let $P = \set{(z_j, z_j')}_{j=1}^{m}$ be a choice of two distinct points of $K$ at height $t_j$, for each $j$.
Let $D_P$ be the chord diagram with these $m$ chords, and $\downarrow_P$ the number of chosen points where $K$ is oriented downward.
The *Kontsevich integral* of $K$ is
$$Z(K) = \sum_{m \ge 0} \frac{1}{(2\pi i)^m} \int_{t_{\min} < t_1 < \cdots < t_m < t_{\max}} \sum_{P} (-1)^{\downarrow_P} D_P \bigwedge_{j=1}^{m} \frac{dz_j - dz_j'}{z_j - z_j'} \in \widehat{\mathcal{A}}.$$

## Remarks

Each integral converges, and $Z(K)$ is unchanged by isotopies that keep the number of critical points of the height fixed.
Let $c$ be the number of critical points and $H$ the unknot embedded with two maxima and two minima.
Then the normalized $\widehat{Z}(K) = Z(H)^{-c/2} Z(K)$, with the product in $\widehat{\mathcal{A}}$, is an isotopy invariant of $K$.
Example: the degree-$0$ part of $Z(K)$ is the empty chord diagram with coefficient $1$, since the term $m = 0$ is the constant $1$.
Non-example: the unnormalized $Z$ is not an invariant, because adding a pair of critical points multiplies it by $Z(H)$, which is not $1$.
Every rational-valued [[finite type invariant|TWH3WVKFF1]] of type $\le n$ factors through $\widehat{Z}$ followed by a linear map on the part of degree $\le n$.
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<document source="https://tamnd.github.io/math2/node/FE1ZZJ8QW2.md" tag="FE1ZZJ8QW2" kind="concept">
# Multiple integral

## Summary

A multiple integral is the integral of a real function of $n$ real variables over a region of $\R^n$.
It extends the one-variable [[Riemann integral|0W29TBTY12]] from intervals to boxes and to more general bounded sets.
The [[double integral|0WAR4F23B4]] and the [[triple integral|0698M7C7AX]] are the cases $n = 2$ and $n = 3$.
Multiple integrals give areas, volumes, masses, centres of mass and probabilities of random vectors.

## Explanations

### L1 human

Cut a region of space into many small boxes.
In each box, multiply a value of the function by the volume of the box, and add up all these products.
When the boxes get smaller and the sums settle to one number, that number is the multiple integral.
For example, the integral of the constant $1$ over the unit cube $[0, 1]^3$ is its volume, $1$.
In practice one often computes a multiple integral as an [[iterated integral|0WZCHE1ZF9]], one variable at a time.
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<document source="https://tamnd.github.io/math2/node/FE83QQS17S.md" tag="FE83QQS17S" kind="statement">
# Weierstrass division theorem

## Statement

Let $n \ge 1$, write $z' = (z_1, \dots, z_{n-1})$, and let $\mathcal{O}_n = \C\{z_1, \dots, z_n\}$ and $\mathcal{O}_{n-1} = \C\{z'\}$ be [[rings of convergent power series|EVPZ7ZG1HB]].
Let $f \in \mathcal{O}_n$ and $d \ge 0$ be such that $f(0, \dots, 0, z_n) = z_n^d\, v(z_n)$ with $v(0) \ne 0$; that is, $f$ is regular of order $d$ in $z_n$.
Then for every $g \in \mathcal{O}_n$ there are unique $q \in \mathcal{O}_n$ and $r \in \mathcal{O}_{n-1}[z_n]$ such that
$$g = q f + r \quad \text{and} \quad \deg_{z_n} r < d.$$

## Remarks

The same statement holds for [[germs of holomorphic functions|7PJ8KVZZCQ]] at $0$, since these form a ring isomorphic to $\mathcal{O}_n$.
One proof first treats a divisor $f = P$ that is a Weierstrass polynomial of degree $d$.
On a small polydisc $\abs{z'} < \varepsilon$, $\abs{z_n} < \delta$, the polynomial $P(z', \cdot)$ has its $d$ roots inside $\abs{z_n} < \delta$ for each $z'$.
The Cauchy integral
$$q(z', z_n) = \frac{1}{2\pi i} \oint_{\abs{w} = \delta} \frac{g(z', w)}{P(z', w)\,(w - z_n)}\, dw$$
defines a holomorphic $q$, and the Cauchy formula for $g$ gives
$$r = g - q P = \frac{1}{2\pi i} \oint_{\abs{w} = \delta} \frac{g(z', w)}{P(z', w)} \cdot \frac{P(z', w) - P(z', z_n)}{w - z_n}\, dw.$$
The last quotient is a polynomial in $z_n$ of degree $d - 1$, so $r$ is a polynomial in $z_n$ of degree less than $d$.
A general $f$ regular of order $d$ is a unit times a Weierstrass polynomial.
This can be shown directly through the symmetric functions of the roots of $f(z', \cdot)$, and it reduces the general case to the first one.
Example: for $n = 1$ and $f = z^d$, the theorem says $g = q z^d + r$, where $r$ is the Taylor polynomial of $g$ of degree $d - 1$.
The regularity hypothesis is needed: for $f = z_1$ in $\mathcal{O}_2$ and any $d$, the series $g = z_2^d$ is not of the form $q z_1 + r$ with $\deg_{z_2} r < d$, since setting $z_1 = 0$ would give $z_2^d = r(0, z_2)$.
Conversely, applying the theorem to $g = z_n^d$ gives the [[Weierstrass preparation theorem|49JSD49P7X]]: $f$ is a unit times the [[Weierstrass polynomial|SDCKNWSWNM]] $z_n^d - r$.
A linear change of coordinates makes any nonzero $f \in \mathcal{O}_n$ regular in $z_n$ of some order, so the hypothesis is mild.

## Explanations

### L1 human

The theorem is a division with remainder, like the division of integers or of polynomials.
One divides a power series $g$ by a power series $f$ that looks, along one axis, like $z_n^d$.
The remainder is then a polynomial in $z_n$ of degree less than $d$, with coefficients that are power series in the other variables.

### L2 human

Division is the engine behind the algebra of the ring $\mathcal{O}_n$.
It implies the preparation theorem, and with it one proves by induction on $n$ that $\mathcal{O}_n$ is Noetherian and factorial.
It also shows that $\mathcal{O}_n/(f)$ is a free module of rank $d$ over $\mathcal{O}_{n-1}$ when $f$ is regular of order $d$.
This is the algebraic form of the local parametrization of hypersurfaces.
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<document source="https://tamnd.github.io/math2/node/FE8EYTGMRS.md" tag="FE8EYTGMRS" kind="definition">
# Iterative root

## Definition

Let $X$ be a set, let $f \colon X \to X$ be a [[function|0EPAT51DXQ]], and let $n \ge 1$ be an integer.
An *iterative root of order $n$* of $f$ is a function $g \colon X \to X$ such that
$$g^n = f,$$
where $g^n = g \circ \cdots \circ g$ is the $n$-th [[iterate|AYVTR2ES5B]] of $g$.

## Remarks

Example: on $X = \R$, the map $g(x) = x + 1$ is an iterative root of order $3$ of $f(x) = x + 3$.
Check: $g^2(x) = x + 2$ and $g^3(x) = x + 3 = f(x)$.

Non-example: $g(x) = 3x$ is not an iterative root of order $3$ of $f(x) = x + 3$, since $g^3(x) = 27x \ne x + 3$.

The roots of order $2$ are the [[functional square roots|E7890PHMGP]].
Roots are rarely unique: both $2x$ and $-2x$ are iterative roots of order $2$ of $4x$.
Some maps have no continuous root of a given order.
If $g \colon \R \to \R$ were continuous with $g^2(x) = -x$, then $g$ would be injective, hence strictly monotone.
Then $g^2$ would be increasing, but $x \mapsto -x$ is decreasing.
For odd $n$ the same map has the root $g(x) = -x$, since $(-1)^n = -1$.
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<document source="https://tamnd.github.io/math2/node/FEAFDTMMW7.md" tag="FEAFDTMMW7" kind="definition">
# Knot

## Definition

Let $S^1$ be the [[circle|060W5CBZP6]] and let $\R^3$ be [[Euclidean space|0XPR1E85W0]] of dimension $3$.

1. A *knot* is a subset $K \subset \R^3$ that is the image of a continuous injective map $f \colon S^1 \to \R^3$.
2. Two knots $K$ and $K'$ are *equivalent* if there is a continuous family of [[homeomorphisms|0QT00PB5G2]] $h_t \colon \R^3 \to \R^3$, $t \in [0, 1]$, with $h_0 = \id$ and $h_1(K) = K'$. Such a family is an *ambient isotopy*.
3. A knot is *tame* if it is equivalent to a knot that is a closed polygon with finitely many edges.

## Remarks

Equivalence of knots is an [[equivalence relation|0G8VEZ3WTK]], and knot theory studies the equivalence classes.
Since $S^1$ is compact, the map $f$ is a homeomorphism onto $K$.
Most texts consider only tame knots.
Every knot that is the image of a smooth embedding is tame, and every tame knot is equivalent to such a knot.
Some texts use the $3$-sphere $S^3$ in place of $\R^3$; this gives the same theory.
Often a knot is also oriented, by choosing a direction along $S^1$.

Example: the circle $\set{(x, y, 0) : x^2 + y^2 = 1}$ is a knot, the image of $t \mapsto (\cos t, \sin t, 0)$, which is injective on $[0, 2\pi)$; it is a tame knot, the [[unknot|1TQ06R4Q66]].
Non-example: the figure-eight curve $t \mapsto (\sin 2t, \sin t, 0)$ is not a knot, because it is not injective: $t = 0$ and $t = \pi$ both map to the origin.
A [[wild knot|QYDQ28X8MA]] satisfies the first clause but not the third.
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<document source="https://tamnd.github.io/math2/node/FEC2XH822R.md" tag="FEC2XH822R" kind="concept">
# j-invariant

## Summary

The $j$-invariant is the function $j = 1728\, E_4^3 / (E_4^3 - E_6^2)$ on the [[upper half-plane|X6XGGH5FPW]], built from [[Eisenstein series|EYREVY31BN]].
It is invariant under the [[modular group|VEKV1EM77B]], holomorphic on $\mathbb{H}$, and has a simple pole at infinity.
It gives a bijection from $\mathrm{SL}_2(\Z) \backslash \mathbb{H}$ onto $\C$.
Two [[elliptic curves|427N6S6QQW]] over $\C$ are isomorphic exactly when they have the same $j$-invariant.

## Explanations

### L1 human

Each point $\tau$ of the upper half-plane describes a lattice in the plane and so an elliptic curve.
Different $\tau$ can give the same curve, namely when they differ by the modular group.
The $j$-invariant is a single complex number that tells exactly which curve one has.
Its expansion $j = q^{-1} + 744 + 196884 q + \cdots$ has coefficients linked to the Monster group by [[monstrous moonshine|E0Y7Y75T6C]].
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<document source="https://tamnd.github.io/math2/node/FEJK846N5X.md" tag="FEJK846N5X" kind="statement">
# Margulis arithmeticity theorem

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite centre and no compact factors, of real rank at least $2$.
Let $\Gamma$ be an irreducible [[lattice|BTYF5Y5PFY]] in $G$: for every noncompact closed normal subgroup $N$ of $G$, the product $\Gamma N$ is dense in $G$.
Then $\Gamma$ is arithmetic.
That is, there are a semisimple [[linear algebraic group|GA81VVQHEP]] $H \subseteq \mathrm{GL}_n$ over $\Q$ and a continuous surjective homomorphism $\phi \colon H(\R)^\circ \to G$ with compact kernel with the following property.
The groups $\phi(H(\Z) \cap H(\R)^\circ)$ and $\Gamma$ are commensurable: their intersection has finite [[index|0DTXBQPA2S]] in both.

## Remarks

The proof rests on the [[Margulis superrigidity theorem|DSKV65NHZZ]].
Superrigidity is applied to $\Gamma$ in real, complex and $p$-adic groups.
It shows that, in suitable coordinates, the matrix entries of $\mathrm{Ad}(\Gamma)$ lie in a number field and have bounded denominators.
From this one builds the $\Q$-group $H$.

Example: $\mathrm{SL}_n(\R)$ with $n \ge 3$ has real rank $n - 1 \ge 2$.
So every lattice in it is commensurable, up to conjugation, with a group built from an algebraic group over $\Q$, such as $\mathrm{SL}_n(\Z)$.

The rank condition cannot be dropped completely.
Gromov and Piatetski-Shapiro constructed non-arithmetic lattices in $\mathrm{SO}(n, 1)$ for every $n \ge 2$.
Mostow, and Deligne and Mostow, found non-arithmetic lattices in $\mathrm{SU}(2, 1)$ and $\mathrm{SU}(3, 1)$.
In real rank $1$, the groups $\mathrm{Sp}(n, 1)$ and the rank one form of $F_4$ still have only arithmetic lattices, by Corlette and Gromov–Schoen.

## Explanations

### L1 human

The simplest way to build a lattice is to take the integer points of a group defined by equations, like $\mathrm{SL}_3(\Z)$ inside $\mathrm{SL}_3(\R)$.
Margulis proved that in the big groups of higher rank, there is no other way.
Every lattice there is built from integer points, after allowing finite index changes and a few standard operations.

### L2 human

Together with the [[Borel–Harish-Chandra theorem|3B451RD8AJ]], this gives a complete description of irreducible lattices in higher rank.
It turns questions about lattices into questions about algebraic groups over number fields.
In rank one, arithmeticity can fail, and hyperbolic manifolds give many non-arithmetic examples.
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<document source="https://tamnd.github.io/math2/node/FEPRJR3MND.md" tag="FEPRJR3MND" kind="concept">
# Eilenberg–Moore category

## Summary

The Eilenberg–Moore category of a [[monad|FF6XXA6D1A]] $T$ on a category $\mathcal{C}$ is the category of [[algebras over|QG1ZXRC8EH]] $T$ and their morphisms.
It comes with a forgetful functor to $\mathcal{C}$ and a free algebra functor, and these form an adjunction whose monad is $T$ again.
It is the terminal adjunction that gives $T$, and so it shows that every monad comes from an adjunction.

## Explanations

### L1 human

Given a monad, collect all the ways of evaluating its formal expressions, that is, all its algebras.
Together with the maps that respect evaluation, they form a new category.
For the list monad on sets, this new category is the category of monoids.
So the Eilenberg–Moore category rebuilds an algebraic category from its monad.
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<document source="https://tamnd.github.io/math2/node/FESTJ0K9TX.md" tag="FESTJ0K9TX" kind="statement">
# Sylow theorems

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a [[prime number|01826KMG2H]], and write $\abs{G} = p^n m$ with $n \ge 0$ and $p \nmid m$.
A [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$ is a subgroup of order $p^n$, and $n_p$ denotes the number of them.

1. $G$ has a Sylow $p$-subgroup.
2. Any two Sylow $p$-subgroups of $G$ are [[conjugate|0H11ZPX4ND]] in $G$, and every [[$p$-subgroup|06QYRTX9KQ]] of $G$ is contained in some Sylow $p$-subgroup.
3. $n_p \equiv 1 \pmod p$, $n_p$ divides $m$, and $n_p = [G : N_G(P)]$ for any Sylow $p$-subgroup $P$, where $N_G(P)$ is the [[normaliser|0YGN3VAA0D]] of $P$.

## Remarks

Each part has its own node: the [[first Sylow theorem|0V98WY94B8]], the [[second Sylow theorem|09GCQ5STVJ]] and the [[third Sylow theorem|0Q8DGAZSJ4]].
The rest of part 3 is in the nodes [[$n_p$ divides the index|0M3PYJ3TSE]] and [[$n_p$ is the index of the normaliser|09ZJ6CAK7S]].
The usual proofs let a group act on a set and count orbits or fixed points.
Part 1 uses the action on subsets of size $p^n$, or induction on $\abs{G}$.
Parts 2 and 3 let a $p$-group act on conjugates or cosets, and count its fixed points modulo $p$.

Example: every group $G$ of order $15 = 3 \cdot 5$ is cyclic.
By part 3, $n_3$ divides $5$ and $n_3 \equiv 1 \pmod 3$, so $n_3 = 1$; and $n_5$ divides $3$ and $n_5 \equiv 1 \pmod 5$, so $n_5 = 1$.
So $G$ has a [[normal|0CB0DX5BVV]] subgroup of order $3$ and one of order $5$.
Their intersection is $\set{e}$, since its order divides both $3$ and $5$.
Hence $G \cong \Z/3\Z \times \Z/5\Z \cong \Z/15\Z$.

The prime power in part 1 matters: the converse of [[Lagrange's theorem|3C53TD]] fails for other divisors, since [[$A_4$ has no subgroup of order $6$|0PDBVPP233]].

## Explanations

### L1 human

Lagrange's theorem says that the order of a subgroup divides the order of the group, but a group need not have a subgroup of each such order.
The Sylow theorems say that this does hold for the largest prime power dividing the order.
Moreover, these largest prime-power subgroups are all alike, and their number obeys strict rules.

### L2 human

The Sylow theorems are the first main tool to study finite groups through their order alone.
A typical use: the counting rules in part 3 force $n_p = 1$, so the Sylow subgroup is normal and the group is not simple.
In this way one shows, for example, that no group of order $20$ and no group of order $pq$, with primes $p < q$, is simple.
The theorems lead to the [[Frattini argument|0QTZDKJEDV]], to [[Hall subgroups|KPWW9X14V2]] in solvable groups, and to the study of how $p$-elements are conjugate, as in the [[focal subgroup theorem|H1BSG0J5G1]].
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# Support of a module

## Summary

The support of a [[module|0NZ8JA25WK]] $M$ over a [[commutative ring|0310REJJY8]] $R$ is the set of [[prime ideals|00MHHZ9SNE]] $\mathfrak p$ at which the [[localization|68E0F8N94E]] $M_{\mathfrak p}$ is nonzero.
It is a subset of the [[spectrum|FB45BHHA2J]] of $R$, and it tells where on that space the module "lives".
For a [[finitely generated module|05QY1VEBYD]], it is the closed set of primes containing the [[annihilator|HB4XDKJTW8]] of $M$.

## Explanations

### L1 human

Think of the prime ideals of $R$ as points and of the module as something spread over these points.
The support is the set of points where the module is not zero.
For the integers, the module $\Z/6\Z$ lives only at the primes $2$ and $3$, since every other prime becomes invertible and kills it.
The [[definition|B0223N93XS]] makes this precise.
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<document source="https://tamnd.github.io/math2/node/FETY7R4X86.md" tag="FETY7R4X86" kind="concept">
# Topological data analysis

## Summary

Topological data analysis studies the shape of data with tools from algebraic topology.
It turns a finite set of points, usually in a [[metric space|0VKD1TP7F3]], into a family of [[simplicial complexes|V8GZ0W1ZZ5]] indexed by a scale.
It then records how topological features such as components and loops appear and disappear as the scale grows.
The main tool is [[persistent homology|S8SNBHVPSP]], whose output, a barcode or persistence diagram, is stable under small perturbations of the data.
It is used to find clusters, loops and voids in data from biology, materials science and image analysis.

## Explanations

### L1 human

Imagine points scattered around a ring.
Draw a ball of radius $r$ around each point and watch what happens as $r$ grows.
For small $r$ the balls are separate dots; then they join into a ring with a hole; for large $r$ the hole fills in.
The hole lives for a long range of $r$, which is a strong sign that the data really lie on a circle.
Topological data analysis measures such long-lived features and ignores the short-lived ones, which are mostly noise.
</document>

<document source="https://tamnd.github.io/math2/node/FETZ0VZGMD.md" tag="FETZ0VZGMD" kind="statement">
# Beck's monadicity theorem

## Statement

Let $U \colon \mathcal{D} \to \mathcal{C}$ be a functor.
A pair of morphisms $f, g \colon d \to d'$ in $\mathcal{D}$ is *$U$-split* if $U(f), U(g)$ have a split coequalizer in $\mathcal{C}$.
A split coequalizer of $p, q \colon x \to y$ is a morphism $e \colon y \to z$ with $s \colon z \to y$ and $t \colon y \to x$ such that $e p = e q$, $e s = \id_z$, $p t = \id_y$ and $q t = s e$.
Then $U$ is [[monadic|DNTDCKZBKK]] if and only if these three conditions hold.

1. $U$ has a [[left adjoint|TZHF6ZM92S]].
2. $U$ reflects isomorphisms: if $U(h)$ is an isomorphism, then $h$ is an isomorphism.
3. Every $U$-split pair in $\mathcal{D}$ has a [[coequalizer|DVSTWGVCSZ]] in $\mathcal{D}$, and $U$ preserves these coequalizers.

## Remarks

A split coequalizer is a coequalizer, and every functor preserves it, because the equations that define it are preserved.
For the proof, let $T = U F$ and let $(A, a)$ be an algebra in the [[Eilenberg–Moore category|FEPRJR3MND]].
The pair $F(a), \varepsilon_{F(A)} \colon F U F (A) \to F(A)$ is $U$-split, and its coequalizer defines a left adjoint of the comparison functor.
Conditions 2 and 3 then make the unit and counit of this adjunction isomorphisms.

Example: the forgetful functor from groups to sets satisfies all three conditions, so it is monadic.
The same holds for rings, modules over a ring, and every variety of universal algebras.

Without condition 2 the conclusion fails.
The forgetful functor from topological spaces to sets has a left adjoint, but a continuous bijection need not be a homeomorphism, and this functor is not monadic.

## Explanations

### L1 human

The theorem gives a practical test for deciding whether a category is algebraic over another one.
One checks that free objects exist, that the forgetful functor detects isomorphisms, and that certain quotients exist and are computed underneath.
When the test passes, the objects are exactly the algebras of a monad.

### L2 human

Beck's theorem is the standard tool for proving monadicity, for example of compact Hausdorff spaces over sets or of modules over a ring.
There is also a variant that uses reflexive coequalizers: a right adjoint that reflects isomorphisms and in whose domain reflexive coequalizers exist and are preserved is monadic.
The dual statement characterizes comonadic functors, and in algebraic geometry it is the basis of descent theory.
In the strict form, $U$ creates coequalizers of $U$-split pairs, and the comparison functor is then an isomorphism.
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<document source="https://tamnd.github.io/math2/node/FEXZC8XKN1.md" tag="FEXZC8XKN1" kind="statement">
# M. Riesz extension theorem

## Statement

Let $E$ be a real [[vector space|000AYPA9AJ]], let $F \subseteq E$ be a [[linear subspace|00YC1D652K]], and let $K \subseteq E$ be a [[convex cone|6VE4KA12VD]].
Let $\varphi \colon F \to \R$ be a [[linear form|07YQ1C78G0]] such that $\varphi(x) \ge 0$ for every $x \in F \cap K$.
Assume that for every $y \in E$ there is $x \in F$ with $y - x \in K$.
Then there is a linear form $\psi \colon E \to \R$ such that $\psi(x) = \varphi(x)$ for all $x \in F$ and $\psi(y) \ge 0$ for all $y \in K$.

## Remarks

The proof extends $\varphi$ one dimension at a time, as in the proof of the [[Hahn–Banach theorem|QB5AE3VYZZ]].
To add a vector $y \notin F$, one needs a value $c = \psi(y)$ with $\sup \varphi(x) \le c \le \inf \varphi(x')$, where $x$ runs over $F$ with $y - x \in K$ and $x'$ runs over $F$ with $x' - y \in K$.
The hypothesis on $F$ makes both sets nonempty, and positivity of $\varphi$ on $F \cap K$ makes the supremum at most the infimum.
Zorn's lemma then gives an extension to all of $E$.
In the moment problem, $E$ is a space of functions containing the polynomials and $K$ is the cone of nonnegative functions in $E$.
There $F$ is the space of polynomials and $\varphi$ is the moment functional $x^n \mapsto m_n$.
The positive extension $\psi$ is then turned into a measure by a representation theorem, which solves the [[Hamburger moment problem|TBNCFXTMMW]] for a positive moment functional.
Without the assumption that $y - x \in K$ for some $x \in F$, an extension may fail to exist.
For example, take $E = \R^2$, $K = \set{(s, t) : t > 0} \cup \set{(s, 0) : s \ge 0}$, $F = \R \times \set{0}$ and $\varphi(s, 0) = s$; then $\varphi \ge 0$ on $F \cap K$, but a positive extension $\psi(s, t) = s + ct$ would need $s + c \ge 0$ for all $s$, which is impossible.

## Explanations

### L1 human

A linear rule that gives nonnegative values on the "positive" vectors of a subspace can be extended to the whole space without losing this property.
The only condition is that the subspace is large enough: every vector must lie above some vector of the subspace.
It is the order version of the Hahn–Banach theorem.

### L2 human

The theorem is the abstract tool behind the existence parts of moment problems.
A sequence $(m_n)$ defines a linear form $L$ on polynomials by $L(x^n) = m_n$.
Suppose that $L(p) \ge 0$ for every polynomial $p \ge 0$.
Then the theorem extends $L$ to a positive linear form on a larger space of functions, such as the continuous functions bounded by a polynomial.
The [[Riesz representation theorem|TTC40N49M6]] then gives a measure with moments $m_n$.
The same argument, with polynomials in several variables and nonnegativity on a closed set, gives the multivariate form of this existence result.
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<document source="https://tamnd.github.io/math2/node/FEZ2HAFZTM.md" tag="FEZ2HAFZTM" kind="definition">
# Ramanujan theta function

## Definition

Let $a, b \in \C \setminus \set{0}$ with $\abs{ab} < 1$.
*Ramanujan's general theta function* is
$$f(a, b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2}\, b^{n(n-1)/2}.$$
Its standard special cases, for $q \in \C$ with $\abs{q} < 1$, are
$$\varphi(q) = f(q, q), \qquad \psi(q) = f(q, q^3), \qquad f(-q) = f(-q, -q^2).$$

## Remarks

The series converges absolutely: for $n \ge 0$ the term is $(ab)^{n(n-1)/2} a^n$, and for $n < 0$ it is $(ab)^{m(m-1)/2} b^{m}$ with $m = -n$, so the terms decay like $\abs{ab}^{n^2/2}$.
Replacing $n$ by $-n$ shows $f(a, b) = f(b, a)$.
The [[Jacobi triple product|JFP2DGMNXT]] gives $f(a, b) = (-a; ab)_\infty\, (-b; ab)_\infty\, (ab; ab)_\infty$ with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].

Example: $\psi(q) = \sum_{n \in \Z} q^{n(n+1)/2 + 3n(n-1)/2} = \sum_{n \in \Z} q^{2n^2 - n}$.
The exponents for $n = 0, 1, -1, 2, -2, \dots$ are $0, 1, 3, 6, 10, \dots$, the triangular numbers, each once, so $\psi(q) = \sum_{k \ge 0} q^{k(k+1)/2}$.
Also $f(-q) = \sum_{n \in \Z} (-1)^n q^{n(3n-1)/2} = (q; q)_\infty$, by [[Euler's pentagonal number theorem|R587SR9GNV]].

Non-example: $f(a, b)$ is not defined for $a = b = 1$, since $\abs{ab} = 1$ and the series $\sum_{n \in \Z} 1$ diverges.

In the notation of the Jacobi theta function, $f(q e^{2iz}, q e^{-2iz}) = \vartheta_3(z, q)$.
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<document source="https://tamnd.github.io/math2/node/FF1JW893K2.md" tag="FF1JW893K2" kind="definition">
# Convolution on the line

## Definition

Let $f, g \colon \R \to \C$ be Lebesgue measurable functions.
Let $x \in \R$ be such that $\int_\R \abs{f(y) \, g(x - y)} \, dy < \infty$.
The *convolution* of $f$ and $g$ at $x$ is the [[Lebesgue integral|DWJF7ABPR1]]
$$(f * g)(x) = \int_\R f(y) \, g(x - y) \, dy.$$
The function $f * g$ is defined on the set of such $x$.

## Remarks

If $f$ and $g$ lie in [[$L^1(\R)$|SASP14MA1Z]], then $f * g$ is defined at almost every $x$, it lies in $L^1(\R)$, and $\norm{f * g}_1 \le \norm{f}_1 \norm{g}_1$.
If $f \in L^p(\R)$ and $g \in L^q(\R)$ with $1/p + 1/q = 1$, then $f * g$ is defined everywhere and is bounded and uniformly continuous.

Example: $f(x) = g(x) = e^{-x^2}$.
Completing the square gives $y^2 + (x - y)^2 = 2(y - x/2)^2 + x^2/2$, so
$$(f * g)(x) = e^{-x^2/2} \int_\R e^{-2(y - x/2)^2} \, dy = \sqrt{\pi/2} \, e^{-x^2/2}.$$
The convolution of two Gaussians is again a Gaussian, wider than both.

Non-example: $f = g = 1$ on $\R$.
Then $\int_\R \abs{f(y) g(x - y)} \, dy = \int_\R 1 \, dy = \infty$ for every $x$, so $f * g$ is defined nowhere.

The integral can also be written $\int_\R f(x - y) g(y) \, dy$, by the substitution $y \mapsto x - y$.
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<document source="https://tamnd.github.io/math2/node/FF27N0BWTP.md" tag="FF27N0BWTP" kind="concept">
# Pseudo-differential operator

## Summary

A pseudo-differential operator is an operator of the form $u \mapsto \int e^{i x \cdot \xi} a(x, \xi) \hat{u}(\xi) \, d\xi / (2\pi)^n$, where the symbol $a$ belongs to a [[symbol class|AM4RNWVV4Q]].
Linear differential operators with smooth bounded coefficients are the case where $a$ is a polynomial in $\xi$.
Pseudo-differential operators form an algebra that also contains approximate inverses of elliptic operators.
They are a basic tool of microlocal analysis and of the theory of linear partial differential equations.

## Explanations

### L1 human

The [[Fourier transform|DPVM34ADZV]] turns differentiation into multiplication: $\partial_x$ becomes multiplication by $i\xi$.
So a differential operator is the same as multiplication by a polynomial in frequency space.
A pseudo-differential operator allows other functions of frequency, such as $\sqrt{1 + \xi^2}$, and lets them depend on position too.
This makes it possible to divide by a symbol, and so to invert an elliptic operator up to a small error.
For example, $(1 - \Delta)^{-1}$ is the pseudo-differential operator with symbol $(1 + \abs{\xi}^2)^{-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/FF2KNFNTB7.md" tag="FF2KNFNTB7" kind="definition">
# Information geometry

## Definition

Let $(X, \mu)$ be a measure space and let $\Theta \subseteq \R^n$ be open.
A *statistical model* is a family $\set{p(\cdot\,; \theta) : \theta \in \Theta}$ of positive [[probability densities|04YG690N5V]] on $X$ with respect to $\mu$ such that:

1. $\theta \mapsto p(x; \theta)$ is smooth for every $x \in X$, and derivatives in $\theta$ may be taken under the integral sign;
2. with $\ell(x; \theta) = \log p(x; \theta)$ and $\partial_i = \partial/\partial\theta^i$, the functions $\partial_1 \ell, \dots, \partial_n \ell$ of $x$ are linearly independent for each $\theta$.

The *Fisher information metric* is the [[Riemannian metric|Q61VFHCH2J]] on $\Theta$ given by
$$g_{ij}(\theta) = \mathbb{E}_\theta\left[\partial_i \ell \; \partial_j \ell\right],$$
where $\mathbb{E}_\theta$ is the [[expected value|0DCT0QZVQ1]] for the density $p(\cdot\,; \theta)$.
For a real number $\alpha$, the *$\alpha$-connection* is the [[affine connection|JX65KSVHSJ]] on $\Theta$ with
$$\Gamma^{(\alpha)}_{ij,k}(\theta) = \mathbb{E}_\theta\left[\left(\partial_i \partial_j \ell + \frac{1 - \alpha}{2}\, \partial_i \ell \; \partial_j \ell\right) \partial_k \ell\right],$$
where $\Gamma^{(\alpha)}_{ij,k} = g\left(\nabla^{(\alpha)}_{\partial_i} \partial_j, \partial_k\right)$.
*Information geometry* is the study of a statistical model through its Fisher metric and its $\alpha$-connections.

## Remarks

Example: the normal densities $p(x; \mu, \sigma) = (2\pi\sigma^2)^{-1/2} e^{-(x - \mu)^2/(2\sigma^2)}$ on $\R$, with $\theta = (\mu, \sigma)$ and $\sigma > 0$.
Here $\partial_\mu \ell = (x - \mu)/\sigma^2$ and $\partial_\sigma \ell = (Z^2 - 1)/\sigma$ with $Z = (x - \mu)/\sigma$ standard normal.
Since $\mathbb{E}[Z^2] = 1$, $\mathbb{E}[Z^4] = 3$ and odd moments vanish, the metric is $(d\mu^2 + 2\, d\sigma^2)/\sigma^2$.
This is a hyperbolic plane of constant curvature $-1/2$.
Non-example: the family $p(x; \theta_1, \theta_2)$ of normal densities with mean $\theta_1 + \theta_2$ and variance $1$ fails clause 2, since $\partial_1 \ell = \partial_2 \ell$; its Fisher matrix has all entries $1$ and is singular.
The $0$-connection is the [[Levi-Civita connection|CAZYGE1T4E]] of $g$.
The $1$-connection and the $(-1)$-connection are called the exponential and the mixture connection, and they are dual to each other with respect to $g$.
The Fisher metric is the Hessian at $\theta' = \theta$ of the [[Kullback–Leibler divergence|PQBW43X5RE]] $D(p_\theta \,\|\, p_{\theta'})$.
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# Plurisubharmonic function

## Summary

A plurisubharmonic function on an open subset of $\C^n$ is an upper semicontinuous function whose restriction to every complex line is [[subharmonic|Z1MHMGFNFW]].
For $C^2$ functions this means that the complex Hessian $\left(\partial^2 u / \partial z_j \partial \bar z_k\right)$ is positive semidefinite, that is, the [[Levi form|80MTNXWW80]] is nonnegative.
If $f$ is [[holomorphic|507C34W0ZT]], then $\log\abs{f}$ is plurisubharmonic, and this is the main source of examples.
Plurisubharmonic functions are the basic objects of pluripotential theory, and they characterise [[pseudoconvexity|AHYZMSBBNH]] and domains of holomorphy.

## Explanations

### L1 human

A subharmonic function of one complex variable lies below the harmonic function with the same values on any circle.
A plurisubharmonic function of several complex variables has this property on every complex line through every point.
Convex functions are examples, but the class is much larger, because only complex directions matter.
For example, $u(z_1, z_2) = \abs{z_1}^2$ is plurisubharmonic on $\C^2$: on each complex line it is the squared modulus of an affine function of one variable.
The function $\log\abs{z_1 z_2 - 1}$ is plurisubharmonic too, and it equals $-\infty$ on the curve $z_1 z_2 = 1$.
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# Monad

## Summary

A monad on a category is an endofunctor $T$ with a unit $\eta \colon \mathrm{Id} \Rightarrow T$ and a multiplication $\mu \colon T T \Rightarrow T$ that satisfy associativity and unit laws.
It is the categorical form of a monoid, with composition of [[functors|28NQ0KRHD3]] in place of multiplication.
Every pair of [[adjoint functors|TZHF6ZM92S]] gives a monad, and every monad comes from an adjunction.
Monads describe algebraic structure: groups, monoids, rings and many other algebraic categories are the algebras of a monad on sets.

## Explanations

### L1 human

Think of $T(X)$ as the set of formal expressions built from elements of $X$.
The unit turns an element into the simplest expression, the element itself.
The multiplication flattens an expression made of expressions into a single expression.
For example, let $T(X)$ be the set of finite lists of elements of $X$.
The unit sends $x$ to the one-element list $(x)$, and the multiplication concatenates a list of lists into one list.
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<document source="https://tamnd.github.io/math2/node/FF8P46NH5S.md" tag="FF8P46NH5S" kind="definition">
# Minimal harmonic function

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a domain.
Let $h$ be a [[positive harmonic function|EYMQ82Z8XJ]] on $\Omega$.
The function $h$ is *minimal* if every [[harmonic function|0DEB0JMN0W]] $v$ on $\Omega$ with $0 \le v \le h$ equals $c h$ for a constant $c \in [0, 1]$.

## Remarks

Some authors also normalise $h(x_0) = 1$ at a fixed reference point $x_0$.
Example: on the unit disc $\mathbb{D}$, let $h(z) = \frac{1 - \abs{z}^2}{\abs{1 - z}^2}$, the [[Poisson kernel|TCYSWCTYYM]] with pole $1$.
If $0 \le v \le h$ is harmonic, the [[Herglotz representation theorem|9ZBAX4ZPX0]] writes $v$ and $h - v$ as Poisson integrals of positive measures $\mu$ and $\nu$ with $\mu + \nu = \delta_1$, the measure of $h$.
Then $\mu$ is a multiple of $\delta_1$, so $v$ is a multiple of $h$, and $h$ is minimal.
Non-example: $h = 1$ on the unit disc is not minimal.
The function $v(z) = \frac{1 + \operatorname{Re} z}{2}$ is harmonic with $0 \le v \le 1$ on $\mathbb{D}$, but it is not constant.
On $\R^n$ every positive harmonic function is constant, so $h = 1$ is minimal there.
The minimal functions with $h(x_0) = 1$ are the extreme points of the convex set of positive harmonic functions with value $1$ at $x_0$.
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<document source="https://tamnd.github.io/math2/node/FF9ZPCZ59K.md" tag="FF9ZPCZ59K" kind="concept">
# Elliptic function

## Summary

An elliptic function is a meromorphic function on the complex plane with two periods that are linearly independent over the real numbers.
It is the same as a meromorphic function on a complex torus $\C / \Lambda$.
Elliptic functions arose from the inversion of elliptic integrals, and they are the analytic side of the theory of elliptic curves.

## Explanations

### L1 human

The sine function repeats when you move by $2\pi$ along the real axis.
An elliptic function repeats along two different directions, so its values are determined by one parallelogram.
Gluing the opposite sides of this parallelogram gives a torus, the surface of a doughnut, and the function lives on that torus.
A nonconstant elliptic function must have poles: without poles it would be bounded, since it is bounded on one parallelogram, and a bounded entire function is constant.
The [[Weierstrass elliptic function|361NAK68QK]] is the basic example, with a double pole at each lattice point.
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<document source="https://tamnd.github.io/math2/node/FFCH15ZNZG.md" tag="FFCH15ZNZG" kind="problem">
# Fourier restriction problem

## Problem

Let $n \ge 2$, let $S \subseteq \R^n$ be a compact smooth hypersurface, and let $\sigma$ be its surface measure.
Let $\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$.
For which pairs of exponents $1 \le p, q \le \infty$ is there a constant $C$ such that
$$\Big( \int_S \abs{\hat f(\xi)}^q \, d\sigma(\xi) \Big)^{1/q} \le C \norm{f}_{L^p(\R^n)}$$
for every [[Schwartz function|CG9JEATW3Q]] $f$ on $\R^n$?

## Remarks

For $p = 1$ the estimate holds for all $q$, since $\abs{\hat f} \le \norm{f}_{L^1}$ everywhere.
For $p = 2$ it fails for every $q$, since $\hat f$ is then an arbitrary $L^2$ function and has no meaningful values on a set of measure zero.
When $S$ is a piece of a hyperplane, it fails for every $p > 1$.
When $S$ has nonzero Gaussian curvature, the [[Stein–Tomas restriction theorem|ZZVWHNFY6F]] gives the estimate for $q = 2$ and $1 \le p \le 2(n+1)/(n+3)$.
For the sphere, the conjectured answer is the [[restriction conjecture|9CE11G48JY]]; it is known for $n = 2$ and open for $n \ge 3$.

## Explanations

### L1 human

The Fourier transform of a general function is only known up to sets of measure zero, so its values on a surface seem to have no meaning.
The problem asks for which $L^p$ spaces the Fourier transform can nevertheless be restricted to a curved surface, with a bound on the result.
The answer depends on the curvature of the surface and is a central open question of harmonic analysis.
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<document source="https://tamnd.github.io/math2/node/FFDYRMG5DP.md" tag="FFDYRMG5DP" kind="definition">
# Sobolev space

## Definition

Let $U \subseteq \R^n$ be open, let $k \ge 0$ be an integer and let $1 \le p \le \infty$.
The *Sobolev space* $W^{k,p}(U)$ is the set of functions $u \in L^p(U)$ with the following property.
For every multi-index $\alpha$ with $\abs{\alpha} \le k$, the [[weak derivative|C3EC148VZD]] $D^\alpha u$ exists and lies in $L^p(U)$.
Its norm is
$$\norm{u}_{W^{k,p}(U)} = \Big( \sum_{\abs{\alpha} \le k} \int_U \abs{D^\alpha u}^p \, dx \Big)^{1/p} \text{ for } p < \infty, \qquad \norm{u}_{W^{k,\infty}(U)} = \sum_{\abs{\alpha} \le k} \operatorname*{ess\,sup}_U \abs{D^\alpha u}.$$
The space $W^{k,p}_0(U)$ is the closure in $W^{k,p}(U)$ of the space $C_c^\infty(U)$ of test functions.

## Remarks

Here $L^p(U)$ is the [[Lp space|SASP14MA1Z]], and functions equal almost everywhere are identified.
Each $W^{k,p}(U)$ is a Banach space, and $H^k(U) = W^{k,2}(U)$ is a Hilbert space with inner product $\sum_{\abs{\alpha} \le k} \int_U D^\alpha u \, D^\alpha v \, dx$.
For $k = 0$ one gets $W^{0,p}(U) = L^p(U)$.
For $1 \le p < \infty$, the smooth functions in $W^{k,p}(U)$ are dense in it (Meyers–Serrin).

Example: $u(x) = \abs{x}$ lies in $W^{1,p}(-1, 1)$ for all $p$.
Indeed $u$ is bounded, and its weak derivative $\sgn x$ is bounded, so both lie in every $L^p(-1, 1)$.

Non-example: $\sgn x$ is bounded, so it lies in every $L^p(-1, 1)$, but it has no weak derivative on $(-1, 1)$.
So it is in no space $W^{1,p}(-1, 1)$.
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<document source="https://tamnd.github.io/math2/node/FFH90QRFFJ.md" tag="FFH90QRFFJ" kind="definition">
# Cellular homology

## Definition

Let $X$ be a [[CW complex|5W95QFHXPR]] with $n$-skeleton $X^n$ for $n \ge 0$, and put $X^{-1} = \emptyset$.
The *cellular chain group* in degree $n$ is the [[relative homology|4BVWH0BN28]] group $C_n^{\mathrm{CW}}(X) = H_n(X^n, X^{n-1})$, taken in [[singular homology|M0Q5CFJ7ZH]] with integer coefficients.
Let $\partial$ be the connecting map of the pair $(X^n, X^{n-1})$, and let $j$ be induced by the inclusion of pairs.
The *cellular boundary map* $d_n \colon C_n^{\mathrm{CW}}(X) \to C_{n-1}^{\mathrm{CW}}(X)$ is the composite
$$H_n(X^n, X^{n-1}) \xrightarrow{\ \partial\ } H_{n-1}(X^{n-1}) \xrightarrow{\ j\ } H_{n-1}(X^{n-1}, X^{n-2}).$$
Then $d_{n-1} \circ d_n = 0$, so these groups and maps form a [[chain complex|WQ82C0M3QB]].
The *cellular [[homology|2J4JJJYP22]]* $H_n^{\mathrm{CW}}(X)$ is the homology of this chain complex in degree $n$.

## Remarks

The group $H_n(X^n, X^{n-1})$ is free abelian with one generator $e_\alpha$ for each $n$-cell $e_\alpha$ of $X$.
In this basis, $d_n(e_\alpha) = \sum_\beta d_{\alpha\beta} e_\beta$, where $d_{\alpha\beta}$ is the [[degree|TMFG2P912R]] of the map $S^{n-1} \to X^{n-1} \to S^{n-1}$ given by the attaching map of $e_\alpha$ followed by the collapse of everything outside $e_\beta$.
The basic theorem is that $H_n^{\mathrm{CW}}(X) \cong H_n(X)$ for all $n$.

Example: the [[real projective plane|1DDYB6VVSR]] $\R P^2$ has one cell in each of the dimensions $0$, $1$, $2$.
The $2$-cell is attached by a map of degree $2$ onto the $1$-skeleton $S^1$, so $d_2 = 2$, and $d_1 = 0$ because the $1$-cell is a loop.
The cellular chain complex is $\Z \xrightarrow{2} \Z \xrightarrow{0} \Z$, which gives $H_0 = \Z$, $H_1 = \Z/2$ and $H_2 = 0$.

Non-example: the definition needs a CW structure, that is, a filtration by skeleta in which each $X^n$ comes from $X^{n-1}$ by attaching $n$-cells.
For an arbitrary increasing chain of subspaces, the groups $H_n(X^n, X^{n-1})$ need not vanish outside degree $n$, and the homology of the resulting complex need not be the homology of $X$.
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<document source="https://tamnd.github.io/math2/node/FFJMRQNF96.md" tag="FFJMRQNF96" kind="definition">
# Lindeberg's condition

## Definition

Let $X_1, X_2, \dots$ be [[independent|0GQ938GDE0]] real [[random variables|0PNRKMRC55]] with finite [[variances|0477R9PYAE]].
Let $\mu_k = E[X_k]$, let $\sigma_k^2 = \operatorname{Var}(X_k)$, and let $s_n^2 = \sigma_1^2 + \cdots + \sigma_n^2$, with $s_n > 0$ for all $n$.
The sequence satisfies *Lindeberg's condition* if, for every $\varepsilon > 0$,
$$\lim_{n \to \infty} \frac{1}{s_n^2} \sum_{k=1}^{n} E\left[(X_k - \mu_k)^2 \, \mathbf{1}_{\set{\abs{X_k - \mu_k} > \varepsilon s_n}}\right] = 0.$$

## Remarks

The Lindeberg central limit theorem states that under this condition $(S_n - \sum_{k \le n} \mu_k) / s_n$ converges in distribution to a standard normal variable, where $S_n = X_1 + \cdots + X_n$.

Example: an i.i.d. sequence with variance $\sigma^2 \in (0, \infty)$ satisfies the condition.
Here $s_n^2 = n \sigma^2$, and the sum equals $\sigma^{-2} E[(X_1 - \mu_1)^2 \mathbf{1}_{\set{\abs{X_1 - \mu_1} > \varepsilon \sigma \sqrt{n}}}]$.
It tends to $0$ by dominated convergence, since $E[(X_1 - \mu_1)^2] < \infty$.

Non-example: let $X_1$ be standard normal and $X_k = 0$ for $k \ge 2$.
Then $s_n = 1$ for all $n$, and the sum equals $E[X_1^2 \mathbf{1}_{\set{\abs{X_1} > \varepsilon}}] > 0$, which does not tend to $0$.
In this case one variable carries all the variance.

The condition implies Feller's condition $\max_{k \le n} \sigma_k^2 / s_n^2 \to 0$.
The Lyapunov condition, $s_n^{-2-\delta} \sum_{k \le n} E[\abs{X_k - \mu_k}^{2+\delta}] \to 0$ for some $\delta > 0$, implies Lindeberg's condition and is often easier to check.
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# Babenko–Beckner inequality

## Statement

Let $n \ge 1$, let $1 < p \le 2$, and let $q$ be the conjugate exponent, $\frac{1}{p} + \frac{1}{q} = 1$.
Put $A_p = \left( p^{1/p} / q^{1/q} \right)^{1/2}$.
Let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\, dx$ be the [[Fourier transform|DPVM34ADZV]].
Then for every $f \in L^1(\R^n) \cap L^p(\R^n)$,
$$\norm{\hat f}_{L^q(\R^n)} \le A_p^n\, \norm{f}_{L^p(\R^n)}.$$
The constant $A_p^n$ is the best possible: equality holds for $f(x) = e^{-\pi \abs{x}^2}$.

## Remarks

The inequality sharpens the [[Hausdorff–Young inequality|HXNJH6W23S]], whose constant is $1$; for $1 < p < 2$ one has $A_p < 1$.
K. I. Babenko proved it when $q$ is an even integer, and W. Beckner proved it for all $p$ in the range.
Beckner's proof goes through a sharp hypercontractive estimate for the Hermite semigroup and a central limit argument.

The equality case can be checked directly with the [[Fourier transform of the Gaussian|ZNEHMR2CCP]]: for $f(x) = e^{-\pi \abs{x}^2}$ one has $\hat f = f$ and $\norm{f}_r = r^{-n/(2r)}$ for every $r > 0$.
So $\norm{\hat f}_q / \norm{f}_p = \left( p^{1/p} / q^{1/q} \right)^{n/2} = A_p^n$.

Differentiating the inequality in $p$ at $p = 2$, where both sides are equal by Plancherel, gives the entropic uncertainty principle.

## Explanations

### L1 human

The Hausdorff–Young inequality says that the Fourier transform does not increase a certain size of a function.
The Babenko–Beckner inequality finds the exact best factor by which it shrinks that size.
The bell curve is the function for which the factor is reached.

### L2 human

This is a typical sharp inequality in analysis, with Gaussians as extremal functions.
The same happens for the logarithmic Sobolev inequality and the hypercontractivity of the Ornstein–Uhlenbeck semigroup, to which it is closely related.
Its derivative at $p = 2$ gives the Hirschman–Beckner entropic uncertainty principle, which in turn implies the Heisenberg inequality.
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<document source="https://tamnd.github.io/math2/node/FFR8NPWG64.md" tag="FFR8NPWG64" kind="statement">
# Convolution property of the Z-transform

## Statement

Let $x = (x_n)_{n \in \Z}$ and $y = (y_n)_{n \in \Z}$ be sequences of complex numbers with [[bilateral Z-transforms|3DBAMXA80B]] $X$ and $Y$.
Let $z$ be a point of both [[regions of convergence|1P8WX3EGHF]].
Then for every $n \in \Z$ the series $(x * y)_n = \sum_{k \in \Z} x_k \, y_{n-k}$ converges absolutely, $z$ lies in the region of convergence of $x * y$, and
$$\mathcal{Z}\{x * y\}(z) = X(z) \, Y(z).$$
If moreover $x_n = y_n = 0$ for all $n < 0$, then $(x * y)_n = \sum_{k=0}^{n} x_k y_{n-k}$ for $n \ge 0$, and the same identity holds for the [[unilateral Z-transforms|EPG6JPVJE7]].

## Remarks

The double series $\sum_{n} \sum_{k} \abs{x_k} \abs{y_{n-k}} \abs{z}^{-n}$ equals $\left(\sum_k \abs{x_k} \abs{z}^{-k}\right)\left(\sum_m \abs{y_m} \abs{z}^{-m}\right)$, which is finite.
So each inner sum converges absolutely, and the terms can be regrouped, which gives the product formula.
In the one-sided case this is the [[Cauchy product|0RVD9R2V99]] of two absolutely convergent series in $z^{-1}$.
Example: for $x_n = y_n = 1$ ($n \ge 0$), $(x * y)_n = n + 1$, and $\left(z/(z-1)\right)^2 = \sum_{n \ge 0} (n+1) z^{-n}$ for $\abs{z} > 1$.
If the two regions do not meet, the [[convolution|2Z29JN3VES]] may fail to exist: for $x_n = y_n = 1$ for all $n \in \Z$, each sum $\sum_k x_k y_{n-k}$ diverges.

## Explanations

### L1 human

Convolution of sequences is the operation that a linear time-invariant filter performs on its input.
The Z-transform turns this operation into ordinary multiplication of functions.
So passing a signal through a filter amounts to multiplying its transform by the transform of the filter.

### L2 human

The transform of the impulse response of a system is its transfer function, and the theorem says that the output transform is the input transform times the transfer function.
It is the discrete counterpart of the convolution theorem for the Laplace transform.
Together with the shift property, it reduces linear constant-coefficient difference equations to algebra.
</document>

<document source="https://tamnd.github.io/math2/node/FFRM041WQX.md" tag="FFRM041WQX" kind="definition">
# Hausdorff content

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]], and let $s > 0$.
For $U \subseteq X$, let $\operatorname{diam} U = \sup \set{d(x, y) : x, y \in U}$, with $\operatorname{diam} \emptyset = 0$.
The *$s$-dimensional Hausdorff content* of a set $E \subseteq X$ is
$$\mathcal{H}^s_\infty(E) = \inf \set{\sum_{i=1}^{\infty} (\operatorname{diam} U_i)^s : E \subseteq \bigcup_{i=1}^{\infty} U_i}.$$
The infimum is over all countable covers $(U_i)$ of $E$ by subsets of $X$.

## Remarks

The map $\mathcal{H}^s_\infty$ is an [[outer measure|BJ314H17HC]] on $X$.
It has the same null sets as the [[Hausdorff measure|Q1TGZK8RN6]] $\mathcal{H}^s$, so $\dim_H E = \inf \set{s > 0 : \mathcal{H}^s_\infty(E) = 0}$, with the convention $\inf \emptyset = \infty$.
Example: in $\R$, $\mathcal{H}^{1}_\infty([0,1]) = 1$.
The cover by $[0,1]$ itself gives at most $1$, and any countable cover has total diameter at least the Lebesgue measure $1$.
Non-example of additivity: in $\R$ with $s = 1/2$, each of $[0,1]$ and $[1,2]$ has content $1$.
Indeed, a cover that uses a set of diameter at least $1$ has total at least $1$.
In a cover with all diameters below $1$, one has $(\operatorname{diam} U_i)^{1/2} \ge \operatorname{diam} U_i$, and the diameters add up to at least $1$.
But the single set $[0,2]$ covers their union, so the union has content at most $\sqrt 2 < 2$.
</document>

<document source="https://tamnd.github.io/math2/node/FFTNS66XG6.md" tag="FFTNS66XG6" kind="example">
# Discontinuous additive functions via a Hamel basis

## Example

View $\R$ as a vector space over the field $\Q$ of rational numbers.
The numbers $1$ and $\sqrt{2}$ are linearly independent over $\Q$, because $\sqrt{2}$ is irrational.
By the [[axiom of choice|0HTGB13FE2]], this independent set extends to a [[basis|0KCQ9Y3QKA]] $B$ of $\R$ over $\Q$, called a Hamel basis.
Every real $x$ is then a unique finite sum $x = \sum_{b \in B} q_b(x)\, b$ with rational coefficients $q_b(x)$, almost all zero.
Define $f \colon \R \to \R$ by $f(x) = q_1(x)$, the coefficient of the basis vector $1$.

Check of additivity: the coefficients of $x + y$ are the sums of the coefficients of $x$ and $y$, so $f(x + y) = f(x) + f(y)$.
So $f$ is an [[additive function|KF719SZD5T]].
It is not of the form $f(x) = cx$: $f(1) = 1$ would force $c = 1$, but $f(\sqrt{2}) = 0 \ne \sqrt{2}$.
Hence $f$ is not [[continuous|0X3WGX95FE]], since every continuous additive function is linear.
One can also see this directly: $f$ takes only rational values, while a continuous function on $\R$ with only rational values is constant, and $f(0) = 0 \ne 1 = f(1)$.

## Explanations

### L1 human

Imagine writing every real number in a strange coordinate system with infinitely many rational coordinates.
The function $f$ just reads off one coordinate.
Reading a coordinate respects addition, so $f$ is additive.
Yet $f$ jumps around wildly and cannot be drawn, because the coordinate system itself comes from the axiom of choice.
</document>

<document source="https://tamnd.github.io/math2/node/FFYQAVT7MB.md" tag="FFYQAVT7MB" kind="concept">
# Picard–Lefschetz theory

## Summary

Picard–Lefschetz theory studies the topology of a complex manifold through a holomorphic function on it with nondegenerate critical points.
It looks at how the fibres change near the critical values.
Its central result, the Picard–Lefschetz formula, describes the [[monodromy|C8HD62AB6Q]] around a critical value in terms of the [[vanishing cycle|M5EK9AKXNR]] $\delta$ of the critical point.
Let $X_t$ be a smooth fibre of complex dimension $n$, near a critical value with a single nondegenerate critical point.
Let $T$ be the monodromy on $H_n(X_t; \Z)$ around this value.
Then $T(a) = a + \varepsilon_n \langle a, \delta \rangle \delta$, where $\langle \cdot, \cdot \rangle$ is the [[intersection form|WK8GKRKAVD]].
Here $\varepsilon_n = \pm 1$ is a sign that depends only on $n$ and on orientation conventions.
Émile Picard developed the case of surfaces and Solomon Lefschetz the general case.
Lefschetz used it to study the topology of algebraic varieties through Lefschetz pencils, together with the [[Lefschetz hyperplane theorem|EX6YSYN7B4]].
Its version in étale cohomology was a main tool in Deligne's proof of the Weil conjectures.

## Explanations

### L1 human

Cut a complex variety into a family of slices given by the level sets of a function.
Most slices look alike, but at finitely many values a slice gets a simple singular point, where a sphere in the slice has shrunk to a point.
Picard–Lefschetz theory tells what happens to the other cycles of a slice when one walks once around such a bad value.
A cycle comes back changed by a multiple of the shrinking sphere, and the multiple is the number of times the cycle crosses that sphere, counted with signs.
For curves, $n = 1$, the change is a [[Dehn twist|W890NRHGS4]] along the vanishing circle.
</document>

<document source="https://tamnd.github.io/math2/node/FFZHNFJCZS.md" tag="FFZHNFJCZS" kind="conjecture">
# Mahler conjecture

## Statement

Let $n \ge 1$ and let $K \subseteq \R^n$ be a compact [[convex body|437FSG3419]] with nonempty interior and $K = -K$.
Let $K^\circ = \set{y \in \R^n : x \cdot y \le 1 \text{ for all } x \in K}$ be its polar body, with $x \cdot y$ the [[dot product|0S52TNNTH9]], and let $\operatorname{vol}$ be [[Lebesgue measure|2Z2QVWZ262]].
The conjecture states that
$$\operatorname{vol}(K) \operatorname{vol}(K^\circ) \ge \frac{4^n}{n!},$$
which is the value for the cube $[-1, 1]^n$.

## Remarks

For the cube, $\operatorname{vol}([-1,1]^n) = 2^n$ and its polar is the cross-polytope of volume $2^n / n!$, which gives $4^n / n!$.
The conjecture is proved for $n = 2$, by Mahler, and for $n = 3$, by Iriyama and Shibata.
It is known for some classes of bodies, such as unconditional bodies and zonoids.
In general the Bourgain–Milman inequality gives $\operatorname{vol}(K) \operatorname{vol}(K^\circ) \ge c^n \frac{4^n}{n!}$ for an absolute constant $c > 0$, and Kuperberg gave an explicit such constant.
The upper bound is the [[Blaschke–Santaló inequality|MNJ030R4CA]].
There is also a version for convex bodies that are not symmetric, in which the simplex is the conjectured minimizer.

## Explanations

### L1 human

Multiply the volume of a symmetric convex shape by the volume of its polar shape.
The result does not change when the shape is stretched.
The largest possible product is reached by balls.
Mahler conjectured that the smallest possible product is reached by the cube, and this is still open in dimension four and higher.
</document>

<document source="https://tamnd.github.io/math2/node/FG33N4ZHDG.md" tag="FG33N4ZHDG" kind="concept">
# Geometric quantization

## Summary

Geometric quantization is a method to build a quantum system from a classical one, using the geometry of a [[symplectic manifold|CAN490EPRT]].
It first attaches to the manifold a complex line bundle with a connection whose curvature is the symplectic form.
Then it cuts down the space of sections of this bundle with a choice of polarization.
The method was developed by Kostant and Souriau in the 1960s and 1970s, and it is closely tied to the orbit method in representation theory.

## Explanations

### L1 human

A classical system lives on a phase space, and its observables are functions there; a quantum system lives in a Hilbert space, and its observables are operators.
Geometric quantization tries to pass from the first to the second in a way that depends only on the geometry, not on chosen coordinates.
Wave functions become sections of a line bundle over phase space.
A polarization says that they should depend on only half of the variables, such as positions and not momenta.
For the sphere with total area $2\pi\hbar k$, the method produces a space of dimension $k + 1$, which is the state space of a quantum spin.
</document>

<document source="https://tamnd.github.io/math2/node/FG5448NZ18.md" tag="FG5448NZ18" kind="statement">
# Cauchy integral formula on a polydisc

## Statement

Let $a \in \C^n$ and $r = (r_1, \dots, r_n)$ with each $r_j > 0$.
Let $P = P(a, r)$ be the [[polydisc|DJBWP7D3M8]] with centre $a$ and polyradius $r$, and let $T = T(a, r)$ be its [[distinguished boundary|A6HN207Y2T]].
Let $f$ be a [[holomorphic function|507C34W0ZT]] on an open set of $\C^n$ that contains the closed polydisc $\overline{P}$.
Then for every $z \in P$,
$$f(z) = \frac{1}{(2\pi i)^n} \int_{T} \frac{f(\zeta)}{(\zeta_1 - z_1) \cdots (\zeta_n - z_n)} \, d\zeta_1 \cdots d\zeta_n,$$
where the integral is the [[torus integral|R49Q9M519Z]] over $T$.

## Remarks

The proof applies the one-variable [[Cauchy integral formula|0DG10TVYPN]] in $z_1$, then in $z_2$ inside the integral, and so on, $n$ times.
Continuity of $f$ on $T$ allows the iterated integrals to be combined into one torus integral.
The formula holds under the weaker hypothesis that $f$ is continuous on $\overline{P}$ and holomorphic on $P$.
Expanding $1/(\zeta_j - z_j)$ in a geometric series in $(z_j - a_j)/(\zeta_j - a_j)$ gives the power series of $f$ at $a$, with coefficients
$$c_\alpha = \frac{1}{(2\pi i)^n} \int_T \frac{f(\zeta)}{(\zeta - a)^{\alpha + \mathbf{1}}} \, d\zeta, \qquad (\zeta - a)^{\alpha + \mathbf{1}} = \prod_{j} (\zeta_j - a_j)^{\alpha_j + 1}.$$
Example: for $f(z) = z_1 z_2$ and $P$ the unit polydisc of $\C^2$, at $z = 0$ the right side is $(2\pi i)^{-2} \int_T d\zeta_1 \, d\zeta_2 = 0 = f(0)$.
Only the values of $f$ on the $n$-dimensional torus $T$ are used, not on the whole $(2n-1)$-dimensional boundary of $P$.

## Explanations

### L1 human

The formula says that a holomorphic function on a polydisc is determined by its values on a thin torus.
Each value inside is an average of boundary values, weighted by a simple kernel.
It is the one-variable Cauchy formula used once in each coordinate.

### L2 human

This is the basic tool of local theory in several variables.
From it one gets that holomorphic functions are locally sums of convergent power series, the Cauchy estimates $\abs{c_\alpha} \le \max_T \abs{f} / r^\alpha$, and the infinite differentiability of holomorphic functions.
Since the kernel is a product of one-variable kernels, it is tied to the product shape of the polydisc.
On other domains one uses integral formulas with other kernels, such as the [[Bochner–Martinelli formula|YMHYAEJQ80]].
</document>

<document source="https://tamnd.github.io/math2/node/FGF275MNQJ.md" tag="FGF275MNQJ" kind="definition">
# Filtered algebra

## Definition

Let $K$ be a [[commutative ring|0GYGHKX834]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $K$ with unit $1$.
An *(increasing, exhaustive) filtration* of $A$ is a family $(F_i)_{i \ge 0}$ of $K$-submodules of $A$ such that:

1. $F_i \subseteq F_{i+1}$ for all $i \ge 0$.
2. $\bigcup_{i \ge 0} F_i = A$.
3. $F_i F_j \subseteq F_{i+j}$ for all $i, j \ge 0$.
4. $1 \in F_0$.

A *filtered algebra* is an algebra together with such a filtration.

## Remarks

The *associated graded algebra* is $\operatorname{gr} A = \bigoplus_{i \ge 0} F_i / F_{i-1}$ with $F_{-1} = 0$; clause 3 makes the product $(a + F_{i-1})(b + F_{j-1}) = ab + F_{i+j-1}$ well defined.
Example: $A = K[x]$ with $F_i$ the polynomials of degree at most $i$.
Clauses 1, 2 and 4 are clear, and clause 3 holds since $\deg(fg) \le \deg f + \deg g$.
Here $\operatorname{gr} A \cong K[x]$.
Non-example: in $K[x]$ take $F_0 = K + Kx$ and $F_i$ the polynomials of degree at most $i + 1$ for $i \ge 1$.
Clauses 1, 2 and 4 hold, but clause 3 fails: $x \cdot x = x^2 \notin F_0$.
Let $U(\mathfrak{g})$ be the universal enveloping algebra of a Lie algebra $\mathfrak{g}$ over a field, with $F_i$ spanned by products of at most $i$ elements of $\mathfrak{g}$.
Then the Poincaré–Birkhoff–Witt theorem gives $\operatorname{gr} U(\mathfrak{g}) \cong S(\mathfrak{g})$.
</document>

<document source="https://tamnd.github.io/math2/node/FGKNAFNHA3.md" tag="FGKNAFNHA3" kind="concept">
# Incidence structure

## Summary

An incidence structure consists of a set of points, a set of lines, and a relation that says which points lie on which lines.
Nothing else is assumed: there are no distances, no angles and no coordinates.
Projective planes, affine planes, block designs and many other combinatorial objects are incidence structures with extra axioms.
It is the basic object of [[incidence geometry|CSXT0BG2CM]].

## Explanations

### L1 human

Take some dots and some curves, and record only which dots sit on which curves.
That record is an incidence structure.
For example, take the $7$ points and $7$ lines of the Fano plane, where each line has $3$ points and each point is on $3$ lines.
Exchanging the roles of points and lines gives the dual structure, which is again an incidence structure.
</document>

<document source="https://tamnd.github.io/math2/node/FGP202VKXW.md" tag="FGP202VKXW" kind="definition">
# Holomorphic vector bundle

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] and let $r \ge 0$ be an integer.
A *holomorphic vector bundle* of rank $r$ over $X$ is a complex manifold $E$ with a [[holomorphic map|3MQTWKBBBT]] $\pi \colon E \to X$ such that:

1. Each fibre $E_x = \pi^{-1}(x)$ has the structure of a complex vector space of dimension $r$.
2. Each point of $X$ has an open neighbourhood $U$ and a biholomorphic map $\varphi \colon \pi^{-1}(U) \to U \times \C^r$ with $\mathrm{pr}_1 \circ \varphi = \pi$.
3. For each $x \in U$, the restriction $\varphi \colon E_x \to \set{x} \times \C^r$ is $\C$-linear.

## Remarks

Given two such trivializations $\varphi_i$ over $U_i$ and $\varphi_j$ over $U_j$, one has $\varphi_i \circ \varphi_j^{-1}(x, v) = (x, g_{ij}(x) v)$ with $g_{ij} \colon U_i \cap U_j \to GL_r(\C)$ holomorphic.
Conversely, holomorphic maps $g_{ij}$ with $g_{ij} g_{jk} = g_{ik}$ on triple overlaps glue the products $U_i \times \C^r$ to a holomorphic vector bundle.
A holomorphic vector bundle is in particular a smooth complex [[vector bundle|P4DZM975BV]].

Example: the holomorphic tangent bundle $T X$ of a complex manifold of dimension $n$ has rank $n$.
A holomorphic chart $z = (z_1, \dots, z_n)$ trivializes it by the frame $\partial/\partial z_1, \dots, \partial/\partial z_n$.
The transition functions are the complex Jacobian matrices of the coordinate changes, which are holomorphic.

Non-example: cover the Riemann sphere by $U_0 = \C$ and $U_1 = \mathbb{P}^1 \setminus \set{0}$, and glue $U_0 \times \C$ and $U_1 \times \C$ over $\C^*$ with $g_{01}(z) = \bar z$.
This gives a smooth complex line bundle, but $g_{01}$ is not holomorphic.
So the trivializations of this atlas are not biholomorphic for any complex structure on the glued space, and clause 2 fails for them.

The sheaf of holomorphic sections of $E$ is a locally free sheaf of $\mathcal{O}_X$-modules of rank $r$, and every such sheaf arises in this way.
</document>

<document source="https://tamnd.github.io/math2/node/FGRPYWW1V1.md" tag="FGRPYWW1V1" kind="definition">
# Sasakian manifold

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]].
The *metric cone* over $M$ is the manifold $C(M) = (0, \infty) \times M$ with the metric $\bar g = dr^2 + r^2 g$, where $r$ is the coordinate on $(0, \infty)$.
The manifold $(M, g)$ is *Sasakian* if $C(M)$ has an integrable almost complex structure $J$ with two properties.
The metric $\bar g$ is a [[Kähler metric|CK1DCPC3D3]] for $J$, and $J$ is invariant under the dilations $(r, x) \mapsto (\lambda r, x)$ for $\lambda > 0$.

## Remarks

Example: the unit sphere $S^{2n+1} \subset \C^{n+1}$ with the round metric.
The map $(r, x) \mapsto r x$ identifies $C(S^{2n+1})$ with $\C^{n+1} \setminus \set{0}$, and it carries $dr^2 + r^2 g$ to the flat metric, which is Kähler.

Non-example: the round sphere $S^2$ is not Sasakian.
Its cone is $\R^3 \setminus \set{0}$, of odd dimension $3$, so it has no almost complex structure at all.

A Sasakian manifold has odd dimension $2n + 1$.
The vector field $\xi = J(r \partial_r)$, restricted to $\set{1} \times M$, is a unit Killing field, the [[Reeb vector field|GGM63KXCF2]] of the contact form $\eta = g(\xi, \cdot)$, which makes $M$ a [[contact manifold|HBQ8BZ7XB5]].
</document>

<document source="https://tamnd.github.io/math2/node/FGSKR6ZS3Q.md" tag="FGSKR6ZS3Q" kind="concept">
# Plurisubharmonic exhaustion function

## Summary

A plurisubharmonic exhaustion function on a complex manifold is a [[plurisubharmonic function|FF2P7ER3QT]] whose sublevel sets are relatively compact.
It tends to $+\infty$ at the "boundary" or "infinity" of the manifold.
Domains in $\C^n$ with such a function are exactly the pseudoconvex domains.
Complex manifolds with a smooth strictly plurisubharmonic one are exactly the Stein manifolds.

## Explanations

### L1 human

An exhaustion function is like a height function on a bowl.
Each region below a given height is a compact piece, and the pieces fill up the whole space as the height grows.
Asking the function to be plurisubharmonic means that it is subharmonic on every complex line, a complex form of convexity.
On $\C^n$ the function $\abs{z}^2$ is a plurisubharmonic exhaustion function: its sublevel sets are balls.
On the unit disc, $-\log(1 - \abs{z}^2)$ is one: it is subharmonic and blows up at the circle.
</document>

<document source="https://tamnd.github.io/math2/node/FGY1XAHVXP.md" tag="FGY1XAHVXP" kind="definition">
# Abel function

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a map.
An *Abel function* of $f$ is a [[function|0EPAT51DXQ]] $\alpha \colon X \to \C$ that satisfies the [[Abel equation|68SHB7Z721]]
$$\alpha(f(x)) = \alpha(x) + 1 \quad \text{for all } x \in X.$$
Many authors also require $\alpha$ to be injective, or, for real $X$, continuous and strictly monotone.

## Remarks

Example: for $f(x) = 3x$ on $X = (0, \infty)$, $\alpha(x) = \log_3 x$ is an Abel function, since $\log_3(3x) = \log_3 x + 1$.
Non-example: for the same $f$, $\alpha(x) = \ln x$ is not an Abel function, since $\ln(3x) = \ln x + \ln 3$ and $\ln 3 \ne 1$.
If $\alpha$ is an injective Abel function and $t$ is real with $\alpha(X) + t \subseteq \alpha(X)$, then $f^t(x) = \alpha^{-1}(\alpha(x) + t)$ satisfies $f^1 = f$ and $f^s \circ f^t = f^{s + t}$, so $(x, t) \mapsto f^t(x)$ solves the translation equation where it is defined.
In the example, $f^t(x) = 3^t x$.
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<document source="https://tamnd.github.io/math2/node/FH0538XCZR.md" tag="FH0538XCZR" kind="concept">
# Trapezoidal rule

## Summary

The trapezoidal rule approximates $\int_a^b f$ by the area of the trapezoid under the chord from $(a, f(a))$ to $(b, f(b))$.
Its composite form splits $[a, b]$ into $N$ equal parts and adds the trapezoids, with error of order $h^2$ for smooth $f$, where $h = (b - a)/N$.
It is the closed [[Newton–Cotes formula|BQPQAY1FDJ]] with two nodes, and it is remarkably accurate for smooth periodic functions over a full period.

## Explanations

### L1 human

Replace the curve over each small piece by the straight line between its endpoints, and add up the areas of the resulting trapezoids.
Each trapezoid has area equal to its width times the average of its two heights.
Halving the width of the pieces roughly cuts the error by four.
For $f(x) = x^2$ on $[0, 1]$ with two pieces, the trapezoids give $3/8$, slightly more than the true value $1/3$, because the parabola bends below its chords.
</document>

<document source="https://tamnd.github.io/math2/node/FH0Z9VHRH7.md" tag="FH0Z9VHRH7" kind="definition">
# Fine topology

## Definition

Let $n \ge 2$.
The *fine topology* on $\R^n$ is the coarsest [[topology|0CQHFPGJ45]] $\mathcal{T}$ on $\R^n$ with the following property.
For every open set $U \subseteq \R^n$ and every [[superharmonic function|X4V2S56YQQ]] $u$ on $U$, the set $U$ belongs to $\mathcal{T}$ and $u \colon U \to (-\infty, +\infty]$ is continuous for $\mathcal{T}$.
Sets in $\mathcal{T}$ are called *finely open*, and limits and continuity for $\mathcal{T}$ are called fine limits and fine continuity.

## Remarks

The fine topology is [[finer|072J68M6HX]] than the Euclidean topology, since every Euclidean open set is finely open by definition.
A set $V$ is finely open if and only if, for every $x \in V$, the complement $\R^n \setminus V$ is [[thin|7DGF6MMEGX]] at $x$.
Example: let $V = \R^3 \setminus \set{k^{-1} e_1 : k \ge 1}$.
The removed set is countable, hence a [[polar set|C95QJ7YH5T]], and a polar set is thin at every point.
So $V$ is finely open, but it is not Euclidean open, because every ball about $0 \in V$ meets the removed points.
Non-example: a closed half-space $H$ is not a fine neighbourhood of a point $x \in \partial H$, since the complementary open half-space is not thin at $x$.
</document>

<document source="https://tamnd.github.io/math2/node/FH250K068Z.md" tag="FH250K068Z" kind="statement">
# Banach–Tarski paradox

## Statement

Assume ZFC, that is, ZF with the [[axiom of choice|0HTGB13FE2]].
Let $B = \set{x \in \R^3 : \norm{x} \le 1}$ be the closed unit ball in [[Euclidean space|0XPR1E85W0]] $\R^3$, and let $B'$ be a translate of $B$ that is disjoint from $B$.
Then there are an integer $n \ge 1$ and a partition of $B$ into disjoint sets $A_1, \ldots, A_n$ with the following property.
There are [[isometries|0878DN3WCS]] $g_1, \ldots, g_n$ of $\R^3$ such that $g_1(A_1), \ldots, g_n(A_n)$ are pairwise disjoint with union $B \cup B'$.

## Remarks

The proof starts with two rotations that generate a [[free group|020Z5SEN27]] $F_2$ of rank $2$ inside the [[special orthogonal group|0CGYNDCWRR]] $SO(3)$.
Apart from the identity, the group $F_2$ splits into four pieces.
Left multiplication by the generators turns two of them into a copy of $F_2$, and the other two into another copy.
The axiom of choice picks one point from each orbit of $F_2$ on the sphere.
This transfers the splitting from the group to the sphere, outside a countable set of fixed points.
Radial segments carry it to the ball minus its centre, and the centre and the countable set are absorbed by a further rotation argument.
The pieces cannot all be Lebesgue measurable, since isometries preserve volume.
Five pieces suffice, and no fewer than five work.
The analogue fails in $\R^1$ and $\R^2$, because their isometry groups are [[amenable|PG8Z6F6DV9]].
So there are finitely additive isometry-invariant measures on all bounded subsets that give the unit ball a positive value.
A stronger form holds in $\R^n$ for each $n \ge 3$: any two bounded subsets with nonempty interior can be cut into finitely many pieces that match by isometries.

## Explanations

### L1 human

You can cut a solid ball into a few pieces.
Moving the pieces only by rotations and translations, you can put them back together as two balls of the same size as the first.
This seems to break the conservation of volume.
There is no contradiction, because the pieces are so scattered that they have no volume at all.
Such pieces can only be shown to exist with the axiom of choice; nobody can draw them.

### L2 human

The theorem shows that there is no finitely additive measure on all subsets of $\R^3$ that is invariant under isometries and gives the unit ball a positive finite value.
This is one reason why measure theory works with a restricted family of measurable sets.
The key algebraic input is a free subgroup of $SO(3)$.
The general idea behind the paradox is the failure of amenability.
A group that contains a free subgroup of rank $2$ is not amenable, and its actions can be paradoxical.
</document>

<document source="https://tamnd.github.io/math2/node/FH2A1PCXNB.md" tag="FH2A1PCXNB" kind="definition">
# Telegrapher's equations

## Definition

Let $L > 0$, $C > 0$, $R \ge 0$ and $G \ge 0$ be constants: the inductance, capacitance, resistance and conductance per unit length of a line.
The *telegrapher's equations* are the system of [[partial differential equations|GC5BDZ6QBC]] for the voltage $V(x, t)$ and the current $I(x, t)$, with $x \in \R$ and $t \in \R$:
$$\frac{\partial V}{\partial x} = -L \frac{\partial I}{\partial t} - R I, \qquad \frac{\partial I}{\partial x} = -C \frac{\partial V}{\partial t} - G V.$$
It is a first-order [[hyperbolic system|92C3S03AQZ]].

## Remarks

For $C^2$ solutions, differentiating the first equation in $x$ and using the second gives the telegraph equation
$$V_{xx} = L C\, V_{tt} + (R C + G L)\, V_t + R G\, V,$$
and the same equation holds for $I$.
Example: with $R = G = 0$, for any $F \in C^1(\R)$ the pair $V = F(x - ct)$, $I = \sqrt{C/L}\, F(x - ct)$ with $c = 1/\sqrt{LC}$ is a solution.
Indeed $V_x = F'$ and $-L I_t = L c \sqrt{C/L}\, F' = F'$, and $I_x = \sqrt{C/L}\, F'$ and $-C V_t = C c F' = \sqrt{C/L}\, F'$.
Non-example: with $R > 0$ and $G = 0$, this undamped pair fails the first equation, since the extra term $-R I$ is nonzero when $F \ne 0$.
When $R = G = 0$ the telegraph equation is the [[wave equation|5EHD2XBQDG]] with speed $1/\sqrt{LC}$.
</document>

<document source="https://tamnd.github.io/math2/node/FH3C0T643F.md" tag="FH3C0T643F" kind="concept">
# Groupoid

## Summary

A groupoid is a [[category|A2DE66937D]] in which every morphism is invertible.
A [[group|G5S1M9]] is the same as a groupoid with one object, so groupoids generalize groups to symmetries between several objects.
The [[fundamental groupoid|ZBHGXJF24E]] of a space, the action groupoid of a [[group action|0WCY42EV9V]] and the groupoid of an [[equivalence relation|0G8VEZ3WTK]] are standard examples.
Groupoids model homotopy 1-types, and they appear in topology, geometry and the theory of stacks.

## Explanations

### L1 human

A group describes the symmetries of one object.
A groupoid describes reversible transformations between several objects, where not every pair of objects needs to be connected.
In the fundamental groupoid, the objects are the points of a space and the arrows are paths up to homotopy, which can be run backwards.
Unlike the fundamental group, it does not need a chosen base point.
</document>

<document source="https://tamnd.github.io/math2/node/FH5WSPD79F.md" tag="FH5WSPD79F" kind="statement">
# Dehn's lemma

## Statement

Let $M$ be a [[3-manifold|R20TB2KEQQ]] and let $D$ be the closed $2$-disk.
Let $f \colon D \to M$ be a piecewise linear map.
Its singular set is the closure of the set of points $x \in D$ such that $f^{-1}(f(x))$ contains a point other than $x$.
If the singular set of $f$ does not meet $\partial D$, then there is a piecewise linear embedding $g \colon D \to M$ with $g = f$ on $\partial D$.

## Remarks

Dehn published a proof in 1910, and Kneser found a gap in it.
Papakyriakopoulos gave a proof in 1957 with his tower construction.
One lifts $f$ through a tower of double coverings of regular neighborhoods of its image, until the singularities can be removed at the top.
Then one pushes the embedded disk back down the tower.
A key application is to knots: a tame [[knot|QYEAZTYGPT]] $K$ in $S^3$ is trivial if and only if its [[knot group|AQVHM3FP0D]] is infinite cyclic.
If the knot group is $\Z$, the longitude of $K$ is null-homotopic in the complement, and Dehn's lemma turns a singular disk bounded by it into an embedded disk.

## Explanations

### L1 human

Suppose a loop in a $3$-dimensional space bounds a disk that may cross itself, but only away from its edge.
Dehn's lemma says that the loop then also bounds a disk that does not cross itself at all.
It turns information about homotopy into information about embedded surfaces.

### L2 human

Dehn's lemma, the loop theorem and the sphere theorem form the classical toolkit of $3$-manifold topology.
They all say that a map that is nontrivial up to homotopy can be replaced by an embedding.
They are used to show that an incompressible surface is the same as a two-sided surface whose inclusion is injective on fundamental groups.
</document>

<document source="https://tamnd.github.io/math2/node/FH92FW76Q0.md" tag="FH92FW76Q0" kind="definition">
# Coercive functional

## Definition

Let $X$ be a real normed space and let $J \colon X \to (-\infty, +\infty]$ be a [[functional|1EQM0DJ4NC]].
The functional $J$ is *coercive* if $J(u) \to +\infty$ as $\norm{u} \to \infty$.
That is, for every $M \in \R$ there is $R > 0$ such that $J(u) > M$ for all $u \in X$ with $\norm{u} > R$.

## Remarks

Equivalently, every sublevel set $\set{u \in X : J(u) \le t}$, $t \in \R$, is bounded.
So a minimising sequence of a coercive functional with $\inf J < +\infty$ is bounded.
In a reflexive [[Banach space|06GZ956657]] it then has a weakly convergent subsequence.

Example: on a Hilbert space $H$ with $f \in H$, let $J(u) = \frac{1}{2}\norm{u}^2 - \ip{f}{u}$.
Then $J(u) \ge \frac{1}{2}\norm{u}^2 - \norm{f} \norm{u}$, which tends to $+\infty$, so $J$ is coercive.

Non-example: on $X = \R$, let $J(u) = e^{u}$.
The values stay below $1$ for all $u < 0$, so $J$ is not coercive, and it has no minimiser.

Some authors use a stronger, quantitative form: $J(u) \ge c \norm{u}^p - C$ for constants $c > 0$, $p \ge 1$ and $C$.
For integral functionals, this follows from a growth bound $L(x, u, \xi) \ge c \abs{\xi}^p - C$ on the integrand together with boundary conditions and the Poincaré inequality.
</document>

<document source="https://tamnd.github.io/math2/node/FH9SJGD1CY.md" tag="FH9SJGD1CY" kind="definition">
# Abel transform

## Definition

Let $f \colon [0, \infty) \to \C$ be measurable, and let $y \ge 0$ be such that $\int_y^\infty \frac{\abs{f(r)}\, r}{\sqrt{r^2 - y^2}}\, dr < \infty$.
The *Abel transform* of $f$ at $y$ is the [[Lebesgue integral|DWJF7ABPR1]]
$$F(y) = 2 \int_y^\infty \frac{f(r)\, r}{\sqrt{r^2 - y^2}}\, dr.$$

## Remarks

The integrand has an integrable singularity at $r = y$, of order $(r - y)^{-1/2}$, when $f$ is bounded near $y$ and $y > 0$.
If $f$ is continuously differentiable and decays fast enough at infinity, the transform can be inverted by
$$f(r) = -\frac{1}{\pi} \int_r^\infty \frac{F'(y)}{\sqrt{y^2 - r^2}}\, dy.$$

Example: $f(r) = e^{-r^2}$.
The substitution $s = r^2 - y^2$, $ds = 2r\, dr$, gives $F(y) = \int_0^\infty \frac{e^{-y^2 - s}}{\sqrt s}\, ds = e^{-y^2}\, \Gamma(1/2) = \sqrt{\pi}\, e^{-y^2}$.

Non-example: $f(r) = 1$ for all $r$.
Then $\frac{r}{\sqrt{r^2 - y^2}} \ge 1$ for $r > y$, so the integral diverges for every $y$ and $f$ has no Abel transform.
</document>

<document source="https://tamnd.github.io/math2/node/FHC702B2R7.md" tag="FHC702B2R7" kind="definition">
# Recurrence and transience of states

## Definition

Let $(X_n)_{n \ge 0}$ be a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable state space $S$.
For a state $i \in S$, write $P_i$ and $E_i$ for probability and expectation when $X_0 = i$.
Let $T_i = \inf\set{n \ge 1 : X_n = i}$ be the first return time to $i$, with $\inf \emptyset = \infty$.

1. The state $i$ is *recurrent* if $P_i(T_i < \infty) = 1$.
2. The state $i$ is *transient* if $P_i(T_i < \infty) < 1$.
3. A recurrent state $i$ is *positive recurrent* if $E_i[T_i] < \infty$, and *null recurrent* if $E_i[T_i] = \infty$.

## Remarks

Let $p^{(n)}_{ii}$ be the diagonal entries of the $n$-th power of the [[transition matrix|0F462493EA]].
Then $i$ is recurrent if and only if $\sum_{n \ge 1} p^{(n)}_{ii} = \infty$, and this sum is the expected number of returns to $i$.
Example: for the simple [[random walk|04VZE3FZXF]] on $\Z$, $p^{(2n)}_{00} = \binom{2n}{n} 4^{-n}$, which behaves like $1/\sqrt{\pi n}$.
The sum diverges, so $0$ is recurrent; it is null recurrent, since this walk has no [[stationary distribution|0KRH5157WP]].
Non-example: for the walk on $\Z$ that steps $+1$ with probability $p \ne 1/2$ and $-1$ with probability $q = 1 - p$, one has $p^{(2n)}_{00} = \binom{2n}{n} (pq)^n \le (4pq)^n$.
Since $4pq < 1$, the sum converges and $0$ is transient, so clause 1 fails.
In an irreducible chain, either all states are transient, or all are null recurrent, or all are positive recurrent.
An irreducible chain has a stationary distribution if and only if its states are positive recurrent.
</document>

<document source="https://tamnd.github.io/math2/node/FHFWE27514.md" tag="FHFWE27514" kind="definition">
# Minimal period

## Definition

Let $X$ be a set, let $f \colon X \to X$ be a map, and let $x \in X$ be a [[periodic point|T75CZ5H47N]] of $f$.
The *minimal period*, or *prime period*, of $x$ is the least integer $n \ge 1$ such that $f^n(x) = x$, where $f^n$ is the $n$-th [[iterate|AYVTR2ES5B]] of $f$.

## Remarks

Let $p$ be the minimal period of $x$.
Then $f^m(x) = x$ holds exactly when $p$ divides $m$: write $m = qp + r$ with $0 \le r < p$; then $x = f^m(x) = f^r(x)$, so $r = 0$ by minimality.
The points $x, f(x), \dots, f^{p-1}(x)$ are distinct, so the orbit of $x$ has exactly $p$ points.
A point has minimal period $1$ exactly when it is a fixed point.

Example: for $f(x) = -x$ on $\R$ and $x = 1$, one has $f(1) = -1 \ne 1$ and $f^2(1) = 1$, so the minimal period is $2$.

Non-example: for the same map, $2$ is not the minimal period of $0$.
It is a period, since $f^2(0) = 0$, but already $f(0) = 0$, so the minimal period of $0$ is $1$.

In Lean's Mathlib, `Function.minimalPeriod f x` is defined for all $x$ and takes the value $0$ when $x$ is not periodic.
</document>

<document source="https://tamnd.github.io/math2/node/FHG1N4EKP4.md" tag="FHG1N4EKP4" kind="statement">
# Chern–Moser normal form

## Statement

Let $n \ge 1$ and let $M$ be a real-analytic real hypersurface in $\C^{n+1}$.
Let $p \in M$, and assume that the [[Levi form|80MTNXWW80]] of $M$ at $p$ is nondegenerate, with $r$ positive and $n - r$ negative eigenvalues.
Write $\langle z, z \rangle = \sum_{j=1}^{n} \varepsilon_j \abs{z_j}^2$, where $\varepsilon_j = 1$ for $j \le r$ and $\varepsilon_j = -1$ for $j > r$, and let $\operatorname{tr} = \sum_{j=1}^{n} \varepsilon_j \frac{\partial^2}{\partial z_j \partial \bar z_j}$.
Then there are local holomorphic coordinates $(z, w) \in \C^n \times \C$ centred at $p$, with $w = u + iv$, in which $M$ is given near $p$ by
$$v = \langle z, z \rangle + \sum_{k, l \ge 2} F_{kl}(z, \bar z, u),$$
where:

1. each $F_{kl}$ is a polynomial of degree $k$ in $z$ and $l$ in $\bar z$, with coefficients real-analytic in $u$, and $F_{lk} = \overline{F_{kl}}$;
2. $\operatorname{tr} F_{22} = 0$, $\operatorname{tr}^2 F_{32} = 0$ and $\operatorname{tr}^3 F_{33} = 0$.

## Remarks

The result is due to S.-S. Chern and J. Moser.
The proof removes, degree by degree, all terms of the defining equation that can be killed by a holomorphic change of coordinates; the remaining terms are the normal form.
The model case is the hyperquadric $v = \langle z, z \rangle$, where all $F_{kl}$ vanish; for $r = n$ it is biholomorphic to the unit sphere minus a point.
The normalising coordinates are not unique: they are determined up to the action of the group of local biholomorphisms that preserve the hyperquadric.
For $n = 1$ the trace conditions force $F_{22} = F_{32} = F_{33} = 0$, so the first term that may survive is $F_{24}$, which carries the invariant found earlier by É. Cartan.
Without the nondegeneracy of the Levi form the quadratic part $\langle z, z \rangle$ is degenerate and this normal form does not apply.

## Explanations

### L1 human

Two hypersurfaces in complex space may look different but be the same after a holomorphic change of coordinates.
The theorem gives a standard way to write such a hypersurface near a point, so that the remaining coefficients are true geometric information.
The leading part is a quadratic "complex paraboloid", and the corrections are put into a reduced shape by fixed trace conditions.

### L2 human

The normal form solves the local equivalence problem for real-analytic Levi-nondegenerate hypersurfaces.
Two such germs are equivalent exactly when their normal forms agree up to the stability group of the hyperquadric.
Chern and Moser also gave a geometric approach that builds a Cartan connection on a bundle over $M$, whose curvature yields the local invariants.
These invariants show for instance that a generic small perturbation of the sphere is not locally biholomorphic to the sphere.
The method is a complex analogue of the use of normal coordinates in Riemannian geometry.
</document>

<document source="https://tamnd.github.io/math2/node/FHGDZWZ2WE.md" tag="FHGDZWZ2WE" kind="concept">
# Schreier–Sims algorithm

## Summary

The Schreier–Sims algorithm takes a few permutations that generate a [[permutation group|ZDJQRKPWGA]] $G$ and computes a base and a strong generating set of $G$.
With this data one can find the [[order|0B85S2V2DR]] of $G$ and test whether a permutation belongs to $G$.
This takes time polynomial in the degree and the number of generators.
It is the basic algorithm of computational group theory, used in systems such as GAP and Magma.
It is named after Otto Schreier and Charles Sims.

## Explanations

### L1 human

A group of permutations of $n$ points can have far more elements than one could ever list, for example $n!$.
The Schreier–Sims algorithm avoids listing them.
It looks first at where the group can send point $1$, then at where the elements fixing point $1$ can send point $2$, and so on.
The order of the group is the product of the sizes of these sets of possible images.
For the six permutations of $\set{1, 2, 3}$: point $1$ can go to $3$ places, and then point $2$ can go to $2$ places, so the order is $3 \cdot 2 = 6$.
</document>

<document source="https://tamnd.github.io/math2/node/FHKGR7MD1A.md" tag="FHKGR7MD1A" kind="definition">
# Harish-Chandra character

## Definition

Let $G$ be a [[real reductive Lie group|42ZCE14AD2]] with a [[Haar measure|VY0HXD3GC4]] $dg$.
Let $\pi$ be an [[admissible representation|SZV8Y806Y7]] of $G$ of finite length on a Hilbert space $H$.
For a [[smooth function|0V33BNKZ9S]] $f \colon G \to \C$ with compact support, let
$$\pi(f) = \int_G f(g) \, \pi(g) \, dg,$$
a bounded operator on $H$.
Harish-Chandra proved that each $\pi(f)$ is a [[trace class operator|Z2CP530WDE]].
The *Harish-Chandra character* of $\pi$ is the distribution
$$\Theta_\pi(f) = \operatorname{tr} \pi(f), \qquad f \in C_c^\infty(G).$$

## Remarks

Example: if $\dim H < \infty$, then $\operatorname{tr} \pi(f) = \int_G f(g) \operatorname{tr} \pi(g) \, dg$ by linearity of the trace.
So $\Theta_\pi$ is the distribution given by the ordinary [[character|RMWMYBY8TC]] $g \mapsto \operatorname{tr} \pi(g)$.

Non-example: for $\dim H = \infty$, the operator $\pi(e) = \id_H$ is not trace class.
So the naive rule $g \mapsto \operatorname{tr} \pi(g)$ does not define a function, and the smearing by $f$ is necessary.

The character $\Theta_\pi$ is invariant under conjugation: $\Theta_\pi(f^x) = \Theta_\pi(f)$ for $f^x(g) = f(x g x^{-1})$.
Two irreducible admissible representations have the same character if and only if they are infinitesimally equivalent.
By [[Harish-Chandra's regularity theorem|M9X7Y3GZZD]], $\Theta_\pi$ is given by a locally integrable function.
The same definition works for $G(F)$ with $F$ a non-archimedean local field, with $f$ locally constant of compact support.
</document>

<document source="https://tamnd.github.io/math2/node/FHNDX22WMK.md" tag="FHNDX22WMK" kind="concept">
# Periodic solution

## Summary

A periodic solution of a differential equation is a solution defined for all times that repeats itself after a fixed time $T > 0$.
For an autonomous system, a periodic solution that is not constant traces a [[periodic orbit|0AA97XA70K]] in phase space.
For an equation whose right-hand side is $T$-periodic in time, such as a periodically forced oscillator, one looks for solutions with the same period $T$.
Periodic solutions model rhythms such as oscillating circuits, heartbeats and planetary motion.

## Explanations

### L1 human

A periodic solution is a motion that comes back to exactly the same state after a fixed time and then repeats forever.
Its graph is a [[periodic function|0ATF6Q0633]] of time.
The function $x(t) = \cos t$ solves $x'' + x = 0$ and repeats after the time $2\pi$.
A pendulum that swings back and forth without friction moves along a periodic solution.
A damped pendulum does not, because its swings get smaller and smaller.
</document>

<document source="https://tamnd.github.io/math2/node/FHQKT6HPW2.md" tag="FHQKT6HPW2" kind="concept">
# Compact quantum group

## Summary

A compact quantum group is a unital C*-algebra with a coassociative comultiplication that satisfies a cancellation condition.
It was introduced by Stanisław Woronowicz as the noncommutative analogue of a compact group, seen through its algebra of continuous functions.
Every compact group gives one, and there are genuinely quantum examples such as $SU_q(2)$.
Much of the theory of compact groups survives: there is a unique Haar state, and finite-dimensional representations behave as in Peter–Weyl theory.

## Explanations

### L1 human

A compact group $G$ can be described by the algebra $C(G)$ of continuous functions on it.
The group law becomes a map $\Delta$ that sends a function $f$ to the function $(s, t) \mapsto f(st)$ of two variables.
A compact quantum group keeps this kind of map but allows the algebra to be noncommutative.
For example, the circle group gives the algebra $C(S^1)$ with $\Delta(z) = z \otimes z$ for the coordinate function $z$.
Deforming such examples with a parameter $q$ gives quantum groups that have no underlying set of points.
</document>

<document source="https://tamnd.github.io/math2/node/FHV05ZK99F.md" tag="FHV05ZK99F" kind="concept">
# Cartier divisor

## Summary

A Cartier divisor on a [[scheme|5A35TYZD02]] is a hypersurface described locally by one equation.
The local equations are given up to multiplication by invertible functions.
Each Cartier divisor determines an [[invertible sheaf|7JCF2TE3V5]], and on an integral scheme every invertible sheaf arises this way.
So the classes of Cartier divisors give the [[Picard group|3Z9DCWYXQZ]] of an integral scheme.
On a Noetherian integral separated scheme whose local rings are factorial, Cartier divisors are the same as [[Weil divisors|Z9DN23DAE2]].
Cartier divisors are the right kind of divisor on singular schemes, where Weil divisors and line bundles no longer match.

## Explanations

### L1 human

On a curve, a divisor is a finite list of points with integer multiplicities.
A Cartier divisor records this through local functions: near each point you give a rational function whose zeros and poles have the desired orders.
Two local functions that differ by a nowhere vanishing function describe the same divisor there.
For example, on the affine line with coordinate $t$, the single function $t^2$ describes the point $0$ counted twice.
The function $t^2(t-1)^{-1}$ describes the point $0$ counted twice minus the point $1$.
</document>

<document source="https://tamnd.github.io/math2/node/FJ1MDBZ680.md" tag="FJ1MDBZ680" kind="concept">
# Schouten–Nijenhuis bracket

## Summary

The Schouten–Nijenhuis bracket extends the [[Lie bracket of vector fields|68FGQEBD6J]] to multivector fields, the sections of the exterior powers of the tangent bundle.
It lowers the total degree by one and obeys graded versions of antisymmetry, the Jacobi identity and the Leibniz rule.
Its main use is in [[Poisson geometry|A2KQ49Y4NF]]: a bivector field $\pi$ defines a Poisson structure exactly when $[\pi, \pi] = 0$.
It also describes the deformation theory of Poisson structures through Poisson cohomology.

## Explanations

### L1 human

A vector field is an arrow at each point; a bivector field is an oriented little parallelogram at each point, and so on.
The Lie bracket combines two vector fields into a third.
The Schouten–Nijenhuis bracket does the same for these higher objects, with signs that keep track of degrees.
For example, the bracket of a vector field with any multivector field is the derivative of the multivector field along the flow of the vector field.
</document>

<document source="https://tamnd.github.io/math2/node/FJ5E6FZVRJ.md" tag="FJ5E6FZVRJ" kind="definition">
# Treewidth

## Definition

Let $G = (V, E)$ be a finite [[graph|0JD5C82QMB]].
A tree decomposition of $G$ is a pair $(T, (X_t)_{t \in V(T)})$ with the following properties.
Here $T$ is a [[tree|0MJ0JS72DZ]], and each $X_t$ is a subset of $V$, called a bag.

1. Every vertex of $G$ lies in some bag.
2. For every edge $uv$ of $G$, some bag contains both $u$ and $v$.
3. For every vertex $v$ of $G$, the nodes $t$ with $v \in X_t$ form a connected subtree of $T$.

The width of the decomposition is $\max_t \abs{X_t} - 1$.
The treewidth $\operatorname{tw}(G)$ is the least width of a tree decomposition of $G$.

## Remarks

Example: the cycle $C_4$ with vertices $1, 2, 3, 4$ in order has the tree decomposition with two adjacent nodes and bags $\set{1, 2, 3}$ and $\set{1, 3, 4}$.
Each vertex is in a bag, each edge lies in a bag, and the nodes containing $1$ or $3$ form the whole tree, so the width is $2$.
Non-example: put the bags $\set{1, 2}, \set{2, 3}, \set{3, 4}, \set{4, 1}$ in this order along a path.
They satisfy clauses 1 and 2 but fail clause 3, because vertex $1$ lies in the first and last bags only.
Every clique of $G$ lies in a single bag, so $\operatorname{tw}(K_n) = n - 1$.
A graph with at least one edge has treewidth $1$ exactly when it is a forest, and the $k \times k$ grid with $k \ge 2$ has treewidth $k$.
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<document source="https://tamnd.github.io/math2/node/FJ79HREXG4.md" tag="FJ79HREXG4" kind="definition">
# Prime model

## Definition

Let $T$ be a [[complete theory|655K8Y7VS8]] in a signature $L$, with infinite models.
A [[model|MVCH6N9SNW]] $M$ of $T$ is a *prime model* of $T$ if $M$ has an [[elementary embedding|CGH8C8EA9N]] into every model $N$ of $T$.

## Remarks

Example: the field $\overline{\Q}$ of algebraic numbers is a prime model of $\mathrm{ACF}_0$, the theory of [[algebraically closed fields|MT8TDVPBXB]] of characteristic $0$.
Every algebraically closed field $K$ of characteristic $0$ contains $\Q$, and hence a copy of the algebraic closure of $\Q$.
This field embedding is elementary, because $\mathrm{ACF}_0$ has quantifier elimination.

Non-example: $\C$ is a model of $\mathrm{ACF}_0$, but it is not prime, since it is uncountable and so cannot embed into the countable model $\overline{\Q}$.

If $L$ is countable, a prime model of $T$ is countable by the downward Löwenheim–Skolem theorem.
Then a model is prime if and only if it is countable and atomic, that is, every tuple of it realizes an isolated type.
Two prime models of $T$ are isomorphic.
Another example is $(\Q, <)$, the prime model of the theory of dense linear orders without endpoints.
</document>

<document source="https://tamnd.github.io/math2/node/FJ92Q6VTJH.md" tag="FJ92Q6VTJH" kind="statement">
# Erdős–Ko–Rado theorem

## Statement

Let $n$ and $k$ be positive integers with $n \ge 2k$.
Let $\mathcal{F}$ be a [[family|0C21Q694AG]] of $k$-element subsets of $\set{1, \ldots, n}$ that is intersecting: $A \cap B \ne \emptyset$ for all $A, B \in \mathcal{F}$.
Then
$$|\mathcal{F}| \le \binom{n - 1}{k - 1}.$$

## Remarks

Katona's proof places $1, \ldots, n$ around a circle in a random order.
An intersecting family contains at most $k$ of the $n$ arcs of $k$ consecutive points, when $n \ge 2k$.
Each $k$-set is such an arc with probability $n / \binom{n}{k}$, so averaging gives $|\mathcal{F}| \cdot n / \binom{n}{k} \le k$, that is, $|\mathcal{F}| \le \frac{k}{n} \binom{n}{k} = \binom{n-1}{k-1}$.

Example: the star of all $k$-sets that contain the element $1$ is intersecting and has exactly $\binom{n-1}{k-1}$ members.
For $n > 2k$ the stars are the only families that attain the bound.
For $n = 2k$ there are others, for example one set from each complementary pair.
The hypothesis $n \ge 2k$ is necessary: if $k < n < 2k$, any two $k$-subsets intersect, and there are $\binom{n}{k} = \frac{n}{k} \binom{n-1}{k-1} > \binom{n-1}{k-1}$ of them.

## Explanations

### L1 human

Suppose every two committees of $k$ people, chosen from $n$ people, must share a member.
How many committees can there be?
The simple answer is to give every committee one fixed common member.
The theorem says that you cannot do better, provided the group is at least twice the committee size.

### L2 human

The theorem is a basic result of extremal set theory, with many extensions: to $t$-intersecting families, to vector spaces, to permutations, and to other structures.
Its circle proof is an averaging argument of the same kind as the proof of the [[LYM inequality|D1GQNNRDDN]].
The [[Kruskal–Katona theorem|G6WNYB0WMY]] gives another proof.
In graph language it gives the independence number of the Kneser graph.
</document>

<document source="https://tamnd.github.io/math2/node/FJ965C4QWA.md" tag="FJ965C4QWA" kind="counterexample">
# Arens–Fort space

## Counterexample

Let $X = \N \times \N$, where $\N = \set{0, 1, 2, \dots}$, and call $\set{m} \times \N$ the $m$-th column.
Declare a set $U \subseteq X$ open if either $(0, 0) \notin U$, or $(0, 0) \in U$ and for all but finitely many $m$ the set $\set{m} \times \N \setminus U$ is finite.
This is the Arens–Fort space.
It is a countable [[Hausdorff|0S1FJ8VEX9]] and [[normal|0NSR9V8TED]] space.
The point $(0, 0)$ lies in the [[closure|0QEAJB91GZ]] of $X \setminus \set{(0, 0)}$.
But no sequence of points different from $(0, 0)$ [[converges|0W4CN9E2W5]] to $(0, 0)$.
So it is not a [[sequential space|WMB2MAEDX4]] and not [[first-countable|0CP5NDD21C]].

## Remarks

Every point other than $(0, 0)$ is isolated, and every open set containing $(0, 0)$ meets $X \setminus \set{(0, 0)}$, so $(0, 0)$ is in the closure.
Let $(p_k)$ be a sequence in $X \setminus \set{(0, 0)}$ with infinitely many distinct terms, and let $S$ be the set of its terms.
Suppose some column contains infinitely many points of $S$, and remove these points from $X$.
The rest of $X$ misses infinitely many points of only one column, so it is an open set containing $(0, 0)$.
Otherwise every column meets $S$ in a finite set, so $X \setminus S$ is an open set containing $(0, 0)$.
In both cases an open set around $(0, 0)$ misses infinitely many terms, so the sequence does not converge to $(0, 0)$.
A sequence with only finitely many distinct terms, all different from $(0, 0)$, stays outside the open set $X$ minus these terms.
Thus $X \setminus \set{(0, 0)}$ is sequentially closed but not closed.
An open set containing $(0, 0)$ is also closed, since its complement misses $(0, 0)$, and each $\set{x}$ with $x \ne (0, 0)$ is open and closed, so the space is regular.
A countable regular space is Lindelöf, and a regular Lindelöf space is normal; this gives normality.

## Explanations

### L1 human

Picture an infinite grid of points with one special corner point.
All grid points other than the corner are open on their own.
To be near the corner, a set must contain almost all of almost every column.
No sequence of grid points can be near the corner in this sense, yet the corner still touches the rest of the grid.
So sequences fail to detect closure in this space.
</document>

<document source="https://tamnd.github.io/math2/node/FJKNJ14AD2.md" tag="FJKNJ14AD2" kind="statement">
# Approximation of a kernel by degenerate kernels

## Statement

Let $a < b$.

1. Let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]], and let $\varepsilon > 0$. Then there is a [[degenerate kernel|6Y1RY5767R]] $K_\varepsilon(x, y) = \sum_{i=1}^{m} a_i(x) b_i(y)$, with polynomials $a_i$ and $b_i$, such that $\abs{K - K_\varepsilon} < \varepsilon$ on $[a, b]^2$.
2. If $K \in L^2([a, b] \times [a, b])$ is a [[square-integrable kernel|WWMWX7AD4V]], then for every $\varepsilon > 0$ there is a degenerate kernel $K_\varepsilon$ with $a_i, b_i \in L^2[a, b]$ such that $\norm{K - K_\varepsilon}_{L^2([a, b]^2)} < \varepsilon$.

## Remarks

Part 1 follows from the [[Stone–Weierstrass theorem|1P8D3MK79H]]: polynomials in $x$ and $y$ are dense in the continuous functions on the square.
Each monomial $x^p y^q$ is a product of a function of $x$ and a function of $y$.
For part 2, let $(e_i)$ be an [[orthonormal basis|0S9GGTR0EX]] of $L^2[a, b]$.
The products $e_i(x) \overline{e_j(y)}$ form an orthonormal basis of $L^2([a, b]^2)$.
The partial sums of the expansion of $K$ in this basis are degenerate kernels converging to $K$.

Example: $K(x, y) = e^{xy}$ on $[0, 1]^2$ is approximated by $\sum_{k=0}^{N} x^k y^k / k!$, with error at most $\sum_{k > N} 1/k!$.

The norm of an integral operator is at most the $L^2$ norm of its kernel, and at most $b - a$ times its sup norm.
So the result says that the integral operator of $K$ is a norm limit of finite-rank operators.
This is the main step in the proof that such operators are compact, and in the reduction of [[Fredholm's theorem|XQ8DYKW4SH]] to linear algebra.

## Explanations

### L1 human

A general kernel mixes the variables $x$ and $y$ in a complicated way.
A degenerate kernel separates them into finitely many products, and equations with such kernels reduce to finite linear systems.
This result says that every reasonable kernel is as close as one wants to a degenerate one.

### L2 human

The approximation is used together with a perturbation argument.
Write $K = K_\varepsilon + (K - K_\varepsilon)$; the small part is inverted by a Neumann series, and the degenerate part leads to a finite linear system.
This yields the Fredholm alternative for square-integrable kernels, and the compactness of their integral operators.
</document>

<document source="https://tamnd.github.io/math2/node/FJKV01J9F1.md" tag="FJKV01J9F1" kind="concept">
# Convergence tests

## Summary

A convergence test is a rule that decides, from a property of the terms, whether a [[series|0JH4MN4K1C]] converges or diverges.
Most series have no closed form for their sum, so such rules are the main way to settle convergence.
Standard examples are the [[comparison test|04Z5G0THN6]], the [[ratio test|0QNY6ZBXPN]], the [[root test|05H4DDDYBZ]] and the [[integral test|019QZ6RRVH]].
The exact definition is in [[convergence test|MA9EMC488R]].

## Explanations

### L1 human

A convergence test is a quick check that tells you whether an infinite sum settles down to a number.
You look at the terms, not at the sum itself.
For example, each term of $\sum 1/2^n$ is half of the term before, and the ratio test then says the series converges.
A test can also stay silent: for $\sum 1/n$ and $\sum 1/n^2$ the ratio test gives no answer.
</document>

<document source="https://tamnd.github.io/math2/node/FJVQH0VPN7.md" tag="FJVQH0VPN7" kind="definition">
# Instanton

## Definition

Let $(X, g)$ be an [[oriented|913WEQERBE]] [[Riemannian manifold|7CBT4SM83Y]] of dimension $4$, and let $\star$ be its [[Hodge star|1Y4WX4XQWJ]], which maps 2-forms to 2-forms.
Let $G$ be a compact [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{g}$, and let $P \to X$ be a [[principal|8E37E74XVP]] $G$-bundle.
Let $A$ be a [[connection|NJ21PN5ZGJ]] on $P$ with [[curvature|ZAJFW1S078]] $F_A$, a 2-form on $X$ with values in the adjoint bundle $P \times_G \mathfrak{g}$.
The connection $A$ is an *instanton*, or anti-self-dual connection, if
$$\star F_A = -F_A.$$

## Remarks

Some authors use the opposite orientation and call self-dual connections, with $\star F_A = F_A$, instantons.
Anti-self-dual connections satisfy the Yang–Mills equations $d_A \star F_A = 0$, because $d_A \star F_A = -d_A F_A = 0$ by the Bianchi identity.
On a compact $X$, they minimize the Yang–Mills functional $\int_X \abs{F_A}^2$ among connections on $P$, the minimum being a topological constant times the absolute value of a characteristic number of $P$.

Example: a flat connection, with $F_A = 0$, is an instanton.
The BPST instanton on $\R^4$ for $G = SU(2)$ has curvature that is nowhere zero and anti-self-dual for one choice of orientation.

Non-example: on the trivial $U(1)$-bundle over $\R^4$ with coordinates $x_1, \dots, x_4$, let $A = i \, x_1 \, dx_2$.
Then $F_A = i \, dx_1 \wedge dx_2$ and $\star F_A = i \, dx_3 \wedge dx_4 \ne -F_A$, so $A$ is not an instanton.
</document>

<document source="https://tamnd.github.io/math2/node/FJZRW9P0K3.md" tag="FJZRW9P0K3" kind="concept">
# Moduli space

## Summary

A moduli space is a geometric space whose points correspond to the isomorphism classes of objects of some kind.
The objects may be curves, vector bundles or subvarieties, for example.
A fine moduli space represents the moduli functor and carries a universal family.
A coarse moduli space only has the right points and is universal for maps out of the functor.
Moduli spaces turn classification problems into geometry, and their geometry, such as dimension and singularities, reflects the deformations of the objects.
Examples are the [[Grassmannian|CYPA882RB9]], the [[Hilbert scheme|XK866CAZHE]] and the [[moduli of algebraic curves|NK9ATH4WB6]].

## Explanations

### L1 human

Some families of objects can be described by a few continuous parameters.
Lines through the origin in the plane are described by one angle, so they form a circle, the projective line.
Elliptic curves over the complex numbers are classified by one complex number, the [[j-invariant|9538W1K1EX]], so they form a line.
A moduli space is the space of these parameters, built so that nearby points are nearby objects.
</document>

<document source="https://tamnd.github.io/math2/node/FJZXB1N7TP.md" tag="FJZXB1N7TP" kind="statement">
# Bott's Morse-theoretic proof of periodicity

## Statement

Let $m \ge 1$.
Give the [[special unitary group|CZCWFPNS4N]] $SU(2m)$ the bi-invariant Riemannian metric induced by $\langle A, B \rangle = \operatorname{Re} \tr(A^* B)$.
Let $\Omega$ be the space of paths in $SU(2m)$ from the identity $I$ to $-I$.

1. The minimal [[geodesics|NSYMQMHPDN]] from $I$ to $-I$ form a space homeomorphic to the complex [[Grassmannian|CYPA882RB9]] $G_m(\C^{2m})$ of $m$-dimensional subspaces of $\C^{2m}$.
2. Every geodesic from $I$ to $-I$ that is not minimal has [[Morse index|MAX2ZYG6H9]] at least $2m + 2$, as a critical point of the energy on $\Omega$.
3. Hence the inclusion of $G_m(\C^{2m})$ into $\Omega$ induces isomorphisms $\pi_i(G_m(\C^{2m})) \cong \pi_i(\Omega) \cong \pi_{i+1}(SU(2m))$ for $0 \le i \le 2m$.
4. Consequently, for the stable [[unitary group|0R9R0Y1HDK]] $U = \bigcup_n U(n)$, there are isomorphisms of [[homotopy groups|P976WYEFMB]] $\pi_{i}(U) \cong \pi_{i+2}(U)$ for all $i \ge 0$.

## Remarks

A geodesic from $I$ in $SU(2m)$ has the form $t \mapsto \exp(t X)$ with $X$ skew-Hermitian.
It ends at $-I$ at time $1$ exactly when the eigenvalues of $X$ are $i \pi k_j$ with odd integers $k_j$ summing to $0$.
The geodesic is minimal exactly when $m$ of the $k_j$ equal $1$ and $m$ equal $-1$.
Such an $X$ is determined by its $m$-dimensional eigenspace for the eigenvalue $i \pi$, which gives the Grassmannian.
The index of a geodesic is computed from the curvature of $SU(2m)$ through the [[Morse index theorem for geodesics|YZ4268SK8G]], and it grows with the $\abs{k_j}$.
Morse theory on the path space then shows that $\Omega$ is obtained from the space of minimal geodesics by attaching cells of dimension at least $2m + 2$.
This gives the third statement.
The last statement follows from the third through the fibration $U(m) \to V_m(\C^{2m}) \to G_m(\C^{2m})$.
Its total space, the Stiefel manifold of orthonormal $m$-frames in $\C^{2m}$, is $2m$-connected.
So $\pi_i(G_m(\C^{2m})) \cong \pi_{i-1}(U(m))$ for $1 \le i \le 2m$, and both sides stabilize as $m$ grows.

In the same way, a chain of eight such steps through symmetric spaces gives the real periodicity $\pi_i(O) \cong \pi_{i+8}(O)$ for the stable orthogonal group $O$.
For example, $U$ is connected and $\pi_1(U) \cong \Z$ by the determinant, so $\pi_{2k+1}(U) \cong \Z$ and $\pi_{2k}(U) = 0$ for all $k \ge 0$.

## Explanations

### L1 human

Bott periodicity says that the homotopy groups of the infinite unitary group repeat with period two.
Bott proved it by looking at geodesics in the group $SU(2m)$ joining the identity to its negative.
The shortest such geodesics form a Grassmannian, and all the longer ones are unstable in many directions.
So, up to a high dimension, the space of all paths looks like the space of shortest geodesics, and this identification gives the periodicity.

### L2 human

The argument is a model application of Morse theory on path spaces.
One computes the space of minimal critical points of the energy and a lower bound for the indices of all other critical points.
Then the path space and the minimal set have the same homotopy groups up to the bound.
The same method, applied to $Sp(n)$ and to symmetric spaces such as $SO(2n)/U(n)$, yields the eight-fold periodicity of the orthogonal group.
Bott periodicity is the foundation of topological K-theory, where it appears as $\tilde{K}(S^{n+2}) \cong \tilde{K}(S^n)$.
</document>

<document source="https://tamnd.github.io/math2/node/FK3QBZAZH7.md" tag="FK3QBZAZH7" kind="concept">
# Lebesgue–Stieltjes integration

## Summary

Lebesgue–Stieltjes integration integrates a function $f$ against a non-decreasing function $g$ on the real line, written $\int f \, dg$.
It is the Lebesgue integral with respect to the Lebesgue–Stieltjes measure of $g$, which gives each interval $(a, b]$ the mass $g(b) - g(a)$.
It extends the Riemann–Stieltjes integral to all measurable functions, and to functions $g$ of bounded variation by splitting $g$ into monotone parts.
It is the language of probability distributions on the line, where $g$ is a cumulative distribution function.

## Explanations

### L1 human

In an ordinary integral each small piece of the line counts with its length.
In a Stieltjes integral it counts with the increase of a given function $g$ over that piece.
Where $g$ is flat, nothing is counted; where $g$ jumps, the point of the jump gets a mass equal to the jump.
For example, if $g$ jumps from $0$ to $1$ at $0$ and is constant elsewhere, then $\int f \, dg = f(0)$.
</document>

<document source="https://tamnd.github.io/math2/node/FK4F02GMVA.md" tag="FK4F02GMVA" kind="definition">
# Yamabe flow

## Definition

Let $M$ be a smooth manifold and let $T > 0$.
Let $(g(t))_{0 \le t < T}$ be a smooth family of [[Riemannian metrics|Q61VFHCH2J]] on $M$.
It solves the *Yamabe flow* if
$$\frac{\partial g}{\partial t} = -R_{g(t)}\, g(t),$$
where $R_{g(t)}$ is the [[scalar curvature|ZHX2VYQDZ3]] of $g(t)$.

Let $M$ be compact.
The *normalized Yamabe flow* is
$$\frac{\partial g}{\partial t} = \left(r_{g(t)} - R_{g(t)}\right) g(t),$$
where $r_{g(t)}$ is the average of $R_{g(t)}$ over $M$ with respect to the volume of $g(t)$.
It preserves the total volume.

## Remarks

Example: on the unit sphere $S^n$ with $n \ge 2$ and round metric $g_0$, put $g(t) = \rho(t) g_0$.
The scalar curvature of $g_0$ is $n(n-1)$, so that of $g(t)$ is $n(n-1)/\rho(t)$.
The equation becomes $\rho'(t) g_0 = -n(n-1) g_0$, which holds for $\rho(t) = 1 - n(n-1)t$ with $0 \le t < 1/(n(n-1))$.
Non-example: the family $g(t) = (1 + t) g_0$ on the flat torus fails, since the scalar curvature is $0$ but the metric changes.
The right side is a multiple of $g$.
So a solution has the form $g(t) = u(\cdot, t)\, g(0)$ for a positive function $u$ on $M \times [0, T)$, and it stays in the conformal class of $g(0)$.
In dimension $2$ the Ricci curvature equals $\tfrac{1}{2} R g$, so the Yamabe flow coincides with the [[Ricci flow|CR14D9YQTY]] $\partial g/\partial t = -2 \operatorname{Ric}$.
</document>

<document source="https://tamnd.github.io/math2/node/FKF9YGBKHF.md" tag="FKF9YGBKHF" kind="object">
# Complex p-adic numbers

## Description

Let $p$ be a prime and let $\Q_p$ be the field of [[p-adic numbers|BT6T702C28]] with the $p$-adic absolute value $\abs{\cdot}_p$.
Let $\overline{\Q}_p$ be an [[algebraic closure|0MCXX29HWY]] of $\Q_p$.
Because $\Q_p$ is complete, $\abs{\cdot}_p$ extends in exactly one way to an absolute value on $\overline{\Q}_p$, and this extension is a [[non-Archimedean absolute value|1H5F1BP6XS]].
The field $\overline{\Q}_p$ is not complete for it.
The field $\C_p$ of *complex $p$-adic numbers* is the [[completion|KFYW0WN9AD]] of $\overline{\Q}_p$ for this absolute value.

The field $\C_p$ is an [[algebraically closed field|0PF6TM8TWT]] and it is complete.
The algebraic closedness follows from [[Krasner's lemma|G8PGN0WQHW]].
One approximates a monic polynomial over $\C_p$ by monic polynomials over $\overline{\Q}_p$, whose roots lie in $\overline{\Q}_p$.
The roots of these approximations can be chosen to form a Cauchy sequence, and its limit in $\C_p$ is a root of the original polynomial.

## Remarks

The set of values $\abs{x}_p$ for $x \in \C_p^\times$ is $p^{\Q} = \set{p^r : r \in \Q}$, the same as for $\overline{\Q}_p$.
The residue field of $\C_p$ is an algebraic closure of the finite field $\F_p$.
The field $\C_p$ plays for $\Q_p$ the role that $\C$ plays for $\R$, but $\C_p$ has infinite degree over $\Q_p$.

## Explanations

### L1 human

To do analysis with $p$-adic numbers one wants a field where every polynomial has a root and every Cauchy sequence has a limit.
Adding all roots of polynomials to $\Q_p$ gives algebraic closedness but loses completeness.
Completing once more gives $\C_p$, and the surprise is that no new polynomial roots are needed after that.
</document>

<document source="https://tamnd.github.io/math2/node/FKPPWF87TR.md" tag="FKPPWF87TR" kind="definition">
# Krylov subspace

## Definition

Let $F$ be $\R$ or $\C$, let $A$ be an $n \times n$ [[matrix|0VD9ZSWWJG]] over $F$, let $b \in F^n$, and let $r \ge 1$ be an integer.
The *Krylov subspace* of order $r$ generated by $A$ and $b$ is the [[linear subspace|00YC1D652K]]
$$\mathcal{K}_r(A, b) = \operatorname{span} \set{b, A b, A^2 b, \dots, A^{r-1} b} \subseteq F^n.$$

## Remarks

Equivalently, $\mathcal{K}_r(A, b) = \set{p(A) b : p \in F[t], \ \deg p < r}$.
The subspaces grow, $\mathcal{K}_1 \subseteq \mathcal{K}_2 \subseteq \cdots$, and their dimensions increase by one until they stop growing.
Once $\mathcal{K}_{r+1} = \mathcal{K}_r$, the subspace is invariant under $A$ and all later ones are equal to it.
This happens at the least degree $d$ of a monic polynomial $p$ with $p(A) b = 0$.
That degree is at most the degree of the [[minimal polynomial|03NDVCSHSB]] of $A$, hence at most $n$.
If $A$ is [[invertible|0EB9N0S33C]], then $A^{-1} b$ lies in $\mathcal{K}_n(A, b)$, since the minimal polynomial of $A$ then has nonzero constant term.
Example: for $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $b = e_2$, one has $A b = e_1$, so $\mathcal{K}_2(A, b) = F^2$.
Non-example: for $A = I$, every Krylov subspace $\mathcal{K}_r(I, b)$ is the line through $b$ or $\set{0}$.
So for $n \ge 2$ the whole space $F^n$ is not a Krylov subspace of the identity.
</document>

<document source="https://tamnd.github.io/math2/node/FKTQ1WBMKM.md" tag="FKTQ1WBMKM" kind="concept">
# Subnormal subgroup

## Summary

A subnormal subgroup is joined to the whole group by a finite chain of subgroups, each normal in the next.
Every normal subgroup is subnormal, but not conversely, because normality is not transitive.
Subnormal subgroups are exactly the subgroups that occur as terms of subnormal series, and they are central in the structure theory of finite groups.

## Explanations

### L1 human

Being normal is a relation between a subgroup and a bigger group.
If $H$ is normal in $K$ and $K$ is normal in $G$, then $H$ need not be normal in $G$.
A subnormal subgroup is one that reaches the whole group in finitely many such normal steps.
For example, in the alternating group $A_4$ the subgroup $\{e, (1\,2)(3\,4)\}$ is normal in the Klein four-group, which is normal in $A_4$.
So it is subnormal in $A_4$, even though it is not normal in $A_4$.
</document>

<document source="https://tamnd.github.io/math2/node/FM0ZZS1PY2.md" tag="FM0ZZS1PY2" kind="concept">
# Multiple gamma function

## Summary

The multiple gamma functions $\Gamma_N(w \mid a_1, \ldots, a_N)$ of E. W. Barnes generalise the [[gamma function|0H5SYBSED9]] to $N$ periods.
They are built from the Barnes zeta function $\sum (w + n_1 a_1 + \cdots + n_N a_N)^{-s}$ by zeta regularisation, as the gamma function is built from the [[Hurwitz zeta function|G4SW7HJG88]].
Shifting $w$ by a period $a_j$ divides $\Gamma_N$ by a multiple gamma function of order $N - 1$.
The [[Barnes G-function|7E2JJWFF00]] is essentially the case $N = 2$ with $a_1 = a_2 = 1$.

## Explanations

### L1 human

The factorial $n!$ multiplies the numbers $1, \ldots, n$, which lie on a line with step $1$.
One can try to multiply all numbers $w + n_1 a_1 + n_2 a_2$ of a lattice in a quadrant, or in a higher-dimensional corner.
This infinite product diverges, but a standard regularisation trick gives it a finite meaning.
The result is the multiple gamma function.
For one period equal to $1$ it gives back $\Gamma(w)$, up to the constant factor $1/\sqrt{2\pi}$.
</document>

<document source="https://tamnd.github.io/math2/node/FM9Y4Y6D8F.md" tag="FM9Y4Y6D8F" kind="concept">
# Stein equation

## Summary

The Stein equation is the linear matrix equation $X - AXA^* = Q$ for an unknown square matrix $X$.
It is the discrete-time form of the [[Lyapunov equation|92GVPZH9FT]].
When every eigenvalue of $A$ has absolute value less than $1$, it has the unique solution $X = \sum_{k \ge 0} A^k Q (A^*)^k$.
This links the stability of the recursion $x_{k+1} = A^* x_k$ with positive definite solutions of a linear equation.

## Explanations

### L1 human

Consider a system that updates its state by a matrix at each time step.
The Stein equation asks for a quadratic energy that drops by a prescribed amount $Q$ at each step.
For one variable with $A = 1/2$ and $Q = 3$, the equation $X - X/4 = 3$ gives $X = 4$.
The solution adds up the contributions of $Q$ over all future steps, which is finite exactly when the system decays.
</document>

<document source="https://tamnd.github.io/math2/node/FMBFW2QXZ3.md" tag="FMBFW2QXZ3" kind="definition">
# Finite-rank operator

## Definition

Let $X$ and $Y$ be [[normed spaces|03J5RAKB30]] over $\mathbb{K} = \R$ or $\C$, and let $T \colon X \to Y$ be a [[bounded linear map|03TM135G0N]].
The map $T$ is a *finite-rank operator* if its [[image|0WSFKKT6FF]] $T(X)$ is a finite-dimensional subspace of $Y$.
Its *rank* is the dimension of $T(X)$.

## Remarks

Equivalently, there are $n \ge 0$, bounded linear functionals $f_1, \ldots, f_n$ on $X$ and vectors $y_1, \ldots, y_n \in Y$ with
$$T x = \sum_{k=1}^{n} f_k(x) \, y_k \quad \text{for all } x \in X.$$
Example: for a nonzero bounded linear functional $f$ on $X$ and a nonzero $y \in Y$, the map $x \mapsto f(x) y$ has image $\mathbb{K} y$, so it has rank $1$.
Non-example: the identity map of $\ell^2$ is bounded, but its image $\ell^2$ is infinite-dimensional, so it is not of finite rank.
For linear maps between finite-dimensional spaces, the rank is the usual [[rank of a linear map|0ZZXSBTR4J]].
A bounded finite-rank operator is compact, because bounded subsets of a finite-dimensional normed space are relatively compact.
On Hilbert spaces, every compact operator is a norm limit of finite-rank operators.
On general Banach spaces this can fail; this is the question of the approximation property.
</document>

<document source="https://tamnd.github.io/math2/node/FMCWXA0WT4.md" tag="FMCWXA0WT4" kind="definition">
# Free lattice

## Definition

Let $X$ be a set.
A *free lattice* on $X$ is a [[lattice|WH4SFAACAS]] $F$ together with a map $\iota \colon X \to F$ with the following property.
For every lattice $L$ and every map $g \colon X \to L$ there is exactly one [[lattice homomorphism|BNVRR925X8]] $h \colon F \to L$ with $h \circ \iota = g$.

## Remarks

A free lattice on $X$ exists and is unique up to a unique isomorphism compatible with the maps from $X$; it is written $\mathrm{FL}(X)$.
One construction takes the set of lattice [[terms|6D3WA7N2W4]] built from the letters of $X$ with $\vee$ and $\wedge$.
It identifies two terms $s$ and $t$ when the identity $s = t$ holds in every lattice.
The map $\iota$ is injective, and $\mathrm{FL}(X)$ is generated by $\iota(X)$ as a lattice.
Example: for $X = \set{x, y}$, $\mathrm{FL}(X) = \set{x, y, x \wedge y, x \vee y}$ with $x \wedge y < x, y < x \vee y$.
Every lattice term in $x$ and $y$ reduces to one of these four by the lattice laws.
These four are distinct because they are distinct in the lattice of subsets of $\set{1, 2}$ under $x \mapsto \set{1}$, $y \mapsto \set{2}$.
Non-example: the two-element chain $\set{x, y}$ with $x < y$, with $\iota$ the identity, is not free on $\set{x, y}$.
The map $g$ with $g(x) = \set{1}$ and $g(y) = \set{2}$ into the subsets of $\set{1, 2}$ has no extension $h$, since a lattice homomorphism is monotone and $\set{1} \not\subseteq \set{2}$.
For $\abs{X} \ge 3$, the free lattice $\mathrm{FL}(X)$ is infinite.
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<document source="https://tamnd.github.io/math2/node/FMFDEE8WN2.md" tag="FMFDEE8WN2" kind="definition">
# Amalgamated free product

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras and let $A$ be an algebra in $\mathcal{V}$.
Let $(B_i)_{i \in I}$ be a family of algebras in $\mathcal{V}$ with [[homomorphisms|KBXSZC6BV1]] $\alpha_i \colon A \to B_i$.
Call a family of homomorphisms $g_i \colon B_i \to D$ *compatible* if $g_i \circ \alpha_i = g_j \circ \alpha_j$ for all $i, j \in I$.
An *amalgamated free product* of the $B_i$ over $A$ is an algebra $C$ in $\mathcal{V}$ with homomorphisms $\beta_i \colon B_i \to C$ such that:

1. the family $(\beta_i)_{i \in I}$ is compatible;
2. for every algebra $D$ in $\mathcal{V}$ and every compatible family $g_i \colon B_i \to D$, there is a unique homomorphism $g \colon C \to D$ with $g \circ \beta_i = g_i$ for all $i$.

It is written $\mathop{\ast}_A B_i$, or $B_1 *_A B_2$ for two factors.

## Remarks

The amalgamated free product exists in every variety and is unique up to a unique compatible isomorphism.
It is the quotient of the [[free product|HHNWHS1XTN]] of the $B_i$ by the smallest congruence that identifies the images of $\alpha_i(a)$ and $\alpha_j(a)$ for all $a \in A$ and $i, j \in I$.
With $A$ the trivial group in the variety of groups, it is the free product.
Usually the $\alpha_i$ are embeddings; whether the $\beta_i$ are then embeddings depends on the variety, and for groups they always are.

Example: in the variety of groups, $\Z/4\Z *_{\Z/2\Z} \Z/6\Z$, with $\Z/2\Z$ mapped to the subgroups of order $2$, is isomorphic to $SL_2(\Z)$.

Non-example: in groups, the direct product $\Z \times \Z$ with the two coordinate inclusions is not $\Z *_{\set{0}} \Z$.
Clause 2 fails for $D = S_3$ and $g_1, g_2$ sending $1$ to two non-commuting transpositions, since the generators of $\Z \times \Z$ commute.

In Mathlib the construction for monoids and groups is `Monoid.PushoutI`.
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<document source="https://tamnd.github.io/math2/node/FMFGTVVR4S.md" tag="FMFGTVVR4S" kind="concept">
# Order type

## Summary

The order type of a totally ordered set is what it shares with every set that is [[order isomorphic|0Y74KN99A5]] to it.
Two ordered sets have the same order type exactly when there is an order isomorphism between them.
Standard order types are $\omega$ for the natural numbers, $\eta$ for the rationals and $\lambda$ for the reals.
The order types of well-ordered sets are the [[ordinal numbers|0RDJK7EQYY]].

## Explanations

### L1 human

The order type forgets what the elements are and keeps only how they are arranged.
The natural numbers $0, 1, 2, \dots$ and the even numbers $0, 2, 4, \dots$ are different sets, but both form one infinite line going up from a first element.
So they have the same order type, $\omega$.
The integers are different in shape: they have no first element, so their order type is not $\omega$.
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<document source="https://tamnd.github.io/math2/node/FMFZQKR71N.md" tag="FMFZQKR71N" kind="definition">
# System of linear integral equations

## Definition

Let $n \ge 1$ and let $\Omega$ be a measurable subset of $\R^m$, for example an interval $[a, b]$.
Let $K_{ij} \colon \Omega \times \Omega \to \C$, for $1 \le i, j \le n$, be [[kernels|GSMBXT7FPK]], let $f_1, \ldots, f_n$ be given functions on $\Omega$, and let $\lambda \in \C$.
A *system of linear integral equations of the second kind* for unknown functions $\varphi_1, \ldots, \varphi_n$ on $\Omega$ is
$$\varphi_i(x) - \lambda \sum_{j=1}^{n} \int_\Omega K_{ij}(x, t)\, \varphi_j(t)\, dt = f_i(x), \quad x \in \Omega, \quad i = 1, \ldots, n.$$
A system *of the first kind* omits the terms $\varphi_i(x)$.
In vector form, with $\varphi = (\varphi_1, \ldots, \varphi_n)$ and the [[matrix|0VD9ZSWWJG]] kernel $K = (K_{ij})$, it reads $\varphi(x) - \lambda \int_\Omega K(x, t)\, \varphi(t)\, dt = f(x)$.

## Remarks

Example: on $\Omega = [0, 1]$ with $n = 2$ and $\lambda = 1$, take $K_{11} = K_{22} = 0$, $K_{12}(x, t) = 1$, $K_{21}(x, t) = x$, $f_1(x) = x$ and $f_2 = 0$.
Put $A = \int_0^1 \varphi_2$ and $B = \int_0^1 \varphi_1$.
The system says $\varphi_1(x) = x + A$ and $\varphi_2(x) = B x$, so $A = B/2$ and $B = \frac{1}{2} + A$, which give $B = 1$ and $A = \frac{1}{2}$.
Check: $\varphi_1(x) = x + \frac{1}{2}$ and $\varphi_2(x) = x$ give $\varphi_1(x) - \int_0^1 \varphi_2 = x$ and $\varphi_2(x) - x \int_0^1 \varphi_1 = x - x = 0$.

Non-example: the pair $\varphi_1(x) - \int_0^1 \varphi_2(t)^2\, dt = x$, $\varphi_2(x) - \int_0^1 \varphi_1(t)\, dt = 0$ is a system of integral equations, but not a linear one, since $\varphi_2$ enters squared.

If $\Omega = [0, 1]$, the substitution $\Phi(x + i - 1) = \varphi_i(x)$ for $x \in [0, 1)$ turns the system into one [[linear integral equation|AYZCKG34QK]] for $\Phi$ on $[0, n)$ with a piecewise defined kernel.
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<document source="https://tamnd.github.io/math2/node/FMHA649HJ9.md" tag="FMHA649HJ9" kind="concept">
# Fano variety

## Summary

A Fano variety is a smooth projective variety whose anticanonical bundle is ample.
Fano varieties are the positively curved end of the classification of varieties, opposite to varieties of general type.
They appear as the general fibres of Mori fibre spaces in the [[minimal model program|Q8CTDCZH01]].
In dimension $2$ they are the del Pezzo surfaces.

## Explanations

### L1 human

The canonical bundle measures how a variety curves, and a Fano variety curves positively in every direction, like a sphere.
Projective space is the main example.
A smooth hypersurface of degree $d$ in $\mathbb{P}^n$ is Fano exactly when $d \le n$: a smooth conic or a smooth cubic surface is Fano, but a smooth plane cubic is not.
Fano varieties have many rational curves, which makes them close to rational varieties.
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<document source="https://tamnd.github.io/math2/node/FMKKQEKNGT.md" tag="FMKKQEKNGT" kind="definition">
# Jensen's functional equation

## Definition

*Jensen's functional equation* is the [[functional equation|SRAN4EKWS0]]
$$f\left(\frac{x + y}{2}\right) = \frac{f(x) + f(y)}{2} \quad \text{for all } x, y \in \R,$$
for an unknown [[function|0EPAT51DXQ]] $f \colon \R \to \R$.
A solution is called a *Jensen function*.

## Remarks

Example: $f(x) = 2x + 3$ is a solution, since both sides equal $x + y + 3$.
Non-example: $f(x) = x^2$ is not a solution, since the right side minus the left side is $(x - y)^2 / 4$, which is positive for $x \ne y$.
If $f$ is a solution, then $A(x) = f(x) - f(0)$ satisfies $A(x/2) = A(x)/2$ and then $A(x + y) = A(x) + A(y)$.
So the solutions are $f = A + b$ with $A$ an [[additive function|KF719SZD5T]] and $b \in \R$.
The same equation makes sense for functions on a real interval or on a convex subset of a vector space.
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<document source="https://tamnd.github.io/math2/node/FMP5AWG6PR.md" tag="FMP5AWG6PR" kind="statement">
# Prime avoidance lemma

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]], let $I$ be an [[ideal|0MYSA6MYNA]] of $R$, and let $P_1, \ldots, P_n$ be finitely many [[prime ideals|06HC6HK0Z8]] of $R$, with $n \ge 1$.
If $I \subseteq P_1 \cup \cdots \cup P_n$, then $I \subseteq P_i$ for some $i$.

## Remarks

The proof is by induction on $n$.
If $I$ is contained in no $P_i$, the induction hypothesis gives, for each $i$, an element $x_i \in I$ that lies in no $P_j$ with $j \ne i$; one may assume $x_i \in P_i$.
Then $y = x_1 + x_2 x_3 \cdots x_n$ lies in $I$ but in none of the $P_i$, using that $P_1$ is prime for the term $x_2 \cdots x_n$.
The same proof works when at most two of the $P_i$ are not prime.
Primality is needed: in $R = \F_2[x, y]/(x, y)^2$, the maximal ideal $(x, y) = \set{0, x, y, x + y}$ is the union of the three ideals $(x)$, $(y)$, $(x + y)$, but it is contained in none of them.
Finiteness is needed: in $k[x, y]$ for a field $k$, every nonzero element of $(x, y)$ has an irreducible factor vanishing at the origin.
So $(x, y)$ is the union of infinitely many principal prime ideals, none of which contains it.

## Explanations

### L1 human

An ideal cannot hide inside a finite union of prime ideals without being inside one of them.
So if an ideal is not contained in any of finitely many primes, one can find a single element of the ideal that avoids all of them at once.
This is the form in which the lemma is used most often.

### L2 human

Prime avoidance is a tool for choosing good elements.
For example, let $M$ be a finitely generated module over a Noetherian ring, and let $I$ be an ideal whose elements are all zero divisors on $M$.
Since [[zero divisors are the union of associated primes|S37DQGW3TZ]], $I$ lies in the union of the finitely many [[associated primes|ERFZ6S8X0X]] of $M$.
So $I$ lies in one associated prime.
In dimension theory the lemma lets one choose elements outside all minimal primes, which is how [[regular sequences|69R5R2FCS5]] and [[systems of parameters|95QTEPMDA4]] are built one element at a time.
If $R$ contains an infinite field, the conclusion holds even when none of the ideals $P_i$ is prime, by a vector space argument.
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# Symbolic power

## Summary

Let $P$ be a prime ideal and $n \ge 1$.
The $n$-th symbolic power $P^{(n)}$ is the set of ring elements that land in $P^n$ after multiplication by some element outside $P$.
It is the smallest [[primary ideal|TMTPWVETQK]] with radical $P$ that contains $P^n$, so it corrects the fact that $P^n$ itself need not be primary.
Symbolic powers appear in the proof of [[Krull's principal ideal theorem|SJZW2HN1SC]].
Over an algebraically closed field, they give the polynomials that vanish to order at least $n$ along a variety.

## Explanations

### L1 human

The ordinary power $P^n$ can contain extra junk located at points other than the generic point of $P$.
The symbolic power throws in everything that belongs to $P^n$ "near" $P$, and so keeps only the part of $P^n$ at $P$.
For a prime generated by some of the variables of a polynomial ring, such as $(x, y)$ in $k[x, y, z]$, nothing changes and $P^{(n)} = P^n$.
For the cone $xy = z^2$ and $P = (x, z)$, the element $x$ lies in $P^{(2)}$ but not in $P^2$.
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<document source="https://tamnd.github.io/math2/node/FMQEG0PVYB.md" tag="FMQEG0PVYB" kind="definition">
# Abel–Jacobi map

## Definition

Let $X$ be a compact [[Riemann surface|5K0N8QMV6Q]] of [[genus|6W8TDV9VVN]] $g \ge 1$.
Let $\omega_1, \ldots, \omega_g$ be a basis of the space of holomorphic [[Abelian differentials|P014C13K5H]] on $X$.
Let $\Lambda \subset \C^g$ be the set of vectors $\left( \int_\gamma \omega_1, \ldots, \int_\gamma \omega_g \right)$, where $\gamma$ runs over the closed loops in $X$.
Then $\Lambda$ is a lattice of rank $2g$ in $\C^g$, and $J(X) = \C^g / \Lambda$ is the [[Jacobian variety|ZQ84M9D9TC]] of $X$.
Fix a base point $P_0 \in X$.
The *Abel–Jacobi map* is
$$A \colon X \to J(X), \qquad A(P) = \left( \int_{P_0}^{P} \omega_1, \ldots, \int_{P_0}^{P} \omega_g \right) \bmod \Lambda,$$
where each integral is taken along the same path from $P_0$ to $P$.
It extends to divisors by $A\left( \sum_i n_i P_i \right) = \sum_i n_i A(P_i)$.

## Remarks

The map is well defined: two paths from $P_0$ to $P$ differ by a closed loop, which changes the vector of integrals by an element of $\Lambda$.
On divisors of degree $0$ the result does not depend on $P_0$.

Example: let $X = \C / \Lambda'$ be a [[complex torus|Y7MA6KBE9M]] with $g = 1$ and $\omega_1 = dz$.
The integral from $P_0$ to $P$ is $P - P_0$ modulo the periods of $dz$, which form $\Lambda'$.
So $A(P) = P - P_0$ is an isomorphism from $X$ onto $J(X) = \C / \Lambda'$.

Non-example: the map $P \mapsto \left( \int_{P_0}^{P} \omega_i \right)_i$ into $\C^g$ itself, without the quotient by $\Lambda$, is not well defined on $X$.
Going once around a loop with a nonzero period changes its value.

Abel's theorem: a divisor $D$ of degree $0$ is the divisor of a meromorphic function if and only if $A(D) = 0$.
Jacobi inversion: the induced map from effective divisors of degree $g$ to $J(X)$ is surjective.
For $g \ge 1$ the map $A \colon X \to J(X)$ is injective, and for $g \ge 2$ it is a holomorphic embedding.
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<document source="https://tamnd.github.io/math2/node/FMVVV8VKHG.md" tag="FMVVV8VKHG" kind="definition">
# Cohomological dimension of a group

## Definition

Let $G$ be a [[group|G5S1M9]].
The *cohomological dimension* $\mathrm{cd}(G)$ is the smallest integer $n \ge 0$ such that $H^k(G, M) = 0$ for every $G$-module $M$ and every $k > n$.
Here $H^k(G, M)$ is the [[cohomology group|YXJXA0Q52R]].
If there is no such integer, then $\mathrm{cd}(G) = \infty$.

## Remarks

Equivalently, $\mathrm{cd}(G)$ is the smallest $n$ such that $\Z$, with trivial action, has a [[projective resolution|HK2AA1YSQ2]] of length $n$ over the [[group ring|4NMTSX0XDS]] $\Z[G]$.
For a subgroup $H$ of $G$, $\mathrm{cd}(H) \le \mathrm{cd}(G)$.
The only group with $\mathrm{cd}(G) = 0$ is the trivial group.

Example: $\mathrm{cd}(\Z) = 1$.
The resolution $0 \to \Z[t, t^{-1}] \xrightarrow{\ t - 1\ } \Z[t, t^{-1}] \to \Z \to 0$ has length $1$, and $H^1(\Z, \Z) \cong \Z$ is not zero.
More generally, $\mathrm{cd}(\Z^n) = n$, and a nontrivial [[free group|0KTW26C80J]] has cohomological dimension $1$.

Non-example: a nontrivial finite [[cyclic group|0KVGH0J5B3]] $C_m$ has $H^{2k}(C_m, \Z) \cong \Z/m\Z \ne 0$ for every $k \ge 1$, so no integer $n$ works and $\mathrm{cd}(C_m) = \infty$.
Since a group containing $C_m$ has dimension at least that of $C_m$, every group of finite cohomological dimension is [[torsion-free|PEMNN32A43]].
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# Pontryagin dual

## Summary

The Pontryagin dual of a locally compact abelian group $G$ is the group $\widehat{G}$ of continuous characters of $G$, with the compact-open topology.
It is again a locally compact abelian group.
The dual is the domain of the Fourier transform on $G$, and the Pontryagin duality theorem says that the dual of $\widehat{G}$ is $G$ again.

## Explanations

### L1 human

The dual group lists all the ways the group can "vibrate": each element of the dual is a wave on $G$.
For the circle, the waves are $e^{in\theta}$ with $n$ an integer, so the dual of the circle is $\Z$.
For the integers, the waves are $n \mapsto z^n$ with $\abs{z} = 1$, so the dual of $\Z$ is the circle.
For the real line, the waves are $x \mapsto e^{i\xi x}$ with $\xi$ real, so $\R$ is its own dual.
Fourier series and the Fourier transform are the cases $G = $ circle and $G = \R$ of one theory built on this dual.
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# Finite type invariant

## Definition

Let $G$ be an abelian group and $v$ a [[knot invariant|GZQ5ZE3X6G]] of oriented [[knots|QYEAZTYGPT]] with values in $G$.
A *singular knot* is an immersion of the oriented circle in $\R^3$ whose only singularities are finitely many transverse double points.
At a double point, let $K_+$ and $K_-$ be its positive and negative resolutions into crossings.
Extend $v$ to singular knots by the rule $v(K_\times) = v(K_+) - v(K_-)$, applied at each double point.
The invariant $v$ is of *type* $\le n$ if its extension vanishes on every singular knot with more than $n$ double points.
It is a *finite type invariant*, or *Vassiliev invariant*, if it is of type $\le n$ for some $n$.

## Remarks

The extension is well defined: resolving the double points in different orders gives the same alternating sum over all $2^k$ resolutions.
The invariants of type $\le 0$ are the constants, and for knots so are those of type $\le 1$.
Example: the coefficient $c_2$ of $z^2$ in the Conway polynomial is of type $\le 2$.
Non-example: $c_2$ is not of type $\le 1$, since it takes the value $1$ on a singular knot with two double points whose chords cross.
After the substitution $t = e^h$, the coefficient of $h^n$ in the [[Jones polynomial|4SPC17FPXE]] is of type $\le n$.
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# Hahn–Kolmogorov theorem

## Statement

Let $X$ be a set, let $\mathcal{A}$ be an [[algebra of sets|PFMGZ7988V]] on $X$, and let $\mu_0 \colon \mathcal{A} \to [0, \infty]$ be a [[premeasure|PV0XXSWFC7]].
For $B \subseteq X$ put
$$\mu^*(B) = \inf \Big\{ \sum_{n=1}^{\infty} \mu_0(A_n) : A_n \in \mathcal{A}, \ B \subseteq \bigcup_{n=1}^{\infty} A_n \Big\}.$$
Then:

1. $\mu^*$ is an [[outer measure|BJ314H17HC]] on $X$ and $\mu^*(A) = \mu_0(A)$ for all $A \in \mathcal{A}$.
2. Every set of $\sigma(\mathcal{A})$, the [[sigma-algebra|0SV8K0TDGY]] generated by $\mathcal{A}$, is Carathéodory measurable for $\mu^*$.
   On $\sigma(\mathcal{A})$, the map $\mu^*$ is a [[measure|PR11BZQQPG]] that extends $\mu_0$.
3. If $\mu_0$ is [[sigma-finite|WEBBT18D4X]], that is, $X = \bigcup_n X_n$ with $X_n \in \mathcal{A}$ and $\mu_0(X_n) < \infty$, then this is the only measure on $\sigma(\mathcal{A})$ that extends $\mu_0$.

## Remarks

Clause 1 uses countable additivity of $\mu_0$: a countable cover of $A \in \mathcal{A}$ by sets of $\mathcal{A}$ can be made disjoint inside $A$, and then its total is at least $\mu_0(A)$.
For clause 2, a set $E \in \mathcal{A}$ splits each $A_n$ of a cover additively, so it splits every $B$; the Carathéodory-measurable sets form a $\sigma$-algebra, so they contain $\sigma(\mathcal{A})$.
Clause 3 follows from the $\pi$-$\lambda$ theorem, applied on each $X_n$.
Example: the finite unions of intervals $(a, b] \cap (0, 1]$ form an algebra on $(0, 1]$, length is a premeasure on it, and the theorem gives Lebesgue measure on the Borel subsets of $(0, 1]$.
The measure $\mu^*$ is in fact defined on all Carathéodory-measurable sets, which may be more than $\sigma(\mathcal{A})$; there it is a complete measure.

## Explanations

### L1 human

Start with sizes for a family of simple sets that is closed under finite unions and complements.
To size any other set, cover it by countably many simple sets as cheaply as possible.
The theorem says that this cheapest cover cost is a true measure on all sets that can be built from the simple ones by countable operations.

### L2 human

This is the algebra form of [[Carathéodory's extension theorem|Q8EXG7EKYW]], with an explicit formula for the extension.
In probability it is used to build measures on sequence spaces and, together with a compactness argument, the measures of the Kolmogorov extension theorem.
The formula for $\mu^*$ also shows that every measurable set can be approximated in measure by countable unions of sets of $\mathcal{A}$.
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<document source="https://tamnd.github.io/math2/node/FN6P3CTPTS.md" tag="FN6P3CTPTS" kind="definition">
# Free category

## Definition

Let $G$ be a [[directed graph|0KFYGSHHF5]], allowed to have loops and several edges between two vertices.
Each edge $e$ of $G$ has a source vertex $s(e)$ and a target vertex $t(e)$.
A *path* from $x$ to $y$ of length $n \ge 1$ is a sequence $(e_1, \dots, e_n)$ of edges with $s(e_1) = x$, $t(e_n) = y$ and $t(e_i) = s(e_{i+1})$ for $1 \le i < n$.
Vertices and edges may repeat, as in a directed [[walk|085CH3CYMX]].
For each vertex $x$ there is also an empty path of length $0$ from $x$ to $x$.
The *free category* $F(G)$ on $G$ is the [[category|A2DE66937D]] whose objects are the vertices of $G$.
Its morphisms $x \to y$ are the paths from $x$ to $y$.
Composition is concatenation of paths, and the identities are the empty paths.

## Remarks

Concatenation is associative and the empty paths are units for it, so $F(G)$ is a category.
It has the universal property of a free object: for every category $\mathcal{C}$, each graph morphism from $G$ to the underlying graph of $\mathcal{C}$ extends to exactly one [[functor|28NQ0KRHD3]] $F(G) \to \mathcal{C}$.

Example: let $G$ have one vertex and one loop $a$.
The paths are $a^n = (a, \dots, a)$ with $n \ge 0$ letters, and $a^m \circ a^n = a^{m + n}$, so $F(G)$ is the one-object category of the monoid $(\N, +)$, the [[free monoid|CV013KHVJG]] on one generator.

Non-example: the category with one object and morphisms $\Z/2\Z$ is generated by one loop $a$, but it satisfies $a \circ a = \id$.
In the free category on one loop, $a \circ a$ has length $2$ and the identity has length $0$, so they differ.
So the category of $\Z/2\Z$ is not free on that graph.
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# Positivstellensatz

## Statement

Let $f_1, \dots, f_m, g \in \R[x_1, \dots, x_n]$ be real [[polynomials|07GNX8GCYP]], and let
$$K = \set{x \in \R^n : f_1(x) \ge 0, \dots, f_m(x) \ge 0}$$
be the basic closed [[semialgebraic set|BX5WSK8ZCT]] they define.
Let $T$ be the preordering generated by $f_1, \dots, f_m$.
It is the set of all sums $\sum_{e \in \set{0,1}^m} \sigma_e f_1^{e_1} \cdots f_m^{e_m}$ in which each $\sigma_e$ is a sum of squares of polynomials.
Then:

1. $K = \emptyset$ if and only if $-1 \in T$.
2. $g > 0$ on $K$ if and only if there are $p, q \in T$ with $pg = 1 + q$.
3. $g \ge 0$ on $K$ if and only if there are $p, q \in T$ and an integer $k \ge 0$ with $pg = g^{2k} + q$.
4. $g = 0$ on $K$ if and only if there is an integer $k \ge 0$ with $-g^{2k} \in T$.

## Remarks

The result is due to Krivine (1964) and Stengle (1974).
The "if" directions are direct: for instance, if $pg = 1 + q$ and $x \in K$, then $p(x) \ge 0$ and $q(x) \ge 0$, so $p(x) g(x) \ge 1$ and $g(x) > 0$.
The "only if" directions are the substance.
They follow from the theory of orderings of rings together with the Tarski–Seidenberg transfer principle.
That principle says that a first-order statement about polynomials holds over $\R$ if and only if it holds over every real closed field.
Example: $K = \set{x \in \R : -x^2 - 1 \ge 0}$ is empty, and the certificate is $-1 = (-x^2 - 1) + x^2 \in T$.
The theorem needs the real field: over $\C$ sign conditions make no sense, and its analogue for equations is [[Hilbert's Nullstellensatz|D85VZWS9QK]].
The denominators $p$ cannot be dropped in general, which is why stronger forms, such as Schmüdgen's theorem that $g \in T$ when $K$ is compact and $g > 0$ on $K$, need extra hypotheses.

## Explanations

### L1 human

How can one prove that a polynomial is positive on a region defined by polynomial inequalities?
The Positivstellensatz says that there is always an algebraic proof: an identity built from squares and from the defining inequalities.
Such an identity can be checked by expanding, without solving anything.

### L2 human

The theorem is the real counterpart of Hilbert's Nullstellensatz, with sums of squares playing the role of the ideal.
It contains Artin's solution of [[Hilbert's seventeenth problem|J45F4F13VB]] as the case $m = 0$ of part 3.
Bounded-degree versions of these certificates can be searched for by semidefinite programming, which is the basis of sum-of-squares methods in polynomial optimization.
</document>

<document source="https://tamnd.github.io/math2/node/FN978TDX67.md" tag="FN978TDX67" kind="conjecture">
# Erdős conjecture on arithmetic progressions

## Statement

Let $A$ be a set of positive integers such that the series $\sum_{a \in A} 1/a$ diverges.
Then for every integer $k \ge 1$, the set $A$ contains an arithmetic progression of length $k$ with nonzero common difference.

## Remarks

The conjecture is open for each $k \ge 4$.
The case $k = 3$ was proved by Bloom and Sisask in 2020, using quantitative bounds in [[Roth's theorem|E7H8NWXESF]].
The set of primes satisfies the hypothesis, since the sum of the reciprocals of the primes diverges.
So the [[Green–Tao theorem|H8BAHWSEMC]] is a special case of the conjecture.
Every set of positive upper density satisfies the hypothesis, so the conjecture would strengthen [[Szemerédi's theorem|B23B0QEMN5]].

## Explanations

### L1 human

The conjecture says that a set of whole numbers that is not too sparse must contain evenly spaced rows of every length.
"Not too sparse" means that the sum of the reciprocals of its elements is infinite.
The primes are an example, and for them the answer is known to be yes.
For general sets of this kind, the question is open beyond rows of three.
</document>

<document source="https://tamnd.github.io/math2/node/FNGWJR2FPK.md" tag="FNGWJR2FPK" kind="concept">
# Method of lines

## Summary

The method of lines solves a time-dependent partial differential equation in two stages.
First only the space variables are discretised, by [[finite differences|X7HYEBA5SA]], finite volumes or finite elements.
This turns the equation into a large system of ordinary differential equations in time.
Then this system is solved with a numerical method for [[initial value problems|0FXDW992DS]].
This separation lets one reuse mature ODE solvers, with their error control and their methods for stiff problems.

## Explanations

### L1 human

Picture the space-time plane with a few vertical lines, one for each grid point in space.
Along each line the unknown is a function of time only, and the PDE couples each line to its neighbours.
So the PDE becomes a system of ODEs, one per line, which a standard ODE solver can handle.
For the [[heat equation|VP2HN86VAG]] $u_t = u_{xx}$, replacing $u_{xx}$ at $x_j$ by $(u_{j+1} - 2u_j + u_{j-1})/\Delta x^2$ gives the system $u_j'(t) = (u_{j+1}(t) - 2u_j(t) + u_{j-1}(t))/\Delta x^2$.
</document>

<document source="https://tamnd.github.io/math2/node/FNJC21BM1B.md" tag="FNJC21BM1B" kind="definition">
# Self-distributive operation

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]] and let $*$ be a [[binary operation|0M418FE7BB]] on $X$.
The operation $*$ is *left self-distributive* if
$$x * (y * z) = (x * y) * (x * z) \quad \text{for all } x, y, z \in X.$$
It is *right self-distributive* if
$$(y * z) * x = (y * x) * (z * x) \quad \text{for all } x, y, z \in X.$$
It is *self-distributive* if it is left or right self-distributive, as fixed by the context, and in other texts if it is both.

## Remarks

Left self-distributivity says that $*$ has the [[distributive property|63JGSKKF7J]] over itself from the left.
Equivalently, each left translation $L_x \colon y \mapsto x * y$ is an endomorphism of $(X, *)$.

Example: on $\Z$ let $x * y = 2x - y$.
Then $x * (y * z) = 2x - (2y - z) = 2x - 2y + z$ and $(x * y) * (x * z) = 2(2x - y) - (2x - z) = 2x - 2y + z$, so $*$ is left self-distributive.

Example: in a group, $x * y = x y x^{-1}$ is left self-distributive, since both sides equal $x y z y^{-1} x^{-1}$.

Non-example: addition on $\Z$.
Here $x + (y + z) = x + y + z$ while $(x + y) + (x + z) = 2x + y + z$, which differ when $x \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/FP32JHA9M8.md" tag="FP32JHA9M8" kind="definition">
# ADE singularity

## Definition

Let $X$ be a complex surface and let $p \in X$ be a [[singular point|D86X15A8WS]].
The point $p$ is an *ADE singularity*, or *Du Val singularity*, if the germ $(X, p)$ is analytically isomorphic to a germ $(\set{f = 0}, 0)$ in $\C^3$.
Here $f$ is one of the following polynomials:

1. $A_n$, for $n \ge 1$: $f = x^2 + y^2 + z^{n+1}$.
2. $D_n$, for $n \ge 4$: $f = x^2 + y^2 z + z^{n-1}$.
3. $E_6$: $f = x^2 + y^3 + z^4$.
4. $E_7$: $f = x^2 + y^3 + y z^3$.
5. $E_8$: $f = x^2 + y^3 + z^5$.

## Remarks

Equivalently, the germ is isomorphic to the quotient $\C^2/G$ at the image of $0$, for a nontrivial finite subgroup $G$ of the [[special linear group|0V9PMN0RBH]] $SL(2, \C)$.
The group is cyclic of order $n + 1$ for $A_n$, and binary dihedral of order $4(n - 2)$ for $D_n$.
For $E_6$, $E_7$, $E_8$ it is binary tetrahedral, octahedral and icosahedral, of orders $24$, $48$ and $120$.
The exceptional curves of the minimal resolution are smooth rational curves with self-intersection $-2$, and their dual graph is the Dynkin diagram of the same name.
ADE singularities are [[rational singularities|98QASBAMEB]], and they are exactly the rational double points.
Example: $G = \set{\pm 1}$ acts on $\C[u, v]$ with invariant ring generated by $a = u^2$, $b = v^2$, $c = uv$, subject to $ab = c^2$.
After the linear change $a = x + iy$, $b = -(x - iy)$, $c = z$, the relation $ab - c^2 = 0$ becomes $x^2 + y^2 + z^2 = 0$, which is the $A_1$ singularity.
Non-example: the cone $x^3 + y^3 + z^3 = 0$ has a singular point at $0$ of multiplicity $3$, while every ADE singularity has multiplicity $2$, so it is not an ADE singularity.
</document>

<document source="https://tamnd.github.io/math2/node/FP37BTP93K.md" tag="FP37BTP93K" kind="concept">
# Gaussian quadrature

## Summary

Gaussian quadrature is a [[quadrature rule|ADSVMCY77A]] with $n$ nodes in which both the nodes and the weights are free.
They are chosen to make the rule exact for all polynomials of degree up to $2n - 1$.
This is the highest [[degree of exactness|ZYT1JNT1Y0]] that an $n$-point rule can reach.
The nodes are the zeros of the degree $n$ [[orthogonal polynomial|8TXASARQN1]] for the weight of the integral, and the weights are positive.
For smooth integrands it converges very fast, and it is one of the standard tools of numerical integration.

## Explanations

### L1 human

A simple rule such as the trapezoidal rule fixes the sample points in advance and only chooses the weights.
Gaussian quadrature also lets the sample points move, which doubles the number of free parameters.
With these extra parameters it can integrate polynomials of about twice the degree exactly.
For example, the two-point rule $\int_{-1}^{1} f(x)\,dx \approx f(-1/\sqrt{3}) + f(1/\sqrt{3})$ is exact for every cubic polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/FP5GVCN8GV.md" tag="FP5GVCN8GV" kind="concept">
# Seminorm

## Summary

A seminorm on a real or complex vector space has all the properties of a [[norm|0V2WP9B1MF]], except that it may vanish on nonzero vectors.
Seminorms are the building blocks of [[locally convex spaces|B6GY2KJMWW]]: a family of seminorms defines a topology, and every locally convex topology arises in this way.
Many function spaces are described by seminorms rather than by one norm.
An example is the space of continuous functions on $\R$ with uniform convergence on compact sets.

## Explanations

### L1 human

A seminorm measures the size of a vector, but it is allowed to be blind to some directions.
It still scales correctly and still obeys the triangle inequality.
On $\R^2$, the function $p(x, y) = \abs{x}$ is a seminorm: it measures only the first coordinate.
It gives $p(0, 5) = 0$ although $(0, 5) \ne 0$, so it is not a norm.
On functions, $p(f) = \abs{f(0)}$ is a seminorm that only looks at the value at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/FPDWWD12FE.md" tag="FPDWWD12FE" kind="counterexample">
# Continuous function with divergent Fourier series

## Counterexample

There is a [[continuous|0X3WGX95FE]] $2\pi$-periodic function $f \colon \R \to \C$ whose [[Fourier series|G12KXDHPPS]] diverges at $0$; in fact $\sup_N \abs{S_N f(0)} = \infty$.

Check: let $C(\mathbb{T})$ be the Banach space of continuous $2\pi$-periodic functions with the supremum norm.
For each $N$, the map $\Lambda_N(f) = S_N f(0) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) D_N(t) \, dt$ is a bounded linear functional on $C(\mathbb{T})$, where $D_N$ is the Dirichlet kernel.
Its norm is the [[Lebesgue constant|RWKQ5EP4YF]] $L_N = \frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{D_N}$, and $L_N \to \infty$.
If $\sup_N \abs{\Lambda_N(f)}$ were finite for every $f \in C(\mathbb{T})$, the [[uniform boundedness principle|YG0042V8CJ]] would give $\sup_N L_N < \infty$, which is false.
So some $f \in C(\mathbb{T})$ has $\sup_N \abs{S_N f(0)} = \infty$, and its Fourier series diverges at $0$.

## Remarks

The [[Baire category theorem|0A7KGAYHGV]] behind the uniform boundedness principle gives more: the set of $f \in C(\mathbb{T})$ with $\sup_N \abs{S_N f(0)} = \infty$ is a dense $G_\delta$ subset of $C(\mathbb{T})$.
Paul du Bois-Reymond gave the first such function in 1873, by an explicit construction.
Continuity is thus not enough for pointwise convergence, while bounded variation or a Dini condition is.
The divergence can happen only on a set of measure zero for $f \in L^2$, by [[Carleson's theorem|FBH6KG484Z]].
By the [[Kahane–Katznelson theorem|42DDDZHP7F]], every set of measure zero is the divergence set of some continuous function.
For integrable functions, [[Kolmogorov's example|25V2MHPHZ7]] diverges everywhere.

## Explanations

### L1 human

One might expect every continuous periodic function to have a convergent Fourier series, but this is false.
The partial sums at a point are averages against a kernel whose total absolute size grows without bound.
A continuous function that oscillates in step with the kernel, at more and more frequencies, makes these averages grow without bound.
</document>

<document source="https://tamnd.github.io/math2/node/FPHNJ5AAYF.md" tag="FPHNJ5AAYF" kind="concept">
# Generalized inverse

## Summary

A generalized inverse of a matrix $A$ is a matrix $G$ with $AGA = A$.
Every matrix over a field has one, even when it is not square or not invertible.
Generalized inverses give solutions of consistent linear systems $Ax = b$ in the form $x = Gb$.
The [[Moore–Penrose inverse|0YY8VX28JR]] is a generalized inverse with three more conditions that make it unique.
The [[Drazin inverse|9FB0AZSEGM]] is a related kind of inverse for square matrices, though it does not satisfy $AGA = A$ in general.

## Explanations

### L1 human

Only some square matrices can be inverted.
A generalized inverse keeps one useful property of an inverse: applied in between, it does not change $A$.
Think of $G$ as a map that sends each vector in the image of $A$ back to one of its preimages.
For example, for $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, every matrix $G = \begin{pmatrix} 1 & a \\ b & c \end{pmatrix}$ satisfies $AGA = A$.
So a generalized inverse exists but is usually not unique.
</document>

<document source="https://tamnd.github.io/math2/node/FPK5ZH7FKC.md" tag="FPK5ZH7FKC" kind="definition">
# Unramified morphism

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]] that is locally [[of finite type|AHTE5DVX61]].
For $x \in X$ and $y = f(x)$, let $\mathfrak{m}_x$ and $\mathfrak{m}_y$ be the maximal ideals of the [[local rings|MZ02BKPHJA]] $\mathcal{O}_{X,x}$ and $\mathcal{O}_{Y,y}$.
Let $\kappa(x)$ and $\kappa(y)$ be the [[residue fields|G51E8MW7J1]].
The morphism $f$ is *unramified at $x$* if the following two conditions hold.

1. $\mathfrak{m}_y \mathcal{O}_{X,x} = \mathfrak{m}_x$.
2. $\kappa(x)$ is a finite [[separable extension|0WCX7NB4N7]] of $\kappa(y)$.

The morphism $f$ is *unramified* if it is unramified at every point of $X$.

## Remarks

An open immersion or a closed immersion is unramified.
For a closed immersion, each $\mathcal{O}_{X,x}$ is a quotient of $\mathcal{O}_{Y,y}$, so both conditions hold with $\kappa(x) = \kappa(y)$.
For a field $k$ of characteristic not $2$, the map $\operatorname{Spec} k[t] \to \operatorname{Spec} k[s]$ given by $s \mapsto t^2$ is not unramified at $t = 0$.
There $\mathfrak{m}_y \mathcal{O}_{X,x} = (t^2)$, which is strictly smaller than $\mathfrak{m}_x = (t)$, so condition 1 fails.
For $f$ locally of finite type, $f$ is unramified exactly when the sheaf of relative [[Kähler differentials|7HCFARHG7D]] $\Omega_{X/Y}$ is zero.
Some authors, as in EGA, also require $f$ to be locally of finite presentation.
The ring version of this notion is an [[unramified morphism of rings|GY3QA6NH0E]].
</document>

<document source="https://tamnd.github.io/math2/node/FPKRV34TBV.md" tag="FPKRV34TBV" kind="definition">
# Causal inference

## Definition

*Causal inference* is the branch of [[statistical inference|JZ2S01BHJJ]] whose targets are causal quantities.
These are the values an outcome would take under interventions that set a treatment or exposure to given values.
In the [[potential outcomes framework|23ATBZMJHR]], a typical target is the average treatment effect $E[Y(1) - Y(0)]$.
Here $Y(w)$ is the outcome a unit would have under treatment level $w$.
A causal quantity is *identified* if, under stated assumptions, it is a function of the distribution of the observed variables.

## Remarks

Causal inference has two steps: identification, which shows that the target is a function of the observed distribution under assumptions, and estimation of that function from data.

Example: in a completely randomised experiment with treatment $W \in \set{0, 1}$, randomization makes $W$ independent of $(Y(0), Y(1))$.
Then $E[Y(1) - Y(0)] = E[Y \mid W = 1] - E[Y \mid W = 0]$, so the average treatment effect is identified and estimated by the difference of group means.

Non-example: estimating $E[Y \mid W = 1] - E[Y \mid W = 0]$ in observational data, with no assumption linking it to $Y(1)$ and $Y(0)$, is a description of an association, not causal inference.
If sicker patients choose the treatment, this difference may even have the opposite sign to the causal effect.
</document>

<document source="https://tamnd.github.io/math2/node/FPMJEX3DSJ.md" tag="FPMJEX3DSJ" kind="statement">
# Myers's theorem

## Statement

Let $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \ge 2$.
Suppose there is a real number $k > 0$ such that the [[Ricci curvature|4GTNM0WHDH]] satisfies $\operatorname{Ric}(v, v) \ge (n-1) k \, g(v, v)$ for every tangent vector $v$.
Then:

1. The [[diameter|0KW3PN547K]] of $M$ for the [[Riemannian distance|B4QR306D8G]] is at most $\pi / \sqrt{k}$.
2. $M$ is compact.
3. The [[fundamental group|S1XK3HXHFX]] of $M$ is finite.

## Remarks

The idea of the proof: let $\gamma$ be a minimizing unit-speed geodesic of length $L > \pi/\sqrt{k}$.
Take an orthonormal frame $E_1, \dots, E_{n-1}$ of parallel fields normal to $\gamma$ and the variation fields $\sin(\pi t / L) E_i$.
By the [[second variation formula|ZB91TH6FKV]], the sum of the second variations of length is negative, so one of them is negative and $\gamma$ is not minimizing, a contradiction.
Part 2 follows from part 1 and the Hopf–Rinow theorem.
For part 3, apply parts 1 and 2 to the [[universal cover|Y1YPE6PM52]] with the pulled-back metric: it is compact, so the covering has finitely many sheets.

Example: the unit sphere $S^n$ has $\operatorname{Ric} = (n-1) g$ and diameter exactly $\pi$, so the bound is sharp.
By a theorem of Cheng, equality of the diameter with $\pi/\sqrt{k}$ holds only for the round sphere of radius $1/\sqrt{k}$.

The uniform lower bound is needed: the paraboloid $z = x^2 + y^2$ in $\R^3$ is complete with positive curvature, but its curvature tends to $0$ at infinity and it is not compact.

## Explanations

### L1 human

If a space is curved positively at least by a fixed amount in every direction, it must close up on itself, like a sphere.
The theorem gives an explicit bound: no two points are farther apart than on the sphere of the same curvature.

### L2 human

Myers's theorem is one of the first results linking a lower bound on Ricci curvature to global topology.
For example, it shows that a compact manifold with infinite fundamental group, such as a torus, carries no metric of positive Ricci curvature.
It has a sharp volume analogue, the Bishop–Gromov inequality, and it is used in general relativity to show that certain spacetimes are closed.
</document>

<document source="https://tamnd.github.io/math2/node/FPS7YFGX32.md" tag="FPS7YFGX32" kind="concept">
# Floquet multiplier

## Summary

The Floquet multipliers of a linear system with periodic coefficients are the eigenvalues of its monodromy matrix, the map that advances solutions by one period.
They play the role that the values $e^{\lambda T}$ play for a system with constant coefficients.
They decide whether solutions grow, decay or stay bounded, and they measure the stability of a [[periodic orbit|0AA97XA70K]] of a nonlinear system.
They are the central notion of [[Floquet theory|BJVZXPB297]].
See the [[definition|GVSY2FZ8YW]].

## Explanations

### L1 human

Watch a solution of a periodic linear system only at the times $0, T, 2T, \ldots$, once per period.
From one snapshot to the next, the state is multiplied by a fixed matrix $M$.
The eigenvalues of $M$ are the Floquet multipliers.
If all have absolute value less than $1$, every solution shrinks to $0$; if one has absolute value greater than $1$, some solution grows without bound.
For the scalar equation $x' = (1 + \cos t)\, x$, one period multiplies $x$ by $e^{2\pi}$, so the multiplier is $e^{2\pi}$.
</document>

<document source="https://tamnd.github.io/math2/node/FPWKGFZE6Q.md" tag="FPWKGFZE6Q" kind="statement">
# Hopf lemma

## Statement

Let $U \subseteq \R^n$ be open, and let
$$L u = \sum_{i,j=1}^{n} a_{ij}(x) \, \partial_i \partial_j u + \sum_{i=1}^{n} b_i(x) \, \partial_i u$$
have bounded coefficients on $U$, with $a_{ij} = a_{ji}$ and $\sum_{i,j} a_{ij}(x) \xi_i \xi_j \ge \theta \abs{\xi}^2$ for some $\theta > 0$ and all $x \in U$, $\xi \in \R^n$.
Let $x_0 \in \partial U$, and assume the interior ball condition at $x_0$: there is an open ball $B \subseteq U$ with $x_0 \in \partial B$.
Let $\nu$ be the outer unit normal of $B$ at $x_0$.
Let $u$ be twice continuously differentiable on $U$ and continuous at $x_0$ on $U \cup \set{x_0}$, with $L u \ge 0$ in $U$ and $u(x) < u(x_0)$ for all $x \in U$.
If the one-sided derivative $\frac{\partial u}{\partial \nu}(x_0) = \lim_{t \to 0^+} \frac{u(x_0) - u(x_0 - t \nu)}{t}$ exists, then
$$\frac{\partial u}{\partial \nu}(x_0) > 0.$$

## Remarks

Proof idea: let $B$ have centre $y$ and radius $R$, and set $v(x) = e^{-\lambda \abs{x - y}^2} - e^{-\lambda R^2}$.
For $\lambda$ large, $L v > 0$ on the annulus $A = \set{R/2 < \abs{x - y} < R}$, and $v = 0$ on the outer sphere.
For small $\varepsilon > 0$, the function $u - u(x_0) + \varepsilon v$ satisfies $L(\cdot) \ge 0$ in $A$ and is at most $0$ on $\partial A$, so it is at most $0$ in $A$ by the weak [[maximum principle|D2JF19HF91]].
At $x_0$ this gives $\frac{\partial u}{\partial \nu}(x_0) \ge -\varepsilon \frac{\partial v}{\partial \nu}(x_0) = 2 \varepsilon \lambda R e^{-\lambda R^2} > 0$.

The interior ball condition is needed.
Let $U$ be the part of the unit disc in the first quadrant, $x_0 = 0$ and $u(x, y) = -xy$.
Then $\Delta u = 0$ and $u < 0 = u(x_0)$ in $U$, but $\nabla u(0) = 0$, so every directional derivative at $0$ vanishes.
At the corner $0$ there is no ball inside $U$ touching $0$.

## Explanations

### L1 human

At a boundary point where a solution of an elliptic equation reaches a strict maximum, the solution cannot approach that value flatly.
It must rise towards the boundary with a strictly positive slope in the normal direction.
For a temperature in equilibrium, the hottest boundary point actually sends heat into the region.

### L2 human

The Hopf lemma is the step that turns the weak maximum principle into the strong one.
If an interior maximum occurred, one would find a small ball touching the set where the maximum is attained.
The lemma would then give a nonzero gradient at a maximum point, which is impossible.
It also gives uniqueness up to constants for the Neumann problem, and it is a basic tool in the moving plane method for proving symmetry of solutions.
The smoothness of the boundary in the form of the interior ball condition holds, for example, for every bounded open set with $C^2$ boundary.
</document>

<document source="https://tamnd.github.io/math2/node/FPXZMN0CH5.md" tag="FPXZMN0CH5" kind="definition">
# Floer homology

## Definition

This defines Hamiltonian Floer homology with coefficients in $\F_2$ in the simplest setting.
Let $(M, \omega)$ be a closed [[symplectic manifold|CAN490EPRT]] such that $\int_{S^2} v^* \omega = 0$ for every smooth map $v \colon S^2 \to M$.
Let $H \colon (\R / \Z) \times M \to \R$ be smooth, write $H_t = H(t, \cdot)$, and let $X_{H_t}$ be the [[Hamiltonian vector field|C5EH3B4BEP]] of $H_t$.

1. Let $\mathcal{P}(H)$ be the set of contractible loops $x \colon \R / \Z \to M$ with $x'(t) = X_{H_t}(x(t))$, the contractible $1$-[[periodic orbits|0AA97XA70K]].
Assume that each of them is nondegenerate: the linearized time-$1$ flow at $x(0)$ does not have $1$ as an eigenvalue.
2. Let $J = (J_t)_{t \in \R / \Z}$ be a family of [[almost complex structures|9SYWRGVTRS]] compatible with $\omega$.
For $x, y \in \mathcal{P}(H)$, let $\mathcal{M}(x, y)$ be the set of smooth maps $u \colon \R \times (\R / \Z) \to M$ with
$$\partial_s u + J_t(u) \left( \partial_t u - X_{H_t}(u) \right) = 0, \qquad \lim_{s \to -\infty} u(s, t) = x(t), \qquad \lim_{s \to +\infty} u(s, t) = y(t),$$
on which $\R$ acts by translation in $s$.
3. For generic $J$, each set $\mathcal{M}(x, y)$ is a smooth manifold. When it has dimension $1$, let $n(x, y) \in \F_2$ be the number modulo $2$ of elements of $\mathcal{M}(x, y) / \R$. Otherwise let $n(x, y) = 0$.
4. Let $CF(H)$ be the $\F_2$-vector space with basis $\mathcal{P}(H)$, and let $\partial \colon CF(H) \to CF(H)$ be the linear map with $\partial x = \sum_{y} n(x, y)\, y$.

Then $\partial \circ \partial = 0$, so $(CF(H), \partial)$ is a [[chain complex|WQ82C0M3QB]].
The *Floer homology* of $(H, J)$ is $HF(H, J) = \ker \partial / \im \partial$.

## Remarks

The solutions $u$ in clause 2 are the negative gradient lines of the action functional on contractible loops, for the $L^2$ metric defined by $J$.
They are [[pseudoholomorphic curves|N469E47V75]] perturbed by $H$.
That $n(x, y)$ is finite and that $\partial^2 = 0$ are theorems of Floer, which rest on compactness of the moduli spaces; the assumption that $\omega = 0$ on spheres rules out sphere bubbling.
The homology does not depend on $(H, J)$ up to isomorphism, and Floer proved that it is isomorphic to the singular homology $H_*(M; \F_2)$.
Since $\dim CF(H)$ is the number of contractible $1$-periodic orbits, this gives the Arnold conjecture lower bound in this setting.
The complex can be graded by the Conley–Zehnder index when the first Chern class also vanishes on spheres.

Example: let $H$ be a $C^2$-small Morse function $f$ independent of $t$.
Then the $1$-periodic orbits are the constant loops at critical points of $f$, and the $t$-independent Floer cylinders are the gradient lines of $f$.
In this case the Floer complex agrees with the Morse complex of $f$.

Signs, and the order of $x$ and $y$, vary with the conventions for $X_H$ and for the action functional.
Lagrangian Floer homology is defined in the same way, with generators the intersection points of two Lagrangian submanifolds and with strips instead of cylinders.
</document>

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# Nishida nilpotence theorem

## Statement

Let $\pi^s_* = \bigoplus_{n \ge 0} \pi^s_n$ be the graded ring of [[stable homotopy groups of spheres|HGZAYPQBXW]], with product given by composition.
Then every element $\alpha \in \pi^s_n$ with $n > 0$ is nilpotent: $\alpha^m = 0$ for some $m \ge 1$.

## Remarks

Nishida proved the theorem in 1973.
His proof uses extended power constructions on spectra and the operations they induce.
For example, the stable Hopf map $\eta \in \pi^s_1$ satisfies $\eta^4 = 0$, while $\eta^3 \ne 0$.
The hypothesis $n > 0$ is necessary: $\pi^s_0 \cong \Z$ is generated by the identity, and no nonzero integer is nilpotent.
The nilpotence theorem of Devinatz, Hopkins and Smith extends this result from the sphere spectrum to all ring spectra.

## Explanations

### L1 human

Stable maps between spheres can be composed, and composing a map of positive degree with itself raises the degree.
The theorem says that if one keeps composing such a map with itself, the result is eventually null-homotopic.
So no element of positive degree can be iterated forever.

### L2 human

The ring $\pi^s_*$ is graded-commutative, and its positive part consists of torsion elements.
In a general graded-commutative ring, torsion elements need not be nilpotent, so the theorem is a real constraint on the stable stems.
It was the starting point for the chromatic picture of stable homotopy theory, where the nilpotence theorem of Devinatz, Hopkins and Smith characterises nilpotent self-maps through complex cobordism.
</document>

<document source="https://tamnd.github.io/math2/node/FQ505YSHNA.md" tag="FQ505YSHNA" kind="definition">
# Semiprime ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
$R$ is *semiprime* if, for every $a \in R$, the condition $aRa = 0$ implies $a = 0$.

## Remarks

Equivalent forms: $R$ has no nonzero [[nilpotent ideal|C2H26BYWHE]]; for every [[two-sided ideal|0DDA6CY7GJ]] $I$, $I^2 = 0$ implies $I = 0$; the [[prime radical|WF117EJ2HF]] of $R$ is $0$.
Example: $\Q \times \Q$ is semiprime: if $a = (a_1, a_2)$ and $aRa = 0$, then taking $r = 1$ gives $a_1^2 = a_2^2 = 0$, so $a = 0$.
Non-example: in the ring $R$ of upper triangular $2 \times 2$ matrices over $\Q$, the matrix unit $a = e_{12}$ is nonzero and $e_{12} R e_{12} = 0$.
Every [[prime ring|Z9TJSZZYG1]] is semiprime.
A commutative ring is semiprime if and only if it is [[reduced|SZHY975FFX]].
</document>

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# Gorenstein ring

## Definition

Let $R$ be a commutative ring and $M$ an $R$-module.
The *injective dimension* $\operatorname{id}_R M$ is the least $n \ge 0$ such that there is an exact sequence $0 \to M \to I^0 \to \cdots \to I^n \to 0$ of $R$-modules.
Here every $I^j$ must be an [[injective module|85H7PRHDPZ]], and $\operatorname{id}_R M = \infty$ if there is no such sequence.
A commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] $R$ is a *Gorenstein local ring* if $\operatorname{id}_R R < \infty$.
A commutative Noetherian ring $R$ is a *Gorenstein ring* if the [[localization|X6JD4AG973]] $R_{\mathfrak{p}}$ is a Gorenstein local ring for every [[prime ideal|00MHHZ9SNE]] $\mathfrak{p}$.

## Remarks

A Gorenstein local ring is [[Cohen–Macaulay|N1QJQEGBFC]], and then $\operatorname{id}_R R = \dim R$.
Equivalently, a Noetherian local ring $(R, \mathfrak{m}, k)$ of dimension $d$ is Gorenstein if and only if $\operatorname{Ext}^i_R(k, R) = 0$ for $i \ne d$ and $\operatorname{Ext}^d_R(k, R) \cong k$.
For an Artinian local ring this says that the socle $(0 :_R \mathfrak{m})$ is a $k$-vector space of dimension $1$, or that $R$ is the [[injective hull|QCJN05GN64]] of $k$ as an $R$-module.
Example: $R = K[x]/(x^n)$ with $n \ge 1$ and $K$ a field is Gorenstein, since its socle is $K x^{n-1}$.
Non-example: $R = K[x,y]/(x^2, xy, y^2)$ is Artinian local with socle $Kx \oplus Ky$ of dimension $2$, so it is not Gorenstein, although it is Cohen–Macaulay.
</document>

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# Conley index

## Definition

Let $X$ be a [[locally compact|0FE2SB8WVJ]] [[metric space|0VKD1TP7F3]] and let $\varphi \colon \R \times X \to X$ be a continuous [[flow|BBZP1R872N]].
For a set $N \subseteq X$, let $\operatorname{Inv}(N) = \set{x \in X : \varphi_t(x) \in N \text{ for all } t \in \R}$, the largest [[invariant set|WWZTW11664]] in $N$.
A compact set $N$ is an *isolating neighbourhood* if $\operatorname{Inv}(N)$ lies in the interior of $N$.
An invariant set $S$ is *isolated* if $S = \operatorname{Inv}(N)$ for some isolating neighbourhood $N$.
An *index pair* for an isolated invariant set $S$ is a pair of compact sets $L \subseteq N \subseteq X$ such that:

1. The closure of $N \setminus L$ is an isolating neighbourhood with $\operatorname{Inv}(\overline{N \setminus L}) = S$.
2. $L$ is positively invariant in $N$: if $x \in L$, $t \ge 0$ and $\varphi_s(x) \in N$ for all $s \in [0, t]$, then $\varphi_t(x) \in L$.
3. $L$ is an exit set for $N$: if $x \in N$ and $\varphi_t(x) \notin N$ for some $t > 0$, then there is $s \in [0, t]$ with $\varphi_s(x) \in L$ and $\varphi_r(x) \in N$ for all $r \in [0, s]$.

Every isolated invariant set has an index pair.
The homotopy type of the [[pointed space|T9DG8DDE75]] $(N/L, [L])$ does not depend on the choice of index pair.
This [[homotopy type|SK0ZJGWXDP]] is the *Conley index* $h(S)$ of $S$.

## Remarks

Here $N/L$ is the [[quotient space|0BR3FY5Q8M]] that collapses $L$ to one point $[L]$; when $L = \emptyset$, one sets $N/L = N \sqcup \set{\ast}$ with base point $\ast$.
Example: let $0$ be a hyperbolic rest point in $\R^n$ with an unstable subspace of dimension $k$.
In adapted coordinates take $N = D^k \times D^{n-k}$, a product of closed balls, and the exit set $L = \partial D^k \times D^{n-k}$.
Then $N/L$ has the homotopy type of $D^k / \partial D^k$, the pointed sphere $S^k$; for an attracting point, $k = 0$ and the index is $S^0$.
Non-example: for the trivial flow $\varphi_t(x) = x$ on $\R$, the invariant set $S = \set{0}$ is not isolated, since every compact $N$ satisfies $\operatorname{Inv}(N) = N$, which is not contained in the interior of $N$.
Ważewski principle: the empty set has index the pointed one-point space, so if $h(S)$ is not that homotopy type, then $S \neq \emptyset$.

## Explanations

### L1 human

The Conley index is the shape left after collapsing the exit of a box around an isolated invariant set.
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<document source="https://tamnd.github.io/math2/node/FQMAKKE8R9.md" tag="FQMAKKE8R9" kind="definition">
# Malliavin calculus

## Definition

Let $T > 0$, let $(B_t)_{0 \le t \le T}$ be a [[Brownian motion|22735FGJV6]] on $(\Omega, \mathcal{F}, P)$, and let $\mathcal{F}$ be generated by $B$.
Let $H = L^2([0, T])$, and for $h \in H$ let $W(h) = \int_0^T h(t) \, dB_t$ be the [[Itô integral|JA95NH206E]].
A *smooth random variable* is $F = f(W(h_1), \ldots, W(h_n))$ with $h_1, \ldots, h_n \in H$ and $f \in C^\infty(\R^n)$, where $f$ and all its partial derivatives have polynomial growth.
Its *Malliavin derivative* is the $H$-valued random variable
$$D_t F = \sum_{i=1}^n \partial_i f(W(h_1), \ldots, W(h_n)) \, h_i(t), \quad 0 \le t \le T.$$
The operator $D$ is closable from $L^2(\Omega)$ to $L^2(\Omega; H)$.
Its closure is again written $D$, and its domain $\mathbb{D}^{1,2}$ is the completion of the smooth random variables for the norm $\norm{F}_{1,2}^2 = E[F^2] + E\left[\norm{DF}_H^2\right]$.
*Malliavin calculus* is the study of $D$, of its adjoint $\delta$, called the Skorokhod integral, and of their applications.

## Remarks

Example: $F = B_T^2 = W(\mathbf{1}_{[0,T]})^2$ is smooth with $f(x) = x^2$, so $D_t F = 2 B_T$ for $0 \le t \le T$.
Non-example: $F = \mathbf{1}_{\set{B_T > 0}}$ is not in $\mathbb{D}^{1,2}$; an indicator of an event $A$ lies in $\mathbb{D}^{1,2}$ only if $P(A)$ is $0$ or $1$.
The basic integration by parts formula is $E[\langle DF, h \rangle_H] = E[F \, W(h)]$ for smooth $F$ and $h \in H$.
The Clark–Ocone formula $F = E[F] + \int_0^T E[D_t F \mid \mathcal{F}_t] \, dB_t$ for $F \in \mathbb{D}^{1,2}$ makes the [[martingale representation theorem|2E8MQXDYNV]] explicit.
The setting is often the [[classical Wiener space|5Z00K7K1HY]] of continuous paths.
</document>

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# q-difference equation

## Summary

A $q$-difference equation relates the values of an unknown function at the points $x, qx, q^2 x, \dots$, where $q \ne 0, 1$ is a fixed number.
It is a difference equation in which the shift $x \mapsto x + 1$ is replaced by the dilation $x \mapsto qx$.
Equivalently, it can be written with the [[q-derivative|PMJT3FZE0Y]] in place of the ordinary derivative.
Such equations govern $q$-analogs of special functions, such as basic hypergeometric series, and they tend formally to differential equations as $q \to 1$.

## Explanations

### L1 human

An ordinary difference equation looks at a function at $x$ and at $x + 1$.
A $q$-difference equation looks at $x$ and at $qx$, a point a fixed factor away.
The simplest one is $y(qx) = y(x)$.
For $q = 2$, any function of $\log_2 x$ that has period $1$ solves it, for example $\cos(2\pi \log_2 x)$ for $x > 0$.
</document>

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# Support of a measure

## Summary

The support of a [[measure|PR11BZQQPG]] on a [[topological space|0CQHFPGJ45]] is the set of points all of whose open neighbourhoods have positive measure.
It is a closed set, and it tells where the measure really lives.
For a [[Radon measure|HZNEQQBCW4]] the complement of the support has measure zero.

## Explanations

### L1 human

Think of a measure as mass spread over a space.
The support is the place where some mass is found near every point, however closely one looks.
Points far from any mass are outside the support.
For example, the unit point mass at $0$ on the real line has support $\set{0}$.
The length measure on the real line has support the whole line, since every open interval has positive length.
</document>

<document source="https://tamnd.github.io/math2/node/FQVVNYV6N8.md" tag="FQVVNYV6N8" kind="statement">
# Courant–Friedrichs–Lewy condition

## Statement

Let $a \in \R$ and consider the [[transport equation|PVD7VZJZ9B]] $u_t + a u_x = 0$ with initial values $u(x, 0) = u_0(x)$.
Fix integers $p, q \ge 0$, real numbers $c_{-p}, \dots, c_q$ and a ratio $\lambda > 0$.
Consider the explicit [[finite difference scheme|X7HYEBA5SA]]
$$u_j^{n+1} = \sum_{k=-p}^{q} c_k\, u_{j+k}^n, \qquad u_j^0 = u_0(j \Delta x),$$
with time step $\Delta t = \lambda \Delta x$.
Suppose that for every smooth $u_0$ with compact support, $u_j^n \to u(x, t)$ whenever $\Delta x \to 0$, $j \Delta x \to x$ and $n \Delta t \to t$.
Then
$$-q \le a \lambda \le p.$$
In particular, for a three-point scheme ($p = q = 1$) convergence requires $\abs{a} \Delta t / \Delta x \le 1$.

## Remarks

Proof idea: $u_j^n$ depends only on the values $u_0(x_i)$ with $x_{j - pn} \le x_i \le x_{j + qn}$.
In the limit this is the interval $[x - p t/\lambda,\ x + q t/\lambda]$, the numerical domain of dependence of $(x, t)$.
The exact solution $u(x, t) = u_0(x - a t)$ depends only on the point $x - at$.
If $x - at$ lies outside the interval, change $u_0$ near $x - at$ only: the limit of the scheme stays the same, but $u(x, t)$ changes.

The condition is necessary but not sufficient.
The forward-time centred-space scheme has $p = q = 1$ and violates no CFL bound when $\abs{a}\lambda \le 1$, yet it diverges for every $\lambda > 0$.
The [[Lax–Friedrichs method|VY3J2NKGKW]] converges exactly when $\abs{a} \lambda \le 1$.
For $a \ge 0$, the upwind scheme $u_j^{n+1} = u_j^n - a\lambda(u_j^n - u_{j-1}^n)$ has $p = 1$ and $q = 0$, and it converges exactly when $a \lambda \le 1$.
For systems, the same argument gives $\Lambda \Delta t \le \Delta x$ for three-point schemes, where $\Lambda$ is the largest characteristic speed.

## Explanations

### L1 human

Information in a wave equation travels at a fixed speed.
An explicit scheme can only pass information one grid cell per time step.
If the time step is too large, the true information has moved further than the scheme can see, and the scheme cannot be right.
So the time step must be small compared with the space step divided by the wave speed.

### L2 human

The condition was found by Courant, Friedrichs and Lewy in 1928, in work on difference equations of mathematical physics.
In practice it sets the largest stable time step of explicit methods, such as $\Delta t \le \Delta x / \Lambda$.
Implicit schemes have a numerical domain of dependence equal to the whole grid, so the argument places no restriction on them.
Sufficient conditions come from stability analysis, and the [[Lax equivalence theorem|D5M7TRD3MY]] links stability with convergence for consistent linear schemes.
In the [[Glimm scheme|Y70AC2X30T]] and [[Godunov's scheme|43HZJ7DRT0]], a CFL bound also keeps neighbouring Riemann problems from interacting.
</document>

<document source="https://tamnd.github.io/math2/node/FQX56M03XD.md" tag="FQX56M03XD" kind="definition">
# Complex dynamics

## Definition

Let $X$ be the [[Riemann sphere|0J78KBPFZB]] $\hat{\C}$ or the complex plane $\C$, and let $f \colon X \to X$ be a [[holomorphic map|0NAHXR1FRH]].
For $n \ge 0$, the [[iterate|AYVTR2ES5B]] $f^n$ is the $n$-fold composition $f \circ \cdots \circ f$, with $f^0 = \id$.
The pair $(X, f)$ is a *holomorphic* [[discrete-time dynamical system|5AA9AJEARJ]].
The *orbit* of a point $z \in X$ is the sequence $z, f(z), f^2(z), \ldots$.
*Complex dynamics* is the study of the orbits of such systems and of how they depend on the point $z$ and on the map $f$.

## Remarks

The standard cases are [[rational functions|0XDK9CGYKJ]] $f \colon \hat{\C} \to \hat{\C}$ of degree at least $2$, polynomials of degree at least $2$, and transcendental [[entire functions|0G4F0XB9X2]] $f \colon \C \to \C$.
Example: $f(z) = z^2 + c$ with $c \in \C$ is a polynomial, hence holomorphic on $\C$, so $(\C, f)$ is a holomorphic dynamical system.
Non-example: $f(z) = \bar z^2$ is not holomorphic, since $\partial f / \partial \bar z = 2 \bar z \ne 0$ for $z \ne 0$, so its iteration is not part of complex dynamics in this sense.
Rational maps of degree $1$ are Möbius maps, whose dynamics is elementary, which is why the degree is usually taken to be at least $2$.
</document>

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# Orthocomplemented lattice

## Definition

Let $L$ be a [[bounded lattice|72V1Z82M5F]] with least element $0$ and greatest element $1$.
An *orthocomplementation* on $L$ is a map $a \mapsto a^\perp$ from $L$ to $L$ such that for all $a, b \in L$:

1. $a \wedge a^\perp = 0$ and $a \vee a^\perp = 1$, so $a^\perp$ is a [[complement|ZT47BK9Y6K]] of $a$;
2. $(a^\perp)^\perp = a$;
3. if $a \le b$, then $b^\perp \le a^\perp$.

An *orthocomplemented lattice*, or *ortholattice*, is a bounded lattice together with an orthocomplementation.

## Remarks

From clauses 2 and 3 follow the De Morgan laws $(a \vee b)^\perp = a^\perp \wedge b^\perp$ and $(a \wedge b)^\perp = a^\perp \vee b^\perp$.

Example: the lattice of closed linear subspaces of a [[Hilbert space|HM2HHE3J51]] $H$, with $M^\perp$ the [[orthogonal complement|0XJXTBCB2T]] of $M$.
Here $M \cap M^\perp = \set{0}$, the closed span of $M$ and $M^\perp$ is $H$, $M^{\perp\perp} = M$ for closed $M$, and $M \subseteq N$ implies $N^\perp \subseteq M^\perp$.

Non-example: the [[diamond lattice M3|JQ6FJH2TP8]] with middle elements $a, b, c$ and the map $a \mapsto b$, $b \mapsto c$, $c \mapsto a$, $0 \mapsto 1$, $1 \mapsto 0$.
Each image is a complement, but $(a^\perp)^\perp = c \ne a$, so clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/FQYSKP388T.md" tag="FQYSKP388T" kind="concept">
# Elliptic hypergeometric series

## Summary

An elliptic hypergeometric series is a series $\sum c_n$ in which the ratio $c_{n+1}/c_n$ is an [[elliptic function|FF9ZPCZ59K]] of $n$.
In ordinary [[hypergeometric series|428M453CR2]] the ratio is a rational function of $n$.
In [[basic hypergeometric series|EB4N9RQR51]] it is a rational function of $q^n$, and the elliptic case is the next step.
The terms are built from theta functions in place of the factors $1 - aq^k$.
Setting the elliptic nome $p$ to $0$ gives back basic hypergeometric series.

## Explanations

### L1 human

Many classical series are built so that each term is the previous one times a simple factor.
For hypergeometric series the factor is a fraction of polynomials in the index $n$.
For elliptic hypergeometric series the factor is a doubly periodic function of $n$, made from theta functions.
In the theta-function notation, putting the extra parameter $p$ equal to $0$ turns every theta factor $\theta(x; p)$ into $1 - x$, and the series becomes a familiar $q$-series.
So elliptic hypergeometric series are a common generalization of the classical and the $q$-series.
</document>

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# Restricted Burnside problem

## Statement

Let $m \ge 1$ and $n \ge 1$ be integers.
Then there are only finitely many finite groups, up to isomorphism, that are generated by $m$ elements and have [[exponent|0ZDW0H0VX4]] dividing $n$.
Equivalently, the free Burnside group $B(m, n)$ has only finitely many [[normal subgroups|0CB0DX5BVV]] of finite [[index|0DTXBQPA2S]].
Then the quotient $R(m, n)$ of $B(m, n)$ by their intersection is finite.
Every finite $m$-generated group of exponent dividing $n$ is a quotient of it.

## Remarks

Kostrikin proved the case where $n$ is prime.
Hall and Higman reduced the general case to prime powers $n = p^k$.
Their reduction assumes the Schreier conjecture on outer automorphism groups of finite simple groups, which follows from the classification of finite simple groups.
Zelmanov solved the prime power case around 1990, for which he received a Fields Medal.
The proofs pass to a Lie ring built from a central series of the group, which satisfies Engel identities.
They show that such Lie rings are nilpotent under the right finiteness conditions.

Example: for $n = 2$, every group of exponent $2$ is abelian, and $R(m, 2) = (\Z/2)^m$ has order $2^m$.

The restriction to finite groups is necessary: the group $B(m, n)$ itself is infinite for $m \ge 2$ and many large $n$, by the negative answer to the bounded [[Burnside problem|XWEMJ7T2XV]].

## Explanations

### L1 human

The Burnside problem asks whether a group with finitely many generators, in which $x^n = 1$ for every $x$, must be finite.
The answer is often no.
The restricted version only looks at the finite groups of this kind.
It says that they cannot be arbitrarily large: the number of generators and the exponent bound their order.

### L2 human

The theorem gives a largest finite group $R(m, n)$ with $m$ generators and exponent dividing $n$, and all others are its quotients.
It is a statement about finite groups, but its proof goes through Lie rings.
Kostrikin and Zelmanov proved that Lie algebras satisfying an Engel condition are locally nilpotent under suitable hypotheses.
It contrasts with the bounded [[Burnside problem|XWEMJ7T2XV]], where the free Burnside group can be infinite while all its finite quotients have bounded order.
</document>

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# Uniform integrability

## Definition

Let $(X, \Sigma, \mu)$ be a [[finite measure|63Z1Q4FS23]] space.
Let $\mathcal{F}$ be a family of real or complex [[integrable functions|X7GTF3D98F]] on $X$.
The family $\mathcal{F}$ is *uniformly integrable* if
$$\lim_{M \to \infty} \ \sup_{f \in \mathcal{F}} \int_{\set{\abs{f} > M}} \abs{f} \, d\mu = 0,$$
where the integrals are [[integrals of non-negative functions|DG76G5Z6XW]].

## Remarks

Example: on $[0, 1]$ with Lebesgue measure, let $\mathcal{F} = \set{f : \abs{f} \le 2}$.
For $M \ge 2$ each set $\set{\abs{f} > M}$ is empty, so the supremum is $0$.

Non-example: on $[0, 1]$, let $f_n = n \mathbf{1}_{(0, 1/n)}$.
For $n > M$ the integral of $f_n$ over $\set{f_n > M}$ equals $1$, so the supremum is $1$ for every $M$.

On a finite measure space the condition is equivalent to two conditions together.
First, $\sup_{f \in \mathcal{F}} \int \abs{f} \, d\mu < \infty$.
Second, for every $\varepsilon > 0$ there is $\delta > 0$ with $\int_A \abs{f} \, d\mu < \varepsilon$ for all $f \in \mathcal{F}$ whenever $\mu(A) < \delta$.
Any family dominated by one integrable function is uniformly integrable.
Bounded families in $L^p$ for some $p > 1$ are uniformly integrable.
On infinite measure spaces other versions of the definition are in use.
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# Topological transitivity

## Summary

A continuous map is topologically transitive when it carries every nonempty open set, at some time, into every other nonempty open set.
On a compact metric space without isolated points this is the same as having a point whose orbit is [[dense|0BSJSX7ME2]].
A transitive system cannot be split into two open invariant pieces, so it is the topological analogue of an [[ergodic transformation|AZRPHS478B]].
Transitivity is one of the three conditions in Devaney's definition of chaos.

## Explanations

### L1 human

Pick any two regions of the space, however small.
A transitive map, applied often enough, moves some point of the first region into the second.
So the motion links every part of the space with every other part.
A rotation of a circle by an irrational angle is transitive: every orbit winds around densely.
The identity map of an interval is not: nothing ever moves from one region to another.
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# Meshfree methods

## Summary

Meshfree methods solve differential equations using only a cloud of nodes, with no mesh that connects them into elements.
Each node carries a shape function built from the positions of nearby nodes, for example by [[moving least squares|T0JZN6WP5X]] or by [[radial basis functions|ZBSXSTQM2J]].
This avoids the cost of [[mesh generation|RTSAFF4KFJ]] and handles large deformations, fragmentation and moving boundaries with ease.
Examples include smoothed-particle hydrodynamics, the element-free Galerkin method and radial basis function collocation.
See the [[definition|30GF7TK8YP]].

## Explanations

### L1 human

A finite element method needs the domain cut into triangles or boxes that fit together, which is hard for complicated or changing shapes.
A meshfree method only scatters points over the domain.
The value of the approximate solution at a place is a weighted mix of the values at nearby points.
If the material tears or splashes, the points just move along, and no mesh has to be rebuilt.
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<document source="https://tamnd.github.io/math2/node/FRCY5BQF6N.md" tag="FRCY5BQF6N" kind="statement">
# Kunz's theorem

## Statement

Let $p$ be a prime and let $R$ be a [[Noetherian ring|0670T59HGE]] of [[characteristic|0F6ES3BRYP]] $p$.
Let $F \colon R \to R$, $F(r) = r^p$, be the [[Frobenius endomorphism|0T3WVTD675]], and view $R$ as an $R$-module through $F$.
Then the following are equivalent:

1. $R$ is a [[flat|73RYYEVDKA]] $R$-module through $F$.
2. $R$ is regular, that is, for every [[prime ideal|00MHHZ9SNE]] $\mathfrak p$ of $R$, the [[localization|4C1AE84N5N]] $R_{\mathfrak p}$ is a [[regular local ring|63XXTF6B5M]].

## Remarks

Both conditions are local, so one may assume that $R$ is local with maximal ideal $\mathfrak m$.
If $R$ is regular, $F$ is a local homomorphism from a regular local ring to a Cohen–Macaulay ring with $\dim R/\mathfrak m^{[p]} = 0$, and such a map is flat ("miracle flatness").
The converse is the deeper direction, and it was the main new point of Kunz's work.

Example: for $R = \F_p[x]$, the image of $F$ is $\F_p[x^p]$.
The ring $R$ is a free module over it with basis $1, x, \ldots, x^{p-1}$, so $F$ is flat, as the theorem predicts for this regular ring.
For $R = \F_p[x, y]/(xy)$, which is not regular at $(x, y)$, the Frobenius map is not flat.

## Explanations

### L1 human

In characteristic $p$, taking $p$-th powers is a ring homomorphism.
Kunz's theorem says that this map is as nice as possible, namely flat, exactly when the ring has no singular points.
So a purely algebraic property of one canonical map detects regularity, the algebraic form of having no singular points.

### L2 human

The theorem is the starting point of the use of Frobenius to study singularities in prime characteristic.
It shows, for instance, that in a regular ring every ideal is tightly closed: from $c x^q \in I^{[q]}$ and flatness one gets $c \in (I^{[q]} : x^q) = (I : x)^{[q]}$, which forces $x \in I$.
Weakening flatness of Frobenius leads to the classes of F-regular, F-rational, F-pure and F-injective rings, see [[tight closure|KJ9MMP99G6]].
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# Simplex category

## Description

For each natural number $n$, let $[n]$ be the [[totally ordered|0FRXBVVR03]] set $\set{0, 1, \dots, n}$ with the usual order.
The *simplex category* $\Delta$ has the sets $[n]$ for $n \ge 0$ as objects.
A morphism $[m] \to [n]$ is an order-preserving ([[monotone|0A7KV2WB7A]]) map, that is, a map $f$ with $f(i) \le f(j)$ whenever $i \le j$.
Composition is the composition of maps.
A composite of order-preserving maps is order-preserving, and identity maps are order-preserving, so $\Delta$ is a [[category|A2DE66937D]].

## Remarks

Every morphism of $\Delta$ is a composite of two kinds of basic maps.
The face map $\delta_i \colon [n-1] \to [n]$, for $0 \le i \le n$, is the injective order-preserving map whose image misses $i$.
The degeneracy map $\sigma_i \colon [n+1] \to [n]$, for $0 \le i \le n$, is the surjective order-preserving map that takes the value $i$ twice.
For example, the map $[2] \to [1]$ with $0 \mapsto 0$, $1 \mapsto 0$, $2 \mapsto 1$ is $\sigma_0$.
One pictures $[n]$ as the standard $n$-simplex with vertices $0, \dots, n$: face maps include a face, and degeneracy maps collapse an edge.
Functors from the opposite category $\Delta^{\mathrm{op}}$ are the [[simplicial objects|WK0A32TZHB]], and functors $\Delta^{\mathrm{op}} \to \mathbf{Set}$ are the [[simplicial sets|WW81ARB8K9]].

## Explanations

### L1 human

The simplex category is a small list of shapes: a point, an edge, a triangle, a tetrahedron, and so on.
The arrows between shapes say how one shape sits in another as a face, or how a shape can be squashed onto a smaller one.
For example, there are two maps $[0] \to [1]$, which pick the two endpoints of an edge, and three maps $[1] \to [1]$: the identity and the two constant maps.
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<document source="https://tamnd.github.io/math2/node/FRKAPMVZ7M.md" tag="FRKAPMVZ7M" kind="definition">
# Riesz measure of a subharmonic function

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]], and let $u$ be [[subharmonic|Z1MHMGFNFW]] on $\Omega$ and not identically $-\infty$ on any connected component of $\Omega$.
Then $u$ is locally integrable on $\Omega$.
Moreover, there is a unique positive [[Radon measure|HZNEQQBCW4]] $\mu_u$ on $\Omega$ such that
$$\int_\Omega u \, \Delta\psi \, d\lambda = \int_\Omega \psi \, d\mu_u \qquad \text{for all } \psi \in C_c^\infty(\Omega),$$
where $\lambda$ is Lebesgue measure and $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
The measure $\mu_u$ is the *Riesz measure* of $u$.
In other words, $\mu_u = \Delta u$ in the sense of distributions.

## Remarks

Some authors normalise differently, for example by $\frac{1}{2\pi}\Delta u$ in the plane, so that $\log\abs{z}$ has Riesz measure $\delta_0$.
If $u$ is of class $C^2$, then $\mu_u = (\Delta u)\, \lambda$.
The function $u$ is [[harmonic|0DEB0JMN0W]] on an open subset $V$ of $\Omega$ if and only if $\mu_u(V) = 0$.

Example: let $u(x) = \abs{x}^2$ on $\R^n$.
Integration by parts gives $\int \abs{x}^2 \Delta\psi \, d\lambda = \int \Delta(\abs{x}^2)\, \psi \, d\lambda = 2n \int \psi \, d\lambda$, so $\mu_u = 2n\,\lambda$.
Example: for $u(z) = \log\abs{z}$ on $\C = \R^2$, Green's formula gives $\mu_u = 2\pi\,\delta_0$.

Non-example: for $v(x) = -\abs{x}^2$, the distribution $\Delta v = -2n\,\lambda$ is a negative measure.
This matches the fact that $v$ is not subharmonic, so the hypothesis of the definition fails and $v$ has no Riesz measure.
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# Wishart distribution

## Definition

Let $p$ and $n$ be positive integers and let $\Sigma$ be a $p \times p$ [[positive definite matrix|0QQCG3JWJ4]].
Let $X_1, \dots, X_n$ be [[independent and identically distributed|0AJN7WHBJZ]] random column vectors in $\R^p$ with the [[multivariate normal distribution|0G558GJQMW]] $\mathcal{N}_p(0, \Sigma)$.
The distribution of the random $p \times p$ matrix
$$S = \sum_{i=1}^n X_i X_i^T$$
is the *Wishart distribution* with scale matrix $\Sigma$ and $n$ degrees of freedom, written $W_p(\Sigma, n)$.

## Remarks

The mean is $E(S) = n \Sigma$.
If $n \ge p$, then $S$ is positive definite with probability $1$.
Its density on positive definite matrices is then
$$f(S) = \frac{\det(S)^{(n - p - 1)/2} \exp\left(-\tfrac{1}{2} \tr(\Sigma^{-1} S)\right)}{2^{np/2} \det(\Sigma)^{n/2} \, \Gamma_p(n/2)}$$
where $\det$ is the [[determinant|0C3CFHDN6R]], $\tr$ is the [[trace|07PTPSZPME]] and $\Gamma_p$ is the multivariate gamma function.

Example: for $p = 1$ and $\Sigma = (\sigma^2)$, $S = \sum_i X_i^2 = \sigma^2 \sum_i (X_i/\sigma)^2$ with $X_i/\sigma$ standard normal, so $W_1(\sigma^2, n)$ is $\sigma^2$ times the [[chi-squared distribution|0RFMMBHFTV]] with $n$ degrees of freedom.

Example: if $Y_1, \dots, Y_m$ are i.i.d. $\mathcal{N}_p(\mu, \Sigma)$ with $m \ge 2$ and $\bar{Y}$ is their mean, then $\sum_{i=1}^m (Y_i - \bar{Y})(Y_i - \bar{Y})^T \sim W_p(\Sigma, m - 1)$.

Non-example: if the $X_i$ have a common mean $\mu \ne 0$, then $E(S) = n\Sigma + n\mu\mu^T \ne n\Sigma$, so $S$ is not $W_p(\Sigma, n)$; its law is a noncentral Wishart distribution.
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# Riesz transform

## Summary

The Riesz transforms $R_1, \ldots, R_n$ are the $n$ [[singular integral operators|HTMCEARGDD]] on $\R^n$ with Fourier multipliers $-i\xi_j/\abs{\xi}$.
They are the higher-dimensional analogue of the [[Hilbert transform|NSCEBXM7ZM]], which is the case $n = 1$.
They express second derivatives through the Laplacian, by $\partial_j\partial_k = -R_j R_k \Delta$, so their $L^p$ bounds give $L^p$ estimates for elliptic equations.
They are bounded on $L^p(\R^n)$ for $1 < p < \infty$ and commute with translations and dilations.

## Explanations

### L1 human

The Hilbert transform on the line keeps the size of each frequency and turns its phase by a quarter turn.
In several variables there is one such operator for each coordinate direction.
The $j$-th Riesz transform weights the frequency $\xi$ by the $j$-th component of its direction $\xi/\abs{\xi}$, and turns the phase by a quarter turn.
Together they turn a function into a vector field, much as the gradient does, but without changing the size of frequencies.
For example, the sum $R_1^2 + \cdots + R_n^2$ is minus the identity, because the squared components of a unit vector add up to $1$.
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# Splitting principle

## Statement

Let $E \to B$ be a complex vector bundle of rank $n$ over a paracompact space $B$.
Then there are a space $F(E)$ and a continuous map $p \colon F(E) \to B$ such that:

1. The pullback $p^* E$ is isomorphic to a [[Whitney sum|BE9NJRVDR3]] $L_1 \oplus \cdots \oplus L_n$ of complex line bundles over $F(E)$.
2. The induced map $p^* \colon H^*(B; \Z) \to H^*(F(E); \Z)$ in [[singular cohomology|6N7BC70CX0]] is injective.

The same holds for a real vector bundle, with real line bundles and with $\Z/2\Z$ coefficients in place of $\Z$.

## Remarks

One can take for $F(E)$ the bundle of complete flags in the fibers of $E$.
It is built in steps.
Over the projective bundle $P(E)$, the pullback of $E$ contains the tautological line bundle and splits it off.
The Leray–Hirsch theorem shows that the pullback in cohomology is injective.
Then one repeats with the complement, of rank $n - 1$.

In use: to prove an identity between [[characteristic classes|7HACJC0JE2]] that is natural and holds in the injective image, it suffices to prove it for sums of line bundles.
For example, the Chern classes of a sum of line bundles are $c(L_1 \oplus \cdots \oplus L_n) = \prod_{i=1}^n (1 + c_1(L_i))$, so $c_k(E)$ pulls back to the $k$-th elementary symmetric polynomial in the classes $x_i = c_1(L_i)$.
The Chern character and the Todd class are defined by symmetric power series in these $x_i$ for this reason.

## Explanations

### L1 human

A vector bundle can be complicated, but a sum of line bundles is easy to handle.
The splitting principle says that any vector bundle becomes such a sum after pulling it back to a suitable bigger space.
The pullback loses no cohomological information, because the map on cohomology is injective.
So a formula about characteristic classes can be checked as if every bundle were a sum of lines.

### L2 human

The splitting principle reduces computations with Chern or Stiefel–Whitney classes of rank $n$ bundles to computations with $n$ first Chern or Stiefel–Whitney classes, the Chern roots.
The Whitney sum formula, the formula for the Chern classes of a tensor product, and the formula $w(T\R P^n) = (1+a)^{n+1}$ all become polynomial identities in these roots.
It is also behind the description of $H^*(BU(n); \Z)$ as the symmetric polynomials in $n$ variables of degree $2$, via the map $BU(1)^n \to BU(n)$.
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# Essential spectrum

## Summary

The essential spectrum of a bounded operator is the part of its spectrum that does not change under compact perturbations.
It consists of the $\lambda$ for which $T - \lambda I$ is not a Fredholm operator.
For a self-adjoint operator on a Hilbert space, it is what remains of the spectrum after removing the isolated eigenvalues of finite multiplicity.

## Explanations

### L1 human

Some points of the spectrum are fragile: an isolated eigenvalue with a finite-dimensional eigenspace can be moved or removed by adding an operator of finite rank.
Other points are robust, because they involve infinitely many directions at once.
The essential spectrum collects the robust points.
For example, every compact operator on an infinite-dimensional space has essential spectrum $\set{0}$, and the identity has essential spectrum $\set{1}$.
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# Subobject

## Summary

A subobject of an object $X$ in a [[category|A2DE66937D]] is an equivalence class of [[monomorphisms|Y2NP1JFR86]] into $X$.
Two monomorphisms are equivalent when each factors through the other.
It is the categorical version of a subset, a subgroup or a subspace, defined only through morphisms.
The subobjects of $X$ are partially ordered by factorisation, and in a topos they are classified by the subobject classifier.

## Explanations

### L1 human

In set theory a subset of $X$ is a smaller set sitting inside $X$.
In a category there is no "inside", so one uses an injective-like arrow $A \to X$ instead, which shows how $A$ sits in $X$.
Two such arrows that differ only by renaming the elements of $A$ give the same subobject.
For example, among sets, the inclusion $\{1, 2\} \to \{1, 2, 3\}$ and the map $\{a, b\} \to \{1, 2, 3\}$ with $a \mapsto 1$, $b \mapsto 2$ represent the same subobject, namely the subset $\{1, 2\}$.
</document>

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# Thom spectrum

## Definition

For $n \ge 0$, let $\gamma_n$ be the [[universal $n$-plane bundle|1HDH716Z24]] over the [[classifying space|WSZ40H5E4T]] $BO(n)$, and let $\varepsilon$ be the trivial line bundle.
Let $(B_n, f_n, g_n)_{n \ge 0}$ be a sequence of spaces $B_n$ with maps $f_n \colon B_n \to BO(n)$ and $g_n \colon B_n \to B_{n+1}$ such that $f_{n+1} \circ g_n$ classifies the [[vector bundle|P4DZM975BV]] $\xi_n \oplus \varepsilon$, where $\xi_n = f_n^* \gamma_n$.
Write $\operatorname{Th}(\xi)$ for the [[Thom space|5MNYT3AFPH]] of a bundle $\xi$.
The *Thom spectrum* of this sequence is the [[spectrum|1GHF4EKBXS]] $M$ with spaces $M_n = \operatorname{Th}(\xi_n)$.
Let $\Sigma$ be the reduced [[suspension|2VD7GR64JJ]].
The structure maps of $M$ are the composites $\Sigma \operatorname{Th}(\xi_n) \cong \operatorname{Th}(\xi_n \oplus \varepsilon) \to \operatorname{Th}(\xi_{n+1})$.
Here the second map is induced by the bundle map $\xi_n \oplus \varepsilon \to \xi_{n+1}$ that covers $g_n$.

## Remarks

Example: $B_n = BO(n)$ with $f_n = \id$ gives the spectrum $MO$.
Here $g_n \colon BO(n) \to BO(n+1)$ classifies $\gamma_n \oplus \varepsilon$, so the condition holds by construction.
Taking $B_n = BSO(n)$ gives $MSO$.
For complex bundles one uses $BU(n)$ and the universal complex bundle, whose Thom space is placed in degree $2n$; this gives the spectrum $MU$ of complex cobordism.

Non-example: keep the spaces $\operatorname{Th}(\gamma_n)$ but take all structure maps constant.
This is still a spectrum, but its structure maps are not induced by bundle maps, so it is not the Thom spectrum $MO$.
Indeed, all its stable homotopy groups vanish, while $\pi_0 MO \cong \Z/2$.

More generally one can form the Thom spectrum of a map $f \colon B \to BO$, taking for $B_n$ the preimage of $BO(n)$ under $f$, after replacing $f$ by a fibration.
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# Logarithm of a matrix

## Summary

A logarithm of a square matrix $A$ is a matrix $B$ whose [[matrix exponential|020M04FN1H]] is $A$.
Over the complex numbers, a matrix has a logarithm exactly when it is [[invertible|0EB9N0S33C]].
The logarithm is not unique, just as for the [[complex logarithm|02QGGD5G87]].
Matrix logarithms pass from a matrix Lie group to its Lie algebra.
They are also used to interpolate between rotations and to find the generator of a linear system from its flow.

## Explanations

### L1 human

For numbers, the logarithm undoes the exponential.
For matrices, a logarithm of $A$ is any matrix $B$ with $e^B = A$.
The rotation of the plane by an angle $\theta$ has the logarithm $\begin{pmatrix} 0 & -\theta \\ \theta & 0 \end{pmatrix}$, an infinitesimal rotation.
Adding $2\pi$ to $\theta$ gives the same rotation and another logarithm, so a logarithm is not unique.
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# Primary obstruction

## Definition

Let $(X, A)$ be a [[relative CW complex|D0DAFG8QK6]], and let $X_k$ be the union of $A$ with the cells of $X$ of dimension at most $k$.
Let $n \ge 1$ and let $Y$ be an [[$(n-1)$-connected|1A943BWBQT]] space; if $n = 1$, assume also that $\pi_1(Y)$ is abelian.
Let $f \colon A \to Y$ be a map.
Since $\pi_k(Y) = 0$ for $k < n$, the map $f$ extends to a map $g \colon X_n \to Y$.
The *primary obstruction* of $f$ is the class
$$\gamma(f) = [c(g)] \in H^{n+1}(X, A; \pi_n(Y)),$$
where $c(g)$ is the obstruction cocycle of $g$ from [[obstruction theory|ZZWHGGFB4P]].

## Remarks

The class $\gamma(f)$ does not depend on the choice of $g$.
Indeed, any two extensions of $f$ to $X_n$ are homotopic relative to $A$ on $X_{n-1}$, since $Y$ is $(n-1)$-connected.
The difference of the two extensions on the $n$-cells changes $c(g)$ by a coboundary.
The map $f$ extends to $X_{n+1}$ if and only if $\gamma(f) = 0$.
If $X \setminus A$ has no cells of dimension above $n + 1$, then $f$ extends to $X$ if and only if $\gamma(f) = 0$.

Example: let $(X, A) = (D^{n+1}, S^n)$, $Y = S^n$ with $n \ge 1$, and $f$ the identity.
Here $H^{n+1}(D^{n+1}, S^n; \pi_n(S^n)) \cong \Z$ and $\gamma(f)$ is a generator, so $f$ does not extend.

Non-example: let $Y$ be not $(n-1)$-connected, for instance $Y = S^1$ with $n = 2$.
Then $f$ need not extend over $X_2$ at all, so no class in degree $n + 1$ is defined from $f$ alone.

The same construction applies to a fibration over a CW complex with $(n-1)$-connected fibre $F$.
It gives the primary obstruction to a section, a class in $H^{n+1}$ of the base with coefficients in $\pi_n(F)$, local in general.
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# Regular scheme

## Definition

A scheme $X$ is *regular* if it is [[locally Noetherian|ND6XEY68P4]] and for every point $x \in X$ the [[stalk|JTAKAPTT8H]] $\mathcal{O}_{X,x}$ is a [[regular local ring|63XXTF6B5M]].
Here a Noetherian local ring $R$ with maximal ideal $\mathfrak{m}$ and residue field $\kappa$ is regular if $\dim_\kappa \mathfrak{m}/\mathfrak{m}^2 = \dim R$, the [[Krull dimension|MVGBS3J1P7]] of $R$.

## Remarks

A regular local ring is an integrally closed domain, so a regular scheme is normal.
Example: $\operatorname{Spec} \Z$ is regular.
Its local rings are $\Q$, of dimension $0$ with $\mathfrak{m} = 0$, and $\Z_{(p)}$, of dimension $1$ with $\mathfrak{m}/\mathfrak{m}^2$ spanned by $p$.
Non-example: the cusp $\operatorname{Spec} k[x, y]/(y^2 - x^3)$ over a field $k$ is not regular at the origin.
There the local ring has dimension $1$, but $\mathfrak{m}/\mathfrak{m}^2$ has basis $x, y$, since $y^2 - x^3 \in \mathfrak{m}^2$; so its dimension is $2 \ne 1$.
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# Homogeneous coordinates

## Summary

Homogeneous coordinates describe a point of the [[projective space|ZT9A6JK8QF]] $\mathbb{P}^n$ by $n + 1$ numbers, not all zero, taken up to a common nonzero factor.
They turn points at infinity into ordinary tuples, so that projective geometry can be done with linear algebra.
Projective transformations become matrices, and curves become zero sets of homogeneous polynomials.
They are standard in computer graphics, where translations and perspective become matrix products.

## Explanations

### L1 human

A point $(x, y)$ of the plane gets three coordinates $[x : y : 1]$, and any multiple such as $[2x : 2y : 2]$ names the same point.
To go back, divide by the last coordinate.
Triples with last coordinate $0$, such as $[1 : 1 : 0]$, cannot be divided back: they are the points at infinity, one for each direction of lines.
So the lines $y = x$ and $y = x + 1$, which are parallel, meet at $[1 : 1 : 0]$.
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# Buekenhout geometry

## Definition

Let $I$ be a finite set, the set of types.
Let $X$ be a set with a reflexive and symmetric relation $*$, called incidence.
Let $\tau \colon X \to I$ be a map, called the type function.
The triple $(X, *, \tau)$ is a *Buekenhout geometry* (or *geometry*) over $I$ if:

1. if $x * y$ and $\tau(x) = \tau(y)$, then $x = y$;
2. every flag, that is, every set of pairwise incident elements of $X$, is contained in a flag $C$ with $\tau(C) = I$.

## Remarks

A flag $C$ with $\tau(C) = I$ is a *chamber*; by clause 1 it has exactly one element of each type.
The number $\abs{I}$ is the rank.
A geometry of rank $2$ is the same as an [[incidence structure|FGKNAFNHA3]] whose flags extend, such as a projective plane or a [[generalized polygon|PCT2HY9AAZ]].
The residue of a flag $F$ is the geometry of the elements incident with all of $F$ and not in $F$, over the types not met by $F$.
The *Buekenhout diagram* records, for each pair of types $i \ne j$, the class of the rank $2$ residues of type $\set{i, j}$.
Tits buildings give geometries whose diagrams are Coxeter diagrams.

Example: let $\mathcal{P}$ be an $n$-dimensional [[projective space|ZT9A6JK8QF]].
Take $X$ to be the set of its subspaces of dimension $0, 1, \dots, n - 1$, let $\tau$ be the dimension, and let $x * y$ mean $x \subseteq y$ or $y \subseteq x$.
Clause 1 holds since two distinct subspaces of the same dimension are not contained in each other.
Clause 2 holds since every chain of subspaces can be refined to a chain with one subspace of each dimension.

Non-example: take the lines of a projective plane, all of one type, with $x * y$ meaning that $x$ and $y$ meet.
Any two lines meet, so two distinct incident elements have the same type and clause 1 fails.
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# Jack function

## Definition

Let $\alpha$ be a positive real number, and let $\Lambda$ be the ring of [[symmetric functions|PAJEHWXW6M]] in variables $x = (x_1, x_2, \dots)$ with real coefficients.
For a [[partition|0JV30XN73Z]] $\lambda$ with $\ell(\lambda)$ parts, let $m_\lambda$ be the monomial symmetric function and $p_\lambda = p_{\lambda_1} p_{\lambda_2} \cdots$ the product of [[power sums|0ZNXZB4HNB]].
Let $z_\lambda = \prod_{i \ge 1} i^{m_i} m_i!$, where $m_i$ is the number of parts of $\lambda$ equal to $i$.
Define a bilinear form on $\Lambda$ by $\langle p_\lambda, p_\mu \rangle_\alpha = \delta_{\lambda\mu}\, z_\lambda\, \alpha^{\ell(\lambda)}$, and write $\mu < \lambda$ for the strict dominance order on partitions of the same number.
The *monic Jack functions* are the unique family $(P_\lambda^{(\alpha)})_\lambda$ in $\Lambda$ such that:

1. $P_\lambda^{(\alpha)} = m_\lambda + \sum_{\mu < \lambda} c_{\lambda\mu} m_\mu$ for some coefficients $c_{\lambda\mu}$;
2. $\langle P_\lambda^{(\alpha)}, P_\mu^{(\alpha)} \rangle_\alpha = 0$ for $\lambda \ne \mu$.

For a box $s$ of the Young diagram of $\lambda$, let $a(s)$ be the number of boxes to its right.
Let $l(s)$ be the number of boxes below $s$.
The *Jack function* is $J_\lambda^{(\alpha)} = c_\lambda(\alpha)\, P_\lambda^{(\alpha)}$ with $c_\lambda(\alpha) = \prod_{s \in \lambda} \big(\alpha\, a(s) + l(s) + 1\big)$.

## Remarks

Example: in degree $2$, $P_{(2)}^{(\alpha)} = m_{(2)} + \frac{2}{1+\alpha} m_{(1,1)}$ and $P_{(1,1)}^{(\alpha)} = m_{(1,1)}$.
Check: $m_{(2)} = p_2$ and $m_{(1,1)} = \frac12 (p_1^2 - p_2)$, so $P_{(2)}^{(\alpha)} = \frac{\alpha}{1+\alpha} p_2 + \frac{1}{1+\alpha} p_1^2$.
With $\langle p_2, p_2 \rangle_\alpha = 2\alpha$ and $\langle p_1^2, p_1^2 \rangle_\alpha = 2\alpha^2$, one gets $\langle P_{(2)}^{(\alpha)}, m_{(1,1)} \rangle_\alpha = \frac12 \left( -\frac{2\alpha^2}{1+\alpha} + \frac{2\alpha^2}{1+\alpha} \right) = 0$.
The normalization gives $J_{(2)}^{(\alpha)} = (1+\alpha) m_{(2)} + 2 m_{(1,1)}$ and $J_{(1,1)}^{(\alpha)} = 2 m_{(1,1)}$.

Non-example: the Schur function $s_{(2)} = m_{(2)} + m_{(1,1)}$ satisfies clause 1 but fails clause 2 for $\alpha \ne 1$, since $\langle s_{(2)}, m_{(1,1)} \rangle_\alpha = \frac{\alpha^2 - \alpha}{2}$.

At $\alpha = 1$ the form is the Hall inner product, $P_\lambda^{(1)} = s_\lambda$, and $J_\lambda^{(1)}$ is $s_\lambda$ times the product of the hook lengths of $\lambda$.
At $\alpha = 2$, $J_\lambda^{(2)}$ are the zonal polynomials.
Existence of the family is a theorem; it is not automatic, because dominance is only a partial order.
Restricting to $n$ variables gives the Jack polynomials $J_\lambda^{(\alpha)}(x_1, \dots, x_n)$, which vanish when $\ell(\lambda) > n$.
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# Jacobi equation

## Summary

The Jacobi equation of a variational problem is the linear second-order differential equation $-(P h')' + Q h = 0$ obtained from the [[second variation|4C3BANJ8AH]] along an extremal.
It is the [[Euler–Lagrange equation|PS0VV149ET]] of the quadratic functional $\int (P h'^2 + Q h^2) \, dx$, and also the linearisation of the Euler–Lagrange equation along the extremal.
Its solutions describe how nearby extremals spread apart, and their zeros define conjugate points.
For the length functional on a Riemannian manifold it becomes the equation of a [[Jacobi field|9AHMM41PJ3]].

## Explanations

### L1 human

Take an extremal curve and wiggle it slightly into a family of nearby extremals.
The small gap between the curve and its neighbours, to first order, solves the Jacobi equation.
For the functional $\int (y'^2 - y^2) \, dx$ the Jacobi equation is $h'' + h = 0$, with solutions $h = c \sin(x - x_0)$.
These solutions vanish again after a distance $\pi$, which is why this extremal stops being a minimiser on intervals longer than $\pi$.
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<document source="https://tamnd.github.io/math2/node/FSGES670AZ.md" tag="FSGES670AZ" kind="statement">
# Hartogs's extension theorem

## Statement

Let $n \ge 2$, let $D \subseteq \C^n$ be a connected [[open set|02BH3MCBQB]], and let $K \subseteq D$ be a [[compact set|0R6EWFKRCF]] such that $D \setminus K$ is [[connected|0SPE8WNWFV]].
Then every [[holomorphic function|507C34W0ZT]] $f$ on $D \setminus K$ is the restriction of a unique holomorphic function $F$ on $D$.

## Remarks

A short proof uses the [[$\bar\partial$ operator|M1VKCW3RFW]].
Choose a smooth function $\chi$ with compact support in $D$ that equals $1$ near $K$, and let $g = (1 - \chi) f$, extended by $0$ on $K$.
The form $\bar\partial g = -f \, \bar\partial \chi$ is smooth with compact support in $\C^n$, and for $n \ge 2$ the equation $\bar\partial u = \bar\partial g$ has a solution $u$ with compact support.
Then $F = g - u$ is holomorphic on $D$.
The function $u$ vanishes on the unbounded component of the complement of the support of $\bar\partial g$.
So $F = f$ on a nonempty open subset of $D \setminus K$, hence on all of $D \setminus K$ by the identity theorem.

The hypothesis $n \ge 2$ is necessary: $1/z$ is holomorphic on $\C \setminus \set{0}$ and does not extend to $\C$.
Connectedness of $D \setminus K$ is also necessary.
In the unit ball of $\C^n$ let $K$ be the sphere $\abs{z} = 1/2$.
The function equal to $0$ inside $K$ and to $1$ outside is holomorphic on the ball minus $K$, but it has no holomorphic extension.

Example: every holomorphic function on the punctured ball $\set{0 < \abs{z} < 1}$ in $\C^2$ extends holomorphically over the origin.

## Explanations

### L1 human

In several complex variables a holomorphic function cannot have a compact hole of singularities.
If it is defined everywhere except on a compact set, it automatically extends across that set.
This is very different from one variable, where $1/z$ has an isolated singularity at $0$.

### L2 human

The theorem is the first sign that domains in $\C^n$, $n \ge 2$, carry hidden constraints for holomorphic functions.
It implies that the zeros and the singularities of holomorphic functions are never isolated when $n \ge 2$.
It leads to the notions of domain of holomorphy and envelope of holomorphy, which describe how far all holomorphic functions on a domain extend.
The proof sketched above shows the role of the compactly supported solution of $\bar\partial u = v$, which exists only when $n \ge 2$.
</document>

<document source="https://tamnd.github.io/math2/node/FSJ9NGRRPA.md" tag="FSJ9NGRRPA" kind="statement">
# Quantifier elimination for real closed fields

## Statement

Let $\mathcal{L} = \set{+, -, \cdot, 0, 1, <}$ be the language of ordered rings.
Let $\mathrm{RCF}$ be the $\mathcal{L}$-[[theory|92SWXST1ZM]] of [[real closed fields|DDY3Q3PJ6W]] with their unique field ordering.
Its axioms are those of an [[ordered field|0P3VCBDEE2]], together with two more.
One axiom says that every positive element has a square root.
One scheme says that every polynomial of odd degree has a root.
Then $\mathrm{RCF}$ admits [[quantifier elimination|7FDDXV3JK7]]: for every $\mathcal{L}$-formula $\varphi(x_1, \dots, x_n)$ there is a quantifier-free $\mathcal{L}$-formula $\psi(x_1, \dots, x_n)$ such that
$$\mathrm{RCF} \vdash \forall x_1 \cdots \forall x_n \, \bigl(\varphi(x_1, \dots, x_n) \leftrightarrow \psi(x_1, \dots, x_n)\bigr).$$

## Remarks

The theorem is due to Tarski.
It suffices to eliminate one existential quantifier in front of a conjunction of polynomial equations and inequalities in one variable $x$.
Such a conjunction has a solution exactly when certain sign conditions on the coefficients hold.
These conditions can be computed, for instance with Sturm sequences and the intermediate value property of polynomials over a real closed field.

Example: for real closed fields,
$$\exists x \, (x^2 + b x + c = 0) \quad \leftrightarrow \quad b^2 - 4c > 0 \ \lor \ b^2 - 4c = 0.$$

Consequences: $\mathrm{RCF}$ is [[complete|655K8Y7VS8]], so every real closed field satisfies the same first-order sentences as $\R$; and $\mathrm{RCF}$ is decidable.
Geometrically, the projection of a semialgebraic set is semialgebraic, which is the [[Tarski–Seidenberg theorem|3W5JGDZAN0]].

The symbol $<$ is necessary.
In the language of rings alone, the set $\set{x \in \R : \exists y \, (x = y^2)} = [0, \infty)$ is definable, but every quantifier-free definable subset of $\R$ in that language is finite or cofinite.

## Explanations

### L1 human

A statement about real numbers that uses "there exists" and "for all", together with addition, multiplication and order, can be rewritten without "there exists" and "for all".
What remains is a combination of polynomial equations and inequalities in the free variables.
For example, "the quadratic $x^2 + bx + c$ has a real root" is the same as "$b^2 - 4c \ge 0$".
So every such question about the real numbers can, in principle, be decided by a computation.

### L2 human

Quantifier elimination is the model-theoretic core of real algebraic geometry.
It shows that the sets definable in $\R$ with $+$, $\cdot$ and $<$ are exactly the semialgebraic sets, and that projections and images under polynomial maps stay in this class.
It gives the completeness of the theory of real closed fields, which transfers results from $\R$ to every real closed field.
This transfer gives a short modern proof of Artin's solution of [[Hilbert's seventeenth problem|J45F4F13VB]].
It also shows that the theory of $\R$ is [[o-minimal|DTWHAW1N0V]]: every definable subset of the line is a finite union of points and intervals.
The analogous result for [[algebraically closed fields|MT8TDVPBXB]] uses only the language of rings.
</document>

<document source="https://tamnd.github.io/math2/node/FSRST924GR.md" tag="FSRST924GR" kind="concept">
# Fuzzy algebraic structure

## Summary

A fuzzy algebraic structure replaces a substructure of an algebra by a fuzzy subset.
A fuzzy subset gives each element a degree of membership between $0$ and $1$.
The fuzzy subset must respect the operations: the result of an operation belongs at least as much as the least member among its arguments.
The first case studied was the fuzzy subgroup of a [[group|G5S1M9]], introduced by Rosenfeld in 1971.
Fuzzy subrings, fuzzy ideals and fuzzy subalgebras of general algebras followed.
A fuzzy subalgebra is determined by its level sets, which form a nested family of ordinary subalgebras, one for each membership level.

## Explanations

### L1 human

An ordinary subgroup either contains an element or does not.
A fuzzy subgroup lets an element belong only partly, with a degree such as $1/2$.
The rule is that combining elements never lowers membership below the weakest of them.
For example, in the integers let even numbers have degree $1$ and odd numbers degree $1/2$.
Adding two numbers of degree at least $1/2$ gives a number of degree at least $1/2$, so this is a fuzzy subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/FSXA3CF365.md" tag="FSXA3CF365" kind="statement">
# Krull–Akizuki theorem

## Statement

Let $A$ be a [[Noetherian|0MVH4Y1CRZ]] [[integral domain|09S3PFBEPM]] of [[Krull dimension|MVGBS3J1P7]] $1$.
Let $K$ be the [[field of fractions|0VF11ZTTMD]] of $A$ and let $L$ be a finite field extension of $K$.
Let $B$ be a ring with $A \subseteq B \subseteq L$.
Then $B$ is Noetherian and has Krull dimension at most $1$.
Moreover, for every nonzero ideal $J$ of $B$, the quotient $B/J$ has finite [[length|CZSCNZH8VY]] as an $A$-module.

## Remarks

The general case reduces to the case $L = K$.
There the key step is the bound $\ell_A(B/aB) \le \ell_A(A/aA)$ for each nonzero $a \in A$, where $\ell_A$ is the length; the right side is finite because $A$ has dimension $1$.
Each nonzero ideal $J$ of $B$ contains a nonzero element $a$ of $A$, so $J/aB$ has finite length, and $J$ is finitely generated.
A main application: the [[integral closure|G0V86SJBQW]] of $A$ in $L$ is a Noetherian integrally closed domain of dimension at most $1$, that is, a Dedekind domain or a field.
For example, with $A = \Z$ and $L$ a number field, this shows that the ring of integers of $L$ is Noetherian.
The ring $B$ need not be a finitely generated $A$-module, so the result is not a consequence of [[Hilbert's basis theorem|0YKA9J3XEE]].

## Explanations

### L1 human

Start with a nice ring of dimension one, such as the integers.
Take any ring that sits between it and a finite extension of its fraction field.
The theorem says that this new ring is still Noetherian and still has dimension at most one.
So rings of integers in number fields keep the finiteness of $\Z$.

### L2 human

The theorem is the standard route to the following fact.
The integral closure of a one-dimensional Noetherian domain in a finite extension of its fraction field is a Dedekind domain, even when it is not a finite module.
It is special to dimension $1$: for any Noetherian domain, the [[Mori–Nagata theorem|J8YYYP5CDN]] only gives a Krull domain, and the integral closure can fail to be Noetherian.
The proof uses [[lengths of modules|CZSCNZH8VY]] and the fact that $A/aA$ is Artinian for $a \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/FSZ6PCKN0F.md" tag="FSZ6PCKN0F" kind="definition">
# Functional completeness

## Definition

Let $A$ be a [[finite set|070R982G4B]] with at least two elements.
A set $F$ of finitary [[operations|EJ8YJVM9BW]] on $A$ is *functionally complete* if the [[clone|PTKR703G6A]] generated by $F$ contains every operation $A^n \to A$ for every $n \ge 1$.

## Remarks

Example: $F = \set{\wedge, \neg}$ on $\set{0, 1}$ is functionally complete.
Check: every Boolean function of $n \ge 1$ variables has a disjunctive normal form, with the constant $0$ written as $x_1 \wedge \neg x_1$.
The disjunctions are removed with $x \vee y = \neg(\neg x \wedge \neg y)$.
Non-example: $F = \set{\wedge, \vee}$ on $\set{0, 1}$ is not functionally complete.
Every operation built from $\wedge$ and $\vee$ maps $(1, \dots, 1)$ to $1$, so $\neg$ is not in the generated clone.
Some texts, especially in logic, allow constants to be used for free; this changes which sets count as complete.
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<document source="https://tamnd.github.io/math2/node/FT1YKVBFC9.md" tag="FT1YKVBFC9" kind="concept">
# Birkhoff interpolation

## Summary

Birkhoff interpolation asks for a polynomial whose derivatives of chosen orders take chosen values at chosen nodes.
Unlike Hermite interpolation, it does not require that all lower derivatives are also prescribed.
It contains [[polynomial interpolation|SS1YMS2NPC]] and [[Hermite interpolation|9Z50M2WPTP]] as special cases.
Unlike these, it does not always have a unique solution, even when the number of conditions equals the number of coefficients.
The main question of the theory is which patterns of conditions, encoded in an incidence matrix, are poised, that is, solvable uniquely for all data.

## Explanations

### L1 human

Suppose you know the value of a quadratic at $-1$ and at $1$ and its slope at $0$.
That is three conditions for three coefficients, so one might expect exactly one answer.
But for $p(x) = c_0 + c_1 x + c_2 x^2$ the slope at $0$ is $c_1$, and the two values only fix $c_0 + c_2$ and $c_1$.
So the slope is either forced or contradicted, and $c_0$ and $c_2$ are not fixed separately.
Birkhoff interpolation studies when such gaps in the conditions are harmless.
The [[definition|AE08JXN7CA]] sets up the incidence matrix.
</document>

<document source="https://tamnd.github.io/math2/node/FT2GJEHWY5.md" tag="FT2GJEHWY5" kind="concept">
# Tropical variety

## Summary

A tropical variety is a polyhedral complex in $\R^n$ attached to an algebraic subvariety of a torus over a field with a [[valuation|B85T5VKR3M]].
It is the set where tropical polynomials are "zero", meaning that the minimum of their linear terms is attained at least twice.
Over an algebraically closed field with a nontrivial valuation, it equals the closure of the image of the variety under the coordinatewise valuation.
This is the fundamental theorem of [[tropical geometry|7YTC3BHK60]].
Tropical varieties keep dimension, degree and intersection data of the original variety in combinatorial form.

## Explanations

### L1 human

Take the line $x + y + 1 = 0$, with $x$ and $y$ in a field of power series in a small parameter $t$.
Record for each solution the orders of vanishing of $x$ and $y$ in $t$.
Two of the three terms $x$, $y$, $1$ must have the same lowest order, or they could not cancel.
So the orders lie on three rays from the origin: one where $x$ and $y$ tie below $0$, and two where one of them ties with $1$.
This tripod is the tropical line, a skeleton of the original line.
</document>

<document source="https://tamnd.github.io/math2/node/FT3ERF0YP9.md" tag="FT3ERF0YP9" kind="definition">
# Hopf algebra structure on cohomology

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $(X, e, \mu)$ be a path-connected [[H-space|EASHTX8EG5]] with the homotopy type of a CW complex.
Assume that each $H^n(X; k)$ is finite-dimensional.
By the Künneth theorem, the cross product gives an isomorphism $H^*(X; k) \otimes_k H^*(X; k) \cong H^*(X \times X; k)$.
The *[[Hopf algebra|7XFYP0N84H]] structure on cohomology* of $X$ consists of:

1. the [[cohomology ring|RDT4785M4D]] $H^*(X; k)$, with the [[cup product|JX47MS1R1H]] as multiplication and $1$ as unit;
2. the comultiplication $\Delta \colon H^*(X; k) \to H^*(X; k) \otimes_k H^*(X; k)$, defined as $\mu^*$ followed by the inverse of the Künneth isomorphism;
3. the counit $H^*(X; k) \to k$ given by restriction to the basepoint $e$.

With these maps, $H^0(X; k) = k$, $\Delta$ is a counital homomorphism of graded algebras, and $\Delta(x) = x \otimes 1 + 1 \otimes x + \sum x'_i \otimes x''_i$ with $0 < \abs{x'_i}, \abs{x''_i} < \abs{x}$ for $\abs{x} > 0$.
If $\mu$ is homotopy associative, $\Delta$ is also coassociative and $H^*(X; k)$ is a connected, graded-commutative Hopf algebra.

## Remarks

In the homotopy associative case the antipode exists automatically, because the bialgebra is connected: $H^0(X; k) = k$.

Example: for $X = S^3$, the unit quaternions, $H^*(S^3; k) = k \oplus k x$ with $\abs{x} = 3$.
By degrees, $\Delta(x) = x \otimes 1 + 1 \otimes x$, and $\Delta(x)^2 = 0 = \Delta(x^2)$, as $x \otimes x$ and $-x \otimes x$ cancel by the sign rule.

Non-example: on $H^*(S^2; \Q) = \Q[x]/(x^2)$ with $\abs{x} = 2$, the only candidate $\Delta(x) = x \otimes 1 + 1 \otimes x$ is not an algebra map, since $\Delta(x)^2 = 2 x \otimes x \ne 0 = \Delta(x^2)$.
This reflects the fact that $S^2$ is not an H-space.

Hopf's theorem: if $k$ has characteristic $0$ and $H^*(X; k)$ is finite-dimensional, then $H^*(X; k)$ is an exterior algebra on finitely many generators of odd degree.
</document>

<document source="https://tamnd.github.io/math2/node/FT9JQA58KJ.md" tag="FT9JQA58KJ" kind="concept">
# Calculus of finite differences

## Summary

The calculus of finite differences studies functions and sequences through their [[finite differences|7JSQZ126BP]] and sums.
It does so in the way that ordinary calculus uses derivatives and integrals.
Its main tools are the [[forward difference operator|8E0EBA8X07]] $\Delta$, the [[shift operator|MBK6AE6734]] $E = I + \Delta$, and the [[indefinite sum|SMAQN2N739]], which inverts $\Delta$.
Many rules of calculus have exact discrete counterparts, such as [[summation by parts|9G82Z17DKA]] and the [[fundamental theorem of the calculus of finite differences|MNSSV4QHD1]].
It is the basis of the theory of difference equations, of interpolation at equally spaced points, and of closed forms for finite sums.

## Explanations

### L1 human

Ordinary calculus works with infinitely small changes; the calculus of finite differences works with steps of size one.
Differences replace derivatives and sums replace integrals.
The powers $x^k$ are replaced by the [[falling factorials|0SJNCJCCQG]] $(x)_k = x(x-1)\cdots(x-k+1)$, because $\Delta (x)_k = k\,(x)_{k-1}$, just as the derivative of $x^k$ is $k x^{k-1}$.
For example, summing this rule gives $\sum_{x=0}^{n-1} x = \frac{(n)_2}{2} = \frac{n(n-1)}{2}$, the discrete analogue of $\int_0^n x\,dx = \frac{n^2}{2}$.
</document>

<document source="https://tamnd.github.io/math2/node/FTAK2F0TR3.md" tag="FTAK2F0TR3" kind="definition">
# Polynomial long division

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $f, g \in R[x]$, where $g$ has [[degree|0DPF6WRZW6]] $m$ and its [[leading coefficient|0P05RJC5XA]] $u$ is a unit of $R$.
Polynomial long division of $f$ by $g$ is the following procedure.

1. Start with $q = 0$ and $r = f$.
2. While $r \ne 0$ and $\deg r \ge m$, let $c$ be the leading coefficient of $r$ and $d = \deg r$, and replace $q$ by $q + u^{-1} c x^{d - m}$ and $r$ by $r - u^{-1} c x^{d - m} g$.
3. When $r = 0$ or $\deg r < m$, stop and return the quotient $q$ and the remainder $r$.

## Remarks

Each pass of step 2 removes the leading term of $r$, so $\deg r$ drops and the procedure stops.
The equation $f = q g + r$ holds at every step, so the output is the unique pair of the [[division algorithm for polynomials|0MES3KS0ZC]].
Example: divide $f = x^3 - 2x^2 - 4$ by $g = x - 3$ in $\Z[x]$.
The steps add $x^2$, then $x$, then $3$ to the quotient, and the remainder is $5$.
Check: $(x - 3)(x^2 + x + 3) + 5 = x^3 - 2x^2 - 9 + 5 = x^3 - 2x^2 - 4$.
Non-example: in $\Z[x]$ the procedure does not apply to $g = 2x$, because the leading coefficient $2$ is not a unit, and the first step for $f = x$ would need the coefficient $\frac{1}{2}$.
Division by a monic $g = x - a$ is often written in a compact table, called synthetic division; it is [[Horner's method|FYBQ3Z52Z7]].

## Explanations

### L1 human

Long division of polynomials repeats one move: kill the highest term of what is left.
It stops when what is left has smaller degree than the divisor.
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<document source="https://tamnd.github.io/math2/node/FTE966EMW8.md" tag="FTE966EMW8" kind="concept">
# Inner measure

## Summary

The inner measure of a set approximates it from inside: it is the largest measure of a measurable set contained in it.
It is the companion of the [[outer measure|BJ314H17HC]], which approximates from outside.
For Lebesgue measure, a set of finite outer measure is measurable exactly when its inner and outer measures agree.
Historically, Lebesgue defined measurability in this way.

## Explanations

### L1 human

To measure an odd-shaped region from inside, fill it with pieces whose size is known, and take the largest total size you can fit.
The outer measure does the opposite: it covers the region and takes the smallest cover.
For a nice region the two numbers agree.
For example, the set of irrational numbers in $[0, 1]$ has inner and outer Lebesgue measure $1$.
For a Vitali set in $[0, 1]$, the inner measure is $0$ and the outer measure is positive.
</document>

<document source="https://tamnd.github.io/math2/node/FTGTM3JB1F.md" tag="FTGTM3JB1F" kind="example">
# Laguerre equation

## Example

Let $\lambda \in \C$. The *Laguerre equation* is
$$z\, y'' + (1 - z)\, y' + \lambda\, y = 0.$$
In normalised form, $p(z) = \frac{1 - z}{z}$ and $q(z) = \frac{\lambda}{z}$.
Then $z p(z) = 1 - z$ and $z^2 q(z) = \lambda z$ are analytic at $0$, so $0$ is a [[regular singular point|H96HXG6CA0]].
The indicial equation is $r (r - 1) + r = r^2 = 0$, a double root $r = 0$.
By [[Fuchs' theorem|6EZ400WM94]], there is one solution $y_1 = \sum_{k \ge 0} a_k z^k$, analytic at $0$, and a second solution containing $y_1 \log z$.
Inserting the series gives $(k + 1)^2 a_{k+1} = (k - \lambda)\, a_k$.
If $\lambda = n$ is a nonnegative integer, the series stops and $y_1$ is a multiple of the Laguerre polynomial $L_n$ of degree $n$.

## Remarks

For $n = 1$, $L_1(z) = 1 - z$: then $y' = -1$ and $y'' = 0$, and $z \cdot 0 + (1 - z)(-1) + (1 - z) = 0$.
At $\infty$, the function $z^2 q(z) = \lambda z$ is unbounded, so $\infty$ is an irregular singular point and the equation is not Fuchsian.
The generalized Laguerre polynomials $L_n^{(\alpha)}$, solutions of $z y'' + (\alpha + 1 - z) y' + n y = 0$, appear in the radial part of the wave functions of the hydrogen atom.

## Explanations

### L1 human

The Laguerre equation has one tame bad point at $0$, where both exponents are equal to $0$.
So one solution is an ordinary power series, and the other one needs a logarithm.
For whole numbers $\lambda = n$, the series solution is a polynomial, the Laguerre polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/FTHFSJ30WJ.md" tag="FTHFSJ30WJ" kind="concept">
# Abscissa of convergence

## Summary

The abscissa of convergence of a [[Laplace transform|0V49JEZW06]] is a number $\sigma_c$ in $[-\infty, +\infty]$.
The Laplace integral converges on the half-plane $\operatorname{Re} s > \sigma_c$ and diverges on the half-plane $\operatorname{Re} s < \sigma_c$.
It plays the role that the radius of convergence plays for a power series.
The same notion exists for [[Dirichlet series|8MWTEF6462]], which are Laplace–Stieltjes transforms of step functions.
On the boundary line $\operatorname{Re} s = \sigma_c$ the integral may converge at some points and diverge at others.

## Explanations

### L1 human

The Laplace integral $\int_0^\infty f(t) e^{-st}\, dt$ converges better when $\operatorname{Re} s$ is larger, because $e^{-st}$ then decays faster.
So the set of $s$ where it converges is a right half-plane, possibly the whole plane or empty.
The abscissa of convergence is the real part of its left edge.
For $f(t) = e^{3t}$ the integral converges exactly when $\operatorname{Re} s > 3$, so the abscissa is $3$.
</document>

<document source="https://tamnd.github.io/math2/node/FTKFCCZ8YA.md" tag="FTKFCCZ8YA" kind="concept">
# Koszul algebra

## Summary

A Koszul algebra is a graded [[associative algebra|3CTW1P4HBP]] over a field whose trivial module has a linear projective resolution.
This means that the $i$-th term of the resolution is generated in degree $i$.
Polynomial algebras and [[exterior algebras|0MDMZ965VE]] are the basic examples, resolved by the [[Koszul complex|YZY11W08SD]].
Koszul algebras are quadratic, and they are the setting of [[Koszul duality|6AW49BZ8TM]].

## Explanations

### L1 human

To understand a graded algebra one often resolves its simplest module, the field $k$, by free modules.
For most algebras the generators of these free modules sit in irregular degrees.
A Koszul algebra is one where the $i$-th step is generated exactly in degree $i$, as tidy as possible.
For the polynomial ring $k[x]$ the resolution is $0 \to k[x] \xrightarrow{x} k[x] \to k \to 0$, with generators in degrees $0$ and $1$.
This tidiness makes many invariants of a Koszul algebra computable from its quadratic relations.
</document>

<document source="https://tamnd.github.io/math2/node/FTM4W0VBC8.md" tag="FTM4W0VBC8" kind="definition">
# Hermitian manifold

## Definition

Let $M$ be a complex manifold, and let $J$ be the integrable [[almost complex structure|9SYWRGVTRS]] given by multiplication by $i$ in holomorphic charts.
A *Hermitian metric* on $M$ is a [[Riemannian metric|Q61VFHCH2J]] $g$ on $M$ such that $g(JX, JY) = g(X, Y)$ for all tangent vectors $X, Y$ at the same point.
A *Hermitian manifold* is a complex manifold with a Hermitian metric.

The *fundamental form* of $g$ is the [[differential $2$-form|0E3G0GRPZ6]] $\omega(X, Y) = g(JX, Y)$.
The metric $g$ is *Kähler* if $d\omega = 0$, where $d$ is the [[exterior derivative|004FRCXH6M]], and then $(M, g)$ is a *Kähler manifold*.

## Remarks

Example: $\C^n = \R^{2n}$ with the Euclidean metric $g_0$ and $J$ the multiplication by $i$.
Multiplication by $i$ is an orthogonal map of $\R^{2n}$, so $g_0(JX, JY) = g_0(X, Y)$.
Its fundamental form is $\omega = \sum_k dx_k \wedge dy_k$, which has constant coefficients, so $d\omega = 0$ and $\C^n$ is Kähler.

Non-example: on $\C = \R^2$ the metric $g = dx^2 + 4\, dy^2$ is Riemannian but not Hermitian.
Here $J\partial_x = \partial_y$, so $g(J\partial_x, J\partial_x) = 4 \ne 1 = g(\partial_x, \partial_x)$.

The form $h = g - i\omega$ is a [[Hermitian form|48P4A1CX85]] on each tangent space viewed as a complex vector space.
Every complex manifold has Hermitian metrics: given any Riemannian metric $g$, the metric $g(X, Y) + g(JX, JY)$ is Hermitian.
The same definition with an almost complex structure that need not be integrable gives *almost Hermitian manifolds*; by the [[Newlander–Nirenberg theorem|VCT848WV1T]] the integrable case is the complex case.
Every complex submanifold of a Kähler manifold is Kähler with the induced metric.
</document>

<document source="https://tamnd.github.io/math2/node/FTNXZ74D8N.md" tag="FTNXZ74D8N" kind="definition">
# Infinitary logic

## Definition

Let $L$ be a signature and let $\kappa \ge \lambda$ be infinite [[cardinals|0EBACQFAZW]].
The formulas of the infinitary logic $L_{\kappa\lambda}$ are generated by the following rules.

1. Each atomic [[formula|HYJ10AF6RT]] of [[first-order logic|086QKPF0A6]] over $L$ is a formula.
2. If $\varphi$ is a formula, then $\neg\varphi$ is a formula.
3. If $\Phi$ is a set of formulas with $\abs{\Phi} < \kappa$ whose free variables together number fewer than $\lambda$, then $\bigwedge \Phi$ and $\bigvee \Phi$ are formulas.
4. If $\varphi$ is a formula and $V$ is a set of variables with $\abs{V} < \lambda$, then $\forall V \, \varphi$ and $\exists V \, \varphi$ are formulas.

Truth in an $L$-[[structure|983GZBQYT4]] is defined as in first-order logic: $\bigwedge \Phi$ holds when every member of $\Phi$ holds, and $\bigvee \Phi$ when some member holds.

## Remarks

The logic $L_{\omega\omega}$ is first-order logic, since then all conjunctions, disjunctions and quantifier blocks are finite.
The logic $L_{\omega_1\omega}$ allows countable conjunctions and disjunctions and finite quantifier blocks.

Example: in the signature of groups, the $L_{\omega_1\omega}$ sentence $\forall x \, \bigvee_{n \ge 1} (x^n = e)$ holds exactly in the torsion groups.
Here $x^n$ is the term $x \cdot x \cdots x$ with $n$ factors.

Non-example: the expression $\forall x_1 \, \forall x_2 \, \forall x_3 \cdots \varphi$ with infinitely many quantified variables is not a formula of $L_{\omega_1\omega}$, because clause 4 allows only finite sets of variables when $\lambda = \omega$.
It is a formula of $L_{\omega_1\omega_1}$ when $\varphi$ is.

The compactness theorem fails for $L_{\omega_1\omega}$, but the downward Löwenheim–Skolem theorem holds for its countable fragments.

## Explanations

### L1 human

Infinitary logic is first-order logic with permission to write infinitely long "and" and "or".
The two cardinals bound how long these may be and how many variables a quantifier may bind at once.
</document>

<document source="https://tamnd.github.io/math2/node/FTQDGFRRS0.md" tag="FTQDGFRRS0" kind="definition">
# Quantum group

## Definition

The term *quantum group* has no single agreed definition.
In the broad sense of Drinfeld, a quantum group is a [[Hopf algebra|7XFYP0N84H]] that is in general neither commutative nor cocommutative.
The base ring is a field or a ring of formal power series in a parameter $h$.
In the narrow sense, it is one of the following families, each depending on a parameter $q$ and recovering a classical object at $q = 1$:

1. the quantized enveloping algebra $U_q(\mathfrak{g})$ of a semisimple Lie algebra $\mathfrak{g}$, a deformation of the universal enveloping algebra;
2. the quantized coordinate algebra $\mathcal{O}_q(G)$ of a semisimple algebraic group $G$, a deformation of the coordinate ring;
3. a [[compact quantum group|FHQKT6HPW2]] in the sense of Woronowicz, a C*-algebraic analogue of the algebra of continuous functions on a compact group.

## Remarks

Example: let $k$ be a field and $q \in k$ nonzero.
The algebra $\mathcal{O}_q(SL_2)$ is generated by $a, b, c, d$ with the relations
$$ab = q\,ba, \quad ac = q\,ca, \quad bd = q\,db, \quad cd = q\,dc, \quad bc = cb,$$
$$ad - da = (q - q^{-1})\,bc, \qquad ad - q\,bc = 1.$$
Its coproduct is matrix multiplication of the generator matrix: $\Delta(a) = a \otimes a + b \otimes c$, $\Delta(b) = a \otimes b + b \otimes d$, $\Delta(c) = c \otimes a + d \otimes c$, $\Delta(d) = c \otimes b + d \otimes d$.
At $q = 1$ the relations say that $a, b, c, d$ commute and $ad - bc = 1$, which is the coordinate ring of $SL_2$.
Non-example in the broad sense: the group algebra $k[G]$ of a group is a Hopf algebra, but it is cocommutative.
So it is a classical object rather than a genuine quantum group.
</document>

<document source="https://tamnd.github.io/math2/node/FTQMQR3YS7.md" tag="FTQMQR3YS7" kind="concept">
# Spectral triple

## Summary

A spectral triple $(A, H, D)$ consists of an algebra $A$ of operators on a Hilbert space $H$ and an unbounded self-adjoint operator $D$ on $H$.
The operator $D$ has compact resolvent, and its commutators with elements of $A$ are bounded.
It is the noncommutative version of a compact Riemannian spin manifold together with its [[Dirac operator|3DXGJTEQZH]].
From a spectral triple one recovers a distance, a dimension and an integral, so it encodes metric geometry in operator language.

## Explanations

### L1 human

On a compact manifold, the Dirac operator is a first-order differential operator that knows the shape of the manifold.
Its eigenvalues grow at a rate fixed by the dimension, and the size of the commutator $[D, f]$ measures how fast a function $f$ changes.
A spectral triple keeps only these features: an algebra standing for the functions, a Hilbert space on which they act, and an operator $D$ standing for the Dirac operator.
The simplest example is the circle: functions on the circle act on $L^2(S^1)$, and $D = -i \, d/d\theta$ has eigenvalues the integers.
The algebra need not be commutative, which allows "spaces" with no points in the usual sense.
</document>

<document source="https://tamnd.github.io/math2/node/FTRW0679E2.md" tag="FTRW0679E2" kind="concept">
# Étale morphism of rings

## Summary

An étale morphism of rings is a map $R \to A$ of commutative rings that is both [[smooth|286G0P1B69]] and [[unramified|GY3QA6NH0E]].
Equivalently, $A$ is finitely presented, flat and unramified over $R$.
It is the algebraic analogue of a local diffeomorphism, or of a covering map.
Over a field $k$, the étale algebras are the finite products of finite separable field extensions of $k$.
Étale maps are the basis of étale cohomology and of the Henselization of a local ring.

## Explanations

### L1 human

An étale map looks, near each point, like an isomorphism to first order, even though it may not be an isomorphism.
A typical example is adjoining a simple root of a polynomial: $R[x]/(f)$ where the derivative $f'$ is a unit.
Over $\Z[\frac12]$, adjoining a square root of $-1$ is étale, while over $\Z$ it is not, because of the prime $2$.
Over a field, an étale algebra is a finite product of separable field extensions, such as $\Q(\sqrt 2) \times \Q$.
Étale maps let one pass to finer "neighbourhoods" than the Zariski topology allows.
</document>

<document source="https://tamnd.github.io/math2/node/FV0E9GEPCD.md" tag="FV0E9GEPCD" kind="definition">
# Birth–death process

## Definition

Let $S = \set{0, 1, 2, \ldots}$, and let $(\lambda_n)_{n \ge 0}$ and $(\mu_n)_{n \ge 1}$ be nonnegative real numbers, with $\mu_0 = 0$.
Let $(X_t)_{t \ge 0}$ be a [[continuous-time Markov chain|TWQFE0FZZS]] on $S$ with generator matrix $Q = (q_{mn})$.
It is a *birth–death process* with birth rates $\lambda_n$ and death rates $\mu_n$ if
$$q_{n, n+1} = \lambda_n, \qquad q_{n, n-1} = \mu_n \ (n \ge 1), \qquad q_{nn} = -(\lambda_n + \mu_n),$$
and $q_{mn} = 0$ when $\abs{m - n} \ge 2$.
A finite version uses $S = \set{0, 1, \ldots, N}$ with $\lambda_N = 0$.

## Remarks

In state $n$ the process waits an exponential time of rate $\lambda_n + \mu_n$, then jumps to $n + 1$ with probability $\lambda_n / (\lambda_n + \mu_n)$ and to $n - 1$ otherwise.
Example: the Poisson process of rate $\lambda$ is the pure birth process with $\lambda_n = \lambda$ and $\mu_n = 0$ for all $n$; each row of $Q$ sums to $0$, as required.
Non-example: a chain on $S$ with $q_{n, n+2} > 0$ for some $n$ is not a birth–death process, because it jumps by two.
Suppose all $\lambda_n > 0$ and all $\mu_n > 0$ for $n \ge 1$, and the chain does not explode.
Then a [[stationary distribution|02KYGSCTCN]] exists if and only if
$$\sum_{n \ge 0} \frac{\lambda_0 \lambda_1 \cdots \lambda_{n-1}}{\mu_1 \mu_2 \cdots \mu_n} < \infty,$$
and then $\pi_n$ is proportional to the $n$-th term of this sum.
It follows from the balance equations $\pi_n \lambda_n = \pi_{n+1} \mu_{n+1}$.
</document>

<document source="https://tamnd.github.io/math2/node/FV1E09CJQN.md" tag="FV1E09CJQN" kind="definition">
# Riccati differential equation of optimal control

## Definition

Let $n, m \ge 1$ and $T > 0$.
Let $A$ be a real $n \times n$ matrix and $B$ a real $n \times m$ matrix.
Let $Q$ and $F$ be [[symmetric|0VR671PHE5]] [[positive semidefinite|01MMVW8CKC]] $n \times n$ matrices, and let $R$ be a symmetric [[positive definite|0KX66CZMBB]] $m \times m$ matrix.
The *Riccati differential equation* of the associated linear-quadratic problem is the terminal value problem
$$-P'(t) = A^{T} P(t) + P(t) A - P(t) B R^{-1} B^{T} P(t) + Q, \qquad P(T) = F,$$
for a differentiable function $P \colon (-\infty, T] \to \R^{n \times n}$ with symmetric values.

## Remarks

Under these hypotheses the solution exists on all of $(-\infty, T]$, and $P(t)$ is symmetric positive semidefinite for every $t$.
Example: $n = m = 1$, $A = 0$, $B = Q = R = 1$ and $F = 0$ give $-P' = 1 - P^2$, $P(T) = 0$.
The solution is $P(t) = \tanh(T - t)$: indeed $P'(t) = -(1 - \tanh^2(T - t)) = -(1 - P(t)^2)$ and $P(T) = 0$.
As $t \to -\infty$, $P(t) \to 1$, the positive solution of the algebraic equation $1 - P^2 = 0$.
Non-example: the linear equation $-P' = A^{T} P + P A + Q$ is a Lyapunov differential equation, not a Riccati equation, since it lacks the quadratic term.
It gives the cost of the zero control; with $A - BK$ in place of $A$ and $Q + K^{T} R K$ in place of $Q$ it gives the cost of a fixed linear feedback $u = -Kx$.
The quadratic term of the Riccati equation accounts for optimizing the control.
For $n = 1$ the equation is a scalar [[Riccati equation|0VT2QK6X6Q]] $P' = q_0 + q_1 P + q_2 P^2$ with constant coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/FVCDGMNSAN.md" tag="FVCDGMNSAN" kind="concept">
# Griffiths positivity

## Summary

Griffiths positivity is a notion of positive curvature for a holomorphic Hermitian vector bundle of any rank.
It asks that the curvature of the [[Chern connection|7RC9CN2Q49]] be positive when evaluated on one tangent vector and one fibre vector at a time.
For line bundles it is the same as being a [[positive line bundle|J45KWFHENP]].
Griffiths positive bundles are ample, and Griffiths asked whether the converse holds; this question is open in general.

## Explanations

### L1 human

For a line bundle, curvature is a single $2$-form, and positivity has one clear meaning.
For a bundle of higher rank, curvature is a matrix of $2$-forms, and there are several reasonable ways to call it positive.
Griffiths positivity is the weakest common one: for each direction on the base and each vector in the fibre, a certain number built from the curvature must be positive.
A direct sum of copies of a positive line bundle, with the sum metric, is Griffiths positive.
</document>

<document source="https://tamnd.github.io/math2/node/FVFFKNKEDF.md" tag="FVFFKNKEDF" kind="concept">
# Entourage

## Summary

An entourage of a [[uniform space|A54T86KS63]] is one of the sets of pairs of points that make up its uniform structure.
Each entourage stands for one degree of closeness: the pairs in it are the pairs of points that are close to that degree.
Entourages replace the numbers $\varepsilon$ of metric spaces in the definitions of uniform continuity, Cauchy filters and completeness.

## Explanations

### L1 human

In a metric space, the pairs $(x, y)$ with distance less than $\varepsilon$ form a set of pairs, a relation on the space.
Such sets, and all larger ones, are the entourages of the metric.
For $\R$ and $\varepsilon = 1$, the entourage is the open band $\set{(x, y) : \abs{x - y} < 1}$ around the diagonal of the plane.
Smaller entourages are thinner bands, and so they say "closer".
</document>

<document source="https://tamnd.github.io/math2/node/FVGB8ZP67B.md" tag="FVGB8ZP67B" kind="concept">
# Topology of pointwise convergence

## Summary

The topology of pointwise convergence lives on a set of functions from a set $X$ to a space $Y$.
In it, convergence means [[pointwise convergence|0GFGK40582]].
It is the [[product topology|0GBG8B49AA]] on $Y^X$, or its restriction to a subset such as the continuous maps.
It is coarser than the [[topology of uniform convergence|YW5P7NEKR4]] and than the [[compact-open topology|RG15F0TWPW]].

## Explanations

### L1 human

Think of a function $f \colon X \to Y$ as a long list of values, one value for each point $x$.
Two functions are close in this topology when they are close at finitely many chosen points.
Nothing is asked at the other points.
For example, the functions $f_n(x) = x^n$ on $[0, 1]$ converge in this topology to the function that is $0$ on $[0, 1)$ and $1$ at $x = 1$.
They do not converge uniformly.
</document>

<document source="https://tamnd.github.io/math2/node/FVGW1381TZ.md" tag="FVGW1381TZ" kind="statement">
# Gage–Hamilton–Grayson theorem

## Statement

Let $\gamma_0$ be a smooth embedded closed [[plane curve|EWS4WFTA7W]] that encloses area $A_0$.
Then the [[curve-shortening flow|DVGFRRYK10]] with initial curve $\gamma_0$ has a unique smooth solution $(\gamma_t)_{0 \le t < T}$ with $T = A_0 / (2\pi)$, and:

1. each $\gamma_t$ is embedded;
2. there is $t_0 < T$ such that $\gamma_t$ bounds a strictly [[convex|02AS45T3H9]] region for all $t \in [t_0, T)$;
3. as $t \to T$, the curves $\gamma_t$ converge to a point $p$. Rescale $\gamma_t$ by the dilation about $p$ that makes the enclosed area equal to $\pi$. These rescaled curves converge smoothly to the unit [[circle|060W5CBZP6]] centered at $p$.

## Remarks

Gage and Hamilton proved in 1986 that convex curves stay convex and shrink to a round point.
Grayson proved in 1987 that every embedded curve becomes convex before it becomes singular, which reduces the general case to the convex one.
The value of $T$ comes from the identity $A'(t) = -\int_{\gamma_t} \kappa \, ds = -2\pi$ for embedded curves, since the total [[curvature|7JQS6VKFEZ]] of a simple closed curve is $2\pi$.
Embeddedness is needed: an immersed figure-eight curve, or a curve with a small loop, can develop a singularity before it shrinks to a point.

Example: a circle of radius $r_0$ encloses area $\pi r_0^2$ and vanishes at time $r_0^2 / 2 = A_0 / (2\pi)$, in agreement with the formula.

## Explanations

### L1 human

Let a simple closed curve in the plane move by curvature, so sharply bent parts move fastest.
However twisted the curve is at the start, it never crosses itself, it becomes convex, and then it shrinks to a point while looking more and more like a circle.
The time it lives is exactly its area divided by $2\pi$.

### L2 human

The theorem is a model for later results on geometric flows, such as Huisken's theorem on convex hypersurfaces under mean curvature flow and Hamilton's work on Ricci flow.
Grayson's step is the hard part, because it must rule out singularities for nonconvex curves.
Later proofs by Huisken and by Andrews and Bryan compare distances along the curve with distances in the plane.
The flow can also be used to prove geometric inequalities and the existence of closed geodesics on surfaces.
</document>

<document source="https://tamnd.github.io/math2/node/FVMNKJ3DT8.md" tag="FVMNKJ3DT8" kind="concept">
# Boolean-valued model

## Summary

A Boolean-valued model is like a first-order structure, but each statement gets a truth value in a complete [[Boolean algebra|0XADJKM699]].
In an ordinary structure the only values are "true" and "false".
When the Boolean algebra has two elements, it is essentially an ordinary structure.
Boolean-valued models of set theory, written $V^B$, give a way to present [[forcing|F2DTWW9HP7]] that needs no countable transitive model.
They were developed by Scott, Solovay and Vopěnka.

## Explanations

### L1 human

In an ordinary structure every statement is either true or false.
In a Boolean-valued model a statement can be "true to some degree", and the degrees form a Boolean algebra.
For example, if the algebra is the algebra of subsets of a set $S$, the truth value of a statement is the set of points of $S$ where it holds.
One can think of the model as a family of ordinary structures indexed by $S$, glued together.
</document>

<document source="https://tamnd.github.io/math2/node/FVPXXFNVQY.md" tag="FVPXXFNVQY" kind="statement">
# Hahn embedding theorem

## Statement

Let $G$ be an [[abelian group|071DDFVG4J]] with a total order that makes it a [[linearly ordered group|1P8087YGYQ]].
Let $\Omega$ be the set of [[Archimedean classes|C8VE2SEVEB]] of the nonzero elements of $G$.
Order $\Omega$ so that $[a] \prec [b]$ when $a$ is infinitely larger than $b$.
Let $H$ be the set of maps $f \colon \Omega \to \R$ whose support $\set{\omega : f(\omega) \ne 0}$ is [[well-ordered|0F1D6PK9RZ]] by $\prec$.
Make $H$ a group by pointwise addition, and set $f > 0$ when $f(\omega_0) > 0$ for the least element $\omega_0$ of the support of $f$.
Then $H$ is a linearly ordered abelian group, and there is an injective [[group homomorphism|0HV9Y7WG5R]] $\varphi \colon G \to H$ with $a \le b$ if and only if $\varphi(a) \le \varphi(b)$.

## Remarks

Hahn proved the theorem in 1907.
For a nonzero $a \in G$, let $C_a$ be the smallest convex subgroup containing $a$ and $D_a$ the largest convex subgroup not containing $a$.
The quotient $C_a / D_a$ depends only on the class of $a$.
This quotient is Archimedean, so [[Hölder's theorem|CDM40MT3HZ]] embeds it in [[the reals|08J2M1JYAG]].
The proofs then glue these embeddings together, using divisible hulls and Zorn's lemma.
Example: $\Z^2$ with the lexicographic order has two nonzero classes, and its image in $H \cong \R^2$ is the obvious one with the lexicographic order.
When $G$ is Archimedean, $\Omega$ has one element and the theorem reduces to Hölder's theorem.

## Explanations

### L1 human

Any way of ordering a commutative group so that addition respects the order can be described with real numbers arranged by size of magnitude.
Each element becomes a list of real coordinates, one for each order of magnitude, read from the largest magnitude down.
Two elements are compared by the first coordinate where they differ.

### L2 human

The theorem says that the groups $H$, called Hahn groups, are universal for linearly ordered abelian groups with a given set of Archimedean classes.
It is the group counterpart of the fact that the [[Hahn series|YQP8V1KKMW]] fields are large enough to contain many valued fields.
</document>

<document source="https://tamnd.github.io/math2/node/FVQWR7NS00.md" tag="FVQWR7NS00" kind="concept">
# Fokker–Planck equation

## Summary

The Fokker–Planck equation is the parabolic equation for the probability density of a [[diffusion process|9HDQ094ADQ]].
Let $X_t$ solve a [[stochastic differential equation|B219G9SHJA]] with drift $\mu$ and diffusion coefficient $\sigma$.
Then its density $p(x, t)$ obeys a linear second-order equation built from $\mu$ and $\sigma\sigma^T$.
Its spatial operator is the adjoint of the [[infinitesimal generator|VPA41PWRR1]] of the process, which is why it is also called the forward Kolmogorov equation.
It links stochastic processes with partial differential equations.

## Explanations

### L1 human

Release many particles that drift with a velocity field and are also kicked by random noise.
The Fokker–Planck equation describes how the cloud of particles moves and spreads.
The drift term transports the density, and the noise term makes it diffuse.
For pure noise in one dimension, $dX = dW$, the equation is $p_t = \frac{1}{2} p_{xx}$, a [[heat equation|VP2HN86VAG]].
</document>

<document source="https://tamnd.github.io/math2/node/FW29W5E80Q.md" tag="FW29W5E80Q" kind="example">
# de Rham curve

## Example

Let $d_0, d_1 \colon \C \to \C$ be two [[contraction mappings|0QN69YA357]] with fixed points $p_0$ and $p_1$, and assume $d_0(p_1) = d_1(p_0)$.
The *de Rham curve* of $(d_0, d_1)$ is the [[continuous|0X3WGX95FE]] map $c \colon [0, 1] \to \C$ that solves the system of [[functional equations|SRAN4EKWS0]]
$$c\left(\frac{x}{2}\right) = d_0(c(x)), \qquad c\left(\frac{1 + x}{2}\right) = d_1(c(x)) \qquad \text{for all } x \in [0, 1].$$
Setting $x = 0$ in the first equation and $x = 1$ in the second gives $c(0) = p_0$ and $c(1) = p_1$.
Both equations give a value at $1/2$: the first with $x = 1$ gives $d_0(p_1)$, the second with $x = 0$ gives $d_1(p_0)$.
This is why the condition $d_0(p_1) = d_1(p_0)$ is needed.

A concrete case: let $a = \frac12 + \frac{\sqrt3}{6} i$, $d_0(z) = a \bar z$ and $d_1(z) = a + (1 - a) \bar z$.
Both maps are contractions with ratio $\abs{a} = \abs{1 - a} = 1/\sqrt3 < 1$.
The fixed point of $d_0$ is $p_0 = 0$, and the fixed point of $d_1$ is $p_1 = 1$, since $d_1(1) = a + 1 - a = 1$.
The condition holds: $d_0(1) = a = d_1(0)$.
The resulting curve from $0$ to $1$ is the Koch curve.

The simplest case is $d_0(z) = z/2$ and $d_1(z) = (z + 1)/2$, for which $c(x) = x$: indeed $c(x/2) = x/2 = d_0(x)$ and $c((1+x)/2) = (1 + x)/2 = d_1(x)$.

## Remarks

The solution exists and is unique.
The system says that $c$ is a fixed point of an operator on the complete metric space of continuous maps $[0, 1] \to \C$ with $c(0) = p_0$ and $c(1) = p_1$.
This operator is a contraction for the uniform metric, so the [[Banach fixed point theorem|0VK6VEH345]] gives exactly one solution.
On dyadic rationals the curve can be computed directly: $c(0) = p_0$, $c(1) = p_1$, $c(1/2) = d_0(p_1)$, and so on.

## Explanations

### L1 human

A de Rham curve is built by a simple rule.
The first half of the curve is a shrunken copy of the whole curve, and so is the second half.
The two shrinking maps say how each copy sits in the plane.
If they shrink without twisting, the curve is a straight segment.
With a reflection and a tilt, the same rule produces the Koch snowflake curve, which has a corner at every scale.
</document>

<document source="https://tamnd.github.io/math2/node/FW2JKAN14B.md" tag="FW2JKAN14B" kind="concept">
# Lambert W function

## Summary

The Lambert W function is the [[inverse|0J2WAR83WP]] of the map $w \mapsto w e^w$, so $W(x)\, e^{W(x)} = x$.
Over the complex numbers it is multivalued, with countably many branches $W_k$; on the reals it has two branches $W_0$ and $W_{-1}$.
It solves many equations that mix exponentials and polynomials, such as $x = e^{-x}$ or $a^x = x$.
It appears in combinatorics, in delay differential equations and in physics.

## Explanations

### L1 human

The function $w e^w$ is not inverted by logarithms or roots, so its inverse gets a name of its own.
To solve $x e^x = 1$, one writes $x = W(1) \approx 0.567$.
To solve $x = e^{-x}$, one multiplies by $e^x$ to get $x e^x = 1$, so the answer is the same number.
On the real line, $w e^w$ has a minimum $-1/e$ at $w = -1$, so the equation $w e^w = x$ has two real solutions for $-1/e < x < 0$, and this is why there are two real branches.
</document>

<document source="https://tamnd.github.io/math2/node/FW57YYXK7Y.md" tag="FW57YYXK7Y" kind="definition">
# Positive element

## Definition

Let $A$ be a C*-algebra.
For $a \in A$, let $\sigma(a)$ be its [[spectrum|EH0CZ19RKP]], computed in the [[unitization|7SB03M850M]] of $A$ when $A$ has no unit.
An element $a \in A$ is *positive*, written $a \ge 0$, if

1. $a$ is self-adjoint, that is, $a^* = a$, and
2. $\sigma(a) \subseteq [0, \infty)$.

## Remarks

The following are equivalent for $a \in A$: $a$ is positive; $a = b^* b$ for some $b \in A$; $a = c^2$ for some self-adjoint $c \in A$.
The positive elements form a closed convex cone $A_+$ with $A_+ \cap (-A_+) = \set{0}$.
For self-adjoint $a, b$ one writes $a \le b$ when $b - a \ge 0$.

Example: in $C(X)$ with $X$ compact Hausdorff, $\sigma(f) = f(X)$, so $f \ge 0$ exactly when $f$ is real-valued with $f(x) \ge 0$ for all $x$.
In $M_n(\C)$ the positive elements are the [[positive semidefinite matrices|01MMVW8CKC]].

Non-example: $a = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ in $M_2(\C)$ has $\sigma(a) = \set{0} \subseteq [0, \infty)$, but $a^* \ne a$, so clause 1 fails.
Also $\operatorname{diag}(1, -1)$ is self-adjoint but has $-1$ in its spectrum, so clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/FW68FVNKMJ.md" tag="FW68FVNKMJ" kind="definition">
# Galois connection between subgroups and subfields

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]], and let $G = \Aut(L/K)$ be the group of field automorphisms of $L$ that fix every element of $K$.
Let $\mathcal{S}$ be the set of subgroups of $G$ and let $\mathcal{F}$ be the set of [[intermediate fields|02NBKVQFRQ]] of $L/K$, both ordered by inclusion.
For $H \in \mathcal{S}$, let $L^H = \set{x \in L : \sigma(x) = x \text{ for all } \sigma \in H}$ be the [[fixed field|0KMJ6919WS]] of $H$.
For $E \in \mathcal{F}$, let $\Aut(L/E) = \set{\sigma \in G : \sigma(x) = x \text{ for all } x \in E}$.
The *[[Galois connection|Q5JHEPRYQ6]] between subgroups and subfields* of $L/K$ is the pair of maps
$$\mathcal{S} \to \mathcal{F},\ H \mapsto L^H \qquad \text{and} \qquad \mathcal{F} \to \mathcal{S},\ E \mapsto \Aut(L/E).$$
It is an antitone Galois connection: for all $H \in \mathcal{S}$ and $E \in \mathcal{F}$,
$$E \subseteq L^H \iff H \subseteq \Aut(L/E).$$

## Remarks

Both sides of the equivalence say that every element of $H$ fixes every element of $E$, which is the check.
From the equivalence, $H \subseteq \Aut(L/L^H)$ and $E \subseteq L^{\Aut(L/E)}$, and each map reverses inclusions.
Example: for $L = \Q(\sqrt{2}, \sqrt{3})$ over $\Q$, the group $G$ has four elements, and the maps are inverse bijections between its five subgroups and the five intermediate fields.
Non-example of a bijection: for $L = \Q(\sqrt[3]{2})$ over $\Q$, the group $G$ is trivial, because $\sqrt[3]{2}$ is the only cube root of $2$ in the real field $L$.
Then the map $E \mapsto \Aut(L/E)$ sends both $\Q$ and $L$ to the trivial group, so it is not injective.
The maps are inverse bijections when $L/K$ is a finite [[Galois extension|0DS2F7GK05]], by the [[fundamental theorem of Galois theory|0T1Q8D0FW8]].
</document>

<document source="https://tamnd.github.io/math2/node/FW83WPH44N.md" tag="FW83WPH44N" kind="definition">
# Kuratowski closure axioms

## Definition

Let $X$ be a set and let $\mathcal{P}(X)$ be its [[power set|00E0YZCERH]].
A map $c \colon \mathcal{P}(X) \to \mathcal{P}(X)$ satisfies the *Kuratowski closure axioms* if, for all subsets $A, B \subseteq X$:

1. $c(\emptyset) = \emptyset$.
2. $A \subseteq c(A)$.
3. $c(c(A)) = c(A)$.
4. $c(A \cup B) = c(A) \cup c(B)$.

Such a map is a *closure operator* on $X$.

## Remarks

Given a closure operator $c$, the sets $F$ with $c(F) = F$ are the [[closed sets|0ZCK8DQYN1]] of a unique topology on $X$.
The [[closure|0QEAJB91GZ]] of each $A$ in this topology is $c(A)$.
Conversely, the closure in any [[topological space|0CQHFPGJ45]] satisfies the four axioms.
So closure operators on $X$ and topologies on $X$ correspond one to one.

Axiom 4 implies that $c$ is monotone: if $A \subseteq B$, then $c(A) \subseteq c(B) = c(A) \cup c(B \setminus A)$.

Example: on $\R$, let $c(A)$ be the set of limits of convergent sequences of points of $A$.
This $c$ is the usual closure, and it satisfies the four axioms.

Non-example: fix a point $x_0$ of a nonempty set $X$ and let $c(A) = A \cup \set{x_0}$.
Axioms 2, 3 and 4 hold, but $c(\emptyset) = \set{x_0} \ne \emptyset$, so axiom 1 fails.
</document>

<document source="https://tamnd.github.io/math2/node/FWF9D69R3F.md" tag="FWF9D69R3F" kind="definition">
# Casson invariant

## Definition

Let $\Sigma$ be an oriented integral [[homology 3-sphere|HK8NS02NNE]], let $K \subset \Sigma$ be a [[knot|QYEAZTYGPT]] and let $n$ be an integer.
Let $\Sigma_{K, 1/n}$ be the result of [[Dehn surgery|ZA405B0Z1S]] on $K$ with coefficient $1/n$; it is again an oriented integral homology $3$-sphere.
Let $\Delta_K(t)$ be the [[Alexander polynomial|JKA25KDPQP]] of $K$, normalised so that $\Delta_K(t^{-1}) = \Delta_K(t)$ and $\Delta_K(1) = 1$.

The *Casson invariant* is the unique function $\lambda$ from oriented integral homology $3$-spheres, up to orientation-preserving homeomorphism, to $\Z$ such that:

1. $\lambda(S^3) = 0$, and
2. for every $\Sigma$, $K$ and $n$ as above,
$$\lambda(\Sigma_{K, 1/(n+1)}) - \lambda(\Sigma_{K, 1/n}) = \tfrac{1}{2} \Delta_K''(1).$$

Casson proved that such a function exists and is unique.

## Remarks

Example: for a trefoil knot $K$ in $S^3$, $\Delta_K(t) = t - 1 + t^{-1}$, so $\Delta_K''(t) = 2 t^{-3}$ and $\tfrac12 \Delta_K''(1) = 1$.
Since $S^3_{K, 1/0} = S^3$, clause 2 with $n = 0$ gives $\lambda(S^3_{K, 1}) = 1$.
For the right-handed trefoil, $S^3_{K, 1}$ is the Poincaré homology sphere with the orientation opposite to that of $\Sigma(2, 3, 5)$, and $\lambda(\Sigma(2, 3, 5)) = -1$.

Further properties: $\lambda(-\Sigma) = -\lambda(\Sigma)$, $\lambda$ is additive under connected sum, and $\lambda(\Sigma)$ is congruent modulo $2$ to the Rokhlin invariant of $\Sigma$.
</document>

<document source="https://tamnd.github.io/math2/node/FWGHMADJR4.md" tag="FWGHMADJR4" kind="concept">
# Schubert calculus

## Summary

Schubert calculus is the computation of intersections of [[Schubert varieties|Z014Q8KPDD]] in [[Grassmannians|CYPA882RB9]] and [[flag varieties|NHRWCQQKTH]].
It works in the [[cohomology ring|RDT4785M4D]], or the Chow ring, where the classes of Schubert varieties form a basis.
The products of these classes are given by the Pieri rule and by Littlewood–Richardson coefficients.
It answers classical questions of [[enumerative geometry|ZPZA1FKZZC]], such as how many lines meet four general lines in space.

## Explanations

### L1 human

Hermann Schubert, in the nineteenth century, counted geometric configurations by turning conditions into symbols and multiplying them.
For example, "a line meets a given line" is one symbol $\sigma_1$, and four such conditions give $\sigma_1^4$.
A set of multiplication rules then turns $\sigma_1^4$ into $2$ times the symbol of a single line, so the answer is $2$.
Today these symbols are cohomology classes of Schubert varieties, and the rules are theorems.
</document>

<document source="https://tamnd.github.io/math2/node/FWGPKXCPB7.md" tag="FWGPKXCPB7" kind="concept">
# Complete category

## Summary

A complete category is a [[category|A2DE66937D]] in which every small diagram has a [[limit|CHS5AFJGWE]].
Products, equalizers, pullbacks and inverse limits then all exist.
The categories of sets, groups, rings, modules and topological spaces are complete.
Completeness is a main hypothesis of the adjoint functor theorems.

## Explanations

### L1 human

A limit is a best way to combine a family of objects and maps into one object, like a product or a set of solutions of equations.
A category is complete when you can always do this, as long as the family is indexed by a set.
The category of sets is complete: the limit is a set of compatible families of elements.
The category of finite sets is not complete, because an infinite product of two-element sets is infinite.
</document>

<document source="https://tamnd.github.io/math2/node/FWJ7CH6WG8.md" tag="FWJ7CH6WG8" kind="definition">
# Almost split sequence

## Definition

Let $R$ be a [[ring|0B1261B5ED]] and let $0 \to L \xrightarrow{f} E \xrightarrow{g} N \to 0$ be a short [[exact sequence|JB7ZBFBA9C]] of left $R$-[[modules|0NZ8JA25WK]].
The sequence is *almost split* if:

1. it does not split;
2. every morphism $u \colon L \to Y$ that is not a split monomorphism factors as $u = u' f$ for some $u' \colon E \to Y$;
3. every morphism $h \colon X \to N$ that is not a split epimorphism factors as $h = g h'$ for some $h' \colon X \to E$.

## Remarks

In an almost split sequence, $L$ and $N$ are [[indecomposable|E2P29GCD9T]] with local endomorphism rings.
Example: let $A$ be the path algebra of $1 \to 2$ over a field $k$, with modules $S_2 = (0 \to k)$, $P_1 = (k \xrightarrow{\id} k)$, $S_1 = (k \to 0)$.
The sequence $0 \to S_2 \to P_1 \to S_1 \to 0$ does not split, since $P_1$ is indecomposable.
For clause 3 it suffices to test indecomposable $X$: maps from $S_1$ are $0$ or isomorphisms, $\Hom(S_2, S_1) = 0$, and maps from the projective $P_1$ lift.
Clause 2 is checked in the same way.
Non-example: the split sequence $0 \to S_2 \to S_2 \oplus S_1 \to S_1 \to 0$ satisfies clauses 2 and 3, since $f$ is a split monomorphism and $g$ a split epimorphism, but it fails clause 1.
For a finite-dimensional algebra, the left end of the almost split sequence ending at $N$ is $\tau N$, the Auslander–Reiten translate.
</document>

<document source="https://tamnd.github.io/math2/node/FWKSHEJ9RP.md" tag="FWKSHEJ9RP" kind="statement">
# Karamata's inequality

## Statement

Let $I$ be an [[interval|0TK8Y3J5RW]] of $\R$ and let $f \colon I \to \R$ be a [[convex function|0VV0HSW1ZS]].
Let $x_1 \ge x_2 \ge \dots \ge x_n$ and $y_1 \ge y_2 \ge \dots \ge y_n$ be numbers in $I$ such that $(x_1, \dots, x_n)$ [[majorizes|5K7VT2VKZT]] $(y_1, \dots, y_n)$, that is,
$$\sum_{i=1}^{k} x_i \ge \sum_{i=1}^{k} y_i \quad \text{for } k = 1, \dots, n - 1, \qquad \sum_{i=1}^{n} x_i = \sum_{i=1}^{n} y_i.$$
Then
$$\sum_{i=1}^{n} f(x_i) \ge \sum_{i=1}^{n} f(y_i).$$

## Remarks

Proof idea: let $c_i$ be the slope of the chord of $f$ between $x_i$ and $y_i$, or a [[subderivative|9C5WXYYQEK]] at $x_i$ when $x_i = y_i$.
Since [[the slopes of a convex function increase|0BY5FTHEEE]] and both sequences decrease, $c_1 \ge c_2 \ge \dots \ge c_n$.
Then $\sum_i \big(f(x_i) - f(y_i)\big) = \sum_i c_i (x_i - y_i)$, and summation by parts writes this as a sum of products of the nonnegative numbers $c_k - c_{k+1}$ with the nonnegative partial sums $\sum_{i \le k} (x_i - y_i)$.

Example: $(2, 0)$ majorizes $(1, 1)$, so $f(2) + f(0) \ge 2 f(1)$, which is midpoint convexity.
More generally $(x_1, \dots, x_n)$ majorizes the constant vector $(\bar x, \dots, \bar x)$ with $\bar x$ the mean, which gives [[Jensen's inequality|0RWMRTPC87]] with equal weights.

The equality of the total sums is needed.
With $n = 1$, $f(x) = -x$, $x_1 = 1$ and $y_1 = 0$, one has $x_1 \ge y_1$ but $f(x_1) = -1 < 0 = f(y_1)$.

## Explanations

### L1 human

Karamata's inequality compares two lists of numbers with the same total.
If one list is more spread out than the other, then summing a convex function over it gives a larger result.
A convex function rewards spreading out, because its graph bends upward.

### L2 human

The inequality says that $x \mapsto \sum_i f(x_i)$ is a [[Schur-convex function|A72XTVF2GK]] whenever $f$ is convex.
It contains Jensen's inequality with equal weights and implies many symmetric inequalities, for instance by comparing a vector with a less spread out one.
A converse also holds: if the conclusion is true for every convex $f$, then $x$ majorizes $y$.
</document>

<document source="https://tamnd.github.io/math2/node/FWPYXMK52W.md" tag="FWPYXMK52W" kind="definition">
# Integral equation of the first kind

## Definition

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ be a [[kernel|GSMBXT7FPK]], and let $f \colon [a, b] \to \C$ be given.
For each $x \in [a, b]$, let $I_x$ be either $[a, b]$ or $[a, x]$.
An *integral equation of the first kind* is a [[linear integral equation|AYZCKG34QK]] of the form
$$\int_{I_x} K(x, s) \varphi(s) \, ds = f(x) \quad \text{for all } x \in [a, b],$$
for the unknown function $\varphi \colon [a, b] \to \C$.

## Remarks

Example: $\int_0^1 (x + s) \varphi(s) \, ds = x$ on $[0, 1]$ is of the first kind.
With $\alpha = \int_0^1 \varphi$ and $\beta = \int_0^1 s \varphi(s) \, ds$ it says $\alpha x + \beta = x$, so $\alpha = 1$ and $\beta = 0$.
The function $\varphi(s) = 4 - 6s$ satisfies both, since $\int_0^1 (4 - 6s) \, ds = 1$ and $\int_0^1 (4s - 6s^2) \, ds = 0$; so do infinitely many others.

Non-example: $\varphi(x) = x + \int_0^1 (x + s) \varphi(s) \, ds$ is not of the first kind, because $\varphi$ also appears outside the integral.

With $I_x = [a, b]$ it is a Fredholm equation of the first kind; with $I_x = [a, x]$ it is a Volterra equation of the first kind.
The example shows that the solution need not be unique.
The right side $f(x) = x^2$ shows that it need not exist, since the left side is always affine in $x$.
</document>

<document source="https://tamnd.github.io/math2/node/FWQCZGXHWQ.md" tag="FWQCZGXHWQ" kind="definition">
# Net

## Definition

Let $(D, \le)$ be a [[directed set|D9RY4667K6]] and let $X$ be a set.
A *net* in $X$ indexed by $D$ is a map $D \to X$, written $(x_d)_{d \in D}$.
Let $X$ be a [[topological space|0CQHFPGJ45]] and let $x \in X$.
The net $(x_d)_{d \in D}$ *converges* to $x$ if for every [[neighbourhood|0ZCY0Z80B6]] $U$ of $x$ there is $d_0 \in D$ such that $x_d \in U$ for all $d \in D$ with $d \ge d_0$.

## Remarks

Example: a sequence $(x_n)_{n \ge 1}$ is a net indexed by $\N$ with its usual order, and the net converges to $x$ exactly when the sequence [[converges|0W4CN9E2W5]] to $x$.

Example: let $A \subseteq X$ and let $x$ be in the [[closure|0QEAJB91GZ]] of $A$.
Let $D$ be the set of neighbourhoods of $x$, ordered by $U \le V$ when $V \subseteq U$; this is directed because $U \cap V$ is a neighbourhood.
For each $U$ choose $x_U \in U \cap A$.
Then $(x_U)$ converges to $x$: for a neighbourhood $U$ take $d_0 = U$, and every $V \ge U$ gives $x_V \in V \subseteq U$.

Non-example: in $\R$, the sequence $x_n = n$ is a net that converges to no point $x$, since the neighbourhood $(x - 1, x + 1)$ contains $x_n$ for at most two values of $n$.

A point lies in the closure of $A$ if and only if some net in $A$ converges to it.
A space is [[Hausdorff|0S1FJ8VEX9]] if and only if every net in it has at most one limit.
</document>

<document source="https://tamnd.github.io/math2/node/FWRJV1ZABC.md" tag="FWRJV1ZABC" kind="definition">
# Kleene's T predicate

## Definition

Fix a [[Gödel numbering|MFJXPAF0R0]] of the programs for [[partial recursive functions|HBK5QHC1GC]] of one argument.
Fix also a coding of finite sequences of natural numbers by natural numbers.
Let $\varphi_e$ denote the partial function computed by the program with number $e$.
*Kleene's T predicate* is the relation $T \subseteq \N^3$ defined as follows.
The triple $(e, x, c)$ is in $T$ if and only if the following holds.
The number $e$ is the number of a program, and $c$ codes a complete halting computation of that program on input $x$.
For each $k \ge 1$ there is a similar relation $T_k(e, x_1, \ldots, x_k, c)$ for programs with $k$ arguments.

## Remarks

Kleene's normal form theorem says two things.
First, $T$ is [[primitive recursive|5NVA2JRHPV]].
Second, there is a primitive recursive function $U$ such that $\varphi_e(x) \simeq U(\mu c \, T(e, x, c))$ for all $e$ and $x$.
Here $\simeq$ means that both sides are undefined, or both are defined and equal.

Example: if $e$ is a program for the successor function, then for each $x$ there is a $c$ with $T(e, x, c)$, and $U(c) = x + 1$ for the least such $c$.

Non-example: if $e$ is a program that never halts, then $T(e, x, c)$ fails for all $x$ and $c$, since no $c$ codes a halting computation.

The halting set $\set{e : \varphi_e(e) \text{ is defined}}$ equals $\set{e : \exists c \, T(e, e, c)}$, so it is $\Sigma^0_1$ in the [[arithmetical hierarchy|7CVJCQKY41]].
</document>

<document source="https://tamnd.github.io/math2/node/FWRW7BYX6T.md" tag="FWRW7BYX6T" kind="definition">
# Variational integrator

## Definition

Let $L \colon \R^d \times \R^d \to \R$, $(q, \dot q) \mapsto L(q, \dot q)$, be a smooth Lagrangian, and fix a step size $h > 0$.
A discrete Lagrangian is a smooth function $L_h \colon \R^d \times \R^d \to \R$ that approximates the action:
$$L_h(q_0, q_1) \approx \int_0^h L(q(t), \dot q(t))\, dt.$$
Here $q$ is the solution of the [[Euler–Lagrange equation|PS0VV149ET]] with $q(0) = q_0$ and $q(h) = q_1$.
The discrete action of a sequence $q_0, \ldots, q_N$ is $S_h = \sum_{k=0}^{N-1} L_h(q_k, q_{k+1})$.
The *variational integrator* defined by $L_h$ is the map $(q_{k-1}, q_k) \mapsto (q_k, q_{k+1})$ given by the discrete Euler–Lagrange equations:
$$D_2 L_h(q_{k-1}, q_k) + D_1 L_h(q_k, q_{k+1}) = 0.$$
Here $D_1$ and $D_2$ denote the gradients with respect to the first and the second argument.
These are the equations $\partial S_h / \partial q_k = 0$ for $1 \le k \le N - 1$, with $q_0$ and $q_N$ fixed.

## Remarks

Example: take $L(q, \dot q) = \frac12 \abs{\dot q}^2 - V(q)$ and
$$L_h(q_0, q_1) = \frac{h}{2} \abs{\frac{q_1 - q_0}{h}}^2 - \frac{h}{2} \left( V(q_0) + V(q_1) \right).$$
Then $D_2 L_h(q_{k-1}, q_k) = \frac{q_k - q_{k-1}}{h} - \frac{h}{2} \nabla V(q_k)$ and $D_1 L_h(q_k, q_{k+1}) = -\frac{q_{k+1} - q_k}{h} - \frac{h}{2} \nabla V(q_k)$.
Their sum vanishes exactly when $q_{k+1} - 2 q_k + q_{k-1} = -h^2 \nabla V(q_k)$, which is the [[Störmer–Verlet method|ESWE8Y3JD4]].
Every variational integrator is symplectic in the variables $(q_k, p_k)$ with $p_k = -D_1 L_h(q_k, q_{k+1})$, when $L_h$ is regular, that is, when this relation can be solved for $q_{k+1}$.
So variational integrators are [[geometric integrators|VQ2BTYFST4]] for symplecticity.
If $L_h$ approximates the exact action to order $r$, then the method has order at least $r$.

## Explanations

### L1 human

Instead of discretising the equation of motion, one discretises the action and asks it to be stationary.
The equations that come out are the numerical method.
</document>

<document source="https://tamnd.github.io/math2/node/FWT0003QS1.md" tag="FWT0003QS1" kind="example">
# Little cubes operad

## Example

Fix $n \ge 1$ and let $I^n = (0, 1)^n$ be the open unit cube.
A *little $n$-cube* is an embedding $I^n \to I^n$ of the form $(t_1, \ldots, t_n) \mapsto (a_1 + b_1 t_1, \ldots, a_n + b_n t_n)$ with all $b_i > 0$.
The space $\mathcal{C}_n(k)$ is the [[topological space|0CQHFPGJ45]] of $k$-tuples of little $n$-cubes with pairwise disjoint images.
The [[symmetric group|0515A6D2Y1]] $\Sigma_k$ acts by permuting the $k$ cubes, and the unit is the identity cube in $\mathcal{C}_n(1)$.
Composition inserts the configuration of $\mathcal{C}_n(j_i)$ into the $i$-th cube of an element of $\mathcal{C}_n(k)$, by composing embeddings.
These data form the *little $n$-cubes operad* $\mathcal{C}_n$.

For a pointed space $X$, the $n$-fold [[loop space|2JFF5F3EGF]] $\Omega^n X$ is an [[algebra|ES9YATDWA3]] over $\mathcal{C}_n$.
View a point of $\Omega^n X$ as a map $\bar{I}^n \to X$ that sends the boundary to the base point.
Given $c = (c_1, \ldots, c_k) \in \mathcal{C}_n(k)$ and loops $f_1, \ldots, f_k$, define $\theta_k(c; f_1, \ldots, f_k)$ to be $f_i \circ c_i^{-1}$ on the image of $c_i$ and the base point elsewhere.
This map is continuous, and the algebra clauses hold because composing cubes corresponds to nesting the pictures.

## Remarks

The space $\mathcal{C}_n(k)$ is homotopy equivalent to the space of ordered configurations of $k$ distinct points in $\R^n$, by taking the centres of the cubes.
In particular $\mathcal{C}_n(2)$ is homotopy equivalent to the sphere $S^{n-1}$.
For $n = 1$ the operations are associative up to homotopy, but not commutative, since $\mathcal{C}_1(2)$ has two contractible components, one for each order.
For $n \ge 2$ the sphere $S^{n-1}$ is connected, which is the reason why $\pi_2$ and higher homotopy groups are abelian.
May's recognition principle says that a group-like $\mathcal{C}_n$-algebra is weakly equivalent to an $n$-fold loop space.
Letting $n \to \infty$ gives an operad $\mathcal{C}_\infty$, which is an [[E-infinity operad|T78YG90VVT]].

## Explanations

### L1 human

Picture a big square with several small squares drawn inside it, none overlapping.
This picture is a rule for combining several maps defined on squares into one: shrink each map into its small square and fill the rest with a constant.
Such pictures can be nested, and that nesting is the operad structure.
In dimension one the small intervals sit in a line and their order matters, which gives associative but not commutative products.
In dimension two or more the small squares can move around each other, which makes the products commutative up to homotopy.
</document>

<document source="https://tamnd.github.io/math2/node/FX11K6XRKV.md" tag="FX11K6XRKV" kind="definition">
# Quasifield

## Definition

Let $Q$ be a set with two binary operations $+$ and $\cdot$, and let $Q^* = Q \setminus \set{0}$.
The triple $(Q, +, \cdot)$ is a *(left) quasifield* if the following hold.

1. $(Q, +)$ is a [[group|G5S1M9]] with identity element $0$.
2. $a \cdot 0 = 0 \cdot a = 0$ for all $a \in Q$.
3. $(Q^*, \cdot)$ is a loop: there is $1 \in Q^*$ with $1 \cdot a = a \cdot 1 = a$ for all $a \in Q^*$, and for $a, b \in Q^*$ the equations $a \cdot x = b$ and $y \cdot a = b$ have unique solutions in $Q^*$.
4. $a \cdot (b + c) = a \cdot b + a \cdot c$ for all $a, b, c \in Q$.
5. For all $a, b, c \in Q$ with $a \ne b$, the equation $a \cdot x = b \cdot x + c$ has exactly one solution $x \in Q$.

A *right quasifield* is defined in the same way with $(b + c) \cdot a = b \cdot a + c \cdot a$ in clause 4 and $x \cdot a = x \cdot b + c$ in clause 5.

## Remarks

Example: every [[division ring|0EA5BTA0CX]] $D$ is a quasifield.
Clauses 1 to 4 are axioms of a division ring.
For clause 5, by distributivity $a x = b x + c$ is equivalent to $(a - b) x = c$, which has the unique solution $x = (a - b)^{-1} c$ since $a \ne b$.

Non-example: the integers $\Z$ with the usual operations.
Clause 3 fails, since $2 \cdot x = 1$ has no solution in $\Z \setminus \set{0}$.

One can prove that the additive group $(Q, +)$ of a quasifield is always abelian.
A quasifield in which both distributive laws hold is a [[semifield|AHH97EJ6VH]], and a quasifield whose multiplication is associative is a [[near-field|DSE72WQ5A2]] (on the matching side).
</document>

<document source="https://tamnd.github.io/math2/node/FX1A4SN6CF.md" tag="FX1A4SN6CF" kind="definition">
# Bayes risk

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a statistical model with a loss function, and let $R(\theta, \delta)$ be the [[risk function|N26TSVRH2S]] of a decision rule $\delta$.
Let $\Theta$ carry a σ-algebra, and let $\pi$ be a probability measure on $\Theta$, the [[prior|00NBBSXRQS]], such that $\theta \mapsto R(\theta, \delta)$ is measurable for the rules considered.
The *Bayes risk* of $\delta$ under $\pi$ is
$$r(\pi, \delta) = \int_\Theta R(\theta, \delta) \, \pi(d\theta) \in [0, \infty].$$
The *Bayes risk of the prior* $\pi$ is $r(\pi) = \inf_\delta r(\pi, \delta)$, the infimum over all decision rules.

## Remarks

The integral is a [[Lebesgue integral|DWJF7ABPR1]] of a nonnegative function, so it always exists, possibly infinite.
By Fubini's theorem, $r(\pi, \delta)$ is also the expected loss when $\theta$ is drawn from $\pi$ and then $X$ from $P_\theta$.

Example: let $X$ be [[binomial|0WCWA1KT3M]] with parameters $n$ and $\theta \in [0, 1]$, with squared error loss and $\delta(X) = X/n$.
Then $R(\theta, \delta) = \theta(1 - \theta)/n$.
For the uniform prior, $r(\pi, \delta) = \frac{1}{n} \int_0^1 \theta(1 - \theta) \, d\theta = \frac{1}{6n}$.

Non-example: the maximum $\sup_\theta R(\theta, \delta) = \frac{1}{4n}$ is not a Bayes risk of $\delta$ for the uniform prior.
It is the worst-case risk used in the minimax principle; it equals $r(\pi, \delta)$ only for priors that put all their mass where the risk is maximal.
</document>

<document source="https://tamnd.github.io/math2/node/FX3NEFYFFN.md" tag="FX3NEFYFFN" kind="concept">
# Irreducible representation

## Summary

An irreducible representation is a nonzero [[group representation|WYGRXCG2RN]] with no proper nonzero [[invariant subspace|0ACX5QT8TE]].
Irreducible representations are the building blocks of representation theory.
For a finite group over the complex numbers, every finite-dimensional representation is a direct sum of irreducible ones, by [[Maschke's theorem|4SBZ4EMPBZ]].

## Explanations

### L1 human

A representation is irreducible when it cannot be cut into smaller pieces that the group preserves.
Think of it as an atom: other representations are built from it, but it is not built from smaller ones.
Every one-dimensional representation is irreducible, because a line has no smaller nonzero subspace.
The rotations of the plane by multiples of $90$ degrees give an irreducible representation of the cyclic group of order $4$ over $\R$.
Indeed, a $90$ degree rotation preserves no line through the origin.
</document>

<document source="https://tamnd.github.io/math2/node/FX40F4A63V.md" tag="FX40F4A63V" kind="statement">
# Rauch comparison theorem

## Statement

Let $M$ and $\bar M$ be [[Riemannian manifolds|7CBT4SM83Y]] of the same dimension $n \ge 2$, and let $a > 0$.
Let $\gamma \colon [0, a] \to M$ and $\bar\gamma \colon [0, a] \to \bar M$ be unit-speed [[geodesics|NSYMQMHPDN]].
Let $J$ along $\gamma$ and $\bar J$ along $\bar\gamma$ be [[Jacobi fields|9AHMM41PJ3]] such that
$$J(0) = 0, \quad \bar J(0) = 0, \quad \ip{J'(0)}{\gamma'(0)} = \ip{\bar J'(0)}{\bar\gamma'(0)}, \quad \abs{J'(0)} = \abs{\bar J'(0)}.$$
Assume that:

1. $\bar\gamma$ has no point $\bar\gamma(t)$ with $0 < t \le a$ [[conjugate|ZPXE5VSZ35]] to $\bar\gamma(0)$.
2. For every $t \in [0, a]$, every 2-plane $\sigma \subset T_{\gamma(t)} M$ containing $\gamma'(t)$ and every 2-plane $\bar\sigma \subset T_{\bar\gamma(t)} \bar M$ containing $\bar\gamma'(t)$, the [[sectional curvatures|1X42JAA349]] satisfy $\bar K(\bar\sigma) \ge K(\sigma)$.

Then $\abs{\bar J(t)} \le \abs{J(t)}$ for all $t \in [0, a]$.

## Remarks

The idea of the proof: for $0 < t \le a$, the quantity $\ip{\bar J'}{\bar J} / \abs{\bar J}^2$ at $t$ equals the index form of $\bar J$ on $[0, t]$ divided by $\abs{\bar J(t)}^2$.
Since $\bar\gamma$ has no conjugate points, $\bar J$ minimizes the index form among fields with the same end values.
Transplanting $J$ to $\bar\gamma$ by a linear isometry along the geodesics and using the curvature inequality shows that the logarithmic derivative of $\abs{J}$ is at least that of $\abs{\bar J}$.
Integration from $0$, where the ratio $\abs{\bar J(t)} / \abs{J(t)}$ tends to $1$, gives the inequality.

Example: take $\bar M = \R^n$, where Jacobi fields with $\bar J(0) = 0$ are $\bar J(t) = t W$ for a constant vector $W$.
If $M$ has sectional curvature $K \le 0$, then $\abs{J(t)} \ge t \abs{J'(0)}$ for normal Jacobi fields, so $J$ never vanishes for $t > 0$ and $M$ has no conjugate points.
This is the key step of the Cartan–Hadamard theorem.

Hypothesis 1 is needed: on the unit sphere, normal Jacobi fields $\sin(t) E(t)$ vanish at $t = \pi$, and comparisons with the sphere hold only up to that first conjugate point.

## Explanations

### L1 human

Jacobi fields measure how fast nearby geodesics spread apart.
The theorem says: where curvature is larger, geodesics from one point spread apart more slowly.
So a space that curves more than another can be compared with it, and distances in it grow more slowly.

### L2 human

Rauch's theorem is the infinitesimal tool behind much of comparison geometry.
Comparing with the space forms of constant curvature gives explicit bounds on Jacobi fields, on conjugate points and on the differential of the exponential map.
It is used in the proofs of the Cartan–Hadamard theorem, of Toponogov's triangle comparison theorem and of the sphere theorem.
</document>

<document source="https://tamnd.github.io/math2/node/FX5PWKCMBQ.md" tag="FX5PWKCMBQ" kind="statement">
# Bipolar theorem

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $(X, Y, \langle \cdot, \cdot \rangle)$ be a [[dual pair|R68HNB5SVR]] over $\mathbb{K}$.
For $A \subseteq X$, its *polar* is $A^\circ = \set{y \in Y : \abs{\langle x, y \rangle} \le 1 \text{ for all } x \in A}$, and for $B \subseteq Y$, its polar is $B^\circ = \set{x \in X : \abs{\langle x, y \rangle} \le 1 \text{ for all } y \in B}$.
Let $A \subseteq X$, and let $C$ be the [[convex hull|0FQSDC2EQ2]] of $\set{\lambda a : a \in A, \lambda \in \mathbb{K}, \abs{\lambda} \le 1}$.
Then the bipolar $A^{\circ\circ} = (A^\circ)^\circ$ is the [[closure|0QEAJB91GZ]] of $C$ in the [[weak topology|P6KHVSTPZV]] $\sigma(X, Y)$.

## Remarks

The set described is the smallest $\sigma(X, Y)$-closed, convex and [[balanced|2TDSHJJYS4]] set containing $A$, the closed absolutely convex hull of $A$.
The bipolar contains $A$ and is convex, balanced and $\sigma(X, Y)$-closed, so it contains this hull.
For the reverse inclusion, take $x_0$ outside the hull.
The continuous dual of $(X, \sigma(X, Y))$ is $Y$, so the Hahn–Banach separation theorem gives $y \in Y$ with $\abs{\langle a, y \rangle} \le 1$ on the hull and $\abs{\langle x_0, y \rangle} > 1$.
Hence $x_0 \notin A^{\circ\circ}$.
Example: take a normed space $E$ with the pair $(E, E')$, and let $A$ be the open unit ball of $E$.
Then $A^\circ$ is the closed unit ball of $E'$, and $A^{\circ\circ}$ is the closed unit ball of $E$, the weak closure of the open ball.
Over $\R$ with one-sided polars $\set{y : \langle x, y \rangle \le 1}$, the bipolar is instead the closed convex hull of $A \cup \set{0}$.

## Explanations

### L1 human

The polar of a set collects all test vectors that keep the set inside the "unit band" $\abs{\langle x, y \rangle} \le 1$.
Taking the polar twice gives back the original set, filled in to be convex, symmetric and closed.
So a closed convex symmetric set is completely described by its polar.

### L2 human

The bipolar theorem is the convex-geometric form of the Hahn–Banach theorem for a dual pair.
It shows that every closed absolutely convex set is an intersection of "slabs" $\set{x : \abs{\langle x, y \rangle} \le 1}$.
It is the main tool in the proof of the [[Mackey–Arens theorem|RKKWS7CDNF]] and in describing equicontinuous sets and bounded sets through polars.
Combined with the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], it describes the closed absolutely convex neighbourhoods of $0$.
Each of them is the polar of its own polar, which is a weak-\* compact absolutely convex set of the dual.
</document>

<document source="https://tamnd.github.io/math2/node/FX88V4XW6F.md" tag="FX88V4XW6F" kind="statement">
# Freyd's adjoint functor theorem

## Statement

Let $\mathcal{D}$ be a [[complete category|FWGPKXCPB7]] whose hom-sets are sets, and let $G \colon \mathcal{D} \to \mathcal{C}$ be a functor.
Then $G$ has a [[left adjoint|TZHF6ZM92S]] if and only if both of these hold:

1. $G$ [[preserves limits|H4MY8JN8W4]] of small diagrams.
2. $G$ satisfies the [[solution set condition|CKTDR0VE8D]].

## Remarks

A right adjoint preserves limits, and the unit $\eta_c$ alone is a solution set, which gives one direction.
For the other, fix $c$ and consider the comma category $c \downarrow G$.
It is complete, because $\mathcal{D}$ is complete and $G$ preserves limits.
The product of a solution set is a weakly initial object, and the joint equalizer of all its endomorphisms is an initial object $(F c, \eta_c)$.
These initial objects define the left adjoint $F$.

Example: the categories of groups, rings, modules and other algebraic structures are complete, and their forgetful functors to sets preserve limits.
A cardinality bound on the subalgebra generated by a set gives the solution set condition.
So the theorem gives free algebras without writing them down.

The solution set condition cannot be dropped.
The forgetful functor from complete Boolean algebras to sets preserves limits, but has no left adjoint.

## Explanations

### L1 human

The theorem gives a test for when a functor has a left adjoint.
The necessary condition, that the functor preserve limits, is almost sufficient.
One only needs an extra size condition, which says that a set of test objects is enough.
It is a general way to show that free objects and similar universal constructions exist.

### L2 human

The theorem is due to Peter Freyd and is often called the general adjoint functor theorem.
In practice, completeness and preservation of limits are easy to check, and the work is in the solution set condition.
The special adjoint functor theorem replaces the solution set condition by conditions on $\mathcal{D}$ alone, such as a cogenerating set.
In locally presentable categories the theorem simplifies further: a functor between them is a right adjoint exactly when it preserves limits and is accessible.
</document>

<document source="https://tamnd.github.io/math2/node/FX8CRCAYXZ.md" tag="FX8CRCAYXZ" kind="example">
# Variety of groups

## Example

Take the signature with a binary operation $\cdot$, a unary operation ${}^{-1}$ and a constant $e$.
The *variety of groups* is the class of algebras $(G, \cdot, {}^{-1}, e)$ of this signature that satisfy the [[identities|PKD440G2XW]]
$$(x \cdot y) \cdot z = x \cdot (y \cdot z), \qquad x \cdot e = e \cdot x = x, \qquad x \cdot x^{-1} = x^{-1} \cdot x = e.$$
These algebras are exactly the [[groups|G5S1M9]], with the inverse map and the identity element listed as operations.
The class is a [[variety|AJNZJD8442]], since it is defined by identities.
In this signature, a subalgebra of a group is a subgroup, a homomorphism is a group homomorphism, and a congruence corresponds to a normal subgroup.

## Remarks

The choice of signature matters.
With the binary operation $\cdot$ alone, the class of groups is not closed under subalgebras.
For example, $(\N, +)$ is closed under $+$ inside the group $(\Z, +)$, but it is not a group.
So the class of groups is not a variety in that signature.
Subvarieties of the variety of groups include the abelian groups, defined by $x \cdot y = y \cdot x$, and the groups of exponent $n$, defined by $x^n = e$.

## Explanations

### L1 human

Groups can be described entirely by equations, once the inverse and the identity element are treated as operations.
This makes the groups a variety, so all the general theorems about varieties apply to them.
For instance, free groups are the free algebras of this variety.
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<document source="https://tamnd.github.io/math2/node/FX92PP3HZR.md" tag="FX92PP3HZR" kind="concept">
# Biholomorphism

## Summary

A biholomorphism is a bijective [[holomorphic map|3MQTWKBBBT]] between open subsets of $\C^n$ whose inverse is also holomorphic.
Two domains related by a biholomorphism are biholomorphically equivalent: they are the same for complex analysis.
In one variable every simply connected proper domain of $\C$ is biholomorphic to the disc, by the [[Riemann mapping theorem|02WHMEG5ZR]].
In several variables this fails: for $n \ge 2$ the [[unit ball|PDQ9KW9FAP]] and the unit [[polydisc|DJBWP7D3M8]] are [[not biholomorphic|1NF8T07H9X]].

## Explanations

### L1 human

A biholomorphism is a change of complex coordinates that can be undone.
It matches the points of two domains one to one, and it carries holomorphic functions on one domain to holomorphic functions on the other.
For example, the map $(z_1, z_2) \mapsto (2 z_1, \ z_2 + z_1^2)$ is a biholomorphism of $\C^2$ onto itself, with inverse $(w_1, w_2) \mapsto (w_1/2, \ w_2 - w_1^2/4)$.
Biholomorphic domains cannot be told apart by complex analysis.
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<document source="https://tamnd.github.io/math2/node/FXCQWFY5RV.md" tag="FXCQWFY5RV" kind="definition">
# Statistical inference

## Definition

Let $\mathcal{P} = \set{P_\theta : \theta \in \Theta}$ be a [[statistical model|JZ6R7J36K5]] on a sample space $S$.
Let $x \in S$ be an observed value of a random element $X$ whose law is $P_\theta$ for an unknown $\theta$.
*Statistical inference* is the derivation, from $x$, of statements about $\theta$ or about a function $\tau(\theta)$.
A rule of inference is a map defined on $S$ that does not depend on $\theta$; its main kinds are these.

1. A point estimate: a value $\hat{\tau}(x)$ of an [[estimator|0TTNRP5PFQ]] of $\tau(\theta)$.
2. A set estimate: a set $C(x)$ of plausible values of $\tau(\theta)$.
3. A test: a decision $\varphi(x) \in \set{0, 1}$ between a hypothesis $\theta \in \Theta_0$ and its alternative $\theta \in \Theta \setminus \Theta_0$.

## Remarks

Example: for $n$ independent $\mathcal{N}(\theta, 1)$ observations, the sample mean $\bar{x}$ is a point estimate of $\theta$, and $[\bar{x} - 1.96/\sqrt{n}, \bar{x} + 1.96/\sqrt{n}]$ is a set estimate of $\theta$.
Both are computed from the data alone, as the definition requires.
Non-example: the rule "report $\theta$" is not a rule of inference, since it uses the unknown parameter.
Inference rules are built from [[statistics|036756NFD6]] and judged either by their behaviour over repeated samples or through a posterior distribution.
</document>

<document source="https://tamnd.github.io/math2/node/FXJ66Z6PQY.md" tag="FXJ66Z6PQY" kind="concept">
# Net of a polyhedron

## Summary

A net of a polyhedron is a flat polygon that folds up into the surface of the polyhedron.
It is made of the faces of the polyhedron, joined along some of their edges.
It arises by cutting the surface along a spanning tree of edges and unfolding it into the plane without overlap.
Nets are used to build paper models, and whether every convex polyhedron has one is the open [[Dürer's conjecture|PEDBPRSV9V]].

## Explanations

### L1 human

Cut a cardboard box along some of its edges and flatten it: the flat piece is a net of the box.
The cross shape made of six squares is a net of the cube, and the cube has $11$ different nets in all.
The cuts must be chosen so that the surface stays in one piece and lies flat without the faces overlapping.
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<document source="https://tamnd.github.io/math2/node/FXN98YEQRZ.md" tag="FXN98YEQRZ" kind="definition">
# Radon measure

## Definition

Let $X$ be a [[Hausdorff space|0S1FJ8VEX9]] and let $\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.
The measure $\mu$ is a *Radon measure* if both of the following hold.

1. $\mu$ is [[locally finite|YQ5C1SRE0W]]: every point has an open neighbourhood of finite measure.
2. $\mu$ is [[inner regular|28E5Q5BM69]]: $\mu(A) = \sup \set{\mu(K) : K \subseteq A, \ K \text{ compact}}$ for every Borel set $A$.

## Remarks

Example: the Dirac measure $\delta_p$ at a point $p$ is a Radon measure.
Every point has a neighbourhood of measure at most $1$; and if $p \in A$ then $K = \{p\}$ is a [[compact|0R6EWFKRCF]] subset of $A$ with $\delta_p(K) = \delta_p(A)$, while if $p \notin A$ both sides are $0$.
Lebesgue measure on $\R^n$ is also a Radon measure.

Non-example: counting measure on $\R$ satisfies clause 2 but not clause 1.

Another common definition, used on locally compact Hausdorff spaces, asks that $\mu$ be finite on compact sets, outer regular on all Borel sets and inner regular on open sets.
On a locally compact Hausdorff space in which every open set is σ-compact, such as $\R^n$, every Borel measure that is finite on compact sets is a Radon measure.
</document>

<document source="https://tamnd.github.io/math2/node/FXSY9SXHFZ.md" tag="FXSY9SXHFZ" kind="definition">
# Discrete wavelet transform

## Definition

Let $\varphi$ be the scaling function of a [[multiresolution analysis|1GHQMBR7DE]] of $L^2(\R)$, with [[refinement equation|YH4BERB952]] $\varphi(x) = \sqrt2 \sum_k h_k \varphi(2x - k)$.
Put $g_k = (-1)^k \overline{h_{1-k}}$ for $k \in \Z$.
For a sequence $a = (a_n)_{n \in \Z} \in \ell^2(\Z)$, one *level* of the discrete wavelet transform gives the two sequences
$$c_k = \sum_{n \in \Z} \overline{h_{n - 2k}} \, a_n, \qquad d_k = \sum_{n \in \Z} \overline{g_{n - 2k}} \, a_n, \qquad k \in \Z.$$
The *discrete wavelet transform* with $J \ge 1$ levels applies one level to $a$, then to the resulting $c$, and so on $J$ times.
Its output is the detail sequences of all levels together with the last coarse sequence.

## Remarks

The map $a \mapsto (c, d)$ is a unitary operator from $\ell^2(\Z)$ onto $\ell^2(\Z) \oplus \ell^2(\Z)$, so the transform is inverted by its adjoint.
If $a_n = \ip{f}{\varphi_{j+1, n}}$ for $f \in L^2(\R)$, where $\varphi_{j,k}(x) = 2^{j/2} \varphi(2^j x - k)$, then $c_k = \ip{f}{\varphi_{j,k}}$ and $d_k = \ip{f}{\psi_{j,k}}$ for the associated orthonormal [[wavelet|4PZ35C0GV3]] $\psi$.
The same name is used for the coefficient family $(\ip{f}{\psi_{j,k}})_{j,k}$ of a function, and for finite signals with periodic or other boundary rules.

Example: for the Haar filter $h_0 = h_1 = 1/\sqrt2$ one gets $g_0 = 1/\sqrt2$ and $g_1 = -1/\sqrt2$.
Then $c_k = (a_{2k} + a_{2k+1})/\sqrt2$ and $d_k = (a_{2k} - a_{2k+1})/\sqrt2$.
For $a = (4, 2, 5, 5)$ (and zeros elsewhere) one level gives $c = (6/\sqrt2, 10/\sqrt2)$ and $d = (2/\sqrt2, 0)$.
Energy is preserved: $16 + 4 + 25 + 25 = 70 = 18 + 50 + 2 + 0$.

Non-example: the pairwise means $(a_{2k} + a_{2k+1})/2$ alone discard information, since $(1, 0)$ and $(0, 1)$ have the same mean; the transform keeps the details $d_k$ as well.
</document>

<document source="https://tamnd.github.io/math2/node/FXT3N6QMEX.md" tag="FXT3N6QMEX" kind="definition">
# Normal coordinates

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ and $p \in M$.
Let $V \subseteq T_pM$ be an open neighbourhood of $0$, star-shaped about $0$.
Assume that the [[exponential map|Y7Z0AGY1A7]] $\exp_p$ restricts to a [[diffeomorphism|0KZZ70ZHXP]] from $V$ onto an open set $U \subseteq M$.
Let $(e_1, \ldots, e_n)$ be an [[orthonormal basis|0FWJ0VW561]] of $T_pM$ for $g_p$, and let $E \colon \R^n \to T_pM$ be $E(x) = \sum_i x^i e_i$.
The *normal coordinates* centred at $p$ are the chart
$$\varphi = E^{-1} \circ (\exp_p|_V)^{-1} \colon U \to \R^n.$$

## Remarks

Example: in $\R^n$ with the Euclidean metric and the standard basis, $\exp_p(v) = p + v$, so the normal coordinates centred at $p$ are $\varphi(q) = q - p$, defined on all of $\R^n$.

Non-example: in $\R^2$ with the Euclidean metric, the chart $\psi(q) = 2(q - p)$ has the form $E^{-1} \circ \exp_p^{-1}$ only for the basis $e_i = \tfrac{1}{2} \varepsilon_i$, where $\varepsilon_1, \varepsilon_2$ is the standard basis.
This basis is orthogonal but not orthonormal, so $\psi$ is not a normal coordinate chart; indeed, in $\psi$ the metric has matrix $\tfrac{1}{4} I$ at $p$, not $I$.

In normal coordinates centred at $p$: $g_{ij}(p) = \delta_{ij}$, $\partial_k g_{ij}(p) = 0$ and $\Gamma^k_{ij}(p) = 0$.
The geodesics through $p$ are the lines $t \mapsto t x$ in these coordinates.
Different orthonormal bases give normal coordinates that differ by an orthogonal matrix.
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<document source="https://tamnd.github.io/math2/node/FY0YG17JBE.md" tag="FY0YG17JBE" kind="definition">
# Autoregressive model

## Definition

A *white noise* with variance $\sigma^2 > 0$ is a sequence $(\varepsilon_t)_{t \in \Z}$ of real random variables with $E[\varepsilon_t] = 0$, $E[\varepsilon_t^2] = \sigma^2$ and $E[\varepsilon_s \varepsilon_t] = 0$ for $s \ne t$.
Let $p \ge 1$, let $c, \phi_1, \dots, \phi_p \in \R$ with $\phi_p \ne 0$, and let $(\varepsilon_t)$ be a white noise.
An *autoregressive model of order $p$*, or AR($p$), is a weakly [[stationary process|DT0QXJE705]] $(X_t)_{t \in \Z}$ that satisfies
$$X_t = c + \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + \varepsilon_t \qquad \text{for all } t \in \Z.$$
Its *autoregressive polynomial* is $\phi(z) = 1 - \phi_1 z - \cdots - \phi_p z^p$.

## Remarks

If $\phi(z) \ne 0$ for all complex $z$ with $\abs{z} = 1$, the equation has exactly one weakly stationary solution.
If moreover $\phi$ has no [[root|019MA1VTR2]] in the closed unit disk $\abs{z} \le 1$, this solution is causal: $X_t = \mu + \sum_{j \ge 0} \psi_j \varepsilon_{t-j}$ with $\sum_j \abs{\psi_j} < \infty$ and $\mu = c / (1 - \phi_1 - \cdots - \phi_p)$.
Many texts include causality in the definition.

Example: for $p = 1$, $c = 0$ and $\abs{\phi_1} < 1$, the process $X_t = \sum_{j \ge 0} \phi_1^j \varepsilon_{t-j}$ converges in mean square and satisfies $X_t = \phi_1 X_{t-1} + \varepsilon_t$.
Its variance is $\sigma^2 / (1 - \phi_1^2)$ and its autocorrelation at lag $h$ is $\phi_1^{\abs{h}}$.

Non-example: for $\phi_1 = 1$ and $c = 0$, the equation $X_t = X_{t-1} + \varepsilon_t$ defines a [[random walk|04VZE3FZXF]], and it has no weakly stationary solution.
Indeed $X_t - X_{t-n} = \varepsilon_{t-n+1} + \cdots + \varepsilon_t$ has variance $n \sigma^2$, which is unbounded in $n$, while for a stationary solution it is at most $4 \operatorname{Var}(X_0)$.
Here $\phi(z) = 1 - z$ has the root $z = 1$ on the unit circle.
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<document source="https://tamnd.github.io/math2/node/FYBQ3Z52Z7.md" tag="FYBQ3Z52Z7" kind="concept">
# Horner's method

## Summary

Horner's method evaluates a polynomial of degree $n$ with $n$ multiplications and $n$ additions.
It rewrites $a_n x^n + \dots + a_0$ as a nested product $a_0 + x(a_1 + x(a_2 + \dots + x a_n))$.
The intermediate values are the coefficients of the quotient by $x - c$, so it also performs synthetic division.
It is the standard way to carry out [[polynomial evaluation|02MAKWDKGT]].

## Explanations

### L1 human

To compute $2x^3 - 6x^2 + 2x - 1$ at $x = 3$, you do not need the powers of $3$.
Start with $2$; multiply by $3$ and add $-6$ to get $0$; multiply by $3$ and add $2$ to get $2$; multiply by $3$ and add $-1$ to get $5$.
So the value is $5$.
Each step is one multiplication and one addition.
The [[definition|Z2EWA10YMB]] gives the recursion.
</document>

<document source="https://tamnd.github.io/math2/node/FYBRP16KAN.md" tag="FYBRP16KAN" kind="concept">
# Concrete category

## Summary

A concrete category is a [[category|A2DE66937D]] $\mathcal{C}$ together with a [[faithful functor|GRG8WEBSH5]] $U$ from $\mathcal{C}$ to the category of sets.
The functor $U$ assigns an underlying set to each object, and morphisms are determined by their underlying functions.
Groups, rings, vector spaces and topological spaces with their usual forgetful functors are concrete categories.
Not every category admits such a functor, so concreteness is a real condition on large categories.

## Explanations

### L1 human

In many categories the objects are sets with extra structure and the arrows are functions that respect the structure.
A concrete category records this through a forgetful functor that remembers only the underlying set and the underlying function.
Being faithful means that two arrows with the same underlying function are the same arrow.
For example, a continuous map between two topological spaces is nothing more than a function with a property.
So topological spaces with the forgetful functor form a concrete category.
</document>

<document source="https://tamnd.github.io/math2/node/FYNFT90NT2.md" tag="FYNFT90NT2" kind="statement">
# Frostman lemma

## Statement

Let $A \subseteq \R^n$ be a Borel set and let $s \ge 0$.
Let $\mathcal{H}^s$ be the $s$-dimensional [[Hausdorff measure|Q1TGZK8RN6]], and let $B(x, r)$ be the closed ball of centre $x$ and radius $r$.
Then $\mathcal{H}^s(A) > 0$ if and only if there is a nonzero [[Radon measure|HZNEQQBCW4]] $\mu$ on $\R^n$ with these two properties:

1. The [[support|FQTXV6BQZG]] of $\mu$ is a [[compact set|0R6EWFKRCF]] contained in $A$.
2. $\mu(B(x, r)) \le r^s$ for every $x \in \R^n$ and every $r > 0$.

## Remarks

The direction from $\mu$ to $\mathcal{H}^s(A) > 0$ is the mass distribution principle.
If $A \subseteq \bigcup_i U_i$, each nonempty $U_i$ lies in a closed ball of radius $\operatorname{diam} U_i$ centred at a point of $U_i$.
So $0 < \mu(A) \le \sum_i \mu(U_i) \le \sum_i (\operatorname{diam} U_i)^s$, and the infimum over covers is at least $\mu(A)$.
For the other direction, with $A$ compact, one builds $\mu$ on dyadic cubes from the finest scale upwards, cutting the mass wherever the bound fails, and takes a weak limit.
The Borel case reduces to the compact case, since a Borel set with $\mathcal{H}^s(A) > 0$ contains a compact set with positive $\mathcal{H}^s$ measure.

As a consequence, $\dim_H A = \sup \set{ s \ge 0 : \text{a measure as above exists} }$ for every Borel set $A$, where $\dim_H$ is the [[Hausdorff dimension|SSNFTHZ6YJ]].

Example: for $A = [0, 1] \subseteq \R$ and $s = 1$, Lebesgue measure restricted to $[0, 1]$ works, since an interval of radius $r$ has length $2r$.
After scaling by $\frac12$, it satisfies $\mu(B(x, r)) \le r$.

## Explanations

### L1 human

A set is at least $s$-dimensional in the sense of Hausdorff measure exactly when one can spread a positive amount of mass on it that is never too concentrated.
Here "not too concentrated" means that a ball of radius $r$ carries at most $r^s$ of the mass.
This turns a question about all covers of the set into the search for one good measure.

### L2 human

Lower bounds for Hausdorff dimension are usually proved by constructing a Frostman measure, since upper bounds come from explicit covers.
The lemma leads to the energy method: if $A$ carries a nonzero measure with finite $t$-energy, then $\dim_H A \ge t$.
It is a standard tool in projection theorems and in the study of Kakeya sets.
It also links Hausdorff dimension to capacity in potential theory.
</document>

<document source="https://tamnd.github.io/math2/node/FYQX9CJAXG.md" tag="FYQX9CJAXG" kind="definition">
# Ext functor

## Definition

Let $R$ be a [[ring|0B1261B5ED]] and let $M$ and $N$ be [[$R$-modules|0NZ8JA25WK]].
Choose a [[projective resolution|HK2AA1YSQ2]] $\cdots \to P_1 \xrightarrow{d_1} P_0 \to M \to 0$ of $M$.
Applying $\Hom_R(-, N)$ to $P_\bullet$ gives a cochain complex
$$0 \to \Hom_R(P_0, N) \xrightarrow{d_1^*} \Hom_R(P_1, N) \xrightarrow{d_2^*} \Hom_R(P_2, N) \to \cdots,$$
where $d_i^*(f) = f \circ d_i$.
For $n \ge 0$, the $n$-th *Ext group* is its cohomology
$$\operatorname{Ext}^n_R(M, N) = \ker d_{n+1}^* / \im d_n^*,$$
with $d_0^* = 0$.

## Remarks

Up to a canonical isomorphism, $\operatorname{Ext}^n_R(M, N)$ does not depend on the choice of the resolution.
One has $\operatorname{Ext}^0_R(M, N) \cong \Hom_R(M, N)$, and the same groups are obtained from an injective resolution of $N$.
If $R$ is commutative, the Ext groups are $R$-modules.
Example: from $0 \to \Z \xrightarrow{\cdot n} \Z \to \Z/n\Z \to 0$ with $n \ge 1$, the complex is $\Z \xrightarrow{\cdot n} \Z$, so $\operatorname{Ext}^0_\Z(\Z/n\Z, \Z) = 0$ and $\operatorname{Ext}^1_\Z(\Z/n\Z, \Z) \cong \Z/n\Z$.
Non-example: the sequence $0 \to \Z/n\Z \xrightarrow{\id} \Z/n\Z \to 0$ is exact but $\Z/n\Z$ is not projective over $\Z$ for $n \ge 2$; using it would give $\operatorname{Ext}^1 = 0$, which is wrong.
A module $M$ is projective if and only if $\operatorname{Ext}^1_R(M, N) = 0$ for all $R$-modules $N$.
</document>

<document source="https://tamnd.github.io/math2/node/FYR6V2NH6C.md" tag="FYR6V2NH6C" kind="statement">
# Phragmén–Lindelöf principle

## Statement

Let $\alpha > \frac{1}{2}$ and let $S = \set{z \in \C \setminus \set{0} : \abs{\arg z} < \frac{\pi}{2\alpha}}$ be the open sector of opening $\pi / \alpha$.
Let $f$ be continuous on the closure $\bar S$ and [[holomorphic|0DG0Z72CQD]] on $S$.
Assume that $\abs{f(z)} \le M$ for each $z$ on the boundary of $S$.
Assume also that there are constants $C > 0$ and $\beta < \alpha$ with $\abs{f(z)} \le C \exp(\abs{z}^{\beta})$ for all $z \in S$.
Then $\abs{f(z)} \le M$ for all $z \in S$.

## Remarks

Proof idea: pick $\gamma$ with $\beta < \gamma < \alpha$ and put $g_\varepsilon(z) = f(z) \exp(-\varepsilon z^{\gamma})$ for $\varepsilon > 0$.
On $\bar S$ one has $\operatorname{Re} z^{\gamma} \ge \cos\left(\frac{\gamma \pi}{2\alpha}\right) \abs{z}^{\gamma}$ with a positive cosine, so $g_\varepsilon \to 0$ at infinity.
The [[maximum modulus principle|048F2P90WR]] on bounded pieces of $S$ gives $\abs{g_\varepsilon} \le M$, and $\varepsilon \to 0$ gives the claim.
The growth condition cannot be relaxed to $\beta = \alpha$.
The function $\exp(z^{\alpha})$ has modulus $\exp(r^{\alpha} \cos(\alpha \theta))$ at $z = r e^{i\theta}$, which is $1$ on the boundary rays and unbounded on the positive axis.
There are similar statements for strips, half-planes and other unbounded domains; the [[Hadamard three-lines theorem|WHS58R43N0]] is one of them.

## Explanations

### L1 human

On a bounded region, the maximum principle says that a holomorphic function is bounded inside by its bound on the boundary.
On an unbounded region such as a sector, this can fail, because the function can grow fast toward infinity.
The Phragmén–Lindelöf principle says that the maximum principle still holds if the function does not grow too fast.
The allowed speed depends on the opening of the sector: the narrower the sector, the faster the allowed growth.

### L2 human

The principle is used throughout the theory of entire functions and in analytic number theory.
It gives convexity bounds for $L$-functions in vertical strips, and it underlies results such as [[Lindelöf's theorem|XBM7J5R1VQ]] and Carlson's theorem.
</document>

<document source="https://tamnd.github.io/math2/node/FYXTVSVHT2.md" tag="FYXTVSVHT2" kind="concept">
# Morita equivalence of Lie groupoids

## Summary

Two [[Lie groupoids|HNVW0E732J]] are Morita equivalent when they describe the same transverse geometry, even if their spaces of objects differ.
Formally, they are joined by a third Lie groupoid that maps to both by weak equivalences, or equivalently by a principal bibundle.
Morita equivalent groupoids have homeomorphic orbit spaces, isomorphic isotropy groups along corresponding orbits and the same invariants such as cohomology and K-theory.
This notion makes Lie groupoids a model for singular spaces such as orbifolds, leaf spaces of foliations and quotients by group actions.

## Explanations

### L1 human

A groupoid can be thought of as a space together with rules saying which points count as the same and in how many ways.
Two groupoids can present the same "quotient" in different ways, for example by using a different set of charts.
Morita equivalence says exactly when two groupoids present the same quotient, with the same symmetries at each point.
For example, let a Lie group act freely and properly on a manifold.
Then the action groupoid is Morita equivalent to the quotient manifold, viewed as a groupoid with only identity arrows.
The idea parallels [[Morita equivalence|RD5WKA1VK7]] of rings, where two rings count as equivalent when their modules behave the same.
</document>

<document source="https://tamnd.github.io/math2/node/FYZ3CJGZYV.md" tag="FYZ3CJGZYV" kind="statement">
# Lindström's theorem

## Statement

Let $\mathcal{L}$ be a regular abstract logic that extends [[first-order logic|086QKPF0A6]].
Regular means that $\mathcal{L}$ is closed under the Boolean connectives, existential quantification, renaming of symbols and relativization.
Suppose that $\mathcal{L}$ has the following two properties.

1. Compactness: a set of $\mathcal{L}$-sentences has a model whenever each of its finite subsets has a model.
2. Downward Löwenheim–Skolem property: each $\mathcal{L}$-sentence with an infinite model has a countable model.

Then each $\mathcal{L}$-sentence is equivalent to a first-order [[sentence|DKFGQTRYZE]], that is, it has the same models.

## Remarks

The proof argues by contradiction with Ehrenfeucht–Fraïssé games.
If an $\mathcal{L}$-sentence $\varphi$ is not first-order, then for each $n$ there are [[structures|983GZBQYT4]] $\mathcal{A}_n \models \varphi$ and $\mathcal{B}_n \models \neg\varphi$ that satisfy the same first-order sentences of quantifier depth $n$.
Compactness and the Löwenheim–Skolem property, applied to an $\mathcal{L}$-sentence coding these pairs and partial isomorphisms between them, give countable $\mathcal{A} \models \varphi$ and $\mathcal{B} \models \neg\varphi$ with a back-and-forth system.
Then $\mathcal{A} \cong \mathcal{B}$, which contradicts $\mathcal{A} \models \varphi$ and $\mathcal{B} \models \neg\varphi$.

Example: second-order logic is a regular extension of first-order logic that is not equivalent to it.
By the theorem, it must lack the [[compactness theorem|1NR3HAWG9D]] or the [[Löwenheim–Skolem theorem|484M0TZ1S7]]; in fact it lacks both.
The infinitary logic $L_{\omega_1\omega}$ has the Löwenheim–Skolem property, so it must fail compactness, and it does.

## Explanations

### L1 human

First-order logic has two famous properties: compactness and the Löwenheim–Skolem property.
Lindström's theorem says that no stronger logic keeps both.
So these two properties characterise first-order logic among all reasonable logics.

### L2 human

The theorem, proved by Per Lindström in 1969, founded abstract model theory, which compares logics by their properties.
It explains why first-order logic is the default logic of model theory: any gain in expressive power costs compactness or the Löwenheim–Skolem theorem.
</document>

<document source="https://tamnd.github.io/math2/node/FYZC634H2D.md" tag="FYZC634H2D" kind="statement">
# Farkas' lemma

## Statement

Let $A$ be a real $m \times n$ [[matrix|0VD9ZSWWJG]] and let $b \in \R^m$.
For vectors, write $v \ge 0$ when every coordinate of $v$ is nonnegative, and let $A^T$ be the [[transpose|09DXWYRD43]] of $A$.
Then exactly one of the following two systems has a solution:

1. $x \in \R^n$ with $A x = b$ and $x \ge 0$;
2. $y \in \R^m$ with $A^T y \ge 0$ and $b^T y < 0$.

## Remarks

Both cannot hold: if $Ax = b$ with $x \ge 0$ and $A^T y \ge 0$, then $b^T y = x^T (A^T y) \ge 0$.
For the other direction, the set $C = \set{Ax : x \ge 0}$ is the [[convex cone|6VE4KA12VD]] generated by the columns of $A$, and it is closed because it is finitely generated.
If $b \notin C$, the hyperplane separation theorem gives a hyperplane through the origin that separates $b$ from $C$; its normal vector $y$ solves the second system.
Example: for $A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ and $b = (1, -1)^T$, the first system has no solution, and $y = (0, 1)^T$ solves the second, since $A^T y = (0, 1)^T \ge 0$ and $b^T y = -1 < 0$.

## Explanations

### L1 human

Either a vector $b$ can be built from the columns of $A$ with nonnegative weights, or there is a simple certificate that it cannot.
The certificate is a direction $y$ in which all columns point "forward" while $b$ points "backward".
Geometrically, a point outside a cone can be cut off from the cone by a flat wall through the apex.

### L2 human

Farkas' lemma is a theorem of the alternative, and it is the basis of the duality theory of [[linear programming|VNMXB0J2KA]].
Many variants follow from it by adding slack variables, for example the version with $Ax \le b$.
[[Gordan's lemma|HEKDP2TDNZ]] and the Minkowski–Weyl theorem belong to the same circle of ideas about finitely generated cones.
</document>

<document source="https://tamnd.github.io/math2/node/FZ25BQTKNT.md" tag="FZ25BQTKNT" kind="statement">
# Semistable reduction theorem

## Statement

Let $R$ be a [[discrete valuation ring|4JF1SGYZ5Y]] with field of fractions $K$.
Let $C$ be a smooth projective geometrically connected [[curve|XJM2MT4S3Y]] over $K$ of [[genus|6W8TDV9VVN]] $g \ge 2$.
Then there are a finite separable extension $K'$ of $K$ and a discrete valuation ring $R' \subset K'$ with $R' \cap K = R$.
Moreover there is a stable curve $\mathcal{C}'$ over $R'$ whose generic fibre is isomorphic to $C \times_K K'$.

Here a *stable curve* over $R'$ is a [[proper|9537RKBZPP]] flat morphism $\mathcal{C}' \to \operatorname{Spec} R'$ with the following properties.
Its geometric fibres are connected reduced curves of [[arithmetic genus|H03GKRVBJ1]] $g$ whose only singularities are ordinary double points.
Each smooth rational component of a fibre meets the other components in at least $3$ points.

## Remarks

The theorem is due to Deligne and Mumford; one of their proofs deduces it from the semistable reduction of the [[Jacobian|ZQ84M9D9TC]] of $C$.
The stable model $\mathcal{C}'$ is unique up to isomorphism.
In terms of [[moduli of curves|NK9ATH4WB6]], it says that the moduli space $\overline{M}_g$ of stable curves satisfies the existence part of the valuative criterion of properness.
The extension of $K$ cannot be dropped in general.
The analogue for elliptic curves shows this: over $K = \Q_p$ with $p \ge 5$, the curve $y^2 = x^3 + p$ has additive reduction, so it has no semistable model over $\Z_p$.
Over $K' = \Q_p(\pi)$ with $\pi^6 = p$, the substitution $x = \pi^2 x'$, $y = \pi^3 y'$ gives $y'^2 = x'^3 + 1$, which has good reduction.

## Explanations

### L1 human

Take a family of smooth curves that degenerates at one point, like a family over a small disc whose central member is singular.
The theorem says that after a finite change of the parameter, one can replace the bad central curve by a mild one.
The mild curve may have several pieces, but they only cross each other in simple nodes.
So degenerations of smooth curves can always be tamed.

### L2 human

The theorem is the key input in the proof that $\overline{M}_g$ is proper.
The base change $K'/K$ is the arithmetic form of taking a root $t = s^n$ of the parameter.
For abelian varieties, the analogue is Grothendieck's semistable reduction theorem.
After a finite extension, the identity component of the special fibre of the [[Néron model|K0EQTBRPE3]] is an extension of an abelian variety by a torus.
</document>

<document source="https://tamnd.github.io/math2/node/FZ4BRYJ0YZ.md" tag="FZ4BRYJ0YZ" kind="concept">
# Pseudospectral optimal control

## Summary

Pseudospectral optimal control is a form of [[direct collocation|T58W34D60G]] that uses a single global polynomial for the state on the whole time interval.
The nodes are the points of a Gaussian quadrature rule, such as the Legendre–Gauss–Lobatto points, and derivatives are taken with a differentiation matrix.
For smooth problems the error decreases faster than any power of the number of nodes, so few nodes give high accuracy.
The method is used in aerospace guidance, for example to plan spacecraft manoeuvres.

## Explanations

### L1 human

Low-order collocation draws a trajectory with many short pieces.
A pseudospectral method draws it with one high-degree polynomial through well-chosen points that cluster near the ends of the interval.
Clustering avoids the wild oscillations that equally spaced points would cause.
With three nodes $-1, 0, 1$, the method uses a parabola and integrates the cost with Simpson's rule.
For smooth problems, adding nodes improves the accuracy very quickly.
</document>

<document source="https://tamnd.github.io/math2/node/FZ64956KX2.md" tag="FZ64956KX2" kind="definition">
# Well-posed problem

## Definition

Let $X$ and $Y$ be [[normed vector spaces|03J5RAKB30]], the space of data and the space of solutions.
Let $D \subseteq Y$ and let $A \colon D \to X$ be a map.
The problem of finding, for given $f \in X$, some $u \in D$ with $A(u) = f$ is *well posed* if:

1. for every $f \in X$ there is a solution $u \in D$ of $A(u) = f$;
2. for every $f \in X$ this solution is unique;
3. the solution map $X \to Y$, $f \mapsto u$, is [[continuous|0X3WGX95FE]].

Otherwise the problem is *ill posed*.

## Remarks

The choice of the spaces $X$ and $Y$ and of their norms is part of the problem.
A problem can be well posed for one choice and ill posed for another.
Example: the [[Dirichlet problem|8D56GQ4AG8]] for the Laplace equation on the closed unit disc $\overline{B}$, with $X = C(\partial B)$ and $Y = C(\overline{B})$ both with the supremum norm.
The Poisson integral gives a solution for each boundary function $\varphi$.
If $u$ and $v$ solve the problem with boundary values $\varphi$ and $\psi$, the [[maximum principle|D2JF19HF91]] applied to $\pm(u - v)$ gives $\max_{\overline{B}} \abs{u - v} \le \max_{\partial B} \abs{\varphi - \psi}$.
This bound gives both uniqueness and continuity, so the problem is well posed.
Non-example: the Cauchy problem for the Laplace equation fails clause 3; see [[Hadamard's example|SZ6MEVM89M]].
The backward heat equation also fails clause 3, and on bounded domains it fails clause 1 for most data.

## Explanations

### L1 human

A well-posed problem has exactly one solution for every datum, and the solution changes little when the datum changes little.
</document>

<document source="https://tamnd.github.io/math2/node/FZ6KV2J5C9.md" tag="FZ6KV2J5C9" kind="definition">
# Matched Z-transform method

## Definition

Let $T > 0$ be a sampling period.
Let $H$ be a rational function of $s$ with nonzero leading constant $c$, [[zeros|00GC8ZQP4Q]] $q_1, \ldots, q_M$ and [[poles|066E7PJCYE]] $p_1, \ldots, p_N$, listed with multiplicity:
$$H(s) = c \, \frac{\prod_{i=1}^{M} (s - q_i)}{\prod_{j=1}^{N} (s - p_j)}.$$
The *matched Z-transform* of $H$ is the following rational function of $z$.
$$H_d(z) = k \, \frac{\prod_{i=1}^{M} (1 - e^{q_i T} z^{-1})}{\prod_{j=1}^{N} (1 - e^{p_j T} z^{-1})}.$$
The constant $k$ is chosen so that $H_d$ and $H$ agree at a chosen reference point.
The usual choice is $H_d(1) = H(0)$ when $H(0)$ is finite and nonzero.

## Remarks

Example: $H(s) = a/(s + a)$ with $a > 0$ has one pole $-a$ and $H(0) = 1$.
Its matched transform is $H_d(z) = k/(1 - e^{-aT} z^{-1})$, and $H_d(1) = 1$ gives $k = 1 - e^{-aT}$.
Non-example: $H(s) = e^{-s}$, a pure delay, is not a rational function, so the method does not apply to it as stated.
A pole $p$ with $\operatorname{Re} p < 0$ goes to $e^{pT}$ with $\abs{e^{pT}} < 1$, so a stable continuous filter gives a stable discrete one.
The map $a \mapsto e^{aT}$ is the relation $z = e^{sT}$ between the [[Z-transform|5FY0JQGHYZ]] of samples and the Laplace transform.
Some versions add zeros at $z = -1$ for the zeros of $H$ at infinity when $N > M$.
</document>

<document source="https://tamnd.github.io/math2/node/FZ6RYHNRWN.md" tag="FZ6RYHNRWN" kind="definition">
# Zariski tangent space

## Definition

Let $k$ be an algebraically closed field, let $X \subseteq k^n$ be an [[affine variety|JERVQT4C12]] with [[ideal|HH6VEVM6Z0]] $I(X)$, and let $p \in X$.
For $f \in k[x_1, \dots, x_n]$, its differential at $p$ is the linear form
$$d_p f(v) = \sum_{i=1}^{n} \frac{\partial f}{\partial x_i}(p) \, v_i, \qquad v \in k^n,$$
with the formal [[partial derivatives|09EGC6F8PT]].
The *Zariski tangent space* of $X$ at $p$ is the linear subspace
$$T_p X = \set{v \in k^n : d_p f(v) = 0 \text{ for all } f \in I(X)}.$$

## Remarks

If $I(X) = (f_1, \dots, f_r)$, then $T_p X$ is the kernel of the [[Jacobian matrix|047NDTEPCV]] $\left(\frac{\partial f_j}{\partial x_i}(p)\right)$, because $d_p(gf) = g(p) \, d_p f$ when $f(p) = 0$.

Intrinsic form: let $\mathfrak m_p$ be the [[maximal ideal|0PV74TR5FP]] of the [[coordinate ring|G9WB27VWP7]] $k[X]$ of functions vanishing at $p$.
Then $T_p X$ is isomorphic to the dual [[vector space|000AYPA9AJ]] of $\mathfrak m_p / \mathfrak m_p^2$.
The same space is the dual of the [[cotangent space|Z1CGB483J6]] of the [[local ring|MZ02BKPHJA]] $k[X]_{\mathfrak m_p}$, so $T_p X$ only depends on a neighbourhood of $p$ and is defined for every variety.

Example: for the parabola $X = V(y - x^2)$ at $p = (a, a^2)$, $d_p(y - x^2) = -2a \, v_1 + v_2$, so $T_p X$ is the line $v_2 = 2a v_1$, of dimension $1$.

Example: for the cusp $X = V(y^2 - x^3)$ at the origin, the polynomial $y^2 - x^3$ generates $I(X)$ and both of its partial derivatives vanish at $(0, 0)$, so $T_{(0,0)} X = k^2$ has dimension $2 > \dim X = 1$.

One always has $\dim T_p X \ge \dim X$.
</document>

<document source="https://tamnd.github.io/math2/node/FZ7237VY65.md" tag="FZ7237VY65" kind="concept">
# Baire property

## Summary

A subset of a topological space has the Baire property if it differs from an [[open set|02BH3MCBQB]] by a [[meagre set|0XGCZ4GDZ6]].
It is the topological analogue of Lebesgue measurability, with meagre sets in the role of null sets.
The sets with the Baire property form a sigma-algebra that contains all Borel sets, and in a Polish space it also contains all [[analytic sets|3RWWZYPAFV]].
Under the axiom of choice there are sets of reals without the Baire property.

## Explanations

### L1 human

Meagre sets are the "topologically small" sets, such as countable sets of reals or the Cantor set.
A set has the Baire property when, after ignoring a small set of errors, it looks like an open set.
For example, the closed interval $[0, 1]$ differs from the open interval $(0, 1)$ only by the two points $0$ and $1$, and a finite set is meagre.
So $[0, 1]$ has the Baire property.
Sets with this property behave well in arguments that use the [[Baire category theorem|0A7KGAYHGV]].
</document>

<document source="https://tamnd.github.io/math2/node/FZ798EZ6M0.md" tag="FZ798EZ6M0" kind="concept">
# Zassenhaus neighbourhood

## Summary

A Zassenhaus neighbourhood of a [[Lie group|7JBRGCVJRB]] $G$ is a neighbourhood $U$ of the identity with the following property.
The elements of any [[discrete subgroup|5M7D1BSHVZ]] that lie in $U$ generate a [[nilpotent group|0Q7MMQHC1P]].
A theorem of Zassenhaus says that every Lie group has such a neighbourhood.
It is the algebraic heart of the [[Margulis lemma|P6WSG7DT4Q]] and of the [[Kazhdan–Margulis theorem|C34RZMKKA4]].
The reason is that commutators of elements near the identity are much closer to the identity than the elements themselves.

## Explanations

### L1 human

In a Lie group, two elements close to the identity almost commute: their commutator is closer still to the identity.
In a discrete subgroup, elements cannot get arbitrarily close to the identity.
So taking commutators of small elements again and again must end at the identity after finitely many steps.
That is what being nilpotent means, and a neighbourhood where this argument works is a Zassenhaus neighbourhood.
In an abelian group like $\R^n$, every neighbourhood works, since all subgroups are abelian.
</document>

<document source="https://tamnd.github.io/math2/node/FZBA1HA9ZM.md" tag="FZBA1HA9ZM" kind="concept">
# Christoffel symbols

## Summary

The Christoffel symbols $\Gamma^k_{ij}$ are the components of an [[affine connection|JX65KSVHSJ]] in a coordinate chart.
They record how the covariant derivative of one coordinate vector field along another fails to vanish.
For the Levi-Civita connection of a Riemannian metric they are given by an explicit formula in the first derivatives of the metric.
They appear in the geodesic equation and in coordinate formulas for curvature.

## Explanations

### L1 human

In curvilinear coordinates, such as polar coordinates in the plane, the coordinate directions change from point to point.
Even a constant vector field then has components that vary.
The Christoffel symbols are the correction numbers that account for this turning of the coordinate axes.
In Cartesian coordinates on the flat plane they are all zero, while in polar coordinates some of them are not, for example $\Gamma^r_{\theta\theta} = -r$.
</document>

<document source="https://tamnd.github.io/math2/node/FZH5T5340Q.md" tag="FZH5T5340Q" kind="statement">
# Prime decomposition of 3-manifolds

## Statement

A closed connected $3$-manifold $P$ is *prime* if $P$ is not homeomorphic to $S^3$ and the following holds.
Whenever $P$ is homeomorphic to a [[connected sum|Y0K0K5NGWW]] $A \# B$ of closed $3$-manifolds, one of $A$ and $B$ is homeomorphic to $S^3$.

Let $M$ be a closed connected [[orientable|702SNFQG0W]] [[3-manifold|R20TB2KEQQ]] not homeomorphic to the [[3-sphere|T62369RFH6]] $S^3$.
Then:

1. $M$ is homeomorphic to a connected sum $P_1 \# \cdots \# P_k$ of finitely many prime $3$-manifolds.
2. The summands are unique.
   If $M$ is also homeomorphic to $Q_1 \# \cdots \# Q_l$ with each $Q_j$ prime, then $k = l$, and after reordering $P_i$ is homeomorphic to $Q_i$ for each $i$.

## Remarks

Existence is due to Kneser and uniqueness to Milnor.
For existence, one cuts $M$ along embedded $2$-spheres that do not bound balls and caps the pieces with balls; Kneser's finiteness argument shows that this process stops.
Uniqueness comes from an innermost-disk argument that makes two systems of splitting spheres disjoint.
A closed orientable prime $3$-manifold is either $S^2 \times S^1$ or irreducible, meaning that every embedded $2$-sphere bounds an embedded $3$-ball.
For example, $\mathbb{RP}^3 \# \mathbb{RP}^3$ has the two prime summands $\mathbb{RP}^3$, and no other decomposition.
Without orientability, uniqueness fails in this form: for a non-orientable $P$, the sums $P \# (S^2 \times S^1)$ and $P \# (S^2 \tilde{\times} S^1)$ are homeomorphic, where $S^2 \tilde{\times} S^1$ is the non-orientable $S^2$-bundle over the circle.

## Explanations

### L1 human

Every closed orientable $3$-manifold breaks up, in exactly one way, into basic pieces glued together by connected sums.
It is the analogue of factoring an integer into primes.
To understand all $3$-manifolds, it is enough to understand the prime ones.

### L2 human

The theorem reduces the classification of closed orientable $3$-manifolds to irreducible ones and $S^2 \times S^1$.
For irreducible manifolds, the next step is the JSJ decomposition along incompressible tori.
These two decompositions are the first stages of the geometrization theorem.
On the algebra side, the fundamental group of a connected sum is the free product of the fundamental groups of the summands.
</document>

<document source="https://tamnd.github.io/math2/node/FZK2M8KGXQ.md" tag="FZK2M8KGXQ" kind="concept">
# Trigonometric series

## Summary

A trigonometric series is an infinite series $\sum_{n \in \Z} c_n e^{inx}$, or $\frac{a_0}{2} + \sum_{n \ge 1} (a_n \cos nx + b_n \sin nx)$, with arbitrary coefficients.
Its partial sums are [[trigonometric polynomials|J3NJQ5FZJ5]].
Every [[Fourier series|G12KXDHPPS]] is a trigonometric series, but not every trigonometric series is the Fourier series of an integrable function.
The study of these series led Cantor to set theory, through the question of uniqueness of the coefficients.

## Explanations

### L1 human

A trigonometric series is an infinite sum of waves with whole-number frequencies, each with its own amplitude.
Nothing says in advance that the sum converges or that it comes from a given function.
For example, $\sum_{n \ge 1} \frac{\sin nx}{n^2}$ converges at every $x$, since its terms are bounded by $1/n^2$.
A Fourier series is the special case where the amplitudes are computed from a function by integrals.
</document>

<document source="https://tamnd.github.io/math2/node/FZM3CTTYVV.md" tag="FZM3CTTYVV" kind="example">
# Peridynamics

## Example

Peridynamics is a nonlocal model of continuum mechanics.
Let a body occupy a region $B \subseteq \R^3$ in its reference configuration, with mass density $\rho(x)$.
Let $u(x, t)$ be the displacement of the point $x$ at time $t$.
Fix a horizon $\delta > 0$ and let $H_x = \set{x' \in B : \abs{x' - x} \le \delta}$.
In the bond-based theory the equation of motion is
$$\rho(x) \, \frac{\partial^2 u}{\partial t^2}(x, t) = \int_{H_x} f\big(u(x', t) - u(x, t), \, x' - x\big) \, dx' + b(x, t).$$
Here $b$ is a given body force density.
The pairwise force function $f(\eta, \xi)$ gives the force per unit volume squared that $x'$ exerts on $x$.
Its arguments are the relative position $\xi = x' - x$ and the relative displacement $\eta$.
It satisfies $f(-\eta, -\xi) = -f(\eta, \xi)$, so that action equals reaction.
The equation contains a time derivative of $u$ and a space integral of $u$, but no space derivative: it is an [[integro-differential equation|YRKW2KX2CW]].

Check: in the prototype microelastic brittle model, $f(\eta, \xi) = c \, s \, \frac{\xi + \eta}{\abs{\xi + \eta}}$ for $\abs{\xi} \le \delta$, where $s = \frac{\abs{\xi + \eta} - \abs{\xi}}{\abs{\xi}}$ is the bond stretch and $c > 0$ is a constant.
A rigid translation $u(x, t) = a$ gives $\eta = 0$ and $s = 0$, so the integral vanishes, and with $b = 0$ the equation reduces to $\ddot u = 0$, which a constant $u$ satisfies.

## Remarks

Since no spatial derivatives of $u$ appear, the equation makes sense for displacements with jumps, so cracks can form and grow without a separate rule for fracture.
In the model above a bond breaks for good once its stretch exceeds a critical value.
When $\delta \to 0$, for smooth displacements and suitable force functions, the model approaches classical linear elasticity.

## Explanations

### L1 human

In classical elasticity a point of a solid feels only its immediate neighbours, through derivatives of the displacement.
In peridynamics each point is linked by springs, called bonds, to all points within a fixed distance.
The force on a point is then an integral over all its bonds, and cracks are simply regions where bonds have broken.
</document>

<document source="https://tamnd.github.io/math2/node/FZNC7ETZA8.md" tag="FZNC7ETZA8" kind="definition">
# Character of an abelian group

## Definition

Let $G$ be an [[abelian group|082GHNPJ53]], and let $\mathbb{T} = \set{z \in \C : \abs{z} = 1}$ be the [[circle group|04EAFVFTF7]].
A *character* of $G$ is a [[group homomorphism|0JYTG5K9Y4]] $\chi \colon G \to \mathbb{T}$, that is, a map with $\chi(x + y) = \chi(x)\chi(y)$ for all $x, y \in G$.
If $G$ is a topological group, a character of $G$ is usually required to be continuous as well.

## Remarks

Example: for $G = \Z/n\Z$ and an integer $j$, the map $\chi_j(k) = e^{2\pi i jk/n}$ is well defined, because $e^{2\pi i j(k + n)/n} = e^{2\pi i jk/n}$.
It is a character, since $\chi_j(k + l) = e^{2\pi i j(k+l)/n} = \chi_j(k)\chi_j(l)$.
Non-example: the map $\Z \to \C$, $k \mapsto 2^k$, satisfies the homomorphism rule, but its values do not have absolute value $1$, so it is not a character.
The characters of $G$ form an abelian group under pointwise multiplication $(\chi\psi)(x) = \chi(x)\psi(x)$, with identity the trivial character $x \mapsto 1$ and inverse $\overline{\chi}$.
This group is the character group, often written $\widehat{G}$.
For a finite group every homomorphism to $\C^\times$ takes values in $\mathbb{T}$, so some texts define characters as homomorphisms to $\C^\times$.
</document>

<document source="https://tamnd.github.io/math2/node/FZRMYTQFRR.md" tag="FZRMYTQFRR" kind="definition">
# Theory

## Definition

Let $L$ be a [[signature|WDVVEYD6AT]].
An $L$-theory is a set $T$ of [[sentences|DKFGQTRYZE]] of [[first-order logic|086QKPF0A6]] in the signature $L$.

## Remarks

Example: in the signature $\set{\cdot, e, {}^{-1}}$, the set of the three sentences $\forall x \, \forall y \, \forall z \, ((x \cdot y) \cdot z = x \cdot (y \cdot z))$, $\forall x \, (x \cdot e = x \land e \cdot x = x)$ and $\forall x \, (x \cdot x^{-1} = e \land x^{-1} \cdot x = e)$ is a theory, the theory of groups.
Each of its elements is a formula without free variables, so it is a set of sentences.

Non-example: the set $\set{x \cdot e = x}$ is not a theory, because the variable $x$ is free in its only formula.

Some authors require a theory to be closed under logical consequence; a set of sentences is then called a set of axioms for the theory it generates.
For an $L$-structure $\mathcal{M}$, the set $\mathrm{Th}(\mathcal{M})$ of all $L$-sentences true in $\mathcal{M}$ is the theory of $\mathcal{M}$.

## Explanations

### L1 human

A theory is any set of sentences; it may be finite or infinite, and it may even be contradictory.
The theory of groups has three sentences.
Peano arithmetic is a theory with infinitely many sentences, one induction axiom for each formula.
</document>

<document source="https://tamnd.github.io/math2/node/FZWAHMFZM3.md" tag="FZWAHMFZM3" kind="statement">
# Gauss's principle of least constraint

## Statement

Consider $N$ particles with masses $m_i > 0$ and applied forces $F_i$, for $i = 1, \dots, N$, subject to ideal constraints that are holonomic or linear in the velocities.
Fix a time $t$ and the positions and velocities of the particles at time $t$.
Call a family $(b_1, \dots, b_N)$ of vectors of $\R^3$ an *admissible acceleration* if it is compatible with the constraints at these positions and velocities.
Define the *constraint* (Gauss's function)
$$Z(b_1, \dots, b_N) = \sum_{i=1}^{N} m_i \,\Big\lvert b_i - \frac{F_i}{m_i} \Big\rvert^2.$$
Then the actual accelerations $a_i = \ddot r_i(t)$ are the unique minimiser of $Z$ among admissible accelerations.

## Remarks

For such constraints the admissible accelerations form an affine subspace $a + W$, where $W$ is the space of virtual displacements.
On it $Z$ is a strictly convex quadratic function, so it has a unique minimiser.
For $b = a + w$ with $w \in W$, one finds $Z(b) - Z(a) = 2\sum_i (m_i a_i - F_i) \cdot w_i + \sum_i m_i \abs{w_i}^2$.
The linear term vanishes by [[d'Alembert's principle|GD776GVZSR]], so $Z(b) > Z(a)$ when $w \neq 0$.
Example: a single free particle has all accelerations admissible, and the minimiser is $b = F/m$, which is Newton's law.

## Explanations

### L1 human

Without constraints, each particle would accelerate as $F_i/m_i$.
Constraints prevent this.
Gauss's principle says that the real motion deviates from the free motion as little as possible, where the deviation is measured by a mass-weighted sum of squares.

### L2 human

Gauss's principle is a minimum principle at each instant, not a stationarity principle over a whole path like [[Hamilton's principle|FAQ9CH2Z76]].
It is a quadratic program in the accelerations, and it handles nonholonomic constraints linear in the velocities with no extra work.
This makes it useful in the numerical simulation of constrained multibody systems.
It is equivalent to d'Alembert's principle for ideal constraints.
</document>

<document source="https://tamnd.github.io/math2/node/FZXRWKD430.md" tag="FZXRWKD430" kind="definition">
# Fredholm map

## Definition

Let $M$ and $N$ be [[Banach manifolds|VB95VVVDM8]] of class $C^k$ with $k \ge 1$, and let $f \colon M \to N$ be a map of class $C^1$.
The map $f$ is a *Fredholm map* if for every $x \in M$ the [[derivative|06EH97V2A7]] $d f_x \colon T_x M \to T_{f(x)} N$ is a [[Fredholm operator|Q5B5YGY7D6]].
This means that the kernel of $d f_x$ is finite-dimensional and its image is closed and of finite codimension.
The *index* of $f$ at $x$ is the [[Fredholm index|ADJZ99KY2Q]]
$$\operatorname{ind}_x f = \dim \ker d f_x - \dim \operatorname{coker} d f_x.$$

## Remarks

The Fredholm operators form an open set on which the index is locally constant, and $x \mapsto d f_x$ is continuous in local charts.
So the index is locally constant in $x$, and on a connected $M$ it is a single integer.


Example: every $C^1$ map $f \colon \R^m \to \R^n$ is Fredholm of index $m - n$, since $\dim \ker A - \dim \operatorname{coker} A = m - n$ for every linear map $A \colon \R^m \to \R^n$ by rank-nullity.

Non-example: the inclusion $\iota \colon \ell^2 \to \ell^2 \times \ell^2$, $x \mapsto (x, 0)$, is smooth and its derivative is injective with closed image, but the cokernel is isomorphic to $\ell^2$, which is infinite-dimensional, so $\iota$ is not Fredholm.

A map of the form $f = L + K$ with $L$ a linear Fredholm operator and $K$ a $C^1$ map whose derivatives are compact operators is Fredholm with the index of $L$.
</document>

<document source="https://tamnd.github.io/math2/node/FZYDPWMS46.md" tag="FZYDPWMS46" kind="definition">
# Integral equation

## Definition

Let $a < b$ and let $\varphi \colon [a, b] \to \C$ be an unknown [[function|0EPAT51DXQ]].
For each $x \in [a, b]$, let $I_x \subseteq [a, b]$ be an interval of integration, either the fixed interval $[a, b]$ or the interval $[a, x]$.
An *integral equation* for $\varphi$ is an equation of the form
$$F\left(x, \varphi(x), \int_{I_x} H\left(x, s, \varphi(s)\right) ds\right) = 0 \quad \text{for all } x \in [a, b],$$
where $F$ and $H$ are given functions.
A *solution* is a function $\varphi$ for which the [[integrals|0W29TBTY12]] exist and the equation holds for every $x \in [a, b]$.

## Remarks

Example: $\varphi(x) = x + \int_0^1 x s \, \varphi(s) \, ds$ on $[0, 1]$ is an integral equation with $I_x = [0, 1]$.
Try $\varphi(x) = c x$: the integral is $c x / 3$, so $c = 1 + c/3$ and $c = 3/2$.
So $\varphi(x) = 3x/2$ is a solution.

Non-example: $\varphi'(x) = \varphi(x)$ is a differential equation, not an integral equation, since $\varphi$ does not appear under an integral.
Its integral form $\varphi(x) = \varphi(0) + \int_0^x \varphi(s) \, ds$ is an integral equation.

The same definition is used with a region of $\R^m$ in place of $[a, b]$, and with vector-valued unknowns.
When $I_x = [a, b]$ the equation is of Fredholm type; when $I_x = [a, x]$ it is of Volterra type.
</document>

<document source="https://tamnd.github.io/math2/node/G003WR6SMX.md" tag="G003WR6SMX" kind="statement">
# Cellular approximation theorem

## Statement

Let $X$ and $Y$ be [[CW complexes|5W95QFHXPR]] with $n$-skeleta $X^n$ and $Y^n$.
A map $g \colon X \to Y$ is *cellular* if $g(X^n) \subseteq Y^n$ for all $n \ge 0$.
Let $f \colon X \to Y$ be a continuous map, and let $A \subseteq X$ be a subcomplex such that $f$ restricted to $A$ is cellular.
Then $f$ is [[homotopic|F2F143KQ2M]] [[relative to|ADDS6HE5SV]] $A$ to a cellular map.

## Remarks

The proof proceeds cell by cell, by induction on the dimension.
The key lemma concerns a map from a closed $k$-disk into a space obtained by attaching an $n$-cell with $n > k$.
Such a map can be deformed so that it misses a point of the open cell, and then pushed off the cell by radial projection.
Taking $A = \emptyset$ shows that every map is homotopic to a cellular map.

Example: give $S^n$ the CW structure with one $0$-cell and one $n$-cell, and let $k < n$.
Every map $S^k \to S^n$ is homotopic to a cellular map, which sends $S^k = (S^k)^k$ into the $k$-skeleton of $S^n$, a single point.
So $\pi_k(S^n) = 0$ for $k < n$.

The theorem also shows that the inclusion $X^n \to X$ induces an isomorphism on $\pi_k$ for $k < n$ and a surjection for $k = n$.

## Explanations

### L1 human

A continuous map can be very wild, for example a curve can fill a whole square.
The theorem says that between CW complexes one can always straighten a map, up to homotopy, so that it sends low-dimensional cells into low-dimensional parts of the target.
This is why a loop in a sphere of dimension $2$ or more can always be shrunk to a point.

### L2 human

Cellular approximation reduces many homotopy questions to combinatorics of cells.
It gives the vanishing of $\pi_k(S^n)$ for $k < n$.
It shows that the fundamental group of a CW complex depends only on its $2$-skeleton.
It also makes the maps induced on cellular chain complexes well defined.
Its simplicial ancestor is the simplicial approximation theorem.
</document>

<document source="https://tamnd.github.io/math2/node/G00C22AEDH.md" tag="G00C22AEDH" kind="concept">
# Simple module

## Summary

A simple module is a nonzero [[module|0NZ8JA25WK]] whose only [[submodules|05WDSN597C]] are zero and the module itself.
Simple modules are the building blocks of modules of finite length: the factors of a [[composition series|0XV2A49X6Q]] are simple.
Over a group algebra, simple modules are the same as [[irreducible representations|FX3NEFYFFN]].
Every simple left module over a ring $R$ is isomorphic to $R/\mathfrak{m}$ for a maximal left ideal $\mathfrak{m}$.

## Explanations

### L1 human

A simple module cannot be broken into a smaller nonzero piece inside it.
It plays the role that prime numbers play for integers, or that simple groups play for finite groups.
For the integers, the simple modules are the groups $\Z/p\Z$ with $p$ prime.
Such a group has $p$ elements, and a subgroup has an order dividing $p$, so it is either zero or everything.
</document>

<document source="https://tamnd.github.io/math2/node/G00EDC2ET4.md" tag="G00EDC2ET4" kind="definition">
# Nonstandard analysis

## Definition

Nonstandard analysis is the part of mathematics that develops analysis with the following tools.

1. A proper extension ${}^*\R$ of the [[real numbers|08J2M1JYAG]], such as the field of [[hyperreal numbers|421QMTFG9T]], that contains nonzero infinitesimals.
2. For each set, function and relation on $\R$, an extension of it to ${}^*\R$.
3. The [[transfer principle|3RT74FQYG1]], under which a first-order statement about $\R$ holds if and only if the corresponding statement about ${}^*\R$ holds.

Notions of analysis are then expressed with infinitesimals and the [[standard part function|1ZBY4W4E81]].

## Remarks

Example: in nonstandard analysis, a function $f \colon \R \to \R$ has [[derivative|0V1EJCT8G6]] $L$ at $x$ if $\operatorname{st}\bigl(({}^*f(x + \varepsilon) - f(x))/\varepsilon\bigr) = L$ for every nonzero infinitesimal $\varepsilon$, where ${}^*f$ is the extension of $f$ to ${}^*\R$.
For $f(x) = x^2$ the quotient is $2x + \varepsilon$, whose standard part is $2x$.

Non-example: smooth infinitesimal analysis also uses infinitesimals, but they are nilpotent and its logic is intuitionistic.
It does not use an extension of $\R$ with transfer, so it is not nonstandard analysis.

The theorems of nonstandard analysis about standard objects can also be proved by ordinary methods, but the nonstandard proofs are often shorter.
</document>

<document source="https://tamnd.github.io/math2/node/G02NBNN6CJ.md" tag="G02NBNN6CJ" kind="statement">
# Koenigs linearization theorem

## Statement

Let $U \subseteq \C$ be an open set containing $0$, and let $f \colon U \to \C$ be [[holomorphic|0NAHXR1FRH]] with $f(0) = 0$.
Let $\lambda = f'(0)$ and assume $\lambda \ne 0$ and $\abs{\lambda} \ne 1$.
Then there is a holomorphic map $\varphi$ on a neighbourhood $V$ of $0$ with $\varphi(0) = 0$ and $\varphi'(0) = 1$ such that
$$\varphi(f(z)) = \lambda \, \varphi(z) \quad \text{for all } z \in V \text{ with } f(z) \in V.$$
The map $\varphi$ is unique.

## Remarks

In the attracting case $0 < \abs{\lambda} < 1$, the map is given by $\varphi(z) = \lim_{n \to \infty} f^n(z) / \lambda^n$.
The limit converges uniformly near $0$, because $f^n(z) = \lambda^n z (1 + O(\abs{z}))$ with errors that shrink geometrically.
The repelling case $\abs{\lambda} > 1$ follows by applying the attracting case to the local inverse of $f$.
So near a [[fixed point|R6QVS6P21N]] that is attracting but not superattracting, or repelling, $f$ is locally [[conjugate|HNXXAPQKG3]] to its linear part $z \mapsto \lambda z$.
Example: $f(z) = z / (2 - z)$ has $f'(0) = 1/2$, and $\varphi(z) = z / (1 - z)$ satisfies $\varphi(f(z)) = \varphi(z) / 2$.
The hypotheses are needed.
For $\lambda = 0$, the map $z \mapsto z^2$ is not conjugate to $z \mapsto 0$, since it is not constant.
For $\lambda = 1$, the map $z \mapsto z + z^2$ is not conjugate to the identity, since it is not the identity.

## Explanations

### L1 human

Near a fixed point, a smooth map looks like multiplication by its derivative there.
The theorem says that for holomorphic maps this is exactly true after a change of coordinates, as long as the derivative has size neither $0$ nor $1$.
In the new coordinate, iterating the map just means multiplying by $\lambda$ again and again.

### L2 human

The function $\varphi$ is called the Koenigs function of $f$ at the fixed point.
In the attracting case it extends to the whole basin of attraction, where it still satisfies $\varphi \circ f = \lambda \varphi$.
The cases left out lead to other theorems: Böttcher's theorem for $\lambda = 0$, the Leau–Fatou flower theorem for roots of unity, and Siegel and Brjuno's small divisor results for other $\abs{\lambda} = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/G03V3D097M.md" tag="G03V3D097M" kind="definition">
# Inversion in a circle

## Definition

Let $O$ be a point of the [[Euclidean plane|0XW7V9ZZ05]] and let $r > 0$.
The *inversion in the [[circle|060W5CBZP6]]* with center $O$ and radius $r$ sends each point $P \ne O$ to the unique point $P'$ such that:

1. $P'$ lies on the ray from $O$ through $P$;
2. $\abs{OP} \cdot \abs{OP'} = r^2$.

## Remarks

In coordinates with $O$ at the origin, the inversion is $x \mapsto r^2 x / \norm{x}^2$.
In complex notation with center $0$, it is $z \mapsto r^2 / \bar{z}$.
Adding a point $\infty$ and setting $O \mapsto \infty$ and $\infty \mapsto O$ extends it to a bijection of the [[Riemann sphere|0J78KBPFZB]].
It is an [[involution|GP56NBN9RY]]: applying it twice gives the identity.
The same formula defines inversion in a sphere of $\R^n$.

Example: for the circle with center $(0, 0)$ and radius $1$, the point $P = (2, 0)$ goes to $P' = (1/2, 0)$.
Check: $P'$ lies on the ray from the origin through $P$, and $\abs{OP} \cdot \abs{OP'} = 2 \cdot \frac{1}{2} = 1$.

Non-example: the map $x \mapsto -r^2 x / \norm{x}^2$ satisfies clause 2 but sends $P$ to the opposite ray, so clause 1 fails.
</document>

<document source="https://tamnd.github.io/math2/node/G05X8BHVB9.md" tag="G05X8BHVB9" kind="definition">
# Ideal quotient

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $I$ and $J$ be [[ideals|0MYSA6MYNA]] of $R$.
The *ideal quotient* of $I$ by $J$ is the set
$$(I : J) = \set{r \in R : rj \in I \text{ for all } j \in J}.$$

## Remarks

$(I : J)$ is an ideal of $R$ and $I \subseteq (I : J)$.
Example: in $\Z$, $(12\Z : 4\Z) = 3\Z$; for $r \in \Z$, $4r \in 12\Z$ holds exactly when $3 \mid r$, and then $rj \in 12\Z$ for every multiple $j$ of $4$.
Non-example: $1 \notin (12\Z : 4\Z)$, since $1 \cdot 4 = 4 \notin 12\Z$.
If $J \subseteq I$, then $(I : J) = R$.
The quotient $(0 : J)$ is the [[annihilator|HB4XDKJTW8]] of $J$.
Other notations are $I : J$ and $(I :_R J)$; the name *colon ideal* is also used.
</document>

<document source="https://tamnd.github.io/math2/node/G0CS8C1FJW.md" tag="G0CS8C1FJW" kind="definition">
# Minimum-variance unbiased estimator

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a family of distributions of the data $X$, and let $g \colon \Theta \to \R$ be the quantity to estimate.
An [[estimator|0TTNRP5PFQ]] $T = T(X)$ is *unbiased* for $g$ if $E_\theta(T) = g(\theta)$ for every $\theta \in \Theta$, that is, its [[bias|01BD2TFFP4]] is zero.
Let $T$ be an unbiased estimator of $g$ with $\operatorname{Var}_\theta(T) < \infty$ for all $\theta$.
It is a *minimum-variance unbiased estimator*, or *uniformly minimum-variance unbiased estimator* (UMVUE), if
$$\operatorname{Var}_\theta(T) \le \operatorname{Var}_\theta(T')$$
for every unbiased estimator $T'$ of $g$ and every $\theta \in \Theta$.

## Remarks

Here $\operatorname{Var}_\theta$ is the [[variance|007QED3E0Y]] when $X \sim P_\theta$.

Example: let $X_1, \dots, X_n$ be i.i.d. [[Bernoulli|0J97P5KZY0]]($p$) with $\Theta = (0, 1)$.
The proportion $\bar{X}$ is unbiased with variance $p(1-p)/n$.
The Fisher information of the sample is $n/(p(1-p))$, so the Cramér–Rao bound for unbiased estimators of $p$ is $p(1-p)/n$.
The estimator $\bar{X}$ attains it at every $p$, so $\bar{X}$ is the UMVUE of $p$.

Non-example: for $n \ge 3$, $T' = (X_1 + X_2)/2$ is unbiased, but its variance $p(1-p)/2$ exceeds $p(1-p)/n$ for every $p$, so it fails the variance inequality.

The UMVUE need not exist, and an estimator that attains the Cramér–Rao bound need not exist even when the UMVUE does.
If $S$ is a complete sufficient statistic and $T$ is unbiased, then $E(T \mid S)$ is the UMVUE; this is the Lehmann–Scheffé theorem.
</document>

<document source="https://tamnd.github.io/math2/node/G0EDYEDG4S.md" tag="G0EDYEDG4S" kind="statement">
# Gauss–Seidel method

## Statement

Let $A$ be a real $n \times n$ matrix with nonzero diagonal entries, and let $b \in \R^n$.
Write $A = L_* + U$, where $L_*$ is the lower triangular part of $A$ with the diagonal, and $U$ is the strictly upper triangular part.
The *Gauss–Seidel method* for the [[linear system|0GT4WWGWH2]] $A x = b$ starts from any $x^{(0)} \in \R^n$ and sets
$$x^{(k+1)} = L_*^{-1} \left( b - U x^{(k)} \right), \quad \text{that is,} \quad x^{(k+1)}_i = \frac{1}{a_{ii}} \Big( b_i - \sum_{j < i} a_{ij} x^{(k+1)}_j - \sum_{j > i} a_{ij} x^{(k)}_j \Big).$$
Suppose that one of the following holds:

1. $A$ is strictly [[diagonally dominant|89P5GTKW87]] by rows;
2. $A$ is symmetric and [[positive definite|0KX66CZMBB]].

Then $A$ is invertible, and for every $b$ and every $x^{(0)}$ the iterates converge to $A^{-1} b$.

## Remarks

The error satisfies $e_{k+1} = G e_k$ with $G = -L_*^{-1} U$, and convergence for all starts means that the [[spectral radius|9V7YQA85PC]] of $G$ is less than $1$.
In case 1, an eigenvector argument bounds each eigenvalue $\lambda$ of $G$ by $\abs{\lambda} \le \max_i \frac{\sum_{j > i} \abs{a_{ij}}}{\abs{a_{ii}} - \sum_{j < i} \abs{a_{ij}}} < 1$.
In case 2, the energy $\norm{e}_A^2 = e^T A e$ strictly decreases along the iteration, by a computation with $L_*^T + L_* - A = D$, where $D$ is the positive diagonal of $A$.
Example: for $A$ with rows $(2, 1)$ and $(1, 2)$ and $b = (3, 3)$, starting at $0$ gives $(1.5, 0.75)$, then $(1.125, 0.9375)$, tending to $(1, 1)$.
The error is multiplied by $1/4$ at each step, the square of the [[Jacobi|B8FVHNXAN1]] factor $1/2$.
Without the hypotheses the method can diverge: for $A$ with rows $(1, 2)$ and $(2, 1)$, which is symmetric but indefinite, $G$ has spectral radius $4$.

## Explanations

### L1 human

Gauss–Seidel is the [[Jacobi method|B8FVHNXAN1]] with one change: each new value is used as soon as it is computed.
One sweeps through the equations in order, and each equation sees the freshest values of the unknowns before it.
This usually converges faster than Jacobi and needs only one copy of the vector in memory.

### L2 human

In matrix form Gauss–Seidel is the [[iterative method|2RPE77ACER]] of the splitting $A = M - N$ with $M = L_*$, so each step is a forward substitution.
For symmetric positive definite $A$ it is a coordinate-wise minimization of the energy $\frac12 x^T A x - b^T x$.
Its sweeps depend on the order of the unknowns, which makes it harder to parallelize than Jacobi.
Successive over-relaxation adds a weight to each update and can converge much faster.
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# Least harmonic majorant

## Summary

The least harmonic majorant of a function is the smallest among its [[harmonic majorants|ZDHPQWWCQW]].
Let $u$ be a [[subharmonic function|Z1MHMGFNFW]] on a domain, with at least one harmonic majorant.
Then $u$ has a least harmonic majorant, which can be built as a limit of Poisson integrals over an exhausting sequence of subdomains.
Take a subharmonic function on a disc that is continuous up to the boundary.
Its least harmonic majorant is the Poisson integral of its boundary values.
It is used in the theory of Hardy spaces and, through its mirror image the greatest harmonic minorant, in the [[Riesz decomposition theorem|CBZ20SS8RQ]].

## Explanations

### L1 human

Among all balanced sheets lying above a sagging sheet, take the lowest one.
It touches the sagging sheet at the edge whenever possible and is pulled as tight as it can be.
For $\abs{z}^2$ on the unit disc, the lowest harmonic function above it is the constant $1$, which is the value of $\abs{z}^2$ on the boundary circle.
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# Wirtinger's inequality for functions

## Statement

Let $f \colon \R \to \R$ be a [[continuously differentiable function|06SKCHDQT5]] that is [[periodic|089KWVYD69]] with period $2\pi$ and satisfies $\int_0^{2\pi} f(x) \, dx = 0$.
Then
$$\int_0^{2\pi} f(x)^2 \, dx \le \int_0^{2\pi} f'(x)^2 \, dx,$$
with equality if and only if $f(x) = \alpha \cos x + \beta \sin x$ for some real numbers $\alpha, \beta$.

## Remarks

Proof idea: expand $f$ in its Fourier series $\sum_k c_k e^{i k x}$.
The mean zero condition gives $c_0 = 0$, and $f'$ has coefficients $i k c_k$.
By [[Parseval's identity|047RFCNVSK]] the two sides are $2\pi \sum_{k \ne 0} \abs{c_k}^2$ and $2\pi \sum_{k \ne 0} k^2 \abs{c_k}^2$, and $k^2 \ge 1$ with equality only for $k = \pm 1$.

The mean zero condition is needed: the constant function $f = 1$ gives $2\pi \le 0$, which is false.

A related form holds on an interval with zero boundary values: if $f \colon [0, \pi] \to \R$ is continuously differentiable with $f(0) = f(\pi) = 0$, then $\int_0^\pi f^2 \le \int_0^\pi f'^2$.

## Explanations

### L1 human

A periodic function with average zero cannot be large without also changing fast.
Wirtinger's inequality makes this precise: the total square of the function is at most the total square of its slope.
The extreme case is a single sine wave of the longest possible period.

### L2 human

This is the simplest Poincaré-type inequality, and the constant $1$ is the smallest nonzero eigenvalue of $-d^2/dx^2$ on the circle of length $2\pi$.
It gives a short proof of the isoperimetric inequality in the plane, by applying it to the coordinates of a closed curve parametrized by arc length.
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# Frattini subgroup

## Definition

Let $G$ be a [[group|G5S1M9]].
A *maximal subgroup* of $G$ is a [[subgroup|0X1JXT47GA]] $M \ne G$ such that no subgroup $H$ satisfies $M < H < G$.
The *Frattini subgroup* $\Phi(G)$ is the [[intersection|0839XNQXDK]] of all maximal subgroups of $G$, with $\Phi(G) = G$ when $G$ has no maximal subgroup.

## Remarks

An element $g \in G$ is a *non-generator* if, for every subset $X \subseteq G$ with $\langle X, g \rangle = G$, already $\langle X \rangle = G$.
Then $\Phi(G)$ is the set of non-generators of $G$.
$\Phi(G)$ is a [[characteristic subgroup|0KABPMRHM2]] of $G$, and if $G$ is finite it is [[nilpotent|0Q7MMQHC1P]].
For a finite [[$p$-group|06QYRTX9KQ]] $P$, $\Phi(P)$ is generated by the $p$-th powers and the commutators, and $P/\Phi(P)$ is an elementary abelian group.

Example: $G = \Z/8\Z$.
Its subgroups form a chain $\set{0} < \set{0, 4} < \set{0, 2, 4, 6} < G$, so the only maximal subgroup is $\set{0, 2, 4, 6}$, and this is $\Phi(G)$.

Example: $G = S_3$.
The maximal subgroups are $A_3$ and the three subgroups of order $2$; their intersection is trivial, so $\Phi(S_3) = \set{e}$.

Example: the group $\Q$ of [[rational numbers|0NCJ5WXN6Z]] under addition has no maximal subgroup, so $\Phi(\Q) = \Q$.

Non-example: in $\Z/8\Z$, the subgroup $\set{0, 4}$ is not the Frattini subgroup, since it is smaller than the intersection of the maximal subgroups.

Do not confuse $\Phi(G)$ with the [[Frattini argument|0QTZDKJEDV]], which is a different result named after the same mathematician.
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# Subobject

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $X$ be an object of $\mathcal{C}$.
For [[monomorphisms|Y2NP1JFR86]] $m \colon A \to X$ and $m' \colon A' \to X$, write $m \le m'$ if there is a [[morphism|859F0TFVA3]] $k \colon A \to A'$ with $m = m' \circ k$, and write $m \equiv m'$ if $m \le m'$ and $m' \le m$.
A *subobject* of $X$ is an equivalence class of monomorphisms into $X$ for the [[equivalence relation|0G8VEZ3WTK]] $\equiv$.

## Remarks

Since $m'$ is a monomorphism, the factor $k$ in $m = m' \circ k$ is unique when it exists.
If $m \equiv m'$, the two factors are inverse isomorphisms, so $m \equiv m'$ holds exactly when $m = m' \circ k$ for an isomorphism $k$.
The relation $\le$ induces a partial order on the subobjects of $X$.

Example: in the category of sets, the monomorphisms are the injective functions.
An injective $m \colon A \to X$ is equivalent to the inclusion of its image $m(A) \subseteq X$, since $m$ is a bijection onto $m(A)$.
So the subobjects of $X$ correspond to the [[subsets|0BJR9QR08E]] of $X$.

Non-example: the map $\{a, b\} \to \{1\}$ that sends both points to $1$ does not represent a subobject of $\{1\}$, since it is not injective and hence not a monomorphism of sets.
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<document source="https://tamnd.github.io/math2/node/G0V86SJBQW.md" tag="G0V86SJBQW" kind="concept">
# Integral closure

## Summary

The integral closure of a ring $A$ in a larger commutative ring $B$ is the set of elements of $B$ that are [[integral|1EHN4KPSFS]] over $A$.
It is a subring of $B$ that contains $A$, and it is the largest subring of $B$ that is [[integral over|B7X0KFY1S1]] $A$.
For an integral domain $A$, its integral closure in its own field of fractions is called its normalization.
The ring of integers of a number field is the integral closure of $\Z$ in that field.

## Explanations

### L1 human

Take all elements of the larger ring that satisfy a monic equation with coefficients in $A$; together they form a ring.
Inside $\Q(\sqrt{5})$, the integral closure of $\Z$ contains $\sqrt{5}$ but also the golden ratio $(1 + \sqrt{5})/2$, a root of $x^2 - x - 1$.
So the "integers" of $\Q(\sqrt{5})$ form the ring $\Z[(1 + \sqrt{5})/2]$, which is larger than $\Z[\sqrt{5}]$.
Passing to the integral closure repairs rings that miss some of their natural elements.
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# Strip of convergence of the bilateral transform

## Definition

Let $f \colon \R \to \C$ be [[integrable|DWJF7ABPR1]] on each bounded interval.
Let
$$\alpha = \inf \left\{ \sigma \in \R : \int_0^\infty \abs{f(t)} e^{-\sigma t} \, dt < \infty \right\}, \qquad \beta = \sup \left\{ \sigma \in \R : \int_{-\infty}^0 \abs{f(t)} e^{-\sigma t} \, dt < \infty \right\},$$
with $\inf \emptyset = +\infty$ and $\sup \emptyset = -\infty$.
If $\alpha < \beta$, the *strip of convergence* of the [[two-sided Laplace transform|V94VYJ2RR1]] of $f$ is the open vertical strip
$$S = \{ s \in \C : \alpha < \operatorname{Re} s < \beta \}.$$
If $\alpha \ge \beta$, the strip of convergence is empty.

## Remarks

The integral defining $\mathcal{B}\{f\}(s)$ converges absolutely for every $s \in S$, and $\mathcal{B}\{f\}$ is [[holomorphic|0DG0Z72CQD]] on $S$.
The integral may also converge on one or both boundary lines; these points are not counted in the open strip.
Example: for $f(t) = e^{-\abs{t}}$, one finds $\alpha = -1$ and $\beta = 1$, so $S = \{-1 < \operatorname{Re} s < 1\}$.
Non-example: for $f = 1$, $\alpha = 0$ and $\beta = 0$, so the strip is empty, and indeed the two-sided integral converges nowhere.
The strip is part of the data of a transform.
The function $1/(1 - s)$ on $\operatorname{Re} s < 1$ inverts to $e^t$ on $t < 0$ and $0$ on $t > 0$, while on $\operatorname{Re} s > 1$ it inverts to $-e^t$ on $t > 0$ and $0$ on $t < 0$.
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# Nevanlinna theory in several variables

## Definition

Let $n \ge 1$, let $\mathbb{P}^n$ be complex [[projective space|ZT9A6JK8QF]], and let $f \colon \C \to \mathbb{P}^n$ be a holomorphic curve.
A *reduced representation* of $f$ is a map $F = (f_0, \ldots, f_n) \colon \C \to \C^{n+1} \setminus \set{0}$ with [[entire|0G4F0XB9X2]] components and $f = [f_0 : \cdots : f_n]$.
It exists, and it is unique up to multiplication by a nowhere-zero entire function.
Let $\norm{\cdot}$ be the Euclidean norm on $\C^{n+1}$ and let $r > 0$.
Nevanlinna theory in several variables studies $f$ through the following functions of $r$.

1. The *characteristic function* $T_f(r) = \dfrac{1}{2\pi} \displaystyle\int_0^{2\pi} \log \norm{F(re^{i\theta})} \, d\theta - \log \norm{F(0)}$.
2. For a hyperplane $H = \set{a_0 x_0 + \cdots + a_n x_n = 0}$ with $a = (a_0, \ldots, a_n) \ne 0$ and $f(\C) \not\subset H$, write $\ip{F}{a} = a_0 f_0 + \cdots + a_n f_n$. The *counting function* $N_f(r, H) = \displaystyle\int_0^r \frac{n(t) - n(0)}{t} \, dt + n(0) \log r$, where $n(t)$ is the number of zeros of $\ip{F}{a}$ in $\set{\abs{z} \le t}$, counted with multiplicity.
3. The *proximity function* $m_f(r, H) = \dfrac{1}{2\pi} \displaystyle\int_0^{2\pi} \log \frac{\norm{F(re^{i\theta})} \, \norm{a}}{\abs{\ip{F(re^{i\theta})}{a}}} \, d\theta$.

The same notions exist for holomorphic maps from $\C^m$ to $\mathbb{P}^n$ or to projective varieties.
They use averages over spheres in $\C^m$, and divisors in place of hyperplanes.

## Remarks

The function $T_f$ does not depend on the reduced representation.
If $F$ is replaced by $gF$ with $g$ entire and nowhere zero, then $\log\abs{g}$ is harmonic, so its mean over the circle of radius $r$ is $\log\abs{g(0)}$.
Jensen's formula applied to $\ip{F}{a}$ gives the *first main theorem* $T_f(r) = m_f(r, H) + N_f(r, H) + O(1)$ as $r \to \infty$.

Example: let $g$ be a nonconstant entire function and $f = [1 : g] \colon \C \to \mathbb{P}^1$, with $F = (1, g)$.
Then $T_f(r) = \frac{1}{2\pi}\int_0^{2\pi} \log\sqrt{1 + \abs{g(re^{i\theta})}^2}\, d\theta + O(1)$.
Since $\log^+ \abs{w} \le \log\sqrt{1 + \abs{w}^2} \le \log^+\abs{w} + \frac12 \log 2$, this differs from the one-variable [[Nevanlinna characteristic|YT6Z0GWS0W]] $T(r, g)$ by a bounded amount.

Non-example: if $f(\C) \subset H$, then $\ip{F}{a} \equiv 0$ and the counting and proximity functions of $H$ are not defined.

Hyperplanes $H_1, \ldots, H_q$ are in *general position* if any $n + 1$ of their coefficient vectors are [[linearly independent|0CRMGPAH83]].
Cartan's second main theorem concerns a linearly nondegenerate $f$, that is, $f(\C)$ lies in no hyperplane, and hyperplanes $H_1, \ldots, H_q$ in general position.
It states that $(q - n - 1)\, T_f(r) \le \sum_{j=1}^q N_f(r, H_j) + O(\log T_f(r) + \log r)$.
The inequality holds for all $r$ outside a set of finite Lebesgue measure.
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# Dilogarithm

## Summary

The dilogarithm is $\operatorname{Li}_2(z) = \sum_{k \ge 1} \frac{z^k}{k^2}$, the [[polylogarithm|96CHDZKGQ0]] of order $2$.
It equals $-\int_0^z \frac{\ln(1 - u)}{u}\, du$ and continues analytically to $\C \setminus [1, \infty)$.
It satisfies remarkable functional equations, such as Euler's reflection formula and the [[five-term relation|6DHQTFAJ03]].
It appears in hyperbolic geometry, through the [[Bloch–Wigner function|Q8K5MWEXB9]], and in algebraic K-theory.

## Explanations

### L1 human

The dilogarithm is the next function after the logarithm in a natural chain: one divides by $u$ and integrates again.
It is not elementary, but it has many exact values.
For example, $\operatorname{Li}_2(1) = \frac{\pi^2}{6}$ and $\operatorname{Li}_2(-1) = -\frac{\pi^2}{12}$.
Its identities, which relate values at points such as $z$, $1 - z$ and $1/z$, have made it a favourite of number theorists and physicists.
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<document source="https://tamnd.github.io/math2/node/G0XVRV6QT5.md" tag="G0XVRV6QT5" kind="concept">
# Barycentric coordinates

## Summary

Barycentric coordinates describe a point of an [[affine space|09XNAPTJQ7]] by weights attached to the vertices of a fixed reference simplex.
The weights add up to $1$, and the point is the [[affine combination|0KJH6SRFZK]] of the vertices with these weights.
They need no origin and no basis, so they are the natural coordinates of affine geometry.
They are widely used for triangles, in computer graphics, and in finite element methods.

## Explanations

### L1 human

Put masses on the three corners of a triangle; their balance point is a point of the plane.
The masses, scaled so that they add up to $1$, are the barycentric coordinates of that point.
The centroid gets the equal weights $(1/3, 1/3, 1/3)$, and each corner gets weight $1$ on itself and $0$ on the others.
A point lies inside the triangle exactly when all three weights are positive.
Negative weights are allowed and give the points outside the triangle.
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# Butterfly effect

## Summary

The butterfly effect is the popular name for sensitive dependence on initial conditions in a dynamical system.
Arbitrarily small changes in the initial state can lead to large differences in the later states.
The name comes from Edward Lorenz, whose computer model of convection, now the [[Lorenz system|WYRC41CQN9]], showed this behaviour.
It is a central ingredient of [[chaos theory|DBKXCGDKTD]] and sets practical limits on long-range prediction, even for fully deterministic systems.

## Explanations

### L1 human

The image is that the flap of a butterfly's wings could, weeks later, change whether a storm forms far away.
The point is not that butterflies cause storms, but that tiny causes can have effects that grow out of all proportion.
A simple model is the doubling map $x \mapsto 2x \bmod 1$ on $[0, 1)$.
Two starting values that differ by $10^{-6}$ differ by about $2^n \cdot 10^{-6}$ after $n$ steps, as long as this is small.
After about $20$ steps the difference is of the same size as the whole interval.
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# Min-max theorem

## Statement

Let $H$ be an infinite-dimensional real or complex [[Hilbert space|HM2HHE3J51]].
Let $A$ be a [[self-adjoint operator|0NEGN5CRSC]] in $H$ that is bounded below and has compact resolvent.
Let $\lambda_1 \le \lambda_2 \le \lambda_3 \le \cdots$ be its [[eigenvalues|0BQAA2BK3Q]], listed with multiplicity.
Let $Q$ be the closed quadratic form of $A$, with form domain $D(Q)$, so that $Q(u) = \ip{Au}{u}$ for $u$ in the domain of $A$.
For $u \in D(Q)$ with $u \ne 0$, let $R(u) = Q(u) / \norm{u}^2$ be the [[Rayleigh quotient|06WHP8SE98]].
Then, for every $k \ge 1$,
$$\lambda_k = \min_{\substack{V \subseteq D(Q) \\ \dim V = k}} \; \max_{u \in V \setminus \set{0}} R(u) = \max_{\substack{W \subseteq H \\ \dim W \le k - 1}} \; \inf_{\substack{u \in D(Q) \setminus \set{0} \\ u \perp W}} R(u),$$
where $V$ and $W$ run over [[linear subspaces|00YC1D652K]] of the given dimensions.

## Remarks

Proof idea: let $(e_j)$ be an orthonormal basis of eigenvectors with $A e_j = \lambda_j e_j$.
On the span of $e_1, \dots, e_k$ the quotient $R$ is at most $\lambda_k$.
Every $k$-dimensional subspace $V$ contains a nonzero vector orthogonal to $e_1, \dots, e_{k-1}$, and there $R \ge \lambda_k$ by expanding in the basis.
The max-min form is proved in the same way, with the orthogonal complement of $W$.

Example: for a bounded open set $\Omega \subseteq \R^n$ and $A = -\Delta$ with Dirichlet condition, $D(Q) = H^1_0(\Omega)$ and $R(u) = \int_\Omega \abs{\nabla u}^2 / \int_\Omega u^2$.
So the [[Dirichlet eigenvalues|4VD8XXQ4P3]] are $\lambda_k(\Omega) = \min_{\dim V = k} \max_{u \in V \setminus \set{0}} R(u)$.
Since $H^1_0(\Omega_1) \subseteq H^1_0(\Omega_2)$ when $\Omega_1 \subseteq \Omega_2$, this gives domain monotonicity: $\lambda_k(\Omega_2) \le \lambda_k(\Omega_1)$.

The compact resolvent is used to have a basis of eigenvectors.
Without it the formulas still hold for the eigenvalues below the bottom of the essential spectrum, and otherwise the min-max values equal the bottom of the essential spectrum.
In finite dimension the result is the [[Courant–Fischer min-max theorem|CJK2HS2508]] for Hermitian matrices.

## Explanations

### L1 human

The eigenvalues of a symmetric operator can be found by optimisation alone.
The lowest one is the smallest value of the energy ratio $R$.
The $k$-th one is the smallest value one can reach for the worst vector of a $k$-dimensional family of trial functions.

### L2 human

The theorem turns questions on eigenvalues into questions on trial functions.
Any $k$-dimensional subspace of trial functions gives the upper bound $\lambda_k \le \max_V R$, which is the basis of the Rayleigh–Ritz and finite element approximations of eigenvalues.
It gives domain monotonicity of Dirichlet eigenvalues, the comparison of Dirichlet and Neumann eigenvalues, and, with Weyl's counting argument, the asymptotic law of eigenvalues.
For the Laplacian, the first eigenvalue is the minimum of the Rayleigh quotient, which is the starting point of the [[Faber–Krahn inequality|E7TY2TPXX7]].
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# Fourier series

## Summary

The Fourier series of a periodic function writes it as a sum of sines and cosines, or of complex exponentials $e^{inx}$.
Its coefficients are computed by integrals.
It turns problems about periodic functions into problems about sequences of coefficients.
Whether and in which sense the series converges back to the function is a central question of harmonic analysis.
[[Parseval's identity|047RFCNVSK]] and [[Fejér's theorem|TPJZCSXSVN]] give two answers.

## Explanations

### L1 human

A musical sound can be split into a fundamental tone and its overtones.
The Fourier series does the same for any periodic function: it is a sum of pure waves whose frequencies are whole multiples of a base frequency.
Each coefficient says how much of one wave is present.
For example, the sawtooth function equal to $x$ on $(-\pi, \pi)$ is $2(\sin x - \frac{1}{2}\sin 2x + \frac{1}{3} \sin 3x - \cdots)$ at every point strictly between $-\pi$ and $\pi$.
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# Smale's problems on dynamics

## Problem

In 1998 Smale published a list of eighteen problems for the twenty-first century.
The following problems of the list concern dynamical systems.

1. Problem 10, the $C^r$ closing lemma. Let $p$ be a non-wandering point of a $C^r$ diffeomorphism $f$ of a compact manifold, with $r \ge 2$. Can $f$ be $C^r$-approximated by diffeomorphisms $g$ for which $p$ is a periodic point?
2. Problem 11: is one-dimensional dynamics generally hyperbolic? In particular, can a complex polynomial be approximated by polynomials of the same degree all of whose critical points tend to attracting periodic orbits?
3. Problem 12, centralisers of diffeomorphisms: can every $C^r$ diffeomorphism of a compact manifold be $C^r$-approximated by one that commutes only with its own powers?
4. Problem 13: Hilbert's sixteenth problem, on the number and position of the limit cycles of polynomial vector fields in the plane.
5. Problem 14: do the Lorenz equations have a strange attractor of the type of the geometric Lorenz model?

## Remarks

Problem 10 is the higher-smoothness version of Pugh's [[closing lemma|E1Z444F6TV]], which settles the case $r = 1$; it remains open in general.
For Problem 11, density of hyperbolicity among real polynomials was proved by Kozlovski, Shen and van Strien.
The real quadratic case had been settled before by Lyubich and by Graczyk and Świątek.
The complex case is the density of hyperbolicity conjecture and is open.
Problem 12 was solved in the $C^1$ topology by Bonatti, Crovisier and Wilkinson.
Problem 13 is open, even for quadratic vector fields.
Problem 14 was solved positively by Tucker with a computer-assisted proof; see [[Smale's fourteenth problem|VDVQD2HFSD]].

## Explanations

### L1 human

Smale chose a few questions that he thought would shape mathematics in the new century, following Hilbert's example of 1900.
Several of them ask how typical systems behave over long times.
Can recurrence be turned into periodicity by tiny smooth changes?
Is chaos rare or common in one-dimensional maps?
Does the famous Lorenz system really have the butterfly-shaped attractor seen on computers?
Some of these questions have been answered, and others are still open.
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# Cramér–von Mises criterion

## Summary

The Cramér–von Mises criterion measures how far an [[empirical distribution function|A05WR6TEK5]] is from a hypothesised [[cumulative distribution function|0H8H7ZTHW2]] by an integrated squared difference.
It gives a goodness-of-fit [[hypothesis test|087P03NEND]] that, like the [[Kolmogorov–Smirnov test|YZYHWC6C5F]], has the same null law for every continuous hypothesised distribution.
Where the Kolmogorov–Smirnov statistic looks at the single largest gap, this criterion adds up the squared gaps over the whole line.
A two-sample version compares two empirical distribution functions.

## Explanations

### L1 human

Picture the staircase of the data and the curve of the expected distribution on one graph.
Instead of looking for the single widest gap, square the gap at every point and average these squares along the curve.
A large average means that the data and the curve disagree over a wide range, not only at one place.
For one observation at $0.5$ tested against the uniform law on $[0, 1]$, the statistic is $1/12$.
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# Spencer cohomology

## Definition

Let $V$ and $W$ be finite-dimensional real [[vector spaces|05EGHEQZAW]], with $\dim V = n$.
Let $S^k = S^k V^* \otimes W$ be the space of $W$-valued [[homogeneous polynomials|2C3PAQ7H50]] of degree $k$ on $V$, with $S^k = 0$ for $k < 0$, and let $\Lambda^j = \Lambda^j V^*$ be the [[exterior power|0MDMZ965VE]].
Choose a basis $e_1, \dots, e_n$ of $V$ with dual basis $e^1, \dots, e^n$, and let $\partial_i$ be the derivative of polynomials in the direction $e_i$.
The *Spencer operator* is the linear map
$$\delta \colon S^k \otimes \Lambda^j \to S^{k-1} \otimes \Lambda^{j+1}, \qquad \delta(P \otimes \omega) = \sum_{i=1}^n \partial_i P \otimes (e^i \wedge \omega).$$
It does not depend on the basis, and $\delta \circ \delta = 0$.

Let $q \ge 1$ and let $g_q \subseteq S^q$ be a linear subspace, the symbol of a system of order $q$.
Put $g_k = S^k$ for $k < q$, and for $k \ge q$ define the prolongations inductively by
$$g_{k+1} = \set{ P \in S^{k+1} : \partial_v P \in g_k \text{ for all } v \in V }.$$
Then $\delta(g_k \otimes \Lambda^j) \subseteq g_{k-1} \otimes \Lambda^{j+1}$, so the spaces $g_k \otimes \Lambda^j$ form [[chain complexes|WQ82C0M3QB]], the *Spencer complexes* of $g_q$.
The *Spencer cohomology* of $g_q$ is
$$H^{k,j}(g_q) = \frac{\ker\big(\delta \colon g_k \otimes \Lambda^j \to g_{k-1} \otimes \Lambda^{j+1}\big)}{\delta(g_{k+1} \otimes \Lambda^{j-1})}.$$

## Remarks

$\delta \circ \delta = 0$ because partial derivatives commute while $e^i \wedge e^l = -e^l \wedge e^i$.
Indexing conventions differ between authors, for instance by shifting $k$.

Example: take $g_q = S^q$, the symbol of the empty system.
Then $g_k = S^k$ for all $k$, and for each $m \ge 1$ the sequence $0 \to S^m \to S^{m-1} \otimes \Lambda^1 \to \dots \to S^0 \otimes \Lambda^m \to 0$ is exact.
It is the degree $m$ part of the de Rham complex of polynomial forms, and it is dual to the [[Koszul complex|YZY11W08SD]].
So $H^{k,j}(S^q) = 0$ whenever $k + j \ge 1$.

Example: for the Laplace equation on $\R^2$, $q = 2$, $W = \R$ and $g_2$ is the space of quadratic forms $a x^2 + b xy + c y^2$ with $a + c = 0$.
Its prolongation $g_k$ is the space of harmonic homogeneous polynomials of degree $k$, which is $2$-dimensional for each $k \ge 1$.

Consider a system of [[partial differential equations|GC5BDZ6QBC]] whose symbol has nonzero Spencer cohomology in the relevant degrees.
Such a system may have hidden integrability conditions, and its power series solutions cannot always be built degree by degree.
</document>

<document source="https://tamnd.github.io/math2/node/G1M4BGC7WK.md" tag="G1M4BGC7WK" kind="definition">
# Method of steps

## Definition

Let $\tau > 0$ and let $f \colon \R \times \R^n \times \R^n \to \R^n$ be continuous.
Consider the [[delay differential equation|39PCDRDB01]] $x'(t) = f(t, x(t), x(t - \tau))$ for $t \ge 0$, with a continuous history $x(\theta) = \phi(\theta)$ for $\theta \in [-\tau, 0]$.
The *method of steps* constructs the solution on the intervals $I_j = [j\tau, (j+1)\tau]$, $j = 0, 1, 2, \ldots$, in order.

1. On $I_0$, $x(t - \tau) = \phi(t - \tau)$ is known, so $x$ solves the [[initial value problem|0FXDW992DS]] $x'(t) = f(t, x(t), \phi(t - \tau))$, $x(0) = \phi(0)$.
2. Suppose $x$ is known on $[-\tau, j\tau]$.
   On $I_j$ the term $x(t - \tau)$ is then known, and $x$ solves $x'(t) = f(t, x(t), x(t - \tau))$ with the value $x(j\tau)$ from the previous step.

Let $f$ be locally Lipschitz in its second argument.
Then each step has a unique local solution, so the solution is unique as long as it exists.

## Remarks

Example: $x'(t) = -x(t - 1)$ with $\phi \equiv 1$.
On $[0, 1]$: $x' = -1$ and $x(0) = 1$, so $x(t) = 1 - t$.
On $[1, 2]$: $x'(t) = -(1 - (t - 1)) = t - 2$ and $x(1) = 0$, so $x(t) = (t - 1)(t - 3)/2$.
Check: the derivative of $(t^2 - 4t + 3)/2$ is $t - 2$, and its value at $t = 1$ is $0$.

Non-example: the method does not apply directly to $x'(t) = \int_{t - 1}^{t} x(s) \, ds$.
There the right side at time $t$ involves $x$ on all of $[t - 1, t]$, including times close to $t$ that are not yet known.
So no step reduces it to an ordinary equation with known data.

The solution is in general less smooth at the points $j\tau$; for the example, $x'$ jumps at $t = 0$ from $0$ to $-1$, and smoothness increases with each step.

## Explanations

### L1 human

Solve one delay-length at a time, using the part already found as the past for the next part.
</document>

<document source="https://tamnd.github.io/math2/node/G1P8EW3Q3E.md" tag="G1P8EW3Q3E" kind="definition">
# Orthomodular lattice

## Definition

Let $L$ be an [[orthocomplemented lattice|J7KN14DYV3]] with orthocomplementation $a \mapsto a^\perp$.
The lattice $L$ is *orthomodular* if for all $a, b \in L$,
$$a \le b \implies b = a \vee (a^\perp \wedge b).$$

## Remarks

An equivalent form is: if $a \le b$ and $a^\perp \wedge b = 0$, then $a = b$.

Example: the lattice of closed subspaces of a [[Hilbert space|HM2HHE3J51]] $H$.
If $M \subseteq N$ are closed subspaces, every $x \in N$ is $x = Px + (x - Px)$, where $P$ is the orthogonal projection onto $M$; then $x - Px \in M^\perp \cap N$.
So $N$ is the closed span of $M$ and $M^\perp \cap N$.
When $H$ is infinite-dimensional, this lattice is orthomodular but not modular.

Non-example: the hexagon ortholattice $O_6$ with elements $0, a, b, b^\perp, a^\perp, 1$ and order generated by $0 < a < b < 1$ and $0 < b^\perp < a^\perp < 1$.
The map exchanging $a$ with $a^\perp$, $b$ with $b^\perp$ and $0$ with $1$ is an orthocomplementation.
Here $a \le b$, but $a^\perp \wedge b = 0$, so $a \vee (a^\perp \wedge b) = a \ne b$.

Every [[Boolean algebra|0XADJKM699]] is orthomodular, and every modular ortholattice is orthomodular.
</document>

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# James's theorem

## Statement

Let $X$ be a [[Banach space|06GZ956657]] over $\R$ or $\C$, and let $X^*$ be its [[continuous dual space|4A82S5FVRT]].
Then $X$ is [[reflexive|NN8RWPC8WG]] if and only if every $f \in X^*$ attains its norm on the closed unit ball.
That is, for every $f \in X^*$ there is $x \in X$ with $\norm{x} \le 1$ and $\abs{f(x)} = \norm{f}$.

## Remarks

The direction "reflexive implies norm attaining" is short.
In a reflexive space the closed unit ball is weakly compact, and $x \mapsto \abs{f(x)}$ is weakly continuous, so it attains its maximum $\norm{f}$.
The converse is the deep part, and its proof by James is long and combinatorial.

The theorem gives a concrete test for non-reflexivity.
On $c_0$, the functional $f(x) = \sum_{n \ge 1} 2^{-n} x_n$ has norm $\sum_n 2^{-n} = 1$.
For $x \in c_0$ with $\norm{x}_\infty \le 1$, some coordinate has $\abs{x_n} < 1$, so $\abs{f(x)} < 1$.
So $f$ does not attain its norm, and $c_0$ is not reflexive.

## Explanations

### L1 human

Every bounded linear functional has a norm, which is the supremum of its values on the unit ball.
The theorem says that this supremum is always a maximum exactly in the reflexive spaces.
So reflexivity, which is defined through the double dual, can be seen by looking at single functionals.

### L2 human

James's theorem is used in optimization and convex analysis, where the existence of maximizers of linear functionals on the unit ball matters.
It contrasts with the Bishop–Phelps theorem, which says that in every Banach space the norm-attaining functionals are dense in $X^*$.
It sits beside the [[Eberlein–Šmulian theorem|2MTN8CQWME]] and the [[Goldstine theorem|K4S9G3AQBM]] as a main characterization of weak compactness and reflexivity.
</document>

<document source="https://tamnd.github.io/math2/node/G1Z3WSTE8Q.md" tag="G1Z3WSTE8Q" kind="concept">
# Fisher information

## Summary

The Fisher information $I(\theta)$ measures how much an observation tells about an unknown parameter $\theta$ of a statistical model.
It is the variance of the score, the derivative in $\theta$ of the log-likelihood.
It bounds the precision of unbiased estimators through the [[Cramér–Rao bound|1JFKTEQVRX]], and its inverse is the asymptotic variance of the [[maximum likelihood estimator|0FREPFKNS7]].

## Explanations

### L1 human

If a small change of the parameter changes the distribution of the data a lot, then the data pin the parameter down well, and the information is large.
If the distribution hardly moves, the information is small.
Picture the log-likelihood as a hill: high information means a sharp peak, low information a flat top.
For one observation of a normal variable with mean $\theta$ and variance $\sigma^2$, the information is $1/\sigma^2$, so less noise means more information.
With $n$ independent observations the information adds up to $n/\sigma^2$.
</document>

<document source="https://tamnd.github.io/math2/node/G206NVQYVD.md" tag="G206NVQYVD" kind="definition">
# Section and retraction

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $f \colon A \to B$ and $r \colon B \to A$ be [[morphisms|859F0TFVA3]] of $\mathcal{C}$ with $r \circ f = 1_A$.
Then $f$ is a *section* of $r$, and $r$ is a *retraction* of $f$.
A morphism that has a retraction is a *split monomorphism*.
A morphism that has a section is a *split epimorphism*.

## Remarks

Every split monomorphism is a [[monomorphism|Y2NP1JFR86]]: if $f \circ g_1 = f \circ g_2$, then $g_1 = r \circ f \circ g_1 = r \circ f \circ g_2 = g_2$.
Dually, every split epimorphism is an [[epimorphism|RVXDSY04KG]].
Example: in the category of sets, the inclusion $f \colon \set{0} \to \set{0, 1}$ and the constant map $r \colon \set{0, 1} \to \set{0}$ satisfy $r \circ f = 1_{\set{0}}$.
Non-example: in the category of [[abelian groups|082GHNPJ53]], the map $f \colon \Z \to \Z$, $f(x) = 2x$, is a monomorphism but not a split monomorphism.
A retraction $r$ would satisfy $r(2) = 1$, so $2 r(1) = 1$ in $\Z$, which is impossible.
A morphism is an isomorphism exactly when it is both a split monomorphism and an epimorphism.
In the category of sets, the statement that every surjection is a split epimorphism is equivalent to the axiom of choice.
</document>

<document source="https://tamnd.github.io/math2/node/G20G1ECN56.md" tag="G20G1ECN56" kind="statement">
# Davis's converse of the Knaster–Tarski theorem

## Statement

Let $L$ be a [[lattice|WH4SFAACAS]].
Then $L$ is a [[complete lattice|D7A0ZGZT5S]] if and only if every [[monotone function|0A7KV2WB7A]] $f \colon L \to L$ has a fixed point, that is, an element $x \in L$ with $f(x) = x$.

## Remarks

One direction is the [[Knaster–Tarski theorem|RHXP3HA64M]]: in a complete lattice, every monotone map has a fixed point, and even a complete lattice of fixed points.
For the converse, if $L$ is not complete, Davis constructs a monotone map from $L$ to $L$ without fixed points.
The lattice $\Z$ with its usual order is not complete, and the map $n \mapsto n + 1$ is monotone without fixed points.
The result was published by Anne C. Davis in 1955.

## Explanations

### L1 human

The Knaster–Tarski theorem says that in a complete lattice every order-preserving map has a fixed point.
Davis showed that this property characterizes complete lattices among all lattices.
So the existence of fixed points is not a lucky extra: it is exactly completeness.

### L2 human

The result gives a test for completeness that does not mention suprema at all.
To show that a lattice is not complete, it is enough to exhibit one monotone map without a fixed point.
A shift $n \mapsto n + 1$ on $\Z$ is such a map.
</document>

<document source="https://tamnd.github.io/math2/node/G20KZ755WW.md" tag="G20KZ755WW" kind="definition">
# Sign test

## Definition

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with a continuous distribution, and let $m$ be its [[median|08W2CW8PKW]].
Let $m_0 \in \R$.
The *sign statistic* is
$$S = \#\set{i : X_i > m_0}.$$
The *sign test* of $H_0 \colon m = m_0$ uses the fact that under $H_0$ the statistic $S$ has the [[binomial distribution|0VRWZDWW6B]] with parameters $n$ and $1/2$.
Against $H_1 \colon m > m_0$ it rejects $H_0$ for large $S$, against $m < m_0$ for small $S$, and against $m \ne m_0$ for $S$ far from $n/2$.

## Remarks

For matched pairs $(Y_i, Z_i)$ one applies the test to the differences $X_i = Y_i - Z_i$ with $m_0 = 0$.
Continuity of the law gives $P(X_i = m_0) = 0$; with real data, observations equal to $m_0$ are usually dropped and $n$ is reduced.

Example: $n = 10$ observations, all above $m_0$, so $S = 10$.
The one-sided p-value is $P(S \ge 10) = 2^{-10} \approx 0.001$ under $H_0$, and $H_0$ is rejected at level $0.05$.

Non-example: let $P(X_i < m_0) = P(X_i > m_0) = 1/4$ and $P(X_i = m_0) = 1/2$.
Then $m_0$ is a median, but the law is not continuous, and $S$ has law $\mathrm{Bin}(n, 1/4)$ instead of $\mathrm{Bin}(n, 1/2)$.

The [[Wilcoxon signed-rank test|38SETDT2HN]] also uses the sizes of the deviations $X_i - m_0$, at the cost of assuming symmetry.
</document>

<document source="https://tamnd.github.io/math2/node/G215XD6D4D.md" tag="G215XD6D4D" kind="concept">
# Quasisymmetric map

## Summary

A quasisymmetric map is an embedding between [[metric spaces|0VKD1TP7F3]] that distorts relative distances in a controlled way.
It may stretch distances a lot, but it sends points in roughly the same relative position to points in roughly the same relative position.
Quasisymmetric self-maps of the real line are the boundary values of [[quasiconformal maps|SB8KG46NHP]] of the upper half-plane.
The notion makes sense in any metric space and is central in the analysis of quasiconformal maps on metric spaces.

## Explanations

### L1 human

Take three points $x, a, b$ with $a$ about as close to $x$ as $b$ is.
A quasisymmetric map keeps this rough balance: the image of $a$ is not much closer or farther from the image of $x$ than the image of $b$ is.
Similarities and bi-Lipschitz maps are quasisymmetric.
The map $x \mapsto e^x$ on the line is not, because it sends a symmetric pair of points around $x$ to points at very unequal distances when the pair is spread out.
</document>

<document source="https://tamnd.github.io/math2/node/G21MZQCME9.md" tag="G21MZQCME9" kind="concept">
# Crossed product algebra

## Summary

A crossed product algebra is a central simple algebra built from a finite [[Galois extension|0QMZAQ6QQ4]] $L/K$.
The extra datum is a $2$-cocycle of its [[Galois group|0XTY86BZKY]] with values in $L^\times$.
It contains $L$ as a maximal subfield, and each Galois automorphism is realized as conjugation by an invertible element.
Every central simple algebra is Brauer equivalent to a crossed product, which gives the isomorphism between the [[Brauer group|BEW48TA9WZ]] and a second [[Galois cohomology|9MPCZ3EAJM]] group.
This algebraic notion is distinct from the [[crossed product|8HG8TS46A6]] of operator algebras, although both twist a ring by a group action.

## Explanations

### L1 human

Start with a Galois extension $L$ of $K$ and its group of symmetries $G$.
Add to $L$ one new symbol $u_\sigma$ for each symmetry $\sigma$, and declare that moving a number past $u_\sigma$ applies $\sigma$ to it.
The products $u_\sigma u_\tau$ are allowed to differ from $u_{\sigma\tau}$ by a nonzero number of $L$, chosen by a rule called a factor set.
For $L = \C$ over $K = \R$ and the right choice of factor set, the result is Hamilton's quaternions, with $u_\sigma = j$.
</document>

<document source="https://tamnd.github.io/math2/node/G24036DJ5K.md" tag="G24036DJ5K" kind="definition">
# Post algebra

## Definition

Let $n \ge 2$ and let $P_n = \set{0, 1, \dots, n - 1}$.
The *$n$-element Post algebra* is $\mathbf{P}_n = (P_n, \vee, \wedge, ', 0, n - 1)$.
Here $x \vee y = \max(x, y)$, $x \wedge y = \min(x, y)$, and $x' = x + 1$ for $x < n - 1$ while $(n - 1)' = 0$.
A *Post algebra of order $n$* is an algebra in the [[variety|AJNZJD8442]] generated by $\mathbf{P}_n$.

## Remarks

Example: $\mathbf{P}_n^2$ is a Post algebra of order $n$, since a variety is closed under direct products.
Non-example: the three-element chain $(\set{0, 1, 2}, \max, \min, ', 0, 2)$ with $x' = 2 - x$ is not a Post algebra of order $3$.
In $\mathbf{P}_3$ the identity $x''' \approx x$ holds, but here $0''' = 2 \ne 0$.
The algebra $\mathbf{P}_n$ is [[primal|XD01Y7B6XW]], because $\max$ and the cyclic negation already generate every operation on $P_n$, a result of Post.
For $n = 2$, $\mathbf{P}_2$ is the two-element Boolean algebra, so Post algebras of order $2$ are Boolean algebras.
Other texts present Post algebras through lattice axioms, as bounded distributive lattices with a chain of constants $e_0 < e_1 < \cdots < e_{n - 1}$.
Post algebras give algebraic semantics for Post's [[many-valued logic|DWYCKAKXZD]].
</document>

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# Modular representation theory

## Summary

Modular representation theory studies [[representations|WYGRXCG2RN]] of a finite group $G$ over a field whose characteristic $p$ divides the order of $G$.
In this case [[Maschke's theorem|4SBZ4EMPBZ]] fails, so representations need not split into irreducible pieces, and the [[group algebra|MNP2HBWB5Q]] is not semisimple.
Brauer developed the theory, with tools such as Brauer characters, blocks, defect groups and the [[Cartan matrix|ZBQ4KRC98K]].
It links ordinary characters in characteristic $0$ with representations over finite fields through reduction modulo $p$.

## Explanations

### L1 human

A representation lets a group act on a vector space by linear maps.
Over the complex numbers, every representation of a finite group breaks into irreducible pieces, like a number into primes.
Over a field such as $\F_p$, where adding $p$ ones gives zero, this breaking can fail when $p$ divides the size of the group.
For the cyclic group of order $p$ acting on $\F_p^2$ by $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the line spanned by the first basis vector is preserved, but no complementary line is.
Modular representation theory is the study of this richer and harder situation.
</document>

<document source="https://tamnd.github.io/math2/node/G2HC1SFQ4P.md" tag="G2HC1SFQ4P" kind="statement">
# Dolbeault isomorphism

## Statement

Let $X$ be a [[complex manifold|JSGWAN2RJV]], let $p, q \ge 0$ be integers, and let $\Omega^p_X$ be the sheaf of holomorphic $p$-forms on $X$.
Then there is a natural isomorphism of complex vector spaces
$$H^q(X, \Omega^p_X) \cong H^{p,q}_{\bar\partial}(X)$$
between the [[sheaf cohomology|PQ6Q143ARJ]] of $\Omega^p_X$ and the [[Dolbeault cohomology|Y3900T2EY5]] of $X$.
More generally, let $E$ be a holomorphic vector bundle on $X$.
Then $H^q(X, \Omega^p_X(E))$ is isomorphic to the cohomology of the complex of smooth $E$-valued $(p,\cdot)$-forms with the operator $\bar\partial_E$.

## Remarks

Let $\mathcal{A}^{p,q}$ be the sheaf of smooth $(p,q)$-forms.
The [[Dolbeault–Grothendieck lemma|5858M5PGN6]] shows that the sequence of sheaves
$$0 \to \Omega^p_X \to \mathcal{A}^{p,0} \xrightarrow{\bar\partial} \mathcal{A}^{p,1} \xrightarrow{\bar\partial} \mathcal{A}^{p,2} \to \cdots$$
is exact.
The sheaves $\mathcal{A}^{p,q}$ are fine, because they are modules over the sheaf of smooth functions, which has partitions of unity, so they have no higher cohomology.
The cohomology of $\Omega^p_X$ is therefore the cohomology of the complex of global sections, which is the Dolbeault complex.
Example: on a compact Riemann surface $X$ of genus $g$, $H^1(X, \mathcal{O}_X) \cong H^{0,1}_{\bar\partial}(X)$, and both have dimension $g$.
For $p = 0$, the theorem identifies the cohomology of the structure sheaf $\mathcal{O}_X = \Omega^0_X$ with the obstruction groups to solving $\bar\partial u = f$.

## Explanations

### L1 human

Sheaf cohomology is defined abstractly and is hard to compute directly.
The Dolbeault isomorphism says that the cohomology of holomorphic forms can be computed with differential forms and the operator $\bar\partial$.
This turns algebraic questions into questions about a partial differential equation.

### L2 human

The isomorphism is the complex analogue of de Rham's theorem, which computes the cohomology of the constant sheaf with smooth forms and the operator $d$.
It lets one prove vanishing theorems for sheaf cohomology with analytic tools, such as Hörmander's $L^2$ estimates or harmonic forms.
On compact Kähler manifolds, Hodge theory represents each class by a harmonic form, and the isomorphism then gives the Hodge decomposition $H^k(X, \C) \cong \bigoplus_{p+q=k} H^q(X, \Omega^p_X)$.
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# Locale

## Summary

A locale is a generalized topological space described only by its lattice of "open sets", which is a [[frame|306PWGH7BF]].
Maps of locales go in the opposite direction to frame homomorphisms, just as a continuous map pulls open sets back.
Every topological space gives a locale, and for [[sober spaces|4N4DNDR22Z]] the space can be recovered from it.
Locales are the objects of [[pointless topology|CYEMGH98WN]].

## Explanations

### L1 human

A continuous map $f \colon X \to Y$ sends points forward but sends open sets backward, by taking preimages.
If one only remembers the open sets, it is natural to describe a map of spaces by a map of open-set lattices going the other way.
A locale is a frame regarded in this reversed way, as a space.
The real line, for instance, gives the locale whose frame is the set of open subsets of $\R$.
There are also nontrivial locales with no points at all, which have no counterpart among ordinary spaces.
</document>

<document source="https://tamnd.github.io/math2/node/G2RYQMGG1R.md" tag="G2RYQMGG1R" kind="conjecture">
# Cycle double cover conjecture

## Statement

Let $G$ be a finite [[graph|00C0BR7D7H]], possibly with multiple edges, that has no [[bridge|0BANA3XYPF]].
The conjecture: there is a finite list of [[cycles|0Z05VCBQ7Y]] of $G$ such that each edge of $G$ lies in exactly two cycles of the list.
The list may contain a cycle more than once.
Such a list is called a cycle double cover of $G$.

## Remarks

The conjecture is open.
It was posed by Szekeres and, independently, by Seymour.
The hypothesis is necessary: a bridge lies on no cycle, so a graph with a bridge has no cycle double cover.
For a bridgeless planar graph, each face boundary of a plane embedding splits into edge-disjoint cycles, and each edge lies on exactly two faces.
So these cycles form a cycle double cover.
The general case reduces to cubic bridgeless graphs, and further to snarks, the cubic bridgeless graphs with no proper edge colouring with $3$ colours.
If a cubic graph has a proper edge colouring with $3$ colours, the unions of two colour classes give three families of disjoint cycles, which together cover each edge twice.

## Explanations

### L1 human

Take a network with no single link whose removal disconnects it.
The conjecture says that one can choose a collection of closed loops in the network that passes over every link exactly twice.
For a map drawn in the plane, the borders of the countries do this.
For general networks, nobody knows a proof, and nobody has found a counterexample.
</document>

<document source="https://tamnd.github.io/math2/node/G2S6H789RG.md" tag="G2S6H789RG" kind="statement">
# Monotone convergence theorem

## Statement

Let $(X, \Sigma, \mu)$ be a measure space.
Let $f_1, f_2, \dots \colon X \to [0, \infty]$ be [[measurable functions|YYNCJSR0PJ]] with $f_1(x) \le f_2(x) \le \cdots$ for every $x \in X$.
Let $f(x) = \lim_{n \to \infty} f_n(x)$, a limit in $[0, \infty]$.
Then $f$ is measurable and
$$\int_X f \, d\mu = \lim_{n \to \infty} \int_X f_n \, d\mu,$$
where the integrals are [[integrals of non-negative functions|DG76G5Z6XW]].

## Remarks

Proof idea: the integrals $\int f_n$ increase and are at most $\int f$, so their limit $L$ satisfies $L \le \int f$.
For the reverse bound, fix a simple $s \le f$ and $c \in (0, 1)$.
The sets $E_n = \set{f_n \ge c s}$ increase to $X$, so $\int f_n \ge c \int_{E_n} s \to c \int s$ by continuity of measure from below.
Letting $c \to 1$ and taking the supremum over $s$ gives $L \ge \int f$.

Example: with $f_n = \min(f, n)$ the theorem computes $\int f$ from bounded functions.
For a series of non-negative measurable functions it gives $\int \sum_k g_k \, d\mu = \sum_k \int g_k \, d\mu$.

Both hypotheses matter.
Without monotonicity, $f_n = \mathbf{1}_{[n, n+1]}$ on $\R$ tends to $0$ pointwise while each integral is $1$.
Without non-negativity, $f_n = -\mathbf{1}_{[n, \infty)}$ increases to $0$ while each integral is $-\infty$.
It holds also when $f_n \le f_{n+1}$ only almost everywhere.
The analogous fact for sequences of numbers is the [[monotone convergence theorem for sequences|0FM6N4R0VB]].

## Explanations

### L1 human

If a sequence of non-negative functions grows at every point, then the integrals grow to the integral of the limit.
No extra bound on the functions is needed.
The value $\infty$ is allowed on both sides.

### L2 human

This is the first of the three convergence theorems of Lebesgue integration.
Fatou's lemma follows from it, and the dominated convergence theorem follows from Fatou's lemma.
It is also the tool to prove that the integral is additive, by approximating with increasing simple functions.
In probability it gives $\mathbb{E}[\sum_k Y_k] = \sum_k \mathbb{E}[Y_k]$ for non-negative random variables.
</document>

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# Bayesian network

## Definition

Let $G = (V, E)$ be a finite [[directed graph|0KFYGSHHF5]] with no directed cycles, that is, a directed acyclic graph.
For $v \in V$ let $\operatorname{pa}(v)$ be the set of parents of $v$, the vertices $u$ with an edge $u \to v$.
Let $X = (X_v)_{v \in V}$ be random variables with a joint density or probability mass function $p$.
Then $X$ is a *Bayesian network* with respect to $G$ if
$$p(x) = \prod_{v \in V} p(x_v \mid x_{\operatorname{pa}(v)})$$
for all $x$, where $p(x_v \mid x_{\operatorname{pa}(v)})$ is the [[conditional density|0ZN9EZTEM6]] of $X_v$ given $X_{\operatorname{pa}(v)}$.

## Remarks

The factorisation is equivalent to the local Markov property.
Each $X_v$ is [[conditionally independent|0THJ7416DX]] of its non-descendants given its parents.
Example: for the chain $A \to B \to C$, the definition asks $p(a, b, c) = p(a) p(b \mid a) p(c \mid b)$.
Dividing by $p(b)$ gives $p(a, c \mid b) = p(a \mid b) p(c \mid b)$, so $A \perp C \mid B$.
Non-example: the graph $A \to B \to A$ has a directed cycle, so it does not define a Bayesian network; a product $p(a \mid b) p(b \mid a)$ is in general not a joint distribution.
Different graphs can give the same model: $A \to B \to C$ and $A \leftarrow B \leftarrow C$ both encode exactly $A \perp C \mid B$.
</document>

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# Nagell–Lutz theorem

## Statement

Let $a, b \in \Z$ with $D = 4a^3 + 27b^2 \ne 0$, and let $E$ be the [[elliptic curve|427N6S6QQW]] $y^2 = x^3 + ax + b$ over $\Q$.
Let $P = (x, y) \in E(\Q)$ be a point of finite [[order|0WE2NHZJ7B]] in the [[group of rational points|P7V1BQJPSS]], with $P \ne O$.
Then $x$ and $y$ are integers, and either $y = 0$ or $y^2$ divides $D$.

## Remarks

The integrality comes from a study of the denominators of points in the $p$-adic filtration: points whose coordinates have $p$ in the denominator form a torsion-free subgroup.
Divisibility follows next: if $P$ has finite order, so does $2P$, and both have integer coordinates; the duplication formula then forces $y^2 \mid D$.
The theorem gives an algorithm for the torsion subgroup: list the finitely many candidates and test each for finite order.
Example: on $y^2 = x^3 + 1$ one has $D = 27$, so a torsion point with $y \ne 0$ has $y \in \set{\pm 1, \pm 3}$.
This gives the points $(0, \pm 1)$ and $(2, \pm 3)$.
With $(-1, 0)$ and $O$ they form a cyclic group of order $6$.
The converse fails: on $y^2 = x^3 + 2$ the point $(-1, 1)$ has integer coordinates and $1 \mid 108$, but it has infinite order.
The hypothesis that $a$ and $b$ are integers is needed; any curve can be scaled to this form by $(x, y) \mapsto (u^2 x, u^3 y)$.

## Explanations

### L1 human

A rational point of finite order on such a curve is very special.
Its coordinates are whole numbers, and the $y$-coordinate is zero or has a square that divides a fixed number.
So the points of finite order can be found by a finite search.

### L2 human

The theorem was found independently by Nagell and Lutz in the 1930s.
It is the standard tool, together with reduction modulo good primes, to compute the torsion subgroup $T$ of $E(\Q)$ in [[Mordell's theorem|BZHGKN6Q3K]].
[[Mazur's torsion theorem|RQS6YHAR8N]] then lists all groups that can occur.
</document>

<document source="https://tamnd.github.io/math2/node/G3A4S7CSB3.md" tag="G3A4S7CSB3" kind="concept">
# Lax pair

## Summary

A Lax pair is a pair of operators $L(t)$ and $P(t)$ such that an evolution equation is equivalent to $\frac{dL}{dt} = [P, L] = PL - LP$.
Then $L(t)$ stays similar to $L(0)$, so its spectrum does not change in time.
The eigenvalues of $L$ are thus conserved quantities.
Lax pairs are the starting point of the [[inverse scattering transform|6GY3X4Y8X5]] and a sign that an equation is integrable.

## Explanations

### L1 human

Imagine a matrix that changes in time but whose eigenvalues never move.
A Lax pair writes an equation in exactly this form: the unknown is hidden in the operator $L$, and the equation says $L$ only rotates.
For the [[Korteweg–De Vries equation|KTT75QC72Z]], $L$ is the operator $-\partial_x^2 + u$, and the KdV evolution keeps its eigenvalues fixed.
This gives infinitely many conserved quantities and explains why [[solitons|63D4AXYCT4]] keep their speeds after they collide.
</document>

<document source="https://tamnd.github.io/math2/node/G3CKBSCM6A.md" tag="G3CKBSCM6A" kind="definition">
# Polynomially convex set

## Definition

Let $K \subseteq \C^n$ be a [[compact set|0R6EWFKRCF]].
Its *polynomially convex hull* is
$$\hat K = \set{z \in \C^n : \abs{p(z)} \le \max_{w \in K} \abs{p(w)} \text{ for every polynomial } p \in \C[z_1, \dots, z_n]},$$
where the polynomials are the [[polynomials in several variables|07GNX8GCYP]] with complex coefficients.
The set $K$ is *polynomially convex* if $\hat K = K$.

## Remarks

Always $K \subseteq \hat K$, and $\hat K$ is compact.
It coincides with the [[holomorphically convex hull|CD6SMMZSW8]] of $K$ in $D = \C^n$, because entire functions are uniform limits of their Taylor polynomials on compact sets.

Example: a compact convex set $K$ is polynomially convex.
If $z \notin K$, there is a complex linear function $\ell$ with $\operatorname{Re} \ell(z) > \max_K \operatorname{Re} \ell$.
Then $f = e^{\ell}$ satisfies $\abs{f(z)} > \max_K \abs{f}$, and the Taylor polynomials of $f$ converge to $f$ uniformly on $K \cup \set{z}$, so some polynomial $p$ has $\abs{p(z)} > \max_K \abs{p}$.

Non-example: the unit circle $K = \set{\abs{z} = 1}$ in $\C$.
By the maximum principle, $\abs{p(0)} \le \max_K \abs{p}$ for every polynomial $p$, so $0 \in \hat K$ but $0 \notin K$.

For $n = 1$, a compact set $K \subseteq \C$ is polynomially convex exactly when $\C \setminus K$ is connected.
</document>

<document source="https://tamnd.github.io/math2/node/G3F2CWQSQH.md" tag="G3F2CWQSQH" kind="example">
# Bounded sequences ℓ∞

## Example

Let $\ell^\infty$ be the set of [[bounded sequences|0A8JNBGT2Z]] $x = (x_n)_{n \ge 1}$ of real (or complex) numbers, with the norm $\norm{x}_\infty = \sup_n \abs{x_n}$.
Then $\ell^\infty$ is a [[sequence space|XYTN19EPF2]] and a [[Banach space|0BDTTM24DP]]; this is the case $p = \infty$ of the [[spaces ℓ^p|08P6T2WH6R]].

Check of completeness: let $(x^{(j)})$ be a Cauchy sequence in $\ell^\infty$.
For each $n$, $\abs{x^{(j)}_n - x^{(i)}_n} \le \norm{x^{(j)} - x^{(i)}}_\infty$, so $(x^{(j)}_n)_j$ is a Cauchy sequence of numbers with a limit $x_n$.
Given $\varepsilon > 0$, choose $J$ with $\norm{x^{(j)} - x^{(i)}}_\infty \le \varepsilon$ for $i, j \ge J$.
Letting $i \to \infty$ gives $\abs{x^{(j)}_n - x_n} \le \varepsilon$ for all $n$ and all $j \ge J$.
So $x$ is bounded and $x^{(j)} \to x$ in norm.

## Remarks

The space $\ell^\infty$ is not [[separable|0Z2P41SVDP]].
The sequences with all terms in $\set{0, 1}$ form an uncountable set, and any two of them are at distance $1$.
So the open balls of radius $1/2$ around them are disjoint, and no countable set meets all of them.
The spaces of convergent sequences and of null sequences are closed subspaces of $\ell^\infty$.
The space $\ell^\infty$ is isometrically isomorphic to the dual of $\ell^1$.

## Explanations

### L1 human

A sequence is bounded when all its terms stay within some fixed distance of $0$.
The bounded sequences form a vector space, and the size of a sequence is the least bound on its terms.
This space is so large that no countable family of sequences can approximate all of its elements.
</document>

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# Morita invariant

## Definition

Let $P$ be a property of [[rings|0B1261B5ED]] with identity.
The property $P$ is a *Morita invariant* if the following holds.
Whenever $R$ and $S$ are [[Morita equivalent|RD5WKA1VK7]] rings, $R$ has $P$ if and only if $S$ has $P$.
More generally, an assignment of an object $X(R)$ to each ring $R$ is a Morita invariant if $X(R) \cong X(S)$ whenever $R$ and $S$ are Morita equivalent.

## Remarks

Example: being left Noetherian is a Morita invariant.
A ring is left Noetherian if and only if every finitely generated left module satisfies the ascending chain condition on submodules.
Finite generation, submodules and chains of submodules are preserved by any equivalence of module categories.
Other Morita invariant properties are being simple, semisimple, left Artinian, primitive, prime, left hereditary, and having a given left global dimension.
The [[center|MVT64B3B6H]] of a ring, the lattice of its two-sided ideals, and its Grothendieck group $K_0$ are Morita invariant objects.
Non-example: commutativity is not a Morita invariant, since a commutative ring $R \ne 0$ is Morita equivalent to $M_2(R)$, which is not commutative.
For the same reason, being a domain and being a local ring are not Morita invariants.
</document>

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# Sketch

## Definition

A *sketch* is a triple $\mathcal{S} = (\mathcal{A}, L, C)$ with these parts.

1. $\mathcal{A}$ is a small [[category|A2DE66937D]].
2. $L$ is a set of [[cones|7TPBQPPN97]] in $\mathcal{A}$, each over a small diagram in $\mathcal{A}$.
3. $C$ is a set of cocones in $\mathcal{A}$, each under a small diagram in $\mathcal{A}$.

Let $\mathcal{K}$ be a category.
A *model* of $\mathcal{S}$ in $\mathcal{K}$ is a [[functor|28NQ0KRHD3]] $M \colon \mathcal{A} \to \mathcal{K}$ with two properties.
$M$ sends each cone in $L$ to a [[limit|CHS5AFJGWE]] cone.
$M$ sends each cocone in $C$ to a [[colimit|BJS1CGTKQ5]] cocone.
A morphism of models is a natural transformation.

## Remarks

The sketch is a *limit sketch* when $C$ is empty.
Some authors use a directed graph with chosen diagrams instead of the category $\mathcal{A}$; this gives the same models.

Example: let $\mathcal{A}$ be the free category on the graph with two objects $X, P$ and three arrows $p_1, p_2, m \colon P \to X$.
Let $L$ contain the single cone $(P; p_1, p_2)$ over the discrete diagram $(X, X)$, and let $C$ be empty.
A model in sets is a functor $M$ with $M(P) \cong M(X) \times M(X)$ through $M(p_1), M(p_2)$.
Then $M(m)$ is a binary operation on $M(X)$, so models are sets with a binary operation.

Non-example: let $M(X) = M(P) = \set{0, 1}$ with $M(p_1) = M(p_2) = M(m) = \id$.
This functor is not a model, since $\set{0, 1}$ with two identity maps is not a product of $\set{0, 1}$ with itself.
Indeed, the pair of maps $(\id, \text{constant } 0)$ does not factor through it.
</document>

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# Spectral space

## Summary

A spectral space is a compact [[sober|4N4DNDR22Z]] topological space whose compact open sets are closed under finite intersections and form a basis of the topology.
The prime spectrum of any [[commutative ring|0310REJJY8]] with the [[Zariski topology|8SQK7RTB63]] is spectral, and by [[Hochster's theorem|C82AXZK2BS]] every spectral space arises in this way.
The compact open sets of a spectral space form a bounded [[distributive lattice|0HM0HQCK6N]], and every bounded distributive lattice arises in this way.
So spectral spaces connect commutative algebra, lattice theory and topology.

## Explanations

### L1 human

The points of a spectral space behave like the prime ideals of a ring: some points are "general" and have other points in their closure.
The space is compact, and it has many open sets that are compact as well, enough to describe the whole topology.
Each closed set that cannot be split into two smaller closed sets has exactly one "generic" point whose closure is that set.
For example, the prime ideals of the integers form a spectral space whose generic point $(0)$ has every point in its closure.
</document>

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# Volterra operator

## Example

The *Volterra operator* is the operator $V$ on the [[Lp space|SASP14MA1Z]] $L^2(0, 1)$ given by
$$(V f)(t) = \int_0^t f(s) \, ds, \quad t \in [0, 1].$$
It is the integral operator of the [[Volterra equation of the first kind|H6ZGEDM4VV]] with kernel $K(t, s) = 1$ for $s \le t$ and $K(t, s) = 0$ for $s > t$.

Check that it is a [[compact operator|M2NEP1K5AE]]: the kernel is square-integrable on $[0, 1]^2$, with $\int_0^1 \int_0^1 \abs{K}^2 = 1/2$.
So $V$ is a [[Hilbert–Schmidt integral operator|CB5G212Q03]], and Hilbert–Schmidt operators are compact.

Check that $V$ has no eigenvalue.
If $V f = 0$, then $\int_0^t f = 0$ for all $t$, so $f = 0$ almost everywhere.
If $V f = \lambda f$ with $\lambda \ne 0$, then $f = \lambda^{-1} V f$ is continuous, hence $V f$ is continuously differentiable, and $f' = f / \lambda$ with $f(0) = 0$; so $f = 0$.

Check that the [[spectrum|BMKATJ802A]] is $\set{0}$.
The power $V^n$ has kernel $(t - s)^{n-1}/(n-1)!$ for $s \le t$, so $\norm{V^n} \le 1/(n-1)!$.
Hence the [[spectral radius|9V7YQA85PC]] $\lim_n \norm{V^n}^{1/n}$ is $0$, while $0$ is in the spectrum because a compact operator on an infinite-dimensional space is not invertible.

## Remarks

The operator norm of $V$ is $2/\pi$.
Its [[singular values|QSJKPKPM7H]] are $2/((2n - 1)\pi)$, $n \ge 1$, which are square-summable but not summable, so $V$ is not a [[trace class operator|Z2CP530WDE]].
The adjoint is $(V^* f)(t) = \int_t^1 f(s) \, ds$, and $V + V^*$ is the rank-one operator $f \mapsto \int_0^1 f$.
The Volterra operator is the standard example of a nonzero quasinilpotent compact operator: its spectrum is the same as that of the zero operator.

## Explanations

### L1 human

The Volterra operator takes a function and returns its running integral from $0$.
It is the most basic integral operator, the continuous version of taking partial sums.
It is a compact operator with no eigenvalues at all, something that cannot happen for a square matrix over the complex numbers.
This is why equations $f - \lambda V f = g$ can be solved for every $\lambda$.
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# Hirzebruch–Riemann–Roch theorem

## Statement

Let $X$ be a [[smooth|EXKGG9X6J0]] [[projective variety|A4H4CZ2V35]] of dimension $n$ over $\C$, and let $\mathcal{E}$ be a [[vector bundle|P4DZM975BV]] of rank $r$ on $X$.
Let $\chi(X, \mathcal{E}) = \sum_{i=0}^{n} (-1)^i \dim_\C H^i(X, \mathcal{E})$, where $H^i$ is [[sheaf cohomology|PQ6Q143ARJ]].
Let $\alpha_1, \dots, \alpha_r$ be the Chern roots of $\mathcal{E}$ and $\beta_1, \dots, \beta_n$ those of the tangent bundle $T_X$.
So the [[Chern classes|PAD37GKDPD]] are the elementary symmetric functions in these roots.
Let $\operatorname{ch}(\mathcal{E}) = \sum_{j=1}^{r} e^{\alpha_j}$ be the Chern character and $\operatorname{td}(T_X) = \prod_{j=1}^{n} \frac{\beta_j}{1 - e^{-\beta_j}}$ the [[Todd class|2YDB5089FR]], both in $H^{*}(X, \Q)$.
Then
$$\chi(X, \mathcal{E}) = \int_X \operatorname{ch}(\mathcal{E}) \, \operatorname{td}(T_X),$$
where $\int_X$ takes the degree $2n$ component and evaluates it on the fundamental class.

## Remarks

Hirzebruch proved the theorem with Thom's cobordism theory: both sides are multiplicative and compatible with the relevant operations, and they agree on enough generators.
Later proofs come from the Grothendieck–Riemann–Roch theorem, applied to $X \to \operatorname{Spec} \C$, and from the Atiyah–Singer index theorem, which extends it to all compact complex manifolds.
The same formula holds for smooth projective varieties over any field, with the Chow ring in place of $H^*(X, \Q)$.

Example: for a curve $X$ of genus $g$ and a line bundle $L$ of degree $d$, $\operatorname{ch}(L) = 1 + c_1(L)$ and $\operatorname{td}(T_X) = 1 + \tfrac{1}{2} c_1(T_X)$ with $\deg c_1(T_X) = 2 - 2g$.
The degree $2$ part is $c_1(L) + \tfrac12 c_1(T_X)$, so $\chi(L) = d + 1 - g$, the classical Riemann–Roch theorem.

For a surface, the case $\mathcal{E} = \mathcal{O}_X$ gives Noether's formula $\chi(\mathcal{O}_X) = \tfrac{1}{12}(c_1^2 + c_2)$.

## Explanations

### L1 human

The Euler characteristic of a vector bundle is an alternating count of the dimensions of its cohomology groups.
The theorem computes this number from topological data alone: the Chern classes of the bundle and of the variety.
So a quantity defined through hard analysis or algebra turns out to depend only on characteristic classes.

### L2 human

In practice $\chi(\mathcal{E})$ is computed by the formula, and a vanishing theorem then kills all but one cohomology group, which gives the dimension of the space of sections.
This is how one finds the dimension of linear systems on surfaces and higher-dimensional varieties.
The theorem generalizes the Riemann–Roch theorem for curves and for surfaces, and it is in turn a special case of both the Grothendieck–Riemann–Roch theorem and the Atiyah–Singer index theorem.
</document>

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# Dirichlet boundary condition

## Definition

Let $a < b$, and consider a second-order ordinary differential equation for $y$ on $[a, b]$.
A *Dirichlet boundary condition* is a pair of conditions
$$y(a) = \alpha, \qquad y(b) = \beta,$$
with given real numbers $\alpha, \beta$.
More generally, let $\Omega \subseteq \R^d$ be a bounded domain, that is, a connected open set, with [[boundary|05D5EGMFYV]] $\partial \Omega$, and let $g \colon \partial \Omega \to \R$ be a given function.
For an equation for $u$ on $\Omega$, the Dirichlet boundary condition is $u(x) = g(x)$ for all $x \in \partial \Omega$.
The condition is *homogeneous* if $\alpha = \beta = 0$, respectively $g = 0$.

## Remarks

A differential equation together with a Dirichlet condition is a Dirichlet problem, a kind of [[boundary value problem|TM8VQJWGHZ]].

Example: $y'' = 0$ on $[0, 1]$ with $y(0) = 1$ and $y(1) = 3$.
The function $y = 1 + 2x$ has $y'' = 0$, $y(0) = 1$ and $y(1) = 3$, so it solves this Dirichlet problem.

Non-example: the conditions $y(0) = 1$ and $y'(1) = 3$ do not form a Dirichlet condition, because the second one prescribes a derivative and not a value.

For the Laplace equation $\Delta u = 0$ on a bounded domain with continuous $g$, a solution that is continuous up to the boundary is unique when it exists, by the maximum principle.
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# Nondegenerate critical point

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$ and let $f \colon M \to \R$ be a [[smooth function|0V33BNKZ9S]].
Let $p$ be a [[critical point|0YTCMFB4HC]] of $f$, so $df_p = 0$.
The *Hessian* of $f$ at $p$ is the [[symmetric bilinear form|0AXZNE5XNH]] on $T_p M$ given by
$$\operatorname{Hess}_p f(v, w) = v\big(W f\big),$$
where $W$ is any vector field with $W(p) = w$.
The critical point $p$ is *nondegenerate* if $\operatorname{Hess}_p f$ is a [[nondegenerate bilinear form|01NHFZF2SK]].
Otherwise $p$ is *degenerate*.

## Remarks

Because $df_p = 0$, the value $v(Wf)$ does not depend on the extension $W$ of $w$, and the form is symmetric.
In a chart $(x_1, \dots, x_n)$ around $p$, $\operatorname{Hess}_p f$ has the matrix $\big(\partial^2 f / \partial x_i \partial x_j (p)\big)$, the [[Hessian matrix|0693Y2METT]], and $p$ is nondegenerate if and only if this matrix is invertible.

Example: $f(x, y) = x^2 + xy + y^2$ on $\R^2$ has the critical point $(0, 0)$ with Hessian matrix $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$.
Its determinant is $3 \ne 0$, so the origin is nondegenerate.

Non-example: $g(x, y) = x^2 + y^4$ has the critical point $(0, 0)$ with Hessian matrix $\begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix}$, which is singular, so the origin is degenerate.

Nondegenerate critical points are isolated among critical points.
A function all of whose critical points are nondegenerate is a [[Morse function|KPQ96640ZJ]].
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# Cohn localization

## Summary

Cohn localization, also called universal localization, inverts a chosen set $\Sigma$ of square matrices over a [[ring|0B1261B5ED]] $R$ in the most general way.
It produces a ring $R_\Sigma$ and a ring homomorphism $R \to R_\Sigma$ over which every matrix in $\Sigma$ becomes invertible, universal among such homomorphisms.
It exists for every ring and every set of matrices, unlike [[Ore localization|SG7AQ9T8HD]], which needs the Ore condition.
P. M. Cohn used it to construct the free field, the universal field of fractions of a free algebra.

## Explanations

### L1 human

For commutative rings, one makes fractions by declaring some elements invertible.
In a noncommutative ring, inverting single elements is not always enough to get a good ring of fractions.
Cohn's idea is to declare whole square matrices invertible, since a field of fractions must make many matrices invertible at once.
The universal localization is the "freest" ring in which the chosen matrices have inverses.
For a set of $1 \times 1$ matrices in a commutative ring, it is the usual [[localization|4C1AE84N5N]].
</document>

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# Lefschetz fixed-point theorem

## Statement

Let $X$ be a [[compact|0TVJZYB6SH]] space that is a [[retract|68PFSGP39B]] of a finite [[simplicial complex|V8GZ0W1ZZ5]].
Examples are compact triangulable spaces and compact smooth manifolds.
Let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]] and let $L(f) = \sum_{k \ge 0} (-1)^k \tr\big(f_{\ast} \colon H_k(X; \Q) \to H_k(X; \Q)\big)$ be its [[Lefschetz number|ZSYCA4EGEZ]].
If $L(f) \neq 0$, then $f$ has a fixed point: there is $x \in X$ with $f(x) = x$.

## Remarks

The idea of the proof: suppose $f$ has no fixed point.
By compactness $d(x, f(x))$ is bounded below, so after fine subdivision $f$ has a simplicial approximation that moves every simplex off itself.
The chain map of this approximation then has zero trace in each degree.
The Hopf trace formula equates the alternating sum of traces on chains with that on [[homology|M0Q5CFJ7ZH]], so $L(f) = 0$.
Example: if $X$ is contractible and compact as above, then $H_0(X; \Q) = \Q$ and the other groups vanish, so $L(f) = 1$ for every $f$ and every continuous self-map has a fixed point.
For $f$ homotopic to the identity, $L(f) = \chi(X)$, the [[Euler characteristic|RNDDWDAFJ5]]; so if $\chi(X) \neq 0$, every map homotopic to the identity has a fixed point.
Compactness is necessary: $\R$ is contractible, so $L(f) = 1$, but the translation $x \mapsto x + 1$ has no fixed point.
The converse fails: $L(f) = 0$ does not prevent fixed points, as the identity of the circle shows.

## Explanations

### L1 human

The Lefschetz number is a whole number computed from how a map acts on the holes of a space, as detected by homology.
The theorem says that if this number is not zero, the map must leave some point where it is.
It turns the geometric question "is there a fixed point?" into an algebraic computation.

### L2 human

The Lefschetz number depends only on the homotopy class of $f$, so the theorem gives fixed points for all maps in a homotopy class at once.
It contains the Brouwer fixed-point theorem, since a closed ball is contractible.
A refined form counts fixed points with indices: for a map with isolated fixed points, the sum of their fixed-point indices equals $L(f)$.
In dynamics, applying it to the iterates $f^n$ gives periodic points, and the numbers $L(f^n)$ enter the Lefschetz zeta function.
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# Bridge number

## Summary

The bridge number of a [[knot|QYEAZTYGPT]] is the least number of local maxima of the height on the knot.
The least is taken over all positions of the knot in space.
It is a [[knot invariant|GZQ5ZE3X6G]] introduced by Horst Schubert.
Knots with bridge number $2$, the two-bridge knots, form a well-understood family that is classified completely.

## Explanations

### L1 human

Lay a knotted rope in space and look at its height above the floor.
The rope goes up and down; count the hilltops.
The bridge number is the smallest number of hilltops you can get by moving the rope around without cutting it.
A plain circle has one hilltop, and nothing smaller is possible.
The [[trefoil knot|VYTYSVHN4A]] needs two hilltops, so its bridge number is $2$.
</document>

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# Erdős–Anning theorem

## Statement

Let $S$ be an infinite set of points in the [[Euclidean plane|0XW7V9ZZ05]].
Assume that the [[Euclidean distance|0NPPGHQNXQ]] between any two points of $S$ is an integer.
Then all points of $S$ lie on one line.

## Remarks

Suppose that $S$ contains three points $A, B, C$ that are not [[collinear|0AFNNTRTJE]].
For any other point $X$ of $S$, the triangle inequality gives $\abs{\,\norm{X - A} - \norm{X - B}\,} \le \norm{A - B}$, and this difference is an integer.
So $X$ lies on one of finitely many [[hyperbolas|0WN5G117YC]] with foci $A$ and $B$, counting degenerate ones such as the perpendicular bisector and parts of the line $AB$.
In the same way $X$ lies on one of finitely many hyperbolas with foci $A$ and $C$, and two such curves with different foci meet in at most $4$ points.
So $S$ is finite, a contradiction.
The hypothesis that $S$ is infinite is needed: the vertices of a $3$-$4$-$5$ right triangle are three non-collinear points with integer distances.
Integer distances cannot be weakened to rational ones: there are infinite non-collinear sets with all distances rational, for example dense subsets of a circle.

## Explanations

### L1 human

If infinitely many points in the plane have whole-number distances between every two of them, then they all sit on one straight line.
Off a line, integer distances are so restrictive that only finitely many points can fit.

### L2 human

Norman Anning and Paul Erdős proved the theorem in 1945.
The proof is a short argument with hyperbolas, using the definition of a hyperbola as the set of points with a fixed difference of distances to two foci.
In contrast, for every $n$ there are $n$ points in the plane, no three on a line, with all distances integers; one can take suitable points on a circle.
</document>

<document source="https://tamnd.github.io/math2/node/G4CA22W2XX.md" tag="G4CA22W2XX" kind="definition">
# Banach lattice

## Definition

Let $E$ be a real [[Banach space|06GZ956657]] with norm $\norm{\cdot}$, and let $\le$ be a [[partial order|0QV2JS9RJ3]] on $E$.
The pair is a *Banach lattice* if the following hold.

1. If $x \le y$, then $x + z \le y + z$ for all $z \in E$.
2. If $x \le y$ and $t \ge 0$ is real, then $t x \le t y$.
3. Every two elements $x, y$ have a least upper bound $x \vee y$ in $E$.
4. If $\abs{x} \le \abs{y}$, then $\norm{x} \le \norm{y}$, where $\abs{x} = x \vee (-x)$.

## Remarks

Clauses 1 to 3 say that $E$ is a [[Riesz space|99709K2NJM]], and clause 4 says that the norm is a lattice norm.

Example: $C(K)$ for a compact Hausdorff space $K$, with $f \le g$ when $f(t) \le g(t)$ for all $t \in K$.
Clauses 1 and 2 are clear, and $f \vee g$ is the pointwise maximum, which is continuous.
Then $\abs{f}$ is the pointwise absolute value, and $\abs{f} \le \abs{g}$ gives $\sup \abs{f} \le \sup \abs{g}$.

Non-example: $\R^2$ with the coordinatewise order and the norm $\norm{(a, b)} = \abs{a} + \abs{a - b}$.
Clauses 1 to 3 hold, and this is a norm.
For $x = (1, 0)$ and $y = (1, 1)$, one has $\abs{x} \le \abs{y}$, but $\norm{x} = 2 > 1 = \norm{y}$, so clause 4 fails.
</document>

<document source="https://tamnd.github.io/math2/node/G4GMSNAX8X.md" tag="G4GMSNAX8X" kind="concept">
# Deficiency indices

## Summary

The deficiency indices of a symmetric operator are two cardinal numbers that measure how far it is from being self-adjoint.
They are the dimensions of the solution spaces of $A^* u = i u$ and $A^* u = -i u$, where $A^*$ is the adjoint.
By von Neumann's theory, they decide whether the operator has self-adjoint extensions and describe all of them.

## Explanations

### L1 human

A symmetric operator, such as a derivative acting on functions that vanish near the boundary, may fail to be self-adjoint because its domain is too small.
The deficiency indices count the missing directions, in two ways that correspond to the two numbers $i$ and $-i$.
If both counts are zero, nothing is missing and the closure is already self-adjoint.
If they are equal, the operator can be made self-adjoint by adding the missing directions in a balanced way; if they differ, it cannot.
For $-i \, d/dx$ on functions vanishing near the ends of $[0,1]$, both indices are $1$, matching a one-parameter family of boundary conditions.
</document>

<document source="https://tamnd.github.io/math2/node/G4GWBYTW7B.md" tag="G4GWBYTW7B" kind="example">
# Abel's integral equation

## Example

Let $b > 0$ and let $f$ be a given function on $[0, b]$.
*Abel's integral equation* is the equation
$$\int_0^x \frac{\varphi(t)}{\sqrt{x - t}} \, dt = f(x), \quad 0 \le x \le b,$$
for an unknown function $\varphi$ on $[0, b]$.
It is a [[Volterra equation of the first kind|H6ZGEDM4VV]], of [[convolution type|SNFYTC1TTA]], with the [[weakly singular kernel|94TNBAAJYX]] $k(x - t) = (x - t)^{-1/2}$.
More generally, Abel's equation with exponent $\alpha \in (0, 1)$ has the kernel $(x - t)^{-\alpha}$.

Check on an instance: take $\varphi = 1$.
Then $\int_0^x (x - t)^{-1/2} \, dt = 2 \sqrt{x}$, so $\varphi = 1$ solves the equation with $f(x) = 2\sqrt{x}$.
Take $\varphi(t) = t$.
The substitution $t = xu$ gives $\int_0^x t (x - t)^{-1/2} \, dt = x^{3/2} \int_0^1 u (1 - u)^{-1/2} \, du = \tfrac43 x^{3/2}$, using the [[Beta function|QHNWVSC6AQ]] value $B(2, \tfrac12) = \tfrac43$.

## Remarks

The kernel is not continuous on the diagonal, so the reduction to an equation of the second kind by differentiation does not apply.
The solution is instead given by the [[Abel inversion formula|TDMXFY06A2]], $\varphi(x) = \frac{1}{\pi} \frac{d}{dx} \int_0^x \frac{f(t)}{\sqrt{x - t}} \, dt$.
For $f(x) = 2\sqrt{x}$ this gives $\frac{1}{\pi} \frac{d}{dx} (\pi x) = 1$, as expected.
Abel met this equation in a mechanical problem that generalizes the [[tautochrone problem|A56P5S5D6Z]].

## Explanations

### L1 human

Abel's equation asks to recover a function from a weighted running integral, where the weight $1/\sqrt{x - t}$ blows up at the current point $x$.
The left side is, up to the factor $\sqrt{\pi}$, the Riemann–Liouville integral of order one half of $\varphi$.
So solving the equation means taking a "half derivative" of $f$.
It is one of the oldest integral equations, and it appears in mechanics, in tomography and in the study of fractional calculus.
</document>

<document source="https://tamnd.github.io/math2/node/G4QDM01P6J.md" tag="G4QDM01P6J" kind="concept">
# Homotopy lifting property

## Summary

A map $p \colon E \to B$ has the homotopy lifting property with respect to a space $X$ if [[homotopies|9YA6S6HD9V]] of maps $X \to B$ lift to $E$.
The lift must start at a given lift of the starting map.
This property defines the fibrations of homotopy theory: [[Hurewicz fibrations|Y773B7EQA7]] have it for all spaces, and [[Serre fibrations|PB9JXQ2KC3]] for all cubes.
It is the formal dual of the [[homotopy extension property|D42X0F1EG1]] that defines [[cofibrations|W5WS5EY8ZM]].

## Explanations

### L1 human

Picture $E$ as a space sitting over $B$, with $p$ the shadow map.
A homotopy in $B$ is a moving shadow, and lifting it means finding a motion in $E$ whose shadow is exactly that motion.
The property asks that such a motion in $E$ exists whenever the starting position is given.
For the covering map $\R \to S^1$, $t \mapsto e^{2\pi i t}$, a path that winds once around the circle lifts to a path in $\R$ that moves from $0$ to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/G4QVDEXYTV.md" tag="G4QVDEXYTV" kind="definition">
# Gabor frame

## Definition

For $x, \omega \in \R$ and $g \in L^2(\R)$, let $T_x g(t) = g(t - x)$ and $M_\omega g(t) = e^{2\pi i \omega t} g(t)$.
Let $g \in L^2(\R)$ be nonzero and let $a, b > 0$.
The *Gabor system* $\mathcal{G}(g, a, b)$ is the family $(M_{bn} T_{ak} g)_{k, n \in \Z}$.
It is a *Gabor frame* if it is a frame of $L^2(\R)$: there are constants $0 < A \le B < \infty$ with
$$A \norm{f}^2 \le \sum_{k, n \in \Z} \abs{\ip{f}{M_{bn} T_{ak} g}}^2 \le B \norm{f}^2 \quad \text{for all } f \in L^2(\R).$$

## Remarks

The coefficients $\ip{f}{M_{bn} T_{ak} g}$ are the values $V_g f(ak, bn)$ of the [[short-time Fourier transform|3DHNP5NEWH]] on the lattice $a\Z \times b\Z$.
If $\mathcal{G}(g, a, b)$ is a frame, then $ab \le 1$; for $ab = 1$ it is a frame only if it is a Riesz basis.
For the Gaussian $g(t) = e^{-\pi t^2}$, the system $\mathcal{G}(g, a, b)$ is a frame exactly when $ab < 1$, a result of Lyubarskii and of Seip.

Example: let $g = 1_{[0, 1)}$ and $a = b = 1$.
The translates $T_k g$ are the indicators of the intervals $[k, k+1)$, which tile $\R$.
On each such interval the functions $e^{2\pi i n t}$, $n \in \Z$, form an orthonormal basis of $L^2([k, k+1))$.
So $\mathcal{G}(g, 1, 1)$ is an orthonormal basis of $L^2(\R)$, hence a Gabor frame with $A = B = 1$.

Non-example: with $g = 1_{[0, 1)}$, $a = 2$ and $b = 1$, every function in the system vanishes on $[1, 2)$.
So a nonzero $f$ supported in $[1, 2)$ has all coefficients $0$, and the lower frame bound fails.
</document>

<document source="https://tamnd.github.io/math2/node/G4R4MWWP3C.md" tag="G4R4MWWP3C" kind="statement">
# Hilbert's inequality

## Statement

Let $n \ge 1$ and let $u_1, \dots, u_n$ be [[complex numbers|09FTAET5Z4]].
Then
$$\Big| \sum_{\substack{1 \le r, s \le n \\ r \ne s}} \frac{u_r \overline{u_s}}{r - s} \Big| \le \pi \sum_{r=1}^{n} \abs{u_r}^2.$$

## Remarks

Proof idea: the function $\varphi(t) = i(\pi - t)$ on $(0, 2\pi)$ has Fourier coefficients $1/k$ for $k \ne 0$ and $0$ for $k = 0$, and $\abs{\varphi} \le \pi$.
So the double sum equals $\frac{1}{2\pi} \int_0^{2\pi} \varphi(t) \, \abs{P(t)}^2 \, dt$ for the trigonometric polynomial $P(t) = \sum_r u_r e^{-i r t}$.
Bounding $\abs{\varphi}$ by $\pi$ and using [[Parseval's identity|047RFCNVSK]] $\frac{1}{2\pi} \int_0^{2\pi} \abs{P}^2 = \sum_r \abs{u_r}^2$ gives the result.

The constant $\pi$ cannot be replaced by a smaller one that works for all $n$.

A closely related inequality, also called Hilbert's inequality, says that for nonnegative $a_m, b_n$,
$$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{a_m b_n}{m + n} \le \pi \Big( \sum_{m=1}^{\infty} a_m^2 \Big)^{1/2} \Big( \sum_{n=1}^{\infty} b_n^2 \Big)^{1/2}.$$

## Explanations

### L1 human

Hilbert's inequality controls a double sum in which each pair of terms is weighted by $1/(r - s)$.
These weights do not shrink fast, and they change sign, so it is not obvious that the sum is small.
The inequality says it is never more than $\pi$ times the total squared size of the numbers.

### L2 human

The matrix with entries $1/(r - s)$ for $r \ne s$ is a discrete version of the Hilbert transform.
The inequality says that this matrix has operator norm at most $\pi$ on $\C^n$ for every $n$, uniformly in $n$.
The proof is a model of a common technique: write a matrix as the Fourier coefficients of a bounded function, then use Parseval's identity.
</document>

<document source="https://tamnd.github.io/math2/node/G4SW7HJG88.md" tag="G4SW7HJG88" kind="concept">
# Hurwitz zeta function

## Summary

The Hurwitz zeta function $\zeta(s, a)$ is the sum of $(n + a)^{-s}$ over all integers $n \ge 0$.
It is a shifted version of the [[Riemann zeta function|1XWBY4N0VT]], which is the case $a = 1$.
Every [[Dirichlet L-function|9D2Z543EJQ]] modulo $q$ is $q^{-s}$ times a finite linear combination of the functions $\zeta(s, r/q)$, so their analytic properties follow from those of $\zeta(s, a)$.

## Explanations

### L1 human

The Riemann zeta function adds up $1/n^s$ for $n = 1, 2, 3, \ldots$.
The Hurwitz zeta function does the same, but starts the counting at a point $a$ instead of at $1$: it adds $1/a^s$, $1/(a+1)^s$, $1/(a+2)^s$, and so on.
For $a = 1$ you get back the Riemann zeta function.
For $a = 1/2$ you add the $s$-th powers of $2, 2/3, 2/5, \ldots$, and this sum equals $(2^s - 1)\zeta(s)$.
Sums over an arithmetic progression, such as $1/1^s + 1/4^s + 1/7^s + \cdots$, are Hurwitz zeta values up to a factor.
</document>

<document source="https://tamnd.github.io/math2/node/G4WSPQJ6W1.md" tag="G4WSPQJ6W1" kind="statement">
# Art gallery theorem

## Statement

Let $P \subset \R^2$ be a simple polygon with $n \ge 3$ vertices.
Say that a point $g \in P$ sees a point $x \in P$ when the segment from $g$ to $x$ lies in $P$.

1. There is a set of at most $\floor{n/3}$ vertices of $P$ such that every point of $P$ is seen by at least one of them.
2. For every $n \ge 3$ there is a simple polygon $P$ with $n$ vertices that no set of fewer than $\floor{n/3}$ points of $P$ sees entirely.

## Remarks

The theorem is due to Václav Chvátal; the short proof below is due to Steve Fisk.
Take a [[triangulation|Q6KKBP3RRP]] of $P$, and consider the graph whose edges are the sides of $P$ and the diagonals used.
This graph can be properly coloured with $3$ colours (its [[chromatic number|0VHJRXNAAS]] is $3$), by removing an ear triangle, colouring the rest, and giving the last vertex the missing colour.
Each triangle then has one vertex of each colour.
The smallest colour class has at most $\floor{n/3}$ vertices, and it contains a vertex of every triangle, which sees the whole triangle because triangles are convex.

For sharpness, a comb with $k$ narrow teeth has $n = 3k$ vertices, and no point sees into two different teeth, so $k = n/3$ guards are needed.

## Explanations

### L1 human

Picture an art gallery whose floor plan is a polygon, and guards who stand still but can turn around.
The theorem says that one guard for every three corners is always enough.
Some galleries, shaped like combs, really need that many.

### L2 human

The proof is a model use of polygon triangulation: cut, colour, and count.
The bound $\floor{n/3}$ is a worst case over all polygons; for a given polygon the minimum number of guards can be much smaller, and computing it is NP-hard.
Many variants exist, such as orthogonal polygons, where $\floor{n/4}$ guards suffice, and polygons with holes.
</document>

<document source="https://tamnd.github.io/math2/node/G51E8MW7J1.md" tag="G51E8MW7J1" kind="concept">
# Residue field of a point

## Summary

The residue field $\kappa(x)$ of a point $x$ of a [[scheme|5A35TYZD02]] is the [[residue field|NJ3APQ8BHH]] of the local ring of the scheme at $x$.
It is the field in which functions near $x$ take their "value at $x$".
Different points of one scheme can have residue fields of different characteristics, as for $\operatorname{Spec} \Z$.

## Explanations

### L1 human

On a variety over the complex numbers, the value of a polynomial at a point is a complex number.
On a scheme, the value of a function at a point lives in the residue field of that point, and this field can change from point to point.
On $\operatorname{Spec} \Z$, an integer $n$ has value $n \bmod p$ at the point $(p)$, in the field with $p$ elements.
At the point $(0)$ its value is $n$ itself, in the field $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/G596MHC68T.md" tag="G596MHC68T" kind="definition">
# Coherent sheaf

## Definition

Let $X$ be a locally [[Noetherian scheme|ND6XEY68P4]].
A [[quasi-coherent sheaf|KPKND0K7W6]] $\mathcal{F}$ on $X$ is *coherent* if $X$ has a cover by affine open sets $U = \operatorname{Spec} A$ with the following property.
For each such $U$, $\mathcal{F}|_U \cong \widetilde{M}$ for a [[finitely generated|05QY1VEBYD]] $A$-module $M$.

## Remarks

On $\mathbb{A}^1_k = \operatorname{Spec} k[t]$ over a field $k$, the skyscraper sheaf $\widetilde{M}$ with $M = k[t]/(t)$ is coherent, since $M$ is generated by one element.
On $\operatorname{Spec} \Z$, the sheaf $\widetilde{\Q}$ is quasi-coherent but not coherent.
For it to be coherent, $\Q = \widetilde{\Q}(\operatorname{Spec} \Z)$ would have to be a finitely generated $\Z$-module, and it is not.
If $\mathcal{F}$ is coherent on a locally Noetherian scheme, then $\mathcal{F}(U)$ is finitely generated over $A$ for every affine open $U = \operatorname{Spec} A$.
On a general ringed space, Serre's definition asks that $\mathcal{F}$ is locally finitely generated and that every finitely generated subsheaf over an open set is locally finitely presented.
On a locally Noetherian scheme the two definitions agree.
</document>

<document source="https://tamnd.github.io/math2/node/G5AN43WE9D.md" tag="G5AN43WE9D" kind="definition">
# Lattice-ordered group

## Definition

Let $(G, \le)$ be an [[ordered group|A49BATWQEH]].
It is a *lattice-ordered group*, or *$\ell$-group*, if the partially ordered set $(G, \le)$ is a [[lattice|WH4SFAACAS]].
That is, any two elements $a, b \in G$ have a least upper bound $a \vee b$ and a greatest lower bound $a \wedge b$ in $G$.

## Remarks

Example: $\Z^2$ under addition with the componentwise order is an $\ell$-group.
Here $(a, b) \vee (c, d) = (\max(a, c), \max(b, d))$ and $(a, b) \wedge (c, d) = (\min(a, c), \min(b, d))$.
Non-example: on $\Z^2$, let $(a, b) \le (c, d)$ when $a \le c$ and $b = d$.
This makes $\Z^2$ an ordered group, but $(0, 0)$ and $(0, 1)$ have no common upper bound, so it is not a lattice.
In an $\ell$-group, multiplication distributes over joins and meets: $c(a \vee b) = ca \vee cb$ and $(a \vee b)c = ac \vee bc$.
The lattice of an $\ell$-group is distributive.
Every $\ell$-group is a [[torsion-free group|VQ4SSKS53D]].
For $a \in G$ one writes $a^+ = a \vee e$, $a^- = a^{-1} \vee e$ and $\abs{a} = a \vee a^{-1}$, and then $a = a^+ (a^-)^{-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/G5FAPAB2G2.md" tag="G5FAPAB2G2" kind="definition">
# Hodge Laplacian

## Definition

Let $(M, g)$ be an oriented [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$, and let $\Omega^k(M)$ be the space of smooth [[differential $k$-forms|0E3G0GRPZ6]].
Let $d \colon \Omega^k(M) \to \Omega^{k+1}(M)$ be the [[exterior derivative|0ZHSTAG8ZY]], and let $\star$ be the [[Hodge star|1Y4WX4XQWJ]].
The *codifferential* $\delta \colon \Omega^k(M) \to \Omega^{k-1}(M)$ is
$$\delta = (-1)^{n(k+1)+1} \star d \star.$$
The *Hodge Laplacian* on $k$-forms is
$$\Delta = d \delta + \delta d \colon \Omega^k(M) \to \Omega^k(M).$$

## Remarks

The sign in $\delta$ makes it the formal adjoint of $d$: $\int_M \ip{d\alpha}{\beta} \, \mathrm{vol} = \int_M \ip{\alpha}{\delta\beta} \, \mathrm{vol}$ when one of the forms has compact support.
Changing the orientation changes $\star$ by a sign twice, so $\delta$ and $\Delta$ are defined on any Riemannian manifold.
Some authors write $\Delta_d$ or $\Delta_H$; it is also called the Laplace–de Rham operator.
It is a nonnegative, formally self-adjoint, elliptic operator of order $2$, and $\Delta = (d + \delta)^2$.

Example: on $\R^n$ with the flat metric, let $f$ be a function.
Then $df = \sum_j \partial_j f \, dx_j$ and $\delta(\sum_j a_j \, dx_j) = -\sum_j \partial_j a_j$, so $\Delta f = \delta df = -\sum_j \partial_j^2 f$.
So on functions $\Delta = -\operatorname{div} \operatorname{grad}$, the [[Laplace–Beltrami operator|F4NWZNTRMD]] with the sign that makes it nonnegative.
On $k$-forms of $\R^n$, $\Delta(\sum_I f_I \, dx_I) = \sum_I (-\sum_j \partial_j^2 f_I) \, dx_I$.

Non-example: the operator $d \delta$ alone is not the Hodge Laplacian; on functions it is zero, since $\delta f = 0$ for a $0$-form.

On a closed manifold, the [[Hodge theorem|A127WFVYKW]] identifies $\ker \Delta$ on $k$-forms with the de Rham cohomology $H^k(M; \R)$.
The [[Weitzenböck identity|BE9GTWMN0B]] writes $\Delta$ as a rough Laplacian plus a curvature term.

## Explanations

### L1 human

The Hodge Laplacian is the combination $d\delta + \delta d$ of the exterior derivative and its adjoint, acting on differential forms.
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<document source="https://tamnd.github.io/math2/node/G5JKJEAKZ9.md" tag="G5JKJEAKZ9" kind="statement">
# Thie's theorem on multiplicity

## Statement

Let $\Omega \subseteq \C^n$ be open, let $A \subseteq \Omega$ be an [[analytic subset|GB7M582M5Y]] of pure [[dimension|AN0BYRKNRA]] $p \ge 1$ and let $x \in A$.
Let $\pi \colon \C^n \to \C^p$ be a generic complex linear map.
Then the restriction of $\pi$ to $A$ near $x$ is a branched covering of a neighbourhood of $\pi(x)$.
Let $m_x(A)$ be its number of sheets, the multiplicity of $A$ at $x$.
Let $\operatorname{vol}_{2p}$ denote the $2p$-dimensional volume of the set of [[regular points|ZW5A88PCPK]] of $A$.
Then
$$\lim_{r \to 0^+} \frac{\operatorname{vol}_{2p}\bigl(A \cap B(x, r)\bigr)}{\pi^p r^{2p} / p!} = m_x(A).$$
In other words, the [[Lelong number|WVB60RKC3Z]] at $x$ of the [[current|360H7MD9RD]] of integration $[A]$ equals $m_x(A)$.

## Remarks

The denominator $\pi^p r^{2p} / p!$ is the volume of a ball of radius $r$ in $\C^p = \R^{2p}$.
The proof uses a generic linear projection $\pi$, under which $A$ covers a neighbourhood of $\pi(x)$ with $m_x(A)$ sheets.
It then compares, as $r \to 0$, the volume of the part of each sheet near $x$ with the volume of its image in $\C^p$.

Example: $A = \set{z_1 z_2 = 0} \subseteq \C^2$ is the union of two lines through $0$.
The set $A \cap B(0, r)$ is two discs of area $\pi r^2$, so the ratio is $2$, and indeed a generic projection to $\C$ has two sheets.
At a regular point the multiplicity is $1$, and the ratio tends to $1$.

## Explanations

### L1 human

Near a singular point, an analytic set may look like several sheets passing through the same point.
The multiplicity counts these sheets.
The theorem says that one can count them by measuring area.
The area of the set inside a small ball is about the number of sheets times the area of a flat disc of the same radius.

### L2 human

The theorem links an algebraic invariant, the multiplicity of the local ring of $A$ at $x$, with a metric invariant, the density of $[A]$.
It is the reason why the points of multiplicity at least $c$ appear as upper level sets of Lelong numbers.
It is also the model case for Lelong numbers of general closed positive currents.
</document>

<document source="https://tamnd.github.io/math2/node/G5MKMRTT62.md" tag="G5MKMRTT62" kind="definition">
# Sequence space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$.
Let $\omega$ be the set of all [[sequences|0QE8STGJH2]] $x = (x_n)_{n \ge 1}$ with terms in $\mathbb{K}$.
With termwise addition $(x + y)_n = x_n + y_n$ and scalar multiplication $(\lambda x)_n = \lambda x_n$, the set $\omega$ is a [[vector space|05EGHEQZAW]] over $\mathbb{K}$.
A *sequence space* is a [[linear subspace|0EPTJA5QV4]] of $\omega$.

## Remarks

Example: the set $\ell^\infty$ of bounded sequences is a sequence space.
It contains the zero sequence, and if $\abs{x_n} \le M$ and $\abs{y_n} \le N$ for all $n$, then $\abs{x_n + y_n} \le M + N$ and $\abs{\lambda x_n} \le \abs{\lambda} M$.
Other examples are $\omega$ itself, the [[spaces ℓ^p|08P6T2WH6R]], the space $c$ of convergent sequences and the space $c_0$ of null sequences.
Another is the space $\varphi$ of sequences with only finitely many nonzero terms.

Non-example: the set of sequences with $x_1 = 1$ is not a sequence space, because it does not contain the zero sequence.

The word is often used for a sequence space together with a norm or a topology, as in the theory of FK-spaces and BK-spaces.
</document>

<document source="https://tamnd.github.io/math2/node/G5PTXQ0QK3.md" tag="G5PTXQ0QK3" kind="proof">
# Proof that groups of prime order are cyclic

## Proof

1. Let $G$ be a group of order $p$, where $p$ is a [[prime number|0SN0MAEKYX]].
2. Since $p \ge 2$, $G$ contains an element $g$ that is not the identity $e$.
3. [[The order of $g$ divides $p$|08V7JHS3QZ]]. Thus the order of $g$ is $1$ or $p$.
4. Only the identity has order $1$, and $g \ne e$. Thus $g$ has order $p$.
5. Thus the [[cyclic subgroup|0YT586TKXF]] $\langle g \rangle$ has $p$ elements, so $\langle g \rangle = G$.
6. Thus $G$ is [[cyclic|0B85Q41RP6]], with generator $g$.

## Idea

A prime has only two divisors. An element that is not the identity cannot have order one, so its order is the full order of the group.
</document>

<document source="https://tamnd.github.io/math2/node/G5PVJMM37Z.md" tag="G5PVJMM37Z" kind="concept">
# Purely transcendental extension

## Summary

A purely transcendental extension of a field $K$ is an extension generated by [[algebraically independent|KV4PT6KW10]] elements.
It is isomorphic to a [[field of rational functions|4V81611P9N]] over $K$ in some set of variables.
Every field extension is an algebraic extension of a purely transcendental one, so these extensions are the building blocks of [[transcendental extensions|12JQCGTQ8W]].

## Explanations

### L1 human

A purely transcendental extension adds free variables and nothing else.
The field $\Q(x, y)$ of fractions of polynomials in $x$ and $y$ is the model example.
Adding $\sqrt 2$ as well gives an extension that is no longer purely transcendental, since $\sqrt 2$ is tied to $\Q$ by the equation $t^2 = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/G5RB8T2WKD.md" tag="G5RB8T2WKD" kind="definition">
# Upwind scheme

## Definition

Let $a \in \R$, and consider the [[transport equation|PVD7VZJZ9B]] $u_t + a u_x = 0$.
Fix $\Delta x > 0$ and $\Delta t > 0$, put $\nu = a \Delta t / \Delta x$, and let $u_j^n$ approximate $u(j \Delta x, n \Delta t)$.
The first-order *upwind scheme* is the explicit [[finite difference|X7HYEBA5SA]] scheme
$$u_j^{n+1} = \begin{cases} u_j^n - \nu \left( u_j^n - u_{j-1}^n \right) & \text{if } a \ge 0, \\ u_j^n - \nu \left( u_{j+1}^n - u_j^n \right) & \text{if } a < 0. \end{cases}$$
Equivalently, with $a^+ = \max(a, 0)$ and $a^- = \min(a, 0)$,
$$u_j^{n+1} = u_j^n - \frac{\Delta t}{\Delta x} \left( a^+ (u_j^n - u_{j-1}^n) + a^- (u_{j+1}^n - u_j^n) \right).$$

## Remarks

Example: let $a > 0$ and $0 \le \nu \le 1$.
Then $u_j^{n+1} = (1 - \nu) u_j^n + \nu u_{j-1}^n$ is a convex combination, so $\min_j u_j^n \le u_j^{n+1} \le \max_j u_j^n$, and the scheme is stable in the maximum norm.
The condition $\nu \le 1$ is the [[Courant–Friedrichs–Lewy condition|FQVVNYV6N8]]: the numerical domain of dependence must contain the characteristic through the point.
Non-example: for $a > 0$, the downwind scheme $u_j^{n+1} = u_j^n - \nu (u_{j+1}^n - u_j^n)$ uses the wrong side.
It multiplies the Fourier mode $e^{i j \xi}$ by $g = 1 + \nu - \nu e^{i\xi}$, with $\abs{g}^2 = 1 + 2\nu(1 + \nu)(1 - \cos \xi) > 1$ for $\xi \ne 0$, so it is unstable for every $\nu > 0$.
For a scalar conservation law $u_t + f(u)_x = 0$, upwinding takes the difference on the side given by the sign of the characteristic speed $f'(u)$.
The scheme is first-order accurate in $\Delta x$ for fixed $\nu$.

## Explanations

### L1 human

The difference is taken towards the side the wave comes from.
This simple rule is what makes the scheme stable.
</document>

<document source="https://tamnd.github.io/math2/node/G5S1M9.md" tag="G5S1M9" kind="concept">
# Group

## Summary

A group is a set $G$ with a binary operation $\cdot$ that has these three properties.
The operation is associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c$ in $G$.
There is an identity element $e$ in $G$ such that $e \cdot a = a \cdot e = a$ for all $a$ in $G$.
Each element $a$ of $G$ has an inverse $a^{-1}$ in $G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$.

The order of a finite group is its number of elements.

## Remarks

[[Lagrange's theorem|3C53TD]] gives a property of the order of a finite group.
</document>

<document source="https://tamnd.github.io/math2/node/G5TWXV565S.md" tag="G5TWXV565S" kind="concept">
# Ramsey theory

## Summary

Ramsey theory studies the statements that every large enough structure, divided into finitely many parts, has a part that contains a given regular substructure.
Its founding result is [[Ramsey's theorem|00TQV3F5S4]] on monochromatic cliques in coloured complete graphs, with the [[Ramsey numbers|0K9QVPJYJF]] as its quantitative side.
In the integers it includes [[Schur's theorem|V6XJQXZ7E8]], [[van der Waerden's theorem|V96FRZ2ME1]] and [[Rado's theorem|BZSGRCTFH4]], and the abstract [[Hales–Jewett theorem|14SGZ5HA4Y]] contains many of these.
Density versions, such as [[Szemerédi's theorem|B23B0QEMN5]], link it to [[additive combinatorics|3NM362A2JG]] and ergodic theory.

## Explanations

### L1 human

Ramsey theory says that complete disorder is impossible.
If you colour the objects of a big enough structure with a few colours, some pattern always appears in a single colour.
The [[pigeonhole principle|0HGVCWQGY3]] is the simplest case: with $n$ colours and $n + 1$ objects, two objects get the same colour.
A famous example: among any six people, there are three who all know each other or three who are all strangers to each other.
The hard questions are how big "big enough" must be, and these numbers are often unknown even for small cases.
</document>

<document source="https://tamnd.github.io/math2/node/G5WT6QN8WR.md" tag="G5WT6QN8WR" kind="concept">
# Convergence in measure

## Summary

A sequence $(f_n)$ of [[measurable functions|YYNCJSR0PJ]] converges in measure to $f$ if bad sets shrink.
For each $\varepsilon > 0$, the set where $f_n$ differs from $f$ by at least $\varepsilon$ must have measure tending to $0$.
It is weaker than uniform convergence and than convergence in $L^p$.
In probability theory it is called convergence in probability.

## Explanations

### L1 human

Convergence in measure does not ask the functions to approach $f$ at every point.
It asks that the region where $f_n$ is far from $f$ shrinks to size $0$.
For example, a narrow bump of height $1$ that sweeps again and again across $[0, 1]$, getting narrower each pass, converges to $0$ in measure.
Yet at each point the bump returns infinitely often, so the sequence converges at no point.
</document>

<document source="https://tamnd.github.io/math2/node/G653QANYYD.md" tag="G653QANYYD" kind="definition">
# Set of uniqueness

## Definition

Let $\mathbb{T} = \R / 2\pi\Z$ be the circle.
A subset $E \subset \mathbb{T}$ is a *set of uniqueness* if the following holds.
Whenever a [[trigonometric series|FZK2M8KGXQ]] $\sum_{n \in \Z} c_n e^{inx}$ with complex coefficients satisfies
$$\lim_{N \to \infty} \sum_{n=-N}^{N} c_n e^{inx} = 0 \quad \text{for every } x \in \mathbb{T} \setminus E,$$
then $c_n = 0$ for all $n \in \Z$.
A subset of $\mathbb{T}$ that is not a set of uniqueness is a *set of multiplicity*.

## Remarks

Sets of uniqueness are often called U-sets, and sets of multiplicity M-sets.
A subset of a set of uniqueness is again a set of uniqueness.

Example: the empty set is a set of uniqueness; this is [[Cantor's uniqueness theorem|NSS3XW4BYZ]].
Every [[countable set|01TQB88H4C]] is also a set of uniqueness, a result of W. H. Young.

Non-example: let $E$ contain an open arc $I$ of $\mathbb{T}$.
Choose a nonzero infinitely differentiable function $g$ on $\mathbb{T}$ that vanishes outside $I$.
Its Fourier series converges absolutely to $g$ at every point, so it converges to $0$ outside $E$.
Its coefficients are not all $0$, because $g \ne 0$, so $E$ is a set of multiplicity.

More generally every set of positive Lebesgue measure is a set of multiplicity, and a Borel set of uniqueness has measure zero.
The converse fails: [[Menshov's example|AW16MYJV5Q]] is a closed set of measure zero that is a set of multiplicity.
The middle-thirds [[Cantor set|0BXNDZFQME]] is a set of uniqueness.
</document>

<document source="https://tamnd.github.io/math2/node/G6FND7X69J.md" tag="G6FND7X69J" kind="concept">
# Coxeter group

## Summary

A Coxeter group is a [[group|G5S1M9]] generated by elements of order $2$.
The only relations prescribe the order of the product of each pair of generators.
The finite Coxeter groups are exactly the finite groups generated by [[reflections|06BRDF3F0E]] of a Euclidean space, such as the [[symmetric groups|0515A6D2Y1]] and the [[dihedral groups|0JZ27DMJ5D]].
Infinite Coxeter groups include the affine Weyl groups and groups generated by reflections in the sides of hyperbolic polygons whose angles have the form $\pi / m$.
They appear in Lie theory, in the theory of buildings and in geometric group theory.

## Explanations

### L1 human

Put two mirrors at an angle of $\pi / m$.
Each reflection undoes itself, so it has order $2$, and the product of the two reflections is a rotation by $2\pi / m$, of order $m$.
The group generated by the two reflections is the dihedral group of order $2m$, the symmetries of a regular $m$-gon.
A Coxeter group is any group built in the same way from several such mirrors, where only these pairwise rules are imposed.
</document>

<document source="https://tamnd.github.io/math2/node/G6GMSEZDB1.md" tag="G6GMSEZDB1" kind="concept">
# Affine group

## Summary

The affine group of an [[affine space|09XNAPTJQ7]] is the group of all its invertible [[affine maps|0JF0TVYGWT]].
On $F^n$ these are the maps $x \mapsto Ax + b$ with $A$ an invertible matrix and $b$ a vector.
It is the [[semidirect product|0H3WZXCZNF]] of the [[translations|087VJM5Q77]] and the [[general linear group|0FCHE902D7]].
In the [[Erlangen program|9QW65PGEPW]], it is the group that defines affine geometry.

## Explanations

### L1 human

An affine map moves the plane so that straight lines stay straight and parallel lines stay parallel.
It may stretch, shear, rotate, reflect and shift, but it may not bend.
For example, $(x, y) \mapsto (2x + y + 1, \, y - 3)$ is an invertible affine map of the plane.
The affine group collects all such maps, with composition as the operation.
</document>

<document source="https://tamnd.github.io/math2/node/G6J21TGRMM.md" tag="G6J21TGRMM" kind="concept">
# Singular perturbation

## Summary

A problem with a small parameter $\varepsilon$ is singularly perturbed when setting $\varepsilon = 0$ does not give a uniformly good approximation of its solutions.
A typical cause is a small parameter in front of the highest derivative.
Then the reduced equation has lower order and cannot meet all boundary or initial conditions.
The solutions then change rapidly in thin regions, such as boundary layers.
Singular perturbations appear in fluid flow at high Reynolds number, chemical kinetics and slow-fast systems.

## Explanations

### L1 human

Sometimes a tiny term in an equation can be dropped without much harm, and sometimes it cannot.
In $\varepsilon y'' + y' = 0$ with $y(0) = 0$ and $y(1) = 1$, dropping $\varepsilon y''$ leaves $y' = 0$, whose solutions are constants.
A constant cannot be $0$ at one end and $1$ at the other.
The true solution is close to $1$ on most of the interval, but it drops steeply to $0$ in a thin layer near $x = 0$.
That thin layer is the mark of a singular perturbation.
</document>

<document source="https://tamnd.github.io/math2/node/G6S707SYF9.md" tag="G6S707SYF9" kind="definition">
# Martin kernel

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a domain that has a [[Green's function|M36SD56MWQ]] $G \colon \Omega \times \Omega \to (0, +\infty]$.
This holds for every domain when $n \ge 3$, and for a plane domain whose complement is not a polar set.
Fix a reference point $x_0 \in \Omega$.
The *Martin kernel* of $\Omega$ with reference point $x_0$ is the function
$$M(x, y) = \frac{G(x, y)}{G(x_0, y)} \qquad \text{for } x \in \Omega,\ y \in \Omega \setminus \set{x_0}.$$

## Remarks

For fixed $y$, the function $M(\cdot, y)$ is positive and [[harmonic|0DEB0JMN0W]] on $\Omega \setminus \set{y}$, and $M(x_0, y) = 1$.
By [[Harnack's inequality|YNH832D50W]], when $y$ stays away from a compact set $K \ni x_0$, the functions $M(\cdot, y)$ are uniformly bounded on $K$.
Let $(y_k)$ be a sequence that leaves each compact subset of $\Omega$.
Then a subsequence of $M(\cdot, y_k)$ converges locally uniformly to a positive harmonic function $h$ with $h(x_0) = 1$.
Example: let $\Omega$ be the unit ball of $\R^n$ and $x_0 = 0$.
If $y_k \to \zeta$ with $\abs{\zeta} = 1$, then $M(x, y_k) \to (1 - \abs{x}^2) / \abs{x - \zeta}^n$, the [[Poisson kernel|TCYSWCTYYM]] for the normalised surface measure.
A change of reference point $x_0$ multiplies $M(\cdot, y)$ by the factor $G(x_0, y) / G(x_0', y)$.
</document>

<document source="https://tamnd.github.io/math2/node/G6WNYB0WMY.md" tag="G6WNYB0WMY" kind="statement">
# Kruskal–Katona theorem

## Statement

Let $k \ge 1$ be an integer.
For a [[family|0C21Q694AG]] $\mathcal{F}$ of $k$-element sets, the *shadow* $\partial \mathcal{F}$ is the family of all $(k-1)$-element sets that are contained in some member of $\mathcal{F}$.
Every positive integer $m$ can be written in exactly one way as
$$m = \binom{a_k}{k} + \binom{a_{k-1}}{k-1} + \cdots + \binom{a_t}{t}$$
with integers $a_k > a_{k-1} > \cdots > a_t \ge t \ge 1$.
If $\mathcal{F}$ is a family of $m$ sets of size $k$, then
$$|\partial \mathcal{F}| \ge \binom{a_k}{k-1} + \binom{a_{k-1}}{k-2} + \cdots + \binom{a_t}{t-1}.$$

## Remarks

The bound is attained by the first $m$ sets of size $k$ of positive integers in the colexicographic order.
In this order, $A$ comes before $B$ when the largest element of $A \mathbin{\triangle} B$ lies in $B$.
The proof shows, with compression operators, that one may assume $\mathcal{F}$ is an initial segment of this order, and then counts.

Example: let $k = 2$ and $m = 3$, so $m = \binom{3}{2}$ and the bound is $\binom{3}{1} = 3$.
A graph with $3$ edges, seen as a family of $2$-sets, covers at least $3$ vertices, and the triangle covers exactly $3$.

A simpler form due to Lovász: if $|\mathcal{F}| = \binom{x}{k}$ for a real number $x \ge k$, then $|\partial \mathcal{F}| \ge \binom{x}{k-1}$, where $\binom{x}{j} = x(x-1)\cdots(x-j+1)/j!$.

## Explanations

### L1 human

Given many sets of the same size, look at all the smaller sets you get by removing one element.
The theorem says exactly how few such smaller sets there can be.
The answer is always attained by packing the sets as tightly as possible, using as few elements as possible.

### L2 human

The theorem characterises the possible $f$-vectors of simplicial complexes, since the faces of a complex form a family closed under taking shadows.
It implies the [[Erdős–Ko–Rado theorem|FJ92Q6VTJH]].
Its numerical form, through the $k$-binomial representation of $m$, reappears in Macaulay's theorem on Hilbert functions and in the [[g-theorem|4CZMWQVC30]] for polytopes.
</document>

<document source="https://tamnd.github.io/math2/node/G6X7SZSNC5.md" tag="G6X7SZSNC5" kind="concept">
# Second-order logic

## Summary

Second-order logic extends [[first-order logic|086QKPF0A6]] with variables for relations and functions on the domain, and with quantifiers over them.
In its standard semantics these variables range over all relations and functions on the domain.
It can describe structures such as the natural numbers up to isomorphism, but it loses the completeness and compactness theorems of first-order logic.

## Explanations

### L1 human

First-order logic can say "for every number $x$", but not "for every set of numbers $X$".
Second-order logic can say both.
With it, the induction principle becomes one sentence: every set of numbers that contains $0$ and is closed under successor contains all numbers.
This one sentence, with the other Peano axioms, pins down the natural numbers completely.
The price is that no proof system can capture all its valid sentences.
</document>

<document source="https://tamnd.github.io/math2/node/G78W36EGMB.md" tag="G78W36EGMB" kind="concept">
# Inner model

## Summary

An inner model is a transitive class of sets that contains all [[ordinals|0YSCSX5YZC]] and satisfies the axioms of [[ZF|0HE5WDQ9FJ]].
It is a smaller universe of sets that lives inside the universe $V$ and has the same ordinals.
Inner models give consistency proofs: a statement true in an inner model is consistent with ZF.
The [[constructible universe|6TF2JBTPPJ]] $L$ is the smallest inner model, and the [[core model|A6ZJ8MSCAY]] is a larger inner model built to hold large cardinals.

## Explanations

### L1 human

Think of the universe of all sets as a large building.
An inner model is a smaller building inside it that has the same height, because it has every ordinal, but fewer rooms on each floor.
Inside the smaller building all the usual rules of set theory still hold.
For example, the constructible sets form such a building, and in it the continuum hypothesis is true.
So the continuum hypothesis cannot be disproved from the usual axioms.
</document>

<document source="https://tamnd.github.io/math2/node/G7D6SPTA2C.md" tag="G7D6SPTA2C" kind="definition">
# Rotation number

## Definition

Let $S^1 = \R / \Z$ be the [[circle|060W5CBZP6]], with projection $\pi \colon \R \to S^1$, and let $f \colon S^1 \to S^1$ be an orientation-preserving [[homeomorphism|0QT00PB5G2]].
A *lift* of $f$ is a continuous map $F \colon \R \to \R$ with $\pi \circ F = f \circ \pi$.
Every such lift is increasing and satisfies $F(x + 1) = F(x) + 1$, and two lifts differ by an integer constant.
The *translation number* of a lift $F$ is the limit
$$\rho(F) = \lim_{n \to \infty} \frac{F^n(x) - x}{n},$$
where $F^n$ is the $n$-th iterate of $F$.
The limit exists and does not depend on $x \in \R$.
The *rotation number* of $f$ is $\rho(f) = \rho(F) - \floor{\rho(F)} \in [0, 1)$, which does not depend on the choice of lift $F$.

## Remarks

Example: the rotation $R_\alpha(x) = x + \alpha$ of $S^1$ has the lift $F(x) = x + \alpha$, and $F^n(x) - x = n\alpha$.
So $\rho(F) = \alpha$, and $\rho(R_\alpha)$ is the fractional part of $\alpha$.

Non-example: the doubling map $x \mapsto 2x$ of $S^1$ is not a homeomorphism.
Its lift $F(x) = 2x$ satisfies $F(x + 1) = F(x) + 2$, and $(F^n(x) - x)/n = (2^n - 1) x / n$ has no finite limit for $x \neq 0$, so no rotation number is defined.

The rotation number is invariant under conjugacy: if $h$ is an orientation-preserving homeomorphism of $S^1$, then $\rho(h^{-1} \circ f \circ h) = \rho(f)$.
It depends continuously on $f$ in the uniform topology.
The fractional part is often left out, and $\rho(f)$ is viewed as an element of $\R / \Z$.
</document>

<document source="https://tamnd.github.io/math2/node/G7JH2NXPC0.md" tag="G7JH2NXPC0" kind="conjecture">
# abc conjecture

## Statement

For a positive integer $n$, let $\operatorname{rad}(n)$ be the product of the distinct [[primes|01826KMG2H]] that divide $n$.
For every real number $\varepsilon > 0$, there are only finitely many triples $(a, b, c)$ of [[coprime|0VZ48N083F]] positive integers with $a + b = c$ and
$$c > \operatorname{rad}(abc)^{1 + \varepsilon}.$$

## Remarks

The conjecture was formulated by Joseph Oesterlé and David Masser in 1985.
Shinichi Mochizuki announced a proof in 2012; it is not accepted by most experts, and the conjecture is generally treated as open.
The analogue for polynomials is a theorem, the [[Mason–Stothers theorem|ZWPWF6NJC3]].
The $\varepsilon$ cannot be removed.
For each $k \ge 1$, take $a = 1$, $c = 2^{6k}$ and $b = c - 1$; since $2^6 = 64 \equiv 1 \pmod 9$, the number $9$ divides $b$, so $\operatorname{rad}(abc) \le 2 \cdot b/3 < c$.
For example, $1 + 8 = 9$ has $\operatorname{rad}(1 \cdot 8 \cdot 9) = 6 < 9$, and $5 + 27 = 32$ has radical $30 < 32$.
The conjecture would imply many results, for instance that the Fermat equation $x^n + y^n = z^n$ has no solutions in positive integers for all large $n$.

## Explanations

### L1 human

The radical of a number keeps each prime factor only once, so $\operatorname{rad}(72) = \operatorname{rad}(2^3 \cdot 3^2) = 6$.
If $a + b = c$ and the three numbers share no factor, then usually $c$ is smaller than the radical of $abc$.
The conjecture says that $c$ can exceed the radical only by a little, and even this happens only rarely in a precise sense.
It expresses a tension between adding numbers and multiplying primes, and it has a remarkable list of consequences.
</document>

<document source="https://tamnd.github.io/math2/node/G7RM16N4PF.md" tag="G7RM16N4PF" kind="concept">
# Quasi-algebraically closed field

## Summary

A field is quasi-algebraically closed, or $C_1$, if every [[homogeneous polynomial|2C3PAQ7H50]] with more variables than its degree has a zero other than the zero vector.
Every [[algebraically closed field|0PF6TM8TWT]] is $C_1$, and so is every [[finite field|0ZXZ82FTX2]], by the [[Chevalley–Warning theorem|08895Q5W96]].
By [[Tsen's theorem|V8HDYN3JMY]], the function field of a curve over an algebraically closed field is $C_1$.
The [[Brauer group|BEW48TA9WZ]] of a $C_1$ field is trivial, which is one reason the notion matters.

## Explanations

### L1 human

Over an algebraically closed field, a polynomial in one variable always has a root.
A $C_1$ field has a weaker property for polynomials in many variables: if a form has many variables compared with its degree, it has a nonzero solution.
For instance, over a finite field every quadratic form in $3$ variables has a nonzero zero.
Over $\R$ this fails: $x^2 + y^2 + z^2$ is zero only at the origin.
So $\R$ is not $C_1$, while the finite fields are.
</document>

<document source="https://tamnd.github.io/math2/node/G7T008BEWY.md" tag="G7T008BEWY" kind="concept">
# Admissible function

## Summary

An admissible function of a variational problem is a function that is allowed to compete for the minimum.
It must lie in the domain of the functional and satisfy every constraint of the problem, such as boundary values or a fixed length.
The set of admissible functions fixes the problem as much as the functional does: a change of the class can change whether a minimiser exists.

## Explanations

### L1 human

Before looking for the best curve, one must say which curves are allowed.
In a problem with fixed ends, only smooth enough curves that start and end at the given points take part.
For example, with ends $y(0) = 0$ and $y(1) = 1$, the curves $y = x$ and $y = x^2$ are admissible, while $y = 2x$ is not, since it ends at height $2$.
</document>

<document source="https://tamnd.github.io/math2/node/G7VE41JHHH.md" tag="G7VE41JHHH" kind="concept">
# Hadamard finite part

## Summary

The Hadamard finite part gives a value to an integral that diverges because the integrand has a non-integrable singularity, such as $\int_{-1}^{1} \frac{\varphi(t)}{t^2}\, dt$.
One cuts out a small neighbourhood of the singularity, computes the integral, and drops the terms that blow up as the neighbourhood shrinks.
It extends the [[Cauchy principal value|0SYR10HD4R]], which handles poles of first order.
Finite part integrals are the integrals in [[hypersingular integral equations|QVRP8FSAZM]].

## Explanations

### L1 human

The integral of $1/t^2$ over $(\varepsilon, 1)$ is $1/\varepsilon - 1$, which grows without bound as $\varepsilon \to 0$.
The finite part throws away the growing piece $1/\varepsilon$ and keeps $-1$.
So a positive function can have a negative finite part integral, and this is not a mistake.
The rule is useful because it agrees with differentiating a principal value integral under the integral sign.
</document>

<document source="https://tamnd.github.io/math2/node/G7VG61YD4J.md" tag="G7VG61YD4J" kind="problem">
# Hausdorff moment problem

## Problem

Let $(m_n)_{n \ge 0}$ be a [[sequence|0G9EKM0E15]] of real numbers.
Decide whether there is a positive Borel [[measure|PR11BZQQPG]] $\mu$ on the closed interval $[0, 1]$ such that
$$m_n = \int_{[0,1]} x^n \, d\mu(x) \quad \text{for all } n \ge 0.$$
The numbers $m_n$ are then the [[moments|01XAHPADBA]] of $\mu$.

## Remarks

The problem is solved.
Write $(\Delta m)_n = m_{n+1} - m_n$ for the difference operator.
Hausdorff proved that such a measure exists if and only if the sequence is completely monotone, that is,
$$(-1)^k (\Delta^k m)_n \ge 0 \quad \text{for all } n, k \ge 0.$$
When the measure exists, it is unique.
Uniqueness holds because polynomials are dense in the continuous functions on $[0, 1]$ by the Weierstrass approximation theorem.
For example, $m_n = 1/(n + 1)$ is the moment sequence of Lebesgue measure on $[0, 1]$.

## Explanations

### L1 human

The moments of a mass distribution on $[0, 1]$ are its total mass, its mean position, its mean squared position, and so on.
The problem asks which lists of numbers can occur as such moments.
Not every list works: for instance, the moments can never increase, since $x^{n+1} \le x^n$ on $[0, 1]$.
Hausdorff found an exact test, a sign condition on all repeated differences of the list, and showed that the list then determines the distribution.
</document>

<document source="https://tamnd.github.io/math2/node/G7WP728SZH.md" tag="G7WP728SZH" kind="concept">
# Projection body

## Summary

Let $K$ be a [[convex body|437FSG3419]] in $\R^n$.
Its projection body $\Pi K$ is a convex body given by its [[support function|1H92CYA4ZC]].
At a unit vector $u$, this function is the volume of the shadow of $K$ on the hyperplane orthogonal to $u$.
It packs all the shadows of $K$ into one convex body, which is always symmetric about the origin.
Projection bodies were introduced by Minkowski and are central in the study of affine isoperimetric inequalities.

## Explanations

### L1 human

Shine light on a solid from direction $u$ and measure the area of its shadow on a wall perpendicular to $u$.
The projection body is the convex shape whose width profile is given by these shadow areas.
A cube of side $1$ has shadow area $\abs{u_1} + \abs{u_2} + \abs{u_3}$ for a unit direction $u$, and its projection body is the cube $[-1, 1]^3$.
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# Klein–Gordon equation

## Definition

Let $n \ge 1$ and $m \ge 0$.
The *Klein–Gordon equation* with mass $m$ is the [[hyperbolic equation|92C3S03AQZ]] for $u \colon \R^n \times \R \to \C$:
$$\frac{\partial^2 u}{\partial t^2} - \Delta u + m^2 u = 0,$$
where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] in $x$.
With the speed of light $c$ and the reduced Planck constant $\hbar$ restored, it reads $\frac{1}{c^2} u_{tt} - \Delta u + \frac{m^2 c^2}{\hbar^2} u = 0$.

## Remarks

Example: for $k \in \R^n$ and $\omega = \sqrt{\abs{k}^2 + m^2}$, the plane wave $u(x, t) = e^{i(k \cdot x - \omega t)}$ is a solution, since $u_{tt} = -\omega^2 u$ and $\Delta u = -\abs{k}^2 u$, so $u_{tt} - \Delta u + m^2 u = (-\omega^2 + \abs{k}^2 + m^2) u = 0$.
Non-example: the plane wave $e^{i(k \cdot x - \abs{k} t)}$ solves the [[wave equation|5EHD2XBQDG]] but not the Klein–Gordon equation with $m > 0$, since then the left side is $m^2 u \ne 0$.
For $m = 0$ the equation is the wave equation with speed $1$.
The conserved energy is $E(t) = \frac{1}{2}\int_{\R^n} \big(\abs{u_t}^2 + \abs{\nabla u}^2 + m^2 \abs{u}^2\big)\, dx$.
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# Formal concept analysis

## Summary

Formal concept analysis studies a binary relation between objects and attributes through the complete lattice of its formal concepts.
A formal concept is a pair of a set of objects and a set of attributes that determine each other exactly.
The resulting concept lattice gives a hierarchy of clusters in the data, and every complete lattice arises in this way.
It is used in data analysis and knowledge representation, and it rests on a [[Galois connection|Q5JHEPRYQ6]] between sets of objects and sets of attributes.

## Explanations

### L1 human

Start with a table: rows are objects, columns are attributes, and a cross means that the object has the attribute.
A formal concept is a block of crosses that cannot be made bigger.
It is a set of objects with all the attributes they share, such that no other object has all these attributes.
For example, take the objects "dog", "cat", "trout" and the attributes "has fur", "lives in water", where the dog and the cat have fur and the trout lives in water.
Then the pair ({dog, cat}, {has fur}) is a formal concept.
The formal concepts, ordered by their sets of objects, form a lattice that shows how the concepts of the data are nested.
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<document source="https://tamnd.github.io/math2/node/G852FHZ3W1.md" tag="G852FHZ3W1" kind="concept">
# Bohr compactification

## Summary

The Bohr compactification of a topological group $G$ is the largest compact Hausdorff group into which $G$ maps continuously.
Every continuous homomorphism from $G$ to a compact Hausdorff group factors through it in a unique way.
For a locally compact abelian group it is the dual of the dual group made discrete.
The continuous functions on it correspond to the almost periodic functions on $G$.

## Explanations

### L1 human

Many groups, like the real line, are not compact, but compact groups are easier to analyse.
The Bohr compactification is the best compact group that "sees" $G$: it keeps exactly the information that compact groups can detect.
For the real line, it keeps all the waves $x \mapsto e^{i\xi x}$ but forgets the topology of the frequencies $\xi$.
The resulting compact group is huge, but the continuous functions on it correspond exactly to the almost periodic functions studied by Harald Bohr, such as $\cos x + \cos(\sqrt{2}x)$.
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<document source="https://tamnd.github.io/math2/node/G87CWTXAWC.md" tag="G87CWTXAWC" kind="definition">
# Quasi-category

## Definition

For $n \ge 0$ let $\Delta^n$ be the standard $n$-simplex, as a [[simplicial set|WW81ARB8K9]].
For $0 \le k \le n$ let $\Lambda^n_k \subset \Delta^n$ be the $k$-th horn: the simplicial subset generated by all faces of $\Delta^n$ except the face opposite the vertex $k$.
The horn is *inner* if $0 < k < n$.
A *quasi-category* is a simplicial set $X$ with the following property.
For every $n \ge 2$, every $0 < k < n$, and every map $\sigma \colon \Lambda^n_k \to X$, there is a map $\tau \colon \Delta^n \to X$ whose restriction to $\Lambda^n_k$ is $\sigma$.

## Remarks

The vertices of $X$ are its objects and the $1$-simplices its morphisms.
For $n = 2$, $k = 1$, the condition says that two morphisms $f \colon x \to y$ and $g \colon y \to z$ have a composite: a $2$-simplex with edges $f$, $g$ and some $h \colon x \to z$.
Example: the [[nerve|RV0MKJ3CQ6]] $N(\mathcal{C})$ of a category $\mathcal{C}$ is a quasi-category, and in it every inner horn has exactly one filler.
Example: every [[Kan complex|FDFYCM3YYC]] is a quasi-category, since it has fillers for all horns, inner or not; the singular simplicial set of a topological space is one.
Non-example: the horn $\Lambda^2_1$ itself is not a quasi-category.
Its identity map $\Lambda^2_1 \to \Lambda^2_1$ does not extend to $\Delta^2$, because every $2$-simplex of $\Lambda^2_1$ is degenerate.
A degenerate $2$-simplex cannot have the two distinct nondegenerate edges $0 \to 1$ and $1 \to 2$ as faces.
Quasi-categories are also called weak Kan complexes or $\infty$-categories.
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# Lifting property

## Definition

Let $i \colon A \to B$ and $p \colon X \to Y$ be morphisms in a [[category|A2DE66937D]] $\mathcal{C}$.
The morphism $i$ has the *left lifting property* with respect to $p$ if this holds.
For all morphisms $u \colon A \to X$ and $v \colon B \to Y$ with $p \circ u = v \circ i$, there is a morphism $d \colon B \to X$ with $d \circ i = u$ and $p \circ d = v$.
In that case $p$ is said to have the *right lifting property* with respect to $i$.
If the morphism $d$ is always unique, $i$ and $p$ are *orthogonal*.

## Remarks

The data $u, v$ form a [[commutative square|WV898P4X16]], and $d$ is a diagonal filler of the square.
A common notation is $i \pitchfork p$, or $i \perp p$ for orthogonality.
Fix a class $M$ of morphisms.
The morphisms with the left lifting property with respect to all members of $M$ contain all isomorphisms and are closed under composition.

Example: in the category of sets, let $i \colon \emptyset \to \set{*}$ and let $p \colon X \to Y$ be surjective.
A square is then just a point $y = v(*)$ of $Y$, and a filler is a point $x$ of $X$ with $p(x) = y$.
It exists since $p$ is surjective.

Non-example: with the same $i$, let $p \colon \emptyset \to \set{*}$.
Take $u$ the empty map and $v = \id$; a filler would be a map $\set{*} \to \emptyset$, which does not exist.

In topology, a Hurewicz fibration is a map with the right lifting property with respect to all inclusions $Z \times \set{0} \to Z \times [0, 1]$; this is the [[homotopy lifting property|G4QDM01P6J]].
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# Exceptional collection

## Summary

An exceptional collection in a linear [[triangulated category|T2V7GJJ2EN]] is a finite ordered list of exceptional objects.
Each of them has only scalar endomorphisms and no self-extensions.
There are no morphisms, in any degree, from later objects to earlier ones.
A full exceptional collection splits the category into a semiorthogonal decomposition whose pieces are copies of the derived category of vector spaces.
The basic example is Beilinson's collection $\mathcal{O}, \mathcal{O}(1), \dots, \mathcal{O}(n)$ on projective space.

## Explanations

### L1 human

An exceptional object behaves like a single point: its only endomorphisms are scalars and it has no deformations or extensions with itself.
An exceptional collection is a list of such objects in which maps only go forward along the list.
When the collection generates everything, the whole category is built from these simple pieces in a triangular way, much like a matrix that is upper triangular.
For projective space, the line bundles $\mathcal{O}, \mathcal{O}(1), \dots, \mathcal{O}(n)$ form such a list, which makes its derived category very concrete.
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<document source="https://tamnd.github.io/math2/node/G8DBMWQSDB.md" tag="G8DBMWQSDB" kind="definition">
# Calderón projector

## Definition

Let $\Omega \subset \R^n$ be a bounded open set with smooth boundary $\Gamma = \partial\Omega$ and outward unit normal $\nu$.
Let $P$ be an [[elliptic|B4TTNHSTRX]] linear differential operator of order $2m$ with smooth coefficients on a neighbourhood $U$ of $\overline{\Omega}$.
Assume $P$ has a [[fundamental solution|K2NK9A8M5F]] $E$ on $U$, that is, $P E = \delta$ in the sense of distributions.
For $u$ smooth up to $\Gamma$, let $\gamma u = (u|_\Gamma, \partial_\nu u|_\Gamma, \ldots, \partial_\nu^{2m-1} u|_\Gamma)$ be its Cauchy data, given by the [[trace operator|Z2T5R88QDB]].
Green's formula for $P$ gives a potential operator $K$, built from $E$, such that $u = K(\gamma u)$ in $\Omega$ whenever $Pu = 0$ in $\Omega$.
Let $\gamma_+$ take the Cauchy data of a function on $\Omega$ as limits from inside.
The *Calderón projector* of $P$ and $\Omega$ is the operator $C = \gamma_+ K$ on $2m$-tuples of functions on $\Gamma$.

## Remarks

$C$ is a $2m \times 2m$ matrix of [[pseudo-differential operators|FF27N0BWTP]] on $\Gamma$, $C^2 = C$, and its range is the space of Cauchy data $\gamma u$ of the solutions of $Pu = 0$ in $\Omega$.
If $E$ is only a parametrix, then $C$ is a projection up to smoothing operators, which is enough for most uses.
Example: for $P = -\Delta$ ($m = 1$), $C$ acts on pairs $(f, g)$.
Its four entries are, up to signs and the factor $\tfrac12 I$, the double layer, single layer, hypersingular and adjoint double layer operators.
A pair $(f, g)$ is fixed by $C$ exactly when $f = u|_\Gamma$ and $g = \partial_\nu u|_\Gamma$ for a [[harmonic function|0DEB0JMN0W]] $u$ on $\Omega$.
Non-example: the identity on pairs $(f, g)$ is a projection, but its range is too large.
On the unit disk it contains $(0, 1)$, which is not Cauchy data of a harmonic $u$, since $u = 0$ on the circle forces $u = 0$.
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# Energy method

## Summary

The energy method is a way to prove uniqueness, stability and a priori bounds for solutions of partial differential equations.
One attaches to a solution a nonnegative quantity, often a physical energy, and shows from the equation that it cannot grow too fast.
For evolution equations this gives control of the solution at time $t$ by its data at time $0$, which is the core of [[well-posedness|W09B9NZ9B5]].
It works for many [[hyperbolic|92C3S03AQZ]] and [[parabolic|1CCTH8X2FT]] equations where no explicit formula is available.

## Explanations

### L1 human

A vibrating string without friction keeps its total energy, kinetic plus elastic.
If two motions start from the same position and velocity, their difference starts with zero energy, and so it keeps zero energy forever, so the two motions agree.
The energy method turns this physical idea into a proof.
One multiplies the equation by a well-chosen quantity, such as $u_t$, integrates over space, and integrates by parts to find how the energy changes in time.
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# Krasner's lemma

## Statement

Let $K$ be a field that is complete for a [[non-Archimedean absolute value|1H5F1BP6XS]] $\abs{\cdot}$.
Let $\overline{K}$ be an [[algebraic closure|0MCXX29HWY]] of $K$, with the unique extension of $\abs{\cdot}$ to $\overline{K}$.
Let $\alpha \in \overline{K}$ be [[separable|0N019F6RE9]] over $K$, and let $\alpha = \alpha_1, \alpha_2, \ldots, \alpha_n$ be the distinct conjugates of $\alpha$ over $K$, with $n \ge 2$.
Let $\beta \in \overline{K}$ satisfy
$$\abs{\beta - \alpha} < \abs{\alpha - \alpha_i} \quad \text{for } i = 2, \ldots, n.$$
Then $K(\alpha) \subseteq K(\beta)$.

## Remarks

Proof idea: let $\sigma$ be an automorphism of $\overline{K}$ that fixes $K(\beta)$.
By uniqueness of the extended absolute value, $\abs{\sigma(x)} = \abs{x}$ for all $x$, so $\abs{\sigma(\alpha) - \beta} = \abs{\sigma(\alpha - \beta)} = \abs{\alpha - \beta}$.
The strong triangle inequality gives $\abs{\sigma(\alpha) - \alpha} \le \max(\abs{\sigma(\alpha) - \beta}, \abs{\beta - \alpha}) < \abs{\alpha - \alpha_i}$ for $i \ge 2$, so $\sigma(\alpha) = \alpha$.
As $\alpha$ is separable and fixed by all such $\sigma$, it lies in $K(\beta)$.
The same proof works for a Henselian valued field, since there too the absolute value extends uniquely.
For $n = 1$ the element $\alpha$ lies in $K$, so the conclusion holds for every $\beta$.

Application: let $f$ be a monic irreducible separable polynomial over $K$, and let $g$ be a monic polynomial over $K$ of the same degree with close enough coefficients.
Then $g$ is irreducible, and each root of $g$ generates the same extension of $K$ as a suitable root of $f$.
This shows that every finite extension of $\Q_p$ is generated by a root of a polynomial with rational coefficients.
It is also the key step in the proof that the [[complex p-adic numbers|FKF9YGBKHF]] form an algebraically closed field.

## Explanations

### L1 human

In a $p$-adic field, let $\beta$ be closer to $\alpha$ than any other conjugate of $\alpha$ is.
Then $\beta$ already "knows" $\alpha$: the field made from $\beta$ contains $\alpha$.
Completeness is needed.
In $\Q$ with the $7$-adic absolute value, rational numbers can be as close as one likes to a square root of $2$.
Yet they generate only $\Q$, which does not contain that square root.

### L2 human

Krasner's lemma is the basic continuity statement for roots in local fields.
With the [[Newton polygon|H7QNCE2QZJ]] and [[Hensel's lemma|010QVB71T2]] it is one of the three main tools for studying extensions of $\Q_p$.
It implies that a finite separable extension of $\Q_p$ does not change when the defining polynomial is perturbed slightly.
With a compactness argument, this shows that $\Q_p$ has only finitely many extensions of each degree.
It also gives that the completion of an algebraic closure of $\Q_p$ is again algebraically closed.
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# Modal logic

## Definition

The modal formulas are built from propositional variables $p, q, \dots$ and the constant $\bot$ with the connectives $\neg$, $\land$, $\lor$, $\to$ and the unary operator $\square$.
The formula $\lozenge\varphi$ abbreviates $\neg\square\neg\varphi$.
A normal modal logic is a set $\Lambda$ of modal formulas with these properties.

1. $\Lambda$ contains every substitution instance of a [[tautology|05006AJ7DZ]] of [[propositional logic|09K9MWD2QQ]].
2. $\Lambda$ contains every formula $\square(\varphi \to \psi) \to (\square\varphi \to \square\psi)$, the axiom $\mathrm{K}$.
3. If $\varphi \in \Lambda$ and $\varphi \to \psi \in \Lambda$, then $\psi \in \Lambda$, the rule of [[modus ponens|0Y0MQYST0M]].
4. If $\varphi \in \Lambda$, then $\square\varphi \in \Lambda$, the rule of necessitation.
5. If $\varphi \in \Lambda$, then each formula obtained from $\varphi$ by substituting formulas for its variables is in $\Lambda$.

The smallest normal modal logic is called $\mathrm{K}$.

## Remarks

Example: the logic $\mathrm{S4}$ is the smallest normal modal logic containing $\square p \to p$ and $\square p \to \square\square p$.
It satisfies the five clauses by construction, as the intersection of all normal modal logics that contain these two formulas.

Non-example: the set of all tautologies of propositional logic, read as modal formulas, is not a normal modal logic.
It fails clause 2, since $\square(p \to q) \to (\square p \to \square q)$ is not an instance of a tautology.

Other standard normal modal logics are $\mathrm{T}$, given by $\square p \to p$, and $\mathrm{S5}$, given by $\square p \to p$ and $\lozenge p \to \square\lozenge p$.
The word "modal logic" also names the whole field that studies such systems, including non-normal ones.

## Explanations

### L1 human

A modal logic is a list of the modal formulas taken as laws.
Every normal modal logic keeps all laws of classical logic and adds the rule that what is proved is necessary.
Extra axioms, such as "what is necessary is true", select the logic that fits a reading of $\square$.
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# Stiffness matrix

## Summary

The stiffness matrix is the matrix of the linear system that the [[finite element method|98GZ9DVHTA]] produces from the bilinear form of a boundary value problem.
Its entries are the values of the form on pairs of basis functions.
Because finite element basis functions have small supports, the matrix is [[sparse|JGACZYEEY7]].
For symmetric coercive problems such as the Poisson equation it is [[symmetric|0VR671PHE5]] and [[positive definite|0KX66CZMBB]].
The name comes from structural mechanics, where the matrix links the displacements of the nodes with the forces on them.
See the [[definition|DPR9DJRPAF]].

## Explanations

### L1 human

Think of a chain of springs whose ends are tied to walls.
If you push one joint, the forces on that joint and on its two neighbours change, and nothing else.
The stiffness matrix records these numbers: row $i$ says how the force at joint $i$ depends on the positions of all joints.
For equal springs in a chain, each row has $2$ on the diagonal and $-1$ next to it, and zeros elsewhere.
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# Constructivism

## Summary

Constructivism is the view in the philosophy of mathematics that a mathematical object exists only when one can construct it.
To prove that something exists, one must give a method to find it, as in a [[constructive proof|01XM7JEKHY]].
As a result, constructive mathematics does not accept the [[law of excluded middle|0RYYK8PG4S]] as a general principle.
Its main schools are Brouwer's intuitionism, Markov's recursive constructive mathematics, and Bishop's constructive analysis.

## Explanations

### L1 human

A constructivist accepts "there is a number with this property" only if you can, at least in principle, produce it.
A [[proof by contradiction|0J0XBKG64S]] that only shows "it cannot be that no such number exists" is not enough.
For example, the classical proof that there are irrational $a, b$ with $a^b$ rational does not say which pair works.
A constructivist asks for a proof that names the pair.
In return, every constructive existence proof contains an algorithm.
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<document source="https://tamnd.github.io/math2/node/G95YDNFXQN.md" tag="G95YDNFXQN" kind="concept">
# Cylinder set measure

## Summary

A cylinder set measure on a [[topological vector space|T32W553W95]] is a consistent family of [[probability measures|059TJN0NZ1]] on its finite-dimensional images.
It assigns a probability to every set that is described by finitely many linear coordinates.
It need not come from a countably additive measure on the whole space.
Cylinder set measures are the raw material for Gaussian measures and white noise in infinite dimensions.

## Explanations

### L1 human

In infinite dimensions one often knows how a random point looks through any finite number of linear measurements, but not yet as a whole.
A cylinder set measure records exactly this: for each finite set of measurements, a probability law of the readings, and the laws fit together.
For example, on a Hilbert space one may ask that every unit vector direction gives a standard normal reading, independent for orthogonal directions.
This is a cylinder set measure, but it is not a true measure on the Hilbert space, because a random point would have infinite length.
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# Kirillov orbit method

## Summary

The orbit method relates the irreducible unitary representations of a Lie group $G$ to the coadjoint orbits.
These are the orbits of $G$ in the dual $\mathfrak g^*$ of its Lie algebra.
Kirillov showed that for connected simply connected nilpotent Lie groups this relation is an exact bijection, built by inducing characters from polarizing subgroups.
For other groups it is a guiding principle rather than a theorem.
It works for exponential solvable groups and gives good heuristics for compact and reductive groups.
It ties representation theory to symplectic geometry, since each coadjoint orbit carries a natural symplectic form.

## Explanations

### L1 human

The dual of the Lie algebra is a vector space on which the group acts, and it breaks into orbits.
The orbit method says: each orbit should give one irreducible representation, and each irreducible representation should come from one orbit.
For the Heisenberg group, the orbits are parallel planes and single points.
The planes give the representations of quantum mechanics, one for each value of Planck's constant, and the points give the one-dimensional characters.
In this way the method is a mathematical version of passing from classical to quantum mechanics.
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# Proof methods

## Summary

This topic collects the standard methods of [[mathematical proof|0B94TKFQJX]], such as direct proof, proof by contradiction, proof by cases and induction.
A proof node names its methods with the method edge, and each target of that edge is a concept in this topic.
The topic lets a reader find all the proofs that use one method.
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# Balayage space

## Summary

A balayage space is a locally compact space with a countable base together with a convex cone of positive lower semicontinuous functions.
This cone plays the role of the positive superharmonic functions.
The axioms are chosen so that [[balayage|NTFY23YDEJ]], the sweeping of functions and measures onto sets, can be carried out.
The notion was developed by Jürgen Bliedtner and Wolfhard Hansen as a common frame for [[harmonic spaces|DWN0XEQHHK]] and for the potential theory of Markov processes.
Its theory includes the fine topology, thin sets, and the Dirichlet problem.

## Explanations

### L1 human

In classical potential theory, sweeping a charge onto a set means replacing it by a charge on the set that produces the same potential outside.
Most of this theory uses only the cone of positive superharmonic functions, not the harmonic functions themselves.
A balayage space starts from such a cone of functions and asks only for the properties needed to sweep.
The positive superharmonic functions on a Euclidean domain give an example, and so do the excessive functions of many Markov processes, such as Brownian motion.
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# Complete local ring

## Summary

A complete local ring is a Noetherian [[local ring|MZ02BKPHJA]] $(R, \mathfrak m)$ that equals its own [[completion|KBN13NPYJQ]] for the $\mathfrak m$-adic topology.
Formal power series rings over a field and the $p$-adic integers are the basic examples.
Complete local rings have a clear structure, given by the [[Cohen structure theorem|N3XH42KVV3]], and every Noetherian local ring embeds into a complete one, its completion.

## Explanations

### L1 human

In a complete local ring, every list of better and better approximations has a limit.
For example, in the formal power series ring $k[\![x]\!]$ the polynomials $1$, $1 + x$, $1 + x + x^2$, $\dots$ approach the series $1 + x + x^2 + \cdots$, which is again an element of the ring.
In the ring of rational functions defined at $0$, many such lists have no limit, so that ring is not complete.
The [[definition|78GT1VFFPR]] makes this precise.
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# Mohr–Mascheroni theorem

## Statement

Let $S$ be a set of points in the Euclidean plane.
Every point that can be constructed from $S$ with [[straightedge and compass|0DAECMZVRT]] can also be constructed from $S$ with compass alone.

## Remarks

A line counts as known in a compass-only construction when two of its points are known; the line itself is never drawn.
The proof shows how to find with compass alone the intersection points of a [[circle|060W5CBZP6]] and a line given by two points.
It also shows how to find the intersection point of two lines given by two points each.
A standard tool is inversion in a circle, which can be done with compass alone and turns lines that miss the centre into circles through the centre.
Georg Mohr published the result in 1672, and Lorenzo Mascheroni found it again independently in 1797.
For example, the midpoint of a segment $AB$ can be found with compass alone, although no line through $A$ and $B$ is ever drawn.

## Explanations

### L1 human

The ruler is not needed in classical constructions.
Anything one can construct with ruler and compass, one can also construct with a compass alone, as long as a line is described by two of its points.
So the field of constructible points is the same for both kinds of tools.

### L2 human

Algebraically, ruler and compass constructions from $0$ and $1$ give exactly the [[constructible numbers|0QNFRBJF4Q]], the smallest subfield of $\C$ closed under square roots and complex conjugation.
The theorem says that compass constructions already give all of them.
The [[Poncelet–Steiner theorem|2P0DP2TK6P]] is the companion result that keeps the ruler and replaces the compass by a single fixed circle.
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# Coordinate ring

## Summary

The coordinate ring of an algebraic set $X \subseteq k^n$ is the ring of polynomial functions on $X$.
It is the quotient of the polynomial ring by the ideal of $X$.
It carries the geometry of $X$: points, closed subsets and morphisms of affine varieties are read off from it.
Also $X$ is irreducible exactly when this ring is an integral domain.

## Explanations

### L1 human

On a set of points cut out by equations, two polynomials give the same function when their difference vanishes there.
The coordinate ring is the ring of these functions.
On the circle $x^2 + y^2 = 1$, the polynomials $x^2$ and $1 - y^2$ give the same function, so they are equal in the coordinate ring.
On the $x$-axis, every polynomial in $x$ and $y$ agrees with a polynomial in $x$ alone, so the coordinate ring is just $k[x]$.
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# System of linear integral equations

## Summary

A system of linear integral equations has several unknown functions $\varphi_1, \ldots, \varphi_n$, and each equation involves integrals of all of them.
In vector notation it is a single [[linear integral equation|AYZCKG34QK]] whose kernel is an $n \times n$ [[matrix|0VD9ZSWWJG]] of functions.
Much of the theory for one equation, such as [[Fredholm theory|242TFD41KZ]], carries over.
The reason is that a system on an interval can be rewritten as one equation on a longer interval.
Systems arise from coupled physical fields, from boundary integral methods for vector problems, and from the factorization of matrix symbols.

## Explanations

### L1 human

Sometimes one unknown function is not enough: two populations, two components of a displacement, two layers of a medium.
Each unknown then satisfies an equation that also contains integrals of the others.
Stacking the unknowns into a vector turns the system into one equation with a matrix kernel.
For example, on $[0, 1]$ the pair $\varphi_1(x) = x + \frac{1}{2}$, $\varphi_2(x) = x$ solves $\varphi_1(x) - \int_0^1 \varphi_2(t)\, dt = x$ and $\varphi_2(x) - x \int_0^1 \varphi_1(t)\, dt = 0$.
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<document source="https://tamnd.github.io/math2/node/G9ZNDNT3YY.md" tag="G9ZNDNT3YY" kind="concept">
# Exact category in the sense of Barr

## Summary

A category is exact in the sense of Barr, or Barr-exact, when it is a [[regular category|2SM3JRJXA4]] with effective equivalence relations.
This means that every internal equivalence relation is the kernel pair of some morphism.
So quotients by equivalence relations exist and behave as in sets.
Every elementary topos, every abelian category, the category of sets and every variety of algebras such as groups or rings are Barr-exact.
This notion is different from the exact categories of Quillen used in algebraic K-theory.

## Explanations

### L1 human

In sets, any equivalence relation on $X$ is the relation "has the same image" for the quotient map $X \to X / {\sim}$.
A Barr-exact category is one where this holds for every equivalence relation expressed inside the category.
For groups, an equivalence relation that is a subgroup of $G \times G$ is the relation "same coset" for a normal subgroup, and the quotient group provides the map.
</document>

<document source="https://tamnd.github.io/math2/node/GA0NCJMZHB.md" tag="GA0NCJMZHB" kind="concept">
# Laguerre geometry

## Summary

Laguerre geometry is the geometry of oriented lines and oriented circles of the plane.
Its transformations preserve oriented contact and map oriented lines to oriented lines.
Points count as circles of radius $0$, and these transformations need not map points to points.
A typical example moves every oriented line parallel to itself by the same distance and adds that distance to every signed radius.
It is a subgeometry of [[Lie sphere geometry|TFQSF9RA1G]], and its abstract incidence version is the [[Laguerre plane|F0JQ1TGKB0]]; it is named after Edmond Laguerre.

## Explanations

### L1 human

Think of lines with arrows on them and circles with a direction of travel.
A directed line and a directed circle touch properly when they are tangent and run the same way at the touching point.
Laguerre geometry keeps only this touching relation and the fact of being a line.
So it may grow every circle by one centimetre and push every line one centimetre sideways at once.
All touching pairs still touch, even though a point has become a small circle.
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<document source="https://tamnd.github.io/math2/node/GA46NC0CJZ.md" tag="GA46NC0CJZ" kind="statement">
# Yule–Walker equations

## Statement

Let $(\varepsilon_t)_{t \in \Z}$ be a white noise with variance $\sigma^2 > 0$.
Let $(X_t)_{t \in \Z}$ be a causal [[autoregressive model|2JC96FMGA1]] of order $p$ with mean $0$,
$$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + \varepsilon_t,$$
so that $X_t = \sum_{j \ge 0} \psi_j \varepsilon_{t-j}$ with $\sum_j \abs{\psi_j} < \infty$.
Let $\gamma(h) = E[X_{t+h} X_t]$ be its autocovariance function.
Then
$$\gamma(h) = \phi_1 \gamma(h-1) + \cdots + \phi_p \gamma(h-p) \quad \text{for } h \ge 1, \qquad \gamma(0) = \phi_1 \gamma(1) + \cdots + \phi_p \gamma(p) + \sigma^2.$$
In matrix form, $\Gamma_p \phi = \gamma_p$ and $\sigma^2 = \gamma(0) - \phi^T \gamma_p$, where $\Gamma_p = [\gamma(i - j)]_{i,j=1}^{p}$, $\phi = (\phi_1, \dots, \phi_p)^T$ and $\gamma_p = (\gamma(1), \dots, \gamma(p))^T$.

## Remarks

Multiply the defining equation by $X_{t-h}$ and take expectations.
For $h \ge 1$, causality makes $X_{t-h}$ a function of $\varepsilon_s$ with $s \le t - h < t$, so $E[\varepsilon_t X_{t-h}] = 0$, which gives the first equations.
For $h = 0$, $E[\varepsilon_t X_t] = E[\varepsilon_t^2] = \sigma^2$, which gives the last one.
The matrix $\Gamma_p$ is a symmetric [[Toeplitz matrix|GKY2WSHE26]], and it is nonsingular because $\gamma(0) > 0$ and $\gamma(h) \to 0$ as $h \to \infty$.
So the autocovariances $\gamma(0), \dots, \gamma(p)$ determine $\phi$ and $\sigma^2$.

Example: for $p = 1$, the equations read $\gamma(1) = \phi_1 \gamma(0)$ and $\gamma(0) = \phi_1 \gamma(1) + \sigma^2$.
Hence $\phi_1 = \rho(1)$ and $\gamma(0) = \sigma^2 / (1 - \phi_1^2)$.

The *Yule–Walker estimator* replaces $\gamma$ by the sample autocovariance $\hat\gamma$ and solves $\hat\Gamma_p \hat\phi = \hat\gamma_p$.
Causality is needed: for the non-causal stationary solution of $X_t = 2 X_{t-1} + \varepsilon_t$, the shock $\varepsilon_t$ is correlated with the past values $X_{t-h}$ and the first equation fails.

## Explanations

### L1 human

In an autoregressive model, each value is a weighted sum of earlier values plus fresh noise.
The Yule–Walker equations say that the correlations of the series obey the same recursion with the same weights.
So once the correlations are known, the weights can be found by solving a small linear system.

### L2 human

The equations give the simplest estimator of the coefficients of an AR($p$) model, computed from the sample [[autocorrelation function|YCZHYY8VE5]].
For a Gaussian causal AR process, this estimator has the same asymptotic distribution as the maximum likelihood estimator.
The Durbin–Levinson algorithm solves the Toeplitz system for $p = 1, 2, 3, \dots$ in turn, and the last coefficient at each order is the partial autocorrelation.
The equations are named after Udny Yule and Gilbert Walker.
</document>

<document source="https://tamnd.github.io/math2/node/GA5PTQGC6G.md" tag="GA5PTQGC6G" kind="concept">
# Differential field

## Summary

A differential field is a [[field|04C5ZBG1WT]] with a [[derivation|3J89REV9WQ]], an additive map that obeys the product rule, see the [[definition|T43TQ817XC]].
It is the algebraic setting for studying differential equations, where solutions are elements of larger differential fields.
The field $\C(x)$ of rational functions with $d/dx$ is the basic example.
[[Differential Galois theory|EEZS4H8QPH]] and the [[Liouville theorem|8VRTPECBPZ]] on elementary integrals are built on differential fields.

## Explanations

### L1 human

Take all fractions of polynomials in $x$, such as $(x^2 + 1)/(x - 3)$, together with the usual rule of differentiation.
This is a field where one can add, multiply, divide and also differentiate, and the derivative of every element is again in the field.
A differential field is any field with such a differentiation rule.
One can then ask whether an equation such as $y' = y$ has a nonzero solution in the field; here it does not, since the nonzero solutions $c e^x$ are not rational functions.
</document>

<document source="https://tamnd.github.io/math2/node/GA7437914V.md" tag="GA7437914V" kind="definition">
# Connection on a principal bundle

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{g}$, and let $\pi : P \to M$ be a smooth [[principal $G$-bundle|8E37E74XVP]] with right action $R_g(u) = u \cdot g$.
For $X \in \mathfrak{g}$, the fundamental vector field $X^\#$ on $P$ is $X^\#_u = \frac{d}{dt}\big|_{t=0} \, u \cdot \exp(tX)$, where $\exp$ is the [[exponential map|H3QX1ZFC8K]].
A *connection form* on $P$ is a smooth $\mathfrak{g}$-valued [[differential $1$-form|0E3G0GRPZ6]] $\omega$ on $P$ such that:

1. $\omega(X^\#) = X$ for every $X \in \mathfrak{g}$.
2. $R_g^* \omega = \mathrm{Ad}(g^{-1}) \, \omega$ for every $g \in G$, where $\mathrm{Ad}$ is the [[adjoint representation|P6ZP0TE07M]].

A *connection* on $P$ is a connection form on $P$.

## Remarks

Equivalently, a connection is an [[Ehresmann connection|TD6VEP3SCE]] $H \subset TP$ with $dR_g(H_u) = H_{u \cdot g}$ for all $u$ and $g$; the two notions correspond by $H = \ker \omega$.
Example: on the trivial bundle $P = M \times G$, let $\theta$ be the left-invariant [[Maurer–Cartan form|P1JVA39H5E]] of $G$, so $\theta_g(v) = dL_{g^{-1}}(v)$, and let $\omega = \mathrm{pr}_2^* \theta$.
Clause 1 holds since $X^\#_{(p, g)} = (0, dL_g X)$, and clause 2 holds since $R_h^* \theta = \mathrm{Ad}(h^{-1}) \theta$.
Non-example: $\omega = 0$ satisfies clause 2 but fails clause 1 when $\dim G > 0$.
For a matrix group the trivial example reads $\omega = g^{-1} dg$.
The difference of two connection forms is a $\mathfrak{g}$-valued $1$-form that vanishes on vertical vectors and satisfies clause 2; so connections on $P$ form an affine space.
</document>

<document source="https://tamnd.github.io/math2/node/GA7Q813F50.md" tag="GA7Q813F50" kind="definition">
# Unconditional convergence

## Definition

Let $X$ be a [[normed vector space|00A0RPAQ8A]] over $\R$ or $\C$, and let $(x_n)_{n \ge 0}$ be a sequence in $X$.
The series $\sum_{n=0}^{\infty} x_n$ converges unconditionally if for every bijection $\sigma \colon \N \to \N$ the [[rearranged series|0HRJ2Y74T3]] $\sum_{n=0}^{\infty} x_{\sigma(n)}$ [[converges|0CVVKY7GY2]] in $X$.

## Remarks

When the series converges unconditionally, all rearrangements have the same sum.
In a Banach space, unconditional convergence is equivalent to [[summability of the family|70HJTK5VZH]] $(x_n)_{n \in \N}$.
Example: in the [[sequence space|08P6T2WH6R]] $\ell^2$ with unit vectors $e_n$, the series $\sum e_n / n$ converges unconditionally.
Indeed, for every finite set $F$ of indices, $\norm{\sum_{n \in F} e_n / n}^2 = \sum_{n \in F} 1/n^2$, so the partial sums of every rearrangement are Cauchy.
This series does not converge absolutely, since $\sum \norm{e_n / n} = \sum 1/n$ diverges.
Non-example: the [[alternating harmonic series|0JQ5Y3WWVE]] converges, but some rearrangements diverge, by the [[Riemann rearrangement theorem|0E7VBYGFF7]].
In $\R^d$, a series converges unconditionally if and only if it converges [[absolutely|0W3K3QPEWT]].

## Explanations

### L1 human

Unconditional convergence asks that the sum not depend on the order of the terms.
In finite dimensions this is the same as absolute convergence, but in spaces of infinite dimension it is weaker.
</document>

<document source="https://tamnd.github.io/math2/node/GA81VVQHEP.md" tag="GA81VVQHEP" kind="concept">
# Linear algebraic group

## Summary

A linear algebraic group is a group of invertible matrices defined by polynomial equations in the entries.
Equivalently, it is an [[algebraic group|GH81SYAPPN]] whose underlying scheme is affine.
Examples are the [[general linear group|0FCHE902D7]], the [[special linear group|0V9PMN0RBH]], the [[orthogonal groups|0TKYAEJ8N7]] and the [[symplectic groups|X12FS2JAP0]].
Their structure theory, through tori, Borel subgroups and root systems, is a large part of modern algebra and number theory.

## Explanations

### L1 human

Think of a group of matrices cut out by equations, such as the matrices $A$ with $A^{T} A = I$, or those with determinant $1$.
Such a group is at the same time a group and a geometric shape in the space of matrices.
Linear algebraic groups are the algebraic counterpart of matrix Lie groups, and they make sense over any field, including finite fields.
Over a finite field they give many of the finite simple groups.
</document>

<document source="https://tamnd.github.io/math2/node/GA8BDBHP4M.md" tag="GA8BDBHP4M" kind="statement">
# Posner's theorem

## Statement

Let $R$ be a [[prime ring|Z9TJSZZYG1]] that satisfies a monic [[polynomial identity|C0EDT81EGH]].
Let $Z$ be the [[center|MVT64B3B6H]] of $R$, and let $S = Z \setminus \set{0}$.

1. $Z$ is a commutative domain, and each element of $S$ is a regular element of $R$.
2. The ring $Q = R S^{-1}$, obtained by inverting the nonzero central elements, is a finite-dimensional [[central simple algebra|5HRKJBAY2V]] over the [[field of fractions|0F2PTGX1A2]] $F$ of $Z$.
3. $Q$ is the [[classical ring of quotients|C5971PGKPH]] of $R$ on both sides.

## Remarks

Posner proved that a prime PI ring has a classical ring of quotients that is a simple Artinian ring finite-dimensional over its centre.
The sharper statement that one only has to invert the nonzero central elements, given in parts 1 and 2, is a later refinement based on [[central polynomials|667YZTHE96]].
One way to see the original statement: a prime PI ring satisfies the conditions of a [[Goldie ring|QFDC3NYHJE]], so [[Goldie's theorem|MDS8655N0Z]] gives a simple Artinian quotient ring $Q \cong M_s(D)$.
The ring $Q$ satisfies the same multilinear identities as $R$, and then [[Kaplansky's theorem|8NE9Q44FEG]] shows that $Q$ is finite-dimensional over its centre.

Example: $R = M_n(\Z)$ is prime and satisfies $s_{2n}$, its centre is $\Z$, and $Q = M_n(\Q)$, a central simple algebra over $\Q$.

The primeness hypothesis is needed for the conclusion that $Q$ is simple.
The ring $\Z \times \Z$ is commutative, hence PI, but it is not prime, and its classical ring of quotients $\Q \times \Q$ is not simple.

## Explanations

### L1 human

A commutative domain sits inside its field of fractions.
Posner's theorem says that a prime ring satisfying a polynomial identity sits, in the same way, inside a matrix-like algebra over a field: a central simple algebra of finite dimension.
One gets there just by dividing by central elements, as for fractions of integers.

### L2 human

Posner's theorem reduces the study of prime PI rings to orders in central simple algebras.
The dimension of $Q$ over $F$ is a square $n^2$, and $n$ is called the PI degree of $R$.
Combined with the theory of [[generic matrix algebras|GJGEVMZ828]], it is the main tool for questions about prime PI rings, such as their dimension theory and their centres.
</document>

<document source="https://tamnd.github.io/math2/node/GAD8ZE5MPV.md" tag="GAD8ZE5MPV" kind="definition">
# Period and index

## Definition

Let $K$ be a field and let $A$ be a [[central simple algebra|5HRKJBAY2V]] over $K$.
By Wedderburn's theorem $A \cong M_m(D)$ for an integer $m \ge 1$ and a [[division ring|0EA5BTA0CX]] $D$ that is central simple over $K$, unique up to isomorphism.

1. The *index* of $A$, written $\operatorname{ind} A$, is the [[degree|NNXSP5KKAC]] of $D$.
2. The *period* (or *exponent*) of $A$, written $\operatorname{per} A$, is the [[order|0WE2NHZJ7B]] of the class $[A]$ in the [[Brauer group|BEW48TA9WZ]] $\mathrm{Br}(K)$.

## Remarks

Basic facts: $\operatorname{per} A$ divides $\operatorname{ind} A$, $\operatorname{ind} A$ divides $\deg A$, and $\operatorname{per} A$ and $\operatorname{ind} A$ have the same prime divisors.
In particular the period is finite, so $\mathrm{Br}(K)$ is a torsion group.

Example: for $\mathbb{H}$ over $\R$, $D = \mathbb{H}$ has degree $2$, so the index is $2$.
The class $[\mathbb{H}]$ is not trivial, and $\mathbb{H} \otimes_\R \mathbb{H} \cong M_4(\R)$, so the period is $2$.

Example: $A = M_n(K)$ has $D = K$, so its index and its period are both $1$.

Over local fields and over global fields, the period equals the index for every central simple algebra.
Over general fields the index can be strictly larger than the period.
</document>

<document source="https://tamnd.github.io/math2/node/GAFQDGF6FM.md" tag="GAFQDGF6FM" kind="concept">
# Pro-object

## Summary

A pro-object in a category $\mathcal{C}$ is a formal cofiltered limit of objects of $\mathcal{C}$.
Pro-objects form the category $\mathrm{Pro}(\mathcal{C})$, which is the dual of the [[Ind-completion|1DKFWGWHH5]] of the opposite category.
They appear in algebraic geometry, in shape theory and in the étale homotopy type.
There a limit that does not exist in $\mathcal{C}$ is kept as a formal system.
The category of profinite sets, that is of Stone spaces, is equivalent to the pro-objects in finite sets.

## Explanations

### L1 human

A pro-object is a system of ever finer approximations, kept as a whole instead of being replaced by its limit.
The finite groups $\Z / p^n \Z$, linked by the reduction maps, form a pro-object in finite groups.
Its limit is the group of $p$-adic integers, but the pro-object remembers the finite stages themselves.
</document>

<document source="https://tamnd.github.io/math2/node/GAJDYXNHN2.md" tag="GAJDYXNHN2" kind="definition">
# Fisher's equation

## Definition

Let $D > 0$ and $r > 0$ be constants.
*Fisher's equation* is the [[reaction–diffusion equation|TAAHWE917M]] for a function $u(x, t)$ of $x \in \R$ and $t > 0$:
$$u_t = D\, u_{xx} + r\, u (1 - u).$$
Here $u_t$ and $u_{xx}$ are [[partial derivatives|09EGC6F8PT]].
The same name is used for $u_t = D \Delta u + r u(1 - u)$ with $x \in \R^n$.

## Remarks

The constants $u \equiv 0$ and $u \equiv 1$ are solutions: both sides vanish.
The state $0$ is unstable and the state $1$ is stable for the reaction $u' = r u(1 - u)$.
A [[travelling wave|M37M411KP7]] is a solution $u(x, t) = U(x - ct)$ with $U(-\infty) = 1$ and $U(+\infty) = 0$.
Such a wave with $0 \le U \le 1$ exists if and only if $c \ge 2\sqrt{rD}$.
A non-example is the heat equation $u_t = D u_{xx}$: it has no reaction term, so it is not Fisher's equation for any $r > 0$.
The rescaling $x \mapsto x\sqrt{r/D}$, $t \mapsto r t$ reduces the equation to $u_t = u_{xx} + u(1 - u)$.
</document>

<document source="https://tamnd.github.io/math2/node/GAMHZY8TWV.md" tag="GAMHZY8TWV" kind="statement">
# Pointwise Kan extension formula

## Statement

Let $\mathcal{A}$ be a small category, let $\mathcal{B}$ be a locally small category, and let $\mathcal{C}$ be a category that has all small [[colimits|BJS1CGTKQ5]].
Let $K \colon \mathcal{A} \to \mathcal{B}$ and $F \colon \mathcal{A} \to \mathcal{C}$ be functors.
For an object $b$ of $\mathcal{B}$, let $(K \downarrow b)$ be the [[comma category|PC6HG1JEGM]] of pairs $(a, f)$ with $a$ in $\mathcal{A}$ and $f \colon K(a) \to b$.
Let $P_b \colon (K \downarrow b) \to \mathcal{A}$ send $(a, f)$ to $a$.
Then the [[left Kan extension|RN9D96ESWN]] of $F$ along $K$ exists, and for each object $b$ of $\mathcal{B}$
$$(\mathrm{Lan}_K F)(b) = \operatorname{colim} \big( F \circ P_b \colon (K \downarrow b) \to \mathcal{C} \big).$$

## Remarks

The comma category $(K \downarrow b)$ is small because $\mathcal{A}$ is small and $\mathcal{B}$ is locally small, so the colimit exists.
A morphism $g \colon b \to b'$ induces a functor $(K \downarrow b) \to (K \downarrow b')$ by composing with $g$, and hence a map between the colimits; this makes $b \mapsto (\mathrm{Lan}_K F)(b)$ a functor.
The component $\eta_a$ is the colimit injection at the object $(a, \id_{K(a)})$.
The universal property is checked by turning a natural transformation $F \Rightarrow G \circ K$ into a compatible family of cocones, one for each $b$.

Dually, suppose that $\mathcal{C}$ has all small [[limits|CHS5AFJGWE]].
Then $(\mathrm{Ran}_K F)(b)$ is the limit of $F$ over the comma category $(b \downarrow K)$ of pairs $(a, f \colon b \to K(a))$.
When $K$ is fully faithful, the unit $\eta \colon F \Rightarrow (\mathrm{Lan}_K F) \circ K$ is an isomorphism.
The reason is that $(a, \id_{K(a)})$ is then a terminal object of $(K \downarrow K(a))$.

Example: view a group $G$ and a subgroup $H$ as one-object categories, and let $K$ be the inclusion.
Let $F$ be a representation of $H$ on a vector space $V$ over a field $k$.
Then the formula gives the induced representation $k[G] \otimes_{k[H]} V$ of $G$.

## Explanations

### L1 human

The formula builds the extended functor one object at a time.
At an object $b$, it collects all values $F(a)$ for which $K(a)$ maps into $b$ and glues them together by a colimit.
So the extension at $b$ is the best approximation of $F$ seen from everything that lies over $b$.

### L2 human

This is the standard way to show that Kan extensions exist, and it is the version used in most applications.
Taking $\mathcal{B} = \mathbf{1}$ recovers the colimit of $F$.
Taking $K$ to be the Yoneda embedding of $\mathcal{A}$ extends $F$ to a colimit-preserving functor on presheaves.
This is the basis of the nerve and realization adjunctions.
A Kan extension computed by this formula is called pointwise.
In the language of coends the formula reads $(\mathrm{Lan}_K F)(b) = \int^{a} \mathcal{B}(K(a), b) \cdot F(a)$.
Here $S \cdot c$ is the coproduct of copies of $c$ indexed by the set $S$.
</document>

<document source="https://tamnd.github.io/math2/node/GARGJYSGFG.md" tag="GARGJYSGFG" kind="statement">
# Poisson summation formula

## Statement

Let $f \colon \R \to \C$ be a function in the [[Schwartz space|CG9JEATW3Q]].
Let $\widehat{f}(\xi) = \int_\R f(x) e^{-2\pi i x \xi} \, dx$ be its [[Fourier transform|DPVM34ADZV]].
Then
$$\sum_{n \in \Z} f(n) = \sum_{k \in \Z} \widehat{f}(k),$$
and both series converge absolutely.

## Remarks

The proof periodises $f$: the function $F(x) = \sum_{n \in \Z} f(x + n)$ is smooth and has period $1$.
Its [[Fourier coefficients|G12KXDHPPS]] are $\int_0^1 F(x) e^{-2\pi i k x} \, dx = \widehat{f}(k)$, so its Fourier series $\sum_k \widehat{f}(k) e^{2\pi i k x}$ converges to $F(x)$, and $x = 0$ gives the formula.
Example: for $t > 0$ and $f(x) = e^{-\pi t x^2}$ one has $\widehat{f}(\xi) = t^{-1/2} e^{-\pi \xi^2 / t}$, and the formula gives $\sum_{n} e^{-\pi t n^2} = t^{-1/2} \sum_{k} e^{-\pi k^2/t}$, the transformation law of the Jacobi theta function.
The Schwartz hypothesis can be weakened, but some decay and regularity of $f$ and $\widehat{f}$ is needed for both sides to converge to the same value.
Let $G$ be a locally compact abelian group and $H$ a discrete subgroup with $G/H$ compact.
A similar formula then holds, with the sum on the right over the characters of $G$ trivial on $H$.
It needs suitable conditions on $f$ and a suitable normalisation of the Haar measures.

## Explanations

### L1 human

Sample a function at the integers and add up the values.
The formula says that one gets the same total by sampling the Fourier transform at the integers and adding those values.
A function that is spread out has a transform that is concentrated, so one side can be hard to compute while the other side is easy.

### L2 human

The formula is the basic link between a lattice and its dual lattice; in $\R^n$ it reads $\sum_{v \in L} f(v) = \frac{1}{\operatorname{vol}(\R^n/L)} \sum_{w \in L^*} \widehat{f}(w)$ for a lattice $L$ with dual lattice $L^*$.
Applied to Gaussians it gives the functional equation of theta functions, which leads to the functional equation of the Riemann zeta function.
In signal processing it underlies the sampling theorem, and in analytic number theory it is used to count lattice points.
</document>

<document source="https://tamnd.github.io/math2/node/GAW3VCFYZV.md" tag="GAW3VCFYZV" kind="definition">
# Germ of a smooth function

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] and let $p \in M$.
Consider pairs $(U, f)$, where $U$ is an open [[neighbourhood|0ZCY0Z80B6]] of $p$ and $f \colon U \to \R$ is a [[smooth function|0V33BNKZ9S]].
Two pairs $(U, f)$ and $(V, g)$ are equivalent if there is an open neighbourhood $W \subseteq U \cap V$ of $p$ with $f = g$ on $W$.
The *germ at $p$* of $(U, f)$ is its equivalence class, written $[f]_p$.
This is the [[germ|VZXF7RNFKS]] of $f$ at $p$ in the general sense, for smooth real functions.

The set of germs at $p$ is written $C^\infty_p(M)$ or $\mathcal{E}_p$.
Sums and products of germs are defined by $[f]_p + [g]_p = [f + g]_p$ and $[f]_p [g]_p = [fg]_p$, with $f + g$ and $fg$ taken on $U \cap V$.

## Remarks

With these operations $\mathcal{E}_p$ is a commutative $\R$-algebra.
It is a [[local ring|MZ02BKPHJA]]: its unique [[maximal ideal|0PV74TR5FP]] $\mathfrak{m}_p$ consists of the germs with $f(p) = 0$.
Indeed, if $f(p) \ne 0$, then $f \ne 0$ on a neighbourhood of $p$ and $[1/f]_p$ is an inverse of $[f]_p$.
The value $f(p)$ and all partial derivatives of $f$ at $p$ in a chart depend only on the germ.

Example: on $M = \R$, let $f(x) = x^2$ on $\R$ and $g(x) = x^2$ on $(-1, 1)$.
They agree on $W = (-1, 1)$, so $[f]_0 = [g]_0$.

Non-example: let $f = 0$ on $\R$, and let $h(x) = e^{-1/x^2}$ for $x \ne 0$ with $h(0) = 0$.
All derivatives of $f$ and $h$ at $0$ agree, but $h(x) \ne 0$ for every $x \ne 0$.
So $f$ and $h$ agree on no neighbourhood of $0$, and $[f]_0 \ne [h]_0$.
A germ is thus finer information than the Taylor series at $p$.
</document>

<document source="https://tamnd.github.io/math2/node/GB0D4XMZV3.md" tag="GB0D4XMZV3" kind="concept">
# Holomorphic fiber bundle

## Summary

A holomorphic fiber bundle is a [[fiber bundle|CX4M8K61AB]] in the category of [[complex manifolds|JSGWAN2RJV]].
The total space, base and fibre are complex manifolds, and the local trivializations are biholomorphic.
[[Holomorphic vector bundles|TKA958Z4KZ]] are the case where the fibre is $\C^r$ and the transition maps are linear.
Other examples are the Hopf map $\C^2 \setminus \set{0} \to \mathbb{P}^1$ and the elliptic fibration of a Hopf surface over $\mathbb{P}^1$.

## Explanations

### L1 human

A holomorphic fiber bundle is a complex manifold that maps onto a base.
Seen from close up over the base, it looks like a product of a piece of the base with a fixed complex manifold, the fibre.
The fibres over different points are all the same complex manifold, but they may be twisted together globally.
For example, sending a nonzero vector of $\C^2$ to the line it spans has every fibre equal to a punctured line $\C^*$.
Globally this is not a product, although locally it is.
</document>

<document source="https://tamnd.github.io/math2/node/GB7K3PTPAR.md" tag="GB7K3PTPAR" kind="problem">
# Matrix completion

## Problem

Let $F$ be a field and let $m, n \ge 1$ be integers.
Let $\Omega$ be a set of positions $(i, j)$ with $1 \le i \le m$ and $1 \le j \le n$.
For each $(i, j) \in \Omega$, let a value $M_{ij} \in F$ be given.
Find an $m \times n$ [[matrix|0VD9ZSWWJG]] $X$ over $F$ such that $X_{ij} = M_{ij}$ for all $(i, j) \in \Omega$ and the [[rank|0ZC9YTXZRY]] of $X$ is as small as possible.

## Remarks

Without more structure the problem has no unique answer: the entries outside $\Omega$ can be anything, so one asks for the completion of lowest rank.
Low-rank completion is NP-hard in general.
Over $\R$, a standard relaxation replaces the rank by the nuclear norm, the sum of the [[singular values|09VMDSRXWJ]] of $X$, which gives a convex problem.
Candès and Recht proved in 2009 that nuclear norm minimization recovers the unknown matrix exactly with high probability, under some conditions.
The matrix must have low rank and satisfy an incoherence condition, and enough entries must be sampled at random.
Other versions ask for a completion that is positive semidefinite, or that has another prescribed property.

## Explanations

### L1 human

Imagine a table of ratings, one row per user and one column per film, where most cells are empty.
Matrix completion asks to fill the empty cells in the simplest way, meaning with a matrix of small rank.
For example, if the known entries are $X_{11} = 1$, $X_{12} = 2$ and $X_{21} = 3$, the completion of rank $1$ is forced: $X_{22} = 6$.
The problem became famous through recommender systems such as the Netflix Prize.
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<document source="https://tamnd.github.io/math2/node/GB7M582M5Y.md" tag="GB7M582M5Y" kind="concept">
# Complex analytic variety

## Summary

A complex analytic variety, or complex analytic set, in a [[complex manifold|JSGWAN2RJV]] $X$ is a closed subset with a local description.
Near each point of $X$ it is the common zero set of finitely many [[holomorphic functions|507C34W0ZT]].
It is the analytic counterpart of an algebraic variety, with polynomials replaced by holomorphic functions.
Analytic varieties may have singular points, and their local structure is studied through [[germs of analytic sets|Q9B74BYZ8C]] and the ring of convergent power series.
Abstract varieties of this kind, defined by gluing local models, are the [[complex analytic spaces|HXSZA44J5J]].

## Explanations

### L1 human

Take a few holomorphic functions on $\C^n$ and look at the points where all of them vanish.
Such a set may be smooth, like the line $\set{z_2 = 0}$ in $\C^2$, or have singular points, like the curve $\set{z_1 z_2 = 0}$, which is two lines crossing at the origin.
An analytic variety is a set that looks like this near every point, even if no single finite family of functions defines it everywhere.
For example, the zero set of $\sin z$ in $\C$, the set of integer multiples of $\pi$, is an analytic variety but not the zero set of a polynomial.
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<document source="https://tamnd.github.io/math2/node/GBN9F8FAQ3.md" tag="GBN9F8FAQ3" kind="definition">
# p-adic integer

## Definition

Let $p$ be a prime number and let $\Q_p$ be the field of [[p-adic numbers|BT6T702C28]], with the [[p-adic absolute value|WGN82TCEF2]] $\abs{\cdot}_p$.
A *$p$-adic integer* is an element $x \in \Q_p$ with $\abs{x}_p \le 1$.
The set of $p$-adic integers is denoted $\Z_p$.

## Remarks

By the strong triangle inequality, $\Z_p$ is a subring of $\Q_p$.
Its units are the $x$ with $\abs{x}_p = 1$, and its only nonzero prime ideal is $p \Z_p$.
Equivalently, $\Z_p$ is the set of sums $\sum_{n \ge 0} a_n p^n$ with digits $a_n \in \set{0, \ldots, p - 1}$.
Equivalently, $\Z_p$ is the inverse limit of the rings $\Z / p^n \Z$ under reduction maps.
Example: $1/2 \in \Z_3$, because $3$ does not divide $2$, so $\abs{1/2}_3 = 1$.
Non-example: $1/3 \notin \Z_3$, because $\abs{1/3}_3 = 3 > 1$.
A rational number lies in $\Z_p$ exactly when its reduced denominator is not divisible by $p$.
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<document source="https://tamnd.github.io/math2/node/GBWQF3HZG0.md" tag="GBWQF3HZG0" kind="concept">
# Ring of differential operators

## Summary

The ring of differential operators $D(A)$ of a commutative algebra $A$ over a field $k$ is a ring of $k$-linear maps $A \to A$.
Its elements behave like linear differential operators of finite order.
Grothendieck's definition uses only commutators with multiplication maps, so it makes sense for any commutative algebra, without coordinates.
For the polynomial ring $k[x_1, \dots, x_n]$ in characteristic $0$ it is the [[Weyl algebra|98QE7KY55W]] $A_n(k)$.
Its modules, called $D$-modules, are a tool for studying linear partial differential equations by algebra.

## Explanations

### L1 human

A differential operator such as $f \mapsto x^2 f'' + 3 f'$ combines multiplications by functions with derivatives.
Commuting such an operator with a multiplication by a function lowers its order by one, by the product rule.
An operator of order $0$ is just a multiplication, and the definition builds the higher orders from there.
For polynomials in one variable over the rationals, every such operator is a finite sum $\sum_{j} a_j(x) \, d^j/dx^j$ with polynomial coefficients $a_j$.
</document>

<document source="https://tamnd.github.io/math2/node/GBXD38Y52R.md" tag="GBXD38Y52R" kind="definition">
# Trigamma function

## Definition

Let $\psi$ be the [[digamma function|HWRRXB0YXS]].
The *trigamma function* is the meromorphic function
$$\psi_1(z) = \psi'(z) = \frac{d^2}{dz^2} \log \Gamma(z), \qquad z \in \C \setminus \set{0, -1, -2, \ldots}.$$

## Remarks

Equivalently, $\psi_1(z) = \sum_{k=0}^\infty \frac{1}{(z+k)^2}$, and $\psi_1$ is the [[polygamma function|3J3QGKPMZQ]] $\psi^{(1)}$.

Example: $\psi_1(1) = \sum_{k \ge 1} k^{-2} = \frac{\pi^2}{6}$.
Also $\psi_1(\frac12) = \sum_{k \ge 0} \frac{4}{(2k+1)^2} = 4 \cdot \frac{\pi^2}{8} = \frac{\pi^2}{2}$.

Non-example of a valid argument: at $z = 0$ the term $k = 0$ is $1/0^2$, so $\psi_1$ has a double pole there, as at every non-positive integer.

The recurrence $\psi_1(z+1) = \psi_1(z) - \frac{1}{z^2}$ and the reflection formula $\psi_1(1-z) + \psi_1(z) = \frac{\pi^2}{\sin^2 \pi z}$ hold for all $z$ that are not integers.
</document>

<document source="https://tamnd.github.io/math2/node/GC5BDZ6QBC.md" tag="GC5BDZ6QBC" kind="concept">
# Partial differential equation

## Summary

A partial differential equation, or PDE, is an equation that relates an unknown function of several variables to its [[partial derivatives|09EGC6F8PT]].
It is the several-variable counterpart of an [[ordinary differential equation|0NT64JV96A]].
PDEs describe heat flow, waves, fluids, electromagnetism, quantum mechanics and many other laws of physics, and they are central objects of geometry and probability.
See the [[definition|PV8DQ41Q55]].

## Explanations

### L1 human

Many quantities depend on several variables at once, such as the temperature at a point of a room at a given time.
A law of nature often says how the rates of change of such a quantity in different directions are linked.
Written as a formula, this law is a partial differential equation.
For example, the temperature $u(t, x)$ in a thin rod obeys $\partial_t u = \partial_x^2 u$, the [[heat equation|VP2HN86VAG]].
Solving the PDE means finding all functions $u$ for which the formula holds.
</document>

<document source="https://tamnd.github.io/math2/node/GC7T4VW3N1.md" tag="GC7T4VW3N1" kind="definition">
# Lie group action

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] with identity $e$ and let $M$ be a [[smooth manifold|F45VQ47GPV]].
A *(left) Lie group action* of $G$ on $M$ is a map $\alpha \colon G \times M \to M$, written $\alpha(g, p) = g \cdot p$, such that:

1. $e \cdot p = p$ for all $p \in M$;
2. $g \cdot (h \cdot p) = (gh) \cdot p$ for all $g, h \in G$ and $p \in M$;
3. $\alpha$ is [[smooth|0V33BNKZ9S]].

## Remarks

Clauses 1 and 2 say that $\alpha$ is a [[group action|0WCY42EV9V]]; clause 3 is the smoothness condition.
Right actions are defined in the same way with $(p \cdot h) \cdot g = p \cdot (hg)$.

Example: $SO(2)$ acts on $\R^2$ by $A \cdot v = Av$.
The identity matrix fixes every vector, $(AB)v = A(Bv)$, and the map $(A, v) \mapsto Av$ is polynomial in the entries, hence smooth.

Non-example: let $G = \R$ act on $M = \R$ by $t \cdot x = (x^3 + t)^{1/3}$, with the real cube root.
This is a group action, since $s \cdot (t \cdot x) = (x^3 + t + s)^{1/3}$, and it is continuous.
It is not smooth: the map $t \mapsto t \cdot 0 = t^{1/3}$ is not differentiable at $t = 0$.

Each $g$ acts by a diffeomorphism of $M$ with inverse given by $g^{-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/GC96MQC0PP.md" tag="GC96MQC0PP" kind="concept">
# Inverse transform

## Summary

The inverse transform of an [[integral transform|YCK709QB06]] $T$ recovers a function $f$ from its image $Tf$.
It exists when $T$ is one-to-one on the class of functions in use.
Often the inverse is itself an integral transform with its own kernel.
Examples are the Fourier inversion formula and the [[Bromwich integral|5KNAXNHRY6]] for the [[inverse Laplace transform|07WBTX1MHA]].
A typical use of a transform is: transform the problem, solve it on the transform side, then apply the inverse transform.

## Explanations

### L1 human

If a transform translates functions into another language, the inverse transform translates back.
For the [[Fourier transform|DPVM34ADZV]], the inverse looks almost like the transform itself: only the sign in the exponential changes.
For the Laplace transform, the inverse needs an integral along a vertical line in the complex plane, so in practice one uses tables of [[transform pairs|ZA1HC10WM6]].
For example, the inverse Laplace transform of $1/(s - 2)$ is $e^{2t}$.
</document>

<document source="https://tamnd.github.io/math2/node/GC9FN7TTBV.md" tag="GC9FN7TTBV" kind="statement">
# Brown representability theorem

## Statement

1. Let $F$ be a contravariant functor from the category of pointed connected [[CW complexes|5W95QFHXPR]] and pointed maps to the category of pointed sets. Assume that:
   - (homotopy) homotopic pointed maps induce the same map;
   - (wedge) for every family $(X_\alpha)$, the map $F(\bigvee_\alpha X_\alpha) \to \prod_\alpha F(X_\alpha)$ induced by the inclusions into the [[wedge sum|5JSWM1HBX2]] is a bijection;
   - (Mayer–Vietoris) let $X = A \cup B$ with $A$, $B$ and $A \cap B$ connected subcomplexes containing the base point. If $a \in F(A)$ and $b \in F(B)$ restrict to the same element of $F(A \cap B)$, then some $x \in F(X)$ restricts to $a$ and to $b$.

   Then $F$ is [[representable|KCCHH87H6E]]. That is, there are a pointed connected CW complex $K$ and an element $u \in F(K)$ with the following property. For every $X$ the map $[X, K]_* \to F(X)$, $f \mapsto F(f)(u)$, is a bijection, where $[X, K]_*$ is the set of pointed homotopy classes of maps.
2. (Neeman's version) Let $\mathcal{T}$ be a compactly generated [[triangulated category|T2V7GJJ2EN]] with all small [[coproducts|4FJ0S71P58]]. Let $H \colon \mathcal{T}^{\mathrm{op}} \to \mathrm{Ab}$ be a cohomological functor that sends coproducts in $\mathcal{T}$ to products. Then $H \cong \Hom_{\mathcal{T}}(-, Y)$ for some object $Y$ of $\mathcal{T}$.

## Remarks

The proof of part 1 builds $K$ by attaching cells, one dimension at a time, to make $[S^n, K]_* \to F(S^n)$ surjective and then injective.
The Mayer–Vietoris axiom is used to extend elements over each new cell.
Each reduced cohomology theory $h^n$ on pointed CW complexes satisfies the axioms in every degree, so $h^n(X) \cong [X, K_n]_*$ for spaces $K_n$, and these spaces form an $\Omega$-[[spectrum|1GHF4EKBXS]].
So every [[generalized cohomology theory|H28PSZR7XG]] is represented by a spectrum.
For ordinary cohomology this gives $H^n(X; G) \cong [X, K(G, n)]_*$ with $K(G, n)$ an [[Eilenberg–MacLane space|AJHEYJE5XY]], see [[representability of cohomology by Eilenberg–MacLane spaces|GPDPYHM3RE]].

## Explanations

### L1 human

Many invariants of spaces assign to each space a set, in a way that pulls back along maps.
Brown's theorem says that, under two natural gluing conditions, such an invariant is always given by maps into one fixed space, the classifying space.
For example, ordinary cohomology in degree $n$ is given by homotopy classes of maps into an Eilenberg–MacLane space.

### L2 human

The theorem is the reason why cohomology theories and spectra are two sides of the same coin in stable homotopy theory.
The wedge axiom is the abstract form of "sums go to products", and the Mayer–Vietoris axiom is a weak gluing condition; neither asks for uniqueness of the glued element.
Neeman's triangulated version is used to construct adjoint functors between derived categories, for example the right adjoint of the derived pushforward in Grothendieck duality.
</document>

<document source="https://tamnd.github.io/math2/node/GC9R87YZ6A.md" tag="GC9R87YZ6A" kind="definition">
# Action-angle coordinates

## Definition

Let $(M, \omega)$ be a [[symplectic manifold|CAN490EPRT]] of dimension $2n$, let $H \colon M \to \R$ be smooth, and let $U \subseteq M$ be open.
Let $\mathbb{T}^n = \R^n / 2\pi \Z^n$ be the $n$-[[torus|03NM7DZXXC]].
*Action-angle coordinates* for $H$ on $U$ are given by a [[diffeomorphism|0KZZ70ZHXP]] $\Psi \colon U \to D \times \mathbb{T}^n$, with $D \subseteq \R^n$ open, written $\Psi = (I_1, \ldots, I_n, \theta_1, \ldots, \theta_n)$, such that:

1. $\omega = \sum_{k=1}^n dI_k \wedge d\theta_k$ on $U$;
2. $H = K(I_1, \ldots, I_n)$ on $U$ for a smooth function $K \colon D \to \R$.

The $I_k$ are the *actions* and the $\theta_k$ are the *angles*.

## Remarks

Here the convention is $\omega = \sum_k dp_k \wedge dq_k$ on $\R^{2n}$, so that the actions play the role of momenta and the angles the role of positions.
In these coordinates the [[Hamiltonian system|0EDKGA2FAK]] reads $I_k' = 0$ and $\theta_k' = \partial K / \partial I_k (I)$, so $\theta(t) = \theta(0) + t \, \nabla K(I)$.

Example: the harmonic oscillator $H = \frac{1}{2}(p^2 + q^2)$ on $\R^2 \setminus \set{0}$.
Put $q = \sqrt{2I} \sin \theta$ and $p = \sqrt{2I} \cos \theta$ with $I > 0$.
A short computation gives $dp \wedge dq = dI \wedge d\theta$, and $H = I$, so both clauses hold with $K(I) = I$.

Non-example: polar coordinates $q = r \sin \theta$, $p = r \cos \theta$ with $r > 0$ give $H = r^2/2$, but $dp \wedge dq = r \, dr \wedge d\theta$, so clause 1 fails.

For an integrable system, the actions can be computed as $I_k = \frac{1}{2\pi} \oint_{\gamma_k} p \, dq$ over basic cycles $\gamma_k$ of an invariant torus, when $\omega = d(\sum_k p_k \, dq_k)$.
</document>

<document source="https://tamnd.github.io/math2/node/GCBY80CSC6.md" tag="GCBY80CSC6" kind="statement">
# Tubular neighborhood theorem

## Statement

Let $M$ be a [[smooth manifold|F45VQ47GPV]] and let $S \subseteq M$ be an [[embedded submanifold|VKFW27WXEZ]] without boundary.
Let $NS = TM|_S / TS$ be the normal bundle of $S$.
It is the quotient of the restriction to $S$ of the [[tangent bundle|WY6K3P4YCG]] of $M$ by the tangent bundle of $S$.
Identify $S$ with the zero section of $NS$.
Then there is an open neighbourhood $U$ of $S$ in $M$ and a [[diffeomorphism|0KZZ70ZHXP]] $\Phi \colon NS \to U$ such that $\Phi(s) = s$ for every $s \in S$.

## Remarks

Such a $U$, with the diffeomorphism $\Phi$, is a *tubular neighbourhood* of $S$.
For $M = \R^n$ the proof is direct: identify $NS$ with the bundle of vectors orthogonal to $S$, and consider $E(s, v) = s + v$.
Its differential along the zero section is an isomorphism.
By the inverse function theorem, $E$ is a local diffeomorphism near the zero section.
An extra argument shows that $E$ is injective, hence a diffeomorphism onto an open set, on a neighbourhood of the zero section that may shrink along $S$.
A fibrewise rescaling of $NS$ onto that neighbourhood finishes the proof.
For a general $M$ one replaces $s + v$ by the exponential map of a Riemannian metric, or embeds $M$ in some $\R^N$.
Example: the circle $S^1 \times \set{0}$ in $\R^3$ has trivial normal bundle, and a solid torus around it is a tubular neighbourhood, diffeomorphic to $S^1 \times \R^2$.
A tubular neighbourhood of the central circle of the open Möbius band is the whole band, which is the total space of the nontrivial line bundle over $S^1$.
Tubular neighbourhoods are unique up to isotopy.

## Explanations

### L1 human

Around a smooth submanifold there is always a thickened copy of it that looks like a bundle of small discs, one disc sticking out at each point.
Around a closed curve in space this is a thin tube, which is where the name comes from.
The theorem lets one study the neighbourhood of a submanifold through linear algebra on its normal bundle.

### L2 human

The theorem is used to cut and paste manifolds along submanifolds, as in surgery, connected sums along submanifolds and Dehn surgery on knots.
It identifies the quotient of $M$ by the complement of a tubular neighbourhood of $S$ with the Thom space of $NS$, which is the key step of the Pontryagin–Thom construction.
It also gives a smooth retraction of a neighbourhood of $S$ onto $S$, which is used to approximate continuous maps by smooth ones.
For manifolds with boundary, the companion result is the collar neighbourhood theorem.
</document>

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# Conjugate function

## Definition

Let $\mathbb{T} = \R / 2\pi\Z$ be the circle, and let $f \colon \mathbb{T} \to \C$ be [[Lebesgue integrable|DWJF7ABPR1]].
The *conjugate function* of $f$ is the function $\tilde f$ on $\mathbb{T}$ given by the [[principal value|0SYR10HD4R]]
$$\tilde f(\theta) = \lim_{\varepsilon \to 0^+} \frac{1}{2\pi} \int_{\varepsilon \le \abs{t} \le \pi} f(\theta - t) \cot\frac{t}{2} \, dt.$$
The limit exists for almost every $\theta \in \mathbb{T}$.

## Remarks

If $f$ is a trigonometric polynomial with [[Fourier series|G12KXDHPPS]] $\sum_k \hat f(k) e^{ik\theta}$, then
$$\tilde f(\theta) = \sum_k (-i \sgn k) \, \hat f(k) e^{ik\theta},$$
with $\sgn 0 = 0$.
For real $f$, let $u$ be the Poisson integral of $f$ in the unit disc, and let $v$ be its [[harmonic conjugate|050F2FMTKF]] with $v(0) = 0$.
Then $v(re^{i\theta}) \to \tilde f(\theta)$ as $r \to 1^-$ for almost every $\theta$.

Example: $f(\theta) = \cos\theta$ has $\tilde f(\theta) = \sin\theta$.
Check: $\cos(\theta - t)\cot(t/2) = \cos\theta \cos t \cot(t/2) + \sin\theta \sin t \cot(t/2)$.
The first term is odd in $t$, so its principal value integral is $0$.
The second uses $\sin t \cot(t/2) = 2\cos^2(t/2) = 1 + \cos t$, whose integral over $[-\pi, \pi]$ is $2\pi$.

Non-example: the integral without the principal value need not exist.
For $f = 1$ near $\theta$ the integrand behaves like $2/t$ near $t = 0$, which is not integrable, while the symmetric limit is $0$ since $\cot(t/2)$ is odd.

The function $\tilde f$ need not be integrable when $f$ is: the operator $f \mapsto \tilde f$ is bounded on $L^p(\mathbb{T})$ only for $1 < p < \infty$.
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<document source="https://tamnd.github.io/math2/node/GCH1Z9YFNG.md" tag="GCH1Z9YFNG" kind="problem">
# Hadamard's maximal determinant problem

## Problem

Let $n \ge 1$ be an integer, and let $\det$ denote the [[determinant|00KDZRW4C0]].
Find the largest value of $\det A$ over all $n \times n$ matrices $A$ whose entries are all $1$ or $-1$.

## Remarks

[[Hadamard's inequality|CGDN1X32NQ]] gives $\abs{\det A} \le n^{n/2}$ for such matrices.
Equality holds exactly when the rows of $A$ are pairwise orthogonal, that is, when $A$ is a Hadamard matrix.
A Hadamard matrix of order $n$ can exist only for $n = 1$, $n = 2$ or $n$ divisible by $4$.
The Hadamard conjecture, that one exists for every multiple of $4$, is open.
For other orders the maximum is smaller than $n^{n/2}$, and it is known only for some values of $n$.

The determinant of an $n \times n$ matrix with entries $\pm 1$ is divisible by $2^{n-1}$.
The maximum for $n$ equals $2^{n-1}$ times the maximal determinant of an $(n-1) \times (n-1)$ matrix with entries $0$ and $1$.
So the problem is the same as the maximal determinant problem for $0/1$ matrices.

Small cases: for $n = 2$, the maximum is $2$, reached by $\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$.
For $n = 3$, the bound $3^{3/2} \approx 5.2$ and the divisibility by $4$ show that the maximum is at most $4$, and $4$ is reached.

## Explanations

### L1 human

Fill a square grid with the numbers $1$ and $-1$.
The problem asks how large its determinant can be.
Geometrically, it asks how large the volume of a box can be when its edges join the centre of a cube to corners of the cube.
The best case is when all edges are at right angles, but this is possible only for some sizes.
</document>

<document source="https://tamnd.github.io/math2/node/GCPQGMJYEV.md" tag="GCPQGMJYEV" kind="concept">
# Skolem normal form

## Summary

A sentence is in Skolem normal form when it is in [[prenex normal form|CS82V969DE]] and has only universal quantifiers.
Every first-order sentence can be turned into one in Skolem normal form by replacing each existential quantifier with a new function symbol.
The new sentence is not equivalent to the old one, but it is satisfiable exactly when the old one is.
This makes Skolem normal form the starting point of [[Herbrand's theorem|VZ21F10WRS]] and of resolution theorem proving.

## Explanations

### L1 human

The sentence "for every $x$ there is a $y$ with $R(x, y)$" says that a suitable $y$ exists for each $x$.
Choose such a $y$ for each $x$ and call it $f(x)$.
Then the sentence becomes "for every $x$, $R(x, f(x))$", which has no "there exists".
The function $f$ is called a Skolem function.
</document>

<document source="https://tamnd.github.io/math2/node/GCPRFTKS4E.md" tag="GCPRFTKS4E" kind="concept">
# Convex independence

## Summary

A family of points is convexly independent if no point of the family lies in the [[convex hull|0FQSDC2EQ2]] of the other points.
A set of points with this property is said to be in convex position.
Then every point of the set is an [[extreme point|3MSGFJM2NX]] of its convex hull.
The notion is the convex analogue of linear and affine independence, and it is central in problems such as the Erdős–Szekeres problem on convex polygons.

## Explanations

### L1 human

Points are in convex position when each of them is a corner of the shape they span.
Stretch a rubber band around the points: every point must touch the band at a corner.
The four corners of a square are convexly independent.
If one adds the center of the square, the new set is not, because the center lies inside the square spanned by the corners.
</document>

<document source="https://tamnd.github.io/math2/node/GCQ23XT6VD.md" tag="GCQ23XT6VD" kind="statement">
# Ostrowski's theorem

## Statement

Let $\abs{\cdot}_*$ be a nontrivial absolute value on the field $\Q$ of rational numbers.
Then $\abs{\cdot}_*$ is equivalent either to the usual [[absolute value|0FFXKTKKQC]] $\abs{\cdot}_\infty$ or to the [[p-adic absolute value|WGN82TCEF2]] $\abs{\cdot}_p$ for exactly one [[prime number|01826KMG2H]] $p$.
Here two absolute values are equivalent when one is a positive real power of the other.

## Remarks

Proof idea: suppose first that $\abs{n}_* > 1$ for some positive integer $n$.
Writing integers in base $b$ and letting the number of digits grow, one shows $\abs{n}_* = n^{\alpha}$ for all positive integers $n$, with a fixed $0 < \alpha \le 1$; so $\abs{\cdot}_* = \abs{\cdot}_\infty^{\alpha}$.
Otherwise $\abs{n}_* \le 1$ for all integers $n$, and the integers with $\abs{n}_* < 1$ form a nonzero prime ideal $p\Z$.
Then $\abs{x}_* = \abs{p}_*^{v_p(x)}$, where $v_p$ is the [[p-adic valuation|03HDF50VN4]], and this is a power of $\abs{x}_p$.

The absolute values $\abs{\cdot}_\infty$ and $\abs{\cdot}_p$ for different $p$ are pairwise inequivalent: for instance $\abs{p}_p < 1$ while $\abs{p}_q = 1$ for a prime $q \ne p$ and $\abs{p}_\infty > 1$.
The hypothesis of nontriviality is needed: the trivial absolute value, equal to $1$ on all $x \ne 0$, is equivalent to none of these.

## Explanations

### L1 human

There are only a few reasonable ways to measure the size of a rational number.
One is the usual size; the others are the $p$-adic sizes, one for each prime $p$, which measure divisibility by $p$.
Up to rescaling, there is nothing else.

### L2 human

The theorem explains why number theory works with the real numbers and the [[p-adic numbers|BT6T702C28]] side by side: these are the completions of $\Q$ for all its [[places|BT5V3TSNY6]].
It leads to the product formula $\abs{x}_\infty \prod_p \abs{x}_p = 1$ for $x \in \Q^\times$, and to the [[adele ring|SEH68BA1QQ]].
For a general number field, the same method shows that every place comes from a real or complex embedding or from a nonzero prime ideal.
</document>

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# Contact structure on a jet bundle

## Summary

The contact structure of a [[jet bundle|HC3VBBCJQP]] is the family of contact forms: the $1$-forms that vanish on every [[prolonged section|BMCCQ8JXNE]].
Their common kernel is a distribution, called the contact or Cartan distribution.
The contact forms tell the holonomic sections apart from all others.
A section of the jet bundle is a prolongation exactly when it pulls back every contact form to zero.
In the simplest case, the first jet bundle of functions of one variable, it is the standard [[contact structure|HBQ8BZ7XB5]] on $\R^3$.

## Explanations

### L1 human

In the jet space with coordinates $(x, u, p)$, the letter $p$ is meant to be the slope $du/dx$.
The contact form $du - p\, dx$ expresses this wish.
Along the curve $x \mapsto (x, s(x), s'(x))$ of a function $s$, this form is zero, since $du = s'(x)\, dx = p\, dx$ there.
Along a curve where $p$ is not the slope, the form is not zero.
So the contact form is a test that a curve in jet space really comes from a function.
</document>

<document source="https://tamnd.github.io/math2/node/GCTN4BZ1ET.md" tag="GCTN4BZ1ET" kind="concept">
# Indeterminate moment problem

## Summary

A [[moment problem|CS7TWHY7JM]] is indeterminate when at least two different measures have the same moments.
It is the failure of [[determinacy|H0GY2J14CA]], and it can only happen when the support is unbounded.
In the indeterminate Hamburger case the solutions form an infinite convex set, described by the Nevanlinna parametrization.
[[Krein's condition|Q401KCEKNZ]] is a test for indeterminacy in terms of a density.

## Explanations

### L1 human

Sometimes two genuinely different distributions have exactly the same mean, the same variance, and the same moments of every order.
Then the moments alone cannot tell them apart, and the moment problem is called indeterminate.
This happens for distributions with very heavy tails, such as the log-normal distribution.
It never happens for distributions on a bounded interval.
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<document source="https://tamnd.github.io/math2/node/GCV10B5HES.md" tag="GCV10B5HES" kind="statement">
# Slutsky's theorem

## Statement

For each $n \ge 1$, let $X_n$ and $Y_n$ be real random variables on one probability space.
Let $X$ be a real random variable and $c \in \R$ a constant.
Assume that $X_n \to X$ [[in distribution|0SRYJGXKR4]] and $Y_n \to c$ [[in probability|0QA1HBD9Z4]].
Then, in distribution:

1. $X_n + Y_n \to X + c$;
2. $Y_n X_n \to c X$;
3. $X_n / Y_n \to X / c$, provided $c \ne 0$, where $X_n / Y_n$ is given any fixed value on the event $\set{Y_n = 0}$.

## Remarks

Proof idea: since the limit of $Y_n$ is a constant, the pair $(X_n, Y_n)$ converges in distribution to $(X, c)$.
Each of the three maps $(x, y) \mapsto x + y$, $xy$ and $x/y$ is continuous at the points $(x, c)$, so the [[continuous mapping theorem|CFRN0G7GDB]] applies.
Example: let $X_1, X_2, \dots$ be i.i.d. with mean $\mu$ and variance $\sigma^2 \in (0, \infty)$, with [[sample mean|03Y6AQ2S0J]] $\bar{X}_n$ and sample standard deviation $S_n$. The [[central limit theorem|0EPZC4Z1FC]] gives $\sqrt{n}(\bar{X}_n - \mu)/\sigma \to N(0, 1)$, and the law of large numbers gives $S_n \to \sigma$ in probability; so $\sqrt{n}(\bar{X}_n - \mu)/S_n \to N(0, 1)$ in distribution.
The limit $c$ must be a constant.
Let $Z$ be standard normal and $X_n = Z$; the variables $Z$ and $-Z$ have the same law.
With $Y_n = -Z$ one gets $X_n + Y_n = 0$, and with $Y_n = Z$ one gets $2Z$.
So convergence in law of both terms does not determine the law of the limit of the sum.
The theorem holds in the same way for random vectors, and for random matrices $Y_n$ that converge in probability to a constant matrix $C$.
For the quotient $Y_n^{-1} X_n$ one needs $C$ to be invertible.

## Explanations

### L1 human

Suppose one sequence settles into a random limit law and another settles down to a fixed number.
Then their sum, product and quotient behave as if the second sequence were already that number.
This is what allows statisticians to replace an unknown parameter by an estimate of it in an approximate normal law.

### L2 human

Slutsky's theorem is used constantly in asymptotic statistics, together with the [[central limit theorem|0EPZC4Z1FC]] and the [[delta method|DMMXBTN6M2]].
It justifies the use of estimated standard errors in confidence intervals and test statistics such as the Student $t$ statistic for large samples.
It follows from the [[continuous mapping theorem|CFRN0G7GDB]] applied to the pair $(X_n, Y_n)$.
Its key point is that joint convergence comes for free when one limit is constant.
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<document source="https://tamnd.github.io/math2/node/GCZ8EEDCHS.md" tag="GCZ8EEDCHS" kind="concept">
# Hyperbolic triangle

## Summary

A hyperbolic triangle is a triangle in the [[hyperbolic plane|GN4N94GQ0Z]]: three points not on one line, joined by geodesic segments.
Its angle sum is always less than $\pi$, and in curvature $-1$ its area equals the defect $\pi - (\alpha + \beta + \gamma)$.
So no hyperbolic triangle has area more than $\pi$.
Unlike in the Euclidean plane, a hyperbolic triangle is determined up to isometry by its three angles.
Its sides and angles are related by the [[hyperbolic law of cosines|7ZYKHQRKSJ]].

## Explanations

### L1 human

Draw a triangle on a saddle-shaped surface, with sides that are the shortest paths between the corners.
The sides bend inward compared with a flat triangle, so the corners become thinner and the three angles add up to less than $180^\circ$.
The bigger the triangle, the more angle it loses, and the lost angle measures its area.
In the limit, the three corners run off to infinity, the angles become $0$ and the area becomes exactly $\pi$; this is an ideal triangle.
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<document source="https://tamnd.github.io/math2/node/GD3Q8G9GCV.md" tag="GD3Q8G9GCV" kind="definition">
# Free probability

## Definition

A *noncommutative probability space* is a pair $(A, \varphi)$ where $A$ is a unital algebra over $\C$ and $\varphi \colon A \to \C$ is a linear functional with $\varphi(1) = 1$.
Elements of $A$ are called noncommutative random variables.
Let $(A_i)_{i \in I}$ be a family of unital subalgebras of $A$.
They are *freely independent*, or *free*, if the following holds.
Whenever $n \ge 1$, $a_j \in A_{i(j)}$ and $\varphi(a_j) = 0$ for all $j$, and consecutive indices differ, $i(1) \ne i(2) \ne \cdots \ne i(n)$, then
$$\varphi(a_1 a_2 \cdots a_n) = 0.$$
Elements of $A$ are free if the unital subalgebras they generate are free.
*Free probability* is the study of noncommutative probability spaces and of free independence.

## Remarks

Example: let $F_2$ be the [[free group|0KTW26C80J]] on generators $a$ and $b$, let $A = \C[F_2]$ be its group algebra, and let $\varphi(\sum_g c_g g) = c_e$, the coefficient of the identity.
Then the subalgebras $A_1 = \C[\langle a \rangle]$ and $A_2 = \C[\langle b \rangle]$ are free.
An element of $A_1$ with $\varphi = 0$ is a combination of powers $a^k$ with $k \ne 0$, and likewise for $A_2$.
A product of such elements with alternating indices is a combination of alternating words $a^{k_1} b^{l_1} a^{k_2} \cdots$ with nonzero exponents.
These are reduced words, hence not the identity, so $\varphi$ of the product is $0$.
The same argument works for any [[free product|TM7QS7P7X9]] of groups.

Non-example: classically independent variables are not free.
Let $A = L^\infty(\Omega, P)$ for a [[probability space|0NFPJVJJHX]] $(\Omega, P)$, $\varphi$ the expectation, and $x, y$ independent with mean $0$ and nonzero variance.
Then $\varphi(xyxy) = \varphi(x^2)\varphi(y^2) \ne 0$, although the indices alternate, so the defining condition fails.

Often $A$ is a C*-algebra or a von Neumann algebra and $\varphi$ is a state, frequently a faithful trace.
Freeness determines all mixed moments of free variables from their separate moments, as independence does classically.
</document>

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# d'Alembert's principle

## Statement

Consider $N$ particles with masses $m_i > 0$, positions $r_i(t) \in \R^3$ and accelerations $a_i = \ddot r_i$, for $i = 1, \dots, N$.
Let $F_i$ be the applied force on particle $i$.
Let the particles be subject to ideal constraints, that is, constraints whose reaction forces do no work on any virtual displacement.
A virtual displacement at time $t$ is a family $(\delta r_1, \dots, \delta r_N)$ of vectors tangent to the constraints at fixed time $t$.
Then the motion satisfies
$$\sum_{i=1}^{N} (F_i - m_i a_i) \cdot \delta r_i = 0$$
for every virtual displacement $(\delta r_1, \dots, \delta r_N)$.

## Remarks

By Newton's second law, $m_i a_i = F_i + R_i$, where $R_i$ is the constraint force on particle $i$.
So the sum equals $-\sum_i R_i \cdot \delta r_i$, which is zero exactly because the constraints are ideal.
The gain is that the unknown constraint forces disappear from the equations.
For a holonomic system with generalised coordinates $q^1, \dots, q^n$, the virtual displacements are $\delta r_i = \sum_k \frac{\partial r_i}{\partial q^k} \delta q^k$ with arbitrary $\delta q^k$.
Inserting them gives Lagrange's equations $\frac{d}{dt}\frac{\partial T}{\partial \dot q^k} - \frac{\partial T}{\partial q^k} = Q_k$, with $T$ the kinetic energy and $Q_k = \sum_i F_i \cdot \frac{\partial r_i}{\partial q^k}$ the generalised forces.
If the applied forces come from a potential $V$, these are the [[Euler–Lagrange equations|PS0VV149ET]] of $L = T - V$.
Example: take a bead of mass $m$ on a fixed smooth horizontal wire along the $x$-axis, under gravity.
The virtual displacements are $(\delta x, 0, 0)$, and the principle gives $m\ddot x = 0$.
The vertical reaction force never enters.
Without the ideal hypothesis the principle fails: sliding friction does work on virtual displacements along the wire.

## Explanations

### L1 human

A constrained system, like a bead on a wire or a pendulum on a rod, feels forces from its constraints.
These forces are hard to know in advance.
Suppose the constraints do no work in the allowed directions.
D'Alembert's principle says that then one may test Newton's law only in the allowed directions, and the constraint forces drop out.

### L2 human

D'Alembert's principle is the bridge from Newton's laws to Lagrangian mechanics.
It extends the principle of virtual work from statics to dynamics by treating $-m_i a_i$ as an extra force, the inertial force.
Integrating it in time over variations with fixed endpoints gives [[Hamilton's principle|FAQ9CH2Z76]] for holonomic systems with a potential.
It also holds for nonholonomic constraints linear in the velocities, where it leads to the equations with multipliers.
</document>

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# Bromwich justification of the Heaviside calculus

## Summary

The Bromwich justification gives the formal expressions $\varphi(p) H$ of the [[Heaviside operational calculus|QQ98C2AXZP]] a precise meaning as contour integrals.
It defines $\varphi(p) H$ by the [[Bromwich integral|5KNAXNHRY6]] of $\varphi(s) e^{st} / s$ along a vertical line in the complex plane.
With this definition, Heaviside's rules become theorems of complex analysis, and his expansion theorem becomes the residue theorem.

## Explanations

### L1 human

Heaviside's symbolic tricks worked but nobody could say why.
The answer was to read each symbolic expression as an integral in the complex plane.
Evaluating that integral by residues reproduces exactly the answers that Heaviside's rules give.
For example, $p^{-1} H$ becomes an integral whose only residue gives the function $t$, the integral of the unit step.
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<document source="https://tamnd.github.io/math2/node/GD8FNE3GTF.md" tag="GD8FNE3GTF" kind="definition">
# Real analytic manifold

## Definition

Let $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.
A *real analytic atlas* on $M$ is an [[atlas|J7P32JQTVE]] $\set{(U_i, \varphi_i)}$ of charts $\varphi_i \colon U_i \to \R^n$ with the following property.
For all $i, j$ the transition map
$$\varphi_j \circ \varphi_i^{-1} \colon \varphi_i(U_i \cap U_j) \to \varphi_j(U_i \cap U_j)$$
is real [[analytic|081WKEDK6C]].
That is, each component of the transition map is given near each point by a convergent power series.
A *real analytic manifold* is a topological manifold $M$ together with a maximal real analytic atlas on $M$.

## Remarks

Example: the sphere $S^n \subseteq \R^{n+1}$ with the two stereographic projections is a real analytic manifold.
The transition map is $y \mapsto y / \abs{y}^2$ on $\R^n \setminus \set{0}$, whose components are quotients of polynomials by a nonvanishing polynomial, hence real analytic.

Non-example: on $\R$ take the identity chart and the chart $\psi$ with $\psi(x) = x$ for $x \le 0$ and $\psi(x) = x + e^{-1/x^2}$ for $x > 0$.
The map $\psi$ is a homeomorphism of $\R$ with positive derivative, and the transition map $\psi$ is smooth, so this atlas makes $\R$ a [[smooth manifold|F45VQ47GPV]].
But $\psi$ is not real analytic at $0$: all its derivatives at $0$ agree with those of $x \mapsto x$, while $\psi(x) \neq x$ for $x > 0$.
So the atlas is not real analytic.

A real analytic manifold is in particular a smooth manifold, since its maximal real analytic atlas lies in a unique maximal smooth atlas.
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# Exponential object

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] with binary [[products|S5744QZV4N]], and let $Y$ and $Z$ be objects of $\mathcal{C}$.
An *exponential object* of $Z$ by $Y$ is an object $Z^Y$ with a morphism $\mathrm{ev} \colon Z^Y \times Y \to Z$ with this [[universal property|EN4DD2FAJH]].
For every object $X$ and every morphism $g \colon X \times Y \to Z$, there is a unique morphism $\lambda g \colon X \to Z^Y$ with $\mathrm{ev} \circ (\lambda g \times \id_Y) = g$.

## Remarks

The morphism $\mathrm{ev}$ is the *evaluation*, and $\lambda g$ is the *transpose* or *curried form* of $g$.
An exponential object is unique up to a unique isomorphism compatible with $\mathrm{ev}$.
If $Z^Y$ exists for all $Z$, then $Z \mapsto Z^Y$ is a functor right adjoint to $- \times Y$.

Example: in the category of sets, let $Z^Y$ be the set of maps $Y \to Z$ and $\mathrm{ev}(h, y) = h(y)$.
For $g \colon X \times Y \to Z$, the map $\lambda g$ sends $x$ to $y \mapsto g(x, y)$.
It is the only map with $\lambda g(x)(y) = g(x, y)$ for all $x, y$.

Non-example: in the category of topological spaces, not every exponential exists.
For the space $\Q$ of rational numbers, the functor $- \times \Q$ has no right adjoint, so some $Z^{\Q}$ do not exist.
For $Y$ locally compact Hausdorff, $Z^Y$ exists and is the space of maps with the [[compact-open topology|RG15F0TWPW]].
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# Uniqueness of identity element

## Statement

The group axioms imply that the identity element is unique; that is, there exists only one identity element: any two identity elements $e$ and $f$ of a group are equal, because the group axioms imply $e=e\cdot f=f$. It is thus customary to speak of *the* identity element of the group.
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<document source="https://tamnd.github.io/math2/node/GDN82Y3ATR.md" tag="GDN82Y3ATR" kind="definition">
# List coloring

## Definition

Let $G = (V, E)$ be a finite [[graph|0JD5C82QMB]].
A list assignment for $G$ is a map $L$ that assigns to each vertex $v$ a finite set $L(v)$ of colours.
An $L$-colouring of $G$ is a map $c$ on $V$ such that:

1. $c(v) \in L(v)$ for every vertex $v$.
2. $c(u) \ne c(v)$ for every edge $uv$.

For an integer $k \ge 1$, the graph $G$ is $k$-choosable if it has an $L$-colouring for every list assignment $L$ with $\abs{L(v)} \ge k$ for all $v$.
The choice number $\operatorname{ch}(G)$ is the least $k$ such that $G$ is $k$-choosable.

## Remarks

If all lists equal $\set{1, \dots, k}$, an $L$-colouring is a proper [[colouring|0874138YP6]] with $k$ colours, so $\operatorname{ch}(G) \ge \chi(G)$, where $\chi$ is the [[chromatic number|0C5M7AM6FH]].
Example: every tree is $2$-choosable.
Colour the vertices in an order in which each vertex after the first has exactly one earlier neighbour.
Each vertex then picks a colour of its list other than the colour of that neighbour.
Non-example: $K_{3,3}$ is not $2$-choosable, although $\chi(K_{3,3}) = 2$.
Give the vertices on each side the lists $\set{1,2}$, $\set{1,3}$ and $\set{2,3}$.
The colours used on one side meet each of these three sets, so they include at least two colours.
Two of these colours form the list of some vertex on the other side, and that vertex fails clause 2.
Thomassen proved that every planar graph is $5$-choosable.
Choice number is also called list chromatic number and written $\chi_\ell(G)$.
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# Grothendieck topology

## Summary

A Grothendieck topology on a [[category|A2DE66937D]] says which families of morphisms into an object count as coverings of that object.
It is stated with [[sieves|CNCXNWKC6R]] and abstracts the open covers of a topological space.
A category with a Grothendieck topology is a [[site|NYE0Z94MME]], and one can define [[sheaves|RT8A8FJ566]] and sheaf cohomology on it.
Grothendieck introduced it to define étale cohomology of schemes, where the "open sets" are étale maps rather than subsets.

## Explanations

### L1 human

On a topological space, a sheaf is defined by gluing data along open covers.
A Grothendieck topology keeps only what is needed for this: a rule saying which collections of arrows into an object cover it.
The rule must contain the trivial cover, be stable when restricted to smaller pieces, and allow covers of covers.
For example, on the open sets of a space $X$, a family of open subsets of $U$ covers $U$ when its union is $U$.
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# Strictly convex space

## Summary

A normed space is strictly convex when its unit sphere contains no line segment.
Equivalently, the midpoint of two distinct unit vectors has norm less than $1$.
Strict convexity gives uniqueness results, such as uniqueness of the nearest point of a convex set and equality cases in the triangle inequality.
It is weaker than [[uniform convexity|BEAMC5011M]].

## Explanations

### L1 human

Look at the boundary of the unit ball of a normed space.
In a strictly convex space this boundary is curved everywhere and has no flat pieces.
The Euclidean disc is strictly convex.
A square is not, because its sides are flat segments.
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# Universal cover

## Definition

Let $X$ be a [[path-connected|0WB6DH499T]] topological space.
A *universal cover* of $X$ is a [[covering map|3XZRRE165N]] $p \colon \tilde X \to X$ such that $\tilde X$ is simply connected.
Here $\tilde X$ is simply connected if it is path-connected and its [[fundamental group|S1XK3HXHFX]] $\pi_1(\tilde X, \tilde x)$ is trivial for one, hence every, point $\tilde x \in \tilde X$.

## Remarks

Example: $p \colon \R \to S^1$, $p(t) = (\cos 2\pi t, \sin 2\pi t)$, is a covering map.
Its total space $\R$ is path-connected and contractible, so $\pi_1(\R) = 1$, and $p$ is a universal cover of the circle.
For $n \ge 2$, the quotient map $S^n \to \R P^n$ is a universal cover, because $S^n$ is simply connected.

Non-example: the map $S^1 \to S^1$, $z \mapsto z^2$, on the unit circle in $\C$ is a covering map.
Its total space $S^1$ has fundamental group $\Z$, so it is not a universal cover.

If $X$ is connected, locally path-connected and [[semilocally simply connected|R3SHZX0NZR]], then a universal cover exists.
If $X$ is locally path-connected, two universal covers of $X$ are isomorphic over $X$.
A universal cover is often written $\tilde X \to X$.
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# Fast wavelet transform

## Definition

Let $\varphi$ be the [[scaling function|YH4BERB952]] of an orthonormal [[multiresolution analysis|1GHQMBR7DE]] of $L^2(\R)$ with real finite filter $(h_n)_{n \in \Z}$, so that
$$\varphi(x) = \sqrt{2} \sum_{n} h_n \, \varphi(2x - n).$$
Put $g_n = (-1)^n h_{1-n}$ and let $\psi(x) = \sqrt{2}\sum_n g_n \varphi(2x - n)$ be the associated wavelet.
For $j, k \in \Z$ write $\varphi_{j,k}(x) = 2^{j/2}\varphi(2^j x - k)$ and $\psi_{j,k}(x) = 2^{j/2}\psi(2^j x - k)$.
For $f \in L^2(\R)$ put $a_{j,k} = \ip{f}{\varphi_{j,k}}$ and $d_{j,k} = \ip{f}{\psi_{j,k}}$.

The *fast wavelet transform* from level $J$ down to level $J_0 < J$ is the algorithm that takes the sequence $(a_{J,k})_k$ and, for $j = J, J-1, \dots, J_0 + 1$, computes
$$a_{j-1,k} = \sum_{m} h_{m-2k} \, a_{j,m}, \qquad d_{j-1,k} = \sum_{m} g_{m-2k} \, a_{j,m}.$$
Its output is $(a_{J_0,k})_k$ together with $(d_{j,k})_k$ for $J_0 \le j < J$.
The inverse fast wavelet transform computes, for $j = J_0 + 1, \dots, J$,
$$a_{j,m} = \sum_{k} h_{m-2k} \, a_{j-1,k} + \sum_{k} g_{m-2k} \, d_{j-1,k}.$$

## Remarks

The formulas follow from $\varphi_{j-1,k} = \sum_m h_{m-2k}\varphi_{j,m}$ and $\psi_{j-1,k} = \sum_m g_{m-2k}\varphi_{j,m}$, which are the refinement equations rescaled.
Each step is a convolution with a filter followed by downsampling by $2$.

Example: for the Haar wavelet, $h_0 = h_1 = 1/\sqrt{2}$, so $g_0 = 1/\sqrt{2}$ and $g_1 = -1/\sqrt{2}$.
One step maps $a_{j,\cdot}$ to $a_{j-1,k} = (a_{j,2k} + a_{j,2k+1})/\sqrt{2}$ and $d_{j-1,k} = (a_{j,2k} - a_{j,2k+1})/\sqrt{2}$, and the inverse step recovers $a_{j,2k}$ and $a_{j,2k+1}$ from these two numbers.

Non-example: if the downsampling is left out and all outputs of the filters are kept, the result is the undecimated (stationary) wavelet transform.
It is redundant: it doubles the number of coefficients at each level, while the fast wavelet transform keeps the total number of coefficients equal to the input length.

With an input of length $N$ and filters of length $L$, the algorithm uses at most about $2LN$ multiplications.

## Explanations

### L1 human

The fast wavelet transform computes all wavelet coefficients by repeatedly filtering and halving.
It never computes an integral: everything comes from the filter coefficients of the scaling function.
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# Ham sandwich theorem

## Statement

Let $d \ge 1$ and let $\mu_1, \dots, \mu_d$ be finite Borel measures on $\R^d$ that are [[absolutely continuous|P0P21KDV6E]] with respect to [[Lebesgue measure|2Z2QVWZ262]].
Then there is an affine hyperplane $H$ of $\R^d$ such that, for each $i$, the two open half-spaces bounded by $H$ have measure $\frac{1}{2} \mu_i(\R^d)$ each.

## Remarks

The usual proof applies the [[Borsuk–Ulam theorem|QHJ0GBQE2E]].
A unit vector $u = (u_0, u_1, \dots, u_d)$ of $\R^{d+1}$ gives the half-space $H_u^+ = \set{x \in \R^d : u_0 + u_1 x_1 + \cdots + u_d x_d > 0}$.
The map $f \colon S^d \to \R^d$ with $f_i(u) = \mu_i(H_u^+) - \mu_i(H_{-u}^+)$ is continuous and odd, so it has a zero $u$.
At such a zero the hyperplane $u_0 + u_1 x_1 + \cdots + u_d x_d = 0$ bisects every $\mu_i$; the points $u = (\pm 1, 0, \dots, 0)$ are not zeros as long as the measures are nonzero.
For $d = 2$: any two bounded regions of the plane, such as two pancakes, can be cut in half at once by one straight line.
Absolute continuity is needed: for $d = 1$ and $\mu_1$ a single point mass, no point splits $\mu_1$ into two open halves of measure $\frac12$ each.

## Explanations

### L1 human

Take a sandwich made of bread, ham and cheese, placed in space in any way.
One straight cut with a knife can halve the bread, the ham and the cheese at the same time.
In the plane, two shapes can always be halved by one straight line.
In general, $d$ objects in $d$-dimensional space can be halved by one flat cut.

### L2 human

The theorem is a standard first application of the [[Borsuk–Ulam theorem|QHJ0GBQE2E]] to combinatorial geometry.
It has a discrete version for finite point sets, where the hyperplane may contain some points and each open side has at most half of each set.
It is used for partitioning point sets in computational geometry, and it is related to the [[necklace splitting problem|WFS3PHFHA1]].
</document>

<document source="https://tamnd.github.io/math2/node/GDY62GEC1H.md" tag="GDY62GEC1H" kind="definition">
# Stochastic geometry

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]], and let $E$ be a space with a geometric structure, usually the [[Euclidean space|0XPR1E85W0]] $\R^d$.
*Stochastic geometry* is the study of [[random elements|0PNRKMRC55]] defined on $(\Omega, \mathcal{F}, P)$ whose values are geometric objects in $E$.
Its basic objects are:

1. [[point processes|DNDKBVHS3H]], that is, random locally finite sets or counting measures in $E$;
2. random closed sets, that is, random elements of the family of closed subsets of $E$;
3. random measures on $E$, and structures built from the above, such as random tessellations, fibre processes and germ-grain models.

The questions concern the laws of these objects, their mean values such as intensities and volume fractions, and their behaviour under motions of $E$.

## Remarks

Example: the [[Poisson point process|0MTNR28K60]] in $\R^d$ is a point process, and the [[Boolean model|NK3CZTVEJD]] built from it is a random closed set; both are central models of the field.

Non-example: a fixed lattice $\Z^d \subset \R^d$ involves no randomness, so on its own it is a question of deterministic geometry.
A randomly shifted lattice $U + \Z^d$, with $U$ uniform on $[0, 1)^d$, is again a point process and belongs to the field.

The term geometric probability is often used for the older part of the subject, which deals with a fixed number of random points, lines or bodies.
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<document source="https://tamnd.github.io/math2/node/GE020MY43M.md" tag="GE020MY43M" kind="definition">
# Schwartz space

## Definition

The *Schwartz space* $\mathcal{S}(\R^n)$ is the set of [[smooth functions|0V33BNKZ9S]] $f \colon \R^n \to \C$ such that
$$p_{\alpha, \beta}(f) = \sup_{x \in \R^n} \abs{x^\alpha \, \partial^\beta f(x)} < \infty$$
for all [[multi-indices|0PJXBR6QA3]] $\alpha, \beta \in \N^n$.
Its elements are called *Schwartz functions* or *rapidly decreasing functions*.
With the family of seminorms $p_{\alpha, \beta}$, the space $\mathcal{S}(\R^n)$ is a [[Fréchet space|3GAA614673]].

## Remarks

Example: the Gaussian $f(x) = e^{-\abs{x}^2}$ is a Schwartz function.
Each derivative $\partial^\beta f$ is a polynomial times $e^{-\abs{x}^2}$, and a polynomial times $e^{-\abs{x}^2}$ is bounded on $\R^n$.
Every [[test function|HS65DW0HXH]] on $\R^n$ is a Schwartz function.
Non-example: $f(x) = 1/(1 + x^2)$ on $\R$ is smooth, but $p_{2, 0}(f) = \sup_x x^2/(1 + x^2) = 1$ while $p_{3, 0}(f) = \sup_x \abs{x}^3/(1 + x^2) = \infty$.
The space is closed under differentiation, under multiplication by polynomials and under the [[Fourier transform|DPVM34ADZV]], which is a bijection of $\mathcal{S}(\R^n)$ onto itself.
Equivalently, $f$ is a Schwartz function if and only if $(1 + \abs{x})^N \abs{\partial^\beta f(x)}$ is bounded for every $N \in \N$ and every $\beta$.

## Explanations

### L1 human

A Schwartz function is a smooth function that, with all its derivatives, decays faster than every inverse power of $\abs{x}$.
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<document source="https://tamnd.github.io/math2/node/GE05N4B84Q.md" tag="GE05N4B84Q" kind="definition">
# Analytic semigroup

## Definition

Let $X$ be a complex [[Banach space|06GZ956657]] and let $(T(t))_{t \ge 0}$ be a [[strongly continuous semigroup|EVF180QJXY]] of bounded operators on $X$.
For $0 < \theta \le \pi/2$, let $\Delta_\theta = \set{z \in \C : z \ne 0, \ \abs{\arg z} < \theta}$.
The semigroup is *analytic* if for some $\theta \in (0, \pi/2]$ there is a family $(T(z))_{z \in \Delta_\theta}$ of bounded operators on $X$ with these properties.
It agrees with the semigroup on $(0, \infty)$, and:

1. $T(z_1 + z_2) = T(z_1)T(z_2)$ for all $z_1, z_2 \in \Delta_\theta$;
2. the map $z \mapsto T(z)$ is [[holomorphic|0NAHXR1FRH]] on $\Delta_\theta$ with values in the bounded operators;
3. $T(z)x \to x$ as $z \to 0$ in $\Delta_\theta$, for every $x \in X$.

## Remarks

Example: let $P \ge 0$ be a self-adjoint operator on a Hilbert space and $T(t) = e^{-tP}$.
By the spectral theorem, $T(z) = e^{-zP}$ is defined for $\operatorname{Re} z > 0$, with $\norm{T(z)} \le 1$.
It satisfies the three clauses with $\theta = \pi/2$, so $e^{-tP}$ is analytic; the heat semigroup is the case $P = -\Delta$.

Non-example: on $L^2(\R)$, let $(T(t)f)(s) = f(s + t)$.
For an analytic semigroup, $T(t)x$ lies in the domain of the [[generator|60HSWZBTK9]] for every $t > 0$ and every $x$.
Here this would mean that every shifted $L^2$ function lies in the Sobolev space $H^1(\R)$, which is false, so no extension satisfying the clauses exists.

The semigroup is a *bounded analytic semigroup* if moreover $\norm{T(z)}$ is bounded on each smaller sector $\Delta_{\theta'}$ with $0 < \theta' < \theta$.
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# Fabry gap theorem

## Statement

Let $(\lambda_k)_{k \ge 1}$ be a strictly increasing sequence of nonnegative integers such that $\lambda_k / k \to \infty$ as $k \to \infty$.
Let $(a_k)$ be complex numbers such that the [[power series|0VP56B4EMX]]
$$f(z) = \sum_{k=1}^{\infty} a_k z^{\lambda_k}$$
has a [[radius of convergence|0660QX1VPR]] $R$ with $0 < R < \infty$.
Then the circle $\abs{z} = R$ is a [[natural boundary|PQFEA12AHA]] of $f$.
That is, $f$ has no [[analytic continuation|0M3YQJ81NB]] to a neighbourhood of any point of this circle.

## Remarks

The condition $\lambda_k / k \to \infty$ says that the exponents that occur have density zero among the nonnegative integers.
It is weaker than the condition $\lambda_{k+1} \ge q \lambda_k$ with $q > 1$ of the [[Hadamard gap theorem|DA8M8VZZHC]], so the Fabry theorem contains the Hadamard theorem.
For example, the series $\sum_{k \ge 1} z^{k^2}$ has radius $1$, and $k^2 / k = k \to \infty$, so the unit circle is a natural boundary; the Hadamard theorem does not apply here, because $(k+1)^2 / k^2 \to 1$.
The gap condition cannot simply be dropped: the series $\sum_{k \ge 0} z^k = 1/(1-z)$ has radius $1$ and continues analytically across every point of the unit circle except $z = 1$.
The proof is harder than for the Hadamard theorem and is not sketched here.

## Explanations

### L1 human

A power series with very many missing powers cannot be extended beyond its disc of convergence at all.
Each point of the boundary circle is a singular point.
"Very many missing" means that among the first $N$ powers only a vanishing proportion appear.
The series $z + z^4 + z^9 + z^{16} + \cdots$ is an example.

### L2 human

Usually a power series has only a few singular points on its circle of convergence, and the function continues across the rest.
Gap theorems show the opposite behaviour for lacunary series.
The density condition on the exponents is the essential hypothesis, and it is much weaker than a fixed ratio between consecutive exponents.
It shows that lacunary series are a simple source of functions with a natural boundary.
</document>

<document source="https://tamnd.github.io/math2/node/GE75S5XZ9T.md" tag="GE75S5XZ9T" kind="definition">
# Polarity

## Definition

Let $V$ be a [[vector space|000AYPA9AJ]] of finite dimension $n + 1 \ge 3$ over a division ring.
Let $\mathcal{L}(V)$ be the set of its [[linear subspaces|00YC1D652K]], which are the subspaces of the [[projective space|ZT9A6JK8QF]] $\mathbb{P}(V)$.
A *correlation* of $\mathbb{P}(V)$ is a bijection $\delta \colon \mathcal{L}(V) \to \mathcal{L}(V)$ such that $U \subseteq W$ if and only if $\delta(W) \subseteq \delta(U)$, for all $U, W \in \mathcal{L}(V)$.
A *polarity* is a correlation $\delta$ with $\delta(\delta(U)) = U$ for all $U \in \mathcal{L}(V)$.

## Remarks

A correlation sends subspaces of dimension $k$ to subspaces of dimension $n + 1 - k$, so it exchanges points and hyperplanes.
For a polarity $\delta$, the hyperplane $\delta(P)$ is the polar of the point $P$, and $P$ is the pole of that hyperplane.

Example: let $F$ be a field, $V = F^{n+1}$ and $\delta(U) = U^\perp = \set{y : x \cdot y = 0 \text{ for all } x \in U}$, with the dot product $x \cdot y = \sum_i x_i y_i$.
If $U \subseteq W$ then $W^\perp \subseteq U^\perp$; since the dot product is a non-degenerate symmetric [[bilinear form|0SQZMXB6GP]], $\dim U^\perp = n + 1 - \dim U$ and $(U^\perp)^\perp = U$.
So $\delta$ is a bijection that reverses inclusion and has $\delta^2 = \id$: it is a polarity.

Non-example: on $F^3$ let $M$ be the matrix with rows $(1, 1, 0)$, $(0, 1, 0)$, $(0, 0, 1)$, and let $\delta(U) = \set{y : x^T M y = 0 \text{ for all } x \in U}$.
This is a correlation, but $\delta(\delta(U)) = M^{-1} M^T U$.
For $U = \langle e_1 \rangle$ one gets $M^T e_1 = (1, 1, 0)$ and $M^{-1} (1, 1, 0) = (0, 1, 0)$, so $\delta^2(\langle e_1 \rangle) = \langle e_2 \rangle \ne \langle e_1 \rangle$, and $\delta$ is not a polarity.

Every correlation of $\mathbb{P}(V)$ comes from a non-degenerate [[sesquilinear form|054YEQ0Y8M]] $f$ by $\delta(U) = \set{y : f(x, y) = 0 \text{ for all } x \in U}$.
It is a polarity exactly when $f$ is reflexive, that is $f(x, y) = 0$ implies $f(y, x) = 0$; up to a scalar, such forms are symmetric, alternating or Hermitian.
</document>

<document source="https://tamnd.github.io/math2/node/GE8MRV41R7.md" tag="GE8MRV41R7" kind="concept">
# Submultiplicative function

## Summary

A submultiplicative function is a nonnegative function $f$ on a [[semigroup|0XMEM406JM]] with $f(xy) \le f(x) f(y)$ for all $x$ and $y$.
It is the multiplicative twin of a [[subadditive function|3X9DD9J3CV]]: where $f$ is positive, $\log f$ is subadditive.
The [[operator norm|06EKW1AKXF]] on square matrices is submultiplicative.
So is the Frobenius norm, a [[matrix norm|54T6Y6DSY4]] that is not an operator norm.
With Fekete's subadditive lemma this gives the existence of limits such as $\lim_n \norm{T^n}^{1/n}$.

## Explanations

### L1 human

A submultiplicative quantity never grows faster under composition than the product of its values.
Think of a machine that stretches vectors by at most a factor $a$ followed by one that stretches by at most $b$: together they stretch by at most $ab$.
For example, the sequence $a_n = \norm{A^n}$ of norms of powers of a matrix $A$ satisfies $a_{m+n} \le a_m a_n$.
So the growth of $\norm{A^n}$ is controlled by a single number, the limit of $a_n^{1/n}$.
</document>

<document source="https://tamnd.github.io/math2/node/GEA8H3228S.md" tag="GEA8H3228S" kind="definition">
# Integral of a non-negative function

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
Let $s = \sum_{i=1}^{n} a_i \mathbf{1}_{A_i}$ be a non-negative [[simple function|7WMBX7M4P6]], with $a_i \in [0, \infty)$ and pairwise disjoint $A_i \in \Sigma$.
Its integral is
$$\int_X s \, d\mu = \sum_{i=1}^{n} a_i \, \mu(A_i),$$
with the convention $0 \cdot \infty = 0$.
Let $f \colon X \to [0, \infty]$ be a [[measurable function|YYNCJSR0PJ]].
Its *integral* is the [[supremum|0BTMQ04556]], in the [[extended real line|0GSZ7B0CGR]],
$$\int_X f \, d\mu = \sup \set{\int_X s \, d\mu : s \text{ simple}, \ 0 \le s \le f}.$$

## Remarks

Example: let $X = \N$ with the counting measure and $f(n) = 2^{-n}$ for $n \ge 1$, $f(0) = 1$.
The simple functions $s_N = f \cdot \mathbf{1}_{\set{0, \dots, N}}$ lie below $f$, and their integrals $\sum_{n=0}^{N} 2^{-n}$ increase to $2$.
Every simple $s \le f$ has integral at most $\sum_n f(n) = 2$, so $\int f \, d\mu = 2$.

Non-example: the formula does not apply to $g(x) = \sin x$ on $\R$, because $g$ takes negative values.
Such functions need the splitting $g = g^+ - g^-$ of the Lebesgue integral.

The value of $\int_X s \, d\mu$ for a simple function does not depend on the way $s$ is written.
For $A \in \Sigma$ one writes $\int_A f \, d\mu = \int_X f \mathbf{1}_A \, d\mu$.
If $f = g$ almost everywhere, then $\int_X f \, d\mu = \int_X g \, d\mu$.
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<document source="https://tamnd.github.io/math2/node/GEFBDG5CJ3.md" tag="GEFBDG5CJ3" kind="statement">
# Steffensen's method

## Statement

Let $f$ be a real function that is twice continuously differentiable on an open interval $I$.
Let $r \in I$ be a simple zero of $f$, that is, $f(r) = 0$ and $f'(r) \ne 0$.
For $x$ with $f(x) \ne 0$, put
$$g(x) = \frac{f(x + f(x)) - f(x)}{f(x)}.$$
*Steffensen's method* is the iteration
$$x_{n+1} = x_n - \frac{f(x_n)}{g(x_n)},$$
stopped when $f(x_n) = 0$.
Then there is $\delta > 0$ such that for every $x_0$ with $\abs{x_0 - r} < \delta$ the iterates are defined and [[converge|YWZQC5E54B]] to $r$ quadratically.

## Remarks

The quantity $g(x)$ is a difference quotient of $f$ with step $h = f(x)$.
Near $r$ the step $f(x)$ is small, and $g(x) = f'(x) + O(f(x))$.
So the method behaves like [[Newton's method|07EXVNF9VP]], with an error in the slope of the same size as the current error, which keeps the order $2$.
Each step costs two evaluations of $f$ and no derivative.
The step is exactly [[Aitken's delta-squared process|WF08BMN6Q6]] applied to the points $x$, $\varphi(x)$, $\varphi(\varphi(x))$ of the fixed-point map $\varphi(x) = x + f(x)$.
The method needs $x_0$ close to $r$: when $f(x_0)$ is large, the point $x_0 + f(x_0)$ can be far away and the slope estimate is useless.

## Explanations

### L1 human

Newton's method needs the derivative.
Steffensen's method replaces it by the slope between $x$ and $x + f(x)$.
Near the root $f(x)$ is tiny, so this slope is very accurate, and the method is as fast as Newton's method.

### L2 human

It is the derivative-free counterpart of Newton's method, with the same quadratic order.
It also shows how convergence acceleration turns a linearly convergent fixed-point iteration into a quadratically convergent one.
Its basin of convergence is often smaller than that of Newton's method.
</document>

<document source="https://tamnd.github.io/math2/node/GEJHRXC176.md" tag="GEJHRXC176" kind="concept">
# Nonlinear Schrödinger equation

## Summary

The nonlinear Schrödinger equation is the [[Schrödinger equation|EPZR6H2WJN]] for a complex wave $u$, with an added term that depends on the size $\abs{u}$.
Its most studied case is the cubic equation $i \partial_t u + \Delta u = \pm \abs{u}^2 u$.
It describes slowly varying wave packets in optics, water waves and Bose–Einstein condensates.
It is a model problem for dispersive equations, and in one space dimension the cubic case is completely integrable.

## Explanations

### L1 human

A wave packet spreads out as it moves, because waves of different frequencies travel at different speeds.
In some media the speed also depends on how strong the wave is, and this adds a nonlinear term.
If the nonlinearity pulls the packet together while dispersion spreads it out, the two effects can balance.
Then a solitary wave forms that keeps its shape, such as $u(x,t) = \sqrt{2}\,\mathrm{sech}(x)\, e^{it}$ for the focusing cubic equation on the line.
Light pulses in optical fibres behave like this.
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# Filtration of a module

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]], let $M$ be an $R$-[[module|0NZ8JA25WK]], and let $I$ be an [[ideal|096Z9MP2ME]] of $R$.
A *filtration* of $M$ is a sequence of [[submodules|05WDSN597C]] $(M_n)_{n \ge 0}$ with $M_0 = M$ and $M_n \supseteq M_{n+1}$ for all $n$.
It is an *$I$-filtration* if moreover $I M_n \subseteq M_{n+1}$ for all $n \ge 0$.
An $I$-filtration is *stable* if there is $n_0$ such that $I M_n = M_{n+1}$ for all $n \ge n_0$.

## Remarks

Example: the $I$-adic filtration $M_n = I^n M$ is a stable $I$-filtration, since $I \cdot I^n M = I^{n+1} M$ for every $n$.
Non-example: for $R = M = \Z$ and $I = 2\Z$, the filtration $M_n = 4^n \Z$ is not a $2\Z$-filtration, since $I M_1 = 8\Z$ is not contained in $M_2 = 16\Z$.
Some authors say $I$-good for stable.
Two stable $I$-filtrations of $M$ have bounded difference: there is $k$ with $M_{n+k} \subseteq M'_n$ and $M'_{n+k} \subseteq M_n$ for all $n$, so they define the same topology.
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# n-term approximation

## Summary

$n$-term approximation approximates a function or vector by a linear combination of at most $n$ elements.
These elements are chosen freely from a fixed family, called a dictionary, such as a basis.
The best error with $n$ terms, $\sigma_n(f)$, measures how well $f$ can be compressed with respect to that dictionary.
The set of $n$-term combinations is not a linear space, so this is a basic model of [[nonlinear approximation|YS1GRQRVY7]].
It underlies wavelet compression, [[sparse approximation|WC8P971XGW]] and greedy algorithms such as [[matching pursuit|8KJ9P9KQ5H]].

## Explanations

### L1 human

In ordinary approximation one fixes in advance which basis functions to use, for example the first $n$ Fourier modes.
In $n$-term approximation one may pick any $n$ basis functions, and the choice depends on the target.
A signal with a few sharp features is well described by a few well-placed [[wavelets|4PZ35C0GV3]], even if they are not the first ones.
For an [[orthonormal basis|0S9GGTR0EX]] of a [[Hilbert space|HM2HHE3J51]] the best choice is simple: keep the $n$ coefficients of largest absolute value.
For example, $f = 3e_1 + e_2 + 2e_3$ in $\R^3$ has best $1$-term approximation $3e_1$, with error $\sqrt{1^2 + 2^2} = \sqrt{5}$.
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# Tanaka's formula

## Statement

Let $(B_t)_{t \ge 0}$ be a standard one-dimensional [[Brownian motion|22735FGJV6]] with $B_0 = 0$, and let $L(t, x)$ be its [[local time|E7ZC459PSA]].
Let $\sgn(y) = 1$ for $y > 0$, $\sgn(y) = -1$ for $y < 0$ and $\sgn(0) = 0$.
Then for every $x \in \R$, almost surely, for all $t \ge 0$,
$$\abs{B_t - x} = \abs{x} + \int_0^t \sgn(B_s - x) \, dB_s + L(t, x),$$
where the integral is an [[Itô integral|JA95NH206E]].

## Remarks

The function $y \mapsto \abs{y - x}$ is not twice differentiable at $x$, so [[Itô's lemma|2J1RVQB0CM]] does not apply to it directly.
Apply Itô's lemma to the convex functions $f_\varepsilon$ that agree with $\abs{y - x}$ outside $(x - \varepsilon, x + \varepsilon)$ and are quadratic with $f_\varepsilon'' = 1/\varepsilon$ inside.
These functions are only $C^1$, but Itô's lemma still holds for them, by smoothing $f_\varepsilon$ further, because $f_\varepsilon''$ is bounded with two jumps.
The second-order term is $\frac{1}{2\varepsilon} \int_0^t \mathbf{1}_{\set{\abs{B_s - x} < \varepsilon}} \, ds$, which tends to $L(t, x)$ as $\varepsilon \to 0$.
The value chosen for $\sgn(0)$ does not matter, because Brownian motion spends a set of times of Lebesgue measure zero at $x$.
Example: taking expectations with $x = 0$ gives $E[L(t, 0)] = E\abs{B_t} = \sqrt{2t/\pi}$.

## Explanations

### L1 human

Tanaka's formula is Itô's chain rule for the absolute value of a Brownian motion.
The kink of $\abs{y}$ at $0$ produces an extra term, the local time at $0$.
This term grows only when the path is at $0$ and pushes $\abs{B}$ up so that it stays nonnegative.

### L2 human

The formula shows that $\abs{B}$ is a semimartingale: a martingale part plus an increasing part $L(\cdot, 0)$.
By [[Lévy's characterization|8CKDVESXT6]], the integral $W_t = \int_0^t \sgn(B_s) \, dB_s$ is a Brownian motion.
Then $\abs{B} = W + L(\cdot, 0)$ and the Skorokhod reflection lemma give $L(t, 0) = \max_{s \le t} (-W_s)$.
This leads to Lévy's identity in law between $\abs{B}$ and $M - B$, where $M$ is the running maximum.
The same idea gives the Itô–Tanaka formula for differences of convex functions.
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# Nielsen fixed-point theory

## Definition

Let $X$ be a compact connected polyhedron, that is, a space homeomorphic to a finite simplicial complex.
Let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]] with fixed point set $\mathrm{Fix}(f) = \set{x \in X : f(x) = x}$.

1. Two fixed points $x, y$ are *Nielsen equivalent* if there is a path $c$ from $x$ to $y$ such that $f \circ c$ is [[path homotopic|CCNKA8C711]] to $c$.
2. The equivalence classes are the *fixed point classes* of $f$; there are finitely many, and each is open and closed in $\mathrm{Fix}(f)$.
3. Each fixed point class $F$ has a *fixed-point index* $i(f, F) \in \Z$.
   For a single fixed point $x$ in an open set of $\R^n$, it is the local degree of $u \mapsto u - f(u)$ at $x$.
4. A class $F$ is *essential* if $i(f, F) \neq 0$.

The *Nielsen number* $N(f)$ is the number of essential fixed point classes of $f$.

## Remarks

The sum of the indices of all fixed point classes is the [[Lefschetz number|ZSYCA4EGEZ]] $L(f)$.
The number $N(f)$ depends only on the homotopy class of $f$, and every map [[homotopic|F2F143KQ2M]] to $f$ has at least $N(f)$ fixed points.

Example: on the circle, let $f(z) = z^k$ with $k \neq 1$.
The fixed points are the $\abs{k - 1}$ roots of $z^{k-1} = 1$; they lie in distinct classes, each of index $\pm 1$, so $N(f) = \abs{1 - k} = \abs{L(f)}$.

Non-example of essentiality: for the identity of $S^1$, the single fixed point class is all of $S^1$.
Its index is $L(\id) = \chi(S^1) = 0$, so it is not essential.
Indeed a small rotation has no fixed point, and $N(\id) = 0$.

For compact manifolds of dimension at least $3$, Wecken's theorem says that some map homotopic to $f$ has exactly $N(f)$ fixed points.
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<document source="https://tamnd.github.io/math2/node/GER28JFXMR.md" tag="GER28JFXMR" kind="definition">
# Pythagorean field

## Definition

Let $F$ be a [[field|0QA8CHH8MG]].
The field $F$ is *Pythagorean* if for all $a, b \in F$ there is $c \in F$ with $a^2 + b^2 = c^2$.

## Remarks

By induction on the number of terms, every finite sum of squares in a Pythagorean field is a square.
Example: $\R$ is Pythagorean, since $a^2 + b^2 \ge 0$ has the real square root $\sqrt{a^2 + b^2}$.
More generally, every [[real closed field|778KFSM09P]] and every [[algebraically closed field|02SJG1HC32]] is Pythagorean.
Non-example: $\Q$ is not Pythagorean, because $1^2 + 1^2 = 2$ is not the square of a rational number.
Non-example: $\F_3$ is not Pythagorean, because $1^2 + 1^2 = 2$ and the squares in $\F_3$ are $0$ and $1$.
</document>

<document source="https://tamnd.github.io/math2/node/GETN4MCAAN.md" tag="GETN4MCAAN" kind="definition">
# Fractional Laplacian

## Definition

Let $n \ge 1$ and $0 < s < 1$.
Let $\mathcal{S}(\R^n)$ be the [[Schwartz space|CG9JEATW3Q]], and let $\hat u(\xi) = \int_{\R^n} e^{-i x \cdot \xi} u(x) \, dx$ be the [[Fourier transform|DPVM34ADZV]] of $u$.
The *fractional Laplacian* of $u \in \mathcal{S}(\R^n)$ is the function $(-\Delta)^s u$ defined by
$$\widehat{(-\Delta)^s u}(\xi) = \abs{\xi}^{2s} \hat u(\xi), \qquad \xi \in \R^n.$$

## Remarks

Equivalently, for $u \in \mathcal{S}(\R^n)$ and $x \in \R^n$,
$$(-\Delta)^s u(x) = C_{n,s} \lim_{\varepsilon \to 0^+} \int_{\abs{x - y} > \varepsilon} \frac{u(x) - u(y)}{\abs{x - y}^{n + 2s}} \, dy, \qquad C_{n,s} = \frac{4^s \, \Gamma(n/2 + s)}{\pi^{n/2} \abs{\Gamma(-s)}}.$$
The integral formula also makes sense for bounded $C^2$ functions.
Example: for $u(x) = \cos(k \cdot x)$ with $k \in \R^n$, the integral formula gives $(-\Delta)^s u = \abs{k}^{2s} u$, in agreement with the multiplier $\abs{\xi}^{2s}$.
Non-example: the exponent $s = 1$ is excluded, since the constant $C_{n,s}$ tends to $0$ as $s \to 1$ and the singular integral diverges; the multiplier $\abs{\xi}^2$ gives the ordinary operator $-\Delta$ instead.
The function $(-\Delta)^s u$ is smooth but in general not in $\mathcal{S}(\R^n)$, since $\abs{\xi}^{2s}$ is not smooth at $\xi = 0$; it decays like $\abs{x}^{-n-2s}$.
</document>

<document source="https://tamnd.github.io/math2/node/GEXSRZ7JEH.md" tag="GEXSRZ7JEH" kind="definition">
# Caustic

## Definition

Let $Q$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.
Let $\pi \colon T^*Q \to Q$ be its cotangent bundle with the canonical symplectic form, and let $L \subset T^*Q$ be a [[Lagrangian submanifold|H8ACPTMX08]].
Let $\Sigma_L \subset L$ be the set of [[Lagrangian singularities|QKPMR9MC9W]] of $L$, the points where the differential of $\pi|_L \colon L \to Q$ has rank less than $n$.
The *caustic* of $L$ is the set
$$C_L = \pi(\Sigma_L) \subset Q$$
of critical values of the Lagrangian map $\pi|_L$.

## Remarks

Example: $Q = \R$ and $L = \set{(p^2, p) : p \in \R} \subset T^*\R = \R^2$.
The map $\pi|_L$ is $p \mapsto p^2$, singular only at $p = 0$, so $\Sigma_L = \set{(0, 0)}$ and $C_L = \set{0}$.
Over $q > 0$ lie two points of $L$, and over $q < 0$ none, so the number of preimages jumps at the caustic.

Non-example: if $L$ is the graph of $dS$ for a smooth function $S$ on $Q$, then $\pi|_L$ is a diffeomorphism, $\Sigma_L$ is empty, and the caustic is empty.

In geometric optics, $Q$ is space and $L$ is the set of positions and directions of the rays of a beam.
Then $C_L$ is the envelope of the family of rays, where the rays focus.
If $L$ is given locally by a generating family $F(q, y)$, the caustic is the [[bifurcation set|JV45S0KQE6]] of the family $y \mapsto F(q, y)$ with $q$ as control.
</document>

<document source="https://tamnd.github.io/math2/node/GF1RK91VED.md" tag="GF1RK91VED" kind="statement">
# Rosenberg's classification of maximal clones

## Statement

Let $A$ be a finite set with $k = \abs{A} \ge 2$.
A [[clone|PTKR703G6A]] on $A$ is *maximal* if it is a coatom of the [[lattice of clones|EX9NCDFYC7]] on $A$.
For a relation $\rho$ on $A$, let $\operatorname{Pol}(\rho)$ be the clone of all [[polymorphisms|A0K9EZRM7E]] of $\rho$.
A relation $\rho \subseteq A^h$ is *totally symmetric* if it is invariant under all permutations of coordinates.
It is *totally reflexive* if it contains every $h$-tuple with two equal entries.
Then a clone $C$ on $A$ is maximal if and only if $C = \operatorname{Pol}(\rho)$ for a relation $\rho$ of one of the following six types.

1. A [[partial order|0QV2JS9RJ3]] on $A$ with a least and a greatest element.
2. The graph $\set{(a, \sigma(a)) : a \in A}$ of a [[permutation|05RWFKVTJ7]] $\sigma$ of $A$ whose cycles all have the same prime length $p$.
3. The relation $\set{(a, b, c, d) \in A^4 : a + b = c + d}$, where $(A, +)$ is an [[elementary abelian|DZSZ8ZS554]] $p$-group for a prime $p$.
4. An [[equivalence relation|0G8VEZ3WTK]] on $A$ other than equality and $A^2$.
5. A central relation $\rho \subseteq A^h$ with $1 \le h \le k - 1$.
   This means that $\rho \ne A^h$, that $\rho$ is totally symmetric and totally reflexive, and that some $c \in A$ satisfies $(c, a_2, \dots, a_h) \in \rho$ for all $a_2, \dots, a_h \in A$.
   For $h = 1$ it means a nonempty proper subset of $A$.
6. An $h$-regular relation with $3 \le h \le k$.
   This is the relation determined by a family $\theta_1, \dots, \theta_m$ of equivalence relations, $m \ge 1$, each with exactly $h$ blocks.
   The family must be such that any choice of one block of each $\theta_i$ has nonempty intersection.
   The relation consists of the $h$-tuples that, for every $i$, have two entries in the same $\theta_i$-block.

## Remarks

The theorem is due to Ivo Rosenberg.
One direction shows that each listed $\operatorname{Pol}(\rho)$ is a proper clone and that adding any operation outside it generates all operations.
The other direction uses the [[Pol–Inv Galois connection|BQ0AC9QX7E]]: a maximal clone is $\operatorname{Pol}(\rho)$ for a single relation $\rho$, which is then reduced to a minimal form.
For $k = 2$ the list gives the five maximal clones of [[Post's lattice|MA20X7JCET]].
Type 1 gives the monotone operations, type 2 the self-dual ones, and type 3 the affine ones.
Type 5 with $h = 1$ gives the operations preserving $\set{0}$ or preserving $\set{1}$.
Types 4 and 6 do not occur for $k = 2$.
On every finite set, there are therefore only finitely many maximal clones.

## Explanations

### L1 human

A maximal clone is a family of operations that is just one step short of everything.
Adding any single missing operation to it lets one build every operation on the set.
The theorem lists all such families on a finite set.
Each is the set of operations that respect one special relation, such as an order or an equivalence.

### L2 human

The classification gives a completeness test.
A set $F$ of operations on a finite set generates all operations exactly when, for each relation $\rho$ of the six types, some member of $F$ fails to preserve $\rho$.
This extends [[Post's functional completeness theorem|WGBR7JHBV4]] from two elements to any finite set.
It is a basic tool for questions about [[functional completeness|TDZVCWYJ9F]] in many-valued logic.
</document>

<document source="https://tamnd.github.io/math2/node/GF3MD842NZ.md" tag="GF3MD842NZ" kind="definition">
# Approximately finite-dimensional C*-algebra

## Definition

Let $A$ be a C*-algebra.
Then $A$ is *approximately finite-dimensional*, or *AF*, if it has finite-dimensional C\*-subalgebras $A_1 \subseteq A_2 \subseteq \cdots$ whose union is dense in $A$.
Equivalently, $A$ is isomorphic to the C\*-algebraic [[direct limit|7SXKJG1Q9B]] of a sequence $A_1 \to A_2 \to \cdots$ of finite-dimensional C\*-algebras and $*$-homomorphisms.

## Remarks

Every finite-dimensional C\*-algebra is isomorphic to a direct sum $M_{n_1}(\C) \oplus \cdots \oplus M_{n_k}(\C)$ of [[matrix rings|0FHVPRAE60]].
An AF algebra in this sense is separable.
Some authors call a nonseparable C\*-algebra AF when every finite subset can be approximated by elements of a finite-dimensional C\*-subalgebra.

Example: the compact operators $K(H)$ on a separable infinite-dimensional Hilbert space $H$ with orthonormal basis $(e_k)$.
Let $A_n$ be the operators that map $\operatorname{span}(e_1, \ldots, e_n)$ into itself and vanish on its orthogonal complement; then $A_n \cong M_n(\C)$ and $A_n \subseteq A_{n+1}$.
The union is dense because the finite-rank operators are dense in $K(H)$, and $P_n T P_n \to T$ for compact $T$, where $P_n$ is the projection onto the span.

Non-example: $C([0, 1])$ is not AF.
A finite-dimensional commutative C\*-algebra is spanned by projections, and the only projections in $C([0,1])$ are $0$ and $1$, since $[0,1]$ is connected.
So every finite-dimensional C\*-subalgebra lies in the constants $\C 1$, which are not dense.
</document>

<document source="https://tamnd.github.io/math2/node/GF40N88TVM.md" tag="GF40N88TVM" kind="definition">
# Homogeneous integral equation

## Definition

Let $a < b$, let $a_0, f \colon [a, b] \to \C$ and a [[kernel|GSMBXT7FPK]] $K \colon [a, b] \times [a, b] \to \C$ be given, and let $\lambda \in \C$.
For each $x \in [a, b]$, let $I_x$ be either $[a, b]$ or $[a, x]$.
A [[linear integral equation|AYZCKG34QK]]
$$a_0(x) \varphi(x) = f(x) + \lambda \int_{I_x} K(x, s) \varphi(s) \, ds \quad \text{for all } x \in [a, b]$$
is *homogeneous* if $f(x) = 0$ for all $x \in [a, b]$, and *inhomogeneous* otherwise.
For an equation of the [[second kind|8KJCN49V64]], the homogeneous equation is
$$\varphi(x) = \lambda \int_{I_x} K(x, s) \varphi(s) \, ds \quad \text{for all } x \in [a, b].$$

## Remarks

Example: $\varphi(x) = \lambda \int_0^1 x s \, \varphi(s) \, ds$ on $[0, 1]$ is homogeneous.
Every solution has the form $\varphi(x) = c x$ with $c = \lambda c / 3$, so there is a nonzero solution exactly when $\lambda = 3$.

Non-example: $\varphi(x) = x + \lambda \int_0^1 x s \, \varphi(s) \, ds$ is inhomogeneous, since its free term $f(x) = x$ is not zero.

The solutions of a homogeneous equation form a vector space, like the solutions of a [[homogeneous system of linear equations|04KGXJG1A2]].
The difference of two solutions of an inhomogeneous equation solves the homogeneous one.
</document>

<document source="https://tamnd.github.io/math2/node/GFADRKMQ51.md" tag="GFADRKMQ51" kind="definition">
# Sequential space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A set $A \subseteq X$ is sequentially closed when the following holds.
If a sequence $(x_n)$ in $A$ [[converges|0W4CN9E2W5]] in $X$ to a point $x$, then $x \in A$.
$X$ is a *sequential space* when every sequentially closed subset of $X$ is [[closed|0ZCK8DQYN1]].

## Remarks

Every closed set is sequentially closed, so in a sequential space the two notions agree.
Equivalently, a set $U$ is [[open|02BH3MCBQB]] if and only if every sequence that converges to a point of $U$ is eventually in $U$.
In a sequential space, a map to any topological space is continuous if and only if it preserves limits of convergent sequences.
Quotients of sequential spaces are sequential, and the sequential spaces are exactly the quotients of first-countable spaces.

Example: every [[first-countable space|0CP5NDD21C]] is sequential.
If $x \in \overline{A}$ and $U_1 \supseteq U_2 \supseteq \cdots$ is a neighbourhood base at $x$, choose $x_n \in A \cap U_n$; then $x_n \to x$, so a sequentially closed $A$ contains $x$.

Non-example: let $X$ be an uncountable set whose closed sets are $X$ and the countable subsets.
A convergent sequence in $X$ is eventually constant, so every subset of $X$ is sequentially closed.
But an uncountable proper subset is not closed, so $X$ is not sequential.
</document>

<document source="https://tamnd.github.io/math2/node/GFC5YKSN5H.md" tag="GFC5YKSN5H" kind="concept">
# Open immersion

## Summary

An open immersion is a [[morphism of schemes|JXRKH6P1Z4]] that identifies its source with an open subscheme of its target.
Open immersions are the maps used to glue schemes and to restrict a scheme to an open subset.
The basic example is the inclusion of a basic open set $D(f) = \operatorname{Spec} A_f$ into $\operatorname{Spec} A$.

## Explanations

### L1 human

An open immersion is the inclusion of an open part of a scheme, carrying the same functions as the big scheme there.
Removing the origin from the line gives an open immersion of $\operatorname{Spec} k[x, x^{-1}]$ into $\operatorname{Spec} k[x]$.
On the open part, the function $1/x$ is allowed, because $x$ is never zero there.
Each chart of projective space is an open immersion of affine space into it.
</document>

<document source="https://tamnd.github.io/math2/node/GFJ20FBYF8.md" tag="GFJ20FBYF8" kind="definition">
# Appell series

## Definition

Let $a, a', b, b', c, c'$ be complex numbers, with every lower parameter $c, c'$ outside $\set{0, -1, -2, \dots}$.
Write $(x)_n$ for the [[Pochhammer symbol|YZHM3HSM0F]].
The four *Appell series* are the double power series in two complex variables $x, y$:

1. $F_1(a; b, b'; c; x, y) = \sum_{m, n \ge 0} \frac{(a)_{m + n} (b)_m (b')_n}{(c)_{m + n}\, m!\, n!} x^m y^n$, convergent for $\abs{x} < 1$ and $\abs{y} < 1$;
2. $F_2(a; b, b'; c, c'; x, y) = \sum_{m, n \ge 0} \frac{(a)_{m + n} (b)_m (b')_n}{(c)_m (c')_n\, m!\, n!} x^m y^n$, convergent for $\abs{x} + \abs{y} < 1$;
3. $F_3(a, a'; b, b'; c; x, y) = \sum_{m, n \ge 0} \frac{(a)_m (a')_n (b)_m (b')_n}{(c)_{m + n}\, m!\, n!} x^m y^n$, convergent for $\abs{x} < 1$ and $\abs{y} < 1$;
4. $F_4(a; b; c, c'; x, y) = \sum_{m, n \ge 0} \frac{(a)_{m + n} (b)_{m + n}}{(c)_m (c')_n\, m!\, n!} x^m y^n$, convergent for $\sqrt{\abs{x}} + \sqrt{\abs{y}} < 1$.

## Remarks

Setting $y = 0$ leaves the terms with $n = 0$, and each series becomes a Gauss [[hypergeometric series|428M453CR2]] in $x$; for instance $F_1(a; b, b'; c; x, 0) = {}_2F_1(a, b; c; x)$.
Example: with $b = c$ and $b' = c'$, the series $F_2$ becomes $\sum_{m, n} \frac{(a)_{m + n}}{m!\, n!} x^m y^n = \sum_{N \ge 0} \frac{(a)_N}{N!} (x + y)^N = (1 - x - y)^{-a}$, by the binomial theorem; this converges when $\abs{x} + \abs{y} < 1$.
Non-example: for $a = 1$ the same series at $x = y = \frac12$ is $\sum_N 1$, which diverges; the point lies on the boundary $\abs{x} + \abs{y} = 1$ of the region of convergence.
For $\operatorname{Re} c > \operatorname{Re} a > 0$, with $\Gamma$ the [[gamma function|0H5SYBSED9]], $F_1$ has Picard's integral representation
$$F_1(a; b, b'; c; x, y) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(c - a)} \int_0^1 t^{a - 1} (1 - t)^{c - a - 1} (1 - xt)^{-b} (1 - yt)^{-b'}\, dt.$$
</document>

<document source="https://tamnd.github.io/math2/node/GFJE824E93.md" tag="GFJE824E93" kind="statement">
# Martin representation theorem

## Statement

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be a domain with a Green's function, and fix $x_0 \in \Omega$.
Let $\Delta$ be the [[Martin boundary|QD02B3DEMR]] of $\Omega$, let $M$ be the [[Martin kernel|1DBC9FYF7M]], and let $\Delta_1 \subseteq \Delta$ be the set of minimal points.
Then $\Delta_1$ is a Borel set.
Let $h$ be a [[positive harmonic function|EYMQ82Z8XJ]] on $\Omega$.
Then there is a unique finite positive [[Borel measure|DYR1QFNGNJ]] $\mu$ on $\Delta$ with $\mu(\Delta \setminus \Delta_1) = 0$ such that
$$h(x) = \int_{\Delta} M(x, \xi)\, d\mu(\xi) \qquad \text{for every } x \in \Omega.$$
Moreover $\mu(\Delta) = h(x_0)$.

## Remarks

Idea of the proof: for an exhaustion of $\Omega$ by smooth subdomains $\Omega_j$, the reduced function of $h$ on $\overline{\Omega_j}$ is a Green potential $\int G(\cdot, y)\, d\nu_j(y)$.
Rewriting it as $\int M(\cdot, y)\, G(x_0, y)\, d\nu_j(y)$ gives measures of mass at most $h(x_0)$ on the compact space $\Omega \cup \Delta$, and a weak-star limit gives a representing measure on $\Delta$.
Choquet's theory of extreme points, or a direct argument of Martin, shows that the measure can be taken on $\Delta_1$, where it is unique.
Example: for the unit ball with $x_0 = 0$, $\Delta = \Delta_1$ is the unit sphere and $M(x, \zeta)$ is the Poisson kernel.
So the theorem becomes the Herglotz representation of positive harmonic functions as Poisson integrals of positive measures.
Uniqueness needs the condition $\mu(\Delta \setminus \Delta_1) = 0$: a non-minimal point $\xi$ gives a function $M(\cdot, \xi)$ that has two representations, by $\delta_\xi$ and by a measure on $\Delta_1$.
The theorem is due to R. S. Martin (1941).

## Explanations

### L1 human

On any reasonable domain, every positive harmonic function is a mixture of special building blocks attached to points of an ideal boundary.
The building blocks are the minimal harmonic functions, and the mixture is described by a unique measure.
For the ball this is the familiar Poisson representation.

### L2 human

The theorem reduces questions about all positive harmonic functions to questions about the minimal ones and about measures on the Martin boundary.
It is the starting point for fine boundary limits, where Naïm's thin sets and Doob's conditioned Brownian motion describe how $h / M(\cdot, \xi)$ behaves near $\xi$.
For Lipschitz domains, the boundary Harnack principle shows that $\Delta = \Delta_1$ is the Euclidean boundary.
The same scheme applies to Markov chains, where it describes all positive harmonic functions of a transition operator.
</document>

<document source="https://tamnd.github.io/math2/node/GFNJKVWQBB.md" tag="GFNJKVWQBB" kind="definition">
# Amenable group action

## Definition

Let $G$ be a group acting on a set $X$ by a [[group action|0WCY42EV9V]] $(g, x) \mapsto g \cdot x$.
Let $\ell^\infty(X)$ be the space of bounded functions $X \to \R$, and for $g \in G$ and $f \in \ell^\infty(X)$ let $(g \cdot f)(x) = f(g^{-1} \cdot x)$.
A [[mean|ZYNY2VFECQ]] on $X$ is a linear map $m \colon \ell^\infty(X) \to \R$ such that:

1. $m(f) \ge 0$ whenever $f \ge 0$.
2. $m(1) = 1$ for the constant function $1$.

The action is *amenable* if there is a mean $m$ on $X$ with $m(g \cdot f) = m(f)$ for all $g \in G$ and $f \in \ell^\infty(X)$.

## Remarks

Example: suppose the action has a finite orbit $O = G \cdot x_0$.
Then $m(f) = \frac{1}{\abs{O}} \sum_{x \in O} f(x)$ is a mean, and it is invariant because each $g \in G$ permutes $O$.
So every action with a finite orbit is amenable, whatever the group.

Non-example: the free group $F_2$ acting on itself by left multiplication is not amenable.
Indeed, an invariant mean for this action is exactly what makes a group [[amenable|CEAT9BESKK]], and $F_2$ is not amenable.

A group $G$ is amenable if and only if its action on itself by left multiplication is amenable.
If $G$ is amenable, every action of $G$ is amenable: for a fixed $x_0 \in X$, an invariant mean $m$ on $G$ gives the invariant mean $f \mapsto m(g \mapsto f(g \cdot x_0))$ on $X$.

The name is also used in other senses.
In ergodic theory, an amenable group action often just means an action of an amenable group.
Zimmer's amenability of a measure-class-preserving action is a different notion: for example, the action of $F_2$ on its boundary is amenable in Zimmer's sense.
</document>

<document source="https://tamnd.github.io/math2/node/GFS95PZ5YK.md" tag="GFS95PZ5YK" kind="statement">
# Kramers–Kronig relations

## Statement

Let $H = \set{z \in \C : \operatorname{Im} z > 0}$ be the upper half-plane.
Let $\chi$ be continuous on the closure $\bar H$ and [[holomorphic|0DG0Z72CQD]] on $H$, with $\chi(z) \to 0$ as $\abs{z} \to \infty$ in $\bar H$, uniformly in the direction.
Write $\chi(\omega) = \chi_1(\omega) + i \chi_2(\omega)$ for real $\omega$, with $\chi_1$ and $\chi_2$ real.
Then for each real $\omega$,
$$\chi_1(\omega) = \frac{1}{\pi} \, \mathrm{p.v.} \int_{-\infty}^{\infty} \frac{\chi_2(\omega')}{\omega' - \omega} \, d\omega', \qquad \chi_2(\omega) = -\frac{1}{\pi} \, \mathrm{p.v.} \int_{-\infty}^{\infty} \frac{\chi_1(\omega')}{\omega' - \omega} \, d\omega'.$$
Here $\mathrm{p.v.}$ is the [[principal value|0V3RDQV3ZX]] at $\omega$ and at infinity: the limit of the integral over $\varepsilon < \abs{\omega' - \omega} < R$ as $\varepsilon \to 0$ and $R \to \infty$.

## Remarks

Proof idea: integrate $\chi(z)/(z - \omega)$ over the boundary of the half-disc of radius $R$ in $H$, with a small half-circle of radius $\varepsilon$ around $\omega$ removed.
By [[Cauchy's theorem|01MPZXYVYF]] the integral is $0$.
The big half-circle contributes at most $\pi \max \abs{\chi} \cdot R/(R - \abs{\omega}) \to 0$, and the small one tends to $-i\pi \chi(\omega)$.
So $\mathrm{p.v.} \int \frac{\chi(\omega')}{\omega' - \omega} \, d\omega' = i\pi \chi(\omega)$, and the real and imaginary parts give the two relations.
Example: $\chi(z) = \frac{1}{z + i}$ gives $\chi_1(\omega) = \frac{\omega}{\omega^2 + 1}$ and $\chi_2(\omega) = -\frac{1}{\omega^2 + 1}$, and these satisfy both relations.
In terms of the [[Hilbert transform|NSCEBXM7ZM]], each of $\chi_1$ and $\chi_2$ determines the other.

## Explanations

### L1 human

In physics, a response function such as the refractive index of a material has a real part and an imaginary part.
Causality, the fact that an effect cannot come before its cause, makes the response holomorphic in the upper half-plane of frequencies.
Then the real part and the imaginary part are not independent: each one can be computed from the other by an integral.
For example, the absorption of light at all frequencies determines the refractive index.

### L2 human

The relations are a direct consequence of Cauchy's integral formula on a half-plane.
They say that the real and imaginary parts of the boundary values of a holomorphic function on $H$ that decays at infinity are Hilbert transforms of each other.
For square-integrable boundary values, Titchmarsh's theorem makes this an equivalence with the Hardy space $H^2$ of the upper half-plane.
They are used to check measured spectra, to recover phases from amplitudes, and as dispersion relations in particle physics.
</document>

<document source="https://tamnd.github.io/math2/node/GFSDMJHHVA.md" tag="GFSDMJHHVA" kind="concept">
# q-analog

## Summary

A q-analog of a formula or a theorem is a version with an extra parameter $q$, see the [[definition|CK80HQNHEK]].
It gives back the original in the limit $q \to 1$.
The basic example is the q-integer $[n]_q = 1 + q + \cdots + q^{n-1}$, which tends to $n$.
From it one builds q-factorials, [[Gaussian binomial coefficients|YS09Z1YNEK]] and q-series.
Good q-analogs often have a meaning of their own, such as counts over [[finite fields|0ZXZ82FTX2]] or generating functions of statistics on [[permutations|05RWFKVTJ7]].

## Explanations

### L1 human

Take a familiar number, such as $3$, and replace it by the polynomial $1 + q + q^2$.
Setting $q = 1$ gives back $3$, so nothing is lost.
But the polynomial carries more information: for instance, it can count objects by size and by an extra feature at the same time.
A q-analog of a theorem is a statement about such polynomials that becomes the original theorem when $q = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/GFTQRDW607.md" tag="GFTQRDW607" kind="definition">
# Morin singularities

## Definition

Let $n \ge 1$ and $1 \le k \le n$.
Let $f \colon (\R^n, 0) \to (\R^n, 0)$ be a smooth map [[germ|VZXF7RNFKS]].
It is a *Morin singularity of order $k$* if there are germs of [[diffeomorphisms|0KZZ70ZHXP]] $\varphi$ and $\psi$ of $(\R^n, 0)$ with this property.
The germ $\psi \circ f \circ \varphi^{-1}$ is the germ at $0$ of
$$(u_1, \ldots, u_{n-1}, t) \mapsto \left(u_1, \ldots, u_{n-1}, \ t^{k+1} + \sum_{i=1}^{k-1} u_i t^i\right).$$

## Remarks

For $k = 1$ the normal form is $(u, t) \mapsto (u, t^2)$, the fold; for $k = 2$ it is $(u_1, \ldots, t) \mapsto (u_1, \ldots, t^3 + u_1 t)$, the cusp; for $k = 3$ it is the swallowtail.
Example: $f(x, y) = (x, y^3 + xy)$ is a Morin singularity of order $2$ with $\varphi$ and $\psi$ the identity.
The local algebra $\mathcal{E}_n / f^*(\mathfrak{m}_n) \mathcal{E}_n$ of a Morin singularity of order $k$ is $\R[t]/(t^{k+1})$, of dimension $k + 1$; this is an invariant of the germ up to the equivalence above.
Non-example: $f(x, y) = (x, y^4)$ has local algebra $\R[y]/(y^4)$ of dimension $4$, so it could only be of order $3$, which needs $n \ge 3$; hence it is not a Morin singularity.
Every Morin singularity is a stable germ of corank one, and its Thom–Boardman symbol is $(1, \ldots, 1, 0)$ with $k$ ones.
Morin proved that a germ with this symbol whose jet is transverse to the Boardman strata is a Morin singularity.
There are analogous normal forms for maps $\R^m \to \R^n$ with $m \ne n$, with extra terms in the remaining variables.
</document>

<document source="https://tamnd.github.io/math2/node/GFVWEATKGE.md" tag="GFVWEATKGE" kind="concept">
# Dimension theory

## Summary

Dimension theory is the part of commutative algebra that studies the [[Krull dimension|MVGBS3J1P7]] of rings and modules.
It also studies related notions such as [[height|87WH65FJB8]], [[system of parameters|95QTEPMDA4]] and [[multiplicity|JX20MNDDXT]].
Its central results are [[Krull's principal ideal theorem|SJZW2HN1SC]] and the [[dimension theorem for local rings|KW3HY77V6D]].
The latter computes the Krull dimension of a Noetherian local ring from its [[Hilbert–Samuel function|BNQVS83GZV]].
It gives an algebraic meaning to the dimension of an algebraic variety.

## Explanations

### L1 human

A line has dimension $1$ and a plane has dimension $2$.
Dimension theory explains what dimension means for an arbitrary commutative ring, where there is no picture to look at.
The main idea is to count the longest chain of prime ideals, and then to show that this count agrees with other natural counts.
For example, the polynomial ring $k[x, y]$ over a field has the chain $0 \subsetneq (x) \subsetneq (x, y)$, and its dimension is $2$, as for the plane.
</document>

<document source="https://tamnd.github.io/math2/node/GFXQ6HEM65.md" tag="GFXQ6HEM65" kind="concept">
# Fredholm map

## Summary

A Fredholm map is a differentiable map between [[Banach manifolds|9K15AMGBV9]] whose derivative at every point is a [[Fredholm operator|8HWGKWD6ZM]].
Its index, the index of the derivative, is locally constant, and it plays the role of the difference of dimensions of source and target.
Nonlinear elliptic equations on compact manifolds give Fredholm maps between Sobolev spaces of maps.
The [[Sard–Smale theorem|Z0NPFYDVP2]] extends Sard's theorem to Fredholm maps, so that the solution spaces of generic perturbed elliptic equations are finite-dimensional manifolds.

## Explanations

### L1 human

A map between finite-dimensional spaces of dimensions $m$ and $n$ has, at a regular point, solution sets of dimension $m - n$.
A Fredholm map is a map between infinite-dimensional spaces that still behaves like this: infinitely many equations in infinitely many unknowns, but with a finite and well-defined difference.
That difference is the index.
For instance, the map $(x_1, x_2, x_3, \dots) \mapsto (x_2, x_3, \dots)$ on $\ell^2$ is linear and Fredholm of index $1$: one unknown is left free.
</document>

<document source="https://tamnd.github.io/math2/node/GFZD5Z7PT2.md" tag="GFZD5Z7PT2" kind="definition">
# Latent variable model

## Definition

Let $\Theta$ be a parameter set.
Let $(X, Z)$ be a pair of random vectors, where $X$ is observed and $Z$ is not.
A *latent variable model* is a family $\set{P_\theta : \theta \in \Theta}$ of [[joint distributions|0DR81R1F2B]] of $(X, Z)$.
The variables $Z$ are the *latent variables*.
The [[statistical model|K3AMMSY8DD]] for the data is the family of [[marginal distributions|0MXT6PRZZ6]] of $X$ under $P_\theta$.
When $P_\theta$ has a density $p_\theta(x, z)$ with respect to a product measure $\mu \otimes \nu$, the observed density is
$$p_\theta(x) = \int p_\theta(x, z) \, d\nu(z).$$

## Remarks

Often the components $X_1, \ldots, X_m$ of $X$ are assumed [[conditionally independent|0THJ7416DX]] given $Z$, the assumption of local independence.
Example: let $Z$ be standard normal and, given $Z$, let $X_1, X_2$ be independent with distribution $N(Z, 1)$.
Then $(X_1, X_2)$ is bivariate normal with variances $2$ and covariance $1$; the dependence of $X_1$ and $X_2$ comes only from $Z$.
Non-example: a regression model of $Y$ given a measured covariate $x$ is not a latent variable model, since all of its variables are observed.
The models are classified by the type of the variables.
Factor analysis has continuous latent and observed variables.
Latent class and mixture models have discrete latent variables.
Item response models have continuous latent and discrete observed variables.
</document>

<document source="https://tamnd.github.io/math2/node/GG6NYPKMHX.md" tag="GG6NYPKMHX" kind="statement">
# Chern–Gauss–Bonnet theorem

## Statement

Let $(M, g)$ be a compact oriented [[Riemannian manifold|7CBT4SM83Y]] without boundary, of even dimension $n = 2m$.
Let $\Omega = (\Omega_{ij})$ be the [[curvature form|ZAJFW1S078]] of the [[Levi-Civita connection|CAZYGE1T4E]] with respect to a local oriented orthonormal frame, a skew-symmetric $n \times n$ matrix of 2-forms.
Then the $n$-form $\operatorname{Pf}(\Omega)$, where $\operatorname{Pf}$ is the [[Pfaffian|NH5HWBPWS9]], does not depend on the frame, and
$$\chi(M) = \frac{1}{(2\pi)^m} \int_M \operatorname{Pf}(\Omega),$$
where $\chi(M)$ is the [[Euler characteristic|RNDDWDAFJ5]] of $M$.

## Remarks

Chern gave an intrinsic proof in 1944: on the unit sphere bundle of $M$, the form $\operatorname{Pf}(\Omega)$ pulled back is exact, $d\Pi$, by a transgression formula.
Take a vector field $X$ with isolated zeros, and integrate over $M$ minus small balls around the zeros.
Stokes' theorem gives the sum of the indices of $X$, which is $\chi(M)$ by the Poincaré–Hopf theorem.

Example: for $m = 1$, $\operatorname{Pf}(\Omega) = \Omega_{12} = K \, dA$ with $K$ the Gaussian curvature, and the formula is the [[Gauss–Bonnet theorem|ESGEHKPD60]] $\int_M K \, dA = 2\pi \chi(M)$.
For $m = 2$ it reads
$$\chi(M) = \frac{1}{32\pi^2} \int_M \bigl(\abs{\operatorname{Rm}}^2 - 4 \abs{\operatorname{Ric}}^2 + R^2\bigr) \, dV,$$
with $\operatorname{Rm}$ the Riemann tensor, $\operatorname{Ric}$ the Ricci tensor and $R$ the scalar curvature.
For the unit sphere $S^4$ the integrand is $24 - 144 + 144 = 24$ and the volume is $8\pi^2/3$, which gives $\chi(S^4) = 2$.

For odd $n$ the Euler characteristic of a closed manifold is $0$, so there is no analogous curvature formula.

## Explanations

### L1 human

Add up the curvature of a closed even-dimensional space over all its points.
The theorem says that the result depends only on the shape up to deformation: it is the Euler characteristic.
However the space is bent, the total curvature stays the same.

### L2 human

The theorem generalizes the Gauss–Bonnet theorem for surfaces to all even dimensions.
In the language of characteristic classes, $\operatorname{Pf}(\Omega / 2\pi)$ represents the Euler class of the tangent bundle in de Rham cohomology.
The theorem then says that the Euler number of the tangent bundle is $\chi(M)$.
It is a model case of Chern–Weil theory and a special case of the Atiyah–Singer index theorem, for the operator $d + \delta$ from even to odd forms.
</document>

<document source="https://tamnd.github.io/math2/node/GG8AYV847V.md" tag="GG8AYV847V" kind="concept">
# Spectral sequence

## Summary

A spectral sequence is a sequence of pages $E_r$ of objects indexed by two integers.
Each page is the [[homology|2J4JJJYP22]] of the previous one for a differential $d_r$.
It is a bookkeeping tool that computes the homology of a filtered complex or a [[double complex|N80ABTQHEQ]] by successive approximations.
Spectral sequences connect derived functors of composites, the homology of fibrations, and many other computations in algebra and topology.

## Explanations

### L1 human

Think of a grid of groups, one at each point of the plane.
Arrows of a given length and direction connect the groups, and taking homology gives a new, smaller grid.
Then one repeats with longer arrows.
In good cases the grid stops changing after a few steps, and the final grid gives the pieces of the answer, like the layers of a cake.
If all the groups lie on one row, then from the second page on every arrow leaves the row, so the grid stops changing at once.
</document>

<document source="https://tamnd.github.io/math2/node/GGBJDS5AV9.md" tag="GGBJDS5AV9" kind="definition">
# Congruence-distributive variety

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
The variety $\mathcal{V}$ is *congruence distributive* if, for every algebra $A$ in $\mathcal{V}$, the [[congruence lattice|REE07CSXHR]] $\operatorname{Con} A$ is a [[distributive lattice|0HM0HQCK6N]].
That is, for all congruences $\alpha, \beta, \gamma$ of $A$,
$$\alpha \wedge (\beta \vee \gamma) = (\alpha \wedge \beta) \vee (\alpha \wedge \gamma).$$

## Remarks

Example: the variety of lattices is congruence distributive.
Every lattice $L$ has the majority term $m(x, y, z) = (x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$, which satisfies $m(x, x, y) = m(x, y, x) = m(y, x, x) = x$, and a variety with a majority term is congruence distributive.
Non-example: the variety of groups is not congruence distributive.
In the Klein four-group $V = \Z/2\Z \times \Z/2\Z$, every subgroup is normal, so $\operatorname{Con} V$ is isomorphic to the lattice of subgroups.
For three distinct subgroups $A, B, C$ of order $2$, $A \wedge (B \vee C) = A \wedge V = A$, while $(A \wedge B) \vee (A \wedge C)$ is the trivial subgroup.
Every congruence-distributive variety is congruence modular.

## Explanations

### L1 human

A variety is congruence distributive when all its algebras have congruence lattices that satisfy the distributive law.
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<document source="https://tamnd.github.io/math2/node/GGCHBSY4HK.md" tag="GGCHBSY4HK" kind="definition">
# Separable closure

## Definition

Let $L/K$ be an [[algebraic extension|09EYMKMSB9]].
An element $\alpha \in L$ is separable over $K$ if its [[minimal polynomial|0GK1WB9R98]] over $K$ has no repeated root in an algebraic closure.
The *separable closure of $K$ in $L$* is the set $L_s$ of elements of $L$ that are separable over $K$.
Let $\overline{K}$ be an [[algebraic closure|0MCXX29HWY]] of $K$.
The *separable closure* of $K$ is the separable closure $K^{\mathrm{sep}}$ of $K$ in $\overline{K}$.

## Remarks

The set $L_s$ is a subfield of $L$ containing $K$, the extension $L_s/K$ is [[separable|0N019F6RE9]], and every separable subextension of $L/K$ lies in $L_s$.
The extension $L/L_s$ is purely inseparable.
The field $K^{\mathrm{sep}}$ is separably closed, and it is unique up to an isomorphism over $K$.

Example: let $K = \Q$ and $L = \overline{\Q}$.
Check: in characteristic $0$ every irreducible polynomial has distinct roots, so every element of $\overline{\Q}$ is separable and $\Q^{\mathrm{sep}} = \overline{\Q}$.

Non-example: let $K = \F_p(t)$ and $L = K(s)$ with $s^p = t$.
The element $s$ is not in the separable closure of $K$ in $L$, since its minimal polynomial $x^p - t = (x - s)^p$ has a repeated root; in fact $L_s = K$.
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<document source="https://tamnd.github.io/math2/node/GGDJN5KXN4.md" tag="GGDJN5KXN4" kind="definition">
# Iterative functional equation

## Definition

Let $X$ and $Y$ be sets.
An *iterative functional equation* for an unknown [[function|0EPAT51DXQ]] $\varphi \colon X \to Y$ is a [[functional equation|SRAN4EKWS0]] in one variable $x \in X$ of the following kind.
In it, $\varphi$ is evaluated at $x$ and at images of $x$ under given maps $f_1, \dots, f_k \colon X \to X$ or under [[iterates|AYVTR2ES5B]] of $\varphi$ itself.
The basic linear form, for $Y = \R$ and given $f \colon X \to X$ and $g, h \colon X \to \R$, is
$$\varphi(f(x)) = g(x)\, \varphi(x) + h(x) \quad \text{for all } x \in X.$$

## Remarks

Example: $\varphi(2x) = \varphi(x) + 1$ on $X = (0, \infty)$ is of the basic linear form with $f(x) = 2x$, $g = 1$, $h = 1$, and $\varphi = \log_2$ solves it.
Non-example: Cauchy's equation $\varphi(x + y) = \varphi(x) + \varphi(y)$ is not iterative, since it involves two independent variables.
The case $g = 1$, $h = 1$ is the [[Abel equation|68SHB7Z721]], and the case $g = s$ constant, $h = 0$ is [[Schröder's equation|VTJV7JVJEA]].
Equations with iterates of the unknown, such as $\varphi(\varphi(x)) = f(x)$, ask for [[iterative roots|27C6J3YNZY]] of $f$.
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<document source="https://tamnd.github.io/math2/node/GGDSF85JPD.md" tag="GGDSF85JPD" kind="concept">
# Partial isometry

## Summary

A partial isometry is a bounded operator on a Hilbert space that is isometric on the orthogonal complement of its kernel.
It maps an initial subspace isometrically onto a final subspace and kills everything orthogonal to the initial subspace.
Partial isometries appear in the [[polar decomposition|0J2C7BDJQA]] $T = V\abs{T}$ of an operator and in the comparison of projections in von Neumann algebras.

## Explanations

### L1 human

A partial isometry first throws away part of a vector and then moves the rest rigidly.
On $\C^2$, the map $(x_1, x_2) \mapsto (0, x_1)$ is an example: it forgets $x_2$ and moves $x_1$ to the second slot without changing its size.
Isometries, unitary operators and orthogonal projections are all partial isometries.
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<document source="https://tamnd.github.io/math2/node/GGE4KYWCNT.md" tag="GGE4KYWCNT" kind="definition">
# Grothendieck topos

## Definition

A *Grothendieck topos* is a category $\mathcal{E}$ that is [[equivalent|K7A04ZBD92]] to the category $\mathrm{Sh}(\mathcal{C}, J)$ of [[sheaves|RT8A8FJ566]] of sets on some small [[site|NYE0Z94MME]] $(\mathcal{C}, J)$.

## Remarks

Example: for a topological space $X$, the category $\mathrm{Sh}(X)$ of sheaves on the site of open sets of $X$ is a Grothendieck topos.
Example: for a small category $\mathcal{C}$, the category of [[presheaves|AND2WHZC8P]] on $\mathcal{C}$ is a Grothendieck topos, for the topology in which only maximal sieves cover.
Non-example: the category of finite sets is an elementary topos but not a Grothendieck topos, since it lacks infinite coproducts, which every Grothendieck topos has.
Every Grothendieck topos is an elementary topos and is complete and cocomplete.
Giraud's theorem gives an intrinsic characterization.
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<document source="https://tamnd.github.io/math2/node/GGEAGW4RRV.md" tag="GGEAGW4RRV" kind="statement">
# Talagrand's concentration inequality

## Statement

Let $n \ge 1$, and for $i = 1, \dots, n$ let $(\Omega_i, P_i)$ be a finite [[probability space|0NFPJVJJHX]] in which every subset is an event.
Let $\Omega = \Omega_1 \times \dots \times \Omega_n$ with the [[product measure|77PV0ZAVXT]] $P = P_1 \otimes \dots \otimes P_n$.
For a nonempty set $A \subseteq \Omega$ and a point $x \in \Omega$, define the convex distance
$$d_T(x, A) = \sup_{\alpha} \ \inf_{y \in A} \sum_{i \,:\, x_i \ne y_i} \alpha_i,$$
where the supremum is over all $\alpha \in [0, \infty)^n$ with $\alpha_1^2 + \dots + \alpha_n^2 = 1$.
Then for every nonempty $A \subseteq \Omega$ and every $t \ge 0$,
$$P(A) \cdot P(\set{x \in \Omega : d_T(x, A) \ge t}) \le e^{-t^2/4}.$$

## Remarks

The theorem holds, with the same proof, for products of general probability spaces, provided the sets involved are measurable.
Talagrand proved the stronger bound $\int_\Omega e^{d_T(x, A)^2 / 4} \, P(dx) \le 1 / P(A)$, by induction on $n$; Markov's inequality applied to $e^{d_T^2/4}$ gives the statement.
The convex distance is at least the normalised Hamming distance $\min_{y \in A} \#\set{i : x_i \ne y_i} / \sqrt{n}$, by the choice $\alpha_i = 1/\sqrt{n}$, so the theorem implies Hamming concentration.
A main consequence concerns convex Lipschitz functions.
Let $X_1, \dots, X_n$ be independent with values in $[0, 1]$, and let $F \colon [0, 1]^n \to \R$ be [[convex|07Q7CNM5WF]] and $1$-[[Lipschitz|0AW1E6YTJY]] for the Euclidean norm.
Then $P(\abs{F(X) - m} \ge t) \le C e^{-c t^2}$ for absolute constants $C, c > 0$, where $m$ is a [[median|07Y2E7P9NP]] of $F(X)$.
The bound does not depend on $n$, while bounded-difference methods would give a bound that gets worse with the dimension for this class.
It is used, for example, to show that the length of the longest increasing subsequence of a random permutation is concentrated around its median.

## Explanations

### L1 human

Take a set $A$ in a product of many independent coordinates that has probability at least one half.
Then almost every point is close to $A$, in a strong sense: changing a few well-chosen coordinates brings it into $A$, whatever weights one puts on the coordinates.
The probability of being at distance $t$ from $A$ drops like $e^{-t^2/4}$, independently of the number of coordinates.

### L2 human

Talagrand's inequality is a central result of the concentration of measure phenomenon in product spaces.
It goes beyond [[McDiarmid's inequality|8APFT1RPM4]] because the distance adapts to each point through the weights $\alpha$, which yields dimension-free bounds for convex Lipschitz functions and for functions with small certificates.
Applications include the largest eigenvalue of random matrices with bounded entries, longest increasing subsequences, and combinatorial optimisation problems on random inputs.
</document>

<document source="https://tamnd.github.io/math2/node/GGEMJF449D.md" tag="GGEMJF449D" kind="statement">
# Nagata's criterion for factoriality

## Statement

Let $A$ be a [[Noetherian|0MVH4Y1CRZ]] [[integral domain|09S3PFBEPM]].
Let $\mathcal{P}$ be a set of [[prime elements|0MNHXBSX3M]] of $A$.
Let $S$ be the [[multiplicative set|AN9SKEZ3VY]] generated by $\mathcal{P}$: the products of a unit of $A$ and finitely many elements of $\mathcal{P}$.
If the [[localization|4C1AE84N5N]] $S^{-1}A$ is a [[unique factorisation domain|06YMXJMXYS]], then $A$ is a unique factorisation domain.

## Remarks

Since $A$ is Noetherian, every nonzero nonunit of $A$ is a product of irreducible elements, so it suffices to show that every irreducible element $a$ of $A$ is prime.
If some prime $p \in \mathcal{P}$ divides $a$, then $a$ is associate to $p$ and so is prime.
Otherwise one checks that $a$ is prime in $S^{-1}A$ and that $aS^{-1}A \cap A = aA$, using that no prime of $\mathcal{P}$ divides $a$.
Then $aA$ is the contraction of a prime ideal, hence prime.
Example: let $A = \Z[x]$ and let $S$ be the set of nonzero integers.
$S$ is generated by the primes $p$, which are prime in $\Z[x]$ because $\Z[x]/(p) \cong \F_p[x]$ is a domain.
Then $S^{-1}A = \Q[x]$ is a principal ideal domain, hence a UFD, so $\Z[x]$ is a UFD.
The hypothesis on $S$ is necessary: for $A = \Z[\sqrt{-5}]$ and $S = A \setminus \set{0}$, $S^{-1}A$ is the field $\Q(\sqrt{-5})$, a UFD, while $A$ is not a UFD.

## Explanations

### L1 human

To prove that a Noetherian domain has unique factorization, it is enough to invert some prime elements and check unique factorization in the bigger ring.
This works because the inverted elements are prime, so they already factor well in $A$ and hide no failure of unique factorization.
So unique factorization can be tested in a simpler ring.

### L2 human

The criterion is a standard way to show that rings given by generators and relations are UFDs.
One finds a prime element $p$ such that $A[1/p]$ is visibly factorial, often a polynomial ring or a Laurent polynomial ring, and concludes for $A$.
It recovers the fact that the [[polynomial ring over a UFD is a UFD|0D6KH8Y59N]] for Noetherian coefficient rings, as in the example with $\Z[x]$.
For a [[Krull domain|CB9WJCN9BM]] $A$, the same idea shows that inverting prime elements does not change the [[divisor class group|WN8PK6RQZR]], since the height-one primes that meet $S$ are principal.
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<document source="https://tamnd.github.io/math2/node/GGFAH1M80J.md" tag="GGFAH1M80J" kind="statement">
# Atiyah–Patodi–Singer index theorem

## Statement

Let $M$ be a compact oriented Riemannian [[manifold with boundary|7DS6B2BFK8]] $Y = \partial M$, whose metric is a product $du^2 + g_Y$ on a collar $[0, 1) \times Y$ of the boundary.
Let $D \colon C^\infty(M; E) \to C^\infty(M; F)$ be a first-order [[elliptic operator|MT9P2X85V8]].
Assume that on the collar it has the form
$$D = \sigma \left( \frac{\partial}{\partial u} + A \right).$$
Here $\sigma \colon E|_Y \to F|_Y$ is a bundle isomorphism and $A$ is a self-adjoint elliptic operator on $Y$, both independent of $u$.
Impose the boundary condition $P_{\ge 0}(s|_Y) = 0$, where $P_{\ge 0}$ is the spectral projection of $A$ onto the span of its eigenvectors with eigenvalue $\ge 0$.
Then $D$ with this boundary condition is Fredholm, and its [[index|44GBFZ16SG]] is
$$\operatorname{ind} D = \int_M \alpha_0 - \frac{h + \eta_A(0)}{2}.$$
Here $\alpha_0$ is the local index density of $D$, the same density as for closed manifolds.
Also $h = \dim \ker A$, and $\eta_A(0)$ is the [[eta invariant|PRJCH788AM]] of $A$.

## Remarks

Idea of the proof: glue the half-infinite cylinder $(-\infty, 0] \times Y$ to $M$ along $Y$.
The boundary condition corresponds to $L^2$ solutions on the elongated manifold, and on the cylinder the heat kernel is computed exactly by separation of variables, which produces the eta invariant.
For operators such as the signature operator there is no elliptic local boundary condition, which is why a global spectral projection is used.
Example: for the signature operator on a $4k$-manifold with boundary, the theorem gives
$$\sigma(M) = \int_M L(p) - \eta(0).$$
Here $L(p)$ is the Hirzebruch $L$-form of the metric, and $\eta(0)$ is the eta invariant of an operator on the even forms of $Y$.
So the [[Hirzebruch signature theorem|TEG1ZX1AGJ]] fails on manifolds with boundary, and the eta invariant measures the defect.

## Explanations

### L1 human

On a closed manifold the index of an elliptic operator is an integral of a curvature expression.
On a manifold with boundary this integral is usually not an integer, so a correction from the boundary is needed.
The correction is the eta invariant, which measures how asymmetric the spectrum of the boundary operator is between positive and negative eigenvalues.

### L2 human

The theorem extends the Atiyah–Singer index theorem to manifolds with boundary.
It introduced the eta invariant into geometry, with uses for the signature defects of $3$-manifolds, for Chern–Simons invariants and for spectral flow.
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<document source="https://tamnd.github.io/math2/node/GGM63KXCF2.md" tag="GGM63KXCF2" kind="concept">
# Reeb vector field

## Summary

The Reeb vector field is the [[vector field|0299VCW1TH]] determined by a contact form on a [[contact manifold|HBQ8BZ7XB5]].
It is transverse to the contact hyperplanes and its flow preserves the contact form.
Reeb flows include geodesic flows on unit cotangent bundles and Hamiltonian flows on energy surfaces of contact type.
Their periodic orbits are the subject of the [[Weinstein conjecture|MP6B64DS31]] and of contact homology.

## Explanations

### L1 human

A contact form picks out a plane at each point, and it also measures how far a vector sticks out of that plane.
The Reeb vector field is the unique direction that sticks out by exactly one unit and along which the twisting of the planes does not change.
On the three-sphere with its standard contact form, the Reeb flow moves every point around a great circle, and these circles are the fibers of the Hopf fibration.
</document>

<document source="https://tamnd.github.io/math2/node/GGZV75ZZMV.md" tag="GGZV75ZZMV" kind="concept">
# Algebraic Riccati equation

## Summary

An algebraic Riccati equation is a quadratic matrix equation for an unknown symmetric matrix $X$, such as $A^T X + XA - XBR^{-1}B^T X + Q = 0$.
It is the steady state of the matrix Riccati differential equation.
Its stabilising solution gives the optimal feedback of the linear-quadratic regulator and the gain of the steady-state Kalman filter.
Without the quadratic term it reduces to a [[Lyapunov equation|92GVPZH9FT]].

## Explanations

### L1 human

A controller wants to steer a linear system to rest while keeping both the error and the effort small.
The best policy feeds back the state through a matrix gain, and that gain comes from a matrix $X$ solving a Riccati equation.
The equation is quadratic in $X$, like $ax^2 + bx + c = 0$ for numbers, so it can have several solutions.
For one variable with $A = 0$ and $B = Q = R = 1$, it reads $1 - x^2 = 0$, and the useful solution is $x = 1$.
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<document source="https://tamnd.github.io/math2/node/GH2SFDQNAE.md" tag="GH2SFDQNAE" kind="definition">
# Digital topology

## Definition

Let $p = (p_1, p_2)$ and $q = (q_1, q_2)$ be distinct points of $\Z^2$.

1. $p$ and $q$ are *$4$-adjacent* if $\abs{p_1 - q_1} + \abs{p_2 - q_2} = 1$.
2. $p$ and $q$ are *$8$-adjacent* if $\max(\abs{p_1 - q_1}, \abs{p_2 - q_2}) = 1$.
3. For $k \in \set{4, 8}$, a *$k$-path* in a set $S \subseteq \Z^2$ is a finite sequence of points of $S$ in which consecutive points are $k$-adjacent.
4. $S$ is *$k$-connected* if any two points of $S$ are joined by a $k$-path in $S$; the maximal $k$-connected subsets of $S$ are its *$k$-components*.

*Digital topology* is the study of such adjacency, connectedness and separation notions on $\Z^2$ and their analogues on $\Z^n$.
For each $k$, the $k$-adjacency makes $S$ a [[graph|00C0BR7D7H]] whose [[connected components|01VVP10MWJ]] are the $k$-components of $S$.

## Remarks

Example: $S = \set{(0, 0), (1, 1)}$ is $8$-connected, since the two points are $8$-adjacent.
Non-example: the same $S$ is not $4$-connected, because $\abs{0 - 1} + \abs{0 - 1} = 2$ and $S$ has no other points.
Let $S = \set{(1, 0), (0, 1), (-1, 0), (0, -1)}$.
With $8$-adjacency for $\Z^2 \setminus S$, the point $(0, 0)$ is $8$-adjacent to $(1, 1) \in \Z^2 \setminus S$, so the complement is $8$-connected and the "curve" $S$ encloses nothing.
With $4$-adjacency for the complement, $(0, 0)$ is a $4$-component by itself, so $S$ separates it from the outside.
For this reason one pairs $8$-adjacency for a set with $4$-adjacency for its complement, or the reverse.
The Khalimsky topology on $\Z$ is a genuine topology: the sets $\set{2n + 1}$ and $\set{2n - 1, 2n, 2n + 1}$ for $n \in \Z$ form a base, and $\Z$ is connected in it.
</document>

<document source="https://tamnd.github.io/math2/node/GH6CXWFNDJ.md" tag="GH6CXWFNDJ" kind="concept">
# Pseudo-Riemannian manifold

## Summary

A pseudo-Riemannian manifold is a [[smooth manifold|F45VQ47GPV]] with a smoothly varying [[nondegenerate|01NHFZF2SK]] [[symmetric bilinear form|0AXZNE5XNH]] on each tangent space, which need not be positive definite.
It generalises a [[Riemannian manifold|7CBT4SM83Y]], where the form is positive definite.
The most important case is a [[Lorentzian manifold|BY4QYYS788]], the setting of general relativity.
Much of Riemannian geometry, such as the [[Levi-Civita connection|CAZYGE1T4E]], geodesics and curvature, works in this setting.

## Explanations

### L1 human

On a Riemannian manifold every nonzero vector has a positive squared length.
A pseudo-Riemannian manifold drops this rule: some vectors may have negative or zero squared length.
What is kept is that the form has no hidden directions: no nonzero vector is orthogonal to every vector.
Minkowski space $\R^4$ with $-dt^2 + dx^2 + dy^2 + dz^2$ is the basic example.
There, a vector such as $(1, 1, 0, 0)$ has squared length $0$ although it is not zero.
</document>

<document source="https://tamnd.github.io/math2/node/GH81SYAPPN.md" tag="GH81SYAPPN" kind="concept">
# Algebraic group

## Summary

An algebraic group is a group whose elements are the points of an algebraic variety, with multiplication and inversion given by polynomial maps.
The two main families are the [[linear algebraic groups|GA81VVQHEP]], such as $\mathrm{GL}_n$, and the [[abelian varieties|R7W03TXRS9]], such as elliptic curves.
[[Chevalley's structure theorem|MZ6XDATQKR]] shows that every smooth connected algebraic group over a perfect field is built from one of each.
Algebraic groups play for algebraic geometry the role that Lie groups play for differential geometry.

## Explanations

### L1 human

The invertible $2 \times 2$ matrices form a group, and both the product and the inverse are given by formulas in the entries.
So the group is also a geometric object, the set where the determinant is nonzero.
An algebraic group is any group of this kind, where the elements form a variety and the group operations are polynomial maps.
Other examples are the numbers under addition, the nonzero numbers under multiplication, and the points of an elliptic curve.
</document>

<document source="https://tamnd.github.io/math2/node/GHEWGCECQJ.md" tag="GHEWGCECQJ" kind="definition">
# Incompressible surface

## Definition

Let $M$ be a compact [[3-manifold|R20TB2KEQQ]].
Let $S \subseteq M$ be a compact connected [[surface|VGJJVBZ1MQ]] that is neither a $2$-sphere nor a closed $2$-disk.
Assume that $S$ is either properly embedded in $M$ (that is, $S \cap \partial M = \partial S$) or contained in $\partial M$.
A *compressing disk* for $S$ is an embedded closed disk $D \subseteq M$ with $D \cap S = \partial D$ such that the curve $\partial D$ does not bound a disk in $S$.
The surface $S$ is *incompressible* if it has no compressing disk, and *compressible* otherwise.

## Remarks

Example: let $M = T^2 \times [0, 1]$ and $S = T^2 \times \set{1/2}$.
Let $D$ be a disk in $M$ with $D \cap S = \partial D$.
The inclusion $S \to M$ is a homotopy equivalence, so the loop $\partial D$, null-homotopic in $M$, is null-homotopic in $S$.
A null-homotopic simple closed curve in a surface bounds a disk in it, so $D$ is not a compressing disk, and $S$ is incompressible.

Non-example: let $M = D^2 \times S^1$ be a solid torus and $S = \partial M$.
The meridian disk $D = D^2 \times \set{1}$ meets $S$ in $\partial D$, and $\partial D$ represents a nonzero class in $\pi_1(S) \cong \Z^2$, so it bounds no disk in $S$.
Hence $D$ is a compressing disk and $S$ is compressible.

If $S$ is two-sided, the [[loop theorem|8YYJBVP68F]] shows that $S$ is incompressible if and only if the inclusion induces an injective map $\pi_1(S) \to \pi_1(M)$.
Some authors also call a $2$-sphere incompressible when it does not bound a $3$-ball in $M$.
</document>

<document source="https://tamnd.github.io/math2/node/GHJC53RFBT.md" tag="GHJC53RFBT" kind="example">
# Hartogs figure

## Example

Let $n \ge 2$ and write a point of $\C^n$ as $z = (z', z_n)$ with $z' \in \C^{n-1}$, and $\abs{z'} = \max_{j < n} \abs{z_j}$.
Let $\Delta^n = \set{z : \abs{z'} < 1, \abs{z_n} < 1}$ be the unit [[polydisc|DJBWP7D3M8]], and fix $0 < r < 1$ and $0 < s < 1$.
The *Hartogs figure* is the connected open set
$$H = \set{z \in \Delta^n : \abs{z'} < r \text{ or } \abs{z_n} > s}.$$
It is the union of a thin column $\set{\abs{z'} < r}$ and a shell $\set{s < \abs{z_n} < 1}$ inside $\Delta^n$.

Every [[holomorphic function|507C34W0ZT]] $f$ on $H$ extends to a holomorphic function on all of $\Delta^n$.
Fix $t$ with $s < t < 1$ and put
$$F(z', z_n) = \frac{1}{2\pi i} \int_{\abs{\zeta} = t} \frac{f(z', \zeta)}{\zeta - z_n} \, d\zeta \quad \text{for } \abs{z'} < 1, \ \abs{z_n} < t.$$
The circle $\abs{\zeta} = t$ lies in the shell, so the integrand is defined, and $F$ is holomorphic by differentiation under the integral sign.
For $\abs{z'} < r$, the one-variable function $\zeta \mapsto f(z', \zeta)$ is holomorphic on the whole unit disc, so the Cauchy integral formula gives $F = f$ there.
The sets where $F$ and $f$ are both defined form a connected open set, so the identity theorem gives $F = f$ on all of it.
Gluing $F$ and $f$ gives the extension to $\Delta^n$.

## Remarks

So $H$ is not a [[domain of holomorphy|4487CSC0MB]]: every holomorphic function on $H$ continues to the larger set $\Delta^n$.
This is the basic instance of [[Hartogs's extension theorem|FSGES670AZ]].
The extra variables are essential: in one variable the shell alone is the annulus $s < \abs{z} < 1$, and $1/z$ is holomorphic there but does not extend to the unit disc.

## Explanations

### L1 human

Picture a solid cylinder made of a thin vertical pole together with a thick outer ring, with the space between them missing.
In several complex variables, every holomorphic function defined on this shape automatically extends to fill the missing part.
The reason is that values on the outer ring determine, through Cauchy's integral, a function on the whole cylinder.
This picture is the starting point for all extension phenomena in several variables.
</document>

<document source="https://tamnd.github.io/math2/node/GHK66XE1S8.md" tag="GHK66XE1S8" kind="statement">
# Higman's lemma

## Statement

Let $(X, \le)$ be a [[well-quasi-ordered|NH25G6GC07]] set, and let $X^*$ be the set of finite words over $X$, the [[free monoid|CV013KHVJG]] on $X$.
For words $u = a_1 \cdots a_m$ and $v = b_1 \cdots b_n$ in $X^*$, write $u \le^* v$ when there are indices $1 \le i_1 < i_2 < \cdots < i_m \le n$ with $a_k \le b_{i_k}$ for $k = 1, \ldots, m$.
Then $(X^*, \le^*)$ is well-quasi-ordered.

## Remarks

The usual proof is the minimal bad sequence argument of Nash-Williams.
Assume a bad sequence exists, and choose one in which each word is as short as possible given the earlier ones.
Remove the first letter of each word; since $X$ is well-quasi-ordered, an infinite subsequence has non-decreasing first letters.
The shortened words of this subsequence, put after the beginning of the minimal sequence, give a bad sequence that contradicts minimality.

Example: take $X = \set{a, b}$ with equality as the order.
Then $u \le^* v$ means that $u$ is a [[subsequence|0F1PR74BRX]] of $v$, a scattered subword.
So in every infinite list of words over two letters, some word is a scattered subword of a later one.

A consequence concerns any language $L \subseteq X^*$ over a finite alphabet.
The set of words having a scattered subword in $L$ is regular, since it is generated by its finitely many minimal elements.

## Explanations

### L1 human

Write down an endless list of words in any finite alphabet.
Higman's lemma says that some word in your list can be found inside a later word by deleting letters.
More generally the letters may come from any well-quasi-ordered set, and a letter may be replaced by a larger one.

### L2 human

The lemma is the first step from simple well-quasi-orders, such as finite sets and $\N$, to orders on structured objects.
Kruskal's tree theorem extends it from words to finite trees, and its proof uses Higman's lemma on the sequences of subtrees.
In computer science it gives termination and decidability results for well-structured transition systems such as lossy channel systems.
</document>

<document source="https://tamnd.github.io/math2/node/GHKYR5PBQC.md" tag="GHKYR5PBQC" kind="statement">
# Going-up theorem

## Statement

Let $A \subseteq B$ be commutative rings with $B$ [[integral over|B7X0KFY1S1]] $A$.
Let $\mathfrak{p}_1 \subseteq \mathfrak{p}_2$ be [[prime ideals|00MHHZ9SNE]] of $A$, and let $\mathfrak{q}_1$ be a prime ideal of $B$ with $\mathfrak{q}_1 \cap A = \mathfrak{p}_1$.
Then there is a prime ideal $\mathfrak{q}_2$ of $B$ with $\mathfrak{q}_1 \subseteq \mathfrak{q}_2$ and $\mathfrak{q}_2 \cap A = \mathfrak{p}_2$.

## Remarks

The ring $B/\mathfrak{q}_1$ is integral over its subring $A/\mathfrak{p}_1$.
The [[lying over theorem|YZ97GKYY3P]] gives a prime of $B/\mathfrak{q}_1$ over $\mathfrak{p}_2/\mathfrak{p}_1$, and its preimage in $B$ is $\mathfrak{q}_2$.

By induction, every chain $\mathfrak{p}_1 \subseteq \cdots \subseteq \mathfrak{p}_n$ of primes of $A$ lifts to a chain $\mathfrak{q}_1 \subseteq \cdots \subseteq \mathfrak{q}_n$ of primes of $B$ with $\mathfrak{q}_i \cap A = \mathfrak{p}_i$.
So the [[Krull dimension|MVGBS3J1P7]] of $B$ is at least that of $A$.

Example: for $\Z \subseteq \Z[i]$, the chain $0 \subseteq 5\Z$ lifts to $0 \subseteq (2 + i)$.

Integrality is needed: for $k[x] \subseteq k[x, x^{-1}]$, the prime $0$ lies under $0$, but no prime of $k[x, x^{-1}]$ lies over $(x)$, since $x$ is a unit there.

## Explanations

### L1 human

If a prime of the big ring sits over a prime of the small ring, then one can climb up along any longer chain downstairs and stay above it upstairs.
In geometry: the image of a closed subset under an integral map is closed, and specializations of points can be followed upstairs.

### L2 human

Going-up and [[incomparability|Q2XBHJ96GV]] together give $\dim B = \dim A$ for every integral extension $A \subseteq B$.
This is the key step in computing the dimension of a finitely generated algebra over a field through [[Noether normalization|7Z28JY8QW4]].
The companion result in the other direction, [[going-down|YE52MQ13MV]], needs extra hypotheses.
</document>

<document source="https://tamnd.github.io/math2/node/GHRPE8CXJ9.md" tag="GHRPE8CXJ9" kind="concept">
# Linear system of divisors

## Summary

A linear system on a projective variety is a family of linearly equivalent effective divisors, parametrized by a projective space.
The complete linear system $\abs{D}$ of a divisor $D$ consists of all effective divisors linearly equivalent to $D$.
It is the projectivization of the space of global sections of $\mathcal{O}_X(D)$.
A linear system without base points gives a morphism to projective space.
Up to a choice of coordinates, every morphism to projective space arises in this way.
They are the basic tool to embed varieties and to study maps between them.

## Explanations

### L1 human

Take all the curves of a fixed kind on a surface, such as the conics in the plane.
Each conic is the zero set of a quadratic polynomial, and two polynomials that differ by a nonzero scalar give the same conic.
So the conics form a projective space, here of dimension $5$, because a quadratic polynomial in three variables has six coefficients.
This family is a linear system: one can add polynomials, so the family is "linear".
The lines in the plane form a smaller linear system of dimension $2$.
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<document source="https://tamnd.github.io/math2/node/GHVW1Z35BS.md" tag="GHVW1Z35BS" kind="concept">
# Hyperbolic 3-manifold

## Summary

A hyperbolic $3$-manifold is a [[3-manifold|R20TB2KEQQ]] with a complete Riemannian metric of constant curvature $-1$.
Equivalently, it is a quotient of hyperbolic $3$-space $\mathbb{H}^3$ by a discrete torsion-free group of isometries, a [[Kleinian group|27QRARKWD5]] in the orientable case.
By the geometrization theorem, hyperbolic manifolds form the largest and richest class of $3$-manifolds.
By [[Mostow rigidity|4BEWKR0EG7]], a hyperbolic metric of finite volume is determined by the fundamental group, so geometric quantities such as volume are topological invariants.

## Explanations

### L1 human

Hyperbolic space is a space where triangles have angle sum less than $\pi$ and where volume grows exponentially with distance.
A hyperbolic $3$-manifold is a space that looks like hyperbolic space near every point.
One can build one by gluing the faces of a polyhedron in hyperbolic space in pairs, so that the angles around each edge add up to a full turn.
The complement of the figure-eight knot in $S^3$ is obtained by gluing two ideal tetrahedra, and it is hyperbolic.
</document>

<document source="https://tamnd.github.io/math2/node/GHXMVWN3W5.md" tag="GHXMVWN3W5" kind="concept">
# Nielsen fixed-point theory

## Summary

Nielsen fixed-point theory studies the least number of fixed points among all maps homotopic to a given self-map $f$ of a space.
It sorts the fixed points of $f$ into classes, using paths and their images, and counts the classes that cannot be removed by a homotopy.
This count is the Nielsen number $N(f)$.
It is a homotopy invariant and a lower bound for the number of fixed points of every map homotopic to $f$.
It refines the [[Lefschetz number|ZSYCA4EGEZ]], which can only show that at least one fixed point exists.

## Explanations

### L1 human

The Lefschetz number can show that a map must have a fixed point, but not how many.
Nielsen theory sorts the fixed points into groups.
Two fixed points are in the same group if the map carries some path joining them to a path that can be deformed back to it.
During this deformation the two ends stay fixed.
Groups with a nonzero total weight can never be removed, so their number is a guaranteed minimum.
The map $z \mapsto z^3$ on the unit circle has two fixed points, $1$ and $-1$, in two essential classes.
So every map homotopic to it has at least two fixed points.
</document>

<document source="https://tamnd.github.io/math2/node/GJ4J77VV60.md" tag="GJ4J77VV60" kind="definition">
# Erdős–Rényi model

## Definition

Let $n \ge 1$ be an integer and let $\mathcal{G}_n$ be the set of all simple [[graphs|00C0BR7D7H]] with vertex set $[n] = \set{1, \ldots, n}$.
Let $N = \binom{n}{2}$ be the number of pairs of distinct vertices; see [[binomial coefficient|0GBZM0MT0F]].

1. For $0 \le p \le 1$, the model $G(n, p)$ is the [[probability measure|059TJN0NZ1]] on $\mathcal{G}_n$ giving each graph with $m$ edges the probability $p^m (1 - p)^{N - m}$.
2. For an integer $M$ with $0 \le M \le N$, the model $G(n, M)$ is the uniform probability measure on the set of graphs in $\mathcal{G}_n$ with exactly $M$ edges.

## Remarks

In $G(n, p)$ the $N$ events "the pair $\set{i, j}$ is an edge" are independent, each of probability $p$.
So the number of edges has the binomial distribution with parameters $N$ and $p$, and its mean is $pN$.

Example: for $n = 3$ and $p = 1/2$, every graph on $\set{1, 2, 3}$ has probability $(1/2)^3 = 1/8$, and these $8$ values sum to $1$.

Non-example: on $\set{1, 2, 3}$, choosing the triangle with probability $1/2$ and the empty graph with probability $1/2$ is not $G(3, p)$ for any $p$.
Its edge probability is $1/2$, but the three edge events are not independent, since the graph with one edge has probability $0$.

A random graph from these models is also written $G(n, p)$ or $G(n, M)$.
For $M$ near $pN$ the two models behave in a similar way.
</document>

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# Prime knot

## Summary

A prime knot is a nontrivial [[knot|QYEAZTYGPT]] that is not the [[knot sum|AGZSZW7GYE]] of two nontrivial knots.
Schubert proved that every nontrivial knot is a knot sum of prime knots in a unique way, up to order.
So prime knots play the role for knots that prime numbers play for integers.
Knot tables list only prime knots.

## Explanations

### L1 human

Some knots are made by tying two simpler knots one after the other on the same string.
A prime knot is a knot that cannot be split like that.
The [[trefoil knot|VYTYSVHN4A]] is prime: it cannot be written as two smaller nontrivial knots.
The granny knot, which is two trefoils in a row, is not prime.
The plain unknotted loop is not counted as prime, just as $1$ is not a prime number.
</document>

<document source="https://tamnd.github.io/math2/node/GJ8VTBFNP2.md" tag="GJ8VTBFNP2" kind="statement">
# von Neumann's minimax theorem

## Statement

Let $A$ be a real $m \times n$ matrix.
For $k \ge 1$ let $\Delta_k = \set{p \in \R^k : p_i \ge 0 \text{ for all } i, \ p_1 + \cdots + p_k = 1}$ be the standard [[simplex|YX1DJK2QAG]] of probability vectors.
Then
$$\max_{x \in \Delta_m} \min_{y \in \Delta_n} x^T A y = \min_{y \in \Delta_n} \max_{x \in \Delta_m} x^T A y.$$

## Remarks

The inequality $\max_x \min_y \le \min_y \max_x$ holds for every function on a product.
The reverse inequality is the content of the theorem.
One proof writes both sides as linear programs that are dual to each other, and uses the strong duality theorem of linear programming.
Another proof separates a convex set from a point with a hyperplane.
The common value is the *value* of the game, and the theorem says that $x^T A y$ has a saddle point on $\Delta_m \times \Delta_n$.

Example: matching pennies, with $A = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$.
For $x = y = (\tfrac12, \tfrac12)$ one gets $x^T A y' = 0$ for every $y'$ and $x'^T A y = 0$ for every $x'$, so the value is $0$.
Mixing is necessary: over pure strategies, that is vertices of the simplices, the row player can only guarantee $\max_i \min_j a_{ij} = -1$, while $\min_j \max_i a_{ij} = 1$.

The theorem extends to a continuous function $f(x, y)$ on $X \times Y$ with $X, Y$ compact convex sets in $\R^m$ and $\R^n$, concave in $x$ and convex in $y$.
Sion's minimax theorem weakens these hypotheses further.

## Explanations

### L1 human

In a two-player zero-sum game, one player's gain is the other's loss.
If the players may randomize, it does not matter who has to announce a strategy first.
The best guaranteed payoff of the first player equals the smallest loss the second player can guarantee.
This common number is the value of the game.

### L2 human

The theorem is the founding result of game theory, proved by John von Neumann in 1928.
It is equivalent to the duality theorem of linear programming, and the value and optimal mixed strategies of a matrix game can be computed by the simplex algorithm.
The general minimax theorems for convex-concave functions, and Sion's version for quasiconvex-quasiconcave functions, all rest on the same convexity.
They are the model for saddle point formulations of duality in convex optimization, such as saddle points of a Lagrangian.
</document>

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# Lattice of subvarieties

## Summary

The lattice of subvarieties of a variety $\mathcal{V}$ is the set of all [[subvarieties|4YSCK1DF26]] of $\mathcal{V}$, ordered by inclusion.
It is a complete lattice, and it is dually isomorphic to the lattice of equational theories that contain the theory of $\mathcal{V}$.
Its shape records how $\mathcal{V}$ can be cut down by further identities, and its study is a large part of the theory of varieties.

## Explanations

### L1 human

Put all the subvarieties of a variety in a diagram, with smaller ones lower down.
The bottom is the trivial variety, of one-element algebras, and the top is the whole variety.
For abelian groups, the subvarieties other than the top are the classes defined by $nx = 0$ for $n \ge 1$.
The class for $n$ lies inside the class for $m$ exactly when $n$ divides $m$.
So this lattice is the divisibility order on the positive integers, with an extra top element.
</document>

<document source="https://tamnd.github.io/math2/node/GJBAX2R2PG.md" tag="GJBAX2R2PG" kind="definition">
# Z-group

## Definition

Let $G$ be a finite [[group|G5S1M9]].
The group $G$ is a *Z-group* when, for every prime $p$, every [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$ is [[cyclic|0KVGH0J5B3]].

## Remarks

Since the Sylow $p$-subgroups of $G$ are conjugate, it suffices to check one of them for each prime $p$ dividing the order of $G$.
Example: a group of square-free order is a Z-group, because each of its Sylow subgroups has prime order and so is cyclic.
In particular $S_3$, of order $6$, is a Z-group.
Non-example: the [[quaternion group|0NRJGW864D]] $Q_8$ is not a Z-group, because it is its own Sylow $2$-subgroup and it is not cyclic.
Every Z-group is [[solvable|0FXJ7YFMMX]], and every subgroup and every quotient of a Z-group is again a Z-group.

## Explanations

### L1 human

A finite group is a Z-group when all its Sylow subgroups are cyclic.
</document>

<document source="https://tamnd.github.io/math2/node/GJD05AZWQN.md" tag="GJD05AZWQN" kind="statement">
# Behnke–Stein theorem

## Statement

Let $X$ be a connected [[Riemann surface|5K0N8QMV6Q]] that is not compact.
Then $X$ is a [[Stein manifold|HMCD0JBFST]].

## Remarks

Heinrich Behnke and Karl Stein proved the theorem in 1949.
The core of the proof is a Runge-type approximation theorem on $X$.
Let $K \subseteq X$ be compact, and assume that $X \setminus K$ has no relatively compact connected component.
Then every function holomorphic near $K$ is a uniform limit on $K$ of holomorphic functions on $X$.
With it one constructs global holomorphic functions that separate points and that are unbounded on any given discrete sequence.
This gives the two conditions in the definition of a Stein manifold.
Example: every connected open subset of a compact Riemann surface, other than the whole surface, carries a nonconstant holomorphic function.
This applies for instance to a torus minus a point.
Noncompactness is necessary: on a compact Riemann surface every holomorphic function is constant.
In several variables the analogue fails: $\C^2 \setminus \set{0}$ is a noncompact complex manifold that is not Stein.
The name Behnke–Stein theorem is also used for an earlier result of the same authors.
It says that the union of an increasing sequence of domains of holomorphy in $\C^n$ is a domain of holomorphy.

## Explanations

### L1 human

On a compact Riemann surface, such as a sphere or a torus, there are no nonconstant holomorphic functions.
The theorem says that as soon as the surface is not compact, there are plenty of them, as many as on a domain in the plane.
So every open Riemann surface behaves, for function theory, like an open subset of $\C$.

### L2 human

The theorem makes the whole theory of Stein manifolds available for open Riemann surfaces.
By Cartan's theorem B, every additive and multiplicative Cousin problem is solvable on an open Riemann surface, because $H^2(X, \Z) = 0$ for a noncompact surface.
So on an open Riemann surface, the theorems of Mittag-Leffler and Weierstrass hold: there are meromorphic functions with prescribed principal parts and holomorphic functions with prescribed zeros.
By the embedding theorem, every open Riemann surface is a closed complex submanifold of $\C^3$.
</document>

<document source="https://tamnd.github.io/math2/node/GJGEVMZ828.md" tag="GJGEVMZ828" kind="concept">
# Generic matrix algebra

## Summary

The generic matrix algebra is the algebra generated by finitely many $n \times n$ matrices whose entries are independent commuting variables.
It is the "most general" algebra generated by $m$ matrices of size $n$.
For every commutative $k$-algebra $C$, the $k$-algebra generated by $m$ matrices in $M_n(C)$ is a quotient of it.
The quotient map substitutes elements of $C$ for the variables.
Over an infinite field $k$, it is the relatively free algebra of the variety of algebras that satisfy all identities of $M_n(k)$.
It is a domain.
It is a basic object in the theory of [[polynomial identity rings|C0EDT81EGH]] and in the invariant theory of matrices.

## Explanations

### L1 human

Fill a $2 \times 2$ matrix with four unknowns and a second one with four other unknowns.
The matrices you can build from these two by adding, multiplying and scaling form the generic matrix algebra.
Any rule that holds in this algebra holds for all pairs of $2 \times 2$ matrices, because you can substitute numbers for the unknowns.
Conversely, a rule that holds for all pairs of $2 \times 2$ matrices over an infinite field holds here, so this algebra records exactly the identities of $2 \times 2$ matrices.
</document>

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# Strict Henselization

## Summary

The strict Henselization of a [[local ring|MZ02BKPHJA]] $R$ is a [[Henselian local ring|RNXPV3KC5J]] over $R$.
Its residue field is a separable closure of the residue field of $R$.
It is universal among local maps from $R$ to strictly Henselian local rings, that is, Henselian local rings with a separably closed residue field.
It refines the [[Henselization|917PGHVZPM]], which keeps the residue field unchanged.
In algebraic geometry the strict Henselizations of the local rings of a scheme are the stalks of its structure sheaf for the étale topology.

## Explanations

### L1 human

The Henselization of a local ring lifts every simple root that exists modulo the maximal ideal.
The strict Henselization goes further: it first enlarges the residue field until every separable polynomial has a root there, and then lifts these roots.
For a field $k$, the strict Henselization is a separable closure of $k$.
For the $p$-adic integers, it is the ring of integers of the maximal unramified extension of the $p$-adic numbers.
It is the ring that sees a point of a space together with all its étale neighbourhoods.
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# Curve complex

## Definition

Let $S$ be a compact connected orientable [[surface|VGJJVBZ1MQ]] of [[genus|NJ787GZS1N]] $g$ with $n$ boundary components, and assume $3g - 3 + n \ge 2$.
A simple closed curve in $S$ is *essential* when it bounds no disc in $S$.
It must also not cobound an annulus with a component of $\partial S$.
The *curve complex* $\mathcal{C}(S)$ is the [[simplicial complex|V8GZ0W1ZZ5]] such that:

1. its vertices are the isotopy classes of essential simple closed curves in $S$;
2. a set of $k + 1$ distinct vertices spans a $k$-simplex if and only if the classes have pairwise disjoint representatives.

## Remarks

A maximal set of pairwise disjoint, pairwise non-isotopic essential curves has $3g - 3 + n$ elements, a pants decomposition, so $\mathcal{C}(S)$ has dimension $3g - 4 + n$.
The [[mapping class group|JV5NYG913V]] acts on $\mathcal{C}(S)$, and under the hypothesis $3g - 3 + n \ge 2$ the complex is connected and has infinite diameter.

Example: for the sphere with $5$ boundary circles, $g = 0$ and $n = 5$, so $3g - 3 + n = 2$ and $\mathcal{C}(S)$ is a graph.
Each essential curve cuts off two of the boundary circles from the other three, and two such curves are joined by an edge when they can be made disjoint.

Non-example of clause 2: on a closed surface of genus $2$, let curves $a$ and $b$ meet transversally in one point.
Their algebraic intersection number is $\pm 1$, so they have no disjoint representatives.
So $a$ and $b$ are not joined by an edge.

For the torus, the once-punctured torus and the four-holed sphere, the definition gives a complex without edges.
For them one joins curves of minimal intersection by edges instead, and the result is the Farey graph.

## Explanations

### L1 human

The vertices are loops on the surface that cannot be shrunk away, counted up to sliding.
Simplices record which loops can coexist without crossings.
The condition on $g$ and $n$ excludes small surfaces where no two different loops are disjoint.
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# Gerstenhaber algebra

## Summary

A Gerstenhaber algebra is a graded commutative algebra with a Lie bracket of degree $-1$ that acts on products by the Leibniz rule.
Murray Gerstenhaber found this structure on the Hochschild cohomology of an associative algebra.
Polyvector fields on a manifold with the [[Schouten–Nijenhuis bracket|FJ1MDBZ680]] form another basic example.
It is the graded analogue of a Poisson algebra, with the bracket shifted by one degree.

## Explanations

### L1 human

A Gerstenhaber algebra carries two operations: a product that is commutative up to signs, and a bracket like the commutator of vector fields.
The bracket lowers the total degree by one.
The two operations fit together by a product rule, just as the derivative of a product follows the Leibniz rule.
For polyvector fields on a manifold, the product is the wedge product and the bracket extends the Lie bracket of vector fields.
The same structure on Hochschild cohomology governs the deformations of an associative algebra.
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# Numerical methods for ordinary differential equations

## Definition

Let $f \colon [t_0, T] \times \R^d \to \R^d$ be continuous and Lipschitz continuous in its second argument.
Then the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$ has a unique solution $y$ on $[t_0, T]$.
For an integer $N \ge 1$ let $h = (T - t_0)/N$ and $t_n = t_0 + nh$ for $n = 0, \ldots, N$.
A *$k$-step method* is a rule that starts from given values $y_0, \ldots, y_{k-1}$.
For $0 \le n \le N - k$ it determines $y_{n+k}$ from $h$, $t_n$, the values $y_n, \ldots, y_{n+k-1}$ and finitely many evaluations of $f$.
It may do so by solving an equation that contains $y_{n+k}$.
It is *one-step* if $k = 1$, *explicit* if $y_{n+k}$ is given by a formula in the earlier values, and *implicit* otherwise.
The method is *convergent* if, for every such $f$ and starting values with $y_i \to y(t_0)$ for $i < k$ as $h \to 0$,
$$\max_{0 \le n \le N} \norm{y(t_n) - y_n} \to 0 \quad \text{as } h \to 0.$$
It is convergent *of order* $p$ if this maximum is $O(h^p)$ for every sufficiently smooth $f$ and starting values with error $O(h^p)$.

## Remarks

Example: the [[Euler method|0JARTXX4XP]] $y_{n+1} = y_n + h f(t_n, y_n)$ is an explicit one-step method, convergent of order $1$ when $f$ is Lipschitz in $y$.
Example: the two-step Adams–Bashforth method $y_{n+2} = y_{n+1} + \frac{h}{2}\big(3 f(t_{n+1}, y_{n+1}) - f(t_n, y_n)\big)$ is explicit with $k = 2$.
Non-example of convergence: the explicit two-step rule $y_{n+2} = -4 y_{n+1} + 5 y_n + h\big(4 f(t_{n+1}, y_{n+1}) + 2 f(t_n, y_n)\big)$ has local truncation error $O(h^4)$ but is not convergent, because the root $-5$ of $\zeta^2 + 4\zeta - 5$ makes perturbations grow like $5^n$.
Many methods use variable steps $h_n = t_{n+1} - t_n$; then $h$ is replaced by $\max_n h_n$.

## Explanations

### L1 human

A numerical method for an initial value problem produces a list of approximate values on a grid of times, each computed from the previous ones.
It is convergent if the approximations get uniformly close to the true solution when the steps get small.
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# Cointegration

## Definition

A real time series is *integrated of order $1$*, written I($1$), if it is not weakly [[stationary|DT0QXJE705]] but its first difference $\Delta x_t = x_t - x_{t-1}$ is.
An example is an [[ARIMA($p$, $1$, $q$) process|KTDVZP4DF2]] whose moving-average polynomial has no root at $1$, such as a random walk.
Let $x_t = (x_{1,t}, \dots, x_{k,t})^T$, with $k \ge 2$, be a vector time series whose components are all I($1$).
The series $x_t$ is *cointegrated* if there is a vector $\beta \in \R^k$, $\beta \ne 0$, such that the [[linear combination|0V8F3E48E6]] $\beta^T x_t$ is weakly stationary.
Such a $\beta$ is a *cointegrating vector*.

## Remarks

The cointegrating vectors, together with $0$, form a linear subspace of $\R^k$; its dimension $r$, with $0 \le r \le k - 1$, is the cointegrating rank.
More generally, components that are I($d$) are cointegrated of order ($d$, $b$) when some nonzero combination is I($d - b$) with $b > 0$.

Example: let $w_t$ be a [[random walk|04VZE3FZXF]] and $e_t$ a white noise independent of it, and set $x_{1,t} = w_t$ and $x_{2,t} = 2 w_t + e_t$.
Both components are I($1$), and $\beta = (2, -1)^T$ gives $\beta^T x_t = -e_t$, which is stationary, so the pair is cointegrated.

Non-example: two independent random walks $w_t$ and $v_t$ with step variance $\sigma^2 > 0$ are not cointegrated.
For $\beta = (a, b)^T \ne 0$, the combination $a w_t + b v_t$ is again a random walk with step variance $(a^2 + b^2) \sigma^2 > 0$, which is not stationary.

In the Engle–Granger method, $\beta$ is estimated by regressing one component on the others.
Then a [[Dickey–Fuller type test|26E0HYRDS8]] is applied to the residuals, with critical values adjusted for the estimation step.
The Granger representation theorem links cointegration with error-correction models.
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# Riemann problem

## Problem

Let $F \colon \R^m \to \R^m$ be a $C^1$ function and consider the system of [[conservation laws|8RA7AMWSAF]] $u_t + F(u)_x = 0$ in one space dimension.
Let $u_l, u_r \in \R^m$ be two constant states.
Find an [[entropy solution|1A01A3EJRX]] $u \colon \R \times (0, \infty) \to \R^m$ with the initial values
$$u(x, 0) = \begin{cases} u_l & \text{if } x < 0, \\ u_r & \text{if } x > 0. \end{cases}$$

## Remarks

The equation and the data are unchanged by the scaling $(x, t) \mapsto (\lambda x, \lambda t)$ for $\lambda > 0$, so one looks for self-similar solutions $u(x, t) = v(x / t)$.

Scalar case with convex flux.
Let $m = 1$ and let $F$ be smooth and uniformly convex, and let $G$ be the inverse function of $F'$.
If $u_l > u_r$, the entropy solution is the [[shock wave|JSDMGGTG7G]] equal to $u_l$ for $x < \sigma t$ and to $u_r$ for $x > \sigma t$, where $\sigma = \dfrac{F(u_l) - F(u_r)}{u_l - u_r}$.
If $u_l < u_r$, the entropy solution is the [[rarefaction wave|MFN4Q5SH8H]] equal to $u_l$ for $x < F'(u_l) t$, to $G(x/t)$ for $F'(u_l) t \le x \le F'(u_r) t$, and to $u_r$ for $x > F'(u_r) t$.

Systems.
Consider a strictly hyperbolic system whose characteristic fields are each genuinely nonlinear or linearly degenerate.
Lax showed that its Riemann problem has a unique solution made of at most $m$ elementary waves when $\abs{u_l - u_r}$ is small.
These waves are shocks, rarefaction waves and contact discontinuities, one for each characteristic family, separated by constant states.
For large data, or for more general systems, solutions may fail to exist or to be unique in this class.
For the Euler equations of gas dynamics, the problem posed by Riemann, the solution is known explicitly up to solving one scalar equation, and it may contain a vacuum.

Solutions of Riemann problems are the building blocks of [[Godunov's scheme|43HZJ7DRT0]] and of the [[Glimm scheme|Y70AC2X30T]].

## Explanations

### L1 human

Put a wall in a tube, with gas at one pressure on the left and another pressure on the right, and remove the wall at time zero.
The Riemann problem asks what happens next.
Waves leave the place of the wall: some are sharp jumps, others are smooth fans that spread out.
Because the start has no length scale, the picture at time $2$ is the picture at time $1$ stretched by a factor $2$.
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# Bang–bang control

## Summary

A bang–bang control only takes extreme values of the allowed range, such as full thrust forward or full thrust backward.
It switches abruptly between these values.
Such controls arise as optimal controls through [[Pontryagin's maximum principle|7RPV2T29CF]] when the dynamics and the cost are linear in the control.
They are typical for minimum-time problems, and they are easy to implement with simple on–off actuators.

## Explanations

### L1 human

To bring a car to a stop at a wall as fast as possible, you accelerate at full power and then brake at full power.
You never use half power: the control jumps from one extreme to the other.
That is a bang–bang control.
A household thermostat that switches a heater fully on or fully off works in the same way.
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# Rank-one convexity

## Summary

A function on $m \times n$ matrices is rank-one convex if it is convex along every line in a rank-one direction $a \otimes b$.
It is the weakest of the convexity notions used in the vectorial calculus of variations.
Polyconvexity implies quasiconvexity, which implies rank-one convexity, and each is implied by ordinary [[convexity|07GR3TNW4H]].
For $C^2$ integrands it is the Legendre–Hadamard condition at every matrix.

## Explanations

### L1 human

Two matrices that differ by a rank-one matrix can be the gradients of one continuous map on the two sides of a flat interface.
Fine layers of two such gradients form a laminate, and a laminate can mimic the average gradient.
Rank-one convexity says that such layering never lowers the energy below its value at the average.
For $2 \times 2$ matrices the determinant is rank-one convex, in fact affine along rank-one lines, but it is not convex.
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<document source="https://tamnd.github.io/math2/node/GKFDKRHEP4.md" tag="GKFDKRHEP4" kind="definition">
# Mixed-type equation

## Definition

Let $\Omega$ be a connected open set in $\R^n$.
Consider a linear second-order partial differential equation in $\Omega$ with real coefficients.
Its type at each point is given by the [[classification of second-order PDEs|5FC6BMM1NA]].
The equation is of *mixed type* in $\Omega$ if both conditions hold:

1. it is [[elliptic|B4TTNHSTRX]] on some nonempty open subset of $\Omega$;
2. it is [[hyperbolic|92C3S03AQZ]] on some nonempty open subset of $\Omega$.

## Remarks

Example: the Tricomi equation $u_{yy} + y \, u_{xx} = 0$ on $\Omega = \R^2$, written as $A u_{xx} + 2B u_{xy} + C u_{yy}$, has $A = y$, $B = 0$ and $C = 1$, so $AC - B^2 = y$.
It is elliptic on the open set $y > 0$ and hyperbolic on the open set $y < 0$, so it is of mixed type.
Non-example: the equation $u_{xx} + (1 + y^2) u_{yy} = 0$ has $AC - B^2 = 1 + y^2 > 0$ everywhere, so it is elliptic on all of $\R^2$ and not of mixed type.
If the coefficients are continuous, the elliptic and hyperbolic regions are open, and in a connected $\Omega$ they are separated by a closed set where the coefficient matrix is degenerate.

## Explanations

### L1 human

An equation is of mixed type when it is elliptic in some region and hyperbolic in another region of the same domain.
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# Tessellation

## Summary

A tessellation, or tiling, is a covering of the plane or of [[Euclidean space|0XPR1E85W0]] by pieces called tiles.
The tiles do not overlap except along their boundaries.
Tilings are the basic objects of the geometry of space filling.
Their symmetries lead to the [[crystallographic groups|7KTJKFQT2N]].
Their non-periodic forms lead to [[aperiodic sets of prototiles|Z3FCJZF6WK]] and to the mathematics of quasicrystals.

## Explanations

### L1 human

Think of the floor of a bathroom covered by tiles with no gaps and no overlaps.
A tessellation is the same idea carried out over the whole infinite plane.
Tiles may touch along their edges and corners, but no two tiles share any inner point.
The simplest example is the grid of unit squares $[a, a+1] \times [b, b+1]$ with $a$ and $b$ integers.
Hexagons, as in a honeycomb of bees, also tile the plane, but regular pentagons do not.
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# Fundamental theorem of projective geometry

## Statement

Let $k$ and $k'$ be [[fields|04C5ZBG1WT]], and let $n \ge 2$.
Let $V$ and $W$ be vector spaces of dimension $n + 1$ over $k$ and $k'$.
Let $f \colon \mathbb{P}(V) \to \mathbb{P}(W)$ be a [[collineation|V7PRPN24CV]] between their [[projective spaces|ZT9A6JK8QF]].
Then there are a field isomorphism $\sigma \colon k \to k'$ and a bijection $T \colon V \to W$ such that:

1. $T(u + v) = T u + T v$ and $T(c v) = \sigma(c) T v$ for all $u, v \in V$ and $c \in k$;
2. $f([v]) = [T v]$ for every nonzero $v \in V$.

In particular, every collineation of $\mathbb{P}^n(\R)$ with $n \ge 2$ is a [[projective transformation|P1MKTF7MWQ]].

## Remarks

A map $T$ with property 1 is called $\sigma$-semilinear.
The proof fixes a frame of $n + 2$ points in general position and its image, and normalizes $T$ on the corresponding basis vectors.
Comparing the points of a line through two frame points with the scalars defines $\sigma$.
The configuration theorems that hold in $\mathbb{P}(V)$ show that $\sigma$ is additive and multiplicative.
For $k = \R$ the identity is the only field automorphism, so $T$ is linear and $f$ is a projective transformation.

The hypothesis $n \ge 2$ is needed: $\mathbb{P}^1(k)$ has only one line, so every bijection of it is a collineation.
The automorphism $\sigma$ cannot be dropped in general: complex conjugation of all coordinates is a collineation of $\mathbb{P}^2(\C)$ that is not a projective transformation.
The theorem also holds for vector spaces over division rings.
The collineation group of $\mathbb{P}^n(k)$, for $n \ge 2$, is therefore the group $\mathrm{P\Gamma L}(n + 1, k)$ of maps induced by semilinear bijections.

## Explanations

### L1 human

A collineation only promises to keep points on a line together.
The theorem says that, in a projective space of dimension at least $2$, this alone forces the map to come from linear algebra, up to a twist of the scalars.
Over the real numbers there is no twist, so every line-preserving bijection of the real projective plane is given by an invertible $3 \times 3$ matrix.

### L2 human

The theorem identifies the symmetry group of the incidence geometry $\mathbb{P}^n(k)$ with $\mathrm{P\Gamma L}(n + 1, k)$, which is $\mathrm{PGL}(n + 1, k)$ extended by the automorphisms of $k$.
Together with the [[Veblen–Young theorem|NPNNXWMVCC]], it shows that the incidence structure of a projective space of dimension at least $3$ determines its coordinate division ring up to isomorphism.
The [[fundamental theorem of affine geometry|W3YYX63R9R]] is the affine counterpart.
It is the reason why, in real projective geometry, one may freely replace "collineation" by "projective transformation".
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# Stiefel–Whitney class

## Definition

Consider real vector bundles $E \to B$ over paracompact spaces $B$.
The *Stiefel–Whitney classes* are the [[characteristic classes|7HACJC0JE2]] $w_i(E) \in H^i(B; \Z/2\Z)$, $i \ge 0$, in [[singular cohomology|6N7BC70CX0]], that satisfy the following axioms.
Write $w(E) = w_0(E) + w_1(E) + w_2(E) + \cdots$ for the total class.

1. $w_0(E) = 1$, and $w_i(E) = 0$ for $i$ greater than the rank of $E$.
2. Naturality: $w_i(f^* E) = f^* w_i(E)$ for every continuous map $f \colon B' \to B$, and isomorphic bundles have equal classes.
3. Whitney product formula: $w(E \oplus E') = w(E) \smile w(E')$ for the [[Whitney sum|BE9NJRVDR3]] of two bundles over the same base, where $\smile$ is the [[cup product|JX47MS1R1H]].
4. Normalization: for the tautological line bundle $\gamma^1_1$ over $\R P^1$, $w_1(\gamma^1_1)$ is the generator of $H^1(\R P^1; \Z/2\Z) \cong \Z/2\Z$.

There is exactly one family of classes with these properties.

## Remarks

Example: a trivial bundle $B \times \R^n$ is pulled back from a point, so by axiom 2 its total class is $w = 1$.
For the tangent bundle of $\R P^n$, one has $w(T\R P^n) = (1 + a)^{n+1}$, where $a$ generates $H^1(\R P^n; \Z/2\Z)$; for $n = 2$ this gives $w = 1 + a + a^2$.

Non-example: the rule $w(E) = 1$ for every bundle satisfies axioms 1 to 3 but fails axiom 4.

A bundle $E$ is orientable exactly when $w_1(E) = 0$.
An orientable bundle admits a spin structure exactly when also $w_2(E) = 0$.
For a smooth manifold $M$, one writes $w_i(M) = w_i(TM)$.
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# Toeplitz matrix

## Summary

A Toeplitz matrix is a matrix that is constant along each diagonal parallel to the main diagonal.
Its entry in row $i$ and column $j$ depends only on the difference $i - j$.
Toeplitz matrices describe discrete convolutions and stationary time series, and linear systems with them can be solved faster than general systems.
They are closely related to [[Hankel matrices|PQ1D8FQB3E]], which are constant along the anti-diagonals.

## Explanations

### L1 human

A Toeplitz matrix is fixed by its first row and its first column; every other entry copies the entry up and to the left of it.
For example, $\begin{pmatrix} a & b & c \\ d & a & b \\ e & d & a \end{pmatrix}$ is Toeplitz.
Such matrices come up whenever a rule acts the same way at every position, like a moving average over a list of numbers.
So an $n \times n$ Toeplitz matrix is described by only $2n - 1$ numbers instead of $n^2$.
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<document source="https://tamnd.github.io/math2/node/GMPJ88PAQE.md" tag="GMPJ88PAQE" kind="definition">
# Automatic group

## Definition

Let $G$ be a [[group|G5S1M9]] and let $A$ be a finite [[generating set|0RP7VWQ0YN]] of $G$ closed under inverses.
For a word $w$ over $A$, write $\bar w$ for the element of $G$ it represents.
Let $\$$ be a new padding symbol.
For words $u, v$ over $A$, write $u$ and $v$ one above the other and fill the end of the shorter one with $\$$.
The result is a word over $(A \cup \set{\$})^2$, the padded pair $(u, v)^{\$}$.
An *automatic structure* on $G$ with respect to $A$ consists of finite automata $W$ over $A$ and $M_x$ over $(A \cup \set{\$})^2$ for $x \in A \cup \set{e}$, such that:

1. Every element of $G$ equals $\bar w$ for at least one word $w$ accepted by $W$.
2. For each $x \in A \cup \set{e}$ and all words $u, v$ accepted by $W$, the automaton $M_x$ accepts $(u, v)^{\$}$ if and only if $\bar u \, x = \bar v$ in $G$.

The group $G$ is *automatic* if it has an automatic structure with respect to some finite generating set.

## Remarks

Equivalently, $G$ is automatic if there is a regular language $L$ over $A$ mapping onto $G$ with the fellow traveller property.
This means there is a constant $k$ such that whenever $u, v \in L$ and $\bar u^{-1} \bar v \in A \cup \set{e}$, the prefixes $u(t)$ and $v(t)$ of length $t$ satisfy $d_A(\overline{u(t)}, \overline{v(t)}) \le k$ for all $t$, in the [[word metric|VCHJVR9NGH]].

Example: $G = \Z^2 = \langle a, b \rangle$ with $A = \set{a, a^{-1}, b, b^{-1}}$ and $L$ the set of words $a^m b^n$, $m, n \in \Z$, which is regular.
Two such words naming elements that differ by one generator differ by one in the exponent of $a$ or of $b$.
Their prefixes stay at distance at most $2$, so $L$ has the fellow traveller property.

Non-example: the integer Heisenberg group is not automatic.
It is finitely presented, but its Dehn function grows like $n^3$, while every automatic group satisfies a quadratic isoperimetric inequality.

Being automatic does not depend on the choice of the finite generating set.
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# Caustic

## Summary

A caustic is the [[envelope|827KR3MVVY]] of a family of rays, the curve or surface where nearby rays focus.
In symplectic terms, project a [[Lagrangian submanifold|H8ACPTMX08]] of a cotangent bundle to the base.
The caustic is the set of critical values of this projection, that is, the image of the [[Lagrangian singularities|QKPMR9MC9W]].
Caustics appear in optics as the bright curves of focused light.
They also appear in the asymptotics of oscillatory integrals and of short-wave solutions of wave equations.
Their generic local shapes, such as folds and cusps, are given by Arnold's classification of Lagrangian singularities.

## Explanations

### L1 human

On a sunny day the inside of a coffee cup shows a bright curve with a sharp point.
Light rays reflected from the curved wall crowd together along this curve, so it is much brighter than its surroundings.
That bright curve is a caustic: the envelope that the reflected rays touch.
A simple case from geometry is the family of normal lines of a plane curve, whose caustic is the [[evolute|Y3ZNV24V2C]] of the curve.
For a parabola the evolute has a cusp, and such cusps are the typical sharp points of caustics.
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# Szegő kernel

## Definition

Let $\Omega \subset \C^n$ be a bounded domain with $C^2$ boundary $\partial\Omega$, and let $\sigma$ be the [[surface measure|065HE5V98E]] on $\partial\Omega$.
Let $H^2(\partial\Omega)$ be the closure in $L^2(\partial\Omega, \sigma)$ of the restrictions to $\partial\Omega$ of the functions [[holomorphic|507C34W0ZT]] on a neighbourhood of $\overline{\Omega}$.
Each $f \in H^2(\partial\Omega)$ is the boundary value of a unique holomorphic function on $\Omega$, also written $f$.
For each $z \in \Omega$, the map $f \mapsto f(z)$ is a bounded linear functional on the Hilbert space $H^2(\partial\Omega)$.
The *Szegő kernel* of $\Omega$ is the function $S \colon \Omega \times \partial\Omega \to \C$ such that, for each $z \in \Omega$, the function $\zeta \mapsto \overline{S(z, \zeta)}$ lies in $H^2(\partial\Omega)$ and
$$f(z) = \int_{\partial\Omega} S(z, \zeta) \, f(\zeta) \, d\sigma(\zeta) \quad \text{for all } f \in H^2(\partial\Omega).$$

## Remarks

By the Riesz representation theorem the function $\overline{S(z, \cdot)}$ exists and is unique.
So $S$ is the reproducing kernel of the [[Hardy space|PNNNFJVF6W]] $H^2(\partial\Omega)$, in the sense of [[reproducing kernel Hilbert spaces|4CK2GSRAGD]].
The kernel extends to $\Omega \times \Omega$, holomorphic in the first variable and antiholomorphic in the second, with $S(z, w) = \overline{S(w, z)}$.

Example: for the unit disc $\mathbb{D} \subset \C$ with arc length, $S(z, \zeta) = \dfrac{1}{2\pi(1 - z\bar\zeta)}$.
Check: for $\abs{\zeta} = 1$ one has $\bar\zeta = 1/\zeta$, so $\dfrac{1}{1 - z\bar\zeta} = \dfrac{\zeta}{\zeta - z}$, and $d\sigma = \abs{d\zeta} = \dfrac{d\zeta}{i\zeta}$.
The formula becomes $f(z) = \dfrac{1}{2\pi i}\displaystyle\oint_{\abs{\zeta} = 1} \dfrac{f(\zeta)}{\zeta - z}\, d\zeta$, which is Cauchy's integral formula.
For the unit ball of $\C^n$ the kernel is $S(z, \zeta) = \dfrac{(n-1)!}{2\pi^n} (1 - \ip{z}{\zeta})^{-n}$, where $\ip{z}{\zeta} = \sum_j z_j \bar\zeta_j$; at $z = 0$ it is $1$ divided by the area $2\pi^n/(n-1)!$ of the sphere.

Non-example: on the unit disc, the Poisson kernel $P(z, \zeta) = \dfrac{1 - \abs{z}^2}{2\pi\abs{\zeta - z}^2}$ also satisfies $f(z) = \int_{\partial\mathbb{D}} P(z, \zeta) f(\zeta) \, d\sigma(\zeta)$ for all $f \in H^2(\partial\mathbb{D})$.
It is not the Szegő kernel, because for $z \ne 0$ the function $\zeta \mapsto P(z, \zeta)$ has nonzero Fourier coefficients of negative index, so it does not lie in $H^2(\partial\mathbb{D})$.

The [[Bergman kernel|8RWDHW0CXC]] is defined in the same way, with $L^2(\Omega)$ in place of $L^2(\partial\Omega, \sigma)$.
</document>

<document source="https://tamnd.github.io/math2/node/GMY8XWETFV.md" tag="GMY8XWETFV" kind="definition">
# Risch algorithm

## Definition

Let $C$ be a field of characteristic $0$ in which the field operations and the test for equality are computable.
Let $F = C(x)(t_1, \dots, t_m)$ be a [[differential field|GA5PTQGC6G]] with [[derivation|3J89REV9WQ]] $'$ such that $x' = 1$ and the constants of $F$ are $C$.
Assume that each $t_i$ is a logarithm, an exponential or an algebraic element over $C(x)(t_1, \dots, t_{i-1})$.
The *Risch algorithm* is the procedure that takes $f \in F$ as input and does one of two things:

1. it returns an element $g$ of an elementary extension of $F$ with $g' = f$, that is, an elementary [[antiderivative|0BYWRWFT4F]] of $f$;
2. it reports, correctly, that no elementary extension of $F$ contains such an element $g$.

## Remarks

The procedure works down the tower.
It writes $f$ in terms of the last generator $t_m$ and uses [[Liouville's theorem|8VRTPECBPZ]] to predict the form of a possible antiderivative.
This reduces the problem to integration and to differential equations in the smaller field $C(x)(t_1, \dots, t_{m-1})$.
The cases of logarithms and exponentials (the transcendental case) are simpler than the algebraic case.

Example: for $F = \Q(x, e^x)$ and $f = x e^x$, the procedure looks for $g = q e^x$ with $q \in \Q(x)$ and $q' + q = x$, finds $q = x - 1$, and returns $g = (x - 1) e^x$.

Example: for $F = \Q(x, e^{x^2})$ and $f = e^{x^2}$, the same reduction asks for $q \in \Q(x)$ with $q' + 2x q = 1$; there is none, so the procedure reports that $e^{x^2}$ has no elementary antiderivative.

Non-example: a table of integrals finds many antiderivatives, but it fails clause 2, because it cannot certify that an integral not in the table is not elementary.

The algorithm needs a way to decide whether a constant expression equals $0$.
For constants built from $\pi$, $e$ and other transcendental numbers this is a hard problem, so in practice the algorithm is only as complete as the zero test for constants.
</document>

<document source="https://tamnd.github.io/math2/node/GMYDMZMN1S.md" tag="GMYDMZMN1S" kind="statement">
# Levi's extension theorem for meromorphic functions

## Statement

Let $n \ge 2$ and write $z = (z', z_n)$ with $z' \in \C^{n-1}$ and $\abs{z'} = \max_{j < n} \abs{z_j}$.
Let $\Delta^n$ be the unit [[polydisc|DJBWP7D3M8]] in $\C^n$, let $0 < r < 1$ and $0 < s < 1$, and let
$$H = \set{z \in \Delta^n : \abs{z'} < r \text{ or } \abs{z_n} > s}.$$
Then every [[meromorphic function|W6FTSF57T1]] on $H$ is the restriction of a unique meromorphic function on $\Delta^n$.

## Remarks

The set $H$ is the [[Hartogs figure|GHJC53RFBT]].
For holomorphic functions the extension is given by a Cauchy integral in $z_n$ over a circle in the shell $s < \abs{z_n} < 1$.
For meromorphic functions one must also control the poles of the one-variable functions $z_n \mapsto f(z', z_n)$ as $z'$ varies, which makes the proof harder.
Uniqueness follows from the identity theorem, since $H$ is a nonempty open subset of the connected set $\Delta^n$.

Example: $f(z) = 1/(z_n - z_1)$ is meromorphic on $H$, and it extends to the meromorphic function $1/(z_n - z_1)$ on $\Delta^n$.
When $r \le s$, its polar set $\set{z_n = z_1}$ passes through the region between the column and the shell.

The hypothesis $n \ge 2$ is needed: in one variable, $e^{1/z}$ is holomorphic on the annulus $s < \abs{z} < 1$ but has no meromorphic extension to the unit disc.

## Explanations

### L1 human

Hartogs's extension theorem says that holomorphic functions on a Hartogs figure fill in the missing region.
Levi's theorem says the same for meromorphic functions, which are allowed to have poles.
The poles can run through the missing region, but the function still extends there as a quotient of holomorphic functions.

### L2 human

The theorem shows that extension phenomena in several variables are not limited to holomorphic functions.
Many continuation arguments in several variables move a Hartogs figure step by step through a domain.
Levi's theorem lets these arguments run for meromorphic functions as well as for holomorphic ones.
It is used, for example, in the study of meromorphic functions on complex manifolds and of the singularities of meromorphic maps.
</document>

<document source="https://tamnd.github.io/math2/node/GMYSQTH3Z7.md" tag="GMYSQTH3Z7" kind="definition">
# Cellular algebra

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] with identity and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $R$.
A *cell datum* for $A$ consists of four pieces of data.
They are a finite partially ordered set $\Lambda$, a finite set $M(\lambda)$ for each $\lambda \in \Lambda$, and an $R$-linear map $* \colon A \to A$.
The fourth is an injective map $(\lambda, S, T) \mapsto C^{\lambda}_{S,T} \in A$, defined for $\lambda \in \Lambda$ and $S, T \in M(\lambda)$.
The data must satisfy:

1. The elements $C^{\lambda}_{S,T}$, for $\lambda \in \Lambda$ and $S, T \in M(\lambda)$, form an $R$-[[basis|0KCQ9Y3QKA]] of $A$.
2. The map $*$ is an anti-involution, that is $(ab)^* = b^* a^*$ and $(a^*)^* = a$ for all $a, b \in A$, and $(C^{\lambda}_{S,T})^* = C^{\lambda}_{T,S}$.
3. For all $a \in A$, $\lambda \in \Lambda$ and $S, T \in M(\lambda)$,
$$a\, C^{\lambda}_{S,T} \equiv \sum_{S' \in M(\lambda)} r_a(S', S)\, C^{\lambda}_{S',T} \pmod{A(< \lambda)},$$
where $r_a(S', S) \in R$ does not depend on $T$, and $A(< \lambda)$ is the $R$-span of the $C^{\mu}_{U,V}$ with $\mu < \lambda$.

The algebra $A$ is *cellular* if it has a cell datum, and the basis in clause 1 is then a *cellular basis*.

## Remarks

Example: $A = M_n(R)$, with $\Lambda = \set{\lambda}$, $M(\lambda) = \set{1, \dots, n}$, $C^{\lambda}_{i,j} = E_{ij}$ and $*$ the transpose.
The check of clause 3: for $a = (a_{kl})$, $a E_{ij} = \sum_k a_{ki} E_{kj}$, whose coefficients $a_{ki}$ do not depend on $j$, and $A(< \lambda) = 0$.
Non-example of a datum: the same basis of $M_2(R)$ with $*$ the identity map fails clause 2, since $(E_{12} E_{21})^* = E_{11}$ but $E_{21}^* E_{12}^* = E_{22}$.
Non-example of an algebra: the upper triangular $2 \times 2$ matrices over a field are not cellular.
Graham and Lehrer showed that the [[Cartan matrix|ZBQ4KRC98K]] of a cellular algebra over a field is symmetric.
This algebra has the Cartan matrix $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$, which is not symmetric.
Applying $*$ to clause 3 gives the mirror condition for right multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/GN02JVADH4.md" tag="GN02JVADH4" kind="definition">
# Stratified sampling

## Definition

Let $U$ be a finite population of $N$ units, and let $U_1, \dots, U_H$ be a [[partition|08QMKTWF9T]] of $U$ into nonempty strata of sizes $N_1, \dots, N_H$.
Choose sample sizes $n_1, \dots, n_H$ with $1 \le n_h \le N_h$.
*Stratified (simple random) sampling* with these sizes draws, for each $h$, a [[simple random sample|Y0PTCJBPEA]] $S_h$ of size $n_h$ from $U_h$.
The $H$ draws are [[independent|0GQ938GDE0]].
The sample is $S = S_1 \cup \cdots \cup S_H$.

## Remarks

For values $y_i$ attached to the units, the stratified estimator of the population mean is $\bar y_{st} = \sum_{h=1}^{H} \frac{N_h}{N} \bar y_h$, where $\bar y_h$ is the sample mean in $S_h$.
It is unbiased, with variance $\sum_h \big(\frac{N_h}{N}\big)^2 \big(1 - \frac{n_h}{N_h}\big) \frac{S_h^2}{n_h}$, where $S_h^2$ is the population variance of the $y_i$ in $U_h$ with divisor $N_h - 1$.
*Proportional allocation* takes $n_h$ proportional to $N_h$.
*Neyman allocation* takes $n_h$ proportional to $N_h S_h$, which minimizes the variance for a fixed total $n = \sum_h n_h$.

Example: $U = \set{1, 2, 3, 4}$ with strata $U_1 = \set{1, 2}$ and $U_2 = \set{3, 4}$, and $n_1 = n_2 = 1$.
The possible samples are $\set{1, 3}$, $\set{1, 4}$, $\set{2, 3}$ and $\set{2, 4}$, each with probability $1/4$.

Non-example: a simple random sample of size $2$ from the whole of $U$.
It can be $\set{1, 2}$, with no unit from $U_2$, so it is not a stratified sample with $n_1 = n_2 = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/GN4N94GQ0Z.md" tag="GN4N94GQ0Z" kind="concept">
# Hyperbolic geometry

## Summary

Hyperbolic geometry is the [[non-Euclidean geometry|SPCYMFN5V2]] in which, through a point off a line, there pass at least two lines that do not meet it.
It can be realised on a surface of constant negative [[Gaussian curvature|SQVQ5DQT6W]].
In it, the angles of a triangle add up to less than $\pi$, and the area of a triangle is determined by its angles.
It appears throughout mathematics, from complex analysis and number theory to the topology of surfaces and 3-manifolds.

## Explanations

### L1 human

Imagine a world where space spreads out faster than in a flat plane, like the frilly edge of a lettuce leaf.
In this world, through a point beside a line you can draw many lines that never meet it.
Triangles are thin, and their angles add up to less than $180^\circ$.
In the [[upper half-plane model|RHKSMPBSQ1]], the lines are vertical half-lines and half-circles that meet the horizontal axis at right angles.
</document>

<document source="https://tamnd.github.io/math2/node/GN6WHHGBAE.md" tag="GN6WHHGBAE" kind="proof">
# Proof of Extension of Lagrange's theorem

## Proof

Let $S$ be a set of coset representatives for $K$ in $H$, so $H = \bigsqcup_{s \in S} sK$ (disjoint union), and $|S|=[H:K]$. For any $a \in G$, left-multiplication-by-$a$ is a bijection $G \to G$, so $aH = \bigsqcup_{s \in S} asK$. Thus each left coset of $H$ decomposes into $[H:K]$ left cosets of $K$. Since $G$ decomposes into $[G:H]$ left cosets of $H$, each of which decomposes into $[H:K]$ left cosets of $K$, the total number $[G:K]$ of left cosets of $K$ in $G$ is $[G:H][H:K]$.
</document>

<document source="https://tamnd.github.io/math2/node/GNDWPYYFW4.md" tag="GNDWPYYFW4" kind="concept">
# IEEE 754 standard

## Summary

IEEE 754 is the technical standard for [[floating-point arithmetic|RTKRSC3KPH]] used by almost all modern processors.
It fixes the binary formats, such as single precision and double precision, and how their bits encode sign, exponent and significand.
It requires the basic operations to be correctly rounded, and it defines special values: signed zeros, infinities, NaN and [[subnormal numbers|DHXXSN12JC]].
Because of it, the same floating-point program gives the same results on different machines, which makes error analysis portable.

## Explanations

### L1 human

Before the standard, each computer maker had its own way of storing real numbers, and the same program could give different answers on different machines.
IEEE 754 is the agreement that settled this.
A double-precision number uses $64$ bits: $1$ bit for the sign, $11$ bits for the exponent and $52$ bits for the fraction.
This gives about $16$ significant decimal digits and numbers up to about $1.8 \times 10^{308}$.
It also says what $1/0$ is (infinity) and what $0/0$ is (NaN, "not a number"), so a computation does not simply crash.
</document>

<document source="https://tamnd.github.io/math2/node/GNEGMN4E26.md" tag="GNEGMN4E26" kind="definition">
# Group law on an elliptic curve

## Definition

Let $K$ be a [[field|04C5ZBG1WT]] of characteristic not $2$ or $3$, and let $E \colon y^2 = x^3 + ax + b$ be an [[elliptic curve|427N6S6QQW]] over $K$, with $a, b \in K$ and $4a^3 + 27b^2 \ne 0$.
Let $E(K)$ be the set of solutions $(x, y) \in K^2$ together with the point at infinity $O$.
The *group law* on $E(K)$ is the operation $+$ defined as follows, for $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ in $E(K)$:

1. $O + P = P + O = P$ for every $P \in E(K)$.
2. If $x_1 = x_2$ and $y_1 = -y_2$, then $P + Q = O$.
3. Otherwise, let $\lambda = \frac{y_2 - y_1}{x_2 - x_1}$ if $x_1 \ne x_2$, and $\lambda = \frac{3x_1^2 + a}{2y_1}$ if $P = Q$.
Then $P + Q = (x_3, y_3)$ with $x_3 = \lambda^2 - x_1 - x_2$ and $y_3 = \lambda(x_1 - x_3) - y_1$.

With this operation, $E(K)$ is an [[abelian group|082GHNPJ53]] with identity $O$, and the inverse of $(x, y)$ is $(x, -y)$.

## Remarks

Geometrically, $P + Q + R = O$ exactly when $P$, $Q$, $R$ are the three intersection points, counted with multiplicity, of $E$ with a line in the projective plane.
The number $\lambda$ is the slope of the line through $P$ and $Q$, or of the tangent at $P$ when $P = Q$.

Example: on $y^2 = x^3 + 1$ over $\Q$, take $P = Q = (2, 3)$.
Then $\lambda = \frac{3 \cdot 4 + 0}{6} = 2$, $x_3 = 4 - 2 - 2 = 0$ and $y_3 = 2(2 - 0) - 3 = 1$, so $2P = (0, 1)$, and indeed $1^2 = 0^3 + 1$.

Non-example: the operation $P * Q$ that sends $P, Q$ to the third intersection point of their line with $E$, without the reflection, is not a group law.
A neutral element $e$ for $*$ would have to lie on the tangent line at every point of $E$.
But only finitely many tangent lines of a smooth cubic pass through a given point.

Associativity is the hard part of the proof that $E(K)$ is a group; it follows from the theorem of Riemann–Roch or from a direct computation with the formulas.
For a general Weierstrass equation the formulas change, but the geometric rule stays the same.
</document>

<document source="https://tamnd.github.io/math2/node/GNH0Y5VDE5.md" tag="GNH0Y5VDE5" kind="statement">
# Reduction of a first-kind Volterra equation to the second kind

## Statement

Let $a < b$, let $\Delta = \set{(t, s) : a \le s \le t \le b}$, and let $K \colon \Delta \to \C$ be continuous, with continuous partial derivative $\partial K / \partial t$ on $\Delta$ and $K(t, t) \ne 0$ for all $t \in [a, b]$.
Let $f \colon [a, b] \to \C$ be continuously differentiable.
Let $x \colon [a, b] \to \C$ be continuous.
Consider the [[Volterra equation of the first kind|H6ZGEDM4VV]]
$$\int_a^t K(t, s) \, x(s) \, ds = f(t), \quad t \in [a, b].$$
Then $x$ solves it if and only if $f(a) = 0$ and $x$ solves the [[Volterra equation of the second kind|ATPSAHYG7C]]
$$x(t) = \frac{f'(t)}{K(t, t)} - \int_a^t \frac{1}{K(t, t)} \frac{\partial K}{\partial t}(t, s) \, x(s) \, ds, \quad t \in [a, b].$$
Consequently, if $f(a) = 0$, the first-kind equation has exactly one continuous solution, and if $f(a) \ne 0$ it has none.

## Remarks

Proof idea: by the Leibniz rule, the derivative of $\int_a^t K(t, s) \, x(s) \, ds$ is $K(t, t) \, x(t) + \int_a^t \partial_t K(t, s) \, x(s) \, ds$.
So differentiating the first-kind equation gives the second-kind equation after division by $K(t, t)$.
Conversely, a solution of the second-kind equation makes both sides of the first-kind equation have the same derivative, and both vanish at $t = a$ when $f(a) = 0$.
The new kernel $-\partial_t K(t, s) / K(t, t)$ is continuous, so [[unique solvability|WG4GVC9HR8]] applies.
Example: $\int_0^t e^{t - s} x(s) \, ds = t$ gives $x(t) = 1 - \int_0^t e^{t - s} x(s) \, ds = 1 - t$, and indeed $\int_0^t e^{t - s} (1 - s) \, ds = t$.
The hypothesis $K(t, t) \ne 0$ matters: for $K(t, s) = t - s$, differentiation gives another first-kind equation $\int_a^t x(s) \, ds = f'(t)$, and one needs a second differentiation.
For the weakly singular kernel $(t - s)^{-1/2}$ the reduction does not apply, and one uses the [[Abel inversion formula|TDMXFY06A2]] instead.

## Explanations

### L1 human

An equation of the first kind hides the unknown inside an integral.
Differentiating with respect to $t$ brings it out, because the derivative of an integral with variable upper limit contains the integrand at that limit.
If the kernel does not vanish on the diagonal, one can divide and obtain an equation of the second kind, which is well understood.

### L2 human

The reduction shows that a first-kind Volterra equation with a good kernel is about as hard as one differentiation.
The price is that the data $f$ must be differentiable, and errors in $f$ are amplified by the derivative, which is the source of ill-posedness in practice.
When $K(t, t) = 0$ but $\partial_t K(t, t) \ne 0$, differentiating twice gives the same kind of reduction, under the corresponding smoothness assumptions and with the conditions $f(a) = f'(a) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/GNWE3XR0ZG.md" tag="GNWE3XR0ZG" kind="concept">
# Prime end

## Summary

Prime ends are the ideal boundary points of a simply connected plane domain, introduced by Constantin Carathéodory.
A prime end is described by a shrinking sequence of cross cuts that close in on the boundary.
For a bounded simply connected domain, a conformal map from the unit disc matches the points of the unit circle with the prime ends.
They are the right notion of boundary when the topological boundary is too irregular for the conformal map to extend continuously.

## Explanations

### L1 human

Imagine the unit disc with a slit cut along a radius.
Seen from inside the domain, a point on the slit can be approached from above or from below, and these feel like two different places.
Prime ends make this precise: each way of closing in on the boundary through smaller and smaller cuts is one prime end.
So each inner point of the slit carries two prime ends, while each point of the circle carries one.
</document>

<document source="https://tamnd.github.io/math2/node/GP56NBN9RY.md" tag="GP56NBN9RY" kind="concept">
# Involution

## Summary

An involution is a function from a set to itself that undoes itself: applying it twice gives back the starting point.
Equivalently, it is a bijection that is its own [[inverse|0J2WAR83WP]].
Negation of numbers, [[complex conjugation|0FTH2E9TAS]], the [[complement|0RKF1ZT5JJ]] of a set and the inverse map of a [[group|G5S1M9]] are involutions.
In algebra the word is also used for an involution that reverses a product, $(xy)^* = y^* x^*$, as in rings with involution.

## Explanations

### L1 human

An involution is an action that, done twice, puts everything back.
Reflecting a picture in a mirror twice gives the original picture.
Changing the sign of a number twice gives the original number: $-(-5) = 5$.
Adding $1$ is not an involution, because doing it twice adds $2$.
</document>

<document source="https://tamnd.github.io/math2/node/GP98BJP4JH.md" tag="GP98BJP4JH" kind="concept">
# Chebyshev series

## Summary

A Chebyshev series expands a function on $[-1, 1]$ as a sum $\sum a_k T_k(x)$ of [[Chebyshev polynomials|M6EAR1JHPP]].
Under the substitution $x = \cos \theta$ it becomes the Fourier cosine series of $f(\cos \theta)$.
Truncated Chebyshev series are cheap to compute and are close to minimax approximations, within a factor that grows like $\log n$.
So they are central in numerical analysis.

## Explanations

### L1 human

A Fourier series writes a periodic function as a sum of cosines and sines.
A Chebyshev series does the same for a function on $[-1, 1]$, using the polynomials $T_k$, which are cosines in disguise.
For example, $x^2 = \frac{1}{2} T_0(x) + \frac{1}{2} T_2(x)$, since $T_0 = 1$ and $T_2 = 2x^2 - 1$.
For smooth functions the coefficients shrink very fast, so a few terms already give a very accurate polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/GP9E4AWJ8W.md" tag="GP9E4AWJ8W" kind="statement">
# Superlinear convergence of the secant method

## Statement

Let $I$ be an open interval, let $f \colon I \to \R$ be [[twice continuously differentiable|0VW90BBSJ4]], and let $r \in I$ with $f(r) = 0$ and $f'(r) \ne 0$.
Let $(x_k)$ be the iterates of the [[secant method|9ZC177PVPC]] from starting points close enough to $r$ that $x_k \to r$, and assume $x_k \ne r$ for all $k$.
Let $e_k = x_k - r$.

1. The convergence is superlinear: $\lim_{k \to \infty} \abs{e_{k+1}}/\abs{e_k} = 0$.
2. If moreover $f''(r) \ne 0$, the convergence has order $\varphi = (1 + \sqrt{5})/2$, the [[golden ratio|0SC8T7548D]]:
$$\lim_{k \to \infty} \frac{\abs{e_{k+1}}}{\abs{e_k}^{\varphi}} = \abs{\frac{f''(r)}{2 f'(r)}}^{\varphi - 1}.$$

## Remarks

The secant method satisfies $e_{k+1} = C_k e_k e_{k-1}$ with $C_k \to C = f''(r)/(2f'(r))$.
Clause 1 follows at once: $\abs{e_{k+1}}/\abs{e_k} = \abs{C_k} \abs{e_{k-1}} \to 0$.
For clause 2, the logarithms $y_k = \log \abs{e_k}$ satisfy approximately $y_{k+1} = y_k + y_{k-1} + \log \abs{C}$.
This is a Fibonacci-type recursion, whose solutions grow like $\varphi^k$.
A careful version of this argument gives the limit; the exponent $\varphi - 1 = 1/\varphi$ comes from $\varphi^2 = \varphi + 1$.
Example: for $f(x) = x^2 - 2$ from $x_0 = 1$, $x_1 = 2$, the errors are about $4.1 \times 10^{-1}$, $5.9 \times 10^{-1}$, $8.1 \times 10^{-2}$, $1.4 \times 10^{-2}$, $4.2 \times 10^{-4}$, $2.1 \times 10^{-6}$, $3.2 \times 10^{-10}$.
Once the errors are small, the number of correct digits grows by a factor of about $1.6$ per step.
At a zero with $f'(r) = 0$ the conclusion fails: for $f(x) = x^2$ the convergence is only linear.

## Explanations

### L1 human

Near a simple zero the secant method improves very fast.
At each step the number of correct digits is multiplied by about $1.618$, the golden ratio.
That is less than the doubling of Newton's method, but each secant step needs only one new function value and no derivative.

### L2 human

In terms of [[rate of convergence|YWZQC5E54B]], the secant method has order $\varphi \approx 1.618$ and Newton's method has order $2$.
One Newton step costs a value of $f$ and a value of $f'$.
If these cost about the same, two secant steps cost as much as one Newton step, and together they have order $\varphi^2 \approx 2.618 > 2$.
This efficiency is why secant-type steps appear in practical solvers such as Brent's method.
The appearance of the golden ratio comes from the recursion $e_{k+1} \approx C e_k e_{k-1}$, whose exponents follow the Fibonacci numbers.
</document>

<document source="https://tamnd.github.io/math2/node/GPBSS1V3SS.md" tag="GPBSS1V3SS" kind="concept">
# Shrinkage estimator

## Summary

A shrinkage estimator is an estimator obtained by pulling a basic estimator toward a fixed target or a pooled value.
Shrinking adds some bias but can lower the variance by more, so the mean squared error goes down.
Examples are the [[James–Stein estimator|709T196XCT]], posterior means under normal priors, and penalized regression such as [[ridge regression|3KZW2930V2]].

## Explanations

### L1 human

Raw estimates computed from little data tend to be too extreme: some are too high and some too low just by chance.
A shrinkage estimator moves each raw estimate part of the way toward a sensible central value.
The less reliable the raw estimate, the more it is moved.
For example, a batting average from only ten games is better predicted by moving it toward the league average than by taking it at face value.
</document>

<document source="https://tamnd.github.io/math2/node/GPDPYHM3RE.md" tag="GPDPYHM3RE" kind="statement">
# Representability of cohomology by Eilenberg–MacLane spaces

## Statement

Let $G$ be an [[abelian group|071DDFVG4J]] and let $n \ge 1$.
Let $K(G, n)$ be an [[Eilenberg–MacLane space|AJHEYJE5XY]] that is a CW complex.
By the Hurewicz theorem and the universal coefficient theorem, $H^n(K(G, n); G) \cong \Hom(\pi_n K(G, n), G) = \Hom(G, G)$; let $\iota \in H^n(K(G, n); G)$ be the class that corresponds to the identity of $G$.
Let $X$ be a connected [[CW complex|5W95QFHXPR]] with a $0$-cell as base point.
Let $[X, K(G, n)]_*$ be the set of base-point-preserving [[homotopy classes|F2F143KQ2M]] of base-point-preserving maps $X \to K(G, n)$.
Then the map
$$[X, K(G, n)]_* \to H^n(X; G), \qquad [f] \mapsto f^*(\iota),$$
is a bijection, natural in $X$, onto the [[singular cohomology|6N7BC70CX0]] group $H^n(X; G)$.

## Remarks

One proof uses obstruction theory.
A map on the $n$-skeleton that extends over the $(n+1)$-skeleton is given, up to homotopy, by a cellular cocycle.
The obstructions to extending it over higher cells lie in groups $H^{k+1}(X; \pi_k K(G, n))$ with $k > n$, which vanish.
Another proof checks that $X \mapsto H^n(X; G)$ satisfies the hypotheses of the Brown representability theorem and identifies the representing space.

Example: $K(\Z, 2) = \C P^\infty$, so $H^2(X; \Z)$ is the set of homotopy classes of maps $X \to \C P^\infty$.
For $X$ a CW complex, this set also classifies complex line bundles on $X$, and the bijection sends a line bundle to its first Chern class.

The CW hypothesis on $X$ matters: for spaces that are not of the homotopy type of a CW complex, singular cohomology need not be represented in this way.
For example, the Warsaw circle has trivial singular cohomology in positive degrees but nontrivial Čech cohomology, and maps into $K(\Z, 1) = S^1$ see the Čech groups.

## Explanations

### L1 human

The $n$-th cohomology of a space with coefficients in $G$ is the same as the set of ways to map the space into one special space $K(G, n)$, up to deformation.
So cohomology classes are maps, and computing cohomology is the same as classifying maps into these spaces.

### L2 human

This theorem turns cohomology into a homotopy-theoretic object.
A natural operation $H^n(-; G) \to H^m(-; G')$ then corresponds, by the Yoneda lemma, to a class in $H^m(K(G, n); G')$, so cohomology operations are computed from the cohomology of Eilenberg–MacLane spaces.
The spaces $K(G, n)$ form an $\Omega$-spectrum, since $\Omega K(G, n+1) \simeq K(G, n)$, and this spectrum represents ordinary cohomology in the stable setting.
</document>

<document source="https://tamnd.github.io/math2/node/GPJC1NY5RE.md" tag="GPJC1NY5RE" kind="statement">
# Continuity principle

## Statement

Let $n \ge 3$ and let $\mu$ be a positive [[Radon measure|HZNEQQBCW4]] on $\R^n$ with compact [[support|FQTXV6BQZG]] $S$.
Let $U^\mu(x) = \int \abs{x - y}^{2-n} \, d\mu(y)$ be its [[Newtonian potential|DCMFQ01QGS]].
Let $x_0 \in S$, and assume that the restriction of $U^\mu$ to $S$ is continuous at $x_0$, with $U^\mu(x_0) < \infty$.
Then $U^\mu$, as a function on all of $\R^n$, is continuous at $x_0$.

## Remarks

This result is also called the Evans–Vasilesco theorem.
Proof idea: split $\mu$ into the part $\mu_1$ on a small ball $B(x_0, \delta)$ and the rest $\mu_2$.
The potential of $\mu_2$ is continuous near $x_0$, since its kernel is continuous there.
For $x$ near $x_0$, let $y$ be a point of $S$ nearest to $x$; every $z \in S$ satisfies $\abs{y - z} \le 2 \abs{x - z}$, so $U^{\mu_1}(x) \le 2^{n-2} U^{\mu_1}(y)$.
The continuity on $S$ and $U^\mu(x_0) < \infty$ make $U^{\mu_1}$ small on $S$ near $x_0$ when $\delta$ is small, hence also off $S$, and lower semicontinuity of $U^\mu$ gives the other inequality.

Example: for the uniform measure $\sigma$ of mass $1$ on the unit sphere in $\R^3$, the potential equals $1$ on the sphere, so its restriction to the support is continuous.
The theorem then says that $U^\sigma$ is continuous on $\R^3$, and indeed $U^\sigma(x) = \min(1, 1/\abs{x})$.

Without the hypothesis on the restriction, the potential can be discontinuous at a point of finite potential.
In $\R^3$, fix a ray from $x_0$ and let $c_k$ be its point at distance $2^{-k}$ from $x_0$.
Let $\sigma_k$ be the uniform probability measure on the sphere of radius $4^{-k}$ centred at $c_k$, and let $\mu = \sum_{k \ge 2} k \, 4^{-k} \sigma_k$.
Then $U^\mu(x_0) \le \sum_k k \, 4^{-k} / (2^{-k} - 4^{-k}) < \infty$.
But $U^\mu(c_k) \ge k$, and $c_k \to x_0$.
The same statement holds for logarithmic potentials in the plane.

## Explanations

### L1 human

A potential is always lower semicontinuous, but it may jump up near points where the charge is concentrated.
The continuity principle says that such jumps can only be seen from the charge itself.
If the potential behaves continuously when one stays on the charge, it behaves continuously when one comes from any direction.

### L2 human

The principle reduces questions of continuity of potentials to the support of the measure.
It is used in the proof of the maximum principle for potentials, to pass from inequalities on the support to inequalities everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/GPK4F1K971.md" tag="GPK4F1K971" kind="definition">
# Conditioned Brownian motion

## Definition

Let $d \ge 1$ and let $D \subseteq \R^d$ be a [[domain|0Q4CBD3HX0]].
Let $p^D_t(x, y)$ be the transition density of [[Brownian motion|22735FGJV6]] in $\R^d$ killed on leaving $D$.
Let $h \colon D \to (0, \infty)$ be a [[positive harmonic function|EYMQ82Z8XJ]] on $D$.
The *h-conditioned Brownian motion* in $D$ is the [[Doob h-transform|ZN7HJNR2AC]] of killed Brownian motion by $h$.
It is the Markov process on $D$ with transition densities
$$p^h_t(x, y) = \frac{h(y)}{h(x)} \, p^D_t(x, y), \qquad t > 0, \ x, y \in D.$$

## Remarks

A positive harmonic function on $D$ is excessive for killed Brownian motion, so the h-transform is defined; the h-process may have a finite lifetime.
Example: $D = (0, \infty)$ and $h(x) = x$.
Here $h$ is positive and harmonic, and the h-process is the three-dimensional Bessel process: Brownian motion conditioned never to hit $0$.
Example: for the unit disc and $h(z) = (1 - \abs{z}^2)/\abs{\zeta - z}^2$, the [[Poisson kernel|TCYSWCTYYM]] with pole at $\zeta$ on the unit circle, the h-process is Brownian motion conditioned to exit the disc at $\zeta$.
Non-example: $h(x) = x$ on $D = \R$ is harmonic but takes negative values, so the formula does not give a family of positive kernels.
More generally, conditioning to exit at a point of the Martin boundary uses the minimal harmonic function with pole at that point.
</document>

<document source="https://tamnd.github.io/math2/node/GPQ5M0SA47.md" tag="GPQ5M0SA47" kind="definition">
# Gelfand pair

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] with a [[Haar measure|VY0HXD3GC4]], and let $K$ be a [[compact|0R6EWFKRCF]] [[subgroup|0X1JXT47GA]] of $G$.
A function $f$ on $G$ is *$K$-bi-invariant* if $f(k_1 g k_2) = f(g)$ for all $g \in G$ and $k_1, k_2 \in K$.
Let $C_c(K \backslash G / K)$ be the space of continuous, compactly supported, $K$-bi-invariant functions on $G$.
It is closed under [[convolution|2Z29JN3VES]].
The pair $(G, K)$ is a *Gelfand pair* if $f_1 * f_2 = f_2 * f_1$ for all $f_1, f_2 \in C_c(K \backslash G / K)$.

## Remarks

Example: $G = \mathrm{SL}_2(\R)$ and $K = \mathrm{SO}(2)$.
Let $\theta(g) = g^{T}$, the transpose, which reverses products and preserves Haar measure.
By the Cartan decomposition $g = k_1 a k_2$ with $a$ diagonal, so $g^{T} = k_2^{-1} a k_1^{-1}$ lies in $K g K$.
So every bi-invariant $f$ satisfies $f \circ \theta = f$, and then $f_1 * f_2 = (f_2 * f_1) \circ \theta = f_2 * f_1$.
This is Gelfand's trick.

Non-example: $G$ a finite non-abelian group and $K = \set{e}$.
Then every function is bi-invariant, and the convolution of the indicators of $\set{a}$ and $\set{b}$ is the indicator of $\set{ab}$.
So the algebra is not commutative when $ab \ne ba$.

Equivalent form: $(G, K)$ is a Gelfand pair exactly when every irreducible unitary representation of $G$ has a space of $K$-fixed vectors of dimension at most $1$.
For finite $G$, this says that the permutation representation of $G$ on functions on $G/K$ is multiplicity-free.
If $(G, K)$ is a Gelfand pair, then $G$ is unimodular.
</document>

<document source="https://tamnd.github.io/math2/node/GPQ9JNSDPA.md" tag="GPQ9JNSDPA" kind="statement">
# Fefferman's expansion of the Bergman kernel

## Statement

Let $n \ge 1$ and let $\Omega \subset \C^n$ be a bounded [[strongly pseudoconvex domain|Z7RVXBA2KB]] with $C^\infty$ boundary.
Let $\rho$ be a $C^\infty$ defining function of $\Omega$: a real [[smooth function|0V33BNKZ9S]] on a neighbourhood of $\overline{\Omega}$ with $\Omega = \set{\rho < 0}$ and $d\rho \ne 0$ on $\partial\Omega$.
Let $K(z) = K(z, z)$ be the [[Bergman kernel|8RWDHW0CXC]] of $\Omega$ on the diagonal.
Then there are functions $\varphi, \psi \in C^\infty(\overline{\Omega})$, with $\varphi > 0$ on $\partial\Omega$, such that
$$K(z) = \frac{\varphi(z)}{(-\rho(z))^{n+1}} + \psi(z) \log(-\rho(z)) \quad \text{for all } z \in \Omega.$$

## Remarks

The value of $\varphi$ on $\partial\Omega$ is an explicit expression in the first and second derivatives of $\rho$, built from the Levi form.
Hörmander had found this leading behaviour before Fefferman.
Fefferman's proof compares $\Omega$ near each boundary point with a model domain osculating it, and controls the error with estimates for the $\bar\partial$-Neumann problem.
Boutet de Monvel and Sjöstrand later gave a microlocal proof that describes the kernel off the diagonal as well.

Example: for the unit ball $B \subset \C^n$, $K(z) = \dfrac{n!}{\pi^n (1 - \abs{z}^2)^{n+1}}$.
With $\rho(z) = \abs{z}^2 - 1$ this is the expansion with $\varphi = n!/\pi^n$ and $\psi = 0$.
The logarithmic term is in general nonzero, and its coefficient $\psi$ on $\partial\Omega$ carries geometric information about the boundary.

Strong pseudoconvexity is needed for the exponent $n + 1$.
For the unit polydisc in $\C^2$, $K(z) = \pi^{-2}(1 - \abs{z_1}^2)^{-2}(1 - \abs{z_2}^2)^{-2}$.
Let $\delta$ be the distance to the boundary.
Near a point with $\abs{z_1} = 1$ and $\abs{z_2} < 1$, the kernel grows only like $\delta^{-2}$, not like $\delta^{-3}$.

## Explanations

### L1 human

The Bergman kernel measures how large a holomorphic function of unit size can be at a point.
Near the boundary of the domain it blows up.
Fefferman's theorem says exactly how it blows up on a smooth strongly pseudoconvex domain: like a power of the distance to the boundary, plus a correction with a logarithm.
The leading part is the same as for the ball, the simplest such domain.

### L2 human

The expansion is the analytic input of [[Fefferman's theorem on the extension of biholomorphisms|N95NQ0G3A6]], through the boundary behaviour of the [[Bergman metric|BB7S4NTZ6V]].
It shows that the [[Bergman projection|6J1X63Z7QZ]] and the Bergman metric of a strongly pseudoconvex domain look, near the boundary, like those of the ball.
The coefficients $\varphi$ and $\psi$ restricted to $\partial\Omega$ are local invariants of the boundary under biholomorphic maps.
This led Fefferman to a programme of computing them from the CR geometry of $\partial\Omega$.
</document>

<document source="https://tamnd.github.io/math2/node/GPQF9GK4T9.md" tag="GPQF9GK4T9" kind="concept">
# Intersection multiplicity

## Summary

Two subvarieties meet along the components of their intersection.
The intersection multiplicity along such a component is a positive integer that says how many times it should be counted.
For curves in the plane it is the dimension of a local ring, and in general it is given by Serre's Tor formula.
With these multiplicities, counts of intersection points become stable under deformation, which is what makes [[Bézout's theorem|KWNVF194D8]] and [[intersection theory|4HV4WTAX9P]] work.

## Explanations

### L1 human

When a line crosses a curve transversally, the crossing point counts once.
When the line is tangent to the curve, two crossings have merged into one point, and the point should count twice.
The intersection multiplicity makes this precise.
For the parabola $y = x^2$ and the line $y = 0$, the only common point is the origin, and its multiplicity is $2$.
Moving the line to $y = \epsilon$ with $\epsilon > 0$ splits it into the two points $x = \pm\sqrt{\epsilon}$.
</document>

<document source="https://tamnd.github.io/math2/node/GPSVJK3R4X.md" tag="GPSVJK3R4X" kind="example">
# Inoue surface

## Example

Let $M$ be a matrix in the [[special linear group|0V9PMN0RBH]] $\mathrm{SL}(3, \Z)$ with one real [[eigenvalue|0BQAA2BK3Q]] $\alpha > 1$ and two non-real eigenvalues $\beta, \bar\beta$.
Let $a = (a_1, a_2, a_3)^T \in \R^3$ be an eigenvector of $M$ for $\alpha$ and $b = (b_1, b_2, b_3)^T \in \C^3$ an eigenvector for $\beta$.
Let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]] and let $G_M$ be the group of holomorphic automorphisms of $\mathbb{H} \times \C$ generated by
$$g_0(w, z) = (\alpha w, \beta z), \qquad g_i(w, z) = (w + a_i, z + b_i) \quad (i = 1, 2, 3).$$
The quotient $S_M = (\mathbb{H} \times \C)/G_M$ is the *Inoue surface* of type $S_M$, a compact complex surface.

Check of the group structure: from $M a = \alpha a$ and $M b = \beta b$ one gets $\alpha a_i = \sum_j m_{ij} a_j$ and $\beta b_i = \sum_j m_{ij} b_j$.
Hence $g_0 g_i g_0^{-1} = g_1^{m_{i1}} g_2^{m_{i2}} g_3^{m_{i3}}$, so the translations form a normal subgroup isomorphic to $\Z^3$, and $G_M$ is a semidirect product $\Z^3 \rtimes \Z$.
The maps $g_i$ keep $\operatorname{Im} w$ fixed, because $a_i$ is real, and $g_0$ multiplies it by $\alpha$; so $G_M$ preserves $\mathbb{H} \times \C$.

A concrete $M$: the companion matrix $M = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$ of $x^3 - x - 1$ has determinant $1$.
The discriminant of $x^3 - x - 1$ is $-4(-1)^3 - 27 = -23 < 0$, so it has one real root $\alpha \approx 1.3247 > 1$ and two non-real roots.

## Remarks

The map $(w, z) \mapsto \log \operatorname{Im} w$ descends to a map $S_M \to \R / (\log\alpha)\Z$, which makes $S_M$ a bundle over a circle with fibre a real $3$-torus.
Its Betti numbers are $b_1 = 1$ and $b_2 = 0$, it contains no compact complex curves, and its Kodaira dimension is $-\infty$, so it is a [[class VII surface|Y4NSSJT4RC]].
Its universal cover is $\mathbb{H} \times \C$, not $\C^2 \setminus \set{0}$, so it is not a [[Hopf surface|PNFNGBXRM3]].
Inoue also introduced two further families, $S^{+}$ and $S^{-}$, built in a similar way from integer matrices.
A theorem of Bogomolov states that a minimal class VII surface with $b_2 = 0$ is a Hopf surface or an Inoue surface.
Li, Yau and Zheng, and later Teleman, gave complete proofs.

## Explanations

### L1 human

An Inoue surface is built from the product of the upper half-plane and the complex line, divided by a group made from an integer $3 \times 3$ matrix.
The group shifts points in three independent directions and stretches by the eigenvalues of the matrix.
The result is a compact complex surface that contains no complex curves at all.
Together with the Hopf surfaces, these surfaces are the class VII surfaces whose second Betti number is zero.
</document>

<document source="https://tamnd.github.io/math2/node/GPWYZX03N8.md" tag="GPWYZX03N8" kind="definition">
# Retract

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] and let $\mathbf{B}$ be a [[subalgebra|7CHG6SD365]] of $\mathbf{A}$ with universe $B$.
The subalgebra $\mathbf{B}$ is a *retract* of $\mathbf{A}$ when there is a [[homomorphism|KBXSZC6BV1]] $r \colon \mathbf{A} \to \mathbf{B}$ with $r(b) = b$ for all $b \in B$.
Such a map $r$ is called a *retraction*.

## Remarks

Up to isomorphism, an algebra $\mathbf{C}$ is a retract of $\mathbf{A}$ when there are homomorphisms $s \colon \mathbf{C} \to \mathbf{A}$ and $r \colon \mathbf{A} \to \mathbf{C}$ with $r \circ s = \id_C$.
Then $s$ is injective and $s(\mathbf{C})$ is a retract of $\mathbf{A}$ in the sense above.

Example: in the group $\Z \times \Z$, the subgroup $\Z \times \set{0}$ is a retract.
The map $r(m, n) = (m, 0)$ is a homomorphism onto $\Z \times \set{0}$ and fixes each $(m, 0)$.

Non-example: the subgroup $2\Z$ of $\Z$ is not a retract.
A homomorphism $r \colon \Z \to 2\Z$ is determined by $x = r(1) \in 2\Z$, and then $r(2) = 2x$.
The condition $r(2) = 2$ forces $x = 1 \notin 2\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/GQ0X3R3WM2.md" tag="GQ0X3R3WM2" kind="concept">
# Toeplitz operator

## Summary

A Toeplitz operator on the [[Hardy space|PNNNFJVF6W]] $H^2$ multiplies a function by a bounded symbol $\varphi$ on the circle and then projects back onto $H^2$.
In the basis $1, z, z^2, \ldots$ its matrix is the infinite [[Toeplitz matrix|GKY2WSHE26]] of the Fourier coefficients of $\varphi$, constant along diagonals.
Toeplitz operators link operator theory with complex analysis: their spectra, Fredholm indices and invertibility are read from the symbol.

## Explanations

### L1 human

Think of $H^2$ as the functions on the circle whose Fourier series has no negative terms.
Multiplying by a function $\varphi$ may create negative terms, so you throw them away.
What remains is the Toeplitz operator $T_\varphi$.
With $\varphi(e^{it}) = e^{it}$ nothing is thrown away and $T_\varphi$ is the shift $f \mapsto z f$.
With $\varphi(e^{it}) = e^{-it}$ the constant term is lost and one gets the backward shift.
</document>

<document source="https://tamnd.github.io/math2/node/GQ3K59MRJP.md" tag="GQ3K59MRJP" kind="concept">
# Lamé equations of elasticity

## Summary

The Lamé equations, also called the Navier–Cauchy equations, govern small deformations of a homogeneous isotropic elastic solid.
Their unknown is the displacement field $u$, which tells how far each point of the body moves.
The material enters through two constants, the Lamé parameters $\lambda$ and $\mu$.
In the static case they form an elliptic system, and the time-dependent version predicts two kinds of elastic waves, pressure waves and shear waves.

## Explanations

### L1 human

Press on a rubber block and every point inside it moves a little.
The Lamé equations say how these small displacements balance the internal stresses and the outside forces.
The parameter $\mu$ measures resistance to shearing and $\lambda$ adds resistance to change of volume.
A displacement that just moves or turns the whole body without bending it costs no stress and solves the equations with no force.
Earthquake waves are solutions of the time-dependent equations: the faster pressure waves arrive first and the shear waves after them.
</document>

<document source="https://tamnd.github.io/math2/node/GQ559QSZ2T.md" tag="GQ559QSZ2T" kind="definition">
# Fourier transform of tempered distributions

## Definition

Let $n \ge 1$, let $\mathcal{S}(\R^n)$ be the [[Schwartz space|CG9JEATW3Q]], and let $\mathcal{S}'(\R^n)$ be the space of [[tempered distributions|AES2T5JPFP]].
For $\varphi \in \mathcal{S}(\R^n)$, let
$$\mathcal{F}\varphi(\xi) = \int_{\R^n} \varphi(x)\, e^{-2\pi i x \cdot \xi}\, dx$$
be its [[Fourier transform|DPVM34ADZV]].
For $T \in \mathcal{S}'(\R^n)$, the *Fourier transform* $\mathcal{F}T$ is the tempered distribution given by
$$\langle \mathcal{F}T, \varphi \rangle = \langle T, \mathcal{F}\varphi \rangle \quad \text{for all } \varphi \in \mathcal{S}(\R^n).$$

## Remarks

The definition makes sense because $\mathcal{F}$ is a continuous linear bijection of $\mathcal{S}(\R^n)$ onto itself.
For an integrable function $f$, viewed as a tempered distribution, $\mathcal{F}f$ is the usual Fourier transform; this follows from Fubini's theorem, $\int (\mathcal{F}f)\varphi = \int f (\mathcal{F}\varphi)$.
On $\mathcal{S}'(\R^n)$, $\mathcal{F}$ is a linear bijection, and $\mathcal{F}(\partial_j T) = 2\pi i \xi_j\, \mathcal{F}T$.

Example: for the [[Dirac delta|0KPHENZ5S7]] $\delta$ at $0$, $\langle \mathcal{F}\delta, \varphi \rangle = \mathcal{F}\varphi(0) = \int \varphi(x)\, dx$.
So $\mathcal{F}\delta$ is the constant function $1$.
By the Fourier inversion formula, $\mathcal{F}1 = \delta$ in the same way.

Non-example: on $\R$, the function $e^{\abs{x}}$ is locally integrable but does not define a tempered distribution.
For the Schwartz function $\varphi(x) = 1/\cosh(x/2)$, the integral $\int e^{\abs{x}} \varphi(x)\, dx$ diverges.
So the formula above does not define a Fourier transform of $e^{\abs{x}}$ as a tempered distribution.
</document>

<document source="https://tamnd.github.io/math2/node/GQB3AS4WKQ.md" tag="GQB3AS4WKQ" kind="definition">
# Honeycomb

## Definition

Let $n \ge 3$.
A *honeycomb* in $\R^n$ is a [[tiling|GKJK293DXS]] of $\R^n$ whose tiles, called *cells*, are $n$-dimensional [[convex polytopes|300K6CB1KH]].
In dimension $3$ the cells are convex [[polyhedra|0325SGSJAR]].

## Remarks

Example: the cubes $[a, a+1] \times [b, b+1] \times [c, c+1]$ with $a, b, c \in \Z$ form the *cubic honeycomb* in $\R^3$.
The cubes are convex polyhedra, every point $(x, y, z)$ lies in the cube with $(a, b, c) = (\floor{x}, \floor{y}, \floor{z})$, and distinct cubes have disjoint interiors.

Non-example: copies of a regular tetrahedron cannot form a honeycomb in $\R^3$.
Take a generic point $p$ on an edge of one cell.
The cells through $p$ each fill either a wedge with the dihedral angle $\alpha = \arccos(1/3)$ or a half-space, and these angles add up to $2\pi$.
So $k \alpha = 2\pi$ or $k \alpha = \pi$ for some integer $k \ge 1$.
Both fail, since $\alpha \approx 70.53^\circ$ and neither $360^\circ$ nor $180^\circ$ is an integer multiple of it.

Many authors also require the honeycomb to be face-to-face, so that two cells meet in a common face of both or not at all.
The term is also used for tilings of hyperbolic space and of spheres by polytopes.
</document>

<document source="https://tamnd.github.io/math2/node/GQCN7KRSDP.md" tag="GQCN7KRSDP" kind="problem">
# Goddard problem

## Problem

A rocket rises vertically from rest, with height $h(t)$, velocity $v(t)$ and mass $m(t)$.
Its thrust $T(t)$ is the control, with $0 \le T(t) \le T_{\max}$.
The motion follows the [[ordinary differential equations|0NT64JV96A]]
$$h' = v, \qquad v' = \frac{T - D(h, v)}{m} - g(h), \qquad m' = -\frac{T}{c},$$
where $D(h, v) \ge 0$ is the aerodynamic drag, $g(h) > 0$ is the gravitational acceleration and $c > 0$ is the exhaust speed.
The initial mass $m(0) = m_0$ and the mass $m_f < m_0$ of the empty rocket are given.
Choose the thrust program $T$ and the final time $t_f$, with $m(t_f) \ge m_f$, so that the final height $h(t_f)$ is as large as possible.

## Remarks

In this optimal control problem the control enters the dynamics linearly.
So the [[Pontryagin maximum principle|7RPV2T29CF]] does not by itself fix the thrust where the switching function vanishes.
With significant drag, optimal solutions can contain a [[singular arc|91RFB97TNV]] on which the thrust takes intermediate values, between arcs of full thrust and of zero thrust.
So the optimal control is not always [[bang-bang|GK1Z3Q1WRZ]], and the problem is a standard test case for singular controls.
Goddard raised the question in his 1919 work on rockets for reaching high altitudes.

## Explanations

### L1 human

A rocket has a fixed amount of fuel and wants to climb as high as possible.
Burning everything at once wastes fuel against air drag at high speed, while burning too slowly wastes fuel just holding the rocket up against gravity.
The best program balances these two losses, and in the optimum the engine may run at partial power for a while.
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# Family-wise error rate

## Summary

When several [[hypothesis tests|087P03NEND]] are done together, the family-wise error rate is the probability of at least one [[type I error|03N8C8ZR4B]] among them.
Controlling it at level $\alpha$ protects against any false rejection in the whole family.
The [[Bonferroni correction|8GRD2TMJRR]] is the simplest procedure that controls it.

## Explanations

### L1 human

If you run $20$ independent tests at level $5\%$ and all null hypotheses are true, the chance of at least one false alarm is $1 - 0.95^{20}$, about $64\%$.
The family-wise error rate is this chance of at least one false alarm.
A multiple testing procedure that controls it keeps this chance below a chosen level, such as $5\%$.
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<document source="https://tamnd.github.io/math2/node/GQFPVXEWK1.md" tag="GQFPVXEWK1" kind="concept">
# Time–frequency analysis

## Summary

Time–frequency analysis studies functions and signals through representations that depend on time and frequency together.
The [[Fourier transform|DPVM34ADZV]] shows which frequencies occur in a signal but not when; time–frequency representations try to show both.
Its main tools are the [[short-time Fourier transform|3DHNP5NEWH]], [[Gabor frames|M9WHY8E2PM]], the [[Wigner distribution|FCHW5QGG5V]] and the [[ambiguity function|CDVA7EJSSD]], and it is close to [[wavelet|4PZ35C0GV3]] analysis.
The [[uncertainty principle|S1FB2K14NV]] limits how sharply a function can be located in time and in frequency at once.

## Explanations

### L1 human

A musical score is a time–frequency picture: time runs from left to right, and the height of each note gives its pitch.
A recording is just a pressure wave in time, and its Fourier transform is a list of pitches with no times.
Time–frequency analysis builds pictures like the score from the recording.
A common method looks at the signal through a short window, takes the Fourier transform of what it sees, and slides the window along.
The price is a trade-off: a shorter window gives better timing but blurrier pitch.
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<document source="https://tamnd.github.io/math2/node/GQG0VMXER2.md" tag="GQG0VMXER2" kind="definition">
# Algebraic dimension

## Definition

Let $X$ be a [[compact|0TVJZYB6SH]], [[connected|0V032VWTHD]] [[complex manifold|JSGWAN2RJV]].
Let $\mathcal{M}(X)$ be the set of [[meromorphic functions|W6FTSF57T1]] on $X$; with pointwise operations it is a field that contains $\C$ as the constant functions.
The *algebraic dimension* of $X$ is
$$a(X) = \operatorname{trdeg}_\C \mathcal{M}(X),$$
the [[transcendence degree|D7TR0YWR9H]] of $\mathcal{M}(X)$ over $\C$.

## Remarks

By [[Siegel's theorem|ENXCWPTG92]], $a(X)$ is an integer with $0 \le a(X) \le \dim X$.

Example: for $X = \mathbb{CP}^n$, the functions $z_1/z_0, \ldots, z_n/z_0$ are meromorphic.
They are algebraically independent over $\C$, because on the chart $\set{z_0 \ne 0} \cong \C^n$ they are the coordinate functions, and a nonzero polynomial does not vanish on all of $\C^n$.
So $a(\mathbb{CP}^n) \ge n$, and by Siegel's bound $a(\mathbb{CP}^n) = n$.

Example: the diagonal [[Hopf surface|PNFNGBXRM3]] has algebraic dimension $1$, and an [[Inoue surface|GPSVJK3R4X]] has algebraic dimension $0$.

Non-example: if $X$ is the disjoint union of two copies of $\mathbb{CP}^1$, the connectedness hypothesis fails.
Then $\mathcal{M}(X)$ is not a field: the function equal to $1$ on one copy and $0$ on the other is a nonzero zero divisor.

The notion extends to irreducible compact complex spaces.
A compact connected complex manifold $X$ with $a(X) = \dim X$ is a [[Moishezon manifold|8ZW9P9JRQD]].
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<document source="https://tamnd.github.io/math2/node/GQJCPFSYVK.md" tag="GQJCPFSYVK" kind="concept">
# Generalized quadrangle

## Summary

A generalized quadrangle is an [[incidence structure|FGKNAFNHA3]] of points and lines with no triangles.
In it, a point off a line is joined to exactly one point of that line.
It is the case $n = 4$ of a [[generalized polygon|PCT2HY9AAZ]].
Examples include the polar spaces of rank $2$ attached to quadrics and to symplectic and Hermitian forms.
They are basic building blocks in finite geometry.

## Explanations

### L1 human

In a generalized quadrangle, two points share at most one line, and there are no triangles.
Instead, given a point and a line not through it, there is exactly one way to walk from the point along a line to the given line.
A simple picture is a grid: the points of a square grid with its rows and columns as lines.
A point not on a given row lies on exactly one column, and that column meets the row in exactly one point.
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<document source="https://tamnd.github.io/math2/node/GQK4DSA11K.md" tag="GQK4DSA11K" kind="concept">
# Relation algebra

## Summary

A relation algebra is a [[Boolean algebra|0XADJKM699]] with two more operations and a constant.
They behave like [[composition|0PYSR20HGF]] and [[converse|054K3HNN5Z]] of [[binary relations|05AVARPXA2]] and like the identity relation.
Alfred Tarski gave the axioms in the 1940s, building on the calculus of relations of De Morgan, Peirce and Schröder.
The main example is the algebra of all binary relations on a set.
Relation algebras give an algebraic form of a fragment of first-order logic.
Tarski and Givant showed that set theory can be formalized in this setting.

## Explanations

### L1 human

Relations such as "is a parent of" can be combined in several ways.
Composing "parent of" with itself gives "grandparent of", and taking its converse gives "child of".
They also form a Boolean algebra: one can take "parent or sibling", "not a parent", and so on.
A relation algebra keeps the laws that these operations obey and studies them abstractly, without a set of people in the background.
</document>

<document source="https://tamnd.github.io/math2/node/GQM8GPBRYN.md" tag="GQM8GPBRYN" kind="definition">
# General linear model

## Definition

Let $n, p, m \ge 1$.
A *general linear model* is a statistical model for a random $n \times m$ response matrix $Y$ of the form
$$Y = X B + U,$$
where:

1. $X$ is a known real $n \times p$ matrix, the *design matrix*;
2. $B$ is an unknown real $p \times m$ matrix of coefficients;
3. $U$ is a random $n \times m$ error matrix whose rows are independent with [[expected value|0WQNRCNY3M]] $0$ and a common [[covariance matrix|0V7G6WMFK0]] $\Sigma$.

Often the rows of $U$ are assumed to follow the [[multivariate normal distribution|0XN3K7RSZY]] $N_m(0, \Sigma)$.

## Remarks

With $m = 1$, the model is $y = X\beta + \varepsilon$, multiple [[linear regression|41FVC4P5JB]], and $\Sigma = \sigma^2$ is a single variance.

Example: the comparison of two group means.
Let observations $1, \dots, n_1$ belong to group $1$ and the others to group $2$, and let $X$ have two columns: the indicator of group $1$ and the indicator of group $2$.
Then $\beta = (\mu_1, \mu_2)^T$, and the model says that $y_i$ has mean $\mu_1$ or $\mu_2$ according to its group, with a common error variance.
This is the model of the two-sample $t$-test.

Non-example: the model $E[y_i] = \exp(x_i^T \beta)$ is not a general linear model.
Its mean is not a linear function of $\beta$, so it cannot be written as the $i$-th entry of $X B$.

When $X$ has rank $p$, the least squares estimate is $\hat B = (X^T X)^{-1} X^T Y$, computed column by column.
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<document source="https://tamnd.github.io/math2/node/GQVAXW5X09.md" tag="GQVAXW5X09" kind="concept">
# Differential graded algebra

## Summary

A differential graded algebra, or dg algebra, is a [[graded algebra|8KB4SQGC0C]] with a differential $d$ of degree one.
The differential squares to zero and obeys a signed Leibniz rule.
Forgetting the product it is a [[chain complex|WQ82C0M3QB]], and its cohomology is again a graded algebra.
The de Rham complex of a smooth manifold, the singular cochains of a space with the cup product, and Koszul complexes are standard examples.
Dg algebras are the basic objects of homological algebra with products and of rational homotopy theory.

## Explanations

### L1 human

Differential forms on a manifold can be multiplied by the wedge product and differentiated by $d$.
Applying $d$ twice always gives zero, and $d$ of a product follows a product rule with a sign.
A differential graded algebra is any algebra with these two features.
A small example is polynomials $f(x)$ in degree $0$ together with the forms $g(x)\,dx$ in degree $1$, with $d f = f'(x)\, dx$.
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<document source="https://tamnd.github.io/math2/node/GR1DDT1B6B.md" tag="GR1DDT1B6B" kind="statement">
# Circle packing theorem

## Statement

Let $G$ be a finite simple [[planar graph|0NC9R64TXP]].

1. There are closed discs $D_v$ in the plane, one for each vertex $v$ of $G$, with pairwise disjoint interiors. For distinct vertices $u$ and $v$, the discs $D_u$ and $D_v$ are tangent if and only if $u$ and $v$ are adjacent.
2. Let $G$ be maximal planar with at least three vertices. Then such a family of discs is unique up to [[Möbius transformations|0SG74WBFA2]] and reflections in lines.

## Remarks

The theorem is also called the Koebe–Andreev–Thurston theorem.
One proof first adds vertices to make $G$ a triangulation.
For a triangulation, the angles of the triangles of centres at each interior vertex must sum to $2\pi$.
This is a system of equations in the radii, and an iterative or variational argument shows that it has a unique normalized solution.
Example: the complete graph $K_4$ is realized by three mutually tangent equal discs together with a fourth small disc in the gap between them, tangent to all three.
The simplicity hypothesis is needed.
Two discs with disjoint interiors touch in at most one point, so a double edge cannot be shown by tangency, and a loop has no meaning.
Conversely, the tangency graph of such a family of discs is always planar, so the theorem characterizes planar graphs.

## Explanations

### L1 human

Draw a planar graph, that is, a network of dots and lines without crossings.
The theorem says that you can replace the dots by coins lying flat on a table so that two coins touch exactly when the dots are joined.
For a triangulation, the picture is essentially unique.
This gives a rigid, geometric way to draw any planar graph.

### L2 human

Thurston suggested that circle packings of fine hexagonal triangulations approximate the [[Riemann mapping|02WHMEG5ZR]] of a domain, and Rodin and Sullivan proved this convergence.
So circle packings give a discrete version of [[conformal maps|0YYT36R4YD]], the subject of discrete conformal geometry.
The theorem also gives a geometric proof of planar separator theorems and of Fáry's theorem that planar graphs can be drawn with straight edges, by joining centres of tangent discs.
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<document source="https://tamnd.github.io/math2/node/GR49AM3PSZ.md" tag="GR49AM3PSZ" kind="concept">
# Interacting particle system

## Summary

An interacting particle system is a continuous-time Markov process on configurations of a lattice or graph, in which each site carries a state.
The state at a site changes at rates that depend on the states of nearby sites.
Each local change is simple, but infinitely many sites interact, and the system can show phase transitions.
Main examples are the [[contact process|9RVKQJQ5JJ]], the [[voter model|YBAE5QCH1A]], the [[exclusion process|TXC6PZT8CF]] and stochastic dynamics of the [[Ising model|R0KGM32QN6]].
They model epidemics, opinions, traffic and magnets.

## Explanations

### L1 human

Put a person at each point of an infinite grid, each healthy or sick.
A sick person recovers after a random time, and a healthy person gets sick at a rate that grows with the number of sick neighbours.
Each person follows simple local rules, yet the whole grid may behave in very different ways: the illness may die out or spread forever.
Interacting particle systems are the models of this kind.
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<document source="https://tamnd.github.io/math2/node/GR4FN6XNWP.md" tag="GR4FN6XNWP" kind="definition">
# Probit model

## Definition

Let $x_1, \dots, x_n \in \R^p$ be known regressor vectors, and let $\Phi$ be the [[cumulative distribution function|0H8H7ZTHW2]] of the standard [[normal distribution|05AQE0QR0D]].
The *probit model* for responses $y_1, \dots, y_n \in \set{0, 1}$ states three things.
The $y_i$ are independent, $y_i$ has the [[Bernoulli distribution|0386Y0Z7Y7]] with parameter $p_i$, and
$$p_i = \Phi(x_i^T \beta)$$
for an unknown $\beta \in \R^p$.

## Remarks

Equivalent latent variable form: let $y_i^* = x_i^T \beta + \varepsilon_i$ with independent $\varepsilon_i \sim N(0, 1)$, and $y_i = 1$ if $y_i^* > 0$ and $y_i = 0$ otherwise.
Then $P(y_i = 1) = P(\varepsilon_i > -x_i^T \beta) = \Phi(x_i^T \beta)$, by the symmetry of the normal distribution.
The error variance is fixed at $1$ because $\beta$ and the scale of $\varepsilon_i$ cannot be identified separately from binary data.

It is the [[generalized linear model|K6C48KMRKC]] with Bernoulli responses and link $g = \Phi^{-1}$, which is not the canonical link.

Example: with $x_i = (1, t_i)$ and $\beta = (-1, 0.5)$, at $t = 2$ one gets $p = \Phi(0) = 1/2$.

Non-example: the model $p_i = 1/(1 + e^{-x_i^T \beta})$ uses the logistic function instead of $\Phi$; it is [[logistic regression|JN1PJ5AJJS]], not the probit model.
Its latent errors are logistic, with variance $\pi^2/3$, so its coefficients are typically larger than probit coefficients by a factor of roughly $1.6$ to $1.8$.
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<document source="https://tamnd.github.io/math2/node/GRBZT7SWK0.md" tag="GRBZT7SWK0" kind="definition">
# Finite element exterior calculus

## Definition

Let $\Omega \subset \R^n$ be a bounded Lipschitz polyhedral domain.
For $0 \le k \le n$, let $H\Lambda^k(\Omega)$ be the [[differential $k$-forms|0E3G0GRPZ6]] $\omega$ with $\omega \in L^2$ and $d\omega \in L^2$.
Here $d$ is the [[exterior derivative|0ZHSTAG8ZY]] in the weak sense.
These [[Hilbert spaces|HM2HHE3J51]] form the $L^2$ de Rham complex $H\Lambda^0 \xrightarrow{d} H\Lambda^1 \xrightarrow{d} \cdots \xrightarrow{d} H\Lambda^n$.
*Finite element exterior calculus* discretizes this complex by a family of spaces $V_h^k \subseteq H\Lambda^k(\Omega)$, indexed by a mesh parameter $h$, with three properties:

1. each $V_h^k$ is a [[finite element|98GZ9DVHTA]] space of piecewise polynomial $k$-forms on a simplicial mesh of $\Omega$;
2. subcomplex: $d V_h^k \subseteq V_h^{k+1}$ for each $k$;
3. bounded cochain projections: there are projections $\pi_h^k$ onto $V_h^k$ with $d \pi_h^k = \pi_h^{k+1} d$.

The projections in clause 3 are required to be bounded in $L^2$ uniformly in $h$.
The method is the [[mixed|87VMQ892VW]] Galerkin method for the [[Hodge Laplacian|PWG3GKZG80]] on these spaces.

## Remarks

The standard families are $\mathcal{P}_r \Lambda^k$ and $\mathcal{P}_r^- \Lambda^k$ of polynomial $k$-forms.
Example: in $\R^3$, take continuous piecewise linear functions, lowest-order Nédélec edge elements, lowest-order Raviart–Thomas face elements, and piecewise constants.
The gradient of a piecewise linear function is piecewise constant with continuous tangential part, so it lies in the Nédélec space.
The other inclusions hold for the same reason, and the canonical interpolation operators commute with $d$.
Under clauses 2 and 3 the discrete complex has the same [[cohomology|GX74C07T0G]] as the continuous one, and discrete Poincaré inequalities hold uniformly in $h$.

Non-example: take continuous piecewise quadratics for $V_h^0$ and lowest-order Nédélec edge elements for $V_h^1$.
The lowest-order Nédélec fields on a tetrahedron have the form $a + b \times x$.
The gradient of a quadratic such as $x_1^2$ is $(2 x_1, 0, 0)$, which is not of this form, so clause 2 fails.
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<document source="https://tamnd.github.io/math2/node/GRC1E284T7.md" tag="GRC1E284T7" kind="definition">
# Spectrum of the Laplace–Beltrami operator

## Definition

Let $(M, g)$ be a closed connected [[Riemannian manifold|7CBT4SM83Y]] and let $\Delta = -\operatorname{div} \operatorname{grad}$ be its [[Laplace–Beltrami operator|F4NWZNTRMD]].
A real number $\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $\Delta$ if $\Delta \phi = \lambda \phi$ for some smooth function $\phi \ne 0$ on $M$.
The set of eigenvalues is discrete, each eigenspace is finite-dimensional, and all eigenvalues are $\ge 0$.
The *spectrum* of $M$ is the sequence
$$0 = \lambda_0 < \lambda_1 \le \lambda_2 \le \cdots$$
of eigenvalues of $\Delta$, each repeated as many times as the dimension of its eigenspace.

## Remarks

The properties follow from the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]] applied to the inverse of $\Delta + 1$, which is compact on $L^2(M)$ by elliptic regularity and Rellich's theorem.
So $\lambda_j \to \infty$, and the normalized eigenfunctions form an orthonormal basis of $L^2(M)$.
The eigenvalue $0$ has multiplicity $1$ because $M$ is connected: $\int_M \phi \Delta \phi \, dv = \int_M \abs{\nabla \phi}^2 dv$ vanishes only for constant $\phi$.
Some authors use the opposite sign $\Delta = \operatorname{div} \operatorname{grad}$, with eigenvalues $\le 0$.

Example: for the round unit sphere $S^2$, the eigenvalues are $k(k+1)$, $k = 0, 1, 2, \ldots$, with multiplicity $2k + 1$, the eigenfunctions being the spherical harmonics of degree $k$.
So the spectrum begins $0, 2, 2, 2, 6, 6, 6, 6, 6, \ldots$.

Non-example: $\R$ is not compact, and the definition fails there.
The equation $-\phi'' = \lambda \phi$ has bounded solutions $e^{\pm i\sqrt{\lambda} x}$ for every $\lambda \ge 0$, but none of them is in $L^2(\R)$.
The spectrum of the Laplacian on $L^2(\R)$ is the continuous set $[0, \infty)$, not a sequence of eigenvalues.
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# Affine group

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $P$ be an [[affine space|09XNAPTJQ7]] over $V$.
The *affine group* $\operatorname{Aff}(P)$ is the set of all bijective [[affine maps|0JF0TVYGWT]] $P \to P$, with composition as the group operation.
For $P = V$ it is written $\operatorname{Aff}(V)$, and for $V = F^n$ it is written $\operatorname{AGL}(n, F)$.

## Remarks

Example: on $\R^2$, the map $f(x) = Ax + b$ with $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $b = (1, 0)$ is in $\operatorname{AGL}(2, \R)$, since $\det A = 1 \ne 0$; its inverse is $x \mapsto A^{-1}(x - b)$.
Non-example: $f(x) = Ax + b$ with $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ is affine but not bijective, so it is not in the affine group.
Each element of $\operatorname{AGL}(n, F)$ has the form $x \mapsto Ax + b$ with $A \in \operatorname{GL}(n, F)$ and $b \in F^n$, in a unique way.
The map to the linear part $A$ is a surjective homomorphism onto the [[general linear group|0FCHE902D7]], whose kernel is the group of [[translations|087VJM5Q77]], isomorphic to $F^n$.
Hence $\operatorname{AGL}(n, F) \cong F^n \rtimes \operatorname{GL}(n, F)$, a [[semidirect product|0H3WZXCZNF]].
It is isomorphic to the group of $(n+1) \times (n+1)$ matrices $\begin{pmatrix} A & b \\ 0 & 1 \end{pmatrix}$ with $A \in \operatorname{GL}(n, F)$.
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# Input-to-state stability

## Summary

Input-to-state stability is a stability notion for systems driven by an input, $\dot x = f(x, u)$.
It asks that the state decays to a size controlled by the size of the input, whatever the initial state.
It merges global asymptotic stability of the unforced system, a strong form of [[Lyapunov stability|0F2SYCKDT2]], with a bounded-input bounded-state property.
It is a basic tool in nonlinear [[control systems|ZPDTVZQN5K]], because it behaves well under cascades and feedback interconnections.

## Explanations

### L1 human

Think of a ball in a bowl that is shaken by an outside force.
Without shaking, the ball rolls back to the bottom.
With shaking, the ball stays close to the bottom, and the distance depends on how hard the shaking is.
For the system $\dot x = -x + u$, the state satisfies $\abs{x(t)} \le e^{-t} \abs{x(0)} + \sup \abs{u}$.
So the effect of the start dies out, and what remains is bounded by the input.
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<document source="https://tamnd.github.io/math2/node/GRG8WEBSH5.md" tag="GRG8WEBSH5" kind="concept">
# Full and faithful functors

## Summary

A [[functor|28NQ0KRHD3]] is faithful when it is injective on each set of morphisms between two fixed objects.
It is full when it is surjective on each such set, and fully faithful when it is both.
A faithful functor loses no information about morphisms, and a fully faithful one identifies the morphisms of its source with those between the images.
Forgetful functors such as groups to sets are faithful but not full.
Fully faithful functors are the embeddings of category theory, and the Yoneda embedding is the main example.
Together with [[essential surjectivity|6ERBSBVF14]] they characterize [[equivalences of categories|K7A04ZBD92]].

## Explanations

### L1 human

Look at all the arrows between two objects, and at all the arrows between their images.
A functor gives a map from the first collection to the second.
It is faithful if different arrows stay different, and full if every arrow between the images comes from some arrow.
Forgetting that a group is a group is faithful, since a homomorphism is determined by its underlying function.
It is not full, because most functions between two groups are not homomorphisms.
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<document source="https://tamnd.github.io/math2/node/GRH8MCVN3N.md" tag="GRH8MCVN3N" kind="concept">
# Residual set

## Summary

A residual set, or comeagre set, in a [[topological space|0CQHFPGJ45]] is a set whose complement is [[meagre|0XGCZ4GDZ6]].
Equivalently, it contains a countable intersection of [[dense|0BSJSX7ME2]] open sets.
In a [[Baire space|0ZH5AJ2ED7]], residual sets are dense, so they are considered "large" sets.
A property is called generic when it holds on a residual set.

## Explanations

### L1 human

A meagre set is a small set, built from countably many thin pieces.
A residual set is what remains after removing such a small set.
For example, the irrational numbers form a residual subset of the real line.
The [[Baire category theorem|0A7KGAYHGV]] says that in a nonempty complete metric space, residual sets are never empty and are in fact dense.
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<document source="https://tamnd.github.io/math2/node/GRMT65QJAY.md" tag="GRMT65QJAY" kind="definition">
# Atomic lattice

## Definition

Let $L$ be a [[lattice|WH4SFAACAS]] with a [[least element|0AANKKNHS4]] $0$.
$L$ is *atomic* if for every $b \in L$ with $b \ne 0$ there is an [[atom|TTVCTW6TNC]] $a$ of $L$ with $a \le b$.

## Remarks

The same definition applies to any partially ordered set with a least element.
Example: every finite lattice $L$ is atomic.
For $b \ne 0$, the set $\set{x \in L : 0 < x \le b}$ is finite and nonempty, so it has a minimal element, and a minimal element of this set is an atom.
Example: $\N_{>0}$ ordered by divisibility is atomic, since every $n > 1$ has a prime divisor.
Non-example: the chain $[0, 1]$ with its usual order is a lattice with least element $0$ and no atoms, so the condition fails for $b = 1$.
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<document source="https://tamnd.github.io/math2/node/GRQJ49XQNR.md" tag="GRQJ49XQNR" kind="concept">
# Szász–Mirakyan operator

## Summary

The Szász–Mirakyan operators approximate functions on the half-line $[0, \infty)$ by series built from the [[Poisson distribution|0XB9GQTN9G]].
They play for $[0, \infty)$ the role that [[Bernstein polynomials|0JT92VK307]] play for $[0, 1]$.
They are [[positive linear operators|AQDXWH4812]], so their convergence follows from Korovkin's theorem.

## Explanations

### L1 human

To approximate $f$ near a point $x \ge 0$, sample $f$ at the grid points $k/n$.
Weight the sample at $k/n$ by the probability that a Poisson random variable with mean $nx$ equals $k$.
The weights concentrate near $k/n \approx x$ as $n$ grows, so the weighted average approaches $f(x)$.
For $f(t) = t$ the average is exactly $x$, because the Poisson variable has mean $nx$.
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<document source="https://tamnd.github.io/math2/node/GRQM6KXA00.md" tag="GRQM6KXA00" kind="definition">
# (∞,1)-category

## Definition

An *$(\infty,1)$-category* is a higher category with $k$-morphisms for every $k \ge 1$, in which every $k$-morphism with $k \ge 2$ is invertible.
This informal notion is made precise by a choice of model; the most used one is the following.
In the quasi-categorical model, an $(\infty,1)$-category is a [[quasi-category|9328WEVJEK]]: a [[simplicial set|WW81ARB8K9]] in which every inner horn $\Lambda^n_k \to X$, $0 < k < n$, extends to the $n$-simplex $\Delta^n$.

## Remarks

Other standard models are:

1. simplicial categories, that is, categories [[enriched|BPH9SJ30Y0]] over simplicial sets, whose hom-objects are all [[Kan complexes|FDFYCM3YYC]];
2. complete Segal spaces, which are certain simplicial objects in simplicial sets;
3. Segal categories.

Each model carries a [[model category|SDTZVZ6B4Q]] structure, and these model categories are connected by Quillen equivalences, so the models give the same homotopy theory of $(\infty,1)$-categories.
Example: the nerve of an ordinary category is a quasi-category, so every category is an $(\infty,1)$-category with only identity higher morphisms.
Example: Kan complexes are the $(\infty,1)$-categories in which every morphism is invertible, the $\infty$-groupoids.
Non-example: $\mathbf{Cat}$, viewed as a strict $2$-category, is not an $(\infty,1)$-category in its own right.
It has non-invertible natural transformations as $2$-morphisms, while the $2$-morphisms of an $(\infty,1)$-category must be invertible.
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# Bessel function of the second kind

## Summary

The Bessel function of the second kind $Y_\nu$ is a second solution of Bessel's equation, independent of the [[Bessel function of the first kind|CZCMRDWDYW]] $J_\nu$.
It is also called the Weber function or the Neumann function, and it is sometimes written $N_\nu$.
Unlike $J_\nu$, it is unbounded near $0$.
Together, $J_\nu$ and $Y_\nu$ span all solutions of [[Bessel's equation|EAV97QH9DF]].

## Explanations

### L1 human

Bessel's equation is of second order, so it needs two independent solutions.
The first kind $J_\nu$ is the one that stays finite at the origin.
The second kind $Y_\nu$ is the other one, and it blows up at the origin.
For example, $Y_0(x)$ behaves like $\frac{2}{\pi} \ln x$ for small $x > 0$.
For large $x$, both $J_\nu$ and $Y_\nu$ oscillate like shifted sine waves of slowly shrinking size.
</document>

<document source="https://tamnd.github.io/math2/node/GRWE9JJ5CT.md" tag="GRWE9JJ5CT" kind="concept">
# Morita invariant

## Summary

A Morita invariant is a property of [[rings|0B1261B5ED]], or an object attached to rings, that is the same for any two [[Morita equivalent|RD5WKA1VK7]] rings.
Properties defined through the category of modules are Morita invariant, such as being semisimple, left Artinian or left Noetherian.
Properties defined through elements, such as commutativity or having no zero divisors, are usually not.
Morita invariants let one replace a ring by a simpler equivalent one, such as $M_n(R)$ by $R$.

## Explanations

### L1 human

Two rings are Morita equivalent when their modules behave the same way.
A Morita invariant is a feature that cannot tell such rings apart.
For example, $\R$ and the ring of $2 \times 2$ real matrices are Morita equivalent, and both are semisimple, so semisimplicity is a candidate invariant, and indeed it is one.
But $\R$ is commutative and the matrix ring is not, so commutativity is not a Morita invariant.
</document>

<document source="https://tamnd.github.io/math2/node/GRZAY7GWC5.md" tag="GRZAY7GWC5" kind="definition">
# Weyl–Titchmarsh theory

## Definition

Let $q \colon [0, \infty) \to \R$ be integrable on each interval $[0, R]$ with $R > 0$.
Assume that the [[Schrödinger operator|89AA6XJY5K]] $-y'' + q y$ is in the [[limit-point case|6J9FXT6HAV]] at $\infty$.
For $z \in \C \setminus \R$, let $\psi(z, \cdot)$ be a nonzero solution of $-y'' + q y = z y$ with $\int_0^\infty \abs{\psi(z, x)}^2 \, dx < \infty$; by the limit-point assumption it is unique up to a constant factor.
The Weyl–Titchmarsh function, or $m$-function, of $q$ is
$$m(z) = \frac{\psi'(z, 0)}{\psi(z, 0)}, \qquad z \in \C \setminus \R.$$
Weyl–Titchmarsh theory is the study of the spectral theory of such operators through the function $m$.

## Remarks

The quotient does not depend on the choice of $\psi$.
Also $\psi(z, 0) \ne 0$, since otherwise the non-real number $z$ would be an eigenvalue of the self-adjoint operator with the Dirichlet condition $y(0) = 0$.
The function $m$ is analytic on $\C \setminus \R$, satisfies $m(\overline{z}) = \overline{m(z)}$, and maps the [[upper half-plane|X6XGGH5FPW]] into itself.
So $m$ is a Herglotz function, and the measure in its [[Herglotz representation|9ZBAX4ZPX0]] is the spectral measure of the operator with the condition $y(0) = 0$.
Example: for $q = 0$, the solution $\psi(z, x) = e^{i\sqrt{z}\, x}$, with the branch of $\sqrt{z}$ that has positive imaginary part, is square integrable on $[0, \infty)$.
So $m(z) = i \sqrt{z}$; for $z = i$ this is $e^{3\pi i/4}$, which lies in the upper half-plane.
Non-example: for $z \in \C \setminus \R$ the solution $e^{-i\sqrt{z}\, x}$ grows exponentially, so it cannot be used for $\psi$.
Some books use the other sign convention, or the boundary condition $y'(0) = h y(0)$ at $0$, which changes $m$ by a Möbius transformation.

## Explanations

### L1 human

The $m$-function compares the slope and the value at $0$ of the one solution that stays square integrable at infinity.
It is a single analytic function, and its boundary behaviour on the real line encodes the spectrum of the operator.
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<document source="https://tamnd.github.io/math2/node/GRZKGHMNTG.md" tag="GRZKGHMNTG" kind="definition">
# Polygamma function

## Definition

Let $\psi$ be the [[digamma function|HWRRXB0YXS]] and let $m \ge 0$ be an integer.
The *polygamma function of order $m$* is the meromorphic function
$$\psi^{(m)}(z) = \frac{d^m}{dz^m} \psi(z) = \frac{d^{m+1}}{dz^{m+1}} \log \Gamma(z), \qquad z \in \C \setminus \set{0, -1, -2, \ldots}.$$

## Remarks

The series $\psi(z) = -\gamma + \sum_{k=0}^\infty \left(\frac{1}{k+1} - \frac{1}{z+k}\right)$, with $\gamma$ the Euler–Mascheroni constant, can be differentiated term by term.
For $m \ge 1$ this gives
$$\psi^{(m)}(z) = (-1)^{m+1}\, m! \sum_{k=0}^\infty \frac{1}{(z+k)^{m+1}} = (-1)^{m+1}\, m!\, \zeta(m+1, z),$$
where $\zeta(s, z)$ is the [[Hurwitz zeta function|G4SW7HJG88]].

Example: for $m = 2$ and $z = 1$ the series gives $\psi^{(2)}(1) = -2 \sum_{k \ge 1} k^{-3} = -2\zeta(3)$.

Non-example of a valid argument: at $z = -n$ with $n \ge 0$ an integer the term $k = n$ of the series is infinite.
There $\psi^{(m)}$ has a pole of order $m + 1$.

The function $\psi^{(0)} = \psi$ is the digamma function and $\psi^{(1)}$ is the [[trigamma function|5VGVPENA0F]].
The recurrence $\psi^{(m)}(z+1) = \psi^{(m)}(z) + (-1)^m\, m!\, z^{-m-1}$ follows by differentiating $\psi(z+1) = \psi(z) + 1/z$ $m$ times.
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<document source="https://tamnd.github.io/math2/node/GRZKVSMJXF.md" tag="GRZKVSMJXF" kind="definition">
# Initial topology

## Definition

Let $X$ be a set, let $I$ be a set, and for each $i \in I$ let $Y_i$ be a [[topological space|0CQHFPGJ45]] and $f_i \colon X \to Y_i$ a map.
The *initial topology* on $X$ for the family $(f_i)_{i \in I}$ is the [[coarsest|072J68M6HX]] topology on $X$ for which every $f_i$ is [[continuous|0VGFHTM75Y]].
Equivalently, it is the topology generated by the [[subbasis|09S8DV34Y0]]
$$\set{f_i^{-1}(V) : i \in I, \ V \text{ open in } Y_i}.$$

## Remarks

The [[preimages|01AHG4G9RT]] $f_i^{-1}(V)$ make each $f_i$ continuous, and any topology that makes each $f_i$ continuous must contain them; this gives the equivalence of the two forms.
Universal property: for a topological space $Z$, a map $g \colon Z \to X$ is continuous for the initial topology if and only if $f_i \circ g$ is continuous for every $i$.

Example: for a subset $X$ of a space $Y$ and the inclusion $\iota \colon X \to Y$, the initial topology is the [[subspace topology|0WS9P12FQR]].
Its open sets are the sets $\iota^{-1}(V) = X \cap V$ with $V$ open in $Y$.
Example: for a product $X = \prod_i Y_i$ and the projections, the initial topology is the [[product topology|0GBG8B49AA]].

Non-example: the discrete topology on $\R$ makes the identity map to $\R$ with its usual topology continuous, but it is not the initial topology for this map.
It is strictly finer than needed: the initial topology here is the usual topology, in which $\set{0}$ is not open.

The initial topology is also called the weak topology or the induced topology for the family.
</document>

<document source="https://tamnd.github.io/math2/node/GS3KT2H645.md" tag="GS3KT2H645" kind="concept">
# Neumann boundary condition

## Summary

A Neumann boundary condition prescribes the derivative of the unknown function in the direction normal to the boundary.
For an ordinary differential equation on $[a, b]$, it fixes $y'(a)$ and $y'(b)$.
For a partial differential equation on a domain, it fixes the outward normal derivative $\frac{\partial u}{\partial \nu}$ on the boundary, which is a flux.
It is one of the three classical conditions in a [[boundary value problem|TM8VQJWGHZ]], with the [[Dirichlet|EETVA6QYX4]] and Robin conditions.
See the [[definition|REH63MEBNC]].

## Explanations

### L1 human

A Neumann condition says how steeply the solution leaves the boundary, not where it is.
For heat flow, it prescribes the heat flux through the boundary.
A perfectly insulated wall lets no heat through, which is the condition $\frac{\partial u}{\partial \nu} = 0$.
For $y'' = 0$ on $[0, 1]$ with $y'(0) = y'(1) = 0$, every constant is a solution, so the solution is not unique.
</document>

<document source="https://tamnd.github.io/math2/node/GS4RC7A7DJ.md" tag="GS4RC7A7DJ" kind="statement">
# Weyl's criterion

## Statement

Let $(x_n)_{n \ge 1}$ be a sequence of real numbers.
Then $(x_n)$ is [[equidistributed modulo one|RT62ZAN3JB]] if and only if, for every nonzero integer $h$,
$$\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} e^{2 \pi i h x_n} = 0.$$
Here $e^{2\pi i t}$ is the [[complex exponential|07GJV36896]].

## Remarks

Equidistribution modulo one is equivalent to $\frac{1}{N} \sum_{n \le N} f(x_n) \to \int_0^1 f(t) \, dt$ for every continuous function $f \colon \R \to \C$ of period $1$.
The functions $t \mapsto e^{2\pi i h t}$ are such functions, with integral $0$ for $h \ne 0$, which gives one direction.
For the other direction, the condition gives the limit for every trigonometric polynomial.
Uniform approximation by the Weierstrass theorem extends it to every continuous periodic $f$.
Finally, squeezing the indicator of an interval between continuous functions gives the counting condition.
Example: for irrational $\alpha$ and $x_n = n \alpha$, the sum is a geometric sum bounded by $2 / \abs{1 - e^{2\pi i h \alpha}}$, so the averages tend to $0$; this proves the [[equidistribution theorem|NCCGNTBEYV]].
For a rational $\alpha = p/q$ the test fails at $h = q$, where every term equals $1$.
The same criterion holds in $\R^k$, with $h$ running over nonzero vectors of $\Z^k$ and $h x_n$ replaced by the dot product $h \cdot x_n$.

## Explanations

### L1 human

To check that a sequence spreads evenly around a circle, it is enough to test it against waves.
For each frequency $h$, average the points $e^{2\pi i h x_n}$ on the unit circle.
The sequence is evenly spread exactly when all these averages tend to zero.

### L2 human

The criterion turns a question about counting points in intervals into estimates of [[exponential sums|97PBMQCPPM]].
It gives a short proof that $(n\alpha)$ is equidistributed for irrational $\alpha$.
With Weyl differencing, it also shows that $p(n)$ is equidistributed for a real polynomial $p$ with an irrational coefficient other than the constant term.
Its proof is a first example of approximating indicator functions by trigonometric polynomials, a basic method of Fourier analysis.
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<document source="https://tamnd.github.io/math2/node/GS8SCGYPX7.md" tag="GS8SCGYPX7" kind="statement">
# Adem relations

## Statement

Let $p$ be a prime and consider operations on $H^*(X; \F_p)$ for spaces $X$.
All [[binomial coefficients|092GEF38YD]] below are read modulo $p$, and $\binom{m}{k} = 0$ when $k < 0$ or $k > m$.

1. For $p = 2$, the [[Steenrod squares|VR7RHP41M0]] satisfy, for all integers $a, b > 0$ with $a < 2b$,
$$Sq^a Sq^b = \sum_{j=0}^{\floor{a/2}} \binom{b - 1 - j}{a - 2j} Sq^{a + b - j} Sq^j.$$
2. For $p$ odd, the [[Steenrod reduced powers|EZYHSE1E8P]] satisfy, for all integers $a, b > 0$ with $a < pb$,
$$P^a P^b = \sum_{j=0}^{\floor{a/p}} (-1)^{a + j} \binom{(p-1)(b - j) - 1}{a - pj} P^{a + b - j} P^j.$$
3. For $p$ odd, with $\beta$ the [[Bockstein homomorphism|H7PCM776GR]] of $0 \to \Z/p \to \Z/p^2 \to \Z/p \to 0$, for all integers $a, b > 0$ with $a \le pb$,
$$P^a \beta P^b = \sum_{j=0}^{\floor{a/p}} (-1)^{a + j} \binom{(p-1)(b - j)}{a - pj} \beta P^{a + b - j} P^j + \sum_{j=0}^{\floor{(a-1)/p}} (-1)^{a + j + 1} \binom{(p-1)(b - j) - 1}{a - pj - 1} P^{a + b - j} \beta P^j.$$

## Remarks

One proof computes the operations on the cohomology of a product of Eilenberg–MacLane spaces, or of symmetric products, where all compositions are visible.
Another proof packages the operations into a generating function and uses the Cartan formula.

Example for $p = 2$: taking $a = b = 1$ gives $Sq^1 Sq^1 = \binom{0}{1} Sq^2 = 0$.
Taking $a = 1, b = 2$ gives $Sq^1 Sq^2 = Sq^3$.
Taking $a = b = 2$ gives $Sq^2 Sq^2 = Sq^3 Sq^1$.

The condition $a < 2b$ is needed: when $a \ge 2b$ the composite $Sq^a Sq^b$ is already admissible and is not rewritten.
The relations, together with $Sq^0 = 1$ (and $P^0 = 1$, $\beta^2 = 0$ for odd $p$), give all relations among composites of these operations.
So the mod $p$ Steenrod algebra is the free associative algebra on the operations modulo these relations.

## Explanations

### L1 human

Steenrod operations can be composed, and some composites are equal to sums of other composites.
The Adem relations list exactly which such equalities hold.
They let one rewrite any composite as a sum of composites of a standard form, called admissible.

### L2 human

The Adem relations make the Steenrod algebra computable: the admissible monomials form a basis, which is the Cartan–Serre basis.
They show that $Sq^n$ is decomposable unless $n$ is a power of $2$, so the algebra is generated by the $Sq^{2^k}$.
This fact is the starting point of the Hopf invariant one problem and of the Adams spectral sequence.
</document>

<document source="https://tamnd.github.io/math2/node/GSFXAV0VD7.md" tag="GSFXAV0VD7" kind="concept">
# Universally measurable set

## Summary

A subset of a measurable space is universally measurable if it is measurable for the completion of every probability measure on the space.
The universally measurable sets form a σ-algebra that contains all the original measurable sets.
In Polish spaces they include the analytic sets, which may fail to be Borel.

## Explanations

### L1 human

A set may fail to be in the original σ-algebra and still be harmless for every measure one could put on the space.
Universally measurable sets are those sets: whatever probability measure one chooses, the set differs from a measurable set by a negligible piece.
Every Borel subset of the real line is universally measurable.
A Vitali set is not, because it is not measurable for Lebesgue measure on $[0, 1]$.
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<document source="https://tamnd.github.io/math2/node/GSMBXT7FPK.md" tag="GSMBXT7FPK" kind="concept">
# Kernel of an integral equation

## Summary

The kernel of a linear [[integral equation|R3SG1SN7JQ]] is the function $K(x, s)$ of two variables that multiplies the unknown function inside the integral.
In $\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds$ it is $K$.
The properties of the kernel, such as continuity, symmetry or the form of its singularity, decide which theory applies to the equation.

## Explanations

### L1 human

The kernel says how strongly the value of the unknown at the point $s$ influences the equation at the point $x$.
It is like an infinite matrix whose rows are indexed by $x$ and whose columns are indexed by $s$.
The integral $\int K(x, s) \varphi(s) \, ds$ is the continuous version of multiplying a matrix by a vector.
For example, in $\varphi(x) = 1 + \int_0^1 (x + s) \varphi(s) \, ds$ the kernel is $K(x, s) = x + s$.
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# Lusternik–Schnirelmann theory of critical points

## Definition

Let $M$ be a compact connected [[smooth manifold|F45VQ47GPV]] without boundary and let $f \colon M \to \R$ be a $C^1$ function.
For a closed subset $A \subseteq M$, let $\operatorname{cat}_M(A)$ be the least $m \ge 0$ such that $m + 1$ open subsets of $M$, each contractible in $M$, cover $A$.
Then $\operatorname{cat}_M(M) = \operatorname{cat}(M)$, the normalized [[Lusternik–Schnirelmann category|W5BHHXCYXA]].
For $k = 0, 1, \dots, \operatorname{cat}(M)$, the *Lusternik–Schnirelmann minimax values* of $f$ are
$$c_k = \inf \set{ \max_{x \in A} f(x) : A \subseteq M \text{ closed}, \ A \ne \emptyset, \ \operatorname{cat}_M(A) \ge k }.$$
*Lusternik–Schnirelmann theory* is the [[critical point theory|9ZQC7ARWTT]] built on these values.
Its basic theorem states:

1. each $c_k$ is a critical value of $f$, and $c_0 \le c_1 \le \cdots \le c_{\operatorname{cat}(M)}$;
2. if $c_k = c_{k+1}$ for some $k$, then $f$ has infinitely many [[critical points|0YTCMFB4HC]] at that level;
3. hence $f$ has at least $\operatorname{cat}(M) + 1$ critical points.

## Remarks

The proof uses a deformation along a pseudo-gradient flow.
If $c$ is not a critical value, the sublevel set $\set{f \le c + \varepsilon}$ can be pushed into $\set{f \le c - \varepsilon}$, and such deformations do not increase category.
If the critical set at level $c_k = c_{k+1}$ were finite, it would have a neighbourhood of category $0$, and subadditivity of category would give a contradiction.
Since one-point sets are allowed for $k = 0$, the first value is $c_0 = \min f$.

Example: for $M = T^2$, $\operatorname{cat}(T^2) = 2$, so every $C^1$ function on the torus has at least three critical points.
This bound is attained by suitable functions, while a Morse function on $T^2$ has at least four.

Non-example: the theory does not apply as stated to $f(x) = \arctan x$ on the non-compact manifold $\R$.
There $\operatorname{cat}(\R) = 0$ would predict one critical point, but $f$ has none, since the infimum defining $c_0$ is not attained.
In infinite dimensions compactness of $M$ is replaced by the [[Palais–Smale condition|4724KRNHAT]] together with completeness and a lower bound on $f$.
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# Hyers–Ulam stability

## Definition

Let $X$ be a nonempty set, let $k \ge 1$, and let $(Y, d)$ be a [[metric space|0VKD1TP7F3]].
Let $\mathcal{F}$ be a set of maps $X \to Y$, and let $E_1, E_2$ assign to each $f \in \mathcal{F}$ a map $X^k \to Y$.
Consider the [[functional equation|SRAN4EKWS0]] $E_1(f)(u) = E_2(f)(u)$ for all $u \in X^k$.
The equation is *Hyers–Ulam stable* in $\mathcal{F}$ if there is a constant $K > 0$ such that the following holds for every $\delta \ge 0$.
If $f \in \mathcal{F}$ satisfies $d(E_1(f)(u), E_2(f)(u)) \le \delta$ for all $u \in X^k$, then there is a solution $g \in \mathcal{F}$ of the equation with $d(f(x), g(x)) \le K \delta$ for all $x \in X$.

## Remarks

Example: the Cauchy equation $f(x + y) = f(x) + f(y)$ for maps from a normed space to a Banach space is Hyers–Ulam stable with $K = 1$; here $k = 2$, $E_1(f)(x, y) = f(x + y)$ and $E_2(f)(x, y) = f(x) + f(y)$.
Non-example: the periodicity equation $f(x + 1) = f(x)$ for maps $\R \to \R$ is not stable.
For $\delta > 0$ the map $f(x) = \delta x$ satisfies $\abs{f(x + 1) - f(x)} = \delta$.
If $g$ is $1$-periodic, then $f(x + n) - g(x + n) = f(x) - g(x) + n\delta$ for every integer $n$, so $f - g$ is unbounded and no $K$ works.
Some authors use a weaker form: for every $\varepsilon > 0$ there is $\delta > 0$ such that a $\delta$-approximate solution lies within $\varepsilon$ of a solution.
In the Hyers–Ulam–Rassias form, the constant bound $\delta$ is replaced by a control function such as $\theta(\norm{x}^p + \norm{y}^p)$.
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# Moments and orthogonal polynomials

## Summary

The [[orthogonal polynomials|8TXASARQN1]] of a measure on the real line depend only on its [[moments|01XAHPADBA]].
Given a moment sequence $(m_n)$ whose [[Hankel matrices|PQ1D8FQB3E]] are positive definite, the monic orthogonal polynomials are explicit quotients of determinants built from the $m_n$.
This link lets one pass between a sequence of numbers, a family of polynomials, a three-term recurrence and a measure.
It is a main tool in the solution of the [[Hamburger moment problem|TBNCFXTMMW]] and in the theory of Gaussian quadrature.

## Explanations

### L1 human

The moments $m_n = \int x^n \, d\mu$ are all one needs to compute integrals of polynomials against $\mu$.
Orthogonality of polynomials is defined by such integrals, so the moments alone determine the orthogonal polynomials.
One can write each orthogonal polynomial as a determinant whose entries are moments, with the powers $1, x, \dots, x^n$ in the last row.
For the uniform weight on $[-1, 1]$ the moments are $2, 0, \tfrac{2}{3}, 0, \tfrac{2}{5}, \dots$, and the determinant formula gives $1$, $x$, $x^2 - \tfrac13$, the monic Legendre polynomials.
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# Dilworth's theorem

## Statement

Let $(P, \le)$ be a finite [[partially ordered set|0QV2JS9RJ3]].
Let $w$ be the largest size of an [[antichain|03HF498P1J]] in $P$.
Then $P$ is the union of $w$ pairwise disjoint [[chains|0S8FE7ZMFM]], and $P$ is not the union of fewer than $w$ chains.

## Remarks

The easy half: a chain meets an antichain in at most one element, so any cover of $P$ by chains uses at least $w$ chains.
The hard half can be proved by induction on $\abs{P}$, by removing a maximal chain or by splitting $P$ along a maximum antichain.
Another proof builds a bipartite graph with two copies of $P$ and an edge from $x$ to $y$ when $x < y$; then [[Kőnig's theorem|0JP0ZF57PN]] on matchings gives the chain partition.
Example: in the set $\set{1, 2, \ldots, 10}$ ordered by divisibility, the set $\set{6, 7, 8, 9, 10}$ is an antichain of size $5$, and $\set{1, 2, 4, 8}$, $\set{3, 6}$, $\set{5, 10}$, $\set{7}$, $\set{9}$ are $5$ chains that cover it.
The number $w$ is called the width of $P$.
The theorem also holds for an infinite partially ordered set of finite width $w$.
The dual result, Mirsky's theorem, says that the largest size of a chain equals the least number of antichains that cover $P$.

## Explanations

### L1 human

In a partial order, some pairs of elements are comparable and others are not.
A chain is a set of elements that can all be lined up, and an antichain is a set of elements no two of which are comparable.
Dilworth's theorem says that the least number of chains needed to cover everything equals the largest number of pairwise incomparable elements.
So a big antichain is the only obstacle to covering with few chains.

### L2 human

Dilworth's theorem is one of the min-max theorems of combinatorics, like Kőnig's theorem and Menger's theorem, and each of these can be derived from the others.
It gives a quick proof of the [[Erdős–Szekeres theorem|0SNJM8HSD3]]: a sequence of $rs + 1$ distinct real numbers has an increasing subsequence of length $r + 1$ or a decreasing subsequence of length $s + 1$.
In algorithms, a minimum chain partition of a finite poset can be computed through a maximum matching in a bipartite graph.
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# Gromov–Hausdorff convergence

## Summary

Gromov–Hausdorff convergence is a notion of convergence for sequences of metric spaces, not of points inside one space.
A sequence of compact metric spaces converges when its distance to the limit in the [[Gromov–Hausdorff space|ETRF2FHAEE]] tends to zero.
It lets one take limits of Riemannian manifolds, and such limits are often singular spaces.
It is a basic tool of metric and synthetic methods in global geometry.

## Explanations

### L1 human

Think of shapes seen from far away, where only distances matter.
Two shapes are close when they can be laid in a common space so that each lies near the other.
A sequence of shapes converges when this closeness tends to zero.
For example, the points $0, \tfrac{1}{n}, \tfrac{2}{n}, \ldots, 1$ form finite spaces that converge to the segment $[0, 1]$.
A thin tube around a circle also converges to the circle when the tube gets thinner.
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# Hyperstructure

## Definition

Let $H$ be a nonempty [[set|0SJ3RX9TXA]] and let $\mathcal{P}^*(H)$ be the set of nonempty subsets of $H$, a part of the [[power set|00E0YZCERH]].
A *hyperoperation* on $H$ is a map $\circ \colon H \times H \to \mathcal{P}^*(H)$ on the [[Cartesian product|09CEDJYBKP]].
For subsets $X, Y \subseteq H$ write $X \circ Y = \bigcup_{x \in X, y \in Y} x \circ y$, and $x \circ Y = \set{x} \circ Y$.
An *(algebraic) hyperstructure* is a set $H$ with one or more hyperoperations, possibly together with ordinary operations.
A set with one hyperoperation $(H, \circ)$ is a *hypergroupoid*, and it is a *hypergroup* if:

1. $(x \circ y) \circ z = x \circ (y \circ z)$ for all $x, y, z \in H$ ([[associativity|0K8F7PVDMX]]);
2. $x \circ H = H \circ x = H$ for all $x \in H$ (reproduction axiom).

A hypergroupoid that satisfies clause 1 is a *semihypergroup*.

## Remarks

Example: $H = \R$ with $x \circ y = [\min(x, y), \max(x, y)]$.
Both sides of clause 1 equal $[\min(x, y, z), \max(x, y, z)]$, since the union of the intervals between $z$ and the points of $[a, b]$ is the interval from $\min(a, z)$ to $\max(b, z)$.
For clause 2, every $y \in \R$ lies in $x \circ y$, so $x \circ \R = \R \circ x = \R$.

Non-example: $H = [0, \infty)$ with $x \circ y = [\max(x, y), \infty)$.
Clause 1 holds, since both sides equal $[\max(x, y, z), \infty)$.
Clause 2 fails: for $x = 1$, $x \circ H = [1, \infty) \ne H$.
So this is a semihypergroup but not a hypergroup.

A group $G$ is a hypergroup with $x \circ y = \set{xy}$.
</document>

<document source="https://tamnd.github.io/math2/node/GT9E1NT6KC.md" tag="GT9E1NT6KC" kind="definition">
# Polar set

## Definition

Let $n \ge 2$ and let $Z \subseteq \R^n$.
The set $Z$ is *polar* if there is a [[subharmonic function|Z1MHMGFNFW]] $u$ on $\R^n$, not identically $-\infty$, such that
$$Z \subseteq \set{ x \in \R^n : u(x) = -\infty }.$$

## Remarks

Example: in $\R^2$, the function $u(x) = \log \abs{x - a}$ is subharmonic, not identically $-\infty$, and equals $-\infty$ exactly at $a$, so $\set{a}$ is polar.
In $\R^n$ with $n \ge 3$ the function $u(x) = -\abs{x - a}^{2-n}$ plays the same role.
A countable union of polar sets is polar, so every countable subset of $\R^n$ is polar.

Non-example: the segment $[0, 1] \times \set{0}$ in $\R^2$ is not polar, because it has positive logarithmic capacity $1/4$.

Polarity is a local property.
A set $Z$ is polar if and only if each point of $Z$ has a neighbourhood $U$ with the following property.
Some subharmonic function on $U$, not identically $-\infty$ on any component, equals $-\infty$ on $Z \cap U$.
For Borel sets, being polar is equivalent to having capacity zero.
A polar set has Lebesgue measure zero.
</document>

<document source="https://tamnd.github.io/math2/node/GTA8ZFGZ3N.md" tag="GTA8ZFGZ3N" kind="statement">
# Universal coefficient theorem

## Statement

Let $R$ be a [[principal ideal domain|0CMD0BCH8T]], let $C_\bullet$ be a [[chain complex|WQ82C0M3QB]] of free $R$-modules, and let $M$ be an $R$-module.
Then for every $n$ there are natural short exact sequences
$$0 \to H_n(C) \otimes_R M \to H_n(C \otimes_R M) \to \operatorname{Tor}_1^R(H_{n-1}(C), M) \to 0,$$
$$0 \to \operatorname{Ext}^1_R(H_{n-1}(C), M) \to H^n(\Hom_R(C, M)) \to \Hom_R(H_n(C), M) \to 0,$$
and both split, though not naturally.
In particular, for a [[topological space|0CQHFPGJ45]] $X$ and an abelian group $G$, applied to the singular chain complex of $X$:
$$H_n(X; G) \cong (H_n(X) \otimes G) \oplus \operatorname{Tor}_1^\Z(H_{n-1}(X), G), \qquad H^n(X; G) \cong \Hom(H_n(X), G) \oplus \operatorname{Ext}^1_\Z(H_{n-1}(X), G).$$

## Remarks

Here $H_n(X)$ is [[singular homology|M0Q5CFJ7ZH]] with integer coefficients and $H^n(X; G)$ is [[singular cohomology|6N7BC70CX0]]; [[Tor|DDQRERAQCV]] and [[Ext|4P7GBWJ4RR]] are the derived functors of $\otimes$ and $\Hom$.
For the proof, the cycles $Z_n$ and boundaries $B_n$ of $C$ are free, since submodules of free modules over a principal ideal domain are free.
So $0 \to Z_\bullet \to C_\bullet \to B_{\bullet-1} \to 0$ splits in each degree.
Apply $- \otimes M$ or $\Hom(-, M)$ and take the long exact sequence; its connecting maps are induced by the inclusions $B_n \to Z_n$.
Their kernels and cokernels are the Tor and Ext terms.
Example: for the real projective plane $P$, $H_0(P) = \Z$, $H_1(P) = \Z/2\Z$, $H_2(P) = 0$.
So $H^1(P; \Z) \cong \Hom(\Z/2\Z, \Z) \oplus \operatorname{Ext}^1(\Z, \Z) = 0$ and $H^2(P; \Z) \cong \Hom(0, \Z) \oplus \operatorname{Ext}^1(\Z/2\Z, \Z) \cong \Z/2\Z$.
This shows that cohomology is not simply the dual of homology: the torsion of $H_1$ moves up to $H^2$.

## Explanations

### L1 human

Homology with integer coefficients determines homology and cohomology with any other coefficients.
The answer is almost the naive one, the tensor product or the dual, plus a correction that comes from torsion.
For the projective plane, the integer homology in degree one is $\Z/2\Z$, and this torsion reappears in integer cohomology one degree higher.

### L2 human

The homology part of the theorem is a special case of the [[Künneth theorem|77K0BZQ4MK]], with the second complex a module in degree $0$.
It is used constantly, for example to compute cohomology from homology.
It also shows that a map inducing isomorphisms on integral homology induces isomorphisms on homology and cohomology with all coefficients.
Over a field $k$, it gives $H^n(X; k) \cong \Hom_k(H_n(X; k), k)$.
</document>

<document source="https://tamnd.github.io/math2/node/GTBMW9QXST.md" tag="GTBMW9QXST" kind="definition">
# Linear discriminant analysis

## Definition

Let $K \ge 2$ be the number of classes.
For $k = 1, \ldots, K$ let $\pi_k > 0$ be prior probabilities with $\pi_1 + \cdots + \pi_K = 1$, and let $\mu_k \in \R^p$ be class means.
Let $\Sigma$ be a [[positive definite|0KX66CZMBB]] $p \times p$ matrix, the common [[covariance matrix|0V7G6WMFK0]].
The *linear discriminant functions* are
$$\delta_k(x) = x^T \Sigma^{-1} \mu_k - \tfrac{1}{2} \mu_k^T \Sigma^{-1} \mu_k + \log \pi_k, \qquad x \in \R^p.$$
*Linear discriminant analysis* assigns $x$ to a class $k$ for which $\delta_k(x)$ is largest.
In practice the parameters are estimated from a labelled training sample.
One uses the class proportions for $\pi_k$, the class sample means for $\mu_k$ and the pooled sample covariance matrix for $\Sigma$.

## Remarks

If the class $k$ has the distribution $N_p(\mu_k, \Sigma)$ and prior probability $\pi_k$, this rule is the Bayes rule.
Indeed, $\delta_k(x)$ differs from $\log(\pi_k f_k(x))$ by a term that does not depend on $k$, where $f_k$ is the density of class $k$.
Each $\delta_k$ is affine in $x$, so the boundary between two classes is a hyperplane.
For two classes, $x$ goes to class $1$ when
$$(\mu_1 - \mu_2)^T \Sigma^{-1} x - \tfrac{1}{2} (\mu_1 - \mu_2)^T \Sigma^{-1} (\mu_1 + \mu_2) > \log(\pi_2 / \pi_1).$$
Example: with $p = 1$, $\mu_1 = 1$, $\mu_2 = -1$, $\Sigma = 1$ and $\pi_1 = \pi_2$, the condition is $2x > 0$, so $x$ goes to class $1$ when $x > 0$.
Non-example: if the normal classes have different covariance matrices $\Sigma_k$, the Bayes rule has quadratic boundaries.
This rule is quadratic discriminant analysis, not a linear rule.
</document>

<document source="https://tamnd.github.io/math2/node/GTBXZXK146.md" tag="GTBXZXK146" kind="concept">
# Friedrichs extension

## Summary

The Friedrichs extension is a canonical [[self-adjoint|0NEGN5CRSC]] extension of a densely defined [[symmetric operator|H971WGMV6C]] that is bounded below.
It is built from the quadratic form of the operator, by completing the domain in the form norm.
It has the same lower bound as the original operator.
It is the standard way to turn a formal differential operator such as $-\Delta + V$ with $V \ge 0$ into a self-adjoint operator.

## Explanations

### L1 human

A symmetric operator is often given only on nice functions, and this domain is too small for the operator to be self-adjoint.
There can be many ways to enlarge the domain, and they give different operators.
When the operator is bounded below, the Friedrichs construction picks one of them in a natural way.
The idea is to use the energy $\ip{Tx}{x}$ to measure functions, and to keep exactly the functions of finite energy.
For $-d^2/dx^2$ on smooth functions with compact support in $(0, 1)$, the Friedrichs extension is the second derivative with the Dirichlet conditions $f(0) = f(1) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/GTFRPF5J2K.md" tag="GTFRPF5J2K" kind="example">
# M/M/1 queue

## Example

Let $\lambda > 0$ and $\mu > 0$.
In the M/M/1 queue of [[queueing theory|P64V0HTE78]], customers arrive as a Poisson process of rate $\lambda$ and there is one server.
Service times are independent [[exponential|0E99F51GVB]] random variables of rate $\mu$.
The number $Q(t)$ of customers in the system is a [[birth–death process|E8DAD7052E]] on $\set{0, 1, 2, \ldots}$.
Its birth rates are $\lambda_n = \lambda$ for $n \ge 0$ and its death rates are $\mu_n = \mu$ for $n \ge 1$.
This holds because the exponential law has no memory.
At each moment, the time to the next arrival and the remaining service time are again exponential, of rates $\lambda$ and $\mu$.

Let $\rho = \lambda / \mu$.
The balance equations $\pi_n \lambda = \pi_{n+1} \mu$ give $\pi_n = \rho^n \pi_0$.
So a [[stationary distribution|02KYGSCTCN]] exists if and only if $\rho < 1$, and then
$$\pi_n = (1 - \rho) \rho^n, \qquad n \ge 0,$$
a geometric law.
The mean number in the system is $L = \sum_n n \pi_n = \rho / (1 - \rho)$.
By [[Little's law|JN2YME6F4A]], the mean time in the system is $W = L / \lambda = 1/(\mu - \lambda)$.
If $\rho \ge 1$ the queue has no stationary distribution: it is null recurrent for $\rho = 1$ and transient for $\rho > 1$.

## Explanations

### L1 human

One clerk serves customers who arrive at random, at $\lambda$ per hour on average, and serves $\mu$ per hour on average when busy.
If $\lambda = 3$ and $\mu = 4$, the clerk is busy three quarters of the time, there are $3$ customers in the system on average, and each spends $1$ hour there on average.
As $\lambda$ comes close to $\mu$, the average line grows without bound.
</document>

<document source="https://tamnd.github.io/math2/node/GTFZ277047.md" tag="GTFZ277047" kind="concept">
# Property (T)

## Summary

A [[locally compact group|AHV05ZBCYW]] has property (T), or Kazhdan's property (T), under the following condition.
Every [[unitary representation|74BPYJNZTM]] that almost has invariant vectors actually has a nonzero invariant vector.
Kazhdan introduced it to show that [[lattices|NS2057MFWP]] in groups such as $\mathrm{SL}_n(\R)$ with $n \ge 3$ are finitely generated.
It is a strong rigidity property, opposite to [[amenability|CEAT9BESKK]]: a locally compact group that is both amenable and has property (T) is compact.
It has applications to expander graphs, to ergodic theory and to the theory of operator algebras.

## Explanations

### L1 human

Let a group act by rotations, or more generally by unitary maps, on a space of possibly infinite dimension.
Suppose some unit vector is moved only a tiny bit by each element of a fixed generating set.
For most groups this tells you nothing, but for groups with property (T) it forces the existence of a vector that is not moved at all.
The integers fail this: shifting a long flat block of ones in $\ell^2(\Z)$ moves it very little, yet no nonzero vector of $\ell^2(\Z)$ is fixed by all shifts.
The groups $\mathrm{SL}_n(\Z)$ with $n \ge 3$ have property (T).
</document>

<document source="https://tamnd.github.io/math2/node/GTG8MJGDKC.md" tag="GTG8MJGDKC" kind="concept">
# Triangular matrix

## Summary

A triangular matrix is a square matrix that is either [[upper triangular|0N4XY7Z3NN]] or [[lower triangular|00J3KZP8Y0]].
All its entries on one side of the main diagonal are zero.
Linear systems with a triangular matrix are solved quickly by substitution.
The [[determinant of a triangular matrix|0K1KKMFHYH]] is the product of its diagonal entries, and these entries are its eigenvalues.
Many factorisations, such as LU, QR and the [[Schur decomposition|V1JG4C7R7B]], produce triangular factors.

## Explanations

### L1 human

Picture a square table of numbers cut along its main diagonal.
A triangular matrix has only zeros in one of the two halves, so its nonzero entries fill a triangle.
For example, $\begin{pmatrix} 1 & 4 \\ 0 & 2 \end{pmatrix}$ is upper triangular and $\begin{pmatrix} 3 & 0 \\ 5 & 1 \end{pmatrix}$ is lower triangular.
In a system with such a matrix, one equation has a single unknown, and each next equation adds only one new unknown.
</document>

<document source="https://tamnd.github.io/math2/node/GTGFXW37YM.md" tag="GTGFXW37YM" kind="definition">
# Normal closure

## Definition

Let $L/K$ be an [[algebraic extension|0PTBM7K0R2]].
A *normal closure* of $L/K$ is a field extension $M/L$ such that:

1. $M/K$ is a [[normal extension|0CBRFQ7DVZ]];
2. no proper subfield of $M$ that contains $L$ is normal over $K$.

## Remarks

A normal closure exists and is unique up to an isomorphism of fields that is the identity on $L$.
Let $L = K(\alpha_1, \ldots, \alpha_n)$ be finite over $K$, and let $f_i$ be the [[minimal polynomial|0GK1WB9R98]] of $\alpha_i$ over $K$.
Then a normal closure is the [[splitting field|0KXQKX9P63]] of $f_1 \cdots f_n$ over $L$, and it is again finite over $K$.
Let $N$ be a normal extension of $K$ that contains $L$.
Inside $N$, the normal closure is the compositum of the fields $\tau(L)$, for all embeddings $\tau \colon L \to N$ over $K$.

Example: for $L = \Q(\sqrt[3]{2})$ over $\Q$, let $M = \Q(\sqrt[3]{2}, \omega)$ with $\omega = e^{2\pi i/3}$.
Check: $M$ is the splitting field of $x^3 - 2$ over $\Q$, so it is normal over $\Q$.
A normal subfield containing $\sqrt[3]{2}$ contains all roots $\sqrt[3]{2}, \omega\sqrt[3]{2}, \omega^2\sqrt[3]{2}$ of its minimal polynomial, hence contains $M$.

Non-example: an algebraic closure $\overline{\Q}$ of $\Q$ contains $L$ and is normal over $\Q$, but it fails clause 2, since $M$ is a proper normal subfield containing $L$.
</document>

<document source="https://tamnd.github.io/math2/node/GTHWVEB1NZ.md" tag="GTHWVEB1NZ" kind="definition">
# Crystallographic group

## Definition

Let $n \ge 1$ and let $E(n)$ be the [[Euclidean group|0H3YY8CPKH]] of all [[isometries|0J1N3WTKM4]] of $\R^n$.
Each element of $E(n)$ is a map $x \mapsto Ax + b$ with $A$ in the [[orthogonal group|0TKYAEJ8N7]] $O(n)$ and $b \in \R^n$, and $E(n)$ carries the topology of $O(n) \times \R^n$.
A subgroup $\Gamma$ of $E(n)$ is a *crystallographic group* of dimension $n$ if:

1. $\Gamma$ is a [[discrete subgroup|5M7D1BSHVZ]] of $E(n)$.
2. The quotient space $\R^n / \Gamma$ of orbits is [[compact|0TVJZYB6SH]].

## Remarks

Example: the group $\Z^n$ of [[translations|087VJM5Q77]] $x \mapsto x + b$ with $b \in \Z^n$.
It is discrete, since distinct elements differ by a translation of length at least $1$.
The quotient $\R^n / \Z^n$ is the image of the compact cube $[0, 1]^n$, so it is compact; it is the $n$-dimensional torus.

Non-example: the translations $(x, y) \mapsto (x + k, y)$ of $\R^2$ with $k \in \Z$.
This group is discrete, but the quotient is an infinite cylinder, which is not compact, so clause 2 fails.

Equivalently, $\Gamma$ acts properly discontinuously on $\R^n$ with a compact fundamental domain.
Crystallographic groups of dimension $3$ are called space groups, and those of dimension $2$ are the [[wallpaper groups|MY3CBHARYQ]].
</document>

<document source="https://tamnd.github.io/math2/node/GTJ6KBD4PX.md" tag="GTJ6KBD4PX" kind="definition">
# Zeta-regularized determinant

## Definition

Let $M$ be a closed smooth manifold of dimension $n$ with a Hermitian vector bundle $E$.
Let $A$ be a formally [[self-adjoint|0NEGN5CRSC]] [[elliptic differential operator|MT9P2X85V8]] of order $m > 0$ on $E$, with nonnegative spectrum.
Let $0 < \mu_1 \le \mu_2 \le \cdots$ be its nonzero eigenvalues, counted with multiplicity.
Its spectral zeta function $\zeta_A(s) = \sum_{j \ge 1} \mu_j^{-s}$ converges for $\operatorname{Re} s > n/m$ and has a meromorphic continuation to $\C$ that is holomorphic at $s = 0$.
The *zeta-regularized determinant* of $A$ is
$$\det{}'_\zeta A = \exp\big( -\zeta_A'(0) \big).$$
If $\ker A = 0$ it is written $\det_\zeta A$.

## Remarks

It is the [[zeta-regularized product|VJCJYXGTDY]] of the nonzero eigenvalues.
For the Laplace–Beltrami operator, $\zeta_A$ is the [[Minakshisundaram–Pleijel zeta function|T1039Q9FY0]].
If $A$ were a positive definite matrix, $-\zeta_A'(0) = \sum_j \log \mu_j$ and the definition would give the usual determinant.
The regularized determinant is not multiplicative in general: $\det_\zeta(AB)$ may differ from $\det_\zeta A \det_\zeta B$ by a multiplicative anomaly.

Example: on $S^1 = \R / 2\pi\Z$ let $A = -d^2/d\theta^2$.
The nonzero eigenvalues are $k^2$ with multiplicity $2$, so $\zeta_A(s) = 2\zeta(2s)$ and $\zeta_A'(0) = 4\zeta'(0) = -2 \log(2\pi)$, using $\zeta'(0) = -\tfrac12 \log(2\pi)$.
Hence $\det{}'_\zeta A = 4\pi^2$.

Non-example: the operator $-d^2/d\theta^2 - c$ with $0 < c < 1$ on $S^1$ has the negative eigenvalue $-c$.
It does not fit the definition, since $\mu^{-s}$ for $\mu < 0$ needs a choice of branch of the logarithm.
</document>

<document source="https://tamnd.github.io/math2/node/GTMT5ZFV00.md" tag="GTMT5ZFV00" kind="definition">
# Submersion

## Definition

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], and let $f \colon M \to N$ be a [[smooth|0V33BNKZ9S]] map.
The map $f$ is a *submersion* if for every point $p \in M$ the [[differential|6P9PXQDSEN]] $df_p \colon T_pM \to T_{f(p)}N$ is surjective.

## Remarks

Equivalently, the rank of $df_p$ equals $\dim N$ at every point $p$; when $M$ is not empty this forces $\dim M \ge \dim N$.
A point $p$ where $df_p$ is surjective is a regular point, so a submersion is a smooth map all of whose points are regular.
Example: the projection $\pi \colon \R^3 \to \R^2$, $\pi(x, y, z) = (x, y)$, is a submersion.
Its differential at every point is the linear map $\pi$ itself, which is onto.
Non-example: $f \colon \R^2 \to \R$, $f(x, y) = x^2 + y^2$, is not a submersion.
At the origin $df_0 = 0$, which is not surjective onto $\R$.
By the constant rank theorem, near each point a submersion has the local form $(x_1, \dots, x_m) \mapsto (x_1, \dots, x_n)$ in suitable charts.
</document>

<document source="https://tamnd.github.io/math2/node/GTN3QQYTT5.md" tag="GTN3QQYTT5" kind="concept">
# Line graph

## Summary

The line graph $L(G)$ of a [[graph|00C0BR7D7H]] $G$ has the edges of $G$ as its vertices.
Two of them are adjacent when they share an endpoint.
It turns questions about edges into questions about vertices.
For example, a matching of $G$ is an independent set of $L(G)$, and an edge colouring of $G$ is a vertex colouring of $L(G)$.
The node [[line graph|CJ57DT86S8]] gives the precise definition.

## Explanations

### L1 human

Draw a dot in the middle of each edge of a graph.
Join two of the new dots when their edges meet at a common vertex.
The new picture is the line graph.
For a star with three arms, the three arms all meet at the centre, so the line graph is a triangle.
</document>

<document source="https://tamnd.github.io/math2/node/GTS394XJ7Y.md" tag="GTS394XJ7Y" kind="definition">
# Cerf theory

## Definition

Let $M$ be a compact [[smooth manifold|F45VQ47GPV]] of dimension $n$ without boundary.
*Cerf theory* is the study of smooth one-parameter families $f_t \colon M \to \R$, $t \in [0, 1]$, of smooth functions, with $(x, t) \mapsto f_t(x)$ smooth.
Its basic notion is a *generic path*, also called a *Cerf path*.
A family $(f_t)$ is a generic path if it satisfies the following conditions.

1. For all but finitely many $t$, $f_t$ is a [[Morse function|KPQ96640ZJ]] with distinct critical values.
2. At each remaining time $t_0$, one of two cases occurs.
3. In the first case, $f_{t_0}$ is Morse, exactly two [[critical points|0YTCMFB4HC]] share a critical value, and these two values cross transversally at $t_0$.
4. In the second case, $f_{t_0}$ has distinct critical values and exactly one degenerate critical point $p_0$.
5. In the second case, near $p_0$ there are local coordinates $(x_1, \dots, x_n)$ and a local time parameter $s$ with $s = 0$ at $t_0$ with the following form.
   For some $0 \le \lambda \le n - 1$ and a smooth function $c$, one has $f_t = c(s) + x_1^3 + s x_1 - x_2^2 - \cdots - x_{\lambda+1}^2 + x_{\lambda+2}^2 + \cdots + x_n^2$.

## Remarks

In item 5, for $s < 0$ there are two critical points near the origin, of indices $\lambda$ and $\lambda + 1$, and for $s > 0$ there are none; this is a *birth–death* singularity.
Cerf's theorem says that generic paths are dense in the space of smooth paths between two given endpoints.
Here the endpoints are Morse functions with distinct critical values.
The *Cerf graphic* of a generic path is the subset $\set{(t, f_t(p)) : p \text{ critical for } f_t}$ of $[0, 1] \times \R$; it shows births, deaths and crossings as cusps and crossings of curves.
Example, in a chart of a $1$-dimensional $M$: $f_t(x) = x^3 - (t - \tfrac{1}{2}) x$ satisfies item 5 at $t_0 = \tfrac{1}{2}$ with $s = -(t - \tfrac{1}{2})$ and $\lambda = 0$.
Non-example: the constant path $f_t(x) = x^3$ fails item 1, because the degenerate critical point $0$ occurs at every time.
</document>

<document source="https://tamnd.github.io/math2/node/GTS4DRRMZR.md" tag="GTS4DRRMZR" kind="concept">
# Free algebra in a variety

## Summary

The free algebra of a [[variety|AJNZJD8442]] $\mathcal{V}$ on a set $X$ is the algebra of $\mathcal{V}$ generated by $X$ with as few relations as possible.
The only relations among the generators are those forced by the identities of $\mathcal{V}$.
Its defining property is that every map from $X$ to an algebra of $\mathcal{V}$ extends to exactly one homomorphism.
Free groups, free abelian groups, polynomial rings and free monoids are free algebras of their varieties.
Every algebra of $\mathcal{V}$ is a quotient of a free algebra, which makes free algebras the starting point of presentations.

## Explanations

### L1 human

Take some letters $x, y, z$ and form all expressions in them that the operations allow.
Declare two expressions equal only when the laws of the variety force them to be equal.
In commutative rings with identity, this gives the polynomial ring $\Z[x, y, z]$: $xy$ equals $yx$, but $x^2$ does not equal $x$.
Because nothing extra was imposed, the letters can be sent anywhere, and the rest of the map is then determined.
</document>

<document source="https://tamnd.github.io/math2/node/GTVCECGH40.md" tag="GTVCECGH40" kind="concept">
# Dunkl operator

## Summary

A Dunkl operator is a [[partial derivative|09EGC6F8PT]] plus a sum of difference terms, one for each reflection of a finite [[reflection group|G6FND7X69J]].
Charles Dunkl introduced these operators and proved that, for a fixed group and multiplicity function, they commute with each other.
They give a theory of special functions attached to root systems.
In it, Dunkl harmonics, the Dunkl transform and Dunkl kernels generalize spherical harmonics, the Fourier transform and the exponential.
They are also used to study the Calogero–Moser–Sutherland quantum systems, whose eigenfunctions in type $A$ are related to the [[Jack functions|WW7H1KNXK8]].

## Explanations

### L1 human

An ordinary derivative measures how a function changes along a direction.
A Dunkl operator adds to this the difference between the function and its mirror image in each mirror of a symmetry group, divided by the distance to that mirror.
On the real line with the single mirror $x \mapsto -x$, it is $T f(x) = f'(x) + k \frac{f(x) - f(-x)}{x}$.
It sends $x^2$ to $2x$, like the derivative, but sends $x^3$ to $(3 + 2k) x^2$.
</document>

<document source="https://tamnd.github.io/math2/node/GTVQFG2SH9.md" tag="GTVQFG2SH9" kind="concept">
# Conformal geometry

## Summary

Conformal geometry studies spaces in which angles can be measured but lengths cannot.
There, a Riemannian metric is known only up to multiplication by a positive function.
Its maps are the [[conformal maps|0YYT36R4YD]], those that preserve angles.
In dimension $2$ there are very many conformal maps, such as all holomorphic functions with nonzero derivative.
In dimension at least $3$, [[Liouville's theorem|NH7J4YQMQC]] reduces them to the Möbius transformations.
The conformal geometry of the sphere is Möbius geometry, the subject of [[inversive geometry|HFBW3SXQVM]].

## Explanations

### L1 human

Imagine a map of the world that may stretch some regions more than others, but always keeps the angles between roads correct.
A conformal geometry remembers exactly this: angles, not distances.
The Mercator map is an example: it enlarges Greenland a lot, yet every angle between two streets is drawn true.
Two shapes count as the same in conformal geometry when one can be bent onto the other by such an angle-keeping map.
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<document source="https://tamnd.github.io/math2/node/GV1HKEAWVN.md" tag="GV1HKEAWVN" kind="concept">
# Dedekind-finite ring

## Summary

A ring is Dedekind-finite when every one-sided inverse is a two-sided inverse: $ab = 1$ forces $ba = 1$.
Such rings are also called directly finite.
Commutative rings, finite rings and matrix rings over fields are Dedekind-finite, while rings of operators on infinite-dimensional spaces need not be.

## Explanations

### L1 human

For square matrices over a field, if $AB = I$ then also $BA = I$.
A ring with this property is Dedekind-finite.
The name recalls Dedekind's definition of a finite set: a set is finite when it is not in bijection with a proper subset.
Consider the ring of linear maps on the space of infinite sequences.
There, "shift right" followed by "shift left" is the identity.
But "shift left" followed by "shift right" loses the first term, so that ring is not Dedekind-finite.
</document>

<document source="https://tamnd.github.io/math2/node/GVCRZH15Y0.md" tag="GVCRZH15Y0" kind="concept">
# Analytic signal

## Summary

The analytic signal of a real function $f$ is the complex function $f + i\, Hf$, where $H$ is the [[Hilbert transform|NSCEBXM7ZM]].
Its Fourier transform vanishes at negative frequencies and is twice that of $f$ at positive frequencies.
It is the boundary value of a [[holomorphic function|0NAHXR1FRH]] on the upper half-plane, which explains the name.
In signal processing it defines the instantaneous amplitude and phase of a real signal.

## Explanations

### L1 human

A real signal, such as a sound wave, contains each frequency twice, once with a positive and once with a negative sign, and the two halves carry the same information.
The analytic signal throws away the negative frequencies and doubles the positive ones.
The result is a complex signal that turns around the origin of the plane.
Its distance from the origin is the envelope of the signal and its angle is the phase.
For example, the analytic signal of $\cos t$ is $e^{it} = \cos t + i \sin t$.
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<document source="https://tamnd.github.io/math2/node/GVMM9GC7CE.md" tag="GVMM9GC7CE" kind="definition">
# Diffusion process

## Definition

Let $d, m \ge 1$, let $b \colon \R^d \to \R^d$ and $\sigma \colon \R^d \to \R^{d \times m}$ be Lipschitz continuous, and let $W$ be an $m$-dimensional [[Wiener process|22735FGJV6]].
The *diffusion process* (Itô diffusion) with drift $b$ and dispersion $\sigma$ started at $x \in \R^d$ is the unique continuous adapted solution $(X_t)_{t \ge 0}$ of
$$dX_t = b(X_t)\, dt + \sigma(X_t)\, dW_t, \qquad X_0 = x.$$
The matrix $a = \sigma \sigma^{\mathsf{T}}$ is its diffusion matrix.

## Remarks

The process is a strong [[Markov process|0B77KZCQ51]], and by [[Itô's lemma|2J1RVQB0CM]] its [[generator|VPA41PWRR1]] acts on twice continuously differentiable functions $f$ with compact support by
$$Lf(x) = \frac12 \sum_{i,j=1}^d a_{ij}(x) \frac{\partial^2 f}{\partial x_i \partial x_j}(x) + \sum_{i=1}^d b_i(x) \frac{\partial f}{\partial x_i}(x).$$
Example: $d = m = 1$, $b(x) = -\theta x$ with $\theta > 0$ and $\sigma(x) = 1$ give the Ornstein–Uhlenbeck process; both coefficients are Lipschitz, and $Lf = \tfrac12 f'' - \theta x f'$.
Non-example: the Poisson process is a Markov process but not a diffusion, because its paths jump.
In a wider sense, a diffusion is any strong Markov process with continuous paths; the Itô diffusions above form the main class.
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<document source="https://tamnd.github.io/math2/node/GVN1MS9PEJ.md" tag="GVN1MS9PEJ" kind="definition">
# Prime ideal in a noncommutative ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
A [[two-sided ideal|0DDA6CY7GJ]] $P$ of $R$ is *prime* if the following hold.

1. $P \ne R$.
2. For all two-sided ideals $A, B$ of $R$, if $AB \subseteq P$, then $A \subseteq P$ or $B \subseteq P$.

Here $AB$ is the ideal generated by the products $ab$ with $a \in A$ and $b \in B$.

## Remarks

Equivalently, $P \ne R$ and, for all $a, b \in R$, $aRb \subseteq P$ implies $a \in P$ or $b \in P$; and $P$ is prime if and only if $R/P$ is a [[prime ring|Z9TJSZZYG1]].
Example: the zero ideal of $M_2(\Q)$ is prime, since the only two-sided ideals are $0$ and $M_2(\Q)$, and $M_2(\Q) \cdot M_2(\Q) \ne 0$.
Non-example: the zero ideal of $\Q \times \Q$ fails clause 2 with $A = \Q \times 0$ and $B = 0 \times \Q$, since $AB = 0$.
A *completely prime* ideal is one with $ab \in P \Rightarrow a \in P$ or $b \in P$; it is prime, but the zero ideal of $M_2(\Q)$ shows the converse fails.
For a commutative ring both notions agree with the usual [[prime ideal|00MHHZ9SNE]].
</document>

<document source="https://tamnd.github.io/math2/node/GVR54SZFX4.md" tag="GVR54SZFX4" kind="concept">
# Smale's degree for proper Fredholm maps

## Summary

Smale's degree is a mod $2$ degree for proper [[Fredholm maps|GFXQ6HEM65]] of index $0$ between connected Banach manifolds.
It counts the preimages of a regular value modulo $2$.
It extends the mod $2$ [[degree of a continuous map|TMFG2P912R]] between compact manifolds of the same dimension to infinite dimensions.
It rests on the [[Sard–Smale theorem|Z0NPFYDVP2]], which supplies regular values.

## Explanations

### L1 human

For a map between two spaces of the same finite dimension, one can pick a typical point of the target and count how many points land on it.
Modulo $2$, this count does not depend on the point chosen, as long as the map is proper.
Smale noticed that the same count works in infinite dimensions when the derivative of the map is a [[Fredholm operator|8HWGKWD6ZM]] of index $0$.
Such an operator behaves like a square matrix up to finite-dimensional errors.
For example, $x \mapsto x^3$ on $\R$ has one preimage over every point, so its degree is $1$.
The map $x \mapsto x^2$ has two preimages over $1$ and none over $-1$, so its degree is $0$.
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<document source="https://tamnd.github.io/math2/node/GVSY2FZ8YW.md" tag="GVSY2FZ8YW" kind="definition">
# Floquet multiplier

## Definition

Let $T > 0$ and let $A \colon \R \to \R^{n \times n}$ be continuous and [[periodic|0ATF6Q0633]] of period $T$.
Consider the linear system $x' = A(t)\, x$ on $\R^n$.
Let $\Phi$ be its [[fundamental matrix|0H1GN6P9R5]] with $\Phi(0) = I$.
The *monodromy matrix* of the system is $M = \Phi(T)$.
A *Floquet multiplier* of the system is an [[eigenvalue|0BQAA2BK3Q]] of $M$, counted with its algebraic multiplicity.

## Remarks

A complex number $\rho$ is a Floquet multiplier if and only if the system has a nonzero solution with $x(t + T) = \rho\, x(t)$ for all $t$.
Every multiplier is nonzero, and by Liouville's formula their product is $\det M = \exp\left(\int_0^T \tr A(t)\, dt\right)$.
A number $\lambda$ with $e^{\lambda T} = \rho$ is called a Floquet exponent.
All solutions tend to $0$ if and only if every multiplier has absolute value less than $1$.

Example: for a constant matrix $A$, one has $\Phi(t) = e^{tA}$, so $M = e^{TA}$ and the multipliers are $e^{\lambda T}$ for the eigenvalues $\lambda$ of $A$.
For the scalar equation $x' = (1 + \cos t)\, x$ with $T = 2\pi$, one has $\Phi(t) = e^{t + \sin t}$, and the only multiplier is $e^{2\pi}$.

Non-example: for $x' = t\, x$, the coefficient is not periodic, so the definition does not apply; the ratio $x(t + T)/x(t) = e^{tT + T^2/2}$ depends on $t$.

For a $T$-periodic orbit $\gamma$ of an autonomous system $x' = f(x)$, the multipliers of $\gamma$ are those of the linear system $y' = Df(\gamma(t))\, y$.
One of them is always $1$, with the eigenvector $f(\gamma(0))$.
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<document source="https://tamnd.github.io/math2/node/GVVG201079.md" tag="GVVG201079" kind="definition">
# Several complex variables

## Definition

Let $n \ge 1$ be an integer.
The space $\C^n$ is the set of $n$-tuples $z = (z_1, \dots, z_n)$ of [[complex numbers|09FTAET5Z4]], with coordinatewise addition and scalar multiplication.
Its norm is $\abs{z} = \left( \abs{z_1}^2 + \cdots + \abs{z_n}^2 \right)^{1/2}$, which makes it the Euclidean space $\R^{2n}$ through $z_j = x_j + i y_j$.
A *function of several complex variables* is a [[function|0XHDTE5YMZ]] $f \colon D \to \C$, where $D$ is a subset of $\C^n$.

## Remarks

Example: $f(z_1, z_2) = e^{z_1} \sin z_2$ is a function on $D = \C^2$.
Non-example: $g(x, y) = x^2 + y^2$ on $\R^2$ is a function of two real variables, not of complex variables.
It becomes a function of the complex variable $z = x + iy$ only when it is rewritten as $\abs{z}^2$ on $\C$, with $n = 1$.
A map $F \colon D \to \C^m$ is the same as $m$ such functions $F_1, \dots, F_m$, its components.
The case $n = 1$ is the function theory of one complex variable.
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<document source="https://tamnd.github.io/math2/node/GW3B4K1WA5.md" tag="GW3B4K1WA5" kind="concept">
# Index of a singular integral operator

## Summary

Consider a singular integral operator $a \varphi + b S \varphi + \int_\Gamma k \varphi$ on a closed curve $\Gamma$.
Its index is the winding number around $0$ of the function $(a - b)/(a + b)$ on $\Gamma$.
It is an integer computed from the coefficients alone, and it does not depend on the regular part $k$.
By the [[Noether theorems|FDC1BEB8FW]], it equals the number of independent solutions of the homogeneous equation minus that of the adjoint homogeneous equation.
So it is the [[Fredholm index|2V9Q4KHY1T]] of the operator.
It decides how far the classical Fredholm alternative fails for the equation.

## Explanations

### L1 human

For a matrix equation, the number of free parameters in the solutions minus the number of conditions on the data is a fixed number, the difference of the dimensions.
For a singular integral equation, both numbers are finite, and their difference is the index.
The surprise is that the index can be read off from a picture.
Follow the complex number $(a(t) - b(t))/(a(t) + b(t))$ as $t$ runs around the curve, and count how many times it turns around $0$.
For example, on the unit circle with $a(t) = (t + 1)/2$ and $b(t) = (t - 1)/2$, the quotient is $1/t$, which turns once clockwise, so the index is $-1$.
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<document source="https://tamnd.github.io/math2/node/GW78A05TNA.md" tag="GW78A05TNA" kind="concept">
# Arithmetical variety

## Summary

An arithmetical variety is a [[variety|AJNZJD8442]] that is both a [[congruence-permutable variety|1B3N9T68YV]] and a [[congruence-distributive variety|9GZ7F6ETC6]].
The name comes from the ring of integers, whose ideals have both properties, as in the Chinese remainder theorem.
Boolean algebras and Heyting algebras form arithmetical varieties.
In an arithmetical variety a form of the Chinese remainder theorem holds for congruences of every algebra.

## Explanations

### L1 human

Congruences on the integers are given by moduli, and they combine very well: any two of them permute, and they obey the distributive law.
An arithmetical variety is one in which every algebra has congruences that behave this well.
Boolean algebras are the main example.
Groups have permuting congruences but not distributive ones, and lattices have the reverse, so neither is arithmetical.
</document>

<document source="https://tamnd.github.io/math2/node/GW8SBF37K1.md" tag="GW8SBF37K1" kind="definition">
# Floquet theory

## Definition

Let $T > 0$ and let $A : \R \to \C^{n \times n}$ be continuous and [[periodic|089KWVYD69]] with period $T$.
*Floquet theory* is the theory of the [[linear system|0A6QZ367GX]] $x' = A(t)\, x$, based on the following objects.

1. For a [[fundamental matrix|0MQ24RBG93]] $\Phi$, the matrix $t \mapsto \Phi(t + T)$ is again a fundamental matrix, so $\Phi(t + T) = \Phi(t)\, M$ for a constant invertible matrix $M = \Phi(0)^{-1} \Phi(T)$, called the monodromy matrix.
2. A Floquet representation of $\Phi$ is a factorization $\Phi(t) = P(t)\, e^{tB}$ with a constant $B \in \C^{n \times n}$ and the [[matrix exponential|0K665H5V1S]] $e^{tB}$. Here $P : \R \to \C^{n \times n}$ is continuously differentiable, $T$-periodic and invertible at every $t$.

Floquet's theorem states that every fundamental matrix has a Floquet representation, with $B$ any matrix such that $e^{TB} = M$.

## Remarks

The eigenvalues of $M$ are the characteristic multipliers; they do not depend on the choice of $\Phi$, since another choice replaces $M$ by a similar matrix.
The change of variables $x = P(t)\, z$ turns the system into $z' = B z$ with constant coefficients.
All solutions tend to $0$ if and only if all multipliers have modulus less than $1$; there is a nonzero $T$-periodic solution if and only if $1$ is a multiplier.
Example: for $n = 1$ and $A(t) = 1 + \cos t$, $T = 2\pi$, the solution is $\Phi(t) = e^{t + \sin t}$, so $M = e^{2\pi}$, $B = 1$ and $P(t) = e^{\sin t}$, which is $2\pi$-periodic.
Non-example: if $A$ is not periodic, the solutions need not have this form.
For $A(t) = 2t$ the solution $e^{t^2}$ is not $P(t) e^{tB}$ with $P$ periodic, since then $\abs{\Phi(t)} \le C e^{t \operatorname{Re} B}$ would grow at most exponentially.
Over $\R$ one can take $B$ real if one allows $P$ to have period $2T$.
</document>

<document source="https://tamnd.github.io/math2/node/GWA0T1WKCN.md" tag="GWA0T1WKCN" kind="concept">
# Global dimension

## Summary

The global dimension of a ring is the supremum of the [[projective dimensions|MR9QCH1PNA]] of all its modules.
It is $0$ for fields and $1$ for principal ideal domains that are not fields.
It is $n$ for a polynomial ring in $n$ variables over a field.
By [[Serre's theorem|93RQRG9R1P]], a Noetherian local ring has finite global dimension exactly when it is a [[regular local ring|63XXTF6B5M]].

## Explanations

### L1 human

The global dimension says how complicated the modules over a ring can be, in the worst case.
A ring of global dimension $0$ is one where every module is projective, as for vector spaces over a field.
Over the integers, every module has a resolution of length at most $1$, so the global dimension of $\Z$ is $1$.
Over $\Z/4\Z$, the module $\Z/2\Z$ needs an infinite resolution, so the global dimension is infinite.
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<document source="https://tamnd.github.io/math2/node/GWCBGYG5RA.md" tag="GWCBGYG5RA" kind="example">
# Space of convergent sequences

## Example

Let $c$ be the set of [[convergent sequences|0W4CN9E2W5]] $x = (x_n)_{n \ge 1}$ of real (or complex) numbers, with the norm $\norm{x}_\infty = \sup_{n} \abs{x_n}$.
Then $c$ is a [[sequence space|XYTN19EPF2]] and a [[Banach space|0BDTTM24DP]].

Check that $c$ is a sequence space: the sum of two convergent sequences converges, and so does a scalar multiple of a convergent sequence.

Check that $c$ is complete: convergent sequences are bounded, so $c$ is a subspace of the Banach space $\ell^\infty$ of [[bounded sequences|G3F2CWQSQH]].
So it suffices to show that $c$ is closed in $\ell^\infty$.
Let $x^{(j)} \in c$ with $\norm{x^{(j)} - x}_\infty \to 0$ for some $x \in \ell^\infty$.
For all $m, n$ and $j$,
$$\abs{x_m - x_n} \le 2 \norm{x - x^{(j)}}_\infty + \abs{x^{(j)}_m - x^{(j)}_n}.$$
Choose $j$ to make the first term small, then use that $x^{(j)}$ is a [[Cauchy sequence|0QZMTN5AXX]].
So $x$ is a Cauchy sequence of numbers, hence convergent, and $x \in c$.

## Remarks

The map $x \mapsto \lim_n x_n$ is a linear functional on $c$ with $\abs{\lim_n x_n} \le \norm{x}_\infty$, so it is continuous.
Its kernel is the [[space of null sequences|WDH6QSH1Z6]] $c_0$.

## Explanations

### L1 human

The convergent sequences form a vector space: you can add them and scale them, and they still converge.
Measuring the size of a sequence by its largest term in absolute value makes this space complete, so limits of convergent sequences, in this size, are again convergent sequences.
</document>

<document source="https://tamnd.github.io/math2/node/GWCY1C1KQV.md" tag="GWCY1C1KQV" kind="statement">
# Bohr's approximation theorem

## Statement

Let $f \colon \R \to \C$ be a [[continuous function|0X3WGX95FE]].
Then $f$ is [[Bohr almost periodic|ED69H4C3PK]] if and only if the following holds.
For every $\varepsilon > 0$ there are $N \ge 1$, reals $\lambda_1, \dots, \lambda_N$ and complex numbers $c_1, \dots, c_N$ such that
$$\sup_{x \in \R} \Big| f(x) - \sum_{k=1}^{N} c_k e^{i \lambda_k x} \Big| < \varepsilon.$$

## Remarks

The sums $\sum c_k e^{i\lambda_k x}$ are called generalized trigonometric polynomials.
One direction is the easier one.
Each $e^{i\lambda x}$ is periodic, and a finite sum of Bohr almost periodic functions is again almost periodic, for example by Bochner's criterion with sequences of translates.
A uniform limit of almost periodic functions is almost periodic, by an $\varepsilon/3$ argument on almost periods.

The other direction is the substance of the theorem.
Every Bohr almost periodic function has a mean value $M(g) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} g(x) \, dx$.
So $f$ has Fourier coefficients $a(\lambda) = M(f(x) e^{-i\lambda x})$, and $a(\lambda) \ne 0$ for at most countably many $\lambda$.
Bohr proved the Parseval identity $M(\abs{f}^2) = \sum_\lambda \abs{a(\lambda)}^2$, and approximating sums can then be built from these frequencies, for example by the Bochner–Fejér summation method.
A modern proof uses the [[Bohr compactification|G852FHZ3W1]] of $\R$ and the [[Stone–Weierstrass theorem|1P8D3MK79H]] or the Peter–Weyl theorem on that compact group.

Example: $\cos x + \cos(\sqrt{2} x) = \frac12 \big( e^{ix} + e^{-ix} + e^{i\sqrt2 x} + e^{-i\sqrt2 x} \big)$ is itself a generalized trigonometric polynomial, hence almost periodic.
The theorem is a uniform analogue for all real frequencies of the density of [[trigonometric polynomials|J3NJQ5FZJ5]] among continuous periodic functions.

## Explanations

### L1 human

A Bohr almost periodic function is one that nearly repeats itself.
The theorem says these are exactly the functions that can be approximated, uniformly on the whole line, by finite sums of waves $e^{i\lambda x}$ with any real frequencies.
So almost periodic functions are built from pure waves, just as periodic functions are built from the harmonics of one basic frequency.

### L2 human

For a continuous periodic function, the Weierstrass approximation theorem for trigonometric polynomials gives uniform approximation by harmonics of the period.
Bohr's theorem removes the restriction that all frequencies are multiples of one frequency.
The approximating frequencies can be taken among the $\lambda$ with $a(\lambda) \ne 0$, which makes the Fourier series of $f$ a faithful description of $f$.
Equivalently, the space of Bohr almost periodic functions is the closure in the uniform norm of the span of the characters $x \mapsto e^{i\lambda x}$ of $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/GWGFT35TCZ.md" tag="GWGFT35TCZ" kind="definition">
# Analysis of variance

## Definition

Let $k \ge 2$ and $n_1, \dots, n_k \ge 1$, with $N = n_1 + \cdots + n_k > k$.
Let $y_{ij}$, for $i = 1, \dots, k$ and $j = 1, \dots, n_i$, be observations of $k$ groups.
Let $\bar y_i = \frac{1}{n_i} \sum_j y_{ij}$ be the mean of group $i$ and $\bar y = \frac{1}{N} \sum_{i,j} y_{ij}$ the overall mean.
The *one-way analysis of variance* is the decomposition
$$\sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij} - \bar y)^2 = \sum_{i=1}^k n_i (\bar y_i - \bar y)^2 + \sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij} - \bar y_i)^2,$$
written $SS_{\mathrm{tot}} = SS_{\mathrm{between}} + SS_{\mathrm{within}}$, together with the statistic
$$F = \frac{SS_{\mathrm{between}} / (k - 1)}{SS_{\mathrm{within}} / (N - k)},$$
defined when $SS_{\mathrm{within}} > 0$.
Consider the model $y_{ij} = \mu_i + \varepsilon_{ij}$ with independent errors $\varepsilon_{ij} \sim N(0, \sigma^2)$.
The *ANOVA $F$-test* of level $\alpha$ rejects the hypothesis $\mu_1 = \cdots = \mu_k$ when $F$ is large.
The threshold is the upper $\alpha$-quantile of the [[F-distribution|PBT8VRJADK]] with $k - 1$ and $N - k$ degrees of freedom.

## Remarks

The decomposition holds because, inside each group, the deviations $y_{ij} - \bar y_i$ sum to zero, so the cross term vanishes.
Under the hypothesis of equal means and [[normal|05AQE0QR0D]] errors, $F$ has the $F$-distribution with $k - 1$ and $N - k$ degrees of freedom.

Example: groups $\{1, 2, 3\}$ and $\{4, 5, 6\}$, so $k = 2$, $N = 6$, $\bar y_1 = 2$, $\bar y_2 = 5$, $\bar y = 3.5$.
Then $SS_{\mathrm{between}} = 3 \cdot 1.5^2 + 3 \cdot 1.5^2 = 13.5$ and $SS_{\mathrm{within}} = 2 + 2 = 4$.
Their sum $17.5$ equals $\sum (y - 3.5)^2 = 2(6.25 + 2.25 + 0.25)$.
So $F = 13.5 / (4/4) = 13.5$ on $1$ and $4$ degrees of freedom.

Non-example: if the groups have very different error variances, the hypothesis of a common $\sigma^2$ fails.
Then $F$ no longer has the $F$-distribution under equal means, and Welch's test is used instead.

The same idea extends to several factors, with one sum of squares for each main effect and interaction.
All these fixed-effects analyses are special cases of the [[general linear model|A06SGFCSCE]].
With $k = 2$, the $F$ statistic is the square of the two-sample $t$ statistic with pooled variance.
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<document source="https://tamnd.github.io/math2/node/GWGGZBPF60.md" tag="GWGGZBPF60" kind="definition">
# Enough projectives

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
The category $\mathcal{C}$ *has enough projectives* if for every object $X$ of $\mathcal{C}$ there are a [[projective object|YXMXAGVC1Q]] $P$ and an [[epimorphism|RVXDSY04KG]] $P \to X$.

## Remarks

Example: the category of modules over a [[ring|0B1261B5ED]] $R$ has enough projectives.
For a module $M$, let $F$ be the [[free module|0DZ9DKAKZR]] with basis $\set{e_m : m \in M}$.
The map $F \to M$ with $e_m \mapsto m$ is surjective, and free modules are [[projective|BFFA03GE3V]].
Non-example: the category of finite abelian groups has no nonzero projective object.
For $n \ge 2$ the quotient map $\Z/n^2\Z \to \Z/n\Z$ does not split, since $\Z/n^2\Z$ is cyclic, so $\Z/n\Z$ is not projective there.
Every nonzero finite abelian group has such a cyclic direct summand, and a direct summand of a projective object is projective.
So no epimorphism from a projective object onto $\Z/2\Z$ exists.
The dual notion is *enough injectives*: every object admits a monomorphism into an [[injective object|5QRFTBF4PG]].
</document>

<document source="https://tamnd.github.io/math2/node/GWJWP187QE.md" tag="GWJWP187QE" kind="statement">
# Hironaka's resolution for complex spaces

## Statement

Let $X$ be a [[reduced complex space|831N2Q4GCB]] whose topology has a countable basis.
Let $X_{\mathrm{reg}}$ and $X_{\mathrm{sing}}$ be its sets of [[regular and singular points|ZW5A88PCPK]].
Then there are a [[complex manifold|JSGWAN2RJV]] $\tilde{X}$ and a [[proper|15XEBGNGHC]] surjective holomorphic map $\pi \colon \tilde{X} \to X$ such that:

1. $\pi$ restricts to a biholomorphic map $\pi^{-1}(X_{\mathrm{reg}}) \to X_{\mathrm{reg}}$;
2. $\pi^{-1}(X_{\mathrm{sing}})$ is [[nowhere dense|0VQYCCQVJ5]] in $\tilde{X}$.

In particular $\pi$ is a [[modification|SB3MY7XQM9]] of $X$, called a resolution of singularities.

## Remarks

Hironaka proved [[resolution of singularities|MB52NDA6WN]] for algebraic varieties over fields of characteristic $0$; the method was then carried over to complex spaces.
The resolution is built from a sequence of [[blow-ups|C4KGYAFPN7]] along smooth centres contained in the singular locus, chosen by a local invariant that drops at each step.
For a non-compact space infinitely many blow-ups may be needed, but they form a locally finite family, and the canonical choice of centres lets local resolutions glue.
The map can also be chosen so that $\pi^{-1}(X_{\mathrm{sing}})$ is a hypersurface with normal crossings.

Example: for a reduced curve, the [[normalization|BP0ZYHGD0S]] is a resolution; for the cusp $\set{z_1^2 = z_2^3}$ it is $t \mapsto (t^3, t^2)$ from $\C$.
For the cone $\set{z_1 z_2 = z_3^2}$ in $\C^3$, one blow-up of the vertex gives a resolution whose exceptional set is a projective line.

## Explanations

### L1 human

A singular complex space can be replaced by a smooth one that maps onto it and agrees with it away from the singular points.
Over the singular points, the smooth space may have whole curves or surfaces instead of single points.
So questions about singular spaces can often be moved to manifolds, where calculus is available.

### L2 human

Resolution is used to define invariants of singularities and to prove local estimates such as the Łojasiewicz inequality.
It also extends integration and Hodge theory to singular spaces, and reduces questions on plurisubharmonic functions and multiplier ideals to normal crossing situations.
The resolution is not unique: blowing up a point of $\tilde{X}$ over $X_{\mathrm{sing}}$ gives another one.
</document>

<document source="https://tamnd.github.io/math2/node/GWKN5VG78A.md" tag="GWKN5VG78A" kind="concept">
# Modular form

## Summary

A modular form of weight $k$ is a [[holomorphic function|0NAHXR1FRH]] on the [[upper half-plane|X6XGGH5FPW]].
It transforms in a fixed way under the [[modular group|VEKV1EM77B]] and stays bounded at infinity.
Its Fourier expansion $\sum_{n \ge 0} a_n q^n$ in $q = e^{2 \pi i \tau}$ often has arithmetic coefficients.
The spaces of modular forms of a given weight are finite-dimensional, so identities between them can be proved by comparing a few coefficients.
Modular forms are central in modern number theory, for example in the proof of Fermat's Last Theorem through the [[modularity theorem|CS1NVJBZMH]].

## Explanations

### L1 human

A modular form is a function with an enormous amount of symmetry.
It is periodic, $f(\tau + 1) = f(\tau)$, and it also changes in a simple way under $\tau \mapsto -1/\tau$.
These symmetries are so strong that, for each weight, the modular forms form a space of small dimension.
The [[Eisenstein series|EYREVY31BN]] $E_4(\tau) = 1 + 240 \sum_{n \ge 1} \sigma_3(n) q^n$ is an example of weight $4$.
Coefficients of modular forms count things, such as the number of ways to write $n$ as a sum of squares.
</document>

<document source="https://tamnd.github.io/math2/node/GWS5QXH7WW.md" tag="GWS5QXH7WW" kind="definition">
# Club set

## Definition

Let $\kappa$ be a limit [[ordinal|0RDJK7EQYY]] of uncountable [[cofinality|NTE5C260S1]], and let $C \subseteq \kappa$.
The set $C$ is a club set in $\kappa$ if both conditions hold:

1. $C$ is closed: for each nonzero limit ordinal $\alpha < \kappa$, if $\sup(C \cap \alpha) = \alpha$, then $\alpha \in C$.
2. $C$ is unbounded: for each $\beta < \kappa$ there is $\gamma \in C$ with $\gamma \ge \beta$.

Here $\sup$ is the [[supremum|0BTMQ04556]] of a set of ordinals.

## Remarks

Example: the set $C$ of limit ordinals below $\omega_1$ is club in $\omega_1$.
It is unbounded, since $\beta + \omega \in C$ for each $\beta < \omega_1$.
It is closed, since a limit of limit ordinals is a limit ordinal.
Non-example: the set of successor ordinals below $\omega_1$ is unbounded but not closed.
Indeed the successor ordinals below $\omega$ have supremum $\omega$, which is not a successor.
Condition 1 says that $C$ is closed in the order topology of $\kappa$.
If $\kappa$ is a regular uncountable cardinal, the intersection of fewer than $\kappa$ club sets in $\kappa$ is club.
For such $\kappa$, the diagonal intersection $\set{\alpha < \kappa : \alpha \in \bigcap_{\xi < \alpha} C_\xi}$ of a sequence $(C_\xi)_{\xi < \kappa}$ of club sets is club.
For such $\kappa$, the range of a [[normal function|PXTF18X433]] from $\kappa$ to $\kappa$ is club, and so is its set of fixed points.
</document>

<document source="https://tamnd.github.io/math2/node/GWYE6745JK.md" tag="GWYE6745JK" kind="statement">
# Second main theorem of Nevanlinna theory

## Statement

Let $f$ be a [[meromorphic function|0NSD1TPHR0]] on $\C$ that is not constant, with [[Nevanlinna characteristic|YT6Z0GWS0W]] $T(r, f)$.
Let $q \ge 3$, and let $a_1, \dots, a_q$ be distinct points of the [[Riemann sphere|0CDMXGPSES]] $\C \cup \set{\infty}$.
For $a$ in the Riemann sphere, let $\overline{N}(r, a)$ be the counting function of the solutions of $f(z) = a$ counted without multiplicity, as in [[Nevanlinna theory|1EHH63VNAD]].
Then there is a set $E \subseteq (0, \infty)$ of finite Lebesgue measure such that
$$(q - 2)\, T(r, f) \le \sum_{j=1}^{q} \overline{N}(r, a_j) + S(r, f) \qquad (r \notin E),$$
where $S(r, f) = O\big(\log T(r, f) + \log r\big)$ as $r \to \infty$ outside $E$.

## Remarks

The proof rests on the lemma on the logarithmic derivative, which says that the mean of $\log^+ \abs{f'/f}$ on the circle $\abs{z} = r$ is small compared with $T(r, f)$ outside an exceptional set.
If $f$ is not rational, then $\log r = o(T(r, f))$, so $S(r, f) = o(T(r, f))$ outside $E$.
If $f$ has finite order, the exceptional set $E$ can be omitted.
For transcendental $f$, dividing by $T(r, f)$ and using the [[first main theorem|QJWCFAK4AK]] gives the defect relation $\sum_a \delta(a) \le 2$, where the sum runs over all points of the Riemann sphere.
For example, $e^z$ has $\delta(0) = \delta(\infty) = 1$, so the relation is sharp.

## Explanations

### L1 human

The first main theorem says that no value is taken too often.
The second main theorem says the opposite for almost all values: a meromorphic function takes nearly every value about as often as its growth allows.
At most two values can be strongly avoided.
This is a quantitative form of Picard's theorem.

### L2 human

[[Picard's little theorem|R2PHM118NG]] follows at once: if a transcendental meromorphic $f$ omitted three values, the right side would be $S(r, f) = o(T(r, f))$ while the left side is at least $T(r, f)$.
The theorem also implies that the set of values $a$ with $\delta(a) > 0$ is countable.
Counting without multiplicity yields results on ramification, such as bounds on the number of totally ramified values.
[[Ahlfors theory|M8611CV76A]] gives a geometric proof of a similar inequality, based on covering surfaces.
</document>

<document source="https://tamnd.github.io/math2/node/GWZN3EKEWD.md" tag="GWZN3EKEWD" kind="statement">
# Stallings theorem about ends of groups

## Statement

Let $G$ be a [[finitely generated group|082DWWXMRJ]] and let $\Gamma$ be its [[Cayley graph|0GYQK9NDJF]] for a finite generating set.
For a finite set $K$ of vertices and edges of $\Gamma$, count the infinite connected components of $\Gamma \setminus K$.
The number of ends $e(G)$ is the supremum of this count over all such $K$.
It does not depend on the generating set.
Then $e(G) > 1$ if and only if one of the following holds:

1. $G$ is an [[amalgamated free product|M848AHBRCM]] $A *_C B$ with $C$ finite, $C \neq A$ and $C \neq B$.
2. $G$ is an [[HNN extension|YEYERRY37K]] $A *_C$ with $C$ a finite [[subgroup|0X1JXT47GA]] of $A$.

## Remarks

The number of ends of a finitely generated group is always $0$, $1$, $2$ or $\infty$.
It is $0$ exactly for finite groups and $2$ exactly for infinite groups with an infinite cyclic subgroup of finite index.

The easy direction: if $G$ splits over a finite subgroup, then $G$ acts on the Bass–Serre tree of the splitting.
Removing a single edge of this tree separates the Cayley graph into at least two infinite parts.
The hard direction, due to Stallings, builds from a set of vertices with finite boundary in $\Gamma$ a tree on which $G$ acts with finite edge stabilisers.

Example: $\Z$ has two ends, since removing one vertex of the line leaves two rays; and $\Z$ is the HNN extension of the trivial group over the trivial subgroup.
Example: the [[free group|0KTW26C80J]] $F_2 = \Z * \Z$ is an amalgam over the trivial group, and it has infinitely many ends.
Example: $\Z^2$ has one end, since the complement of a finite set in the grid has one infinite component; so $\Z^2$ does not split over a finite subgroup.

## Explanations

### L1 human

Remove a big finite chunk from the Cayley graph of a group and count the infinite pieces that remain; the largest such count is the number of ends.
The line has two ends, the plane has one.
The theorem says that a group has more than one end exactly when it can be cut, algebraically, into two pieces glued along a finite group.
So a picture at infinity detects an algebraic splitting.

### L2 human

The theorem is a cornerstone of [[Bass–Serre theory|6M731X66RD]] and of [[geometric group theory|2T3218RXTD]].
Since the number of ends is a [[quasi-isometry|6FKYSH729T]] invariant, splitting over a finite subgroup is a quasi-isometry invariant property.
It implies that a finitely generated torsion-free group with a free subgroup of finite index is itself free.
Iterating splittings over finite groups leads to the accessibility theorem of Dunwoody for finitely presented groups.
</document>

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# Singularity theory

## Definition

Let $f \colon (\C^n, 0) \to (\C, 0)$ be a [[germ|VZXF7RNFKS]] of a [[holomorphic function|0NAHXR1FRH]] with $f(0) = 0$.
The point $0$ is a *singular point* of $f$ if it is a [[critical point|0YTCMFB4HC]], that is, if $\partial f/\partial x_i (0) = 0$ for $i = 1, \ldots, n$.
Two such germs $f$ and $g$ are *right equivalent* if $f = g \circ \varphi$ for a germ $\varphi \colon (\C^n, 0) \to (\C^n, 0)$ of a biholomorphic map.
*Singularity theory* studies singular germs of functions, maps and varieties up to such equivalences, and their invariants.

## Remarks

The same definitions apply to real smooth germs, with diffeomorphisms in place of biholomorphic maps.
Example: $f = x^2 + y^2$ has a singular point at $0$, since both partial derivatives vanish there.
It is right equivalent to $g = xy$ via $\varphi(x, y) = (x + iy, x - iy)$, since $(x + iy)(x - iy) = x^2 + y^2$.
Non-example: $f = x + y^2$ has $\partial f/\partial x = 1$ at $0$, so $0$ is not a singular point.
By the holomorphic Morse lemma, every germ with a nondegenerate critical point at $0$ is right equivalent to $x_1^2 + \cdots + x_n^2$.
An isolated singularity has a finite Milnor number $\mu(f) = \dim_\C \C\{x_1, \ldots, x_n\} / (\partial f/\partial x_1, \ldots, \partial f/\partial x_n)$, which is invariant under right equivalence.
</document>

<document source="https://tamnd.github.io/math2/node/GX74C07T0G.md" tag="GX74C07T0G" kind="concept">
# de Rham cohomology

## Summary

De Rham cohomology is a cohomology theory of [[smooth manifolds|F45VQ47GPV]] built from [[differential forms|0E3G0GRPZ6]].
Its groups measure how far closed forms, those with $d\omega = 0$, are from being exact, of the form $\omega = d\eta$.
They are real vector spaces that are diffeomorphism invariants, and by the [[de Rham theorem|K7Z2PQZ4WH]] they agree with singular cohomology with real coefficients.
This links calculus on a manifold to its topology.

## Explanations

### L1 human

In calculus, a vector field with zero curl on all of space is a gradient, but on the plane with a hole removed this can fail.
The swirling field around the hole has zero curl but is not a gradient: its line integral around the hole is not zero.
De Rham cohomology counts such obstructions in every degree.
For the punctured plane, there is exactly one independent obstruction in degree $1$, which detects the single hole.
</document>

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# Image of J

## Summary

The image of J is the image of the stable [[J-homomorphism|YYJPRD1HYW]] $J \colon \pi_r(SO) \to \pi_r^s$ inside the [[stable homotopy groups of spheres|HGZAYPQBXW]].
It is the part of the stable stems that comes from rotations, and it is known completely.
In degree $4k - 1$ it is cyclic of order equal to the denominator of $B_{2k}/4k$, where $B_{2k}$ is a [[Bernoulli number|0KC00K7KY4]].
It is the most regular and best understood part of the stable homotopy groups of spheres.

## Explanations

### L1 human

The stable homotopy groups of spheres look chaotic, but a part of them follows a clear rule.
That part comes from twisting spheres by rotations, and it is called the image of J.
Its sizes are given by Bernoulli numbers.
For instance, it has $24$ elements in degree $3$ and $240$ elements in degree $7$.
The rest of the stable groups is much harder to understand.
</document>

<document source="https://tamnd.github.io/math2/node/GX9Q2PYNZE.md" tag="GX9Q2PYNZE" kind="example">
# Sinai billiard

## Example

Let $\mathbb{T}^2 = \R^2 / \Z^2$ be the flat [[torus|03NM7DZXXC]], and let $D \subset \mathbb{T}^2$ be the image of a closed disc of radius $r$ with $0 < r < 1/2$.
The *Sinai billiard* is the [[billiard|XWMKYE038F]] on the table $Q = \mathbb{T}^2 \setminus \operatorname{int} D$.
Equivalently, it is the billiard in the unit square with a disc of radius $r$ removed from its centre.
A particle leaving through one side of the square comes back through the opposite side.
The table fits the definition: away from $\partial D$ the particle moves in straight lines, and on the circle $\partial D$ it reflects.
The boundary $\partial D$ is convex when seen from the disc, so it curves away from the table: the billiard is *dispersing*.

## Remarks

Sinai proved in 1970 that this billiard flow is ergodic and [[mixing|S3RJQ9CX2P]] with respect to the Liouville measure.
The mechanism is dispersion: a narrow beam of parallel trajectories hitting a convex obstacle spreads out after the reflection, which gives a positive [[Lyapunov exponent|NJ1FJSNK6A]].
Unfolding the torus gives the periodic Lorentz gas: a particle moving among discs placed at the points of $\Z^2$.

## Explanations

### L1 human

Put a round post in the middle of a square billiard table whose sides wrap around, as in a video game.
Each time the ball hits the post, nearby paths are pushed apart, like light reflected by a convex mirror.
After many bounces the ball forgets where it started, and in the long run it spends equal time in regions of equal area.
It was one of the first billiards proved to be chaotic in this strong sense.
</document>

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# Closed graph theorem

## Statement

Let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\R$ or $\C$, and let $T \colon X \to Y$ be a [[linear map|0NRYHJPQG2]].
Give $X \times Y$ the norm $\norm{(x, y)} = \norm{x} + \norm{y}$.
Then $T$ is [[continuous|03TM135G0N]] if and only if its [[graph|0JBJYT69ZV]] $\Gamma(T) = \set{(x, Tx) : x \in X}$ is [[closed|0ZCK8DQYN1]] in $X \times Y$.

## Remarks

If $T$ is continuous, its graph is closed because $Y$ is Hausdorff.
Conversely, if $\Gamma(T)$ is closed, it is a Banach space, and the projection $(x, Tx) \mapsto x$ is a continuous linear bijection from $\Gamma(T)$ onto $X$.
The [[bounded inverse theorem|MR5FJXR348]] makes its inverse $x \mapsto (x, Tx)$ continuous, and so $T$ is continuous.

In practice the closed graph condition is easier to check: it says that if $x_n \to x$ and $T x_n \to y$, then $y = Tx$.
One does not have to show that $T x_n$ converges.
Completeness is necessary.
Let $X = C^1[0, 1]$ and $Y = C[0, 1]$, both with the sup norm, and let $T f = f'$.
If $f_n \to f$ and $f_n' \to g$ uniformly, then $f$ is differentiable with $f' = g$, so the graph is closed.
But $T$ is not continuous, since $f_n(t) = t^n$ has $\norm{f_n} = 1$ and $\norm{f_n'} = n$; here $X$ is not complete.

## Explanations

### L1 human

To show that a linear map between complete normed spaces is continuous, it is enough to show that its graph is a closed set.
This means: whenever inputs converge and outputs converge, the limit output is the image of the limit input.
That is often much easier than finding a bound directly.

### L2 human

The theorem is used to prove automatic continuity: for example, a linear map $T$ defined on a whole Hilbert space with $\ip{Tx}{y} = \ip{x}{Ty}$ for all $x, y$ is bounded (Hellinger–Toeplitz theorem).
It is also used to show that a projection onto a closed subspace along a closed complement is bounded, see [[complemented subspace|Q0TY362K0V]].
Unbounded operators in quantum mechanics are typically closed operators defined only on dense subspaces, which is consistent with the theorem.
</document>

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# Obstacle problem

## Problem

Let $\Omega \subseteq \R^n$ be a bounded open set with Lipschitz boundary.
Let $\psi \colon \overline{\Omega} \to \R$ be a continuous function in the [[Sobolev space|M4CEZ5K9HP]] $H^1(\Omega)$ with $\psi \le 0$ on $\partial \Omega$, the obstacle.
Let
$$K = \set{v \in H^1_0(\Omega) : v \ge \psi \text{ almost everywhere in } \Omega}.$$
Find $u \in K$ that minimizes the Dirichlet energy
$$D(v) = \frac{1}{2} \int_\Omega \abs{\nabla v}^2 \, dx$$
over all $v \in K$.

## Remarks

The set $K$ is nonempty, closed and [[convex|02AS45T3H9]] in $H^1_0(\Omega)$, since it contains $\max(\psi, 0)$.
So the problem has exactly one solution, by the Stampacchia theorem for [[variational inequalities|6QS7X5GVMY]] and the Poincaré inequality.
Equivalently, $u \in K$ satisfies $\int_\Omega \nabla u \cdot \nabla (v - u) \, dx \ge 0$ for all $v \in K$.
When $u$ is regular enough, this is the complementarity system $u \ge \psi$, $\Delta u \le 0$, and $\Delta u = 0$ where $u > \psi$.
So $u$ is [[harmonic|0DEB0JMN0W]] in the open set $\set{u > \psi}$ and equals $\psi$ on the contact set $\set{u = \psi}$.
The boundary of the contact set inside $\Omega$ is not known in advance, so this is a [[free boundary problem|YR42E2W19Q]].
If $\psi$ is smooth, the solution has bounded second derivatives locally, and this is the best regularity possible in general, since $\Delta u$ jumps across the free boundary.
In dimension one with $\Omega = (-1, 1)$, the solution is the smallest concave function above $\psi$ that vanishes at $\pm 1$: it is linear off the contact set.

## Explanations

### L1 human

Stretch an elastic membrane over a frame and push an obstacle up from below.
The membrane rests on the obstacle in some places and is free elsewhere.
Where it is free, it takes the shape of least tension, a harmonic function.
The problem is to find the shape of the membrane and the region where it touches the obstacle.
That region is not known at the start: finding its edge is part of the problem.
</document>

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# Nullary operation

## Definition

Let $A$ be a set and let $A^0 = \set{()}$ be the set whose only element is the empty tuple $()$.
A *nullary operation* on $A$ is a [[finitary operation|EJ8YJVM9BW]] of arity $0$, that is, a function $c \colon A^0 \to A$.
The element $c(()) \in A$ is the *constant* given by $c$.

## Remarks

The map $c \mapsto c(())$ is a bijection from the set of nullary operations on $A$ to $A$.
So a nullary operation is usually written as an element of $A$.

Example: in $(\Z, +, 0)$ the constant $0$ is the nullary operation $c$ with $c(()) = 0$.

Non-example: on the empty set there is no nullary operation, because a function from the one-element set $A^0$ to $\emptyset$ does not exist.
This is why an algebra with a constant has a nonempty universe.
</document>

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# Bayesian information criterion

## Summary

The Bayesian information criterion (BIC), also called the Schwarz criterion, scores a fitted model by its maximized [[likelihood|0VW9E4QXR6]] with a penalty of $\ln n$ per parameter.
Among candidate models, one prefers the model with the smallest BIC.
It comes from a large-sample approximation of the marginal likelihood of a model in [[Bayesian inference|0JWRJX2RMP]].
Its penalty is heavier than that of the [[Akaike information criterion|CMAX3KF9H0]] once $n \ge 8$.

## Explanations

### L1 human

Like the AIC, the BIC trades the fit of a model against its number of parameters.
Its price for each parameter grows slowly with the sample size $n$, as $\ln n$.
So with much data, the BIC prefers simpler models than the AIC does.
For $n = 100$ observations, each parameter costs about $4.6$ units instead of $2$.
</document>

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# Cyclic algebra

## Definition

Let $L/K$ be a [[cyclic extension|VFE5B71E7Y]] of degree $n$, and let $\sigma$ be a generator of its Galois group.
Let $a \in K$ be nonzero.
The *cyclic algebra* $(L/K, \sigma, a)$ is the $L$-vector space with basis $1, u, u^2, \dots, u^{n-1}$, with the multiplication determined by
$$u \, x = \sigma(x) \, u \quad (x \in L), \qquad u^n = a.$$

## Remarks

It is the [[crossed product algebra|G21MZQCME9]] for the factor set $f(\sigma^i, \sigma^j) = 1$ if $i + j < n$ and $f(\sigma^i, \sigma^j) = a$ if $i + j \ge n$, for $0 \le i, j < n$.
So it is a [[central simple algebra|5HRKJBAY2V]] over $K$ of degree $n$, split by $L$.
It is isomorphic to $M_n(K)$ if and only if $a$ is a [[norm|KRNFEZWPY0]] from $L$ to $K$, that is, $a = N_{L/K}(b)$ for some $b \in L$.

Example: $L = \C$, $K = \R$, $\sigma$ complex conjugation and $a = -1$ give $\mathbb{H}$.
Since $N_{\C/\R}(z) = \abs{z}^2 \ge 0$, the number $-1$ is not a norm, and $\mathbb{H}$ is not split, as expected.
With $a = 1 = N_{\C/\R}(1)$ one gets $M_2(\R)$.

Non-example: if $L/K$ is Galois but not cyclic, such as $\Q(\sqrt{2}, \sqrt{3})/\Q$, there is no single generator $\sigma$, and the construction does not apply.

If the characteristic of $K$ is not $2$, the quaternion algebra $\left(\frac{a, b}{K}\right)$ is the cyclic algebra $(K(\sqrt{a})/K, \sigma, b)$ when $a$ is not a square in $K$.
</document>

<document source="https://tamnd.github.io/math2/node/GXX2WPVRJ2.md" tag="GXX2WPVRJ2" kind="statement">
# Integration in the transform variable

## Statement

Let $f \colon [0, \infty) \to \C$ be piecewise continuous and of [[exponential order|TKZAV3KR86]] $a$, that is, $\abs{f(t)} \le M e^{a t}$ for all $t \ge 0$.
Assume that $\int_0^1 \abs{f(t)} / t \, dt < \infty$, which holds for example when $\lim_{t \to 0^+} f(t)/t$ exists.
Let $F$ be the [[Laplace transform|05XT5AVJBK]] of $f$.
Then for every real $s > a$,
$$\mathcal{L}\left\{\frac{f(t)}{t}\right\}(s) = \int_s^\infty F(\sigma) \, d\sigma,$$
where the integral on the right is taken along the real axis and converges.

## Remarks

Write $\int_s^\infty F(\sigma) \, d\sigma = \int_s^\infty \int_0^\infty f(t) e^{-\sigma t} \, dt \, d\sigma$ and swap the integrals with [[Fubini's theorem|0R0T1VWH5X]].
The swap is allowed because $\int_0^\infty \abs{f(t)} e^{-s t} / t \, dt < \infty$ under the hypotheses.
The inner integral is $\int_s^\infty e^{-\sigma t} \, d\sigma = e^{-s t} / t$.
Example: with $f(t) = \sin t$, one gets $\mathcal{L}\{\sin t / t\}(s) = \int_s^\infty d\sigma / (\sigma^2 + 1) = \pi/2 - \arctan s$.
The hypothesis at $0$ is needed: for $f = 1$, the function $1/t$ has no Laplace transform, and $\int_s^\infty d\sigma / \sigma$ diverges.

## Explanations

### L1 human

Dividing a function by $t$ corresponds to integrating its transform from $s$ out to infinity.
It is the reverse of the rule that multiplying by $t$ corresponds to differentiating the transform.

### L2 human

The rule gives transforms of functions such as $\sin t / t$ and $(1 - e^{-t})/t$ that are hard to compute directly.
Letting $s \to 0^+$ in the example, with a limit argument, recovers the Dirichlet integral $\int_0^\infty \sin t / t \, dt = \pi / 2$.
Compare [[differentiation in the transform variable|VEPJN1NK3C]].
</document>

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# Klein quartic

## Description

The *Klein quartic* is the plane curve
$$X = \set{ [x : y : z] \in \mathbb{P}^2(\C) : x^3 y + y^3 z + z^3 x = 0 }.$$
It is a smooth curve of degree $4$, so it is a compact [[Riemann surface|5K0N8QMV6Q]] of [[genus|6W8TDV9VVN]] $\frac{(4-1)(4-2)}{2} = 3$ by the [[genus–degree formula|MQCBFACHXH]].

Two automorphisms can be checked by hand.
Let $\zeta = e^{2\pi i / 7}$.
The map $[x : y : z] \mapsto [\zeta^4 x : \zeta^2 y : \zeta z]$ preserves $X$, because $x^3 y$ is multiplied by $\zeta^{12+2} = 1$, $y^3 z$ by $\zeta^{6+1} = 1$, and $z^3 x$ by $\zeta^{3+4} = 1$.
The cyclic shift $[x : y : z] \mapsto [y : z : x]$ permutes the three monomials, so it also preserves $X$.

The full group of holomorphic automorphisms of $X$ is isomorphic to $\mathrm{PSL}(2, \F_7)$, of order $168$.
This equals the bound $84(g - 1) = 168$ of [[Hurwitz's automorphisms theorem|3GNKSCQSCD]] for $g = 3$.
Up to isomorphism, $X$ is the only compact Riemann surface of genus $3$ with $168$ automorphisms.

## Explanations

### L1 human

The Klein quartic is a surface with three holes that is as symmetric as a surface of its genus can be.
No compact Riemann surface of genus $3$ has more than $168$ symmetries, and this one has exactly $168$.
It can be built by gluing $24$ regular heptagons, with three at each corner, in the hyperbolic plane.
Its equation $x^3 y + y^3 z + z^3 x = 0$ is short, and one can see some symmetries directly: a cyclic shift of the variables, and a scaling by seventh roots of unity.
</document>

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# Unramified morphism of rings

## Summary

An unramified morphism of rings is a map $R \to A$ of commutative rings that makes $A$ a [[finitely generated|AC5TFFQT9M]] $R$-algebra.
Moreover its module of [[Kähler differentials|7HCFARHG7D]] $\Omega_{A/R}$ must be zero.
It generalizes finite separable field extensions and unramified extensions of rings of integers in number theory.
Locally, take a prime $\mathfrak q$ of $A$ and the prime $\mathfrak p$ of $R$ below it.
Then $\mathfrak p$ generates the maximal ideal of $A_{\mathfrak q}$, and the residue field extension is finite and separable.
An unramified map that is also flat and finitely presented is [[étale|FTRW0679E2]].

## Explanations

### L1 human

In number theory, a prime $p$ ramifies in a larger ring of integers when it becomes a square, or a higher power, of a prime upstairs.
In $\Z[i]$, the prime $2$ ramifies, since $2 = -i(1 + i)^2$.
An unramified map is one where nothing like this happens at any prime.
The test is short: the differentials of $A$ over $R$ must all vanish.
For $\Z[i] = \Z[x]/(x^2 + 1)$, the differentials are $\Z[i]/(2i)$, which is nonzero exactly because of the prime $2$.
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# Quantum dilogarithm

## Definition

Let $q \in \C$ with $\abs{q} < 1$.
The *quantum dilogarithm* is the function of $x \in \C$ given by the infinite [[q-Pochhammer symbol|Y1X8ZZQ7VM]]
$$(x; q)_\infty = \prod_{n=0}^{\infty} (1 - x q^n).$$
For $\abs{x} < 1$ it is related to the series
$$\operatorname{Li}_2(x; q) = \sum_{n=1}^{\infty} \frac{x^n}{n (1 - q^n)}$$
by $(x; q)_\infty = \exp\big(-\operatorname{Li}_2(x; q)\big)$, and $\operatorname{Li}_2(x; q)$ is also called the quantum dilogarithm.

## Remarks

The product converges for every $x$, because $\sum_n \abs{x q^n} < \infty$, so $(x; q)_\infty$ is an entire function of $x$; for $q \neq 0$ its zeros are the points $x = q^{-n}$, $n \ge 0$.
The relation with the series comes from $\log(1 - x q^n) = -\sum_{k \ge 1} x^k q^{nk} / k$ and summing the geometric series in $n$.

Check of the classical limit: for fixed $x$ with $\abs{x} < 1$ and real $q \to 1^-$, one has $(1 - q) / (1 - q^n) \to 1/n$, so $(1 - q) \operatorname{Li}_2(x; q) \to \sum_{n \ge 1} x^n / n^2 = \operatorname{Li}_2(x)$, the [[dilogarithm|G0X0NWSDB7]].

Non-example: for $\abs{q} = 1$ and $x \neq 0$ the product $\prod_{n \ge 0} (1 - x q^n)$ does not converge, since its factors do not tend to $1$.
For $\abs{q} = 1$ other versions are used, such as the non-compact quantum dilogarithm of Faddeev, given by an integral.

Euler's expansion gives $\frac{1}{(x; q)_\infty} = \sum_{n \ge 0} \frac{x^n}{(q; q)_n}$ for $\abs{x} < 1$.
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# De Bruijn sequence

## Description

Let $k \ge 1$ and $n \ge 1$ be integers, and let $A$ be an alphabet with $k$ letters.
A *de Bruijn sequence* of order $n$ over $A$ is a cyclic word $w = a_0 a_1 \cdots a_{k^n - 1}$ of length $k^n$ with the following property.
Every word of length $n$ over $A$ occurs exactly once as a cyclic factor $a_i a_{i+1} \cdots a_{i+n-1}$, with indices taken modulo $k^n$.
Such sequences exist for all $k$ and $n$.

## Remarks

Example: for $k = 2$ and $n = 3$, the word $00010111$ is a de Bruijn sequence.
Its eight cyclic factors of length $3$, starting at positions $0$ to $7$, are $000, 001, 010, 101, 011, 111, 110, 100$.
These are all the $2^3 = 8$ binary words of length $3$, each once.

Existence follows from graph theory.
The de Bruijn graph is the [[directed graph|0KFYGSHHF5]] whose vertices are the words of length $n - 1$, and in which each word $a_1 \cdots a_n$ is an edge from $a_1 \cdots a_{n-1}$ to $a_2 \cdots a_n$.
Each vertex has in-degree and out-degree $k$, and the graph is strongly connected.
By the directed form of [[Euler's theorem|020422MDQF]], it has an [[Eulerian circuit|05NWGBHKSZ]].
Reading the last letters of the edges along this circuit gives a de Bruijn sequence.

The number of distinct de Bruijn sequences of order $n$ over $k$ letters, counted as cyclic words, is $(k!)^{k^{n-1}} / k^n$.
For $k = 2$ and $n = 3$ this is $2^4 / 8 = 2$: the words $00010111$ and $00011101$.

## Explanations

### L1 human

A de Bruijn sequence is a shortest circular string that contains every possible block of a given length.
Think of a lock with a keypad that opens as soon as the last $n$ digits pressed are the code.
Typing a de Bruijn sequence tries every code while pressing far fewer keys than typing each code separately.
For binary codes of length $3$, the circular string $00010111$ contains all eight codes.
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# Virtually Haken conjecture

## Statement

Let $M$ be a compact connected [[orientable|702SNFQG0W]] irreducible [[3-manifold|R20TB2KEQQ]] with infinite [[fundamental group|S1XK3HXHFX]], where irreducible means that every embedded $2$-sphere bounds a $3$-ball.
Then $M$ has a finite [[covering space|3XZRRE165N]] that is a [[Haken manifold|7DP7H8M5B2]].

## Remarks

Agol completed the proof in 2012.
By the [[geometrization theorem|B27HB43K86]], the main case is that of a closed hyperbolic $3$-manifold.
Kahn and Markovic showed that its fundamental group contains many quasi-Fuchsian surface subgroups, and Bergeron and Wise deduced that it acts properly and cocompactly on a CAT(0) cube complex.
Agol then proved that such a cubulated hyperbolic group is virtually special in the sense of Haglund and Wise.
Wise's theory then gives a finite cover with positive first Betti number, which is Haken.
For example, Burton, Rubinstein and Tillmann showed that the [[Seifert–Weber space|FBVEG8HVTP]] is not Haken, so the theorem gives it a finite cover that is Haken.

## Explanations

### L1 human

Many closed $3$-dimensional spaces contain no essential surface, so they cannot be cut into simpler pieces.
The theorem says that one can always unwrap such a space a finite number of times to get a bigger space that does contain one.
So the cutting techniques for Haken manifolds apply, after passing to a finite cover.

### L2 human

Agol's work proves much more: every closed hyperbolic $3$-manifold has a finite cover that fibers over the circle, and its fundamental group is large and linear over $\Z$.
These results answer several of Thurston's questions on $3$-manifolds from 1982.
They also show that the fundamental groups of such manifolds are residually finite in a strong form, being virtually subgroups of right-angled Artin groups.
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# Conformal radius

## Summary

The conformal radius of a simply connected domain at a point measures the size of the domain as seen from that point.
It is the factor by which the Riemann map from the unit disc stretches lengths at the centre.
It is comparable to the distance to the boundary, up to a factor $4$.
It also appears as the constant term of the Green's function at its pole.

## Explanations

### L1 human

Map the unit disc onto a region so that the centre goes to a chosen point and the map has positive derivative there.
The derivative at the centre is the conformal radius.
For a disc of radius $R$ and its centre, the map $w \mapsto Rw$ works, so the conformal radius is $R$.
From a point near the edge of a region, the region looks small, and the conformal radius is small too.
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# Broyden's method

## Statement

Let $U \subseteq \R^n$ be open and let $F \colon U \to \R^n$ be continuously differentiable, with [[Jacobian matrix|047NDTEPCV]] $J(x)$.
Let $x^* \in U$ satisfy $F(x^*) = 0$, let $J(x^*)$ be [[invertible|0EB9N0S33C]], and let $J$ be [[Lipschitz continuous|0AW1E6YTJY]] near $x^*$.
*Broyden's method* starts from $x_0 \in U$ and an invertible $n \times n$ matrix $B_0$, and for $k \ge 0$ sets
$$x_{k+1} = x_k - B_k^{-1} F(x_k), \qquad s_k = x_{k+1} - x_k, \qquad y_k = F(x_{k+1}) - F(x_k),$$
$$B_{k+1} = B_k + \frac{(y_k - B_k s_k) \, s_k^T}{s_k^T s_k}.$$
Then there are $\varepsilon > 0$ and $\delta > 0$ such that, if $\norm{x_0 - x^*} < \varepsilon$ and $\norm{B_0 - J(x^*)} < \delta$, the iterates are defined and $x_k \to x^*$ superlinearly:
$$\lim_{k \to \infty} \frac{\norm{x_{k+1} - x^*}}{\norm{x_k - x^*}} = 0,$$
whenever $x_k \ne x^*$ for all $k$.

## Remarks

The update makes $B_{k+1}$ satisfy the secant equation $B_{k+1} s_k = y_k$, the analogue of the [[secant method|9ZC177PVPC]] slope in several variables.
Among all matrices that satisfy it, $B_{k+1}$ is the closest to $B_k$ in the [[Frobenius norm|JT8Q204E4R]], and $B_{k+1} - B_k$ has rank at most $1$.
By the Sherman–Morrison formula, $B_{k+1}^{-1}$ is obtained from $B_k^{-1}$ in $O(n^2)$ operations, so no linear system has to be solved from scratch.
The proof first shows that the errors decrease linearly while $B_k$ stays close to $J(x^*)$, and then that $(B_k - J(x^*)) s_k / \norm{s_k} \to 0$, which is the Dennis–Moré condition for superlinear [[convergence|YWZQC5E54B]].
The matrices $B_k$ need not converge to $J(x^*)$.
For $n = 1$ the method is the secant method.

## Explanations

### L1 human

[[Newton's method|07EXVNF9VP]] for a system of equations needs the full matrix of derivatives at every step, which can be expensive.
Broyden's method guesses this matrix once and then corrects the guess cheaply after each step, using only the change in the function values.
It is slower than Newton's method near the solution, but each step is much cheaper.

### L2 human

Broyden's method is the first and simplest quasi-Newton method for nonlinear systems.
Its update is the model for the BFGS and DFP updates in optimization, which keep the matrix symmetric.
In practice $B_0$ is often a finite-difference Jacobian, and a line search or trust region makes the method converge from farther away.
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# Krohn–Rhodes theory

## Definition

The basic notions of Krohn–Rhodes theory are the following.
Let $S$ and $T$ be [[semigroups|0XMEM406JM]].

1. $S$ *divides* $T$ if $S$ is the image of a homomorphism from a subsemigroup of $T$ onto $S$.
2. Let $(X, A)$ and $(Y, B)$ be [[transformation semigroups|7ZB7MRKPQ9]]. Their *wreath product* $(X, A) \wr (Y, B)$ is the transformation semigroup on $X \times Y$ consisting of all maps $(x, y) \mapsto (\varphi(y)(x), \, b(y))$, where $b \in B$ and $\varphi \colon Y \to A$ is any function.
3. The *flip-flop* is the transformation monoid on $\set{0, 1}$ consisting of the identity and the two constant maps.
4. A *Krohn–Rhodes decomposition* of a finite semigroup $S$ is a division of $S$ into the semigroup of an iterated wreath product $(X_1, A_1) \wr \cdots \wr (X_k, A_k)$.
   Each factor $(X_i, A_i)$ is the flip-flop or a finite [[simple group|0YMYMNQ0QZ]] acting on itself by left multiplication.

## Remarks

The set in clause 2 is closed under composition: applying $(\varphi', b')$ and then $(\varphi, b)$ gives the map of the same form with the function $y \mapsto \varphi(b'(y)) \circ \varphi'(y)$ and the element $b \circ b'$.
In the wreath product, the coordinate in $Y$ moves on its own, and it controls how the coordinate in $X$ moves; this is a cascade of two machines.

The Krohn–Rhodes theorem says that every finite semigroup $S$ has a Krohn–Rhodes decomposition in which each simple group divides $S$.

Example: a finite group $G$ with a composition series has a decomposition into its composition factors, with no flip-flops.
For $G = \Z/4\Z$, two copies of $\Z/2\Z$ suffice, as in the carry rule of a two-digit binary counter.

Non-example: the wreath product of two copies of $\Z/2\Z$ contains no flip-flop factor, and its semigroup is a finite group of bijections of $X \times Y$.
Every subsemigroup of a finite group is a group, and so is every homomorphic image of a group.
So the flip-flop, which is not a group, does not divide it, and every decomposition of a finite semigroup that is not a group needs at least one flip-flop.
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# Pseudo-Anosov map

## Summary

A pseudo-Anosov map is a homeomorphism of a surface that stretches by a factor $\lambda > 1$ along one singular [[foliation|98HTB7MSPX]].
It contracts by $1/\lambda$ along a transverse one.
It generalizes the [[Anosov diffeomorphisms|RCQH8P6DPQ]] of the torus to surfaces of higher genus.
The price is a finite set of singular points where the foliations have three or more prongs.
By the Nielsen–Thurston classification, pseudo-Anosov maps are the typical elements of the [[mapping class group|JV5NYG913V]].
They have rich dynamics: positive entropy $\log \lambda$, dense periodic points, and they are the monodromies of hyperbolic 3-manifolds fibering over the circle.

## Explanations

### L1 human

Take a square torus and apply the matrix $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$: it stretches along one direction and squeezes along a perpendicular one, and wraps around.
On a surface with two or more holes, like a pretzel, there is no such linear map.
But there are maps that stretch and squeeze in the same way everywhere except at a few special points.
Near such a point the stretching lines meet in a star with three or more arms, like the lines on a monkey saddle.
These maps are the pseudo-Anosov maps, and they mix the surface as thoroughly as possible.
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# Quasi-Frobenius ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
The ring $R$ is *quasi-Frobenius* if:

1. $R$ is left Noetherian, that is, $R$ is a [[Noetherian module|AK1ZRCXR32]] as a left module over itself;
2. $R$ is [[left self-injective|KQKJENEMSN]].

## Remarks

The definition is left-right symmetric: a ring satisfying the two clauses is also right Noetherian and right self-injective, and it is left and right Artinian.
Example: $\Z/n\Z$ for $n \ge 1$.
It is finite, so clause 1 holds, and it is self-injective by Baer's criterion, so clause 2 holds.
Non-example: $\Z$ is Noetherian but not self-injective, since the map $2\Z \to \Z$, $2m \mapsto m$, does not extend to $\Z$; clause 2 fails.
Non-example: $k[x, y]/(x^2, xy, y^2)$ over a field $k$ is Artinian, hence Noetherian, but not self-injective, so clause 2 fails.
Equivalent forms: $R$ is quasi-Frobenius if and only if every projective left $R$-module is injective, if and only if every injective left $R$-module is projective.
Every [[Frobenius algebra|8EVP1K2CZH]] is quasi-Frobenius.
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# Rational surface

## Definition

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]], and let $\mathbb{P}^2$ be the [[projective plane|ZT9A6JK8QF]] over $k$.
An [[algebraic surface|KFTECAATKG]] $X$ over $k$ is *rational* if there is a [[birational map|KA5890R9YM]] $X \dashrightarrow \mathbb{P}^2$.

## Remarks

Equivalently, the function field of $X$ is isomorphic to the field $k(s, t)$ of rational functions in two variables.
Example: $\mathbb{P}^1 \times \mathbb{P}^1$ is rational: on the open subsets $\mathbb{A}^1 \times \mathbb{A}^1$ and $\mathbb{A}^2$ the identity $(s, t) \mapsto (s, t)$ is an isomorphism, and it extends to a birational map $\mathbb{P}^1 \times \mathbb{P}^1 \dashrightarrow \mathbb{P}^2$.
Non-example: for an elliptic curve $E$, the surface $E \times \mathbb{P}^1$ is not rational, because the irregularity $q = h^1(\mathcal{O})$ is a birational invariant, equal to $0$ for $\mathbb{P}^2$ and to $1$ for $E \times \mathbb{P}^1$.
Every smooth rational surface is obtained from $\mathbb{P}^2$ or from a Hirzebruch surface by a finite sequence of blow-ups of points.
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# Almost periodic function

## Summary

An almost periodic function is a function that nearly repeats itself.
For any tolerance there are many shifts that change it by less than the tolerance.
On the real line, the uniformly almost periodic functions are exactly the uniform limits of finite sums $\sum c_k e^{i\lambda_k x}$ with arbitrary real frequencies $\lambda_k$.
They extend the theory of Fourier series from [[periodic functions|0ATF6Q0633]] to functions such as $\cos x + \cos(\sqrt{2} x)$.
The notion was created by Harald Bohr, and it extends to abelian groups, where it is linked to the [[Bohr compactification|G852FHZ3W1]].

## Explanations

### L1 human

The function $\cos x + \cos(\sqrt{2} x)$ is not periodic, because the two periods $2\pi$ and $\sqrt{2} \cdot 2\pi$ have an irrational ratio.
Still, one can find shifts $\tau$ for which both $\tau / 2\pi$ and $\sqrt{2}\tau / 2\pi$ are very close to whole numbers.
Shifting by such a $\tau$ changes the function by very little everywhere.
Functions with plenty of such near-periods are called almost periodic.
There are several precise versions: uniform ones due to Bohr, and mean-square ones due to Stepanov, Weyl and Besicovitch.
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# Inverse problem

## Definition

Let $X$ and $Y$ be [[normed vector spaces|03J5RAKB30]], or more generally metric spaces, and let $F \colon X \to Y$ be a map, called the forward map.
The elements of $X$ are the unknown parameters, and the elements of $Y$ are the data.
The *inverse problem* for $F$ is: given $y \in Y$, find $x \in X$ with $F(x) = y$.
In practice only noisy data $y^\delta$ with $\norm{y^\delta - y} \le \delta$ are known, and one seeks approximations of $x$ that converge to it as $\delta \to 0$.

## Remarks

The inverse problem is [[well-posed|W09B9NZ9B5]] in the sense of Hadamard when $F$ is a bijection with continuous inverse; otherwise it is ill-posed.
Example: let $X = Y = C[0, 1]$ with the maximum norm and $(Fx)(t) = \int_0^t x(s) \, ds$.
The inverse problem is differentiation: $x = y'$ when $y = Fx$.
It is ill-posed, since $y_n(t) = n^{-1} \sin(n^2 t)$ satisfies $\norm{y_n} \le n^{-1} \to 0$ while $\norm{y_n'} = n \to \infty$.
Non-example: computing $y = Fx$ from a given $x$ is the forward problem, not an inverse problem; here it is stable, since $\norm{Fx} \le \norm{x}$.
Standard regularization methods replace $F^{-1}$ by a family of stable maps, such as Tikhonov regularization $x_\alpha = \operatorname{argmin}_x \left( \norm{Fx - y^\delta}^2 + \alpha \norm{x}^2 \right)$ in Hilbert spaces.
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# Cheeger constant

## Definition

Let $G = (V, E)$ be a finite [[graph|0JD5C82QMB]] with $\abs{V} \ge 2$.
For $S \subseteq V$, let $\partial S$ be the set of edges with one end in $S$ and the other end in $V \setminus S$.
The Cheeger constant of $G$ is
$$h(G) = \min \set{ \frac{\abs{\partial S}}{\abs{S}} : S \subseteq V, \ 1 \le \abs{S} \le \frac{\abs{V}}{2} }.$$

## Remarks

Example: in the complete graph $K_n$ with $n \ge 2$, a set $S$ of size $k$ has $k(n - k)$ edges leaving it, so the ratio is $n - k$.
The minimum over $1 \le k \le n/2$ gives $h(K_n) = n - \floor{n/2} = \ceil{n/2}$.
For the path with $n$ vertices, the first $\floor{n/2}$ vertices have one edge leaving them, so $h \le 1/\floor{n/2}$.
The bound $\abs{S} \le \abs{V}/2$ cannot be dropped: for $S = V$ the ratio would be $0$ for every graph.
$h(G) > 0$ if and only if $G$ is [[connected|0FFAYNMAJZ]].
If $G$ is $d$-regular and $\lambda_2$ is the second largest eigenvalue of its adjacency matrix, then $\frac{d - \lambda_2}{2} \le h(G) \le \sqrt{2d(d - \lambda_2)}$.
The notation $h(G)$ and the names edge expansion and isoperimetric number are also used.
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# Lusin's theorem

## Statement

Let $n \ge 1$, let $\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$, and let $A \subseteq \R^n$ be a Lebesgue measurable set with $\lambda(A) < \infty$.
Let $f \colon A \to \R$ be a [[measurable function|YYNCJSR0PJ]].
Then for every $\varepsilon > 0$ there is a [[compact set|0R6EWFKRCF]] $K \subseteq A$ with $\lambda(A \setminus K) < \varepsilon$ such that the restriction $f|_K$ is [[continuous|0X3WGX95FE]].

## Remarks

One proof approximates $f$ by simple functions and uses that each measurable set is nearly closed.
For $f = \mathbf{1}_B$, choose closed $F_1 \subseteq B$ and $F_2 \subseteq A \setminus B$ with small leftover measure; $f$ is constant on each of the disjoint closed sets $F_1$ and $F_2$, so it is continuous on $F_1 \cup F_2$.
For general $f$, take a sequence of simple functions that converges to $f$ pointwise, and use Egorov's theorem to make the convergence uniform on a large compact set.
Example: $f = \mathbf{1}_{\Q}$ on $A = [0, 1]$ is discontinuous at every point, but its restriction to a compact set $K$ of irrational numbers with $\lambda(K) > 1 - \varepsilon$ is the constant $0$.
The conclusion is about $f|_K$, not about $f$: the points of $K$ need not be points of continuity of $f$.
By the Tietze extension theorem, there is also a continuous $g \colon \R^n \to \R$ with $g = f$ on $K$.
The same result holds for every Radon measure on a locally compact Hausdorff space, for functions on sets of finite measure.

## Explanations

### L1 human

A measurable function can be very irregular, such as a function that jumps between $0$ and $1$ on the rational and irrational numbers.
Lusin's theorem says that after removing a set of small measure, what is left of the function is continuous.
So every measurable function is continuous up to an arbitrarily small set.

### L2 human

This is the second of Littlewood's three principles.
It explains why continuous functions with compact support are dense in $L^p$ spaces for $1 \le p < \infty$, together with the density of simple functions.
It is a basic tool for passing from statements about continuous functions to statements about measurable ones.
In the other direction, a function with this property for every $\varepsilon$ is measurable, so the property characterises measurability on sets of finite measure.
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# Knot invariant

## Definition

Let $\mathcal{K}$ be the set of [[knots|QYEAZTYGPT]] in $\R^3$, and let $S$ be a set.
A *knot invariant* with values in $S$ is a map $f \colon \mathcal{K} \to S$ such that $f(K) = f(K')$ whenever the knots $K$ and $K'$ are equivalent.

## Remarks

Equivalently, a knot invariant is a map on the set of equivalence classes of knots.
The values may be numbers, polynomials, groups up to isomorphism, or homology theories up to isomorphism.
Example: the [[crossing number|SWJZ4QRF9W]] is a knot invariant with values in $\N$, because it is a minimum taken over the diagrams of all knots equivalent to $K$.
Non-example: the number of crossings of one chosen diagram of $K$ is not an invariant of $K$.
Indeed, a twist in a diagram of the unknot adds a crossing without changing the knot.
An invariant is *complete* if $f(K) = f(K')$ implies that $K$ and $K'$ are equivalent.
The same notion is used for links.
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# Knot invariant

## Summary

A knot invariant is a quantity attached to each [[knot|QYEAZTYGPT]] that takes the same value on equivalent knots.
Invariants are the main tool to prove that two knots are different: if an invariant differs, the knots are not equivalent.
Examples are the [[crossing number|SWJZ4QRF9W]], the [[knot group|AQVHM3FP0D]], [[tricolorability|KNAGK0W65G]], the [[Alexander polynomial|JKA25KDPQP]] and the [[Jones polynomial|4SPC17FPXE]].

## Explanations

### L1 human

To tell two knots apart, it is not enough to fail to untangle one into the other.
Instead one computes something from each knot that does not change when the knot is moved.
If the two results differ, the knots are surely different.
For example, the trefoil can be coloured with three colours by a certain rule and the unknot cannot, so the trefoil is really knotted.
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# Sobolev embedding theorem

## Statement

Let $n \ge 1$ and $1 \le p < \infty$.
Let $W^{1,p}(\R^n)$ be the [[Sobolev space|M4CEZ5K9HP]] of functions $u \in L^p(\R^n)$ whose first [[weak derivatives|C3EC148VZD]] lie in $L^p(\R^n)$.

1. If $p < n$, let $p^* = np/(n - p)$. There is a constant $C = C(n, p)$ such that $\norm{u}_{L^{p^*}} \le C \norm{\nabla u}_{L^p}$ for all $u \in W^{1,p}(\R^n)$. So $W^{1,p}(\R^n)$ embeds continuously in the [[Lp space|SASP14MA1Z]] $L^{p^*}(\R^n)$.
2. If $p > n$, let $\gamma = 1 - n/p$. Each $u \in W^{1,p}(\R^n)$ agrees almost everywhere with a bounded function that is [[Hölder continuous|012PBA0FMD]] with exponent $\gamma$. Its supremum norm and Hölder constant are at most $C(n, p) \norm{u}_{W^{1,p}}$.

## Remarks

Part 1 is the Gagliardo–Nirenberg–Sobolev inequality, and part 2 is Morrey's inequality.
For $p < n$, the space $W^{1,p}(\R^n)$ also embeds in $L^q(\R^n)$ for every $q$ between $p$ and $p^*$, by interpolation.

The exponent $p^*$ is forced by scaling.
For $u_\lambda(x) = u(\lambda x)$, one has $\norm{u_\lambda}_{L^q} = \lambda^{-n/q} \norm{u}_{L^q}$ and $\norm{\nabla u_\lambda}_{L^p} = \lambda^{1 - n/p} \norm{\nabla u}_{L^p}$.
The inequality of part 1 can hold for all $\lambda > 0$ only if $-n/q = 1 - n/p$, that is $q = p^*$.

For $p = n \ge 2$, functions in $W^{1,n}(\R^n)$ need not be bounded.
On the ball $\abs{x} < 1/2$ in $\R^2$, the function $\log \log (1/\abs{x})$ is in $W^{1,2}$ but unbounded near $0$.

## Explanations

### L1 human

If a function and its derivative are both small in an average sense, then the function cannot be too large anywhere on a big set.
The theorem says how much better the function is than its derivative.
With enough integrability of the derivative compared with the dimension, the function is even continuous.

### L2 human

Sobolev embeddings are the main tool to get regularity of weak solutions of partial differential equations.
One first finds a solution in a Sobolev space by energy methods, then uses embeddings to show that it is continuous or smooth.
On bounded domains with regular boundary the same embeddings hold, and the Rellich–Kondrachov theorem makes them compact for $q < p^*$.
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# Bergman space

## Definition

Let $\Omega$ be a [[domain|0EWMDD6C14]] in $\C$, let $dA$ be area measure on $\Omega$, and let $0 < p < \infty$.
The *Bergman space* $A^p(\Omega)$ is the set of [[holomorphic functions|0NAHXR1FRH]] $f \colon \Omega \to \C$ such that
$$\norm{f}_{A^p} = \left( \int_\Omega \abs{f(z)}^p \, dA(z) \right)^{1/p} < \infty.$$

## Remarks

The sub-mean value property of $\abs{f}^p$ gives $\abs{f(z)}^p \le \frac{1}{\pi \delta^2}\int_{\abs{w - z} < \delta} \abs{f(w)}^p \, dA(w)$ whenever the disc of radius $\delta$ about $z$ lies in $\Omega$.
So convergence in $A^p$ implies locally uniform convergence, and for $p \ge 1$ the space $A^p(\Omega)$ is a closed subspace of $L^p(\Omega)$, hence a Banach space.
On the [[unit disc|8G5ABXRK85]] $\mathbb{D}$, the monomials satisfy $\int_{\mathbb{D}} \abs{z}^{2n}\, dA = \int_0^{2\pi}\int_0^1 r^{2n+1}\, dr\, d\theta = \pi/(n + 1)$, so $\norm{\sum a_n z^n}_{A^2}^2 = \pi \sum_n \abs{a_n}^2/(n + 1)$.
Example: $g(z) = 1/(1 - z) = \sum z^n$ on $\mathbb{D}$ has $\abs{g(z)}^p = \abs{1 - z}^{-p}$, which is area-integrable near $z = 1$ exactly when $p < 2$; so $g \in A^p(\mathbb{D})$ for $p < 2$.
Non-example: $g \notin A^2(\mathbb{D})$, as also seen from $\pi \sum 1/(n + 1) = \infty$.
Some authors use the normalized area measure $dA/\pi$ on $\mathbb{D}$, which multiplies the norm by a constant.
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<document source="https://tamnd.github.io/math2/node/H03BGDDF0N.md" tag="H03BGDDF0N" kind="definition">
# Sinc function

## Definition

The *sinc function* is the function $\operatorname{sinc} \colon \R \to \R$ given by
$$\operatorname{sinc} x = \begin{cases} \dfrac{\sin x}{x} & \text{if } x \ne 0, \\ 1 & \text{if } x = 0, \end{cases}$$
where $\sin$ is the [[sine function|08WFZP6P89]].
The *normalised sinc function* is $x \mapsto \operatorname{sinc}(\pi x)$.

## Remarks

The value at $0$ makes the function continuous, since $\frac{\sin x}{x} \to 1$ as $x \to 0$.
In fact $\operatorname{sinc} x = \sum_{k \ge 0} \frac{(-1)^k x^{2k}}{(2k+1)!}$, so it extends to an entire even function.

Example: $\operatorname{sinc}(\pi/2) = \frac{1}{\pi/2} = \frac{2}{\pi}$, and the zeros of $\operatorname{sinc}$ are the points $k\pi$ with $k$ a non-zero integer.

Non-example: the function equal to $\frac{\sin x}{x}$ for $x \ne 0$ and to $0$ at $x = 0$ is not the sinc function; it fails the second clause and is not continuous at $0$.

The improper integral $\int_{-\infty}^{\infty} \operatorname{sinc} x\, dx = \pi$ converges, but $\operatorname{sinc}$ is not Lebesgue integrable on $\R$.
Some authors, mainly in signal processing, write $\operatorname{sinc}$ for the normalised version; Mathlib's `Real.sinc` is the unnormalised one above.
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<document source="https://tamnd.github.io/math2/node/H03GKRVBJ1.md" tag="H03GKRVBJ1" kind="concept">
# Arithmetic genus

## Summary

The arithmetic genus of a projective variety is an integer computed from the Euler characteristic of its structure sheaf.
For a connected reduced projective curve over an algebraically closed field, it is the dimension of the first cohomology of the structure sheaf.
It is constant in flat families, so it behaves well under degeneration, unlike the geometric genus.
For smooth projective curves it agrees with the usual [[genus|6W8TDV9VVN]].

## Explanations

### L1 human

The ordinary genus of a curve counts holes, but it can jump when a smooth curve degenerates to a singular one.
The arithmetic genus is a version of the genus that does not jump.
A smooth plane cubic has genus $1$; when it degenerates to the nodal cubic $y^2 = x^2(x + 1)$, the geometric genus drops to $0$, but the arithmetic genus stays $1$.
In a sense, the node keeps track of the lost hole.
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<document source="https://tamnd.github.io/math2/node/H0DB07WM20.md" tag="H0DB07WM20" kind="statement">
# Faltings's theorem

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]], and let $C$ be a smooth projective geometrically irreducible [[curve|XJM2MT4S3Y]] over $K$ of [[genus|6W8TDV9VVN]] $g \ge 2$.
Then the set $C(K)$ of points of $C$ defined over $K$ is finite.

## Remarks

The statement was conjectured by Mordell in 1922 and proved by Faltings in 1983; it was long called the Mordell conjecture.
Faltings's proof goes through the Tate conjecture for abelian varieties over number fields.
It also proves the Shafarevich conjecture: there are finitely many abelian varieties of given dimension with good reduction outside a given finite set of primes, up to isomorphism.
A construction of Parshin passes from curves to abelian varieties.
Vojta later gave a different proof by Diophantine approximation, simplified by Bombieri.
The genus hypothesis is necessary.
A curve of genus $0$ with one rational point is isomorphic to $\mathbb{P}^1$ and has infinitely many rational points.
An elliptic curve, of genus $1$, has infinitely many rational points when the rank in [[Mordell's theorem|BZHGKN6Q3K]] is positive.
Example: for $n \ge 4$, the Fermat curve $x^n + y^n = z^n$ has genus $(n-1)(n-2)/2 \ge 3$, so it has only finitely many rational points.
The proofs are not effective: they bound the number of points but give no algorithm to find them all.

## Explanations

### L1 human

A polynomial equation in two variables describes a curve, and the shape of the curve over the complex numbers has a number of holes, its genus.
Faltings's theorem says that once the genus is at least two, the equation has only finitely many rational solutions.
So the shape of a curve controls how many rational points it can have.

### L2 human

A curve of genus $0$ has no rational point or infinitely many, and for genus $1$ with a point the points form a finitely generated group.
Faltings's theorem completes this qualitative picture of rational points on curves.
For integral points on affine curves, the analogous finiteness is [[Siegel's theorem on integral points|4KTBRP27P6]], which already applies from genus $1$.
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# Determinate moment problem

## Summary

A [[moment problem|CS7TWHY7JM]] is determinate when exactly one measure has the given moments.
Then the moments determine the measure, and in probability the moments determine the law of a random variable.
This justifies the method of moments in limit theorems.
If the moments of a sequence of laws converge to those of a determinate law, the laws converge to it.
Determinacy holds on bounded sets, and on unbounded sets under growth conditions such as [[Carleman's condition|RVZ831CNYW]].

## Explanations

### L1 human

Knowing all moments of a distribution does not always tell you the distribution.
When it does, the moment problem is called determinate.
Every distribution on a bounded interval is determined by its moments.
So is the normal distribution, because its moments do not grow too fast.
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# Exponential integrator

## Statement

Let $A$ be a real $d \times d$ matrix, let $N \colon \R^d \to \R^d$ be continuous, and consider the [[initial value problem|0FXDW992DS]] $y' = A y + N(y)$, $y(t_0) = y_0$.
Put $\varphi_1(z) = (e^z - 1)/z$ for $z \ne 0$ and $\varphi_1(0) = 1$, so that $\varphi_1(hA) = \sum_{k \ge 0} \frac{(hA)^k}{(k+1)!}$.

1. The exact solution satisfies the variation-of-constants formula
$$y(t_0 + h) = e^{hA} y_0 + \int_0^h e^{(h - s)A} N(y(t_0 + s))\, ds,$$
where $e^{hA}$ is the [[matrix exponential|020M04FN1H]].
2. The exponential Euler method is the step
$$y_{n+1} = e^{hA} y_n + h\, \varphi_1(hA)\, N(y_n).$$
If $N$ is constant, it gives the exact solution at every step, for every $h > 0$.

## Remarks

Clause 1 follows by differentiating $e^{-tA} y(t)$.
For clause 2, if $N \equiv b$, the integral in clause 1 equals $\int_0^h e^{(h-s)A} b\, ds = h \varphi_1(hA) b$, which is the method.
In general, the exponential Euler method replaces $N(y(t_0 + s))$ by $N(y_n)$ in the integral; it has order $1$ when $N$ is Lipschitz and the solution is smooth.
Example: for the scalar equation $y' = -100 y + 1$ the method reproduces $y(t) = 1/100 + (y_0 - 1/100) e^{-100 t}$ exactly, whatever the step size.
Higher-order exponential Runge–Kutta and Rosenbrock-type methods use the functions $\varphi_k(z) = \sum_{j \ge 0} z^j / (j + k)!$.

## Explanations

### L1 human

Many equations split into a simple linear part, which may be very fast, and a slower remainder.
An exponential integrator solves the linear part exactly with a matrix exponential and approximates only the remainder.
So the fast part does not force tiny time steps.
When the remainder is constant, the method is exact.

### L2 human

Exponential integrators are a tool for [[stiff equations|RR5FCG92PN]] where the stiffness sits in a known linear operator, such as the discretised Laplacian of a semilinear parabolic equation.
They are explicit in $N$ and do not need the solution of nonlinear equations.
They need products of $\varphi_k(hA)$ with vectors, computed with Krylov methods, Padé approximants or contour integrals.
Their error analysis for parabolic problems uses that the bounds do not depend on the norm of $A$.
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<document source="https://tamnd.github.io/math2/node/H0NDB85RCD.md" tag="H0NDB85RCD" kind="concept">
# Conditioned Brownian motion

## Summary

A conditioned Brownian motion in a domain is the Doob h-transform of Brownian motion killed on leaving the domain, for a positive harmonic function $h$.
Take for $h$ the Poisson kernel or the Martin kernel with pole at a boundary point.
Then the process is Brownian motion conditioned to leave the domain at that point.
Conditioned Brownian motion gives probabilistic proofs of boundary limit theorems, such as Fatou-type theorems and the relative Fatou theorem.

## Explanations

### L1 human

Run a Brownian particle inside a region, but only keep the runs that leave the region at one chosen boundary point.
The resulting random motion is drawn towards that point and eventually exits there.
A standard example is Brownian motion on the half-line conditioned never to reach $0$: it is pushed away from $0$ and drifts off to infinity.
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# Partially well-ordered set

## Summary

A partially well-ordered set is a [[partially ordered set|0QV2JS9RJ3]] whose order is a [[well-quasi-ordering|NH25G6GC07]].
Such an order is also called a well-partial-order.
It generalizes a [[well-order|01CGSJ0GNQ]]: the order need not be total, but it is well-founded and every antichain is finite.
One also says that a subset of a preordered set is partially well-ordered when the restricted order has this property.

## Explanations

### L1 human

A well-order lines elements up so that every descent stops.
A partial well-order allows elements that cannot be compared, but only finitely many at a time in any antichain, and still no endless descent.
The set $\N \times \N$ ordered coordinatewise is an example: $(1, 2)$ and $(2, 1)$ are incomparable, yet every antichain is finite.
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<document source="https://tamnd.github.io/math2/node/H0T4RFG2AD.md" tag="H0T4RFG2AD" kind="definition">
# Steenrod square

## Definition

Write $H^*(X, A)$ for [[singular cohomology|6N7BC70CX0]] of a pair of spaces with coefficients in $\Z/2\Z$.
The *Steenrod squares* are the [[cohomology operations|K8T15QHN3W]]
$$\operatorname{Sq}^i \colon H^n(X, A) \to H^{n+i}(X, A), \quad i \ge 0, \ n \ge 0,$$
defined for all pairs $(X, A)$, that satisfy:

1. Naturality: $f^* \operatorname{Sq}^i = \operatorname{Sq}^i f^*$ for every map of pairs $f$.
2. Additivity: $\operatorname{Sq}^i(x + y) = \operatorname{Sq}^i x + \operatorname{Sq}^i y$.
3. Cartan formula: $\operatorname{Sq}^k(x \smile y) = \sum_{i + j = k} \operatorname{Sq}^i x \smile \operatorname{Sq}^j y$, with $\smile$ the [[cup product|JX47MS1R1H]].
4. Stability: $\operatorname{Sq}^i$ commutes with the connecting homomorphism $\delta \colon H^n(A) \to H^{n+1}(X, A)$.
5. $\operatorname{Sq}^0$ is the identity.
6. If $x$ has degree $n$, then $\operatorname{Sq}^n x = x \smile x$, and $\operatorname{Sq}^i x = 0$ for $i > n$.

There is exactly one family of operations with these properties.

## Remarks

The total square is $\operatorname{Sq} = \operatorname{Sq}^0 + \operatorname{Sq}^1 + \operatorname{Sq}^2 + \cdots$; on each class only finitely many terms are nonzero.
The square $\operatorname{Sq}^1$ is the [[Bockstein homomorphism|H7PCM776GR]] of the sequence $0 \to \Z/2\Z \to \Z/4\Z \to \Z/2\Z \to 0$.

Example: in $H^*(\R P^\infty) = (\Z/2\Z)[a]$ with $\deg a = 1$, one has $\operatorname{Sq} a = a + a^2$ by clauses 5 and 6.
By the Cartan formula, $\operatorname{Sq}(a^n) = (a + a^2)^n$, so $\operatorname{Sq}^i(a^n) = \binom{n}{i} a^{n+i}$.
For instance $\operatorname{Sq}^1(a^2) = 2a^3 = 0$ and $\operatorname{Sq}^1(a^3) = a^4$.

Non-example: the family with $\operatorname{Sq}^0 = \id$ and $\operatorname{Sq}^i = 0$ for all $i \ge 1$ satisfies clauses 1 to 5 but fails clause 6, since $\operatorname{Sq}^1 a$ would be $0$ instead of $a^2$ in $H^2(\R P^\infty)$.
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# Paratopological group

## Summary

A paratopological group is a [[group|G5S1M9]] with a topology in which multiplication is jointly continuous, while inversion need not be continuous.
It sits between a [[semitopological group|5WGMRS4JSE]] and a [[topological group|92700QAB5Y]].
The standard example that is not a topological group is the real line under addition with the lower limit topology.
Much of the theory asks which extra conditions force inversion to be continuous.

## Explanations

### L1 human

In a paratopological group, multiplying nearby elements gives nearby results, but taking inverses may tear the space apart.
Picture the real line where the basic open sets are the half-open intervals $[a, b)$.
Adding two numbers still behaves well: if $x$ and $y$ move a little to the right, so does $x + y$.
But the map $x \mapsto -x$ turns intervals $[a, b)$ into intervals $(-b, -a]$, which are not open in this topology.
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<document source="https://tamnd.github.io/math2/node/H11X2TCMAZ.md" tag="H11X2TCMAZ" kind="definition">
# Painlevé transcendents

## Definition

Let $\alpha, \beta, \gamma, \delta$ be complex constants, and let $y$ be a function of a complex variable $t$.
The six *Painlevé equations* are the [[ordinary differential equations|0NT64JV96A]]:

1. $\mathrm{P_I}$: $y'' = 6y^2 + t$;
2. $\mathrm{P_{II}}$: $y'' = 2y^3 + ty + \alpha$;
3. $\mathrm{P_{III}}$: $y'' = \frac{(y')^2}{y} - \frac{y'}{t} + \frac{\alpha y^2 + \beta}{t} + \gamma y^3 + \frac{\delta}{y}$;
4. $\mathrm{P_{IV}}$: $y'' = \frac{(y')^2}{2y} + \frac32 y^3 + 4ty^2 + 2(t^2 - \alpha) y + \frac{\beta}{y}$;
5. $\mathrm{P_V}$: $y'' = \left( \frac{1}{2y} + \frac{1}{y - 1} \right) (y')^2 - \frac{y'}{t} + \frac{(y - 1)^2}{t^2} \left( \alpha y + \frac{\beta}{y} \right) + \frac{\gamma y}{t} + \frac{\delta y (y + 1)}{y - 1}$;
6. $\mathrm{P_{VI}}$: $y'' = \frac12 \left( \frac1y + \frac{1}{y - 1} + \frac{1}{y - t} \right) (y')^2 - \left( \frac1t + \frac{1}{t - 1} + \frac{1}{y - t} \right) y' + \frac{y (y - 1)(y - t)}{t^2 (t - 1)^2} \left( \alpha + \frac{\beta t}{y^2} + \frac{\gamma (t - 1)}{(y - 1)^2} + \frac{\delta t (t - 1)}{(y - t)^2} \right)$.

A *Painlevé transcendent* is a solution of one of these six equations.

## Remarks

Each equation has the [[Painlevé property|8EV1G2Y6CC]]: the only movable singularities of its solutions are poles.
Example: for $\alpha = 1$, the function $y = -\frac1t$ solves $\mathrm{P_{II}}$.
Check: $y'' = -\frac{2}{t^3}$, and $2y^3 + ty + 1 = -\frac{2}{t^3} - 1 + 1 = -\frac{2}{t^3}$.
This solution is rational, so for special parameters some Painlevé transcendents are classical functions; the generic solution is not.
Non-example: a solution of $y'' = 6y^2$ is not a Painlevé transcendent, since this equation is not one of the six; its nonconstant solutions are translates of the Weierstrass function $\wp(t; 0, g_3)$.
Each of $\mathrm{P_I}$ to $\mathrm{P_V}$ is a limit of $\mathrm{P_{VI}}$ under a suitable coalescence of singular points and parameters.
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<document source="https://tamnd.github.io/math2/node/H129B6DA0F.md" tag="H129B6DA0F" kind="concept">
# Remez algorithm

## Summary

The Remez algorithm is an iterative method that computes the [[minimax approximation|ASWF03X5AM]] of a continuous function by polynomials of a given degree.
Each round has two steps.
First it fits a polynomial whose error has equal size and alternating signs on $n + 2$ reference points.
Then it moves the reference points to the places where the error is largest.
It was introduced by Evgeny Remez in 1934 and is a standard way to design polynomial and rational approximations in numerical libraries.

## Explanations

### L1 human

The best uniform approximation has an error curve that wiggles up and down with equal height at $n + 2$ places.
The algorithm guesses those places, forces the error to have equal height there, and then looks where the error is really largest.
It moves the guessed places there and repeats.
For a straight-line approximation of $x^2$ on $[-1, 1]$, starting from the points $-1, 0, 1$ already gives the answer $1/2$ in one step.
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# Width of a poset

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
The *width* of $P$ is the supremum of the cardinalities $\abs{A}$ of the [[antichains|03HF498P1J]] $A \subseteq P$.

## Remarks

Example: the set $\set{1, \ldots, 10}$ ordered by divisibility has width $5$.
The set $\set{6, 7, 8, 9, 10}$ is an antichain, and the five chains $\set{1, 2, 4, 8}$, $\set{3, 6}$, $\set{5, 10}$, $\set{7}$, $\set{9}$ cover the set, so no antichain has more than $5$ elements.
Non-example: the set $\set{2, 3, 4}$ has $3$ elements but is not an antichain there, since $2 \mid 4$, so it does not show that the width is at least $3$.
A nonempty poset has width $1$ exactly when it is a total order.
The width of an infinite poset can be infinite even when each antichain is finite.
[[Dilworth's theorem|GT224XDA4E]] says that a finite poset of width $w$ is a union of $w$ chains and of no fewer.
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# Spread

## Summary

A spread of a [[projective space|ZT9A6JK8QF]] is a family of subspaces of one fixed dimension that partitions the points.
Every point lies in exactly one member.
The best known case is a spread of lines of three-dimensional space.
Spreads matter because they build [[translation planes|JZ5J6REAMD]], through the André–Bruck–Bose construction.

## Explanations

### L1 human

Imagine filling all of space with lines so that no two lines meet and every point is on one line.
Such a family is a line spread.
Over the finite field with $q$ elements, three-dimensional projective space has $(q^2 + 1)(q + 1)$ points and each line has $q + 1$ of them, so a line spread has $q^2 + 1$ lines.
One such spread comes from viewing pairs of elements of the field with $q^2$ elements as a four-dimensional space over the field with $q$ elements.
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# Focal subgroup theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a [[prime number|01826KMG2H]], and let $P$ be a [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$.
Let $G'$ be the [[commutator subgroup|0ASGSTXSBW]] of $G$.
The *focal subgroup* of $P$ in $G$ is [[generated|0RP7VWQ0YN]] by all $x^{-1} y$ with $x, y \in P$ and $y$ [[conjugate|0DXD4QSZ8A]] to $x$ in $G$:
$$\operatorname{Foc}_G(P) = \langle x^{-1} y : x, y \in P, \ y = g^{-1} x g \text{ for some } g \in G \rangle.$$
Then
$$P \cap G' = \operatorname{Foc}_G(P).$$

## Remarks

The inclusion $\operatorname{Foc}_G(P) \subseteq P \cap G'$ is easy, because $x^{-1} g^{-1} x g$ is a commutator.
The other inclusion uses the [[transfer|3R2Y906QAQ]] $V \colon G \to P/P'$ and a formula for $V(x)$ with $x \in P$ in terms of conjugates of powers of $x$ that lie in $P$.

Example: $G = S_3$.
For $p = 3$, $P = A_3$, and $(1\,2\,3)$ is conjugate to $(1\,3\,2)$, so $\operatorname{Foc}_G(P)$ contains $(1\,2\,3)^{-1}(1\,3\,2) = (1\,2\,3)$ and equals $A_3 = P \cap G'$.
For $p = 2$, $P = \set{e, (1\,2)}$, the only element of $P$ conjugate to $(1\,2)$ is $(1\,2)$ itself, so $\operatorname{Foc}_G(P) = \set{e}$; indeed $P \cap A_3 = \set{e}$.

As a consequence, the largest abelian $p$-group that is a quotient of $G$ is isomorphic to $P / \operatorname{Foc}_G(P)$.
So the abelian $p$-quotients of $G$ can be read off from $P$ and the way $G$ conjugates the elements of $P$.

## Explanations

### L1 human

The commutator subgroup measures how far a group is from being abelian.
The theorem says that its part inside a Sylow subgroup $P$ is generated by the "jumps" between elements of $P$ that the whole group makes conjugate.
So a question about the whole group reduces to a question about $P$ and conjugation.

### L2 human

The focal subgroup theorem is a central result of transfer theory.
It tells when a finite group has a nontrivial abelian $p$-group as a quotient: exactly when $\operatorname{Foc}_G(P) \ne P$.
Like Burnside's normal $p$-complement theorem, it is proved with the transfer, and it leads to the study of fusion, that is, of which elements of $P$ are conjugate in $G$.
The [[Z* theorem|2XCJ8NNMXP]] is another result about fusion, for involutions.
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# Jacobian criterion

## Statement

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $P$ be a [[prime ideal|00MHHZ9SNE]] of the [[polynomial ring|0SDC934R52]] $k[x_1, \ldots, x_n]$, generated by $f_1, \ldots, f_t$.
Let $A = k[x_1, \ldots, x_n]/P$ and let $d$ be its [[Krull dimension|MVGBS3J1P7]].
Let $a \in k^n$ be a point with $f_1(a) = \cdots = f_t(a) = 0$, and let $\mathfrak m$ be the maximal ideal of $A$ generated by the classes of $x_1 - a_1, \ldots, x_n - a_n$.
Let $J(a)$ be the $t \times n$ [[Jacobian matrix|047NDTEPCV]] $\left( \frac{\partial f_i}{\partial x_j}(a) \right)$ of formal partial derivatives.
Then $\rank J(a) \le n - d$, and the [[localization|4C1AE84N5N]] $A_{\mathfrak m}$ is a [[regular local ring|63XXTF6B5M]] if and only if $\rank J(a) = n - d$.

## Remarks

Proof idea: let $M$ be the maximal ideal of $a$ in $k[x_1, \ldots, x_n]$.
By Taylor expansion at $a$, the image of $P$ in the $n$-dimensional space $M/M^2$ is spanned by the linear parts of the $f_i$, so it has dimension $\rank J(a)$.
Hence $\dim_k \mathfrak m/\mathfrak m^2 = n - \rank J(a)$.
Since $A$ is a finitely generated domain over $k$, the local ring $A_{\mathfrak m}$ has dimension $d$, and $\dim_k \mathfrak m/\mathfrak m^2 \ge d$ with equality exactly when $A_{\mathfrak m}$ is regular.

Example: for the cusp $f = y^2 - x^3$ in $k[x, y]$, one has $d = 1$ and $J = (-3x^2, \; 2y)$.
At $a = (0, 0)$ the rank is $0 \ne 1$, so the local ring at the origin is not regular.
At $a = (1, 1)$ the matrix is $(-3, 2)$, which has rank $1$ in every characteristic, so the local ring there is regular.

The hypothesis that the $f_i$ generate $P$ is needed, not only that they have the same zeros.
For $f = x^2$ in $k[x]$, the zero set is the point $0$ and the ring $k[x]/(x)$ is a field, hence regular, but $J(0) = (0)$ has rank $0 \ne 1 - 0$.

## Explanations

### L1 human

A point of a curve or surface defined by polynomials is smooth when the equations cut it out cleanly to first order.
The test is linear algebra: compute the matrix of partial derivatives at the point and its rank.
If the rank is as large as possible, namely the number of variables minus the dimension, the point is nonsingular.
For the cusp $y^2 = x^3$, all partial derivatives vanish at the origin, which is the sharp point.

### L2 human

The criterion turns the abstract notion of a [[regular local ring|63XXTF6B5M]] into a computation.
It identifies the Zariski cotangent space $\mathfrak m/\mathfrak m^2$ with the cokernel of the transposed Jacobian matrix at the point.
The set of singular points is therefore cut out by the minors of size $n - d$ of the Jacobian matrix together with $P$, so it is a closed subset.
In the language of [[Kähler differentials|7HCFARHG7D]], the matrix presents $\Omega_{A/k}$.
The criterion says that $A$ is [[smooth|286G0P1B69]] over $k$ near $a$ exactly when this module is free of rank $d$ near $a$.
Over a field that is not perfect, regularity and smoothness differ, and the rank condition then characterizes smoothness rather than regularity.
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# Viscosity solution

## Summary

A viscosity solution is a notion of weak solution for fully nonlinear first-order and second-order equations.
Such equations, for example Hamilton–Jacobi equations, need not have classical solutions.
It replaces derivatives of the solution, which may not exist, by derivatives of smooth test functions that touch the solution from above or from below.
The notion was introduced by Crandall and Lions, and for many equations it gives existence, uniqueness and stability under uniform limits.
Its name comes from the vanishing viscosity method.
In this method a small multiple of the Laplacian is added to the equation and then sent to zero.

## Explanations

### L1 human

Think of a function with a sharp corner, like a roof.
It has no derivative at the ridge, so it cannot satisfy a differential equation there in the usual sense.
The viscosity approach tests the function with smooth curves that touch it from above or from below at a point.
It asks the equation to hold as an inequality for the slopes of those curves.
For the equation $\abs{u'} = 1$ on $(-1, 1)$ with $u = 0$ at both ends, this singles out the roof $u(x) = 1 - \abs{x}$ and rejects the valley $\abs{x} - 1$.
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# Liouville–Neumann series

## Statement

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]] with $M = \max \abs{K}$, and let $K_n$ be its [[iterated kernels|9QNBFERMHD]].
Let $f \colon [a, b] \to \C$ be continuous and let $\lambda \in \C$ with $\abs{\lambda} M (b - a) < 1$.
Then the series
$$\varphi(x) = f(x) + \sum_{n=1}^{\infty} \lambda^n \int_a^b K_n(x, s) f(s) \, ds$$
converges absolutely and uniformly on $[a, b]$.
Its sum $\varphi$ is the unique continuous solution of the [[integral equation of the second kind|8KJCN49V64]]
$$\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds, \quad x \in [a, b].$$

## Remarks

Proof idea: $\abs{K_n} \le M^n (b - a)^{n-1}$, so the $n$-th term is bounded by $\max \abs{f} \cdot (\abs{\lambda} M (b - a))^n$, and the [[Weierstrass M-test|0NX5NWAFC9]] gives uniform convergence.
Substituting the series into the equation and integrating term by term shows that $\varphi$ is a solution.
If $\psi$ is a second continuous solution, then $\max \abs{\varphi - \psi} \le \abs{\lambda} M (b - a) \max \abs{\varphi - \psi}$, so $\varphi = \psi$.
This is the [[Neumann series|8QJ91709V2]] $(I - \lambda K)^{-1} = \sum_{n \ge 0} \lambda^n K^n$ for the integral operator on continuous functions with the maximum norm.

Example: for $K = 1$ on $[0, 1]$ and $\abs{\lambda} < 1$, all $K_n = 1$, and the series gives $\varphi = f + \frac{\lambda}{1 - \lambda} \int_0^1 f$.
The condition on $\lambda$ cannot be dropped in general: for $\lambda = 1$ the equation $\varphi = f + \int_0^1 \varphi$ has no solution when $\int_0^1 f \ne 0$.
For Volterra kernels the bound improves to $\abs{K_n(x, s)} \le M^n (x - s)^{n-1} / (n-1)!$, and the series converges for every $\lambda$.

## Explanations

### L1 human

The equation says that the unknown is $f$ plus a small correction computed from the unknown itself.
Start from $f$, compute the correction, add it, and repeat.
When $\lambda$ is small, each round changes the answer less, and the rounds add up to the exact solution.

### L2 human

The partial sums of the series are the iterates $\varphi_0 = f$, $\varphi_{k} = f + \lambda K \varphi_{k-1}$ of the method of successive approximations.
Collecting the terms gives the resolvent kernel $R = \sum_{n \ge 1} \lambda^{n-1} K_n$ and the formula $\varphi = f + \lambda \int R f$.
The series in $\lambda$ converges for $\abs{\lambda}$ below the smallest absolute value of a characteristic value of $K$, and for all $\lambda$ if there is none.
Beyond that disc one needs Fredholm determinants or the theory of compact operators.
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# Quermassintegral

## Summary

The quermassintegrals $W_0(K), \dots, W_n(K)$ of a convex body $K$ in $\R^n$ are [[mixed volumes|QB3Q5FBJ08]] of $K$ and the unit ball $B$.
In $W_j(K)$, the body $K$ is taken $n - j$ times and $B$ is taken $j$ times.
They are the coefficients of the [[Steiner formula|F02XA3MYX8]] for the volume of the parallel body $K + tB$.
They include the volume, a multiple of the surface area and a multiple of the mean width.
Each one is an average of volumes of projections of $K$ onto subspaces.

## Explanations

### L1 human

The quermassintegrals are a list of $n + 1$ numbers that measure a convex shape in different ways.
In the plane they are the area, half the perimeter, and the constant $\pi$.
In space they are the volume, one third of the surface area, a multiple of the mean width, and $\frac43 \pi$.
For a ball of radius $r$ in $\R^n$, the $j$-th quermassintegral is $\kappa_n r^{n-j}$, where $\kappa_n$ is the volume of the unit ball.
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# Analytic function between Banach spaces

## Summary

A map between Banach spaces is analytic at a point if near that point it is the sum of a convergent power series.
The terms of the series are continuous homogeneous polynomials, given by continuous multilinear maps.
This extends real and complex analytic functions of finitely many variables to infinite dimension.
Over the complex numbers, analytic maps are the same as holomorphic maps.

## Explanations

### L1 human

An analytic function of one variable is a convergent power series $\sum a_n x^n$.
For a map on a space of vectors, the monomial $a_n x^n$ is replaced by an expression of degree $n$ in the vector, such as $A(x, x, \ldots, x)$ for a multilinear map $A$.
A basic example is $(1 - x)^{-1} = 1 + x + x^2 + \cdots$ for a square matrix $x$ of small norm.
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# Character of an abelian group

## Summary

A character of an abelian group is a group homomorphism from the group to the circle group of complex numbers of absolute value $1$.
The characters of a group form a group under pointwise multiplication, the character group.
Characters are the "pure waves" of Fourier analysis on groups: every reasonable function on the group is built from them.

## Explanations

### L1 human

A character labels each element of the group by a point on the unit circle, so that adding elements multiplies the labels.
On the cyclic group $\Z/n\Z$, the map $k \mapsto e^{2\pi i k/n}$ is a character: going once around the group goes once around the circle.
More generally $k \mapsto e^{2\pi i jk/n}$ is a character for each $j$, and these are all of them.
On the real line, the maps $x \mapsto e^{i\xi x}$ for real $\xi$ are the continuous characters, and they are the waves used in the Fourier transform.
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# Convergence of an infinite product

## Definition

Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, and consider the [[infinite product|RZ0Z2BTMT4]] $\prod_{n=1}^{\infty} a_n$.
The product converges if there is an index $m$ such that:

1. $a_n \ne 0$ for all $n \ge m$;
2. the partial products $q_N = \prod_{n=m}^{N} a_n$ [[converge|0AZ3R52593]] to a limit $q$ as $N \to \infty$;
3. $q \ne 0$.

The value of the product is then $\prod_{n=1}^{\infty} a_n = a_1 \cdots a_{m-1} \, q$.
A product that does not converge diverges.

## Remarks

The value does not depend on the choice of $m$.
If the product converges, then $a_n \to 1$, since $a_n = q_n / q_{n-1}$ for $n > m$.
Example: $\prod_{n=2}^{\infty} (1 - 1/n^2)$ converges with value $\tfrac12$.
With $m = 2$, the partial products are $\prod_{n=2}^{N} \frac{(n-1)(n+1)}{n^2} = \frac{N+1}{2N}$, which tend to $\tfrac12 \ne 0$.
Non-example: $\prod_{n=2}^{\infty} (1 - 1/n)$ has partial products $1/N \to 0$, so it fails clause 3 for every $m$ and diverges.
The product $0 \cdot 2 \cdot 2 \cdot 2 \cdots$ also diverges, because its tail products are growing powers of $2$ with no limit.

## Explanations

### L1 human

A convergent product may contain a few zero factors, but the rest must multiply to a nonzero limit.
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# Integrally closed domain

## Summary

An integrally closed domain, or normal domain, is an [[integral domain|0QVD7WBFGN]] that contains every element of its [[field of fractions|0F2PTGX1A2]] that is [[integral|1EHN4KPSFS]] over it.
Every [[unique factorisation domain|0W4X9G6263]] is integrally closed, and so is every Dedekind domain and every valuation ring.
The [[going-down theorem|YE52MQ13MV]] holds over integrally closed domains.
In geometry, integrally closed domains are the coordinate rings of normal varieties, whose singular points form a set of codimension at least two.

## Explanations

### L1 human

A fraction that satisfies a monic equation with coefficients in the ring should already be in the ring.
For the integers this is the familiar fact that a rational root of $x^n + a_{n-1} x^{n-1} + \cdots + a_0$ with integer coefficients is an integer.
The ring $k[t^2, t^3]$ of polynomials without a linear term fails this test.
The fraction $t = t^3 / t^2$ is a root of $x^2 - t^2$, a monic polynomial over the ring, but $t$ itself is not in the ring.
Geometrically this ring describes the curve $y^2 = x^3$, which has a cusp at the origin.
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# Functional square root

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a [[function|0EPAT51DXQ]].
A *functional square root* of $f$ is a function $g \colon X \to X$ such that
$$g(g(x)) = f(x) \quad \text{for all } x \in X.$$
So it is an [[iterative root|27C6J3YNZY]] of order $2$ of $f$.

## Remarks

Example: on $X = (0, \infty)$, let $f(x) = \frac{x}{1 + 2x}$ and $g(x) = \frac{x}{1 + x}$.
Check: $g(g(x)) = \frac{x/(1+x)}{1 + x/(1+x)} = \frac{x}{1 + 2x} = f(x)$.

Non-example: $g(x) = \frac{x}{2}$ is not a functional square root of $f(x) = \frac{x}{2}$, since $g(g(x)) = \frac{x}{4} \ne \frac{x}{2}$ for $x \ne 0$.

Common notations are $g = f^{1/2}$ and $g = f^{[1/2]}$; the notation hides the fact that roots are not unique.
For instance $x \mapsto \pi - x$ and $x \mapsto x$ are both functional square roots of the identity of $\R$.
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# Generalized cohomology theory

## Summary

A generalized (or extraordinary) cohomology theory is a contravariant functor from pairs of spaces to graded abelian groups.
It satisfies all the [[Eilenberg–Steenrod axioms|YHK51GHK2T]] except the dimension axiom.
So the cohomology of a point may be nonzero in many degrees.
Topological K-theory, complex cobordism and stable cohomotopy are the main examples, and each such theory is represented by a spectrum.

## Explanations

### L1 human

Ordinary cohomology gives a point only one nonzero group, in degree $0$.
Some useful invariants of spaces behave exactly like cohomology in every other way, but give a point a whole sequence of nonzero groups.
These are the generalized cohomology theories.
For example, complex K-theory, built from vector bundles, assigns to a point the integers in every even degree and zero in every odd degree.
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# Painlevé property

## Definition

Let $U \subseteq \C$ be a domain, and consider an [[ordinary differential equation|0NT64JV96A]]
$$y^{(n)} = F\left( z, y, y', \ldots, y^{(n-1)} \right),$$
where $F$ is rational in $y, y', \ldots, y^{(n-1)}$ with coefficients holomorphic in $z \in U$.
Consider the singularities of solutions, reached by [[analytic continuation|0M3YQJ81NB]].
A singularity is *fixed* if its position is determined by the equation alone, such as a singular point of the coefficients.
Otherwise it is *movable*: its position depends on the initial conditions.
The equation has the *Painlevé property* if every movable singularity of every solution is a [[pole|066E7PJCYE]].

## Remarks

For the first, second and fourth Painlevé equations, the only fixed singular point is $\infty$, and every solution is a [[meromorphic|0ZZ7ZYDR8W]] function on all of $\C$.
For the other three, solutions can still branch at the fixed singular points, such as $0$, $1$ and $\infty$ for the sixth equation.
Some authors only require that no movable singularity is a branch point.

Example: the Riccati equation $y' = y^2$ has the solutions $y = 0$ and $y = \frac{1}{c - z}$, $c \in \C$.
The only singularity is the simple pole at $z = c$, which moves with $c$, so this movable singularity is a pole.
More generally, every Riccati equation $y' = a(z) + b(z) y + c(z) y^2$ has the Painlevé property.

Non-example: $y' = y^3$ has the solutions $y^2 = \frac{1}{2 (c - z)}$, so $y = \pm \left( 2 (c - z) \right)^{-1/2}$.
The point $z = c$ is a movable branch point and not a pole, so the property fails.

The first Painlevé equation $y'' = 6 y^2 + z$ has the Painlevé property, and its solutions are not expressible by classical functions in general.
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# Geometric Brownian motion

## Definition

Let $(W_t)_{t \ge 0}$ be a [[Wiener process|552AWX3QXN]], and let $\mu \in \R$, $\sigma > 0$ and $s_0 > 0$.
The *geometric Brownian motion* with drift $\mu$, volatility $\sigma$ and initial value $s_0$ is the process $(S_t)_{t \ge 0}$ given by
$$S_t = s_0 \exp\Bigl(\bigl(\mu - \tfrac{\sigma^2}{2}\bigr) t + \sigma W_t\Bigr).$$

## Remarks

By [[Itô's lemma|2J1RVQB0CM]], $S$ is the unique strong solution of $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$ with $S_0 = s_0$.
Check: with $f(t, x) = s_0 \exp((\mu - \sigma^2/2) t + \sigma x)$, Itô's lemma gives $dS_t = (\mu - \sigma^2/2) S_t \, dt + \sigma S_t \, dW_t + \tfrac{1}{2} \sigma^2 S_t \, dt$, which simplifies to the equation above.
The arithmetic process $s_0 + \mu t + \sigma W_t$ is not a geometric Brownian motion: it takes negative values with positive probability, while $S_t > 0$ always.
For each $t$, $\log S_t$ is normal with mean $\log s_0 + (\mu - \sigma^2/2) t$ and variance $\sigma^2 t$, so $S_t$ is [[log-normal|4W40NHWPQT]].
Its mean is $E[S_t] = s_0 e^{\mu t}$, and for $\mu = 0$ the process is a martingale.
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# Kan complex

## Definition

For $n \ge 1$ and $0 \le k \le n$, the *horn* $\Lambda^n_k$ is the simplicial subset of the standard $n$-simplex $\Delta^n$ generated by all faces $d_i \Delta^n$ with $i \ne k$.
A [[simplicial set|WW81ARB8K9]] $X$ is a *Kan complex* if for all $n \ge 1$ and $0 \le k \le n$, every map of simplicial sets $\Lambda^n_k \to X$ extends to a map $\Delta^n \to X$.

## Remarks

Equivalently, $X$ is a Kan complex when the map $X \to \Delta^0$ is a Kan fibration.
A map $\Lambda^n_k \to X$ is the same as a list of simplices $x_i \in X_{n-1}$ for $i \ne k$ with $d_i x_j = d_{j-1} x_i$ whenever $i < j$ and $i, j \ne k$.
A filler is an $x \in X_n$ with $d_i x = x_i$ for $i \ne k$.

Example: for a topological space $T$, the [[singular simplicial set|RWBMXQW6S5]] $\mathrm{Sing}(T)$ is a Kan complex.
A map $\Lambda^n_k \to \mathrm{Sing}(T)$ is a continuous map from the geometric horn $\abs{\Lambda^n_k}$ to $T$, and $\abs{\Lambda^n_k}$ is a retract of the topological simplex $\abs{\Delta^n}$, so the map extends.

Example: the [[nerve|RV0MKJ3CQ6]] of a [[groupoid|FH3C0T643F]] is a Kan complex.

Non-example: $\Delta^1$, the nerve of the poset $0 < 1$, is not a Kan complex.
The map $\Lambda^2_0 \to \Delta^1$ sending the vertices $0, 1, 2$ to $0, 1, 0$ is defined, since the edges $0 \to 1$ and $0 \to 2$ go to $0 \le 1$ and $0 \le 0$.
A filler would be an order-preserving map $[2] \to [1]$ with values $0, 1, 0$, which does not exist.
More generally, the nerve of a category is a Kan complex if and only if the category is a groupoid.
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# Projective space

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $n \ge 0$ be an integer.
On $k^{n+1} \setminus \set{0}$, let $x \sim y$ when $y = \lambda x$ for some $\lambda \in k \setminus \set{0}$; this is an [[equivalence relation|0G8VEZ3WTK]].
The *projective $n$-space* over $k$ is the set of equivalence classes
$$\mathbb{P}^n = \mathbb{P}^n(k) = \left(k^{n+1} \setminus \set{0}\right) / \sim.$$
The class of $(x_0, \dots, x_n)$ is written $[x_0 : \dots : x_n]$, and the $x_i$ are its homogeneous coordinates.

## Remarks

Equivalently, $\mathbb{P}^n$ is the set of one-dimensional [[linear subspaces|00YC1D652K]] of the [[vector space|000AYPA9AJ]] $k^{n+1}$.
More generally, $\mathbb{P}(V)$ denotes the set of one-dimensional subspaces of a vector space $V$.

Example: $[1 : 2 : 3] = [2 : 4 : 6]$ in $\mathbb{P}^2(\Q)$, because $(2, 4, 6) = 2 \cdot (1, 2, 3)$.
Over $\R$ with its usual topology, $\mathbb{P}^n(\R)$ is the [[real projective space|1DDYB6VVSR]].

Non-example: $(0 : 0 : 0)$ is not a point of $\mathbb{P}^2$, since the zero vector is excluded.

For each $i$, the set $U_i = \set{x_i \ne 0}$ is in bijection with $k^n$ through $[x_0 : \dots : x_n] \mapsto (x_0/x_i, \dots, \widehat{x_i/x_i}, \dots, x_n/x_i)$, and the $U_i$ cover $\mathbb{P}^n$.
The complement of $U_0$ is $\set{x_0 = 0}$, a copy of $\mathbb{P}^{n-1}$, the hyperplane at infinity.
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# Trigonometric functions

## Summary

The trigonometric functions are the six functions sine, cosine, tangent, cotangent, secant and cosecant of a real angle.
[[Sine and cosine|08WFZP6P89]] give the coordinates of a point on the unit circle.
The other four are quotients built from them, such as the [[tangent|05CG0S20S3]] $\tan = \sin / \cos$.
They are periodic, they describe rotations and oscillations, and they are the building blocks of Fourier series.
In analysis they are defined by power series, which makes the geometric facts into theorems.

## Explanations

### L1 human

Walk an angle $x$ (in radians) around the circle of radius $1$ starting from the point $(1, 0)$.
The point you reach is $(\cos x, \sin x)$.
The tangent $\tan x = \sin x / \cos x$ is the slope of the line from the centre to that point.
The secant, cosecant and cotangent are the reciprocals $1/\cos x$, $1/\sin x$ and $\cos x / \sin x$.
For example, at $x = \pi/4$ the point is $(\frac{\sqrt2}{2}, \frac{\sqrt2}{2})$, so $\tan(\pi/4) = 1$.
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# Geometric optics

## Definition

Let $\Omega \subseteq \R^n$ be open and let $c \colon \Omega \to (0, \infty)$ be smooth.
Consider the reduced [[wave equation|5EHD2XBQDG]] $\Delta u + \frac{\omega^2}{c(x)^2} u = 0$ at a high frequency $\omega = 1/\varepsilon$.
The *geometric optics ansatz* is a formal series
$$u(x) = e^{i\varphi(x)/\varepsilon} \sum_{k=0}^{\infty} \varepsilon^k a_k(x),$$
with a real phase $\varphi$ and amplitudes $a_k$.
Inserting it into the equation and setting the coefficient of each power of $\varepsilon$ to zero gives:

1. the [[eikonal equation|HGMX1QGTAZ]] $\abs{\nabla \varphi}^2 = c(x)^{-2}$, from the power $\varepsilon^{-2}$;
2. the [[transport equation|PVD7VZJZ9B]] $2\nabla\varphi \cdot \nabla a_0 + (\Delta\varphi)\, a_0 = 0$, from the power $\varepsilon^{-1}$;
3. the transport equations $2\nabla\varphi \cdot \nabla a_k + (\Delta\varphi)\, a_k = i\, \Delta a_{k-1}$ for $k \ge 1$.

*Geometric optics* is the approximation of $u$ by such a series, with $\varphi$ and the $a_k$ solving these equations.
The rays are the integral curves of the [[gradient|0ABK9V9Y64]] $\nabla\varphi$.

## Remarks

Example: with $c \equiv 1$, the phase $\varphi(x) = e \cdot x$ with a unit vector $e$ satisfies the eikonal equation, and $a_0 = 1$, $a_k = 0$ for $k \ge 1$ satisfy the transport equations.
The series reduces to the exact solution $e^{i e \cdot x/\varepsilon}$, a plane wave whose rays are straight lines in the direction $e$.
Non-example: $\varphi(x) = 2\, e \cdot x$ with $c \equiv 1$ and $\abs{e} = 1$ fails clause 1, since $\abs{\nabla \varphi}^2 = 4$.
The eikonal equation usually has smooth solutions only locally; where rays cross, at caustics, the approximation breaks down and other ansätze are needed.
The same procedure applies to time-dependent hyperbolic equations, with phases $\varphi(x, t)$.
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# Bogdanov–Takens bifurcation

## Definition

Let $f \colon \R^2 \times \R^2 \to \R^2$ be [[smooth|0V33BNKZ9S]], and consider the family of planar systems $x' = f(x, \mu)$ with a parameter $\mu \in \R^2$.
Let $x_0$ be an [[equilibrium point|0G4X36AE0X]] of the system with $\mu = \mu_0$, and let $A$ be the [[Jacobian matrix|047NDTEPCV]] of $x \mapsto f(x, \mu_0)$ at $x_0$.
The family has a *Bogdanov–Takens bifurcation* at $(x_0, \mu_0)$ if:

1. $0$ is an [[eigenvalue|0BQAA2BK3Q]] of $A$ of algebraic multiplicity $2$, that is, $\tr A = 0$ and $\det A = 0$;
2. $A \ne 0$, so that $A$ is similar to $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$;
3. two nondegeneracy conditions on the quadratic terms of $f(\cdot, \mu_0)$ at $x_0$ hold;
4. the map from $\mu$ to the unfolding parameters of the normal form is a local diffeomorphism at $\mu_0$.

## Remarks

Under these conditions, the family is locally topologically equivalent, near $(x_0, \mu_0)$, to the normal form
$$y_1' = y_2, \qquad y_2' = \beta_1 + \beta_2 y_1 + y_1^2 + s\, y_1 y_2, \qquad s = \pm 1,$$
with parameters $(\beta_1, \beta_2)$ near $0$.
In the $(\beta_1, \beta_2)$-plane, a saddle-node curve, a Hopf curve and a homoclinic bifurcation curve start at the origin.

Example: the normal form itself at $\beta = 0$ has the equilibrium $0$ with Jacobian $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, so clauses 1 and 2 hold.

Non-example: if $A = 0$, then $0$ is a double eigenvalue but clause 2 fails; this degenerate case needs more parameters to unfold.
In higher dimensions, the same definition applies to the system reduced to a two-dimensional centre manifold.
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# Exponential map of a Lie group

## Summary

Let $G$ be a [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{g} = T_eG$.
The exponential map sends $X \in \mathfrak{g}$ to the value at time $1$ of the [[one-parameter subgroup|YHA64AP1HN]] with initial velocity $X$.
For matrix groups it is the [[matrix exponential|020M04FN1H]].
It restricts to a diffeomorphism from a neighbourhood of $0$ onto a neighbourhood of the identity.
So it carries information from the Lie algebra to the group.
It is the main tool for passing between Lie groups and Lie algebras, and it differs in general from the [[exponential map|Y7Z0AGY1A7]] of Riemannian geometry.

## Explanations

### L1 human

Start at the identity of the group and move with a constant velocity $X$, in the sense that the motion is a one-parameter subgroup.
Where you are after one unit of time is $\exp(X)$.
On the circle group, with velocity $\theta$, you end at the point $e^{i\theta}$, which is the ordinary exponential function.
For a square matrix $A$, $\exp(A)$ is the series $I + A + A^2/2! + A^3/3! + \cdots$.
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# Ergodic transformation

## Definition

Let $(X, \mathcal{A}, \mu)$ be a [[probability space|0NFPJVJJHX]], and let $T \colon X \to X$ be a [[measure-preserving transformation|556WBDAHNP]].
A set $A \in \mathcal{A}$ is *$T$-invariant* if $T^{-1}(A) = A$.
The map $T$ is *ergodic* if every $T$-invariant set $A \in \mathcal{A}$ satisfies $\mu(A) = 0$ or $\mu(A) = 1$.

## Remarks

Equivalent forms: every measurable $f \colon X \to \R$ with $f \circ T = f$ almost everywhere is constant almost everywhere.
One can also require $\mu(T^{-1}(A) \mathbin{\triangle} A) = 0$ instead of $T^{-1}(A) = A$; this gives the same class of maps.
Example: the rotation $T(x) = x + \alpha \bmod 1$ on $[0,1)$ with Lebesgue measure and $\alpha$ irrational.
If $A$ is invariant, the Fourier coefficients of $f = 1_A$ satisfy $\hat f(k) e^{2 \pi i k \alpha} = \hat f(k)$.
Since $k \alpha \notin \Z$ for $k \ne 0$, all $\hat f(k)$ with $k \ne 0$ vanish, so $f$ is constant almost everywhere and $\mu(A) \in \set{0, 1}$.
Non-example: for $\alpha = 1/2$ the set $A = [0, 1/4) \cup [1/2, 3/4)$ is invariant with $\mu(A) = 1/2$.
The identity map is not ergodic on $[0,1]$ either, since every measurable set is invariant.
The definition extends to $\sigma$-finite measures: then an invariant set must be null or have null complement.
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# Congruence relation

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.
A *congruence* on $\mathbf{A}$ is a relation $\theta \subseteq A \times A$ such that:

1. $\theta$ is an [[equivalence relation|0G8VEZ3WTK]] on $A$;
2. for each fundamental operation $f$ of $\mathbf{A}$, of arity $n$, and all $a_1, b_1, \dots, a_n, b_n \in A$ with $(a_i, b_i) \in \theta$ for each $i$, also $(f(a_1, \dots, a_n), f(b_1, \dots, b_n)) \in \theta$.

## Remarks

Clause 2 is called the compatibility property.
One often writes $a \mathrel{\theta} b$ or $a \equiv b \pmod{\theta}$ for $(a, b) \in \theta$.
Every algebra has the trivial congruences $\Delta_A = \set{(a, a) : a \in A}$ and $\nabla_A = A \times A$.

Example: on the ring $(\Z, +, \cdot, -, 0, 1)$, fix $n \ge 1$ and let $a \mathrel{\theta} b$ when $n$ divides $a - b$.
If $n$ divides $a - a'$ and $b - b'$, it divides $(a + b) - (a' + b')$, $-a - (-a')$ and $ab - a'b' = a(b - b') + (a - a')b'$.

Non-example: on the group $(\Z, +, -, 0)$, let $a \mathrel{\theta} b$ when $a$ and $b$ are both positive, both negative, or both $0$.
This is an equivalence relation, but $1 \mathrel{\theta} 2$ while $1 + (-1) = 0$ and $2 + (-1) = 1$ are not related.

In a group, the congruences correspond to the normal subgroups via $\theta \mapsto \set{a : a \mathrel{\theta} e}$.
In a ring, they correspond to the two-sided ideals via $\theta \mapsto \set{a : a \mathrel{\theta} 0}$.
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<document source="https://tamnd.github.io/math2/node/H408H4WHBA.md" tag="H408H4WHBA" kind="definition">
# Linear logic

## Definition

The formulas of propositional linear logic are built from propositional variables with these connectives and constants.

1. Multiplicatives: the conjunction $A \otimes B$ (tensor), the disjunction "$A$ par $B$", and the units $1$ and $\bot$.
2. Additives: the conjunction $A \mathbin{\&} B$ (with), the disjunction $A \oplus B$ (plus), and the units $\top$ and $0$.
3. Exponentials: $!A$ (of course) and $?A$ (why not).
4. Linear implication: $A \multimap B$.

Linear logic is the set of sequents $\Gamma \vdash \Delta$, with $\Gamma$ and $\Delta$ finite lists of formulas, derivable in Girard's [[sequent calculus|W72CA3R9MP]].
That calculus has the logical rules of each connective and the exchange rule.
It has weakening and contraction only for formulas $!A$ on the left and $?A$ on the right.

## Remarks

The two conjunctions differ in how they treat the context.
The right rule of $\otimes$ derives $\Gamma, \Gamma' \vdash A \otimes B, \Delta, \Delta'$ from $\Gamma \vdash A, \Delta$ and $\Gamma' \vdash B, \Delta'$, so the context is split.
The right rule of $\&$ derives $\Gamma \vdash A \mathbin{\&} B, \Delta$ from $\Gamma \vdash A, \Delta$ and $\Gamma \vdash B, \Delta$, so the context is shared.

Example: $A \vdash A \mathbin{\&} A$ is derivable for a variable $A$: apply the $\&$ rule to two copies of the axiom $A \vdash A$.

Non-example: $A \vdash A \otimes A$ is not derivable for a variable $A$.
By cut elimination, a derivation can be taken cut-free, and then its last rule is the $\otimes$ rule.
That rule would need $A$ for one premise and the empty context for the other, and $\vdash A$ is not derivable.
In classical [[propositional logic|09K9MWD2QQ]], $A \to A \land A$ holds, which shows the role of contraction.

## Explanations

### L1 human

Linear logic counts its hypotheses.
The sequent $A \vdash B$ reads "one copy of $A$ can be turned into one copy of $B$".
The exponential $!A$ marks a hypothesis that may be copied or discarded freely, and so recovers ordinary reasoning when needed.
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# Perfect set property

## Summary

A subset of a [[Polish space|584GXXY73M]] has the perfect set property if it is either countable or contains a nonempty [[perfect set|628242KN0T]].
A nonempty perfect subset of a Polish space has the [[cardinality of the continuum|0WC28W9DNM]].
So an uncountable set with this property is as large as the whole real line.
Thus the [[continuum hypothesis|0MN2MER9WN]] holds for every class of sets that has the property.
Closed sets, Borel sets and [[analytic sets|3RWWZYPAFV]] have the perfect set property, but the axiom of choice gives sets of reals without it.

## Explanations

### L1 human

The property says that a set of reals cannot be uncountable in a thin or scattered way.
Either the set can be listed as a sequence, or it contains a whole copy of the Cantor set.
For example, the interval $[0, 1]$ is uncountable and is itself perfect: it is closed and none of its points is isolated.
The set of rational numbers is countable, so it has the property for the other reason.
Descriptive set theory asks which definable sets have the property, and the answer depends on the axioms one adopts.
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# Weak Lp space

## Summary

The weak $L^p$ space $L^{p,\infty}(\mu)$ contains the measurable functions whose [[distribution function|HQVXKEMN5N]] decays at least as fast as $t^{-p}$.
It contains [[$L^p(\mu)$|SASP14MA1Z]], by Chebyshev's inequality.
It is the natural target space for operators that fail to be bounded on $L^1$, such as the [[Hardy–Littlewood maximal function|VX1A56P5KQ]].
It is the [[Lorentz space|3240RP14MT]] $L^{p,q}$ with $q = \infty$.

## Explanations

### L1 human

A function is in $L^p$ when $\abs{f}^p$ has finite integral.
Weak $L^p$ asks only that the set where $\abs{f}$ exceeds $t$ has measure at most $C t^{-p}$.
This allows functions that just barely fail to be in $L^p$.
On $\R$, the function $1/\abs{x}$ is not integrable near $0$, but the set where it exceeds $t$ is an interval of length $2/t$, so it lies in weak $L^1$.
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# Empirical likelihood

## Definition

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] random vectors in $\R^d$, and let $\mu \in \R^d$.
The *empirical likelihood ratio* for the mean $\mu$ is
$$\mathcal{R}(\mu) = \max\Big\{ \prod_{i=1}^{n} n w_i \;:\; w_i \ge 0, \ \sum_{i=1}^{n} w_i = 1, \ \sum_{i=1}^{n} w_i X_i = \mu \Big\},$$
with $\mathcal{R}(\mu) = 0$ when no weights satisfy the constraints.
More generally, for a parameter $\theta$ defined by $E[m(X, \theta)] = 0$, the constraint $\sum_i w_i X_i = \mu$ is replaced by $\sum_i w_i m(X_i, \theta) = 0$.

## Remarks

The factor $n$ makes the unconstrained maximum equal to $1$: without the mean constraint, $\prod w_i$ is largest at $w_i = 1/n$.
Feasible weights exist exactly when $\mu$ lies in the [[convex hull|0TCHAS04WK]] of $X_1, \dots, X_n$.
When $\mu$ is in the interior of the hull, the maximiser is $w_i = \frac{1}{n(1 + \lambda^{T}(X_i - \mu))}$, where $\lambda \in \R^d$ solves $\sum_i \frac{X_i - \mu}{1 + \lambda^{T}(X_i - \mu)} = 0$.
Owen's theorem: if $\mu_0 = E[X_1]$ and the covariance matrix of $X_1$ is finite and nonsingular, then $-2 \ln \mathcal{R}(\mu_0)$ converges in distribution to the [[chi-squared distribution|0RFMMBHFTV]] with $d$ degrees of freedom.
So $\set{\mu : -2\ln\mathcal{R}(\mu) \le c}$, with $c$ the $(1-\alpha)$-quantile of $\chi^2_d$, is an asymptotic confidence region of level $1 - \alpha$.

Example: $\mu = \bar X$.
The weights $w_i = 1/n$ satisfy the constraint, so $\mathcal{R}(\bar X) = 1$, the largest possible value.

Non-example: data $(1, 2, 3)$ in $\R$ and $\mu = 5$.
No weighted average of $1, 2, 3$ equals $5$, because $5$ is outside the convex hull $[1, 3]$, so $\mathcal{R}(5) = 0$.
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# Continuous wavelet transform

## Definition

Let $\psi \in L^2(\R)$ be a nonzero function that satisfies the [[admissibility condition|2BBB8NY5ZA]]
$$C_\psi = \int_\R \frac{\abs{\hat\psi(\xi)}^2}{\abs{\xi}} \, d\xi < \infty,$$
where $\hat\psi(\xi) = \int_\R \psi(t) e^{-2\pi i t \xi} \, dt$ is the [[Fourier transform|DPVM34ADZV]].
Let $f \in L^2(\R)$ (see [[Lp space|SASP14MA1Z]]).
Let $a \ne 0$ (the scale) and $b$ (the position) be real numbers.
The *continuous wavelet transform* of $f$ with respect to $\psi$ is
$$W_\psi f(a, b) = \frac{1}{\sqrt{\abs{a}}} \int_\R f(t) \, \overline{\psi\!\left(\frac{t - b}{a}\right)} \, dt.$$

## Remarks

The integral converges by the Cauchy–Schwarz inequality, and $\abs{W_\psi f(a, b)} \le \norm{f}_2 \norm{\psi}_2$.
Taking the Fourier transform in $b$ and using the Plancherel theorem gives Calderón's identity
$$\int_{\R \setminus \set{0}} \int_\R \abs{W_\psi f(a, b)}^2 \, db \, \frac{da}{a^2} = C_\psi \norm{f}_2^2.$$
So $f$ can be recovered from $W_\psi f$, in a weak sense, by $f = C_\psi^{-1} \int \int W_\psi f(a, b) \, \abs{a}^{-1/2} \psi((\cdot - b)/a) \, db \, da / a^2$.
Many authors take only $a > 0$; then the constant changes and, for real $\psi$, one obtains $\tfrac12 C_\psi$.

Example: the Mexican hat $\psi(t) = (1 - t^2) e^{-t^2/2}$ has $\hat\psi(\xi)$ equal to a constant times $\xi^2 e^{-2\pi^2 \xi^2}$, so $C_\psi < \infty$ and $W_\psi$ is defined.

Non-example: for the Gaussian $\psi(t) = e^{-\pi t^2}$, $\hat\psi(0) = 1$, and $C_\psi$ diverges near $\xi = 0$.
The formula for $W_\psi f$ still makes sense, but Calderón's identity fails, and this choice is excluded by the definition.
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# Littlewood–Paley square function

## Summary

The Littlewood–Paley square function of a function $f$ on $\R^n$ is $Sf = \left( \sum_{j \in \Z} \abs{\Delta_j f}^2 \right)^{1/2}$, where the $\Delta_j f$ are the pieces of the [[Littlewood–Paley decomposition|D1PAT0K446]].
By Plancherel's theorem its $L^2$ norm is comparable to that of $f$, and by the [[Littlewood–Paley inequality|6GYQNB2862]] the same holds in $L^p$ for $1 < p < \infty$.
It turns the frequency information of $f$ into a pointwise quantity that can be estimated with real-variable tools.
It is a model for many other square functions, such as the [[Lusin area integral|KHY35P4AKF]].

## Explanations

### L1 human

Split $f$ into octaves of frequency, look at the size of each octave at a point $x$, and combine these sizes by a square root of a sum of squares.
The result is a new nonnegative function $Sf$, which ignores the phases of the pieces and only remembers their sizes.
Changing the signs of the pieces does not change $Sf$, which explains why such sign changes do not change the $L^p$ norm much.
The square root of a sum of squares is used because it matches the $L^2$ norm exactly when the pieces are orthogonal.
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# Topological field

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] with a [[topology|0CNQXVBHCX]].
Give $K \times K$ the product topology and $K^\times = K \setminus \set{0}$ the subspace topology.
The field $K$ is a *topological field* if the following maps are [[continuous|0P27W9RW0G]]:

1. addition $K \times K \to K$, $(x, y) \mapsto x + y$;
2. negation $K \to K$, $x \mapsto -x$;
3. multiplication $K \times K \to K$, $(x, y) \mapsto xy$;
4. inversion $K^\times \to K^\times$, $x \mapsto x^{-1}$.

## Remarks

Clauses 1 to 3 say that $K$ is a [[topological ring|TSF8RZDD5B]].
Many authors also require the topology to be Hausdorff and not discrete.
Example: $\R$ with the usual topology is a topological field.
Clauses 1 to 3 are the limit rules for sums, negatives and products, and clause 4 holds because $x \mapsto 1/x$ is continuous on $\R \setminus \set{0}$.
More generally, every field with an absolute value, with the topology of the metric $d(x, y) = \abs{x - y}$, is a topological field.
Non-example: $\R$ with the lower limit topology, generated by the intervals $[a, b)$, is not a topological field.
Negation fails clause 2: the preimage of the open set $[0, 1)$ under $x \mapsto -x$ is $(-1, 0]$, which is not open.
Every field is a topological field for the discrete topology and for the indiscrete topology.
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# Monoid object

## Definition

Let $(\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)$ be a [[monoidal category|M3MSQ3NKAE]].
A *monoid object* in $\mathcal{C}$ is an object $M$ together with morphisms $\mu \colon M \otimes M \to M$ (multiplication) and $\eta \colon I \to M$ (unit) such that:

1. associativity: $\mu \circ (\mu \otimes \id_M) = \mu \circ (\id_M \otimes \mu) \circ \alpha_{M,M,M}$ as morphisms $(M \otimes M) \otimes M \to M$;
2. left unit: $\mu \circ (\eta \otimes \id_M) = \lambda_M$ as morphisms $I \otimes M \to M$;
3. right unit: $\mu \circ (\id_M \otimes \eta) = \rho_M$ as morphisms $M \otimes I \to M$.

A morphism of monoid objects $(M, \mu, \eta) \to (M', \mu', \eta')$ is a morphism $f \colon M \to M'$ with $f \circ \mu = \mu' \circ (f \otimes f)$ and $f \circ \eta = \eta'$.

## Remarks

Example: in $(\mathbf{Set}, \times, \set{*})$, a monoid object is a set $M$ with $\mu(a, b) = ab$ and $\eta(*) = e$; the three axioms say $(ab)c = a(bc)$, $ea = a$ and $ae = a$, so it is an ordinary [[monoid|0N6R902FG1]].
Example: in $(\mathbf{Ab}, \otimes, \Z)$, a monoid object is a ring, since a bilinear multiplication is a morphism $R \otimes R \to R$.
Non-example: in $(\mathbf{Set}, \times, \set{*})$, the set $\Z$ with $\mu(a, b) = a - b$ and $\eta(*) = 0$ fails associativity, because $(1 - 1) - 1 = -1$ while $1 - (1 - 1) = 1$.
A monoid object in a symmetric monoidal category is *commutative* if $\mu \circ \gamma_{M,M} = \mu$, where $\gamma$ is the symmetry.
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# Unit-regular ring

## Summary

A ring $R$ is unit-regular if every element $a$ has a unit $u$ with $aua = a$.
This strengthens [[von Neumann regularity|DXX7GNY3HE]], where $u$ is only required to be some element of $R$.
Unit-regular rings are [[Dedekind-finite|GV1HKEAWVN]], and for a vector space $V$ the ring of endomorphisms is unit-regular exactly when $V$ is finite-dimensional.
The notion matters in the study of cancellation of modules and direct sum decompositions.

## Explanations

### L1 human

In a von Neumann regular ring each element $a$ has a "partial inverse" $x$ with $axa = a$.
In a unit-regular ring the partial inverse can be chosen to be a genuinely invertible element.
For a square matrix $A$ over a field, one can write $A = PEQ$ with $P, Q$ invertible and $E$ diagonal with entries $0$ and $1$, and then $u = Q^{-1}P^{-1}$ works.
For infinite-dimensional spaces this fails: the shift operator that moves every basis vector one step forward has no such invertible partner.
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# Power iteration

## Summary

Power iteration, or the power method, finds the [[eigenvalue|0BQAA2BK3Q]] of largest absolute value of a matrix and a corresponding [[eigenvector|0N6960W0BV]].
It multiplies a starting vector by the matrix again and again and rescales it.
It needs only matrix-vector products, so it works for very large sparse matrices, and it is the basis of the PageRank computation.
Its speed depends on the ratio between the two largest absolute values of the eigenvalues.

## Explanations

### L1 human

Applying a matrix stretches each eigenvector direction by its eigenvalue.
After many applications, the direction with the largest stretch wins over all others.
For $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ and $x_0 = (1, 0)$, the vectors $A x_0, A^2 x_0, A^3 x_0$ are $(2, 1)$, $(5, 4)$, $(14, 13)$.
They point closer and closer to $(1, 1)$, the eigenvector for the largest eigenvalue $3$.
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# Preservation of limits

## Summary

A [[functor|28NQ0KRHD3]] preserves limits when it sends every limit cone to a limit cone.
Such a functor is also called continuous.
Forgetful functors from algebraic categories to sets preserve limits, and right adjoint functors always do.
Preservation of limits is the main condition in the adjoint functor theorems.

## Explanations

### L1 human

A limit is built from a diagram, like a product or a set of solutions.
A functor preserves it when doing the construction first and then applying the functor gives the same as applying the functor first.
For example, the underlying set of a product of two groups is the product of the underlying sets.
So the forgetful functor from groups to sets preserves products.
It does not preserve coproducts, because the free product of two groups is much bigger than the disjoint union of the sets.
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# Fibred category

## Definition

Let $p \colon \mathcal{E} \to \mathcal{B}$ be a [[functor|28NQ0KRHD3]].
A morphism $\varphi \colon x \to y$ of $\mathcal{E}$ is *cartesian* if this holds.
For every morphism $\psi \colon z \to y$ of $\mathcal{E}$ and every morphism $g \colon p(z) \to p(x)$ of $\mathcal{B}$ with $p(\varphi) \circ g = p(\psi)$, there is a unique $\chi \colon z \to x$ with $\varphi \circ \chi = \psi$ and $p(\chi) = g$.
The functor $p$ is a *fibration*, and $\mathcal{E}$ is *fibred* over $\mathcal{B}$, if this holds.
For every object $y$ of $\mathcal{E}$ and every morphism $f \colon b \to p(y)$ of $\mathcal{B}$, there is a cartesian morphism $\varphi \colon x \to y$ with $p(\varphi) = f$.

## Remarks

The cartesian morphism $\varphi$ over $f$ is unique up to a unique isomorphism over $\id_b$, and $x$ is often written $f^* y$.
The fibre over $b$ is the subcategory of objects over $b$ and morphisms over $\id_b$.
Fibrations over $\mathcal{B}$ correspond to pseudofunctors from $\mathcal{B}^{\mathrm{op}}$ to categories, by the Grothendieck construction.

Example: let $\mathcal{B}$ have [[pullbacks|Q05K04QJ71]], let $\mathcal{E}$ be the arrow category whose objects are morphisms $e \colon a \to b$ of $\mathcal{B}$, and let $p$ send $e$ to its codomain $b$.
Given $e \colon a \to b$ and $f \colon b' \to b$, the pullback square of $e$ along $f$ is a morphism of $\mathcal{E}$ over $f$.
The universal property of the pullback says exactly that it is cartesian, so $p$ is a fibration.

Non-example: let $\mathcal{B}$ be the category with objects $0, 1$ and one non-identity arrow $f \colon 0 \to 1$.
Let $\mathcal{E}$ have one object $y$ and only its identity, and let $p(y) = 1$.
No object of $\mathcal{E}$ lies over $0$, so there is no morphism over $f$ ending at $y$, and $p$ is not a fibration.
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# Wiener–Hopf method

## Summary

The Wiener–Hopf method solves [[Wiener–Hopf equations|QYY4NMDY5H]] and boundary value problems whose conditions change type across a half-line.
It takes [[Fourier transforms|DPVM34ADZV]], so that functions supported on a half-line become functions holomorphic in a half-plane.
The key step factors the symbol of the problem into two factors.
One is holomorphic and nonzero in the upper half-plane, the other in the lower half-plane.
An argument with [[Liouville's theorem|0EHKVRF4J6]] then pins down the solution.
It was introduced by Norbert Wiener and Eberhard Hopf for integral equations on a half-line.
It is a standard tool in diffraction theory, radiative transfer and fracture mechanics.

## Explanations

### L1 human

A function that vanishes on the negative half-line has a Fourier transform that extends nicely to the upper half of the complex plane.
A function that vanishes on the positive half-line extends to the lower half.
The method rewrites the equation so that one side extends to the upper half-plane and the other side to the lower half-plane.
Since both sides are equal on the real line, together they form one function on the whole plane, which is then a polynomial by Liouville's theorem.
Reading off this polynomial gives the solution.
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<document source="https://tamnd.github.io/math2/node/H4QSPK0XTZ.md" tag="H4QSPK0XTZ" kind="proof">
# Proof of the first isomorphism theorem

## Proof

1. Let $\varphi : G \to H$ be a group homomorphism with kernel $K$. [[$K$ is a normal subgroup|0C9ZKYSPYR]] of $G$, so the quotient group $G/K$ exists.
2. Define $\psi : G/K \to \varphi(G)$ by $\psi(gK) = \varphi(g)$.
3. The map $\psi$ is well defined. If $gK = hK$, then $g^{-1}h \in K$, so $\varphi(g)^{-1}\varphi(h) = e$ and $\varphi(g) = \varphi(h)$.
4. The map $\psi$ is a homomorphism, because $\psi(gK\,hK) = \psi(ghK) = \varphi(gh) = \varphi(g)\varphi(h)$.
5. The map $\psi$ is onto the [[image|0112DTS464]] $\varphi(G)$, because each $\varphi(g)$ is $\psi(gK)$.
6. If $\psi(gK) = e$, then $\varphi(g) = e$, so $g \in K$ and $gK = K$. Thus the kernel of $\psi$ is trivial, and [[$\psi$ is injective|0N2P0XRTDF]].
7. Thus $\psi$ is an [[isomorphism|085PND4KNT]], and $G/K \cong \varphi(G)$.

## Idea

Send each coset of the kernel to the common image of its elements. This map is well defined, respects products, and has a trivial kernel.
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<document source="https://tamnd.github.io/math2/node/H4XGGXDTKN.md" tag="H4XGGXDTKN" kind="definition">
# Clone

## Definition

Let $A$ be a nonempty set.
For $n \ge 1$, let $O_A^{(n)}$ be the set of all $n$-ary [[operations|EJ8YJVM9BW]] $A^n \to A$, and let $O_A = \bigcup_{n \ge 1} O_A^{(n)}$.
A *clone* on $A$ is a subset $C \subseteq O_A$ such that:

1. $C$ contains every [[projection|769B35T6WG]] $\pi^n_k$ with $1 \le k \le n$.
2. $C$ is closed under [[composition|0THBK9V825]]: if $f \in C$ is $m$-ary and $g_1, \dots, g_m \in C$ are $n$-ary, then the $n$-ary operation
$$h(a_1, \dots, a_n) = f\bigl(g_1(a_1, \dots, a_n), \dots, g_m(a_1, \dots, a_n)\bigr)$$
is in $C$.

## Remarks

Example: the set of all operations $f(x_1, \dots, x_n) = c_1 x_1 + \cdots + c_n x_n$ on $\Z$ with integer coefficients $c_i$ is a clone.
Check: $\pi^n_k$ has $c_k = 1$ and the other $c_i = 0$, and substituting linear forms into a linear form gives a linear form.
Non-example: the set of all operations $f(x_1, \dots, x_n) = x_1 + \cdots + x_n$ on $\Z$, one for each $n \ge 1$, together with all projections, is not a clone.
It contains $f(x_1, x_2) = x_1 + x_2$, so a composition $f(f(\pi^2_1, \pi^2_2), \pi^2_1)$ gives $2x_1 + x_2$, which is not in the set.
The set of all [[term operations|N9M2N27END]] of an algebra $(A, F)$ is the smallest clone containing $F$, the clone generated by $F$.
Some authors also allow nullary operations in clones.
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# Asymptotic scale

## Definition

Let $D$ be a subset of $\R$ or of $\C$ and let $x_0$ be a limit point of $D$, possibly $\infty$.
An *asymptotic scale* (or *asymptotic sequence*) at $x_0$ is a sequence of functions $\varphi_0, \varphi_1, \varphi_2, \dots \colon D \to \C$ such that:

1. each $\varphi_n$ has no zero in $D \cap V$ for some punctured neighbourhood $V$ of $x_0$;
2. for each $n \ge 0$, $\varphi_{n+1}(x) = o\big(\varphi_n(x)\big)$ as $x \to x_0$ in $D$, in [[little o notation|00T5EPFQB9]].

## Remarks

Example: $\varphi_n(x) = (x - x_0)^n$ on $D = \R$ at a finite $x_0$.
These have no zero for $x \ne x_0$, and $\varphi_{n+1}(x) / \varphi_n(x) = x - x_0 \to 0$.

Mixed scales such as $1, \frac{\log x}{x}, \frac1x, \frac{\log x}{x^2}, \frac{1}{x^2}, \dots$ at $\infty$ are also asymptotic scales.

Non-example: $\varphi_n(x) = x^{-n}$ is not an asymptotic scale at $x_0 = 0$, since $\varphi_{n+1}(x)/\varphi_n(x) = 1/x$ does not tend to $0$ there; it is a scale at $\infty$.

Non-example: $\varphi_n(x) = \sin(x)\, x^{-n}$ at $\infty$ on $D = (1, \infty)$ fails clause 1, since every $\varphi_n$ has zeros at $x = k\pi$ for all large $k$.

Some authors drop clause 1 and write the condition 2 as $\varphi_{n+1} = o(\varphi_n)$ in the sense of an inequality $\abs{\varphi_{n+1}} \le \varepsilon \abs{\varphi_n}$ near $x_0$.
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<document source="https://tamnd.github.io/math2/node/H5145REFMW.md" tag="H5145REFMW" kind="concept">
# Frobenius element

## Summary

Let $L/K$ be a Galois extension of number fields, and let $\mathfrak{P}$ be a prime of $L$ that is unramified over $K$.
Let $q$ be the size of the residue field of the prime of $K$ below $\mathfrak{P}$.
The Frobenius element of $\mathfrak{P}$ is the unique element of the [[decomposition group|7X92G8CQ33]] that acts on the residue field as the [[Frobenius automorphism|0CGCPX8Z70]] $x \mapsto x^q$.
Its conjugacy class depends only on the prime of $K$ below $\mathfrak{P}$.
Frobenius elements encode how primes split, and they are the input of the Artin map in [[class field theory|BEKCEYHCHX]] and of [[Chebotarev's density theorem|X8VKW5Y1CZ]].

## Explanations

### L1 human

Over a finite field with $q$ elements, raising to the power $q$ is a symmetry of every larger finite field.
A Frobenius element lifts this symmetry from the residue field to the number field itself.
In $\Q(i)/\Q$, the Frobenius element at an odd prime $p$ is the identity if $p \equiv 1 \pmod 4$ and complex conjugation if $p \equiv 3 \pmod 4$.
This matches whether $p$ splits or stays prime in $\Z[i]$.
So one group element records the splitting of a prime.
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<document source="https://tamnd.github.io/math2/node/H51S5V6XYJ.md" tag="H51S5V6XYJ" kind="definition">
# One-parameter subgroup

## Definition

Let $G$ be a [[topological group|92700QAB5Y]].
A *one-parameter subgroup* of $G$ is a continuous [[group homomorphism|0JYTG5K9Y4]] $\gamma \colon \R \to G$ from the additive group of real numbers.
So $\gamma$ is continuous and
$$\gamma(s + t) = \gamma(s)\gamma(t) \quad \text{for all } s, t \in \R.$$

## Remarks

Despite the name, a one-parameter subgroup is the map $\gamma$, not its image.

Example: for a real $n \times n$ matrix $A$, the map $\gamma(t) = e^{tA}$ into $GL_n(\R)$, given by the [[matrix exponential|020M04FN1H]], is continuous, and $e^{(s + t)A} = e^{sA} e^{tA}$ because $sA$ and $tA$ commute.

Non-example: $\gamma(t) = \begin{pmatrix} 1 & t \\ 0 & 1 + t^2 \end{pmatrix}$ is a continuous curve in $GL_2(\R)$ with $\gamma(0) = I$, but it is not a homomorphism.
The $(2,2)$ entry of $\gamma(1)\gamma(1)$ is $4$, while that of $\gamma(2)$ is $5$.

If $G$ is a [[Lie group|7JBRGCVJRB]], every one-parameter subgroup is smooth.
Then $\gamma \mapsto \gamma'(0)$ is a bijection from the one-parameter subgroups of $G$ onto the tangent space $T_eG$.
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<document source="https://tamnd.github.io/math2/node/H56XFFM23H.md" tag="H56XFFM23H" kind="definition">
# Convex combination

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]] and let $x_1, \dots, x_k$ be points of $V$, with $k \ge 1$.
A *convex combination* of $x_1, \dots, x_k$ is a point of the form
$$\lambda_1 x_1 + \cdots + \lambda_k x_k,$$
where the real numbers $\lambda_1, \dots, \lambda_k$ satisfy $\lambda_i \ge 0$ for each $i$ and $\lambda_1 + \cdots + \lambda_k = 1$.

## Remarks

Example: in $\R^2$, the point $(1, 1)$ is a convex combination of $(0, 0)$, $(3, 0)$ and $(0, 3)$, with weights $1/3$, $1/3$, $1/3$.
The weights are nonnegative and add up to $1$, and $\tfrac13 (0,0) + \tfrac13 (3,0) + \tfrac13 (0,3) = (1, 1)$.

Non-example: $2 \cdot (1, 0) + (-1) \cdot (0, 0) = (2, 0)$ uses weights that add up to $1$, so it is an [[affine combination|0KJH6SRFZK]], but the weight $-1$ is negative, so it is not a convex combination.

A convex combination of two points $x, y$ is a point $(1 - t) x + t y$ with $0 \le t \le 1$.
The same definition works over any ordered field in place of $\R$.
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<document source="https://tamnd.github.io/math2/node/H5765CJ7VB.md" tag="H5765CJ7VB" kind="definition">
# Algebra over an operad

## Definition

Let $(\mathcal{C}, \otimes, \mathbb{1})$ be a [[symmetric monoidal category|5S7ER95XAC]], such as sets or topological spaces with the cartesian product.
An *operad* $\mathcal{O}$ in $\mathcal{C}$ consists of objects $\mathcal{O}(n)$ for $n \ge 0$ and the following data.
There is a right [[action|0WCY42EV9V]] of the [[symmetric group|0515A6D2Y1]] $\Sigma_n$ on each $\mathcal{O}(n)$, and a unit $\eta \colon \mathbb{1} \to \mathcal{O}(1)$.
There are composition maps, which are associative, unital and equivariant:
$$\gamma \colon \mathcal{O}(k) \otimes \mathcal{O}(j_1) \otimes \cdots \otimes \mathcal{O}(j_k) \to \mathcal{O}(j_1 + \cdots + j_k).$$
An *algebra over $\mathcal{O}$*, or *$\mathcal{O}$-algebra*, is an object $A$ of $\mathcal{C}$ with maps
$$\theta_n \colon \mathcal{O}(n) \otimes A^{\otimes n} \to A \qquad (n \ge 0).$$
These maps must satisfy three conditions, written here with elements as for sets or spaces:

1. (associativity) $\theta_j(\gamma(c; d_1, \ldots, d_k); \mathbf{a}_1, \ldots, \mathbf{a}_k) = \theta_k(c; \theta_{j_1}(d_1; \mathbf{a}_1), \ldots, \theta_{j_k}(d_k; \mathbf{a}_k))$, where $j = j_1 + \cdots + j_k$ and $\mathbf{a}_i$ is a $j_i$-tuple of elements of $A$;
2. (unit) $\theta_1(1; a) = a$, where $1 = \eta(\ast) \in \mathcal{O}(1)$;
3. (equivariance) $\theta_n(c \cdot \sigma; a_1, \ldots, a_n) = \theta_n(c; a_{\sigma^{-1}(1)}, \ldots, a_{\sigma^{-1}(n)})$ for $\sigma \in \Sigma_n$.

In a general $\mathcal{C}$ the three clauses are the commutativity of the corresponding diagrams.

## Remarks

Equivalently, an $\mathcal{O}$-algebra structure on $A$ is a morphism of operads $\mathcal{O} \to \mathrm{End}_A$, where $\mathrm{End}_A(n)$ is the internal hom from $A^{\otimes n}$ to $A$, when $\mathcal{C}$ is closed.

Example: the commutative operad in sets has $\mathcal{O}(n) = \set{\ast}$ for every $n$.
An algebra is a set $A$ with one map $\theta_n \colon A^n \to A$ for each $n$.
Equivariance makes $\theta_2(a, b) = \theta_2(b, a)$, the associativity clause makes $\theta_2$ associative, and $\theta_0$ picks a unit.
So the algebras are exactly the commutative [[monoids|0N6R902FG1]].
Likewise the associative operad, with $\mathcal{O}(n) = \Sigma_n$, has the monoids as its algebras.

Non-example: take $A = \Z$ and try to make it an algebra over the commutative operad with $\theta_2(a, b) = a - b$.
Equivariance fails for the transposition, since $1 - 0 \ne 0 - 1$.
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<document source="https://tamnd.github.io/math2/node/H58AGW8ZHF.md" tag="H58AGW8ZHF" kind="definition">
# Elementary equivalence

## Definition

Let $L$ be a signature and let $\mathcal{M}$ and $\mathcal{N}$ be $L$-[[structures|983GZBQYT4]].
The structures $\mathcal{M}$ and $\mathcal{N}$ are elementarily equivalent, written $\mathcal{M} \equiv \mathcal{N}$, when for every $L$-[[sentence|DKFGQTRYZE]] $\varphi$,
$$\mathcal{M} \models \varphi \iff \mathcal{N} \models \varphi.$$

## Remarks

Equivalently, $\mathcal{M} \equiv \mathcal{N}$ when $\mathrm{Th}(\mathcal{M}) = \mathrm{Th}(\mathcal{N})$, where $\mathrm{Th}$ is the set of sentences true in a structure.

Example: $(\Q, <) \equiv (\R, <)$.
Both are dense linear orders without endpoints, and the theory of dense linear orders without endpoints is complete.

Non-example: $(\Z, <) \not\equiv (\Q, <)$.
The sentence $\forall x \, \forall y \, (x < y \to \exists z \, (x < z \land z < y))$ is true in $(\Q, <)$ and false in $(\Z, <)$, with $x = 0$ and $y = 1$.

Isomorphic structures are elementarily equivalent.
The example shows that the converse fails, since $\Q$ and $\R$ have different cardinalities.

## Explanations

### L1 human

Elementary equivalence says that first-order logic cannot tell two structures apart.
It is weaker than isomorphism: by the [[Löwenheim–Skolem theorem|484M0TZ1S7]], an infinite structure in a countable signature is elementarily equivalent to structures of every infinite size.
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<document source="https://tamnd.github.io/math2/node/H5AQZ3336P.md" tag="H5AQZ3336P" kind="definition">
# K3 surface

## Definition

Let $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field, with structure sheaf $\mathcal{O}_X$ and [[canonical bundle|W05VYY4HBR]] $\omega_X$.
$X$ is a *K3 surface* if:

1. $\omega_X \cong \mathcal{O}_X$, and
2. the [[sheaf cohomology|PQ6Q143ARJ]] group $H^1(X, \mathcal{O}_X)$ is $0$.

## Remarks

Over $\C$, the same two conditions define K3 surfaces among all compact complex surfaces; such a surface need not be projective.
A K3 surface has Hodge numbers $h^{2,0} = 1$ and $h^{1,1} = 20$, second Betti number $22$ and topological Euler characteristic $24$.
Example: a smooth quartic $X \subset \mathbb{P}^3$ has $\omega_X \cong \mathcal{O}_X(4 - 4) = \mathcal{O}_X$ by adjunction, and $H^1(X, \mathcal{O}_X) = 0$ from the exact sequence $0 \to \mathcal{O}_{\mathbb{P}^3}(-4) \to \mathcal{O}_{\mathbb{P}^3} \to \mathcal{O}_X \to 0$ and the vanishing of $H^1$ and $H^2$ of line bundles on $\mathbb{P}^3$.
Non-example: an abelian surface, such as $E \times E$ for an elliptic curve $E$, has trivial canonical bundle but $\dim H^1(X, \mathcal{O}_X) = 2$, so it fails clause 2.
Other examples are double covers of $\mathbb{P}^2$ branched along a smooth sextic and the Kummer surfaces, obtained from abelian surfaces by dividing by $x \mapsto -x$ and resolving the $16$ singular points.
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# Weighted least squares

## Summary

Weighted least squares fits a regression model by minimising a sum of squared residuals in which each observation has its own positive weight.
It is the case of [[generalized least squares|P142WXJCF3]] in which the errors are uncorrelated but [[heteroscedastic|D8VX6XQ3TT]].
With weights inversely proportional to the error variances, it gives the best linear unbiased estimator.
It is also the inner step of iteratively reweighted least squares, used to fit [[generalized linear models|K6C48KMRKC]] and robust regressions.

## Explanations

### L1 human

Some data points are more precise than others.
Weighted least squares lets the precise points pull harder on the fitted line.
Each squared residual is multiplied by a weight, and a point with weight $2$ counts as much as two copies of the same point.
For a model that is just a constant, weighted least squares gives the weighted average $\sum_i w_i y_i / \sum_i w_i$.
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<document source="https://tamnd.github.io/math2/node/H5CW67CCST.md" tag="H5CW67CCST" kind="definition">
# Many-one reduction

## Definition

Let $A$ and $B$ be subsets of $\N$.
A *many-one reduction* from $A$ to $B$ is a total [[computable function|KZSB5E471K]] $f \colon \N \to \N$ such that for every $x \in \N$,
$$x \in A \iff f(x) \in B.$$
The set $A$ is *many-one reducible* to $B$, written $A \le_m B$, if such an $f$ exists.
The sets $A$ and $B$ are *many-one equivalent*, written $A \equiv_m B$, if $A \le_m B$ and $B \le_m A$.

## Remarks

Example: let $E$ be the set of even numbers and let $f(x) = 0$ for even $x$ and $f(x) = 1$ for odd $x$.
Then $f$ is total and computable, and $x \in E$ holds exactly when $f(x) \in \set{0}$, so $E \le_m \set{0}$.

Non-example: the empty set is not many-one reducible to $\N$.
For any $f$, the point $0 \notin \emptyset$ would need $f(0) \notin \N$, which is impossible.

The relation $\le_m$ is reflexive and transitive.
If $A \le_m B$ and $B$ is a [[computable set|PPGD5KW8JC]], then $A$ is computable.
If $A \le_m B$ and $B$ is [[computably enumerable|TN9G24VABN]], then $A$ is computably enumerable.
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# Zonotope

## Definition

Let $d \ge 1$ and $m \ge 1$, and let $a_1, b_1, \dots, a_m, b_m$ be points of $\R^d$.
A *zonotope* is a set of the form
$$Z = [a_1, b_1] + [a_2, b_2] + \cdots + [a_m, b_m],$$
the [[Minkowski sum|7GK5417BJF]] of the [[line segments|VQ4Q2V45RP]] $[a_i, b_i]$.

## Remarks

Equivalently, $Z = \set{c + \lambda_1 v_1 + \cdots + \lambda_m v_m : 0 \le \lambda_i \le 1}$ with $c = a_1 + \cdots + a_m$ and $v_i = b_i - a_i$.
So $Z$ is the image of the cube $[0,1]^m$ under the affine map $\lambda \mapsto c + \sum_i \lambda_i v_i$, and $Z$ is a [[convex polytope|300K6CB1KH]].
Every zonotope is centrally symmetric about the point $c + \tfrac12 (v_1 + \cdots + v_m)$.

Example: in $\R^2$, take $v_1 = (1, 0)$, $v_2 = (\tfrac12, \tfrac{\sqrt3}{2})$, $v_3 = (-\tfrac12, \tfrac{\sqrt3}{2})$ and $c = 0$.
The sum of the three segments $[0, v_i]$ is a regular hexagon with side length $1$.

Non-example: a triangle is not a zonotope, because it is not centrally symmetric.
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# Incidence geometry

## Definition

*Incidence geometry* is the study of [[incidence structures|FGKNAFNHA3]] $(P, L, I)$ and of their properties that are preserved by isomorphisms.
An isomorphism of incidence structures is a pair of bijections, on points and on lines, that preserves incidence in both directions.
The structures studied are usually required to satisfy incidence axioms, such as:

1. any two distinct points lie on at most one common line (a *partial linear space*);
2. any two distinct points lie on exactly one common line (a *linear space*).

## Remarks

Example: the statement "two distinct lines of a linear space meet in at most one point" belongs to incidence geometry.
Check: if two distinct lines had two common points, these points would lie on two lines, against clause 2.

Non-example: the statement "the angles of a triangle add up to $\pi$" is not a statement of incidence geometry, since angles are not defined from points, lines and incidence alone.

Main classes are [[affine planes|EC7XT2ZM5S]], [[projective planes|DANFNN8C4Y]], projective spaces, polar spaces and generalized polygons.
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<document source="https://tamnd.github.io/math2/node/H5FQQWDJRV.md" tag="H5FQQWDJRV" kind="example">
# Calabi–Eckmann manifold

## Example

Let $p, q \ge 0$ be integers and let $\tau \in \C$ with $\operatorname{Im} \tau \ne 0$.
Let $W = (\C^{p+1} \setminus \set{0}) \times (\C^{q+1} \setminus \set{0})$, and let the additive group $\C$ act on $W$ by
$$t \cdot (u, v) = (e^{t} u, \; e^{\tau t} v).$$
The quotient $X_{p,q} = W / \C$ is a *Calabi–Eckmann manifold*, a compact [[complex manifold|JSGWAN2RJV]] of dimension $p + q + 1$.

Check that $X_{p,q}$ is diffeomorphic to $S^{2p+1} \times S^{2q+1}$: each orbit meets $S^{2p+1} \times S^{2q+1}$ in exactly one point.
Write $t = x + iy$.
The conditions $\abs{e^t u} = 1$ and $\abs{e^{\tau t} v} = 1$ read $x = -\log\abs{u}$ and $x \operatorname{Re}\tau - y \operatorname{Im}\tau = -\log\abs{v}$.
Since $\operatorname{Im}\tau \ne 0$, they have exactly one solution $(x, y)$.
The action is free and proper and is by holomorphic maps, so the quotient is a complex manifold of dimension $(p + 1) + (q + 1) - 1$.

## Remarks

The map $[u, v] \mapsto ([u], [v])$ makes $X_{p,q}$ a holomorphic [[fibre bundle|CX4M8K61AB]] over $\mathbb{CP}^p \times \mathbb{CP}^q$ (see [[complex projective space|7GJ4STCNNM]]) whose fibres are [[elliptic curves|427N6S6QQW]].
For $p = q = 0$ one gets an elliptic curve, and for $p = 0$ a [[Hopf manifold|Z8R2D33WSM]] diffeomorphic to $S^1 \times S^{2q+1}$.
For $p, q \ge 1$ the manifold $S^{2p+1} \times S^{2q+1}$ is simply connected and has $H^2 = 0$, so $X_{p,q}$ is not [[Kähler|X60JG9AEQJ]]: a compact Kähler manifold has a nonzero class in $H^2$.
So Calabi and Eckmann's construction gives simply connected compact complex manifolds that are not Kähler, for example on $S^3 \times S^3$.

## Explanations

### L1 human

A product of two odd-dimensional spheres, such as $S^3 \times S^3$, has no obvious complex structure, since neither factor is even-dimensional.
Calabi and Eckmann found one by viewing each sphere as a circle bundle over a complex projective space and combining the two circles into a complex torus.
The result is a compact complex manifold that cannot carry a Kähler metric.
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<document source="https://tamnd.github.io/math2/node/H5G6TGRQH1.md" tag="H5G6TGRQH1" kind="example">
# Lattice of subspaces

## Example

Let $V$ be a [[vector space|000AYPA9AJ]] over a field $F$, and let $L(V)$ be the set of all [[linear subspaces|00YC1D652K]] of $V$, ordered by inclusion.
The meet of two subspaces $U$ and $W$ is $U \cap W$, and their join is the sum $U + W = \set{u + w : u \in U, w \in W}$.
For any family $(U_i)_{i \in I}$ of subspaces, the intersection $\bigcap_i U_i$ and the span of $\bigcup_i U_i$ are subspaces.
They are the greatest lower bound and the least upper bound of the family.
So $L(V)$ is a [[complete lattice|D7A0ZGZT5S]], with least element $\set{0}$ and greatest element $V$.
It is a [[modular lattice|QX7WZ92QAK]]: if $U \subseteq W$, then $U + (X \cap W) = (U + X) \cap W$ for every subspace $X$.
The inclusion $U + (X \cap W) \subseteq (U + X) \cap W$ holds because $U \subseteq W$.
For the other inclusion, an element $w = u + x$ of the right side with $u \in U$, $x \in X$ has $x = w - u \in W$, so $x \in X \cap W$.

## Remarks

The lattice $L(V)$ is not distributive once $\dim V \ge 2$.
In $V = \R^2$, take three distinct lines $a$, $b$, $c$ through $0$.
Then $a \cap (b + c) = a \cap \R^2 = a$, while $(a \cap b) + (a \cap c) = \set{0}$.
When $V$ is finite-dimensional, the subspaces of dimension $1$ and $2$ are the points and lines of the projective space of $V$, and $L(V)$ encodes its incidence geometry.

## Explanations

### L1 human

Take all the subspaces of a vector space: the zero space, the lines through the origin, the planes through the origin, and so on up to the whole space.
Order them by inclusion.
Any two subspaces have a largest common subspace, their intersection, and a smallest subspace containing both, their sum.
For two distinct lines in $\R^3$, the intersection is $\set{0}$ and the sum is the plane they span.
This ordered set is the basic example of a modular lattice that is not distributive, and it is the algebraic skeleton of projective geometry.
</document>

<document source="https://tamnd.github.io/math2/node/H5G7CNYZEB.md" tag="H5G7CNYZEB" kind="statement">
# Dvoretzky–Kiefer–Wolfowitz inequality

## Statement

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with [[cumulative distribution function|0MXMS3AE13]] $F$.
Let $F_n(x) = \frac{1}{n} \#\set{i : X_i \le x}$ be their [[empirical distribution function|A05WR6TEK5]].
Then for every $\varepsilon > 0$,
$$P\Big( \sup_{x \in \R} \abs{F_n(x) - F(x)} > \varepsilon \Big) \le 2 e^{-2 n \varepsilon^2}.$$

## Remarks

Dvoretzky, Kiefer and Wolfowitz proved in 1956 a bound of the form $C e^{-2n\varepsilon^2}$ with an unspecified constant $C$.
Massart showed in 1990 that $C = 2$ works, and this constant cannot be lowered.
The reduction to the uniform law on $[0, 1]$ uses the quantile transform.
The variable $X_i$ has the same law as $Q(U_i)$, for $U_i$ uniform and $Q$ the quantile function of $F$.
The supremum can only shrink under this change.
The hard part is the sharp tail bound for the uniform case.

Example: a confidence band.
Given $\alpha \in (0, 1)$, put $\varepsilon = \sqrt{\ln(2/\alpha) / (2n)}$, so that $2e^{-2n\varepsilon^2} = \alpha$.
Then the band $F_n(x) \pm \varepsilon$ contains the whole graph of $F$ with probability at least $1 - \alpha$.
For $n = 1000$ and $\alpha = 0.05$, this gives $\varepsilon \approx 0.043$.

The bound needs no assumption on $F$: no continuity and no moments.
It does need independence and a common law; for dependent data $F_n$ can stay far from $F$.

## Explanations

### L1 human

The empirical distribution function of a sample is the natural guess for the true distribution function.
This inequality says how good the guess is, uniformly over all $x$ at once.
The chance that the guess is off by more than $\varepsilon$ anywhere decays like $e^{-2n\varepsilon^2}$, whatever the distribution is.
So a sample of a few thousand points pins down the whole distribution function to within a few hundredths.

### L2 human

The inequality is a quantitative form of the [[Glivenko–Cantelli theorem|E7B78X780X]].
Since $\sum_n 2e^{-2n\varepsilon^2} < \infty$, the Borel–Cantelli lemma gives $\sup_x \abs{F_n - F} \to 0$ almost surely.
It also bounds the tail of the [[Kolmogorov–Smirnov|YZYHWC6C5F]] statistic: $P(\sqrt{n} D_n > t) \le 2e^{-2t^2}$ for every $n$, which matches the leading term of the limiting Kolmogorov law.
It is a common tool in learning theory and in the analysis of nonparametric estimators, because it is non-asymptotic.
</document>

<document source="https://tamnd.github.io/math2/node/H5GB19W2H3.md" tag="H5GB19W2H3" kind="concept">
# Elliptic geometry

## Summary

Elliptic geometry is the [[non-Euclidean geometry|SPCYMFN5V2]] of constant positive curvature.
In it, any two points determine exactly one line, and any two lines meet in exactly one point.
Its standard model is the [[real projective space|1DDYB6VVSR]], seen as the sphere with antipodal points identified.
There are no parallel lines, and the angle sum of a triangle exceeds $\pi$.
Together with Euclidean and [[hyperbolic geometry|GN4N94GQ0Z]] it is one of the three classical geometries of constant curvature.

## Explanations

### L1 human

Start with [[spherical geometry|SQ9RQVANMG]] on a globe, where the lines are great circles.
There, two great circles always cross twice, at opposite points of the globe.
Now declare each pair of opposite points to be a single point.
Then two lines cross exactly once, and two points lie on exactly one line, as in the plane, yet there are still no parallels.
The resulting geometry is elliptic geometry; it is finite in size, and no two points are more than a quarter of a great circle apart.
</document>

<document source="https://tamnd.github.io/math2/node/H5P06V330V.md" tag="H5P06V330V" kind="definition">
# Picard–Vessiot extension

## Definition

Let $(K, \partial)$ be a [[differential field|GA5PTQGC6G]] of characteristic $0$, where $\partial$ is a [[derivation|3J89REV9WQ]] of $K$.
Let $C = \set{c \in K : \partial c = 0}$ be its field of constants, and assume that $C$ is [[algebraically closed|0PF6TM8TWT]].
Let
$$L(y) = y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y$$
with $a_0, \dots, a_{n-1} \in K$, where $y^{(k)}$ denotes $\partial^k y$.
A *Picard–Vessiot extension* of $K$ for $L$ is a differential [[field extension|0EQWVHF3W5]] $E \supseteq K$ such that:

1. There are $y_1, \dots, y_n \in E$ with $L(y_i) = 0$ for each $i$ that are linearly independent over the constants of $E$.
2. $E$ is generated as a field over $K$ by $y_1, \dots, y_n$ and all their derivatives, that is, $E = K(y_i^{(k)} : 1 \le i \le n, \ 0 \le k \le n - 1)$.
3. The field of constants of $E$ is $C$.

## Remarks

Under these hypotheses a Picard–Vessiot extension for $L$ exists, and it is unique up to an isomorphism of differential fields over $K$.
The group of automorphisms of $E$ over $K$ that commute with $\partial$ is the differential Galois group of $L$; it is a linear algebraic group over $C$.

Example: take $K = \C(x)$ with $\partial = d/dx$, so $C = \C$, and $L(y) = y' - y$.
The field $E = \C(x, e^x)$ is a Picard–Vessiot extension: $y_1 = e^x$ solves $L(y) = 0$, it generates $E$ over $K$, and the constants of $\C(x, e^x)$ are only the complex numbers.

Non-example: for the same $K$ and $L$, the field $E' = \C(x, e^x, e^{x^2})$ contains the solution $e^x$ and has constant field $\C$, but it fails clause 2.
It is strictly larger than the field generated by the solutions and their derivatives, because $e^{x^2}$ does not lie in $\C(x, e^x)$.

The extension is an analogue of the [[splitting field|0KXQKX9P63]] of a polynomial, and it is the basic object of [[differential Galois theory|EEZS4H8QPH]].
</document>

<document source="https://tamnd.github.io/math2/node/H5QDJACKMS.md" tag="H5QDJACKMS" kind="definition">
# Solution set condition

## Definition

Let $G \colon \mathcal{D} \to \mathcal{C}$ be a [[functor|28NQ0KRHD3]] between [[categories|A2DE66937D]].
$G$ satisfies the *solution set condition* if the following holds for every object $c$ of $\mathcal{C}$.
There are a set $I$, objects $d_i$ of $\mathcal{D}$ and [[morphisms|859F0TFVA3]] $f_i \colon c \to G d_i$ for $i \in I$ with this property.
Let $d$ be any object of $\mathcal{D}$ and $h \colon c \to G d$ any morphism.
Then $h = G(t) \circ f_i$ for some $i \in I$ and some morphism $t \colon d_i \to d$.

## Remarks

The family $(f_i)_{i \in I}$ is called a *solution set* for $c$.
The factorization need not be unique, and $I$ must be a set, not a proper class.

Example: let $G$ be the forgetful functor from groups to sets, and let $c = X$ be a set.
Every map $h \colon X \to G H$ lands in the subgroup $H'$ generated by $h(X)$, which has at most $\max(\abs{X}, \aleph_0)$ elements.
Each such $H'$ is isomorphic to a group whose underlying set is a subset of $X \sqcup \N$.
All such groups, with all maps from $X$ into them, form a solution set for $X$.

Non-example: the forgetful functor from complete Boolean algebras to sets fails it, since it preserves limits but has no left adjoint.
Indeed, there is no free complete Boolean algebra on a countably infinite set.
</document>

<document source="https://tamnd.github.io/math2/node/H5RC67SD24.md" tag="H5RC67SD24" kind="concept">
# Ramification

## Summary

Let $L$ be a finite extension of a number field $K$.
A prime ideal $\mathfrak{p}$ of $K$ ramifies in $L$ when its factorisation in $L$ has a repeated prime factor.
The exponent of a prime factor is its ramification index.
Only finitely many primes ramify, namely those dividing the relative discriminant.
Over $\Q$ these are the primes dividing the [[discriminant|E0VZTNPJXA]] of $L$.
Ramification is one part of the [[splitting of prime ideals|B7KNZCBFYT]], and the word comes from the branching of covering maps of Riemann surfaces.

## Explanations

### L1 human

When a prime number is factored in a larger ring of integers, the factors may come with exponents bigger than $1$.
This is ramification: two or more factors have merged into one.
In $\Z[i]$, the prime $2$ factors as $-i(1+i)^2$, so $2$ ramifies with index $2$.
Every other prime number is unramified in $\Z[i]$.
A picture: the map $z \mapsto z^2$ on the complex plane is two-to-one except at $0$, where the two preimages come together; this point is a branch point.
</document>

<document source="https://tamnd.github.io/math2/node/H5V4099GN7.md" tag="H5V4099GN7" kind="definition">
# Stokes phenomenon

## Definition

A *sector* at $\infty$ is a set $S = \set{z \in \C : \abs{z} > R,\ \alpha < \arg z < \beta}$ with $R > 0$ and $\alpha < \beta$.
Let $f$ be [[holomorphic|0NAHXR1FRH]] on $S$, and let $m \ge 1$ be an integer.
Let $q$ be a polynomial in $z^{1/m}$, let $\mu \in \C$, and let $\sum_{k \ge 0} c_k z^{-k/m}$ be a formal series with $c_0 \ne 0$.
The function $f$ has the *asymptotic expansion* $f(z) \sim e^{q(z)} z^{\mu} \sum_{k \ge 0} c_k z^{-k/m}$ in $S$ if, for each $N \ge 0$ and each closed subsector $S'$ of $S$,
$$z^{N/m} \left( e^{-q(z)} z^{-\mu} f(z) - \sum_{k=0}^{N-1} c_k z^{-k/m} \right) \to c_N \quad \text{as } z \to \infty \text{ in } S'.$$
Let $f$ be a solution of a linear differential equation with an [[irregular singular point|AT1W4MMTYR]] at $\infty$, [[continued analytically|0M3YQJ81NB]] to a large sector.
The equation shows the *Stokes phenomenon* if some such solution has an asymptotic expansion of this form in a sector $S_1$ with the following property.
In an adjacent overlapping sector $S_2$, the expansion of the same solution is a different combination of such series.
So no single expansion of the solution holds on $S_1 \cup S_2$.

## Remarks

Consider the rays where a small term $e^{q_j(z)}$ is maximally subdominant compared with another term $e^{q_i(z)}$.
Its coefficient may jump there, and these rays are called Stokes lines in one common convention; other authors call them anti-Stokes lines.
Consider a system $Y' = A(z) Y$ where $A$ has a pole at $\infty$.
On two overlapping sectors, the actual solutions asymptotic to a fixed formal solution differ by a constant matrix, the *Stokes matrix*.
The phenomenon occurs exactly when some Stokes matrix is not the identity.
At a regular singular point it does not occur: the convergent Frobenius series hold in all directions.

Example: the Airy function satisfies $\operatorname{Ai}(z) \sim \frac{e^{-\zeta}}{2 \sqrt{\pi}\, z^{1/4}}$ with $\zeta = \frac{2}{3} z^{3/2}$ in the sector $\abs{\arg z} < \pi$.
On the negative real axis, however, $\operatorname{Ai}(-x) \sim \frac{1}{\sqrt{\pi}\, x^{1/4}} \sin\left( \frac{2}{3} x^{3/2} + \frac{\pi}{4} \right)$ for $x \to +\infty$, which is a sum of two exponential terms $e^{\pm i \frac{2}{3} x^{3/2}}$.

Non-example: the function $e^{z}$ solves $y' = y$, which has an irregular singular point at $\infty$, and $e^{z} \sim e^{z} \cdot 1$ holds in every sector; this solution shows no change of expansion.
</document>

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# Separable degree

## Summary

The separable degree $[L:K]_s$ of an algebraic extension $L/K$ counts the embeddings of $L$ into an [[algebraically closed field|0PF6TM8TWT]].
Only the embeddings that extend a fixed embedding of $K$ are counted.
For a finite extension it divides the [[degree|0EAJ2XXADC]] $[L:K]$, and the quotient is a power of the characteristic exponent, called the inseparable degree.
It equals $[L:K]$ exactly when $L/K$ is [[separable|0WCX7NB4N7]], and it equals the degree of the [[separable closure|JJ91A1V8NQ]] of $K$ in $L$.
It measures how much of an extension is visible to Galois theory.

## Explanations

### L1 human

The degree of an extension counts dimensions.
The separable degree counts the distinct ways to place the extension inside a big field such as $\C$.
For $\Q(\sqrt{2})$ there are two ways, sending $\sqrt{2}$ to $\sqrt{2}$ or to $-\sqrt{2}$, and the degree is also $2$.
In characteristic $p$, the extension of $\F_p(t)$ by a $p$-th root of $t$ has degree $p$ but only one placement, because $x^p - t$ has only one root.
So its separable degree is $1$.
</document>

<document source="https://tamnd.github.io/math2/node/H6GTJRG26G.md" tag="H6GTJRG26G" kind="concept">
# Weak lower semicontinuity

## Summary

A functional on a Banach space is weakly lower semicontinuous if it cannot jump up in a weak limit.
That is, $J(u) \le \liminf_k J(u_k)$ whenever $u_k \to u$ weakly.
It is [[semicontinuity|RN4G8SGP6P]] with respect to the [[weak topology|P6KHVSTPZV]], usually in its sequential form.
Together with coercivity it is the main hypothesis of the [[direct method|TE379HBTY0]].
For integral functionals it is closely tied to convexity of the integrand in the gradient variable.

## Explanations

### L1 human

Weak convergence allows a sequence to oscillate faster and faster while its average settles down.
The functions $\sin(kx)$ on $[0, 2\pi]$ converge weakly to $0$, even though their size does not shrink.
A weakly lower semicontinuous energy may lose value in such a limit, but it cannot gain value.
The norm itself is a typical example: the limit $0$ has norm $0$, below the norms of $\sin(kx)$.
</document>

<document source="https://tamnd.github.io/math2/node/H6KS0E9MF5.md" tag="H6KS0E9MF5" kind="statement">
# Convergence of approximations for second-kind equations

## Statement

Let $X$ be a [[Banach space|06GZ956657]], let $K \colon X \to X$ be a [[compact operator|M2NEP1K5AE]] such that $I - K$ is injective, and let $f \in X$.
Let $\varphi \in X$ be the solution of $\varphi - K\varphi = f$.
Let $P_n \colon X \to X$ be bounded linear projections onto finite-dimensional subspaces $X_n$ such that $P_n \psi \to \psi$ for every $\psi \in X$.
Then there is $n_0$ such that for every $n \ge n_0$ the [[projection method|2AFNVJ54G7]] equation $\varphi_n - P_n K \varphi_n = P_n f$ has a unique solution $\varphi_n \in X$, which lies in $X_n$.
Moreover there is a constant $M$, independent of $n$ and $f$, such that
$$\norm{\varphi_n - \varphi} \le M \norm{P_n \varphi - \varphi} \quad \text{for all } n \ge n_0,$$
and in particular $\varphi_n \to \varphi$.

## Remarks

The solution $\varphi$ exists because, for compact $K$, injectivity of $I - K$ implies that $I - K$ is invertible with bounded inverse (Riesz theory, the [[Fredholm alternative|4HH8C9V8JQ]]).
Since $P_n \psi \to \psi$ for each $\psi$, the norms $\norm{P_n}$ are bounded by the [[uniform boundedness principle|YG0042V8CJ]], so the convergence is uniform on compact sets.
The set $\set{K\psi : \norm{\psi} \le 1}$ is relatively compact, hence $\norm{P_n K - K} \to 0$.
A perturbation argument then shows that $I - P_n K$ is invertible for large $n$, with $\norm{(I - P_n K)^{-1}} \le M$.
Finally $(I - P_n K)(\varphi_n - \varphi) = P_n \varphi - \varphi$, which gives the error bound; and if $\varphi_n - P_nK\varphi_n = P_n f$ then $\varphi_n = P_n(f + K\varphi_n) \in X_n$.

Example: for a continuous kernel on $[a, b]$, collocation with piecewise linear interpolation on grids with mesh tending to $0$ satisfies the hypothesis in $C[a, b]$, since $P_n \psi \to \psi$ uniformly for continuous $\psi$.
The error is then of the order of the interpolation error of the exact solution, for instance $O(h^2)$ for a twice continuously differentiable $\varphi$ and mesh size $h$.

The hypothesis of pointwise convergence $P_n \psi \to \psi$ is needed.
For collocation with polynomial interpolation at equidistant points in $C[a, b]$ it fails, since the norms of the interpolation operators grow without bound, and the theorem gives nothing.

## Explanations

### L1 human

The theorem says that for equations of the form "unknown minus a smoothing integral equals data", projection methods work as soon as the projections themselves approximate every function well.
The error of the computed solution is then at most a fixed multiple of the best error that the projections can achieve on the exact solution.
So the quality of the method is reduced to a question of approximation theory.

### L2 human

The key step is that a compact operator turns pointwise convergence of $P_n$ into norm convergence of $P_n K$.
This is why second-kind equations with compact $K$ are so well suited to numerical methods, in contrast to first-kind equations, which are ill-posed.
In a Hilbert space with orthogonal projections, this gives the convergence of the Galerkin method for second-kind equations.
For the Nyström method, $K$ is replaced by quadrature operators $K_n$ that do not converge in norm, and the same conclusion needs [[collectively compact operator approximation|AEZE6B2XXY]].
</document>

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# Mutually orthogonal Latin squares

## Summary

Two [[Latin squares|CDVRMVED2V]] of the same order are orthogonal if, when one is laid over the other, every ordered pair of symbols appears exactly once.
A family of Latin squares is mutually orthogonal if every two of them are orthogonal.
For $n \ge 2$, there are at most $n - 1$ mutually orthogonal Latin squares of order $n$.
A family of $n - 1$ of them exists exactly when a [[finite projective plane|2HWB7P1DZN]] of order $n$ exists.
The node [[mutually orthogonal Latin squares|KHD2QMNR7R]] gives the precise definition.

## Explanations

### L1 human

Imagine a $3 \times 3$ field planted with three crops and treated with three fertilisers.
Each crop should appear once per row and column, and so should each fertiliser.
If in addition every crop meets every fertiliser exactly once, the two Latin squares are orthogonal.
Such a pair exists for every order except $2$ and $6$.
</document>

<document source="https://tamnd.github.io/math2/node/H6WXC7Q71A.md" tag="H6WXC7Q71A" kind="concept">
# Ordered ring

## Summary

An ordered ring is a commutative [[ring|0B1261B5ED]] with a [[total order|0FRXBVVR03]] that is preserved by addition and for which products of nonnegative elements are nonnegative.
The integers are the basic example, and every [[ordered field|0P3VCBDEE2]] is an ordered ring.
In an ordered ring every square is nonnegative, and the usual rules for inequalities hold.

## Explanations

### L1 human

The integers can be added, multiplied and compared, and the familiar rules link these operations.
Adding the same number to both sides of an inequality keeps it true.
The product of two nonnegative numbers is nonnegative.
An ordered ring is any ring with an order that obeys these rules, such as the integers, the rationals or the polynomials with the right order.
</document>

<document source="https://tamnd.github.io/math2/node/H6ZGEDM4VV.md" tag="H6ZGEDM4VV" kind="concept">
# Volterra equation of the first kind

## Summary

A Volterra equation of the first kind is an equation $f(t) = \int_a^t K(t, s) \, x(s) \, ds$ in which the unknown function $x$ appears only under the integral.
It is the [[Volterra integral equation|DND8CBEVXT]] analogue of an [[integral equation of the first kind|ZDS77Q0F06]].
Solving it is like inverting an integration, so it behaves like differentiation and is in general ill-posed.
When the kernel is smooth and does not vanish on the diagonal, it can be reduced to an equation of the second kind.

## Explanations

### L1 human

The data $f(t)$ is an accumulated quantity: a weighted total of the unknown $x$ over all past times $s \le t$.
The task is to recover the rate $x$ from the totals.
With $K = 1$ and $a = 0$, the equation $f(t) = \int_0^t x(s) \, ds$ has the solution $x = f'$, provided $f$ is differentiable and $f(0) = 0$.
For instance, $f(t) = t^2$ gives $x(t) = 2t$.
A small wiggle in $f$ can produce a large change in $f'$, which is why such equations are delicate.
</document>

<document source="https://tamnd.github.io/math2/node/H70CF6YBY1.md" tag="H70CF6YBY1" kind="definition">
# Carleman matrix

## Definition

Let $K$ be a field and let $f(x) = \sum_{k \ge 0} f_k x^k$ be a [[formal power series|033WGN00BF]] over $K$.
For integers $j, k \ge 0$, let $M[f]_{jk}$ be the coefficient of $x^k$ in the power $f(x)^j$, with $f(x)^0 = 1$.
The *Carleman matrix* of $f$ is the infinite [[matrix|0VD9ZSWWJG]] $M[f] = (M[f]_{jk})_{j, k \ge 0}$.
So
$$f(x)^j = \sum_{k \ge 0} M[f]_{jk} \, x^k \quad \text{for every } j \ge 0.$$

## Remarks

Example: for $f(x) = ax$ with $a \in K$, one has $f(x)^j = a^j x^j$, so $M[f]$ is diagonal with entries $1, a, a^2, \ldots$.
For $f(x) = x$ it is the identity matrix.

Non-example: the diagonal matrix with entries $1, 2, 3, \ldots$ is not a Carleman matrix.
Its row $1$ would force $f(x) = 2x$, and then row $2$ would have to list the coefficients of $4x^2$, not of $3x^2$.

If $g(0) = 0$, then $M[f \circ g] = M[f] \, M[g]$, a product of infinite matrices in which each entry is a finite sum.
Check: $f(g(x))^j = \sum_k M[f]_{jk} \, g(x)^k = \sum_k M[f]_{jk} \sum_l M[g]_{kl} x^l$, and $M[g]_{kl} = 0$ for $k > l$ because $g(x)^k$ starts at $x^k$.
In particular, if $f(0) = 0$, then $M[f^n] = M[f]^n$ for the [[iterates|AYVTR2ES5B]] of $f$, and $M[f]$ is upper triangular with diagonal entries $f_1^j$.
The transpose of $M[f]$ is the Bell matrix of $f$, which reverses the order of products.
</document>

<document source="https://tamnd.github.io/math2/node/H77NYP0C68.md" tag="H77NYP0C68" kind="definition">
# Logarithmic integral

## Definition

Let $\ln$ be the [[natural logarithm|04DXAFN2ES]].
For a real number $x > 0$ with $x \ne 1$, the *logarithmic integral* is
$$\operatorname{li}(x) = \int_0^x \frac{dt}{\ln t},$$
where for $x > 1$ the integral is the [[Cauchy principal value|0SYR10HD4R]] at $t = 1$:
$$\operatorname{li}(x) = \lim_{\varepsilon \to 0^+} \left( \int_0^{1 - \varepsilon} \frac{dt}{\ln t} + \int_{1 + \varepsilon}^x \frac{dt}{\ln t} \right).$$
The *offset logarithmic integral* is $\operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2) = \int_2^x \frac{dt}{\ln t}$ for $x \ge 2$.

## Remarks

The integrand tends to $0$ as $t \to 0^+$, so there is no problem at the lower end.

Example: the substitution $t = e^u$ gives $\operatorname{li}(x) = \operatorname{Ei}(\ln x)$, where $\operatorname{Ei}$ is the [[exponential integral|EV6ZDQ0NDY]].
The function $\operatorname{li}$ has exactly one positive zero, at $x = \mu \approx 1.4513$, the Ramanujan–Soldner constant, and $\operatorname{li}(2) \approx 1.045$.

Non-example: without the principal value the integral diverges for $x > 1$, because $\frac{1}{\ln t} \sim \frac{1}{t - 1}$ near $t = 1$ and $\int \frac{dt}{\abs{t - 1}}$ diverges.

The [[prime number theorem|0EHBY5Y84N]] is equivalent to $\pi(x) \sim \operatorname{li}(x)$, where $\pi$ is the [[prime-counting function|0RQSNVTFVZ]].
Integration by parts gives $\operatorname{li}(x) \sim \frac{x}{\ln x}$ as $x \to \infty$.
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<document source="https://tamnd.github.io/math2/node/H78RXJC1K1.md" tag="H78RXJC1K1" kind="statement">
# Layer cake representation

## Statement

Let $(X, \Sigma, \mu)$ be a [[σ-finite|WEBBT18D4X]] measure space.
Let $f \colon X \to [0, \infty]$ be a measurable function with [[distribution function|HQVXKEMN5N]] $d_f(t) = \mu(\set{f > t})$.
Then
$$\int_X f \, d\mu = \int_0^\infty d_f(t) \, dt,$$
where the left side is the [[integral of a non-negative function|DG76G5Z6XW]] and the right side is taken with [[Lebesgue measure|2Z2QVWZ262]].
More generally, for $0 < p < \infty$,
$$\int_X f^p \, d\mu = p \int_0^\infty t^{p-1} d_f(t) \, dt.$$

## Remarks

The name comes from the pointwise identity $f(x) = \int_0^\infty \mathbf{1}_{\set{f > t}}(x) \, dt$.
It writes $f$ as a stack of thin horizontal layers, one for each height $t$.
Integrating over $X$ and exchanging the two integrals by Tonelli's theorem gives the formula.
The second formula follows in the same way from $f(x)^p = \int_0^{f(x)} p t^{p-1} \, dt$.

Example: for $f(x) = x$ on $[0, 1]$ with Lebesgue measure, $d_f(t) = 1 - t$ for $0 \le t \le 1$ and $0$ after.
Both sides equal $1/2$.

For a non-negative random variable $Y$ the first formula reads $\mathbb{E}[Y] = \int_0^\infty P(Y > t) \, dt$.

## Explanations

### L1 human

To find the volume under a graph, one can slice it vertically or horizontally.
The layer cake formula slices horizontally.
At each height $t$ it measures the size of the region where the function is above $t$, and then adds these sizes over all heights.

### L2 human

The formula turns integrals of $f$ into integrals of its distribution function.
This is why the $L^p$ norms depend only on $d_f$, so equimeasurable functions and decreasing rearrangements have the same norms.
It is the basic tool behind weak $L^p$ estimates, the Marcinkiewicz interpolation theorem and the bounds for the Hardy–Littlewood maximal function.
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# Compactly generated space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *compactly generated* if it satisfies the following condition.
A subset $A \subseteq X$ is closed whenever $u^{-1}(A)$ is closed in $K$ for every [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]] $K$ and every [[continuous map|0VGFHTM75Y]] $u \colon K \to X$.

## Remarks

If $X$ is Hausdorff, the condition is equivalent to the classical one.
A subset $A \subseteq X$ is closed whenever $A \cap C$ is closed in $C$ for every compact subset $C \subseteq X$.

Example: every [[metric space|0VKD1TP7F3]] is compactly generated.
If $A$ is not closed, there is a sequence $a_n \in A$ converging to some $x \notin A$.
The set $K = \set{x} \cup \set{a_n : n \ge 1}$ is compact, and $A \cap K$ is not closed in $K$.

Non-example: an uncountable set $X$ with the cocountable topology.
Its compact subsets are finite, so every test map $u$ has finite image.
Finite subsets of $X$ are closed and discrete, so every $u^{-1}(A)$ is closed.
But not every subset of $X$ is closed.
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# Analytic geometry

## Summary

Analytic geometry is geometry done with coordinates: points become tuples of numbers, and figures become sets of solutions of equations.
Through a [[Cartesian coordinate system|0RWKYX8339]], lines are given by linear equations, circles and [[conic sections|00HJCMC2CD]] by quadratic ones, and distances by the formula $\sqrt{(x_1 - y_1)^2 + \cdots + (x_n - y_n)^2}$.
Geometric questions then become algebraic computations, which can be carried out systematically.
It underlies much of calculus, linear algebra and algebraic geometry, and it is usually traced to Descartes and Fermat.

## Explanations

### L1 human

Put a grid on the plane, so that every point gets an address $(x, y)$.
Then a line is the set of points whose addresses satisfy an equation like $y = 2x + 1$, and a circle of radius $5$ around the origin is $x^2 + y^2 = 25$.
To find where the line meets the circle, one solves the two equations together, with no ruler or compass needed.
This turns pictures into algebra, and algebra into pictures.
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<document source="https://tamnd.github.io/math2/node/H7PCM776GR.md" tag="H7PCM776GR" kind="concept">
# Bockstein homomorphism

## Summary

The Bockstein homomorphism is the connecting map in the long exact sequence of (co)homology that comes from a short exact sequence of coefficient groups.
It raises cohomological degree by one.
For the sequence $0 \to \Z/p \to \Z/p^2 \to \Z/p \to 0$ it gives a natural operation on mod $p$ cohomology that detects $p$-torsion in integral cohomology.

## Explanations

### L1 human

Cohomology with coefficients in $\Z/p$ loses some information about integral cohomology.
The Bockstein homomorphism recovers part of it by asking whether a mod $p$ class lifts to a mod $p^2$ class.
When it does not lift, the Bockstein gives a nonzero class one degree higher.
For the real projective plane, the generator of $H^1(\R P^2; \Z/2)$ does not lift to $\Z/4$ coefficients, and its Bockstein is the generator of $H^2(\R P^2; \Z/2)$.
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# Newton polygon

## Summary

The Newton polygon of a polynomial over a valued field is a broken line built from its coefficients $a_i$, see the [[definition|D186VNHFZJ]].
It is the lower boundary of the [[convex hull|0FQSDC2EQ2]] of the points $(i, v(a_i))$.
Its slopes and their lengths give the [[valuations|B85T5VKR3M]] of the roots and how many roots have each valuation.
It is a quick tool for factoring polynomials over $p$-adic fields and for finding [[Puiseux series|SENTYNSKSF]] solutions of algebraic equations.

## Explanations

### L1 human

Plot one point for each coefficient: the horizontal position is the power of $X$, the height is how divisible the coefficient is by $p$.
Stretch a string under these points from the leftmost to the rightmost point and pull it tight from below.
The broken line you get is the Newton polygon.
Each straight piece with slope $s$ and width $\ell$ says: exactly $\ell$ roots have valuation $-s$.
For $X^2 + X + p$ the pieces have slopes $-1$ and $0$, so one root is divisible by $p$ and the other is not.
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<document source="https://tamnd.github.io/math2/node/H7RAXJV9CP.md" tag="H7RAXJV9CP" kind="statement">
# Jacobson radical of an Artinian ring is nilpotent

## Statement

Let $R$ be a left [[Artinian ring|ZV04WWYE2Y]], that is, a ring with identity whose [[left ideals|5N2CJZ47D4]] satisfy the descending chain condition.
Let $J = J(R)$ be its [[Jacobson radical|XA0DCNH7WY]], the intersection of all maximal left ideals of $R$.
Then $J$ is a [[nilpotent ideal|C2H26BYWHE]]: there is an integer $n \ge 1$ with $J^n = 0$.

## Remarks

The chain $J \supseteq J^2 \supseteq J^3 \supseteq \cdots$ becomes constant, say $J^n = J^{n+1} = I$, and one shows $I = 0$.
If $I \ne 0$, choose a left ideal $L$ minimal among those with $IL \ne 0$, and $x \in L$ with $Ix \ne 0$.
Minimality gives $L = Rx$ and, since $I(Ix) = I^2 x = Ix \ne 0$, also $Ix = Rx$.
So $x = yx$ for some $y \in I \subseteq J$; as $1 - y$ is a [[unit|019NFGFPR5]], $x = 0$, a contradiction.
Example: for the ring of upper triangular $2 \times 2$ matrices over a field, $J$ is the set of strictly upper triangular matrices and $J^2 = 0$.
The Artinian hypothesis is needed: the local ring $\Z_{(p)}$ is Noetherian, but $J = p\Z_{(p)}$ is not nilpotent, since $p^n \ne 0$ for all $n$.

## Explanations

### L1 human

In a ring with the descending chain condition, the Jacobson radical is "small" in a strong sense.
Multiplying enough elements of it together always gives zero.
So the radical consists of nilpotent elements, and it is even killed by a fixed power.

### L2 human

This is a key step in the structure theory of Artinian rings.
With the [[Artin–Wedderburn theorem|R1F271180N]], which describes $R/J$ as a product of matrix rings over division rings, it shows that every left Artinian ring is [[semiprimary|XVP517QRH5]].
It is also used in the proof of the [[Hopkins–Levitzki theorem|W29AA0266Z]], which says that a left Artinian ring is left Noetherian.
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<document source="https://tamnd.github.io/math2/node/H7RX5729ZB.md" tag="H7RX5729ZB" kind="concept">
# Partial algebra

## Summary

A partial algebra is a set with operations that need not be defined for every choice of arguments.
It generalises the [[algebraic structure|QJGPT2W0FT]] of universal algebra, where every operation is total.
Partial algebras model structures such as fields, where the inverse of $0$ is not defined, and categories, where two arrows compose only when they match.

## Explanations

### L1 human

In an ordinary algebra you can combine any elements and always get an answer.
In a partial algebra some combinations are simply not allowed.
A field is the classic example: you can divide by any element except $0$.
Another example is the set of natural numbers with subtraction, where $3 - 5$ has no value in $\N$.
The theory of partial algebras asks which tools of universal algebra still work when operations have gaps.
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# Lions–Stampacchia theorem

## Statement

Let $H$ be a real [[Hilbert space|HM2HHE3J51]] and let $K \subseteq H$ be a nonempty [[closed|0ZCK8DQYN1]] [[convex set|02AS45T3H9]].
Let $a \colon H \times H \to \R$ be a bounded [[coercive bilinear form|86KRHS6X22]], not necessarily symmetric: there are constants $M, \alpha > 0$ with $\abs{a(u, v)} \le M \norm{u} \norm{v}$ and $a(v, v) \ge \alpha \norm{v}^2$ for all $u, v \in H$.
Let $\ell$ be an element of the [[continuous dual space|4A82S5FVRT]] $H^*$.
Then there is exactly one $u \in K$ such that
$$a(u, v - u) \ge \ell(v - u) \quad \text{for all } v \in K.$$
If moreover $a$ is symmetric, then $u$ is the unique minimizer over $K$ of the function $v \mapsto \tfrac12 a(v, v) - \ell(v)$.

## Remarks

By the Riesz representation theorem there are a bounded linear operator $A$ on $H$ and a vector $f \in H$ with $a(u, v) = \ip{Au}{v}$ and $\ell(v) = \ip{f}{v}$.
The inequality then says that $u = P_K\big(u - \rho (Au - f)\big)$ for any $\rho > 0$, where $P_K$ is the projection onto $K$ given by the [[Hilbert projection theorem|J5YNTHQMFX]].
For $0 < \rho < 2\alpha / M^2$ the map $v \mapsto P_K\big(v - \rho (Av - f)\big)$ is a contraction of $K$, so the Banach fixed-point theorem gives existence and uniqueness.
Uniqueness also follows directly: if $u_1, u_2$ are two solutions, adding the two inequalities gives $a(u_1 - u_2, u_1 - u_2) \le 0$, hence $u_1 = u_2$.

For $K = H$ the inequality becomes the equation $a(u, w) = \ell(w)$ for all $w \in H$, and the theorem reduces to the [[Lax–Milgram theorem|85KK79FCN5]].
Example: the [[obstacle problem|GXN2NEHC8X]].
Let $\Omega \subseteq \R^n$ be bounded and open, $H = H^1_0(\Omega)$, $a(u, v) = \int_\Omega \nabla u \cdot \nabla v \, dx$, and $K = \set{v \in H : v \ge \psi \text{ a.e.}}$ for an obstacle $\psi$ such that $K$ is nonempty.
The theorem gives a unique $u \in K$ that minimizes the Dirichlet energy among functions lying above $\psi$.
Coercivity cannot be dropped: for $H = \R$, $K = [0, \infty)$, $a = 0$ and $\ell(v) = v$, the inequality $0 \ge v - u$ for all $v \ge 0$ has no solution.

## Explanations

### L1 human

Many equilibrium problems with a one-sided constraint, such as a membrane pushed against an obstacle, lead to an inequality rather than an equation.
The theorem says that, under natural conditions, such an inequality has exactly one solution.
The conditions are a closed convex set of allowed states and an energy-like form that grows at least quadratically.

### L2 human

This is the basic existence theorem for [[variational inequalities|6QS7X5GVMY]] and the constrained version of the Lax–Milgram theorem.
In the symmetric case it is a statement about minimizing a strictly convex quadratic function over a closed convex set.
The nonsymmetric case needs the fixed-point argument, since no energy is available.
It applies to obstacle problems, to contact problems in elasticity such as the Signorini problem, and, after a change of unknown, to some free boundary problems.
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# Bayesian information criterion

## Definition

Let $M$ be a parametric statistical model with $k$ free real parameters, and let $x$ be data of $n$ observations.
Let $\hat L$ be the maximum of the [[likelihood function|05FSTXM92W]] of $M$ at $x$, the value at the [[maximum likelihood estimate|0FREPFKNS7]].
The *Bayesian information criterion* of $M$ at $x$ is
$$\mathrm{BIC} = k \ln n - 2 \ln \hat L.$$
Among several candidate models for the same data, the BIC rule selects a model with the smallest BIC.

## Remarks

Under regularity conditions, $-\mathrm{BIC}/2$ approximates the logarithm of the marginal likelihood of $M$, up to a term that stays bounded as $n \to \infty$.
So differences of BIC approximate $-2$ times the logarithm of a Bayes factor.
Example: with $n = 100$, a model with $k = 2$ and $\ln \hat L = -100$ has $\mathrm{BIC} = 2 \ln 100 + 200 \approx 209.2$.
A model with $k = 3$ and $\ln \hat L = -98$ has $\mathrm{BIC} \approx 209.8$, so the BIC rule keeps the smaller model, while the AIC values $204$ and $202$ favour the larger one.
Non-example: $2k - 2 \ln \hat L$ is the [[Akaike information criterion|CMAX3KF9H0]], not the BIC; the two agree only when $\ln n = 2$.
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# Étale morphism

## Summary

An étale morphism of schemes is a morphism that is flat, unramified and locally of finite presentation.
It is the algebraic analogue of a local diffeomorphism, or of a covering map in topology.
Étale morphisms are the open sets of the étale topology, which gives étale cohomology and the étale fundamental group.

## Explanations

### L1 human

The Zariski topology has very few open sets, so the inverse function theorem fails for it.
Étale morphisms replace the missing small open sets: they are maps that look locally like an isomorphism on tangent spaces.
For example, over a field of characteristic not $2$, the squaring map $t \mapsto t^2$ from the punctured line to itself is étale.
It is a two-sheeted covering, like the map $z \mapsto z^2$ on the punctured complex plane.
At the origin, the full squaring map of the line is not étale, because there the two sheets meet.
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# Deformation theory

## Definition

Let $k$ be a field and let $X$ be a [[scheme|5A35TYZD02]] over $k$.
Let $\mathrm{Art}_k$ be the category of local [[Artinian|ZV04WWYE2Y]] $k$-algebras $A$ with residue field $k$.
For $A \in \mathrm{Art}_k$, a *deformation of $X$ over $A$* is a pair $(\mathcal{X}, \phi)$ where:

1. $\mathcal{X}$ is a scheme with a [[flat|5M6KS8F47R]] morphism $\mathcal{X} \to \operatorname{Spec} A$;
2. $\phi \colon \mathcal{X} \times_{\operatorname{Spec} A} \operatorname{Spec} k \to X$ is an isomorphism of $k$-schemes, where the left side is a [[fiber product|1DZMT4S9FB]].

Two deformations $(\mathcal{X}, \phi)$ and $(\mathcal{X}', \phi')$ are isomorphic if there is an isomorphism $\mathcal{X} \to \mathcal{X}'$ over $A$ compatible with $\phi$ and $\phi'$.
The *deformation functor* of $X$ is $\operatorname{Def}_X \colon \mathrm{Art}_k \to \mathrm{Set}$, sending $A$ to the set of isomorphism classes of deformations of $X$ over $A$.
A *first-order deformation* is a deformation over the dual numbers $k[\varepsilon] = k[\varepsilon]/(\varepsilon^2)$.
Deformation theory is the study of $\operatorname{Def}_X$ and of the analogous functors for other objects, such as sheaves, morphisms and subschemes.

## Remarks

Let $X$ be smooth over $k$ with tangent sheaf $T_X$.
Then $\operatorname{Def}_X(k[\varepsilon]) \cong H^1(X, T_X)$, in [[sheaf cohomology|PQ6Q143ARJ]].
The obstructions to lifting a deformation over $A$ to a small extension of $A$ lie in $H^2(X, T_X)$.
Example: the trivial deformation $X \times_k \operatorname{Spec} A$ with the identity $\phi$ satisfies both clauses, since a base change of $\operatorname{Spec} A \to \operatorname{Spec} k$ is flat.
For $X = \mathbb{P}^1$, $H^1(\mathbb{P}^1, T) = H^1(\mathbb{P}^1, \mathcal{O}(2)) = 0$, so every first-order deformation of $\mathbb{P}^1$ is trivial.
Non-example: let $A = k[\varepsilon]$ and $\mathcal{X} = \operatorname{Spec} k[\varepsilon, x]/(\varepsilon x)$.
Its special fiber is $\operatorname{Spec} k[x] = \mathbb{A}^1_k$, so clause 2 holds.
But $\mathcal{X}$ is not flat over $A$, since $x$ is killed by $\varepsilon$ without being a multiple of $\varepsilon$, so clause 1 fails.
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# Lagrangian submanifold

## Summary

A Lagrangian submanifold of a symplectic manifold is a submanifold of half the dimension on which the symplectic form vanishes.
Lagrangian submanifolds are the largest submanifolds on which the form vanishes identically.
They include the zero section and the graphs of closed $1$-forms in cotangent bundles, and the graphs of [[symplectomorphisms|A8TS2SNX9Z]].
They are central objects of symplectic topology, Floer theory and mirror symmetry.

## Explanations

### L1 human

A symplectic form measures areas of small surfaces inside a manifold of even dimension $2n$.
A Lagrangian submanifold has dimension $n$ and is so placed that every small surface inside it has zero symplectic area.
In the phase plane with position $q$ and momentum $p$, every curve is Lagrangian.
In $\R^4$ with coordinates $(q_1, q_2, p_1, p_2)$, the plane where both momenta are zero is Lagrangian.
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# Green–Tao theorem

## Statement

For every integer $k \ge 1$, there are positive integers $a$ and $d$ such that the $k$ numbers $a, a + d, a + 2d, \dots, a + (k-1)d$ are all [[prime numbers|01826KMG2H]].

## Remarks

Green and Tao proved more: every subset $A$ of the set $\mathcal{P}$ of primes with
$$\limsup_{N \to \infty} \frac{\abs{A \cap \set{1, \dots, N}}}{\abs{\mathcal{P} \cap \set{1, \dots, N}}} > 0$$
contains arithmetic progressions of every length.
[[Szemerédi's theorem|B23B0QEMN5]] does not apply directly, because the primes have density zero by the [[prime number theorem|0EHBY5Y84N]].
The proof is a transference principle.
A relative Szemerédi theorem holds for dense subsets of a pseudorandom set.
The primes sit densely inside such a set, built from sieve weights of Goldston and Yıldırım.
For example, $5, 11, 17, 23, 29$ is a progression of five primes with difference $6$.
The primes contain no infinite arithmetic progression: for $a \ge 2$ the term $a + ad = a(1 + d)$ is composite, and for $a = 1$ the first term is not prime.

## Explanations

### L1 human

You can find evenly spaced prime numbers in long rows, as long as you like.
For example, $3, 5, 7$ is a row of three and $5, 11, 17, 23, 29$ is a row of five.
The theorem says rows of every length exist, even though primes get rarer.

### L2 human

The Green–Tao theorem combines [[additive combinatorics|3NM362A2JG]] with analytic number theory.
Its idea is that Szemerédi's theorem survives inside sparse but random-looking sets.
It is a special case of the open [[Erdős conjecture on arithmetic progressions|FN978TDX67]], since the sum of the reciprocals of the primes diverges.
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# Spread

## Definition

Let $\mathcal{P}$ be a [[projective space|ZT9A6JK8QF]] and let $k \ge 0$ be an integer.
A *$k$-spread* of $\mathcal{P}$ is a set $\mathcal{S}$ of $k$-dimensional [[subspaces|2RD6T5KG59]] such that every point lies in exactly one member of $\mathcal{S}$.

## Remarks

A $1$-spread is a *line spread*.
In the finite space $\mathrm{PG}(n, q)$ over the [[finite field|0ZXZ82FTX2]] $\F_q$, a $k$-spread exists if and only if $k + 1$ divides $n + 1$.

Example: let $V = \F_{q^2}^2$, which is a vector space of dimension $4$ over $\F_q$, so its $1$-dimensional $\F_q$-subspaces are the points of $\mathrm{PG}(3, q)$.
The $1$-dimensional $\F_{q^2}$-subspaces of $V$ are $2$-dimensional over $\F_q$, so they are lines of $\mathrm{PG}(3, q)$.
Every nonzero vector of $V$ spans exactly one $1$-dimensional $\F_{q^2}$-subspace, so these $q^2 + 1$ lines form a line spread, called the regular (or Desarguesian) spread.

Non-example: two lines of $\mathrm{PG}(3, q)$ that meet in a point $P$ cannot both belong to a spread, since $P$ would lie in two members.

Given a line spread of $\mathrm{PG}(3, q)$, the André–Bruck–Bose construction produces a [[translation plane|JZ5J6REAMD]] of order $q^2$.
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# Integral operator

## Summary

An integral operator is a [[linear map|0NRYHJPQG2]] between spaces of functions that acts by integration against a kernel: $(Tf)(x) = \int K(x, y) f(y)\, dy$.
Integral operators are the infinite-dimensional analogue of matrices.
They appear as [[integral transforms|YCK709QB06]], as solution operators of differential equations, and in [[Volterra integral equations|DND8CBEVXT]] and Fredholm integral equations.
With a square-integrable kernel they are [[compact operators|M2NEP1K5AE]], which gives them a spectral theory close to that of matrices.

## Explanations

### L1 human

A matrix turns a vector $y$ into the vector with entries $\sum_j K_{ij} y_j$.
An integral operator does the same with functions, and the sum becomes an integral.
The simplest example is $(Vf)(x) = \int_0^x f(y)\, dy$ on $[0, 1]$, which takes a function to its running total.
Its kernel is $1$ when $y < x$ and $0$ otherwise, a "triangular matrix" with infinitely many entries.
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# Deformation of algebras

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$ with product $(a, b) \mapsto ab$.
Let $A[\![t]\!]$ be the $k[\![t]\!]$-module of [[formal power series|033WGN00BF]] with coefficients in $A$.
A *formal deformation* of $A$ is a $k[\![t]\!]$-bilinear associative product $*$ on $A[\![t]\!]$ of the form
$$a * b = ab + t\,\mu_1(a, b) + t^2 \mu_2(a, b) + \cdots \quad (a, b \in A),$$
where each $\mu_i \colon A \times A \to A$ is $k$-bilinear.
Two formal deformations $*$ and $*'$ are *equivalent* if there is a $k[\![t]\!]$-linear map $T = \id + t T_1 + t^2 T_2 + \cdots$, with each $T_i \colon A \to A$ $k$-linear, such that $T(a * b) = T(a) *' T(b)$ for all $a, b$.

## Remarks

Comparing the coefficients of $t$ in $(a * b) * c = a * (b * c)$ gives
$$a \mu_1(b, c) - \mu_1(ab, c) + \mu_1(a, bc) - \mu_1(a, b) c = 0,$$
so $\mu_1$ is a Hochschild $2$-cocycle, and equivalent deformations have cohomologous $\mu_1$.
Example: for any $T = \id + t T_1 + \cdots$, the product $a * b = T^{-1}(T(a) T(b))$ is a formal deformation, since it is the original product carried over by $T$; it is called trivial.
Non-example: on $A = k[x]$, let $\mu_1(f, g) = f(0) g(0)$ and $a * b = ab + t \mu_1(a, b)$.
For $a = x$, $b = c = 1$ the left side of the cocycle identity is $x \cdot 1 - 0 + 0 - 0 = x \ne 0$, so $*$ is not associative.
If the Hochschild cohomology $HH^2(A, A)$ is zero, then every formal deformation of $A$ is trivial, and $A$ is called rigid.
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# Cokernel

## Summary

In a category with zero morphisms, the cokernel of a morphism $f$ is the universal morphism $q$ with $q \circ f = 0$.
It is the [[coequalizer|DVSTWGVCSZ]] of $f$ and the zero morphism, and the notion dual to the [[kernel|S417YNEJ6M]].
For modules, the cokernel of $f \colon X \to Y$ is the quotient map $Y \to Y / f(X)$.
The cokernel measures how far a morphism is from being surjective.

## Explanations

### L1 human

The image of a linear map is the part of the target that the map reaches.
The cokernel is what is left when one divides the target by the image: it records the directions the map misses.
A map onto its target has zero cokernel.
For multiplication by $n$ on $\Z$, the image is $n\Z$ and the cokernel is $\Z / n\Z$.
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# Complete integral

## Summary

A complete integral of a first-order [[partial differential equation|GC5BDZ6QBC]] in $n$ variables is a family of solutions depending on $n$ independent parameters.
From such a family one builds many more solutions by taking envelopes of subfamilies.
Complete integrals are the classical route to solving [[Hamilton–Jacobi equations|5TRXQKP7DY]] in mechanics and Clairaut-type equations.
See the [[definition|KEC5TGS9WM]].

## Explanations

### L1 human

For a first-order equation in $n$ variables, one often finds a simple family of solutions, such as planes with a free slope.
If the family has $n$ genuinely different parameters, it is called a complete integral.
For example, every plane $u = a_1 x_1 + a_2 x_2 + a_1^2 + a_2^2$ solves the equation $u = x \cdot \nabla u + \abs{\nabla u}^2$.
Bending these planes together, by taking the envelope of a one-parameter subfamily, produces new curved solutions.
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# Fujiki class C

## Summary

Fujiki class $\mathcal{C}$ is the class of compact complex manifolds that are bimeromorphic to compact [[Kähler manifolds|X60JG9AEQJ]].
It contains all compact Kähler manifolds and all [[Moishezon manifolds|8ZW9P9JRQD]].
Unlike the Kähler property, membership in class $\mathcal{C}$ is invariant under bimeromorphic maps and under images of holomorphic maps.
Manifolds in class $\mathcal{C}$ keep much of the topology of Kähler manifolds: their cohomology has a Hodge decomposition, and their odd Betti numbers are even.

## Explanations

### L1 human

Kähler manifolds are compact complex manifolds with a particularly nice kind of metric.
Blowing up or down, or other birational changes, can destroy that metric, even though the manifold stays almost the same.
Class $\mathcal{C}$ is the bigger class one gets by allowing such changes.
It is the natural home for results about Kähler manifolds that only depend on the manifold up to birational change.
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<document source="https://tamnd.github.io/math2/node/H924SAD12G.md" tag="H924SAD12G" kind="statement">
# Dvoretzky–Rogers theorem

## Statement

Let $X$ be an infinite-dimensional [[Banach space|0BDTTM24DP]].
Let $(c_n)$ be a sequence of nonnegative real numbers with $\sum_n c_n^2 < \infty$.
Then there is a sequence $(x_n)$ in $X$ with $\norm{x_n} = c_n$ for all $n$ such that $\sum_n x_n$ [[converges unconditionally|2ME9M61TJK]].

## Remarks

Taking $c_n = 1/n$ gives a series in $X$ that converges unconditionally but not [[absolutely|0W3K3QPEWT]].
In finite dimensions every unconditionally convergent series converges absolutely.
So a Banach space is finite-dimensional if and only if unconditional and absolute convergence of series coincide in it.
In a Hilbert space, every unconditionally convergent series satisfies $\sum \norm{x_n}^2 < \infty$, so the condition on $(c_n)$ cannot be weakened in general.
The proof uses the Dvoretzky–Rogers lemma, a statement about the geometry of finite-dimensional normed spaces.

## Explanations

### L1 human

In ordinary space, a series that converges in every order also converges when you replace each vector by its length.
The theorem says that this fails in every infinite-dimensional Banach space.
There are always series whose sum does not depend on the order, although the lengths of the terms add up to infinity.

### L2 human

The theorem says that absolute and unconditional convergence agree only in finite dimensions.
It shows that the example of $\sum e_n / n$ in $\ell^2$ is not special to Hilbert space.
Its proof works inside large finite-dimensional subspaces of $X$, a typical method of local Banach space theory.
</document>

<document source="https://tamnd.github.io/math2/node/H96HXG6CA0.md" tag="H96HXG6CA0" kind="concept">
# Regular singular point

## Summary

A regular singular point of a linear differential equation is a [[singular point|QK2J038VB4]] where the coefficients blow up only mildly.
In the normalised equation, the coefficient of $y^{(n-k)}$ has a pole of order at most $k$ there.
At such a point the solutions grow at most like a power of the distance to the point.
The [[Frobenius method|W8ZWXC2HQS]] then finds solutions of the form $(z - z_0)^r$ times a power series.
Bessel, Legendre and hypergeometric equations all have regular singular points.
See the [[definition|BMD7KQQA9C]].

## Explanations

### L1 human

Some singular points are tame and some are wild.
At a tame one, the solutions behave like powers $z^r$, perhaps with a logarithm, times nice series.
The model is the Euler equation $z^2 y'' + a z y' + b y = 0$, solved by $y = z^r$ whenever $r(r - 1) + a r + b = 0$.
At a wild, irregular point, solutions can behave like $e^{1/z}$ instead.
</document>

<document source="https://tamnd.github.io/math2/node/H971WGMV6C.md" tag="H971WGMV6C" kind="concept">
# Symmetric operator

## Summary

A symmetric operator is a densely defined linear operator $T$ on a Hilbert space with $\ip{Tx}{y} = \ip{x}{Ty}$ for all $x, y$ in its domain.
For bounded operators defined on the whole space, symmetric is the same as self-adjoint.
For unbounded operators, such as differential operators, symmetry is weaker than self-adjointness.
The gap between the two is the subject of the theory of self-adjoint extensions.

## Explanations

### L1 human

A symmetric operator can move from one side of the inner product to the other, as a symmetric matrix does.
Think of $-i \tfrac{d}{dx}$ acting on smooth functions that vanish near the ends of an interval: integration by parts moves the derivative across the inner product with no boundary terms.
The operator is symmetric, but it is only defined on a small set of functions.
Choosing a larger domain on which it becomes self-adjoint is a real question, and the answer depends on boundary conditions.
</document>

<document source="https://tamnd.github.io/math2/node/H9F9G4V1EM.md" tag="H9F9G4V1EM" kind="statement">
# Double centralizer theorem

## Statement

Let $K$ be a field and let $A$ be a [[central simple algebra|5HRKJBAY2V]] over $K$.
Let $B$ be a $K$-subalgebra of $A$ that is a [[simple ring|62M0NC3E2T]].
For a subset $S$ of $A$, let $C_A(S) = \set{a \in A : as = sa \text{ for all } s \in S}$ be its centralizer.
Then:

1. $C_A(B)$ is a simple $K$-algebra.
2. $\dim_K B \cdot \dim_K C_A(B) = \dim_K A$.
3. $C_A(C_A(B)) = B$.
4. If $B$ is central simple over $K$, then $A \cong B \otimes_K C_A(B)$.

## Remarks

The proof views $A$ as a module over the simple algebra $B \otimes_K A^{\mathrm{op}}$.
Then $C_A(B)$ is the endomorphism ring of this module, a matrix algebra over a division algebra.
Counting dimensions gives clause 2.
Since $B \subseteq C_A(C_A(B))$ and clause 2 applied twice gives equal dimensions, clause 3 follows.
Example: let $A = M_n(K)$ and $B = K[x]/(p)$, embedded through the companion matrix $C$ of an irreducible polynomial $p$ of degree $n$.
Then $C_A(B)$ consists of the polynomials in $C$, so it equals $B$, and indeed $n \cdot n = n^2$.
Example: for $A = M_2(K) \otimes_K M_3(K)$ and $B = M_2(K) \otimes 1$, the centralizer is $1 \otimes M_3(K)$, and $4 \cdot 9 = 36$.
Simplicity of $B$ is needed for clause 2: in $A = M_3(K)$ let $B$ be spanned by $I$ and the matrix unit $E_{13}$, a subalgebra with the nilpotent ideal $K E_{13}$.
A matrix commutes with $E_{13}$ exactly when its entries in positions $(2,1)$, $(3,1)$, $(3,2)$ vanish and its entries in positions $(1,1)$ and $(3,3)$ agree, so $\dim_K C_A(B) = 5$ and $2 \cdot 5 \ne 9$.

## Explanations

### L1 human

Inside a central simple algebra, take a simple subalgebra and collect everything that commutes with it.
The theorem says that the sizes of the two pieces multiply to the size of the whole algebra.
Doing it twice gives back the subalgebra one started with.

### L2 human

The main use is to subfields: if $L$ is a subfield of $A$ containing $K$, then $[L : K] \cdot \dim_K C_A(L) = \dim_K A$, so $[L : K]$ divides the degree of $A$.
When $C_A(L) = L$, this gives $[L : K]^2 = \dim_K A$, so $L$ is a maximal subfield and splits $A$.
Clause 4 shows that a central simple subalgebra always splits off as a tensor factor.
With the Skolem–Noether theorem it underlies the crossed product description of the Brauer group.
</document>

<document source="https://tamnd.github.io/math2/node/H9FSFB7T28.md" tag="H9FSFB7T28" kind="concept">
# Z-transform as a generating function

## Summary

The [[unilateral Z-transform|EPG6JPVJE7]] of a sequence is its ordinary [[generating function|0PHAC833JF]] evaluated at $1/z$.
So the two notions carry the same information and obey the same algebra, with the variable inverted.
Engineers use the Z-transform in the variable $z$, and combinatorialists use generating functions in $w = 1/z$.
Results proved on one side, such as rules for shifts, convolution and partial fractions, transfer to the other at once.

## Explanations

### L1 human

A generating function packs a sequence $x_0, x_1, \ldots$ into $x_0 + x_1 w + x_2 w^2 + \cdots$.
The Z-transform packs it into $x_0 + x_1 z^{-1} + x_2 z^{-2} + \cdots$.
These are the same series with $w = 1/z$.
For the constant sequence $1, 1, 1, \ldots$, the generating function is $1/(1 - w)$ and the Z-transform is $z/(z - 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/H9H80DM3SX.md" tag="H9H80DM3SX" kind="concept">
# Wiener amalgam space

## Summary

A Wiener amalgam space measures a function with one norm locally and another norm globally.
The basic example $W(L^p, \ell^q)$ takes the $L^p$ norm of the function on each unit cube and then the $\ell^q$ norm of the resulting sequence.
Such spaces separate the local regularity of a function from its decay at infinity.
They go back to Norbert Wiener, were developed in general by Hans Feichtinger, and are used in [[time–frequency analysis|GQFPVXEWK1]] and sampling theory.

## Explanations

### L1 human

Cut the line or the space into unit cubes.
On each cube, measure the function with one ruler, for instance its largest value.
Then add up these local sizes over all cubes with a second ruler.
A function with tall but narrow spikes can fail the local test, and a function with slow decay can fail the global test.
For example, $1/(1 + \abs{x})$ on the line has local maxima about $1/\abs{k}$ on the cube near $k$, so its sizes have finite sum of squares but infinite plain sum.
</document>

<document source="https://tamnd.github.io/math2/node/H9KTH7B380.md" tag="H9KTH7B380" kind="statement">
# Berry–Esseen theorem

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|00MRN0FARV]] real random variables with $E[X_1] = 0$, $\sigma^2 = E[X_1^2] > 0$ and $\rho = E[\abs{X_1}^3] < \infty$.
Let $F_n$ be the [[cumulative distribution function|0H8H7ZTHW2]] of $(X_1 + \cdots + X_n) / (\sigma \sqrt{n})$, and let $\Phi$ be the cumulative distribution function of the standard [[normal distribution|05AQE0QR0D]].
Then there is an absolute constant $C$, independent of $n$ and of the law of $X_1$, such that for all $n \ge 1$,
$$\sup_{x \in \R} \abs{F_n(x) - \Phi(x)} \le \frac{C \rho}{\sigma^3 \sqrt{n}}.$$

## Remarks

The theorem gives a rate for the [[central limit theorem|0EPZC4Z1FC]], which by itself only says that the left side tends to $0$.
The classical proof compares characteristic functions and uses Esseen's smoothing inequality, which bounds the distance between two distribution functions by an integral of the difference of their characteristic functions.
The best value of $C$ is not known; it is known that one may take $C < 1/2$.

The rate $n^{-1/2}$ cannot be improved in general.
Let $X_k$ be $1$ or $-1$ with probability $1/2$ each, so $\sigma = \rho = 1$.
For even $n$, the variable $S_n = X_1 + \cdots + X_n$ takes the value $0$ with probability $\binom{n}{n/2} 2^{-n}$, which is of order $n^{-1/2}$.
So $F_n$ has a jump of that size at $0$, while $\Phi$ is continuous, and the supremum is at least half the jump.

## Explanations

### L1 human

The central limit theorem says that a standardised sum of many independent copies of a variable has nearly a normal distribution.
The Berry–Esseen theorem says how near: the error is at most a constant times $1/\sqrt{n}$.
The bound depends on the law of the variable only through the ratio of its third absolute moment to the cube of its standard deviation.

### L2 human

The bound turns the normal approximation into a quantitative statement, which matters in statistics when the sample size is fixed.
It needs a finite third [[moment|01XAHPADBA]]; with only a finite variance, the convergence in the central limit theorem can be arbitrarily slow.
There are versions for independent variables that are not identically distributed, with $\sum_k E[\abs{X_k}^3] / s_n^3$ in place of $\rho / (\sigma^3 \sqrt{n})$, and multivariate versions.
Stein's method gives another proof and extends to dependent variables.
</document>

<document source="https://tamnd.github.io/math2/node/H9S7RMBMFD.md" tag="H9S7RMBMFD" kind="concept">
# Galois ring

## Summary

A Galois ring is a finite [[commutative ring|0310REJJY8]] that extends $\Z/p^n\Z$ in the same way as a [[finite field|0ZXZ82FTX2]] $\F_{p^r}$ extends $\F_p$.
Galois rings are [[local rings|MZ02BKPHJA]], and their residue fields are finite fields.
They contain both the rings $\Z/p^n\Z$ and the finite fields as special cases.
They are used in the theory of codes over $\Z/4\Z$ and in the study of [[finite rings|X6713ZP2NH]].

## Explanations

### L1 human

A finite field $\F_{p^r}$ is built from arithmetic modulo a prime $p$ by adding a root of an irreducible polynomial.
A Galois ring does the same over arithmetic modulo a prime power $p^n$.
For example, take arithmetic modulo $4$ and add a symbol $x$ with $x^2 = -x - 1$.
The result has $16$ elements $a + bx$ with $a, b \in \set{0, 1, 2, 3}$.
Reducing everything modulo $2$ gives the field $\F_4$ with four elements.
</document>

<document source="https://tamnd.github.io/math2/node/H9X4FG08JT.md" tag="H9X4FG08JT" kind="definition">
# Equilibrium measure

## Definition

Let $K$ be a [[compact set|0R6EWFKRCF]] in $\C$.
For a Borel [[probability measure|059TJN0NZ1]] $\mu$ on $K$, let
$$I(\mu) = \int_K \int_K \log\frac{1}{\abs{z - w}} \, d\mu(z) \, d\mu(w)$$
be its logarithmic [[energy|SFSRQCM3EG]], with values in $(-\infty, \infty]$.
Let $V(K) = \inf_\mu I(\mu)$, the infimum over all Borel probability measures $\mu$ on $K$.
Assume that $V(K) < \infty$, that is, $K$ has positive [[logarithmic capacity|DBQXRTC0VN]].
An *equilibrium measure* of $K$ is a Borel probability measure $\nu$ on $K$ with $I(\nu) = V(K)$.

## Remarks

Such a measure exists, because the probability measures on $K$ form a weak-star compact set and $I$ is lower semicontinuous for this topology.
It is unique, because the energy is strictly convex on probability measures of finite energy.
So one speaks of *the* equilibrium measure of $K$.

Example: let $K$ be the unit circle and let $\nu$ be normalized arc length $d\theta / 2\pi$.
For $\abs{z} \le 1$ the mean value $\frac{1}{2\pi} \int_0^{2\pi} \log \abs{z - e^{i\theta}} \, d\theta$ equals $\log \max(\abs{z}, 1) = 0$.
So the potential of $\nu$ vanishes on $K$, and $I(\nu) = 0 = V(K)$, which agrees with capacity $1$.

Non-example: the unit point mass $\delta_1$ is a probability measure on the unit circle, but $I(\delta_1) = \log(1/0) = \infty \ne V(K)$.

In $\R^n$ with $n \ge 3$ the same definition is used with the Newtonian kernel $\abs{x - y}^{2-n}$ in place of $\log(1/\abs{z - w})$.
</document>

<document source="https://tamnd.github.io/math2/node/HA1KJ3B9DX.md" tag="HA1KJ3B9DX" kind="concept">
# Cross-validation

## Summary

Cross-validation estimates how well a fitted model predicts new data by repeatedly fitting it on part of the data and testing it on the rest.
Each observation is used for testing exactly once, with a model that did not see it.
It is the standard tool to choose tuning parameters, such as a bandwidth or a penalty, and to compare predictive models.

## Explanations

### L1 human

Checking a model on the same data that built it is too optimistic, like grading students on questions they saw in advance.
Cross-validation hides a part of the data, fits the model on the rest, and checks the predictions on the hidden part.
In $5$-fold cross-validation, the data are cut into $5$ pieces and each piece is hidden once.
The average error over the $5$ rounds estimates the error on new data.
</document>

<document source="https://tamnd.github.io/math2/node/HA5YJM70TF.md" tag="HA5YJM70TF" kind="definition">
# Yang–Mills equations

## Definition

Let $(M, g)$ be an oriented [[Riemannian manifold|Q61VFHCH2J]] of dimension $n$.
Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and let $P \to M$ be a [[principal $G$-bundle|8E37E74XVP]].
Let $A$ be a [[connection|NJ21PN5ZGJ]] on $P$ with [[curvature|ZAJFW1S078]] $F_A$, a $2$-form on $M$ with values in the adjoint bundle $\mathrm{ad}\, P = P \times_G \mathfrak{g}$.
Let $d_A$ be the exterior [[covariant derivative|5FZDQQHQKM]] of $A$ on $\mathrm{ad}\, P$-valued forms, and let $\star$ be the [[Hodge star operator|1Y4WX4XQWJ]] of $g$.
The connection $A$ satisfies the *Yang–Mills equations* if
$$d_A \star F_A = 0.$$
Such a connection is called a *Yang–Mills connection*.

## Remarks

The equation is equivalent to $d_A^* F_A = 0$, where $d_A^*$ is the formal adjoint of $d_A$.
Every connection also satisfies the Bianchi identity $d_A F_A = 0$.
For compact $M$, the Yang–Mills connections are the critical points of the Yang–Mills functional $\mathrm{YM}(A) = \tfrac{1}{2} \int_M \abs{F_A}^2 \, d\mathrm{vol}_g$, where the norm uses an invariant inner product on $\mathfrak{g}$.

Example: a flat connection, with $F_A = 0$, satisfies the equations.
On a four-manifold, a connection with $\star F_A = \pm F_A$, an instanton or anti-instanton, also satisfies them, because $d_A \star F_A = \pm d_A F_A = 0$ by the Bianchi identity.

Non-example: let $M = \R^4$ with the Euclidean metric and $G = U(1)$, so that $d_A = d$, and take $A = i x_1^2 \, dx_2$.
Then $F_A = 2i x_1 \, dx_1 \wedge dx_2$, $\star F_A = 2i x_1 \, dx_3 \wedge dx_4$, and $d \star F_A = 2i \, dx_1 \wedge dx_3 \wedge dx_4 \ne 0$.

For $G = U(1)$ and a Lorentzian metric the equations are Maxwell's equations in vacuum.
</document>

<document source="https://tamnd.github.io/math2/node/HA63ZVJFVZ.md" tag="HA63ZVJFVZ" kind="definition">
# Monge point

## Definition

Let $n \ge 2$ and let $\Delta$ be an $n$-[[simplex|YX1DJK2QAG]] in $\R^N$ with vertices $v_0, \ldots, v_n$.
Let $o$ be its circumcentre, the centre of its [[circumsphere|MRC4WMVTAH]].
Let $g = \frac{1}{n+1}(v_0 + \cdots + v_n)$ be its [[centroid|0Y5RHNRKJZ]].
The *Monge point* of $\Delta$ is
$$m = o + \frac{n + 1}{n - 1} \, (g - o).$$

## Remarks

For a tetrahedron, $n = 3$ and $m = 2g - o$, the reflection of $o$ in $g$.
In that case $m$ is the common point of the six planes through the midpoint of an edge perpendicular to the opposite edge.

For a triangle, $n = 2$ and $m = 3g - 2o$, which is the [[orthocentre|0WB619CA0V]].
Example: the triangle with vertices $(0, 0)$, $(2, 0)$ and $(0, 2)$ has $o = (1, 1)$ and $g = (\frac{2}{3}, \frac{2}{3})$.
Then $m = (1, 1) + 3 \cdot (-\frac{1}{3}, -\frac{1}{3}) = (0, 0)$, the vertex of the right angle, which is indeed the orthocentre.

Non-example: for a segment, $n = 1$, the factor $\frac{n+1}{n-1}$ is undefined, so a $1$-simplex has no Monge point.

For a regular tetrahedron, $g = o$, so the Monge point is the centre.
</document>

<document source="https://tamnd.github.io/math2/node/HA79YFZAQR.md" tag="HA79YFZAQR" kind="concept">
# Ordered semigroup

## Summary

An ordered semigroup is a [[semigroup|0XMEM406JM]] with a [[partial order|0QV2JS9RJ3]] that is preserved by multiplication on both sides.
It is the most general of the ordered algebraic structures with one associative operation.
Ordered monoids and [[ordered groups|KRDR04RQ7N]] are special cases.
Ordered semigroups appear in the theory of formal languages, of semirings and of residuated structures.

## Explanations

### L1 human

The natural numbers under addition are an ordered semigroup: if $a \le b$, then $a + c \le b + c$.
The positive integers under multiplication are another: if $a \le b$, then $ac \le bc$.
In general, an ordered semigroup is any associative operation together with a way to compare elements that the operation respects.
</document>

<document source="https://tamnd.github.io/math2/node/HABCJ4Y0SR.md" tag="HABCJ4Y0SR" kind="statement">
# Hodge decomposition

## Statement

Let $X$ be a compact [[Kähler manifold|X60JG9AEQJ]] of complex dimension $n$, and let $0 \le k \le 2n$.
Let $\Omega^p_X$ be the sheaf of holomorphic $p$-forms on $X$.
Let $H^{p,q}(X)$ be the space of classes in $H^k(X, \C)$ represented by a closed form of type $(p, q)$.
Then:

1. The [[singular cohomology|6N7BC70CX0]] with complex coefficients decomposes as a direct sum
$$H^k(X, \C) = \bigoplus_{p + q = k} H^{p,q}(X).$$
2. Complex conjugation on $H^k(X, \C) = H^k(X, \R) \otimes \C$ maps $H^{p,q}(X)$ onto $H^{q,p}(X)$.
3. There are natural isomorphisms $H^{p,q}(X) \cong H^q(X, \Omega^p_X)$ with [[sheaf cohomology|PQ6Q143ARJ]], which is Dolbeault cohomology.

## Remarks

The idea of the proof: by the [[Hodge theorem|A127WFVYKW]], each class in [[de Rham cohomology|GX74C07T0G]] has a unique harmonic representative for the Laplacian $\Delta_d$.
The Kähler identities give $\Delta_d = 2\Delta_{\bar\partial}$, so $\Delta_d$ preserves the type $(p, q)$ of forms, and the harmonic part of a form splits by type.
The $\Delta_{\bar\partial}$-harmonic $(p,q)$-forms represent Dolbeault cohomology, which gives part 3; part 2 holds because $\Delta_d$ is a real operator.

Example: for a compact Riemann surface of genus $g$, $H^1(X, \C) = H^{1,0} \oplus H^{0,1}$ with $H^{1,0}$ the $g$-dimensional space of holomorphic $1$-forms, so $b_1 = 2g$.

Consequence: the Hodge numbers $h^{p,q} = \dim H^{p,q}$ satisfy $h^{p,q} = h^{q,p}$, so every odd Betti number $b_{2m+1} = \sum_{p+q = 2m+1} h^{p,q}$ is even.
The Kähler hypothesis is needed: the Hopf surface, diffeomorphic to $S^1 \times S^3$, is a compact complex manifold with $b_1 = 1$, so no such decomposition with symmetry exists on it.

## Explanations

### L1 human

The cohomology of a compact Kähler manifold, such as a smooth projective complex variety, splits into pieces labelled by pairs $(p, q)$.
The piece $(p, q)$ comes from forms with $p$ holomorphic and $q$ antiholomorphic differentials.
The pieces $(p, q)$ and $(q, p)$ are mirror images under complex conjugation.

### L2 human

The decomposition makes $H^k(X, \Z)$ modulo torsion into a Hodge structure of weight $k$, which is the starting point of period maps and of the Torelli problem.
Combined with the hard Lefschetz theorem and the Hodge–Riemann bilinear relations, it gives polarized Hodge structures.
It also links topology to holomorphic invariants: for example $h^{1,0}$, the number of independent holomorphic $1$-forms, equals half the first Betti number.
</document>

<document source="https://tamnd.github.io/math2/node/HACWFM9JXE.md" tag="HACWFM9JXE" kind="definition">
# Elliptic integral

## Definition

Let $R(x, y)$ be a [[rational function|0XDK9CGYKJ]] of two variables, and let $P$ be a polynomial of degree $3$ or $4$ with no repeated roots.
An *elliptic integral* is an integral of the form
$$f(x) = \int_c^x R\big(t, \sqrt{P(t)}\big) \, dt,$$
where $c$ is a constant and $\sqrt{\cdot}$ is a [[square root|0F7FQS6XY6]].

Let $0 \le k < 1$, let $\varphi$ be real, and let $n$ be a real parameter with $n \sin^2\theta \ne 1$ for $\theta$ between $0$ and $\varphi$.
The *incomplete elliptic integrals in Legendre form* of the first, second and third kind are
$$F(\varphi, k) = \int_0^\varphi \frac{d\theta}{\sqrt{1 - k^2 \sin^2\theta}}, \qquad E(\varphi, k) = \int_0^\varphi \sqrt{1 - k^2 \sin^2\theta} \, d\theta,$$
$$\Pi(n; \varphi, k) = \int_0^\varphi \frac{d\theta}{(1 - n \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}}.$$
The number $k$ is the *modulus*.

## Remarks

The substitution $t = \sin\theta$ turns $F(\varphi, k)$ into $\int_0^{\sin\varphi} dt / \sqrt{(1 - t^2)(1 - k^2 t^2)}$, an elliptic integral with $P(t) = (1 - t^2)(1 - k^2 t^2)$, which has the four distinct roots $\pm 1$ and $\pm 1/k$ when $0 < k < 1$.
Example: the ellipse $x = a\sin\theta$, $y = b\cos\theta$ with $0 < b < a$ has arc length element $\sqrt{a^2\cos^2\theta + b^2\sin^2\theta}\,d\theta = a\sqrt{1 - e^2\sin^2\theta}\,d\theta$, where $e^2 = 1 - b^2/a^2$; so its arc length from $\theta = 0$ to $\theta = \varphi$ is $a E(\varphi, e)$.
Non-example: $\int_0^x dt / \sqrt{1 - t^2} = \arcsin x$ is not an elliptic integral, since $P(t) = 1 - t^2$ has degree $2$.
Every elliptic integral is a sum of elementary functions and constant multiples of integrals of the three Legendre kinds.
The values at $\varphi = \pi/2$ are the complete integrals $K(k) = F(\pi/2, k)$, [[of the first kind|RM54CPPE12]], and $E(k) = E(\pi/2, k)$, [[of the second kind|VS2PSX14C8]].
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<document source="https://tamnd.github.io/math2/node/HASA2XSG66.md" tag="HASA2XSG66" kind="statement">
# Interpolation error formula

## Statement

Let $n \ge 0$, let $a < b$, and let $f \colon [a, b] \to \R$ be $n + 1$ times continuously differentiable.
Let $x_0, \dots, x_n \in [a, b]$ be pairwise distinct, and let $p$ be the [[interpolation polynomial|SS1YMS2NPC]] of degree at most $n$ with $p(x_i) = f(x_i)$ for all $i$.
Then for every $x \in [a, b]$ there is $\xi$ in the smallest closed interval that contains $x, x_0, \dots, x_n$ such that
$$f(x) - p(x) = \frac{f^{(n+1)}(\xi)}{(n + 1)!} \prod_{i=0}^{n} (x - x_i).$$

## Remarks

If $x$ is a node, both sides are $0$.
Otherwise set $\omega(t) = \prod_i (t - x_i)$ and choose the constant $c$ so that $g(t) = f(t) - p(t) - c \, \omega(t)$ vanishes at $t = x$.
Then $g$ has $n + 2$ distinct zeros, and $n + 1$ applications of [[Rolle's theorem|0194DY35BG]] give a zero $\xi$ of the [[derivative|0V1EJCT8G6]] $g^{(n+1)} = f^{(n+1)} - c \, (n + 1)!$.
This gives $c = f^{(n+1)}(\xi)/(n + 1)!$.

Example: take $f = \exp$ on $[0, 1]$ with the nodes $0$ and $1$.
At $x = 1/2$ the error of the linear interpolant is $\frac{e^{\xi}}{2} \cdot \frac{1}{2} \cdot \left(-\frac{1}{2}\right)$ for some $\xi \in [0, 1]$.
So its absolute value is at most $e/8$.

For $n = 0$ the formula is the mean value theorem.
It is the analogue of [[Taylor's theorem|03QHBVW4TT]] with the Lagrange remainder, with $n + 1$ separate points in place of one point of high contact.

## Explanations

### L1 human

When a polynomial is fitted through some points of a smooth curve, it matches the curve exactly at those points and is off somewhere else.
The formula says how far off it is.
The error is small when the curve does not bend too much, measured by a high derivative, and when the point is close to the nodes.

### L2 human

Taking the maximum over $[a, b]$ gives $\norm{f - p}_\infty \le \frac{\norm{f^{(n+1)}}_\infty}{(n + 1)!} \max_{x \in [a, b]} \abs{\omega(x)}$.
The first factor depends only on $f$, the second only on the nodes.
[[Chebyshev nodes|HYK8J56PSH]] make the second factor as small as possible on an interval.
When the derivatives of $f$ grow fast, the bound can fail to tend to $0$, which is what happens in [[Runge's phenomenon|ZMT8JF21R6]].
In terms of [[divided differences|N5ZHV4HWDC]], for $x$ not a node the error equals $f[x_0, \dots, x_n, x] \, \omega(x)$.
</document>

<document source="https://tamnd.github.io/math2/node/HAT8HG0RCK.md" tag="HAT8HG0RCK" kind="statement">
# Annulus theorem

## Statement

Let $n \ge 2$.
Let $\Sigma_1, \Sigma_2 \subset \R^n$ be disjoint locally flat embedded $(n-1)$-spheres, that is, images of [[topological embeddings|00ZG9Y9TR4]] $S^{n-1} \times [-1, 1] \to \R^n$ restricted to $S^{n-1} \times \set{0}$.
For $i = 1, 2$, let $U_i$ be the bounded component of $\R^n \setminus \Sigma_i$, and suppose $\Sigma_1 \subset U_2$.
Then the region $\overline{U_2} \setminus U_1$ is [[homeomorphic|0QT00PB5G2]] to the annulus $S^{n-1} \times [0, 1]$.

## Remarks

Each $\overline{U_i}$ is a closed ball by the generalized [[Schoenflies theorem|3VX06XRVZG]].
This does not give the statement, because the inner ball may sit inside the outer one in a complicated way.
For $n \le 3$ the theorem is classical; in dimension $3$ it rests on Moise's work on triangulations.
For $n \ge 5$ it was proved by Robion Kirby in 1969, using the torus trick, and for $n = 4$ by Frank Quinn in 1982.
An equivalent form is the stable homeomorphism theorem: every orientation-preserving homeomorphism of $\R^n$ is a finite composition of homeomorphisms that are the identity on some open set.
The local flatness hypothesis is necessary: wild spheres such as the Alexander horned sphere give counterexamples in dimension $3$.

## Explanations

### L1 human

Put one ball inside a larger ball, both possibly deformed but with tame surfaces.
The theorem says that the shell between them is always a standard thick spherical shell, like the peel of an orange.
In the plane, this says that the ring between two nested loops is a deformed round ring.

### L2 human

The annulus theorem was long called the annulus conjecture, and its proof was a turning point in the theory of topological manifolds.
It is needed to show that the connected sum of oriented topological manifolds is well defined.
It is also a step in Kirby and Siebenmann's study of triangulations and piecewise linear structures on topological manifolds.
Kirby's torus trick, introduced for this proof, became a standard tool of the subject.
</document>

<document source="https://tamnd.github.io/math2/node/HATVWNZNMR.md" tag="HATVWNZNMR" kind="concept">
# Lambert series

## Summary

A Lambert series is a series of the form $\sum_{n \ge 1} a_n \frac{q^n}{1 - q^n}$ in a variable $q$ with $\abs{q} < 1$.
Expanding each fraction as a geometric series turns it into a power series whose coefficients are divisor sums $\sum_{d \mid m} a_d$.
This makes Lambert series a bridge between [[arithmetic functions|09115NQZ07]] and power series, used in the theory of partitions, modular forms and sums of squares.

## Explanations

### L1 human

Each term $q^n / (1 - q^n)$ is the geometric series $q^n + q^{2n} + q^{3n} + \cdots$, which collects the powers of $q$ whose exponent is a multiple of $n$.
Adding these up with weights $a_n$, the power $q^m$ receives the weight $a_d$ once for each divisor $d$ of $m$.
For example, with all $a_n = 1$, the coefficient of $q^m$ is the number of divisors of $m$, so $\sum_n q^n/(1 - q^n) = q + 2q^2 + 2q^3 + 3q^4 + \cdots$.
</document>

<document source="https://tamnd.github.io/math2/node/HAZ0NKCAXS.md" tag="HAZ0NKCAXS" kind="definition">
# Traveling wave

## Definition

Consider a [[partial differential equation|GC5BDZ6QBC]] for a function $u \colon \R \times \R \to \R^m$ of a space variable $x$ and a time variable $t$.
A *traveling wave* solution is a solution of the form
$$u(x, t) = U(x - c t),$$
where $c \in \R$ is the *speed* and $U \colon \R \to \R^m$ is the *profile*.
When the limits $U(\pm\infty)$ exist and differ, the wave is called a *front*.
When they exist and are equal and $U$ is not constant, it is called a *pulse*.

## Remarks

In $\R^n$ one also considers plane traveling waves $u(x, t) = U(x \cdot e - ct)$ for a unit vector $e$.
Substituting $\xi = x - ct$ turns the equation into an ordinary differential equation for $U(\xi)$, in which $c$ is an unknown parameter.

Example: the viscous [[Burgers' equation|NEGHXN8RE1]] $u_t + u u_x = \nu u_{xx}$, with $\nu > 0$, has the front $u = c\left(1 - \tanh\frac{c(x - ct)}{2\nu}\right)$ for every $c > 0$.
Check: with $U(\xi) = c(1 - \tanh \tfrac{c\xi}{2\nu})$, the equation becomes $-cU' + UU' = \nu U''$.
Integrating once with $U(+\infty) = 0$ gives $\nu U' = \tfrac12 U^2 - cU$, which this $U$ satisfies.
The front goes from $2c$ to $0$, and its speed $c$ is the average of the two states, as for a shock.

Non-example: $u(x, t) = e^{-t} \sin x$ solves the heat equation $u_t = u_{xx}$, but it is not a traveling wave.
For a traveling wave, $\sup_x \abs{u(x, t)} = \sup \abs{U}$ does not depend on $t$, while here it equals $e^{-t}$.

## Explanations

### L1 human

A traveling wave is a fixed shape that slides along the line at constant speed.
</document>

<document source="https://tamnd.github.io/math2/node/HB0MJCV482.md" tag="HB0MJCV482" kind="statement">
# Christoffel–Darboux formula

## Statement

Let $\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\R$ with infinite support and all moments finite.
Let $(p_n)_{n \ge 0}$ be the orthonormal polynomials for $\mu$, with $\deg p_n = n$ and leading coefficient $k_n > 0$.
So $\int_\R p_m p_n\, d\mu$ is $1$ if $m = n$ and $0$ otherwise.
Then for every $n \ge 0$ and all real $x \ne y$,
$$\sum_{j=0}^{n} p_j(x)\, p_j(y) = \frac{k_n}{k_{n+1}} \cdot \frac{p_{n+1}(x)\, p_n(y) - p_n(x)\, p_{n+1}(y)}{x - y},$$
and for every real $x$,
$$\sum_{j=0}^{n} p_j(x)^2 = \frac{k_n}{k_{n+1}} \left( p_{n+1}'(x)\, p_n(x) - p_n'(x)\, p_{n+1}(x) \right).$$

## Remarks

For orthonormal polynomials the three-term recurrence reads $x p_j = \alpha_{j+1} p_{j+1} + \beta_j p_j + \alpha_j p_{j-1}$ with $\alpha_j = \frac{k_{j-1}}{k_j}$.
Multiplying it at $x$ by $p_j(y)$, subtracting the same with $x$ and $y$ exchanged, and summing over $j$ makes the sum telescope to the right-hand side.
The second formula is the limit $y \to x$ of the first.
Example: let $n = 0$ and let $\mu$ be a probability measure with mean $m$ and variance $s^2$.
Then $p_0 = 1$ and $p_1 = \frac{x - m}{s}$, and the right-hand side is $\frac{1}{1/s} \cdot \frac{(x - m)/s - (y - m)/s}{x - y} = 1$, which matches the left-hand side.
The left-hand side $K_n(x, y)$ is the reproducing kernel of the polynomials of degree at most $n$ for the [[inner product|0NB0AWGJBW]] $\int f g\, d\mu$.

## Explanations

### L1 human

Adding up the products $p_j(x) p_j(y)$ for all degrees up to $n$ looks like a long computation.
The formula says that the sum collapses to an expression in only the last two polynomials.
This is the polynomial analogue of the closed form for the Dirichlet kernel in Fourier series.

### L2 human

The kernel $K_n(x, y)$ gives the orthogonal projection onto polynomials of degree at most $n$, so the formula controls partial sums of orthogonal expansions.
The diagonal form shows $p_{n+1}' p_n - p_n' p_{n+1} > 0$, which implies that the zeros of $p_n$ and $p_{n+1}$ are real, simple and interlace.
The numbers $\frac{1}{K_n(x, x)}$ are the Christoffel function, whose values at the zeros of $p_{n+1}$ are the Gaussian quadrature weights.
In random matrix theory the kernel $K_n$ describes the correlations of eigenvalues of unitary invariant ensembles.
</document>

<document source="https://tamnd.github.io/math2/node/HB14PWFCS6.md" tag="HB14PWFCS6" kind="definition">
# Envelope

## Definition

Let $J \subseteq \R$ be an open interval, let $U \subseteq \R^2$ be open, and let $F : U \times J \to \R$ be continuously differentiable.
For $c \in J$, let $C_c = \set{(x, y) \in U : F(x, y, c) = 0}$, a one-parameter family of curves.
An *envelope* of the family is a [[regular curve|06JMR2XWXW]] $\gamma : J \to U$ such that for each $c \in J$:

1. $\gamma(c)$ lies on $C_c$, that is, $F(\gamma(c), c) = 0$;
2. the gradient $\nabla_{x,y} F(\gamma(c), c)$ is nonzero;
3. $\gamma$ is tangent to $C_c$ at $\gamma(c)$, that is, $\nabla_{x,y} F(\gamma(c), c) \cdot \gamma'(c) = 0$.

## Remarks

Differentiating $F(\gamma(c), c) = 0$ and using clause 3 gives $\partial F / \partial c\, (\gamma(c), c) = 0$, where $\partial F/\partial c$ is the [[partial derivative|05AXG3QRKR]] in $c$.
So the points of an envelope satisfy the two equations $F = 0$ and $\partial F / \partial c = 0$.
Eliminating $c$ gives the discriminant set, which contains the envelope but may also contain singular points of the curves.
Example: $F(x, y, c) = x \cos c + y \sin c - 1$, the lines at distance $1$ from the origin.
Then $\gamma(c) = (\cos c, \sin c)$ satisfies $F = \cos^2 c + \sin^2 c - 1 = 0$, and $\gamma'(c) = (-\sin c, \cos c)$ is orthogonal to $\nabla F = (\cos c, \sin c)$; so the unit circle is the envelope.
Non-example: the concentric circles $x^2 + y^2 - c^2 = 0$, $c > 0$, have no envelope, since $\partial F / \partial c = -2c \ne 0$ and clause 3 can never hold together with clause 1.
For a regular curve $\beta$, the family of its [[tangent lines|01R1HW3Y42]], with parameter $c$ the parameter of $\beta$, has $\beta$ itself as an envelope.
</document>

<document source="https://tamnd.github.io/math2/node/HB2K72PSZ6.md" tag="HB2K72PSZ6" kind="statement">
# Gromov's systolic inequality

## Statement

Let $n \ge 1$ be an integer.
Let $M$ be a closed connected manifold of dimension $n$ with fundamental group $\pi$.
Let $f \colon M \to K(\pi, 1)$ be a map that induces an isomorphism of fundamental groups.
Let $A = \Z$ if $M$ is orientable and $A = \Z/2\Z$ otherwise.
Call $M$ *essential* if $f$ sends the fundamental class of $M$ to a nonzero class of $H_n(K(\pi, 1); A)$.

There is a constant $C_n > 0$, depending only on $n$, with the following property.
For every essential $M$ of dimension $n$ and every [[Riemannian metric|Q61VFHCH2J]] $g$ on $M$,
$$\operatorname{sys}(M, g)^n \le C_n \operatorname{vol}(M, g),$$
where $\operatorname{sys}$ is the [[systole|PZCJV40M4K]].

## Remarks

Gromov's proof goes through the filling radius.
One embeds $M$ isometrically into the Banach space $L^\infty(M)$ by the Kuratowski embedding.
For essential $M$ the filling radius is at least $\operatorname{sys}(M, g)/6$.
On the other hand, a filling inequality bounds the filling radius by a constant times $\operatorname{vol}(M, g)^{1/n}$.

The tori $T^n$, the real projective spaces $\R P^n$ and all closed aspherical manifolds are essential.
For $T^2$ the inequality is a weak form of [[Loewner's torus inequality|1Y9X1YW6GH]].

The hypothesis that $M$ is essential cannot be dropped.
On $S^2 \times S^1$ take the product of a round metric of small area $\varepsilon$ on $S^2$ and a circle of length $1$.
Every noncontractible loop winds around the circle factor, so the systole is $1$, while the volume $\varepsilon$ tends to $0$.

## Explanations

### L1 human

For many manifolds, the shortest loop that cannot be shrunk to a point controls the volume.
If every such loop has length at least $L$, then the volume is at least a fixed multiple of $L^n$.
The theorem gives this for a large class of manifolds, the essential ones, in every dimension.

### L2 human

This is the higher-dimensional extension of Loewner's and Pu's inequalities in [[systolic geometry|PZCJV40M4K]].
The constant $C_n$ from the proof is not optimal, and optimal constants are unknown in most cases.
The notion of essential manifold is topological: it asks that the fundamental class survive in the homology of the classifying space of the fundamental group.
</document>

<document source="https://tamnd.github.io/math2/node/HB4XDKJTW8.md" tag="HB4XDKJTW8" kind="concept">
# Annihilator

## Summary

Let $M$ be a module over a commutative ring $R$.
The annihilator of $M$ is the set of elements of $R$ that send every element of $M$ to zero.
It is an ideal of $R$.
It measures how far the action of $R$ on $M$ is from being faithful.
It is used to study the [[support of a module|FETMR0M0M8]] and the [[associated primes|ERFZ6S8X0X]].
It differs from the [[annihilator of a subspace|0ASHNYF7AG]] in linear algebra, which lives in the dual space.

## Explanations

### L1 human

The annihilator of a module collects the ring elements that "kill" the whole module.
For the $\Z$-module $\Z/6\Z$, multiplying by $6$, $12$ or $-18$ gives zero on every element, and multiplying by any integer outside $6\Z$ does not.
So the annihilator of $\Z/6\Z$ is $6\Z$.
For a single element $m$, its annihilator is the set of $r$ with $rm = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/HBBE0C1NFD.md" tag="HBBE0C1NFD" kind="definition">
# Measurable cardinal

## Definition

Let $\kappa$ be a [[cardinal|0EBACQFAZW]].
An [[ultrafilter|J64Y3JC0RE]] $U$ on $\kappa$ is *$\kappa$-complete* if $\bigcap \mathcal{F} \in U$ for every family $\mathcal{F} \subseteq U$ with $0 < \abs{\mathcal{F}} < \kappa$.
It is *nonprincipal* if it contains no set $\set{\alpha}$ with $\alpha < \kappa$.
The cardinal $\kappa$ is *measurable* if:

1. $\kappa$ is [[uncountable|0W7FBYS1F1]].
2. There is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$.

## Remarks

The cardinal $\aleph_0$ satisfies clause 2: every ultrafilter is closed under finite intersections, and nonprincipal ultrafilters on $\omega$ exist by the axiom of choice; it fails only clause 1.
The cardinal $\aleph_1$ fails clause 2, because every measurable cardinal is inaccessible, and $\aleph_1$ is not.
Equivalent form: $\kappa$ is measurable if and only if there is an elementary embedding $j \colon V \to M$, with $M$ a transitive class, whose critical point is $\kappa$.
The critical point condition means that $j(\alpha) = \alpha$ for $\alpha < \kappa$ and $j(\kappa) > \kappa$.
The embedding comes from the ultrapower of $V$ by $U$, and conversely $U = \set{X \subseteq \kappa : \kappa \in j(X)}$.
Scott proved that if a measurable cardinal exists, then $V \neq L$.
</document>

<document source="https://tamnd.github.io/math2/node/HBDEPTRF11.md" tag="HBDEPTRF11" kind="concept">
# Solvmanifold

## Summary

A solvmanifold is a homogeneous space $G/H$ of a connected [[solvable Lie group|9CNSVFQ801]] $G$ by a closed subgroup $H$.
Tori, nilmanifolds and many mapping tori of torus maps are solvmanifolds.
The compact ones of the form $G/\Gamma$, with $\Gamma$ a lattice, are natural test spaces in geometry and dynamics.
For example, they give examples of symplectic manifolds without Kähler structure.

## Explanations

### L1 human

A solvable group is one built from commutative groups in finitely many steps.
A solvmanifold is a space on which such a group acts, so that any point can be moved to any other point.
The line is an example: the maps $x \mapsto ax + b$ already move any point of the line to any other, and their group is solvable.
The torus $\R^n / \Z^n$ is a compact example, with the group $\R^n$ acting by translations.
</document>

<document source="https://tamnd.github.io/math2/node/HBFHDH4EJX.md" tag="HBFHDH4EJX" kind="statement">
# Urysohn metrization theorem

## Statement

Let $X$ be a [[topological space|0CQHFPGJ45]].
Assume that $X$ is [[second-countable|03PPNCM9GG]], [[regular|03HMJASQBQ]] and [[Hausdorff|0S1FJ8VEX9]].
Then $X$ is [[metrizable|091GJ3FYD7]]: there is a metric on $X$ whose open sets are exactly the open sets of $X$.

## Remarks

A second-countable regular space is normal, so [[Urysohn's lemma|01EY3VR7ZD]] applies to it.
With a countable base one picks countably many continuous functions $f_n \colon X \to [0, 1]$ that separate points from closed sets.
The map $x \mapsto (f_n(x))_n$ is then an embedding of $X$ into the [[Hilbert cube|JKMMJ8RFC6]] $[0,1]^{\N}$, which is metrizable.
For example, every subspace of $\R^n$ satisfies the hypotheses, and so does every compact Hausdorff space with a countable base.
Second countability cannot be dropped: the [[Sorgenfrey line|Y4T3GVK8GV]] is regular and Hausdorff but not metrizable.
Regularity cannot be dropped either: there are second-countable Hausdorff spaces that are not regular, and a metrizable space is always regular.
The converse holds only in part: a metrizable space is regular and Hausdorff, and it is second-countable exactly when it is separable.

## Explanations

### L1 human

A metric is a way to measure distances, and many spaces are given only by their open sets.
The theorem says when distances can be put back in.
It is enough that the space has a countable family of basic open sets and that points and closed sets can be kept apart by open sets.
So a large class of spaces that look abstract can be treated as metric spaces.

### L2 human

The theorem is the first answer to the metrization problem: which topological spaces come from a metric.
The proof is an embedding into the Hilbert cube, a standard method that also shows each such space is a subspace of one fixed compact metric space.
It characterizes separable metrizable spaces as the second-countable regular Hausdorff spaces.
The [[Nagata–Smirnov metrization theorem|KB11T6389G]] and the [[Bing metrization theorem|RGCZ2P9QRM]] remove the countability condition and give a full characterization.
</document>

<document source="https://tamnd.github.io/math2/node/HBGCDP2BRN.md" tag="HBGCDP2BRN" kind="concept">
# Pseudocompact space

## Summary

A pseudocompact space is a [[topological space|0CQHFPGJ45]] on which every continuous real-valued function is bounded.
It keeps one of the most used consequences of [[compactness|0TVJZYB6SH]], the boundedness of continuous functions, without asking for finite subcovers.
Every compact space and every [[countably compact space|S3N3CHZV89]] is pseudocompact.
For [[metric spaces|0VKD1TP7F3]] the notion gives nothing new: a metric space is pseudocompact exactly when it is compact.

## Explanations

### L1 human

On a closed bounded interval, a continuous function cannot run off to infinity.
A pseudocompact space is a space where this good behaviour holds for every continuous function into the real line.
The real line itself is not pseudocompact, because the function $f(x) = x$ is continuous and unbounded.
The interval $[0, 1]$ is pseudocompact, because it is compact.
</document>

<document source="https://tamnd.github.io/math2/node/HBHNB63J7R.md" tag="HBHNB63J7R" kind="statement">
# Continuity of measure from below and above

## Statement

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]], and let $A_1, A_2, \ldots$ be sets in $\Sigma$.

1. (Continuity from below.) If $A_1 \subseteq A_2 \subseteq \cdots$, then $\mu\left(\bigcup_{n=1}^{\infty} A_n\right) = \lim_{n \to \infty} \mu(A_n)$.
2. (Continuity from above.) If $A_1 \supseteq A_2 \supseteq \cdots$ and $\mu(A_1) < \infty$, then $\mu\left(\bigcap_{n=1}^{\infty} A_n\right) = \lim_{n \to \infty} \mu(A_n)$.

The [[limits|05M5SWSZZD]] are taken in $[0, \infty]$.

## Remarks

For part 1, put $B_1 = A_1$ and $B_n = A_n \setminus A_{n-1}$ for $n \ge 2$.
The sets $B_n$ are pairwise disjoint, their union is $\bigcup_n A_n$, and $B_1 \cup \cdots \cup B_n = A_n$.
[[Countable additivity|D7KB9HFCGZ]] gives $\mu(\bigcup_n A_n) = \sum_n \mu(B_n)$, and the partial sums of this series are the numbers $\mu(A_n)$.
For part 2, apply part 1 to the increasing sets $A_1 \setminus A_n$, and subtract from the finite number $\mu(A_1)$.

For Lebesgue measure $\lambda$ on $\R$ and a point $x$, the sets $[x - 1/n, x + 1/n]$ decrease to $\set{x}$ and have measure $2/n$.
Part 2 gives $\lambda(\set{x}) = 0$.

The hypothesis $\mu(A_1) < \infty$ in part 2 is needed.
For Lebesgue measure, the sets $A_n = [n, \infty)$ decrease to $\varnothing$, but $\lambda(A_n) = \infty$ for every $n$.

## Explanations

### L1 human

If sets grow and fill out a limit set, their measures grow to the measure of the limit set.
If sets shrink to a limit set, their measures shrink to its measure, provided one starts from a set of finite measure.
So measures behave well under monotone limits.

### L2 human

For a finitely additive measure on a σ-algebra, continuity from below is equivalent to countable additivity.
So it is a useful way to check that a set function is a measure.
It is the set version of the monotone convergence theorem.
For [[probability measures|0X8HBKCSYB]] both parts hold without extra hypothesis, since every set has finite measure.
Continuity of measures underlies the Borel–Cantelli lemmas and the right-continuity of distribution functions.
</document>

<document source="https://tamnd.github.io/math2/node/HBK5QHC1GC.md" tag="HBK5QHC1GC" kind="concept">
# General recursive function

## Summary

The general recursive functions, also called $\mu$-recursive functions, are the functions on the natural numbers obtained from the [[primitive recursive functions|5NVA2JRHPV]] by adding unbounded search.
The search operator $\mu$ returns the least number at which a given function takes the value $0$, and may run forever.
So the class contains partial functions, and the total ones among them are exactly the total [[computable functions|KZSB5E471K]].
This class was one of the first precise definitions of computability, and it agrees with computability by [[Turing machines|758EYCN27B]].

## Explanations

### L1 human

Primitive recursion is like a for-loop: the number of steps is known before the loop starts.
Unbounded search is like a while-loop: keep trying $y = 0, 1, 2, \ldots$ until a condition holds.
A while-loop may never stop, and then the function has no value at that input.
For example, searching for the least $y$ with $y^2 \ge x$ always stops and gives the rounded-up square root of $x$.
Adding this one tool to bounded loops gives every computable function.
</document>

<document source="https://tamnd.github.io/math2/node/HBQ8BZ7XB5.md" tag="HBQ8BZ7XB5" kind="concept">
# Contact manifold

## Summary

A contact manifold is an odd-dimensional [[smooth manifold|F45VQ47GPV]] with a field of hyperplanes.
This field is as far as possible from being tangent to a family of hypersurfaces.
Contact geometry is the odd-dimensional sister of symplectic geometry.
Level sets of energy in phase space, unit cotangent bundles and spaces of first-order jets carry natural contact structures.
It is the setting of geometric optics, thermodynamics and Legendrian knot theory.

## Explanations

### L1 human

At each point of a three-dimensional space, pick a plane through that point.
If the planes fit together as the tangent planes of a stack of surfaces, the field is integrable.
A contact structure is the opposite, a field of planes that twists everywhere.
In $\R^3$ the planes spanned by $\partial_y$ and $\partial_x + y\,\partial_z$ form such a field: moving in the $y$ direction, the planes turn about the $y$-axis.
No surface can be tangent to these planes on an open set.
</document>

<document source="https://tamnd.github.io/math2/node/HBSWH80PVB.md" tag="HBSWH80PVB" kind="definition">
# Bounded lattice

## Definition

A *bounded lattice* is a [[lattice|WH4SFAACAS]] $(L, \le)$ that has a [[greatest element|0XR2QK49V6]] and a [[least element|0AANKKNHS4]].
The greatest element is written $1$ or $\top$, and the least element is written $0$ or $\bot$.

Algebraic form: a bounded lattice is a lattice $(L, \vee, \wedge)$ with two elements $0, 1 \in L$ such that $x \vee 0 = x$ and $x \wedge 1 = x$ for all $x \in L$.

## Remarks

The two forms agree: $x \vee 0 = x$ for all $x$ says $0 \le x$ for all $x$, and $x \wedge 1 = x$ says $x \le 1$.
Then also $x \wedge 0 = 0$ and $x \vee 1 = 1$.
Example: the power set of a set $X$, ordered by $\subseteq$, is a bounded lattice with $0 = \emptyset$ and $1 = X$.
Every subset $A$ satisfies $\emptyset \subseteq A \subseteq X$.
Non-example: $(\Z, \le)$ is a lattice with $x \vee y = \max(x, y)$ and $x \wedge y = \min(x, y)$, but it has no greatest element.
Every finite nonempty lattice $L$ is bounded, with $1 = \bigvee L$ and $0 = \bigwedge L$, the join and the meet of all its elements.
</document>

<document source="https://tamnd.github.io/math2/node/HC3BQBQECH.md" tag="HC3BQBQECH" kind="definition">
# Boolean matrix

## Definition

Let $m, n \ge 1$ be integers.
A *Boolean matrix* of size $m \times n$ is an $m \times n$ [[matrix|0VD9ZSWWJG]] $A = (a_{ij})$ with every entry $a_{ij} \in \set{0, 1}$.
Let $A$ be a Boolean matrix of size $m \times n$ and $B$ one of size $n \times p$.
Their *Boolean product* $A \odot B$ is the $m \times p$ Boolean matrix with entries
$$(A \odot B)_{ik} = \max_{1 \le j \le n} \min(a_{ij}, b_{jk}),$$
that is, $(A \odot B)_{ik} = 1$ exactly when there is some $j$ with $a_{ij} = b_{jk} = 1$.

## Remarks

Equivalently, the entries are computed in the Boolean semiring $\set{0, 1}$ with $1 + 1 = 1$.
For finite sets $X = \set{x_1, \ldots, x_m}$ and $Y = \set{y_1, \ldots, y_n}$, a [[relation|05AVARPXA2]] $R \subseteq X \times Y$ has the Boolean matrix with $a_{ij} = 1$ exactly when $(x_i, y_j) \in R$.
If $S \subseteq Y \times Z$ has matrix $B$, then the [[composite relation|0PYSR20HGF]] "first $R$, then $S$" has matrix $A \odot B$.
Example: for $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, $A \odot A = A$, since the entry $(1,2)$ is $\max(\min(1,1), \min(1,1)) = 1$ and the entry $(2,1)$ is $\max(\min(0,1), \min(1,0)) = 0$; the ordinary product would give the entry $2$ at $(1,2)$.
Non-example: $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ is not a Boolean matrix, because the entry $2$ is not in $\set{0, 1}$.
</document>

<document source="https://tamnd.github.io/math2/node/HC3T4EY6WZ.md" tag="HC3T4EY6WZ" kind="statement">
# Riemann localization principle

## Statement

Let $f, g \colon \R \to \C$ be [[periodic functions|0ATF6Q0633]] of period $2\pi$ that are integrable on $[-\pi, \pi]$.
Let $S_N f$ and $S_N g$ be the partial sums of their [[Fourier series|G12KXDHPPS]].
Let $x_0 \in \R$ and $\delta > 0$, and suppose that $f(x) = g(x)$ for almost every $x$ in $(x_0 - \delta, x_0 + \delta)$.
Then
$$\lim_{N \to \infty} \big(S_N f(x_0) - S_N g(x_0)\big) = 0.$$
In particular the Fourier series of $f$ converges at $x_0$ if and only if that of $g$ does, and then to the same limit.

## Remarks

Put $h = f - g$; then $h$ vanishes near $x_0$, and $S_N h(x_0) = \frac{1}{2\pi} \int_{\delta \le \abs{t} \le \pi} h(x_0 - t) D_N(t) \, dt$ with the [[Dirichlet kernel|3JY4BVE4DW]] $D_N$.
On $\delta \le \abs{t} \le \pi$ the factor $\frac{1}{\sin(t/2)}$ is bounded, so $t \mapsto \frac{h(x_0 - t)}{\sin(t/2)}$ is integrable there.
The [[Riemann–Lebesgue lemma|WVKDC011KP]] applied to $\sin((N + \frac{1}{2}) t)$ gives $S_N h(x_0) \to 0$.
The same argument shows that $S_N h \to 0$ uniformly on every closed interval inside the interval where $h$ vanishes.
Example: a function equal to $x^2$ near $x_0 = 1$ has a Fourier series converging to $1$ at $x_0$, whatever its values far from $x_0$.
By contrast, each Fourier coefficient depends on the values of $f$ on the whole period.

## Explanations

### L1 human

Whether a Fourier series converges at a point depends only on the function near that point.
Changing the function far away changes every coefficient, but these changes cancel in the limit at the point.

### L2 human

Localization reduces convergence questions to local behaviour, which is why the Dini and Dirichlet–Jordan tests involve only values of $f$ near $x$.
It contrasts with the global nature of the coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/HC3VBBCJQP.md" tag="HC3VBBCJQP" kind="concept">
# Jet bundle

## Summary

The $k$-th jet bundle of a [[fiber bundle|CX4M8K61AB]] $E \to M$ is the manifold of all [[$k$-jets|1QE0Z0756D]] of local sections of $E$.
A point of it records a point of $M$, a value in the fiber and all derivatives up to order $k$ of a section there.
It is the natural home of differential equations on bundles: an equation of order $k$ is a subset of the $k$-th jet bundle.
Jet bundles are used in the calculus of variations, in the geometric theory of partial differential equations and in field theory.

## Explanations

### L1 human

For a function $u(x)$ of one variable, the first jet bundle has three coordinates: the point $x$, the value $u$ and the slope $u_x$.
Every function $u$ gives a curve $x \mapsto (x, u(x), u'(x))$ in this three-dimensional space.
A first-order differential equation such as $u_x = u$ is just a surface in that space.
Solving the equation means finding functions whose curves stay in that surface.
Higher jet bundles add coordinates for higher derivatives.
</document>

<document source="https://tamnd.github.io/math2/node/HC5NDQ3DTS.md" tag="HC5NDQ3DTS" kind="statement">
# Capelli's theorem

## Statement

Let $K$ be a [[field|0QA8CHH8MG]], let $a \in K$ and let $n \ge 2$ be an integer.
The polynomial $x^n - a$ is [[irreducible|00X2A9G9W3]] in $K[x]$ if and only if both of the following hold:

1. for every prime $p$ dividing $n$, $a$ is not of the form $b^p$ with $b \in K$;
2. if $4$ divides $n$, then $a$ is not of the form $-4 c^4$ with $c \in K$.

## Remarks

The conditions are necessary: if $a = b^p$ with $p \mid n$, then $y^p - b^p$ has the factor $y - b$, and $y = x^{n/p}$ gives a factor of $x^n - a$.
If $a = -4c^4$ and $4 \mid n$, the identity $y^4 + 4c^4 = (y^2 + 2cy + 2c^2)(y^2 - 2cy + 2c^2)$ with $y = x^{n/4}$ gives a factorisation.
The proof of sufficiency first treats $n$ prime with a norm argument, then prime powers, and combines them with the multiplicativity of degrees in towers of extensions.
Example: $x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2)$ over $\Q$, although $-4$ is not a square in $\Q$; this is the case $c = 1$ of clause 2.
Example: $x^n - 2$ is irreducible over $\Q$ for every $n \ge 1$.
For $n$ odd, clause 2 is empty and the test reduces to $p$-th powers for the primes $p \mid n$.

## Explanations

### L1 human

The polynomial $x^n - a$ factors in two obvious ways: when $a$ is a perfect $p$-th power for a prime $p$ dividing $n$, and in one special case built from $x^4 + 4$.
Capelli's theorem says that there are no other ways.

### L2 human

The theorem is the basic tool to study binomial equations $x^n = a$ and the radical extensions $K(\sqrt[n]{a})$.
It gives the degree $[K(\sqrt[n]{a}) : K] = n$ when the two clauses hold.
This is used in the theory of [[radical extensions|V74WMGQQ9D]] and of Kummer extensions.
</document>

<document source="https://tamnd.github.io/math2/node/HCD6W2XK6W.md" tag="HCD6W2XK6W" kind="statement">
# Zariski's main theorem

## Statement

Let $R$ be a commutative ring and let $S$ be a commutative $R$-algebra of [[finite type|AC5TFFQT9M]].
Let $S'$ be the [[integral closure|G0V86SJBQW]] of $R$ in $S$.
So $S'$ is the set of elements of $S$ that are integral over the image of $R$.
Let $\mathfrak{q}$ be a [[prime ideal|06HC6HK0Z8]] of $S$ and let $\mathfrak{p}$ be its preimage in $R$.
Assume that $\mathfrak{q}$ is both minimal and maximal among the prime ideals of $S$ whose preimage in $R$ is $\mathfrak{p}$.
Then there is an element $g \in S'$ with $g \notin \mathfrak{q}$ such that the inclusion $S' \to S$ induces an isomorphism of [[localizations|4C1AE84N5N]] $S'_g \cong S_g$.

## Remarks

The hypothesis says that $S$ is quasi-finite at $\mathfrak{q}$: the point $\mathfrak{q}$ is isolated in its fibre over $\mathfrak{p}$.
This is Grothendieck's algebraic form of the theorem.
In words, near a point where it is quasi-finite, a finite type algebra looks like a localization of an integral algebra.
In geometric language, a quasi-finite separated morphism of finite type between Noetherian schemes factors as an open immersion followed by a finite morphism.
Zariski's original form follows: a birational morphism of varieties with finite fibres onto a normal variety is an open immersion.
For example, $k[x, x^{-1}]$ is quasi-finite over $k[x]$ at every prime, and indeed it is the localization of $k[x]$ at $g = x$.
Without the hypothesis the conclusion fails: for $R = k$ a field, $S = k[x]$ and $\mathfrak{q} = (x)$, the prime $\mathfrak{q}$ contains $(0)$, which also lies over $(0)$; here $S' = k$, and $k_g = k$ is never $k[x]_g$.

## Explanations

### L1 human

A map of algebraic objects is quasi-finite at a point when, near that point, the point is alone in its fibre.
The theorem says that such a map is, near that point, just an open piece of a finite map.
So finiteness of fibres, which looks like a weak condition, forces strong structure.

### L2 human

The theorem has many forms: the algebraic one stated here, the factorisation of quasi-finite separated morphisms, and Zariski's statement on birational maps to normal varieties.
It is a basic tool in algebraic geometry, for example in the study of étale and unramified morphisms.
The role of [[integral closure|G0V86SJBQW]] is central: the finite algebra in the factorisation is built from the integral closure of $R$ in $S$.
</document>

<document source="https://tamnd.github.io/math2/node/HCJ2ETX89Z.md" tag="HCJ2ETX89Z" kind="concept">
# Eigenvalue algorithm

## Summary

An eigenvalue algorithm is a numerical procedure that computes [[eigenvalues|0BQAA2BK3Q]], and often [[eigenvectors|0N6960W0BV]], of a square matrix.
By the [[Abel–Ruffini theorem|163A2WQQJX]], the eigenvalues of a general matrix of size at least $5$ cannot be written with radicals.
So such algorithms are iterative and produce approximations.
They are central in numerical linear algebra, with uses from vibration analysis to search engines and quantum chemistry.

## Explanations

### L1 human

An eigenvector of a matrix is a direction that the matrix only stretches, and the eigenvalue is the stretch factor.
For a $2 \times 2$ matrix one can find the eigenvalues by solving a quadratic equation, but for large matrices no formula exists.
Eigenvalue algorithms instead improve a guess step by step until it is accurate enough.
The simplest one, [[power iteration|H4MQD841BR]], multiplies a vector by the matrix again and again; the vector turns toward the direction of the largest stretch.
For example, repeatedly applying $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ to $(1, 1)$ gives $(2^k, 1)$, which points more and more along $(1, 0)$, the eigenvector for eigenvalue $2$.
</document>

<document source="https://tamnd.github.io/math2/node/HCJK9SV43D.md" tag="HCJK9SV43D" kind="definition">
# Hilbert transform

## Definition

Let $1 \le p < \infty$ and let $f \in L^p(\R)$, a function in the [[$L^p$ space|SASP14MA1Z]] of $\R$.
The *Hilbert transform* of $f$ is the function
$$Hf(x) = \frac{1}{\pi}\, \lim_{\varepsilon \to 0^+} \int_{\abs{x - y} > \varepsilon} \frac{f(y)}{x - y}\, dy,$$
the [[Cauchy principal value|0SYR10HD4R]] of $\frac{1}{\pi} \int_\R \frac{f(y)}{x - y}\, dy$.
The limit exists for almost every $x \in \R$.

## Remarks

Example: for $f(x) = \frac{1}{1 + x^2}$, one has $Hf(x) = \frac{x}{1 + x^2}$.
Check: $F(z) = \frac{i}{z + i}$ is holomorphic on the upper half-plane and tends to $0$ at infinity.
On $\R$ it equals $\frac{1}{1 + x^2} + i \frac{x}{1 + x^2}$, so its imaginary part is the [[harmonic conjugate|050F2FMTKF]] of its real part.

Non-example: the integral is singular at $y = x$, so the principal value is needed.
The integral $\int_\R \frac{f(y)}{x - y}\, dy$ diverges absolutely at every $x$ near which $f$ is continuous and nonzero.

On the [[Fourier transform|DPVM34ADZV]] side, $\widehat{Hf}(\xi) = -i \sgn(\xi)\, \hat f(\xi)$ for $f \in L^2(\R)$.
Hence $H$ is an isometry of $L^2(\R)$ with $H^2 = -\id$.
For $1 < p < \infty$, $H$ is a bounded operator on $L^p(\R)$, a theorem of Marcel Riesz; for $p = 1$ it is only of weak type $(1, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/HCQWPAHYAT.md" tag="HCQWPAHYAT" kind="concept">
# Cubic Hermite spline

## Summary

A cubic Hermite spline is a piecewise cubic function that matches prescribed values and prescribed first derivatives at given nodes.
On each interval between nodes it is the cubic [[Hermite interpolant|9Z50M2WPTP]] of the data at the two ends.
It is continuously differentiable, and each piece depends only on local data, which makes it the basis of many curve tools in graphics and animation.

## Explanations

### L1 human

On each gap between two nodes, prescribe a height and a slope at both ends.
Exactly one cubic matches these four numbers.
A cubic Hermite spline joins these cubics end to end.
Since neighbouring pieces share the value and the slope at each node, the curve has no corners.
For example, data $p_0 = 0$, $m_0 = 0$ at $x = 0$ and $p_1 = 1$, $m_1 = 0$ at $x = 1$ give the smooth step $3x^2 - 2x^3$.
</document>

<document source="https://tamnd.github.io/math2/node/HCTNBZZQ0W.md" tag="HCTNBZZQ0W" kind="object">
# Zero sharp

## Description

Let $L$ be the [[constructible universe|6TF2JBTPPJ]].
Suppose that there is a closed unbounded class $I$ of [[ordinals|0YSCSX5YZC]] with the following property for every uncountable cardinal $\kappa$.
The set $I \cap \kappa$ has order type $\kappa$ and is a set of indiscernibles for the structure $(L_\kappa, \in)$.
Every element of $L_\kappa$ is definable in $L_\kappa$ from finitely many elements of $I \cap \kappa$.
Such a class is unique when it exists, and its elements are the *Silver indiscernibles*.
In this case *zero sharp* is the set
$$0^\sharp = \set{\ulcorner \varphi \urcorner : L_{\aleph_\omega} \models \varphi[\aleph_1, \ldots, \aleph_n]}.$$
Here $\varphi(x_1, \ldots, x_n)$ runs over the formulas of set theory, and $\ulcorner \varphi \urcorner$ is the Gödel number of $\varphi$.
One says that $0^\sharp$ exists when such a class $I$ exists.

## Remarks

Since $0^\sharp$ is a set of natural numbers, it is often treated as a real number.
If $0^\sharp$ exists, then every uncountable cardinal of $V$ is a Silver indiscernible, and hence an inaccessible cardinal in $L$.
If $0^\sharp$ exists, then $0^\sharp \notin L$, so $V \neq L$.
Silver proved that $0^\sharp$ exists if there is a [[measurable cardinal|AKBWT323RQ]].
If $0^\sharp$ does not exist, then [[Jensen's covering theorem|RJE1W5NRE3]] shows that $L$ is close to $V$.
If ZFC is consistent, it does not prove that $0^\sharp$ exists, since $0^\sharp$ does not exist in $L$.

## Explanations

### L1 human

Zero sharp is a single set of natural numbers that, if it exists, describes all truths about the constructible universe.
It exists only if the full universe is much richer than the constructible universe.
Its existence marks a sharp line: either the constructible universe is close to everything, or zero sharp exists and the constructible universe is very thin.
Its existence follows from large cardinals such as a measurable cardinal, but it cannot be proved from the usual axioms.
</document>

<document source="https://tamnd.github.io/math2/node/HD4EESS2DP.md" tag="HD4EESS2DP" kind="definition">
# Todd class

## Definition

Let $X$ be a smooth [[quasi-projective variety|EQSHJPE7H8]] over a field, with rational [[Chow ring|4HV4WTAX9P]] $CH^\bullet(X)_\Q = CH^\bullet(X) \otimes \Q$.
Let $E$ be a [[vector bundle|P4DZM975BV]] of rank $r$ on $X$ with [[Chern classes|PAD37GKDPD]] $c_1, \ldots, c_r$.
Write formally $\sum_{i=0}^r c_i(E) t^i = \prod_{j=1}^r (1 + \alpha_j t)$, where $\alpha_1, \ldots, \alpha_r$ are the *Chern roots* of $E$.
The *Todd class* of $E$ is
$$\operatorname{td}(E) = \prod_{j=1}^r \frac{\alpha_j}{1 - e^{-\alpha_j}} \in CH^\bullet(X)_\Q,$$
where each factor is expanded as the power series $\frac{x}{1 - e^{-x}} = 1 + \frac{x}{2} + \frac{x^2}{12} - \frac{x^4}{720} + \cdots$.
The product is symmetric in the $\alpha_j$, so it is a polynomial in $c_1, \ldots, c_r$ in each degree, and only finitely many degrees are nonzero.
The Todd class of $X$ is $\operatorname{td}(X) = \operatorname{td}(T_X)$, for the tangent bundle $T_X$.

## Remarks

In low degrees,
$$\operatorname{td}(E) = 1 + \frac{c_1}{2} + \frac{c_1^2 + c_2}{12} + \frac{c_1 c_2}{24} + \cdots.$$
The Todd class is multiplicative: $\operatorname{td}(E) = \operatorname{td}(E') \operatorname{td}(E'')$ for an exact sequence $0 \to E' \to E \to E'' \to 0$.
Example: for a smooth projective curve $X$ of genus $g$, $\operatorname{td}(X) = 1 + \frac{1}{2} c_1(T_X)$, and $\deg c_1(T_X) = 2 - 2g$, so the degree of the part in $CH^1$ is $1 - g$.
Non-example: the power series $1 + x$ also gives a multiplicative class, but it is the total Chern class, not the Todd class.
The Todd series, with its coefficient $\frac{1}{12}$ of $x^2$, is the one that makes the Hirzebruch–Riemann–Roch formula hold.
</document>

<document source="https://tamnd.github.io/math2/node/HD5KCKVPS8.md" tag="HD5KCKVPS8" kind="definition">
# Lambda calculus

## Definition

Fix a countably infinite set $V$ of variables.
The *lambda terms* are the strings defined by these rules:

1. every variable $x \in V$ is a term;
2. if $M$ is a term and $x \in V$, then $(\lambda x.\, M)$ is a term, called an *abstraction*;
3. if $M$ and $N$ are terms, then $(M\, N)$ is a term, called an *application*.

In $\lambda x.\, M$ the variable $x$ is bound in $M$.
Terms that differ only in the names of bound variables are identified ($\alpha$-equivalence).
Let $M[x := N]$ be the term obtained from $M$ by replacing the free occurrences of $x$ by $N$.
Before the replacement, bound variables of $M$ are renamed so that no free variable of $N$ gets bound.
The *untyped lambda calculus* is the set of terms together with *$\beta$-reduction*.
This is the relation generated by the rule
$$(\lambda x.\, M)\, N \to_\beta M[x := N],$$
applied to any subterm of a term.

## Remarks

Example: $(\lambda x.\, x\, x)(\lambda y.\, y)$ is a term by rules 1 to 3, and
$$(\lambda x.\, x\, x)(\lambda y.\, y) \to_\beta (\lambda y.\, y)(\lambda y.\, y) \to_\beta \lambda y.\, y.$$
The last term has no further reduction, so it is a *normal form*.

Non-example: the string $\lambda x$ alone is not a term, since rule 2 requires a body $M$ after the dot.

The term $\Omega = (\lambda x.\, x\, x)(\lambda x.\, x\, x)$ reduces only to itself, so it has no normal form.
By the Church–Rosser theorem, a term has at most one normal form up to $\alpha$-equivalence.
</document>

<document source="https://tamnd.github.io/math2/node/HD74ZBGCJ2.md" tag="HD74ZBGCJ2" kind="definition">
# Energy method

## Definition

Let $u$ be a solution of an evolution equation on $\Omega \times [0, T]$, with $\Omega \subseteq \R^n$ open.
The *energy method* is the following kind of argument.

1. Choose an energy $E(t) \ge 0$, an integral over $\Omega$ of a nonnegative quadratic expression in $u(\cdot, t)$ and its derivatives.
2. Differentiate $E$ in $t$, use the equation, and integrate by parts with the boundary conditions. The aim is a bound $E'(t) \le C E(t) + F(t)$ with a constant $C \ge 0$ and a known function $F \ge 0$.
3. Conclude by Grönwall's inequality that $E(t) \le e^{Ct}\big(E(0) + \int_0^t F(s)\, ds\big)$ for $0 \le t \le T$.

Apply the argument to the difference of two solutions of a linear problem, with $F = 0$.
It gives uniqueness and continuous dependence on the data, as required for a [[well-posed problem|W09B9NZ9B5]].

## Remarks

Example: let $\Omega$ be bounded with smooth boundary and let $u \in C^2(\overline{\Omega} \times [0, T])$ solve the [[wave equation|5EHD2XBQDG]] $u_{tt} = \Delta u$ with $u = 0$ on $\partial\Omega$.
Take $E(t) = \frac{1}{2}\int_\Omega \big(u_t^2 + \abs{\nabla u}^2\big)\, dx$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
Then $E'(t) = \int_\Omega (u_t u_{tt} + \nabla u \cdot \nabla u_t)\, dx = \int_\Omega u_t (u_{tt} - \Delta u)\, dx = 0$, since the boundary term contains $u_t = 0$ on $\partial\Omega$.
So $E$ is constant, and two solutions with the same data and boundary values coincide.
For the heat equation $u_t = \Delta u$ with $u = 0$ on $\partial\Omega$, the energy $E(t) = \frac{1}{2}\int_\Omega u^2\, dx$ satisfies $E'(t) = -\int_\Omega \abs{\nabla u}^2\, dx \le 0$.
Non-example: for the backward heat equation $u_t = -\Delta u$ the same computation gives $E'(t) = \int_\Omega \abs{\nabla u}^2\, dx$, which is not bounded by $C E(t)$, so step 2 fails; indeed that problem is ill-posed.
</document>

<document source="https://tamnd.github.io/math2/node/HD8CK28AJC.md" tag="HD8CK28AJC" kind="definition">
# Algebraic lattice

## Definition

Let $L$ be a [[complete lattice|D7A0ZGZT5S]].
It is an *algebraic lattice*, or compactly generated, if every $x \in L$ is the [[supremum|0R3H5PANRQ]] of the set $\set{c \in L : c \le x,\ c \text{ compact}}$ of [[compact elements|9F7TYES4KE]] below it.

## Remarks

Example: the lattice of subgroups of a group $G$, whose compact elements are the finitely generated subgroups.
Each subgroup $H$ is the join of the cyclic subgroups $\langle h \rangle$ for $h \in H$.
Non-example: the closed subsets of $[0, 1]$, ordered by inclusion, form a complete lattice in which the join of a family is the closure of its union.
A nonempty closed set $F$ is not compact: pick $p \in F$; the closed sets $C_n = [0, 1] \setminus (p - 1/n, p + 1/n)$ form a directed family whose join is $[0, 1] \supseteq F$, yet $p \notin C_n$ for every $n$.
So the only compact element is $\varnothing$, and no nonempty closed set is a join of compact elements.
Every algebraic lattice is a continuous lattice.
Some authors do not require completeness and speak of algebraic posets or algebraic dcpos instead.
</document>

<document source="https://tamnd.github.io/math2/node/HDD0EB48AB.md" tag="HDD0EB48AB" kind="definition">
# Hilbert metric

## Definition

Let $n \ge 1$ and let $\Omega \subseteq \R^n$ be a bounded open [[convex set|02AS45T3H9]].
Write $\abs{pq}$ for the [[Euclidean distance|0NPPGHQNXQ]] of points $p$ and $q$.
Let $x, y \in \Omega$ be distinct.
The line through $x$ and $y$ meets the boundary of $\Omega$ in exactly two points $a$ and $b$.
Name them so that $a, x, y, b$ lie on the line in this order.
The *Hilbert metric* on $\Omega$ is
$$d_\Omega(x, y) = \frac{1}{2} \log \frac{\abs{ay} \, \abs{xb}}{\abs{ax} \, \abs{yb}},$$
with $d_\Omega(x, x) = 0$.

## Remarks

The map $d_\Omega$ is a [[metric|063W5JMY85]] on $\Omega$; the triangle inequality uses the convexity of $\Omega$.

Example: let $\Omega = (-1, 1) \subset \R$ and $0 < y < 1$.
For $x = 0$ the boundary points are $a = -1$ and $b = 1$, so $d_\Omega(0, y) = \frac{1}{2} \log \frac{1 + y}{1 - y}$, which tends to $\infty$ as $y \to 1$.

Non-example: the open half-plane $\set{(u, v) \in \R^2 : v > 0}$ is convex but not bounded.
A horizontal line in it does not meet the boundary at all, so the formula has no meaning.

Some authors omit the factor $\frac{1}{2}$.
When $\Omega$ is the open unit ball, $d_\Omega$ is the hyperbolic distance of the [[Beltrami–Klein model|1WN9B18H86]].
</document>

<document source="https://tamnd.github.io/math2/node/HDFM8ZY7XG.md" tag="HDFM8ZY7XG" kind="statement">
# Bernstein's theorem on inverse approximation

## Statement

Let $f \colon \R \to \C$ be continuous and $2\pi$-periodic, let $r \ge 0$ be an integer, and let $0 < \alpha < 1$.
For $n \ge 1$, let $E_n(f)$ be the uniform distance from $f$ to the [[trigonometric polynomials|J3NJQ5FZJ5]] of degree at most $n$.
Assume that there is a constant $M$ with $E_n(f) \le M n^{-(r + \alpha)}$ for all $n \ge 1$.
Then $f$ has $r$ continuous derivatives, and $f^{(r)}$ is [[Hölder continuous|012PBA0FMD]] with exponent $\alpha$.

## Remarks

Proof idea: let $T_k$ be a best approximation of degree $2^k$ and write $f = T_0 + \sum_{k \ge 0} (T_{k+1} - T_k)$.
Each block $T_{k+1} - T_k$ has degree at most $2^{k+1}$ and size $O(2^{-k(r + \alpha)})$.
[[Bernstein's inequality|F0YGC89S9R]] bounds its $j$-th derivative by $O(2^{k(j - r - \alpha)})$, so the series of $r$-th derivatives converges uniformly.
Splitting the sum at $2^k \approx 1/\abs{x - y}$ gives the Hölder bound for $f^{(r)}$.

Example: if $E_n(f) \le M n^{-1/2}$ for all $n$, then $\abs{f(x) - f(y)} \le M' \abs{x - y}^{1/2}$.

The hypothesis $\alpha < 1$ is necessary.
For $\alpha = 1$ and $r = 0$, the condition $E_n(f) = O(1/n)$ does not imply that $f$ is Lipschitz.
It characterises the larger Zygmund class, defined by $\abs{f(x + h) - 2f(x) + f(x - h)} = O(h)$.

## Explanations

### L1 human

If a function can be approximated very well by trigonometric polynomials of low degree, it must be smooth.
The faster the error goes down, the more derivatives the function has.
This is the reverse of Jackson's inequality.

### L2 human

With Jackson's inequality it gives a two-way bridge: for $0 < \alpha < 1$, $E_n(f) = O(n^{-r-\alpha})$ exactly when $f^{(r)}$ is Hölder continuous with exponent $\alpha$.
So smoothness can be read off from approximation rates, which is the starting point of approximation spaces and Besov spaces.
The same telescoping argument over dyadic degrees appears in wavelet and Littlewood–Paley theory.
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<document source="https://tamnd.github.io/math2/node/HDJ1GKG0NA.md" tag="HDJ1GKG0NA" kind="concept">
# Banach lattice

## Summary

A Banach lattice is a [[Banach space|06GZ956657]] with an order that is compatible with the vector operations and with the norm.
Every two elements have a largest lower bound and a least upper bound, and a smaller absolute value gives a smaller norm.
The spaces $L^p$, $C(K)$ and $\ell^p$ with the pointwise order are the main examples.
Banach lattices are the setting for positive operators, which appear in probability, in partial differential equations and in economics.

## Explanations

### L1 human

In many function spaces you can compare two functions point by point and take their pointwise maximum.
A Banach lattice keeps exactly this structure, in an abstract form, together with a complete norm.
The rule linking order and norm says that if $\abs{f} \le \abs{g}$ everywhere, then $f$ is not longer than $g$.
For example, in $C([0,1])$ the function $\max(\sin, \cos)$ is again continuous, and its norm is at most $1$ because both $\abs{\sin}$ and $\abs{\cos}$ are at most $1$.
</document>

<document source="https://tamnd.github.io/math2/node/HE7MP4HP6Q.md" tag="HE7MP4HP6Q" kind="statement">
# Structure theorem for Artinian rings

## Statement

Let $R$ be a commutative [[Artinian ring|ZV04WWYE2Y]].
Then $R$ has only finitely many [[maximal ideals|09J6KGT65G]] $\mathfrak{m}_1, \dots, \mathfrak{m}_n$.
Each [[localization|X6JD4AG973]] $R_{\mathfrak{m}_i}$ is an Artinian [[local ring|MZ02BKPHJA]], and the natural map $R \to R_{\mathfrak{m}_1} \times \cdots \times R_{\mathfrak{m}_n}$ is a ring isomorphism.

## Remarks

In an Artinian ring the Jacobson radical $\mathfrak{m}_1 \cap \cdots \cap \mathfrak{m}_n$ is nilpotent, so $\mathfrak{m}_1^k \cdots \mathfrak{m}_n^k = 0$ for some $k$.
The ideals $\mathfrak{m}_i^k$ are pairwise coprime, so the [[Chinese remainder theorem|0W97J8QED7]] gives $R \cong \prod_i R/\mathfrak{m}_i^k$.
One then checks that $R/\mathfrak{m}_i^k \cong R_{\mathfrak{m}_i}$.
For example, $\Z/12\Z \cong \Z/4\Z \times \Z/3\Z$, and these factors are the localizations at the maximal ideals $(2)$ and $(3)$.
The decomposition is unique: in any decomposition of $R$ as a finite [[product of rings|0MNZCN4BQK]] of local rings, the factors are, up to isomorphism and order, the rings $R_{\mathfrak{m}_i}$.

## Explanations

### L1 human

Every commutative Artinian ring splits into finitely many pieces, one for each maximal ideal.
Each piece is local: it has only one maximal ideal.
So to understand Artinian rings, it is enough to understand Artinian local rings.

### L2 human

Together with the [[Akizuki–Hopkins–Levitzki theorem|C7X657TF19]], this reduces many questions about zero-dimensional Noetherian rings to the local case.
Geometrically, the spectrum of an Artinian ring is a finite discrete set of points, and the ring is the product of the "fat points" at each of them.
An Artinian local ring is the kind of ring met as $R/I$ for a Noetherian local ring $R$ and an ideal $I$ primary to the maximal ideal.
</document>

<document source="https://tamnd.github.io/math2/node/HE8F7KBF42.md" tag="HE8F7KBF42" kind="concept">
# Abel–Jacobi map

## Summary

The Abel–Jacobi map sends a point of a compact [[Riemann surface|5K0N8QMV6Q]] of genus $g \ge 1$ to its [[Jacobian variety|ZQ84M9D9TC]], a [[complex torus|Y7MA6KBE9M]] of dimension $g$.
It is defined by integrating a basis of holomorphic [[Abelian differentials|P014C13K5H]] from a fixed base point.
Abel's theorem says that it detects which divisors of degree $0$ are divisors of meromorphic functions.
It is the basic bridge between a curve and its Jacobian.

## Explanations

### L1 human

Pick a base point on the surface and a point $P$, and integrate the $g$ holomorphic differentials along a path from the base point to $P$.
This gives $g$ complex numbers, but they depend on the path.
Two paths differ by a loop, and the integrals over loops form a lattice, so the answer is well defined modulo that lattice.
For a torus $\C / \Lambda$ with the differential $dz$, the map is just $P \mapsto P - P_0$, so the torus is its own Jacobian.
</document>

<document source="https://tamnd.github.io/math2/node/HEB1VQW0DQ.md" tag="HEB1VQW0DQ" kind="concept">
# Constructive analysis

## Summary

Constructive analysis is mathematical analysis carried out under the principles of [[constructivism|G958EQ8CWW]].
[[Real numbers|08J2M1JYAG]] are given by [[Cauchy sequences|0PWX18FM4G]] of rationals with an explicit rate of convergence, and every existence proof must give an approximation method.
Bishop's book "Foundations of Constructive Analysis" of 1967 showed that a large part of classical analysis can be developed in this way.
Its theorems are also valid classically, and they carry algorithms.

## Explanations

### L1 human

In constructive analysis, a real number is something you can compute to any accuracy you ask for.
Some classical facts then need care.
For example, you cannot always decide whether a real number is zero, since you may never see a nonzero digit.
So the [[intermediate value theorem|00KAFBR32E]] is replaced by a version that finds a point where the function is as close to zero as you like.
Results like this are often what a computation can actually deliver.
</document>

<document source="https://tamnd.github.io/math2/node/HEE922EHQ8.md" tag="HEE922EHQ8" kind="concept">
# Residually finite group

## Summary

A [[group|G5S1M9]] is residually finite if every element other than the identity survives in some finite quotient.
So the group can be studied through its finite images, which separate all its elements.
[[Free groups|0KTW26C80J]], finitely generated abelian groups and finitely generated subgroups of $GL(n, \C)$ are residually finite.
A finitely generated residually finite group is [[Hopfian|PHEN4E1JPT]], and a finitely presented residually finite group has solvable [[word problem|PBGTTAVMMT]].

## Explanations

### L1 human

To see that an element $g$ is not the identity, map the group onto a finite group and look at the image of $g$.
In a residually finite group this always works for a suitable finite group.
For example, the integer $n \ne 0$ survives in $\Z / (\abs{n} + 1)$, since $0 < \abs{n} < \abs{n} + 1$.
The group of rational numbers fails badly: it has no finite quotient except the trivial one.
</document>

<document source="https://tamnd.github.io/math2/node/HEGTKZ640Y.md" tag="HEGTKZ640Y" kind="definition">
# Hausdorff summation

## Definition

Let $(\mu_n)_{n \ge 0}$ be a sequence of real numbers.
Define its iterated differences by $\Delta^0 \mu_k = \mu_k$ and $\Delta^{j+1} \mu_k = \Delta^j \mu_k - \Delta^j \mu_{k+1}$, so that
$$\Delta^j \mu_k = \sum_{i=0}^{j} (-1)^i \binom{j}{i} \mu_{k+i}.$$
The *Hausdorff matrix* of $(\mu_n)$ is the lower triangular matrix $H_\mu = (h_{nk})$ with
$$h_{nk} = \binom{n}{k} \Delta^{n-k} \mu_k \quad (0 \le k \le n), \qquad h_{nk} = 0 \quad (k > n).$$
A [[sequence|0QE8STGJH2]] $(s_k)$ of complex numbers is *Hausdorff summable* by $H_\mu$ to $s$ if $t_n = \sum_{k=0}^{n} h_{nk} s_k \to s$ as $n \to \infty$.
A series is Hausdorff summable if the sequence of its partial sums is.

## Remarks

Example: for $\mu_n = \frac{1}{n+1} = \int_0^1 t^n \, dt$ one gets $\Delta^{n-k} \mu_k = \int_0^1 t^k (1 - t)^{n-k} \, dt = \frac{k! \, (n-k)!}{(n+1)!}$.
So $h_{nk} = \frac{1}{n+1}$, and $t_n = \frac{s_0 + \cdots + s_n}{n+1}$ is the Cesàro mean.
Similarly, $\mu_n = 2^{-n}$ gives $h_{nk} = \binom{n}{k} 2^{-n}$, the Euler $(E, 1)$ means.

Non-example of regularity: for $\mu_0 = 1$ and $\mu_n = 0$ for $n \ge 1$, one has $\Delta^{n-k} \mu_k = 0$ for $k \ge 1$ and $\Delta^n \mu_0 = 1$.
So $t_n = s_0$ for all $n$, and the method sends every sequence to its first term.

Hausdorff showed that $H_\mu$ is regular if and only if $\mu_n = \int_0^1 t^n \, d\chi(t)$ for a function $\chi$ of bounded variation on $[0, 1]$ with $\chi(0) = 0$, $\chi(1) = 1$ and $\chi$ continuous at $0$.
In the non-example, $\chi$ jumps from $0$ to $1$ at $0$, which violates the last condition.
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# Gordan's lemma

## Statement

Let $n \ge 1$ and let $v_1, \dots, v_k \in \Z^n$.
Let $C = \set{\lambda_1 v_1 + \cdots + \lambda_k v_k : \lambda_1, \dots, \lambda_k \ge 0}$ be the rational polyhedral [[convex cone|6VE4KA12VD]] they generate in $\R^n$.
Then the [[monoid|0N6R902FG1]] $C \cap \Z^n$, with the addition of $\Z^n$, is finitely generated.

## Remarks

Let $P = \set{t_1 v_1 + \cdots + t_k v_k : 0 \le t_i < 1}$.
It is a bounded set, so $P \cap \Z^n$ is finite.
Every $x \in C \cap \Z^n$ can be written $x = \sum_i t_i v_i$ with $t_i \ge 0$, and then $x = \sum_i \floor{t_i} v_i + p$ with $p = \sum_i (t_i - \floor{t_i}) v_i \in P \cap \Z^n$.
So the finite set $\set{v_1, \dots, v_k} \cup (P \cap \Z^n)$ generates $C \cap \Z^n$.

Example: $v_1 = (1, 0)$ and $v_2 = (1, 2)$ in $\Z^2$.
Here $P \cap \Z^2 = \set{(0,0), (1,1)}$, so $C \cap \Z^2$ is generated by $(1,0)$, $(1,1)$, $(1,2)$.
The point $(1,1)$ is needed: it lies in $C$ but is not a combination of $(1,0)$ and $(1,2)$ with nonnegative integer coefficients.

Rationality is needed.
For the cone $C = \set{(x, y) : x \ge 0, \ 0 \le y \le \sqrt{2}\, x}$ the monoid $C \cap \Z^2$ is not finitely generated.
Finitely many generators in $C \cap \Z^2$ only produce points of slope at most $q$, the largest slope of a generator, and $q < \sqrt{2}$ since $q$ is rational.
But $C$ contains lattice points $(b, a)$ with $q < a/b < \sqrt{2}$.

## Explanations

### L1 human

Take a cone in space whose edges point along whole-number directions.
The lattice points inside the cone can be added together, and the lemma says that finitely many of them generate all the others by addition.
The generators are the edge vectors together with the few lattice points in a small box spanned by them.

### L2 human

The lemma is the combinatorial input of the theory of affine toric varieties.
For a rational cone $\sigma$, the semigroup algebra of $\sigma^\vee \cap M$ is then finitely generated, and its spectrum is the [[toric variety|YWB981MEF2]] of $\sigma$.
The minimal generating set of $C \cap \Z^n$ for a pointed cone is its Hilbert basis.
In invariant theory the same statement is used to show that rings of invariants of tori are finitely generated.
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<document source="https://tamnd.github.io/math2/node/HEKS0NMYEQ.md" tag="HEKS0NMYEQ" kind="definition">
# Exponential sum

## Definition

For a real number $x$, write $e(x) = e^{2\pi i x}$, where $e^{z}$ is the [[complex exponential|07GJV36896]].
Let $A$ be a finite set of integers, and let $f \colon A \to \R$ be a function.
The *exponential sum* of $f$ over $A$ is the complex number
$$S = \sum_{n \in A} e(f(n)).$$
Most often $A$ is an interval $\set{n \in \Z : M < n \le M + N}$.

## Remarks

Each term has absolute value $1$, so $\abs{S} \le \#A$; this is the trivial bound.
Example: for $f(n) = n\alpha$ with $\alpha \in \R \setminus \Z$ and $A = \set{1, \dots, N}$, the sum is geometric and
$$\abs{S} = \abs{\frac{e(N\alpha) - 1}{e(\alpha) - 1}} \le \frac{1}{\abs{\sin \pi \alpha}},$$
a bound independent of $N$.
Non-example: $\sum_{n=1}^{N} 2^n$ is a sum of values of an exponential function, but it is not an exponential sum in this sense.
Its terms have absolute value larger than $1$, so they are not of the form $e(f(n))$ with $f$ real.
When $f$ is a polynomial, $S$ is called a Weyl sum.
Sums with weights, $\sum_{n \in A} a_n e(f(n))$ with complex $a_n$, are also called exponential sums.
</document>

<document source="https://tamnd.github.io/math2/node/HEKY61PKA9.md" tag="HEKY61PKA9" kind="definition">
# Isomorphism of topological groups

## Definition

Let $G$ and $H$ be [[topological groups|92700QAB5Y]].
An *isomorphism of topological groups* from $G$ to $H$ is a map $f \colon G \to H$ that is both a [[group isomorphism|08MG7H8JW5]] and a [[homeomorphism|0QT00PB5G2]].
Equivalently, $f$ is a bijective [[continuous group homomorphism|9J5PQMG3HJ]] whose inverse $f^{-1} \colon H \to G$ is continuous.
The groups $G$ and $H$ are *isomorphic* if such a map exists.

## Remarks

Example: $\log \colon (\,(0, \infty), \cdot\,) \to (\R, +)$.
It is bijective, $\log(xy) = \log x + \log y$, and both $\log$ and its inverse $\exp$ are continuous.
Non-example: the identity map from $(\R, +)$ with the discrete topology to $(\R, +)$ with the usual topology.
It is a continuous bijective homomorphism, but its inverse is not continuous, since $\set{0}$ is open in the discrete topology and not in the usual one.
</document>

<document source="https://tamnd.github.io/math2/node/HEMXNPQBSV.md" tag="HEMXNPQBSV" kind="definition">
# Abscissa of convergence

## Definition

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$ with $T > 0$.
The *abscissa of convergence* of the [[Laplace transform|05XT5AVJBK]] of $f$ is
$$\sigma_c = \inf \set{ \sigma \in \R : \int_0^\infty f(t) e^{-\sigma t}\, dt \text{ converges} } \in [-\infty, +\infty],$$
where convergence is that of the [[improper integral|0VCTBJMEJF]] $\lim_{T \to \infty} \int_0^T f(t) e^{-\sigma t}\, dt$, and $\inf \emptyset = +\infty$.

## Remarks

If the Laplace integral converges at a complex number $s_0$, it converges at every $s$ with $\operatorname{Re} s > \operatorname{Re} s_0$.
Hence the integral converges for $\operatorname{Re} s > \sigma_c$ and diverges for $\operatorname{Re} s < \sigma_c$.
Example: for $f(t) = e^{at}$ with $a \in \R$, $\int_0^T e^{(a - \sigma)t}\, dt$ has a finite limit exactly when $\sigma > a$, so $\sigma_c = a$.
Non-example of a finite value: for $f(t) = e^{t^2}$ the integral diverges for every real $\sigma$, so $\sigma_c = +\infty$; for $f(t) = e^{-t^2}$ it converges for every real $\sigma$, so $\sigma_c = -\infty$.
For $f = 1$, $\sigma_c = 0$, and on the line $\operatorname{Re} s = 0$ the integral diverges at every point.
</document>

<document source="https://tamnd.github.io/math2/node/HEV7170SEK.md" tag="HEV7170SEK" kind="definition">
# Steenrod algebra

## Definition

Consider symbols $\operatorname{Sq}^i$ of degree $i$, for $i \ge 0$.
The mod $2$ *Steenrod algebra* $\mathcal{A}_2$ is the [[graded|8KB4SQGC0C]] associative algebra over $\Z/2\Z$ generated by them, subject to these relations.

1. $\operatorname{Sq}^0 = 1$.
2. The Adem relations hold:
$$\operatorname{Sq}^a \operatorname{Sq}^b = \sum_{j=0}^{\floor{a/2}} \binom{b - 1 - j}{a - 2j} \operatorname{Sq}^{a + b - j} \operatorname{Sq}^j \quad \text{for all } 0 < a < 2b,$$
where the [[binomial coefficients|092GEF38YD]] are taken modulo $2$.

## Remarks

The algebra acts on the mod $2$ cohomology of every space, with $\operatorname{Sq}^i$ acting as the [[Steenrod square|VR7RHP41M0]] of the same name.
It is isomorphic to the algebra of all stable cohomology operations on mod $2$ cohomology.

Example: for $a = b = 1$, the sum has the single term $j = 0$ with coefficient $\binom{0}{1} = 0$, so $\operatorname{Sq}^1 \operatorname{Sq}^1 = 0$.
For $a = 1$ and $b = 2$, the coefficient is $\binom{1}{1} = 1$, so $\operatorname{Sq}^1 \operatorname{Sq}^2 = \operatorname{Sq}^3$.
Thus $\operatorname{Sq}^3$ is not needed as a generator; the algebra is generated by the $\operatorname{Sq}^{2^k}$, $k \ge 0$.

Non-example: the free associative algebra over $\Z/2\Z$ on the symbols $\operatorname{Sq}^i$ is not the Steenrod algebra, since in it $\operatorname{Sq}^1 \operatorname{Sq}^1 \ne 0$.

A basis of $\mathcal{A}_2$ as a vector space is given by the admissible monomials $\operatorname{Sq}^{i_1} \cdots \operatorname{Sq}^{i_r}$ with $i_s \ge 2 i_{s+1}$ for all $s$ and $i_r \ge 1$, together with $1$.
</document>

<document source="https://tamnd.github.io/math2/node/HEVF97NZ14.md" tag="HEVF97NZ14" kind="definition">
# Acyclic object

## Definition

Let $\mathcal{A}$ and $\mathcal{B}$ be [[abelian categories|Y9CZ3MTNF1]], and assume that $\mathcal{A}$ has enough injectives.
Let $F \colon \mathcal{A} \to \mathcal{B}$ be a left [[exact|S1G61TDW0K]] [[additive functor|KAN07YC9T2]].
An object $X$ of $\mathcal{A}$ is *$F$-acyclic* if $R^n F(X) = 0$ for every $n \ge 1$, where $R^n F$ are the right [[derived functors|NPMC3K6J2F]] of $F$.
Dually, for a right exact $F$ on a category with enough projectives, $X$ is $F$-acyclic if $L_n F(X) = 0$ for every $n \ge 1$.

## Remarks

Example: every injective object $I$ is $F$-acyclic, because $0 \to I \to I \to 0$ is an injective resolution, and applying $F$ gives a complex with no cohomology in degrees $n \ge 1$.
Non-example: for $F = \Hom_\Z(\Z/2\Z, -)$ on abelian groups, the group $\Z$ is not $F$-acyclic, since $R^1 F(\Z) = \operatorname{Ext}^1_\Z(\Z/2\Z, \Z) \cong \Z/2\Z \ne 0$.
For $F = - \otimes_R N$, every [[flat module|73RYYEVDKA]] is $F$-acyclic, since $\operatorname{Tor}_n^R(M, N) = 0$ for flat $M$ and $n \ge 1$.
If $X \to J^\bullet$ is a resolution by $F$-acyclic objects, then $R^n F(X) \cong H^n(F(J^\bullet))$ for every $n$.
</document>

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# Milnor K-theory

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $K^\times$ be its [[group of units|05THZKR9AN]], an [[abelian group|071DDFVG4J]].
Let $n \ge 2$.
Let $T_n$ be the [[tensor product|F0HQXQ0HCD]] over $\Z$ of $n$ copies of $K^\times$.
Let $R_n \subseteq T_n$ be the subgroup generated by all $a_1 \otimes \cdots \otimes a_n$ with $a_i + a_j = 1$ for some $i < j$.
The *$n$-th Milnor K-group* of $K$ is the quotient
$$K_n^M(K) = T_n / R_n.$$
Also $K_0^M(K) = \Z$ and $K_1^M(K) = K^\times$.
The class of $a_1 \otimes \cdots \otimes a_n$ is the *symbol* $\set{a_1, \ldots, a_n}$, and the group law of $K_n^M(K)$ is written additively.

## Remarks

Equivalently, $K_*^M(K) = \bigoplus_{n \ge 0} K_n^M(K)$ is the quotient of the tensor algebra of $K^\times$ over $\Z$ by the two-sided ideal generated by the elements $a \otimes (1 - a)$ with $a \ne 0, 1$.
The relation $\set{a, 1 - a} = 0$ is the Steinberg relation.
Consequences: $\set{a, -a} = 0$ and $\set{a, b} = -\set{b, a}$ for all $a, b \in K^\times$.
Example: in $K_2^M(\Q)$, $\set{2, -1} = 0$ since $2 + (-1) = 1$.
Non-example: the element $2 \otimes (-1)$ of $\Q^\times \otimes_\Z \Q^\times$ is not zero.
Indeed $\Q^\times$ is the product of $\set{\pm 1}$ and the free abelian group on the primes.
So $2 \otimes (-1)$ generates a direct summand $\langle 2 \rangle \otimes \set{\pm 1} \cong \Z/2\Z$, where $\langle 2 \rangle$ is the cyclic group generated by $2$.
Thus the tensor product alone is not $K_2^M(\Q)$.
For a finite field $\F_q$, $K_n^M(\F_q) = 0$ for $n \ge 2$.
By a theorem of Matsumoto, $K_2^M(K)$ equals Quillen's $K_2(K)$.
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# C* identity

## Definition

Let $A$ be a complex [[Banach algebra|9KYWN94QDN]] with an [[involution|GP56NBN9RY]] $a \mapsto a^*$, that is, a map with $(a + b)^* = a^* + b^*$, $(\lambda a)^* = \overline{\lambda} a^*$, $(ab)^* = b^* a^*$ and $a^{**} = a$ for $a, b \in A$ and $\lambda \in \C$.
The norm of $A$ satisfies the *C\* identity* if
$$\norm{a^* a} = \norm{a}^2 \quad \text{for all } a \in A.$$
A Banach algebra with involution whose norm satisfies the C\* identity is a C*-algebra.

## Remarks

The identity implies $\norm{a^*} = \norm{a}$: from $\norm{a}^2 = \norm{a^* a} \le \norm{a^*} \norm{a}$ one gets $\norm{a} \le \norm{a^*}$, and the same with $a^*$ in place of $a$.

Example: let $X$ be a compact Hausdorff space and $A = C(X)$ with the sup norm and $f^* = \overline{f}$.
Then $\norm{\overline{f} f}_\infty = \max_x \abs{f(x)}^2 = \norm{f}_\infty^2$.
The algebra of bounded operators on a Hilbert space with the adjoint also satisfies it: $\norm{Tx}^2 = \ip{T^* T x}{x} \le \norm{T^* T} \norm{x}^2$ gives $\norm{T}^2 \le \norm{T^* T} \le \norm{T}^2$.

Non-example: $A = \C^2$ with coordinatewise operations, $(x, y)^* = (\overline{x}, \overline{y})$, and the norm $\norm{(x, y)} = \abs{x} + \abs{y}$.
It is a Banach algebra with involution, but for $a = (1, 1)$ one has $\norm{a^* a} = \norm{(1, 1)} = 2$, while $\norm{a}^2 = 4$.
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<document source="https://tamnd.github.io/math2/node/HF00FDAW2F.md" tag="HF00FDAW2F" kind="definition">
# Ultrafilter on a Boolean algebra

## Definition

Let $B$ be a [[Boolean algebra|0XADJKM699]] with order $\le$.
A *filter* of $B$ is a subset $F \subseteq B$ such that $1 \in F$, $a \wedge b \in F$ for all $a, b \in F$, and $b \in F$ whenever $a \in F$ and $a \le b$.
It is *proper* if $0 \notin F$.
An *ultrafilter* of $B$ is a proper filter $U$ such that for every proper filter $F$ with $U \subseteq F$, one has $F = U$.

## Remarks

The filters of $B$ are the nonempty [[order filters|Z15FXM9WM1]] of $B$ seen as a lattice.
For a proper filter $U$, these are equivalent.
(a) $U$ is an ultrafilter.
(b) For every $a \in B$, exactly one of $a$ and $\neg a$ lies in $U$.
(c) $U$ is prime: $a \vee b \in U$ implies $a \in U$ or $b \in U$.
Ultrafilters $U$ correspond to homomorphisms $\chi \colon B \to \set{0, 1}$, by $U = \chi^{-1}(1)$.

Example: in the power set of a set $X$, for $x \in X$, the family $\set{A \subseteq X : x \in A}$ is an ultrafilter.
It is a proper filter, and for each $A$, either $x \in A$ or $x \in X \setminus A$.

Non-example: in the power set of $\set{1, 2}$, the family $F = \set{\set{1, 2}}$ is a proper filter but not an ultrafilter.
It contains neither $\set{1}$ nor its complement $\set{2}$, and it lies in the larger proper filter $\set{\set{1}, \set{1, 2}}$.
</document>

<document source="https://tamnd.github.io/math2/node/HF32CBN5MA.md" tag="HF32CBN5MA" kind="concept">
# Sphere packing

## Summary

A sphere packing is an arrangement of balls in space that do not overlap.
The classical question asks for the largest fraction of space that a packing of equal balls can fill, its [[packing density|4P5PSWADJD]].
The answer is known in dimensions $1$, $2$, $3$, $8$ and $24$.
In the plane it is the hexagonal packing, and in space the face-centred cubic packing of the [[Kepler conjecture|AFMBQT06ZV]].
In dimensions $8$ and $24$ it is given by the [[E8 lattice|2CXH65EKY7]] and the [[Leech lattice|XETV3657DG]].
Sphere packings are linked with error-correcting codes, lattices, number theory and crystallography.

## Explanations

### L1 human

Think of oranges stacked in a crate.
A sphere packing is any way of placing equal balls so that none of them overlap; they may touch.
Stacking layers of oranges so that each orange sits in a hollow of the layer below fills about $74\%$ of space.
The same question makes sense in the plane with coins, and in higher dimensions with balls that cannot be drawn but can be described by coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/HF8AJZP6PG.md" tag="HF8AJZP6PG" kind="definition">
# Mahalanobis distance

## Definition

Let $X$ be a random vector in $\R^p$ with [[mean|0WQNRCNY3M]] $\mu$ and [[covariance matrix|0V7G6WMFK0]] $\Sigma$, and assume that $\Sigma$ is [[positive definite|0KX66CZMBB]].
The *Mahalanobis distance* of a point $x \in \R^p$ from the distribution of $X$ is
$$D_M(x) = \sqrt{(x - \mu)^T \Sigma^{-1} (x - \mu)}.$$
The Mahalanobis distance between two points $x, y \in \R^p$ with respect to $\Sigma$ is
$$d_M(x, y) = \sqrt{(x - y)^T \Sigma^{-1} (x - y)}.$$

## Remarks

Example: let $\mu = 0$ and $\Sigma = \operatorname{diag}(4, 1)$.
Then $\Sigma^{-1} = \operatorname{diag}(1/4, 1)$, so $D_M(2, 0) = \sqrt{4/4} = 1$ and $D_M(0, 1) = 1$, while $D_M(0, 2) = 2$.
Non-example: if $\Sigma = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, then $\Sigma$ is not invertible, so the formula is not defined.
If $\Sigma = A A^T$ with $A$ invertible, then $d_M(x, y) = \abs{A^{-1} x - A^{-1} y}$, the Euclidean distance after the change of variables $x \mapsto A^{-1} x$.
So $d_M$ is a metric on $\R^p$.
If $\Sigma$ is the identity, $d_M$ is the Euclidean distance.
If $X$ is multivariate normal, then $D_M(X)^2$ has the chi-squared distribution with $p$ degrees of freedom.
</document>

<document source="https://tamnd.github.io/math2/node/HFBST11CT6.md" tag="HFBST11CT6" kind="concept">
# Unit in the last place

## Summary

The unit in the last place, or ulp, of a [[floating-point|RTKRSC3KPH]] number is the value of its last significant digit position.
It is the spacing between neighbouring floating-point numbers at that size.
Errors of computed results are often stated in ulps.
For example, a result correctly rounded to nearest is within half an ulp of the exact value.
The ulp of $1$ is the [[machine epsilon|JA3QXVF2B8]].

## Explanations

### L1 human

If a ruler is marked in millimetres, one millimetre is the smallest step it can show.
The ulp is that smallest step for a floating-point number.
In decimal with three significant digits, the number $3.14$ has ulp $0.01$, while $314$ has ulp $1$.
So the ulp grows with the size of the number, which is why floating-point accuracy is relative and not absolute.
</document>

<document source="https://tamnd.github.io/math2/node/HFBW3SXQVM.md" tag="HFBW3SXQVM" kind="concept">
# Inversive geometry

## Summary

Inversive geometry studies the plane with one point at infinity added.
It keeps the properties of figures that are preserved by inversions in circles and reflections in lines.
In it, lines count as circles through the point at infinity, and inversions map circles to circles.
Its transformations are the Möbius transformations and their composites with complex conjugation.
So it is the geometry of the Riemann sphere in the sense of the Erlangen program.

## Explanations

### L1 human

Inversive geometry treats lines and circles alike: a line is a circle that passes through a point at infinity.
Its basic moves are reflections in lines and inversions in circles, which turn circles into circles and keep angles between curves.
Lengths and straightness are not kept, so problems about tangent circles can often be simplified by an inversion that turns some circles into lines.
For example, two circles touching at a point $P$ become two parallel lines after an inversion centered at $P$.
</document>

<document source="https://tamnd.github.io/math2/node/HFEA4FKGW1.md" tag="HFEA4FKGW1" kind="statement">
# Poincaré–Birkhoff–Witt theorem

## Statement

Let $k$ be a [[field|04C5ZBG1WT]] and let $\mathfrak g$ be a Lie algebra over $k$.
Let $(x_i)_{i \in I}$ be a [[basis|0KCQ9Y3QKA]] of $\mathfrak g$ indexed by a totally ordered set $I$.
Let $U(\mathfrak g)$ be the [[universal enveloping algebra|SCZ8PRVY00]] of $\mathfrak g$ with its canonical map $\iota \colon \mathfrak g \to U(\mathfrak g)$.
Then the ordered monomials
$$\iota(x_{i_1}) \iota(x_{i_2}) \cdots \iota(x_{i_m}), \qquad m \ge 0, \quad i_1 \le i_2 \le \cdots \le i_m,$$
form a basis of $U(\mathfrak g)$ over $k$, where the monomial with $m = 0$ is $1$.

## Remarks

That the ordered monomials span is easy: any product of the $\iota(x_i)$ can be reordered.
Each swap of neighbours changes it by a product of lower length, by the relation $xy - yx = [x, y]$.
Linear independence is the hard part.
A standard proof builds an action of $\mathfrak g$ on the polynomial ring $S(\mathfrak g)$ that sends each ordered monomial applied to $1$ to the matching polynomial monomial.
The Jacobi identity is used to check that this is an action.
In particular $\iota$ is injective, so every Lie algebra over a field is a Lie subalgebra of an associative algebra with the commutator bracket.
Equivalently, the [[associated graded algebra|8HFAEGFQT1]] of $U(\mathfrak g)$ for its [[filtration|R1PTHQACWM]] by length of products is isomorphic to the [[symmetric algebra|PBGF9TXMB7]] $S(\mathfrak g)$.
Example: for $\mathfrak{sl}_2$ with basis $f < h < e$, the elements $f^a h^b e^c$ with $a, b, c \ge 0$ form a basis of $U(\mathfrak{sl}_2)$.
Over a commutative ring instead of a field, the theorem holds when $\mathfrak g$ is free as a module, but it can fail in general.

## Explanations

### L1 human

The enveloping algebra is defined by generators and relations, and such definitions can collapse.
The theorem says that nothing collapses here: the enveloping algebra is exactly as large as a polynomial ring in a basis of the Lie algebra.
You may write every element as a combination of monomials with the basis elements in a fixed order, and this way of writing it is unique.

### L2 human

The theorem is the basic structural fact about $U(\mathfrak g)$.
It shows that $\mathfrak g$ embeds in $U(\mathfrak g)$, that $U(\mathfrak g)$ has no zero divisors, and that $U(\mathfrak g)$ is Noetherian when $\mathfrak g$ is finite-dimensional, by passing to the polynomial ring $S(\mathfrak g)$.
For a subalgebra $\mathfrak h \subseteq \mathfrak g$, it shows that $U(\mathfrak g)$ is a free $U(\mathfrak h)$-module, which is the key input for Verma modules and induced representations.
It also gives the vector space isomorphism $S(\mathfrak g) \to U(\mathfrak g)$ by symmetrisation in characteristic $0$.
</document>

<document source="https://tamnd.github.io/math2/node/HFJCAPWWD4.md" tag="HFJCAPWWD4" kind="concept">
# Geometric graph

## Summary

A geometric graph is a [[graph|00C0BR7D7H]] drawn in the plane with its vertices as points and its edges as straight line segments, which may cross.
Geometric graph theory asks how the positions of the points constrain the graph.
A typical question is how many edges force two edges to cross.
A geometric graph without crossings is a straight-line drawing of a [[planar graph|0NC9R64TXP]].
The subject sits between graph theory and [[discrete geometry|Q0RW86P85F]].

## Explanations

### L1 human

Mark some dots on paper and join some pairs of dots with a ruler.
The result is a geometric graph: the dots are vertices and the straight strokes are edges.
For example, the four corners of a square joined by its four sides and two diagonals give a geometric graph of the complete graph $K_4$ with one crossing.
Placing one corner inside the triangle of the other three gives a drawing of $K_4$ with no crossing.
So the same abstract graph can have different geometric drawings with different properties.
</document>

<document source="https://tamnd.github.io/math2/node/HFQSSB1Z23.md" tag="HFQSSB1Z23" kind="concept">
# Diamond principle

## Summary

The diamond principle, written $\diamondsuit$, is a combinatorial axiom about $\omega_1$ introduced by Ronald Jensen.
It says that a sequence of guesses, one at each countable ordinal, correctly predicts every subset of $\omega_1$ on a [[stationary set|YE6WZR46RQ]] of ordinals.
It holds in the [[constructible universe|6TF2JBTPPJ]], and it implies the [[continuum hypothesis|0MN2MER9WN]].
It is used to build objects such as Suslin trees, and the [[precise definition|J8GYFAA70H]] is given separately.

## Explanations

### L1 human

Imagine an oracle that, at each countable stage $\alpha$, makes one guess $A_\alpha$ about what a hidden set $A \subseteq \omega_1$ looks like below $\alpha$.
Diamond says that there is an oracle so good that, for every possible hidden set, its guesses are right at a large set of stages.
This is surprising, because there are $2^{\aleph_1}$ subsets of $\omega_1$, while each stage gets only one guess.
Such an oracle lets one kill every possible obstacle during a construction of length $\omega_1$, and that is how it is used.
</document>

<document source="https://tamnd.github.io/math2/node/HFSM647VBR.md" tag="HFSM647VBR" kind="definition">
# Jones polynomial

## Definition

Let $\mathcal{L}$ be the set of isotopy classes of oriented tame [[links|1A5WRCNTM3]] in $S^3$.
Let $L_+$, $L_-$ and $L_0$ be oriented links with diagrams that agree outside a small disk.
Inside the disk, $L_+$ has a positive crossing, $L_-$ has a negative crossing, and $L_0$ has the two strands resolved without crossing, respecting the orientation.
The *Jones polynomial* is the unique map $V \colon \mathcal{L} \to \Z[t^{1/2}, t^{-1/2}]$ such that:

1. $V(O) = 1$ for the [[unknot|1TQ06R4Q66]] $O$;
2. $t^{-1} V(L_+) - t \, V(L_-) = (t^{1/2} - t^{-1/2}) \, V(L_0)$ for every such triple.

## Remarks

Existence is the main theorem; uniqueness holds because changing crossings in any diagram reaches an unlink, and the relation computes the rest.
Example: a one-crossing diagram of the unknot gives $L_+ = L_- = O$ and $L_0$ the two-component unlink $O \sqcup O$.
The relation gives $t^{-1} - t = (t^{1/2} - t^{-1/2}) V(O \sqcup O)$, so $V(O \sqcup O) = -(t^{1/2} + t^{-1/2})$.
For the trefoils, $V = t + t^3 - t^4$ for one and $V = t^{-1} + t^{-3} - t^{-4}$ for its mirror image.
Non-example: the map $L \mapsto 1$ satisfies clause 1 but not clause 2, since $t^{-1} - t \ne t^{1/2} - t^{-1/2}$.
In terms of the [[Kauffman bracket|JYKVYS4N32]], $V_L(t) = (-A^3)^{-w(D)} \langle D \rangle$ at $A = t^{-1/4}$, where $w(D)$ is the writhe of a diagram $D$ of $L$.
Mirror image changes $V_L(t)$ to $V_L(t^{-1})$.
</document>

<document source="https://tamnd.github.io/math2/node/HFTBG5PMXC.md" tag="HFTBG5PMXC" kind="concept">
# Model theory

## Summary

Model theory is the branch of mathematical logic that studies formal [[theories|92SWXST1ZM]] together with their [[models|MVCH6N9SNW]].
It asks which classes of [[structures|983GZBQYT4]] a set of axioms describes, and which subsets of a structure a formal language can define.
Its tools, such as the [[compactness theorem|1NR3HAWG9D]], have applications in algebra, number theory and geometry.

## Explanations

### L1 human

Model theory looks at mathematics from two sides at once: the sentences you write and the objects that make them true.
For example, the field axioms are sentences, and the fields $\Q$, $\R$ and $\C$ are objects that satisfy them.
A typical question is: if two structures satisfy the same sentences, how alike must they be?
Another is: which sets of points can a formula describe inside a given structure?
</document>

<document source="https://tamnd.github.io/math2/node/HG4S5HNWXF.md" tag="HG4S5HNWXF" kind="definition">
# Continuum

## Definition

A *continuum* is a nonempty [[metric space|0VKD1TP7F3]] that is [[compact|0TVJZYB6SH]] and [[connected|0V032VWTHD]].
A continuum is *nondegenerate* if it has more than one point.
A *subcontinuum* of a continuum $X$ is a subset of $X$ that is a continuum in the induced metric.

## Remarks

The closed interval $[0, 1]$ is a continuum: it is compact by the Heine–Borel theorem and connected because it is an [[interval|0Q1D85B8G6]].
The set $[0, 1] \cup [2, 3]$ is compact but not connected, so it is not a continuum.
Some authors define a continuum as a nonempty compact connected [[Hausdorff space|0S1FJ8VEX9]], and call the metric ones metric continua.
The intersection of a decreasing sequence of continua is a continuum.

## Explanations

### L1 human

A continuum is a compact space in one piece.
Arcs, circles and filled squares are continua.
</document>

<document source="https://tamnd.github.io/math2/node/HG65C9R1SP.md" tag="HG65C9R1SP" kind="statement">
# Hardy–Littlewood–Sobolev inequality

## Statement

Let $n \ge 1$, let $0 < \alpha < n$, and let $I_\alpha$ be the [[fractional integral|P1VFBJV1VN]] of order $\alpha$ on $\R^n$.
Let $1 \le p < q < \infty$ with
$$\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}.$$

1. If $p > 1$, there is a constant $C = C(n, \alpha, p)$ such that $\norm{I_\alpha f}_{L^q(\R^n)} \le C \norm{f}_{L^p(\R^n)}$ for all $f \in L^p(\R^n)$.
2. If $p = 1$, so that $q = n/(n - \alpha)$, then $I_\alpha$ is of [[weak type|9534R6VMY0]] $(1, q)$: there is $C = C(n, \alpha)$ with $\abs{\set{x : \abs{I_\alpha f(x)} > \lambda}} \le \left( C \norm{f}_{L^1} / \lambda \right)^q$ for all $f \in L^1(\R^n)$ and $\lambda > 0$.

Here $L^p(\R^n)$ is the [[Lp space|SASP14MA1Z]] for Lebesgue measure.

## Remarks

A short proof is Hedberg's.
Fix $x$ and $R > 0$, and split the integral defining $I_\alpha \abs{f}(x)$ into $\abs{x - y} < R$ and $\abs{x - y} \ge R$.
The near part is at most $C R^\alpha Mf(x)$, where $M$ is the [[Hardy–Littlewood maximal function|VX1A56P5KQ]], by summing over dyadic shells.
The far part is at most $C R^{-n/q} \norm{f}_p$, by [[Hölder's inequality|JQ9WXC729D]].
Choosing $R$ to balance the two terms gives $\abs{I_\alpha f(x)} \le C \, Mf(x)^{p/q} \norm{f}_p^{1 - p/q}$, and the [[maximal inequality|BF7T7VE57V]] for $M$ gives both parts.

The relation between $p$, $q$ and $\alpha$ is forced by scaling.
If $f_\delta(x) = f(\delta x)$, then $I_\alpha f_\delta(x) = \delta^{-\alpha} I_\alpha f(\delta x)$, and an inequality $\norm{I_\alpha f}_q \le C \norm{f}_p$ for all $f$ and all $\delta > 0$ needs $\frac{1}{q} = \frac{1}{p} - \frac{\alpha}{n}$.
At $p = 1$ the strong bound fails: if $f \ge 0$ is integrable with $\int f > 0$, then $I_\alpha f(x)$ behaves like $c \abs{x}^{\alpha - n}$ at infinity, which is in weak $L^{q}$ but not in $L^{q}$ for $q = n/(n - \alpha)$.

## Explanations

### L1 human

Fractional integration of order $\alpha$ smooths a function by $\alpha$ derivatives.
The inequality says that this smoothing also improves integrability: an $L^p$ function becomes an $L^q$ function with a larger exponent $q$.
The gain $\frac{1}{p} - \frac{1}{q} = \frac{\alpha}{n}$ is exactly what dimensional analysis allows.

### L2 human

The inequality implies the [[Sobolev inequality|B03CA692K3]] for $1 < p < n$.
A smooth function $u$ with compact support satisfies $\abs{u(x)} \le C \, I_1(\abs{\nabla u})(x)$, and the case $\alpha = 1$ then gives $\norm{u}_{np/(n - p)} \le C \norm{\nabla u}_p$.
There is also a bilinear form: $\abs{\iint f(x) g(y) \abs{x - y}^{-\lambda} \, dx \, dy} \le C \norm{f}_p \norm{g}_r$ when $0 < \lambda < n$, $1 < p, r < \infty$ and $\frac{1}{p} + \frac{1}{r} + \frac{\lambda}{n} = 2$.
Lieb found the sharp constant of this bilinear form in the case $p = r$.
</document>

<document source="https://tamnd.github.io/math2/node/HG7HC57NDM.md" tag="HG7HC57NDM" kind="definition">
# Nakayama algebra

## Definition

Let $K$ be a field and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $K$.
A module is *uniserial* if it has a unique [[composition series|0XV2A49X6Q]], equivalently if its submodules are totally ordered by inclusion.
The algebra $A$ is a *Nakayama algebra* if every [[indecomposable|E2P29GCD9T]] [[projective|BFFA03GE3V]] left $A$-module and every indecomposable projective right $A$-module is [[uniserial|Z1045A0V29]].

## Remarks

Example: $A = K[x]/(x^n)$ with $n \ge 1$.
The only indecomposable projective module on either side is $A$ itself, and its submodules are the ideals $(x^i)$ for $0 \le i \le n$, which form a chain.
Example: the algebra of upper triangular $n \times n$ matrices over $K$ is a Nakayama algebra; its indecomposable projective left modules are its columns.
Non-example: $A = K[x, y]/(x, y)^2$, with basis $1, x, y$.
The module $A$ is indecomposable and projective, but $Kx$ and $Ky$ are submodules neither of which contains the other, so $A$ is not uniserial.
Over a Nakayama algebra every indecomposable module is uniserial and is a quotient of an indecomposable projective module.
In the language of rings, a Nakayama algebra is a finite-dimensional algebra that is a [[serial ring|TW3ESQB3GQ]] on both sides.
</document>

<document source="https://tamnd.github.io/math2/node/HGEX40P0Z0.md" tag="HGEX40P0Z0" kind="concept">
# Bézout domain

## Summary

A Bézout domain is an [[integral domain|09S3PFBEPM]] in which every finitely generated ideal is [[principal|00A2EDFT9Q]].
In such a ring [[Bézout's identity|0DQVHMYAGA]] holds: a gcd of $a$ and $b$ can be written as $ax + by$.
Every [[principal ideal domain|0RZYDVZ475]] is a Bézout domain, and a Noetherian Bézout domain is a principal ideal domain.
Bézout domains are [[GCD domains|ACDEK1GD9G]] and [[Prüfer domains|2RDAA4PCK8]].

## Explanations

### L1 human

In the integers, $\gcd(a, b)$ can always be written as $ax + by$ for some integers $x, y$.
A Bézout domain is a ring where this works for every pair of elements, even if the ring has ideals that are not finitely generated.
The ring of entire functions on $\C$ and the ring of all algebraic integers are Bézout domains that are not principal ideal domains.
The polynomial ring $\Z[x]$ is not Bézout: the ideal generated by $2$ and $x$ is not principal.
</document>

<document source="https://tamnd.github.io/math2/node/HGF11AHX5R.md" tag="HGF11AHX5R" kind="statement">
# Biconjugate gradient method

## Statement

Let $A$ be a real [[invertible|0EB9N0S33C]] $n \times n$ matrix, let $b \in \R^n$, and let $x_0 \in \R^n$ with $r_0 = b - A x_0$.
Choose a *shadow residual* $\tilde{r}_0 \in \R^n$ with $\tilde{r}_0^T r_0 \ne 0$, and set $p_0 = r_0$ and $\tilde{p}_0 = \tilde{r}_0$.
The *biconjugate gradient method* (BiCG) computes, for $k = 0, 1, \dots$,
$$\alpha_k = \frac{\tilde{r}_k^T r_k}{\tilde{p}_k^T A p_k}, \quad x_{k+1} = x_k + \alpha_k p_k, \quad r_{k+1} = r_k - \alpha_k A p_k, \quad \tilde{r}_{k+1} = \tilde{r}_k - \alpha_k A^T \tilde{p}_k,$$
$$\beta_k = \frac{\tilde{r}_{k+1}^T r_{k+1}}{\tilde{r}_k^T r_k}, \quad p_{k+1} = r_{k+1} + \beta_k p_k, \quad \tilde{p}_{k+1} = \tilde{r}_{k+1} + \beta_k \tilde{p}_k,$$
where $A^T$ is the [[transpose|09DXWYRD43]].
Suppose that the denominators $\tilde{p}_k^T A p_k$ and $\tilde{r}_k^T r_k$ are nonzero up to step $m$.
Then in exact arithmetic, for $i, j \le m$:

1. $r_k = b - A x_k$, and $x_k \in x_0 + \mathcal{K}_k(A, r_0)$, where $\mathcal{K}_k$ is the [[Krylov subspace|QB45PCGQPV]];
2. (biorthogonality) $\tilde{r}_i^T r_j = 0$ for $i \ne j$;
3. (biconjugacy) $\tilde{p}_i^T A p_j = 0$ for $i \ne j$.

## Remarks

Clauses 2 and 3 say that $r_k$ is orthogonal to $\mathcal{K}_k(A^T, \tilde{r}_0)$, a Petrov–Galerkin condition.
Suppose no breakdown occurs up to step $n$.
Then $\tilde{r}_0, \dots, \tilde{r}_{n-1}$ are linearly independent, since $\tilde{r}_i^T r_j$ is zero for $i \ne j$ and nonzero for $i = j$.
So $r_n$ is orthogonal to all of $\R^n$, and $r_n = 0$.
Example: if $A$ is symmetric and $\tilde{r}_0 = r_0$, then $\tilde{r}_k = r_k$ and $\tilde{p}_k = p_k$ for all $k$.
Then BiCG is the [[conjugate gradient method|AFXZGQB4EN]], which never breaks down when $A$ is [[positive definite|0KX66CZMBB]].
For a general $A$ the method can break down when $\tilde{r}_k^T r_k = 0$ with $r_k \ne 0$, and its residual norms can oscillate wildly.

## Explanations

### L1 human

Conjugate gradients rely on symmetry to keep short recurrences.
BiCG regains short recurrences for a nonsymmetric matrix by running a second, shadow iteration with the transpose alongside the first.
Each new residual is made orthogonal to the earlier shadow residuals, instead of to its own earlier residuals as in conjugate gradients.
The price is two matrix-vector products per step, one with $A$ and one with $A^T$, and less robust convergence.

### L2 human

BiCG is the linear-system counterpart of the nonsymmetric Lanczos process.
Unlike GMRES it has no minimization property, but its memory and work per step do not grow with $k$.
Its weaknesses, breakdowns and erratic residuals, and the need for $A^T$, motivated the transpose-free variants CGS and BiCGSTAB and the smoother QMR method.
It is one of the Krylov [[iterative methods|2RPE77ACER]] for nonsymmetric systems.
</document>

<document source="https://tamnd.github.io/math2/node/HGJVM49PA4.md" tag="HGJVM49PA4" kind="concept">
# Double-negation translation

## Summary

A double-negation translation is a map from formulas to formulas that embeds classical logic into [[intuitionistic logic|Q11N8X99FK]].
It inserts [[negations|0XR7A7RSXB]] $\neg \neg$ at chosen places, so that a formula is classically provable exactly when its translation is intuitionistically provable.
The best known versions are due to Kolmogorov, to Gödel and Gentzen, and to Kuroda.
It shows that [[Peano arithmetic|0QFT7Z76N5]] is consistent if [[Heyting arithmetic|CK8ZQRN183]] is.

## Explanations

### L1 human

Intuitionistic logic rejects some classical reasoning, such as "$A$ or not $A$".
But it accepts "it is not the case that neither $A$ nor not $A$", that is, $\neg \neg (A \vee \neg A)$.
The translation rewrites each classical claim in such a doubly negated, weaker form.
Then every classical proof becomes an intuitionistic proof of the weaker claim.
So a contradiction in classical reasoning would give a contradiction in intuitionistic reasoning too.
</document>

<document source="https://tamnd.github.io/math2/node/HGK7R92QH7.md" tag="HGK7R92QH7" kind="definition">
# Hilbert class field

## Definition

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] and fix an algebraic closure $\overline{K}$ of $K$.
A finite extension $L/K$ is *unramified at every place* if two conditions hold.
Every nonzero prime ideal of $\mathcal{O}_K$ is [[unramified|H5RC67SD24]] in $L$.
Every real embedding of $K$ extends only to real embeddings of $L$.
Consider the finite [[Galois extensions|0QMZAQ6QQ4]] $L/K$ in $\overline{K}$ with [[abelian|082GHNPJ53]] Galois group that are unramified at every place.
The *Hilbert class field* of $K$ is the compositum $H \subseteq \overline{K}$ of all these fields $L$.

## Remarks

By class field theory, $H/K$ is a finite abelian extension, unramified at every place, and the [[Artin map|7938EKMMVR]] gives an isomorphism from the [[ideal class group|WQ46BD3ARY]] $\mathrm{Cl}(K)$ onto $\Gal(H/K)$.
So $[H : K] = h_K$, and a prime ideal of $\mathcal{O}_K$ splits completely in $H$ if and only if it is principal.
Every ideal of $\mathcal{O}_K$ becomes principal in $\mathcal{O}_H$ (the principal ideal theorem).

Example: $K = \Q(\sqrt{-5})$ and $L = K(i) = \Q(\sqrt{-5}, i)$.
The three quadratic subfields of $L$ are $\Q(i)$, $\Q(\sqrt{5})$ and $\Q(\sqrt{-5})$, with [[discriminants|E0VZTNPJXA]] $-4$, $5$ and $-20$, so $d_L = (-4) \cdot 5 \cdot (-20) = 400 = d_K^2$.
Hence the relative discriminant of $L/K$ has norm $1$ and no finite prime ramifies; $K$ has no real embedding, so $L/K$ is unramified at every place.
Since $h_K = 2 = [L : K]$, the field $L$ is the Hilbert class field of $K$.

Non-example: $L' = \Q(\sqrt{-5}, \sqrt{2})$ is abelian of degree $2$ over $K$ but not unramified.
Its quadratic subfields have discriminants $8$, $-20$ and $-40$, so $d_{L'} = 6400 = 16 \, d_K^2$, and some prime of $K$ above $2$ ramifies in $L'$.
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<document source="https://tamnd.github.io/math2/node/HGKN8E09RS.md" tag="HGKN8E09RS" kind="concept">
# Finite volume method

## Summary

The finite volume method discretizes a [[conservation law|8RA7AMWSAF]] by dividing the domain into small control volumes and keeping the average of the solution in each.
The [[divergence theorem|0JX86PSSA9]] turns the change of each average into a sum of fluxes through the faces of the volume.
Each flux is computed once and used by the two volumes that share the face, so the scheme conserves the total amount exactly.
This makes it the standard method for shocks in gas dynamics and for transport in fluid mechanics; [[Godunov's scheme|43HZJ7DRT0]] is a classical example.
See the [[definition|CAB1ARHH31]].

## Explanations

### L1 human

Split a pipe into short sections and keep track of how much water each section holds.
The amount in a section changes only by what flows in or out through its two ends.
Whatever leaves one section through a wall enters the neighbour through the same wall, so no water is created or lost.
A finite volume method computes these flows from the amounts in the neighbouring sections and updates the amounts step by step.
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<document source="https://tamnd.github.io/math2/node/HGMX1QGTAZ.md" tag="HGMX1QGTAZ" kind="concept">
# Eikonal equation

## Summary

The eikonal equation $\abs{\nabla u} = f$ says that the gradient of the unknown function has a prescribed length at each point.
It is a stationary [[Hamilton–Jacobi equation|5TRXQKP7DY]].
Its solutions measure travel times: with $f = 1$ and $u = 0$ on the boundary of a domain, the [[viscosity solution|H1ERJ3T75T]] is the distance to the boundary.
It is the basic equation of [[geometric optics|90F3NWRHVD]], where the level sets of $u$ are the wave fronts.
See the [[definition|X8CF1K8QB4]].

## Explanations

### L1 human

Light a fire along the edge of a field, and suppose that the fire spreads with speed $1/f(x)$ at the point $x$.
Let $u(x)$ be the time at which the fire reaches $x$.
Moving one step in the best direction raises the arrival time at rate $f(x)$, and this is what $\abs{\nabla u} = f$ says.
For a disc of radius $1$ with $f = 1$, the fire burns inward from the circle, and $u(x) = 1 - \abs{x}$.
At the centre the fronts from all sides meet, and $u$ has a corner there.
</document>

<document source="https://tamnd.github.io/math2/node/HGPH5QWVC2.md" tag="HGPH5QWVC2" kind="concept">
# Periodic solutions of integral equations

## Summary

A periodic solution of an [[integral equation|R3SG1SN7JQ]] is a solution defined on the whole real line that repeats after a fixed period $T > 0$.
One looks for such solutions when the data of the equation, its forcing term and its kernel, are themselves $T$-periodic in time.
Equations with infinite memory, where the integral runs over the whole past, are the natural setting.
Periodic solutions describe the steady oscillations of systems with memory under periodic forcing, much as [[periodic solutions|FHNDX22WMK]] do for differential equations.

## Explanations

### L1 human

Think of a system that is pushed by a force that repeats every day, and that remembers its own past.
After a long time one expects the system to settle into a daily rhythm.
That rhythm is a periodic solution of the integral equation that models the system.
For example, $x(t) = \sin t$ solves $x(t) = \frac{1}{2}(\sin t + \cos t) + \int_{-\infty}^t e^{-(t - s)} x(s) \, ds$, and it repeats after the time $2\pi$.
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<document source="https://tamnd.github.io/math2/node/HGZAYPQBXW.md" tag="HGZAYPQBXW" kind="concept">
# Stable homotopy groups of spheres

## Summary

The stable homotopy groups of spheres $\pi_k^s$ are the groups $\pi_{n+k}(S^n)$ for $n$ large, where they no longer depend on $n$.
The [[Freudenthal suspension theorem|7DM2KSS2DB]] shows that the suspension maps become isomorphisms once $n \ge k + 2$.
Together they form a graded ring, the coefficient ring of stable homotopy theory, and they are the homotopy groups of the sphere spectrum.
Their computation is a central problem of algebraic topology, studied with the Adams spectral sequence and chromatic methods.

## Explanations

### L1 human

The homotopy groups of spheres change wildly as the dimensions change.
If both dimensions are raised together by the same amount, however, the groups settle down after a while.
These settled values are the stable groups.
The first ones are $\Z$, $\Z/2$, $\Z/2$, $\Z/24$, $0$, $0$, $\Z/2$, $\Z/240$, and their pattern is still not understood in general.
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<document source="https://tamnd.github.io/math2/node/HGZMZRSS5D.md" tag="HGZMZRSS5D" kind="statement">
# Docquier–Grauert theorem

## Statement

Let $X$ be a [[Stein manifold|HMCD0JBFST]] and let $D \subseteq X$ be an open subset.
Suppose that $D$ is locally Stein in $X$.
This means that every point $p \in \partial D$ has an open neighbourhood $U$ in $X$ such that $U \cap D$ is Stein.
Then $D$ is a Stein manifold.

## Remarks

For $X = \C^n$ this is Oka's solution of the Levi problem: a locally pseudoconvex domain in $\C^n$ is a domain of holomorphy.
The proof follows the same pattern.
Using that $X$ embeds in some $\C^N$ and has a holomorphic retraction from a neighbourhood, one builds a distance-to-the-boundary function $\delta$ on $D$.
The local Stein condition makes $-\log \delta$ [[plurisubharmonic|FF2P7ER3QT]] near $\partial D$.
Adding a strictly plurisubharmonic exhaustion of $X$ gives a plurisubharmonic exhaustion function of $D$, and Grauert's solution of the Levi problem then shows that $D$ is Stein.
The same holds for unbranched Riemann domains spread over a Stein manifold, which is the form in which Docquier and Grauert proved it.
Example: if $D_1, D_2$ are Stein open subsets of $X$, then $D_1 \cap D_2$ is locally Stein, hence Stein.
The hypothesis that $X$ is Stein cannot be dropped.
If $X$ is the Riemann sphere and $D = X$, then $\partial D$ is empty, so $D$ is locally Stein, but $D$ is compact and not Stein.

## Explanations

### L1 human

Being Stein is a global property of a complex manifold, roughly saying that it has many holomorphic functions.
The theorem says that, inside a Stein manifold, it is enough to check this property near each boundary point of an open set.
A local test at the boundary then gives a global conclusion.

### L2 human

The theorem shows that the class of Stein open sets of a Stein manifold is described by a local boundary condition, pseudoconvexity, just as for domains in $\C^n$.
It is used to prove that intersections, and suitable unions and limits, of Stein domains are Stein.
It reduces many questions on Stein manifolds to the case of domains in $\C^n$ through the embedding theorem.
</document>

<document source="https://tamnd.github.io/math2/node/HH2EM3DPNK.md" tag="HH2EM3DPNK" kind="statement">
# Gabrielov's theorem of the complement

## Statement

Let $X \subseteq \R^n$ be a [[subanalytic set|K1HTQF26MA]].
Then the complement $\R^n \setminus X$ is subanalytic.

## Remarks

For [[semianalytic sets|E1V22QQHYJ]] the analogous fact is immediate, since the complement of a set defined by equations and inequalities is again so defined.
A subanalytic set is locally a projection, and the complement of a projection is not visibly a projection.
Overcoming this is the content of the theorem; Gabrielov's proof studies the fibres of projections of semianalytic sets.
For example, if $A \subseteq \R^{n+m}$ is bounded and semianalytic and $\pi$ forgets the last $m$ coordinates, then $\R^n \setminus \pi(A)$ is subanalytic.
With the theorem, the subanalytic subsets of $\R^n$ form a Boolean algebra, and the closure, interior and boundary of a subanalytic set are subanalytic.

## Explanations

### L1 human

Subanalytic sets are built by taking shadows of sets cut out by analytic equations and inequalities.
The theorem says that the region outside such a shadow is again a set of the same kind.
So one can take complements freely and never leave the class.

### L2 human

In semialgebraic geometry, closure under projection is the [[Tarski–Seidenberg theorem|3W5JGDZAN0]], and closure under complement is trivial.
For subanalytic sets the roles are reversed: projections are built in, and complements are the hard part.
Gabrielov's theorem is the starting point of the result that the real field with restricted analytic functions is an [[o-minimal|DTWHAW1N0V]] structure.
</document>

<document source="https://tamnd.github.io/math2/node/HH35EXPKEC.md" tag="HH35EXPKEC" kind="definition">
# Partial linear space

## Definition

Let $(P, L, I)$ be an [[incidence structure|FGKNAFNHA3]] with point set $P$, line set $L$ and incidence relation $I \subseteq P \times L$.
It is a *partial linear space* if:

1. any two distinct points are incident with at most one common line;
2. every line is incident with at least two points.

It is a *linear space* if in clause 1 "at most one" is replaced by "exactly one".

## Remarks

Example: the $3 \times 3$ grid, with $P = \set{1, 2, 3}^2$ and $L$ the three rows $\set{i} \times \set{1, 2, 3}$ and the three columns $\set{1, 2, 3} \times \set{j}$.
A row and a column meet in one point, two rows or two columns are disjoint, so two distinct points share at most one line; each line has three points.

Non-example: $P = \set{a, b, c}$ with lines $\set{a, b}$ and $\set{a, b, c}$.
The points $a$ and $b$ lie on two lines, so clause 1 fails.

Clause 1 is equivalent to its dual: two distinct lines meet in at most one point.
Partial linear spaces are also called semilinear spaces or near-linear spaces.
</document>

<document source="https://tamnd.github.io/math2/node/HH6VEVM6Z0.md" tag="HH6VEVM6Z0" kind="concept">
# Ideal of an algebraic set

## Summary

The ideal of a subset $X$ of affine space $k^n$ is the set of all polynomials that vanish at every point of $X$.
It is the algebraic shadow of $X$ and is always a radical ideal.
Over an algebraically closed field, Hilbert's Nullstellensatz says that this gives a bijection from algebraic sets to radical ideals.
This turns geometry into algebra.

## Explanations

### L1 human

Given a set of points, collect every polynomial that is zero on all of them.
These polynomials form an ideal: sums of such polynomials and multiples of them still vanish on the set.
For the $x$-axis in the plane, the ideal consists of all polynomials divisible by $y$.
For the single point $(1, 2)$, the ideal is generated by $x - 1$ and $y - 2$.
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<document source="https://tamnd.github.io/math2/node/HHBGB5R5S2.md" tag="HHBGB5R5S2" kind="concept">
# Absolutely convergent Fourier series

## Summary

A [[Fourier series|G12KXDHPPS]] converges absolutely when the sum of the absolute values of its coefficients is finite.
Then the series converges uniformly to a continuous function, which agrees with the original function almost everywhere.
If the function is continuous, it equals the sum of its Fourier series at every point.
The functions with this property form the [[Wiener algebra|5H8XESYCHQ]].
Criteria such as [[Bernstein's theorem|NQC6CRDWFW]] tell which smoothness forces absolute convergence.

## Explanations

### L1 human

A Fourier series writes a periodic function as a sum of waves with amplitudes $c_n$.
If the amplitudes are so small that $\sum \abs{c_n}$ is finite, the waves can never add up to anything wild.
The partial sums then approach the function uniformly, with no Gibbs overshoot.
For example, the triangle wave $\abs{x}$ on $[-\pi, \pi]$ has coefficients of size about $1/n^2$, so its Fourier series converges absolutely.
</document>

<document source="https://tamnd.github.io/math2/node/HHKKPD81J2.md" tag="HHKKPD81J2" kind="concept">
# Fermi–Dirac integral

## Summary

The complete Fermi–Dirac integral $F_j(x)$ is the integral $\frac{1}{\Gamma(j+1)} \int_0^\infty \frac{t^j}{e^{t-x} + 1} \, dt$.
It arises in the statistical physics of fermions, such as electrons in metals and semiconductors, where $x$ plays the role of a reduced chemical potential.
It is a [[polylogarithm|96CHDZKGQ0]] in disguise: $F_j(x) = -\operatorname{Li}_{j+1}(-e^x)$.

## Explanations

### L1 human

In a gas of electrons, the chance that a state of energy $t$ is filled is $1/(e^{t - x} + 1)$, a smooth step that falls from $1$ to $0$ near $t = x$.
Weighting this step by a power $t^j$ and integrating over all energies gives the Fermi–Dirac integral.
For $j = 0$ the integral can be done by hand and gives $\log(1 + e^x)$.
For very negative $x$ the step is almost an exponential, and $F_j(x)$ is close to $e^x$.
</document>

<document source="https://tamnd.github.io/math2/node/HHKSWC8Y6D.md" tag="HHKSWC8Y6D" kind="definition">
# Principal symbol of an operator on a manifold

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] with cotangent bundle $\pi \colon T^*M \to M$, and let $E$ and $F$ be smooth [[vector bundles|P4DZM975BV]] over $M$.
Let $P \colon \Gamma(E) \to \Gamma(F)$ be a [[differential operator|CGBX1N9V8H]] of order at most $m$.
Let $x \in M$, $\xi \in T^*_x M$ and $e \in E_x$.
Choose a smooth function $f$ on $M$ with $f(x) = 0$ and $df(x) = \xi$, and a section $u \in \Gamma(E)$ with $u(x) = e$.
The *principal symbol* of order $m$ of $P$ is the bundle map $\sigma_m(P) \colon \pi^* E \to \pi^* F$ over $T^*M$ given by
$$\sigma_m(P)(x, \xi) \, e = \frac{1}{m!} \, P(f^m u)(x).$$

## Remarks

The right side does not depend on the choices of $f$ and $u$.
Take a chart and trivialisations in which $P = \sum_{\abs{\alpha} \le m} A_\alpha(x) \partial^\alpha$.
Then, with $\xi = \sum_j \xi_j \, dx_j$,
$$\sigma_m(P)(x, \xi) = \sum_{\abs{\alpha} = m} A_\alpha(x) \, \xi^\alpha.$$
Indeed, at $x$ only the terms in which all $m$ derivatives fall on distinct factors $f$ survive, and they give $m! \, \xi^\alpha$.
So $\sigma_m(P)$ agrees with the [[principal symbol|F31T650DKY]] on open subsets of $\R^n$, and it is homogeneous of degree $m$ in $\xi$.
Many texts include a factor $i^m$, defining the symbol through $D_j = -i \partial_j$; this does not change where the symbol is invertible.
The symbol is multiplicative: $\sigma_{m + l}(P Q) = \sigma_m(P) \, \sigma_l(Q)$.

Example: for the [[exterior derivative|0ZHSTAG8ZY]] $d$ on $k$-forms, $d(f \omega)(x) = df(x) \wedge \omega(x) + f(x) \, d\omega(x) = \xi \wedge \omega(x)$.
So $\sigma_1(d)(x, \xi) \omega = \xi \wedge \omega$.
For the [[Laplace–Beltrami operator|F4NWZNTRMD]] on functions of a Riemannian manifold, with the sign convention $\Delta = -\operatorname{div} \operatorname{grad} = -\sum_{j,k} g^{jk} \partial_j \partial_k + (\text{lower order})$, one gets $\sigma_2(\Delta)(x, \xi) = -\abs{\xi}_g^2$, or $\abs{\xi}^2_g$ with the factor $i^2$.

Non-example: the full expression $\sum_{\abs{\alpha} \le m} A_\alpha(x) \xi^\alpha$, with the lower-order terms kept, is not the principal symbol.
It depends on the chart: for $P = \partial_x^2$ on $\R$ and the new coordinate $y = e^x$, one has $P = y^2 \partial_y^2 + y \partial_y$, whose first-order term is not there in the old coordinate.

## Explanations

### L1 human

The principal symbol keeps the top-order derivatives of an operator, with each derivative replaced by a component of a covector.
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# Ordered vector space

## Summary

An ordered vector space is a real [[vector space|000AYPA9AJ]] with a compatible [[partial order|0QV2JS9RJ3]].
The order is preserved by adding a vector and by multiplying by a nonnegative scalar.
Such an order is the same as a [[positive cone|Z344RY5XX1]], the set of vectors that are at least $0$.
Spaces of real functions, of sequences and of self-adjoint operators carry natural orders of this kind.
Ordered vector spaces are the setting of positive linear maps, and [[Riesz spaces|99709K2NJM]] are the case where the order is a lattice.

## Explanations

### L1 human

In $\R^2$ say that $(a, b) \le (c, d)$ when $a \le c$ and $b \le d$.
Moving both points by the same vector keeps this relation, and so does stretching both by a positive factor.
Some pairs of vectors are not comparable, such as $(1, 0)$ and $(0, 1)$.
An ordered vector space is any real vector space with an order that behaves like this.
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<document source="https://tamnd.github.io/math2/node/HHNJT9CBEJ.md" tag="HHNJT9CBEJ" kind="definition">
# Term rewriting for equational theories

## Definition

Let $\sigma$ be a signature and $X$ a set of variables.
Let $T(X)$ be the set of terms of $\sigma$ over $X$, as in the [[term algebra|3YYKGWJJND]].
A *term rewriting system* is a set $R$ of pairs $(l, r)$ of terms, written $l \to r$ and called rules.
Each rule must satisfy two conditions: $l$ is not a variable, and every variable of $r$ occurs in $l$.
For terms $s, t$, one writes $s \to_R t$ if there are a rule $l \to r$ in $R$ and a [[substitution|NCDZ969V7X]] $\tau$ with the following property.
The term $s$ contains $\tau(l)$ as a subterm, and $t$ is obtained from $s$ by replacing one occurrence of it by $\tau(r)$.
Let $\leftrightarrow_R^*$ be the smallest equivalence relation that contains $\to_R$.
The system $R$ *presents* the [[equational theory|F1M51B4MDG]] $E$ if, for all terms $s, t$, one has $s \leftrightarrow_R^* t$ if and only if $s \approx t$ belongs to $E$.
The system $R$ is *convergent* if:

1. it is terminating: there is no infinite chain $t_0 \to_R t_1 \to_R t_2 \to_R \cdots$;
2. it is confluent: if $t \to_R^* u$ and $t \to_R^* v$, then $u \to_R^* w$ and $v \to_R^* w$ for some term $w$.

## Remarks

By [[Birkhoff's completeness theorem|WDHWCQ6Y2Y]], a system $R$ presents the theory axiomatised by the [[identities|PKD440G2XW]] $\set{l \approx r : (l \to r) \in R}$.
If $R$ is convergent, each term $t$ has a unique normal form $\operatorname{nf}(t)$, a term with no further rewrite step.
Then $s \approx t$ is in the theory if and only if $\operatorname{nf}(s) = \operatorname{nf}(t)$.
For a finite convergent system, this decides the equational theory.
Example: $R = \set{f(f(x)) \to x}$ is convergent: each step shortens the term, and the only overlap, in $f(f(f(x)))$, rewrites to $f(x)$ in both ways.
So $f^m(x) \approx f^n(x)$ holds in the theory of an involution exactly when $m$ and $n$ have the same parity.
Non-example: $R = \set{x \cdot y \to y \cdot x}$ presents the theory of a commutative operation but is not terminating, since $a \cdot b \to b \cdot a \to a \cdot b \to \cdots$.
The Knuth–Bendix completion procedure tries to turn a set of identities into an equivalent convergent system.
</document>

<document source="https://tamnd.github.io/math2/node/HHNR31W3WD.md" tag="HHNR31W3WD" kind="statement">
# Differentiation and multiplication under the Fourier transform

## Statement

Let $n \ge 1$, let $f \in L^1(\R^n)$, and let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\, dx$ be its [[Fourier transform|DPVM34ADZV]].
Fix an index $j \in \set{1, \dots, n}$.

1. If the function $x \mapsto x_j f(x)$ lies in [[$L^1(\R^n)$|SASP14MA1Z]], then $\hat f$ has a continuous partial derivative in $\xi_j$, and $\partial_{\xi_j} \hat f(\xi)$ is the Fourier transform of $-2\pi i x_j f(x)$.
2. If $f$ is continuously differentiable and $\partial_j f$ lies in $L^1(\R^n)$, then the Fourier transform of $\partial_j f$ is $\xi \mapsto 2\pi i \xi_j \hat f(\xi)$.

## Remarks

For part 1, differentiate under the integral sign.
The difference quotients of $e^{-2\pi i x \cdot \xi}$ in $\xi_j$ are bounded by $2\pi \abs{x_j}$, so [[dominated convergence|BYNV84PF86]] applies with dominating function $2\pi \abs{x_j f(x)}$.
For part 2, integrate by parts in $x_j$.
On almost every line parallel to the $j$-th axis, $f$ and $\partial_j f$ are integrable, so $f$ has limits at both ends, and these limits are $0$; the boundary terms vanish.

Example: for $f(x) = e^{-\pi x^2}$ on $\R$, part 2 gives $\widehat{f'}(\xi) = 2\pi i \xi e^{-\pi \xi^2}$, and part 1 gives $\frac{d}{d\xi} e^{-\pi \xi^2} = -2\pi \xi e^{-\pi \xi^2}$; both agree with $f' = -2\pi x f$.

Iterating, if $f$ lies in the Schwartz space then $\widehat{\partial^\alpha f}(\xi) = (2\pi i \xi)^\alpha \hat f(\xi)$ and $\partial^\alpha \hat f$ is the transform of $(-2\pi i x)^\alpha f$ for every multi-index $\alpha$.
The integrability hypotheses cannot be dropped: if $x_j f$ is not integrable, $\hat f$ need not be differentiable, as for $f(x) = 1/(1 + x^2)$ on $\R$, whose transform $\pi e^{-2\pi \abs{\xi}}$ has a corner at $0$.

## Explanations

### L1 human

The Fourier transform swaps two operations.
Taking a derivative of $f$ becomes multiplying $\hat f$ by $2\pi i \xi$.
Multiplying $f$ by $x$ becomes, up to a constant, taking a derivative of $\hat f$.

### L2 human

These rules turn a linear differential operator with constant coefficients $P(\partial)$ into multiplication by the polynomial $P(2\pi i \xi)$.
This is how the transform solves the heat, wave and Schrödinger equations on $\R^n$.
They also give the duality between smoothness and decay: the more derivatives $f$ has in $L^1$, the faster $\hat f$ decays, and the faster $f$ decays, the smoother $\hat f$ is.
This duality is the reason the Schwartz space is mapped onto itself.
</document>

<document source="https://tamnd.github.io/math2/node/HHNWHS1XTN.md" tag="HHNWHS1XTN" kind="concept">
# Free product of algebras

## Summary

The free product of a family of algebras in a [[variety|AJNZJD8442]] is the most general algebra of the variety generated by images of the members.
It is the coproduct in the category of algebras of the variety.
The [[free product of groups|TM7QS7P7X9]] and the free product of monoids are the best known cases.

## Explanations

### L1 human

To form a free product, put the algebras together and let their elements combine with each other only as the laws of the variety demand.
No other relation between elements of different factors is added.
In the variety of groups, the free product of two copies of $\Z$ is the free group on two generators.
In the variety of abelian groups, the free product of $\Z$ and $\Z$ is $\Z \times \Z$, because the laws of the variety force the two generators to commute.
</document>

<document source="https://tamnd.github.io/math2/node/HHR8HP3R6G.md" tag="HHR8HP3R6G" kind="statement">
# Variational principle for topological entropy

## Statement

Let $X$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
Let $\mathcal{M}(f)$ be the set of $f$-[[invariant|B232EZ1FMB]] Borel [[probability measures|059TJN0NZ1]] on $X$.
For $\mu \in \mathcal{M}(f)$, let $h_\mu(f)$ be the [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] of $f$ with respect to $\mu$.
Then the [[topological entropy|BDZYBXBFDE]] satisfies
$$h_{\mathrm{top}}(f) = \sup_{\mu \in \mathcal{M}(f)} h_\mu(f).$$
The supremum may be taken over the ergodic measures in $\mathcal{M}(f)$ only.

## Remarks

The set $\mathcal{M}(f)$ is nonempty by the [[Krylov–Bogolyubov theorem|1DHZ81S0YE]].
The inequality $h_\mu(f) \le h_{\mathrm{top}}(f)$ follows by comparing a finite measurable partition with an open cover, and Misiurewicz gave a short proof of the reverse inequality.
For the reverse inequality one builds, from large $(n, \varepsilon)$-separated sets, averaged measures whose weak-$\ast$ limit is invariant and has entropy close to $h_{\mathrm{top}}(f)$.
The restriction to ergodic measures follows from the ergodic decomposition, since $h_\mu$ is the average of the entropies of the ergodic components.
Example: for the full shift on $k$ symbols, $h_{\mathrm{top}} = \log k$, and the Bernoulli measure giving each symbol mass $1/k$ has entropy $\log k$, so the supremum is attained.
The supremum is not always attained; a measure that attains it is called a measure of maximal entropy.

## Explanations

### L1 human

There are two ways to measure how chaotic a system is.
Topological entropy counts distinguishable orbits, while measure-theoretic entropy measures the unpredictability of typical orbits for a given statistics.
The variational principle says that the first is the best possible value of the second.

### L2 human

The theorem lets one compute topological entropy from a well-chosen invariant measure, and conversely bounds all measure-theoretic entropies by a topological quantity.
It is the starting point of the thermodynamic formalism, where entropy is replaced by pressure and the maximizing measures become equilibrium states.
Measures of maximal entropy exist when $\mu \mapsto h_\mu(f)$ is upper semicontinuous, for example for expansive homeomorphisms.
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# Taylor term

## Definition

Let $\mathcal{V}$ be a [[variety of algebras|AJNZJD8442]] and let $n \ge 2$.
A *Taylor term* for $\mathcal{V}$ is an $n$-ary [[term|N9M2N27END]] $t(x_1, \dots, x_n)$ such that:

1. $t$ is [[idempotent|9FS1JFZSX4]], that is, $\mathcal{V}$ satisfies $t(x, x, \dots, x) \approx x$.
2. For each $i \in \set{1, \dots, n}$, $\mathcal{V}$ satisfies an [[identity|PKD440G2XW]] $t(u_1, \dots, u_n) \approx t(v_1, \dots, v_n)$ with $u_i = x$ and $v_i = y$.
   Here every $u_j$ and every $v_j$ is one of the two variables $x$ and $y$.

An algebra has a Taylor term if the variety it generates has one.

## Remarks

Example: a semilattice term $x \wedge y$ is a Taylor term of the variety of semilattices.
Check: $x \wedge x \approx x$, and the identity $x \wedge y \approx y \wedge x$ has $x$ on the left and $y$ on the right in place $1$.
Swapping the names of $x$ and $y$ in it gives the required identity for place $2$.
Non-example: in the variety of sets with no operations, the term $x_1$ is idempotent but fails clause 2 for $i = 1$.
An identity with $x_1$ on both sides reads $u_1 \approx v_1$, which is false for $u_1 = x$ and $v_1 = y$.
Every [[majority term|9879ZMW5AA]] and every [[Mal'cev term|Y203PPTS61]] is a Taylor term.
For a majority term $m$, the identity $m(x, x, y) \approx m(y, x, x)$ serves places $1$ and $3$, and $m(x, y, x) \approx m(x, x, y)$ handles place $2$ after renaming.
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<document source="https://tamnd.github.io/math2/node/HJ2QSEMHMC.md" tag="HJ2QSEMHMC" kind="definition">
# LF-space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a vector space over $\mathbb{K}$.
Suppose $E = \bigcup_{n \ge 1} E_n$, where $E_1 \subseteq E_2 \subseteq \cdots$ are linear subspaces.
Suppose each $E_n$ carries a topology making it a [[Fréchet space|3GAA614673]], and each inclusion $E_n \to E_{n+1}$ is continuous.
The *inductive limit topology* on $E$ is the locally convex topology defined as follows.
A set is a [[neighbourhood|0ZCY0Z80B6]] of $0$ if it contains a [[convex|02AS45T3H9]] [[balanced|2TDSHJJYS4]] set $U$ with the following property.
For every $n$, the set $U \cap E_n$ is a neighbourhood of $0$ in $E_n$.
A [[locally convex space|B6GY2KJMWW]] is an *LF-space* if it is Hausdorff and arises in this way.

## Remarks

The inductive limit topology is the finest locally convex topology on $E$ for which all inclusions $E_n \to E$ are continuous.
Many authors require the limit to be *strict*: each $E_n$ carries the topology induced by $E_{n+1}$.
Consider a strict limit with each $E_n$ closed in $E_{n+1}$.
Then $E$ is Hausdorff and complete, and it induces the given topology on each $E_n$.
A set is bounded in $E$ if and only if it lies in some $E_n$ and is bounded there.
Example: let $\Omega \subseteq \R^d$ be open, and let $K_1 \subseteq K_2 \subseteq \cdots$ be compact sets with union $\Omega$, each inside the interior of the next.
Let $\mathcal{D}_{K_n}$ be the Fréchet space of smooth functions supported in $K_n$.
Then $C_c^\infty(\Omega) = \bigcup_n \mathcal{D}_{K_n}$ is a strict LF-space.
Non-example: the space $c_{00}$ of finitely supported sequences with the sup norm is not an LF-space, since every LF-space is barrelled and this space is not.
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# Sumudu transform

## Definition

Let $f \colon [0, \infty) \to \C$ be continuous, and assume that $f$ is of [[exponential order|TKZAV3KR86]]: there are constants $M > 0$ and $c \ge 0$ with $\abs{f(t)} \le M e^{ct}$ for all $t \ge 0$.
The *Sumudu transform* of $f$ is the function $G$ given by
$$G(u) = \int_0^\infty f(ut)\, e^{-t}\, dt$$
for $0 \le u < 1/c$, and for all $u \ge 0$ when $c = 0$.

## Remarks

The integral converges absolutely, since $\abs{f(ut)} e^{-t} \le M e^{-(1 - cu) t}$ and $cu < 1$.
At $u = 0$ one gets $G(0) = f(0)$.
For $u > 0$ the substitution $s = ut$ gives $G(u) = \frac{1}{u} \int_0^\infty f(s) e^{-s/u}\, ds = \frac{1}{u} F\big(\frac{1}{u}\big)$, where $F$ is the [[Laplace transform|0V49JEZW06]] of $f$.
So every rule for the Laplace transform has a Sumudu counterpart; for example, the transform of $f'$ is $\frac{G(u) - f(0)}{u}$ when $f'$ is continuous and of exponential order.

Example: $f(t) = e^{at}$ with $a \in \R$.
Then $G(u) = \int_0^\infty e^{-(1 - au) t}\, dt = \frac{1}{1 - au}$ for $0 \le u$ and $au < 1$.
Also $f(t) = t^n$ gives $G(u) = u^n \int_0^\infty t^n e^{-t}\, dt = n!\, u^n$.

Non-example: $f(t) = e^{t^2}$ is not of exponential order, and $\int_0^\infty e^{u^2 t^2 - t}\, dt$ diverges for every $u > 0$.
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# Jordan–Dedekind chain condition

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
For $x \le y$ in $P$, let $[x, y] = \set{z \in P : x \le z \le y}$.
A finite [[chain|0S8FE7ZMFM]] $z_0 < z_1 < \cdots < z_n$ has length $n$.
The poset $P$ satisfies the *Jordan–Dedekind chain condition* when the following holds for all $x \le y$ in $P$.
All [[maximal chains|3AX549FFFM]] of $[x, y]$ are finite and have the same length.

## Remarks

Example: the positive divisors of $12$ under divisibility satisfy the condition.
A maximal chain of $[a, b]$ multiplies by one prime at each step, so its length is the number of prime factors of $b / a$ counted with multiplicity.
Non-example: the pentagon with $0 < a < b < 1$ and $0 < c < 1$ fails it, since $0 < a < b < 1$ and $0 < c < 1$ are maximal chains of $[0, 1]$ with lengths $3$ and $2$.
Let $P$ be finite with a least element $\hat{0}$.
Then $P$ satisfies the condition if and only if $P$ has a rank function.
In that case, setting $\rho(x)$ equal to the length of any maximal chain of $[\hat{0}, x]$ gives a rank function.
Every modular lattice of finite length satisfies the condition.
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# Fraïssé limit

## Definition

Let $L$ be a countable signature.
The *age* of an [[$L$-structure|983GZBQYT4]] $M$ is the class of all finitely generated $L$-structures that embed into $M$.
Let $\mathcal{K}$ be a class of finitely generated $L$-structures, closed under isomorphism.
A *Fraïssé limit* of $\mathcal{K}$ is an $L$-structure $M$ such that:

1. $M$ is [[countable|01TQB88H4C]];
2. the age of $M$ is $\mathcal{K}$;
3. $M$ is ultrahomogeneous: every isomorphism between two finitely generated [[substructures|676H7EGTKJ]] of $M$ extends to an [[automorphism|0K1SCPBNKJ]] of $M$.

## Remarks

Fraïssé's theorem: suppose $\mathcal{K}$ is nonempty and has countably many isomorphism types.
Suppose it has the hereditary property, the joint embedding property and the [[amalgamation property|XSCSX39GEZ]].
Then $\mathcal{K}$ has a Fraïssé limit, and any two Fraïssé limits of $\mathcal{K}$ are isomorphic.
Conversely, the age of a countable ultrahomogeneous structure has these properties.

Example: $(\Q, <)$ is the Fraïssé limit of the class of finite linear orders.
Every finite linear order embeds in $\Q$, so clause 2 holds.
For clause 3, an order isomorphism between finite subsets of $\Q$ extends piecewise linearly to an order automorphism of $\Q$.

Non-example: $(\Z, <)$ is countable and its age is again the class of finite linear orders, but it fails clause 3.
The map $0 \mapsto 0$, $1 \mapsto 2$ is an isomorphism between the substructures $\set{0, 1}$ and $\set{0, 2}$.
It has no extension to an automorphism, because no integer lies strictly between $0$ and $1$, while $1$ lies strictly between $0$ and $2$.
</document>

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# Relative homotopy

## Definition

Let $X$ and $Y$ be topological spaces and let $A \subseteq X$.
Let $f, g \colon X \to Y$ be continuous maps with $f(a) = g(a)$ for all $a \in A$.
A *homotopy from $f$ to $g$ relative to $A$* is a [[homotopy|9YA6S6HD9V]] $H$ from $f$ to $g$ with $H(a, t) = f(a)$ for all $a \in A$ and $t \in [0, 1]$.
The maps $f$ and $g$ are *homotopic relative to $A$*, written $f \simeq g \ (\mathrm{rel}\ A)$, if such a homotopy exists.

## Remarks

For $A = \emptyset$ this is an ordinary homotopy.
Being homotopic relative to $A$ is an equivalence relation on the continuous maps $X \to Y$ that agree with a given map on $A$.
Example: for $X = [0, 1]$, $A = \set{0, 1}$, $Y = \R^2$, and two paths $f, g$ with $f(0) = g(0)$ and $f(1) = g(1)$, the map $H(s, t) = (1 - t) f(s) + t g(s)$ is a homotopy relative to $A$, since $H(0, t) = f(0)$ and $H(1, t) = f(1)$.
Non-example: in $\R^2 \setminus \set{0}$, the upper and lower half circles from $(1, 0)$ to $(-1, 0)$ are homotopic as maps, since $[0, 1]$ is contractible, but they are not homotopic relative to $\set{0, 1}$.
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# Minkowski plane

## Definition

A *Minkowski plane* is an [[incidence structure|FGKNAFNHA3]] $(\mathcal{P}, \mathcal{Z}, \in)$ of points and circles, where circles are subsets of $\mathcal{P}$.
It comes with two equivalence relations $\parallel_+$ and $\parallel_-$ on $\mathcal{P}$.
Two points are *parallel* if they are $(+)$-parallel or $(-)$-parallel.
The following conditions hold:

1. for any points $P, Q$ there is exactly one point $R$ with $P \parallel_+ R$ and $R \parallel_- Q$;
2. for every circle $z$, every point $P$ and each sign $\pm$, exactly one point of $z$ is $(\pm)$-parallel to $P$;
3. any three pairwise non-parallel points lie on exactly one circle;
4. for every circle $z$, every point $P \in z$ and every point $Q \notin z$ not parallel to $P$, there is exactly one circle $z'$ with $P, Q \in z'$ and $z \cap z' = \set{P}$;
5. every circle has at least three points, and there is a circle $z$ and a point not on $z$.

## Remarks

Example: let $L = \R \cup \set{\infty}$ be the real projective line.
The classical real Minkowski plane has points $L \times L$, with $(x_1, y_1) \parallel_+ (x_2, y_2)$ when $x_1 = x_2$ and $(x_1, y_1) \parallel_- (x_2, y_2)$ when $y_1 = y_2$.
Its circles are the graphs of the maps $x \mapsto (ax + b)/(cx + d)$ of $L$ with $ad - bc \ne 0$.
Clause 1 holds with $R = (x_1, y_2)$, and clause 2 holds because each circle is the graph of a bijection of $L$.
Clause 3 holds because for distinct $x_1, x_2, x_3$ and distinct $y_1, y_2, y_3$ there is exactly one such map with $x_i \mapsto y_i$.

Non-example: the same points and parallelisms with the graphs of all bijections of $L$ as circles fail clause 3, since many bijections take three given values at three given points.

Equivalently, the classical plane comes from a hyperboloid of one sheet.
Its circles are the plane sections that contain no line of the hyperboloid, and its two families of lines give the parallel classes.
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# Diagram

## Summary

A diagram of shape $J$ in a [[category|A2DE66937D]] $\mathcal{C}$ is a [[functor|28NQ0KRHD3]] from an index category $J$ to $\mathcal{C}$.
It picks out a family of objects and morphisms of $\mathcal{C}$, arranged in the pattern given by $J$, with the compositions that $J$ prescribes.
Diagrams are the input of limits and colimits: products, equalizers, pullbacks and their duals are limits and colimits of diagrams of particular shapes.

## Explanations

### L1 human

A diagram is a picture of objects and arrows drawn inside a category, following a fixed template.
The template $J$ says how many objects there are and which arrows join them, and the functor says which actual objects and arrows fill the template.
For example, the template with two objects and two parallel arrows $\bullet \rightrightarrows \bullet$ picks out a pair of maps $f, g \colon X \to Y$.
The template with two objects and no arrows between them picks out just a pair of objects.
</document>

<document source="https://tamnd.github.io/math2/node/HJPPRZ29YE.md" tag="HJPPRZ29YE" kind="concept">
# Stochastic ordering

## Summary

A stochastic ordering compares two random variables through their distributions, not through their values.
The basic one is the usual stochastic order.
In it, $X$ is stochastically smaller than $Y$ when $X$ is less likely than $Y$ to exceed every threshold.
Equivalently, there is a [[coupling|HTR7J0W66V]] of the two laws in which $X \le Y$ holds pointwise.
Other orders, such as the convex order and the hazard rate order, compare variability or ageing.
They are used in reliability theory, queueing, economics and decision theory.

## Explanations

### L1 human

One lottery is stochastically better than another if, for every amount of money, it is at least as likely to pay that amount or more.
For example, a waiting time with an [[exponential|0E99F51GVB]] law of rate $2$ is stochastically smaller than one with rate $1$, because $e^{-2t} \le e^{-t}$ for all $t \ge 0$.
Two bell curves centred at $0$ with different spreads are not comparable in this order.
Each one is more likely than the other to exceed some thresholds.
</document>

<document source="https://tamnd.github.io/math2/node/HJVM390354.md" tag="HJVM390354" kind="concept">
# Hidden Markov model

## Summary

A hidden Markov model is a pair of processes: a [[Markov chain|0G48PYM96Q]] of hidden states and a sequence of observations.
Each observation depends only on the hidden state at the same time.
Only the observations are seen, and the task is to infer the hidden states or the parameters of the model.
Hidden Markov models are used in speech recognition, bioinformatics and signal processing, with efficient algorithms such as the forward algorithm and the Viterbi algorithm.

## Explanations

### L1 human

Think of a casino that secretly switches between a fair die and a loaded die from time to time.
The switching follows a Markov chain, but one sees only the numbers that come up.
From a long run of sixes, one can guess that the loaded die is probably in use.
A hidden Markov model is the precise version of this situation: hidden states that evolve at random, and noisy clues about them.
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# Fraïssé limit

## Summary

A Fraïssé limit of a class of finitely generated structures is a countable structure that is as symmetric as possible.
Its finitely generated substructures are exactly the members of the class, up to isomorphism.
Here "as symmetric as possible" means ultrahomogeneous: every isomorphism between finitely generated substructures extends to an automorphism.
Fraïssé's theorem gives existence and uniqueness up to isomorphism under four conditions.
The class must have the hereditary, joint embedding and [[amalgamation|XSCSX39GEZ]] properties, and only countably many isomorphism types.
The rational numbers with their order and the [[Rado graph|RMXR0FEF54]] are the standard examples.

## Explanations

### L1 human

Start with a collection of small pieces, such as all finite linear orders or all finite graphs.
The Fraïssé limit is one countable object into which every piece fits, and in which any two copies of the same piece look alike from the outside.
For finite linear orders the limit is $(\Q, <)$.
Every finite order sits inside $\Q$, and any order-preserving map between two finite subsets of $\Q$ extends to an order-preserving bijection of all of $\Q$.
</document>

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# Berlekamp's algorithm

## Definition

Let $\F_q$ be the [[finite field|08RTDHX1SK]] with $q$ elements and let $f \in \F_q[x]$ be a [[monic|0BDX4MBKT4]] square-free polynomial of degree $n \ge 1$.
Let $V$ be the $\F_q$-vector space of polynomials of degree less than $n$.
The Berlekamp subalgebra of $f$ is
$$B = \set{g \in V : g^q \equiv g \pmod{f}}.$$
Berlekamp's algorithm factors $f$ as follows.

1. Compute the matrix $Q$ of the $\F_q$-linear map $V \to V$, $h \mapsto h^q \bmod f$, and a basis of $B$ as the [[null space|0PQ2GGY1QV]] of $Q - I$.
2. If $\dim B = 1$, then $f$ is irreducible.
3. Otherwise take $g \in B$ of positive degree; then $f = \prod_{s \in \F_q} \gcd(f, g - s)$, and at least two of these factors have positive degree.
4. Apply step 3 to the factors, with other elements of $B$, until there are $\dim B$ factors.

## Remarks

By the [[Chinese remainder theorem|0W97J8QED7]], $B \cong \F_q^k$, where $k$ is the number of irreducible factors of $f$, so $\dim B = k$.
Step 3 holds because $g^q - g = \prod_{s \in \F_q} (g - s)$ and $f$ divides $g^q - g$; the factors $g - s$ are pairwise coprime, and none of them is divisible by $f$.
Example: $f = x^2 + 1$ over $\F_5$; since $x^5 \equiv x \pmod{f}$, $B = V$ has dimension $2$, and $g = x$ gives $\gcd(f, x - 2) = x - 2$ and $\gcd(f, x - 3) = x - 3$.
Non-example: $f = x^2 + 1$ over $\F_3$; since $x^3 \equiv -x \pmod{f}$, $(a + bx)^3 \equiv a - bx$, so $B$ is the constants, $\dim B = 1$, and $f$ is irreducible.
The algorithm needs $f$ square-free; a repeated factor is first removed with $\gcd(f, f')$.
The running time is polynomial in $n$ and $q$, so it suits small fields; for large $q$ randomised variants are used.

## Explanations

### L1 human

The algorithm computes the $q$-th power map on remainders modulo $f$ and finds its fixed points by Gaussian elimination.
Each fixed point that is not a constant lets one split $f$ with greatest common divisors.
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# Convex cone

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]].
A subset $C \subseteq V$ is a *convex cone* if for all $x, y \in C$ and all real numbers $\alpha > 0$ and $\beta > 0$,
$$\alpha x + \beta y \in C.$$

## Remarks

Equivalently, $C$ is closed under addition and under multiplication by positive scalars.
Equivalently again, $C$ is [[convex|02AS45T3H9]] and $t x \in C$ for all $x \in C$ and $t > 0$.

Example: the nonnegative orthant $\R^n_{\ge 0} = \set{x \in \R^n : x_i \ge 0 \text{ for all } i}$.
If $x, y$ have nonnegative coordinates and $\alpha, \beta > 0$, then each coordinate $\alpha x_i + \beta y_i$ is nonnegative.

Non-example: the closed unit disc in $\R^2$ is convex, but $(1, 0)$ lies in it and $2 \cdot (1, 0) = (2, 0)$ does not.

Conventions differ: many authors also allow $\alpha = \beta = 0$, so that every nonempty convex cone contains $0$.
A convex cone that contains $0$ is called a pointed cone in some texts.
The same definition works over any ordered field in place of $\R$.
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# Cauchy filter

## Definition

Let $(X, \mathcal{U})$ be a [[uniform space|A54T86KS63]] and let $\mathcal{F}$ be a [[filter|15PF73BKY6]] on $X$.
The filter $\mathcal{F}$ is a *Cauchy filter* if for every [[entourage|FVFFKNKEDF]] $U \in \mathcal{U}$ there is a set $A \in \mathcal{F}$ with $A \times A \subseteq U$.
The uniform space is *complete* if every Cauchy filter on $X$ converges to a point of $X$.

## Remarks

The neighbourhood filter $\mathcal{N}_x$ of a point $x$ is a Cauchy filter.
Given an entourage $U$, choose a symmetric entourage $V$ with $V \circ V \subseteq U$ and let $A = V[x]$, a member of $\mathcal{N}_x$.
For $y, z \in A$ one has $(y, x) \in V$ and $(x, z) \in V$, so $(y, z) \in V \circ V \subseteq U$.
On $\R$ with its usual uniformity, the filter of sets that contain some interval $[n, \infty)$ is not a Cauchy filter.
For $U = \set{(x, y) : \abs{x - y} < 1}$, every member contains two points at distance $1$, so no member $A$ has $A \times A \subseteq U$.
In a metric space, a sequence is a [[Cauchy sequence|0PWX18FM4G]] if and only if the filter of its tails is a Cauchy filter.
Every convergent filter is a Cauchy filter.

## Explanations

### L1 human

A Cauchy filter is a filter with members as small as one likes, in the sense of the uniform structure.
It is the right replacement for Cauchy sequences in spaces where sequences are not enough.
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# Tomographic reconstruction

## Problem

Let $f$ be an unknown function on $\R^n$, $n \ge 2$, with compact support.
Given the [[Radon transform|VJNZJ352PC]] $R f$, or the X-ray transform of $f$, on a set of lines or hyperplanes, possibly with measurement errors, recover $f$.
In practice the data are values $R f(\theta_j, s_k)$ at finitely many directions $\theta_j$ and offsets $s_k$.
The task is then to compute a good approximation of $f$.

## Remarks

With complete and exact data in $\R^n$, the problem has a unique solution, given by the [[Radon inversion formula|HN93ST5CBT]]; Radon gave the inversion in the plane in 1917.
The problem is an [[inverse problem|PCMW7KN5P4]] that is not [[well-posed|W09B9NZ9B5]] in $L^2$: the inversion contains the filter $\abs{\sigma}^{n-1}$, which amplifies high-frequency noise, so some regularization is needed.
With finitely many directions, uniqueness fails: for any finite set of directions there is a nonzero smooth function with compact support whose projections in these directions all vanish.
The main algorithms are [[filtered back-projection|SJQDDY9FMX]], Fourier reconstruction based on the [[projection-slice theorem|J7VA040MC3]], and iterative algebraic methods such as the Kaczmarz method.
Limited-angle data and interior data lead to harder versions of the problem, with weaker stability.

## Explanations

### L1 human

A CT scanner sends X-rays through a body from many directions and measures how much each ray is weakened.
Tomographic reconstruction is the task of computing a picture of the inside of the body from these measurements.
Mathematically, one must recover a function from its integrals along lines.
It can be done exactly when all lines are known, but real data are finite and noisy, so the methods must balance sharpness against noise.
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# Projective resolution

## Summary

A projective resolution of a module $M$ is an [[exact sequence|JB7ZBFBA9C]] of [[projective modules|BFFA03GE3V]] that ends in $M$.
It replaces $M$ by a complex of simple pieces that has the same homological information.
Every module has a projective resolution, and the derived functors such as [[Ext|4P7GBWJ4RR]] and [[Tor|DDQRERAQCV]] are computed from it.

## Explanations

### L1 human

Write $M$ as a quotient of a free module, then describe the relations by another free module, then the relations among the relations, and so on.
The chain of free modules that you get is a free resolution, a special kind of projective resolution.
For example, $\Z/n\Z$ is the quotient of $\Z$ by the multiples of $n$, and there are no further relations.
So $0 \to \Z \xrightarrow{n} \Z \to \Z/n\Z \to 0$ is a free resolution of $\Z/n\Z$.
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# Lie product formula

## Statement

Let $n \ge 1$ and let $A$ and $B$ be $n \times n$ complex [[matrices|0JXZKX526R]].
Then
$$e^{A + B} = \lim_{m \to \infty} \left( e^{A/m} e^{B/m} \right)^m,$$
where $e^X$ is the [[matrix exponential|020M04FN1H]] and the [[limit|05M5SWSZZD]] is taken entry by entry.

## Remarks

Let $X_m = e^{A/m} e^{B/m}$ and $Y_m = e^{(A + B)/m}$.
Both equal $I + (A + B)/m + O(1/m^2)$, so $\norm{X_m - Y_m} = O(1/m^2)$.
The identity $X^m - Y^m = \sum_{k=0}^{m-1} X^k (X - Y) Y^{m-1-k}$ and the bound $\norm{X_m}, \norm{Y_m} \le e^{(\norm{A} + \norm{B})/m}$ give $\norm{X_m^m - Y_m^m} = O(1/m)$, and $Y_m^m = e^{A + B}$.
If $AB = BA$, then $e^{A/m} e^{B/m} = e^{(A + B)/m}$ and the formula holds for every $m$.
In general $e^{A + B} \ne e^A e^B$: for $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, $e^A e^B = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$, while $e^{A + B} = \begin{pmatrix} \cosh 1 & \sinh 1 \\ \sinh 1 & \cosh 1 \end{pmatrix}$.

## Explanations

### L1 human

For numbers, $e^{a + b} = e^a e^b$, but for matrices this fails when $A$ and $B$ do not commute.
The formula repairs this: alternate many tiny steps of $A$ and of $B$, and the result approaches $e^{A + B}$.
It is like walking diagonally by taking many small steps east and north in turn.

### L2 human

The formula is the basis of splitting methods: to solve $x' = (A + B)x$, one alternates the flows of $x' = Ax$ and $x' = Bx$, which may be easier to compute.
In a [[Lie group|7JBRGCVJRB]] it shows that the sum in the Lie algebra corresponds to a limit of products in the group.
Trotter extended it to certain unbounded operators on Hilbert spaces, where it is called the Trotter product formula.
It is a key step in the proof of the [[Golden–Thompson inequality|Z7R754JC7N]].
</document>

<document source="https://tamnd.github.io/math2/node/HK8339ECNN.md" tag="HK8339ECNN" kind="statement">
# Shannon–McMillan–Breiman theorem

## Statement

Let $(X, \mathcal{B}, \mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \colon X \to X$ be an [[ergodic|AZRPHS478B]] [[measure-preserving transformation|556WBDAHNP]].
Let $\mathcal{P}$ be a finite measurable [[partition|08QMKTWF9T]] of $X$.
Let $h(T, \mathcal{P})$ be the entropy of $T$ with respect to $\mathcal{P}$, as in the definition of the [[Kolmogorov–Sinai entropy|X1D1C7VH6P]].
For $n \ge 1$ and $x \in X$, let $\mathcal{P}_n(x)$ be the set of the partition $\bigvee_{i=0}^{n-1} T^{-i}\mathcal{P}$ that contains $x$.
Then
$$\lim_{n \to \infty} -\frac{1}{n} \log \mu(\mathcal{P}_n(x)) = h(T, \mathcal{P})$$
for [[almost every|RARTMA0ZEC]] $x \in X$, and also in $L^1(\mu)$.

## Remarks

The set $\mathcal{P}_n(x)$ consists of the points whose first $n$ positions $x, Tx, \dots, T^{n-1}x$ lie in the same sets of $\mathcal{P}$ as those of $x$.
The function $-\log \mu(\mathcal{P}_n(x))$ is the information carried by these $n$ symbols, and its integral is $H(\bigvee_{i=0}^{n-1} T^{-i}\mathcal{P})$.
So the convergence in $L^1$ refines the definition of $h(T, \mathcal{P})$, and the theorem says that the information per symbol is almost surely constant.
The proof writes the information as a Birkhoff sum of conditional information functions, plus an error, and applies Birkhoff's ergodic theorem together with the martingale convergence theorem.
Shannon proved convergence in probability for ergodic Markov sources, McMillan proved $L^1$ convergence for stationary ergodic sources, and Breiman proved almost sure convergence.
Example: for the Bernoulli scheme $B(p_1, \dots, p_k)$ with the partition by the zeroth coordinate, $\mu(\mathcal{P}_n(x)) = p_{x_0} \cdots p_{x_{n-1}}$, and the theorem is the strong law of large numbers for the i.i.d. variables $-\log p_{x_i}$, whose mean is $-\sum p_i \log p_i$.
Without ergodicity the limit still exists almost everywhere, but it is an invariant function whose integral is $h(T, \mathcal{P})$.

## Explanations

### L1 human

Watch a long stretch of output of a stationary random source, such as a long English text.
The theorem says that almost every long stretch of $n$ symbols has probability close to $e^{-nh}$, where $h$ is the entropy of the source.
So the likely messages all have about the same probability, and there are about $e^{nh}$ of them.

### L2 human

This is the asymptotic equipartition property, the basis of data compression: about $nh$ nats, or $nh / \log 2$ bits, suffice to encode a typical message of length $n$.
In ergodic theory it gives a local, pointwise meaning to the [[Kolmogorov–Sinai entropy|X1D1C7VH6P]].
It is a key tool in the proofs of the Ornstein isomorphism theorem and of Sinai's factor theorem, and it has analogues for actions of amenable groups.
</document>

<document source="https://tamnd.github.io/math2/node/HK8NS02NNE.md" tag="HK8NS02NNE" kind="concept">
# Homology sphere

## Summary

A homology $n$-sphere is a closed $n$-manifold with the same integral homology groups as the [[$n$-sphere|T62369RFH6]].
The first example that is not a sphere is the Poincaré homology sphere, a $3$-manifold with a nontrivial fundamental group of order $120$.
Homology spheres show that homology alone does not determine a manifold, and they were central in the history of the Poincaré conjecture.

## Explanations

### L1 human

Homology counts holes of each dimension.
A homology sphere has exactly the holes of an ordinary sphere and no others.
Still, it can differ from the sphere: it may contain loops that cannot be shrunk to a point, though each such loop has trivial homology class.
The Poincaré homology sphere, obtained by gluing opposite faces of a solid dodecahedron with a twist, is the classical example.
</document>

<document source="https://tamnd.github.io/math2/node/HKB6KES6PK.md" tag="HKB6KES6PK" kind="definition">
# Kernel of an integral transform

## Definition

Let $I$ and $J$ be intervals of $\R$, and let $T$ be an [[integral transform|YCK709QB06]] given by
$$(Tf)(u) = \int_I f(t)\, K(t, u)\, dt \quad (u \in J)$$
for a measurable function $K \colon I \times J \to \C$.
The function $K$ is the *kernel* (or *nucleus*) of $T$.

## Remarks

Example: the Laplace transform $F(s) = \int_0^\infty f(t) e^{-st}\, dt$ has kernel $K(t, s) = e^{-st}$ on $[0, \infty) \times (0, \infty)$; this is read off by comparing with the formula above.
Non-example: the function $f$ itself is not a kernel of $T$; the kernel does not depend on the input $f$, only on the transform.
Suppose the transform is defined on all bounded functions with compact support.
Then the kernel is determined by the transform up to changes on a set of measure zero in $I \times J$.
A kernel is *symmetric* when $I = J$ and $K(t, u) = K(u, t)$.
</document>

<document source="https://tamnd.github.io/math2/node/HKPHRSN3ER.md" tag="HKPHRSN3ER" kind="definition">
# Legendre transformation

## Definition

Let $I$ be an open [[interval|0TK8Y3J5RW]] of $\R$, and let $f \colon I \to \R$ be a [[differentiable function|0A61B60B5R]] whose derivative $f'$ is [[strictly increasing|011EJQWWJS]] on $I$.
Let $J = f'(I)$; it is an interval, and $f' \colon I \to J$ is a bijection with inverse $g \colon J \to I$.
The *Legendre transform* of $f$ is the function $f^* \colon J \to \R$ given by
$$f^*(p) = p \, g(p) - f(g(p)).$$
The map $f \mapsto f^*$ is the *Legendre transformation*.

## Remarks

For $p \in J$ one has $f^*(p) = \sup_{x \in I} \big(p x - f(x)\big)$, and the [[supremum|0BTMQ04556]] is attained at $x = g(p)$.
Indeed $x \mapsto p x - f(x)$ has derivative $p - f'(x)$, which is positive for $x < g(p)$ and negative for $x > g(p)$.
The function $f^*$ is differentiable with $(f^*)' = g = (f')^{-1}$, and $(f^*)^* = f$.
The supremum formula, taken over all $x$, defines the convex conjugate of any function, which extends this definition.

Example: $f(x) = e^x$ on $\R$.
Here $f'(x) = e^x$ is strictly increasing, $J = (0, \infty)$ and $g(p) = \ln p$.
So $f^*(p) = p \ln p - p$ for $p > 0$.

Non-example: $f(x) = x^3$ on $\R$.
Its derivative $3x^2$ is not strictly increasing, since $3 \cdot (-1)^2 = 3 \cdot 1^2$, so the definition does not apply.
</document>

<document source="https://tamnd.github.io/math2/node/HKRN90DCWK.md" tag="HKRN90DCWK" kind="statement">
# Kolmogorov criterion for best approximation

## Statement

Let $K$ be a [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]], and let $C(K)$ be the space of continuous functions $K \to \R$ with the supremum norm.
Let $V \subseteq C(K)$ be a [[linear subspace|00YC1D652K]], let $f \in C(K)$ and $p^* \in V$, and let
$$E = \set{x \in K : \abs{f(x) - p^*(x)} = \norm{f - p^*}}$$
be the set of extreme points of the error.
Then $p^*$ is a [[best approximation|7N4988HN3P]] of $f$ from $V$ if and only if
$$\min_{x \in E} \big( f(x) - p^*(x) \big) \, p(x) \le 0 \quad \text{for every } p \in V.$$

## Remarks

The idea of the proof: suppose some $p \in V$ has $(f - p^*) p > 0$ on $E$.
By compactness this product is positive on an open set $U \supseteq E$, and $\abs{f - p^*}$ stays below $\norm{f - p^*} - \eta$ for some $\eta > 0$ off $U$.
Then $\norm{f - p^* - t p} < \norm{f - p^*}$ for small $t > 0$, so $p^*$ is not best.
Conversely, if $q \in V$ has $\norm{f - q} < \norm{f - p^*}$, then $p = q - p^*$ satisfies $(f - p^*) p = (f - p^*)^2 - (f - p^*)(f - q) > 0$ on $E$.

Example: for $f(x) = x^2$ on $[0, 1]$, $V$ the lines and $p^*(x) = x - 1/8$, the set $E$ is $\set{0, 1/2, 1}$ with error signs $+, -, +$.
A line $p$ with $p(0) > 0$, $p(1/2) < 0$ and $p(1) > 0$ would have two zeros, so it does not exist, and the criterion shows $p^*$ is best.

For complex-valued functions the same criterion holds with $\operatorname{Re} \big( \overline{(f(x) - p^*(x))} \, p(x) \big)$ in place of the product.

## Explanations

### L1 human

A candidate approximation is best unless some correction pushes in the right direction at every point where the error is worst.
The criterion checks exactly this: there must be no $p$ that has the same sign as the error at all the worst points.

### L2 human

The criterion holds for every linear subspace, not only for polynomials or Haar subspaces.
For Haar subspaces it turns into the alternation property of the equioscillation theorem, because a function of a Haar subspace can follow any sign pattern with few sign changes.
It is a form of the first-order optimality condition for the convex function $p \mapsto \norm{f - p}$.
Similar criteria hold in other normed spaces with the norm's subdifferential in place of the product.
</document>

<document source="https://tamnd.github.io/math2/node/HKZABKRPN6.md" tag="HKZABKRPN6" kind="concept">
# Cohomology of Lie groups

## Summary

The cohomology of a [[Lie group|7JBRGCVJRB]] $G$ is the [[singular cohomology|6N7BC70CX0]] of its underlying manifold, with the extra structure given by the group multiplication.
The multiplication makes it a Hopf algebra, which strongly restricts its form.
For a compact connected group, real cohomology can be computed from invariant differential forms, that is, from the cohomology of the Lie algebra.
This turns a topological question into a question of linear algebra.

## Explanations

### L1 human

Cohomology counts the holes of a space in each dimension.
A Lie group is a space, so one can ask about its holes.
The circle has one hole of dimension $1$, and the group $\mathrm{SU}(2)$, a $3$-sphere, has one hole of dimension $3$.
Because a Lie group looks the same at every point, its holes can be detected by forms that are invariant under the group.
Those forms are determined by their values at the identity, where they become algebra on the Lie algebra.
</document>

<document source="https://tamnd.github.io/math2/node/HM23BV561Z.md" tag="HM23BV561Z" kind="statement">
# Thom's cobordism theorem

## Statement

Let $\Omega_n^O$ be the group of [[cobordism|T8BVGPPSZP]] classes of closed smooth $n$-manifolds.
Let $\Omega_*^O = \bigoplus_{n \ge 0} \Omega_n^O$ be the unoriented [[cobordism ring|C02WQY819R]], with product induced by the Cartesian product of manifolds.
For $k \ge 1$, let $MO(k)$ be the [[Thom space|5MNYT3AFPH]] of the [[universal bundle|1HDH716Z24]] $\gamma^k$ of rank $k$ over $BO(k)$.

1. For $k \ge n + 2$, there is an isomorphism $\Omega_n^O \cong \pi_{n+k}(MO(k))$ with the [[homotopy group|P976WYEFMB]] of $MO(k)$.
2. The ring $\Omega_*^O$ is a polynomial ring over $\Z/2\Z$ with one generator $x_i$ in each degree $i \ge 2$ that is not of the form $2^j - 1$:
$$\Omega_*^O \cong (\Z/2\Z)[x_2, x_4, x_5, x_6, x_8, x_9, \ldots].$$
3. Two closed smooth $n$-manifolds are cobordant if and only if all their [[Stiefel–Whitney numbers|JQCK7A210W]] agree.

## Remarks

The isomorphism in part 1 is the Pontryagin–Thom construction.
A closed $n$-manifold is embedded in $\R^{n+k}$, and a tubular neighbourhood of it is mapped to the Thom space of its normal bundle.
Collapsing the rest of the sphere $S^{n+k}$ gives an element of $\pi_{n+k}(MO(k))$, and transversality gives the inverse.
Parts 2 and 3 then follow from a computation of the homotopy of $MO(k)$ in the stable range by mod $2$ cohomology and Steenrod squares.

Example: in low degrees, $\Omega_1^O = 0$, $\Omega_2^O \cong \Z/2\Z$ generated by $\R P^2$, $\Omega_3^O = 0$ and $\Omega_4^O \cong (\Z/2\Z)^2$, spanned by the classes $x_2^2$ and $x_4$.
For even $i$, the generator $x_i$ can be taken to be $\R P^i$.

The result was proved by René Thom in 1954.

## Explanations

### L1 human

Deciding whether a manifold is the boundary of another manifold sounds like a hard geometric problem.
Thom's theorem turns it into a computation in homotopy theory, and then solves it completely.
In the end, whether a manifold bounds is decided by a finite list of numbers modulo $2$, its Stiefel–Whitney numbers.

### L2 human

The Pontryagin–Thom construction is the model for many later results that turn geometric questions about manifolds into stable homotopy groups of Thom spectra.
The same method computes the oriented cobordism ring through $MSO(k)$, and complex cobordism through $MU(k)$, whose homotopy is described by [[Quillen's theorem|4E518M24MA]].
Part 3 combines Pontryagin's observation that cobordant manifolds have the same Stiefel–Whitney numbers with Thom's converse.
</document>

<document source="https://tamnd.github.io/math2/node/HM2HHE3J51.md" tag="HM2HHE3J51" kind="concept">
# Hilbert space

## Summary

A Hilbert space is an [[inner product space|0QVNCJXRY3]] that is complete for the [[norm induced by its inner product|0MFJJV97RJ]].
So it is a [[Banach space|06GZ956657]] whose geometry has angles and orthogonality, as in Euclidean space.
Every Hilbert space has an orthonormal basis, and every closed subspace has an orthogonal complement.
Every continuous linear functional is the inner product with a fixed vector.
Hilbert spaces are the setting of Fourier series, of spectral theory and of quantum mechanics.

## Explanations

### L1 human

A Hilbert space is a space of vectors, possibly of infinite dimension, where lengths and angles work as in the plane.
The theorem of Pythagoras and orthogonal projections still make sense.
Completeness means that a sequence whose terms get closer and closer always has a limit in the space.
The main example is $\ell^2$, the sequences $(x_1, x_2, \dots)$ with $\sum \abs{x_n}^2$ finite.
Another is the space $L^2$ of square-integrable functions, where Fourier series live.
</document>

<document source="https://tamnd.github.io/math2/node/HM74NCW5JX.md" tag="HM74NCW5JX" kind="definition">
# Prony's method

## Definition

Let $n \ge 1$, and suppose the numbers $f(0), f(1), \dots, f(2n - 1)$ are samples of $f(k) = \sum_{j=1}^{n} c_j z_j^k$, where $z_1, \dots, z_n$ are distinct nonzero complex numbers and $c_1, \dots, c_n$ are nonzero complex numbers.
*Prony's method* recovers the $z_j$ and $c_j$ in three steps.

1. Solve the [[linear system|0JTRMBQ1SA]] $\sum_{m=0}^{n-1} a_m f(k + m) = -f(k + n)$ for $k = 0, \dots, n - 1$ for $a_0, \dots, a_{n-1}$.
2. Compute the [[roots|019MA1VTR2]] $z_1, \dots, z_n$ of the polynomial $P(z) = z^n + a_{n-1} z^{n-1} + \cdots + a_0$.
3. Solve the Vandermonde system $\sum_{j=1}^{n} c_j z_j^k = f(k)$ for $k = 0, \dots, n - 1$ for $c_1, \dots, c_n$.

## Remarks

The coefficients of $P(z) = \prod_j (z - z_j)$ satisfy step 1, since $f$ obeys the [[linear recurrence|0NDVHBVBXB]] with characteristic polynomial $P$.
The matrix $H = (f(k + m))_{k, m = 0}^{n-1}$ of step 1 is a [[Hankel matrix|PQ1D8FQB3E]].
It factors as $H = V^T C V$ with the Vandermonde matrix $V = (z_j^m)$ and $C = \operatorname{diag}(c_j)$.
So $H$ is invertible under the hypotheses, and step 1 has a unique solution.
Example: samples $2, 3, 5, 9$ with $n = 2$ give $2 a_0 + 3 a_1 = -5$ and $3 a_0 + 5 a_1 = -9$, so $a_0 = 2$, $a_1 = -3$, $P(z) = (z - 1)(z - 2)$, and then $c_1 = c_2 = 1$: $f(k) = 1 + 2^k$.
Non-example: $f(k) = k$ is not such a sum with $n = 1$; the step 1 equation $a_0 f(0) = -f(1)$ reads $0 = -1$ and has no solution.
With noisy data or more samples, one solves steps 1 and 3 in the least squares sense.
The method is then sensitive to noise, and variants such as the matrix pencil method improve it.
</document>

<document source="https://tamnd.github.io/math2/node/HM9HBVSM54.md" tag="HM9HBVSM54" kind="definition">
# Cluster algebra

## Definition

This is the coefficient-free, skew-symmetrizable version.
Let $n \ge 1$ and let $\mathcal{F} = \Q(u_1, \dots, u_n)$ be the field of [[rational functions|0XDK9CGYKJ]] in $n$ indeterminates.
An integer $n \times n$ [[matrix|0VD9ZSWWJG]] $B = (b_{ij})$ is *skew-symmetrizable* if there are positive integers $d_1, \dots, d_n$ with $d_i b_{ij} = -d_j b_{ji}$ for all $i, j$.
A *seed* is a pair $(\mathbf{x}, B)$, where $\mathbf{x} = (x_1, \dots, x_n)$ is a free generating set of $\mathcal{F}$ over $\Q$ and $B$ is skew-symmetrizable.
For $k \in \set{1, \dots, n}$, the *mutation* of $(\mathbf{x}, B)$ at $k$ is the seed $(\mathbf{x}', B')$ given as follows, where $[a]_+ = \max(a, 0)$:

1. $x'_i = x_i$ for $i \ne k$, and $x'_k x_k = \prod_{i} x_i^{[b_{ik}]_+} + \prod_{i} x_i^{[-b_{ik}]_+}$.
2. $b'_{ij} = -b_{ij}$ if $i = k$ or $j = k$.
3. $b'_{ij} = b_{ij} + \tfrac{1}{2}\left(\abs{b_{ik}} b_{kj} + b_{ik} \abs{b_{kj}}\right)$ otherwise.

Fix an initial seed $(\mathbf{x}, B)$.
The *cluster variables* are the entries of all seeds reached from it by finite sequences of mutations, and each such $\mathbf{x}'$ is a *cluster*.
The *cluster algebra* $\mathcal{A}(B)$ is the subring of $\mathcal{F}$ generated by all cluster variables.

## Remarks

Example: $n = 2$ and $B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, which is skew-symmetric.
Mutation at $1$ gives $x'_1 = (1 + x_2)/x_1$, since $b_{21} = -1$ and $b_{11} = 0$.
Alternating mutations give the cluster variables $x_1$, $x_2$, $(x_2 + 1)/x_1$, $(x_1 + x_2 + 1)/(x_1 x_2)$, $(x_1 + 1)/x_2$, five in all.
This is the cluster algebra of type $A_2$.
Non-example: $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is not skew-symmetrizable, because $d_1 b_{12} = -d_2 b_{21}$ would give $d_1 = -d_2$ with $d_1, d_2 > 0$.
Mutation is an involution: mutating twice at the same $k$ returns the original seed.
By the Laurent phenomenon of Fomin and Zelevinsky, every cluster variable is a Laurent polynomial with integer coefficients in the initial cluster.
The general definition also allows coefficients from a semifield.
</document>

<document source="https://tamnd.github.io/math2/node/HMC733EFMD.md" tag="HMC733EFMD" kind="statement">
# Abel–Plana formula

## Statement

Let $f$ be continuous on the closed half-plane $\set{z \in \C : \operatorname{Re} z \ge 0}$ and [[holomorphic|0NAHXR1FRH]] on the open half-plane $\operatorname{Re} z > 0$.
Assume there are constants $C > 0$ and $\varepsilon > 0$ with $\abs{f(z)} \le C / \abs{z}^{1 + \varepsilon}$ for all $z \ne 0$ with $\operatorname{Re} z \ge 0$.
Then
$$\sum_{n=0}^{\infty} f(n) = \int_0^\infty f(x)\,dx + \frac{f(0)}{2} + i \int_0^\infty \frac{f(it) - f(-it)}{e^{2\pi t} - 1}\,dt.$$

## Remarks

Proof idea: the kernels $1/(e^{\mp 2\pi i z} - 1)$ have simple poles at the integers.
Apply the residue theorem to $f$ times such a kernel on large rectangles in the upper and the lower quarter planes, with one side on the imaginary axis.
The residues at the integers produce the sum, the decay of $f$ kills the far sides, and the sides on the imaginary axis give the last integral.

Example: $f(z) = 1/(z + 1)^2$ satisfies the hypotheses with $\varepsilon = 1$, since $\abs{z + 1} \ge \abs{z}$ when $\operatorname{Re} z \ge 0$.
Here $f(it) - f(-it) = -4it/(1 + t^2)^2$, so the formula reads
$$\frac{\pi^2}{6} = 1 + \frac12 + 4 \int_0^\infty \frac{t}{(1 + t^2)^2 (e^{2\pi t} - 1)}\,dt.$$
Applied to $f(z) = (z + a)^{-s}$ it gives an integral formula for the [[Hurwitz zeta function|G4SW7HJG88]].

Some decay or growth condition is needed for both sides to make sense: for $f = 1$ the series and the integral $\int_0^\infty f$ both diverge.

## Explanations

### L1 human

The formula compares a sum $f(0) + f(1) + f(2) + \cdots$ with the integral $\int_0^\infty f$.
The difference is half of the first term plus a correction that only uses the values of $f$ on the imaginary axis.
This correction is often small and easy to estimate, because of the rapidly growing factor $e^{2\pi t}$.
So the formula turns hard sums into integrals.

### L2 human

The Abel–Plana formula is an exact alternative to the [[Euler–Maclaurin formula|9P839VB8S7]] for functions that extend holomorphically to the right half-plane.
Euler–Maclaurin gives an asymptotic series with Bernoulli numbers, while Abel–Plana gives one convergent integral.
Expanding $f(it) - f(-it)$ in powers of $t$ and integrating term by term recovers the Euler–Maclaurin terms, since $\int_0^\infty t^{2k-1} / (e^{2\pi t} - 1)\,dt$ is a multiple of a Bernoulli number.
It is a standard tool for the zeta functions and for the analytic continuation of series in physics, such as the computation of Casimir energies.
</document>

<document source="https://tamnd.github.io/math2/node/HMCD0JBFST.md" tag="HMCD0JBFST" kind="concept">
# Stein manifold

## Summary

A Stein manifold is a complex manifold with enough global holomorphic functions: they separate points and give compact holomorphically convex hulls of compact sets.
Stein manifolds are the complex analogue of affine varieties, and they are exactly the closed complex submanifolds of some $\C^N$.
On them the main problems of several complex variables, such as the Cousin problems, are governed by topology, through Cartan's theorems A and B.

## Explanations

### L1 human

A compact complex manifold has no nonconstant global holomorphic functions, so global function theory there is empty.
Stein manifolds are at the opposite end: global holomorphic functions are plentiful, as on $\C^n$ itself.
Examples are $\C^n$, every domain of holomorphy in $\C^n$, every open Riemann surface, and every closed complex submanifold of $\C^n$, such as the curve $\set{(z, w) \in \C^2 : w^2 = z^3 - z}$.
They are the natural spaces on which to do holomorphic function theory in several variables.
</document>

<document source="https://tamnd.github.io/math2/node/HMDGNS1V2T.md" tag="HMDGNS1V2T" kind="concept">
# Fixed-point property of posets

## Summary

A [[partially ordered set|0QV2JS9RJ3]] has the fixed-point property if every order-preserving map from it to itself fixes some element.
It is the order-theoretic counterpart of the [[fixed-point property of topological spaces|JQDHHFXBF3]].
Every [[complete lattice|D7A0ZGZT5S]] has it by the [[Knaster–Tarski theorem|RHXP3HA64M]], and among lattices exactly the complete ones have it.
Deciding which finite posets have the fixed-point property is a hard combinatorial question.

## Explanations

### L1 human

Move the elements of an ordering around without ever reversing a comparison.
The ordering has the fixed-point property if, however one does this, some element stays in place.
Any finite chain has it: if the smallest element moves, it moves up, and following the moves upward one must stop somewhere.
An ordering of two incomparable elements does not have it, since swapping them reverses no comparison and moves both.
</document>

<document source="https://tamnd.github.io/math2/node/HMF8G4EBDX.md" tag="HMF8G4EBDX" kind="definition">
# Transcendental extension

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
The extension $L/K$ is *transcendental* if it is not [[algebraic|09EYMKMSB9]], that is, if some element of $L$ is [[transcendental|00SW8B91E7]] over $K$.

## Remarks

Example: $\R/\Q$ is transcendental, because $\pi$ is a [[transcendental number|079EGSD32Y]].
Example: for a field $K$, the extension $K(t)/K$ of rational functions is transcendental, since $g(t) \ne 0$ for every nonzero $g \in K[x]$.
Non-example: $\Q(\sqrt 2)/\Q$ is not transcendental, because it is finite and hence algebraic.
Every field extension $L/K$ is the composite of a purely transcendental step $K \subseteq K(B)$ and an algebraic step $K(B) \subseteq L$, where $B$ is a transcendence basis.
</document>

<document source="https://tamnd.github.io/math2/node/HMRP135M6K.md" tag="HMRP135M6K" kind="concept">
# Holonomy groupoid of a foliation

## Summary

The holonomy groupoid of a [[foliation|98HTB7MSPX]] is a [[Lie groupoid|HNVW0E732J]] whose objects are the points of the manifold.
Its arrows are paths inside leaves, where two paths count as equal when they have the same holonomy.
Its orbits are the leaves, and its isotropy groups are the holonomy groups of the leaves.
It replaces the leaf space, which is often a badly behaved topological space, by a smooth object.
It is the starting point of the noncommutative geometry of foliations, through its convolution algebra.

## Explanations

### L1 human

A foliation cuts a manifold into leaves, like the pages of a book, but the leaves can wind around in complicated ways.
To compare two points of the same leaf one walks from one to the other inside the leaf.
Walking along a path also drags nearby leaves along, and the way they are moved is the holonomy of the path.
The holonomy groupoid has an arrow for each way of walking from one point to another.
Two walks count as the same when they move nearby leaves in the same way.
For the foliation of the plane by horizontal lines, there is exactly one arrow between two points on the same line and none between points on different lines.
</document>

<document source="https://tamnd.github.io/math2/node/HMTCW4EAMW.md" tag="HMTCW4EAMW" kind="definition">
# Inversion of rational transforms by partial fractions

## Definition

Let $P$ and $Q$ be complex polynomials with $Q \ne 0$ and $\deg P < \deg Q$, and let $F = P/Q$ be the [[rational function|0WCZH4AFMT]].
Let $a_1, \ldots, a_r$ be the distinct roots of $Q$, with multiplicities $m_1, \ldots, m_r$.
The [[partial fraction decomposition|030VFEREKG]] writes $F$ uniquely as
$$F(s) = \sum_{j=1}^{r} \sum_{k=1}^{m_j} \frac{c_{j k}}{(s - a_j)^k}$$
with complex constants $c_{j k}$.
*Inversion by partial fractions* is the computation of the [[inverse Laplace transform|0JRW4X1MDC]] of $F$ as
$$f(t) = \sum_{j=1}^{r} \sum_{k=1}^{m_j} c_{j k} \, \frac{t^{k-1}}{(k-1)!} \, e^{a_j t}, \qquad t \ge 0.$$
It rests on the identity $\mathcal{L}\{t^{k-1} e^{a t}/(k-1)!\}(s) = 1/(s - a)^k$ for $\operatorname{Re} s > \operatorname{Re} a$.

## Remarks

The identity follows from $\mathcal{L}\{t^{k-1}\}(s) = (k-1)!/s^k$ and the [[frequency shift theorem|CD35MSBNE7]].
Example: $F(s) = \frac{s + 3}{(s + 1)(s + 2)} = \frac{2}{s + 1} - \frac{1}{s + 2}$, so $f(t) = 2 e^{-t} - e^{-2t}$; check $f(0) = 1 = \lim_{s \to \infty} s F(s)$.
Non-example: $F(s) = s/(s + 1)$ has $\deg P = \deg Q$, so the method does not apply as stated.
Indeed $F(s) \to 1$ as $s \to \infty$, while the Laplace transform of a function [[tends to $0$ at infinity|N2JFN9VHD9]]; here $F = 1 - 1/(s+1)$ is the transform of the distribution $\delta - e^{-t}$.
For real $P$ and $Q$, the complex roots come in conjugate pairs, and the two terms of a simple pair combine into $e^{\alpha t}(A \cos \beta t + B \sin \beta t)$.
The coefficients of simple roots are quickly found with the [[Heaviside cover-up method|SRQ5P3077W]].
</document>

<document source="https://tamnd.github.io/math2/node/HN0AXS6VSG.md" tag="HN0AXS6VSG" kind="definition">
# Dirichlet series

## Definition

Let $a \colon \set{1, 2, 3, \dots} \to \C$ be a function, called an arithmetic function.
For $s \in \C$ and an integer $n \ge 1$, let $n^{-s} = e^{-s \log n}$, using the [[complex exponential|07GJV36896]].
The *Dirichlet series* of $a$ is the [[series|0NM626FH6Q]]
$$F(s) = \sum_{n=1}^{\infty} \frac{a(n)}{n^s},$$
considered as a function of $s$ at the points where it converges.

## Remarks

Let $\sigma_a$ be the infimum of the real $\sigma$ such that $\sum_n \abs{a(n)} n^{-\sigma}$ converges, the abscissa of [[absolute convergence|0YBDSJARWN]].
The series converges absolutely for $\operatorname{Re} s > \sigma_a$ and defines a holomorphic function there.
It also has an abscissa of convergence $\sigma_c$, with $\sigma_c \le \sigma_a \le \sigma_c + 1$.
Example: for $a(n) = (-1)^{n+1}$, the series $\sum (-1)^{n+1} n^{-s}$ converges for $\operatorname{Re} s > 0$ and converges absolutely for $\operatorname{Re} s > 1$, so $\sigma_c = 0$ and $\sigma_a = 1$.
Non-example: the power series $\sum_{n} a_n z^n$ is not a Dirichlet series, since its terms are powers $z^n$ of the variable rather than $n^{-s}$.
If $F$ and $G$ are the Dirichlet series of $a$ and $b$, both absolutely convergent at $s$, then $F(s) G(s)$ is the Dirichlet series of the [[Dirichlet convolution|037F589D1D]] $a * b$ at $s$.
For example $\sum_n \mu(n) n^{-s} = 1/\zeta(s)$ for $\operatorname{Re} s > 1$, with $\mu$ the [[Möbius function|0FANE7P0DY]].
</document>

<document source="https://tamnd.github.io/math2/node/HN5A4M2GE8.md" tag="HN5A4M2GE8" kind="definition">
# Minkowski sum of polytopes

## Definition

Let $P$ and $Q$ be [[convex polytopes|300K6CB1KH]] in $\R^n$.
Their *Minkowski sum* is the set
$$P + Q = \set{p + q : p \in P, \ q \in Q}.$$

## Remarks

The set $P + Q$ is again a convex polytope.
If $P$ has vertex set $V$ and $Q$ has vertex set $W$, then $P + Q$ is the [[convex hull|0FQSDC2EQ2]] of the finite set $\set{v + w : v \in V, \ w \in W}$.
For a linear functional $u$, the face of $P + Q$ on which $u$ is largest is the sum of the faces of $P$ and of $Q$ on which $u$ is largest.
So each vertex of $P + Q$ is $v + w$ for a unique vertex $v$ of $P$ and a unique vertex $w$ of $Q$.

Example: let $T$ be the triangle with vertices $(0,0)$, $(1,0)$, $(0,1)$ in $\R^2$, and let $-T$ be its image under $x \mapsto -x$.
The sums of vertices of $T$ and $-T$ are the differences of vertices of $T$, namely $(0,0)$, $(\pm 1, 0)$, $(0, \pm 1)$, $(1,-1)$ and $(-1,1)$.
Their convex hull is the hexagon with vertices $(1,0)$, $(1,-1)$, $(0,-1)$, $(-1,0)$, $(-1,1)$, $(0,1)$, so $T + (-T)$ is this hexagon.

Non-example: the closed unit disk $D$ is convex but not a polytope, and $[0,1]^2 + D$ is a square with rounded corners, which is not a polytope.
So the closure of polytopes under the sum needs both summands to be polytopes.

The operation is commutative and associative, and $\set{0}$ is its neutral element.
It is the general [[Minkowski addition|SXS6D87PHE]] of sets, restricted to polytopes.
</document>

<document source="https://tamnd.github.io/math2/node/HN5XRJTHND.md" tag="HN5XRJTHND" kind="definition">
# Nevanlinna theory

## Definition

Let $f$ be a [[meromorphic function|0NSD1TPHR0]] on $\C$ that is not constant, and write $\log^+ x = \max(\log x, 0)$ for $x \ge 0$, with $\log^+ 0 = 0$.
Nevanlinna theory studies $f$ through the following functions of $r > 0$, for $a$ in the [[Riemann sphere|0CDMXGPSES]] $\C \cup \set{\infty}$.

1. The number $n(t, a)$ of solutions of $f(z) = a$ in the disc $\abs{z} \le t$, counted with multiplicity; for $a = \infty$ these are the poles of $f$. The number $\overline{n}(t, a)$ counts the same points without multiplicity.
2. The *counting function* $N(r, a) = \int_0^r \dfrac{n(t, a) - n(0, a)}{t} \, dt + n(0, a) \log r$, and $\overline{N}(r, a)$ defined in the same way from $\overline{n}$.
3. The *proximity function* $m(r, a) = \dfrac{1}{2\pi} \displaystyle\int_0^{2\pi} \log^+ \frac{1}{\abs{f(re^{i\theta}) - a}} \, d\theta$ for $a \in \C$, and $m(r, \infty) = \dfrac{1}{2\pi} \displaystyle\int_0^{2\pi} \log^+ \abs{f(re^{i\theta})} \, d\theta$.
4. The [[Nevanlinna characteristic|YT6Z0GWS0W]] $T(r, f) = m(r, \infty) + N(r, \infty)$.
5. The *defect* $\delta(a) = \liminf_{r \to \infty} \dfrac{m(r, a)}{T(r, f)}$, using the [[limit inferior|0H4R3RESGV]].

## Remarks

Example: for $f(z) = e^z$ there are no poles and no zeros, so $N(r, \infty) = N(r, 0) = 0$.
The proximity functions are $m(r, \infty) = m(r, 0) = \frac{1}{2\pi}\int_0^{2\pi} \max(r \cos\theta, 0) \, d\theta = r/\pi$, so $T(r, e^z) = r/\pi$ and $\delta(0) = \delta(\infty) = 1$.
For a value $a \ne 0, \infty$, the solutions of $e^z = a$ have $n(t, a) \sim t/\pi$, so $N(r, a) \sim r/\pi$ and $\delta(a) = 0$.
By the [[first main theorem|QJWCFAK4AK]], $\delta(a) = 1 - \limsup_{r \to \infty} N(r, a)/T(r, f)$ when $f$ is transcendental, and $0 \le \delta(a) \le 1$.
A value that $f$ omits has defect $1$.
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<document source="https://tamnd.github.io/math2/node/HN93ST5CBT.md" tag="HN93ST5CBT" kind="statement">
# Radon inversion formula

## Statement

Let $n \ge 2$ and let $f$ be in the [[Schwartz space|GE020MY43M]] $\mathcal{S}(\R^n)$.
Let $R$ be the [[Radon transform|VJNZJ352PC]] and $R^*$ the [[dual Radon transform|938SXZQS8G]].
For a function $g(\theta, s)$ on $S^{n-1} \times \R$, let $\Lambda g$ be defined by filtering in $s$ with the multiplier $\abs{\sigma}^{n-1}$:
$$(\Lambda g)(\theta, s) = \frac{1}{2\pi} \int_\R \abs{\sigma}^{n-1}\, \hat g(\theta, \sigma)\, e^{i s \sigma}\, d\sigma, \qquad \hat g(\theta, \sigma) = \int_\R g(\theta, s)\, e^{-i s \sigma}\, ds.$$
Then, for every $x \in \R^n$,
$$f(x) = \frac{1}{2} (2\pi)^{1-n}\, R^* (\Lambda R f)(x).$$
If $n$ is odd, $\Lambda = (-1)^{(n-1)/2} \partial_s^{n-1}$, and the formula reads
$$f(x) = \frac{(-1)^{(n-1)/2}}{2 (2\pi)^{n-1}} \int_{S^{n-1}} \left( \partial_s^{n-1} R f \right)(\theta, x \cdot \theta)\, d\theta.$$

## Remarks

The proof starts from the Fourier inversion formula $f(x) = (2\pi)^{-n} \int_{\R^n} \hat f(\xi)\, e^{i x \cdot \xi}\, d\xi$ in polar coordinates $\xi = \sigma\theta$.
Writing the integral over $\sigma \in \R$ and $\theta \in S^{n-1}$ counts each $\xi$ twice, which gives the factor $\frac12$ and the weight $\abs{\sigma}^{n-1}$.
The [[projection-slice theorem|J7VA040MC3]] replaces $\hat f(\sigma\theta)$ by the Fourier transform of $R_\theta f$, and the remaining $\theta$-integral is $R^*$.

Example: for $n = 3$ the formula is $f(x) = -\frac{1}{8\pi^2} \int_{S^2} (\partial_s^2 R f)(\theta, x \cdot \theta)\, d\theta$, so $f(x)$ depends only on the integrals of $f$ over planes close to $x$.

For even $n$ the multiplier $\abs{\sigma}^{n-1}$ is not a polynomial, $\Lambda$ involves the [[Hilbert transform|NSCEBXM7ZM]] in $s$, and the formula is not local: $f(x)$ depends on integrals over all hyperplanes.

## Explanations

### L1 human

The Radon transform of a function lists its integrals over all hyperplanes, and the formula gets the function back.
First each profile is sharpened by a filter that boosts high frequencies.
Then all sharpened profiles are smeared back over space and added up.
The result is exactly the original function.

### L2 human

The formula is the continuous model of [[filtered back-projection|SJQDDY9FMX]], the standard reconstruction algorithm of computed tomography.
The filter $\abs{\sigma}^{n-1}$ grows at high frequencies, which shows why inversion amplifies noise and why practical filters cut it off.
In odd dimensions the inversion is local, a fact used in three-dimensional imaging; in even dimensions, such as the plane, it is not.
The same formula, read with the Riesz potential, extends to wider classes of functions and to distributions.
</document>

<document source="https://tamnd.github.io/math2/node/HNCY0067F1.md" tag="HNCY0067F1" kind="statement">
# Kronecker's lemma

## Statement

Let $(x_n)_{n \ge 1}$ be a real sequence such that the series $\sum_{n=1}^{\infty} x_n$ [[converges|0CVVKY7GY2]].
Let $(b_n)_{n \ge 1}$ be a [[monotone|0KN22AAW5R]] sequence of real numbers with $0 < b_1 \le b_2 \le \cdots$ and $b_n \to \infty$.
Then
$$\lim_{n \to \infty} \frac{1}{b_n} \sum_{k=1}^{n} b_k x_k = 0.$$

## Remarks

Let $r_k = \sum_{j > k} x_j$ be the tails of the series, so $x_k = r_{k-1} - r_k$ with $r_k \to 0$.
Summation by parts writes $\frac{1}{b_n} \sum_{k \le n} b_k x_k$ as $b_1 r_0 / b_n - r_n$ plus a weighted average of $r_1, \ldots, r_{n-1}$ with weights $(b_{k+1} - b_k)/b_n$.
These weights are nonnegative, their sum is at most $1$, and each fixed weight tends to $0$, so the average tends to $0$ because $r_k \to 0$.
Example: with $b_n = n$, if $\sum y_n / n$ converges, then $(y_1 + \cdots + y_n)/n \to 0$.

## Explanations

### L1 human

If a series converges, its terms become small on average.
Kronecker's lemma makes this precise: even after multiplying the terms by growing weights $b_k$, the weighted sum grows more slowly than $b_n$.

### L2 human

The main use is in probability.
To prove a strong law of large numbers, one shows that $\sum X_n / n$ converges almost surely, and Kronecker's lemma with $b_n = n$ then gives $(X_1 + \cdots + X_n)/n \to 0$.
The proof is a summation by parts followed by an averaging argument, as in the [[Stolz–Cesàro theorem|0Q3AKZV74D]].
</document>

<document source="https://tamnd.github.io/math2/node/HNGT8S7JAG.md" tag="HNGT8S7JAG" kind="definition">
# Carlson symmetric forms

## Definition

Let $x, y, z$ be nonnegative real numbers, at most one of which is $0$, and let $p > 0$.
The *Carlson symmetric forms* are [[improper integrals|05KQ3F99BQ]].
They are
$$R_F(x, y, z) = \frac{1}{2} \int_0^\infty \frac{dt}{\sqrt{(t + x)(t + y)(t + z)}},$$
$$R_J(x, y, z, p) = \frac{3}{2} \int_0^\infty \frac{dt}{(t + p)\sqrt{(t + x)(t + y)(t + z)}},$$
together with the special cases
$$R_C(x, y) = R_F(x, y, y), \qquad R_D(x, y, z) = R_J(x, y, z, z),$$
where for $R_C$ one takes $x \ge 0$ and $y > 0$, and for $R_D$ one takes $z > 0$.

## Remarks

The integrals converge: near $t = \infty$ the integrand of $R_F$ behaves like $t^{-3/2}$, and near $t = 0$ at most one factor vanishes, which gives at worst $t^{-1/2}$.
$R_F$ is symmetric in $x, y, z$, $R_J$ in $x, y, z$, and $R_D$ in $x, y$.
Example: $R_F(x, x, x) = \frac12\int_0^\infty (t + x)^{-3/2}\,dt = x^{-1/2}$ for $x > 0$.
Non-example: $R_F(0, 0, 1)$ is not defined, since two arguments are $0$ and the integrand $1/(t\sqrt{t + 1})$ is not integrable at $t = 0$.
With $k' = \sqrt{1 - k^2}$ and $0 \le k < 1$, the complete [[elliptic integrals|NXMCYGCB59]] are
$$K(k) = R_F(0, k'^2, 1), \qquad E(k) = R_F(0, k'^2, 1) - \frac{k^2}{3} R_D(0, k'^2, 1).$$
$R_C$ is elementary; for $0 \le x < y$ it equals $\arccos(\sqrt{x/y}) / \sqrt{y - x}$.
</document>

<document source="https://tamnd.github.io/math2/node/HNK1Y62G7V.md" tag="HNK1Y62G7V" kind="concept">
# Filtration

## Summary

A filtration is an increasing family $(\mathcal{F}_t)_{t \in T}$ of sub-$\sigma$-algebras of the $\sigma$-algebra of a probability space, indexed by time.
The $\sigma$-algebra $\mathcal{F}_t$ models the information available at time $t$, and information is never lost as time goes on.
Filtrations are the setting for [[adapted processes|YRNFE016FF]], [[stopping times|WND87FBAKC]] and [[martingales|0D0GBNDKDH]].

## Explanations

### L1 human

Imagine watching a game of chance round by round.
After each round you know more than before: all earlier results and the new one.
A filtration records this growing knowledge: at each time it lists the events whose occurrence you can already decide.
For repeated coin tosses, the information at time $n$ consists of the events that depend only on the first $n$ tosses.
</document>

<document source="https://tamnd.github.io/math2/node/HNRM1CW4JK.md" tag="HNRM1CW4JK" kind="definition">
# Absolutely monotonic function

## Definition

Let $I \subseteq \R$ be an open [[interval|0Q1D85B8G6]] and let $f \colon I \to \R$ be [[infinitely differentiable|0V33BNKZ9S]].
The function $f$ is *absolutely monotonic* on $I$ if $f^{(k)}(x) \ge 0$ for every integer $k \ge 0$ and every $x \in I$.
Here $f^{(0)} = f$ and $f^{(k)}$ is the [[derivative of order $k$|04TKTF8KDK]].
The function $f$ is *completely monotone* on $I$ if $(-1)^k f^{(k)}(x) \ge 0$ for every $k \ge 0$ and every $x \in I$.

## Remarks

Example: $f(x) = e^{x}$ on $\R$ is absolutely monotonic, since $f^{(k)}(x) = e^x > 0$ for all $k$.
Likewise $g(x) = e^{-x}$ is completely monotone, since $(-1)^k g^{(k)}(x) = e^{-x} > 0$.
Non-example: $f(x) = x^3$ on $(-1, 1)$ fails at $k = 0$, since $f(-\tfrac{1}{2}) < 0$, although $f^{(3)} = 6 > 0$.
The function $f$ is absolutely monotonic on $I$ if and only if $x \mapsto f(-x)$ is completely monotone on $-I$.
On an interval that contains an endpoint, such as $[0, \infty)$, one asks the inequalities on the interior and continuity of $f$ at the endpoint.

## Explanations

### L1 human

Every derivative, starting with the function itself, is nonnegative.
So the function is nonnegative, increasing, convex, and its convexity keeps increasing, without end.
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<document source="https://tamnd.github.io/math2/node/HNTE5FPS14.md" tag="HNTE5FPS14" kind="concept">
# Ovoid

## Summary

An ovoid is a set of points of a [[projective space|ZT9A6JK8QF]] that looks like a sphere from the point of view of incidence.
Every line meets it in at most two points, and at each of its points the tangent lines fill a hyperplane.
Ovoids in three-dimensional space are the spatial analogue of [[ovals|7B03TB64N9]] in a plane.
They give Möbius planes and generalized quadrangles, and finite ovoids are a main topic of Galois geometry.

## Explanations

### L1 human

Think of a sphere in space, completed with points at infinity.
A line pierces a sphere in at most two points, and the lines that just touch the sphere at a point make up its tangent plane.
An ovoid is any set of points with these two properties, defined only through points, lines and planes.
Over a finite field with $q$ elements, an ovoid of three-dimensional projective space has exactly $q^2 + 1$ points.
</document>

<document source="https://tamnd.github.io/math2/node/HNVW0E732J.md" tag="HNVW0E732J" kind="concept">
# Lie groupoid

## Summary

A Lie groupoid is a [[groupoid|FH3C0T643F]] whose objects and arrows each form a [[smooth manifold|F45VQ47GPV]].
All its structure maps are smooth, and the source and target maps are submersions.
It generalises [[Lie groups|7JBRGCVJRB]], which are Lie groupoids with a single object, and also smooth group actions, foliations and principal bundles.
Lie groupoids give a common language for symmetries that vary from point to point, and they have Lie algebroids as their infinitesimal versions.

## Explanations

### L1 human

A group describes symmetries of one object; a groupoid describes symmetries between many objects.
An arrow goes from one object to another, and two arrows can be composed when they meet end to end.
In a Lie groupoid both the objects and the arrows form smooth spaces, and composing arrows is a smooth operation.
For example, take as objects the points of a manifold $M$, and as arrows from $x$ to $y$ the pairs $(y, x)$.
Composing $(z, y)$ with $(y, x)$ gives $(z, x)$.
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<document source="https://tamnd.github.io/math2/node/HNW4WRSEEX.md" tag="HNW4WRSEEX" kind="concept">
# Riesz projector

## Summary

The Riesz projector of an operator $T$ for an isolated part of its spectrum is a projection.
It is obtained by integrating the resolvent of $T$ around that part.
It splits the space into two closed invariant subspaces on which $T$ has the separated pieces of the spectrum.
It generalises the projection onto a generalised eigenspace of a matrix along the other generalised eigenspaces.

## Explanations

### L1 human

For a matrix, one can split the space into the generalised eigenspace of one eigenvalue and the sum of the others.
The Riesz projector does this for operators, for any part of the spectrum that is separated from the rest by a gap.
It is computed by a contour integral around that part, as in Cauchy's integral formula.
For the diagonal matrix with entries $1$ and $2$, the Riesz projector for $\set{1}$ keeps the first coordinate and kills the second.
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<document source="https://tamnd.github.io/math2/node/HNXXAPQKG3.md" tag="HNXXAPQKG3" kind="concept">
# Topological conjugacy

## Summary

Two maps are topologically conjugate when a [[homeomorphism|0QT00PB5G2]] between their spaces turns one map into the other.
Conjugate systems have the same dynamics up to a continuous change of coordinates: fixed points, periodic orbits, dense orbits and attractors correspond.
Topological conjugacy is the basic notion of "same system" in topological dynamics, and many classification results, such as structural stability, are stated with it.
If the homeomorphism is only required to be a continuous surjection, one gets the weaker notion of a [[semiconjugacy|AEXJNRBASV]].

## Explanations

### L1 human

Imagine two maps that do the same thing, but one is drawn on a rubber sheet that has been stretched.
A conjugacy is the stretching that matches them up, step by step.
For example, $f(x) = x/2$ and $g(x) = x/8$ on the real line are conjugate through $h(x) = x^3$.
Indeed, $h(f(x)) = x^3/8 = g(h(x))$, so moving by $f$ and then changing coordinates is the same as changing coordinates and then moving by $g$.
</document>

<document source="https://tamnd.github.io/math2/node/HP56DJECMT.md" tag="HP56DJECMT" kind="statement">
# Hilbert–Serre theorem

## Statement

Let $k$ be a [[field|04C5ZBG1WT]] and let $S = k[x_1, \dots, x_r]$ be the [[polynomial ring|07GNX8GCYP]] with each $x_i$ of degree $1$.
Let $M = \bigoplus_{n \in \Z} M_n$ be a [[finitely generated|05QY1VEBYD]] [[graded|8KB4SQGC0C]] $S$-module, and let $H_M(n) = \dim_k M_n$.

1. The Hilbert series $\sum_{n \in \Z} H_M(n)\, t^n$ equals $Q(t)/(1 - t)^r$ for some Laurent polynomial $Q \in \Z[t, t^{-1}]$.
2. There is a unique polynomial $P_M \in \Q[t]$ such that $H_M(n) = P_M(n)$ for all sufficiently large $n$, and $P_M$ is zero or has degree at most $r - 1$.

## Remarks

The proof goes by induction on $r$.
Multiplication by $x_r$ gives an exact sequence of graded modules $0 \to K(-1) \to M(-1) \to M \to C \to 0$, where $K$ and $C$ are killed by $x_r$ and so are finitely generated graded modules over $k[x_1, \dots, x_{r-1}]$.
Taking dimensions degree by degree gives $(1 - t)$ times the Hilbert series of $M$ in terms of those of $C$ and $K$, and the induction hypothesis finishes the proof.
Part 2 follows from part 1 by expanding $1/(1-t)^r = \sum_{n \ge 0} \binom{n + r - 1}{r - 1} t^n$.
Example: for $M = S$ one has $Q = 1$ and $H_S(n) = \binom{n + r - 1}{r - 1}$ for $n \ge 0$.
When the Krull dimension of $M$ is at least $1$, the degree of $P_M$ is one less than this dimension.

## Explanations

### L1 human

A graded module splits into pieces of degree $0, 1, 2, \dots$, and each piece is a finite-dimensional vector space.
The theorem says that the dimensions of these pieces are eventually given by a polynomial in the degree.
So an infinite list of numbers is controlled by finitely many coefficients.

### L2 human

The polynomial $P_M$ is the [[Hilbert polynomial|RMH4FBCFDA]] of $M$.
In projective geometry, applied to the coordinate ring of a projective variety, it gives the dimension and the degree of the variety.
The local analogue is the existence of the Hilbert–Samuel polynomial, which is proved by applying this theorem to the associated graded ring; it underlies the [[dimension theorem for local rings|KW3HY77V6D]].
</document>

<document source="https://tamnd.github.io/math2/node/HP5VJ9EBST.md" tag="HP5VJ9EBST" kind="concept">
# Fredholm minors

## Summary

The Fredholm minors of a continuous kernel $K$ are entire functions of the parameter $\lambda$.
They are built like the [[Fredholm determinant|PPX0AK81QV]], from determinants of values of $K$, but with some rows and columns held at fixed points.
They are the integral analogues of the minors and cofactors of the matrix $I - \lambda A$.
The first minor divided by the Fredholm determinant is the [[resolvent kernel|ARTJV1XCRX]].
So the minors give the explicit solution of the [[Fredholm equation of the second kind|DEYYNVCV9J]].
At a zero of the determinant, the first minor that is not identically zero yields the solutions of the homogeneous equation.

## Explanations

### L1 human

For a square matrix, Cramer's rule writes the inverse as cofactors divided by the determinant.
Fredholm minors are the cofactors of an integral operator.
The first minor $D(x, y; \lambda)$ depends on two points $x$ and $y$, and $D(x, y; \lambda) / D(\lambda)$ is the kernel of the inverse operation.
For $K(x, y) = xy$ on $[0, 1]$ one finds $D(x, y; \lambda) = xy$, so the resolvent kernel is $xy / (1 - \lambda/3)$.
</document>

<document source="https://tamnd.github.io/math2/node/HP97SGN9A3.md" tag="HP97SGN9A3" kind="definition">
# Askey–Wilson polynomials

## Definition

Let $q \in \C$ with $0 < \abs{q} < 1$, and let $a, b, c, d \in \C$ with $a \ne 0$.
Write $(c_1, \dots, c_k; q)_n = (c_1; q)_n \cdots (c_k; q)_n$ with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].
For an integer $n \ge 0$ and $x = \cos\theta$, the *Askey–Wilson polynomial* of degree $n$ is
$$p_n(x; a, b, c, d \mid q) = a^{-n} (ab, ac, ad; q)_n\; {}_4\phi_3 \left( \begin{matrix} q^{-n},\ abcd\, q^{n-1},\ a e^{i\theta},\ a e^{-i\theta} \\ ab,\ ac,\ ad \end{matrix} ; q, q \right),$$
where ${}_4\phi_3$ is the terminating [[basic hypergeometric series|EB4N9RQR51]].

## Remarks

Since $(a e^{i\theta}, a e^{-i\theta}; q)_k = \prod_{j=0}^{k-1} (1 - 2 a q^j x + a^2 q^{2j})$, each term is a polynomial in $x$, and $p_n$ is a polynomial of degree at most $n$ with leading coefficient $2^n (abcd\, q^{n-1}; q)_n$.
The factor $a^{-n} (ab, ac, ad; q)_n$ cancels the denominators, and $p_n$ is symmetric in $a, b, c, d$.

Example: $n = 1$.
The ${}_4\phi_3$ has two terms, and since $\frac{(1 - q^{-1})\, q}{1 - q} = -1$,
$$p_1(x) = a^{-1} \left[ (1 - ab)(1 - ac)(1 - ad) - (1 - abcd)(1 - 2ax + a^2) \right] = 2(1 - abcd)\, x + (abc + abd + acd + bcd) - (a + b + c + d).$$

Orthogonality: if $0 < q < 1$ and $a, b, c, d$ are real with absolute values less than $1$, then for $m \ne n$
$$\int_{-1}^{1} p_m(x)\, p_n(x)\, w(x)\, \frac{dx}{\sqrt{1 - x^2}} = 0, \qquad w(\cos\theta) = \frac{(e^{2i\theta}, e^{-2i\theta}; q)_\infty}{(a e^{i\theta}, a e^{-i\theta}, b e^{i\theta}, b e^{-i\theta}, c e^{i\theta}, c e^{-i\theta}, d e^{i\theta}, d e^{-i\theta}; q)_\infty}.$$

Non-example: for parameters as in the orthogonality statement, $x^2$ is not $p_2$, since $p_2$ is orthogonal to the constant $p_0 = 1$ for the positive weight $w$, while $\int x^2 w(x) \frac{dx}{\sqrt{1-x^2}} > 0$.
</document>

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# Generalized hypergeometric function

## Definition

Let $p, q \ge 0$ be integers, let $a_1, \ldots, a_p \in \C$, and let $b_1, \ldots, b_q \in \C \setminus \set{0, -1, -2, \ldots}$.
Let $(x)_n$ be the [[Pochhammer symbol|YZHM3HSM0F]].
The *generalized hypergeometric series* is the [[power series|0MZB06ST2B]]
$${}_pF_q(a_1, \ldots, a_p; b_1, \ldots, b_q; z) = \sum_{n=0}^{\infty} \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \, \frac{z^n}{n!}.$$
The *generalized hypergeometric function* is its sum where it converges, together with its analytic continuation.

## Remarks

Assume that no $a_i$ is a non-positive integer, so that the series does not terminate.
By the ratio test, its [[radius of convergence|0N288P2TB5]] is:

1. infinite when $p \le q$, so the function is entire;
2. equal to $1$ when $p = q + 1$, and then the function continues analytically to $\C \setminus [1, \infty)$;
3. equal to $0$ when $p > q + 1$.

Examples: ${}_0F_0(;;z) = e^z$ and ${}_1F_0(a;;z) = (1 - z)^{-a}$ for $\abs{z} < 1$.
Also $J_\nu(x) = \frac{(x/2)^\nu}{\Gamma(\nu + 1)} \, {}_0F_1(;\nu + 1; -x^2/4)$ for the Bessel function of the first kind.

Non-example: ${}_2F_0(1, 1;; z) = \sum_n n! \, z^n$ is a formal series with radius of convergence $0$; it is only an asymptotic series and defines no function near $0$.

A lower parameter equal to an upper one cancels: ${}_{p+1}F_{q+1}(a, c; b, c; z) = {}_pF_q(a; b; z)$, with $a$ and $b$ standing for the lists of parameters.
</document>

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# Minimal set

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $T \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
A subset $M \subseteq X$ is a *minimal set* for $T$ if:

1. $M$ is nonempty and closed.
2. $M$ is [[invariant|WWZTW11664]]: $T(M) \subseteq M$.
3. No nonempty closed set $N \subsetneq M$ satisfies $T(N) \subseteq N$.

The system $(X, T)$ is *minimal* if $X$ itself is a minimal set.

## Remarks

Equivalent form: a nonempty closed invariant set $M$ is minimal if and only if, for every $x \in M$, the forward orbit $\set{T^n(x) : n \ge 0}$ is [[dense|0BSJSX7ME2]] in $M$.
Indeed, the closure of a forward orbit is closed and invariant by continuity of $T$.
If $X$ is nonempty and [[compact|0TVJZYB6SH]], Zorn's lemma applied to the nonempty closed invariant sets gives at least one minimal set.
For a continuous action of a group $G$, a minimal set is a nonempty closed $G$-invariant set with no proper nonempty closed $G$-invariant subset.
The action is minimal when every orbit is dense.
Example: for $T(x) = x + \alpha \bmod 1$ on the circle $\R / \Z$ with $\alpha$ irrational, the points $n \alpha \bmod 1$ are dense, so every orbit is dense and the circle is minimal.
Non-example: for $\alpha = 1/2$, the circle is not minimal, since the closed invariant set $\set{0, 1/2}$ is a proper nonempty subset; clause 3 fails.

## Explanations

### L1 human

A minimal set is closed, is mapped into itself, and cannot be shrunk to a smaller closed set with the same property.
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<document source="https://tamnd.github.io/math2/node/HPMAJV2RDY.md" tag="HPMAJV2RDY" kind="concept">
# No small subgroups

## Summary

A [[topological group|92700QAB5Y]] has no small subgroups if some neighbourhood of the identity contains no subgroup other than the trivial one.
Every [[Lie group|7JBRGCVJRB]] has this property.
For locally compact groups the converse holds as well, by the work of Gleason and Yamabe.
This is the key to the solution of Hilbert's fifth problem.

## Explanations

### L1 human

In the real line, the interval $(-1, 1)$ contains no subgroup except $\set{0}$: any subgroup containing a number $x \ne 0$ also contains $x, 2x, 3x, \dots$, which eventually leave the interval.
Groups built from smooth spaces all behave this way near the identity.
In the $p$-adic integers, by contrast, every small ball around $0$ is itself a subgroup.
So "no small subgroups" is a simple test that separates smooth groups from groups of this other kind.
</document>

<document source="https://tamnd.github.io/math2/node/HPW5F0GQ4B.md" tag="HPW5F0GQ4B" kind="concept">
# Distance geometry

## Summary

Distance geometry studies sets of points through the distances between them alone.
Its central question asks when given distances between some pairs of points can be realized by actual points in a Euclidean space.
Tools include the [[Cayley–Menger determinant|KGC5CYDF53]], [[Euclidean distance matrices|1BCJ7A73KS]] and the theory of [[rigid frameworks|V5YS9BZX0N]].
It has applications to molecular shape, sensor networks and the design of structures.

## Explanations

### L1 human

Suppose you know how far apart some cities are, but you have no map.
Distance geometry asks whether such a map exists and whether it is unique.
For example, three distances $3$, $4$ and $5$ fit a right triangle in the plane.
The distances $1$, $1$ and $3$ fit nothing, because $3 > 1 + 1$ breaks the [[triangle inequality|01M6CW9VM6]].
</document>

<document source="https://tamnd.github.io/math2/node/HPX85PA8NX.md" tag="HPX85PA8NX" kind="concept">
# Desarguesian plane

## Summary

A Desarguesian plane is a [[projective plane|DANFNN8C4Y]] in which [[Desargues's theorem|5Z19Y02MME]] holds.
These are exactly the planes $\mathbb{P}^2(D)$ built from a division ring $D$, so they are the planes that come from linear algebra.
Every finite Desarguesian plane is $\mathbb{P}^2(\F_q)$ for a finite field $\F_q$.
Planes in which the theorem fails are called [[non-Desarguesian planes|HYE5FT4M34]].

## Explanations

### L1 human

Some projective planes are built from numbers, by taking lines through the origin in a three-dimensional space.
Others are built by more exotic rules.
Desargues's theorem about two triangles in perspective is the test that tells them apart: it holds exactly in the planes built from numbers.
The real projective plane and the Fano plane with $7$ points are Desarguesian.
</document>

<document source="https://tamnd.github.io/math2/node/HPZHRZ0502.md" tag="HPZHRZ0502" kind="statement">
# Plancherel theorem

## Statement

Let $n \ge 1$, and let $L^p(\R^n)$ denote the [[Lebesgue spaces|SASP14MA1Z]] with respect to Lebesgue measure.
For $f \in L^1(\R^n)$ let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi} \, dx$ be its [[Fourier transform|DPVM34ADZV]].

1. If $f \in L^1(\R^n) \cap L^2(\R^n)$, then $\hat f \in L^2(\R^n)$ and
$$\int_{\R^n} \abs{\hat f(\xi)}^2 \, d\xi = \int_{\R^n} \abs{f(x)}^2 \, dx.$$
2. The map $f \mapsto \hat f$ on $L^1(\R^n) \cap L^2(\R^n)$ extends uniquely to a [[unitary operator|MM0EJ2GDE4]] $\mathcal F$ of the [[Hilbert space|HM2HHE3J51]] $L^2(\R^n)$ onto itself.

## Remarks

Proof idea: for Schwartz functions the identity follows from the multiplication formula $\int \hat f \, \overline{\hat g} = \int f \, \overline{g}$ and Fourier inversion.
Schwartz functions are dense in $L^1 \cap L^2$ for the norm $\norm{f}_1 + \norm{f}_2$, which gives part 1.
Since $L^1 \cap L^2$ is dense in $L^2$, the isometry extends uniquely to $L^2$.
The extension is onto because its range is closed and contains the Schwartz space.

For $f \in L^2$ the transform is computed as the $L^2$ limit $\mathcal F f = \lim_{R \to \infty} \int_{\abs{x} \le R} f(x) e^{-2\pi i x \cdot \xi} \, dx$, because the integral itself may diverge.
The inner product is also preserved: $\int \mathcal F f \, \overline{\mathcal F g} = \int f \, \overline{g}$ for $f, g \in L^2(\R^n)$.

Example: for $f = \mathbf{1}_{[-1/2, 1/2]}$ on $\R$, $\hat f(\xi) = \frac{\sin \pi \xi}{\pi \xi}$ and $\int \abs{f}^2 = 1$.
So the theorem gives $\int_\R \left( \frac{\sin \pi \xi}{\pi \xi} \right)^2 d\xi = 1$.

The analogue for Fourier series is [[Parseval's identity|047RFCNVSK]].
The theorem generalizes to [[locally compact abelian groups|V9K9TG0K3J]] and, with representations in place of characters, to [[more general groups|DH3PYQJXH0]].

## Explanations

### L1 human

The energy of a signal, the integral of its squared size, is the same whether one computes it from the signal or from its frequency content.
So the Fourier transform preserves lengths and angles, like a rotation in infinite dimensions.
This lets one define the Fourier transform of every square-integrable function, even when the defining integral does not converge.

### L2 human

Plancherel's theorem makes $L^2$ the natural home of the Fourier transform.
It is the starting point of the $L^p$ theory, through interpolation with the trivial bound from $L^1$ to $L^\infty$.
In quantum mechanics it says that the position and momentum representations of a wave function carry the same total probability.
It also underlies the definition of Sobolev spaces $H^s$ by weights on the Fourier side.
</document>

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# Projective conic

## Definition

Let $F$ be a [[field|04C5ZBG1WT]] and let $\mathbb{P}^2(F)$ be the [[projective plane|ZT9A6JK8QF]] over $F$, with homogeneous coordinates $[x : y : z]$.
Let $Q$ be a nonzero [[quadratic form|0RBQS1JJCR]] on $F^3$.
The *projective conic* of $Q$ is the set
$$C(Q) = \set{[x : y : z] \in \mathbb{P}^2(F) : Q(x, y, z) = 0}.$$
The form $Q$ and its conic are *non-degenerate* if no invertible linear change of coordinates turns $Q$ into a form in at most two variables.

## Remarks

Example: $Q(x, y, z) = xz - y^2$.
Its points are $[1 : t : t^2]$ for $t \in F$ and the point $[0 : 0 : 1]$: if $x \ne 0$ scale to $x = 1$ and get $z = y^2$; if $x = 0$ then $y = 0$.
It is non-degenerate.
If $Q$ involved at most two variables in some coordinates, there would be a nonzero vector $v$ with $Q(u + v) = Q(u)$ for all $u$.
Here $Q(u + v) - Q(u) = u_x v_z + u_z v_x - 2 u_y v_y + Q(v)$, and this vanishes for all $u$ only if $v_x = v_z = 0$ and $Q(v) = -v_y^2 = 0$, so $v = 0$.
If $F = \F_q$, this conic has $q + 1$ points.

Non-example of non-degeneracy: $Q(x, y, z) = xy$ gives the union of the two lines $x = 0$ and $y = 0$.
It involves only $x$ and $y$, so it is degenerate.

Over $\R$, the non-degenerate form $x^2 + y^2 + z^2$ has an empty conic, so the form carries more information than the point set.
Over an algebraically closed field, or over $\R$ when the conic has a point, all non-degenerate conics are equivalent under projective transformations.
The affine [[conic sections|00HJCMC2CD]] are the parts of projective conics off the line at infinity.
</document>

<document source="https://tamnd.github.io/math2/node/HQ7KMZYPSQ.md" tag="HQ7KMZYPSQ" kind="concept">
# Primitive polynomial over a finite field

## Summary

A primitive polynomial over a [[finite field|0ZXZ82FTX2]] $\F_q$ is the minimal polynomial of a [[primitive element|00QSK85Y2X]] of some extension $\F_{q^m}$.
With such a polynomial, every nonzero element of $\F_{q^m}$ is a power of one root, which makes multiplication easy to compute.
Primitive polynomials over $\F_2$ give linear feedback shift registers of maximal period.
This notion differs from the [[primitive polynomial|0TXNRQZY0H]] of ring theory, whose coefficients have no common factor.

## Explanations

### L1 human

Take a root $\alpha$ of an irreducible polynomial and look at its powers $\alpha, \alpha^2, \alpha^3, \dots$.
The polynomial is primitive when these powers run through every nonzero element of the field before they come back to $1$.
Over $\F_2$, the polynomial $x^2 + x + 1$ is primitive: its root $\omega$ gives $\omega$, $\omega^2 = \omega + 1$ and $\omega^3 = 1$, which are the three nonzero elements of $\F_4$.
</document>

<document source="https://tamnd.github.io/math2/node/HQFZQ0F991.md" tag="HQFZQ0F991" kind="concept">
# Regular measure

## Summary

A regular measure on a topological space is a Borel measure that is both [[inner regular|28E5Q5BM69]] and [[outer regular|33C1KEK3SC]].
Each measurable set can then be approximated from inside by compact sets and from outside by open sets.
Regularity ties the measure to the topology and allows integrable functions to be approximated by continuous functions.
Conventions differ between authors: some approximate from inside by closed sets rather than compact sets.

## Explanations

### L1 human

For a regular measure, every set of finite measure can be squeezed between a compact set inside it and an open set around it.
The measure between the two can be made as small as one likes.
Length on the real line is the main example.
For the interval $[0, 1)$, the compact sets $[0, 1 - \varepsilon]$ inside and the open sets $(-\varepsilon, 1)$ outside have lengths as close to $1$ as one likes.
</document>

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# Sperner's lemma

## Statement

Let $n \ge 1$ and let $\Delta$ be an $n$-simplex in $\R^n$ with vertices $v_0, \ldots, v_n$.
Let $T$ be a triangulation of $\Delta$ by a finite [[simplicial complex|V8GZ0W1ZZ5]].
Let $\lambda$ be a colouring of the vertices of $T$ by $\set{0, 1, \ldots, n}$.
Assume the following for every set $S \subseteq \set{0, \ldots, n}$.
Each vertex of $T$ in the face $\operatorname{conv}\set{v_i : i \in S}$ of $\Delta$ has a colour in $S$, where $\operatorname{conv}$ is the [[convex hull|0FQSDC2EQ2]].
Then the number of $n$-simplices of $T$ whose vertices carry all $n + 1$ colours is odd; in particular there is at least one.

## Remarks

The proof is by induction on $n$ with a parity count.
Call an $(n-1)$-face of $T$ a door if its vertices carry exactly the colours $0, \ldots, n-1$.
Count pairs (an $n$-simplex of $T$, a door in it).
A fully coloured simplex has one door, and any other simplex has zero or two.
An interior door lies in two simplices, while a boundary door lies in one.
All boundary doors lie in the face $\operatorname{conv}\set{v_0, \ldots, v_{n-1}}$, where their number is odd by induction; so the number of fully coloured simplices is odd.

Example: for $n = 1$, subdivide a segment from $v_0$ to $v_1$ and colour $v_0$ with $0$, $v_1$ with $1$, and the inner points freely.
The colour changes an odd number of times along the segment, so an odd number of small segments have both colours.

The boundary condition is needed: if every vertex of $T$ gets the colour $0$, there is no fully coloured simplex.

## Explanations

### L1 human

Cut a triangle into small triangles and colour every corner red, green or blue.
Colour the three big corners with the three colours, and colour each point on a big side with one of the two colours at the ends of that side.
Then some small triangle has all three colours, and in fact the number of such triangles is odd.

### L2 human

Sperner's lemma is a discrete form of the [[Brouwer fixed-point theorem|N7AAJ2MW5T]]: applying it to finer and finer triangulations and passing to a limit gives a fixed point.
Its constructive proof by following doors gives algorithms for approximate fixed points and for economic equilibria.
It is also used for fair division problems, such as envy-free cake cutting.
</document>

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# Algebraic space

## Definition

Let $S$ be a [[scheme|5A35TYZD02]], and let $\mathcal{C}$ be the category of schemes over $S$ with the fppf topology.
For an $S$-scheme $U$, let $h_U$ be its [[functor of points|SWWW9X4A3N]].
An *algebraic space* over $S$ is a functor $F \colon \mathcal{C}^{\mathrm{op}} \to \mathrm{Sets}$ such that:

1. $F$ is a [[sheaf|TB6TM6R4MD]] for the fppf topology.
2. For all $S$-schemes $T_1, T_2$ and maps $h_{T_1} \to F$, $h_{T_2} \to F$, the [[fiber product|1DZMT4S9FB]] $h_{T_1} \times_F h_{T_2}$ is isomorphic to $h_T$ for some scheme $T$.
3. There are an $S$-scheme $U$ and a map $h_U \to F$ with the following property.
For every $S$-scheme $T$ and map $h_T \to F$, the induced morphism $U \times_F T \to T$ is surjective and [[étale|H84VHC8PX2]].

## Remarks

Every $S$-scheme $X$ gives an algebraic space $h_X$.
Clause 1 holds because representable functors are fppf sheaves, clause 2 because $h_{T_1} \times_{h_X} h_{T_2} = h_{T_1 \times_X T_2}$, and clause 3 holds with $U = X$.
A functor that is not a sheaf is not an algebraic space.
For example, the constant presheaf with value $\set{0, 1}$ fails clause 1: on the disjoint union of two copies of $S$ it should have four sections, but it has two.
Clause 2 says that the diagonal $F \to F \times F$ is representable by schemes, and clause 3 then makes sense.
Some authors use the étale topology in clause 1 and add a quasi-compactness condition on the diagonal, as in Knutson's book.
</document>

<document source="https://tamnd.github.io/math2/node/HQMNCB9FZ5.md" tag="HQMNCB9FZ5" kind="definition">
# Fréchet–Urysohn space

## Definition

A [[topological space|0CQHFPGJ45]] $X$ is a *Fréchet–Urysohn space* when the following holds.
For every $A \subseteq X$ and every point $x$ in the [[closure|0QEAJB91GZ]] $\overline{A}$, there is a sequence $(x_n)$ in $A$ that [[converges|0W4CN9E2W5]] to $x$.

## Remarks

A Fréchet–Urysohn space is [[sequential|WMB2MAEDX4]]: if $A$ is sequentially closed and $x \in \overline{A}$, a sequence in $A$ converges to $x$, so $x \in A$.
Every subspace of a Fréchet–Urysohn space is Fréchet–Urysohn.
A sequential space is Fréchet–Urysohn if and only if each of its subspaces is sequential.

Example: every [[first-countable space|0CP5NDD21C]] is Fréchet–Urysohn.
For $x \in \overline{A}$ and a decreasing neighbourhood base $U_1 \supseteq U_2 \supseteq \cdots$ at $x$, pick $x_n \in A \cap U_n$; then $x_n \to x$.

Non-example: let $X$ be an uncountable set whose closed sets are $X$ and the countable subsets, and let $A \subset X$ with $A$ and $X \setminus A$ uncountable.
Then $\overline{A} = X$, but the only convergent sequences are eventually constant.
So a point $x \notin A$ is in $\overline{A}$ but is not the limit of a sequence in $A$.

The name Fréchet space is also used, but it has other meanings.
</document>

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# ADE singularity

## Summary

The ADE singularities, also called Du Val singularities, simple surface singularities or rational double points, are the simplest singular points of complex surfaces.
They are the quotients $\C^2/G$ by finite subgroups $G$ of the [[special linear group|0V9PMN0RBH]] $SL(2, \C)$.
They come in two infinite families $A_n$ and $D_n$ and three exceptional cases $E_6$, $E_7$, $E_8$, matching the simply laced Dynkin diagrams.
The match is visible in the minimal resolution, whose exceptional curves meet in the pattern of the Dynkin diagram.

## Explanations

### L1 human

The simplest example is the cone $x^2 + y^2 + z^2 = 0$, the $A_1$ singularity, which is the quotient of the plane by $v \mapsto -v$.
Blowing up the tip of the cone replaces it by one smooth rational curve.
In general the $A_n$ singularity is resolved by a chain of $n$ such curves, each meeting the next one, just like the Dynkin diagram $A_n$.
The same ADE list classifies finite rotation groups in three dimensions and simple Lie algebras of simply laced type.
This is why these singularities are a crossroads of mathematics.
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<document source="https://tamnd.github.io/math2/node/HQTPVQS7TY.md" tag="HQTPVQS7TY" kind="definition">
# Comodule

## Definition

Let $K$ be a field and let $(C, \Delta, \varepsilon)$ be a [[coalgebra|A06RA8KB0T]] over $K$.
A *right $C$-comodule* is a [[vector space|000AYPA9AJ]] $M$ over $K$ with a [[linear map|0NRYHJPQG2]] $\rho \colon M \to M \otimes C$, the coaction, such that:

1. $(\id_M \otimes \Delta) \circ \rho = (\rho \otimes \id_C) \circ \rho$ as maps $M \to M \otimes C \otimes C$.
2. $(\id_M \otimes \varepsilon) \circ \rho = \id_M$, where $M \otimes K$ is identified with $M$.

A *left $C$-comodule* is defined in the same way with a map $M \to C \otimes M$.

## Remarks

Example: $C$ is a right comodule over itself with $\rho = \Delta$; clauses 1 and 2 are coassociativity and the counit law.
Example: let $C = K[S]$ with $\Delta(s) = s \otimes s$ and $\varepsilon(s) = 1$, and let $M = \bigoplus_{s \in S} M_s$ be a vector space graded by $S$.
Then $\rho(m) = m \otimes s$ for $m \in M_s$ is a coaction: both sides of clause 1 give $m \otimes s \otimes s$, and clause 2 gives $\varepsilon(s) m = m$.
Every right $K[S]$-comodule arises in this way.
Non-example: for $M \ne 0$, the zero map $\rho = 0$ satisfies clause 1 but fails clause 2.
A morphism of comodules is a linear map $f \colon M \to N$ with $\rho_N \circ f = (f \otimes \id_C) \circ \rho_M$.
</document>

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# Distribution function of a measurable function

## Summary

The distribution function of a measurable function $f$ records, for each level $t \ge 0$, the measure of the set where $\abs{f}$ is larger than $t$.
It forgets where the large values of $f$ sit and keeps only how much room they take.
Many quantities, such as the $L^p$ norms, depend on $f$ only through its distribution function.
It is the starting point for the layer cake formula, rearrangements, weak $L^p$ spaces and interpolation.

## Explanations

### L1 human

Think of the graph of $\abs{f}$ as a mountain range and of $t$ as the height of a flood.
The distribution function at $t$ is the total length, area or volume of the land that stays above the water.
As the water rises, this amount can only shrink.
For example, for $f(x) = x$ on $[0, 1]$ with length as the measure, the set where $f > t$ is $(t, 1]$, so the distribution function is $1 - t$ for $0 \le t \le 1$ and $0$ for $t \ge 1$.
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<document source="https://tamnd.github.io/math2/node/HR2GM9DSDJ.md" tag="HR2GM9DSDJ" kind="definition">
# Hunt process

## Definition

Let $E$ be a locally compact Hausdorff space with a countable base, and let $E_\Delta = E \cup \set{\Delta}$ be its one-point compactification; the point $\Delta$ is the cemetery.
Let $X = (X_t)_{t \ge 0}$ be a [[stochastic process|0QH5ENKZM1]] with values in $E_\Delta$, adapted to a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_t)$, with a family of probability measures $(P^x)_{x \in E_\Delta}$.
$X$ is a *Hunt process* if:

1. $X$ is a Markov process with respect to $(\mathcal{F}_t)$ under each $P^x$, with $P^x(X_0 = x) = 1$, and $\Delta$ is absorbing: $X_t = \Delta$ implies $X_s = \Delta$ for all $s \ge t$.
2. Each path $t \mapsto X_t$ is right-continuous on $[0, \infty)$ and has left limits in $E_\Delta$ on $(0, \infty)$.
3. $X$ has the strong [[Markov property|0B77KZCQ51]] at every [[stopping time|WND87FBAKC]] of $(\mathcal{F}_t)$, where $(\mathcal{F}_t)$ is right-continuous and suitably completed.
4. $X$ is quasi-left-continuous: if $T_n$ are stopping times increasing to $T$, then $X_{T_n} \to X_T$ almost surely on $\set{T < \infty}$, under each $P^x$.

## Remarks

The time $\zeta = \inf\set{t : X_t = \Delta}$ is the lifetime of the process.
Example: [[Brownian motion|22735FGJV6]] in $\R^d$ is a Hunt process, with $\zeta = \infty$; more generally every [[Feller process|4HS3C12472]] with right-continuous paths having left limits is a Hunt process.
Non-example: on $E = \R$, let the process move deterministically at unit speed, except that it jumps up by $1$ when it reaches the point $1$ from below.
It is a strong Markov process with right-continuous paths, but clause 4 fails.
Started at $0$, the stopping times $T_n = 1 - 1/n$ increase to $T = 1$, while $X_{T_n} \to 1$ and $X_T = 2$.
Quasi-left-continuity allows jumps only at totally inaccessible times, such as the jump times of a Poisson process.
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<document source="https://tamnd.github.io/math2/node/HR5MB30MV3.md" tag="HR5MB30MV3" kind="concept">
# Structure theory of totally disconnected locally compact groups

## Summary

Every [[locally compact group|AHV05ZBCYW]] $G$ has a closed connected normal subgroup, its [[identity component|SDQTNBDJ2H]] $G^\circ$.
The quotient $G/G^\circ$ is a [[totally disconnected|AW3J9AQSA0]] locally compact group.
Connected locally compact groups are close to Lie groups by the solution of Hilbert's fifth problem.
So the study of general locally compact groups leads to the structure theory of totally disconnected ones.
Its basic tools are [[van Dantzig's theorem|4E4RXKV9W3]] on compact open subgroups and the [[Cayley–Abels graphs|QWG66JE1JE]] of compactly generated groups.
Another is the scale function of an automorphism, introduced by George Willis.

## Explanations

### L1 human

Totally disconnected locally compact groups include all discrete groups, the $p$-adic groups such as $\mathrm{GL}_n(\Q_p)$, and the automorphism groups of locally finite trees.
Such a group has many small compact open subgroups, which play the role that small neighbourhoods play in Lie groups.
A compactly generated group of this kind acts nicely on a connected graph of finite degree, so it can be studied like a finitely generated group through its geometry.
The structure theory asks how such groups are built from simpler pieces, such as simple groups and compact groups.
</document>

<document source="https://tamnd.github.io/math2/node/HR720TJZZR.md" tag="HR720TJZZR" kind="problem">
# Linear-quadratic regulator

## Problem

Let $A$ be a real $n \times n$ matrix and $B$ a real $n \times m$ matrix.
Let $Q$ and $F$ be symmetric [[positive semidefinite|01MMVW8CKC]] $n \times n$ matrices, and let $R$ be a symmetric [[positive definite|0KX66CZMBB]] $m \times m$ matrix.
Let $T > 0$ and $x_0 \in \R^n$.
The *finite-horizon linear-quadratic regulator problem* is to minimize
$$J(u) = \int_0^T \big( x(t)^{T} Q x(t) + u(t)^{T} R u(t) \big) \, dt + x(T)^{T} F x(T)$$
over all controls $u \in L^2([0, T]; \R^m)$, where $x$ solves $x' = A x + B u$, $x(0) = x_0$.

## Remarks

The problem is solved.
Let $P$ be the solution of the [[Riccati differential equation|Y4V154JWR7]] $-P' = A^{T} P + P A - P B R^{-1} B^{T} P + Q$ with $P(T) = F$.
Then the unique optimal control is given by the linear feedback $u^*(t) = -R^{-1} B^{T} P(t) x^*(t)$, and the optimal cost is $x_0^{T} P(0) x_0$.
The proof completes the square: along any trajectory, $\frac{d}{dt} (x^{T} P x) = -x^{T} Q x - u^{T} R u + (u + R^{-1} B^{T} P x)^{T} R (u + R^{-1} B^{T} P x)$.
Integrating from $0$ to $T$ gives $J(u) = x_0^{T} P(0) x_0 + \int_0^T \abs{R^{1/2}(u + R^{-1} B^{T} P x)}^2 \, dt$, which is smallest exactly for the feedback above.
For the infinite horizon $T = \infty$ and $F = 0$, assume $(A, B)$ stabilizable and $(C, A)$ detectable, where $Q = C^{T} C$.
Then the [[algebraic Riccati equation|GGZV75ZZMV]] $A^{T} P + P A - P B R^{-1} B^{T} P + Q = 0$ has a unique symmetric positive semidefinite solution $P$.
The feedback $u = -R^{-1} B^{T} P x$ is optimal and makes the closed loop asymptotically stable.

## Explanations

### L1 human

The linear-quadratic regulator steers a linear system toward zero while keeping both the deviation and the control effort small.
The cost charges squares of both, weighted by the matrices $Q$ and $R$.
The answer is remarkably simple: the best control is a fixed matrix times the current state, computed once from a Riccati equation.
It is one of the most used results of control theory, from aircraft autopilots to robots.
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# Rayleigh quotient for symmetric kernels

## Summary

Let $K$ be a Hermitian kernel with integral operator $T_K$.
The Rayleigh quotient of $K$ at a nonzero function $\varphi$ is the real number $\ip{T_K \varphi}{\varphi} / \ip{\varphi}{\varphi}$.
It is the integral analogue of the [[Rayleigh quotient|06WHP8SE98]] of a Hermitian matrix.
Its extreme values and critical points give the eigenvalues and eigenfunctions of $T_K$, which turns eigenvalue problems for integral equations into problems of optimization.

## Explanations

### L1 human

The Rayleigh quotient measures how much the integral operator stretches a function in its own direction.
The function that is stretched the most is an eigenfunction, and the amount of stretching is the largest eigenvalue.
For $K(x, y) = xy$ on $[0, 1]$ and $\varphi = 1$, the quotient is $1/4$, while for the eigenfunction $\varphi(x) = x$ it reaches the maximum $1/3$.
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# Determinate moment problem

## Definition

Let $K$ be a closed subset of $\R$.
Let $\mu$ be a positive Borel [[measure|PR11BZQQPG]] on $\R$ with support in $K$, such that every [[moment|01XAHPADBA]] $m_n = \int x^n \, d\mu(x)$, $n \ge 0$, is finite.
The [[moment problem|CS7TWHY7JM]] for $\mu$ on $K$ is *determinate* if the following holds.
Every positive Borel measure $\nu$ with support in $K$ and $\int x^n \, d\nu(x) = m_n$ for all $n \ge 0$ equals $\mu$.

## Remarks

Example: the standard normal distribution is determinate on $K = \R$.
Its moments are $m_{2n} = (2n - 1)!! \le (2n)^n$ and $m_{2n+1} = 0$, so $\sum_n m_{2n}^{-1/(2n)}$ diverges, and Carleman's condition applies.
Example: every measure with support in a compact set $K$ is determinate on $K$, by the density of polynomials in $C(K)$.
Non-example: the log-normal distribution with density $f(x) = (x \sqrt{2\pi})^{-1} e^{-(\ln x)^2/2}$ on $(0, \infty)$ is not determinate, since $f(x)(1 + \sin(2\pi \ln x))$ is a different density with the same moments.
Determinacy depends on $K$: a measure on $[0, \infty)$ can be determinate among measures on $[0, \infty)$ (Stieltjes determinate) but not among measures on $\R$ (Hamburger determinate).
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# Kobayashi hyperbolic manifold

## Summary

A complex manifold is Kobayashi hyperbolic when its [[Kobayashi pseudodistance|DY8QQ5GTTX]] is a true distance, that is, it separates points.
Bounded domains in $\C^n$ and compact Riemann surfaces of genus at least $2$ are hyperbolic.
The spaces $\C^n$, the projective spaces and the complex tori are not.
A hyperbolic manifold admits no nonconstant holomorphic map from $\C$, and for compact manifolds this property characterises hyperbolicity by [[Brody's lemma|VEV5E2077C]].
Hyperbolicity is linked with the geometry of the canonical bundle and with the arithmetic of varieties, through the [[Kobayashi conjecture|1X3EV2TZNN]] and the [[Green–Griffiths–Lang conjecture|AD9GD9EM9R]].

## Explanations

### L1 human

A complex manifold is hyperbolic when holomorphic discs inside it cannot be too large.
Then the Kobayashi construction, which measures distances with chains of discs, sees every pair of distinct points at a positive distance.
In $\C$ there are discs of every radius, so all Kobayashi distances are $0$, and $\C$ is not hyperbolic.
The unit disc is hyperbolic, and its Kobayashi distance is the usual hyperbolic distance.
A domain such as $\C$ minus two points is also hyperbolic, which is one way to see the little Picard theorem.
</document>

<document source="https://tamnd.github.io/math2/node/HS0R4H5SAW.md" tag="HS0R4H5SAW" kind="definition">
# Generalized Boolean algebra

## Definition

Let $L$ be a [[distributive lattice|0T36P6TXQ6]] with meet $\wedge$, join $\vee$ and a least element $0$.
The lattice $L$ is a *generalized Boolean algebra* if for all $a, b \in L$ there is an element $x \in L$ such that
$$(a \wedge b) \vee x = a \quad \text{and} \quad (a \wedge b) \wedge x = 0.$$
The element $x$ is unique by distributivity.
It is the *relative complement* of $b$ in $a$, written $a \setminus b$.

## Remarks

Example: the finite subsets of $\N$, ordered by inclusion, form a generalized Boolean algebra.
For finite sets $a$ and $b$, the set $x = a \setminus b$ is finite, $(a \cap b) \cup x = a$, and $(a \cap b) \cap x = \emptyset$.
This lattice has no greatest element, so it is not a [[Boolean algebra|06GKBTAA5E]].
Non-example: the chain $0 < 1 < 2$ is a distributive lattice with least element $0$, but for $a = 2$ and $b = 1$ there is no $x$ with $1 \vee x = 2$ and $1 \wedge x = 0$.
Indeed $1 \vee x = 2$ forces $x = 2$, and then $1 \wedge x = 1 \ne 0$.
Equivalently, a distributive lattice with $0$ is a generalized Boolean algebra if and only if each interval $[0, a]$ is a Boolean algebra.
A generalized Boolean algebra with a greatest element is the same as a Boolean algebra.
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<document source="https://tamnd.github.io/math2/node/HS2NTCMZ4T.md" tag="HS2NTCMZ4T" kind="concept">
# Minkowski content

## Summary

The Minkowski content measures the $m$-dimensional size of a subset of $\R^n$ through the volume of its thin neighbourhoods.
One thickens the set by $r$, takes the [[Lebesgue measure|2Z2QVWZ262]] of the result, and divides by the volume of an $(n-m)$-dimensional ball of radius $r$.
For nice sets it gives length, area and volume, and for rough sets it is linked to the box-counting dimension.

## Explanations

### L1 human

To find the length of a curve in the plane, paint a thin band of width $2r$ around it and measure the area of the band.
The area is close to $2r$ times the length, so dividing by $2r$ recovers the length as $r$ shrinks.
For a segment of length $L$, the band is a rectangle of area $2rL$ with two half-discs at the ends, and the ratio tends to $L$.
The same idea measures surfaces in space through thin shells.
</document>

<document source="https://tamnd.github.io/math2/node/HS5ZEM7TWA.md" tag="HS5ZEM7TWA" kind="definition">
# Bifurcation diagram

## Definition

Let $I \subseteq \R$ be an interval of parameters and $X$ a state space.
Let $(\phi_\mu)_{\mu \in I}$ be a one-parameter family of dynamical systems on $X$, flows or maps.
For each $\mu$, let $A_\mu \subseteq X$ be a chosen invariant set of $\phi_\mu$.
Typical choices are the set of [[equilibria|0N8NGKDZDZ]] and periodic points, or the [[attractor|7FWX24X5MW]] reached from a fixed initial point.
The *bifurcation diagram* of the family for this choice is the set
$$\set{(\mu, x) \in I \times X : x \in A_\mu}.$$

## Remarks

Example: for $\dot x = \mu x - x^3$ on $\R$, take $A_\mu$ to be the set of equilibria.
The equilibria solve $x(\mu - x^2) = 0$, so the diagram is the line $x = 0$ together with the parabola $\mu = x^2$.
The branches meet at $(0, 0)$, the [[pitchfork bifurcation|P88X34MSVV]] of [[bifurcation theory|D9TW46P19Y]].
Usually stable branches are drawn solid and unstable ones dashed.
For the [[logistic map|QN9X9BNKTX]], one takes for $A_\mu$ the limit points of the orbit of a typical initial point.
Non-example: a plot of a single orbit $x_n$ against the time $n$, at one parameter value, is a time series, not a bifurcation diagram.
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<document source="https://tamnd.github.io/math2/node/HS65DW0HXH.md" tag="HS65DW0HXH" kind="concept">
# Bump function

## Summary

A bump function is a [[smooth function|0V33BNKZ9S]] that vanishes outside a bounded region.
Such functions exist because some smooth functions are not analytic.
They are used to cut off, localize and glue functions.
Examples are smooth partitions of unity, mollification, and test functions in the theory of distributions.

## Explanations

### L1 human

Picture a smooth hill that rises from flat ground and returns to flat ground, with no corners anywhere.
Outside the hill, the function is exactly zero.
A polynomial cannot do this: a polynomial that is zero on an interval is zero everywhere.
The standard formula is $e^{-1/(1 - x^2)}$ for $\abs{x} < 1$ and $0$ elsewhere.
</document>

<document source="https://tamnd.github.io/math2/node/HS7GJ1C86M.md" tag="HS7GJ1C86M" kind="definition">
# Jónsson terms

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras and let $n \ge 1$.
*Jónsson terms* for $\mathcal{V}$ are ternary terms $d_0, d_1, \dots, d_n$ such that $\mathcal{V}$ satisfies the following [[identities|PKD440G2XW]]:

1. $d_0(x, y, z) = x$ and $d_n(x, y, z) = z$.
2. $d_i(x, y, x) = x$ for every $i \le n$.
3. $d_i(x, x, z) = d_{i+1}(x, x, z)$ for every even $i < n$.
4. $d_i(x, z, z) = d_{i+1}(x, z, z)$ for every odd $i < n$.

## Remarks

Jónsson's theorem says that $\mathcal{V}$ is a [[congruence-distributive variety|9GZ7F6ETC6]] if and only if $\mathcal{V}$ has Jónsson terms for some $n$.
Example: lattices have Jónsson terms with $n = 2$: $d_0 = x$, $d_1 = (x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$, $d_2 = z$.
The checks are $d_1(x, y, x) = x$, $d_0(x, x, z) = x = d_1(x, x, z)$, and $d_1(x, z, z) = z = d_2(x, z, z)$.
More generally, a majority term $m$ always gives Jónsson terms $x, m, z$.
Non-example: for groups, the terms $x$, $x y^{-1} z$, $z$ are not Jónsson terms.
Clause 2 fails for the middle term: in the group $\Z$, the value $x - y + x$ differs from $x$ when $y \ne x$.

## Explanations

### L1 human

Jónsson terms are a chain of three-input formulas that witness, by equations alone, that a variety has distributive congruence lattices.
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<document source="https://tamnd.github.io/math2/node/HS9VYE19Z2.md" tag="HS9VYE19Z2" kind="definition">
# Castelnuovo–Mumford regularity

## Definition

Let $k$ be a field and let $S = k[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]] graded by total degree.
For an integer $j$, let $S(-j)$ be $S$ with grading shifted so that $S(-j)_d = S_{d - j}$.
Let $M \ne 0$ be a [[finitely generated|05QY1VEBYD]] [[graded|8KB4SQGC0C]] $S$-module.
Let
$$0 \to F_p \to \cdots \to F_1 \to F_0 \to M \to 0$$
be a minimal graded free [[resolution|HK2AA1YSQ2]] of $M$, with $F_i = \bigoplus_j S(-j)^{\beta_{i,j}}$.
The numbers $\beta_{i,j}$ are the graded [[Betti numbers|3B1743XS7S]] of $M$.
The *Castelnuovo–Mumford regularity* of $M$ is
$$\operatorname{reg} M = \max \set{ j - i : \beta_{i,j} \ne 0 }.$$

## Remarks

Let $f \in S$ be homogeneous of degree $d \ge 1$ and $M = S/(f)$.
The minimal resolution is $0 \to S(-d) \to S \to M \to 0$, so $\beta_{0,0} = 1$ and $\beta_{1,d} = 1$.
Hence $\operatorname{reg} M = \max(0, d - 1) = d - 1$.

Minimality of the resolution is needed.
Take $S = k[x]$ and $M = S/(x)$, so that $\operatorname{reg} M = 0$.
Add the exact summand $S(-5) \to S(-5)$ in homological degrees $1$ and $0$.
The result is a non-minimal resolution with the pair $(i, j) = (0, 5)$, and it would give the wrong value $5$.

An equivalent form uses [[local cohomology|6JB30BXDA2]] with respect to $\mathfrak{m} = (x_1, \ldots, x_n)$: $\operatorname{reg} M = \max_i \set{ a_i(M) + i }$, where $a_i(M)$ is the largest degree $d$ with $H^i_{\mathfrak{m}}(M)_d \ne 0$, and the maximum is over the $i$ with $H^i_{\mathfrak{m}}(M) \ne 0$.
For a nonzero proper homogeneous ideal $I$, one has $\operatorname{reg} I = \operatorname{reg}(S/I) + 1$.
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# Cochran's theorem

## Statement

Let $X = (X_1, \dots, X_n)^T$ be a random vector with the [[multivariate normal distribution|0G558GJQMW]] $\mathcal{N}_n(0, I_n)$, that is, $X_1, \dots, X_n$ are independent standard normal variables.
Let $A_1, \dots, A_k$ be real [[symmetric|0N28QN4CYA]] $n \times n$ matrices with $A_1 + \cdots + A_k = I_n$.
Let $r_i$ be the [[rank|0EWQEYRVA8]] of $A_i$ and let $Q_i = X^T A_i X$ be the corresponding [[quadratic form|0RBQS1JJCR]], so that $X^T X = Q_1 + \cdots + Q_k$.
Then the following are equivalent.

1. $r_1 + \cdots + r_k = n$.
2. The random variables $Q_1, \dots, Q_k$ are [[independent|0E9502NT92]], and each $Q_i$ has the [[chi-squared distribution|0RFMMBHFTV]] with $r_i$ degrees of freedom.

## Remarks

The proof of 1 implies 2 rests on a fact of linear algebra.
If symmetric matrices sum to $I_n$ and their ranks sum to $n$, then each $A_i$ is an [[idempotent matrix|0K7TD9BV2D]] and $A_i A_j = 0$ for $i \ne j$.
Then $\R^n$ is the orthogonal direct sum of the images of the $A_i$.
In an orthonormal basis adapted to this sum, each $Q_i$ is a sum of $r_i$ squares of distinct coordinates of a standard normal vector.
This gives independence and the chi-squared laws.
For 2 implies 1, the sum of the independent $Q_i$ is chi-squared with $\sum r_i$ degrees of freedom.
But $X^T X$ is chi-squared with $n$ degrees of freedom, so $\sum r_i = n$.

Example: with $J$ the $n \times n$ matrix of ones, take $A_1 = \frac{1}{n} J$ and $A_2 = I_n - \frac{1}{n} J$, of ranks $1$ and $n - 1$.
Then $Q_1 = n \bar{X}^2$ and $Q_2 = \sum_i (X_i - \bar{X})^2$ are independent, with $\chi^2$ laws with $1$ and $n - 1$ degrees of freedom.
This decomposition is the basis of the analysis of variance.

Without normality the conclusion fails: for independent variables with another distribution, the quadratic forms are in general neither chi-squared nor independent.

## Explanations

### L1 human

The squared length of a standard normal vector can be split into pieces along perpendicular subspaces, like Pythagoras' theorem.
Cochran's theorem says that the pieces are then independent, and each piece is chi-squared with as many degrees of freedom as the dimension of its subspace.
The dimensions must add up to the full dimension, and that is exactly the condition of the theorem.

### L2 human

In linear models, sums of squares such as the regression sum of squares and the residual sum of squares are quadratic forms in normal errors.
Cochran's theorem shows that they are independent chi-squared variables, so their scaled ratio has an F-distribution.
This is why F-tests in the analysis of variance and in regression have exact null distributions.
The independence of the sample mean and the sample variance of a normal sample is the case of two pieces.
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# Freedman's classification of simply connected 4-manifolds

## Statement

A *unimodular form* is a [[symmetric bilinear form|0AXZNE5XNH]] $Q \colon \Z^n \times \Z^n \to \Z$ whose matrix has determinant $\pm 1$.
It is *even* if $Q(x, x)$ is even for all $x \in \Z^n$, and *odd* otherwise.

1. For every unimodular form $Q$ there is a closed, simply connected, topological [[4-manifold|8H58Q6YJCH]] whose [[intersection form|WK8GKRKAVD]] is isomorphic to $Q$.
2. Two closed, simply connected, topological $4$-manifolds are [[homeomorphic|0QT00PB5G2]] if and only if their intersection forms are isomorphic and their [[Kirby–Siebenmann invariants|E4NJER1RFQ]] in $\Z/2$ are equal.
3. If $Q$ is even, there is exactly one such manifold with form $Q$ up to homeomorphism. If $Q$ is odd, there are exactly two, one with each value of the Kirby–Siebenmann invariant.

## Remarks

Michael Freedman proved the theorem in 1982.
The main step is his proof that Casson handles are homeomorphic to standard open handles, which makes the topological h-cobordism theorem hold for simply connected $4$-manifolds.
For an even form, the Kirby–Siebenmann invariant equals the signature divided by $8$, modulo $2$, so it is determined by $Q$.
The manifold with odd form and Kirby–Siebenmann invariant $1$ admits no smooth structure.
For example, the form $(1)$ is realized by the complex projective plane $\C P^2$ and by a fake $\C P^2$ that is not homeomorphic to it.
The [[$E_8$ manifold|BNDRN0G9BR]] realizes the $E_8$ form.
The case $Q = 0$ of rank $0$ is the four-dimensional topological Poincaré conjecture: a homotopy $4$-sphere is homeomorphic to $S^4$.

## Explanations

### L1 human

In dimension four, the way surfaces cross inside a manifold is recorded by a square table of integers, the intersection form.
Freedman's theorem says that, for manifolds without loops that cannot be contracted, this table and one extra bit decide the manifold up to continuous deformation.
Every possible table occurs.

### L2 human

The theorem reduces the topological classification of closed simply connected $4$-manifolds to the algebra of unimodular forms over $\Z$.
Indefinite forms are classified by rank, signature and type, so these manifolds are well understood.
The smooth picture is very different: [[Donaldson's theorem|34B7K77P31]] and [[Rokhlin's theorem|12BXVH5W5H]] show that many of Freedman's manifolds have no smooth structure.
Together the two results give [[exotic $\R^4$|X44PNC38E2]].
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# Schauder estimates

## Statement

Let $n \ge 1$, let $0 < \alpha < 1$, and let $U \subseteq \R^n$ be a bounded open set.
Let
$$L u = \sum_{i,j=1}^{n} a_{ij}(x) \, \partial_i \partial_j u + \sum_{i=1}^{n} b_i(x) \, \partial_i u + c(x) \, u,$$
where $a_{ij} = a_{ji}$, $b_i$ and $c$ are functions on $U$.
Let $\lambda > 0$ and $\Lambda$ be constants with $\sum_{i,j} a_{ij}(x) \xi_i \xi_j \ge \lambda \abs{\xi}^2$ for all $x \in U$ and $\xi \in \R^n$.
Assume that every coefficient has $C^{0,\alpha}(\overline{U})$ norm at most $\Lambda$.
Here $C^{k,\alpha}$ denotes the [[Hölder space|P101AX1ZSC]] of functions whose derivatives up to order $k$ are bounded and [[Hölder continuous|012PBA0FMD]] with exponent $\alpha$.
Its norm is written $\norm{\cdot}_{C^{k,\alpha}}$.

1. (Interior estimate.) Let $u \in C^{2,\alpha}(U)$ be bounded with $f = L u \in C^{0,\alpha}(\overline{U})$, and let $V$ be an open set whose closure is a compact subset of $U$. Then
$$\norm{u}_{C^{2,\alpha}(\overline{V})} \le C \big( \sup_U \abs{u} + \norm{f}_{C^{0,\alpha}(\overline{U})} \big),$$
where $C$ depends only on $n$, $\alpha$, $\lambda$, $\Lambda$, $U$ and $V$.
2. (Global estimate.) Assume moreover that $U$ has $C^{2,\alpha}$ boundary. Let $u \in C^{2,\alpha}(\overline{U})$, let $f = L u$, and let $\varphi \in C^{2,\alpha}(\overline{U})$ with $u = \varphi$ on $\partial U$. Then
$$\norm{u}_{C^{2,\alpha}(\overline{U})} \le C \big( \sup_U \abs{u} + \norm{\varphi}_{C^{2,\alpha}(\overline{U})} + \norm{f}_{C^{0,\alpha}(\overline{U})} \big),$$
where $C$ depends only on $n$, $\alpha$, $\lambda$, $\Lambda$ and $U$.

## Remarks

Proof idea: first treat the Laplacian, where $u$ is given by convolution of $f$ with the fundamental solution.
Potential estimates show that the second derivatives of the Newtonian potential of a $C^{0,\alpha}$ function are $C^{0,\alpha}$.
A linear change of variables gives the same bound for operators with constant coefficients.
For variable coefficients one freezes them at a point.
On a small ball, $L$ is close to a constant-coefficient operator, and the error is absorbed using the smallness of the Hölder seminorm of the coefficients on small balls.
The global estimate follows by flattening the boundary locally and treating half-balls in the same way.

Example: for $L = \Delta$, the estimate says that a harmonic function is controlled in $C^{2,\alpha}$ on interior sets by its supremum.
It also says that a solution of Poisson's equation $\Delta u = f$ with $f \in C^{0,\alpha}$ has second derivatives in $C^{0,\alpha}$.

The restriction $0 < \alpha < 1$ is needed.
For $\alpha = 0$ the estimate fails: there are continuous $f$ for which $\Delta u = f$ has no solution with continuous second derivatives.
For $\alpha = 1$ it also fails: a Lipschitz $f$ need not give $u$ with Lipschitz second derivatives.

Together with the method of continuity, the global estimate gives existence of $C^{2,\alpha}(\overline{U})$ solutions of the Dirichlet problem for $L$ with $c \le 0$, starting from the case of the Laplacian.

## Explanations

### L1 human

The Schauder estimates say that solutions of elliptic equations are two derivatives smoother than the right-hand side, when smoothness is measured with Hölder norms.
If the data are slightly better than continuous, the second derivatives of the solution are slightly better than continuous too, with a bound on how much.
This is a quantitative form of the idea that elliptic equations smooth things out.

### L2 human

The Schauder estimates are a priori estimates: they bound the $C^{2,\alpha}$ norm of a solution that is already known to exist.
Combined with the maximum principle, which gives uniqueness and bounds $\sup \abs{u}$ when $c \le 0$, they give existence of classical solutions through the method of continuity or the Leray–Schauder theory.
Applied to derivatives of $u$, they give higher regularity: if the coefficients and $f$ are in $C^{k,\alpha}$, then $u$ is in $C^{k+2,\alpha}$ inside $U$.
They are the Hölder counterpart of the $L^p$ theory of Calderón and Zygmund.
They are also a key step in nonlinear problems such as Hilbert's nineteenth problem, once the De Giorgi–Nash–Moser theorem has given Hölder continuity of the coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/HSQJSK4E23.md" tag="HSQJSK4E23" kind="definition">
# Additive functor

## Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be [[preadditive categories|SC6FD5F4J0]].
A [[functor|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathcal{D}$ is *additive* if $F(f + g) = F(f) + F(g)$ for all objects $X, Y$ of $\mathcal{C}$ and all $f, g \colon X \to Y$.
Equivalently, each map $\Hom_{\mathcal{C}}(X, Y) \to \Hom_{\mathcal{D}}(FX, FY)$, $f \mapsto F(f)$, is a homomorphism of abelian groups.

## Remarks

An additive functor sends zero morphisms to zero morphisms and zero objects to zero objects.
If $\mathcal{C}$ and $\mathcal{D}$ are [[additive categories|B2VP96NJSK]], a functor is additive if and only if it preserves binary [[biproducts|BZEJKTQ76T]].

Example: for a fixed [[abelian group|082GHNPJ53]] $A$, the functor $\Hom(A, -)$ from abelian groups to abelian groups is additive.
For $f, g \colon X \to Y$ and $u \colon A \to X$, one has $(f + g) \circ u = f \circ u + g \circ u$.
Likewise the [[tensor product|F0HQXQ0HCD]] functor $- \otimes A$ is additive.

Non-example: the functor $X \mapsto X \otimes X$, $f \mapsto f \otimes f$, on abelian groups is not additive.
On $X = \Z$, with $\Z \otimes \Z = \Z$, the morphism $\id + \id = 2$ goes to multiplication by $4$, while $F(\id) + F(\id)$ is multiplication by $2$.
</document>

<document source="https://tamnd.github.io/math2/node/HSRNND3TG5.md" tag="HSRNND3TG5" kind="statement">
# Universal approximation theorem

## Statement

Let $\sigma \colon \R \to \R$ be [[continuous|0X3WGX95FE]], and let $n \ge 1$.
Let $\mathcal{N}_\sigma$ be the set of all functions $g \colon \R^n \to \R$ of the form
$$g(x) = \sum_{i=1}^{N} c_i \, \sigma(w_i \cdot x + b_i).$$
Here $N \ge 1$, $c_i, b_i \in \R$ and $w_i \in \R^n$.
These are the functions computed by a neural network with one hidden layer and activation $\sigma$.
Then the following are equivalent.

1. For every [[compact set|0R6EWFKRCF]] $K \subseteq \R^n$, the restrictions of the functions in $\mathcal{N}_\sigma$ to $K$ are [[dense|0BSJSX7ME2]] in $C(K)$ with the supremum norm.
2. $\sigma$ is not a polynomial.

## Remarks

The idea of the proof: if $\sigma$ is a polynomial of degree $d$, every $g \in \mathcal{N}_\sigma$ is a polynomial of degree at most $d$.
These polynomials do not form a dense set.
Conversely, by smoothing one may assume $\sigma$ is smooth; then difference quotients in the weight show that $t \mapsto t^k \sigma^{(k)}(w t + b)$ is a limit of elements of $\mathcal{N}_\sigma$ in one variable.
Since $\sigma$ is not a polynomial, for each $k$ there is a $b$ with $\sigma^{(k)}(b) \ne 0$, so all monomials $t^k$ are limits, and the Weierstrass theorem applies in one variable.
The case of $n$ variables reduces to one variable because the ridge functions $h(w \cdot x)$ span a dense set in $C(K)$.

Example: the logistic sigmoid $\sigma(t) = 1 / (1 + e^{-t})$ and the rectifier $\sigma(t) = \max(0, t)$ are not polynomials.
So networks with one hidden layer and either activation give [[uniform approximations|XAARVNKXF1]] of every continuous function on every compact set.

The theorem says nothing about how large $N$ must be for a given accuracy, and it does not say how to find the weights.
The sigmoid case was proved by George Cybenko in 1989, and the characterization for continuous $\sigma$ by Moshe Leshno, Vladimir Lin, Allan Pinkus and Shimon Schocken in 1993.

## Explanations

### L1 human

A neural network with one hidden layer adds up many copies of one fixed function $\sigma$, each stretched, shifted and pointed in some direction.
The theorem says that such sums can imitate any continuous function on a bounded closed region as closely as you want.
The only condition on $\sigma$ is that it is not a polynomial.

### L2 human

This is a density theorem in the same family as the Weierstrass and Stone–Weierstrass theorems, with ridge functions $\sigma(w \cdot x + b)$ as building blocks.
It justifies neural networks as a flexible class of approximants but gives no rates.
Rates depend on the smoothness of the target and on the dimension, as in classical approximation theory.
Related results treat deep networks of bounded width and approximation in $L^p$ norms.
</document>

<document source="https://tamnd.github.io/math2/node/HSSC2ZZWMY.md" tag="HSSC2ZZWMY" kind="definition">
# Stochastic approximation

## Definition

Let $h \colon \R^d \to \R^d$ be an unknown function.
Assume that at each point $\theta \in \R^d$ one can observe a [[random variable|0DKRR22AEQ]] whose mean is $h(\theta)$.
A *stochastic approximation algorithm* is a recursion
$$\theta_{n+1} = \theta_n + a_n Y_{n+1}, \qquad n \ge 0,$$
in which:

1. $\theta_0 \in \R^d$ is an initial point, possibly random.
2. $(a_n)_{n \ge 0}$ is a [[sequence|0QE8STGJH2]] of positive step sizes.
3. $Y_{n+1}$ is a noisy observation at $\theta_n$: with $\mathcal{F}_n = \sigma(\theta_0, Y_1, \dots, Y_n)$, the [[conditional expectation|0YAMJRK04E]] satisfies $E[Y_{n+1} \mid \mathcal{F}_n] = h(\theta_n)$.

The aim is that $\theta_n$ converges to a zero of $h$.

## Remarks

Example: let $Z_1, Z_2, \dots$ be independent with common mean $m$, and take $h(\theta) = m - \theta$, $Y_{n+1} = Z_{n+1} - \theta_n$ and $a_n = 1/(n+1)$, with $\theta_0 = 0$.
Then $E[Y_{n+1} \mid \mathcal{F}_n] = m - \theta_n$, and by induction $\theta_n = (Z_1 + \cdots + Z_n)/n$, the sample mean, which tends to the zero $m$ of $h$.

Non-example: the deterministic Newton iteration $\theta_{n+1} = \theta_n - h(\theta_n)/h'(\theta_n)$ is not a stochastic approximation algorithm, since it needs exact values of $h$ and $h'$ rather than noisy observations, so clause 3 fails.

The usual conditions on the step sizes are $\sum_n a_n = \infty$ and $\sum_n a_n^2 < \infty$.
The first condition lets the iterates travel any distance, and the second makes the accumulated noise $\sum_n a_n (Y_{n+1} - h(\theta_n))$ converge.
Writing the noise as a martingale difference sequence, the iterates follow, in the long run, the ordinary differential equation $\dot\theta = h(\theta)$.
</document>

<document source="https://tamnd.github.io/math2/node/HSTFKC0Y39.md" tag="HSTFKC0Y39" kind="definition">
# Fisher information

## Definition

Let $\set{f(\cdot\,; \theta) : \theta \in \Theta}$ be a [[parametric family|MSC0MR8VMH]] of densities with respect to a fixed measure, where $\Theta \subseteq \R$ is an open interval.
Assume that $\theta \mapsto \log f(x; \theta)$ is differentiable for every $x$ with $f(x; \theta) > 0$.
Let $X$ have density $f(\cdot\,; \theta)$, and let
$$s(\theta; X) = \frac{\partial}{\partial \theta} \log f(X; \theta)$$
be the [[score|CP97GRVTQ0]].
The *Fisher information* at $\theta$ is
$$I(\theta) = E_\theta\big[ s(\theta; X)^2 \big] \in [0, \infty].$$
For $\Theta \subseteq \R^d$ open, the *Fisher information matrix* is the $d \times d$ matrix with entries $I_{ij}(\theta) = E_\theta[ s_i(\theta; X) \, s_j(\theta; X) ]$, where $s_i = \partial \log f / \partial \theta_i$.

## Remarks

Under the usual regularity conditions, differentiation under the integral sign gives $E_\theta[s(\theta; X)] = 0$.
Then $I(\theta) = \operatorname{Var}_\theta(s(\theta; X))$, the [[variance|0477R9PYAE]] of the score.
If moreover $\log f$ is twice differentiable, then $I(\theta) = -E_\theta\big[\partial^2_\theta \log f(X; \theta)\big]$.
For $n$ independent observations with the same density, the information is $n I(\theta)$.

Example: let $X$ be [[normal|05AQE0QR0D]] with mean $\theta$ and known variance $\sigma^2$.
Then $\log f(x; \theta) = -(x - \theta)^2 / (2\sigma^2) + c$, so $s = (X - \theta)/\sigma^2$.
Hence $I(\theta) = E_\theta[(X - \theta)^2] / \sigma^4 = 1/\sigma^2$.

Non-example of the regular behaviour: the uniform densities $f(x; \theta) = 1/\theta$ on $(0, \theta)$.
Here $s = -1/\theta$ wherever it is defined, so $E_\theta[s^2] = 1/\theta^2$, but $E_\theta[s] = -1/\theta \ne 0$.
The support depends on $\theta$, differentiation under the integral fails, and the usual properties of $I(\theta)$ do not hold.
</document>

<document source="https://tamnd.github.io/math2/node/HSV1J7BJXX.md" tag="HSV1J7BJXX" kind="concept">
# Fibration and cofibration

## Summary

In a [[model category|SDTZVZ6B4Q]], the fibrations and the cofibrations are two of the three distinguished classes of morphisms, besides the weak equivalences.
They abstract the [[Serre fibrations|PB9JXQ2KC3]] and the [[cofibrations|W5WS5EY8ZM]] of topology: maps along which homotopies lift, and inclusions along which homotopies extend.
Each class is determined by the other together with the weak equivalences, through lifting properties.
Fibrant and cofibrant objects are the good objects on which homotopy classes of maps compute the homotopy category.

## Explanations

### L1 human

A fibration is a map along which paths and deformations can be lifted, like the projection of a product $X \times F \to X$.
A cofibration is a map that behaves like the inclusion of a nice subspace, such as a circle as the boundary of a disc.
A model category specifies which maps play these two roles.
Every map can be split into a cofibration followed by a fibration, one of them being a weak equivalence.
This lets one replace objects by better ones without changing their homotopy type.
</document>

<document source="https://tamnd.github.io/math2/node/HSWWF35BEX.md" tag="HSWWF35BEX" kind="statement">
# Path lifting property

## Statement

Let $p \colon E \to X$ be a [[covering map|3XZRRE165N]], let $I = [0, 1]$, let $\gamma \colon I \to X$ be a continuous path, and let $e_0 \in E$ with $p(e_0) = \gamma(0)$.
Then there is a unique continuous path $\tilde\gamma \colon I \to E$ with $\tilde\gamma(0) = e_0$ and $p \circ \tilde\gamma = \gamma$.

## Remarks

For existence, cover $X$ by evenly covered open sets.
By the Lebesgue number lemma, there is a subdivision $0 = s_0 < s_1 < \cdots < s_k = 1$ of the [[interval|0Q1D85B8G6]] with each $\gamma([s_{j-1}, s_j])$ in one evenly covered set $U_j$.
The lift is built step by step: on $[s_{j-1}, s_j]$ it is $(p|_{V})^{-1} \circ \gamma$, where $V$ is the sheet over $U_j$ that contains the point already reached.
Uniqueness holds because the set where two lifts agree is open and closed in the connected interval $I$, and it contains $0$.
Example: for $p(t) = e^{2\pi i t}$ from $\R$ to $S^1$ and $\gamma(s) = e^{2\pi i n s}$, the lift starting at $0$ is $\tilde\gamma(s) = ns$, which ends at $n$.
The covering hypothesis is needed: for the local homeomorphism $q = p|_{(0, 2)} \colon (0, 2) \to S^1$, the loop $\gamma(s) = e^{-2\pi i s}$ with $e_0 = 1$ has no lift.
A lift in $(0, 2)$ is also a lift in $\R$, so by uniqueness it would be $s \mapsto 1 - s$, whose value at $s = 1$ is $0 \notin (0, 2)$.

## Explanations

### L1 human

A covering map is like a projection from many stacked floors to a ground floor.
If you walk along a path on the ground floor and choose a floor to start on, there is exactly one way to walk upstairs that stays above your path.
This lets one read off information about loops downstairs from where the lifted path ends upstairs.

### L2 human

The path lifting property, together with its extension to homotopies, is the main tool of covering space theory.
It shows that $p_* \colon \pi_1(E, e_0) \to \pi_1(X, p(e_0))$ is injective, and it computes the [[fundamental group of the circle|6SQXPE0JPN]] through the covering $\R \to S^1$.
The endpoint $\tilde\gamma(1)$ of the lift of a loop defines an action of $\pi_1(X, x_0)$ on the fibre $p^{-1}(x_0)$, which leads to the classification of covering spaces.
</document>

<document source="https://tamnd.github.io/math2/node/HSYGHAY6WE.md" tag="HSYGHAY6WE" kind="definition">
# Scott domain

## Definition

Let $D$ be a [[directed-complete partial order|7TSXP95QN6]].
It is a *Scott domain* if:

1. $D$ is algebraic: for every $x \in D$, the set of [[compact elements|9F7TYES4KE]] $c \le x$ is [[directed|D9RY4667K6]] and has [[supremum|0R3H5PANRQ]] $x$;
2. $D$ is bounded complete: every subset of $D$ that has an [[upper bound|0S60DVMVPN]] in $D$ has a supremum in $D$.

## Remarks

Taking the empty subset in clause 2 gives a least element when $D$ is nonempty; some authors require $D$ nonempty, and some also require countably many compact elements.
Example: the partial functions from $\N$ to $\N$, ordered by extension.
The compact elements are the finite partial functions, and every partial function is the union of its finite restrictions.
A family with a common extension consists of compatible functions, and their union is its supremum.
Non-example: take the poset with elements $\bot, a, b, c, d$, where $\bot$ is below all, and $a, b$ are each below both $c$ and $d$.
It is a finite dcpo, but $\set{a, b}$ has the upper bounds $c$ and $d$ and no least one.
So it is not bounded complete.
Every algebraic lattice is a Scott domain.
</document>

<document source="https://tamnd.github.io/math2/node/HT4Q9K3P2Z.md" tag="HT4Q9K3P2Z" kind="concept">
# de Casteljau's algorithm

## Summary

De Casteljau's algorithm computes a point of a [[Bézier curve|NEXBTGYYRG]] by repeated linear interpolation between control points.
It also evaluates polynomials in [[Bernstein–Bézier form|Z5Y0VH10YR]].
It uses only [[convex combinations|21SXYJYC59]], so it is numerically stable.
The intermediate points also give the control points of the two halves of the curve, which is the basis of subdivision methods in computer-aided design.

## Explanations

### L1 human

Draw the polygon through the control points and mark the point a fraction $t$ of the way along each edge.
Join the marked points to get a polygon with one fewer vertex, and repeat.
After as many rounds as the degree, a single point is left, and it is the point of the curve at parameter $t$.
For three points and $t = 1/2$, one takes the two edge midpoints and then the midpoint between them.
</document>

<document source="https://tamnd.github.io/math2/node/HTDYK465JA.md" tag="HTDYK465JA" kind="definition">
# Lusternik–Schnirelmann category

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]].
An open subset $U \subseteq X$ is *contractible in $X$* if the inclusion $U \to X$ is [[null-homotopic|SMEV5BSQ5S]].
The *Lusternik–Schnirelmann category* $\operatorname{cat}(X)$ is the least integer $m \ge 0$ such that there is an [[open cover|049PXYHW1T]] $X = U_0 \cup U_1 \cup \cdots \cup U_m$ by $m + 1$ open sets, each contractible in $X$.
If no such finite cover exists, $\operatorname{cat}(X) = \infty$.

## Remarks

This is the normalized convention; some authors count the open sets instead, which adds $1$.
If $X$ is a path-connected space, $R$ is a commutative ring, and $H^*(X; R)$ has $m$ elements of positive degree with nonzero cup product, then $\operatorname{cat}(X) \ge m$.
The category is a homotopy invariant.

Example: for $n \ge 1$, $\operatorname{cat}(S^n) = 1$.
The sets $U_0 = S^n \setminus \set{N}$ and $U_1 = S^n \setminus \set{S}$, where $N$ and $S$ are the poles, cover $S^n$ and are homeomorphic to $\R^n$, hence contractible, hence contractible in $S^n$.
And $\operatorname{cat}(S^n) \ne 0$, since the identity of $S^n$ is not null-homotopic.

Non-example: the cover of $S^1$ by the single open set $U_0 = S^1$ does not show $\operatorname{cat}(S^1) = 0$, since the inclusion $S^1 \to S^1$ is the identity, which is not null-homotopic.

Other values: the torus $T^n = (S^1)^n$ and the real projective space $\R P^n$ both have category $n$.
</document>

<document source="https://tamnd.github.io/math2/node/HTF9E5W7EZ.md" tag="HTF9E5W7EZ" kind="definition">
# Curry–Howard correspondence

## Definition

Consider intuitionistic propositional logic in [[natural deduction|6KT3BJZ7D1]], with [[implication|0ZNX6DQ3EJ]], [[conjunction|0M28ASWG8T]] and [[disjunction|0H5GKDTB01]].
Consider the simply typed [[lambda calculus|BD756KR8FM]] with function types $A \to B$, product types $A \times B$ and sum types $A + B$.
The *Curry–Howard correspondence* is the map between the two systems given by these clauses.

1. A formula is read as a type: $A \to B$ as the function type, $A \wedge B$ as $A \times B$, and $A \vee B$ as $A + B$.
2. An assumption $A$ with label $x$ is read as a variable $x : A$.
3. Each introduction rule is read as a term constructor: $\to$-introduction as $\lambda$-abstraction, $\wedge$-introduction as pairing, and $\vee$-introduction as an injection.
4. Each elimination rule is read as a destructor: $\to$-elimination as application, $\wedge$-elimination as a projection, and $\vee$-elimination as case analysis.

Under this map, a derivation of $B$ from assumptions $A_1, \dots, A_n$ corresponds to a term $M$ with $x_1 : A_1, \dots, x_n : A_n \vdash M : B$, and conversely.
A detour in a derivation is an introduction followed at once by an elimination.
It corresponds to a redex, and removing the detour corresponds to one reduction step, such as $(\lambda x.\, M) N \to M[N/x]$.

## Remarks

Example: the term $\lambda f.\, \lambda x.\, f\, (f\, x)$ has type $(A \to A) \to (A \to A)$.
Read as a derivation, it uses $\to$-elimination twice and $\to$-introduction twice, and it proves the formula $(A \to A) \to (A \to A)$.
Non-example: Peirce's law $((A \to B) \to A) \to A$ is a classical tautology but has no closed term of its type in the simply typed lambda calculus.
This matches the fact that it is not provable in [[intuitionistic logic|Q11N8X99FK]].
The correspondence extends to quantifiers with dependent types, and to classical logic with control operators.
</document>

<document source="https://tamnd.github.io/math2/node/HTJ03RFKFH.md" tag="HTJ03RFKFH" kind="concept">
# Honeycomb

## Summary

A honeycomb is a filling of three-dimensional space, or of a higher-dimensional space, by [[polyhedra|0325SGSJAR]] or polytopes without gaps or overlaps.
It is the analogue in space of a [[tessellation|GKJK293DXS]] of the plane by polygons.
Honeycombs occur in crystallography, in [[sphere packing|HF32CBN5MA]] and in the study of regular and uniform polytopes.

## Explanations

### L1 human

Stack equal cubes like boxes in a warehouse, filling space completely in every direction.
The result, the cubic honeycomb, is the simplest honeycomb.
The cells may also be other solids, such as truncated octahedra or hexagonal prisms.
Regular tetrahedra alone cannot fill space: their dihedral angle of about $70.5^\circ$ does not divide $360^\circ$.
</document>

<document source="https://tamnd.github.io/math2/node/HTK7A4JRW7.md" tag="HTK7A4JRW7" kind="statement">
# Crandall–Lions uniqueness theorem

## Statement

Let $n \ge 1$, let $T > 0$, and let $H \colon \R^n \times \R^n \to \R$ be continuous, written $H(x, p)$.
Assume that there is a constant $C$ such that, for all $x, y, p, q \in \R^n$,
$$\abs{H(x, p) - H(x, q)} \le C \abs{p - q}, \qquad \abs{H(x, p) - H(y, p)} \le C \abs{x - y} (1 + \abs{p}).$$
Let $g \colon \R^n \to \R$ be bounded and [[uniformly continuous|0M4JJT780X]].
Consider the Cauchy problem for the [[Hamilton–Jacobi equation|5TRXQKP7DY]]
$$u_t + H(x, D_x u) = 0 \quad \text{in } \R^n \times (0, T), \qquad u(x, 0) = g(x) \quad \text{for } x \in \R^n.$$
Then this problem has at most one solution $u \colon \R^n \times [0, T] \to \R$ in the following sense.
The function $u$ is bounded and uniformly continuous, satisfies the initial condition, and is a [[viscosity solution|H1ERJ3T75T]] of the equation in $\R^n \times (0, T)$.

## Remarks

The theorem is a consequence of a comparison principle: if $u$ is a bounded uniformly continuous subsolution and $v$ a bounded uniformly continuous supersolution with $u(\cdot, 0) \le v(\cdot, 0)$, then $u \le v$.
The proof doubles the variables.
One maximises $u(x, t) - v(y, s) - \abs{x - y}^2 / \varepsilon - \abs{t - s}^2 / \varepsilon$ over pairs of points, after adding small extra terms that force a maximum to exist on the unbounded domain.
At the maximum point the two halves of the penalty serve as test functions for $u$ and for $v$, and the two inequalities of the definition are subtracted.
The Lipschitz hypotheses on $H$ make the error terms tend to $0$ with $\varepsilon$, which gives a contradiction if $u - v$ had a positive supremum.

Example: for $H(x, p) = \abs{p}$ the hypotheses hold with $C = 1$.
The unique bounded uniformly continuous viscosity solution is given by the Hopf–Lax formula $u(x, t) = \min_{\abs{y - x} \le t} g(y)$.

The viscosity sense is needed.
For $H(x, p) = \abs{p}$ and $g = 0$, both $u = 0$ and $w(x, t) = \min(0, \abs{x} - t)$ (with $n = 1$) are Lipschitz and satisfy the equation almost everywhere.
Only $u = 0$ is a viscosity solution: at a point $(0, t)$ the function $w - \varphi$ with $\varphi(x, s) = -s$ has a local minimum, while $\varphi_t + \abs{\varphi_x} = -1 < 0$.

## Explanations

### L1 human

Hamilton–Jacobi equations usually have no smooth solutions, and they have many solutions that hold only almost everywhere.
Viscosity solutions are a way to choose the right one.
This theorem says the choice is never ambiguous: given the initial data, there is only one viscosity solution.

### L2 human

Uniqueness is what makes viscosity solutions useful in control theory.
To show that a value function equals a given formula, it is enough to check that both are viscosity solutions of the same Hamilton–Jacobi–Bellman equation with the same data.
The same doubling-of-variables argument gives comparison for stationary equations such as $u + H(x, Du) = 0$, and, with more work, for second-order degenerate elliptic equations.
Existence comes separately, for instance from the value function of a control problem or from the vanishing viscosity method.
</document>

<document source="https://tamnd.github.io/math2/node/HTK9WQZJ1G.md" tag="HTK9WQZJ1G" kind="concept">
# Chain map

## Summary

A chain map is a morphism between two [[chain complexes|WQ82C0M3QB]]: a family of maps $f_n \colon C_n \to D_n$, one in each degree, that commute with the differentials.
Chain complexes and chain maps form a category, which is abelian when the underlying category is abelian.
Every chain map induces maps on [[homology|2J4JJJYP22]], and continuous maps of spaces induce chain maps of singular chains.

## Explanations

### L1 human

A chain complex is a row of objects with downward maps.
A chain map from one such row to another is a ladder of vertical maps such that every square in the ladder commutes.
Going down then across gives the same result as going across then down.
For example, multiplication by $2$ in every degree is a chain map from any complex of abelian groups to itself.
Because the squares commute, a chain map sends cycles to cycles and boundaries to boundaries.
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<document source="https://tamnd.github.io/math2/node/HTMCEARGDD.md" tag="HTMCEARGDD" kind="concept">
# Singular integral operator

## Summary

A singular integral operator is an integral operator $Tf(x) = \int K(x, y)f(y) \, dy$ whose kernel has a non-integrable singularity on the diagonal $x = y$, of size $\abs{x - y}^{-n}$ in $\R^n$.
The integral exists only as a principal value, thanks to cancellation in the kernel.
The Hilbert transform and the Riesz transforms are the basic examples.
Calderón–Zygmund theory shows that such operators are bounded on $L^p$ for $1 < p < \infty$, which is central in harmonic analysis and elliptic PDE.

## Explanations

### L1 human

Many natural operators, such as the map from the Laplacian of a function to its second derivatives, are given by integrals against kernels that blow up like $1/\abs{x}^n$.
Such an integral diverges if taken with absolute values, so it is not an ordinary integral.
It still makes sense because the kernel takes positive and negative values in balanced amounts near the singularity.
For example, the Hilbert transform on the line integrates $f(y)$ against $1/(\pi(x - y))$, which is positive on one side of $x$ and negative on the other.
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<document source="https://tamnd.github.io/math2/node/HTR7J0W66V.md" tag="HTR7J0W66V" kind="concept">
# Coupling

## Summary

A coupling of two [[probability measures|059TJN0NZ1]] $\mu$ and $\nu$ is a joint law of a pair $(X, Y)$.
In it, $X$ has law $\mu$ and $Y$ has law $\nu$.
The two marginals are fixed, but the dependence between $X$ and $Y$ is free to choose.
A clever choice turns a comparison of two laws into a comparison of two random variables on one space.
Couplings are a main tool for bounds in [[total variation distance|3SZESKX3N5]], for convergence of [[Markov chains|0G48PYM96Q]], for [[stochastic ordering|HJPPRZ29YE]] and in optimal transport.

## Explanations

### L1 human

Two dice are a coupling of two uniform laws on $\set{1, \dots, 6}$; they can be thrown independently, or glued so that both always show the same face.
Both choices give the right law for each die, but very different pairs.
Take a coin with heads probability $p$ and one with $q \ge p$, and one uniform number $U$ in $[0, 1]$.
Say that coin one shows heads when $U < p$ and coin two when $U < q$.
Then each coin has the right law, and coin two shows heads whenever coin one does, which proves at once that coin two is more likely to show heads.
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<document source="https://tamnd.github.io/math2/node/HTWCTA2MTJ.md" tag="HTWCTA2MTJ" kind="statement">
# Jenkins–Traub algorithm

## Statement

Let $p$ be a [[monic polynomial|0BDX4MBKT4]] of degree $n \ge 2$ with [[complex|09FTAET5Z4]] coefficients and $p(0) \ne 0$.
Given complex shifts $s_0, s_1, \dots$ with $p(s_\lambda) \ne 0$, define polynomials $H^{(\lambda)}$ of degree at most $n - 1$ by $H^{(0)} = p'$ and
$$H^{(\lambda + 1)}(z) = \frac{1}{z - s_\lambda} \left( H^{(\lambda)}(z) - \frac{H^{(\lambda)}(s_\lambda)}{p(s_\lambda)} \, p(z) \right).$$
Write $\bar H^{(\lambda)}$ for $H^{(\lambda)}$ divided by its leading coefficient, when this is defined.
The *Jenkins–Traub algorithm* runs three stages:

1. no shift: $s_\lambda = 0$ for $0 \le \lambda < M$;
2. fixed shift: $s_\lambda = s$ for $M \le \lambda < L$, where $s$ is a fixed complex number;
3. variable shift: $s_L = s - p(s) / \bar H^{(L)}(s)$ and $s_{\lambda + 1} = s_\lambda - p(s_\lambda) / \bar H^{(\lambda + 1)}(s_\lambda)$ for $\lambda \ge L$.

Suppose $p$ has a [[root|019MA1VTR2]] $\alpha_1$ with $\abs{s - \alpha_1} < \abs{s - \alpha}$ for every other root $\alpha$ of $p$.
Then $\bar H^{(L)}(z) \to p(z)/(z - \alpha_1)$ as $L \to \infty$, coefficient by coefficient, and for $L$ large enough the variable shifts $s_\lambda$ of stage 3 converge to $\alpha_1$.

## Remarks

Write $p(z) = \prod_j (z - \alpha_j)$ over the roots, so that $H^{(0)} = p' = \sum_j p(z)/(z - \alpha_j)$.
One checks by induction that $H^{(\lambda)}$ stays a combination of the polynomials $p(z)/(z - \alpha_j)$, and that each shift multiplies the coefficient of the $j$-th one by $1/(\alpha_j - s_\lambda)$.
With a fixed shift $s$, the term of the root nearest to $s$ grows fastest, so the normalized $\bar H^{(\lambda)}$ tends to $p(z)/(z - \alpha_1)$.
Then $p(z)/\bar H^{(\lambda)}(z)$ is close to $z - \alpha_1$, and the stage 3 step behaves like [[Newton's method|07EXVNF9VP]] with an ever better model; its convergence is very fast.
In practice $s$ is taken at a random angle on a circle whose radius is a lower bound for the moduli of the roots.
Stage 2 is ended by a test on the shifts.
After a root is found, it is divided out and the algorithm runs again on the quotient.
A variant for real polynomials works with quadratic factors and real arithmetic.

## Explanations

### L1 human

The Jenkins–Traub algorithm finds the roots of a polynomial one after the other, roughly in order of increasing size.
It builds a sequence of auxiliary polynomials that slowly forget all roots except the one nearest to a chosen point.
Once that root dominates, a fast Newton-like phase homes in on it.

### L2 human

It was for a long time the standard black-box polynomial root finder in numerical libraries, under the name CPOLY for the complex version.
The stages give a global convergence guarantee that Newton's method alone lacks.
Finding the small roots first makes the deflation steps numerically stable.
</document>

<document source="https://tamnd.github.io/math2/node/HTWWJNSZNQ.md" tag="HTWWJNSZNQ" kind="example">
# Dyadic transformation

## Example

The *dyadic transformation*, or *doubling map*, is
$$T \colon [0, 1) \to [0, 1), \qquad T(x) = 2x \bmod 1.$$

It is a [[measure-preserving transformation|556WBDAHNP]] for [[Lebesgue measure|2Z2QVWZ262]] $\lambda$.
Check on an interval $[a, b) \subseteq [0, 1)$:
$$T^{-1}[a, b) = \Big[\frac{a}{2}, \frac{b}{2}\Big) \cup \Big[\frac{a+1}{2}, \frac{b+1}{2}\Big),$$
a disjoint union of two intervals of length $(b - a)/2$, so $\lambda(T^{-1}[a, b)) = b - a$.

Write $x = \sum_{n \ge 1} x_n 2^{-n}$ in binary, with digits $x_n \in \set{0, 1}$ not eventually all equal to $1$.
Then $T$ deletes the first digit: $T(x) = \sum_{n \ge 1} x_{n+1} 2^{-n}$.
Under $\lambda$ the digits are independent fair coin tosses.
So the binary expansion is an isomorphism, up to null sets, from $T$ to the one-sided [[Bernoulli scheme|D33JK3AXRZ]] $B(1/2, 1/2)$.
Hence $T$ is [[mixing|S3RJQ9CX2P]], and its [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] is $\log 2$.

The map is not invertible: each point has two preimages.

## Explanations

### L1 human

Double the number and throw away the whole part.
In binary, this is the same as shifting the digits one step to the left and forgetting the first digit.
So the future of a point is written in its digits, and for a typical number these digits look like fair coin tosses.
That is why such a simple formula gives a fully random system.
</document>

<document source="https://tamnd.github.io/math2/node/HV003CB8EP.md" tag="HV003CB8EP" kind="example">
# Umbilic catastrophes

## Example

The *umbilic catastrophes* are the three elementary catastrophes with two state variables $(x, y) \in \R^2$.
In their standard forms they are:

1. the *hyperbolic umbilic* $V = x^3 + y^3 + a xy + b x + c y$, with controls $(a, b, c) \in \R^3$;
2. the *elliptic umbilic* $V = \tfrac{1}{3}x^3 - x y^2 + a(x^2 + y^2) + b x + c y$, with controls $(a, b, c) \in \R^3$;
3. the *parabolic umbilic* $V = x^2 y + y^4 + a x^2 + b y^2 + c x + d y$, with controls $(a, b, c, d) \in \R^4$.

When all controls vanish, each $V$ has a [[critical point|0YTCMFB4HC]] at the origin where the whole [[Hessian matrix|0693Y2METT]] is zero.
For the hyperbolic umbilic, the Hessian of $x^3 + y^3$ is $\begin{pmatrix} 6x & 0 \\ 0 & 6y \end{pmatrix}$, which is $0$ at the origin.
For the elliptic umbilic, the Hessian of $\tfrac13 x^3 - x y^2$ is $\begin{pmatrix} 2x & -2y \\ -2y & -2x \end{pmatrix}$, again $0$ at the origin.
For the parabolic umbilic, the Hessian of $x^2 y + y^4$ is $\begin{pmatrix} 2y & 2x \\ 2x & 12y^2 \end{pmatrix}$, again $0$ at the origin.
So the degeneracy has corank $2$: no change of coordinates can split off a nondegenerate square in one of the variables.
This separates the umbilics from the fold, cusp, swallowtail and butterfly, which have corank $1$ and need only one state variable.

For the elliptic umbilic, the slices $a = \text{const} \ne 0$ of the [[bifurcation set|JV45S0KQE6]] are curves with three cusps.

## Explanations

### L1 human

The fold, cusp, swallowtail and butterfly describe a ball rolling along a line.
The umbilics describe a ball rolling on a surface, so the potential has two state variables.
At the most degenerate setting the surface is flat to second order in every direction, like the monkey saddle $\tfrac13 x^3 - x y^2$ of the elliptic umbilic.
Small changes of the controls break this point into several ordinary critical points, in a few typical patterns.
Thom's list of seven elementary catastrophes contains exactly three of these: the hyperbolic, elliptic and parabolic umbilics.
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<document source="https://tamnd.github.io/math2/node/HV2QBDHTDD.md" tag="HV2QBDHTDD" kind="concept">
# Schreier coset graph

## Summary

Let $G$ be a [[group|G5S1M9]], $H$ a [[subgroup|0X1JXT47GA]] and $S$ a set of elements of $G$.
The Schreier coset graph is a [[directed graph|0KFYGSHHF5]] whose vertices are the [[right cosets|0NE1DK7H4T]] of $H$, with an edge labelled $s$ from $Hg$ to $Hgs$.
For $H = \set{1}$ it is the [[Cayley graph|0GYQK9NDJF]] of $G$ with respect to $S$.
It shows how the generators permute the cosets of $H$.
It is used in coset enumeration and to find generators of subgroups, as in the Reidemeister–Schreier method.

## Explanations

### L1 human

Group the elements of $G$ into the right cosets of $H$, and draw one dot for each coset.
Each generator $s$ moves a coset $Hg$ to the coset $Hgs$; draw an arrow for this move.
For example, in $\Z$ with $H = 3\Z$ and the generator $1$, there are three dots $3\Z$, $1 + 3\Z$ and $2 + 3\Z$, and the arrows form a triangle.
Paths in the graph that start and end at the dot $H$ spell words that represent elements of $H$.
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<document source="https://tamnd.github.io/math2/node/HV6YBS0YDH.md" tag="HV6YBS0YDH" kind="concept">
# Algebraic variety

## Summary

An algebraic variety is the central object of algebraic geometry.
It is a geometric space that is locally the set of solutions of a system of polynomial equations.
In the classical setting over an algebraically closed field, the varieties are the affine, quasi-affine, projective and quasi-projective varieties.
In modern language, a variety is an integral separated scheme of finite type over a field.
This allows varieties that do not sit inside any projective space.
Conventions differ between books, mainly on whether a variety must be irreducible.

## Explanations

### L1 human

A variety is a shape described by polynomial equations, like a line, a circle, a sphere or the curve $y^2 = x^3 + x$.
Locally it always looks like the solution set of equations in some $k^n$, and it may be completed by points at infinity in projective space.
The word usually means a single piece: the union of two crossing lines is an algebraic set with two varieties as its pieces.
Varieties are studied with algebra, through the rings of polynomial functions that live on them.
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<document source="https://tamnd.github.io/math2/node/HVA7TFK9XR.md" tag="HVA7TFK9XR" kind="definition">
# Topological quantum field theory

## Definition

Let $n \ge 0$ be an integer and let $k$ be a field.
All manifolds below are smooth, compact and [[oriented|702SNFQG0W]].
For closed $n$-manifolds $\Sigma_0$ and $\Sigma_1$, a *cobordism* from $\Sigma_0$ to $\Sigma_1$ is an $(n+1)$-[[manifold with boundary|7DS6B2BFK8]] $M$ with an orientation-preserving [[diffeomorphism|0Z1Y7JHGBR]] $\partial M \to \overline{\Sigma_0} \sqcup \Sigma_1$.
Here $\overline{\Sigma}$ is $\Sigma$ with the opposite orientation.
An $(n+1)$-dimensional *topological quantum field theory* $Z$ over $k$ assigns two kinds of data.
To each closed $n$-manifold $\Sigma$ it assigns a finite-dimensional [[vector space|05EGHEQZAW]] $Z(\Sigma)$ over $k$.
To each cobordism $M$ from $\Sigma_0$ to $\Sigma_1$ it assigns a [[linear map|0RJF668P1T]] $Z(M) \colon Z(\Sigma_0) \to Z(\Sigma_1)$.
These data satisfy the following axioms.

1. Each orientation-preserving diffeomorphism $f \colon \Sigma \to \Sigma'$ gives a linear isomorphism $Z(f) \colon Z(\Sigma) \to Z(\Sigma')$, with $Z(f' \circ f) = Z(f') \circ Z(f)$.
   Diffeomorphic cobordisms, by a diffeomorphism compatible with the boundaries, give equal maps after these isomorphisms.
2. Let $M$ be a cobordism from $\Sigma_0$ to $\Sigma_1$, and $M'$ one from $\Sigma_1$ to $\Sigma_2$.
   Let $M' \cup_{\Sigma_1} M$ be the cobordism obtained by gluing them along $\Sigma_1$.
   Then $Z(M' \cup_{\Sigma_1} M) = Z(M') \circ Z(M)$.
3. For each $\Sigma$, the cylinder $\Sigma \times [0, 1]$ gives the identity map of $Z(\Sigma)$.
4. The empty $n$-manifold gives $Z(\emptyset) = k$, and disjoint unions give natural isomorphisms $Z(\Sigma \sqcup \Sigma') \cong Z(\Sigma) \otimes Z(\Sigma')$ under which $Z(M \sqcup M') = Z(M) \otimes Z(M')$, where $\otimes$ is the [[tensor product|0G31VJ5YS5]] over $k$.

## Remarks

A closed $(n+1)$-manifold $M$ is a cobordism from $\emptyset$ to $\emptyset$, so $Z(M)$ is a linear map $k \to k$, that is, a number $Z(M) \in k$.
By clause 1 this number is a diffeomorphism invariant of $M$.
The axioms imply that $Z(\overline{\Sigma})$ is the [[dual space|0M856Q1AMZ]] of $Z(\Sigma)$.
In the language of categories, a topological quantum field theory is a symmetric monoidal functor from the category of $(n+1)$-dimensional cobordisms to the category of vector spaces over $k$.
Example: let $n = 0$ and let $V$ be a finite-dimensional vector space.
Set $Z(+) = V$ for the positive point and $Z(-) = V^*$, and extend to finite sets of points by tensor products.
An arc from $\emptyset$ to $\set{+, -}$ gives the element of $V \otimes V^*$ that corresponds to $\id_V$, an arc from $\set{+, -}$ to $\emptyset$ gives evaluation $V \otimes V^* \to k$, and straight arcs give identities.
These choices satisfy the axioms, and the circle, which is the composite of the two arcs, gets the number $Z(S^1) = \dim V$.
Non-example: set $Z(\Sigma) = k$ for all $\Sigma$ and let every cobordism act as multiplication by $2$.
Then the cylinder $\Sigma \times [0, 1]$ gives multiplication by $2$, not the identity, so clause 3 fails.
In dimension $1 + 1$, topological quantum field theories correspond to commutative Frobenius algebras over $k$.
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<document source="https://tamnd.github.io/math2/node/HVF403PQ4H.md" tag="HVF403PQ4H" kind="concept">
# Initial algebra semantics

## Summary

Initial algebra semantics gives meaning to syntax through the universal property of an initial algebra.
The terms of a signature form the [[term algebra|3YYKGWJJND]], which is an [[initial algebra|R76EC56V9R]]: it has exactly one [[homomorphism|KBXSZC6BV1]] to every algebra of the signature.
Choosing a semantic algebra then fixes the meaning of every term, and the meaning is compositional because the map is a homomorphism.
The same idea, with equations added, gives the standard meaning of an algebraic specification and of an [[abstract data type|86144NHH59]].

## Explanations

### L1 human

A program or a formula is built from small pieces by fixed rules.
To give it a meaning, it is enough to say what each basic symbol means and how each rule combines meanings.
The meaning of the whole then follows by working up from the pieces, and there is only one way to do it.
For example, the term $s(s(0)) + s(0)$ in the symbols $0$, $s$, $+$ gets the value $3$ once $0$ means zero, $s$ means "add one" and $+$ means addition.
Initial algebra semantics is the precise form of this recipe.
</document>

<document source="https://tamnd.github.io/math2/node/HVMYKZ2PB5.md" tag="HVMYKZ2PB5" kind="definition">
# Algebraic group

## Definition

Let $k$ be a field.
An *algebraic group* over $k$ is a [[group scheme|TJ04VQP4NB]] $G$ over $k$ whose underlying scheme is [[of finite type|AHTE5DVX61]] over $k$.
So $G$ comes with morphisms $m \colon G \times_k G \to G$, $i \colon G \to G$ and $e \colon \operatorname{Spec} k \to G$.
They satisfy associativity, the unit law and the inverse law as identities of morphisms.

## Remarks

Many authors also require $G$ to be smooth, or to be a variety; such groups are called group varieties.
Over a field of characteristic $0$, every algebraic group is smooth.

Example: the general linear group $\mathrm{GL}_n = \operatorname{Spec} k[x_{ij}, 1/\det]$, with matrix multiplication and the inverse matrix.
The entries of a product and of an inverse are polynomials in the $x_{ij}$ and $1/\det$, so these are morphisms, and the group axioms hold.

Non-example: the constant group scheme $\Z_k = \coprod_{n \in \Z} \operatorname{Spec} k$ is a group scheme over $k$.
It has infinitely many connected components, so it is not of finite type over $k$.

The group of points $G(k)$ is an abstract group, and $T \mapsto G(T)$ is a functor from $k$-schemes to groups.
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# Kronecker's approximation theorem

## Statement

For a real number $x$, let $\{x\} = x - \floor{x}$ be its fractional part, where $\floor{x}$ is the [[floor|08GR54CKEA]] of $x$.
Let $k \ge 1$ and let $\theta_1, \ldots, \theta_k$ be real numbers such that $1, \theta_1, \ldots, \theta_k$ are [[linearly independent|0EVHJFTG9Z]] over $\Q$.
Then the set
$$\set{(\{n\theta_1\}, \ldots, \{n\theta_k\}) : n \ge 1 \text{ an integer}}$$
is [[dense|0D198GPMTJ]] in $[0, 1]^k$.
For $k = 1$: if $\theta$ is an [[irrational number|080T2TXVQV]], then the fractional parts $\{n\theta\}$, $n \ge 1$, are dense in $[0, 1]$.

## Remarks

For $k = 1$, fix $\varepsilon > 0$.
The pigeonhole principle, as in [[Dirichlet's approximation theorem|04JAY9DGB7]], gives an $m \ge 1$ with $0 < \{m\theta\} < \varepsilon$ or $1 - \varepsilon < \{m\theta\} < 1$.
The value $\{m\theta\}$ is never $0$, since $\theta$ is irrational.
The multiples of $m$ then move around the circle in steps shorter than $\varepsilon$, so they come within $\varepsilon$ of every point.
In general dimension, the theorem follows from the $k$-dimensional form of Weyl's criterion, which gives even equidistribution in $[0,1]^k$.
The independence hypothesis is necessary.
Suppose $a_0 + a_1 \theta_1 + \cdots + a_k \theta_k = 0$ with integers $a_i$ not all $0$; then some $a_i$ with $i \ge 1$ is nonzero.
All points then satisfy $a_1 \{n\theta_1\} + \cdots + a_k \{n\theta_k\} \in \Z$, so they lie on finitely many hyperplanes and are not dense.
Example: $1, \sqrt{2}, \sqrt{3}$ are linearly independent over $\Q$, so the points $(\{n\sqrt{2}\}, \{n\sqrt{3}\})$ are dense in the unit square.

## Explanations

### L1 human

Walk around a circle of length $1$ in steps of an irrational length $\theta$.
You never come back to the start, and in the long run you pass as close as you like to every point of the circle.
Kronecker's theorem says the same for several walks at once, as long as the step lengths have no rational relation between them.

### L2 human

The theorem is the qualitative form of the [[equidistribution theorem|NCCGNTBEYV]] and its higher-dimensional versions.
It is a statement about the orbit of a rotation of the torus $\R^k / \Z^k$: under the independence condition, the orbit of $0$ is dense.
Its higher-dimensional form is used in dynamics and in the theory of almost periodic functions.
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<document source="https://tamnd.github.io/math2/node/HWC3SQGCQ0.md" tag="HWC3SQGCQ0" kind="definition">
# Kissing number

## Definition

Let $n \ge 1$.
A *kissing configuration* in the [[Euclidean space|0XPR1E85W0]] $\R^n$ is a finite set $X \subset \R^n$ such that:

1. $\abs{x} = 2$ for every $x \in X$;
2. $\abs{x - y} \ge 2$ for all distinct $x, y \in X$, where $\abs{\cdot}$ is the [[Euclidean distance|0NPPGHQNXQ]].

The *kissing number* $\tau(n)$ is the largest size of a kissing configuration in $\R^n$.

## Remarks

The conditions say that the [[closed balls|0Q0H13G8ZB]] of radius $1$ centred at the points of $X$ touch the unit ball at $0$ and have pairwise disjoint interiors.
Example: in $\R^2$, the six points $2(\cos(k\pi/3), \sin(k\pi/3))$ for $k = 0, \dots, 5$ have norm $2$, and neighbours are at distance exactly $2$, so they form a kissing configuration; hence $\tau(2) \ge 6$.
Non-example: seven points of norm $2$ in $\R^2$ cannot form one.
Two of them make an angle less than $60^\circ$ at $0$, and are then at distance less than $2$.
Equivalently, $\tau(n)$ is the largest number of points on the unit sphere $S^{n-1}$ with pairwise angles at least $60^\circ$.
The *lattice kissing number* is the largest size of such a configuration when $X$ is the set of shortest nonzero vectors of a lattice, scaled to length $2$.
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# Griffiths positivity

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $n$ and let $E \to X$ be a [[holomorphic vector bundle|TKA958Z4KZ]] of rank $r$ with a [[Hermitian metric|Q6X6W2N29G]] $h$.
Let $\Theta$ be the curvature of the [[Chern connection|7RC9CN2Q49]] of $(E, h)$.
Near a point $x$, choose local coordinates $z_1, \dots, z_n$ and a frame $e_1, \dots, e_r$ of $E$ that is orthonormal at $x$, and write
$$i \Theta(x) = i \sum_{j,k=1}^{n} \sum_{\lambda,\mu=1}^{r} c_{jk\lambda\mu}\, dz_j \wedge d\bar z_k \otimes e_\lambda^* \otimes e_\mu.$$
The bundle $(E, h)$ is *Griffiths positive* if at every point $x \in X$
$$\sum_{j,k,\lambda,\mu} c_{jk\lambda\mu}\, \xi_j \bar\xi_k v_\lambda \bar v_\mu > 0$$
for all nonzero $\xi = (\xi_1, \dots, \xi_n) \in \C^n$ and all nonzero $v = (v_1, \dots, v_r) \in \C^r$.

## Remarks

Intrinsically, the condition says that $h(i\Theta(\xi, \bar\xi) v, v) > 0$ for every nonzero tangent vector $\xi \in T_x X$ and every nonzero $v \in E_x$.
A holomorphic vector bundle is called Griffiths positive if it admits a metric with this property.
For $r = 1$ the condition is exactly that of a [[positive line bundle|J45KWFHENP]].

Example: let $L$ be a positive line bundle with metric $h_L$ and curvature $i\Theta_L = i \sum a_{jk} dz_j \wedge d\bar z_k$, and let $E = L^{\oplus r}$ with the direct sum metric.
Then $c_{jk\lambda\mu} = a_{jk} \delta_{\lambda\mu}$, and the sum equals $\left(\sum a_{jk} \xi_j \bar \xi_k\right) \abs{v}^2 > 0$.

Non-example: $E = L \oplus \mathcal{O}_X$ with the sum of the metric of $L$ and the trivial metric.
The curvature vanishes on the second summand, so the sum is $0$ for $v = (0, 1)$, and the condition fails.

Quotient bundles of Griffiths positive bundles are Griffiths positive, and Griffiths positive bundles on compact manifolds are ample.
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# Gluing of schemes

## Definition

Let $I$ be an index set, and for each $i \in I$ let $X_i$ be a [[scheme|5A35TYZD02]].
For all $i, j \in I$, let $U_{ij} \subseteq X_i$ be an open subscheme with $U_{ii} = X_i$.
Let $\varphi_{ij} \colon U_{ij} \to U_{ji}$ be isomorphisms of schemes such that:

1. $\varphi_{ii}$ is the identity of $X_i$ for every $i$.
2. $\varphi_{ij}(U_{ij} \cap U_{ik}) = U_{ji} \cap U_{jk}$ for all $i, j, k$.
3. $\varphi_{ik} = \varphi_{jk} \circ \varphi_{ij}$ on $U_{ij} \cap U_{ik}$ for all $i, j, k$ (the cocycle condition).

A *gluing* of these data is a scheme $X$ with [[open immersions|GFC5YKSN5H]] $\psi_i \colon X_i \to X$ such that:

1. the open sets $\psi_i(X_i)$ cover $X$;
2. $\psi_i(U_{ij}) = \psi_i(X_i) \cap \psi_j(X_j)$ for all $i, j$;
3. $\psi_i = \psi_j \circ \varphi_{ij}$ on $U_{ij}$ for all $i, j$.

Such a gluing always exists, and it is unique up to a unique isomorphism compatible with the maps $\psi_i$.

## Remarks

Example: let $k$ be a field, $X_1 = \operatorname{Spec} k[t]$, $X_2 = \operatorname{Spec} k[s]$, $U_{12} = D(t)$ and $U_{21} = D(s)$.
Let $\varphi_{12}$ be the isomorphism given by the ring map $k[s, s^{-1}] \to k[t, t^{-1}]$, $s \mapsto t^{-1}$, and $\varphi_{21} = \varphi_{12}^{-1}$.
With only two indices the cocycle condition holds, and the gluing is the projective line $\mathbb{P}^1_k$.
If instead $\varphi_{12}$ is given by $s \mapsto t$ on $D(t)$, the gluing is the affine line with a doubled origin, which is a scheme but not a separated one.
Non-example: with $\varphi_{12}$ as in the first example, take $\varphi_{21}$ to be given by $t \mapsto 2s^{-1}$ over a field $k$ with $2 \ne 0$.
The composite $\varphi_{21} \circ \varphi_{12}$ is given by $t \mapsto 2t$, so it is not $\varphi_{11} = \id$, and clause 3 fails for $i = k = 1$, $j = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/HWHNY5854A.md" tag="HWHNY5854A" kind="concept">
# Mellin transform method

## Summary

The Mellin transform method solves integral equations on $(0, \infty)$ whose kernel depends only on the ratio $x / t$ of the variables.
Such an integral is a [[Mellin convolution|A2PMJG7J68]], and the [[Mellin transform|N5PVGRJE1T]] turns it into a product.
The integral equation becomes an algebraic equation in a vertical strip of the complex plane, and the [[Mellin inversion theorem|XJWC13NY9E]] brings the solution back.
The method covers all kernels that are homogeneous of degree $-1$, such as $\frac{1}{x + t}$.
Such kernels arise in problems with scaling symmetry, such as wedges, cones and cracks.

## Explanations

### L1 human

The [[Fourier transform method|NRG03YP2PW]] works when the kernel is unchanged by shifts $x \mapsto x + c$.
The Mellin method works when the kernel is unchanged by stretches $x \mapsto c x$.
The substitution $x = e^{u}$ turns stretches into shifts, so the Mellin method is the Fourier method in logarithmic coordinates.
For example, $\int_0^\infty \frac{\varphi(t)}{x + t} \, dt = f(x)$ has a kernel that depends only on $x / t$ after taking out a factor $\frac{1}{t}$.
After the Mellin transform it reads $\frac{\pi}{\sin \pi s} \Phi(s) = F(s)$, which is solved by division.
</document>

<document source="https://tamnd.github.io/math2/node/HWP11G3HTN.md" tag="HWP11G3HTN" kind="concept">
# Monte Carlo method

## Summary

A Monte Carlo method computes a deterministic quantity approximately by drawing random samples.
In its basic form the quantity is the [[expected value|0WQNRCNY3M]] of a [[random variable|0PNRKMRC55]].
It is estimated by the average of many independent copies.
The [[law of large numbers|0ZVEAYVRC3]] makes the estimate converge.
The [[central limit theorem|0EPZC4Z1FC]] shows that the error decreases like $1/\sqrt{n}$ for $n$ samples, whatever the dimension.
This makes Monte Carlo methods the tool of choice for integrals in high dimension and for complex stochastic models.

## Explanations

### L1 human

To estimate the area of a pond, throw many stones at random into a square field around it and count the fraction that land in the water.
That fraction times the area of the field is close to the area of the pond, and closer when more stones are thrown.
This is a Monte Carlo method: a number that has nothing random about it is found by repeated random experiments.
For example, if $10\,000$ random points in the unit square give $7\,850$ points with $x^2 + y^2 \le 1$, the estimate of $\pi/4$ is $0.785$, so $\pi \approx 3.14$.
The price is slow accuracy: to gain one more decimal digit one needs about a hundred times more samples.
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<document source="https://tamnd.github.io/math2/node/HWRRXB0YXS.md" tag="HWRRXB0YXS" kind="concept">
# Digamma function

## Summary

The digamma function $\psi$ is the logarithmic derivative of the [[gamma function|0H5SYBSED9]], $\psi(z) = \Gamma'(z)/\Gamma(z)$.
It turns the product rule $\Gamma(z+1) = z\,\Gamma(z)$ into the sum rule $\psi(z+1) = \psi(z) + 1/z$.
At positive integers it gives the [[harmonic numbers|0NANDX7V02]] shifted by the [[Euler–Mascheroni constant|00C5P9JZBK]].
It is the first member of the family of [[polygamma functions|3J3QGKPMZQ]].

## Explanations

### L1 human

The gamma function grows very fast, so it is often easier to look at its logarithm.
The digamma function is the slope of $\log \Gamma$.
Because $\Gamma(n+1) = n!$, the slope of $\log \Gamma$ changes by $1/n$ when one steps from $n$ to $n+1$.
So the values at whole numbers add up the fractions $1, \frac12, \frac13, \ldots$.
For example, $\psi(1) = -\gamma \approx -0.5772$ and $\psi(4) = 1 + \frac12 + \frac13 - \gamma$.
</document>

<document source="https://tamnd.github.io/math2/node/HWY72ZK273.md" tag="HWY72ZK273" kind="concept">
# Special functions

## Summary

Special functions are the named functions of analysis that occur again and again in mathematics and physics, beyond the elementary functions.
Examples are the [[gamma function|0H5SYBSED9]], the [[hypergeometric function|463X5V31D9]], the [[Bessel functions|CZCMRDWDYW]], the [[Legendre polynomials|QFYGJVA996]] and the [[elliptic integrals|NXMCYGCB59]].
The term has no formal definition: a function counts as special when it has an established name, notation and theory.
Their study collects identities, integral representations, series, asymptotic formulas and differential equations that make these functions usable in computation.

## Explanations

### L1 human

Polynomials, exponentials, logarithms and trigonometric functions are the everyday functions of school mathematics.
Many problems lead to functions that are not of this kind, but appear so often that they get a name, like a new word in a language.
For example, the area under the bell curve $e^{-x^2}$ from $0$ to $x$ cannot be written with elementary functions, so it gets its own name, the error function.
Tables, formulas and computer libraries then treat these functions as well as the familiar ones.
Many special functions solve a simple differential equation or extend a sequence such as the factorials to non-integer arguments.
</document>

<document source="https://tamnd.github.io/math2/node/HX2DMKTT3Y.md" tag="HX2DMKTT3Y" kind="definition">
# Stability condition

## Definition

Let $\mathcal{D}$ be an essentially small [[triangulated category|T2V7GJJ2EN]] with shift functor $[1]$, and let $K(\mathcal{D})$ be its Grothendieck group.
A *stability condition* on $\mathcal{D}$ is a pair $(Z, \mathcal{P})$ with the following data.
The central charge $Z \colon K(\mathcal{D}) \to \C$ is a [[group homomorphism|0HV9Y7WG5R]].
The slicing $\mathcal{P}$ assigns to each $\phi \in \R$ a full additive subcategory $\mathcal{P}(\phi)$ of $\mathcal{D}$.
These data must satisfy:

1. If $E$ is a nonzero object of $\mathcal{P}(\phi)$, then $Z(E) = m(E) e^{i \pi \phi}$ for some real number $m(E) > 0$.
2. $\mathcal{P}(\phi + 1) = \mathcal{P}(\phi)[1]$ for all $\phi \in \R$.
3. If $\phi_1 > \phi_2$ and $A_j$ is an object of $\mathcal{P}(\phi_j)$ for $j = 1, 2$, then $\Hom(A_1, A_2) = 0$.
4. Each nonzero object $E$ of $\mathcal{D}$ has a Harder–Narasimhan filtration. This means real numbers $\phi_1 > \phi_2 > \cdots > \phi_n$ and morphisms $0 = E_0 \to E_1 \to \cdots \to E_n = E$. Each step sits in a [[distinguished triangle|ADKK56VZHW]] $E_{j-1} \to E_j \to A_j \to E_{j-1}[1]$ with $A_j$ a nonzero object of $\mathcal{P}(\phi_j)$.

Here $Z(E)$ means $Z$ applied to the class of $E$ in $K(\mathcal{D})$.
The nonzero objects of $\mathcal{P}(\phi)$ are the semistable objects of phase $\phi$.

## Remarks

Example: let $k$ be a field and let $\mathcal{D}$ be the bounded derived category of finite-dimensional vector spaces over $k$.
Each object is a finite direct sum of shifts $k[n]$, and $\Hom(k[a], k[b]) = 0$ for $a \ne b$.
Put $Z(E) = -\chi(E)$, where $\chi(E) = \sum_i (-1)^i \dim H^i(E)$.
Let $\mathcal{P}(\phi)$ be the finite direct sums of copies of $k[\phi - 1]$ when $\phi$ is an integer, and $0$ otherwise.
Then $Z(k[n]) = (-1)^{n+1} = e^{i \pi (n+1)}$, so clause 1 holds, and clause 2 holds since $k[n][1] = k[n+1]$.
Clause 3 holds because $\Hom(k[a], k[b]) = 0$ for $a > b$.
Clause 4 holds by splitting an object into its shifted summands, taken in order of decreasing phase.

Non-example: keep the same $Z$ but move every slice down by one, so that $k$ lies in $\mathcal{P}(0)$.
Then $Z(k) = -1$ while $e^{i \pi \cdot 0} = 1$, so clause 1 fails.

Many authors add a finiteness hypothesis, such as local finiteness or the support property, so that the set of stability conditions becomes a complex manifold.
Equivalently, a stability condition is a bounded t-structure with heart $\mathcal{A}$ together with a stability function on $\mathcal{A}$ that has the Harder–Narasimhan property.
</document>

<document source="https://tamnd.github.io/math2/node/HX6SS7WYQV.md" tag="HX6SS7WYQV" kind="definition">
# Invariant measure

## Definition

Let $(X, \mathcal{A})$ be a measurable space and let $T \colon X \to X$ be a [[measurable map|YYNCJSR0PJ]].
A [[measure|PR11BZQQPG]] $\mu$ on $(X, \mathcal{A})$ is *$T$-invariant* if
$$\mu(T^{-1}(A)) = \mu(A) \quad \text{for every } A \in \mathcal{A}.$$
Let a family of measurable maps be given, such as a flow $(\varphi_t)$ or a group action.
Then $\mu$ is *invariant* if it is invariant under each map of the family.

## Remarks

Equivalently, the [[pushforward measure|3N66TM3TSQ]] $T_* \mu$ equals $\mu$, or $T$ is a [[measure-preserving transformation|556WBDAHNP]] of $(X, \mathcal{A}, \mu)$ to itself.
For an integrable function $g$, invariance gives $\int g \circ T \, d\mu = \int g \, d\mu$.
The invariant probability measures of $T$ form a convex set, whose extreme points are the ergodic ones.

Example: the doubling map $T(x) = 2x \bmod 1$ on $[0, 1)$ and Lebesgue measure $\lambda$.
For an interval $[a, b) \subseteq [0, 1)$, one has $T^{-1}([a, b)) = [a/2, b/2) \cup [(a+1)/2, (b+1)/2)$, of length $b - a$.
Since intervals generate the Borel sets and are closed under intersection, $\lambda$ is $T$-invariant.

Non-example: the map $T(x) = x^2$ on $[0, 1]$ with Lebesgue measure $\lambda$.
One has $T^{-1}([0, 1/4]) = [0, 1/2]$, so $\lambda(T^{-1}([0, 1/4])) = 1/2 \ne 1/4$, and $\lambda$ is not $T$-invariant.
The point masses at $0$ and at $1$ are $T$-invariant, since $0$ and $1$ are fixed points.
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<document source="https://tamnd.github.io/math2/node/HX72KT9D0G.md" tag="HX72KT9D0G" kind="concept">
# Extremal length

## Summary

The extremal length of a family of curves in a plane domain measures how long the curves are compared with the area they sweep out.
It is defined by testing all possible ways of measuring length with a weight function $\rho$, and it is invariant under conformal maps.
Introduced by Beurling and developed by Ahlfors, it is a flexible tool for estimates in geometric function theory and for the theory of quasiconformal maps.
Its reciprocal is the [[conformal modulus|2FQAKBVBX5]] of the family.

## Explanations

### L1 human

Think of a rectangle of width $a$ and height $b$, and of all the curves that cross it from left to right.
These curves are long when $a$ is large and plentiful when $b$ is large, and the ratio $a/b$ captures both effects; it is the extremal length of the family.
Stretching the rectangle by a common factor does not change $a/b$, and in fact no conformal map changes the extremal length.
The definition replaces the Euclidean length by any weighted length and takes the most favourable weight.
</document>

<document source="https://tamnd.github.io/math2/node/HXNJH6W23S.md" tag="HXNJH6W23S" kind="statement">
# Hausdorff–Young inequality

## Statement

Let $n \ge 1$, let $1 \le p \le 2$, and let $q$ be the conjugate exponent, $\frac{1}{p} + \frac{1}{q} = 1$.
Let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\, dx$ be the [[Fourier transform|DPVM34ADZV]] of $f \in L^1(\R^n)$.
Then for every $f \in L^1(\R^n) \cap L^p(\R^n)$,
$$\norm{\hat f}_{L^q(\R^n)} \le \norm{f}_{L^p(\R^n)}.$$
Consequently the Fourier transform extends uniquely to a bounded linear map from [[$L^p(\R^n)$|SASP14MA1Z]] to $L^q(\R^n)$ of norm at most $1$.

## Remarks

The two endpoint cases are known: for $p = 1$, $\abs{\hat f(\xi)} \le \norm{f}_1$ for every $\xi$, and for $p = 2$, the [[Plancherel theorem|HPZHRZ0502]] gives $\norm{\hat f}_2 = \norm{f}_2$.
The [[Riesz–Thorin theorem|AGRKRG1TMJ]] interpolates between them and gives the inequality for $1 < p < 2$.

The constant $1$ is not sharp for $1 < p < 2$; the sharp constant is given by the Babenko–Beckner inequality.
The restriction $p \le 2$ is necessary: for $p > 2$ there are functions in $L^p(\R^n)$ whose Fourier transform, as a tempered distribution, is not a function.

There is a version for Fourier series on the circle $\mathbb{T} = \R / \Z$ with normalized measure: for $1 \le p \le 2$ and $f \in L^p(\mathbb{T})$, $\left(\sum_{k \in \Z} \abs{\hat f(k)}^q\right)^{1/q} \le \norm{f}_{L^p(\mathbb{T})}$.

Example: for $f(x) = e^{-\pi x^2}$ on $\R$, $\hat f = f$, and $\norm{f}_r = r^{-1/(2r)}$ for every $r > 0$.
So $\norm{\hat f}_q / \norm{f}_p = \left( p^{1/p} / q^{1/q} \right)^{1/2}$, and the theorem says that this number is at most $1$.

## Explanations

### L1 human

The Fourier transform cannot make a function much bigger in a suitable average sense.
If $f$ is $p$-th power integrable with $p$ between $1$ and $2$, then its transform is $q$-th power integrable for the conjugate exponent $q$, with norm no larger.

### L2 human

The inequality is the first example of an $L^p$ estimate obtained by interpolation, and it is the model for the Riesz–Thorin method.
It shows that the Fourier transform is defined as a function on $L^p$ for $1 \le p \le 2$.
It is used in restriction theory, in the theory of Fourier multipliers and in the entropic uncertainty principle.
On locally compact abelian groups the same statement holds, with the dual group in place of the frequency space.
</document>

<document source="https://tamnd.github.io/math2/node/HXSK0BNBNE.md" tag="HXSK0BNBNE" kind="statement">
# Stewart's theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane, and let $D$ be a point on the segment $BC$.
Write $a = BC$, $b = CA$, $c = AB$, $m = BD$, $n = DC$ and $d = AD$ for the [[distances|0NPPGHQNXQ]].
Then
$$b^2 m + c^2 n = a \left( d^2 + m n \right).$$

## Remarks

If $D$ lies strictly between $B$ and $C$, the angles $\angle ADB$ and $\angle ADC$ add up to $\pi$, so their cosines are opposite.
Apply the [[law of cosines|060ZVZBV3E]] to the triangles $ABD$ and $ADC$ at the vertex $D$.
Multiply the first equation by $n$, the second by $m$, and add: the cosine terms cancel, and $m + n = a$ gives the formula.
If $D = B$ or $D = C$, the formula is direct.

Example: if $D$ is the midpoint of $BC$, then $m = n = a/2$, and the formula becomes $d^2 = \frac{b^2 + c^2}{2} - \frac{a^2}{4}$.
This is [[Apollonius's theorem|0NAKHQ9XYY]] on the length of a [[median|0XW8HX8VEV]].

For a point $D$ on the line $BC$ outside the segment, the formula holds with signed lengths $m$ and $n$.

## Explanations

### L1 human

Draw a segment from one corner of a triangle to any point of the opposite side.
Stewart's theorem gives the length of that segment from the three sides and the two pieces into which it cuts the opposite side.
No angles are needed.

### L2 human

Stewart's theorem is the general form of several familiar length formulas.
With the midpoint it gives the length of a median.
With the foot of an angle bisector, located by the [[angle bisector theorem|0859R2V6ES]], it gives the length of the bisector.
</document>

<document source="https://tamnd.github.io/math2/node/HXSZA44J5J.md" tag="HXSZA44J5J" kind="concept">
# Complex analytic space

## Summary

A complex analytic space, or complex space, is a [[locally ringed space|B7FPT8JTXR]] that is locally modelled on an analytic set.
The model is the common zero set of finitely many [[holomorphic functions|507C34W0ZT]] on an open subset of $\C^n$.
Its structure sheaf is the quotient of the sheaf of holomorphic functions by the ideal that these functions generate.
Complex manifolds and [[complex analytic varieties|GB7M582M5Y]] are complex spaces.
The structure sheaf may have nilpotent elements, which record multiplicities, as for the double point $z^2 = 0$.
Complex spaces are the analytic counterpart of schemes and the setting of the global theory of several complex variables.

## Explanations

### L1 human

A complex manifold looks locally like an open subset of $\C^n$.
A complex space may look locally like the solution set of holomorphic equations, which may have singular points.
An example is the curve $z_2^2 = z_1^3$ with its cusp at the origin.
The space remembers not only the points but also the equations, through the functions that live on it.
So the equation $z^2 = 0$ in $\C$ gives a single point that remembers it was counted twice.
</document>

<document source="https://tamnd.github.io/math2/node/HY7VRFBAP9.md" tag="HY7VRFBAP9" kind="conjecture">
# Arnold conjecture

## Statement

Let $(M, \omega)$ be a closed [[symplectic manifold|CAN490EPRT]].
Let $H \colon [0, 1] \times M \to \R$ be smooth, let $X_{H_t}$ be the [[Hamiltonian vector field|C5EH3B4BEP]] of $H_t = H(t, \cdot)$, and let $\phi \colon M \to M$ be the time-$1$ map of the flow of $X_{H_t}$.
Such a [[symplectomorphism|A8TS2SNX9Z]] $\phi$ is called a Hamiltonian diffeomorphism.
Assume that every fixed point $x$ of $\phi$ is nondegenerate: $1$ is not an eigenvalue of $d\phi_x$.
Then the number of fixed points of $\phi$ is at least the minimal number of critical points of a [[Morse function|KPQ96640ZJ]] on $M$.

## Remarks

The homological version of the conjecture replaces the right side by the sum of the [[Betti numbers|ST71ZV3FES]] $\sum_k \dim H_k(M; \F)$ for a field $\F$.
By the Morse inequalities this sum is at most the number of critical points of any Morse function, so the homological version is weaker.
A degenerate version, without the nondegeneracy assumption, asks for at least as many fixed points as the minimal number of critical points of an arbitrary smooth function.

For $\phi = \id$ generated by $H = 0$ the hypothesis fails, and indeed every point is fixed.
If $H$ does not depend on $t$ and is a $C^2$-small Morse function, the fixed points of $\phi$ are exactly the critical points of $H$.
Taking $H$ with the minimal number of critical points shows that the bound is sharp.

Conley and Zehnder proved the conjecture for the standard torus $\mathbb{T}^{2n}$ in 1983.
Floer proved the homological version for monotone symplectic manifolds by building [[Floer homology|NCKFPAGHXF]], whose chain complex is generated by the fixed points of $\phi$.
The homological version with rational coefficients was proved for all closed symplectic manifolds in the late 1990s, by Fukaya and Ono and by Liu and Tian, among others.
The version with the minimal number of critical points of a Morse function is open in general.

## Explanations

### L1 human

A symplectic manifold is the phase space of a mechanical system.
A Hamiltonian diffeomorphism is what happens to the whole phase space after running the system for one unit of time.
The conjecture says that such a map must have many fixed points: at least as many as a smooth function on the space must have critical points.
For a two-dimensional torus, a generic such map has at least four fixed points, much more than topology alone would force.
The conjecture led to Floer homology, one of the main tools of modern symplectic geometry.
</document>

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# Strip of convergence of the bilateral transform

## Summary

The strip of convergence of a [[two-sided Laplace transform|6J2AGD0BYY]] is the vertical strip $\alpha < \operatorname{Re} s < \beta$ in the complex plane where the defining integral converges absolutely.
The left edge $\alpha$ is set by the growth of $f$ as $t \to +\infty$, and the right edge $\beta$ by its growth as $t \to -\infty$.
Inside the strip the transform is holomorphic.
The strip must be given along with the formula, since one formula can be the transform of different functions on different strips.

## Explanations

### L1 human

The two-sided transform needs $f(t) e^{-s t}$ to be small at both ends of the time axis.
A large $\operatorname{Re} s$ helps for positive times and hurts for negative times, so only a band of values of $\operatorname{Re} s$ works.
For $e^{-\abs{t}}$ the band is $-1 < \operatorname{Re} s < 1$.
The formula $1/(1 - s)$ is the transform of $e^{t}$ for $t < 0$ (and $0$ for $t > 0$) on $\operatorname{Re} s < 1$, but of $-e^{t}$ for $t > 0$ (and $0$ for $t < 0$) on $\operatorname{Re} s > 1$.
</document>

<document source="https://tamnd.github.io/math2/node/HYE5FT4M34.md" tag="HYE5FT4M34" kind="example">
# Non-Desarguesian plane

## Example

A *non-Desarguesian plane* is a [[projective plane|DANFNN8C4Y]] in which [[Desargues's theorem|5Z19Y02MME]] does not hold.
A concrete one is the projective completion of the [[Moulton plane|3V9GYYXM1H]].
Check: the Moulton plane has triangles $ABC$ and $A'B'C'$ in perspective from a point $O$.
In it $AB \parallel A'B'$ and $AC \parallel A'C'$, but $BC$ is not parallel to $B'C'$.
Add the line at infinity.
Then $AB \cap A'B'$ and $AC \cap A'C'$ lie on the line at infinity, but $BC \cap B'C'$ does not, so the three points are not collinear.

So this plane is not a [[Desarguesian plane|HPX85PA8NX]], and it is not isomorphic to $\mathbb{P}^2(D)$ for any division ring $D$.

## Remarks

Every projective plane of order at most $8$ is Desarguesian.
The smallest non-Desarguesian planes have order $9$.
There are exactly four projective planes of order $9$, and three of them are non-Desarguesian: the Hall plane, its dual and the Hughes plane.
The [[octonionic projective plane|S26GYRVDR5]] is a non-Desarguesian plane that is still a [[Moufang plane|7F4HSNJJH8]].
Every projective plane can be coordinatized by a [[planar ternary ring|Q0NXXFADT9]], which in a non-Desarguesian plane does not come from a division ring.

## Explanations

### L1 human

A projective plane only has to obey a few rules about points and lines.
One might expect that these rules force the plane to look like the usual plane built from numbers.
Non-Desarguesian planes show that this is false: they obey the rules, yet Desargues's theorem on triangles fails in them.
So no system of coordinates from a field, or even from a noncommutative division ring, can describe them.
</document>

<document source="https://tamnd.github.io/math2/node/HYJ10AF6RT.md" tag="HYJ10AF6RT" kind="concept">
# Well-formed formula

## Summary

A well-formed formula, or simply formula, is a string of symbols built by the grammar of a formal language.
In first-order logic, formulas are built from atomic formulas with [[connectives|QS0XVAN9BG]] and quantifiers.
Formulas are the objects that proofs manipulate and that [[structures|983GZBQYT4]] make true or false.
Only well-formed strings have a meaning, so the definition is the first step of every formal logic.

## Explanations

### L1 human

Not every string of symbols makes sense: $\forall x \, (x < y)$ does, but $\forall \land x <$ does not.
The rules for formulas are like grammar rules for sentences.
They start with simple claims such as $x = y$ or $x < y$, and allow you to combine claims with "not", "and", "for all" and so on.
A formula with free variables, such as $x < y$, is true or false only once values for $x$ and $y$ are given.
</document>

<document source="https://tamnd.github.io/math2/node/HYK8J56PSH.md" tag="HYK8J56PSH" kind="concept">
# Chebyshev nodes

## Summary

Chebyshev nodes are the zeros of a [[Chebyshev polynomial|M6EAR1JHPP]], possibly moved to an interval $[a, b]$ by an affine map.
They crowd near the ends of the interval.
They are the standard choice of points for [[polynomial interpolation|SS1YMS2NPC]], because they avoid the large errors that equally spaced points cause.

## Explanations

### L1 human

Take points spaced evenly around the upper half of a circle and drop them straight down onto the diameter.
The shadows are the Chebyshev nodes: sparse in the middle and dense near the two ends.
For $n = 2$ the nodes in $[-1, 1]$ are $\pm 1/\sqrt{2}$, about $\pm 0.707$.
Interpolating at these points keeps the polynomial under control near the ends, where equally spaced points let it oscillate wildly.
</document>

<document source="https://tamnd.github.io/math2/node/HYP9AZHD06.md" tag="HYP9AZHD06" kind="statement">
# Iterated Laplace transform equals Stieltjes transform

## Statement

Let $f \colon (0, \infty) \to \C$ be locally integrable, and let $x > 0$ satisfy
$$\int_0^\infty \frac{\abs{f(t)}}{x + t}\, dt < \infty.$$
Let $(\mathcal{L} f)(s) = \int_0^\infty e^{-st} f(t)\, dt$ denote the [[Laplace transform|0V49JEZW06]].
Then $(\mathcal{L} f)(s)$ converges absolutely for every $s > 0$, the function $\mathcal{L} f$ is integrable against $e^{-xs}\, ds$ on $(0, \infty)$, and
$$\big(\mathcal{L}(\mathcal{L} f)\big)(x) = \int_0^\infty \frac{f(t)}{x + t}\, dt,$$
the [[Stieltjes transform|RG1HFAYD98]] of $f$ at $x$.

## Remarks

The proof is one application of Fubini's theorem.
For $x, t > 0$ one has $\int_0^\infty e^{-xs} e^{-st}\, ds = \frac{1}{x + t}$.
So $\int_0^\infty \int_0^\infty e^{-xs} e^{-st} \abs{f(t)}\, dt\, ds = \int_0^\infty \frac{\abs{f(t)}}{x + t}\, dt < \infty$ by Tonelli, and Fubini allows the exchange of the two integrals.
The inner integral $\int_0^\infty e^{-st} \abs{f(t)}\, dt$ is then finite for almost every $s$, and since it decreases in $s$ it is finite for every $s > 0$.
The hypothesis cannot be dropped: for $f = 1$ the right side diverges for every $x$, while $\mathcal{L} f(s) = 1/s$ is not integrable against $e^{-xs}$ near $s = 0$.

Example: $f(t) = t^{-1/2}$.
Then $\mathcal{L} f(s) = \sqrt{\pi / s}$ and $\mathcal{L}(\mathcal{L} f)(x) = \sqrt{\pi}\, \Gamma(1/2)\, x^{-1/2} = \pi\, x^{-1/2}$.
Directly, the substitution $t = x u^2$ gives $\int_0^\infty \frac{t^{-1/2}}{x + t}\, dt = \frac{2}{\sqrt x} \int_0^\infty \frac{du}{1 + u^2} = \frac{\pi}{\sqrt x}$.

## Explanations

### L1 human

Applying the Laplace transform twice gives another well-known transform, the Stieltjes transform.
The reason is the elementary integral $\int_0^\infty e^{-(x+t)s}\, ds = 1/(x + t)$.

### L2 human

This identity lets one move facts between the two transforms.
Properties of the Stieltjes transform, such as analyticity in $x$ and its behaviour as $x \to 0^+$ or $x \to \infty$, follow from the corresponding properties of the Laplace transform applied twice.
Conversely, inversion formulas for the [[Laplace transform|0V49JEZW06]], such as [[Post's inversion formula|D6QKPBS2EN]], can be applied twice to recover $f$ from its Stieltjes transform.
</document>

<document source="https://tamnd.github.io/math2/node/HYRFMXAJVW.md" tag="HYRFMXAJVW" kind="concept">
# Laguerre series expansion of the Laplace transform

## Summary

The Laguerre series expansion writes the [[Laplace transform|0V49JEZW06]] $F(s)$ of a function, times $s + 1/2$, as a power series in $w = (s - 1/2)/(s + 1/2)$.
Here $w$ is a Möbius transformation of the transform variable $s$.
Its coefficients are the coefficients of the function in the orthonormal basis of Laguerre functions $e^{-t/2} L_n(t)$ of $L^2(0, \infty)$.
Here $L_n$ are the [[Laguerre polynomials|WAMRHWW82R]].
The expansion converts Laplace inversion into the computation of Taylor coefficients of an analytic function on the unit disk.
It is the basis of Weeks' method for the numerical inversion of the Laplace transform.

## Explanations

### L1 human

The Laplace transform of each Laguerre function is very simple: a power of $(s - 1/2)$ divided by a power of $(s + 1/2)$.
So if a function is a sum of Laguerre functions, its Laplace transform is a sum of these simple terms.
The change of variable $w = (s - 1/2)/(s + 1/2)$ makes this sum an ordinary power series in $w$.
For example, $f(t) = e^{-t/2}$ is the first Laguerre function, and its transform $1/(s + 1/2)$ is the first term of the series.
To recover $f$ from its transform, one reads off the power series coefficients in $w$.
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<document source="https://tamnd.github.io/math2/node/HYSCKJN3MG.md" tag="HYSCKJN3MG" kind="definition">
# Poisson bracket

## Definition

Let $(M, \omega)$ be a [[symplectic manifold|CAN490EPRT]].
For a smooth function $f \colon M \to \R$, let $X_f$ be its [[Hamiltonian vector field|C5EH3B4BEP]], so $\iota_{X_f} \omega = df$.
The *Poisson bracket* of smooth functions $f, g \colon M \to \R$ is the smooth function
$$\{f, g\} = \omega(X_f, X_g).$$

## Remarks

Equivalently $\{f, g\} = df(X_g) = X_g(f) = -X_f(g)$.
The bracket is bilinear and antisymmetric, satisfies the Leibniz rule $\{f, gh\} = \{f, g\} h + g \{f, h\}$, and satisfies the Jacobi identity because $d\omega = 0$.
With these conventions $X_{\{f, g\}} = -[X_f, X_g]$.

Example: on $\R^{2n}$ with $\omega = \sum_i dq_i \wedge dp_i$,
$$\{f, g\} = \sum_{i=1}^n \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right).$$
For instance $X_{p_j} = \partial / \partial q_j$, so $\{q_i, p_j\} = X_{p_j}(q_i) = \delta_{ij}$, while $\{q_i, q_j\} = \{p_i, p_j\} = 0$.

Non-example: on $\R^4$ the closed $2$-form $dq_1 \wedge dp_1$ is degenerate, so it does not define a bracket in this way: the function $q_2$ has no Hamiltonian vector field for it.

Along an integral curve $\gamma$ of $X_H$, a function $g$ satisfies $\frac{d}{dt} g(\gamma(t)) = \{g, H\}(\gamma(t))$.
Some authors use the opposite sign convention for $\{f, g\}$.
</document>

<document source="https://tamnd.github.io/math2/node/HYVFPWS9ER.md" tag="HYVFPWS9ER" kind="concept">
# Cospherical points

## Summary

A set of points is cospherical if all of its points lie on one [[sphere|P9CM4GQ2T0]].
In the plane this is the same as being [[concyclic|049DCJZ448]].
Four points of space that do not lie in one plane are always cospherical, which gives every tetrahedron a circumscribed sphere.

## Explanations

### L1 human

Points are cospherical when one ball can be chosen whose surface passes through all of them.
The eight corners of a cube are cospherical: they all lie at the same distance from the centre of the cube.
Three distinct points on one straight line are never cospherical in the plane, since no circle meets a line in three points.
</document>

<document source="https://tamnd.github.io/math2/node/HZ17ES9PDY.md" tag="HZ17ES9PDY" kind="concept">
# Dialectica interpretation

## Summary

The Dialectica interpretation translates [[Heyting arithmetic|CK8ZQRN183]] into Gödel's system T.
This is a quantifier-free theory of computable functionals of finite type.
Gödel published it in 1958 in the journal Dialectica.
Each formula becomes a formula of the form $\exists x\, \forall y\, A_D(x, y)$, and a proof yields a term of system T that gives the witness $x$.
It gives a relative consistency proof of arithmetic and is the main tool for extracting bounds from proofs.

## Explanations

### L1 human

The interpretation reads each statement as a game between two players.
One player chooses $x$, the other answers with $y$, and the first wins if the simple condition $A_D(x, y)$ holds.
A proof of the statement gives a winning strategy, written as a program built from [[primitive recursion|5NVA2JRHPV]] in higher types.
For example, a proof of $\forall n\, \exists m\, (m > n)$ yields the program $n \mapsto n + 1$.
In this way, numbers and bounds can be read off from proofs.
</document>

<document source="https://tamnd.github.io/math2/node/HZ5J8GGS16.md" tag="HZ5J8GGS16" kind="statement">
# Associativity of convolution

## Statement

Let $n \ge 1$, and let $f, g, h \in L^1(\R^n)$.
Then the [[convolutions|2Z29JN3VES]] $(f * g) * h$ and $f * (g * h)$ are both defined almost everywhere, they lie in [[$L^1(\R^n)$|SASP14MA1Z]], and
$$(f * g) * h = f * (g * h)$$
almost everywhere.

## Remarks

Both sides are formally the double integral $\iint f(y) g(z - y) h(x - z) \, dy \, dz$.
[[Tonelli's theorem|067SYD2J4B]] shows that $\iiint \abs{f(y) g(z - y) h(x - z)} \, dy \, dz \, dx = \norm{f}_1 \norm{g}_1 \norm{h}_1 < \infty$, so the double integral converges absolutely for almost every $x$.
[[Fubini's theorem|0R0T1VWH5X]] then allows the order of integration to be swapped, and the substitution $z = y + w$ turns one side into the other.

Example: on $\R$ with $f = g = h = \mathbf{1}_{[0, 1]}$, both sides equal the convolution of the tent function $\mathbf{1}_{[0,1]} * \mathbf{1}_{[0,1]}$ with $\mathbf{1}_{[0, 1]}$, a piecewise quadratic spline supported on $[0, 3]$.

Integrability matters.
Take $f = 1$, $g = \mathbf{1}_{[0,1]} - \mathbf{1}_{[1,2]}$ and $h = \mathbf{1}_{[0,\infty)}$ on $\R$; only $g$ is integrable.
Then $f * g = \int_\R g = 0$, so $(f * g) * h = 0$.
But $g * h$ is the tent function equal to $x$ on $[0, 1]$, to $2 - x$ on $[1, 2]$ and to $0$ elsewhere, so $f * (g * h) = 1$.

## Explanations

### L1 human

When you convolve three functions, you may first combine the first two or the last two.
The result is the same.
So an expression like $f * g * h$ needs no brackets.

### L2 human

Associativity makes $L^1(\R^n)$ an associative algebra under convolution, in fact a commutative Banach algebra.
It lets one define [[convolution powers|JGR1E7Q9KQ]] $f^{*k}$ without brackets.
In probability it says that the distribution of $X + Y + Z$ for independent $X, Y, Z$ does not depend on how the sum is grouped.
</document>

<document source="https://tamnd.github.io/math2/node/HZ5V3M1F54.md" tag="HZ5V3M1F54" kind="definition">
# n-connected space

## Definition

Let $n \ge 0$ be an integer, and let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *$n$-connected* if both of the following hold:

1. $X$ is nonempty and [[path-connected|0WB6DH499T]];
2. $\pi_k(X, x) = 0$ for every $x \in X$ and every $k$ with $1 \le k \le n$, where $\pi_k$ is the [[homotopy group|P976WYEFMB]].

## Remarks

Equivalently, $X$ is nonempty and for $0 \le k \le n$ every continuous map $S^k \to X$ extends to a continuous map $D^{k+1} \to X$.
So $0$-connected means path-connected, and $1$-connected means simply connected.
By convention, $(-1)$-connected means nonempty.

Example: the sphere $S^m$ is $(m-1)$-connected, since $\pi_k(S^m) = 0$ for $k < m$.

Non-example: $S^m$ is not $m$-connected, because $\pi_m(S^m) \cong \Z$, so clause 2 fails for $k = m$.

A pair $(X, A)$ is called $n$-connected if every path component of $X$ meets $A$ and $\pi_k(X, A, a) = 0$ for $1 \le k \le n$ and all $a \in A$.
</document>

<document source="https://tamnd.github.io/math2/node/HZCH4KNGY2.md" tag="HZCH4KNGY2" kind="definition">
# Milnor number

## Definition

Let $\mathcal{O}_{n+1} = \C\{z_0, \dots, z_n\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]] in $n + 1$ variables.
Let $f \in \mathcal{O}_{n+1}$ with $f(0) = 0$.
The *Jacobian ideal* of $f$ is the [[ideal|096Z9MP2ME]] $J(f) = \left( \frac{\partial f}{\partial z_0}, \dots, \frac{\partial f}{\partial z_n} \right)$ of $\mathcal{O}_{n+1}$.
The *Milnor number* of $f$ at $0$ is
$$\mu(f) = \dim_\C \mathcal{O}_{n+1} / J(f),$$
a non-negative integer or $\infty$.

## Remarks

The quotient $\mathcal{O}_{n+1} / J(f)$ is called the *local algebra* or *Milnor algebra* of $f$.
The number $\mu(f)$ is finite if and only if $f$ has at most an [[isolated singularity|VTGYEES3VS]] at $0$, and $\mu(f) = 0$ exactly when $0$ is not a critical point.

Example: $f(x, y) = x^p + y^q$ with integers $p, q \ge 2$.
Then $J(f) = (x^{p-1}, y^{q-1})$, and the monomials $x^a y^b$ with $0 \le a \le p - 2$ and $0 \le b \le q - 2$ form a basis of the quotient.
So $\mu(f) = (p - 1)(q - 1)$.
For $f(z) = z_0^2 + \cdots + z_n^2$, $J(f)$ is the maximal ideal and $\mu(f) = 1$.

Non-example of a finite Milnor number: $f(x, y) = x^2 y$.
Here $J(f) = (2xy, x^2)$, and the powers $y, y^2, y^3, \dots$ are linearly independent modulo $J(f)$, so $\mu(f) = \infty$.
The singularity is not isolated: the gradient vanishes on the whole line $x = 0$.

Let $\nabla f = (\partial f / \partial z_0, \dots, \partial f / \partial z_n)$.
For an isolated singularity, $\mu(f)$ is also the topological degree of the map $z \mapsto \nabla f(z) / \norm{\nabla f(z)}$ from a small sphere about $0$ to the unit sphere.
It is also the rank of the middle homology of the Milnor fibre.
</document>

<document source="https://tamnd.github.io/math2/node/HZKFNZTHZS.md" tag="HZKFNZTHZS" kind="definition">
# Smooth dynamical system

## Definition

Let $M$ be a [[smooth manifold|WBZHXGKNSJ]] and let $r \ge 1$ be an integer or $\infty$.
A *smooth dynamical system of class $C^r$* on $M$ is one of the following.

1. Discrete time: a $C^r$ [[diffeomorphism|0Z1Y7JHGBR]] $f \colon M \to M$, with the action $n \mapsto f^n$ of $\Z$ on $M$.
2. Continuous time: a $C^r$ map $\varphi \colon \R \times M \to M$, written $\varphi_t(x) = \varphi(t, x)$, such that $\varphi_0 = \id_M$ and $\varphi_{s+t} = \varphi_s \circ \varphi_t$ for all $s, t \in \R$.

## Remarks

In case 2 each $\varphi_t$ is a $C^r$ diffeomorphism with inverse $\varphi_{-t}$, and $\varphi$ is called a $C^r$ flow.
It is the flow of the vector field $X(x) = \frac{d}{dt}\varphi_t(x)\big|_{t=0}$, which is of class $C^{r-1}$.
Conversely, on a compact manifold every $C^r$ vector field generates a complete $C^r$ flow.
Some authors also allow non-invertible $C^r$ maps $f \colon M \to M$ in case 1.
Example: on the circle $M = \R/\Z$, the rotation $f(x) = x + \alpha \bmod 1$ is a $C^\infty$ diffeomorphism, and $\varphi_t(x) = x + t \bmod 1$ is a $C^\infty$ flow.
Non-example: $f(x) = 2x \bmod 1$ on the circle is smooth but two-to-one, so it is not a diffeomorphism and fails case 1 in this definition.
</document>

<document source="https://tamnd.github.io/math2/node/HZM2ASCDY4.md" tag="HZM2ASCDY4" kind="concept">
# Ornstein–Uhlenbeck process

## Summary

The Ornstein–Uhlenbeck process is a Brownian motion with a pull back towards a long-run mean.
It solves the [[Langevin equation|ZW2RBS9VTR]] $dX_t = \theta(\mu - X_t) \, dt + \sigma \, dW_t$ with $\theta > 0$.
It is Gaussian, Markov and, from a suitable start, stationary, and it models velocities of particles, interest rates and mean-reverting quantities.

## Explanations

### L1 human

Think of a particle tied to the point $\mu$ by a spring, and kicked around at random.
The farther it strays, the harder the spring pulls it back, so it never wanders off for good.
In the long run its position is normally distributed around $\mu$, with spread set by the balance of kicks $\sigma$ and spring strength $\theta$.
Unlike Brownian motion, whose spread grows with time, this process settles into a steady random state.
</document>

<document source="https://tamnd.github.io/math2/node/HZNEQQBCW4.md" tag="HZNEQQBCW4" kind="concept">
# Radon measure

## Summary

A Radon measure on a Hausdorff topological space is a Borel measure that is [[locally finite|YQ5C1SRE0W]] and [[inner regular|28E5Q5BM69]] with respect to compact sets.
Radon measures are the measures that fit well with the topology.
On a locally compact Hausdorff space, the Riesz–Markov–Kakutani theorem represents each positive linear functional on continuous functions of compact support by such a measure.
Lebesgue measure, Haar measures and the usual measures of probability on Polish spaces are Radon measures.

## Explanations

### L1 human

A Radon measure is a measure that gives finite mass to small regions and can be read off from compact sets.
Length on the real line is the basic example.
Such a measure is determined by how it integrates continuous functions that vanish outside a bounded region.
This links measure theory with functional analysis.
</document>

<document source="https://tamnd.github.io/math2/node/HZRC5N7M9E.md" tag="HZRC5N7M9E" kind="definition">
# Chebyshev system

## Definition

Let $K$ be a [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]] with at least $n$ points, and let $u_1, \dots, u_n \colon K \to \R$ be [[continuous functions|0X3WGX95FE]].
The family $(u_1, \dots, u_n)$ is a *Chebyshev system* (or *Haar system*) on $K$ if the following holds.
Every linear combination $c_1 u_1 + \cdots + c_n u_n$ with $(c_1, \dots, c_n) \ne 0$ has at most $n - 1$ zeros in $K$.
An $n$-dimensional subspace of $C(K)$ is a *Haar subspace* if it has a basis that is a Chebyshev system.

## Remarks

Equivalently, the [[determinant|00KDZRW4C0]] $\det \big( u_i(x_j) \big)_{i, j = 1}^{n}$ is nonzero for all distinct $x_1, \dots, x_n \in K$.
Equivalently, for all distinct $x_1, \dots, x_n \in K$ and all $y_1, \dots, y_n \in \R$ there is exactly one combination $u = \sum c_i u_i$ with $u(x_j) = y_j$ for all $j$.

Example: $1, x, \dots, x^{n-1}$ on $[a, b]$.
A nonzero polynomial of degree at most $n - 1$ has at most $n - 1$ roots.

Non-example: $1, x^2$ on $[-1, 1]$, with $n = 2$.
The combination $x^2 - 1$ is nonzero but has two zeros, $\pm 1$, more than $n - 1 = 1$.

The trigonometric polynomials of degree at most $m$ form a Haar subspace of dimension $2m + 1$ on the circle $\R / 2\pi\Z$: a nonzero one has at most $2m$ zeros in a period.
</document>

<document source="https://tamnd.github.io/math2/node/HZTRH10Z8Y.md" tag="HZTRH10Z8Y" kind="example">
# Smith–Volterra–Cantor set

## Example

Start with $C_0 = [0, 1]$.
At step $n \ge 1$, the set $C_{n-1}$ is a union of $2^{n-1}$ disjoint closed intervals.
Remove from the middle of each of them an open interval of length $4^{-n}$, and call the result $C_n$.
The Smith–Volterra–Cantor set is $S = \bigcap_{n \ge 0} C_n$.
Then $S$ is a [[closed set|0ZCK8DQYN1]], it is [[nowhere dense|0VQYCCQVJ5]] in $\R$, and its [[Lebesgue measure|2Z2QVWZ262]] is $1/2$.

## Remarks

Step $1$ removes $(3/8, 5/8)$ and leaves $[0, 3/8] \cup [5/8, 1]$.
By induction, each of the $2^n$ intervals of $C_n$ has length $2^{-n-1} + 2^{-2n-1}$.
This is more than $4^{-n-1}$, so each removal is possible.
The total length removed is
$$\sum_{n=1}^{\infty} 2^{n-1} \cdot 4^{-n} = \frac{1}{4} \sum_{n=0}^{\infty} 2^{-n} = \frac{1}{2},$$
so $\lambda(S) = 1 - 1/2 = 1/2$.
The set $S$ is closed as an intersection of closed sets.
These lengths tend to $0$, so $S$ contains no interval; a closed set with empty interior is nowhere dense.
So a nowhere dense set can have positive measure, unlike the middle-thirds [[Cantor set|0BXNDZFQME]], which has measure $0$.
The indicator function of $S$ is bounded and discontinuous at every point of $S$, a set of positive measure, so it is not Riemann integrable.

## Explanations

### L1 human

The ordinary Cantor set removes a third of each interval at every step and ends with total length $0$.
The Smith–Volterra–Cantor set removes much smaller gaps, of length $1/4$, then $1/16$, then $1/64$, and so on.
The gaps still reach everywhere, so the set contains no interval at all.
Yet the gaps only add up to length $1/2$, so the remaining dust of points still has total length $1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/HZX6Q514X8.md" tag="HZX6Q514X8" kind="concept">
# Herman ring

## Summary

A Herman ring is a component of the [[Fatou set|C0N0XGC6GK]] of a rational map that is a topological annulus.
On it some iterate of the map acts as an irrational rotation.
It is one of the four types of periodic Fatou components, next to attracting basins, parabolic basins and [[Siegel discs|KNDYA9T8ZK]].
Unlike a Siegel disc, it has no fixed point at its centre.
Polynomials never have Herman rings, so they appear only for rational maps that are not polynomials.

## Explanations

### L1 human

Picture a ring, like a washer, that the map turns around and around by an angle that is an irrational fraction of a full turn.
Points travel forever on closed loops that circle the hole of the ring.
The hole and the outside of the ring both contain points of the Julia set.
Michael Herman showed that such rings really occur for rational maps.
</document>

<document source="https://tamnd.github.io/math2/node/HZXSESHC8P.md" tag="HZXSESHC8P" kind="concept">
# Dual abelian variety

## Summary

The dual of an [[abelian variety|R7W03TXRS9]] $A$ is an abelian variety $A^\vee$ of the same dimension.
Its points are the line bundles on $A$ that are algebraically equivalent to zero.
It is the identity component of the [[Picard group|3Z9DCWYXQZ]] of $A$, made into a variety.
Duality is an involution: the dual of $A^\vee$ is $A$ again.
Polarizations of $A$ are certain maps $A \to A^\vee$, and the Weil pairing comes from duality.

## Explanations

### L1 human

On an abelian variety one can look at all line bundles, the algebraic analogues of twisted strips over the space.
Those that can be deformed continuously to the trivial one form a group that is itself an abelian variety.
This new abelian variety is the dual.
For an elliptic curve $E$ with origin $O$, the point $P$ corresponds to the line bundle of the divisor $P - O$, so $E$ is its own dual.
In higher dimension $A$ and $A^\vee$ are isogenous but need not be isomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/HZY2EWWKB5.md" tag="HZY2EWWKB5" kind="definition">
# Translation plane

## Definition

Let $\Pi$ be a [[projective plane|DANFNN8C4Y]] and let $\ell$ be a line of $\Pi$.
An *elation with axis $\ell$* is a [[collineation|V7PRPN24CV]] $\alpha$ of $\Pi$ with two properties.
It fixes every point of $\ell$, and for some point $C$ on $\ell$ it maps every line through $C$ to itself.
The plane $\Pi$ is a *translation plane with respect to $\ell$* if the following holds.
For any two points $P$ and $Q$ not on $\ell$, some elation with axis $\ell$ maps $P$ to $Q$.
The line $\ell$ is then a *translation line* of $\Pi$.

## Remarks

Example: let $F$ be a field and let $\Pi$ be the projective plane over $F$, with $\ell$ the line at infinity.
For $(a, b) \in F^2$, the translation $(x, y) \mapsto (x + a, y + b)$ of $F^2$ maps lines to parallel lines, so it extends to a collineation of $\Pi$ that fixes every point at infinity.
If $(a, b) \ne (0, 0)$, it maps each line of direction $(a, b)$ to itself.
So it fixes every line through the point at infinity $C$ of that direction, and it is an elation with axis $\ell$.
The translation by $Q - P$ maps $P$ to $Q$, so $\Pi$ is a translation plane with respect to $\ell$.

Non-example: the Hughes plane of order $9$ is a projective plane that is not a translation plane with respect to any of its lines.

When $\Pi$ is a translation plane with respect to $\ell$, the elations with axis $\ell$ form an abelian group that acts regularly on the points off $\ell$.
The translation planes are exactly the planes coordinatized by [[quasifields|JQ22QH86DE]].
</document>

<document source="https://tamnd.github.io/math2/node/J00G7MTBFB.md" tag="J00G7MTBFB" kind="definition">
# Order of a difference equation

## Definition

Let $X$ be a set and let $(u_n)_{n \ge 0}$ denote an unknown sequence in $X$.
Consider a difference equation of the form
$$F(n, u_n, u_{n+1}, \dots, u_{n+k}) = 0 \qquad (n \ge 0),$$
where $k \ge 0$ is an integer and $F$ is a given function.
The equation has *order* $k$ if $F$ depends on both its second argument $u_n$ and its last argument $u_{n+k}$.
For a [[recurrence relation|07TZV87Y0T]] $u_{n+k} = f(n, u_n, \dots, u_{n+k-1})$ in which $f$ depends on $u_n$, the order is $k$.

## Remarks

Example: $u_{n+3} = u_{n+1} + 2u_n$ has order $3$.
The indices that occur are $n$, $n+1$ and $n+3$, and the largest minus the smallest is $3$.
Non-example: $u_{n+3} = 2u_{n+2}$ looks like order $3$, but $u_n$ does not occur.
After the substitution $v_n = u_{n+2}$ it reads $v_{n+1} = 2v_n$, which has order $1$.
For a linear equation $a_k(n) u_{n+k} + \dots + a_0(n) u_n = b(n)$ the order is $k$ when neither $a_k$ nor $a_0$ is the zero sequence.
</document>

<document source="https://tamnd.github.io/math2/node/J00Z252GRE.md" tag="J00Z252GRE" kind="statement">
# Erdős–Ginzburg–Ziv theorem

## Statement

Let $n$ be a positive integer.
Then among any $2n - 1$ integers, not necessarily distinct, there are $n$ whose sum is divisible by $n$.

## Remarks

The proof first treats a prime $n = p$, using the Chevalley–Warning theorem or the [[Cauchy–Davenport theorem|F7K7XFACSG]].
For composite $n = ab$, the cases $a$ and $b$ imply the case $n$.
One repeatedly extracts $2b - 1$ disjoint groups of $a$ elements, each with sum divisible by $a$.
Then one applies the case $b$ to the $2b - 1$ group sums divided by $a$.
The number $2n - 1$ cannot be lowered.
The $2n - 2$ integers consisting of $n - 1$ zeros and $n - 1$ ones have no $n$ elements with sum divisible by $n$.
Indeed, any $n$ of them contain $j$ ones with $1 \le j \le n - 1$, so their sum $j$ is not a multiple of $n$.
For $n = 2$: among any three integers, two have the same parity, so their sum is even.

## Explanations

### L1 human

Write down $2n - 1$ whole numbers.
You can always choose exactly $n$ of them whose sum is a multiple of $n$.
For example, among any $5$ integers, some $3$ add up to a multiple of $3$.

### L2 human

The theorem is the starting point of zero-sum theory in [[additive combinatorics|3NM362A2JG]].
In the language of the [[integers modulo $n$|03VFZ2WETT]], every sequence of $2n - 1$ elements of $\Z/n\Z$ has a subsequence of length $n$ with sum $0$.
Related invariants, such as the Davenport constant, ask for the shortest length that forces a nonempty zero-sum subsequence.
</document>

<document source="https://tamnd.github.io/math2/node/J037NW1Q38.md" tag="J037NW1Q38" kind="definition">
# Class operators H S P

## Definition

Let $K$ be a class of algebras of a fixed signature $\sigma$.

1. $I(K)$ is the class of algebras [[isomorphic|PZMGQMDN25]] to a member of $K$.
2. $H(K)$ is the class of algebras $B$ such that there is a surjective [[homomorphism|KBXSZC6BV1]] $A \to B$ for some $A \in K$.
3. $S(K)$ is the class of algebras isomorphic to a [[subalgebra|7CHG6SD365]] of a member of $K$.
4. $P(K)$ is the class of algebras isomorphic to a [[direct product|WJ73XYBNE4]] $\prod_{i \in I} A_i$ with all $A_i \in K$ and $I$ any index set.

For operators $O_1, O_2$, the composite $O_1 O_2$ is defined by $O_1 O_2(K) = O_1(O_2(K))$.
One writes $O_1 \le O_2$ when $O_1(K) \subseteq O_2(K)$ for every class $K$.
A class $K$ is *closed under* $O$ when $O(K) \subseteq K$.

## Remarks

The empty product, with $I = \emptyset$, is the one-element algebra, so $P(K)$ always contains a trivial algebra.
Example: for $K = \set{\Z}$ in the signature of groups, $\Z / 2\Z \in H(K)$, by reduction modulo $2$, and $\Z^2 \in P(K)$.
Non-example: $\Z / 2\Z \notin S(K)$, because every nontrivial subgroup of $\Z$ is infinite.
Each of $H$, $S$ and $P$ is extensive, monotone and idempotent on classes closed under $I$.
The standard composition rules are $SH \le HS$, $PS \le SP$ and $PH \le HP$.
</document>

<document source="https://tamnd.github.io/math2/node/J03P12X1X4.md" tag="J03P12X1X4" kind="definition">
# Laguerre geometry

## Definition

Work in the Euclidean plane $\R^2$ with the [[dot product|0S52TNNTH9]].

1. A *cycle* is a pair $(c, r)$ with $c \in \R^2$ and $r \in \R$. For $r \ne 0$ it is the circle with centre $c$ and radius $\abs{r}$, oriented by the sign of $r$. For $r = 0$ it is the point $c$.
2. A *spear* is an oriented [[line|0MP9BCF6DP]], given by a unit vector $N \in \R^2$ and $h \in \R$ as the line $\set{x : x \cdot N = h}$ with the orientation determined by $N$.
3. A cycle $(c, r)$ and a spear $(N, h)$ are in *oriented contact* when $c \cdot N - h = r$.

A *Laguerre transformation* is a pair of bijections, one of the set of spears and one of the set of cycles.
It is required that a cycle and a spear are in oriented contact if and only if their images are.
*Laguerre geometry* is the study of the properties of spears and cycles that are invariant under all Laguerre transformations.

## Remarks

Two cycles $(c_1, r_1)$ and $(c_2, r_2)$ touch with matching orientations when $\abs{c_1 - c_2} = \abs{r_1 - r_2}$.
In the cyclographic model, the cycle $(c, r)$ is the point $(c, r) \in \R^3$, and this condition says that the difference of the two points is a null vector of the form $x^2 + y^2 - z^2$.
Example: for $t \in \R$, the map sending the cycle $(c, r)$ to $(c, r + t)$ and the spear $(N, h)$ to $(N, h - t)$ is a Laguerre transformation.
Check: $c \cdot N - (h - t) = r + t$ holds exactly when $c \cdot N - h = r$, and both maps are bijections.
It sends each point $(c, 0)$ to a circle of radius $\abs{t}$, so it is not a map of points.
Non-example: inversion in the unit circle is not a Laguerre transformation in this sense.
It sends the lines not through the origin to circles, so it does not map spears to spears.
Euclidean isometries and dilations $x \mapsto \lambda x$, $\lambda > 0$, acting on centres and radii, are also Laguerre transformations.
</document>

<document source="https://tamnd.github.io/math2/node/J05VKG4XQB.md" tag="J05VKG4XQB" kind="conjecture">
# Schiffer's conjecture

## Statement

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a bounded, simply connected domain with smooth boundary $\partial\Omega$ and outer unit normal $\nu$.
Suppose that for some $\lambda > 0$ and some constant $c$ there is a nonzero function $u$ with
$$\Delta u + \lambda u = 0 \text{ in } \Omega, \qquad \frac{\partial u}{\partial \nu} = 0 \text{ on } \partial\Omega, \qquad u = c \text{ on } \partial\Omega,$$
where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
The conjecture states that $\Omega$ is then a ball.

## Remarks

The boundary value problem is overdetermined: $u$ is a Neumann eigenfunction that is also constant on the boundary.
Every ball admits such a solution: a radial Neumann eigenfunction is constant on each sphere centred at the centre of the ball.
By a result of Williams (1976), for these domains the conjecture is equivalent to the [[Pompeiu problem|M1KYAQMF9W]], that only balls fail the Pompeiu property.
Consider the analogous problem for a Dirichlet eigenfunction with constant normal derivative.
By Serrin's theorem on overdetermined problems, when the Dirichlet eigenfunction is positive, the domain must be a ball.

## Explanations

### L1 human

Take a membrane with a free edge and look at one of its vibration patterns.
Suppose that along the whole edge the membrane sits at one common height.
Schiffer guessed that this can only happen when the membrane is round.
</document>

<document source="https://tamnd.github.io/math2/node/J0DHYQ0MDE.md" tag="J0DHYQ0MDE" kind="concept">
# Lattice-ordered group

## Summary

A lattice-ordered group, or $\ell$-group, is an [[ordered group|KRDR04RQ7N]] whose order is a [[lattice|WH4SFAACAS]].
Any two elements have a least upper bound and a greatest lower bound, and multiplication preserves the order on both sides.
Every linearly ordered group is an $\ell$-group, and so are groups of real-valued functions with the pointwise order.
In an $\ell$-group each element splits into a positive part and a negative part, which gives an absolute value and a rich arithmetic of inequalities.

## Explanations

### L1 human

Think of real functions on a set, added pointwise and compared pointwise.
For two such functions $f$ and $g$ you can form the pointwise maximum and the pointwise minimum.
So the order is not total, but every pair of elements still has a best upper bound and a best lower bound.
A lattice-ordered group is a group with an order of this kind.
Each $f$ is the difference $f^+ - f^-$ of its positive part $\max(f, 0)$ and its negative part $\max(-f, 0)$.
</document>

<document source="https://tamnd.github.io/math2/node/J0EVKMWT94.md" tag="J0EVKMWT94" kind="statement">
# Tait conjectures

## Statement

A diagram of a [[knot|QYEAZTYGPT]] is *alternating* if the crossings alternate between over and under along the knot.
It is *reduced* if no crossing can be removed by turning one side of the diagram over, that is, it has no nugatory crossing.
Let $K$ be a knot that has an alternating diagram.

1. Every reduced alternating diagram of $K$ has $c(K)$ crossings, where $c(K)$ is the [[crossing number|SWJZ4QRF9W]] of $K$.
2. Any two reduced alternating diagrams of $K$ have the same writhe.
3. If $K$ is prime, then any two reduced alternating diagrams of $K$ are related by a finite sequence of flypes.

## Remarks

Part 1 was proved in 1987 by Kauffman, Murasugi and Thistlethwaite, independently, with the [[Jones polynomial|4SPC17FPXE]].
Part 2 was proved soon after, also with polynomial invariants.
The key fact is that the span of the Jones polynomial is at most the number of crossings of any connected diagram, with equality for reduced alternating diagrams.
Part 3, the flyping conjecture, was proved by Menasco and Thistlethwaite.
Example: the standard diagram of the [[figure-eight knot|C01XBGAHC6]] is reduced and alternating, so its crossing number is $4$ and its writhe $0$ is the same in every reduced alternating diagram.
The hypothesis that the diagram is reduced is needed: adding a nugatory kink keeps a diagram alternating but adds a crossing.
Part 2 shows that an alternating knot with a reduced alternating diagram of nonzero writhe is not isotopic to its mirror image.

## Explanations

### L1 human

Peter Guthrie Tait made knot tables in the nineteenth century and guessed some rules about alternating drawings.
In such a drawing, travelling along the knot you go over, under, over, under, and so on.
The first rule says that a clean alternating drawing already uses the fewest possible crossings.
The second says that all clean alternating drawings of one knot have the same twist count.
The third says that they are all related by a simple local flip, the flype.

### L2 human

The theorems make the tabulation of alternating knots possible: crossing number is read off any reduced alternating diagram, and diagrams can be compared through flypes.
The proof of part 1 is an early triumph of the Jones polynomial and of the [[Kauffman bracket|JYKVYS4N32]] state sum.
For a reduced alternating diagram, the extreme states of the bracket each contribute a single monomial that cannot cancel.
The writhe result gives a quick test for chirality of alternating knots.
For non-alternating knots, finding the crossing number is much harder.
</document>

<document source="https://tamnd.github.io/math2/node/J0JY4AG3E6.md" tag="J0JY4AG3E6" kind="concept">
# Method of steps

## Summary

The method of steps solves a [[delay differential equation|39PCDRDB01]] with a constant delay one interval at a time.
On each interval of the length of the delay, the delayed term is already known, so the equation becomes an [[ordinary differential equation|0NT64JV96A]].
It proves existence and uniqueness of solutions forward in time, and it gives explicit solutions in simple cases.

## Explanations

### L1 human

If the delay is one second, then during the first second the delayed values are just the given history.
So one can solve an ordinary equation for the first second.
Then this answer becomes the history for the next second, and so on.
For $x'(t) = -x(t - 1)$ with $x = 1$ before time $0$, the first step gives $x(t) = 1 - t$ on $[0, 1]$.
The second step gives $x(t) = (t - 1)(t - 3)/2$ on $[1, 2]$.
</document>

<document source="https://tamnd.github.io/math2/node/J0K6T02FQQ.md" tag="J0K6T02FQQ" kind="concept">
# Bachmann's reflection geometry

## Summary

Bachmann's reflection geometry builds plane [[absolute geometry|VKFPCSRNDH]] from a [[group|G5S1M9]] generated by reflections.
Lines are replaced by the reflections in them, and points by the half-turns about them.
Incidence and perpendicularity become algebraic relations between involutions, the elements of order $2$ of the group.
The resulting structures are called metric planes, and they include Euclidean, hyperbolic and elliptic planes over many fields.
Friedrich Bachmann developed this approach in his book "Aufbau der Geometrie aus dem Spiegelungsbegriff".

## Explanations

### L1 human

In the Euclidean plane, every line gives a reflection and every point gives a half-turn.
A point lies on a line exactly when the half-turn and the reflection, done one after the other, give a reflection again.
Two lines are perpendicular exactly when the product of their reflections is a half-turn.
Bachmann turned these facts around: start with a group and a set of its involutions, call them lines, and write axioms in the group.
Theorems of geometry then become computations with reflections, as in the three reflections theorem.
</document>

<document source="https://tamnd.github.io/math2/node/J0MHS4ZCKR.md" tag="J0MHS4ZCKR" kind="definition">
# Fourier series method

## Definition

Let $L > 0$ and let $A$ be a linear differential operator in $x$, with boundary conditions at $0$ and $L$.
Consider the linear evolution equation $u_t = A u$ for $u \colon [0, L] \times [0, \infty) \to \R$.
Assume that $L^2(0, L)$ has an [[orthonormal basis|0S9GGTR0EX]] $(\varphi_k)_{k \ge 1}$ of functions that satisfy the boundary conditions and $A \varphi_k = -\mu_k \varphi_k$ with $\mu_k \in \R$.
Typically the $\varphi_k$ are sines or cosines, so that expansions in them are [[Fourier series|G12KXDHPPS]].
The *Fourier series method* for the initial value $u(\cdot, 0) = f$ consists of three steps.

1. Expand $f = \sum_k b_k \varphi_k$ with $b_k = \int_0^L f \varphi_k \, dx$.
2. Seek $u(x, t) = \sum_k c_k(t) \varphi_k(x)$, which turns the equation into the ordinary differential equations $c_k' = -\mu_k c_k$ with $c_k(0) = b_k$.
3. Solve them, which gives $u(x, t) = \sum_k b_k e^{-\mu_k t} \varphi_k(x)$.

The same steps apply to second-order equations in time, such as the wave equation $u_{tt} = A u$, with two initial values.

## Remarks

Example: the [[heat equation|VP2HN86VAG]] $u_t = u_{xx}$ on $(0, \pi)$ with $u(0, t) = u(\pi, t) = 0$.
Here $\varphi_k(x) = \sqrt{2/\pi}\, \sin(kx)$ and $\mu_k = k^2$, since $(\sin kx)'' = -k^2 \sin kx$.
For $f(x) = \sin x + \sin 3x$, the method gives $u(x, t) = e^{-t} \sin x + e^{-9t} \sin 3x$.
Check: $u_t = -e^{-t} \sin x - 9 e^{-9t} \sin 3x = u_{xx}$, and $u$ vanishes at $0$ and $\pi$.

Non-example: the nonlinear equation $u_t = u_{xx} + u^2$ with the same boundary conditions does not decouple.
The product of two sine modes contains other modes, so the coefficient equations form an infinite coupled system.

Each term $c_k(t) \varphi_k(x)$ is a product solution, so the method is [[separation of variables|47NFRVB564]] followed by superposition.
For $f \in L^2(0, L)$ and $\mu_k \ge 0$, the series for $u$ converges in $L^2$ for each $t \ge 0$.

## Explanations

### L1 human

The method writes the initial state as a sum of basic shapes that the equation only rescales in time.
Each shape is then evolved by a simple exponential factor, and the pieces are added back together.
</document>

<document source="https://tamnd.github.io/math2/node/J0N4J606X0.md" tag="J0N4J606X0" kind="concept">
# Hadamard matrix

## Summary

A Hadamard matrix is a square [[matrix|0VD9ZSWWJG]] with entries $\pm 1$ whose rows are pairwise orthogonal.
Their determinant has absolute value $n^{n/2}$, the largest possible for a real $n \times n$ matrix with entries in $[-1, 1]$.
Hadamard matrices give error-correcting codes, block designs and experimental designs.
The node [[Hadamard matrix|DM59GHSMAJ]] gives the precise definition.

## Explanations

### L1 human

Take a square grid of plus and minus signs.
It is a Hadamard matrix if any two rows agree in exactly half of the places and disagree in the other half.
The smallest interesting one is the $2 \times 2$ grid with rows $+\,+$ and $+\,-$.
Repeating this pattern in blocks gives Hadamard matrices of sizes $4, 8, 16$ and so on.
</document>

<document source="https://tamnd.github.io/math2/node/J0NR3YRW07.md" tag="J0NR3YRW07" kind="definition">
# Numerical analysis

## Definition

*Numerical analysis* is the branch of mathematics that studies algorithms for computing approximate values of quantities defined by problems of continuous mathematics.
Its objects are the following.

1. A *problem*: a map $f$ from data $x$ to an exact answer $y = f(x)$, such as a root or an integral.
2. A *numerical method*: a finite procedure, or a [[sequence|0G9EKM0E15]] of them, that computes an approximation $\tilde y$ of $y$ from $x$ by arithmetic operations.
3. An *error analysis*: a bound on the error $\tilde y - y$, or a proof that the approximations have [[limit|05M5SWSZZD]] $y$, together with the cost of the method.

## Remarks

Example: the problem is to compute $\sqrt{2}$, and the method is the iteration $x_{k+1} = (x_k + 2/x_k)/2$ from $x_0 = 1$.
The error analysis shows that $x_k \to \sqrt{2}$ and that the error is roughly squared at each step.
Non-example: the symbolic simplification $\int_0^1 2x \, dx = 1$ by an antiderivative is exact computation, not an approximation, so it is not a numerical method in this sense.
The error of a method has two parts: the error of the method in exact arithmetic, and the error caused by rounding in a computer.
</document>

<document source="https://tamnd.github.io/math2/node/J0SV0PCDBW.md" tag="J0SV0PCDBW" kind="definition">
# Region of convergence

## Definition

Let $x = (x_n)_{n \in \Z}$ be a sequence of complex numbers, with [[Z-transform|5FY0JQGHYZ]] $X(z) = \sum_{n \in \Z} x_n z^{-n}$.
The *region of convergence* of $X$ is the set
$$\mathrm{ROC}(x) = \set{ z \in \C \setminus \set{0} : \sum_{n \in \Z} \abs{x_n} \abs{z}^{-n} < \infty }.$$

## Remarks

Let $r = \limsup_{n \to \infty} \abs{x_n}^{1/n}$ and $1/R = \limsup_{n \to \infty} \abs{x_{-n}}^{1/n}$, with the conventions $1/0 = \infty$ and $1/\infty = 0$.
By the root test, the open [[annulus|0CW52NV5TC]] $r < \abs{z} < R$ lies in the region, and no point with $\abs{z} < r$ or $\abs{z} > R$ does.
This is the [[radius of convergence|0N288P2TB5]] argument applied to each half of the [[Laurent series|0GXMPDNYX3]].
Points on the circles $\abs{z} = r$ and $\abs{z} = R$ may or may not belong to it.
Example: $x_n = 1$ for $n \ge 0$ and $x_n = 0$ for $n < 0$ has region $\abs{z} > 1$; the point $z = 1$ fails because $\sum_{n \ge 0} 1$ diverges.
Non-example: $z = 1/2$ is not in the region of $x_n = 2^n$ ($n \ge 0$), since $\sum 2^n 2^n$ diverges.
Some texts add the point $z = \infty$ to the region of a sequence with $x_n = 0$ for $n < 0$, or $z = 0$ for a sequence with $x_n = 0$ for $n > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/J10YF5CXX9.md" tag="J10YF5CXX9" kind="definition">
# Graphical model

## Definition

Let $G = (V, E)$ be a finite [[graph|00C0BR7D7H]], directed or undirected, and let $X = (X_v)_{v \in V}$ be a family of [[random variables|0PNRKMRC55]] indexed by the vertices.
A *Markov property* for $G$ is a list of [[conditional independence|0THJ7416DX]] statements $X_A \perp X_B \mid X_S$, for subsets $A, B, S \subseteq V$, that is read off the graph.
The *graphical model* of $G$ is the set of all [[joint distributions|0DR81R1F2B]] of $X$ that satisfy the Markov property of $G$.
For a directed acyclic graph it is called a Bayesian network, and for an undirected graph a Markov random field.

## Remarks

Here $X_A = (X_v)_{v \in A}$.
Example: the undirected path $1 - 2 - 3$ gives the model of all distributions with $X_1 \perp X_3 \mid X_2$.
A Markov chain $X_1, X_2, X_3$ lies in this model.
Non-example: let $X_1, X_2$ be independent fair coins and $X_3 = X_1$.
Then $X_1$ and $X_3$ are dependent given $X_2$, so this distribution is not in the model of the path.
A complete graph imposes no condition, so its model contains all joint distributions.
Often the model is restricted further to a parametric family, such as the Gaussian graphical models.
</document>

<document source="https://tamnd.github.io/math2/node/J1E6BEFAPZ.md" tag="J1E6BEFAPZ" kind="problem">
# Illumination problem

## Problem

Let $n \ge 1$, and let $K \subseteq \R^n$ be a [[convex body|437FSG3419]].
A nonzero vector $u$ *illuminates* a [[boundary|05D5EGMFYV]] point $x$ of $K$ if the ray $\set{x + t u : t > 0}$ meets the [[interior|0RZH6W98HK]] of $K$.
The *illumination number* $I(K)$ is the least number of vectors such that every boundary point of $K$ is illuminated by one of them.
Show that $I(K) \le 2^n$, with equality only when $K$ is a parallelotope.

## Remarks

Boltyanski proved that $I(K)$ equals the least number of smaller homothetic copies $x_i + \lambda K$, with $0 < \lambda < 1$, that cover $K$.
So the problem is equivalent to the [[Hadwiger covering conjecture|SVP75YY386]], and it is open for $n \ge 3$.
Example: for a disc, a vector $u$ illuminates the boundary point with outer normal $\nu$ exactly when $u \cdot \nu < 0$.
Three vectors at angles $2\pi/3$ illuminate every boundary point, while two vectors never suffice, so $I = 3$.
For the square, each corner needs its own direction, so $I = 4 = 2^2$.

## Explanations

### L1 human

Imagine a convex solid lit by floodlights placed very far away, each shining in one fixed direction.
A point of the surface counts as lit if the light, coming in, can pass through it into the solid.
How many floodlights are needed to light the whole surface?
A round ball needs few, a cube needs one for each corner, and the conjecture says that no convex body in $n$ dimensions needs more than $2^n$.
</document>

<document source="https://tamnd.github.io/math2/node/J1EZZN14WC.md" tag="J1EZZN14WC" kind="definition">
# Jackknife

## Definition

Let $x_1, \dots, x_n$ be data with $n \ge 2$, and let $\hat\theta = T_n(x_1, \dots, x_n)$ be a [[statistic|036756NFD6]] defined for every sample size.
For $i = 1, \dots, n$, let $\hat\theta_{(i)} = T_{n-1}(x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_n)$ be the leave-one-out value, and let $\hat\theta_{(\cdot)} = \frac{1}{n} \sum_{i=1}^{n} \hat\theta_{(i)}$.
The *jackknife* gives:

1. the bias estimate $\widehat{\mathrm{bias}} = (n - 1)\big(\hat\theta_{(\cdot)} - \hat\theta\big)$;
2. the bias-corrected estimate $\hat\theta_{\mathrm{jack}} = \hat\theta - \widehat{\mathrm{bias}} = n\hat\theta - (n - 1)\hat\theta_{(\cdot)}$;
3. the variance estimate $\widehat{\operatorname{Var}}_{\mathrm{jack}} = \frac{n - 1}{n} \sum_{i=1}^{n} \big(\hat\theta_{(i)} - \hat\theta_{(\cdot)}\big)^2$.

## Remarks

If the [[bias|01BD2TFFP4]] of $\hat\theta$ has the form $a/n + O(1/n^2)$, the bias of $\hat\theta_{\mathrm{jack}}$ is $O(1/n^2)$.

Example: for the sample mean $\hat\theta = \bar x$, one has $\hat\theta_{(i)} = (n \bar x - x_i)/(n - 1)$, so $\hat\theta_{(i)} - \hat\theta_{(\cdot)} = (\bar x - x_i)/(n-1)$.
Then $\widehat{\operatorname{Var}}_{\mathrm{jack}} = \frac{1}{n(n-1)} \sum_i (x_i - \bar x)^2 = s^2/n$, the usual estimate of the [[variance|007QED3E0Y]] of $\bar x$, with $s^2$ the [[sample variance|0JN9P4A944]].
For the plug-in variance $\hat\theta = \frac1n \sum_i (x_i - \bar x)^2$, the bias-corrected estimate is exactly $s^2 = \frac{1}{n-1}\sum_i (x_i - \bar x)^2$.

Non-example: for the sample median the variance estimate in clause 3 is not consistent.
The median is not a smooth function of the data, and its leave-one-out values take only two or three distinct values.
</document>

<document source="https://tamnd.github.io/math2/node/J1SGNEWKWS.md" tag="J1SGNEWKWS" kind="definition">
# P-adic dynamics

## Definition

Let $p$ be a prime.
Let $K$ be a field that is complete for a [[non-Archimedean absolute value|1H5F1BP6XS]] $\abs{\cdot}$ extending the $p$-adic absolute value of $\Q$.
Examples are $\Q_p$, its finite extensions, and $\C_p$.
A *$p$-adic dynamical system* is a pair $(X, \varphi)$ in which:

1. $X$ is either the projective line $\mathbb{P}^1(K)$ or a subset of $K$, such as a closed disc;
2. $\varphi \colon X \to X$ is given by a rational function in $K(z)$ when $X = \mathbb{P}^1(K)$, or by a power series with coefficients in $K$ that converges on $X$ otherwise.

*$p$-adic dynamics* is the study of the [[iterates|AYVTR2ES5B]] $\varphi^n$ and of the orbits $\set{\varphi^n(x) : n \in \N}$ of such systems, with respect to the topology of $\abs{\cdot}$.

## Remarks

Example: let $X = \Z_p$ and $\varphi(x) = x + 1$.
It maps $\Z_p$ into $\Z_p$, and it is a polynomial, so $(\Z_p, \varphi)$ is a $p$-adic dynamical system.
It is an isometry, since $\abs{\varphi(x) - \varphi(y)} = \abs{x - y}$.
The orbit of $0$ is $\N$, which is dense in $\Z_p$.

Non-example: let $\varphi(x) = x / p$ on $X = \Z_p$.
Then $\varphi(1) = 1/p$ has absolute value $p > 1$, so $\varphi$ does not map $X$ into $X$, and clause 2 fails.

A second example: $\varphi(z) = z^2$ on the open unit disc $\set{x \in \C_p : \abs{x} < 1}$.
Since $\abs{\varphi^n(x)} = \abs{x}^{2^n}$, every orbit tends to the attracting fixed point $0$.
</document>

<document source="https://tamnd.github.io/math2/node/J1TZHNRV02.md" tag="J1TZHNRV02" kind="definition">
# Pseudorandom number generator

## Definition

A *pseudorandom number generator* is a tuple $(S, s_0, T, U, G)$ where:

1. $S$ is a finite set, the state space;
2. $s_0 \in S$ is the initial state, or seed;
3. $T \colon S \to S$ is a map, the transition function;
4. $U$ is a set, the output space, usually $[0, 1)$ or a set of integers $\set{0, \ldots, m-1}$;
5. $G \colon S \to U$ is a map, the output function.

The generator produces the states $s_i = T(s_{i-1})$ for $i \ge 1$ and the output sequence $u_i = G(s_i)$ for $i \ge 0$.

## Remarks

Example: $S = \set{0, \ldots, 7}$, $T(s) = (5s + 1) \bmod 8$, $U = [0, 1)$, $G(s) = s/8$ and $s_0 = 0$; all five clauses hold, and the outputs are $0, 0.125, 0.75, 0.875, 0.5, 0.625, 0.25, 0.375$, then repeat.
Non-example: the sequence of decimal digits of $\pi$ is not eventually periodic, because $\pi$ is irrational.
It is not the output sequence of any such tuple, since a finite state space as in clause 1 forces the outputs to be eventually periodic.
Since $S$ is finite, the sequence $(s_i)$ is eventually periodic, with period at most $\abs{S}$; if $T$ is a bijection it is purely periodic.
The definition is purely structural; the quality of a generator is judged by its period, by the uniformity of tuples $(u_i, \ldots, u_{i+k-1})$, and by statistical tests.
Generators for cryptography must in addition be unpredictable from past outputs, which is a stronger requirement.
</document>

<document source="https://tamnd.github.io/math2/node/J1YDW3BE56.md" tag="J1YDW3BE56" kind="concept">
# Mathematical formulation of quantum mechanics

## Summary

The mathematical formulation of quantum mechanics describes a physical system by a complex [[Hilbert space|HM2HHE3J51]].
States are unit vectors or density operators, and measurable quantities are [[self-adjoint operators|0NEGN5CRSC]].
Measurement probabilities come from the spectral theorem through [[projection-valued measures|ATGT25E5FF]], and time evolution is a one-parameter unitary group generated by the Hamiltonian.
Dirac and von Neumann set out this framework around 1930, and it is also stated as the Dirac–von Neumann axioms.
It made quantum theory a part of functional analysis and gave rise to operator algebras.

## Explanations

### L1 human

In classical mechanics, the state of a system is a point, and a quantity such as energy is a function of that point.
In quantum mechanics, the state is a vector in a space of possibly infinite dimension, and a quantity is an operator that acts on these vectors.
The possible results of measuring the quantity are the points of the spectrum of the operator, and the chances of each result are computed from the state vector.
For example, a spin-$\tfrac12$ particle lives in $\C^2$, and its spin along one axis is the matrix $\operatorname{diag}(1, -1)$, with possible results $+1$ and $-1$.
Because operators need not commute, some pairs of quantities cannot both have sharp values, which is the source of the uncertainty principle.
</document>

<document source="https://tamnd.github.io/math2/node/J28WVG6KPT.md" tag="J28WVG6KPT" kind="definition">
# Double layer potential

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]] whose [[boundary|05D5EGMFYV]] $\partial\Omega$ is a $C^1$ hypersurface.
Let $\sigma$ be surface measure on $\partial\Omega$ and $\nu(y)$ the outward unit normal at $y \in \partial\Omega$.
Let $\Phi$ be the [[fundamental solution of the Laplacian|BSJ8X2XRN9]], normalised by $-\Delta\Phi = \delta_0$.
Let $\varphi \colon \partial\Omega \to \R$ be continuous.
The *double layer potential* with density $\varphi$ is the function
$$D\varphi(x) = \int_{\partial\Omega} \frac{\partial \Phi(x - y)}{\partial \nu(y)}\, \varphi(y)\, d\sigma(y), \qquad x \in \R^n \setminus \partial\Omega,$$
where $\dfrac{\partial \Phi(x - y)}{\partial \nu(y)} = \nu(y) \cdot \nabla_y \big[\Phi(x - y)\big]$ is the derivative in $y$ along the outward normal.

## Remarks

The integral is a [[surface integral|065HE5V98E]] with an integrand that is smooth in $x$ off $\partial\Omega$, and $D\varphi$ is [[harmonic|0DEB0JMN0W]] on $\R^n \setminus \partial\Omega$.
For $n = 3$, $\Phi(x) = 1/(4\pi\abs{x})$ and the kernel is $\dfrac{\nu(y) \cdot (x - y)}{4\pi \abs{x - y}^3}$.

Example (Gauss's integral): let $\varphi \equiv 1$.
By the [[divergence theorem|0JX86PSSA9]], $D1(x) = -1$ for $x \in \Omega$ and $D1(x) = 0$ for $x \notin \overline{\Omega}$.
For the unit ball in $\R^3$ and $x = 0$ this is a direct check: $\nu(y) = y$, so the kernel is $y \cdot (-y)/(4\pi) = -1/(4\pi)$, and the sphere has area $4\pi$.

Non-example: replacing the normal derivative of $\Phi$ by $\Phi$ itself gives the [[single layer potential|9T6ZEV4125]] instead.
For the unit ball in $\R^3$ with $\varphi \equiv 1$, that potential equals $1$ inside the ball, while the double layer potential equals $-1$ there.
</document>

<document source="https://tamnd.github.io/math2/node/J29JWVDSEJ.md" tag="J29JWVDSEJ" kind="definition">
# Skew polynomial ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity and let $\sigma \colon R \to R$ be a [[ring endomorphism|0VTH8KPCG2]] with $\sigma(1) = 1$.
The *skew polynomial ring* $R[x; \sigma]$ is the [[Ore extension|VF3PV13BA1]] $R[x; \sigma, 0]$.
So it is a free left $R$-module with basis $1, x, x^2, \dots$, it contains $R$ as a subring, and $x a = \sigma(a) x$ for all $a \in R$.

## Remarks

The product of monomials is $(a x^i)(b x^j) = a \sigma^i(b) x^{i+j}$.
Example: let $R = \C$ and let $\sigma$ be complex conjugation.
Then $x z = \bar z x$ for $z \in \C$, and $x^2 z = x \bar z x = z x^2$, so $x^2$ commutes with all of $\C$ and with $x$; it lies in the center.
Example: for a field $k$, $q \in k$ nonzero, $R = k[y]$ and $\sigma(y) = q y$, the ring $R[x; \sigma]$ has $x y = q y x$; it is the quantum plane.
Non-example: the first Weyl algebra, with $x y = y x + 1$, is an Ore extension $k[y][x; \id, d/dy]$ with a nonzero derivation, so it is not of the form $R[x; \sigma]$ with this $R$ and this $x$.
When $\sigma = \id$ the ring $R[x; \sigma]$ is the ordinary polynomial ring $R[x]$.
Some books call every Ore extension $R[x; \sigma, \delta]$ a skew polynomial ring.
</document>

<document source="https://tamnd.github.io/math2/node/J2GWK27Y15.md" tag="J2GWK27Y15" kind="definition">
# Quantile regression

## Definition

Let $\tau \in (0, 1)$.
The *check function* is $\rho_\tau \colon \R \to \R$, $\rho_\tau(u) = u\big(\tau - \mathbf{1}\set{u < 0}\big)$, so $\rho_\tau(u) = \tau u$ for $u \ge 0$ and $\rho_\tau(u) = (\tau - 1)u$ for $u < 0$.
Let $(x_1, y_1), \dots, (x_n, y_n)$ be data with $x_i \in \R^p$ and $y_i \in \R$.
The *linear quantile regression* estimate at level $\tau$ is any minimiser
$$\hat\beta(\tau) \in \arg\min_{\beta \in \R^p} \sum_{i=1}^{n} \rho_\tau\big(y_i - x_i^{T} \beta\big).$$
It estimates $\beta(\tau)$ in the model in which the conditional [[quantile function|0B3Q3XDM5X]] of $Y$ given $X = x$ at level $\tau$ is $x^{T}\beta(\tau)$.

## Remarks

For a real random variable $Y$, the minimisers of $b \mapsto E[\rho_\tau(Y - b)]$ are exactly the $\tau$-quantiles of $Y$ (when $E\abs{Y} < \infty$).
This is why the check [[loss|3BTG6VVCGZ]] targets quantiles, just as squared loss targets the mean.
The minimisation is a linear program.
When $x_1, \dots, x_n$ span $\R^p$, some minimiser fits at least $p$ data points exactly.
For $\tau = 1/2$, $\rho_{1/2}(u) = \abs{u}/2$, and the method is least absolute deviations regression.

Example: the intercept-only model, $p = 1$ and $x_i = 1$, with $y = (1, 2, 3, 4, 5)$ and $\tau = 1/2$.
The function $b \mapsto \sum_i \abs{y_i - b}/2$ is minimised at $b = 3$, the sample median.

Non-example: minimising $\sum_i (y_i - x_i^{T}\beta)^2$ is [[linear regression|41FVC4P5JB]] by least squares.
Its loss is quadratic, not a check function, and it estimates the conditional mean.
For skewed data the conditional mean and the conditional median differ, so the two fits differ.
</document>

<document source="https://tamnd.github.io/math2/node/J2KM2VYA3J.md" tag="J2KM2VYA3J" kind="concept">
# Steiner conic

## Summary

A Steiner conic is a conic built synthetically from two pencils of lines.
Take two points $U$ and $V$ and a projective map from the lines through $U$ to the lines through $V$.
The points where each line meets its image form the Steiner conic.
In a projective plane over a field, Steiner conics are exactly the non-degenerate [[projective conics|R3TMVHTP3A]], so this gives a definition of conics with no equations.
The construction goes back to Jakob Steiner.

## Explanations

### L1 human

Pin one ruler at a point $U$ and another at a point $V$.
Turn them together by a fixed rule, so that each position of the first ruler decides the position of the second.
Mark where the two rulers cross.
When the rule is a projective map, and the line $UV$ is not sent to itself, the marks trace out a conic through $U$ and $V$.
</document>

<document source="https://tamnd.github.io/math2/node/J2T75ATRV5.md" tag="J2T75ATRV5" kind="concept">
# Join and meet

## Summary

In a [[partially ordered set|0QV2JS9RJ3]], the join of two elements is their least upper bound and the meet is their greatest lower bound.
They are written $x \vee y$ and $x \wedge y$.
Join and meet are the [[supremum|0R3H5PANRQ]] and the [[infimum|0QF1MZ9FRG]] of a two-element set, seen as binary operations.
They are the basic operations of lattice theory: a poset in which all joins and meets of pairs exist is a lattice.

## Explanations

### L1 human

Think of the elements of a poset as points of a diagram, with larger elements drawn higher.
The join of $x$ and $y$ is the lowest point that lies above both; the meet is the highest point below both.
For sets ordered by inclusion, the join of $A$ and $B$ is $A \cup B$ and the meet is $A \cap B$.
For positive integers ordered by divisibility, the join of $4$ and $6$ is $\lcm(4, 6) = 12$ and the meet is $\gcd(4, 6) = 2$.
In some posets a join or a meet can fail to exist.
</document>

<document source="https://tamnd.github.io/math2/node/J2TX4TDPCX.md" tag="J2TX4TDPCX" kind="concept">
# Enough projectives

## Summary

A category has enough projectives when every object is a quotient of a [[projective object|YXMXAGVC1Q]].
In an [[abelian category|Y9CZ3MTNF1]] with enough projectives, every object has a [[projective resolution|HK2AA1YSQ2]].
This is the condition that makes left [[derived functors|NPMC3K6J2F]] such as Tor well defined.

## Explanations

### L1 human

Projective objects are the easy building blocks of a category, like free modules among all modules.
Having enough projectives means that every object can be reached from such a block by a surjection.
Then one can describe any object by blocks, then the relations by blocks, and so on.
For modules over a ring $R$, every module $M$ is a quotient of the free module with one basis vector for each element of $M$.
So the category of $R$-modules has enough projectives.
</document>

<document source="https://tamnd.github.io/math2/node/J2XK2WAHBG.md" tag="J2XK2WAHBG" kind="definition">
# Sparse approximation

## Definition

Let $A$ be an $m \times N$ real or complex [[matrix|0VD9ZSWWJG]], usually with $m < N$, let $b \in \mathbb{K}^m$ with $\mathbb{K} = \R$ or $\C$, and let $\norm{\cdot}_2$ be the Euclidean norm.
For $x \in \mathbb{K}^N$, let $\norm{x}_0$ be the number of nonzero entries of $x$; the vector $x$ is *$s$-sparse* if $\norm{x}_0 \le s$.
*Sparse approximation* is either of the two following problems.

1. Given an integer $s \ge 1$, find an $s$-sparse $x$ that minimizes $\norm{Ax - b}_2$.
2. Given a tolerance $\varepsilon \ge 0$, find $x$ with $\norm{Ax - b}_2 \le \varepsilon$ that minimizes $\norm{x}_0$.

For $\varepsilon = 0$, the second problem asks for the sparsest solution of the [[system of linear equations|0GT4WWGWH2]] $Ax = b$.

## Remarks

The count $\norm{x}_0$ is not a norm, since $\norm{2x}_0 = \norm{x}_0$; the notation is standard anyway.
Example: let $A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$, $b = (1, 1)$ and $\varepsilon = 0$.
The vector $x = (0, 0, 1)$ solves $Ax = b$ with $\norm{x}_0 = 1$, and no $x$ with $\norm{x}_0 = 0$ works since $b \ne 0$, so it solves problem 2.
Non-example: $x = (1, 1, 0)$ also solves $Ax = b$, but $\norm{x}_0 = 2$ is not minimal, so it is not a solution of problem 2.
Replacing $\norm{x}_0$ by the convex norm $\norm{x}_1 = \sum_j \abs{x_j}$ gives basis pursuit, a linear program; a penalized form of it is the [[Lasso|KZ226116HW]].
</document>

<document source="https://tamnd.github.io/math2/node/J32951135H.md" tag="J32951135H" kind="concept">
# Order complex

## Summary

The order complex of a [[partially ordered set|0QV2JS9RJ3]] $P$ is a [[simplicial complex|V8GZ0W1ZZ5]] with the elements of $P$ as vertices.
Its faces are the finite [[chains|0S8FE7ZMFM]] of $P$.
It turns a poset into a topological space, its [[geometric realization|TG6WHHR7CF]].
So one can speak of the homotopy type and the homology of a poset.
This is the basis of poset topology.
For example, take an open interval of a finite poset.
By a theorem of Philip Hall, the reduced Euler characteristic of its order complex equals the value of the [[Möbius function|2Y75S8NG5B]] on that interval.
The order complex of the face poset of a simplicial complex is its [[barycentric subdivision|J5DZEYCEX2]].

## Explanations

### L1 human

Draw a point for each element of the poset.
Join two points by an edge when they are comparable, fill in a triangle for each chain of three elements, a tetrahedron for each chain of four, and so on.
The resulting shape is the order complex.
For a chain $a < b < c$ the shape is a full triangle.
For the poset with $a, b < c, d$ and no other relations, the four edges $ac$, $ad$, $bc$, $bd$ form a square loop, which is a circle.
</document>

<document source="https://tamnd.github.io/math2/node/J332CQ8SPB.md" tag="J332CQ8SPB" kind="definition">
# Discontinuous Galerkin method

## Definition

Let $\Omega \subset \R^d$ be a bounded polygonal domain and let $F \colon \R \to \R^d$ be a $C^1$ flux.
Consider the scalar [[conservation law|8RA7AMWSAF]] $u_t + \nabla \cdot F(u) = 0$ on $\Omega$.
Let $\mathcal{T}_h$ be a mesh of $\Omega$ and $k \ge 0$ an integer.
Let $V_h$ be the functions that are [[polynomials|07GNX8GCYP]] of degree at most $k$ on each element $K$, with no continuity condition between elements.
Let $\hat H(a, b, n)$ be a numerical flux, such as a [[Riemann solver|ZY4GY93C4B]], with $\hat H(a, a, n) = F(a) \cdot n$ and $\hat H(a, b, n) = -\hat H(b, a, -n)$.
The *discontinuous Galerkin method* seeks $u_h(t) \in V_h$ such that, for every $K \in \mathcal{T}_h$ and every polynomial $v$ of degree at most $k$ on $K$,
$$\int_K \partial_t u_h \, v \, dx - \int_K F(u_h) \cdot \nabla v \, dx + \int_{\partial K} \hat H(u_h^-, u_h^+, n_K) \, v \, ds = 0.$$
Here $n_K$ is the outward unit normal of $K$, and $u_h^-$ and $u_h^+$ are the traces of $u_h$ from inside and outside $K$.

## Remarks

The equation is the weak form on one element, obtained by multiplying by $v$ and integrating by parts once.
The boundary flux $F(u) \cdot n_K$ is replaced by $\hat H$, because $u_h$ has two values on $\partial K$.
Since the test functions live on one element, the mass matrix is block diagonal and explicit time stepping is cheap.
Taking $v = 1$ on $K$ shows that the integral of $u_h$ over $K$ changes only through the fluxes on $\partial K$.

Example: for $k = 0$ and $d = 1$, with cells $K_i$ of width $\Delta x$, $u_h = u_i$ on $K_i$.
Here $\nabla v = 0$, and the conservation property of $\hat H$ turns the flux on the left face into $-\hat H(u_{i-1}, u_i)$, with $\hat H(a, b)$ short for $\hat H(a, b, +1)$.
The equation reads $\Delta x \, u_i' + \hat H(u_i, u_{i+1}) - \hat H(u_{i-1}, u_i) = 0$, a [[finite volume|HGKN8E09RS]] scheme.

Non-example: the continuous [[Galerkin method|Y5B0PEBKE8]] with continuous piecewise polynomials uses a space with continuity across elements and no numerical flux.
It does not have the element-by-element form above.
</document>

<document source="https://tamnd.github.io/math2/node/J39DGKG9A7.md" tag="J39DGKG9A7" kind="definition">
# Lipschitz class

## Definition

Let $I \subseteq \R$ be an interval, let $0 < \alpha \le 1$, and let $M > 0$.
The class $\mathrm{Lip}_M \alpha$ on $I$ is the set of functions $f \colon I \to \R$ such that
$$\abs{f(x) - f(y)} \le M \abs{x - y}^\alpha \quad \text{for all } x, y \in I.$$
The *Lipschitz class* $\mathrm{Lip}\,\alpha$ on $I$ is the union of the classes $\mathrm{Lip}_M \alpha$ over all $M > 0$.

## Remarks

Example: $f(x) = \sqrt{x}$ on $[0, 1]$ lies in $\mathrm{Lip}_1 \tfrac12$.
For $0 \le y \le x$ one has $\sqrt{x} - \sqrt{y} \le \sqrt{x - y}$, since squaring gives $x + y - 2\sqrt{xy} \le x - y$, that is $y \le \sqrt{xy}$.

Non-example: the same $f$ is not in $\mathrm{Lip}\,1$, because $\abs{f(x) - f(0)} / \abs{x - 0} = x^{-1/2}$ is unbounded as $x \to 0$.

In terms of the [[modulus of continuity|AXCJBSQZWF]] $\omega(f, \delta)$, the condition $f \in \mathrm{Lip}_M \alpha$ says $\omega(f, \delta) \le M \delta^\alpha$ for all $\delta > 0$.
$\mathrm{Lip}\,1$ is the class of [[Lipschitz continuous|0QNVQ4VD9T]] functions, and $\mathrm{Lip}\,\alpha$ for $\alpha < 1$ is the class of [[Hölder continuous|012PBA0FMD]] functions of exponent $\alpha$.
The restriction $\alpha \le 1$ is natural: for $\alpha > 1$ the condition forces $f' = 0$, so only constant functions remain.
</document>

<document source="https://tamnd.github.io/math2/node/J3CT5D7FSZ.md" tag="J3CT5D7FSZ" kind="statement">
# Janson inequality

## Statement

Let $\Gamma$ be a finite set, and let $R \subseteq \Gamma$ be a random subset that contains each $r \in \Gamma$ independently with probability $p_r$.
Let $A_1, \dots, A_n$ be subsets of $\Gamma$, and let $B_i$ be the [[event|031TPJVS74]] that $A_i \subseteq R$.
Write $i \sim j$ if $i \ne j$ and $A_i \cap A_j \ne \emptyset$.
Let
$$\mu = \sum_{i=1}^n \Pr(B_i), \qquad \Delta = \sum_{(i, j) : i \sim j} \Pr(B_i \cap B_j),$$
where the second sum runs over ordered pairs.
Then
$$\Pr\Big(\bigcap_{i=1}^n \overline{B_i}\Big) \le e^{-\mu + \Delta/2}.$$

## Remarks

If moreover $\Delta \ge \mu$, a further argument gives the bound $e^{-\mu^2/(2\Delta)}$.
In the other direction, the [[Harris inequality|W1YH3YD3S4]] gives $\Pr\big(\bigcap_i \overline{B_i}\big) \ge \prod_i (1 - \Pr(B_i))$.
So when $\Delta$ is small and each $\Pr(B_i)$ is small, the probability that no $B_i$ occurs is close to $e^{-\mu}$, as for independent events.
For example, in the [[random graph|TNT3DG0HRJ]] $G(n, p)$, take $\Gamma$ to be the set of edges of $K_n$ and the $A_i$ to be the edge sets of triangles.
Then $\mu = \binom{n}{3} p^3$.
When $p = c/n$ for a constant $c > 0$, one has $\mu \to c^3/6$ and $\Delta \to 0$ as $n \to \infty$.
With the lower bound, the probability that $G(n, p)$ has no triangle tends to $e^{-c^3/6}$.
The proof writes the probability as a product of the factors $\Pr\big(\overline{B_i} \mid \bigcap_{j < i} \overline{B_j}\big)$.
It bounds each conditional probability of $B_i$ from below, using the Harris inequality for increasing events.

## Explanations

### L1 human

Many small "bad" patterns may appear in a random set.
If the patterns rarely overlap, the chance that none of them appears is close to what it would be if they were independent.
The Janson inequality makes this precise, with an error term that measures the overlaps.

### L2 human

The Janson inequality is a standard tool in the theory of random graphs and the [[probabilistic method|3B55HNS754]].
It gives exponentially small upper bounds on the probability that a random graph contains no copy of a given subgraph.
Compared with the second moment method, it gives a much sharper bound on the probability of the lower tail.
</document>

<document source="https://tamnd.github.io/math2/node/J3D1T7F0CB.md" tag="J3D1T7F0CB" kind="definition">
# Markov chain mixing time

## Definition

Let $S$ be a finite set and let $P$ be the [[transition matrix|0F462493EA]] of an irreducible and aperiodic [[Markov chain|033TJBZDG0]] on $S$.
Let $\pi$ be its [[stationary distribution|0KRH5157WP]].
The *total variation distance* between probability distributions $\mu$ and $\nu$ on $S$ is
$$\norm{\mu - \nu}_{TV} = \max_{A \subseteq S} \abs{\mu(A) - \nu(A)} = \frac12 \sum_{x \in S} \abs{\mu(x) - \nu(x)}.$$
For an integer $t \ge 0$ let $d(t) = \max_{x \in S} \norm{P^t(x, \cdot) - \pi}_{TV}$, where $P^t(x, \cdot)$ is row $x$ of $P^t$.
For $\varepsilon \in (0, 1)$, the *mixing time* is
$$t_{\mathrm{mix}}(\varepsilon) = \min\set{t \ge 0 : d(t) \le \varepsilon},$$
and $t_{\mathrm{mix}} = t_{\mathrm{mix}}(1/4)$.

## Remarks

The minimum exists because $d(t) \to 0$ by the [[convergence theorem|B1E1ES1BYD]] for ergodic chains.
Example: on $S = \set{1, 2}$ let every entry of $P$ be $1/2$.
Then $\pi = (1/2, 1/2)$, $d(0) = 1/2$ and $P^t(x, \cdot) = \pi$ for $t \ge 1$, so $t_{\mathrm{mix}} = 1$.
Non-example: for $P = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ the chain is periodic, $d(t) = 1/2$ for all $t$, and $t_{\mathrm{mix}}(\varepsilon)$ does not exist for $\varepsilon < 1/2$.
The function $d$ is nonincreasing, and $t_{\mathrm{mix}}(\varepsilon) \le \ceil{\log_2 (1/\varepsilon)} \, t_{\mathrm{mix}}$.
</document>

<document source="https://tamnd.github.io/math2/node/J3FQ13SP85.md" tag="J3FQ13SP85" kind="definition">
# Homogenization

## Definition

Let $Y = [0, 1)^n$ be the unit cell and let $a = (a_{ij}) \colon \R^n \to \R^{n \times n}$ be measurable and $Y$-periodic.
Assume $a$ is uniformly elliptic: for constants $0 < \alpha \le \beta$, $a(y)\xi \cdot \xi \ge \alpha \abs{\xi}^2$ and $\abs{a(y)\xi} \le \beta \abs{\xi}$ for all $y, \xi \in \R^n$.
Let $\Omega \subseteq \R^n$ be a bounded open set and $f \in L^2(\Omega)$.
For $\varepsilon > 0$, let $u_\varepsilon \in H^1_0(\Omega)$ be the [[weak solution|SJZX81A783]] of the [[elliptic|B4TTNHSTRX]] problem
$$-\operatorname{div}\big(a(x/\varepsilon) \nabla u_\varepsilon\big) = f \text{ in } \Omega, \qquad u_\varepsilon = 0 \text{ on } \partial\Omega.$$
For $j = 1, \dots, n$, the *cell problem* asks for a $Y$-periodic $\chi_j \in H^1_{\mathrm{loc}}(\R^n)$ with mean $0$ over $Y$ and
$$-\operatorname{div}_y\big(a(y)(e_j + \nabla_y \chi_j)\big) = 0 \text{ in } \R^n,$$
where $e_j$ is the $j$-th unit vector.
The *homogenized matrix* is the constant matrix $a^0$ with entries
$$a^0_{ij} = \int_Y e_i \cdot a(y)\big(e_j + \nabla_y \chi_j(y)\big) \, dy,$$
and the *homogenized problem* is $-\operatorname{div}(a^0 \nabla u_0) = f$ in $\Omega$, $u_0 = 0$ on $\partial\Omega$.
*Homogenization* is the passage from the problems for $u_\varepsilon$ to the homogenized problem as $\varepsilon \to 0$.

## Remarks

The basic theorem of periodic homogenization states that $a^0$ is again uniformly elliptic and that $u_\varepsilon \to u_0$ weakly in the [[Sobolev space|M4CEZ5K9HP]] $H^1_0(\Omega)$ and strongly in $L^2(\Omega)$.
The formula for $a^0$ comes from the two-scale expansion $u_\varepsilon(x) \approx u_0(x) + \varepsilon \sum_j \chi_j(x/\varepsilon) \partial_j u_0(x)$.

Example: in one dimension, let $a$ be a $1$-periodic function with $\alpha \le a \le \beta$.
The cell problem gives $a(y)(1 + \chi') = c$ for a constant $c$, and the mean-zero condition $\int_0^1 \chi' = 0$ gives $c = \big(\int_0^1 a^{-1} \, dy\big)^{-1}$.
So $a^0 = \big(\int_0^1 a^{-1} \, dy\big)^{-1}$ is the harmonic mean of $a$.
For $a$ equal to $1$ on half the period and $3$ on the other half, $a^0 = 3/2$, while the arithmetic mean is $2$.

Non-example: the naive guess $a^0 = \int_Y a \, dy$, the arithmetic mean, is wrong in general, as the example shows; the weak limit of $a(x/\varepsilon) \nabla u_\varepsilon$ is not the product of the weak limits.

## Explanations

### L1 human

Homogenization replaces a finely striped material by one uniform material that behaves the same at large scale.
The effective coefficients come from solving one small problem on a single period of the stripes.
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<document source="https://tamnd.github.io/math2/node/J3JV10S9YX.md" tag="J3JV10S9YX" kind="definition">
# Gowers norm

## Definition

Let $G$ be a finite [[abelian group|082GHNPJ53]], let $f \colon G \to \C$ be a function, and let $k \ge 1$ be an integer.
For $\omega = (\omega_1, \dots, \omega_k) \in \set{0, 1}^k$ and $h = (h_1, \dots, h_k) \in G^k$, write $\omega \cdot h = \omega_1 h_1 + \dots + \omega_k h_k$ and $\abs{\omega} = \omega_1 + \dots + \omega_k$.
Let $\mathcal{C}$ be [[complex conjugation|0FTH2E9TAS]], so $\mathcal{C}^m z$ is $z$ for even $m$ and $\bar z$ for odd $m$.
The *Gowers $U^k$ norm* of $f$ is the nonnegative real number $\norm{f}_{U^k}$ with
$$\norm{f}_{U^k}^{2^k} = \frac{1}{\abs{G}^{k+1}} \sum_{x \in G} \sum_{h \in G^k} \prod_{\omega \in \set{0,1}^k} \mathcal{C}^{\abs{\omega}} f(x + \omega \cdot h).$$

## Remarks

The sum on the right is a nonnegative real number, so the root is defined.
For $k \ge 2$ the map $f \mapsto \norm{f}_{U^k}$ is a [[norm|0V2WP9B1MF]] on the space of functions $G \to \C$.
For $k = 2$, with the Fourier transform $\hat f(\chi) = \abs{G}^{-1} \sum_x f(x) \overline{\chi(x)}$, one has $\norm{f}_{U^2}^4 = \sum_\chi \abs{\hat f(\chi)}^4$.
For a character $\chi$ of $G$, each product $\chi(x) \overline{\chi(x + h_1)} \, \overline{\chi(x + h_2)} \, \chi(x + h_1 + h_2)$ equals $1$, so $\norm{\chi}_{U^2} = 1$.
For $k = 1$ the formula gives $\norm{f}_{U^1} = \abs{\abs{G}^{-1} \sum_x f(x)}$, which is only a seminorm.
A nontrivial character $\chi$ has $\norm{\chi}_{U^1} = 0$ but $\chi \ne 0$, so $U^1$ fails positive definiteness.
The norms increase with $k$: $\norm{f}_{U^k} \le \norm{f}_{U^{k+1}}$.
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# Decreasing rearrangement

## Definition

Let $(X, \Sigma, \mu)$ be a measure space and let $f$ be a real or complex measurable function on $X$.
Let $d_f(s) = \mu(\set{\abs{f} > s})$ be the [[distribution function|HQVXKEMN5N]] of $f$.
The *decreasing rearrangement* of $f$ is the function $f^* \colon [0, \infty) \to [0, \infty]$ given by the [[infimum|04KFTA34C5]]
$$f^*(t) = \inf \set{s \ge 0 : d_f(s) \le t},$$
with the convention $\inf \varnothing = \infty$.

## Remarks

Example: let $X = [0, 1]$ with Lebesgue measure and $f(x) = x$.
Then $d_f(s) = 1 - s$ for $0 \le s \le 1$ and $d_f(s) = 0$ for $s \ge 1$.
For $0 \le t < 1$ the condition $d_f(s) \le t$ means $s \ge 1 - t$, so $f^*(t) = 1 - t$.
For $t \ge 1$ every $s \ge 0$ works, so $f^*(t) = 0$.

Non-example: the function $f(x) = x$ on $[0, 1]$ is not its own decreasing rearrangement, since it is increasing.
A function $g \ge 0$ on $[0, \infty)$ is its own rearrangement exactly when it is non-increasing and right-continuous.

The function $f^*$ is non-increasing and right-continuous.
It has the same distribution function as $\abs{f}$ for Lebesgue measure on $[0, \infty)$.
Hence $\int_0^\infty (f^*)^p \, dt = \int_X \abs{f}^p \, d\mu$ for $0 < p < \infty$.
The *symmetric decreasing rearrangement* of a function on $\R^n$ is a related notion: it is radial, decreasing in $\abs{x}$, and equimeasurable with $\abs{f}$.
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# Involute

## Definition

Let $\gamma \colon I \to \R^n$ be a [[regular|06JMR2XWXW]] curve of class $C^2$, parametrised by [[arc length|00R1JDDV5X]] $s$ on an interval $I$, and let $c \in \R$.
The *involute* of $\gamma$ with parameter $c$ is the curve
$$J(s) = \gamma(s) + (c - s) \, \gamma'(s), \qquad s \in I.$$
Here $\gamma'(s)$ is the unit [[tangent vector|0D98771SM0]], so $J(s)$ lies on the tangent line at $\gamma(s)$, at distance $\abs{c - s}$ from $\gamma(s)$.

## Remarks

Example: for the unit circle $\gamma(s) = (\cos s, \sin s)$ and $c = 0$, one gets $J(s) = (\cos s + s \sin s, \sin s - s \cos s)$.
Check: $J'(s) = (s \cos s, s \sin s)$, which is orthogonal to $\gamma'(s) = (-\sin s, \cos s)$.
In general $J'(s) = (c - s) \gamma''(s)$, so $J$ meets each tangent line of $\gamma$ at a right angle, and $J$ is singular at $s = c$.
Non-example: the circle of radius $2$ about the origin is not an involute of the unit circle.
Indeed, it meets each tangent line of the unit circle at an angle that is not a right angle.
For a plane curve with nonzero curvature, the [[evolute|Y3ZNV24V2C]] of each involute is the original curve, on the part where $s \ne c$.
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<document source="https://tamnd.github.io/math2/node/J3NJQ5FZJ5.md" tag="J3NJQ5FZJ5" kind="concept">
# Trigonometric polynomial

## Summary

A trigonometric polynomial is a finite linear combination of the functions $e^{inx}$, $n \in \Z$, or equivalently of $1$, $\cos nx$ and $\sin nx$.
Trigonometric polynomials are the partial sums of [[Fourier series|G12KXDHPPS]].
They are dense in the continuous $2\pi$-periodic functions, so they play for periodic functions the role that ordinary polynomials play on a closed interval.

## Explanations

### L1 human

A trigonometric polynomial is a sum of finitely many pure waves, each with a whole-number frequency.
Its graph repeats itself after every interval of length $2\pi$.
An example is $T(x) = 1 + 2\cos x - 3 \sin 2x$, which has degree $2$, the largest frequency present.
Writing $z = e^{ix}$ turns it into a sum of powers $z^n$ with $n$ positive or negative, which explains the word polynomial.
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# Matrix rings over simple rings are simple

## Statement

Let $R$ be a ring with $1$ and let $n \ge 1$ be an integer.
Every [[two-sided ideal|0CD87SFM2N]] of the [[matrix ring|0FHVPRAE60]] $M_n(R)$ has the form $M_n(I)$ for a unique two-sided ideal $I$ of $R$.
In particular, if $R$ is a [[simple ring|62M0NC3E2T]], then $M_n(R)$ is a simple ring.

## Remarks

Let $J$ be a two-sided ideal of $M_n(R)$ and let $I$ be the set of all entries of all matrices in $J$.
With the matrix units $E_{kl}$, one has $E_{1i} A E_{j1} = a_{ij} E_{11}$, and multiplying by scalar matrices and matrix units moves an entry to any position.
From this, $I$ is a two-sided ideal of $R$ and $J = M_n(I)$.
For example, the ideals of $M_2(\Z)$ are the sets $M_2(m\Z)$ for $m \ge 0$.
If $R$ is simple, the only choices are $I = 0$ and $I = R$, so $M_n(R)$ has only the ideals $0$ and $M_n(R)$.

## Explanations

### L1 human

Square matrices whose entries come from a simple ring again form a simple ring.
So from one simple ring one gets an infinite family of simple rings, one for each size.
In particular the matrix rings over a field or a division ring are simple.

### L2 human

The bijection $I \mapsto M_n(I)$ between ideals is one instance of Morita equivalence: $R$ and $M_n(R)$ have equivalent module categories and the same lattice of two-sided ideals.
Combined with Wedderburn's theorem it shows that the simple Artinian rings are exactly the rings $M_n(D)$ with $D$ a division ring.
Note that $M_n(R)$ has many one-sided ideals even when $R$ is a field, such as the matrices whose only nonzero column is the first one.
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<document source="https://tamnd.github.io/math2/node/J45F4F13VB.md" tag="J45F4F13VB" kind="problem">
# Hilbert's seventeenth problem

## Problem

Let $n \ge 1$ and let $f \in \R[x_1, \dots, x_n]$ be a polynomial with real coefficients such that $f(a) \ge 0$ for every $a \in \R^n$.
Is $f$ a sum of squares of [[rational functions|0WCZH4AFMT]], that is, are there $g_1, \dots, g_m \in \R(x_1, \dots, x_n)$ with $f = g_1^2 + \cdots + g_m^2$?

## Remarks

Emil Artin answered the question positively in 1927.
His proof uses the theory of [[formally real fields|X32TSWJSYT]] and [[real closures|778KFSM09P]].
If $f$ were not a sum of squares, some ordering of $\R(x_1, \dots, x_n)$ would make $f$ negative.
Passing to a real closure then yields a real point where $f < 0$.
Rational functions are needed: a nonnegative polynomial need not be a sum of squares of polynomials.
An example is the Motzkin polynomial $x^4 y^2 + x^2 y^4 - 3x^2 y^2 + 1$, which is not a sum of squares of polynomials.
It is nonnegative by the inequality of arithmetic and geometric means applied to $x^4 y^2$, $x^2 y^4$ and $1$.
Hilbert showed that every nonnegative polynomial is a sum of squares of polynomials exactly in these cases: one variable, degree $2$, or two variables and degree $4$.
Pfister showed that $2^n$ squares of rational functions always suffice.

## Explanations

### L1 human

A sum of squares of real numbers is never negative.
Hilbert asked for the converse for polynomials: if a polynomial never takes a negative value, can one see this by writing it as a sum of squares?
With fractions allowed, the answer is yes, and a certificate of positivity always exists.
With polynomials only, the answer is no in general.
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<document source="https://tamnd.github.io/math2/node/J45KWFHENP.md" tag="J45KWFHENP" kind="concept">
# Positive line bundle

## Summary

A holomorphic line bundle is positive when it has a Hermitian metric whose [[Chern connection|7RC9CN2Q49]] has positive curvature.
Then $i$ times the curvature is a Kähler form, so a compact complex manifold with a positive line bundle is a [[Kähler manifold|X60JG9AEQJ]].
By the [[Kodaira embedding theorem|7G7CSM2NZ9]], on a compact complex manifold the positive line bundles are exactly the [[ample line bundles|RRK8P6TXG2]].
Positivity is the analytic hypothesis of the Kodaira and Kodaira–Nakano vanishing theorems.

## Explanations

### L1 human

The curvature of a line bundle with a metric is a $2$-form, and "positive" means that this form is like a Kähler form at every point.
The model is the line bundle $\mathcal{O}(1)$ on projective space, whose curvature is the Fubini–Study form.
Positive bundles have many holomorphic sections in their high powers, enough to embed the manifold in projective space.
The trivial bundle on a compact manifold is never positive: its curvature can be made zero, but never strictly positive everywhere.
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# Wythoff construction

## Summary

The Wythoff construction builds a polytope from a finite reflection group and a chosen point.
One takes a point in the fundamental region of the group and reflects it in the mirrors over and over.
The convex hull of all the images is the polytope.
Choosing which mirrors the point lies on gives the regular polytopes, their truncations and most other [[uniform polytopes|BZXHGNGQAX]] with the same symmetry.
It ties the geometry of polytopes to the combinatorics of [[Coxeter groups|G6FND7X69J]].

## Explanations

### L1 human

Think of a kaleidoscope: a few mirrors meeting at angles like $90^\circ$ or $60^\circ$.
Put a single dot between the mirrors and look at all its reflections; then join up the dots into a solid.
If the dot sits on some of the mirrors, its images merge, and one gets for instance a cube or an octahedron.
If it sits at a well-chosen place between the mirrors, one gets a truncated shape such as the regular octagon from the mirrors of a square.
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<document source="https://tamnd.github.io/math2/node/J4TDK6Z4XA.md" tag="J4TDK6Z4XA" kind="statement">
# Dispersive estimate for the Schrödinger equation

## Statement

Let $n \ge 1$.
For $f \in L^2(\R^n)$ and $t \in \R$, let $e^{it\Delta} f$ be the solution at time $t$ of the free [[Schrödinger equation|EPZR6H2WJN]] $i \partial_t u + \Delta u = 0$ with $u(0) = f$.
It is defined by the [[Fourier transform|DPVM34ADZV]] $\hat f(\xi) = \int_{\R^n} f(x) e^{-i x \cdot \xi} \, dx$ as $\widehat{e^{it\Delta} f}(\xi) = e^{-it\abs{\xi}^2} \hat f(\xi)$.
Then for all $f \in L^1(\R^n) \cap L^2(\R^n)$ and all $t \ne 0$,
$$\norm{e^{it\Delta} f}_{L^\infty(\R^n)} \le (4\pi \abs{t})^{-n/2} \norm{f}_{L^1(\R^n)}.$$

## Remarks

For $t \ne 0$ the solution operator has an explicit kernel:
$$e^{it\Delta} f(x) = (4\pi i t)^{-n/2} \int_{\R^n} e^{i \abs{x - y}^2 / (4t)} f(y) \, dy,$$
with the principal branch of the square root.
The kernel has modulus $(4\pi\abs{t})^{-n/2}$ at every point, and the estimate follows at once.
The formula for the kernel comes from the Fourier transform of the complex Gaussian $e^{-it\abs{\xi}^2}$, or from the heat kernel by replacing $t$ by $it$.

The solution also conserves the $L^2$ norm: $\norm{e^{it\Delta} f}_{L^2} = \norm{f}_{L^2}$, by Plancherel's theorem.
Interpolation between this and the dispersive estimate with the Riesz–Thorin theorem gives, for $2 \le r \le \infty$ and $1/r + 1/r' = 1$,
$$\norm{e^{it\Delta} f}_{L^r} \le (4\pi\abs{t})^{-n(1/2 - 1/r)} \norm{f}_{L^{r'}}.$$

The decay rate $\abs{t}^{-n/2}$ cannot be improved.
For the Gaussian $f(x) = e^{-\abs{x}^2}$ one computes $\abs{e^{it\Delta} f(0)} = (1 + 16 t^2)^{-n/4}$, which is of exact order $\abs{t}^{-n/2}$ for large $\abs{t}$.
The $L^1$ norm on the right cannot be replaced by the $L^2$ norm.
For fixed $t$, the operator $e^{it\Delta}$ maps $L^2(\R^n)$ onto $L^2(\R^n)$, which contains unbounded functions, so no bound of $\norm{e^{it\Delta} f}_{L^\infty}$ by a multiple of $\norm{f}_{L^2}$ holds.

## Explanations

### L1 human

A free quantum particle described by the Schrödinger equation spreads out over time.
Waves of different frequencies travel at different speeds, so an initially concentrated wave packet disperses.
The estimate says how fast: in $n$ dimensions, the peak height of the wave decays like $\abs{t}^{-n/2}$, as long as the initial wave has finite total mass in $L^1$.

### L2 human

The dispersive estimate is the pointwise decay input for the [[Strichartz estimates|4PDKJ8N1S9]].
Combined with the $TT^*$ argument and the Hardy–Littlewood–Sobolev inequality, it gives space-time integrability of $e^{it\Delta} f$ for $L^2$ data.
These bounds are the basic tool of the local and global well-posedness theory of nonlinear Schrödinger equations.
The same pattern applies to other [[dispersive equations|AYMJ2NFR4W]] such as the wave and Airy equations, with different decay rates.
One proves a pointwise decay bound from an oscillatory kernel and interpolates it with energy conservation.
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# Descartes' theorem

## Statement

Let $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$ be four [[circles|060W5CBZP6]] in the Euclidean plane such that each two of them are tangent and the six points of tangency are distinct.
Let $r_i$ be the radius of $\Gamma_i$.
Put $k_i = 1/r_i$ if no other circle lies inside $\Gamma_i$, and $k_i = -1/r_i$ if the other three circles lie inside $\Gamma_i$.
Then
$$(k_1 + k_2 + k_3 + k_4)^2 = 2\left(k_1^2 + k_2^2 + k_3^2 + k_4^2\right).$$

## Remarks

The numbers $k_i$ are the signed curvatures of the circles.
The statement extends to the case where one circle is a straight line, with curvature $0$.
Solving the quadratic equation for $k_4$ gives
$$k_4 = k_1 + k_2 + k_3 \pm 2\sqrt{k_1 k_2 + k_2 k_3 + k_3 k_1},$$
and the two signs correspond to the two circles that touch $\Gamma_1, \Gamma_2, \Gamma_3$.
Example: an outer circle of radius $1$ ($k_1 = -1$) holds two circles of radius $1/2$ ($k_2 = k_3 = 2$).
The formula gives $k_4 = 3$, and indeed $(-1 + 2 + 2 + 3)^2 = 36 = 2(1 + 4 + 4 + 9)$.
One proof applies an inversion that sends two of the circles to parallel lines and computes directly.

## Explanations

### L1 human

Four circles that all touch each other are tightly linked.
If you know the sizes of three of them, the theorem tells you the size of the fourth.
It is cleanest when written with curvatures, the reciprocals of the radii.

### L2 human

The theorem is the basic step in the construction of Apollonian circle packings.
Starting from four mutually tangent circles, the second root of the quadratic replaces one circle by a new one, and the process repeats.
If the first four curvatures are integers, all curvatures in the packing are integers, which makes these packings an object of number theory.
It is a special case of the [[problem of Apollonius|F9410EB0N1]] where the given circles touch each other.
</document>

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# Zonal spherical harmonics

## Definition

Let $d \ge 2$ and $\ell \ge 0$ be integers, and let $\sigma$ be the surface measure on the unit sphere $S^{d-1} \subset \R^d$.
Let $\mathcal{H}_\ell$ be the space of [[spherical harmonics|3F32Y21HP4]] of degree $\ell$ on $S^{d-1}$.
Equip $\mathcal{H}_\ell$ with the [[inner product|0NB0AWGJBW]] $\ip{f, g} = \int_{S^{d-1}} f \, \overline{g} \, d\sigma$.
Let $y \in S^{d-1}$.
The *zonal spherical harmonic* of degree $\ell$ with pole $y$ is the unique $Z_y^{(\ell)} \in \mathcal{H}_\ell$ such that
$$Y(y) = \int_{S^{d-1}} Y(x) \, \overline{Z_y^{(\ell)}(x)} \, d\sigma(x) \quad \text{for all } Y \in \mathcal{H}_\ell.$$

## Remarks

It exists and is unique because $Y \mapsto Y(y)$ is a linear functional on the finite-dimensional inner product space $\mathcal{H}_\ell$.
If $Y_1, \dots, Y_N$ is an orthonormal basis of $\mathcal{H}_\ell$, then $Z_y^{(\ell)}(x) = \sum_{j=1}^{N} Y_j(x) \overline{Y_j(y)}$.
It is real-valued, and $Z_{gy}^{(\ell)}(gx) = Z_y^{(\ell)}(x)$ for every rotation $g$, so it depends only on $x \cdot y$.
Example: for $d = 3$ and $\ell = 1$, $Z_y^{(1)}(x) = \frac{3}{4\pi} \, x \cdot y$.
Check: every $Y \in \mathcal{H}_1$ has the form $Y(x) = x \cdot a$ with $a \in \C^3$, and $\int_{S^2} (x \cdot a)(x \cdot y) \, d\sigma(x) = \frac{4\pi}{3} \, a \cdot y$, so the integral returns $a \cdot y = Y(y)$.
Non-example: the function $x \mapsto x \cdot y$ itself is zonal and lies in $\mathcal{H}_1$, but it fails the reproducing identity, since it returns $\frac{4\pi}{3} Y(y)$ instead of $Y(y)$.
In general $Z_y^{(\ell)}(x) = \frac{N}{\sigma(S^{d-1})} \frac{C_\ell^{(\lambda)}(x \cdot y)}{C_\ell^{(\lambda)}(1)}$ for $d \ge 3$, with $\lambda = \frac{d - 2}{2}$, $N = \dim \mathcal{H}_\ell$ and $C_\ell^{(\lambda)}$ the [[Gegenbauer polynomial|5TRFA6F79B]].
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# Quasi-compactness of the spectrum

## Statement

Let $R$ be a [[commutative ring|0310REJJY8]].
Then the [[spectrum|FB45BHHA2J]] $\operatorname{Spec} R$ with the [[Zariski topology|8SQK7RTB63]] is quasi-compact: every cover of $\operatorname{Spec} R$ by open sets has a finite subcover.

## Remarks

It is enough to treat covers by [[basic open sets|N6JX9BCKQP]] $D(f_i) = \set{\mathfrak p : f_i \notin \mathfrak p}$, $i \in I$.
These cover $\operatorname{Spec} R$ exactly when no prime ideal contains all $f_i$.
Since every proper [[ideal|096Z9MP2ME]] lies in a [[maximal ideal|0PV74TR5FP]], this means that the $f_i$ generate the unit ideal.
Then $1 = a_1 f_{i_1} + \cdots + a_n f_{i_n}$ for finitely many indices, and $D(f_{i_1}), \ldots, D(f_{i_n})$ already cover.

Example: $D(2)$ and $D(3)$ cover $\operatorname{Spec} \Z$, because $1 = 3 - 2$.

The word "quasi-compact" is used because the space is usually not [[Hausdorff|0S1FJ8VEX9]], and some authors reserve "[[compact|0TVJZYB6SH]]" for Hausdorff spaces.
The same argument shows that each basic open set $D(f)$ is quasi-compact, since $D(f) \cong \operatorname{Spec} R_f$.

## Explanations

### L1 human

However you cover the spectrum of a ring with open sets, a finite number of them already do the job.
The reason is algebraic: covering the space means producing $1$ from the given elements, and any such expression uses only finitely many of them.

### L2 human

Quasi-compactness of $\operatorname{Spec} R$ is the algebraic shadow of the fact that $1$ is a finite sum.
It is used to pass from local to global data in finitely many steps.
For example, it lets one glue finitely many local constructions, or bound a quantity over a finite cover.
In scheme theory, it says that affine schemes are quasi-compact.
It is one of the conditions in [[Hochster's theorem on spectral spaces|C82AXZK2BS]].
</document>

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# Strongly regular graph

## Definition

Let $v, k, \lambda, \mu$ be non-negative integers.
A finite [[graph|0JD5C82QMB]] $G$ is strongly regular with parameters $(v, k, \lambda, \mu)$, written $\mathrm{srg}(v, k, \lambda, \mu)$, if:

1. $G$ has $v$ vertices and is $k$-[[regular|0KA4S8R39M]].
2. Every two adjacent vertices have exactly $\lambda$ common [[neighbours|01T3TMGTTE]].
3. Every two distinct non-adjacent vertices have exactly $\mu$ common neighbours.

## Remarks

Many authors also require that $G$ is neither complete nor edgeless, so that both conditions 2 and 3 apply to some pair.
Counting the edges between the neighbours of a vertex $x$ and the vertices not adjacent to $x$ gives $k(k - \lambda - 1) = (v - k - 1)\mu$.
Example: the [[Petersen graph|0NR95CTFCZ]] is an $\mathrm{srg}(10, 3, 0, 1)$: it is $3$-regular, it has no triangles, so $\lambda = 0$, and two non-adjacent vertices have exactly one common neighbour.
Check of the identity: $3 \cdot (3 - 0 - 1) = 6 = (10 - 3 - 1) \cdot 1$.
Non-example: the [[cycle graph|0QQ0F12962]] $C_6$ is $2$-regular, but two vertices at distance $2$ have one common neighbour while two opposite vertices have none, so clause 3 fails.
The [[complement|0RNMP6VCR2]] of an $\mathrm{srg}(v, k, \lambda, \mu)$ is an $\mathrm{srg}(v, v - k - 1, v - 2 - 2k + \mu, v - 2k + \lambda)$.
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# Classification of second-order PDEs

## Definition

Let $\Omega$ be an open set in $\R^n$ and consider a [[linear PDE|6DZETZ5T9G]] of order $2$ in $\Omega$.
$$\sum_{i,j=1}^{n} a_{ij}(x) \, \partial_i \partial_j u + \sum_{i=1}^{n} b_i(x) \, \partial_i u + c(x) \, u = f(x).$$
Its coefficients are real and $a_{ij} = a_{ji}$, so that $A(x) = (a_{ij}(x))$ is a real [[symmetric matrix|0VR671PHE5]].
At a point $x \in \Omega$ the equation is called:

1. *elliptic* if all [[eigenvalues|0BQAA2BK3Q]] of $A(x)$ are nonzero and have the same sign;
2. *hyperbolic* if all eigenvalues of $A(x)$ are nonzero, and exactly one of them has a sign opposite to the others;
3. *ultrahyperbolic* if all eigenvalues are nonzero, and at least two are positive and at least two are negative;
4. *parabolic* if exactly one eigenvalue of $A(x)$ is zero and the others are nonzero and have the same sign.

The equation has one of these types in $\Omega$ if it has that type at every point of $\Omega$.

## Remarks

The quadratic form $\xi \mapsto \sum a_{ij}(x) \xi_i \xi_j$ is the [[principal symbol|F31T650DKY]] at $x$, so the type depends only on the second-order part.
In two variables, write the second-order part as $A u_{xx} + 2B u_{xy} + C u_{yy}$.
The equation is elliptic where $AC - B^2 > 0$, hyperbolic where $AC - B^2 < 0$, and parabolic where $AC - B^2 = 0$ and $(A, B, C) \ne (0, 0, 0)$.
Example: the [[Tricomi equation|21DD2334NR]] $u_{yy} + y \, u_{xx} = 0$ has $AC - B^2 = y$, so it is elliptic for $y > 0$, hyperbolic for $y < 0$ and parabolic on the line $y = 0$.
Non-example: for $n = 3$, a matrix $A(x)$ with eigenvalues $1, 0, 0$ fits none of the four clauses; such points are degenerate.
The type is invariant under a smooth change of variables, by [[Sylvester's law of inertia|0WFED3K3DQ]] applied to the transformed coefficient matrix.

## Explanations

### L1 human

The type of a second-order equation is read off from the signs of the eigenvalues of its matrix of second-order coefficients.
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<document source="https://tamnd.github.io/math2/node/J56JP23J57.md" tag="J56JP23J57" kind="definition">
# Kantorovich polynomials

## Definition

Let $f \colon [0, 1] \to \R$ be Lebesgue integrable and let $n \ge 0$ be an integer.
The *Kantorovich polynomial* of degree $n$ of $f$ is
$$K_n(f)(x) = (n + 1) \sum_{k=0}^{n} \binom{n}{k} x^k (1 - x)^{n-k} \int_{k/(n+1)}^{(k+1)/(n+1)} f(t) \, dt, \qquad x \in [0, 1].$$

## Remarks

Let $F(x) = \int_0^x f(t) \, dt$ and let $B_{n+1}$ be the [[Bernstein polynomial|0JT92VK307]] operator of degree $n + 1$.
Then $K_n(f) = (B_{n+1} F)'$, by the formula for the derivative of a Bernstein polynomial.
Each $K_n$ is a positive linear operator.
For $f \in L^p[0, 1]$ with $1 \le p < \infty$, $K_n(f) \to f$ in the [[$L^p$ norm|SASP14MA1Z]], and for continuous $f$ the convergence is uniform.

Example: for $f(t) = t$, the integral over $[k/(n+1), (k+1)/(n+1)]$ is $\frac{2k + 1}{2(n+1)^2}$.
Hence $K_n(f)(x) = \sum_{k} \binom{n}{k} x^k (1 - x)^{n-k} \frac{2k + 1}{2(n + 1)} = \frac{nx + 1/2}{n + 1}$, which tends to $x$.

Non-example: the Bernstein polynomial $B_n f(x) = \sum_k \binom{n}{k} x^k (1 - x)^{n-k} f(k/n)$ is not a Kantorovich polynomial.
It uses the point values $f(k/n)$, so it changes when $f$ is changed on the null set $\set{0, 1/n, \dots, 1}$, while $K_n(f)$ does not.
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<document source="https://tamnd.github.io/math2/node/J58ED1GV6N.md" tag="J58ED1GV6N" kind="example">
# SL(2,C)

## Example

The group $SL(2, \C)$ is the [[special linear group|0V9PMN0RBH]] of complex $2 \times 2$ matrices with determinant $1$:
$$SL(2, \C) = \set{\begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \C, \ ad - bc = 1}.$$
It is cut out of $M_2(\C) = \C^4$ by the single polynomial equation $ad - bc = 1$, whose differential is nonzero at every solution.
So it is a complex [[Lie group|7JBRGCVJRB]] of complex dimension $3$, that is, of real dimension $6$.

It acts on the [[Riemann sphere|0J78KBPFZB]] $\C \cup \set{\infty}$ by [[Möbius transformations|0SG74WBFA2]] $z \mapsto (az + b)/(cz + d)$.
Check: the matrix $g$ acts as the identity map exactly when $(az + b)/(cz + d) = z$ for all $z$, which forces $b = c = 0$ and $a = d$, and then $ad = 1$ gives $g = \pm I$.
So the group of Möbius transformations is $PSL(2, \C) = SL(2, \C)/\set{\pm I}$.

## Remarks

Every $g \in SL(2, \C)$ has a unique polar decomposition $g = u p$ with $u \in$ [[SU(2)|E5P8AG6CK7]] and $p$ a positive definite [[Hermitian matrix|0P17NGQQMQ]] of determinant $1$.
These $p$ form a space homeomorphic to $\R^3$, so $SL(2, \C)$ is homeomorphic to $S^3 \times \R^3$.
Hence it is connected and simply connected, and $SU(2)$ is a maximal compact subgroup.
Its Lie algebra $\mathfrak{sl}(2, \C)$ is the complex simple Lie algebra of type $A_1$.
$SL(2, \C)$ acts on the Hermitian $2 \times 2$ matrices by $X \mapsto g X g^*$, preserving $\det X$, which is a quadratic form of signature $(1, 3)$.
This gives a double cover of the identity component of the Lorentz group, so $SL(2, \C)$ is the spin group $\mathrm{Spin}(1, 3)$.

## Explanations

### L1 human

$SL(2, \C)$ is the complex version of the group of $2 \times 2$ matrices with determinant $1$.
Each of its elements moves the points of the extended complex plane by a fractional linear map, and two matrices $g$ and $-g$ give the same map.
These maps send circles to circles, and they are exactly the orientation-preserving conformal symmetries of the sphere.
In physics the same group describes how spinors change under the rotations and boosts of special relativity.
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<document source="https://tamnd.github.io/math2/node/J58ERN8KK8.md" tag="J58ERN8KK8" kind="definition">
# Simple module

## Definition

Let $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].
The module $M$ is *simple* if:

1. $M \ne 0$;
2. the only [[submodules|0RGZ10DFPG]] of $M$ are $0$ and $M$.

## Remarks

Equivalently, $M \ne 0$ and $M = Rm$ for every nonzero $m \in M$.
Example: for a prime $p$, the $\Z$-module $\Z/p\Z$ is simple.
The check: its submodules are subgroups, whose orders divide $p$, so they are $0$ or $\Z/p\Z$.
Example: for a field $K$, the column space $K^n$ is a simple $M_n(K)$-module, since any nonzero vector is sent to any other vector by some matrix.
Non-example: the $\Z$-module $\Z$ is not simple, since $2\Z$ is a submodule different from $0$ and $\Z$; clause 2 fails.
The zero module is not simple, by clause 1.
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<document source="https://tamnd.github.io/math2/node/J58YA8QNNX.md" tag="J58YA8QNNX" kind="statement">
# Jump relations for layer potentials

## Statement

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a bounded open set whose [[boundary|05D5EGMFYV]] $\partial\Omega$ is a $C^2$ hypersurface.
Let $\sigma$ be surface measure on $\partial\Omega$ and $\nu$ the outward unit normal.
Let $\Phi$ be the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] with $-\Delta\Phi = \delta_0$, and let $\varphi \colon \partial\Omega \to \R$ be continuous.
Let $S\varphi$ be the [[single layer potential|9T6ZEV4125]] and $D\varphi$ the [[double layer potential|EG8Z3FWV9A]] with density $\varphi$.
For $x \in \partial\Omega$ put
$$K\varphi(x) = \int_{\partial\Omega} \frac{\partial \Phi(x - y)}{\partial \nu(y)}\, \varphi(y)\, d\sigma(y), \qquad K'\varphi(x) = \int_{\partial\Omega} \frac{\partial \Phi(x - y)}{\partial \nu(x)}\, \varphi(y)\, d\sigma(y);$$
both integrals converge absolutely.
Then:

1. $S\varphi$ is continuous on $\R^n$.
2. The restriction of $D\varphi$ to $\Omega$ extends continuously to $\overline{\Omega}$, with boundary values $D_-\varphi = K\varphi - \tfrac12 \varphi$ on $\partial\Omega$.
The restriction of $D\varphi$ to $\R^n \setminus \overline{\Omega}$ extends continuously to $\R^n \setminus \Omega$, with boundary values $D_+\varphi = K\varphi + \tfrac12 \varphi$ on $\partial\Omega$.
3. For $x \in \partial\Omega$ the limits
$$\partial_\nu S_\pm\varphi(x) = \lim_{h \to 0^+} \nu(x) \cdot \nabla S\varphi\big(x \pm h\,\nu(x)\big)$$
exist uniformly on $\partial\Omega$, and $\partial_\nu S_\pm \varphi = K'\varphi \mp \tfrac12 \varphi$.

## Remarks

Proof idea: for $\varphi \equiv 1$, the divergence theorem gives Gauss's integral, $D1 = -1$ in $\Omega$, $D1 = 0$ outside $\overline{\Omega}$ and $K1 = -\tfrac12$ on $\partial\Omega$.
For general $\varphi$, write $D\varphi(x) = D\big(\varphi - \varphi(x_0)\big)(x) + \varphi(x_0)\, D1(x)$ for $x$ near $x_0 \in \partial\Omega$.
The first term is continuous across $\partial\Omega$ at $x_0$, because the $C^2$ boundary makes the kernel weakly singular and the density vanishes at $x_0$.
So the whole jump comes from $\varphi(x_0)\, D1$, and the normal derivative of $S\varphi$ is handled in the same way.

The jumps can be read off: $D_+\varphi - D_-\varphi = \varphi$ and $\partial_\nu S_-\varphi - \partial_\nu S_+\varphi = \varphi$.
Example: for the unit ball in $\R^3$ and $\varphi \equiv 1$, $S1 = 1$ inside and $S1 = 1/\abs{x}$ outside.
So $\partial_\nu S_-1 = 0$ and $\partial_\nu S_+1 = -1$ on the sphere, and both equal $K'1 \mp \tfrac12$ with $K'1 = -\tfrac12$.

## Explanations

### L1 human

A layer of charge on a surface gives a potential that is continuous across the surface, but whose slope jumps by the charge density.
A layer of dipoles gives a potential that itself jumps across the surface, by the dipole density.
The jump is split evenly: half of it on each side of the value on the surface.

### L2 human

These formulas are the bridge between the Laplace equation in a domain and integral equations on its boundary.
If one looks for the solution of the [[Dirichlet problem|8D56GQ4AG8]] as $D\varphi$, part 2 turns the boundary condition into the equation $K\varphi - \tfrac12\varphi = f$; this is the boundary integral method.
For the Neumann problem one uses $S\varphi$ and part 3 in the same way.
Because $K$ and $K'$ are compact operators on $C(\partial\Omega)$ and adjoint to each other for the pairing of $L^2(\partial\Omega)$, the [[Fredholm alternative|4HH8C9V8JQ]] then governs solvability.
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# Barycentric subdivision

## Summary

The barycentric subdivision of a [[simplicial complex|V8GZ0W1ZZ5]] cuts every simplex into smaller simplices whose vertices are barycentres of faces.
It does not change the underlying space, but it makes the simplices smaller: repeated subdivision makes their diameters tend to $0$.
This is the key step in the simplicial approximation theorem and in the proof of excision for singular homology.

## Explanations

### L1 human

Take a triangle and mark the midpoint of each edge and the centre of the triangle.
Join the centre to the three corners and the three midpoints.
The triangle is now cut into six smaller triangles, and each small triangle has one corner, one midpoint and the centre as its vertices.
An interval is cut at its midpoint into two halves in the same way.
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<document source="https://tamnd.github.io/math2/node/J5HQWT3ZDA.md" tag="J5HQWT3ZDA" kind="definition">
# Lattice ideal

## Definition

Let $(L, \vee, \wedge)$ be a [[lattice|WH4SFAACAS]] with its order $\le$.
An *ideal* of $L$ is a subset $I \subseteq L$ such that:

1. $I$ is nonempty;
2. if $x \in I$, $y \in L$ and $y \le x$, then $y \in I$;
3. if $x, y \in I$, then $x \vee y \in I$.

The ideal $I$ is *proper* if $I \ne L$.

## Remarks

For $a \in L$, the set $\downarrow a = \set{x \in L : x \le a}$ is an ideal, the *principal ideal* generated by $a$.
In a finite lattice every ideal is principal, generated by the join of all its elements.
Example: in the power set of $\N$ ordered by $\subseteq$, the finite subsets form an ideal.
It is nonempty, every subset of a finite set is finite, and the union of two finite sets is finite.
It is not principal, since no finite set contains all finite sets.
Non-example: in the power set of $\set{1, 2}$, the set $\set{\emptyset, \set{1}, \set{2}}$ satisfies clauses 1 and 2 but not clause 3, since $\set{1} \cup \set{2} = \set{1, 2}$ is missing.
The same notion makes sense in any [[partially ordered set|0QV2JS9RJ3]], with clause 3 replaced by: any two elements of $I$ have an upper bound in $I$.
In a lattice the two versions agree.
</document>

<document source="https://tamnd.github.io/math2/node/J5QPP8REV6.md" tag="J5QPP8REV6" kind="definition">
# Coequalizer

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $f, g \colon X \to Y$ be two [[morphisms|859F0TFVA3]] of $\mathcal{C}$.
A *coequalizer* of $f$ and $g$ is an object $Q$ with a morphism $q \colon Y \to Q$ such that:

1. $q \circ f = q \circ g$;
2. for every object $Z$ and every morphism $k \colon Y \to Z$ with $k \circ f = k \circ g$, there is exactly one morphism $u \colon Q \to Z$ with $u \circ q = k$.

## Remarks

A coequalizer is the [[colimit|BJS1CGTKQ5]] of the diagram formed by $f$ and $g$, and it is an [[equalizer|CTJ5AD8ST5]] in the opposite category.
It is an epimorphism, by the dual of the argument for equalizers.

Example: in the category of sets, let $\sim$ be the smallest [[equivalence relation|0G8VEZ3WTK]] on $Y$ with $f(x) \sim g(x)$ for all $x \in X$, let $Q = Y/{\sim}$ and let $q$ be the quotient map.
Clause 1 holds by construction.
If $k \circ f = k \circ g$, the relation "$k(y) = k(y')$" is an equivalence relation containing all pairs $(f(x), g(x))$, so it contains $\sim$, and $u([y]) = k(y)$ is well defined; it is unique because $q$ is surjective.
In the category of abelian groups the coequalizer of $f$ and $g$ is $Y / \im(f - g)$ with the quotient map.

Non-example: for $X = \{*\}$, $Y = \{0, 1, 2\}$, $f(*) = 0$ and $g(*) = 1$, the constant map $q \colon Y \to \{*\}$ satisfies clause 1 but not clause 2.
The map $k \colon Y \to \{0, 2\}$ with $k(0) = k(1) = 0$ and $k(2) = 2$ has $k \circ f = k \circ g$ but does not factor through a one-point set.
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# Pettis lemma

## Statement

Let $G$ be a [[topological group|92700QAB5Y]] with identity $e$.
Let $A \subseteq G$ be a set that has the [[Baire property|FZ7237VY65]] and is not [[meagre|0XGCZ4GDZ6]].
Then the set $A^{-1} A = \set{a^{-1} b : a, b \in A}$ is a [[neighbourhood|0ZCY0Z80B6]] of $e$.

## Remarks

Since $A$ has the Baire property and is not meagre, there is a nonempty open set $U$ such that $A$ differs from $U$ by a meagre set.
Since $G$ is not meagre, the Banach category theorem and homogeneity show that no nonempty open subset of $G$ is meagre.
For $g$ near $e$, the open set $U \cap gU$ is nonempty, so it is not meagre.
The set $A \cap gA$ differs from it by a meagre set, so $A \cap gA$ is not empty and $g \in A A^{-1}$.
Applying this to $A^{-1}$, which also has the Baire property and is not meagre, gives the statement for $A^{-1}A$.
It is the category analogue of the Steinhaus theorem on sets of positive measure.
A standard application: a homomorphism between Polish groups that is Baire measurable is continuous.
The hypothesis that $A$ is not meagre is needed.
In $\R$ the set $\Z$ is closed, so it has the Baire property, but it is meagre, and $\Z - \Z = \Z$ is not a neighbourhood of $0$.

## Explanations

### L1 human

A set in a group can be large in a topological sense, called not meagre, even when it has no interior.
The lemma says that such a set, if it is reasonably well behaved, produces a whole neighbourhood of the identity when you take all quotients $a^{-1} b$ of its elements.
So largeness of $A$ turns into an open set inside $A^{-1} A$.

### L2 human

The lemma is the main tool of automatic continuity results in descriptive set theory.
To show a homomorphism $f$ is continuous at $e$, take a neighbourhood $V$ of the identity in the target and a smaller symmetric $W$ with $W^{-1} W \subseteq V$.
If $f^{-1}(W)$ has the Baire property and is not meagre, the lemma shows that $f^{-1}(V)$ contains a neighbourhood of $e$.
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# Properly discontinuous action

## Summary

An action of a group by homeomorphisms is properly discontinuous when each compact set meets only finitely many of its translates.
On a locally compact Hausdorff space, it is the same as a proper action of the group with the discrete topology.
A free properly discontinuous action of a group on a manifold gives a quotient manifold, and the quotient map is a [[covering map|3XZRRE165N]].
Examples include lattices of translations acting on $\R^n$, Fuchsian groups acting on the hyperbolic plane, and deck groups acting on universal covers.

## Explanations

### L1 human

A group acts properly discontinuously if, from the point of view of any bounded region, only finitely many group elements move it to a nearby place.
Integer translations of the plane tile it by unit squares, and a bounded piece of the plane meets only finitely many of its shifted copies.
Gluing points that differ by a group element then gives a nice space, here a torus.
An irrational rotation of the circle is the opposite case: every arc meets infinitely many of its rotated copies, and the quotient is a bad space.
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<document source="https://tamnd.github.io/math2/node/J5XQRP1SNQ.md" tag="J5XQRP1SNQ" kind="concept">
# Automatic group

## Summary

An automatic group is a finitely generated group whose elements can be named by words of a regular language.
The naming must allow finite automata to check when two named elements differ by a generator.
The idea comes from the work of Cannon, Epstein, Thurston and others on computing with groups.
Automatic groups are finitely presented, their word problem can be solved in quadratic time, and their [[Dehn function|Q4HGMRA9JQ]] is at most quadratic.
[[Hyperbolic groups|JWJ8YPR7EB]], $\Z^n$, braid groups and many [[Coxeter groups|G6FND7X69J]] are automatic.

## Explanations

### L1 human

A finite automaton is a machine with finitely many states that reads a word one letter at a time.
In an automatic group, such a machine can recognise a chosen list of normal-form words, one or more for each element.
Further machines, reading two words side by side, can tell whether the elements they name differ by one generator.
In $\Z^2$ with generators $a$ and $b$, the words $a^m b^n$ with $m, n \in \Z$ form such a list.
This makes many computations in the group fast and mechanical.
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# Resolvent

## Summary

The resolvent of an operator $T$ is the operator-valued function $\lambda \mapsto (T - \lambda I)^{-1}$, defined on the resolvent set, the complement of the spectrum.
It is an analytic function of $\lambda$, and its singularities are exactly the points of the spectrum.
Through the resolvent, complex analysis becomes a tool of operator theory: contour integrals of the resolvent give projections and functions of $T$.

## Explanations

### L1 human

For a number $t$, the function $\lambda \mapsto 1/(t - \lambda)$ blows up only at $\lambda = t$.
The resolvent does the same for an operator: it inverts $T - \lambda I$ wherever this is possible.
Where the inverse stops existing, the resolvent has a singularity, and these points form the spectrum.
For a diagonal matrix with entries $1$ and $2$, the resolvent is the diagonal matrix with entries $1/(1 - \lambda)$ and $1/(2 - \lambda)$.
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<document source="https://tamnd.github.io/math2/node/J5YNTHQMFX.md" tag="J5YNTHQMFX" kind="statement">
# Hilbert projection theorem

## Statement

Let $H$ be a real or complex [[Hilbert space|HM2HHE3J51]].
Let $C$ be a nonempty [[closed|0ZCK8DQYN1]] [[convex|02AS45T3H9]] subset of $H$, and let $x \in H$.
Then there is exactly one $y \in C$ such that
$$\norm{x - y} = \inf_{z \in C} \norm{x - z}.$$
This point $y$ is the only point of $C$ with $\operatorname{Re} \ip{x - y}{z - y} \le 0$ for all $z \in C$.

## Remarks

The idea of the proof: let $d$ be the infimum and take $z_n \in C$ with $\norm{x - z_n} \to d$.
The [[parallelogram law|026HPGYGS4]] gives $\norm{z_n - z_m}^2 = 2 \norm{x - z_n}^2 + 2 \norm{x - z_m}^2 - 4 \norm{x - (z_n + z_m)/2}^2$.
By convexity the midpoint lies in $C$, so the last norm is at least $d$, and $(z_n)$ is a Cauchy sequence.
Completeness and closedness give a limit $y \in C$, and the same identity gives uniqueness.

When $C$ is a closed linear subspace $M$, the condition becomes $x - y \perp M$.
Then $y$ is the [[orthogonal projection|0B8KZRT28M]] of $x$ on $M$, and $H = M \oplus M^\perp$ with the [[orthogonal complement|0XJXTBCB2T]] $M^\perp$.
For finite-dimensional $M$ this is the [[closest point property|0MCDDSCB9Z]].

The hypotheses matter.
If $C$ is not closed, such as an open ball not containing $x$, there may be no nearest point.
If $C$ is not convex, such as a circle in $\R^2$ and $x$ its center, the nearest point need not be unique.
In the Banach space $\R^2$ with the max norm, the nearest point of the line $\set{(t, 1)}$ to $0$ is not unique.

## Explanations

### L1 human

From any point there is exactly one nearest point in a closed convex set.
The segment from the point to its nearest point meets the set at a right angle or wider.
This is what makes orthogonal projection work in infinite dimensions.

### L2 human

The theorem is the base of Hilbert space geometry.
It gives the orthogonal decomposition $H = M \oplus M^\perp$, the existence of orthogonal projections and the Riesz representation theorem.
In optimisation and approximation it shows that least squares problems over closed convex sets have unique solutions.
</document>

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# Discrete harmonic function

## Definition

Let $G = (V, E)$ be a [[graph|00C0BR7D7H]] in which every vertex has finitely many neighbours.
Let $c \colon V \times V \to [0, \infty)$ be a symmetric weight with $c(x, y) > 0$ if and only if $\set{x, y} \in E$, and let $c(x) = \sum_{y} c(x, y)$.
A function $f \colon V \to \R$ is *harmonic at* a vertex $x$ with $c(x) > 0$ if
$$f(x) = \frac{1}{c(x)} \sum_{y \in V} c(x, y) f(y),$$
or equivalently if $\sum_{y \in V} c(x, y) \big(f(y) - f(x)\big) = 0$.
For $A \subseteq V$, the function $f$ is *harmonic on* $A$ if it is harmonic at every vertex of $A$.
With unit weights on all edges, harmonicity at $x$ says that $f(x)$ is the average of $f$ over the neighbours of $x$.

## Remarks

Example: on the graph $\Z$ with edges $\set{n, n + 1}$ and unit weights, $f(n) = 3n + 2$ is harmonic on $\Z$, since $\tfrac12\big(f(n - 1) + f(n + 1)\big) = \tfrac12(6n + 4) = f(n)$.
Non-example: $g(n) = n^2$ is harmonic at no vertex, since $\tfrac12\big((n - 1)^2 + (n + 1)^2\big) = n^2 + 1$.
The expression $\sum_y c(x, y)(f(y) - f(x))$ is minus the weighted [[graph Laplacian|TN6M6DVZRP]] applied to $f$ at $x$, so the harmonic functions on $V$ form the kernel of the Laplacian.
For the random walk with transition probabilities $p(x, y) = c(x, y)/c(x)$, the function $f$ is harmonic at $x$ exactly when $f(x)$ is the expected value of $f$ after one step from $x$.
This is the discrete form of the mean value property of a [[harmonic function|0DEB0JMN0W]].
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# Ultrafilter

## Summary

An ultrafilter on a set $X$ is a [[filter|15PF73BKY6]] on $X$ that is maximal: no strictly larger filter on $X$ exists.
Equivalently, for every subset $A$ of $X$, either $A$ or its complement belongs to it.
Every filter lies in an ultrafilter, by [[Zorn's lemma|04RXTQA5ZA]].
Ultrafilters describe [[compactness|0TVJZYB6SH]], build the Stone–Čech compactification, and give ultraproducts in logic.

## Explanations

### L1 human

An ultrafilter makes a firm decision about every subset: either the set is "big" or its complement is.
The simplest ultrafilters pick a point $a$ and call a set big when it contains $a$.
On an infinite set there are also ultrafilters that are not of this kind.
One can show that they exist, using the axiom of choice, but one cannot write one down explicitly.
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# Approximately additive function

## Summary

An approximately additive function is a map $f$ from an abelian group to a normed space.
The norm of its defect $f(x + y) - f(x) - f(y)$ is at most a fixed $\delta$.
It is an [[additive function|KF719SZD5T]] up to a uniform error.
[[Hyers' theorem|VFRJ0P2P15]] says that an approximately additive map from a normed space to a Banach space lies within $\delta$ of a unique additive map.
Such maps are also called $\delta$-additive maps.

## Explanations

### L1 human

An additive function turns sums into sums exactly.
An approximately additive function does so with an error that never exceeds a fixed amount.
The floor function is an example: $\floor{x + y} - \floor{x} - \floor{y}$ is always $0$ or $1$.
It stays within distance $1$ of the additive function $x \mapsto x$.
</document>

<document source="https://tamnd.github.io/math2/node/J6H7YPRZG7.md" tag="J6H7YPRZG7" kind="definition">
# Sine and cosine series

## Definition

Let $f \colon [0, \pi] \to \R$ be Lebesgue [[integrable|X7GTF3D98F]].
For $n \ge 0$ put
$$a_n = \frac{2}{\pi} \int_0^\pi f(x) \cos nx \, dx, \qquad b_n = \frac{2}{\pi} \int_0^\pi f(x) \sin nx \, dx.$$
The *Fourier cosine series* of $f$ is the [[trigonometric series|FZK2M8KGXQ]] $\frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos nx$.
The *Fourier sine series* of $f$ is the trigonometric series $\sum_{n=1}^{\infty} b_n \sin nx$.

## Remarks

The cosine series is the [[Fourier series|G12KXDHPPS]] of the even $2\pi$-periodic function equal to $f$ on $[0, \pi]$.
The sine series is the Fourier series of the odd $2\pi$-periodic function equal to $f$ on $(0, \pi)$.
Example: for $f(x) = x$ on $[0, \pi]$, integration by parts gives $b_n = \frac{2(-1)^{n+1}}{n}$, so the sine series is $2 \sum_{n \ge 1} \frac{(-1)^{n+1}}{n} \sin nx$.
For the same $f$, $a_0 = \pi$, $a_n = 0$ for even $n \ge 2$ and $a_n = -\frac{4}{\pi n^2}$ for odd $n$.
Non-example: $f(x) = 1/x$ on $(0, \pi]$ is not integrable on $[0, \pi]$, so its sine and cosine coefficients are not defined.
</document>

<document source="https://tamnd.github.io/math2/node/J6JP5CJNPX.md" tag="J6JP5CJNPX" kind="concept">
# Fisher's exact test

## Summary

Fisher's exact test is a test of independence of two binary factors in a $2 \times 2$ [[contingency table|PANSMHE0SW]].
It conditions on the row and column totals, and then the count in one cell has a [[hypergeometric distribution|0X3Y0ZSKNF]] under the null hypothesis.
The [[p-value|0GGKJGZF6D]] is computed exactly from this law, so the test is valid for small samples where the chi-squared approximation fails.

## Explanations

### L1 human

Ronald Fisher's lady tasting tea example illustrates the test.
A lady claims she can tell whether milk or tea was poured first, and she gets $8$ cups, $4$ of each kind.
If she is guessing, the chance that she picks the right $4$ cups is $1/\binom{8}{4} = 1/70$, about $0.014$.
So a fully correct answer is strong evidence against pure guessing.
</document>

<document source="https://tamnd.github.io/math2/node/J6MTQSZVEM.md" tag="J6MTQSZVEM" kind="definition">
# Outer regular measure

## Definition

Let $X$ be a topological space and let $\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.
A Borel set $A \subseteq X$ is *outer regular* for $\mu$ if
$$\mu(A) = \inf \set{\mu(U) : U \supseteq A, \ U \text{ open}}.$$
The measure $\mu$ is *outer regular* if every Borel set is outer regular for $\mu$.

## Remarks

Every [[open set|02BH3MCBQB]] is outer regular, by taking $U = A$.

Example: Lebesgue measure $\lambda$ on the Borel sets of $\R$ is outer regular.
For a Borel set $A$ and $\varepsilon > 0$, the outer measure construction gives intervals $I_k$ covering $A$ with $\sum_k \lambda(I_k) < \lambda(A) + \varepsilon$.
Enlarging each $I_k$ slightly to an open interval gives an open $U \supseteq A$ with $\lambda(U) < \lambda(A) + 2\varepsilon$.

Non-example: counting measure $\mu$ on $\R$ is not outer regular.
The set $\{0\}$ has $\mu(\{0\}) = 1$, but every open set containing $0$ is infinite and has measure $\infty$.
</document>

<document source="https://tamnd.github.io/math2/node/J6NMBX3E55.md" tag="J6NMBX3E55" kind="statement">
# Birkhoff's ergodic theorem

## Statement

Let $(X, \mathcal{A}, \mu)$ be a [[probability space|0NFPJVJJHX]], let $T \colon X \to X$ be a [[measure-preserving transformation|556WBDAHNP]], and let $f \colon X \to \R$ be [[integrable|X7GTF3D98F]].
Let $A_n f = \frac{1}{n} \sum_{k=0}^{n-1} f \circ T^k$ be the [[Birkhoff averages|8TD2KCVH5N]] of $f$, and let $\mathcal{I} = \set{A \in \mathcal{A} : T^{-1}(A) = A}$ be the $\sigma$-algebra of invariant sets.
Then:

1. $A_n f(x)$ converges as $n \to \infty$ for [[almost every|RARTMA0ZEC]] $x \in X$, to the [[conditional expectation|0TTDGTEFMN]] $f^* = \mathbb{E}(f \mid \mathcal{I})$;
2. the limit satisfies $f^* \circ T = f^*$ almost everywhere and $\int_X f^* \, d\mu = \int_X f \, d\mu$;
3. $A_n f \to f^*$ also in $L^1(\mu)$;
4. if $T$ is [[ergodic|AZRPHS478B]], then $f^* = \int_X f \, d\mu$ almost everywhere.

## Remarks

The usual proof combines the [[maximal ergodic theorem|V7PA1V3N0K]] with a density argument.
For functions of the form $h + g - g \circ T$, with $h$ invariant and $g$ bounded, the averages converge everywhere, because the sums of $g - g \circ T$ telescope.
The maximal inequality bounds $\sup_n \abs{A_n f}$ in measure by $\norm{f}_1$, and this lets convergence pass to all integrable $f$.
For $f \in L^2(\mu)$, convergence in $L^2$ is the content of the [[mean ergodic theorem|PQPDQCCD02]].

Example: let $T(x) = 2x \bmod 1$ on $[0, 1)$ with Lebesgue measure, which is ergodic, and let $f$ be the indicator of $[1/2, 1)$.
Then $f(T^k(x))$ is the $(k+1)$-st binary digit of $x$, so the theorem says that almost every $x$ has, in the limit, half of its binary digits equal to $1$.

Applied to a sequence of independent identically distributed random variables, viewed as a shift on a product space, the theorem gives the [[strong law of large numbers|0DFWHQ839T]].

The measure must be invariant: for $T(x) = x/2$ on $[0, 1]$ with Lebesgue measure, which is not $T$-invariant, and $f(x) = x$, the averages tend to $0$ everywhere, while $\int_0^1 f \, dx = 1/2$, so conclusion 2 fails.

## Explanations

### L1 human

Follow one orbit for a long time and average a quantity along it: this is a time average.
Average the same quantity over the whole space: this is a space average.
For an ergodic system the two agree for almost every starting point.
So watching one typical trajectory long enough reveals the statistics of the whole system.

### L2 human

The theorem is the basic convergence result of ergodic theory.
Without ergodicity, the limit is still defined almost everywhere, but it can differ on different invariant pieces, and it equals the conditional expectation on the invariant sets.
It is used to prove the strong law of large numbers, the normality of almost every real number, and statistical properties of billiards and geodesic flows.
The same statement holds for measure-preserving flows, with sums replaced by integrals over time.
Kingman's subadditive ergodic theorem extends it from sums $S_n f$ to subadditive sequences, and the ergodic decomposition reduces the non-ergodic case to the ergodic one.
</document>

<document source="https://tamnd.github.io/math2/node/J6X53EPYR5.md" tag="J6X53EPYR5" kind="example">
# Stefan problem as a variational inequality

## Example

Take the one-dimensional one-phase [[Stefan problem|3B0HB3DHW1]] in units with thermal diffusivity and conductivity equal to $1$ and latent heat $L > 0$.
The liquid occupies $0 < x < s(t)$, with $s(0) = 0$ and $s$ increasing.
The end $x = 0$ is heated, with a given temperature $u(0, t) > 0$, and the temperature $u$ satisfies:
$$\partial_t u = \partial_x^2 u \ \text{ for } 0 < x < s(t), \qquad u(s(t), t) = 0, \qquad L s'(t) = -\partial_x u(s(t), t).$$
Extend $u$ by $0$ to the solid region $x > s(t)$, and define the integrated temperature
$$w(x, t) = \int_0^t u(x, \tau) \, d\tau.$$
Then $w$ solves the obstacle-type system
$$w \ge 0, \qquad \partial_t w - \partial_x^2 w + L \ge 0, \qquad w \, \big(\partial_t w - \partial_x^2 w + L\big) = 0.$$

Check in the liquid: fix $x > 0$ and let $\theta(x)$ be the time with $s(\theta(x)) = x$, so $u(x, \tau) = 0$ for $\tau \le \theta(x)$.
Then $w(x, t) = \int_{\theta(x)}^t u(x, \tau) \, d\tau$ for $t > \theta(x)$.
Differentiating twice in $x$ and using $u(x, \theta(x)) = 0$ gives $\partial_x^2 w = \int_{\theta(x)}^t \partial_x^2 u \, d\tau - \partial_x u(x, \theta(x)) \, \theta'(x)$.
The Stefan condition and $s'(\theta(x)) \, \theta'(x) = 1$ give $-\partial_x u(x, \theta(x)) \, \theta'(x) = L$.
The heat equation turns the integral into $u(x, t) = \partial_t w$, so $\partial_t w - \partial_x^2 w = -L$ where $w > 0$.
In the solid, $w = 0$, so $\partial_t w - \partial_x^2 w + L = L > 0$.

## Remarks

For each time $t$, the system is equivalent to a [[variational inequality|6QS7X5GVMY]] over the convex set $K = \set{v \ge 0}$:
$$\int \big(\partial_t w - \partial_x^2 w + L\big) (v - w) \, dx \ge 0 \quad \text{for all } v \in K.$$
Indeed, where $w > 0$ the first factor vanishes, and where $w = 0$ both factors are nonnegative.
So the free boundary $x = s(t)$ becomes the boundary of the set $\set{w > 0}$, as in the [[obstacle problem|GXN2NEHC8X]], and it no longer appears explicitly among the unknowns.
This change of unknown is due to Georges Duvaut, and it allows existence and regularity results for the Stefan problem through the theory of variational inequalities.

## Explanations

### L1 human

In a melting problem, the moving boundary between ice and water is unknown, which makes the equations hard to handle.
The trick is to add up the temperature over time at each place.
The new function is zero where the ice has not melted yet and positive in the water.
It solves a problem of the same type as a membrane pressed against an obstacle, where the contact set is found as part of the solution.
</document>

<document source="https://tamnd.github.io/math2/node/J6X65C2W4P.md" tag="J6X65C2W4P" kind="definition">
# Generated subalgebra

## Definition

Let $\mathbf{A}$ be a [[structure|983GZBQYT4]] with domain $A$ for a signature $\sigma$, and let $X \subseteq A$.
A subset $B \subseteq A$ is *closed* if it contains $c^{\mathbf{A}}$ for each constant symbol $c$ and $f^{\mathbf{A}}(b_1, \dots, b_n) \in B$ for each $n$-ary function symbol $f$ and all $b_1, \dots, b_n \in B$.
The set generated by $X$ is
$$\operatorname{Sg}(X) = \bigcap \set{B \subseteq A : X \subseteq B \text{ and } B \text{ is closed}},$$
the [[intersection|0ZNMMA9TF5]] of all closed subsets that contain $X$.
When $\operatorname{Sg}(X)$ is nonempty, the *subalgebra generated by $X$* is the [[substructure|676H7EGTKJ]] of $\mathbf{A}$ with domain $\operatorname{Sg}(X)$.

## Remarks

The set $A$ itself is closed and an intersection of closed sets is closed, so $\operatorname{Sg}(X)$ is the smallest closed subset containing $X$.
For finitary operations, $\operatorname{Sg}(X) = \bigcup_{k \ge 0} X_k$, where $X_0 = X$ and $X_{k+1}$ is $X_k$ together with the constants and all values $f^{\mathbf{A}}(b_1, \dots, b_n)$ with $b_1, \dots, b_n \in X_k$.

Example: in $(\Z, +, -, 0)$, $\operatorname{Sg}(\set{4, 6}) = 2\Z$.
The set $2\Z$ is closed and contains $4$ and $6$; and any closed set containing $4$ and $6$ contains $6 - 4 = 2$, hence all multiples of $2$.

Non-example: the set $4\Z \cup 6\Z$ contains $\set{4, 6}$ and $0$ but is not $\operatorname{Sg}(\set{4, 6})$, because it is not closed: $4 + 6 = 10$ is not in it.

When $\sigma$ has no constants, $\operatorname{Sg}(\emptyset) = \emptyset$.
</document>

<document source="https://tamnd.github.io/math2/node/J70H5D30DC.md" tag="J70H5D30DC" kind="statement">
# Grothendieck duality

## Statement

Let $f \colon X \to Y$ be a [[proper morphism|9537RKBZPP]] of [[Noetherian schemes|ND6XEY68P4]].
Let $D^+_{qc}(X)$ be the bounded below [[derived category|5ST7NVJFSM]] of $\mathcal{O}_X$-modules with [[quasi-coherent|KPKND0K7W6]] cohomology sheaves, and similarly for $Y$.
Let $Rf_* \colon D^+_{qc}(X) \to D^+_{qc}(Y)$ be the right [[derived functor|NPMC3K6J2F]] of $f_*$.
Then $Rf_*$ has a right adjoint $f^! \colon D^+_{qc}(Y) \to D^+_{qc}(X)$: for $\mathcal{F} \in D^+_{qc}(X)$ and $\mathcal{G} \in D^+_{qc}(Y)$ there is a natural isomorphism
$$\Hom_{D(Y)}(Rf_* \mathcal{F}, \mathcal{G}) \cong \Hom_{D(X)}(\mathcal{F}, f^! \mathcal{G}).$$
If moreover $f$ is smooth of relative dimension $n$, then $f^! \mathcal{G} \cong Lf^* \mathcal{G} \otimes \omega_{X/Y}[n]$, where $\omega_{X/Y} = \bigwedge^n \Omega_{X/Y}$ is the top exterior power of the [[sheaf of relative differentials|6VC6HQTFSA]].

## Remarks

Grothendieck's theory of duality was first written up in Hartshorne's book Residues and Duality.
In a later approach by Neeman, a right adjoint of $Rf_*$ on the unbounded category $D_{qc}(X)$ exists by Brown representability.
The main content is then the explicit form of $f^!$ and of the trace map $Rf_* f^! \to \mathrm{id}$.

Example: let $Y = \operatorname{Spec} k$ and $X$ a smooth projective variety of dimension $n$ over $k$.
Then $f^! k = \omega_X[n]$, and taking $\mathcal{F}$ to be a vector bundle placed in degree $-i$ gives Serre duality $H^{n-i}(X, \mathcal{E}^\vee \otimes \omega_X) \cong H^i(X, \mathcal{E})^\vee$.

For a finite morphism $f$, the right adjoint is $f^! \mathcal{G} = R\mathcal{H}om_{\mathcal{O}_Y}(f_* \mathcal{O}_X, \mathcal{G})$, viewed as a complex of $\mathcal{O}_X$-modules.

## Explanations

### L1 human

Grothendieck duality is a relative version of Serre duality.
Instead of one variety over a field, it treats a family of varieties over a base, and it works with complexes of sheaves instead of single sheaves.
It says that pushing sheaves forward along a proper map has a partner that pulls them back in a twisted way, and the twist is by the relative canonical bundle.

### L2 human

The formula $f^! = Lf^* \otimes \omega_{X/Y}[n]$ for smooth $f$ and the description of $f^!$ for finite $f$ together compute $f^!$ for projective morphisms, by factoring through a closed immersion into projective space.
The complex $f^! \mathcal{O}_Y$ is the relative dualizing complex; for flat $f$ of finite type it is concentrated in one degree exactly when the fibers of $f$ are Cohen–Macaulay.
Grothendieck duality underlies the theory of dualizing complexes, local duality and the Grothendieck–Riemann–Roch formalism of pushforwards.
</document>

<document source="https://tamnd.github.io/math2/node/J72DEZHNAW.md" tag="J72DEZHNAW" kind="conjecture">
# Inscribed square problem

## Statement

Let $C \subset \R^2$ be a Jordan curve, that is, a [[closed curve|045YFKRW0P]] in the plane that does not meet itself.
Then $C$ contains four points that are the vertices of a square.

## Remarks

The conjecture is open in general.
It is known for many classes of curves, among them convex curves, polygons, and curves that are smooth enough.
For smooth Jordan curves, Joshua Greene and Andrew Lobb proved more: such a curve contains the vertices of a rectangle of every prescribed aspect ratio.
The difficulty for general continuous curves is that limits of inscribed squares of approximating smooth curves may shrink to a single point.

## Explanations

### L1 human

Draw any closed loop on paper that never crosses itself, however wiggly.
The question is whether one can always find four points on the loop that form the corners of a square.
For a circle this is easy, and for many nice curves it has been proved, but for arbitrary continuous loops nobody knows.
</document>

<document source="https://tamnd.github.io/math2/node/J7410PG2DR.md" tag="J7410PG2DR" kind="concept">
# Robust statistics

## Summary

Robust statistics studies statistical procedures whose results change little when the data depart slightly from the model assumptions.
Such departures include outliers or a heavier tail than assumed.
Two main measures of robustness are the [[breakdown point|BRGGGZVQDS]] and the [[influence function|EB2QJQNE5Z]].
The first says how much bad data an estimator can absorb, and the second how much a single point can move it.
Typical robust procedures are the median, trimmed means and other [[L-estimators|E3C4F0RKJY]], and Huber's [[M-estimators|4503R8BVF8]].
The field was shaped by Tukey, Huber and Hampel from the 1960s on.

## Explanations

### L1 human

The average of a list of incomes can jump enormously when one billionaire joins the list.
The median barely moves.
Robust statistics prefers methods like the median, which give sensible answers even if a few data points are wrong or extreme.
It also studies the price of this safety, since robust methods are often a little less precise when the data really are clean.
</document>

<document source="https://tamnd.github.io/math2/node/J76687HP2E.md" tag="J76687HP2E" kind="statement">
# Sierpiński's theorem on non-atomic measures

## Statement

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] such that $\mu$ is [[non-atomic|SV96KN4WR9]].
Let $A \in \Sigma$ with $\mu(A) < \infty$, and let $b$ be a real number with $0 \le b \le \mu(A)$.
Then there is a set $B \in \Sigma$ with $B \subseteq A$ and $\mu(B) = b$.

## Remarks

The first step shows that a set of positive finite measure contains measurable subsets of arbitrarily small positive measure.
This uses the absence of atoms to split a set again and again into pieces of positive measure, keeping the smaller piece.
The second step builds $B$ by adding, at each stage, a subset of nearly the largest size that keeps the total at most $b$.
Continuity of measure then shows that the limit set has measure exactly $b$.

For Lebesgue measure on $[0, 1]$ the conclusion is easy: take $B = [0, b]$.
The theorem matters for abstract non-atomic measures, such as the product measure of fair coin tosses on $\set{0, 1}^{\N}$.

The hypothesis that $\mu$ is non-atomic is needed.
For the [[Dirac measure|0ACRGYNHSR]] $\delta_x$ and $A = \set{x}$, every measurable subset of $A$ has measure $0$ or $1$, so no subset has measure $1/2$.

## Explanations

### L1 human

A measure without lumps can be cut into a piece of any size one likes.
If a set has measure $5$, one can find a part of it with measure exactly $\pi$, or $0.3$, or any number between $0$ and $5$.
Lumpy measures, such as point masses, cannot be cut like this.

### L2 human

As a corollary, a finite non-atomic measure takes every value in $[0, \mu(X)]$, so its range is an interval.
The Lyapunov convexity theorem extends this to finite non-atomic vector measures with values in $\R^n$: their range is compact and convex.
The theorem is used in probability to build events of any prescribed probability, and in economics and game theory for fair division of non-atomic resources.
</document>

<document source="https://tamnd.github.io/math2/node/J7GYAF4FGT.md" tag="J7GYAF4FGT" kind="definition">
# Multivariate division algorithm

## Definition

Let $k$ be a [[field|04C5ZBG1WT]], let $S = k[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]], and fix a [[monomial order|E874NARWPV]] on $S$.
For nonzero $h \in S$, let $\operatorname{LT}(h)$ be its leading term.
Let $f \in S$ and let $(g_1, \ldots, g_s)$ be an ordered list of nonzero elements of $S$.
The *multivariate division algorithm* computes $q_1, \ldots, q_s, r \in S$ as follows.

1. Start with $q_1 = \cdots = q_s = 0$, $r = 0$ and $p = f$.
2. While $p \ne 0$: if some $\operatorname{LT}(g_i)$ divides $\operatorname{LT}(p)$, take the least such $i$, and replace $q_i$ by $q_i + \operatorname{LT}(p)/\operatorname{LT}(g_i)$ and $p$ by $p - (\operatorname{LT}(p)/\operatorname{LT}(g_i)) \, g_i$.
3. Otherwise, replace $r$ by $r + \operatorname{LT}(p)$ and $p$ by $p - \operatorname{LT}(p)$.
4. When $p = 0$, return $q_1, \ldots, q_s$ and the *remainder* $r$.

## Remarks

The algorithm stops, because each step makes the leading monomial of $p$ strictly smaller, or makes $p$ zero, and a monomial order is a well-order.
The output satisfies $f = q_1 g_1 + \cdots + q_s g_s + r$.
No monomial of $r$ is divisible by any $\operatorname{LT}(g_i)$, and the leading monomial of $q_i g_i$ is at most that of $f$ whenever $q_i g_i \ne 0$.

Example: in $k[x, y]$ with the lexicographic order $x > y$, divide $f = x^2 y + x y^2 + y^2$ by $(g_1, g_2) = (xy - 1, \; y^2 - 1)$.
The steps give $q_1 = x + y$, $q_2 = 1$ and $r = x + y + 1$; check: $(x + y)(xy - 1) + (y^2 - 1) + x + y + 1 = x^2 y + x y^2 + y^2$.
With the list in the order $(y^2 - 1, \; xy - 1)$, the algorithm gives the quotients $x + 1$ for $y^2 - 1$ and $x$ for $xy - 1$, and the remainder $2x + 1$.
So in general the remainder depends on the order of the list.
For a [[Gröbner basis|C83KPT47R0]], the remainder does not depend on the order, and it is zero exactly when $f$ lies in the ideal $(g_1, \ldots, g_s)$.
</document>

<document source="https://tamnd.github.io/math2/node/J7K42A9F8A.md" tag="J7K42A9F8A" kind="definition">
# Genus of a curve

## Definition

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $C$ be a smooth projective connected [[curve|XJM2MT4S3Y]] over $k$.
Let $\Omega_C$ be its [[sheaf of differentials|6VC6HQTFSA]].
The *genus* of $C$ is
$$g(C) = \dim_k H^0(C, \Omega_C),$$
the dimension of the space of regular differential $1$-forms on $C$.
Let $C$ be an integral curve, possibly singular or not projective.
The *geometric genus* of $C$ is the genus of the smooth projective curve with the same function field.
This curve is the [[normalization|M2FAMD9HZN]] of a projective closure of $C$.

## Remarks

For a smooth projective connected curve, Serre duality gives $g(C) = \dim_k H^1(C, \mathcal{O}_C)$, so the genus equals the [[arithmetic genus|H03GKRVBJ1]].
Example: on $\mathbb{P}^1$ one has $\Omega = \mathcal{O}(-2)$, which has no nonzero global sections, so $g(\mathbb{P}^1) = 0$.
Example: the smooth projective curve $E$ with affine equation $y^2 = x^3 - x$, in characteristic not $2$, has the form $\omega = dx/y$, regular and nowhere zero on $E$, and $g(E) = 1$.
Non-example of the equality with the arithmetic genus: the nodal cubic $y^2 = x^2(x + 1)$ is singular, its normalization is $\mathbb{P}^1$, so its geometric genus is $0$, while its arithmetic genus is $1$.
Over $\C$, the genus equals the topological genus of $C(\C)$ as a closed orientable surface.
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<document source="https://tamnd.github.io/math2/node/J7KN14DYV3.md" tag="J7KN14DYV3" kind="concept">
# Orthocomplemented lattice

## Summary

An orthocomplemented lattice, or ortholattice, is a [[complemented lattice|ZT47BK9Y6K]] with a chosen complement $a^\perp$ for each element $a$.
The map $a \mapsto a^\perp$ must reverse the order and be its own inverse.
The main examples are [[Boolean algebras|0XADJKM699]], with the Boolean complement.
Another is the lattice of closed subspaces of a [[Hilbert space|HM2HHE3J51]], with the [[orthogonal complement|0XJXTBCB2T]].
The De Morgan laws hold in every ortholattice.
Ortholattices are the starting point of the lattice-theoretic approach to [[quantum logic|2E9DMB7A6Q]].

## Explanations

### L1 human

In a general complemented lattice an element can have many complements, with no preferred one.
An orthocomplementation picks one complement for each element in a coherent way, like taking the orthogonal complement of a subspace.
Taking it twice gives back the element, and a bigger element gets a smaller complement.
For a line $\ell$ through the origin in $\R^3$, the chosen complement is the plane perpendicular to $\ell$.
</document>

<document source="https://tamnd.github.io/math2/node/J7NBAJMQND.md" tag="J7NBAJMQND" kind="definition">
# Projective variety

## Definition

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]], let $n \ge 0$, and let $\mathbb{P}^n$ be the [[projective space|ZT9A6JK8QF]] over $k$.
For a set $T$ of [[homogeneous polynomials|2C3PAQ7H50]] in $k[x_0, \dots, x_n]$, let
$$V(T) = \set{[a] \in \mathbb{P}^n : f(a) = 0 \text{ for all } f \in T}.$$
This is well defined, because $f(\lambda a) = \lambda^d f(a)$ when $f$ is homogeneous of degree $d$.
A *projective algebraic set* is a set of the form $V(T)$; these are the closed sets of the Zariski topology on $\mathbb{P}^n$.
A *projective variety* is a projective algebraic set that is [[irreducible|SY2MWC434Z]] in this topology.

## Remarks

The ideal of a projective algebraic set $X$ is the [[homogeneous ideal|6ZBNGS0P98]] generated by the homogeneous polynomials that vanish on $X$.
Then $X$ is irreducible if and only if this ideal is prime.

Example: the plane conic $C = V(xz - y^2) \subseteq \mathbb{P}^2$ is a projective variety.
Its polynomial $xz - y^2$ is irreducible, so it generates a prime ideal, and $C$ is the image of $\mathbb{P}^1$ under $[s : t] \mapsto [s^2 : st : t^2]$.

Non-example: $V(xy) \subseteq \mathbb{P}^2$ is a projective algebraic set but not a projective variety.
It is the union of the two lines $V(x)$ and $V(y)$.

The intersection of $X$ with a chart $U_i = \set{x_i \ne 0} \cong k^n$ is an [[algebraic set|VHFMQQYSTN]] in $k^n$, defined by setting $x_i = 1$ in the equations; when nonempty, it is an [[affine variety|JERVQT4C12]].
Some authors do not require irreducibility.
</document>

<document source="https://tamnd.github.io/math2/node/J7P32JQTVE.md" tag="J7P32JQTVE" kind="concept">
# Atlas

## Summary

An atlas on a [[topological space|0CQHFPGJ45]] is a family of charts whose domains cover the space.
A chart is a homeomorphism from an open piece of the space onto an open subset of $\R^n$.
Through an atlas, questions about the space can be turned into questions about open sets of $\R^n$.
Extra structures on manifolds, such as smooth or piecewise linear structures, are defined by special atlases.
In such an atlas, the changes of chart belong to a chosen class of maps.

## Explanations

### L1 human

The word comes from a book of maps of the Earth.
Each page shows one region of the globe on a flat sheet, and the pages overlap at their edges.
No single flat page can show the whole sphere without tearing it, but a collection of pages covers everything.
For the circle, two charts suffice: one for the circle without its north pole and one for the circle without its south pole.
Where two charts overlap, a map called the transition map converts positions on one chart into positions on the other.
</document>

<document source="https://tamnd.github.io/math2/node/J7VA040MC3.md" tag="J7VA040MC3" kind="statement">
# Projection-slice theorem

## Statement

Let $n \ge 2$ and let $f \in L^1(\R^n)$.
Let $\hat f(\xi) = \int_{\R^n} f(x)\, e^{-i x \cdot \xi}\, dx$ be its [[Fourier transform|DPVM34ADZV]].
Let $\theta$ be in the unit [[sphere|P9CM4GQ2T0]] $S^{n-1}$, and let $R_\theta f(s) = R f(\theta, s)$ be the [[Radon transform|VJNZJ352PC]] of $f$ in the direction $\theta$.
Then $R_\theta f \in L^1(\R)$, and for every $\sigma \in \R$,
$$\int_\R R_\theta f(s)\, e^{-i s \sigma}\, ds = \hat f(\sigma \theta).$$

## Remarks

The proof is one application of Fubini's theorem: write $x = s\theta + y$ with $s \in \R$ and $y \in \theta^\perp$, so that $x \cdot \sigma\theta = s\sigma$ and $dx = ds\, dy$.

Example: for $f(x) = e^{-\abs{x}^2}$ one has $R_\theta f(s) = \pi^{(n-1)/2} e^{-s^2}$, whose one-dimensional Fourier transform is $\pi^{n/2} e^{-\sigma^2/4}$.
This is $\hat f(\sigma\theta) = \pi^{n/2} e^{-\abs{\sigma\theta}^2/4}$, as the theorem says.

In words: the one-dimensional Fourier transform of a projection is the restriction of the $n$-dimensional Fourier transform to a line through the origin.
Since these lines cover $\R^n$, the theorem shows at once that $R f$ determines $f$ for $f \in L^1(\R^n)$, by injectivity of the Fourier transform on $L^1$.

## Explanations

### L1 human

Project a two-dimensional picture onto a line by adding up the values along each perpendicular line.
The theorem says that the frequencies of this one-dimensional profile are exactly the frequencies of the whole picture along that direction.
Projections in all directions therefore give all the frequencies of the picture, and hence the picture itself.

### L2 human

The theorem is the bridge between the [[Radon transform|VJNZJ352PC]] and Fourier analysis.
Combined with Fourier inversion in polar coordinates, it gives the [[Radon inversion formula|HN93ST5CBT]] and the ramp filter $\abs{\sigma}^{n-1}$ of [[filtered back-projection|SJQDDY9FMX]].
It also shows that the Radon transform is injective on $L^1(\R^n)$ and explains the moment conditions in the [[range conditions for the Radon transform|WASXFTTKK6]].
In practice it suggests Fourier reconstruction: compute the Fourier transforms of the projections, interpolate them onto a grid, and invert one $n$-dimensional Fourier transform.
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<document source="https://tamnd.github.io/math2/node/J7YS0DA3HQ.md" tag="J7YS0DA3HQ" kind="concept">
# Solvable by radicals

## Summary

A polynomial is solvable by radicals when all its roots can be written from its coefficients with the four field operations and $n$-th roots.
Formally, its splitting field lies in a [[radical extension|V74WMGQQ9D]] of the base field.
In characteristic $0$, [[Galois' theorem|VHCWCFA3MX]] says this holds exactly when the [[Galois group of the polynomial|0JZY3R908H]] is [[solvable|0P3KZK0FTV]].

## Explanations

### L1 human

The quadratic formula shows that every equation $ax^2 + bx + c = 0$ is solvable by radicals: its roots are $(-b \pm \sqrt{b^2 - 4ac})/(2a)$.
The [[cubic formula|FB6J5WAT5V]] and the [[quartic formula|4CDFMKKZ01]] do the same for degrees $3$ and $4$.
The [[Abel–Ruffini theorem|163A2WQQJX]] says that no such formula exists for the general equation of degree $5$.
Some quintics are still solvable by radicals, such as $x^5 - 2 = 0$, whose roots are $\sqrt[5]{2}$ times the fifth roots of unity.
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<document source="https://tamnd.github.io/math2/node/J7Z1H328BC.md" tag="J7Z1H328BC" kind="concept">
# Fourier transform as Laplace transform on the imaginary axis

## Summary

The [[Fourier transform|DPVM34ADZV]] of a function on $\R$ is the [[two-sided Laplace transform|6J2AGD0BYY]] of the same function evaluated on the imaginary axis.
With $\hat f(\xi) = \int f(x) e^{-2\pi i x \xi} \, dx$, this reads $\hat f(\xi) = \mathcal{B}f(2\pi i \xi)$.
More generally, the values of $\mathcal{B}f$ on a vertical line $\operatorname{Re} s = \sigma$ are the Fourier transform of $e^{-\sigma x} f(x)$.
This lets results on one transform pass to the other, such as inversion formulas, Plancherel's theorem and the use of analytic continuation.

## Explanations

### L1 human

The Laplace transform uses the weights $e^{-sx}$ with a complex number $s$.
When $s$ is purely imaginary, these weights are the pure oscillations $e^{-i\omega x}$ of the Fourier transform.
So the Fourier transform is the Laplace transform seen along one vertical line of the complex plane.
For example, $f(x) = e^{-\abs{x}}$ has $\mathcal{B}f(s) = 2/(1 - s^2)$ for $\abs{\operatorname{Re} s} < 1$, and putting $s = 2\pi i \xi$ gives $\hat f(\xi) = 2/(1 + 4\pi^2 \xi^2)$.
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# Cayley–Klein metric

## Summary

A Cayley–Klein metric is a distance on a region of [[projective space|ZT9A6JK8QF]].
It is built from a fixed [[quadric|Q3Q4FF23VG]], the absolute, through the logarithm of a [[cross-ratio|09ASMRMD67]].
Different choices of the absolute give [[hyperbolic|GN4N94GQ0Z]] and [[elliptic|H5GB19W2H3]] geometry, and Euclidean geometry appears as a limit case.
The construction of Cayley and Klein showed that these metric geometries all live inside [[projective geometry|N32Z68F0D3]].

## Explanations

### L1 human

Fix an ellipse in the plane and look only at the points inside it.
To measure the distance between two inside points, draw the line through them and see where it hits the ellipse.
The distance is half the logarithm of a ratio of four lengths along that line, the cross-ratio.
Points near the ellipse are very far from everything, so the ellipse acts as a horizon at infinity.
With a circle as the absolute, this gives the [[Beltrami–Klein model|1WN9B18H86]] of the hyperbolic plane.
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<document source="https://tamnd.github.io/math2/node/J80FB2BWNF.md" tag="J80FB2BWNF" kind="statement">
# Kac's lemma

## Statement

Let $(X, \mathcal{B}, \mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \colon X \to X$ be an [[ergodic|AZRPHS478B]] [[measure-preserving transformation|556WBDAHNP]].
Let $A \in \mathcal{B}$ with $\mu(A) > 0$, and for $x \in A$ let
$$r_A(x) = \min\set{n \ge 1 : T^n x \in A}$$
be the first return time of $x$ to $A$, which is finite for almost every $x \in A$.
Then
$$\int_A r_A \, d\mu = 1, \qquad \text{that is,} \qquad \frac{1}{\mu(A)} \int_A r_A \, d\mu = \frac{1}{\mu(A)}.$$

## Remarks

The return time is finite almost everywhere on $A$ by the [[Poincaré recurrence theorem|TVYJPBHD0S]].
For invertible $T$, let $A_n = \set{x \in A : r_A(x) = n}$.
The sets $T^j A_n$ with $n \ge 1$ and $0 \le j < n$ are pairwise disjoint, and their union is invariant up to a null set and contains $A$, so by ergodicity it has measure $1$.
Hence $1 = \sum_{n \ge 1} n \, \mu(A_n) = \int_A r_A \, d\mu$.
The non-invertible case follows by passing to the natural extension.
Example: for the rotation $x \mapsto x + 1/q \bmod 1$ on $\set{0, 1/q, \dots, (q-1)/q}$ with the uniform measure and $A$ a single point, the return time is $q$ and $\int_A r_A \, d\mu = q \cdot \frac{1}{q} = 1$.
Without ergodicity the integral equals the measure of the set of points whose orbit meets $A$, which can be less than $1$.
For the identity map, $r_A = 1$ on $A$ and $\int_A r_A \, d\mu = \mu(A)$.

## Explanations

### L1 human

The rarer a set of states, the longer it takes to come back to it, and the lemma gives the exact rate.
If a set has probability $\mu(A)$, then starting from that set, the average waiting time to return is $1/\mu(A)$.
For example, if a fair die is rolled again and again, a six comes back after $6$ rolls on average.

### L2 human

Kac's lemma explains why Poincaré recurrence does not contradict the second law of thermodynamics.
For a macroscopic system, the set of states near a given low-entropy state has astronomically small measure.
So the mean return time is astronomically long.
It is also the basis of the induced transformation $T_A(x) = T^{r_A(x)} x$ on $A$, which preserves the normalized measure on $A$ and is ergodic when $T$ is.
For an ergodic Markov chain started at a state $s$ with stationary probability $\pi(s) > 0$, it gives the classical formula $1/\pi(s)$ for the expected return time to $s$.
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# Rayleigh quotient for symmetric kernels

## Definition

Let $a < b$ and let $K \in L^2([a, b] \times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]] that is [[Hermitian|2ZB3NFZ94T]]: $K(y, x) = \overline{K(x, y)}$ for almost all $x, y$.
Let $T_K$ be its [[Hilbert–Schmidt integral operator|CB5G212Q03]] on $L^2[a, b]$, and let $\ip{u}{v} = \int_a^b u(x) \overline{v(x)} \, dx$.
The *Rayleigh quotient* of $K$ at a function $\varphi \in L^2[a, b]$ that is not $0$ almost everywhere is
$$R_K(\varphi) = \frac{\ip{T_K \varphi}{\varphi}}{\ip{\varphi}{\varphi}} = \frac{\int_a^b \int_a^b K(x, y) \, \varphi(y) \, \overline{\varphi(x)} \, dx \, dy}{\int_a^b \abs{\varphi(x)}^2 \, dx}.$$

## Remarks

Since $T_K$ is self-adjoint, $R_K(\varphi)$ is a real number, and $R_K(c \varphi) = R_K(\varphi)$ for every nonzero $c \in \C$.
If $\varphi$ is an eigenfunction with eigenvalue $\mu$, then $R_K(\varphi) = \mu$.

Example: for $K(x, y) = xy$ on $[0, 1]$ and $\varphi = 1$, the numerator is $\int_0^1 \int_0^1 xy \, dx \, dy = 1/4$ and the denominator is $1$, so $R_K(1) = 1/4$.
For $\varphi(x) = x$, the numerator is $\left(\int_0^1 x^2 \, dx\right)^2 = 1/9$ and the denominator is $1/3$, so $R_K(\varphi) = 1/3$, the eigenvalue of $x$.

Non-example: the constant kernel $K(x, y) = i$ is not Hermitian, since $\overline{K(x, y)} = -i \ne K(y, x)$.
The same formula at $\varphi = 1$ on $[0, 1]$ gives $i$, which is not real, so it is not a Rayleigh quotient in the sense above.

For a Hermitian matrix $A$, the same formula with sums instead of integrals is the [[Rayleigh quotient|06WHP8SE98]] $x^* A x / x^* x$.
</document>

<document source="https://tamnd.github.io/math2/node/J85FZEQG38.md" tag="J85FZEQG38" kind="concept">
# Local field

## Summary

A local field is a field with a nontrivial absolute value for which it is locally compact.
The Archimedean local fields are $\R$ and $\C$.
The others are the finite extensions of the fields $\Q_p$ of [[p-adic numbers|BT6T702C28]] and the fields $\F_q((t))$ of Laurent series over finite fields.
They are the completions of [[global fields|634XKY8RZ0]].
Number theory studies a global field one place at a time through its local fields.

## Explanations

### L1 human

A local field is a field where one can measure size, where Cauchy sequences converge, and where closed balls are compact.
The real numbers are the most familiar example.
The $p$-adic numbers $\Q_p$ are another: there, a ball such as $\Z_p$ is compact, like a closed interval.
Studying an equation in all local fields is a first test for whether it has rational solutions.
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<document source="https://tamnd.github.io/math2/node/J87MYGKPDD.md" tag="J87MYGKPDD" kind="definition">
# Heun function

## Definition

Let $a \in \C \setminus \set{0, 1}$ and let $q, \alpha, \beta, \gamma, \delta, \epsilon$ be complex numbers with
$$\epsilon = \alpha + \beta + 1 - \gamma - \delta.$$
*Heun's equation* is the [[linear differential equation|0QQRP17Z7S]]
$$\frac{d^2 w}{dz^2} + \left( \frac{\gamma}{z} + \frac{\delta}{z - 1} + \frac{\epsilon}{z - a} \right) \frac{dw}{dz} + \frac{\alpha \beta z - q}{z (z - 1)(z - a)}\, w = 0.$$
Assume $\gamma \notin \set{0, -1, -2, \dots}$.
The *local Heun function* $Hl(a, q; \alpha, \beta, \gamma, \delta; z)$ is the unique solution of Heun's equation that is analytic at $z = 0$ and takes the value $1$ there.

## Remarks

Heun's equation is a [[Fuchsian equation|ZBVH2YE50B]] with [[regular singular points|H96HXG6CA0]] $0, 1, a, \infty$.
Every Fuchsian equation with four regular singular points takes this form after moving the singular points there and making one exponent $0$ at each finite point.
The exponents are $\set{0, 1 - \gamma}$ at $0$, $\set{0, 1 - \delta}$ at $1$, $\set{0, 1 - \epsilon}$ at $a$ and $\set{\alpha, \beta}$ at $\infty$.
The relation $\epsilon = \alpha + \beta + 1 - \gamma - \delta$ is Fuchs' relation: the sum of all exponents equals $2$.
The number $q$ is the accessory parameter; it is not determined by the exponents.
The power series of $Hl$ at $0$ converges in the disc $\abs{z} < \min(1, \abs{a})$.
Example: for $q = 0$ and $\alpha = 0$, the constant $w = 1$ is a solution, since $w' = w'' = 0$ and $\alpha\beta z - q = 0$; so $Hl(a, 0; 0, \beta, \gamma, \delta; z) = 1$.
Non-example: for $q = 1$ and $\alpha = 0$, the constant $1$ is not a solution, since the equation leaves the term $\frac{-1}{z (z - 1)(z - a)} \ne 0$.
A *Heun function* in the narrow sense is a local solution that is analytic at two singular points at once, for instance at $0$ and $1$.
This happens only for special values of $q$.
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# Runge–Kutta method for stochastic differential equations

## Statement

Let $W$ be a one-dimensional [[Wiener process|22735FGJV6]].
Consider the scalar Itô [[stochastic differential equation|B219G9SHJA]]
$$dX_t = a(X_t)\, dt + b(X_t)\, dW_t, \qquad X_0 = x_0 \in \R, \qquad 0 \le t \le T.$$
Assume, as for the [[Milstein method|C5WJT0NWGV]], that $a$ and $b$ are twice continuously differentiable with bounded first and second derivatives.
Assume also that $b b'$ is globally Lipschitz continuous.
Let $h = T/N$, $t_n = n h$ and $\Delta W_n = W_{t_{n+1}} - W_{t_n}$.
The derivative-free scheme of strong order $1$ sets $Y_0 = x_0$,
$$\Upsilon_n = Y_n + a(Y_n)\, h + b(Y_n) \sqrt{h},$$
$$Y_{n+1} = Y_n + a(Y_n)\, h + b(Y_n)\, \Delta W_n + \frac{b(\Upsilon_n) - b(Y_n)}{2 \sqrt{h}} \left( (\Delta W_n)^2 - h \right).$$
Then there is a constant $C$, independent of $N$, such that
$$\max_{0 \le n \le N} \left( \mathbb{E} \abs{X_{t_n} - Y_n}^2 \right)^{1/2} \le C h.$$

## Remarks

By Taylor's theorem, $\frac{b(\Upsilon_n) - b(Y_n)}{\sqrt{h}} = b'(Y_n)\, b(Y_n) + O(\sqrt{h})$.
So the scheme differs from the Milstein scheme by a term of size $O(\sqrt{h}) \cdot O(h)$ with mean zero, and it keeps the strong order $1$.
As in deterministic [[Runge–Kutta methods|7DBPJKYC6C]], extra evaluations of a coefficient replace its derivative.
A naive transfer of a deterministic Runge–Kutta method to the stochastic case does not in general give a convergent or high-order scheme, because Itô calculus changes the order conditions.
Example: for $dX = \sigma X\, dW$, one has $\Upsilon_n = Y_n (1 + \sigma \sqrt{h})$, and the correction term equals $\frac{\sigma^2}{2} Y_n ((\Delta W_n)^2 - h)$, exactly the Milstein term.

## Explanations

### L1 human

The Milstein method for random differential equations needs the derivative of the noise coefficient.
This scheme avoids the derivative: it evaluates the coefficient at a second, slightly shifted point and uses the difference.
It keeps the accuracy of Milstein's method, so halving the step roughly halves the pathwise error.

### L2 human

Stochastic Runge–Kutta methods are built to match the Itô–Taylor expansion up to a given order, the way deterministic ones match the Taylor expansion.
Their order conditions are described with rooted trees, as for deterministic methods, but with several kinds of nodes for the drift and the noise.
For several noise sources, strong order $1$ again needs iterated integrals or a commutativity condition on the noise coefficients.
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# Subvariety

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras of signature $\sigma$.
A *subvariety* of $\mathcal{V}$ is a variety $\mathcal{W}$ of signature $\sigma$ such that $\mathcal{W} \subseteq \mathcal{V}$.

## Remarks

If $\mathcal{V} = \operatorname{Mod}(\Sigma)$, the subvarieties of $\mathcal{V}$ are exactly the classes $\operatorname{Mod}(\Sigma \cup \Gamma)$ for sets $\Gamma$ of [[identities|PKD440G2XW]] of signature $\sigma$.
Example: abelian groups form a subvariety of the variety of groups, since they are the groups that satisfy $xy \approx yx$.
Non-example: the class of finite groups is contained in the variety of groups but is not a subvariety.
It is not closed under products: $\Z$ is a subgroup of $\prod_{n \ge 1} \Z / n\Z$, generated by the element $(1, 1, \ldots)$.
Every variety has itself and the trivial variety, of one-element algebras, as subvarieties.
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<document source="https://tamnd.github.io/math2/node/J8GYFAA70H.md" tag="J8GYFAA70H" kind="definition">
# Diamond principle

## Definition

Let $\omega_1$ be the least uncountable ordinal.
The diamond principle $\diamondsuit$ is the following statement.
There is a [[sequence|0QE8STGJH2]] $(A_\alpha)_{\alpha < \omega_1}$ with $A_\alpha \subseteq \alpha$ for each $\alpha$, such that for each $A \subseteq \omega_1$ the set
$$\set{\alpha < \omega_1 : A \cap \alpha = A_\alpha}$$
is [[stationary|6FDYJVG1CK]] in $\omega_1$.
Such a sequence is called a $\diamondsuit$-sequence.

## Remarks

Example: Jensen proved that $\diamondsuit$ holds in the constructible universe $L$, so it is consistent with ZFC if ZFC is consistent.
Non-example: the sequence with $A_\alpha = \varnothing$ for all $\alpha$ is not a $\diamondsuit$-sequence.
For $A = \omega_1$, the set $\set{\alpha : \alpha \cap \omega_1 = \varnothing} = \set{0}$ is not stationary.
$\diamondsuit$ implies the continuum hypothesis: each $A \subseteq \omega$ equals $A \cap \alpha = A_\alpha$ for some $\alpha > \omega$, so there are at most $\aleph_1$ subsets of $\omega$.
$\diamondsuit$ implies that there is a Suslin tree, and so it settles Suslin's problem negatively.
More generally, $\diamondsuit_\kappa(S)$ is defined for a regular uncountable cardinal $\kappa$ and a stationary $S \subseteq \kappa$, with guesses only at points of $S$.
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# Metacompact space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A family $\mathcal{V}$ of subsets of $X$ is *point-finite* if each $x \in X$ belongs to only finitely many members of $\mathcal{V}$.
The space $X$ is *metacompact* if every [[open cover|049PXYHW1T]] $\mathcal{U}$ of $X$ has a point-finite open refinement.
That is, there is a point-finite open cover $\mathcal{V}$ such that each $V \in \mathcal{V}$ lies in some $U \in \mathcal{U}$.

## Remarks

Example: every [[paracompact space|8R29G6NG9Q]] is metacompact.
A [[locally finite|DZ297MMGV8]] family is point-finite, since a neighbourhood of $x$ meeting finitely many members also shows that $x$ lies in finitely many.

Non-example: the space $[0, \omega_1)$ of countable ordinals with the order topology is not metacompact.
It is [[countably compact|S3N3CHZV89]] but not compact.
A countably compact metacompact space is always compact.
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# Generalized quadrangle

## Definition

Let $s \ge 1$ and $t \ge 1$ be integers.
A *generalized quadrangle of order $(s, t)$* is an [[incidence structure|FGKNAFNHA3]] $(P, L, I)$ of points and lines such that:

1. each point lies on exactly $t + 1$ lines, and two distinct points lie on at most one common line;
2. each line contains exactly $s + 1$ points, and two distinct lines meet in at most one common point;
3. if a point $p$ is not on a line $\ell$, exactly one line through $p$ meets $\ell$.

## Remarks

Counting gives $(s + 1)(st + 1)$ points and $(t + 1)(st + 1)$ lines.
A generalized quadrangle is the same as a [[generalized polygon|PCT2HY9AAZ]] with $n = 4$.
The finite [[polar spaces|Z7T0V2VAWJ]] of rank $2$ are generalized quadrangles.

Example: the $(s + 1) \times (s + 1)$ grid, with the rows and columns as lines, has order $(s, 1)$.
Clauses 1 and 2 are clear.
For clause 3, let $p$ be off the row $\ell$.
The column through $p$ meets $\ell$ in one point $q$, and the row through $p$ is disjoint from $\ell$, so exactly one line through $p$ meets $\ell$.
The same check works with rows and columns exchanged.

Non-example: in a projective plane of order $n \ge 2$, every line through a point $p$ off $\ell$ meets $\ell$, so $n + 1 > 1$ lines through $p$ meet $\ell$ and clause 3 fails.

The symplectic quadrangle $W(q)$, formed by the points of $\mathrm{PG}(3, q)$ and the totally isotropic lines of a symplectic form, has order $(q, q)$.
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# Mori–Nagata theorem

## Statement

Let $A$ be a [[Noetherian|0MVH4Y1CRZ]] [[integral domain|09S3PFBEPM]] with [[field of fractions|0VF11ZTTMD]] $K$.
Then the [[integral closure|G0V86SJBQW]] of $A$ in $K$ is a [[Krull domain|CB9WJCN9BM]].

## Remarks

A Noetherian integrally closed domain is a Krull domain, so the theorem says that integral closure keeps part of the Noetherian property, even when it loses the rest.
The integral closure of a Noetherian domain need not be Noetherian, and Nagata gave such examples.
So the class of Krull domains is the natural home for integral closures of Noetherian domains.
In dimension $1$, the [[Krull–Akizuki theorem|FSXA3CF365]] gives more: the integral closure is Noetherian.
For example, the integral closure of $\Z[\sqrt{-3}]$ in $\Q(\sqrt{-3})$ is $\Z[(1 + \sqrt{-3})/2]$, a Dedekind domain and so a Krull domain.

## Explanations

### L1 human

Making a ring integrally closed can destroy the Noetherian property.
The theorem says that something useful survives.
The closed ring is a Krull domain, which means that it still has a good theory of divisors, as in a ring with unique factorisation up to a class group.

### L2 human

Krull domains are the domains that are intersections of discrete valuation rings with a finiteness condition, and they carry a divisor class group.
The theorem lets one use divisor theory on the normalisation of any Noetherian domain, for example on the normalisation of the coordinate ring of a singular affine variety.
Compare the [[Krull–Akizuki theorem|FSXA3CF365]], which in dimension one gives a Noetherian normalisation.
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# Lax–Wendroff method

## Definition

Fix $\Delta x > 0$ and $\Delta t > 0$, put $\lambda = \Delta t / \Delta x$, and let $u_j^n$ approximate $u(j \Delta x, n \Delta t)$.

1. For the [[transport equation|PVD7VZJZ9B]] $u_t + a u_x = 0$ with $a \in \R$ and $\nu = a \lambda$, the *Lax–Wendroff method* is the explicit [[finite difference|X7HYEBA5SA]] scheme
$$u_j^{n+1} = u_j^n - \frac{\nu}{2} \left( u_{j+1}^n - u_{j-1}^n \right) + \frac{\nu^2}{2} \left( u_{j+1}^n - 2 u_j^n + u_{j-1}^n \right).$$
2. For a [[conservation law|8RA7AMWSAF]] $u_t + f(u)_x = 0$ with a smooth flux $f$, its two-step (Richtmyer) form is
$$u_{j+1/2}^{n+1/2} = \frac{u_j^n + u_{j+1}^n}{2} - \frac{\lambda}{2} \left( f(u_{j+1}^n) - f(u_j^n) \right), \qquad u_j^{n+1} = u_j^n - \lambda \left( f(u_{j+1/2}^{n+1/2}) - f(u_{j-1/2}^{n+1/2}) \right).$$

## Remarks

For $f(u) = a u$, the two-step form reduces to the scheme of clause 1.
The first step is a [[Lax–Friedrichs|VY3J2NKGKW]] step over half a time step on the staggered grid.
Example: for clause 1, the Fourier mode $e^{i j \xi}$ is multiplied by $g = 1 - i \nu \sin \xi - \nu^2 (1 - \cos \xi)$, and a short computation gives $\abs{g}^2 = 1 - 4 \nu^2 (1 - \nu^2) \sin^4 (\xi / 2)$.
So $\abs{g} \le 1$ for all $\xi$ exactly when $\abs{\nu} \le 1$, and the scheme is stable under this condition.
Taylor expansion shows that the local truncation error is $O(\Delta x^3)$ for fixed $\nu$, so the method is second-order accurate.
Non-example: dropping the last term gives the forward-time centred-space scheme, which is only first order in time and has $\abs{g}^2 = 1 + \nu^2 \sin^2 \xi > 1$ for $\sin \xi \ne 0$, so it is unstable.

## Explanations

### L1 human

The scheme is the second-order Taylor step in time, with the time derivatives traded for space derivatives through the equation.
The last term adds just enough smoothing to make the centred scheme stable.
</document>

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# Dirichlet series as Mellin transforms

## Definition

Let $a \colon \set{1, 2, 3, \dots} \to \C$ and let $F(s) = \sum_{n \ge 1} a(n) n^{-s}$ be its [[Dirichlet series|8MWTEF6462]].
Let $\sigma_a$ be its abscissa of [[absolute convergence|0YBDSJARWN]], and assume $\sigma_a < \infty$.
The *Mellin representations* of $F$ are the two identities below, which hold for every $s \in \C$ with $\operatorname{Re} s > \max(\sigma_a, 0)$.

1. With $f_a(x) = \sum_{n \ge 1} a(n) e^{-nx}$ for $x > 0$,
$$\Gamma(s) F(s) = \int_0^\infty x^{s-1} f_a(x) \, dx = \mathcal{M}f_a(s).$$
2. With the summatory function $A(x) = \sum_{n \le x} a(n)$ for $x > 0$,
$$\frac{F(s)}{s} = \int_1^\infty A(x) \, x^{-s-1} \, dx = \mathcal{M}A(-s).$$

Here $\mathcal{M}$ is the [[Mellin transform|N5PVGRJE1T]] and $\Gamma$ is the [[gamma function|0H5SYBSED9]].

## Remarks

For identity 1, $\int_0^\infty x^{s-1} e^{-nx} \, dx = \Gamma(s) n^{-s}$ when $\operatorname{Re} s > 0$, and the sum of the absolute values $\Gamma(\operatorname{Re} s) \sum_n \abs{a(n)} n^{-\operatorname{Re} s}$ is finite when $\operatorname{Re} s > \sigma_a$, so sum and integral may be exchanged.
For identity 2, $A(x) = 0$ for $x < 1$, and $\int_n^\infty x^{-s-1} \, dx = n^{-s}/s$; exchanging sum and integral is justified the same way.
Example: $a(n) = 1$ gives $\sigma_a = 1$, $f_a(x) = 1/(e^{x} - 1)$ and $A(x) = \floor{x}$, so
$$\Gamma(s) \zeta(s) = \int_0^\infty \frac{x^{s-1}}{e^{x} - 1} \, dx, \qquad \frac{\zeta(s)}{s} = \int_1^\infty \floor{x} \, x^{-s-1} \, dx, \qquad \operatorname{Re} s > 1.$$
Non-example: at $s = 1$ with $a(n) = 1$, the condition $\operatorname{Re} s > \sigma_a$ fails, and indeed $\int_0^\infty dx/(e^x - 1)$ diverges at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/J9367QFD3G.md" tag="J9367QFD3G" kind="statement">
# Rodrigues' formula

## Statement

Let $(I, w, \sigma)$ be one of the following triples:

1. $I = (-1, 1)$, $w(x) = (1 - x)^\alpha (1 + x)^\beta$ with $\alpha, \beta > -1$, and $\sigma(x) = 1 - x^2$.
2. $I = (0, \infty)$, $w(x) = x^\alpha e^{-x}$ with $\alpha > -1$, and $\sigma(x) = x$.
3. $I = \R$, $w(x) = e^{-x^2}$, and $\sigma(x) = 1$.

For $n \ge 0$ let
$$q_n(x) = \frac{1}{w(x)}\, \frac{d^n}{dx^n} \left( w(x)\, \sigma(x)^n \right), \qquad x \in I.$$
Then $q_n$ is a polynomial of [[degree|0P4JRMSWNH]] exactly $n$, and $\int_I q_m(x)\, q_n(x)\, w(x)\, dx = 0$ for $m \ne n$.
So $(q_n)$ is, up to constant factors, the family of [[classical orthogonal polynomials|47WJ34NC9K]] for $w$.

## Remarks

That $q_n$ is a polynomial follows by induction: each derivative of $w \sigma^n$ is $w \sigma^{n-k}$ times a polynomial of degree at most $k$.
This holds because $\sigma w' / w$ is a polynomial of degree at most $1$.
For orthogonality, take $m < n$ and integrate $\int_I x^m\, \frac{d^n}{dx^n}(w \sigma^n)\, dx$ by parts $n$ times.
The boundary terms vanish because $w \sigma^n$ and its first $n - 1$ derivatives vanish at the ends of $I$, and the final integrand contains $\frac{d^n}{dx^n} x^m = 0$.
The usual normalisations are $P_n^{(\alpha,\beta)} = \frac{(-1)^n}{2^n n!}\, q_n$ (Jacobi), $L_n^{(\alpha)} = \frac{1}{n!}\, q_n$ (Laguerre), and $H_n = (-1)^n q_n$ (Hermite).
Example: in the Hermite case with $n = 2$, $\frac{d^2}{dx^2} e^{-x^2} = (4x^2 - 2) e^{-x^2}$, so $q_2 = 4x^2 - 2 = H_2$.
Example: in the Jacobi case with $\alpha = \beta = 0$, the formula gives the Legendre polynomials $P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n$.

## Explanations

### L1 human

Each classical family of orthogonal polynomials comes from one recipe.
Multiply the weight by a power of a simple polynomial, differentiate $n$ times, and divide by the weight again.
For the Legendre polynomials this means differentiating $(x^2 - 1)^n$ exactly $n$ times.
The formula makes orthogonality almost automatic, through integration by parts.

### L2 human

Rodrigues' formula is the quickest way to compute norms, leading coefficients and generating functions of the classical families.
With Cauchy's integral formula it gives contour integral representations, from which generating functions such as $\sum_n P_n(x)\, t^n = (1 - 2xt + t^2)^{-1/2}$ follow.
The existence of such a formula characterises the classical families among orthogonal polynomials, together with Bochner's characterisation by differential equations.
The polynomial $\sigma$ is the same one that appears in the second-order differential equation $\sigma y'' + \tau y' + \lambda_n y = 0$ satisfied by $q_n$.
</document>

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# Inherently nonfinitely based algebra

## Definition

Let $A$ be a finite algebra of a finite signature with only function symbols.
The algebra $A$ is *inherently nonfinitely based* if every [[locally finite variety|8QPW65PZVW]] that contains $A$ is not [[finitely based|YGYZAQ12MZ]].

## Remarks

The [[variety|AJNZJD8442]] generated by a finite algebra is locally finite, so an inherently nonfinitely based finite algebra is not finitely based.
Let $A$ be inherently nonfinitely based and let $B$ be a finite algebra whose variety contains $A$.
Then $B$ is inherently nonfinitely based as well, since every locally finite variety containing $B$ contains $A$.

Example: the six-element Brandt monoid $B_2^1$, as a semigroup, is inherently nonfinitely based, by a theorem of Sapir.

Non-example: no finite group $G$ is inherently nonfinitely based.
The variety generated by $G$ is locally finite and contains $G$, and it is finitely based by the [[Oates–Powell theorem|5RFPT46DXX]].
</document>

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# Prékopa–Leindler inequality

## Statement

Let $n \ge 1$ and $0 < t < 1$.
Let $f, g, h \colon \R^n \to [0, \infty)$ be Lebesgue [[measurable functions|YYNCJSR0PJ]] such that
$$h((1-t) x + t y) \ge f(x)^{1-t} g(y)^{t} \quad \text{for all } x, y \in \R^n.$$
Then
$$\int_{\R^n} h \, dx \ge \left(\int_{\R^n} f \, dx\right)^{1-t} \left(\int_{\R^n} g \, dx\right)^{t}.$$

## Remarks

The integrals are [[Lebesgue integrals|DWJF7ABPR1]] and may be infinite.
The proof for $n = 1$ uses a transport map between $f$ and $g$ after normalisation, and the general case follows by induction on $n$ with Fubini's theorem.
Example: let $A, B$ be nonempty compact sets and $f = 1_A$, $g = 1_B$, $h = 1_{(1-t)A + tB}$.
The hypothesis holds, and the conclusion gives $\lambda((1-t) A + t B) \ge \lambda(A)^{1-t} \lambda(B)^{t}$.
This multiplicative form implies the [[Brunn–Minkowski theorem|9B02RPN7XD]] by scaling $A$ and $B$.
It is a reversed form of Hölder's inequality: Hölder bounds $\int f^{1-t} g^t$ from above, while here a function above the "sup-convolution" is bounded from below.

## Explanations

### L1 human

If a function $h$ is large at each weighted midpoint of points where $f$ and $g$ are large, then the integral of $h$ is large too.
It is the functional version of the Brunn–Minkowski inequality for volumes of sums of sets.

### L2 human

A main consequence is that log-concave functions and measures are stable: marginals and convolutions of log-concave densities are log-concave.
This makes the inequality a basic tool in convex geometry and in the concentration of measure, for example for Gaussian measures.
</document>

<document source="https://tamnd.github.io/math2/node/J9HA54ACME.md" tag="J9HA54ACME" kind="concept">
# Pointed cone

## Summary

A pointed cone is a [[convex cone|6VE4KA12VD]] that contains the origin.
Equivalently, it is a nonempty set closed under sums and under multiplication by nonnegative scalars.
So it behaves like a subspace in which only nonnegative coefficients are allowed.
The word is not used uniformly: some authors call a cone pointed when it contains no line through the origin, a property also called salient.

## Explanations

### L1 human

A pointed cone is a wedge that includes its tip at the origin.
You may add vectors of the cone, stretch them by any factor $t \ge 0$, and shrink them all the way to $0$.
The closed first quadrant of the plane is a pointed cone.
The open first quadrant, without its two boundary half-lines and without the origin, is a convex cone that is not pointed in this sense.
</document>

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# Directly indecomposable algebra

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] whose universe has at least two elements.
The algebra $\mathbf{A}$ is *directly indecomposable* when it is not isomorphic to any [[direct product|WJ73XYBNE4]] $\mathbf{B} \times \mathbf{C}$.
Here $\mathbf{B}$ and $\mathbf{C}$ range over algebras of the same signature with at least two elements each.

## Remarks

Equivalently, the only [[factor congruences|6N268PHKQA]] of $\mathbf{A}$ are $\Delta_A$ and $\nabla_A$.
Every finite algebra is isomorphic to a direct product of finitely many directly indecomposable algebras, by induction on the number of elements.

Example: an algebra $\mathbf{A}$ with a prime number $p$ of elements is directly indecomposable.
If $\mathbf{A} \cong \mathbf{B} \times \mathbf{C}$, then $p = \abs{B} \cdot \abs{C}$, so $\abs{B} = 1$ or $\abs{C} = 1$.

Non-example: the group $\Z/6\Z$ is not directly indecomposable.
The map $k \mapsto (k \bmod 2, k \bmod 3)$ is an isomorphism onto $\Z/2\Z \times \Z/3\Z$, and both factors have at least two elements.

Every subdirectly irreducible algebra is directly indecomposable, but not conversely: the ring $\Z$ is directly indecomposable and not subdirectly irreducible.
</document>

<document source="https://tamnd.github.io/math2/node/J9QYGM8AKF.md" tag="J9QYGM8AKF" kind="definition">
# Henselization

## Definition

Let $(R, \mathfrak{m})$ be a [[local ring|MZ02BKPHJA]].
A *Henselization* of $R$ is a local ring $R^h$ together with a [[local homomorphism|P8YTMGEN9R]] $i \colon R \to R^h$ such that:

1. $R^h$ is a [[Henselian local ring|RNXPV3KC5J]].
2. For every Henselian local ring $S$ and every local homomorphism $f \colon R \to S$, there is exactly one local homomorphism $g \colon R^h \to S$ with $g \circ i = f$.

## Remarks

A Henselization exists for every local ring, and the universal property makes it unique up to a unique isomorphism.
One construction takes the direct limit of the localizations of [[étale|FTRW0679E2]] $R$-algebras at primes over $\mathfrak{m}$ with trivial residue field extension.
The map $i$ is [[faithfully flat|C31PYYPRGC]], the ideal $\mathfrak{m} R^h$ is the maximal ideal of $R^h$, and $i$ induces an isomorphism of residue fields.
If $R$ is Noetherian, then $R^h$ is Noetherian and has the same [[completion|KBN13NPYJQ]] as $R$.

Example: if $R$ is already Henselian, then $R$ with the identity map is a Henselization, because $g = f$ is the only choice in clause 2.

Non-example: $\Z_{(5)}$ with the identity map is not its own Henselization, because it fails clause 1.
The polynomial $x^2 - 6$ has the simple root $1$ modulo $5$, but $6$ is not a square in $\Q$, so the root does not lift to $\Z_{(5)}$.
</document>

<document source="https://tamnd.github.io/math2/node/J9SQCM1DWM.md" tag="J9SQCM1DWM" kind="definition">
# Starred transform

## Definition

Let $x \colon [0, \infty) \to \C$ be a function and let $T > 0$ be a sampling period.
The impulse-sampled signal is
$$x^*(t) = \sum_{n = 0}^{\infty} x(nT) \, \delta(t - nT),$$
where $\delta$ is the [[Dirac delta|0KPHENZ5S7]].
The *starred transform* of $x$ is the [[Laplace transform|0V49JEZW06]] of $x^*$:
$$X^*(s) = \sum_{n = 0}^{\infty} x(nT) \, e^{-nTs},$$
for the complex $s$ where this series converges absolutely.

## Remarks

The impulse-sampled signal is $x$ times a [[Dirac comb|7KHGQFM8A9]] of period $T$, restricted to $t \ge 0$.
If $X(z)$ is the [[unilateral Z-transform|EPG6JPVJE7]] of the samples $x_n = x(nT)$, then $X^*(s) = X(e^{sT})$.
Example: for $x(t) = e^{-at}$ with $a \in \C$, $X^*(s) = \sum_{n \ge 0} e^{-n T (s + a)} = 1/(1 - e^{-T(s + a)})$ for $\operatorname{Re}(s + a) > 0$.
Non-example: for $x(t) = e^{t^2}$ the terms $e^{n^2 T^2 - nTs}$ do not tend to $0$ for any $s$, so the series converges nowhere and $x$ has no starred transform.
$X^*$ is periodic in the imaginary direction: $X^*(s + 2\pi i/T) = X^*(s)$, since $e^{-nT \cdot 2\pi i/T} = 1$.
</document>

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# Langlands program

## Statement

The following is the global reciprocity conjecture for $\mathrm{GL}_n$, one central part of the program.
Let $F$ be an [[algebraic number field|MWBDFS2KJ5]], let $G_F$ be its absolute Galois group, let $n \ge 1$, and let $\ell$ be a prime.
Fix a field isomorphism $\iota \colon \overline{\Q}_\ell \to \C$.
Let $\rho \colon G_F \to \mathrm{GL}_n(\overline{\Q}_\ell)$ be an irreducible [[Galois representation|JQEYEQJY2G]].
Assume that $\rho$ is unramified at all but finitely many [[places|BT5V3TSNY6]] of $F$ and de Rham at the places above $\ell$.
Then there is a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb{A}_F)$, where $\mathbb{A}_F$ is the [[adele ring|SEH68BA1QQ]] of $F$, with the following property.
For all but finitely many finite places $v$ of $F$, both $\rho$ and $\pi_v$ are unramified at $v$.
Let $\mathrm{Frob}_v$ be a [[Frobenius element|H5145REFMW]] at $v$.
Then the Satake parameters of $\pi_v$ are, up to the standard normalising twist, the images under $\iota$ of the eigenvalues of $\rho(\mathrm{Frob}_v)$.

## Remarks

The Langlands program is a web of conjectures proposed by Robert Langlands from 1967 on.
It relates Galois representations, [[automorphic forms|3C73HND6XA]] and representations of reductive groups over local and global fields.
Besides reciprocity, its main parts are the principle of functoriality, which predicts transfers of automorphic representations between different reductive groups, and the local Langlands correspondence.
For $n = 1$, reciprocity is a reformulation of [[class field theory|BEKCEYHCHX]] in terms of [[Hecke characters|R65XKHN4D1]].
The [[modularity theorem|CS1NVJBZMH]] for elliptic curves over $\Q$ and [[Serre's modularity conjecture|RWXXNZNQYJ]] are proved cases for $n = 2$ and $F = \Q$.
The local Langlands correspondence for $\mathrm{GL}_n$ over a $p$-adic field was proved by Harris and Taylor and, independently, by Henniart.
In general, both reciprocity and functoriality remain open.

## Explanations

### L1 human

The Langlands program predicts a dictionary between two worlds.
On one side are symmetries of the solutions of polynomial equations, recorded by Galois groups and their representations.
On the other side are highly symmetric functions from analysis, the automorphic forms.
The dictionary says that the counts of solutions modulo primes, on the first side, match the coefficients of functions on the second side.
Proved pieces of this dictionary, such as the modularity of elliptic curves, led to the proof of Fermat's Last Theorem.
</document>

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# Schauder's theorem on adjoints of compact operators

## Statement

Let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\R$ or $\C$, with [[continuous duals|4A82S5FVRT]] $X'$ and $Y'$.
Let $T \colon X \to Y$ be a [[bounded linear map|03TM135G0N]], and let $T' \colon Y' \to X'$, $T'(g) = g \circ T$, be its [[dual operator|VCHS7PH0NF]].
Then $T$ is a [[compact operator|M2NEP1K5AE]] if and only if $T'$ is a compact operator.

## Remarks

For the forward direction, let $K$ be the closure of the image of the unit ball of $X$; it is a compact metric space.
The functionals in the unit ball of $Y'$, restricted to $K$, form a bounded and equicontinuous family of continuous functions on $K$.
By the Arzelà–Ascoli theorem, every sequence $(g_n)$ in this ball has a subsequence that converges uniformly on $K$, and then $(T' g_n)$ converges in $X'$.
For the converse, apply the forward direction to $T'$: the bidual $T''$ is compact, and $T$ is its restriction to $X$ under the canonical isometric embeddings $X \subseteq X''$ and $Y \subseteq Y''$.
In a Hilbert space $H$, the same theorem says that a bounded operator $T$ is compact if and only if its Hilbert space adjoint $T^*$ is compact.

## Explanations

### L1 human

A compact operator squeezes the unit ball into a set that is almost finite-dimensional.
Every operator $T$ has a partner $T'$ that acts on linear functionals, the dual operator.
The theorem says that $T$ squeezes in this strong way exactly when its partner does.

### L2 human

The theorem lets one move compactness between a space and its dual.
It is a key step in the Riesz–Schauder theory of compact operators: it shows that $T - \lambda$ and $T' - \lambda$ have kernels of the same finite dimension for $\lambda \ne 0$.
It also shows that the compact operators on a Hilbert space form a closed ideal stable under the adjoint, which is why the quotient by them is a C*-algebra.
</document>

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# Weak-type estimate

## Definition

Let $(X, \mu)$ and $(Y, \nu)$ be [[measure spaces|P15VQS5DJ9]] and let $1 \le p \le \infty$ and $1 \le q < \infty$.
Let $T$ be an operator that maps each $f$ in the [[Lp space|SASP14MA1Z]] $L^p(X, \mu)$ to a [[measurable function|YYNCJSR0PJ]] $Tf$ on $Y$.
The operator $T$ is of *weak type $(p, q)$* if there is a constant $C$ such that
$$\nu(\set{y \in Y : \abs{Tf(y)} > \lambda}) \le \left( \frac{C \norm{f}_{L^p}}{\lambda} \right)^q$$
for every $f \in L^p(X, \mu)$ and every $\lambda > 0$.
One also says that $T$ satisfies a weak-type $(p, q)$ estimate.

## Remarks

Equivalently, $\norm{Tf}_{q, \infty} \le C \norm{f}_p$, where $\norm{\cdot}_{q,\infty}$ is the quasinorm of the [[weak Lp space|H4C888M5AZ]] $L^{q,\infty}(\nu)$.
The operator is of *strong type $(p, q)$* if $\norm{Tf}_q \le C \norm{f}_p$.
By Chebyshev's inequality $\nu(\set{\abs{g} > \lambda}) \le \lambda^{-q} \norm{g}_q^q$, strong type $(p, q)$ implies weak type $(p, q)$ with the same constant.
By convention, weak type $(p, \infty)$ means strong type $(p, \infty)$.

Example: let $X = (0, 1)$ and $Y = (0, \infty)$ with Lebesgue measure, and let $Tf(y) = c_f / y$ with $c_f = \int_0^1 f$.
Then $\set{y : \abs{Tf(y)} > \lambda} = (0, \abs{c_f}/\lambda)$, of measure $\abs{c_f}/\lambda \le \norm{f}_1/\lambda$.
So $T$ is of weak type $(1, 1)$ with $C = 1$, but not of strong type $(1, 1)$, because $\int_0^\infty dy/y = \infty$ when $c_f \neq 0$.

Non-example: on the same spaces, with $Y = (0, 1)$, let $Tf(y) = c_f / y^2$.
For $f = 1$ one has $c_f = 1$ and $\set{Tf > \lambda} = (0, \lambda^{-1/2})$ for $\lambda > 1$.
The ratio $\lambda^{-1/2} / (C/\lambda) = \lambda^{1/2}/C$ is unbounded as $\lambda \to \infty$, so $T$ is not of weak type $(1, 1)$.
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# Dilogarithm

## Definition

The *dilogarithm* is the function $\operatorname{Li}_2 \colon \C \setminus (1, \infty) \to \C$ given by
$$\operatorname{Li}_2(z) = -\int_0^z \frac{\log(1 - u)}{u}\, du.$$
Here $\log$ is the principal branch of the [[complex logarithm|02QGGD5G87]].
The path of integration lies in $\C \setminus [1, \infty)$, except possibly for its endpoint $z = 1$.
The function is holomorphic on $\C \setminus [1, \infty)$ and continuous at $z = 1$.

## Remarks

For $\abs{z} \le 1$ it agrees with the series $\sum_{k \ge 1} \frac{z^k}{k^2}$, so it is the [[polylogarithm|96CHDZKGQ0]] $\operatorname{Li}_2$.
The integral does not depend on the path, because $\C \setminus [1, \infty)$ is simply connected and the integrand is holomorphic there (the singularity at $u = 0$ is removable).

Example: $\operatorname{Li}_2(1) = \sum_{k \ge 1} k^{-2} = \frac{\pi^2}{6}$, and $\operatorname{Li}_2(-1) = \sum_{k \ge 1} \frac{(-1)^k}{k^2} = -\frac{\pi^2}{12}$.
Euler's reflection formula $\operatorname{Li}_2(z) + \operatorname{Li}_2(1 - z) = \frac{\pi^2}{6} - \log z \log(1 - z)$ at $z = \frac12$ gives $\operatorname{Li}_2(\frac12) = \frac{\pi^2}{12} - \frac{(\ln 2)^2}{2}$.

Non-example of a valid argument: for real $x > 1$ the point $x$ lies on the cut $[1, \infty)$.
The limits of $\operatorname{Li}_2$ from above and below the cut differ by $2\pi i \ln x$, so $\operatorname{Li}_2$ has no continuous extension there.
</document>

<document source="https://tamnd.github.io/math2/node/J9ZY817SRC.md" tag="J9ZY817SRC" kind="example">
# Abel equation of the first kind

## Example

Let $f_0, f_1, f_2, f_3$ be continuous functions on an open interval $I$, with $f_3$ not identically zero.
The *Abel equation of the first kind* is the first-order [[ordinary differential equation|0P97V20RW7]]
$$y' = f_3(x)\, y^3 + f_2(x)\, y^2 + f_1(x)\, y + f_0(x).$$
Its right side is a cubic polynomial in $y$.
When $f_3 = 0$ it would be a [[Riccati equation|0E1ZZBHZ3Z]], so the Abel equation is the next case after the Riccati equation.

Instance: $y' = y^3$, with $f_3 = 1$ and $f_0 = f_1 = f_2 = 0$.
It is [[separable|0V4E623THY]], and its nonzero solutions are $y = \pm (C - 2x)^{-1/2}$ for $x < C/2$.
Check: for $y = (C - 2x)^{-1/2}$, $y' = -\tfrac12 (C - 2x)^{-3/2} \cdot (-2) = (C - 2x)^{-3/2} = y^3$.
The solution tends to infinity as $x \to C/2$, so it exists only on the interval $(-\infty, C/2)$, which is bounded on the right.

## Explanations

### L1 human

The Abel equation of the first kind says that the slope of $y$ is a cubic polynomial in $y$ whose coefficients depend on $x$.
Unlike linear equations and many Riccati equations, it usually has no solution in closed form.
The example $y' = y^3$ shows a typical nonlinear effect: solutions can blow up in finite time.
</document>

<document source="https://tamnd.github.io/math2/node/JA29ETCM02.md" tag="JA29ETCM02" kind="concept">
# Riemann curvature tensor

## Summary

The Riemann curvature tensor of an [[affine connection|JX65KSVHSJ]], usually the Levi-Civita connection of a Riemannian metric, measures how far second covariant derivatives fail to commute.
For the Levi-Civita connection, it vanishes exactly when the manifold is locally isometric to Euclidean space.
Sectional, Ricci and scalar curvature are all obtained from it.
In general relativity it describes the tidal effects of gravity.

## Explanations

### L1 human

On a flat sheet of paper, carrying an arrow around a closed loop without turning it brings the arrow back unchanged.
On a sphere it does not: the arrow comes back rotated.
The Riemann curvature tensor measures this rotation for very small loops, in every pair of directions at every point.
For a round sphere of radius $r$ the effect is the same in every direction and is proportional to $1/r^2$.
For a cylinder it is zero, because a cylinder can be unrolled flat.
</document>

<document source="https://tamnd.github.io/math2/node/JA3QXVF2B8.md" tag="JA3QXVF2B8" kind="concept">
# Machine epsilon

## Summary

Machine epsilon measures the relative precision of a system of [[floating-point arithmetic|RTKRSC3KPH]].
It is the gap between $1$ and the next larger floating-point number, or, in another common convention, half of this gap.
The half-gap, called the unit roundoff, bounds the relative error of rounding a real number to the nearest floating-point number.
Almost every error bound in numerical analysis is stated in terms of it.

## Explanations

### L1 human

A computer stores numbers with a fixed number of binary digits.
Machine epsilon tells how fine this grid of numbers is near $1$.
In double precision it is $2^{-52} \approx 2.2 \times 10^{-16}$, so a computer can tell $1$ from $1 + 2^{-52}$ but rounds $1 + 2^{-54}$ back to $1$.
Since the grid scales with the size of the number, the same relative accuracy of about $16$ decimal digits holds for big and small numbers alike.
</document>

<document source="https://tamnd.github.io/math2/node/JA83J040Q2.md" tag="JA83J040Q2" kind="definition">
# Nilmanifold

## Definition

Let $N$ be a connected [[nilpotent Lie group|DK87YN458D]] and let $H$ be a closed subgroup of $N$.
The [[homogeneous space|85TWR6E75P]] $N/H$ is a *nilmanifold*.
When $N$ is simply connected and $H = \Gamma$ is a [[lattice|BTYF5Y5PFY]], a discrete subgroup with $N/\Gamma$ compact, the space $N/\Gamma$ is a *compact nilmanifold*.

## Remarks

Example: the [[Heisenberg group|ZZYCV5EX9J]] $N$ of real matrices $\begin{pmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{pmatrix}$ and the subgroup $\Gamma$ of those with $x, y, z \in \Z$.
The group $N$ is nilpotent, since its commutator subgroup is the center $\set{x = y = 0}$.
It is diffeomorphic to $\R^3$, so it is simply connected.
The subgroup $\Gamma$ is discrete, and each element of $N$ is an element with $x, y, z \in [0, 1)$ times an element of $\Gamma$, so $N/\Gamma$ is compact.

Non-example for the compact case: let $\Z$ be the integer points on the first axis of $\R^2$.
Then $\R^2/\Z$ is a nilmanifold, but $\Z$ is not a lattice in $\R^2$, and the cylinder $\R^2/\Z$ is not compact.

Each nilmanifold is a [[solvmanifold|HBDEPTRF11]].
Many texts use the word nilmanifold only for compact nilmanifolds.
By a theorem of Malcev, a simply connected nilpotent Lie group has a lattice exactly when its Lie algebra has a basis with rational structure constants.
</document>

<document source="https://tamnd.github.io/math2/node/JA95NH206E.md" tag="JA95NH206E" kind="concept">
# Itô calculus

## Summary

Itô calculus is the calculus of integrals with respect to [[Brownian motion|22735FGJV6]] and, more generally, [[semimartingales|4EJB6E2AZK]].
Its central object is the Itô integral $\int_0^t H_s \, dB_s$ of an adapted process $H$.
Its chain rule is [[Itô's lemma|2J1RVQB0CM]], which has an extra second-order term coming from the [[quadratic variation|X4DHMZD5JA]] of the path.
It is the language of stochastic differential equations and of continuous-time models in finance and physics.

## Explanations

### L1 human

A Brownian path is too rough to have a derivative, so the usual integral $\int H \, dB$ cannot be defined path by path.
Itô's idea is to approximate by sums $\sum H_{t_i} (B_{t_{i+1}} - B_{t_i})$ that always use the value of $H$ at the left end of each interval.
Such a sum is a fair bet: the stake is fixed before the random increment is seen.
The limit of these sums is the Itô integral.
For example, $\int_0^t B_s \, dB_s = \tfrac12 (B_t^2 - t)$, and the term $-t/2$ shows that the rules of ordinary calculus change.
</document>

<document source="https://tamnd.github.io/math2/node/JAFPWZDM95.md" tag="JAFPWZDM95" kind="concept">
# Isotonic regression

## Summary

Isotonic regression fits a non-decreasing sequence to data by [[least squares|0BYTQ4X1HC]].
It is used when a response is known to increase with a covariate, as for dose and response.
No formula for the trend is assumed.
The fit is the projection of the data onto a closed convex cone, so it exists and is unique.
It can be computed quickly by the pool adjacent violators algorithm.

## Explanations

### L1 human

Suppose a measurement should grow with the dose, but noise makes some values go down.
Isotonic regression finds the staircase that never goes down and stays as close as possible to the data.
Whenever a value is higher than the next one, the two merge into a block that takes their average.
Blocks keep merging in this way until no drop remains.
For the data $1, 3, 2$, the drop from $3$ to $2$ is replaced by $2.5, 2.5$, giving $1, 2.5, 2.5$.
</document>

<document source="https://tamnd.github.io/math2/node/JB02WZSDDW.md" tag="JB02WZSDDW" kind="definition">
# Hilbert dimension

## Definition

Let $H$ be a real or complex [[Hilbert space|HM2HHE3J51]].
An *orthonormal basis* of $H$ is an [[orthonormal family|0RW9TNNT87]] $(e_i)_{i \in I}$ in $H$ that is maximal: no nonzero vector of $H$ is orthogonal to all $e_i$.
All orthonormal bases of $H$ have the same [[cardinality|0G0ASKNA6Z]].
The *Hilbert dimension* of $H$ is this cardinality $\abs{I}$.

## Remarks

Example: in $\ell^2$, the unit vectors $e_n$, $n \ge 1$, form an orthonormal basis.
They are orthonormal, and if $\ip{x}{e_n} = x_n = 0$ for all $n$ then $x = 0$.
So the Hilbert dimension of $\ell^2$ is $\aleph_0$, the cardinality of $\N$.

Non-example: the family $(e_n)_{n \ge 2}$ in $\ell^2$ is orthonormal but not maximal, since $e_1$ is orthogonal to all of it.
So it is not an orthonormal basis, though its cardinality is also $\aleph_0$.

For every orthonormal basis, each $x \in H$ is the sum $x = \sum_{i \in I} \ip{x}{e_i} e_i$, with only countably many nonzero terms.
For finite-dimensional $H$, the Hilbert dimension is the usual dimension.
For $H = L^2([0, 2\pi])$, the functions $e^{inx} / \sqrt{2\pi}$, $n \in \Z$, form an orthonormal basis, so $L^2([0, 2\pi])$ and $\ell^2$ are isometrically isomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/JB7S6WPZ77.md" tag="JB7S6WPZ77" kind="concept">
# Continuous functional calculus

## Summary

Let $a$ be a normal element of a C*-algebra, and let $f$ be a continuous function on the [[spectrum|EH0CZ19RKP]] of $a$.
The continuous functional calculus gives a meaning to $f(a)$.
It extends the evaluation of polynomials at $a$ to all continuous functions, and it respects sums, products, adjoints and norms.
It is built from the [[commutative Gelfand–Naimark theorem|NT564Y315M]].
It is used to define square roots, absolute values, positive and negative parts, and exponentials of operators.

## Explanations

### L1 human

For a diagonal matrix, applying a function means applying it to each diagonal entry.
A matrix that can be diagonalized by a unitary change of basis can be handled the same way: diagonalize, apply the function to the eigenvalues, and change back.
The continuous functional calculus does this for normal elements of any C\*-algebra, such as normal operators on an infinite-dimensional Hilbert space.
For example, a positive operator has a unique positive square root, given by $f(t) = \sqrt{t}$.
</document>

<document source="https://tamnd.github.io/math2/node/JB7ZBFBA9C.md" tag="JB7ZBFBA9C" kind="concept">
# Exact sequence

## Summary

An exact sequence is a chain of modules and linear maps in which the image of each map equals the [[kernel|0W88CEGQP1]] of the next.
Short exact sequences $0 \to A \to B \to C \to 0$ describe $B$ as built from a submodule $A$ and a quotient $C$.
Exact sequences are the language of homological algebra: one asks which functors, such as $\Hom$ and the [[tensor product|F0HQXQ0HCD]], preserve them.
The same notion makes sense for groups and in any abelian category.

## Explanations

### L1 human

An exact sequence is a row of modules connected by maps, where what one map produces is exactly what the next map kills.
So nothing is lost and nothing extra appears between two steps.
The basic example is $0 \to \Z \xrightarrow{2} \Z \to \Z/2\Z \to 0$.
Here multiplication by $2$ is injective, its image $2\Z$ is exactly the set of integers that become $0$ modulo $2$, and reduction modulo $2$ is onto.
</document>

<document source="https://tamnd.github.io/math2/node/JB8GQBVW6Q.md" tag="JB8GQBVW6Q" kind="statement">
# Kojima–Schur theorem

## Statement

Let $A = (a_{nk})_{n, k \ge 0}$ be an infinite complex matrix.
Every [[convergent sequence|0AZ3R52593]] $s$ is [[$A$-summable|RKFB6AHPBE]], that is, $(As)_n = \sum_k a_{nk} s_k$ is defined for all $n$ and converges as $n \to \infty$, if and only if:

1. $\sup_{n} \sum_{k=0}^{\infty} \abs{a_{nk}} < \infty$;
2. for every $k$, the limit $\alpha_k = \lim_{n \to \infty} a_{nk}$ exists;
3. the limit $\alpha = \lim_{n \to \infty} \sum_{k=0}^{\infty} a_{nk}$ exists.

In that case $\sum_k \abs{\alpha_k} < \infty$, and for every convergent $s$ with limit $L$,
$$\lim_{n \to \infty} (As)_n = \alpha L + \sum_{k=0}^{\infty} \alpha_k (s_k - L).$$

## Remarks

Such matrices are called *conservative*.
Conditions 2 and 3 are needed because the unit sequences $e_k$ and the constant sequence $1$ are convergent; condition 1 follows from the uniform boundedness principle.
The limit formula comes from writing $s = L \cdot \mathbf{1} + (s - L \cdot \mathbf{1})$, where the second sequence tends to $0$.
Example: let $a_{nk} = 1$ if $k = 0$, $a_{nk} = \frac{1}{n+1}$ if $1 \le k \le n$, and $a_{nk} = 0$ otherwise.
Then $\alpha_0 = 1$, $\alpha_k = 0$ for $k \ge 1$ and $\alpha = 2$.
The formula gives $\lim (As)_n = 2L + (s_0 - L) = s_0 + L$, which one can also check directly.
With $\alpha_k = 0$ for all $k$ and $\alpha = 1$ the limit formula gives $L$, and one recovers the [[Silverman–Toeplitz theorem|S4J7SRPXMZ]].

## Explanations

### L1 human

The theorem tells exactly which infinite matrices turn every convergent sequence into a convergent sequence.
It also gives the new limit in terms of the old limit and of the limits of the columns.

### L2 human

Conservative matrices are exactly the bounded linear maps from the space of convergent sequences to itself that are given by matrices.
The new limit is a continuous linear functional of $s$, and the formula above is its general form.
Regular matrices are the conservative ones for which this functional is the ordinary limit.
</document>

<document source="https://tamnd.github.io/math2/node/JB8QMT8DCH.md" tag="JB8QMT8DCH" kind="concept">
# Inverse Fourier transform

## Summary

The inverse Fourier transform takes a function of frequency and builds a function of position by superposing the waves $e^{2\pi i x \cdot \xi}$ with these amplitudes.
It has the same form as the [[Fourier transform|DPVM34ADZV]], with the opposite sign in the exponent.
The [[Fourier inversion theorem|T4Z8YQZKZV]] says that it undoes the Fourier transform, so a function can be recovered from its frequency content.

## Explanations

### L1 human

The Fourier transform measures how much of each frequency a signal contains.
The inverse Fourier transform goes back: given the amount of each frequency, it adds up all the corresponding waves to rebuild the signal.
The two operations look almost the same, with only a sign changed in the exponent.
For example, the bell curve $e^{-\pi \abs{\xi}^2}$ is sent to the same bell curve $e^{-\pi \abs{x}^2}$.
</document>

<document source="https://tamnd.github.io/math2/node/JB9HJ9CPEN.md" tag="JB9HJ9CPEN" kind="definition">
# Dense order

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]], and write $x < y$ for $x \le y$ and $x \ne y$.
The order is *dense* when, for all $x, y \in P$ with $x < y$, there is $z \in P$ with $x < z < y$.

## Remarks

The order of $\Q$ is dense: for rationals $x < y$, the rational $z = (x + y)/2$ satisfies $x < z < y$.
The order of $\Z$ is not dense: for $x = 0$ and $y = 1$ there is no integer strictly between.
A dense order has no [[covering pairs|EHXHC4ZNPA]].
A poset with at most one element is dense, since there is no pair $x < y$ to check.
The term is used mostly for [[total orders|0316AYH1SC]], where it is a basic property in the study of [[order types|28PE4GRFD7]].
The order topology of a dense total order with more than one element has no isolated points.

## Explanations

### L1 human

Dense means that any two different elements have room between them.
The rationals are dense; the integers are not.
</document>

<document source="https://tamnd.github.io/math2/node/JBJJHSW72S.md" tag="JBJJHSW72S" kind="definition">
# Hybrid system

## Definition

Several formal models of hybrid systems are in use; a common one is the hybrid automaton.
A *hybrid automaton* with continuous state space $\R^n$ consists of the following data.

1. A finite set $Q$ of discrete states, called modes.
2. For each mode $q \in Q$, a [[vector field|0299VCW1TH]] $f_q \colon \R^n \to \R^n$ and a set $D_q \subseteq \R^n$, the domain of $q$.
3. A set $E \subseteq Q \times Q$ of edges, that is, allowed mode switches.
4. For each edge $e \in E$, a guard $G_e \subseteq \R^n$ and a reset map $R_e$ that gives a nonempty set $R_e(x) \subseteq \R^n$ for each $x \in G_e$.
5. A set $\mathrm{Init} \subseteq Q \times \R^n$ of initial states.

An *execution* is a finite or infinite sequence of intervals $[\tau_0, \tau_0'], [\tau_1, \tau_1'], \ldots$ with $\tau_i' = \tau_{i+1}$.
On the interval $[\tau_i, \tau_i']$ it has a mode $q_i$ and a curve $x_i$ that solves $x' = f_{q_i}(x)$ and stays in $D_{q_i}$.
The start satisfies $(q_0, x_0(\tau_0)) \in \mathrm{Init}$.
At each switch, $e = (q_i, q_{i+1}) \in E$, $x_i(\tau_i') \in G_e$ and $x_{i+1}(\tau_{i+1}) \in R_e(x_i(\tau_i'))$.
A *hybrid system* is a system modelled by such data, or by an equivalent formalism with flows and jumps.

## Remarks

The discrete part alone is a finite-state machine, and the continuous part in one mode is an [[ordinary differential equation|0NT64JV96A]].
Example: the bouncing ball has one mode and the state $x = (h, v)$ of height and velocity.
Its vector field is $f(h, v) = (v, -g)$ with $g > 0$, and its domain is $h \ge 0$.
It has one edge from the mode to itself, with guard $\set{h = 0, v \le 0}$ and reset $R(0, v) = \set{(0, -c v)}$, where $0 \le c < 1$.
Between impacts the ball follows a parabola, and each impact multiplies its speed by $c$.
For $0 < c < 1$ the impacts accumulate at a finite time; this Zeno behaviour has no analogue for ordinary differential equations.
Non-example: data with infinitely many modes, one for each integer, fail clause 1, which asks for a finite set of modes; such models need a more general framework.
</document>

<document source="https://tamnd.github.io/math2/node/JBSSXP4FFY.md" tag="JBSSXP4FFY" kind="example">
# Category of modules

## Example

Let $R$ be a [[ring|0B1261B5ED]] with identity.
The *category of left $R$-modules* $R\text{-}\mathbf{Mod}$ has the left [[$R$-modules|0NZ8JA25WK]] as objects and the $R$-linear maps as morphisms, with composition of maps.
A map $f \colon M \to N$ is $R$-linear when $f(x + y) = f(x) + f(y)$ and $f(rx) = r f(x)$ for all $x, y \in M$ and $r \in R$.
It is an [[abelian category|Y9CZ3MTNF1]].
Each $\Hom_R(M, N)$ is an abelian group under pointwise addition, and composition is bilinear.
The zero module is a zero object, and the direct sum $M \oplus N$ is both product and coproduct.
The kernel of $f \colon M \to N$ is the submodule $\set{x \in M : f(x) = 0}$ and the cokernel is $N / f(M)$.
As in abelian groups, each injective linear map is the kernel of its cokernel and each surjective linear map is the cokernel of its kernel.

## Remarks

For $R = \Z$ one gets the [[category of abelian groups|99ZVSM254K]], and for a field $R = k$ the category of vector spaces over $k$.
The module $R$ itself is a [[generator|CK08JW32WB]] and a [[projective object|YXMXAGVC1Q]], and the category has [[enough projectives|J2TX4TDPCX]] since every module is a quotient of a free module.
It also has enough injective objects, which is the basis for derived functors such as $\operatorname{Ext}$ and $\operatorname{Tor}$.
Two rings whose module categories are equivalent are called Morita equivalent; for example $R$ and the ring of $n \times n$ matrices over $R$ are Morita equivalent.

## Explanations

### L1 human

The category of modules over a ring gathers all modules and all linear maps between them.
It behaves much like linear algebra over a field, except that modules need not have bases.
For example, over $R = \Z$ the module $\Z/2\Z$ is not free, yet it is the cokernel of the map $\Z \to \Z$ given by multiplication by $2$.
</document>

<document source="https://tamnd.github.io/math2/node/JBVKKDXETM.md" tag="JBVKKDXETM" kind="definition">
# Equivalence of categories

## Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
A [[functor|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathcal{D}$ is an *equivalence of categories* if there are a functor $G \colon \mathcal{D} \to \mathcal{C}$ and [[natural isomorphisms|KYCPCC57FT]] $\eta \colon 1_{\mathcal{C}} \Rightarrow G \circ F$ and $\varepsilon \colon F \circ G \Rightarrow 1_{\mathcal{D}}$.
Then $G$ is a *quasi-inverse* of $F$, and $\mathcal{C}$ and $\mathcal{D}$ are *equivalent*, written $\mathcal{C} \simeq \mathcal{D}$.

## Remarks

If moreover $G \circ F = 1_{\mathcal{C}}$ and $F \circ G = 1_{\mathcal{D}}$, then $F$ is an *isomorphism of categories*, a much stronger condition.
Example: let $K$ be a field and let $\mathbf{Mat}_K$ have the natural numbers as objects, the $n \times m$ matrices over $K$ as morphisms $m \to n$, and matrix multiplication as composition.
The functor $\mathbf{Mat}_K \to$ (finite-dimensional [[vector spaces|000AYPA9AJ]] over $K$) sending $n$ to $K^n$ and a matrix to its linear map is an equivalence.
A quasi-inverse chooses a basis of each space $V$ and sends $V$ to $\dim V$ and a linear map to its matrix in the chosen bases.
Non-example: let $\mathbf{1}$ have one object and one morphism, and let $\mathbf{2}$ have two objects and only identity morphisms.
No functor $F \colon \mathbf{1} \to \mathbf{2}$ is an equivalence.
Both objects $X$ of $\mathbf{2}$ would need an isomorphism $F(G(X)) \cong X$.
But $F(G(X))$ is the single object in the image of $F$, and the two objects of $\mathbf{2}$ are not isomorphic.
Equivalence of categories is reflexive, symmetric and transitive.
</document>

<document source="https://tamnd.github.io/math2/node/JBWB5H8JZG.md" tag="JBWB5H8JZG" kind="statement">
# Riesz's subsequence theorem

## Statement

Let $(X, \Sigma, \mu)$ be a measure space, and let $f_1, f_2, \ldots$ and $f$ be measurable functions from $X$ to $\R$.
If $f_n \to f$ [[in measure|G5WT6QN8WR]], then there is a [[subsequence|0F1PR74BRX]] $(f_{n_k})$ such that $f_{n_k}(x) \to f(x)$ for [[almost every|RARTMA0ZEC]] $x \in X$.

## Remarks

Choose $n_1 < n_2 < \cdots$ with $\mu(E_k) < 2^{-k}$, where $E_k = \set{x : \abs{f_{n_k}(x) - f(x)} \ge 2^{-k}}$.
Let $E = \bigcap_{j} \bigcup_{k \ge j} E_k$; then $\mu(E) \le \sum_{k \ge j} 2^{-k} = 2^{1-j}$ for every $j$, so $\mu(E) = 0$.
For $x \notin E$, one has $\abs{f_{n_k}(x) - f(x)} < 2^{-k}$ for all large $k$, so $f_{n_k}(x) \to f(x)$.
Example: the sliding indicator functions $f_{2^k + j} = \mathbf{1}_{[j 2^{-k}, (j+1) 2^{-k}]}$ on $[0, 1]$ converge to $0$ in measure but at no point; the subsequence $f_{2^k}$, the indicators of $[0, 2^{-k}]$, converges to $0$ at every $x \ne 0$.
The passage to a subsequence is necessary, as this example shows.

## Explanations

### L1 human

Convergence in measure is weak: it allows the sequence to fail to converge at every point.
This theorem says that one can always pick out a subsequence that does converge, except on a set of measure zero.

### L2 human

The result is a standard tool to pass from integral estimates to pointwise statements.
For example, if $f_n \to f$ in $L^p$, then $f_n \to f$ in measure, so a subsequence converges to $f$ almost everywhere; this is a step in the proof that $L^p$ is complete.
On a space of finite measure, it gives a characterisation of convergence in measure.
There, $f_n \to f$ in measure if and only if every subsequence has a further subsequence converging to $f$ almost everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/JBZMRZY4YF.md" tag="JBZMRZY4YF" kind="definition">
# Field homomorphism

## Definition

Let $K$ and $L$ be [[fields|0QA8CHH8MG]].
A field homomorphism is a map $f \colon K \to L$ such that:

1. $f(a + b) = f(a) + f(b)$ for all $a, b \in K$;
2. $f(ab) = f(a) f(b)$ for all $a, b \in K$;
3. $f(1) = 1$.

## Remarks

So a field homomorphism is a [[ring homomorphism|0GGE0Z0JQ1]] between two fields.
Example: complex conjugation $f(a + bi) = a - bi$ on $\C$ satisfies the three clauses, since conjugation respects sums and products and fixes $1$.
Non-example: $f(a) = 2a$ on $\Q$ is additive, but $f(1) = 2 \ne 1$ and $f(1 \cdot 1) = 2 \ne 4 = f(1) f(1)$.
Every field homomorphism is injective, by [[ring homomorphisms from a field are injective|09N7DZ7SDB]].
So $f(K)$ is a [[subfield|07QY5RPFBR]] of $L$ isomorphic to $K$, and $f$ is also called an embedding of $K$ into $L$.
Since $f(n \cdot 1) = n \cdot 1$ and $f$ is injective, $K$ and $L$ have the same [[characteristic|1ZB13N78XW]].
A bijective field homomorphism is a field isomorphism.

## Explanations

### L1 human

A field homomorphism is a map that keeps sums, products and the number $1$.
Because a field has inverses, such a map cannot collapse anything.
</document>

<document source="https://tamnd.github.io/math2/node/JC0SZT6453.md" tag="JC0SZT6453" kind="definition">
# Young measure

## Definition

Let $n, m \ge 1$, and let $\Omega \subseteq \R^n$ be a Lebesgue measurable set.
Let $C_0(\R^m)$ be the space of continuous functions $\varphi \colon \R^m \to \R$ that tend to $0$ at infinity.
A *Young measure* on $\Omega$ with values in $\R^m$ is a family $(\nu_x)_{x \in \Omega}$ of Borel [[probability measures|059TJN0NZ1]] on $\R^m$.
It must satisfy one condition: for every $\varphi \in C_0(\R^m)$ the function
$$x \mapsto \int_{\R^m} \varphi \, d\nu_x$$
is Lebesgue [[measurable|YYNCJSR0PJ]] on $\Omega$.

## Remarks

A sequence of measurable $u_k \colon \Omega \to \R^m$ *generates* $(\nu_x)$ if $\varphi(u_k) \to \int \varphi \, d\nu_x$ weakly-$*$ in $L^\infty(\Omega)$ for every $\varphi \in C_0(\R^m)$.
The fundamental theorem on Young measures says that every sequence bounded in $L^\infty(\Omega; \R^m)$ has a subsequence that generates a Young measure.
Example: a measurable $u \colon \Omega \to \R^m$ gives the Young measure $\nu_x = \delta_{u(x)}$, since $\int \varphi \, d\nu_x = \varphi(u(x))$ is measurable.
Example: $u_k(x) = \sgn(\sin(2 \pi k x))$ on $(0,1)$ generates the constant family $\nu_x = \frac12 (\delta_{-1} + \delta_1)$.
Non-example: if $g \colon \Omega \to \R^m$ is not measurable, then $\nu_x = \delta_{g(x)}$ is not a Young measure.
Some open $U$ has a non-measurable preimage, and $1_U$ is an increasing limit of functions $\varphi_j \in C_0(\R^m)$, so some $\varphi_j \circ g$ is not measurable.
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<document source="https://tamnd.github.io/math2/node/JC4J24MF93.md" tag="JC4J24MF93" kind="concept">
# Order dimension

## Summary

The order dimension of a [[partially ordered set|0QV2JS9RJ3]] is the least number of [[linear extensions|W8DZ5KGGHN]] whose intersection is the given order.
It measures how far the order is from being total: total orders have dimension $1$.
Equivalently, it is the least $d$ such that the poset embeds in a product of $d$ chains with the [[product order|K0AWJK6NZH]].

## Explanations

### L1 human

Imagine ranking a set of objects by several criteria, such as price and quality.
Say that one object beats another when it is better in every criterion.
The dimension of a partial order is the least number of criteria that produce exactly this order.
For points in the plane with $(a, b) \le (c, d)$ when $a \le c$ and $b \le d$, two criteria suffice, so the dimension is at most $2$.
</document>

<document source="https://tamnd.github.io/math2/node/JC9R1073M9.md" tag="JC9R1073M9" kind="definition">
# Flag manifold

## Definition

Let $G$ be a connected complex [[semisimple Lie group|XJS07AJ50Y]].
Let $B$ be a [[Borel subgroup|5JSYA1JNCC]] of $G$, a maximal connected solvable closed complex subgroup.
Let $P$ be a closed complex subgroup of $G$ with $B \subseteq P$, a parabolic subgroup.
The *flag manifold* of $G$ and $P$ is the [[homogeneous space|85TWR6E75P]] $G/P$, a complex manifold with the holomorphic action of $G$ by left translation.
When $P = B$, the space $G/B$ is the *complete* (or full) flag manifold of $G$.

## Remarks

Example: $G = \mathrm{SL}_n(\C)$ with $B$ the upper triangular matrices of determinant $1$.
Then $G$ acts transitively on complete flags $V_1 \subset \cdots \subset V_{n-1}$ in $\C^n$ with $\dim V_i = i$, and $B$ is the stabilizer of the standard flag $V_i = \operatorname{span}(e_1, \dots, e_i)$.
So $G/B$ is the space of complete flags, of complex dimension $n(n-1)/2$.
The subgroup $P$ of matrices with first column in $\C e_1$ contains $B$, and $G/P$ is the projective space $\C P^{n-1}$ of lines in $\C^n$.

Non-example: $G = \mathrm{SL}_2(\C)$ and $H$ the diagonal subgroup.
Then $H$ is closed, but it does not contain any Borel subgroup, since a Borel subgroup of $\mathrm{SL}_2(\C)$ has complex dimension $2$ and $H$ has dimension $1$.
The quotient $G/H$ is not compact, while every flag manifold is compact.

A flag manifold $G/P$ is compact, simply connected and projective algebraic.
For a maximal compact subgroup $K$ of $G$, the group $K$ acts transitively on $G/P$, so $G/P = K/(K \cap P)$.
For $K$ compact connected and $T$ a maximal torus, $K/T$ is the complete flag manifold of the complexification of $K$.
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<document source="https://tamnd.github.io/math2/node/JCCTFJV9CN.md" tag="JCCTFJV9CN" kind="definition">
# Homotopy type theory

## Definition

Start from [[intuitionistic type theory|KCV8RKBATN]] with intensional identity types, dependent function and pair types, and a hierarchy of universes.
*Homotopy type theory* is this theory extended by two principles.

1. The *univalence axiom*: for each universe $U$ and all types $A, B : U$, the canonical map $\mathrm{idtoeqv} \colon (A =_U B) \to (A \simeq B)$ is an equivalence.
2. *Higher inductive types*: types given by generators for points and also for paths. An example is the circle $S^1$, generated by a point $\mathrm{base} : S^1$ and a path $\mathrm{loop} : \mathrm{base} =_{S^1} \mathrm{base}$.

Here $A \simeq B$ is the type of equivalences, that is, of functions $f \colon A \to B$ that have a two-sided inverse up to identification.
The theory is read homotopically.
A type is a space, a term $a : A$ is a point, and a term $p : a =_A b$ is a path from $a$ to $b$.

## Remarks

Example: in homotopy type theory one proves that the type of loops $\mathrm{base} =_{S^1} \mathrm{base}$ is equivalent to the integers $\Z$.
This matches the classical fact that the fundamental group of the circle is $\Z$.
Non-example: intuitionistic type theory with extensional identity types, where any two terms of $a =_A b$ are equal, is not compatible with univalence.
The type $\mathbf{2}$ of booleans has two self-equivalences, the identity and the swap.
With univalence they give two paths $\mathbf{2} = \mathbf{2}$, and extensional identity types make these paths equal.
Then the identity and the swap would be equal, which is false.
Univalence implies function extensionality: two [[functions|0XHDTE5YMZ]] that agree at each point are equal.
</document>

<document source="https://tamnd.github.io/math2/node/JCG0MRFGC7.md" tag="JCG0MRFGC7" kind="definition">
# Clean ring

## Definition

Let $R$ be a [[ring|0894J980RK]] with unit $1$.
An element $a \in R$ is *clean* if $a = e + u$ for an [[idempotent|0X4JKZXJ86]] $e$ (so $e^2 = e$) and a [[unit|019NFGFPR5]] $u$ of $R$.
The ring $R$ is *clean* if every element of $R$ is clean.

## Remarks

Example: a local ring $R$, one whose non-units form a two-sided ideal $\mathfrak{m}$.
If $a$ is a unit, then $a = 0 + a$.
If $a \in \mathfrak{m}$, then $a - 1 \notin \mathfrak{m}$, since otherwise $1 \in \mathfrak{m}$; so $a - 1$ is a unit and $a = 1 + (a - 1)$.
Non-example: the ring $\Z$, whose idempotents are $0, 1$ and units are $\pm 1$.
The sums $e + u$ are $-1, 0, 1, 2$, so $3$ is not clean.
Every homomorphic image of a clean ring is clean, since idempotents and units map to idempotents and units.
</document>

<document source="https://tamnd.github.io/math2/node/JCMTFY6HMA.md" tag="JCMTFY6HMA" kind="concept">
# Resolutivity of boundary functions

## Summary

A function on the boundary of a bounded open set is resolutive when the Perron method gives a well-defined harmonic function from it.
This function is then the generalised solution of the [[Dirichlet problem|8D56GQ4AG8]] with that boundary function as data.
The Perron method squeezes the solution between an upper envelope built from [[superharmonic functions|X4V2S56YQQ]] and a lower envelope built from [[subharmonic functions|Z1MHMGFNFW]].
Resolutivity says that the two envelopes meet and give a [[harmonic function|0DEB0JMN0W]].
It is the right notion of "solvable data" for the generalised Dirichlet problem of the [[Perron method|AEZ7PPCNQS]], and it holds for every continuous boundary function.

## Explanations

### L1 human

Imagine trying to fit a soap film to a wire frame whose height is given by a function on the frame.
From above, press down with every "superharmonic lid" that stays above the frame; from below, push up with every "subharmonic floor" that stays below it.
If the lowest lid and the highest floor coincide, the data is resolutive, and the common surface is the answer.
On the unit disc, every continuous boundary function is resolutive, and the answer is its Poisson integral.
A boundary function that is $+\infty$ everywhere is not resolutive, because both envelopes are then $+\infty$.
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<document source="https://tamnd.github.io/math2/node/JCMXX84BWT.md" tag="JCMXX84BWT" kind="definition">
# Relaxation oscillation

## Definition

Let $f, g \colon \R^2 \to \R$ be [[smooth|0V33BNKZ9S]], and for $\varepsilon > 0$ consider the planar slow–fast [[autonomous system|0673D85QKZ]]
$$\varepsilon x' = f(x, y), \qquad y' = g(x, y).$$
Here $x$ is the fast variable and $y$ is the slow variable.
The *critical manifold* is the curve $S = \set{(x, y) : f(x, y) = 0}$.
Its points with $\partial f / \partial x < 0$ form the attracting part of $S$, and its points with $f = \partial f / \partial x = 0$ are the fold points.
A *relaxation oscillation* is a family of [[periodic orbits|0AA97XA70K]] $\Gamma_\varepsilon$, for small $\varepsilon > 0$, such that, as $\varepsilon \to 0$:

1. $\Gamma_\varepsilon$ converges, in the Hausdorff distance, to a closed curve $\Gamma_0$;
2. $\Gamma_0$ is a union of arcs of the attracting part of $S$, traversed in the direction of the slow flow $y' = g(x, y)$;
3. the other pieces of $\Gamma_0$ are horizontal segments $y = \text{const}$ that start at fold points;
4. $\Gamma_0$ contains at least one horizontal segment.

## Remarks

Along the arcs of $S$ the motion takes a time of order $1$, and along the horizontal segments, the jumps, it takes a time of order $\varepsilon$.

Example: the van der Pol equation $u'' - \mu(1 - u^2) u' + u = 0$ with large $\mu$.
With $F(x) = x^3/3 - x$, $\varepsilon = 1/\mu^2$ and a rescaled time, it becomes $\varepsilon x' = y - F(x)$, $y' = -x$.
The critical manifold is the cubic $y = F(x)$; its attracting part is $\abs{x} > 1$ and its fold points are $(\pm 1, \mp 2/3)$.
The limit curve $\Gamma_0$ runs down the right branch from $(2, 2/3)$ to $(1, -2/3)$, jumps to $(-2, -2/3)$, runs up the left branch to $(-1, 2/3)$, and jumps back to $(2, 2/3)$.
The period of the van der Pol oscillation is about $(3 - 2 \ln 2)\, \mu$ for large $\mu$.

Non-example: the harmonic oscillator $x' = y$, $y' = -x$ has periodic orbits, but no small parameter and no fast variable, so it is not of the slow–fast form.

Some authors use the term more loosely, for any oscillation with slow phases and fast jumps.
</document>

<document source="https://tamnd.github.io/math2/node/JCNCD8PSGF.md" tag="JCNCD8PSGF" kind="example">
# Laplace transforms of sine and cosine

## Example

Let $\omega$ be a real number.
The [[Laplace transforms|05XT5AVJBK]] of [[sine and cosine|0WB57D9488]] are
$$\mathcal{L}\{\cos \omega t\}(s) = \frac{s}{s^2 + \omega^2}, \qquad \mathcal{L}\{\sin \omega t\}(s) = \frac{\omega}{s^2 + \omega^2} \quad (\operatorname{Re} s > 0).$$
Check: write $\cos \omega t = \frac{1}{2}(e^{i\omega t} + e^{-i\omega t})$ and $\sin \omega t = \frac{1}{2i}(e^{i\omega t} - e^{-i\omega t})$.
By the [[transform of the exponential|X5JYH26J3M]], $\mathcal{L}\{e^{\pm i\omega t}\}(s) = 1/(s \mp i\omega)$ for $\operatorname{Re} s > 0$.
By [[linearity|SHX9JPKJBE]],
$$\mathcal{L}\{\cos \omega t\}(s) = \frac{1}{2}\left(\frac{1}{s - i\omega} + \frac{1}{s + i\omega}\right) = \frac{s}{s^2 + \omega^2},$$
$$\mathcal{L}\{\sin \omega t\}(s) = \frac{1}{2i}\left(\frac{1}{s - i\omega} - \frac{1}{s + i\omega}\right) = \frac{\omega}{s^2 + \omega^2}.$$
For $\omega \ne 0$ both transforms have poles at $s = \pm i\omega$ on the imaginary axis, which matches the abscissa of convergence $0$ of these bounded, non-decaying functions.

## Explanations

### L1 human

Oscillations of angular frequency $\omega$ show up on the transform side as a pair of poles at $\pm i\omega$.
The cosine starts at $1$ and the sine starts at $0$, which is why the cosine has $s$ in the numerator and the sine has the constant $\omega$.
These two lines of the table are what makes the Laplace transform useful for springs and electric circuits.
</document>

<document source="https://tamnd.github.io/math2/node/JCSAC9Y1CV.md" tag="JCSAC9Y1CV" kind="concept">
# Idempotent semiring

## Summary

An idempotent semiring, also called a dioid, is a [[semiring|D31S3GR51T]] in which $a + a = a$ for every element $a$.
In such a semiring the addition defines a natural partial order, with $a \le b$ when $a + b = b$, and addition becomes the join of this order.
The Boolean semiring, the tropical semirings and the semiring of formal languages are examples.
Idempotent semirings are the algebra of shortest paths, of tropical geometry and of [[Kleene algebras|BK4DDS24XC]].

## Explanations

### L1 human

In ordinary arithmetic, $1 + 1 = 2$.
In an idempotent semiring, adding something to itself changes nothing, like taking the larger of a number and itself.
A good example uses "maximum" as addition and ordinary addition as multiplication: the "sum" of $3$ and $3$ is $3$.
Because of this, the addition behaves like choosing the best of several options, which is why these semirings fit optimisation problems.
</document>

<document source="https://tamnd.github.io/math2/node/JCSR7HJ7PC.md" tag="JCSR7HJ7PC" kind="concept">
# Haar measure on a Lie group

## Summary

On a [[Lie group|7JBRGCVJRB]], the [[Haar measure|VY0HXD3GC4]] can be written down explicitly as the measure defined by a nonzero left-invariant differential form of top degree.
This gives a smooth density in every chart, so integrals over the group can be computed with ordinary calculus.
The left and right Haar measures agree exactly when $\abs{\det \operatorname{Ad}(g)} = 1$ for all $g$, which holds for compact, abelian and semisimple groups.
It is the basic tool for averaging over a group, as in the proof that representations of compact groups are unitarisable.

## Explanations

### L1 human

A Haar measure is a way to measure the size of sets of group elements that does not change when the set is moved by the group.
On a Lie group one gets it by fixing a small volume element at the identity and carrying it to every other point by left multiplication.
On the circle group this gives the arc length $d\theta$.
On the positive real numbers under multiplication it gives $dx/x$, which is unchanged by the substitution $x \mapsto cx$.
</document>

<document source="https://tamnd.github.io/math2/node/JDDH4DAT2W.md" tag="JDDH4DAT2W" kind="statement">
# Symmetry of the Green function

## Statement

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a bounded connected [[open set|02BH3MCBQB]] that has a [[Green's function|M36SD56MWQ]] $G$ for the Laplacian with zero boundary values.
Then
$$G(x, y) = G(y, x) \quad \text{for all } x, y \in \Omega \text{ with } x \ne y.$$

## Remarks

Proof idea for a smooth boundary: fix $x \ne y$ and apply Green's second identity to $v(z) = G(z, x)$ and $w(z) = G(z, y)$ on $\Omega$ with small balls around $x$ and $y$ removed.
Both functions are harmonic there and vanish on $\partial\Omega$, so only the small spheres contribute.
As the radii tend to $0$, the sphere around $x$ gives $G(x, y)$ and the sphere around $y$ gives $G(y, x)$, up to the same constant, and the two must be equal.
For a general domain one approximates it from inside by smooth domains, whose Green's functions increase to $G$.
Example: on the unit disc, $G(z, w) = \log\abs{\frac{1 - \bar w z}{z - w}}$, and this is symmetric because $\abs{1 - \bar w z} = \abs{1 - \bar z w}$.

## Explanations

### L1 human

Put a unit charge at $y$ in a grounded box and measure the potential at $x$.
Then put the charge at $x$ and measure at $y$.
The two numbers are the same.
This reciprocity is the symmetry of the Green's function.

### L2 human

Symmetry means that the Green operator $f \mapsto \int_\Omega G(\cdot, y) f(y) \, dy$ is self-adjoint on $L^2(\Omega)$, which matches the self-adjointness of the Dirichlet Laplacian.
It lets one treat $G(x, \cdot)$ as a function of the second variable, for example to see that $G(x, y)$ is harmonic in $y$ away from $x$.
For operators that are not self-adjoint, the Green's function of the adjoint operator appears instead, and symmetry fails.
</document>

<document source="https://tamnd.github.io/math2/node/JDDXZ89N33.md" tag="JDDXZ89N33" kind="concept">
# Abelian scheme

## Summary

An abelian scheme over a base scheme $S$ is a family of [[abelian varieties|R7W03TXRS9]] parametrized by $S$.
It is a smooth proper [[group scheme|TJ04VQP4NB]] over $S$ with geometrically connected fibres.
Abelian schemes describe abelian varieties with good reduction, and they are the objects classified by moduli spaces of abelian varieties.

## Explanations

### L1 human

An abelian variety is a single geometric object with an addition law.
An abelian scheme lets this object vary with a parameter, like a whole family of elliptic curves.
Over the integers, the parameter is a prime $p$, and the fibre at $p$ is the reduction of the variety modulo $p$.
For example, the curve $y^2 = x^3 - x$ gives an abelian scheme over $\Z[1/2]$, because it stays smooth modulo every odd prime.
</document>

<document source="https://tamnd.github.io/math2/node/JDGQ135YK2.md" tag="JDGQ135YK2" kind="statement">
# Carathéodory's criterion

## Statement

Let $n \ge 1$ and let $\lambda^*$ be Lebesgue [[outer measure|BJ314H17HC]] on [[Euclidean space|0XPR1E85W0]] $\R^n$.
That is, $\lambda^*(A)$ is the infimum of $\sum_k \operatorname{vol}(R_k)$ over all countable covers of $A$ by open boxes $R_k$.
Call $E \subseteq \R^n$ Lebesgue measurable if for every $\varepsilon > 0$ there is an [[open set|02BH3MCBQB]] $U \supseteq E$ with $\lambda^*(U \setminus E) < \varepsilon$.
Then $E$ is Lebesgue measurable if and only if
$$\lambda^*(A) = \lambda^*(A \cap E) + \lambda^*(A \setminus E) \quad \text{for every } A \subseteq \R^n.$$

## Remarks

The inequality $\le$ always holds by subadditivity, so only $\ge$ has content, and only for $A$ with $\lambda^*(A) < \infty$.
If $E$ is Lebesgue measurable, approximate $E$ from outside by an open set, and use that open sets split every $A$ additively.
Conversely, apply the splitting identity to bounded parts of $E$ and to open covers chosen close to $\lambda^*$.
For example, a closed box $E$ splits every $A$ additively, in line with its being measurable.
The criterion is useful because the splitting condition makes sense for any outer measure $\mu^*$ on any set $X$.
For any outer measure, the sets $E$ with $\mu^*(A) = \mu^*(A \cap E) + \mu^*(A \setminus E)$ for all $A \subseteq X$ form a $\sigma$-algebra, and $\mu^*$ restricted to it is a complete measure.

## Explanations

### L1 human

A set is well behaved for length, area or volume exactly when it cuts every other set cleanly.
Cutting cleanly means the outer measure of any set $A$ is the sum of the outer measures of the part of $A$ inside $E$ and the part outside $E$.
This test needs no topology, so it can be used in any setting where an outer measure is available.

### L2 human

In a measure theory course this splitting condition is often taken as the definition of a measurable set.
With it, Carathéodory's construction turns an outer measure into a complete measure; this gives Lebesgue measure, Lebesgue–Stieltjes measures and Hausdorff measures.
The same construction is the heart of the proof of the [[Carathéodory extension theorem|Q8EXG7EKYW]].
The theorem above shows that for $\R^n$ this abstract notion agrees with the geometric one of approximation by open sets.
</document>

<document source="https://tamnd.github.io/math2/node/JDH0THTVGM.md" tag="JDH0THTVGM" kind="concept">
# Householder transformation

## Summary

A Householder transformation is the [[reflection|06BRDF3F0E]] of $\R^n$ or $\C^n$ in a hyperplane through the origin.
Its matrix is $I - 2 v v^*$ for a unit vector $v$ normal to the hyperplane.
It is an [[orthogonal|0W02QESPJQ]] or unitary matrix, and in $\R^n$ it can send any given vector to any other vector of the same length.
This makes it the basic tool for creating zeros in a matrix in a stable way.
It is used in the [[QR decomposition|0ERWG4NXQ9]] and in the reduction to Hessenberg or tridiagonal form.

## Explanations

### L1 human

Think of a mirror through the origin.
The Householder transformation reflects every vector in this mirror: the part along the mirror stays, and the part perpendicular to it flips sign.
By placing the mirror halfway between two vectors of the same length, one can swap them.
For example, the mirror between $(3, 4)$ and $(5, 0)$ in the plane sends $(3, 4)$ to $(5, 0)$, so the second entry becomes zero.
</document>

<document source="https://tamnd.github.io/math2/node/JDM2M1FRY6.md" tag="JDM2M1FRY6" kind="concept">
# Strongly measurable function

## Summary

A strongly measurable function is a function from a measurable space to a Banach space that is a pointwise limit of simple measurable functions.
It is the right notion of measurability for functions with values in an infinite-dimensional space.
The Bochner integral is built for strongly measurable functions, and the Pettis measurability theorem describes them through linear functionals.

## Explanations

### L1 human

A simple function takes only finitely many values, each on a measurable piece of the space.
A strongly measurable function is one that such simple functions can approximate as closely as one wants at every point.
For real-valued functions this is the same as ordinary measurability.
For example, a continuous function $f \colon [0, 1] \to E$ into a Banach space $E$ is strongly measurable.
To see it, cut $[0, 1]$ into $n$ intervals and take on each interval the value of $f$ at its left end.
These step functions converge to $f$ at every point as $n$ grows.
</document>

<document source="https://tamnd.github.io/math2/node/JE1Z920QKT.md" tag="JE1Z920QKT" kind="problem">
# Three-body problem

## Problem

Let $G > 0$ be the gravitational constant and let $m_1, m_2, m_3 > 0$ be masses.
Let the positions $x_1(t), x_2(t), x_3(t) \in \R^3$ satisfy Newton's equations
$$x_i'' = \sum_{j \ne i} \frac{G m_j (x_j - x_i)}{\norm{x_j - x_i}^3}, \qquad i = 1, 2, 3.$$
Given distinct initial positions and initial velocities, describe the solution for all time.

## Remarks

This is the case $n = 3$ of the [[n-body problem|62SNWC5E45]].
Bruns showed in 1887 that the ten classical first integrals are the only independent algebraic ones.
Poincaré found that the problem is not integrable and that its orbits can behave in a very complicated way, near homoclinic orbits; this was the beginning of chaos theory.
Sundman found in 1912 a convergent power series solution for all initial conditions with nonzero angular momentum, but it converges too slowly to be of practical use.

Special periodic solutions exist.
Euler found solutions with the three bodies on a rotating line, and Lagrange found solutions with the three bodies at the vertices of a rotating equilateral triangle.
For three equal masses there is a solution in which the bodies chase each other along a figure-eight curve.
Moore found it numerically in 1993, and Chenciner and Montgomery proved its existence in 2000.

In the planar circular restricted three-body problem one body has negligible mass and moves in the plane of the other two, which move on circles.
In a rotating frame it is a [[Hamiltonian system|0EDKGA2FAK]] with two degrees of freedom.

## Explanations

### L1 human

Three bodies, such as the Sun, the Earth and the Moon, attract each other by gravity.
The problem asks how they move.
Unlike the case of two bodies, there is no general formula: tiny changes of the starting positions can lead to completely different motions.
Yet there are beautiful special motions, like three bodies at the corners of a rotating triangle, and the search for them shaped the study of [[ordinary differential equations|0NT64JV96A]].
</document>

<document source="https://tamnd.github.io/math2/node/JEAYPXPD87.md" tag="JEAYPXPD87" kind="definition">
# Akaike information criterion

## Definition

Let $M$ be a parametric statistical model with $k$ free real parameters, and let $x$ be data.
Let $\hat L$ be the maximum over the parameters of the [[likelihood function|05FSTXM92W]] of $M$ at $x$, the value at the [[maximum likelihood estimate|0FREPFKNS7]].
The *Akaike information criterion* of $M$ at $x$ is
$$\mathrm{AIC} = 2k - 2 \ln \hat L.$$
Among several candidate models for the same data, the AIC rule selects a model with the smallest AIC.

## Remarks

Only differences of AIC between models fitted to the same data have a meaning.
Example: a model with $k = 2$ and $\ln \hat L = -100$ has $\mathrm{AIC} = 204$.
A larger model with $k = 3$ and $\ln \hat L = -99.5$ has $\mathrm{AIC} = 205$, so the AIC rule keeps the smaller model.
Non-example: $-2 \ln \hat L$ alone, without the term $2k$, is not the AIC; it always favours the larger of two nested models.
For small samples a corrected version, AICc, adds a further penalty.
The [[Bayesian information criterion|GXP6H9VGRB]] uses the penalty $k \ln n$ in place of $2k$.
</document>

<document source="https://tamnd.github.io/math2/node/JEBP2FYZFM.md" tag="JEBP2FYZFM" kind="statement">
# Lomonosov's invariant subspace theorem

## Statement

Let $X$ be an infinite-dimensional complex [[Banach space|06GZ956657]] and let $K \colon X \to X$ be a nonzero [[compact operator|M2NEP1K5AE]].

1. There is a closed linear subspace $M$ of $X$ with $M \ne \set{0}$ and $M \ne X$ such that $A(M) \subseteq M$ for every bounded linear operator $A \colon X \to X$ with $AK = KA$.
2. In particular, every bounded linear operator $T \colon X \to X$ with $TK = KT$ has a nontrivial closed [[invariant subspace|0ACX5QT8TE]].

## Remarks

Proof idea: let $\mathcal{A}$ be the algebra of operators that commute with $K$, and suppose that it has no common nontrivial closed invariant subspace.
Then the orbit $\set{Ay : A \in \mathcal{A}}$ of every $y \ne 0$ is dense.
A compactness argument with finitely many $A_i \in \mathcal{A}$ and a partition of unity builds a continuous map of a closed ball into itself.
The [[Schauder fixed-point theorem|DCPFVVZZ54]] gives a fixed point.
The fixed point is an eigenvector, with eigenvalue $1$, of a compact operator $L$ that commutes with $K$.
So $K$ has an eigenvalue $\lambda$ on the finite-dimensional space $\ker(L - I)$.
Then $\ker(K - \lambda I)$ is a common invariant subspace, a contradiction.

The infinite dimension is needed in part 1.
On $\C^n$ with $n \ge 2$, the identity $K = I$ is compact and commutes with every operator, and the operators on $\C^n$ have no common nontrivial invariant subspace.

Example: the Volterra operator $(Vf)(x) = \int_0^x f(t)\,dt$ on $L^2[0,1]$ is compact, so every operator that commutes with $V$ has a nontrivial closed invariant subspace.

## Explanations

### L1 human

Compact operators are the operators that are closest to matrices.
The theorem says that a nonzero compact operator always shares a proper closed invariant subspace with every operator that commutes with it.
So any operator that commutes with a nonzero compact operator escapes the [[invariant subspace problem|5RF055GK4N]].

### L2 human

Before this theorem, the invariant subspace problem was known for compact operators and for polynomially compact operators, with long proofs.
Lomonosov's short fixed-point argument covers all of these at once, and in arbitrary complex Banach spaces.
It does not settle the general problem: there are operators on Hilbert space that commute with no nonzero compact operator.
</document>

<document source="https://tamnd.github.io/math2/node/JEBXXWHW8M.md" tag="JEBXXWHW8M" kind="statement">
# Morita equivalence of a ring with its matrix rings

## Statement

Let $R$ be a [[ring|0B1261B5ED]] with identity, let $n \ge 1$ be an integer, and let $M_n(R)$ be the [[ring of $n \times n$ matrices|0FHVPRAE60]] over $R$.
Then $R$ and $M_n(R)$ are [[Morita equivalent|RD5WKA1VK7]].
An equivalence from left $R$-[[modules|0NZ8JA25WK]] to left $M_n(R)$-modules sends $N$ to the module $N^n$ of columns of length $n$ over $N$.
On $N^n$ matrices act by matrix multiplication.

## Remarks

A quasi-inverse functor sends a left $M_n(R)$-module $L$ to $e_{11} L$, where $e_{11}$ is the matrix with $1$ in position $(1, 1)$ and $0$ elsewhere.
It is a left $R$-module through the scalar matrices.
The matrix units $e_{ij}$ give mutually inverse isomorphisms between $e_{11} L$ and $e_{jj} L$, and $L = e_{11} L \oplus \cdots \oplus e_{nn} L$, which shows that the two functors are inverse up to natural isomorphism.
In the language of [[progenerators|ZQ9ABBKF1N]], $P = R^n$ is a progenerator of right $R$-modules and its endomorphism ring is $M_n(R)$.
Example: a module over $M_2(\Z)$ is the same as a pair of copies of one abelian group $N$, glued by the matrix units, and the module is $N^2$.
So the ideal theory, the Artinian and Noetherian conditions and the homological dimensions of $M_n(R)$ match those of $R$.
Being commutative is not shared: $M_2(R)$ is not commutative for $R \ne 0$.

## Explanations

### L1 human

The ring of $n \times n$ matrices over $R$ looks much bigger than $R$, but its modules carry no new information.
Each module over the matrix ring is just $n$ stacked copies of a module over $R$.
So anything about modules over $R$ can be read off from modules over the matrix ring, and back.

### L2 human

This is the first example of Morita equivalence and the model for the general theory.
For a division ring $D$, it shows that $M_n(D)$, like $D$, has exactly one simple module up to isomorphism, namely $D^n$.
Morita's theorem says that every ring Morita equivalent to $R$ is of the form $e M_n(R) e$ for some $n$ and some idempotent $e$ with $M_n(R) e M_n(R) = M_n(R)$.
So matrix rings and full corners of them give all rings with the same module category as $R$.
</document>

<document source="https://tamnd.github.io/math2/node/JEFBKDG84Z.md" tag="JEFBKDG84Z" kind="statement">
# Schauder fixed-point theorem for integral equations

## Statement

Let $a < b$ and $R > 0$.
Let $K \colon [a, b] \times [a, b] \times [-R, R] \to \R$ and $f \colon [a, b] \to \R$ be [[continuous|0X3WGX95FE]], and let $M = \max \abs{K}$.
Assume that
$$\max_{x \in [a, b]} \abs{f(x)} + (b - a)\, M \le R.$$
Then the [[Urysohn equation|WZ4Z26XSVC]]
$$\varphi(x) = f(x) + \int_a^b K\bigl(x, t, \varphi(t)\bigr)\, dt, \quad x \in [a, b],$$
has a continuous solution $\varphi$ with $\abs{\varphi(x)} \le R$ for all $x \in [a, b]$.

## Remarks

Let $B$ be the set of continuous functions on $[a, b]$ with values in $[-R, R]$; it is closed, bounded and convex in $C[a, b]$ with the sup norm.
Let $T\varphi$ be the right side of the equation.
The hypothesis gives $T(B) \subseteq B$, and the uniform continuity of $K$ on its compact domain shows that $T$ is continuous and that $T(B)$ is equicontinuous.
By the [[Arzelà–Ascoli theorem|0381S8Z5P0]], $T(B)$ is relatively compact.
The [[Schauder fixed-point theorem|DCPFVVZZ54]], in its form for continuous maps of a closed convex set into a compact subset, gives a fixed point.

The theorem gives existence but not uniqueness.
Example: on $[0, 1]$ take $f = 0$, $R = 1$ and $K(x, t, u) = \sqrt{\abs{u}}$, so $M = 1$ and the hypothesis $0 + 1 \le 1$ holds.
Both $\varphi = 0$ and $\varphi = 1$ are solutions, since $\int_0^1 \sqrt{1}\, dt = 1$.
Here $K$ is not Lipschitz in $u$ near $0$, so the [[Banach fixed point theorem|0VK6VEH345]] does not apply.

The size condition cannot be dropped: for $f = 0$ and $K(x, t, u) = 1 + u^2$ on $[0, 1]$, any solution would be a constant $c = \int_0^1 (1 + c^2)\, dt = 1 + c^2$, which has no real root.

## Explanations

### L1 human

To solve an integral equation, one looks for a function that the right side of the equation leaves unchanged.
If the right side maps a ball of continuous functions into itself and squeezes it into a compact set, Schauder's theorem says that such a function exists.
The size condition makes sure the ball is mapped into itself.
The theorem says nothing about how many solutions there are, and there can be several.

### L2 human

This is the model for existence proofs with compactness, as opposed to proofs with contractions.
Compactness comes from the smoothing effect of integration and the Arzelà–Ascoli theorem; continuity of the kernel is all that is needed.
Peano's existence theorem for ordinary differential equations is the Volterra analogue.
Sometimes no invariant ball is easy to find.
Then [[Schaefer's fixed point theorem|7H3YSSDEBK]] or the [[Leray–Schauder method|QGRBR04JJN]] replaces it by an a priori bound for a family of equations.
</document>

<document source="https://tamnd.github.io/math2/node/JEMH9P1NVX.md" tag="JEMH9P1NVX" kind="definition">
# Normal ring

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]].
The ring $R$ is *normal* if for every [[prime ideal|00MHHZ9SNE]] $\mathfrak{p}$ of $R$, the [[localization|X6JD4AG973]] $R_{\mathfrak{p}}$ is an [[integrally closed domain|H20P600HQS]].

## Remarks

Example: $\Z \times \Z$ is normal.
Its prime ideals are $\mathfrak{q} \times \Z$ and $\Z \times \mathfrak{q}$ for a prime ideal $\mathfrak{q}$ of $\Z$, and the localizations are $\Z_{\mathfrak{q}}$, which are integrally closed domains.
It is not a domain, since $(1, 0) \cdot (0, 1) = 0$.

Non-example: $R = k[x, y]/(y^2 - x^3)$ over a field $k$ is not normal.
At the prime $\mathfrak{p} = (x, y)$, the element $y/x$ of the fraction field of $R_{\mathfrak{p}}$ is a root of $z^2 - x$, but it is not in $R_{\mathfrak{p}}$.

An integral domain is normal if and only if it is integrally closed.
A Noetherian ring is normal if and only if it is a finite product of integrally closed domains.
Some authors use the word "normal ring" only for integrally closed domains.
</document>

<document source="https://tamnd.github.io/math2/node/JEPDM5B5FC.md" tag="JEPDM5B5FC" kind="statement">
# Spectral theory of compact operators

## Statement

Let $X$ be a complex [[Banach space|06GZ956657]], let $T \colon X \to X$ be a [[compact operator|M2NEP1K5AE]], and let $\sigma(T)$ be its [[spectrum|BMKATJ802A]].
Then the following hold.

1. Every $\lambda \in \sigma(T)$ with $\lambda \ne 0$ is an [[eigenvalue|0BQAA2BK3Q]] of $T$, and its eigenspace $\ker(T - \lambda I)$ is finite-dimensional.
2. For $\lambda \ne 0$, the [[image|0WSFKKT6FF]] of $T - \lambda I$ is closed, and its codimension in $X$ equals $\dim \ker(T - \lambda I)$.
3. The set $\sigma(T)$ is finite or countable, and $0$ is its only possible accumulation point.
4. If $X$ is infinite-dimensional, then $0 \in \sigma(T)$.

## Remarks

The key tool is the Riesz lemma: in an infinite-dimensional normed space, the unit ball contains a sequence of unit vectors at mutual distance at least $1/2$.
If $\ker(T - \lambda I)$ were infinite-dimensional for some $\lambda \ne 0$, such a sequence in it would be mapped to $\lambda$ times itself, which has no convergent subsequence, against compactness.
A similar argument shows that, for each $\varepsilon > 0$, only finitely many eigenvalues satisfy $\abs{\lambda} \ge \varepsilon$, which gives part 3.
Part 2 is the Fredholm alternative: for $\lambda \ne 0$, the equation $Tx - \lambda x = y$ is solvable for every $y$ as soon as the homogeneous equation has only the solution $x = 0$.
Part 4 holds because a compact operator on an infinite-dimensional space cannot have a bounded inverse.
Example: on $\ell^2$, the diagonal operator $T(x_n) = (x_n / n)$ is compact, and $\sigma(T) = \set{1/n : n \ge 1} \cup \set{0}$, where $0$ is not an eigenvalue.
The compactness hypothesis matters: the identity on an infinite-dimensional space has the eigenvalue $1$ with infinite-dimensional eigenspace.

## Explanations

### L1 human

Compact operators are the infinite-dimensional operators that behave most like matrices.
This theorem says that their spectrum looks like that of a matrix, apart from the point $0$.
Away from $0$, the spectrum consists of eigenvalues with finitely many independent eigenvectors each, and these eigenvalues can only pile up at $0$.

### L2 human

This is the Riesz–Schauder theory, the abstract form of Fredholm's results on integral equations.
It applies to integral operators with continuous kernels, which are compact on spaces of continuous or square-integrable functions.
For compact self-adjoint operators on a Hilbert space, it combines with orthogonality to give the spectral theorem for compact self-adjoint operators.
In the language of Fredholm theory, part 2 says that $T - \lambda I$ is a Fredholm operator of index $0$ for each $\lambda \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/JEQVZXHE6A.md" tag="JEQVZXHE6A" kind="statement">
# Newman's theorem on rational approximation of the absolute value

## Statement

Let $n \ge 5$ be an integer, let $\xi = e^{-1/\sqrt{n}}$, and let
$$p(x) = \prod_{k=0}^{n-1} (x + \xi^k), \qquad r(x) = x \, \frac{p(x) - p(-x)}{p(x) + p(-x)}.$$
Then $r$ is a [[rational function|0WCZH4AFMT]] whose numerator and denominator have degree at most $n + 1$, its denominator has no zero on $[-1, 1]$, and
$$\max_{-1 \le x \le 1} \big| \abs{x} - r(x) \big| \le 3 e^{-\sqrt{n}}.$$

## Remarks

For $0 \le x \le 1$ one has $p(x) > 0$ and $\abs{p(-x)} \le p(x)$, so the denominator is positive there, and by evenness on all of $[-1, 1]$.
Then $x - r(x) = 2x\, \frac{p(-x)}{p(x) + p(-x)}$, and the proof bounds the ratio $\abs{p(-x)/p(x)} = \prod_k \abs{\xi^k - x}/(\xi^k + x)$ separately for $x$ near $0$ and for $x$ in $[\xi^n, 1]$, where some factor $\abs{\xi^k - x}/(\xi^k + x)$ is small.
Newman also showed a lower bound of the form $e^{-c\sqrt{n}}$ for every rational approximation of type $(n, n)$, so the rate $e^{-c\sqrt{n}}$ is the true order.
By contrast, the best uniform approximation of $\abs{x}$ on $[-1, 1]$ by polynomials of degree $n$ has error of order $1/n$, a result of S. N. Bernstein.
Herbert Stahl later found the exact asymptotics of the best rational error, $8 e^{-\pi \sqrt{n}}$.
Example: for $n = 100$ the bound is $3 e^{-10} \approx 1.4 \cdot 10^{-4}$, while the best polynomial approximation of degree $100$ has error about $2.8 \cdot 10^{-3}$, roughly $0.28/n$.

## Explanations

### L1 human

The function $\abs{x}$ has a corner at $0$, and polynomials, which are smooth, approximate it poorly near the corner.
Donald J. Newman showed in 1964 that quotients of polynomials do much better, with an error that shrinks like $e^{-\sqrt{n}}$ instead of like $1/n$.
The trick uses the points $\xi^k = e^{-k/\sqrt{n}}$, which crowd geometrically toward the corner at $0$.

### L2 human

This theorem started the modern study of [[rational approximation|EHFQK1WXKS]] of nonsmooth functions.
It shows that rational functions are far stronger than polynomials near singularities, which is why rational methods are used for functions with corners, branch points and boundary layers.
The geometric clustering of poles near the singularity reappears in many later constructions, for example in rational methods for problems on domains with corners.
</document>

<document source="https://tamnd.github.io/math2/node/JERVQT4C12.md" tag="JERVQT4C12" kind="concept">
# Affine variety

## Summary

An affine variety is an irreducible algebraic set in affine space $k^n$ over an algebraically closed field $k$.
Affine varieties are the local building blocks of all algebraic varieties.
An affine variety corresponds to a prime ideal of the polynomial ring, and it is determined up to isomorphism by its coordinate ring.
Some authors drop the irreducibility condition and call every algebraic set an affine variety.

## Explanations

### L1 human

An affine variety is the solution set of polynomial equations that cannot be broken into smaller solution sets.
The parabola $y = x^2$ in the plane is one: it is a single piece.
The set $xy = 0$ is not, since it splits into the two axes.
Lines, planes, smooth conics and the whole space $k^n$ are all affine varieties.
</document>

<document source="https://tamnd.github.io/math2/node/JEWP2J397B.md" tag="JEWP2J397B" kind="definition">
# Grothendieck group

## Definition

Let $M$ be a commutative [[monoid|0N6R902FG1]], written additively.
A *Grothendieck group* of $M$ is an [[abelian group|082GHNPJ53]] $K(M)$ with a monoid homomorphism $i \colon M \to K(M)$ with this universal property.
For every abelian group $A$ and monoid homomorphism $f \colon M \to A$, there is exactly one group homomorphism $g \colon K(M) \to A$ with $g \circ i = f$.

## Remarks

It exists and is unique up to unique isomorphism.
One construction: on $M \times M$ put $(m, n) \sim (m', n')$ if $m + n' + p = m' + n + p$ for some $p \in M$.
The classes form an abelian group under componentwise addition, $i(m)$ is the class of $(m, 0)$, and the class of $(m, n)$ is $i(m) - i(n)$.
Example: for $M = \N$ under addition, the inclusion $\N \to \Z$ is a Grothendieck group, since a homomorphism $f$ from $\N$ extends uniquely by $g(k) = k f(1)$.
Non-example: the inclusion $\N \to \Q$ is not, because the identity map $\N \to \N \subseteq \Z$ has no extension to a group homomorphism $\Q \to \Z$.
The map $i$ is injective exactly when $M$ is cancellative; for $M = \N \cup \set{\infty}$ with $n + \infty = \infty$, one gets $K(M) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/JEXQ0MFJ0Q.md" tag="JEXQ0MFJ0Q" kind="definition">
# Airy function

## Definition

For real $x$, the *Airy functions* of the first and second kind are
$$\operatorname{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos\left( \frac{t^3}{3} + x t \right) dt,$$
$$\operatorname{Bi}(x) = \frac{1}{\pi} \int_0^\infty \left( \exp\left( -\frac{t^3}{3} + x t \right) + \sin\left( \frac{t^3}{3} + x t \right) \right) dt,$$
where the oscillatory integrals are improper Riemann integrals.
Both extend to entire functions of $z \in \C$ that solve the [[Airy equation|PNESAD6R5T]] $y'' = z y$.

## Remarks

The two functions are fixed by their initial values $\operatorname{Ai}(0) = \frac{1}{3^{2/3}\,\Gamma(2/3)}$, $\operatorname{Ai}'(0) = -\frac{1}{3^{1/3}\,\Gamma(1/3)}$, $\operatorname{Bi}(0) = \frac{1}{3^{1/6}\,\Gamma(2/3)}$, $\operatorname{Bi}'(0) = \frac{3^{1/6}}{\Gamma(1/3)}$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
Their Wronskian is $\operatorname{Ai}(z) \operatorname{Bi}'(z) - \operatorname{Ai}'(z) \operatorname{Bi}(z) = \frac{1}{\pi}$, so they are independent.
Example: $\operatorname{Ai}$ is the solution that tends to $0$ as $x \to +\infty$, where $\operatorname{Ai}(x) \sim \frac{e^{-\zeta}}{2\sqrt{\pi}\, x^{1/4}}$ with $\zeta = \frac{2}{3} x^{3/2}$; for $x > 0$ it equals $\frac{1}{\pi} \sqrt{\frac{x}{3}}\, K_{1/3}(\zeta)$ with the [[modified Bessel function|2ET068BZZH]] $K_{1/3}$.
A non-example: $\operatorname{Ai} + \operatorname{Bi}$ solves the equation but is not $\operatorname{Ai}$, since it grows like $\operatorname{Bi}$ as $x \to +\infty$.
For $x \to -\infty$, $\operatorname{Ai}(x) \sim \frac{1}{\sqrt{\pi}\, \abs{x}^{1/4}} \sin\left( \frac{2}{3} \abs{x}^{3/2} + \frac{\pi}{4} \right)$.
</document>

<document source="https://tamnd.github.io/math2/node/JFJYSSWSSR.md" tag="JFJYSSWSSR" kind="definition">
# Minimal clone

## Definition

Let $A$ be a set with at least two elements, and let $J_A$ be the clone of all [[projections|769B35T6WG]] on $A$.
A [[clone|PTKR703G6A]] $C$ on $A$ is *minimal* if $C \ne J_A$ and the only clones $D$ with $J_A \subseteq D \subseteq C$ are $J_A$ and $C$.
Equivalently, $C$ is an atom of the [[lattice of clones|EX9NCDFYC7]] on $A$.

## Remarks

A clone $C \ne J_A$ is minimal exactly when every non-projection operation in $C$ generates $C$.
Example: on $\set{0, 1}$, the clone generated by $\wedge$ is minimal.
Check: its $n$-ary members are the operations $x_{i_1} \wedge \cdots \wedge x_{i_r}$, and identifying variables in one with $r \ge 2$ distinct variables gives $x \wedge y$ back.
Non-example: the clone generated by $\wedge$ and $\vee$ on $\set{0, 1}$ is not minimal.
It properly contains the clone generated by $\wedge$, which is not the projection clone.
On $\set{0, 1}$ there are exactly seven minimal clones, generated by the constant $0$, the constant $1$, $\neg$, $\wedge$, $\vee$, the majority operation, and $x + y + z$ modulo $2$.
</document>

<document source="https://tamnd.github.io/math2/node/JFKYYK2CEB.md" tag="JFKYYK2CEB" kind="concept">
# CR function

## Summary

A CR function on a [[CR manifold|ZC78JKZX9S]] is a function annihilated by the anti-holomorphic tangent vectors of the CR structure, see the [[definition|Z23FCT4WTZ]].
On a real hypersurface of $\C^n$, these are the solutions of the [[tangential Cauchy–Riemann equations|X7GP9BECD8]], and the restriction of any holomorphic function is one.
The [[Lewy extension theorem|V2YCGHD719]] shows that on a strongly pseudoconvex hypersurface, CR functions extend locally holomorphically to one side.
So they are locally boundary values of holomorphic functions.

## Explanations

### L1 human

On a surface in $\C^n$ one cannot ask a function to be holomorphic, since the surface is not a complex space.
One can still ask it to satisfy the Cauchy–Riemann equations in the directions that the surface allows.
These are the CR functions.
Restrictions of holomorphic functions are the main examples, and in good cases every CR function is of this kind.
</document>

<document source="https://tamnd.github.io/math2/node/JFP2DGMNXT.md" tag="JFP2DGMNXT" kind="statement">
# Jacobi triple product

## Statement

Let $q, z \in \C$ with $\abs{q} < 1$ and $z \ne 0$.
Then
$$\prod_{m=1}^{\infty} \left(1 - q^{2m}\right) \left(1 + z q^{2m-1}\right) \left(1 + z^{-1} q^{2m-1}\right) = \sum_{n=-\infty}^{\infty} q^{n^2} z^n.$$
Equivalently, with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]], $(q; q)_\infty\, (z; q)_\infty\, (q/z; q)_\infty = \sum_{n \in \Z} (-1)^n q^{\binom{n}{2}} z^n$.

## Remarks

Proof idea: call the product $F(z)$; it is holomorphic on $\C \setminus \set{0}$, so it has a Laurent expansion $\sum_n c_n z^n$.
The product satisfies $F(q^2 z) = (q z)^{-1} F(z)$, which forces $c_n = q^{n^2} c_0$.
The constant $c_0 = 1$ can be found by specializing $z$, for instance with Gauss's or Euler's identities, or by a limit argument.
Example: substituting $q \mapsto q^{3/2}$ and $z \mapsto -q^{1/2}$ in the first form gives [[Euler's pentagonal number theorem|R587SR9GNV]] $(q; q)_\infty = \sum_{n \in \Z} (-1)^n q^{n(3n-1)/2}$.
With $z = 1$ the first form gives $\sum_{n \in \Z} q^{n^2} = \prod_{m \ge 1} (1 - q^{2m})(1 + q^{2m-1})^2$, a product for a [[theta function|2F5CYQJVAF]].
The condition $\abs{q} < 1$ is needed for the [[infinite product|RZ0Z2BTMT4]] and the series to converge.

## Explanations

### L1 human

The triple product identity says that a certain infinite product equals a sum of powers $q^{n^2}$, one for each integer $n$.
The sum side is a theta function, so the identity turns theta functions into products.
Products are good for counting partitions, sums are good for computing, and the identity lets one pass between them.

### L2 human

It is the basic identity behind the product formulas for the Jacobi theta functions and the [[Dedekind eta function|MX3CVYMNDV]].
It is a limiting case of [[Ramanujan's 1psi1 summation|RK6DCJVJCR]].
Together with its relative, the [[quintuple product identity|PZR6RC79ED]], it gives many partition identities, and it is a step in proofs of the [[Rogers–Ramanujan identities|M24H4GJPPW]].
In Lie theory it is the Macdonald identity for the affine root system of type $A_1^{(1)}$.
</document>

<document source="https://tamnd.github.io/math2/node/JFX8ASQS3E.md" tag="JFX8ASQS3E" kind="concept">
# Tetrahedron

## Summary

A tetrahedron is a [[polyhedron|0325SGSJAR]] with four triangular faces, six edges and four vertices.
It is the convex hull of four points of space that do not lie in one plane.
It is the three-dimensional [[simplex|YX1DJK2QAG]], the simplest solid, and it plays the role in space that the triangle plays in the plane.

## Explanations

### L1 human

A tetrahedron is a pyramid on a triangular base.
Take four points that do not lie on one flat sheet and join each pair by a straight edge: the solid you get has four triangle faces.
The corners $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$ give a tetrahedron of volume $1/6$.
The regular tetrahedron, with all four faces equilateral, is one of the five Platonic solids.
</document>

<document source="https://tamnd.github.io/math2/node/JG7WB75BHC.md" tag="JG7WB75BHC" kind="statement">
# Correspondence theorem for congruences

## Statement

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$ and let $\theta$ be a [[congruence|DCSXXT4SER]] on $\mathbf{A}$.
For a congruence $\psi \supseteq \theta$ on $\mathbf{A}$, put
$$\psi/\theta = \set{(a/\theta, b/\theta) : (a, b) \in \psi}.$$
Then $\psi \mapsto \psi/\theta$ is an isomorphism from the interval $\set{\psi \in \operatorname{Con} \mathbf{A} : \theta \subseteq \psi}$ of the [[congruence lattice|REE07CSXHR]] onto $\operatorname{Con}(\mathbf{A}/\theta)$.

## Remarks

Proof idea.
Let $\nu \colon \mathbf{A} \to \mathbf{A}/\theta$ be the natural map of the [[quotient algebra|N5KE7FH6CX]].
For $\psi \supseteq \theta$, the relation $\psi/\theta$ is a congruence, and its preimage under $\nu \times \nu$ is $\psi$ again, because $\psi$ contains $\theta$.
Conversely, the preimage of any congruence on $\mathbf{A}/\theta$ is a congruence on $\mathbf{A}$ containing $\theta$.
These two maps are inverse to each other and preserve inclusion, so they are lattice isomorphisms.

Example: in the group $\Z$, take $\theta$ to be congruence modulo $12$.
The congruences above $\theta$ are congruence modulo $d$ for the divisors $d$ of $12$, and they match the six subgroups of $\Z/12\Z$.

The hypothesis $\psi \supseteq \theta$ is needed.
For $\psi$ not containing $\theta$, the relation $\psi/\theta$ can fail to be transitive, and different $\psi$ can give the same relation.

## Explanations

### L1 human

Gluing elements of an algebra in two steps is the same as gluing them in one step.
So the ways to glue the quotient $\mathbf{A}/\theta$ further are exactly the gluings of $\mathbf{A}$ that are coarser than $\theta$.
The theorem says this matching keeps the order, so the two lattices have the same shape.

### L2 human

This is the general form of the correspondence theorems for groups, rings and modules, where congruences are replaced by normal subgroups, ideals and submodules.
Together with the second isomorphism theorem $(\mathbf{A}/\theta)/(\psi/\theta) \cong \mathbf{A}/\psi$, it shows that a quotient of a quotient is again a quotient.
It is used to prove that $\mathbf{A}/\theta$ is [[simple|9CJCXNYNK8]] exactly when $\theta$ is a maximal proper congruence, and in the proof of [[Birkhoff's subdirect representation theorem|CZF5696TPY]].
</document>

<document source="https://tamnd.github.io/math2/node/JGACZYEEY7.md" tag="JGACZYEEY7" kind="concept">
# Sparse matrix

## Summary

A sparse matrix is a matrix in which most entries are zero.
Such matrices are stored by listing only the nonzero entries with their positions, which saves memory.
Algorithms that skip the zeros can be much faster than algorithms for general, dense matrices.
Large sparse matrices arise from networks, from discretised differential equations and from many problems in data science.

## Explanations

### L1 human

Think of a matrix that records which of a million people know each other.
Each person knows only a few hundred others, so almost all of the trillion entries are zero.
Storing all entries would be wasteful; it is enough to store, for each nonzero entry, its row, its column and its value.
The [[tridiagonal|DARX1KZRHZ]] and [[band|Z58FRZCF8P]] matrices are simple examples of sparse matrices.
</document>

<document source="https://tamnd.github.io/math2/node/JGBW4KVMN8.md" tag="JGBW4KVMN8" kind="definition">
# Homology manifold

## Definition

Let $n \ge 0$ be an integer and let $X$ be a [[locally compact|0FE2SB8WVJ]] [[Hausdorff space|0S1FJ8VEX9]].
The space $X$ is a *homology $n$-manifold* if, for every point $x \in X$, the local [[relative homology|4BVWH0BN28]] groups satisfy
$$H_i(X, X \setminus \set{x}; \Z) \cong \begin{cases} \Z & \text{if } i = n, \\ 0 & \text{if } i \neq n. \end{cases}$$

## Remarks

Several variants are in use: some authors also ask that $X$ be finite-dimensional or an absolute neighbourhood retract, or use sheaf-theoretic homology or coefficients in a ring other than $\Z$.

Example: a [[topological $n$-manifold|QQQ9B1DK4P]] $M$ is a homology $n$-manifold.
By excision, $H_i(M, M \setminus \set{x}) \cong H_i(\R^n, \R^n \setminus \set{0})$, and the long exact sequence of the pair gives $H_i(\R^n, \R^n \setminus \set{0}) \cong \tilde{H}_{i-1}(S^{n-1})$, which is $\Z$ for $i = n$ and $0$ otherwise.

Non-example: let $X$ be the wedge of two circles and $x$ the wedge point.
Then $X \setminus \set{x}$ is a disjoint union of two open arcs, and the long exact sequence gives $H_1(X, X \setminus \set{x}) \cong \Z^3$, so $X$ is not a homology $1$-manifold.

The suspension of the Poincaré homology sphere is a homology $4$-manifold that is not a topological manifold.
The reason is that a small neighbourhood of a suspension point, with that point removed, is not simply connected.
</document>

<document source="https://tamnd.github.io/math2/node/JGF3WBT4GC.md" tag="JGF3WBT4GC" kind="definition">
# Content

## Definition

Let $X$ be a set and let $\mathcal{S}$ be a [[semiring of sets|PYH82FMXQ2]] on $X$.
More generally, $\mathcal{S}$ may be any family of subsets of $X$ that contains $\varnothing$.
A *content* on $\mathcal{S}$ is a map $\mu \colon \mathcal{S} \to [0, \infty]$, with values in the [[extended real line|0GSZ7B0CGR]], such that:

1. $\mu(\varnothing) = 0$.
2. If $A_1, \ldots, A_n \in \mathcal{S}$ are pairwise disjoint and $A_1 \cup \cdots \cup A_n \in \mathcal{S}$, then
$$\mu(A_1 \cup \cdots \cup A_n) = \mu(A_1) + \cdots + \mu(A_n).$$

## Remarks

Example: on the semiring of bounded intervals $[a, b)$ in $\R$, length $\mu([a, b)) = b - a$ is a content.
Indeed, if $[a, b)$ is a disjoint union of finitely many such intervals, their lengths add up to $b - a$.
Non-example: on the same semiring, $\mu([a, b)) = (b - a)^2$ fails clause 2, since $[0, 2) = [0, 1) \cup [1, 2)$ but $4 \ne 1 + 1$.
On a ring of sets a content is monotone and finitely subadditive.
A content on a ring is called a [[finitely additive measure|KFWQ24D9M6]] by many authors.
A content that is countably additive is a [[premeasure|PV0XXSWFC7]].
The word content also names a set function used to build Haar measure.
It is defined on the compact subsets of a space; it is monotone and finitely subadditive, and additive on disjoint compact sets.
</document>

<document source="https://tamnd.github.io/math2/node/JGQM4K9H8T.md" tag="JGQM4K9H8T" kind="definition">
# Free algebra

## Definition

Let $k$ be a [[commutative ring|0310REJJY8]] with identity and let $X$ be a set.
Let $X^*$ be the [[free monoid|CV013KHVJG]] on $X$, with product given by concatenation.
Its elements are the finite words $x_1 x_2 \cdots x_n$ with $x_i \in X$, including the empty word $1$.
The *free algebra* $k\langle X \rangle$ is the free $k$-module with basis $X^*$, with the multiplication that extends concatenation of words $k$-bilinearly:
$$\Big(\sum_{u} a_u \, u\Big)\Big(\sum_{v} b_v \, v\Big) = \sum_{u, v} a_u b_v \, uv.$$
It is an [[associative algebra|3CTW1P4HBP]] over $k$ with identity the empty word.

## Remarks

Universal property: for every associative $k$-algebra $A$ with identity and every map $f \colon X \to A$, there is a unique homomorphism of $k$-algebras $\bar f \colon k\langle X \rangle \to A$ with $\bar f(x) = f(x)$ for $x \in X$.
It sends a word $x_1 \cdots x_n$ to $f(x_1) \cdots f(x_n)$.

Example: for $X = \set{x}$, the words are the powers $x^n$, and $k\langle x \rangle$ is the [[polynomial ring|0SDC934R52]] $k[x]$.

Non-example: for $X = \set{x, y}$, the polynomial ring $k[x, y]$ is not free on $X$.
The map $x \mapsto e_{12}$, $y \mapsto e_{21}$ into $2 \times 2$ matrices over $k$ has no extension to a homomorphism from $k[x, y]$, because $e_{12} e_{21} = e_{11} \ne e_{22} = e_{21} e_{12}$ while $xy = yx$ in $k[x, y]$.

The free algebra is the [[monoid ring|1HM7ZYMPD7]] $k[X^*]$.
It is isomorphic to the [[tensor algebra|P4J5JZDW0B]] $T(V)$ of the free $k$-module $V$ with basis $X$, graded by word length.
For $\abs{X} \ge 2$ and $k$ a field, it is a domain that is not commutative.
</document>

<document source="https://tamnd.github.io/math2/node/JGR1E7Q9KQ.md" tag="JGR1E7Q9KQ" kind="concept">
# Convolution power

## Summary

The $k$-th convolution power $f^{*k}$ of a function or measure $f$ is the [[convolution|2Z29JN3VES]] of $k$ copies of $f$.
It is the analogue of the ordinary power $x^k$ when the product is convolution.
If $f$ is the density of a random variable $X$, then $f^{*k}$ is the density of the sum of $k$ independent copies of $X$.
Under the Fourier transform, $f^{*k}$ becomes the ordinary power $\hat f^{\,k}$.

## Explanations

### L1 human

Convolving a function with itself again and again gives its convolution powers.
For the uniform density on $[0, 1]$, the second power is a tent on $[0, 2]$, the third is a rounded hump on $[0, 3]$ made of quadratic pieces, and so on.
The higher powers look more and more like a bell curve, which is the central limit theorem in action.
</document>

<document source="https://tamnd.github.io/math2/node/JGT7T3A9PD.md" tag="JGT7T3A9PD" kind="statement">
# Fáry's theorem

## Statement

Let $G$ be a finite simple [[planar graph|0NC9R64TXP]].
Then $G$ has an [[embedding|CG91WNMG2T]] in the plane in which every edge is a straight line segment.

## Remarks

The idea of the proof is induction on the number of vertices.
Add edges until $G$ is maximal planar, so that every face, including the outer one, is a triangle.
For at least four vertices, Euler's formula gives a vertex $v$ of degree at most $5$ that is not on the outer triangle.
Remove $v$, triangulate the face it leaves, and draw the smaller graph with straight edges by induction.
Delete the added diagonals; the face left by $v$ is a polygon with at most $5$ sides.
Such a polygon always contains a point that sees all its corners, so $v$ can be placed there and joined by straight segments.

The theorem is named after István Fáry; Klaus Wagner had proved it earlier, and Sherman Stein proved it independently.
The result needs $G$ to be simple: a multigraph with two parallel edges cannot be drawn with both edges straight.

## Explanations

### L1 human

If a network can be drawn on paper without crossing lines, using curvy lines if needed, then it can also be drawn without crossings using only straight lines.
So curves give no extra power for planar drawings.

### L2 human

Fáry's theorem lets one treat planar graphs as plane [[geometric graphs|HFJCAPWWD4]].
Stronger versions exist: every planar graph with $n$ vertices has a straight-line drawing with vertices on an $O(n) \times O(n)$ integer grid.
By the Koebe–Andreev–Thurston theorem, every planar graph is the contact graph of a family of discs with disjoint interiors.
Tutte's spring embedding gives a straight-line drawing of every $3$-connected planar graph with convex faces.
</document>

<document source="https://tamnd.github.io/math2/node/JGTGXJB7V5.md" tag="JGTGXJB7V5" kind="definition">
# Quantum enveloping algebra

## Definition

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan matrix $(a_{ij})_{1 \le i, j \le n}$.
Let $d_1, \ldots, d_n$ be positive integers with $d_i a_{ij} = d_j a_{ji}$ for all $i, j$.
Let $q \in \C$ be nonzero and not a [[root of unity|0S61N75H45]], and put $q_i = q^{d_i}$.
For an integer $m \ge 0$ put $[m]_{q} = \frac{q^m - q^{-m}}{q - q^{-1}}$, $[m]_q! = [1]_q [2]_q \cdots [m]_q$, and $\left[{m \atop r}\right]_q = \frac{[m]_q!}{[r]_q!\,[m-r]_q!}$.
The *quantum enveloping algebra* $U_q(\mathfrak{g})$ is the $\C$-algebra with generators $E_i, F_i, K_i, K_i^{-1}$ for $1 \le i \le n$ and relations:

1. $K_i K_i^{-1} = K_i^{-1} K_i = 1$ and $K_i K_j = K_j K_i$.
2. $K_i E_j K_i^{-1} = q_i^{a_{ij}} E_j$ and $K_i F_j K_i^{-1} = q_i^{-a_{ij}} F_j$.
3. $E_i F_j - F_j E_i = \delta_{ij} \dfrac{K_i - K_i^{-1}}{q_i - q_i^{-1}}$.
4. For $i \ne j$, with $m = 1 - a_{ij}$: $\sum_{r=0}^{m} (-1)^r \left[{m \atop r}\right]_{q_i} E_i^{m-r} E_j E_i^{r} = 0$, and the same relation with every $E$ replaced by $F$.

It is a [[Hopf algebra|7XFYP0N84H]] with $\Delta(K_i) = K_i \otimes K_i$, $\Delta(E_i) = E_i \otimes 1 + K_i \otimes E_i$ and $\Delta(F_i) = F_i \otimes K_i^{-1} + 1 \otimes F_i$.

## Remarks

Example: for $\mathfrak{g} = \mathfrak{sl}_2$ the Cartan matrix is $(2)$, $d_1 = 1$, and there is no relation 4.
So $U_q(\mathfrak{sl}_2)$ is generated by $E, F, K^{\pm 1}$ with $K E K^{-1} = q^2 E$, $K F K^{-1} = q^{-2} F$ and $EF - FE = (K - K^{-1})/(q - q^{-1})$.
Non-example: the value $q = 1$ is excluded, since the denominator $q_i - q_i^{-1}$ in relation 3 vanishes.
The classical [[universal enveloping algebra|SCZ8PRVY00]] $U(\mathfrak{g})$ is recovered only through a limit, or through a version over formal power series in $h$ with $q = e^h$.
Sign and normalization conventions differ between textbooks; the coproduct is sometimes replaced by its opposite.
</document>

<document source="https://tamnd.github.io/math2/node/JGVDA1N106.md" tag="JGVDA1N106" kind="definition">
# Stably finite ring

## Definition

Let $R$ be a ring with $1$.
For $n \ge 1$, let $M_n(R)$ be the [[matrix ring|0FHVPRAE60]] of $n \times n$ matrices over $R$.
The ring $R$ is *stably finite* if $M_n(R)$ is [[Dedekind-finite|GV1HKEAWVN]] for every $n \ge 1$.
That is, for all $n \ge 1$ and $A, B \in M_n(R)$, the equation $AB = I_n$ implies $BA = I_n$.

## Remarks

Example: every commutative ring $R$ is stably finite.
If $AB = I_n$ in $M_n(R)$, then $\det(A) \det(B) = 1$, so $A$ has the inverse $\det(A)^{-1} \operatorname{adj}(A)$, and $BA = A^{-1}(AB)A = I_n$.

Non-example: the ring $\End(V)$ of linear maps on an infinite-dimensional vector space $V$ is not stably finite, since already $M_1(\End(V)) = \End(V)$ contains $a, b$ with $ab = 1 \ne ba$.

Taking $n = 1$ shows that a stably finite ring is Dedekind-finite.
The converse fails: there are Dedekind-finite rings $R$ for which $M_2(R)$ is not Dedekind-finite.
Equivalently, $R$ is stably finite if for every $n \ge 1$ the left module $R^n$ is not isomorphic to $R^n \oplus N$ for a nonzero module $N$.
A nonzero stably finite ring has invariant basis number.
</document>

<document source="https://tamnd.github.io/math2/node/JH4BN4W27Z.md" tag="JH4BN4W27Z" kind="definition">
# Strong maximal function

## Definition

Let $n \ge 1$ and let $\abs{R}$ be the [[Lebesgue measure|2Z2QVWZ262]] of a set $R \subseteq \R^n$.
A *rectangle* here is a set $R = I_1 \times \cdots \times I_n$, where each $I_k$ is a bounded interval of positive length.
Let $f$ be a locally [[integrable|X7GTF3D98F]] function on $\R^n$.
The *strong maximal function* of $f$ is the function $M_S f \colon \R^n \to [0, \infty]$ given by
$$M_S f(x) = \sup_{R \ni x} \frac{1}{\abs{R}} \int_R \abs{f(y)} \, dy,$$
where the supremum is over all such rectangles $R$ that contain $x$.

## Remarks

Each ball of radius $r$ lies in a cube of side $2r$, which is a rectangle of comparable volume.
So $Mf \le C_n M_S f$ for the [[Hardy–Littlewood maximal function|VX1A56P5KQ]] $Mf$, with $C_n$ depending only on $n$.
Let $M_k$ be the one-dimensional maximal function in the $k$-th variable, with the other variables fixed.
Then $M_S f \le M_1 M_2 \cdots M_n f$ almost everywhere, and the one-dimensional bounds give $\norm{M_S f}_p \le C \norm{f}_p$ for $1 < p \le \infty$.
For $n \ge 2$ the operator $M_S$ is not of weak type $(1, 1)$.
It satisfies the weaker bound $\abs{\set{M_S f > \lambda}} \le C \int \frac{\abs{f}}{\lambda} \left( 1 + \log^+ \frac{\abs{f}}{\lambda} \right)^{n - 1}$, due to Jessen, Marcinkiewicz and Zygmund.

Example: let $n = 2$, $f = \mathbf{1}_{[0, 1]^2}$ and $x = (a, 1/2)$ with $a > 1$.
The rectangle $R = [0, a] \times [0, 1]$ contains $x$ and has $\frac{1}{\abs{R}} \int_R f = \frac{1}{a}$.
Any rectangle $R' = [s, t] \times J$ containing $x$ and meeting the square has $s \le 1 < a \le t$.
Its average of $f$ is at most $(1 - \max(s, 0))/(t - s)$, which is at most $1/a$.
So $M_S f(x) = 1/a$.

Non-example: a square centred at $x$ and rotated by $45$ degrees is not a rectangle in the sense above.
Its sides are not parallel to the axes.
Allowing rectangles in all directions gives a different operator, which is unbounded on $L^p(\R^2)$ for every $p < \infty$, because of Besicovitch sets.
</document>

<document source="https://tamnd.github.io/math2/node/JH6Y2E1ZCN.md" tag="JH6Y2E1ZCN" kind="statement">
# Catalan's conjecture

## Statement

Let $x$, $y$, $a$, $b$ be [[natural numbers|0MWKJ44V5A]] with $x > 1$, $y > 1$, $a > 1$ and $b > 1$.
If $x^a - y^b = 1$, then $x = 3$, $a = 2$, $y = 2$ and $b = 3$.

## Remarks

The result is also called Mihăilescu's theorem: Eugène Catalan conjectured it in 1844, and Preda Mihăilescu proved it in 2002.
Robert Tijdeman had shown in 1976, with [[Baker's theorem|WMBJSNG57F]] on linear forms in logarithms, that the [[Diophantine equation|08B1RTW7VZ]] has only finitely many solutions.
Mihăilescu's proof reduces to odd prime exponents and then uses the arithmetic of [[cyclotomic fields|NN77VYSGXR]].
Equivalently: $8$ and $9$ are the only two consecutive integers that are both perfect powers.
The hypothesis $a, b > 1$ is needed, since for $b = 1$ every $x^a - 1$ is a valid $y$.
Euler had settled the special case $x^2 - y^3 = 1$, whose only positive solution is $(3, 2)$.

## Explanations

### L1 human

Perfect powers are numbers such as $4, 8, 9, 16, 25, 27, 32$.
Among them, $8$ and $9$ sit right next to each other.
Catalan guessed that this never happens again, and Mihăilescu proved it about one and a half centuries later.

### L2 human

The equation $x^a - y^b = 1$ is an exponential Diophantine equation, with unknown exponents.
Tijdeman's finiteness result is typical of Baker's method: it bounds the solutions, but the bound was far too large to search.
Mihăilescu's proof avoids computation and works with units and class groups of cyclotomic fields, a theme shared with work on Fermat's Last Theorem.
The more general Pillai conjecture, that each positive integer occurs only finitely often as a difference of perfect powers, remains open.
</document>

<document source="https://tamnd.github.io/math2/node/JHKSF3XFY2.md" tag="JHKSF3XFY2" kind="concept">
# Potential of a measure

## Summary

The potential of a measure $\mu$ with respect to a kernel $k$ is the function $x \mapsto \int k(x, y) \, d\mu(y)$.
It describes the field of a distribution of mass or charge $\mu$.
Here $k(x, y)$ is the effect at $x$ of a unit mass at $y$.
With the [[Newtonian kernel|6TJTWQ5HCR]] one gets the Newtonian potential, with the [[logarithmic kernel|VDZ5FKYC2T]] the logarithmic potential, and with $\abs{x - y}^{\alpha - n}$ the Riesz potentials.
Energy, capacity, equilibrium measures and balayage are all defined through potentials of measures.

## Explanations

### L1 human

Spread some mass over space, perhaps unevenly.
Each small piece of mass creates a small potential around it, given by the kernel.
The potential of the whole measure is the sum of all these small contributions, which is an integral.
For a single point mass at $a$, the potential is just $k(x, a)$.
</document>

<document source="https://tamnd.github.io/math2/node/JHQNT58FEH.md" tag="JHQNT58FEH" kind="concept">
# Shilov boundary

## Summary

Let $A$ be a commutative unital Banach algebra.
Its Shilov boundary is the smallest closed subset of the [[character space|M7JVFJ3MMX]] on which every [[Gelfand transform|VWNSHKJT92]] attains its maximum modulus.
It is an abstract version of the maximum modulus principle of complex analysis.
For the disc algebra it is the unit circle, and for $C(X)$ it is all of $X$.

## Explanations

### L1 human

A holomorphic function on the closed unit disc takes its largest absolute value on the boundary circle.
So to know how large such a function can be, it suffices to look at the circle.
The Shilov boundary is the smallest closed set with this property, defined for any commutative Banach algebra with unit.
For the disc algebra the circle cannot be made smaller: for each point of the circle there is a function in the algebra that peaks exactly there.
</document>

<document source="https://tamnd.github.io/math2/node/JJ0PQZ7VZX.md" tag="JJ0PQZ7VZX" kind="definition">
# Circle packing

## Definition

Let $(C_i)_{i \in I}$ be a family of [[circles|060W5CBZP6]] of positive radius in the plane $\R^2$, and let $D_i$ be the closed disc bounded by $C_i$.
The family is a *circle packing* if the discs $D_i$ have pairwise disjoint interiors.
Its *tangency graph* has vertex set $I$, with an edge between distinct $i$ and $j$ when $C_i$ and $C_j$ are tangent.
A packing of *equal circles* is one in which all radii are equal.

## Remarks

Example: the circles of radius $1$ centred at $(0, 0)$ and $(2, 0)$.
The distance between centres is $2$, the sum of the radii, so the [[discs|0Q0H13G8ZB]] meet in the single point $(1, 0)$ and their interiors are disjoint.
The tangency graph is one edge.
Non-example: circles of radius $1$ centred at $(0, 0)$ and $(1, 0)$; the distance $1$ is less than $2$, so the open discs overlap.
The same notion is used on the sphere and in the hyperbolic plane.
A packing of equal circles of radius $r$ is the same as a sphere packing in $\R^2$ with radius $r$.
</document>

<document source="https://tamnd.github.io/math2/node/JJ2G94HHJ6.md" tag="JJ2G94HHJ6" kind="definition">
# Calderón–Zygmund operator

## Definition

Let $n \ge 1$.
A *Calderón–Zygmund operator* is a linear operator $T \colon L^2(\R^n) \to L^2(\R^n)$ such that:

1. $T$ is bounded on the [[$L^2$ space|SASP14MA1Z]], that is $\norm{Tf}_{L^2} \le C\norm{f}_{L^2}$ for some constant $C$ and all $f \in L^2(\R^n)$;
2. there is a [[Calderón–Zygmund kernel|78N6AXP9VV]] $K$ such that, for every $f \in L^2(\R^n)$ with compact support,
$$Tf(x) = \int_{\R^n} K(x, y) f(y) \, dy \quad \text{for almost every } x \notin \operatorname{supp} f.$$

## Remarks

Example: the Hilbert transform $H$ on $\R$ is a Calderón–Zygmund operator.
It is bounded on $L^2(\R)$ with norm $1$, since its multiplier $-i\sgn\xi$ has modulus $1$.
For $x$ outside the support of $f$, the principal value is an ordinary integral, so $Hf(x) = \int \frac{f(y)}{\pi(x - y)} \, dy$, and $\frac{1}{\pi(x - y)}$ is a Calderón–Zygmund kernel.

Non-example: on $\R$, let $Tf = f * \mathbf{1}_{[0, 1]}$, so $Tf(x) = \int \mathbf{1}_{[0, 1]}(x - y) f(y) \, dy$.
It is bounded on $L^2(\R)$ by Young's inequality, so clause 1 holds.
Clause 2 fails, because its kernel $\mathbf{1}_{[0, 1]}(x - y)$ jumps at $x - y = 1$.
For $y = 0$, $x = 1 - \eta$ and $x' = 1 + \eta$, the difference of the kernel values is $1$, while the bound $A(2\eta)^\delta (1 - \eta)^{-1-\delta}$ tends to $0$ as $\eta \to 0$.

The kernel does not determine the operator: the identity and the zero operator both have kernel $0$ off the diagonal.
Clause 2 is only required off the support of $f$, since near the diagonal the integral need not converge absolutely.
The [[singular integral operators|HTMCEARGDD]] with smooth mean-zero angular part are Calderón–Zygmund operators.
</document>

<document source="https://tamnd.github.io/math2/node/JJ5XQEVBZ4.md" tag="JJ5XQEVBZ4" kind="definition">
# Microlocal analysis

## Definition

Let $X$ be an open subset of $\R^n$, or a smooth manifold, with cotangent bundle $T^*X$.
*Microlocal analysis* studies distributions and linear operators on $X$ through properties localized at points $(x, \xi) \in T^*X \setminus 0$.
Here $x$ is a position and $\xi \neq 0$ is a frequency direction.
A property holds *microlocally* near $(x_0, \xi_0)$ if it holds after cutting off near $x_0$ and keeping only frequencies in a conic neighbourhood of $\xi_0$.
The basic notions are:

1. The [[wave front set|3T5Q9WVQX8]] $\mathrm{WF}(u) \subset T^*X \setminus 0$ of a distribution $u$: the set of $(x_0, \xi_0)$ near which $u$ fails to be smooth microlocally.
2. Properly supported [[pseudo-differential operators|FF27N0BWTP]], which act microlocally and satisfy $\mathrm{WF}(Pu) \subseteq \mathrm{WF}(u)$.

## Remarks

Example: let $u = H(x_1)$ on $\R^2$, where $H$ is the Heaviside function.
Then $u$ is smooth off the line $x_1 = 0$, and $\mathrm{WF}(u) = \set{(x, \xi) : x_1 = 0, \ \xi = (\xi_1, 0), \ \xi_1 \neq 0}$.
The check: take a cutoff $\chi(x)$ near a point of the line.
The [[Fourier transform|DPVM34ADZV]] of $\chi u$ decays rapidly in every closed cone avoiding the $\xi_1$-axis, but only like $\abs{\xi_1}^{-1}$ along it.
Non-example: classical local regularity, such as "$u$ is smooth near $x_0$", is not a microlocal statement, since it does not distinguish frequency directions.
The projection of $\mathrm{WF}(u)$ to $X$ is the singular support: $\pi(\mathrm{WF}(u)) = \operatorname{sing\,supp} u$.
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# Zero-dimensional space

## Summary

A zero-dimensional space is a [[topological space|0CQHFPGJ45]] that has a base of [[clopen sets|DQ4E55T3BW]].
In terms of dimension theory, a nonempty space with this property has small [[inductive dimension|MKS9H497AM]] zero.
Examples include discrete spaces, the rational numbers and the [[Cantor set|0BXNDZFQME]].
Zero-dimensional $T_0$ spaces are [[totally disconnected|0RJSBR9GW1]], and for compact Hausdorff spaces the two notions agree.

## Explanations

### L1 human

In a zero-dimensional space, every small neighbourhood can be chosen with no boundary at all.
Such a set is both open and closed, so the space falls apart into tiny separate pieces.
The rational numbers are like this: the set of rationals between two irrational numbers is both open and closed in $\Q$.
The real line is not, because the only sets that are both open and closed in $\R$ are $\emptyset$ and $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/JJ91A1V8NQ.md" tag="JJ91A1V8NQ" kind="concept">
# Separable closure

## Summary

The separable closure of $K$ in an algebraic extension $L$ is the set of elements of $L$ that are [[separable|0WCX7NB4N7]] over $K$.
It is the largest separable subextension of $L/K$.
The separable closure of a field $K$, written $K^{\mathrm{sep}}$, is the separable closure of $K$ in an [[algebraic closure|0X6AB5P3G1]]; it is a [[separably closed field|8B57B7EGK4]].
Its automorphism group over $K$ is the [[absolute Galois group|PGKWBJDTKX]] of $K$.
For a [[perfect field|VWBVKT794W]] the separable closure is the whole algebraic closure.

## Explanations

### L1 human

Inside a big field, collect all the elements whose minimal polynomials over $K$ have no repeated roots.
These elements form a field, and it is the separable closure.
In characteristic $0$ every polynomial behaves well, so nothing is lost and one gets the algebraic closure.
In characteristic $p$, elements such as a $p$-th root of $t$ over $\F_p(t)$ are left out.
The separable closure is the part of the algebraic closure that Galois theory sees.
</document>

<document source="https://tamnd.github.io/math2/node/JJA90EB2PS.md" tag="JJA90EB2PS" kind="concept">
# Doubly stochastic matrix

## Summary

A doubly stochastic matrix is a square [[nonnegative matrix|JPQCCFCY8X]] in which every row and every column sums to $1$.
[[Permutation matrices|08SP3DF8NB]] are doubly stochastic, and by the [[Birkhoff–von Neumann theorem|3KY1RMEDMJ]] every doubly stochastic matrix is a convex combination of them.
These matrices describe averaging processes and appear in [[majorization|5K7VT2VKZT]], assignment problems and Markov chains with a uniform stationary distribution.

## Explanations

### L1 human

Imagine $n$ glasses of water and a rule that pours fixed fractions of each glass into the others.
Suppose that no water is lost, and that glasses which start equally full stay equally full.
Then the table of fractions is doubly stochastic.
Each row and each column of the table adds up to $1$.
For example, $\frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ mixes two glasses evenly.
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# Runge–Kutta methods

## Definition

Let $f \colon \R \times \R^n \to \R^n$, and consider the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$.
Let $s \ge 1$ be an integer and let $a_{ij}$, $b_i$, $c_i$ be real numbers for $i, j = 1, \ldots, s$.
The *$s$-stage Runge–Kutta method* with these coefficients and step size $h > 0$ computes approximations $y_k \approx y(t_k)$ at $t_k = t_0 + k h$ by
$$K_i = f\Big( t_k + c_i h, \; y_k + h \sum_{j=1}^{s} a_{ij} K_j \Big), \quad i = 1, \ldots, s, \qquad y_{k+1} = y_k + h \sum_{i=1}^{s} b_i K_i.$$
The method is *explicit* if $a_{ij} = 0$ for all $j \ge i$, and *implicit* otherwise.

## Remarks

The coefficients are written in the [[Butcher tableau|RH8BD7PW5F]] with the $c_i$ in a column, the matrix $(a_{ij})$ beside it, and the $b_i$ as a bottom row.
Usually $c_i = \sum_j a_{ij}$, and the method is consistent, of order at least $1$, exactly when $\sum_i b_i = 1$.
Example: $s = 1$, $a_{11} = 0$, $b_1 = 1$, $c_1 = 0$ gives $y_{k+1} = y_k + h f(t_k, y_k)$, the [[Euler method|0JARTXX4XP]].
The classical method has $s = 4$, $c = (0, \tfrac12, \tfrac12, 1)$, $a_{21} = a_{32} = \tfrac12$, $a_{43} = 1$, the other $a_{ij} = 0$, and $b = (\tfrac16, \tfrac13, \tfrac13, \tfrac16)$; it has order $4$.
For $y' = \lambda y$ it gives $y_{k+1} = \left( 1 + z + \frac{z^2}{2} + \frac{z^3}{6} + \frac{z^4}{24} \right) y_k$ with $z = h \lambda$, the Taylor polynomial of $e^z$ of degree $4$.
Non-example: the two-step formula $y_{k+2} = y_k + 2 h f(t_{k+1}, y_{k+1})$ is not a Runge–Kutta method, since $y_{k+2}$ depends on two earlier values and not only on $y_{k+1}$; it is a linear multistep method.
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# k-vertex-connected graph

## Definition

Let $k$ be a positive integer and let $G = (V, E)$ be a finite [[graph|0JD5C82QMB]].
The graph $G$ is $k$-vertex-connected, or $k$-connected, if:

1. $V$ has more than $k$ elements.
2. For each set $X \subseteq V$ with fewer than $k$ elements, the [[induced subgraph|0V05FC4GKQ]] $G - X$ on $V \setminus X$ is [[connected|0FFAYNMAJZ]].

The vertex connectivity $\kappa(G)$ is the largest integer $k$ such that $G$ is $k$-connected, and $\kappa(G) = 0$ if there is no such $k$.

## Remarks

A graph is $1$-connected if and only if it is connected and has at least $2$ vertices.
A $k$-connected graph is also $j$-connected for each $j \le k$.
Example: the [[cycle graph|0QQ0F12962]] $C_n$ with $n \ge 3$ is $2$-connected: it has $n > 2$ vertices, and removing one vertex leaves a path, which is connected.
It is not $3$-connected, since removing two non-adjacent vertices disconnects it when $n \ge 4$, and $C_3$ has only $3$ vertices.
Non-example: a [[tree|0MJ0JS72DZ]] with at least $3$ vertices is not $2$-connected, because removing a vertex that is not a leaf disconnects it.
The [[complete graph|03WKB2H474]] $K_n$ has $\kappa(K_n) = n - 1$; clause 1 is what prevents it from being $n$-connected.
Every $k$-connected graph has minimum degree at least $k$.
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<document source="https://tamnd.github.io/math2/node/JJP4JFH3C6.md" tag="JJP4JFH3C6" kind="concept">
# Induced representation

## Summary

Induction builds a [[representation|WYGRXCG2RN]] of a group $G$ from a representation of a [[subgroup|0X1JXT47GA]] $H$.
The induced representation is a sum of copies of the given space, one for each left coset of $H$, which $G$ permutes.
Induction is adjoint to restriction, by [[Frobenius reciprocity|PAP76H0ND9]].
Many representations of finite groups are constructed this way.

## Explanations

### L1 human

Suppose you know how a subgroup $H$ acts on a space $W$.
Take one copy of $W$ for each coset of $H$ in $G$, and let $G$ shuffle the copies while acting inside them through $H$.
The result is a representation of the whole group.
For instance, inducing the trivial representation of $H$ gives the permutation representation of $G$ on the cosets $G/H$.
Inducing from the trivial subgroup gives the regular representation, where $G$ acts on its own group algebra.
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<document source="https://tamnd.github.io/math2/node/JJQEGRK2DG.md" tag="JJQEGRK2DG" kind="concept">
# Scorer's function

## Summary

Scorer's functions $\operatorname{Gi}$ and $\operatorname{Hi}$ are particular solutions of the inhomogeneous Airy equations $y'' - x y = \mp\frac{1}{\pi}$.
They are built from the same integrals as the [[Airy functions|1N5D4PN92W]], split into two pieces.
Their sum is the Airy function $\operatorname{Bi}$.
They appear when a turning-point problem has a constant forcing term.

## Explanations

### L1 human

The Airy function $\operatorname{Bi}$ is an integral of a sum of two terms, one exponential and one oscillating.
Taking each term alone gives Scorer's two functions, $\operatorname{Hi}$ and $\operatorname{Gi}$.
Each piece alone no longer solves the Airy equation, but it solves it up to a constant.
For example, $\operatorname{Hi}(0) = \frac{1}{\pi} \int_0^\infty e^{-t^3/3}\, dt$, a positive number.
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<document source="https://tamnd.github.io/math2/node/JJSXXJQGEH.md" tag="JJSXXJQGEH" kind="example">
# Spin group

## Example

Let $n \ge 2$ and let $\mathrm{Cl}_n$ be the real [[Clifford algebra|A38B1QFK20]] of $\R^n$ with the relations $v^2 = -\abs{v}^2$ for $v \in \R^n$.
The *spin group* $\mathrm{Spin}(n)$ is the set of products $u_1 u_2 \cdots u_{2k}$ in $\mathrm{Cl}_n$ of an even number of unit vectors $u_i \in \R^n$, with the Clifford product.
It is a group: the product of two such elements is again one.
The inverse of $u_1 \cdots u_{2k}$ is $u_{2k} \cdots u_1$, because $u_i u_i = -1$ and the number of factors is even.

For $x \in \mathrm{Spin}(n)$ and $v \in \R^n$, the element $x v x^{-1}$ lies again in $\R^n$.
Check for one unit vector $u$: $u^{-1} = -u$, and $uv = -vu$ when $v \perp u$.
So $-u v u^{-1}$ equals $v$ for $v \perp u$ and $-u$ for $v = u$, and $v \mapsto -u v u^{-1}$ is the reflection $r_u$ of $\R^n$ in the hyperplane $u^\perp$.
For $x = u_1 \cdots u_{2k}$ the signs cancel, and $v \mapsto x v x^{-1}$ is the product $r_{u_1} \cdots r_{u_{2k}}$ of $2k$ reflections, which lies in [[SO(n)|0CGYNDCWRR]].

This gives a homomorphism $\rho \colon \mathrm{Spin}(n) \to SO(n)$, $\rho(x)(v) = x v x^{-1}$.
It is surjective, because every element of $SO(n)$ is a product of an even number of reflections, and its kernel is $\set{1, -1}$.

## Remarks

$\mathrm{Spin}(n)$ is a compact connected Lie group, and $\rho$ is a two-sheeted [[covering map|3XZRRE165N]].
For $n \ge 3$ the group $\mathrm{Spin}(n)$ is simply connected, so it is the universal cover of $SO(n)$.
For $n = 2$ the group $\mathrm{Spin}(2)$ is a circle, and $\rho$ is the double cover $z \mapsto z^2$ of the circle $SO(2)$.
In low dimensions there are isomorphisms $\mathrm{Spin}(3) \cong$ [[SU(2)|E5P8AG6CK7]], $\mathrm{Spin}(4) \cong SU(2) \times SU(2)$, $\mathrm{Spin}(5) \cong \mathrm{Sp}(2)$ and $\mathrm{Spin}(6) \cong SU(4)$.
The same construction for a quadratic form of signature $(p, q)$ gives the groups $\mathrm{Spin}(p, q)$.

## Explanations

### L1 human

The spin group is a twofold cover of the rotation group: each rotation corresponds to exactly two elements of $\mathrm{Spin}(n)$, which differ by a sign.
It is built from products of unit vectors in the Clifford algebra, in which a vector squares to minus its length squared.
Each unit vector stands for a reflection, and a product of an even number of reflections is a rotation.
Physicists need the spin group because electrons and other spin $1/2$ particles change sign under a full turn, so they transform under $\mathrm{Spin}(3)$ rather than $SO(3)$.
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# Eckart–Young theorem

## Statement

Let $A$ be an $m \times n$ complex matrix with [[singular value decomposition|000NPTMX6J]] $A = \sum_{i=1}^{r} \sigma_i u_i v_i^*$.
Here $r$ is the [[rank|0ZC9YTXZRY]] of $A$, $\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_r > 0$ are its [[singular values|09VMDSRXWJ]], and $u_1, \dots, u_r$ and $v_1, \dots, v_r$ are orthonormal families in $\C^m$ and $\C^n$.
Let $0 \le k < r$ and put $A_k = \sum_{i=1}^{k} \sigma_i u_i v_i^*$.
Then for every $m \times n$ complex matrix $B$ of rank at most $k$:

1. $\norm{A - B}_2 \ge \norm{A - A_k}_2 = \sigma_{k+1}$, where $\norm{\cdot}_2$ is the [[operator norm|06EKW1AKXF]] for the Euclidean norms.
2. $\norm{A - B}_F \ge \norm{A - A_k}_F = \left(\sigma_{k+1}^2 + \cdots + \sigma_r^2\right)^{1/2}$, where $\norm{\cdot}_F$ is the [[Frobenius norm|JT8Q204E4R]].

## Remarks

For the operator norm, the kernel of $B$ has dimension at least $n - k$, so it meets the span of $v_1, \dots, v_{k+1}$ in a unit vector $x$.
Then $\norm{(A - B)x} = \norm{Ax} \ge \sigma_{k+1}$.
The Frobenius case follows from Weyl's inequalities for singular values, $\sigma_{i+j-1}(A) \le \sigma_i(A - B) + \sigma_j(B)$, applied with $j = k+1$.
For example, if $A = \operatorname{diag}(3, 2, 1)$, the best rank one approximation is $\operatorname{diag}(3, 0, 0)$, with operator norm error $2$ and Frobenius error $\sqrt{5}$.
The minimiser $A_k$ is unique in the Frobenius norm when $\sigma_k > \sigma_{k+1}$; in the operator norm it is not unique in general.
The same result holds for real matrices with real singular value decompositions.

## Explanations

### L1 human

The theorem says how to approximate a matrix as well as possible by a simpler matrix of low rank.
Write the matrix as a sum of rank one pieces from its singular value decomposition, and keep the largest ones.
The error is controlled by the singular values that were thrown away.

### L2 human

This is the theoretical basis of principal component analysis and of truncated SVD compression.
Mirsky extended it to every unitarily invariant norm, so it is often called the Eckart–Young–Mirsky theorem.
It also shows that $\sigma_{k+1}$ is the distance from $A$ to the set of matrices of rank at most $k$.
In particular, for a square invertible $A$, the smallest singular value is the distance to the singular matrices.
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<document source="https://tamnd.github.io/math2/node/JJWETW5AT8.md" tag="JJWETW5AT8" kind="concept">
# Yang–Mills equations

## Summary

The Yang–Mills equations are the field equations of a gauge theory: they are nonlinear partial differential equations for a [[connection|NJ21PN5ZGJ]] on a [[principal bundle|8E37E74XVP]].
They are the Euler–Lagrange equations of the Yang–Mills functional, the integral of the squared norm of the [[curvature|ZAJFW1S078]].
For the circle group they reduce to Maxwell's equations in vacuum, and for non-abelian groups they describe the strong and weak forces.
In dimension four their self-dual solutions, the instantons, led to deep results on the topology of four-manifolds.

## Explanations

### L1 human

In electromagnetism the field is described by a potential, and the field strength measures how the potential twists.
Gauge theories replace numbers in this picture by elements of a matrix group, so twists in different directions no longer commute.
The Yang–Mills equations say that the field strength is as balanced as possible, in the sense that it is a critical point of its total energy.
The simplest solutions are flat connections, whose field strength is zero everywhere.
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<document source="https://tamnd.github.io/math2/node/JJZ5X28NNR.md" tag="JJZ5X28NNR" kind="definition">
# Weyl algebra

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $n \ge 1$ be an integer.
Let $k\langle x_1, \dots, x_n, y_1, \dots, y_n \rangle$ be the [[free algebra|2P3Y2FRCSZ]] on $2n$ generators.
Let $I$ be the [[two-sided ideal|0CD87SFM2N]] generated by the elements
$$x_i x_j - x_j x_i, \qquad y_i y_j - y_j y_i, \qquad y_i x_j - x_j y_i - \delta_{ij}$$
for $1 \le i, j \le n$, where $\delta_{ij}$ is $1$ if $i = j$ and $0$ otherwise.
The *$n$-th Weyl algebra* is the [[quotient ring|0076HWHGHQ]] $A_n(k) = k\langle x_1, \dots, x_n, y_1, \dots, y_n \rangle / I$.

## Remarks

The monomials $x_1^{a_1} \cdots x_n^{a_n} y_1^{b_1} \cdots y_n^{b_n}$ with $a_i, b_i \ge 0$ form a basis of $A_n(k)$ over $k$.
Example: in $A_1(k)$, with $y x = x y + 1$, one gets $y x^2 = (x y + 1) x = x (y x) + x = x(xy + 1) + x = x^2 y + 2x$.
This matches the rule $\frac{d}{dx}(x^2 f) = x^2 f' + 2 x f$ for the operators of multiplication by $x$ and differentiation.
Non-example: the commutative polynomial ring $k[x, y]$ satisfies the first two kinds of relations but has $y x - x y = 0 \ne 1$.
If $k$ has characteristic $0$, then $A_n(k)$ is a simple Noetherian domain with center $k$.
If $k$ has characteristic $p > 0$, then $x_1^p$ lies in the center of $A_n(k)$, and $A_n(k)$ is not simple.
Equivalently, $A_1(k)$ is the Ore extension $k[x][y; \id, d/dx]$.
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<document source="https://tamnd.github.io/math2/node/JK0S0WYN01.md" tag="JK0S0WYN01" kind="definition">
# Proper morphism

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].
The morphism $f$ is *proper* if it satisfies the following three conditions.

1. $f$ is [[separated|X30G4V7VAJ]].
2. $f$ is [[of finite type|AHTE5DVX61]].
3. $f$ is [[universally closed|W0B0BBAH35]].

## Remarks

Let $k$ be a field.
The structure morphism $\mathbb{P}^n_k \to \operatorname{Spec} k$ of projective $n$-space is proper.
It is separated and of finite type, and it is universally closed by the main theorem of elimination theory, which can also be derived from the [[valuative criterion|QZN6P7AG33]].
The structure morphism $\mathbb{A}^1_k \to \operatorname{Spec} k$ is separated and of finite type but not proper.
After base change along $\mathbb{A}^1_k \to \operatorname{Spec} k$ it becomes the projection $\mathbb{A}^2_k \to \mathbb{A}^1_k$, $(x, y) \mapsto x$.
This projection maps the closed curve $xy = 1$ onto $\mathbb{A}^1_k \setminus \set{0}$, which is not closed, so clause 3 fails.
Closed immersions and finite morphisms are proper, and proper morphisms are stable under composition and base change.
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<document source="https://tamnd.github.io/math2/node/JKA25KDPQP.md" tag="JKA25KDPQP" kind="concept">
# Alexander polynomial

## Summary

The Alexander polynomial is a [[knot invariant|GZQ5ZE3X6G]] that assigns to each [[knot|QYEAZTYGPT]] a Laurent polynomial with integer coefficients.
It was the first polynomial knot invariant, found by James Waddell Alexander II in the 1920s.
It can be computed from a Seifert matrix, from a knot diagram, or from the homology of the infinite cyclic cover of the knot complement.
It detects many knots but not all; some nontrivial knots have Alexander polynomial $1$.

## Explanations

### L1 human

The Alexander polynomial turns a knot into a polynomial such as $t - 1 + t^{-1}$.
If two knots have different polynomials, they are different knots.
The [[unknot|1TQ06R4Q66]] has polynomial $1$, while the [[trefoil knot|VYTYSVHN4A]] has $t - 1 + t^{-1}$, so the trefoil is really knotted.
The polynomial records how the loops on a surface bounded by the knot wind around each other.
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# Group C*-algebra

## Summary

The group C*-algebra $C^*(G)$ of a [[locally compact group|AHV05ZBCYW]] $G$ is a completion of the [[group algebra|6GD25S53ND]] $L^1(G)$.
It uses the largest norm coming from [[unitary representations|74BPYJNZTM]] of $G$.
It is a C*-algebra whose nondegenerate representations correspond exactly to the unitary representations of $G$.
The reduced group C*-algebra $C^*_r(G)$ uses the left regular representation alone.
These algebras let one apply the theory of operator algebras to groups, and they are central in noncommutative geometry and the Baum–Connes conjecture.

## Explanations

### L1 human

The integrable functions on a group form an algebra, but its norm is not well suited to operator theory.
One fixes this by measuring each function through how large an operator it becomes in every unitary representation, and then filling in the gaps.
For the integers, this turns absolutely summable sequences into all continuous functions on the circle, via Fourier series.
So $C^*(\Z)$ is the algebra $C(\mathbb{T})$ of continuous functions on the circle.
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# Stable map

## Definition

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], and give the set $C^\infty(M, N)$ of smooth maps the [[Whitney $C^\infty$ topology|7PN9MD1PQB]].
Two smooth maps $f, g \colon M \to N$ are *equivalent* if there are [[diffeomorphisms|0KZZ70ZHXP]] $\varphi \colon M \to M$ and $\psi \colon N \to N$ with $g = \psi \circ f \circ \varphi^{-1}$.
A smooth map $f \colon M \to N$ is *stable* if there is a neighbourhood $U$ of $f$ in $C^\infty(M, N)$ such that every $g \in U$ is equivalent to $f$.

## Remarks

Example: the height function $f \colon S^1 \to \R$, $f(x, y) = y$, on the unit circle is stable.
Its critical points $(0, \pm 1)$ are nondegenerate with distinct values $\pm 1$.
Any $g$ close enough to $f$ in the $C^2$ sense also has exactly two nondegenerate critical points, a maximum and a minimum.
Their values are distinct, and such a $g$ is equivalent to $f$.
Non-example: a constant map $c \colon S^1 \to \R$ is not stable.
The maps $c + \varepsilon f$ come arbitrarily close to $c$ and have nonzero differential at some points, while every map equivalent to $c$ is constant.
For a compact manifold $M$, a function $f \colon M \to \R$ is stable if and only if all its critical points are nondegenerate and its critical values are pairwise distinct.
The equivalence used here is often called $\mathcal{A}$-equivalence, or left-right equivalence.
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<document source="https://tamnd.github.io/math2/node/JKCFHKT13F.md" tag="JKCFHKT13F" kind="definition">
# Fourier–Stieltjes transform

## Definition

Let $n \ge 1$ and let $\mu$ be a finite [[complex measure|NY6TYHRGMS]] on the [[Borel sets|0WQXY32Q8D]] of $\R^n$.
The *Fourier–Stieltjes transform* of $\mu$ is the function $\hat\mu \colon \R^n \to \C$ given by
$$\hat\mu(\xi) = \int_{\R^n} e^{-2\pi i x \cdot \xi}\, d\mu(x), \qquad \xi \in \R^n.$$

## Remarks

The integral exists because the integrand has modulus $1$ and $\mu$ has finite total variation $\norm{\mu}$.
So $\abs{\hat\mu(\xi)} \le \norm{\mu}$, and $\hat\mu$ is uniformly continuous on $\R^n$.

Example: the Dirac measure $\delta_a$ at a point $a \in \R^n$ is a finite measure, and $\hat\delta_a(\xi) = e^{-2\pi i a \cdot \xi}$; in particular $\hat\delta_0 = 1$.

Example: if $d\mu = f\, dx$ with $f \in L^1(\R^n)$, then $\hat\mu$ is the usual [[Fourier transform|DPVM34ADZV]] $\hat f$.

Non-example: Lebesgue measure on $\R$ is not finite, and $\int_\R e^{-2\pi i x \xi}\, dx$ diverges for every $\xi$; its transform exists only as a tempered distribution.

Unlike the transform of an integrable function, $\hat\mu$ need not tend to $0$ at infinity, as the example $\delta_0$ shows.
The transform is injective: $\hat\mu = 0$ implies $\mu = 0$.
It turns convolution of measures into products, $\widehat{\mu * \nu} = \hat\mu\, \hat\nu$.
For a probability measure $\mu$ on $\R$, the characteristic function $\varphi(t) = \int e^{itx}\, d\mu(x)$ equals $\hat\mu(-t / (2\pi))$.
On $\R$, writing $\mu$ through a function $F$ of bounded variation, the transform is the Stieltjes integral $\int e^{-2\pi i x \xi}\, dF(x)$.
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# Chudnovsky algorithm

## Summary

The Chudnovsky algorithm computes the digits of [[pi|0C1VYVNV2N]] by summing a fast-converging series for $1/\pi$ found by David and Gregory Chudnovsky.
Each term of the series adds about fourteen correct decimal digits.
Evaluated with binary splitting and fast multiplication, it has been used for many record computations of $\pi$.
It belongs to the same family as [[Ramanujan's series for 1/pi|8PZPMF2QJD]].

## Explanations

### L1 human

To compute millions of digits of $\pi$ one needs a formula whose terms shrink very quickly.
The Chudnovsky series is such a formula: each term is about $10^{-14}$ times the previous one.
The terms are fractions of factorials and powers of the integer $640320$, so the sum can be organised as exact integer arithmetic until the final division.
The magic number $640320$ comes from the theory of elliptic curves and modular functions.
</document>

<document source="https://tamnd.github.io/math2/node/JKMMJ8RFC6.md" tag="JKMMJ8RFC6" kind="object">
# Hilbert cube

## Description

Let $\ell^2$ be the space of real sequences $x = (x_n)_{n \ge 1}$ with $\sum_n x_n^2 < \infty$ and the norm $\norm{x}_2 = (\sum_n x_n^2)^{1/2}$.
The *Hilbert cube* is the subset
$$H = \set{x \in \ell^2 : 0 \le x_n \le 1/n \text{ for all } n \ge 1}$$
with the metric induced by $\norm{\cdot}_2$.

## Remarks

The map $(t_n)_{n \ge 1} \mapsto (t_n / n)_{n \ge 1}$ is a [[homeomorphism|0QT00PB5G2]] from $[0, 1]^{\N}$, with the [[product topology|0GBG8B49AA]], onto $H$.
So $H$ is [[compact|0TVJZYB6SH]], for example by [[Tychonoff's theorem|0JTRSJKCFB]].
It is also convex in $\ell^2$.

Every [[separable|0Z2P41SVDP]] metrizable space is homeomorphic to a subspace of $H$.
This is the embedding behind the [[Urysohn metrization theorem|HBFHDH4EJX]].

The Hilbert cube has the [[fixed-point property|JQDHHFXBF3]]: every continuous map $H \to H$ has a fixed point.
This follows from the Schauder fixed-point theorem, since $H$ is a compact convex subset of a Banach space.

## Explanations

### L1 human

The Hilbert cube is an infinite-dimensional cube whose sides get shorter and shorter: $1$, $1/2$, $1/3$, and so on.
Because the sides shrink, the cube stays compact, unlike the unit ball of an infinite-dimensional space.
It is big enough to contain a copy of every separable metric space, up to homeomorphism.
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<document source="https://tamnd.github.io/math2/node/JKPSBMX80N.md" tag="JKPSBMX80N" kind="definition">
# Tate–Shafarevich group

## Definition

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] with algebraic closure $\overline{K}$ and absolute [[Galois group|0XTY86BZKY]] $G_K = \Gal(\overline{K}/K)$, and let $E$ be an [[elliptic curve|427N6S6QQW]] over $K$.
For each [[place|BT5V3TSNY6]] $v$ of $K$, let $K_v$ be the completion of $K$ at $v$.
Fix an embedding $\overline{K} \to \overline{K_v}$; it identifies $G_{K_v}$ with a decomposition subgroup of $G_K$.
Let $H^1(K, E) = H^1(G_K, E(\overline{K}))$ be the first [[Galois cohomology|9MPCZ3EAJM]] group, and similarly $H^1(K_v, E) = H^1(G_{K_v}, E(\overline{K_v}))$.
Let the map below have as components the restrictions, with the product over all places $v$ of $K$.
The *Tate–Shafarevich group* of $E$ over $K$ is
$$\text{Ш}(E/K) = \ker\left( H^1(K, E) \to \prod_{v} H^1(K_v, E) \right).$$

## Remarks

The group $H^1(K, E)$ classifies principal homogeneous spaces of $E$: curves $C$ over $K$ with a simply transitive action of $E$ defined over $K$.
A class is trivial exactly when $C$ has a point over $K$, so $\text{Ш}(E/K)$ consists of the classes of such curves $C$ that have a point over every $K_v$.

Example: Selmer's curve $C \colon 3x^3 + 4y^3 + 5z^3 = 0$ in $\mathbb{P}^2_{\Q}$ has points over $\R$ and over every $\Q_p$, but no point over $\Q$.
It is a principal homogeneous space for its Jacobian $E \colon x^3 + y^3 + 60z^3 = 0$, so it gives a nonzero element of $\text{Ш}(E/\Q)$.

Non-example: let $C$ be a principal homogeneous space of $E$ over $\Q$ with $C(\R) = \emptyset$.
Its class in $H^1(\Q, E)$ has nonzero restriction at the real place, so it is not in $\text{Ш}(E/\Q)$.

The group $\text{Ш}(E/K)$ is a torsion abelian group, and each of its $n$-torsion subgroups is finite, by the finiteness of the [[Selmer group|36YNSTESMQ]].
It is conjectured to be finite; when it is finite, the Cassels–Tate pairing shows that its order is a perfect square.
The letter Ш is the Cyrillic letter sha.
</document>

<document source="https://tamnd.github.io/math2/node/JKQP7AFZ0E.md" tag="JKQP7AFZ0E" kind="definition">
# Separable Banach space

## Definition

Let $X$ be a [[Banach space|06GZ956657]] over $\R$ or $\C$.
The space $X$ is *separable* if it is [[separable|0Z2P41SVDP]] as a metric space.
That is, there is a countable subset $D \subseteq X$ that is [[dense|0BSJSX7ME2]] in $X$.

## Remarks

Equivalently, there is a countable set $S \subseteq X$ whose linear span is dense in $X$.
Indeed, finite combinations of elements of $S$ with rational (or Gaussian rational) coefficients then form a countable dense set.
Example: $\ell^1$ is separable.
The finitely supported sequences with rational entries form a countable set $D$.
For $x \in \ell^1$ and $\varepsilon > 0$, choose $N$ with $\sum_{n > N} \abs{x_n} < \varepsilon/2$.
Then choose rational $q_n$ with $\sum_{n \le N} \abs{x_n - q_n} < \varepsilon/2$.
Then $(q_1, \ldots, q_N, 0, \ldots) \in D$ is within $\varepsilon$ of $x$.
Non-example: $\ell^\infty$ is not separable.
For each subset $A$ of $\N$ its indicator sequence $\mathbf{1}_A$ lies in $\ell^\infty$, and $\norm{\mathbf{1}_A - \mathbf{1}_B}_\infty = 1$ for $A \ne B$.
The open balls of radius $1/2$ around these uncountably many points are disjoint, so no countable set meets them all.
If the dual $X^*$ is separable, then $X$ is separable; the converse fails, as $X = \ell^1$ with $X^* = \ell^\infty$ shows.
</document>

<document source="https://tamnd.github.io/math2/node/JKXTT55WCV.md" tag="JKXTT55WCV" kind="definition">
# End and coend

## Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be [[categories|A2DE66937D]], and let $S \colon \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathcal{D}$ be a [[functor|28NQ0KRHD3]] on the [[product|AFNE9C4NH6]] of the [[opposite category|DAMK8XE1FR]] with $\mathcal{C}$.

A *wedge* to $S$ is an object $w$ of $\mathcal{D}$ with morphisms $e_c \colon w \to S(c, c)$ for all objects $c$.
They must satisfy $S(c, f) \circ e_c = S(f, c') \circ e_{c'}$ for every morphism $f \colon c \to c'$ of $\mathcal{C}$.
An *end* of $S$ is a wedge $(E, e)$ such that every wedge $(w, e')$ factors as $e'_c = e_c \circ u$ for a unique $u \colon w \to E$.

A *cowedge* from $S$ is an object $w$ with morphisms $\omega_c \colon S(c, c) \to w$.
They must satisfy $\omega_c \circ S(f, c) = \omega_{c'} \circ S(c', f)$ for every $f \colon c \to c'$.
A *coend* of $S$ is a cowedge $(E, \omega)$ such that every cowedge $(w, \omega')$ factors as $\omega'_c = u \circ \omega_c$ for a unique $u \colon E \to w$.

## Remarks

The end is written $\int_{c} S(c, c)$ and the coend $\int^{c} S(c, c)$.
Both are unique up to a unique isomorphism.
For $\mathcal{C}$ small and $\mathcal{D}$ complete, the end is the equalizer of two maps $\prod_c S(c, c) \rightrightarrows \prod_{f \colon c \to c'} S(c, c')$.

Example: let $F, G \colon \mathcal{C} \to \mathcal{E}$ be functors with $\mathcal{C}$ small and $\mathcal{E}$ locally small, and let $S(c, c') = \Hom_{\mathcal{E}}(F c, G c')$ in $\mathbf{Set}$.
A wedge from a one-point set is a family $\alpha_c \colon F c \to G c$ with $G f \circ \alpha_c = \alpha_{c'} \circ F f$, that is, a natural transformation.
So $\int_c \Hom(F c, G c)$ is the set of natural transformations $F \Rightarrow G$.

Non-example: in the same setting, the set $\prod_c \Hom(F c, G c)$ of all families, with its projections, is in general not a wedge.
The wedge condition would require every family $(\alpha_c)$ to be natural.
It fails as soon as some family has $G f \circ \alpha_c \neq \alpha_{c'} \circ F f$ for some $f \colon c \to c'$.
</document>

<document source="https://tamnd.github.io/math2/node/JM98RN8HVF.md" tag="JM98RN8HVF" kind="definition">
# Lambert quadrilateral

## Definition

Let $ABCD$ be a convex quadrilateral in an [[absolute plane|VKFPCSRNDH]].
It is a *Lambert quadrilateral* if three of its [[angles|0GJAJC6TNZ]], say $\angle DAB$, $\angle ABC$ and $\angle BCD$, are right angles.

## Remarks

Example: in $\R^2$, the rectangle with vertices $(0,0)$, $(3,0)$, $(3,1)$, $(0,1)$ is a Lambert quadrilateral, since all four of its angles are right.
Non-example: the quadrilateral with vertices $(0,0)$, $(2,0)$, $(2,1)$, $(0,3)$ has right angles at $(0,0)$ and $(2,0)$ only, so it is not a Lambert quadrilateral.
The fourth angle $\angle CDA$ is right in the Euclidean plane and acute in the hyperbolic plane.
</document>

<document source="https://tamnd.github.io/math2/node/JMBXBG860R.md" tag="JMBXBG860R" kind="definition">
# Ordered ring

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $\le$ be a [[total order|0316AYH1SC]] on $R$.
The pair $(R, \le)$ is an *ordered ring* if for all $a, b, c \in R$:

1. if $a \le b$, then $a + c \le b + c$;
2. if $0 \le a$ and $0 \le b$, then $0 \le ab$.

## Remarks

So an ordered ring is a commutative [[partially ordered ring|A0NP6HQNPX]] whose order is total.
Example: $\Z$ with the usual order is an ordered ring, and so is every [[ordered field|010JCV9J4E]].
Example: $\Z[x]$ is an ordered ring if $p > 0$ means that the leading coefficient of $p$ is positive.
In any ordered ring, $a^2 \ge 0$ for all $a$: if $a \ge 0$ this is clause 2, and if $a < 0$ then $-a > 0$ by clause 1 and $a^2 = (-a)^2$.
Non-example: the ring $\Z[i]$ of Gaussian integers has no such order.
Squares would be nonnegative, so $1 = 1^2 \ge 0$ and $-1 = i^2 \ge 0$, and adding gives $0 \ge 1$; with $1 \ge 0$ this forces $1 = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/JMCKACGCQ0.md" tag="JMCKACGCQ0" kind="statement">
# Four lemma

## Statement

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], such as the category of modules over a ring.
Consider a [[commutative diagram|WV898P4X16]]
$$\begin{array}{ccccccc} A & \to & B & \to & C & \to & D \\ \downarrow{\scriptstyle \alpha} & & \downarrow{\scriptstyle \beta} & & \downarrow{\scriptstyle \gamma} & & \downarrow{\scriptstyle \delta} \\ A' & \to & B' & \to & C' & \to & D' \end{array}$$
in $\mathcal{A}$ whose two rows are [[exact|JB7ZBFBA9C]].

1. If $\alpha$ is an [[epimorphism|RVXDSY04KG]] and $\beta$ and $\delta$ are [[monomorphisms|Y2NP1JFR86]], then $\gamma$ is a monomorphism.
2. If $\delta$ is a monomorphism and $\alpha$ and $\gamma$ are epimorphisms, then $\beta$ is an epimorphism.

## Remarks

Part 1 is a diagram chase.
Let the row maps be $r \colon A \to B$, $s \colon B \to C$, $t \colon C \to D$ and $r', s', t'$ below.
If $\gamma(x) = 0$ for $x \in C$, then $\delta(t(x)) = t'(\gamma(x)) = 0$, so $t(x) = 0$ and $x = s(y)$ for some $y \in B$.
Then $s'(\beta(y)) = 0$, so $\beta(y) = r'(u')$ with $u' = \alpha(u)$ for some $u \in A$; now $\beta(y - r(u)) = 0$, so $y = r(u)$ and $x = s(r(u)) = 0$.
Part 2 is the dual statement, obtained by reversing all arrows.
Example: take $A = A' = 0$, so that the rows are exact sequences $0 \to B \to C \to D$ and $0 \to B' \to C' \to D'$.
If $\beta$ and $\delta$ are monomorphisms, part 1 shows that $\gamma$ is a monomorphism.

## Explanations

### L1 human

The four lemma is the five lemma split into two halves.
One half says when the third vertical map is one-to-one, and the other half says when the second vertical map is onto.
Each half needs only four columns of the ladder.

### L2 human

The two parts are proved by element chasing in modules, and hold in every abelian category by the [[Freyd–Mitchell embedding theorem|8V296XR2M3]] or by [[pseudo-elements|1WV2THA6NY]].
Applying part 1 to the left four columns and part 2 to the right four columns of a five-column ladder gives the five lemma.
The four lemma is useful when only one of injectivity or surjectivity is needed, so fewer hypotheses have to be checked.
</document>

<document source="https://tamnd.github.io/math2/node/JMFHAMX0JG.md" tag="JMFHAMX0JG" kind="definition">
# Tarski's axioms

## Definition

The language has variables for points and two relations.
The ternary relation $B(x, y, z)$ reads "$y$ lies [[between|CE8GX6ADZS]] $x$ and $z$".
The quaternary relation $xy \equiv zw$ reads "the segment $xy$ is congruent to the segment $zw$".
*Tarski's axioms* are the following first-order sentences, each universally closed.

1. Reflexivity of congruence: $xy \equiv yx$.
2. Identity of congruence: if $xy \equiv zz$, then $x = y$.
3. Transitivity of congruence: if $xy \equiv zu$ and $xy \equiv vw$, then $zu \equiv vw$.
4. Identity of betweenness: if $B(x, y, x)$, then $x = y$.
5. Pasch: if $B(x, u, z)$ and $B(y, v, z)$, then there is $a$ with $B(u, a, y)$ and $B(v, a, x)$.
6. Continuity, a schema: let $\varphi(x)$ and $\psi(y)$ be formulas in which $a$ and $b$ are not free. If some $a$ has $B(a, x, y)$ whenever $\varphi(x)$ and $\psi(y)$, then some $b$ has $B(x, b, y)$ whenever $\varphi(x)$ and $\psi(y)$.
7. Lower dimension: there are $a, b, c$ such that none of $B(a, b, c)$, $B(b, c, a)$, $B(c, a, b)$ holds.
8. Upper dimension: if $xu \equiv xv$, $yu \equiv yv$, $zu \equiv zv$ and $u \ne v$, then $B(x, y, z)$ or $B(y, z, x)$ or $B(z, x, y)$.
9. Euclid: if $B(x, u, t)$, $B(y, u, z)$ and $x \ne u$, then there are $v, w$ with $B(x, y, v)$, $B(x, z, w)$ and $B(v, t, w)$.
10. Five segments: if $x \ne y$, $B(x, y, z)$, $B(x', y', z')$, $xy \equiv x'y'$, $yz \equiv y'z'$, $xu \equiv x'u'$ and $yu \equiv y'u'$, then $zu \equiv z'u'$.
11. Segment construction: for all $x, y, a, b$ there is $z$ with $B(x, y, z)$ and $yz \equiv ab$.

## Remarks

Example: the plane $\R^2$, with $B(x, y, z)$ meaning that $y$ is on the closed segment $[x, z]$ and $xy \equiv zw$ meaning $\abs{x - y} = \abs{z - w}$, satisfies all eleven axioms.
Non-example: the line $\R$ with the same relations fails the lower dimension axiom, since of any three reals one lies between the other two.
The models are exactly the planes $K^2$ over [[real closed fields|DDY3Q3PJ6W]] $K$, with the relations defined as in $\R^2$.
The [[theory|92SWXST1ZM]] they define is [[complete|655K8Y7VS8]] and decidable.
Axiom 9 is a form of the parallel postulate; without it one gets an axiom system for absolute geometry.
</document>

<document source="https://tamnd.github.io/math2/node/JMJEVS7S1V.md" tag="JMJEVS7S1V" kind="definition">
# Characteristic variety

## Definition

Let $P$ be a [[linear differential operator|6DZETZ5T9G]] of order $m$ on an open set $\Omega \subseteq \R^n$, with [[principal symbol|F31T650DKY]] $\sigma_m(P)$.
The *characteristic variety* of $P$ is the set
$$\operatorname{Char}(P) = \set{(x, \xi) \in \Omega \times (\R^n \setminus \set{0}) : \sigma_m(P)(x, \xi) = 0}.$$
A nonzero vector $\xi$ is *characteristic* for $P$ at $x$ if $(x, \xi) \in \operatorname{Char}(P)$.
A $C^1$ hypersurface $S \subseteq \Omega$ is *noncharacteristic* at a point $x \in S$ if a [[normal vector|0WQJJCCTC8]] $\nu$ of $S$ at $x$ satisfies $\sigma_m(P)(x, \nu) \ne 0$.

## Remarks

Example: for the heat operator $P = \partial_t - \partial_x^2$ on $\R^2$ with variables $(t, x)$, the order is $2$ and $\sigma_2(P)((t, x), (\tau, \xi)) = -\xi^2$.
So $\operatorname{Char}(P)$ consists of the covectors with $\xi = 0$, that is, $(\tau, 0)$ with $\tau \ne 0$.
The lines $t = \text{const}$ have normal $(1, 0)$, so they are characteristic for the heat operator.
Non-example: for the same operator the lines $x = \text{const}$ have normal $(0, 1)$, and $\sigma_2 = -1 \ne 0$ there, so they are noncharacteristic.
The set $\operatorname{Char}(P)$ is conic: if $(x, \xi)$ lies in it, so does $(x, \lambda \xi)$ for every real $\lambda \ne 0$.
The operator is [[elliptic|B4TTNHSTRX]] exactly when $\operatorname{Char}(P)$ is empty.

## Explanations

### L1 human

The characteristic variety is the zero set of the principal symbol, with the zero covector removed.
</document>

<document source="https://tamnd.github.io/math2/node/JMST03Q37Q.md" tag="JMST03Q37Q" kind="definition">
# Jacobi equation

## Definition

Let $a < b$ and let $L \colon [a, b] \times \R \times \R \to \R$, written $L(x, y, p)$, be of class $C^3$.
Let $J(y) = \int_a^b L(x, y, y') \, dx$ be the [[integral functional|JRQZ803Q5Y]] of $L$.
Let $y_0 \in C^2([a, b])$ be a solution of the [[Euler–Lagrange equation|PS0VV149ET]] of $J$.
Put $P(x) = L_{pp}(x, y_0(x), y_0'(x))$ and $Q(x) = L_{yy}(x, y_0(x), y_0'(x)) - \frac{d}{dx} L_{yp}(x, y_0(x), y_0'(x))$.
The *Jacobi equation* of $J$ along $y_0$ is the linear [[ordinary differential equation|0NT64JV96A]]
$$-\frac{d}{dx}\big( P(x) \, h'(x) \big) + Q(x) \, h(x) = 0$$
for an unknown function $h$ on $[a, b]$.

## Remarks

For $h \in C^1([a, b])$ with $h(a) = h(b) = 0$, the [[second variation|4C3BANJ8AH]] of $J$ at $y_0$ equals $\int_a^b (P h'^2 + Q h^2) \, dx$.
The Jacobi equation is the Euler–Lagrange equation of this quadratic functional.
It is also the linearisation of the Euler–Lagrange equation of $J$ at $y_0$.
If $y(x, c)$ is a smooth family of extremals with $y(x, 0) = y_0(x)$, then $h = \partial y / \partial c$ at $c = 0$ solves it.

Example: for $L = p^2 - y^2$ and $y_0 = 0$, one has $P = 2$ and $Q = -2$.
The Jacobi equation is $h'' + h = 0$; the function $h = \sin x$ satisfies it, since $h'' = -\sin x = -h$.

Non-example: for the same $L$, the function $h = x$ does not solve it, since $-2h'' - 2h = -2x \ne 0$ for $x \ne 0$.
It does solve the Jacobi equation $h'' = 0$ of $L = p^2$, whose extremals are straight lines.

When $P > 0$ on $[a, b]$, the equation has a unique solution for given $h(a)$ and $h'(a)$.
For curves $y \colon [a, b] \to \R^n$ the same construction gives a linear system with matrices $P$ and $Q$.
</document>

<document source="https://tamnd.github.io/math2/node/JMTJG4B94M.md" tag="JMTJG4B94M" kind="definition">
# Kloosterman sum

## Definition

Let $m \ge 1$ be an integer, and let $a, b \in \Z$.
For a real $t$, write $e(t) = e^{2\pi i t}$.
For an integer $x$ coprime to $m$, let $\bar{x}$ be an integer with $x \bar{x} \equiv 1 \pmod{m}$.
The *Kloosterman sum* is
$$K(a, b; m) = \sum_{\substack{x \bmod m \\ \gcd(x, m) = 1}} e\!\left(\frac{a x + b \bar{x}}{m}\right),$$
where $x$ runs over a set of representatives of the invertible classes of the [[integers modulo m|03VFZ2WETT]].

## Remarks

The value does not depend on the choice of representatives or of $\bar{x}$, since the phase is defined modulo $1$.
Example: $m = 5$, $a = b = 1$.
The pairs $(x, \bar{x})$ are $(1, 1), (2, 3), (3, 2), (4, 4)$, so the phases are $2/5, 1, 1, 8/5$ and
$$K(1, 1; 5) = e(2/5) + 2 + e(3/5) = 2 + 2\cos(4\pi/5) \approx 0.382.$$
Non-example: the sum $\sum_{x = 0}^{m-1} e((a x + b \bar{x})/m)$ over all residues is not defined, since $\bar{x}$ does not exist when $\gcd(x, m) > 1$.
The sum is real, because replacing $x$ by $-x$ conjugates each term.
If $\gcd(a, m) = 1$, then $K(a, b; m) = K(1, ab; m)$.
For $b = 0$, the sum $K(a, 0; m)$ is the Ramanujan sum $c_m(a)$.
It is an [[exponential sum|97PBMQCPPM]] in which the phase function is $x \mapsto (ax + b\bar{x})/m$.
</document>

<document source="https://tamnd.github.io/math2/node/JMVHZ6HDNZ.md" tag="JMVHZ6HDNZ" kind="statement">
# Legendre–Hadamard condition

## Statement

Let $\Omega \subseteq \R^n$ be a bounded open set and let $m \ge 1$.
Let $L \colon \overline{\Omega} \times \R^m \times \R^{m \times n} \to \R$, written $L(x, u, \xi)$ with $\xi = (\xi^i_\alpha)$, be of class $C^2$.
Let $J(u) = \int_\Omega L(x, u, \nabla u) \, dx$ be the [[integral functional|JRQZ803Q5Y]] on $C^1(\overline{\Omega}; \R^m)$.
Let $u_0 \in C^1(\overline{\Omega}; \R^m)$ be a [[weak local minimiser|WMPCR723MJ]] of $J$ among functions with the same boundary values.
Then for every $x \in \Omega$, every $a \in \R^m$ and every $b \in \R^n$,
$$\sum_{i, j = 1}^{m} \sum_{\alpha, \beta = 1}^{n} \frac{\partial^2 L}{\partial \xi^i_\alpha \, \partial \xi^j_\beta}\big(x, u_0(x), \nabla u_0(x)\big) \, a_i a_j b_\alpha b_\beta \ge 0.$$

## Remarks

The [[second variation|4C3BANJ8AH]] at $u_0$ is nonnegative for all test functions $\varphi \in C^1_c(\Omega; \R^m)$.
One tests it with $\varphi(x) = \varepsilon \, \eta(x) \, a \sin(b \cdot x / \varepsilon)$ for a fixed smooth cut-off function $\eta$ supported near $x_0$.
As $\varepsilon \to 0$, only the term quadratic in $\nabla \varphi$ survives, and its limit is a positive multiple of the stated quadratic form at $x_0$ weighted by $\eta^2$.

The condition asks for positivity only on rank-one matrices $a \otimes b = (a_i b_\alpha)$, not on all of $\R^{m \times n}$.
For $m = 1$ or $n = 1$ every matrix is of the form $a \otimes b$.
Then the condition says that the Hessian of $L$ in $\xi$ is positive semidefinite, as in the one-dimensional Legendre condition.

Example: for $m = n = 2$ and $L(\xi) = \det \xi$, the quadratic form is $2 \det(a \otimes b) = 0$, so the condition holds although the Hessian of $\det$ is indefinite.
For a $C^2$ integrand, the condition at every $\xi$ is equivalent to rank-one convexity of $\xi \mapsto L(x, u, \xi)$.

## Explanations

### L1 human

For maps with several components and several variables, a minimiser need not make the integrand convex in all gradient directions.
It only has to be convex along very special directions, the rank-one matrices $a \otimes b$.
These are the directions in which a gradient can oscillate quickly, like a fine fold or a laminate.

### L2 human

The Legendre–Hadamard condition is the necessary condition from the second variation for vector-valued problems in several variables.
When it holds with a strict inequality and a uniform constant, the Euler–Lagrange system is called strongly elliptic, which is the basic hypothesis of linear elliptic theory for systems.
It is the infinitesimal form of rank-one convexity, which in turn is implied by Morrey's quasiconvexity.
</document>

<document source="https://tamnd.github.io/math2/node/JMWHKWD7EX.md" tag="JMWHKWD7EX" kind="example">
# Hopf link

## Example

Let $C_1 = \set{(x, y, 0) \in \R^3 : x^2 + y^2 = 1}$ and $C_2 = \set{(x, 0, z) \in \R^3 : (x - 1)^2 + z^2 = 1}$.
Each of them is a round [[circle|060W5CBZP6]].
They are disjoint: a common point would have $y = 0$ and $z = 0$, so $x = \pm 1$ from $C_1$ and $x \in \set{0, 2}$ from $C_2$, which is impossible.
So $C_1 \cup C_2$ is a [[link|1A5WRCNTM3]] with $2$ components, the *Hopf link*.
The circle $C_2$ passes once through the flat disk bounded by $C_1$, at the point $(0, 0, 0)$, and it meets that disk nowhere else.
Hence the [[linking number|SFJ329YB9F]] of the two components is $\pm 1$, the sign depending on the orientations.
The unlink of two components has linking number $0$, so the Hopf link is not equivalent to it: the two rings cannot be pulled apart.

## Remarks

Each component is an [[unknot|1TQ06R4Q66]].
The Hopf link has a diagram with $2$ crossings, and it is the only nontrivial link with at most $2$ crossings.
Its complement in $S^3$ is homeomorphic to $T^2 \times (0, 1)$, the product of a torus and an open interval, so its link group is $\Z \times \Z$.
Any two distinct fibres of the [[Hopf fibration|M5AKNYC4QQ]] $S^3 \to S^2$ form a Hopf link, which gives the name.
It is the torus link of type $(2, 2)$, a two-component relative of the [[torus knots|NS60VV786W]].

## Explanations

### L1 human

The Hopf link is two rings hooked together, like two neighbouring links of a chain.
Each ring on its own is unknotted, but they cannot be separated without cutting one of them.
It is the simplest link that is really linked.
</document>

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# Logistic regression

## Summary

Logistic regression models the probability of a binary outcome as the logistic function of a linear combination of regressors.
Equivalently, the log-odds of the outcome is linear in the regressors.
So each coefficient is the change in log-odds for a unit change of its regressor.
It is the [[generalized linear model|K6C48KMRKC]] for [[Bernoulli|0386Y0Z7Y7]] responses with the canonical logit link, and it is fitted by [[maximum likelihood|0FREPFKNS7]].
It is one of the most used models in medicine, the social sciences and machine learning, where it is a basic linear classifier.

## Explanations

### L1 human

Many questions have a yes/no answer: does a patient recover, does a customer buy.
A straight line can predict values below $0$ or above $1$, which make no sense as probabilities.
Logistic regression bends the line into an S-shaped curve that always stays between $0$ and $1$.
If the model says $\log(p / (1 - p)) = -2 + 0.5 x$, then at $x = 4$ the odds are $e^0 = 1$, so the probability is $1/2$.
Each extra unit of $x$ multiplies the odds by $e^{0.5} \approx 1.65$.
</document>

<document source="https://tamnd.github.io/math2/node/JN2YME6F4A.md" tag="JN2YME6F4A" kind="statement">
# Little's law

## Statement

Consider a system, empty at time $0$, that customers enter at times $0 \le a_1 \le a_2 \le \cdots$ and leave at times $d_k \ge a_k$, with $a_k \to \infty$.
Let $W_k = d_k - a_k$ be the time customer $k$ spends in the system, let $A(t) = \#\set{k : a_k \le t}$, and let $L(t) = \#\set{k : a_k \le t < d_k}$ be the number in the system at time $t$.
Assume that the limits
$$\lambda = \lim_{t \to \infty} \frac{A(t)}{t} \quad \text{and} \quad W = \lim_{n \to \infty} \frac1n \sum_{k=1}^n W_k$$
exist and are finite.
Then the limit $L = \lim_{t \to \infty} \frac1t \int_0^t L(s)\, ds$ exists and
$$L = \lambda W.$$

## Remarks

The proof compares two ways of computing the total time spent in the system up to time $t$.
The integral $\int_0^t L(s)\, ds$ lies between the sum of the $W_k$ over customers who have left by time $t$ and the sum over customers who have arrived by time $t$.
Dividing by $t$, both bounds tend to $\lambda W$; the upper bound needs the fact that $W_n / a_n \to 0$, which follows from the two hypotheses.
Example: in the stable [[M/M/1 queue|GTFRPF5J2K]] with $\lambda = 3$ and $\mu = 4$, the mean number in the system is $L = 3$, so the mean time in the system is $W = L / \lambda = 1$.
The law needs no assumption on the arrival law, the service law, the number of servers or the order of service.

## Explanations

### L1 human

Average number of customers inside equals arrival rate times average time each one stays.
If $10$ people per minute enter a museum and each stays $30$ minutes, there are about $300$ people inside.
It is a piece of accounting, so it holds for almost any system with a stable long-run behaviour.

### L2 human

In [[queueing theory|P64V0HTE78]] one usually computes one of $L$ and $W$ from a model and gets the other one free.
For the M/M/1 queue, the stationary law gives $L = \rho/(1 - \rho)$, and Little's law gives $W = 1/(\mu - \lambda)$.
Applied to the waiting room alone, it links the mean queue length and the mean waiting time before service.
In the stochastic setting it holds for stationary systems with expectations in place of time averages.
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# Differential equations in Banach spaces

## Summary

A differential equation in a [[Banach space|06GZ956657]] has an unknown $u(t)$ that is a vector of an infinite-dimensional space, such as a space of functions.
The basic example is the abstract Cauchy problem $u'(t) = A u(t)$, $u(0) = u_0$, with a linear operator $A$.
Many partial differential equations of evolution type, such as the heat equation, take this form when the space variables are absorbed into the state space.
The linear theory rests on [[strongly continuous semigroups|EVF180QJXY]] of operators.

## Explanations

### L1 human

The heat in a rod is a whole temperature profile, a function of position.
As time passes, the profile changes, and its rate of change is given by its second derivative in space.
So one can view the heat equation as an ordinary differential equation for one moving point in a space of functions.
When $A$ is a bounded operator, the solution is $u(t) = e^{tA} u_0$, given by the exponential series, just as for matrices.
For the heat equation $A$ is unbounded, and this is where the theory becomes deeper than for ordinary differential equations.
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<document source="https://tamnd.github.io/math2/node/JN73KE96YB.md" tag="JN73KE96YB" kind="concept">
# Topological modular forms

## Summary

Topological modular forms is a [[ring spectrum|BAW0T68GWK]], written $\mathrm{tmf}$, that collects all [[elliptic cohomology|XH68V2BP4K]] theories into one universal object.
Goerss, Hopkins and Miller built it as the global sections of a sheaf of $E_\infty$ ring spectra on the moduli stack of elliptic curves.
Its homotopy groups map to the ring of integral [[modular forms|GWKN5VG78A]], and this map is an isomorphism after tensoring with $\Q$.
The torsion in $\pi_* \mathrm{tmf}$ detects many elements of the stable homotopy groups of spheres, and $\mathrm{tmf}$ receives the Witten genus of string manifolds.

## Explanations

### L1 human

Each elliptic curve gives an elliptic cohomology theory, and modular forms are functions on the space of all elliptic curves.
Topological modular forms glue all the elliptic cohomology theories together in a homotopy-theoretic way.
The result is a single cohomology theory whose coefficients are "modular forms, plus extra torsion".
For example, $\pi_3 \mathrm{tmf} \cong \Z/24$, which matches the stable homotopy group $\pi_3^s \cong \Z/24$ of the sphere.
No classical modular form explains this group, so $\mathrm{tmf}$ sees more than modular forms do.
</document>

<document source="https://tamnd.github.io/math2/node/JNB1VM75XP.md" tag="JNB1VM75XP" kind="definition">
# Polycyclic group

## Definition

Let $G$ be a [[group|G5S1M9]].
The group $G$ is *polycyclic* when it has a [[subnormal series|0ZZR8D7Y1Z]]
$$\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$$
in which each factor $G_i/G_{i-1}$ is [[cyclic|0KVGH0J5B3]].

## Remarks

Equivalently, $G$ is [[solvable|0FXJ7YFMMX]] and every subgroup of $G$ is [[finitely generated|082DWWXMRJ]].
The number of infinite cyclic factors does not depend on the series; it is the Hirsch length of $G$.
Example: $\Z^2$ is polycyclic, with the series $0 \le \Z \times 0 \le \Z^2$ whose factors are both isomorphic to $\Z$; its Hirsch length is $2$.
Non-example: the additive group $\Q$ is abelian, hence solvable, but it is not polycyclic.
A polycyclic group is generated by lifts of generators of its factors, so it is finitely generated, while $\Q$ is not finitely generated.
Every finite solvable group is polycyclic, and every [[supersolvable group|AWWM9RWETX]] is polycyclic.

## Explanations

### L1 human

A group is polycyclic when it can be built from finitely many cyclic layers, each normal in the next.
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# Fundamental lemma of the calculus of variations

## Statement

Let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]], and let $C^\infty_c(\Omega)$ be the set of smooth functions $\varphi \colon \Omega \to \R$ whose support is a compact subset of $\Omega$.
Let $f \colon \Omega \to \R$ be locally [[integrable|X7GTF3D98F]], that is, integrable on every compact subset of $\Omega$.
If
$$\int_\Omega f(x) \varphi(x) \, dx = 0 \quad \text{for every } \varphi \in C^\infty_c(\Omega),$$
then $f = 0$ [[almost everywhere|RARTMA0ZEC]] on $\Omega$.
In particular, if $f$ is continuous, then $f(x) = 0$ for every $x \in \Omega$.

## Remarks

For continuous $f$ the proof is short.
If $f(x_0) > 0$, then $f > 0$ on a ball $B$ around $x_0$; a [[bump function|HS65DW0HXH]] $\varphi \ge 0$ supported in $B$ with $\varphi(x_0) > 0$ gives $\int_\Omega f \varphi \, dx > 0$, a contradiction.
For locally integrable $f$, one convolves $f$ with a smooth mollifier: the hypothesis makes every mollified function vanish, and these converge to $f$ in $L^1$ on compact sets.

The lemma is the step that turns the vanishing of the [[first variation|5B8NFYB2XQ]] into the [[Euler–Lagrange equation|PS0VV149ET]].
After an integration by parts, the first variation has the form $\int_\Omega E(x) \varphi(x) \, dx$, where $E$ is the Euler–Lagrange expression; the lemma gives $E = 0$.

The class of test functions must be large enough.
If only $\varphi$ with $\int_\Omega \varphi \, dx = 0$ were allowed, every constant $f$ would satisfy the hypothesis; this case is the [[du Bois-Reymond lemma|AATQKCWAAC]].

## Explanations

### L1 human

If a function gives zero when it is averaged against every smooth bump, then the function is zero.
A bump placed where the function is positive would otherwise give a positive answer.
It is the reason why "zero for all variations" can be turned into an equation that holds at every point.

### L2 human

In a derivation of the Euler–Lagrange equation one first finds $\int_a^b \big(L_z - \frac{d}{dx} L_p\big)\eta \, dx = 0$ for all $\eta$ that vanish at the ends.
The lemma then gives the pointwise equation $L_z - \frac{d}{dx} L_p = 0$.
In distribution theory the lemma says that a locally integrable function is determined by the distribution it defines.
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<document source="https://tamnd.github.io/math2/node/JNR9YJTT54.md" tag="JNR9YJTT54" kind="example">
# Milnor's isospectral flat tori

## Example

Let $\Gamma \subset \R^n$ be a [[lattice|WN9V7WSJKF]], and let $T_\Gamma = \R^n / \Gamma$ be the flat [[torus|03NM7DZXXC]] with the metric induced from $\R^n$.
Let $\Gamma^* = \set{v \in \R^n : \ip{v}{\gamma} \in \Z \text{ for all } \gamma \in \Gamma}$ be the dual lattice.
For each $\gamma^* \in \Gamma^*$, the function $x \mapsto e^{2\pi i \ip{\gamma^*}{x}}$ is an eigenfunction of the Laplacian with eigenvalue $4\pi^2 \abs{\gamma^*}^2$.
These functions form an orthogonal basis of $L^2(T_\Gamma)$.
So the [[spectrum|B8TYKV7AA1]] of $T_\Gamma$ is determined by the numbers $N_{\Gamma^*}(r) = \#\set{\gamma^* \in \Gamma^* : \abs{\gamma^*}^2 = r}$.

Take $n = 16$, $\Gamma_1 = E_8 \oplus E_8$, with $E_8$ the [[E8 lattice|2CXH65EKY7]], and $\Gamma_2 = D_{16}^+$.
Both are even unimodular lattices, so $\Gamma_i^* = \Gamma_i$.
The theta series $\sum_{\gamma \in \Gamma_i} q^{\abs{\gamma}^2/2}$ of an even unimodular lattice of rank $16$ is a [[modular form|GWKN5VG78A]] of weight $8$ for $SL_2(\Z)$ with constant term $1$.
That space of modular forms has dimension $1$, so the two theta series are equal, and $N_{\Gamma_1}(r) = N_{\Gamma_2}(r)$ for every $r$.
Hence $T_{\Gamma_1}$ and $T_{\Gamma_2}$ are [[isospectral|18MQEGN13R]].

They are not isometric.
An isometry of flat tori lifts to an affine isometry of $\R^{16}$ whose linear part maps $\Gamma_1$ onto $\Gamma_2$.
But the vectors of squared length $2$ generate a sublattice of index $1$ in $E_8 \oplus E_8$, and a sublattice of index $2$ in $D_{16}^+$, namely $D_{16}$.

## Explanations

### L1 human

A flat torus is a box whose opposite faces are glued, and its frequencies are given by the lengths of the vectors of a lattice.
Milnor noticed in 1964 that two special lattices in $16$ dimensions have exactly the same number of vectors of each length, although they are shaped differently.
So the two tori sound the same but are not the same shape.
This was the first example showing that one cannot always hear the shape of a manifold.
</document>

<document source="https://tamnd.github.io/math2/node/JNTTAHTPDD.md" tag="JNTTAHTPDD" kind="concept">
# Mixed model

## Summary

A mixed model is a linear model with both fixed effects and random effects.
Fixed effects are unknown constants, and random effects are random variables whose covariance is to be estimated.
Random effects describe correlated groups of observations that share a source of variation.
Examples are repeated measurements on the same person or pupils in the same school.
It extends the [[general linear model|A06SGFCSCE]], and its covariance parameters are usually estimated by maximum likelihood or restricted maximum likelihood.
Mixed models are standard for longitudinal, clustered and hierarchical data.

## Explanations

### L1 human

Measure the blood pressure of ten patients five times each.
The five values of one patient are closer to each other than to the values of other patients, because each patient has a personal level.
A mixed model gives each patient a random personal shift around the population mean, plus independent measurement noise.
The fixed part answers questions about the population, and the variance of the random shifts says how much patients differ.
</document>

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# Triangulation conjecture

## Problem

Let $n \ge 0$ and let $M$ be an $n$-dimensional [[topological manifold|QQQ9B1DK4P]].
Does $M$ admit a [[triangulation|YH170WBQ6M]]?
That is, is there a [[simplicial complex|V8GZ0W1ZZ5]] $K$ whose geometric realization $\abs{K}$ is [[homeomorphic|0QT00PB5G2]] to $M$?
The triangulation conjecture asserts that the answer is yes for every manifold.

## Remarks

The question was raised by Hellmuth Kneser in 1926.
The conjecture holds in dimensions $n \le 3$.
For $n \le 1$ this is elementary, for $n = 2$ it is [[Radó's theorem|C9X05EDS5F]], and for $n = 3$ it follows from [[Moise's theorem|SV0816DJP9]].
It fails in dimension $4$: the closed [[$E_8$ manifold|BNDRN0G9BR]] of Michael Freedman is not triangulable, which follows from properties of the [[Casson invariant|33GY6P5A4T]].
It fails in every dimension $n \ge 5$.
Galewski and Stern, and independently Matumoto, reduced this case to a question about [[homology 3-spheres|HK8NS02NNE]] and homology cobordism.
Ciprian Manolescu answered that question in 2013 with $\mathrm{Pin}(2)$-equivariant Seiberg–Witten Floer homology, which gives closed non-triangulable manifolds of each dimension $n \ge 5$.
The conjecture asks for any simplicial triangulation, not one that is combinatorial.
The stronger question whether every manifold has a piecewise linear structure was settled negatively earlier, by Kirby and Siebenmann in 1969, through the [[Kirby–Siebenmann class|E4NJER1RFQ]].
Every smooth manifold is triangulable, so the counterexamples are manifolds without a smooth structure.

## Explanations

### L1 human

A triangulation cuts a space into points, segments, triangles, tetrahedra and their higher analogues, glued along common faces.
Surfaces and three-dimensional manifolds can always be cut up this way.
The conjecture said that every manifold, in every dimension, can be cut up like this.
It turned out to be false: in dimension four and in every higher dimension there are manifolds that admit no such decomposition at all.
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# Symbolic method

## Summary

The symbolic method translates a description of a combinatorial class into an equation for its [[generating function|0PHAC833JF]], see the [[definition|39HWNYW5EF]].
Each construction, such as disjoint union, pair or sequence, has a fixed rule that acts on generating functions.
Unlabelled classes lead to ordinary generating functions, and labelled classes lead to [[exponential generating functions|0JHXPVFKT0]].
It is the first part of [[analytic combinatorics|YJK71RKVEB]], whose second part extracts asymptotics from the equations.

## Explanations

### L1 human

A binary tree is either empty, or a root with a left and a right binary tree.
The symbolic method turns this sentence directly into the equation $B(z) = 1 + z B(z)^2$.
Here "or" becomes $+$, "with" becomes a product, and the root becomes a factor $z$.
Solving the equation gives the [[Catalan numbers|0YWZJEABB2]] as the coefficients of $B(z)$.
The point is that one never counts by hand: the grammar of the objects does the work.
</document>

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# Accelerated failure time model

## Summary

An accelerated failure time model says that covariates stretch or shrink the time scale of a lifetime.
The [[survival function|QTM1HZM53Q]] for covariates $x$ is the baseline survival function evaluated at a rescaled time $t e^{-\beta^T x}$.
Equivalently, $\log T$ is a linear function of the covariates plus an error term, so the model is a linear regression for log-times.
It is the main alternative to the [[proportional hazards model|N029V3JX60]] in [[survival analysis|443ZWDX88P]]; the Weibull family is the only one that satisfies both.

## Explanations

### L1 human

The common saying that one dog year equals seven human years is an accelerated failure time idea.
Dogs go through the same life stages as people, only seven times faster.
In such a model a treatment does not add a fixed risk; it makes the clock run slower or faster.
For example, a coefficient with $e^{\beta} = 2$ means that every lifetime quantile, such as the median, is doubled.
</document>

<document source="https://tamnd.github.io/math2/node/JP9VD777AE.md" tag="JP9VD777AE" kind="definition">
# Inverse transform

## Definition

Let $T$ be an [[integral transform|YCK709QB06]], defined on a class $A$ of functions, and assume that $T$ is injective on $A$.
The *inverse transform* of $T$ is the map $T^{-1} \colon T(A) \to A$ that sends $F$ to the unique $f \in A$ with $Tf = F$.
It is the [[inverse function|0W766JAPSN]] of $T$ viewed as a bijection from $A$ onto $T(A)$.

## Remarks

Example: the Fourier transform $\hat f(\xi) = \int_\R f(x) e^{-2\pi i x \xi}\, dx$ is a bijection of the Schwartz space onto itself, and its inverse is $f(x) = \int_\R \hat f(\xi) e^{2\pi i x \xi}\, d\xi$.
So here the inverse transform is again an integral transform, with kernel $e^{2\pi i x \xi}$.
Non-example: the transform $Tf(u) = \int_0^1 f(t)\, dt$ with kernel $K(t, u) = 1$ on $[0,1] \times \R$ has no inverse transform on the continuous functions, since $f(t) = 0$ and $f(t) = 2t - 1$ both have $Tf = 0$.
When the inverse is an integral transform with kernel $H$, one writes $f(t) = \int F(u)\, H(u, t)\, du$, and $(f, F)$ is a transform pair.
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# Skorokhod's representation theorem

## Statement

Let $S$ be a [[separable|0Z2P41SVDP]] [[metric space|0VKD1TP7F3]] with its Borel σ-algebra.
Let $\mu, \mu_1, \mu_2, \dots$ be [[probability measures|059TJN0NZ1]] on $S$ such that $\mu_n$ [[converges weakly|MBW9JCGZC1]] to $\mu$.
Then there are a probability space $(\Omega, \mathcal{F}, P)$ and random variables $X, X_1, X_2, \dots$ on it with values in $S$ with the following properties.
The variable $X$ has law $\mu$, each $X_n$ has law $\mu_n$, and $X_n \to X$ [[almost surely|0AAXWGT5TD]].

## Remarks

On $S = \R$ there is a short proof: let $U$ be uniform on $(0, 1)$, and set $X_n = F_n^{-1}(U)$ and $X = F^{-1}(U)$, where $F_n^{-1}$ and $F^{-1}$ are the [[quantile functions|0X60VE98HN]] of $\mu_n$ and $\mu$.
Weak convergence implies $F_n^{-1}(u) \to F^{-1}(u)$ at each continuity point $u$ of $F^{-1}$, and $F^{-1}$ is monotone, so it has only countably many discontinuities, a null set for $U$.
The general proof builds the variables from fine partitions of $S$ into small sets whose boundaries have $\mu$-measure $0$.
The theorem gives quick proofs of facts about weak convergence: for example, if $X_n \to X$ in distribution and $h$ is continuous, then $h(X_n) \to h(X)$ in distribution, which is the [[continuous mapping theorem|CFRN0G7GDB]].
It also gives $E[\abs{X}] \le \liminf_n E[\abs{X_n}]$ for real variables, by Fatou's lemma.
The new variables only have the right laws; the original random variables, if any, need not converge almost surely.

## Explanations

### L1 human

Convergence in distribution is a weak notion: it only says that the laws get close, not the random values.
Skorokhod's theorem says that one can always rebuild the random variables, with the same laws, on a new space where the values themselves converge.
This lets one use strong tools for almost sure convergence when only convergence in law is known.

### L2 human

The theorem is a standard device in weak convergence theory.
It appears in proofs of the [[continuous mapping theorem|CFRN0G7GDB]] and of results on uniform integrability and convergence of moments.
It is also used for stochastic processes with values in function spaces.
In the real case it is just the quantile coupling.
It should not be confused with Skorokhod's embedding theorem, which embeds a random walk into a Brownian motion.
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# Einstein–Hilbert action

## Summary

The Einstein–Hilbert action is the functional on Lorentzian metrics of a four-dimensional spacetime given by the integral of the [[scalar curvature|ZHX2VYQDZ3]] against the volume form.
Its critical points are exactly the solutions of the vacuum [[Einstein field equations|3MCZ97SW6V]].
It casts general relativity as a variational principle, like the other field theories of physics.

## Explanations

### L1 human

In general relativity, gravity is the curvature of spacetime.
The Einstein–Hilbert action adds up the curvature over all of spacetime into one number.
The shapes of spacetime that make this number stationary, under small changes of the metric, are those allowed by Einstein's equations without matter.
For example, flat Minkowski space, the spacetime of special relativity, has no curvature at all and is such a stationary shape.
Adding the action of matter to it gives the full Einstein equations, with matter as the source of curvature.
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# Poincaré map

## Definition

Let $U$ be an open subset of $\R^n$ with $n \ge 2$ and let $f \colon U \to \R^n$ be continuously differentiable.
Let $\varphi$ be the [[flow|02903SSA48]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.
Let $\Gamma$ be a [[periodic orbit|0AA97XA70K]] of period $T > 0$ and let $x_0 \in \Gamma$.
Let $H$ be a hyperplane through $x_0$ that does not contain the direction $f(x_0)$.
A transversal section at $x_0$ is the intersection $\Sigma$ of $H$ with a small open ball around $x_0$.
Then there are an open neighbourhood $V$ of $x_0$ in $\Sigma$ and a continuously differentiable function $\tau \colon V \to (0, \infty)$ with $\tau(x_0) = T$.
For each $x \in V$, the time $\tau(x)$ is the first time near $T$ at which the solution through $x$ returns to $\Sigma$.
The Poincaré map of $\Gamma$ at $\Sigma$ is
$$P \colon V \to \Sigma, \qquad P(x) = \varphi_{\tau(x)}(x).$$

## Remarks

The existence of $\tau$ follows from the implicit function theorem, because $f(x_0)$ is not parallel to $\Sigma$.
The point $x_0$ is a fixed point of $P$, and fixed points of $P$ correspond to periodic orbits near $\Gamma$.
If the eigenvalues of the derivative $DP(x_0)$ all have absolute value less than $1$, the orbit $\Gamma$ attracts nearby solutions.
Example: in polar coordinates let $r' = r(1 - r^2)$ and $\theta' = 1$, with $\Gamma$ the unit circle and $\Sigma$ the positive $x$-axis near $(1, 0)$.
Every solution returns to $\Sigma$ after the time $2\pi$, and solving the equation for $r$ gives $P(r) = \left(1 + (r^{-2} - 1) e^{-4\pi}\right)^{-1/2}$.
Then $P(1) = 1$ and $P'(1) = e^{-4\pi} < 1$, so the circle is attracting.
Non-example: in the same system, the vertical line $x = 1$ near $(1, 0)$ does not give a transversal section at $(1, 0)$, since $f(1, 0) = (0, 1)$ points along that line.

## Explanations

### L1 human

The Poincaré map records where a solution crosses a fixed screen after one trip around a closed orbit.
It turns the study of the orbit into the study of a fixed point of one map.
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# Regular and singular points of an analytic set

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] and let $A$ be an [[analytic subset|VMDTGTMN9P]] of $X$.
A point $a \in A$ is a *regular point* of $A$ if it has an open neighbourhood $U$ in $X$ with the following property.
The set $A \cap U$ is a complex [[submanifold|VKFW27WXEZ]] of $U$.
A point $a \in A$ is a *singular point* of $A$ if it is not a regular point.
The sets of regular and singular points are written $\operatorname{reg} A$ and $\operatorname{sing} A$.

## Remarks

Example: for the cusp $A = \set{z_1^2 = z_2^3}$ in $\C^2$, every point $a \ne 0$ is regular, since the gradient $(2 z_1, -3 z_2^2)$ of $f = z_1^2 - z_2^3$ is nonzero there and the holomorphic implicit function theorem applies.
The origin is singular.
If $A$ were a complex submanifold near $0$, it would have dimension $1$ and be a graph over its tangent line at $0$.
The secant directions from $0$ to $(t^3, t^2)$ tend to the $z_2$-axis, so this tangent line is the $z_2$-axis, but the projection $(t^3, t^2) \mapsto t^2$ to it is two-to-one near $0$.
Regularity is a property of the set, not of the equations.
The set $A = \set{z_1^2 = 0}$ in $\C^2$ is the line $\set{z_1 = 0}$, so all its points are regular, although the gradient of $z_1^2$ vanishes on $A$.
The set $\operatorname{reg} A$ is open in $A$ by definition, and it is dense in $A$.
The set $\operatorname{sing} A$ is a closed analytic subset of $X$ that is nowhere dense in $A$; this is a theorem, not a direct consequence of the definition.
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# Nonnegative matrix

## Summary

A nonnegative matrix is a real matrix whose entries are all greater than or equal to zero.
Transition matrices of [[Markov chains|0G48PYM96Q]], adjacency matrices of graphs and input-output tables in economics are nonnegative.
The [[Perron–Frobenius theorem|62NS7PV1M9]] describes the largest eigenvalue of such a matrix and its eigenvector.

## Explanations

### L1 human

A nonnegative matrix is a table of numbers with no negative entry.
Such tables often count things or give probabilities, and these are never negative.
For example, $\begin{pmatrix} 0 & 2 \\ 1 & 3 \end{pmatrix}$ is nonnegative, while $\begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$ is not.
Do not confuse this with a positive semidefinite matrix: the condition is on the entries, not on the eigenvalues.
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# Derived set

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $A \subseteq X$.
The *derived set* of $A$ is the set $A'$ of all [[limit points|0E3X7TXS67]] of $A$.
So $x \in A'$ if and only if every [[neighbourhood|0ZCY0Z80B6]] of $x$ contains a point of $A$ different from $x$.

## Remarks

The [[closure|0QEAJB91GZ]] of $A$ is $\overline{A} = A \cup A'$.
In a [[T1 space|0NW6CH38YR]], the derived set $A'$ is always [[closed|0ZCK8DQYN1]].

Example: in $\R$, let $A = \set{1/n : n \ge 1}$.
Each $1/n$ has a neighbourhood that meets $A$ only in $1/n$, so $1/n \notin A'$.
Every interval $(-\varepsilon, \varepsilon)$ contains $1/n$ for large $n$, so $0 \in A'$.
Thus $A' = \set{0}$.

Example: the derived set of $\Q$ in $\R$ is $\R$, and the derived set of $\Z$ in $\R$ is $\emptyset$.

Non-example: in $\R$, the point $2$ is not in the derived set of $A = \set{2} \cup [3, 4]$.
The neighbourhood $(1, 5/2)$ of $2$ contains no point of $A$ other than $2$, so the condition fails.
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<document source="https://tamnd.github.io/math2/node/JQ22QH86DE.md" tag="JQ22QH86DE" kind="concept">
# Quasifield

## Summary

A quasifield is a set with an addition and a multiplication that behaves like a [[division ring|0EA5BTA0CX]].
The difference is that multiplication need not be associative and distributes only on one side.
Quasifields are the coordinate systems of [[translation planes|JZ5J6REAMD]]: every quasifield gives a translation plane and every translation plane comes from a quasifield.
[[Semifields|AHH97EJ6VH]], finite [[near-fields|DSE72WQ5A2]] and division rings are special quasifields.

## Explanations

### L1 human

In a field one can add, subtract, multiply, divide, and the usual rules hold.
A quasifield keeps the addition and the possibility to divide, but allows the multiplication to break associativity and one of the two distributive laws.
Such loose arithmetic is still enough to draw a plane in which lines have equations like $y = x \cdot m + k$.
The Hall plane of order $9$ comes from a quasifield with $9$ elements that is not a field.
</document>

<document source="https://tamnd.github.io/math2/node/JQ6FJH2TP8.md" tag="JQ6FJH2TP8" kind="example">
# Diamond lattice M3

## Example

The *diamond lattice* $M_3$ has five elements $0, a, b, c, 1$.
Its order is $0 < a < 1$, $0 < b < 1$ and $0 < c < 1$, with $a$, $b$, $c$ pairwise incomparable.
The join of any two distinct elements among $a, b, c$ is $1$ and their meet is $0$, so $M_3$ is a [[lattice|WH4SFAACAS]].

$M_3$ is a [[modular lattice|QX7WZ92QAK]].
If $u \le v$ and $u = v$, or $u = 0$, or $v = 1$, the modular law $u \vee (x \wedge v) = (u \vee x) \wedge v$ reduces to a trivial identity; these cases cover all comparable pairs.
For example, with $u = 0$ both sides equal $x \wedge v$, and with $v = 1$ both sides equal $u \vee x$.

$M_3$ is not a [[distributive lattice|0HM0HQCK6N]].
Indeed $a \wedge (b \vee c) = a \wedge 1 = a$, while $(a \wedge b) \vee (a \wedge c) = 0 \vee 0 = 0$.

## Remarks

$M_3$ is isomorphic to the lattice of subspaces of the plane $\F_2^2$ over the field with two elements: the zero subspace, the three lines and the whole plane.
By the [[M3–N5 characterization of distributivity|C6GH4VHAMQ]], a lattice is distributive exactly when it has no sublattice isomorphic to $M_3$ or to the [[pentagon lattice N5|DHC8TZXGYS]].

## Explanations

### L1 human

Draw a diamond with an extra middle point: a bottom, a top and three points side by side in between.
Each middle point meets each other one at the bottom and joins it at the top.
So $a$ meets "$b$ or $c$" in $a$ itself, but $a$ meets $b$ and meets $c$ only in the bottom.
This is the simplest place where meet does not distribute over join, although the weaker modular law still holds.
</document>

<document source="https://tamnd.github.io/math2/node/JQ9WXC729D.md" tag="JQ9WXC729D" kind="statement">
# Hölder's inequality

## Statement

Let $(S, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
Let $p, q \in (1, \infty)$ with $\frac{1}{p} + \frac{1}{q} = 1$.
Let $f, g \colon S \to \R$ be [[measurable functions|YYNCJSR0PJ]].
Then, with the [[integrals|DWJF7ABPR1]] taking values in $[0, \infty]$ and the convention $0 \cdot \infty = 0$,
$$\int_S \abs{f g} \, d\mu \le \Big( \int_S \abs{f}^p \, d\mu \Big)^{1/p} \Big( \int_S \abs{g}^q \, d\mu \Big)^{1/q}.$$

## Remarks

Proof idea: if a factor on the right is $0$ or $\infty$ the claim is easy.
Otherwise scale $f$ and $g$ so that both factors are $1$, apply [[Young's inequality|07VHQ31EYJ]] $ab \le \frac{a^p}{p} + \frac{b^q}{q}$ pointwise to $\abs{f(s)}$ and $\abs{g(s)}$, and integrate.
When the right side is finite, equality holds if and only if there are constants $\alpha, \beta \ge 0$, not both $0$, with $\alpha \abs{f}^p = \beta \abs{g}^q$ almost everywhere.

With the counting measure on $\set{1, \dots, n}$ the statement becomes [[Hölder's inequality for sums|0BKDAGMGX9]].
For $p = q = 2$ it is the [[Cauchy–Schwarz inequality|0RFWW56Y22]] for integrals.
The same bound holds for $p = 1$ and $q = \infty$, with the essential supremum of $\abs{g}$ in place of the second factor.

## Explanations

### L1 human

Hölder's inequality bounds the integral of a product by the sizes of the two factors.
The size of $f$ is measured with the exponent $p$ and the size of $g$ with the matching exponent $q$.
It is the main tool for showing that a product of two functions is integrable.

### L2 human

In the language of [[Lp spaces|SASP14MA1Z]], it says $\norm{f g}_1 \le \norm{f}_p \norm{g}_q$.
So each $g \in L^q$ gives a bounded linear functional on $L^p$, which is the start of the duality between $L^p$ and $L^q$.
It is also the key step in the proof of the [[Minkowski inequality|N4QZS912MT]], the triangle inequality in $L^p$.
</document>

<document source="https://tamnd.github.io/math2/node/JQAB8D3FFN.md" tag="JQAB8D3FFN" kind="definition">
# Reflective subcategory

## Definition

Let $\mathcal{C}$ be a category and let $\mathcal{B}$ be a [[full subcategory|V3BFGKBJF7]] of $\mathcal{C}$ with inclusion functor $I \colon \mathcal{B} \to \mathcal{C}$.
$\mathcal{B}$ is *reflective* in $\mathcal{C}$ if $I$ has a [[left adjoint|TZHF6ZM92S]] $R \colon \mathcal{C} \to \mathcal{B}$.
The functor $R$ is called the *reflector*.

## Remarks

Equivalently, every object $c$ of $\mathcal{C}$ has a morphism $\eta_c \colon c \to R c$ with $R c$ in $\mathcal{B}$.
Every morphism $c \to b$ with $b$ in $\mathcal{B}$ must factor through $\eta_c$ in exactly one way.
Some authors also ask $\mathcal{B}$ to be closed under isomorphic objects.
The dual notion is a *coreflective* subcategory.

Example: abelian groups form a reflective subcategory of groups, with reflector $R(H) = H/[H, H]$ and $\eta_H$ the quotient map.
Every homomorphism $H \to A$ with $A$ abelian kills all commutators $x y x^{-1} y^{-1}$, so it factors uniquely through $H/[H, H]$.

Non-example: finite groups do not form a reflective subcategory of groups.
A reflective subcategory closed under isomorphic objects contains every limit, formed in $\mathcal{C}$, of a diagram in $\mathcal{B}$.
But the product of countably many copies of $\Z/2\Z$ is an infinite group.
</document>

<document source="https://tamnd.github.io/math2/node/JQBG1GT0XW.md" tag="JQBG1GT0XW" kind="definition">
# Machine epsilon

## Definition

Consider a [[floating-point|RTKRSC3KPH]] system with base $\beta$ and precision $p$.
The *machine epsilon* is
$$\varepsilon_{\mathrm{mach}} = \beta^{1-p},$$
the distance from $1$ to the next larger floating-point number.
The *unit roundoff* of the system with rounding to nearest is
$$u = \tfrac12 \beta^{1-p}.$$
Some authors call $u$ the machine epsilon instead.

## Remarks

Example: in IEEE binary64, $\beta = 2$ and $p = 53$, so $\varepsilon_{\mathrm{mach}} = 2^{-52} \approx 2.22 \times 10^{-16}$ and $u = 2^{-53} \approx 1.11 \times 10^{-16}$.
Check: the floating-point numbers in $[1, 2)$ are $1 + k \, 2^{-52}$ for $k = 0, 1, \ldots, 2^{52} - 1$, so the successor of $1$ is $1 + 2^{-52}$.
Non-example: the smallest positive floating-point number, about $4.9 \times 10^{-324}$ in binary64, is not the machine epsilon; it measures the range, not the relative precision.
If $x$ lies in the range of normalized numbers and $\mathrm{fl}(x)$ is the nearest floating-point number, then $\mathrm{fl}(x) = x(1 + \delta)$ with $\abs{\delta} \le u$.
</document>

<document source="https://tamnd.github.io/math2/node/JQCK7A210W.md" tag="JQCK7A210W" kind="concept">
# Stiefel–Whitney class

## Summary

The Stiefel–Whitney classes $w_i(E) \in H^i(B; \Z/2\Z)$ are mod $2$ [[characteristic classes|7HACJC0JE2]] of real vector bundles $E \to B$.
Four axioms determine them: naturality, the [[Whitney sum|BE9NJRVDR3]] formula, a normalization, and vanishing above the rank.
The first class detects orientability and the second detects the existence of a spin structure.
Stiefel–Whitney numbers of manifolds decide when two closed manifolds are cobordant.

## Explanations

### L1 human

A real vector bundle can be twisted in ways that ordinary integer invariants do not see.
Stiefel–Whitney classes count such twists modulo $2$, as cohomology classes with values $0$ or $1$.
The first one answers a yes-or-no question: can the fibers be oriented consistently along every loop?
For the Möbius band over the circle the answer is no, and $w_1$ is nonzero; for the cylinder it is yes, and $w_1 = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/JQDHHFXBF3.md" tag="JQDHHFXBF3" kind="concept">
# Fixed-point property

## Summary

A topological space has the fixed-point property if every continuous map from the space to itself fixes at least one point.
It is a topological invariant, so it can tell spaces apart.
The [[Brouwer fixed-point theorem|N7AAJ2MW5T]] says that every closed ball in $\R^n$ has this property.
Fixed-point results of this kind are a main tool for proving that equations have solutions.

## Explanations

### L1 human

Imagine moving every point of a space to some point of the same space, without tearing it.
The space has the fixed-point property if, however you do this, some point stays where it was.
The interval $[0, 1]$ has it: the graph of a continuous map $f \colon [0, 1] \to [0, 1]$ must cross the diagonal line $y = x$.
A circle does not have it, since a rotation by a small angle moves every point.
The real line does not have it either, since $x \mapsto x + 1$ moves every point.
</document>

<document source="https://tamnd.github.io/math2/node/JQEYEQJY2G.md" tag="JQEYEQJY2G" kind="concept">
# Galois representation

## Summary

A Galois representation is a continuous [[representation|WYGRXCG2RN]] of the absolute [[Galois group|0XTY86BZKY]] of a field on a finite-dimensional vector space or free module.
The field is usually a number field or a local field.
Typical coefficients are the complex numbers, finite fields and $\ell$-adic numbers.
Galois representations arise from the torsion points of [[elliptic curves|427N6S6QQW]], from the cohomology of varieties, and from [[modular forms|GWKN5VG78A]].
Matching them with automorphic objects is the content of the [[Langlands program|J9T2M084W5]].

## Explanations

### L1 human

The absolute Galois group of $\Q$ records all the symmetries of all algebraic numbers at once; it is far too large to study directly.
A Galois representation looks at this group through matrices: each symmetry becomes an invertible matrix, compatibly with composition.
For example, the symmetries act on the roots of unity of order $\ell^k$ for all $k$, and this gives the cyclotomic character, a $1 \times 1$ representation.
The torsion points of an elliptic curve give a $2 \times 2$ representation, which encodes how many points the curve has modulo each prime.
</document>

<document source="https://tamnd.github.io/math2/node/JQFAE9S2F4.md" tag="JQFAE9S2F4" kind="concept">
# Veblen function

## Summary

The Veblen functions $\varphi_\gamma$ are a hierarchy of [[normal functions|ERTAM0SFVB]] on the [[ordinals|0YSCSX5YZC]], each one listing the fixed points of the ones before.
They start from $\varphi_0(\alpha) = \omega^\alpha$, and they give names to large countable ordinals such as the epsilon numbers and the Feferman–Schütte ordinal $\Gamma_0$.
The [[precise definition|6QFEVWKE25]] is by transfinite recursion on the index $\gamma$.
They are a basic tool of ordinal notations and of ordinal analysis in proof theory.

## Explanations

### L1 human

Start with the function $\alpha \mapsto \omega^\alpha$.
Its fixed points, the ordinals with $\omega^\alpha = \alpha$, are the epsilon numbers, and listing them gives a new function $\varphi_1$.
The fixed points of $\varphi_1$ give $\varphi_2$, and so on, also through infinite stages.
Each new function grows so fast that its values outrun everything the earlier functions can name.
For instance, $\varphi_1(0) = \varepsilon_0$, the first ordinal with $\omega^{\varepsilon_0} = \varepsilon_0$.
</document>

<document source="https://tamnd.github.io/math2/node/JQWXE54TVM.md" tag="JQWXE54TVM" kind="definition">
# Dual abelian variety

## Definition

Let $A$ be an [[abelian variety|R7W03TXRS9]] over a field $k$, with unit $e \in A(k)$.
For a point $x$ of $A$, let $t_x \colon A \to A$ be translation by $x$.
Let $\bar{k}$ be an algebraic closure of $k$.
Let $\operatorname{Pic}^0(A)$ be the subgroup of the [[Picard group|3Z9DCWYXQZ]] of classes of [[line bundles|7JCF2TE3V5]] $L$ with $t_x^{*} L \cong L$ for all $x \in A(\bar{k})$.
The *dual abelian variety* of $A$ is an abelian variety $A^\vee$ over $k$ with a line bundle $P$ on $A \times_k A^\vee$, the Poincaré bundle, such that:

1. $P$ restricted to $\set{e} \times A^\vee$ is trivial;
2. $P$ restricted to $A \times \set{y}$ lies in $\operatorname{Pic}^0$ for every point $y$ of $A^\vee$;
3. $(A^\vee, P)$ is universal, in the following sense.
   Let $T$ be a $k$-scheme and $L$ a line bundle on $A \times_k T$.
   Assume $L$ is trivial on $\set{e} \times T$ and in $\operatorname{Pic}^0$ on each fibre $A \times \set{t}$.
   Then there is a unique morphism $f \colon T \to A^\vee$ with $L \cong (\id_A \times f)^{*} P$.

## Remarks

The pair $(A^\vee, P)$ exists and is unique up to unique isomorphism.
When $k$ is algebraically closed, clause 3 with $T = \operatorname{Spec} k$ gives a bijection $A^\vee(k) \cong \operatorname{Pic}^0(A)$.
The dimension of $A^\vee$ equals the dimension of $A$, and $(A^\vee)^\vee \cong A$.

Example: let $E$ be an elliptic curve over an algebraically closed field with origin $O$.
The map $P \mapsto \mathcal{O}_E(P - O)$ is a bijection from $E(k)$ onto the line bundles of degree $0$, and these are exactly the translation-invariant ones.
So $E^\vee \cong E$.

Non-example: the full Picard group of $E$ is not the dual.
The line bundle $\mathcal{O}_E(O)$ has degree $1$, and $t_x^{*} \mathcal{O}_E(O) \cong \mathcal{O}_E(O)$ fails for every $x \ne O$, so it is not in $\operatorname{Pic}^0(E)$.

Over $\C$, if $A = V / \Lambda$, then $A^\vee$ is the quotient of the space of $\C$-antilinear forms $f$ on $V$ by the lattice of those $f$ with $\operatorname{Im} f(\Lambda) \subseteq \Z$.
</document>

<document source="https://tamnd.github.io/math2/node/JQYCTY3Q8V.md" tag="JQYCTY3Q8V" kind="concept">
# Homoclinic bifurcation

## Summary

A homoclinic bifurcation is a global bifurcation in which, at one parameter value, an orbit leaves a saddle equilibrium and returns to it.
This orbit is a [[homoclinic orbit|6DBSQKZK4W]].
As the parameter crosses this value, the loop breaks, and periodic orbits can be created or destroyed near it.
In the plane, under nondegeneracy conditions, a single [[limit cycle|0AAQ73WRFB]] is born from the loop.
In three dimensions, near a saddle-focus, the bifurcation can produce chaos, as the [[Shilnikov theorem|Q134D9EMAZ]] shows.

## Explanations

### L1 human

A saddle point has a path leading out of it and a path leading into it.
Usually these paths miss each other, but at a special parameter value the outgoing path loops around and comes back in.
That loop is fragile and breaks as soon as the parameter moves.
When it breaks, a closed orbit that circles around near the old loop can appear, with a period that becomes infinite as the parameter approaches the special value.
</document>

<document source="https://tamnd.github.io/math2/node/JR0G4HN9EA.md" tag="JR0G4HN9EA" kind="definition">
# Simplicial complex

## Definition

Let $V$ be a [[finite set|070R982G4B]].
An *(abstract) simplicial complex* on the vertex set $V$ is a family $\Delta$ of subsets of $V$ such that:

1. If $\sigma \in \Delta$ and $\tau \subseteq \sigma$, then $\tau \in \Delta$.
2. $\set{v} \in \Delta$ for every $v \in V$.

The elements of $\Delta$ are its *faces*; a face $\sigma$ has dimension $\abs{\sigma} - 1$, and the dimension of $\Delta$ is the largest dimension of a face.
The maximal faces are called *facets*.

## Remarks

Example: on $V = \set{1, 2, 3}$, let $\Delta$ be the family of all subsets of $V$ except $V$ itself.
A subset of a set with at most $2$ elements has at most $2$ elements, so clause 1 holds.
Clause 2 holds, and $\Delta$ is the boundary of a triangle, of dimension $1$.

Non-example: on $V = \set{1, 2, 3}$, the family $\set{\emptyset, \set{1}, \set{2}, \set{3}, \set{1, 2}, \set{1, 2, 3}}$ fails clause 1, since $\set{1, 3} \subseteq \set{1, 2, 3}$ is missing.

A nonempty complex contains the empty face, of dimension $-1$.
Some authors allow an infinite vertex set, keeping every face finite.
A geometric realization places the vertices as affinely independent points in some $\R^N$ and replaces each face by the convex hull of its points.
</document>

<document source="https://tamnd.github.io/math2/node/JR0N0BDR3B.md" tag="JR0N0BDR3B" kind="definition">
# NIP theory

## Definition

Let $T$ be a [[complete theory|655K8Y7VS8]] in a first-order language $L$.
Let $\varphi(\bar x; \bar y)$ be an $L$-[[formula|HYJ10AF6RT]] whose free variables are split into two tuples $\bar x$ and $\bar y$.
The formula $\varphi$ has the *independence property* in $T$ if the following holds for every $n \in \N$.
There are a [[model|MVCH6N9SNW]] $M$ of $T$, tuples $\bar a_1, \ldots, \bar a_n$ in $M$, and tuples $\bar b_J$ in $M$ for each $J \subseteq \set{1, \ldots, n}$.
For all $i$ and $J$, $M \models \varphi(\bar a_i; \bar b_J)$ holds if and only if $i \in J$.
The theory $T$ is *NIP* if no $L$-formula has the independence property in $T$.

## Remarks

Example: the theory of dense linear orders without endpoints is NIP.
By quantifier elimination, a formula $\varphi(x; \bar y)$ defines, for each parameter $\bar b$, a union of at most $k$ points and intervals.
The endpoints are entries of $\bar b$, and $k$ depends only on $\varphi$.
Such a family cannot cut out all $2^n$ subsets of a set of $n$ points once $n$ is large.
It is a standard fact that it suffices to check formulas with $\bar x$ a single variable.

Non-example: the theory of the [[Rado graph|RMXR0FEF54]] is not NIP.
For the formula $x E y$, take distinct vertices $a_1, \ldots, a_n$; by the extension property, for each $J$ there is a vertex $b_J$ adjacent to $a_i$ exactly when $i \in J$.

Equivalently, $T$ is NIP if for every formula $\varphi(\bar x; \bar y)$ and every model $M$ of $T$, the family $\set{\varphi(M; \bar b) : \bar b \in M}$ has finite [[VC dimension|KCFXR9K3GD]].
Every stable theory and every o-minimal theory is NIP.
</document>

<document source="https://tamnd.github.io/math2/node/JR2BWZEWH1.md" tag="JR2BWZEWH1" kind="statement">
# Ultraparallel theorem

## Statement

Let $\ell$ and $m$ be two lines of the [[hyperbolic plane|GN4N94GQ0Z]] that do not meet and do not share an ideal endpoint (ultraparallel lines).
Then there is exactly one line that meets both $\ell$ and $m$ at right angles.

## Remarks

Proof idea in the [[Poincaré half-plane model|RHKSMPBSQ1]]: after an isometry, $\ell$ is the imaginary axis and $m$ is the semicircle with centre $c > 0$ and radius $r$ on the real axis.
Since $m$ neither meets $\ell$ nor ends at $0$ or $\infty$, one has $0 < r < c$.
A line meeting $\ell$ at a right angle is a semicircle centred at $0$, of radius $R$ say.
It meets $m$ at a right angle exactly when $R^2 + r^2 = c^2$, so $R = \sqrt{c^2 - r^2}$ is the only choice.
The common perpendicular realizes the distance between $\ell$ and $m$, and the distance between points of $\ell$ and $m$ grows on both sides of it.
Both hypotheses are needed.
Two lines that meet have no common perpendicular, since with it they would bound a triangle with two right angles and a third positive angle, against the angle sum.
Two lines with a common ideal endpoint (limiting parallels) have no common perpendicular either, and the distance between them tends to $0$ toward that endpoint.

## Explanations

### L1 human

In hyperbolic geometry, two lines that never meet come in two kinds.
Some approach each other forever in one direction; the others separate in both directions.
For the second kind, there is exactly one bridge between them that is perpendicular to both, and it marks the place where the lines are closest.

### L2 human

In the Euclidean plane two parallel lines have infinitely many common perpendiculars, all of the same length.
In the hyperbolic plane the ultraparallel case has exactly one, and the limiting parallel case has none.
This gives a clean classification of pairs of distinct lines: intersecting, limiting parallel, or ultraparallel, the last characterized by a unique common perpendicular.
It is used to define the distance between ultraparallel lines, and to build right-angled hexagons and pants decompositions of hyperbolic surfaces.
</document>

<document source="https://tamnd.github.io/math2/node/JR2M58RCW3.md" tag="JR2M58RCW3" kind="statement">
# Â-genus as the index of the Dirac operator

## Statement

Let $M$ be a closed Riemannian manifold of dimension $4k$ with a [[spin structure|RSPZ9TCFQ8]], and let $S = S^+ \oplus S^-$ be its complex spinor bundle.
Let $D^+ \colon C^\infty(M; S^+) \to C^\infty(M; S^-)$ be the [[Dirac operator|QMRAVX14BJ]] restricted to positive spinors.
Let $p_i \in H^{4i}(M; \Q)$ be the [[Pontryagin classes|DN11FKWN63]] of $TM$.
Let $\hat A_k(p_1, \ldots, p_k)$ be the degree $4k$ part of the multiplicative sequence with characteristic power series $\dfrac{\sqrt{z}/2}{\sinh(\sqrt{z}/2)}$.
Then the [[analytic index|44GBFZ16SG]] of $D^+$ is
$$\operatorname{ind} D^+ = \hat A(M) := \langle \hat A_k(p_1, \ldots, p_k), [M] \rangle.$$

## Remarks

The theorem is the case of the Atiyah–Singer index theorem for the spin Dirac operator.
In the cohomological formula, the Chern character of the spinor symbol combined with the Todd class leaves exactly the $\hat A$-class.
The first terms are $\hat A_1 = -\tfrac{1}{24} p_1$ and $\hat A_2 = \tfrac{1}{5760}(-4 p_2 + 7 p_1^2)$.
Consequence: the $\hat A$-genus of a closed spin manifold is an integer, while for general manifolds it is only rational.
The spin hypothesis is necessary: $\C P^2$ is not spin, and $\hat A(\C P^2) = -\tfrac{1}{24} \cdot 3 = -\tfrac18$.
Example: a K3 surface is spin with $p_1 = -48$, so $\hat A = 2$, and the Dirac operator has index $2$.
With the Lichnerowicz formula $D^2 = \nabla^*\nabla + \tfrac{s}{4}$, where $s$ is the scalar curvature, the theorem shows that a closed spin manifold with $\hat A(M) \ne 0$ has no metric of positive scalar curvature.

## Explanations

### L1 human

The $\hat A$-genus is a number computed from the curvature, or the Pontryagin classes, of a manifold.
A priori it is a fraction.
On a spin manifold it counts the harmonic positive spinors minus the harmonic negative spinors, so it is an integer.

### L2 human

On an even-dimensional spin manifold, the K-theory Thom isomorphism shows that every class in $K(T^*M)$ is a difference of symbol classes of Dirac operators twisted by vector bundles.
Twisting by a complex vector bundle $W$ gives $\operatorname{ind} D^+_W = \langle \hat A(TM) \operatorname{ch}(W), [M] \rangle$, where $\operatorname{ch}$ is the Chern character.
So on spin manifolds this special case already contains the general index theorem.
The vanishing of $\hat A(M)$ under positive scalar curvature, due to Lichnerowicz, began the study of obstructions to positive scalar curvature.
</document>

<document source="https://tamnd.github.io/math2/node/JR4QSEFXWF.md" tag="JR4QSEFXWF" kind="definition">
# Dirichlet character

## Definition

Let $q \ge 1$ be an integer.
A *Dirichlet character modulo $q$* is a function $\chi \colon \Z \to \C$ such that:

1. $\chi(n + q) = \chi(n)$ for all $n \in \Z$.
2. $\chi(n) = 0$ if $\gcd(n, q) > 1$, and $\chi(n) \ne 0$ if $\gcd(n, q) = 1$.
3. $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \in \Z$.

The *principal character* $\chi_0$ modulo $q$ has $\chi_0(n) = 1$ for $n$ [[coprime|0VZ48N083F]] to $q$ and $\chi_0(n) = 0$ otherwise.

## Remarks

Example: for $q = 4$, let $\chi(n) = 0$ for even $n$, $\chi(n) = 1$ for $n \equiv 1$ and $\chi(n) = -1$ for $n \equiv 3 \pmod 4$.
Clauses 1 and 2 are clear.
For clause 3 it suffices to check odd classes: $1 \cdot 1 \equiv 1$, $1 \cdot 3 \equiv 3$ and $3 \cdot 3 \equiv 1 \pmod 4$ match $1 \cdot 1 = 1$, $1 \cdot (-1) = -1$ and $(-1)(-1) = 1$.
Non-example: for $q = 3$, the function $f(n) = r$, where $r \in \set{0, 1, 2}$ is the remainder of $n$ modulo $3$, satisfies clauses 1 and 2 but not clause 3, since $f(2 \cdot 2) = f(4) = 1 \ne 4 = f(2) f(2)$.
Equivalently, a Dirichlet character modulo $q$ is a [[group homomorphism|0JYTG5K9Y4]] $(\Z/q\Z)^\times \to \C^\times$ from the unit group of the [[integers modulo q|03VFZ2WETT]], extended by $0$ on the classes not coprime to $q$.
Hence each value $\chi(n)$ with $\gcd(n, q) = 1$ is a $\varphi(q)$-th root of unity, and there are exactly $\varphi(q)$ characters modulo $q$, where $\varphi$ is [[Euler's totient function|0DAMHCT17R]].
Each Dirichlet character is a [[completely multiplicative function|0RXKW044R4]].
The Legendre symbol $n \mapsto (n/p)$ is a Dirichlet character modulo an odd prime $p$.
</document>

<document source="https://tamnd.github.io/math2/node/JR7PBS4WAR.md" tag="JR7PBS4WAR" kind="definition">
# Besicovitch almost periodic function

## Definition

Let $1 \le p < \infty$.
For a function $g \colon \R \to \C$ whose $p$-th power is locally integrable, put
$$\norm{g}_{B^p} = \Big( \limsup_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} \abs{g(x)}^p \, dx \Big)^{1/p} \in [0, \infty].$$
A *generalized trigonometric polynomial* is a function $P(x) = \sum_{k=1}^{N} c_k e^{i\lambda_k x}$ with $N \ge 1$, $\lambda_k \in \R$ and $c_k \in \C$.
Let $f \colon \R \to \C$ be a function with $\abs{f}^p$ locally integrable.
It is *Besicovitch almost periodic* of order $p$, or $B^p$ almost periodic, if the following holds.
For every $\varepsilon > 0$ there is a generalized trigonometric polynomial $P$ with $\norm{f - P}_{B^p} < \varepsilon$.

## Remarks

The quantity $\norm{\cdot}_{B^p}$ is a seminorm, not a norm.
It vanishes on every function in the [[Lebesgue space|SASP14MA1Z]] $L^p(\R)$, and the space of $B^p$ classes is obtained by identifying functions $f, g$ with $\norm{f - g}_{B^p} = 0$.
Every [[Bohr almost periodic function|ED69H4C3PK]] is $B^p$ almost periodic for every $p$, since $\norm{g}_{B^p} \le \sup \abs{g}$.
For $p = 2$, each $B^2$ almost periodic $f$ has a mean value of $\abs{f}^2$ and Fourier coefficients $a(\lambda)$, nonzero for at most countably many $\lambda$, with $\lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} \abs{f}^2 = \sum_\lambda \abs{a(\lambda)}^2$.

Example: $f = 1_{[0,1]}$ is $B^p$ almost periodic for every $p$.
Indeed $\frac{1}{2T} \int_{-T}^{T} \abs{f}^p = \frac{1}{2T}$ for $T \ge 1$, so $\norm{f - 0}_{B^p} = 0$ and the polynomial $P = 0$ works.
This $f$ is not continuous, so it is not Bohr almost periodic.

Non-example: $f(x) = x$ is not $B^p$ almost periodic.
Every generalized trigonometric polynomial $P$ is bounded, say by $C$, so $\abs{f - P} \ge \abs{x} - C$ and $\norm{f - P}_{B^p} = \infty$.
</document>

<document source="https://tamnd.github.io/math2/node/JR83AKA184.md" tag="JR83AKA184" kind="definition">
# Serre's twisting sheaf

## Definition

Let $S = \bigoplus_{d \ge 0} S_d$ be a [[graded ring|CAPQBQDRA7]], and let $X = \operatorname{Proj} S$ be given by the [[Proj construction|A39XQHCVNW]].
For an integer $n$, let $S(n)$ be the graded $S$-module with $S(n)_d = S_{n + d}$, where $S_e = 0$ for $e < 0$.
Let $M$ be a graded $S$-module, and let $f \in S$ be homogeneous of positive degree.
Let $\widetilde{M}$ be the sheaf on $X$ with sections $M_{(f)}$ over $D_+(f)$, where $M_{(f)}$ is the degree zero part of $M_f$.
The *twisting sheaf* of degree $n$ is the [[quasi-coherent sheaf|KPKND0K7W6]] $\mathcal{O}_X(n) = \widetilde{S(n)}$.
*Serre's twisting sheaf* is $\mathcal{O}_X(1)$.

## Remarks

Let $S = k[x_0, \dots, x_r]$ over a field $k$, with $r \ge 1$, so that $X = \mathbb{P}^r_k$.
Then $\mathcal{O}_X(n)$ is an [[invertible sheaf|7JCF2TE3V5]], and its global sections are the homogeneous polynomials of degree $n$ for $n \ge 0$, and $0$ for $n < 0$.
On $D_+(x_i)$, the section $x_i^n$ generates $\mathcal{O}_X(n)$, which gives the local isomorphism with $\mathcal{O}_X$.
If $S$ is generated by $S_1$ as an $S_0$-algebra, then each $\mathcal{O}_X(n)$ is invertible and $\mathcal{O}_X(n) \otimes \mathcal{O}_X(m) \cong \mathcal{O}_X(n + m)$.
Without that hypothesis $\mathcal{O}_X(1)$ need not be invertible.
Take $S = k[x_0, x_1, x_2]$ with $x_0, x_1$ of degree $1$ and $x_2$ of degree $2$, and the point $p$ where $x_0 = x_1 = 0$.
On $D_+(x_2)$, the sections of $\mathcal{O}_X(1)$ need the two generators $x_0$ and $x_1$ near $p$, so $\mathcal{O}_X(1)$ is not invertible at $p$.
For a quasi-coherent sheaf $\mathcal{F}$, one writes $\mathcal{F}(n) = \mathcal{F} \otimes \mathcal{O}_X(n)$, the twist of $\mathcal{F}$ by $n$.
</document>

<document source="https://tamnd.github.io/math2/node/JRAEY07Y4G.md" tag="JRAEY07Y4G" kind="statement">
# Lê–Ramanujam theorem

## Statement

Let $n \ne 2$ be a non-negative integer.
Let $F(z, t)$ be a holomorphic function on a neighbourhood of $(0, 0)$ in $\C^{n+1} \times \C$, and write $f_t(z) = F(z, t)$.
Assume that, for all $t$ in a neighbourhood of $0$, $f_t(0) = 0$ and $f_t$ has an [[isolated singularity|VTGYEES3VS]] at $0$.
Assume also that the [[Milnor number|1ZE51P7GRQ]] $\mu(f_t)$ does not depend on $t$.
Then, for all $t$ small enough, there is a [[homeomorphism|0QT00PB5G2]] $\phi$ between neighbourhoods of $0$ in $\C^{n+1}$ with $\phi(0) = 0$ and $\phi(f_0^{-1}(0)) = f_t^{-1}(0)$.
In words, a $\mu$-constant family of isolated hypersurface singularities has constant topological type.

## Remarks

Lê Dũng Tráng and C. P. Ramanujam proved the theorem in 1976.
The proof shows that the Milnor fibrations of $f_0$ and $f_t$ are related by an h-cobordism, and then applies the h-cobordism theorem of Smale.
That theorem needs the cobordism to have dimension at least $6$, which is why the case $n = 2$, surfaces in $\C^3$, is excluded; this case is still open.
The converse holds: the Milnor number is a topological invariant of the hypersurface germ, so a family of constant topological type has constant $\mu$.

Example: $f_t(x, y) = x^4 + y^4 + t x^2 y^2$ for $t$ near $0$.
For $t \ne \pm 2$ the quartic form $f_t$ is a product of four distinct linear factors, so $f_t$ has an isolated singularity with $\mu(f_t) = 9$.
The theorem says that all these germs are topologically the same as $x^4 + y^4$, four lines through the origin.

## Explanations

### L1 human

Deform a singular hypersurface by changing a parameter, and keep track of the Milnor number, which is an integer computed by algebra.
If this number never jumps, the theorem says that the shape of the singularity, as seen by topology, does not change at all.
So a single integer controls the topology in families, except for surfaces in $\C^3$, where the method breaks down and the question is still open.

### L2 human

The theorem is a central result of equisingularity theory, which asks when a family of singularities should count as having the same singularity.
For plane curves, $n = 1$, constant $\mu$ is equivalent to constant topological type, and to constant Puiseux characteristics.
Later work showed that a $\mu$-constant family is in fact topologically trivial as a family when $n \ne 2$.
The Milnor number is upper semicontinuous in families, so it can only jump up at special parameter values.
</document>

<document source="https://tamnd.github.io/math2/node/JRBRNZT680.md" tag="JRBRNZT680" kind="definition">
# Set theory

## Definition

*Set theory* is the branch of mathematical logic whose objects of study are [[sets|0SJ3RX9TXA]] and the membership relation $\in$ between an [[element|0EZ758XE73]] and a set.
An *axiomatic set theory* is a theory in [[first-order logic|086QKPF0A6]] with axioms about $\in$.
Its language has the binary relation symbol $\in$, and possibly further symbols.
The standard axiomatic set theory is ZFC, [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with the axiom of choice.

## Remarks

Example: ZFC is an axiomatic set theory.
Its language has the single relation symbol $\in$, and its axioms, such as the [[axiom of extensionality|0TB16T2C97]], speak only about $\in$ and equality.
Non-example: Peano arithmetic is not a set theory in this sense, because its language has $0$, successor, addition and multiplication, and its objects are natural numbers.
Other axiomatic set theories include Zermelo set theory, von Neumann–Bernays–Gödel set theory, Morse–Kelley set theory and Quine's New Foundations.
Questions such as the [[continuum hypothesis|0MN2MER9WN]] belong to set theory and are independent of ZFC, if ZFC is consistent.
</document>

<document source="https://tamnd.github.io/math2/node/JRGADFNBV2.md" tag="JRGADFNBV2" kind="statement">
# Cheeger's inequality for the first eigenvalue

## Statement

Let $M$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] without boundary, of dimension $n \ge 2$.
Consider the compact smooth hypersurfaces $\Sigma$ that divide $M$ into two disjoint open sets $A$ and $B$ with $M = A \cup \Sigma \cup B$.
Let $\operatorname{area}(\Sigma)$ be the $(n-1)$-dimensional volume of $\Sigma$.
The *Cheeger constant* of $M$ is
$$h(M) = \inf_{\Sigma} \frac{\operatorname{area}(\Sigma)}{\min(\operatorname{vol}(A), \operatorname{vol}(B))}.$$
Let $\lambda_1$ be the smallest nonzero eigenvalue of the [[Laplace–Beltrami operator|F4NWZNTRMD]] of $M$.
Then
$$\lambda_1 \ge \frac{h(M)^2}{4}.$$

## Remarks

The proof starts from the variational characterisation of $\lambda_1$ as the infimum of $\int_M \abs{\nabla f}^2 / \int_M f^2$ over nonzero smooth functions $f$ with $\int_M f = 0$.
One applies the coarea formula to $g^2$, where $g$ is the positive part of an eigenfunction shifted so that its support has at most half the volume.
The bound from $h(M)$ on each level set, together with the Cauchy–Schwarz inequality, gives the factor $1/4$.
The inequality shows that a small $\lambda_1$ forces a small Cheeger constant, that is, a narrow bottleneck.
A dumbbell, two round spheres joined by a thin tube, has a small Cheeger constant and also a small $\lambda_1$.
The reverse bound needs more hypotheses: Buser proved an upper bound for $\lambda_1$ in terms of $h(M)$ under a lower bound on the Ricci curvature.
The [[Cheeger constant of a graph|Y7S6W12N64]] satisfies a discrete analogue of the inequality.

## Explanations

### L1 human

The Cheeger constant measures the narrowest bottleneck of a shape: the smallest cut that splits it into two big pieces.
The first eigenvalue measures how fast heat evens out over the shape.
Cheeger's inequality says that a shape without narrow bottlenecks must have a large first eigenvalue.
So geometry, the size of cuts, controls analysis, the rate of mixing.

### L2 human

The inequality gives a lower bound on $\lambda_1$ from purely geometric data, without assumptions on curvature.
Its discrete version for graphs is a basic tool for expander graphs, where the spectral gap and the edge expansion control each other.
It complements curvature bounds such as the [[Lichnerowicz–Obata theorem|TK1PT9FDV0]], which need a positive lower bound on the Ricci curvature.
</document>

<document source="https://tamnd.github.io/math2/node/JRMWDJCMCE.md" tag="JRMWDJCMCE" kind="definition">
# Conway polynomial

## Definition

Let $p$ be a prime.
Order the [[monic|0BDX4MBKT4]] polynomials of degree $n$ over $\F_p$ as follows: write such a polynomial as
$$x^n - a_{n-1} x^{n-1} + a_{n-2} x^{n-2} - \cdots + (-1)^n a_0$$
with $a_i \in \set{0, 1, \dots, p - 1}$, and compare the sequences $(a_{n-1}, a_{n-2}, \dots, a_0)$ lexicographically.
The *Conway polynomial* $C_{p,n}$ is defined by induction on $n \ge 1$.
It is the least monic polynomial $f$ of degree $n$ over $\F_p$ for this order such that:

1. $f$ is a [[primitive polynomial|HQ7KMZYPSQ]] over $\F_p$;
2. for every divisor $m$ of $n$ with $m < n$, the polynomial $f(x)$ divides $C_{p,m}\left(x^{(p^n - 1)/(p^m - 1)}\right)$.

## Remarks

Clause 2 says: if $\alpha$ is a root of $C_{p,n}$, then $\alpha^{(p^n - 1)/(p^m - 1)}$ is a root of $C_{p,m}$.
So the copies of $\F_{p^m}$ inside the models of $\F_{p^n}$ match for all $m$ dividing $n$.
Example: $C_{2,3} = x^3 + x + 1$.
The primitive cubics over $\F_2$ are $x^3 + x + 1$ and $x^3 + x^2 + 1$, with sequences $(0, 1, 1)$ and $(1, 0, 1)$.
Clause 2 holds for both, since $C_{2,1} = x + 1$ has root $1$ and every power of $1$ is $1$.
Non-example: $x^4 + x^3 + x^2 + x + 1$ is irreducible over $\F_2$ but fails clause 1, so it is not $C_{2,4}$, which is $x^4 + x + 1$.
For $p = 5$, $C_{5,1} = x - 2 = x + 3$, since $1$ is not a primitive root modulo $5$ and $2$ is.
</document>

<document source="https://tamnd.github.io/math2/node/JRQZ803Q5Y.md" tag="JRQZ803Q5Y" kind="concept">
# Integral functional with a Lagrangian density

## Summary

An integral functional assigns to a function $u$ the integral of an expression $L(x, u, \nabla u)$ built from the point, the value of $u$ and its derivatives.
The function $L$ is the Lagrangian density.
Most problems of the [[calculus of variations|PQ25NHV0KS]] concern such functionals, for example length, area, energy and action.
Their smooth critical points solve the [[Euler–Lagrange equation|PS0VV149ET]].

## Explanations

### L1 human

Many quantities are computed by adding up small local contributions.
The length of a graph adds up small pieces $\sqrt{1 + y'(x)^2} \, dx$, and an energy adds up local energy densities.
The rule that gives the local contribution from the position, the value and the slope is the Lagrangian density $L$.
For $L(x, z, p) = \sqrt{1 + p^2}$ on $[0, 1]$, the function $y(x) = x$ gets the value $\int_0^1 \sqrt{2} \, dx = \sqrt{2}$, the length of the diagonal of the unit square.
</document>

<document source="https://tamnd.github.io/math2/node/JRTB9SHAFD.md" tag="JRTB9SHAFD" kind="definition">
# Reduced complex space

## Definition

Let $(X, \mathcal{O}_X)$ be a [[complex analytic space|HXSZA44J5J]].
It is *reduced* if, for every $x \in X$, the [[stalk|JTAKAPTT8H]] $\mathcal{O}_{X,x}$ is a [[reduced ring|SZHY975FFX]], that is, it has no nonzero nilpotent elements.

## Remarks

Example: the local model in $\C^2$ given by $f = z_1 z_2$ is reduced.
Its stalk at $0$ is $\C\{z_1, z_2\}/(z_1 z_2)$, and if $g^k \in (z_1 z_2)$ then $z_1 \mid g^k$ and $z_2 \mid g^k$ in the factorial ring $\C\{z_1, z_2\}$, so $z_1 \mid g$ and $z_2 \mid g$, hence $z_1 z_2 \mid g$.
At the other points the stalk is a ring of convergent power series, which is reduced.

Non-example: the local model in $\C$ given by $f = z^2$ is not reduced, since the class of $z$ in $\C\{z\}/(z^2)$ is nonzero and has square $0$.

Equivalently, $X$ is reduced if every section of $\mathcal{O}_X$ whose value vanishes at every point of its domain is zero.
For any complex space $X$, the quotient of $\mathcal{O}_X$ by its sheaf of nilpotent germs defines a reduced complex space $X_{\mathrm{red}}$ on the same topological space.
</document>

<document source="https://tamnd.github.io/math2/node/JS0YH5E9ZR.md" tag="JS0YH5E9ZR" kind="definition">
# Kernel pair

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $f \colon X \to Y$ be a [[morphism|859F0TFVA3]] of $\mathcal{C}$.
A *kernel pair* of $f$ is an object $K$ with morphisms $p_1, p_2 \colon K \to X$ such that $(K, p_1, p_2)$ is a [[pullback|Q05K04QJ71]] of $f$ and $f$.
That is:

1. $f \circ p_1 = f \circ p_2$.
2. For every object $W$ and morphisms $q_1, q_2 \colon W \to X$ with $f \circ q_1 = f \circ q_2$, there is a unique $u \colon W \to K$ with $p_1 \circ u = q_1$ and $p_2 \circ u = q_2$.

## Remarks

Example: in the category of sets, $K = \set{(x, x') \in X \times X : f(x) = f(x')}$ with the two projections is a kernel pair.
Clause 1 holds by the definition of $K$, and for clause 2 the map $u$ is $w \mapsto (q_1(w), q_2(w))$.
For a group homomorphism $f$ with kernel $N$, this $K$ is the subgroup of pairs $(x, x')$ with $x^{-1} x' \in N$.

Non-example: in the category of sets, if $f$ is not injective, then $(X, \id_X, \id_X)$ satisfies clause 1 but not clause 2.
Take $W$ a point and $q_1, q_2$ picking $x \neq x'$ with $f(x) = f(x')$; no $u$ has $u = q_1$ and $u = q_2$.

A morphism $f$ is a monomorphism if and only if $(X, \id_X, \id_X)$ is a kernel pair of $f$.
</document>

<document source="https://tamnd.github.io/math2/node/JS2CCR3RNS.md" tag="JS2CCR3RNS" kind="statement">
# Poincaré inequality

## Statement

Let $n \ge 1$ and let $U \subset \R^n$ be a bounded open set.

1. Let $1 \le p < \infty$. There is a constant $C = C(n, p, U)$ such that $\norm{u}_{L^p(U)} \le C \norm{\nabla u}_{L^p(U)}$ for all $u$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}_0(U)$.
2. Assume moreover that $U$ is connected and has $C^1$ boundary, and let $1 \le p \le \infty$. There is a constant $C = C(n, p, U)$ such that
$$\norm{u - u_U}_{L^p(U)} \le C \norm{\nabla u}_{L^p(U)} \quad \text{for all } u \in W^{1,p}(U),$$
where $u_U = \frac{1}{\abs{U}} \int_U u \, dx$ is the mean value of $u$ on $U$.

## Remarks

For part 1 one can take $U$ inside a slab $\set{0 < x_1 < d}$.
For smooth $u$ with compact support, $u(x) = \int_0^{x_1} \partial_1 u \, dt$, and [[Hölder's inequality|JQ9WXC729D]] gives $\norm{u}_{L^p} \le d \norm{\partial_1 u}_{L^p}$; density extends this to $W^{1,p}_0(U)$.
Part 2 is proved by contradiction.
If it failed, there would be functions with mean $0$, $L^p$ norm $1$ and gradients tending to $0$.
The [[Rellich–Kondrachov theorem|YM3JZ34PME]] would give a limit with zero gradient, hence constant on the connected set $U$.
This limit has mean $0$, so it is $0$, which contradicts the norm $1$.

The hypotheses are needed.
In part 2 one must subtract the mean, since constant functions have zero gradient.
Connectedness is needed: on two disjoint intervals, a function equal to $1$ on one and $-1$ on the other has mean $0$, zero gradient and nonzero norm.

## Explanations

### L1 human

A function that vanishes on the boundary of a bounded region cannot be large inside unless its slope is large somewhere.
The Poincaré inequality says that the size of the function is bounded by the size of its slope, up to a constant that depends on the region.
For functions without boundary values, the same holds after subtracting the average value.

### L2 human

The inequality shows that $\norm{\nabla u}_{L^2}$ is a norm on $H^1_0(U)$ equivalent to the full $H^1$ norm.
This is exactly the coercivity needed to apply the Lax–Milgram theorem to the Dirichlet problem for $-\Delta u = f$.
The best constant for $p = 2$ in part 1 is $1 / \sqrt{\lambda_1}$, where $\lambda_1$ is the first Dirichlet eigenvalue of $-\Delta$ on $U$.
</document>

<document source="https://tamnd.github.io/math2/node/JS3695KMCC.md" tag="JS3695KMCC" kind="definition">
# Unimodular matrix

## Definition

Let $n \ge 1$ and let $A$ be an $n \times n$ [[square matrix|0JXZKX526R]] with entries in $\Z$.
The matrix $A$ is *unimodular* when its [[determinant|00KDZRW4C0]] is $1$ or $-1$.

## Remarks

Example: $\begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$ is unimodular, because its determinant is $4 - 3 = 1$.
Non-example: $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ is not unimodular, because its determinant is $2$; its inverse $\begin{pmatrix} 1/2 & 0 \\ 0 & 1 \end{pmatrix}$ is not an integer matrix.
An integer square matrix is unimodular exactly when it is [[invertible|0EB9N0S33C]] and its inverse has integer entries.
This follows from $\det(A)\det(A^{-1}) = 1$ and from the formula of the inverse by the adjugate.
More generally, over a commutative ring $R$, a square matrix is called unimodular when its determinant is a [[unit|0XVEZDE9J8]] of $R$.
</document>

<document source="https://tamnd.github.io/math2/node/JS7BWX3HW3.md" tag="JS7BWX3HW3" kind="definition">
# Substructure

## Definition

Let $L$ be a [[signature|WDVVEYD6AT]] and let $\mathcal{M}$ and $\mathcal{N}$ be $L$-[[structures|983GZBQYT4]] with domains $M$ and $N$.
The structure $\mathcal{N}$ is a substructure of $\mathcal{M}$, written $\mathcal{N} \subseteq \mathcal{M}$, when the following hold.

1. $N$ is a [[subset|0BJR9QR08E]] of $M$.
2. For each $n$-ary function symbol $f$, the map $f^{\mathcal{N}}$ is the restriction of $f^{\mathcal{M}}$ to $N^n$.
3. For each $n$-ary relation symbol $R$, $R^{\mathcal{N}} = R^{\mathcal{M}} \cap N^n$.
4. For each constant symbol $c$, $c^{\mathcal{N}} = c^{\mathcal{M}}$.

## Remarks

Clause 2 forces $N$ to be closed under each $f^{\mathcal{M}}$, and clause 4 forces $N$ to contain each $c^{\mathcal{M}}$.
Conversely, a nonempty subset of $M$ with these two closure properties is the domain of exactly one substructure of $\mathcal{M}$.

Example: in the signature $\set{+, -, 0}$, the set $2\Z$ of even integers is closed under $+$ and $-$ and contains $0$.
So it carries a substructure of $(\Z, +, -, 0)$.

Non-example: the set $\N$ is not the domain of a substructure of $(\Z, +, -, 0)$, because $-1 \notin \N$ although $1 \in \N$.
In the smaller signature $\set{+, 0}$, the set $\N$ is the domain of a substructure of $(\Z, +, 0)$.

## Explanations

### L1 human

A substructure is a part of a structure that is a structure on its own, with the same operations and relations.
In the signature of groups, the substructures of a group are its subgroups.
</document>

<document source="https://tamnd.github.io/math2/node/JSDMGGTG7G.md" tag="JSDMGGTG7G" kind="concept">
# Shock wave

## Summary

In the theory of [[conservation laws|8RA7AMWSAF]], a shock wave is a jump discontinuity of a weak solution that moves along a curve in spacetime.
Its speed is fixed by the [[Rankine–Hugoniot conditions|S4XMYJ3V42]], and an entropy condition decides which jumps are physically admissible.
Shocks form in finite time from smooth data in nonlinear equations such as [[Burgers' equation|NEGHXN8RE1]], so they are unavoidable in the theory.
They model the sudden jumps of pressure and density in gas dynamics, such as the shock in front of a supersonic aircraft.
See the [[definition|A5GDAAYNM0]].

## Explanations

### L1 human

On a road, fast cars catch up with slow cars ahead of them, and a sharp edge of a traffic jam forms.
Behind the edge the traffic is dense, in front of it the road is nearly free, and the edge itself moves.
This moving jump is a shock wave.
For Burgers' equation with value $1$ on the left and $0$ on the right, the jump moves to the right at speed $1/2$, the average of the two values.
</document>

<document source="https://tamnd.github.io/math2/node/JSGWAN2RJV.md" tag="JSGWAN2RJV" kind="concept">
# Complex manifold

## Summary

A complex manifold is a space that looks locally like an open subset of $\C^n$, glued together by holomorphic maps.
It is the setting for complex analysis beyond domains in $\C^n$, just as a [[smooth manifold|F45VQ47GPV]] is the setting for calculus.
[[Riemann surfaces|5K0N8QMV6Q]] are the complex manifolds of dimension $1$; complex projective space and complex tori are basic compact examples.
Compact complex manifolds behave very differently from smooth ones: for instance, on a compact connected complex manifold the only global holomorphic functions are the constants.

## Explanations

### L1 human

A complex manifold is built from pieces of $\C^n$ that are glued along overlaps.
The gluing rules must be holomorphic, so that the notion of a holomorphic function makes sense on the whole space.
The Riemann sphere is an example: it is made of two copies of the complex plane, glued by $z \mapsto 1/z$ away from $0$.
Every complex manifold of dimension $n$ is also a smooth manifold of real dimension $2n$, but a smooth manifold can carry many complex structures, or none.
</document>

<document source="https://tamnd.github.io/math2/node/JSNNSNB9MJ.md" tag="JSNNSNB9MJ" kind="concept">
# Analytic semigroup

## Summary

An analytic semigroup is a [[strongly continuous semigroup|EVF180QJXY]] $(T(t))_{t \ge 0}$ that extends holomorphically to complex times $z$.
The allowed $z$ fill a sector of the complex plane around the positive real axis.
Such semigroups have a strong smoothing effect: $T(t)x$ lies in the domain of every power of the generator for $t > 0$.
They are the abstract model of parabolic equations such as the heat equation.
Their generators are characterised by resolvent estimates on a sector, which makes them the sectorial operators.

## Explanations

### L1 human

The heat equation smooths: even a rough initial temperature becomes smooth instantly.
An analytic semigroup is a semigroup with this smoothing property, expressed by the fact that time $t$ may be replaced by a complex number $z$ with small argument.
For a bounded operator $B$, the formula $e^{zB}$ makes sense for every complex $z$, so $e^{tB}$ is analytic.
The heat semigroup $e^{t\Delta}$ on $L^2(\R^n)$ is analytic, with $z$ allowed in the whole right half-plane.
The shift $f(s) \mapsto f(s + t)$ is not analytic: it moves a rough function without smoothing it.
</document>

<document source="https://tamnd.github.io/math2/node/JSQ840425F.md" tag="JSQ840425F" kind="definition">
# Sphere

## Definition

Let $n \ge 1$ and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]] on the [[Euclidean space|0XPR1E85W0]] $\R^n$.
Let $c \in \R^n$ be a point and let $r > 0$ be a real number.
The *sphere* with centre $c$ and radius $r$ is the set
$$S(c, r) = \set{x \in \R^n : d(x, c) = r}.$$

## Remarks

In coordinates, $S(c, r)$ is the set of solutions of $(x_1 - c_1)^2 + \cdots + (x_n - c_n)^2 = r^2$.
The ordinary sphere is the case $n = 3$, and the case $n = 2$ is a [[circle|060W5CBZP6]].
The unit sphere $S(0, 1)$ in $\R^{n+1}$ is the [[n-sphere|T62369RFH6]] $S^n$.
In any [[metric space|0VKD1TP7F3]] $(X, d)$, the same formula defines the sphere $S(c, r)$; Mathlib's `EuclideanGeometry.Sphere` records a centre and a radius in this way.

Example: the set $x^2 + y^2 + z^2 - 2x = 0$ in $\R^3$ is $S((1, 0, 0), 1)$.
Indeed, the equation is $(x - 1)^2 + y^2 + z^2 = 1$.

Non-example: for $r = 0$ the set $\set{x : d(x, c) = 0}$ is the single point $c$.
It fails the condition $r > 0$, so it is not a sphere.
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# Modulation space

## Summary

A modulation space $M^{p,q}(\R^d)$ is a space of tempered distributions defined by the size of their [[short-time Fourier transform|3DHNP5NEWH]].
The exponent $p$ measures integrability in the time variable and $q$ in the frequency variable.
Modulation spaces are the natural function spaces of [[time–frequency analysis|GQFPVXEWK1]].
They are used to study pseudo-differential operators, Gabor expansions and dispersive equations.

## Explanations

### L1 human

A piece of music is described by which notes sound at which times.
The short-time Fourier transform of a signal gives such a description: a number for each time $x$ and each frequency $\omega$.
A modulation space collects the signals whose time–frequency picture has finite size in a mixed norm.
For $p = q = 2$ the size is the usual energy, and the space is $L^2(\R^d)$.
A Gaussian pulse lies in every modulation space, since its time–frequency picture is again a Gaussian bump.
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# Growth rate of a group

## Definition

Let $G$ be a [[finitely generated group|082DWWXMRJ]] and let $S$ be a finite [[generating set|0RP7VWQ0YN]] of $G$.
The *word length* $\abs{g}_S$ of $g \in G$ is the least $n$ such that $g = s_1 \cdots s_n$ with each $s_i \in S \cup S^{-1}$.
The *growth function* of $G$ with respect to $S$ is
$$\gamma_S(n) = \abs{\set{g \in G : \abs{g}_S \le n}}, \quad n \in \N.$$
For functions $f, h \colon \N \to \N$, write $f \preceq h$ if there is a constant $C \ge 1$ with $f(n) \le h(Cn)$ for all $n$, and $f \sim h$ if $f \preceq h$ and $h \preceq f$.
The *growth rate* of $G$ is the $\sim$-class of $\gamma_S$.

1. $G$ has *polynomial growth* if $\gamma_S(n) \le C n^d$ for some constants $C, d > 0$ and all $n \ge 1$.
2. $G$ has *exponential growth* if $\gamma_S(n) \ge a^n$ for some $a > 1$ and all $n$.
3. $G$ has *intermediate growth* if it has neither polynomial nor exponential growth.

## Remarks

The class of $\gamma_S$ does not depend on $S$.
If $S'$ is another finite generating set, each element of $S'$ is a word of length at most some $C$ in $S$, so $\gamma_{S'}(n) \le \gamma_S(Cn)$.
Hence the three types above do not depend on $S$.

Example: $G = \Z^2$ with $S = \set{(1, 0), (0, 1)}$.
The elements of length at most $n$ are the $(a, b)$ with $\abs{a} + \abs{b} \le n$, so $\gamma_S(n) = 2n^2 + 2n + 1$, and $\Z^2$ has polynomial growth.

Example: the [[free group|0KTW26C80J]] on $\set{a, b}$.
There are $4 \cdot 3^{k-1}$ reduced words of length $k \ge 1$, so $\gamma_S(n) = 2 \cdot 3^n - 1$, and the growth is exponential.

Non-example: the additive group $\Q$ is not finitely generated, so the definition does not apply; with an infinite generating set the balls may be infinite.

Since $\gamma_S(m + n) \le \gamma_S(m) \gamma_S(n)$, the limit $\lim_{n \to \infty} \gamma_S(n)^{1/n}$ exists, and $G$ has exponential growth exactly when this limit is greater than $1$.
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# Frobenius norm

## Summary

The Frobenius norm of a matrix is the square root of the sum of the squares of the absolute values of its entries.
It is the Euclidean length of the matrix seen as a long vector, and it comes from the inner product $\ip{A}{B} = \tr(B^* A)$.
It is a submultiplicative [[matrix norm|54T6Y6DSY4]] that is easy to compute.
It equals the square root of the sum of the squares of the [[singular values|09VMDSRXWJ]].

## Explanations

### L1 human

Write all entries of a matrix in one long row and measure the length of that row as in the theorem of Pythagoras.
That length is the Frobenius norm.
For $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ it is $\sqrt{1 + 4 + 9 + 16} = \sqrt{30}$.
It is used to measure how far one matrix is from another, for example in least squares fitting and low rank approximation.
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# Log-gamma function

## Summary

The log-gamma function is the logarithm of the [[gamma function|0H5SYBSED9]]: $\ln\Gamma(x)$ for real $x > 0$, and its holomorphic extension $\operatorname{Log}\Gamma$ to the slit plane $\C \setminus (-\infty, 0]$.
It is used because $\Gamma$ grows faster than any exponential, so its logarithm is easier to compute and to estimate.
Its derivative is the [[digamma function|HWRRXB0YXS]], and its asymptotic expansion is [[Stirling's series|K29QYWB6A5]].

## Explanations

### L1 human

The gamma function grows so fast that its values overflow quickly; for example $\Gamma(171)$ is already larger than $10^{306}$.
Taking the logarithm turns this huge growth into a gentle one, roughly like $x \ln x$.
Products of gamma values become sums of log-gamma values, which is often simpler.
For example $\ln\Gamma(1) = \ln\Gamma(2) = 0$, and $\ln\Gamma(\tfrac12) = \tfrac12 \ln \pi$.
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# Riemann–Stieltjes integral

## Summary

The Riemann–Stieltjes integral $\int_a^b f \, dg$ integrates a function $f$ against another function $g$, the integrator.
Its sums weight each value of $f$ by the change of $g$ over a subinterval instead of by the length of the subinterval.
With $g(x) = x$ it is the [[Riemann integral|0QM4TXT3DR]].
It handles sums and integrals in one formula.
It is the setting of the Riesz representation of continuous linear functionals on $C[a, b]$, and of expectations written with a [[cumulative distribution function|0MXMS3AE13]].

## Explanations

### L1 human

In an ordinary integral every piece of the $x$-axis counts by its length.
In a Stieltjes integral a piece counts by how much $g$ grows on it, so $g$ decides where the mass sits.
If $g$ jumps by $1$ at the point $c$ and is constant elsewhere, then $\int_a^b f \, dg = f(c)$ for continuous $f$: the integral just reads off a value.
If $g$ is a step function with several jumps, the integral becomes a finite sum.
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# Bergman metric

## Definition

Let $\Omega \subseteq \C^n$ be a bounded domain.
Let $K(z, w)$ be its [[Bergman kernel|8RWDHW0CXC]], the reproducing kernel of the [[Bergman space|2D619WTPYC]] $A^2(\Omega)$ of holomorphic functions in $L^2(\Omega)$.
The *Bergman metric* of $\Omega$ is the [[Hermitian metric|CK1DCPC3D3]]
$$ds^2 = \sum_{j,k=1}^n g_{j\bar k}(z) \, dz_j \, d\bar z_k, \qquad g_{j\bar k}(z) = \frac{\partial^2}{\partial z_j \partial \bar z_k} \log K(z, z) .$$

## Remarks

On a bounded domain $K(z, z) > 0$, so $\log K(z, z)$ is defined and smooth.
The matrix $(g_{j\bar k})$ is positive definite at every point; the bounded functions $1$ and $z_j$ in $A^2(\Omega)$ give this positivity.
If $F \colon \Omega_1 \to \Omega_2$ is a biholomorphism, then $K_1(z, w) = \det F'(z) \, K_2(F(z), F(w)) \, \overline{\det F'(w)}$, and the factors $\det F'$ disappear under $\partial\bar\partial \log$, so $F$ is an isometry.
Example: for the unit disc, $K(z, w) = \frac{1}{\pi (1 - z \bar w)^2}$, so $\log K(z, z) = -\log \pi - 2 \log(1 - \abs{z}^2)$ and $g_{1\bar 1} = \frac{2}{(1 - \abs{z}^2)^2}$.
Example: for the unit ball of $\C^n$, the Bergman metric is $n + 1$ times the metric with potential $-\log(1 - \abs{z}^2)$.
Non-example: for $\Omega = \C^n$ the Bergman space is $\set{0}$, so $K \equiv 0$ and the definition does not apply.
The Bergman metric is a Kähler metric, since it has a global potential.
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# Many-one reduction

## Summary

A many-one reduction from a set $A$ to a set $B$ of natural numbers is a total [[computable function|KZSB5E471K]] $f$ with $x \in A$ exactly when $f(x) \in B$.
It turns every question about membership in $A$ into one question about membership in $B$.
Many-one reductions compare how hard decision problems are, and they are the standard tool for proving that a problem is undecidable.

## Explanations

### L1 human

Think of $f$ as a translator: it rewrites each question about $A$ as a question about $B$ with the same answer.
If someone could answer questions about $B$, they could then answer questions about $A$.
So $A$ is at most as hard as $B$.
For example, the set of even numbers reduces to the set $\set{0}$ by sending each even number to $0$ and each odd number to $1$.
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# Stalk

## Summary

The stalk of a sheaf at a point is the set of germs of sections at that point.
A germ is a section defined near the point, where two sections are identified when they agree on some smaller neighbourhood.
Stalks capture the local behaviour of a sheaf at a single point.
Many properties of sheaves and of maps between them, such as being zero, injective or surjective, can be checked stalk by stalk.

## Explanations

### L1 human

Zoom in on one point of a space and look only at what a sheaf does very close to it.
Two sections that agree on some small enough neighbourhood are counted as the same.
For the sheaf of continuous functions, the stalk at a point is the set of germs of continuous functions there.
For holomorphic functions near $0$ in $\C$, a germ is the same as a power series with a positive radius of convergence.
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# Strong dual space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, let $E$ be a topological vector space over $\mathbb{K}$, and let $E'$ be its [[continuous dual space|4A82S5FVRT]].
For each [[von Neumann bounded set|3BAJ8ZTVPX]] $B \subseteq E$, put $p_B(f) = \sup_{x \in B} \abs{f(x)}$ for $f \in E'$.
The *strong dual space* $E'_b$ of $E$ is $E'$ with the [[topology generated by the seminorms|ZVFXQ7S6FY]] $p_B$, for all bounded subsets $B$ of $E$.

## Remarks

Each $p_B$ is finite, because a continuous linear form maps bounded sets to bounded subsets of $\mathbb{K}$.
A net $(f_i)$ converges to $f$ in $E'_b$ if and only if $f_i \to f$ uniformly on every bounded subset of $E$.
The topology is also written $\beta(E', E)$.
Example: let $E$ be a normed space with closed unit ball $B_1$.
Every bounded set lies in some $r B_1$, and $p_{r B_1}(f) = r \norm{f}$, so the strong dual topology is the topology of the dual norm.
Non-example: for an infinite-dimensional Banach space $E$, the weak-\* topology $\sigma(E', E)$ uses only the finite sets $B$.
It is not the strong topology, since its neighbourhoods of $0$ are not norm bounded.
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# Local trivialization

## Summary

A local trivialization of a map $\pi \colon E \to B$ over an open set $U$ is a homeomorphism $\pi^{-1}(U) \to U \times F$ over $U$.
It says that over $U$ the map $\pi$ looks exactly like the projection of a product.
A [[fiber bundle|CX4M8K61AB]] is a map that has a local trivialization around every point of the base.
Comparing two local trivializations on an overlap gives the transition functions, which encode how the bundle twists.

## Explanations

### L1 human

A local trivialization is a way to straighten out a piece of a bundle so that it becomes a plain product.
On the Möbius band over the circle, the strip above any proper arc of the circle looks like an ordinary rectangle, arc times segment.
Choosing such an identification for one arc is a local trivialization.
No single identification works over the whole circle, because of the twist.
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# Linearized stability theorem for maps

## Statement

Let $U \subseteq \R^k$ be open, let $f \colon U \to \R^k$ be continuously differentiable, and let $x^* \in U$ be a [[fixed point|R6QVS6P21N]] of $f$.
Let $J = Df(x^*)$ be the [[Jacobian matrix|047NDTEPCV]] of $f$ at $x^*$, and consider the difference equation $x_{n+1} = f(x_n)$.

1. If every [[eigenvalue|0BQAA2BK3Q]] $\lambda$ of $J$ satisfies $\abs{\lambda} < 1$, then $x^*$ is locally asymptotically [[stable|CHEQ235HW9]].
   Equivalently, the hypothesis is that the [[spectral radius|9V7YQA85PC]] of $J$ is less than $1$.
2. If some eigenvalue $\lambda$ of $J$ satisfies $\abs{\lambda} > 1$, then $x^*$ is unstable.

## Remarks

For part 1, choose a norm on $\R^k$ in which $\norm{J} < 1$; such a norm exists because the spectral radius is below $1$.
By continuity of $Df$, the map $f$ is a contraction with ratio $q < 1$ on a small closed ball around $x^*$ in this norm.
So iterates stay in the ball and $\norm{x_n - x^*} \le q^n \norm{x_0 - x^*}$.
Part 2 needs a finer argument along the direction of the expanding eigenvalues.
For the logistic map $f(x) = r x (1 - x)$ with $1 < r < 3$, the fixed point $x^* = 1 - 1/r$ has $f'(x^*) = 2 - r$, of modulus less than $1$, so it is asymptotically stable.
When some eigenvalue has modulus exactly $1$ and none has modulus greater than $1$, the theorem says nothing.
For example, $x \mapsto x - x^3$ and $x \mapsto x + x^3$ both have derivative $1$ at $0$, but $0$ is asymptotically stable for the first and unstable for the second.

## Explanations

### L1 human

To know whether a fixed point of a map attracts nearby points, replace the map by its best linear approximation there.
If the linear part shrinks every direction, the fixed point attracts.
If it stretches some direction, nearby points are pushed away.
This reduces a nonlinear question to computing eigenvalues of one matrix.

### L2 human

This is the main tool for the local stability of equilibria of nonlinear difference equations.
For a scalar equation it reduces to $\abs{f'(x^*)} < 1$ for stability and $\abs{f'(x^*)} > 1$ for instability.
For higher-order scalar equations one writes the system in vector form and applies the theorem to the companion-type Jacobian, whose characteristic polynomial can be tested by the Jury criterion.
When no eigenvalue has modulus $1$, the fixed point is hyperbolic.
Then the [[Hartman–Grobman theorem|0S8WX8A4V9]] gives the stronger conclusion that $f$ is locally topologically conjugate to $J$ when $f$ is a local diffeomorphism.
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# Fréchet algebra

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$.
A *Fréchet algebra* over $\mathbb{K}$ is an [[associative algebra|3CTW1P4HBP]] $A$ over $\mathbb{K}$ with a topology such that:

1. $A$ with this topology is a [[Fréchet space|3GAA614673]], that is, a complete metrizable locally convex space;
2. the multiplication $A \times A \to A$, $(a, b) \mapsto ab$, is jointly continuous.

Let $(p_n)_{n \ge 1}$ be an increasing sequence of [[seminorms|FP5GVCN8GV]] defining the topology.
Then condition 2 holds if and only if for each $n$ there are $m \ge n$ and $C_n > 0$ with $p_n(ab) \le C_n \, p_m(a) \, p_m(b)$ for all $a, b \in A$.

## Remarks

The algebra is called *locally multiplicatively convex* (or m-convex) if the topology is given by seminorms with $p_n(ab) \le p_n(a) p_n(b)$.
Some authors include this condition in the definition.
Every [[Banach algebra|9KYWN94QDN]] is a Fréchet algebra, with the single seminorm given by the norm.

Example: $A = C(\R)$, the continuous complex functions on $\R$, with $p_n(f) = \max_{\abs{x} \le n} \abs{f(x)}$.
Each $p_n$ is a seminorm with $p_n(fg) \le p_n(f) p_n(g)$, and the sequence is increasing.
Convergence for all $p_n$ is uniform convergence on compact sets, and a locally uniform limit of continuous functions is continuous, so $A$ is complete.
So $C(\R)$ is an m-convex Fréchet algebra.

Non-example: the polynomials $\C[x]$ with the same seminorms satisfy every condition except completeness in clause 1.
The Taylor polynomials of $e^x$ form a Cauchy sequence for every $p_n$, but their limit $e^x$ is not a polynomial.
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# Riemann–Liouville integral

## Summary

The Riemann–Liouville integral of order $\alpha > 0$ is the integral operator $I_a^\alpha f(x) = \frac{1}{\Gamma(\alpha)} \int_a^x (x - t)^{\alpha - 1} f(t) \, dt$.
For a whole number $\alpha = n$ it is the $n$-fold repeated integral, by [[Cauchy's formula for repeated integration|EYXG11V76E]].
It is the basic operator of [[fractional calculus|T73XK15SA2]], and fractional derivatives are defined from it.

## Explanations

### L1 human

Integrating $n$ times from $a$ is the same as one integral with the weight $(x - t)^{n-1} / (n - 1)!$.
Replace $n$ by any positive real $\alpha$ and the factorial by the [[gamma function|0H5SYBSED9]], and the formula still makes sense.
This gives integrals of order $1/2$, $3/2$, and so on.
For example, with $a = 0$, the integral of order $\alpha$ of $1$ is $x^\alpha / \Gamma(\alpha + 1)$.
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# Von Neumann–Bernays–Gödel set theory

## Summary

Von Neumann–Bernays–Gödel set theory, NBG, is an axiomatic [[set theory|RTHHCX84HN]] whose objects are [[classes|Y48FA69563]].
In it, sets are the classes that are members of other classes.
It proves the same statements about sets as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with choice.
Unlike ZFC, it can be given by finitely many axioms.
It lets one speak directly about large collections, such as the class of all sets or of all groups.

## Explanations

### L1 human

In ZFC, a phrase such as "the collection of all sets" is only a way of talking.
NBG makes such collections into real objects, called classes.
A class that is small enough to belong to another class is a set, and the others are proper classes.
Proper classes cannot be elements, so Russell's paradox does not arise.
Category theory often uses this setting to talk about the category of all sets.
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# Normal family

## Summary

A normal family is a family of holomorphic functions in which every sequence has a subsequence that converges uniformly on compact sets, see the [[definition|QWSD2JDRP0]].
It is the complex-analytic version of a relatively compact set of functions.
[[Montel's theorem|ZVFVBVJMCB]] gives simple tests for normality.
Normal families are a key tool in the proof of the [[Riemann mapping theorem|02WHMEG5ZR]] and in complex dynamics.

## Explanations

### L1 human

In a finite set, every sequence repeats some element infinitely often.
A normal family is a family of functions with a similar property: from any sequence one can pick a subsequence that settles down to a limit.
The limit is reached uniformly on each closed and bounded piece of the domain, though maybe not on the whole domain.
For example, the powers $z, z^2, z^3, \dots$ on the unit disc form a normal family, since they tend to $0$ on each smaller disc.
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# Elliptic gamma function

## Summary

The elliptic gamma function $\Gamma(z; p, q)$ is a two-parameter generalisation of the [[q-gamma function|68P2VKSEP5]].
It is a double infinite product, and shifting $z$ to $qz$ multiplies it by a theta function in the base $p$.
In this way it stands to theta functions as the gamma function stands to linear functions.
It is the basic object of elliptic hypergeometric series and integrals.

## Explanations

### L1 human

The gamma function turns multiplication by $x$ into a shift: $\Gamma(x + 1) = x\, \Gamma(x)$.
The $q$-gamma function does the same with $q$-numbers in place of $x$.
The elliptic gamma function goes one step further and uses theta functions, which are infinite products with a second parameter $p$.
When $p = 0$ the theta function becomes the simple factor $1 - z$ and the theory falls back to the $q$-world.
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# Hewitt–Savage zero–one law

## Statement

Let $(S, \mathcal{S})$ be a measurable space, and let $X_1, X_2, \dots$ be [[independent and identically distributed|00MRN0FARV]] random variables with values in $S$.
Let $B \subseteq S^{\N}$ be a measurable set for the product σ-algebra.
Assume that $B$ is symmetric: for every [[permutation|05RWFKVTJ7]] $\pi$ of $\N$ that moves only finitely many indices, $(x_n)_n \in B$ if and only if $(x_{\pi(n)})_n \in B$.
Then the [[event|031TPJVS74]] $A = \set{(X_1, X_2, \dots) \in B}$ has $P(A) \in \set{0, 1}$.

## Remarks

Proof idea: approximate $A$ by an event $A_n$ that depends only on $X_1, \dots, X_n$.
Let $A_n'$ be the image of $A_n$ under the permutation that swaps the blocks $1, \dots, n$ and $n+1, \dots, 2n$.
By symmetry $A_n'$ approximates $A$ as well, and $A_n, A_n'$ are independent with equal probability.
So $P(A) = \lim P(A_n \cap A_n') = \lim P(A_n)^2 = P(A)^2$.
Every tail event is symmetric, so the theorem extends [[Kolmogorov's zero–one law|073Q8Z94KG]] in the i.i.d. case.
Example: let $X_i$ be real and $S_n = X_1 + \dots + X_n$. The event $\set{S_n \in I \text{ for infinitely many } n}$, for an interval $I$, is symmetric, because a finite permutation leaves $S_n$ unchanged for all large $n$. It is not a tail event in general, since changing $X_1$ shifts all $S_n$.
The identical distribution is needed: if $X_1$ is a fair coin and $X_n = 0$ for $n \ge 2$, then $\set{\text{some } X_n = 1}$ is symmetric and has probability $\frac{1}{2}$.

## Explanations

### L1 human

Take an infinite sequence of independent draws from one distribution.
Any event that does not care about the order of finitely many draws is either almost certain or almost impossible.
For instance, whether a random walk returns to its start infinitely often has probability $0$ or $1$.

### L2 human

The theorem is used to show that quantities such as $\limsup_n S_n$ for a random walk are almost surely constant, and to obtain the recurrence dichotomy for random walks.
Compared with [[Kolmogorov's zero–one law|073Q8Z94KG]], it needs identical distribution but covers more events, namely all events of the exchangeable σ-algebra.
In the language of [[de Finetti's theorem|W19129PVKQ]], it says that an i.i.d. sequence is an exchangeable sequence whose mixing measure is a point mass.
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# Lie subgroup

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]].
A *Lie subgroup* of $G$ is a [[subgroup|0X1JXT47GA]] $H$ of $G$ together with a topology and a smooth structure on $H$ such that:

1. $H$ with this structure is a Lie group;
2. the inclusion map $H \to G$ is an [[immersion|6RW68JGBHC]].

If moreover the inclusion is a homeomorphism onto $H$ with the subspace topology, then $H$ is an *embedded Lie subgroup*.
Equivalently, $H$ is an embedded [[submanifold|VKFW27WXEZ]] of $G$.

## Remarks

Example: in the torus $T = S^1 \times S^1$, let $\alpha$ be irrational and $H = \set{(e^{it}, e^{i\alpha t}) : t \in \R}$, with the topology and smooth structure of $\R$ transported by $t \mapsto (e^{it}, e^{i\alpha t})$.
This map is an injective homomorphism, since $\alpha$ is irrational, and its derivative $(i e^{it}, i\alpha e^{i\alpha t})$ never vanishes, so it is an immersion.
So $H$ is a Lie subgroup, but not an embedded one: $H$ is dense in $T$.

Non-example: the circle $C = \set{z \in \C : \abs{z} = 2}$ is an embedded submanifold of the Lie group $\C^\times$, but it is not a subgroup, since $2 \cdot 2 = 4 \notin C$.

Some authors use "Lie subgroup" only for embedded Lie subgroups, and call the general ones immersed or virtual Lie subgroups.
An embedded Lie subgroup is always closed in $G$.
</document>

<document source="https://tamnd.github.io/math2/node/JTZK8DZ257.md" tag="JTZK8DZ257" kind="concept">
# Perturbation theory of linear operators

## Summary

Perturbation theory of linear operators studies how the properties of an operator $T$ change under replacement by $T + V$.
Here $V$ is small in a suitable sense.
The main questions are whether closedness, self-adjointness, the [[spectrum|BMKATJ802A]] and the eigenvalues survive the change, and how they move.
The basic notion of smallness for unbounded operators is relative boundedness.
The theory underlies much of mathematical quantum mechanics, where a Hamiltonian is a free part plus an interaction.

## Explanations

### L1 human

Often one understands a simple operator well, such as the Laplacian, and wants to study a more complicated one obtained by adding a term.
Perturbation theory says when the new operator still behaves like the old one.
For matrices, this is the study of how [[eigenvalues move|VR89YHC0TX]] when the entries change a little.
For differential operators, the added term may be unbounded, so one measures it against the original operator instead of against the identity.
For example, a bounded real-valued potential $V$ added to the Laplacian on $\R^n$ never destroys self-adjointness.
</document>

<document source="https://tamnd.github.io/math2/node/JV24HVZPJ0.md" tag="JV24HVZPJ0" kind="statement">
# Lindström–Gessel–Viennot lemma

## Statement

Let $G$ be a finite acyclic [[directed graph|0KFYGSHHF5]] whose edges carry weights in a [[commutative ring|0310REJJY8]].
The weight $\omega(P)$ of a directed [[path|03NAMV1DHJ]] $P$ is the product of the weights of its edges.
For vertices $a, b$, let $e(a, b) = \sum_P \omega(P)$, the sum over all paths $P$ from $a$ to $b$.
Let $a_1, \dots, a_n$ and $b_1, \dots, b_n$ be vertices of $G$ and let $M$ be the $n \times n$ matrix with entries $M_{ij} = e(a_i, b_j)$.
Let $\mathcal{F}$ be the set of all tuples $(\sigma, P_1, \dots, P_n)$, where $\sigma$ is a permutation of $\set{1, \dots, n}$ and $P_1, \dots, P_n$ are pairwise vertex-disjoint paths.
Here $P_i$ goes from $a_i$ to $b_{\sigma(i)}$ for each $i$.
Then
$$\det M = \sum_{(\sigma, P_1, \dots, P_n) \in \mathcal{F}} \sgn(\sigma) \prod_{i=1}^{n} \omega(P_i).$$

## Remarks

Expand the [[determinant|00KDZRW4C0]] by the Leibniz formula: it is a signed sum over all families of paths $P_i$ from $a_i$ to $b_{\sigma(i)}$, disjoint or not.
On the families that meet, define an involution as follows.
Take the smallest $i$ such that $P_i$ meets another path, and the first vertex $v$ on $P_i$ that lies on another path.
Take the smallest $j \ne i$ with $v \in P_j$, and swap the parts of $P_i$ and $P_j$ after $v$.
The involution keeps the product of weights and changes the [[sign|02DKV69ZH0]] of $\sigma$, so these terms cancel.

Often the vertices are placed so that only $\sigma = \id$ admits disjoint paths.
Then $\det M$ is the weighted number of families of disjoint paths from $a_i$ to $b_i$.

Example: in the grid $\Z^2$ with unit steps to the east and to the north and all weights $1$, take $a_1 = (0, 1)$, $a_2 = (1, 0)$, $b_1 = (1, 2)$, $b_2 = (2, 1)$.
The path counts are binomial coefficients, $M = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$, and $\det M = 3$.
Of the $4$ pairs of paths $a_1 \to b_1$, $a_2 \to b_2$, exactly one meets (both pass through $(1, 1)$), and no disjoint pair goes $a_1 \to b_2$, $a_2 \to b_1$; so there are $3$ disjoint pairs.

## Explanations

### L1 human

Counting families of paths that never touch each other is hard to do directly.
The lemma says that a determinant does it for you: the entries count single paths, and the signs in the determinant cancel every family in which paths touch.
The trick is to swap the tails of two touching paths at their first meeting point.

### L2 human

The lemma is a standard way to turn a determinant into a count, or a count into a determinant.
It proves the Jacobi–Trudi formula for Schur functions, where semistandard tableaux become families of disjoint lattice paths.
It also counts plane partitions in a box and rhombus tilings of hexagons.
It is due to Lindström, and Gessel and Viennot brought it into enumerative combinatorics.
</document>

<document source="https://tamnd.github.io/math2/node/JV45S0KQE6.md" tag="JV45S0KQE6" kind="concept">
# Bifurcation set

## Summary

The bifurcation set of a smooth family of functions is a set of parameter values.
It contains the values at which some function of the family has a degenerate critical point.
It is the central object of [[catastrophe theory|6FKJ6SXP3G]].
Off this set the critical points move smoothly with the parameters, and across it they are created or destroyed.
For the cusp family $x^4/4 + a x^2/2 + b x$ it is the cusp curve $4a^3 + 27b^2 = 0$ in the $(a, b)$-plane.
For a family of potentials it describes where sudden jumps of the equilibrium can start.

## Explanations

### L1 human

Imagine a landscape that changes shape as you turn some knobs.
For most knob settings every valley and every hilltop is a clean, rounded one.
At special settings a valley and a hilltop merge into a flat spot and then vanish.
The bifurcation set is the map of all such special settings in the space of knobs.
Take the family $x^3 + a x$ with one knob $a$.
A valley and a hilltop exist for $a < 0$, they merge at $a = 0$, and none exist for $a > 0$.
So the bifurcation set is the single point $a = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/JV4H0R6GZM.md" tag="JV4H0R6GZM" kind="definition">
# Local trivialization

## Definition

Let $E$, $B$ and $F$ be [[topological spaces|0CQHFPGJ45]] and let $\pi \colon E \to B$ be a [[continuous map|0X3WGX95FE]].
Let $U$ be an open subset of $B$, and give $U \times F$ the [[product topology|0GBG8B49AA]].
A *local trivialization* of $\pi$ over $U$ with fiber $F$ is a [[homeomorphism|0QT00PB5G2]]
$$\varphi \colon \pi^{-1}(U) \to U \times F$$
such that $\operatorname{pr}_1 \circ \varphi = \pi$ on $\pi^{-1}(U)$, where $\operatorname{pr}_1 \colon U \times F \to U$ is the first projection.

## Remarks

Example: let $E = \R \times S^1$, $B = S^1$ and $\pi(s, z) = z$.
Then $\varphi(s, z) = (z, s)$ over $U = S^1$ is a local trivialization with fiber $\R$, since it is a homeomorphism and its first coordinate is $\pi(s, z) = z$.

Non-example: for the double cover $\pi \colon S^1 \to S^1$, $\pi(z) = z^2$, there is no local trivialization over $U = S^1$ with fiber a two-point space, since $\pi^{-1}(S^1) = S^1$ is connected and $S^1 \times F$ is not.
Over each proper open arc, however, such a trivialization exists.

The map $\pi$ is a [[fiber bundle|CX4M8K61AB]] with fiber $F$ exactly when the open sets carrying a local trivialization cover $B$.
If $\varphi_U$ and $\varphi_V$ are local trivializations, then $\varphi_U \circ \varphi_V^{-1}$ on $(U \cap V) \times F$ has the form $(b, f) \mapsto (b, g_{UV}(b)(f))$, and the maps $g_{UV}$ are the transition functions.
</document>

<document source="https://tamnd.github.io/math2/node/JV5NYG913V.md" tag="JV5NYG913V" kind="concept">
# Mapping class group

## Summary

The mapping class group of a surface is the [[group|G5S1M9]] of its orientation-preserving [[homeomorphisms|0QT00PB5G2]] up to deformation.
Two homeomorphisms give the same element when one can be continuously deformed into the other.
It records the symmetries of the surface that cannot be undone by a deformation.
The mapping class group of the torus is $SL(2, \Z)$, and the [[braid groups|BNR4H5GCTP]] are mapping class groups of punctured disks.
For closed orientable surfaces it is generated by finitely many [[Dehn twists|W890NRHGS4]], by the [[Dehn–Lickorish theorem|BG3EH18KRE]], and it acts on Teichmüller space.

## Explanations

### L1 human

Take a surface, such as the surface of a doughnut, and map it to itself without tearing, in a way that keeps its orientation.
Many such maps are just small wiggles that can be undone continuously; these are ignored.
What remains are the genuinely different ways to map the surface to itself.
On the torus, cutting along a circle around the tube, twisting one side by a full turn and gluing back gives a map that cannot be undone.
The twists along two such circles, one around the tube and one along it, generate the group.
The mapping class group of the torus is the group $SL(2, \Z)$ of integer $2 \times 2$ matrices with determinant $1$.
</document>

<document source="https://tamnd.github.io/math2/node/JVDEWP6MWG.md" tag="JVDEWP6MWG" kind="definition">
# Hyperplane arrangement

## Definition

Let $K$ be a field and $d \ge 1$.
An affine hyperplane in $K^d$ is a set $\set{x \in K^d : a_1 x_1 + \cdots + a_d x_d = b}$ with $(a_1, \dots, a_d) \ne 0$ and $b \in K$; it is an [[affine subspace|0PYW26VNT6]] of dimension $d - 1$.
A *hyperplane arrangement* in $K^d$ is a finite set $\mathcal{A}$ of affine hyperplanes in $K^d$.

## Remarks

The arrangement is *central* if all its hyperplanes have a common point.
Its *intersection poset* $L(\mathcal{A})$ is the set of all nonempty intersections of subsets of $\mathcal{A}$, including $K^d$ itself as the empty intersection, [[partially ordered|0QV2JS9RJ3]] by reverse inclusion.
For $K = \R$, the *regions* of $\mathcal{A}$ are the connected components of $\R^d \setminus \bigcup_{H \in \mathcal{A}} H$; each is an open convex set.

Example: in $\R^2$, the three lines $x = 0$, $y = 0$ and $x + y = 1$ form an arrangement.
It is not central, since the three lines have no common point.
They cut the plane into $7$ regions, one of them the bounded open triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$.

Non-example: the unit circle together with the line $y = 0$ is not a hyperplane arrangement, since the circle is not a hyperplane.
An infinite family such as the lines $x = k$, $k \in \Z$, is not one either, since an arrangement is finite.
</document>

<document source="https://tamnd.github.io/math2/node/JW431VDN08.md" tag="JW431VDN08" kind="concept">
# Geometric measure theory

## Summary

Geometric measure theory studies the geometry of sets and measures in Euclidean space and in manifolds with the tools of measure theory.
Its basic objects are [[Hausdorff measure|Q1TGZK8RN6]], [[rectifiable sets|7N0GFMW05R]], [[currents|360H7MD9RD]] and [[varifolds|N3QA80M44M]], which act as surfaces that may be singular, have multiplicities, or arise as limits.
It provides compactness theorems for such generalised surfaces, which give existence results for variational problems such as [[Plateau's problem|DVYA6GD8RN]] in every dimension and codimension.
It also studies the regularity of the minimisers and the structure of their singular sets.

## Explanations

### L1 human

Soap films can have edges where three sheets meet, and limits of smooth surfaces can develop folds and corners.
Classical calculus works with smooth surfaces and struggles with such objects.
Geometric measure theory replaces a surface by something more flexible, such as a measure that records area, so that limits always exist.
For example, the graph of the function $\abs{x}$ has a corner, but it still has a well-defined length and an approximate tangent line at almost every point.
Once a best generalised surface is found, one proves that it is smooth except on a small set.
</document>

<document source="https://tamnd.github.io/math2/node/JW5PA9XA7F.md" tag="JW5PA9XA7F" kind="example">
# Haar wavelet

## Example

The *Haar wavelet* is the function $\psi \colon \R \to \R$ with
$$\psi(x) = \begin{cases} 1 & 0 \le x < \tfrac12, \\ -1 & \tfrac12 \le x < 1, \\ 0 & \text{otherwise.} \end{cases}$$
It is an orthonormal [[wavelet|4PZ35C0GV3]]: the functions $\psi_{j,k}(x) = 2^{j/2} \psi(2^j x - k)$, $j, k \in \Z$, form an orthonormal basis of $L^2(\R)$.

Check of orthonormality.
$\psi_{j,k}$ is supported on the dyadic interval $I_{j,k} = [k 2^{-j}, (k+1) 2^{-j})$ and $\norm{\psi_{j,k}}_2^2 = 2^j \cdot 2^{-j} = 1$.
Two dyadic intervals are either disjoint or nested.
If they are disjoint, the product $\psi_{j,k} \psi_{j',k'}$ is $0$.
If $I_{j',k'}$ lies strictly inside $I_{j,k}$, then it lies inside one half of $I_{j,k}$, where $\psi_{j,k}$ is constant, and $\psi_{j',k'}$ has integral $0$.

Completeness comes from the Haar [[multiresolution analysis|1GHQMBR7DE]], whose space $V_j$ consists of the functions in $L^2(\R)$ that are constant on the intervals $I_{j,k}$.
The [[scaling function|YH4BERB952]] is $\varphi = 1_{[0, 1)}$, with $\varphi(x) = \varphi(2x) + \varphi(2x - 1)$ and $\psi(x) = \varphi(2x) - \varphi(2x - 1)$.
On $[0, 1]$, the constant $1$ together with the $\psi_{j,k}$ with $j \ge 0$ and $0 \le k < 2^j$ is the [[Haar system|N0B51FAA9S]].

The Haar wavelet has exactly one [[vanishing moment|7QERXM0C52]]: $\int \psi = 0$, while $\int_\R x \psi(x) \, dx = \tfrac18 - \tfrac38 = -\tfrac14$.
It is discontinuous, so for smooth functions its expansions converge slowly compared with smoother wavelets.

## Explanations

### L1 human

The Haar wavelet is a single up-step followed by a down-step.
Its squeezed and shifted copies work at every scale and place.
Each one measures how the average of a function on the left half of a small interval differs from its average on the right half.
Alfréd Haar used these functions in 1910, long before the word wavelet existed.
It is the first member of the family of Daubechies wavelets.
</document>

<document source="https://tamnd.github.io/math2/node/JW71J3AKY4.md" tag="JW71J3AKY4" kind="conjecture">
# Zimmer's conjecture

## Statement

Let $n \ge 3$ and let $\Gamma$ be a [[lattice|BTYF5Y5PFY]] in the [[special linear group|0V9PMN0RBH]] $\mathrm{SL}_n(\R)$, for example $\Gamma = \mathrm{SL}_n(\Z)$.
Let $M$ be a compact [[smooth manifold|F45VQ47GPV]] with $\dim M < n - 1$, and let $\mathrm{Diff}(M)$ be its group of [[diffeomorphisms|0KZZ70ZHXP]].
Then every group homomorphism $\rho \colon \Gamma \to \mathrm{Diff}(M)$ has finite image.

## Remarks

This is the case of $\mathrm{SL}_n(\R)$ of a more general conjecture of Zimmer for lattices in higher-rank simple Lie groups, with a dimension bound depending on the group.
Zimmer stated conjectures of this kind in his 1986 address to the International Congress of Mathematicians, in several versions, some of them only for actions preserving a volume form.
The bound $n - 1$ is sharp: $\mathrm{SL}_n(\Z)$ acts with infinite image on the real projective space of dimension $n - 1$.
Aaron Brown, David Fisher and Sebastián Hurtado proved it, with $C^2$ regularity, for cocompact lattices in $\mathrm{SL}_n(\R)$.
In a paper published in Inventiones Mathematicae in 2020 they proved it for actions by $C^2$ diffeomorphisms of finite-index subgroups of $\mathrm{SL}_n(\Z)$ with $n > 3$.
The general conjecture for lattices in other higher-rank groups is still studied.

## Explanations

### L1 human

The group $\mathrm{SL}_n(\Z)$ of integer matrices with determinant $1$ is a very rigid infinite group when $n \ge 3$.
The conjecture says it cannot act in an interesting way by smooth symmetries on a compact space whose dimension is too small.
Any such action must factor through a finite group.
So a big rigid group needs room to move, and a low-dimensional manifold does not give it enough.
</document>

<document source="https://tamnd.github.io/math2/node/JWB6B13Q8Z.md" tag="JWB6B13Q8Z" kind="concept">
# Factorization of polynomials

## Summary

Factorization of polynomials writes a polynomial as a product of [[irreducible polynomials|03FKX7Z5DC]].
Over a field the result is unique up to order and constant factors, because $K[x]$ is a [[unique factorisation domain|0W4X9G6263]].
The answer depends on the field: $x^2 + 1$ is irreducible over $\R$ but splits over $\C$.
Computing factorizations is a central task of computer algebra, with different methods over finite fields, over $\Q$ and over number fields.

## Explanations

### L1 human

Factoring a polynomial is like factoring an integer into primes.
The irreducible polynomials play the role of the primes.
For example, over $\Q$ one has $x^4 - 1 = (x - 1)(x + 1)(x^2 + 1)$, and none of the three factors can be split further over $\Q$.
Over $\C$ the last factor splits as $(x - i)(x + i)$.
The [[definition|EFDBS6ZME8]] states the form of the result.
</document>

<document source="https://tamnd.github.io/math2/node/JWCDDMJE3K.md" tag="JWCDDMJE3K" kind="statement">
# Open embedding theorem for Hilbert manifolds

## Statement

Let $H$ be an infinite-dimensional [[separable|0Z2P41SVDP]] real [[Hilbert space|HM2HHE3J51]].
Let $M$ be a smooth [[Hilbert manifold|R8DMMW0E9M]] modelled on $H$ whose underlying topological space is metrizable and separable.
Then $M$ is [[diffeomorphic|0KZZ70ZHXP]] to an [[open subset|02BH3MCBQB]] of $H$.

## Remarks

The theorem is due to Eells and Elworthy.
The idea is to embed $M$ as a closed submanifold of $H$ and to take an open tubular neighbourhood $U$ of it.
Then $U$ is a bundle over $M$ with Hilbert space fibres, which turns out to be trivial, so $U$ is diffeomorphic to $M \times H$.
Finally, in this separable infinite-dimensional setting $M \times H$ is diffeomorphic to $M$.
Example: the unit sphere $S$ of $H$ is a Hilbert manifold of this kind, and it is in fact diffeomorphic to $H$ itself, which is the open subset $H \subseteq H$.
The infinite dimension is needed: the sphere $S^n$ is a compact $n$-manifold, and a non-empty open subset of $\R^n$ is never compact.
So $S^n$ is not diffeomorphic to an open subset of $\R^n$.


## Explanations

### L1 human

In finite dimension, a sphere cannot be flattened into a piece of the plane; it is closed up on itself.
In a separable infinite-dimensional Hilbert space there is so much room that every manifold of the same kind fits inside the space as an open region.
So every separable Hilbert manifold is, up to diffeomorphism, just an open subset of the space it is modelled on.

### L2 human

The theorem reduces questions about separable Hilbert manifolds to questions about open subsets of $H$.
It is a key step in the classification of such manifolds: two of them are diffeomorphic as soon as they are homotopy equivalent.
This is very different from finite dimension, where homotopy equivalent manifolds are often not diffeomorphic.
It is one reason why infinite-dimensional manifolds of maps, such as path spaces, are studied mainly through their homotopy type.
</document>

<document source="https://tamnd.github.io/math2/node/JWH2KYCGBQ.md" tag="JWH2KYCGBQ" kind="statement">
# Shift property of the Z-transform

## Statement

Let $x = (x_n)_{n \in \Z}$ be a sequence of complex numbers and let $k$ be an integer.

1. Bilateral form. Let $y_n = x_{n - k}$ for all $n \in \Z$. Then $y$ and $x$ have the same [[region of convergence|1P8WX3EGHF]], and on it the [[bilateral Z-transforms|3DBAMXA80B]] satisfy $Y(z) = z^{-k} X(z)$.
2. Unilateral delay. Suppose $x_n = 0$ for $n < 0$ and $k \ge 0$. Then the [[unilateral Z-transform|EPG6JPVJE7]] of $(x_{n - k})_{n \ge 0}$ is $z^{-k} X(z)$.
3. Unilateral advance. Let $k \ge 1$ and let $X$ be the unilateral Z-transform of $(x_n)_{n \ge 0}$. Then the unilateral Z-transform of $(x_{n + k})_{n \ge 0}$ is
$$z^{k} \left( X(z) - \sum_{j=0}^{k-1} x_j z^{-j} \right),$$
on the same set of $z$.

## Remarks

Each clause is a change of summation index.
For clause 1, $\sum_n x_{n-k} z^{-n} = z^{-k} \sum_m x_m z^{-m}$ with $m = n - k$, and the same computation with absolute values shows that the regions agree.
For clause 3, $\sum_{n \ge 0} x_{n+k} z^{-n} = z^{k} \sum_{m \ge k} x_m z^{-m}$, and the sum over $m \ge k$ is $X(z)$ minus its first $k$ terms.
Example: for $x_n = 1$ ($n \ge 0$), $X(z) = z/(z-1)$, and clause 3 with $k = 1$ gives $z (z/(z-1) - 1) = z/(z-1)$, as it must, since the shifted sequence is again constant.
Clause 2 needs $x_n = 0$ for $n < 0$; otherwise the terms $x_{-k}, \ldots, x_{-1}$ also enter the unilateral transform of the delayed sequence.

## Explanations

### L1 human

Delaying a sequence by one step multiplies its Z-transform by $z^{-1}$.
Advancing it by one step multiplies by $z$, but in the one-sided case the first value falls off and must be subtracted.
This is why $z^{-1}$ is called the unit delay in signal processing.

### L2 human

Clause 3 is the discrete analogue of $\mathcal{L}\{f'\}(s) = s F(s) - f(0)$ for the Laplace transform.
It is what makes the Z-transform solve linear difference equations with constant coefficients: each shifted unknown becomes $z^k$ times the transform, plus a polynomial built from the initial values.
Clause 1 shows that a time-invariant system acts on transforms by multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/JWJ8YPR7EB.md" tag="JWJ8YPR7EB" kind="concept">
# Hyperbolic group

## Summary

A hyperbolic group is a [[finitely generated group|082DWWXMRJ]] whose [[Cayley graph|0GYQK9NDJF]] looks like a tree at large scale: its geodesic triangles are uniformly thin.
The notion was introduced by Gromov.
It covers free groups, fundamental groups of closed hyperbolic manifolds, and, in a precise sense, most finitely presented groups.
Hyperbolic groups are finitely presented, have a solvable word problem and conjugacy problem, and have a linear [[Dehn function|Q4HGMRA9JQ]].

## Explanations

### L1 human

In a tree, the three sides of a triangle meet in a tripod, so each side runs right along the other two.
In the hyperbolic plane the same is nearly true: triangles are thin, with every side close to the union of the other two.
A group is hyperbolic if its Cayley graph has this thin-triangle property, with a fixed bound on the thickness.
The free group on two generators is the simplest infinite example, since its Cayley graph is a tree.
The grid $\Z^2$ is not hyperbolic: its large triangles are fat, like triangles in the flat plane.
</document>

<document source="https://tamnd.github.io/math2/node/JWK49HMHYT.md" tag="JWK49HMHYT" kind="concept">
# Affine independence

## Summary

Points of an [[affine space|09XNAPTJQ7]] are affinely independent if none of them is an [[affine combination|0KJH6SRFZK]] of the others.
Equivalently, the vectors from one of the points to the others are [[linearly independent|0CRMGPAH83]].
Affinely independent points are the vertices of a simplex.
In an $n$-dimensional space, $n + 1$ of them form an affine frame, in which barycentric coordinates are defined.

## Explanations

### L1 human

Two points are affinely independent when they are different.
Three points are affinely independent when they do not lie on one line, so they span a triangle.
Four points are affinely independent when they do not lie in one plane, so they span a tetrahedron.
In general, affinely independent points are in "general position": no one of them lies in the flat spanned by the others.
</document>

<document source="https://tamnd.github.io/math2/node/JWNRB34ZGV.md" tag="JWNRB34ZGV" kind="statement">
# Fischer decomposition

## Statement

Let $n \ge 2$ and $m \ge 2$ be integers, and write $\abs{x}^2 = x_1^2 + \cdots + x_n^2$ for $x \in \R^n$.
Let $p$ be a [[homogeneous polynomial|2C3PAQ7H50]] of degree $m$ in $x_1, \dots, x_n$ with real or complex coefficients.
Then there are unique homogeneous polynomials $h$ of degree $m$ and $q$ of degree $m - 2$ such that $h$ is [[harmonic|MVYFSYMECG]] and
$$p = h + \abs{x}^2 q.$$
Repeating this with $q$ gives the unique expansion
$$p = h_m + \abs{x}^2 h_{m-2} + \abs{x}^4 h_{m-4} + \cdots.$$
Here each $h_j$ is a homogeneous harmonic polynomial of degree $j$, and the sum ends at $j = 0$ or $j = 1$.

## Remarks

Proof idea: on the space $P_m$ of homogeneous polynomials of degree $m$, use the inner product $\langle p, q \rangle = p(\partial) \bar q$, where $p(\partial)$ replaces each $x_i$ by $\partial / \partial x_i$.
For this inner product, multiplication by $\abs{x}^2$ from $P_{m-2}$ to $P_m$ is the adjoint of the [[Laplacian|0E0B6RA8K0]] from $P_m$ to $P_{m-2}$.
So the harmonic polynomials of degree $m$, which form the kernel of the Laplacian, are the orthogonal complement of $\abs{x}^2 P_{m-2}$ in $P_m$.
Example: $x_1^2 = \left(x_1^2 - \tfrac{1}{n}\abs{x}^2\right) + \abs{x}^2 \cdot \tfrac{1}{n}$, and the first term is harmonic since its Laplacian is $2 - \tfrac{2n}{n} = 0$.
Restricted to the unit sphere, where $\abs{x}^2 = 1$, the expansion shows that every polynomial is a finite sum of spherical harmonics.

## Explanations

### L1 human

Every homogeneous polynomial splits into a harmonic part plus a multiple of $\abs{x}^2$.
Repeating the split writes the polynomial as harmonic pieces times powers of $\abs{x}^2$.
On the unit sphere the powers of $\abs{x}^2$ equal $1$, so only harmonic pieces remain.

### L2 human

The decomposition gives the dimension of the space of homogeneous harmonic polynomials of degree $m$: it is $\dim P_m - \dim P_{m-2}$, which is $2m + 1$ in $\R^3$.
It shows that the restrictions of polynomials to the sphere are dense in the continuous functions on the sphere, and that they are sums of spherical harmonics.
This leads to the decomposition of $L^2$ of the sphere into spherical harmonics, which underlies the [[expansion in spherical harmonics|YTPE0GBKWW]].
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# Bergman kernel

## Definition

Let $\Omega$ be a [[domain|0EWMDD6C14]] in $\C$, and let $A^2(\Omega)$ be its [[Bergman space|GZV7QFA0RZ]] with the [[inner product|0QVNCJXRY3]] $\ip{f}{g} = \int_\Omega f \bar g \, dA$.
For each $w \in \Omega$, the map $f \mapsto f(w)$ is a bounded linear functional on the Hilbert space $A^2(\Omega)$.
By the Riesz representation theorem there is a unique $K_w \in A^2(\Omega)$ with
$$f(w) = \ip{f}{K_w} = \int_\Omega f(z)\, \overline{K_w(z)}\, dA(z) \quad \text{for all } f \in A^2(\Omega).$$
The *Bergman kernel* of $\Omega$ is the function $K \colon \Omega \times \Omega \to \C$ given by $K(z, w) = K_w(z)$.

## Remarks

The kernel satisfies $K(w, z) = \overline{K(z, w)}$, and $K(w, w) = \norm{K_w}^2 > 0$ when $A^2(\Omega) \ne \set{0}$.
For any [[orthonormal basis|0S9GGTR0EX]] $(e_n)$ of $A^2(\Omega)$, $K(z, w) = \sum_n e_n(z)\overline{e_n(w)}$.
Example: on the unit disc the functions $e_n(z) = \sqrt{(n + 1)/\pi}\, z^n$ form an orthonormal basis, so $K(z, w) = \frac{1}{\pi}\sum_{n \ge 0} (n + 1)(z\bar w)^n = \frac{1}{\pi(1 - z\bar w)^2}$.
Non-example: the function $\frac{1}{1 - z\bar w}$ is not the Bergman kernel of the disc.
It is the Szegő kernel, which reproduces values for the inner product of the Hardy space $H^2$ with normalized arc length, not for the area inner product.
For instance, with $w = 0$ it is the constant $1$, while $\int_{\mathbb{D}} f \cdot 1 \, dA = \pi f(0)$, not $f(0)$.
If $\varphi \colon \Omega_1 \to \Omega_2$ is biholomorphic, then $K_1(z, w) = \varphi'(z) K_2(\varphi(z), \varphi(w)) \overline{\varphi'(w)}$.
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# Steenrod algebra

## Summary

The mod $2$ Steenrod algebra $\mathcal{A}_2$ is the graded algebra over $\Z/2\Z$ generated by the [[Steenrod squares|VR7RHP41M0]], subject to the [[Adem relations|GS8SCGYPX7]].
It is the algebra of all stable [[cohomology operations|K8T15QHN3W]] on mod $2$ cohomology.
So the mod $2$ cohomology of every space is a module over it.
For each odd prime $p$ there is an analogous algebra built from the reduced powers and the Bockstein.
The Steenrod algebra is a Hopf algebra, and it is the input of the [[Adams spectral sequence|DD4D0EKX74]], which computes stable homotopy groups.

## Explanations

### L1 human

Steenrod squares can be composed, and compositions of them are again cohomology operations.
The Steenrod algebra collects all such compositions and their sums.
The squares do not commute freely: certain compositions can be rewritten in terms of others, and the Adem relations list the rules.
For example, applying $\operatorname{Sq}^1$ twice always gives zero.
</document>

<document source="https://tamnd.github.io/math2/node/JX20MNDDXT.md" tag="JX20MNDDXT" kind="concept">
# Multiplicity of a local ring

## Summary

The multiplicity of a Noetherian [[local ring|MZ02BKPHJA]] $(R, \mathfrak m)$ of dimension $d$ is the positive integer $e(R)$ such that the [[Hilbert–Samuel function|BNQVS83GZV]] $\ell(R/\mathfrak m^n)$ grows like $e(R) n^d / d!$.
It measures how singular the corresponding point is.
A regular local ring has multiplicity $1$, and a plane curve singularity has multiplicity equal to the lowest degree of a term in its equation.

## Explanations

### L1 human

At a smooth point a curve looks like a single line, and at a node it looks like two lines crossing.
The multiplicity counts how many sheets come together at the point, in an algebraic way.
For the cusp $y^2 = x^3$ the lowest degree term is $y^2$, of degree $2$, and the multiplicity is $2$.
The [[definition|B7CV5DZPRK]] makes this precise.
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# Cartier divisor

## Definition

Let $X$ be a [[scheme|5A35TYZD02]] with structure sheaf $\mathcal{O}_X$.
For an open set $U \subseteq X$, let $S(U)$ be the set of $s \in \mathcal{O}_X(U)$ whose germ $s_x$ is a non-zero-divisor in $\mathcal{O}_{X,x}$ for every $x \in U$.
The *sheaf of total quotient rings* $\mathcal{K}_X$ is the [[sheaf|TB6TM6R4MD]] associated to the presheaf $U \mapsto S(U)^{-1}\mathcal{O}_X(U)$.
For an integral scheme it is the constant sheaf of the function field; in general it is built from the [[total ring of fractions|K5Z67HR9ET]].
Let $\mathcal{K}_X^*$ and $\mathcal{O}_X^*$ be the sheaves of [[units|07C6ECNJET]] of $\mathcal{K}_X$ and of $\mathcal{O}_X$.
A *Cartier divisor* on $X$ is a global section of the quotient sheaf $\mathcal{K}_X^* / \mathcal{O}_X^*$.

Equivalently, a Cartier divisor is given by an [[open cover|049PXYHW1T]] $(U_i)$ of $X$ and sections $f_i \in \mathcal{K}_X^*(U_i)$ such that $f_i / f_j \in \mathcal{O}_X^*(U_i \cap U_j)$ for all $i, j$.
Two such families $(U_i, f_i)$ and $(V_k, g_k)$ define the same divisor when $f_i / g_k \in \mathcal{O}_X^*(U_i \cap V_k)$ for all $i, k$.

## Remarks

The Cartier divisors form an abelian group, written additively, with $(U_i, f_i) + (U_i, g_i) = (U_i, f_i g_i)$.
A divisor is *principal* if it is given by one global $f \in \mathcal{K}_X^*(X)$, and *effective* if all $f_i$ lie in $\mathcal{O}_X(U_i)$.

Example: on $X = \mathbb{A}^1_k = \operatorname{Spec} k[t]$, the single pair $(X, t)$ is an effective principal Cartier divisor, the point $0$.
The compatibility condition holds because there is only one open set, and $t/t = 1$ is a unit.

Non-example: on $\mathbb{A}^1_k$ take $U = X$ with $f_U = t$ and $V = X \setminus \set{0}$ with $f_V = t - 1$.
On $U \cap V = V$, the ratio $t/(t-1)$ is not a unit, since it has a pole at $t = 1$, so the compatibility condition fails.

The divisor $D = (U_i, f_i)$ defines the invertible subsheaf $\mathcal{O}_X(D) \subseteq \mathcal{K}_X$ generated by $f_i^{-1}$ on $U_i$.
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<document source="https://tamnd.github.io/math2/node/JX47MS1R1H.md" tag="JX47MS1R1H" kind="concept">
# Cup product

## Summary

The cup product is a multiplication $H^k(X; R) \times H^l(X; R) \to H^{k+l}(X; R)$ on [[singular cohomology|6N7BC70CX0]] with coefficients in a commutative ring $R$.
It is associative, has a unit, and is graded commutative.
It makes cohomology into a ring, the [[cohomology ring|RDT4785M4D]], which carries much more information than the cohomology groups alone.

## Explanations

### L1 human

Cohomology classes can be multiplied, and the degrees add.
Take a $k$-cochain and an $l$-cochain, and a simplex of dimension $k + l$.
Test the first on the front face of dimension $k$, test the second on the back face of dimension $l$, and multiply the results.
On the torus, the two classes that measure winding around each circle multiply to a class that measures the whole surface.
On the wedge of two circles and a sphere the same groups appear, but every product of two degree $1$ classes is zero.
So the cup product tells the two spaces apart.
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<document source="https://tamnd.github.io/math2/node/JX65KSVHSJ.md" tag="JX65KSVHSJ" kind="concept">
# Affine connection

## Summary

An affine connection on a smooth manifold is a [[connection|5ZYRRVE8BW]] on its tangent bundle.
It lets one differentiate vector fields along vector fields, and so defines parallel transport of tangent vectors, geodesics, torsion and curvature.
On a Riemannian manifold there is a preferred affine connection, the Levi-Civita connection.
The name comes from the idea that a connection identifies nearby tangent spaces as pieces of one affine space.

## Explanations

### L1 human

In the plane, comparing arrows at two points is easy: slide one arrow to the other point without turning it.
On a curved surface no such sliding is given in advance.
An affine connection is a rule that tells how to differentiate one field of arrows along another, and so how to slide arrows along curves.
In the plane the usual rule takes the directional derivative of each component of the vector field.
Straight lines are then the curves whose velocity does not change, and on a curved surface the same idea gives geodesics.
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<document source="https://tamnd.github.io/math2/node/JX6CHG7HWV.md" tag="JX6CHG7HWV" kind="definition">
# Field of rational functions

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $K[x]$ be the [[polynomial ring|0Q9EZS90GY]] in one variable $x$ over $K$.
The field of rational functions $K(x)$ is the [[field of fractions|0VF11ZTTMD]] of $K[x]$.
Its elements are the fractions $p/q$ with $p, q \in K[x]$ and $q \ne 0$, where $p/q = p'/q'$ if and only if $p q' = p' q$.
More generally, for variables $x_1, \dots, x_n$, the field $K(x_1, \dots, x_n)$ is the field of fractions of $K[x_1, \dots, x_n]$.

## Remarks

Example: $\frac{x^2 - 1}{x - 1} = \frac{x + 1}{1}$ in $\Q(x)$, since $(x^2 - 1) \cdot 1 = (x + 1)(x - 1)$.
Non-example: no element $p/q$ of $\Q(x)$ satisfies $(p/q)^2 = x$, since $p^2 = x q^2$ would give $2 \deg p = 1 + 2 \deg q$, which is impossible.
So $K(x) \supset K$ is a field extension in which $x$ is [[transcendental|0NQMZ89X4A]] over $K$.
An element of $K(x)$ is a formal fraction, not a function on $K$: over $\F_2$, the fraction $\frac{1}{x^2 + x}$ is a nonzero element, although $x^2 + x$ vanishes at every point of $\F_2$.
Over $\R$ or $\C$, the elements correspond to [[rational functions|0WCZH4AFMT]] in the sense of analysis.

## Explanations

### L1 human

You build $K(x)$ from polynomials in the same way as you build the rational numbers from the integers.
Every nonzero fraction of polynomials has an inverse, obtained by turning it upside down.
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<document source="https://tamnd.github.io/math2/node/JX775JPB0N.md" tag="JX775JPB0N" kind="concept">
# Box–Jenkins method

## Summary

The Box–Jenkins method is an iterative procedure for building an [[ARIMA model|KTDVZP4DF2]] of a time series and using it to forecast.
It cycles through model identification, parameter estimation and diagnostic checking until the fitted model describes the data adequately.
George Box and Gwilym Jenkins set it out in their 1970 book *Time Series Analysis: Forecasting and Control*.

## Explanations

### L1 human

The method is a recipe a forecaster follows step by step.
First, look at plots of the data and of its autocorrelations to guess a simple model.
Second, fit that model to the data.
Third, check whether what the model leaves unexplained looks like pure noise; if not, go back and try another model.
For example, a sample partial autocorrelation that cuts off after lag $2$ suggests trying an AR($2$) model.
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<document source="https://tamnd.github.io/math2/node/JXBSN6E2X5.md" tag="JXBSN6E2X5" kind="definition">
# Opposite structure

## Definition

Let $\ast$ be a [[binary operation|0M418FE7BB]] on a set $S$.
The *opposite* of $(S, \ast)$ is $(S, \ast^{\mathrm{op}})$, where $a \ast^{\mathrm{op}} b = b \ast a$ for all $a, b \in S$.
For a structure with several operations, such as a ring, the opposite structure reverses the chosen multiplication and keeps all other operations and constants.

## Remarks

Example: if $(G, \cdot)$ is a [[group|G5S1M9]], then $(G, \cdot^{\mathrm{op}})$ is a group with the same identity and the same inverses.
Associativity holds since $(a \cdot^{\mathrm{op}} b) \cdot^{\mathrm{op}} c = c \cdot b \cdot a = a \cdot^{\mathrm{op}} (b \cdot^{\mathrm{op}} c)$.
The map $x \mapsto x^{-1}$ is an isomorphism from $G$ to its opposite, because $(xy)^{-1} = y^{-1} x^{-1}$.

Non-example: for $2 \times 2$ real matrices, the opposite multiplication is not the same operation as the usual one, because the matrices do not all commute.
The transpose $A \mapsto A^T$ is still an isomorphism from the matrix ring to its opposite, because $(AB)^T = B^T A^T$.

Common notations are $S^{\mathrm{op}}$ and $S^{\circ}$.
The opposite of the opposite is the original structure.
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# Young's convolution inequality

## Statement

Let $n \ge 1$ and consider [[Lp spaces|SASP14MA1Z]] on $\R^n$ with [[Lebesgue measure|2Z2QVWZ262]].
Let $1 \le p, q, r \le \infty$ with
$$\frac{1}{p} + \frac{1}{q} = 1 + \frac{1}{r}.$$
Let $f \in L^p(\R^n)$ and $g \in L^q(\R^n)$.
Then the [[convolution|2Z29JN3VES]]
$$(f * g)(x) = \int_{\R^n} f(y) \, g(x - y) \, dy$$
converges absolutely for almost every $x \in \R^n$, the function $f * g$ lies in $L^r(\R^n)$, and
$$\norm{f * g}_r \le \norm{f}_p \norm{g}_q.$$

## Remarks

The case $p = 1$ and $q = r$ follows from Minkowski's inequality for integrals: $f * g$ is an average of translates of $g$, each of norm $\norm{g}_q$.
The case $r = \infty$, with $p$ and $q$ conjugate, is Hölder's inequality applied for each fixed $x$; then $f * g$ is defined everywhere.
The general case follows from these two by the [[Riesz–Thorin theorem|AGRKRG1TMJ]], or directly by a three-factor Hölder inequality.

Example: with $p = q = 1$, one gets $r = 1$ and $\norm{f * g}_1 \le \norm{f}_1 \norm{g}_1$.
So $L^1(\R^n)$ is a Banach algebra under convolution.

The exponent relation is forced by scaling.
If $f_\lambda(x) = f(\lambda x)$, then $\norm{f_\lambda}_p = \lambda^{-n/p} \norm{f}_p$ and $f_\lambda * g_\lambda = \lambda^{-n} (f * g)_\lambda$.
An inequality $\norm{f * g}_r \le C \norm{f}_p \norm{g}_q$ for all $\lambda > 0$ therefore requires $1 + 1/r = 1/p + 1/q$.

The same inequality holds with the same proof on any unimodular locally compact group with Haar measure.
The constant $1$ is not sharp on $\R^n$ when $1 < p, q, r < \infty$; the sharp constant is known.

## Explanations

### L1 human

Convolution smears one function out by another, like blurring a picture with a lens.
Young's inequality says how big the blurred result can be in terms of the sizes of the two ingredients.
The exponents must fit together in one specific way, which a change of scale explains.

### L2 human

Young's inequality is the basic estimate for convolution operators $f \mapsto f * g$ with a fixed kernel $g$.
With $g \in L^1$ it shows that mollification $f * \varphi_\varepsilon$ is bounded on every $L^p$, which is the first step of approximation by smooth functions.
With $g$ a power such as $\abs{x}^{-\alpha}$, which is not in any $L^q(\R^n)$, Young's inequality does not apply, and one needs the Hardy–Littlewood–Sobolev inequality, which uses weak $L^q$ instead.
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<document source="https://tamnd.github.io/math2/node/JXFHBY77RN.md" tag="JXFHBY77RN" kind="concept">
# Littlewood's three principles

## Summary

Littlewood's three principles are rules of thumb for working with [[Lebesgue measure|2Z2QVWZ262]] on the real line.
Every measurable set is nearly a finite union of intervals, and every [[measurable function|YYNCJSR0PJ]] is nearly continuous.
Every convergent sequence of measurable functions is nearly uniformly convergent.
Each principle has a precise form, and the last two are [[Lusin's theorem|GZMAH9W2SK]] and [[Egorov's theorem|762Y2NYA7W]].
They let one transfer arguments for simple objects to general measurable ones.

## Explanations

### L1 human

Measurable sets and functions can look wild, but up to a small error they behave like familiar ones.
The word "nearly" always means: after throwing away a set of arbitrarily small measure.
For example, the set of rational numbers in $[0, 1]$ looks scattered everywhere, but its measure is $0$, so it is nearly the empty set.
Its indicator function is discontinuous everywhere, yet it is nearly the zero function, which is continuous.
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# Minimal variety

## Summary

A minimal variety is a nontrivial [[variety|AJNZJD8442]] of algebras whose only proper [[subvariety|4YSCK1DF26]] is the trivial variety.
Minimal varieties are the atoms of the [[lattice of subvarieties|GJAYCVYZYF]] of a variety.
Every nontrivial variety contains at least one minimal variety.

## Explanations

### L1 human

A minimal variety is as small as a variety can be without being trivial.
If one adds to its defining identities any identity that does not already hold in it, only one-element algebras are left.
The variety of Boolean algebras is an example.
Each Boolean algebra with at least two elements contains the two-element Boolean algebra, and that algebra generates the whole variety.
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<document source="https://tamnd.github.io/math2/node/JXK6NRDH4V.md" tag="JXK6NRDH4V" kind="problem">
# Banach measure problem

## Problem

Let $n \ge 1$.
Is there a function $\mu$, defined on all subsets of $\R^n$, with values in $[0, \infty]$, such that:

1. $\mu$ is finitely additive: $\mu(A \cup B) = \mu(A) + \mu(B)$ for disjoint $A, B \subseteq \R^n$.
2. $\mu$ is invariant under isometries: $\mu(g(A)) = \mu(A)$ for every isometry $g$ of $\R^n$, such as a [[translation|087VJM5Q77]] or a [[rotation|0HFREYRSZP]].
3. $\mu([0, 1]^n) = 1$.

## Remarks

Such a function is a [[finitely additive measure|KFWQ24D9M6]] on the power set.
The countably additive version has a negative answer, by the Vitali set.
Hausdorff showed with his paradox on the sphere that the answer is negative for $n \ge 3$.
Banach showed in 1923 that the answer is positive for $n = 1$ and $n = 2$; such a function can even be chosen to agree with [[Lebesgue measure|2Z2QVWZ262]] on Lebesgue measurable sets.
These functions are called Banach measures, and they are not unique.
The difference between the dimensions is explained by amenability.
The isometry groups of $\R$ and $\R^2$ are amenable, and the rotation group of $\R^3$ contains a free group on two generators.

## Explanations

### L1 human

The question asks whether every set of points, however wild, can be given a length, an area or a volume.
The size should add up for two pieces and should not change when a set is moved rigidly.
On the line and in the plane this is possible, but in space of dimension three or more it is not.
In space, the Banach–Tarski paradox even cuts a ball into finitely many pieces that can be reassembled into two balls of the same size.
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# Morphism of schemes

## Summary

A morphism of schemes is a morphism of [[locally ringed spaces|B7FPT8JTXR]] between [[schemes|5A35TYZD02]].
It is a continuous map together with a compatible way to pull back functions.
Between [[affine schemes|B3V3XJ2W8G]], morphisms are the same as ring homomorphisms in the opposite direction.
Schemes with these morphisms form a category, the setting for almost all constructions of algebraic geometry.

## Explanations

### L1 human

A map between two geometric objects should send points to points and also let functions be pulled back.
For schemes, the second part is part of the data: a function $g$ on an open set of the target becomes a function on its preimage.
The local condition says that if $g$ vanishes at $f(x)$, then its pullback vanishes at $x$.
For example, the ring inclusion $\Z \to \Z[i]$ gives a morphism $\operatorname{Spec} \Z[i] \to \operatorname{Spec} \Z$, which sends a prime ideal of $\Z[i]$ to its intersection with $\Z$.
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<document source="https://tamnd.github.io/math2/node/JXRXV92WN5.md" tag="JXRXV92WN5" kind="definition">
# Oval

## Definition

Let $\Pi$ be a [[projective plane|DANFNN8C4Y]].
A line is a *tangent* of a set $O$ of points at $P \in O$ if it meets $O$ only in $P$.
A set $O$ of points of $\Pi$ is an *oval* if:

1. every line of $\Pi$ meets $O$ in at most two points;
2. for every point $P \in O$ there is exactly one tangent of $O$ at $P$.

## Remarks

If $\Pi$ is finite of [[order|K2YVMTGTBB]] $n$, a set $O$ is an oval if and only if $O$ has $n + 1$ points and no three of them on a line.
Indeed, suppose $O$ has no three points on a line, and let $P \in O$.
Of the $n + 1$ lines through $P$, exactly $\abs{O} - 1$ meet $O$ again.
So $P$ has $n + 2 - \abs{O}$ tangents, and this equals $1$ exactly when $\abs{O} = n + 1$.

Example: in $PG(2, q)$ the non-degenerate [[conic|R3TMVHTP3A]] $xz = y^2$ has the $q + 1$ points $[1 : t : t^2]$, $t \in \F_q$, and $[0 : 0 : 1]$.
A line $ax + by + cz = 0$, with $(a, b, c) \ne 0$, meets it in the points $[1 : t : t^2]$ with $a + bt + ct^2 = 0$, and in $[0 : 0 : 1]$ exactly when $c = 0$.
If $c \ne 0$, the polynomial has at most two roots.
If $c = 0$, it has at most one root, and the line also contains $[0 : 0 : 1]$.
In both cases the line meets the conic in at most two points.
So the conic is a set of $q + 1$ points with no three on a line, hence an oval.

Non-example: three non-collinear points $A, B, C$ in $PG(2, 3)$.
No line contains all three, so clause 1 holds.
Through $A$ there are $4$ lines, and only $AB$ and $AC$ meet the set again.
So $A$ has two tangents, and clause 2 fails.
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# Frobenius group

## Summary

A Frobenius group is a finite [[transitive|02GK6HZYBT]] permutation group in which some non-identity element fixes a point, but no non-identity element fixes two points.
The elements that fix no point, together with the identity, form a normal subgroup, the Frobenius kernel, by [[Frobenius's theorem|XK7H3YXBQT]].
The group is then a [[semidirect product|0H3WZXCZNF]] of the kernel and a point [[stabiliser|0AYMP41THN]], the Frobenius complement.
Frobenius groups appear throughout finite group theory and character theory.

## Explanations

### L1 human

Take the three corners of a triangle and all six ways to permute them.
A reflection keeps one corner in place and swaps the other two; a rotation moves every corner.
No symmetry other than the identity keeps two corners in place.
This is the smallest Frobenius group.
Its rotations, together with the identity, form a subgroup, and that is the pattern in every Frobenius group.
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# Pell numbers

## Example

The *Pell numbers* $P_n$ are the [[solution|06MJC8YDYR]] of
$$P_{n+2} = 2 P_{n+1} + P_n \qquad (n \ge 0)$$
with $P_0 = 0$ and $P_1 = 1$.
The first terms are $0, 1, 2, 5, 12, 29, 70, 169, \dots$.
The characteristic polynomial $X^2 - 2X - 1$ has roots $1 \pm \sqrt{2}$, which gives
$$P_n = \frac{(1 + \sqrt{2})^n - (1 - \sqrt{2})^n}{2\sqrt{2}}.$$
Check: the formula solves the recurrence, since each root $q$ satisfies $q^2 = 2q + 1$; it gives $P_0 = 0$ and $P_1 = 2\sqrt{2}/(2\sqrt{2}) = 1$.
They form the [[Lucas sequence|DP39F5KQ2W]] $U_n(2, -1)$.

Let $H_n$ be the solution of the same recurrence with $H_0 = H_1 = 1$, so $H_n = 1, 1, 3, 7, 17, 41, 99, \dots$.
Then $H_n^2 - 2 P_n^2 = (-1)^n$.
For example $3^2 - 2 \cdot 2^2 = 1$ and $7^2 - 2 \cdot 5^2 = -1$.
So the pairs $(H_n, P_n)$ give solutions of the [[Pell equations|0Z0NE3J6JX]] $x^2 - 2y^2 = \pm 1$, and $H_n / P_n$ tends to $\sqrt{2}$.

## Explanations

### L1 human

Pell numbers grow like the Fibonacci numbers, but each term is twice the previous one plus the one before.
Their ratio $P_{n+1}/P_n$ tends to $1 + \sqrt{2}$, about $2.414$.
The fractions $1/1, 3/2, 7/5, 17/12, 41/29, \dots$ built from them are better and better approximations of $\sqrt{2}$.
For instance $17/12 \approx 1.41667$ while $\sqrt{2} \approx 1.41421$.
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# Multiply transitive group action

## Definition

Let a group $G$ [[act|0WCY42EV9V]] on a set $\Omega$, and let $k$ be a positive integer with $k \le \abs{\Omega}$.
The action is *$k$-transitive* if for all $k$-tuples $(x_1, \dots, x_k)$ and $(y_1, \dots, y_k)$ of pairwise distinct elements of $\Omega$ there is $g \in G$ with $g x_i = y_i$ for $i = 1, \dots, k$.
The action is *sharply $k$-transitive* if, moreover, this element $g$ is unique for all such pairs of $k$-tuples.

## Remarks

A $1$-transitive action is a [[transitive action|02GK6HZYBT]].
A $k$-transitive action is $j$-transitive for every $j \le k$.
For $\abs{\Omega} \ge 2$, an action is $2$-transitive if and only if it is transitive and the [[stabiliser|0AYMP41THN]] $G_\omega$ of one point $\omega$ is transitive on $\Omega \setminus \set{\omega}$.

Example: the [[symmetric group|0515A6D2Y1]] $S_n$ is $n$-transitive on $\set{1, \dots, n}$, since any bijection between two orderings of distinct points extends to a permutation.
For $n \ge 3$, the [[alternating group|0Z5905Q73D]] $A_n$ is $(n-2)$-transitive.
Indeed, let $\sigma \in S_n$ do the job, and let $\tau$ swap the two points not among $y_1, \dots, y_{n-2}$.
Then $\tau \sigma$ also does the job, and one of $\sigma$ and $\tau \sigma$ is even.
For a [[finite field|0ZXZ82FTX2]] $\F_q$, the maps $x \mapsto ax + b$ with $a \ne 0$ act sharply $2$-transitively on $\F_q$.

Non-example: the eight symmetries of a square, acting on its corners $1, 2, 3, 4$ in cyclic order.
The action is transitive, but no symmetry sends the pair $(1, 2)$ of neighbouring corners to the pair $(1, 3)$ of opposite corners, so it is not $2$-transitive.

Each $2$-transitive action is [[primitive|Y10S0751KG]].
Camille Jordan showed that for $k \ge 4$ the only finite sharply $k$-transitive groups are $S_k$, $S_{k+1}$, $A_{k+2}$, and the [[Mathieu groups|T1KR4KPX8P]] $M_{11}$ for $k = 4$ and $M_{12}$ for $k = 5$.
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# Unit and counit

## Definition

Let $F \colon \mathcal{C} \to \mathcal{D}$ be [[left adjoint|TZHF6ZM92S]] to $G \colon \mathcal{D} \to \mathcal{C}$ by the natural bijections $\varphi_{c, d} \colon \Hom_{\mathcal{D}}(F c, d) \to \Hom_{\mathcal{C}}(c, G d)$.

1. The *unit* of the adjunction is the family $\eta_c = \varphi_{c, F c}(\id_{F c}) \colon c \to G F c$ for objects $c$ of $\mathcal{C}$.
2. The *counit* of the adjunction is the family $\varepsilon_d = \varphi_{G d, d}^{-1}(\id_{G d}) \colon F G d \to d$ for objects $d$ of $\mathcal{D}$.

## Remarks

Both families are [[natural transformations|KYCPCC57FT]], $\eta \colon \mathrm{Id}_{\mathcal{C}} \Rightarrow G F$ and $\varepsilon \colon F G \Rightarrow \mathrm{Id}_{\mathcal{D}}$.
They satisfy the *triangle identities*
$$\varepsilon_{F c} \circ F(\eta_c) = \id_{F c}, \qquad G(\varepsilon_d) \circ \eta_{G d} = \id_{G d}.$$
They recover $\varphi$ by $\varphi(g) = G(g) \circ \eta_c$ for $g \colon F c \to d$.

Example: for the free vector space functor $F$ over a field $k$ and the forgetful functor $G$, the unit $\eta_X \colon X \to G F X$ sends $x$ to the basis vector $e_x$.
The counit $\varepsilon_V \colon F G V \to V$ sends the basis vector $e_v$ to the vector $v$.
Then $\varepsilon_{F X}(F(\eta_X)(e_x)) = \varepsilon_{F X}(e_{e_x}) = e_x$, which checks the first triangle identity.

Non-example: if the characteristic of $k$ is not $2$, the natural maps $\eta'_X(x) = 2 e_x$ are not a unit for this counit.
Here $\varepsilon_{F X} \circ F(\eta'_X)$ is multiplication by $2$, so the first triangle identity fails.
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# Fourier series method

## Summary

The Fourier series method solves linear partial differential equations on bounded intervals or boxes by expanding the unknown in a [[Fourier series|G12KXDHPPS]].
Each term of the series is a sine or cosine mode, and the equation acts on each mode separately.
The partial differential equation then becomes a family of ordinary differential equations for the coefficients.
Fourier introduced the method to solve the [[heat equation|VP2HN86VAG]], and it led to the theory of Fourier series.
See the [[definition|J0MHS4ZCKR]].

## Explanations

### L1 human

A vibrating string or a heated rod can be described as a sum of simple waves, each with its own frequency.
Under the heat equation or the wave equation, each simple wave evolves on its own, in a way that is easy to compute.
So one splits the initial state into simple waves, lets each of them evolve, and adds them up again.
For the heat equation on $(0, \pi)$ with zero boundary values, the mode $\sin(kx)$ just decays like $e^{-k^2 t}$.
High frequencies decay fastest, which is why heat smooths out sharp temperature differences.
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<document source="https://tamnd.github.io/math2/node/JYKVYS4N32.md" tag="JYKVYS4N32" kind="concept">
# Kauffman bracket

## Summary

The Kauffman bracket is a Laurent polynomial in one variable $A$ attached to an unoriented diagram of a [[link|1A5WRCNTM3]].
It is computed by smoothing every crossing in the two possible ways and counting the circles that result.
It is unchanged by the second and third [[Reidemeister moves|T3HAGWYKK7]] and changes by a simple factor under the first.
After a correction by the writhe, it gives the [[Jones polynomial|4SPC17FPXE]].

## Explanations

### L1 human

At each crossing of a knot drawing you can cut and reconnect the strands in two ways, so that the crossing disappears.
Doing this at every crossing leaves a set of disjoint circles.
The bracket adds up all the ways to do this, with a weight that depends on how many smoothings of each type you chose and how many circles you got.
It is a recipe that turns any diagram into a polynomial by simple counting.
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<document source="https://tamnd.github.io/math2/node/JYRZ927BMQ.md" tag="JYRZ927BMQ" kind="statement">
# Jensen's inequality for integrals

## Statement

Let $(X, \Sigma, \mu)$ be a measure space with $\mu$ a [[probability measure|0VGMEBVW1M]], that is, $\mu(X) = 1$.
Let $-\infty \le a < b \le \infty$ and let $g \colon X \to (a, b)$ be an [[integrable function|X7GTF3D98F]].
Let $\varphi \colon (a, b) \to \R$ be a [[convex function|0VV0HSW1ZS]].
Then $\int_X g \, d\mu$ lies in $(a, b)$, and the negative part of $\varphi \circ g$ is integrable.
Moreover,
$$\varphi\Big( \int_X g \, d\mu \Big) \le \int_X \varphi \circ g \, d\mu,$$
where the integrals are [[Lebesgue integrals|DWJF7ABPR1]] and the right side may be $+\infty$.

## Remarks

Proof idea: let $m = \int g \, d\mu$.
A convex function on an open interval has a supporting line at $m$: there is $c \in \R$ with $\varphi(y) \ge \varphi(m) + c (y - m)$ for all $y \in (a, b)$.
Put $y = g(x)$ and integrate over $X$; the term $c(g - m)$ has integral $0$ because $\mu(X) = 1$.

Example: with $\varphi(y) = y^2$ it gives $(\int g \, d\mu)^2 \le \int g^2 \, d\mu$.
In probability this is $\operatorname{Var}(Y) \ge 0$.
With $\varphi = \exp$ it gives $\exp(\int g \, d\mu) \le \int e^{g} \, d\mu$.

The normalisation $\mu(X) = 1$ is needed.
For $\mu = 2 \cdot$ Lebesgue measure on $[0, 1]$, $g = 1$ and $\varphi(y) = y^2$, the left side is $4$ and the right side is $2$.
When $\mu$ is the probability measure with masses $w_i$ at finitely many points, the statement is the finite [[Jensen's inequality|0RWMRTPC87]].

## Explanations

### L1 human

The value of a convex function at an average is at most the average of its values.
Here the average is an integral against a probability measure, so it covers both weighted sums and continuous averages.
The graph of a convex function bends upwards, which is why averaging after applying the function gives the larger result.

### L2 human

In probability the inequality reads $\varphi(\mathbb{E}[Y]) \le \mathbb{E}[\varphi(Y)]$ for an integrable random variable $Y$.
It yields the comparison $\norm{f}_p \le \norm{f}_q$ for $p \le q$ on probability spaces, the AM–GM inequality, and the non-negativity of the Kullback–Leibler divergence.
There is also a version for conditional expectations, used in martingale theory.
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# Infinite product

## Summary

An infinite product $\prod_{n=1}^{\infty} a_n$ is the multiplicative analogue of a [[series|0NM626FH6Q]]: it is the sequence of partial products $a_1 a_2 \cdots a_N$.
Its value, when it [[converges|PCYRWA7SYB]], is the limit of these partial products.
Infinite products express many classical functions and constants, such as the [[Wallis product|0XCYA4YERS]] for $\pi$ and the [[Euler product formula|0C3PN4075R]] for the zeta function.
The exact definition is in [[infinite product|RZ0Z2BTMT4]].

## Explanations

### L1 human

An infinite product multiplies infinitely many numbers, one after another.
As with infinite sums, one multiplies the first $N$ factors and looks at what happens as $N$ grows.
For the product to settle down to a nonzero number, the factors must get closer and closer to $1$.
For example, $(1 - \tfrac14)(1 - \tfrac19)(1 - \tfrac1{16}) \cdots$ has partial product $(N+1)/(2N)$ up to the factor $1 - 1/N^2$, and these approach $\tfrac12$.
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# Siggers term

## Summary

A Siggers term is an idempotent term of arity $6$ that satisfies one fixed identity in three variables, $s(x, y, x, z, y, z) \approx s(y, x, z, x, z, y)$.
Every Siggers term is a [[Taylor term|QEAMRNRJJD]].
Conversely, a finite algebra that has a Taylor term also has a Siggers term, so one explicit identity replaces the whole family of Taylor conditions.
This makes the Siggers term a convenient test in the algebraic theory of constraint satisfaction problems.

## Explanations

### L1 human

Taylor terms come in many shapes and arities, which makes them hard to search for.
A Siggers term has one fixed shape: six inputs and one balanced equation in three letters.
For finite algebras, looking for this single shape is enough to decide whether some Taylor term exists.
For example, in a semilattice the term $x_1 \wedge x_2 \wedge \cdots \wedge x_6$ is a Siggers term, because both sides of the identity become $x \wedge y \wedge z$.
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# Perverse sheaf

## Definition

Let $X$ be a complex [[algebraic variety|HV6YBS0YDH]], with its classical topology.
A [[sheaf|TB6TM6R4MD]] of finite-dimensional $\C$-vector spaces on $X$ is *constructible* if it is locally constant on each piece of some finite partition of $X$.
The pieces must be locally closed subvarieties.
Let $D^b(X)$ be the bounded [[derived category|5ST7NVJFSM]] of sheaves of $\C$-vector spaces on $X$.
Let $D^b_c(X)$ be its full subcategory of objects $K$ whose cohomology sheaves $\mathcal{H}^i(K)$ are constructible.
Let $\mathbb{D} \colon D^b_c(X) \to D^b_c(X)$ be Verdier duality.
An object $K$ of $D^b_c(X)$ is a *perverse sheaf* if, for every integer $i$:

1. (support) $\dim_\C \operatorname{supp} \mathcal{H}^{-i}(K) \le i$;
2. (cosupport) $\dim_\C \operatorname{supp} \mathcal{H}^{-i}(\mathbb{D} K) \le i$.

Here $\operatorname{supp}$ is the closure of the set of points where the stalk is nonzero, and the empty set has dimension $-\infty$.

## Remarks

The same definition works for complex analytic spaces, with partitions into analytic strata, for example [[Whitney stratifications|3F84AMVJNV]], and for other coefficient fields.

Example: let $X$ be smooth and connected of dimension $d$, and $K = \C_X[d]$, the constant sheaf placed in degree $-d$.
Its only cohomology sheaf is $\mathcal{H}^{-d}(K) = \C_X$, with support of dimension $d$, so clause 1 holds.
Verdier duality gives $\mathbb{D}(\C_X) = \C_X[2d]$ on a smooth $X$, so $\mathbb{D} K = \C_X[d]$ and clause 2 holds too.

Non-example: for $d \ge 1$, the constant sheaf $\C_X$ in degree $0$ is not perverse.
Its cohomology sheaf $\mathcal{H}^0(\C_X) = \C_X$ has support of dimension $d > 0$, so clause 1 fails for $i = 0$.

The skyscraper sheaf $\C_p$ at a point, in degree $0$, is perverse, and so is the intersection cohomology complex of any closed irreducible subvariety, suitably shifted.
Perverse sheaves form an abelian category in which every object has finite length.
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# Favard's theorem on trigonometric approximation

## Statement

Let $r \ge 1$ and $n \ge 1$ be integers, and let $K_r$ be the [[Favard constant|9K1W9Q49NR]].
Let $f \colon \R \to \R$ be $2\pi$-[[periodic|0ATF6Q0633]], with $f^{(r-1)}$ absolutely continuous and $\abs{f^{(r)}} \le 1$ almost everywhere.
Then there is a real [[trigonometric polynomial|J3NJQ5FZJ5]] $T$ of degree at most $n - 1$ such that
$$\max_{x \in \R} \abs{f(x) - T(x)} \le \frac{K_r}{n^r}.$$
The constant $K_r$ is sharp.
For each $n$, some $f$ of this class has distance exactly $K_r / n^r$ from the set of all such $T$.

## Remarks

The extremal function is the $r$-fold periodic integral of the square wave $\sgn \sin(nx)$, normalised to have mean $0$.
Its error of best approximation equals $K_r / n^r$, which explains the alternating series in the definition of $K_r$: they come from the Fourier series of the square wave.
The upper bound comes from explicit linear approximation methods, so it is attained by a linear operator.

Example: $r = 1$, $n = 1$.
A periodic function with $\abs{f'} \le 1$ has $\max f - \min f \le \pi$, so the best constant has error at most $\pi/2 = K_1$.
The triangle wave equal to $\abs{x}$ on $[-\pi, \pi]$ reaches this bound.

## Explanations

### L1 human

This is the best possible form of Jackson's inequality for functions with a bounded $r$-th derivative.
It says exactly how well trigonometric polynomials of degree below $n$ can approximate every such function at once.

### L2 human

Jackson's inequality gives the rate $n^{-r}$ with some constant; Favard's theorem gives the exact worst-case constant.
It is a model result for the theory of widths and optimal recovery, where one asks for the smallest error over a whole class.
The same constants appear in sharp inequalities between a function and its derivatives, such as the Kolmogorov inequalities.
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# Statistical inference

## Summary

Statistical inference is the process of drawing conclusions about an unknown probability distribution from data generated by it.
It works inside a [[statistical model|K3AMMSY8DD]] and produces estimates, confidence sets, tests or predictions.
The two main schools are [[frequentist inference|YDSNHKM3EG]] and [[Bayesian inference|0JWRJX2RMP]].

## Explanations

### L1 human

You see a sample and want to say something about the whole population or process behind it.
Statistical inference turns the sample into a statement, such as "the average height is about $172$ cm, give or take $2$ cm".
It also says how much trust such a statement deserves.
For example, from $1000$ voters in a poll you infer the share of a party among all voters.
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# Symmetric monoidal category

## Definition

A *symmetric monoidal category* is a [[braided monoidal category|VT8SFWT0XW]] $(\mathcal{C}, \otimes, I, \alpha, \lambda, \rho, \gamma)$ such that for all objects $A, B$ of $\mathcal{C}$
$$\gamma_{B,A} \circ \gamma_{A,B} = \id_{A \otimes B}.$$

## Remarks

Under this condition the two hexagon identities of the braiding are equivalent, so one of them suffices.
Example: $\mathbf{Vect}_k$ with $\gamma(v \otimes w) = w \otimes v$; applying the swap twice returns $v \otimes w$.
Example: $\Z/2$-graded vector spaces with the Koszul sign $\gamma(v \otimes w) = (-1)^{\abs{v}\abs{w}}\, w \otimes v$ on homogeneous elements; the signs cancel when the swap is applied twice.
Non-example: $\Z/3$-graded complex vector spaces with $\gamma(v \otimes w) = q^{\abs{v}\abs{w}}\, w \otimes v$, where $q = e^{2\pi i/3}$.
This is a braiding because $(a, b) \mapsto q^{ab}$ is a bicharacter of $\Z/3$, but applying it twice multiplies $v \otimes w$ by $q^{2\abs{v}\abs{w}}$, which is not $1$ when $\abs{v} = \abs{w} = 1$.
Symmetric monoidal categories form the special case of [[monoidal categories|M3MSQ3NKAE]] that model commutative operations.
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# Translation plane

## Summary

A translation plane is a [[projective plane|DANFNN8C4Y]] with a special line $\ell$.
The collineations that act like translations with respect to $\ell$ can move any point off $\ell$ to any other.
Removing $\ell$ leaves an affine plane with a full group of translations, as in the real plane.
Translation planes are exactly the planes coordinatized by [[quasifields|JQ22QH86DE]], and they form the largest well-understood family of non-Desarguesian planes.

## Explanations

### L1 human

In the ordinary plane one can slide every figure by a fixed vector, and these slides carry any point to any other point.
A translation plane is a projective plane that keeps this property for one chosen line, which plays the role of the line at infinity.
Each slide fixes every point at infinity and moves every line to a parallel line.
The real projective plane is a translation plane, and so is every finite plane $PG(2, q)$.
There are also translation planes that are not of this form, such as the Hall plane of order $9$.
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# Statistical model

## Definition

Let $S$ be a set, the sample space, with a [[sigma-algebra|04WSW8T3SY]] $\mathcal{S}$ on $S$.
A *statistical model* on $(S, \mathcal{S})$ is a pair $(S, \mathcal{P})$, where $\mathcal{P}$ is a set of [[probability measures|0VGMEBVW1M]] on $(S, \mathcal{S})$.
The model is often written with an index set $\Theta$ as $\mathcal{P} = \set{P_\theta : \theta \in \Theta}$, and $\Theta$ is the parameter space.

## Remarks

Example: $S = \set{0, 1}^n$ with all subsets measurable, and $P_p(x) = p^{k}(1 - p)^{n - k}$ where $k = x_1 + \dots + x_n$, for $p \in [0, 1]$.
Each $P_p$ is a probability measure, because the binomial theorem gives $\sum_{x} P_p(x) = (p + 1 - p)^n = 1$.
This is the model of $n$ independent tosses of a coin with unknown chance $p$ of heads.
Non-example: the set of measures $\set{c \cdot P_{1/2} : c > 0}$ is not a statistical model, because for $c \ne 1$ the measure $c \cdot P_{1/2}$ has total mass $c$, so it is not a probability measure.
The data are modelled as a value $x$ of a random element $X$ with law in $\mathcal{P}$.
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# Group extension

## Definition

Let $N$ and $Q$ be [[groups|G5S1M9]].
An *extension* of $Q$ by $N$ is a group $G$ together with two [[group homomorphisms|0JYTG5K9Y4]] $\iota \colon N \to G$ and $\pi \colon G \to Q$ such that:

1. $\iota$ is [[injective|09NWETVTNY]].
2. $\pi$ is surjective.
3. The [[image|0S21GNW9ZB]] of $\iota$ equals the kernel of $\pi$.

In other words, $1 \to N \xrightarrow{\iota} G \xrightarrow{\pi} Q \to 1$ is a short exact sequence of groups.
The extension is *split* if there is a homomorphism $s \colon Q \to G$ with $\pi \circ s = \id_Q$.
The extension is *central* if $\iota(N)$ lies in the [[centre|0DZ93493VN]] of $G$.

## Remarks

In an extension, $\iota(N)$ is a normal subgroup of $G$ and $G / \iota(N) \cong Q$.
Some authors call the same object an extension of $N$ by $Q$, so the order of the words varies between books.

Example: $G = S_3$, $N = A_3$ with the inclusion $\iota$, and $Q = \set{\pm 1}$ with $\pi$ the sign map.
The inclusion is injective, the sign map is onto, and the kernel of the sign map is $A_3$.
The extension is split: $s(-1) = (1\ 2)$ is a homomorphism with $\pi \circ s = \id$.

Non-example: $\iota \colon \Z \to \Z$, $n \mapsto 4n$, and $\pi \colon \Z \to \Z/2\Z$, reduction modulo $2$.
Clauses 1 and 2 hold, but the image $4\Z$ of $\iota$ is not the kernel $2\Z$ of $\pi$, so clause 3 fails.

The extension $0 \to 2\Z/4\Z \to \Z/4\Z \to \Z/2\Z \to 0$ does not split, because the only element of order $2$ in $\Z/4\Z$ lies in $2\Z/4\Z$.
A split extension is the same as a [[semidirect product|0H3WZXCZNF]] $N \rtimes Q$.
When $N$ is abelian, the extensions of $Q$ by $N$ that induce a given action of $Q$ on $N$ are classified up to equivalence by the second cohomology group $H^2(Q, N)$.
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# Fourier–Mukai transform

## Definition

Let $k$ be a field and let $X$ and $Y$ be smooth [[projective varieties|A4H4CZ2V35]] over $k$.
Let $D^b(X)$ be the [[bounded derived category of coherent sheaves|2VP2D51KB8]] on $X$, and let $p \colon X \times Y \to X$ and $q \colon X \times Y \to Y$ be the projections.
For an object $P$ of $D^b(X \times Y)$, the *Fourier–Mukai transform* with kernel $P$ is the functor
$$\Phi_P \colon D^b(X) \to D^b(Y), \qquad \Phi_P(E) = Rq_*\bigl(p^* E \otimes^L P\bigr).$$
Here $p^*$ is the pullback, which is exact since $p$ is flat.
The symbols $\otimes^L$ and $Rq_*$ denote the derived tensor product and the derived pushforward, which are [[derived functors|NPMC3K6J2F]].

## Remarks

$\Phi_P$ is an exact functor of triangulated categories, and composites of Fourier–Mukai transforms are again Fourier–Mukai transforms, with kernel given by a convolution of the kernels.
Example: for $Y = X$ and $P = \mathcal{O}_\Delta$, the structure sheaf of the diagonal $\Delta \subseteq X \times X$, one has $p^* E \otimes^L \mathcal{O}_\Delta \cong \delta_* E$ for the diagonal embedding $\delta$, and $q \circ \delta = \id_X$, so $\Phi_{\mathcal{O}_\Delta} \cong \id$.
For an abelian variety $A$ with dual $\hat A$ and Poincaré line bundle $\mathcal{P}$ on $A \times \hat A$, Mukai proved that $\Phi_{\mathcal{P}} \colon D^b(A) \to D^b(\hat A)$ is an equivalence.
An example that is not an equivalence when $\dim Y > 0$: for $P = \mathcal{O}_{X \times Y}$, flat base change gives $\Phi_P(E) \cong R\Gamma(X, E) \otimes_k \mathcal{O}_Y$.
So every object goes to a sum of shifts of $\mathcal{O}_Y$, and the structure sheaf of a closed point of $Y$ is not in the image.
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# Picard group

## Definition

Let $(X, \mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]].
The *Picard group* $\operatorname{Pic}(X)$ is the set of isomorphism classes of [[invertible sheaves|7JCF2TE3V5]] on $X$, with the operation
$$[\mathcal{L}] \cdot [\mathcal{M}] = [\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{M}].$$
It is an [[abelian group|082GHNPJ53]] with identity $[\mathcal{O}_X]$ and inverse $[\mathcal{L}]^{-1} = [\mathcal{L}^\vee]$, where $\mathcal{L}^\vee = \mathcal{H}om(\mathcal{L}, \mathcal{O}_X)$.

## Remarks

For a field $k$ and $r \ge 1$, $\operatorname{Pic}(\mathbb{P}^r_k) \cong \Z$, with $n \mapsto [\mathcal{O}(n)]$.
For $X = \operatorname{Spec} A$, $\operatorname{Pic}(X)$ is the [[Picard group of the ring|DZW58R6T2Q]] $A$.
If $A$ is a principal ideal domain, every invertible sheaf on $\operatorname{Spec} A$ is trivial, so $\operatorname{Pic}(\operatorname{Spec} A) = 0$.
There is a natural isomorphism $\operatorname{Pic}(X) \cong H^1(X, \mathcal{O}_X^\times)$ with the first cohomology of the sheaf of units.
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# Kontsevich quantization formula

## Statement

Let $U \subseteq \R^d$ be open and let $\pi^{ij} = -\pi^{ji}$, $1 \le i, j \le d$, be smooth functions on $U$.
Assume that $\{f, g\} = \sum_{i,j} \pi^{ij} \partial_i f \, \partial_j g$ satisfies the Jacobi identity.
So $U$ is a [[Poisson manifold|A2KQ49Y4NF]] with Poisson bivector $\pi$.
For $n \ge 0$, let $G_n$ be the set of admissible graphs.
Such a graph has aerial vertices $1, \ldots, n$, ground vertices $L$ and $R$, and $2n$ directed edges.
Each aerial vertex is the source of exactly two edges, labeled first and second.
No edge goes from a vertex to itself, and no two edges have the same source and target.
To $\Gamma \in G_n$ attach:

1. the bidifferential operator $B_\Gamma$. Place a copy of $(\pi^{ij})$ at each aerial vertex, $f$ at $L$ and $g$ at $R$. Let each edge act as a partial derivative $\partial_i$ on the object at its target, and sum over all indices;
2. the weight $w_\Gamma$ defined below.

Let $C_n$ be the space of configurations of $n$ distinct points in the upper half-plane $\mathbb{H}$, and put $L = 0$ and $R = 1$.
Let $e_k^1$ and $e_k^2$ be the edges leaving vertex $k$.
For an edge $e$, let $\phi_e$ be the angle at the source of $e$, in the hyperbolic metric of $\mathbb{H}$.
It goes from the geodesic to $\infty$ to the geodesic to the target of $e$.
Then
$$w_\Gamma = \frac{1}{n! \, (2\pi)^{2n}} \int_{C_n} \bigwedge_{k=1}^n \left( d\phi_{e_k^1} \wedge d\phi_{e_k^2} \right).$$

Let $C^\infty(U)[\![\hbar]\!]$ be the space of [[formal power series|033WGN00BF]] in $\hbar$ with coefficients in $C^\infty(U)$.
Extend the following product $\hbar$-bilinearly to it:
$$f \star g = \sum_{n \ge 0} \hbar^n \sum_{\Gamma \in G_n} w_\Gamma \, B_\Gamma(f, g).$$
Then $\star$ is associative, and $f \star g = fg + \hbar \{f, g\} + O(\hbar^2)$.
So $\star$ is a [[deformation quantization|3MKSJ1J04K]] of $(U, \pi)$.

## Remarks

Kontsevich obtained the formula from his formality theorem: there is an $L_\infty$ quasi-isomorphism from polyvector fields to polydifferential operators on $\R^d$.
The formula is its explicit form, and associativity follows from Stokes' theorem on compactified configuration spaces, where the boundary terms either cancel or vanish by the Jacobi identity $[\pi, \pi] = 0$.
For a constant Poisson bivector only graphs without edges between aerial vertices contribute, and the formula gives the Moyal product.
With the formality theorem and a globalization argument, Kontsevich showed that every Poisson manifold has a deformation quantization.
He also showed that equivalence classes of star products correspond bijectively to equivalence classes of formal Poisson structures.

## Explanations

### L1 human

In quantum mechanics observables do not commute, while in classical mechanics they do.
A deformation quantization replaces the ordinary product of functions by a new associative product.
The new product depends on a small parameter $\hbar$, Planck's constant, and its first correction is the Poisson bracket.
Kontsevich's formula writes such a product explicitly for any Poisson structure on a piece of $\R^d$, as a sum over pictures of graphs with numerical weights given by integrals.

### L2 human

Before Kontsevich, star products were known for symplectic manifolds, by work of De Wilde and Lecomte and of Fedosov, but not for general Poisson manifolds.
The weights $w_\Gamma$ are hard to compute in general.
Cattaneo and Felder later explained the formula as a perturbative expansion of a topological field theory, the Poisson sigma model, on a disc.
</document>

<document source="https://tamnd.github.io/math2/node/JZNQAN00ZJ.md" tag="JZNQAN00ZJ" kind="concept">
# Torsion tensor

## Summary

The torsion tensor of an [[affine connection|JX65KSVHSJ]] measures how far covariant derivatives are from being symmetric in the two vector fields.
The Lie bracket is subtracted to make this a tensor.
It is a tensor field of type $(1, 2)$, antisymmetric in its two arguments.
Connections with zero torsion are called torsion-free or symmetric; the Levi-Civita connection is the unique torsion-free connection compatible with a given Riemannian metric.
Geometrically, torsion describes how infinitesimal parallelograms built by parallel transport fail to close.

## Explanations

### L1 human

Start at a point, move a little in direction $X$, then a little in direction $Y$, carrying each direction along with the connection.
Do the same in the other order.
For a torsion-free connection the two endpoints agree to second order, so the small parallelogram closes.
Torsion measures the small gap that remains.
The flat connection of the plane has no torsion, while a connection that twists directions as you move, like a screw, can have torsion.
</document>

<document source="https://tamnd.github.io/math2/node/JZNQTTMG58.md" tag="JZNQTTMG58" kind="concept">
# Torsion group

## Summary

A torsion group, also called a periodic group, is a [[group|G5S1M9]] in which every element has finite [[order|0WE2NHZJ7B]].
Every finite group is a torsion group, but there are also infinite ones.
The Burnside problem asks whether a finitely generated torsion group must be finite; the answer is no in general.
For an abelian group, the elements of finite order form the [[torsion subgroup|0DBWXQRXQJ]].

## Explanations

### L1 human

In a torsion group, if you take any element and multiply it by itself again and again, you come back to the identity after finitely many steps.
A clock is a good picture: adding hours, you always return to 12.
The group $\Q/\Z$ of fractions modulo $1$ is an infinite example.
The fraction $\tfrac{3}{7}$ added to itself $7$ times gives $3$, which is $0$ modulo $1$.
The integers are not a torsion group, because adding $1$ to itself never returns to $0$.
</document>

<document source="https://tamnd.github.io/math2/node/JZTCEHFVS2.md" tag="JZTCEHFVS2" kind="statement">
# Existence of limits from products and equalizers

## Statement

Let $\mathcal{C}$ be a [[category|A2DE66937D]] that has [[equalizers|CTJ5AD8ST5]] of all pairs of parallel morphisms.

1. If $\mathcal{C}$ has all small [[products|S5744QZV4N]], then $\mathcal{C}$ is [[complete|FWGPKXCPB7]].
2. If $\mathcal{C}$ has all finite products, then $\mathcal{C}$ is [[finitely complete|PYSBNWCHCF]].

## Remarks

Let $F \colon J \to \mathcal{C}$ be a diagram with $J$ small, and write $u \colon j \to k$ for the morphisms of $J$.
Form the products $P = \prod_{j} F(j)$ over the objects and $Q = \prod_{u \colon j \to k} F(k)$ over the morphisms.
Two morphisms $s, t \colon P \to Q$ are given on the factor of $u \colon j \to k$ by $F(u) \circ \pi_j$ and by $\pi_k$.
The equalizer of $s$ and $t$ is a [[limit|CHS5AFJGWE]] of $F$.
If $J$ is finite, only finite products are used.

The converse is clear, since products and equalizers are limits.
For example, in the category of groups the limit of a diagram is the subgroup of the product cut out by the equations $F(u)(x_j) = x_k$.
Dually, a category with all small coproducts and all coequalizers is cocomplete.

## Explanations

### L1 human

Every limit can be built from two simple ones.
First take the product of all the objects of the diagram.
Then cut it down with an equalizer to the families that agree with all the maps.
So to show that a category has all limits, it is enough to check products and equalizers.

### L2 human

This is the usual way to prove that a category is complete.
For sets, groups, rings and topological spaces the products and equalizers are easy to write down, and the theorem does the rest.
The same construction shows that a functor which preserves products and equalizers preserves all limits.
It also gives the finite version: a terminal object, binary products and equalizers give all finite limits.
</document>

<document source="https://tamnd.github.io/math2/node/K00NJY6C33.md" tag="K00NJY6C33" kind="statement">
# Dimension of the solution space of a linear recurrence

## Statement

Let $K$ be a field, let $d \ge 0$ be an integer, and let $c_0, \dots, c_{d-1} \in K$.
Let $E$ be the [[linear recurrence with constant coefficients|081THG13JZ]] $u_{n+d} = c_0 u_n + c_1 u_{n+1} + \dots + c_{d-1} u_{n+d-1}$ $(n \ge 0)$.
Then the [[space of solutions|05BDYVQQ0T]] of $E$ is a $K$-vector space of [[dimension|05G2G0P224]] $d$.

## Remarks

The map $u \mapsto (u_0, \dots, u_{d-1})$ from the space of solutions to $K^d$ is linear.
It is injective, because the recurrence computes every later term from the first $d$ terms, and surjective, because any choice of first terms extends to a solution.
So it is an isomorphism, and the dimension is $d$.
Example: for the Fibonacci recurrence $u_{n+2} = u_{n+1} + u_n$ over $\R$, the space has dimension $2$.
A basis is given by the solutions that start $(1, 0)$ and $(0, 1)$, or by $\varphi^n$ and $\psi^n$ with $\varphi, \psi = \frac{1 \pm \sqrt{5}}{2}$.
The same argument works for a linear difference equation with variable coefficients $a_k(n) u_{n+k} + \dots + a_0(n) u_n = 0$, provided $a_k(n) \ne 0$ for all $n$.

## Explanations

### L1 human

A linear recurrence of order $d$ has as many independent solutions as it has starting values.
Any solution is a combination of $d$ basic solutions.
So to solve such a recurrence completely, it is enough to find $d$ solutions that are independent.

### L2 human

The theorem is the discrete analogue of the fact that a linear differential equation of order $d$ has a $d$-dimensional space of solutions.
In practice, one finds a basis from the roots of the [[characteristic polynomial|0BWDJGQYST]].
By the theorem on [[geometric solutions|096SD4ERQH]], each root $q$ gives the solution $q^n$.
In characteristic $0$, a nonzero root $q$ of multiplicity $m$ gives the solutions $n^j q^n$ for $j < m$.
When $c_0 \ne 0$ and the characteristic polynomial splits over $K$, these $d$ sequences are independent, so the dimension count shows they form a basis.
The general solution of an inhomogeneous equation is one particular solution plus an element of this space.
</document>

<document source="https://tamnd.github.io/math2/node/K02CEYQBCX.md" tag="K02CEYQBCX" kind="definition">
# Distinguished boundary of a polydisc

## Definition

Let $P(a, r)$ be the [[polydisc|DJBWP7D3M8]] in $\C^n$ with centre $a = (a_1, \dots, a_n)$ and polyradius $r = (r_1, \dots, r_n)$, each $r_j > 0$.
Its *distinguished boundary* is the set
$$T(a, r) = \set{z \in \C^n : \abs{z_j - a_j} = r_j \text{ for } j = 1, \dots, n}.$$

## Remarks

The set $T(a, r)$ is the product of the $n$ [[circles|060W5CBZP6]] $\abs{z_j - a_j} = r_j$, so it is a real torus of dimension $n$.
It is also called the *skeleton* of the polydisc, and it is written $\partial_0 P(a, r)$ in some texts.
Example: $(1, -i)$ lies in the distinguished boundary of the unit polydisc of $\C^2$, since $\abs{1} = 1$ and $\abs{-i} = 1$.
Non-example: $(1, 0)$ lies in the topological boundary of the unit polydisc but not in its distinguished boundary, since $\abs{0} \ne 1$.
For $n = 1$ the distinguished boundary is the whole boundary circle.
If $f$ is continuous on the closed polydisc and holomorphic on $P(a, r)$, then $\max \abs{f}$ over the closed polydisc equals $\max \abs{f}$ over $T(a, r)$.
</document>

<document source="https://tamnd.github.io/math2/node/K04XRMH820.md" tag="K04XRMH820" kind="concept">
# Keller–Segel model

## Summary

The Keller–Segel model is a system of parabolic equations for chemotaxis, the motion of cells toward a chemical signal that they produce themselves.
It was proposed by Keller and Segel in 1970 to describe the aggregation of slime mould amoebae.
The cells diffuse and drift up the gradient of the chemical, and this self-attraction can make the cell density concentrate.
In two dimensions the total mass decides between global existence and blow-up in finite time.
This makes the model a standard test case for blow-up in parabolic systems.

## Explanations

### L1 human

Imagine cells that release a scent and also like to move toward where the scent is strong.
Random motion spreads the cells out, while the scent pulls them together.
If there are few cells, spreading wins and the cells stay spread out.
If there are many, attraction wins and the cells collapse into a dense cluster.
</document>

<document source="https://tamnd.github.io/math2/node/K07QH8EKD0.md" tag="K07QH8EKD0" kind="statement">
# Hammersley–Clifford theorem

## Statement

Let $G = (V, E)$ be a finite undirected [[graph|00C0BR7D7H]], and let $\mathcal{C}$ be the set of its [[cliques|0BYX1D7C2E]].
For each $v \in V$ let $S_v$ be a finite set, and let $X = (X_v)_{v \in V}$ be a random vector with values in $\prod_{v \in V} S_v$.
Assume that its probability mass function $p$ is strictly positive: $p(x) > 0$ for every $x$.
For $C \subseteq V$ write $x_C = (x_v)_{v \in C}$.
Then the following are equivalent.

1. $X$ is a [[Markov random field|6W69H3A3G5]] with respect to $G$.
2. There are functions $\psi_C > 0$, one for each $C \in \mathcal{C}$, such that for all $x$,
$$p(x) = \prod_{C \in \mathcal{C}} \psi_C(x_C).$$

## Remarks

That 2 implies 1 is direct.
Let $S$ separate $A$ from $B$, and enlarge $A$ and $B$ so that $A \cup B \cup S = V$ while $S$ still separates them.
No clique meets both $A$ and $B$, so each clique lies in $A \cup S$ or in $B \cup S$.
So $p$ is a function of $(x_A, x_S)$ times a function of $(x_B, x_S)$, which gives $X_A \perp X_B \mid X_S$.
For 1 implies 2, one writes $\log p$ as a sum of terms indexed by the subsets of $V$, by Möbius inversion.
The pairwise Markov property then shows that the terms of the non-cliques vanish.
This second step needs positivity, since it takes logarithms and divides by $p$.
The direction 2 implies 1 holds without positivity, but the direction 1 implies 2 can fail.
There are Markov random fields with zero probabilities on a cycle of four vertices that do not factorise over the cliques.

Example: for the Ising model on $G$, $p(x) \propto \exp\bigl(\beta \sum_{\set{u, v} \in E} x_u x_v + h \sum_v x_v\bigr)$ with $x_v \in \set{-1, 1}$.
This is a product of positive factors on edges and vertices, which are cliques, so the Ising model is a Markov random field.

## Explanations

### L1 human

The theorem says two descriptions of a random field agree.
One description lists independence facts: each variable cares only about its neighbours.
The other gives a formula: the probability is a product of terms, each involving only a group of mutually adjacent variables.
This holds as long as no configuration has probability zero.

### L2 human

The theorem is why Markov random fields are written as Gibbs distributions $p(x) \propto \exp(-\sum_C V_C(x_C))$ with clique potentials $V_C$.
Conversely, any such energy defines a Markov random field, and in Gibbs sampling the update of $x_v$ then depends only on the neighbours of $v$.
The same statement holds for positive densities with respect to a product measure, for example for Gaussian graphical models.
There the missing edges of the graph are the zero entries of the inverse covariance matrix.
For directed graphs, the analogous factorisation is the definition of a [[Bayesian network|21JT983G60]].
</document>

<document source="https://tamnd.github.io/math2/node/K0AWJK6NZH.md" tag="K0AWJK6NZH" kind="concept">
# Product order

## Summary

The product order on a [[Cartesian product|09CEDJYBKP]] of partially ordered sets compares tuples coordinate by coordinate.
One tuple is below another when each of its coordinates is below the matching coordinate.
It is the product in the category of posets, and products of chains are the model spaces for [[order dimension|JC4J24MF93]].

## Explanations

### L1 human

Compare two students by their marks in two subjects.
Say that the first is behind the second when the first scored at most as well in both subjects.
A student with marks $(70, 80)$ is behind one with $(75, 90)$, but $(70, 90)$ and $(75, 80)$ cannot be compared.
So the product of two total orders is usually not a total order.
</document>

<document source="https://tamnd.github.io/math2/node/K0EQTBRPE3.md" tag="K0EQTBRPE3" kind="concept">
# Néron model

## Summary

Let $R$ be a [[discrete valuation ring|MGM3CNJW9V]] with fraction field $K$.
The Néron model of an [[abelian variety|R7W03TXRS9]] over $K$ is its best smooth model over $R$.
It is a smooth [[group scheme|TJ04VQP4NB]] of finite type over $R$, characterized by a universal property for maps from smooth $R$-schemes.
André Néron proved that it always exists.
Its special fibre describes the reduction of the abelian variety, even when the reduction is bad.

## Explanations

### L1 human

An abelian variety given by equations with rational coefficients can be reduced modulo a prime, but the reduction may become singular.
One would like a version over the $p$-adic integers that is as good as possible.
The Néron model is that version: it is smooth, and every rational point extends to a point of the model.
When the variety has good reduction, the Néron model is just the abelian scheme given by good equations.
</document>

<document source="https://tamnd.github.io/math2/node/K0GVMM3YPF.md" tag="K0GVMM3YPF" kind="concept">
# Dirac bundle

## Summary

A Dirac bundle over a Riemannian manifold is a bundle of [[Clifford modules|31Q0RDPZ00]] with a metric and a connection that are compatible with Clifford multiplication.
These data are exactly what is needed to define a [[Dirac operator|3DXGJTEQZH]] that is formally self-adjoint and satisfies a general Bochner–Weitzenböck formula.
Spinor bundles, the bundle of differential forms and the Clifford bundle itself are Dirac bundles.
The notion gives one framework for the de Rham, signature, Dolbeault and spin Dirac operators used in index theory.

## Explanations

### L1 human

A Dirac operator needs three ingredients.
They are a way to multiply sections by tangent vectors with the Clifford rule, a way to measure lengths of sections, and a way to differentiate them.
A Dirac bundle is a vector bundle carrying all three, fitted together so that the multiplication preserves lengths and the differentiation respects both.
For example, differential forms with multiplication $v \cdot \omega = v^\flat \wedge \omega - \iota_v \omega$ form a Dirac bundle.
Its Dirac operator is $d + d^*$, the square root of the Hodge Laplacian.
</document>

<document source="https://tamnd.github.io/math2/node/K0KF9KMF5W.md" tag="K0KF9KMF5W" kind="definition">
# Autocorrelation function

## Definition

Let $(X_t)_{t \in \Z}$ be a real weakly [[stationary process|DT0QXJE705]]: $E[X_t^2] < \infty$, the mean $E[X_t] = \mu$ does not depend on $t$, and the [[covariance|0KBFAWYRTE]] $\operatorname{Cov}(X_{t+h}, X_t)$ depends only on $h$.
The *autocovariance function* is $\gamma(h) = \operatorname{Cov}(X_{t+h}, X_t)$ for $h \in \Z$.
If $\gamma(0) > 0$, the *autocorrelation function* is
$$\rho(h) = \frac{\gamma(h)}{\gamma(0)}, \qquad h \in \Z.$$
It is the [[correlation coefficient|0DNN9B6Q2F]] of $X_{t+h}$ and $X_t$.
For observations $x_1, \dots, x_n$ of a [[time series|EA3MF4VTEH]] with mean $\bar x$, the *sample autocovariance* and *sample autocorrelation* at lag $\abs{h} < n$ are
$$\hat\gamma(h) = \frac1n \sum_{t=1}^{n - \abs{h}} (x_{t + \abs{h}} - \bar x)(x_t - \bar x), \qquad \hat\rho(h) = \frac{\hat\gamma(h)}{\hat\gamma(0)}.$$

## Remarks

Every autocorrelation function satisfies $\rho(0) = 1$, $\rho(-h) = \rho(h)$ and $\abs{\rho(h)} \le 1$.
It is nonnegative definite: $\sum_{i,j} a_i a_j \rho(t_i - t_j) \ge 0$ for all real $a_i$ and integers $t_i$, since this sum is $\operatorname{Var}(\sum_i a_i X_{t_i}) / \gamma(0)$.

Example: let $(\varepsilon_t)$ be uncorrelated with mean $0$ and variance $\sigma^2 > 0$, and let $X_t = \varepsilon_t + \varepsilon_{t-1}$.
Then $\gamma(0) = 2\sigma^2$, $\gamma(\pm 1) = \sigma^2$ and $\gamma(h) = 0$ for $\abs{h} \ge 2$, so $\rho(\pm 1) = 1/2$ and $\rho(h) = 0$ for $\abs{h} \ge 2$.

Non-example: for a random walk $S_t = \varepsilon_1 + \cdots + \varepsilon_t$, the variance $t \sigma^2$ depends on $t$, so the process is not weakly stationary and $\rho$ is not defined.

The divisor $n$ in $\hat\gamma$, rather than $n - \abs{h}$, makes the sample autocovariance nonnegative definite.
In signal processing, autocorrelation often means the unnormalized $E[X_{t+h} X_t]$.
</document>

<document source="https://tamnd.github.io/math2/node/K0NRSZCCW7.md" tag="K0NRSZCCW7" kind="definition">
# Complex geodesic

## Definition

Let $\mathbb{D}$ be the unit disc in $\C$ with its Poincaré distance $\rho$.
Let $X$ be a connected [[complex manifold|JSGWAN2RJV]] with [[Kobayashi pseudodistance|DY8QQ5GTTX]] $k_X$, normalised so that $k_{\mathbb{D}} = \rho$.
A *complex geodesic* of $X$ is a [[holomorphic map|3MQTWKBBBT]] $\varphi \colon \mathbb{D} \to X$ such that
$$k_X(\varphi(a), \varphi(b)) = \rho(a, b) \quad \text{for all } a, b \in \mathbb{D}.$$

## Remarks

Since holomorphic maps decrease Kobayashi distances, the inequality $\le$ always holds; the condition is the reverse inequality.
Some authors use the Carathéodory distance $c_X$ in place of $k_X$; on bounded convex domains the two notions agree, since $c_X = k_X$ there.
Example: in the unit ball $B \subset \C^n$, $\varphi(\zeta) = (\zeta, 0, \dots, 0)$ is a complex geodesic.
Check: the projection $\pi(z) = z_1$ maps $B$ into $\mathbb{D}$ and $\pi \circ \varphi = \id$, so $\rho(a, b) = \rho(\pi\varphi(a), \pi\varphi(b)) \le k_B(\varphi(a), \varphi(b)) \le \rho(a, b)$.
Non-example: $\psi(\zeta) = (\zeta / 2, 0, \dots, 0)$ is holomorphic into $B$, but $k_B(\psi(a), \psi(b)) = \rho(a/2, b/2) < \rho(a, b)$ for $a \ne b$, by the same argument and the Schwarz–Pick lemma for $\zeta \mapsto \zeta/2$.
A complex geodesic is injective, since $\rho(a, b) > 0$ for $a \ne b$.
</document>

<document source="https://tamnd.github.io/math2/node/K0V6T18367.md" tag="K0V6T18367" kind="concept">
# Advanced Z-transform

## Summary

The advanced Z-transform is a variant of the [[Z-transform|5FY0JQGHYZ]] of sampled signals.
It samples a continuous-time signal at the instants $kT + m$, shifted by a part $m$ of the sampling period $T$.
It is $F(z, m) = \sum_{k \ge 0} f(kT + m) z^{-k}$, with $0 \le m \le T$.
It lets one describe delays that are not whole multiples of the sampling period, and the behaviour of a sampled system between sampling instants.
For fixed $m$ it has the usual properties of the Z-transform.

## Explanations

### L1 human

Sampling a signal at $0, T, 2T, \ldots$ only shows it at those instants.
The advanced Z-transform samples it at $m, T + m, 2T + m, \ldots$ instead, for a chosen offset $m$.
Letting $m$ run from $0$ to $T$ shows the signal at every time.
For $f(t) = e^{-at}$ one gets $F(z, m) = e^{-am} z/(z - e^{-aT})$.
</document>

<document source="https://tamnd.github.io/math2/node/K10YADF5XV.md" tag="K10YADF5XV" kind="statement">
# Borel–Weil–Bott theorem

## Statement

Let $G$ be a connected and simply connected complex [[semisimple Lie group|XJS07AJ50Y]], and let $T \subseteq G$ be a maximal torus.
Let $\Phi$ be the roots of $G$ with respect to $T$, with a set $\Phi^+$ of positive roots, and let $\rho = \frac{1}{2}\sum_{\alpha \in \Phi^+}\alpha$.
Let $B \supseteq T$ be the [[Borel subgroup|5JSYA1JNCC]] whose Lie algebra is spanned by the Lie algebra of $T$ and the root spaces of the negative roots.
Let $W$ be the [[Weyl group|EZGTEK3HQN]] with length function $\ell$, and write $w \cdot \mu = w(\mu + \rho) - \rho$.
Let $\mu$ be a character of $T$, extended to $B$.
Let $L_\mu$ be the holomorphic line bundle on the [[flag variety|NHRWCQQKTH]] $G/B$ defined as follows.
Its sections over an open set $U \subseteq G/B$ are the holomorphic $f$ on the preimage of $U$ in $G$ with $f(gb) = \mu(b)^{-1} f(g)$ for $b \in B$.
The group $G$ acts on the [[sheaf cohomology|PQ6Q143ARJ]] groups $H^i(G/B, L_\mu)$ by left translation.

1. If $\mu$ is dominant, then $H^0(G/B, L_\mu)$ is the [[irreducible|FX3NEFYFFN]] representation of highest weight $\mu$, and $H^i(G/B, L_\mu) = 0$ for $i > 0$.
2. If $\mu + \rho$ is orthogonal to some root, then $H^i(G/B, L_\mu) = 0$ for all $i$.
3. Otherwise there is a unique $w \in W$ such that $\lambda = w \cdot \mu$ is dominant; then $H^{\ell(w)}(G/B, L_\mu)$ is the irreducible representation of highest weight $\lambda$, and $H^i(G/B, L_\mu) = 0$ for $i \ne \ell(w)$.

## Remarks

Part 1 is the Borel–Weil theorem, and parts 2 and 3 are Bott's extension; part 1 is the case $w = 1$ of part 3.
A short proof due to Demazure uses fibrations of $G/B$ with fibre $\mathbb{P}^1$ to compare $L_\mu$ with $L_{s \cdot \mu}$ for simple reflections $s$.
This reduces the computation to the case of $SL(2,\C)$.
Example: for $G = SL(2, \C)$, $G/B = \mathbb{P}^1$ and $L_n \cong \mathcal{O}(n)$ for $n \in \Z$, with $\rho = 1$.
For $n \ge 0$, $H^0(\mathbb{P}^1, \mathcal{O}(n))$ is the space of homogeneous polynomials of degree $n$ in two variables, the irreducible representation of dimension $n + 1$.
For $n = -1$, $\mu + \rho = 0$, and indeed all cohomology of $\mathcal{O}(-1)$ vanishes; for $n = -m - 2$ with $m \ge 0$, $w$ is the reflection, $w \cdot \mu = m$, and $H^1(\mathbb{P}^1, \mathcal{O}(-m-2))$ has dimension $m + 1$.
Conventions differ: with $B$ built from the positive roots, or with the opposite sign in $f(gb) = \mu(b)^{-1}f(g)$, one obtains dual representations instead.
Since $G/B = K/T$ for a maximal compact subgroup $K$, the theorem also realises all irreducible representations of compact connected simply connected Lie groups.

## Explanations

### L1 human

Every finite-dimensional irreducible holomorphic representation of a simply connected complex semisimple group, or of a compact connected Lie group, can be built geometrically.
One takes the flag variety, a compact complex manifold attached to the group, and a line bundle on it chosen from a weight.
The holomorphic sections of the bundle form the representation with that highest weight.
Bott's part says what happens for the other weights: at most one cohomology group survives, and it is again irreducible.

### L2 human

The theorem is a geometric companion of the [[Weyl character formula|230F5FXPMQ]]: applying the holomorphic Lefschetz fixed point formula to the action of $T$ on $G/B$ gives the character formula.
It is the model for many later constructions of representations from geometry, such as geometric representation theory and the Beilinson–Bernstein localisation.
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<document source="https://tamnd.github.io/math2/node/K14ZBKYT81.md" tag="K14ZBKYT81" kind="definition">
# Lie group

## Definition

A *real Lie group* is a [[group|G5S1M9]] $G$ that is also a finite-dimensional [[smooth manifold|F45VQ47GPV]], such that:

1. the multiplication map $\mu \colon G \times G \to G$, $\mu(g, h) = gh$, is [[smooth|0V33BNKZ9S]];
2. the inversion map $\iota \colon G \to G$, $\iota(g) = g^{-1}$, is smooth.

A *complex Lie group* is defined in the same way with a complex manifold and holomorphic maps.
A *homomorphism of Lie groups* is a group homomorphism that is a smooth map.

## Remarks

Example: the circle $S^1 = \set{z \in \C : \abs{z} = 1}$ with complex multiplication.
It is a smooth submanifold of $\C = \R^2$, and in the angle coordinate the product and the inverse are $(\theta, \varphi) \mapsto \theta + \varphi$ and $\theta \mapsto -\theta$, which are smooth.

Example: $GL_n(\R)$ is an open subset of $\R^{n^2}$, and the product of matrices is polynomial in the entries.
By Cramer's rule, the inverse is a rational function of the entries with nonvanishing denominator $\det$.

Non-example: the additive group $\Q$ with the topology induced from $\R$ is a topological group but not a Lie group.
It is not locally Euclidean, since no point of $\Q$ has a neighbourhood homeomorphic to an open set of some $\R^n$.

The two clauses can be merged: $G$ is a Lie group exactly when the map $(g, h) \mapsto g h^{-1}$ is smooth.
The dimension of $G$ is its dimension as a manifold.
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# Local parametrization theorem

## Statement

Let $A_0$ be an [[irreducible analytic germ|7VPJWF0AZ7]] at $0 \in \C^n$.
Then there are an integer $d$ with $0 \le d \le n$ and linear coordinates $z = (z', z'')$ on $\C^n$ with $z' \in \C^d$ and $z'' \in \C^{n-d}$.
There are also open polydiscs $U' \subseteq \C^d$ and $U'' \subseteq \C^{n-d}$ centred at $0$.
With $U = U' \times U''$ and $\pi(z', z'') = z'$, the following properties hold.

1. There is a representative $A$ of $A_0$ that is an [[analytic subset|VMDTGTMN9P]] of $U$, and $\pi \colon A \to U'$ is a surjective [[proper map|15XEBGNGHC]] with finite fibres.
2. There are an analytic subset $\Delta \subsetneq U'$ and an integer $k \ge 1$ such that $\pi \colon A \setminus \pi^{-1}(\Delta) \to U' \setminus \Delta$ is a $k$-sheeted [[covering map|3XZRRE165N]].
3. The set $A \setminus \pi^{-1}(\Delta)$ is a connected complex [[submanifold|VKFW27WXEZ]] of $U$ of dimension $d$, and it is dense in $A$.

## Remarks

The integer $d$ is the dimension of the germ $A_0$, and the coordinates can be taken generic: a dense open set of linear coordinate systems works.
The proof uses the ideal $\mathfrak{p}$ of $A_0$, which is prime, and Noether normalization in the analytic setting.
After a generic change of coordinates, the Weierstrass preparation theorem makes $\mathcal{O}_{n,0}/\mathfrak{p}$ a finite extension of $\mathcal{O}_{d,0}$, the germs in $z'$ alone.
Here $d$ is the largest number of variables for which $\mathcal{O}_{d,0} \to \mathcal{O}_{n,0}/\mathfrak{p}$ is injective.
A primitive element of the fraction field extension and its discriminant give the set $\Delta$ and the number $k$ of sheets.
Example: for the cusp $A = \set{z_1^2 = z_2^3}$ in $\C^2$, take $z' = z_2$ and $\pi(z_1, z_2) = z_2$.
Over $z_2 \ne 0$ the fibre consists of the two square roots of $z_2^3$, so $\Delta = \set{0}$ and $k = 2$; the set $A \setminus \set{0}$ is the image of the punctured disc under $t \mapsto (t^3, t^2)$, hence connected.
Irreducibility is needed for the connectedness in part 3.
For $A = \set{z_1 z_2 = 0}$ and any linear projection to a line with finite fibres on $A$, the complement of $\pi^{-1}(\Delta)$ has a component in each axis.

## Explanations

### L1 human

The theorem describes what an irreducible analytic set looks like near a point.
After a good choice of coordinates, it is a finite branched cover of a ball in $\C^d$.
Over most points of the ball there are exactly $k$ points of the set, and these sheets come together only over a thin set.
So locally an analytic set is a many-valued graph over a $d$-dimensional ball.

### L2 human

The local parametrization is the basic structure theorem of local analytic geometry.
It gives the dimension of an analytic set and shows that the regular points are dense.
It also yields the Rückert Nullstellensatz, and it is the analytic counterpart of Noether normalization for affine varieties.
The number $k$ of sheets for generic coordinates is the multiplicity of the germ at $0$.
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# Motive

## Summary

A motive is an object of a category built from algebraic varieties and [[algebraic cycles|9VFWPZEVGN]] on their products.
Grothendieck proposed this category as a universal cohomology theory for varieties.
Every good cohomology theory, such as [[ℓ-adic cohomology|E1DVHN06CR]], [[algebraic de Rham cohomology|13F5JZPDD0]] or singular cohomology, should factor through the category of motives.
Pure motives over a field are well defined, with morphisms given by correspondences modulo an equivalence relation such as [[rational equivalence|K76N026JFK]].
Many of their expected properties depend on the [[standard conjectures|D8GA2R3QT5]].

## Explanations

### L1 human

Different cohomology theories of the same variety give groups that look alike, with the same dimensions and similar structures.
Grothendieck asked for a single object behind all of them, the motive of the variety, of which each cohomology is a "realization".
Motives can be split into pieces even when the variety cannot.
For example, the motive of the projective line is the sum of the motive of a point and the Lefschetz motive $\mathbb{L}$.
The piece $\mathbb{L}$ accounts for the cohomology in degree $2$.
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<document source="https://tamnd.github.io/math2/node/K1DX0S4797.md" tag="K1DX0S4797" kind="example">
# Category of topological spaces

## Example

The *category of topological spaces* $\mathbf{Top}$ has the [[topological spaces|0CQHFPGJ45]] as objects and the [[continuous maps|0VGFHTM75Y]] as morphisms.
Composition is the composition of maps, and the identity of $X$ is $\id_X$.
This is a [[category|A2DE66937D]].
If $f \colon X \to Y$ and $g \colon Y \to Z$ are continuous and $U \subseteq Z$ is open, then $(g \circ f)^{-1}(U) = f^{-1}(g^{-1}(U))$ is open, so $g \circ f$ is continuous.
The identity map is continuous, and composition of maps is associative.

## Remarks

The isomorphisms of $\mathbf{Top}$ are the homeomorphisms.
A continuous bijection need not be an isomorphism.
The identity map from $\R$ with the discrete topology to $\R$ with the usual topology is continuous and bijective, but its inverse is not continuous.
$\mathbf{Top}$ is a [[complete category|FWGPKXCPB7]] and is also cocomplete: the product is the product topology, and the coproduct is the disjoint union topology.
The forgetful functor to sets has a left adjoint (the discrete topology) and a right adjoint (the indiscrete topology).
$\mathbf{Top}$ is not [[Cartesian closed|BFBNQ36QS8]]; for this reason algebraic topology often works in the full subcategory of compactly generated spaces.

## Explanations

### L1 human

The category of topological spaces gathers all spaces and all continuous maps between them.
Its isomorphisms are the homeomorphisms, the maps that topology regards as "the same shape".
For example, the map $t \mapsto (\cos 2\pi t, \sin 2\pi t)$ from $[0, 1]$ to the circle is a morphism, though not an isomorphism, since it glues the two ends of the interval.
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<document source="https://tamnd.github.io/math2/node/K1HTQF26MA.md" tag="K1HTQF26MA" kind="concept">
# Subanalytic set

## Summary

A subanalytic set is a subset of a [[real analytic manifold|Z7H98E1NHP]] that is locally the image of a bounded [[semianalytic set|E1V22QQHYJ]] under a coordinate [[projection|033DA1WE8Y]].
The class is larger than the class of semianalytic sets, because projections of semianalytic sets need not be semianalytic.
Subanalytic sets are closed under finite unions, intersections, complements and images under proper analytic maps.
They play in real analytic geometry the role that [[semialgebraic sets|BX5WSK8ZCT]] play in real algebraic geometry.

## Explanations

### L1 human

A semianalytic set is cut out locally by finitely many equations and inequalities between real analytic functions.
Shadows of such sets, obtained by forgetting some coordinates, can be more complicated.
Subanalytic sets are exactly these shadows, taken locally and of bounded pieces.
For example, the image of the bounded set $\set{(x, t) : x = t^2,\ 0 < t < 1}$ under $(x, t) \mapsto x$ is the interval $(0, 1)$, which is subanalytic.
In spite of the larger class, subanalytic sets still behave well: near each point they have only finitely many connected components.
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<document source="https://tamnd.github.io/math2/node/K1SQAAN8D3.md" tag="K1SQAAN8D3" kind="definition">
# Product category

## Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be [[categories|A2DE66937D]].
The *product category* $\mathcal{C} \times \mathcal{D}$ is defined as follows:

1. its objects are the pairs $(A, B)$ with $A$ an object of $\mathcal{C}$ and $B$ an object of $\mathcal{D}$;
2. a [[morphism|859F0TFVA3]] $(A, B) \to (A', B')$ is a pair $(f, g)$ with $f \colon A \to A'$ in $\mathcal{C}$ and $g \colon B \to B'$ in $\mathcal{D}$;
3. composition is $(f', g') \circ (f, g) = (f' \circ f, g' \circ g)$, and the identity of $(A, B)$ is $(1_A, 1_B)$.

## Remarks

The category axioms hold because they hold in each coordinate.
There are projection functors $\mathcal{C} \times \mathcal{D} \to \mathcal{C}$ and $\mathcal{C} \times \mathcal{D} \to \mathcal{D}$.
Example: a [[monoid|0N6R902FG1]] $M$ is a category with one object; the product of the one-object categories of $M$ and $N$ is the one-object category of the product monoid $M \times N$.
Indeed its only object is the pair of the two objects, and its morphisms are the pairs $(m, n)$ multiplied coordinatewise.
The set of objects of $\mathcal{C} \times \mathcal{D}$ is the [[Cartesian product|09CEDJYBKP]] of the sets of objects when both categories are small.
A morphism $(f, g)$ is an isomorphism exactly when $f$ and $g$ are both isomorphisms.
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# Calderón–Zygmund lemma

## Statement

Let $n \ge 1$, let $f \in L^1(\R^n)$ be an integrable function, and let $\alpha > 0$.
Then there are disjoint sets $F$ and $U$ with $\R^n = F \cup U$ such that:

1. $\abs{f(x)} \le \alpha$ for almost every $x \in F$;
2. $U$ is a countable union of closed cubes $Q_k$ with sides parallel to the axes and pairwise disjoint interiors;
3. for every $k$,
$$\alpha < \frac{1}{\abs{Q_k}}\int_{Q_k} \abs{f(x)} \, dx \le 2^n\alpha,$$
where $\abs{Q_k}$ is the [[Lebesgue measure|2Z2QVWZ262]] of $Q_k$.

## Remarks

The proof is a stopping-time argument on dyadic cubes.
Cut $\R^n$ into a grid of cubes so large that the average of $\abs{f}$ on each is at most $\alpha$.
Divide each cube into $2^n$ subcubes by halving its sides, keep the subcubes on which the average exceeds $\alpha$, and continue dividing the others.
A kept cube has a parent with average at most $\alpha$, which gives the bound $2^n\alpha$, and the Lebesgue differentiation theorem gives $\abs{f} \le \alpha$ almost everywhere on the points never kept.

Clause 3 gives $\abs{U} = \sum_k \abs{Q_k} \le \alpha^{-1}\norm{f}_{L^1}$.
Setting $g = f$ on $F$ and $g$ equal to the average of $f$ on each $Q_k$, and $b = f - g$, one obtains the Calderón–Zygmund decomposition $f = g + b$.
Here $\abs{g} \le 2^n\alpha$ almost everywhere, and $b$ has mean zero on each cube $Q_k$ and vanishes off $U$.

Example: for $n = 1$, $f = \mathbf{1}_{[0, 1]}$ and $\alpha = 1/2$, one may take $U = [0, 1]$ and $F = \R \setminus [0, 1]$; the average of $\abs{f}$ on $[0, 1]$ is $1$, which lies in $(1/2, 1]$.

## Explanations

### L1 human

The lemma splits space into a part where $f$ is small, at most $\alpha$, and a collection of cubes where $f$ is large on average, but not too large.
The large part occupies little volume, at most $\norm{f}_{L^1}/\alpha$.
It is the basic tool for studying functions that are only integrable, without any better bound.

### L2 human

The lemma, through the decomposition $f = g + b$, is the key step in proving that [[singular integral operators|HTMCEARGDD]] are of weak type $(1, 1)$.
The good part $g$ is bounded and integrable, so it lies in $L^2$, where the operator is bounded.
The bad part $b$ has mean zero on each cube, and the smoothness of the kernel controls the operator on it away from the cubes.
The same stopping-time idea gives the weak type $(1, 1)$ bound for the [[Hardy–Littlewood maximal function|VX1A56P5KQ]] and the John–Nirenberg inequality.
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# Discrete-time algebraic Riccati equation

## Summary

The discrete-time algebraic Riccati equation is the nonlinear matrix equation $X = A^T X A - A^T X B (R + B^T X B)^{-1} B^T X A + Q$ for an unknown symmetric matrix $X$.
It is the discrete-time form of the [[algebraic Riccati equation|GGZV75ZZMV]].
Its stabilising solution gives the optimal state feedback for the linear-quadratic regulator of the system $x_{k+1} = A x_k + B u_k$.
A dual form of it, with $A$ and $B$ replaced by $A^T$ and $C^T$, gives the steady-state error covariance of the Kalman filter.

## Explanations

### L1 human

A controller acts on a system once per time step and wants to keep both the state and its own effort small over all future steps.
The best policy multiplies the current state by a fixed gain matrix, and the gain is computed from the solution $X$ of this equation.
The matrix $X$ measures the total future cost that starts from each state.
For one variable with $A = B = Q = R = 1$, the equation becomes $X^2 - X - 1 = 0$, and the useful solution is the golden ratio $(1 + \sqrt{5})/2$.
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# Spherical Bessel functions

## Definition

Let $n \ge 0$ be an integer and let $x > 0$.
Let $J_\nu$ and $Y_\nu$ be the [[Bessel functions of the first kind|CZCMRDWDYW]] and of the [[second kind|GRSNDJG363]].
The *spherical Bessel functions* of order $n$ are
$$j_n(x) = \sqrt{\frac{\pi}{2x}}\, J_{n + 1/2}(x), \qquad y_n(x) = \sqrt{\frac{\pi}{2x}}\, Y_{n + 1/2}(x).$$

## Remarks

They solve $x^2 y'' + 2x y' + \left( x^2 - n(n+1) \right) y = 0$, which is the radial equation of $\Delta u + u = 0$ in spherical coordinates.
Example: $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$, so $j_0(x) = \frac{\sin x}{x}$; one checks $x^2 j_0'' + 2x j_0' + x^2 j_0 = x (x j_0)'' + x^2 j_0 = x (\sin x)'' + x \sin x = 0$.
Likewise $y_0(x) = -\frac{\cos x}{x}$.
In general $j_n(x) = (-x)^n \left( \frac{1}{x} \frac{d}{dx} \right)^n \frac{\sin x}{x}$.
The spherical Hankel functions are $h^{(1)}_n = j_n + i y_n$ and $h^{(2)}_n = j_n - i y_n$; for instance $h^{(1)}_0(x) = -i \frac{e^{ix}}{x}$.
A non-example: $J_n(x)$ for an integer $n$ is a cylindrical, not a spherical, Bessel function; it is not elementary.
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# Baire function

## Summary

Baire functions are the functions obtained from [[continuous functions|0P27W9RW0G]] by taking [[pointwise limits|056FSKG32J]] of sequences, again and again.
They are sorted into Baire classes indexed by the countable ordinals.
Class $0$ is the continuous functions, and class $1$ is the pointwise limits of continuous functions.
On $\R$, the Baire functions are exactly the Borel measurable functions.
The classes give a fine measure of how far a function is from being continuous.

## Explanations

### L1 human

Pointwise limits of continuous functions need not be continuous.
For example, $x^n$ on $[0, 1]$ tends to a function that is $0$ before $1$ and $1$ at $1$.
Such limits form Baire class $1$.
Limits of those can be even rougher, and they form class $2$, and the process continues.
</document>

<document source="https://tamnd.github.io/math2/node/K290RBT8GB.md" tag="K290RBT8GB" kind="concept">
# Struve function

## Summary

The Struve function $\mathbf{H}_\nu$ is a particular solution of Bessel's equation with a power of $z$ on the right-hand side.
It complements the [[Bessel functions|CZCMRDWDYW]], which solve the equation with right-hand side $0$.
It appears in acoustics, for the radiation of a vibrating piston, and in problems of water waves and optics.

## Explanations

### L1 human

Bessel's equation without a right-hand side describes free vibrations of a round object.
Adding a forcing term that is a power of $z$ gives a new equation, and the Struve function is its standard solution.
Every other solution is the Struve function plus a combination of Bessel functions.
For example, $\mathbf{H}_{1/2}(x) = \sqrt{\frac{2}{\pi x}}\,(1 - \cos x)$ for $x > 0$.
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# Stirling's series

## Example

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], let $\ln \Gamma$ be its principal branch on $\C \setminus (-\infty, 0]$, and let $B_{2k}$ be the [[Bernoulli numbers|0RAS24E1R5]].
Fix $\delta$ with $0 < \delta < \pi$.
As $z \to \infty$ in the sector $\abs{\arg z} \le \pi - \delta$,
$$\ln \Gamma(z) \sim \left(z - \tfrac{1}{2}\right) \ln z - z + \tfrac{1}{2} \ln(2\pi) + \sum_{k=1}^{\infty} \frac{B_{2k}}{2k(2k-1) \, z^{2k-1}}.$$
This is an [[asymptotic expansion|BJQTEPEZ9G]] in the scale $z \ln z, z, \ln z, 1, z^{-1}, z^{-3}, z^{-5}, \ldots$.
With $B_2 = \frac{1}{6}$, $B_4 = -\frac{1}{30}$ and $B_6 = \frac{1}{42}$, the first correction terms are
$$\frac{1}{12 z} - \frac{1}{360 z^3} + \frac{1}{1260 z^5}.$$

## Remarks

The series diverges for every $z$.
Indeed $\abs{B_{2k}}$ grows like $2 (2k)! / (2\pi)^{2k}$, so the terms grow like a factorial divided by a power of $z$.
For real $z = x > 0$, the error after the term in $z^{-(2N-1)}$ has the sign of the first omitted term and is smaller in absolute value.
For $x = 10$, the first correction $\frac{1}{120}$ already gives $\ln \Gamma(10) = \ln(9!)$ with an error below $\frac{1}{360\,000} < 3 \cdot 10^{-6}$.
Taking the exponential of the first terms gives [[Stirling's formula|0KC0F51T6A]] $n! \sim \sqrt{2\pi n} \, (n/e)^n$.

## Explanations

### L1 human

Stirling's series is a sharpened Stirling formula for the logarithm of the factorial.
Each extra term makes the estimate better for large $z$, up to a point.
If one keeps adding terms forever, the sum blows up, which shows that a useful asymptotic expansion can diverge.
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# Directed-complete partial order

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
It is a *directed-complete partial order*, or *dcpo*, if every [[directed subset|D9RY4667K6]] $D \subseteq P$ has a [[supremum|0R3H5PANRQ]] $\bigvee D$ in $P$.
Here $D$ is directed if it is nonempty and every two elements of $D$ have an upper bound in $D$.

## Remarks

A dcpo with a least element $\bot$ is called pointed, and is often simply called a cpo.
Example: the set of partial functions from $\N$ to $\N$, ordered by inclusion of graphs.
If $D$ is a directed family, any two members agree where both are defined, so $\bigcup D$ is again a partial function and is the supremum.
Non-example: $\N$ with its usual order is not a dcpo, because $\N$ itself is directed and has no upper bound.
Every finite poset is a dcpo, since a finite directed set has a greatest element.
Equivalently, by a theorem that uses the axiom of choice, a poset is a dcpo if and only if every nonempty chain has a supremum.
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<document source="https://tamnd.github.io/math2/node/K2N123JM35.md" tag="K2N123JM35" kind="definition">
# Strong amalgamation property

## Definition

Fix a signature and let $\mathcal{K}$ be a class of algebras of this signature, closed under isomorphism.
The class $\mathcal{K}$ has the *strong amalgamation property* if the following holds.
For all $A, B, C \in \mathcal{K}$ and all [[embeddings|SCRB74SWQN]] $f \colon A \to B$ and $g \colon A \to C$, there are $D \in \mathcal{K}$ and embeddings $f' \colon B \to D$ and $g' \colon C \to D$ such that:

1. $f' \circ f = g' \circ g$;
2. $f'(B) \cap g'(C) = f'(f(A))$.

## Remarks

Clause 1 alone is the [[amalgamation property|XSCSX39GEZ]], so the strong amalgamation property implies the amalgamation property.
Clause 2 says that an element of $B$ and an element of $C$ have the same image in $D$ only if both come from the same element of $A$.

Example: the class of groups has the strong amalgamation property.
Take $D = B *_A C$, the free product of $B$ and $C$ amalgamated over $A$.
By the normal form theorem for amalgamated free products, $B$ and $C$ embed in $D$ and their images meet exactly in the image of $A$.

Non-example: the class of fields has the amalgamation property but fails clause 2.
Let $A = \Q$ and $B = C = \Q(\sqrt{2})$, with $f = g$ the inclusion.
In any field $D$ with embeddings $f', g'$ that agree on $\Q$, both $f'(\sqrt{2})$ and $g'(\sqrt{2})$ are roots of $x^2 - 2$, so $g'(\sqrt{2}) = \pm f'(\sqrt{2})$.
Hence $g'(\sqrt{2})$ lies in $f'(B) \cap g'(C)$ but not in the image of $\Q$.
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# Fundamental solution

## Summary

A fundamental solution of a linear differential operator $P$ is a distribution $E$ with $P E = \delta$, the Dirac delta at the origin.
For an operator with constant coefficients, convolving $E$ with a source $f$ gives a solution of $P u = f$.
Fundamental solutions are the Green's functions of the whole space, and they underlie potential theory, the [[heat kernel|KD404MF042]] and the theory of wave propagation.
See the [[definition|ZY724M2NKQ]].

## Explanations

### L1 human

A fundamental solution is the response of a physical system to a point source.
For gravity or electrostatics in space, the potential of a point mass is proportional to $1/\abs{x}$.
Up to a constant factor, this function is the fundamental solution of the Laplace operator.
A general mass distribution is a sum of point masses, so its potential is a sum, or integral, of shifted copies of $1/\abs{x}$.
That is why one fundamental solution solves the equation for all sources.
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# Order of a zero or pole

## Definition

Let $U \subseteq \C$ be an open set, let $f$ be a [[meromorphic function|0NSD1TPHR0]] on $U$, and let $a \in U$.
Assume that $f$ is not identically $0$ on any punctured disc around $a$.
The *order* of $f$ at $a$, written $\operatorname{ord}_a f$, is the unique integer $n$ with the following property.
There are $r > 0$ and a [[holomorphic|0DG0Z72CQD]] function $g$ on $\set{z : \abs{z - a} < r}$ with $g(a) \neq 0$ and $f(z) = (z - a)^n g(z)$ for $0 < \abs{z - a} < r$.
If $f$ is identically $0$ near $a$, one sets $\operatorname{ord}_a f = \infty$.

## Remarks

If $n > 0$, then $a$ is a zero of [[order|064N002TX4]] $n$; if $n < 0$, then $a$ is a pole of [[order|0YK27TJ4WF]] $-n$; if $n = 0$, then $f(a)$ is defined and nonzero.
Equivalently, $n$ is the least index with a nonzero coefficient in the [[Laurent series|0X3XD2ATB0]] of $f$ at $a$.
Example: $f(z) = \dfrac{\sin z}{z^3}$ has $\operatorname{ord}_0 f = -2$, since $\sin z = z \, h(z)$ with $h$ holomorphic and $h(0) = 1$.
Non-example: $e^{1/z}$ is not meromorphic near $0$ and has no order there.
Its Laurent series at $0$ has infinitely many nonzero terms of negative index.
The order satisfies $\operatorname{ord}_a(fg) = \operatorname{ord}_a f + \operatorname{ord}_a g$ and $\operatorname{ord}_a(f + g) \ge \min(\operatorname{ord}_a f, \operatorname{ord}_a g)$.
Some authors call it the valuation of $f$ at $a$.
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# Order of a projective plane

## Summary

The order of a [[finite projective plane|2HWB7P1DZN]] is the number $n$ such that every line has $n + 1$ points.
It determines all the basic counts: each point is on $n + 1$ lines, and there are $n^2 + n + 1$ points and as many lines.
The plane $PG(2, q)$ over the field with $q$ elements has order $q$, and which other orders occur is a central open question of finite geometry.

## Explanations

### L1 human

In a finite projective plane all lines have the same size, so one number describes the size of the whole plane.
By convention it is one less than the number of points on a line.
The [[Fano plane|QQ8GW1ASZM]] has three points on each line, so its order is $2$.
The minus one makes the order of the plane over a field with $q$ elements equal to $q$.
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# Sublinear function

## Summary

A sublinear function on a real vector space is a real-valued map that is positively homogeneous and subadditive.
Norms, seminorms and linear functionals are all sublinear.
Sublinear functions are the dominating functions in the [[Hahn–Banach theorem|QB5AE3VYZZ]], and the [[Minkowski functional|PAX9SHDHDW]] of a convex set is a basic example.

## Explanations

### L1 human

A sublinear function behaves like a length that may be lopsided.
Scaling a vector by a positive number scales the value by the same number.
The value of a sum is at most the sum of the values, as in the triangle inequality.
On the plane, $p(x_1, x_2) = \max(x_1, x_2)$ is sublinear but is not a norm, because it can be negative.
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# Frobenius norm

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, let $m, n \ge 1$, and let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $\mathbb{K}$.
The *Frobenius norm* of $A$ is
$$\norm{A}_F = \sqrt{\sum_{i=1}^{m} \sum_{j=1}^{n} \abs{a_{ij}}^2}.$$

## Remarks

Equivalently, $\norm{A}_F = \sqrt{\tr(A^* A)}$, where $A^*$ is the [[conjugate transpose|0DXKE10PHF]] and $\tr$ is the [[trace|0XC1RDBY7A]].
If $\sigma_1, \ldots, \sigma_r$ are the nonzero [[singular values|09VMDSRXWJ]] of $A$, then $\norm{A}_F^2 = \sigma_1^2 + \cdots + \sigma_r^2$.
Example: for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, $\norm{A}_F = \sqrt{30}$.
The Frobenius norm is a submultiplicative [[matrix norm|54T6Y6DSY4]], by the Cauchy–Schwarz inequality, and $\norm{UAV}_F = \norm{A}_F$ for unitary $U$ and $V$.
It is not an operator norm when $n \ge 2$: every operator norm gives $\norm{I_n} = 1$, but $\norm{I_n}_F = \sqrt{n}$.
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# Statistical model

## Summary

A statistical model is a set of probability distributions that are candidates for the law of the observed data.
It fixes what is assumed about the data before any inference is made.
Every method of [[statistical inference|JZ2S01BHJJ]] starts from a model, and its guarantees hold only under the laws in that model.

## Explanations

### L1 human

You do not know exactly how your data were produced, but you know some things.
A statistical model lists all the laws that you still regard as possible.
For example, toss a coin $10$ times.
A natural model says that the number of heads is binomial with $10$ trials and some success chance $p$ between $0$ and $1$.
The data then help you decide which $p$ is plausible.
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# Jacobi eigenvalue algorithm

## Statement

Let $A \in \R^{n \times n}$, $n \ge 2$, be [[symmetric|0VR671PHE5]], and let $\mathrm{off}(A)^2 = \sum_{i \ne j} a_{ij}^2$.
For indices $p < q$ with $a_{pq} \ne 0$, let $J = J(p, q, \theta)$ be the rotation in the $(p, q)$ coordinate plane, where $\theta \in [-\pi/4, \pi/4]$ solves $\cot 2\theta = (a_{qq} - a_{pp})/(2 a_{pq})$.
Then $A' = J^T A J$ is symmetric, has the same [[eigenvalues|0BQAA2BK3Q]] as $A$, satisfies $a'_{pq} = 0$, and
$$\mathrm{off}(A')^2 = \mathrm{off}(A)^2 - 2 a_{pq}^2.$$
The *classical Jacobi method* chooses at each step a pair $(p, q)$ with $\abs{a_{pq}}$ maximal among the off-diagonal entries and sets $A_{k+1} = J_k^T A_k J_k$, starting from $A_0 = A$.
Then
$$\mathrm{off}(A_{k+1})^2 \le \Bigl( 1 - \frac{2}{n(n-1)} \Bigr) \mathrm{off}(A_k)^2,$$
so $A_k$ converges to a [[diagonal matrix|044JNC7ZAA]] whose diagonal entries are the eigenvalues of $A$.

## Remarks

The rotation $J$ is an [[orthogonal matrix|0W02QESPJQ]], so $A'$ is [[similar|0E89HQATM8]] to $A$ and the [[Frobenius norm|JT8Q204E4R]] $\norm{A'}_F = \norm{A}_F$ is unchanged.
On the $2 \times 2$ block in rows and columns $p, q$, the choice of $\theta$ diagonalizes the block.
So the sum of squares of that block's diagonal grows by $2 a_{pq}^2$.
The other entries in rows and columns $p, q$ are only rotated, so their total square sum does not change.
If $\abs{a_{pq}}$ is maximal among the $n(n-1)$ off-diagonal entries, then $a_{pq}^2 \ge \mathrm{off}(A)^2 / (n(n-1))$, which gives the linear bound.
The diagonal entries of $A_k$ approach the eigenvalues because each eigenvalue lies within $\mathrm{off}(A_k)$ of some diagonal entry, by Weyl's inequality.

Example: for $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$, $\cot 2\theta = 0$ gives $\theta = \pi/4$, and one rotation gives $\mathrm{diag}(1, 3)$.
In practice the local convergence is quadratic once the off-diagonal part is small.
The product $J_0 J_1 \cdots J_k$ of the rotations converges to an orthogonal matrix whose columns are eigenvectors.

## Explanations

### L1 human

The Jacobi method diagonalizes a symmetric matrix by a sequence of plane rotations.
Each rotation kills one off-diagonal entry; later rotations may make it nonzero again, but the total size of the off-diagonal part always drops.
After enough rotations the matrix is nearly diagonal and the eigenvalues can be read off the diagonal.

### L2 human

The method dates from Jacobi in 1846 and was revived on early computers.
It is slower than the [[QR algorithm|B6F8YWHYZ5]] on a serial computer.
But it computes small eigenvalues of well-scaled positive definite matrices to high relative accuracy, and it is easy to parallelize.
Cyclic variants sweep through all pairs in a fixed order instead of searching for the largest entry, and they also converge.
A one-sided version applied to $A^T A$ implicitly computes the [[singular value decomposition|000NPTMX6J]].
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# Percus–Yevick approximation

## Example

Consider a classical fluid in $\R^3$ with number density $\rho$, inverse temperature $\beta$ and a pair potential $u(r)$.
Let $g$ be the radial distribution function, $h = g - 1$, and let $c$ be the direct correlation function of the [[Ornstein–Zernike equation|29T4NC87AR]].
The Percus–Yevick approximation is the closure
$$c(r) = g(r) \left(1 - e^{\beta u(r)}\right).$$
With the cavity function $y(r) = g(r) e^{\beta u(r)}$ and the Mayer function $f(r) = e^{-\beta u(r)} - 1$ it reads $c(r) = f(r) \, y(r)$.
Inserted in the Ornstein–Zernike equation it gives
$$y(r_{12}) = 1 + \rho \int_{\R^3} f(r_{13}) \, y(r_{13}) \, h(r_{23}) \, dx_3, \qquad h = (1 + f) \, y - 1,$$
where $r_{ij} = \abs{x_i - x_j}$; this is a [[nonlinear integral equation|BSCYBP6AS6]] for $y$, quadratic in $y$.

Check of the second form: $f y = (e^{-\beta u} - 1) g e^{\beta u} = g (1 - e^{\beta u})$.

## Remarks

For hard spheres of diameter $\sigma$, $g(r) = 0$ for $r < \sigma$ and the closure gives $c(r) = 0$ for $r > \sigma$.
In this case the equation has a known closed-form solution, in which $c$ is a cubic polynomial in $r$ for $r < \sigma$.
With the packing fraction $\eta = \frac{\pi}{6} \rho \sigma^3$, the compressibility route gives the equation of state $\frac{\beta P}{\rho} = \frac{1 + \eta + \eta^2}{(1 - \eta)^3}$, and the virial route gives $\frac{1 + 2\eta + 3\eta^2}{(1 - \eta)^2}$.
The two routes differ because the closure is an approximation.

## Explanations

### L1 human

The Ornstein–Zernike equation alone has two unknown functions, so a second rule is needed to close it.
The Percus–Yevick rule says that the direct correlation is zero wherever the particles do not interact.
For hard balls this rule leads to a formula for the pressure that is close to computer simulations at moderate densities.
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# Van Wijngaarden transformation

## Summary

The van Wijngaarden transformation accelerates an [[alternating series|0QED0GPYKY]] by averaging its partial sums again and again.
Each new row of the table holds the means of neighbouring entries of the row above.
It is a variant of the [[Euler transform|WT0HACKQV7]] that is easy to carry out by hand or by machine.

## Explanations

### L1 human

The partial sums of an alternating series jump above and below the sum.
The average of two consecutive partial sums lies between them and is usually much closer to the sum.
Averaging the averages improves the values again.
For the Leibniz series $1 - \frac13 + \frac15 - \cdots = \frac{\pi}{4}$, three rounds of averaging of the first four partial sums give about $0.7905$, against $0.7854$ for $\frac{\pi}{4}$ and $0.7238$ for the fourth partial sum.
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# Satisfiability

## Definition

1. A formula $\varphi$ of [[propositional logic|09K9MWD2QQ]] is *satisfiable* if some truth assignment to its variables gives $\varphi$ the value true.
2. Let $\sigma$ be a signature and $\varphi$ a first-order [[formula|HYJ10AF6RT]] over $\sigma$. Then $\varphi$ is *satisfiable* if there are a $\sigma$-[[structure|983GZBQYT4]] $M$ and an assignment $s$ of elements of $M$ to the variables with $M \models \varphi[s]$.
3. A set $\Gamma$ of formulas is *satisfiable* if one interpretation makes all formulas of $\Gamma$ true at once.

A formula or set that is not satisfiable is *unsatisfiable*.

## Remarks

The formula $(p \lor q) \land \neg p$ is satisfiable: the assignment $p = \mathsf{F}$, $q = \mathsf{T}$ makes both conjuncts true.
The set $\set{p, \; p \to q, \; \neg q}$ is unsatisfiable: if $p$ and $p \to q$ are true then $q$ is true, so $\neg q$ is false.
Each of its three formulas is satisfiable on its own, so clause 3 is stronger than satisfiability of each member.
A formula $\varphi$ is valid, for example a [[tautology|05006AJ7DZ]], exactly when $\neg \varphi$ is unsatisfiable.
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# Berkovich space dynamics

## Definition

Let $K$ be an algebraically closed field that is complete for a nontrivial [[non-Archimedean absolute value|1H5F1BP6XS]] $\abs{\cdot}$, such as [[$\C_p$|FKF9YGBKHF]].
The *Berkovich affine line* $\mathbb{A}^1_{\mathrm{Berk}}$ is the set of maps $x \colon K[T] \to [0, \infty)$, written $f \mapsto [f]_x$, such that:

1. $[f g]_x = [f]_x [g]_x$ for all $f, g \in K[T]$;
2. $[f + g]_x \le \max([f]_x, [g]_x)$ for all $f, g \in K[T]$;
3. $[c]_x = \abs{c}$ for every constant $c \in K$.

It carries the weakest topology for which $x \mapsto [f]_x$ is continuous for every $f \in K[T]$.
The *Berkovich projective line* is $\mathbb{P}^1_{\mathrm{Berk}} = \mathbb{A}^1_{\mathrm{Berk}} \cup \set{\infty}$, the one-point compactification.
Each point $a \in K$ gives the point $[f]_a = \abs{f(a)}$, so $\mathbb{P}^1(K)$ is a dense subset of $\mathbb{P}^1_{\mathrm{Berk}}$.
Every [[rational map|0XDK9CGYKJ]] $\varphi \in K(z)$ of degree $d \ge 1$ extends uniquely to a continuous map $\varphi \colon \mathbb{P}^1_{\mathrm{Berk}} \to \mathbb{P}^1_{\mathrm{Berk}}$; for a polynomial $\varphi$ it is given by $[f]_{\varphi(x)} = [f \circ \varphi]_x$.
*Berkovich space dynamics* is the study of the [[iterates|AYVTR2ES5B]] $\varphi^n$ of this extended map on $\mathbb{P}^1_{\mathrm{Berk}}$, and more generally of endomorphisms of Berkovich analytic spaces.

## Remarks

Example: the *Gauss point* $\zeta$ is given by $[\sum_i c_i T^i]_\zeta = \max_i \abs{c_i}$.
It is multiplicative by Gauss's lemma, satisfies the strong triangle inequality, and restricts to $\abs{\cdot}$ on constants, so it lies in $\mathbb{A}^1_{\mathrm{Berk}}$.
It is the supremum norm on the closed unit disc, and it is not of the form $[f]_a$ for any $a \in K$.
For $\varphi(z) = z^2$, the series $f(T^2)$ has the same coefficients as $f(T)$, so $[f]_{\varphi(\zeta)} = [f]_\zeta$ and $\zeta$ is a fixed point.

Non-example: $[f] = \max(\abs{f(0)}, \abs{f(1)})$ satisfies clauses 2 and 3 but not clause 1.
Indeed $[T] = 1$ and $[T - 1] = 1$, but $[T(T - 1)] = 0$.

Berkovich showed that $\mathbb{P}^1_{\mathrm{Berk}}$ is compact, Hausdorff and path-connected.
The Berkovich line is the analytification of $\mathbb{P}^1$; a related but different notion of analytic space is the [[rigid analytic space|YAH6MZVG7K]].
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# Higher-order logic

## Definition

The relational types are defined inductively.

1. $\iota$ is a type, the type of individuals.
2. If $\tau_1, \dots, \tau_n$ are types with $n \ge 1$, then $(\tau_1, \dots, \tau_n)$ is a type, the type of $n$-ary relations between objects of types $\tau_1, \dots, \tau_n$.

Higher-order logic over a signature $L$ has, for each type $\tau$, variables of type $\tau$ and quantifiers $\forall$ and $\exists$ over them.
Its formulas are built as in [[first-order logic|086QKPF0A6]], with the atomic formula $X(t_1, \dots, t_n)$ whenever $X$ has type $(\tau_1, \dots, \tau_n)$ and each $t_i$ has type $\tau_i$.
In the standard semantics over an $L$-[[structure|983GZBQYT4]] with domain $M$, the domains $D_\tau$ are given by $D_\iota = M$ and $D_{(\tau_1, \dots, \tau_n)} = \mathcal{P}(D_{\tau_1} \times \dots \times D_{\tau_n})$.
Here $\mathcal{P}$ is the [[power set|00E0YZCERH]] and $\times$ the [[Cartesian product|09CEDJYBKP]].

## Remarks

The order of a type is $1$ for $\iota$ and one more than the largest order of $\tau_1, \dots, \tau_n$ for $(\tau_1, \dots, \tau_n)$.
Logic of order $k$ allows quantifiers only over variables whose type has order at most $k$.
First-order logic is the case $k = 1$, and second-order logic is the case $k = 2$.

Example: a variable $\mathcal{X}$ of type $((\iota))$ ranges over sets of subsets of $M$, and $\forall \mathcal{X} \, \exists Y \, \forall x \, (Y(x) \leftrightarrow \exists Z \, (\mathcal{X}(Z) \land Z(x)))$ is a third-order sentence.
It says that each family of subsets has a union, and it is true in every structure.

Non-example: the expression $X(X)$ is not a formula, because no type $\tau$ equals a type $(\tau)$ of relations on objects of type $\tau$.

Many presentations instead use the simply typed lambda calculus, with function types and a type of truth values; this is Church's simple theory of types.

## Explanations

### L1 human

Higher-order logic sorts all objects into levels: individuals, relations on individuals, relations on those relations, and so on.
A variable can range over any one level, and a relation may only be applied to objects of the level below it.
This rule blocks paradoxes such as a set that contains itself.
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# Hyperbolic knot

## Summary

A hyperbolic knot is a [[knot|QYEAZTYGPT]] whose complement in the $3$-sphere carries a complete hyperbolic metric of finite volume.
By Mostow–Prasad rigidity this metric is unique, so its volume and other geometric quantities are knot invariants.
Thurston showed that every nontrivial knot is a [[torus knot|NS60VV786W]], a [[satellite knot|A1E8TDEVJ9]] or a hyperbolic knot, and exactly one of these.
Most knots in the tables are hyperbolic.

## Explanations

### L1 human

Remove a knot from space and study the space that is left over.
For a hyperbolic knot, that leftover space can be given the geometry of the hyperbolic plane, raised to three dimensions, in which triangles have angle sums below $180$ degrees.
This geometry is rigid, so its total volume is a number attached to the knot.
The [[figure-eight knot|C01XBGAHC6]] is hyperbolic, and its complement has volume about $2.0299$.
The trefoil is not hyperbolic, because it is a torus knot.
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# Ribbon Hopf algebra

## Summary

A ribbon Hopf algebra is a [[quasitriangular Hopf algebra|NCRPWS9W5R]] with a central invertible element, the ribbon element, that behaves like a full twist.
Its finite-dimensional modules form a ribbon category: a [[braided monoidal category|VT8SFWT0XW]] with duals and compatible twists.
In such a category every framed knot or link diagram coloured by modules gives a well-defined morphism.
So each ribbon Hopf algebra gives invariants of knots and links.
Suitable versions of the [[quantum enveloping algebras|1V8H34F6N9]] are ribbon, and the [[Jones polynomial|4SPC17FPXE]] arises this way from $U_q(\mathfrak{sl}_2)$.

## Explanations

### L1 human

Think of a ribbon, a flat band, rather than a thin string.
A ribbon can be twisted by a full turn, and that twist cannot be undone by sliding the band around.
In a ribbon Hopf algebra a special element $v$ records this full twist.
The braiding from the R-matrix records how two ribbons cross, and $v$ records how one ribbon twists.
Together they give exactly the bookkeeping needed to turn pictures of tangled bands into algebra.
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# Schröder's equation

## Definition

Let $X$ be a set, let $f \colon X \to X$ be a [[function|0EPAT51DXQ]], and let $\lambda \in \C$.
*Schröder's equation* for $f$ and $\lambda$ is the [[functional equation|SRAN4EKWS0]]
$$\Psi(f(x)) = \lambda \, \Psi(x) \quad \text{for all } x \in X,$$
in which the unknown is a function $\Psi \colon X \to \C$.
In the main case, $X$ is a neighbourhood of a [[fixed point|R6QVS6P21N]] $0$ of a holomorphic map $f$, and $\lambda = f'(0)$.
Then one asks for a holomorphic $\Psi$ with $\Psi(0) = 0$ and $\Psi'(0) = 1$.

## Remarks

Example: $f(x) = x^2 + 2x$ on $X = (-1, \infty)$, with $\lambda = 2$.
The fixed point is $0$, and $f'(0) = 2$.
The function $\Psi(x) = \log(1 + x)$ is a solution with $\Psi(0) = 0$ and $\Psi'(0) = 1$, since $\Psi(f(x)) = \log\big((1+x)^2\big) = 2 \log(1 + x)$.

Non-example: for the same $f$, the function $\Psi(x) = x$ is not a solution, since $f(x) = x^2 + 2x \ne 2x$ for $x \ne 0$.

If $\Psi$ is invertible, the equation says $f = \Psi^{-1} \circ (\lambda \Psi)$, so $f^n = \Psi^{-1} \circ (\lambda^n \Psi)$ for every $n \ge 0$.
The function $\Psi$ is called a Schröder function of $f$.
If $\Psi$ is a solution, so is $c \Psi$ for every constant $c$; the condition $\Psi'(0) = 1$ fixes this freedom.
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# Dense order

## Summary

An order is dense when between any two elements with $x < y$ there is a third element $z$ with $x < z < y$.
The rational numbers and the real numbers are dense, while the integers are not.
By [[Cantor's isomorphism theorem|ABN5PTXEHY]], every countable dense total order without endpoints, and with at least one element, looks exactly like the rationals.

## Explanations

### L1 human

In a dense order there are no gaps of the "next element" kind: any two distinct elements have something strictly between them.
Between the fractions $1/3$ and $1/2$ lies their average $5/12$.
Between the integers $1$ and $2$ there is no integer, so the integers are not dense.
Repeating the argument, between two rationals there are infinitely many rationals.
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# Field of extremals

## Summary

A field of extremals is a family of solutions of the [[Euler–Lagrange equation|PS0VV149ET]] that covers a region of the $(x, y)$ plane.
Exactly one curve of the family passes through each point of the region.
The slopes of the curves define a slope function $p(x, y)$ on the region.
Embedding an extremal in such a field is the key step of the Weierstrass theory of sufficient conditions for a strong minimum.
It is the classical counterpart of the method of characteristics for the Hamilton–Jacobi equation.

## Explanations

### L1 human

Think of the lines of a flow that fill a strip without crossing, like parallel straight lines.
For the length functional $\int \sqrt{1 + y'^2} \, dx$ the extremals are straight lines, and the lines $y = x + c$ for all real $c$ form a field covering the whole plane.
Each point lies on exactly one of these lines, and the slope function is the constant $p = 1$.
A pencil of lines through one point outside a region also forms a field on that region; this is a central field.
When a minimiser is surrounded by such a field, one can compare it with every nearby curve, not only with smooth small perturbations.
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# Bramble–Hilbert lemma

## Statement

Let $\Omega \subset \R^n$ be a bounded open set of diameter $d$.
Assume $\Omega$ is star-shaped with respect to every point of an open ball $B \subset \Omega$ of radius $\rho > 0$.
Let $m \ge 1$ be an integer and $1 \le p \le \infty$.
Let $u$ be in the [[Sobolev space|M4CEZ5K9HP]] $W^{m,p}(\Omega)$ and let $0 \le k \le m$.
Let $\abs{u}_{W^{k,p}(\Omega)}$ be the [[seminorm|4KX81Q85DT]] formed from the $L^p$ norms of the [[weak derivatives|C3EC148VZD]] of $u$ of order exactly $k$.
Then there is a constant $C$, depending only on $n$, $m$, $p$ and $d / \rho$, with the following property.
For every $u \in W^{m,p}(\Omega)$ there is a [[polynomial|07GNX8GCYP]] $q$ of degree less than $m$ such that
$$\abs{u - q}_{W^{k,p}(\Omega)} \le C \, d^{\,m - k} \, \abs{u}_{W^{m,p}(\Omega)} \quad \text{for } k = 0, 1, \ldots, m.$$

## Remarks

One can take for $q$ the averaged Taylor polynomial of $u$ of order $m$.
It is the Taylor polynomial of degree $m - 1$ about a point $y$, averaged over $y \in B$ with a smooth weight.
The remainder of this averaged Taylor formula is an integral operator applied to the derivatives of order $m$ of $u$, and its bounds give the estimate.
The scaling $d^{\,m-k}$ follows by mapping $\Omega$ to a set of diameter $1$.

The right side involves only the seminorm of order $m$, with no lower-order terms.
Every polynomial of degree less than $m$ has $\abs{q}_{W^{m,p}} = 0$, so such a bound can only control $u$ up to these polynomials, which is why $q$ appears.

Example: for $m = 1$ the lemma gives a constant $c$ with $\norm{u - c}_{L^p(\Omega)} \le C d \, \norm{\nabla u}_{L^p(\Omega)}$, a Poincaré-type inequality.

In finite element analysis, the lemma is applied on a reference element to the error $u - I u$ of an interpolation operator $I$ that reproduces polynomials of degree less than $m$.
Since $u - I u = (u - q) - I(u - q)$, the error is bounded by the seminorm $\abs{u}_{W^{m,p}}$, and scaling back to an element of size $h$ gives the factor $h^{m-k}$.

## Explanations

### L1 human

A function whose derivatives of order $m$ are small is close to a polynomial of degree less than $m$.
The lemma makes this precise in Sobolev norms, with a constant that only depends on the shape of the domain.
It is the basic tool behind the error estimates of the finite element method.

### L2 human

For a triangle of size $h$ and piecewise linear interpolation, the lemma with $m = 2$ gives errors of order $h^2$ in $L^2$ and of order $h$ in $H^1$, for $u \in H^2$.
The constant depends on $d / \rho$, which is why finite element meshes must avoid very thin elements.
Combined with Céa's lemma, it turns approximation by polynomials into convergence rates of the finite element method.
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# Multidimensional transform

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be a measurable set, and let $U$ be a set.
Let $K \colon U \times \Omega \to \C$ be a function, the [[kernel|9M0NNNFJGM]].
Let $f \colon \Omega \to \C$.
Its *multidimensional transform* with kernel $K$ is
$$(Tf)(u) = \int_\Omega K(u, x)\, f(x)\, dx,$$
a [[multiple integral|FE1ZZJ8QW2]], defined at those $u \in U$ where the integral converges absolutely.
For functions on $\Z^n$ the integral is replaced by a sum over $\Z^n$.

## Remarks

The transform is *separable* when $\Omega = \Omega_1 \times \cdots \times \Omega_n$, $U = U_1 \times \cdots \times U_n$ and $K(u, x) = K_1(u_1, x_1) \cdots K_n(u_n, x_n)$.
Then, by Fubini's theorem, $T$ is the composition of the one-variable transforms with kernels $K_j$ applied in each variable.

Example: $\Omega = U = \R^n$ and $K(\xi, x) = e^{-i \xi \cdot x}$ give the $n$-dimensional [[Fourier transform|DPVM34ADZV]], a separable transform since $e^{-i \xi \cdot x} = \prod_j e^{-i \xi_j x_j}$.

Non-example: the [[Radon transform|VJNZJ352PC]] sends a function on $\R^n$ to its integrals over hyperplanes.
It is not of the form above with a function $K$, since each value is an integral over a set of measure zero.
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# Goldstine theorem

## Statement

Let $X$ be a [[normed vector space|03J5RAKB30]] over $\R$ or $\C$, with closed unit ball $B_X$.
Let $J \colon X \to X^{**}$ be the [[canonical embedding into the bidual|QKR3D65TTH]].
Let $B_{X^{**}}$ be the closed unit ball of $X^{**}$.
Then $J(B_X)$ is [[dense|0BSJSX7ME2]] in $B_{X^{**}}$ for the weak* topology of $X^{**}$, the [[weak topology|P6KHVSTPZV]] defined by the functionals $\xi \mapsto \xi(f)$ for $f \in X^*$.

## Remarks

The weak* closure $C$ of $J(B_X)$ is convex and lies in $B_{X^{**}}$, since $B_{X^{**}}$ is weak*-closed.
If some $\xi \in B_{X^{**}}$ were not in $C$, the [[Hahn–Banach separation theorem|6GWMGFGFJJ]] in $X^{**}$ with the weak* topology would give $f \in X^*$ with $\operatorname{Re} \xi(f) > \sup_{x \in B_X} \operatorname{Re} f(x) = \norm{f}$.
This contradicts $\norm{\xi} \le 1$.

For $X = c_0$, with $X^{**} = \ell^\infty$, the theorem says that every sequence $y$ with $\sup_n \abs{y_n} \le 1$ is a weak* limit of elements of the unit ball of $c_0$.
Indeed the truncations $(y_1, \ldots, y_n, 0, 0, \ldots)$ converge to $y$ against every element of $\ell^1$.

## Explanations

### L1 human

A normed space may be strictly smaller than its double dual.
The theorem says it is still large in a weak sense.
Every element of the unit ball of the double dual can be approximated, in the weak* topology, by elements of the unit ball of the space itself.
So the double dual adds only limit points of a loose kind.

### L2 human

Combined with the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], the theorem shows that a Banach space is [[reflexive|NN8RWPC8WG]] exactly when its closed unit ball is weakly compact.
It is also the key step in the proof of the [[Milman–Pettis theorem|AJ5RG9XQYZ]].
In general the approximation needs nets, not sequences; for separable $X^*$ sequences suffice.
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<document source="https://tamnd.github.io/math2/node/K4XSSPWHCP.md" tag="K4XSSPWHCP" kind="concept">
# Calderón projector

## Summary

The Calderón projector of an elliptic differential operator $P$ on a domain $\Omega$ is a projection acting on boundary data.
Its range is exactly the set of Cauchy data on $\partial\Omega$ of the solutions of $Pu = 0$ in $\Omega$.
It is a matrix of [[pseudo-differential operators|FF27N0BWTP]] on the boundary.
It reduces boundary value problems in $\Omega$ to equations on $\partial\Omega$, and it is the tool behind the general theory of [[elliptic boundary value problems|C1F7T523G0]].

## Explanations

### L1 human

A solution of an elliptic equation inside a region leaves a trace on the boundary: its values and its normal derivatives there.
Not every list of boundary data comes from a solution, because the interior equation imposes a hidden relation between them.
The Calderón projector is a projection that keeps the boundary data of solutions unchanged and sends every list of boundary data to the data of some solution.
For the Laplacian in the plane, the data are a pair $(f, g)$ of the boundary values and the normal derivative.
The projector fixes the pairs $(u|_{\partial\Omega}, \partial_\nu u|_{\partial\Omega})$ of [[harmonic functions|0DEB0JMN0W]] $u$ and sends every other pair to such a pair.
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<document source="https://tamnd.github.io/math2/node/K5C968HMSN.md" tag="K5C968HMSN" kind="concept">
# Affine differential geometry

## Summary

Affine differential geometry studies curves and surfaces through the properties that do not change under volume-preserving affine maps of space.
Lengths and angles are not preserved by such maps, so the Euclidean curvature of a curve or surface is not an invariant here.
Instead the theory builds its own invariants.
Examples are the affine arc length of a curve and the affine normal and Blaschke metric of a hypersurface.
It is an instance of the idea that each group of transformations has its own geometry.

## Explanations

### L1 human

Stretch a drawing in one direction and squeeze it in another so that areas stay the same.
Circles turn into ellipses, and lengths change, but some features of curves survive.
Affine differential geometry studies exactly these surviving features.
For example, every parabola looks the same as every other parabola from this point of view, just as every circle looks the same in Euclidean geometry up to size.
</document>

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# Brauer–Fowler theorem

## Statement

An involution of a group is an element of [[order|0WE2NHZJ7B]] $2$.
Let $n$ be a positive integer.
Then, up to isomorphism, there are only finitely many finite [[simple groups|0YMYMNQ0QZ]] $G$ that contain an involution $t$ whose [[centraliser|02YYV36BFM]] $C_G(t)$ has order $n$.

## Remarks

The proof bounds the order of $G$ by a function of $n$ alone, using counting arguments with involutions and their products.
In particular, for a given finite group $H$, only finitely many finite simple groups have an involution with centraliser isomorphic to $H$.
The bound obtained is far too large to be used directly.
But the finiteness result showed that a simple group can in principle be recognised from the centraliser of an involution.

## History

The theorem was proved by Richard Brauer and K. A. Fowler in 1955.

## Explanations

### L1 human

An involution is an element that undoes itself.
The theorem says that knowing the elements that commute with one involution limits a simple group to finitely many possibilities.

### L2 human

By the [[Feit–Thompson theorem|6VK9RSG13Q]], every non-abelian finite simple group has involutions, so the theorem applies to all of them.
It started the programme of characterising simple groups by the centralisers of their involutions.
This programme became a central part of the [[classification of finite simple groups|06J6P03MG7]]; for example, several [[sporadic groups|PZ49Z4CBFZ]] were first predicted from a possible centraliser of an involution.
</document>

<document source="https://tamnd.github.io/math2/node/K5G3VB3DEZ.md" tag="K5G3VB3DEZ" kind="definition">
# Magma

## Definition

A *magma* is a pair $(M, \cdot)$, where $M$ is a [[set|0SJ3RX9TXA]] and $\cdot \colon M \times M \to M$ is a [[binary operation|0M418FE7BB]] on $M$.
A *homomorphism* of magmas $(M, \cdot)$ and $(N, *)$ is a map $h \colon M \to N$ with $h(x \cdot y) = h(x) * h(y)$ for all $x, y \in M$.

## Remarks

Example: $(\Z, -)$ is a magma, since the difference of two integers is an integer.
It is not associative: $(1 - 1) - 1 = -1$ but $1 - (1 - 1) = 1$.

Non-example: $(\N, -)$ is not a magma, since $1 - 2 \notin \N$, so subtraction is not a binary operation on $\N$.

The empty set with the empty operation is a magma.
The word *groupoid* is an older name for a magma.
</document>

<document source="https://tamnd.github.io/math2/node/K5H0RT71A3.md" tag="K5H0RT71A3" kind="definition">
# Weak convergence of measures

## Definition

Let $X$ be a [[metric space|0VKD1TP7F3]] with its [[Borel σ-algebra|0WQXY32Q8D]].
Let $\mu$ and $\mu_1, \mu_2, \dots$ be [[probability measures|059TJN0NZ1]] on $X$.
The sequence $(\mu_n)$ *converges weakly* to $\mu$ if
$$\lim_{n \to \infty} \int_X f \, d\mu_n = \int_X f \, d\mu$$
for every bounded [[continuous function|0X3WGX95FE]] $f \colon X \to \R$.
The integrals are [[Lebesgue integrals|DWJF7ABPR1]].

## Remarks

Common notations are $\mu_n \Rightarrow \mu$ and $\mu_n \xrightarrow{w} \mu$.
Some authors call the same notion narrow convergence.
Weak limits are unique on a metric space.

Example: let $\delta_a$ be the [[Dirac measure|0ACRGYNHSR]] at $a$ on $\R$.
Then $\delta_{1/n}$ converges weakly to $\delta_0$.
Indeed $\int f \, d\delta_{1/n} = f(1/n)$, and $f(1/n) \to f(0)$ because $f$ is continuous.

Non-example: $\delta_n$ does not converge weakly to $\delta_0$.
Take $f(x) = 1/(1 + x^2)$, which is bounded and continuous.
Then $\int f \, d\delta_n = 1/(1 + n^2) \to 0$, but $\int f \, d\delta_0 = 1$.

Random variables converge in distribution exactly when their laws converge weakly.

## Explanations

### L1 human

Weak convergence asks that every bounded continuous average converges.
It allows mass to shift slightly, which makes it the natural notion for limit theorems in probability.
</document>

<document source="https://tamnd.github.io/math2/node/K5RXH32SE7.md" tag="K5RXH32SE7" kind="concept">
# Dirichlet series as Mellin transforms

## Summary

A [[Dirichlet series|8MWTEF6462]] $F(s) = \sum_{n \ge 1} a(n) n^{-s}$ can be written as a [[Mellin transform|N5PVGRJE1T]] in two standard ways.
First, $\Gamma(s) F(s)$ is the Mellin transform of the exponential sum $\sum_n a(n) e^{-nx}$.
Second, $F(s)/s$ is the Mellin transform at $-s$ of the summatory function $A(x) = \sum_{n \le x} a(n)$.
These identities let one study Dirichlet series with the tools of integral transforms: analytic continuation, inversion formulas such as Perron's formula, and asymptotics through residues.

## Explanations

### L1 human

The basic fact is $\int_0^\infty x^{s-1} e^{-nx} \, dx = \Gamma(s) n^{-s}$.
So a sum of exponentials $\sum a(n) e^{-nx}$ has a Mellin transform equal to $\Gamma(s)$ times the Dirichlet series with the same coefficients.
For $a(n) = 1$ the exponential sum is $1/(e^x - 1)$, and the identity becomes $\int_0^\infty x^{s-1}/(e^x - 1) \, dx = \Gamma(s)\zeta(s)$ for $\operatorname{Re} s > 1$.
Going back by Mellin inversion recovers information on the coefficients from the analytic behaviour of $F$.
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# Spectral theory

## Summary

Spectral theory studies linear operators through their spectra, the sets of numbers $\lambda$ for which $T - \lambda I$ fails to be invertible.
It extends the theory of eigenvalues and diagonalisation of matrices to operators on infinite-dimensional spaces.
Its central results, the spectral theorems, describe self-adjoint and normal operators as multiplication operators, and they are the mathematical basis of quantum mechanics.

## Explanations

### L1 human

A symmetric matrix can be described completely by its eigenvalues and eigenvectors: in the right coordinates, it just stretches each axis by a number.
Spectral theory asks how much of this survives for operators on spaces of functions or sequences.
For example, the operator that multiplies a square-integrable function $f(x)$ on $[0, 1]$ by $x$ has no eigenvectors at all, yet its spectrum is the whole interval $[0, 1]$.
Spectral theory gives the language to say in what sense this operator still "stretches by $x$ at the point $x$".
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<document source="https://tamnd.github.io/math2/node/K5Z67HR9ET.md" tag="K5Z67HR9ET" kind="concept">
# Total ring of fractions

## Summary

The total ring of fractions of a [[commutative ring|0310REJJY8]] $R$ is the [[localization|4C1AE84N5N]] of $R$ at the set of elements that are not [[zero divisors|0NNRDW10RZ]].
It is the largest localization in which $R$ still embeds.
For an [[integral domain|0QVD7WBFGN]], it is the [[field of fractions|0F2PTGX1A2]].

## Explanations

### L1 human

To build fractions from a ring, you want to divide by as many elements as possible.
You cannot divide by a zero divisor $s$, with $sa = 0$ for some $a \ne 0$, without making $a$ equal to zero.
So the safe choice is to allow every element that is not a zero divisor as a denominator.
For the integers this gives the rational numbers.
The [[definition|8MGH8PGJWR]] states it.
</document>

<document source="https://tamnd.github.io/math2/node/K6C48KMRKC.md" tag="K6C48KMRKC" kind="concept">
# Generalized linear model

## Summary

A generalized linear model relates the mean of a response to a linear combination of regressors through a link function.
The response distribution is taken from an [[exponential family|EP9VPZ0S6G]].
It unifies [[linear regression|41FVC4P5JB]], [[logistic regression|JN1PJ5AJJS]], [[Poisson regression|STHTW16717]] and the [[probit model|XCR3V8SJ01]] in one framework, with one fitting method and one theory of inference.
The framework was introduced by John Nelder and Robert Wedderburn in 1972.
Its coefficients are estimated by [[maximum likelihood|0FREPFKNS7]], usually with iteratively reweighted least squares.

## Explanations

### L1 human

Ordinary regression predicts a number that can be any real value, with normal noise.
Many responses are not like that: a yes/no answer, a count, a waiting time.
A generalized linear model keeps the simple linear formula $x^T \beta$, but passes it through a link function that fits the range of the mean.
It also uses a noise distribution that suits the data.
For counts, for example, the logarithm of the mean is linear in the regressors and the counts are Poisson.
</document>

<document source="https://tamnd.github.io/math2/node/K6EV57QFYG.md" tag="K6EV57QFYG" kind="statement">
# Strang splitting

## Statement

Let $A$ and $B$ be complex $d \times d$ matrices and let $e^{M}$ denote the [[matrix exponential|020M04FN1H]].

1. As $h \to 0$,
$$e^{hA/2}\, e^{hB}\, e^{hA/2} - e^{h(A + B)} = O(h^3).$$
2. For each $t > 0$, as $n \to \infty$,
$$\left( e^{tA/(2n)}\, e^{tB/n}\, e^{tA/(2n)} \right)^n - e^{t(A + B)} = O(n^{-2}).$$

## Remarks

For clause 1, expand both sides in powers of $h$: the terms of order $0$, $1$ and $2$ agree, because the symmetric product gives $\frac{h^2}{2}\left(\frac{A^2}{4} + \frac{A^2}{4} + B^2 + AB + BA + \frac{A^2}{2}\right) = \frac{h^2}{2}(A + B)^2$.
For clause 2, write the difference of $n$-th powers as a telescoping sum of $n$ terms.
Each term is a local error $O(n^{-3})$ multiplied by matrices of norm at most $e^{t(\norm{A} + \norm{B})}$.
The [[Lie–Trotter product formula|KH7EPKDCH3]] uses $e^{tA/n} e^{tB/n}$ instead, and its error is only $O(n^{-1})$; the symmetric arrangement gains one order.
If $AB = BA$, both splittings are exact.
In applications, $e^{hA}$ and $e^{hB}$ are replaced by the flows $\varphi^A$ and $\varphi^B$ of $u' = A(u)$ and $u' = B(u)$ for nonlinear operators.
The composition $\varphi^A_{h/2} \circ \varphi^B_h \circ \varphi^A_{h/2}$ is still a method of order $2$ for smooth problems.

## Explanations

### L1 human

Some evolution equations are a sum of two parts, each easy to solve alone.
Strang splitting advances the first part half a step, the second part a full step, and the first part another half step.
Doing the halves on both sides makes the error much smaller than doing one part after the other.

### L2 human

Strang splitting is used for reaction-diffusion equations, for the Schrödinger equation (kinetic and potential parts, solved by FFT and pointwise), and for advection in several dimensions.
Consecutive half steps of $A$ merge, so the cost is nearly that of the first-order Lie–Trotter splitting.
The leapfrog method for Hamiltonian systems is the Strang splitting of the kinetic and potential flows.
</document>

<document source="https://tamnd.github.io/math2/node/K6G0DRQ3CP.md" tag="K6G0DRQ3CP" kind="concept">
# Value function

## Summary

The value function of an optimal control problem gives the optimal cost as a function of the initial time and the initial state.
It is the central object of dynamic programming: it satisfies the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]], in the [[viscosity|H1ERJ3T75T]] sense when it is not smooth.
Its gradient, where it exists, is the costate of [[Pontryagin's maximum principle|7RPV2T29CF]] up to sign.
In economics the same notion appears as the indirect utility or the optimal value of a maximization problem.

## Explanations

### L1 human

Imagine a map that shows, at each place and each time, the least cost to finish a journey from there.
That map is the value function.
If you know it, you can always choose the move that leads to the cheapest remaining cost.
For example, if you can walk at speed at most $1$ and pay the distance to the origin that is left at time $T$, the value at $(t, x)$ is $\max(\abs{x} - (T - t), 0)$.
</document>

<document source="https://tamnd.github.io/math2/node/K6PPGRT5E1.md" tag="K6PPGRT5E1" kind="definition">
# Catenary ring

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]].
A chain of [[prime ideals|00MHHZ9SNE]] $\mathfrak p_0 \subsetneq \mathfrak p_1 \subsetneq \cdots \subsetneq \mathfrak p_n$ has length $n$.
It is saturated if no prime ideal $\mathfrak q$ satisfies $\mathfrak p_{i-1} \subsetneq \mathfrak q \subsetneq \mathfrak p_i$ for some $i$.
The ring $R$ is *catenary* if the following holds for all prime ideals $\mathfrak p \subseteq \mathfrak q$ of $R$.
Every saturated chain of prime ideals from $\mathfrak p$ to $\mathfrak q$ is finite, and all such chains have the same length.

## Remarks

Every principal ideal domain is catenary.
Its nonzero primes are maximal, so the saturated chains are $(0) \subsetneq \mathfrak m$ and the trivial chains, and between two given primes they all have the same length.
Nagata constructed Noetherian local domains that are not catenary: in such a ring there are primes $\mathfrak p \subseteq \mathfrak q$ joined by saturated chains of two different lengths.
Many authors include the condition that $R$ is [[Noetherian|0670T59HGE]] in the definition.
A stronger notion is that of a [[universally catenary ring|5WAFPTH09V]].
</document>

<document source="https://tamnd.github.io/math2/node/K6QV4WSTXX.md" tag="K6QV4WSTXX" kind="concept">
# Linear recurrence with polynomial coefficients

## Summary

A linear recurrence with polynomial coefficients relates consecutive terms of a sequence linearly, with coefficients that are polynomials in the index $n$.
It generalizes the [[linear recurrences with constant coefficients|0NDVHBVBXB]] and covers many sequences of combinatorics, such as factorials, Catalan numbers and harmonic numbers.
The sequences that satisfy such a recurrence are the [[holonomic sequences|NWKX329T01]], and their generating functions are [[holonomic functions|7H44JD9NQD]].
Computer algebra uses these recurrences as finite descriptions of infinite sequences.

## Explanations

### L1 human

In the Fibonacci rule $u_{n+2} = u_{n+1} + u_n$, the coefficients are fixed numbers.
Here the coefficients may grow with $n$, like the factor $n + 1$ in $u_{n+1} = (n+1) u_n$.
That rule, started at $u_0 = 1$, produces the factorials $1, 1, 2, 6, 24, \dots$.
The rule is still linear: the terms of the sequence are only multiplied by known quantities and added.
</document>

<document source="https://tamnd.github.io/math2/node/K6R7GHB3VS.md" tag="K6R7GHB3VS" kind="statement">
# Reflection principle

## Statement

Let $(W_t)_{t \ge 0}$ be a [[Wiener process|552AWX3QXN]], let $a > 0$, and let $\tau_a = \inf\set{t \ge 0 : W_t = a}$ be the first hitting time of $a$, a [[stopping time|WND87FBAKC]].

1. The process $\widetilde{W}$ defined by $\widetilde{W}_t = W_t$ for $t \le \tau_a$ and $\widetilde{W}_t = 2a - W_t$ for $t > \tau_a$ is again a Wiener process.
2. For every $t > 0$,
$$P\Bigl(\sup_{0 \le s \le t} W_s \ge a\Bigr) = 2\, P(W_t \ge a).$$

## Remarks

Part 1 follows from the strong Markov property: after $\tau_a$, the process $W_{\tau_a + s} - a$ is a Wiener process independent of the past, and so is its negative.
For part 2, the event $\{\sup_{s \le t} W_s \ge a\}$ equals $\{\tau_a \le t\}$, and it splits into $\{W_t \ge a\}$ and $\{\tau_a \le t, W_t < a\}$.
By part 1 the second event has the same probability as $\{\widetilde{W}_t > a\} = \{W_t > a\}$, and $P(W_t = a) = 0$.
As a consequence, $\sup_{s \le t} W_s$ has the same law as $\abs{W_t}$, and $P(\tau_a \le t) = 2\bigl(1 - \Phi(a/\sqrt{t})\bigr)$, where $\Phi$ is the standard normal distribution function.
Differentiating gives the density $\frac{a}{\sqrt{2\pi t^3}} e^{-a^2/(2t)}$ of $\tau_a$, which shows $\tau_a < \infty$ almost surely but $E[\tau_a] = \infty$.

## Explanations

### L1 human

Once a Brownian path touches the level $a$, it is equally likely to end above or below $a$.
Reflecting the rest of the path in the line $y = a$ turns one kind of path into the other.
So the chance that the path ever reaches $a$ by time $t$ is twice the chance that it ends above $a$ at time $t$.

### L2 human

The principle gives the joint law of $W_t$ and its running maximum, and the law of first passage times.
These are used to price barrier options and to study boundary crossing in sequential statistics.
The same reflection idea for simple random walks gives the ballot theorem and counts of lattice paths.
</document>

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# Fixed-point iteration

## Summary

Fixed-point iteration computes a solution of $f(x) = x$ by applying $f$ again and again to a starting point.
If the resulting sequence converges and $f$ is continuous, its limit is a fixed point of $f$.
The [[Banach fixed point theorem|0VK6VEH345]] guarantees convergence for a contraction on a complete metric space.
Many numerical methods, such as [[Newton's method|07EXVNF9VP]], are fixed-point iterations.

## Explanations

### L1 human

Pick a number, apply a function to it, apply the function to the result, and keep going.
If the numbers settle down, they settle at a point that the function does not move.
Try it with the cosine key of a calculator in radians: starting from $1$, the values approach about $0.739085$, the only real solution of $\cos x = x$.
Iteration is a simple way to solve an equation when one can rewrite it as $x = f(x)$.
</document>

<document source="https://tamnd.github.io/math2/node/K7106VBEBF.md" tag="K7106VBEBF" kind="definition">
# Dehn function

## Definition

Let $G$ be a [[finitely presented group|TJKANTKASZ]] with a finite [[presentation|0JGEQXQ4AC]] $\langle S \mid R \rangle$, and let $F(S)$ be the [[free group|0KTW26C80J]] on $S$.
Let $w$ be a word over $S \cup S^{-1}$ that represents the identity of $G$.
Its *area* $\mathrm{Area}(w)$ is the least $N \ge 0$ such that $w$ is a product of $N$ [[conjugates|0DXD4QSZ8A]] of relators and their inverses in $F(S)$.
This means an equality $w = \prod_{i=1}^{N} u_i \, r_i^{\pm 1} \, u_i^{-1}$ in $F(S)$, with words $u_i$ and relators $r_i \in R$.
The *Dehn function* of the presentation is the function $\delta \colon \N \to \N$ given by
$$\delta(n) = \max \set{\mathrm{Area}(w) : \abs{w} \le n, \ \bar w = 1 \text{ in } G}.$$

## Remarks

For functions $f, g \colon \N \to \N$, write $f \preceq g$ if there is $C \ge 1$ with $f(n) \le C g(Cn + C) + Cn + C$ for all $n$, and $f \simeq g$ if $f \preceq g \preceq f$.
The Dehn functions of two finite presentations of the same group are $\simeq$-equivalent, so one speaks of the Dehn function of $G$.

Example: for $\Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle$, the word $a^n b^n a^{-n} b^{-n}$ of length $4n$ has area $n^2$, and every null-homotopic word of length $n$ has area at most a constant times $n^2$; so $\delta(n) \simeq n^2$.

Example: free groups and finite groups have Dehn function $\simeq n$.

Non-example: the [[lamplighter group|4GNPFMT7DZ]] is finitely generated but not finitely presented, so it has no Dehn function in this sense.

A finitely presented group has a solvable word problem if and only if its Dehn function is bounded above by a recursive function.
The [[Baumslag–Solitar group|429JJ5ZNBY]] $\langle a, b \mid b a b^{-1} = a^2 \rangle$ has an exponential Dehn function.
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# Monomial order

## Definition

Let $k$ be a field and let $n \ge 1$.
The monomials of the [[polynomial ring|0SDC934R52]] $k[x_1, \ldots, x_n]$ are the products $x^a = x_1^{a_1} \cdots x_n^{a_n}$ with $a \in \N^n$.
A *monomial order* is a relation $>$ on the set of monomials such that:

1. $>$ is a strict [[total order|0FRXBVVR03]].
2. If $x^a > x^b$, then $x^a x^c > x^b x^c$ for every $c \in \N^n$.
3. $>$ is a [[well-order|01CGSJ0GNQ]]: every nonempty set of monomials has a smallest element.

## Remarks

Given clauses 1 and 2, clause 3 is equivalent to $x^a > 1$ for every $a \ne 0$; this uses Dickson's lemma.
Equivalently, a monomial order is a total order on $\N^n$ that is compatible with addition and has $0$ as its least element.
For a nonzero polynomial $f$, the leading monomial $\operatorname{LM}(f)$ is the largest monomial that occurs in $f$, and the leading term $\operatorname{LT}(f)$ is that monomial with its coefficient.

Example: the [[lexicographic order|0B68KNPGCG]]: $x^a > x^b$ if the first nonzero entry of $a - b$ is positive.
It is total, it is compatible since $(a + c) - (b + c) = a - b$, and it is a well-order on $\N^n$.

Non-example: for $n = 1$, order the monomials by $x^a > x^b$ if $a < b$.
This is a total order compatible with multiplication, but $x > x^2 > x^3 > \cdots$ is an infinite descending chain, so clause 3 fails.
</document>

<document source="https://tamnd.github.io/math2/node/K7131908S7.md" tag="K7131908S7" kind="definition">
# Relatively free algebra

## Definition

Let $A$ be an algebra, and let $V(A)$ be the [[variety|AJNZJD8442]] generated by $A$.
The algebra $A$ is *relatively free* if, for some subset $X \subseteq A$, it is a [[free algebra|GTS4DRRMZR]] of $V(A)$ on $X$ via the inclusion map.

## Remarks

Equivalently, $A$ has a [[generating set|2TQD8NMG9P]] $X$ such that every map $X \to A$ extends to an endomorphism of $A$.
This follows from the fact that a universal property for the class $\set{A}$ carries over to $V(A) = HSP(A)$, by [[Tarski's formula|M4ZHB3GXBP]].
Example: the free group $F_2$ on $\set{a, b}$ is relatively free, since every map $\set{a, b} \to F_2$ extends to an endomorphism.
Example: $\Z / n\Z$ with $X = \set{1}$ is relatively free; it is the free algebra on one generator of the variety of abelian groups satisfying $nx \approx 0$.
Non-example: the symmetric group $S_3$ is not relatively free.
Every generating set $X$ of $S_3$ contains a transposition $t$, since the even permutations form a proper subgroup.
A map $X \to S_3$ that sends $t$ to an element of order $3$ cannot extend to an endomorphism, since $t^2 = e$ would force the image $u$ to satisfy $u^2 = e$.
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<document source="https://tamnd.github.io/math2/node/K76033B5G1.md" tag="K76033B5G1" kind="definition">
# Navier–Stokes equations

## Definition

Let $\Omega \subseteq \R^3$ be an open set, let $T > 0$, let $\nu > 0$ be a constant, and let $f \colon \Omega \times (0, T) \to \R^3$ be a given force.
The *incompressible Navier–Stokes equations* are the system
$$\partial_t u + (u \cdot \nabla) u = \nu \Delta u - \nabla p + f, \qquad \nabla \cdot u = 0,$$
for an unknown [[vector field|0299VCW1TH]] $u \colon \Omega \times (0, T) \to \R^3$, the velocity, and an unknown function $p \colon \Omega \times (0, T) \to \R$, the pressure.
Here $\nabla p$ is the [[gradient|0ABK9V9Y64]] of $p$ in $x$, and $\nabla \cdot u$ is the [[divergence|0MWV6ZTK6S]] of $u$.
The [[Laplacian|0E0B6RA8K0]] $\Delta$ acts on each component, and $(u \cdot \nabla) u$ has components $\sum_{j=1}^3 u_j \partial_j u_i$.
The constant $\nu$ is the kinematic viscosity, and the density is normalised to $1$.

## Remarks

Example: on $\Omega = \R^3$ with $f = 0$, the shear flow $u(x, y, z, t) = (y, 0, 0)$ with constant pressure is a solution.
Indeed $\partial_t u = 0$, $(u \cdot \nabla) u = y \, \partial_x u = 0$, $\Delta u = 0$, $\nabla p = 0$ and $\nabla \cdot u = 0$.

Non-example: $u = (x, 0, 0)$ with any pressure is not a solution, because $\nabla \cdot u = 1 \ne 0$.

The same equations are studied in two dimensions and on the torus.
For a compressible fluid the density $\rho$ becomes an unknown, with the continuity equation $\partial_t \rho + \nabla \cdot (\rho u) = 0$ in place of $\nabla \cdot u = 0$.
Setting $\nu = 0$ gives the Euler equations of an ideal fluid.
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<document source="https://tamnd.github.io/math2/node/K768V9ZS7R.md" tag="K768V9ZS7R" kind="definition">
# Perfectoid space

## Definition

Let $p$ be a prime.
Let $K$ be a field, complete for a [[non-archimedean absolute value|1H5F1BP6XS]] $\abs{\cdot}$, and let $\mathcal{O}_K = \set{x : \abs{x} \le 1}$.
The field $K$ is a *perfectoid field* if:

1. The value group $\abs{K^\times}$ is not discrete.
2. The residue field of $\mathcal{O}_K$ has characteristic $p$.
3. The [[Frobenius map|0T3WVTD675]] $x \mapsto x^p$ on $\mathcal{O}_K / p\mathcal{O}_K$ is surjective.


Let $K$ be a perfectoid field.
A *perfectoid $K$-algebra* is a Banach $K$-algebra $R$ such that the subring $R^\circ$ of power-bounded elements is bounded and the Frobenius map $R^\circ / p R^\circ \to R^\circ / p R^\circ$ is surjective.
A *perfectoid affinoid $K$-algebra* is a pair $(R, R^+)$ with $R$ a perfectoid $K$-algebra and $R^+ \subseteq R^\circ$ an open and integrally closed subring.
A *perfectoid space* over $K$ is an adic space over $K$ that is covered by open subspaces isomorphic to $\operatorname{Spa}(R, R^+)$ with $(R, R^+)$ perfectoid affinoid.

## Remarks

Example: the [[completion|W168DS966V]] $K$ of $\Q_p(p^{1/p^\infty}) = \bigcup_n \Q_p(p^{1/p^n})$ is a perfectoid field.
Its value group is $p^{\Z[1/p]}$, which is not discrete, and every element of $\mathcal{O}_K / p$ is a polynomial in the $p^{1/p^n}$, each of which is the $p$-th power of $p^{1/p^{n+1}}$.
The completed algebraic closure $\C_p$ of the [[$p$-adic numbers|BT6T702C28]] is also perfectoid.

Non-example: $\Q_p$ is not perfectoid.
The Frobenius map on $\Z_p / p = \F_p$ is surjective, but the value group $p^{\Z}$ is discrete, so the first condition fails.

The tilt of $K$ is $K^\flat$, the fraction field of $\varprojlim_{x \mapsto x^p} \mathcal{O}_K / p$; it is a perfectoid field of characteristic $p$.
Scholze's tilting equivalence identifies perfectoid spaces over $K$ with perfectoid spaces over $K^\flat$.
</document>

<document source="https://tamnd.github.io/math2/node/K76N026JFK.md" tag="K76N026JFK" kind="concept">
# Rational equivalence

## Summary

Rational equivalence is the finest of the usual equivalence relations on [[algebraic cycles|9VFWPZEVGN]].
Two cycles are rationally equivalent when their difference is a sum of divisors of rational functions on subvarieties.
The quotient of the group of cycles by rational equivalence is the [[Chow group|EBEF6PEK4N]].

## Explanations

### L1 human

On a curve, the zeros of a rational function minus its poles form a cycle of points that one wants to count as zero.
Rational equivalence declares all such cycles to be zero, and does the same on every subvariety of every dimension.
On the projective line, the function $(x - a)/(x - b)$ has a zero at $a$ and a pole at $b$, so any two points are rationally equivalent.
On an elliptic curve, two distinct points are never rationally equivalent, so the relation sees the geometry of the curve.
</document>

<document source="https://tamnd.github.io/math2/node/K784BA2WHV.md" tag="K784BA2WHV" kind="concept">
# Walsh–Hadamard transform

## Summary

The Walsh–Hadamard transform takes a vector of length $2^m$ to the vector of its correlations with all $\pm 1$ patterns given by a Sylvester [[Hadamard matrix|J0N4J606X0]].
It is the discrete Fourier transform of the group $(\Z/2\Z)^m$, and its rows are sampled [[Walsh functions|F12A5QKW8Z]].
A fast algorithm computes it with about $m 2^m$ additions and subtractions, and no multiplications.
It is used in signal processing, coding theory, the analysis of Boolean functions and quantum computing.

## Explanations

### L1 human

Take a list of $2^m$ numbers.
For each pattern of plus and minus signs from a fixed family of $2^m$ patterns, add up the numbers with those signs.
The resulting list of sums is the transform.
For $m = 1$ the patterns are $(+, +)$ and $(+, -)$, so $(a, b)$ goes to $(a + b, a - b)$.
Doing the same step on pairs, then on pairs of pairs, and so on, gives the whole transform quickly.
</document>

<document source="https://tamnd.github.io/math2/node/K7A04ZBD92.md" tag="K7A04ZBD92" kind="concept">
# Equivalence of categories

## Summary

An equivalence of categories is a [[functor|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathcal{D}$ that has a functor $G \colon \mathcal{D} \to \mathcal{C}$ in the other direction.
The composites $G \circ F$ and $F \circ G$ must be naturally isomorphic to the identity functors.
It is the right notion of "sameness" for categories: it ignores how many copies of each object up to isomorphism a category has.
A functor is an equivalence exactly when it is [[fully faithful|GRG8WEBSH5]] and [[essentially surjective|6ERBSBVF14]], by [[this theorem|WA60RW9E04]].
Every category is equivalent to its [[skeleton|A49QJYG669]].

## Explanations

### L1 human

Two categories are equivalent when they have the same objects up to isomorphism and the same arrows between them.
Isomorphism of categories asks for more: an exact match of objects, which is rarely useful.
For example, the category of finite-dimensional real vector spaces has a huge number of objects.
The category with one object $n$ for each natural number, and with $m \times n$ real matrices as the arrows from $n$ to $m$, has very few.
Yet the two are equivalent, which is why linear maps between finite-dimensional spaces can be studied as matrices.
</document>

<document source="https://tamnd.github.io/math2/node/K7CNWXF9SN.md" tag="K7CNWXF9SN" kind="concept">
# Set of uniqueness

## Summary

A set of uniqueness is a subset $E$ of the circle with the following property.
A [[trigonometric series|FZK2M8KGXQ]] that converges to $0$ at every point outside $E$ must have all coefficients equal to $0$.
A set without this property is a set of multiplicity.
[[Cantor's uniqueness theorem|NSS3XW4BYZ]] says that the empty set is a set of uniqueness.
The question of which exceptional sets are allowed led Cantor to his theory of sets and ordinals.
The study of these sets connects harmonic analysis with measure theory, number theory and descriptive set theory.

## Explanations

### L1 human

Suppose a sum of waves is known to be $0$ everywhere except on a set $E$, where nothing is known.
If $E$ is a set of uniqueness, this already forces every wave in the sum to vanish.
Small sets, such as finite or countable sets, are sets of uniqueness.
Large sets, such as intervals, are not.
Surprisingly, Menshov found a closed set of length zero that is not a set of uniqueness, so being small in length is not enough.
</document>

<document source="https://tamnd.github.io/math2/node/K7H05NZEX8.md" tag="K7H05NZEX8" kind="definition">
# Limit-point and limit-circle

## Definition

Let $-\infty \le a < b \le \infty$.
Let $p, q, w \colon (a, b) \to \R$ be measurable with $p > 0$ and $w > 0$ almost everywhere, and with $1/p$, $q$ and $w$ locally integrable on $(a, b)$.
For $\lambda \in \C$ consider the [[Sturm–Liouville|7WV6B2502K]] equation
$$-(p y')' + q y = \lambda w y \quad \text{on } (a, b).$$
Its solutions are the functions $y$ such that $y$ and $p y'$ are locally absolutely continuous and the equation holds almost everywhere.
Fix $c \in (a, b)$.
The endpoint $b$ is in the limit-circle case if there is a $\lambda \in \C$ such that every solution $y$ of the equation satisfies
$$\int_c^b \abs{y(x)}^2 w(x) \, dx < \infty.$$
Otherwise $b$ is in the limit-point case.
The same definition applies to the endpoint $a$ with the interval $(a, c)$.

## Remarks

The definition does not depend on $c$.
Weyl's alternative says that if every solution is square integrable near $b$ for one $\lambda \in \C$, then this holds for every $\lambda \in \C$.
For $\lambda \in \C \setminus \R$ there is always at least one nonzero solution that is square integrable near $b$.
So $b$ is limit-point exactly when, for such $\lambda$, this solution is unique up to a constant factor.
Example: for $p = w = 1$, $q = 0$ on $(0, \infty)$ and $\lambda = 0$, the solutions are $y = c_1 + c_2 x$, and $y = 1$ is not square integrable near $\infty$.
By Weyl's alternative, $\infty$ is in the limit-point case.
Non-example of limit-point: a regular endpoint, such as $b < \infty$ with $1/p$, $q$, $w$ integrable near $b$, is always in the limit-circle case, since every solution then extends continuously to $b$.
For $p = w = 1$ and $q(x) = \kappa / x^2$ on $(0, 1]$, the endpoint $0$ is in the limit-point case exactly when $\kappa \ge 3/4$.
If $p = w = 1$ and $q$ is bounded below near $b = \infty$, then $\infty$ is in the limit-point case.

## Explanations

### L1 human

Look at all solutions near the endpoint and ask whether they are square integrable there.
If all of them are, the endpoint is limit-circle; if some are not, it is limit-point.
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<document source="https://tamnd.github.io/math2/node/K7MR77CHP6.md" tag="K7MR77CHP6" kind="statement">
# Poincaré–Birkhoff theorem

## Statement

Let $A = (\R / \Z) \times [0, 1]$ be the closed [[annulus|0CW52NV5TC]], with the area measure $d\theta \, dr$ given by [[Lebesgue measure|2Z2QVWZ262]].
Let $f \colon A \to A$ be a [[homeomorphism|0QT00PB5G2]] such that:

1. $f$ preserves area.
2. $f$ maps each boundary circle $(\R / \Z) \times \set{0}$ and $(\R / \Z) \times \set{1}$ to itself.
3. $f$ twists the boundary circles in opposite directions: some lift $F \colon \R \times [0, 1] \to \R \times [0, 1]$ of $f$ to the [[universal cover|Y1YPE6PM52]], written $F(x, r) = (x + g(x, r), h(x, r))$, satisfies $g(x, 0) < 0 < g(x, 1)$ for all $x \in \R$.

Then $f$ has at least two fixed points.

## Remarks

Poincaré stated the theorem shortly before his death and checked many special cases, and Birkhoff published a proof in 1913.
Birkhoff's proof works with the lift $F$ on the strip $\R \times [0, 1]$.
The twist condition makes the horizontal displacement $g$ change sign between the two boundaries.
Area preservation then rules out the ways in which $F$ could still avoid a fixed point.
Applied to iterates $f^q$ and lifts shifted by integers $p$, the theorem gives periodic orbits of every rotation number $p/q$ between the rotation numbers of the two boundaries.
Area preservation is necessary: the map with lift $F(x, r) = (x + r - 1/2, r^2)$ satisfies 2 and 3, but a fixed point would need $r^2 = r$ and $r - 1/2 \in \Z$, which is impossible.
This map is not area preserving, since its Jacobian determinant is $2r$.

## Explanations

### L1 human

Picture a ring that is twisted so that its inner edge turns one way and its outer edge turns the other way.
If the twist does not squeeze or stretch any area, then at least two points of the ring end up exactly where they started.
The theorem is a basic tool for finding periodic motions in mechanics.

### L2 human

Area-preserving twist maps of an annulus arise as return maps of Hamiltonian systems with two degrees of freedom, for example near an elliptic periodic orbit or in billiards.
The theorem then produces infinitely many periodic orbits, one family for each rational rotation number in the twist interval.
It is a forerunner of symplectic topology: Arnold's conjecture on fixed points of Hamiltonian diffeomorphisms was proposed as a higher-dimensional generalisation.
</document>

<document source="https://tamnd.github.io/math2/node/K7N565NKCV.md" tag="K7N565NKCV" kind="concept">
# Supporting hyperplane

## Summary

A supporting hyperplane of a set $S$ in $\R^n$ is a hyperplane that has all of $S$ on one side.
It also touches $S$ at a point of the [[boundary|05D5EGMFYV]] of $S$.
By the [[supporting hyperplane theorem|DR9BXFWMQB]], a [[convex set|02AS45T3H9]] has a supporting hyperplane through each of its boundary points.
Supporting hyperplanes link convex sets with linear functions: they describe where a linear function is largest on the set.

## Explanations

### L1 human

Put a convex object on a table.
The table top is a supporting plane: the object lies entirely above it and touches it.
In the plane, a supporting line of a disc is a tangent line, and a supporting line of a square through a corner can be tilted in many ways.
For the square $[0, 1]^2$, the line $x = 1$ is a supporting line touching the right side.
</document>

<document source="https://tamnd.github.io/math2/node/K7TYSAWGPC.md" tag="K7TYSAWGPC" kind="definition">
# Wiman–Valiron theory

## Definition

Let $f(z) = \sum_{n \ge 0} a_n z^n$ be a transcendental [[entire function|0419794R0T]], that is, an entire function that is not a polynomial.
Wiman–Valiron theory studies $f$ through the following functions of $r > 0$:

1. the *maximum term* $\mu(r) = \max_{n \ge 0} \abs{a_n} r^n$;
2. the *central index* $\nu(r)$, the largest integer $n$ with $\abs{a_n} r^n = \mu(r)$.

## Remarks

The maximum exists because $\abs{a_n} r^n \to 0$ as $n \to \infty$ for every $r$, by convergence of the [[Taylor series|0225QPWRRE]].
The function $\nu(r)$ is nondecreasing, and $\nu(r) \to \infty$ because $f$ is not a polynomial.
Example: for $f = e^z$, $a_n = 1/n!$ and the ratio of consecutive terms is $r/(n+1)$, so $\nu(r) = \lfloor r \rfloor$ and $\mu(r) = r^{\nu(r)}/\nu(r)!$.
Non-example: for a polynomial of degree $d$, $\nu(r) = d$ for all large $r$, so it stays bounded; this is why the theory requires $f$ transcendental.
The Cauchy estimates give $\mu(r) \le M(r)$, where $M(r) = \max_{\abs{z} = r} \abs{f(z)}$.
The main result states that, for $r$ outside a set $E$ with $\int_E dr/r < \infty$ and $z_r$ with $\abs{z_r} = r$ and $\abs{f(z_r)} = M(r)$, one has $f^{(k)}(z_r) = \left(\nu(r)/z_r\right)^k f(z_r)\,(1 + o(1))$ as $r \to \infty$, for each fixed $k$.
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# Credible interval

## Definition

Let $\theta$ be a real parameter, and let $\Pi(\cdot \mid x)$ be its [[posterior distribution|0KJ7ZS8AJA]] given the observed data $x$.
Let $\alpha \in (0, 1)$.
A *credible interval* of level $1 - \alpha$ is an interval $[a, b]$, with $a$ and $b$ depending only on $x$, such that
$$\Pi\left([a, b] \mid x\right) \ge 1 - \alpha.$$
The *equal-tailed* credible interval has as endpoints the $\alpha/2$ and the $1 - \alpha/2$ quantiles of the posterior.
Suppose the posterior has a unimodal density $\pi(\cdot \mid x)$.
The *highest posterior density* interval is then the set $\set{\theta : \pi(\theta \mid x) \ge c}$, with $c$ chosen so that its posterior probability is $1 - \alpha$.

## Remarks

Example: with a uniform prior on $p \in (0, 1)$ and $n$ successes in $n$ Bernoulli trials, the posterior is $\mathrm{Beta}(n + 1, 1)$, with distribution function $p^{n + 1}$ on $[0, 1]$.
So $[\alpha^{1/(n+1)}, 1]$ has posterior probability $1 - \alpha$, and it is a credible interval of level $1 - \alpha$.
Non-example: a frequentist [[confidence interval|0K4NMMJBG2]] is defined by its coverage over repeated samples, not by a posterior probability.
For the same data it can differ from every credible interval, and it need not have posterior probability $1 - \alpha$.
In a parameter space of higher dimension the same notion is called a credible region or credible set.
</document>

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# de Rham theorem

## Statement

Let $M$ be a [[smooth manifold|F45VQ47GPV]].
Let $\omega$ be a closed $k$-form on $M$ and let $c$ be a smooth singular $k$-cycle in $M$.
Put $\langle [\omega], [c] \rangle = \int_c \omega$.
Let $H^k_{\mathrm{dR}}(M)$ be the [[de Rham cohomology|GX74C07T0G]] of $M$ and $H^k(M; \R)$ its [[singular cohomology|6N7BC70CX0]] with real coefficients.
For every $k \ge 0$, this pairing induces an isomorphism of real vector spaces
$$H^k_{\mathrm{dR}}(M) \cong H^k(M; \R).$$

## Remarks

The map is well defined by the [[generalised Stokes theorem|0HPHKCGEQ0]].
It gives that the integral of an exact form over a cycle is $0$, and so is the integral of a closed form over a boundary.
One proof shows that both sides satisfy the Mayer–Vietoris sequence and agree on convex open subsets of $\R^n$.
On such sets the Poincaré lemma gives the de Rham side.
Then one inducts over a good cover of $M$.
Another proof says that both sides compute the sheaf cohomology of the constant sheaf $\R$ on $M$.
The isomorphism also takes the wedge product of forms to the [[cup product|JX47MS1R1H]], so it is an isomorphism of graded rings.

Example: on the circle $S^1$ with angle coordinate $\theta$, the closed form $d\theta$ is not exact.
Its integral over the circle is $2\pi \neq 0$, so its class corresponds to a nonzero class in $H^1(S^1; \R) \cong \R$.

The coefficients must be real (or contain $\R$): differential forms do not see torsion, and $H^k(M; \Z)$ can have torsion, as for the real projective plane.

## Explanations

### L1 human

On a smooth space there are two ways to count holes.
One uses calculus: differential forms whose derivative is zero but which are not derivatives themselves.
The other uses topology: cycles that are not boundaries.
The de Rham theorem says that the two counts agree, and the matching is given by integration.

### L2 human

The theorem lets you compute topological invariants of a manifold with calculus, and conversely.
For example, it shows that the de Rham cohomology is a topological invariant, though it is defined with smooth structure.
It is the model for comparison theorems in other settings, such as the comparison between algebraic de Rham cohomology and singular cohomology of a complex variety.
Together with Poincaré duality and Hodge theory it is a basic tool of global analysis on manifolds.
</document>

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# Ricci curvature

## Definition

Let $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] with curvature $R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$ (see [[Riemann curvature tensor|JA29ETCM02]]).
The *Ricci curvature* of $(M, g)$ is the $(0,2)$-tensor field $\operatorname{Ric}$ defined, for $p \in M$ and $u, v \in T_pM$, by
$$\operatorname{Ric}(u, v) = \tr\bigl( w \mapsto R(w, u)v \bigr),$$
the [[trace|07PTPSZPME]] of the linear map $T_pM \to T_pM$, $w \mapsto R(w, u)v$.

## Remarks

If $e_1, \ldots, e_n$ is a $g$-orthonormal basis of $T_pM$ in the Riemannian case, then $\operatorname{Ric}(u, v) = \sum_i g(R(e_i, u)v, e_i)$, and for a unit vector $v = e_1$ this gives $\operatorname{Ric}(v, v) = \sum_{i \ge 2} K(e_i, v)$, a sum of [[sectional curvatures|1X42JAA349]].
In coordinates, $R_{ij} = \sum_k R^k{}_{kij}$ where $R(\partial_k, \partial_i)\partial_j = \sum_l R^l{}_{kij} \partial_l$.
The Ricci tensor is symmetric, by the first Bianchi identity.

Example: on the unit sphere $S^n$, $R(w, u)v = g(u, v)w - g(w, v)u$, whose trace in $w$ is $n \, g(u, v) - g(u, v)$, so $\operatorname{Ric} = (n - 1)g$.

Non-example: $\operatorname{Ric}(u, v) = \tr(w \mapsto R(u, v)w)$ is a different trace, and it is identically zero because $R(u, v)$ is skew-adjoint; it is not the Ricci curvature.

A metric with $\operatorname{Ric} = \lambda g$ for a constant $\lambda$ is an [[Einstein metric|41S2EZ0KR3]].
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# Seifert surface

## Summary

A Seifert surface of a [[knot|QYEAZTYGPT]] or [[link|1A5WRCNTM3]] is a compact, connected, orientable surface in space whose boundary is the knot or link.
Every oriented link has one, by an algorithm of Herbert Seifert from 1934.
The least [[genus|NJ787GZS1N]] of a Seifert surface of a knot is the genus of the knot, a [[knot invariant|GZQ5ZE3X6G]].
Seifert surfaces are also used to define the Alexander polynomial and the signature of a knot.

## Explanations

### L1 human

Dip a knotted wire loop into soap solution: the soap film that forms is a surface whose edge is the knot.
A Seifert surface is such a surface with two distinct sides, so that it can be painted in two colours.
For a round circle, a flat disk works.
For more complicated knots, the surface needs handles, and the least number of handles needed is the genus of the knot.
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# Miquel's theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane.
Let $A'$, $B'$, $C'$ be points on the lines $BC$, $CA$, $AB$, each different from the vertices of $ABC$.
Then the three [[circles|060W5CBZP6]] through $A, B', C'$, through $B, C', A'$ and through $C, A', B'$ have a common point.

## Remarks

The three points on each circle are not collinear, since for example $B'$ and $C'$ lie on two different lines through $A$ and differ from $A$.
For a proof, let $M$ be the second common point of the first two circles, or their point of tangency.
Chasing directed angles in the cyclic quadrilaterals $AB'MC'$ and $BC'MA'$ shows that $CA'MB'$ is cyclic too.
The common point is called the Miquel point.
Example: if $A'$, $B'$, $C'$ are the midpoints of the sides, the Miquel point is the [[circumcentre|0F98FZK14A]] $O$.
Indeed, $\angle OB'A$ and $\angle OC'A$ are right angles, so $O$ lies on the circle with diameter $AO$ through $A$, $B'$, $C'$.

## Explanations

### L1 human

Mark one point on each side of a triangle.
Each corner and the two marked points next to it determine a circle.
The three circles you get always pass through one common point.

### L2 human

Miquel's theorem is a standard exercise in angle chasing with cyclic quadrilaterals.
It is the starting point of a family of results on Miquel points of complete quadrilaterals.
</document>

<document source="https://tamnd.github.io/math2/node/K8BK5SKY1Y.md" tag="K8BK5SKY1Y" kind="definition">
# j-invariant

## Definition

Let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]], let $E_4$ and $E_6$ be the normalized [[Eisenstein series|EYREVY31BN]] of weights $4$ and $6$, and let $\Delta = (E_4^3 - E_6^2)/1728$.
The *$j$-invariant* is the function $j \colon \mathbb{H} \to \C$ given by
$$j(\tau) = \frac{E_4(\tau)^3}{\Delta(\tau)} = \frac{1728\, E_4(\tau)^3}{E_4(\tau)^3 - E_6(\tau)^2}.$$

## Remarks

The [[cusp form|A50Y012RBP]] $\Delta$ has no zeros on $\mathbb{H}$, so $j$ is holomorphic on $\mathbb{H}$.
Both $E_4^3$ and $\Delta$ have weight $12$, so $j(\gamma \tau) = j(\tau)$ for every $\gamma$ in the [[modular group|VEKV1EM77B]].
With $q = e^{2 \pi i \tau}$, $j(\tau) = q^{-1} + 744 + 196884 q + \cdots$, so $j$ is a [[meromorphic|0ZZ7ZYDR8W]] modular function with a simple pole at infinity.
Example: $E_6(i) = 0$, so $j(i) = 1728$; and $E_4(e^{2 \pi i/3}) = 0$, so $j(e^{2 \pi i /3}) = 0$.
Non-example: $j$ is not a [[modular form|GWKN5VG78A]] of weight $0$, because it is unbounded as $\operatorname{Im} \tau \to \infty$.
For the elliptic curve $y^2 = x^3 + ax + b$, the $j$-invariant is $1728 \cdot 4a^3 / (4a^3 + 27b^2)$.
</document>

<document source="https://tamnd.github.io/math2/node/K8EBASR5VG.md" tag="K8EBASR5VG" kind="problem">
# Optimal transport and the Monge problem

## Problem

Let $X$ and $Y$ be complete separable metric spaces.
Let $\mu$ be a [[Borel|B91V9BKZMR]] [[probability measure|059TJN0NZ1]] on $X$ and $\nu$ a Borel probability measure on $Y$.
Let $c \colon X \times Y \to [0, \infty]$ be a lower semicontinuous cost function.

1. Monge problem: minimize $\int_X c(x, T(x)) \, d\mu(x)$ over all Borel maps $T \colon X \to Y$ with $T_\# \mu = \nu$, where $T_\# \mu$ is the [[pushforward measure|3N66TM3TSQ]] $T_\#\mu(B) = \mu(T^{-1}(B))$.
2. Kantorovich problem: minimize $\int_{X \times Y} c \, d\pi$ over all transport plans, the Borel probability measures $\pi$ on $X \times Y$ with marginals $\mu$ and $\nu$.

Decide when minimizers exist, when they are unique, and when an optimal plan is given by a map.

## Remarks

Monge posed the problem in 1781, in his memoir on cuttings and fillings, for the cost $c(x, y) = \abs{x - y}$ in $\R^3$.
His aim was to move a pile of soil into a hole of the same volume with least total work.
The Monge problem may have no admissible map at all: if $\mu$ is a Dirac mass and $\nu$ is not, no $T$ satisfies $T_\#\mu = \nu$.
Kantorovich introduced the relaxed problem in 1942; under the hypotheses above it always has a minimizer, by weak compactness of the set of plans and lower semicontinuity of the cost.
For $X = Y = \R^n$, $c(x, y) = \abs{x - y}^2$, $\mu$ absolutely continuous and both measures with finite second moments, Brenier's theorem gives a unique optimal plan, induced by the map $T = \nabla \varphi$ for a convex function $\varphi$.
Then $\varphi$ solves a [[Monge–Ampère equation|14AKAC0S0T]] in a weak sense, and the optimal cost defines the squared [[Wasserstein distance|VEY29CS26X]] $W_2(\mu, \nu)^2$.
For Monge's original cost $\abs{x - y}$ in $\R^n$ with $\mu$ absolutely continuous and both measures with finite first moments, optimal maps exist but are in general not unique.
Evans and Gangbo proved existence under extra hypotheses in 1999, and later work removed them.

## Explanations

### L1 human

You have a pile of sand and a hole of the same volume, and you want to fill the hole while moving the sand as little as possible in total.
Monge asked for the best way to decide where each grain goes.
Kantorovich allowed each grain to be split among several destinations, which makes the problem much easier to solve.
Today optimal transport gives a way to measure distances between probability distributions, with uses in geometry, PDE, economics and machine learning.
</document>

<document source="https://tamnd.github.io/math2/node/K8JG3C40HZ.md" tag="K8JG3C40HZ" kind="definition">
# Homotopy category of chain complexes

## Definition

Let $\mathcal{A}$ be an [[additive category|B2VP96NJSK]].
The *homotopy category of chain complexes* $K(\mathcal{A})$ is the [[category|A2DE66937D]] defined as follows.

1. Its objects are the [[chain complexes|WQ82C0M3QB]] in $\mathcal{A}$.
2. For complexes $C$ and $D$, the morphisms $C \to D$ are the [[chain maps|HTK9WQZJ1G]] $C \to D$ modulo [[chain homotopy|4EBHAWJJ8C]]. Here $f \sim g$ if there are morphisms $s_n \colon C_n \to D_{n+1}$ with $f_n - g_n = d_{n+1} s_n + s_{n-1} d_n$ for all $n$.
3. Composition is induced by the composition of chain maps.

## Remarks

Chain homotopy is an equivalence relation compatible with sums and composition, so $K(\mathcal{A})$ is again additive and composition is well defined.
Example: in $K$ of abelian groups, the complex $C = (0 \to \Z \xrightarrow{\id} \Z \to 0)$, in degrees $1$ and $0$, is isomorphic to $0$, since $s_0 = \id \colon C_0 \to C_1$ gives $\id_C = d s + s d$.
Non-example: the complex $\cdots \to \Z/4\Z \xrightarrow{\cdot 2} \Z/4\Z \xrightarrow{\cdot 2} \Z/4\Z \to \cdots$ is exact, but it is not isomorphic to $0$ in $K$: a contraction would split each short exact sequence $0 \to 2\Z/4\Z \to \Z/4\Z \to 2\Z/4\Z \to 0$, and $\Z/4\Z \not\cong \Z/2\Z \oplus \Z/2\Z$.
If $\mathcal{A}$ is abelian, homology is a functor on $K(\mathcal{A})$.
Bounded variants $K^b(\mathcal{A})$, $K^+(\mathcal{A})$ and $K^-(\mathcal{A})$ use only complexes that are zero in all but finitely many degrees, in low degrees, or in high degrees.
</document>

<document source="https://tamnd.github.io/math2/node/K8P15XTH1M.md" tag="K8P15XTH1M" kind="concept">
# Sketch

## Summary

A sketch is a small category together with chosen cones and cocones in it.
A model of a sketch is a functor that turns the chosen cones into limits and the chosen cocones into colimits.
Sketches give a way to describe mathematical theories, such as the theory of groups or of categories, by diagrams instead of formulas.
The categories of models of sketches in sets are exactly the accessible categories.

## Explanations

### L1 human

A sketch is a blueprint for a kind of structure.
It names some objects and arrows, and it says that certain objects must be products, pullbacks, sums and so on.
A model is an actual structure in sets that follows the blueprint.
For example, a sketch with an object $X$, an object $P$ declared to be $X \times X$, and an arrow $m \colon P \to X$ has as models the sets with a binary operation.
</document>

<document source="https://tamnd.github.io/math2/node/K8PHD2NAW9.md" tag="K8PHD2NAW9" kind="definition">
# q-Pochhammer symbol

## Definition

Let $a$ and $q$ be [[complex numbers|09FTAET5Z4]], and let $n \ge 0$ be an integer.
The $q$-Pochhammer symbol is
$$(a; q)_n = \prod_{k=0}^{n-1} (1 - a q^k),$$
with $(a; q)_0 = 1$.
If $\abs{q} < 1$, the infinite $q$-Pochhammer symbol is the infinite product
$$(a; q)_\infty = \prod_{k=0}^{\infty} (1 - a q^k).$$

## Remarks

For $\abs{q} < 1$ the product defining $(a; q)_\infty$ converges, since $\sum_k \abs{a q^k} = \abs{a} / (1 - \abs{q}) < \infty$; it is zero exactly when some factor is zero.
Example: $(q; q)_2 = (1 - q)(1 - q^2) = 1 - q - q^2 + q^3$, by the definition with $a = q$ and $n = 2$.
Non-example: for $\abs{q} > 1$ and $a \ne 0$, the factors $1 - a q^k$ do not tend to $1$, so the infinite product does not [[converge|H1Z2TKY3EQ]] and $(a; q)_\infty$ is not defined.
The name comes from the limit $\frac{(q^x; q)_n}{(1 - q)^n} \to x(x+1)\cdots(x+n-1)$ as $q \to 1$, the rising factorial or Pochhammer symbol.
The [[generating function for partitions|0578PA53VA]] reads $\sum_{n \ge 0} p(n) q^n = 1/(q; q)_\infty$ for $\abs{q} < 1$.

## Explanations

### L1 human

The symbol $(a; q)_n$ is shorthand for the product of $n$ factors $1 - a$, $1 - aq$, up to $1 - aq^{n-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/K8RE4YQ1S9.md" tag="K8RE4YQ1S9" kind="definition">
# Finitary operation

## Definition

Let $A$ be a set and let $n \ge 0$ be an integer.
Write $A^n$ for the [[Cartesian product|09CEDJYBKP]] of $n$ copies of $A$, with $A^0 = \set{()}$ the one-element set whose element is the empty tuple.
An *$n$-ary operation* on $A$ is a [[function|0EPAT51DXQ]] $f \colon A^n \to A$.
A *finitary operation* on $A$ is an $n$-ary operation on $A$ for some integer $n \ge 0$, called its [[arity|MSXX2P86CY]].

## Remarks

Example: the map $f \colon \Z^3 \to \Z$, $f(x, y, z) = x - y + z$, is a finitary operation of arity $3$ on $\Z$, because $x - y + z$ is an integer for all integers $x, y, z$.

Non-example: division $(x, y) \mapsto x / y$ is not a binary operation on $\Q$, because it is not defined when $y = 0$, so it is not a function on all of $\Q^2$.
On $\Q \setminus \set{0}$ it is a binary operation.

The requirement that the values lie in $A$ is often called closure of $A$ under $f$.
Some authors use the word operation only for finitary operations.
</document>

<document source="https://tamnd.github.io/math2/node/K8T15QHN3W.md" tag="K8T15QHN3W" kind="concept">
# Cohomology operation

## Summary

A cohomology operation turns cohomology classes of one fixed degree into classes of another degree.
It does so on every space at once, compatibly with all continuous maps.
Examples are the cup square $x \mapsto x \smile x$, the [[Bockstein homomorphism|H7PCM776GR]] and the [[Steenrod squares|VR7RHP41M0]].
Operations give extra structure on cohomology that can distinguish spaces and maps with the same cohomology groups or rings.
By [[representability of cohomology|GPDPYHM3RE]], operations from degree $n$ with coefficients $G$ correspond to cohomology classes of the [[Eilenberg–MacLane space|AJHEYJE5XY]] $K(G, n)$.

## Explanations

### L1 human

Cohomology attaches groups to spaces, and maps between spaces give maps between these groups.
A cohomology operation is a recipe that works on the cohomology of every space and commutes with all these induced maps.
Squaring a class with the cup product is such a recipe, because induced maps respect cup products.
Operations are useful because a map must respect them, which can rule out maps that the groups alone allow.
</document>

<document source="https://tamnd.github.io/math2/node/K8YRDXE4WC.md" tag="K8YRDXE4WC" kind="definition">
# Pseudogroup

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A *pseudogroup* on $X$ is a set $\Gamma$ of [[homeomorphisms|0QT00PB5G2]] $f \colon U \to V$ between open subsets $U, V$ of $X$ with these properties.

1. The identity map of $X$ is in $\Gamma$.
2. If $f \colon U \to V$ is in $\Gamma$, then $f^{-1} \colon V \to U$ is in $\Gamma$.
3. If $f \colon U \to V$ and $h \colon U' \to V'$ are in $\Gamma$, then the composite $h \circ f$, defined on the open set $f^{-1}(V \cap U')$, is in $\Gamma$.
4. If $f \colon U \to V$ is in $\Gamma$ and $W \subseteq U$ is open, then the restriction $f|_W \colon W \to f(W)$ is in $\Gamma$.
5. Let $f \colon U \to V$ be a homeomorphism between open subsets of $X$, and let $U = \bigcup_i U_i$ be a cover by open sets. If each restriction $f|_{U_i}$ is in $\Gamma$, then $f$ is in $\Gamma$.

## Remarks

Example: the set of all $C^\infty$ diffeomorphisms between open subsets of $\R^n$ is a pseudogroup on $\R^n$.
Each clause holds because being a diffeomorphism is preserved by inversion, composition and restriction, and it is a local property, which gives clause 5.
The same holds for real-analytic diffeomorphisms, for biholomorphic maps between open subsets of $\C^n$ and for symplectomorphisms between open subsets of $\R^{2n}$.

Non-example: on $\R$, let $\Gamma$ be the set of restrictions to open sets of the bijective affine maps $x \mapsto ax + b$, $a \ne 0$.
It satisfies clauses 1 to 4 but fails clause 5.
The map on $(0, 1) \cup (2, 3)$ that is the identity on $(0, 1)$ and $x \mapsto x + 10$ on $(2, 3)$ is affine on each piece but is not the restriction of a single affine map.

A pseudogroup generated by a set of local homeomorphisms is the smallest pseudogroup containing them.
A $\Gamma$-structure on a space, such as a smooth structure, is an atlas whose transition maps lie in $\Gamma$.

## Explanations

### L1 human

A pseudogroup is a set of local symmetries that can be composed, inverted, shrunk and patched together.
</document>

<document source="https://tamnd.github.io/math2/node/K91VB1DXPR.md" tag="K91VB1DXPR" kind="definition">
# Banach analytic space

## Definition

A *model* is a triple $(U, F, f)$ where $E$ and $F$ are complex [[Banach spaces|06GZ956657]], $U \subseteq E$ is open and $f \colon U \to F$ is a [[holomorphic map|E8N3EC0YH5]].
Its underlying space is $Z = f^{-1}(0)$ with the topology induced from $U$.
For each complex Banach space $G$, the model defines a [[sheaf|TB6TM6R4MD]] $\mathcal{H}_Z(G)$ on $Z$.
Its stalk at $z \in Z$ is a quotient of the space of germs at $z$ of holomorphic maps $g \colon U \to G$.
Let $L(F, G)$ be the Banach space of continuous linear maps $F \to G$.
One divides by the germs $x \mapsto \varphi(x)\bigl(f(x)\bigr)$, where $\varphi$ is a germ at $z$ of a holomorphic map $U \to L(F, G)$.
These sheaves are the holomorphic maps from $Z$ to $G$.

A *Banach analytic space* is a [[Hausdorff space|0S1FJ8VEX9]] $X$ with the following data and property.
For each complex Banach space $G$ there is a sheaf $\mathcal{H}_X(G)$ of complex vector spaces on $X$, depending functorially on $G$.
Its sections play the role of holomorphic maps to $G$.
Each point of $X$ has an open neighbourhood isomorphic to the underlying space of a model, compatibly with the sheaves $\mathcal{H}(G)$ for all $G$.

## Remarks

Example: if $E = \C^n$ and $F = \C^k$, then a model is a local model of a [[complex analytic space|HXSZA44J5J]], and the sheaf $\mathcal{H}_Z(\C)$ is its structure sheaf.
In this case $\mathcal{H}_Z(G)$ is determined by $\mathcal{H}_Z(\C)$, so complex analytic spaces are Banach analytic spaces.
Also every open subset $U$ of a complex Banach space is a Banach analytic space, given by the model $(U, \set{0}, 0)$.

Non-example: the triple $(\C, \C, f)$ with $f(z) = \bar{z}$ is not a model, because $f$ is not holomorphic.

The definition uses all Banach spaces $G$, not only $G = \C$.
When $F$ is infinite-dimensional, the map $f$ is not given by finitely many scalar equations, and scalar functions alone do not record the structure that $f$ defines.
The notion is due to Douady.
</document>

<document source="https://tamnd.github.io/math2/node/K964YTWETD.md" tag="K964YTWETD" kind="definition">
# Cumulant

## Definition

Let $X$ be a real [[random variable|0PNRKMRC55]] such that $M(t) = E[e^{tX}]$ is finite for all $t$ in an open interval containing $0$.
Let $K(t) = \log M(t)$ be its [[cumulant generating function|0N2SP6K9ZC]] on that interval.
For $n \ge 1$, the *$n$-th cumulant* of $X$ is
$$\kappa_n(X) = K^{(n)}(0),$$
so that $K(t) = \sum_{n \ge 1} \kappa_n \frac{t^n}{n!}$ near $0$.

## Remarks

Writing $\mu = E[X]$ and $\mu_k = E[(X - \mu)^k]$, the first cumulants are $\kappa_1 = \mu$, $\kappa_2 = \mu_2 = \operatorname{Var}(X)$, $\kappa_3 = \mu_3$ and $\kappa_4 = \mu_4 - 3 \mu_2^2$.
If $X$ and $Y$ are independent, then $K_{X+Y} = K_X + K_Y$, so $\kappa_n(X + Y) = \kappa_n(X) + \kappa_n(Y)$.
Also $\kappa_n(cX) = c^n \kappa_n(X)$ for real $c$, and $\kappa_n(X + c) = \kappa_n(X)$ for $n \ge 2$.
Example: if $X$ is normal with mean $\mu$ and variance $\sigma^2$, then $K(t) = \mu t + \sigma^2 t^2 / 2$, so $\kappa_1 = \mu$, $\kappa_2 = \sigma^2$ and $\kappa_n = 0$ for $n \ge 3$.
Example: if $X$ is Poisson with mean $\lambda$, then $K(t) = \lambda(e^t - 1)$, so $\kappa_n = \lambda$ for all $n$.
Non-example: a Cauchy variable has $E[e^{tX}] = \infty$ for all $t \ne 0$, so the hypothesis fails and it has no cumulants.
When only $E[\abs{X}^n] < \infty$ is known, one defines $\kappa_1, \dots, \kappa_n$ instead from the expansion of $\log E[e^{itX}]$ at $0$, as $i^k \kappa_k$ for the coefficient of $t^k/k!$; the two definitions agree when both apply.
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<document source="https://tamnd.github.io/math2/node/K99YMX66XD.md" tag="K99YMX66XD" kind="definition">
# Fundamental class

## Definition

Let $M$ be a closed [[topological manifold|QQQ9B1DK4P]] of dimension $n$, that is, a [[compact|0TVJZYB6SH]] manifold without boundary, and let $x \mapsto \mu_x$ be an [[orientation|913WEQERBE]] of $M$.
A *fundamental class* of $M$ is a class $[M] \in H_n(M; \Z)$ in [[singular homology|M0Q5CFJ7ZH]] with the following property.
For every $x \in M$, the map $H_n(M) \to H_n(M, M \setminus \set{x})$ induced by inclusion sends $[M]$ to $\mu_x$.

## Remarks

For a closed oriented $n$-manifold a fundamental class exists and is unique.
If $M$ is also connected, then $H_n(M; \Z) \cong \Z$, generated by $[M]$.
With $\Z/2$ coefficients every closed manifold, orientable or not, has a unique fundamental class in $H_n(M; \Z/2)$.
The groups $H_n(M, M \setminus \set{x})$ are [[relative homology|4BVWH0BN28]] groups.

Example: for $M = S^1$ with its counterclockwise orientation, the loop $\sigma \colon [0, 1] \to S^1$, $\sigma(t) = e^{2\pi i t}$, is a singular $1$-cycle.
Its class generates $H_1(S^1) \cong \Z$ and restricts to the counterclockwise local generator at each point, so $[S^1] = [\sigma]$.

Non-example: the compactness hypothesis is needed, since $H_n(\R^n) = 0$ and so $\R^n$ has no fundamental class.
The orientation hypothesis is needed with $\Z$ coefficients, since $H_2(\R P^2; \Z) = 0$ for the [[real projective plane|1DDYB6VVSR]].
</document>

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# Conformal modulus

## Definition

Let $\Omega \subseteq \C$ be an [[open set|01ZQQDCERP]], and let $\Gamma$ be a family of locally [[rectifiable curves|0C28RT5BG7]] in $\Omega$.
A Borel measurable function $\rho \colon \Omega \to [0, \infty]$ is *admissible* for $\Gamma$ if $\int_\gamma \rho \, \abs{dz} \ge 1$ for every $\gamma \in \Gamma$.
The *conformal modulus* of $\Gamma$ is
$$M(\Gamma) = \inf_{\rho} \int_\Omega \rho(x + iy)^2 \, dx\, dy \in [0, \infty],$$
the infimum taken over all admissible $\rho$.

## Remarks

The modulus is the reciprocal of the [[extremal length|HX72KT9D0G]]: $M(\Gamma) = 1/\lambda(\Gamma)$, with $1/0 = \infty$ and $1/\infty = 0$.
It is invariant under conformal maps of $\Omega$.
Example: for an [[annulus|0AJ0YVK743]] $\set{r < \abs{z} < R}$ and the family $\Gamma$ of curves in it that join the two boundary circles, $\rho(z) = \frac{1}{\abs{z}\log(R/r)}$ is admissible, and $M(\Gamma) = \frac{2\pi}{\log(R/r)}$.
The check of the lower bound integrates the inequality $\int_r^R \rho(te^{i\theta})\, dt \ge 1$ over $\theta$ and applies the Cauchy–Schwarz inequality.
Non-example: $\rho = \frac{1}{2\abs{z}\log(R/r)}$ is not admissible, since along a radial segment its integral is $\frac{1}{2}$.
A quadrilateral is a Jordan domain $Q$ with four marked boundary points, which cut the boundary into four sides.
Its modulus is the modulus of the family of curves in $Q$ that join one pair of opposite sides.
Map $Q$ conformally onto a rectangle $(0, a) \times (0, b)$, with the marked points at the corners and the chosen sides vertical; then the modulus is $b/a$.
Conventions for ring domains differ: the modulus of $\set{r < \abs{z} < R}$ is often defined as $\frac{1}{2\pi}\log(R/r)$, which is the extremal length of the family above, or as $\log(R/r)$.
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<document source="https://tamnd.github.io/math2/node/K9DSR21DQC.md" tag="K9DSR21DQC" kind="concept">
# Hypoelliptic operator

## Summary

A hypoelliptic operator is a linear differential operator $P$ with smooth coefficients such that every distribution $u$ with $Pu$ smooth is itself smooth.
So $P$ cannot create singularities in its solutions that are not already in the right-hand side.
[[Elliptic operators|XY8GEQSWPR]] with smooth coefficients are hypoelliptic, and so is the heat operator, which is not elliptic.
Hörmander characterized hypoelliptic operators with constant coefficients and gave a wide class of hypoelliptic sums of squares of vector fields.

## Explanations

### L1 human

Some equations smooth things out: any solution of $\Delta u = 0$, even a very rough one at first sight, turns out to be infinitely differentiable.
Hypoelliptic operators are those with this smoothing property, whatever smooth right-hand side one puts in.
The heat operator $\partial_t - \Delta$ is one: temperature profiles are smooth even if the operator only involves one time derivative.
The wave operator is not: a sharp wave front $u(x, t) = H(x - t)$, with $H$ the jump function, solves the one-dimensional wave equation and keeps its jump forever.
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<document source="https://tamnd.github.io/math2/node/K9JAYYA0WS.md" tag="K9JAYYA0WS" kind="statement">
# Fractional Helly theorem

## Statement

Let $d \ge 1$ be an integer and let $\alpha \in (0, 1]$.
There is a number $\beta = \beta(d, \alpha) > 0$ with the following property.
Let $\mathcal{F}$ be a finite family of $n \ge d + 1$ [[convex sets|02AS45T3H9]] in $\R^d$.
Assume that at least $\alpha \binom{n}{d+1}$ of the subfamilies of $\mathcal{F}$ with $d + 1$ members have a common point.
Then some point of $\R^d$ lies in at least $\beta n$ members of $\mathcal{F}$.

## Remarks

The theorem is due to Katchalski and Liu.
One can take $\beta = 1 - (1 - \alpha)^{1/(d+1)}$, a value found by Kalai, and this value is the best possible.
A proof looks at the lexicographically smallest points of the nonempty intersections of $d$-element subfamilies and counts how often each such point is caught by the $(d+1)$-subfamilies.
For $\alpha = 1$ every $d + 1$ members meet, and [[Helly's theorem|F1W3QRBNGJ]] gives a point in all $n$ members.
The constant $\beta$ depends only on $d$ and $\alpha$, not on $n$ or on the sets.

## Explanations

### L1 human

Helly's theorem says: if every $d + 1$ convex sets of a family meet, then all of them meet.
The fractional version asks for less and gives less.
If only a fixed fraction of the $(d+1)$-tuples meet, then a fixed fraction of all the sets still share a point.
So "many local meetings" force "one big meeting".

### L2 human

The fractional Helly theorem is the key step in the proof of the $(p, q)$-theorem of Alon and Kleitman.
It is a basic tool in the theory of geometric transversals.
Together with [[Helly's theorem|F1W3QRBNGJ]] and the [[centerpoint theorem|BWEGTCW5RJ]], it belongs to the core of combinatorial convexity.
</document>

<document source="https://tamnd.github.io/math2/node/K9SBQZMWB9.md" tag="K9SBQZMWB9" kind="statement">
# Pick's theorem

## Statement

Let $P$ be a simple [[polygon|8VEFWD3EJB]] in the plane whose vertices all have integer coordinates.
Let $i$ be the number of points of $\Z^2$ in the interior of $P$ and $b$ the number of points of $\Z^2$ on its boundary.
Then the area $A$ of the region bounded by $P$ is
$$A = i + \frac{b}{2} - 1.$$

## Remarks

Both sides are additive when two lattice polygons are glued along a common edge, once the shared boundary points are counted correctly.
So it suffices to check the formula for lattice triangles, and then for triangles with no lattice points other than their vertices, which have area $1/2$.
Example: the triangle with vertices $(0, 0), (2, 0), (0, 2)$ has area $2$, $b = 6$ and $i = 0$, and $0 + 3 - 1 = 2$.
Simplicity matters: for a region with holes the formula changes by a term for each hole.
There is no direct analogue in dimension $3$: the tetrahedra with vertices $(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, r)$ have no other lattice points but volume $r/6$.
The theorem is due to Georg Pick.

## Explanations

### L1 human

Draw a polygon on squared paper with its corners on grid points.
To find its area, count the grid points strictly inside and the grid points on the edges.
The area is the inside count plus half the edge count, minus one.

### L2 human

Pick's theorem links a continuous quantity, area, to a count of lattice points.
It is the two-dimensional seed of Ehrhart theory, which counts lattice points in dilates of lattice polytopes.
It also gives a quick proof that a lattice triangle with no other lattice points has area $1/2$, which is used in the study of Farey sequences.
</document>

<document source="https://tamnd.github.io/math2/node/K9VZKJACKT.md" tag="K9VZKJACKT" kind="definition">
# Combinatorial species

## Definition

Let $\mathbb{B}$ be the [[category|A2DE66937D]] whose objects are the [[finite sets|0BBTH3YSZA]] and whose morphisms are the bijections.
A *combinatorial species* is a [[functor|28NQ0KRHD3]] $F \colon \mathbb{B} \to \mathbb{B}$.
Thus $F$ assigns:

1. to each finite set $U$ a finite set $F[U]$, whose elements are the *$F$-structures* on $U$;
2. to each bijection $\sigma \colon U \to V$ a bijection $F[\sigma] \colon F[U] \to F[V]$, the *transport* along $\sigma$;
3. in such a way that $F[\id_U] = \id_{F[U]}$ and $F[\tau \circ \sigma] = F[\tau] \circ F[\sigma]$ for bijections $\sigma \colon U \to V$ and $\tau \colon V \to W$.

## Remarks

The *exponential generating function* of $F$ is $F(x) = \sum_{n \ge 0} \abs{F[\set{1, \ldots, n}]}\, \frac{x^n}{n!}$.

Example: the species $L$ of linear orders assigns to $U$ the set of total orders on $U$.
A bijection $\sigma$ transports an order $u_1 < \cdots < u_n$ to $\sigma(u_1) < \cdots < \sigma(u_n)$, and the functor rules hold.
Since $\abs{L[\set{1, \ldots, n}]} = n!$, its generating function is $\frac{1}{1 - x}$.
The species $E$ of sets, with $E[U] = \set{U}$, has generating function $e^x$.

Non-example: let $F[U] = \set{U}$ if $U \subseteq \N$ and $F[U] = \emptyset$ otherwise.
For a bijection $\sigma$ from $\set{1}$ to $\set{a}$ with $a \notin \N$, clause 2 needs a bijection from a one-element set to the empty set, which does not exist.
So $F$ is not a species.

Sum, product, composition and derivative of species correspond to the same operations on exponential generating functions.
Some authors let $F$ take values in the category of finite sets and all functions; this gives the same notion, since a functor sends bijections to bijections.
</document>

<document source="https://tamnd.github.io/math2/node/K9XQKT69WF.md" tag="K9XQKT69WF" kind="definition">
# Heap

## Definition

A *heap* is a [[set|0SJ3RX9TXA]] $H$ with a [[ternary operation|WC47MFR6GW]] $[\cdot, \cdot, \cdot] \colon H \times H \times H \to H$ such that, for all $a, b, c, d, e \in H$:

1. $[\,[a, b, c], d, e] = [a, b, [c, d, e]\,] = [a, [d, c, b], e]$ (para-associativity);
2. $[a, a, b] = b$ and $[b, a, a] = b$ (identity law).

## Remarks

Example: a [[group|G5S1M9]] $G$ with $[x, y, z] = x y^{-1} z$.
For clause 1, all three expressions equal $a b^{-1} c d^{-1} e$; the third one because $(d c^{-1} b)^{-1} = b^{-1} c d^{-1}$.
For clause 2, $a a^{-1} b = b$ and $b a^{-1} a = b$.

Non-example: $\Z$ with $[x, y, z] = x + y + z$.
Clause 1 holds, since all three expressions equal $a + b + c + d + e$.
Clause 2 fails: $[0, 1, 1] = 2 \ne 0$.

Conversely, let $H$ be a nonempty heap and fix $e \in H$.
Then $x \cdot y = [x, e, y]$ makes $H$ a group with identity $e$ and inverse $x^{-1} = [e, x, e]$, and the heap operation is recovered as $[x, y, z] = x y^{-1} z$.
Different choices of $e$ give isomorphic groups.
Some authors allow the empty heap, which corresponds to no group.
</document>

<document source="https://tamnd.github.io/math2/node/KA0YDB6QXE.md" tag="KA0YDB6QXE" kind="concept">
# Profinite group

## Summary

A profinite group is a [[topological group|92700QAB5Y]] that is an [[inverse limit|DZB378CNCK]] of finite groups.
Equivalently, it is a compact, Hausdorff and totally disconnected topological group.
Profinite groups record a whole compatible family of finite quotients at once.
The main examples are the [[p-adic integers|V6TJAFZ4MD]], the profinite completion $\hat\Z$ of $\Z$, and Galois groups of infinite Galois extensions with the Krull topology.

## Explanations

### L1 human

The integers modulo $p$, modulo $p^2$, modulo $p^3$, and so on, form a tower of finite groups, each mapping onto the previous one by reduction.
A compatible choice of one residue at each level is a $p$-adic integer, and these choices form a group.
This group is profinite: it is built entirely from finite pieces, and it carries a topology in which two elements are close when they agree at many levels.
In the same way, the Galois group of all algebraic numbers over $\Q$ is assembled from the finite Galois groups of number fields.
</document>

<document source="https://tamnd.github.io/math2/node/KA3V66S6F5.md" tag="KA3V66S6F5" kind="definition">
# Palais–Smale compactness condition

## Definition

Let $X$ be a real [[Banach space|06GZ956657]] with [[continuous dual space|4A82S5FVRT]] $X^*$.
Let $I \colon X \to \R$ be continuously [[Fréchet differentiable|06EH97V2A7]], with derivative $I'(u) \in X^*$ at $u$.
A *Palais–Smale sequence* for $I$ is a sequence $(u_n)$ in $X$ such that $(I(u_n))$ is bounded and $\norm{I'(u_n)}_{X^*} \to 0$.
The functional $I$ satisfies the *Palais–Smale condition* if every Palais–Smale sequence for $I$ has a convergent subsequence.

## Remarks

Example: on a real Hilbert space $H$, let $I(u) = \tfrac12 \norm{u}^2$.
Under the identification of $H^*$ with $H$, the derivative is $I'(u) = u$, so a Palais–Smale sequence satisfies $u_n \to 0$, and the condition holds.
Non-example: $I(x) = e^x$ on $\R$.
The sequence $x_n = -n$ has $0 < I(x_n) \le 1$ and $I'(x_n) = e^{-n} \to 0$, but no subsequence converges.
A common variant at a level $c \in \R$, written $(PS)_c$, asks only for sequences with $I(u_n) \to c$ and $I'(u_n) \to 0$.
If $I$ satisfies the condition, then the set of critical points at any level $c$ is compact.
</document>

<document source="https://tamnd.github.io/math2/node/KA4MW1FRZ4.md" tag="KA4MW1FRZ4" kind="example">
# Random walk as a difference equation

## Example

Let $\xi_1, \xi_2, \dots$ be [[independent and identically distributed|00MRN0FARV]] random variables with $P(\xi_n = 1) = P(\xi_n = -1) = \tfrac{1}{2}$.
The simple symmetric [[random walk|04VZE3FZXF]] on $\Z$ is the solution of the stochastic difference equation
$$S_{n+1} = S_n + \xi_{n+1}, \qquad S_0 = 0.$$
It has the form $X_{n+1} = f(X_n, \xi_{n+1})$ with $f(x, e) = x + e$, so it is a [[stochastic difference equation|WPYG78BSD5]].
Solving the recursion gives $S_n = \xi_1 + \cdots + \xi_n$.
Taking expectations in the equation gives the deterministic difference equation $E[S_{n+1}] = E[S_n]$, so $E[S_n] = 0$.
Squaring gives $S_{n+1}^2 = S_n^2 + 2 S_n \xi_{n+1} + 1$, and since $\xi_{n+1}$ is independent of $S_n$ with mean $0$, $E[S_{n+1}^2] = E[S_n^2] + 1$, so $E[S_n^2] = n$.

Probabilities of the walk also solve difference equations.
Fix an integer $N \ge 1$ and let $p_j$ be the probability that the walk started at $j \in \set{0, \dots, N}$ reaches $N$ before $0$.
Conditioning on the first step gives $p_j = \tfrac{1}{2} p_{j+1} + \tfrac{1}{2} p_{j-1}$ for $0 < j < N$, with $p_0 = 0$ and $p_N = 1$.
This linear recurrence has characteristic polynomial $(z - 1)^2$, so $p_j = a + bj$, and the boundary values give $p_j = j/N$.

## Explanations

### L1 human

Flip a fair coin at each step and move one unit up for heads and one unit down for tails.
Your position follows the rule: new position equals old position plus a random step.
That rule is a difference equation with a random term.
It tells you right away that you stay at $0$ on average while your typical distance grows like $\sqrt{n}$.
It also gives the gambler's ruin rule: starting with $j$ euros and playing fair one-euro bets, you reach $N$ euros before going broke with probability $j/N$.
</document>

<document source="https://tamnd.github.io/math2/node/KA5890R9YM.md" tag="KA5890R9YM" kind="concept">
# Birational map

## Summary

A birational map between two varieties is a rational map that has a rational inverse.
It is an isomorphism between nonempty open subsets, so birational varieties look the same away from lower-dimensional subsets.
Two varieties are birational exactly when their function fields are isomorphic over the base field.
The classification of varieties up to birational equivalence is the goal of birational geometry.

## Explanations

### L1 human

Two shapes are birational if, after removing some small pieces from each, what is left is the same shape.
A smooth conic in the plane, such as $x^2 + y^2 = 1$, is birational to the line.
Each line through the point $(-1, 0)$ with slope $t$ meets the circle in one more point, and this matches points of the circle with numbers $t$.
Only the point $(-1, 0)$ itself is left out, since it would need the vertical line through it, which has no slope.
So the circle and the line are birational.
The plane and the projective plane are another birational pair: they agree once the line at infinity is removed, yet they are not isomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/KA95J9V6AS.md" tag="KA95J9V6AS" kind="concept">
# Stochastic Volterra equation

## Summary

A stochastic Volterra equation is a [[Volterra integral equation|DND8CBEVXT]] driven by noise.
It has the form $X(t) = X_0(t) + \int_0^t a(t, s, X(s)) \, ds + \int_0^t b(t, s, X(s)) \, dW(s)$, where the last integral is an Itô integral against a [[Wiener process|22735FGJV6]].
When the coefficients $a$ and $b$ do not depend on $t$ and $X_0$ is constant, it reduces to a [[stochastic differential equation|B219G9SHJA]].
The dependence on $t$ lets the noise and the drift have memory.
Such memory is used in models of rough volatility in finance and of viscoelastic materials under random forcing.

## Explanations

### L1 human

A stochastic differential equation moves a quantity by small random kicks whose effect lasts forever and unchanged.
In a stochastic Volterra equation the effect of a kick received at time $s$ may change with the elapsed time.
For example, it may fade away.
So the system remembers its past noise in a weighted way.
The process $\int_0^t (t - s)^{H - 1/2} \, dW(s)$ with $H > 0$ is a simple example: recent kicks are weighted differently from old ones.
</document>

<document source="https://tamnd.github.io/math2/node/KAAJ0JJ2GF.md" tag="KAAJ0JJ2GF" kind="statement">
# Pfaff and Euler transformations

## Statement

Let $a, b, c \in \C$ with $c \notin \set{0, -1, -2, \ldots}$, and let ${}_2F_1$ be the [[hypergeometric function|463X5V31D9]] on $\C \setminus [1, \infty)$, with principal powers.
Then for all $z \in \C \setminus [1, \infty)$:

1. (Pfaff transformations)
$${}_2F_1(a, b; c; z) = (1 - z)^{-a} \, {}_2F_1\left(a, c - b; c; \frac{z}{z - 1}\right) = (1 - z)^{-b} \, {}_2F_1\left(c - a, b; c; \frac{z}{z - 1}\right).$$
2. (Euler transformation)
$${}_2F_1(a, b; c; z) = (1 - z)^{c - a - b} \, {}_2F_1(c - a, c - b; c; z).$$

## Remarks

The map $z \mapsto \frac{z}{z - 1}$ sends $\C \setminus [1, \infty)$ onto itself, so the right sides make sense.
For $\operatorname{Re} c > \operatorname{Re} b > 0$, the first Pfaff formula follows from [[Euler's integral representation|812THX456R]] by the substitution $t \mapsto 1 - t$, and the general case by analytic continuation in the parameters.
The second Pfaff formula is the first with $a$ and $b$ exchanged, and applying the two Pfaff formulas one after the other gives the Euler transformation.

Example: for $b = c$, the Euler transformation reads $(1 - z)^{-a} = (1 - z)^{-a} \, {}_2F_1(c - a, 0; c; z)$, which is true since ${}_2F_1(\cdot, 0; c; z) = 1$.

The Pfaff transformation turns the series at $z$ into a series at $\frac{z}{z - 1}$, which converges for $\operatorname{Re} z < \frac12$; this gives the continuation of ${}_2F_1$ to that half-plane.

## Explanations

### L1 human

The same hypergeometric function can be written in several different ways, with changed parameters, a changed variable, and a power of $1 - z$ in front.
These formulas let one pass from a series that converges slowly to one that converges fast, or to a region where the original series does not converge at all.

### L2 human

Together with the identity, the Pfaff and Euler transformations form a group of four expressions for the same solution of the hypergeometric equation.
Combined with the solutions at the other singular points, they give [[Kummer's 24 solutions|7QFW771QMA]].
The Euler transformation also explains the factor $(1 - z)^{c - a - b}$ that governs the behaviour of ${}_2F_1$ near $z = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/KAEDCQ8RM6.md" tag="KAEDCQ8RM6" kind="concept">
# Ordinal arithmetic

## Summary

Ordinal arithmetic is the addition, multiplication and exponentiation of [[ordinal numbers|0YSCSX5YZC]].
The operations extend those of the natural numbers to transfinite ordinals, and they are [[defined|2C7MBG93AZ]] by transfinite recursion or by order types of well-ordered sets.
They are not commutative, and they are the language of ordinal notations, such as the [[Cantor normal form|6GD30FGYDZ]].

## Explanations

### L1 human

Think of an ordinal as a queue of positions.
Adding $\alpha + \beta$ means placing the queue $\beta$ after the queue $\alpha$.
Multiplying $\alpha \cdot \beta$ means taking $\beta$ copies of $\alpha$, one after the other.
Order matters: $1 + \omega$ puts one extra point before an infinite queue, which gives again a queue of type $\omega$.
But $\omega + 1$ puts a point after all the natural numbers, which is a longer queue.
So $1 + \omega = \omega$ while $\omega + 1 > \omega$.
</document>

<document source="https://tamnd.github.io/math2/node/KAN07YC9T2.md" tag="KAN07YC9T2" kind="concept">
# Additive functor

## Summary

An additive functor is a [[functor|28NQ0KRHD3]] between [[preadditive categories|SC6FD5F4J0]] that respects the addition of morphisms.
Between additive categories, these are exactly the functors that preserve finite biproducts.
Hom functors, tensor products with a fixed module and derived functors are additive.
Additivity is the basic compatibility needed to apply a functor to chain complexes and exact sequences.

## Explanations

### L1 human

In a preadditive category one can add arrows between the same objects.
An additive functor sends a sum of arrows to the sum of their images, just as a linear map sends a sum of vectors to the sum of their images.
For example, tensoring with a fixed abelian group $M$ sends a homomorphism $f$ to $f \otimes \id_M$, and $(f + g) \otimes \id_M = f \otimes \id_M + g \otimes \id_M$.
Additive functors take direct sums to direct sums, so they can be computed piece by piece.
</document>

<document source="https://tamnd.github.io/math2/node/KAT7K2NE9Z.md" tag="KAT7K2NE9Z" kind="concept">
# Computational geometry

## Summary

Computational geometry is the study of algorithms and data structures for geometric objects such as points, segments, polygons and polyhedra.
Typical problems are computing a [[convex hull|0FQSDC2EQ2]], a [[polygon triangulation|Q6KKBP3RRP]], a [[Voronoi diagram|QDNEBBVZ84]] or a [[Delaunay triangulation|259Y10BYJC]], and finding intersections among many segments.
Its results are measured by running time and memory in terms of the input size, and it relies on combinatorial bounds from [[discrete geometry|Q0RW86P85F]].
Applications include computer graphics, geographic information systems, robotics and computer-aided design.

## Explanations

### L1 human

Computational geometry asks how a computer can answer geometric questions quickly.
Given a million points on a map, which is the nearest post office to each house, and which points form the outer boundary of the set?
A naive method may compare every pair of points, which takes about $n^2$ steps for $n$ points.
Clever methods often need only about $n \log n$ steps, which is a huge saving for large $n$.
</document>

<document source="https://tamnd.github.io/math2/node/KB11T6389G.md" tag="KB11T6389G" kind="statement">
# Nagata–Smirnov metrization theorem

## Statement

Let $X$ be a [[topological space|0CQHFPGJ45]].
A family of subsets of $X$ is *$\sigma$-locally finite* if it is a union of countably many [[locally finite families|DZ297MMGV8]].
Then $X$ is [[metrizable|091GJ3FYD7]] if and only if $X$ is [[regular|03HMJASQBQ]], [[Hausdorff|0S1FJ8VEX9]] and has a $\sigma$-locally finite base.

## Remarks

For a metric space, [[Stone's theorem|7QFKD5JXH6]] gives, for each $n$, a locally finite open refinement $\mathcal{B}_n$ of the cover by balls of radius $1/n$.
The union of the $\mathcal{B}_n$ is a $\sigma$-locally finite base.
For the converse one shows that $X$ is normal and builds countably many locally finite families of continuous functions to $[0,1]$.
These functions embed $X$ into a metrizable product space.
A countable base is $\sigma$-locally finite, since each one-element family is locally finite.
So the [[Urysohn metrization theorem|HBFHDH4EJX]] is a special case.

## Explanations

### L1 human

The theorem gives an exact test for when a topology comes from a distance.
Countably many basic open sets are no longer needed.
It is enough that the basic open sets come in countably many layers, and that near each point only finitely many sets of each layer are present.

### L2 human

Together with the [[Bing metrization theorem|RGCZ2P9QRM]], this is the general solution of the metrization problem.
Its proof uses [[Urysohn's lemma|01EY3VR7ZD]] to build the continuous functions of the embedding.
A useful consequence is that a paracompact Hausdorff space that is locally metrizable is metrizable.
</document>

<document source="https://tamnd.github.io/math2/node/KB76923A1M.md" tag="KB76923A1M" kind="concept">
# Sheaf cohomology in topology

## Summary

Sheaf cohomology assigns groups $H^q(X, F)$ to a topological space $X$ and a [[sheaf|TB6TM6R4MD]] $F$ of abelian groups on $X$.
They are the right [[derived functors|NPMC3K6J2F]] of the global sections functor.
It measures the failure of local sections to glue into global sections.
With a constant sheaf it recovers ordinary cohomology on reasonable spaces, and with other sheaves it gives local coefficients, Borel–Moore theory and Poincaré–Verdier duality.
The general notion is [[sheaf cohomology|PQ6Q143ARJ]] in algebraic geometry and topology.

## Explanations

### L1 human

A sheaf is a rule that tells you what the allowed local data are on each open piece of a space, such as continuous functions or locally constant functions.
Often local data agree on overlaps but still do not come from one global object.
Sheaf cohomology counts such obstructions.
For example, on the unit circle in $\C$, a continuous square root of $z$ can be chosen on each small arc.
But these choices cannot be made to fit together around the whole circle.
This obstruction is a nonzero class in $H^1$ of the circle with coefficients in $\Z/2$.
</document>

<document source="https://tamnd.github.io/math2/node/KB792QTG4F.md" tag="KB792QTG4F" kind="statement">
# Existence and uniqueness of Haar measure

## Statement

Let $G$ be a [[locally compact|AHV05ZBCYW]] Hausdorff [[topological group|92700QAB5Y]].

1. There is a nonzero [[Radon measure|HZNEQQBCW4]] $\mu$ on the Borel sets of $G$ that is left-invariant: $\mu(gA) = \mu(A)$ for every $g \in G$ and every Borel set $A \subseteq G$.
2. If $\nu$ is another nonzero left-invariant Radon measure on $G$, then $\nu = c \mu$ for some constant $c > 0$.

## Remarks

Such a measure is a left [[Haar measure|VY0HXD3GC4]] of $G$.
One proof compares the size of a compact set $K$ with the number of translates of a small open set needed to cover it.
These ratios have a limit point, by a compactness argument, which gives an invariant content and then a Radon measure.
Another proof builds an invariant positive linear functional on compactly supported continuous functions and applies the [[Riesz–Markov–Kakutani representation theorem|DWFPDF8XTB]].
Uniqueness follows from a Fubini argument applied to $\mu \otimes \nu$ and the map $(x, y) \mapsto (x, xy)$.

Example: on $G = (\R^n, +)$, Lebesgue measure is a Haar measure, so every translation-invariant Radon measure on $\R^n$ is a multiple of Lebesgue measure.
On the circle group, the Haar measures are the multiples of arc length.
On a discrete group, the Haar measures are the multiples of counting measure.

Local compactness enters the proof through compact sets with nonempty interior, which give the comparison of sizes.

## Explanations

### L1 human

Every locally compact group has a natural notion of volume that does not change when sets are moved by the group.
It is unique up to the choice of a unit.
For the real line it is ordinary length, and for a finite group it is counting.

### L2 human

Haar measure lets one integrate over a group, which is the base of harmonic analysis on groups.
It gives the group algebra $L^1(G)$, the Peter–Weyl theorem for compact groups and Pontryagin duality for abelian groups.
For a compact group the measure is finite and is usually normalized to total mass $1$.
Left and right Haar measures may differ, and their ratio is the modular function of $G$.
</document>

<document source="https://tamnd.github.io/math2/node/KBN13NPYJQ.md" tag="KBN13NPYJQ" kind="concept">
# Completion of a ring

## Summary

The completion of a [[commutative ring|0310REJJY8]] $R$ with respect to an [[ideal|096Z9MP2ME]] $I$ is the ring $\hat R = \varprojlim R/I^n$.
Its elements are compatible sequences of residues modulo the powers of $I$.
It is the completion of $R$ for the [[$I$-adic topology|2VF2TSJ3PY]], in the same way as the real numbers complete the rationals.
The $p$-adic integers and the formal power series rings arise in this way.
Completion is a basic tool to study a local ring near its closed point.

## Explanations

### L1 human

Knowing an integer modulo $2$, modulo $4$, modulo $8$, and so on, determines it.
A compatible list of such residues need not come from an integer, and the completion adds all such lists as new elements.
For $\Z$ and the prime $p$ one gets the [[$p$-adic integers|V6TJAFZ4MD]].
For $k[x]$ and the ideal $(x)$ one gets the [[formal power series|033WGN00BF]] $k[\![x]\!]$, where an element is known by its truncations modulo every power of $x$.
The [[definition|NS84D2920R]] makes this precise.
</document>

<document source="https://tamnd.github.io/math2/node/KBPJH3J56K.md" tag="KBPJH3J56K" kind="concept">
# Hilbert C*-module

## Summary

A Hilbert C*-module over a C*-algebra $A$ is a module over $A$ with an inner product.
The values of this inner product lie in $A$ instead of in $\C$.
It generalizes a Hilbert space, which is the case $A = \C$.
Hilbert modules are the basic tool of Kasparov's KK-theory and of Morita equivalence of C*-algebras.

## Explanations

### L1 human

In a Hilbert space the inner product of two vectors is a complex number.
In a Hilbert C*-module the "numbers" are replaced by elements of a C*-algebra, for example by continuous functions on a space.
A typical picture is a continuous family of Hilbert spaces indexed by points of a space, with a pointwise inner product.
The simplest example is the algebra $A$ itself, with $\ip{a}{b} = a^* b$.
</document>

<document source="https://tamnd.github.io/math2/node/KBSHB2N0VG.md" tag="KBSHB2N0VG" kind="definition">
# Progenerator

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity and let $P$ be a left $R$-[[module|0NZ8JA25WK]].
The module $P$ is a *generator* if every left $R$-module is a homomorphic image of a direct sum of copies of $P$.
The module $P$ is a *progenerator* if all three of these hold:

1. $P$ is [[finitely generated|05QY1VEBYD]].
2. $P$ is [[projective|BFFA03GE3V]].
3. $P$ is a generator.

## Remarks

A module $P$ is a generator if and only if $R$ is a direct summand of $P^n$ for some $n \ge 1$.
Example: $P = R^n$ with $n \ge 1$ is free of finite rank, hence finitely generated and projective, and $R$ is a direct summand of it, so $P$ is a progenerator.
Non-example: let $k$ be a field, $R = k \times k$ and $P = k \times 0 = R(1, 0)$.
It is finitely generated and projective, as a direct summand of $R$.
It is not a generator: every map $P \to 0 \times k$ is zero, since it sends $(1, 0) = (1, 0)^2$ to $(1, 0) \cdot m = 0$; so $0 \times k$ is not a quotient of a sum of copies of $P$.
Over a commutative ring $R$, a finitely generated projective module is a generator if and only if it is faithful, that is, its rank is positive at every prime ideal.
The ring $\End_R(P)$ of a progenerator $P$ is [[Morita equivalent|RD5WKA1VK7]] to $R$.
</document>

<document source="https://tamnd.github.io/math2/node/KBX8RBZ626.md" tag="KBX8RBZ626" kind="statement">
# Isoperimetric inequality

## Statement

Let $n \ge 2$, and let $\omega_n$ be the volume of the unit ball of $\R^n$.
Let $\Omega \subseteq \R^n$ be a bounded open set with $C^1$ boundary.
Let $\abs{\Omega}$ be its [[Lebesgue measure|2Z2QVWZ262]] and let $\abs{\partial \Omega}$ be the [[surface area|0S29Y4ZW0Y]] of its boundary.
Then
$$\abs{\partial \Omega}^n \ge n^n \omega_n \abs{\Omega}^{n-1},$$
and equality holds if and only if $\Omega$ is a ball.

## Remarks

The inequality follows from the [[Brunn–Minkowski theorem|9B02RPN7XD]].
Let $B$ be the closed unit ball and $\bar\Omega$ the closure of $\Omega$.
Then $\abs{\bar\Omega + \varepsilon B} \ge (\abs{\Omega}^{1/n} + \varepsilon \omega_n^{1/n})^n$.
For a $C^1$ boundary, $\abs{\partial \Omega}$ is the limit of $(\abs{\bar\Omega + \varepsilon B} - \abs{\Omega}) / \varepsilon$ as $\varepsilon \to 0$, and this limit is at least $n \omega_n^{1/n} \abs{\Omega}^{(n-1)/n}$.
Raising to the power $n$ gives the inequality; the equality case needs a separate argument.
For $n = 2$ it reads $L^2 \ge 4 \pi A$ for a region of area $A$ bounded by curves of total length $L$.
Example: a square of side $1$ has $L = 4$ and $A = 1$, and indeed $16 \ge 4\pi$.

## Explanations

### L1 human

Among all shapes with a given volume, the ball has the smallest surface.
Equivalently, among all shapes with a given surface area, the ball holds the most volume.
This is why soap bubbles are round: surface tension makes the surface as small as possible.

### L2 human

The inequality has a version for sets of finite perimeter, which is the natural setting in geometric measure theory.
It is equivalent to the sharp Sobolev inequality $\norm{u}_{n/(n-1)} \le C_n \norm{\nabla u}_1$ for smooth compactly supported $u$, through the coarea formula.
In the plane it is often proved with Fourier series or by Steiner symmetrisation.
</document>

<document source="https://tamnd.github.io/math2/node/KBXSZC6BV1.md" tag="KBXSZC6BV1" kind="concept">
# Homomorphism

## Summary

A homomorphism is a map between two [[structures|983GZBQYT4]] of the same [[signature|WDVVEYD6AT]] that preserves the operations, the constants and the relations.
[[Group homomorphisms|0JYTG5K9Y4]], [[ring homomorphisms|0VTH8KPCG2]] and [[linear maps|0NRYHJPQG2]] are all homomorphisms for suitable signatures.
Homomorphisms are the maps used to compare structures: [[isomorphisms|PZMGQMDN25]], [[embeddings|SCRB74SWQN]] and [[quotient maps|N5KE7FH6CX]] are special homomorphisms.

## Explanations

### L1 human

A homomorphism translates one structure into another without breaking its rules.
If $h$ is a homomorphism for addition, then computing $a + b$ and then applying $h$ gives the same as applying $h$ to $a$ and $b$ and then adding.
[[Reduction of integers modulo a fixed number|BV08DGBD7C]] is the standard example: the remainder of a sum is determined by the remainders of the summands.
The map $x \mapsto x + 1$ on the integers is [[not a homomorphism|3A3JJPJHQR]] for addition, since it does not send $0$ to $0$.
</document>

<document source="https://tamnd.github.io/math2/node/KC22CWGGAE.md" tag="KC22CWGGAE" kind="statement">
# Cartan–Brauer–Hua theorem

## Statement

Let $D$ be a [[division ring|0WZXK05ME7]], and let $K \subseteq D$ be a division [[subring|0859D58WPD]] of $D$.
Assume that $x K x^{-1} \subseteq K$ for every nonzero $x \in D$.
Then $K = D$ or $K$ is contained in the [[center|MVT64B3B6H]] $Z(D) = \set{z \in D : zd = dz \text{ for all } d \in D}$.

## Remarks

A short proof works with two conjugates.
Take $a \in K$ and $b \in D \setminus K$; then $b \ne 0$ and $1 + b \ne 0$, so $k_1 = b a b^{-1}$ and $k_2 = (1+b) a (1+b)^{-1}$ lie in $K$.
Expanding $(1+b)a = k_2(1+b)$ and using $ba = k_1 b$ gives $(k_1 - k_2) b = k_2 - a$, so $k_1 = k_2$ (otherwise $b \in K$), hence $a = k_2 = k_1$ and $ab = ba$.
So every element of $K$ commutes with $D \setminus K$; if $K \ne D$, every element of $K$ is a difference of two elements of $D \setminus K$, so it also commutes with $K$, and $K \subseteq Z(D)$.
Example: in the real quaternions $\mathbb{H}$, the subfield $\C = \R + \R i$ is neither $\mathbb{H}$ nor central, and indeed $(1+j)\, i \,(1+j)^{-1} = -k \notin \C$.

## Explanations

### L1 human

Conjugating by $x$ means replacing $y$ by $x y x^{-1}$.
The theorem says that a division subring which is stable under all conjugations is either everything or sits inside the center.
So a skew field has no "normal" sub-skew-fields other than the trivial ones.

### L2 human

The theorem is a division-ring analogue of the fact that a normal subgroup is a union of conjugacy classes.
It is often applied with $K$ the subring generated by a conjugation-invariant set, such as the center together with some commutators.
Its proof is elementary, in contrast with deeper results such as [[Wedderburn's little theorem|0RRNVX1VKA]], and it has a companion in [[Hua's identity|AWS7XTZ49C]], another elementary tool for division rings.
</document>

<document source="https://tamnd.github.io/math2/node/KC7BVZZZ2W.md" tag="KC7BVZZZ2W" kind="definition">
# Postnikov tower

## Definition

Let $X$ be a [[path-connected space|0WB6DH499T]] with base point, and let $\pi_k$ denote the $k$-th [[homotopy group|P976WYEFMB]].
A *Postnikov tower* of $X$ is a sequence of pointed path-connected spaces $X_n$, $n \ge 1$, with pointed maps $f_n \colon X \to X_n$ and $p_{n} \colon X_{n} \to X_{n-1}$ for $n \ge 2$, such that:

1. $p_n \circ f_n = f_{n-1}$ for all $n \ge 2$.
2. $f_n$ induces an isomorphism $\pi_k(X) \to \pi_k(X_n)$ for all $k \le n$.
3. $\pi_k(X_n) = 0$ for all $k > n$.

## Remarks

Every path-connected CW complex has a Postnikov tower, built by attaching cells of dimension at least $n + 2$ to $X$ to kill $\pi_k$ for $k > n$.
Up to homotopy, $p_n$ can be taken to be a fibration, and then its fibre is an [[Eilenberg–MacLane space|AJHEYJE5XY]] $K(\pi_n(X), n)$.
The map from $X$ to the homotopy limit of the tower is a weak homotopy equivalence.

Example: let $X = K(G, m)$.
Take $X_n$ a point for $n < m$ and $X_n = X$ for $n \ge m$, with the evident maps.
Clause 2 holds since $\pi_k(X) = 0$ for $k < m$, and clause 3 holds since $\pi_k(X) = 0$ for $k > m$.

Non-example: for $X = S^2$, the choice $X_2 = S^2$ with $f_2 = \id$ fails clause 3, since $\pi_3(S^2) \cong \Z \ne 0$.
A valid choice is $X_2 = \C P^\infty$, a $K(\Z, 2)$.
</document>

<document source="https://tamnd.github.io/math2/node/KCBFS516DE.md" tag="KCBFS516DE" kind="definition">
# Multivariate formal power series

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $n \ge 1$.
For $\alpha = (\alpha_1, \dots, \alpha_n) \in \N^n$ write $X^\alpha = X_1^{\alpha_1} \cdots X_n^{\alpha_n}$.
A *formal power series* in $X_1, \dots, X_n$ over $R$ is a family $(a_\alpha)_{\alpha \in \N^n}$ of elements of $R$, written $\sum_{\alpha} a_\alpha X^\alpha$.
The set $R[\![X_1, \dots, X_n]\!]$ of these series is a commutative ring with
$$\sum_\alpha a_\alpha X^\alpha + \sum_\alpha b_\alpha X^\alpha = \sum_\alpha (a_\alpha + b_\alpha) X^\alpha, \qquad \Big(\sum_\alpha a_\alpha X^\alpha\Big)\Big(\sum_\alpha b_\alpha X^\alpha\Big) = \sum_\alpha \Big(\sum_{\beta + \gamma = \alpha} a_\beta b_\gamma\Big) X^\alpha.$$

## Remarks

The inner sum in the product is finite, since each $\alpha$ has finitely many decompositions $\alpha = \beta + \gamma$ in $\N^n$.
For $n = 1$ this is the ring of [[formal power series|033WGN00BF]] in one variable, and $R[\![X_1, \dots, X_n]\!] \cong R[\![X_1, \dots, X_{n-1}]\!][\![X_n]\!]$.
Example: in $\Z[\![X, Y]\!]$ the series $f = \sum_{k \ge 0} (X + Y)^k$ satisfies $(1 - X - Y) f = 1$, because the coefficient of each monomial of degree $k \ge 1$ cancels between $(X + Y)^k$ and $(X + Y)(X + Y)^{k-1}$.
Non-example: $\sum_{k \ge 0} X^{-k} Y^k$ is not a formal power series in $X, Y$, since the exponents must lie in $\N^n$.
A series is a [[unit|0XVEZDE9J8]] if and only if its constant coefficient $a_{(0, \dots, 0)}$ is a unit of $R$.
</document>

<document source="https://tamnd.github.io/math2/node/KCCFZTP3MM.md" tag="KCCFZTP3MM" kind="statement">
# Van Vleck's theorem

## Statement

Let $0 < \varepsilon < \pi/2$ and let $a_1, a_2, \dots$ be complex numbers with $\abs{\arg a_n} < \pi/2 - \varepsilon$ for every $n$.
Let $f_n$ be the $n$-th [[convergent|0SRWNR42BQ]] of the [[continued fraction|0BK6JBK58W]]
$$f = \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cdots}}}.$$
Then:

1. every $f_n$ is finite and satisfies $\abs{\arg f_n} < \pi/2 - \varepsilon$;
2. the sequences $(f_{2n})$ and $(f_{2n+1})$ of even and odd convergents both converge;
3. the continued fraction converges, that is, $(f_n)$ has a finite limit, if and only if $\sum_{n \ge 1} \abs{a_n}$ diverges.

## Remarks

Item 1 follows by induction from the tail: the open sector $S = \set{z \ne 0 : \abs{\arg z} < \pi/2 - \varepsilon}$ is closed under addition, and $z \mapsto 1/z$ maps $S$ onto itself.
Here $\arg$ is the [[argument|070PFSYJJR]] with values in $(-\pi, \pi]$.
Example: $a_n = 1$ for all $n$ lies in the sector and $\sum \abs{a_n} = \infty$, so the continued fraction converges; its value is $\frac{\sqrt5 - 1}{2}$.
If $a_n = 2^{-n}$, the sum $\sum \abs{a_n}$ is finite, and by item 3 the even and odd convergents tend to different limits.
For positive real $a_n$ the theorem contains the classical criterion that $\cfrac{1}{a_1 + \cfrac{1}{a_2 + \cdots}}$ converges exactly when $\sum a_n$ diverges.

## Explanations

### L1 human

The theorem treats continued fractions whose partial denominators all point into a fixed wedge around the positive real axis.
Then the fraction converges exactly when these denominators are not too small in total, that is, when the sum of their sizes is infinite.

### L2 human

The sector condition keeps all convergents away from $0$ and $\infty$, so only the oscillation between even and odd convergents can prevent convergence.
The divergence of $\sum \abs{a_n}$ is what forces the even and odd limits to meet.
Compare the [[Śleszyński–Pringsheim theorem|NEC48RHNN0]], where large partial denominators give convergence.
</document>

<document source="https://tamnd.github.io/math2/node/KCCHH87H6E.md" tag="KCCHH87H6E" kind="concept">
# Representable functor

## Summary

A set-valued [[functor|28NQ0KRHD3]] is representable when it is naturally isomorphic to a [[hom functor|VJXS8AK67E]] $\Hom(A, -)$ or $\Hom(-, A)$ for some object $A$, called a representing object.
Representability is the formal language of universal properties: products, kernels, free objects and many other constructions are representing objects of suitable functors.
The representing object is unique up to unique isomorphism.

## Explanations

### L1 human

Some ways of producing sets from objects are secretly of the form "all arrows from a fixed object $A$".
For example, in the category of sets, consider the functor that sends a set $X$ to the set of its subsets.
It is the same as "all functions from $X$ to the two-point set $\{0, 1\}$", because a subset is described by its indicator function.
So the power set functor, viewed as contravariant, is represented by $\{0, 1\}$.
Recognising a functor as representable tells you that one object and one special element carry all the information.
</document>

<document source="https://tamnd.github.io/math2/node/KCDT9Q3JGQ.md" tag="KCDT9Q3JGQ" kind="statement">
# Sphere eversion

## Statement

Let $S^2 \subseteq \R^3$ be the unit [[sphere|T62369RFH6]], let $\iota \colon S^2 \to \R^3$ be the inclusion and let $\alpha \colon S^2 \to \R^3$ be the map $\alpha(x) = -x$.
Then there is a smooth map $H \colon S^2 \times [0, 1] \to \R^3$ such that each $H_t = H(\cdot, t)$ is an [[immersion|6RW68JGBHC]], $H_0 = \iota$ and $H_1 = \alpha$.

## Remarks

Such a family $H$ is a regular homotopy that turns the sphere inside out: the outward normal of $\iota$ corresponds to the inward normal of $\alpha$.
The surfaces $H_t$ may pass through themselves, but they may never develop a crease or a pinch.
Stephen Smale proved this as a consequence of his classification of immersions of spheres: regular homotopy classes of immersions $S^2 \to \R^3$ correspond to $\pi_2(SO(3))$, which is the trivial group.
The proof was not constructive; explicit eversions were later found, for example by Bernard Morin, and by William Thurston with the method of corrugations.
The analogue for circles fails: by the Whitney–Graustein theorem, immersions $S^1 \to \R^2$ are classified by the turning number, and the standard circle and its reversal have turning numbers $1$ and $-1$.

## Explanations

### L1 human

Imagine a sphere made of a material that can pass through itself but cannot be creased.
The theorem says that such a sphere can be turned inside out by a continuous motion.
The result is surprising, because the obvious attempt, pushing two opposite caps through each other, leaves a sharp ring that cannot be smoothed away.

### L2 human

Sphere eversion is the standard example of the power of the Smale–Hirsch theorem and of the homotopy principle.
It shows that a regular homotopy can connect immersions that look very different.
The only invariants of immersions $S^2 \to \R^3$ up to regular homotopy are homotopy-theoretic, and here they vanish.
Its failure for circles in the plane shows that the outcome depends on the dimensions through the relevant homotopy group.
</document>

<document source="https://tamnd.github.io/math2/node/KCEVPEV70X.md" tag="KCEVPEV70X" kind="definition">
# Semialgebraic set

## Definition

Let $n \ge 1$ and let $\R[x_1, \dots, x_n]$ be the ring of [[real polynomials|07GNX8GCYP]] in $n$ variables.
A subset $S \subseteq \R^n$ is *semialgebraic* if it is a finite union of sets of the form
$$\set{x \in \R^n : f(x) = 0,\ g_1(x) > 0, \dots, g_k(x) > 0},$$
with $k \ge 0$ and $f, g_1, \dots, g_k \in \R[x_1, \dots, x_n]$.

## Remarks

Equivalently, the semialgebraic subsets of $\R^n$ form the smallest family of subsets that is closed under finite unions, finite intersections and complements.
This family must contain all sets $\set{x : g(x) > 0}$ with $g$ a polynomial.
The same definition works over any [[real closed field|DDY3Q3PJ6W]] in place of $\R$.

Example: the closed unit disc $D = \set{(x, y) : x^2 + y^2 \le 1}$ is semialgebraic.
It is the union of $\set{1 - x^2 - y^2 = 0}$ (with $k = 0$) and $\set{0 = 0,\ 1 - x^2 - y^2 > 0}$.

Non-example: $\Z \subseteq \R$ is not semialgebraic.
A semialgebraic subset of $\R$ is a finite union of points and open intervals.
This holds because a nonzero polynomial in one variable has finitely many roots and constant sign between them.
But $\Z$ is infinite and discrete.
</document>

<document source="https://tamnd.github.io/math2/node/KCFXR9K3GD.md" tag="KCFXR9K3GD" kind="concept">
# VC dimension

## Summary

The VC dimension (Vapnik–Chervonenkis dimension) of a [[family of sets|0C21Q694AG]] is the size of the largest set on which it cuts out every subset.
It measures the complexity of the family as a single number.
By the [[Sauer–Shelah lemma|Y4FQT5WPQJ]], a family of finite VC dimension $d$ picks out only polynomially many subsets, about $n^d$, of an $n$-point set.
This is why VC dimension controls sample sizes in statistical learning theory.

## Explanations

### L1 human

Think of each set in the family as a rule that says "yes" to some points and "no" to others.
A set of points is shattered when the rules together can produce every possible yes/no pattern on it.
The VC dimension is the largest number of points that can be shattered.
For example, intervals on the real line can shatter any two points but no three points $a < b < c$, because no interval contains $a$ and $c$ but not $b$.
So intervals have VC dimension $2$.
</document>

<document source="https://tamnd.github.io/math2/node/KCP52TPREW.md" tag="KCP52TPREW" kind="definition">
# Lie groupoid

## Definition

A *Lie groupoid* $G_1 \rightrightarrows G_0$ is a [[groupoid|FH3C0T643F]] with set of objects $G_0$ and set of arrows $G_1$, with source and target maps $s, t \colon G_1 \to G_0$, such that:

1. $G_0$ and $G_1$ are [[smooth manifolds|F45VQ47GPV]];
2. $s$ and $t$ are [[smooth|0V33BNKZ9S]] submersions;
3. the composition $m \colon G_1 \times_{G_0} G_1 \to G_1$, defined on pairs $(h, g)$ with $s(h) = t(g)$, is smooth;
4. the unit map $u \colon G_0 \to G_1$, sending $x$ to the identity arrow at $x$, and the inversion $i \colon G_1 \to G_1$ are smooth.

Since $s$ and $t$ are submersions, the set $G_1 \times_{G_0} G_1 = \set{(h, g) \in G_1 \times G_1 : s(h) = t(g)}$ is an embedded submanifold of $G_1 \times G_1$, so clause 3 makes sense.

## Remarks

Example: a [[Lie group|7JBRGCVJRB]] $G$ is a Lie groupoid $G \rightrightarrows \set{*}$ with one object; the maps $s$ and $t$ to a point are submersions, and composition is the group product.

Example: for a smooth [[action|96B4E2GPKJ]] of a Lie group $G$ on a manifold $M$, the action groupoid has $G_0 = M$, $G_1 = G \times M$, $s(g, x) = x$, $t(g, x) = g \cdot x$, and $(h, g \cdot x)(g, x) = (hg, x)$.

Non-example: the compact group $(\Z/2)^{\N}$, viewed as a groupoid with one object, is not a Lie groupoid.
Its space of arrows is not a manifold: it is totally disconnected but has no isolated points, so it is not locally Euclidean of any dimension.

Some authors allow $G_1$ to be a non-Hausdorff manifold, which is needed for groupoids arising from foliations.
</document>

<document source="https://tamnd.github.io/math2/node/KCV4BDWZ87.md" tag="KCV4BDWZ87" kind="statement">
# Contraction mapping principle for integral equations

## Statement

Let $a < b$, let $f \colon [a, b] \to \R$ be continuous, and let $H \colon [a, b] \times [a, b] \times \R \to \R$ be continuous.
Assume that $H$ satisfies a [[Lipschitz condition|0AW1E6YTJY]] in its third variable: there is $L \ge 0$ with
$$\abs{H(x, s, u) - H(x, s, v)} \le L \abs{u - v} \quad \text{for all } x, s \in [a, b] \text{ and } u, v \in \R.$$
Let $\lambda \in \R$ with $\abs{\lambda} L (b - a) < 1$.
Then the [[nonlinear integral equation|BSCYBP6AS6]]
$$\varphi(x) = f(x) + \lambda \int_a^b H\left(x, s, \varphi(s)\right) ds, \quad x \in [a, b],$$
has exactly one continuous solution $\varphi \colon [a, b] \to \R$.
For every continuous $\varphi_0$, the functions $\varphi_{k+1}(x) = f(x) + \lambda \int_a^b H(x, s, \varphi_k(s)) \, ds$ converge to $\varphi$ uniformly on $[a, b]$.

## Remarks

Proof idea: let $T$ be the right side, viewed as a map of $C[a, b]$ with the maximum norm.
This space is a complete metric space, by the [[completeness of the space of bounded continuous functions|09GRCZVR5E]].
The map $T$ sends continuous functions to continuous functions, since $H$ is uniformly continuous on compact sets.
The Lipschitz condition gives $\max \abs{T \varphi - T \psi} \le \abs{\lambda} L (b - a) \max \abs{\varphi - \psi}$, so $T$ is a [[contraction|0QN69YA357]].
The [[Banach fixed point theorem|0VK6VEH345]] gives the unique fixed point and the convergence of the [[fixed-point iteration|K6ZF91051B]].

Example: the linear case $H(x, s, u) = K(x, s) u$ with a continuous kernel $K$ has $L = \max \abs{K}$, and the theorem gives unique solvability of $\varphi = f + \lambda \int_a^b K \varphi$ for $\abs{\lambda} \max \abs{K} (b - a) < 1$.

The smallness condition matters: for $H(x, s, u) = u$ on $[0, 1]$ and $\lambda = 1$, so $\abs{\lambda} L (b - a) = 1$, the equation $\varphi = f + \int_0^1 \varphi$ has no solution when $\int_0^1 f \ne 0$.
For the Volterra form, with $\int_a^x$ in place of $\int_a^b$, the conclusion holds for every $\lambda$, because some power of $T$ is a contraction.
The same proof works for complex-valued functions.

## Explanations

### L1 human

If the integral term reacts only weakly to changes of the unknown, the equation has exactly one solution.
One finds it by guessing any function and feeding it into the right side again and again.
The guesses get closer to each other at a fixed rate and converge to the solution.

### L2 human

This is the standard first existence theorem for nonlinear Fredholm-type equations of Urysohn form.
The error after $k$ steps is at most $q^k (1 - q)^{-1} \max \abs{\varphi_1 - \varphi_0}$ with $q = \abs{\lambda} L (b - a)$, so the method also gives numerical bounds.
The same argument, applied on a short time interval, proves the Picard–Lindelöf theorem for ordinary differential equations through their integral form.
For larger $\lambda$ one needs other tools, such as Schauder's fixed point theorem or Fredholm theory in the linear case.
</document>

<document source="https://tamnd.github.io/math2/node/KCV8RKBATN.md" tag="KCV8RKBATN" kind="definition">
# Intuitionistic type theory

## Definition

*Intuitionistic type theory*, or Martin-Löf type theory, is the dependent [[type theory|42A2M9S2C1]] with the following parts.

1. Four forms of judgment in a context $\Gamma$. They say that $A$ is a type, or that $A$ and $B$ are equal types. They also say that $a$ has type $A$, written $a : A$, or that $a$ and $b$ are equal terms of type $A$.
2. Dependent function types $\prod_{x : A} B(x)$, whose terms are functions $\lambda x.\, b(x)$ with $b(x) : B(x)$ for each $x : A$.
3. Dependent pair types $\sum_{x : A} B(x)$, whose terms are pairs $(a, b)$ with $a : A$ and $b : B(a)$.
4. Identity types $\mathrm{Id}_A(a, b)$ for $a, b : A$, with the reflexivity term $\mathrm{refl}_a : \mathrm{Id}_A(a, a)$ and an induction rule.
5. Base types: the empty type $\mathbf{0}$, the unit type $\mathbf{1}$, the type $\mathbf{2}$ of booleans and the natural numbers $\N$. Each has introduction rules and an induction rule.
6. Universes $U_0 : U_1 : \cdots$, types whose terms are codes of types.

Each type former comes with formation, introduction, elimination and computation rules.

## Remarks

Under the [[Curry–Howard correspondence|HTF9E5W7EZ]], $\prod_{x : A} B(x)$ reads as "for all $x$ in $A$, $B(x)$", $\sum_{x : A} B(x)$ as "there is $x$ in $A$ with $B(x)$", and $\mathbf{0}$ as falsity.
Example: the term $\lambda n.\, (n + 1, \mathrm{refl}_{n+1})$ has type $\prod_{n : \N} \sum_{m : \N} \mathrm{Id}_{\N}(m, n + 1)$, so it proves that every natural number has a successor.
Non-example: there is no closed term of type $\prod_{A : U_0} (A + (A \to \mathbf{0}))$, so the [[intuitionistic logic|Q11N8X99FK]] of the theory does not include excluded middle.
Martin-Löf gave several versions, with intensional or extensional identity types.
</document>

<document source="https://tamnd.github.io/math2/node/KCVCPGX7ZC.md" tag="KCVCPGX7ZC" kind="statement">
# Bergman's gap theorem

## Statement

Let $k$ be a [[field|04C5ZBG1WT]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$.
Then the [[Gelfand–Kirillov dimension|MDVSY2NMD2]] of $A$ is not in the open interval $(1, 2)$.
Together with the elementary fact that it is not in $(0, 1)$, this gives
$$\operatorname{GKdim} A \in \set{0} \cup \set{1} \cup [2, \infty].$$

## Remarks

The gap between $0$ and $1$ is easy: if a finitely generated algebra is infinite-dimensional, then $V^n \subsetneq V^{n+1}$ for all $n$, so $\dim V^n \ge n + 1$.
The gap between $1$ and $2$ is harder.
One reduces to algebras given by monomial relations, whose growth is governed by counting words in the generators.
Then a combinatorial argument on words applies.
If the number of nonzero words of some length $n$ is at most $n$, then the number of nonzero words of each length stays bounded.
So the growth is at most linear.
Since the dimension of a general algebra is the supremum over its finitely generated subalgebras, the gaps pass to all algebras.

Example: $k[x]$ has Gelfand–Kirillov dimension $1$, and the [[polynomial ring|0SDC934R52]] $k[x, y]$ has dimension $2$.
So both ends of the gap $(1, 2)$ are attained.
No further gap exists: every real number $r \ge 2$ is the Gelfand–Kirillov dimension of some finitely generated algebra.
This contrasts with the [[Krull dimension|MVGBS3J1P7]] of commutative rings, which takes only integer values or $\infty$.

## Explanations

### L1 human

The Gelfand–Kirillov dimension measures how fast an algebra grows, and one might expect any value to be possible.
Bergman's theorem says that values strictly between $1$ and $2$ never occur.
If the growth of an algebra is measured by an exponent $d$, so that it grows roughly like $n^d$, then $d$ is $0$, $1$, or at least $2$.

### L2 human

The theorem is a statement about the [[growth|EHY4PYSMGZ]] of algebras, and its heart is combinatorics on words.
It shows that Gelfand–Kirillov dimension behaves like an integer-valued dimension at the bottom of the scale, while above $2$ every real value can occur.
It is a basic input in the study of algebras of small Gelfand–Kirillov dimension, such as algebras of dimension $1$ and $2$, which are often compared with curves and surfaces.
</document>

<document source="https://tamnd.github.io/math2/node/KCVKGSYJR9.md" tag="KCVKGSYJR9" kind="definition">
# Ridge regression

## Definition

Let $X$ be a real $n \times p$ matrix, let $y \in \R^n$, and let $\lambda > 0$.
The *ridge regression estimator* with parameter $\lambda$ is the minimiser of
$$\norm{y - X b}^2 + \lambda \norm{b}^2$$
over $b \in \R^p$, where $\norm{\cdot}$ is the Euclidean norm.
It equals
$$\hat\beta_\lambda = (X^T X + \lambda I_p)^{-1} X^T y.$$

## Remarks

The matrix $X^T X + \lambda I_p$ is [[positive definite|0KX66CZMBB]] for $\lambda > 0$, so the minimiser exists and is unique for every $X$, whatever its rank.
The formula comes from setting the gradient $-2 X^T (y - X b) + 2 \lambda b$ to zero.

Example: take $p = 1$, $X = (1, 2)^T$, $y = (1, 2)^T$ and $\lambda = 1$.
Then $X^T X = 5$ and $X^T y = 5$, so $\hat\beta_1 = 5/6$, while the [[ordinary least squares|YM708KHV2W]] slope is $1$.

Non-example: with $\lambda = 0$ the problem is plain [[least squares|0BYTQ4X1HC]], and if $X^T X$ is singular, for example when two columns of $X$ are equal, the minimiser is not unique.

In practice the regressors are usually standardised and the intercept is not penalised.
For each $\lambda > 0$ there is $t > 0$ such that $\hat\beta_\lambda$ also minimises $\norm{y - Xb}^2$ under the constraint $\norm{b}^2 \le t$.
In a normal linear model, $\hat\beta_\lambda$ is the posterior mode for a centred normal prior on $\beta$.
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<document source="https://tamnd.github.io/math2/node/KD404MF042.md" tag="KD404MF042" kind="concept">
# Heat kernel

## Summary

The heat kernel is the solution of the [[heat equation|VP2HN86VAG]] that starts from a unit amount of heat concentrated at one point.
On $\R^n$ it is the Gaussian $\Phi(x, t) = (4 \pi t)^{-n/2} e^{-\abs{x}^2 / (4t)}$.
Every bounded continuous initial temperature evolves by convolution with it.
On domains and on Riemannian manifolds the heat kernel encodes the spectrum of the Laplacian.
It is a basic tool in geometric analysis and index theory.

## Explanations

### L1 human

Touch a cold infinite metal plate with a hot needle for an instant and watch the heat spread.
At each later time the temperature is a bell-shaped curve centred at the touched point, getting wider and flatter as time goes on.
The total amount of heat stays the same, so the area under the bell is always one.
Any initial temperature is a sum of many such touches, so its evolution is the matching sum of bells.
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<document source="https://tamnd.github.io/math2/node/KD9BP7YJK5.md" tag="KD9BP7YJK5" kind="definition">
# Approximation property

## Definition

Let $X$ be a [[Banach space|06GZ956657]] over $\R$ or $\C$.
A [[continuous linear map|03TM135G0N]] $T \colon X \to X$ has *finite rank* if its image $T(X)$ is finite-dimensional.
The space $X$ has the *approximation property* if the following holds.
For every [[compact|0R6EWFKRCF]] set $K \subseteq X$ and every $\varepsilon > 0$ there is a continuous linear map $T \colon X \to X$ of finite rank such that $\norm{Tx - x} < \varepsilon$ for all $x \in K$.

## Remarks

Example: let $X$ have a Schauder basis $(e_n)$, and let $P_n$ be the projection $\sum_k a_k e_k \mapsto \sum_{k \le n} a_k e_k$.
Each $P_n$ is continuous (a theorem of Banach) and has finite rank, $P_n x \to x$ for every $x$, and $\sup_n \norm{P_n} < \infty$ by the [[uniform boundedness principle|YG0042V8CJ]].
Pointwise convergence of a uniformly bounded sequence of operators is uniform on compact sets, so the $P_n$ witness the approximation property.
In particular $\ell^p$ for $1 \le p < \infty$ and $c_0$ have it.
Every [[Hilbert space|HM2HHE3J51]] has it too, using orthogonal projections onto finite-dimensional subspaces, even when it has no countable basis.
Non-example: the space built in [[Enflo's counterexample|ZKP5KGFWCK]] fails the definition for some compact set $K$ and some $\varepsilon > 0$.
Grothendieck's equivalent form: $X$ has the approximation property if and only if for every Banach space $Y$, every compact operator $Y \to X$ is a norm limit of finite-rank operators.
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<document source="https://tamnd.github.io/math2/node/KDEVJ69P10.md" tag="KDEVJ69P10" kind="example">
# Cubic surface

## Example

A *cubic surface* is the zero set in [[projective space|ZT9A6JK8QF]] $\mathbb{P}^3$ of a homogeneous polynomial of degree $3$ in four variables.
A concrete smooth one is the Fermat cubic over $\C$,
$$X = \set{[x : y : z : w] \in \mathbb{P}^3 : x^3 + y^3 + z^3 + w^3 = 0}.$$
It is smooth: the partial derivatives $3x^2, 3y^2, 3z^2, 3w^2$ vanish together only at $x = y = z = w = 0$, which is not a point of $\mathbb{P}^3$.
By the [[adjunction formula|RYB6F7JFGV]], a smooth surface of degree $3$ in $\mathbb{P}^3$ has canonical divisor $K_X = (3 - 4) H = -H$, where $H$ is a plane section.
So $-K_X = H$ is ample, and $X$ is a [[del Pezzo surface|AV2ZAAZXN9]] of degree $K_X^2 = H^2 = 3$.

## Remarks

The Fermat cubic contains the line $\set{x = -y, \ z = -w}$, since then $x^3 + y^3 = 0$ and $z^3 + w^3 = 0$.
More generally, for cube roots of unity $\zeta, \zeta'$ the lines $\set{x = -\zeta y, \ z = -\zeta' w}$ lie on $X$, which gives $9$ lines.
The other two ways of pairing the four variables give $9$ more each.
So there are $27$ lines in all, as the [[twenty-seven lines theorem|VQK5HQFYEM]] predicts.
Every smooth cubic surface over an algebraically closed field is the blow-up of $\mathbb{P}^2$ at six points in general position, so it is a rational surface.
The Clebsch surface, given by $\sum x_i = 0$ and $\sum x_i^3 = 0$ in $\mathbb{P}^4$, is a smooth cubic surface on which all $27$ lines are defined over $\R$.

## Explanations

### L1 human

A cubic surface is the set of solutions of one cubic equation in four homogeneous coordinates.
The Fermat cubic $x^3 + y^3 + z^3 + w^3 = 0$ is the most symmetric example.
It is smooth, it can be parametrized by the plane, and it contains exactly $27$ straight lines, which can be written down by hand.
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<document source="https://tamnd.github.io/math2/node/KDRZKC9T1V.md" tag="KDRZKC9T1V" kind="statement">
# Density of trigonometric polynomials in continuous functions

## Statement

Let $f \colon \R \to \C$ be a [[continuous function|0X3WGX95FE]] that is [[periodic|0ATF6Q0633]] with period $2\pi$, and let $\varepsilon > 0$.
Then there is a [[trigonometric polynomial|J3NJQ5FZJ5]] $T$ such that
$$\sup_{x \in \R} \abs{f(x) - T(x)} < \varepsilon.$$
Equivalently, the trigonometric polynomials are dense in the space $C(\mathbb{T})$ of continuous functions on $\mathbb{T} = \R / 2\pi\Z$ with the supremum norm.

## Remarks

One proof uses the [[Stone–Weierstrass theorem|1P8D3MK79H]] on the compact space $\mathbb{T}$.
The trigonometric polynomials form a subalgebra of $C(\mathbb{T})$ that contains the constants, is closed under complex conjugation, and separates points, since $e^{ix}$ is injective on $\mathbb{T}$.
Another proof is [[Fejér's theorem|TPJZCSXSVN]]: the Cesàro means of the Fourier series of $f$ are trigonometric polynomials that converge to $f$ [[uniformly|08VZ57PT26]].
The partial sums of the Fourier series of $f$ itself need not converge uniformly, and need not converge at all.
So the approximating polynomial is in general not a Fourier partial sum.
The periodicity hypothesis is necessary: every trigonometric polynomial $T$ satisfies $T(-\pi) = T(\pi)$, so the function $x$ on $[-\pi, \pi]$ cannot be approximated uniformly on that interval.
A consequence is that a continuous $2\pi$-periodic function whose Fourier coefficients all vanish is identically zero.

## Explanations

### L1 human

Every continuous periodic function can be imitated, as closely as one likes and at every point at once, by a finite sum of waves.
The number of waves needed grows as the required precision grows.
This is the periodic version of the Weierstrass theorem that continuous functions on an interval are uniform limits of polynomials.

### L2 human

The result is the basic density fact of Fourier analysis on the circle.
Since $C(\mathbb{T})$ is dense in $L^p(\mathbb{T})$ for $1 \le p < \infty$, the trigonometric polynomials are also dense in these spaces.
For $p = 2$ this gives the completeness of the Fourier monomials, so that they form an orthonormal basis of $L^2(\mathbb{T})$.
It also gives uniqueness: two continuous periodic functions with the same Fourier coefficients are equal.
The rate of approximation in terms of the modulus of continuity of $f$ is the subject of Jackson's theorems.
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<document source="https://tamnd.github.io/math2/node/KDT3E0VE0K.md" tag="KDT3E0VE0K" kind="statement">
# Jordan decomposition of a function of bounded variation

## Statement

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function.
Then $f$ has [[bounded variation|A8251BHCMP]] on $[a, b]$ if and only if there are nondecreasing functions $g, h \colon [a, b] \to \R$ with $f = g - h$.
Moreover, when $f$ has bounded variation, one can take $g(x) = V_a^x(f)$ and $h(x) = V_a^x(f) - f(x)$, where $V_a^x(f)$ is the [[total variation|8QA5DEWNYD]] of $f$ on $[a, x]$ and $V_a^a(f) = 0$.

## Remarks

A nondecreasing function has bounded variation, and a difference of two functions of bounded variation has bounded variation, which gives one direction.
For the other direction, $g(x) = V_a^x(f)$ is nondecreasing because variation is additive over adjacent intervals.
For $x < y$ one has $f(y) - f(x) \le \abs{f(y) - f(x)} \le V_x^y(f) = g(y) - g(x)$, so $h = g - f$ is nondecreasing too.
For example, $f(x) = \sin x$ on $[0, 2\pi]$ is the difference of $g(x) = V_0^x(\sin)$ and $g - \sin$.
The decomposition is not unique: one may add the same nondecreasing function to $g$ and to $h$.

## Explanations

### L1 human

A function of bounded variation may go up and down, but it can always be split into a part that only goes up and a part that only goes down.
This reduces many questions about such functions to [[monotone functions|0WFF6AZPBP]], which are much easier to handle.

### L2 human

With the decomposition, results about monotone functions pass to functions of bounded variation.
By [[Lebesgue's theorem on monotone functions|1Y4N1DDQNR]], a function of bounded variation is differentiable almost everywhere.
It also has one-sided limits at every point and at most countably many discontinuities.
The theorem is the one-variable model of the [[Jordan decomposition of a signed measure|WAGJVK2NWE]].
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# C* identity

## Summary

The C* identity is the condition $\norm{a^* a} = \norm{a}^2$ on a Banach algebra with an [[involution|GP56NBN9RY]].
A Banach $*$-algebra whose norm satisfies it is a C*-algebra.
This one equation ties the norm to the algebra so tightly that the algebraic structure determines the norm.
Indeed, $\norm{a}$ is the square root of the spectral radius of $a^* a$.

## Explanations

### L1 human

For a complex number, $\abs{\overline{z} z} = \abs{z}^2$.
The C* identity asks the same of a whole algebra of objects, such as matrices or operators, where $a^*$ plays the role of the complex conjugate.
For a matrix with the operator norm and the conjugate transpose as $a^*$, the identity holds.
Algebras with this property behave like algebras of operators on a Hilbert space, and in fact every one of them is such an algebra.
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# Isomonodromic deformation

## Summary

An isomonodromic deformation is a family of linear differential equations whose [[monodromy|C8HD62AB6Q]] stays the same.
The family depends on parameters, such as the positions of the singular points.
Keeping the monodromy fixed forces the coefficients to satisfy nonlinear differential equations in the parameters.
For Fuchsian systems these are the Schlesinger equations, and in the simplest nontrivial case they reduce to the sixth Painlevé equation.
This gives one explanation of why the Painlevé equations have the [[Painlevé property|8EV1G2Y6CC]], and it links them to random matrices and integrable systems.
See the [[definition|SVDH9C5MCK]].

## Explanations

### L1 human

Take a linear equation with a few singular points and move the singular points a little.
In general the monodromy, the way solutions get mixed around the points, changes.
One can ask to move the points and adjust the coefficients at the same time so that the monodromy does not change.
The coefficients then follow a nonlinear equation in the positions of the points.
Many famous nonlinear equations arise this way.
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<document source="https://tamnd.github.io/math2/node/KDZN92Z8JR.md" tag="KDZN92Z8JR" kind="definition">
# Automatic differentiation

## Definition

Let $U \subseteq \R^n$ be open and let $f \colon U \to \R^m$ be given by an *evaluation trace*.
Its first variables $v_{1-n}, \ldots, v_0$ are the inputs $x_1, \ldots, x_n$.
The others are given for $k = 1, \ldots, \ell$ by
$$v_k = \varphi_k\big( (v_j)_{j \in P_k} \big),$$
where $P_k \subseteq \set{1 - n, \ldots, k - 1}$ and each $\varphi_k$ is a differentiable elementary function, such as $+$, $\times$, $\sin$ or $\exp$.
The outputs $y_1, \ldots, y_m$ are some of the $v_k$.

1. *Forward mode* takes a direction $\dot x \in \R^n$, sets $\dot v_{i - n} = \dot x_i$, and computes along with each $v_k$ the value $\dot v_k = \sum_{j \in P_k} \frac{\partial \varphi_k}{\partial v_j} \dot v_j$.
2. *Reverse mode* first evaluates the trace and takes $\bar y \in \R^m$.
It sets the adjoint $\bar v$ of each output to its entry of $\bar y$ and all other adjoints to $0$.
Then for $k = \ell, \ell - 1, \ldots, 1$ it adds $\bar v_k \frac{\partial \varphi_k}{\partial v_j}$ to $\bar v_j$ for each $j \in P_k$.

*Automatic differentiation* is the computation of derivatives of $f$ by these propagation rules.
Forward mode returns $\dot y = J_f(x) \dot x$, where $J_f(x)$ is the [[Jacobian matrix|047NDTEPCV]].
Reverse mode returns $\bar x = J_f(x)^{T} \bar y$, where $\bar x_i = \bar v_{i-n}$.

## Remarks

Example: $f(x_1, x_2) = x_1 x_2 + \sin x_1$ has the trace $v_1 = v_{-1} v_0$, $v_2 = \sin v_{-1}$, $v_3 = v_1 + v_2$.
Forward mode at $(x_1, x_2) = (1, 2)$ with $\dot x = (1, 0)$ gives $\dot v_1 = 2$, $\dot v_2 = \cos 1$, $\dot v_3 = 2 + \cos 1$, which is the [[partial derivative|09EGC6F8PT]] $\partial f / \partial x_1 = x_2 + \cos x_1$ at $(1, 2)$.
Reverse mode with $\bar y = 1$ gives $\bar v_1 = \bar v_2 = 1$, then $\bar v_0 = v_{-1} = 1$ and $\bar v_{-1} = v_0 + \cos v_{-1} = 2 + \cos 1$, the full gradient in one backward sweep.
Non-example: the difference quotient $\frac{f(x + h e_1) - f(x)}{h}$ is not automatic differentiation; it does not use the trace and has a truncation error of order $h$.
Forward mode can be described with dual numbers $a + b\varepsilon$, $\varepsilon^2 = 0$: evaluating the program on $x + \dot x \varepsilon$ gives $f(x) + (J_f(x) \dot x) \varepsilon$.

## Explanations

### L1 human

Automatic differentiation follows the program step by step and differentiates each step by the rules of calculus.
Forward mode answers how all outputs change when one input direction changes; reverse mode answers how one output depends on all inputs at once.
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# Padé table

## Definition

Let $f(z) = \sum_{k \ge 0} c_k z^k$ be a [[formal power series|072W67KFYK]] with complex coefficients.
For integers $m, n \ge 0$ let $[m/n]_f$ be the [[Padé approximant|75XCD68557]] of $f$ with numerator degree at most $m$ and denominator degree at most $n$.
The *Padé table* of $f$ is the doubly infinite array $\big([m/n]_f\big)_{m, n \ge 0}$.
The table is *normal* if no rational function occurs at two different positions $(m, n)$.

## Remarks

The column $n = 0$ consists of the [[Taylor polynomials|0HFSFVMWQB]] of $f$.
By a theorem of Padé, the positions where a given rational function occurs form a square block, possibly infinite.
Example: for $f = e^z$ the table is normal, and $[1/1]_f = (2 + z)/(2 - z)$.
Non-example of normality: for $f = 1/(1 - z) = \sum z^k$, one has $[m/n]_f = 1/(1 - z)$ for all $m \ge 0$ and $n \ge 1$, since $(1 - z) f - 1 = 0$; so this table is not normal.
Orientation conventions differ: some authors put $m$ along rows and $n$ along columns, others the reverse.
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<document source="https://tamnd.github.io/math2/node/KE4HEWEG38.md" tag="KE4HEWEG38" kind="definition">
# Periodogram

## Definition

Let $x_1, \dots, x_n$ be observations of a real [[time series|EA3MF4VTEH]].
The *periodogram* of the data is the function
$$I_n(\omega) = \frac{1}{n} \abs{\sum_{t=1}^{n} x_t e^{-i t \omega}}^2, \qquad \omega \in [-\pi, \pi].$$
It is usually evaluated at the *Fourier frequencies* $\omega_j = 2\pi j / n$ with $-\pi < \omega_j \le \pi$, where it equals $\frac1n \abs{\hat x_j}^2$ for the [[discrete Fourier transform|TST0GH66A5]] $\hat x_j = \sum_{t=1}^n x_t e^{-2\pi i j t / n}$.

## Remarks

Some authors divide by $2\pi n$ instead of $n$, so that the periodogram estimates the spectral density $f(\omega) = \frac{1}{2\pi} \sum_h \gamma(h) e^{-i h \omega}$ directly.
At every Fourier frequency $\omega_j \ne 0$, the periodogram equals $\sum_{\abs{h} < n} \hat\gamma(h) e^{-i h \omega_j}$, where $\hat\gamma$ is the sample autocovariance from the [[autocorrelation function|YCZHYY8VE5]].
So the periodogram is the Fourier transform of the sample autocovariance.

Example: let $n = 12$ and $x_t = \cos(2\pi t / 4)$, a cycle of period $4$, so the frequency is $\omega_3 = 2\pi \cdot 3/12$.
Writing $\cos u = (e^{iu} + e^{-iu})/2$ and using $\sum_{t=1}^{12} e^{2\pi i k t / 12} = 0$ for $k \not\equiv 0 \pmod{12}$, one gets $\sum_t x_t e^{-i t \omega_3} = 12/2 = 6$.
Hence $I_{12}(\omega_3) = 36/12 = 3$, while $I_{12}(\omega_j) = 0$ at all other Fourier frequencies in $[0, \pi]$.

Non-example: the function $\omega \mapsto \abs{\sum_t x_t e^{-it\omega}}$ without the square is not the periodogram; it is the amplitude spectrum of the data, and its square divided by $n$ is the periodogram.

For a stationary process with summable autocovariances, $E[I_n(\omega)] \to 2\pi f(\omega)$ for $\omega \ne 0$.
But the variance of $I_n(\omega)$ does not tend to $0$, so the periodogram is not a consistent estimator of the spectral density.
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# Preconditioner

## Summary

A preconditioner for a linear system $A x = b$ is an invertible matrix $P$ that is close to $A$ in a suitable sense.
Systems $P z = r$ must be cheap to solve.
One replaces the system by $P^{-1} A x = P^{-1} b$, or by a similar transformed system.
The new matrix has a smaller [[condition number|AB3S6MHJ8X]] or better clustered eigenvalues.
Iterative methods then converge in far fewer steps.
For large sparse systems, the choice of preconditioner usually matters more than the choice of iterative method.

## Explanations

### L1 human

An iterative solver is slow when the matrix is badly scaled or ill-conditioned.
A preconditioner is a cheap rough approximation of the matrix that is easy to invert.
Applying its inverse first turns the problem into a nicer one with the same solution.
The simplest example is the diagonal of $A$: dividing each equation by its diagonal entry often helps already.
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<document source="https://tamnd.github.io/math2/node/KEARMKNZ1Q.md" tag="KEARMKNZ1Q" kind="concept">
# Functional data analysis

## Summary

Functional data analysis is the branch of statistics in which each observation is a whole function, such as a curve, a surface or an image.
The functions are modelled as random elements of an infinite-dimensional space, usually the [[Hilbert space|HM2HHE3J51]] $L^2$ of square-integrable functions.
Its basic tools are the mean function, the covariance operator and functional [[principal component analysis|ABXD13M340]], which rests on the [[Karhunen–Loève theorem|5CN7KY8MWP]].
It also has regression models in which the predictor or the response is a function.

## Explanations

### L1 human

Record the height of a hundred children at every age from birth to eighteen.
Each child gives a growth curve, and it is natural to treat each curve as one data point.
Functional data analysis computes the average curve, describes the main ways in which curves differ from it, and relates curves to other variables.
For example, the first principal component of growth curves may describe children who are taller at every age, and the second may describe the timing of the growth spurt.
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<document source="https://tamnd.github.io/math2/node/KEC5TGS9WM.md" tag="KEC5TGS9WM" kind="definition">
# Complete integral

## Definition

Let $U \subseteq \R^n$ and $A \subseteq \R^n$ be open sets, and let $F \colon \R^n \times \R \times U \to \R$ be a function.
Consider the first-order [[partial differential equation|GC5BDZ6QBC]] $F(\nabla u, u, x) = 0$ for $u \colon U \to \R$.
A *complete integral* of this equation is a function $u \colon U \times A \to \R$, written $u(x; a)$, such that:

1. $u$ is $C^2$ in $(x, a)$;
2. for each $a \in A$, the function $x \mapsto u(x; a)$ solves the equation on $U$;
3. for all $(x, a) \in U \times A$, the $n \times (n + 1)$ matrix
$$\big( D_a u \;\big|\; D^2_{xa} u \big) = \begin{pmatrix} u_{a_1} & u_{x_1 a_1} & \cdots & u_{x_n a_1} \\ \vdots & \vdots & & \vdots \\ u_{a_n} & u_{x_1 a_n} & \cdots & u_{x_n a_n} \end{pmatrix}$$
of [[partial derivatives|09EGC6F8PT]] has rank $n$.

## Remarks

Clause 3 says that $u$ depends on all $n$ parameters independently, so that no parameter can be removed by a change of the others.

Example: Clairaut's equation $x \cdot \nabla u + f(\nabla u) = u$, with $f \colon \R^n \to \R$ of class $C^2$, has the complete integral $u(x; a) = a \cdot x + f(a)$ for $a \in \R^n$.
Check: $\nabla u = a$, so $x \cdot a + f(a) = u$, and clause 2 holds.
For clause 3, $D^2_{xa} u$ is the identity matrix, so the matrix has rank $n$.

Non-example: for the equation $\partial_1 u = 0$ on $\R^2$, the family $u(x; a) = a_1 + a_2$ satisfies clauses 1 and 2 but not clause 3.
Indeed both rows of the matrix equal $(1, 0, 0)$, so its rank is $1$; the family really has only one parameter $a_1 + a_2$.
By contrast, $u(x; a) = a_1 + a_2 x_2$ is a complete integral of the same equation: the rows are $(1, 0, 0)$ and $(x_2, 0, 1)$.

Envelopes of subfamilies of a complete integral give further solutions, called general integrals, and the envelope of the whole family, when it exists, is a singular integral.
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# Euclidean geometry

## Definition

Let $n \ge 1$ and let $E^n$ be the [[Euclidean space|0XPR1E85W0]] of dimension $n$, that is, $\R^n$ with the distance $d(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}$.
The [[Euclidean group|0H3YY8CPKH]] is the group of all distance-preserving bijections of $E^n$.
*Euclidean geometry* of dimension $n$ is the study of the properties of subsets of $E^n$ that are invariant under the Euclidean group.
Synthetically, plane and solid Euclidean geometry is the theory given by [[Hilbert's axioms|KM1FB8XZ2R]] in dimension $2$ and $3$.

## Remarks

Example: the property "the three points $A$, $B$, $C$ form a right triangle with the right angle at $B$" is a property of Euclidean geometry.
Indeed, an isometry preserves distances, so it preserves the equation $d(A,B)^2 + d(B,C)^2 = d(A,C)^2$.
Non-example: the property "the segment $AB$ is horizontal" is not a property of Euclidean geometry, since a rotation moves a horizontal segment to a slanted one.
The two descriptions agree: every model of Hilbert's axioms of the plane is isomorphic to $\R^2$ with its usual lines, betweenness and congruence.
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# Freiman's theorem

## Statement

A generalized arithmetic progression of dimension $d$ in $\Z$ is a set
$$P = \set{a + x_1 q_1 + \dots + x_d q_d : x_i \in \Z, \ 0 \le x_i < L_i \text{ for } 1 \le i \le d},$$
where $a, q_1, \dots, q_d$ are integers and $L_1, \dots, L_d$ are positive integers.
Let $K \ge 1$ be a real number.
Then there are numbers $d(K)$ and $f(K)$, depending only on $K$, with the following property.
Every finite set $A \subseteq \Z$ with $\abs{A + A} \le K \abs{A}$ is contained in a generalized arithmetic progression $P$ of dimension at most $d(K)$ with $\abs{P} \le f(K) \abs{A}$.

## Remarks

Here $A + A$ is the [[sumset|XNBN31HQWB]] of $A$ with itself.
Ruzsa's proof has four steps.
First, the [[Plünnecke–Ruzsa inequality|EBEVWSRD2E]] bounds $\abs{2A - 2A}$.
Second, a large part of $A$ is modelled by a set in a cyclic group $\Z/N\Z$ with $N$ not much larger than $\abs{A}$.
Third, Bogolyubov's lemma finds a large Bohr set inside the set $2A' - 2A'$ of the model $A'$.
Fourth, geometry of numbers finds a progression inside the Bohr set, and a covering argument finishes.
A converse holds: if $A$ is a large subset of a proper progression $P$ of dimension $d$, then $\abs{A + A} \le \abs{P + P} \le 2^d \abs{P}$, so $A$ has small doubling.
Green and Ruzsa extended the theorem to all abelian groups, with coset progressions in place of progressions.

## Explanations

### L1 human

Some sets of numbers have very few different pairwise sums compared with their size.
Arithmetic progressions like $3, 7, 11, 15, \dots$ are the main example.
Freiman's theorem says that every such set is a large piece of a "multi-dimensional" arithmetic progression.
So few sums means a lot of hidden regular structure.

### L2 human

Freiman's theorem is the central inverse theorem of [[additive combinatorics|3NM362A2JG]].
Good bounds on $d(K)$ and $f(K)$ matter in applications, and finding the right ones is related to the polynomial Freiman–Ruzsa conjecture.
It is used in Gowers' proof of [[Szemerédi's theorem|B23B0QEMN5]], together with the [[Balog–Szemerédi–Gowers theorem|W5BZZ59CVM]].
</document>

<document source="https://tamnd.github.io/math2/node/KET2GSPZFG.md" tag="KET2GSPZFG" kind="definition">
# Localization

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $S \subseteq R$ be a [[multiplicative set|AN9SKEZ3VY]].
On $R \times S$, let $(a, s) \sim (b, t)$ if there is $u \in S$ with $u(at - bs) = 0$.
This is an [[equivalence relation|0G8VEZ3WTK]].
Write $a/s$ for the class of $(a, s)$.
The *localization* of $R$ at $S$ is the set $S^{-1}R$ of these classes with the operations
$$\frac{a}{s} + \frac{b}{t} = \frac{at + bs}{st}, \qquad \frac{a}{s} \cdot \frac{b}{t} = \frac{ab}{st}.$$
These operations are well defined and make $S^{-1}R$ a commutative ring with zero $0/1$ and identity $1/1$.
The *canonical map* is the [[ring homomorphism|0VTH8KPCG2]] $\lambda \colon R \to S^{-1}R$, $a \mapsto a/1$.

## Remarks

Universal property: $\lambda(s)$ is a [[unit|0XVEZDE9J8]] for each $s \in S$.
For each ring homomorphism $f \colon R \to B$ to a commutative ring $B$ such that $f(s)$ is a unit for all $s \in S$, there is a unique ring homomorphism $g \colon S^{-1}R \to B$ with $g \circ \lambda = f$, namely $g(a/s) = f(a) f(s)^{-1}$.

The kernel of $\lambda$ is the set of $a \in R$ with $ua = 0$ for some $u \in S$.
So $\lambda$ is injective when $S$ contains no [[zero divisors|0NNRDW10RZ]], and $S^{-1}R = 0$ when $0 \in S$.

Example: for $R = \Z$ and $S = \Z \setminus \set{0}$, the localization is $\Q$, the [[field of fractions|0F2PTGX1A2]] of $\Z$.
For $S = \set{1, 2, 4, 8, \ldots}$, it is the ring $\Z[1/2]$ of rationals $a / 2^k$.
For an element $f$, the localization at $\set{1, f, f^2, \ldots}$ is written $R_f$ or $R[1/f]$.

Non-example: the factor $u$ cannot be dropped.
Take $R = \Z/6\Z$ and $S = \set{1, 3}$, and consider the naive relation "$at = bs$".
It relates $(2, 1)$ and $(0, 3)$, because $2 \cdot 3 = 0 = 0 \cdot 1$ in $R$.
It relates $(0, 3)$ and $(0, 1)$.
It does not relate $(2, 1)$ and $(0, 1)$, because $2 \ne 0$.
So the naive relation is not transitive.
With the factor $u = 3$, one has $3(2 \cdot 1 - 0 \cdot 1) = 0$, so $2/1 = 0/1$ in $S^{-1}R$.

When $\mathfrak p$ is a [[prime ideal|00MHHZ9SNE]] and $S = R \setminus \mathfrak p$, the localization is the [[localization at a prime ideal|X6JD4AG973]] $R_{\mathfrak p}$.
</document>

<document source="https://tamnd.github.io/math2/node/KEXJX0YDT4.md" tag="KEXJX0YDT4" kind="problem">
# Zermelo's navigation problem

## Problem

A boat moves in the plane with constant speed $V > 0$ relative to the water.
The water moves with a velocity field $w = (w_1, w_2) \colon \R^2 \to \R^2$ of class $C^1$ with $\abs{w} < V$ everywhere.
The captain controls the heading $\theta(t)$, so that the position $(x(t), y(t))$ obeys the [[ordinary differential equations|0NT64JV96A]]
$$x' = V \cos \theta + w_1(x, y), \qquad y' = V \sin \theta + w_2(x, y).$$
Given two points $P$ and $Q$, find a heading $t \mapsto \theta(t)$ that brings the boat from $P$ to $Q$ in the least time.

## Remarks

It is a [[time-optimal control|YTR959CYMX]] problem, and the [[Pontryagin maximum principle|7RPV2T29CF]] gives necessary conditions on optimal headings.
Zermelo derived a differential equation for the optimal heading along the path, known as Zermelo's navigation formula.
For a constant current $w$, the optimal heading is constant and the boat moves along the straight segment from $P$ to $Q$.
One aims so that the sum of $V(\cos\theta, \sin\theta)$ and $w$ points from $P$ to $Q$.
The condition $\abs{w} < V$ ensures that the boat can make headway in every direction.

## Explanations

### L1 human

A boat crosses a river or a sea with currents, and the captain wants to arrive as soon as possible.
Pointing straight at the target is not best, because the current pushes the boat off course.
The problem asks how to steer at each moment, and it is one of the first problems of optimal control.
</document>

<document source="https://tamnd.github.io/math2/node/KEXKKBDXR3.md" tag="KEXKKBDXR3" kind="definition">
# Equational basis

## Definition

Let $\sigma$ be a signature with only function symbols and let $\mathcal{K}$ be a class of algebras of signature $\sigma$, with [[equational theory|F1M51B4MDG]] $\operatorname{Id}(\mathcal{K})$.
A set $\Sigma$ of [[identities|PKD440G2XW]] over $\sigma$ is an *equational basis* of $\mathcal{K}$ if the following hold:

1. $\Sigma \subseteq \operatorname{Id}(\mathcal{K})$;
2. every identity in $\operatorname{Id}(\mathcal{K})$ is satisfied by every algebra that satisfies all identities of $\Sigma$.

An equational basis of a single algebra $A$ is an equational basis of $\set{A}$.

## Remarks

Equivalently, the class of all algebras satisfying $\Sigma$ is the [[variety|AJNZJD8442]] generated by $\mathcal{K}$.
By Birkhoff's completeness theorem, clause 2 can be read syntactically: every identity of $\mathcal{K}$ is derivable from $\Sigma$ in equational logic.
The whole theory $\operatorname{Id}(\mathcal{K})$ is always an equational basis of $\mathcal{K}$, so the interest lies in small bases.

Example: in the signature $(\cdot, {}^{-1}, e)$, the set $\set{x(yz) \approx (xy)z, \ x e \approx x, \ x x^{-1} \approx e}$ is an equational basis of the class of groups.
Clause 1 holds because every group satisfies these laws, and clause 2 holds because every algebra satisfying them is a group.

Non-example: in the same signature, $\set{x(yz) \approx (xy)z}$ is not an equational basis of the class of groups, since clause 2 fails.
Let $A = \set{0, 1}$ with $a \cdot b = 0$ for all $a, b$, with $a^{-1} = a$, and with $e = 1$.
Then $A$ satisfies associativity, since all products are $0$, but $1 \cdot e = 0 \ne 1$, so $A$ does not satisfy $x e \approx x$.
</document>

<document source="https://tamnd.github.io/math2/node/KEY8MM7F28.md" tag="KEY8MM7F28" kind="definition">
# Trace operator

## Definition

Let $U \subset \R^n$ be a bounded open set with $C^1$ boundary $\partial U$, carrying its surface measure, and let $1 \le p < \infty$.
The *trace operator* is the unique bounded linear map
$$T \colon W^{1,p}(U) \to L^p(\partial U)$$
such that $T u = u|_{\partial U}$ for every $u \in W^{1,p}(U)$ that is continuous on $\overline{U}$.
Here $W^{1,p}(U)$ is the [[Sobolev space|M4CEZ5K9HP]].
The function $T u$ is the *trace* of $u$ on $\partial U$.

## Remarks

The operator exists and is unique.
For $u$ smooth up to the boundary, the divergence theorem gives $\norm{u|_{\partial U}}_{L^p(\partial U)} \le C \norm{u}_{W^{1,p}(U)}$ with $C$ depending on $p$ and $U$.
Such functions are dense in $W^{1,p}(U)$, so restriction extends uniquely to a [[continuous linear map|03TM135G0N]] on all of $W^{1,p}(U)$.

Example: on $U = (0, 1)$ the boundary is $\set{0, 1}$, and every $u \in W^{1,p}(0, 1)$ has a continuous representative.
Its trace is the pair of values $(u(0), u(1))$ of that representative.

Non-example: on $U = (0, 1)$, restriction to the boundary is not bounded for the $L^p$ norm alone.
The functions $u_k(x) = (1 - x)^k$ satisfy $u_k(0) = 1$ while $\norm{u_k}_{L^p(0,1)} \to 0$, so no bounded map $L^p(0, 1) \to \R^2$ extends restriction.
This is why the definition uses $W^{1,p}(U)$.

The kernel of $T$ is $W^{1,p}_0(U)$.
For $1 < p < \infty$, the image of $T$ is the [[fractional Sobolev space|N6NX1VT837]] $W^{1 - 1/p, p}(\partial U)$, a result of Gagliardo.
</document>

<document source="https://tamnd.github.io/math2/node/KF06HRCM4N.md" tag="KF06HRCM4N" kind="statement">
# Differentiable dependence on parameters

## Statement

Let $D \subseteq \R \times \R^n \times \R^k$ be open and let $f : D \to \R^n$, $(t, x, \lambda) \mapsto f(t, x, \lambda)$, be continuous with continuous partial derivatives with respect to $x$ and $\lambda$.
For $(t_0, x_0, \lambda) \in D$, let $t \mapsto \varphi(t; t_0, x_0, \lambda)$ be the [[maximal solution|019VSXSFMQ]] of $x' = f(t, x, \lambda)$ with $x(t_0) = x_0$.
Let $\Omega$ be the set of $(t, t_0, x_0, \lambda)$ for which $\varphi(t; t_0, x_0, \lambda)$ is defined.
Then:

1. The set $\Omega$ is open and $\varphi$ is [[continuously differentiable|06SKCHDQT5]] on $\Omega$.
2. The matrix $J(t) = \partial \varphi / \partial x_0$ solves the variational equation $J' = A(t)\, J$ with $J(t_0) = I$, where $A(t) = \partial_x f(t, \varphi(t), \lambda)$ is the [[Jacobian matrix|047NDTEPCV]] along the solution.
3. The matrix $K(t) = \partial \varphi / \partial \lambda$ solves $K' = A(t)\, K + \partial_\lambda f(t, \varphi(t), \lambda)$ with $K(t_0) = 0$.
4. The derivative with respect to the initial time is $\partial \varphi / \partial t_0 = -J(t)\, f(t_0, x_0, \lambda)$.

## Remarks

Proof idea: the parameter $\lambda$ can be absorbed into the state by adding the equation $\lambda' = 0$, so it suffices to treat initial values.
One shows that the difference quotients $(\varphi(t; t_0, x_0 + h e_i) - \varphi(t; t_0, x_0))/h$ converge, using the mean value theorem and [[Grönwall's inequality|07EJHQYR9A]] to compare them with the solution of the variational equation.
Example: for $x' = \lambda x$, $\varphi(t; 0, x_0, \lambda) = x_0 e^{\lambda t}$, and $K = \partial \varphi / \partial \lambda = t x_0 e^{\lambda t}$ indeed solves $K' = \lambda K + x_0 e^{\lambda t}$ with $K(0) = 0$.
If $f$ is only Lipschitz in $x$, the solution depends [[continuously|0G3DBXRRQJ]] on the data but need not be differentiable.
For $x' = \abs{x}$, the map $x_0 \mapsto \varphi(1; 0, x_0)$ is $e x_0$ for $x_0 \ge 0$ and $e^{-1} x_0$ for $x_0 < 0$, not differentiable at $0$.
If $f$ is of class $C^r$, then $\varphi$ is of class $C^r$.

## Explanations

### L1 human

If the equation and its parameters change a little, the solution changes a little, and in a smooth way.
The theorem also tells how fast the solution reacts: the rates of change solve a linear equation along the solution.
This is how one measures the sensitivity of a model to its inputs.

### L2 human

The variational equation is the linearization of the system along a given solution.
It is the basic tool for stability of periodic orbits, for Lyapunov exponents and for perturbation methods.
The result also shows that the flow of a $C^1$ vector field is a $C^1$ map, with derivative a [[fundamental matrix|0MQ24RBG93]] of the variational equation.
</document>

<document source="https://tamnd.github.io/math2/node/KF26RN3E8P.md" tag="KF26RN3E8P" kind="definition">
# L-curve

## Definition

Let $A \in \R^{m \times n}$, $b \in \R^m$ and $L \in \R^{p \times n}$ with $\ker A \cap \ker L = \set{0}$.
For $\lambda > 0$ let $x_\lambda$ be the unique minimizer of the [[Tikhonov|M68V0M5A3R]] functional
$$\norm{A x - b}_2^2 + \lambda^2 \norm{L x}_2^2.$$
The *L-curve* is the plane curve
$$\lambda \mapsto \bigl( \log \norm{A x_\lambda - b}_2, \ \log \norm{L x_\lambda}_2 \bigr), \qquad \lambda > 0,$$
defined where both norms are positive.
The *L-curve criterion* chooses $\lambda$ at the point of maximal curvature of this curve.

## Remarks

The residual norm $\norm{A x_\lambda - b}$ is non-decreasing in $\lambda$, and the seminorm $\norm{L x_\lambda}$ is non-increasing.
So the curve runs from upper left (small $\lambda$, noise-dominated) to lower right (large $\lambda$, over-smoothed).
Example: for $A = L = I$ and $b \ne 0$, one gets $x_\lambda = b/(1 + \lambda^2)$.
Then $\norm{A x_\lambda - b} = \lambda^2 \norm{b}/(1 + \lambda^2)$ and $\norm{x_\lambda} = \norm{b}/(1 + \lambda^2)$, so the first increases and the second decreases with $\lambda$, as stated.
Non-example: the curve $\lambda \mapsto (\lambda, \norm{A x_\lambda - b})$ is not the L-curve, since it plots the parameter itself instead of the solution seminorm.
Some authors use the squared norms or linear axes, but the corner is clearest on log-log axes.
The criterion is a heuristic: it is not a convergent [[regularization|WTY54BFC46]] parameter choice in the worst-case sense, yet it works well on many practical problems.

## Explanations

### L1 human

The L-curve shows the trade-off between fitting the data and keeping the solution tame, one point for each value of the regularization parameter.
Its corner marks a good compromise.
</document>

<document source="https://tamnd.github.io/math2/node/KF719SZD5T.md" tag="KF719SZD5T" kind="concept">
# Additive function

## Summary

An additive function is a function $f$ with $f(x + y) = f(x) + f(y)$ for all $x$ and $y$, that is, a solution of [[Cauchy's functional equation|ZBMEVWN100]].
Between abelian groups it is the same thing as a [[group homomorphism|0JYTG5K9Y4]].
Every linear map is additive, and on the real line the continuous additive functions are exactly $x \mapsto cx$.
With the axiom of choice there are also wild additive functions on $\R$ whose graphs are dense in the plane.
In number theory the phrase "additive function" has another meaning, for arithmetic functions with $f(mn) = f(m) + f(n)$ when $m$ and $n$ are coprime.

## Explanations

### L1 human

An additive function turns sums into sums: the output for $x + y$ is the output for $x$ plus the output for $y$.
Doubling, $f(x) = 2x$, is additive, because $2(x + y) = 2x + 2y$.
Squaring is not, because $(x + y)^2$ is not $x^2 + y^2$ in general.
Adding a constant is not either: $f(x) = x + 1$ gives $f(0 + 0) = 1$ but $f(0) + f(0) = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/KF7HK5Z9XW.md" tag="KF7HK5Z9XW" kind="definition">
# Strassen algorithm

## Definition

Let $R$ be a [[ring|0B1261B5ED]], let $n = 2^m$ with $m \ge 0$, and let $A$, $B$ be $n \times n$ [[matrices|0VD9ZSWWJG]] over $R$.
The *Strassen algorithm* computes $C = AB$ as follows.
If $n = 1$, it returns the product of the two entries.
Otherwise it splits $A$, $B$ and $C$ into four blocks of size $n/2$, as in $A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}$.
It computes seven products, each by a recursive call:
$$\begin{aligned}
M_1 &= (A_{11} + A_{22})(B_{11} + B_{22}), & M_2 &= (A_{21} + A_{22}) B_{11}, \\
M_3 &= A_{11}(B_{12} - B_{22}), & M_4 &= A_{22}(B_{21} - B_{11}), \\
M_5 &= (A_{11} + A_{12}) B_{22}, & M_6 &= (A_{21} - A_{11})(B_{11} + B_{12}), \\
M_7 &= (A_{12} - A_{22})(B_{21} + B_{22}).
\end{aligned}$$
Then it returns
$$C_{11} = M_1 + M_4 - M_5 + M_7, \quad C_{12} = M_3 + M_5, \quad C_{21} = M_2 + M_4, \quad C_{22} = M_1 - M_2 + M_3 + M_6.$$

## Remarks

The formulas never use commutativity of $R$, which is why they can be applied to blocks.
With $T(n)$ the number of ring operations, $T(n) = 7\,T(n/2) + 18 (n/2)^2$, so $T(n) = O(n^{\log_2 7})$ in [[big O notation|0TGV80E6A4]].
Matrices of other sizes are padded with zero rows and columns up to the next power of $2$.
Example: for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$, the products are $M_1 = 65$, $M_2 = 35$, $M_3 = -2$, $M_4 = 8$, $M_5 = 24$, $M_6 = 22$, $M_7 = -30$, and they give $C = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$, which equals $AB$.
Non-example: the schoolbook recursion with the eight block products $A_{i1} B_{1j} + A_{i2} B_{2j}$ satisfies $T(n) = 8\,T(n/2) + O(n^2)$, which gives only $O(n^3)$.
In practice one switches to the ordinary method below some block size, because the extra additions make the recursion slower for small matrices.
</document>

<document source="https://tamnd.github.io/math2/node/KFCXGKK288.md" tag="KFCXGKK288" kind="concept">
# Gale diagram

## Summary

The Gale diagram, or Gale transform, turns $n$ points spanning $\R^d$ into $n$ vectors in $\R^{n-d-1}$.
The new vectors encode the affine dependences of the old points.
The faces of the [[convex hull|0FQSDC2EQ2]] of the points can be read off from them.
Suppose a [[convex polytope|300K6CB1KH]] has only a few more vertices than its dimension.
Then its Gale diagram lives in a very low dimension and can be drawn.
This makes Gale diagrams a main tool for classifying such polytopes and for building polytopes with unexpected properties.

## Explanations

### L1 human

A polytope in high dimension with few corners is hard to picture directly.
The Gale diagram trades the dimension of the space for the number of extra corners.
For example, a $d$-dimensional polytope with $d + 3$ corners becomes a set of vectors in the plane.
Questions about faces of the polytope then become questions about which vectors surround the origin.
</document>

<document source="https://tamnd.github.io/math2/node/KFHYQM8T7E.md" tag="KFHYQM8T7E" kind="statement">
# Jacobi–Anger expansion

## Statement

Let $J_n$ denote the [[Bessel function of the first kind|CZCMRDWDYW]] of integer order $n$.
For all $z \in \C$ and $\theta \in \R$,
$$e^{iz \cos\theta} = \sum_{n=-\infty}^{\infty} i^n J_n(z)\, e^{in\theta}, \qquad e^{iz \sin\theta} = \sum_{n=-\infty}^{\infty} J_n(z)\, e^{in\theta},$$
and both series converge absolutely and uniformly in $\theta$.

## Remarks

The generating function of the Bessel functions is $\exp\left( \frac{z}{2} \left( t - \frac{1}{t} \right) \right) = \sum_{n \in \Z} J_n(z)\, t^n$ for $t \ne 0$, obtained by multiplying the series of $e^{zt/2}$ and $e^{-z/(2t)}$.
Putting $t = e^{i\theta}$ gives the second formula, since $\frac{1}{2}(t - t^{-1}) = i\sin\theta$.
Replacing $\theta$ by $\theta + \frac{\pi}{2}$ turns $\sin\theta$ into $\cos\theta$ and gives the first formula.
So the expansion is the [[Fourier series|G12KXDHPPS]] of the periodic function $\theta \mapsto e^{iz\sin\theta}$, and its Fourier coefficients are $J_n(z)$.
Example: at $\theta = 0$ the second formula gives $1 = \sum_{n} J_n(z) = J_0(z) + 2\sum_{k \ge 1} J_{2k}(z)$, using $J_{-n} = (-1)^n J_n$.

## Explanations

### L1 human

A plane wave $e^{iz\cos\theta}$ can be broken up into circular pieces, one for each whole number $n$.
The size of the $n$-th piece is the Bessel function $J_n(z)$.
This is why Bessel functions appear whenever a straight wave meets something round.

### L2 human

Taking real and imaginary parts gives expansions of $\cos(z\sin\theta)$ and $\sin(z\sin\theta)$ used in frequency modulation, where the strengths of the sidebands are Bessel values.
Reading off a Fourier coefficient gives Bessel's integral for $J_n$.
In two-dimensional scattering it expands an incident plane wave in cylindrical waves, the first step of the partial-wave method.
Multiplying two such expansions gives addition theorems for Bessel functions.
</document>

<document source="https://tamnd.github.io/math2/node/KFTECAATKG.md" tag="KFTECAATKG" kind="concept">
# Algebraic surface

## Summary

An algebraic surface is an [[algebraic variety|HV6YBS0YDH]] of [[dimension|36X2WTHYCB]] two.
Over the complex numbers a smooth projective algebraic surface is a compact complex manifold of complex dimension two, so of real dimension four.
Surfaces are the first case where birational geometry is subtle, and their classification by Enriques and Kodaira is a model for higher dimensions.

## Explanations

### L1 human

A curve is described by one equation in two variables, and a surface by one equation in three variables, such as $x^2 + y^2 + z^2 = 1$.
Locally a surface looks like a piece of the plane, so it has two directions of freedom.
Over the complex numbers each of these directions counts twice, so a complex surface is a four-dimensional real object.
Examples are the projective plane, the product of two curves, and the zero set of a polynomial in projective three-space.
</document>

<document source="https://tamnd.github.io/math2/node/KFVY0RPGF2.md" tag="KFVY0RPGF2" kind="concept">
# Witt group

## Summary

The Witt group of a field $F$ of characteristic not $2$ is an abelian group built from the nondegenerate [[quadratic forms|0RBQS1JJCR]] over $F$.
Two forms give the same element when they differ only by hyperbolic planes, and the group operation is the orthogonal sum.
It collects the essential information about quadratic forms over $F$ into one algebraic object.
With the tensor product of forms it becomes a ring, the Witt ring.

## Explanations

### L1 human

Quadratic forms can be added by placing them side by side in separate variables.
Some forms, the hyperbolic ones such as $x^2 - y^2$, carry no interesting information, so they are treated as zero.
After this, every form has a negative, namely the form with all signs reversed, and one gets a group.
For the real numbers, the class of a form is given by its signature, the number of plus signs minus the number of minus signs.
So the Witt group of $\R$ is $\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/KFW9841GQK.md" tag="KFW9841GQK" kind="definition">
# Spherical function

## Definition

Let $(G, K)$ be a [[Gelfand pair|W38ZDXDVYC]], and let $dk$ be the [[Haar measure|VY0HXD3GC4]] on $K$ of total mass $1$.
A *spherical function* for $(G, K)$ is a [[continuous function|0P27W9RW0G]] $\varphi \colon G \to \C$ such that:

1. $\varphi(k_1 g k_2) = \varphi(g)$ for all $g \in G$ and $k_1, k_2 \in K$;
2. $\varphi(e) = 1$;
3. $\int_K \varphi(x k y) \, dk = \varphi(x) \varphi(y)$ for all $x, y \in G$.

## Remarks

Example: $G = \R$ and $K = \set{0}$, a Gelfand pair since $\R$ is abelian.
Clause 1 is empty, and clause 3 reads $\varphi(x + y) = \varphi(x)\varphi(y)$.
So the spherical functions are $\varphi(x) = e^{s x}$ with $s \in \C$, and clause 2 holds since $e^0 = 1$.
The positive-definite ones are the unitary characters $e^{i \xi x}$ with $\xi \in \R$.

Non-example: $\varphi(x) = 2 e^{x}$ on $\R$ fails clause 2, and also clause 3, since $\varphi(x + y) \ne \varphi(x)\varphi(y)$.

For $(\mathrm{SO}(3), \mathrm{SO}(2))$, the spherical functions are $g \mapsto P_\ell(\cos \theta(g))$, where $\theta(g)$ is the angle between $g e_3$ and $e_3$.
Here $P_\ell$ is the Legendre polynomial of degree $\ell \ge 0$.

Equivalent form: a continuous $K$-bi-invariant $\varphi \ne 0$ is spherical exactly when $f \mapsto \int_G f(g) \varphi(g^{-1}) \, dg$ is multiplicative on $C_c(K \backslash G / K)$.
If $\pi$ is an irreducible unitary representation with a unit $K$-fixed vector $v$, then $\varphi(g) = \ip{\pi(g) v}{v}$ is a positive-definite spherical function.
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<document source="https://tamnd.github.io/math2/node/KFWQ24D9M6.md" tag="KFWQ24D9M6" kind="concept">
# Finitely additive measure

## Summary

A finitely additive measure is a function on an algebra of sets with values in $[0, \infty]$.
It is $0$ on the empty set and adds over finite disjoint unions.
It is weaker than a measure, which must add over countable disjoint unions.
Finitely additive measures appear in the construction of measures, in densities on the integers and in the duals of spaces of bounded functions.

## Explanations

### L1 human

A finitely additive measure gives sizes to sets so that the size of two separate pieces together is the sum of their sizes.
Nothing is required for infinitely many pieces.
For example, on the natural numbers, give size $0$ to finite sets and size $1$ to sets with finite complement.
Two disjoint pieces cannot both have finite complement, so sizes add.
But $\N$ is the union of the singletons $\set{n}$, each of size $0$, so the rule fails for infinitely many pieces.
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<document source="https://tamnd.github.io/math2/node/KFY58YKR3N.md" tag="KFY58YKR3N" kind="concept">
# Transcritical bifurcation

## Summary

A transcritical bifurcation is a change in a family of dynamical systems at which two branches of [[equilibrium points|0N8NGKDZDZ]] cross and exchange their stability.
Both equilibria exist on both sides of the critical parameter.
It is typical when one equilibrium is forced to exist for all parameter values, such as the zero population in a model of population growth.
See the [[definition|1K8Y6BP097]].

## Explanations

### L1 human

The model is $x' = \mu x - x^2$, with the equilibria $x = 0$ and $x = \mu$.
For $\mu < 0$ the equilibrium $0$ is stable and $\mu$ is unstable.
For $\mu > 0$ the roles are swapped: $0$ is unstable and $\mu$ is stable.
At $\mu = 0$ the two equilibria meet and pass through each other, trading their stability.
In a population model, this is the threshold at which a species can start to grow from small numbers.
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<document source="https://tamnd.github.io/math2/node/KFYW0WN9AD.md" tag="KFYW0WN9AD" kind="definition">
# Completion of a valued field

## Definition

Let $K$ be a field with an absolute value $\abs{\cdot}$, that is, a [[normed field|ZFW9Q3WTAQ]], with the metric $d(x, y) = \abs{x - y}$.
A *completion* of $(K, \abs{\cdot})$ is a field $\widehat{K}$ with an absolute value $\norm{\cdot}$, together with a field homomorphism $\iota \colon K \to \widehat{K}$, such that:

1. $\norm{\iota(x)} = \abs{x}$ for all $x \in K$.
2. $\widehat{K}$ is a [[complete metric space|08EWT7J0SC]] for the metric $(a, b) \mapsto \norm{a - b}$.
3. The image $\iota(K)$ is dense in $\widehat{K}$.

## Remarks

A completion exists and is unique up to a unique isomorphism of fields that preserves the absolute values and commutes with the maps from $K$.
One construction: the [[Cauchy sequences|0QZMTN5AXX]] in $K$ form a ring under termwise operations.
The sequences tending to $0$ form a maximal ideal, and the quotient is a field.
The absolute value of the class of $(x_n)$ is $\lim_n \abs{x_n}$.
As a metric space it is the [[completion of the metric space|0G4J7Z7WPJ]] $K$.
Example: $\R$ with the usual absolute value and the inclusion $\Q \to \R$ is a completion of $\Q$; clause 3 holds because every real number is a limit of rationals.
Example: for a prime $p$, the [[p-adic numbers|BT6T702C28]] $\Q_p$ are the completion of $\Q$ for the [[p-adic absolute value|WGN82TCEF2]].
Non-example: the inclusion $\Q \to \C$ satisfies clauses 1 and 2 for the usual absolute values but fails clause 3, since the closure of $\Q$ in $\C$ is $\R$.
For a [[non-Archimedean absolute value|1H5F1BP6XS]], the completion has the same value group and the same residue field as $K$.
</document>

<document source="https://tamnd.github.io/math2/node/KGC5CYDF53.md" tag="KGC5CYDF53" kind="concept">
# Cayley–Menger determinant

## Summary

The Cayley–Menger determinant of $n + 1$ points is a [[determinant|00KDZRW4C0]] built only from the squares of their pairwise distances.
Up to a constant factor, it equals the square of the $n$-dimensional volume of the [[simplex|YX1DJK2QAG]] with these points as vertices.
It lets one compute volumes, and test whether given distances fit points of Euclidean space, without coordinates.
[[Heron's formula|0FRBZ2X4HJ]] for the area of a triangle is its case $n = 2$.

## Explanations

### L1 human

If you know the six edge lengths of a tetrahedron, you know its shape, so you should be able to compute its volume.
The Cayley–Menger determinant does exactly this.
Put the squared distances in a table, add a border of ones, and take the determinant.
A suitable constant times this number is the squared volume.
For three points on a line the determinant is zero, since the flat triangle they form has area zero.
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<document source="https://tamnd.github.io/math2/node/KGJD4A4ATY.md" tag="KGJD4A4ATY" kind="concept">
# Outer function

## Summary

An outer function is a special holomorphic function on the [[unit disc|8G5ABXRK85]].
It is determined, up to a constant of modulus $1$, by the modulus of its boundary values.
It is the exponential of the holomorphic function whose real part is the Poisson integral of the logarithm of that boundary modulus.
Outer functions are the other half of the [[inner-outer factorization|X6ZQPAQF76]] of functions in [[Hardy spaces|PNNNFJVF6W]].

## Explanations

### L1 human

Let $\varphi$ be a positive weight on the circle whose logarithm is integrable.
There is a holomorphic function with $\abs{F} = \varphi$ on the circle that is as large as possible inside the disc.
It has no zeros, and $\log \abs{F}$ is the harmonic function with boundary values $\log \varphi$.
That function is the outer function with modulus $\varphi$.
For example, $F(z) = 2 + z$ is outer: it has no zeros in the closed disc.
By contrast $z$ is not outer; it is [[inner|NN8XH5RJYB]], and the outer function with the same boundary modulus is the constant $1$.
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<document source="https://tamnd.github.io/math2/node/KGNCXDK4D5.md" tag="KGNCXDK4D5" kind="definition">
# Linear integral equation

## Definition

Let $a < b$, let $a_0, f \colon [a, b] \to \C$ be given functions, let $K \colon [a, b] \times [a, b] \to \C$ be a given [[kernel|GSMBXT7FPK]], and let $\lambda \in \C$.
For each $x \in [a, b]$, let $I_x$ be either $[a, b]$ or $[a, x]$.
A *linear integral equation* for the unknown function $\varphi \colon [a, b] \to \C$ is an [[integral equation|R3SG1SN7JQ]] of the form
$$a_0(x) \varphi(x) = f(x) + \lambda \int_{I_x} K(x, s) \varphi(s) \, ds \quad \text{for all } x \in [a, b].$$

## Remarks

Example: $\varphi(x) = \sin x + \int_0^{\pi} \cos(x - s) \varphi(s) \, ds$ is linear, with $a_0 = 1$, $f = \sin$, $K(x, s) = \cos(x - s)$ and $\lambda = 1$.

Non-example: $\varphi(x) = 1 + \int_0^1 x s \, \varphi(s)^2 \, ds$ is not linear, because $\varphi$ enters through $\varphi(s)^2$.
If $\varphi$ solves it, then $2\varphi$ does not solve the equation with $f$ replaced by $2$.

The equation is of the first kind when $a_0 = 0$ and of the second kind when $a_0 = 1$.
It is of the third kind when $a_0$ is a [[continuous function|0X3WGX95FE]] that vanishes somewhere but not everywhere.
If $a_0$ has no zeros, dividing by $a_0$ gives an equation of the second kind.
In operator notation the equation reads $a_0 \varphi = f + \lambda K \varphi$, where $K$ also denotes the integral operator.
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# Injective algebra

## Summary

An algebra in a class of algebras is injective when every homomorphism into it from a subalgebra extends to the whole algebra.
It is the universal-algebra form of an injective object, with embeddings in the role of monomorphisms.
In the variety of abelian groups the injective algebras are the divisible groups, and in modules they are the [[injective modules|85H7PRHDPZ]].
Injective algebras and injective hulls are tools for embedding algebras into well-behaved ones.

## Explanations

### L1 human

An injective algebra is a target that never blocks an extension.
If a map is defined on part of a bigger algebra, it can always be extended to all of it.
For example, any homomorphism from the subgroup $\Z$ of the rational numbers into $\Q$ extends to a homomorphism from $\Q$ to $\Q$: $n \mapsto nq$ extends to $x \mapsto xq$.
The group $\Z$ itself is not injective: the identity map $\Z \to \Z$ has no extension to $\Q$.
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# Unique ergodicity

## Definition

Let $X$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]] and let $T \colon X \to X$ be a continuous map.
A [[Borel|DYR1QFNGNJ]] [[probability measure|059TJN0NZ1]] $\mu$ on $X$ is $T$-[[invariant|B232EZ1FMB]] if $\mu(T^{-1}A) = \mu(A)$ for every Borel set $A \subseteq X$.
The map $T$ is *uniquely ergodic* if there is exactly one $T$-invariant Borel probability measure on $X$.

## Remarks

By the [[Krylov–Bogolyubov theorem|1DHZ81S0YE]] the set of invariant Borel probability measures is never empty, so unique ergodicity asks that it has a single element.

Example: let $X = \R/\Z$, let $\alpha$ be irrational and let $T(x) = x + \alpha \bmod 1$.
If $\mu$ is invariant, its Fourier coefficients $\hat\mu(h) = \int e^{-2\pi i h x} \, d\mu(x)$ satisfy $\hat\mu(h) = e^{-2\pi i h \alpha} \hat\mu(h)$.
For $h \ne 0$ the factor $e^{-2\pi i h \alpha}$ is not $1$, so $\hat\mu(h) = 0$.
A measure on the circle is determined by its Fourier coefficients, so $\mu$ is Lebesgue measure, and $T$ is uniquely ergodic.

Non-example: the doubling map $T(x) = 2x \bmod 1$ on $\R/\Z$ is continuous and fixes $0$.
Both the point mass at $0$ and Lebesgue measure are invariant, so $T$ is not uniquely ergodic.

Equivalent form: $T$ is uniquely ergodic if and only if, for every continuous $f \colon X \to \R$, the averages $\frac{1}{N} \sum_{n=0}^{N-1} f(T^n x)$ converge uniformly in $x$ to a constant.
The constant is then $\int f \, d\mu$ for the unique invariant measure $\mu$.
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# Mosco convergence

## Summary

Mosco convergence is a convergence of functionals on a normed space that combines the weak and the strong topology.
It asks for the lower bound inequality of [[Γ-convergence|NW9MM370A5]] along weakly convergent sequences and for recovery sequences that converge strongly.
It was introduced by Umberto Mosco to study the stability of variational inequalities when the constraint sets change.
In finite dimension it is the same as Γ-convergence; in general it implies Γ-convergence for both the weak and the norm topology.
The [[definition|RNDKD74V17]] gives the precise conditions.

## Explanations

### L1 human

In an infinite-dimensional space, bounded sequences often have no strongly convergent subsequence, but they have weakly convergent ones.
So a useful lower bound must hold along weakly convergent sequences, while a useful approximation should be as good as possible, that is strong.
Mosco convergence asks for exactly this pair of conditions.
For closed convex sets $K_n$ it says two things.
Weak limits of points of $K_n$ lie in the limit set $K$, and each point of $K$ is a strong limit of points of $K_n$.
For example, the segments $K_n = [0, 1 - 1/n]$ in $\R$ converge in this sense to $[0, 1]$.
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# Orbifold

## Definition

Let $n \ge 0$ be an integer and let $X$ be a [[Hausdorff|0MXB4CKHFW]] and [[second-countable|00RE1R4W7N]] topological space.
An *orbifold chart* of dimension $n$ on an open set $U \subseteq X$ is a triple $(\tilde U, G, \varphi)$ with these parts.
The set $\tilde U$ is a connected open subset of $\R^n$.
The group $G$ is finite and has a [[faithful|0B0WE1GKTK]] [[action|0YSE802YCQ]] on $\tilde U$ by diffeomorphisms.
The map $\varphi \colon \tilde U \to U$ is continuous, and $\varphi(g x) = \varphi(x)$ for all $g \in G$ and $x \in \tilde U$.
The induced map from the [[quotient space|0V8046R0P9]] $\tilde U / G$ to $U$ is a [[homeomorphism|05GZ11PWRY]].
An *embedding* of a chart $(\tilde U_i, G_i, \varphi_i)$ into a chart $(\tilde U_j, G_j, \varphi_j)$ is a smooth embedding $\psi \colon \tilde U_i \to \tilde U_j$ with $\varphi_j \circ \psi = \varphi_i$.
An *orbifold atlas* of dimension $n$ on $X$ is a family of charts $(\tilde U_i, G_i, \varphi_i)$ with these properties:

1. The sets $U_i = \varphi_i(\tilde U_i)$ cover $X$.
2. For any two charts and any point $x \in U_i \cap U_j$, there is a chart $(\tilde U_k, G_k, \varphi_k)$ of the atlas with $x \in U_k \subseteq U_i \cap U_j$.
3. Whenever $U_i \subseteq U_j$, there is an embedding of the chart $(\tilde U_i, G_i, \varphi_i)$ into the chart $(\tilde U_j, G_j, \varphi_j)$.

An atlas $\mathcal V$ *refines* an atlas $\mathcal U$ if every chart of $\mathcal V$ has an embedding into some chart of $\mathcal U$.
Two atlases are equivalent if they have a common refinement.
An $n$-dimensional *orbifold* is a space $X$ as above together with an equivalence class of $n$-dimensional orbifold atlases.

## Remarks

An embedding $\psi$ of charts induces an injective [[group homomorphism|0HV9Y7WG5R]] $f \colon G_i \to G_j$ with $\psi(g x) = f(g) \psi(x)$ for all $g \in G_i$ and $x \in \tilde U_i$.
Let $x = \varphi(\tilde x)$ for a chart $(\tilde U, G, \varphi)$.
Up to isomorphism, the stabilizer of $\tilde x$ in $G$ does not depend on the chart; it is the *local group* of $x$.
The point $x$ is *singular* when its local group is not trivial.
Example: let $m \ge 2$ and let $G = \Z / m\Z$ act on $\R^2$ by rotations through multiples of $2\pi / m$.
The quotient $X = \R^2 / G$ is Hausdorff and second countable.
The single chart $(\R^2, G, \varphi)$, with the quotient map $\varphi$, is an atlas: clauses 2 and 3 only ask for the chart itself.
So $X$ is an orbifold, the cone with one cone point of order $m$.
Non-example: let the group $\Q$ act on $\R$ by translations.
Every orbit is dense, so the quotient $\R / \Q$ carries the trivial topology and is not Hausdorff; it is not an orbifold.
A [[smooth manifold|R2TQ82PCNB]] is an orbifold all of whose local groups are trivial.
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# Lie–Trotter product formula

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]].
Let $A$ and $B$ be [[self-adjoint operators|0NEGN5CRSC]] on $H$, with domains $D(A)$ and $D(B)$.
Assume that $A + B$ is [[essentially self-adjoint|DT49GKR9AS]] on $D(A) \cap D(B)$, and let $C$ be its closure.
Then for every $t \in \R$ and every $x \in H$,
$$e^{itC}x = \lim_{n \to \infty} \left( e^{itA/n} e^{itB/n} \right)^n x,$$
where the unitary groups $e^{itA}$, $e^{itB}$ and $e^{itC}$ are those of [[Stone's theorem|DVGQB92HPJ]].

## Remarks

For $n \times n$ matrices this is the [[Lie product formula|HK3BRMK912]], and the proof idea is the same.
The difference $e^{isA}e^{isB} - e^{isC}$ is $o(s)$ as $s \to 0$ when applied to a vector of $D(A) \cap D(B)$.
A telescoping sum turns $n$ such errors of size $o(1/n)$ into a total error that tends to $0$.
In infinite dimensions, the convergence is in the strong sense, vector by vector, and the essential self-adjointness of $A + B$ is used to pass from $D(A) \cap D(B)$ to general $x$.
A similar formula holds for the semigroups $e^{-tA}$ and $e^{-tB}$ with $t \ge 0$, when $A$ and $B$ are bounded below and $A + B$ is essentially self-adjoint on $D(A) \cap D(B)$.

Example: in quantum mechanics with $A = -\Delta$ and $B = V$ a suitable potential, $e^{-itA/n}$ acts by a Fourier multiplier and $e^{-itV/n}$ by multiplication.
The formula approximates the Schrödinger evolution by alternating free motion and potential kicks; this is the basis of split-step methods.

## Explanations

### L1 human

For numbers, $e^{a + b} = e^{a}e^{b}$, but for operators that do not commute this fails.
The product formula repairs this: alternating many small steps of $A$ and of $B$ gives the evolution of $A + B$ in the limit.
It is like crossing a field diagonally by many tiny steps east and north in turn.

### L2 human

The result is due to Trotter, who extended Lie's formula for matrices to unbounded operators.
It is used to build the Feynman–Kac formula: for $-\Delta + V$, the alternating products become integrals over paths of Brownian motion.
In numerical analysis it justifies operator splitting, where a hard evolution is approximated by easy ones.
The hypothesis that $A + B$ is essentially self-adjoint on $D(A) \cap D(B)$ ensures that the limit is the evolution of a well-defined Hamiltonian.
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<document source="https://tamnd.github.io/math2/node/KHA4WMRDKY.md" tag="KHA4WMRDKY" kind="definition">
# Instrumental variables estimation

## Definition

Consider the linear model $y_i = x_i^T \beta + \varepsilon_i$, $i = 1, \dots, n$, with $x_i \in \R^k$, $\beta \in \R^k$ unknown and $E[\varepsilon_i] = 0$, where $E[x_i \varepsilon_i] \ne 0$ is allowed.
A random vector $z_i \in \R^L$ with $L \ge k$ is a vector of *instruments* if:

1. (exogeneity) $E[z_i \varepsilon_i] = 0$;
2. (relevance) the $L \times k$ matrix $E[z_i x_i^T]$ has [[rank|0ZC9YTXZRY]] $k$.

Let $X$, $Z$ and $y$ stack the $x_i^T$, $z_i^T$ and $y_i$ as rows, and assume $Z^T Z$ and $X^T P_Z X$ are invertible, where $P_Z = Z (Z^T Z)^{-1} Z^T$.
The *two-stage least squares* or *instrumental variables estimator* is
$$\hat\beta_{\mathrm{IV}} = (X^T P_Z X)^{-1} X^T P_Z y.$$
When $L = k$ and $Z^T X$ is invertible, it reduces to $\hat\beta_{\mathrm{IV}} = (Z^T X)^{-1} Z^T y$.

## Remarks

The name two-stage least squares comes from its computation.
First regress each column of $X$ on $Z$, which gives $\hat X = P_Z X$, the [[orthogonal projection|0B8KZRT28M]] of $X$ onto the column space of $Z$.
Then regress $y$ on $\hat X$ by [[ordinary least squares|YM708KHV2W]].
Under clauses 1 and 2, for independent and identically distributed observations with finite second moments, $\hat\beta_{\mathrm{IV}}$ is [[consistent|0AS800VAYX]] for $\beta$, while ordinary least squares in general is not.

Example: $k = L = 1$, no intercept, $y_i = \beta x_i + \varepsilon_i$.
Exogeneity gives $E[z_i y_i] = \beta E[z_i x_i]$, so $\beta = E[z_i y_i] / E[z_i x_i]$.
The estimator $\sum_i z_i y_i / \sum_i z_i x_i$ is the sample analogue.

Non-example: if $z_i$ is independent of $x_i$ and $E[z_i] = 0$, then $E[z_i x_i] = 0$ and clause 2 fails; the ratio above has a denominator tending to $0$ and does not identify $\beta$.
Instruments that satisfy clause 2 only barely, called weak instruments, give unreliable estimates in finite samples.
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# Compressed sensing

## Definition

Let $m < N$ and $s \ge 1$ be integers and let $A$ be a real $m \times N$ [[matrix|0VD9ZSWWJG]], called the *measurement matrix*.
A vector $x \in \R^N$ is *$s$-sparse* if it has at most $s$ nonzero entries.
*Compressed sensing* is the problem of recovering an unknown $s$-sparse vector $x$ from its measurement vector $y = Ax \in \R^m$.
Its standard method is *basis pursuit*: output a solution of
$$\min\set{\norm{z}_1 : z \in \R^N, \ Az = y}, \qquad \norm{z}_1 = \textstyle\sum_{j=1}^N \abs{z_j} .$$
The matrix $A$ *allows recovery of $s$-sparse vectors* if every $s$-sparse $x$ is the unique solution of this problem with $y = Ax$.

## Remarks

It is the case $\varepsilon = 0$ of [[sparse approximation|WC8P971XGW]], restricted to data that come from a sparse vector.
Suppose every $2s$ columns of $A$ are linearly independent.
Then an $s$-sparse $x$ is the only $s$-sparse solution of $Az = Ax$, since two such solutions differ by a $2s$-sparse vector in the kernel of $A$.
A matrix with the [[restricted isometry property|B0K18PWN2A]] of order $2s$ and constant $\delta_{2s} < \sqrt{2} - 1$ allows recovery of $s$-sparse vectors by basis pursuit.
Example: $A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$, $s = 1$, $x = (0, 0, 2)$, $y = (2, 2)$.
The solutions of $Az = y$ are $z = (2 - t, 2 - t, t)$, and $\norm{z}_1 = 2\abs{2 - t} + \abs{t}$ is minimal exactly at $t = 2$, so basis pursuit returns $x$.
Non-example: for $A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and $s = 1$, the $1$-sparse vectors $e_1$ and $e_2$ give the same $y = (1, 0)$, so no method can recover both.
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# Mutually orthogonal Latin squares

## Definition

Let $n$ be a positive integer.
Let $L = (L_{ij})$ and $M = (M_{ij})$ be [[Latin squares|44HTG620GB]] of order $n$, on symbol sets $S$ and $T$ with $n$ elements each.
The squares $L$ and $M$ are orthogonal if the $n^2$ ordered pairs $(L_{ij}, M_{ij})$, for $1 \le i, j \le n$, are pairwise distinct.
A family of Latin squares of order $n$ is mutually orthogonal if each two distinct squares of the family are orthogonal.

## Remarks

Since $S \times T$ has $n^2$ elements, orthogonality means that every pair in $S \times T$ occurs exactly once.
Example of order $3$: for $i, j \in \Z/3\Z$, let $L_{ij} = i + j$ and $M_{ij} = 2i + j$.
If $(L_{ij}, M_{ij}) = (s, t)$, then $i = t - s$ and $j = 2s - t$, so each pair $(s, t)$ comes from exactly one cell, and $L$ and $M$ are orthogonal.
Non-example: a Latin square of order $n \ge 2$ is never orthogonal to itself, because only the $n$ pairs $(s, s)$ occur.
There are no orthogonal Latin squares of order $2$, and there is no orthogonal pair of order $6$; this answers [[Euler's thirty-six officers problem|WXJAVQKFR6]].
For a prime power $q$ and the [[finite field|08RTDHX1SK]] $\F_q$, the squares $L^{(a)}_{xy} = ax + y$ for $a \in \F_q \setminus \set{0}$ form $q - 1$ mutually orthogonal Latin squares of order $q$.
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# Curve complex

## Summary

The curve complex of a [[surface|VGJJVBZ1MQ]] is a [[simplicial complex|V8GZ0W1ZZ5]].
Its vertices are the isotopy classes of essential simple closed curves, and its simplices are sets of curves that can be made pairwise disjoint.
The [[mapping class group|JV5NYG913V]] acts on it by simplicial automorphisms.
Harvey introduced it, and Masur and Minsky proved that it is hyperbolic in the sense of Gromov.
It is a central tool in the study of mapping class groups and of Teichmüller space.

## Explanations

### L1 human

Draw all the essentially different loops on a surface, one dot for each.
Join two dots by an edge when the two loops can be drawn without crossing.
Fill in a triangle when three loops can be drawn pairwise without crossing, and so on.
The resulting shape is infinite and tangled, but it records how all loops on the surface sit with respect to each other.
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# Barycentric coordinates

## Definition

Let $A$ be an [[affine space|09XNAPTJQ7]] of dimension $n$ over a field $k$.
Let $p_0, \dots, p_n$ be [[affinely independent|JWK49HMHYT]] points of $A$.
For every point $x$ of $A$ there is a unique tuple $(\lambda_0, \dots, \lambda_n) \in k^{n+1}$ such that
$$\lambda_0 + \cdots + \lambda_n = 1 \quad \text{and} \quad x = \lambda_0 p_0 + \cdots + \lambda_n p_n,$$
where the right side is the [[affine combination|0KJH6SRFZK]] $p_0 + \sum_{i=1}^{n} \lambda_i (p_i - p_0)$.
The numbers $\lambda_0, \dots, \lambda_n$ are the *barycentric coordinates* of $x$ with respect to $p_0, \dots, p_n$.

## Remarks

Example: in $\R^2$ take $p_0 = (0, 0)$, $p_1 = (1, 0)$ and $p_2 = (0, 1)$.
The point $(x, y)$ has barycentric coordinates $(1 - x - y, x, y)$.
Check: the weights add up to $1$, and $(1 - x - y)(0, 0) + x(1, 0) + y(0, 1) = (x, y)$.
The centroid $(1/3, 1/3)$ gets the coordinates $(1/3, 1/3, 1/3)$.

Non-example: the points $(0, 0)$, $(1, 0)$, $(2, 0)$ of $\R^2$ are collinear, so they are not affinely independent.
For them the coordinates are not unique, and the point $(0, 1)$ has none at all.

When $k$ is $\R$, a point lies in the closed simplex with vertices $p_0, \dots, p_n$ exactly when all its barycentric coordinates are nonnegative.
Homogeneous barycentric coordinates drop the condition $\sum \lambda_i = 1$: any tuple with nonzero sum is accepted, and two tuples that differ by a nonzero factor give the same point.
</document>

<document source="https://tamnd.github.io/math2/node/KHJ2G2R5G7.md" tag="KHJ2G2R5G7" kind="concept">
# Strong operator topology

## Summary

The strong operator topology is the topology of pointwise convergence on the bounded operators on a Hilbert space.
A net $T_i$ converges to $T$ when $T_i x \to T x$ in norm for every vector $x$.
It lies between the [[weak operator topology|4XM1VN1C1N]] and the norm topology.
It is the natural topology for limits such as partial sums of orthogonal projections and for one-parameter unitary groups.

## Explanations

### L1 human

Two operators are close in the strong operator topology when they move each of finitely many chosen vectors to nearby places.
This is weaker than being close in norm, which asks for the same accuracy on all unit vectors at once.
For example, let $P_n$ keep the first $n$ coordinates of a sequence and set the rest to $0$.
For each fixed sequence $x$ in $\ell^2$, the truncation $P_n x$ tends to $x$, so $P_n \to I$ strongly.
But $P_n$ never comes close to $I$ in norm, since $I - P_n$ still keeps the $(n+1)$-st basis vector unchanged.
</document>

<document source="https://tamnd.github.io/math2/node/KHMT0E4BRK.md" tag="KHMT0E4BRK" kind="concept">
# Nichols algebra

## Summary

The Nichols algebra $\mathfrak{B}(V)$ of a braided vector space $V$ is a graded braided Hopf algebra generated by $V$.
Its only primitive elements are those of $V$.
It is the quotient of the [[tensor algebra|P4J5JZDW0B]] $T(V)$ by the kernels of the quantum symmetrizers.
These symmetrizers are built from the action of the [[braid group|BNR4H5GCTP]] on $V^{\otimes n}$.
Over a field of characteristic $0$, for the ordinary flip it is the symmetric algebra, and for minus the flip it is the [[exterior algebra|0MDMZ965VE]].
The positive part of a [[quantum enveloping algebra|1V8H34F6N9]] is a Nichols algebra, and Nichols algebras are central in the classification of pointed Hopf algebras.

## Explanations

### L1 human

Start with a vector space and a rule $c$ for swapping two vectors, which may twist them in the process.
Form all words in the vectors, and declare a word zero whenever "symmetrizing it with the twisted swap" gives zero.
What remains is the Nichols algebra.
With the ordinary swap this gives polynomials, where $xy = yx$, and with minus the swap it gives the exterior algebra, where $xy = -yx$.
For a single vector $x$ whose swap multiplies by a root of unity $q$ of order $N$, the result is $k[x]/(x^N)$.
</document>

<document source="https://tamnd.github.io/math2/node/KHN8H3SNSM.md" tag="KHN8H3SNSM" kind="statement">
# Abel transform as the projection of a radial function

## Statement

Let $f \colon [0, \infty) \to \C$ be continuous with $\int_0^\infty \abs{f(r)}\, dr < \infty$.
Let $g(x, y) = f\big(\sqrt{x^2 + y^2}\big)$ be the radial function on $\R^2$ with profile $f$.
Then for every $y \in \R$ the integral $\int_\R g(x, y)\, dx$ converges absolutely, and
$$\int_\R g(x, y)\, dx = F(\abs{y}),$$
where $F$ is the [[Abel transform|QMW1MWQH6J]] of $f$.

## Remarks

Since $g(x, y)$ is even in $x$, the line integral is $2 \int_0^\infty f\big(\sqrt{x^2 + y^2}\big)\, dx$.
For $y \ne 0$, the substitution $r = \sqrt{x^2 + y^2}$, with $dx = \frac{r\, dr}{\sqrt{r^2 - y^2}}$, maps $x \in (0, \infty)$ onto $r \in (\abs{y}, \infty)$ and gives the formula; for $y = 0$ it is immediate.
The absolute convergence follows the same way: $\frac{r}{\sqrt{r^2 - y^2}}$ is integrable near $r = \abs{y}$ and tends to $1$ as $r \to \infty$, while $f$ is bounded near $\abs{y}$ and integrable.
By rotation invariance, the integral of $g$ along any line at distance $\abs{y}$ from the origin is also $F(\abs{y})$; this is the [[Radon transform|VJNZJ352PC]] of $g$.

Example: for $f(r) = e^{-r^2}$ the line integral is $\int_\R e^{-x^2 - y^2}\, dx = \sqrt{\pi}\, e^{-y^2}$, which is the Abel transform of $f$ at $\abs{y}$.

## Explanations

### L1 human

A round object seen from the side looks the same from every direction.
The Abel transform gives what one sees: the total along each line of sight as a function of the distance of the line from the centre.
This statement is the computation behind that picture.

### L2 human

The statement identifies the Abel transform with the two-dimensional Radon transform restricted to radial functions.
Together with the projection-slice theorem, it shows that the Fourier transform of the Abel transform of $f$ is the [[Hankel transform|2P9YWDGKC4]] of order $0$ of $f$, up to constants.
This chain is used in tomography of axially symmetric objects, where only one projection is needed.
</document>

<document source="https://tamnd.github.io/math2/node/KHRTBP37VD.md" tag="KHRTBP37VD" kind="concept">
# Laplace–Stieltjes transform

## Summary

The Laplace–Stieltjes transform of a function $g$ on $[0, \infty)$ is $\int_0^\infty e^{-s t} \, dg(t)$, a [[Riemann–Stieltjes integral|JT9AFHG0NA]] against $g$ instead of an ordinary integral of $g$.
It extends the [[Laplace transform|0V49JEZW06]]: when $g$ has a density $f$, it is the Laplace transform of $f$.
It also handles jumps of $g$, that is, point masses.
In probability, applied to the distribution function of a nonnegative random variable $X$, it gives $\mathbb{E}[e^{-s X}]$.

## Explanations

### L1 human

Think of $g$ as describing how mass is laid out along the half-line: $g(t)$ is the total mass in $[0, t]$.
The transform weights each bit of mass at position $t$ by $e^{-s t}$ and adds up.
A single point mass at $a$ gives just $e^{-a s}$.
A uniform spread of mass with density $1$ gives $1/s$, the ordinary Laplace transform of the constant $1$.
</document>

<document source="https://tamnd.github.io/math2/node/KHTCR3VFST.md" tag="KHTCR3VFST" kind="definition">
# Finitely presented group

## Definition

Let $G$ be a [[group|G5S1M9]].
For a set $S$ and a subset $R$ of the [[free group|0KTW26C80J]] $F(S)$, let $N(R)$ be the [[normal closure|6XM4CCVTKE]] of $R$ in $F(S)$.
The group $G$ is *finitely presented* if it is [[isomorphic|08MG7H8JW5]] to $\langle S \mid R \rangle = F(S) / N(R)$ for some finite $S$ and finite $R$.

## Remarks

The pair $(S, R)$ is a finite [[presentation|0JGEQXQ4AC]] of $G$.
Example: $\Z^2 \cong \langle a, b \mid a b a^{-1} b^{-1} \rangle$; the map $a \mapsto (1, 0)$, $b \mapsto (0, 1)$ induces the isomorphism, since the quotient is abelian and generated by two elements, and $\Z^2$ is the free abelian group on them.
Every finite group $G$ is finitely presented, with $S = G$ and $R = \set{ g h k^{-1} : g, h, k \in G, \ gh = k \text{ in } G}$.
Non-example: the additive group $\Q$ is not finitely presented; it fails already because it is not [[finitely generated|082DWWXMRJ]], since a subgroup generated by finitely many fractions has bounded denominators.
A finitely generated group need not be finitely presented; the [[lamplighter group|4GNPFMT7DZ]] $\Z/2 \wr \Z$ is such an example.
If $G$ is finitely presented, then for every finite generating set $S$ of $G$ there is a finite set of relations $R$ with $G \cong \langle S \mid R \rangle$.
</document>

<document source="https://tamnd.github.io/math2/node/KHXYWTNN2R.md" tag="KHXYWTNN2R" kind="statement">
# Ill-posedness of first-kind equations

## Statement

Let $X$ and $Y$ be [[normed vector spaces|03J5RAKB30]] with $X$ infinite-dimensional, and let $K \colon X \to Y$ be a [[compact linear operator|M2NEP1K5AE]].
Then there is no constant $c > 0$ with $\norm{K x} \ge c \norm{x}$ for all $x \in X$.
Consequently, the equation of the first kind $K x = y$ is not [[well posed|W09B9NZ9B5]].
That is, $K$ is not injective, or $K$ is not surjective onto $Y$, or the inverse $K^{-1} \colon K(X) \to X$ is not continuous.

## Remarks

Proof idea: if $\norm{K x} \ge c \norm{x}$ for all $x$, then $K$ is injective and $K^{-1}$ is bounded on $K(X)$ with norm at most $1/c$.
The identity of $X$ equals $K^{-1} K$, the product of a bounded and a compact operator, so it is compact.
Then the closed unit ball of $X$ is compact, which contradicts [[Riesz's lemma|097C7KCAT5]] since $X$ is infinite-dimensional.

The main application is to [[integral equations of the first kind|ZDS77Q0F06]].
Let $k$ be continuous on $[0, 1]^2$ and $(K x)(t) = \int_0^1 k(t, s) \, x(s) \, ds$ on $X = Y = L^2(0, 1)$; this operator is compact.
For $x_n(s) = \sin(n \pi s)$ one has $\norm{x_n}_2 = 1/\sqrt{2}$, but $(K x_n)(t) \to 0$ uniformly in $t$ by the [[Riemann–Lebesgue lemma|WVKDC011KP]] and the uniform continuity of $k$.
So the data $y + K x_n$ are close to $y$ while the solutions $x + x_n$ stay far from $x$.

The hypothesis that $X$ is infinite-dimensional is needed.
On a finite-dimensional space every linear operator is compact, and an invertible matrix gives a well-posed equation.

## Explanations

### L1 human

An integral operator with a smooth kernel averages out fast oscillations.
So two very different functions, one with an added fast wiggle, are sent to almost the same image.
Going backwards, from the image to the function, therefore amplifies small errors in the data without bound.
This is why first-kind integral equations, unlike second-kind ones, are ill posed.

### L2 human

The theorem explains why [[inverse problems|PCMW7KN5P4]] modelled by first-kind equations, such as deblurring an image or recovering a source from far-field measurements, need regularization.
Its quantitative form uses the [[singular values|QSJKPKPM7H]] $\sigma_n \to 0$ of $K$: the error in the component along the $n$-th singular vector is multiplied by $1/\sigma_n$.
[[Picard's criterion|MPB3QF09E5]] describes exactly which data lie in the range, and [[Tikhonov regularization|M68V0M5A3R]] gives stable approximate solutions.
By contrast, a second-kind equation $x - K x = y$ with compact $K$ is well posed as soon as it is uniquely solvable, by the [[Fredholm alternative|4HH8C9V8JQ]].
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<document source="https://tamnd.github.io/math2/node/KHY35P4AKF.md" tag="KHY35P4AKF" kind="concept">
# Lusin area integral

## Summary

The Lusin area integral is a square function built from the harmonic extension $u$ of a function $f$ on $\R^n$ to the upper half-space.
At a point $x$ it integrates $\abs{\nabla u}^2$, suitably weighted, over the cone of points $(y, t)$ with $\abs{y - x} < t$.
For $1 < p < \infty$ its $L^p$ norm is comparable to that of $f$, as for the [[Littlewood–Paley square function|H4E1PW67Z1]].
It was introduced by Lusin for holomorphic functions on the disc, and it plays a key role in the real-variable theory of Hardy spaces.

## Explanations

### L1 human

Extend $f$ to a harmonic function $u$ one dimension up, using the Poisson kernel.
Over each point $x$ stand an upside-down cone with vertex at $x$, and measure how much $u$ varies inside the cone.
The total variation, measured by a weighted integral of $\abs{\nabla u}^2$, is the area integral at $x$.
For a holomorphic function $F$ on the unit disc, one integrates $\abs{F'}^2$ over a cone-shaped region.
The result is the area of the image of that region, counted with multiplicity, which explains the name.
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<document source="https://tamnd.github.io/math2/node/KHZ89X6Y8R.md" tag="KHZ89X6Y8R" kind="definition">
# Words and profiles

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]] with [[chromatic number|0VHJRXNAAS]] $\chi(G)$. Let $\mathcal C$ be a set of colors with $|\mathcal C|=\chi(G)$.

**Words and profiles** are defined as follows.

A word on $G$ is any map $w:V(G)\to\mathcal C$. The word need not be a proper [[colouring|0NCPXH194Y]]: adjacent vertices may get the same color.

The profile of a word $w$ is the vector $\mathbf m(w)=(m_X(w))_{X\in\mathcal C}$. Its entry $m_X(w)=|w^{-1}(X)|$ is the number of vertices of color $X$.

Vectors are written in bold, and inequalities between vectors hold in every coordinate. The vector $\mathbf e_X$ has entry $1$ at $X$ and $0$ elsewhere. The vector $\mathbf 1$ has every entry equal to $1$.

Let $\mathcal P(G)$ be the set of proper colorings of $G$ with palette $\mathcal C$. Since $|\mathcal C|=\chi(G)$, each coloring in $\mathcal P(G)$ uses every color.

A profile $\mathbf m$ is feasible if $\mathbf m=\mathbf m(w)$ for some $w\in\mathcal P(G)$.
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<document source="https://tamnd.github.io/math2/node/KJ11DAXMR0.md" tag="KJ11DAXMR0" kind="concept">
# Étale fundamental group

## Summary

The étale fundamental group of a connected scheme classifies its finite étale covers, as the [[fundamental group|S1XK3HXHFX]] of a space classifies its [[covering spaces|3XZRRE165N]].
It is a [[profinite group|KA0YDB6QXE]].
For the spectrum of a field it is the [[absolute Galois group|DPFMXS71D1]].
For a complex variety it is the profinite completion of the topological fundamental group.
It unites Galois theory and the theory of coverings in one framework, [[Grothendieck's Galois theory|CX7BDQ47ZK]].

## Explanations

### L1 human

A loop in a space can be detected by the covers it produces: going around the loop permutes the sheets of a cover.
In algebraic geometry one cannot draw loops, but one can still study finite covers.
The étale fundamental group is the group of symmetries of all finite covers at once.
For the punctured complex line, the finite covers are $z \mapsto z^n$, and the group is the profinite completion $\widehat{\Z}$ of $\Z$.
For a field $k$, the finite covers are the finite separable field extensions, so the group is the Galois group of the separable closure.
</document>

<document source="https://tamnd.github.io/math2/node/KJ2984R97Y.md" tag="KJ2984R97Y" kind="definition">
# Volterra equation of convolution type

## Definition

Let $T > 0$ or $T = \infty$, let $k \colon (0, T) \to \C$ be locally integrable, let $f \colon [0, T) \to \C$ be a given function, and let $\lambda \in \C$.
A *Volterra equation of convolution type* is a [[Volterra integral equation|DND8CBEVXT]] whose kernel is a [[difference kernel|B5AB451TB7]] $K(t, s) = k(t - s)$.
In the second kind it reads
$$x(t) = f(t) + \lambda \int_0^t k(t - s) \, x(s) \, ds, \quad 0 \le t < T,$$
and in the first kind it reads
$$\int_0^t k(t - s) \, x(s) \, ds = f(t), \quad 0 \le t < T.$$

## Remarks

The integral is the [[convolution|2Z29JN3VES]] $(k * x)(t) = \int_0^t k(t - s) \, x(s) \, ds$ of functions on $[0, \infty)$, so the second kind is written $x = f + \lambda \, k * x$.
Example: $x(t) = 1 + \int_0^t e^{-(t - s)} x(s) \, ds$, with $k(u) = e^{-u}$.
The function $x(t) = 1 + t$ solves it: $\int_0^t e^{-(t - s)} (1 + s) \, ds = t$, by the substitution $u = t - s$ and $\int_0^t e^{-u}(1 + t - u) \, du = t$.
Non-example: $x(t) = 1 + \int_0^t t s \, x(s) \, ds$ has kernel $K(t, s) = ts$, which is not a function of $t - s$ alone.
The [[Laplace transform method|EHGECREYB4]] solves convolution equations on $[0, \infty)$ when $k$ and $f$ have Laplace transforms.
</document>

<document source="https://tamnd.github.io/math2/node/KJ2NZCB7TH.md" tag="KJ2NZCB7TH" kind="definition">
# Free object

## Definition

Let $\mathcal{C}$ be a [[concrete category|FYBRP16KAN]] with forgetful functor $U \colon \mathcal{C} \to \mathbf{Set}$, and let $X$ be a set.
A *free object* on $X$ is an object $A$ of $\mathcal{C}$ with a map of sets $i \colon X \to U A$ with this [[universal property|EN4DD2FAJH]].
For every object $B$ of $\mathcal{C}$ and every map $f \colon X \to U B$, there is a unique morphism $g \colon A \to B$ with $U(g) \circ i = f$.

## Remarks

A free object on $X$ is unique up to a unique isomorphism compatible with $i$.
If every set has a free object $F(X)$, then $F$ is a functor and is left adjoint to $U$.

Example: in the category of groups, the free group $F(X)$ of reduced words in $X \cup X^{-1}$, with $i$ sending $x$ to the one-letter word, is a free object.
Given $f \colon X \to U B$, the morphism $g$ sends a reduced word $x_1^{\epsilon_1} \cdots x_n^{\epsilon_n}$ to $f(x_1)^{\epsilon_1} \cdots f(x_n)^{\epsilon_n}$, and it is the only homomorphism extending $f$.

Non-example: the group $\Z/2\Z$ with $i$ sending the point of $X = \set{x}$ to $1$ is not free on $X$.
For $B = \Z$ and $f(x) = 1$, there is no homomorphism $g \colon \Z/2\Z \to \Z$ with $g(1) = 1$, since $2 g(1) = g(0) = 0$.
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<document source="https://tamnd.github.io/math2/node/KJ9MMP99G6.md" tag="KJ9MMP99G6" kind="concept">
# Tight closure

## Summary

Tight closure is an operation on ideals of Noetherian rings of prime characteristic $p$, defined with the [[Frobenius endomorphism|0T3WVTD675]].
It was introduced by Melvin Hochster and Craig Huneke.
The tight closure $I^*$ of an ideal $I$ contains $I$ and is contained in the integral closure of $I$.
In a regular ring, $I^* = I$.
It gives short proofs of results such as the [[Hochster–Roberts theorem|56107GMFYV]] and the Briançon–Skoda theorem.
It also leads to the classes of F-regular and F-rational rings.

## Explanations

### L1 human

In characteristic $p$, raising to the $p$-th power is a ring homomorphism, and one can raise elements to the powers $q = p, p^2, p^3, \ldots$.
An element $x$ is in the tight closure of $I$ when its powers $x^q$ almost lie in the ideal generated by the $q$-th powers of elements of $I$.
"Almost" means: up to one fixed multiplier $c$ that is not too degenerate.
Elements of $I$ are in it, and so are elements that differ from $I$ by something "infinitely small", such as a nilpotent element.
In a polynomial ring, nothing else is added: every ideal is tightly closed.
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<document source="https://tamnd.github.io/math2/node/KJ9ZQWYBY6.md" tag="KJ9ZQWYBY6" kind="concept">
# Observed information

## Summary

The observed information is minus the second derivative of the log-likelihood of the data actually observed.
It is usually evaluated at the [[maximum likelihood estimate|0FREPFKNS7]].
It is a data-based counterpart of the [[Fisher information|G1Z3WSTE8Q]], whose expected value it equals under regularity conditions.
It gives standard errors for maximum likelihood estimates.

## Explanations

### L1 human

Draw the log-likelihood as a curve with a peak at the best estimate.
The observed information is the curvature at that peak.
A sharp peak means the data strongly favour a narrow range of values, so the estimate is precise.
For example, with Poisson counts of total $t$ in $n$ observations, the observed information at the estimate $t/n$ is $n^2/t$.
</document>

<document source="https://tamnd.github.io/math2/node/KJB47D7Z05.md" tag="KJB47D7Z05" kind="definition">
# Functor

## Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be [[categories|A2DE66937D]].
A *functor* $F \colon \mathcal{C} \to \mathcal{D}$ consists of:

1. for each object $A$ of $\mathcal{C}$, an object $F(A)$ of $\mathcal{D}$;
2. for each [[morphism|859F0TFVA3]] $f \colon A \to B$ of $\mathcal{C}$, a morphism $F(f) \colon F(A) \to F(B)$ of $\mathcal{D}$;

such that:

1. $F(1_A) = 1_{F(A)}$ for every object $A$ of $\mathcal{C}$;
2. $F(g \circ f) = F(g) \circ F(f)$ for all $f \colon A \to B$ and $g \colon B \to C$ in $\mathcal{C}$.

## Remarks

Such a functor is also called *covariant*.
Functors compose, and every category has an identity functor.
A functor sends isomorphisms to isomorphisms: if $g \circ f = 1_A$ and $f \circ g = 1_B$, then $F(g) \circ F(f) = 1_{F(A)}$ and $F(f) \circ F(g) = 1_{F(B)}$.
Example: the forgetful functor from [[groups|G5S1M9]] to sets sends a group to its underlying set and a [[group homomorphism|0JYTG5K9Y4]] to the same map.
Both conditions hold because identities and composites of homomorphisms are the identity and composite maps.
Non-example: sending each group $G$ to its centre $Z(G)$ and each homomorphism to its restriction does not define a functor from groups to groups.
Let $f \colon \Z/2 \to S_3$ be the inclusion onto the subgroup generated by a transposition.
It maps the centre $\Z/2$ outside the trivial centre of $S_3$.
So restriction gives no morphism $Z(\Z/2) \to Z(S_3)$, and the second item of the data fails.
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<document source="https://tamnd.github.io/math2/node/KJF8Y19BNV.md" tag="KJF8Y19BNV" kind="statement">
# Routh–Hurwitz stability criterion

## Statement

Let $p(s) = a_0 s^n + a_1 s^{n-1} + \cdots + a_{n-1} s + a_n$ be a polynomial of degree $n \ge 1$ with real coefficients and $a_0 > 0$.
Put $a_k = 0$ for $k < 0$ and for $k > n$.
Let $H$ be the $n \times n$ Hurwitz matrix with entries $H_{ij} = a_{2j - i}$ for $1 \le i, j \le n$, and let $\Delta_k$ be its leading principal [[minor|0BT9BYWVRY]] of order $k$.
Then every [[root|019MA1VTR2]] of $p$ has negative real part if and only if $\Delta_k > 0$ for $k = 1, \ldots, n$.

## Remarks

The rows of $H$ are $(a_1, a_3, a_5, \ldots)$, $(a_0, a_2, a_4, \ldots)$, $(0, a_1, a_3, \ldots)$, $(0, a_0, a_2, \ldots)$, and so on.
One proof counts the roots of $p$ in the right half-plane by following the argument of $p(i\omega)$ as $\omega$ runs over $\R$.
Routh's array computes the same information by a Euclidean algorithm on the even and odd parts of $p$.

For $n = 2$ and $p(s) = s^2 + a_1 s + a_2$, the conditions are $a_1 > 0$ and $a_1 a_2 > 0$, that is, $a_1 > 0$ and $a_2 > 0$.
For $n = 3$, they are $a_1 > 0$, $a_1 a_2 - a_0 a_3 > 0$ and $a_3 > 0$, since $\Delta_3 = a_3 \Delta_2$.
Positive coefficients are necessary but not sufficient when $n \ge 3$.
For $p(s) = s^3 + s^2 + s + 1 = (s + 1)(s^2 + 1)$, all coefficients are positive, but $\Delta_2 = 1 - 1 = 0$, and indeed $\pm i$ are roots.

## Explanations

### L1 human

To know whether a linear system returns to rest, one must know whether all roots of its characteristic polynomial lie in the left half-plane.
The criterion answers this from the coefficients alone, by checking that some determinants are positive.
No root has to be computed.

### L2 human

Applied to the [[characteristic polynomial|0VSSPAFM2C]] of a real matrix $A$, the criterion tests whether $A$ is [[Hurwitz-stable|9A3TBBHRFA]].
With the [[linearisation|09ZWYRQ41J]] of a nonlinear system, it gives a test for the stability of an equilibrium that uses only arithmetic on the entries of the Jacobian matrix.
It is the standard tool in control engineering for stability with respect to parameters, since the conditions are explicit inequalities.
The analogous test for roots inside the unit disc, used for discrete-time systems, is the Schur–Cohn or Jury test.
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# Short exact sequence

## Summary

A short exact sequence is an [[exact sequence|JB7ZBFBA9C]] $0 \to A \to B \to C \to 0$ whose first and last terms are zero.
It says that $A$ is a subobject of $B$ and that $C$ is the corresponding quotient $B/A$.
Short exact sequences are the basic input of homological algebra: functors are tested on them, and each gives a long exact sequence of derived functors.

## Explanations

### L1 human

A short exact sequence describes an object $B$ as built from a piece $A$ and what is left over, $C$.
The first map puts $A$ inside $B$ without collapsing anything.
The second map sends $B$ onto $C$ and kills exactly the copy of $A$.
For example, $0 \to \Z \xrightarrow{2} \Z \to \Z/2\Z \to 0$ shows the integers built from the even integers and the two classes "even" and "odd".
The middle object is not always the direct sum of the outer two, as this example shows.
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<document source="https://tamnd.github.io/math2/node/KJNSKV75VQ.md" tag="KJNSKV75VQ" kind="concept">
# Completion of a measure

## Summary

The completion of a measure space adds to the σ-algebra every set that differs from a measurable set by a subset of a null set.
The measure extends to these sets in only one way, and the result is the smallest complete extension.
Lebesgue measure on the Lebesgue measurable sets is the completion of Lebesgue measure on the Borel sets.

## Explanations

### L1 human

Some measures leave out subsets of negligible sets.
The completion puts them back, and gives each new set the measure of the measurable set it is almost equal to.
Nothing of substance changes: the new sets differ from old ones only by negligible pieces.
This is how one passes from Borel sets on the line to all Lebesgue measurable sets.
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# Projective morphism

## Definition

Let $Y$ be a scheme and let $n \ge 0$ be an integer.
Let $\mathbb{P}^n_{\Z} = \operatorname{Proj} \Z[x_0, \dots, x_n]$ be [[projective space|ZT9A6JK8QF]] over $\Z$, given by the [[Proj construction|A39XQHCVNW]].
Let $\mathbb{P}^n_Y = \mathbb{P}^n_{\Z} \times_{\operatorname{Spec} \Z} Y$ be the [[fiber product|1DZMT4S9FB]], with projection $p \colon \mathbb{P}^n_Y \to Y$.
A [[morphism of schemes|JXRKH6P1Z4]] $f \colon X \to Y$ is *projective* if $f = p \circ i$ for some $n \ge 0$ and some [[closed immersion|WWRWYSJDET]] $i \colon X \to \mathbb{P}^n_Y$.

## Remarks

For a field $k$ and a homogeneous polynomial $F \in k[x_0, x_1, x_2]$ of positive degree, the curve $X = V(F) \subseteq \mathbb{P}^2_k$ is projective over $\operatorname{Spec} k$.
Indeed, $X \to \mathbb{P}^2_k$ is a closed immersion and $\mathbb{P}^2_k \to \operatorname{Spec} k$ is the projection $p$.
The affine line $\mathbb{A}^1_k \to \operatorname{Spec} k$ is not projective.
A projective morphism is [[proper|9537RKBZPP]], and $\mathbb{A}^1_k$ is not proper over $k$.
This is the definition in Hartshorne's textbook.
In EGA the definition is more general: $X$ is a closed subscheme of the projective bundle of a quasi-coherent $\mathcal{O}_Y$-module of finite type.
The two definitions agree when $Y$ is affine.
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# Proper forcing axiom

## Definition

Partial orders, dense sets and filters are as in [[Martin's axiom|Z8XMCCZFA4]].
For an uncountable set $\lambda$, let $[\lambda]^{\omega}$ be the set of countable subsets of $\lambda$.
A set $C \subseteq [\lambda]^\omega$ is club if both conditions hold.
It contains the union of each increasing sequence $x_0 \subseteq x_1 \subseteq \cdots$ of its members.
Each countable subset of $\lambda$ is contained in a member of $C$.
A set $S \subseteq [\lambda]^\omega$ is stationary if it meets each club subset of $[\lambda]^\omega$.
A partial order $P$ is proper if [[forcing|F2DTWW9HP7]] with $P$ preserves stationarity: for each uncountable $\lambda$, each stationary $S \subseteq [\lambda]^\omega$ stays stationary in the generic extension.
The proper forcing axiom, PFA, is the following statement.
Let $P$ be a proper partial order, and let $\mathcal{D}$ be a family of at most $\aleph_1$ dense subsets of $P$.
Then some filter on $P$ meets every member of $\mathcal{D}$.

## Remarks

Example: each ccc partial order is proper, so PFA implies $\mathrm{MA}_{\aleph_1}$.
Each countably closed partial order is proper too.
Non-example: the partial order that collapses $\omega_1$ to be countable, by finite partial functions from $\omega$ to $\omega_1$, is not proper.
Proper forcings preserve $\omega_1$, while this one makes $\omega_1$ countable.
Baumgartner proved that PFA is consistent relative to a supercompact cardinal.
PFA implies $2^{\aleph_0} = \aleph_2$, and it implies that every two $\aleph_1$-dense sets of reals are order isomorphic.
These notions extend the [[stationary sets|6FDYJVG1CK]] of a cardinal to sets of countable subsets.
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# Pasch's axiom

## Statement

Let $A$, $B$ and $C$ be three non-collinear points of a plane.
Let $\ell$ be a [[line|0MP9BCF6DP]] of that plane that passes through none of them.
Suppose that $\ell$ contains a point [[between|CE8GX6ADZS]] $A$ and $B$.
Then $\ell$ contains a point between $A$ and $C$ or a point between $B$ and $C$.

## Remarks

Moritz Pasch stated it as an axiom in 1882, after he noticed that Euclid uses it without saying so.
In Hilbert's system it is one of the axioms of order, and it cannot be proved from the other axioms of incidence and order.
From the axioms of order one can also prove that $\ell$ cannot meet all three sides of the [[triangle|0CTNFGYMM2]] $ABC$.
In $\R^2$ it holds: write $\ell = \set{x : f(x) = 0}$ with $f$ affine, so that $f(A)$ and $f(B)$ have opposite signs.
Then $f(C)$ has the sign opposite to one of them, and $f$ vanishes between $C$ and that point.
It is the basis of the plane separation theorem: a line divides the rest of the plane into two half-planes.

## Explanations

### L1 human

If a straight line goes into a triangle through one side, it has to come out through another side.
A picture makes this obvious, but it does not follow from Euclid's postulates.
So modern axiom systems state it as an axiom.

### L2 human

Pasch's axiom is what one uses to prove that a line splits the plane into two sides, and that the inside of a triangle or an angle is well defined.
Many proofs in Euclid use it without saying so.
The crossbar theorem, which says that a ray inside an angle of a triangle meets the opposite side, rests on it.
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# Alexander duality

## Statement

Let $n \ge 1$ and let $K$ be a nonempty, [[compact|0TVJZYB6SH]], locally contractible proper subspace of the [[sphere|T62369RFH6]] $S^n$.
Then for every integer $i$ there is an isomorphism
$$\tilde H_i(S^n \setminus K; \Z) \cong \tilde H^{\,n-i-1}(K; \Z),$$
where $\tilde H_i$ is [[reduced homology|NJQMWVK3QK]] and $\tilde H^{\,j}$ is reduced [[singular cohomology|6N7BC70CX0]].

## Remarks

The proof uses a duality between the cohomology of a compact subset of a manifold and the homology of the manifold relative to its complement.
This duality is proved like [[Poincaré duality|9VY90HJVZX]].
The long exact sequence of the pair $(S^n, S^n \setminus K)$ then gives the result.
For an arbitrary compact $K$, the statement still holds with Čech cohomology in place of singular cohomology.
With singular cohomology, local contractibility cannot be dropped: the Warsaw circle in $S^2$ has the singular cohomology of a point but separates $S^2$ into two components.

Example: if $K \subset S^n$ is homeomorphic to $S^{n-1}$, then $\tilde H_0(S^n \setminus K) \cong \tilde H^{\,n-1}(S^{n-1}) \cong \Z$.
So the complement has exactly two path components, which is the Jordan–Brouwer separation theorem.

Example: for a knot $K \subset S^3$, that is, a subspace homeomorphic to $S^1$, $H_1(S^3 \setminus K) \cong \tilde H^1(S^1) \cong \Z$, whatever the knot is.

## Explanations

### L1 human

The holes in the complement of a shape in a sphere are determined by the holes of the shape itself, with dimensions flipped.
A circle drawn on the $2$-sphere cuts it into two pieces.
A knotted circle in $3$-space always has a complement with exactly one independent loop around it, however complicated the knot.

### L2 human

Alexander duality computes the homology of complements of embedded subsets, such as knot complements and complements of embedded spheres.
It gives a quick proof of the Jordan–Brouwer separation theorem in all dimensions.
Its proof is a direct application of the duality theory of manifolds, and its Čech version is a standard reason to introduce Čech cohomology.
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# Kuratowski embedding

## Statement

Let $(X, d)$ be a nonempty [[metric space|0VKD1TP7F3]].
Let $\ell^\infty(X)$ be the [[Banach space|06GZ956657]] of bounded functions $X \to \R$ with the norm $\norm{f}_\infty = \sup_{y \in X} \abs{f(y)}$.
Fix a point $x_0 \in X$.
For $x \in X$, define $\Phi(x) \colon X \to \R$ by
$$\Phi(x)(y) = d(x, y) - d(x_0, y).$$
Then each $\Phi(x)$ lies in $\ell^\infty(X)$, and $\Phi \colon X \to \ell^\infty(X)$ is an [[isometry|02HPP5VCZH]] onto its image.
That is, $\norm{\Phi(x) - \Phi(x')}_\infty = d(x, x')$ for all $x, x' \in X$.

## Remarks

By the triangle inequality, $\abs{d(x, y) - d(x', y)} \le d(x, x')$ for every $y$.
With $x' = x_0$, this shows that $\Phi(x)$ is bounded by $d(x, x_0)$.
In general, it gives $\norm{\Phi(x) - \Phi(x')}_\infty \le d(x, x')$.
Taking $y = x'$ gives the value $d(x, x')$, so equality holds.

If $X$ is bounded, the simpler map $x \mapsto d(x, \cdot)$ already works.
If $X$ is [[separable|0Z2P41SVDP]], the same idea with a countable dense subset embeds $X$ isometrically in the sequence space $\ell^\infty$.

## Explanations

### L1 human

Every metric space, however abstract, can be placed inside a space of real functions without changing any distance.
Each point $x$ is recorded by the list of its distances to all other points.
This turns questions about metric spaces into questions about normed vector spaces.

### L2 human

Since $\ell^\infty(X)$ is complete, the closure of $\Phi(X)$ is a complete metric space containing $X$ densely.
This gives a quick construction of the [[completion of a metric space|06RW4RKWH7]].
The embedding also lets one take convex hulls and straight segments of points of $X$ inside a Banach space.
For compact $X$ and $Y$, it shows that the distance on the [[Gromov–Hausdorff space|ETRF2FHAEE]] can be computed inside the single space $Z = \ell^\infty$.
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# Sub-Riemannian manifold

## Definition

Let $M$ be a connected [[smooth manifold|F45VQ47GPV]] with [[tangent bundle|WY6K3P4YCG]] $TM$.
A *sub-Riemannian manifold* is a triple $(M, H, g)$ such that:

1. $H \subseteq TM$ is a smooth subbundle, the horizontal distribution;
2. $H$ is bracket-generating: at each $p \in M$, the vector fields tangent to $H$ and their iterated [[Lie brackets|68FGQEBD6J]] span $T_pM$;
3. $g$ assigns to each $p \in M$ an [[inner product|0NB0AWGJBW]] $g_p$ on the fibre $H_p$, depending smoothly on $p$.

A horizontal curve is an absolutely continuous curve $\gamma$ with $\gamma'(t) \in H_{\gamma(t)}$ for almost every $t$.
The sub-Riemannian distance $d(p, q)$ is the infimum of $\int \sqrt{g(\gamma', \gamma')} \, dt$ over the horizontal curves from $p$ to $q$.

## Remarks

Example: the Heisenberg group is $\R^3$ with $H$ spanned by $X = \partial_x - \tfrac{y}{2} \partial_z$ and $Y = \partial_y + \tfrac{x}{2} \partial_z$, with $X, Y$ orthonormal.
Here $[X, Y] = \partial_z$, so $X, Y, [X, Y]$ span $T_p\R^3$ at each point, and clause 2 holds.

Non-example: on $\R^3$, let $H$ be spanned by $\partial_x$ and $\partial_y$.
All brackets of fields tangent to $H$ stay in $H$, so clause 2 fails; horizontal curves stay in the planes $z = \text{const}$, and points with different $z$ are at infinite distance.

When $H = TM$, the definition gives a Riemannian manifold.
Some authors do not require clause 2 in the definition and add it as a hypothesis when needed.
</document>

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# Partition of unity

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
For a function $\rho \colon X \to \R$, its support $\operatorname{supp} \rho$ is the closure of $\set{x \in X : \rho(x) \neq 0}$.
A *partition of unity* on $X$ is a family $(\rho_i)_{i \in I}$ of [[continuous functions|0X3WGX95FE]] $\rho_i \colon X \to [0, 1]$ such that:

1. the family of supports $(\operatorname{supp} \rho_i)_{i \in I}$ is [[locally finite|DZ297MMGV8]]: each point of $X$ has a neighbourhood that meets only finitely many of them;
2. $\sum_{i \in I} \rho_i(x) = 1$ for every $x \in X$.

Let $(U_i)_{i \in I}$ be an [[open cover|049PXYHW1T]] of $X$ with the same index set.
The partition of unity is *subordinate* to $(U_i)_{i \in I}$ if $\operatorname{supp} \rho_i \subset U_i$ for every $i$.
If $X$ is a [[smooth manifold|F45VQ47GPV]] and every $\rho_i$ is smooth, it is a *smooth partition of unity*.

## Remarks

Example on $\R$ with the cover $U_1 = (-\infty, 1)$ and $U_2 = (-1, \infty)$.
Let $\rho_1(x) = 1$ for $x \le -1/2$, $\rho_1(x) = 1/2 - x$ for $-1/2 \le x \le 1/2$, and $\rho_1(x) = 0$ for $x \ge 1/2$, and let $\rho_2 = 1 - \rho_1$.
Both are continuous with values in $[0, 1]$, the sum is $1$, and $\operatorname{supp} \rho_1 = (-\infty, 1/2] \subset U_1$ and $\operatorname{supp} \rho_2 = [-1/2, \infty) \subset U_2$.
Replacing the linear piece by a smooth monotone transition gives a smooth partition of unity.

Non-example: the constant functions $\rho_1 = \rho_2 = 1/4$ on $\R$ are continuous with locally finite supports, but their sum is $1/2$, so clause 2 fails.

Clause 1 makes the sum in clause 2 a finite sum near each point, so it is continuous, or smooth in the smooth case.
Every paracompact Hausdorff space has a partition of unity subordinate to any open cover.
Every smooth manifold has a smooth partition of unity subordinate to any open cover; the proof uses bump functions and second countability.

## Explanations

### L1 human

A partition of unity is a family of continuous functions with values in $[0,1]$ that add up to $1$, where near each point only finitely many of them are nonzero.
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# Free distributive lattice

## Definition

Let $S$ be a set.
Let $F$ be a [[distributive lattice|0HM0HQCK6N]] and let $\iota \colon S \to F$ be a map.
The pair $(F, \iota)$ is a *free distributive lattice* on $S$ if the following holds.
For every distributive lattice $L$ and every map $g \colon S \to L$, there is exactly one [[lattice homomorphism|BNVRR925X8]] $\bar g \colon F \to L$ with $\bar g \circ \iota = g$.

## Remarks

It exists for every $S$ and is unique up to a unique isomorphism compatible with $\iota$.
For nonempty $S$, every element of $F$ is a finite join of finite nonempty meets of generators $\iota(s)$.
For finite $S$ with $n \ge 1$ elements, $F$ is isomorphic to the lattice of non-constant monotone Boolean functions of $n$ variables, so it has $M(n) - 2$ elements, with $M(n)$ the [[Dedekind number|481ZPQ0GC3]].
If one also asks for $0$ and $1$ and homomorphisms that preserve them, the free bounded distributive lattice on $n$ generators has $M(n)$ elements.

Example: on $S = \set{x, y}$, the lattice $\set{x \wedge y, x, y, x \vee y}$ with $x \wedge y < x, y < x \vee y$ is free.
A map $g$ into $L$ extends by $x \wedge y \mapsto g(x) \wedge g(y)$ and $x \vee y \mapsto g(x) \vee g(y)$, and this is forced.

Non-example: the two-element chain $0 < 1$ with $\iota(x) = 0$ and $\iota(y) = 1$ is distributive and generated by the image of $\iota$, but it is not free.
The map $g(x) = 1$, $g(y) = 0$ into the same chain would need a homomorphism sending $0$ to $1$ and $1$ to $0$, which does not preserve the order.
</document>

<document source="https://tamnd.github.io/math2/node/KKK6EHQQ22.md" tag="KKK6EHQQ22" kind="definition">
# Laguerre series expansion of the Laplace transform

## Definition

Let $L_n$ be the [[Laguerre polynomial|WAMRHWW82R]] of degree $n$, and let $\ell_n(t) = e^{-t/2} L_n(t)$ for $t \ge 0$.
The functions $\ell_0, \ell_1, \dots$ form an orthonormal basis of the [[space|SASP14MA1Z]] $L^2(0, \infty)$.
Let $f \in L^2(0, \infty)$, with coefficients $a_n = \int_0^\infty f(t)\, \ell_n(t)\, dt$, and let $F$ be its [[Laplace transform|0V49JEZW06]].
The *Laguerre series expansion* of $F$ is
$$F(s) = \sum_{n=0}^{\infty} a_n \frac{(s - 1/2)^n}{(s + 1/2)^{n+1}}, \qquad \operatorname{Re} s > 0.$$

## Remarks

The expansion holds because the Laplace transform of $\ell_n$ is $(s - 1/2)^n / (s + 1/2)^{n+1}$, and because $F(s)$ is the $L^2$ inner product of $f$ with $e^{-\bar{s} t}$, which lies in $L^2(0, \infty)$ when $\operatorname{Re} s > 0$.
With $w = (s - 1/2)/(s + 1/2)$, which maps the half-plane $\operatorname{Re} s > 0$ onto the unit disk, it reads $(s + 1/2) F(s) = \sum_{n} a_n w^n$.
The coefficients satisfy $\sum_n \abs{a_n}^2 = \norm{f}_2^2 < \infty$.

Example: $f = \ell_1$, that is $f(t) = e^{-t/2}(1 - t)$.
Then $a_1 = 1$ and all other $a_n = 0$.
The check: $\int_0^\infty e^{-st} e^{-t/2}(1 - t)\, dt = \frac{1}{s + 1/2} - \frac{1}{(s + 1/2)^2} = \frac{s - 1/2}{(s + 1/2)^2}$.

Non-example: $f(t) = 1$ has the Laplace transform $1/s$, but $f \notin L^2(0, \infty)$, so the coefficients $a_n$ are not defined by the formula above.

Applying the same expansion to $e^{-\sigma t} f(bt)$ for parameters $\sigma$ and $b > 0$ gives the scaled variants used in practice.
</document>

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# Logical consequence

## Definition

Let $\Gamma$ be a set of [[formulas|HYJ10AF6RT]] and $\varphi$ a formula, all over the same signature.
Then $\varphi$ is a *logical consequence* of $\Gamma$, written $\Gamma \models \varphi$, if every interpretation that makes every formula of $\Gamma$ true also makes $\varphi$ true.
In [[propositional logic|09K9MWD2QQ]] an interpretation is a truth assignment to the variables.
In first-order logic it is a [[structure|983GZBQYT4]] together with an assignment of its elements to the variables.

## Remarks

$\set{p, \; p \to q} \models q$: in every assignment where $p$ and $p \to q$ are true, $q$ is true by the truth table of $\to$.
$\set{p \to q, \; q} \not\models p$: the assignment $p = \mathsf{F}$, $q = \mathsf{T}$ makes both premises true and $p$ false.
$\emptyset \models \varphi$ means that $\varphi$ is valid, written $\models \varphi$.
$\Gamma \models \varphi$ holds exactly when $\Gamma \cup \set{\neg \varphi}$ is not [[satisfiable|QN7M1JCGT9]].
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# Aronszajn tree

## Description

A tree, in set theory, is a [[partially ordered set|011C29Q1S1]] $(T, <)$ in which, for each $t \in T$, the set $\set{s \in T : s < t}$ is [[well-ordered|0F1D6PK9RZ]].
The height of $t$ is the order type of this set, and level $\alpha$ of $T$ is the set of elements of height $\alpha$.
A branch is a maximal linearly ordered subset of $T$.
An Aronszajn tree is a tree with these three properties:

1. Its height is $\omega_1$, the least [[uncountable|0X55M9XET4]] ordinal: the levels $\alpha < \omega_1$ are nonempty, and level $\omega_1$ is empty.
2. Each level is [[countable|0ZG61TFBES]].
3. It has no branch of length $\omega_1$, that is, no uncountable branch.

## Remarks

Aronszajn trees exist in ZFC, by a construction of Nachman Aronszajn.
One construction uses bounded, strictly increasing sequences of rational numbers, each of countable successor length, ordered by extension.
Levels are chosen by recursion on $\alpha < \omega_1$, so that each level is countable and each node has extensions at all higher levels with only slightly larger supremum.
An uncountable branch would give a strictly increasing sequence of rationals of length $\omega_1$, which is impossible since $\mathbb{Q}$ is countable.
This shows that König's lemma, which gives an infinite branch in each infinite finitely branching tree, has no analogue at $\omega_1$.
An Aronszajn tree in which every antichain is countable is a Suslin tree, and its existence is independent of ZFC.

## Explanations

### L1 human

Picture a tree that grows upward through $\omega_1$ levels, with only countably many nodes on each level.
You might expect some path to climb all the way to the top.
An Aronszajn tree is a tree of this kind in which every path stops at some countable level.
It shows that infinite combinatorics at $\omega_1$ behaves very differently from the countable case.
</document>

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# Milne problem

## Example

Let $E_1(x) = \int_1^\infty e^{-x\mu}\, \mu^{-1}\, d\mu$ for $x > 0$ be the [[exponential integral|EV6ZDQ0NDY]], and more generally $E_n(x) = \int_1^\infty e^{-x\mu}\, \mu^{-n}\, d\mu$.
The *Milne problem* asks for a nonzero function $J$ on $[0, \infty)$ of at most linear growth with
$$J(\tau) = \frac{1}{2} \int_0^\infty E_1(\abs{\tau - t})\, J(t)\, dt, \quad \tau \ge 0.$$
This is a homogeneous [[Wiener–Hopf equation|QYY4NMDY5H]] with kernel $k(x) = \frac{1}{2} E_1(\abs{x})$.
In astrophysics $\tau$ is the optical depth in a plane-parallel grey atmosphere in radiative equilibrium, and $J$ is the mean intensity.

Check of the main features.
The kernel has total mass $1$: $\int_{\R} \frac{1}{2} E_1(\abs{x})\, dx = \int_1^\infty \mu^{-2}\, d\mu = 1$.
So on the whole line every linear function is a solution, and the cut at $\tau = 0$ makes the problem nontrivial.
Using $E_n' = -E_{n-1}$ for $n \ge 2$, one computes for $\tau \ge 0$
$$\frac{1}{2} \int_0^\infty E_1(\abs{\tau - t})\, dt = 1 - \frac{1}{2} E_2(\tau), \qquad \frac{1}{2} \int_0^\infty E_1(\abs{\tau - t})\, t\, dt = \tau + \frac{1}{2} E_3(\tau).$$
Hence the Eddington guess $J = \tau + \frac{2}{3}$ gives a right side that differs from $J$ by $\frac{1}{2} E_3(\tau) - \frac{1}{3} E_2(\tau)$.
This error tends to $0$ as $\tau \to \infty$, but at $\tau = 0$ it is $\frac{1}{4} - \frac{1}{3} = -\frac{1}{12}$, since $E_2(0) = 1$ and $E_3(0) = \frac{1}{2}$.

## Remarks

The exact solution, up to a constant factor, is $J(\tau) = \tau + q(\tau)$, where $q$ is the Hopf function.
It is found by the [[Wiener–Hopf method|H4NXEY6QTZ]].
The function $q$ increases from $q(0) = 1/\sqrt{3}$ to a limit $q(\infty) \approx 0.7104$.
The Fourier symbol of the kernel is $\frac{\arctan \xi}{\xi}$, which equals $1$ at $\xi = 0$, so $1 - \hat k$ vanishes at $0$; this zero is why the solution grows linearly.

## Explanations

### L1 human

Light travels through the outer layers of a star, is absorbed and is re-emitted in all directions.
In equilibrium, the light at each depth is an average of the light at all other depths, weighted by how much of it survives the trip.
Deep inside, the intensity grows linearly with depth, but near the surface there is a correction because no light comes in from outside.
The Milne problem asks for this correction, and its exact answer was one of the first successes of the Wiener–Hopf method.
</document>

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# Hilbert's axioms

## Summary

Hilbert's axioms are the axiom system for [[Euclidean geometry|KRZD96GSVC]] that David Hilbert gave in his book *Grundlagen der Geometrie* of 1899.
They speak of points, lines and planes and of the relations of incidence, [[betweenness|CE8GX6ADZS]] and congruence, and they fall into five groups.
They fill the gaps of [[Euclid's Elements|83WHDAV68J]].
They made it possible to prove that geometry is consistent if the real numbers are.
They also made it possible to prove the independence of single axioms, such as the [[parallel postulate|2PC3XY0PYZ]].

## Explanations

### L1 human

Euclid's proofs sometimes use facts that are clear from the picture but not from his axioms.
Hilbert wrote down a complete list of rules, so that every proof can be checked without looking at a picture.
For example, his axiom of Pasch says that a line that enters a triangle through one side must leave it through another side.
Hilbert said that one could replace "points, lines and planes" by "tables, chairs and beer mugs" and the proofs would still work.
</document>

<document source="https://tamnd.github.io/math2/node/KM59PWHJHH.md" tag="KM59PWHJHH" kind="example">
# Hurwitz quaternion order

## Example

Let $\eta = 2\cos(2\pi/7)$ and let $K = \Q(\eta)$, the real cubic subfield of the seventh cyclotomic field, an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\Z[\eta]$.
The number $\eta$ is a root of $t^3 + t^2 - 2t - 1$.
Let $D$ be the [[quaternion algebra|ZWXW0T3EAC]] $(\eta, \eta)_K$, with $K$-basis $1, i, j, ij$ and relations $i^2 = j^2 = \eta$ and $ij = -ji$.
Put $\tau = 1 + \eta + \eta^2$ and $j' = \tfrac{1}{2}(1 + \eta i + \tau j)$.
The *Hurwitz quaternion order* is the $\Z[\eta]$-[[order|1ZBX4TKVZ9]]
$$\mathcal{Q}_{\mathrm{Hur}} = \Z[\eta][i, j, j']$$
generated by $i$, $j$ and $j'$.
It is a maximal order of $D$.

A check that $j'$ is integral: an element $a + bi + cj + d\,ij$ of $D$ has reduced trace $2a$ and reduced norm $a^2 - \eta b^2 - \eta c^2 + \eta^2 d^2$.
So $j'$ has reduced trace $1$ and reduced norm $\tfrac{1}{4}(1 - \eta^3 - \eta\tau^2)$.
Using $\eta^3 = -\eta^2 + 2\eta + 1$ one finds $\eta \tau^2 = \eta^2 + 10\eta + 4$, so the reduced norm is $\tfrac{1}{4}(-4 - 12\eta) = -1 - 3\eta \in \Z[\eta]$.
Hence $j'$ satisfies the monic equation $x^2 - x - 1 - 3\eta = 0$ over $\Z[\eta]$.

## Remarks

The group of elements of reduced norm $1$ in $\mathcal{Q}_{\mathrm{Hur}}$, modulo $\set{\pm 1}$, is isomorphic to the $(2,3,7)$ triangle group.
Quotients of the hyperbolic plane by its principal congruence subgroups give Hurwitz surfaces, the compact Riemann surfaces with the largest possible automorphism group for their genus.

## Explanations

### L1 human

This is a specific ring of quaternion-like numbers with coefficients from the field generated by $2\cos(2\pi/7)$.
It plays the role that the integers play inside the rationals, and it cannot be enlarged to a bigger such ring.
Its unit group acts on the hyperbolic plane like a very symmetric tiling by triangles with angles $\pi/2$, $\pi/3$ and $\pi/7$.
This is why it is used to build the most symmetric Riemann surfaces, such as the Klein quartic.
</document>

<document source="https://tamnd.github.io/math2/node/KM8Q5HZ48G.md" tag="KM8Q5HZ48G" kind="statement">
# Douglas–Radó solution of Plateau's problem

## Statement

Let $n \ge 2$ and let $\Gamma \subseteq \R^n$ be a [[rectifiable|0HATW2VWAZ]] Jordan curve, that is, a curve of finite length homeomorphic to a circle.
Let $D$ be the open unit disc in $\R^2$, with coordinates $(u, v)$, and let $\overline{D}$ be its closure.
Let $\mathcal{C}(\Gamma)$ be the set of continuous maps $Y \colon \overline{D} \to \R^n$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,2}(D; \R^n)$.
The restriction of each such $Y$ to $\partial D$ must be a monotone parametrization of $\Gamma$.
Then there is a map $X \in \mathcal{C}(\Gamma)$ such that:

1. $X$ restricted to $\partial D$ is a homeomorphism onto $\Gamma$;
2. $X$ is [[harmonic|0DEB0JMN0W]] in $D$;
3. $X$ is weakly [[conformal|0YYT36R4YD]] in $D$: $\abs{X_u}^2 = \abs{X_v}^2$ and $X_u \cdot X_v = 0$;
4. the area $\int_D \abs{X_u \wedge X_v} \, du \, dv$ of $X$ is the least area of all maps in $\mathcal{C}(\Gamma)$.

## Remarks

Jesse Douglas and Tibor Radó found the theorem independently, around 1930.
The idea is to minimize the Dirichlet energy $\frac12 \int_D (\abs{Y_u}^2 + \abs{Y_v}^2)$ instead of the area.
The energy is easier to control, and it is at least the area, with equality for conformal maps.
The energy is invariant under the conformal maps of the disc, so one fixes the images of three boundary points.
Then the Courant–Lebesgue lemma makes a minimizing sequence equicontinuous on the boundary, and a minimizer exists.
A minimizer of the energy is harmonic, and varying the parametrization of the disc shows that it is weakly conformal, so it also minimizes area.
Points where $X_u = 0$ are isolated branch points, and away from them the image is a [[minimal surface|41X0YW2R9N]].
For example, if $\Gamma$ is a planar circle, the solution parametrizes the flat disc that it bounds.
The theorem controls only disc-type surfaces: a curve may bound a surface of another topological type with smaller area.

## Explanations

### L1 human

A wire loop dipped in soapy water holds a film of least area.
Douglas and Radó proved that for every loop of finite length a least-area surface shaped like a disc exists.
Their surface is given by a parametrization that is both harmonic and angle-preserving.

### L2 human

The theorem answers [[Plateau's problem|DVYA6GD8RN]] for disc-type surfaces, and it is a model of the direct method of the calculus of variations.
The trick of replacing area by the Dirichlet energy reappears in the theory of harmonic maps and of minimal surfaces in Riemannian manifolds.
Later work studied the branch points: for area-minimizing solutions in $\R^3$ there are no interior branch points.
Geometric measure theory treats the problem in other dimensions and without fixing the topology, with currents and varifolds.
</document>

<document source="https://tamnd.github.io/math2/node/KMAEWSVGRS.md" tag="KMAEWSVGRS" kind="statement">
# Riemann–Hilbert correspondence

## Statement

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $d$ and let $\mathcal{D}_X$ be its sheaf of holomorphic linear [[differential operators|GBWQF3HZG0]].
Let $D^b_{rh}(\mathcal{D}_X)$ be the bounded [[derived category|5ST7NVJFSM]] of left $\mathcal{D}_X$-modules whose cohomology modules are regular holonomic.
Let $D^b_c(X)$ be the bounded derived category of sheaves of $\C$-vector spaces on $X$ with constructible cohomology sheaves.
For $\mathcal{M}$ in $D^b_{rh}(\mathcal{D}_X)$, let $\operatorname{DR}(\mathcal{M}) = \omega_X \otimes^{L}_{\mathcal{D}_X} \mathcal{M}$ be its de Rham complex, where $\omega_X = \Omega^d_X$ is the canonical sheaf, the complex
$$\mathcal{M} \to \Omega^1_X \otimes \mathcal{M} \to \cdots \to \Omega^d_X \otimes \mathcal{M}$$
placed in degrees $-d, \dots, 0$.
Then:

1. $\operatorname{DR}$ is an equivalence of triangulated categories $D^b_{rh}(\mathcal{D}_X) \to D^b_c(X)$;
2. it restricts to an equivalence between the abelian category of regular holonomic $\mathcal{D}_X$-modules and the category of [[perverse sheaves|5WHXRZDZFK]] on $X$.

## Remarks

Masaki Kashiwara and Zoghman Mebkhout proved the theorem independently around 1980.
Holonomic means that the characteristic variety of the module, a subvariety of the cotangent bundle, has dimension $d$.
Regular is a growth condition on solutions near singularities.
It extends the notion of a [[regular singular point|H96HXG6CA0]] of an ordinary differential equation.

The simplest case: a holomorphic vector bundle $E$ with a flat connection $\nabla$ is a regular holonomic $\mathcal{D}_X$-module.
Its de Rham complex is quasi-isomorphic to the local system of flat sections $\ker \nabla$, shifted by $d$.
On this part, the correspondence is the classical equivalence between flat bundles and local systems, that is, representations of the fundamental group when $X$ is connected.

There is also an algebraic version, for a smooth complex algebraic variety $X$ and algebraic regular holonomic modules, where regularity includes a condition at infinity.
There regularity is needed.
On the affine line, the algebraic connections $d$ and $d - dz$ on the trivial bundle are not isomorphic, because an isomorphism would be multiplication by $e^{z}$, which is not a polynomial.
Yet their analytic flat sections, the constants and the functions $c \, e^{z}$, form isomorphic local systems.
The connection $d - dz$ has an irregular singular point at infinity.

## Explanations

### L1 human

Systems of linear differential equations and topological data about loops are two descriptions of the same thing.
Given equations, follow their solutions around loops and record how they change; this is the monodromy.
Following Riemann, Hilbert asked whether every monodromy comes from equations with mild singularities.
The modern theorem says that, for equations with regular singularities, the dictionary between equations and topology is perfect, even in many variables and with complicated singularities.

### L2 human

The theorem gives an answer to a modern form of [[Hilbert's twenty-first problem|7RHW0KA5RY]].
Its classical forerunner, due to Deligne, treats algebraic vector bundles with regular singular connections on smooth varieties, and matches them with local systems.
In the Kashiwara–Mebkhout form, the category of perverse sheaves belongs to topology and the category of D-modules to analysis.
Many results move across.
For example, the intersection cohomology complex corresponds to a minimal extension of a D-module.
It is a basic tool of geometric representation theory, for example in the proof of the Kazhdan–Lusztig conjecture.
</document>

<document source="https://tamnd.github.io/math2/node/KMBHAYAGZ9.md" tag="KMBHAYAGZ9" kind="definition">
# Fréchet mean

## Definition

Let $(M, d)$ be a [[metric space|0VKD1TP7F3]] and let $X$ be a random element of $M$ with $E[d(X, q)^2] < \infty$ for some $q \in M$.
The *Fréchet function* of $X$ is $F(p) = E[d(X, p)^2]$ for $p \in M$, an [[expected value|0WQNRCNY3M]].
A *Fréchet mean* of $X$ is a point $\mu \in M$ with
$$F(\mu) = \inf_{p \in M} F(p).$$
For points $x_1, \dots, x_n \in M$ and weights $w_i > 0$, a *Fréchet mean* is a minimiser of $p \mapsto \sum_{i=1}^{n} w_i \, d(x_i, p)^2$; with equal weights it is the *sample Fréchet mean*.

## Remarks

The condition on $q$ and the triangle inequality give $F(p) < \infty$ for every $p$.
The set of Fréchet means may be empty or have several points.
On a complete Riemannian manifold a local minimiser of $F$ is called a Karcher mean.
Replacing $d^2$ by $d$ gives the Fréchet median.

Example: in $M = \R^k$ with the Euclidean distance, $\sum_i \norm{x_i - p}^2 = \sum_i \norm{x_i - \bar x}^2 + n \norm{\bar x - p}^2$, where $\bar x$ is the [[sample mean|03Y6AQ2S0J]].
So the sample Fréchet mean is unique and equals $\bar x$.

Non-uniqueness: on the unit sphere with the great-circle distance, take two antipodal points $x$ and $-x$.
If $p$ is at distance $\theta$ from $x$, it is at distance $\pi - \theta$ from $-x$, and $\theta^2 + (\pi - \theta)^2$ is smallest exactly at $\theta = \pi/2$.
So every point of the equator between them is a Fréchet mean, and the "midpoint" is not a single point.

Non-example: if $X$ is real with the Cauchy distribution, then $E[(X - q)^2] = \infty$ for every $q \in \R$.
The moment condition fails, $F$ is infinite everywhere, and $X$ has no Fréchet mean, just as it has no ordinary mean.
</document>

<document source="https://tamnd.github.io/math2/node/KMNV11B91V.md" tag="KMNV11B91V" kind="example">
# Reuleaux triangle

## Example

Let $w > 0$, and let $A$, $B$, $C$ be the vertices of an equilateral triangle of side $w$ in the plane.
The *Reuleaux triangle* is the intersection of the three closed discs of radius $w$ centred at $A$, $B$ and $C$.
Its boundary consists of three arcs of [[circles|060W5CBZP6]] of radius $w$: the arc from $B$ to $C$ centred at $A$, and the two similar arcs.
Each arc has angle $\pi/3$.

Its boundary is a [[curve of constant width|B4YD1ZXRVH]] $w$.
Check: take a unit vector $u$ and the two supporting lines orthogonal to $u$.
The arc centred at $A$ consists of the points $A + w v$ with $v$ in an angle $V_A$ of size $\pi/3$.
Its outer normal at $A + w v$ is $v$.
The outer normals of the body at the vertex $A$ are the vectors $-v$ with $v \in V_A$.
These six angles of size $\pi/3$ cover all directions.
So for some vertex, say $A$, one of $u$ and $-u$ lies in $V_A$ and the other is an outer normal at $A$.
Then one supporting line passes through $A$ and the other through a point $A \pm w u$ of the arc centred at $A$.
The distance between the two lines is $w$.

The perimeter is $3 \cdot w \cdot \pi/3 = \pi w$.
The area is the sum of three circular sectors of angle $\pi/3$ minus twice the triangle: $3 \cdot \frac{\pi w^2}{6} - 2 \cdot \frac{\sqrt{3}}{4} w^2 = \frac{\pi - \sqrt{3}}{2} w^2$.

## Explanations

### L1 human

Draw an equilateral triangle and replace each side by an arc of a circle centred at the opposite corner.
The result is a rounded triangle that has the same width in every direction, just like a coin.
It can turn between two parallel boards that stay at a fixed distance, and it is used in some coin shapes and drill designs.
Among all shapes of a given constant width it has the smallest area.
</document>

<document source="https://tamnd.github.io/math2/node/KMVW45B0TT.md" tag="KMVW45B0TT" kind="statement">
# Hermite–Minkowski theorem

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] of degree $n \ge 2$ with [[discriminant|E0VZTNPJXA]] $d_K$.
Let $r_2$ be the number of pairs of complex conjugate embeddings of $K$.

1. (Minkowski) The discriminant satisfies
$$\abs{d_K} \ge \left(\frac{\pi}{4}\right)^{2 r_2} \left(\frac{n^n}{n!}\right)^2 > 1.$$
In particular $\abs{d_K} \ge 3$, and some prime number [[ramifies|H5RC67SD24]] in $K$.
2. (Hermite) For each real number $D$, there are only finitely many number fields $K \subset \C$ with $\abs{d_K} \le D$.

## Remarks

Part 1 follows from the [[Minkowski bound|1TQ2F9CFVE]]: every ideal class contains a nonzero integral ideal of norm at least $1$ and at most $M_K$, so $M_K \ge 1$.
Since $2 r_2 \le n$ and $\pi/4 < 1$, the right side is at least $(\pi/4)^n (n^n/n!)^2$, which is $\pi^2/4 > 2$ for $n = 2$ and increases with $n$.
A prime $p$ ramifies in $K$ exactly when $p$ divides $d_K$, so $\abs{d_K} > 1$ gives a ramified prime.

For part 2, the bound of part 1 first bounds the degree $n$.
Then [[Minkowski's theorem|0K387S2PKQ]] on convex bodies gives a primitive element of $K$ whose conjugates are bounded in terms of $D$.
So its minimal polynomial has bounded integer coefficients, and there are finitely many such polynomials.

Example: $\Q(\sqrt{-3})$ has $\abs{d_K} = 3$, so the bound $\abs{d_K} \ge 3$ for $n = 2$ is sharp.

## Explanations

### L1 human

Every number field other than $\Q$ has a discriminant of absolute value larger than $1$.
So in every proper extension of $\Q$ some prime number breaks into repeated factors.
Also, only finitely many number fields have a discriminant below a given size.

### L2 human

Part 1 says that $\Q$ has no nontrivial extension that is unramified at all primes.
Many other number fields do have such extensions, and the abelian ones are described by the [[Hilbert class field|WWCJ7CR2R3]].
Part 2 makes tables of number fields ordered by discriminant possible, since each range contains finitely many fields.
</document>

<document source="https://tamnd.github.io/math2/node/KN2PXSCT9J.md" tag="KN2PXSCT9J" kind="conjecture">
# Sidorenko's conjecture

## Statement

Let $H$ and $G$ be finite simple [[graphs|00C0BR7D7H]], with $G$ having at least one vertex.
Let $\hom(H, G)$ be the number of [[graph homomorphisms|052JJ9YVNM]] from $H$ to $G$, and let
$$t(H, G) = \frac{\hom(H, G)}{|V(G)|^{|V(H)|}}$$
be the homomorphism density.
Let $K_2$ be the graph with one edge, so that $t(K_2, G) = 2 |E(G)| / |V(G)|^2$.
Let $H$ be a [[bipartite graph|0ZVQKTRKVQ]] with $e(H)$ edges.
Then for every graph $G$,
$$t(H, G) \ge t(K_2, G)^{e(H)}.$$

## Remarks

The conjecture is open.
It is known for trees, for even cycles, for complete bipartite graphs and for hypercube graphs.
It is also known for every bipartite graph that has a vertex adjacent to all vertices of the other part.

The right side is the density of $H$ in a random graph with the same edge density.
So the conjecture says that random graphs minimise the density of $H$ among graphs of a given edge density.
Bipartiteness is necessary.
If $H$ has an odd cycle and $G$ is the complete bipartite graph $K_{n,n}$, then $t(H, G) = 0$, while $t(K_2, G) = 1/2$.
For $H$ a path with two edges, the conjecture is the inequality $\frac{1}{n} \sum_v d(v)^2 \ge \left(\frac{1}{n}\sum_v d(v)\right)^2$ for the degrees $d(v)$ of $G$, which holds by the Cauchy–Schwarz inequality.

## Explanations

### L1 human

Fix a small pattern $H$ with no odd cycle, such as a square.
Among all graphs with a given proportion of edges, which one has the fewest copies of $H$?
Sidorenko's conjecture says that a random graph is always at least as good as anything else: no clever arrangement of the edges can beat it.
</document>

<document source="https://tamnd.github.io/math2/node/KNAGK0W65G.md" tag="KNAGK0W65G" kind="concept">
# Tricolorability

## Summary

A [[knot|QYEAZTYGPT]] is tricolorable if the arcs of a diagram can be colored with three colors under a simple rule.
At each crossing the three meeting arcs must show one color or all three colors, and at least two colors must appear.
Whether this is possible does not depend on the chosen diagram, so tricolorability is a property detected by a [[knot invariant|GZQ5ZE3X6G]].
It is the simplest tool to prove that a knot is really knotted.

## Explanations

### L1 human

Draw a knot and break each under-strand at the crossings, so the drawing falls into arcs.
Color each arc red, green or blue.
The rule is that at every crossing you see either one color or all three colors.
The plain circle has a single arc, so it can only use one color.
The [[trefoil knot|VYTYSVHN4A]] has three arcs, and giving them three different colors satisfies the rule at all three crossings.
So the trefoil is tricolorable and the circle is not, which proves that the trefoil cannot be untied.
</document>

<document source="https://tamnd.github.io/math2/node/KNBQ68XG2M.md" tag="KNBQ68XG2M" kind="example">
# Whitney umbrella

## Example

The *Whitney umbrella* is the surface in $\R^3$ with equation
$$x^2 = y^2 z.$$
It contains the image of the smooth map $f \colon \R^2 \to \R^3$, $f(u, v) = (uv, u, v^2)$, since $(uv)^2 = u^2 \cdot v^2$.

Check of the singularity: the [[differential|6P9PXQDSEN]] of $f$ has matrix
$$\begin{pmatrix} v & u \\ 1 & 0 \\ 0 & 2v \end{pmatrix}.$$
Its first two rows show that it has rank $2$ unless $u = 0$.
For $u = 0$, its first and third rows show that it has rank $2$ unless $v = 0$.
So the origin is the only [[singular point|1JG3HFWX3T]] of $f$, and there the rank is $1$.

The points $(0, 0, z)$ with $z > 0$ have the two preimages $(0, \pm \sqrt{z})$, so the image crosses itself along the positive $z$-axis.
The origin has the single preimage $(0, 0)$ and is called the pinch point.
The points $(0, 0, z)$ with $z < 0$ satisfy the equation but are not in the image of $f$; they form the handle of the umbrella.

## Remarks

The map germ of $f$ at the origin is the cross-cap.
Whitney showed that, besides immersion germs, the cross-cap is the only singularity of a generic smooth map from a surface into $3$-space, and that it is stable.

## Explanations

### L1 human

Take a sheet, pinch one point and let the sheet pass through itself along a line starting at that point.
The result looks like an umbrella that has been folded and pushed through itself, with a stick, the handle, sticking out the other side.
The pinch point is where the map from the plane fails to be an immersion.
A generic smooth map of a surface into space has only isolated pinch points of this kind as singular points.
</document>

<document source="https://tamnd.github.io/math2/node/KNDSJAYMSY.md" tag="KNDSJAYMSY" kind="concept">
# Unital magma

## Summary

A unital magma is a [[magma|9689SFPVE9]] with an [[identity element|07WYQPB8E3]], an element that leaves every element unchanged when combined with it on either side.
The identity of a unital magma is unique.
A [[monoid|0N6R902FG1]] is exactly an associative unital magma, and loops in the theory of quasigroups are unital magmas with unique solutions of equations.

## Explanations

### L1 human

A unital magma has a neutral element $e$: combining $e$ with any $x$, in either order, gives back $x$.
No other rule is asked for.
For example, the integers with $x * y = x + y + xy$ have neutral element $0$, since $0 * y = y$ and $x * 0 = x$.
If two elements $e$ and $e'$ were both neutral, then $e = e * e' = e'$, so there is only one.
</document>

<document source="https://tamnd.github.io/math2/node/KNDXDE7K8M.md" tag="KNDXDE7K8M" kind="concept">
# Ringed space

## Summary

A ringed space is a topological space together with a sheaf of commutative rings on it.
The rings play the role of the rings of functions on the open sets.
Manifolds with their smooth functions, complex manifolds with their holomorphic functions, and varieties with their regular functions are all ringed spaces.
Schemes are defined as ringed spaces of a special kind, so the notion is the starting point of modern algebraic geometry.

## Explanations

### L1 human

Many kinds of geometry fix a space and say which functions on it are allowed: continuous, smooth, holomorphic or polynomial.
A ringed space packages exactly this: a space, and for each open set a ring of allowed functions on it.
For example, $\R^n$ with the smooth functions on each open set is a ringed space.
The point of view lets one compare different geometries in one framework, and it allows "functions" that are not functions on points at all, as in scheme theory.
</document>

<document source="https://tamnd.github.io/math2/node/KNDYA9T8ZK.md" tag="KNDYA9T8ZK" kind="concept">
# Siegel disc

## Summary

A Siegel disc is a component of the [[Fatou set|C0N0XGC6GK]] of a rational map on which an iterate acts as an irrational rotation.
The rotation is seen after a conformal change of coordinates.
It surrounds a neutral periodic point whose multiplier is $e^{2\pi i \theta}$ with $\theta$ irrational.
Whether such a disc exists depends on the arithmetic of $\theta$, through small divisor problems.
Siegel discs are one of the four types of periodic Fatou components.

## Explanations

### L1 human

Imagine a disc that the map turns by a fixed angle at each step, like a wheel.
The angle is an irrational fraction of a full turn, so no point ever returns exactly to where it started.
Each point travels forever along a closed curve around a centre point.
For $z \mapsto e^{2\pi i \theta} z + z^2$ with $\theta$ the golden mean, such a turning disc exists around $0$.
Its boundary is a complicated fractal curve.
</document>

<document source="https://tamnd.github.io/math2/node/KNJA2NEZ9F.md" tag="KNJA2NEZ9F" kind="concept">
# Fundamental domain

## Summary

A fundamental domain for a [[group action|0WCY42EV9V]] on a space is a subset that contains one point of each [[orbit|0YQPZB5Q1G]].
Often this is only asked up to a negligible boundary.
Its translates by the group tile the space.
Fundamental domains turn questions about the quotient space into questions about a concrete piece of the original space.
For instance, they compute the volume of a quotient by a lattice.

## Explanations

### L1 human

The integers act on the real line by shifting, and the interval $[0, 1)$ contains exactly one point from each set $x + \Z$.
Its shifts $[n, n+1)$ cover the line without overlap, like tiles.
In the plane, the unit square plays the same role for shifts by $\Z^2$.
Studying the quotient, here a circle or a torus, is the same as studying one tile with its edges glued.
The modular group acting on the upper half-plane has a famous fundamental domain bounded by two vertical lines and an arc of the unit circle.
</document>

<document source="https://tamnd.github.io/math2/node/KNP5ZEQ458.md" tag="KNP5ZEQ458" kind="definition">
# H-representation and V-representation

## Definition

Let $n \ge 1$, and let $P \subseteq \R^n$ be a [[convex polytope|300K6CB1KH]].

1. A *V-representation* of $P$ is a finite set $V \subseteq \R^n$ with $P = \mathrm{conv}(V)$, the [[convex hull|0FQSDC2EQ2]] of $V$.
2. An *H-representation* of $P$ is a pair $(A, b)$ of a real $m \times n$ matrix $A$ and a vector $b \in \R^m$ with $P = \set{x \in \R^n : A x \le b}$. Here $\le$ is taken coordinatewise.

Equivalently, an H-representation is a finite family of closed half-spaces whose intersection is $P$.

## Remarks

Example: for $P = [0, 1]^2$, the set $V = \set{(0, 0), (1, 0), (0, 1), (1, 1)}$ is a V-representation, and the inequalities $-x_1 \le 0$, $x_1 \le 1$, $-x_2 \le 0$, $x_2 \le 1$ form an H-representation.
Non-example: the inequalities $-x_1 \le 0$, $-x_2 \le 0$ alone are not an H-representation of $[0, 1]^2$; they define the unbounded quadrant, which contains $(2, 0) \notin P$.
Representations are not unique, since points and inequalities can be added without changing $P$.
The smallest V-representation is the set of vertices of $P$, and for a full-dimensional $P$ an irredundant H-representation has exactly one inequality for each facet, unique up to positive scaling.
Every convex polytope has both kinds of representation; this is the Minkowski–Weyl theorem.
</document>

<document source="https://tamnd.github.io/math2/node/KNYSF300GC.md" tag="KNYSF300GC" kind="definition">
# Incomplete beta function

## Definition

Let $a > 0$ and $b > 0$ be real numbers and let $0 \le x \le 1$.
The *incomplete beta function* is
$$B(x; a, b) = \int_0^x t^{a-1} (1-t)^{b-1}\, dt.$$
The *regularised incomplete beta function* is $I_x(a, b) = B(x; a, b)/B(a, b)$, where $B(a, b) = B(1; a, b)$ is the [[beta function|QHNWVSC6AQ]].

## Remarks

Example: for $a = 2$ and $b = 1$, $B(x; 2, 1) = \int_0^x t\, dt = \frac{x^2}{2}$ and $B(2, 1) = \frac12$, so $I_x(2, 1) = x^2$.

Non-example of a valid argument: for $a = 0$ the integrand is $t^{-1}(1-t)^{b-1}$, whose integral diverges at $0$, so $B(x; 0, b)$ is not defined for $x > 0$.

The function $x \mapsto I_x(a, b)$ increases from $0$ to $1$ and is the distribution function of the [[beta distribution|0EW824Z7G2]] with parameters $a$ and $b$.
The substitution $t \mapsto 1 - t$ gives the symmetry $I_x(a, b) = 1 - I_{1-x}(b, a)$.
For positive integers $a$ and $n$ with $a \le n$, $I_p(a, n - a + 1)$ is the probability of at least $a$ successes in $n$ independent trials with success probability $p$.
</document>

<document source="https://tamnd.github.io/math2/node/KP1PP7A6JR.md" tag="KP1PP7A6JR" kind="concept">
# Quasi-Frobenius ring

## Summary

A quasi-Frobenius ring is a [[ring|0B1261B5ED]] that is Noetherian and [[self-injective|KQKJENEMSN]] on one side; it is then Artinian and self-injective on both sides.
Over such a ring a module is projective if and only if it is injective.
[[Frobenius algebras|8EVP1K2CZH]], group algebras of finite groups over fields and the rings $\Z/n\Z$ are quasi-Frobenius.
The class generalizes Frobenius algebras from algebras over a field to arbitrary rings.

## Explanations

### L1 human

For most rings, projective modules and injective modules are very different kinds of modules.
In a quasi-Frobenius ring they are the same modules.
The rings $\Z/n\Z$ are examples: they are finite, hence Noetherian, and they are injective over themselves.
The ring $\Z$ is not quasi-Frobenius: $\Z$ is projective over itself but not injective.
</document>

<document source="https://tamnd.github.io/math2/node/KP23SF2Z0Q.md" tag="KP23SF2Z0Q" kind="statement">
# Koebe's theorem on the mean value property

## Statement

Let $U \subseteq \R^n$ be an open set and let $u \colon U \to \R$ be [[continuous|0X3WGX95FE]].
Assume that for every $x \in U$ there is $\rho(x) > 0$ such that
$$u(x) = \frac{1}{\abs{\partial B(x, r)}} \int_{\partial B(x, r)} u \, dS \quad \text{for all } 0 < r < \rho(x),$$
where $\abs{\partial B(x, r)}$ is the area of the sphere $\partial B(x, r)$.
Then $u$ is [[harmonic|0DEB0JMN0W]] on $U$.

## Remarks

The radius $\rho(x)$ may depend on the point, so the mean value property is only asked for small spheres; this strengthens the [[converse of the mean value property|CQ7E0V6T2G]].
Proof idea: fix a ball $B$ with closure in $U$, and let $h$ be the harmonic function on $B$ with boundary values $u$ given by the [[Poisson integral formula|PCZBE7EX01]].
The difference $v = u - h$ is continuous on $\overline{B}$, vanishes on $\partial B$, and has the local mean value property inside $B$.
Suppose $v$ had a positive maximum $m$, and choose a point $x$ of the compact set $\set{v = m}$ farthest from the centre of $B$.
On each small sphere about $x$, $v \le m$, with $v < m$ on an open piece of the sphere.
So the average is less than $m = v(x)$, a contradiction.
The same applies to $-v$, so $v = 0$ and $u = h$ is harmonic on $B$.
Example: for $u(x) = \abs{x}^{2-n}$ on $U = \R^n \setminus \set{0}$ with $n \ge 3$, one may take $\rho(x) = \abs{x}$.
The mean value property on these spheres follows from the potential of a uniform spherical shell, and the theorem then shows that $u$ is harmonic.

## Explanations

### L1 human

To know that a continuous function is harmonic, it is enough to check the averaging property on tiny spheres.
Each point may come with its own size limit for the spheres.
This makes harmonicity a purely local property that can be tested by averages.

### L2 human

The proof combines the solvability of the Dirichlet problem on balls with a maximum principle that only uses small spheres.
This style of argument, comparing a function with the harmonic function that has the same boundary values, reappears in Perron's method and in the theory of subharmonic functions.
Even weaker versions exist, where the mean value property is asked only for one sequence of radii tending to $0$ at each point; the same proof works for them.
</document>

<document source="https://tamnd.github.io/math2/node/KP27TDCZPJ.md" tag="KP27TDCZPJ" kind="statement">
# Nonexistence of a projective plane of order 10

## Statement

There is no [[projective plane|DANFNN8C4Y]] of [[order|K2YVMTGTBB]] $10$.

## Remarks

The [[Bruck–Ryser–Chowla theorem|R43K0GTTDS]] does not exclude order $10$: since $10 \equiv 2 \pmod 4$, it requires $10$ to be a sum of two squares, and $10 = 1^2 + 3^2$.
The proof uses coding theory.
The incidence vectors of the lines of a putative plane of order $10$ span a binary linear code of length $111$.
Its weight enumerator is determined by the numbers of codewords of a few small weights.
A series of large computer searches showed that these numbers lead to a contradiction.
The final search was completed by Clement Lam, Larry Thiel and Stanley Swiercz, and announced in 1989.

As a consequence, there is no set of $9$ [[mutually orthogonal Latin squares|H6TN6BADVY]] of order $10$, since such a set would give a plane of order $10$.
The proof depends on computer calculations, which have since been checked independently by other computations.

## Explanations

### L1 human

A projective plane of order $10$ would have $111$ points and $111$ lines, with $11$ points on each line.
The obvious necessary conditions do not rule it out.
It took years of reasoning and computer search to show that no such plane exists.

### L2 human

Order $10$ is the smallest order that is not a prime power and is not excluded by the Bruck–Ryser–Chowla theorem.
Orders $2, 3, 4, 5, 7, 8, 9$ are prime powers and have planes, and order $6$ is excluded by Bruck–Ryser–Chowla.
The next open case is order $12$.
The result is a famous early example of a theorem whose proof rests on a massive computation.
</document>

<document source="https://tamnd.github.io/math2/node/KPCM3KY0MR.md" tag="KPCM3KY0MR" kind="definition">
# Character table

## Definition

Let $G$ be a finite [[group|G5S1M9]].
Let $\chi_1, \ldots, \chi_r$ be the distinct irreducible complex [[characters|RMWMYBY8TC]] of $G$, and let $C_1, \ldots, C_s$ be the [[conjugacy classes|0DEBXAJ5SA]] of $G$, with $g_j \in C_j$.
The *character table* of $G$ is the $r \times s$ matrix with entries $\chi_i(g_j)$.
By convention $\chi_1$ is the trivial character and $C_1 = \set{1}$.

## Remarks

The table is square: $r = s$.
Example: the symmetric group $S_3$ has three classes, of the identity, the transpositions and the $3$-cycles.
Its three irreducible characters are the trivial one, the sign, and a character of degree $2$.
The last one is the permutation character on three points minus the trivial character.
Its table is
$$\begin{array}{c|ccc} & 1 & (1\,2) & (1\,2\,3) \\ \hline \chi_1 & 1 & 1 & 1 \\ \chi_2 & 1 & -1 & 1 \\ \chi_3 & 2 & 0 & -1 \end{array}$$
The check: $1^2 + 1^2 + 2^2 = 6 = \abs{S_3}$, and the rows are orthogonal for the weighted inner product of the [[Schur orthogonality relations|AY0B20QC04]].
Non-example: the $2 \times 3$ matrix with rows $\chi_1, \chi_2$ only is not the character table of $S_3$, because it omits the irreducible character $\chi_3$.
Non-isomorphic groups can have the same character table: the dihedral group of order $8$ and the quaternion group do.
</document>

<document source="https://tamnd.github.io/math2/node/KPCN3KHG50.md" tag="KPCN3KHG50" kind="concept">
# Differentially closed field

## Summary

A differentially closed field is a [[differential field|GA5PTQGC6G]] in which every differential equation that can have a solution in some extension already has one.
It is the analogue, for differential equations, of an [[algebraically closed field|0PF6TM8TWT]] for polynomial equations.
In characteristic $0$, the differentially closed fields form an elementary class whose theory, called $\mathrm{DCF}_0$, is a central example in [[model theory|HFTBG5PMXC]].

## Explanations

### L1 human

In an algebraically closed field, every nonconstant polynomial has a root.
A differentially closed field asks the same of equations that involve an unknown $y$ and its derivatives $y', y'', \dots$.
Such fields are huge and abstract: no field of familiar functions, such as $\C(x)$, is differentially closed.
For instance, $\C(x)$ contains no solution of $y' = 1/x$, since $\log x$ is not a rational function.
Every differential field of characteristic $0$ sits inside a differentially closed one, much as every field sits inside an algebraically closed one.
</document>

<document source="https://tamnd.github.io/math2/node/KPJED63FT6.md" tag="KPJED63FT6" kind="definition">
# Growth of algebras

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $A$ be a finitely generated [[associative algebra|3CTW1P4HBP]] over $k$ with identity.
Let $V \subseteq A$ be a finite-dimensional [[subspace|000AYPA9AJ]] that contains $1$ and generates $A$ as an algebra.
For $n \ge 0$, let $V^n$ be the span of all products $v_1 v_2 \cdots v_n$ with $v_i \in V$, with $V^0 = k$.
The *growth function* of $A$ with respect to $V$ is
$$d_V(n) = \dim_k V^n.$$
For functions $f, g \colon \N \to \N$, write $f \preceq g$ if there is an integer $c \ge 1$ with $f(n) \le g(cn)$ for all $n$, and $f \sim g$ if $f \preceq g$ and $g \preceq f$.
The *growth* of $A$ is the class of $d_V$ under $\sim$.

## Remarks

The growth does not depend on $V$.
If $W$ is another such subspace, then $W \subseteq V^c$ for some $c$, because $W$ is finite-dimensional and $\bigcup_n V^n = A$; so $d_W(n) \le d_V(cn)$, and symmetrically.

The algebra $A$ has *polynomial growth* if $d_V(n) \le C n^r$ for some constants $C, r$.
It has *exponential growth* if $d_V(n) \ge a^n$ for some $a > 1$ and all $n$.
Otherwise it has *intermediate growth*.

Example: for $A = k[x, y]$ and $V = k + kx + ky$, the space $V^n$ has the basis of monomials $x^i y^j$ with $i + j \le n$.
So $d_V(n) = (n+1)(n+2)/2$, and $A$ has polynomial growth.

Non-example of the setting: the field $k(x)$ of rational functions is not a finitely generated $k$-algebra, so no growth function is defined for it in this way.

For a finitely generated [[group|4CMRM3975W]] $G$ with a finite symmetric generating set $S$, take $A = k[G]$, the [[group algebra|MNP2HBWB5Q]], and $V$ the span of $\set{1} \cup S$.
Then $V^n$ has the basis of group elements of word length at most $n$, so the growth of $k[G]$ is the growth of $G$.
</document>

<document source="https://tamnd.github.io/math2/node/KPKND0K7W6.md" tag="KPKND0K7W6" kind="concept">
# Quasi-coherent sheaf

## Summary

A quasi-coherent sheaf on a scheme is a sheaf of modules that is locally given by a module over the coordinate ring.
On an affine scheme $\operatorname{Spec} A$, quasi-coherent sheaves are the same as $A$-modules.
They form the basic category of linear objects on a scheme, closed under kernels, cokernels, direct sums and tensor products.

## Explanations

### L1 human

An affine scheme is described by a ring $A$, and many linear objects on it are described by $A$-modules.
A quasi-coherent sheaf is what one gets by gluing such modules over the affine pieces of a scheme.
For example, any ideal of $A$ gives a quasi-coherent sheaf on $\operatorname{Spec} A$, and so does any vector space over a field $k$ on the point $\operatorname{Spec} k$.
Most sheaves of modules that arise in algebraic geometry are quasi-coherent.
</document>

<document source="https://tamnd.github.io/math2/node/KPNY26EKNP.md" tag="KPNY26EKNP" kind="example">
# Associahedron

## Example

Let $n \ge 1$ and let $Q$ be a convex polygon with $n + 3$ vertices.
A diagonal of $Q$ joins two non-adjacent vertices.
The *associahedron* $K$ of dimension $n$ is a [[convex polytope|300K6CB1KH]] whose face lattice is described as follows.

1. The nonempty faces of $K$ correspond to the sets of pairwise non-crossing diagonals of $Q$, ordered by reverse inclusion. A $k$-face corresponds to $n - k$ diagonals.
2. The [[vertices|R7V3S9V891]] correspond to the [[triangulations|Q6KKBP3RRP]] of $Q$, the maximal sets of $n$ non-crossing diagonals.
3. The [[facets|DPDXN2FJC4]] correspond to the single diagonals of $Q$.
4. Two vertices are joined by an edge when the triangulations differ by a flip. A flip removes one diagonal and inserts the other diagonal of the quadrilateral that appears.

Each triangulation has $n$ diagonals and each can be flipped, so every vertex lies in exactly $n$ edges, and $K$ is a [[simple polytope|7E6FXTGH2Y]].
The number of vertices is the [[Catalan number|0YWZJEABB2]] $C_{n+1}$, the number of triangulations of an $(n+3)$-gon.

## Remarks

For $n = 2$, $Q$ is a pentagon with $5$ triangulations and $5$ diagonals, and $K$ is a pentagon.
For $n = 3$, $Q$ is a hexagon with $14$ triangulations and $9$ diagonals, and $K$ is a three-dimensional polytope with $14$ vertices and $9$ facets ($3$ squares and $6$ pentagons).
The same lattice also describes the ways to put brackets in a product of $n + 2$ factors, which is where the name comes from.
Several explicit realizations by coordinates are known.

## Explanations

### L1 human

Draw a hexagon and cut it into triangles with straight cuts between corners.
There are $14$ ways to do it.
Changing one cut at a time moves you between them, and these moves form the edges of a solid with $14$ corners.
That solid is the associahedron; in general its corners count the triangulations of a polygon.
</document>

<document source="https://tamnd.github.io/math2/node/KPQ96640ZJ.md" tag="KPQ96640ZJ" kind="concept">
# Morse function

## Summary

A Morse function is a smooth real-valued function on a [[smooth manifold|F45VQ47GPV]] all of whose [[critical points|0YTCMFB4HC]] are nondegenerate.
This means that the [[Hessian|0693Y2METT]] at each critical point is invertible.
Near each critical point it has a simple quadratic normal form, given by the Morse lemma.
Morse functions exist on every smooth manifold and are generic among smooth functions.
They are the basic tool of Morse theory, which reads the topology of a manifold off the critical points of such a function.

## Explanations

### L1 human

Think of the height above the ground on a hilly landscape, or on a doughnut standing upright on a table.
Its critical points are the peaks, the pits and the passes, where the ground is flat.
A Morse function is one where each such flat spot is a clean peak, pit or saddle, not a flat plateau or a degenerate shape like a monkey saddle.
For the upright doughnut, the height has four critical points: a bottom, two saddles, and a top.
</document>

<document source="https://tamnd.github.io/math2/node/KPWW9X14V2.md" tag="KPWW9X14V2" kind="concept">
# Hall subgroup

## Summary

A Hall subgroup of a finite group is a [[subgroup|0X1JXT47GA]] whose order is [[coprime|0VZ48N083F]] to its [[index|0DTXBQPA2S]].
Hall subgroups generalise [[Sylow subgroups|0053CQE9Y2]] from one prime to a set of primes.
In a [[solvable group|0FXJ7YFMMX]] they always exist and behave like Sylow subgroups, by [[Hall's theorem|8Z8PE69SJS]]; in other groups they may fail to exist.
They are named after Philip Hall.

## Explanations

### L1 human

A Sylow subgroup collects all the factors of one prime in the order of a group.
A Hall subgroup does the same for a chosen set of primes.
For example, the group $A_5$ has order $60 = 2^2 \cdot 3 \cdot 5$.
Its subgroup $A_4$ has order $12 = 2^2 \cdot 3$ and index $5$, and $12$ and $5$ have no common factor, so $A_4$ is a Hall subgroup for the primes $2$ and $3$.
But $A_5$ has no subgroup of order $15$, so there is no Hall subgroup for the primes $3$ and $5$.
</document>

<document source="https://tamnd.github.io/math2/node/KPYFGB342X.md" tag="KPYFGB342X" kind="definition">
# Hecke character

## Definition

Let $K$ be a [[number field|MWBDFS2KJ5]], let $I_K$ be its idele group and let $C_K = I_K / K^\times$ be its [[idele class group|9GHD9MXNYV]].
A *Hecke character* of $K$ is a continuous [[group homomorphism|0JYTG5K9Y4]]
$$\chi \colon I_K \to \C^\times$$
such that $\chi(a) = 1$ for every $a \in K^\times$.
Equivalently, it is a continuous homomorphism $C_K \to \C^\times$.

## Remarks

Example: the idele norm $\abs{x} = \prod_v \abs{x_v}_v$ is continuous and multiplicative, and $\abs{a} = 1$ for $a \in K^\times$ by the product formula; so it is a Hecke character, and so is $\abs{\cdot}^s$ for every $s \in \C$.
Non-example: for $K = \Q$, the map $x \mapsto \abs{x_\infty}$ that keeps only the real component is a continuous homomorphism, but it sends the rational number $2$ to $2 \neq 1$.
Some authors call $\chi$ a Hecke character only when $\abs{\chi(x)} = 1$ for all $x$, and call the general case a quasi-character or Größencharakter.
Every Hecke character is a unitary one times $\abs{\cdot}^\sigma$ for a real number $\sigma$.
The Hecke characters of $\Q$ of finite order correspond to the primitive [[Dirichlet characters|7XB5VJKDPG]].
</document>

<document source="https://tamnd.github.io/math2/node/KPZ1S446MN.md" tag="KPZ1S446MN" kind="definition">
# Mellin transform

## Definition

Let $f \colon (0, \infty) \to \C$ be measurable and let $s \in \C$.
For $x > 0$, let $x^{s-1} = e^{(s-1)\log x}$ be the [[complex power|0RNGDMHT7M]] with the real logarithm.
Assume that $f$ is [[Mellin convergent|SGZZ4Y0D36]] at $s$, that is, $x \mapsto x^{s-1} f(x)$ is integrable on $(0, \infty)$.
The *Mellin transform* of $f$ at $s$ is the [[Lebesgue integral|DWJF7ABPR1]]
$$\mathcal{M}f(s) = \int_0^\infty x^{s-1} f(x) \, dx.$$
As a function, $\mathcal{M}f$ is defined on the set of $s$ at which $f$ is Mellin convergent.

## Remarks

Since $\abs{x^{s-1}} = x^{\operatorname{Re} s - 1}$, this set depends only on $\operatorname{Re} s$ and is a union of vertical lines; it is a vertical strip, possibly empty or degenerate.
Example: $f(x) = e^{-x}$.
Near $0$, $\abs{x^{s-1} e^{-x}} \le x^{\operatorname{Re} s - 1}$, which is integrable on $(0, 1)$ when $\operatorname{Re} s > 0$; near $\infty$, the factor $e^{-x}$ beats every power.
So $\mathcal{M}f(s) = \Gamma(s)$ for $\operatorname{Re} s > 0$, the [[gamma function|0H5SYBSED9]].
Non-example: $f(x) = 1$.
Integrability of $x^{\operatorname{Re} s - 1}$ on $(0, 1)$ needs $\operatorname{Re} s > 0$, and on $(1, \infty)$ it needs $\operatorname{Re} s < 0$, so $f$ is Mellin convergent at no $s$.
Other notations are $\tilde f(s)$, $F(s)$ and $\set{\mathcal{M} f}(s)$.
Some authors call the open strip of convergence the fundamental strip and write it $\langle \alpha, \beta \rangle$.
</document>

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# Free functor

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras, regarded as a category with homomorphisms, and let $U \colon \mathcal{V} \to \mathbf{Set}$ be its [[forgetful functor|WV3BPYCHVG]].
A *free functor* for $\mathcal{V}$ is a functor $F \colon \mathbf{Set} \to \mathcal{V}$ that is [[left adjoint|TZHF6ZM92S]] to $U$.
Explicitly, $F$ comes with maps $\eta_X \colon X \to U F(X)$, natural in $X$, with the following property.
For every algebra $A \in \mathcal{V}$ and every map $g \colon X \to U(A)$, there is a unique homomorphism $\bar g \colon F(X) \to A$ with $U(\bar g) \circ \eta_X = g$.

## Remarks

Such a functor exists for every variety: $F(X)$ is the [[free algebra|GTS4DRRMZR]] of $\mathcal{V}$ on $X$, and $F$ sends a map $h \colon X \to Y$ to the unique homomorphism $F(X) \to F(Y)$ that extends $\eta_Y \circ h$.
It is unique up to a unique natural isomorphism, as is any left adjoint.
More generally, a left adjoint of any forgetful functor $U \colon \mathcal{C} \to \mathcal{D}$ is called a free functor.

Example: for the variety of monoids, $F(X)$ is the monoid $X^*$ of finite words in $X$ under concatenation, and $\eta_X$ sends $x$ to the one-letter word $x$.
A map $g \colon X \to M$ into a monoid extends to $\bar g(x_1 \cdots x_n) = g(x_1) \cdots g(x_n)$, and this is the only homomorphism extending $g$.

Non-example: the functor from sets to groups that sends every set to the trivial group is not a free functor.
For $X = \set{x}$ and the map $x \mapsto 1$ into $\Z$, the only homomorphism from the trivial group to $\Z$ sends everything to $0$, so it does not extend $g$.
</document>

<document source="https://tamnd.github.io/math2/node/KQ0ABB8N82.md" tag="KQ0ABB8N82" kind="definition">
# Holomorphic dynamics in several variables

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $k \ge 2$, and let $f \colon X \to X$ be a [[holomorphic map|3MQTWKBBBT]].
For $m \ge 0$, the [[iterate|AYVTR2ES5B]] $f^m$ is the $m$-fold composition of $f$, with $f^0 = \id$.
The *orbit* of $x \in X$ is the sequence $x, f(x), f^2(x), \ldots$.
*Holomorphic dynamics in several variables* is the study of the orbits of such a [[discrete-time dynamical system|5AA9AJEARJ]] $(X, f)$.

Let $X$ be compact.
The *Fatou set* of $f$ is the set of points $x \in X$ with a neighbourhood $U$ on which the iterates $(f^m|_U)_{m \ge 0}$ form a [[normal family|JTVWRXXAKM]].
This means that every sequence of these iterates has a subsequence converging uniformly on compact subsets of $U$.
The *Julia set* of $f$ is the complement of the Fatou set.

## Remarks

The main cases are $X = \mathbb{P}^k$ with $f$ a holomorphic endomorphism of algebraic degree $d \ge 2$, and $X = \C^2$ with $f$ a polynomial automorphism.

Example: $f[z_0 : \cdots : z_k] = [z_0^d : \cdots : z_k^d]$ with $d \ge 2$ is a holomorphic self-map of $\mathbb{P}^k$.
It is well defined because the $z_j^d$ vanish together only when all $z_j$ vanish.
If one coordinate $z_i$ has strictly larger modulus than all the others, then the ratios $(z_j/z_i)^{d^m}$ tend to $0$ uniformly near $[z]$.
So the orbit tends to the coordinate point with $1$ in place $i$, the iterates converge near $[z]$, and $[z]$ lies in the Fatou set.

Example: a [[Hénon map|KS4FJ6H5JH]] $(x, y) \mapsto (p(x) - a y, x)$ with $a \ne 0$ and $\deg p \ge 2$ is a holomorphic automorphism of $\C^2$.

Non-example: the map $(z, w) \mapsto (\bar z^2, w)$ of $\C^2$ is smooth but not holomorphic, so its iteration is not part of holomorphic dynamics.

For $k = 1$ the same definitions give one-variable [[complex dynamics|BQQZ0ZCGQF]].
</document>

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# Wiener–Ikehara theorem

## Statement

Let $A \colon [0, \infty) \to [0, \infty)$ be a nondecreasing [[monotone function|0WFF6AZPBP]].
Assume that the [[Laplace transform|05XT5AVJBK]]
$$f(s) = \int_0^\infty A(x) e^{-xs} \, dx$$
converges for every complex $s$ with $\operatorname{Re} s > 1$.
Assume also that there is a real number $c \ge 0$ such that $f(s) - \frac{c}{s - 1}$ extends to a continuous function on the closed half-plane $\operatorname{Re} s \ge 1$.
Then
$$\lim_{x \to \infty} e^{-x} A(x) = c.$$

## Remarks

The proof is Fourier-analytic.
One integrates $f(s) - \frac{c}{s-1}$ on vertical lines against a Fejér-type kernel with compactly supported Fourier transform, lets $\operatorname{Re} s \to 1^+$, and uses the Riemann–Lebesgue lemma; monotonicity of $A$ then removes the smoothing.

Example: the [[prime number theorem|0EHBY5Y84N]].
Let $\psi(y) = \sum_{n \le y} \Lambda(n)$ with the [[von Mangoldt function|087EMZTHWN]] $\Lambda$, and put $A(x) = \psi(e^x)$.
Then $\int_0^\infty \psi(e^x) e^{-xs} \, dx = -\frac{\zeta'(s)}{s \, \zeta(s)}$ for $\operatorname{Re} s > 1$.
This differs from $\frac{1}{s - 1}$ by a function continuous on $\operatorname{Re} s \ge 1$, because $\zeta$ has a simple pole at $1$ and [[does not vanish on the line|CXES7915R7]] $\operatorname{Re} s = 1$.
So $\psi(y)/y \to 1$, which is equivalent to the prime number theorem.

The hypothesis is only continuity up to the boundary line, not analytic continuation beyond it.

## Explanations

### L1 human

The theorem reads the growth of a nondecreasing function off its Laplace transform.
If the transform behaves like $\frac{c}{s-1}$ near the line $\operatorname{Re} s = 1$ and has no other trouble on that line, then the function grows like $c e^x$.
Applied to the primes, it turns the fact that the zeta function has no zeros on the line $\operatorname{Re} s = 1$ into the prime number theorem.

### L2 human

In Dirichlet series form: if $a_n \ge 0$ and $F(s) = \sum a_n n^{-s}$ converges for $\operatorname{Re} s > 1$ with $F(s) - \frac{c}{s-1}$ continuous on $\operatorname{Re} s \ge 1$, then $\sum_{n \le y} a_n \sim c\,y$.
This follows by applying the theorem to $A(x) = \sum_{n \le e^x} a_n$.
The theorem is a special case of Wiener's general Tauberian theory; Newman's later contour-integral proof of the prime number theorem uses a closely related Tauberian argument.
</document>

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# Matrix pencil

## Summary

A matrix pencil is a family of matrices $A - \lambda B$ that depends linearly on a parameter $\lambda$.
Its eigenvalues are the values of $\lambda$ for which $A - \lambda B$ is singular, which leads to the generalised eigenvalue problem $Av = \lambda Bv$.
Pencils arise in vibration problems, in differential-algebraic equations and in control theory, and they are classified by the Kronecker canonical form.
More general pencils allow higher powers of $\lambda$; the linear ones are the most common.

## Explanations

### L1 human

Take two matrices $A$ and $B$ of the same size and look at all the combinations $A - \lambda B$ as the number $\lambda$ varies.
The ordinary eigenvalue problem is the case $B = I$.
In mechanics, $A$ may describe stiffness and $B$ mass, and the eigenvalues of the pencil give the squares of the natural frequencies.
For $A = \operatorname{diag}(1, 2)$ and $B = I$, the matrix $A - \lambda B$ is singular exactly for $\lambda = 1$ and $\lambda = 2$.
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<document source="https://tamnd.github.io/math2/node/KQB7WTZ1TX.md" tag="KQB7WTZ1TX" kind="statement">
# Regular value theorem

## Statement

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] of dimensions $m$ and $n$, and let $f \colon M \to N$ be a smooth map.
A point $y \in N$ is a *regular value* of $f$ if the [[differential|6P9PXQDSEN]] $df_x \colon T_xM \to T_yN$ is surjective for every $x \in f^{-1}(y)$.
Let $y$ be a regular value of $f$ with $f^{-1}(y)$ not empty.
Then the [[level set|0B2RZ0778N]] $f^{-1}(y)$ is an [[embedded submanifold|VKFW27WXEZ]] of $M$ of dimension $m - n$.
Its tangent space at each point $x \in f^{-1}(y)$ is $\ker df_x$.

## Remarks

The set of points where $df_x$ is surjective is open, so $f$ is a [[submersion|BMY55EW3RF]] on a neighbourhood of $f^{-1}(y)$.
In charts the claim becomes the [[implicit function theorem|0BXBADR94C]].
Near each $x$ there are coordinates in which $f$ is the projection onto the first $n$ coordinates, and the fibre is then a coordinate slice.
Example: $f \colon \R^{n+1} \to \R$, $f(x) = \abs{x}^2$, has $df_x(v) = 2 \langle x, v \rangle$, which is onto for $x \neq 0$.
So $1$ is a regular value and the unit sphere $S^n = f^{-1}(1)$ is a submanifold of dimension $n$.
The hypothesis is needed: $0$ is not a regular value of $g(x, y) = xy$, and $g^{-1}(0)$ is the union of the two axes, which is not a submanifold.

## Explanations

### L1 human

Solve an equation $f(x) = y$ on a smooth space.
If the equation is "non-degenerate" at every solution, the solutions form a smooth space again, of dimension equal to the number of unknowns minus the number of equations.
This is how one shows that spheres, many matrix groups and other familiar shapes are manifolds.

### L2 human

The theorem is the main tool for producing examples of manifolds.
For instance, the orthogonal group $O(n)$ is the fibre over the identity of the map $A \mapsto A^T A$ into symmetric matrices.
The identity is a regular value, so $O(n)$ is a manifold of dimension $n(n-1)/2$.
By [[Sard's theorem|Z7R8J71SJN]] almost every value is regular, so the theorem applies to almost every fibre.
It is the special case of transversality to a point, and the same proof gives that the preimage of a submanifold under a transverse map is a submanifold.
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<document source="https://tamnd.github.io/math2/node/KQG5GXJ5MB.md" tag="KQG5GXJ5MB" kind="statement">
# Parareal

## Statement

Consider an [[initial value problem|0FXDW992DS]] with state space $\R^d$ on a time interval $[0, T]$, cut into $N$ slices by times $0 = T_0 < T_1 < \cdots < T_N = T$.
For $n = 0, \ldots, N - 1$, let $\mathcal{F}_n \colon \R^d \to \R^d$ (fine) and $\mathcal{G}_n \colon \R^d \to \R^d$ (coarse) be maps, called propagators.
They approximate the solution at $T_{n+1}$ of the problem started at $T_n$.
Let $Y_0 = y_0$ and $Y_{n+1} = \mathcal{F}_n(Y_n)$ be the sequential fine solution.
The Parareal iteration sets $U_0^k = y_0$ for all $k \ge 0$, $U_{n+1}^0 = \mathcal{G}_n(U_n^0)$, and
$$U_{n+1}^{k+1} = \mathcal{G}_n(U_n^{k+1}) + \mathcal{F}_n(U_n^k) - \mathcal{G}_n(U_n^k), \qquad k \ge 0,\ 0 \le n \le N - 1.$$
Then $U_n^k = Y_n$ for all $k \ge 0$ and $0 \le n \le k$.
In particular, after $N$ iterations the iterates equal the fine solution at every time $T_n$.

## Remarks

The proof is an induction on $k$.
For $k = 0$ only $n = 0$ is concerned, and $U_0^0 = y_0 = Y_0$.
If the claim holds for $k$, then $U_0^{k+1} = Y_0$, and if $U_n^{k+1} = Y_n$ with $n \le k$, then also $U_n^k = Y_n$, so the iteration gives $U_{n+1}^{k+1} = \mathcal{G}_n(Y_n) + \mathcal{F}_n(Y_n) - \mathcal{G}_n(Y_n) = Y_{n+1}$.
The result needs no property of $\mathcal{F}_n$ and $\mathcal{G}_n$; it guarantees termination but no speed-up.
In practice the iteration is stopped after $K$ iterations with $K$ much smaller than $N$, when $\mathcal{G}_n$ is accurate and stable enough for the iterates to converge fast.

## Explanations

### L1 human

Parareal is a way to use many processors on a problem that seems purely sequential: marching forward in time.
A cheap coarse solver first gives rough values at the start of each time slice.
Then all slices are solved accurately at the same time, in parallel, from these rough starting values.
A cheap sequential sweep corrects the starting values, and the process repeats.
The statement says that the correct values spread forward by at least one slice per iteration.
So the iteration reaches the fine solution after at most as many iterations as there are slices.

### L2 human

The expensive fine solves $\mathcal{F}_n(U_n^k)$ for all $n$ are independent, so they run in parallel; only the coarse sweep is sequential.
The parallel speed-up is bounded by about $N/K$, so the method pays off only when it converges in few iterations.
It works well for diffusive problems and less well for hyperbolic or highly oscillatory ones.
Parareal can be seen as a multiple shooting method in which a Newton step is approximated with the coarse propagator.
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<document source="https://tamnd.github.io/math2/node/KQKJENEMSN.md" tag="KQKJENEMSN" kind="concept">
# Self-injective ring

## Summary

A left self-injective ring is a [[ring|0B1261B5ED]] that is an [[injective module|85H7PRHDPZ]] when viewed as a left module over itself.
Rings $\Z/n\Z$, [[Frobenius algebras|8EVP1K2CZH]] and products of fields are self-injective, while $\Z$ is not.
A left Noetherian left self-injective ring is exactly a [[quasi-Frobenius ring|KP1PP7A6JR]].

## Explanations

### L1 human

Every ring acts on itself by multiplication, so it is a module over itself.
Injective modules are those into which maps can always be extended from a submodule to a bigger module.
A self-injective ring is one for which the ring itself has this extension property.
For $\Z/6\Z$ this holds: any map from an ideal of $\Z/6\Z$ into $\Z/6\Z$ is multiplication by some element.
For $\Z$ it fails: the map $2\Z \to \Z$ sending $2$ to $1$ cannot be extended to $\Z$, since $1$ would have to go to $1/2$.
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<document source="https://tamnd.github.io/math2/node/KQMJ7M57ER.md" tag="KQMJ7M57ER" kind="concept">
# Kolmogorov backward equation

## Summary

The Kolmogorov backward equation is a parabolic partial differential equation satisfied by expectations $u(t, x) = E[f(X_t) \mid X_0 = x]$ of a [[diffusion process|9HDQ094ADQ]].
Its operator is the [[infinitesimal generator|VPA41PWRR1]] of the diffusion, acting on the starting point $x$.
It links stochastic differential equations with partial differential equations and is the basis of many probabilistic formulas for solutions of such equations.

## Explanations

### L1 human

Start a random process at the point $x$ and ask for the average value of $f(X_t)$ at a later time $t$.
The answer is a function $u(t, x)$ of the time and the starting point.
The backward equation says how $u$ changes with $t$: by the drift and the diffusion acting on the starting point.
For a Brownian motion, $u$ solves the heat equation $\partial_t u = \tfrac12 \partial_{xx} u$.
For instance, $f(y) = y^2$ gives $u(t, x) = x^2 + t$, and both sides equal $1$.
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<document source="https://tamnd.github.io/math2/node/KQSVGPG3W0.md" tag="KQSVGPG3W0" kind="definition">
# Monodromy of the hypergeometric equation

## Definition

Let $a, b, c \in \C$, let $X = \C \setminus \set{0, 1}$, and fix a base point $z_0 \in X$.
Let $V$ be the two-dimensional complex vector space of solutions of the [[hypergeometric differential equation|W8VE25R5A7]] on a small disc around $z_0$.
For a loop $\gamma$ in $X$ based at $z_0$, [[analytic continuation|0M3YQJ81NB]] of each solution along $\gamma$ gives a solution near $z_0$ again.
This defines a linear map $M_\gamma \colon V \to V$ that depends only on the homotopy class of $\gamma$.
The *monodromy representation* of the equation is the [[group representation|WYGRXCG2RN]]
$$\rho \colon \pi_1(X, z_0) \to \mathrm{GL}(V), \qquad [\gamma] \mapsto M_\gamma,$$
of the [[fundamental group|S1XK3HXHFX]], with the convention that makes $\rho$ a homomorphism.
Its image is the *monodromy group* of the equation, defined up to conjugation in $\mathrm{GL}_2(\C)$.

## Remarks

The group $\pi_1(X, z_0)$ is free on the classes $\gamma_0$, $\gamma_1$ of simple loops around $0$ and $1$, so $\rho$ is determined by the two matrices $M_{\gamma_0}$ and $M_{\gamma_1}$.
The eigenvalues of $M_{\gamma_0}$ are $1$ and $e^{2\pi i(1 - c)}$, and those of $M_{\gamma_1}$ are $1$ and $e^{2\pi i(c - a - b)}$.
A loop around $\infty$, positively oriented in the coordinate $1/z$, has eigenvalues $e^{2\pi i a}$ and $e^{2\pi i b}$.

Example: for $a = b = 1$, $c = 2$, the solutions near $z_0$ are spanned by $w_1 = 1/z$ and $w_2 = \log(1 - z)/z$, as a direct check in the equation shows.
Both are single-valued near $0$, so $M_{\gamma_0}$ is the identity.
Around $1$, $\log(1 - z)$ changes by $\pm 2\pi i$, so in the basis $(w_1, w_2)$ the matrix $M_{\gamma_1}$ is $\begin{pmatrix} 1 & \pm 2\pi i \\ 0 & 1 \end{pmatrix}$, with the sign fixed by the orientation.
Here the monodromy group is infinite cyclic.

Non-example: continuation along a path from $z_0$ to a different point $z_1$ does not give an element of the monodromy group.
It maps solutions near $z_0$ to solutions near $z_1$, which is a different space.

This is a special case of the general notion of [[monodromy|C8HD62AB6Q]] of a linear differential equation.
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<document source="https://tamnd.github.io/math2/node/KQWJ8SGJEZ.md" tag="KQWJ8SGJEZ" kind="definition">
# Tangential Cauchy–Riemann equations

## Definition

Let $M$ be a smooth real [[submanifold|VKFW27WXEZ]] of $\C^n$.
For $p \in M$, let $T^{0,1}_p M = (T_p M \otimes \C) \cap T^{0,1}_p \C^n$, where $T_p M$ is the [[tangent space|WY6K3P4YCG]] and $T^{0,1}_p \C^n$ is spanned by $\partial / \partial \bar z_1, \dots, \partial / \partial \bar z_n$.
A function $f \colon M \to \C$ of class $C^1$ satisfies the *tangential Cauchy–Riemann equations* if $X f = 0$ for every $p \in M$ and every $X \in T^{0,1}_p M$.

## Remarks

These are the [[Cauchy–Riemann equations|9PT772R94A]] restricted to the directions in which a function on $M$ can be differentiated.
When $T^{0,1} M$ has constant rank, it is the conjugate of a CR structure on $M$, and the solutions are the [[CR functions|JFKYYK2CEB]]; one writes the equations as $\bar\partial_b f = 0$.

For a hypersurface $M = \set{\rho = 0}$, with $\rho$ real and $d\rho \ne 0$ on $M$, the space $T^{0,1}_p M$ is spanned by the operators
$$L_{jk} = \frac{\partial \rho}{\partial \bar z_j} \frac{\partial}{\partial \bar z_k} - \frac{\partial \rho}{\partial \bar z_k} \frac{\partial}{\partial \bar z_j}, \quad 1 \le j < k \le n,$$
so the equations are $L_{jk} f = 0$.
Each $L_{jk}$ is tangent to $M$ because $L_{jk} \rho = 0$.

Example: on $S^3 \subset \C^2$, with $\rho = \abs{z_1}^2 + \abs{z_2}^2 - 1$, there is one equation $L_{12} f = 0$ with $L_{12} = z_1 \, \partial / \partial \bar z_2 - z_2 \, \partial / \partial \bar z_1$.
The function $f = z_1 z_2$ is a solution, since it does not depend on $\bar z_1, \bar z_2$.

Non-example: $f = \abs{z_1}^2$ on $S^3$.
Then $L_{12} f = -z_2 z_1$, which is nonzero at points where $z_1 z_2 \ne 0$, so $f$ does not satisfy the equations.

For a real curve $M$ in $\C$, $T^{0,1} M = 0$, so every function satisfies the tangential equations.
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# Structural stability

## Definition

Let $M$ be a compact smooth manifold without boundary, such as a circle, a sphere or a torus.
Let $\mathfrak{X}^1(M)$ be the space of [[continuously differentiable|0VW90BBSJ4]] [[vector fields|0299VCW1TH]] on $M$, with the $C^1$ topology.
Two vector fields $X$ and $Y$ on $M$ are *topologically equivalent* if there is a [[homeomorphism|0QT00PB5G2]] $h \colon M \to M$ with two properties.
It maps each [[trajectory|0BVT6VSB3S]] of $X$ onto a trajectory of $Y$.
It preserves the direction of time along trajectories.
A vector field $X \in \mathfrak{X}^1(M)$ is *structurally stable* if it has a neighbourhood $\mathcal{N}$ in $\mathfrak{X}^1(M)$ such that every $Y \in \mathcal{N}$ is topologically equivalent to $X$.

## Remarks

Example: on the circle $M = \R / 2\pi\Z$, the field $X(\theta) = \sin \theta$ has the two zeros $0$ and $\pi$, with $X'(0) = 1$ and $X'(\pi) = -1$.
A field $Y$ that is $C^1$-close to $X$ still has exactly two zeros, near $0$ and $\pi$, with derivatives of the same signs.
So $Y$ has one repelling and one attracting rest point joined by two arcs, and a homeomorphism of the circle carries the trajectories of $X$ to those of $Y$.
Hence $X$ is structurally stable.

Non-example: $X(\theta) = 1 - \cos \theta$ has the single zero $0$.
For $\varepsilon > 0$, the field $1 - \cos \theta + \varepsilon$ has no zero, and it is $C^1$-close to $X$ for small $\varepsilon$.
A homeomorphism maps rest points to rest points, so these fields are not equivalent to $X$, and $X$ is not structurally stable.

For flows on a compact surface, structural stability is closely tied to hyperbolicity of the rest points and periodic orbits and to the absence of trajectories joining saddles.
For a region in the plane, the definition is adapted by asking the vector field to point inward on the boundary.
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<document source="https://tamnd.github.io/math2/node/KR1F39C0SP.md" tag="KR1F39C0SP" kind="definition">
# Omega-complete partial order

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
An *$\omega$-chain* in $P$ is a sequence $(x_n)_{n \in \N}$ with $x_0 \le x_1 \le x_2 \le \cdots$.
The poset $P$ is an *omega-complete partial order*, or *$\omega$-cpo*, if every $\omega$-chain in $P$ has a [[supremum|0R3H5PANRQ]] $\bigvee_n x_n$ in $P$.

## Remarks

Some authors also require a least element.
Every [[directed-complete partial order|7TSXP95QN6]] is an $\omega$-cpo, since the set of terms of an $\omega$-chain is directed.
Example: $\N \cup \set{\infty}$ with $n \le \infty$ for all $n$; an $\omega$-chain is either eventually constant, with supremum its final value, or unbounded in $\N$, with supremum $\infty$.
Non-example: $\N$ itself, since the chain $0 \le 1 \le 2 \le \cdots$ has no supremum.
The converse fails for the set $\omega_1$ of countable ordinals with their usual order.
It is an $\omega$-cpo, because a countable set of countable ordinals has a countable supremum.
But the directed set $\omega_1$ itself has no supremum in $\omega_1$.
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<document source="https://tamnd.github.io/math2/node/KR4KX354JN.md" tag="KR4KX354JN" kind="definition">
# Symmetric operator

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]].
Let $T \colon D(T) \to H$ be a [[densely defined|BCJHNHKQFG]] linear operator, where the domain $D(T)$ is a dense linear subspace of $H$.
The operator $T$ is *symmetric* when
$$\ip{Tx}{y} = \ip{x}{Ty} \quad \text{for all } x, y \in D(T).$$

## Remarks

Example: on $H = L^2(0, 1)$ let $D(T)$ be the smooth functions with compact support in $(0, 1)$ and $Tf = -i f'$.
For $f, g \in D(T)$, integration by parts with vanishing boundary terms gives $\int_0^1 -i f' \bar g = \int_0^1 f \, \overline{-i g'}$, so $T$ is symmetric.

Non-example: on the same domain, $Sf = f'$ satisfies $\ip{Sf}{g} = -\ip{f}{Sg}$, so $\ip{Sf}{g} \ne \ip{f}{Sg}$ whenever $\ip{f}{Sg} \ne 0$; it is skew-symmetric, not symmetric.

An equivalent form: $T$ is symmetric if and only if its [[adjoint|86B7HF79PQ]] $T^*$ extends $T$, written $T \subseteq T^*$.
$T$ is self-adjoint when moreover $T^* = T$, including equality of domains.
By the [[Hellinger–Toeplitz theorem|ZXJ7FJ3WCE]], a symmetric operator with $D(T) = H$ is bounded, and then it is self-adjoint.
Some authors, and Mathlib, call a linear map $T \colon E \to E$ on an inner product space symmetric when the identity holds on all of $E$, without a domain.
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<document source="https://tamnd.github.io/math2/node/KR5Q7WMAVC.md" tag="KR5Q7WMAVC" kind="concept">
# Bateman transform

## Summary

The Bateman transform builds solutions of [[Laplace's equation|S8RG6N1PRT]] in four real variables from holomorphic functions of three complex variables.
It also gives solutions of the [[wave equation|5EHD2XBQDG]] in three space dimensions.
The solution is a contour integral in one complex variable $\zeta$ of a holomorphic function evaluated at two linear expressions in the coordinates.
It is named after Harry Bateman and extends Whittaker's integral formula for harmonic functions in three variables.
It is a forerunner of the Penrose transform of twistor theory.

## Explanations

### L1 human

Holomorphic functions of one variable give harmonic functions of two variables for free: their real parts.
The Bateman transform does something similar one level up.
It plugs carefully chosen combinations of four real coordinates into a holomorphic function and integrates along a loop.
The combinations are chosen so that the Laplacian of the result vanishes automatically, whatever the holomorphic function is.
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<document source="https://tamnd.github.io/math2/node/KR7SWBYHRQ.md" tag="KR7SWBYHRQ" kind="concept">
# Normal ring

## Summary

A normal ring is a commutative ring whose [[localization|X6JD4AG973]] at every prime ideal is an [[integrally closed domain|H20P600HQS]].
It is the local form of integral closedness, and it allows rings that are not domains, such as products of integrally closed domains.
A Noetherian normal ring is a finite product of integrally closed domains.
Normal rings are the rings of functions on normal schemes, and normality is described by [[Serre's criterion|W27Z60Q1PF]].

## Explanations

### L1 human

Being integrally closed is a property of a domain, but a space can have several pieces, and its ring of functions is then not a domain.
Normality asks for integral closedness near each point, by looking at the ring of germs of functions there.
The ring $\Z \times \Z$ is normal, as it describes two disjoint copies of the same well-behaved space.
The ring $k[x, y]/(y^2 - x^3)$ of the cusp curve is not normal, because it fails at the cusp.
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<document source="https://tamnd.github.io/math2/node/KR80YEKVWB.md" tag="KR80YEKVWB" kind="statement">
# Independence of sample mean and sample variance for normal samples

## Statement

Let $n \ge 2$, and let $X_1, \dots, X_n$ be [[independent and identically distributed|0AJN7WHBJZ]] with the [[normal distribution|07RXBVTVZB]] $\mathcal{N}(\mu, \sigma^2)$, where $\sigma > 0$.
Let $\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$ be the [[sample mean|0QNRBHWV35]] and $S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2$ the [[sample variance|0JN9P4A944]].
Then:

1. $\bar{X}$ and $S^2$ are [[independent|0E9502NT92]];
2. $\bar{X} \sim \mathcal{N}(\mu, \sigma^2/n)$;
3. $(n - 1) S^2/\sigma^2$ has the [[chi-squared distribution|0RFMMBHFTV]] with $n - 1$ degrees of freedom.

## Remarks

Put $Z_i = (X_i - \mu)/\sigma$, which are independent standard normal.
[[Cochran's theorem|HSBWJSBH0D]] with $A_1 = \frac{1}{n} J$ and $A_2 = I_n - \frac{1}{n} J$, where $J$ is the matrix of ones, shows that $n \bar{Z}^2$ and $\sum_i (Z_i - \bar{Z})^2 = (n-1)S^2/\sigma^2$ are independent, with $\chi^2$ laws with $1$ and $n - 1$ degrees of freedom.
Independence of $\bar{X}$ itself follows because $\bar{Z}$ is a function of the projection $A_1 Z$, which is independent of $A_2 Z$.
Another proof uses an orthogonal matrix whose first row is $(1, \dots, 1)/\sqrt{n}$.

At work: the statistic $T = \sqrt{n}(\bar{X} - \mu)/S$ is a standard normal variable divided by $\sqrt{V/(n-1)}$, with $V \sim \chi^2_{n-1}$ independent of it.
So $T$ has the [[Student t-distribution|0NQ5798SB1]] with $n - 1$ degrees of freedom.
This gives the one-sample $t$-test and the $t$ confidence interval for $\mu$.

The normality hypothesis is necessary: for i.i.d. samples from a nondegenerate distribution with finite variance and $n \ge 2$, independence of $\bar{X}$ and $S^2$ holds only if the distribution is normal.
For example, for Bernoulli variables, $\bar{X} = 0$ forces $S^2 = 0$, while $S^2 > 0$ is possible when $0 < \bar{X} < 1$.

## Explanations

### L1 human

For normal data, knowing the average of the sample tells you nothing about how spread out the sample is.
This is special to the normal distribution.
It is the reason why the $t$ statistic has a simple exact distribution.

### L2 human

The result is the case of two pieces of Cochran's theorem.
It underlies the exact $t$-test, the $\chi^2$ confidence interval for $\sigma^2$, and through them much of classical small-sample inference.
The analogous result for multivariate normal samples says that the sample mean vector and the sample covariance matrix are independent, the latter with a Wishart distribution.
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<document source="https://tamnd.github.io/math2/node/KRDR04RQ7N.md" tag="KRDR04RQ7N" kind="concept">
# Ordered group

## Summary

An ordered group, or partially ordered group, is a [[group|G5S1M9]] with a [[partial order|0QV2JS9RJ3]] that is preserved by multiplication on both sides.
The order is then determined by its positive cone, the set of elements that are at least the identity.
Ordered groups include the additive groups of ordered fields, groups of functions with the pointwise order, and lattice-ordered groups such as groups of real functions.
They are the common setting of the theory of ordered algebraic structures.

## Explanations

### L1 human

In the integers, if $a \le b$ then $a + c \le b + c$: adding the same number to both sides keeps an inequality.
An ordered group is a group with an order that has this property.
The order does not need to compare every two elements.
For example, in the plane one can say $(a, b) \le (c, d)$ when $a \le c$ and $b \le d$.
Then $(1, 0)$ and $(0, 1)$ are not comparable, but adding a vector to both sides still keeps every inequality.
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<document source="https://tamnd.github.io/math2/node/KREYT31SS1.md" tag="KREYT31SS1" kind="statement">
# Viscosity solution of the Hamilton–Jacobi–Bellman equation

## Statement

Let $U \subseteq \R^m$ be a nonempty [[compact set|0R6EWFKRCF]] and $T > 0$.
Let $f \colon [0, T] \times \R^n \times U \to \R^n$ and $L \colon [0, T] \times \R^n \times U \to \R$ be bounded and continuous.
Assume that both are Lipschitz in $x$, uniformly in $(t, u)$, and let $g \colon \R^n \to \R$ be bounded and Lipschitz.
Let $V$ be the [[value function|K6G0DRQ3CP]] of the problem of minimizing $\int_t^T L(s, y, u) \, ds + g(y(T))$ subject to $y' = f(s, y, u)$, $y(t) = x$.
Let $H(t, x, p) = \max_{u \in U} \big( -p \cdot f(t, x, u) - L(t, x, u) \big)$.
Then $V$ is bounded and Lipschitz on $[0, T] \times \R^n$, $V(T, \cdot) = g$, and $V$ is a [[viscosity solution|H1ERJ3T75T]] of the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]]
$$-V_t + H(t, x, \nabla_x V) = 0 \quad \text{in } (0, T) \times \R^n.$$
That is, for every $C^1$ function $\varphi$ on $(0, T) \times \R^n$ and every point $(t_0, x_0) \in (0, T) \times \R^n$:

1. if $V - \varphi$ has a local maximum at $(t_0, x_0)$, then $-\varphi_t(t_0, x_0) + H(t_0, x_0, \nabla_x \varphi(t_0, x_0)) \le 0$;
2. if $V - \varphi$ has a local minimum at $(t_0, x_0)$, then $-\varphi_t(t_0, x_0) + H(t_0, x_0, \nabla_x \varphi(t_0, x_0)) \ge 0$.

## Remarks

The proof uses the dynamic programming principle.
If $V - \varphi$ has a local maximum at $(t_0, x_0)$, use a constant control $u$ on $[t_0, t_0 + h]$: the principle gives $\varphi(t_0, x_0) - \varphi(t_0 + h, y(t_0 + h)) \le \int_{t_0}^{t_0 + h} L \, ds$.
Dividing by $h$ and letting $h \to 0$ gives $-\varphi_t - \nabla_x \varphi \cdot f(t_0, x_0, u) - L(t_0, x_0, u) \le 0$ for every $u \in U$, which is condition 1.
Condition 2 uses nearly optimal controls in the same way, with the compactness of $U$.
Example: for $f = u$, $U = [-1, 1]$, $L = 0$ and $g(x) = \abs{x}$, the value function $\max(\abs{x} - (T - t), 0)$ is not differentiable, yet it is a viscosity solution of $-V_t + \abs{V_x} = 0$.
Here $g$ is unbounded, but one checks the two conditions directly at the corners.
By the comparison principle of Crandall and Lions, $V$ is the only bounded uniformly continuous viscosity solution with $V(T, \cdot) = g$.
So the equation characterizes the value function even when no classical solution exists.

## Explanations

### L1 human

The optimal cost of a control problem should satisfy the Hamilton–Jacobi–Bellman equation, but it often has corners where derivatives do not exist.
This theorem says that it still satisfies the equation in the weak sense of viscosity solutions.
Together with uniqueness, it shows that the equation determines the optimal cost completely.

### L2 human

The theorem is the bridge between optimal control and the theory of viscosity solutions.
Smooth test functions $\varphi$ touching $V$ from above or below replace the missing derivatives of $V$.
Combined with the comparison principle, it gives convergence proofs for numerical schemes, which only need to be monotone, stable and consistent.
It also explains why the verification theorem for smooth solutions is not enough in general.
</document>

<document source="https://tamnd.github.io/math2/node/KRF03HSD44.md" tag="KRF03HSD44" kind="definition">
# Hazard function

## Definition

Let $T$ be a random variable with values in $[0, \infty)$ that has a [[probability density function|0ZY9DX8FBF]] $f$, and let $S$ be its [[survival function|QTM1HZM53Q]].
The *hazard function* of $T$ is defined for all $t \ge 0$ with $S(t) > 0$ by
$$h(t) = \frac{f(t)}{S(t)}.$$
The *cumulative hazard function* is $H(t) = \int_0^t h(u) \, du$.

## Remarks

At every point $t$ where $f$ is continuous, the hazard is the limit of a [[conditional probability|0990W5PN13]] per unit time:
$$h(t) = \lim_{\Delta \downarrow 0} \frac{P(t \le T < t + \Delta \mid T \ge t)}{\Delta}.$$
Since $h = -S'/S = -(\log S)'$ there, one gets $S(t) = e^{-H(t)}$, so $h$ determines the distribution of $T$.

Example: for the [[exponential distribution|0E99F51GVB]] with rate $\lambda$, $f(t) = \lambda e^{-\lambda t}$ and $S(t) = e^{-\lambda t}$, so $h(t) = \lambda$ is constant and $H(t) = \lambda t$.
For the Weibull distribution with shape $k > 0$ and scale $\lambda > 0$, $S(t) = e^{-(t/\lambda)^k}$ and $h(t) = (k/\lambda)(t/\lambda)^{k-1}$, which increases when $k > 1$ and decreases when $k < 1$.

Non-example: if $T$ takes only the values $1$ and $2$, each with probability $1/2$, then $T$ has no density and the formula does not apply.
For discrete times one uses instead the discrete hazard $P(T = t \mid T \ge t)$, here $1/2$ at $t = 1$ and $1$ at $t = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/KRJX5RA0PW.md" tag="KRJX5RA0PW" kind="concept">
# Unbounded operator

## Summary

An unbounded operator is a linear map defined on a linear subspace of a normed space, its domain, with values in another normed space.
Despite the name, it need not be unbounded: the term means that boundedness is not assumed.
Differential operators such as $f \mapsto f'$ and the position and momentum operators of quantum mechanics are the main examples.
The domain is part of the data, and two operators with the same formula but different domains are different.

## Explanations

### L1 human

Differentiation is a linear operation on functions, but it cannot be applied to every square-integrable function, and it can turn a small function into a huge one.
For example, $\sin(nx)/n$ is small on $[0, \pi]$, but its derivative $\cos(nx)$ is not small.
An unbounded operator handles this by fixing a domain, a subspace of functions where the operation is defined.
On this domain the map is linear, but no bound $\norm{Tx} \le C \norm{x}$ needs to hold.
</document>

<document source="https://tamnd.github.io/math2/node/KRK48SX454.md" tag="KRK48SX454" kind="definition">
# Area-minimizing current

## Definition

Let $1 \le m \le n$, and let $T$ be an [[integral current|EG4AJVEFT2]] of dimension $m$ on $\R^n$, with mass $\mathbf{M}(T)$ and boundary $\partial T$.
The current $T$ is *area-minimizing*, or *mass-minimizing*, if
$$\mathbf{M}(T) \le \mathbf{M}(S)$$
for every integral $m$-current $S$ on $\R^n$ with $\partial S = \partial T$.

## Remarks

A local version is used for currents in an open set $U$.
There one compares $T$ with $S$ only when $T - S$ is the boundary of an integral current with compact support in $U$.
For a compact oriented $C^1$ submanifold, the mass of the associated [[current|360H7MD9RD]] is its area.
So an area-minimizing submanifold has least area among all oriented competitors with the same boundary, smooth or not.

Example: let $D$ be the closed unit disc in the plane $x_3 = 0$ of $\R^3$, oriented by $e_1 \wedge e_2$, and let $\omega = dx_1 \wedge dx_2$, a closed [[differential form|0E3G0GRPZ6]] of comass $1$.
For every integral $2$-current $S$ with $\partial S = \partial [\![D]\!]$, one has $S - [\![D]\!] = \partial R$ for an integral $3$-current $R$, since every integral cycle in $\R^3$ bounds.
Then $S(\omega) - [\![D]\!](\omega) = R(d\omega) = 0$ by the definition of the boundary, as in [[Stokes' theorem|0AKRKVHW94]].
So $\mathbf{M}(S) \ge S(\omega) = [\![D]\!](\omega) = \pi = \mathbf{M}([\![D]\!])$, and $[\![D]\!]$ is area-minimizing.

Non-example: the upper unit hemisphere, oriented so that its boundary is $\partial [\![D]\!]$, has mass $2\pi > \pi$, so it is not area-minimizing.

## Explanations

### L1 human

A current is area-minimizing if no other integral current with the same boundary has smaller mass.
</document>

<document source="https://tamnd.github.io/math2/node/KRM89R0T28.md" tag="KRM89R0T28" kind="statement">
# McKenzie's undecidability of the finite basis problem

## Statement

There is no algorithm that decides, for an arbitrary finite algebra $A$ of finite signature given by its operation tables, whether $A$ is [[finitely based|YGYZAQ12MZ]].
More precisely, there is an effective construction that assigns a finite algebra $A(T)$ of finite signature to each [[Turing machine|758EYCN27B]] $T$.
It has this property: $T$ halts, started on the empty tape, if and only if $A(T)$ is finitely based.

## Remarks

The second sentence implies the first, because the [[halting problem|Z7TN2B35B0]] is undecidable.
So this theorem gives a negative answer to [[Tarski's finite basis problem|AB22FWBQ0M]].
McKenzie published the proof in 1996.
McKenzie built his algebra from a simpler finite algebra that encodes the computation of $T$.
The proof relates the halting of $T$ to the sizes of the subdirectly irreducible algebras in the variety generated by this simpler algebra.
McKenzie also used the simpler algebra to show that it is undecidable whether a finite algebra generates a variety with a finite residual bound.
Willard later proved that the simpler algebra itself is finitely based if and only if $T$ halts, which gives a shorter route to the theorem.

## Explanations

### L1 human

Some finite algebras have all their laws captured by a finite list, and others do not.
The theorem says that no computer program can sort finite algebras into these two kinds.
It is proved by hiding a computer program inside a finite algebra so that the algebra has a finite list of laws exactly when the program stops.

### L2 human

The theorem closes a question that had driven much of the work on finite bases since the 1950s.
It shows that results such as Baker's finite basis theorem, which give finite bases under structural hypotheses, cannot be replaced by a general decision procedure.
The construction of $A(T)$ links equational logic with residual bounds of varieties, which is also the setting of Park's conjecture.
</document>

<document source="https://tamnd.github.io/math2/node/KRNFEZWPY0.md" tag="KRNFEZWPY0" kind="concept">
# Field norm

## Summary

Let $L/K$ be a [[finite extension|0C1Y04SZ1W]] and let $\alpha \in L$.
Multiplication by $\alpha$ is a $K$-linear map of $L$, and the field norm $N_{L/K}(\alpha)$ is its [[determinant|0X0HQYSZ0K]].
It is a multiplicative map $L \to K$, and it is nonzero exactly on the nonzero elements.
For a separable extension it is the product of the images of $\alpha$ under all embeddings of $L$ into an algebraic closure.
Norms measure the size of algebraic numbers, detect units in rings of integers, and appear in [[Hilbert's Theorem 90|YFF8B5A0QG]].

## Explanations

### L1 human

Multiplying by a fixed element $\alpha$ stretches the field $L$, seen as a vector space over $K$.
The norm is the determinant of this stretching.
For the Gaussian numbers $a + bi$ over $\Q$, it is $a^2 + b^2$, the product of $a + bi$ and its conjugate $a - bi$.
So $N(3 + 4i) = 25$.
The rule $N(\alpha\beta) = N(\alpha) N(\beta)$ explains why a product of sums of two squares is again a sum of two squares.
</document>

<document source="https://tamnd.github.io/math2/node/KRTDAZP29B.md" tag="KRTDAZP29B" kind="definition">
# Noncommutative ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
The ring $R$ is *noncommutative* if there are elements $a, b \in R$ with $ab \ne ba$.

## Remarks

Example: the ring $M_2(\R)$ of $2 \times 2$ real matrices is noncommutative.
With $E_{12}$ the matrix with $1$ in position $(1,2)$ and $0$ elsewhere, and $E_{21}$ defined likewise, one has $E_{12} E_{21} = E_{11} \ne E_{22} = E_{21} E_{12}$.
Non-example: the ring $\Z$ is not noncommutative, since $ab = ba$ for all integers $a, b$.
Many authors use "noncommutative ring" for a ring that is not assumed to be commutative.
In that use, commutative rings are allowed as special cases.
</document>

<document source="https://tamnd.github.io/math2/node/KRWE7P7FHP.md" tag="KRWE7P7FHP" kind="definition">
# Associativity equation

## Definition

Let $X$ be a set.
The *associativity equation* on $X$ is the [[functional equation|SRAN4EKWS0]]
$$F(F(x, y), z) = F(x, F(y, z)) \quad \text{for all } x, y, z \in X,$$
for an unknown [[binary operation|0M418FE7BB]] $F \colon X \times X \to X$.
A solution is an associative operation on $X$.

## Remarks

Example: $F(x, y) = x + y + xy$ on $X = (-1, \infty)$ is a solution, since both sides equal $(1 + x)(1 + y)(1 + z) - 1$.
Non-example: $F(x, y) = x - y$ on $\R$ is not a solution, since $F(F(0, 0), 1) = -1$ while $F(0, F(0, 1)) = 1$.
Every operation $F(x, y) = g^{-1}(g(x) + g(y))$, with $g \colon X \to \R$ injective and $g(X)$ closed under addition, is a solution.
On a real interval, the continuous solutions that are strictly increasing in each variable are all of this form, with $g$ continuous and strictly monotone.
</document>

<document source="https://tamnd.github.io/math2/node/KRX3JWDV1B.md" tag="KRX3JWDV1B" kind="definition">
# Aperiodic set of prototiles

## Definition

Let $n \ge 1$ and let $S$ be a finite set of subsets of $\R^n$, called *prototiles*, possibly together with matching rules on how they may meet.
A tiling of $\R^n$ *by* $S$ is a [[tiling|GKJK293DXS]] in which every tile is congruent to a member of $S$ and the matching rules hold.
The set $S$ is *aperiodic* if:

1. there is a tiling of $\R^n$ by $S$;
2. no tiling of $\R^n$ by $S$ is [[periodic|WFRGSBFZGV]].

## Remarks

Example: the two rhombi of the [[Penrose tiling|MBE83FTRBG]], with their matching rules, form an aperiodic set of prototiles in $\R^2$.

Non-example: the set $S = \set{[0,1]^2}$ consisting of the unit square is not aperiodic.
Clause 1 holds, but the standard grid of unit squares is a periodic tiling by $S$, so clause 2 fails.

A tiling with no nonzero period is called *non-periodic*.
A set of prototiles is aperiodic exactly when it admits tilings and all of them are non-periodic.
The first aperiodic set, found by Robert Berger in 1966, consisted of a very large number of square tiles with coloured edges.
</document>

<document source="https://tamnd.github.io/math2/node/KRZD96GSVC.md" tag="KRZD96GSVC" kind="concept">
# Euclidean geometry

## Summary

Euclidean geometry is the geometry of the [[Euclidean plane|0XW7V9ZZ05]] and of [[Euclidean space|0XPR1E85W0]]: the study of points, lines, distances and angles in a flat space.
It can be built synthetically from axioms, as in [[Euclid's Elements|83WHDAV68J]] and [[Hilbert's axioms|KM1FB8XZ2R]], or analytically with coordinates in $\R^n$.
Its characteristic axiom is the [[parallel postulate|2PC3XY0PYZ]]; dropping it leads to [[absolute geometry|VKFPCSRNDH]] and to [[non-Euclidean geometry|SPCYMFN5V2]].

## Explanations

### L1 human

Euclidean geometry is the geometry of a flat sheet of paper that goes on forever.
Straight lines, circles, triangles and their measures are its objects.
Through a point outside a line there is exactly one line parallel to it, and the angles of a triangle add up to $180^\circ$.
For example, a triangle with sides $3$, $4$ and $5$ has a right angle, by the [[Pythagorean theorem|0VP3A1290K]].
</document>

<document source="https://tamnd.github.io/math2/node/KS3KDRRJ58.md" tag="KS3KDRRJ58" kind="definition">
# Self-similar process

## Definition

Let $(X_t)_{t \ge 0}$ be a real-valued [[stochastic process|0QH5ENKZM1]] and let $H > 0$.
The process is *self-similar with Hurst index $H$* if for every $a > 0$ the processes $(X_{at})_{t \ge 0}$ and $(a^H X_t)_{t \ge 0}$ have the same [[finite-dimensional distributions|Y0CN7V3B7E]].
That is, for all $n \ge 1$ and $t_1, \dots, t_n \ge 0$, the vectors $(X_{a t_1}, \dots, X_{a t_n})$ and $(a^H X_{t_1}, \dots, a^H X_{t_n})$ have the same law.

## Remarks

The [[Wiener process|22735FGJV6]] $W$ is self-similar with $H = 1/2$.
Check: $(W_{at})_{t \ge 0}$ and $(\sqrt{a}\, W_t)_{t \ge 0}$ are both centred Gaussian processes with covariance $a \min(s, t)$, so they have the same finite-dimensional distributions.
Brownian motion with drift, $X_t = W_t + t$, is not self-similar for any $H$.
Indeed $E[X_{at}] = at$ and $E[a^H X_t] = a^H t$ force $H = 1$, while $\operatorname{Var}(X_{at}) = at$ and $\operatorname{Var}(a^H X_t) = a^{2H} t$ force $H = 1/2$.
One often writes $X_{at} \overset{d}{=} a^H X_t$, where equality in law is meant for the whole process.
Taking $t = 0$ gives $X_0 \overset{d}{=} a^H X_0$ for all $a > 0$, which forces $X_0 = 0$ almost surely.
</document>

<document source="https://tamnd.github.io/math2/node/KS4FJ6H5JH.md" tag="KS4FJ6H5JH" kind="example">
# Hénon map

## Example

For parameters $a, b$, the *Hénon map* is
$$H_{a,b}(x, y) = (1 - a x^2 + y, \; b x).$$
Hénon studied it for real $a, b$ as a model of a dissipative system; for $a = 1.4$ and $b = 0.3$ its orbits accumulate numerically on a [[strange attractor|9WKCZGG906]].

For complex $a, b$ with $b \ne 0$, $H_{a,b}$ is a holomorphic automorphism of $\C^2$.
Check: if $(u, v) = H_{a,b}(x, y)$, then $x = v/b$ and $y = u - 1 + a v^2/b^2$, so $H_{a,b}$ is a bijection with polynomial inverse
$$H_{a,b}^{-1}(u, v) = \left(\frac{v}{b}, \; u - 1 + \frac{a v^2}{b^2}\right).$$
Its Jacobian matrix is $\begin{pmatrix} -2ax & 1 \\ b & 0 \end{pmatrix}$, with [[Jacobian determinant|0ZCSMYHTBP]] $-b$, a nonzero constant.
For real $a, b$ it multiplies area in $\R^2$ by $\abs{b}$, so for $\abs{b} < 1$ it contracts area.
As a map of $\C^2$ it multiplies volume by $\abs{b}^2$.

## Remarks

In complex dynamics one often writes generalized Hénon maps as $(x, y) \mapsto (p(x) - a y, x)$ with $p$ a polynomial of degree $d \ge 2$ and $a \ne 0$.
Friedland and Milnor showed that every polynomial automorphism of $\C^2$ is conjugate either to a map of a simple triangular form or to a composition of generalized Hénon maps.
So Hénon maps are the main objects of [[holomorphic dynamics in several variables|6WCRMYEYF9]] on $\C^2$.
For them one studies the sets $K^\pm$ of points whose forward or backward orbits stay bounded, and the Julia sets $J^\pm = \partial K^\pm$.

## Explanations

### L1 human

The Hénon map folds the plane and squeezes it, then applies the same rule again.
Unlike the logistic map, it can be undone: from the image point one recovers the starting point by a formula.
Even so, repeated application mixes points in a chaotic way, and for some parameters the orbits pile up on a fractal set.
Taken over the complex numbers, it is the simplest interesting invertible holomorphic map of two-dimensional complex space.
</document>

<document source="https://tamnd.github.io/math2/node/KS91STZCGM.md" tag="KS91STZCGM" kind="statement">
# Gibbs sampling

## Statement

Let $S = S_1 \times \cdots \times S_d$ be a product of countable sets, and let $\pi$ be a probability distribution on $S$.
For $x \in S$ and $1 \le i \le d$, write $x_{-i}$ for $x$ with its $i$-th coordinate removed, and let $Z_i(x) = \sum_{z \in S, \, z_{-i} = x_{-i}} \pi(z)$.
Let $K_i$ be the [[transition matrix|0F462493EA]] on $S$ with
$$K_i(x,y) = \frac{\pi(y)}{Z_i(x)} \quad \text{if } y_{-i} = x_{-i} \text{ and } Z_i(x) > 0,$$
$K_i(x,y) = 0$ for the other $y$ when $Z_i(x) > 0$, and $K_i(x, \cdot) = \delta_x$ when $Z_i(x) = 0$.
So $K_i$ redraws the $i$-th coordinate from its [[conditional distribution|0ZN9EZTEM6]] under $\pi$ given the others.
Then:

1. each $K_i$ satisfies [[detailed balance|CTGN6CK1ZS]] with respect to $\pi$, that is, $\pi(x) K_i(x,y) = \pi(y) K_i(y,x)$ for all $x, y \in S$;
2. $\pi$ is a [[stationary distribution|0KRH5157WP]] of the systematic-scan Gibbs sampler $K_1 K_2 \cdots K_d$ and of the random-scan Gibbs sampler $\frac{1}{d} \sum_{i=1}^{d} K_i$.

## Remarks

For clause 1, if $y_{-i} = x_{-i}$ and $Z_i(x) > 0$, then $Z_i(y) = Z_i(x)$ and both sides equal $\pi(x) \pi(y) / Z_i(x)$.
If $Z_i(x) = 0$ then $\pi(x) = \pi(y) = 0$ for all such $y$, and in all other cases both sides are $0$.
Summing detailed balance over $x$ gives $\pi K_i = \pi$, and clause 2 follows because a product or an average of kernels that fix $\pi$ also fixes $\pi$.
The same argument works for densities on $\R^d$, with sums replaced by integrals.
Stationarity alone does not give convergence of the chain to $\pi$.
If $S$ is finite and $\pi(x) > 0$ for every $x$, every entry of $K_1 \cdots K_d$ is positive, so the chain is irreducible and aperiodic and converges to $\pi$ by the [[convergence theorem|B1E1ES1BYD]].
Without positivity this can fail: for $\pi$ uniform on $\set{(0,0), (1,1)} \subset \set{0,1}^2$, each $K_i$ fixes every state of positive mass, so the chain never moves.

## Explanations

### L1 human

Gibbs sampling draws random samples from a joint distribution of many variables when sampling them all at once is hard.
It updates one variable at a time, drawing it from its distribution given the current values of all the others.
Repeating these simple updates builds a random walk through the space.
The result says that this walk leaves the target distribution unchanged.
So, when the walk mixes well, the visited states are samples from the target.

### L2 human

Gibbs sampling is the [[Markov chain Monte Carlo|ZGNBR2SPZ0]] method of choice when the full conditional distributions are easy to sample.
This is the case in many Bayesian hierarchical models with conjugate priors and in lattice models such as the Ising model.
Each update is a Metropolis–Hastings step whose proposal is the full conditional, so its acceptance probability is always $1$.
The systematic-scan sampler is in general not reversible, though each $K_i$ is; the random-scan sampler is reversible.
Strong dependence between coordinates slows mixing, since moves along single coordinates then make little progress; blocking several coordinates together is the usual remedy.
</document>

<document source="https://tamnd.github.io/math2/node/KSA55H8Q8N.md" tag="KSA55H8Q8N" kind="definition">
# Turing pattern

## Definition

Let $m \ge 2$, let $D$ be a diagonal $m \times m$ matrix with positive diagonal entries, and let $f \colon \R^m \to \R^m$ be smooth.
Consider the [[reaction–diffusion system|TAAHWE917M]] $\partial_t u = D \Delta u + f(u)$ for $u \colon \R^n \times [0, \infty) \to \R^m$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
Let $u^* \in \R^m$ satisfy $f(u^*) = 0$ and let $J$ be the [[Jacobian matrix|047NDTEPCV]] of $f$ at $u^*$.
The uniform state $u^*$ is *Turing unstable*, or has a *diffusion-driven instability*, if:

1. every [[eigenvalue|0BQAA2BK3Q]] of $J$ has negative real part, so $u^*$ is stable without diffusion;
2. there is $k > 0$ such that $J - k^2 D$ has an eigenvalue with positive real part.

A *Turing pattern* is a stationary, spatially non-constant solution near $u^*$ that bifurcates from a Turing unstable uniform state.
It is typically periodic, with a wave number $k$ as in clause 2.

## Remarks

The perturbation $u = u^* + e^{\lambda t} e^{i q \cdot x} w$ with a wave vector $q \in \R^n$, $\abs{q} = k$, solves the linearised system exactly when $\lambda$ is an eigenvalue of $J - k^2 D$ with eigenvector $w$, which explains clause 2.
Example: $m = 2$, $J = \begin{pmatrix} 1 & -1 \\ 3 & -2 \end{pmatrix}$, $D = \operatorname{diag}(1, 10)$.
Then $\tr J = -1 < 0$ and $\det J = 1 > 0$, so clause 1 holds.
For $k^2 = 0.3$, $\det(J - k^2 D) = (0.7)(-5) + 3 = -0.5 < 0$, so $J - k^2 D$ has a positive real eigenvalue and clause 2 holds.
Non-example: if $D$ is a multiple $d I$ of the identity, the eigenvalues of $J - k^2 D$ are those of $J$ shifted by $-d k^2$, so clause 2 fails whenever clause 1 holds.
For $m = 2$, a necessary condition is that the two diffusion coefficients differ.
</document>

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# Versal unfolding

## Summary

An unfolding of a function [[germ|7SWSRGDT4A]] is a family of germs depending on parameters that reduces to the given germ when the parameters are zero.
An unfolding is versal if every other unfolding can be obtained from it by changes of a simple kind.
These are a change of parameters, a parameter-dependent change of coordinates and the addition of a function of the parameters.
So a versal unfolding contains, up to equivalence, every way of perturbing the germ.
A versal unfolding with the least number of parameters is called universal or miniversal, and that number is the [[codimension|B4FRQMWQKP]] of the germ.
Versal unfoldings are the normal forms of [[catastrophe theory|6FKJ6SXP3G]].

## Explanations

### L1 human

Take a degenerate function, such as $x^3$, whose flat point at $0$ is destroyed by almost any small change.
An unfolding lists the changes by parameters, for example $x^3 + a x + b x^2 + c x^4$.
A versal unfolding is a short list that is already enough.
For $x^3$, the one-parameter family $x^3 + a x$ captures every possible perturbation up to a change of coordinates and an added constant.
Indeed, the term $b x^2$ can be removed by shifting $x$, which only changes $a$ and adds a constant.
The term $c x^4$ can be removed by a smooth change of coordinates near $0$.
</document>

<document source="https://tamnd.github.io/math2/node/KSN3AJ6EJD.md" tag="KSN3AJ6EJD" kind="definition">
# Localization of a category

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $W$ be a class of morphisms of $\mathcal{C}$.
A *localization of $\mathcal{C}$ at $W$* is a category $\mathcal{C}[W^{-1}]$ together with a [[functor|28NQ0KRHD3]] $\gamma \colon \mathcal{C} \to \mathcal{C}[W^{-1}]$ such that:

1. $\gamma(w)$ is an isomorphism for every $w \in W$;
2. for every functor $F \colon \mathcal{C} \to \mathcal{D}$ such that $F(w)$ is an isomorphism for every $w \in W$, there is a unique functor $G \colon \mathcal{C}[W^{-1}] \to \mathcal{D}$ with $F = G \circ \gamma$.

## Remarks

By the universal property, a localization is unique up to a unique isomorphism of categories compatible with $\gamma$.
It always exists if size issues are ignored.
Its objects are those of $\mathcal{C}$, and its morphisms are equivalence classes of zigzags of morphisms of $\mathcal{C}$ whose backward arrows are in $W$.
The hom-sets of this construction may fail to be small; conditions such as a calculus of fractions or a model structure give control over them.
Many authors use a weaker universal property, asking that precomposition with $\gamma$ be an equivalence between functor categories onto the functors that invert $W$.

Example: let $\mathcal{C}$ have two objects $0$ and $1$ and one non-identity morphism $a \colon 0 \to 1$, and let $W = \set{a}$.
The category $\mathcal{L}$ with objects $0, 1$ and exactly one morphism between any two objects is a localization, with $\gamma$ the identity on objects.
Indeed $\gamma(a)$ is invertible, and a functor $F$ inverting $a$ factors uniquely by sending the morphism $1 \to 0$ of $\mathcal{L}$ to $F(a)^{-1}$.

Non-example: the identity functor $\mathcal{C} \to \mathcal{C}$ satisfies clause 2 for the functors that invert $a$, but it fails clause 1, since $a$ is not an isomorphism in $\mathcal{C}$.
</document>

<document source="https://tamnd.github.io/math2/node/KSPDXANYC0.md" tag="KSPDXANYC0" kind="definition">
# Extreme set

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]] and let $B \subseteq A \subseteq V$.
The set $B$ is an *extreme set* of $A$ if the following holds.
For all $x_1, x_2 \in A$ and $x \in B$ with $x \in (x_1, x_2)$, both $x_1$ and $x_2$ lie in $B$.
Here $(x_1, x_2)$ is the [[open segment|CCZA91NE5X]] between $x_1$ and $x_2$.

## Remarks

A point $x \in A$ is an [[extreme point|3MSGFJM2NX]] of $A$ exactly when $\set{x}$ is an extreme set of $A$.
The sets $\emptyset$ and $A$ are always extreme sets of $A$, and an extreme set of an extreme set of $A$ is an extreme set of $A$.

Example: for $A = [0, 1]^2$, the bottom edge $B = [0, 1] \times \set{0}$ is an extreme set.
Let $x \in B$ lie in $(x_1, x_2)$ with $x_1, x_2 \in A$.
The second coordinates of $x_1, x_2$ are $\ge 0$, and an average of them with positive weights is $0$.
So both are $0$, and $x_1, x_2 \in B$.

Non-example: the diagonal $D = \set{(s, s) : 0 \le s \le 1}$ of the same square.
The point $(1/2, 1/2) \in D$ lies in the open segment between $(1, 0)$ and $(0, 1)$, which are in $A$ but not in $D$.
</document>

<document source="https://tamnd.github.io/math2/node/KST0YNAP55.md" tag="KST0YNAP55" kind="definition">
# Measure algebra

## Definition

A *measure algebra* is a pair $(\mathfrak{A}, \mu)$ such that:

1. $\mathfrak{A}$ is a [[Boolean algebra|0XADJKM699]] in which every countable subset has a supremum.
2. $\mu \colon \mathfrak{A} \to [0, \infty]$ satisfies $\mu(0) = 0$.
3. $\mu\left(\bigvee_{k} a_k\right) = \sum_k \mu(a_k)$ for every sequence $(a_k)_{k \in \N}$ of pairwise disjoint elements, that is, $a_j \wedge a_k = 0$ for $j \ne k$.
4. $\mu(a) > 0$ for every $a \ne 0$.

## Remarks

Example: let $(X, \Sigma, \nu)$ be a [[measure space|P15VQS5DJ9]] and $\mathcal{N} = \set{E \in \Sigma : \nu(E) = 0}$.
Call $E, F \in \Sigma$ equivalent when $\nu(E \mathbin{\triangle} F) = 0$.
The classes form a Boolean algebra $\Sigma / \mathcal{N}$ with countable suprema given by countable unions, and $\mu([E]) = \nu(E)$ is well defined.
Clauses 2 and 3 come from the measure $\nu$, and clause 4 holds since $[E] \ne 0$ means $\nu(E) > 0$.

Non-example: the [[sigma-algebra|0SV8K0TDGY]] of Borel subsets of $\R$ with Lebesgue measure satisfies clauses 1 to 3 but not 4.
The set $\set{0}$ is a nonzero element of measure $0$.

The quotient in the example is called the measure algebra of $(X, \Sigma, \nu)$.
</document>

<document source="https://tamnd.github.io/math2/node/KSW18V47ZD.md" tag="KSW18V47ZD" kind="statement">
# Kakeya problem over finite fields

## Statement

Let $\F_q$ be a [[finite field|0ZXZ82FTX2]] with $q$ elements and let $n \ge 1$.
A *Kakeya set* in $\F_q^n$ is a set $K \subseteq \F_q^n$ such that for every $x \in \F_q^n \setminus \set{0}$ there is $y \in \F_q^n$ with $\set{y + t x : t \in \F_q} \subseteq K$.
Then every Kakeya set $K$ in $\F_q^n$ satisfies
$$\abs{K} \ge \binom{q + n - 1}{n} \ge \frac{q^n}{n!}.$$

## Remarks

This is Dvir's theorem, which answered the finite field analogue of the [[Kakeya problem|6PEN7AKC7M]] asked by Thomas Wolff.
The proof uses the polynomial method.
If $\abs{K} < \binom{q + n - 1}{n}$, the number of monomials of degree at most $q - 1$ in $n$ variables, then some nonzero [[polynomial|07GNX8GCYP]] $P$ of degree $d \le q - 1$ vanishes on $K$.
Let $P_d$ be the top-degree homogeneous part of $P$.
Restricting $P$ to a line $y + t x$ inside $K$ gives a polynomial in $t$ of degree at most $d < q$ with $q$ roots, so it is zero.
Its coefficient of $t^d$ is $P_d(x)$, so $P_d(x) = 0$.
So $P_d$ vanishes on all of $\F_q^n$, which is impossible for a nonzero polynomial of degree less than $q$ in each variable.

For $n = 1$ the bound gives $\abs{K} \ge q$, and indeed $K$ must be the whole line $\F_q$.
For $n = 2$ it gives $\abs{K} \ge q(q + 1)/2$.

## Explanations

### L1 human

A Kakeya set is a set that contains a full line pointing in every direction.
Over the real numbers such sets can have measure zero, which is surprising.
Over a finite field the theorem says the opposite: such a set must contain a fixed positive fraction, about $1/n!$, of all the points of the space.

### L2 human

Dvir's proof is a few lines long and made the polynomial method a standard tool in combinatorics.
The idea is that a set too small cannot avoid being the zero set of a low-degree polynomial, while lines in all directions force the polynomial to vanish identically.
The constant $1/n!$ has since been improved, but the order $q^n$ is optimal.
The finite field result does not solve the Euclidean Kakeya conjecture, whose difficulty comes from the multiscale structure of $\R^n$.
</document>

<document source="https://tamnd.github.io/math2/node/KT0380H39M.md" tag="KT0380H39M" kind="definition">
# Level-set method

## Definition

Let $\Gamma(t) \subset \R^d$, $t \ge 0$, be a family of closed hypersurfaces that bound open sets $\Omega(t)$.
Let $V(x, t)$ be a given speed of $\Gamma(t)$ in its outward normal direction.
The *level-set method* represents and evolves $\Gamma(t)$ as follows.

1. Representation: choose a continuous function $\phi \colon \R^d \times [0, \infty) \to \R$ with $\Omega(t) = \set{x : \phi(x, t) < 0}$ and $\Gamma(t) = \set{x : \phi(x, t) = 0}$, the zero [[level set|0B2RZ0778N]] of $\phi(\cdot, t)$.
2. Evolution: with $V$ extended near $\Gamma(t)$, $\phi$ solves the level-set equation
$$\phi_t + V \abs{\nabla \phi} = 0.$$
This is a [[Hamilton–Jacobi equation|5TRXQKP7DY]], understood in the [[viscosity sense|H1ERJ3T75T]] where $\phi$ is not smooth.
3. Discretization: the equation is solved on a fixed grid, usually with upwind finite differences. The interface $\Gamma(t)$ is read off as the zero set of the discrete $\phi$.

If the interface is carried by a velocity field $v(x, t)$, the evolution equation is $\phi_t + v \cdot \nabla \phi = 0$, which is the case $V = v \cdot n$.

## Remarks

Where $\nabla \phi \ne 0$, the outward unit normal is $n = \nabla \phi / \abs{\nabla \phi}$, with $\nabla$ the [[gradient|0ABK9V9Y64]].
The [[divergence|0MWV6ZTK6S]] $\kappa = \nabla \cdot n$ is the sum of the principal curvatures of $\Gamma$, a multiple of its [[mean curvature|PA78NFEPCF]], and it is used for motion by curvature with $V = -\kappa$.
One usually keeps $\phi$ close to a signed distance function, with $\abs{\nabla \phi} = 1$, by periodic reinitialization.

Example: in $\R^2$ let $V = 1$ and $\phi(x, t) = \abs{x} - (r_0 + t)$ with $r_0 > 0$.
For $x \ne 0$ one has $\phi_t = -1$ and $\abs{\nabla \phi} = 1$, so $\phi_t + V \abs{\nabla \phi} = 0$, and $\Gamma(t)$ is the circle of radius $r_0 + t$.

Non-example: $\phi(x, t) = \abs{x} - r_0$ represents a circle but fails clause 2 for $V = 1$, since $\phi_t + \abs{\nabla \phi} = 1 \ne 0$.
Its zero set does not move.
</document>

<document source="https://tamnd.github.io/math2/node/KT2GNAY91E.md" tag="KT2GNAY91E" kind="definition">
# Countably generated structure

## Definition

Let $L$ be a [[signature|WDVVEYD6AT]] and let $\mathcal{M}$ be an [[$L$-structure|983GZBQYT4]] with universe $M$.
For $X \subseteq M$, let $\operatorname{Sg}(X)$ denote the [[substructure generated by|2TQD8NMG9P]] $X$.
It is the smallest subset of $M$ that contains $X$ and the constants and is closed under the functions of $\mathcal{M}$.
The structure $\mathcal{M}$ is *countably generated* when there is a [[countable set|01TQB88H4C]] $X \subseteq M$ with $\operatorname{Sg}(X) = M$.

## Remarks

Here countable means finite or countably infinite, so every [[finitely generated algebra|WR4WF1K2KW]] is countably generated.

Example: the group $(\Q, +, -, 0)$ is countably generated.
Take $X = \set{1/n : n \ge 1}$; then $p/q = p \cdot (1/q)$ is a sum of copies of $1/q$ or of its negative, so $\operatorname{Sg}(X) = \Q$.

Non-example: the group $(\R, +, -, 0)$ is not countably generated.
For a countable $X$, each element of $\operatorname{Sg}(X)$ is the value of a term at finitely many elements of $X$.
There are only countably many such values, so $\operatorname{Sg}(X)$ is countable, while $\R$ is not.

The same counting shows that, when $L$ is countable, an $L$-structure is countably generated if and only if its universe is countable.
</document>

<document source="https://tamnd.github.io/math2/node/KT50SMH3EC.md" tag="KT50SMH3EC" kind="object">
# Epsilon nought

## Description

Ordinal exponentiation is defined by $\omega^0 = 1$, $\omega^{\alpha + 1} = \omega^\alpha \cdot \omega$, and $\omega^\lambda = \sup_{\alpha < \lambda} \omega^\alpha$ for a limit ordinal $\lambda$.
An *epsilon number* is an [[ordinal|0RDJK7EQYY]] $\varepsilon$ with $\omega^\varepsilon = \varepsilon$.
The ordinal $\varepsilon_0$, read "epsilon nought", is the least epsilon number.
It is the limit of the sequence
$$\omega, \quad \omega^\omega, \quad \omega^{\omega^\omega}, \quad \dots,$$
that is, $\varepsilon_0 = \sup \set{\omega_n : n \in \N}$ with $\omega_0 = \omega$ and $\omega_{n+1} = \omega^{\omega_n}$.

Check: let $\varepsilon = \sup_n \omega_n$.
The map $\alpha \mapsto \omega^\alpha$ is continuous at limits, so $\omega^\varepsilon = \sup_n \omega^{\omega_n} = \sup_n \omega_{n+1} = \varepsilon$.
If $\beta$ is an epsilon number, then $\omega \le \beta$, and induction on $n$ gives $\omega_n \le \beta$, so $\varepsilon \le \beta$.
Since each $\omega_n$ is a [[countable|01TQB88H4C]] ordinal, $\varepsilon_0$ is a countable union of countable sets, hence countable.

## Remarks

Every ordinal $\alpha$ with $0 < \alpha < \varepsilon_0$ has a Cantor normal form $\omega^{\beta_1} c_1 + \dots + \omega^{\beta_k} c_k$ with $\alpha > \beta_1 > \dots > \beta_k$ and positive integers $c_i$.
Since each exponent is smaller than $\alpha$, the exponents can be written in the same form again, and the process ends.
This gives a finite notation for each ordinal below $\varepsilon_0$.
The ordinal $\varepsilon_0$ is the proof-theoretic ordinal of Peano arithmetic.

## Explanations

### L1 human

Start with $\omega$, the first infinite ordinal, and keep raising $\omega$ to the power of what you have.
The tower $\omega^{\omega^{\omega^{\cdots}}}$ never ends, and $\varepsilon_0$ is the first ordinal past all of its stages.
It is the first ordinal that does not change when you raise $\omega$ to it.
Every smaller ordinal can be written as a finite expression built from $\omega$, sums and powers.
</document>

<document source="https://tamnd.github.io/math2/node/KT9WJFRQ3F.md" tag="KT9WJFRQ3F" kind="problem">
# Zarankiewicz problem

## Problem

Let $m, n, s, t$ be positive integers with $s \le m$ and $t \le n$.
Call an $m \times n$ matrix with entries in $\set{0, 1}$ *$(s, t)$-free* if no $s$ rows and $t$ columns cross only in entries equal to $1$.
Let $z(m, n; s, t)$ be the largest number of entries equal to $1$ in an $(s, t)$-free $m \times n$ matrix.
Determine $z(m, n; s, t)$, or its order of growth.

## Remarks

Equivalently, consider [[bipartite graphs|0ZVQKTRKVQ]] with parts $A$ and $B$ of sizes $m$ and $n$.
Then $z(m, n; s, t)$ is the largest number of edges of such a graph with no [[complete bipartite|0V0GC7BQAN]] subgraph $K_{s,t}$ that has $s$ vertices in $A$ and $t$ in $B$.
It is closely tied to the [[extremal number|DRGS5TT784]] $\mathrm{ex}(n, K_{s,t})$.

The Kővári–Sós–Turán theorem gives the upper bound $z(n, n; t, t) = O(n^{2 - 1/t})$ for fixed $t$.
For $s = t = 2$ the order $n^{3/2}$ is correct.
If $q$ is a prime power and $n = q^2 + q + 1$, then $z(n, n; 2, 2) = (q + 1)(q^2 + q + 1)$, and the incidence matrix of a [[finite projective plane|2HWB7P1DZN]] of order $q$ attains this value.
The order $n^{2 - 1/t}$ is also known to be correct for $t = 3$ and, for $z(n, n; s, t)$, when $s$ is large enough compared with $t$, by algebraic constructions called norm graphs.
In general the problem is open; for example, the order of growth of $z(n, n; 4, 4)$ is not known.

## Explanations

### L1 human

Fill a square grid with as many ones as you can, but never let $s$ rows and $t$ columns cross in ones only.
How many ones can you place?
For $2$ rows and $2$ columns the forbidden pattern is a rectangle with ones at its four corners, and the answer grows like $n^{3/2}$.
For larger forbidden blocks the exact growth rate is one of the main open questions of extremal graph theory.
</document>

<document source="https://tamnd.github.io/math2/node/KTBBV459TV.md" tag="KTBBV459TV" kind="definition">
# Embedded prime

## Definition

Let $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $I \ne R$ be an [[ideal|0MYSA6MYNA]] of $R$.
An [[associated prime|1RJJ33KKSS]] $P$ of $R/I$ is an *embedded prime* of $I$ if there is another associated prime $P'$ of $R/I$ with $P' \subsetneq P$.
An associated prime that is not embedded is called a minimal or isolated prime of $I$.

## Remarks

Example: for $I = (x^2, xy)$ in $k[x, y]$ over a field $k$, the associated primes are $(x)$ and $(x, y)$, and $(x) \subsetneq (x, y)$, so $(x, y)$ is embedded.
Non-example: $(x)$ is not embedded for the same $I$, because no associated prime is strictly smaller.
The minimal primes of $I$ in this sense are exactly the [[minimal prime ideals|S2D8823P8B]] over $I$.
In a [[primary decomposition|M2VPVJH33K]], the components at embedded primes are not unique, while those at minimal primes are.
The same terms apply to the associated primes of a module.

## Explanations

### L1 human

An embedded prime is an associated prime that sits strictly above another associated prime.
</document>

<document source="https://tamnd.github.io/math2/node/KTDVZP4DF2.md" tag="KTDVZP4DF2" kind="concept">
# Autoregressive integrated moving average

## Summary

An autoregressive integrated moving average model, or ARIMA($p$, $d$, $q$) model, is a time series model whose $d$-th differences form an [[ARMA($p$, $q$) process|B1HNX0YT0Z]].
It extends ARMA models to non-stationary series with trends or random-walk behaviour.
The word "integrated" refers to undoing the differencing by summation.
ARIMA models are among the most widely used models for forecasting economic and business series.

## Explanations

### L1 human

Many series, such as a stock price index, do not hover around a fixed level; they drift and wander.
The changes from one step to the next, however, often look stable.
ARIMA models fit an ARMA model to these changes, and then add the changes back up to forecast the series itself.
The simplest case is the random walk $X_t = X_{t-1} + \varepsilon_t$, which is ARIMA($0$, $1$, $0$): its changes $X_t - X_{t-1} = \varepsilon_t$ are pure noise.
</document>

<document source="https://tamnd.github.io/math2/node/KTFG1S2SMG.md" tag="KTFG1S2SMG" kind="statement">
# Milnor fibration theorem

## Statement

Let $f \colon U \to \C$ be a [[holomorphic function|507C34W0ZT]] on an open neighbourhood $U$ of $0$ in $\C^{n+1}$, with $f(0) = 0$ and $f$ not identically zero.
For $\varepsilon > 0$, let $S_\varepsilon$ be the [[sphere|P9CM4GQ2T0]] of radius $\varepsilon$ about $0$ and let $K_\varepsilon = S_\varepsilon \cap f^{-1}(0)$, the [[link|C831AN3RFM]] of $f^{-1}(0)$ at $0$.
Then there is $\varepsilon_0 > 0$ such that, for every $0 < \varepsilon \le \varepsilon_0$:

1. the map $\phi = f / \abs{f} \colon S_\varepsilon \setminus K_\varepsilon \to S^1$ is a smooth locally trivial [[fibre bundle|CX4M8K61AB]];
2. each fibre $F_\theta = \phi^{-1}(e^{i\theta})$ is a smooth manifold of real dimension $2n$, called the *Milnor fibre* of $f$ at $0$.

## Remarks

Milnor proved this in his 1968 book on singular points of complex hypersurfaces.
The proof shows, with the curve selection lemma, that for small $\varepsilon$ the map $\phi$ has no critical points on $S_\varepsilon \setminus K_\varepsilon$.
One then builds a vector field on $S_\varepsilon \setminus K_\varepsilon$ that $\phi$ sends to the unit tangent field of $S^1$ and that is controlled near $K_\varepsilon$.
Its flow carries each fibre onto the others.
An equivalent form, used often, is the fibration $f \colon B_\varepsilon \cap f^{-1}(\partial D_\delta) \to \partial D_\delta$, where $B_\varepsilon$ is the closed ball and $D_\delta$ a disc of radius $0 < \delta \ll \varepsilon$.

Example: for $f(z_0, z_1) = z_0 z_1$ on $\C^2$, the link is a pair of circles forming a Hopf link in $S^3$.
The fibre in the tube form is $\set{z_0 z_1 = \delta} \cap B_\varepsilon$, an annulus, which has the homotopy type of a circle.

The radius must be small.
On a large sphere, $f^{-1}(0)$ may meet the sphere non-transversally or other critical points of $f$ may intervene, and the conclusion can fail.

## Explanations

### L1 human

Take a hypersurface with a singular point and look at it through a small sphere around that point.
Away from the hypersurface, the argument of $f$ is an angle, so it gives a map from the rest of the sphere to the circle.
Milnor's theorem says that this map is a fibration: all its fibres look the same.
The sphere minus the link is swept out by them as the angle turns once around the circle.
The common fibre, the Milnor fibre, is a smooth model of the hypersurface near the singular point, slightly pushed off the singular value.

### L2 human

The Milnor fibration is the starting point of the topology of hypersurface singularities.
Its fibre carries the [[Milnor number|1ZE51P7GRQ]], and for an isolated singularity the [[fibre is a bouquet of spheres|X5HMTTGB62]].
Going once around the circle gives a diffeomorphism of the fibre, whose action on homology is the monodromy studied in the [[monodromy theorem|DQP63REKEK]].
For plane curves, $n = 1$, the link is a link in $S^3$ and the Milnor fibre is a Seifert surface of it, which connects singularity theory with knot theory.
</document>

<document source="https://tamnd.github.io/math2/node/KTGAS9DGRX.md" tag="KTGAS9DGRX" kind="problem">
# Class number problem

## Problem

For each positive integer $h$, determine the complete list of imaginary [[quadratic fields|7SV4WA92SM]] $\Q(\sqrt{-d})$, with $d$ a positive squarefree integer, whose [[class number|KTKK9B1X41]] equals $h$.

## Remarks

The problem goes back to Gauss, who stated it in the language of [[binary quadratic forms|SBDSYA037Q]].
It has three parts: show that the class number tends to infinity as $d \to \infty$, so that each list is finite; make this effective; and compute the lists.

Status:
the class number of $\Q(\sqrt{-d})$ tends to infinity with $d$ (Heilbronn), but this proof gives no effective bound.
The case $h = 1$ is the [[Stark–Heegner theorem|KZTFH832H0]], with nine fields, and the case $h = 2$ was solved by Baker and by Stark.
Goldfeld, combined with the theorem of Gross and Zagier, gave an effective lower bound for the class number, which reduces each $h$ to a finite computation.
Watkins completed the lists for all $h \le 100$.

The analogue for real quadratic fields is open: it is not known whether infinitely many real quadratic fields have class number $1$.

## Explanations

### L1 human

The class number of a number field measures how badly unique factorisation fails in its integers.
The problem asks, for each value of the class number, which imaginary quadratic fields have it.
For class number $1$ the answer is nine fields; for larger class numbers the lists are known up to $100$ by computer-assisted proofs.
</document>

<document source="https://tamnd.github.io/math2/node/KTKK9B1X41.md" tag="KTKK9B1X41" kind="concept">
# Class number

## Summary

The class number $h_K$ of a [[number field|MWBDFS2KJ5]] $K$ is the order of its [[ideal class group|WQ46BD3ARY]].
It is always finite, and $h_K = 1$ exactly when the ring of integers $\mathcal{O}_K$ is a principal ideal domain, equivalently a unique factorisation domain.
Class numbers appear in the [[class number formula|55XX5A7XZV]], and the question of which fields have a given class number is the [[class number problem|KTGAS9DGRX]].

## Explanations

### L1 human

The class number counts how many essentially different kinds of ideals a number ring has.
If it is $1$, every ideal is generated by one number, and numbers factor into primes in only one way, as in the ordinary integers.
For example, the Gaussian integers $\Z[i]$ have class number $1$.
The ring $\Z[\sqrt{-5}]$ has class number $2$, and there $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ factors in two ways.
</document>

<document source="https://tamnd.github.io/math2/node/KTN8HTSY3R.md" tag="KTN8HTSY3R" kind="concept">
# dg category

## Summary

A dg category, or differential graded category, is a category whose morphism sets are [[chain complexes|WQ82C0M3QB]] and whose composition is compatible with the differentials.
It is a category [[enriched|BPH9SJ30Y0]] over chain complexes of modules over a commutative ring.
Dg categories serve as enhancements of triangulated categories: they keep homotopical information that the [[triangulated category|T2V7GJJ2EN]] loses, such as functorial cones.
They are a main tool in noncommutative algebraic geometry and in the study of derived categories of sheaves.

## Explanations

### L1 human

In an ordinary linear category, the maps between two objects form a vector space.
In a dg category they form a chain complex, with maps of each degree and a differential that sends a map of degree $n$ to a map of degree $n + 1$.
The main example has chain complexes as objects and all graded maps between them as morphisms.
Taking cohomology in degree $0$ of each morphism complex gives an ordinary category, here the [[homotopy category of chain complexes|9Y5Y92R4D5]].
A dg category with one object is the same thing as a differential graded algebra.
</document>

<document source="https://tamnd.github.io/math2/node/KTP37HF2RZ.md" tag="KTP37HF2RZ" kind="concept">
# Lamé function

## Summary

A Lamé function is a solution of Lamé's equation $w'' + \left( h - \nu(\nu + 1) k^2 \operatorname{sn}^2(z, k) \right) w = 0$, where $\operatorname{sn}$ is a Jacobi [[elliptic function|FF9ZPCZ59K]].
The equation arises when [[Laplace's equation|S8RG6N1PRT]] is separated in ellipsoidal coordinates, and its solutions are also called ellipsoidal harmonics.
When $\nu = n$ is a nonnegative integer, special values of $h$ give Lamé polynomials, which play for ellipsoids the role that Legendre polynomials play for spheres.
The equation is a special case of Heun's equation and a periodic Schrödinger equation with an elliptic potential.

## Explanations

### L1 human

Lamé's equation is like the equation $w'' + h w = 0$ of a vibrating string, but with an extra term that oscillates periodically along $z$.
The oscillating term is a square of an elliptic sine, which is periodic like the sine but has a shape controlled by a modulus $k$.
For degree $\nu = 1$ the three functions $\operatorname{sn} z$, $\operatorname{cn} z$ and $\operatorname{dn} z$ are solutions, for $h = 1 + k^2$, $h = 1$ and $h = k^2$ respectively.
These are the simplest Lamé polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/KTQC33ZJTN.md" tag="KTQC33ZJTN" kind="definition">
# Regular local ring

## Definition

Let $(R, \mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] with residue field $k = R/\mathfrak m$, and let $d = \dim R$ be its [[Krull dimension|MVGBS3J1P7]].
The ring $R$ is *regular* if $\mathfrak m$ can be generated by $d$ elements.
Equivalently, $R$ is regular if its [[embedding dimension|DP3KSCRKFP]] $\dim_k \mathfrak m / \mathfrak m^2$ equals $d$.

## Remarks

A set of $d$ generators of $\mathfrak m$ in a regular local ring is called a *regular system of parameters*.

Example: a field is regular, with $d = 0$ and $\mathfrak m = 0$.
A Noetherian local domain of dimension $1$ is regular exactly when $\mathfrak m$ is principal, that is, when it is a [[discrete valuation ring|MGM3CNJW9V]], such as $\Z_{(p)}$.
For a field $k$, the rings $k[x_1, \ldots, x_n]_{(x_1, \ldots, x_n)}$ and $k[\![x_1, \ldots, x_n]\!]$ have dimension $n$ and maximal ideal generated by the $n$ elements $x_1, \ldots, x_n$, so they are regular.

Non-example: $R = k[x]/(x^2)$ has dimension $0$, but its maximal ideal $(x) \ne 0$ needs one generator, so it is not regular.
The cusp $k[x, y]_{(x, y)}/(y^2 - x^3)$ has dimension $1$ and embedding dimension $2$, so it is not regular.
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# Korteweg–De Vries equation

## Summary

The Korteweg–De Vries equation, or KdV equation, is the nonlinear [[partial differential equation|GC5BDZ6QBC]] $u_t + u_{xxx} + 6 u u_x = 0$ for a real function $u(x,t)$.
It models long waves of small amplitude in shallow water.
It has solitary wave solutions that keep their shape, and it is the first equation that was solved by the inverse scattering transform.
It has infinitely many conserved quantities and is a central example of a completely integrable system.

## Explanations

### L1 human

Imagine a single smooth hump of water moving along a narrow canal without changing its shape.
Two effects act on it: the higher parts of the wave move faster, which steepens the front, and dispersion spreads the wave out.
The KdV equation contains exactly these two effects, the term $6 u u_x$ and the term $u_{xxx}$.
When they balance, one gets the soliton $u(x,t) = \tfrac{c}{2} \mathrm{sech}^2\!\big(\tfrac{\sqrt{c}}{2}(x - ct)\big)$ with speed $c > 0$.
Taller solitons are narrower and move faster.
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<document source="https://tamnd.github.io/math2/node/KTVJR95GSF.md" tag="KTVJR95GSF" kind="definition">
# Spline

## Definition

Let $a < b$, let $k \ge 1$, and let $a = t_0 < t_1 < \cdots < t_k = b$ be points called *knots*.
Let $n \ge 1$ be an integer.
A *spline of degree $n$* with these knots is a function $s \colon [a, b] \to \R$ such that:

1. for each $i = 1, \dots, k$, the restriction of $s$ to $[t_{i-1}, t_i]$ is a [[polynomial function|1G0B73WPX8]] of degree at most $n$;
2. $s$ is $n - 1$ times continuously differentiable on $[a, b]$.

## Remarks

Splines of degree $0$ are usually taken to be step functions, constant on each $[t_{i-1}, t_i)$, with jumps allowed at the knots.
For $n = 1$, clause 2 asks that $s$ is [[continuous|0X3WGX95FE]], so $s$ is a broken line.
The splines of degree $n$ with fixed knots form a vector space of dimension $n + k$.
Some authors allow lower smoothness at some knots; this is encoded by repeating knots.

Example: with knots $-1 < 0 < 1$ and $n = 3$, the function $s(x) = 0$ for $x \le 0$ and $s(x) = x^3$ for $x \ge 0$ is a cubic spline.
Each piece is a polynomial of degree at most $3$.
At $0$ the values, first [[derivatives|0V1EJCT8G6]] and second derivatives of both pieces are $0$, so $s$ is twice continuously differentiable.

Non-example: with the same knots, $s(x) = 0$ for $x \le 0$ and $s(x) = x^2$ for $x \ge 0$ is not a cubic spline.
Its second derivative jumps from $0$ to $2$ at $0$, so clause 2 fails.
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# Estimating equations

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a family of distributions on a set $\mathcal{X}$, with $\Theta \subseteq \R^k$.
An *estimating function* is a map $g \colon \mathcal{X} \times \Theta \to \R^k$.
It is *unbiased* if $E_\theta[g(X, \theta)] = 0$ for every $\theta \in \Theta$, where $X \sim P_\theta$ and $E_\theta$ is the [[expectation|0DCT0QZVQ1]].
For [[i.i.d.|0AJN7WHBJZ]] observations $X_1, \dots, X_n$, the *estimating equation* is
$$\sum_{i=1}^n g(X_i, \theta) = 0,$$
and an [[estimator|0TTNRP5PFQ]] $\hat{\theta}_n$ that solves it is an estimating-equation estimator.

## Remarks

Example: the [[score|CP97GRVTQ0]] $g(x, \theta) = \nabla_\theta \log f(x; \theta)$ is an unbiased estimating function under the usual regularity conditions, and its estimating equation gives the likelihood equations.

Example: in the [[method of moments|09J1H07VAH]] for a Poisson mean, $g(x, \lambda) = x - \lambda$ has $E_\lambda[X - \lambda] = 0$, and the equation $\sum_i (X_i - \lambda) = 0$ gives $\hat{\lambda} = \bar{X}$.

Non-example: for the Poisson mean, $g(x, \lambda) = x - 2\lambda$ has $E_\lambda[g(X, \lambda)] = -\lambda \ne 0$, so it is not unbiased, and its root $\bar{X}/2$ converges to $\lambda/2$, not to $\lambda$.

Estimators of this kind are the $\psi$-type [[M-estimators|4503R8BVF8]].
Under regularity conditions, $\sqrt{n}(\hat{\theta}_n - \theta) \xrightarrow{d} \mathcal{N}(0, A^{-1} B A^{-T})$ with $A = E_\theta[\nabla_\theta g(X, \theta)]$ and $B = E_\theta[g(X, \theta) g(X, \theta)^T]$.
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# Denjoy–Carleman theorem

## Statement

Let $a < b$ and let $M = (M_k)_{k \ge 0}$ be a sequence of positive real numbers with $M_0 = 1$.
Assume that $M$ is logarithmically convex: $M_k^2 \le M_{k-1} M_{k+1}$ for all $k \ge 1$.
Let $C^M([a, b])$ be the class of smooth functions $f$ on $[a, b]$ with $\abs{f^{(k)}} \le A B^k M_k$ on $[a, b]$ for some constants $A, B > 0$ and all $k \ge 0$.
Then the following three conditions are equivalent.

1. The class $C^M([a, b])$ is [[quasi-analytic|YF1KAWB5MB]].
2. $\sum_{k=1}^{\infty} M_k^{-1/k} = \infty$.
3. $\sum_{k=1}^{\infty} \dfrac{M_{k-1}}{M_k} = \infty$.

## Remarks

When the series converge, one builds a nonzero function of the class that vanishes outside a small interval.
A standard construction is an infinite convolution of normalized indicator functions of short intervals, whose lengths are related to the ratios $M_{k-1}/M_k$.
Carleman's proof of the direction from divergence to quasi-analyticity uses complex analysis.
Without log-convexity, condition 1 is still equivalent to condition 2 with $M_k^{1/k}$ replaced by $\inf_{j \ge k} M_j^{1/j}$.
Condition 3 then has to be applied to the largest log-convex minorant of $M$ instead of $M$.

Example: for $M_k = k!$, the sequence is log-convex since $(k!)^2 \le (k-1)! \, (k+1)!$.
The ratios $M_{k-1}/M_k = 1/k$ have a divergent sum, so the class is quasi-analytic; this is the class of [[analytic functions|0JD5ZNW0F4]] on $[a, b]$.
For $M_k = (k!)^s$ with $s > 1$, the ratios are $k^{-s}$, and the sum converges.
So these classes contain nonzero [[bump functions|HS65DW0HXH]].

## Explanations

### L1 human

Analytic functions are determined by their derivatives at one point, while general smooth functions are not.
The theorem finds the exact border between these two behaviours.
If the derivatives of the functions in a class may grow only a little faster than $k!$, uniqueness survives.
If they may grow a bit faster still, bump functions appear and uniqueness is lost.

### L2 human

The test is a single series, much like a convergence test for ordinary series.
Divergence of $\sum M_{k-1}/M_k$ means that the allowed growth is slow, and the class behaves like the analytic functions.
Convergence means that cut-off functions exist in the class, so it supports partitions of unity, as needed in the theory of ultradistributions.
The case $M_k = (k!)^s$ gives the Gevrey classes, which appear in the study of partial differential equations.
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# Algebraic independence

## Summary

A set of elements of a field extension is algebraically independent if no nonzero polynomial relation with coefficients in the base field holds among them.
It is the analogue for polynomials of [[linear independence|0CRMGPAH83]] for linear combinations.
A single element is algebraically independent exactly when it is [[transcendental|0NQMZ89X4A]].
Maximal algebraically independent sets are [[transcendence bases|E45PA267S4]].

## Explanations

### L1 human

Two numbers are algebraically dependent when some polynomial equation ties them together.
For example, $\pi$ and $\pi^2$ are dependent over $\Q$, since $y - x^2 = 0$ at $x = \pi$, $y = \pi^2$.
In the field $\Q(x, y)$ of rational functions in two variables, $x$ and $y$ are independent: no nonzero polynomial in them is zero.
Whether $\pi$ and $e$ are algebraically independent over $\Q$ is not known.
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# Adjoint functors

## Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be [[categories|A2DE66937D]], and let $F \colon \mathcal{C} \to \mathcal{D}$ and $G \colon \mathcal{D} \to \mathcal{C}$ be [[functors|28NQ0KRHD3]].
An *adjunction* between $F$ and $G$ is a family of bijections
$$\varphi_{c, d} \colon \Hom_{\mathcal{D}}(F c, d) \to \Hom_{\mathcal{C}}(c, G d),$$
one for each object $c$ of $\mathcal{C}$ and each object $d$ of $\mathcal{D}$.
The family must be natural in $c$ and in $d$.
Naturality means $\varphi(k \circ g \circ F h) = G k \circ \varphi(g) \circ h$.
This must hold for all $h \colon c' \to c$, $g \colon F c \to d$ and $k \colon d \to d'$.
If an adjunction exists, $F$ is *left adjoint* to $G$ and $G$ is *right adjoint* to $F$, written $F \dashv G$.

## Remarks

Equivalently, $\varphi$ is a [[natural isomorphism|KYCPCC57FT]] between the two [[hom functors|VJXS8AK67E]] $\Hom_{\mathcal{D}}(F -, -)$ and $\Hom_{\mathcal{C}}(-, G -)$ from $\mathcal{C}^{\mathrm{op}} \times \mathcal{D}$ to sets.

Example: let $F$ send a set $X$ to the free group $F(X)$, and let $G$ send a group to its underlying set.
A homomorphism $g \colon F(X) \to H$ is determined by its values on the generators, and these values can be any elements of $H$.
So $\varphi(g) = g|_X$ is a bijection, and it is natural in $X$ and $H$.

Non-example: the forgetful functor $G$ from groups to sets has no right adjoint.
Indeed, a left adjoint preserves colimits, hence the initial object.
But $G$ sends the trivial group, which is initial, to a one-point set, which is not initial in sets.

Another standard example is $- \times Y \dashv (-)^Y$ in sets, where $Z^Y$ is the set of maps $Y \to Z$.
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# Dedekind eta function

## Definition

Let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]], and for $\tau \in \mathbb{H}$ let $q = e^{2 \pi i \tau}$, so that $\abs{q} < 1$.
The *Dedekind eta function* is the function $\eta \colon \mathbb{H} \to \C$ given by
$$\eta(\tau) = e^{\pi i \tau / 12} \prod_{n = 1}^{\infty} (1 - q^n).$$

## Remarks

The product converges absolutely and locally uniformly, since $\sum_n \abs{q}^n < \infty$.
So $\eta$ is [[holomorphic|0NAHXR1FRH]] on $\mathbb{H}$, and it has no zeros there.
It satisfies $\eta(\tau + 1) = e^{\pi i / 12} \eta(\tau)$ and $\eta(-1/\tau) = \sqrt{-i\tau}\, \eta(\tau)$, with the principal branch of the square root.
Example: $\eta(\tau)^{24} = q \prod_{n \ge 1} (1 - q^n)^{24} = \Delta(\tau)$, the [[cusp form|A50Y012RBP]] of weight $12$.
Non-example: $\eta$ itself is not invariant under $\tau \mapsto \tau + 1$, because of the factor $e^{\pi i / 12}$, so it is not a modular form for $\mathrm{SL}_2(\Z)$.
By the [[pentagonal number theorem|PMYDHW00KF]], $\prod_{n \ge 1} (1 - q^n) = \sum_{m \in \Z} (-1)^m q^{m(3m - 1)/2}$.
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# Neville's algorithm

## Definition

Let $K$ be a [[field|04C5ZBG1WT]], let $n \ge 0$, let $x_0, \dots, x_n \in K$ be pairwise distinct, let $y_0, \dots, y_n \in K$, and let $x \in K$.
*Neville's algorithm* computes the numbers $p_{i,j}$ for $0 \le i \le j \le n$ by the recursion

1. $p_{i,i} = y_i$ for $0 \le i \le n$;
2. for $i < j$,
$$p_{i,j} = \frac{(x - x_i) \, p_{i+1,j} - (x - x_j) \, p_{i,j-1}}{x_j - x_i},$$

in order of increasing $j - i$, and returns $p_{0,n}$.

## Remarks

By induction on $j - i$, the number $p_{i,j}$ is the value at $x$ of the [[interpolation polynomial|SS1YMS2NPC]] of degree at most $j - i$ through $(x_i, y_i), \dots, (x_j, y_j)$.
Indeed, the same recursion with $x$ as a variable gives a polynomial of degree at most $j - i$.
At $x = x_i$ it gives $p_{i,j-1}(x_i) = y_i$, and at $x = x_j$ it gives $p_{i+1,j}(x_j) = y_j$.
At an inner node $x_k$ both terms give $y_k$.
So the output $p_{0,n}$ is the value at $x$ of the interpolation polynomial of all the data.

Example: data $(0, 1)$, $(1, 3)$, $(2, 7)$ and $x = 3$.
Then $p_{0,1} = \frac{3 \cdot 3 - 2 \cdot 1}{1} = 7$, $p_{1,2} = \frac{2 \cdot 7 - 1 \cdot 3}{1} = 11$ and $p_{0,2} = \frac{3 \cdot 11 - 1 \cdot 7}{2} = 13$.
This agrees with the interpolation polynomial $x^2 + x + 1$ at $3$.

Non-example: if two nodes coincide, $x_i = x_j$ with $i < j$, the recursion divides by $x_j - x_i = 0$, so the algorithm is not defined.
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# Bloch's theorem

## Statement

Let $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ be the [[unit disc|8G5ABXRK85]].
There is an absolute constant $b > 0$ with the following property.
Let $f \colon \mathbb{D} \to \C$ be any [[holomorphic|0DG0Z72CQD]] function with $f'(0) = 1$.
Then there is an open set $\Omega \subseteq \mathbb{D}$ such that $f$ is [[injective|0PVG1CRAJN]] on $\Omega$ and $f(\Omega)$ is an open disc of radius $b$.

## Remarks

The supremum of all such $b$ is the Bloch constant $B$, one of [[Landau's constants|AC3FNJ8Y1Y]].
Lars Ahlfors proved $B \ge \sqrt{3}/4$, and it is known that $B < 0.472$; the exact value is unknown.
One proof first replaces $f$ by $f(rz)/r$ for some $r < 1$, so that $f$ is holomorphic on the closed disc.
Then one picks a point where $(1 - \abs{z}^2)\abs{f'(z)}$ is maximal and rescales around it, which gives a function whose derivative is bounded on a disc of fixed size.
A bounded derivative with $\abs{f'(0)} = 1$ forces $f$ to be injective on a smaller disc of explicit radius, and the image of that disc contains a disc of explicit radius.
The theorem gives a short proof of [[Picard's little theorem|R2PHM118NG]].
The normalisation $f'(0) = 1$ is needed: the functions $f(z) = \varepsilon z$ only cover discs of radius $\varepsilon$.

## Explanations

### L1 human

Take any holomorphic function on the unit disc whose derivative at the centre is $1$.
Its image may fold over itself in a complicated way.
Bloch's theorem says that, somewhere inside the image, there is always a round disc of a fixed size that is covered exactly once.
The size of this disc does not depend on the function.

### L2 human

The theorem is a covering theorem with no assumption of injectivity on $f$.
For injective $f$, the [[Koebe quarter theorem|WX0HMW4GT7]] gives much more: a disc of radius $1/4$ around $f(0)$.
Bloch's theorem is scale invariant in the right way, so applying it to $f(rz)$ with large $r$ shows that a nonconstant entire function covers schlicht discs of every radius.
This is the key step in the proof of Picard's little theorem by Bloch's method.
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# Dimension theorem for local rings

## Statement

Let $(R, \mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]].
For $n \ge 0$ let $\chi(n) = \ell(R/\mathfrak m^n)$ be the [[length|CZSCNZH8VY]] of $R/\mathfrak m^n$; this is the [[Hilbert–Samuel function|BNQVS83GZV]] of $R$.
Then $\chi(n)$ agrees with a polynomial in $n$ for all large $n$, and the following three integers are equal:

1. the [[Krull dimension|MVGBS3J1P7]] $\dim R$;
2. the least number $\delta(R)$ of elements of $\mathfrak m$ that generate an [[$\mathfrak m$-primary ideal|TMTPWVETQK]];
3. the degree $d(R)$ of the polynomial that agrees with $\chi(n)$ for large $n$.

## Remarks

The usual proof shows $\delta(R) \ge d(R) \ge \dim R \ge \delta(R)$.
The first inequality holds because the Hilbert–Samuel polynomial of an ideal generated by $s$ elements has degree at most $s$, and its degree does not depend on the $\mathfrak m$-primary ideal.
The second goes by induction on $d(R)$.
Take a chain of primes $\mathfrak p_0 \subsetneq \cdots \subsetneq \mathfrak p_r$ and an element $x \in \mathfrak p_1 \setminus \mathfrak p_0$.
Then the ring $R/(\mathfrak p_0 + (x))$ has a smaller $d$ and still contains a chain of length $r - 1$.
The third goes by building, with the [[prime avoidance lemma|FMP5AWG6PR]], elements $x_1, \dots, x_{\dim R}$ such that each $x_i$ lies outside every prime of height $i - 1$ that contains $x_1, \dots, x_{i-1}$.
For example, for $R = k[x, y]_{(x, y)}$ with $k$ a field, $\chi(n) = n(n+1)/2$, of degree $2$, the ideal $\mathfrak m = (x, y)$ needs two generators, and $\dim R = 2$.
Consequences: $\dim R \le \dim_k \mathfrak m/\mathfrak m^2$ with $k = R/\mathfrak m$, so $\dim R$ is finite, and [[Krull's height theorem|TBRTZMHFTV]] follows.

## Explanations

### L1 human

There are three natural ways to measure the size of a local ring.
One counts the longest chain of prime ideals.
One counts how many equations are needed to cut out the point.
One measures how fast the ring grows modulo higher and higher powers of the maximal ideal.
The theorem says that the three counts always give the same number.

### L2 human

The theorem turns the order-theoretic Krull dimension into something computable.
The number $\delta(R)$ gives upper bounds, since any $\mathfrak m$-primary ideal with $s$ generators shows $\dim R \le s$.
The degree $d(R)$ connects dimension with [[Hilbert polynomials|RMH4FBCFDA]] and with the [[multiplicity|JX20MNDDXT]] of the ring.
A list of $\dim R$ elements that generates an $\mathfrak m$-primary ideal is a [[system of parameters|95QTEPMDA4]].
</document>

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# Algorithmic randomness

## Summary

Algorithmic randomness makes precise which individual infinite binary sequences are random.
The standard notion is Martin-Löf randomness.
It calls a sequence random if it passes every effective statistical test, that is, if it avoids every effectively given set of measure zero.
Almost every sequence, in the sense of the uniform measure, is random in this sense, while no computable sequence is.
The notion connects computability, probability and [[Kolmogorov complexity|8PY3K6NCKF]].

## Explanations

### L1 human

Probability theory says that a fair coin produces "typical" sequences, but it does not say which single sequence is random.
The sequence $000\ldots$ is as likely as any other, yet nobody would call it random.
Algorithmic randomness says a sequence is random if no computer program can find a pattern in it that a fair coin would rarely produce.
Equivalently, its first $n$ bits cannot be compressed much, for every $n$.
</document>

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# Real closure

## Definition

A field $R$ is *real closed* if $R$ is [[formally real|X32TSWJSYT]] and no [[algebraic extension|09EYMKMSB9]] $R' \ne R$ of $R$ is formally real.
Let $(F, \le)$ be an [[ordered field|010JCV9J4E]].
A *real closure* of $(F, \le)$ is an algebraic extension $R$ of $F$ such that:

1. $R$ is real closed;
2. the ordering of $R$, in which the nonnegative elements are the squares, restricts to $\le$ on $F$.

## Remarks

A real closed field has exactly one ordering, given by $a \ge 0$ if and only if $a$ is a square, so clause 2 makes sense.
Equivalently, $R$ is real closed if and only if two things hold.
It has an ordering in which every positive element is a square, and every polynomial of odd degree over $R$ has a root in $R$.
Every ordered field has a real closure, and it is unique up to a unique isomorphism over $F$ that respects the orderings.
Example: the field of real [[algebraic numbers|0MB4TGMXG6]] is a real closure of $\Q$.
Non-example: $\R$ is real closed, but it is not a real closure of $\Q$, because it is not algebraic over $\Q$.
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# Ring of sets

## Summary

A ring of sets is a nonempty family of sets that is closed under unions and differences of two members.
It is then closed under intersections and symmetric differences too.
Rings of sets are a common domain for contents and premeasures before they are extended to measures.

## Explanations

### L1 human

A ring of sets is a family of sets in which one can glue two members together and cut one member out of another.
Unlike an algebra of sets, it need not contain a largest set.
For example, the finite subsets of the natural numbers form a ring: the union and the difference of two finite sets are finite.
The name comes from algebra: with symmetric difference as addition and intersection as multiplication, the family is a ring.
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# Stanley–Wilf conjecture

## Statement

Let $\sigma$ be a [[permutation|05RWFKVTJ7]] of $\set{1, \ldots, k}$ for some $k \ge 1$.
For $n \ge 1$, let $\operatorname{Av}_n(\sigma)$ be the set of permutations of $\set{1, \ldots, n}$ that avoid $\sigma$ as a [[pattern|1H8Z1WMG23]].
Then there is a constant $c_\sigma$ such that $\abs{\operatorname{Av}_n(\sigma)} \le c_\sigma^{\,n}$ for all $n \ge 1$.

## Remarks

The conjecture was proved by Adam Marcus and Gábor Tardos in 2004.
They proved the Füredi–Hajnal conjecture on $0$-$1$ matrices that avoid a permutation matrix, and Martin Klazar had shown earlier that it implies the Stanley–Wilf conjecture.
The bound means that the limit $L(\sigma) = \lim_{n \to \infty} \abs{\operatorname{Av}_n(\sigma)}^{1/n}$ is finite; Arratia showed that this limit exists.
The number $L(\sigma)$ is the Stanley–Wilf limit of $\sigma$.

Example: for $\sigma = 123$, $\abs{\operatorname{Av}_n(123)}$ is the [[Catalan number|0YWZJEABB2]] $C_n \le 4^n$, so $c_{123} = 4$ works and $L(123) = 4$.

Without a forbidden pattern the statement fails, since there are $n!$ permutations of $\set{1, \ldots, n}$, and $n!$ grows faster than $c^n$ for each constant $c$.

## Explanations

### L1 human

Forbidding a single pattern is a strong restriction on a long permutation.
The theorem makes this precise: the number of permutations that avoid a fixed pattern grows only exponentially in $n$.
Without restriction there are $n!$ permutations, which grow much faster.

### L2 human

For patterns of length $3$ the counts are Catalan numbers, but for longer patterns exact formulas are rare.
The theorem says that every pattern class has finite exponential growth rate, the Stanley–Wilf limit.
The proof by Marcus and Tardos is short and goes through extremal problems for $0$-$1$ matrices.
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# Dirichlet space

## Summary

The Dirichlet space $\mathcal{D}$ is the Hilbert space of [[holomorphic functions|0NAHXR1FRH]] $f$ on the unit disc whose derivative is square-integrable with respect to area.
The Dirichlet integral $\int_{\mathbb{D}} \abs{f'}^2 \, dA$ is the area of the image $f(\mathbb{D})$, counted with multiplicity.
In terms of Taylor coefficients, $f = \sum a_n z^n$ lies in $\mathcal{D}$ exactly when $\sum n \abs{a_n}^2 < \infty$, so $\mathcal{D}$ is contained in the [[Hardy space|PNNNFJVF6W]] $H^2$.
It is a model space in operator theory and is linked to logarithmic [[capacity|DBQXRTC0VN]] on the circle.

## Explanations

### L1 human

A holomorphic function on the disc maps the disc onto some region, possibly covering parts of it several times.
The function is in the Dirichlet space when the total area covered, counting overlaps, is finite.
For example, any one-to-one map of the disc onto a bounded region is in the Dirichlet space.
The function $\log(1 - z)$ is in the Hardy space $H^2$ but not in the Dirichlet space, because its image has infinite area.
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# Thom transversality theorem

## Statement

Let $X$ and $Y$ be [[smooth manifolds|F45VQ47GPV]], let $k \ge 0$, and let $J^k(X, Y)$ be the [[jet bundle|HC3VBBCJQP]] of $k$-jets of smooth maps from $X$ to $Y$.
For a smooth map $f \colon X \to Y$, let $j^k f \colon X \to J^k(X, Y)$ be its $k$-jet extension, which sends $x$ to the $k$-jet of $f$ at $x$.
Let $W$ be an [[embedded submanifold|VKFW27WXEZ]] of $J^k(X, Y)$.
Give $C^\infty(X, Y)$ the Whitney $C^\infty$ topology.
Then the set of maps $f \in C^\infty(X, Y)$ such that $j^k f$ is [[transverse|A2PDZ8GA9C]] to $W$ is a [[residual set|GRH8MCVN3N]] in $C^\infty(X, Y)$, and in particular it is dense.

## Remarks

For $k = 0$ one has $J^0(X, Y) = X \times Y$, and with $W = X \times Z$ the theorem says that maps transverse to a submanifold $Z \subseteq Y$ are generic.
The space $C^\infty(X, Y)$ with the Whitney topology is a Baire space, so a residual set is dense.
The proof reduces to the parametric transversality theorem.
If a smooth family $F \colon X \times S \to Y$ is transverse to $Z$, then $F(\cdot, s)$ is transverse to $Z$ for almost every $s$.
This follows from Sard's theorem applied to the projection $F^{-1}(Z) \to S$.
Locally one adds polynomial terms of degree at most $k$ to $f$, which moves $j^k f$ in all directions of the fibre.
Example: let $k = 1$ and let $W$ be the set of $1$-jets whose linear part has rank $r$.
The theorem shows that for a generic $f$ the set of points where $df$ has rank $r$ is a submanifold of the expected codimension.
This is how Whitney's results on generic singularities are obtained.

## Explanations

### L1 human

A typical smooth map is in general position with respect to any fixed condition on its derivatives.
Whatever submanifold one fixes in the space of jets, most maps meet it cleanly, and one can always reach such a map by an arbitrarily small change.
So in proofs one may assume that the maps one works with are transverse to whatever one likes.

### L2 human

The theorem is the source of most genericity statements in differential topology and singularity theory.
With $k = 0$ it gives transversality of maps to submanifolds, which is used to define intersection numbers and the Pontryagin–Thom construction.
With $k = 1$ it shows that generic maps have only mild singularities, for instance that Morse functions are dense and that generic maps $M^n \to \R^{2n}$ are immersions.
With higher $k$ it is the starting point of the classification of stable singularities.
</document>

<document source="https://tamnd.github.io/math2/node/KWNVF194D8.md" tag="KWNVF194D8" kind="statement">
# Bézout's theorem

## Statement

Let $k$ be an [[algebraically closed field|02SJG1HC32]].
Let $C$ and $D$ be projective [[plane curves|EWS4WFTA7W]] in the [[projective plane|ZT9A6JK8QF]] $\mathbb{P}^2_k$.
Suppose they are defined by homogeneous polynomials of degrees $d$ and $e$ with no common factor of positive degree.
Then $C \cap D$ is finite, and
$$\sum_{P \in C \cap D} i_P(C, D) = d e,$$
where $i_P(C, D)$ is the [[intersection multiplicity|GPQF9GK4T9]] of $C$ and $D$ at $P$.

## Remarks

One proof uses the resultant of the two polynomials with respect to one variable, after a change of coordinates.
It is a homogeneous polynomial of degree $de$ in the other two variables, and its roots, with multiplicity, correspond to the intersection points.
In [[intersection theory|4HV4WTAX9P]] the proof is one line: $[C] = d h$ and $[D] = e h$ in $CH^\bullet(\mathbb{P}^2) = \Z[h]/(h^3)$, so $[C] \cdot [D] = de \, h^2$, and $h^2$ is the class of a point.
Example: a line and a smooth conic meet in two points, or in one point of multiplicity $2$ when the line is tangent.
Each hypothesis is needed.
Over $\R$, the circle $x^2 + y^2 = z^2$ and the line $x = 2z$ have no common real point, but they meet in two complex points.
In the affine plane, two parallel lines do not meet; in $\mathbb{P}^2$ they meet at a point at infinity.
Without multiplicities, a tangent line meets a conic in only one point.
If the polynomials have a common factor, the curves share a component and meet in infinitely many points.
More generally, $n$ hypersurfaces of degrees $d_1, \ldots, d_n$ in $\mathbb{P}^n_k$ with finite intersection meet in $d_1 \cdots d_n$ points counted with multiplicity.

## Explanations

### L1 human

Two curves in the plane, given by equations of degrees $d$ and $e$ with no common factor, meet in exactly $de$ points.
This is true once three adjustments are made: complex points are allowed, points at infinity are included, and tangency points are counted several times.
For example, two conics without a common component meet in four points, counted properly.

### L2 human

Bézout's theorem is the first result of [[intersection theory|4HV4WTAX9P]], and the model for the general principle that intersection numbers depend only on the classes of the cycles.
It gives the bound of at most $de$ common points for any two plane curves without common component, over any field.
Together with the genus formula it is the basic tool of the classical theory of plane curves.
</document>

<document source="https://tamnd.github.io/math2/node/KWQYX7B6P5.md" tag="KWQYX7B6P5" kind="definition">
# Angle of parallelism

## Definition

Let $\mathbb{H}^2$ be the [[hyperbolic plane|GN4N94GQ0Z]] of constant curvature $-1$.
Let $\ell$ be a line and $P$ a point not on $\ell$.
Let $Q$ be the foot of the perpendicular from $P$ to $\ell$, and let $a = d(P, Q) > 0$.
A ray from $P$ meets $\ell$ when its [[angle|0GJAJC6TNZ]] with the ray $PQ$ is small.
Fix a side of the line $PQ$, and for $\theta \in [0, \pi/2]$ let $r_\theta$ be the ray from $P$ on that side at angle $\theta$ with $PQ$.
The *angle of parallelism* $\Pi(a)$ is the supremum of the angles $\theta$ such that $r_\theta$ meets $\ell$.
The ray at angle exactly $\Pi(a)$ does not meet $\ell$; it is a *limiting parallel* to $\ell$ through $P$.

## Remarks

The value does not depend on the side of $PQ$, nor on $\ell$ and $P$, only on $a$, and Lobachevsky's formula gives
$$\tan \frac{\Pi(a)}{2} = e^{-a}, \qquad \text{equivalently} \qquad \cos \Pi(a) = \tanh a, \quad \sin \Pi(a) = \frac{1}{\cosh a}.$$
Example: in the [[Beltrami–Klein model|1WN9B18H86]], take $P$ the centre and $\ell$ the chord $x = r$ with $0 < r < 1$.
Lines through the centre are diameters, so the foot is $Q = (r, 0)$ and $a = \operatorname{artanh} r$.
The limiting parallels are the chords from $P$ to the endpoints $(r, \pm\sqrt{1 - r^2})$, which make angle $\theta$ with $\cos \theta = r$ at the centre; angles at the centre are Euclidean in this model.
So $\cos \Pi(a) = r = \tanh a$, as the formula says.
Non-example: in the Euclidean plane every ray from $P$ at angle less than $\pi/2$ with $PQ$ meets $\ell$, and the ray at angle $\pi/2$ is the unique parallel.
There the supremum is $\pi/2$ for every $a$, so the angle carries no information about distance.
In curvature $-k^2$, the formula holds with $a$ replaced by $ka$.
</document>

<document source="https://tamnd.github.io/math2/node/KWRF0DEYA2.md" tag="KWRF0DEYA2" kind="statement">
# Kontinuitätssatz

## Statement

Let $D \subseteq \C^n$ be a connected open set that is a [[domain of holomorphy|4487CSC0MB]].
Let $\bar\Delta$ be the closed unit disc in $\C$, with boundary circle $\partial\Delta$.
For $j = 1, 2, \dots$ let $\varphi_j \colon \bar\Delta \to D$ be a continuous map that is holomorphic on the open disc.
If there is a [[compact set|0R6EWFKRCF]] $K \subseteq D$ with $\varphi_j(\partial\Delta) \subseteq K$ for all $j$, then there is a compact set $L \subseteq D$ with $\varphi_j(\bar\Delta) \subseteq L$ for all $j$.

## Remarks

Let $f$ be holomorphic on $D$.
By the maximum principle applied to $f \circ \varphi_j$, the values of $\abs{f}$ on $\varphi_j(\bar\Delta)$ are at most $\max_K \abs{f}$.
So every disc lies in the holomorphically convex hull $\hat K_D$, which is compact by the [[Cartan–Thullen theorem|1XCBCB3MD7]].
Take $L = \hat K_D$.

Example: the Hartogs figure $H = \set{(z_1, z_2) : \abs{z_1} < 1, \abs{z_2} < 1, \ \abs{z_1} < r \text{ or } \abs{z_2} > s}$ fails the conclusion.
Fix $\rho$ with $s < \rho < 1$, choose $c_j$ with $\abs{c_j} < r$ and $c_j \to c$ with $\abs{c} = r$, and let $\varphi_j(\zeta) = (c_j, \rho\zeta)$.
These discs lie in $H$, and their boundaries lie in the compact set $\set{\abs{z_1} \le r, \abs{z_2} = \rho} \subseteq H$.
But the centres $(c_j, 0)$ tend to $(c, 0) \notin H$, so no compact $L \subseteq H$ contains all the discs.
Hence $H$ is not a domain of holomorphy.

The converse also holds: an open set with the disc property of the statement is pseudoconvex, and hence a domain of holomorphy by the solution of the Levi problem.

## Explanations

### L1 human

Think of a sequence of holomorphic discs inside a domain whose rims stay in a fixed compact region.
In a domain of holomorphy the discs themselves cannot escape to the boundary either.
If a domain allows the discs to run out while their rims stay inside, then holomorphic functions get trapped and must extend.

### L2 human

The continuity principle is one of the oldest forms of pseudoconvexity.
It is a convenient practical test.
A single family of discs whose rims stay inside but whose centres leave the domain shows that the domain is not a domain of holomorphy.
The proof above shows how it follows from holomorphic convexity and the maximum principle.
</document>

<document source="https://tamnd.github.io/math2/node/KWS2STNJAA.md" tag="KWS2STNJAA" kind="definition">
# Clifford module

## Definition

Let $V$ be a finite-dimensional real vector space with a [[quadratic form|0RBQS1JJCR]] $q$.
Let $\mathrm{Cl}(V, q)$ be the [[Clifford algebra|A38B1QFK20]] of $(V, q)$, with the convention that $v \cdot v = -q(v) \cdot 1$ for $v \in V$.
Let $\mathbb{K}$ be $\R$ or $\C$.
A *Clifford module* over $\mathrm{Cl}(V, q)$ is a $\mathbb{K}$-vector space $W$ together with a homomorphism of real algebras $\rho \colon \mathrm{Cl}(V, q) \to \End_{\mathbb{K}}(W)$ that sends $1$ to $\id_W$.

## Remarks

By the universal property of the Clifford algebra, giving $\rho$ is the same as giving a linear map $c \colon V \to \End_{\mathbb{K}}(W)$ with
$$c(v)^2 = -q(v) \, \id_W \quad \text{for all } v \in V.$$
Authors who use the convention $v \cdot v = q(v) \cdot 1$ get the same theory with $q$ replaced by $-q$.
A Clifford module is irreducible if it has no proper nonzero invariant subspace; when $q$ is nondegenerate, every finite-dimensional Clifford module is a direct sum of irreducible ones.

Example: let $V = \R$ with $q(x) = x^2$, let $W = \C$ as a real vector space and let $c(x)$ be multiplication by $ix$.
Then $c(x)^2 = (ix)^2 = -x^2 = -q(x)$, so $W$ is a Clifford module, and in fact $\mathrm{Cl}(V, q) \cong \C$.

Non-example: with the same $V$ and $q$, let $W = \R$ and $c(x) = x \, \id_W$.
Then $c(1)^2 = 1 \ne -1$, so this is not a Clifford module.
In fact no real vector space of dimension $1$ is a Clifford module here, since no real number squares to $-1$.

## Explanations

### L1 human

A Clifford module is a vector space on which each vector $v$ acts by a linear map whose square is the scalar $-q(v)$.
</document>

<document source="https://tamnd.github.io/math2/node/KX0B3AWD5H.md" tag="KX0B3AWD5H" kind="statement">
# Toponogov's theorem

## Statement

Let $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \ge 2$ whose [[sectional curvature|1X42JAA349]] satisfies $K \ge k$ for a real number $k$.
Let $M_k^2$ be the simply connected complete surface of constant curvature $k$, a [[space form|1VM1AQT956]].
Let $p \in M$, and let $\gamma_1, \gamma_2$ be minimizing unit-speed [[geodesics|NSYMQMHPDN]] starting at $p$, of lengths $\ell_1, \ell_2$, that make the angle $\alpha \in [0, \pi]$ at $p$.
If $k > 0$, assume $\ell_1, \ell_2 \le \pi / \sqrt{k}$.
Let $\bar\gamma_1, \bar\gamma_2$ be unit-speed geodesics in $M_k^2$ starting at a point $\bar p$, of lengths $\ell_1, \ell_2$, that make the angle $\alpha$ at $\bar p$.
Then
$$d\bigl(\gamma_1(\ell_1), \gamma_2(\ell_2)\bigr) \le \bar d\bigl(\bar\gamma_1(\ell_1), \bar\gamma_2(\ell_2)\bigr),$$
where $d$ is the [[Riemannian distance|B4QR306D8G]] of $M$ and $\bar d$ that of $M_k^2$.

## Remarks

This is the hinge form of the theorem.
An equivalent form compares the angles of a geodesic triangle with minimizing sides in $M$ with those of the triangle with the same side lengths in $M_k^2$.
Each angle in $M$ is at least the corresponding angle in $M_k^2$.
The proof starts from the Rauch comparison theorem, which gives the result for short hinges, and extends it to all hinges by a subdivision argument that uses completeness.

Example: if $K \ge 0$, compare with the Euclidean plane.
Then for two minimizing geodesics of lengths $\ell_1, \ell_2$ from $p$ at angle $\alpha$, the endpoints are at distance at most $\sqrt{\ell_1^2 + \ell_2^2 - 2 \ell_1 \ell_2 \cos\alpha}$, the value given by the law of cosines.
On the round sphere, for $0 < \alpha < \pi$, the distance is strictly smaller, since triangles on a sphere are fatter than flat triangles.

## Explanations

### L1 human

On a space that curves at least as much as a model surface, triangles are "fatter" than in the model.
Two paths that leave a point at a given angle end up closer together than they would in the model.

### L2 human

Toponogov's theorem turns a lower curvature bound into a statement about distances only, without derivatives.
This is why it is the basis of many global results such as the soul theorem and the Grove–Shiohama diameter sphere theorem.
Its conclusion is also used as the definition of a lower curvature bound for metric spaces, which leads to Alexandrov spaces.
</document>

<document source="https://tamnd.github.io/math2/node/KX2XY7S3X6.md" tag="KX2XY7S3X6" kind="statement">
# Spectral radius formula

## Statement

Let $X \ne \set{0}$ be a complex [[Banach space|06GZ956657]], and let $T \colon X \to X$ be a [[continuous linear map|03TM135G0N]] with [[spectrum|BMKATJ802A]] $\sigma(T)$.
Then $\sigma(T)$ is nonempty and compact, and its *spectral radius* $r(T) = \max \set{\abs{\lambda} : \lambda \in \sigma(T)}$ satisfies
$$r(T) = \lim_{n \to \infty} \norm{T^n}^{1/n} = \inf_{n \ge 1} \norm{T^n}^{1/n},$$
where $\norm{\cdot}$ is the [[operator norm|06EKW1AKXF]].

## Remarks

Proof idea: if $\lambda \in \sigma(T)$, then $\lambda^n \in \sigma(T^n)$, and the [[Neumann series|8QJ91709V2]] shows $\abs{\lambda}^n \le \norm{T^n}$; this gives $r(T) \le \inf_n \norm{T^n}^{1/n}$.
For $\abs{\lambda} > \norm{T}$, the [[resolvent|J5YNNKSB9P]] has the expansion $R(\lambda, T) = -\sum_{n \ge 0} T^n / \lambda^{n+1}$.
The resolvent is analytic on $\abs{\lambda} > r(T)$, so this Laurent series converges there, and the uniform boundedness principle with the Cauchy estimates gives $\limsup_n \norm{T^n}^{1/n} \le r(T)$.
Nonemptiness of $\sigma(T)$ follows from Liouville's theorem applied to $\lambda \mapsto f(R(\lambda, T))$ for continuous linear functionals $f$.

Example: the Volterra operator $(Vf)(x) = \int_0^x f(t) \, dt$ on $C[0, 1]$ with the supremum norm satisfies $\norm{V^n} = 1/n!$, attained at $f = 1$.
So $r(V) = \lim (1/n!)^{1/n} = 0$ and $\sigma(V) = \set{0}$, although $V \ne 0$.

The field must be $\C$.
The rotation $T(x, y) = (-y, x)$ of the real plane has $\norm{T^n} = 1$ for all $n$, but $T - \lambda I$ is invertible for every real $\lambda$, so its real spectrum is empty.

For matrices this is [[Gelfand's formula|3KX0KQ7V05]] for the [[spectral radius|9V7YQA85PC]].
The same statement, with the same proof, holds for every element of a unital complex Banach algebra.

## Explanations

### L1 human

The spectral radius is the size of the largest point in the spectrum.
The formula says it can be read off from the norms of the powers $T^n$: these grow, or shrink, like $r(T)^n$.
So the long-run growth rate of $T^n$ is fixed by the spectrum, even though a single norm $\norm{T}$ can be much larger than $r(T)$.

### L2 human

The formula shows that $T^n \to 0$ in norm if and only if $r(T) < 1$, and that $\sum T^n$ converges in that case, which sharpens the Neumann series.
It also shows that $r(T)$ depends only on the algebra, not on the choice of an equivalent norm.
A quasinilpotent operator, with $r(T) = 0$, need not be nilpotent, as the Volterra operator shows.
For normal operators on a Hilbert space, $\norm{T^{2^k}} = \norm{T}^{2^k}$, so $r(T) = \norm{T}$; this is a key step in the theory of C*-algebras.
</document>

<document source="https://tamnd.github.io/math2/node/KX3QRA3CEP.md" tag="KX3QRA3CEP" kind="concept">
# Exponential of a bounded operator

## Summary

The exponential of a bounded operator $T$ on a Banach space is the operator $e^T = \sum_{n \ge 0} T^n / n!$.
It extends the [[matrix exponential|020M04FN1H]] to infinite dimensions.
The family $t \mapsto e^{tT}$ solves the linear differential equation $x' = Tx$ in the Banach space and is the simplest example of an operator semigroup.

## Explanations

### L1 human

The exponential of a number can be computed by the series $1 + t + t^2/2 + t^3/6 + \cdots$.
The same series makes sense when $t$ is replaced by a bounded operator, because the factorials make the terms shrink fast.
If $T$ multiplies the $n$-th coordinate of a sequence by $\lambda_n$, with $(\lambda_n)$ bounded, then $e^T$ multiplies it by $e^{\lambda_n}$.
This turns a rule of instantaneous change, $x' = Tx$, into the state at every later time, $x(t) = e^{tT} x(0)$.
</document>

<document source="https://tamnd.github.io/math2/node/KX4152NVW8.md" tag="KX4152NVW8" kind="statement">
# Rayleigh monotonicity law

## Statement

Let $G = (V, E)$ be a finite connected graph, and let $r, r' \colon E \to (0, \infty)$ be two resistance functions.
So $(G, 1/r)$ and $(G, 1/r')$ are [[electrical networks|T7KRE7J3B4]].
Suppose that $r(e) \le r'(e)$ for every edge $e \in E$.
Then for all vertices $a \ne b$,
$$R_{\mathrm{eff}}(a, b) \le R'_{\mathrm{eff}}(a, b),$$
where $R_{\mathrm{eff}}$ and $R'_{\mathrm{eff}}$ are the [[effective resistances|XAE5YJ3H68]] for $r$ and for $r'$.

## Remarks

By [[Thomson's principle|8BPMHA5MT4]], each effective resistance is the minimum energy of a unit flow from $a$ to $b$.
For every unit flow $\theta$, the energy $\sum_e r(e) \theta(e)^2$ is at most $\sum_e r'(e) \theta(e)^2$.
So the minimum for $r$ is at most the minimum for $r'$.
Example: the path $a - m - b$ with unit resistances has $R_{\mathrm{eff}}(a, b) = 2$.
Adding an edge from $a$ to $b$ of resistance $1$ gives $R_{\mathrm{eff}}(a, b) = \tfrac23$.
This agrees with the law, viewing the new edge as the limit of an edge of very large resistance whose resistance is lowered.
Two consequences are standard: removing an edge, as long as the graph stays connected, does not decrease effective resistances.
Gluing two vertices into one, which amounts to giving a resistance $0$ to an edge between them, does not increase effective resistances.

## Explanations

### L1 human

If some wires in a network get worse, the network as a whole cannot get better.
Raising the resistance of any wire can only raise the resistance between any two points.
Cutting a wire is an extreme case of raising its resistance, and joining two points by a perfect wire is an extreme case of lowering it.
The law sounds obvious, but its proof needs a variational principle such as Thomson's principle.

### L2 human

The law is the main tool for comparing random walks on different graphs.
To show that a walk on a graph is recurrent, one may glue vertices together, which lowers resistances, and show that the resistance to infinity is still infinite.
To show transience, one may delete edges, which raises resistances, and find a subgraph, such as a tree, with finite resistance to infinity.
Combined with the criterion that relates recurrence to the resistance to infinity, this proves the transience of $\Z^3$ from the transience of a suitable subgraph.
</document>

<document source="https://tamnd.github.io/math2/node/KXC9KHWVGE.md" tag="KXC9KHWVGE" kind="definition">
# Convolution

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] with a left [[Haar measure|VY0HXD3GC4]] $\mu$.
Let $f, h \colon G \to \C$ be functions in [[$L^1(G)$|SASP14MA1Z]], the space of $\mu$-integrable functions.
The *convolution* of $f$ and $h$ is the function $f * h$ on $G$ given by
$$(f * h)(x) = \int_G f(y) \, h(y^{-1} x) \, d\mu(y)$$
for every $x \in G$ at which the integral converges absolutely.

## Remarks

The integral converges absolutely for almost every $x$, and $f * h$ lies in $L^1(G)$ with $\norm{f * h}_1 \le \norm{f}_1 \norm{h}_1$.
Convolution is associative, and it is commutative when $G$ is abelian.

Example: $G = (\R, +)$ with Lebesgue measure, $f = h = \mathbf{1}_{[0, 1]}$.
Then $(f * h)(x) = \int_\R \mathbf{1}_{[0,1]}(y) \mathbf{1}_{[0,1]}(x - y) \, dy$ is the length of $[0, 1] \cap [x - 1, x]$.
This equals $x$ for $0 \le x \le 1$, $2 - x$ for $1 \le x \le 2$, and $0$ otherwise: a triangle function.

Non-example: for $f = h = 1$ on $\R$, the defining integral $\int_\R 1 \, dy$ diverges at every $x$, so the convolution is not defined; the hypothesis $f, h \in L^1$ fails.

On $(\Z, +)$ with counting measure, $(f * h)(n) = \sum_{m \in \Z} f(m) h(n - m)$, the Cauchy product of coefficient sequences.
The convolution of probability distributions is the special case of measures, which describes sums of independent random variables.
</document>

<document source="https://tamnd.github.io/math2/node/KXJG5VV340.md" tag="KXJG5VV340" kind="concept">
# Frattini subgroup

## Summary

The Frattini subgroup $\Phi(G)$ of a group $G$ is the intersection of all maximal subgroups of $G$.
It is exactly the set of non-generators: elements that can be dropped from any generating set without changing the group generated.
It is a [[characteristic subgroup|0KABPMRHM2]], and for a finite group it is [[nilpotent|0Q7MMQHC1P]].
For a finite [[p-group|06QYRTX9KQ]] $P$, the quotient $P/\Phi(P)$ is elementary abelian, and its dimension is the minimal number of generators of $P$.

## Explanations

### L1 human

Some elements of a group are useless for generating it.
If a set of elements together with such an element generates the whole group, the set already generates it without that element.
The Frattini subgroup collects all these useless elements.
In the cyclic group $\Z/8\Z$, the even residues are useless: any generating set must contain an odd residue, which alone generates the group.
So the Frattini subgroup of $\Z/8\Z$ is $\set{0, 2, 4, 6}$.
</document>

<document source="https://tamnd.github.io/math2/node/KXS5DQDFYD.md" tag="KXS5DQDFYD" kind="concept">
# Dedekind zeta function

## Summary

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]].
Its Dedekind zeta function $\zeta_K(s)$ is the sum of $N(\mathfrak{a})^{-s}$ over the nonzero ideals $\mathfrak{a}$ of the ring of integers.
For $K = \Q$ it is the [[Riemann zeta function|1XWBY4N0VT]].
It has an Euler product over the prime ideals and a functional equation.
It extends to a meromorphic function on the whole complex plane with a single simple pole at $s = 1$.
Its residue at $s = 1$ is given by the [[class number formula|55XX5A7XZV]].
This formula ties together the class number, the [[regulator|RWK0HNF1NA]] and the [[discriminant|E0VZTNPJXA]].

## Explanations

### L1 human

The Riemann zeta function adds up $1/n^s$ over the positive integers.
The Dedekind zeta function does the same for a number field, with ideals in place of integers and their norms in place of $n$.
For the Gaussian integers, the ideals of norm $5$ are $(2 + i)$ and $(2 - i)$, so the term $5^{-s}$ appears twice.
The analytic behaviour of this function encodes how the prime ideals of the field are distributed.
</document>

<document source="https://tamnd.github.io/math2/node/KXTQ10RDSX.md" tag="KXTQ10RDSX" kind="definition">
# Reduced ring

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
$R$ is *reduced* if it has no nonzero [[nilpotent elements|0N9R5NN6TW]].
That is, for all $r \in R$ and all integers $n \ge 1$, if $r^n = 0$ then $r = 0$.

## Remarks

Example: $\Z/6\Z$ is reduced; if $r^n \equiv 0 \pmod 6$, then $2$ and $3$ divide $r$, so $r \equiv 0$.
Non-example: $\Z/4\Z$ is not reduced, since $2^2 = 0$ and $2 \ne 0$.
Equivalent forms: the [[nilradical|QQDN6EYQNJ]] of $R$ is zero; the zero ideal is a [[radical ideal|S4GQN5V54X]]; for all $r \in R$, $r^2 = 0$ implies $r = 0$.
Every [[integral domain|09S3PFBEPM]] is reduced, and so is every product of integral domains.
The same condition defines reduced rings that are not commutative.
</document>

<document source="https://tamnd.github.io/math2/node/KY7SB854VH.md" tag="KY7SB854VH" kind="concept">
# Face of a convex set

## Summary

A face of a [[convex set|02AS45T3H9]] $C$ is a convex subset $F$ of $C$ with the following property.
Any segment of $C$ whose inside meets $F$ lies entirely in $F$.
In other words, a face is a convex [[extreme set|WD3RJZC5A7]] of $C$.
For a polytope, the nonempty faces are the vertices, edges and higher-dimensional faces, together with the polytope itself.
The zero-dimensional faces of $C$ are exactly its [[extreme points|3MSGFJM2NX]].

## Explanations

### L1 human

The faces of a cube are the cube itself, its six square sides, its twelve edges, its eight corners, and the empty set.
Each is a flat part of the boundary, or the whole cube, that cannot be entered from outside along a straight line through its inside.
If a segment of the cube passes through the middle of an edge, the segment must run along that edge.
For a disc, the faces are the empty set, the disc and its single boundary points.
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<document source="https://tamnd.github.io/math2/node/KY9ZXCHCVX.md" tag="KY9ZXCHCVX" kind="definition">
# Superadditive function

## Definition

Let $(A, +)$ be a [[semigroup|0XMEM406JM]], for example $[0, \infty)$, $\R$ or the positive integers under addition.
A [[function|0EPAT51DXQ]] $f \colon A \to \R$ is *superadditive* if
$$f(x + y) \ge f(x) + f(y) \quad \text{for all } x, y \in A.$$

## Remarks

Example: $f(x) = x^2$ on $[0, \infty)$ is superadditive.
Check: $(x + y)^2 = x^2 + y^2 + 2xy \ge x^2 + y^2$, since $xy \ge 0$.

Non-example: $f(x) = \sqrt{x}$ on $[0, \infty)$ is not superadditive, since $\sqrt{2} < 2 = \sqrt{1} + \sqrt{1}$.

A function $f$ is superadditive if and only if $-f$ is [[subadditive|3X9DD9J3CV]].
Every convex function $f \colon [0, \infty) \to \R$ with $f(0) \le 0$ is superadditive.
For a superadditive sequence, $a_{m+n} \ge a_m + a_n$, the limit of $a_n / n$ exists in $(-\infty, \infty]$ and equals $\sup_n a_n / n$.
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<document source="https://tamnd.github.io/math2/node/KYCPCC57FT.md" tag="KYCPCC57FT" kind="concept">
# Natural transformation

## Summary

Let $F, G \colon \mathcal{C} \to \mathcal{D}$ be two [[functors|28NQ0KRHD3]].
A natural transformation $\alpha \colon F \Rightarrow G$ is a family of morphisms $\alpha_A \colon F(A) \to G(A)$, one for each object $A$.
The family must commute with the images of all morphisms of $\mathcal{C}$.
It makes precise the idea of a construction that is "natural", that is, defined in the same way for all objects without arbitrary choices.
The canonical map from a vector space to its double dual is natural.
By contrast, an isomorphism of a finite-dimensional space with its dual depends on a choice of basis.
Natural transformations are the morphisms of [[functor categories|1X0WKFQX8K]] and enter the definitions of [[equivalence of categories|K7A04ZBD92]] and of [[adjoint functors|TZHF6ZM92S]].

## Explanations

### L1 human

Two functors give two pictures of the same category inside another category.
A natural transformation is a way of moving from the first picture to the second, one object at a time.
"Natural" means that it does not matter whether one first moves along an arrow and then switches pictures, or first switches and then moves.
For example, every vector space $V$ maps to its double dual by sending $v$ to "evaluate at $v$".
This rule uses no basis, and it is compatible with every linear map, so it is natural.
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<document source="https://tamnd.github.io/math2/node/KYF5SCYC64.md" tag="KYF5SCYC64" kind="definition">
# Renewal theory

## Definition

Let $T_1, T_2, \ldots$ be [[independent and identically distributed|00MRN0FARV]] random variables with values in $[0, \infty)$ and $P(T_1 = 0) < 1$.
Let $S_0 = 0$ and $S_n = T_1 + \cdots + T_n$ for $n \ge 1$; these are the renewal times.
The *renewal process* with holding times $T_i$ is the process
$$N(t) = \sup \set{n \ge 0 : S_n \le t}, \qquad t \ge 0.$$
Its *renewal function* is $m(t) = \mathbb{E}[N(t)]$.
Renewal theory is the study of $N$, of $m$, and of processes that start afresh at the times $S_n$.

## Remarks

The condition $P(T_1 = 0) < 1$ makes $S_n \to \infty$ almost surely, so $N(t)$ is finite for each $t$.
Example: if the $T_i$ are [[exponential|0E99F51GVB]] with rate $\lambda$, then $N$ is the Poisson process of rate $\lambda$, and $m(t) = \lambda t$.
Non-example: if $T_1, T_2, \ldots$ are not identically distributed, for instance $T_i = 2^{-i}$, then $N$ is not a renewal process, and here $N(t) = \infty$ for $t \ge 1$.
Let $\mu = \mathbb{E}[T_1] \in (0, \infty]$ be the [[expected value|0WQNRCNY3M]] of a holding time.
The elementary renewal theorem says $m(t)/t \to 1/\mu$ as $t \to \infty$, with $1/\infty = 0$; the strong law gives $N(t)/t \to 1/\mu$ almost surely.
</document>

<document source="https://tamnd.github.io/math2/node/KYGE7SPYA6.md" tag="KYGE7SPYA6" kind="definition">
# Topological transformation group

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] and let $X$ be a [[topological space|0CQHFPGJ45]].
A *topological transformation group* is a triple $(G, X, \alpha)$ where $\alpha \colon G \times X \to X$ is a [[group action|0WCY42EV9V]] of $G$ on $X$, written $\alpha(g, x) = g \cdot x$, such that $\alpha$ is [[continuous|0VGFHTM75Y]].
Here $G \times X$ carries the [[product topology|0GBG8B49AA]].
One also says that $G$ acts continuously on $X$.

## Remarks

Example: the group $\mathrm{SO}(3)$ of rotations of $\R^3$, with the topology from the matrix entries, acting on the unit sphere $S^2$ by $(A, x) \mapsto Ax$.
The map is given by polynomials in the entries of $A$ and the coordinates of $x$, so it is continuous.
Non-example: $(\R, +)$ with the usual topology acting on $\R$ with the discrete topology by $(t, x) \mapsto t + x$.
This is a group action, but it is not continuous: for fixed $x$, the map $t \mapsto t + x$ from $\R$ to the discrete space is not continuous.
For each $g \in G$ the map $x \mapsto g \cdot x$ is a homeomorphism of $X$, with inverse $x \mapsto g^{-1} \cdot x$.
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<document source="https://tamnd.github.io/math2/node/KYSR7NHNFQ.md" tag="KYSR7NHNFQ" kind="statement">
# Hadamard factorization theorem

## Statement

Let $f$ be an [[entire function|0419794R0T]] that is not identically zero and has finite [[order|SHF8Z7GMFX]] $\rho$.
Let $p$ be the integer with $p \le \rho < p + 1$.
Let $m \ge 0$ be the order of the [[zero|0CEGEASY49]] of $f$ at $0$.
Let $(a_n)$ be the nonzero zeros of $f$, repeated according to multiplicity.
Let $E_p(z) = (1 - z) \exp(z + z^2/2 + \cdots + z^p/p)$, with $E_0(z) = 1 - z$.
Then $\sum_n \abs{a_n}^{-(p+1)} < \infty$.
Moreover, there is a polynomial $g$ of degree at most $p$ such that
$$f(z) = z^m e^{g(z)} \prod_{n} E_p\!\left(\frac{z}{a_n}\right)$$
for all $z \in \C$, where the product converges uniformly on compact sets.

## Remarks

The proof first bounds the number $n(r)$ of zeros in the disc $\abs{z} \le r$ by [[Jensen's formula|497K99BQ4Y]]: $n(r) = O(r^{\rho + \varepsilon})$ for every $\varepsilon > 0$.
This gives the convergence of $\sum_n \abs{a_n}^{-(p+1)}$, so the [[Weierstrass product|ACSF58HYDH]] with the fixed index $p$ converges.
The quotient of $f$ by $z^m$ times the product is $e^{g}$ with $g$ entire.
Lower bounds for the product away from its zeros show that $g$ is a polynomial of degree at most $\rho$.
For example, $\sin(\pi z)$ has order $1$, so $p = 1$ and $\sin(\pi z) = \pi z \prod_{n \ne 0} (1 - z/n) e^{z/n}$.
The finite order hypothesis is necessary: $f(z) = e^{e^z}$ has no zeros and infinite order, and $f = e^{g}$ with $g(z) = e^z$, which is not a polynomial.

## Explanations

### L1 human

The theorem refines the [[Weierstrass factorization theorem|11VC51YHQJ]] for entire functions that do not grow too fast.
For such a function, all convergence factors can be taken of the same small degree.
The leftover zero-free factor is the exponential of a polynomial, not of an arbitrary entire function.
So the zeros determine the function up to a few constants.

### L2 human

A useful consequence: an entire function of finite order that is not an integer has infinitely many zeros.
Indeed, with finitely many zeros the factorization gives a polynomial times $e^{g}$ with $g$ a polynomial, whose order is an integer.
Another consequence is that an entire function of finite order without zeros is $e^{g}$ with $g$ a polynomial.
The theorem is a basic tool for the Riemann zeta function, through the product formula of the function $\xi$, and for the [[Laguerre–Pólya class|X9SRCEENV3]].
</document>

<document source="https://tamnd.github.io/math2/node/KYVS4PSDBQ.md" tag="KYVS4PSDBQ" kind="definition">
# Heegaard Floer homology

## Definition

Let $Y$ be a closed oriented [[3-manifold|R20TB2KEQQ]].
Let $(\Sigma, \alpha, \beta, z)$ be a pointed Heegaard diagram of $Y$ coming from a [[Heegaard splitting|EDE9919B5R]].
Here $\Sigma$ is a closed oriented surface of genus $g$.
The families $\alpha = (\alpha_1, \ldots, \alpha_g)$ and $\beta = (\beta_1, \ldots, \beta_g)$ consist of disjoint simple closed curves on $\Sigma$.
The base point $z \in \Sigma$ lies off all of these curves.
Assume the diagram is admissible, and fix a generic path of almost complex structures on the symmetric product $\mathrm{Sym}^g(\Sigma)$.

1. The tori $\mathbb{T}_\alpha = \alpha_1 \times \cdots \times \alpha_g$ and $\mathbb{T}_\beta = \beta_1 \times \cdots \times \beta_g$ are submanifolds of $\mathrm{Sym}^g(\Sigma)$ that meet transversally in finitely many points.
2. The chain complex $\widehat{CF}(\Sigma, \alpha, \beta, z)$ is the vector space over $\F_2$ with basis $\mathbb{T}_\alpha \cap \mathbb{T}_\beta$.
3. Let $x$ and $y$ be basis elements.
Consider the holomorphic disks from $x$ to $y$ with Maslov index $1$ whose domain avoids $z$.
The coefficient of $y$ in $\partial x$ is their number modulo $2$.
Disks that differ by a translation of the parameter are counted once.

The homology $\widehat{HF}(Y)$ of this complex is the *hat version of Heegaard Floer homology* of $Y$.
The versions $HF^-$, $HF^\infty$ and $HF^+$ are defined in the same way over $\F_2[U]$ or $\F_2[U, U^{-1}]$.
In them the disks may cover $z$, and the power of $U$ records how many times they do.

## Remarks

Ozsváth and Szabó proved that $\partial^2 = 0$ and that the homology does not depend, up to isomorphism, on the diagram and the other choices, so it is an invariant of $Y$.
For $Y = S^3$, the genus $1$ diagram with $\alpha$ and $\beta$ meeting in one point gives $\widehat{HF}(S^3) \cong \F_2$.
For a rational homology sphere $Y$, the dimension of $\widehat{HF}(Y)$ is at least the order of $H_1(Y; \Z)$; the spaces with equality are called L-spaces, and lens spaces are examples.
With suitable orientation choices the construction also works over $\Z$.
It is a form of Lagrangian [[Floer homology|NCKFPAGHXF]] for the pair $(\mathbb{T}_\alpha, \mathbb{T}_\beta)$.
</document>

<document source="https://tamnd.github.io/math2/node/KYW1PV0SX5.md" tag="KYW1PV0SX5" kind="definition">
# Supermartingale

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]], let $T \subseteq [0, \infty)$ be an index set, and let $(\mathcal{F}_t)_{t \in T}$ be a [[filtration|HNK1Y62G7V]].
A *supermartingale* with respect to $(\mathcal{F}_t)$ is a family $(X_t)_{t \in T}$ of real random variables such that:

1. $(X_t)$ is [[adapted|YRNFE016FF]]: $X_t$ is $\mathcal{F}_t$-measurable for each $t \in T$.
2. $E[\abs{X_t}] < \infty$ for each $t \in T$.
3. $E[X_t \mid \mathcal{F}_s] \le X_s$ almost surely for all $s \le t$ in $T$, where $E[\cdot \mid \mathcal{F}_s]$ is the [[conditional expectation|0YAMJRK04E]].

## Remarks

Let $\xi_1, \xi_2, \dots$ be independent with $P(\xi_i = 1) = p$ and $P(\xi_i = -1) = 1 - p$, where $p \le 1/2$, and let $\mathcal{F}_n = \sigma(\xi_1, \dots, \xi_n)$.
Then $S_n = \xi_1 + \dots + \xi_n$ is a supermartingale: it is adapted, $\abs{S_n} \le n$, and $E[S_{n+1} \mid \mathcal{F}_n] = S_n + (2p - 1) \le S_n$.
For $p = 1/2$, the process $S_n^2$ is adapted and integrable but fails clause 3, since $E[S_{n+1}^2 \mid \mathcal{F}_n] = S_n^2 + 1$.
$(X_t)$ is a supermartingale if and only if $(-X_t)$ is a [[submartingale|EQ055QMACW]].
A process is a [[martingale|0CDVK9RQBH]] if and only if it is both a submartingale and a supermartingale.
Taking expectations in clause 3 shows that $t \mapsto E[X_t]$ is non-increasing.
</document>

<document source="https://tamnd.github.io/math2/node/KZ00AZ2HFM.md" tag="KZ00AZ2HFM" kind="definition">
# CAT(k) space

## Definition

Let $k$ be a real number.
Let $M_k$ be the complete simply connected surface of constant curvature $k$.
It is the round sphere of radius $1/\sqrt{k}$ if $k > 0$, the [[Euclidean plane|0XPR1E85W0]] if $k = 0$, and the hyperbolic plane of curvature $k$ if $k < 0$.
Let $D_k = \pi/\sqrt{k}$ if $k > 0$ and $D_k = \infty$ if $k \le 0$.

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]].
A *geodesic segment* in $X$ is a map $c \colon [0, \ell] \to X$ with $d(c(s), c(t)) = \abs{s - t}$ for all $s, t$.
A *geodesic triangle* consists of three points of $X$ and a geodesic segment between each pair of them.
A *comparison triangle* for it is a [[triangle|0CTNFGYMM2]] in $M_k$ with the same three side lengths.
Each point $p$ on a side has a comparison point $\bar p$ on the matching side, at the same distances from the ends.

The space $X$ is a *CAT($k$) space* if:

1. any two points of $X$ at distance less than $D_k$ are joined by a geodesic segment, and
2. for every geodesic triangle of perimeter less than $2 D_k$ and all points $p, q$ on its sides, $d(p, q) \le d_{M_k}(\bar p, \bar q)$.

## Remarks

Example: $\R^n$ with the Euclidean distance is CAT(0).
Its geodesic triangles are flat triangles, and a comparison triangle in $M_0$ is congruent to it, so equality holds in clause 2.

Non-example: the unit round sphere $S^2$ is not CAT(0).
Take the triangle with vertices on the equator at longitudes $0$, $\pi/2$ and the north pole; its sides have length $\pi/2$.
The midpoints of the two sides through the pole are at spherical distance $\pi/3$.
The comparison triangle in $M_0$ is equilateral with side $\pi/2$, so the comparison points are at distance $\pi/4 < \pi/3$.
So clause 2 fails; the sphere is CAT(1).

A CAT($k$) space is also CAT($k'$) for every $k' \ge k$.
A complete simply connected Riemannian manifold is CAT($k$) for some $k \le 0$ exactly when its [[sectional curvature|1X42JAA349]] is at most $k$.
Metric trees are CAT($k$) for every $k$.
</document>

<document source="https://tamnd.github.io/math2/node/KZ226116HW.md" tag="KZ226116HW" kind="concept">
# Lasso

## Summary

The lasso, short for least absolute shrinkage and selection operator, fits a linear model by penalised least squares.
The penalty is a multiple of the sum of the absolute values of the coefficients.
Because of the corners of the $\ell^1$ ball, the penalty sets some coefficients exactly to zero.
So the lasso performs estimation and variable selection at the same time.
It was introduced in statistics by Robert Tibshirani in 1996 and is a basic tool for regression with many more regressors than observations.
It is a [[shrinkage estimator|GPBSS1V3SS]], and the [[elastic net|TKQPE0Q05Y]] combines it with [[ridge regression|3KZW2930V2]].

## Explanations

### L1 human

The lasso charges a price for every unit of coefficient size, whatever the coefficient.
A regressor whose contribution to the fit is smaller than that price is dropped completely: its coefficient becomes zero.
The result is a model that uses only a few regressors, which is easier to read.
When the regressors are orthonormal, the lasso takes each least squares coefficient, moves it towards zero by the penalty level, and sets it to zero if it would cross zero.
For example, with penalty level $1$, least squares coefficients $3$, $-0.5$ and $-2$ become $2$, $0$ and $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/KZ3YVM10SP.md" tag="KZ3YVM10SP" kind="definition">
# Peetre K-functional

## Definition

Let $(X_0, X_1)$ be a compatible couple of [[Banach spaces|06GZ956657]].
So both are continuously embedded in a common Hausdorff topological vector space, and the sum $X_0 + X_1$ is defined.
For $x \in X_0 + X_1$ and $t > 0$, the *Peetre K-functional* is the [[infimum|04KFTA34C5]]
$$K(t, x) = K(t, x; X_0, X_1) = \inf \set{ \norm{x_0}_{X_0} + t \norm{x_1}_{X_1} : x = x_0 + x_1, \ x_0 \in X_0, \ x_1 \in X_1 }.$$

## Remarks

Example: take $X_0 = X_1 = \R$ with the absolute value.
For a split $x = x_0 + x_1$ the cost is $\abs{x_0} + t \abs{x_1} \ge \min(1, t) (\abs{x_0} + \abs{x_1}) \ge \min(1, t) \abs{x}$, and putting all of $x$ in the cheaper space attains this, so $K(t, x) = \min(1, t) \abs{x}$.

Non-example: the K-functional is only defined on $X_0 + X_1$.
For $X_0 = X_1 = \ell^1$ inside the space of all sequences, the sequence $(1, 1, 1, \dots)$ has no split at all, so $K(t, \cdot)$ is not defined there.

For each $x$, the function $t \mapsto K(t, x)$ is nondecreasing and concave, and $K(1, x)$ is the norm of $x$ in $X_0 + X_1$.
For $0 < \theta < 1$ and $1 \le q \le \infty$, the real [[interpolation space|VX1R6FN6PS]] $(X_0, X_1)_{\theta, q}$ consists of the $x$ for which $t \mapsto t^{-\theta} K(t, x)$ lies in $L^q((0, \infty), dt / t)$.
In approximation theory one takes $X_0 = L^p[a, b]$ and $X_1$ the Sobolev space $W^{k, p}[a, b]$, measured by the seminorm $\norm{g^{(k)}}_p$.
Then the quantity $K(t^k, f)$ is bounded above and below by constant multiples of the [[modulus of smoothness|CRPZ1RVNKQ]] $\omega_k(f, t)_p$ for small $t$.
</document>

<document source="https://tamnd.github.io/math2/node/KZCXHTEYK7.md" tag="KZCXHTEYK7" kind="concept">
# Quillen equivalence

## Summary

A Quillen [[equivalence|K7A04ZBD92]] is a [[Quillen adjunction|SY38F954FE]] between two [[model categories|SDTZVZ6B4Q]] that induces an equivalence of their homotopy categories.
It is the right notion of "same homotopy theory" for model categories.
The basic example is the adjunction between simplicial sets and topological spaces given by geometric realization and the singular complex.
It shows that both categories model the same homotopy theory of spaces.

## Explanations

### L1 human

A model category is a setting where some maps are declared to be weak equivalences, as in homotopy theory.
Two different settings can describe the same homotopy theory: for example, spaces built from simplices and general topological spaces.
A Quillen equivalence is a pair of functors back and forth that preserves the model structures and becomes an equivalence once weak equivalences are inverted.
Through it, one can switch to whichever setting is most convenient for a problem, without losing homotopical information.
</document>

<document source="https://tamnd.github.io/math2/node/KZDMD3SDJA.md" tag="KZDMD3SDJA" kind="definition">
# Anderson–Darling test

## Definition

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with [[empirical distribution function|A05WR6TEK5]] $F_n$.
Let $F_0$ be a fixed continuous [[cumulative distribution function|0MXMS3AE13]].
The *Anderson–Darling statistic* is
$$A^2 = n \int_{-\infty}^{\infty} \frac{\big(F_n(x) - F_0(x)\big)^2}{F_0(x)\big(1 - F_0(x)\big)} \, dF_0(x).$$
The *Anderson–Darling test* of $H_0 \colon$ "the $X_i$ have distribution function $F_0$" rejects $H_0$ when $A^2$ is large.
The critical value comes from the null law of $A^2$.

## Remarks

With the [[order statistics|58NHGZDSRE]] $X_{(1)} \le \dots \le X_{(n)}$ and $U_i = F_0(X_{(i)})$,
$$A^2 = -n - \frac{1}{n} \sum_{i=1}^{n} (2i - 1) \big( \ln U_i + \ln(1 - U_{n+1-i}) \big).$$
Under $H_0$ the law of $A^2$ does not depend on $F_0$.
When parameters of $F_0$ are estimated from the data, for example in a test of normality, other tables of critical values are used.

Example: $n = 1$, $X_1 = 1/2$ and $F_0(x) = x$ on $[0, 1]$.
The integral is $\int_0^{1/2} \frac{x}{1 - x} \, dx + \int_{1/2}^{1} \frac{1 - x}{x} \, dx = 2\big(\ln 2 - \tfrac12\big) = 2\ln 2 - 1$.
The formula gives $-1 - 2\ln(1/2) = 2 \ln 2 - 1 \approx 0.386$, the same value.

Non-example: the [[Cramér–von Mises statistic|G14XJXPKFS]] $n \int (F_n - F_0)^2 \, dF_0$ lacks the weight $1/(F_0(1 - F_0))$, so it is a different statistic that gives less weight to the tails.
</document>

<document source="https://tamnd.github.io/math2/node/KZHNA19FSS.md" tag="KZHNA19FSS" kind="statement">
# Borel's lemma

## Statement

Let $(a_n)_{n \ge 0}$ be any sequence of real numbers.
Then there is a [[smooth function|0BE3QDH73A]] $f \colon \R \to \R$ such that $f^{(n)}(0) = a_n$ for every $n \ge 0$.

## Remarks

Idea of the proof: choose a [[bump function|HS65DW0HXH]] $\chi$ equal to $1$ on $[-1/2, 1/2]$ and $0$ outside $[-1, 1]$.
Set $f(x) = \sum_{n \ge 0} \frac{a_n}{n!} x^n \chi(\lambda_n x)$ with numbers $\lambda_n \ge 1$ that grow fast.
The $n$-th term agrees with $\frac{a_n}{n!} x^n$ near $0$.
For $k < n$, its $k$-th derivative has supremum at most $2^{-n}$ when $\lambda_n$ is large enough.
Then every series of derivatives converges uniformly, so [[term-by-term differentiation|0CS42KEM5C]] gives $f^{(n)}(0) = a_n$.

The construction also gives $f$ with compact support.
Example: for $a_n = (n!)^2$, the [[Taylor series|0225QPWRRE]] of $f$ at $0$ is $\sum n! \, x^n$, which converges only at $x = 0$.

## Explanations

### L1 human

Any list of numbers can be the list of derivatives at $0$ of some smooth function.
There is no growth condition, unlike for analytic functions, whose derivatives at a point can grow at most like $C R^n n!$.
So every formal power series is the Taylor series of some smooth function.

### L2 human

The lemma says that the map sending a smooth function to its Taylor series at $0$, a [[formal power series|072W67KFYK]], is onto.
It is far from one-to-one: the [[smooth non-analytic function|09RTYYHXKA]] equal to $e^{-1/x}$ for $x > 0$ and to $0$ for $x \le 0$ has zero Taylor series at $0$.
Together these facts show how loosely the Taylor series controls a smooth function.
The lemma has versions in several variables and with parameters, and it is used in the study of singularities and of partial differential equations.
</document>

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# Harmonic map

## Summary

A harmonic map is a smooth map between [[Riemannian manifolds|7CBT4SM83Y]] that is a critical point of the energy.
The energy is the integral of the squared size of the differential.
Harmonic maps include [[harmonic functions|0DEB0JMN0W]], [[geodesics|NSYMQMHPDN]], isometries and holomorphic maps between Kähler manifolds.
They satisfy a nonlinear elliptic system and are a central object of geometric analysis.

## Explanations

### L1 human

Think of a rubber sheet $M$ stretched over a curved surface $N$.
The energy measures how much the sheet is stretched.
A harmonic map is a position of the sheet in equilibrium, where small changes do not change the stretching to first order.
A map from a circle that runs along a great circle of a sphere at constant speed is a harmonic map.
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# Initial value theorem

## Statement

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$.
Assume that the Laplace integral of $f$ converges at some real number $s_0$, and let $F$ be the [[Laplace transform|05XT5AVJBK]] of $f$.
Assume that the one-sided [[limit|0TXTSEMHR5]] $f(0^+) = \lim_{t \to 0^+} f(t)$ exists and is finite.
Then
$$\lim_{s \to +\infty} s F(s) = f(0^+),$$
where $s$ tends to $+\infty$ through real values.

## Remarks

Since $s \int_0^\infty e^{-s t} \, dt = 1$, the difference $s F(s) - f(0^+)$ is $s \int_0^\infty e^{-s t} (f(t) - f(0^+)) \, dt$.
The weight $s e^{-s t}$ has total mass $1$ and concentrates near $t = 0$ as $s \to \infty$.
On $[0, \delta]$ the integrand is small by the existence of $f(0^+)$; on $[\delta, \infty)$ the contribution tends to $0$ by the convergence at $s_0$.
Example: for $f(t) = \cos t$, $s F(s) = s^2 / (s^2 + 1) \to 1 = f(0)$.
The finiteness of $f(0^+)$ matters: for $f(t) = t^{-1/2}$ one has $F(s) = \sqrt{\pi / s}$, so $s F(s) = \sqrt{\pi s} \to +\infty$.

## Explanations

### L1 human

The behaviour of the transform for very large $s$ tells the starting value of the function.
One can read $f(0^+)$ from $F$ without inverting the transform.

### L2 human

In control and circuit theory one uses it to check the value of a response just after a switch is closed.
It is the simplest Abelian theorem for the Laplace transform: behaviour of $f$ near $0$ gives behaviour of $F$ near $\infty$.
Its companion is the [[final value theorem|BDHADVMJ8G]], which links $f$ near $\infty$ with $F$ near $0$.
</document>

<document source="https://tamnd.github.io/math2/node/KZSB5E471K.md" tag="KZSB5E471K" kind="concept">
# Computable function

## Summary

A computable function is a function on the natural numbers whose values can be calculated by an algorithm.
Formally, it is a partial function computed by a [[Turing machine|758EYCN27B]], and this class equals the class of partial [[general recursive functions|HBK5QHC1GC]].
Computable functions are the central objects of [[computability theory|X14Y880PN5]].
Some functions, such as the one that tells whether a program halts, are well defined but not computable.

## Explanations

### L1 human

A function is computable if there is a mechanical procedure that, given an input, eventually produces the output.
Addition, multiplication and the test "is $n$ prime?" are computable: there are clear step-by-step recipes for them.
A partial computable function may have no value at some inputs, because the procedure runs forever there.
The surprise of the 1930s was that some precisely defined functions have no such procedure at all.
</document>

<document source="https://tamnd.github.io/math2/node/KZTFH832H0.md" tag="KZTFH832H0" kind="statement">
# Stark–Heegner theorem

## Statement

Let $d$ be a positive squarefree integer.
The imaginary [[quadratic field|7SV4WA92SM]] $\Q(\sqrt{-d})$ has [[class number|KTKK9B1X41]] $1$ if and only if
$$d \in \set{1, 2, 3, 7, 11, 19, 43, 67, 163}.$$

## Remarks

Equivalently, the fundamental discriminants $D < 0$ with class number $1$ are $-3, -4, -7, -8, -11, -19, -43, -67, -163$.
Class number $1$ means that the ring of integers is a principal ideal domain, or, equivalently for rings of integers, a [[unique factorisation domain|0W4X9G6263]].
Gauss's work on [[binary quadratic forms|SBDSYA037Q]] led him to expect this list; the theorem settles the case $h = 1$ of the [[class number problem|KTGAS9DGRX]].
Heegner gave a proof using modular functions and [[complex multiplication|D3HEJTVY7G]], which was at first not accepted.
Baker, with linear forms in logarithms, and Stark, with a method close to Heegner's, later gave complete proofs.

Example: $\Q(\sqrt{-163})$ has class number $1$.
A related fact is that $n^2 + n + 41$ is prime for $n = 0, 1, \ldots, 39$; this works because $163 = 4 \cdot 41 - 1$.
By contrast $\Q(\sqrt{-5})$ has class number $2$, as $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ shows.

## Explanations

### L1 human

In most rings of algebraic integers, numbers can factor into irreducibles in several different ways.
For imaginary quadratic fields, the theorem gives the complete list of the nine fields where factorisation is unique.
The largest one comes from $\sqrt{-163}$.

### L2 human

The main difficulty is to rule out one possible extra field with a huge discriminant.
Lower bounds for class numbers of the form $h(-d) \to \infty$ were known, but they were not effective, so they could not exclude a single exception.
The theorem is closely related to the near-integer $e^{\pi \sqrt{163}}$, which comes from the $j$-invariant at a point of complex multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/M00S55MA1T.md" tag="M00S55MA1T" kind="definition">
# Analytical hierarchy

## Definition

Work in the language of [[second-order arithmetic|DC60FYCSPM]], which has number variables and set variables.
It is interpreted in the standard model, where set variables range over all subsets of $\N$.
A formula is *arithmetical* if it has no quantifiers over sets; it may have free set variables.
The classes of formulas $\Sigma^1_n$ and $\Pi^1_n$ are defined by induction on $n$.

1. $\Sigma^1_0 = \Pi^1_0$ is the class of arithmetical formulas.
2. A formula is $\Sigma^1_{n+1}$ if it has the form $\exists X_1 \cdots \exists X_k \, \psi$ with $\psi$ in $\Pi^1_n$.
3. A formula is $\Pi^1_{n+1}$ if it has the form $\forall X_1 \cdots \forall X_k \, \psi$ with $\psi$ in $\Sigma^1_n$.

A set $A \subseteq \N$ is $\Sigma^1_n$, or $\Pi^1_n$, if a formula of that class defines it in the standard model.
The formula must have one free number variable and no other free variables.
A set is $\Delta^1_n$ if it is both $\Sigma^1_n$ and $\Pi^1_n$, and *analytical* if it is $\Sigma^1_n$ for some $n$.

## Remarks

Example: let $W$ be the set of $e$ such that program $e$ computes the characteristic function of a well-ordering of a subset of $\N$.
"Every nonempty set $X$ in the field of the ordering has a least element" is a universal set quantifier over an arithmetical formula.
So $W$ is $\Pi^1_1$; in fact it is $\Pi^1_1$-complete, so it is not $\Sigma^1_1$.

Non-example: the set of Gödel numbers of second-order sentences true in the standard model is not analytical, by Tarski's undefinability argument.

Every [[arithmetical|7CVJCQKY41]] set is $\Delta^1_1$, and the inclusion is strict.
By a theorem of Kleene, the $\Delta^1_1$ sets are exactly the [[hyperarithmetical sets|NS1HWGTD8E]].
</document>

<document source="https://tamnd.github.io/math2/node/M03K2ZWNST.md" tag="M03K2ZWNST" kind="definition">
# Pluricomplex Green function

## Definition

Let $\Omega \subseteq \C^n$ be a domain and let $w \in \Omega$.
Let $\mathcal{F}_w$ be the set of [[plurisubharmonic functions|FF2P7ER3QT]] $u$ on $\Omega$ such that:

1. $u < 0$ on $\Omega$;
2. there are a neighbourhood $U$ of $w$ and a constant $C$ with $u(\zeta) \le \log\abs{\zeta - w} + C$ for $\zeta \in U$.

The *pluricomplex Green function* of $\Omega$ with pole at $w$ is the function $g_\Omega(\cdot, w) \colon \Omega \to [-\infty, 0]$ given by
$$g_\Omega(z, w) = \sup\set{u(z) : u \in \mathcal{F}_w} .$$

## Remarks

Example: for the unit ball $\Omega = \set{\abs{z} < 1}$ and $w = 0$, $g_\Omega(z, 0) = \log\abs{z}$.
Check: $\log\abs{z}$ belongs to $\mathcal{F}_0$.
Conversely, let $u \in \mathcal{F}_0$ and let $\xi$ be a unit vector.
The function $\lambda \mapsto u(\lambda \xi) - \log\abs{\lambda}$ is subharmonic on $0 < \abs{\lambda} < 1$ and bounded above near $0$, so it extends across $0$.
Its boundary values are at most $0$, so $u(\lambda \xi) \le \log\abs{\lambda}$ by the maximum principle.
Non-example: $u(z) = \abs{z}^2 - 1$ is negative and plurisubharmonic on the ball but is bounded near $0$, so it fails clause 2.
For $n = 1$ and a bounded regular domain, $g_\Omega(z, w)$ is the classical Green function with the sign convention $g_\Omega(z, w) \sim \log\abs{z - w}$ near $w$.
Holomorphic maps decrease it: if $f \colon \Omega_1 \to \Omega_2$ is holomorphic, then $g_{\Omega_2}(f(z), f(w)) \le g_{\Omega_1}(z, w)$.
If $\Omega$ is bounded, then $g_\Omega(\cdot, w)$ is plurisubharmonic and belongs to $\mathcal{F}_w$.
</document>

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# Perron integral

## Summary

The Perron integral squeezes the integral of a function between the increments of major and minor functions.
The lower derivatives of a major function lie above the function, and the upper derivatives of a minor function lie below it.
It integrates every derivative, and it gives the same integrable functions and the same values as the [[Henstock–Kurzweil integral|FCGHXDGVV9]].
It extends the [[Lebesgue integral|DWJF7ABPR1]].

## Explanations

### L1 human

The Darboux integral traps the area between step functions lying above and below the graph.
The Perron integral traps the integral instead between functions whose slopes lie above and below the function.
A function $U$ whose slope is everywhere at least $f$ should grow at least as much as the integral of $f$.
A function $V$ whose slope is at most $f$ should grow at most as much.
If these bounds meet, the common value is the Perron integral.
</document>

<document source="https://tamnd.github.io/math2/node/M092V97ZET.md" tag="M092V97ZET" kind="definition">
# Spearman's rank correlation coefficient

## Definition

Let $(x_1, y_1), \dots, (x_n, y_n)$ be $n \ge 2$ pairs of real numbers, with $x_1, \dots, x_n$ distinct and $y_1, \dots, y_n$ distinct.
Let $r_i$ be the rank of $x_i$ among $x_1, \dots, x_n$ and $s_i$ the rank of $y_i$ among $y_1, \dots, y_n$, with rank $1$ for the smallest.
*Spearman's rank correlation coefficient* is the sample [[correlation coefficient|04CTY6XYWN]] of the pairs $(r_i, s_i)$.
With $d_i = r_i - s_i$ it equals
$$\rho_s = 1 - \frac{6 \sum_{i=1}^{n} d_i^2}{n(n^2 - 1)}.$$

## Remarks

The closed formula holds because the ranks of each variable are the numbers $1, \dots, n$, whose mean and variance are known.
For random variables $X$ and $Y$ with continuous [[distribution functions|0MXMS3AE13]] $F$ and $G$, the population version is the correlation of $F(X)$ and $G(Y)$, which equals $12 \, E[F(X) G(Y)] - 3$.

Example: $x = (1, 2, 3)$ and $y = (1, 3, 2)$.
The ranks are $r = (1, 2, 3)$ and $s = (1, 3, 2)$, so $d = (0, -1, 1)$ and $\rho_s = 1 - 6 \cdot 2 / (3 \cdot 8) = 1/2$.
For $y = (1, 4, 9)$ instead, $d = 0$ and $\rho_s = 1$.

Non-example: for data with ties, such as $y = (1, 1, 2)$, the ranks are not defined by the clause above.
The usual fix gives tied values the average of their ranks and takes the Pearson correlation; the formula with $\sum d_i^2$ is then no longer exact.

The [[Kendall rank correlation coefficient|DYFY16SFHZ]] is another rank correlation, based on pairs of observations.
</document>

<document source="https://tamnd.github.io/math2/node/M0B618E2P6.md" tag="M0B618E2P6" kind="problem">
# Homicidal chauffeur problem

## Problem

Two players move in the plane.
The pursuer $P$ is a car: it moves with constant speed $w_1 > 0$ and has minimum turning radius $R > 0$.
Its position $(x_1, y_1)$ and heading $\theta$ obey
$$x_1' = w_1 \sin \theta, \qquad y_1' = w_1 \cos \theta, \qquad \theta' = \frac{w_1}{R} \phi, \qquad \phi(t) \in [-1, 1].$$
The evader $E$ moves with speed at most $w_2$, where $0 < w_2 < w_1$, in any direction: its position $(x_2, y_2)$ obeys $x_2' = w_2 \sin \psi$, $y_2' = w_2 \cos \psi$ with $\psi(t)$ free.
Let $\ell > 0$ be the capture radius; capture happens at the first time the distance between $P$ and $E$ is at most $\ell$.
The problem is a [[differential game|THSHGNJNPS]] between $P$ and $E$.
It asks for which parameters $w_1, w_2, R, \ell$ and initial positions the pursuer can force capture in finite time against every play of the evader.
It also asks for the optimal strategies when the pursuer minimises and the evader maximises the time to capture.

## Remarks

Isaacs used the problem as a model example in his theory of differential games.
By using coordinates attached to the car, the game reduces to a two-dimensional state, the position of the evader relative to the pursuer.
The evader is slower but more agile, so the car cannot simply drive at it: when it comes close, the evader can step aside inside the turning circle.
Isaacs found an explicit condition on the ratios $w_2 / w_1$ and $\ell / R$ under which capture is possible from every initial position.
The optimal strategies of the time-to-capture game turned out to involve several kinds of singular curves in the reduced state plane.

## Explanations

### L1 human

A driver in a car tries to run over a pedestrian in a large empty parking lot.
The car is faster, but it cannot turn sharply, while the pedestrian can change direction at will.
The question is when the driver can be sure to succeed, and how both should move.
It is a classical test problem for the theory of pursuit and evasion.
</document>

<document source="https://tamnd.github.io/math2/node/M0G4AASSG7.md" tag="M0G4AASSG7" kind="statement">
# Viète's formula

## Statement

Let $a_1 = \sqrt{2}$ and $a_{n+1} = \sqrt{2 + a_n}$ for $n \ge 1$.
Then the infinite product $\prod_{n=1}^{\infty} a_n / 2$ [[converges|H1Z2TKY3EQ]], and
$$\prod_{n=1}^{\infty} \frac{a_n}{2} = \frac{\sqrt2}{2} \cdot \frac{\sqrt{2 + \sqrt2}}{2} \cdot \frac{\sqrt{2 + \sqrt{2 + \sqrt2}}}{2} \cdots = \frac{2}{\pi},$$
where $\pi$ is the number [[pi|0C1VYVNV2N]].

## Remarks

By induction, $a_n = 2 \cos(\pi / 2^{n+1})$, using $2\cos(\theta/2) = \sqrt{2 + 2\cos\theta}$ for $0 \le \theta \le \pi$.
The double-angle formula $\sin 2t = 2 \sin t \cos t$, applied $N$ times, gives
$$\sin x = 2^N \sin(x / 2^N) \prod_{n=1}^{N} \cos(x / 2^n).$$
For $x = \pi/2$ this says $\prod_{n=1}^{N} \cos(\pi / 2^{n+1}) = \frac{1}{2^N \sin(\pi / 2^{N+1})}$.
Since $2^{N+1} \sin(\pi / 2^{N+1}) \to \pi$, the right side tends to $2/\pi$.
The formula appeared in work of François Viète published in 1593.
It converges fast: each factor adds about $0.6$ correct decimal digits of $\pi$.

## Explanations

### L1 human

Viète's formula writes $2/\pi$ as an infinite product of nested square roots of $2$.
It comes from inscribing regular polygons with $4, 8, 16, \ldots$ sides in a circle and comparing their areas.
As the number of sides doubles, the polygons fill the circle, and the ratio of areas gives the next factor.

### L2 human

The proof uses only the [[sine and cosine|08WFZP6P89]] double-angle formula and the limit $\sin(t)/t \to 1$.
The same computation gives the product $\frac{\sin x}{x} = \prod_{n \ge 1} \cos(x / 2^n)$ for every $x \ne 0$.
Viète's formula is one of the first infinite products in mathematics, and the [[Wallis product|0XCYA4YERS]] is another classical product for $\pi$.
</document>

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# Backward differentiation formula

## Definition

Let $f \colon \R \times \R^n \to \R^n$, and consider the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$, with step size $h > 0$ and $t_j = t_0 + j h$.
Let $k \ge 1$, and suppose that $y_{j-k+1}, \ldots, y_j$ are known.
Let $P$ be the [[interpolation polynomial|SS1YMS2NPC]] of degree at most $k$ with $P(t_{j+1-i}) = y_{j+1-i}$ for $i = 0, \ldots, k$.
The *$k$-step backward differentiation formula* determines $y_{j+1}$ by the condition
$$P'(t_{j+1}) = f(t_{j+1}, y_{j+1}).$$
With the [[backward difference operator|MK5JGVDPS9]] $\nabla y_{j+1} = y_{j+1} - y_j$, this condition reads
$$\sum_{i=1}^{k} \frac{1}{i} \nabla^i y_{j+1} = h f(t_{j+1}, y_{j+1}).$$

## Remarks

It is an implicit [[linear multistep method|BX4D85A79P]] in which only $\beta_k \ne 0$.
Example: for $k = 1$ the formula is $y_{j+1} - y_j = h f(t_{j+1}, y_{j+1})$, the backward Euler method.
For $k = 2$, $\nabla y_{j+1} + \frac12 \nabla^2 y_{j+1} = \frac32 y_{j+1} - 2 y_j + \frac12 y_{j-1}$, so the formula is $y_{j+1} - \frac43 y_j + \frac13 y_{j-1} = \frac23 h f(t_{j+1}, y_{j+1})$.
Non-example: the Adams–Bashforth formula $y_{j+1} - y_j = h \left( \frac32 f_j - \frac12 f_{j-1} \right)$ is a linear multistep method but not a BDF, since it uses $f$ at old points and not at $t_{j+1}$.
The $k$-step formula has order $k$; it is zero-stable for $1 \le k \le 6$ and not zero-stable for $k \ge 7$; the formulas with $k = 1, 2$ are A-stable.
</document>

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# Widder's theorem

## Statement

Let $n \ge 1$ and $T > 0$.
Let $u \colon \R^n \times [0, T) \to \R$ be continuous, with $\partial_t u$ and the second partial derivatives in $x$ continuous on $\R^n \times (0, T)$.
Assume that $u$ solves the [[heat equation|VP2HN86VAG]] $\partial_t u = \Delta u$ on $\R^n \times (0, T)$ and that $u \ge 0$.
Let $g(x) = u(x, 0)$.
Then for all $x \in \R^n$ and $0 < t < T$,
$$u(x, t) = \int_{\R^n} \Phi(x - y, t) \, g(y) \, dy,$$
where $\Phi$ is the [[heat kernel|KD404MF042]] of $\R^n$.

## Remarks

In particular, a nonnegative solution with initial value $0$ is identically $0$, so nonnegative solutions of the Cauchy problem are unique.
No growth condition at infinity is needed, which is the point of the theorem.
The sign condition cannot be dropped: [[Tychonoff's example|RN4J24CD83]] is a nonzero solution with initial value $0$, and it changes sign.
One half of the proof is a comparison on large balls with the maximum principle, which gives $u \ge \int \Phi(x - y, t) g(y) \, dy$.
The main work is to show that the difference, a nonnegative solution with initial value $0$, vanishes.
David Widder proved the theorem for $n = 1$; the version stated holds in every dimension.

## Explanations

### L1 human

Temperatures measured from absolute zero are never negative.
Widder's theorem says that for such physical temperatures on all of space, the initial temperature determines the future completely.
The future is then given by the usual formula, which spreads the initial heat with Gaussian bells.

### L2 human

Without a sign condition, uniqueness for the Cauchy problem needs a growth bound such as $\abs{u} \le A e^{a \abs{x}^2}$.
Widder's theorem replaces this bound by positivity, which is often easier to check in applications, for example for probability densities.
It is related to the parabolic Harnack inequality for nonnegative solutions.
</document>

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# Primitive ideal

## Summary

A left primitive ideal of a ring is the annihilator of a simple left module.
Equivalently, it is a two-sided ideal $P$ such that $R/P$ is a [[primitive ring|8F6EHDVJ62]].
The [[Jacobson radical|XA0DCNH7WY]] of a ring is the intersection of its primitive ideals.
Every primitive ideal is [[prime|E8K2MN5V1E]], and in a commutative ring the primitive ideals are the maximal ideals.

## Explanations

### L1 human

Each simple module of a ring is killed by some elements of the ring.
The set of these elements is a primitive ideal.
In $\Z$ the simple modules are $\Z/p\Z$, and the primitive ideals are the ideals $p\Z$ for primes $p$.
Primitive ideals play for noncommutative rings the role that maximal ideals play for commutative rings.
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# Singular homology

## Summary

Singular homology assigns to every topological space $X$ a sequence of abelian groups $H_n(X)$, built from all continuous maps of simplices into $X$.
It is defined for every space without any choice, it is functorial, and it is invariant under homotopy equivalence.
It satisfies the Eilenberg–Steenrod axioms, and on CW complexes and simplicial complexes it agrees with cellular and simplicial homology, which makes it computable.

## Explanations

### L1 human

Singular homology looks at all the ways of placing points, paths, triangles and higher simplices into a space, however crumpled.
It forms formal sums of them and keeps those sums that have no boundary, modulo those that bound something bigger.
What remains counts holes: $H_0$ counts path components, $H_1$ detects loops that do not bound, and so on.
For example, the circle has $H_0 \cong \Z$, $H_1 \cong \Z$ and all higher groups zero.
</document>

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# Oka manifold

## Definition

Let $Y$ be a [[complex manifold|JSGWAN2RJV]] with a distance function $d$ that induces its topology.
The manifold $Y$ has the *convex approximation property* if the following holds for every $n \ge 1$.
Let $K \subseteq \C^n$ be a [[compact|0R6EWFKRCF]] convex set, let $U$ be an open neighbourhood of $K$, and let $f \colon U \to Y$ be a [[holomorphic map|3MQTWKBBBT]].
Then for every $\varepsilon > 0$ there is a holomorphic map $F \colon \C^n \to Y$ with $d(F(z), f(z)) < \varepsilon$ for all $z \in K$.

The manifold $Y$ is an *Oka manifold* if it has the convex approximation property.

## Remarks

The property does not depend on the choice of $d$, since any two such distances are uniformly equivalent on a compact neighbourhood of the compact set $f(K)$.
Example: $Y = \C^m$ is Oka.
A compact convex set is polynomially convex.
So by the [[Oka–Weil theorem|7AWZJNQGT2]] each component of $f$ is a uniform limit on $K$ of polynomials, and polynomial maps $F$ approximate $f$ on $K$.
Non-example: the unit disc $D \subseteq \C$ is not Oka.
Take $n = 1$, $K = \set{\abs{z} \le 1/2}$ and $f(z) = z$; any holomorphic $F \colon \C \to D$ is bounded, hence constant by Liouville's theorem, and a constant cannot be closer than $1/2$ to $z$ at both $z = 1/2$ and $z = -1/2$.
Forstnerič proved that the convex approximation property is equivalent to the full Oka principle for maps from Stein manifolds to $Y$, with approximation and interpolation; this justifies the definition.
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# Grigorchuk group

## Description

Let $\set{0, 1}^*$ be the set of finite words in the letters $0$ and $1$, the vertices of the infinite rooted binary [[tree|062D9KXJB1]].
Define four maps $a, b, c, d$ of $\set{0, 1}^*$ to itself by recursion on the length of a word $w$.
Each map sends the empty word to the empty word, and:
$$a(0w) = 1w, \quad a(1w) = 0w,$$
$$b(0w) = 0\,a(w), \quad b(1w) = 1\,c(w),$$
$$c(0w) = 0\,a(w), \quad c(1w) = 1\,d(w),$$
$$d(0w) = 0w, \quad d(1w) = 1\,b(w).$$
Each map is a bijection that preserves the length of words and the tree structure.
The *Grigorchuk group* $\Gamma$ is the [[group|G5S1M9]] generated by $a, b, c, d$ under composition.

## Remarks

Each generator has order $2$, and $\set{1, b, c, d}$ is a group isomorphic to $\Z/2 \times \Z/2$ with $bc = d$.
$\Gamma$ is infinite, finitely generated, and every element has order a power of $2$, so it answers the general [[Burnside problem|XWEMJ7T2XV]] in the negative.
It is [[residually finite|HEE922EHQ8]]: the action on the words of length $n$ gives finite quotients that separate all elements.
It has [[growth|4CMRM3975W]] faster than every polynomial and slower than every exponential; it was the first group known with this intermediate growth.
It is [[amenable|CEAT9BESKK]] and not finitely presented.

## Explanations

### L1 human

The Grigorchuk group is a group of symmetries of an infinite binary tree, generated by four explicit moves.
The move $a$ swaps the two halves of the tree.
The moves $b$, $c$ and $d$ act on the two halves separately, each time by $a$ or by another of $b$, $c$, $d$, in a repeating pattern.
From these simple rules comes a group with properties that were long thought hard to find together.
It is infinite, every element repeated enough times gives the identity, and its size grows at an unusual speed.
</document>

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# Analytic signal

## Definition

Let $f \in L^2(\R)$ be real-valued, an element of the [[$L^2$ space|SASP14MA1Z]].
Let $Hf$ be its [[Hilbert transform|NSCEBXM7ZM]].
The *analytic signal* of $f$ is the complex-valued function
$$f_a = f + i\, Hf.$$

## Remarks

With the [[Fourier transform|DPVM34ADZV]] $\hat f(\xi) = \int_\R f(x) e^{-2\pi i x \xi}\, dx$, one has $\widehat{Hf}(\xi) = -i \sgn(\xi) \hat f(\xi)$, so
$$\widehat{f_a}(\xi) = \big(1 + \sgn(\xi)\big) \hat f(\xi),$$
which is $2 \hat f(\xi)$ for $\xi > 0$ and $0$ for $\xi < 0$.
Since $f$ is real, $f = \operatorname{Re} f_a$, and $f_a$ is determined by $f$.
One writes $f_a(t) = A(t) e^{i\phi(t)}$ with $A = \abs{f_a}$ the instantaneous amplitude and $\phi$ the instantaneous phase.

Example: $f(x) = \frac{1}{1 + x^2}$.
Its Hilbert transform is $\frac{x}{1 + x^2}$, so $f_a(x) = \frac{1 + ix}{1 + x^2} = \frac{1}{1 - ix} = \frac{i}{x + i}$.
The function $z \mapsto \frac{i}{z + i}$ is holomorphic on the upper half-plane, as the definition predicts.

Non-example: the clause "real-valued" matters.
For the complex function $g = \frac{i}{x + i}$ one has $Hg = -i g$, so $g + i Hg = 2g \ne g$, and the real part of $g + iHg$ is $2 \operatorname{Re} g$, not $\operatorname{Re} g$.
</document>

<document source="https://tamnd.github.io/math2/node/M13HAQSHEM.md" tag="M13HAQSHEM" kind="statement">
# Lie group–Lie algebra correspondence

## Statement

Let $G$ and $H$ be real [[Lie groups|7JBRGCVJRB]] with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$, and [[exponential maps|H3QX1ZFC8K]] $\exp$.

1. For every Lie group homomorphism $\varphi \colon G \to H$, the differential $d\varphi_e \colon \mathfrak{g} \to \mathfrak{h}$ is a Lie algebra homomorphism, and $\varphi(\exp X) = \exp(d\varphi_e X)$ for all $X \in \mathfrak{g}$.
2. If $G$ is connected, a Lie group homomorphism $\varphi \colon G \to H$ is determined by $d\varphi_e$.
3. If $G$ is connected and simply connected, each Lie algebra homomorphism $\psi \colon \mathfrak{g} \to \mathfrak{h}$ is $d\varphi_e$ for a unique homomorphism $\varphi \colon G \to H$.
4. The map $K \mapsto \operatorname{Lie}(K)$ is a bijection from the connected [[Lie subgroups|NM8XCPS1Q3]] $K$ of $G$ onto the Lie subalgebras of $\mathfrak{g}$.
5. Every finite-dimensional real Lie algebra is the Lie algebra of a connected and simply connected Lie group, unique up to isomorphism.

Here simply connected means that the [[fundamental group|S1XK3HXHFX]] is trivial.

## Remarks

Part 1 follows from the chain rule, since $\varphi$ maps one-parameter subgroups to one-parameter subgroups.
Part 2 holds because a connected Lie group is generated by any neighbourhood of $e$, and $\exp$ covers a neighbourhood of $e$.
Part 3 is proved by integrating the graph of $\psi$, a subalgebra of $\mathfrak{g} \oplus \mathfrak{h}$, to a connected Lie subgroup of $G \times H$ by part 4.
Its projection to $G$ is a covering map, hence an isomorphism since $G$ is simply connected.
Part 4 rests on the Frobenius theorem applied to the left-invariant distribution spanned by a subalgebra.
Part 5 is [[Lie's third theorem|P6QNEK44RX]].
Simple connectivity in part 3 is necessary.
The identity map of $\R$ is the differential of no homomorphism from the circle group $\R / \Z$ to $\R$, since the only such continuous homomorphism is trivial.

## Explanations

### L1 human

Every Lie group has a Lie algebra, a linear object that records how the group looks near its identity.
This theorem says that, for connected groups, almost all information about homomorphisms and subgroups can be read off the Lie algebra.
For simply connected groups the dictionary is perfect: Lie groups and Lie algebras then carry the same information.

### L2 human

In practice one computes with Lie algebras and lifts the answers to groups.
For example, a representation of $\mathfrak{su}(2)$ gives a representation of the simply connected group $SU(2)$ by part 3, while for $SO(3)$ only some of them lift.
Parts 3 and 5 together say that $G \mapsto \mathfrak{g}$ is an equivalence of categories between connected simply connected real Lie groups and finite-dimensional real Lie algebras.
A connected Lie group with Lie algebra $\mathfrak{g}$ is the quotient of the simply connected one by a discrete central subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/M1ABX18FC4.md" tag="M1ABX18FC4" kind="statement">
# Jordan–Hölder theorem for modules

## Statement

Let $R$ be a [[ring|0894J980RK]] and let $M$ be a left [[module|09VDS2W1E8]] over $R$.
A simple module is a nonzero module with no submodules other than $0$ and itself.
A composition series of $M$ is a chain of [[submodules|0RGZ10DFPG]] $0 = M_0 \subsetneq M_1 \subsetneq \cdots \subsetneq M_n = M$ in which each quotient $M_i / M_{i-1}$ is simple.
Let $0 = M_0 \subsetneq \cdots \subsetneq M_n = M$ and $0 = N_0 \subsetneq \cdots \subsetneq N_m = M$ be two composition series of $M$.
Then $n = m$, and there is a permutation $\sigma$ of $\set{1, \dots, n}$ with $M_i / M_{i-1} \cong N_{\sigma(i)} / N_{\sigma(i) - 1}$ for each $i$.

## Remarks

The proof is by induction on the shortest length of a composition series, comparing the two series through the submodules $M_{n-1}$, $N_{m-1}$ and $M_{n-1} \cap N_{m-1}$.
So the number $n$ and the simple factors, counted with multiplicity, depend only on $M$; the number $n$ is the [[length|CZSCNZH8VY]] of $M$.
For example, $\Z/12\Z$ has the composition series $0 \subset 6\Z/12\Z \subset 3\Z/12\Z \subset \Z/12\Z$ and $0 \subset 6\Z/12\Z \subset 2\Z/12\Z \subset \Z/12\Z$, with factors $\Z/2\Z, \Z/2\Z, \Z/3\Z$ in different orders.
A module that has a composition series is exactly a module of finite length, and then every strict chain of submodules can be refined to a composition series.
The theorem for groups is the [[Jordan–Hölder theorem|0E088ZP533]]; both are cases of the Jordan–Hölder theorem for groups with operators, and the same proof works.

## Explanations

### L1 human

A module of finite length can be cut into simple layers in many ways.
The theorem says that the number of layers and the layers themselves are always the same, up to order.

### L2 human

The theorem makes the [[length of a module|CZSCNZH8VY]] well defined, and it lets one speak of the multiplicity of a simple module in $M$.
Lengths of modules such as $R/I$, for a Noetherian local ring $R$, are the input of Hilbert–Samuel functions and multiplicities.
In representation theory, the simple factors of a representation are its irreducible constituents, which are well defined by this theorem.
</document>

<document source="https://tamnd.github.io/math2/node/M1KYAQMF9W.md" tag="M1KYAQMF9W" kind="problem">
# Pompeiu problem

## Problem

Let $n \ge 2$ and let $K \subseteq \R^n$ be a bounded set with nonempty interior.
The set $K$ has the *Pompeiu property* if the only continuous function $f \colon \R^n \to \C$ with
$$\int_{\sigma(K)} f \, dx = 0 \quad \text{for every rigid motion } \sigma \text{ of } \R^n$$
is $f = 0$.
Here a rigid motion is a composition of a rotation and a translation, and the integral is against [[Lebesgue measure|2Z2QVWZ262]].
Determine which sets have the Pompeiu property; in particular, decide whether every bounded simply connected Lipschitz domain without the Pompeiu property is a ball.

## Remarks

A ball does not have the Pompeiu property.
For the unit ball in $\R^n$ and a suitable nonzero vector $a$, the function $f(x) = \sin(a \cdot x)$ has integral $0$ over every translate of the ball, and rotations map balls to balls.
Squares and other polygons are known to have the Pompeiu property.
By a theorem of Williams (1976), for smooth contractible domains the conjectured answer is equivalent to [[Schiffer's conjecture|J05VKG4XQB]] on an overdetermined Neumann eigenvalue problem.

## Explanations

### L1 human

Slide and rotate a fixed shape all over the plane, and suppose a function averages to zero on every copy of the shape.
Must the function be zero everywhere?
For a square the answer is yes.
For a disc the answer is no: a suitable wave $\sin(a x)$ averages to zero on every disc of the given radius.
Pompeiu's problem asks whether the disc is the only shape for which this happens.
</document>

<document source="https://tamnd.github.io/math2/node/M1VKCW3RFW.md" tag="M1VKCW3RFW" kind="concept">
# Dbar operator

## Summary

The $\bar\partial$ operator, read "d-bar", sends a [[$(p, q)$-form|DC81TCR98Y]] to a $(p, q+1)$-form by differentiating its coefficients in the conjugate variables $\bar z_1, \dots, \bar z_n$.
On functions it is $\bar\partial f = \sum_j \frac{\partial f}{\partial \bar z_j} \, d\bar z_j$, and $\bar\partial f = 0$ is exactly the system of [[Cauchy–Riemann equations in several variables|9PT772R94A]].
It satisfies $\bar\partial \circ \bar\partial = 0$ and splits the [[exterior derivative|0ZHSTAG8ZY]] as $d = \partial + \bar\partial$.
It generalises the one-variable [[Cauchy–Riemann operator|PPSSTHTEDE]], and it is the basic operator of Dolbeault cohomology and of the $L^2$ methods of several complex variables.

## Explanations

### L1 human

The exterior derivative $d$ measures how a function or form changes in all real directions.
Using complex coordinates, this change splits into a part along $z_1, \dots, z_n$ and a part along $\bar z_1, \dots, \bar z_n$.
The $\bar\partial$ operator keeps only the second part.
A function is holomorphic exactly when this part is zero.
For example, $\bar\partial(z_1 \bar z_2) = z_1 \, d\bar z_2$, which is not zero, so $z_1 \bar z_2$ is not holomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/M1WQTWFNNQ.md" tag="M1WQTWFNNQ" kind="statement">
# Cobordism hypothesis

## Statement

Let $n \ge 1$ and let $\mathcal{C}$ be a [[symmetric monoidal|5S7ER95XAC]] $(\infty, n)$-category.
Let $\mathrm{Bord}_n^{\mathrm{fr}}$ be the symmetric monoidal $(\infty, n)$-category of $n$-framed [[bordisms|T8BVGPPSZP]].
Its objects are $n$-framed $0$-manifolds.
For $1 \le k \le n$, its $k$-morphisms are $n$-framed $k$-dimensional bordisms between $(k-1)$-dimensional bordisms.
Its higher morphisms are diffeomorphisms and their isotopies.
The monoidal product is disjoint union.
Let $\mathrm{Fun}^{\otimes}(\mathrm{Bord}_n^{\mathrm{fr}}, \mathcal{C})$ be the $\infty$-groupoid of symmetric monoidal functors from $\mathrm{Bord}_n^{\mathrm{fr}}$ to $\mathcal{C}$.
Let $(\mathcal{C}^{\mathrm{fd}})^{\simeq}$ be the $\infty$-groupoid of fully [[dualizable objects|8A7YXMW727]] of $\mathcal{C}$.
Then evaluation at the point with its standard $n$-framing is an equivalence
$$\mathrm{Fun}^{\otimes}\left(\mathrm{Bord}_n^{\mathrm{fr}}, \mathcal{C}\right) \simeq \left(\mathcal{C}^{\mathrm{fd}}\right)^{\simeq}.$$

## Remarks

A symmetric monoidal functor $Z \colon \mathrm{Bord}_n^{\mathrm{fr}} \to \mathcal{C}$ is a fully extended framed [[topological quantum field theory|DAP92B80J2]].
The theorem says that such a theory is determined by its value $Z(\mathrm{pt})$ on a point, and that every fully dualizable object occurs.
The point is fully dualizable in the bordism category, because bending a bordism gives the duality data; this is why $Z(\mathrm{pt})$ must be fully dualizable.
The hypothesis cannot be dropped: an object of $\mathcal{C}$ that is not fully dualizable is not the value on a point of any such theory.
Baez and Dolan proposed the hypothesis in 1995 in the language of $n$-categories.
Lurie published a detailed outline of a proof in 2009 in the language of $(\infty, n)$-categories.
For $n = 1$ and $\mathcal{C}$ the category of vector spaces over a field, the fully dualizable objects are the finite-dimensional spaces $V$.
The theory with $Z(\mathrm{pt}) = V$ assigns to a circle the number $\dim V$, the trace of the identity of $V$.
There are versions for other tangential structures, such as orientations, in which framings are replaced by the structure and the object carries a homotopy fixed point structure.

## Explanations

### L1 human

A topological field theory assigns algebraic data to manifolds and to the ways of gluing them.
The cobordism hypothesis says that, in the most local version, all the data are decided by what the theory assigns to a single point.
Everything else is built by gluing points into lines, lines into surfaces, and so on.
The only condition on the data at the point is that it be finite in a strong algebraic sense, called full dualizability.

### L2 human

Classical two-dimensional field theories are classified by commutative Frobenius algebras, which uses the decomposition of surfaces into pairs of pants.
The cobordism hypothesis extends such classifications to all dimensions by requiring the theory to be defined down to points, using higher categories.
It ties [[higher category theory|3A1NMGJKJ3]] to manifold topology: the framed bordism $(\infty, n)$-category is the free symmetric monoidal $(\infty, n)$-category with duals on one object.
Its proof outline relies on Morse theory to decompose bordisms into elementary handles.
</document>

<document source="https://tamnd.github.io/math2/node/M218MWZ62X.md" tag="M218MWZ62X" kind="statement">
# Inverse function theorem in Banach spaces

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\mathbb{K}$.
Let $U \subseteq X$ be open, and let $f \colon U \to Y$ be of class $C^1$.
This means that $f$ has a [[Fréchet derivative|06EH97V2A7]] $Df(x)$ at each $x \in U$ and that $x \mapsto Df(x)$ is continuous for the [[operator norm|06EKW1AKXF]].
Let $a \in U$, and suppose that $Df(a) \colon X \to Y$ is bijective.
Then there are open sets $V \subseteq U$ with $a \in V$ and $W \subseteq Y$ with $f(a) \in W$ such that $f$ maps $V$ bijectively onto $W$.
The inverse $g \colon W \to V$ is of class $C^1$, and $Dg(f(x)) = Df(x)^{-1}$ for each $x \in V$.
If $f$ is of class $C^k$ with $k \ge 1$, then so is $g$.

## Remarks

By the [[bounded inverse theorem|MR5FJXR348]], the inverse of the bijective continuous linear map $Df(a)$ is continuous.
To solve $f(x) = y$ for $y$ near $f(a)$, one looks for a fixed point of $x \mapsto x - Df(a)^{-1}(f(x) - y)$.
Near $a$ this map is a contraction, because $Df(x)$ is close to $Df(a)$; the [[Banach fixed point theorem|0VK6VEH345]] gives existence and uniqueness.
Smoothness of the inverse follows from the formula for $Dg$ and the continuity of inversion of operators.

Example: on $X = Y = C([0,1])$ with the sup norm, let $f(u) = u + u^3$, computed pointwise.
Then $Df(0) = \id$, so each $v$ close to $0$ equals $u + u^3$ for exactly one $u$ close to $0$, and $u$ depends smoothly on $v$.

The hypothesis $C^1$ cannot be weakened to differentiability at each point.
On $\R$, the map $f(x) = x + 2x^2 \sin(1/x)$, $f(0) = 0$, has $f'(0) = 1$ but is not injective on any neighbourhood of $0$.
For $X = Y = \R^n$, the theorem is the [[inverse function theorem|0K9HSJFCMJ]] of several-variable calculus.

## Explanations

### L1 human

If the linear approximation of a smooth map at a point is invertible, the map itself is invertible near that point.
This holds even when the spaces are spaces of functions, as long as they are complete normed spaces.
So a nonlinear equation $f(x) = y$ can be solved for $y$ near $f(a)$ when its linearization at $a$ can be solved.

### L2 human

The theorem is the main tool of local nonlinear analysis.
It gives the [[implicit function theorem in Banach spaces|BTM6WJ0A52]], the local structure of Banach manifolds and the local existence for many nonlinear differential equations.
Its proof needs completeness, through the contraction principle.
In Fréchet spaces, such as $C^\infty$ function spaces, it fails in general, and the Nash–Moser theorem replaces it under extra hypotheses.
</document>

<document source="https://tamnd.github.io/math2/node/M24H4GJPPW.md" tag="M24H4GJPPW" kind="statement">
# Rogers–Ramanujan identities

## Statement

For an integer $n \ge 0$, let $(q; q)_n = (1 - q)(1 - q^2) \cdots (1 - q^n)$, the [[q-Pochhammer symbol|Y1X8ZZQ7VM]], with $(q; q)_0 = 1$.
In the ring of [[formal power series|033WGN00BF]] $\Z[\![q]\!]$, the following two identities hold.

1. $$\sum_{n=0}^{\infty} \frac{q^{n^2}}{(q; q)_n} = \prod_{n=0}^{\infty} \frac{1}{(1 - q^{5n+1})(1 - q^{5n+4})}.$$
2. $$\sum_{n=0}^{\infty} \frac{q^{n^2 + n}}{(q; q)_n} = \prod_{n=0}^{\infty} \frac{1}{(1 - q^{5n+2})(1 - q^{5n+3})}.$$

## Remarks

In terms of [[integer partitions|0JV30XN73Z]], the identities say the following for each $m \ge 0$.
The partitions of $m$ whose parts differ pairwise by at least $2$ are as many as the partitions of $m$ into parts congruent to $1$ or $4$ modulo $5$.
Consider the partitions of $m$ whose parts differ by at least $2$ and are all at least $2$.
They are as many as the partitions of $m$ into parts congruent to $2$ or $3$ modulo $5$.
The term $q^{n^2} / (q; q)_n$ counts the partitions of the first kind with exactly $n$ parts, since $n^2 = 1 + 3 + \cdots + (2n - 1)$ is the smallest such sum.

Example: for $m = 6$, the partitions with parts differing by at least $2$ are $6$, $5 + 1$ and $4 + 2$.
The partitions into parts from $\set{1, 4, 6, 9, \ldots}$ are $6$, $4 + 1 + 1$ and $1 + 1 + 1 + 1 + 1 + 1$.
Both counts are $3$.

A standard proof transforms the sum side and then applies the [[Jacobi triple product|JFP2DGMNXT]] to reach the product side.
The identities were found by Leonard James Rogers and later found again by Srinivasa Ramanujan.

## Explanations

### L1 human

Count the ways to write a number as a sum of parts that are far apart, each two parts differing by at least $2$.
Count also the ways to write it as a sum of parts that end in $1, 4, 6$ or $9$.
The theorem says that the two counts always agree.
Nothing in the two rules suggests the connection, which is why the identities are famous.

### L2 human

The identities are the prototype of partition identities of Rogers–Ramanujan type, where a difference condition on the parts matches a congruence condition.
Their sum sides are basic hypergeometric series and their product sides are, up to a power of $q$, quotients of theta functions.
They appear in the theory of modular forms, in the hard hexagon model of statistical mechanics, and in the representation theory of affine Lie algebras.
Compare with [[Euler's partition theorem|0XFZ1BMF5Q]], where parts are distinct on one side and odd on the other.
</document>

<document source="https://tamnd.github.io/math2/node/M28SB04806.md" tag="M28SB04806" kind="definition">
# Envelope of holomorphy

## Definition

Let $D \subseteq \C^n$ be a domain, that is, a nonempty connected open set.
A *holomorphic extension* of $D$ is a triple $(X, \pi, \alpha)$ with two properties.

1. $(X, \pi)$ is a [[Riemann domain over $\C^n$|A15BF6QHDR]] and $\alpha \colon D \to X$ is a [[holomorphic map|3MQTWKBBBT]] with $\pi \circ \alpha = \id_D$.
2. For every [[holomorphic function|507C34W0ZT]] $f$ on $D$ there is a holomorphic function $F$ on $X$ with $F \circ \alpha = f$.

An *envelope of holomorphy* of $D$ is a holomorphic extension $(E, \pi, \alpha)$ of $D$ with the following property.
For every holomorphic extension $(X, \rho, \beta)$ of $D$ there is a holomorphic map $\tau \colon X \to E$ with $\pi \circ \tau = \rho$ and $\tau \circ \beta = \alpha$.

## Remarks

The function $F$ in clause 2 is unique, because $X$ is connected and $\alpha(D)$ is a nonempty open subset of $X$.
Every domain $D$ has an envelope of holomorphy, and it is unique up to an isomorphism of Riemann domains compatible with the maps from $D$.
It can be built from the germs of holomorphic functions obtained from $D$ by analytic continuation.

Example: for the [[Hartogs figure|GHJC53RFBT]] $H$ in $\C^n$, the polydisc $\Delta^n$ with the inclusion map is an envelope of holomorphy.
Every holomorphic function on $H$ extends to $\Delta^n$; and $\Delta^n$ is a domain of holomorphy, which gives the maximality.

Non-example: for $D = \set{z \in \C^2 : 0 < \abs{z} < 1}$, the triple $(D, \id, \id)$ is a holomorphic extension but not an envelope.
The extension given by the ball $\set{\abs{z} < 1}$ admits no map $\tau$ into $D$ over $\C^2$, since $\tau$ would have to be the identity on the ball.

A domain $D$ is a domain of holomorphy exactly when $(D, \id, \id)$ is its envelope of holomorphy.
</document>

<document source="https://tamnd.github.io/math2/node/M2900DMEW6.md" tag="M2900DMEW6" kind="example">
# Indefinite orthogonal group

## Example

Let $p, q \ge 0$ be integers with $n = p + q \ge 1$, and let
$$\eta = \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}.$$
The matrix $\eta$ defines the [[quadratic form|0RBQS1JJCR]] $Q(x) = x^T \eta x = x_1^2 + \cdots + x_p^2 - x_{p+1}^2 - \cdots - x_n^2$ on $\R^n$, of [[signature|07EZA80NEH]] $(p, q)$.
The *indefinite orthogonal group* is
$$O(p, q) = \set{A \in GL_n(\R) : A^T \eta A = \eta},$$
the group of linear maps of $\R^n$ that preserve $Q$.
It is a closed subgroup of $GL_n(\R)$, so it is a [[Lie group|7JBRGCVJRB]].

Check with $p = q = 1$: for every $t \in \R$, the matrix
$$A_t = \begin{pmatrix} \cosh t & \sinh t \\ \sinh t & \cosh t \end{pmatrix}$$
satisfies $A_t^T \eta A_t = \begin{pmatrix} \cosh^2 t - \sinh^2 t & 0 \\ 0 & \sinh^2 t - \cosh^2 t \end{pmatrix} = \eta$, so $A_t \in O(1, 1)$.
Its entries are unbounded as $t \to \infty$, so $O(1, 1)$ is not compact.

## Remarks

For $q = 0$ the group $O(n, 0)$ is the compact [[orthogonal group|0TKYAEJ8N7]] $O(n)$.
The dimension of $O(p, q)$ is $n(n - 1)/2$, as for $O(n)$.
$O(p, q)$ and $O(q, p)$ are isomorphic, since replacing $\eta$ by $-\eta$ does not change the group.
When $p$ and $q$ are both positive, $O(p, q)$ is not compact and has four connected components.
Its subgroup $SO(p, q)$ of elements with determinant $1$ has two components, and the identity component is written $SO^+(p, q)$.
A maximal compact subgroup of $O(p, q)$ is $O(p) \times O(q)$.
The Lorentz group of special relativity is $O(1, 3)$.

## Explanations

### L1 human

The ordinary orthogonal group keeps the usual squared length $x_1^2 + \cdots + x_n^2$.
The indefinite orthogonal group keeps a "length" in which some squares count with a minus sign.
Then the circles of constant length become hyperbolas, and the rotations become hyperbolic rotations built from $\cosh$ and $\sinh$.
These maps can stretch vectors arbitrarily far along the hyperbolas, so the group is no longer compact.
</document>

<document source="https://tamnd.github.io/math2/node/M2F4N31JQ2.md" tag="M2F4N31JQ2" kind="concept">
# Dolbeault operator

## Summary

The Dolbeault operator $\bar\partial$ on a [[complex manifold|JSGWAN2RJV]] is a part of the [[exterior derivative|0ZHSTAG8ZY]].
It is the part that raises the antiholomorphic degree of a complex [[differential form|0E3G0GRPZ6]] by one.
A function is [[holomorphic|0NAHXR1FRH]] exactly when $\bar\partial f = 0$, so $\bar\partial$ is the Cauchy–Riemann operator of complex geometry.
It satisfies $\bar\partial^2 = 0$, and the cohomology of the resulting Dolbeault complex computes the sheaf cohomology of holomorphic forms.
The Dolbeault complex is an [[elliptic complex|M8GMH8GQR6]], and its index is the holomorphic Euler characteristic given by the Riemann–Roch theorems.

## Explanations

### L1 human

In one complex variable $z = x + iy$, a function is holomorphic when it satisfies the Cauchy–Riemann equations.
These can be written as one equation $\partial f / \partial \bar z = 0$, where $\partial / \partial \bar z = \frac{1}{2}(\partial_x + i \partial_y)$.
The Dolbeault operator packages this derivative as $\bar\partial f = \frac{\partial f}{\partial \bar z} \, d\bar z$, and extends it to forms and to several variables.
For example $\bar\partial z = 0$, since $z$ is holomorphic, while $\bar\partial \bar z = d\bar z \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/M2FAMD9HZN.md" tag="M2FAMD9HZN" kind="concept">
# Normalization

## Summary

The normalization of an [[integral scheme|17NVR4XBHK]] $X$ is a [[normal scheme|EK261G1XP9]] $\tilde{X}$ with a morphism $\tilde{X} \to X$.
It is built by taking the [[integral closure|G0V86SJBQW]] of each affine coordinate ring in its fraction field.
It is the best normal approximation of $X$: every dominant morphism from a normal integral scheme to $X$ factors through it.
For curves it resolves singularities, which makes it a basic tool in the study of singular curves.

## Explanations

### L1 human

Normalization repairs the mildest singularities of a space by adding the functions that "should" be there.
The cusp $y^2 = x^3$ is traced by $t \mapsto (t^2, t^3)$.
The function $t = y/x$ is missing on the cusp, and adding it gives back the ordinary line with coordinate $t$.
So the normalization of the cusp is the line, mapped onto the cusp by $t \mapsto (t^2, t^3)$.
A curve that crosses itself is pulled apart at the crossing in the same way.
</document>

<document source="https://tamnd.github.io/math2/node/M2FSQNBJCT.md" tag="M2FSQNBJCT" kind="definition">
# Heaviside cover-up method

## Definition

Let $P$ and $Q$ be complex polynomials with $\deg P < \deg Q$, and let $F = P/Q$ be the [[rational function|0WCZH4AFMT]].
Let $a$ be a [[root|00EHX6FWC0]] of $Q$ of multiplicity $m \ge 1$, and write $Q(s) = (s - a)^m R(s)$ with $R$ a polynomial and $R(a) \ne 0$.
In the [[partial fraction decomposition|030VFEREKG]] of $F$, let $c$ be the coefficient of $1/(s - a)^m$.
The *Heaviside cover-up method* computes this coefficient as
$$c = \frac{P(a)}{R(a)} = \Big[ (s - a)^m F(s) \Big]_{s = a}.$$
When $a$ is a simple root, this equals $P(a)/Q'(a)$.

## Remarks

The formula holds because $(s - a)^m F(s)$ equals $c$ plus terms that vanish at $s = a$.
Example: for $F(s) = \frac{1}{(s - 1)(s - 2)(s - 3)}$, cover-up gives the coefficients $\frac{1}{(1-2)(1-3)} = \tfrac12$, $\frac{1}{(2-1)(2-3)} = -1$ and $\frac{1}{(3-1)(3-2)} = \tfrac12$.
Its inverse Laplace transform is therefore $\tfrac12 e^{t} - e^{2t} + \tfrac12 e^{3t}$.
Non-example: for a repeated root the method gives only the top coefficient.
For $\frac{1}{s (s - 1)^2} = \frac{A}{s} + \frac{B}{s - 1} + \frac{C}{(s - 1)^2}$, cover-up gives $A = 1$ and $C = 1$.
The true value $B = -1$ does not come from covering up; one finds it by comparing coefficients or by differentiating $(s - 1)^2 F(s)$ at $s = 1$.
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# Variance reduction

## Definition

Let $q$ be a real number.
Let $Y$ be a random variable with finite [[variance|0477R9PYAE]] and [[expected value|0WQNRCNY3M]] $\mathbb E Y = q$, used in a plain [[Monte Carlo method|HWP11G3HTN]] for $q$.
A *variance reduction* for this method is a random variable $Z$ such that:

1. $\mathbb E Z = q$;
2. $\operatorname{Var} Z < \operatorname{Var} Y$;
3. independent copies of $Z$ can be generated.

The Monte Carlo method then uses the average $\frac1n \sum_{i=1}^n Z_i$ of independent copies of $Z$ instead of the average of copies of $Y$.

## Remarks

Example: $q = \int_0^1 e^u\, du = e - 1$ and $Y = e^U$ with $U$ uniform on $(0, 1)$, so $\operatorname{Var} Y = \frac{e^2 - 1}{2} - (e - 1)^2 \approx 0.2420$.
Take $Z = \frac12 \bigl(e^U + e^{1 - U}\bigr)$.
Clause 1 holds because $1 - U$ is also uniform, clause 3 is clear, and a direct computation gives $\operatorname{Var} Z = \frac{e^2 - 1 + 2e}{4} - (e - 1)^2 \approx 0.0039$, so clause 2 holds.
Non-example: $Z = 1 + U + \frac12 U^2$, the Taylor polynomial of $e^U$, has variance about $0.189 < 0.2420$, but $\mathbb E Z = \frac53 \ne e - 1$, so clause 1 fails and the estimate is biased.
A fair comparison also counts the cost.
If one copy of $Z$ costs $c$ times as much as one copy of $Y$, the reduction pays off when $c \operatorname{Var} Z < \operatorname{Var} Y$.
In the example $c = 2$ and $2 \cdot 0.0039 < 0.2420$.
Some techniques, such as stratified sampling, change the whole estimator rather than one sample.
For them one compares the variances of two estimators of the same cost.
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# Compact operator

## Summary

A compact operator is a linear map between Banach spaces that sends bounded sets to relatively compact sets.
Compact operators are the operators closest to finite-dimensional linear algebra: their nonzero spectrum consists of eigenvalues of finite multiplicity, which can accumulate only at $0$.
They include the [[finite-rank operators|Z8FABSYQTS]] and their norm limits, and integral operators with continuous kernels.
They are the setting of the Fredholm alternative and of the spectral theorem for compact self-adjoint operators.

## Explanations

### L1 human

A compact operator squeezes the unit ball, which in infinite dimensions is very large, into a set that is almost finite-dimensional.
Take the operator on sequences that multiplies the $n$-th entry by $1/n$.
It flattens the directions far out in the sequence more and more.
So, up to any given accuracy, the image of the unit ball is small in all but finitely many directions.
The identity on an infinite-dimensional space is not compact: it leaves the unit ball as large as it was.
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# Finsler manifold

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] with [[tangent bundle|WY6K3P4YCG]] $TM$, and write $0$ for the zero section.
A *Finsler function* on $M$ is a continuous function $F \colon TM \to [0, \infty)$ such that:

1. $F$ is smooth on $TM \setminus 0$;
2. $F$ is positively homogeneous: $F(\lambda v) = \lambda F(v)$ for all $v \in TM$ and all $\lambda > 0$;
3. $F$ is strongly convex: for each $p \in M$ and nonzero $v \in T_pM$, the [[Hessian|0SCSQWXRA4]] at $v$ of $\tfrac{1}{2} F^2$ on $T_pM$ is [[positive definite|0QQCG3JWJ4]].

A *Finsler manifold* is a pair $(M, F)$ of a smooth manifold and a Finsler function on it.

## Remarks

Example: if $g$ is a Riemannian metric on $M$, then $F(v) = \sqrt{g(v, v)}$ is a Finsler function.
It is smooth away from $0$, homogeneous of degree $1$, and the Hessian of $\tfrac{1}{2}F^2 = \tfrac{1}{2} g(v, v)$ on $T_pM$ is $g_p$ itself, which is positive definite.

Non-example: on $\R^2$, $F(x, v) = \abs{v_1} + \abs{v_2}$ is positively homogeneous, but it is not smooth along the lines $v_1 = 0$ and $v_2 = 0$, so clause 1 fails.

Clauses 2 and 3 imply that $F(v) > 0$ for $v \ne 0$ and that $F$ satisfies the triangle inequality on each tangent space.
The function $F$ need not be symmetric: $F(-v) = F(v)$ is not required, and Randers metrics are standard examples with $F(-v) \ne F(v)$.
The length of a piecewise smooth curve $\gamma$ is $\int F(\gamma'(t)) \, dt$.
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# Depth

## Summary

The depth of a module $M$ along an ideal $I$ is the common length of the maximal [[regular sequences|69R5R2FCS5]] on $M$ inside $I$.
For a Noetherian local ring it is bounded by the [[Krull dimension|MVGBS3J1P7]], and the rings where equality holds are the [[Cohen–Macaulay rings|N1QJQEGBFC]].
Depth can be computed with the [[Ext functor|4P7GBWJ4RR]], with [[Koszul complexes|YZY11W08SD]] or with [[local cohomology|6JB30BXDA2]].

## Explanations

### L1 human

Depth counts how many independent equations one can impose, one after the other, without any of them being a zero divisor.
Dimension counts how many independent directions a space has.
For nice rings, like polynomial rings, the two numbers agree.
For the ring $K[x,y]/(x^2, xy)$ near the origin, the dimension is $1$ but the depth is $0$, because every element vanishing at the origin is killed by $x$.
</document>

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# Clausen's formula

## Statement

Let $a$ and $b$ be complex numbers such that neither $a + b + \frac12$ nor $2a + 2b$ lies in $\set{0, -1, -2, \dots}$.
Then for every complex $x$ with $\abs{x} < 1$,
$$\left[ {}_2F_1\left( a, b; a + b + \tfrac12; x \right) \right]^2 = {}_3F_2\left( \begin{matrix} 2a,\ 2b,\ a + b \\ a + b + \frac12,\ 2a + 2b \end{matrix} ; x \right),$$
where ${}_2F_1$ is the [[hypergeometric function|463X5V31D9]] and ${}_3F_2$ is the [[generalized hypergeometric series|VDBAW9PN88]].

## Remarks

One proof compares differential equations.
If $y$ solves a second order linear equation, then $y^2$ solves a third order linear equation, the symmetric square.
For the hypergeometric equation with $c = a + b + \frac12$, the symmetric square turns out to be the generalized hypergeometric equation of the ${}_3F_2$ on the right.
Both sides are analytic at $0$ with value $1$, and that equation has only one such solution, so they agree.
Check of the first coefficient: on the left, the coefficient of $x$ is $\frac{2ab}{a + b + \frac12}$.
On the right it is $\frac{2a \cdot 2b \cdot (a + b)}{(a + b + \frac12)(2a + 2b)} = \frac{2ab}{a + b + \frac12}$, as it should be.
For $a = b = \frac14$ the formula reads ${}_2F_1(\frac14, \frac14; 1; x)^2 = {}_3F_2(\frac12, \frac12, \frac12; 1, 1; x)$.
The relation $c = a + b + \frac12$ between the parameters is what makes the symmetric square equation hypergeometric, and the proof uses it.

## Explanations

### L1 human

Clausen's formula says that the square of a certain hypergeometric function is again a hypergeometric function, with one more parameter.
Squaring a power series normally mixes the coefficients in a complicated way.
Here the coefficients of the square come out as a simple ratio of rising factorials.

### L2 human

The formula links two levels of the hypergeometric hierarchy, ${}_2F_1$ and ${}_3F_2$.
It gives positivity results: when the ${}_2F_1$ is real, the ${}_3F_2$ on the right is a square, hence nonnegative.
It is also used to turn identities for squares of elliptic integrals into series for $1/\pi$.
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# Primary decomposition

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $I$ be an [[ideal|0MYSA6MYNA]] of $R$.
A *primary decomposition* of $I$ is an expression $I = Q_1 \cap \cdots \cap Q_n$, with $n \ge 1$, in which each $Q_i$ is a [[primary ideal|V026AAV8E6]].
The decomposition is *minimal* if these two conditions hold:

1. The [[radicals|437YRCNYRJ]] $\sqrt{Q_1}, \dots, \sqrt{Q_n}$ are pairwise distinct.
2. For each $i$, the intersection of the $Q_j$ with $j \ne i$ is not contained in $Q_i$.

## Remarks

Example: in $k[x, y]$ over a field $k$, $(x^2, xy) = (x) \cap (x^2, y)$, where $(x)$ is prime and $(x^2, y)$ is $(x, y)$-primary; the radicals differ and neither term contains the other, so the decomposition is minimal.
Non-example: $(6) = (2) \cap (6)$ in $\Z$ is not a primary decomposition, because $(6)$ is not primary.
The decomposition $(x^2, xy) = (x) \cap (x^2, xy, y^2)$ is another minimal one, so the primary components need not be unique.
In the same way one defines a primary decomposition of a submodule $N$ of a module $M$, with primary submodules.

## Explanations

### L1 human

A primary decomposition writes an ideal as an intersection of finitely many primary ideals, like writing a number as a product of prime powers.
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<document source="https://tamnd.github.io/math2/node/M2W79VF0TN.md" tag="M2W79VF0TN" kind="statement">
# Palais–Smale theory of critical points

## Statement

Let $M$ be a $C^\infty$ [[Hilbert manifold|R8DMMW0E9M]] with a complete Riemannian metric.
Let $f \colon M \to \R$ be a $C^2$ function that is bounded below and satisfies the [[Palais–Smale condition|4724KRNHAT]].
This means that every sequence $(p_n)$ with $(f(p_n))$ bounded and $\norm{\nabla f(p_n)} \to 0$ has a convergent subsequence.
For $c \in \R$ let $M^c = \set{p \in M : f(p) \le c}$.

1. (Deformation) If $a < b$ and $f^{-1}[a, b]$ contains no critical point of $f$, then $M^a$ is a strong [[deformation retract|YW8HBW1GJX]] of $M^b$.
2. (Minimum) The infimum of $f$ is attained, and every point where it is attained is a critical point of $f$.
3. (Lusternik–Schnirelmann bound) The number of critical points of $f$ is at least $\operatorname{cat}(M) + 1$, where $\operatorname{cat}$ is the normalized [[Lusternik–Schnirelmann category|W5BHHXCYXA]].

## Remarks

The proof of the deformation statement follows the flow of $-\nabla f / \norm{\nabla f}^2$ on $f^{-1}[a, b]$.
The Palais–Smale condition gives a positive lower bound for $\norm{\nabla f}$ there, and completeness of the metric lets the flow run until it reaches $M^a$.
For the minimum, suppose $c = \inf f$ were not a critical value.
Then the condition would give $\varepsilon > 0$ with no critical value in $[c, c + \varepsilon]$.
The same flow, extended below $c$, would push the nonempty set $M^{c + \varepsilon}$ into a sublevel set $M^{c'}$ with $c' < c$, which is empty.
For the third statement, the minimax values $c_k = \inf \set{\sup_A f : A \subseteq M \text{ closed}, \ \operatorname{cat}_M(A) \ge k}$ for $k = 0, \dots, \operatorname{cat}(M)$ are critical values, as in [[Lusternik–Schnirelmann theory|QARH15D2PN]], and equal values force infinitely many critical points.

Example: the energy functional on paths with fixed endpoints in a compact Riemannian manifold, or on its free loop space, satisfies these hypotheses.
This gives existence of geodesics and is the setting of the [[Lyusternik–Fet theorem|EXZA8ZA8P6]].
Without the Palais–Smale condition the conclusions fail: $f(x) = e^x$ on $\R$ is bounded below and has no critical point, so its infimum $0$ is not attained.
The [[mountain pass theorem|EPXNCSP2MN]] is another critical point theorem that rests on the same condition.

## Explanations

### L1 human

On a compact manifold, every continuous function has a minimum, and topology forces further critical points.
In infinite dimensions this fails, because a minimizing sequence can run off to infinity.
Palais and Smale found one condition on the function that restores what is needed: almost-critical sequences must have convergent subsequences.
With it, the classical statements about minima, deformations and the number of critical points hold again.

### L2 human

These results are the base of variational methods for nonlinear partial differential equations and for geodesics.
To apply them, one checks the Palais–Smale condition for a concrete functional, typically with Sobolev embeddings and compactness theorems.
The deformation statement is the engine of all minimax arguments, including the [[mountain pass theorem|EPXNCSP2MN]] and the Lusternik–Schnirelmann bound.
The same results hold for $C^1$ functions on complete Finsler manifolds modelled on Banach spaces, with a pseudo-gradient vector field in place of the gradient.
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# Ax–Grothendieck theorem

## Statement

Let $K$ be an [[algebraically closed field|0PF6TM8TWT]], let $n \ge 1$, and let $f_1, \dots, f_n \in K[x_1, \dots, x_n]$ be [[polynomials|1G0B73WPX8]].
Let $F \colon K^n \to K^n$ be the map $F(a) = (f_1(a), \dots, f_n(a))$.
If $F$ is [[injective|09NWETVTNY]], then $F$ is [[surjective|01RVCMSN5N]].

## Remarks

For fixed $n$ and a bound $d$ on the degrees of the $f_i$, the statement is a single first-order sentence in the language of rings.
It holds for every [[finite field|0ZXZ82FTX2]], because an injective map from a finite set to itself is surjective.
It also holds in the algebraic closure $\overline{\F_p}$, the union of the finite fields $\F_{p^k}$.
Given an injective $F$ over $\overline{\F_p}$ and a point $b$, the coefficients of the $f_i$ and the coordinates of $b$ lie in one finite field $\F_{p^k}$.
The map $F$ sends $\F_{p^k}^n$ injectively into itself, so it is onto $\F_{p^k}^n$, and $b$ has a preimage.
By the [[Lefschetz principle|SKTMN14X5P]], it then holds in every algebraically closed field of characteristic $0$, such as $\C$.

Example: over $\C$, the map $F(x, y) = (x, y + x^2)$ is injective, and it is surjective, since $(a, b) = F(a, b - a^2)$.

The hypothesis that $K$ is algebraically closed is necessary.
Over $\Q$, the polynomial map $x \mapsto x^3$ is injective but not surjective, since $2$ is not the cube of a rational number.

## Explanations

### L1 human

Take a map from $n$-dimensional space to itself given by polynomials, over the complex numbers.
If the map never sends two different points to the same point, then it reaches every point.
For finite sets this is obvious, and the theorem carries this simple fact over to the complex numbers.

### L2 human

The theorem was proved by James Ax with model theory and by Alexander Grothendieck with algebraic geometry.
The model-theoretic proof is the standard showcase of the Lefschetz principle: a statement checked over finite fields transfers to $\C$.
The converse fails: a surjective polynomial map need not be injective, as $z \mapsto z^2$ on $\C$ shows.
It is related to the open Jacobian conjecture, which asks whether a polynomial map $\C^n \to \C^n$ with constant nonzero Jacobian determinant has a polynomial inverse.
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# Dual operator

## Definition

Let $X$ and $Y$ be [[normed spaces|03J5RAKB30]] over $\R$ or $\C$, with [[continuous dual spaces|4A82S5FVRT]] $X'$ and $Y'$.
Let $T \colon X \to Y$ be a [[continuous linear map|03TM135G0N]].
The *dual operator* (or *transpose*, or *Banach space adjoint*) of $T$ is the map
$$T' \colon Y' \to X', \qquad T'(f) = f \circ T.$$

## Remarks

The map $T'$ is well defined since $f \circ T$ is linear and continuous, and $T'$ is linear and bounded.
By the [[Hahn–Banach theorem|QB5AE3VYZZ]], its [[operator norm|06EKW1AKXF]] is $\norm{T'} = \norm{T}$.
It is the restriction of the algebraic [[dual map|0AZ5ARP287]] to continuous functionals; other notations are $T^*$ and $T^t$.

Example: for $1 < p < \infty$ and $1/p + 1/q = 1$, identify the dual of $\ell^p$ with $\ell^q$ via $f(x) = \sum_n f_n x_n$.
Let $S$ be the right shift on $\ell^p$.
Then $f(Sx) = \sum_{n \ge 2} f_n x_{n-1} = \sum_{m \ge 1} f_{m+1} x_m$, so $S' f = (f_2, f_3, \ldots)$ is the left shift on $\ell^q$.

Non-example: the rule $f \mapsto f \circ T$ does not give a map $Y' \to X'$ when $T$ is not continuous.
Let $X = Y$ be the space of polynomials on $[0, 1]$ with the supremum norm, $T p = p'$, and $f(p) = p(0)$.
The polynomials $p_n(x) = x (1 - x)^n$ have $\norm{p_n} \le 1/(n+1)$, but $(f \circ T)(p_n) = p_n'(0) = 1$, so $f \circ T$ is not continuous.

The rule $(ST)' = T' S'$ holds, and $\ker T'$ is the annihilator of the range of $T$.
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# Zeros of Bessel functions

## Summary

The zeros of a [[Bessel function of the first kind|CZCMRDWDYW]] $J_\nu$ with $\nu > -1$ are real.
Its positive zeros form an increasing sequence $j_{\nu,1} < j_{\nu,2} < \cdots$ tending to infinity.
These numbers fix the frequencies of a vibrating circular drum and the modes of heat flow in a cylinder.
They obey interlacing and spacing rules, and they are the subject of [[Bourget's hypothesis|38D9RPAXQ8]].

## Explanations

### L1 human

The graph of $J_0(x)$ for $x > 0$ looks like a damped cosine, so it crosses the axis again and again.
The crossing points are the zeros $j_{0,1} \approx 2.405$, $j_{0,2} \approx 5.520$, and so on.
Far out they are spaced by almost exactly $\pi$, like the zeros of a cosine.
A drum of radius $1$ fixed at its rim vibrates in its lowest radially symmetric mode with frequency proportional to $j_{0,1}$.
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# Green's function for the Laplacian

## Summary

The Green's function of a domain for the [[Laplacian|0E0B6RA8K0]] is the potential of a unit point charge in the domain.
The boundary of the domain is held at potential zero.
In a plane domain $\Omega$ with pole $w$, it is a function $g_\Omega(z, w)$ that behaves like $\log(1/\abs{z - w})$ near $w$.
It is [[harmonic|0DEB0JMN0W]] elsewhere and vanishes on the boundary.
It solves the Poisson equation with zero boundary values: $\Delta_z g_\Omega(z, w) = -2\pi \delta_w$.
In free space $\R^3$ the corresponding function is $1/(4\pi\abs{x - y})$, the fundamental solution of the three-variable Laplace equation.
Green's functions give integral formulas for the [[Dirichlet problem|8D56GQ4AG8]] and are invariant under [[conformal maps|0YYT36R4YD]] in the plane.

## Explanations

### L1 human

Place a single point charge inside a grounded metal box.
The electric potential it creates inside the box is the Green's function, with the charge's position as the pole.
Near the charge the potential is huge, and on the walls of the box it is zero.
For the unit disc with the pole at the centre, the Green's function is $\log(1/\abs{z})$, which is $0$ on the unit circle.
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# Traveling wave

## Summary

A traveling wave is a solution of an evolution equation that moves at constant speed without changing its shape.
In one space dimension it has the form $u(x, t) = U(x - ct)$, with a profile $U$ and a speed $c$.
Substituting this form turns the partial differential equation into an ordinary differential equation for $U$.
Traveling waves describe fronts in reaction–diffusion equations, viscous shock profiles and [[solitons|63D4AXYCT4]].
See the [[definition|HAZ0NKCAXS]].

## Explanations

### L1 human

Think of a surfer riding a wave: in the frame that moves with the surfer, the water surface does not change.
A traveling wave is a solution that looks frozen when one moves along with it at the right speed.
The simplest case is the [[transport equation|PVD7VZJZ9B]] $u_t + a u_x = 0$, whose solutions $u = g(x - at)$ all travel at speed $a$.
In nonlinear equations only special shapes travel unchanged, and finding them is a question about an ordinary differential equation.
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# Curvature of a curve

## Definition

Let $n \ge 2$, let $I \subseteq \R$ be an interval, and let $\gamma \colon I \to \R^n$ be a [[regular|06JMR2XWXW]] [[differentiable curve|C57T562NKB]] of class $C^2$.
Let $T(t) = \gamma'(t) / \norm{\gamma'(t)}$ be its unit [[tangent vector|0D98771SM0]].
The *curvature* of $\gamma$ at $t \in I$ is
$$\kappa(t) = \frac{\norm{T'(t)}}{\norm{\gamma'(t)}}.$$
If $\gamma$ is parametrised by [[arc length|00R1JDDV5X]] $s$, this reads $\kappa(s) = \norm{\gamma''(s)}$.

## Remarks

Example: the circle $\gamma(t) = (r \cos t, r \sin t)$ with $r > 0$ has $\norm{\gamma'(t)} = r$ and $T(t) = (-\sin t, \cos t)$, so $\norm{T'(t)} = 1$ and $\kappa(t) = 1/r$.
Non-example: for $\gamma(t) = (t^2, t^3)$ the formula is not defined at $t = 0$, because $\gamma'(0) = 0$; the curve is not regular there.
The curvature does not depend on the regular parametrisation chosen.
In $\R^3$ it is given by $\kappa = \norm{\gamma' \times \gamma''} / \norm{\gamma'}^3$, where $\times$ is the [[cross product|0SEDGXG9S3]].
For a plane curve $\gamma = (x, y)$ one also uses the *signed curvature*
$$k = \frac{x' y'' - y' x''}{(x'^2 + y'^2)^{3/2}},$$
which is positive where the curve turns left; then $\kappa = \abs{k}$.
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# Family-wise error rate

## Definition

Let $H_1, \dots, H_m$ be null hypotheses about the law $P$ of the data, tested by a procedure that rejects a random subset of them.
Let $I_0(P) \subseteq \set{1, \dots, m}$ be the set of indices $i$ such that $H_i$ is true under $P$.
Let $V$ be the number of rejected $H_i$ with $i \in I_0(P)$, the number of [[type I errors|0R41PVFA39]].
The *family-wise error rate* is $\mathrm{FWER} = P(V \ge 1)$.
The procedure controls the FWER *strongly* at level $\alpha$ if $P(V \ge 1) \le \alpha$ for every $P$ in the model.
It controls the FWER *weakly* at level $\alpha$ if this holds for every $P$ under which all $H_i$ are true.

## Remarks

Strong control implies weak control.
Example: if $m$ tests of exact level $\alpha$ are independent and all null hypotheses are true, then $\mathrm{FWER} = 1 - (1 - \alpha)^m$.
So testing each hypothesis at level $\alpha$ does not control the FWER at level $\alpha$ for $m \ge 2$; this is the non-example.
The [[Bonferroni correction|8GRD2TMJRR]] tests each hypothesis at level $\alpha/m$ and gives strong control.
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# Huntington's axioms

## Summary

Huntington's axioms are a short system of axioms for [[Boolean algebras|0XADJKM699]] in terms of one binary operation $+$ and one unary operation $n$.
Besides commutativity and associativity of $+$, they need only Huntington's equation $n(n(x) + y) + n(n(x) + n(y)) = x$.
Edward Huntington gave this system in 1933, after an earlier set of postulates from 1904.
The question whether Huntington's equation can be replaced by a similar equation of Herbert Robbins became the [[Robbins conjecture|X74S395RZD]].

## Explanations

### L1 human

Read $+$ as "or" and $n$ as "not".
Then Huntington's equation says: "($x$ and not $y$) or ($x$ and $y$)" is the same as $x$, written with "or" and "not" only.
It is surprising that this single law, together with the rules for reordering and regrouping "or", already gives all of Boolean algebra.
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# Locally finite family

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $(A_i)_{i \in I}$ be a family of subsets of $X$.
The family is *locally finite* if every point $x \in X$ has a [[neighbourhood|0ZCY0Z80B6]] $N$ such that the set $\set{i \in I : N \cap A_i \ne \emptyset}$ is finite.

## Remarks

Example: in $\R$, the family $([n, n + 1])_{n \in \Z}$ is locally finite.
For $x \in \R$, the neighbourhood $(x - 1, x + 1)$ meets $[n, n + 1]$ only when $x - 2 < n < x + 1$, which holds for at most three integers $n$.

Non-example: in $\R$, the family $((0, 1/n))_{n \ge 1}$ is not locally finite.
Every neighbourhood of $0$ contains an interval $(-\varepsilon, \varepsilon)$, which meets every $(0, 1/n)$.

If $(A_i)_{i \in I}$ is locally finite, then the [[closure|0QEAJB91GZ]] of $\bigcup_i A_i$ equals $\bigcup_i \overline{A_i}$.
In particular, the union of a locally finite family of [[closed sets|0ZCK8DQYN1]] is closed.
</document>

<document source="https://tamnd.github.io/math2/node/M3MSQ3NKAE.md" tag="M3MSQ3NKAE" kind="concept">
# Monoidal category

## Summary

A monoidal category is a [[category|A2DE66937D]] with a tensor product of objects and a unit object, associative and unital up to coherent isomorphism.
It is the categorical version of a [[monoid|0N6R902FG1]], one level up.
Sets with the Cartesian product, vector spaces with the [[tensor product|0BF2DE2SYK]], and endofunctors with composition are monoidal categories.
Monoidal categories are the setting for monoid objects, enriched categories, duals, and the string diagrams of quantum algebra and computer science.

## Explanations

### L1 human

A monoidal category is a category where two objects can be put side by side to give a new object.
Putting $A$ next to $B$, and then next to $C$, gives the same result as putting $A$ next to the pair $B, C$, but only up to a chosen isomorphism.
There is also an "empty" object $I$ that changes nothing when placed next to another.
For example, for sets the product $A \times B$ is the side-by-side object, and a one-point set is the empty object, since $\set{*} \times A$ is in bijection with $A$.
</document>

<document source="https://tamnd.github.io/math2/node/M3PFV1BX9V.md" tag="M3PFV1BX9V" kind="statement">
# Complete class theorem

## Statement

Let $\set{P_\theta : \theta \in \Theta}$ be a statistical model with a loss function, where $\Theta = \set{\theta_1, \dots, \theta_k}$ is finite.
Allow randomized decision rules, and assume that every rule has finite [[risk|N26TSVRH2S]] at every $\theta_i$.
Then every [[admissible|DSTF6X8DRJ]] rule $\delta$ is a [[Bayes rule|PP1A0JB47Q]] for some prior: there are weights $\pi_1, \dots, \pi_k \ge 0$ with $\sum_i \pi_i = 1$ such that
$$\sum_{i=1}^{k} \pi_i R(\theta_i, \delta) = \inf_{\delta'} \sum_{i=1}^{k} \pi_i R(\theta_i, \delta').$$

## Remarks

Consider the risk set $S = \set{(R(\theta_1, \delta), \dots, R(\theta_k, \delta))}$ in $\R^k$, over all randomized rules $\delta$.
Mixing two rules mixes their risk points, so $S$ is a [[convex set|02AS45T3H9]].
If $\delta$ is admissible, its risk point $x$ meets the convex set $\set{y : y_i \le x_i \text{ for all } i}$ in $S$ only at points equal to $x$.
A separating hyperplane through $x$ then has a normal vector with nonnegative entries, and normalizing it gives the prior $\pi$.
The minimized quantity is the [[Bayes risk|VEBDNCJJTN]] of $\pi$.

A class $\mathcal{C}$ of rules is called *complete* if every rule outside $\mathcal{C}$ is dominated by some rule in $\mathcal{C}$.
Complete class theorems are results of this kind, which show that the Bayes rules, or their limits, form a complete class.
For infinite $\Theta$, similar results hold under extra conditions, with limits of Bayes rules or generalized Bayes rules in place of Bayes rules.

Randomization matters in the proof: without it, the risk set need not be convex.

## Explanations

### L1 human

Any rule that cannot be beaten everywhere is, secretly, the best rule for some way of weighting the possible truths.
So when the set of possible truths is finite, a search for good rules can be restricted to Bayes rules.

### L2 human

The theorem gives a frequentist reason to use Bayesian methods: admissibility, a purely frequentist property, forces a rule to be Bayes.
It reduces the search for admissible rules to the study of Bayes rules for all priors.
The proof is a direct application of the separating hyperplane theorem to the convex risk set.
In infinite-dimensional problems the analogues are more delicate, and admissible rules are typically limits of Bayes rules.
</document>

<document source="https://tamnd.github.io/math2/node/M3S9JK4RPK.md" tag="M3S9JK4RPK" kind="definition">
# Convex join

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]] and let $A, B \subseteq V$.
The *convex join* of $A$ and $B$ is the set
$$\operatorname{join}(A, B) = \bigcup_{a \in A,\ b \in B} [a, b],$$
where $[a, b]$ is the [[line segment|VQ4Q2V45RP]] from $a$ to $b$.

## Remarks

Example: in $\R^2$, let $A = \set{(0, 0)}$ and $B = [(1, 0), (1, 1)]$.
The join is the union of the segments from the origin to the points $(1, s)$ with $0 \le s \le 1$, which is the closed triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$.

If $A$ and $B$ are nonempty [[convex sets|02AS45T3H9]], then $\operatorname{join}(A, B)$ equals the [[convex hull|0FQSDC2EQ2]] of $A \cup B$.
Indeed, a convex combination of points of $A \cup B$ can be regrouped as $(1 - t) a + t b$ with $a \in A$ and $b \in B$.

Non-example of that identity: if $A = \emptyset$ and $B = \set{b}$, then $\operatorname{join}(A, B) = \emptyset$, while the convex hull of $A \cup B$ is $\set{b}$.
So the nonemptiness hypothesis cannot be dropped.
</document>

<document source="https://tamnd.github.io/math2/node/M3T81YNSKC.md" tag="M3T81YNSKC" kind="concept">
# Action-angle coordinates

## Summary

Action-angle coordinates are canonical coordinates $(I, \theta)$ for an [[integrable system|N0N7CMRCMH]] in which the Hamiltonian depends on the actions $I$ only.
In these coordinates the actions are constant and the angles turn at constant speed, so the equations of motion are solved at once.
They exist near compact regular level sets by the [[Liouville–Arnold theorem|PNQCW85HTJ]].
They are the natural coordinates for perturbation theory, such as [[KAM theory|3ZZSXKM16Z]].

## Explanations

### L1 human

Think of a motion that winds around a doughnut.
The actions say which doughnut you are on, and they never change.
The angles say where you are on that doughnut, and they increase at steady rates.
For the harmonic oscillator $H = \frac{1}{2}(p^2 + q^2)$, the action is the energy and the angle is the phase of the oscillation, which grows linearly in time.
</document>

<document source="https://tamnd.github.io/math2/node/M3WQTN3FH2.md" tag="M3WQTN3FH2" kind="concept">
# Nevanlinna class

## Summary

The Nevanlinna class $N$ is the space of [[holomorphic functions|0DG0Z72CQD]] on the [[unit disc|8G5ABXRK85]] whose positive part of $\log \abs{f}$ has bounded averages on circles.
Its elements are exactly the quotients of two bounded holomorphic functions with zero-free denominator, the functions of bounded type.
It contains every [[Hardy space|PNNNFJVF6W]] $H^p$, and its nonzero elements still have radial limits almost everywhere and zeros satisfying the Blaschke condition.

## Explanations

### L1 human

Hardy spaces control the size of $\abs{f}^p$ on circles.
The Nevanlinna class only controls the size of $\log \abs{f}$ where $\abs{f} > 1$, which is a much weaker condition.
For example, $1/(1 - z)$ is not bounded but lies in $N$, since it is $1$ divided by the bounded zero-free function $1 - z$.
Even $\exp\left(\frac{1}{1 - z}\right)$ lies in $N$.
The quantity $\frac{1}{2\pi} \int \log^+ \abs{f(r e^{it})} \, dt$ is the proximity part of the [[Nevanlinna characteristic|YT6Z0GWS0W]], which explains the name.
</document>

<document source="https://tamnd.github.io/math2/node/M3XJ4ZKYGT.md" tag="M3XJ4ZKYGT" kind="definition">
# Place (number field)

## Definition

Let $K$ be a field.
An *absolute value* on $K$ is a map $\abs{\cdot} \colon K \to \R_{\ge 0}$ such that for all $x, y \in K$:

1. $\abs{x} = 0$ if and only if $x = 0$;
2. $\abs{xy} = \abs{x}\abs{y}$;
3. $\abs{x + y} \le \abs{x} + \abs{y}$.

It is *trivial* if $\abs{x} = 1$ for all $x \ne 0$.
Two absolute values $\abs{\cdot}_1$ and $\abs{\cdot}_2$ are *equivalent* if there is a real $c > 0$ with $\abs{x}_2 = \abs{x}_1^{c}$ for all $x \in K$.
A *place* of an [[algebraic number field|MWBDFS2KJ5]] $K$ is an equivalence class of nontrivial absolute values on $K$.

## Remarks

Equivalent absolute values define the same topology on $K$.
A place is *infinite* (archimedean) if the values $\abs{n \cdot 1}$ for $n \in \N$ are unbounded, and *finite* (non-archimedean) otherwise.

Let $K$ be a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings.
Each embedding $\sigma \colon K \to \C$ gives the absolute value $\abs{x}_\sigma = \abs{\sigma(x)}$, and conjugate embeddings give the same one.
These give the $r_1 + r_2$ infinite places.
Each nonzero [[prime ideal|00MHHZ9SNE]] $\mathfrak{p}$ of $\mathcal{O}_K$ gives $\abs{x}_{\mathfrak{p}} = N(\mathfrak{p})^{-v_{\mathfrak{p}}(x)}$ for $x \ne 0$.
Here $v_{\mathfrak{p}}(x)$ is the exponent of $\mathfrak{p}$ in the factorisation of $x\mathcal{O}_K$, and $N(\mathfrak{p})$ is the [[norm|M54ADM0CVF]].
These give all finite places.
Every place arises in one of these two ways.

Example: on $\Q$, the [[p-adic absolute value|WGN82TCEF2]] $\abs{x}_p = p^{-v_p(x)}$ satisfies the three clauses, even with $\abs{x + y}_p \le \max(\abs{x}_p, \abs{y}_p)$, and it is nontrivial since $\abs{p}_p = 1/p$; so it defines a place of $\Q$.
By [[Ostrowski's theorem|GCQ23XT6VD]], the places of $\Q$ are these and the place of the usual absolute value.

Non-example: on $\Q$, the map $x \mapsto \abs{x}^2$ fails clause 3, since $\abs{1 + 1}^2 = 4 > 1^2 + 1^2$.
So it is not an absolute value in this sense, although it is a power of one.
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<document source="https://tamnd.github.io/math2/node/M3YXA82ZFX.md" tag="M3YXA82ZFX" kind="concept">
# Morita duality

## Summary

A Morita duality between two [[rings|0B1261B5ED]] $R$ and $S$ is a contravariant equivalence between suitable categories of left $R$-modules and right $S$-modules.
It is given by a bimodule $U$ through the functors $\Hom(-, U)$.
It generalizes the duality $V \mapsto V^*$ between finite-dimensional vector spaces.
Matlis duality over complete Noetherian local rings and the duality $\Hom(-, R)$ over quasi-Frobenius rings are the main examples.
Morita characterized such dualities by the bimodule $U$, which must be an [[injective|85H7PRHDPZ]] cogenerator on both sides.

## Explanations

### L1 human

For finite-dimensional vector spaces, taking the dual space reverses all arrows and taking it twice gives back the space.
A Morita duality is the same phenomenon for modules over a ring.
One fixes a module $U$ that plays the role of the field and sends a module $M$ to the module of maps from $M$ to $U$.
For finitely generated modules, doing this twice gives back $M$.
Not every ring admits such a duality; it forces strong finiteness conditions on the ring.
</document>

<document source="https://tamnd.github.io/math2/node/M435YDQWQ4.md" tag="M435YDQWQ4" kind="definition">
# Bessel potential

## Definition

Let $n \ge 1$ and $\alpha > 0$.
Use the [[Fourier transform|DPVM34ADZV]] $\hat g(\xi) = \int_{\R^n} g(x) e^{-2\pi i x \cdot \xi} \, dx$.
Let $G_\alpha$ be the integrable function on $\R^n$ whose Fourier transform is
$$\widehat{G_\alpha}(\xi) = (1 + 4 \pi^2 \abs{\xi}^2)^{-\alpha/2}.$$
For $f \in L^p(\R^n)$ with $1 \le p \le \infty$, the *Bessel potential* of order $\alpha$ of $f$ is the [[convolution|2Z29JN3VES]]
$$J_\alpha f = G_\alpha * f.$$

## Remarks

The kernel $G_\alpha$ exists, is positive, and has integral $\widehat{G_\alpha}(0) = 1$.
So $J_\alpha$ is a bounded operator on each $L^p$ with norm at most $1$, by Young's inequality.
On Schwartz functions $J_\alpha = (I - \Delta)^{-\alpha/2}$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]], since $-\Delta$ has symbol $4 \pi^2 \abs{\xi}^2$.
The semigroup law $J_\alpha J_\beta = J_{\alpha + \beta}$ holds for all $\alpha, \beta > 0$.
Near $0$, $G_\alpha(x)$ is comparable to $\abs{x}^{\alpha - n}$ when $0 < \alpha < n$, and $G_\alpha(x)$ decays like $e^{-c\abs{x}}$ for some $c > 0$ as $\abs{x} \to \infty$.

Example: for $n = 1$ and $\alpha = 2$, the kernel is $G_2(x) = \frac12 e^{-\abs{x}}$.
Check: $\int_{\R} \frac12 e^{-\abs{x}} e^{-2\pi i x \xi} \, dx = \frac12 \cdot \frac{2}{1 + 4\pi^2 \xi^2} = (1 + 4\pi^2 \xi^2)^{-1}$.
So $u = J_2 f$ solves $u - u'' = f$ for nice $f$.

Non-example: the kernel $\abs{x}^{\alpha - n}$ of the Riesz potential is not the Bessel kernel, since it is not integrable on $\R^n$, while $G_\alpha$ must be integrable with integral $1$.
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# Topological entropy

## Definition

Let $(X, d)$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
For an integer $n \ge 1$, define the metric $d_n(x, y) = \max_{0 \le i < n} d(f^i(x), f^i(y))$ on $X$.
For $\varepsilon > 0$, a set $E \subseteq X$ is *$(n, \varepsilon)$-separated* if $d_n(x, y) > \varepsilon$ for all distinct $x, y \in E$.
Let $s(n, \varepsilon)$ be the largest size of an $(n, \varepsilon)$-separated set, which is finite by compactness.
The *topological entropy* of $f$ is
$$h_{\mathrm{top}}(f) = \lim_{\varepsilon \to 0} \limsup_{n \to \infty} \frac{1}{n} \log s(n, \varepsilon) \in [0, \infty],$$
where $\log$ is the [[natural logarithm|04DXAFN2ES]].

## Remarks

The limit in $\varepsilon$ exists because the $\limsup$ grows as $\varepsilon$ decreases.
The value depends only on the topology, not on the metric.
Adler, Konheim and McAndrew defined it on any compact space through the growth of joins of [[open covers|049PXYHW1T]], and this gives the same number.
Topologically conjugate maps have equal entropy, and $h_{\mathrm{top}}(f^k) = k \, h_{\mathrm{top}}(f)$ for $k \ge 1$.
Example: the full one-sided [[shift|A1XE59ZHNM]] $\sigma$ on $\set{0, \ldots, k-1}^{\N}$ with $d(x, y) = 2^{-\min\set{i : x_i \neq y_i}}$.
Then $d_n(x, y) > 2^{-m}$ exactly when $x$ and $y$ differ at some coordinate $i < n + m - 1$, so $s(n, 2^{-m}) = k^{n + m - 1}$ and $h_{\mathrm{top}}(\sigma) = \log k$.
A map with zero entropy: an isometry has $d_n = d$, so $s(n, \varepsilon)$ does not depend on $n$ and $h_{\mathrm{top}} = 0$.

## Explanations

### L1 human

Topological entropy is the exponential growth rate of the number of orbit segments that can be told apart at a fixed small resolution.
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<document source="https://tamnd.github.io/math2/node/M4606WJDYG.md" tag="M4606WJDYG" kind="definition">
# Determinantal ideal

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]], let $M$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] with entries in $R$, and let $t$ be an integer with $1 \le t \le \min(m, n)$.
The *determinantal ideal* $I_t(M)$ is the [[ideal|096Z9MP2ME]] of $R$ generated by all $t \times t$ [[minors|0BT9BYWVRY]] of $M$.
When $k$ is a field, $R = k[x_{ij} : 1 \le i \le m, \; 1 \le j \le n]$ is the [[polynomial ring|0SDC934R52]] and $X = (x_{ij})$ is the generic matrix, $I_t(X)$ is the *generic determinantal ideal*.

## Remarks

When $k$ is algebraically closed, the zero set of $I_t(X)$ in $k^{mn}$ is the set of $m \times n$ matrices of [[rank|0ZC9YTXZRY]] less than $t$.
For the generic matrix, $I_t(X)$ is a [[prime ideal|00MHHZ9SNE]] of [[height|87WH65FJB8]] $(m - t + 1)(n - t + 1)$, and $R/I_t(X)$ is [[Cohen–Macaulay|N1QJQEGBFC]].

Example: for $m = 2$, $n = 3$ and $t = 2$, the ideal $I_2(X)$ is generated by the three minors $x_{11} x_{22} - x_{12} x_{21}$, $x_{11} x_{23} - x_{13} x_{21}$ and $x_{12} x_{23} - x_{13} x_{22}$.
Its height is $(2 - 2 + 1)(3 - 2 + 1) = 2$.

Non-example: in the same ring, the principal ideal $(x_{11} x_{22} - x_{12} x_{21})$ is generated by one $2 \times 2$ minor only.
It has height $1$ by Krull's principal ideal theorem, so it is not $I_2(X)$, which has height $2$.
</document>

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# Sobolev space

## Summary

A Sobolev space $W^{k,p}(U)$ consists of the functions in the [[Lp space|SASP14MA1Z]] $L^p(U)$ whose [[weak derivatives|C3EC148VZD]] up to order $k$ also lie in $L^p(U)$.
With the sum of the $L^p$ norms of the derivatives it is a [[Banach space|06GZ956657]], and for $p = 2$ it is a [[Hilbert space|HM2HHE3J51]], written $H^k(U)$.
Sobolev spaces are the main setting of the modern theory of partial differential equations, because weak solutions are found in them by functional analysis.
Embedding theorems then translate the integrability of derivatives into integrability or continuity of the function itself.

## Explanations

### L1 human

A Sobolev space measures a function together with its derivatives, but in an averaged sense rather than point by point.
A function belongs to $W^{1,2}$ when the function and its slope have finite energy $\int u^2 + \abs{\nabla u}^2$.
Corners are allowed, but jumps are not.
For example, $\abs{x}$ on $(-1, 1)$ lies in $W^{1,p}$ for every $p$, while the sign function on $(-1, 1)$ lies in no $W^{1,p}$.
</document>

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# Rees matrix semigroup

## Definition

Let $G$ be a [[group|G5S1M9]], let $I$ and $\Lambda$ be nonempty [[sets|0SJ3RX9TXA]], and let $P = (p_{\lambda i})_{\lambda \in \Lambda, i \in I}$ be a $\Lambda \times I$ matrix with entries in $G$.
The *Rees matrix semigroup* $\mathcal{M}(G; I, \Lambda; P)$ is the set $I \times G \times \Lambda$ with the product
$$(i, g, \lambda)(j, h, \mu) = (i, \, g \, p_{\lambda j} \, h, \, \mu).$$
The matrix $P$ is called the sandwich matrix.

## Remarks

The product is associative: both ways of bracketing $(i, g, \lambda)(j, h, \mu)(k, f, \nu)$ give $(i, \, g p_{\lambda j} h p_{\mu k} f, \, \nu)$.
So $\mathcal{M}(G; I, \Lambda; P)$ is a [[semigroup|0XMEM406JM]].
It is simple: given $x = (i, g, \lambda)$ and $y = (j, h, \mu)$, one can solve $(j, a, \lambda) \, x \, (i, b, \mu) = y$ for $a, b \in G$.
Rees's theorem: a semigroup is completely simple if and only if it is isomorphic to some $\mathcal{M}(G; I, \Lambda; P)$.

Example: if $I = \Lambda = \set{1}$ and $P = (e)$, then $(1, g, 1) \mapsto g$ is an isomorphism onto $G$.
Example: if $G$ is trivial, $I = \set{1, 2}$ and $\Lambda = \set{1}$, the product is $(i, \lambda)(j, \mu) = (i, \mu) = (i, 1)$, so $xy = x$ for all $x, y$: a left zero semigroup.

Non-example: the monoid $(\N, +)$ is not isomorphic to any Rees matrix semigroup over a group.
It has the proper ideal $\set{1, 2, 3, \dots}$, while a Rees matrix semigroup has no proper two-sided ideal.
</document>

<document source="https://tamnd.github.io/math2/node/M4G4RXASTY.md" tag="M4G4RXASTY" kind="concept">
# Brun sieve

## Summary

The Brun sieve is a combinatorial sieve method of Viggo Brun.
It counts the integers of a set that have no prime factor below a bound $z$.
It uses a truncated inclusion–exclusion, which gives upper and lower bounds with a controllable error.
It was the first sieve to give nontrivial results on problems such as twin primes.
There the [[sieve of Eratosthenes|0XM8ZA9RNM]] with full inclusion–exclusion produces too large an error.
With it Brun proved that the sum of the reciprocals of the twin primes converges.

## Explanations

### L1 human

To count numbers with no small prime factor, inclusion–exclusion subtracts the multiples of each prime, adds back the multiples of each product of two primes, and so on.
With many primes this formula has a huge number of terms, and the small rounding errors add up to something larger than the answer.
Brun's idea is to stop the formula early.
Stopping after an even number of steps gives an upper bound, stopping after an odd number gives a lower bound.
For example, the numbers up to $10$ with no factor $2$ or $3$ are $1, 5, 7$; stopping after the first step gives the lower bound $10 - 5 - 3 = 2 \le 3$.
</document>

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# Noetherian topological space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *Noetherian* if it satisfies the descending chain condition on [[closed sets|0ZCK8DQYN1]].
That is, for every sequence $C_1 \supseteq C_2 \supseteq C_3 \supseteq \cdots$ of closed subsets of $X$, there is $n$ with $C_m = C_n$ for all $m \ge n$.

## Remarks

Equivalent forms: every ascending chain of open subsets stabilises.
Also equivalent: every nonempty family of closed subsets has a minimal element.
Also equivalent: every subset of $X$ is [[compact|0TVJZYB6SH]] in the subspace topology.

Example: any set $X$ with the [[cofinite topology|0B906ZBTMZ]].
A proper closed subset is finite, so a strictly descending chain of closed sets has finitely many steps after its first proper member.

Non-example: $\R$ with the usual topology.
The closed sets $C_n = [n, \infty)$ satisfy $C_1 \supsetneq C_2 \supsetneq \cdots$, so the chain never stabilises.
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# Ultrametric space

## Definition

Let $X$ be a set.
An *ultrametric* on $X$ is a [[metric|0VKD1TP7F3]] $d \colon X \times X \to [0, \infty)$ that satisfies the strong triangle inequality
$$d(x, z) \le \max(d(x, y), d(y, z)) \quad \text{for all } x, y, z \in X.$$
An *ultrametric space* is a set with an ultrametric.

## Remarks

The strong triangle inequality implies the usual one, since $\max(a, b) \le a + b$ for $a, b \ge 0$.

Example: fix a prime $p$ and let $v_p$ be the [[p-adic valuation|03HDF50VN4]] on $\Q$.
Put $d(x, y) = p^{-v_p(x - y)}$ for $x \ne y$ and $d(x, x) = 0$.
Since $v_p(a + b) \ge \min(v_p(a), v_p(b))$, applying it to $a = x - y$ and $b = y - z$ gives the strong triangle inequality.

Non-example: the usual metric $d(x, y) = \abs{x - y}$ on $\R$.
Here $d(0, 2) = 2$, while $\max(d(0, 1), d(1, 2)) = 1$.

In an ultrametric space, if $d(x, y) \ne d(y, z)$, then $d(x, z) = \max(d(x, y), d(y, z))$.
Every open ball $B(a, r)$ equals $B(b, r)$ for each $b \in B(a, r)$.
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<document source="https://tamnd.github.io/math2/node/M4KG2KBKS9.md" tag="M4KG2KBKS9" kind="definition">
# Lamé function

## Definition

Let $k \in (0, 1)$, and let $\operatorname{sn}(z, k)$, $\operatorname{cn}(z, k)$ and $\operatorname{dn}(z, k)$ be the Jacobi [[elliptic functions|FF9ZPCZ59K]] of modulus $k$.
Let $\nu$ and $h$ be complex numbers.
*Lamé's equation* is the [[linear differential equation|0QQRP17Z7S]]
$$\frac{d^2 w}{dz^2} + \left( h - \nu(\nu + 1) k^2 \operatorname{sn}^2(z, k) \right) w = 0.$$
A *Lamé function* is a solution $w$ of Lamé's equation.
Let $\nu = n$ be a nonnegative integer.
A *Lamé polynomial* of degree $n$ is a nonzero solution of the form $\operatorname{sn}^{e_1} z\, \operatorname{cn}^{e_2} z\, \operatorname{dn}^{e_3} z \cdot P(\operatorname{sn}^2 z)$.
Here $e_1, e_2, e_3 \in \set{0, 1}$ and $P$ is a polynomial.

## Remarks

Example: for $\nu = 1$ and $h = 1 + k^2$, the function $w = \operatorname{sn} z$ is a Lamé polynomial.
Check: from $\operatorname{sn}' = \operatorname{cn} \operatorname{dn}$, $\operatorname{cn}^2 = 1 - \operatorname{sn}^2$ and $\operatorname{dn}^2 = 1 - k^2 \operatorname{sn}^2$, one gets $\operatorname{sn}'' = -(1 + k^2) \operatorname{sn} + 2k^2 \operatorname{sn}^3$, so $w'' + (1 + k^2 - 2k^2 \operatorname{sn}^2) w = 0$.
In the same way $\operatorname{cn} z$ is a solution for $h = 1$ and $\operatorname{dn} z$ for $h = k^2$.
Non-example: for $\nu = 1$ and $h = 0$, the function $\operatorname{sn} z$ is not a solution, since the left side equals $-(1 + k^2) \operatorname{sn} z$, which is not identically $0$.
For each nonnegative integer $n$ there are exactly $2n + 1$ values of $h$ that admit a Lamé polynomial of degree $n$.
This matches the dimension $2n + 1$ of the space of spherical harmonics of degree $n$.
An equivalent form of the equation uses the Weierstrass function $\wp$ in place of $k^2 \operatorname{sn}^2$, and in the variable $\operatorname{sn}^2 z$ it becomes a special case of Heun's equation.
</document>

<document source="https://tamnd.github.io/math2/node/M4N5FNNYCF.md" tag="M4N5FNNYCF" kind="definition">
# Cokernel

## Definition

Let $\mathcal{C}$ be a category with [[zero morphisms|YF9F9EG6M3]], and let $f \colon X \to Y$ be a morphism of $\mathcal{C}$.
A *cokernel* of $f$ is an object $Q$ with a morphism $q \colon Y \to Q$ such that:

1. $q \circ f = 0$;
2. for every morphism $u \colon Y \to T$ with $u \circ f = 0$ there is a unique morphism $v \colon Q \to T$ with $u = v \circ q$.

## Remarks

Equivalently, $q$ is a [[coequalizer|DVSTWGVCSZ]] of $f$ and $0 \colon X \to Y$; it is the [[kernel|S417YNEJ6M]] of $f$ in the opposite category.
A cokernel is unique up to a unique isomorphism, is written $Y \to \operatorname{coker} f$, and is an [[epimorphism|RVXDSY04KG]].

Example: in the category of [[modules|0NZ8JA25WK]] over a ring, the quotient map $q \colon Y \to Y / f(X)$ is a cokernel of $f$.
If $u \circ f = 0$, then $u$ vanishes on $f(X)$ and so induces a unique $v \colon Y / f(X) \to T$ with $u = v \circ q$.
For multiplication by $n \ge 1$ on the [[abelian group|082GHNPJ53]] $\Z$, the cokernel is the reduction $\Z \to \Z / n\Z$.

Non-example: for multiplication by $2$ on $\Z$, the map $\Z \to 0$ satisfies clause 1 but not clause 2.
The reduction $u \colon \Z \to \Z/2$ satisfies $u \circ 2 = 0$ but does not factor through the zero group, since $u \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/M4RJHZ3KNA.md" tag="M4RJHZ3KNA" kind="example">
# Total variation denoising

## Example

Let $\Omega \subseteq \R^n$ be a bounded open set, let $f \in L^2(\Omega)$ be a noisy image, and let $\lambda > 0$.
For $u \in L^2(\Omega)$ let
$$TV(u) = \sup \Big\{ \int_\Omega u \operatorname{div} \varphi \, dx : \varphi \in C^1_c(\Omega; \R^n), \ \abs{\varphi(x)} \le 1 \text{ for all } x \Big\},$$
the [[total variation|8QA5DEWNYD]] of $u$, which equals $\int_\Omega \abs{\nabla u} \, dx$ for smooth $u$.
*Total variation denoising*, in the model of Rudin, Osher and Fatemi, replaces $f$ by the minimizer of
$$E(u) = TV(u) + \frac{\lambda}{2} \int_\Omega (u - f)^2 \, dx, \qquad u \in L^2(\Omega).$$
The functional $E$ is lower semicontinuous, coercive and strictly [[convex|07Q7CNM5WF]] on the [[Lp space|SASP14MA1Z]] $L^2(\Omega)$, so it has exactly one minimizer.

A worked case on $\Omega = (0, 1)$: let $h > 0$ and $f = h$ on $(0, \tfrac12)$, $f = 0$ on $(\tfrac12, 1)$, a single edge of height $h$.
Assume $\lambda \ge 4/h$ and set $\delta = 2/\lambda$.
The minimizer is $u = h - \delta$ on $(0, \tfrac12)$ and $u = \delta$ on $(\tfrac12, 1)$.

Check: let $w(x) = 2x$ on $[0, \tfrac12]$ and $w(x) = 2 - 2x$ on $[\tfrac12, 1]$.
Then $\abs{w} \le 1$, $w(0) = w(1) = 0$, $w' = \lambda (f - u)$, and $\int_0^1 u w' \, dx = h - 2\delta = TV(u)$.
For every $v$ one has $TV(v) \ge \int_0^1 v w' \, dx$, and expanding the squares gives $E(v) - E(u) \ge \frac{\lambda}{2} \int_0^1 (v - u)^2 \, dx \ge 0$.

## Remarks

The edge survives at the same place, and its height drops from $h$ to $h - 4/\lambda$.
A quadratic penalty $\int \abs{\nabla u}^2$ instead of $TV(u)$ would smear the jump into a smooth ramp.
If $\lambda < 4/h$, the minimizer is the constant $h/2$, so a weak fidelity term erases small edges.

## Explanations

### L1 human

A noisy photo has many small random oscillations.
Total variation denoising looks for an image close to the photo whose total amount of up-and-down change is small.
Noise has a lot of such change and is removed.
A sharp edge costs only its height, not its steepness, so edges are kept, though with a little less contrast.
In the example, a step of height $h$ comes out as a step of height $h - 4/\lambda$.
</document>

<document source="https://tamnd.github.io/math2/node/M4RMZNGZ2P.md" tag="M4RMZNGZ2P" kind="concept">
# Chebyshev set

## Summary

A Chebyshev set in a normed space is a set from which every point has exactly one [[best approximation|7N4988HN3P]].
On a Chebyshev set the map sending each point to its nearest point, the metric projection, is well defined.
Closed convex sets in Hilbert spaces are the main examples.
The question of which Chebyshev sets are convex is a classical problem of geometry of normed spaces.

## Explanations

### L1 human

Stand anywhere in the plane and look for the nearest point of a set.
If there is always exactly one nearest point, the set is a Chebyshev set.
A line, a filled disc or a filled square in the plane has this property.
A circle does not: from its centre all points of the circle are equally near.
</document>

<document source="https://tamnd.github.io/math2/node/M4TXKR87H8.md" tag="M4TXKR87H8" kind="definition">
# Schur stable polynomial

## Definition

Let $P(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_0$ be a polynomial of degree $n \ge 1$ with complex coefficients.
The polynomial $P$ is *Schur stable* if every complex [[root|019MA1VTR2]] $\lambda$ of $P$ satisfies $\abs{\lambda} < 1$.

## Remarks

Example: $P(z) = z^2 - z + \tfrac{1}{2}$ has roots $\tfrac{1}{2} \pm \tfrac{1}{2} i$.
Each has modulus $\sqrt{1/4 + 1/4} = 1/\sqrt{2} < 1$, so $P$ is Schur stable.

Non-example: $P(z) = z^2 - 1$ has roots $1$ and $-1$.
They have modulus exactly $1$, so the strict inequality fails and $P$ is not Schur stable.

Some authors call such polynomials simply *stable* in a discrete-time context, or say that $P$ has all its zeros in the open unit disc.
The polynomial $P$ is Schur stable if and only if its reversed polynomial $z^n P(1/z)$ has all its roots outside the closed unit disc, provided $a_0 \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/M4VJHPZJMQ.md" tag="M4VJHPZJMQ" kind="statement">
# Time shift theorem

## Statement

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$, and let $F$ be its [[Laplace transform|05XT5AVJBK]].
Let $a \ge 0$ be real, and let $u$ be the [[Heaviside step function|09C4JP7Y0M]].
Let $g(t) = u(t - a) f(t - a)$, that is, $g(t) = 0$ for $0 \le t < a$ and $g(t) = f(t - a)$ for $t \ge a$.
If the Laplace integral of $f$ converges at the complex number $s$, then so does the Laplace integral of $g$, and
$$\mathcal{L}\{g\}(s) = e^{-a s} F(s).$$

## Remarks

The substitution $t = \tau + a$ gives $\int_a^\infty f(t - a) e^{-s t} \, dt = e^{-a s} \int_0^\infty f(\tau) e^{-s \tau} \, d\tau$.
Example: with $f = 1$ one gets $\mathcal{L}\{u(t - a)\}(s) = e^{-a s} / s$ for $\operatorname{Re} s > 0$.
A rectangular pulse $u(t - 1) - u(t - 2)$ has transform $(e^{-s} - e^{-2 s}) / s$.
The condition $a \ge 0$ matters: for $a < 0$ the shifted function would need values of $f$ at negative times, which the one-sided transform does not see.

## Explanations

### L1 human

Delaying a signal by $a$ units of time multiplies its transform by $e^{-a s}$.
The step function makes sure that nothing happens before the delay is over.

### L2 human

This rule is also called the second shifting theorem.
It is how one handles forcing terms that switch on and off in the [[Laplace method for initial value problems|0K3HFZJK5Q]].
Read backwards, a factor $e^{-a s}$ in a transform means: invert the rest, then delay by $a$.
Compare the [[frequency shift theorem|CD35MSBNE7]], where a factor $e^{a t}$ becomes a shift in $s$.
</document>

<document source="https://tamnd.github.io/math2/node/M4ZHB3GXBP.md" tag="M4ZHB3GXBP" kind="statement">
# Tarski's HSP formula

## Statement

Let $K$ be a class of algebras of a fixed signature.
Let $V(K)$ be the smallest class that contains $K$ and is closed under the [[class operators|S7MJVSP79N]] $H$, $S$ and $P$.
Then
$$V(K) = HSP(K).$$
By Birkhoff's HSP theorem, $HSP(K)$ is then the [[variety|AJNZJD8442]] generated by $K$.

## Remarks

Each of $H$, $S$ and $P$ maps $V(K)$ into itself, so $HSP(K) \subseteq V(K)$.
It remains to show that $HSP(K)$ is closed under $H$, $S$ and $P$.
This follows from the rules $SH \le HS$, $PS \le SP$, $PH \le HP$ and from the idempotence of each operator.
For example, $S\,HSP \le HS\,SP = HSP$ and $P\,HSP \le HP\,SP \le HSP\,P = HSP$.
Example: for $K = \set{\Z}$ in the signature of groups, $HSP(K)$ is the class of all abelian groups.
Indeed, every abelian group is a quotient of a free abelian group, and a free abelian group embeds in a power of $\Z$.

## Explanations

### L1 human

To build the smallest variety containing some given algebras, it is enough to apply three steps once each.
First take products, then subalgebras of those products, then homomorphic images.
No further rounds of the three constructions produce anything new.

### L2 human

The formula gives the standard way to find the members of $V(A)$ for one algebra $A$.
Each member of $V(A)$ is a homomorphic image of a subalgebra of a power $A^I$.
In particular, the free algebras of $V(A)$ are subalgebras of powers of $A$, which is the starting point of the study of free spectra.
In the congruence-distributive case, Jónsson's lemma sharpens the formula for the subdirectly irreducible members.
</document>

<document source="https://tamnd.github.io/math2/node/M4ZJ8QV3MR.md" tag="M4ZJ8QV3MR" kind="statement">
# CORDIC

## Statement

Let $n \ge 1$, let $a_i = \arctan 2^{-i}$ for $i \ge 0$ (see [[arctangent|0CE768XNVP]]), and let $K_n = \prod_{i=0}^{n-1} (1 + 2^{-2i})^{-1/2}$.
Let $\theta$ be a real number with $\abs{\theta} \le \sum_{i=0}^{n-1} a_i$.
The *CORDIC* iteration in rotation mode starts from $x_0 = K_n$, $y_0 = 0$, $z_0 = \theta$ and, for $i = 0, \ldots, n-1$, sets $\sigma_i = 1$ if $z_i \ge 0$ and $\sigma_i = -1$ otherwise, and
$$x_{i+1} = x_i - \sigma_i 2^{-i} y_i, \qquad y_{i+1} = y_i + \sigma_i 2^{-i} x_i, \qquad z_{i+1} = z_i - \sigma_i a_i.$$
Then, in exact arithmetic,

1. $x_n = \cos(\theta - z_n)$ and $y_n = \sin(\theta - z_n)$;
2. $\abs{z_n} \le a_{n-1} < 2^{-(n-1)}$;
3. hence $\abs{x_n - \cos\theta} < 2^{-(n-1)}$ and $\abs{y_n - \sin\theta} < 2^{-(n-1)}$.

## Remarks

Each step multiplies the vector $(x_i, y_i)$ by the matrix with rows $(1, -\sigma_i t)$ and $(\sigma_i t, 1)$, $t = 2^{-i}$, which is $\sqrt{1 + t^2}$ times the [[rotation|0HFREYRSZP]] by the angle $\sigma_i \arctan t$.
After $n$ steps the scale factors multiply to $1/K_n$, which cancels $x_0 = K_n$, and the total angle is $\sum_i \sigma_i a_i = \theta - z_n$; this is clause 1.
For clause 2, concavity of $\arctan$ on $[0, \infty)$ gives $a_i \le 2 a_{i+1}$, hence $a_i \le a_{i+1} + \cdots + a_{n-1} + a_{n-1}$ by induction.
With this, the bound $\abs{z_i} \le a_i + \cdots + a_{n-1} + a_{n-1}$ passes from $z_i$ to $z_{i+1} = z_i - \sigma_i a_i$, and at $i = n$ it gives $\abs{z_n} \le a_{n-1}$.
Clause 3 follows because $\cos$ and $\sin$ are $1$-Lipschitz.
Since $a_0 + a_1 + a_2 + a_3 \approx 1.618 > \pi/2$, for $n \ge 4$ the method covers all $\theta \in [-\pi/2, \pi/2]$; other angles are first reduced by symmetries.
Multiplication by $2^{-i}$ is a shift in binary, so each step needs only shifts, additions and a table lookup of $a_i$.

## Explanations

### L1 human

CORDIC computes sine and cosine without any multiplication.
It turns a vector towards the target angle by smaller and smaller fixed angles: $45°$, about $26.6°$, about $14°$, and so on.
Each turn goes to the left or to the right.
These special angles are chosen so that each turn needs only a binary shift and an addition.
Each step gains about one binary digit of accuracy.
This made it the method of choice in pocket calculators and in hardware without a multiplier.

### L2 human

The same iteration in vectoring mode drives $y$ to $0$ instead of $z$, and then computes $\arctan(y_0/x_0)$ and, up to the known constant factor, the length $\sqrt{x_0^2 + y_0^2}$.
Replacing $\arctan 2^{-i}$ by $\operatorname{artanh} 2^{-i}$ and rotations by hyperbolic rotations gives the hyperbolic functions, the exponential and the logarithm; in that case some steps must be repeated to ensure convergence.
The constant $K_n$ tends to about $0.6073$ and is applied once at the start, not at each step.
</document>

<document source="https://tamnd.github.io/math2/node/M5196PBDCA.md" tag="M5196PBDCA" kind="definition">
# j-invariant of an elliptic curve

## Definition

Let $K$ be a field, and let $E$ be an elliptic curve over $K$.
Let $y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$ be a [[Weierstrass equation|Z3C8B6QTS6]] of $E$, with quantities $b_2, b_4$ and discriminant $\Delta \ne 0$.
Put $c_4 = b_2^2 - 24 b_4$.
The *$j$-invariant* of $E$ is
$$j(E) = \frac{c_4^3}{\Delta} \in K.$$

## Remarks

If the characteristic of $K$ is not $2$ or $3$ and $E$ is $y^2 = x^3 + Ax + B$, then $c_4 = -48A$ and $\Delta = -16(4A^3 + 27B^2)$, so
$$j(E) = 1728 \, \frac{4A^3}{4A^3 + 27B^2}.$$
The value does not depend on the chosen Weierstrass equation of $E$.
Over an [[algebraically closed field|0PF6TM8TWT]], two [[elliptic curves|427N6S6QQW]] are isomorphic if and only if they have the same $j$-invariant.
Moreover, every element of the field is the $j$-invariant of some elliptic curve.
Example: $y^2 = x^3 - x$ has $A = -1$ and $B = 0$, so $j = 1728 \cdot (-4)/(-4) = 1728$.
Non-example: $y^2 = x^3$ has $\Delta = 0$, so it is not an elliptic curve and $c_4^3/\Delta$ is not defined.
Over a field that is not algebraically closed, curves with the same $j$-invariant can fail to be isomorphic: $y^2 = x^3 - x$ and $y^2 = x^3 + x$ both have $j = 1728$ but are not isomorphic over $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/M53B55WESR.md" tag="M53B55WESR" kind="definition">
# Inverse limit of algebras

## Definition

Let $(I, \le)$ be a directed partially ordered set and fix a signature.
An *inverse system* of algebras over $I$ consists of:

1. an algebra $A_i$ for each $i \in I$;
2. a [[homomorphism|KBXSZC6BV1]] $\varphi_{ij} \colon A_j \to A_i$ for each pair $i \le j$;
3. the conditions $\varphi_{ii} = \id_{A_i}$ and $\varphi_{ij} \circ \varphi_{jk} = \varphi_{ik}$ for all $i \le j \le k$.

The *inverse limit* $\varprojlim A_i$ of the system is the set of all $(a_i)_{i \in I}$ in the [[direct product|WJ73XYBNE4]] $\prod_{i \in I} A_i$ such that $\varphi_{ij}(a_j) = a_i$ for all $i \le j$.

## Remarks

The inverse limit is closed under the operations of the product, so it is a [[subalgebra|7CHG6SD365]] of $\prod_i A_i$; it can be empty when the signature has no constant symbols.
The projections $\pi_i \colon \varprojlim A_j \to A_i$ satisfy $\varphi_{ij} \circ \pi_j = \pi_i$.
For every algebra $B$ with homomorphisms $\psi_i \colon B \to A_i$ satisfying $\varphi_{ij} \circ \psi_j = \psi_i$, there is a unique homomorphism $\psi \colon B \to \varprojlim A_i$ with $\pi_i \circ \psi = \psi_i$, namely $\psi(b) = (\psi_i(b))_i$.

Example: let $p$ be a prime, let $I = \N$, let $A_n = \Z/p^n\Z$ as rings, and let $\varphi_{mn}$ be reduction modulo $p^m$ for $m \le n$.
Reduction modulo $p^n$ followed by reduction modulo $p^m$ is reduction modulo $p^m$, so clause 3 holds.
The inverse limit is the ring $\Z_p$ of $p$-adic integers.

Non-example: on $I = \set{1, 2, 3}$ with the usual order, take $A_i = \Z/2\Z$ as groups, $\varphi_{ii} = \varphi_{12} = \varphi_{23} = \id$ and $\varphi_{13} = 0$.
Then $\varphi_{12} \circ \varphi_{23} = \id \ne \varphi_{13}$, so clause 3 fails and the data is not an inverse system.

Some authors allow any partially ordered set, or any small category, as the index; the construction is the same.
</document>

<document source="https://tamnd.github.io/math2/node/M5437FP73G.md" tag="M5437FP73G" kind="statement">
# Arnold's classification of simple critical points

## Statement

Let $f \colon (\C^n, 0) \to (\C, 0)$ be a [[germ|VZXF7RNFKS]] of a [[holomorphic function|0NAHXR1FRH]] with a [[critical point|0YTCMFB4HC]] at $0$.
Call $f$ *simple* if, in a jet space of high enough order, some neighbourhood of $f$ meets only finitely many right equivalence classes.
Equivalently, $f$ has [[modality|8CFTJP1WN8]] $0$.
Then $f$ is simple if and only if it has the following form in some holomorphic coordinates $(x, y, z_1, \ldots, z_{n-2})$ centred at $0$.
It is one of the following germs plus $z_1^2 + \cdots + z_{n-2}^2$, with the obvious changes when $n \le 2$:

1. $A_k$, $k \ge 1$: $x^{k+1} + y^2$;
2. $D_k$, $k \ge 4$: $x^2 y + y^{k-1}$;
3. $E_6$: $x^3 + y^4$;
4. $E_7$: $x^3 + x y^3$;
5. $E_8$: $x^3 + y^5$.

## Remarks

For $n = 1$ only the germs $A_k$, written $x^{k+1}$, occur.
In each case the index is the Milnor number, $\dim_\C \mathcal{O}_n / J(f)$, where $\mathcal{O}_n$ is the ring of convergent power series and $J(f)$ is the ideal of partial derivatives.
The proof proceeds by the corank and the lowest-order terms of $f$, using [[finite determinacy|EGPP69XA4F]] to reduce to polynomials.
It shows that every germ outside the list deforms, by arbitrarily small perturbations, into members of a unimodal family.
These families are $x^3 + y^3 + z^3 + a xyz$, $x^4 + y^4 + a x^2 y^2$ and $x^3 + y^6 + a x^2 y^2$, with suitable conditions on $a$.
Example: $x^2 y + y^3$ is a $D_4$ singularity; its cubic part factors into three distinct lines through $0$.
Over $\R$ the same list holds up to signs of the terms.
For $n = 3$ the zero sets of these germs are the [[ADE singularities|HQNXQBZ6C7]] of surfaces.

## Explanations

### L1 human

Critical points of functions of several variables can be very complicated.
Arnold asked which of them are not part of a continuous family of different shapes, and found a short list.
The list is labelled by the same letters $A$, $D$, $E$ as the simply laced Dynkin diagrams of Lie theory.
Everything else comes in families that depend on at least one continuous parameter.

### L2 human

The theorem is the start of Arnold's classification of critical points by modality, which continues with the unimodal and bimodal families.
The ADE pattern links it with finite subgroups of $SL(2, \C)$, simple Lie algebras and the Du Val surface singularities.
The real forms of these germs with small codimension are the organising centres of the elementary catastrophes in [[Thom's classification|23RVD8K8RB]].
</document>

<document source="https://tamnd.github.io/math2/node/M54ADM0CVF.md" tag="M54ADM0CVF" kind="concept">
# Ideal norm

## Summary

Let $\mathfrak{a}$ be a nonzero ideal in the ring of integers $\mathcal{O}_K$ of a [[number field|MWBDFS2KJ5]].
The norm of $\mathfrak{a}$ is the number of elements of the quotient ring $\mathcal{O}_K / \mathfrak{a}$.
It is multiplicative, and for a principal ideal it is the absolute value of the field norm of a generator.
It measures the size of an ideal.
It is used in the [[finiteness of the class number|F99RM1T02C]] and in the definition of the [[Dedekind zeta function|KXS5DQDFYD]].

## Explanations

### L1 human

In the integers, the ideal $n\Z$ has $\abs{n}$ residue classes, so its size is $\abs{n}$.
The ideal norm extends this to other number rings: it counts the residue classes modulo the ideal.
For example, in the Gaussian integers $\Z[i]$ the ideal generated by $1 + i$ has two classes, so its norm is $2$, and indeed $(1 + i)(1 - i) = 2$.
A smaller ideal has more residue classes: $4\Z$ lies inside $2\Z$ and has norm $4$ instead of $2$.
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<document source="https://tamnd.github.io/math2/node/M58WRZ7PFM.md" tag="M58WRZ7PFM" kind="concept">
# Local coefficient system

## Summary

A local coefficient system on a space $X$ gives an abelian group for each point of $X$.
These groups may be transported along paths.
The transport depends only on the homotopy class of the path.
Homology and cohomology with local coefficients generalise ordinary homology and cohomology with a fixed coefficient group.
They are needed for Poincaré duality on non-orientable manifolds, for obstruction theory and for the Serre spectral sequence of a fibration with non-simply-connected base.

## Explanations

### L1 human

Ordinary cohomology attaches the same group, say $\Z$, to every point of a space.
A local system attaches a copy of a group to each point, but going around a loop may come back with a twist.
For example, on a circle, attach $\Z$ to each point and let one trip around the circle act by $n \mapsto -n$.
Locally nothing has changed, but globally the coefficients are twisted, as the Möbius band is a twisted version of a cylinder.
</document>

<document source="https://tamnd.github.io/math2/node/M5910N7YF0.md" tag="M5910N7YF0" kind="concept">
# Final functor

## Summary

A final functor $F \colon \mathcal{C} \to \mathcal{D}$ is a [[functor|28NQ0KRHD3]] such that restricting a diagram on $\mathcal{D}$ along $F$ does not change its [[colimit|BJS1CGTKQ5]].
It is the categorical form of a cofinal subset of a [[directed set|D9RY4667K6]], as in [[cofinality|BAWHE1W6MH]].
Final functors let one compute a colimit over a smaller or simpler index category.
Some authors call them cofinal functors; the dual notion is an initial functor.

## Explanations

### L1 human

To find the limit of a sequence, you may look only at its terms with even index, since they go just as far out.
A final functor is the same idea for colimits in a category.
For example, if the index category has a terminal object, the colimit of any diagram is simply its value at that object.
This is because the functor that picks out the terminal object is final.
</document>

<document source="https://tamnd.github.io/math2/node/M5AKNYC4QQ.md" tag="M5AKNYC4QQ" kind="example">
# Hopf fibration

## Example

Let $S^3 = \set{(z_0, z_1) \in \C^2 : \abs{z_0}^2 + \abs{z_1}^2 = 1}$, and identify $\R^3$ with $\C \times \R$.
Define $h \colon S^3 \to \R^3$ by
$$h(z_0, z_1) = \bigl(2 z_0 \overline{z_1}, \ \abs{z_0}^2 - \abs{z_1}^2\bigr).$$
The image lies in the unit sphere $S^2$, because
$$\abs{2 z_0 \overline{z_1}}^2 + \bigl(\abs{z_0}^2 - \abs{z_1}^2\bigr)^2 = \bigl(\abs{z_0}^2 + \abs{z_1}^2\bigr)^2 = 1.$$
Two points of $S^3$ have the same image if and only if one is $(\lambda z_0, \lambda z_1)$ for the other $(z_0, z_1)$ and some $\lambda \in \C$ with $\abs{\lambda} = 1$.
So each fibre of $h \colon S^3 \to S^2$ is a circle.
The map $h$ is a fibre bundle with fibre $S^1$, hence a [[Serre fibration|PB9JXQ2KC3]].
Equivalently, $h$ is the quotient map $S^3 \to \C P^1 \cong S^2$, $(z_0, z_1) \mapsto [z_0 : z_1]$.

## Remarks

The [[long exact sequence of homotopy groups|E0RV5D1GQ0]] of $S^1 \to S^3 \to S^2$ gives $\pi_2(S^2) \cong \Z$ and $\pi_3(S^2) \cong \pi_3(S^3) \cong \Z$.
The class of $h$ generates $\pi_3(S^2)$, and $h$ has [[Hopf invariant|TEGTSY7XMX]] $\pm 1$.
Any two fibres are linked circles in $S^3$, with linking number $\pm 1$.
Similar fibrations $S^3 \to S^7 \to S^4$ and $S^7 \to S^{15} \to S^8$ come from the quaternions and the octonions.

## Explanations

### L1 human

The Hopf fibration splits the $3$-sphere into circles, one circle over each point of the ordinary sphere.
No two of these circles are separate: every pair is linked like two rings of a chain.
It was the first example of a map from a sphere to a lower-dimensional sphere that cannot be deformed to a constant map.
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<document source="https://tamnd.github.io/math2/node/M5D95FN7G3.md" tag="M5D95FN7G3" kind="concept">
# Almost split sequence

## Summary

An almost split sequence, or Auslander–Reiten sequence, is a non-split short [[exact sequence|JB7ZBFBA9C]] $0 \to L \to E \to N \to 0$ of modules that is as close to splitting as possible.
Every map into $N$ that is not a split epimorphism lifts to $E$.
Dually, every map out of $L$ that is not a split monomorphism extends to $E$.
Over an Artin algebra, every indecomposable non-projective module $N$ is the end of one, unique up to isomorphism.
These sequences are the building blocks of [[Auslander–Reiten theory|2VY5M3M91Z]].

## Explanations

### L1 human

A short exact sequence glues a module $L$ and a module $N$ into a bigger module $E$.
If it splits, $E$ is just $L$ and $N$ side by side; if not, the glueing is genuine.
An almost split sequence is a genuine glueing that splits as soon as it is pulled back along any map into $N$ that is not a split epimorphism.
In this sense it is the smallest possible non-trivial extension ending at $N$.
For the path algebra of $1 \to 2$, the sequence $0 \to S_2 \to P_1 \to S_1 \to 0$ is almost split.
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<document source="https://tamnd.github.io/math2/node/M5EK9AKXNR.md" tag="M5EK9AKXNR" kind="concept">
# Vanishing cycle

## Summary

A vanishing cycle is a homology class in a smooth fibre of a holomorphic function, see the [[definition|QEZFT0EM0G]].
It shrinks to a point as the fibre approaches a critical fibre.
Near a nondegenerate critical point it is represented by an embedded sphere of the middle dimension.
Vanishing cycles generate the homology of the [[Milnor fibre|X5HMTTGB62]] of an isolated singularity, and [[Picard–Lefschetz theory|FFYQAVT7MB]] computes the monodromy in terms of them.
The idea was later turned into the vanishing cycle functor on sheaves, used in étale cohomology and in the theory of [[perverse sheaves|5WHXRZDZFK]].

## Explanations

### L1 human

Consider the curves $x^2 + y^2 = t$ in $\C^2$ for small $t$.
For $t \ne 0$ each curve is a smooth tube, and it contains the circle of real points of radius $\sqrt{t}$ when $t > 0$.
As $t$ goes to $0$, this circle shrinks to the origin, where the curve becomes two crossing lines.
The circle is the vanishing cycle: a loop that disappears when the fibre becomes singular.
</document>

<document source="https://tamnd.github.io/math2/node/M5FDV1TMBG.md" tag="M5FDV1TMBG" kind="definition">
# Torsion group

## Definition

Let $G$ be a [[group|G5S1M9]].
$G$ is a *torsion group* (or *periodic group*) if every element of $G$ has finite [[order|0WE2NHZJ7B]].
That is, for every $g \in G$ there is an integer $n \ge 1$ with $g^n = e$.

## Remarks

Example: the group $\Q/\Z$, the [[rational numbers|0NCJ5WXN6Z]] modulo the [[integers|0C4JS9RF5M]] under addition.
If $x = \tfrac{a}{b} + \Z$ with $a, b \in \Z$ and $b \ge 1$, then $b x = a + \Z = 0$, so $x$ has finite order.
The Prüfer group is another infinite example.

Non-example: the group $\Z$ under addition.
For $n \ne 0$ and $k \ge 1$, $k n \ne 0$, so $n$ has infinite order.

A group of finite [[exponent|0ZDW0H0VX4]] is a torsion group, but $\Q/\Z$ is a torsion group of infinite exponent.
A finitely generated abelian torsion group is finite.
For nonabelian groups this fails: there are finitely generated infinite torsion groups, such as the Grigorchuk group; see the Burnside problem.
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<document source="https://tamnd.github.io/math2/node/M5N08HD75V.md" tag="M5N08HD75V" kind="problem">
# Hilbert's fourteenth problem

## Problem

Let $k$ be a [[field|04C5ZBG1WT]] and let $K$ be a field with $k \subseteq K \subseteq k(x_1, \ldots, x_n)$.
Is the $k$-algebra $K \cap k[x_1, \ldots, x_n]$ always finitely generated?

## Remarks

The motivating case is a [[ring of invariants|6VTR5EBNYC]]: if a group $G$ acts linearly on $k^n$, then $k[x_1, \ldots, x_n]^G = k(x_1, \ldots, x_n)^G \cap k[x_1, \ldots, x_n]$.
The answer is no in general.
Nagata gave a counterexample in 1959, as the ring of invariants of a linear action of a group on a polynomial ring.
The answer is yes in several important cases.
If $G$ is a [[reductive group|2XS84PWE3S]], the ring of invariants of a rational action on $k[x_1, \ldots, x_n]$ is finitely generated.
In characteristic $0$ this goes back to Hilbert's finiteness theorem of 1890.
Zariski proved that the answer is yes when $K$ has [[transcendence degree|D7TR0YWR9H]] at most $2$ over $k$.
Weitzenböck proved that, in characteristic $0$, the invariants of a linear action of the additive group of $k$ are finitely generated.

## Explanations

### L1 human

Hilbert asked whether the polynomials that stay unchanged under a group of linear substitutions can always be built from finitely many of them.
For many groups, such as finite groups and the general linear group, they can.
Nagata showed that for some groups they cannot, so the answer to Hilbert's question in its general form is negative.
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# Bessel polynomials

## Summary

The Bessel polynomials $y_n$ are polynomials with positive integer coefficients that arise from the [[modified Bessel functions|2ET068BZZH]] of half-integer order.
They solve a second-order differential equation with polynomial coefficients.
They are orthogonal for a weight on a circle in the complex plane, not on an interval of the real line.
Their reversed forms are the denominators of Bessel filters in electrical engineering.

## Explanations

### L1 human

Bessel functions of half-integer order can be written with exponentials and finitely many powers of $1/x$.
The polynomials that appear in this way are the Bessel polynomials.
The first ones are $y_0 = 1$, $y_1 = 1 + x$ and $y_2 = 1 + 3x + 3x^2$.
Engineers use them to design filters that delay all frequencies of a signal by almost the same time.
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<document source="https://tamnd.github.io/math2/node/M5YRA6CK4G.md" tag="M5YRA6CK4G" kind="concept">
# Hölder summation

## Summary

Hölder summation is a [[summability method|8X25Q4XYMN]] that averages the partial sums of a series, then averages these averages.
It repeats the averaging a fixed number $k$ of times.
For $k = 1$ it is [[Cesàro summation|N8ZGG7EW6J]] of order $1$.
For each $k$, the Hölder method of order $k$ sums the same series to the same values as the Cesàro method of order $k$.
Repeated averaging can tame series whose partial sums oscillate with growing size.

## Explanations

### L1 human

One round of averaging turns the running totals $1, 0, 1, 0, \dots$ of $1 - 1 + 1 - \cdots$ into a sequence that tends to $1/2$.
Some series need more rounds.
For $1 - 2 + 3 - 4 + \cdots$, one round of averaging gives a sequence that still jumps between about $1/2$ and $0$.
A second round averages this out and gives $1/4$.
Hölder summation of order $k$ simply means $k$ rounds of averaging.
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<document source="https://tamnd.github.io/math2/node/M6311DWJ1E.md" tag="M6311DWJ1E" kind="concept">
# Matrix summability method

## Summary

A matrix summability method uses an infinite matrix $A = (a_{nk})$ to turn a sequence $(s_k)$ into the weighted sums $\sum_k a_{nk} s_k$.
The method then takes the limit of these sums.
Most classical [[summability methods|8X25Q4XYMN]], such as [[Cesàro summation|N8ZGG7EW6J]], Nörlund means and Euler summation, are of this kind.
Their properties can be read off from the entries of the matrix, as in the [[Silverman–Toeplitz theorem|S4J7SRPXMZ]] and the [[Kojima–Schur theorem|JB8GQBVW6Q]].

## Explanations

### L1 human

Each row of the matrix is a recipe for averaging the terms of the sequence with certain weights.
Row $n$ produces one number, and the method looks at what these numbers do as $n$ grows.
The Cesàro matrix has row $n$ equal to $\frac{1}{n+1}, \dots, \frac{1}{n+1}$ followed by zeros, so it takes plain averages of the first $n + 1$ terms.
Applied to $1, 0, 1, 0, \dots$ it gives averages that tend to $\frac12$.
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<document source="https://tamnd.github.io/math2/node/M665JWCEXY.md" tag="M665JWCEXY" kind="definition">
# Maxwell's equations

## Definition

Let $\varepsilon_0 > 0$ and $\mu_0 > 0$ be constants, the vacuum permittivity and permeability, and put $c = 1 / \sqrt{\varepsilon_0 \mu_0}$.
Let $\rho \colon \R^3 \times \R \to \R$ be a given charge density and $J \colon \R^3 \times \R \to \R^3$ a given current density.
*Maxwell's equations* in vacuum are the system
$$\nabla \cdot E = \frac{\rho}{\varepsilon_0}, \qquad \nabla \cdot B = 0, \qquad \nabla \times E = - \partial_t B, \qquad \nabla \times B = \mu_0 J + \mu_0 \varepsilon_0 \, \partial_t E,$$
for unknown [[vector fields|0299VCW1TH]] $E, B \colon \R^3 \times \R \to \R^3$, the electric and the magnetic field.
Here $\nabla \cdot$ is the [[divergence|0MWV6ZTK6S]] and $\nabla \times$ is the [[curl|0NF6GQNXMY]] in the space variable.

## Remarks

Example: with $\rho = 0$ and $J = 0$, for $E_0 \in \R$, $k > 0$ and $\omega = ck$, the plane wave
$$E = \big(0, E_0 \cos(kx - \omega t), 0\big), \qquad B = \big(0, 0, \tfrac{E_0}{c} \cos(kx - \omega t)\big)$$
is a solution.
Both fields depend only on $x$ and point across the $x$ direction, so both divergences vanish.
One has $(\nabla \times E)_z = \partial_x E_y = -k E_0 \sin(kx - \omega t) = -\partial_t B_z$, and $(\nabla \times B)_y = -\partial_x B_z = \tfrac{k E_0}{c} \sin(kx - \omega t) = \tfrac{1}{c^2} \partial_t E_y$.

Non-example: $E = (-y, x, 0)$ and $B = 0$ with $\rho = 0$ and $J = 0$ is not a solution, because $\nabla \times E = (0, 0, 2)$ while $-\partial_t B = 0$.

The first two equations are Gauss's laws, the third is Faraday's law, and the fourth is the Ampère–Maxwell law.
Taking the divergence of the fourth equation gives the continuity equation $\partial_t \rho + \nabla \cdot J = 0$ as a condition on the sources.
In matter one also uses the fields $D$ and $H$, related to $E$ and $B$ by constitutive relations.
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# Higher homotopy groups are abelian

## Statement

Let $(X, x_0)$ be a pointed space, and let $n \ge 2$.
Then the [[homotopy group|P976WYEFMB]] $\pi_n(X, x_0)$ is an [[abelian group|082GHNPJ53]].

## Remarks

Represent two elements by maps $f, g \colon I^n \to X$ that send $\partial I^n$ to $x_0$.
Up to homotopy, $f$ can be shrunk to a small cube and $g$ to another small cube, with the map equal to $x_0$ elsewhere.
Because $n \ge 2$, there is room in a second coordinate direction to slide the two cubes around each other, which turns $f + g$ into $g + f$.
Algebraically, the sums along the first and the second coordinate are two operations with a common unit that commute with each other.
The [[Eckmann–Hilton argument|EJJYCPWHPA]] shows that they agree and are commutative.
The hypothesis $n \ge 2$ is necessary: the fundamental group of the wedge of two circles is the free group on two generators, which is not abelian.

## Explanations

### L1 human

Loops in a space can be combined, and the order may matter, as in a figure eight.
For spheres of dimension two or more, there is always room to move one past the other, so the order never matters.

### L2 human

The result explains why $\pi_n$ for $n \ge 2$ behaves much like homology, and why the [[Hurewicz theorem|ZA11N5XPZM]] can compare them directly.
The fundamental group still acts on $\pi_n(X, x_0)$, so $\pi_n$ is a module over the group ring $\Z[\pi_1(X, x_0)]$.
The same sliding argument shows that $\pi_1$ of a topological group, or of any H-space, is abelian.
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# Tikhonov regularization

## Summary

Tikhonov regularization is a method to solve an ill-posed linear equation $K x = y$ in a stable way.
Instead of solving the equation, it minimizes $\norm{K x - y}^2 + \alpha \norm{x}^2$ for a small parameter $\alpha > 0$.
The minimizer always exists, is unique, and depends continuously on the data $y$.
It is the standard tool for [[inverse problems|PCMW7KN5P4]] such as first-kind integral equations, and in statistics it is known as [[ridge regression|3KZW2930V2]].

## Explanations

### L1 human

When small errors in the data cause huge errors in the solution, solving the equation exactly is useless.
Tikhonov's idea is to accept a small misfit in the equation in exchange for a solution that is not too large.
The parameter $\alpha$ sets the balance: a larger $\alpha$ gives a smaller, more stable solution that fits the data less well.
For a single equation $k x = y$ with a number $k \ne 0$, the regularized solution is $x = k y / (k^2 + \alpha)$.
It is close to $y / k$ when $\alpha$ is much smaller than $k^2$, and it stays moderate when $k$ is tiny.
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<document source="https://tamnd.github.io/math2/node/M6BN1K0B30.md" tag="M6BN1K0B30" kind="concept">
# Projective line over a ring

## Summary

The projective line over a [[ring|0B1261B5ED]] $R$ extends the usual projective line over a field to any associative ring with identity.
Its points are the pairs $(a, b) \in R^2$ that can be completed to an invertible $2 \times 2$ matrix, taken up to multiplication on the left by units.
The group $\mathrm{GL}(2, R)$ acts on it.
The construction gives models of circle geometries, such as the Möbius plane from the complex numbers and the Minkowski plane from the split-complex numbers.

## Explanations

### L1 human

Over a field, a point of the projective line is a ratio $a : b$ with $a$ and $b$ not both zero.
Over a ring one must be more careful, because a nonzero element need not be invertible.
The rule is that $(a, b)$ must be the first row of an invertible matrix, and two pairs give the same point when one is a unit times the other.
Over the integers, this gives the reduced fractions $a / b$ together with $1 / 0 = \infty$, that is, the rationals with a point at infinity.
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# Chebyshev polynomials

## Description

The Chebyshev polynomials of the first kind are the polynomials $T_n \in \Z[x]$, $n \ge 0$, given by
$$T_0 = 1, \qquad T_1 = x, \qquad T_{n+1} = 2x \, T_n - T_{n-1} \quad (n \ge 1).$$
They are characterised by $T_n(\cos \theta) = \cos(n \theta)$ for all real $\theta$.
The Chebyshev polynomials of the second kind $U_n$ satisfy the same recursion with $U_0 = 1$ and $U_1 = 2x$, and $U_n(\cos \theta) \sin \theta = \sin((n + 1) \theta)$.

## Remarks

The first values are $T_2 = 2x^2 - 1$, $T_3 = 4x^3 - 3x$ and $T_4 = 8x^4 - 8x^2 + 1$.
Check for $T_2$: $\cos(2\theta) = 2 \cos^2 \theta - 1$.
For $n \ge 1$, $T_n$ has degree $n$ and leading coefficient $2^{n-1}$.
Its roots are the $n$ numbers $\cos \frac{(2k - 1)\pi}{2n}$, $k = 1, \dots, n$, all in $(-1, 1)$.
On $[-1, 1]$ one has $\abs{T_n(x)} \le 1$, and the $T_n$ are orthogonal for the weight $(1 - x^2)^{-1/2}$: $\int_{-1}^{1} T_m(x) T_n(x) (1 - x^2)^{-1/2} \, dx = 0$ for $m \ne n$.
They also satisfy $T_m(T_n(x)) = T_{mn}(x)$.
Their roots are good nodes for [[polynomial interpolation|SS1YMS2NPC]] on $[-1, 1]$.

## Explanations

### L1 human

The formula for $\cos(n\theta)$ in terms of $\cos \theta$ is a polynomial, and this polynomial is $T_n$.
For example, $\cos(3\theta) = 4\cos^3 \theta - 3 \cos \theta$, so $T_3 = 4x^3 - 3x$.
Because cosines stay between $-1$ and $1$, these polynomials wave between $-1$ and $1$ on the interval $[-1, 1]$.
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# Analytical hierarchy

## Summary

The analytical hierarchy classifies sets of natural numbers by the quantifiers over sets of numbers needed to define them in [[second-order arithmetic|DC60FYCSPM]].
Its levels are the classes $\Sigma^1_n$, $\Pi^1_n$ and $\Delta^1_n$.
It extends the [[arithmetical hierarchy|7CVJCQKY41]], whose sets all sit in $\Delta^1_1$.
It is the effective, or lightface, counterpart of the [[projective hierarchy|8DZTNVTGY2]] of descriptive set theory.

## Explanations

### L1 human

Arithmetic can talk about numbers; second-order arithmetic can also talk about infinite sets of numbers.
A definition that says "there is a set $X$ such that ..." or "for every set $X$ ..." can describe far more complicated sets.
For example, "this computable ordering of numbers has no infinite descending sequence" says something about all infinite sequences.
The analytical hierarchy counts how many alternations of such set quantifiers a definition needs.
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# Cyclic polytope

## Summary

A cyclic polytope is the [[convex hull|0FQSDC2EQ2]] of $n \ge d + 1$ distinct points on the moment curve $t \mapsto (t, t^2, \dots, t^d)$ in $\R^d$.
Its combinatorial type depends only on the number $n$ of points and the dimension $d$, and it is denoted $C(n, d)$.
Cyclic polytopes are [[simplicial|YVJ15PGKDM]] and [[neighborly|9V4PR1RVXW]].
They have the largest number of faces of each dimension among all $d$-polytopes with $n$ vertices, which is the content of the [[upper bound theorem|BH8R1M1KVF]].

## Explanations

### L1 human

Pick points on a curve that bends in every direction of space, such as $(t, t^2, t^3)$, and take the solid they span.
The curve bends so much that many sets of points become faces.
In four dimensions, any two of the points are joined by an edge of the solid.
That is far more edges than a three-dimensional solid could have with the same number of corners.
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# Bohr almost periodic function

## Definition

Let $f \colon \R \to \C$ be a [[continuous function|0X3WGX95FE]].
For $\varepsilon > 0$, a number $\tau \in \R$ is an *$\varepsilon$-almost period* of $f$ if
$$\sup_{x \in \R} \abs{f(x + \tau) - f(x)} \le \varepsilon.$$
A set $T \subseteq \R$ is *relatively dense* if there is $L > 0$ such that every closed interval of length $L$ contains a point of $T$.
The function $f$ is *Bohr almost periodic* if for every $\varepsilon > 0$ the set of $\varepsilon$-almost periods of $f$ is relatively dense.

## Remarks

Every Bohr almost periodic function is bounded and uniformly continuous.
Bochner proved that $f$ is Bohr almost periodic if and only if every sequence of translates of $f$ has a uniformly convergent subsequence.
So the definition agrees with the general [[almost periodic function|GZBSV8DYGS]] on the group $\R$.

Example: let $f$ be continuous with period $p > 0$.
Every multiple $kp$, $k \in \Z$, satisfies $f(x + kp) = f(x)$, so it is an $\varepsilon$-almost period for every $\varepsilon > 0$.
Every closed interval of length $p$ contains a multiple of $p$, so $f$ is Bohr almost periodic.

Non-example: $f(x) = \arctan x$ is continuous, but not Bohr almost periodic.
For $\tau > 0$ the difference $\arctan(x + \tau) - \arctan x$ is largest at $x = -\tau/2$, where it equals $2 \arctan(\tau/2)$, and the case $\tau < 0$ is symmetric.
So for $0 < \varepsilon < \pi$ the $\varepsilon$-almost periods are the $\tau$ with $\abs{\tau} \le 2\tan(\varepsilon/2)$.
This set is bounded, hence not relatively dense.
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# Analytic index

## Definition

Let $M$ be a closed smooth manifold and let $E$, $F$ be smooth complex [[vector bundles|P4DZM975BV]] over $M$.
Let $P \colon C^\infty(M; E) \to C^\infty(M; F)$ be an [[elliptic differential operator|MT9P2X85V8]].
Then $\ker P$ and $\operatorname{coker} P = C^\infty(M; F) / P(C^\infty(M; E))$ are finite-dimensional.
The *analytic index* of $P$ is the integer
$$\operatorname{ind} P = \dim \ker P - \dim \operatorname{coker} P.$$

## Remarks

If $E$ and $F$ carry Hermitian metrics and $M$ carries a smooth positive density, then $\operatorname{coker} P$ is isomorphic to $\ker P^*$, where $P^*$ is the formal adjoint, so $\operatorname{ind} P = \dim \ker P - \dim \ker P^*$.
Let $m$ be the order of $P$ and let $H^s$ denote Sobolev spaces.
For every real $s$, the extension $P \colon H^{s}(M; E) \to H^{s - m}(M; F)$ is a [[Fredholm operator|8HWGKWD6ZM]].
Its [[Fredholm index|2V9Q4KHY1T]] equals $\operatorname{ind} P$.
The index depends only on the homotopy class of the principal symbol of $P$ through elliptic symbols.

Example: on $S^1 = \R / 2\pi\Z$, let $P = d/d\theta$ on complex functions.
The kernel is the constant functions, of dimension $1$.
The image is the functions with $\int_0^{2\pi} f \, d\theta = 0$, so the cokernel has dimension $1$, and $\operatorname{ind} P = 0$.

Non-example: the zero operator $C^\infty(S^1) \to C^\infty(S^1)$, viewed as an operator of order $1$, is not elliptic, since its principal symbol vanishes.
Its kernel and cokernel are infinite-dimensional, so it has no analytic index.
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# Character space

## Summary

The character space of a commutative complex [[Banach algebra|9KYWN94QDN]] $A$ is the set of nonzero algebra homomorphisms $A \to \C$, with the weak-* topology.
It is also called the maximal ideal space or the spectrum of $A$.
For unital $A$, the characters correspond to the [[maximal ideals|0PV74TR5FP]] of $A$.
It is a locally compact Hausdorff space, compact when $A$ has a unit.
It is the domain of the [[Gelfand transform|VWNSHKJT92]], which turns elements of $A$ into continuous functions.

## Explanations

### L1 human

A character is a way to turn each element of the algebra into a single complex number while respecting sums and products.
For the algebra of continuous functions on a compact space $X$, the characters are exactly the evaluations $f \mapsto f(x)$ at points $x \in X$.
So the character space of $C(X)$ is $X$ itself.
For a general commutative Banach algebra, the character space is the space of "points" on which its elements become functions.
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<document source="https://tamnd.github.io/math2/node/M7RPDE39FM.md" tag="M7RPDE39FM" kind="definition">
# Subshift of finite type

## Definition

Let $A$ be a finite set, called the alphabet.
Let $A^{\Z}$ be the set of all sequences $x = (x_i)_{i \in \Z}$ with $x_i \in A$, with the [[product topology|0Z57EC6YX3]] of the discrete topologies.
Let $\sigma \colon A^{\Z} \to A^{\Z}$ be the shift map, $\sigma(x)_i = x_{i+1}$.
A word $w = w_1 \cdots w_n$ over $A$ occurs in $x$ if $x_{i+1} \cdots x_{i+n} = w$ for some $i \in \Z$.
For a set $F$ of words, let $X_F$ be the set of all $x \in A^{\Z}$ in which no word of $F$ occurs.
A subset $X \subseteq A^{\Z}$ is a *subshift of finite type* if $X = X_F$ for some finite set $F$ of words.

## Remarks

Each $X_F$ is closed and satisfies $\sigma(X_F) = X_F$, so it is a [[shift space|A1XE59ZHNM]].
Example: the golden mean shift is $X_F$ with $A = \set{0, 1}$ and $F = \set{11}$.
It is a subshift of finite type because $F$ is finite.
Non-example: the even shift is the set of $x \in \set{0,1}^{\Z}$ in which between any two $1$s there is an even number of $0$s.
It is a shift space, but it is not of finite type.
Each word $1 0^{2n+1} 1$ is forbidden, while all its proper subwords occur in allowed sequences, so no finite list of forbidden words can exclude all of them.
Every subshift of finite type is conjugate to a vertex shift.
Given a finite set $B$ and a square $0$-$1$ matrix $M$ indexed by $B$, this is the set of $x \in B^{\Z}$ with $M_{x_i x_{i+1}} = 1$ for all $i$.
The same definition with $A^{\Z^d}$ and finite forbidden patterns gives the multidimensional subshifts of finite type.
</document>

<document source="https://tamnd.github.io/math2/node/M7XKVNATFA.md" tag="M7XKVNATFA" kind="statement">
# Backward Euler method

## Statement

Let $f \colon [t_0, T] \times \R^d \to \R^d$ be continuous and [[Lipschitz|0AW1E6YTJY]] in $y$ with constant $L$, and consider the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$.
The *backward Euler method* with step $h > 0$ sets $t_{n+1} = t_n + h$ and defines $y_{n+1}$ by the implicit equation
$$y_{n+1} = y_n + h \, f(t_{n+1}, y_{n+1}).$$
Then:

1. if $hL < 1$, the equation has exactly one solution $y_{n+1}$ for each $y_n$;
2. if the exact solution $y$ is of class $C^3$, the [[local truncation error|T220HF0562]] is $y(t_{n+1}) - y(t_n) - h \, y'(t_{n+1}) = -\frac{h^2}{2} y''(t_n) + O(h^3)$, so the method has order $1$;
3. for the test equation $y' = \lambda y$ with $\lambda \in \C$, it gives $y_{n+1} = R(h\lambda) y_n$ with $R(z) = \frac{1}{1 - z}$, and $\abs{R(z)} < 1$ whenever $\operatorname{Re} z < 0$.

## Remarks

Clause 1: the map $z \mapsto y_n + h f(t_{n+1}, z)$ is a contraction with constant $hL < 1$, so the [[Banach fixed point theorem|0VK6VEH345]] gives a unique fixed point, which fixed-point iteration finds.
Clause 2 follows by Taylor expansion of $y(t_n)$ and $y'(t_{n+1})$ about $t_n$.
Clause 3: if $\operatorname{Re} z < 0$ then $\abs{1 - z} > 1$; so the numerical solution decays whenever the exact one does, for every step size, and the method is A-stable.
Example: for $y' = -1000 y$ and $h = 0.1$, the explicit Euler method multiplies by $1 - 100 = -99$ at each step and explodes, while backward Euler multiplies by $\frac{1}{101}$ and decays like the true solution.
For stiff problems $hL$ is large, and the equation is solved by Newton's method rather than by fixed-point iteration.
So the restriction $hL < 1$ of clause 1 is a feature of that proof, not of the method.

## Explanations

### L1 human

The ordinary Euler method uses the slope at the start of a step; the backward Euler method uses the slope at the end of the step.
Since that endpoint is not yet known, each step requires solving an equation.
The reward is stability: the method does not blow up on problems whose solutions decay very fast, even with large steps.

### L2 human

Backward Euler is the simplest implicit method and the prototype of methods for [[stiff equations|RR5FCG92PN]].
It is both the one-stage Radau IIA Runge–Kutta method and the first backward differentiation formula (BDF1).
Its stability function $R(z) = 1/(1-z)$ tends to $0$ as $z \to -\infty$, a property called L-stability, which damps fast transients strongly.
Its accuracy is only first order, so in practice higher-order implicit methods such as the BDF family or the [[Gauss–Legendre methods|D99FZQ2SQT]] are used, with backward Euler as their simplest relative.
</document>

<document source="https://tamnd.github.io/math2/node/M848AHBRCM.md" tag="M848AHBRCM" kind="concept">
# Amalgamated free product

## Summary

An amalgamated free product glues several algebras of a [[variety|AJNZJD8442]] along a common algebra that maps into each of them.
It is the most general algebra of the variety that receives maps from all the pieces.
In it, the images of the common part are identified.
In category terms it is the colimit of the diagram formed by the maps out of the common algebra.
For two pieces it is the pushout.
For groups, amalgamated free products are central in combinatorial group theory and in the Seifert–van Kampen theorem.

## Explanations

### L1 human

Start with two algebras $B$ and $C$ that both receive a map from a third algebra $A$.
To get the amalgamated free product, form the [[free product|HHNWHS1XTN]] of $B$ and $C$.
Then force the two images of each element of $A$ to be equal.
For example, glue the cyclic groups $\Z/4\Z$ and $\Z/6\Z$ along their subgroups of order $2$.
The result is a group that contains both, in which the two elements of order $2$ coincide; it is isomorphic to $SL_2(\Z)$.
</document>

<document source="https://tamnd.github.io/math2/node/M84YQMSQZX.md" tag="M84YQMSQZX" kind="definition">
# Image of J

## Definition

Let $SO = \operatorname{colim}_n SO(n)$ be the infinite [[special orthogonal group|0CGYNDCWRR]], and let $r \ge 0$.
Let $J \colon \pi_r(SO) \to \pi_r^s$ be the stable [[J-homomorphism|YYJPRD1HYW]] into the [[stable homotopy group of spheres|HGZAYPQBXW]] $\pi_r^s$.
The *image of J* in degree $r$ is the subgroup $\im J \subseteq \pi_r^s$.

## Remarks

By Bott periodicity, for $r \ge 1$ the group $\pi_r(SO)$ is $\Z/2$ if $r \equiv 0, 1 \pmod 8$, $\Z$ if $r \equiv 3, 7 \pmod 8$, and $0$ otherwise.
For $r = 4k - 1$, the image is cyclic of order the denominator of $B_{2k}/4k$, where $B_{2k}$ is the [[Bernoulli number|0KC00K7KY4]].
For $r \equiv 0, 1 \pmod 8$ with $r \ge 1$, $J$ is injective and the image is $\Z/2$.

Example: in degree $3$, $B_2 = 1/6$ gives $B_2 / 4 = 1/24$, and the image of J is all of $\pi_3^s \cong \Z/24$.

Non-example: in degree $2$, $\pi_2(SO) = 0$, so the image of J is $0$; the nonzero element $\eta^2$ of $\pi_2^s \cong \Z/2$ is not in the image of J.

J. F. Adams determined the image, with the upper bound depending on the Adams conjecture, which was later proved by Quillen and by Sullivan.
</document>

<document source="https://tamnd.github.io/math2/node/M8611CV76A.md" tag="M8611CV76A" kind="concept">
# Ahlfors theory

## Summary

Ahlfors theory is a geometric counterpart of [[Nevanlinna theory|1EHH63VNAD]], due to Lars Ahlfors.
It views a [[meromorphic function|0NSD1TPHR0]] on a disc as a surface spread over the [[Riemann sphere|0CDMXGPSES]].
It compares the area of this covering surface with the length of its boundary.
When the boundary is short compared with the area, the surface covers each region of the sphere about equally often.
Its best-known consequence is the Ahlfors five islands theorem.

## Explanations

### L1 human

Think of the image of a disc under a meromorphic function as a sheet of paper laid many times over a sphere, with folds and branch points.
The average number of layers is the area of the sheet divided by the area of the sphere.
Ahlfors showed that, apart from a contribution of the boundary, the number of layers over any small region is close to this average.
He also showed that the boundary contribution is small for a suitable sequence of radii.
This gives Picard-type theorems with a clear geometric meaning.
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<document source="https://tamnd.github.io/math2/node/M8DEV2YM2S.md" tag="M8DEV2YM2S" kind="definition">
# Energy drift

## Definition

Let $H \colon \R^{2d} \to \R$ be smooth, and consider the [[Hamiltonian system|0EDKGA2FAK]] $y' = J \nabla H(y)$, where $J = \begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}$; its exact solutions satisfy $H(y(t)) = H(y(0))$.
Let a numerical method with fixed step size $h > 0$ produce the sequence $y_0, y_1, y_2, \ldots$ from $y_0$.
The energy error at step $n$ is $e_n = H(y_n) - H(y_0)$.
The method shows *energy drift* on this problem if the energy error has a systematic trend in time.
This means that $\abs{e_n}$ is not bounded uniformly in $n$ but grows with the time $t_n = n h$.

## Remarks

Example: for $H(q, p) = \frac12 (q^2 + p^2)$, the explicit Euler method $q_{n+1} = q_n + h p_n$, $p_{n+1} = p_n - h q_n$ gives $q_{n+1}^2 + p_{n+1}^2 = (1 + h^2)(q_n^2 + p_n^2)$.
So $H(y_n) = (1 + h^2)^n H(y_0)$, and the energy error grows exponentially in $n$: the method has energy drift.
Non-example: the symplectic Euler method $p_{n+1} = p_n - h q_n$, $q_{n+1} = q_n + h p_{n+1}$ is a linear map of determinant $1$ and trace $2 - h^2$.
For $0 < h < 2$ its eigenvalues are distinct and of modulus $1$, so the sequence $y_n$ is bounded and so is $e_n$.
There is no drift, only a bounded oscillation of size $O(h)$.
More generally, let $H$ be analytic and use a [[symplectic integrator|B4C25SXFRE]] of order $p$.
If the numerical solution stays in a compact set, backward error analysis gives $e_n = O(h^p)$ over times exponentially long in $1/h$.
In floating-point arithmetic, round-off adds a slow random-walk drift even to such methods.

## Explanations

### L1 human

Energy drift means that the computed energy wanders away from its true constant value with a trend over time, not just with small bounded wiggles.
It is the usual way to judge whether an integrator is fit for long simulations of conservative systems.
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<document source="https://tamnd.github.io/math2/node/M8GMH8GQR6.md" tag="M8GMH8GQR6" kind="concept">
# Elliptic complex

## Summary

An elliptic complex is a sequence of [[differential operators between vector bundles|CGBX1N9V8H]] whose consecutive composites vanish.
Its [[principal symbols|S8GC34D5ZG]] must form an exact sequence at each nonzero covector.
It generalises an [[elliptic operator|MT9P2X85V8]], which is an elliptic complex of length two.
On a compact manifold its cohomology groups are finite-dimensional, and its index is their alternating sum of dimensions.
The [[de Rham complex|GX74C07T0G]] and the Dolbeault complex are the basic examples.

## Explanations

### L1 human

Some important operators come in chains, where applying two consecutive operators always gives zero, as $d \circ d = 0$ for the exterior derivative.
Then one studies the cohomology: what the next operator kills, modulo what the previous one produces.
An elliptic complex is a chain of this kind that is elliptic in the sense that, at top order and in every nonzero frequency direction, the chain is exact.
This makes the cohomology finite-dimensional on a closed manifold.
For the de Rham complex, this cohomology is the de Rham cohomology, and its alternating sum of dimensions is the Euler characteristic.
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<document source="https://tamnd.github.io/math2/node/M8NA9X8BTV.md" tag="M8NA9X8BTV" kind="definition">
# Modular form

## Definition

Let $k$ be an integer and let $\mathbb{H} = \set{\tau \in \C : \operatorname{Im} \tau > 0}$ be the [[upper half-plane|X6XGGH5FPW]].
A *modular form of weight $k$* for the [[modular group|VEKV1EM77B]] $\mathrm{SL}_2(\Z)$ is a function $f \colon \mathbb{H} \to \C$ such that:

1. $f$ is [[holomorphic|0NAHXR1FRH]] on $\mathbb{H}$;
2. $f\left(\frac{a\tau + b}{c\tau + d}\right) = (c\tau + d)^k f(\tau)$ for all $\tau \in \mathbb{H}$ and all $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\Z)$;
3. $f$ is bounded on the region $\set{\tau \in \mathbb{H} : \operatorname{Im} \tau \ge 1}$.

## Remarks

Condition 2 with $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ gives $f(\tau + 1) = f(\tau)$, so $f(\tau) = \sum_{n \in \Z} a_n q^n$ with $q = e^{2 \pi i \tau}$.
Condition 3 then says $a_n = 0$ for $n < 0$; this is "holomorphic at infinity".
Example: every constant function is a modular form of weight $0$.
Example: the [[Eisenstein series|EYREVY31BN]] $G_4$ is a nonzero modular form of weight $4$.
Non-example: the [[j-invariant|FEC2XH822R]] satisfies 1 and 2 with $k = 0$, but $j = q^{-1} + 744 + \cdots$ is unbounded as $\operatorname{Im} \tau \to \infty$, so 3 fails.
If $k$ is odd, the matrix $-I$ gives $f = (-1)^k f$, so $f = 0$.
For a [[congruence subgroup|1H6R856P3C]] $\Gamma$, condition 2 is required only for matrices in $\Gamma$.
Condition 3 is then required at every cusp: $(c\tau + d)^{-k} f(\gamma \tau)$ is bounded on $\operatorname{Im} \tau \ge 1$ for every $\gamma \in \mathrm{SL}_2(\Z)$.
The modular forms of weight $k$ form a finite-dimensional complex vector space $M_k$.
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<document source="https://tamnd.github.io/math2/node/M8Q24XB0AF.md" tag="M8Q24XB0AF" kind="concept">
# Opposite ring

## Summary

The opposite ring of a [[ring|0B1261B5ED]] $R$ has the same elements and the same addition as $R$, with the factors of every product swapped.
A ring equals its opposite exactly when it is commutative.
The opposite ring turns right modules into left modules and back.
So results proved for one side carry over to the other.
It also appears in the endomorphism ring of a ring viewed as a module over itself.

## Explanations

### L1 human

Take a ring and read every product backwards: the new product of $a$ and $b$ is the old product $ba$.
The result is again a ring, called the opposite ring.
For commutative rings nothing changes.
For matrices over a commutative ring such as $\R$, the transpose turns $AB$ into $B^T A^T$, so the transpose map identifies the matrix ring with its opposite.
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<document source="https://tamnd.github.io/math2/node/M8TAKF85NT.md" tag="M8TAKF85NT" kind="definition">
# Wandering set

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
A set $W \subseteq X$ is a *wandering set* for $f$ if $f^n(W) \cap W = \emptyset$ for every integer $n \ge 1$.
A point $x \in X$ is a *wandering point* if it has an [[open|02BH3MCBQB]] neighbourhood $U$ that is a wandering set.

## Remarks

If $f$ is a [[homeomorphism|0QT00PB5G2]], the images $f^n(W)$, $n \in \Z$, of a wandering set are pairwise disjoint.
The set of wandering points is open.
In measurable dynamics, let $T$ be a measurable map of a [[measure space|P15VQS5DJ9]] $(X, \mathcal{A}, \mu)$.
A measurable set $W$ of positive measure is called wandering when the sets $T^{-n}(W)$, $n \ge 0$, are pairwise disjoint.
Example: for $f(x) = x + 1$ on $\R$, the set $W = (0, 1)$ has $f^n(W) = (n, n+1)$, which is disjoint from $W$ for $n \ge 1$; so every point of $\R$ is wandering.
Non-example: for $f(x) = x/2$ on $\R$, any open interval $U$ containing $0$ satisfies $0 \in f^n(U) \cap U$ for all $n$, so $U$ is not wandering and $0$ is not a wandering point.

## Explanations

### L1 human

A wandering set never overlaps with any of its later images.
A wandering point is a point with a small open neighbourhood that wanders away and never comes back to meet itself.
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<document source="https://tamnd.github.io/math2/node/M8W64BRCQH.md" tag="M8W64BRCQH" kind="definition">
# Wiener–Hopf equation

## Definition

Let $k \in L^1(\R)$, a function in the [[$L^1$ space|SASP14MA1Z]] of $\R$, and let $f$ be a given function on $[0, \infty)$.
A *Wiener–Hopf equation of the second kind* for an unknown function $\varphi$ on $[0, \infty)$ is
$$\varphi(x) - \int_0^\infty k(x - t)\, \varphi(t)\, dt = f(x), \quad x \ge 0.$$
A *Wiener–Hopf equation of the first kind* is
$$\int_0^\infty k(x - t)\, \varphi(t)\, dt = f(x), \quad x \ge 0.$$
The kernel $k(x - t)$ is a [[difference kernel|B5AB451TB7]], and both $x$ and $t$ range over the half-line $[0, \infty)$.

## Remarks

The unknown is usually sought in $L^p(0, \infty)$ with $1 \le p \le \infty$, or in a space of bounded or of polynomially growing functions.
Kernels such as $\frac{1}{2} E_1(\abs{x})$, with $E_1$ the [[exponential integral|EV6ZDQ0NDY]], are also used; this one is in $L^1(\R)$.

Example: $k(x) = \frac{1}{2} e^{-\abs{x}}$ and $f(x) = \frac{1}{2} e^{-x}$.
The constant $\varphi = 1$ is a solution.
Check: $\int_0^\infty \frac{1}{2} e^{-\abs{x - t}}\, dt = \frac{1}{2}(1 - e^{-x}) + \frac{1}{2} = 1 - \frac{1}{2} e^{-x}$, so $\varphi - \int_0^\infty k(x - t) \varphi(t)\, dt = \frac{1}{2} e^{-x}$.
The function $1 + x$ solves the homogeneous equation with the same kernel, so this equation has more than one solution among functions of linear growth.

Non-example: $\varphi(x) - \int_{-\infty}^{\infty} k(x - t)\, \varphi(t)\, dt = f(x)$ for all $x \in \R$ is a convolution equation on the whole line.
It is not a Wiener–Hopf equation, and the Fourier transform solves it directly when $1 - \hat k$ has no zeros.

The integral operator $\varphi \mapsto \int_0^\infty k(x - t) \varphi(t)\, dt$ is the continuous analogue of a Toeplitz matrix, and it is called a Wiener–Hopf operator.
</document>

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# Product sigma-algebra

## Definition

Let $I$ be a nonempty index set and, for each $i \in I$, let $(X_i, \Sigma_i)$ be a [[measurable space|N4KDT83V3S]].
Let $X = \prod_{i \in I} X_i$ be the [[Cartesian product|09CEDJYBKP]], and let $\pi_i \colon X \to X_i$ be the coordinate projections.
The *product σ-algebra* $\bigotimes_{i \in I} \Sigma_i$ is the [[σ-algebra|0SV8K0TDGY]] on $X$ generated by the sets $\pi_i^{-1}(A)$ with $i \in I$ and $A \in \Sigma_i$.
Equivalently, it is the smallest σ-algebra on $X$ for which every $\pi_i$ is [[measurable|YYNCJSR0PJ]].

## Remarks

For two factors, $\Sigma_1 \otimes \Sigma_2$ is generated by the rectangles $A_1 \times A_2$ with $A_1 \in \Sigma_1$ and $A_2 \in \Sigma_2$, since $A_1 \times A_2 = \pi_1^{-1}(A_1) \cap \pi_2^{-1}(A_2)$.
For general $I$, it is generated by the cylinder sets $\prod_{i \in J} A_i \times \prod_{i \notin J} X_i$ with $J \subseteq I$ finite.

Example: $\mathcal{B}(\R) \otimes \mathcal{B}(\R) = \mathcal{B}(\R^2)$, where $\mathcal{B}$ denotes the [[Borel σ-algebra|0WQXY32Q8D]].
Each rectangle of Borel sets is Borel in $\R^2$, which gives one inclusion.
Each open set of $\R^2$ is a countable union of open rectangles with rational corners, which gives the other.

Non-example: let $\mathcal{L}$ be the σ-algebra of Lebesgue measurable subsets of $\R$.
Then $\mathcal{L} \otimes \mathcal{L}$ is strictly smaller than the σ-algebra of Lebesgue measurable subsets of $\R^2$.
For a non-measurable $N \subseteq \R$, the set $\set{0} \times N$ is a Lebesgue null set in $\R^2$, but its section at $0$ is $N \notin \mathcal{L}$, and every set in $\mathcal{L} \otimes \mathcal{L}$ has measurable sections.

A function $f \colon Z \to X$ is measurable for $\bigotimes_i \Sigma_i$ if and only if every coordinate $\pi_i \circ f$ is measurable.
</document>

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# Nonlinear potential theory

## Summary

Nonlinear potential theory extends [[potential theory|9PM8Q2E30Q]] from the Laplace equation to nonlinear equations, above all the [[p-Laplace equation|XJMYGY0ZZF]].
Its objects are [[p-harmonic functions|7BKQT71FYZ]], $p$-superharmonic functions, $p$-capacities and nonlinear potentials such as the [[Wolff potential|EBD8TWFNS0]].
Without linearity there is no superposition, so the tools are variational methods, comparison principles and estimates instead of Green's functions.
Many classical results survive, including a Wiener criterion for regular boundary points.

## Explanations

### L1 human

Classical potential theory studies harmonic functions, which minimize the energy $\int \abs{\nabla u}^2$.
Nonlinear potential theory studies functions that minimize energies such as $\int \abs{\nabla u}^p$ with $p \ne 2$.
For these equations the sum of two solutions is no longer a solution, so a potential cannot be built by adding up the effects of point charges.
Even so, one can define the size of a set, its capacity, and decide which boundary points are well behaved, much as in the classical case.
The Wiener criterion for the $p$-Laplacian was proved sufficient by Maz'ya and necessary by Kilpeläinen and Malý.
</document>

<document source="https://tamnd.github.io/math2/node/M92S0Y15N1.md" tag="M92S0Y15N1" kind="statement">
# Crank–Nicolson method

## Statement

Let $\kappa > 0$ and consider the [[heat equation|VP2HN86VAG]] $u_t = \kappa u_{xx}$ on $0 < x < 1$, $t > 0$, with $u(0, t) = u(1, t) = 0$ and a given initial value $u(x, 0)$.
Let $M \ge 1$, $\Delta x = 1/(M + 1)$, $x_j = j \Delta x$, $\Delta t > 0$, $t_n = n \Delta t$ and $r = \kappa \Delta t / \Delta x^2$.
For a grid vector $v = (v_0, \ldots, v_{M+1})$ with $v_0 = v_{M+1} = 0$, put $(\delta^2 v)_j = (v_{j+1} - 2 v_j + v_{j-1}) / \Delta x^2$.
The Crank–Nicolson method computes $u^{n+1}$ from $u^n$ by
$$\frac{u_j^{n+1} - u_j^n}{\Delta t} = \frac{\kappa}{2} \left( (\delta^2 u^{n+1})_j + (\delta^2 u^n)_j \right), \qquad 1 \le j \le M.$$

1. Each step has a unique solution $u^{n+1}$, given by a tridiagonal linear system.
2. The method is unconditionally stable: $\norm{u^{n+1}}_{2} \le \norm{u^n}_2$ for every $r > 0$, where $\norm{v}_2^2 = \Delta x \sum_{j=1}^M v_j^2$.
3. Assume the exact solution has bounded derivatives $u_{ttt}$, $u_{xxtt}$ and $u_{xxxx}$. Then the local truncation error is $O(\Delta t^2 + \Delta x^2)$. On each interval $[0, T]$ the error $\max_{n \Delta t \le T} \norm{u(\cdot, t_n) - u^n}_2$ is also $O(\Delta t^2 + \Delta x^2)$.

## Remarks

The vectors $s^{(k)}_j = \sin(k \pi x_j)$, $1 \le k \le M$, are eigenvectors of $\delta^2$ with eigenvalues $-\frac{4}{\Delta x^2} \sin^2 \frac{k \pi \Delta x}{2}$, and they are orthogonal.
In this basis each step multiplies the $k$-th component by $g_k = \frac{1 - 2 r \sigma_k}{1 + 2 r \sigma_k}$ with $\sigma_k = \sin^2 \frac{k \pi \Delta x}{2} \in (0, 1)$, and $\abs{g_k} < 1$ for all $r > 0$; this gives clauses 1 and 2.
Clause 3 follows from Taylor expansion about $(x_j, t_n + \Delta t/2)$ and from stability, as in the [[Lax equivalence theorem|D5M7TRD3MY]].
The method is the trapezoidal rule applied to the [[method of lines|FNGWJR2FPK]] semi-discretisation $u' = \kappa \delta^2 u$.
For large $r$, $g_k$ is close to $-1$ for the high frequencies, so rough initial data produce slowly decaying oscillations: the method is A-stable but not L-stable.

## Explanations

### L1 human

The Crank–Nicolson method advances the heat equation in time by averaging the explicit and the implicit [[finite difference|X7HYEBA5SA]] formulas.
This averaging centres the scheme at the half step, which makes it second-order accurate in time as well as in space.
It is stable for every time step, unlike the explicit method, which needs $r \le 1/2$.
The price is a tridiagonal system to solve at each step, which costs only a few operations per grid point.

### L2 human

It is the default second-order method for diffusion problems, including the Black–Scholes equation in finance.
For non-smooth data it is often started with a few backward Euler steps (Rannacher smoothing) to damp the oscillating modes.
In several space dimensions the systems are no longer tridiagonal, which motivates alternating-direction implicit methods.
</document>

<document source="https://tamnd.github.io/math2/node/M9AQ5HKGP4.md" tag="M9AQ5HKGP4" kind="concept">
# Analysis of variance

## Summary

Analysis of variance, or ANOVA, is a family of methods that split the total variation of a response into parts.
Each part is due to one source, such as treatment groups or random error.
Its best known use is to test whether several group means are equal.
The test compares the variation between groups with the variation within groups through an [[F-test|8MTVCMAATD]].
It was developed by Ronald Fisher for agricultural experiments and is a central tool of [[design of experiments|2TE42ZWJ3E]].
The fixed-effects models of ANOVA are special cases of the [[general linear model|A06SGFCSCE]].

## Explanations

### L1 human

Suppose three fertilisers are each tried on several plots and the yields vary.
Some of the variation is because the fertilisers differ, and some is just plot-to-plot noise.
ANOVA measures both: how far the group averages are from the overall average, and how far the plots are from their own group average.
If the first is large compared with the second, the fertilisers probably differ.
For the groups $\{1, 2, 3\}$ and $\{4, 5, 6\}$, the between-group sum of squares is $13.5$ and the within-group sum is $4$, which together make the total $17.5$.
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# Homoclinic bifurcation

## Definition

Let $\dot x = F(x, \mu)$, $\mu \in \R$, be a smooth one-parameter family of vector fields on $\R^n$.
Assume that for $\mu$ near $\mu_0$ the system has a [[hyperbolic equilibrium|01TPMXM6NM]] $x_\mu$ of saddle type, depending smoothly on $\mu$.
The family has a *homoclinic bifurcation* at $\mu_0$ if the following hold.

1. At $\mu = \mu_0$ there is a [[homoclinic orbit|6DBSQKZK4W]] $\Gamma$ to $x_{\mu_0}$: an orbit with $x(t) \to x_{\mu_0}$ as $t \to +\infty$ and as $t \to -\infty$.
2. For $\mu \ne \mu_0$ close to $\mu_0$ there is no homoclinic orbit to $x_\mu$ near $\Gamma$.

## Remarks

The orbit $\Gamma$ lies in both the stable and the unstable manifold of $x_{\mu_0}$, given by the [[stable manifold theorem|WF22KK7C3R]].
It is a global bifurcation in the sense of [[bifurcation theory|D9TW46P19Y]], since it depends on a whole orbit and not only on the equilibrium.
In the plane, let $\lambda_1 < 0 < \lambda_2$ be the eigenvalues at the saddle and $\sigma = \lambda_1 + \lambda_2$.
If $\sigma \ne 0$ and the loop splits transversally in $\mu$, a unique [[limit cycle|0AAQ73WRFB]] appears near $\Gamma$ on one side of $\mu_0$, stable if $\sigma < 0$ and unstable if $\sigma > 0$.
Example: for $\dot x = y$, $\dot y = x - x^2 + \mu y$, let $H = y^2/2 - x^2/2 + x^3/3$.
Along solutions $\dot H = \mu y^2$.
At $\mu = 0$ the level set $H = 0$ contains a homoclinic loop to the saddle $(0, 0)$, in the half-plane $x > 0$.
For $\mu \ne 0$, $H$ is strictly monotone along nonconstant orbits, so no orbit can leave and return to the saddle, and the loop is broken.
Here $\sigma = \mu$ vanishes at $\mu_0 = 0$, so this example is degenerate and no limit cycle is born.
Non-example: the constant family $\dot x = y$, $\dot y = x - x^2$, which does not depend on $\mu$, fails clause 2, since the loop persists for every $\mu$.
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# Parabolic subgroup

## Definition

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center.
Let $G = KAN$ be an [[Iwasawa decomposition|4YDR9Q36DS]], with $\mathfrak{a}$ the Lie algebra of $A$.
Let $M = Z_K(\mathfrak{a})$ be the set of $k \in K$ with $\mathrm{Ad}(k) H = H$ for all $H \in \mathfrak{a}$.
The group $P_0 = MAN$ is the *minimal parabolic subgroup*.
A *parabolic subgroup* of $G$ is a closed subgroup $P$ of $G$ such that $g P_0 g^{-1} \subseteq P$ for some $g \in G$.

## Remarks

Example: $G = \mathrm{SL}_2(\R)$, with $K = \mathrm{SO}(2)$, $A$ the positive diagonal matrices and $N$ the upper unipotent matrices.
Then $M = \set{\pm 1}$, and $P_0 = MAN$ is the group of all upper triangular matrices in $\mathrm{SL}_2(\R)$.
So $P_0$ is parabolic, with $g = e$.

Non-example: $\mathrm{SO}(2)$ in $\mathrm{SL}_2(\R)$ is closed, but it is not parabolic.
It is compact, so each subgroup of it is relatively compact, while each conjugate of $P_0$ contains a conjugate of $A \cong \R$ and is not.

In $\mathrm{SL}_n(\R)$, the minimal parabolic subgroup is the group of upper triangular matrices, and the parabolic subgroups are the conjugates of the block upper triangular subgroups.
Equivalently, $P$ is parabolic when it is the normalizer in $G$ of a subalgebra $\mathfrak{q}$ of $\mathfrak{g}$ whose complexification contains a Borel subalgebra of $\mathfrak{g}_\C$.

For complex or algebraic groups, a parabolic subgroup is a closed subgroup containing a [[Borel subgroup|5JSYA1JNCC]].
The phrase also has a different meaning for Coxeter groups: a subgroup generated by a subset of the simple reflections.
</document>

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# Unbounded operator

## Definition

Let $X$ and $Y$ be [[normed spaces|03J5RAKB30]] over $\R$ or $\C$.
An *unbounded operator* (or *linear operator*) $T$ from $X$ to $Y$ is a pair consisting of
1. a [[linear subspace|00YC1D652K]] $D(T)$ of $X$, the *domain* of $T$, and
2. a [[linear map|0NRYHJPQG2]] $T \colon D(T) \to Y$.

Two such operators $S$ and $T$ are equal when $D(S) = D(T)$ and $Sx = Tx$ for all $x \in D(T)$.
The operator $T$ is an *extension* of $S$, written $S \subseteq T$, when $D(S) \subseteq D(T)$ and $Tx = Sx$ for all $x \in D(S)$.

## Remarks

Example: on $X = Y = L^2(0, 1)$, let $D(T)$ be the space of continuously differentiable functions on $[0, 1]$, and let $Tf = f'$.
This is linear, so $T$ is an unbounded operator.
It is not bounded: $f_n(x) = \sin(n \pi x)$ has $\norm{f_n} = 1/\sqrt{2}$, while $\norm{T f_n} = n \pi / \sqrt{2}$.

Non-example: the rule $f \mapsto f'$ is not an operator with domain all of $L^2(0, 1)$, since many elements of $L^2(0, 1)$ have no derivative in $L^2(0, 1)$.

Every [[continuous linear map|03TM135G0N]] $X \to Y$ is an unbounded operator with domain $X$; the word "unbounded" means "not necessarily bounded".
Sums and products are defined on the natural domains: $D(S + T) = D(S) \cap D(T)$ and $D(ST) = \set{x \in D(T) : Tx \in D(S)}$.
</document>

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# Statistical learning theory

## Definition

*Statistical learning theory* is the study of the following setting.

1. There are an input space $\mathcal X$, an output space $\mathcal Y$ and an unknown probability distribution $P$ on $\mathcal X \times \mathcal Y$.
2. A [[loss function|3BTG6VVCGZ]] $L \colon \mathcal Y \times \mathcal Y \to [0, \infty)$ measures the cost $L(f(x), y)$ of predicting $f(x)$ when the truth is $y$.
3. The *risk* of a measurable function $f \colon \mathcal X \to \mathcal Y$ is $R(f) = E[L(f(X), Y)]$, where $(X, Y)$ has law $P$.
4. A learning algorithm maps a sample $(X_1, Y_1), \dots, (X_n, Y_n)$ of [[independent and identically distributed|00MRN0FARV]] pairs with law $P$ to a function $\hat f_n$.
   Usually $\hat f_n$ lies in a fixed class $\mathcal F$.

Its questions are how fast $R(\hat f_n)$ approaches $\inf_{f \in \mathcal F} R(f)$ or the best possible risk, uniformly over the distributions $P$ in some class.

## Remarks

A central tool is the uniform deviation $\sup_{f \in \mathcal F} \abs{R_n(f) - R(f)}$, where $R_n(f) = \frac1n \sum_{i=1}^{n} L(f(X_i), Y_i)$ is the empirical risk; it controls [[empirical risk minimization|56NVR0SS61]].
For binary classification with the $0$-$1$ loss, this deviation tends to $0$ uniformly over all $P$ if and only if $\mathcal F$ has finite [[VC dimension|KCFXR9K3GD]].

Example: binary classification with $\mathcal Y = \set{0, 1}$, the $0$-$1$ loss $L(a, y) = \mathbf{1}\set{a \ne y}$ and $\mathcal F$ the half-planes in $\mathcal X = \R^2$ fits the setting; then $R(f)$ is the probability of a misclassification.

Non-example: clustering points with no labels does not fit clauses 2 and 3 as stated, since there is no output to predict.
It belongs to unsupervised learning.
</document>

<document source="https://tamnd.github.io/math2/node/M9SW4NVP2H.md" tag="M9SW4NVP2H" kind="concept">
# Sylvester equation

## Summary

The Sylvester equation is the linear matrix equation $AX + XB = C$, in which the matrices $A$, $B$, $C$ are given and the matrix $X$ is unknown.
It has a unique solution exactly when $A$ and $-B$ have no common [[eigenvalue|0BQAA2BK3Q]].
It appears in block diagonalisation, in control theory, and as the general form of the [[Lyapunov equation|92GVPZH9FT]].

## Explanations

### L1 human

The equation asks for a matrix $X$ such that multiplying it by $A$ on the left and by $B$ on the right, and adding, gives $C$.
Although $X$ is a matrix, the equation is linear in its entries, so it is a large system of ordinary linear equations.
For $1 \times 1$ matrices it reads $ax + xb = c$, with the solution $x = c / (a + b)$ when $a + b \ne 0$.
The condition $a + b \ne 0$ becomes the condition on eigenvalues for larger matrices.
</document>

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# Gabor frame

## Summary

A Gabor frame is a frame of $L^2(\R)$ made of time shifts and frequency shifts of one window function $g$.
The shifts run over a lattice $a\Z \times b\Z$ of the time–frequency plane.
It gives a stable, discrete version of the [[short-time Fourier transform|3DHNP5NEWH]].
A function is determined by the samples of $V_g f$ on the lattice, and can be rebuilt from them.
A Gabor frame can exist only when $ab \le 1$.
At $ab = 1$, the [[Balian–Low theorem|XVVHKCYF2Z]] shows that the window cannot be well localized in both time and frequency.

## Explanations

### L1 human

Pick one short wave packet, say a bell curve.
Make copies of it shifted in time by multiples of $a$ and shifted in pitch by multiples of $b$.
If these copies are dense enough in the time–frequency plane, every signal can be written as a stable combination of them.
This is how one turns a continuous time–frequency picture into a grid of numbers that still holds all the information.
</document>

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# Harish-Chandra's regularity theorem

## Statement

Let $G$ be a connected [[real reductive Lie group|42ZCE14AD2]] with finite center and Haar measure $dg$.
Let $G'$ be the open dense subset of regular semisimple elements of $G$.
Let $\mathfrak Z$ be the center of the [[universal enveloping algebra|SCZ8PRVY00]] of the complexified Lie algebra of $G$.
It acts on distributions on $G$ as bi-invariant differential operators.
Let $\Theta$ be a distribution on $G$ such that:

1. $\Theta$ is invariant under conjugation by $G$;
2. there is a homomorphism $\chi \colon \mathfrak Z \to \C$ with $z \Theta = \chi(z) \Theta$ for all $z \in \mathfrak Z$.

Then there is a locally integrable function $F$ on $G$, real analytic on $G'$ and constant on [[conjugacy classes|0DEBXAJ5SA]], with the following property.
For every smooth compactly supported $f$ on $G$, one has $\Theta(f) = \int_G f(g) F(g) \, dg$.
In particular, the [[Harish-Chandra character|F2E3317P78]] $\Theta_\pi$ of an irreducible [[admissible representation|SZV8Y806Y7]] $\pi$ is given by such a function.

## Remarks

The character of an irreducible admissible representation satisfies the hypotheses, because $\mathfrak Z$ acts on such a representation by scalars, its infinitesimal character.
The proof first treats invariant eigendistributions on the Lie algebra, using the radial parts of invariant differential operators along Cartan subalgebras.
It then transfers the result to the group through the exponential map near each semisimple element.
On $G'$ the eigen-equations become an elliptic system, which gives analyticity there; the hard part is the local integrability across the singular set $G \setminus G'$.
For a finite-dimensional representation, $F$ is the ordinary character $g \mapsto \operatorname{tr} \pi(g)$, which is analytic on all of $G$.
For infinite-dimensional representations $F$ is usually singular on $G \setminus G'$, so analyticity on all of $G$ cannot be expected.

## Explanations

### L1 human

The character of an infinite-dimensional representation is a priori only a generalised function.
The theorem says that it is in fact an honest function, smooth away from a thin bad set, and still integrable near that set.
This makes it possible to write characters by explicit formulas, as for finite groups.

### L2 human

The theorem is the starting point of Harish-Chandra's harmonic analysis on real reductive groups.
Explicit formulas for the function $F$ on each Cartan subgroup lead to the construction of the discrete series and to the Plancherel formula.
An analogue holds for reductive groups over $p$-adic fields, also due to Harish-Chandra, where the proof is different.
</document>

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# Fisher's inequality

## Statement

Let $v, k, \lambda$ be positive integers with $k < v$.
Let $(X, \mathcal{B})$ be a $2$-$(v, k, \lambda)$ [[block design|TGZATGBHTH]] with $b$ blocks.
Then $b \ge v$.

## Remarks

Proof idea: let $r$ be the number of blocks through a point, and let $N$ be the $v \times b$ [[incidence matrix|9DNFN8M83X]] of points against blocks.
Then $N N^{T} = (r - \lambda) I + \lambda J$, where $I$ is the identity matrix and $J$ is the all-ones matrix of size $v$.
Its [[determinant|0C3CFHDN6R]] is $rk (r - \lambda)^{v - 1}$.
From $r(k - 1) = \lambda(v - 1)$ and $k < v$ one gets $r > \lambda$, so the determinant is not zero.
So $N$ has rank $v$, and a $v \times b$ matrix of rank $v$ needs $b \ge v$.
Example: the [[Fano plane|QQ8GW1ASZM]] is a $2$-$(7, 3, 1)$ design with $b = v = 7$, so the bound is sharp.
The designs with $b = v$ are the symmetric designs; the [[Bruck–Ryser–Chowla theorem|R43K0GTTDS]] restricts their parameters.
The hypothesis $k < v$ is needed.
If $k = v$, every block is the whole set $X$, and $\lambda$ copies of $X$ form a design with $b = \lambda$ blocks, which can be less than $v$.

## Explanations

### L1 human

In a balanced design where every two points share the same number of blocks, there are at least as many blocks as points.
So one cannot save on blocks: balance costs at least one block per point.

### L2 human

Fisher's inequality is a necessary condition for the existence of a design, beyond the divisibility conditions $bk = vr$ and $r(k-1) = \lambda(v-1)$.
Its proof by the rank of the incidence matrix is a model of the [[linear algebra method|DMYGZ3YZ6V]] in combinatorics.
The equality case $b = v$ leads to symmetric designs, such as [[finite projective planes|2HWB7P1DZN]] and the designs built from [[difference sets|MG1KM4S1BV]].
</document>

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# Grünwald–Letnikov derivative

## Summary

The Grünwald–Letnikov derivative of real order $\alpha$ is a limit of weighted sums of values of the function.
It extends the [[difference quotient|AZBDB5F4M4]] that defines ordinary derivatives.
The weights are generalised [[binomial coefficients|0GBZM0MT0F]] $\binom{\alpha}{k}$ with alternating signs.
For whole orders it gives the usual derivatives, and for smooth enough functions it agrees with the Riemann–Liouville derivative of [[fractional calculus|T73XK15SA2]].
Its discrete form is the basis of many numerical methods for fractional differential equations.

## Explanations

### L1 human

The first derivative is the limit of $(f(x) - f(x - h))/h$, and the second derivative is the limit of $(f(x) - 2f(x - h) + f(x - 2h))/h^2$.
The coefficients $1, -1$ and $1, -2, 1$ are binomial coefficients with alternating signs.
Binomial coefficients $\binom{\alpha}{k}$ make sense for any real $\alpha$, so the same recipe defines a derivative of order $\alpha$.
For non-integer $\alpha$ the sum does not stop, so the derivative at $x$ depends on all earlier values of $f$ back to a starting point.
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# Plurisubharmonic exhaustion function

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]], for example a domain in $\C^n$.
A *plurisubharmonic exhaustion function* on $X$ is a function $\varphi \colon X \to [-\infty, \infty)$ such that:

1. $\varphi$ is [[plurisubharmonic|FF2P7ER3QT]] on $X$.
2. $\varphi$ is an exhaustion: for every $c \in \R$, the set $\set{x \in X : \varphi(x) < c}$ is relatively compact in $X$, that is, its closure in $X$ is [[compact|0R6EWFKRCF]].

It is *smooth and strictly plurisubharmonic* if moreover $\varphi$ is $C^\infty$ and its [[Levi form|80MTNXWW80]] is positive definite at every point, in local holomorphic coordinates.

## Remarks

Example: on $\C^n$, $\varphi(z) = \abs{z}^2$ is a smooth strictly plurisubharmonic exhaustion function.
Its Levi form is $\abs{w}^2$, and its sublevel sets are open balls.
Non-example: on the unit ball $B \subseteq \C^n$, $\varphi(z) = \abs{z}^2$ is still strictly plurisubharmonic but is not an exhaustion, since $\set{\varphi < 2} = B$ is not relatively compact in $B$.
On $B$, the function $-\log(1 - \abs{z}^2)$ is a smooth strictly plurisubharmonic exhaustion function.
For a domain $\Omega \subsetneq \C^n$, an equivalent form of clause 2 for continuous $\varphi$ is that $\varphi(z) \to +\infty$ as $z$ tends to $\partial\Omega$ or, when $\Omega$ is unbounded, as $\abs{z} \to \infty$.
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# Post's lattice

## Description

*Post's lattice* is the [[lattice of clones|EX9NCDFYC7]] on the two-element set $\set{0, 1}$, that is, the set of all [[clones|PTKR703G6A]] of [[Boolean functions|0X05GS525E]] ordered by inclusion.
Emil Post gave a complete description of it, published in 1941.
Its main features are these.

1. It is countably infinite.
2. Every clone on $\set{0, 1}$ is generated by finitely many Boolean functions.
3. It has exactly five maximal clones, that is, coatoms below the clone of all Boolean functions.
   They consist of the operations that preserve $0$, that preserve $1$, that are monotone, that are self-dual, and that are affine.
4. Its least element is the clone of projections.

Check of one maximal clone: the monotone operations form a clone, since projections are monotone and a composition of monotone operations is monotone.
It is a proper clone, since negation $\neg$ is not monotone.

## Explanations

### L1 human

Post's lattice is a complete map of all the ways to pick a family of logic gates and see what circuits they can build.
Each point in the map is a closed family of Boolean functions, and lines go from smaller families to larger ones.
The map is infinite but countable, and every family in it is generated by finitely many gates.
Reading off the five families just below the top gives [[Post's functional completeness theorem|WGBR7JHBV4]].
</document>

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# Biconjugate gradient stabilized method

## Statement

Let $A$ be a real [[invertible|0EB9N0S33C]] $n \times n$ matrix, let $b \in \R^n$, and let $x_0 \in \R^n$ with $r_0 = b - A x_0$.
Choose $\hat{r}_0 \in \R^n$ with $\hat{r}_0^T r_0 \ne 0$, and set $p_0 = r_0$.
The *biconjugate gradient stabilized method* (BiCGSTAB) computes, for $k = 0, 1, \dots$,
$$v_k = A p_k, \quad \alpha_k = \frac{\hat{r}_0^T r_k}{\hat{r}_0^T v_k}, \quad s_k = r_k - \alpha_k v_k, \quad t_k = A s_k, \quad \omega_k = \frac{t_k^T s_k}{t_k^T t_k},$$
$$x_{k+1} = x_k + \alpha_k p_k + \omega_k s_k, \quad r_{k+1} = s_k - \omega_k t_k,$$
$$\beta_k = \frac{\hat{r}_0^T r_{k+1}}{\hat{r}_0^T r_k} \cdot \frac{\alpha_k}{\omega_k}, \quad p_{k+1} = r_{k+1} + \beta_k (p_k - \omega_k v_k).$$
Suppose that all denominators are nonzero up to step $m$.
Then in exact arithmetic, for $k \le m$:

1. $r_k = b - A x_k$, and $r_k = Q_k(A) P_k(A) r_0$, where $P_k$ is the residual polynomial of the [[biconjugate gradient method|HGF11AHX5R]] with shadow residual $\hat{r}_0$, and $Q_k(t) = \prod_{j=0}^{k-1} (1 - \omega_j t)$;
2. each $\omega_k$ minimizes $\norm{s_k - \omega t_k}_2$ over $\omega \in \R$;
3. each step uses two products with $A$ and none with $A^T$.

## Remarks

The residual polynomial $P_k$ of BiCG has degree $k$ and value $1$ at $0$, so $r_k$ comes from a polynomial of degree $2k$ in the [[Krylov subspace|QB45PCGQPV]] $\mathcal{K}_{2k+1}(A, r_0)$.
The recurrences come from writing the BiCG coefficients in terms of the inner products $\hat{r}_0^T Q_k(A) P_k(A) r_0$, which needs no $A^T$ because $\hat{r}_0^T Q_k(A) = (Q_k(A^T) \hat{r}_0)^T$.
The factor $Q_k$ is a product of local one-step minimal residual steps, which smooths the erratic convergence of BiCG and of the method CGS.
Example: if $r_0$ is an eigenvector of $A$ for $\lambda$, then $v_0 = \lambda r_0$, $\alpha_0 = 1/\lambda$ and $s_0 = 0$, so $x_0 + \alpha_0 p_0$ already solves the system.
In that case $t_0 = 0$ and $\omega_0$ is undefined, and an implementation stops when $s_k$ is small.
The method can still break down, when $\hat{r}_0^T r_k = 0$ or $\omega_k = 0$.
It can stagnate when the $\omega_k$ are close to $0$, which tends to happen when $A$ has eigenvalues with large imaginary parts.

## Explanations

### L1 human

BiCGSTAB is a fix of the biconjugate gradient method for nonsymmetric systems.
It avoids the transpose of the matrix, and after each BiCG-like step it adds a small step that reduces the residual as much as possible.
The result usually converges more smoothly than BiCG and costs little memory.

### L2 human

BiCGSTAB, due to van der Vorst, is with restarted [[GMRES|A3FR6PNEPB]] one of the default Krylov [[iterative methods|2RPE77ACER]] for nonsymmetric sparse systems.
Its memory and work per step are fixed, in contrast with full GMRES, but it has no global minimization property.
It is used with a [[preconditioner|KE98SW3SBD]], typically an incomplete LU factorization.
Variants such as BiCGSTAB($\ell$) use minimal residual polynomials of higher degree to handle complex eigenvalues better.
</document>

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# Cesàro means of Fourier series

## Summary

The Cesàro means of a [[Fourier series|G12KXDHPPS]] are the averages of its first $N$ partial sums.
They equal the convolution of the function with the [[Fejér kernel|DCCA2T9GKH]], a positive [[summability kernel|1QC5206K07]].
For this reason they behave much better than the partial sums: by [[Fejér's theorem|TPJZCSXSVN]] they converge uniformly for every continuous periodic function.
They are the [[Cesàro mean|08R2V9GZF7]] of the sequence of partial sums.

## Explanations

### L1 human

The partial sums of a Fourier series can oscillate and overshoot, and for some continuous functions they do not converge at all.
Averaging the first $N$ partial sums smooths out these oscillations.
In terms of the waves, the average keeps each frequency $k$ with the weight $1 - \abs{k}/N$, so high frequencies are faded out gently instead of being cut off.
For $f(x) = \cos x$ the $N$-th mean is $\frac{N-1}{N} \cos x$, which tends to $\cos x$.
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# Convergence tests

## Definition

Let $P$ be a property of real or complex sequences $(a_n)$.
The property $P$ is a convergence test if every [[series|0JH4MN4K1C]] $\sum a_n$ whose terms have $P$ is [[convergent|0CVVKY7GY2]].
It is a divergence test if every series $\sum a_n$ whose terms have $P$ is [[divergent|0NBRP6093V]].

## Remarks

Example: the property "there are $q < 1$ and $N$ with $\abs{a_{n+1}} \le q \abs{a_n}$ for all $n \ge N$" is a convergence test.
Indeed $\abs{a_n} \le \abs{a_N} q^{n - N}$ for $n \ge N$, and the [[comparison test|04Z5G0THN6]] with a [[geometric series|01X5CVV1Q5]] gives convergence.
Non-example: the property "$a_n \to 0$" is not a convergence test.
The terms $1/n$ of the [[harmonic series|0CXTMAY5PX]] tend to $0$, but the series diverges.
Its negation "$a_n \not\to 0$" is a divergence test.
A test is inconclusive on a series when the series has neither the property of the test nor the property of its companion divergence test.

## Explanations

### L1 human

A convergence test is a sufficient condition: when it holds, convergence is certain.
When it does not hold, nothing follows unless another test applies.
</document>

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# Schläfli symbol

## Summary

The Schläfli symbol is a short notation $\set{p, q, r, \dots}$ for a [[regular polytope|F60PD912D5]].
The first entry gives the polygons that form the two-dimensional faces.
Each further entry says how many of the previous pieces meet around a face of lower dimension.
The symbol determines the regular polytope up to similarity, and reversing it gives the dual polytope.
It is the main bookkeeping tool in the [[classification of regular polytopes|XYCD53PB5K]].

## Explanations

### L1 human

The symbol $\set{4, 3}$ means: faces are squares, and three of them meet at each corner; this is the cube.
The symbol $\set{3, 4}$ means triangles with four at each corner; this is the octahedron.
In four dimensions, $\set{4, 3, 3}$ means cubes as the three-dimensional cells, with three cubes around each edge; this is the tesseract.
Reading the symbol backwards gives the dual shape, as the cube and the octahedron show.
</document>

<document source="https://tamnd.github.io/math2/node/MAHHKQST2A.md" tag="MAHHKQST2A" kind="concept">
# Uniform convergence of series

## Summary

A [[series of functions|0Y3KSN2E9X]] converges uniformly on a set when its partial sums converge [[uniformly|08VZ57PT26]] on that set.
Equivalently, the tails of the series become small at the same rate at every point.
This is the mode of convergence that lets one carry continuity, integrals and, with extra hypotheses, derivatives from the terms to the sum.

## Explanations

### L1 human

A series of functions is, at each point, an ordinary series of numbers.
At each point the leftover tail after $N$ terms shrinks, but the speed may depend on the point.
Uniform convergence asks for one $N$ that makes the tail small at all points at once.
For example, $\sum_{n \ge 0} x^n$ converges uniformly on $[0, 1/2]$, because the tail after $N$ terms is at most $2 \cdot 2^{-(N+1)}$ there.
On $[0, 1)$ it converges at each point but not uniformly, since near $x = 1$ the tail stays large.
</document>

<document source="https://tamnd.github.io/math2/node/MAJ3DH7MZS.md" tag="MAJ3DH7MZS" kind="concept">
# First fundamental form

## Summary

The first fundamental form of a surface in $\R^3$ is the inner product of $\R^3$ restricted to the tangent planes of the surface.
In a local parametrisation it is given by three functions $E$, $F$, $G$.
It lets one measure lengths of curves, angles and areas on the surface without leaving it.
Properties that depend only on it form the intrinsic geometry of the surface.

## Explanations

### L1 human

An ant living on a surface can measure distances and angles with a tiny ruler and protractor, but it cannot see the surrounding space.
The first fundamental form is exactly the information that the ant's ruler gives.
On a map of the Earth with longitude and latitude, it tells how many kilometres one degree of longitude is at each latitude.
Near the poles one degree of longitude is short, and the first fundamental form records this.
</document>

<document source="https://tamnd.github.io/math2/node/MAKCYGGERD.md" tag="MAKCYGGERD" kind="concept">
# Laplace transform on a convex cone

## Summary

The Laplace transform on a convex cone integrates a function on a closed [[convex cone|6VE4KA12VD]] $\Gamma \subseteq \R^n$ against $e^{-\ip{y}{x}}$.
Here $y$ runs over the interior of the [[dual cone|QFY1JR1TK4]].
For the orthant $\Gamma = [0, \infty)^n$ it is the [[multidimensional Laplace transform|QT3M720TCV]].
For a function of moderate growth, the transform extends to a holomorphic function on a tube domain over the interior of the dual cone.
On the cone of positive definite matrices it gives the multivariate gamma function, and it is a basic tool of analysis on symmetric cones.

## Explanations

### L1 human

The ordinary Laplace transform lives on the half-line $[0, \infty)$, and the exponential $e^{-yx}$ decays there when $y > 0$.
In several variables the half-line is replaced by a cone, such as a quarter plane or the set of positive definite matrices.
The exponential $e^{-\ip{y}{x}}$ decays along the whole cone exactly when $y$ points into the dual cone.
For example, on the quarter plane with $f = 1$ the transform is $1/(y_1 y_2)$ for $y_1, y_2 > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/MAM2MDG0XG.md" tag="MAM2MDG0XG" kind="statement">
# Coordinatization of projective planes

## Statement

Let $(R, T)$ be a [[planar ternary ring|Q0NXXFADT9]], with distinguished elements $0$ and $1$.
Let $\Pi(R, T)$ be the [[incidence structure|FGKNAFNHA3]] defined as follows.
Its points are the pairs $(x, y)$ with $x, y \in R$, the symbols $(m)$ with $m \in R$, and one more symbol $(\infty)$.
Its lines are the symbols $[m, k]$ with $m, k \in R$, the symbols $[c]$ with $c \in R$, and one more symbol $[\infty]$.
The line $[m, k]$ consists of the point $(m)$ and the points $(x, y)$ with $y = T(x, m, k)$.
The line $[c]$ consists of the point $(\infty)$ and the points $(c, y)$ with $y \in R$.
The line $[\infty]$ consists of the points $(m)$ with $m \in R$ and the point $(\infty)$.

1. For every planar ternary ring $(R, T)$, the structure $\Pi(R, T)$ is a [[projective plane|DANFNN8C4Y]].
2. For every projective plane $\Pi$, there is a planar ternary ring $(R, T)$ such that $\Pi$ is isomorphic to $\Pi(R, T)$.

## Remarks

For the second part, choose four points $O, X, Y, I$ of $\Pi$ with no three on a line.
The set $R$ is the set of points of the line $OI$ other than its point on the line $XY$.
The point $O$ plays the role of $0$ and $I$ the role of $1$.
The points off $XY$ get coordinates $(x, y)$ by projecting from $X$ and $Y$ onto $OI$, and $T(x, m, k)$ is read off from the line of slope $m$ through the point $(0, k)$.
The planar ternary ring depends on the choice of the four points, and different choices can give rings that are not isomorphic.
Marshall Hall Jr. introduced planar ternary rings for this purpose.

Example: for a field $F$ and $T(x, m, k) = xm + k$, the plane $\Pi(F, T)$ is the projective plane over $F$, with $[m, k]$ the line $y = mx + k$ and $[\infty]$ the line at infinity.
The plane $\Pi$ satisfies [[Desargues's theorem|5Z19Y02MME]] if and only if, for some (equivalently every) choice of the four points, $T(x, m, k) = xm + k$ for a [[division ring|0EA5BTA0CX]] structure on $R$.
If $\Pi$ is finite of order $n$, then $R$ has exactly $n$ elements.

## Explanations

### L1 human

A projective plane is defined only by points, lines and which points lie on which lines.
The theorem says that one can always put coordinates on it, as in school geometry with $x$ and $y$.
The price is that the arithmetic of the coordinates may be strange: in place of $y = mx + k$ one has a single three-place operation $y = T(x, m, k)$.
Conversely, any such operation with the right rules builds a projective plane.

### L2 human

The theorem turns questions about projective planes into questions about algebraic systems.
Geometric properties of the plane match algebraic properties of the ring: Desargues's theorem matches a division ring, and Pappus's theorem matches a field.
Planes coordinatized by quasifields, semifields and near-fields give the main families of non-Desarguesian planes.
In the finite case it gives a way to search for planes of a given order by searching for ternary operations on a set of that size.
</document>

<document source="https://tamnd.github.io/math2/node/MAM9H3SY3Y.md" tag="MAM9H3SY3Y" kind="definition">
# Polynomial function

## Definition

A function $f \colon \R \to \R$ is a *polynomial function* if there are an integer $n \ge 0$ and real numbers $a_0, a_1, \dots, a_n$ such that
$$f(x) = a_0 + a_1 x + \dots + a_n x^n \quad \text{for all } x \in \R.$$

## Remarks

The coefficients are unique: if two such formulas give the same function, their difference is a polynomial with infinitely many roots, so all its coefficients are $0$.
So each polynomial function comes from exactly one element of the [[polynomial ring|0Q9EZS90GY]] $\R[x]$, and its [[degree|0DPF6WRZW6]] is well defined.
The same definition works on an interval of $\R$, or with $\R$ replaced by $\C$.
Every polynomial function is [[continuous|0P27W9RW0G]] and differentiable, and its derivative is the polynomial function $a_1 + 2 a_2 x + \dots + n a_n x^{n-1}$.

Example: $f(x) = (x - 1)^2$ is a polynomial function, since $(x - 1)^2 = 1 - 2x + x^2$ for all $x$, with $n = 2$, $a_0 = 1$, $a_1 = -2$, $a_2 = 1$.

Non-example: $f(x) = \abs{x}$ is not a polynomial function.
If it were equal to a polynomial $p$, then $p(x) - x$ would vanish for all $x > 0$, so $p(x) = x$; but then $p(-1) = -1 \ne \abs{-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/MAP5J6HK0T.md" tag="MAP5J6HK0T" kind="concept">
# Saddle-node bifurcation

## Summary

A saddle-node bifurcation is a change in a family of dynamical systems at which two [[equilibrium points|0N8NGKDZDZ]] meet and disappear.
On one side of the critical parameter there are two equilibria, one stable and one unstable, and on the other side there are none nearby.
It is the most common way in which equilibria are created or destroyed as a parameter varies.
It is also called a fold bifurcation or a tangent bifurcation.
See the [[definition|TB7NBCCXJ1]].

## Explanations

### L1 human

The model is $x' = \mu - x^2$ on the line.
For $\mu < 0$ the right side is always negative, so there is no equilibrium.
At $\mu = 0$ the single equilibrium $x = 0$ appears.
For $\mu > 0$ it splits into a stable equilibrium at $\sqrt{\mu}$ and an unstable one at $-\sqrt{\mu}$.
Read backwards in $\mu$, the two equilibria collide and annihilate each other.
</document>

<document source="https://tamnd.github.io/math2/node/MAQKH72P6V.md" tag="MAQKH72P6V" kind="concept">
# Additive energy

## Summary

The additive energy of two finite sets $A$ and $B$ in an abelian group counts the quadruples $(a_1, a_2, b_1, b_2)$ with $a_1 + b_1 = a_2 + b_2$.
It measures how often sums repeat, so it is large when the [[sumset|XNBN31HQWB]] $A + B$ is small.
It is a more robust measure of additive structure than the size of the sumset.
The [[Balog–Szemerédi–Gowers theorem|W5BZZ59CVM]] turns large energy into a small sumset for large subsets.

## Explanations

### L1 human

Add each element of $A$ to each element of $B$ and look for coincidences among the sums.
The additive energy counts these coincidences, including trivial ones.
For an arithmetic progression many sums coincide, so the energy is large.
For a set like $\set{1, 2, 4, 8}$, almost all sums are different, so the energy is as small as possible.
</document>

<document source="https://tamnd.github.io/math2/node/MAVAK2WWXK.md" tag="MAVAK2WWXK" kind="concept">
# Lyapunov–Schmidt reduction

## Summary

The Lyapunov–Schmidt reduction turns a nonlinear equation $F(x, \lambda) = 0$ between infinite-dimensional spaces into an equivalent system of finitely many equations in finitely many unknowns.
It works near a solution where the derivative of $F$ is a Fredholm operator, which is close to being invertible.
The part of the equation where the derivative is invertible is solved by the [[implicit function theorem|0BXBADR94C]].
What remains is a small bifurcation equation, which describes how solutions branch as the parameter $\lambda$ varies.
See the [[definition|CC666JT89X]].

## Explanations

### L1 human

Many problems ask for the solutions of an equation that depends on a parameter, such as the shape of a beam under a load.
Usually the solution changes smoothly with the parameter, but at special values several solutions can appear.
The Lyapunov–Schmidt method splits the unknown into a few "dangerous" directions and all the others.
The other directions can be solved for, so only the few dangerous directions are left.
The problem then becomes a question about a small number of equations, which can often be solved by hand.
</document>

<document source="https://tamnd.github.io/math2/node/MAX2ZYG6H9.md" tag="MAX2ZYG6H9" kind="concept">
# Morse index

## Summary

The Morse index of a [[critical point|0YTCMFB4HC]] is the number of independent directions in which the function decreases to second order.
It is the largest dimension of a subspace on which the [[Hessian|0693Y2METT]] is negative definite.
In Morse theory, passing a [[nondegenerate critical point|MGPPCDFQDG]] of index $\lambda$ attaches a cell of dimension $\lambda$ to the sublevel set.

## Explanations

### L1 human

At the bottom of a bowl no direction goes down, so the index is $0$.
At a mountain pass one direction goes down on each side, so the index is $1$.
At the top of a hill in a surface every direction goes down, so the index is $2$.
For example, $x^2 - y^2 - z^2$ on $\R^3$ has a critical point of index $2$ at the origin, since it decreases along the $y$ and $z$ axes.
</document>

<document source="https://tamnd.github.io/math2/node/MB1K62QDM4.md" tag="MB1K62QDM4" kind="definition">
# Krull dimension of a noncommutative ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] and let $M$ be a right $R$-[[module|0NZ8JA25WK]].
The *Krull dimension* $\operatorname{Kdim} M$, when it exists, is an [[ordinal|0YSCSX5YZC]] or $-1$, defined by transfinite recursion as follows.

1. $\operatorname{Kdim} M = -1$ if and only if $M = 0$.
2. For an ordinal $\alpha \ge 0$, $\operatorname{Kdim} M \le \alpha$ if, for every descending chain $M_0 \supseteq M_1 \supseteq M_2 \supseteq \cdots$ of [[submodules|05WDSN597C]] of $M$, there is $N$ such that $\operatorname{Kdim}(M_i / M_{i+1}) < \alpha$ for all $i \ge N$.
3. $\operatorname{Kdim} M = \alpha$ if $\operatorname{Kdim} M \le \alpha$ and $\operatorname{Kdim} M \not\le \beta$ for every $\beta < \alpha$.

If no ordinal $\alpha$ satisfies $\operatorname{Kdim} M \le \alpha$, then $M$ has no Krull dimension.
The *right Krull dimension* of $R$ is $\operatorname{Kdim} R_R$, the Krull dimension of $R$ as a right module over itself.

## Remarks

From clause 2 with $\alpha = 0$, a module has Krull dimension $0$ exactly when it is nonzero and [[Artinian|YQNZTQGP1X]]: the factors of every descending chain are eventually $0$.

Example: $\operatorname{Kdim} \Z_\Z = 1$.
Take a descending chain $M_0 \supseteq M_1 \supseteq \cdots$ of subgroups of $\Z$.
If every $M_i = a_i\Z$ is nonzero, every factor $M_i / M_{i+1}$ is finite, hence Artinian.
Otherwise some $M_i = 0$, and all later factors are $0$.
So $\operatorname{Kdim} \Z \le 1$, and $\Z$ is not Artinian, so $\operatorname{Kdim} \Z = 1$.

Non-example: an infinite-dimensional vector space $V$ over a field has no Krull dimension.
Splitting a basis into infinitely many infinite parts gives a descending chain $V \supsetneq M_1 \supsetneq M_2 \supsetneq \cdots$ in which every factor is infinite-dimensional.
So an induction on $\alpha$ shows that $\operatorname{Kdim} V \le \alpha$ fails for every ordinal $\alpha$.

Every [[Noetherian module|AK1ZRCXR32]] has a Krull dimension.
For a commutative Noetherian ring, the Krull dimension defined here equals the classical [[Krull dimension|MVGBS3J1P7]] given by chains of prime ideals.
The definition is the deviation of the partially ordered set of submodules of $M$.
</document>

<document source="https://tamnd.github.io/math2/node/MB52NDA6WN.md" tag="MB52NDA6WN" kind="statement">
# Resolution of singularities

## Statement

Let $k$ be a field of characteristic $0$ and let $X$ be an integral [[algebraic variety|HV6YBS0YDH]] over $k$.
Then there is a nonsingular variety $\tilde{X}$ over $k$ with a [[proper|9537RKBZPP]] [[birational|KA5890R9YM]] morphism $\pi \colon \tilde{X} \to X$.
Moreover $\pi$ can be chosen to be an isomorphism over the set of nonsingular points of $X$.

## Remarks

This is Hironaka's theorem.
The morphism $\pi$ can be chosen as a composition of [[blow-ups|C4KGYAFPN7]] along nonsingular centres that lie over the [[singular points|D86X15A8WS]] of $X$.
The proof is an induction on dimension: one restricts to a hypersurface of maximal contact and uses an invariant of the singularity that drops after each blow-up.
For curves, in any characteristic, the [[normalization|M2FAMD9HZN]] is a resolution.
Example: the cusp $y^2 = x^3$ is resolved by $\mathbb{A}^1 \to X$, $t \mapsto (t^2, t^3)$, which is bijective and an isomorphism away from $t = 0$.
In characteristic $p > 0$ the statement is known in dimension at most $3$, and it is open in general.

## Explanations

### L1 human

A singular variety has points where it is not smooth, such as a cusp or a place where it crosses itself.
The theorem says that one can always replace it by a smooth variety that differs from it only over the bad points.
The smooth variety maps onto the old one and keeps all its good points unchanged.

### L2 human

Resolution lets one prove results for smooth varieties and then transfer them to singular ones.
It is used to define invariants of singularities, such as rational singularities, and to compactify smooth varieties by a divisor with simple normal crossings.
Resolution by blow-ups is also a basic step of the minimal model program, which starts from a smooth model of a birational class.
</document>

<document source="https://tamnd.github.io/math2/node/MBASZABTJ1.md" tag="MBASZABTJ1" kind="concept">
# Hamiltonian path

## Summary

A Hamiltonian path in a graph is a [[path|03NAMV1DHJ]] that visits every vertex exactly once.
It is the open form of a [[Hamiltonian cycle|0HF30P75G4]].
Deciding whether a graph has a Hamiltonian path is NP-complete, and no simple criterion like Euler's parity test for [[Eulerian trails|05NWGBHKSZ]] is known.
The node [[Hamiltonian path|AT8GWPWQWT]] gives the precise definition.

## Explanations

### L1 human

A Hamiltonian path is a tour that visits every city exactly once, without needing to come back home.
On a chessboard, a knight's tour that visits every square once is a Hamiltonian path in the knight's move graph.
The path graph $1 - 2 - 3 - 4$ is itself a Hamiltonian path, while a star with three arms has none.
</document>

<document source="https://tamnd.github.io/math2/node/MBE83FTRBG.md" tag="MBE83FTRBG" kind="example">
# Penrose tiling

## Example

A *Penrose tiling* is a [[tiling|GKJK293DXS]] of the plane by copies of two prototiles that obey matching rules.
Roger Penrose introduced these tilings in the 1970s.
In the rhombus version, the prototiles are two rhombi with sides of length $1$.
The thick rhombus has angles $72^\circ$ and $108^\circ$, and the thin rhombus has angles $36^\circ$ and $144^\circ$.
The matching rules mark the edges of the rhombi and allow two tiles to meet along an edge only when the marks agree.
In the kite and dart version, the two prototiles are a convex kite and a non-convex dart, again with matching rules.

The two prototiles with their matching rules admit tilings of the whole plane, and none of these tilings is [[periodic|WFRGSBFZGV]].
So each version is an [[aperiodic set of prototiles|Z3FCJZF6WK]].
In every Penrose tiling the ratio of the number of thick to thin rhombi, counted in large disks, tends to the [[golden ratio|0SC8T7548D]] $\varphi = (1 + \sqrt{5})/2$.
Since $\varphi$ is irrational, this ratio cannot come from a periodic tiling, whose ratio would be the rational ratio within one period.

## Remarks

The matching rules are needed.
Without them, a thick rhombus is already a parallelogram that tiles the plane periodically, and a kite and a dart fit together into a rhombus that tiles periodically.
There are uncountably many different Penrose tilings, and every finite patch of one of them appears in every other one.
Some Penrose tilings have a global fivefold rotational symmetry about one point, which no periodic tiling of the plane can have.

## Explanations

### L1 human

A Penrose tiling covers the plane with just two shapes, yet the pattern never repeats.
You can slide it in any direction, and it never lands exactly on itself.
Still, it is highly ordered: patches with fivefold symmetry, like small stars, appear again and again.
The two kinds of tiles occur in the proportion of the golden ratio, about $1.618$ to $1$.
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<document source="https://tamnd.github.io/math2/node/MBF660WMYJ.md" tag="MBF660WMYJ" kind="statement">
# Duality of Lp spaces

## Statement

Let $(\Omega, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $1 \le p < \infty$.
Let $L^p(\mu)$ be the [[Lp space|SASP14MA1Z]] of real or complex functions.
Let $q$ be the conjugate exponent, with $1/p + 1/q = 1$ and $q = \infty$ when $p = 1$.
For $g \in L^q(\mu)$, let $\Phi(g)$ be the functional $f \mapsto \int_\Omega f g \, d\mu$ on $L^p(\mu)$.

1. If $1 < p < \infty$, then $\Phi$ is a [[linear isometry|07SJXTWVQW]] of $L^q(\mu)$ onto the [[continuous dual|4A82S5FVRT]] of $L^p(\mu)$.
2. If $p = 1$ and $\mu$ is [[σ-finite|WEBBT18D4X]], then $\Phi$ is a linear isometry of $L^\infty(\mu)$ onto the continuous dual of $L^1(\mu)$.

## Remarks

By [[Hölder's inequality|JQ9WXC729D]], $\abs{\Phi(g)(f)} \le \norm{f}_p \norm{g}_q$, so $\Phi(g)$ is bounded with norm at most $\norm{g}_q$.
The test function $f = \abs{g}^{q-1} \overline{\sgn g}$ gives equality when $1 < p < \infty$.
Surjectivity is the hard part: for σ-finite $\mu$ it follows from the Radon–Nikodym theorem, and for $1 < p < \infty$ uniform convexity of $L^p$ removes the σ-finiteness.

For counting measure on $\N$, the theorem says that the dual of $\ell^p$ is $\ell^q$, and the dual of $\ell^1$ is $\ell^\infty$.
The case $p = \infty$ is excluded: the dual of $\ell^\infty$ is strictly larger than $\ell^1$.
For example, a Banach limit is a bounded functional on $\ell^\infty$ that is not given by any sequence in $\ell^1$.
For $p = 1$, the σ-finite hypothesis cannot be dropped in general.

## Explanations

### L1 human

Every continuous linear way to turn an $L^p$ function into a number is integration against a fixed function from $L^q$.
That function is unique, and its size is exactly the size of the functional.
This turns abstract functionals into concrete functions.

### L2 human

The theorem shows that $L^p$ is reflexive for $1 < p < \infty$, since the dual of $L^q$ is again $L^p$.
In the case $p = 2$ it is a special case of the Riesz representation theorem for Hilbert spaces.
It is used to identify weak limits, to define weak solutions of equations and to prove inequalities by duality, where one computes $\norm{f}_p$ as a supremum of $\abs{\int f g}$ over $\norm{g}_q \le 1$.
</document>

<document source="https://tamnd.github.io/math2/node/MBK6AE6734.md" tag="MBK6AE6734" kind="concept">
# Shift operator

## Summary

A shift operator moves the argument of a function or the index of a sequence by a fixed amount.
On functions of a real variable it sends $f$ to $x \mapsto f(x + h)$; on sequences it sends $(u_n)$ to $(u_{n+1})$.
It is the building block of the operator calculus of finite differences, since the forward difference is the shift minus the identity.
In functional analysis the shift on sequence spaces is a basic example of a linear operator.

## Explanations

### L1 human

Shifting a sequence means dropping its first term and moving every other term one place to the left.
The sequence $1, 4, 9, 16, \dots$ becomes $4, 9, 16, 25, \dots$.
For a function, shifting by $h$ moves its graph $h$ units to the left.
A linear recurrence such as $u_{n+2} = u_{n+1} + u_n$ can be written with the shift $E$ as $(E^2 - E - I) u = 0$, a polynomial in $E$ applied to $u$.
</document>

<document source="https://tamnd.github.io/math2/node/MBK7NFE74S.md" tag="MBK7NFE74S" kind="concept">
# Rng

## Summary

A rng, also called a non-unital ring, is a structure with all the axioms of a [[ring|0B1261B5ED]] except the existence of a multiplicative identity.
The name is "ring" without the letter "i", for "identity".
Rngs arise naturally: ideals, rings of functions with compact support, and many algebras in analysis have no unit element.
Every rng embeds as an ideal in a ring with identity, so much of ring theory still applies.

## Explanations

### L1 human

A rng has addition and multiplication with the usual rules, but there may be no number $1$ that leaves every element unchanged.
The even integers $2\Z$ are the standard example: they are closed under addition and multiplication, but $1$ is not even.
No even integer $e$ satisfies $e \cdot 2 = 2$, so $2\Z$ has no identity.
</document>

<document source="https://tamnd.github.io/math2/node/MBMHS6FQF6.md" tag="MBMHS6FQF6" kind="statement">
# Łojasiewicz inequality

## Statement

Let $U \subseteq \R^n$ be open and let $f \colon U \to \R$ be a real [[analytic function|081WKEDK6C]].

1. Let $Z = f^{-1}(0)$, assume $Z \neq \emptyset$, and let $K \subseteq U$ be [[compact|0R6EWFKRCF]].
Then there are constants $C > 0$ and $\alpha > 0$ such that $\abs{f(x)} \ge C \operatorname{dist}(x, Z)^{\alpha}$ for all $x \in K$.
2. Let $a \in U$.
Then there are a neighbourhood $W \subseteq U$ of $a$ and constants $C > 0$ and $\theta \in (0, 1)$ such that $\abs{f(x) - f(a)}^{\theta} \le C \norm{\nabla f(x)}$ for all $x \in W$.

## Remarks

Part 1 is the distance inequality and part 2 is the gradient inequality.
Łojasiewicz proved them with his theory of semianalytic sets.
A later route uses resolution of singularities to make $f$ locally a monomial times a unit, where both inequalities are direct.

Analyticity is necessary.
The smooth function with $f(x) = e^{-1/x^2}$ for $x \neq 0$ and $f(0) = 0$ has $Z = \set{0}$.
But $\abs{f(x)} / \abs{x}^{\alpha} \to 0$ as $x \to 0$ for every $\alpha > 0$, so part 1 fails on $K = [-1, 1]$.

For $f(x) = x^2$ on $\R$ and $a = 0$, part 2 holds with $\theta = 1/2$ and $C = 1/2$, since $\abs{x^2}^{1/2} = \abs{x} = \tfrac12 \abs{f'(x)}$.

## Explanations

### L1 human

An analytic function cannot become small without being close to its zero set: it is at least a power of the distance to its zeros.
In the same spirit, near a critical point the gradient cannot be much smaller than the change in the value of the function.
Both say that analytic functions are never infinitely flat.

### L2 human

The gradient inequality is the key to the convergence of gradient flows.
If $f$ is real analytic and $x(t)$ is a bounded trajectory of $x' = -\nabla f(x)$, the inequality bounds the length of the trajectory, so $x(t)$ converges to a single critical point.
For merely smooth functions a bounded gradient trajectory can accumulate on a whole circle of critical points.
The distance inequality is used to divide distributions by real analytic functions.
</document>

<document source="https://tamnd.github.io/math2/node/MBQS4B0TPA.md" tag="MBQS4B0TPA" kind="definition">
# Index theory

## Definition

Let $M$ be a closed smooth manifold, let $E$ and $F$ be complex vector bundles over $M$, and let $P \colon C^\infty(M; E) \to C^\infty(M; F)$ be an [[elliptic differential operator|MT9P2X85V8]].
Its [[analytic index|44GBFZ16SG]] $\operatorname{ind} P = \dim \ker P - \dim \operatorname{coker} P$ is a finite integer.
*Index theory* is the part of global analysis that computes such indices, and their refinements, in terms of topological data.
Its main tool is the [[topological index|QDCD33CS21]], which depends only on the homotopy class of the principal symbol of $P$.

## Remarks

The Atiyah–Singer index theorem states that the analytic index and the topological index of $P$ are equal.
Refinements replace the integer index by an element of a representation ring (equivariant index) or of a K-theory group of a parameter space (families index).
Other refinements add correction terms on manifolds with boundary.

Example: on the circle $S^1 = \R / 2\pi\Z$, the operator $P = d/d\theta$ on complex functions is elliptic.
Its kernel is the constants and its image is the functions of mean zero, so $\operatorname{ind} P = 1 - 1 = 0$.

Non-example: the dimension of the kernel alone is not an index quantity, because it is not stable.
For $P_c = d/d\theta + c$ with $c \in \C$, $\dim \ker P_c$ is $1$ when $c \in i\Z$ and $0$ otherwise, while $\operatorname{ind} P_c = 0$ for all $c$.
</document>

<document source="https://tamnd.github.io/math2/node/MBRSCQRGGH.md" tag="MBRSCQRGGH" kind="statement">
# q-Vandermonde identity

## Statement

Let $n \ge 0$ be an integer, let $q \in \C \setminus \set{0}$ with $q^k \ne 1$ for $1 \le k \le n$, and let $a, c \in \C$ with $a \ne 0$ and $(c; q)_n \ne 0$.
Then the terminating [[basic hypergeometric series|EB4N9RQR51]] satisfy
$${}_2\phi_1 \left( \begin{matrix} a, q^{-n} \\ c \end{matrix} ; q, q \right) = \frac{(c/a; q)_n}{(c; q)_n}\, a^n$$
and
$${}_2\phi_1 \left( \begin{matrix} a, q^{-n} \\ c \end{matrix} ; q, \frac{c q^n}{a} \right) = \frac{(c/a; q)_n}{(c; q)_n},$$
where $(\cdot\,; q)_n$ is the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].

## Remarks

Both sides are rational functions, and the series stop at index $n$ since $(q^{-n}; q)_k = 0$ for $k > n$.
Proof idea: the second form is the case $b = q^{-n}$ of the $q$-Gauss sum, which follows from [[Heine's transformation|A93A5VATSD]].
The first form is the second one with the order of summation reversed.
Example: $n = 1$.
The first sum is $1 + \frac{(1 - a)(1 - q^{-1})}{(1 - c)(1 - q)}\, q = 1 - \frac{1 - a}{1 - c} = \frac{a - c}{1 - c}$, and the right side is $\frac{(1 - c/a)\, a}{1 - c} = \frac{a - c}{1 - c}$.
In terms of [[Gaussian binomial coefficients|YS09Z1YNEK]] it reads $\binom{m+n}{k}_q = \sum_{j} \binom{m}{j}_q \binom{n}{k-j}_q\, q^{(m-j)(k-j)}$.
At $q = 1$ this becomes [[Vandermonde's identity|01JADB9VW5]].

## Explanations

### L1 human

Vandermonde's identity counts the ways to choose $n$ people from a group of men and women by splitting according to the number of men.
The $q$-Vandermonde identity is the same sum with each binomial coefficient replaced by its $q$-analogue and suitable powers of $q$.
It is the simplest closed-form sum of a terminating basic hypergeometric series.

### L2 human

Its Gaussian binomial form can also be proved, for $q$ a prime power, by counting subspaces of $\F_q^{m+n}$.
It is the $q$-analogue of the [[Chu–Vandermonde identity|4YK46R1QDR]] for ${}_2F_1$, and it is a terminating case of the $q$-Gauss sum.
It is used to evaluate connection coefficients between $q$-orthogonal polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/MBVJ4JQXC5.md" tag="MBVJ4JQXC5" kind="definition">
# Galois connection

## Definition

Let $(P, \le)$ and $(Q, \le)$ be [[partially ordered sets|011C29Q1S1]].
A (monotone) *Galois connection* between $P$ and $Q$ is a pair of maps $f \colon P \to Q$ and $g \colon Q \to P$ such that, for all $x \in P$ and $y \in Q$,
$$f(x) \le y \iff x \le g(y).$$
The map $f$ is the *lower adjoint* and $g$ is the *upper adjoint* of the connection.

## Remarks

Let $i \colon \Z \to \R$ be the inclusion and $\lfloor \cdot \rfloor \colon \R \to \Z$ the [[floor function|0YVRNEDSVG]].
Then $i(n) \le x$ if and only if $n \le \lfloor x \rfloor$, so $(i, \lfloor \cdot \rfloor)$ is a Galois connection.
On $\R$, the pair $f(x) = x + 1$, $g(y) = y$ is not a Galois connection: for $x = 0$ and $y = 1/2$, $x \le g(y)$ holds but $f(x) \le y$ fails.
In a Galois connection both maps are [[monotone|0WFF6AZPBP]], $x \le g(f(x))$ and $f(g(y)) \le y$, and $f \circ g \circ f = f$ and $g \circ f \circ g = g$.
Each adjoint determines the other: $g(y)$ is the greatest $x$ with $f(x) \le y$.
The lower adjoint preserves all existing suprema, and the upper adjoint preserves all existing infima.
An *antitone* Galois connection uses order-reversing maps with $y \le f(x)$ if and only if $x \le g(y)$; it is a monotone connection between $P$ and the dual of $Q$.

## Explanations

### L1 human

In a Galois connection, the question "is $f(x)$ below $y$?" has the same answer as "is $x$ below $g(y)$?".
So $f$ and $g$ behave like inverses as far as the order can tell.
</document>

<document source="https://tamnd.github.io/math2/node/MBW9JCGZC1.md" tag="MBW9JCGZC1" kind="concept">
# Weak convergence of measures

## Summary

A sequence of [[probability measures|059TJN0NZ1]] on a [[metric space|0VKD1TP7F3]] converges weakly when the integrals of every bounded continuous function converge.
It is the mode of convergence behind [[convergence in distribution|0SRYJGXKR4]] of random variables and the central limit theorem.
Mass may move a little in the limit, so the measure of a fixed set need not converge.

## Explanations

### L1 human

Two measures are close in this sense if they look alike to every smooth test, even if they put mass on slightly different points.
Picture a point mass at $1/n$ sliding towards $0$.
For every continuous bounded $f$, the value $f(1/n)$ tends to $f(0)$, so these point masses converge weakly to the point mass at $0$.
Yet the set $\set{0}$ has mass $0$ for every $n$ and mass $1$ in the limit.
Weak convergence only sees the overall shape, not exact sets.
</document>

<document source="https://tamnd.github.io/math2/node/MC25MTN2N7.md" tag="MC25MTN2N7" kind="concept">
# Relational clone

## Summary

A relational clone on a set $A$ is a set of finitary relations on $A$ that is closed under primitive positive definitions.
It must also contain the equality relation and the empty relation.
Primitive positive definitions are those that use only conjunction, existential quantification and equality.
Relational clones are dual to [[clones|PTKR703G6A]] of operations.
On a finite set, the operators $\operatorname{Pol}$ and $\operatorname{Inv}$ give a bijection between them.
Another common name is co-clone.

## Explanations

### L1 human

A relational clone is a family of relations that is closed under the simplest ways of building new relations from old ones.
One may combine relations with "and", hide some coordinates with "there exists", and identify coordinates with "equals".
For example, from the order $\le$ on a set one can define $\set{(x, z) : \exists y \, (x \le y \wedge y \le z)}$, and a relational clone containing $\le$ contains this relation too.
The set of all relations preserved by a fixed set of operations is always a relational clone.
</document>

<document source="https://tamnd.github.io/math2/node/MC4EFHWRJG.md" tag="MC4EFHWRJG" kind="definition">
# Kleinian group

## Definition

Let $\mathrm{PSL}_2(\C) = \mathrm{SL}_2(\C) / \set{\pm I}$, where [[SL(2,C)|J58ED1GV6N]] has the topology of a subset of $\C^4$ and $\mathrm{PSL}_2(\C)$ has the quotient topology.
A *Kleinian group* is a [[discrete subgroup|5M7D1BSHVZ]] of $\mathrm{PSL}_2(\C)$.

## Remarks

The group $\mathrm{PSL}_2(\C)$ acts on the [[Riemann sphere|0J78KBPFZB]] by [[Möbius transformations|0SG74WBFA2]] and on hyperbolic $3$-space $\mathbb H^3$ by isometries.
A subgroup is Kleinian if and only if its action on $\mathbb H^3$ is properly discontinuous.
Some authors also require that the group act properly discontinuously on a nonempty open subset of the Riemann sphere.

Example: the Picard group $\mathrm{PSL}_2(\Z[i])$, of matrices with Gaussian integer entries, is Kleinian.
Distinct Gaussian integers are at distance at least $1$, so a small neighbourhood of $\pm I$ contains no other element of the group.

Non-example: the group $\mathrm{PSL}_2(\Z[\sqrt2])$ is not Kleinian.
It contains the translations $z \mapsto z + a + b\sqrt2$ with $a, b \in \Z$, and their translation lengths come arbitrarily close to $0$ without being $0$.
</document>

<document source="https://tamnd.github.io/math2/node/MC4S4WCGJS.md" tag="MC4S4WCGJS" kind="definition">
# Gibbs measure

## Definition

Let $A$ be a $k \times k$ matrix with entries in $\set{0, 1}$.
Let $\Sigma_A \subseteq \set{1, \dots, k}^{\N}$ be the one-sided [[subshift of finite type|624XDXX8FN]] of the sequences $x = (x_0, x_1, \dots)$ with $A_{x_i x_{i+1}} = 1$ for all $i$.
Let $\sigma \colon \Sigma_A \to \Sigma_A$ be the shift, $(\sigma x)_i = x_{i+1}$.
Let $\varphi \colon \Sigma_A \to \R$ be continuous, and write $S_n \varphi(x) = \sum_{j=0}^{n-1} \varphi(\sigma^j x)$ for its [[Birkhoff sum|8TD2KCVH5N]].
For $x \in \Sigma_A$, the cylinder $[x_0 \dots x_{n-1}]$ is the set of $y \in \Sigma_A$ with $y_i = x_i$ for $0 \le i < n$.
Let $\mu$ be a $\sigma$-[[invariant|B232EZ1FMB]] [[probability measure|059TJN0NZ1]] on $\Sigma_A$.
It is a *Gibbs measure* for $\varphi$ if there are $C \ge 1$ and $P \in \R$ with
$$\frac{1}{C} \le \frac{\mu([x_0 \dots x_{n-1}])}{\exp\left(-nP + S_n \varphi(x)\right)} \le C$$
for every $x \in \Sigma_A$ and every $n \ge 1$.

## Remarks

Example: on the full shift $\set{1, \dots, k}^{\N}$ take $\varphi = 0$, and let $\mu$ be the uniform [[Bernoulli measure|D33JK3AXRZ]], under which the coordinates are independent and uniform.
Then $\mu([x_0 \dots x_{n-1}]) = k^{-n}$ and $S_n \varphi = 0$, so the ratio is $1$ with $P = \log k$, and $C = 1$ works.

Non-example: for the same shift and $\varphi = 0$, the Dirac measure at the constant sequence $(1, 1, 1, \dots)$ is invariant.
It gives measure $0$ to the cylinder $[2]$, so the lower bound fails.

The constant $P$ is then the topological pressure of $\varphi$.
For a topologically mixing $\Sigma_A$ and a [[Hölder continuous|012PBA0FMD]] $\varphi$, a Gibbs measure exists, is unique, and is the unique equilibrium state of $\varphi$ (Bowen).

In statistical mechanics, consider a finite system with states $s$ and energies $E(s)$.
The Gibbs measure at inverse temperature $\beta$ gives $s$ the probability $e^{-\beta E(s)} / Z$, where $Z = \sum_s e^{-\beta E(s)}$ is the [[partition function|0CV69WDRXW]].
For infinite lattice systems, Gibbs measures are defined instead by the Dobrushin–Lanford–Ruelle condition on conditional probabilities, and there may be several of them.
</document>

<document source="https://tamnd.github.io/math2/node/MC5CGVRFJM.md" tag="MC5CGVRFJM" kind="concept">
# Banach–Lie group

## Summary

A Banach–Lie group is a group that is also a smooth [[Banach manifold|9K15AMGBV9]] in which multiplication and inversion are smooth.
It generalises a finite-dimensional [[Lie group|7JBRGCVJRB]] to infinite dimension.
Its Lie algebra is a Banach–Lie algebra, and, unlike for more general infinite-dimensional groups, the exponential map is still a local diffeomorphism at $0$.
Basic examples are the group of invertible bounded operators on a Banach space and the unitary group of a Hilbert space with the norm topology.


## Explanations

### L1 human

A Lie group is a group whose elements form a smooth space, such as the rotations of space.
A Banach–Lie group is the same idea when the group has infinitely many directions.
For example, the invertible bounded operators on the Hilbert space $\ell^2$ form a group, and near the identity they look like a ball of the space of all bounded operators.
Many tools of finite-dimensional Lie theory, such as the exponential map, keep working in this setting.
</document>

<document source="https://tamnd.github.io/math2/node/MC9HE0M2RG.md" tag="MC9HE0M2RG" kind="definition">
# Chebyshev functions

## Definition

Let $x$ be a real number.
The *first Chebyshev function* is
$$\vartheta(x) = \sum_{p \le x} \log p,$$
where $p$ runs over the [[prime numbers|01826KMG2H]] at most $x$ and $\log$ is the [[natural logarithm|04DXAFN2ES]].
The *second Chebyshev function* is
$$\psi(x) = \sum_{\substack{p, k \\ p^k \le x}} \log p = \sum_{1 \le n \le x} \Lambda(n).$$
Here the first sum runs over the pairs of a prime $p$ and an integer $k \ge 1$.
The symbol $\Lambda$ denotes the [[von Mangoldt function|0NK8JTE1NN]].

## Remarks

Example: $\vartheta(10) = \log 2 + \log 3 + \log 5 + \log 7 = \log 210$.
The prime powers up to $10$ are $2, 4, 8, 3, 9, 5, 7$, so $\psi(10) = \log(2^3 \cdot 3^2 \cdot 5 \cdot 7) = \log 2520$.
In general $\psi(x) = \log \operatorname{lcm}(1, 2, \dots, \floor{x})$ for $x \ge 1$.
Non-example: $\sum_{n \le 10} \log n = \log 10!$ is not $\psi(10)$, since it also has terms such as $\log 6$ for $n$ that are not prime powers.
The two functions are related by $\psi(x) = \sum_{k \ge 1} \vartheta(x^{1/k})$, and $\psi(x) - \vartheta(x) = O(\sqrt{x} \log x)$.
Both are $0$ for $x < 2$.
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<document source="https://tamnd.github.io/math2/node/MC9MX40Z7S.md" tag="MC9MX40Z7S" kind="concept">
# Discriminator variety

## Summary

A discriminator variety is a [[variety|AJNZJD8442]] generated by algebras that share one term acting as the [[ternary discriminator|72PG6XCQ7D]] on each of them.
Boolean algebras, the varieties generated by a finite field, and the varieties generated by a quasi-primal algebra are examples.
Discriminator varieties are arithmetical, and each of their subdirectly irreducible members is simple.
Their algebras can be studied as Boolean products of simple algebras, much as Boolean algebras are studied through Stone spaces.

## Explanations

### L1 human

In some algebras one can write, with the given operations, a formula that tests whether two elements are equal.
A discriminator variety is built from algebras that all use the same formula for this test.
The two-element Boolean algebra is the basic example: the test is "if $x$ and $y$ agree, output $z$, else output $x$", and Boolean operations can express it.
Having such a test makes the variety behave much like the variety of Boolean algebras.
</document>

<document source="https://tamnd.github.io/math2/node/MCCXNPGGSH.md" tag="MCCXNPGGSH" kind="definition">
# Cesàro means of Fourier series

## Definition

Let $f \colon \R \to \C$ be a $2\pi$-periodic function that is integrable on $[-\pi, \pi]$, with [[Fourier coefficients|5NN2YZZR3V]] $\hat f(k)$, $k \in \Z$.
For $n \ge 0$ let $S_n f(x) = \sum_{k=-n}^{n} \hat f(k) e^{ikx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]].
For $N \ge 1$, the $N$-th *Cesàro mean* (or Fejér mean) of the Fourier series of $f$ is
$$\sigma_N f(x) = \frac{1}{N} \sum_{n=0}^{N-1} S_n f(x).$$

## Remarks

Counting how often each frequency occurs in $S_0 f, \ldots, S_{N-1} f$ gives
$$\sigma_N f(x) = \sum_{\abs{k} < N} \left( 1 - \frac{\abs{k}}{N} \right) \hat f(k) e^{ikx} = \frac{1}{2\pi} \int_{-\pi}^{\pi} F_N(t) f(x - t) \, dt,$$
where $F_N$ is the [[Fejér kernel|DCCA2T9GKH]].
So $\sigma_N f$ is a trigonometric polynomial of degree at most $N - 1$.
It is the [[Cesàro mean|08R2V9GZF7]] of order $1$ of the sequence $(S_n f(x))_{n \ge 0}$.

Example: for $f(x) = \cos x$, $S_0 f = 0$ and $S_n f = \cos x$ for $n \ge 1$.
So $\sigma_N f = \frac{N-1}{N} \cos x$, which tends to $f$.

Non-example: the Abel–Poisson means $\sum_k r^{\abs{k}} \hat f(k) e^{ikx}$ also average the Fourier series, but with the weights $r^{\abs{k}}$ instead of $1 - \abs{k}/N$; they are a different method, the [[Abel–Poisson summation|RZZ78GXP5F]].
</document>

<document source="https://tamnd.github.io/math2/node/MCDP25YMSE.md" tag="MCDP25YMSE" kind="definition">
# Linear continuum

## Definition

Let $L$ be a set with more than one element and let $<$ be a [[total order|0FRXBVVR03]] on $L$.
The ordered set $(L, <)$ is a *linear continuum* if:

1. Every nonempty subset of $L$ that is bounded above has a least upper bound, a [[supremum|0R3H5PANRQ]], in $L$.
2. For all $x, y \in L$ with $x < y$, there is $z \in L$ with $x < z < y$.

## Remarks

The real line $\R$ is a linear continuum: clause 1 is the completeness of $\R$, and for clause 2 one takes $z = (x + y)/2$.
The integers $\Z$ satisfy clause 1, but they fail clause 2, since there is no integer strictly between $1$ and $2$.
The rationals $\Q$ satisfy clause 2 but fail clause 1: the set $\set{q \in \Q : q^2 < 2}$ is bounded above but has no least upper bound in $\Q$.
The square $[0, 1] \times [0, 1]$ with the dictionary order is a linear continuum that is not order isomorphic to an interval of $\R$.
A linear continuum and each of its intervals are [[connected|0V032VWTHD]] in the order topology.

## Explanations

### L1 human

A linear continuum is an ordered set with no gaps and no jumps.
"No gaps" means that bounded sets have a least upper bound, and "no jumps" means that between two elements there is always a third one.
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<document source="https://tamnd.github.io/math2/node/MCDZP31ZF3.md" tag="MCDZP31ZF3" kind="concept">
# Finite geometry

## Summary

A finite geometry is a geometric system, such as a projective or affine plane or space, with only finitely many points.
The main examples are the projective and affine spaces over [[finite fields|0ZXZ82FTX2]], but there are also many finite planes that do not come from fields.
Finite geometry links geometry with combinatorics, algebra and coding theory, through objects such as [[block designs|4JCSS2Z1BA]], ovals and spreads.

## Explanations

### L1 human

Ordinary geometry has infinitely many points, but the rules of incidence also make sense for finite sets.
The smallest example is the [[Fano plane|QQ8GW1ASZM]]: seven points and seven lines, each line with three points, and any two points on exactly one line.
Counting arguments then replace measurements of length and angle.
Typical questions are how many points such a geometry can have and which sets of points avoid having three on a line.
</document>

<document source="https://tamnd.github.io/math2/node/MCEE53VGBQ.md" tag="MCEE53VGBQ" kind="definition">
# Category

## Definition

A *category* $\mathcal{C}$ consists of the following data:

1. a class $\mathrm{Ob}(\mathcal{C})$, whose members are called the objects of $\mathcal{C}$;
2. for each pair of objects $A, B$, a class $\Hom_{\mathcal{C}}(A, B)$ of morphisms from $A$ to $B$, written $f \colon A \to B$, with these classes pairwise disjoint;
3. for each triple of objects $A, B, C$, a composition map $\Hom_{\mathcal{C}}(B, C) \times \Hom_{\mathcal{C}}(A, B) \to \Hom_{\mathcal{C}}(A, C)$, written $(g, f) \mapsto g \circ f$;
4. for each object $A$, an identity morphism $1_A \in \Hom_{\mathcal{C}}(A, A)$.

These data must satisfy two axioms:

1. associativity: $h \circ (g \circ f) = (h \circ g) \circ f$ for all $f \colon A \to B$, $g \colon B \to C$ and $h \colon C \to D$;
2. identity: $f \circ 1_A = f = 1_B \circ f$ for all $f \colon A \to B$.

## Remarks

The category is *locally small* when every $\Hom_{\mathcal{C}}(A, B)$ is a [[set|0BSJ7RQ9BW]], and *small* when moreover $\mathrm{Ob}(\mathcal{C})$ is a set.
Example: $\mathbf{Set}$ has sets as objects and [[functions|0EPAT51DXQ]] $A \to B$ as morphisms, with the usual composition and the identity functions.
Composition of functions is associative and the identity functions are neutral, so both axioms hold.
Non-example: take sets as objects and, as morphisms $A \to B$, the functions $A \to B$ whose image has at most one point.
A composite of two such functions again has at most one point in its image, so composition is defined.
The identity function of a set with two elements is not such a morphism, so clause 4 fails.
Many authors write $\mathcal{C}(A, B)$ or $\hom(A, B)$ for $\Hom_{\mathcal{C}}(A, B)$, and $gf$ for $g \circ f$.
</document>

<document source="https://tamnd.github.io/math2/node/MCPFA4DC9Z.md" tag="MCPFA4DC9Z" kind="definition">
# Unknotting number

## Definition

Let $K$ be a tame [[knot|QYEAZTYGPT]] in $\R^3$, and let $D$ be a diagram of $K$.
A *crossing change* at a crossing of $D$ exchanges the over-strand and the under-strand at that crossing.
Let $u(D)$ be the least number of crossings of $D$ whose simultaneous change gives a diagram of the [[unknot|1TQ06R4Q66]].
The *unknotting number* of $K$ is
$$u(K) = \min \set{u(D) : D \text{ is a diagram of } K}.$$

## Remarks

The number $u(D)$ is finite: changing suitable crossings of any diagram gives a diagram of the unknot.
Example: the standard three-crossing diagram of the [[trefoil knot|VYTYSVHN4A]] becomes a diagram of the unknot after one crossing change.
The trefoil is not the unknot, so its unknotting number is $1$.
Non-example of the minimum: a single diagram may need more changes than $u(K)$, so $u(D)$ for one fixed diagram $D$ is only an upper bound for $u(K)$.
A knot has $u(K) = 0$ if and only if it is the unknot.
The unknotting number of the [[torus knot|NS60VV786W]] $T(p, q)$ with coprime $p, q \ge 2$ is $(p-1)(q-1)/2$; this is the [[Milnor conjecture|5SDA7ZXT95]].
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<document source="https://tamnd.github.io/math2/node/MCW8V4RFYJ.md" tag="MCW8V4RFYJ" kind="definition">
# Complete atomic Boolean algebra

## Definition

Let $B$ be a Boolean algebra.
An [[atom|TTVCTW6TNC]] of $B$ is an element $a \ne 0$ such that there is no $b$ with $0 < b < a$.
The algebra $B$ is a *complete atomic Boolean algebra* if:

1. $B$ is a [[complete Boolean algebra|75BPV95AAS]].
2. $B$ is [[atomic|5SE8EMSDNY]]: for every $b \ne 0$ in $B$ there is an atom $a$ with $a \le b$.

## Remarks

If $B$ is complete and atomic, with set of atoms $A$, then $b \mapsto \set{a \in A : a \le b}$ is an isomorphism from $B$ onto the power set of $A$.
So the complete atomic Boolean algebras are, up to isomorphism, the power set algebras.

Example: the power set of a set $X$.
It is complete, and every nonempty $Y \subseteq X$ contains a singleton $\set{x}$, which is an atom.

Non-example: the algebra of [[regular open sets|833XAA2Q65]] of $\R$ is complete but has no atom at all.
Every nonempty regular open set contains an open interval, which contains two disjoint smaller open intervals, and these are regular open and strictly smaller.
So clause 2 fails.

In Mathlib the notion is `CompleteAtomicBooleanAlgebra`.
</document>

<document source="https://tamnd.github.io/math2/node/MCXT7ZW329.md" tag="MCXT7ZW329" kind="definition">
# Iterated function

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a [[map|0EPAT51DXQ]].
The *iterates* of $f$ are the maps $f^n \colon X \to X$ for $n \in \N$, defined recursively by
$$f^0 = \id_X, \qquad f^{n+1} = f \circ f^n,$$
where $\circ$ is [[function composition|0THBK9V825]].
The map $f^n$ is the *$n$-th iterate* of $f$.

## Remarks

By induction, $f^m \circ f^n = f^{m + n}$ and $(f^m)^n = f^{mn}$ for all $m, n \in \N$.
If $f$ is a bijection, one also sets $f^{-n} = (f^{-1})^n$, and the rule $f^m \circ f^n = f^{m+n}$ then holds for all $m, n \in \Z$.

Example: for $f(x) = x^2$ on $\R$, one has $f^2(x) = (x^2)^2 = x^4$ and in general $f^n(x) = x^{2^n}$.
The check is the induction step $f^{n+1}(x) = f(x^{2^n}) = x^{2^{n+1}}$.

Non-example: for $g(x) = x + 1$ on $\R$, the pointwise square $x \mapsto g(x)^2 = (x+1)^2$ is not the second iterate, which is $g^2(x) = x + 2$.
The notation $f^n$ can mean either object, so texts state which one is meant; for example $\sin^2 x$ usually means $(\sin x)^2$.
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<document source="https://tamnd.github.io/math2/node/MD1QFAEFF0.md" tag="MD1QFAEFF0" kind="definition">
# Torus knot

## Definition

Let $p$ and $q$ be coprime integers.
The *$(p, q)$-torus knot* $T(p, q)$ is the image of the map $f \colon \R / 2\pi\Z \to \R^3$,
$$f(t) = \big( (2 + \cos qt) \cos pt, \ (2 + \cos qt) \sin pt, \ -\sin qt \big).$$
A [[knot|QYEAZTYGPT]] is a *torus knot* if it is isotopic to $T(p, q)$ for some coprime $p, q$ with $\abs{p} \ge 2$ and $\abs{q} \ge 2$.

## Remarks

The image lies on the standard [[torus|03NM7DZXXC]] obtained by revolving the circle $(x - 2)^2 + z^2 = 1$ about the $z$-axis.
The map $f$ is injective because $p$ and $q$ are coprime.
Example: $T(2, 3)$ is the [[trefoil knot|VYTYSVHN4A]]; here $\abs{p}, \abs{q} \ge 2$, so it is a torus knot in the sense above.
Non-example: $T(1, q)$ is isotopic to the [[unknot|1TQ06R4Q66]], so the bound $\abs{p} \ge 2$ fails and it is not counted as a torus knot.
$T(p, q)$ is isotopic to $T(q, p)$, and $T(p, -q)$ is the mirror image of $T(p, q)$.
For $p, q \ge 2$ the Seifert genus of $T(p, q)$ is $(p-1)(q-1)/2$.
</document>

<document source="https://tamnd.github.io/math2/node/MD624VBPHA.md" tag="MD624VBPHA" kind="definition">
# Crystalline cohomology

## Definition

Let $k$ be a perfect field of characteristic $p > 0$, let $W = W(k)$ be its ring of [[Witt vectors|C9Z22J9SQE]], and let $W_n = W/p^n W$.
Let $X$ be a [[scheme|5A35TYZD02]] that is [[smooth|EXKGG9X6J0]] and [[proper|9537RKBZPP]] over $k$.
A *divided power structure* on an ideal $I$ of a ring is a family of maps $\gamma_m \colon I \to I$, $m \ge 1$, with the identities of $x \mapsto x^m/m!$.
The ideal $pW_n$ of $W_n$ has a canonical one.
The *crystalline site* $\mathrm{Cris}(X/W_n)$ is the [[site|NYE0Z94MME]] defined as follows.

1. Its objects are triples $(U, T, \delta)$ with $U \subset X$ Zariski open and $U \hookrightarrow T$ a closed immersion into a $W_n$-scheme $T$ with ideal $I$.
   Here $\delta$ is a divided power structure on $I$ compatible with the one on $pW_n$.
2. Its morphisms are pairs of compatible maps $U \to U'$ and $T \to T'$ that respect the divided powers.
3. Its coverings are the families $\set{(U_i, T_i, \delta_i) \to (U, T, \delta)}$ in which the maps $T_i \to T$ are open immersions that cover $T$.

The *structure sheaf* $\mathcal{O}_{X/W_n}$ sends $(U, T, \delta)$ to $\Gamma(T, \mathcal{O}_T)$.
The *crystalline cohomology* of $X$ is
$$H^i_{\mathrm{cris}}(X/W) = \varprojlim_n H^i\big(\mathrm{Cris}(X/W_n), \mathcal{O}_{X/W_n}\big),$$
an [[inverse limit|DZB378CNCK]] of sheaf cohomology groups on these sites.

## Remarks

Each $H^i_{\mathrm{cris}}(X/W)$ is a finitely generated $W$-module, and it vanishes for $i > 2 \dim X$.
If $\mathfrak{X}$ is a smooth proper scheme over $W$ with special fiber $X$, then $H^i_{\mathrm{cris}}(X/W) \cong H^i_{\mathrm{dR}}(\mathfrak{X}/W)$, the [[algebraic de Rham cohomology|13F5JZPDD0]] of the lift.
Example: for $X = \mathbb{P}^1_k$, the lift $\mathbb{P}^1_W$ gives $H^0_{\mathrm{cris}} \cong H^2_{\mathrm{cris}} \cong W$ and $H^1_{\mathrm{cris}} = 0$.
Non-example: replacing $W$ by $k$, that is, taking only $n = 1$, gives the de Rham cohomology $H^i_{\mathrm{dR}}(X/k)$.
This is a vector space over a field of characteristic $p$, so it cannot give coefficients in a field of characteristic $0$.
After tensoring with the fraction field $K$ of $W$, crystalline cohomology is a Weil cohomology theory, with the Frobenius of $X$ acting semilinearly.
</document>

<document source="https://tamnd.github.io/math2/node/MD99PYBCE6.md" tag="MD99PYBCE6" kind="concept">
# Harmonic map heat flow

## Summary

The harmonic map heat flow deforms a map between [[Riemannian manifolds|7CBT4SM83Y]] in the direction of its [[tension field|W8F8S8DX5X]].
It is the gradient flow of the [[energy|E6HDE1DFXD]], so the energy decreases along it, and its stationary solutions are the [[harmonic maps|KZJYG9DS3M]].
James Eells and Joseph Sampson introduced it to find harmonic maps in a given homotopy class.
For compact manifolds and a target of nonpositive sectional curvature, they proved that it exists for all time.
It then converges to a harmonic map along a sequence of times.
In general the flow can develop singularities, and its analysis led to the study of bubbling.

## Explanations

### L1 human

Start with any map and let it relax, the way a hot metal plate relaxes to an even temperature.
At each moment every point moves in the direction that lowers the stretching energy fastest.
For maps into the real line, this is exactly the heat equation, and the map tends to a harmonic function.
For a closed curve wrapped around a cylinder, the flow straightens the curve into a circle of the cylinder, a closed geodesic, which cannot be shortened further.
</document>

<document source="https://tamnd.github.io/math2/node/MDH71ZZ8XX.md" tag="MDH71ZZ8XX" kind="concept">
# Local homeomorphism

## Summary

A local homeomorphism is a map between [[topological spaces|0CQHFPGJ45]] that is a [[homeomorphism|0QT00PB5G2]] near each point, onto an open set.
It need not be injective or surjective as a whole.
Covering maps and the projections of étale spaces are local homeomorphisms, and every local homeomorphism is [[continuous|0VGFHTM75Y]] and [[open|0YCG648KDD]].

## Explanations

### L1 human

A local homeomorphism looks like a perfect copy when you only look at a small piece of the space.
Globally it may wrap the space around many times.
The map that winds the real line around the circle, sending $t$ to the point at angle $2\pi t$, is the standard example.
Each short interval goes onto an arc without any folding, but the whole line covers the circle infinitely often.
</document>

<document source="https://tamnd.github.io/math2/node/MDH7VXDCVF.md" tag="MDH7VXDCVF" kind="concept">
# Proper group action

## Summary

A continuous action of a topological group $G$ on a space $X$ is proper when the map $(g, x) \mapsto (g \cdot x, x)$ is a [[proper map|15XEBGNGHC]].
Informally, group elements that move a compact set back onto itself form a compact set.
Proper actions have Hausdorff orbit spaces, closed orbits and compact stabilizers.
They are the right setting for quotients such as homogeneous spaces $G/H$ and for the action of discrete groups on manifolds.

## Explanations

### L1 human

Think of a group moving the points of a space around.
The action is proper when only a "small", compact, part of the group can bring a given bounded region back near itself.
Integer translations of the real line are proper: to move $[0, 1]$ to a place where it meets itself, only the shifts by $-1$, $0$ and $1$ work.
Scaling the line by positive numbers is not proper, because every scaling keeps the point $0$ in place.
</document>

<document source="https://tamnd.github.io/math2/node/MDHWF76S1Z.md" tag="MDHWF76S1Z" kind="statement">
# Grötzsch's theorem

## Statement

Let $G$ be a finite [[planar graph|029JQEK15B]] that contains no triangle, that is, no cycle of length $3$.
Then $G$ is [[$3$-colourable|0Y7VDAX8Q3]].

## Remarks

An easy weaker bound comes from [[Euler's formula|0W1C598HMS]].
A triangle-free planar graph with $n \ge 3$ vertices has at most $2n - 4$ edges, so it has a vertex of degree at most $3$.
Induction then gives a colouring with $4$ colours.
Getting down to $3$ colours needs a finer induction, which proves a stronger statement about extending colourings from a short precoloured face.
Herbert Grötzsch proved the theorem in 1959, and Carsten Thomassen later gave shorter proofs.
Both hypotheses are needed.
Without the triangle condition the bound fails: $K_4$ is planar and needs $4$ colours.
Without planarity it fails too: the Grötzsch graph, with $11$ vertices, is triangle-free and needs $4$ colours.
For example, every planar bipartite graph is triangle-free, and indeed $2$-colourable.

## Explanations

### L1 human

The four colour theorem says that $4$ colours suffice for every planar graph.
If the graph has no triangles, Grötzsch's theorem says that $3$ colours suffice.
Triangles are the main reason a planar graph might need a fourth colour.

### L2 human

The theorem is a sharp statement about the interplay of girth and colouring on surfaces.
Grünbaum and Aksenov extended it to planar graphs with at most three triangles.
The precoloured-face strengthening used in the proof is typical of colouring proofs for planar graphs, as in the proof that planar graphs are $5$-choosable.
</document>

<document source="https://tamnd.github.io/math2/node/MDM8F48X18.md" tag="MDM8F48X18" kind="concept">
# Padé table

## Summary

The Padé table of a power series is the array of all its [[Padé approximants|1F16G4V64X]] $[m/n]$.
It is indexed by the numerator degree $m$ and the denominator degree $n$.
Its first column holds the Taylor polynomials, and its diagonal holds the diagonal approximants $[n/n]$.
Equal entries occur in square blocks, and the table is called normal when every entry occurs only once.
The table organizes convergence questions and links Padé approximants with [[continued fractions|04P2RJ4WPS]].

## Explanations

### L1 human

For a given function there is one best matching quotient of polynomials for each choice of the two degrees.
Writing these quotients in a grid, by degree of the top and degree of the bottom, gives the Padé table.
Moving down a column adds terms to the top; moving along a row allows more poles.
For $1/(1-z)$ every entry with a denominator of degree at least $1$ is $1/(1-z)$ itself, a single huge block.
</document>

<document source="https://tamnd.github.io/math2/node/MDPRVV9DN9.md" tag="MDPRVV9DN9" kind="definition">
# Flat morphism

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].
For a point $x \in X$, let $y = f(x)$, and let $f^\# \colon \mathcal{O}_{Y,y} \to \mathcal{O}_{X,x}$ be the induced map of [[stalks|JTAKAPTT8H]].
These stalks are [[local rings|MZ02BKPHJA]], and $f^\#$ makes $\mathcal{O}_{X,x}$ a module over $\mathcal{O}_{Y,y}$.
The morphism $f$ is *flat at $x$* if $\mathcal{O}_{X,x}$ is a [[flat module|73RYYEVDKA]] over $\mathcal{O}_{Y,y}$.
The morphism $f$ is *flat* if it is flat at every point of $X$.

## Remarks

For a ring $A$, the morphism $\mathbb{A}^n_A \to \operatorname{Spec} A$ is flat.
Each local ring of $\mathbb{A}^n_A$ is a localization of $A[x_1, \dots, x_n]$, which is a free $A$-module, and localizations of flat modules are flat.
For a field $k$, the closed immersion $\operatorname{Spec} k \to \operatorname{Spec} k[t]$ given by $t \mapsto 0$ is not flat.
Its stalk map is $k[t]_{(t)} \to k$, and $k$ is not flat over $k[t]_{(t)}$, since multiplication by $t$ is injective on $k[t]_{(t)}$ but becomes zero on $k$.
For a morphism of affine schemes $\operatorname{Spec} B \to \operatorname{Spec} A$, flatness is equivalent to flatness of $B$ as an $A$-module.
</document>

<document source="https://tamnd.github.io/math2/node/MDS8655N0Z.md" tag="MDS8655N0Z" kind="statement">
# Goldie's theorem

## Statement

Let $R$ be a ring with identity.

1. $R$ is [[semiprime|5F0YMCV50R]] and right [[Goldie|QFDC3NYHJE]] if and only if $R$ has a [[semisimple|NY25HXA478]] right [[classical ring of quotients|C5971PGKPH]].
2. $R$ is [[prime|Z9TJSZZYG1]] and right Goldie if and only if $R$ has a [[simple|62M0NC3E2T]] [[Artinian|20QEAM8AX2]] right classical ring of quotients.

## Remarks

For the forward direction, one shows that in a semiprime right Goldie ring a right ideal is [[essential|8MS51FASMX]] exactly when it contains a regular element.
From this one gets that the regular elements satisfy the right [[Ore condition|TJP4ARC2X7]], so $Q$ exists.
One also gets that $Q$ has no proper essential right ideals, which forces $Q$ to be semisimple.
For the converse, the Goldie conditions and semiprimeness pass from the semisimple ring $Q$ down to its right order $R$.

Example: a commutative integral domain $R$ is prime and Goldie, and its classical ring of quotients is its field of fractions, a simple Artinian ring.
Example: every semiprime right [[Noetherian ring|0670T59HGE]] has a semisimple right classical ring of quotients, since right Noetherian rings are right Goldie.
So a Weyl algebra over a field, which is a Noetherian domain, has a division ring of quotients.

The semiprime hypothesis is needed.
The ring $\Z/4\Z$ is Noetherian, hence Goldie, but it is not semiprime, because the ideal $2\Z/4\Z$ has square $0$.
Its regular elements are $1$ and $3$, which are already units, so its classical ring of quotients is $\Z/4\Z$ itself, which is not semisimple.

## Explanations

### L1 human

Every integral domain sits inside a field of fractions, and this is how one builds $\Q$ from $\Z$.
Goldie's theorem is the noncommutative version.
A ring with no nilpotent ideals and with mild finiteness conditions sits inside a ring of fractions, and this ring of fractions is as nice as possible.
By the [[Artin–Wedderburn theorem|R1F271180N]], it is a finite product of matrix rings over division rings.

### L2 human

Goldie's theorem is the starting point of the theory of noncommutative Noetherian rings.
It lets one study a prime Noetherian ring $R$ through its quotient ring $Q \cong M_n(D)$, with $D$ a division ring, and define invariants such as the Goldie rank $n$.
It applies to enveloping algebras of finite-dimensional Lie algebras over a field and to Weyl algebras, which are Noetherian domains.
</document>

<document source="https://tamnd.github.io/math2/node/MDVSY2NMD2.md" tag="MDVSY2NMD2" kind="concept">
# Gelfand–Kirillov dimension

## Summary

The Gelfand–Kirillov dimension of an algebra over a field is the exponent of the polynomial rate at which the algebra grows.
It is defined through the [[growth|EHY4PYSMGZ]] of finite-dimensional subspaces under multiplication, so it makes sense for noncommutative algebras.
For a finitely generated commutative algebra it equals the [[Krull dimension|MVGBS3J1P7]], and the $n$-th [[Weyl algebra|98QE7KY55W]] has Gelfand–Kirillov dimension $2n$.
It is a basic invariant in noncommutative ring theory and in the theory of $D$-modules.

## Explanations

### L1 human

Take a finite set of generators and look at all products of at most $n$ of them.
If the dimension of their span grows roughly like $n^d$, the algebra has Gelfand–Kirillov dimension $d$.
The polynomial ring in $d$ variables has dimension $d$, just like the space it describes.
An algebra whose span grows exponentially, like the free algebra on two letters, has infinite Gelfand–Kirillov dimension.
</document>

<document source="https://tamnd.github.io/math2/node/MDWN6E58AW.md" tag="MDWN6E58AW" kind="concept">
# Cauchy's power equation

## Summary

Cauchy's power equation is the [[functional equation|SRAN4EKWS0]] $f(xy) = f(x) f(y)$ for an unknown function $f \colon (0, \infty) \to \R$.
It asks which functions turn products into products, that is, which functions are multiplicative.
Its continuous solutions are the zero function and the power functions $f(x) = x^c$ with $c \in \R$.
It is the fourth of Cauchy's basic equations, beside the additive, exponential and logarithmic ones.

## Explanations

### L1 human

Scaling the input by a factor scales the output by a matching factor.
The area of a square of side $x$ is $x^2$, and doubling the side multiplies the area by $2^2 = 4$.
The function $f(x) = x^2$ thus satisfies $f(xy) = f(x) f(y)$.
This equation is the rule behind scaling laws in geometry and physics.
</document>

<document source="https://tamnd.github.io/math2/node/ME4ZSRAT5Z.md" tag="ME4ZSRAT5Z" kind="statement">
# Raghunathan's conjecture

## Statement

Let $G$ be a connected [[Lie group|7JBRGCVJRB]], and let $\Gamma$ be a [[lattice|BTYF5Y5PFY]] in $G$.
Let $H$ be a connected closed subgroup of $G$ generated by one-parameter subgroups $(u_t)$ such that each $\mathrm{Ad}(u_t)$ is unipotent.
Then for every $x \in G/\Gamma$ there is a closed subgroup $L$ of $G$ containing $H$ with the following property.
The [[closure|0QEAJB91GZ]] of the [[orbit|0YQPZB5Q1G]] $Hx$ equals the orbit $Lx$, and $Lx$ carries an $L$-invariant probability measure.

## Remarks

The statement was conjectured by M. S. Raghunathan and proved by Marina Ratner in 1991.
Her proof derives it from her classification of the invariant measures of unipotent flows, one of [[Ratner's theorems|3MXR03096V]].
Before that, Margulis studied $H = \mathrm{SO}(2, 1)^\circ$ acting on $\mathrm{SL}_3(\R)/\mathrm{SL}_3(\Z)$.
He proved that every relatively compact $H$-orbit there is closed, and deduced the Oppenheim conjecture.

Example: for $G = \R^2$, $\Gamma = \Z^2$ and $H$ a line through $0$ of slope $\alpha$, all elements are unipotent.
If $\alpha$ is rational, each orbit of $H$ on the torus is a closed circle, and $L = H$.
If $\alpha$ is irrational, each orbit is dense, and $L = G$.

The unipotent hypothesis is necessary: orbits of the diagonal group in $\mathrm{SL}_2(\R)/\mathrm{SL}_2(\Z)$ can have closures that are not orbits of a subgroup.

## Explanations

### L1 human

Lines of rational slope on a torus close up, and lines of irrational slope fill the torus.
The conjecture says that the same simple dichotomy, in a more general form, holds for orbits of unipotent groups on all spaces of the form $G/\Gamma$.
The closure of an orbit is never a messy set: it is always a smooth homogeneous piece of the space.

### L2 human

The theorem is the topological shadow of Ratner's measure classification.
It is the main tool for questions about the values of polynomials at integer points.
The first such question was the Oppenheim conjecture.
Let $Q$ be a nondegenerate indefinite real quadratic form in $n \ge 3$ variables, not a multiple of a form with rational coefficients.
Then the values $Q(\Z^n)$ are dense in $\R$.
The theorem shows that orbit closures of unipotent flows behave like closures of linear flows on tori, but on noncommutative spaces.
</document>

<document source="https://tamnd.github.io/math2/node/ME71ZQE49J.md" tag="ME71ZQE49J" kind="definition">
# Commutative magma

## Definition

Let $(M, \cdot)$ be a [[magma|9689SFPVE9]].
It is a *commutative magma* if $x \cdot y = y \cdot x$ for all $x, y \in M$, that is, if the operation has the [[commutativity|099E72KTF8]] property.

## Remarks

Example: $\Q$ with $x \circ y = (x + y)/2$.
The value is rational, and $(x + y)/2 = (y + x)/2$.
The magma is not associative: $(0 \circ 4) \circ 8 = 5$ but $0 \circ (4 \circ 8) = 3$.

Non-example: $(\Z, -)$, since $1 - 0 = 1$ but $0 - 1 = -1$.

In Mathlib this structure is the class `CommMagma`.
</document>

<document source="https://tamnd.github.io/math2/node/ME8GM0VTWP.md" tag="ME8GM0VTWP" kind="statement">
# Prime ideal theorem for distributive lattices

## Statement

Let $L$ be a [[distributive lattice|0HM0HQCK6N]].
Let $I$ be an [[ideal|5GW2T2AM1A]] of $L$ and let $F$ be a [[filter|Z15FXM9WM1]] of $L$ with $I \cap F = \emptyset$.
Then there is a [[prime ideal|QKWVXJWHP6]] $P$ of $L$ such that $I \subseteq P$ and $P \cap F = \emptyset$.

## Remarks

The usual proof applies [[Zorn's lemma|04RXTQA5ZA]] to the set of ideals that contain $I$ and are disjoint from $F$, ordered by inclusion.
Let $P$ be a maximal one, and suppose $x \wedge y \in P$ with $x \notin P$ and $y \notin P$.
By maximality, the ideals generated by $P \cup \set{x}$ and by $P \cup \set{y}$ meet $F$, so $f \le p \vee x$ and $g \le q \vee y$ for some $f, g \in F$ and $p, q \in P$.
Then $f \wedge g \le (p \vee q) \vee (x \wedge y)$ by distributivity, so $f \wedge g \in P \cap F$, a contradiction.

Example: in the lattice of divisors of $12$, with $\gcd$ as meet and $\lcm$ as join, take $I = \set{1, 2}$ and $F = \set{3, 6, 12}$.
The set $P = \set{1, 2, 4}$ of divisors not divisible by $3$ is a prime ideal with $I \subseteq P$ and $P \cap F = \emptyset$.

Distributivity is necessary.
In the [[diamond lattice M3|JQ6FJH2TP8]] with middle elements $a, b, c$, the proper ideals are $\set{0}$ and $\set{0, x}$ for a middle element $x$.
None of them is prime: for instance $a \wedge b = 0$, while neither $a$ nor $b$ lies in $\set{0, c}$.
So the ideal $\set{0}$ and the filter $\set{1}$ admit no separating prime ideal.

The [[Boolean prime ideal theorem|STPBXBKSZP]] is the special case of Boolean algebras.
Over ZF set theory, the two statements are equivalent, and both are strictly weaker than the [[axiom of choice|0HTGB13FE2]].

## Explanations

### L1 human

An ideal is a family of "small" elements and a filter is a family of "large" elements.
If they do not overlap, the theorem finds a prime ideal that keeps all the small ones and avoids all the large ones.
Prime ideals are the "points" of a distributive lattice, so the theorem says there are enough points to separate things.

### L2 human

The theorem is the key step in representing a distributive lattice as a lattice of sets: map $a$ to the set of prime filters that contain it.
Separating $a \not\le b$ by a prime ideal containing $b$ and missing $a$ shows that this map is injective.
This is the basis of Stone's representation theorem for distributive lattices and of [[Priestley duality|SA22SEC75K]].
It needs a form of choice, and over ZF it is equivalent to the [[Boolean prime ideal theorem|STPBXBKSZP]].
</document>

<document source="https://tamnd.github.io/math2/node/MEBR2NYA1S.md" tag="MEBR2NYA1S" kind="definition">
# Congruence-modular variety

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
The variety $\mathcal{V}$ is *congruence modular* if, for every algebra $A$ in $\mathcal{V}$, the [[congruence lattice|REE07CSXHR]] $\operatorname{Con} A$ is a [[modular lattice|QX7WZ92QAK]].
That is, for all congruences $\alpha, \beta, \gamma$ of $A$ with $\alpha \le \gamma$,
$$\alpha \vee (\beta \wedge \gamma) = (\alpha \vee \beta) \wedge \gamma.$$

## Remarks

Example: the variety of groups is congruence modular, since the normal subgroups of a group form a modular lattice.
More generally, every congruence-permutable variety and every congruence-distributive variety is congruence modular.
Non-example: the variety of sets, with no operations, is not congruence modular.
The congruences of a set are all its equivalence relations.
On $\set{1, 2, 3, 4}$, let $\alpha$ have the classes $\set{1, 2}, \set{3}, \set{4}$, let $\gamma$ have the classes $\set{1, 2}, \set{3, 4}$, and let $\beta$ have the classes $\set{1, 3}, \set{2, 4}$.
Then $\alpha \le \gamma$, $\beta \wedge \gamma$ is the identity relation and $\alpha \vee \beta$ is the full relation.
So $\alpha \vee (\beta \wedge \gamma) = \alpha$, while $(\alpha \vee \beta) \wedge \gamma = \gamma \ne \alpha$.

## Explanations

### L1 human

A variety is congruence modular when all its algebras have congruence lattices that satisfy the modular law.
</document>

<document source="https://tamnd.github.io/math2/node/MEC3ZMC4F1.md" tag="MEC3ZMC4F1" kind="concept">
# Elementary class

## Summary

An elementary class is the class of all [[models|MVCH6N9SNW]] of a first-order [[theory|92SWXST1ZM]] in a fixed signature.
Groups, rings, fields of characteristic $0$ and dense linear orders are elementary classes.
An elementary class is closed under [[elementary equivalence|P3B1V182N4]] and under ultraproducts.
By the [[compactness theorem|1NR3HAWG9D]], many natural classes, such as torsion groups, are not elementary.
Universal classes, universal Horn classes, quasivarieties and varieties are elementary classes with axioms of special shapes.

## Explanations

### L1 human

Pick a list of first-order axioms, such as the group axioms.
The elementary class of the list is everything that satisfies all of them.
First-order axioms may quantify over elements but not over sets of elements or over natural numbers.
So "every element has finite order" cannot be written as one first-order axiom, and torsion groups are not an elementary class.
</document>

<document source="https://tamnd.github.io/math2/node/MECCMT61DV.md" tag="MECCMT61DV" kind="definition">
# Markov chain Monte Carlo

## Definition

Let $S$ be a countable set, let $\pi$ be a probability distribution on $S$, and let $f \colon S \to \R$ be bounded.
A *Markov chain Monte Carlo method* for estimating $\pi(f) = \sum_{x \in S} f(x) \pi(x)$ consists of:

1. a [[transition matrix|0F462493EA]] $P$ on $S$ such that $\pi$ is a [[stationary distribution|0KRH5157WP]] of $P$ and the chain with transition matrix $P$ is irreducible;
2. a [[Markov chain|033TJBZDG0]] $(X_k)_{k \ge 0}$ with transition matrix $P$, simulated from some starting state;
3. the estimator $\hat{\pi}_n(f) = \frac{1}{n} \sum_{k=1}^{n} f(X_k)$, computed for a large $n$.

## Remarks

By the ergodic theorem for irreducible chains with a stationary distribution, $\hat{\pi}_n(f) \to \pi(f)$ almost surely as $n \to \infty$, for every starting state.
The usual way to obtain clause 1 is to impose [[detailed balance|CTGN6CK1ZS]] for $\pi$, as the [[Metropolis–Hastings algorithm|MKHF449E22]] does.
In practice the first steps, the burn-in, are often discarded, and if the chain is also aperiodic then the law of $X_k$ tends to $\pi$ by the [[convergence theorem|B1E1ES1BYD]].
Example: on $S = \Z$ with $\pi(x) \propto e^{-x^2}$, propose $x \pm 1$ with probability $1/2$ each and accept with probability $\min(1, \pi(y)/\pi(x))$.
This chain is irreducible and satisfies detailed balance for $\pi$.
Non-example: the simple [[random walk|04VZE3FZXF]] on the path $\set{1, \ldots, m}$, which moves to a uniform neighbour, has stationary law proportional to the degrees.
So its averages estimate expectations under that law, not under the uniform law, and it fails clause 1 for uniform $\pi$ when $m \ge 3$.
</document>

<document source="https://tamnd.github.io/math2/node/MEKFR4Q617.md" tag="MEKFR4Q617" kind="object">
# Suslin line

## Description

A *Suslin line* is a nonempty [[totally ordered set|0316AYH1SC]] $(L, <)$ with these properties.

1. $L$ has no least element and no greatest element.
2. The order is [[dense|JB9HJ9CPEN]].
3. The order is complete: each nonempty subset of $L$ that is bounded above has a least upper bound.
4. Each family of pairwise disjoint nonempty open intervals of $L$ is countable.
5. $L$ is not [[separable|0GKJ78948V]]: no countable subset of $L$ meets every nonempty open interval.

## Remarks

The real line satisfies clauses 1 to 4 but not clause 5, so $\R$ is not a Suslin line.
In fact, a total order with clauses 1 to 3 and separability is order isomorphic to $\R$.
So a Suslin line is exactly a counterexample to [[Suslin's problem|D1T39KHPDN]].
The existence of a Suslin line is independent of ZFC, if ZFC is consistent.
The [[diamond principle|HFQSSB1Z23]] implies that a Suslin line exists.
[[Martin's axiom|Z8XMCCZFA4]] together with the failure of the continuum hypothesis implies that no Suslin line exists.
A Suslin line exists if and only if a Suslin tree exists.

## Explanations

### L1 human

A Suslin line would be a line that looks almost exactly like the real line, but is not the real line.
It has no ends and no gaps, and one can only fit countably many disjoint intervals into it.
Yet it has no countable set of points that is spread through all of it, as the rationals are spread through the reals.
Whether such a line exists cannot be decided from the usual axioms of set theory.
</document>

<document source="https://tamnd.github.io/math2/node/MF3BWZ9YSW.md" tag="MF3BWZ9YSW" kind="concept">
# Urysohn space

## Summary

An Urysohn space is a topological space in which any two distinct points have disjoint closed neighbourhoods.
This separation axiom, also written $T_{2\frac12}$, lies between [[Hausdorff|0S1FJ8VEX9]] and regular Hausdorff.
Every Urysohn space is Hausdorff, and every regular Hausdorff space is Urysohn.
Some authors use the name for the stronger notion of a completely Hausdorff space.

## Explanations

### L1 human

In a Hausdorff space, two points can be put in two open sets that do not meet.
An Urysohn space asks a little more: the closures of these open sets must not meet either.
So there is a real gap between the two points, not just a thin border.
Every metric space is an Urysohn space.
For points $x \ne y$ at distance $r$, the closed balls of radius $r / 3$ around them are disjoint neighbourhoods.
</document>

<document source="https://tamnd.github.io/math2/node/MF3GV42CG6.md" tag="MF3GV42CG6" kind="definition">
# Nonnegative matrix

## Definition

Let $m, n \ge 1$ and let $A = (a_{ij})$ be a real $m \times n$ [[matrix|0VD9ZSWWJG]].
The matrix $A$ is *nonnegative* if $a_{ij} \ge 0$ for all $i$ and $j$.
The matrix $A$ is *positive* if $a_{ij} > 0$ for all $i$ and $j$.

## Remarks

One writes $A \ge 0$ for a nonnegative matrix and $A > 0$ for a positive matrix.
More generally, $A \ge B$ means that $A - B \ge 0$ entry by entry.
Example: $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$ is positive, hence nonnegative.
It is not [[positive semidefinite|01MMVW8CKC]], because it has the eigenvalue $-1$.
Non-example: $\begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$ is not nonnegative, because its entry in row $1$ and column $2$ is $-1$.
Products and sums of nonnegative matrices are nonnegative.
</document>

<document source="https://tamnd.github.io/math2/node/MF4N3VGGFV.md" tag="MF4N3VGGFV" kind="concept">
# Group extension

## Summary

A group extension is a group $G$ with a [[normal subgroup|0CB0DX5BVV]] $N$ and a [[quotient group|0KYX2N69J0]] $G/N$ isomorphic to a given group $Q$.
It is the way to build a group out of two smaller pieces.
The extension problem asks for all groups $G$ that can be built from given $N$ and $Q$.
Every finite group is built from simple groups by repeated extensions, along a [[composition series|0XV2A49X6Q]].

## Explanations

### L1 human

A group extension glues a group $N$ and a group $Q$ together into a bigger group $G$.
Inside $G$ you find a copy of $N$ as a normal subgroup, and when you collapse that copy to a point, what is left is $Q$.
The same two pieces can be glued in different ways.
For example, both $\Z/4\Z$ and $\Z/2\Z \times \Z/2\Z$ have a subgroup of order $2$ with quotient of order $2$.
So both are extensions of a group of order $2$ by a group of order $2$, but they are not isomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/MF56GK7QY2.md" tag="MF56GK7QY2" kind="statement">
# Nested intervals theorem

## Statement

For each $n \ge 1$, let $a_n \le b_n$ be real numbers and let $I_n = [a_n, b_n]$ be the closed [[interval|0TK8Y3J5RW]].
Assume that $I_{n+1} \subseteq I_n$ for all $n \ge 1$.
Then the intersection $\bigcap_{n \ge 1} I_n$ is not empty.
If moreover $b_n - a_n \to 0$, then the intersection consists of exactly one point.

## Remarks

The left endpoints $a_n$ increase and are bounded above by $b_1$.
By the [[completeness axiom|0AXWET80YW]], they have a [[supremum|0BTMQ04556]] $s$.
Each $b_m$ is an upper bound of all $a_n$, so $a_n \le s \le b_m$ for all $n$ and $m$, and $s$ lies in every $I_n$.
If two points $s < t$ lie in every $I_n$, then $b_n - a_n \ge t - s > 0$ for all $n$, so the lengths cannot tend to $0$.

Each hypothesis is needed.
The open intervals $(0, 1/n)$ are nested and nonempty, but no real number lies in all of them.
The closed unbounded sets $[n, \infty)$ are nested, but their intersection is empty.
Over $\Q$, the intervals $\{q \in \Q : a_n \le q \le b_n\}$ with $a_n, b_n$ the decimal approximations of $\sqrt 2$ from below and above have empty intersection.

The theorem is the case $X = \R$ of [[Cantor's intersection theorem|0EZBG8SKJM]].

## Explanations

### L1 human

Take a closed interval, then a smaller closed interval inside it, then a smaller one inside that, and so on forever.
Some point stays inside all of them.
If the lengths shrink to zero, the intervals close in on a single point.
For example, the intervals $[0, 1/n]$ all contain $0$, and $0$ is the only common point.

### L2 human

The theorem is one form of the [[completeness of the real numbers|E3CPZNJZNG]], and in an Archimedean ordered field it is equivalent to the least-upper-bound property.
It drives bisection proofs: halve an interval, keep a half with a desired property, and repeat.
This is how one proves the Bolzano–Weierstrass theorem and the intermediate value theorem, and how Cantor showed that $\R$ is uncountable.
</document>

<document source="https://tamnd.github.io/math2/node/MF77N00SK8.md" tag="MF77N00SK8" kind="statement">
# Compactness of integral operators

## Statement

Let $a < b$.

1. Let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]].
Then the operator $(T f)(x) = \int_a^b K(x, y) f(y) \, dy$ maps $C[a, b]$ to itself and is a [[compact operator|M2NEP1K5AE]] on $C[a, b]$ with the sup norm.
2. Let $(X, \mu)$ be a $\sigma$-finite measure space and let $K \in L^2(X \times X, \mu \otimes \mu)$.
Then the [[Hilbert–Schmidt integral operator|CB5G212Q03]] $(T f)(x) = \int_X K(x, y) f(y) \, d\mu(y)$ is a compact operator on $L^2(X, \mu)$, and $\norm{T} \le \norm{K}_{L^2(X \times X)}$.

## Remarks

For part 1, let $f$ run over the unit ball of $C[a, b]$.
Then $\abs{Tf(x)} \le (b - a) \max \abs{K}$, and $\abs{Tf(x) - Tf(x')} \le (b - a) \max_y \abs{K(x, y) - K(x', y)}$, which is small when $x$ and $x'$ are close, by uniform continuity of $K$.
So the image of the unit ball is bounded and equicontinuous, and the [[Arzelà–Ascoli theorem|0381S8Z5P0]] shows that it is relatively compact.

For part 2, the Cauchy–Schwarz inequality gives $\abs{Tf(x)}^2 \le \int_X \abs{K(x, y)}^2 \, d\mu(y) \cdot \norm{f}^2$, and integrating in $x$ gives the norm bound.
[[Approximating $K$ by degenerate kernels|FJKNJ14AD2]] in $L^2$ then writes $T$ as a norm limit of [[finite-rank operators|Z8FABSYQTS]], and such a limit is compact.

Example: the Volterra operator $(Vf)(x) = \int_0^x f(y) \, dy$ on $L^2[0, 1]$ has the square-integrable kernel equal to $1$ for $y \le x$ and $0$ otherwise, so it is compact.

The integrability of the kernel matters.
On $L^2(\R)$, the convolution $(Tf)(x) = \int_\R e^{-\abs{x - y}} f(y) \, dy$ has a kernel that is not in $L^2(\R^2)$.
This $T$ is not compact: it commutes with translations, so the images of the translates of a fixed nonzero function have no convergent subsequence.

## Explanations

### L1 human

An integral operator averages a function against a kernel.
Averaging smooths things out, so the outputs of all functions of size at most one form a tame family, without wild oscillations.
Compactness is the precise form of this: every sequence of outputs has a convergent subsequence.

### L2 human

Compactness is what makes integral equations of the second kind behave like finite linear systems.
Once the operator is known to be compact, [[Fredholm's theorem|XQ8DYKW4SH]] and the spectral theory of compact operators apply, and for symmetric kernels so does the [[Hilbert–Schmidt theorem|A1G25R99PY]].
</document>

<document source="https://tamnd.github.io/math2/node/MF89BJZ8KS.md" tag="MF89BJZ8KS" kind="definition">
# Projective algebra

## Definition

Fix a signature and let $\mathcal{K}$ be a class of algebras of this signature.
An algebra $A \in \mathcal{K}$ is *projective* in $\mathcal{K}$ if the following holds.
For all $B, C \in \mathcal{K}$, every surjective [[homomorphism|KBXSZC6BV1]] $g \colon B \to C$ and every homomorphism $f \colon A \to C$, there is a homomorphism $h \colon A \to B$ with $g \circ h = f$.

## Remarks

In a variety, epimorphisms of the category of algebras need not be surjective.
So this notion can differ from a projective object of the category; universal algebra uses the surjective version.
In a variety, an algebra is projective exactly when it is a [[retract|68PFSGP39B]] of a [[free algebra|GTS4DRRMZR]].

Example: every free algebra $F$ on a set $X$ in a variety is projective.
Given $g \colon B \to C$ surjective and $f \colon F \to C$, choose $b_x \in B$ with $g(b_x) = f(x)$ for each $x \in X$.
The homomorphism $h \colon F \to B$ with $h(x) = b_x$ satisfies $g \circ h = f$, since both sides agree on the generators $X$.
In the variety of modules over a ring, the projective algebras are the [[projective modules|BFFA03GE3V]].

Non-example: $\Z/2\Z$ is not projective in the variety of abelian groups.
Take $g \colon \Z \to \Z/2\Z$ the reduction map and $f = \id$.
Every homomorphism $h \colon \Z/2\Z \to \Z$ is zero, because $\Z$ has no element of order $2$, so $g \circ h = 0 \ne f$.
</document>

<document source="https://tamnd.github.io/math2/node/MFB1Q9JTPD.md" tag="MFB1Q9JTPD" kind="concept">
# Multicategory

## Summary

A multicategory is like a [[category|A2DE66937D]], except that a morphism has a finite list of objects as its source and one object as its target.
Such morphisms, called multimorphisms, compose by plugging outputs into inputs.
Multilinear maps between vector spaces are the basic example.
A multicategory with one object is the same as a non-symmetric operad, and every monoidal category gives a multicategory.

## Explanations

### L1 human

Many natural maps take several inputs at once, such as the multiplication of two numbers or the dot product of two vectors.
A category only records maps with one input, so it cannot see this directly.
A multicategory records maps with any number of inputs and one output.
For example, a bilinear map $V \times W \to U$ is a morphism from the list $(V, W)$ to $U$.
Feeding the outputs of two such maps into a third gives a map with four inputs.
</document>

<document source="https://tamnd.github.io/math2/node/MFDEMR9GMM.md" tag="MFDEMR9GMM" kind="definition">
# Configuration

## Definition

Let $p, \ell, \gamma, \pi$ be positive integers.
A *configuration* $(p_\gamma\ \ell_\pi)$ is a finite [[incidence structure|FGKNAFNHA3]] of points and lines such that:

1. there are $p$ points and $\ell$ lines;
2. each point lies on exactly $\gamma$ lines;
3. each line contains exactly $\pi$ points;
4. two distinct points lie on at most one common line.

## Remarks

Clause 4 says that the structure is a [[partial linear space|CX07CWPCDX]]; it is equivalent to saying that two distinct lines meet in at most one point.
Counting incident point-line pairs in two ways gives $p \gamma = \ell \pi$.
When $p = \ell$, and hence $\gamma = \pi = k$, the configuration is *symmetric* and is written $(p_k)$.
A *geometric* configuration is one whose points and lines are points and lines of the real (or another) projective plane, with incidence as in the plane.

Example: the complete quadrilateral, made of $4$ lines in the real plane with no three concurrent and their $6$ intersection points, is a $(6_2\ 4_3)$ configuration.
Each line meets the other three lines in $3$ distinct points, each point lies on $2$ of the lines, and $6 \cdot 2 = 4 \cdot 3$.

Non-example: a triangle together with one median has $4$ points and $4$ lines.
The vertex the median starts from lies on $3$ of the lines, but each other point lies on only $2$, so clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/MFJSS5HVA0.md" tag="MFJSS5HVA0" kind="concept">
# Volterra equations with completely monotone kernels

## Summary

These are [[Volterra equations of convolution type|SNFYTC1TTA]] whose kernel $k$ is completely monotone on $(0, \infty)$: it is infinitely differentiable and its derivatives alternate in sign.
By [[Bernstein's theorem on monotone functions|ZBS1JA62QP]], such a kernel is a superposition $k(t) = \int e^{-st} \, d\mu(s)$ of decaying exponentials with a positive measure $\mu$.
The class contains the exponential kernels and the Abel kernels $t^{\alpha - 1}$ with $0 < \alpha < 1$.
It is a natural setting for positivity and decay results for solutions.
Such kernels model fading memory, for instance in linear viscoelasticity.

## Explanations

### L1 human

A completely monotone kernel describes a memory that fades with time in the most regular way possible.
It is positive, decreasing, convex, and so on for all higher derivatives.
Each such kernel is a mixture of simple exponential memories $e^{-st}$ with different decay rates $s$.
For example, $k(t) = 1/\sqrt{t}$ is completely monotone: it is the mixture $\frac{1}{\sqrt{\pi}} \int_0^\infty e^{-st} s^{-1/2} \, ds$.
So Abel's integral equation is an equation of this type.
</document>

<document source="https://tamnd.github.io/math2/node/MFJXPAF0R0.md" tag="MFJXPAF0R0" kind="concept">
# Gödel numbering

## Summary

A Gödel numbering assigns to each symbol, formula or proof of a formal language a natural number, its code, in an effective and injective way.
It lets statements about formulas and proofs become statements about numbers.
Gödel introduced it to prove his incompleteness theorems, where a theory of arithmetic talks about its own provability through codes.

## Explanations

### L1 human

Give each symbol a number, for example $1$ for "$0$", $2$ for "$=$", and so on.
A formula is a list of symbols, so it becomes a list of numbers.
A list of numbers can be packed into one number, for example as a product of prime powers.
Then "this formula is provable" becomes a property of a number, which arithmetic can talk about.
Any computer that stores text as bits does the same thing: each file is, in the end, one large number.
</document>

<document source="https://tamnd.github.io/math2/node/MFM9ZXWMHA.md" tag="MFM9ZXWMHA" kind="concept">
# Regular variation

## Summary

A positive function $f$ on $(0, \infty)$ is regularly varying if $f(ax)/f(x)$ has a positive finite limit as $x \to \infty$ for every $a > 0$.
The limit is then always a power $a^\rho$, and $f(x) = x^\rho L(x)$ with $L$ a [[slowly varying function|EHJ7D5CRZS]].
So regularly varying functions behave like powers up to a slowly varying factor.
They are the natural comparison functions in [[Karamata's Tauberian theorem|DBEGF38V9H]].

## Explanations

### L1 human

A regularly varying function scales almost like a power of $x$ when $x$ is large.
If $f(x) = x^2 \log x$, then doubling $x$ multiplies $f$ by about $4$, and the error in this "about" goes away as $x$ grows.
The number $\rho$ in $a^\rho$ is the index; here it is $2$.
The exponential function is not regularly varying, because doubling $x$ squares $e^x$, which is far more than any fixed factor.
</document>

<document source="https://tamnd.github.io/math2/node/MFN4Q5SH8H.md" tag="MFN4Q5SH8H" kind="concept">
# Rarefaction wave

## Summary

A rarefaction wave is a continuous solution of a [[conservation law|8RA7AMWSAF]] that spreads out from a point like a fan.
In the fan, the unknown takes every value between two states.
It is self-similar: it depends only on $x/t$.
It is the admissible solution when the characteristics on the two sides of a jump move apart.
In that case a [[shock wave|JSDMGGTG7G]] would violate the entropy condition.
Together with shocks and contact discontinuities, rarefaction waves are the building blocks of solutions of the [[Riemann problem|GJWKX22VVE]].
See the [[definition|MM6TXKN90V]].

## Explanations

### L1 human

When a traffic light turns green, the dense queue of cars does not leave as one block.
The front cars drive off fast, the cars behind start later, and the queue thins out gradually.
The density goes smoothly from jammed to free over a stretch of road that grows with time.
This spreading fan is a rarefaction wave.
For Burgers' equation with value $0$ on the left and $1$ on the right, it is $u(x, t) = x/t$ for $0 \le x \le t$.
</document>

<document source="https://tamnd.github.io/math2/node/MFNNZQ484T.md" tag="MFNNZQ484T" kind="statement">
# Strong perfect graph theorem

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]], and let $\overline{G}$ be its [[complement|0RNMP6VCR2]].
An odd hole of a graph is an [[induced subgraph|0V05FC4GKQ]] that is a [[cycle|07CT9RDY7Q]] of odd length at least $5$.
Then $G$ is [[perfect|1YZ7NV161A]] if and only if neither $G$ nor $\overline{G}$ has an odd hole.

## Remarks

The complement of an odd hole is called an odd antihole, and graphs with neither are called Berge graphs.
The easy direction holds because an odd hole $C_{2k+1}$ with $k \ge 2$ has $\omega = 2$ and $\chi = 3$, and its complement has $\omega = k$ and $\chi = k + 1$.
The hard direction was conjectured by Claude Berge and proved by Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, with the proof published in 2006.
Their proof shows that every Berge graph lies in one of a few basic classes of perfect graphs or admits one of a few kinds of decomposition.
A smallest Berge graph that is not perfect can have neither property.
The theorem implies the weak perfect graph theorem, since the condition is symmetric in $G$ and $\overline{G}$.

## Explanations

### L1 human

A graph is perfect when, in every part of it, the colours needed equal the size of the largest clique.
The theorem says there are only two kinds of obstruction: an odd cycle of length at least five with no chords, or the complement of one.
If you cannot find either inside a graph, the graph is perfect.

### L2 human

The theorem turns a condition about all induced subgraphs into a finite list of forbidden patterns.
It was one of the main open problems of graph theory for about forty years.
Later work showed that Berge graphs, and hence perfect graphs, can be recognized in polynomial time.
</document>

<document source="https://tamnd.github.io/math2/node/MG1KM4S1BV.md" tag="MG1KM4S1BV" kind="concept">
# Difference set

## Summary

A difference set is a subset $D$ of a finite group with a balance property.
Every non-identity element of the group arises as a "difference" $d_1 d_2^{-1}$ of elements of $D$ the same number $\lambda$ of times.
The translates of a difference set form a symmetric [[block design|4JCSS2Z1BA]] on which the group acts regularly.
Difference sets give many projective planes, Hadamard matrices and sequences with good correlation.
The node [[difference set|D1QA6N9EWJ]] gives the precise definition.

## Explanations

### L1 human

Take the numbers $1, 2, 4$ in arithmetic modulo $7$.
The differences between two of them are $\pm 1$, $\pm 2$ and $\pm 3$, and these are each nonzero residue modulo $7$ exactly once.
So $\set{1, 2, 4}$ "sees" every nonzero distance exactly once.
Shifting it by $0, 1, \dots, 6$ gives seven triples that form a copy of the [[Fano plane|QQ8GW1ASZM]].
</document>

<document source="https://tamnd.github.io/math2/node/MG20HMMYKY.md" tag="MG20HMMYKY" kind="statement">
# Borel fixed-point theorem

## Statement

Let $k$ be an algebraically closed field.
Let $G$ be a connected [[solvable|0FXJ7YFMMX]] [[linear algebraic group|GA81VVQHEP]] over $k$.
Let $X$ be a nonempty complete [[variety|HV6YBS0YDH]] over $k$ with an [[action|RW85C34GQW]] of $G$.
Then $G$ has a [[fixed point|0RRWNMYRXC]] in $X$.

## Remarks

The proof goes by induction on $\dim G$.
The derived group $G'$ is connected, normal and of smaller dimension, so its fixed-point set $Y = X^{G'}$ is nonempty, and it is closed, hence complete.
The group $G$ acts on $Y$, and an orbit of minimal dimension $G \cdot y$ is closed, so it is complete.
The stabilizer $G_y$ contains $G'$, so it is normal and $G / G_y$ is an affine group.
The orbit map gives a bijective morphism from $G / G_y$ onto the complete orbit $G \cdot y$, and one checks that $G / G_y$ is then complete as well.
A connected variety that is both affine and complete is a point, so the orbit is a point and $y$ is fixed.

Example: a connected solvable closed subgroup $H$ of $\mathrm{GL}_n$ acts on the variety of full flags in $k^n$, which is complete.
The theorem gives a flag fixed by $H$, and this proves the Lie–Kolchin theorem: $H$ is conjugate to a subgroup of the group $B$ of invertible upper triangular matrices.

Completeness is needed: $\mathbb{G}_m$ acts on $\mathbb{A}^1 \setminus \set{0}$ by multiplication without a fixed point.
Solvability is needed: $\mathrm{SL}_2$ acts on $\mathbb{P}^1$ transitively, so without a fixed point.
Connectedness is needed: the group $\Z / 2$ swaps the two points of a two-point variety.

## Explanations

### L1 human

A connected solvable group acting on a complete variety always leaves some point where it is.
Complete varieties, like projective space, have no "escape to infinity", so the group cannot push every point away.
The theorem is the algebraic analogue of fixed-point theorems for compact spaces.

### L2 human

The theorem is the basic tool for Borel subgroups.
Applied to a Borel subgroup $B$ acting on $G / P$, it shows that $B$ is conjugate into every parabolic subgroup $P$.
It gives the conjugacy of all [[Borel subgroups|5JSYA1JNCC]], and with it the conjugacy of maximal tori.
It also gives the Lie–Kolchin theorem on triangularizing connected solvable matrix groups.
</document>

<document source="https://tamnd.github.io/math2/node/MG2NCWHQF2.md" tag="MG2NCWHQF2" kind="concept">
# Binomial proportion confidence interval

## Summary

A binomial proportion confidence interval is a [[confidence interval|0S6T9KQNC0]] for the success probability $p$ of a [[binomial distribution|0WCWA1KT3M]].
It is computed from the number of successes in $n$ trials.
Because the data are discrete, no interval has coverage exactly $1 - \alpha$ for every $p$.
So there are several standard intervals, such as the Wald, Wilson and Clopper–Pearson intervals, with different trade-offs between coverage and length.

## Explanations

### L1 human

You toss a coin $n$ times, see $x$ heads, and want a range of plausible values for the true probability of heads.
The simplest answer is the observed proportion plus or minus about two standard errors.
For $x = 40$ heads in $n = 100$ tosses, this gives roughly $0.40 \pm 0.10$, so the interval $[0.30, 0.50]$ at level about $95\%$.
This simple interval behaves badly when $p$ is near $0$ or $1$, which is why better intervals exist.
</document>

<document source="https://tamnd.github.io/math2/node/MG3X8Y1VT4.md" tag="MG3X8Y1VT4" kind="concept">
# Space of bounded continuous functions

## Summary

For a topological space $X$, the space $C_b(X)$ consists of the bounded continuous scalar functions on $X$, with the supremum norm.
It is a [[Banach space|06GZ956657]]: see [[completeness of the space of bounded continuous functions|09GRCZVR5E]] and the example of the [[sup norm|0PARCKE8WP]].
Convergence in this norm is uniform convergence on $X$.
When $X$ is compact, every continuous function is bounded and $C_b(X)$ is the classical space $C(X)$.

## Explanations

### L1 human

Think of all continuous functions on $X$ whose graphs stay inside some horizontal band.
The distance between two of them is the largest vertical gap between their graphs.
For $X = \R$, the functions $\sin$ and $\arctan$ belong to $C_b(\R)$, but $x \mapsto x$ does not.
A limit of such functions, uniform over all of $X$, is again bounded and continuous.
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<document source="https://tamnd.github.io/math2/node/MG4RNMA8AR.md" tag="MG4RNMA8AR" kind="concept">
# Bernstein–Gelfand–Ponomarev reflection functors

## Summary

The reflection functors of Bernstein, Gelfand and Ponomarev pass between [[representations of quivers|TXVWFWP6VQ]] that differ by reversing all arrows at one sink or source.
On dimension vectors they act by the simple reflections of the Weyl group of the quiver.
They send each indecomposable representation, other than one simple representation, to an indecomposable one.
They gave a short proof of [[Gabriel's theorem|FC78F2E87E]] and were an early model of tilting theory.

## Explanations

### L1 human

A sink of a quiver is a vertex where all arrows arrive and none leaves.
Reversing all arrows at a sink gives a new quiver, in which that vertex is a source.
The reflection functor turns each representation of the old quiver into one of the new quiver, by replacing the space at the sink by a kernel.
Take an indecomposable representation other than the simple one at the sink.
Then the dimension at that vertex changes by the rule of a reflection: the new dimension is the sum of the neighbouring dimensions minus the old one.
Repeating this moves representations around until they become simple, which is how one classifies them.
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# Hermite polynomials

## Description

The (physicists') Hermite polynomials are the polynomials $H_n \in \Z[x]$, $n \ge 0$, given by
$$H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}.$$
Equivalently, $H_0 = 1$, $H_1 = 2x$ and $H_{n+1} = 2x \, H_n - 2n \, H_{n-1}$ for $n \ge 1$.
The probabilists' Hermite polynomials $\operatorname{He}_n$ use the weight $e^{-x^2/2}$ instead: $\operatorname{He}_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2/2}$.

## Remarks

The first values are $H_2 = 4x^2 - 2$, $H_3 = 8x^3 - 12x$, and $\operatorname{He}_2 = x^2 - 1$, $\operatorname{He}_3 = x^3 - 3x$.
Check for $H_2$ with the recursion: $2x \cdot 2x - 2 \cdot 1 \cdot 1 = 4x^2 - 2$.
The polynomial $H_n$ has degree $n$ and leading coefficient $2^n$, while $\operatorname{He}_n$ is monic.
The two families are related by $H_n(x) = 2^{n/2} \operatorname{He}_n(\sqrt{2} \, x)$.
They are orthogonal on $\R$: $\int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} \, dx = \sqrt{\pi} \, 2^n n! \, \delta_{mn}$.
They satisfy $H_n' = 2n \, H_{n-1}$ and $\operatorname{He}_n' = n \operatorname{He}_{n-1}$.
In quantum mechanics, the eigenfunctions of the harmonic oscillator are $H_n(x) e^{-x^2/2}$ up to scaling.

## Explanations

### L1 human

Differentiate the bell curve $e^{-x^2}$ again and again.
Each derivative is a polynomial times $e^{-x^2}$, and up to sign these polynomials are the Hermite polynomials.
They are the natural polynomials to use when the weight is a Gaussian bell curve.
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<document source="https://tamnd.github.io/math2/node/MGBKF52TQE.md" tag="MGBKF52TQE" kind="concept">
# Cauchy-type integral

## Summary

A Cauchy-type integral is a function $\Phi(z) = \frac{1}{2\pi i} \int_\Gamma \frac{\varphi(\tau)}{\tau - z} \, d\tau$ built from a curve $\Gamma$ and a function $\varphi$ on it.
It has the form of [[Cauchy's integral formula|0DG10TVYPN]], but $\varphi$ is arbitrary and need not be the boundary value of a holomorphic function.
It is a [[holomorphic function|0NAHXR1FRH]] off $\Gamma$, and in general it jumps across $\Gamma$ by exactly $\varphi$.
This makes it the basic tool for boundary value problems of Riemann–Hilbert type and for singular integral equations.

## Explanations

### L1 human

Cauchy's formula recovers a holomorphic function inside a curve from its values on the curve.
A Cauchy-type integral uses the same formula with any function $\varphi$ on the curve.
The result is still holomorphic on both sides of the curve, but the inside and outside functions need not fit together.
The amount by which they differ across the curve is $\varphi$ itself.
For example, on the unit circle with $\varphi(\tau) = \tau + \tau^{-1}$, the integral equals $z$ inside and $-1/z$ outside, and the difference at $\tau$ on the circle is $\tau + \tau^{-1}$.
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<document source="https://tamnd.github.io/math2/node/MGCVQXXQJH.md" tag="MGCVQXXQJH" kind="definition">
# Lindenbaum–Tarski algebra

## Definition

Let $P$ be a set of propositional variables.
Let $F$ be the set of formulas of classical [[propositional logic|09K9MWD2QQ]] over $P$ built with $\wedge$, $\vee$, $\neg$, $\top$ and $\bot$.
Let $T \subseteq F$ be a theory.
For $\varphi, \psi \in F$, put $\varphi \sim_T \psi$ if $T \vdash \varphi \leftrightarrow \psi$.
This is an [[equivalence relation|0G8VEZ3WTK]] on $F$; write $[\varphi]$ for the class of $\varphi$.
The *Lindenbaum–Tarski algebra* of $T$ is the [[quotient set|0011MTQMR6]] $F / {\sim_T}$ with the operations
$$[\varphi] \wedge [\psi] = [\varphi \wedge \psi], \quad [\varphi] \vee [\psi] = [\varphi \vee \psi], \quad \neg [\varphi] = [\neg \varphi],$$
and the constants $0 = [\bot]$ and $1 = [\top]$.

## Remarks

The operations are well defined because provable equivalence is compatible with the connectives, and the result is a [[Boolean algebra|0XADJKM699]].
Its order is given by $[\varphi] \le [\psi]$ if and only if $T \vdash \varphi \to \psi$.

Example: for $P = \set{p}$ and $T = \emptyset$, every formula is equivalent to one of $\bot$, $p$, $\neg p$, $\top$, and no two of these are equivalent, as their truth tables differ.
So the algebra is the four-element Boolean algebra with atoms $[p]$ and $[\neg p]$.

Non-example: identify formulas only when they are equal as strings.
Then the classes $[p \wedge q]$ and $[q \wedge p]$ differ, so $\wedge$ is not commutative and the result is not a Boolean algebra.

The same construction for intuitionistic logic gives a Heyting algebra.
A theory $T$ is consistent if and only if $0 \ne 1$ in its algebra.
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# Alexander–Spanier cohomology

## Summary

Alexander–Spanier cohomology is a cohomology theory for topological spaces built from arbitrary functions on tuples of points, taken modulo functions that vanish near the diagonal.
It does not use simplices or maps from a model space, so it behaves well on spaces with bad local structure.
On paracompact Hausdorff spaces it agrees with Čech cohomology and with sheaf cohomology of a constant sheaf.
On manifolds and CW complexes it agrees with [[singular cohomology|6N7BC70CX0]].

## Explanations

### L1 human

Singular cohomology probes a space with maps of triangles into it.
On wild spaces such maps may miss features that you can see by eye.
Alexander–Spanier cohomology instead looks at functions of several points that only matter when the points are close together.
For example, the Warsaw circle is a circle with a wild oscillating piece.
It has no first singular cohomology, but its Alexander–Spanier cohomology sees the loop, as it does for an ordinary circle.
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<document source="https://tamnd.github.io/math2/node/MGM3CNJW9V.md" tag="MGM3CNJW9V" kind="concept">
# Discrete valuation ring

## Summary

A discrete valuation ring is a [[principal ideal domain|0CMD0BCH8T]] with exactly one nonzero [[prime ideal|00MHHZ9SNE]].
It is the ring of elements of nonnegative value for a [[discrete valuation|A2KKZAV29W]] on its field of fractions.
Discrete valuation rings are among the simplest rings that are not fields: their ideals form a single chain of powers of one element.
They describe the local behaviour of curves at smooth points and of number rings at a prime.

## Explanations

### L1 human

In a discrete valuation ring there is one special element $\pi$, a uniformizer, and every nonzero element is a unit times a power of $\pi$.
The power of $\pi$ measures "how divisible" an element is.
The main example is the ring $\Z_{(p)}$ of fractions $a/b$ with $p \nmid b$, where $\pi = p$.
For instance, $12/5 = 3/5 \cdot 2^2$ in $\Z_{(2)}$, so its value is $2$.
Another example is the ring of formal power series $k[\![t]\!]$, where $\pi = t$ and the value of a series is its lowest degree.
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<document source="https://tamnd.github.io/math2/node/MGPPCDFQDG.md" tag="MGPPCDFQDG" kind="concept">
# Nondegenerate critical point

## Summary

A nondegenerate critical point of a smooth function is a [[critical point|0YTCMFB4HC]] at which the second derivative, the [[Hessian|0693Y2METT]], is a nondegenerate bilinear form.
Such points are isolated, and the [[Morse lemma|ERSD8D4HWX]] shows that near one of them the function is a sum of squares with signs.
A function whose critical points are all nondegenerate is a [[Morse function|KPQ96640ZJ]].

## Explanations

### L1 human

At a critical point the graph of a function is flat to first order.
The point is nondegenerate when the function still curves in every direction, up or down, to second order.
For example, $x^2 + y^2$ and $x^2 - y^2$ have nondegenerate critical points at the origin: a bowl and a saddle.
The function $x^3$ has a degenerate critical point at $0$, because its second derivative vanishes there and the graph is flat to second order.
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# Selberg sieve

## Definition

Let $\mathcal{A}$ be a finite set of integers, $\mathcal{P}$ a set of primes, $z \ge 2$, and $P(z)$, $\mathcal{A}_d$ and $S(\mathcal{A}, \mathcal{P}, z)$ as for the [[Brun sieve|M4G4RXASTY]].
Let $D \ge 1$.
Let $(\lambda_d)$ be real numbers indexed by the divisors $d$ of $P(z)$ such that:

1. $\lambda_1 = 1$;
2. $\lambda_d = 0$ for $d > D$.

The *Selberg sieve* is the upper bound
$$S(\mathcal{A}, \mathcal{P}, z) \le \sum_{a \in \mathcal{A}} \Big( \sum_{d \mid \gcd(a, P(z))} \lambda_d \Big)^{2} = \sum_{d_1, d_2 \mid P(z)} \lambda_{d_1} \lambda_{d_2}\, \#\mathcal{A}_{\lcm(d_1, d_2)},$$
together with the choice of the weights $\lambda_d$ that minimises the main term of the right side.
That main term is a quadratic form in the $\lambda_d$ once $\#\mathcal{A}_d$ is written as $X g(d) + r_d$ with a multiplicative function $g$ and remainders $r_d$.

## Remarks

The inequality holds termwise: if $\gcd(a, P(z)) = 1$, the inner sum is $\lambda_1 = 1$; otherwise the square is $\ge 0$.
Example: take $\mathcal{A} = \set{1, \ldots, 10}$, $P(z) = 6$, $D = 3$, $\lambda_1 = 1$, $\lambda_2 = \lambda_3 = -1/2$ and $\lambda_6 = 0$.
The inner sum is $1$ for the three $a$ coprime to $6$, and $1/2$ for the six $a$ divisible by exactly one of $2$ and $3$.
It is $0$ for $a = 6$.
So the bound is $3 + 6 \cdot \tfrac14 = 4.5$, and indeed $S(\mathcal{A}, \mathcal{P}, z) = 3 \le 4.5$.
Non-example: weights with $\lambda_1 = 0$ fail clause 1, and then $a = 1$ contributes $0$ instead of $1$, so the inequality can fail.
With $0 < g(p) < 1$ for $p \mid P(z)$, the optimal weights give the main term $X / G$ with
$$G = \sum_{\substack{d \mid P(z) \\ d \le D}} \prod_{p \mid d} \frac{g(p)}{1 - g(p)},$$
and they satisfy $\abs{\lambda_d} \le 1$.
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# Dirichlet form

## Summary

A Dirichlet form is a closed, symmetric, nonnegative bilinear form on a space $L^2(X, m)$ that does not increase under truncation of functions.
The model is the classical [[Dirichlet energy|YG3W8CBB3E]] $\int \abs{\nabla u}^2 \, dx$.
Dirichlet forms correspond one to one to symmetric Markovian semigroups on $L^2(X, m)$.
By a theorem of Fukushima, a regular Dirichlet form is the energy of a symmetric [[Hunt process|Z7T669BQ9B]], so the theory joins analysis and probability.

## Explanations

### L1 human

Think of a function $u$ as a temperature profile and of $\mathcal{E}(u, u)$ as the amount of tension in it.
The energy $\int \abs{u'}^2 \, dx$ of a function on an interval is the basic example.
The key rule is that cutting a function off at $0$ and at $1$ never increases its energy.
This rule links the form to a random motion: it says that heat flow keeps temperatures between their bounds.
Dirichlet forms also make sense on graphs and fractals, where there are no derivatives.
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# Simplex algorithm

## Summary

The simplex algorithm, introduced by George Dantzig, solves [[linear programs|VNMXB0J2KA]].
It walks from vertex to adjacent vertex of the feasible polyhedron along edges, never decreasing the objective, until no neighbouring vertex is better.
In practice it is fast.
But for the usual pivot rules there are inputs, such as the Klee–Minty cubes, on which it takes exponentially many steps.
The question of how few steps suffice in the worst case is linked to the diameter of polytope graphs and to the [[Hirsch conjecture|ZBVJ4B41AX]].

## Explanations

### L1 human

Picture the allowed region of a linear program as a many-sided gem, and the objective as height.
Start at one corner, look at the edges going up, and walk along one of them to the next corner.
Repeat until every edge at the current corner goes down; that corner is the highest point of the gem.
Because the gem is convex, a corner with no upward edge is the top.
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<document source="https://tamnd.github.io/math2/node/MH6CSWD7F0.md" tag="MH6CSWD7F0" kind="concept">
# Boltzmann equation

## Summary

The Boltzmann equation describes a dilute gas through the density $f(t, x, v)$ of particles at position $x$ with velocity $v$.
Particles move freely in straight lines and change velocity only in binary collisions, which the collision operator accounts for.
It links the motion of individual molecules with the equations of fluid dynamics.
Boltzmann's H-theorem shows that a certain entropy cannot decrease, which explains the approach to equilibrium.

## Explanations

### L1 human

A gas consists of a huge number of molecules flying around and bumping into each other.
Instead of following each molecule, one keeps track of how many molecules are near each point with each velocity.
Between collisions this distribution just drifts along with the molecules.
Collisions reshuffle velocities but keep the total momentum and energy.
Over time the gas settles into the Maxwell distribution, the bell-shaped spread of velocities of a gas at rest at a fixed temperature.
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<document source="https://tamnd.github.io/math2/node/MH7MEB0MRR.md" tag="MH7MEB0MRR" kind="statement">
# Morse inequalities

## Statement

Let $M$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary, of dimension $n$, and let $f \colon M \to \R$ be a [[Morse function|KPQ96640ZJ]].
For $0 \le k \le n$, let $c_k$ be the number of critical points of $f$ of index $k$.
Let $F$ be a field, and let $b_k = \dim_F H_k(M; F)$ be the $k$-th [[Betti number|ST71ZV3FES]] of $M$ over $F$, where $H_k$ is [[singular homology|M0Q5CFJ7ZH]].
Then:

1. (weak Morse inequalities) $c_k \ge b_k$ for every $k$;
2. (strong Morse inequalities) $\sum_{i=0}^{k} (-1)^{k-i} c_i \ge \sum_{i=0}^{k} (-1)^{k-i} b_i$ for every $k$;
3. $\sum_{k=0}^{n} (-1)^k c_k = \sum_{k=0}^{n} (-1)^k b_k = \chi(M)$, where $\chi(M)$ is the [[Euler characteristic|RNDDWDAFJ5]] of $M$.

## Remarks

The idea of the proof: as $c$ increases, the sublevel set $\set{f \le c}$ changes up to homotopy only when $c$ crosses a critical value.
At such a value it changes by attaching one cell of dimension $k$ for each critical point of index $k$.
So $M$ is homotopy equivalent to a CW complex with $c_k$ cells of dimension $k$.
The inequalities follow from the cellular chain complex of this CW complex.
The strong inequalities imply the weak ones, by adding the inequalities for $k$ and $k - 1$.
Item 3 is the Euler characteristic of the CW complex, computed once from cells and once from homology.
Example: the height function on an upright torus has $c_0 = 1$, $c_1 = 2$, $c_2 = 1$, and the torus has $b_0 = 1$, $b_1 = 2$, $b_2 = 1$, so all inequalities are equalities.
Consequence: every Morse function on a closed orientable surface of genus $g$ has at least $2g + 2$ critical points.

## Explanations

### L1 human

A Morse function needs at least as many critical points of each index as the manifold has holes of that dimension.
For example, any nice height function on a doughnut has at least one bottom, one top, and two saddles.
Topology forces critical points to exist.

### L2 human

The inequalities are the first quantitative link between the analysis of a function and the topology of its domain.
They can be used in two directions.
They bound the number of critical points of a function, as in Morse's results on geodesics.
They also bound Betti numbers, once one constructs a function with few critical points.
A Morse function for which all inequalities are equalities is called perfect.
Morse homology refines the inequalities by building a chain complex on the critical points whose homology is $H_*(M)$.
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# Filtered back-projection

## Definition

Let $f$ be in the Schwartz space $\mathcal{S}(\R^2)$, with [[Radon transform|VJNZJ352PC]] $R f$, and write $R_\theta f(s) = R f(\theta, s)$.
Let $\Phi \colon \R \to \R$ be an even bounded function with $\Phi(\sigma) = 0$ for $\abs{\sigma} > 1$, continuous at $0$, with $\Phi(0) = 1$.
For $b > 0$, let $w_b$ be the filter with Fourier transform $\hat w_b(\sigma) = \abs{\sigma}\, \Phi(\sigma/b)$.
Its *filtered back-projection* reconstruction with filter $w_b$ is
$$f_b(x) = \frac{1}{4\pi} R^* \left( w_b * R_\theta f \right)(x) = \frac{1}{4\pi} \int_{S^1} (w_b * R_\theta f)(x \cdot \theta)\, d\theta,$$
where $*$ is [[convolution|KXC9KHWVGE]] in the variable $s$ and $R^*$ is the [[dual Radon transform|938SXZQS8G]].

## Remarks

The Fourier convention is $\hat g(\sigma) = \int g(s)\, e^{-is\sigma}\, ds$.
The projection-slice theorem gives $\hat f_b(\xi) = \Phi(\abs{\xi}/b)\, \hat f(\xi)$, so $f_b \to f$ as $b \to \infty$, uniformly on $\R^2$.
With $\Phi$ the indicator of $[-1, 1]$ the filter is the Ram–Lak filter, a band-limited ramp.

Example: if $\Phi$ is the indicator of $[-1, 1]$ and $\hat f$ vanishes for $\abs{\xi} > b$, then $\hat f_b = \hat f$ and the reconstruction is exact: $f_b = f$.

Non-example: back-projection without filter, $\frac{1}{4\pi} R^* R f$, is not a reconstruction of $f$.
It corresponds to $\hat w(\sigma) = 1$ instead of $\abs{\sigma}$, and its Fourier transform is $\hat f(\xi)/\abs{\xi}$, a blurred image.

In practice $R f$ is known only at finitely many angles and offsets, and the integral and the convolution are replaced by sums.
</document>

<document source="https://tamnd.github.io/math2/node/MHF34RK5ZT.md" tag="MHF34RK5ZT" kind="definition">
# Chain map

## Definition

Let $\mathcal{A}$ be an [[additive category|B2VP96NJSK]] and let $(C_\bullet, d^C)$ and $(D_\bullet, d^D)$ be [[chain complexes|WQ82C0M3QB]] in $\mathcal{A}$.
A *chain map* $f \colon C_\bullet \to D_\bullet$ is a family of [[morphisms|859F0TFVA3]] $f_n \colon C_n \to D_n$, one for each integer $n$, such that
$$d^D_n \circ f_n = f_{n-1} \circ d^C_n \quad \text{for all } n \in \Z.$$

## Remarks

Chain maps are composed degreewise, and the identity chain map has $f_n = 1_{C_n}$.
With these, chain complexes in $\mathcal{A}$ form a category, often written $\mathrm{Ch}(\mathcal{A})$.
A chain map sends $\ker d^C_n$ into $\ker d^D_n$ and $\im d^C_{n+1}$ into $\im d^D_{n+1}$.

Example: let $C_\bullet = D_\bullet$ be the complex with $C_1 = C_0 = \Z$, $d_1$ multiplication by $3$, and $C_n = 0$ otherwise.
Then $f_1 = f_0 = $ multiplication by $2$ is a chain map, since $3 \cdot 2x = 2 \cdot 3x$ for $x \in \Z$.

Non-example: in the same complex, take $f_1 = 1_{\Z}$ and $f_0 = 0$.
Then $d_1 \circ f_1$ is multiplication by $3$ but $f_0 \circ d_1 = 0$, so the square in degree $1$ does not commute.
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# Predictor–corrector method

## Statement

Let $f \colon \R \times \R^d \to \R^d$ be sufficiently smooth, and consider the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$.
Let $P$ be an explicit [[linear multistep method|BX4D85A79P]] of order $p^* \ge 1$, the predictor.
Let $C$ be an implicit linear multistep method of order $p \ge 1$, the corrector.
One step of the PECE mode with step size $h$ does the following:

1. P: compute a predicted value $y_{n+1}^{[0]}$ with $P$.
2. E: evaluate $f_{n+1}^{[0]} = f(t_{n+1}, y_{n+1}^{[0]})$.
3. C: compute $y_{n+1}$ with the formula of $C$, where $f_{n+1}^{[0]}$ replaces the unknown value $f(t_{n+1}, y_{n+1})$.
4. E: evaluate $f_{n+1} = f(t_{n+1}, y_{n+1})$ for use in later steps.

Then the PECE method is an explicit method of order $\min(p, p^* + 1)$.
If $p^* \ge p$, its principal local truncation error equals that of the corrector.

## Remarks

The corrector is applied once instead of being iterated to convergence, so no nonlinear equation is solved.
The error of the predicted value is $O(h^{p^*+1})$.
It enters the corrected value multiplied by $h$ times a coefficient of the corrector, so it adds $O(h^{p^*+2})$ to the local error.
Comparing this with the local error $O(h^{p+1})$ of the corrector gives the order $\min(p, p^*+1)$.
Example: the [[Euler method|0JARTXX4XP]] ($p^* = 1$) as predictor and the trapezoidal rule ($p = 2$) as corrector give [[Heun's method|RJEK60G28F]], of order $\min(2, 2) = 2$.
When $p^* = p$, the difference $y_{n+1} - y_{n+1}^{[0]}$ gives an estimate of the local error (Milne's device), which is used for [[adaptive step size|6MREWBZR7R]] control.

## Explanations

### L1 human

An implicit method is often more accurate and more stable, but its formula contains the unknown new value on both sides.
A predictor-corrector method first guesses the new value with a cheap explicit formula.
It then plugs the guess into the implicit formula once, which corrects it.
The result is explicit, and it keeps the order of the corrector when the guess is good enough.

### L2 human

The Adams–Bashforth–Moulton pairs are the standard example: an Adams–Bashforth predictor with an Adams–Moulton corrector of the same order.
The PECE method has the same order as the corrector, but its stability region is different and is in general smaller than that of the corrector.
So predictor-corrector methods are not suited to [[stiff equations|RR5FCG92PN]], where one solves the implicit equations with Newton's method instead.
</document>

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# Link of a singularity

## Definition

Let $V$ be a complex [[analytic set|3RWWZYPAFV]] in an open subset of $\C^N$ and let $p \in V$.
For $\varepsilon > 0$, let $S_\varepsilon(p)$ be the [[sphere|P9CM4GQ2T0]] of radius $\varepsilon$ centred at $p$, and let $K_\varepsilon = V \cap S_\varepsilon(p)$.
There is $\varepsilon_0 > 0$ such that, for $0 < \varepsilon < \varepsilon' \le \varepsilon_0$, the spaces $K_\varepsilon$ and $K_{\varepsilon'}$ are [[homeomorphic|0QT00PB5G2]].
The *link* of $V$ at $p$ is $K_\varepsilon$ for any such $\varepsilon$, defined up to homeomorphism.

## Remarks

For $\varepsilon$ small, $V \cap B_\varepsilon(p)$, where $B_\varepsilon(p)$ is the closed ball, is homeomorphic to the cone over $K_\varepsilon$ with vertex $p$; this is the conic structure theorem.
If $V$ is smooth at $p$ of complex dimension $d$, the link is a sphere $S^{2d - 1}$.
If $V = f^{-1}(0) \subset \C^{n+1}$ with an isolated singularity at $p$, the link is a smooth closed manifold of dimension $2n - 1$, and Milnor showed that it is $(n-2)$-connected.

Example: $V = \set{x^2 = y^3} \subset \C^2$ at $0$.
The map $t \mapsto (t^3, t^2)$ is a bijection from $\C$ onto $V$.
On the sphere $S_\varepsilon$, the points of $V$ are $(t^3, t^2)$ with $\abs{t}$ fixed, which trace the torus knot of type $(2, 3)$, the trefoil.

Non-example of the condition on $\varepsilon$: $V = \set{x y (y - 1) = 0} \subset \C^2$ at $0$.
For $\varepsilon < 1$, $K_\varepsilon$ consists of the two circles cut out by $x = 0$ and $y = 0$, a Hopf link.
For $\varepsilon > 1$, the line $y = 1$ also meets $S_\varepsilon$ in a third circle, so this sphere is too large to give the link.
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# Surgery theory

## Summary

Surgery is an operation that cuts out a thickened sphere $S^p \times D^q$ from a manifold and glues in $D^{p+1} \times S^{q-1}$ along the common boundary.
The result is a new manifold, [[cobordant|T8BVGPPSZP]] to the old one.
Surgery theory uses this operation to decide whether a space is homotopy equivalent to a manifold.
It also classifies the manifolds within a homotopy type.
In dimensions at least $5$ it reduces these questions to homotopy theory and to algebra in the surgery obstruction groups.

## Explanations

### L1 human

Surgery is a cut-and-paste move on a manifold.
In a surface, cut out two small discs and glue in a tube between the two holes: the surface gains a handle.
This is surgery on a $0$-sphere, the pair of centres of the discs.
Repeating such moves, one can kill unwanted loops and holes and steer a manifold towards a desired shape.
Surgery theory is the systematic study of when this steering succeeds.
</document>

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# Loop space

## Definition

Let $(X, x_0)$ be a [[pointed space|T9DG8DDE75]] and let $I = [0, 1]$.
The *loop space* of $(X, x_0)$ is the set
$$\Omega X = \set{\gamma \colon I \to X \text{ continuous} : \gamma(0) = \gamma(1) = x_0},$$
with the [[subspace topology|0WS9P12FQR]] of the [[compact-open topology|RG15F0TWPW]] on the set of continuous maps $I \to X$.
Its base point is the constant loop at $x_0$.

## Remarks

Concatenation $(\alpha \cdot \beta)(s) = \alpha(2s)$ for $s \le 1/2$ and $\beta(2s - 1)$ for $s \ge 1/2$ is a continuous product on $\Omega X$ that is associative and unital up to homotopy.
The set of path components $\pi_0(\Omega X)$ is in bijection with $\pi_1(X, x_0)$, and $\pi_n(\Omega X) \cong \pi_{n+1}(X, x_0)$ for $n \ge 1$.
For pointed spaces $A$ and $X$, pointed homotopy classes of maps $\Sigma A \to X$ from the reduced [[suspension|Q7RFNJH76Q]] correspond bijectively to pointed homotopy classes of maps $A \to \Omega X$.
Example: if $X$ is the one-point space, then $\Omega X$ is a single point, the constant loop.
Non-example: the free loop space of all continuous maps $S^1 \to X$ with no condition at a base point is a different space; it contains loops that miss $x_0$.
</document>

<document source="https://tamnd.github.io/math2/node/MHSBP40HDC.md" tag="MHSBP40HDC" kind="statement">
# Shift operator as an exponential of differentiation

## Statement

Let $f \colon \R \to \C$ be given by a power series $f(x) = \sum_{n \ge 0} c_n x^n$ that converges for every $x \in \R$; for example, $f$ may be a polynomial.
For $h \in \R$, let $T_h f$ be the shifted function $(T_h f)(x) = f(x + h)$, and let $D = d/dx$.
Then for all $x, h \in \R$
$$(T_h f)(x) = \sum_{n=0}^\infty \frac{h^n}{n!} \, (D^n f)(x).$$
In operator notation, $T_h = e^{hD}$ on such functions.

## Remarks

The right side is the [[Taylor series|0Z49YH6ZWY]] of $f$ at $x$, evaluated at the increment $h$.
A power series that converges on all of $\R$ is an [[analytic function|081WKEDK6C]] whose Taylor series at every point converges to it on all of $\R$, which gives the identity.
For a polynomial of degree $d$ the sum stops at $n = d$, and the identity is exact algebra.

Example: $f(x) = x^2$.
Then $f + h f' + \frac{h^2}{2} f'' = x^2 + 2hx + h^2 = (x + h)^2$.

The hypothesis of analyticity is needed.
For the [[smooth non-analytic function|09RTYYHXKA]] $f(x) = e^{-1/x^2}$ for $x \ne 0$, $f(0) = 0$, all derivatives vanish at $0$, so the right side at $x = 0$ is $0$, while $f(h) > 0$ for $h \ne 0$.

## Explanations

### L1 human

Knowing all derivatives of a nice function at one point determines its values nearby.
The formula packages this as "shifting by $h$ equals the exponential of $h$ times differentiation".
It is Taylor's formula written as an identity between operators.

### L2 human

Since $e^{hD} e^{kD} = e^{(h+k)D}$, the identity explains why shifts compose by adding the steps.
In the calculus of finite differences, the forward difference $\Delta = T_1 - \id$ becomes $e^D - 1$, and on polynomials this gives formal identities such as $D = \log(1 + \Delta) = \Delta - \Delta^2/2 + \Delta^3/3 - \cdots$, a finite sum on each polynomial.
In quantum mechanics the same identity says that the momentum operator generates translations.
</document>

<document source="https://tamnd.github.io/math2/node/MHSKT2Z4MX.md" tag="MHSKT2Z4MX" kind="statement">
# Birkhoff–Kakutani theorem

## Statement

Let $G$ be a [[Hausdorff topological group|7GKCGQ7215]].
The following are equivalent.

1. $G$ is [[metrizable|091GJ3FYD7]].
2. $G$ is [[first-countable|0CP5NDD21C]], that is, the identity of $G$ has a countable base of neighbourhoods.
3. There is a [[metric|0VKD1TP7F3]] $d$ on $G$ that induces the topology of $G$ and is left-invariant: $d(gx, gy) = d(x, y)$ for all $g, x, y \in G$.

## Remarks

The implications from 3 to 1 and from 1 to 2 are immediate.
For the main step from 2 to 3, choose a base $U_0 \supseteq U_1 \supseteq \cdots$ of symmetric neighbourhoods of the identity with $U_{n+1}^3 \subseteq U_n$.
These sets are used to build a function that plays the role of $d(e, x)$, and then $d(x, y)$ is defined from $x^{-1} y$.
Example: the $p$-adic integers $\Z_p$ have the countable base $p^n \Z_p$ at $0$, and $d(x, y) = p^{-v_p(x - y)}$ is an invariant metric, where $v_p$ is the $p$-adic valuation.
The Hausdorff hypothesis is needed: a group with two or more elements and the indiscrete topology is first-countable but not metrizable.
Without first countability the conclusion fails: an uncountable product of copies of $\Z/2\Z$ is a compact Hausdorff group that is not metrizable.

## Explanations

### L1 human

For a topological group, having a distance function that describes its topology costs very little.
It is enough that the identity has a countable list of shrinking neighbourhoods that capture all of its neighbourhoods.
Then one can even choose the distance so that multiplying on the left by a fixed element does not change distances.

### L2 human

The theorem turns questions about first-countable Hausdorff groups into questions about metric spaces, so sequences suffice to describe the topology.
It is a basic tool for Polish groups and for second-countable locally compact groups, where one often fixes a compatible left-invariant metric.
A left-invariant metric need not be complete even when the group is complete for its two-sided uniformity, which is why the theory of Polish groups distinguishes these notions.
</document>

<document source="https://tamnd.github.io/math2/node/MHSMQ097GP.md" tag="MHSMQ097GP" kind="concept">
# Bochner integral

## Summary

The Bochner integral extends the Lebesgue integral to functions with values in a Banach space.
It is defined as a limit of integrals of simple functions, approximated in the mean.
A strongly measurable function is Bochner integrable exactly when the integral of its norm is finite.
It keeps the main theorems of Lebesgue integration, such as dominated convergence.
It is the standard integral for vector-valued functions in analysis and partial differential equations.

## Explanations

### L1 human

For a function with vector values that takes finitely many values, the integral is a weighted sum of vectors.
A general function is approached by such simple functions, closer and closer on average.
The Bochner integral is the limit of their integrals.
For example, a continuous curve $t \mapsto u(t)$ in a Banach space, defined on $[0, 1]$, has a Bochner integral $\int_0^1 u(t) \, dt$, the limit of its Riemann sums.
</document>

<document source="https://tamnd.github.io/math2/node/MJ48S56VQ7.md" tag="MJ48S56VQ7" kind="definition">
# Hankel operator

## Definition

Let $a = (a_n)_{n \ge 0}$ be a sequence of complex numbers.
Let $\ell^2 = \ell^2(\set{0, 1, 2, \ldots})$ be the [[space of square-summable sequences|08P6T2WH6R]] with standard basis $(e_k)_{k \ge 0}$.
The *Hankel operator* $\Gamma_a \colon \ell^2 \to \ell^2$ is the [[bounded linear operator|03TM135G0N]] with the [[Hankel matrix|PQ1D8FQB3E]] $(a_{j+k})_{j, k \ge 0}$, that is,
$$\ip{\Gamma_a e_k}{e_j} = a_{j+k} \quad \text{for all } j, k \ge 0,$$
when such a bounded operator exists.

## Remarks

Example: the Hilbert matrix, with $a_n = 1/(n + 1)$, defines a bounded Hankel operator; Hilbert's inequality gives $\norm{\Gamma_a} = \pi$.

Non-example: the identity matrix is not a Hankel matrix.
Its entries at $(1, 1)$ and $(0, 2)$ have the same index sum $2$, but they are $1$ and $0$.

Nehari's theorem: $\Gamma_a$ is bounded if and only if there is an essentially bounded function $\psi$ on the unit circle with $\hat\psi(n) = a_n$ for all $n \ge 0$.
Then $\norm{\Gamma_a}$ is the least $\norm{\psi}_\infty$ over such $\psi$, where $\hat\psi(n)$ is the $n$-th Fourier coefficient.
Kronecker's theorem: $\Gamma_a$ has finite rank if and only if the power series $\sum_{n \ge 0} a_n z^n$ is a rational function.
In the function form, for a bounded symbol $\varphi$ on the circle, the Hankel operator is $H_\varphi f = (I - P)(\varphi f)$ from $H^2$ to $L^2 \ominus H^2$, with $P$ the orthogonal projection onto $H^2$.
</document>

<document source="https://tamnd.github.io/math2/node/MJ864PBRRJ.md" tag="MJ864PBRRJ" kind="statement">
# Casus irreducibilis

## Statement

Let $K$ be a [[subfield|07QY5RPFBR]] of $\R$, and let $f \in K[x]$ be a cubic polynomial that is [[irreducible|00X2A9G9W3]] over $K$ and has three real roots.
Then no root of $f$ lies in a [[radical extension|V74WMGQQ9D]] $R$ of $K$ with $R \subseteq \R$.

## Remarks

The three roots are distinct, since an irreducible polynomial over a field of characteristic $0$ has no repeated roots.
For the idea of the proof, adjoin first the square root of the discriminant, which is real.
Then over every real field $F$ that contains this square root and over which $f$ is irreducible, adjoining one root of $f$ adjoins all three roots.
Take a real tower of steps adjoining $q$-th roots, $q$ prime, whose last step $E \subseteq E(\alpha)$ with $\alpha^q \in E$ first makes $f$ reducible.
Then $3$ divides $[E(\alpha) : E]$, which is $1$ or $q$, so $q = 3$ and $E(\alpha)/E$ is normal of degree $3$.
Hence $x^3 - \alpha^3$ splits in $E(\alpha)$, so this real field contains a nonreal cube root of unity, which is impossible.
Example: $f = x^3 - 3x + 1$ over $\Q$ is irreducible, since $\pm 1$ are not roots, and its roots $2\cos(2\pi/9)$, $2\cos(4\pi/9)$, $2\cos(8\pi/9)$ are real.
So every expression of these roots by radicals passes through nonreal numbers, as in the [[cubic formula|FB6J5WAT5V]].
Irreducibility is necessary: $x^3 - x = x(x - 1)(x + 1)$ has three real roots, all in $\Q$.

## Explanations

### L1 human

The cubic formula sometimes takes the square root of a negative number even when all three answers are real.
The casus irreducibilis says this detour is unavoidable when the cubic does not factor.
No formula built only from real roots of real numbers can give the roots.

### L2 human

For a real cubic with three distinct real roots, the discriminant is positive, so the square root in the [[cubic formula|FB6J5WAT5V]] is of a negative number.
The theorem shows that the problem is not the formula but the numbers themselves.
The roots can still be written with real numbers using trigonometry, for example as $2\sqrt{-p/3}\cos\theta$ for suitable angles $\theta$ when the cubic is $t^3 + pt + q$.
</document>

<document source="https://tamnd.github.io/math2/node/MJBXNNFQGW.md" tag="MJBXNNFQGW" kind="definition">
# Tight and overtwisted contact structures

## Definition

Let $M$ be a [[$3$-manifold|R20TB2KEQQ]] and let $\xi$ be a contact structure on $M$, so that $(M, \xi)$ is a [[contact manifold|HBQ8BZ7XB5]].
An *overtwisted disc* in $(M, \xi)$ is an embedded closed disc $D \subset M$ such that:

1. the boundary $\partial D$ is a [[Legendrian|Q2DMSTWZYZ]] curve, that is, tangent to $\xi$;
2. for every $x \in \partial D$, the plane $\xi_x$ equals the tangent plane $T_x D$.

The contact structure $\xi$ is *overtwisted* if $(M, \xi)$ contains an overtwisted disc, and *tight* otherwise.

## Remarks

Example: on $\R^3$ with cylindrical coordinates $(r, \theta, z)$, the form $\alpha = \cos r\, dz + r \sin r\, d\theta$ is a contact form.
The disc $D = \set{z = 0, \ r \le \pi}$ is overtwisted: on its boundary $r = \pi$ one has $\alpha = -dz$, so $\xi = \ker dz$ is the horizontal plane, which is $T_x D$, and the boundary circle is tangent to it.
So this contact structure is overtwisted.

Non-example: the standard contact structure $\ker(dz - y\,dx)$ on $\R^3$ is tight, by Bennequin's theorem, so no disc satisfies both clauses.

Clause 2 says that the framing of $\partial D$ given by the contact planes agrees with the framing given by $D$.
In other words, the Legendrian unknot $\partial D$ has Thurston–Bennequin number $0$.
Eliashberg proved that two overtwisted contact structures on a closed $3$-manifold are isotopic if and only if they are homotopic as plane fields.
He also proved that every homotopy class of plane fields contains an overtwisted contact structure.
Tight contact structures are much rarer; for example, $S^3$ has a unique tight contact structure up to isotopy, also by Eliashberg.
</document>

<document source="https://tamnd.github.io/math2/node/MJDV3PFBRT.md" tag="MJDV3PFBRT" kind="definition">
# I-adic topology

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be an $R$-[[module|0NZ8JA25WK]].
The *$I$-adic topology* on $M$ is the following [[topology|0CQHFPGJ45]].
A subset $U \subseteq M$ is open if and only if for every $x \in U$ there is $n \ge 0$ with $x + I^n M \subseteq U$.
The $I$-adic topology on $R$ is the case $M = R$.

## Remarks

The cosets $x + I^n M$, for $x \in M$ and $n \ge 0$, form a base of this topology; each of them is open and also closed.
Addition on $M$ and multiplication $R \times M \to M$ are continuous, so $R$ becomes a topological ring and $M$ a topological module.
The topology is Hausdorff if and only if $\bigcap_n I^n M = 0$; then $d(x, y) = 2^{-\nu(x - y)}$, with $\nu(z)$ the largest $n$ such that $z \in I^n M$ and $\nu(0) = \infty$, is a metric for it.
Example: on $\Z$ with $I = (2)$, the sequence $2^n$ tends to $0$, since $2^m \in 2^n \Z$ for all $m \ge n$.
Non-example: the set $\set{0}$ is not open in $\Z$ for this topology, since no $2^n \Z$ is contained in $\set{0}$.
So the $2$-adic topology on $\Z$ is not discrete, unlike the topology that $\Z$ gets from $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/MJF9M19E3E.md" tag="MJF9M19E3E" kind="concept">
# Relational structure

## Summary

A relational structure is a set together with a family of relations on it and no operations.
It is a [[structure|983GZBQYT4]] for a [[signature|WDVVEYD6AT]] that has only relation symbols.
Graphs, partially ordered sets and sets with an equivalence relation are relational structures.
They are the other extreme from [[algebraic structures|QJGPT2W0FT]], which have only operations.

## Explanations

### L1 human

A relational structure is a set where some tuples of elements are marked as related.
A graph is the simplest example: the set is the set of vertices, and two vertices are related when there is an edge between them.
A [[partial order|0QV2JS9RJ3]] such as $\le$ on the integers is another example with one binary relation.
Nothing is computed in a relational structure; one only asks whether given elements are related.
</document>

<document source="https://tamnd.github.io/math2/node/MJGPW5EVB3.md" tag="MJGPW5EVB3" kind="statement">
# Ekeland's variational principle

## Statement

Let $(X, d)$ be a [[complete metric space|0BY52JW6NK]].
Let $f \colon X \to (-\infty, +\infty]$ be [[lower semicontinuous|RN4G8SGP6P]], bounded below, and not identically $+\infty$.
Let $\varepsilon > 0$ and $\lambda > 0$, and let $x_0 \in X$ satisfy $f(x_0) \le \inf_X f + \varepsilon$.
Then there is a point $v \in X$ such that:

1. $f(v) \le f(x_0)$;
2. $d(x_0, v) \le \lambda$;
3. $f(w) > f(v) - \dfrac{\varepsilon}{\lambda} d(v, w)$ for every $w \in X$ with $w \ne v$.

## Remarks

The proof builds a sequence $x_0, x_1, x_2, \ldots$.
Each $x_{n+1}$ is chosen in the set of points $y$ with $f(y) \le f(x_n) - (\varepsilon / \lambda) d(x_n, y)$, with $f(x_{n+1})$ close to the infimum of $f$ on that set.
These sets are closed and nested, their diameters tend to $0$, and completeness gives a limit point $v$, which has the three properties.
Taking $\lambda = 1$ on a Banach space with $f$ Gateaux differentiable gives a point $v$ with $f(v) \le \inf f + \varepsilon$ and $\norm{f'(v)} \le \varepsilon$: an almost minimizer with small derivative.
With this, a $C^1$ function that is bounded below and satisfies the [[Palais–Smale condition|4724KRNHAT]] attains its minimum.
Completeness is needed: on $X = (0, 1]$ with $f(x) = x$, $\varepsilon = 1/2$ and $\lambda = 1$, every $v$ violates clause 3 for any $w < v$, because $f(w) = v - (v - w) < v - \tfrac12 (v - w)$.

## Explanations

### L1 human

A function that is bounded below need not have a minimum point.
Ekeland's principle says that one can always find a point that is almost as good as the infimum and that is a strict minimum of a slightly tilted function.
The tilt is a cone of slope $\varepsilon / \lambda$ around the point.
So near-minimizers can always be taken to be "stable" in a precise sense.

### L2 human

The principle is a basic tool of nonsmooth analysis and of critical point theory.
It produces Palais–Smale sequences for functionals bounded below, and it gives short proofs of the Caristi fixed point theorem and of the Bishop–Phelps theorem.
Conversely, a metric space in which the conclusion holds for all such functions is complete, so the principle characterizes completeness.
</document>

<document source="https://tamnd.github.io/math2/node/MJMB7SH7SD.md" tag="MJMB7SH7SD" kind="statement">
# Theorem of the cube

## Statement

Let $k$ be an algebraically closed field.
Let $X$ and $Y$ be complete [[varieties|HV6YBS0YDH]] over $k$, and let $Z$ be a variety over $k$.
Let $x_0 \in X$, $y_0 \in Y$ and $z_0 \in Z$ be closed points.
Let $L$ be a [[line bundle|7JCF2TE3V5]] on $X \times Y \times Z$.
Assume that the restrictions of $L$ to $\set{x_0} \times Y \times Z$, to $X \times \set{y_0} \times Z$ and to $X \times Y \times \set{z_0}$ are trivial.
Then $L$ is trivial.

## Remarks

The standard proof uses the seesaw principle, which detects triviality of a line bundle on a product from its restrictions to fibres.
It combines this with a reduction to the case where $Z$ is a curve.

Applied to an [[abelian variety|R7W03TXRS9]] $A$, the theorem gives a formula on $A \times A \times A$.
Let $p_i$ be the projections and $p_{ij} = p_i + p_j$, $p_{123} = p_1 + p_2 + p_3$ the sum maps to $A$.
For every line bundle $L$ on $A$, the line bundle
$$p_{123}^{*} L \otimes p_{12}^{*} L^{-1} \otimes p_{13}^{*} L^{-1} \otimes p_{23}^{*} L^{-1} \otimes p_1^{*} L \otimes p_2^{*} L \otimes p_3^{*} L$$
is trivial.
Let $t_x$ be translation by a point $x$ of $A$.
Pulling back along maps into $A \times A \times A$ gives the theorem of the square: $t_{x+y}^{*} L \otimes L \cong t_x^{*} L \otimes t_y^{*} L$ for all points $x, y$.
It also gives $[n]^{*} L \cong L^{(n^2 + n)/2} \otimes [-1]^{*} L^{(n^2 - n)/2}$ for multiplication by $n$.
So for a symmetric line bundle, with $[-1]^{*} L \cong L$, one gets $[n]^{*} L \cong L^{n^2}$.

## Explanations

### L1 human

A line bundle on a product of three spaces is trivial as soon as it is trivial on three coordinate "faces" through a common corner.
The name comes from this picture of a cube with three faces through one vertex.
For abelian varieties this turns into identities for line bundles under addition of points.

### L2 human

The theorem of the cube is a basic tool in the algebraic theory of abelian varieties.
The theorem of the square that follows from it gives a homomorphism $A \to \operatorname{Pic}(A)$, $x \mapsto t_x^{*} L \otimes L^{-1}$, which lands in the points of the [[dual abelian variety|HZXSESHC8P]].
For an ample $L$ this map is an isogeny, the polarization attached to $L$.
The formula $[n]^{*} L \cong L^{n^2}$ for symmetric $L$ is a key step in computing the degree of multiplication by $n$, and in the theory of heights.
</document>

<document source="https://tamnd.github.io/math2/node/MJP3GW3HD7.md" tag="MJP3GW3HD7" kind="definition">
# Stone–Čech compactification

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A *Stone–Čech compactification* of $X$ is a pair $(\beta X, \iota)$ with the following properties.

1. $\beta X$ is a [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]].
2. $\iota \colon X \to \beta X$ is a [[continuous map|0VGFHTM75Y]].
3. For every compact Hausdorff space $K$ and every continuous map $f \colon X \to K$, there is a unique continuous map $g \colon \beta X \to K$ with $g \circ \iota = f$.

## Remarks

Such a pair exists for every $X$ and is unique up to a unique homeomorphism compatible with $\iota$.
The map $\iota$ is an embedding with dense image exactly when $X$ is a [[Tychonoff space|C5N2J2K6Z3]].
Then $(\beta X, \iota)$ is a [[compactification|Q6XNC8JAZD]] of $X$.

Example: if $X$ is compact Hausdorff, then $\beta X = X$ with $\iota$ the identity.
For a continuous $f \colon X \to K$, the map $g = f$ is the required extension, and it is unique.

Non-example: the one-point compactification $\N \cup \set{\infty}$ of $\N$ is not $\beta \N$.
The map $f \colon \N \to \set{0, 1}$, $f(n) = n \bmod 2$, has no continuous extension.
An extension would need $g(\infty) = \lim g(2n) = 0$ and $g(\infty) = \lim g(2n + 1) = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/MJQARBFG10.md" tag="MJQARBFG10" kind="concept">
# Fourier–Motzkin elimination

## Summary

Fourier–Motzkin elimination removes one variable from a system of linear inequalities, as Gaussian elimination does for equations.
The new system has a solution exactly when the old one has, and its solutions are the projections of the old solutions.
Repeating it decides whether a system of linear inequalities is solvable.
It gives short proofs of [[Farkas' lemma|FYZC634H2D]] and of the fact that projections of polyhedra are polyhedra.
The number of inequalities can grow very fast, so it is mainly a theoretical tool.

## Explanations

### L1 human

To get rid of $y$, sort the inequalities into those that bound $y$ from above, those that bound it from below, and those without $y$.
A value of $y$ exists exactly when every lower bound is at most every upper bound.
So replace all inequalities with $y$ by the comparisons "lower bound $\le$ upper bound".
For $0 \le y \le 1 - x$ this gives $0 \le 1 - x$, that is $x \le 1$.
</document>

<document source="https://tamnd.github.io/math2/node/MK3ASPEN8A.md" tag="MK3ASPEN8A" kind="concept">
# Carlson symmetric forms

## Summary

The Carlson symmetric forms $R_F$, $R_J$, $R_C$ and $R_D$ are a standard set of [[elliptic integrals|NXMCYGCB59]] that are symmetric in their main arguments.
Every elliptic integral can be written with them, and in particular the Legendre integrals $F$, $E$ and $\Pi$.
Their symmetry gives a duplication theorem that leads to fast and stable numerical algorithms, which is why software libraries use them.

## Explanations

### L1 human

Legendre's forms of the elliptic integrals depend on an angle and a modulus, and the formulas that transform them are lopsided.
Carlson's forms put the three roots of the cubic under the square root on an equal footing, so permuting $x$, $y$, $z$ does not change the value.
For example $R_F(x, y, z) = \frac{1}{2}\int_0^\infty dt / \sqrt{(t + x)(t + y)(t + z)}$, and $R_F(x, x, x) = 1/\sqrt{x}$.
The complete integral of the first kind is $K(k) = R_F(0, 1 - k^2, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/MK4FDH3EDM.md" tag="MK4FDH3EDM" kind="definition">
# Dickey–Fuller test

## Definition

Let $y_0, y_1, \dots, y_n$ follow the [[autoregressive model|2JC96FMGA1]] $y_t = \rho y_{t-1} + u_t$.
Here the $u_t$ are independent and identically distributed with mean $0$ and variance $\sigma^2 > 0$.
Write it as $\Delta y_t = \delta y_{t-1} + u_t$ with $\Delta y_t = y_t - y_{t-1}$ and $\delta = \rho - 1$.
Let $\hat\delta$ be the [[ordinary least squares|YM708KHV2W]] estimate of $\delta$ in the regression of $\Delta y_t$ on $y_{t-1}$.
Let $\operatorname{se}(\hat\delta)$ be its usual standard error, and let $\tau = \hat\delta / \operatorname{se}(\hat\delta)$.
The *Dickey–Fuller test* is the [[test|087P03NEND]], based on $\tau$, of
$$H_0 \colon \delta = 0 \ (\text{a unit root}) \qquad \text{against} \qquad H_1 \colon \delta < 0 \ (\text{stationarity}).$$
It rejects $H_0$ when $\tau$ is below a critical value of the Dickey–Fuller distribution.

## Remarks

Under $H_0$, the series $y_t$ is a [[random walk|04VZE3FZXF]], an [[ARIMA($0$, $1$, $0$) process|KTDVZP4DF2]], and the usual central limit theory fails.
If $y_0 = 0$, then as $n \to \infty$
$$\tau \to \frac{W(1)^2 - 1}{2 \left( \int_0^1 W(r)^2 \, dr \right)^{1/2}}$$
in distribution, where $W$ is a standard [[Wiener process|22735FGJV6]].
This limit is skewed to the left, so its critical values lie well below those of the standard normal distribution; they are tabulated.
Variants add a constant, or a constant and a linear trend, to the regression, and each variant has its own limiting distribution and critical values.
The *augmented* Dickey–Fuller test adds lagged differences $\Delta y_{t-1}, \dots, \Delta y_{t-k}$ to the regression, to allow for autocorrelated errors.

Example: for a series with $\hat\delta = -0.2$ and $\operatorname{se}(\hat\delta) = 0.05$, the statistic is $\tau = -4$.
This lies far in the left tail of the Dickey–Fuller distribution, so the unit root hypothesis is rejected at the usual levels.

Non-example: comparing $\tau$ with the critical values of the Student $t$ distribution is not the Dickey–Fuller test.
That rule rejects a true unit root far too often, because the null distribution of $\tau$ is not a $t$ distribution.
</document>

<document source="https://tamnd.github.io/math2/node/MK5JGVDPS9.md" tag="MK5JGVDPS9" kind="concept">
# Backward difference operator

## Summary

The backward difference operator $\nabla_h$ sends a function $f$ to the function $x \mapsto f(x) - f(x - h)$.
It looks one step back instead of one step ahead.
It is natural in time-stepping and interpolation schemes that use only values already computed.

## Explanations

### L1 human

The backward difference compares the value now with the value one step earlier.
If $u_n$ is the balance of an account at the end of month $n$, then $\nabla u_n = u_n - u_{n-1}$ is the change during month $n$.
For the squares $0, 1, 4, 9, 16$ the backward differences at $n = 1, 2, 3, 4$ are $1, 3, 5, 7$.
It is the forward difference read from the other end of the step: $\nabla_h f(x) = \Delta_h f(x - h)$.
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# Poisson boundary

## Definition

Let $\Gamma$ be a countable group and let $\mu$ be a [[probability measure|059TJN0NZ1]] on $\Gamma$.
It defines the [[random walk|WAS9EANSJJ]] $X_n = g_1 g_2 \cdots g_n$ with independent steps $g_k$ of law $\mu$.
A function $f \colon \Gamma \to \R$ is $\mu$-*harmonic* if $f(g) = \sum_{h \in \Gamma} \mu(h) f(gh)$ for every $g \in \Gamma$.
Let $H^\infty(\Gamma, \mu)$ be the space of bounded $\mu$-harmonic functions.
A *Poisson boundary* of $(\Gamma, \mu)$ is a standard probability space $(B, \nu)$ with a measurable action of $\Gamma$ such that:

1. $\nu$ is $\mu$-stationary: $\sum_{h \in \Gamma} \mu(h)\, h_* \nu = \nu$.
2. The Poisson transform $\varphi \mapsto f_\varphi$, with $f_\varphi(g) = \int_B \varphi(g \xi) \, d\nu(\xi)$, is a bijection from $L^\infty(B, \nu)$ onto $H^\infty(\Gamma, \mu)$.

## Remarks

Clause 1 makes each $f_\varphi$ harmonic: $\sum_h \mu(h) \int_B \varphi(gh\xi) \, d\nu(\xi) = \int_B \varphi(g\xi) \, d\Big(\sum_h \mu(h) h_* \nu\Big)(\xi) = f_\varphi(g)$.
The Poisson boundary exists and is unique up to a $\Gamma$-equivariant measure isomorphism.
One construction takes the space of ergodic components of the shift on the space of sample paths of the walk, with the image of the path measure.
Example: let $\Gamma = \Z^d$ and let the support of $\mu$ generate $\Z^d$ as a group.
By the Choquet–Deny theorem the bounded harmonic functions are constant, so a one-point space is a Poisson boundary.
Non-example: for the free group $F_2$ on $a, b$ and $\mu$ uniform on $\set{a, a^{-1}, b, b^{-1}}$, a one-point space fails clause 2, since there are nonconstant bounded harmonic functions.
In that case a Poisson boundary is the space of infinite reduced words in $a^{\pm 1}, b^{\pm 1}$ with the law of the limit of $X_n$.
For the simple random walk on a Cayley graph, a trivial Poisson boundary is the same as the [[Liouville property|Y1EGR8DF84]] of the graph.
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# Optimal experimental design

## Definition

Let $\mathcal{X}$ be a compact set, the *design region*, and let $f \colon \mathcal{X} \to \R^p$ be continuous.
Consider the linear model $y = f(x)^T \beta + \varepsilon$ with unknown $\beta \in \R^p$ and uncorrelated errors of mean $0$ and common variance $\sigma^2$.
An *approximate design* is a probability measure $\xi$ on $\mathcal{X}$ with finite support, and its *information matrix* is
$$M(\xi) = \int_{\mathcal{X}} f(x) f(x)^T \, d\xi(x) = \sum_{x \in \operatorname{supp} \xi} \xi(\set{x}) f(x) f(x)^T.$$
A design $\xi^*$ is:

1. *D-optimal* if it maximises $\det M(\xi)$ over all approximate designs;
2. *A-optimal* if $M(\xi^*)$ is invertible and it minimises $\tr M(\xi)^{-1}$ over the designs with $M(\xi)$ invertible;
3. *E-optimal* if it maximises the smallest [[eigenvalue|0BQAA2BK3Q]] of $M(\xi)$;
4. *G-optimal* if it minimises $\max_{x \in \mathcal{X}} f(x)^T M(\xi)^{-1} f(x)$ over the designs with $M(\xi)$ invertible.

## Remarks

For an exact [[design|2TE42ZWJ3E]] with $n$ runs at $x_1, \dots, x_n$ and design matrix $X$ with rows $f(x_i)^T$, one has $M = \frac{1}{n} X^T X$.
If $M$ is invertible, the least squares estimator has [[covariance matrix|0V7G6WMFK0]] $\frac{\sigma^2}{n} M^{-1}$.
So a D-optimal design minimises the volume of the confidence ellipsoid for $\beta$ under normal errors.

Example: $\mathcal{X} = [-1, 1]$ and $f(x) = (1, x)^T$, a straight line.
With moments $m_1 = \int x \, d\xi$ and $m_2 = \int x^2 \, d\xi$, one gets $\det M(\xi) = m_2 - m_1^2$, the variance of $x$ under $\xi$.
On $[-1, 1]$ this variance is at most $1$, with equality exactly for the design with mass $1/2$ at $-1$ and $1/2$ at $1$, which is therefore D-optimal.

Non-example: the design with all mass at $x = 0$ has $M = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, which is singular, so $\det M = 0$ and the slope cannot be estimated; it fails clause 1.
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# Aubin–Lions lemma

## Statement

Let $X_0 \subseteq X \subseteq X_1$ be three [[Banach spaces|06GZ956657]] such that $X_0$ and $X_1$ are [[reflexive|NN8RWPC8WG]], the inclusion $X_0 \to X$ is compact, and the inclusion $X \to X_1$ is continuous.
Let $T > 0$ and $1 < p, q < \infty$.
Let $L^p(0, T; X_0)$ be the [[Lebesgue space|SASP14MA1Z]] of [[Bochner-integrable|MHSMQ097GP]] functions with values in $X_0$, and let $u'$ be the weak time derivative.
Let $W = \set{ u \in L^p(0, T; X_0) : u' \in L^q(0, T; X_1) }$.
Give $W$ the norm $\norm{u}_W = \norm{u}_{L^p(0, T; X_0)} + \norm{u'}_{L^q(0, T; X_1)}$.
Then the inclusion of $W$ into $L^p(0, T; X)$ is compact.

## Remarks

Proof idea: by Ehrling's lemma, for every $\eta > 0$ there is $C_\eta$ with $\norm{v}_X \le \eta \norm{v}_{X_0} + C_\eta \norm{v}_{X_1}$ for all $v \in X_0$.
A bounded set in $W$ is bounded in $L^p(0, T; X_0)$ and equicontinuous in time with values in $X_1$, because of the bound on $u'$.
This gives a subsequence that converges in $L^p(0, T; X_1)$, and Ehrling's lemma upgrades the convergence to $L^p(0, T; X)$.

Example: let $\Omega \subseteq \R^n$ be a bounded Lipschitz domain, $X_0 = H^1(\Omega)$, $X = L^2(\Omega)$ and $X_1 = (H^1(\Omega))'$, the dual space.
The [[Rellich–Kondrachov theorem|YM3JZ34PME]] gives the compact inclusion.
So a sequence of approximate solutions of a parabolic equation, bounded in $L^2(0, T; H^1(\Omega))$ with time derivatives bounded in $L^2(0, T; (H^1)')$, has a subsequence that converges strongly in $L^2(0, T; L^2(\Omega))$.
This strong convergence lets one pass to the limit in nonlinear terms, as in the construction of [[Leray weak solutions|DHHR5PZZ7E]] of the Navier–Stokes equations.

Both hypotheses are needed.
If $X_0 = X = X_1$ is an infinite-dimensional Hilbert space, the constant functions $u_k(t) = e_k$, for an orthonormal sequence $(e_k)$, are bounded in $W$ with $u_k' = 0$, but no subsequence converges in $L^p(0, T; X)$.
If the bound on $u'$ is dropped, the functions $u_k(t) = \sin(kt) v$ with a fixed $v \ne 0$ are bounded in $L^p(0, T; X_0)$, but they converge weakly to $0$ and not strongly.

## Explanations

### L1 human

Bounds in space alone or in time alone do not give compactness for functions of time and space.
The lemma says that a bound on spatial derivatives together with a weak bound on time derivatives is enough to extract a strongly convergent subsequence.

### L2 human

The lemma is the standard compactness tool for evolution equations.
One builds approximate solutions, for example by the Galerkin method.
Energy estimates bound them in $L^2(0, T; H^1)$ and their time derivatives in a dual space.
The lemma then gives strong convergence in $L^2(0, T; L^2)$.
It plays for time-dependent problems the role that the Rellich–Kondrachov theorem plays for stationary ones.
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# Orientation of a manifold

## Definition

Let $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $n \ge 1$, and use integral [[relative homology|4BVWH0BN28]].
For each $x \in M$, the [[excision theorem|SGHMSBHHA0]] gives $H_n(M, M \setminus \set{x}) \cong H_n(\R^n, \R^n \setminus \set{0}) \cong \Z$.
A *local orientation* of $M$ at $x$ is a choice of generator $\mu_x$ of $H_n(M, M \setminus \set{x})$.
An *orientation* of $M$ is a function $x \mapsto \mu_x$ assigning a local orientation to each point, with the following local consistency.
Each $x \in M$ has a chart neighbourhood $U \cong \R^n$ containing an open ball $B$ of finite radius about $x$, and a class $\mu_B \in H_n(M, M \setminus B)$.
For every $y \in B$, the image of $\mu_B$ in $H_n(M, M \setminus \set{y})$ is $\mu_y$.
$M$ is *orientable* if it has an orientation.

## Remarks

A connected orientable manifold has exactly two orientations.
For a [[smooth manifold|F45VQ47GPV]], an orientation in this sense is the same as a choice of [[orientation|0QJJAQ6ZYZ]] of each tangent space $T_x M$ that is locally given by a chart.
It is also the same as an atlas whose transition maps have positive Jacobian determinant.
Replacing $\Z$ by a commutative ring $R$ gives $R$-orientations; every manifold has a unique $\Z/2$-orientation.

Example: the [[sphere|T62369RFH6]] $S^n$ is orientable.
Fix a generator $\mu \in H_n(S^n) \cong \Z$; its image in each $H_n(S^n, S^n \setminus \set{y})$ is a generator, and one takes this image as $\mu_y$.
The condition holds with $\mu_B$ the image of $\mu$ in $H_n(S^n, S^n \setminus B)$ for every chart ball $B$.

Non-example: the open [[Möbius strip|04AJE0M271]] is not orientable.
Following a local orientation once around the central circle returns the opposite generator, so no continuous choice exists.
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# Kolmogorov quotient

## Summary

The Kolmogorov quotient of a topological space identifies any two points that no open set can tell apart.
The result is a [[T0 space|07SG12TQH3]], and its open sets correspond exactly to the open sets of the original space.
It lets one pass to a $T_0$ space without losing any topological information.
For a pseudometric space it gives the associated metric space.

## Explanations

### L1 human

In some spaces, two different points have exactly the same neighbourhoods.
No open set contains one of them without the other, so the topology cannot see the difference.
The Kolmogorov quotient glues such twin points into one point.
For example, on $\R^2$ with the pseudometric $d((x_1, y_1), (x_2, y_2)) = \abs{x_1 - x_2}$, points on the same vertical line are twins.
Gluing them gives the real line.
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# Metropolis–Hastings algorithm

## Summary

The Metropolis–Hastings algorithm is a [[Markov chain Monte Carlo|ZGNBR2SPZ0]] method.
It turns an arbitrary proposal mechanism into a [[Markov chain|0G48PYM96Q]] with a prescribed stationary distribution $\pi$.
At each step a candidate move is proposed and then accepted or rejected with a probability chosen to enforce [[detailed balance|CTGN6CK1ZS]].
Only ratios $\pi(y)/\pi(x)$ enter, so the normalising constant of $\pi$ is never needed.

## Explanations

### L1 human

The chain sits at a state and suggests a new state at random.
If the new state is more likely under $\pi$, the move is usually taken; if it is less likely, the move is taken only sometimes.
Rejected moves keep the chain where it is.
This balance of moves up and down in probability makes the chain spend time in each state in proportion to $\pi$.
For instance, with a symmetric proposal and $\pi(y) = \pi(x)/2$, a move from $x$ to $y$ is accepted half the time.
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# Inductive dimension

## Summary

The inductive dimensions of a topological space are defined by induction from the dimension of boundaries.
A space has dimension at most $n$ when points, or closed sets, have small neighbourhoods whose boundaries have dimension at most $n - 1$.
There are two versions, the small inductive dimension $\operatorname{ind}$ and the large inductive dimension $\operatorname{Ind}$.
For separable metric spaces both agree with the [[Lebesgue covering dimension|A3W7RGXXK9]].

## Explanations

### L1 human

A solid ball in space can be cut out from its surroundings by its surface, which is two-dimensional.
A disc in the plane is cut out by a circle, which is one-dimensional, and an interval on a line is cut out by two points, which are zero-dimensional.
The inductive dimension turns this into a definition: the dimension of a space is one more than the dimension of the walls one needs to cut out small pieces.
The empty set has dimension $-1$ to start the induction.
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# Exceptional Lie group E8

## Example

The compact Lie group $E_8$ is the compact, connected and simply connected [[simple Lie group|AM7YJP8AR6]] whose Lie algebra has the root system of type $E_8$.
The root system $E_8$ lives in $\R^8$ and consists of the $240$ vectors of the following two kinds:

1. the $112$ vectors with two entries $\pm 1$ and six entries $0$;
2. the $128$ vectors $(\pm \tfrac12, \dots, \pm \tfrac12)$ with an even number of minus signs.

Every root has squared length $2$.
Check of the count: there are $\binom{8}{2} \cdot 4 = 112$ vectors of the first kind and $2^8 / 2 = 128$ of the second kind.

The dimension of $E_8$ is the rank plus the number of roots, $8 + 240 = 248$.

## Remarks

$E_8$ is the largest of the five exceptional compact simple Lie groups $G_2, F_4, E_6, E_7, E_8$, see [[G2|YX5FFSCR8P]] for the smallest.
Its Dynkin diagram is a tree with $8$ nodes and branch lengths $1$, $2$ and $4$ from the branch point.
The compact group $E_8$ has trivial centre, so it is at the same time simply connected and adjoint.
Its smallest nontrivial representation is the adjoint representation, of dimension $248$.
The $240$ roots are the shortest nonzero vectors of the [[E8 lattice|2CXH65EKY7]], which is spanned by them.
The Weyl group of $E_8$ has order $696\,729\,600$.
The name $E_8$ also denotes the complex simple Lie group of this type and its real forms.

## Explanations

### L1 human

$E_8$ is the largest and most intricate of the exceptional Lie groups, the ones that do not belong to the infinite families of matrix groups.
It has dimension $248$, yet it is built from a very symmetric arrangement of $240$ vectors in eight-dimensional space.
The lattice that these vectors span gives the densest packing of equal balls in eight dimensions.
The group appears in many places, from the classification of lattices to string theory.
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# Unification

## Definition

Let $\sigma$ be a signature with only function symbols, let $X$ be a set of variables, and let $s, t$ be [[terms|6D3WA7N2W4]] over $\sigma$ and $X$.
A *unifier* of $s$ and $t$ is a [[substitution|NCDZ969V7X]] $\theta$ with $s\theta = t\theta$, equality meaning identical terms.
A unifier $\theta$ is a *most general unifier* if the following holds for every unifier $\eta$ of $s$ and $t$.
There is a substitution $\rho$ with $x\eta = (x\theta)\rho$ for every variable $x$.
*Unification* is the problem of deciding whether two given terms have a unifier and of finding a most general one.

## Remarks

If $s$ and $t$ have a unifier, then they have a most general unifier, unique up to renaming of variables, and Robinson's algorithm finds it.
The same notions apply to finite sets of pairs of terms, which are unified simultaneously.
In $E$-unification one asks instead for $s\theta \approx t\theta$ to follow from a set $E$ of identities.

Example: $s = f(x, b)$ and $t = f(a, y)$, with constants $a, b$, have the unifier $\theta = \set{x \mapsto a, \ y \mapsto b}$.
Indeed $s\theta = f(a, b) = t\theta$, and $\theta$ is most general because every unifier must send $x$ to $a$ and $y$ to $b$.

Non-example: $x$ and $f(x)$ have no unifier.
For every substitution $\theta$, the term $f(x)\theta = f(x\theta)$ has one more symbol than $x\theta$, so they are never identical.
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# Moving-average model

## Definition

Let $(\varepsilon_t)_{t \in \Z}$ be a white noise with variance $\sigma^2 > 0$: the $\varepsilon_t$ have mean $0$, variance $\sigma^2$ and are pairwise uncorrelated.
Let $q \ge 1$, and let $\mu, \theta_1, \dots, \theta_q \in \R$ with $\theta_q \ne 0$.
A *moving-average model of order $q$*, or MA($q$), is the [[time series|EA3MF4VTEH]] model
$$X_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \cdots + \theta_q \varepsilon_{t-q}, \qquad t \in \Z.$$
Its *moving-average polynomial* is $\theta(z) = 1 + \theta_1 z + \cdots + \theta_q z^q$.

## Remarks

Every MA($q$) process is weakly stationary with mean $\mu$.
With $\theta_0 = 1$, its autocovariance is $\gamma(h) = \sigma^2 \sum_{j=0}^{q - \abs{h}} \theta_j \theta_{j + \abs{h}}$ for $\abs{h} \le q$ and $\gamma(h) = 0$ for $\abs{h} > q$.
So its [[autocorrelation function|YCZHYY8VE5]] vanishes beyond lag $q$.
The model is *invertible* when $\theta$ has no [[root|019MA1VTR2]] in the closed unit disk.
Then $\varepsilon_t$ is a convergent linear combination of $X_t, X_{t-1}, \dots$, and the process is an [[autoregressive model|2JC96FMGA1]] of infinite order.

Example: for MA($1$), $X_t = \varepsilon_t + \theta \varepsilon_{t-1}$ has $\gamma(0) = \sigma^2 (1 + \theta^2)$, $\gamma(\pm 1) = \theta \sigma^2$ and $\gamma(h) = 0$ otherwise.
Hence $\rho(1) = \theta / (1 + \theta^2)$, which always lies in $[-1/2, 1/2]$.

Non-example: the process $X_t = \sum_{j \ge 0} 0.5^j \varepsilon_{t-j}$ is not an MA($q$) for any finite $q$, because its autocorrelation $0.5^{\abs{h}}$ is nonzero at every lag.

The name has nothing to do with the moving average used to smooth data, although both are weighted sums over a window.
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# Borel functional calculus

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]], and let $N$ be a bounded normal operator on $H$ with [[spectrum|BMKATJ802A]] $\sigma(N)$.
Let $E$ be the [[projection-valued measure|ATGT25E5FF]] of $N$ given by the [[spectral theorem for bounded normal operators|X71XRPJR7D]], and write $E_{x,y}(B) = \ip{E(B) x}{y}$.
For a bounded [[Borel|0WQXY32Q8D]] measurable function $f \colon \sigma(N) \to \C$, let $f(N)$ be the unique bounded operator on $H$ with
$$\ip{f(N) x}{y} = \int_{\sigma(N)} f(\lambda) \, d E_{x,y}(\lambda) \quad \text{for all } x, y \in H.$$
The map $f \mapsto f(N)$, from the bounded Borel functions on $\sigma(N)$ to the bounded operators on $H$, is the *Borel functional calculus* of $N$.

## Remarks

The map $f \mapsto f(N)$ is linear and multiplicative, sends $\overline{f}$ to $f(N)^*$, and satisfies $\norm{f(N)} \le \sup_{\sigma(N)} \abs{f}$.
For $f(\lambda) = \lambda$ it gives $N$, and on continuous functions it agrees with the [[continuous functional calculus|JB7S6WPZ77]].
If $f_n \to f$ pointwise with $\sup_n \sup_{\sigma(N)} \abs{f_n} < \infty$, then $f_n(N) x \to f(N) x$ for every $x \in H$.
Example: for a Borel set $B$, the indicator function $1_B$ gives $1_B(N) = E(B)$, the spectral projection of $N$ for $B$.
For instance, if $N$ is the multiplication by $t$ on $L^2([0,1])$, then $f(N)$ is the multiplication by $f(t)$.
Non-example: let $N$ be the diagonal operator $N e_n = e_n / n$ on $\ell^2$, so $\sigma(N) = \set{1/n : n \ge 1} \cup \set{0}$.
The function $f(\lambda) = 1/\lambda$ for $\lambda \ne 0$, $f(0) = 0$, is Borel but not bounded, and the formula would give $f(N) e_n = n e_n$, which is not a bounded operator.
Changing $f$ on a set $B$ with $E(B) = 0$ does not change $f(N)$.
For unbounded self-adjoint operators, the same formula defines $f(A)$ for every bounded Borel function on $\R$, and for unbounded $f$ on a suitable dense domain.
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# Unitary operator

## Summary

A unitary operator is a bounded linear operator $U$ on a Hilbert space with $U^*U = UU^* = I$.
Equivalently, it is a surjective linear map that preserves the inner product.
Unitary operators are the isomorphisms of Hilbert spaces, so two operators that differ by conjugation with a unitary operator have the same properties.
They extend [[unitary matrices|0AR68JTMDT]] to infinite dimensions and describe the time evolution in quantum mechanics.

## Explanations

### L1 human

A unitary operator is a rigid motion of a Hilbert space that can be undone.
It keeps all lengths and all angles, and every vector is hit by it.
The Fourier transform on $L^2(\R)$, suitably normalized, is a unitary operator.
A simpler example is the shift on two-sided sequences, $(Ux)_n = x_{n-1}$ on $\ell^2(\Z)$: it moves every entry one step to the right and is undone by the shift to the left.
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# Cohn localization

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity and let $\Sigma$ be a set of square [[matrices|0VD9ZSWWJG]] over $R$, of possibly different sizes.
A [[ring homomorphism|0VTH8KPCG2]] $f \colon R \to S$ is *$\Sigma$-inverting* if for every $A \in \Sigma$ the matrix $f(A)$, obtained by applying $f$ to each entry, is invertible over $S$.
The *Cohn localization* (or *universal localization*) of $R$ at $\Sigma$ is a ring $R_\Sigma$ with a $\Sigma$-inverting homomorphism $\lambda \colon R \to R_\Sigma$ with this property.
Every $\Sigma$-inverting homomorphism $f \colon R \to S$ factors as $f = g \circ \lambda$ for exactly one ring homomorphism $g \colon R_\Sigma \to S$.

## Remarks

$R_\Sigma$ exists for every $R$ and $\Sigma$, and it is unique up to a unique isomorphism over $R$.
It can be built by adjoining, for each $n \times n$ matrix $A \in \Sigma$, $n^2$ new generators forming a matrix $\bar A$, with the relations $A \bar A = \bar A A = I_n$.
The map $\lambda$ need not be injective, and $R_\Sigma$ may be the zero ring.
Example: let $R$ be [[commutative|0310REJJY8]] and let $\Sigma$ be the set of $1 \times 1$ matrices $(s)$ with $s$ in a multiplicative set $S$.
A homomorphism is $\Sigma$-inverting exactly when it sends each $s \in S$ to a unit, so $R_\Sigma$ is the [[localization|4C1AE84N5N]] $S^{-1}R$.
More generally, for a right denominator set $S$, that is, a multiplicative set satisfying the right Ore condition and right reversibility, $R_\Sigma$ is the [[Ore localization|SG7AQ9T8HD]] $R S^{-1}$.
Non-example: the identity map $\Z \to \Z$ is not $\Sigma$-inverting for $\Sigma = \set{(2)}$, since $2$ is not a unit in $\Z$; the localization in this case is $\Z[1/2]$.
For the [[free algebra|2P3Y2FRCSZ]] $k\langle X \rangle$ over a field $k$ and $\Sigma$ the set of full matrices, $R_\Sigma$ is a division ring, the free field.
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# Linear continuum

## Summary

A linear continuum is a [[totally ordered set|0FRXBVVR03]] with more than one element that has no gaps and no jumps.
Every nonempty subset bounded above has a [[supremum|0R3H5PANRQ]], and between any two elements lies a third.
The real line is the model example.
In the order topology, every linear continuum is [[connected|0V032VWTHD]], and so are its intervals.

## Explanations

### L1 human

A linear continuum is an ordered line without holes.
The rational numbers have holes, for example at $\sqrt{2}$, and the integers have jumps, from $1$ straight to $2$.
The real numbers have neither, so they form a linear continuum.
This is exactly what makes the real line connected and makes the intermediate value theorem true.
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# Numerical diffusion

## Definition

Let $a \in \R$, and consider a consistent linear [[finite difference|X7HYEBA5SA]] scheme for the [[transport equation|PVD7VZJZ9B]] $u_t + a u_x = 0$, with mesh sizes $\Delta x$, $\Delta t$ and fixed ratio $\nu = a \Delta t / \Delta x$.
Its modified equation is the partial differential equation that the numerical solution satisfies to higher order in $\Delta x$.
It is found by Taylor expansion of the scheme, with time derivatives replaced by space derivatives.
Suppose the modified equation has the form
$$u_t + a u_x = D\, u_{xx} + \sum_{k \ge 3} c_k\, \partial_x^k u.$$
The coefficient $D$ is the *numerical diffusion* coefficient of the scheme.
The scheme is said to have numerical diffusion when $D > 0$.

## Remarks

Example: for the [[upwind scheme|MTCRZC5QKT]] $u_j^{n+1} = u_j^n - \nu (u_j^n - u_{j-1}^n)$ with $a > 0$, Taylor expansion gives $u_t + \frac{\Delta t}{2} u_{tt} + a u_x - \frac{a \Delta x}{2} u_{xx} = O(\Delta x^2)$.
Using $u_{tt} = a^2 u_{xx} + O(\Delta x)$ gives $D = \frac{a \Delta x}{2} (1 - \nu)$, which is positive for $0 < \nu < 1$.
For the [[Lax–Friedrichs method|VY3J2NKGKW]] the same computation gives $D = \frac{\Delta x^2}{2 \Delta t} (1 - \nu^2)$, which is larger than for upwind when $0 < \nu < 1$.
Non-example: for the [[Lax–Wendroff method|YHZ9BCNDRZ]] the coefficient of $u_{xx}$ is zero to this order, so $D = 0$.
Its leading error is a third-derivative term, which is dispersive (wiggles), not diffusive (smearing).
A negative $D$ means anti-diffusion, as for the downwind scheme $u_j^{n+1} = u_j^n - \nu (u_{j+1}^n - u_j^n)$ with $a > 0$, which is unstable.

## Explanations

### L1 human

Numerical diffusion is the hidden diffusion term that a scheme adds to the equation it solves.
Its coefficient is read off from the modified equation.
</document>

<document source="https://tamnd.github.io/math2/node/MM3TQ1WJWF.md" tag="MM3TQ1WJWF" kind="definition">
# Nilpotent ideal

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $I$ be a [[left ideal|5N2CJZ47D4]], right ideal or [[two-sided ideal|0DDA6CY7GJ]] of $R$.
For $n \ge 1$, let $I^n$ be the additive subgroup of $R$ generated by all products $a_1 a_2 \cdots a_n$ with $a_1, \dots, a_n \in I$.
The ideal $I$ is *nilpotent* if $I^n = 0$ for some $n \ge 1$.

## Remarks

Every nilpotent ideal is a [[nil ideal|NZ1G9WRWFB]], since $I^n = 0$ gives $a^n = 0$ for all $a \in I$.
Example: let $R$ be the ring of upper triangular $2 \times 2$ matrices over a field $K$ and let $I$ be the set of its matrices with zero diagonal.
$I$ is a two-sided ideal, and the product of two matrices of the form $\begin{pmatrix} 0 & x \\ 0 & 0 \end{pmatrix}$ is zero, so $I^2 = 0$.
Non-example: let $R = \Z[x_1, x_2, \dots] / (x_1, x_2^2, x_3^3, \dots)$ and let $I$ be the ideal generated by the images of all $x_k$.
$I$ is nil, since it is generated by nilpotent elements of a commutative ring.
It is not nilpotent, because for every $n$ the product $x_{n+1}^{n}$ of $n$ elements of $I$ is nonzero in $R$.
</document>

<document source="https://tamnd.github.io/math2/node/MM6TXKN90V.md" tag="MM6TXKN90V" kind="definition">
# Rarefaction wave

## Definition

Let $F \colon \R \to \R$ be a $C^2$ function with $F'' > 0$, and consider the scalar [[conservation law|8RA7AMWSAF]] $u_t + F(u)_x = 0$ on $\R \times (0, \infty)$.
Let $G$ be the inverse function of the strictly increasing function $F'$.
Let $u_l < u_r$ be two real numbers.
The *centered rarefaction wave* from $u_l$ to $u_r$ is the function
$$u(x, t) = \begin{cases} u_l & \text{if } x < F'(u_l)\, t, \\ G(x/t) & \text{if } F'(u_l)\, t \le x \le F'(u_r)\, t, \\ u_r & \text{if } x > F'(u_r)\, t. \end{cases}$$

## Remarks

The function is continuous, since $G(F'(u_l)) = u_l$ and $G(F'(u_r)) = u_r$.
Inside the fan, $F'(u) = x/t$, so $u_t = -\tfrac{x}{t^2} G'(x/t)$ and $u_x = \tfrac1t G'(x/t)$.
Hence $u_t + F'(u)\, u_x = 0$ there, and $u$ is a [[weak solution|SJZX81A783]] with the jump initial values $u_l$ for $x < 0$ and $u_r$ for $x > 0$.
It depends only on $x/t$, so it is self-similar.

Example: for [[Burgers' equation|NEGHXN8RE1]], $F(u) = u^2/2$, so $F'(u) = u$ and $G$ is the identity.
The rarefaction wave from $0$ to $1$ is $u = 0$ for $x < 0$, $u = x/t$ for $0 \le x \le t$, and $u = 1$ for $x > t$.
Check: $u_t + u u_x = -x/t^2 + (x/t)(1/t) = 0$ inside the fan.

Non-example: if $u_l > u_r$, then $F'(u_l) t > F'(u_r) t$ and the middle interval is empty.
The formula then gives no fan, and the admissible solution is a [[shock wave|JSDMGGTG7G]] instead.

## Explanations

### L1 human

A rarefaction wave is a fan of straight lines that spreads a jump into a smooth ramp.
Along each line through the origin with slope $x/t = c$, the solution takes the value whose characteristic speed is $c$.
So the solution moves continuously from the left state to the right state.
</document>

<document source="https://tamnd.github.io/math2/node/MME2BSPTZG.md" tag="MME2BSPTZG" kind="definition">
# Schwarz triangle function

## Definition

Let $a, b, c \in \C$, and let $w_1, w_2$ be two linearly independent solutions of the [[hypergeometric differential equation|W8VE25R5A7]] on the upper half-plane $\mathbb{H} = \set{z \in \C : \operatorname{Im} z > 0}$.
A *Schwarz triangle function* is the ratio
$$s(z) = \frac{w_2(z)}{w_1(z)}, \qquad z \in \mathbb{H}.$$
When $c \notin \Z$, the standard choice uses the two solutions at $0$:
$$s(z) = \frac{z^{1-c} \, {}_2F_1(a - c + 1, b - c + 1; 2 - c; z)}{{}_2F_1(a, b; c; z)},$$
with the principal branch of $z^{1-c}$ and the [[hypergeometric function|463X5V31D9]] ${}_2F_1$.

## Remarks

Changing the pair of solutions replaces $s$ by $(\alpha s + \beta)/(\gamma s + \delta)$ with $\alpha\delta - \beta\gamma \neq 0$, so $s$ is well defined up to a Möbius transformation.

Let $a, b, c$ be real and set $\lambda = 1 - c$, $\mu = c - a - b$, $\nu = a - b$.
If $0 \le \lambda, \mu, \nu < 1$, then $s$ is a [[conformal map|0YYT36R4YD]] from $\mathbb{H}$ onto a triangle bounded by circular arcs or segments.
Its angles are $\pi\lambda$, $\pi\mu$, $\pi\nu$ at the images of $0$, $1$, $\infty$.

Example: for $a = b = \frac12$, $c = 1$ all three angles are $0$.
Here $c \in \Z$, so the ratio is formed with the logarithmic second solution.
Up to a Möbius transformation it equals $\tau = i \, K(1 - z)/K(z)$.
Here $K(m) = \frac{\pi}{2} {}_2F_1(\frac12, \frac12; 1; m)$ is the complete elliptic integral of the first kind.
Its inverse is the modular lambda function.

Non-example: a ratio $w_2 / w_1$ of two proportional solutions is constant, so it is not a triangle function; independence of $w_1$ and $w_2$ is needed.
</document>

<document source="https://tamnd.github.io/math2/node/MMEX2JZWGR.md" tag="MMEX2JZWGR" kind="definition">
# Kovacic's algorithm

## Definition

Let $r \in \C(x)$ be a rational function and consider the equation $y'' = r\, y$ over the differential field $\C(x)$.
Let $G$ be its [[differential Galois group|RW8DQQYJMR]], the group of differential automorphisms of a [[Picard–Vessiot extension|H5P06V330V]].
Then $G$ is an algebraic subgroup of $\mathrm{SL}_2(\C)$, and exactly one of the following holds.

1. $G$ is conjugate to a group of upper triangular matrices; then there is a solution $y = e^{\int \omega}$ with $\omega \in \C(x)$.
2. Case 1 fails and $G$ is conjugate to a group of matrices that are diagonal or anti-diagonal. Then some solution is $y = e^{\int \omega}$ with $\omega$ algebraic of degree $2$ over $\C(x)$.
3. Cases 1 and 2 fail and $G$ is finite. Then all solutions are algebraic, and some solution is $y = e^{\int \omega}$ with $\omega$ algebraic of degree $4$, $6$ or $12$.
4. $G = \mathrm{SL}_2(\C)$; then the equation has no nonzero [[Liouvillian|R80FA7993W]] solution.

*Kovacic's algorithm* tests cases 1, 2 and 3 in turn.
In each case it builds finitely many candidate exponents from the expansions of $r$ at its poles and at infinity.
Then it searches for $\omega$ by solving for a polynomial of bounded degree.
It returns a Liouvillian solution, or the answer that none exists.

## Remarks

Every equation $y'' + a(x)\, y' + b(x)\, y = 0$ with $a, b \in \C(x)$ can be brought to the reduced form: $z = y\, e^{\frac12 \int a}$ satisfies $z'' = r z$ with $r = \tfrac14 a^2 + \tfrac12 a' - b$.
Example: for $r = 1$, the solution $y = e^{x}$ has $\omega = y'/y = 1 \in \C(x)$, so the equation falls in case 1.
Non-example: for the Airy equation, $r = x$, the algorithm finds no candidate in cases 1 to 3, so $G = \mathrm{SL}_2(\C)$ and no nonzero solution is Liouvillian.
In case 3 the group modulo $\pm 1$ is the tetrahedral, octahedral or icosahedral group, and these correspond to the degrees $4$, $6$ and $12$.
Jerald Kovacic published the algorithm in 1986.
</document>

<document source="https://tamnd.github.io/math2/node/MMHWS2TC2A.md" tag="MMHWS2TC2A" kind="problem">
# Danzer set

## Problem

Let $d \ge 2$.
A set $D \subseteq \R^d$ is a *Danzer set* if it meets every [[convex body|437FSG3419]] in $\R^d$ of volume $1$, where volume is [[Lebesgue measure|2Z2QVWZ262]].
Is there a Danzer set $D$ of bounded density?
This means that there is a constant $C$ such that every ball of radius $r \ge 1$ contains at most $C r^d$ points of $D$.

## Remarks

Ludwig Danzer asked the question; it is listed as a problem of Danzer in the proceedings of a convexity colloquium held in Copenhagen in 1965.
The question is open.
Bambah and Woods showed in 1971 that a finite union of lattices is never a Danzer set.
They also constructed Danzer sets with at most $C r^d \log^{d-1} r$ points in a ball of radius $r$, by overlaying rectangular grids of different shapes.
Solomon and Weiss gave a construction with $O(r^d \log r)$ points, and they showed that the vertex sets of substitution tilings, such as Penrose tilings, are not Danzer sets.

## Explanations

### L1 human

Can one scatter points in the plane, not too densely, so that every convex shape of area one contains at least one of them?
Very long and thin shapes are the difficulty: they can slip between the points of any grid.
Adding a little more than a constant density of points is known to be enough.
Whether bounded density suffices is open.
</document>

<document source="https://tamnd.github.io/math2/node/MNAWWP9J1X.md" tag="MNAWWP9J1X" kind="statement">
# Density of the graph of a discontinuous additive function

## Statement

Let $f \colon \R \to \R$ be an [[additive function|KF719SZD5T]], that is, $f(x + y) = f(x) + f(y)$ for all $x, y \in \R$.
Assume that there is no constant $c$ with $f(x) = cx$ for all $x \in \R$.
Then the [[graph|0JBJYT69ZV]] $\set{(x, f(x)) : x \in \R}$ is a [[dense subset|0BSJSX7ME2]] of the plane $\R^2$.

## Remarks

Additivity gives $f(qx) = q f(x)$ for all rational $q$ and real $x$.
Since $f$ is not linear, there are nonzero $x_1, x_2$ with $f(x_1)/x_1 \ne f(x_2)/x_2$.
Then the vectors $v_1 = (x_1, f(x_1))$ and $v_2 = (x_2, f(x_2))$ are linearly independent in $\R^2$.
The graph contains every point $q_1 v_1 + q_2 v_2$ with $q_1, q_2 \in \Q$, because $f(q_1 x_1 + q_2 x_2) = q_1 f(x_1) + q_2 f(x_2)$.
These points are dense in $\R^2$, since $v_1, v_2$ form a basis of $\R^2$ and $\Q$ is dense in $\R$.
A typical instance is the [[coordinate function of a Hamel basis|FFTNS66XG6]].

## Explanations

### L1 human

A line through the origin is a thin set in the plane.
A non-linear additive function is the opposite: its graph comes arbitrarily close to every point of the plane.
So such a function is as far from a drawable curve as possible.

### L2 human

This proposition explains why any mild regularity makes an additive function linear.
A dense graph meets every small rectangle, so the function is unbounded above and below on every interval.
Hence boundedness on one interval, as in [[Darboux's theorem|2F30ZHN1RT]], already rules out the non-linear case.
The proof only uses that $f$ is $\Q$-linear, which every additive function on $\R$ is.
</document>

<document source="https://tamnd.github.io/math2/node/MNJ030R4CA.md" tag="MNJ030R4CA" kind="statement">
# Blaschke–Santaló inequality

## Statement

Let $n \ge 1$ and let $K \subseteq \R^n$ be a compact [[convex body|437FSG3419]] with nonempty interior that is symmetric about the origin, that is, $K = -K$.
Let $K^\circ = \set{y \in \R^n : x \cdot y \le 1 \text{ for all } x \in K}$ be its polar body, where $x \cdot y$ is the [[dot product|0S52TNNTH9]].
Let $\operatorname{vol}$ be [[Lebesgue measure|2Z2QVWZ262]] and $\kappa_n$ the volume of the closed unit ball $B$ of $\R^n$.
Then
$$\operatorname{vol}(K) \operatorname{vol}(K^\circ) \le \kappa_n^2,$$
with equality if and only if $K$ is an ellipsoid centered at the origin.

## Remarks

The product $\operatorname{vol}(K) \operatorname{vol}(K^\circ)$ is the Mahler volume of $K$; it does not change when $K$ is replaced by $AK$ for an invertible linear map $A$, because $(AK)^\circ = (A^T)^{-1} K^\circ$.
A standard proof uses Steiner symmetrization: symmetrizing $K$ in a hyperplane does not decrease the Mahler volume, and suitable sequences of symmetrizations converge to a ball.
The inequality holds without symmetry if the origin is replaced by the Santaló point of $K$, the unique interior point $z$ that minimizes $\operatorname{vol}((K - z)^\circ)$.

Example: $K = [-1, 1]^2$ has $K^\circ = \set{(y_1, y_2) : \abs{y_1} + \abs{y_2} \le 1}$, of area $2$.
So the Mahler volume is $4 \cdot 2 = 8$, which is less than $\pi^2 \approx 9.87$.

Some centering is needed: for $K = [-1, 1]^2 - (1 - \varepsilon, 0)$, the origin is at distance $\varepsilon$ from the boundary, and $\operatorname{vol}(K^\circ) \to \infty$ as $\varepsilon \to 0$.

## Explanations

### L1 human

The polar body $K^\circ$ is a kind of reciprocal of $K$: when $K$ gets bigger, $K^\circ$ gets smaller.
The product of their volumes is unchanged by stretching.
The Blaschke–Santaló inequality says that this product is largest for balls and ellipsoids.

### L2 human

The inequality is a basic tool of asymptotic convex geometry and has functional forms for log-concave functions.
The reverse question, how small $\operatorname{vol}(K) \operatorname{vol}(K^\circ)$ can be, is the [[Mahler conjecture|FFZHNFJCZS]].
For symmetric $K$, the Bourgain–Milman inequality gives a lower bound $c^n \kappa_n^2$ with an absolute constant $c > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/MNMJC8YVBS.md" tag="MNMJC8YVBS" kind="concept">
# Tightness of measures

## Summary

A family of [[probability measures|059TJN0NZ1]] on a topological space is tight if all of them live on one [[compact set|0R6EWFKRCF]] up to a small mass.
The mass left out can be made as small as one likes.
Tightness rules out mass escaping to infinity.
By [[Prokhorov's theorem|D6FMEQYA76]], on a [[Polish space|584GXXY73M]] a family is tight if and only if every sequence in it has a [[weakly convergent|MBW9JCGZC1]] subsequence.
So proofs of [[convergence in distribution|0SRYJGXKR4]] often have two steps: prove tightness, then identify the limit.

## Explanations

### L1 human

Think of each measure as a pile of sand of total mass $1$ spread on the line.
The family is tight if one fixed bounded interval holds almost all the sand of every pile, say all but $1\%$, and the same works for any small percentage.
The normal laws with means between $-1$ and $1$ form a tight family.
The point masses at $1, 2, 3, \dots$ do not: the sand runs off to infinity, and no single interval catches it.
</document>

<document source="https://tamnd.github.io/math2/node/MNP2HBWB5Q.md" tag="MNP2HBWB5Q" kind="concept">
# Group algebra

## Summary

The group algebra $F[G]$ of a [[group|G5S1M9]] $G$ over a field $F$ is the vector space with basis $G$.
It is a [[ring|0B1261B5ED]], with the group multiplication extended linearly.
Its modules are exactly the [[representations|WYGRXCG2RN]] of $G$ over $F$.
So representation theory of groups becomes module theory of a single ring.
Over a commutative ring $R$ in place of $F$, the same construction gives the group ring $R[G]$.

## Explanations

### L1 human

Take formal sums like $2g + 3h - k$ of group elements with coefficients in a field.
Add them term by term, and multiply them by expanding and using the group law, so $(ag)(bh) = ab \, (gh)$.
For the cyclic group $\set{1, t}$ of order $2$, the product $(1 + t)(1 - t)$ equals $1 - t^2 = 1 - 1 = 0$.
This shows the group algebra can have zero divisors even though the field has none.
</document>

<document source="https://tamnd.github.io/math2/node/MNPWBBNCAD.md" tag="MNPWBBNCAD" kind="definition">
# Large deviations theory

## Definition

Let $\mathcal{X}$ be a [[topological space|0CQHFPGJ45]] with its Borel $\sigma$-algebra.
Let $(\mu_n)_{n \ge 1}$ be a sequence of [[probability measures|059TJN0NZ1]] on $\mathcal{X}$, and let $I \colon \mathcal{X} \to [0, \infty]$ be a [[rate function|140N2PQSTE]].
For $A \subseteq \mathcal{X}$, write $I(A) = \inf_{x \in A} I(x)$, with $I(\emptyset) = \infty$.
The sequence $(\mu_n)$ satisfies the *large deviation principle* with rate function $I$ if both of the following hold.

1. For every [[closed set|0ZCK8DQYN1]] $F \subseteq \mathcal{X}$, $\displaystyle \limsup_{n \to \infty} \frac{1}{n} \log \mu_n(F) \le -I(F)$.
2. For every [[open set|02BH3MCBQB]] $G \subseteq \mathcal{X}$, $\displaystyle \liminf_{n \to \infty} \frac{1}{n} \log \mu_n(G) \ge -I(G)$.

## Remarks

Here $\log 0 = -\infty$.
A sequence of random elements $Z_n$ of $\mathcal{X}$ satisfies the principle when their laws do.
The factor $1/n$ is the speed; some applications use another speed $1/\varepsilon$ or $1/a_n$ with $a_n \to \infty$.
If $A$ is a Borel set whose interior $A^\circ$ and closure $\bar A$ satisfy $I(A^\circ) = I(\bar A)$, then $\frac{1}{n} \log \mu_n(A) \to -I(A)$.

Example: let $\mu_n$ be the normal law with mean $0$ and variance $1/n$ on $\R$, and let $I(x) = x^2/2$.
For an interval $[a, \infty)$ with $a > 0$, the Gaussian tail estimate gives $\frac{1}{n} \log \mu_n([a, \infty)) \to -a^2/2 = -I([a, \infty))$.
The same tail estimate gives both bounds for all closed and open sets, so $(\mu_n)$ satisfies the principle with this $I$.

Non-example: the same sequence does not satisfy the principle with $I(x) = x^2$.
The upper bound fails for the closed set $F = [1, \infty)$.
Indeed $\frac{1}{n} \log \mu_n(F) \to -1/2$, while $-I(F) = -1 < -1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/MNQXM3PX4R.md" tag="MNQXM3PX4R" kind="example">
# Rössler attractor

## Example

Let $a, b, c$ be real parameters.
The Rössler system is the [[autonomous system|0673D85QKZ]] in $\R^3$
$$x' = -y - z, \qquad y' = x + a y, \qquad z' = b + z (x - c).$$
Otto Rössler proposed it in 1976 with $a = b = 0.2$ and $c = 5.7$.
For these values, numerical solutions approach a bounded set with a folded band shape, the Rössler attractor.

The [[equilibrium points|0G4X36AE0X]] are found as follows.
The equation $y' = 0$ gives $x = -a y$, and $x' = 0$ gives $z = -y$.
Then $z' = 0$ becomes $a y^2 + c y + b = 0$.
So for $a \ne 0$ and $c^2 > 4ab$ there are exactly two equilibria, with $y = \left(-c \pm \sqrt{c^2 - 4ab}\right)/(2a)$.

## Remarks

The only nonlinear term is $z x$, so the system is simpler than the Lorenz system, which has two.
In the $(x, y)$-plane, the first two equations give a spiral that moves outward when $0 < a < 2$.
When $x$ exceeds $c$, the variable $z$ grows fast and folds the solution back toward the centre.
This stretching and folding is the mechanism of the chaotic behaviour seen in simulations.

## Explanations

### L1 human

The Rössler system is a three-dimensional flow built to show chaos with as few terms as possible.
A solution spirals outward in a plane, then is lifted up and thrown back to the middle.
Repeating this stretch and fold mixes nearby solutions, like kneading dough.
</document>

<document source="https://tamnd.github.io/math2/node/MNSSV4QHD1.md" tag="MNSSV4QHD1" kind="statement">
# Fundamental theorem of the calculus of finite differences

## Statement

Let $G$ be an additive abelian group, let $F \colon \Z \to G$ be a function, and let $\Delta F(k) = F(k + 1) - F(k)$ be its [[forward difference|TXTR68V4R1]].
Then for all integers $a \le b$,
$$\sum_{k=a}^{b-1} \Delta F(k) = F(b) - F(a).$$
Equivalently, if $F$ is an [[indefinite sum|5VK9NZM0R7]] of $f \colon \Z \to G$, then $\sum_{k=a}^{b-1} f(k) = F(b) - F(a)$.

## Remarks

The proof is a telescoping sum: $\sum_{k=a}^{b-1} (F(k+1) - F(k))$ cancels to $F(b) - F(a)$, as in a [[telescoping series|0GXKM1R7E4]].
For $a = b$ both sides are $0$.
Example: $\Delta (k)_3 = 3 (k)_2$ for the falling factorial $(k)_3 = k(k-1)(k-2)$, so $\sum_{k=0}^{n-1} k(k-1) = \frac{n(n-1)(n-2)}{3}$.
A related identity, with $\Delta$ of step $h$ on any additive commutative monoid, is $\sum_{k=0}^{n-1} f(x + kh) = \sum_{j=0}^{n-1} \binom{n}{j+1} \Delta_h^j f(x)$, obtained by summing [[Newton's forward difference formula|DDHCMKHD2T]].

## Explanations

### L1 human

Adding up all the steps of a function gives its total change.
This is the discrete version of $\int_a^b F'(x)\,dx = F(b) - F(a)$.
To sum $f(a) + \dots + f(b-1)$, it is enough to find one function $F$ whose steps are the values of $f$.

### L2 human

The theorem reduces the evaluation of finite sums to finding indefinite sums, just as the [[fundamental theorem of calculus|0NHGY0N0JB]] reduces integrals to antiderivatives.
A table of differences, such as $\Delta (k)_m = m (k)_{m-1}$ for [[falling factorials|0SJNCJCCQG]] and $\Delta c^k = (c - 1) c^k$, becomes a table of sums.
To sum $k^2$, write $k^2 = (k)_2 + (k)_1$, which gives $\sum_{k=0}^{n-1} k^2 = \frac{(n)_3}{3} + \frac{(n)_2}{2}$.
[[Summation by parts|9G82Z17DKA]] is the discrete analogue of integration by parts and is proved from the same telescoping.
</document>

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# Simpson's rule

## Summary

Simpson's rule approximates $\int_a^b f$ by integrating the parabola through the values of $f$ at $a$, at the midpoint, and at $b$, which gives $\frac{b - a}{6}\big(f(a) + 4 f(m) + f(b)\big)$.
It is the closed [[Newton–Cotes formula|BQPQAY1FDJ]] with three nodes, and it is exact for all cubic polynomials.
Its composite form has error of order $h^4$ for smooth integrands, which makes it a standard simple method.

## Explanations

### L1 human

Instead of joining samples by straight lines, Simpson's rule fits a parabola through three equally spaced samples and takes the area under it.
The middle sample gets weight four times that of the ends, because the parabola's area depends most on the height in the middle.
For $f(x) = x^3$ on $[0, 1]$ it gives $\frac{1}{6}\big(0 + 4 \cdot \frac{1}{8} + 1\big) = \frac{1}{4}$, exactly the true value.
Halving the step size reduces the error by a factor of about sixteen.
</document>

<document source="https://tamnd.github.io/math2/node/MNXKTXJ35G.md" tag="MNXKTXJ35G" kind="concept">
# Helmholtz equation

## Summary

The Helmholtz equation is the equation $\Delta u + k^2 u = 0$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $k$ is a constant called the wave number.
It describes waves of a single frequency, and it arises when the time variable is separated in the [[wave equation|5EHD2XBQDG]].
It appears in acoustics, electromagnetism, quantum mechanics and scattering theory.
For $k = 0$ it becomes Laplace's equation.

## Explanations

### L1 human

Think of a drum or an organ pipe that vibrates at one fixed pitch.
At each point the motion goes up and down in time like $\cos(\omega t)$, and only the amplitude $u(x)$ changes from place to place.
The amplitude then satisfies the Helmholtz equation, with $k$ proportional to the frequency.
On a line, $u(x) = \sin(kx)$ is a solution: its second derivative is $-k^2 \sin(kx)$, which cancels $k^2 u$.
</document>

<document source="https://tamnd.github.io/math2/node/MNY9YSTZAY.md" tag="MNY9YSTZAY" kind="definition">
# Cluster analysis

## Definition

Let $x_1, \ldots, x_n$ be data points and let $d(x_i, x_j) \ge 0$ be a dissimilarity with $d(x_i, x_i) = 0$ and $d(x_i, x_j) = d(x_j, x_i)$.
A (hard) *clustering* of the data into $K$ clusters is a [[partition|08QMKTWF9T]] of $\set{1, \ldots, n}$ into $K$ non-empty sets $C_1, \ldots, C_K$.
*Cluster analysis* is the task of choosing a clustering in which dissimilarities within clusters are small and dissimilarities between clusters are large.
A method of cluster analysis makes this precise.
It may minimise an objective function over all clusterings, or follow a rule that merges or splits clusters.
A *hierarchical clustering* is a nested sequence of clusterings, from $n$ singletons to one cluster.

## Remarks

Example: the points $0, 1, 10, 11$ on the real line with $d(x, y) = \abs{x - y}$ have the natural clustering $\set{0, 1}$, $\set{10, 11}$: within-cluster distances are $1$ and between-cluster distances are at least $9$.
Non-example: the sets $\set{0, 1}$ and $\set{1, 10, 11}$ are not a clustering, because the point $1$ lies in two sets, so they do not form a partition.
Soft or fuzzy clusterings relax the partition to membership weights in $[0, 1]$ that add up to $1$ for each point.
Single linkage merges at each step the two clusters with the smallest minimum distance; complete linkage uses the maximum distance instead.
</document>

<document source="https://tamnd.github.io/math2/node/MNZPMSSPD8.md" tag="MNZPMSSPD8" kind="definition">
# Trigonometric series

## Definition

Let $(c_n)_{n \in \Z}$ be a sequence of complex numbers.
The *trigonometric series* with coefficients $c_n$ is the formal [[series|0NM626FH6Q]] of functions of $x \in \R$
$$\sum_{n=-\infty}^{\infty} c_n e^{inx},$$
where $e^{inx}$ is the [[complex exponential|07GJV36896]].
Its $N$-th partial sum is the [[trigonometric polynomial|J3NJQ5FZJ5]] $S_N(x) = \sum_{n=-N}^{N} c_n e^{inx}$.
The series *converges* at $x$ when $\lim_{N \to \infty} S_N(x)$ exists.

## Remarks

In real form the series is $\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx)$, with $a_n = c_n + c_{-n}$ and $b_n = i(c_n - c_{-n})$, and its partial sums are the same.
Example: $\sum_{n=1}^{\infty} \frac{\cos nx}{n^2}$ is a trigonometric series with $c_0 = 0$ and $c_n = c_{-n} = \frac{1}{2n^2}$ for $n \ge 1$; it converges at every $x$ by comparison with $\sum 1/n^2$.
Non-example: the power series $\sum_{n \ge 0} x^n$ is a series of functions, but its terms are powers of $x$ and not multiples of $e^{inx}$, so it is not a trigonometric series.
A trigonometric series is the [[Fourier series|G12KXDHPPS]] of an integrable function only for special coefficients.
For instance, $\sum_{n \ge 2} \frac{\sin nx}{\log n}$ converges at every $x$ but is not the Fourier series of a Lebesgue integrable function.
</document>

<document source="https://tamnd.github.io/math2/node/MP3F49QT3C.md" tag="MP3F49QT3C" kind="problem">
# Group isomorphism problem

## Problem

The *group isomorphism problem* asks for an algorithm that takes two finite [[presentations|0JGEQXQ4AC]] of groups and decides whether the groups they define are [[isomorphic|08MG7H8JW5]].
The problem may also be posed inside a class of [[finitely presented groups|TJKANTKASZ]]: the input is two presentations of groups of the class.

## Remarks

The problem is unsolvable in general.
By the Adian–Rabin theorem, there is no algorithm that decides whether a finite presentation defines the trivial group.
Since the trivial group has the finite presentation $\langle \ \mid \ \rangle$, no algorithm decides isomorphism of finitely presented groups.

The problem is solvable for some classes.
For finitely generated abelian groups given by finite presentations, the Smith normal form of the relation matrix gives the invariant factors.
Two such groups are isomorphic exactly when the invariants agree.
It is also solvable for [[hyperbolic groups|JWJ8YPR7EB]].
It is one of the three decision problems, with the [[word problem|PBGTTAVMMT]] and the [[conjugacy problem|4AF7AF7690]], that Max Dehn posed.

For finite groups given by their multiplication tables the question is always decidable, and the interest is in the running time; this is a different, computational problem.

## Explanations

### L1 human

The same group can be written with very different generators and rules.
For example, $\langle a, b \mid a^2, b^3, a b a^{-1} b^{-1} \rangle$ and $\langle c \mid c^6 \rangle$ both define the cyclic group of order $6$, with $c = ab$.
The isomorphism problem asks whether a computer can always tell when two descriptions define the same group.
In general it cannot: no program can even decide whether a given description defines the group with one element.
</document>

<document source="https://tamnd.github.io/math2/node/MP5C9T2EYG.md" tag="MP5C9T2EYG" kind="statement">
# Galois group of a finite field extension

## Statement

Let $q$ be a prime power, let $n \ge 1$, and let $\F_q \subseteq \F_{q^n}$ be [[finite fields|08RTDHX1SK]] of orders $q$ and $q^n$.
Then $\F_{q^n}$ is a [[Galois extension|0DS2F7GK05]] of $\F_q$.
Its [[Galois group|0XVR72M51M]] is [[cyclic|0B85Q41RP6]] of order $n$, generated by the map $\varphi(x) = x^q$.

## Remarks

The map $\varphi$ is a power of the [[Frobenius automorphism|0VP4TBVJ19]], so it is a field automorphism of $\F_{q^n}$.
It fixes $\F_q$, because [[every element of $\F_q$ satisfies $a^q = a$|0QWR8WTZEW]].
If $\varphi^k = \id$ with $k \ge 1$, then every element of $\F_{q^n}$ is a root of $x^{q^k} - x$, so $q^n \le q^k$ and $k \ge n$; and $\varphi^n = \id$.
So $\varphi$ has order $n$, and since [[the Galois group has order at most the degree|0QM98DNWVW]] $n$, the group is $\langle \varphi \rangle$ and the extension is Galois.
For example, $\Gal(\F_4/\F_2) = \set{\id, \varphi}$ with $\varphi(x) = x^2$, and $\varphi$ swaps the two roots $\omega$ and $\omega + 1$ of $x^2 + x + 1$.

## Explanations

### L1 human

A finite field has very few symmetries over a smaller finite field.
All of them are powers of one map, raising to the $q$-th power.
So the symmetry group is as simple as possible: a cycle of length $n$.

### L2 human

With the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], the subgroups of a cyclic group of order $n$ match the fields between $\F_q$ and $\F_{q^n}$.
This gives again that these fields are the $\F_{q^m}$ with $m$ dividing $n$, as in [[subfields of a finite field|08YERZDM9X]].
The absolute Galois group of $\F_q$ is the profinite completion $\widehat{\Z}$ of $\Z$, topologically generated by $\varphi$.
The Frobenius element of a prime in algebraic number theory comes from this theorem applied to residue fields.
</document>

<document source="https://tamnd.github.io/math2/node/MP6B64DS31.md" tag="MP6B64DS31" kind="conjecture">
# Weinstein conjecture

## Statement

Let $M$ be a closed smooth manifold of odd dimension.
Let $\alpha$ be a contact form on $M$, so that $(M, \ker \alpha)$ is a [[contact manifold|HBQ8BZ7XB5]].
Then the [[Reeb vector field|GGM63KXCF2]] $R_\alpha$ of $\alpha$ has at least one [[periodic orbit|0AA97XA70K]].
That is, there are a point $x \in M$ and a time $T > 0$ such that the flow of $R_\alpha$ maps $x$ to itself at time $T$.

## Remarks

Weinstein stated the conjecture for hypersurfaces of contact type in symplectic manifolds.
He was motivated by Rabinowitz's theorem on periodic orbits on star-shaped energy surfaces in $\R^{2n}$.
The form above is the standard intrinsic version.
Viterbo proved it for hypersurfaces of contact type in $\R^{2n}$ in 1987.
Hofer proved it for $S^3$ and for closed contact $3$-manifolds that are overtwisted or have $\pi_2 \ne 0$, using pseudoholomorphic curves in the symplectization.
Taubes proved it for all closed $3$-manifolds in 2007, using Seiberg–Witten Floer homology.
In dimension $5$ and higher it is open in general, although it is known in many cases.

## Explanations

### L1 human

On a closed contact manifold, a contact form determines a flow, the Reeb flow.
The conjecture says that this flow always has a closed orbit, a trajectory that comes back to where it started.
For energy surfaces of a mechanical system, this means a periodic motion with that energy.
</document>

<document source="https://tamnd.github.io/math2/node/MP8VSZK19A.md" tag="MP8VSZK19A" kind="concept">
# Percolation threshold

## Summary

The percolation threshold of a graph, also called its critical probability, is a value $p_c$ of the parameter $p$ of [[percolation|BH7YPXCYDK]].
At $p_c$ an infinite open cluster starts to appear.
Below $p_c$ all open clusters are finite almost surely, and above $p_c$ there is an infinite open cluster almost surely.
It depends on the graph and on the choice of bond or site percolation.
Exact values are known only for a few graphs, such as $p_c = 1/2$ for bond percolation on the square lattice.

## Explanations

### L1 human

Open each edge of a large network with probability $p$.
For small $p$ the open pieces are small islands; for large $p$ they join into one huge network.
The threshold is the value of $p$ at which this switch happens.
On a tree in which every vertex has $3$ neighbours, the threshold is $1/2$.
Each step away from the root offers $2$ new edges, so the cluster can grow forever only if $2p$, the mean number of new open edges, exceeds $1$.
</document>

<document source="https://tamnd.github.io/math2/node/MP8X04H1ZD.md" tag="MP8X04H1ZD" kind="definition">
# Function approximation

## Definition

Let $S$ be a set and let $X$ be a [[normed vector space|03J5RAKB30]] of [[functions|0EPAT51DXQ]] on $S$.
Let $A \subseteq X$ be a nonempty subset, the approximating class.
Let $f \in X$ and $\varepsilon > 0$.
An *approximation of $f$ from $A$ within $\varepsilon$* is a function $g \in A$ with $\norm{f - g} \le \varepsilon$.
The function $f$ *can be approximated arbitrarily well from $A$* if such a $g$ exists for every $\varepsilon > 0$.
Equivalently, $f$ lies in the closure of $A$.

## Remarks

Example: take $X = C[0, 1]$ with the supremum norm, $A$ the polynomials of degree at most $2$, and $f(x) = e^x$.
The function $g(x) = 1 + x + x^2/2$ is an approximation within $0.22$.
Indeed $e^x - g(x)$ is increasing on $[0, 1]$, since its derivative $e^x - 1 - x$ is nonnegative, so its maximum is $e - 2.5 < 0.22$.

Non-example: on $[-1, 1]$ with the supremum norm, the sign function cannot be approximated within any $\varepsilon < 1$ by continuous functions.
Let $g$ be continuous with $g(0) = c$.
Near $0$ the gaps $\abs{1 - g(x)}$ for $x > 0$ and $\abs{-1 - g(x)}$ for $x < 0$ come close to $\abs{1 - c}$ and $\abs{1 + c}$.
One of these two numbers is at least $1$.

The choice of norm matters: the sign function can be approximated arbitrarily well by continuous functions in the $L^2$ norm.
</document>

<document source="https://tamnd.github.io/math2/node/MPB3QF09E5.md" tag="MPB3QF09E5" kind="statement">
# Picard's criterion for first-kind equations

## Statement

Let $X$ and $Y$ be [[Hilbert spaces|HM2HHE3J51]] and let $K \colon X \to Y$ be a [[compact linear operator|M2NEP1K5AE]] with [[adjoint|6HSJBJ8TTQ]] $K^*$.
Let $(\sigma_n, v_n, u_n)_{n \in N}$ be a singular system of $K$, with $N$ finite or $N = \set{1, 2, \dots}$.
This means that the $\sigma_n > 0$ are the nonzero [[singular values|QSJKPKPM7H]] of $K$, and $K v_n = \sigma_n u_n$ and $K^* u_n = \sigma_n v_n$.
It also means that $(v_n)$ is an orthonormal basis of $\ker(K)^\perp$ and $(u_n)$ is an orthonormal basis of the closure of $K(X)$.
Let $y \in Y$.
Then the equation $K x = y$ has a solution $x \in X$ if and only if

1. $y \in \ker(K^*)^\perp$, and
2. $\displaystyle \sum_{n \in N} \frac{\abs{\ip{y}{u_n}}^2}{\sigma_n^2} < \infty$.

In this case $x^\dagger = \sum_{n \in N} \sigma_n^{-1} \ip{y}{u_n} v_n$ is a solution, and it is the unique solution of smallest norm.

## Remarks

Proof idea: if $K x = y$, then $y \perp \ker(K^*)$ and $\ip{y}{u_n} = \ip{x}{K^* u_n} = \sigma_n \ip{x}{v_n}$, so the sum in clause 2 equals $\sum_n \abs{\ip{x}{v_n}}^2 \le \norm{x}^2$ by Bessel's inequality.
Conversely, clause 2 makes the series for $x^\dagger$ converge.
Then $K x^\dagger = \sum_n \ip{y}{u_n} u_n$ is the orthogonal projection of $y$ onto the closure of $K(X)$, which is $\ker(K^*)^\perp$.
Clause 1 makes it equal to $y$.
The solution $x^\dagger$ lies in $\ker(K)^\perp$, so it has the smallest norm among all solutions.

Example: the Volterra operator $(V x)(t) = \int_0^t x(s) \, ds$ on $L^2(0, 1)$ is injective with dense range.
Its singular system is $\sigma_n = \frac{2}{(2n - 1)\pi}$, $v_n(s) = \sqrt{2} \cos\big((n - \tfrac{1}{2}) \pi s\big)$ and $u_n(t) = \sqrt{2} \sin\big((n - \tfrac{1}{2}) \pi t\big)$.
So $V x = y$ is solvable if and only if $\sum_n (2n - 1)^2 \abs{\ip{y}{u_n}}^2 < \infty$, a condition on the decay of the coefficients of $y$ that expresses the differentiability of $y$.

## Explanations

### L1 human

A compact operator shrinks the component of $x$ along the $n$-th singular direction by the factor $\sigma_n$, and these factors tend to zero.
To undo it one must divide the components of the data by $\sigma_n$.
Picard's criterion says that the equation can be solved exactly when the result of these divisions still has finite energy.
Data whose components do not decay fast enough, for example noisy data, are not in the range.

### L2 human

The criterion is the infinite-dimensional form of solving a linear system by the singular value decomposition, see [[singular value decomposition|000NPTMX6J]].
It shows the [[ill-posedness of first-kind equations|KHXYWTNN2R]] in a quantitative way: an error of size $\delta$ in the coefficient $\ip{y}{u_n}$ becomes an error $\delta / \sigma_n$ in the solution.
The faster the singular values decay, the smoother the data must be, and the more ill posed the problem.
Regularization methods such as truncated singular value expansion and Tikhonov regularization replace the factors $1/\sigma_n$ by bounded filter factors.
The map $y \mapsto x^\dagger$ is the [[Moore–Penrose inverse|0YY8VX28JR]] of $K$ on its natural domain.
</document>

<document source="https://tamnd.github.io/math2/node/MPE4AWEP6K.md" tag="MPE4AWEP6K" kind="concept">
# Conservative system

## Summary

A conservative system is a measurable dynamical system in which every set of positive measure returns to itself with positive measure.
It is the opposite of a dissipative system, in which some mass escapes and never comes back.
On a finite measure space every [[measure-preserving transformation|556WBDAHNP]] is conservative, by the [[Poincaré recurrence theorem|TVYJPBHD0S]].
The notion matters most for infinite measures and for maps that only respect null sets.
In physics the word refers to systems without friction, such as Hamiltonian systems, whose flows preserve phase-space volume.

## Explanations

### L1 human

Picture a system that moves mass around without creating or destroying it, and without letting it run away for good.
Any blob of positive size that one follows will, at some later step, overlap its starting position again.
A rotation of a circle is conservative: an arc keeps coming back to overlap itself.
The translation $x \mapsto x + 1$ on the real line is not: the interval $[0, 1)$ moves off to the right and never overlaps itself again.
</document>

<document source="https://tamnd.github.io/math2/node/MPFD24TYW2.md" tag="MPFD24TYW2" kind="concept">
# Hyperbolic set

## Summary

A hyperbolic set of a diffeomorphism is a compact invariant set with a splitting of the tangent space at each of its points.
One part is uniformly contracted and the other is uniformly expanded.
It extends the notion of a hyperbolic fixed point from one point to a whole set, possibly a Cantor set or the whole manifold.
Hyperbolic sets have stable and unstable manifolds, the shadowing property and robustness under perturbation.
They are the setting of uniformly hyperbolic dynamics, which includes [[Anosov diffeomorphisms|RCQH8P6DPQ]] and the [[Smale horseshoe|N63Z6EA6YS]].

## Explanations

### L1 human

Picture a map that, near some set of points, squeezes in one direction and stretches in another, at every point of the set and at a uniform rate.
Such a set is hyperbolic.
Nearby points drift apart along the stretched direction, so the motion on a large hyperbolic set can be chaotic, yet the clear geometry makes it possible to analyse.
The simplest case is a single saddle point, such as the origin for the map $(x, y) \mapsto (x/2, 2y)$.
See the [[definition|NY9MCZXY6N]].
</document>

<document source="https://tamnd.github.io/math2/node/MPN10SB26R.md" tag="MPN10SB26R" kind="concept">
# Autoregressive conditional heteroskedasticity

## Summary

An autoregressive conditional heteroskedasticity (ARCH) model is a [[time series|EA3MF4VTEH]] model with a changing variance.
The [[conditional variance|0VGPX25NXQ]] of today's shock depends on the squares of past shocks.
It describes series whose volatility changes over time and comes in clusters, which is typical of financial returns.
The model was introduced by Robert Engle, and its extension GARCH also lets past conditional variances enter the equation.
It is a basic example of [[heteroscedasticity|D8VX6XQ3TT]] in time.

## Explanations

### L1 human

Stock prices move calmly for a while and then wildly for a while.
An ARCH model captures this: after a big move, the next move is likely to be big too, in either direction.
The size of the noise today is computed from how large the recent noises were.
For example, in an ARCH(1) model the variance today is $\alpha_0 + \alpha_1 \varepsilon_{t-1}^2$, so a large surprise yesterday raises the uncertainty today.
</document>

<document source="https://tamnd.github.io/math2/node/MPNGD4V271.md" tag="MPNGD4V271" kind="statement">
# Cartan–Serre finiteness theorem

## Statement

Let $X$ be a [[complex analytic space|HXSZA44J5J]] whose underlying topological space is [[compact|0TVJZYB6SH]], and let $\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.
Then for every integer $q \ge 0$ the [[sheaf cohomology|PQ6Q143ARJ]] group $H^q(X, \mathcal{F})$ is a finite-dimensional complex vector space.

## Remarks

The proof uses two finite coverings of $X$ by open [[Stein spaces|Z4AMJS5KA6]], one relatively compact in the other.
By [[Cartan's theorem B|R7VKRV8AGQ]] both coverings compute $H^q(X, \mathcal{F})$ by Čech cochains, which form Fréchet spaces.
The restriction map from the larger covering to the smaller one is a compact operator that induces an isomorphism on cohomology.
A theorem of Schwartz on compact perturbations of surjective maps between Fréchet spaces then gives finite dimension.

Example: on the projective line $\mathbb{P}^1$ with $\mathcal{F} = \mathcal{O}$, one has $H^0 = \C$, the constants, and $H^1 = 0$.

Compactness is necessary.
On $X = \C$, the space $H^0(\C, \mathcal{O})$ of entire functions is infinite-dimensional, since it contains $1, z, z^2, \dots$.

## Explanations

### L1 human

On a compact complex space, the spaces of global holomorphic sections of a coherent sheaf, and the higher cohomology groups that measure obstructions, are all finite-dimensional.
So each of them is described by a single number, its dimension.
Compactness is what keeps these spaces small.

### L2 human

The theorem is the source of the numerical invariants of compact complex spaces, such as the dimensions of spaces of sections of line bundles and the arithmetic genus.
It makes the Euler characteristic $\sum_q (-1)^q \dim H^q(X, \mathcal{F})$ of a coherent sheaf well defined on a compact complex space, the quantity computed by the Riemann–Roch theorem.
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<document source="https://tamnd.github.io/math2/node/MPR6MH4YSQ.md" tag="MPR6MH4YSQ" kind="concept">
# Height of a poset

## Summary

The height of a [[partially ordered set|0QV2JS9RJ3]] is the largest number of elements of a [[chain|0S8FE7ZMFM]] in it.
It is the number of elements in the longest sequence of elements each strictly above the previous one.
By [[Mirsky's theorem|5VPXR3RDF1]], for a finite poset it is also the least number of antichains that cover the poset.

## Explanations

### L1 human

Draw a poset with smaller elements lower down.
The height counts the elements on the tallest path that only goes upward.
In the divisors of $12$ ordered by divisibility, the path $1, 2, 4, 12$ has $4$ elements, and no chain is longer, so the height is $4$.
An antichain has height $1$.
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<document source="https://tamnd.github.io/math2/node/MPRCFT30Z7.md" tag="MPRCFT30Z7" kind="definition">
# Oscillatory solution of a difference equation

## Definition

Let $(x_n)_{n \ge n_0}$ be a [[real sequence|04CC6K5VDT]] that is a solution of a difference equation and is not identically zero.
The solution is *oscillatory* if for every integer $N \ge n_0$ there is an integer $n \ge N$ with
$$x_n \, x_{n+1} \le 0.$$
Otherwise it is *nonoscillatory*, that is, it is eventually positive or eventually negative.

## Remarks

Example: $x_n = (-1)^n$ solves $x_{n+1} = -x_n$.
For every $n$ one has $x_n x_{n+1} = -1 \le 0$, so the solution is oscillatory.

Non-example: $x_n = 2^{-n}$ solves $x_{n+1} = x_n / 2$.
Here $x_n x_{n+1} > 0$ for all $n$, so the condition fails already for $N = n_0$, and the solution is nonoscillatory.

One also speaks of oscillation about a value $x^*$, often an equilibrium, meaning that $(x_n - x^*)$ is oscillatory.
For the linear equation $x_{n+1} = a x_n$ with $a \ne 0$, every nonzero solution is oscillatory exactly when $a < 0$.
</document>

<document source="https://tamnd.github.io/math2/node/MPSZJ3E38R.md" tag="MPSZJ3E38R" kind="definition">
# Perron method

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]] with [[boundary|05D5EGMFYV]] $\partial\Omega$, and let $f \colon \partial\Omega \to \R$ be a bounded function.
The *Perron family* of $f$ is the set $\mathcal{S}_f$ of all [[subharmonic functions|Z1MHMGFNFW]] $v$ on $\Omega$ such that
$$\limsup_{x \to \xi, \, x \in \Omega} v(x) \le f(\xi) \quad \text{for every } \xi \in \partial\Omega.$$
The *Perron solution* of $f$ is the function $H_f \colon \Omega \to \R$ given by the [[supremum|0BTMQ04556]]
$$H_f(x) = \sup_{v \in \mathcal{S}_f} v(x).$$
The *Perron method* is the construction of $H_f$ as a candidate solution of the [[Dirichlet problem|8D56GQ4AG8]] with boundary data $f$.

## Remarks

The family $\mathcal{S}_f$ is not empty, since it contains the constant $\inf f$, and each $v \in \mathcal{S}_f$ is at most $\sup f$ by the maximum principle; so $H_f$ is finite.
Perron's theorem says that $H_f$ is harmonic on $\Omega$.
Its proof replaces a function of the family by its Poisson integral on a small ball, which keeps it in the family and makes it larger.
Example: on the unit disc with $f(\xi) = \xi_1$, the function $v(x) = x_1$ belongs to $\mathcal{S}_f$, and the maximum principle gives $v \le x_1$ for every $v \in \mathcal{S}_f$, so $H_f(x) = x_1$.
Non-example: $v(x) = x_1 + \frac12$ is subharmonic but not in $\mathcal{S}_f$, since its boundary values exceed $f$ by $\frac12$.
A boundary point $\xi$ at which $H_f(x) \to f(\xi)$ for every continuous $f$ is called a regular boundary point.
</document>

<document source="https://tamnd.github.io/math2/node/MPYQ39P01F.md" tag="MPYQ39P01F" kind="statement">
# Pólya enumeration theorem

## Statement

Let $G$ be a finite group that [[acts|0WCY42EV9V]] on a finite set $X$ with $\abs{X} = d$.
For $g \in G$ and $1 \le j \le d$, let $c_j(g)$ be the number of [[cycles|0YWDAZPRPQ]] of length $j$ of the permutation $x \mapsto g x$ of $X$.
The *cycle index* of the action is the polynomial
$$Z_G(t_1, \ldots, t_d) = \frac{1}{\abs{G}} \sum_{g \in G} t_1^{c_1(g)} t_2^{c_2(g)} \cdots t_d^{c_d(g)}.$$
Let $C$ be a finite set of colours, and let $G$ act on the set $C^X$ of maps $f \colon X \to C$ by $(g f)(x) = f(g^{-1} x)$.

1. The number of [[orbits|0YQPZB5Q1G]] of $G$ on $C^X$ is $Z_G(m, m, \ldots, m)$, where $m = \abs{C}$.
2. Treat the colours $c \in C$ as commuting variables, and give $f$ the weight $\prod_{x \in X} f(x)$, which is constant on orbits. Then the sum of the weights of the orbits is $Z_G(p_1, p_2, \ldots, p_d)$ with $p_j = \sum_{c \in C} c^j$.

## Remarks

Proof idea: by [[Burnside's lemma|0Z47M1B2VS]], the number of orbits is the average number of colourings fixed by an element $g$.
A colouring is fixed by $g$ exactly when it is constant on each cycle of $g$, so $g$ fixes $m^{c_1(g) + \cdots + c_d(g)}$ colourings.
The weighted form follows from a weighted Burnside lemma in the same way, since a cycle of length $j$ with colour $c$ contributes $c^j$.

Example: colour the $4$ beads of a necklace black or white, up to rotation.
The rotation group $C_4$ has cycle index $\frac{1}{4}(t_1^4 + t_2^2 + 2 t_4)$.
With $m = 2$ this gives $\frac{1}{4}(16 + 4 + 4) = 6$ necklaces.
With colours $b$ and $w$, the weighted form gives $b^4 + b^3 w + 2 b^2 w^2 + b w^3 + w^4$, so $2$ of the necklaces have two beads of each colour.

The group acts on the positions $X$, not on the colours.
If one also allows permutations of the colours, the count changes, and the theorem has to be replaced by a version for the product group.

## Explanations

### L1 human

Pólya's theorem counts colourings when two colourings that differ by a symmetry count as the same.
Typical examples are necklaces up to rotation, or colourings of the faces of a cube up to turning the cube.
One first records how each symmetry moves the positions, in the cycle index.
Then one substitutes the number of colours, or the colours themselves, and reads off the answer.

### L2 human

The theorem refines [[Burnside's lemma|0Z47M1B2VS]] by keeping track of how many times each colour is used.
It counts unlabelled structures, such as isomorphism classes of graphs or chemical isomers, by letting the symmetric group act on positions.
In the [[symbolic method|JP11D90EJC]], the multiset and cycle constructions for unlabelled classes are instances of the same idea.
The cycle index of the symmetric group $S_d$ acting on $d$ points is $\sum_{\lambda} p_\lambda / z_\lambda$, a sum over the partitions $\lambda$ of $d$.
Here $p_\lambda = \prod_j t_j^{m_j}$ and $z_\lambda = \prod_j j^{m_j} m_j!$, where $m_j$ is the number of parts of $\lambda$ equal to $j$.
With $t_j$ read as power sums, this links the theorem to [[symmetric functions|PAJEHWXW6M]].
</document>

<document source="https://tamnd.github.io/math2/node/MQ9A7TZEQ0.md" tag="MQ9A7TZEQ0" kind="definition">
# Simple Lie group

## Definition

Let $G$ be a connected real or complex [[Lie group|7JBRGCVJRB]], and let $\mathfrak{g}$ be its Lie algebra.
The group $G$ is a *simple Lie group* if $\mathfrak{g}$ is a simple Lie algebra, that is:

1. $\mathfrak{g}$ is not abelian, and
2. the only ideals of $\mathfrak{g}$ are $0$ and $\mathfrak{g}$.

## Remarks

Example: $G = SU(2)$, with Lie algebra $\mathfrak{su}(2) \cong \R^3$ and the cross product as bracket.
The bracket is not identically zero, so clause 1 holds.
If an ideal $I$ contains $x \ne 0$, it contains $x \times y$ for all $y$, and these vectors fill the plane $x^\perp$.
The cross product of two independent vectors of that plane is a nonzero multiple of $x$, so $I = \R^3$, and clause 2 holds.

Non-example: $G = U(n)$ with $n \ge 2$.
Its Lie algebra $\mathfrak{u}(n)$ of skew-Hermitian matrices contains the ideal $i\R\, I_n$ of scalar matrices, which is neither $0$ nor $\mathfrak{u}(n)$, so clause 2 fails.
For $n = 1$ the group $U(1)$ is abelian, so clause 1 fails.

For connected $G$, the connected normal Lie subgroups of $G$ correspond to the ideals of $\mathfrak{g}$.
So $G$ is simple exactly when it is not abelian and has no connected normal Lie subgroups other than $\set{e}$ and $G$.
A simple Lie group may have a nontrivial discrete centre, so it need not be a simple group in the sense of abstract group theory.
Some authors also require the centre to be trivial or finite.
</document>

<document source="https://tamnd.github.io/math2/node/MQA1VVCJ63.md" tag="MQA1VVCJ63" kind="statement">
# Bernstein–Doetsch theorem

## Statement

Let $I \subseteq \R$ be an open [[interval|0Q1D85B8G6]], and let $f \colon I \to \R$ be [[midpoint convex|1TQ1JQXEWP]].
Assume that $f$ is bounded above on some nonempty open interval $J \subseteq I$.
Then $f$ is [[continuous|0X3WGX95FE]] on $I$, and $f$ is [[convex|07Q7CNM5WF]].

## Remarks

The idea of the proof has three steps.
First, midpoint convexity and induction give $f(qx + (1 - q)y) \le q f(x) + (1 - q) f(y)$ for all dyadic rationals $q \in [0, 1]$, and this carries the upper bound from $J$ to a neighbourhood of every point of $I$.
Second, an upper bound $M$ near a point $x_0$ gives a lower bound there, from $f(x_0) \le \frac{f(x_0 + h) + f(x_0 - h)}{2}$.
Third, with $f$ locally bounded, the inequality with $q = 2^{-n}$ gives $\abs{f(x_0 + 2^{-n} h) - f(x_0)} \le 2^{-n} C$ for small $h$, which yields continuity; a continuous midpoint convex function is convex.

The boundedness hypothesis cannot be dropped.
A discontinuous additive function $a \colon \R \to \R$, built from a Hamel basis, is midpoint convex but not continuous, and it is unbounded above on every interval.
Example of use: a midpoint convex function on $(0, 1)$ with values in $[0, 1]$ is automatically convex.
The theorem holds in the same form for midpoint convex functions on an open convex subset of $\R^n$.

## Explanations

### L1 human

Midpoint convexity is a weak form of convexity that only looks at midpoints.
On its own it allows very wild functions.
The theorem says that the wildness disappears as soon as the function is bounded above on one small interval: then the function is continuous and truly convex.

### L2 human

The theorem is the convexity analogue of the classical results on Cauchy's equation, where an additive function bounded on an interval is linear.
It reduces many regularity questions to finding an upper bound on some interval.
For instance, Sierpiński's theorem obtains such a bound for measurable functions, and so shows that measurable midpoint convex functions are convex.
</document>

<document source="https://tamnd.github.io/math2/node/MQCBFACHXH.md" tag="MQCBFACHXH" kind="statement">
# Genus–degree formula

## Statement

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $C \subset \mathbb{P}^2_k$ be a smooth [[plane curve|EWS4WFTA7W]] of degree $d \ge 1$ in the [[projective plane|ZT9A6JK8QF]].
Then the [[genus|6W8TDV9VVN]] of $C$ is
$$g(C) = \frac{(d - 1)(d - 2)}{2}.$$

## Remarks

One proof uses the adjunction formula: the canonical bundle of $C$ is $\mathcal{O}_C(d - 3)$, of degree $d(d - 3)$, and this equals $2g - 2$.
Another uses the exact sequence $0 \to \mathcal{O}_{\mathbb{P}^2}(-d) \to \mathcal{O}_{\mathbb{P}^2} \to \mathcal{O}_C \to 0$.
It gives $\chi(\mathcal{O}_C) = 1 - (d - 1)(d - 2)/2$, so the [[arithmetic genus|H03GKRVBJ1]] of every plane curve of degree $d$, smooth or not, is $(d - 1)(d - 2)/2$.
Example: lines and conics have genus $0$, smooth cubics have genus $1$, and smooth quartics have genus $3$.
Smoothness is needed: the nodal cubic $y^2 z = x^2(x + z)$ has degree $3$ but geometric genus $0$.
For an irreducible plane curve whose singular points $p$ are ordinary with multiplicities $r_p$, the geometric genus is $(d - 1)(d - 2)/2 - \sum_p r_p (r_p - 1)/2$.

## Explanations

### L1 human

For a smooth curve in the plane, the degree alone fixes the genus.
Degree $1$ and $2$ give spheres, degree $3$ gives a torus, and degree $4$ gives a surface with three handles.
The genus grows roughly like half the square of the degree.

### L2 human

The formula shows that not every genus occurs for smooth plane curves: the values are $0, 1, 3, 6, 10, \ldots$.
So a curve of genus $2$ is never a smooth plane curve, although it can be drawn in the plane with singular points.
The formula is a special case of the adjunction formula, and its singular version is the basic tool to compute the genus of a plane curve from its equation.
</document>

<document source="https://tamnd.github.io/math2/node/MR2JWBCZQZ.md" tag="MR2JWBCZQZ" kind="concept">
# Parallelohedron

## Summary

A parallelohedron is a convex polytope whose translated copies fill space face to face, with no rotations allowed.
In the plane the parallelohedra are the parallelograms and the centrally symmetric hexagons.
In space there are five combinatorial types, found by Fedorov: the parallelepiped, the hexagonal prism, the rhombic dodecahedron, the elongated dodecahedron and the truncated octahedron.
The [[Voronoi cell|V1R1R88GV6]] of any lattice is a parallelohedron, and [[Voronoi's conjecture|F03A4MKA5V]] asks whether every parallelohedron arises this way up to an affine map.

## Explanations

### L1 human

Think of tiles that you may slide but never turn.
A parallelohedron is a convex tile of this kind that fills the whole plane or space, with each face of one tile matched exactly with a face of a neighbour.
Bricks in a wall laid in a straight grid are an example, and so are hexagonal floor tiles.
A regular triangle is not one: copies obtained only by sliding all point the same way and leave gaps.
</document>

<document source="https://tamnd.github.io/math2/node/MR4D29XAPB.md" tag="MR4D29XAPB" kind="definition">
# Galerkin method

## Definition

Let $H$ be a real [[Hilbert space|HM2HHE3J51]], and let $B \colon H \times H \to \R$ be a bounded [[bilinear form|0SQZMXB6GP]].
Let $f$ be a bounded linear functional on $H$, an element of the [[continuous dual space|4A82S5FVRT]].
Consider the problem of finding $u \in H$ with $B(u, v) = f(v)$ for all $v \in H$.
Let $V_N$ be a finite-dimensional subspace of $H$.
The *Galerkin method* replaces this problem by the following one: find $u_N \in V_N$ such that
$$B(u_N, v) = f(v) \quad \text{for all } v \in V_N.$$
The element $u_N$ is the *Galerkin approximation* of $u$ in $V_N$.

## Remarks

If $\varphi_1, \ldots, \varphi_N$ is a basis of $V_N$ and $u_N = \sum_j c_j \varphi_j$, the problem is the linear system $\sum_{j=1}^{N} B(\varphi_j, \varphi_i) \, c_j = f(\varphi_i)$ for $i = 1, \ldots, N$.
If $B$ is [[coercive|86KRHS6X22]] with constant $\alpha$ and bounded with constant $M$, the [[Lax–Milgram theorem|85KK79FCN5]] applied on $V_N$ gives a unique $u_N$.
Subtracting the two problems gives Galerkin orthogonality: $B(u - u_N, v) = 0$ for all $v \in V_N$.
From it follows Céa's lemma: $\norm{u - u_N} \le \frac{M}{\alpha} \inf_{v \in V_N} \norm{u - v}$.

Example: for $-u'' = 1$ on $(0, 1)$ with $u(0) = u(1) = 0$, take $H = H^1_0(0, 1)$, $B(u, v) = \int_0^1 u' v' \, dx$, $f(v) = \int_0^1 v \, dx$ and $V_1$ spanned by $\varphi(x) = x(1 - x)$.
Then $B(\varphi, \varphi) = \int_0^1 (1 - 2x)^2 \, dx = 1/3$ and $f(\varphi) = 1/6$, so $u_1 = \frac{1}{2} x(1 - x)$.
This is the exact solution, as Céa's lemma predicts, since the exact solution lies in $V_1$.

Non-example: choosing $u_N \in V_N$ so that $B(u_N, w) = f(w)$ for all $w$ in a different subspace $W_N \ne V_N$ is not the Galerkin method in the sense above.
It is called a Petrov–Galerkin method, and its analysis needs an inf-sup condition instead of coercivity.
</document>

<document source="https://tamnd.github.io/math2/node/MR5FJXR348.md" tag="MR5FJXR348" kind="statement">
# Bounded inverse theorem

## Statement

Let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\R$ or $\C$.
Let $T \colon X \to Y$ be a [[continuous linear map|03TM135G0N]] that is [[bijective|0GG9TVW90M]].
Then the [[inverse|0J2WAR83WP]] $T^{-1} \colon Y \to X$ is continuous, that is, there is a constant $C \ge 0$ with $\norm{T^{-1} y} \le C \norm{y}$ for all $y \in Y$.

## Remarks

The theorem follows from the open mapping theorem: a surjective continuous linear map between Banach spaces sends open sets to open sets.
That theorem is proved with the [[Baire category theorem|0A7KGAYHGV]] applied to the cover $Y = \bigcup_k \overline{T(B(0, k))}$.
For a bijection, openness of $T$ is exactly continuity of $T^{-1}$.

A common application: if a vector space is complete for two norms $\norm{\cdot}_1$ and $\norm{\cdot}_2$ with $\norm{x}_2 \le c \norm{x}_1$, then the two norms are equivalent.
Indeed, the identity map is a continuous bijection from the first norm to the second.
Completeness of $Y$ is necessary.
On $\ell^1$ the identity map from $(\ell^1, \norm{\cdot}_1)$ to $(\ell^1, \norm{\cdot}_\infty)$ is a continuous linear bijection, because $\norm{x}_\infty \le \norm{x}_1$.
Its inverse is not bounded: $x = e_1 + \cdots + e_n$ has $\norm{x}_\infty = 1$ and $\norm{x}_1 = n$.
Here $(\ell^1, \norm{\cdot}_\infty)$ is not complete.

## Explanations

### L1 human

If a continuous linear map between complete normed spaces is one-to-one and onto, then undoing it is also continuous.
So such a map is an isomorphism of Banach spaces, with no extra checks.
Small changes in the output come from small changes in the input.

### L2 human

The theorem is used to prove stability of solutions: if an equation $Tx = y$ has exactly one solution for each $y$, the solution depends continuously on $y$.
It is equivalent, by short arguments, to the open mapping theorem and to the [[closed graph theorem|GXM8BX8YM7]].
Together with the [[uniform boundedness principle|YG0042V8CJ]] these results all rest on the [[Baire category theorem|0A7KGAYHGV]].
</document>

<document source="https://tamnd.github.io/math2/node/MR5HNV0Q07.md" tag="MR5HNV0Q07" kind="statement">
# Baer's criterion

## Statement

Let $R$ be a ring with identity and let $E$ be a left [[$R$-module|0NZ8JA25WK]].
Then $E$ is an [[injective module|85H7PRHDPZ]] if and only if the following holds.
For every left [[ideal|096Z9MP2ME]] $I$ of $R$, every $R$-[[linear map|0NRYHJPQG2]] $f \colon I \to E$ extends to an $R$-linear map $g \colon R \to E$.

## Remarks

The condition says that for every such $f$ there is $e \in E$ with $f(a) = ae$ for all $a \in I$; take $e = g(1)$.
One direction is the definition applied to the inclusion $I \subseteq R$.
For the other, let $M \subseteq N$ and $f \colon M \to E$; by [[Zorn's lemma|04RXTQA5ZA]] there is a maximal extension $(M', f')$ of $f$ to a submodule $M' \subseteq N$.
If $x \in N$ is not in $M'$, then $I = \set{a \in R : ax \in M'}$ is a left ideal, the map $a \mapsto f'(ax)$ extends to $R$ by hypothesis, and this extends $f'$ to $M' + Rx$, a contradiction; so $M' = N$.

Example: over a [[principal ideal domain|0CMD0BCH8T]] $R$, a divisible module $E$ is injective.
A nonzero ideal is $I = (r)$ with $r \ne 0$, and a map $f \colon I \to E$ is fixed by $f(r)$; choose $e$ with $re = f(r)$, then $a \mapsto ae$ extends $f$.
For instance $\Q$ and $\Q/\Z$ are injective $\Z$-modules.

## Explanations

### L1 human

To check that a module is injective, one should extend maps from every submodule of every module, which is a lot of work.
Baer's criterion says that it is enough to extend maps defined on ideals of the ring into the whole ring.
So injectivity only has to be tested against one module, the ring itself.

### L2 human

The criterion is the main tool for proving that a given module is injective.
Over a principal ideal domain it shows that injective means divisible, which gives the injective $\Z$-modules $\Q$ and $\Q/\Z$.
From this one shows that every module embeds in an injective one, first for abelian groups and then for modules over any ring via $\Hom_\Z(R, -)$.
This existence result is what makes injective resolutions, and so right derived functors such as $\operatorname{Ext}$, available.
</document>

<document source="https://tamnd.github.io/math2/node/MR9QCH1PNA.md" tag="MR9QCH1PNA" kind="concept">
# Projective dimension

## Summary

The projective dimension of a module $M$ is the length of a shortest [[projective resolution|HK2AA1YSQ2]] of $M$.
It is $0$ exactly for the nonzero [[projective modules|BFFA03GE3V]], so it measures how far $M$ is from being projective.
Over a Noetherian local ring it is linked to the [[depth|M2RADBFW99]] by the [[Auslander–Buchsbaum formula|SQB74J5E8Q]].

## Explanations

### L1 human

Some modules are free or projective; they need no relations at all.
Others need relations, then relations among relations, and so on.
The projective dimension counts how many steps this takes in the best case, or is infinite if it never stops.
Over $\Z$, the module $\Z/6\Z$ has projective dimension $1$: one relation, $6 = 0$, and nothing more.
</document>

<document source="https://tamnd.github.io/math2/node/MRBGM52H94.md" tag="MRBGM52H94" kind="definition">
# Wiener–Hopf method

## Definition

Consider the [[Wiener–Hopf equation|QYY4NMDY5H]] $\varphi(x) - \int_0^\infty k(x - t)\, \varphi(t)\, dt = f(x)$ for $x \ge 0$, with $k \in L^1(\R)$.
Write $\hat g(\xi) = \int_{\R} g(x)\, e^{i \xi x}\, dx$ for the [[Fourier transform|DPVM34ADZV]].
For this convention, the transform of a function supported in $[0, \infty)$ is [[holomorphic|0NAHXR1FRH]] in the upper half-plane $\operatorname{Im} \xi > 0$.
The transform of a function supported in $(-\infty, 0]$ is holomorphic in the lower half-plane.
The *Wiener–Hopf method* is the following procedure.

1. Extend $\varphi$ and $f$ by $0$ to $x < 0$, and let $\psi(x) = \int_0^\infty k(x - t) \varphi(t)\, dt$ for $x < 0$ and $\psi(x) = 0$ for $x \ge 0$. Then $\varphi - k * \varphi = f - \psi$ on all of $\R$.
2. Take Fourier transforms: $\bigl(1 - \hat k(\xi)\bigr)\, \hat\varphi(\xi) = \hat f(\xi) - \hat\psi(\xi)$, where $\hat\varphi$ and $\hat f$ are upper functions and $\hat\psi$ is a lower function.
3. Factor the symbol as $1 - \hat k = a_- \, a_+$. Here $a_+$ and $1/a_+$ are holomorphic and bounded in the upper half-plane. Also $a_-$ and $1/a_-$ are holomorphic and bounded in the lower half-plane.
4. Split $\hat f / a_- = g_+ + g_-$ into an upper and a lower part, for example with Cauchy integrals.
5. The equation becomes $a_+ \hat\varphi - g_+ = g_- - \hat\psi / a_-$. The left side extends to the upper half-plane and the right side to the lower half-plane, so together they define an entire function. Growth bounds and [[Liouville's theorem|0EHKVRF4J6]] show that it is a polynomial, often $0$.
6. Then $\hat\varphi = (g_+ + P)/a_+$ for that polynomial $P$, and $\varphi$ is recovered by the inverse Fourier transform.

## Remarks

The factorization in step 3 exists when $1 - \hat k$ has no zeros on $\R$ and its winding number about $0$ is zero.
Otherwise rational factors such as $\bigl((\xi - i)/(\xi + i)\bigr)^\nu$ appear, see [[Wiener–Hopf factorization|BYKP1V2AQN]].
The same steps apply to boundary value problems for partial differential equations, where step 1 comes from a transform in one variable along a half-line boundary.

Example: for $k(x) = \frac{1}{2} e^{-\abs{x}}$ one has $\hat k(\xi) = \frac{1}{1 + \xi^2}$, so $1 - \hat k(\xi) = \frac{\xi^2}{\xi^2 + 1} = \frac{\xi}{\xi - i} \cdot \frac{\xi}{\xi + i}$.
The factor $\xi/(\xi + i)$ is holomorphic in the upper half-plane and $\xi/(\xi - i)$ in the lower half-plane.
Both vanish at the real point $0$.
This zero of the symbol on $\R$ allows the solution $1 + x$ of the homogeneous equation, which grows linearly.

Non-example: a direct Fourier transform of the equation, without step 1, does not work, because the integral covers only $t \ge 0$ and the equation holds only for $x \ge 0$.
</document>

<document source="https://tamnd.github.io/math2/node/MRBRH9CH7E.md" tag="MRBRH9CH7E" kind="definition">
# Filter

## Definition

Let $X$ be a set.
A *filter* on $X$ is a family $\mathcal{F}$ of subsets of $X$ such that:

1. $X \in \mathcal{F}$.
2. $\emptyset \notin \mathcal{F}$.
3. If $A \in \mathcal{F}$ and $A \subseteq B \subseteq X$, then $B \in \mathcal{F}$.
4. If $A \in \mathcal{F}$ and $B \in \mathcal{F}$, then $A \cap B \in \mathcal{F}$.

## Remarks

Some authors, and Mathlib, drop clause 2.
Then the only family that satisfies clauses 1, 3 and 4 and contains $\emptyset$ is the [[power set|00E0YZCERH]] of $X$.
A filter in the sense above is then called a *proper* filter.

Example: for a point $x$ of a [[topological space|0CQHFPGJ45]] $X$, the [[neighbourhoods|0ZCY0Z80B6]] of $x$ form a filter $\mathcal{N}(x)$, the *neighbourhood filter* of $x$.
Clause 2 holds because each neighbourhood contains $x$, and clause 4 holds because the intersection of two neighbourhoods is a neighbourhood.

Example: on an infinite set $X$, the sets with finite complement form a filter, the *Fréchet filter*.
Clause 2 holds because $X$ is infinite, and clause 4 holds because the union of two finite sets is finite.

Example: for $a \in X$, the family $\set{A \subseteq X : a \in A}$ is a filter, the *principal filter* at $a$.

Non-example: on $\N$, the family of sets that contain $0$ or contain $1$ satisfies clauses 1, 2 and 3.
It fails clause 4: it contains $\set{0}$ and $\set{1}$, but not $\set{0} \cap \set{1} = \emptyset$.

A nonempty family $\mathcal{B}$ of nonempty subsets of $X$ is a *filter base* if the intersection of any two members contains a member.
The supersets of members of a filter base form a filter.
</document>

<document source="https://tamnd.github.io/math2/node/MRC4WMVTAH.md" tag="MRC4WMVTAH" kind="concept">
# Circumsphere

## Summary

The circumsphere of a [[simplex|YX1DJK2QAG]] is the unique [[sphere|P9CM4GQ2T0]] in its affine span that passes through all its vertices.
Its centre is the circumcentre and its radius the circumradius.
For a triangle it is the [[circumcircle|03VZJE6C8C]], and for a [[tetrahedron|JFX8ASQS3E]] it is a sphere through the four vertices.
The word is also used for a sphere through all vertices of a polyhedron, which exists only for some polyhedra.

## Explanations

### L1 human

Through any three points that are not on a line there is exactly one circle.
In space, through any four points not in a plane there is exactly one sphere.
That sphere is the circumsphere of the tetrahedron with these four corners.
For the cube with corners $(\pm 1, \pm 1, \pm 1)$, the sphere of radius $\sqrt{3}$ about the origin passes through all eight corners.
</document>

<document source="https://tamnd.github.io/math2/node/MRCBJ0WDR0.md" tag="MRCBJ0WDR0" kind="definition">
# Generalized method of moments

## Definition

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] observations with values in a set $\mathcal{X}$, and let $\Theta \subseteq \R^k$ be a parameter set.
Let $g \colon \mathcal{X} \times \Theta \to \R^m$ with $m \ge k$ be a moment function such that the true parameter $\theta_0$ satisfies $E[g(X_1, \theta_0)] = 0$, where $E$ is the [[expectation|0DCT0QZVQ1]].
Let $\hat{g}_n(\theta) = \frac{1}{n}\sum_{i=1}^n g(X_i, \theta)$, and let $W_n$ be an $m \times m$ [[positive semidefinite|0880F9EQCP]] weight matrix computed from the data.
The *generalized method of moments estimator* is an [[estimator|0TTNRP5PFQ]]
$$\hat{\theta}_n \in \operatorname*{arg\,min}_{\theta \in \Theta} \; \hat{g}_n(\theta)^T \, W_n \, \hat{g}_n(\theta).$$

## Remarks

If $m = k$ and the equations $\hat{g}_n(\theta) = 0$ have a solution, then that solution is the estimator for every positive definite $W_n$.
This is the [[method of moments|09J1H07VAH]], a case of [[estimating equations|WTXFYA093D]].
When $m > k$ the model is over-identified, and the choice of $W_n$ matters.
Under regularity conditions, the choice $W_n \to S^{-1}$ in probability, with $S = E[g(X_1, \theta_0) g(X_1, \theta_0)^T]$ nonsingular, gives the smallest asymptotic variance, $(G^T S^{-1} G)^{-1}$ with $G = E[\nabla_\theta g(X_1, \theta_0)]$.

Example: for [[Poisson|0VEFSSKX18]]($\lambda$) data, take $g(x, \lambda) = (x - \lambda, \; x^2 - \lambda - \lambda^2)^T$, with $m = 2 > k = 1$.
Both components have mean zero at the true $\lambda$, since $E X = \lambda$ and $E X^2 = \lambda + \lambda^2$.

Non-example: in a normal model with the two parameters $(\mu, \sigma^2)$, take the single condition $E[X - \mu] = 0$, so $m = 1 < k = 2$.
The clause $m \ge k$ fails, and the objective does not depend on $\sigma^2$, so it cannot determine $\sigma^2$.
</document>

<document source="https://tamnd.github.io/math2/node/MRDZAR4X0N.md" tag="MRDZAR4X0N" kind="definition">
# Bounded mean oscillation

## Definition

Let $n \ge 1$ and let $f \colon \R^n \to \C$ be locally integrable for [[Lebesgue measure|2Z2QVWZ262]] on [[Euclidean space|0XPR1E85W0]].
Let $Q \subseteq \R^n$ be a cube with sides parallel to the axes, and let $\abs{Q}$ be its volume.
Let $f_Q = \frac{1}{\abs{Q}} \int_Q f \, dx$ be the mean of $f$ on $Q$, a [[Lebesgue integral|DWJF7ABPR1]].
The function $f$ has *bounded mean oscillation* if
$$\norm{f}_{\mathrm{BMO}} = \sup_Q \frac{1}{\abs{Q}} \int_Q \abs{f(x) - f_Q} \, dx < \infty,$$
where the supremum is over all such cubes $Q$.
The space of these functions is denoted $\mathrm{BMO}(\R^n)$.

## Remarks

$\norm{\cdot}_{\mathrm{BMO}}$ vanishes exactly on the constant functions, so it is a norm on BMO modulo constants, which is a Banach space.
The same definition with arcs of the unit circle in place of cubes gives $\mathrm{BMO}(\mathbb{T})$.
Every bounded function lies in BMO with $\norm{f}_{\mathrm{BMO}} \le 2 \sup \abs{f}$.
The John–Nirenberg inequality says that a BMO function is exponentially integrable on each cube after subtracting its mean.

Example: $f(x) = \log \abs{x}$ on $\R$ lies in BMO.
For $\lambda > 0$ one has $f(\lambda x) = f(x) + \log \lambda$, and adding a constant does not change the mean oscillation.
So it suffices to bound the mean oscillation on intervals of length $1$, where a direct estimate works.

Non-example: $f(x) = x$ on $\R$.
On $Q = [0, L]$ one has $f_Q = L/2$ and $\frac{1}{L} \int_0^L \abs{x - L/2} \, dx = L/4$, which is unbounded in $L$.
</document>

<document source="https://tamnd.github.io/math2/node/MREZVW7JTW.md" tag="MREZVW7JTW" kind="statement">
# Grauert's semicontinuity theorem

## Statement

Let $f \colon X \to Y$ be a [[proper|15XEBGNGHC]] holomorphic map of [[complex analytic spaces|HXSZA44J5J]] and let $\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.
Assume that $\mathcal{F}$ is flat over $Y$: for each $x \in X$ the stalk $\mathcal{F}_x$ is a flat module over $\mathcal{O}_{Y, f(x)}$.
For $y \in Y$ let $\mathfrak{m}_y \subseteq \mathcal{O}_{Y,y}$ be the maximal ideal.
Let $X_y = f^{-1}(y)$ be the fibre with the structure sheaf $\mathcal{O}_X / \mathfrak{m}_y \mathcal{O}_X$.
Let $\mathcal{F}(y) = \mathcal{F} / \mathfrak{m}_y \mathcal{F}$, a coherent sheaf on $X_y$.
For an integer $q \ge 0$ let $h^q(y) = \dim_\C H^q(X_y, \mathcal{F}(y))$.
Then:

1. for every $q \ge 0$ and every integer $k$, the set $\set{y \in Y : h^q(y) \ge k}$ is an [[analytic subset|GB7M582M5Y]] of $Y$; in particular $h^q$ is upper [[semicontinuous|RN4G8SGP6P]];
2. the [[Euler characteristic|RNDDWDAFJ5]] $y \mapsto \sum_{q \ge 0} (-1)^q h^q(y)$ is locally constant on $Y$.

## Remarks

Each $h^q(y)$ is finite by the [[Cartan–Serre finiteness theorem|MPNGD4V271]], since $X_y$ is compact.
The proof uses [[Grauert's direct image theorem|ZNCF16AENG]] in a refined form: locally on $Y$, the cohomology of all fibres is computed by one bounded complex of finite free $\mathcal{O}_Y$-modules.
Ranks of matrices of holomorphic functions can only drop on analytic subsets, which gives part 1, and the alternating sum of the ranks of the free modules gives part 2.

Example: let $E$ be a compact Riemann surface of genus $1$ and let $\mathcal{P}$ be the Poincaré line bundle on $E \times E$.
For $y \in E$, its restriction to $E \times \set{y}$ is a line bundle $L_y$ of degree $0$ on $E$, trivial exactly when $y = 0$.
For the projection $f \colon E \times E \to E$ onto the second factor, the fibre over $y$ is $E \times \set{y}$, and one gets $h^0(y) = h^1(y) = 1$ for $y = 0$ and $h^0(y) = h^1(y) = 0$ otherwise.
The dimensions jump up on the analytic set $\set{0}$, and the Euler characteristic is constantly $0$.

## Explanations

### L1 human

Consider a family of compact complex spaces with a coherent sheaf on each of them, varying holomorphically.
The dimensions of the cohomology groups may jump at special members of the family, but they can only jump up, and only on an analytic subset of the parameter space.
Their alternating sum does not change at all.

### L2 human

The theorem is a basic tool in deformation theory.
If $h^q$ is constant and $Y$ is reduced, then $R^q f_* \mathcal{F}$ is locally free and its fibre at $y$ is $H^q(X_y, \mathcal{F}(y))$.
This base change property is used to build vector bundles over moduli spaces.
It is the analytic counterpart of the semicontinuity theorem for flat proper morphisms of schemes.
</document>

<document source="https://tamnd.github.io/math2/node/MRJ2FHX6K1.md" tag="MRJ2FHX6K1" kind="example">
# Hypernetted-chain equation

## Example

Consider a classical fluid in $\R^3$ with number density $\rho$, inverse temperature $\beta$ and pair potential $u(r)$.
Let $g$ be the radial distribution function, $h = g - 1$, and $c$ the direct correlation function of the [[Ornstein–Zernike equation|29T4NC87AR]].
The hypernetted-chain closure is
$$g(r) = \exp\big(-\beta u(r) + h(r) - c(r)\big), \quad \text{equivalently} \quad c(r) = h(r) - \ln g(r) - \beta u(r).$$
Together with the Ornstein–Zernike equation
$$h(r_{12}) = c(r_{12}) + \rho \int_{\R^3} c(r_{13}) \, h(r_{23}) \, dx_3,$$
where $r_{ij} = \abs{x_i - x_j}$, it forms a closed [[nonlinear integral equation|BSCYBP6AS6]] for $h$: substitute $c = h - \ln(1 + h) - \beta u$.

Check at low density: as $\rho \to 0$ the Ornstein–Zernike equation gives $c = h$ to leading order.
Then the closure gives $g = e^{-\beta u}$, the exact low-density limit of the radial distribution function.

## Remarks

The closure is usually solved numerically by iteration, with the convolution done by the fast Fourier transform.
The [[Percus–Yevick approximation|K3E536PSRJ]] is usually more accurate for hard spheres, while the hypernetted-chain equation is usually preferred for long-range potentials such as the Coulomb potential.

## Explanations

### L1 human

The equation describes how particles in a liquid arrange around each other.
It combines the exact Ornstein–Zernike relation with an approximate rule that leaves out a certain class of terms, the so-called bridge diagrams, in an expansion of the correlations.
The resulting equation is nonlinear because the rule involves a logarithm.
</document>

<document source="https://tamnd.github.io/math2/node/MRKVM2VVNN.md" tag="MRKVM2VVNN" kind="example">
# Gauss map

## Example

The *Gauss map* is
$$G \colon [0, 1) \to [0, 1), \qquad G(x) = \frac{1}{x} - \floor{\frac{1}{x}} \text{ for } x \ne 0, \qquad G(0) = 0.$$
If $x = [0; a_1, a_2, a_3, \dots]$ is the [[continued fraction expansion|04VA28F3HB]] of an irrational $x \in (0, 1)$, then $a_1 = \floor{1/x}$ and $G(x) = [0; a_2, a_3, \dots]$.
So $G$ shifts the continued fraction digits, as the doubling map shifts binary digits.

The *Gauss measure* is the probability measure
$$\mu(A) = \frac{1}{\log 2} \int_A \frac{dx}{1 + x}.$$
It is $G$-invariant; check on $A = [0, t)$ with $0 < t \le 1$.
The preimage $G^{-1}[0, t)$ is, up to the point $0$, the disjoint union over $n \ge 1$ of the intervals $\big(\frac{1}{n + t}, \frac{1}{n}\big]$.
The measure of the $n$-th interval is $\frac{1}{\log 2}\big(\log\frac{n+1}{n} - \log\frac{n+t+1}{n+t}\big)$.
The partial sum up to $N$ telescopes to $\frac{1}{\log 2}\big(\log(N+1) - \log(N+1+t) + \log(1 + t)\big)$, which tends to $\frac{\log(1+t)}{\log 2} = \mu([0, t))$.

The map $G$ is [[ergodic|AZRPHS478B]] for $\mu$, and $\mu$ is equivalent to [[Lebesgue measure|2Z2QVWZ262]].
Hence for Lebesgue almost every $x$, the digit $k$ appears among the $a_n$ with frequency $\frac{1}{\log 2}\log\frac{(k+1)^2}{k(k+2)}$, the Gauss–Kuzmin distribution.
The [[transfer operator|CNZV00JQRC]] of $G$ for Lebesgue measure is the Gauss–Kuzmin–Wirsing operator, $(\mathcal{L}\varphi)(x) = \sum_{n \ge 1} \frac{1}{(x+n)^2} \varphi\big(\frac{1}{x+n}\big)$, and the density $\frac{1}{(1+x)\log 2}$ is its fixed point.

## Explanations

### L1 human

The Gauss map takes a number between $0$ and $1$, flips it to $1/x$, and keeps only the fractional part.
Doing this again and again reads off the digits of the continued fraction of the number, one at a time.
Gauss found the measure that this map preserves, a density that is a bit larger near $0$ than near $1$.
Ergodic theory then tells how often each digit appears in the continued fraction of a typical number: the digit $1$ appears about $41.5\%$ of the time.
</document>

<document source="https://tamnd.github.io/math2/node/MRRQEEP3T8.md" tag="MRRQEEP3T8" kind="definition">
# Iwasawa theory

## Definition

Let $p$ be a prime number, and let $K$ be an [[algebraic number field|MWBDFS2KJ5]].
Let $\Z_p$ be the [[p-adic integers|V6TJAFZ4MD]] with their additive group and $p$-adic topology.
A *$\Z_p$-extension* of $K$ is a [[Galois extension|0QMZAQ6QQ4]] $K_\infty/K$ inside a fixed algebraic closure such that the following holds.
Its [[Galois group|0XTY86BZKY]], with the Krull topology, is isomorphic to $\Z_p$ as a topological group.
For $n \ge 0$, the $n$-th *layer* $K_n$ is the fixed field of the closed subgroup corresponding to $p^n \Z_p$.
Then $\Gal(K_n/K) \cong \Z/p^n\Z$ and $K_\infty = \bigcup_n K_n$.
The *Iwasawa algebra* is $\Lambda = \Z_p[\![T]\!]$, the ring of formal power series over $\Z_p$.
Fix a topological generator $\gamma$ of $\Gal(K_\infty/K)$.
Then $\Lambda$ acts on each compact $\Z_p$-module with a continuous action of $\Gal(K_\infty/K)$, with $T$ acting as $\gamma - 1$.
Iwasawa theory is the study of such $\Lambda$-modules attached to $\Z_p$-extensions.
Let $A_n$ be the $p$-part of the [[ideal class group|WQ46BD3ARY]] of $K_n$.
The main such module is $X = \varprojlim_n A_n$, the inverse limit under the norm maps.

## Remarks

Iwasawa's theorem: there are integers $\lambda \ge 0$, $\mu \ge 0$ and $\nu$ such that $\abs{A_n} = p^{\mu p^n + \lambda n + \nu}$ for all sufficiently large $n$.
For a finite abelian extension $K$ of $\Q$ and its cyclotomic $\Z_p$-extension, Ferrero and Washington proved $\mu = 0$.

Example: let $p$ be odd.
The Galois group of $\bigcup_n \Q(\zeta_{p^n})$ over $\Q$ is $\Z_p^\times \cong \mu_{p-1} \times (1 + p\Z_p)$.
As a topological group, $1 + p\Z_p$ is isomorphic to $\Z_p$.
So the fixed field of the factor $\mu_{p-1}$ is a $\Z_p$-extension of $\Q$, the *cyclotomic* $\Z_p$-extension.
Its layer $K_1$ is the unique subfield of degree $p$ of the [[cyclotomic field|NN77VYSGXR]] $\Q(\zeta_{p^2})$.

Non-example: for $p$ odd, $\bigcup_n \Q(\zeta_{p^n})$ itself is not a $\Z_p$-extension of $\Q$.
Its Galois group $\Z_p^\times$ has the nontrivial finite subgroup $\mu_{p-1}$, while $\Z_p$ has no nonzero element of finite order.
It is, however, a $\Z_p$-extension of $\Q(\zeta_p)$.
</document>

<document source="https://tamnd.github.io/math2/node/MRXBC4ZNRR.md" tag="MRXBC4ZNRR" kind="statement">
# Doob's martingale convergence theorems

## Statement

Let $(X_n)_{n \ge 0}$ be a sequence of real random variables, adapted to a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_n)_{n \ge 0}$.
A family $(Y_i)$ of random variables is uniformly integrable if $\sup_i E\bigl[\abs{Y_i} \mathbf{1}\{\abs{Y_i} > K\}\bigr] \to 0$ as $K \to \infty$.

1. If $(X_n)$ is a [[submartingale|EQ055QMACW]] and $\sup_n E[X_n^+] < \infty$, then $X_n$ converges [[almost surely|0AAXWGT5TD]] to a random variable $X_\infty$ with $E[\abs{X_\infty}] < \infty$.
2. If $(X_n)$ is a non-negative [[supermartingale|KYW1PV0SX5]], then $X_n$ converges almost surely to a random variable $X_\infty$ with $E[X_\infty] \le E[X_0]$.
3. Let $(X_n)$ be a [[martingale|0CDVK9RQBH]].
These are equivalent: $(X_n)$ is uniformly integrable; $X_n$ converges almost surely and in $L^1$; $X_n = E[X \mid \mathcal{F}_n]$ for an integrable $X$.
In this case $X_n = E[X_\infty \mid \mathcal{F}_n]$ for all $n$, where $X_\infty$ is the limit.

## Remarks

The proof of part 1 uses [[Doob's upcrossing inequality|10HFW1GRV2]].
The bound on $E[X_n^+]$ makes the number of upcrossings of each rational interval finite almost surely, so the limit exists in $[-\infty, \infty]$.
Fatou's lemma shows that the limit is finite and integrable.
Part 2 follows from part 1 applied to $-X_n$, and part 3 uses that uniform integrability upgrades almost sure convergence to $L^1$ convergence.
Example: for independent $\xi_i$ with $P(\xi_i = 0) = P(\xi_i = 2) = 1/2$, the product $M_n = \xi_1 \cdots \xi_n$ is a non-negative martingale with $E[M_n] = 1$.
It converges almost surely to $0$ but not in $L^1$, so it is not uniformly integrable, which shows that part 3 needs its hypothesis.
The bound in part 1 is needed: the simple symmetric random walk is a martingale with $E[\abs{S_n}] \to \infty$, and it does not converge.
The same statements hold in continuous time for submartingales with right-continuous paths.

## Explanations

### L1 human

A fair or favourable game whose size stays under control must settle down to a limit.
A gambler who keeps playing fair games with a fortune that cannot go negative ends up with a fortune that stops changing.
But the limit may lose mass: the expected value can drop in the limit unless the process is uniformly integrable.

### L2 human

These theorems are the main reason martingales are useful in analysis and probability.
They give short proofs of the strong law of large numbers, of [[Lévy's upward theorem|W01FD2A8NK]] and of Kolmogorov's zero–one law.
They also give almost sure convergence of likelihood ratios, of branching process normalisations and of the Radon–Nikodym derivatives on finer and finer partitions.
The $L^p$ version, for $p > 1$, says that a martingale bounded in $L^p$ converges almost surely and in $L^p$, by [[Doob's martingale inequality|4G53E7BRXF]].
</document>

<document source="https://tamnd.github.io/math2/node/MRYFDAG5G8.md" tag="MRYFDAG5G8" kind="concept">
# Duality in order theory

## Summary

Every [[partially ordered set|011C29Q1S1]] has a dual, obtained by reversing the order.
Each order-theoretic notion has a dual notion: upper bound and lower bound, supremum and infimum, greatest and least element.
The duality principle says that a statement true for all posets stays true when every notion in it is replaced by its dual.
This halves the work in many proofs.

## Explanations

### L1 human

Turn a picture of an order upside down, and you get another order: its dual.
What was the top is now the bottom, and what was above is now below.
So every theorem about orders comes with a mirror theorem for free.
For example, the divisors of $12$ ordered by divisibility have greatest element $12$; in the dual order, $12$ is the least element.
</document>

<document source="https://tamnd.github.io/math2/node/MS263NBTRR.md" tag="MS263NBTRR" kind="concept">
# Lambert quadrilateral

## Summary

A Lambert quadrilateral is a quadrilateral with three right angles.
Johann Heinrich Lambert studied it in the 1760s in an attempt to prove the [[parallel postulate|2PC3XY0PYZ]].
Its fourth angle is right in [[Euclidean geometry|KRZD96GSVC]] and acute in [[hyperbolic geometry|GN4N94GQ0Z]].
Half of a [[Saccheri quadrilateral|5SJVH0BHJE]], cut along the line joining the midpoints of base and summit, is a Lambert quadrilateral.

## Explanations

### L1 human

Draw a shape with four sides in which three corners are square corners.
On a flat sheet the fourth corner is forced to be square too, so you get a rectangle.
In hyperbolic geometry the fourth corner is sharper than a right angle.
Lambert worked out many consequences of a sharp fourth corner and noticed that they look like geometry on a sphere of imaginary radius.
</document>

<document source="https://tamnd.github.io/math2/node/MS5PXY4D9Y.md" tag="MS5PXY4D9Y" kind="definition">
# Unary algebra

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]] and let $I$ be an index set.
A *unary algebra* is a pair $(A, (f_i)_{i \in I})$, where each $f_i \colon A \to A$ is a [[unary operation|Q2W26ZY10B]] on $A$.
It is *mono-unary* when $I$ has exactly one element.
A *homomorphism* of unary algebras $(A, (f_i))$ and $(B, (g_i))$ with the same index set is a [[function|0EPAT51DXQ]] $h \colon A \to B$ with $h(f_i(a)) = g_i(h(a))$ for all $i \in I$ and $a \in A$.

## Remarks

Example: $(\Z, s)$ with $s(n) = n + 1$ is a mono-unary algebra, since $s$ maps $\Z$ to $\Z$.
The map $h \colon \Z \to \Z/2\Z$, $h(n) = n \bmod 2$, is a homomorphism to $(\Z/2\Z, t)$ with $t(x) = x + 1$, because $h(n + 1) = h(n) + 1$.

Non-example: $(\N, p)$ with the predecessor $p(n) = n - 1$ is not a unary algebra, since $p(0) = -1 \notin \N$; the predecessor is only a partial operation on $\N$.

A unary algebra $(A, (f_i)_{i \in I})$ is the same as an action of the free monoid $I^*$ on $A$: a word $i_1 i_2 \cdots i_k$ acts by $f_{i_k} \circ \cdots \circ f_{i_1}$ in one common convention.
</document>

<document source="https://tamnd.github.io/math2/node/MS8G1FD8FD.md" tag="MS8G1FD8FD" kind="definition">
# Ruelle zeta function

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a map.
Assume that the set $\operatorname{Fix}(f^n)$ of [[fixed points|R6QVS6P21N]] of $f^n$ is finite for every $n \ge 1$.
Let $\varphi \colon X \to \C$ be a function, called the weight.
The *Ruelle zeta function* of $(f, \varphi)$ is the [[formal power series|033WGN00BF]]
$$\zeta_{f, \varphi}(z) = \exp\left( \sum_{n=1}^{\infty} \frac{z^n}{n} \sum_{x \in \operatorname{Fix}(f^n)} \prod_{k=0}^{n-1} \varphi(f^k x) \right).$$

## Remarks

Example: let $f$ be the shift on the full shift $\set{0, 1}^{\N}$ and let $\varphi = c$ be a constant.
The shift $f^n$ has $2^n$ fixed points, each of weight $c^n$, so $\zeta_{f, \varphi}(z) = \exp \sum_{n \ge 1} (2cz)^n / n = 1/(1 - 2cz)$.

Non-example: for the identity map of the circle the set $\operatorname{Fix}(f)$ is infinite, so the inner sum is not defined.

With $\varphi = 1$ one gets the [[Artin–Mazur zeta function|2A5Z21CE4P]] of $f$.
More generally the weight may take values in $d \times d$ matrices, and then the product is replaced by the trace of the ordered product $\varphi(f^{n-1} x) \cdots \varphi(x)$.
If $f$ is the shift on a [[subshift of finite type|624XDXX8FN]] with transition matrix $A$ and $\varphi(x) = w(x_0, x_1)$ depends only on the first two symbols, then $\zeta_{f, \varphi}(z) = 1/\det(I - z B)$, where $B_{ij} = A_{ij} w(i, j)$.
</document>

<document source="https://tamnd.github.io/math2/node/MSC0MR8VMH.md" tag="MSC0MR8VMH" kind="concept">
# Parametric model

## Summary

A parametric model is a [[statistical model|K3AMMSY8DD]] whose laws are described by finitely many real numbers, the parameters.
Inference in such a model reduces to learning a point of a subset of $\R^k$.
Most classical theory, such as maximum likelihood and the Cramér–Rao bound, is developed for parametric models.

## Explanations

### L1 human

A parametric model is a family of distributions that you can tune with a few knobs.
The normal distributions form such a family: two knobs, the mean $\mu$ and the variance $\sigma^2$, fix the whole bell curve.
Learning from data then means finding good settings of the knobs.
A model with "any distribution at all" has no finite set of knobs, and it is called nonparametric.
</document>

<document source="https://tamnd.github.io/math2/node/MSDC626JK5.md" tag="MSDC626JK5" kind="definition">
# Random integral equation

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]] and let $[a, b]$ be a compact interval.
Let $k \colon [a, b] \times [a, b] \times \Omega \to \R$ and $h \colon [a, b] \times \Omega \to \R$ be [[measurable functions|YYNCJSR0PJ]].
Here the domains carry the product of the Borel $\sigma$-algebras and $\mathcal{F}$.
The *random (linear Fredholm) integral equation* with random kernel $k$ and random free term $h$ is
$$x(t, \omega) = h(t, \omega) + \int_a^b k(t, s, \omega) \, x(s, \omega) \, ds, \qquad t \in [a, b], \ \omega \in \Omega.$$
A *random solution* is a measurable function $x \colon [a, b] \times \Omega \to \R$ with the following property.
For $P$-almost every $\omega$, the integral converges absolutely and the equation holds, for every $t \in [a, b]$.

## Remarks

Random Volterra equations, with $\int_a^t$ in place of $\int_a^b$, and nonlinear random equations such as $x(t, \omega) = h(t, \omega) + \int_a^b k(t, s, \omega) f(s, x(s, \omega)) \, ds$ are defined in the same way.
A random solution is a [[stochastic process|0QH5ENKZM1]] indexed by $t \in [a, b]$.
The integral is an ordinary Lebesgue integral, computed separately for each $\omega$.

Example: $[a, b] = [0, 1]$, $Y$ a random variable with values in $[0, 1]$, $k(t, s, \omega) = Y(\omega)/2$ and $h = 1$.
A constant $x(t, \omega) = c(\omega)$ solves the equation when $c = 1 + Y c / 2$, that is, $c = 2/(2 - Y)$.
This $c$ is measurable, so $x(t, \omega) = 2/(2 - Y(\omega))$ is a random solution.

Non-example: the equation $x(t) = 1 + \int_0^t x(s) \, dW(s)$, with $W$ a [[Wiener process|22735FGJV6]], is not a random integral equation in this sense.
Almost every path of $W$ has unbounded variation, so the integral cannot be computed path by path as a Lebesgue–Stieltjes integral.
It is an Itô integral, and the equation is a [[stochastic Volterra equation|KA95J9V6AS]].

## Explanations

### L1 human

It is a family of integral equations, one for each outcome $\omega$ of a random experiment, solved outcome by outcome.
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<document source="https://tamnd.github.io/math2/node/MSMD2BS9C5.md" tag="MSMD2BS9C5" kind="definition">
# Fourier integral operator

## Definition

Let $n \ge 1$ and $m \in \R$.
Let $\varphi \colon \R^n \times (\R^n \setminus \set{0}) \to \R$ be a smooth function such that:

1. $\varphi(x, t\xi) = t \, \varphi(x, \xi)$ for all $t > 0$, $x \in \R^n$ and $\xi \ne 0$.
2. $\det \big( \partial^2 \varphi / \partial x_j \partial \xi_k \big)_{j,k} \ne 0$ everywhere.

Let $a \in S^m(\R^n \times \R^n)$ be a symbol in the [[symbol class|AM4RNWVV4Q]] $S^m = S^m_{1,0}$ with $a(x, \xi) = 0$ for $\abs{\xi} \le 1$.
For $u$ in the [[Schwartz space|CG9JEATW3Q]] $\mathcal{S}(\R^n)$ with [[Fourier transform|DPVM34ADZV]] $\hat{u}$, put
$$T u(x) = \frac{1}{(2\pi)^n} \int_{\R^n} e^{i \varphi(x, \xi)} a(x, \xi) \, \hat{u}(\xi) \, d\xi.$$
The operator $T$ is a *Fourier integral operator* of order $m$ with *phase function* $\varphi$ and *amplitude* $a$.

## Remarks

Example: the phase $\varphi(x, \xi) = x \cdot \xi$ satisfies both conditions, with mixed Hessian the identity matrix, and gives the [[pseudo-differential operator|FF27N0BWTP]] $\mathrm{Op}(a)$.
For fixed $t \in \R$, the phase $\varphi(x, \xi) = x \cdot \xi + t \abs{\xi}$ also satisfies both conditions, and with a symbol $a$ equal to $1$ for $\abs{\xi} \ge 2$ it gives $e^{i t \sqrt{-\Delta}}$ up to a smoothing operator.

Non-example: $\varphi(x, \xi) = x \cdot \xi + \abs{\xi}^2$ fails the first condition, since it is not homogeneous of degree one in $\xi$; it is the phase of a Schrödinger evolution.

The general definition, due to Hörmander, allows phases $\varphi(x, y, \theta)$ with extra variables and works on manifolds.
There a Fourier integral operator is associated with a canonical relation, a [[Lagrangian submanifold|H8ACPTMX08]] of the product of the cotangent bundles, and it moves wave front sets along that relation.
</document>

<document source="https://tamnd.github.io/math2/node/MSN9TFS59W.md" tag="MSN9TFS59W" kind="definition">
# Uniformly most powerful test

## Definition

Let $X$ be data whose law $P_\theta$ depends on $\theta \in \Theta$, let $\Theta_0$ and $\Theta_1$ be disjoint subsets of $\Theta$, and let $\alpha \in (0, 1)$.
A (possibly randomized) test $\varphi$ of $H_0 \colon \theta \in \Theta_0$ against $H_1 \colon \theta \in \Theta_1$ has level $\alpha$ if $E_\theta[\varphi(X)] \le \alpha$ for every $\theta \in \Theta_0$.
A test $\varphi^*$ is *uniformly most powerful* (UMP) at level $\alpha$ if:

1. $\varphi^*$ has level $\alpha$;
2. for every test $\varphi$ of level $\alpha$ and every $\theta \in \Theta_1$, $E_\theta[\varphi^*(X)] \ge E_\theta[\varphi(X)]$.

## Remarks

Here $E_\theta[\varphi(X)]$ is the [[power function|4RJSN0QARD]] of $\varphi$.
Example: let $X_1, \dots, X_n$ be i.i.d. $\mathcal{N}(\theta, 1)$, $H_0 \colon \theta \le 0$, $H_1 \colon \theta > 0$.
The test that rejects when $\sqrt{n}\, \bar X > z_{1 - \alpha}$, the $1 - \alpha$ quantile of $\mathcal{N}(0, 1)$, is UMP at level $\alpha$.
For each $\theta_1 > 0$ it is the most powerful test of $\theta = 0$ against $\theta = \theta_1$, by the [[Neyman–Pearson lemma|0EE19CFEEF]], and its power function is increasing.
Non-example: for $H_0 \colon \theta = 0$ against $H_1 \colon \theta \ne 0$ in the same model, no test satisfies clause 2.
The one-sided test above is most powerful against each $\theta_1 > 0$, but at each $\theta_1 < 0$ the mirror one-sided test has strictly more power.
</document>

<document source="https://tamnd.github.io/math2/node/MSPNPVP9NK.md" tag="MSPNPVP9NK" kind="statement">
# Compactness of families of harmonic functions

## Statement

Let $U \subseteq \R^n$ be an open set and let $\mathcal{F}$ be a family of [[harmonic functions|0DEB0JMN0W]] on $U$.
Assume that $\mathcal{F}$ is uniformly bounded on each compact subset of $U$.
That is, for each compact $K \subseteq U$ there is $M_K$ with $\abs{u(x)} \le M_K$ for all $u \in \mathcal{F}$ and $x \in K$.
Then every sequence in $\mathcal{F}$ has a subsequence that converges uniformly on each compact subset of $U$, and the limit is harmonic on $U$.

## Remarks

Proof idea: the [[gradient estimates|N76R7JWMNJ]] bound $\abs{\nabla u}$ on each compact set, uniformly for $u \in \mathcal{F}$, so the family is [[equicontinuous|0MW2WXTHMC]] on compact sets.
The [[Arzelà–Ascoli theorem|0381S8Z5P0]] on an exhaustion of $U$ by compact sets, with a diagonal argument, gives a subsequence converging uniformly on compact sets.
The limit is continuous and inherits the mean value property, so it is harmonic by the [[converse of the mean value property|CQ7E0V6T2G]].
The derivatives of all orders then converge uniformly on compact sets as well.
For nonnegative harmonic functions on a connected $U$, it is enough that the family is bounded at a single point, by [[Harnack's inequality|YNH832D50W]]; this is closely related to [[Harnack's principle|P15K8RKHDV]].
The boundedness hypothesis is needed: the constants $u_k = k$ have no convergent subsequence.
Example: the family $\set{e^{t x} \cos(t y) : \abs{t} \le 1}$ on $\R^2$ is uniformly bounded on compact sets, so every sequence in it has a locally uniformly convergent subsequence.

## Explanations

### L1 human

A collection of harmonic functions that stays within fixed bounds on each bounded region cannot oscillate wildly.
From any sequence in it, one can extract a subsequence that settles down to a limit, and that limit is again harmonic.

### L2 human

This is the harmonic analogue of Montel's theorem on normal families of holomorphic functions.
It is the compactness tool behind existence proofs in potential theory, such as the construction of Green's functions by exhaustion and of extremal harmonic functions.
On a connected domain, Harnack's inequality upgrades boundedness at one point to local boundedness for positive families.
So the positive harmonic functions with $u(x_0) = 1$ form a compact set, and this compactness underlies the Martin boundary.
</document>

<document source="https://tamnd.github.io/math2/node/MSRAD0X7H4.md" tag="MSRAD0X7H4" kind="concept">
# Willmore flow

## Summary

The Willmore flow is the gradient flow of the [[Willmore energy|R7WFJGJBV4]] for closed surfaces immersed in $\R^3$.
Each point of the surface moves along the normal with a speed given by a fourth-order expression in the surface.
In this way the energy decreases as fast as possible.
Its stationary surfaces are the Willmore surfaces, such as round spheres.
It is a tool for finding surfaces that minimize bending energy.

## Explanations

### L1 human

The Willmore energy measures how much a closed surface bends, by integrating the square of its [[mean curvature|PA78NFEPCF]].
A round sphere has the least energy, $4\pi$, among all closed surfaces.
The Willmore flow deforms a surface step by step so that this energy drops.
A bumpy sphere tends to smooth out toward a round shape, while a round sphere does not move at all.
Unlike [[mean curvature flow|2YYV01C7QQ]], the flow does not simply shrink surfaces, because the energy does not change under scaling.
</document>

<document source="https://tamnd.github.io/math2/node/MSX3VKM9Z5.md" tag="MSX3VKM9Z5" kind="statement">
# Oka coherence theorem

## Statement

Let $\Omega \subseteq \C^n$ be open and let $\mathcal{O}_\Omega$ be the sheaf of [[holomorphic functions|507C34W0ZT]] on $\Omega$.
Then $\mathcal{O}_\Omega$ is a [[coherent analytic sheaf|DSJET02WPR]] of $\mathcal{O}_\Omega$-modules.
Equivalently, let $U \subseteq \Omega$ be open and let $f_1, \dots, f_q$ be holomorphic functions on $U$.
Then the sheaf of relations
$$\mathcal{R}(f_1, \dots, f_q) = \set{(g_1, \dots, g_q) \in \mathcal{O}_U^q : g_1 f_1 + \dots + g_q f_q = 0}$$
is of finite type: near each point of $U$ it is generated by finitely many of its sections.

## Remarks

The proof is by induction on $n$.
By the Weierstrass preparation theorem, after a linear change of coordinates one may take the $f_i$ to be Weierstrass polynomials in $z_n$ with coefficients holomorphic in $z' = (z_1, \dots, z_{n-1})$.
With the [[Weierstrass division theorem|FE83QQS17S]], relations among polynomials in $z_n$ of bounded degree are then reduced to relations over $\mathcal{O}_{\C^{n-1}}$, where the induction hypothesis applies.

Example: on $\C^2$ the relations of $f_1 = z_1$ and $f_2 = z_2$ are generated at every point by the single relation $(z_2, -z_1)$.

A consequence is that the structure sheaf $\mathcal{O}_X$ of every [[complex analytic space|HXSZA44J5J]] is coherent.
Indeed, locally it is a quotient of $\mathcal{O}_\Omega$ by an ideal sheaf generated by finitely many sections.
The real analogue fails: the sheaf of real analytic functions on $\R^3$ has non-coherent ideal sheaves, such as the ideal of germs vanishing on Cartan's umbrella $\set{x^3 = z(x^2 + y^2)}$.

## Explanations

### L1 human

Take finitely many holomorphic functions and look at all the ways to combine them into zero with holomorphic coefficients.
At a single point there are finitely many basic combinations from which all others follow.
Oka's theorem says that these basic combinations can be chosen so that they work at all nearby points at once.

### L2 human

Coherence is what makes sheaf theory effective in several complex variables.
Kernels, images and cokernels of maps between finite sums of $\mathcal{O}$ are coherent, so one can apply Cartan's theorems A and B and the finiteness theorems to them.
Oka's theorem is the base case, and Cartan's coherence of ideal sheaves of analytic sets and Oka's coherence of the normalization sheaf build on it.
</document>

<document source="https://tamnd.github.io/math2/node/MSXT31P944.md" tag="MSXT31P944" kind="definition">
# Zeros of Bessel functions

## Definition

Let $\nu > -1$ be real and let $J_\nu$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $\nu$.
The positive zeros of $J_\nu$ form an infinite set with no finite accumulation point.
For $k \ge 1$, the $k$-th *zero* $j_{\nu,k}$ of $J_\nu$ is the $k$-th smallest element of this set, so that
$$0 < j_{\nu,1} < j_{\nu,2} < j_{\nu,3} < \cdots, \qquad J_\nu(j_{\nu,k}) = 0.$$

## Remarks

For $\nu > -1$, every zero of $z^{-\nu} J_\nu(z)$ in $\C$ is real and simple, so the $j_{\nu,k}$ and their negatives are all of them.
Example: $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$, so $j_{1/2,k} = k\pi$; the check is $\sin(k\pi) = 0$ and $\sin x \ne 0$ on $(0, \pi)$.
Numerical values: $j_{0,1} \approx 2.4048$ and $j_{1,1} \approx 3.8317$.
The zeros of $J_\nu$ and $J_{\nu+1}$ interlace: $j_{\nu,1} < j_{\nu+1,1} < j_{\nu,2} < j_{\nu+1,2} < \cdots$.
For fixed $\nu$, $j_{\nu,k+1} - j_{\nu,k} \to \pi$ as $k \to \infty$, and $j_{\nu,k} = \left( k + \frac{\nu}{2} - \frac{1}{4} \right)\pi + O\left( \frac{1}{k} \right)$ (McMahon's expansion).
A non-example: $0$ is a zero of $J_\nu$ for $\nu > 0$, but it is not one of the $j_{\nu,k}$, which are positive by definition.
</document>

<document source="https://tamnd.github.io/math2/node/MSXX2P86CY.md" tag="MSXX2P86CY" kind="concept">
# Arity

## Summary

The arity of an operation, a function or a relation is the number of arguments it takes.
An operation of arity $n$ is called $n$-ary: nullary for $0$, unary for $1$, binary for $2$ and ternary for $3$.
Arities are part of every [[signature|WDVVEYD6AT]], and they tell what kind of interpretation each symbol gets in a [[structure|983GZBQYT4]].

## Explanations

### L1 human

The arity counts the inputs.
Negation $x \mapsto -x$ takes one number, so its arity is $1$.
Addition $(x, y) \mapsto x + y$ takes two numbers, so its arity is $2$.
The relation "$x$ lies between $y$ and $z$" compares three points, so its arity is $3$.
A constant such as $0$ takes no input at all and is seen as an operation of arity $0$.
</document>

<document source="https://tamnd.github.io/math2/node/MSYQ5YEPN5.md" tag="MSYQ5YEPN5" kind="definition">
# Erlangen program

## Definition

Let $X$ be a set and let $G$ be a [[group|G5S1M9]] with a [[transitive action|0WCY42EV9V]] on $X$.
In the sense of the *Erlangen program*, the pair $(X, G)$ is a *geometry*.
A property of subsets of $X$ belongs to the geometry $(X, G)$ if it is invariant under $G$.
This means that, for each subset $S$ and each $g \in G$, it holds for $S$ exactly when it holds for $g(S)$.
Two subsets of $X$ are *equivalent* in the geometry if some $g \in G$ maps one onto the other.

## Remarks

Example: $X = \R^2$ with $G$ the affine group gives affine geometry.
The property "the points $a, b, c$ are collinear" belongs to it, since affine maps send lines onto lines.
Non-example: the property "the segment $ab$ has length $1$" does not belong to affine geometry, since the map $x \mapsto 2x$ is affine and doubles lengths.
Since $G$ acts transitively, $X$ is a [[homogeneous space|85TWR6E75P]] $G/H$, where $H$ is the stabilizer of a point.
</document>

<document source="https://tamnd.github.io/math2/node/MT5JZ63RD4.md" tag="MT5JZ63RD4" kind="concept">
# Mittag-Leffler summation

## Summary

Mittag-Leffler summation is a [[summability method|8X25Q4XYMN]] that divides the $k$-th term of a series by $\Gamma(1 + \alpha k)$ and lets $\alpha$ decrease to $0$.
Here $\Gamma$ is the [[gamma function|0H5SYBSED9]].
For $\alpha = 1$ the divided series is an exponential series, and the method is related to [[Borel summation|8DJPSYSHGJ]].
Applied to the Taylor series of an analytic function, it sums the series in the [[Mittag-Leffler star|C21ECQ1H84]] of the function.

## Explanations

### L1 human

Dividing the $k$-th term by $\Gamma(1 + \alpha k)$ shrinks the late terms strongly, since the gamma function grows like a factorial.
So the divided series converges for many divergent series.
As $\alpha$ goes to $0$, $\Gamma(1 + \alpha k)$ tends to $\Gamma(1) = 1$ for each $k$, and the division disappears.
The Mittag-Leffler sum is the limit of the divided sums.
</document>

<document source="https://tamnd.github.io/math2/node/MT8TDVPBXB.md" tag="MT8TDVPBXB" kind="concept">
# Model theory of algebraically closed fields

## Summary

This part of model [[theory|92SWXST1ZM]] studies the first-order theory $\mathrm{ACF}$ of [[algebraically closed fields|0PF6TM8TWT]] in the language of rings.
It also studies the completions $\mathrm{ACF}_p$, one for each characteristic $p$.
The theory $\mathrm{ACF}$ admits [[quantifier elimination|7FDDXV3JK7]].
Each $\mathrm{ACF}_p$ is [[complete|655K8Y7VS8]] and decidable, and categorical in every uncountable cardinality.
These facts give the [[Lefschetz principle|SKTMN14X5P]] and short proofs of results in algebra, such as the [[Ax–Grothendieck theorem|M2XAN2HGZH]].
It is also the basic example of a [[strongly minimal theory|PNGY60EBDJ]].

## Explanations

### L1 human

Model theory asks which statements about a mathematical structure can be written with "for all", "there exists", and the symbols $+$, $\cdot$, $0$, $1$, $=$.
For algebraically closed fields, every such statement is equivalent to a statement without "for all" and "there exists", that is, to a combination of polynomial equations and inequations.
For example, "there is $x$ with $a x^2 + b x + c = 0$" is equivalent to "$a \ne 0$, or $b \ne 0$, or $c = 0$".
As a consequence, a statement of this kind that holds in the complex numbers holds in every algebraically closed field of characteristic $0$.
This lets one move results from $\C$ to other fields and back.
</document>

<document source="https://tamnd.github.io/math2/node/MT9P2X85V8.md" tag="MT9P2X85V8" kind="concept">
# Elliptic operator on a compact manifold

## Summary

An elliptic operator on a compact manifold is a [[differential operator between vector bundles|CGBX1N9V8H]] whose [[principal symbol|S8GC34D5ZG]] is invertible at every nonzero covector.
It extends the notion of an [[elliptic operator|XY8GEQSWPR]] on a domain of $\R^n$ to sections of bundles on a closed manifold.
On a compact manifold such an operator is Fredholm between suitable Sobolev spaces, so it has a finite index.
The Hodge Laplacian, the operator $d + d^*$ on forms, Dirac operators and the operator $\bar\partial + \bar\partial^*$ built from the Dolbeault operator are the main examples.
The Atiyah–Singer index theorem computes their indices.

## Explanations

### L1 human

Ellipticity says that the operator controls every derivative of its input, in every direction, at the top order.
On a compact manifold there is no boundary and no infinity where things can escape.
Together these two facts make an elliptic operator nearly invertible: only finitely many independent inputs are killed and only finitely many independent outputs are missed.
For example, the Laplacian on the circle kills only the constant functions, and its image consists of the functions with average zero.
</document>

<document source="https://tamnd.github.io/math2/node/MTCRZC5QKT.md" tag="MTCRZC5QKT" kind="concept">
# Upwind scheme

## Summary

An upwind scheme is a [[finite difference|X7HYEBA5SA]] or finite volume scheme for advection problems.
Its spatial differences are taken on the side from which the information comes.
For the [[transport equation|PVD7VZJZ9B]] $u_t + a u_x = 0$ with $a > 0$, waves move to the right.
So the scheme uses the values at the point and at its left neighbour.
Following the direction of propagation gives stable schemes under the [[Courant–Friedrichs–Lewy condition|FQVVNYV6N8]], while the centred or the opposite choice leads to instability.
The first-order upwind scheme is the basic building block of Godunov-type methods for conservation laws.

## Explanations

### L1 human

Think of smoke carried by the wind: what happens here next depends on what is upwind now, not downwind.
An upwind scheme updates each grid value using its neighbour on the upwind side.
For $u_t + u_x = 0$ with time step equal to half the grid spacing, the new value is the average of the old value and its left neighbour.
This makes the scheme stable and free of wiggles, though it smears sharp fronts a little.
</document>

<document source="https://tamnd.github.io/math2/node/MTEW1M61QK.md" tag="MTEW1M61QK" kind="concept">
# Global analysis

## Summary

Global analysis is the study of differential equations, variational problems and analysis on [[smooth manifolds|F45VQ47GPV]] as whole objects.
It looks at properties that depend on the entire manifold and not only on one chart.
Its tools include [[differential operators on vector bundles|CGBX1N9V8H]], [[spaces of smooth maps|BW12J14X0Z]], [[critical point theory|9ZQC7ARWTT]] and [[index theory|675MAXBQN6]].
It links analysis with the topology of manifolds, for instance through Morse theory and the Atiyah–Singer index theorem.

## Explanations

### L1 human

Classical calculus studies functions near one point or on a piece of flat space.
Global analysis asks what happens on a curved space, such as a sphere or a torus, when the whole space is taken into account.
A typical question is how many solutions an equation has on the whole space, or how many critical points a function must have.
For example, every smooth function on a sphere has at least two critical points, a maximum and a minimum, because the sphere is compact.
The shape of the space thus forces facts about the analysis on it.
</document>

<document source="https://tamnd.github.io/math2/node/MTM92ND7HE.md" tag="MTM92ND7HE" kind="definition">
# Mikusiński operational calculus

## Definition

Let $\mathcal{M}$ be the [[Mikusiński field of operators|82YAXZMMQ1]].
It contains the complex numbers and the continuous functions on $[0, \infty)$, with the [[convolution on the half-line|FCKX8KFVGN]] as product.
Let $l$ be the constant function $1$ and $s = 1/l \in \mathcal{M}$.
The *Mikusiński operational calculus* is the method of solving linear problems in $\mathcal{M}$ with the following rules.

1. For every $n \ge 1$ and every $n$ times continuously differentiable $f$ on $[0, \infty)$, $f^{(n)} = s^n f - s^{n-1} f(0) - s^{n-2} f'(0) - \dots - f^{(n-1)}(0)$ in $\mathcal{M}$.
2. For every complex $\alpha$, the operator $1/(s - \alpha)$ is the function $e^{\alpha t}$, and more generally $1/(s - \alpha)^{k}$ is the function $t^{k-1} e^{\alpha t}/(k-1)!$ for $k \ge 1$.
3. A [[linear differential equation|0QQRP17Z7S]] with constant coefficients and given initial values is rewritten by rule 1 as a linear equation for $y \in \mathcal{M}$. This equation is solved in the field $\mathcal{M}$, and the solution is identified as a function by rule 2 and partial fractions.

## Remarks

Rule 1 follows by induction from $s f = f' + f(0)$, which is the identity $l * f' = f - f(0) l$ divided by $l$.
Rule 2 for $k = 1$ follows from rule 1: for $f = e^{\alpha t}$, $s f - 1 = \alpha f$, so $(s - \alpha) f = 1$.
The solution found in step 3 is the unique solution, because $\mathcal{M}$ is a field and the embedding of continuous functions is injective.
The rules agree with those of the Laplace transform method, with $s$ in the role of the transform variable, but they apply to all continuous functions, without growth conditions.

Example: $y'' + y = 0$, $y(0) = 1$, $y'(0) = 0$.
Rule 1 gives $s^2 y - s + y = 0$, so $y = s/(s^2 + 1) = \frac{1}{2}\left(\frac{1}{s - i} + \frac{1}{s + i}\right)$.
By rule 2, $y = \frac{1}{2}(e^{it} + e^{-it}) = \cos t$.

Non-example: the operator $s$ itself is not a continuous function, so rule 2 has no analogue for $s$.
If $s = g$ for a continuous $g$, then $l * g = l \cdot s = 1$, so $l * g * l = l$ as functions.
The left side is a convolution of continuous functions, so it vanishes at $t = 0$, while $l(0) = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/MTQ3SGX56W.md" tag="MTQ3SGX56W" kind="definition">
# Fully nonlinear PDE

## Definition

A [[partial differential equation|GC5BDZ6QBC]] $F\left(D^m u, \dots, D u, u, x\right) = 0$ of [[order|E7JDFDPM1X]] $m$ is *fully nonlinear* if it is not [[quasilinear|7X9N9H3RK6]].
That is, $F$ cannot be written as an affine function of the derivatives $D^m u$ of order $m$ whose coefficients depend only on $D^{m-1} u, \dots, D u, u, x$.

## Remarks

Example: the Monge–Ampère equation $\det D^2 u = f(x)$ in $\R^n$ with $n \ge 2$ is fully nonlinear of order $2$.
For $n = 2$ it reads $u_{xx} u_{yy} - u_{xy}^2 = f$, which contains the product $u_{xx} u_{yy}$ of two second derivatives.
Non-example: the minimal surface equation is not fully nonlinear, because it is affine in the second derivatives with coefficients depending on $\nabla u$ only.
For first-order equations, the eikonal equation $\abs{\nabla u}^2 = 1$ is fully nonlinear.

## Explanations

### L1 human

A PDE is fully nonlinear when its highest derivatives enter in a genuinely nonlinear way.
</document>

<document source="https://tamnd.github.io/math2/node/MTV9TFKF3X.md" tag="MTV9TFKF3X" kind="concept">
# Algebraic space

## Summary

An algebraic space is a generalization of a scheme that is glued from schemes along étale maps instead of open immersions.
Formally it is a sheaf on the category of schemes that is the quotient of a scheme by an étale equivalence relation.
Algebraic spaces were introduced by Michael Artin, and they arise as quotients and moduli spaces that do not exist as schemes.

## Explanations

### L1 human

A scheme is glued from affine pieces along open sets.
An algebraic space is glued from schemes along étale maps, which are like local isomorphisms but not open inclusions.
This extra freedom lets one form quotients of schemes by free actions of finite groups, and other constructions, that are not schemes.
Every scheme is an algebraic space, and locally in the étale sense every algebraic space looks like a scheme.
</document>

<document source="https://tamnd.github.io/math2/node/MTWNAJMH1D.md" tag="MTWNAJMH1D" kind="statement">
# Galois correspondence for coverings

## Statement

Let $X$ be a [[path-connected|0WB6DH499T]], [[locally path-connected|1ZVR8BRK65]] and [[semilocally simply connected|R3SHZX0NZR]] space, and let $x_0 \in X$.
For a [[covering map|3XZRRE165N]] $p \colon (\tilde X, \tilde x_0) \to (X, x_0)$ with $\tilde X$ path-connected, let $H_p = p_* \pi_1(\tilde X, \tilde x_0)$ in the [[fundamental group|S1XK3HXHFX]] $\pi_1(X, x_0)$.
Let $G(p)$ be the [[deck transformation group|BH32RQBX6A]] of $p$.
Then:

1. For two such coverings $p_1$ and $p_2$, there is a map $f \colon (\tilde X_1, \tilde x_1) \to (\tilde X_2, \tilde x_2)$ with $p_2 \circ f = p_1$ if and only if $H_{p_1} \subseteq H_{p_2}$.
2. The group $G(p)$ acts transitively on the fibre $p^{-1}(x_0)$ if and only if $H_p$ is a [[normal subgroup|0CB0DX5BVV]] of $\pi_1(X, x_0)$.
3. The group $G(p)$ is isomorphic to $N(H_p)/H_p$, where $N(H_p)$ is the [[normaliser|0YGN3VAA0D]] of $H_p$ in $\pi_1(X, x_0)$.
4. In particular, if $H_p$ is normal, then $G(p) \cong \pi_1(X, x_0)/H_p$, a [[quotient group|0KYX2N69J0]], and $G(p) \cong \pi_1(X, x_0)$ when $\tilde X$ is simply connected.

## Remarks

The first clause is the lifting criterion applied to $p_1$ and $p_2$.
For the third clause, a loop $\gamma$ at $x_0$ lifts to a path from $\tilde x_0$ to a point $\tilde x_1$ of the fibre.
A deck transformation sending $\tilde x_0$ to $\tilde x_1$ exists exactly when $[\gamma]$ normalises $H_p$, and it is unique, which gives a surjection $N(H_p) \to G(p)$ with kernel $H_p$.
For the covering $S^1 \to S^1$, $z \mapsto z^n$, the subgroup is $n\Z \subseteq \Z$, which is normal, and the deck group is $\Z / n\Z$, acting by rotations through multiples of $2\pi / n$.
Together with the [[classification of covering spaces|YY277RBF8Z]], these facts form a dictionary between coverings and subgroups.

## Explanations

### L1 human

Coverings of a space behave like field extensions in Galois theory.
Bigger subgroups of the fundamental group belong to smaller coverings, and one covering sits over another exactly when the subgroups are nested.
The most symmetric coverings belong to normal subgroups, and their symmetry group is the quotient group.

### L2 human

The universal cover plays the part of the algebraic closure, and the fundamental group plays the part of the absolute Galois group.
Normal coverings correspond to Galois extensions, and the deck group corresponds to the Galois group.
The analogy becomes an exact theorem in Grothendieck's theory of the étale fundamental group, where finite étale coverings replace topological coverings.
In practice, the statement computes deck groups and tells which quotients of $\pi_1(X)$ arise as symmetry groups of coverings.
</document>

<document source="https://tamnd.github.io/math2/node/MTZCFEV37Y.md" tag="MTZCFEV37Y" kind="concept">
# Complex powers of elliptic operators

## Summary

Complex powers $A^s$, $s \in \C$, of a positive [[elliptic operator|MT9P2X85V8]] $A$ on a closed manifold generalise the powers $\lambda^s$ of a positive number.
Seeley showed that they are [[pseudo-differential operators|FF27N0BWTP]] of order $ms$, where $m$ is the order of $A$, depending holomorphically on $s$.
Their traces define the spectral zeta function $\zeta_A(s) = \operatorname{tr} A^{-s}$, which extends meromorphically to $\C$ and is regular at $s = 0$.
This is the basis of [[zeta function regularization|VJCJYXGTDY]] of determinants of Laplacians and of analytic torsion.

## Explanations

### L1 human

For a positive matrix with eigenvalues $\lambda_j$, the power $A^s$ is the matrix with the same eigenvectors and eigenvalues $\lambda_j^s$.
An elliptic operator such as a Laplacian on a closed manifold has infinitely many eigenvalues tending to infinity, and one can define $A^s$ in the same way.
For example, on the circle the operator $1 - d^2/dx^2$ multiplies $e^{inx}$ by $1 + n^2$, and $A^s$ multiplies it by $(1 + n^2)^s$.
When the real part of $s$ is negative enough, these powers are trace class, and their traces, continued to all $s$, give a zeta function attached to the operator.
Its derivative at $0$ defines a regularised determinant, a finite substitute for the infinite product of the eigenvalues.
</document>

<document source="https://tamnd.github.io/math2/node/MV1CFTH98T.md" tag="MV1CFTH98T" kind="statement">
# Zaslavsky's theorem

## Statement

Let $\mathcal{A}$ be a [[hyperplane arrangement|N602W0WAVT]] in $\R^d$.
Let $L(\mathcal{A})$ be its intersection poset: the nonempty intersections of subsets of $\mathcal{A}$, with $\R^d$ included.
It is [[ordered|0QV2JS9RJ3]] by reverse inclusion, so that $\R^d$ is the least element.
For $x \in L(\mathcal{A})$ let $\mu(x) = \mu(\R^d, x)$, where $\mu$ is the [[Möbius function|2Y75S8NG5B]] of $L(\mathcal{A})$.
The characteristic polynomial of $\mathcal{A}$ is
$$\chi_{\mathcal{A}}(t) = \sum_{x \in L(\mathcal{A})} \mu(x) \, t^{\dim x}.$$
Let $r(\mathcal{A})$ be the number of regions of $\mathcal{A}$, the connected components of $\R^d \setminus \bigcup_{H \in \mathcal{A}} H$, and let $b(\mathcal{A})$ be the number of bounded regions.
Let $\rank \mathcal{A}$ be the dimension of the span of the normal vectors of the hyperplanes of $\mathcal{A}$.
Then:

1. $r(\mathcal{A}) = (-1)^d \chi_{\mathcal{A}}(-1) = \sum_{x \in L(\mathcal{A})} \abs{\mu(x)}$.
2. $b(\mathcal{A}) = (-1)^{\rank \mathcal{A}} \chi_{\mathcal{A}}(1)$.

## Remarks

The theorem is due to Thomas Zaslavsky, in 1975.
Proof idea: fix $H \in \mathcal{A}$, and let $\mathcal{A}'$ be $\mathcal{A}$ without $H$ and $\mathcal{A}''$ the arrangement induced on $H$.
Each region of $\mathcal{A}'$ is either untouched by $H$ or cut in two, and the cut regions correspond to regions of $\mathcal{A}''$, so $r(\mathcal{A}) = r(\mathcal{A}') + r(\mathcal{A}'')$.
The characteristic polynomial satisfies the matching recursion $\chi_{\mathcal{A}} = \chi_{\mathcal{A}'} - \chi_{\mathcal{A}''}$, and induction on the number of hyperplanes finishes the proof; the second count is similar.

Example: $n$ lines in general position in $\R^2$, no two parallel and no three through a point.
Here $\chi(t) = t^2 - n t + \binom{n}{2}$, so $r = 1 + n + \binom{n}{2}$ and $b = 1 - n + \binom{n}{2} = \binom{n-1}{2}$.
For $n = 3$ this gives $7$ regions, $1$ of them bounded.

Example: the braid arrangement of the hyperplanes $x_i = x_j$, $1 \le i < j \le n$, in $\R^n$ has $\chi(t) = t(t-1)\cdots(t-n+1)$, so it has $n!$ regions, one for each ordering of the coordinates.

## Explanations

### L1 human

Cut space with finitely many flat walls; how many pieces do you get?
Zaslavsky's theorem says that the answer depends only on how the walls meet, not on their exact positions.
You record which intersections occur, compute one polynomial from this data, and evaluate it at $-1$ to count all pieces and at $1$ to count the bounded ones.

### L2 human

The theorem turns a geometric counting problem into a computation in the intersection poset, through its [[Möbius function|2Y75S8NG5B]].
For a central arrangement, $L(\mathcal{A})$ is a geometric lattice, and $\chi_{\mathcal{A}}$ is the characteristic polynomial of the associated [[matroid|PX31ZNYS3Z]].
The graphic arrangement of a graph $G$ on $n$ vertices has the hyperplanes $x_i = x_j$ for the edges $ij$.
Its characteristic polynomial is the [[chromatic polynomial|0B621RBVS9]] $\chi_G$ of $G$.
So the theorem says that $(-1)^n \chi_G(-1)$ counts the acyclic orientations of $G$, a result of Stanley.
When the hyperplanes have integer equations, $\chi_{\mathcal{A}}(q)$ counts the points of $\F_q^d$ off the reduced hyperplanes for every large enough prime $q$, which gives another route to computing it.
</document>

<document source="https://tamnd.github.io/math2/node/MV28Z5N9XX.md" tag="MV28Z5N9XX" kind="statement">
# Smith normal form

## Statement

Let $R$ be a [[principal ideal domain|0CMD0BCH8T]], and let $A$ be an $m \times n$ matrix over $R$.
Then there are an [[invertible|0EB9N0S33C]] $m \times m$ matrix $P$ and an invertible $n \times n$ matrix $Q$ over $R$, an integer $r \ge 0$, and nonzero $d_1, \dots, d_r \in R$ such that:

1. $PAQ$ is the $m \times n$ matrix with diagonal entries $d_1, \dots, d_r, 0, \dots, 0$ and all other entries $0$;
2. $d_i$ [[divides|0PG2H7WKGG]] $d_{i+1}$ for $1 \le i < r$.

The number $r$ is the rank of $A$, and the elements $d_1, \dots, d_r$ are unique up to multiplication by units of $R$.

## Remarks

Over a Euclidean domain such as $\Z$ or $F[x]$, the proof is an algorithm.
Row and column operations move an entry of smallest size to the top left corner.
Division with remainder then reduces the rest of its row and column.
This is repeated until the corner entry divides all other entries.
Then the process continues on the remaining block.
Uniqueness follows because $d_1 d_2 \cdots d_k$ is a [[greatest common divisor|0C8WYW4KYQ]] of the $k \times k$ [[minors|0BT9BYWVRY]] of $A$, and this gcd does not change under multiplication by invertible matrices.

Example over $\Z$: for $A = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix}$, the gcd of the entries is $2$ and $\det A = -8$.
So $d_1 = 2$ and $d_1 d_2 = 8$ up to sign, and the Smith normal form is $\operatorname{diag}(2, 4)$.

The hypothesis that $R$ is a principal ideal domain is needed.
Over $\Z[x]$, the $1 \times 2$ matrix $(2 \;\; x)$ has no Smith normal form: $d_1$ would generate the ideal $(2, x)$, which is not principal.

The $d_i$ are the [[invariant factors|0WHCA4SVP2]] of $A$.

## Explanations

### L1 human

Any matrix over the integers can be brought to diagonal form by invertible integer row and column operations.
Moreover, each diagonal entry divides the next one, and these entries do not depend on the choice of operations.
They are a complete set of invariants of the matrix up to such operations.

### L2 human

The Smith normal form of a presentation matrix gives the structure theorem for finitely generated modules over a principal ideal domain.
For $R = \Z$ this is the [[structure theorem for finitely generated abelian groups|0RY0B72XAS]].
Applied to $xI - M$ over $F[x]$, it gives the invariant factors of a square matrix $M$ over a field $F$, hence the rational canonical form.
Two square matrices $M$ and $N$ over $F$ are similar if and only if $xI - M$ and $xI - N$ have the same Smith normal form.
</document>

<document source="https://tamnd.github.io/math2/node/MV3PHCJ6CE.md" tag="MV3PHCJ6CE" kind="definition">
# Plücker coordinates

## Definition

Let $F$ be a [[field|04C5ZBG1WT]] and let $L$ be a line of the [[projective space|ZT9A6JK8QF]] $\mathbb{P}^3(F)$.
Let $x = (x_0, x_1, x_2, x_3)$ and $y = (y_0, y_1, y_2, y_3)$ be [[linearly independent|0CRMGPAH83]] vectors of $F^4$ with $L$ the line through $[x]$ and $[y]$.
For $0 \le i < j \le 3$ put
$$p_{ij} = x_i y_j - x_j y_i.$$
The *Plücker coordinates* of $L$ are the point
$$[p_{01} : p_{02} : p_{03} : p_{12} : p_{13} : p_{23}] \in \mathbb{P}^5(F).$$

## Remarks

The $p_{ij}$ are the $2 \times 2$ minors of the matrix with rows $x$ and $y$.
If $x, y$ are replaced by another basis $x' = a x + b y$, $y' = c x + d y$ of the same plane, each $p_{ij}$ is multiplied by the [[determinant|00KDZRW4C0]] $ad - bc \ne 0$, so the point of $\mathbb{P}^5(F)$ depends only on $L$.
The coordinates satisfy the Plücker relation
$$p_{01} p_{23} - p_{02} p_{13} + p_{03} p_{12} = 0.$$
In the language of the [[exterior algebra|0MDMZ965VE]], the $p_{ij}$ are the coordinates of $x \wedge y$ in the basis $e_i \wedge e_j$ of $\Lambda^2 F^4$.

Example: for $x = (1, 0, 0, 0)$ and $y = (0, 1, 0, 0)$, one gets $p_{01} = 1$ and all other $p_{ij} = 0$, so the line has Plücker coordinates $[1 : 0 : 0 : 0 : 0 : 0]$, and the relation holds since every product in it is $0$.

Non-example: if $y = 2x$, every $p_{ij}$ is $0$, which is not a point of $\mathbb{P}^5(F)$; the vectors must be linearly independent to span a line.
</document>

<document source="https://tamnd.github.io/math2/node/MV45MQ12YE.md" tag="MV45MQ12YE" kind="definition">
# Gauss–Kronrod quadrature formula

## Definition

Let $n \ge 1$ and let $x_1, \dots, x_n$ be the nodes of the $n$-point [[Gauss–Legendre rule|RGGXDRCJVE]] on $[-1, 1]$.
The $(2n + 1)$-point *Gauss–Kronrod quadrature formula* is a [[quadrature rule|ADSVMCY77A]]
$$\int_{-1}^{1} f(x)\,dx \approx \sum_{i=1}^{n} a_i f(x_i) + \sum_{j=1}^{n+1} b_j f(y_j)$$
in which:

1. the nodes $x_1, \dots, x_n$ are the Gauss–Legendre nodes;
2. the $n + 1$ new nodes $y_1, \dots, y_{n+1}$ and the $2n + 1$ weights $a_i, b_j$ are free parameters;
3. these parameters are chosen so that the rule is exact for every polynomial of degree at most $3n + 1$.

## Remarks

The rule has $3n + 2$ free parameters, which matches the $3n + 2$ conditions of exactness for $1, x, \dots, x^{3n+1}$.
The new nodes are the zeros of the Stieltjes polynomial $E_{n+1}$, the polynomial of degree $n + 1$ with $\int_{-1}^{1} E_{n+1}(x) P_n(x) x^k\,dx = 0$ for $k = 0, \dots, n$, where $P_n$ is the Legendre polynomial.
For the Legendre weight these zeros are real, lie in $(-1, 1)$, and interlace with the Gauss nodes.
In practice $\abs{G_n - K_{2n+1}}$, the difference of the Gauss value and the Kronrod value, serves as an error estimate.

Example: $n = 1$.
The Gauss node is $x_1 = 0$, and the rule $\int_{-1}^{1} f \approx \frac{8}{9} f(0) + \frac{5}{9} f(-\sqrt{3/5}) + \frac{5}{9} f(\sqrt{3/5})$ keeps it and adds two nodes.
It is exact for $1$ ($\frac89 + \frac{10}{9} = 2$), for $x^2$ ($\frac{10}{9} \cdot \frac35 = \frac23$), for $x^4$ ($\frac{10}{9} \cdot \frac{9}{25} = \frac25$), and for odd powers by symmetry, so up to degree $3n + 1 = 4$.
In this smallest case the Kronrod rule coincides with the $3$-point Gauss–Legendre rule.

Non-example: for $n = 2$ the $5$-point Gauss–Legendre rule is not a Gauss–Kronrod formula.
Its nodes are $0$ and the zeros of $63x^4 - 70x^2 + 15$, approximately $\pm 0.538$ and $\pm 0.906$, so they do not contain the $2$-point Gauss nodes $\pm 1/\sqrt{3} \approx \pm 0.577$.
</document>

<document source="https://tamnd.github.io/math2/node/MV58A1F52E.md" tag="MV58A1F52E" kind="problem">
# Busemann–Petty problem

## Problem

Let $n \ge 2$ and let $K, L \subseteq \R^n$ be compact [[convex bodies|437FSG3419]] with nonempty interior, both symmetric about the origin.
Assume that for every hyperplane $H$ through the origin,
$$\operatorname{vol}_{n-1}(K \cap H) \le \operatorname{vol}_{n-1}(L \cap H),$$
where $\operatorname{vol}_{n-1}$ is the $(n-1)$-dimensional [[Lebesgue measure|2Z2QVWZ262]] on $H$.
Does it follow that $\operatorname{vol}_n(K) \le \operatorname{vol}_n(L)$?

## Remarks

The problem is solved: the answer is yes for $n \le 4$ and no for $n \ge 5$.
For $n = 2$ the answer is yes: a chord through the center in direction $u$ has length $2\rho(u)$, where $\rho$ is the radial function, so the hypothesis gives $K \subseteq L$.
Lutwak showed that the answer in $\R^n$ is yes if and only if every origin-symmetric convex body in $\R^n$ is an [[intersection body|Y4ZE8FVZ8D]].
Counterexamples in high dimensions were found first by Larman and Rogers, and the remaining dimensions were settled through work of Ball, Bourgain, Giannopoulos, Papadimitrakis, Gardner and Zhang, among others.
A unified Fourier-analytic solution in all dimensions was given by Gardner, Koldobsky and Schlumprecht.

## Explanations

### L1 human

Suppose that every slice through the center of one symmetric convex body is smaller than the matching slice of another body.
It seems obvious that the first body must then have smaller volume.
Surprisingly, this is true only up to dimension four: in dimension five and higher there are counterexamples.
</document>

<document source="https://tamnd.github.io/math2/node/MV7ADC2DEH.md" tag="MV7ADC2DEH" kind="statement">
# McShane–Whitney extension

## Statement

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]], let $A \subseteq X$ be nonempty, and let $L \ge 0$.
Let $f \colon A \to \R$ satisfy $\abs{f(a) - f(b)} \le L \, d(a, b)$ for all $a, b \in A$.
For $x \in X$ set
$$F(x) = \inf_{a \in A} \bigl( f(a) + L \, d(x, a) \bigr), \qquad G(x) = \sup_{a \in A} \bigl( f(a) - L \, d(x, a) \bigr).$$
Then:

1. $F$ and $G$ are real-valued, equal $f$ on $A$, and satisfy the [[Lipschitz condition|0AW1E6YTJY]] with constant $L$ on $X$.
2. Every function $h \colon X \to \R$ that equals $f$ on $A$ and is Lipschitz with constant $L$ satisfies $G \le h \le F$ on $X$.

## Remarks

Proof idea: fix $a_0 \in A$; for $a \in A$, $f(a) + L d(x, a) \ge f(a_0) - L d(a, a_0) + L d(x, a) \ge f(a_0) - L d(x, a_0)$, so the [[infimum|0QF1MZ9FRG]] is finite.
For $x \in A$ the same inequality with $a_0 = x$ shows $F(x) \ge f(x)$, and the choice $a = x$ gives $F(x) \le f(x)$.
Each function $x \mapsto f(a) + L d(x, a)$ is $L$-Lipschitz, and an infimum of $L$-Lipschitz functions that is finite is $L$-Lipschitz.
Example: on $X = \R$ with $A = \set{0, 1}$, $f(0) = 0$, $f(1) = 1$ and $L = 1$, both $F$ and $G$ equal $x$ on $[0, 1]$, while $F(x) = \abs{x}$ and $G(x) = x$ for $x \le 0$, and $F(x) = x$ and $G(x) = 1 - \abs{x - 1}$ for $x \ge 1$.
For maps into $\R^m$ with $m \ge 2$, the same constant needs the [[Kirszbraun theorem|2ZM39HSFTQ]].

## Explanations

### L1 human

A real function that is Lipschitz on part of a metric space extends to the whole space with the same Lipschitz constant.
The formula is explicit: take the lowest of the cones $f(a) + L d(x, a)$ that sit on the graph of $f$.
The highest of the downward cones $f(a) - L d(x, a)$ gives a second extension.
These two extensions are the largest and the smallest possible ones.

### L2 human

The result needs no structure on $X$ beyond the metric, unlike the vector-valued [[Kirszbraun theorem|2ZM39HSFTQ]].
It shows, for example, that every Lipschitz function on a closed set of $\R^n$ extends to $\R^n$, which is useful before applying [[Rademacher's theorem|QEY26RT19R]].
</document>

<document source="https://tamnd.github.io/math2/node/MVCENBR5M1.md" tag="MVCENBR5M1" kind="definition">
# Mellin transform method

## Definition

For a function $g$ on $(0, \infty)$ write $\mathcal{M}g(s) = \int_0^\infty x^{s - 1} g(x) \, dx$ for its [[Mellin transform|N5PVGRJE1T]].
Let $k$ and $f$ be given functions on $(0, \infty)$, let $\lambda \in \C$, and consider the [[integral equation|R3SG1SN7JQ]]
$$\varphi(x) - \lambda \int_0^\infty k\!\left(\frac{x}{t}\right) \varphi(t) \, \frac{dt}{t} = f(x), \quad x > 0,$$
or its first-kind form without the term $\varphi(x)$.
Assume that $\mathcal{M}k$ and $\mathcal{M}f$ converge absolutely in a common strip $a < \operatorname{Re} s < b$.
The *Mellin transform method* consists of the following steps.

1. Apply $\mathcal{M}$; since the integral is the [[Mellin convolution|A2PMJG7J68]] of $k$ and $\varphi$, the equation becomes $(1 - \lambda \mathcal{M}k(s)) \, \mathcal{M}\varphi(s) = \mathcal{M}f(s)$ in the strip.
2. Solve for $\mathcal{M}\varphi(s) = \mathcal{M}f(s) / (1 - \lambda \mathcal{M}k(s))$ (for the first kind, $\mathcal{M}\varphi = \mathcal{M}f / \mathcal{M}k$).
3. Recover $\varphi$ by the [[Mellin inversion formula|XJWC13NY9E]] $\varphi(x) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} x^{-s} \, \mathcal{M}\varphi(s) \, ds$ with $a < c < b$.

## Remarks

Step 1 rests on the identity $\mathcal{M}\left[\int_0^\infty k(x/t) \varphi(t) \, dt/t\right](s) = \mathcal{M}k(s) \, \mathcal{M}\varphi(s)$, which follows from the substitution $x = t u$.

Example: the first-kind equation $\int_0^\infty e^{-x/t} \varphi(t) \, \frac{dt}{t} = f(x)$ with $f(x) = \int_0^\infty e^{-x/t - t} \, \frac{dt}{t}$.
Here $k(u) = e^{-u}$ and $\mathcal{M}k(s) = \Gamma(s)$ for $\operatorname{Re} s > 0$.
By step 1 applied to $e^{-t}$, $\mathcal{M}f(s) = \Gamma(s)^2$, so step 2 gives $\mathcal{M}\varphi(s) = \Gamma(s)$ and step 3 gives $\varphi(t) = e^{-t}$.

A kernel $h(x, t)$ that is homogeneous of degree $-1$, that is $h(cx, ct) = c^{-1} h(x, t)$ for all $c > 0$, can be written $h(x, t) = \frac{1}{t} h\!\left(\frac{x}{t}, 1\right)$, so the method applies to $\int_0^\infty h(x, t) \varphi(t) \, dt$.
For $h(x, t) = \frac{1}{x + t}$ one gets $k(u) = \frac{1}{1 + u}$ and $\mathcal{M}k(s) = \frac{\pi}{\sin \pi s}$ for $0 < \operatorname{Re} s < 1$.

Non-example: the kernel $e^{-(x - t)^2}$ on $(0, \infty)$ is not a function of $x / t$ times $\frac{1}{t}$, so the Mellin transform does not turn the equation into a product.

The substitution $x = e^{u}$, $t = e^{v}$ turns the equation into a [[convolution integral equation|EY75YD988Z]] on $\R$, and the method into the [[Fourier transform method|NRG03YP2PW]].
</document>

<document source="https://tamnd.github.io/math2/node/MVCH6N9SNW.md" tag="MVCH6N9SNW" kind="concept">
# Model

## Summary

A model of a [[theory|92SWXST1ZM]] is a [[structure|983GZBQYT4]] in which every sentence of the theory is true.
Models give meaning to axioms: the models of the group axioms are exactly the groups.
A theory with a model is consistent, and by [[Gödel's completeness theorem|WQ1ZE255SF]] every consistent first-order theory has a model.

## Explanations

### L1 human

A theory is a list of rules, and a model is a concrete world in which all the rules hold.
The rules "$<$ is a strict linear order with no least and no greatest element" have the integers with their usual order as a model.
The rational numbers with their usual order are another model of the same rules.
The set $\set{1, 2, 3}$ with its usual order is not a model, because $1$ is a least element.
</document>

<document source="https://tamnd.github.io/math2/node/MVGBS3J1P7.md" tag="MVGBS3J1P7" kind="concept">
# Krull dimension

## Summary

The Krull dimension of a commutative ring is the largest length of a strictly increasing chain of [[prime ideals|00MHHZ9SNE]].
It is the algebraic notion of dimension: the coordinate ring of an affine variety has Krull dimension equal to the dimension of the variety.
Fields have dimension $0$, and principal ideal domains that are not fields, such as $\Z$, have dimension $1$.
The polynomial ring in $n$ variables over a field has dimension $n$.

## Explanations

### L1 human

Points, curves and surfaces have dimensions $0$, $1$ and $2$.
In algebra, a point corresponds to a maximal ideal, and a bigger irreducible object corresponds to a smaller prime ideal.
The Krull dimension counts the longest staircase of prime ideals, each strictly inside the next.
In $k[x, y]$ the staircase $(0) \subset (x) \subset (x, y)$ goes from the whole plane, to a line, to a point, and has $2$ steps.
</document>

<document source="https://tamnd.github.io/math2/node/MVHCYMKFY9.md" tag="MVHCYMKFY9" kind="statement">
# Bockstein spectral sequence

## Statement

Let $p$ be a prime and let $X$ be a space whose integral [[singular homology|M0Q5CFJ7ZH]] groups $H_n(X; \Z)$ are finitely generated for every $n$.
Then there is a [[spectral sequence|GG8AYV847V]] $(E^r_n, d^r)_{r \ge 1}$ with $d^r \colon E^r_n \to E^r_{n-1}$ such that:

1. $E^1_n = H_n(X; \F_p)$, the [[homology with coefficients|XT7VWNTKGB]] in $\F_p$, and $d^1$ is the [[Bockstein homomorphism|H7PCM776GR]] of $0 \to \Z/p \to \Z/p^2 \to \Z/p \to 0$;
2. $E^\infty_n = E^r_n$ for $r$ large, and $E^\infty_n \cong \big(H_n(X; \Z) / \mathrm{torsion}\big) \otimes \F_p$;
3. each direct summand $\Z/p^r$ of $H_{n-1}(X; \Z)$ gives classes in $E^r_n$ and $E^r_{n-1}$ joined by a nonzero $d^r$.
All nonzero $d^r$ arise in this way.

## Remarks

The spectral sequence comes from an [[exact couple|VXDPP4EJPG]] with $D = H_*(X; \Z)$ and $E = H_*(X; \F_p)$.
The map $D \to D$ is multiplication by $p$, and the other two maps come from the long exact sequence of $0 \to \Z \xrightarrow{p} \Z \to \Z/p \to 0$.

Example: for $X = \R P^2$ and $p = 2$, one has $H_1(X; \Z) = \Z/2$ and $H_2(X; \Z) = 0$.
Here $E^1_1 = E^1_2 = \F_2$ and $d^1 \colon E^1_2 \to E^1_1$ is an isomorphism, so $E^2_n = 0$ for $n = 1, 2$, which matches $E^\infty$ since the free part of $H_n(X; \Z)$ vanishes for $n = 1, 2$.

The finite generation hypothesis ensures that the spectral sequence stops at a finite page in each degree.
For a summand such as $\Z[1/p]$ or $\Z/p^\infty$ the description in terms of $E^\infty$ changes.

## Explanations

### L1 human

Mod $p$ homology is easy to compute but forgets how large the torsion of integral homology is.
The Bockstein spectral sequence recovers this: a summand $\Z/p^r$ shows up as a pair of mod $p$ classes that survive exactly $r$ pages before cancelling.
What survives to the end is the free part of integral homology.

### L2 human

The spectral sequence is used to detect higher torsion, for instance in the homology of Lie groups, loop spaces and Eilenberg–MacLane spaces.
In cohomology it is multiplicative, and the higher Bocksteins interact with Steenrod operations, which gives restrictions on how torsion can appear.
It is one of the simplest examples of a spectral sequence built from an exact couple.
</document>

<document source="https://tamnd.github.io/math2/node/MVT64B3B6H.md" tag="MVT64B3B6H" kind="concept">
# Center of a ring

## Summary

The center of a [[ring|0B1261B5ED]] is the set of elements that commute with every element of the ring.
It is a commutative subring, and the ring is commutative exactly when its center is the whole ring.
Every ring is an algebra over its center, which is why the center is the natural base ring in the study of noncommutative rings.
It is the ring analogue of the [[centre of a group|0DZ93493VN]].

## Explanations

### L1 human

Some elements of a ring can be moved past any other element in a product without changing it.
These elements form the center.
In the ring of $n \times n$ real matrices, only the multiples of the identity matrix commute with every matrix, so the center is a copy of $\R$.
In the [[quaternions|0GE4DVQKR8]], the center consists of the real numbers.
</document>

<document source="https://tamnd.github.io/math2/node/MVYFSYMECG.md" tag="MVYFSYMECG" kind="concept">
# Harmonic polynomial

## Summary

A harmonic polynomial is a polynomial in $n$ real variables whose Laplacian is the zero polynomial.
Harmonic polynomials are the simplest [[harmonic functions|0DEB0JMN0W]] on all of $\R^n$.
The homogeneous ones of degree $k$ form a finite-dimensional space, and their restrictions to the unit sphere are the [[spherical harmonics|3F32Y21HP4]].
Every harmonic function on a ball is a locally uniform limit of harmonic polynomials, so they play the role that monomials play for power series.

## Explanations

### L1 human

Take a polynomial such as $x^2 - y^2$ and add up its second derivatives in each variable: $2 - 2 = 0$.
When that sum is zero, the polynomial is called harmonic.
In the plane, the real and imaginary parts of $(x + iy)^k$ give two harmonic polynomials of each degree $k \ge 1$, for example $x^2 - y^2$ and $2xy$.
The polynomial $x^2 + y^2$ is not harmonic, since the sum of its second derivatives is $4$.
</document>

<document source="https://tamnd.github.io/math2/node/MVYTPJ3V05.md" tag="MVYTPJ3V05" kind="definition">
# Sets of capacity zero

## Definition

Let $n \ge 2$ and let $E \subseteq \R^n$.
For $n = 2$, identify $\R^2$ with $\C$ and let $\operatorname{cap}$ be the [[logarithmic capacity|DBQXRTC0VN]].
For $n \ge 3$, let $\operatorname{cap}$ be the Newtonian [[capacity|4YZVFGJVSZ]].
The set $E$ has *capacity zero* if $\operatorname{cap}(K) = 0$ for every [[compact set|0R6EWFKRCF]] $K \subseteq E$.

## Remarks

This is the condition that the inner capacity of $E$ vanishes.
For Borel sets it is equivalent to the vanishing of the outer capacity, by Choquet's capacitability theorem.

Example: in $\C$, a finite set $K$ has capacity zero, since every probability measure on $K$ has an atom and so has infinite logarithmic energy.
Every compact subset of a countable set is countable, and a countable compact set carries no atomless probability measure, so countable subsets of $\C$ also have capacity zero.

Non-example: the segment $[0, 1]$ in $\C$ does not have capacity zero, since it is compact with logarithmic capacity $1/4$.

A Borel set has capacity zero if and only if it is a [[polar set|C95QJ7YH5T]].
In $\R^n$, $n \ge 3$, a compact set with finite $(n-2)$-dimensional Hausdorff measure has capacity zero.
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<document source="https://tamnd.github.io/math2/node/MW198BJJNK.md" tag="MW198BJJNK" kind="definition">
# Singular control

## Definition

Let $a < b$, $T > 0$, and let $f_0, f_1 \colon \R^n \to \R^n$ and $L_0, L_1 \colon \R^n \to \R$ be of class $C^1$.
Consider the [[control system|ZPDTVZQN5K]] $x' = f_0(x) + u f_1(x)$ with scalar control $u(t) \in [a, b]$ and running cost $\int_0^T (L_0(x) + u L_1(x)) \, dt$.
Let $(x, u, p, \lambda_0)$ be an extremal, that is, a quadruple that satisfies the conclusions of [[Pontryagin's maximum principle|7RPV2T29CF]].
The [[Hamiltonian|RACVWG5QTK]] along it is $H = H_0 + u \Phi$ with $H_0 = p \cdot f_0(x) - \lambda_0 L_0(x)$, where the *switching function* is
$$\Phi(t) = p(t) \cdot f_1(x(t)) - \lambda_0 L_1(x(t)).$$
The control $u$ is *singular* on an interval $J \subseteq [0, T]$ of positive length if $\Phi(t) = 0$ for all $t \in J$.
An extremal arc on such an interval $J$ is a *singular arc*.

## Remarks

Where $\Phi(t) > 0$ the maximum condition forces $u(t) = b$, and where $\Phi(t) < 0$ it forces $u(t) = a$.
Where $\Phi$ vanishes only at isolated times, the control is [[bang–bang|GK1Z3Q1WRZ]].
Example: minimize $\int_0^2 x^2 \, dt$ for $x' = u$, $x(0) = 1$, $u \in [-1, 1]$, with free final state.
The optimal control is $u = -1$ on $[0, 1]$ and $u = 0$ on $[1, 2]$, with $x(t) = 1 - t$ on $[0, 1]$ and $x = 0$ on $[1, 2]$.
Here $H = p u - x^2$, $\Phi = p$, and the [[costate equation|WS9MRAZMFW]] is $p' = 2x$ with $p(2) = 0$.
So $p = 0$ on $[1, 2]$, and the control is singular there, with the interior value $u = 0$.
Non-example: in the same problem, the control is not singular on $[0, 1)$, since $p(t) = -(1 - t)^2 < 0$ there.
To find a singular control, one differentiates $\Phi$ in time until $u$ appears and solves for $u$.
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<document source="https://tamnd.github.io/math2/node/MW1QAB5AN4.md" tag="MW1QAB5AN4" kind="definition">
# Runge domain

## Definition

Let $D \subseteq \C^n$ be an [[open set|02BH3MCBQB]].
Then $D$ is a *Runge domain* if the following holds for every [[holomorphic function|507C34W0ZT]] $f$ on $D$.
For every compact set $K \subseteq D$ and every $\varepsilon > 0$ there is a [[polynomial|07GNX8GCYP]] $p$ in $z_1, \dots, z_n$ with $\abs{f(z) - p(z)} < \varepsilon$ for all $z \in K$.

Equivalently, the polynomials are dense in the space of holomorphic functions on $D$ for the topology of [[uniform convergence|08VZ57PT26]] on compact subsets.

## Remarks

Example: the unit polydisc $\Delta^n$.
Every holomorphic $f$ on $\Delta^n$ is the sum of its Taylor series at $0$, and this series converges uniformly on each compact subset of $\Delta^n$; its partial sums are polynomials.
More generally every convex open set is a Runge domain, by the [[Oka–Weil theorem|7AWZJNQGT2]] applied to the convex hull of $K$, which is compact and polynomially convex.

Non-example: $D = \set{z \in \C^n : 1 < \abs{z_1} < 2}$ and $f(z) = 1/z_1$.
On the compact circle $K = \set{\abs{z_1} = 3/2, \ z_2 = \cdots = z_n = 0}$, every polynomial $p$ satisfies $\int_{\abs{\zeta} = 3/2} p(\zeta, 0, \dots, 0) \, d\zeta = 0$, while the same integral of $f$ is $2\pi i$.
So $f$ is not a uniform limit of polynomials on $K$.

For $n = 1$, [[Runge's theorem|6DG4CTGXJD]] gives a complete answer.
An open set $D \subseteq \C$ is a Runge domain exactly when every connected component of $D$ is simply connected.
</document>

<document source="https://tamnd.github.io/math2/node/MW5P4P4S9A.md" tag="MW5P4P4S9A" kind="definition">
# Locally finite measure

## Definition

Let $X$ be a topological space and let $\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.
The measure $\mu$ is *locally finite* if every point $x \in X$ has an [[open|02BH3MCBQB]] neighbourhood $U$ with $\mu(U) < \infty$.

## Remarks

Example: Lebesgue measure $\lambda$ on $\R^n$ is locally finite.
The open ball of radius $1$ around $x$ has finite measure.

Non-example: counting measure on $\R$ is not locally finite.
Every open neighbourhood of a point is infinite, so it has measure $\infty$.

A locally finite measure is finite on every [[compact set|0R6EWFKRCF]] $K$: finitely many neighbourhoods of finite measure cover $K$.
Conversely, if $X$ is a Hausdorff [[locally compact space|0FE2SB8WVJ]] and $\mu$ is finite on compact sets, then $\mu$ is locally finite, since each point has a compact neighbourhood.
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<document source="https://tamnd.github.io/math2/node/MW5R19P2X4.md" tag="MW5R19P2X4" kind="example">
# Lippmann–Schwinger equation

## Example

Consider the stationary [[Schrödinger equation|EPZR6H2WJN]] $-\Delta \psi + V \psi = k^2 \psi$ on $\R^3$, with a real potential $V$ of compact support, in units with $\hbar^2 / 2m = 1$.
Let $k > 0$ and let $\mathbf{k} \in \R^3$ with $\abs{\mathbf{k}} = k$ be the direction and wavenumber of an incoming plane wave.
The scattering solution $\psi$ solves the Lippmann–Schwinger equation
$$\psi(x) = e^{i \mathbf{k} \cdot x} - \frac{1}{4\pi} \int_{\R^3} \frac{e^{ik\abs{x - y}}}{\abs{x - y}} \, V(y) \, \psi(y) \, dy, \quad x \in \R^3.$$
It is an [[integral equation of the second kind|8KJCN49V64]] for $\psi$, with the [[weakly singular kernel|94TNBAAJYX]] $-\frac{1}{4\pi} \frac{e^{ik\abs{x - y}}}{\abs{x - y}} V(y)$.

Check: the function $G(x) = -\frac{e^{ik\abs{x}}}{4\pi \abs{x}}$ is the outgoing [[Green's function|SH6JY9RN2E]] of the Helmholtz operator, $(\Delta + k^2) G = \delta_0$.
Since $(\Delta + k^2) e^{i\mathbf{k} \cdot x} = 0$, applying $\Delta + k^2$ to the right side gives $V \psi$, which is the Schrödinger equation $(\Delta + k^2)\psi = V \psi$.
The choice of $e^{+ik\abs{x - y}}$ makes the scattered part $\psi - e^{i \mathbf{k} \cdot x}$ an outgoing spherical wave at infinity.

## Remarks

In operator form the equation reads $\psi = \phi + G_0 V \psi$ with $\phi$ the incoming wave and $G_0$ the convolution with $G$.
Iterating it gives the [[Born series|2PCVRG9NBN]], and its first term beyond $\phi$ is the Born approximation.

## Explanations

### L1 human

A wave hits an obstacle described by $V$ and scatters.
The equation says that the total wave is the incoming wave plus the waves sent out by each point of the obstacle.
Each point sends out a wave with strength equal to the potential times the total wave there.
Since the unknown wave appears on both sides, it is an integral equation rather than a formula.
</document>

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# Extended finite element method

## Summary

The extended finite element method, or XFEM, enriches a standard [[finite element|98GZ9DVHTA]] space with extra functions.
These carry known features of the solution, such as a jump across a crack.
The enrichment is built with the [[partition of unity|NH739XXVXK]] formed by the usual shape functions.
So cracks, interfaces and inclusions can cut through elements, and the mesh does not need to follow them.
It was developed in fracture mechanics by Ted Belytschko and coworkers, and is often combined with the [[level-set method|7KWR08FJET]] to describe the moving crack.
See the [[definition|35KXRGMAHT]].

## Explanations

### L1 human

A crack in a plate makes the displacement jump across the crack faces.
Ordinary finite elements are continuous, so they need a mesh that is cut exactly along the crack, and a new mesh whenever the crack grows.
XFEM keeps a fixed mesh and adds a jump function, such as a step that is $0$ on one side and $1$ on the other, near the crack.
The method then finds how much of this step each node needs.
</document>

<document source="https://tamnd.github.io/math2/node/MWBDFS2KJ5.md" tag="MWBDFS2KJ5" kind="concept">
# Algebraic number field

## Summary

An algebraic number field, or number field, is a [[field extension|0EQWVHF3W5]] of the rational numbers of finite degree.
Number fields are the main objects of algebraic number theory.
Each number field $K$ has a ring of integers $\mathcal{O}_K$, a [[Dedekind domain|7NPY2CN5KB]] that plays the role of $\Z$ inside $\Q$.
The failure of unique factorisation in $\mathcal{O}_K$ is measured by the [[ideal class group|WQ46BD3ARY]].

## Explanations

### L1 human

Start with the fractions $\Q$ and adjoin a root of a polynomial, for example $i = \sqrt{-1}$ or $\sqrt{2}$.
The numbers you can then form with $+$, $-$, $\times$ and $\div$ make up a number field.
For example, $\Q(i)$ consists of the numbers $a + bi$ with $a, b$ rational.
It is a two-dimensional space over $\Q$, with basis $1, i$.
Number theory in such fields explains facts about ordinary integers, such as which primes are sums of two squares.
</document>

<document source="https://tamnd.github.io/math2/node/MWDB7RACWQ.md" tag="MWDB7RACWQ" kind="example">
# Real projective plane

## Example

The *real projective plane* $\mathbb{RP}^2$ is the set of 1-dimensional linear subspaces of $\R^3$.
A point is written in [[homogeneous coordinates|FS9FRJ17D2]] $[x : y : z]$ with $(x, y, z) \ne (0, 0, 0)$, where $[x : y : z] = [\lambda x : \lambda y : \lambda z]$ for every $\lambda \ne 0$.
A line is the set of points of a 2-dimensional linear subspace of $\R^3$, that is a set $\set{[x : y : z] : ax + by + cz = 0}$ with $(a, b, c) \ne (0, 0, 0)$.

Check of the axioms of a [[projective plane|DANFNN8C4Y]]:

1. Two distinct points are two distinct lines through $0$ in $\R^3$; they span exactly one plane through $0$, so they lie on exactly one line.
2. Two distinct lines are two distinct planes through $0$; their intersection is exactly one line through $0$, so they meet in exactly one point.
3. The four points $[1 : 0 : 0]$, $[0 : 1 : 0]$, $[0 : 0 : 1]$, $[1 : 1 : 1]$ have no three on a line, since any three of the vectors are linearly independent.

## Remarks

The points with $z \ne 0$ form a copy of the plane $\R^2$ through $[x : y : z] \mapsto (x/z, y/z)$.
The points with $z = 0$ form the line at infinity; two parallel lines of $\R^2$ meet there, at the point given by their common direction.
As a topological space, $\mathbb{RP}^2$ is the sphere $S^2$ with antipodal points identified, a special case of [[real projective space|1DDYB6VVSR]].
It is a compact surface that is not orientable: it contains a [[Möbius strip|04AJE0M271]], and it is obtained by gluing a disc to the boundary circle of a Möbius strip.
Its [[Euler characteristic|RNDDWDAFJ5]] is $1$, and its fundamental group has order $2$.
Desargues's theorem holds in $\mathbb{RP}^2$, as in every projective plane over a field.

## Explanations

### L1 human

The real projective plane is the ordinary plane with one extra point added for each direction, where parallel lines going that way meet.
These extra points form one more line, the line at infinity.
With them, any two lines meet in exactly one point, with no exceptions for parallels.
A picture of it: the set of lines through your eye, each line being one point of the projective plane.
As a surface it is strange: it cannot be built in ordinary space without crossing itself, and it has only one side.
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<document source="https://tamnd.github.io/math2/node/MWGG1P8WDC.md" tag="MWGG1P8WDC" kind="definition">
# Sum-free set

## Definition

Let $G$ be an [[abelian group|082GHNPJ53]] written additively, and let $A$ be a subset of $G$.
The set $A$ is *sum-free* if there are no $x, y, z \in A$, not necessarily distinct, with $x + y = z$.

## Remarks

Equivalently, $(A + A) \cap A = \emptyset$, where $A + A$ is the [[sumset|XNBN31HQWB]].
The set of odd integers is sum-free in $\Z$: if $x$ and $y$ are odd, then $x + y$ is even, so it is not in the set.
The set $\set{1, 2, 4}$ is not sum-free: with $x = y = 2$ one gets $x + y = 4$, and repeated elements are allowed.
For a positive integer $N$, the upper half $\set{n : N/2 < n \le N}$ of $\set{1, \dots, N}$ is sum-free, since any sum of two of its elements exceeds $N$.
The set $\set{0}$ is never sum-free, since $0 + 0 = 0$; so a sum-free subset of a group does not contain $0$.
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<document source="https://tamnd.github.io/math2/node/MWMFQR8SMA.md" tag="MWMFQR8SMA" kind="problem">
# Congruent number problem

## Problem

A positive integer $n$ is a *congruent number* if there is a right triangle with rational side lengths and area $n$.
The problem is to decide, for a given positive integer $n$, whether $n$ is a congruent number.

## Remarks

The problem reduces to [[square-free|08X0VCR2NW]] $n$, since $n$ is congruent if and only if $nk^2$ is congruent for every positive integer $k$.
The number $6$ is congruent, by the triangle with sides $3, 4, 5$, which is a [[Pythagorean triple|0M7ARED5Q7]].
The number $5$ is congruent: the sides $3/2$, $20/3$, $41/6$ satisfy $(3/2)^2 + (20/3)^2 = 1681/36 = (41/6)^2$, and the area is $\frac12 \cdot \frac32 \cdot \frac{20}{3} = 5$.
The number $1$ is not congruent; this is [[Fermat's right triangle theorem|086BE0J3FZ]].
A positive integer $n$ is congruent if and only if the [[elliptic curve|427N6S6QQW]] $y^2 = x^3 - n^2 x$ has a rational point with $y \ne 0$.
Equivalently, this curve has positive rank.
Jerrold Tunnell gave in 1983 an explicit criterion by counting integer solutions of certain ternary quadratic equations.
The criterion is necessary for $n$ to be congruent, and its sufficiency follows from the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].
So the problem is open in general.

## Explanations

### L1 human

The question is simple: which whole numbers can be the area of a right triangle whose three sides are fractions?
For $6$ the familiar $3$-$4$-$5$ triangle works, and for $5$ a triangle with sides $3/2$, $20/3$, $41/6$ works.
For $1$, $2$ and $3$ no such triangle exists.
The surprising part is that the question is really about rational points on an elliptic curve.
A complete answer is tied to one of the major open conjectures of number theory.
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<document source="https://tamnd.github.io/math2/node/MWSD85BB4X.md" tag="MWSD85BB4X" kind="definition">
# Computably enumerable set

## Definition

Let $A \subseteq \N$ be a [[subset|0BJR9QR08E]] of the natural numbers.
The set $A$ is *computably enumerable*, or *recursively enumerable*, if $A$ is the domain of a partial [[computable function|KZSB5E471K]] $f \colon \N \rightharpoonup \N$.
That is, $A = \set{n \in \N : f(n) \text{ is defined}}$.

## Remarks

The following conditions on $A \subseteq \N$ are equivalent.

1. $A$ is computably enumerable.
2. $A$ is empty or the range of a total computable function $\N \to \N$.
3. There is a computable set $R \subseteq \N^2$ such that $A = \set{n : \exists m \, (n, m) \in R}$.

Example: the halting set $K = \set{e : \text{program } e \text{ halts on input } e}$ is computably enumerable.
It is the domain of the partial computable function that, on input $e$, runs program $e$ on input $e$ and then outputs $0$.

Non-example: the complement $\N \setminus K$ is not computably enumerable.
Otherwise $K$ and its complement would both be computably enumerable, and then $K$ would be [[computable|PPGD5KW8JC]], which is false.
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<document source="https://tamnd.github.io/math2/node/MX16DQ5Q8G.md" tag="MX16DQ5Q8G" kind="concept">
# Suspension flow

## Summary

The suspension flow of an invertible map turns a discrete-time system into a continuous-time flow.
Each point moves up at unit speed along a vertical segment whose height is given by a positive roof function.
At the top it jumps to the base above the image point.
The original map is recovered as the first-return map of the flow to the base.
Suspensions with constant roof $1$ give the natural flow on the mapping torus.
Suspensions with general roofs are a standard way to build examples of flows and to reduce flows to maps.

## Explanations

### L1 human

Picture each point $x$ of a space sitting at the bottom of a vertical stick of height $r(x)$.
A particle starts at the bottom of the stick, climbs at speed one, and when it reaches the top it is moved to the bottom of the stick over $T(x)$.
Watching only the moments when particles are at the bottom shows the map $T$ step by step.
For example, with $T$ a rotation of a circle and $r = 1$, the space of all sticks is a torus, and the flow is a linear flow winding around it.
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<document source="https://tamnd.github.io/math2/node/MX26N71ZD3.md" tag="MX26N71ZD3" kind="definition">
# Commuting matrices

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $n \ge 1$.
Two $n \times n$ matrices $A$ and $B$ over $R$ *commute* if
$$AB = BA,$$
where the products are [[matrix products|032Q9FB8MT]].
A family of $n \times n$ matrices is *commuting* if each two of its members commute.

## Remarks

Example: $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 5 \\ 0 & 3 \end{pmatrix}$ over $\R$ commute, since $AB = BA = \begin{pmatrix} 3 & 11 \\ 0 & 3 \end{pmatrix}$.
Non-example: $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ do not commute, since $AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ but $BA = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$.
The condition is often written $[A, B] = 0$, where $[A, B] = AB - BA$ is the commutator of $A$ and $B$.
The matrices that commute with a fixed matrix $A$ form the centraliser of $A$ in the matrix ring; it contains every polynomial in $A$.
Over an algebraically closed field, a commuting family of matrices has a common [[eigenvector|0N6960W0BV]].
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# Piecewise linear manifold

## Definition

Let $n \ge 0$ and let $U$ and $V$ be [[open subsets|02BH3MCBQB]] of $\R^n$.
A map $f \colon U \to V$ is *piecewise linear* if $U$ is a union of a locally finite family of [[simplices|YX1DJK2QAG]] in $\R^n$ with $f$ affine on each.
Any two simplices of the family must meet in a common face or not at all.
Let $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.
A *PL atlas* on $M$ is an [[atlas|J7P32JQTVE]] of dimension $n$ all of whose transition maps are piecewise linear.
Two PL atlases are *equivalent* if their union is a PL atlas.
A *piecewise linear manifold* is a topological manifold together with an equivalence class of PL atlases, called a *PL structure*.

## Remarks

The boundary $Q$ of the square $[0,1]^2$ is a PL manifold of dimension $1$.
For each vertex $v$, let $U_v$ be the union of $v$ and the two open edges at $v$.
Unfold $U_v$ onto an open interval of $\R$ by a map that sends $v$ to $0$ and is affine on each of the two edges.
These four charts cover $Q$, and each transition map is affine on each of finitely many intervals, so it is piecewise linear.

The two stereographic charts $x/(1 - y)$ and $x/(1 + y)$ of the unit circle $S^1$ do not form a PL atlas.
Their transition map $t \mapsto 1/t$ on $\R \setminus \set{0}$ is not affine on any interval.
The circle does have a PL structure, carried over from $Q$ by a homeomorphism $Q \to S^1$.

Every smooth manifold has a PL structure, unique up to PL homeomorphism, by a theorem of J. H. C. Whitehead.
Every topological manifold of dimension at most $3$ has a unique PL structure, while in higher dimensions a manifold may have none or several.
Equivalently, a PL manifold of dimension $n$ is the realization of a combinatorial manifold.
This is a simplicial complex in which the link of each vertex is PL homeomorphic to the boundary of an $n$-simplex.
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# Dedekind eta function

## Summary

The Dedekind eta function is $\eta(\tau) = e^{\pi i \tau / 12} \prod_{n \ge 1} (1 - e^{2 \pi i n \tau})$ on the [[upper half-plane|X6XGGH5FPW]].
It is holomorphic and never zero there, and it transforms like a modular form of weight $1/2$, up to $24$-th roots of unity.
Its $24$-th power is the cusp form $\Delta$ of weight $12$.
Its reciprocal is, up to a factor, the generating function of the [[partition function|0CV69WDRXW]].

## Explanations

### L1 human

The eta function is an infinite product $q^{1/24}(1 - q)(1 - q^2)(1 - q^3) \cdots$ with $q = e^{2 \pi i \tau}$.
Expanding $1/\prod (1 - q^n)$ counts integer partitions, so eta is tied to counting partitions.
Under $\tau \mapsto -1/\tau$ it changes only by the simple factor $\sqrt{-i\tau}$.
This hidden symmetry is the source of the Hardy–Ramanujan formula for the number of partitions.
</document>

<document source="https://tamnd.github.io/math2/node/MX6157RAVA.md" tag="MX6157RAVA" kind="statement">
# Bernstein inequalities

## Statement

Let $X_1, \dots, X_n$ be [[independent|0GQ938GDE0]] real random variables with $E[X_i] = 0$ for each $i$.
Assume that there is a constant $M > 0$ with $\abs{X_i} \le M$ almost surely for each $i$.
Let $\sigma^2 = \sum_{i=1}^n E[X_i^2]$.
Then for every $t \ge 0$,
$$P\left(\sum_{i=1}^n X_i \ge t\right) \le \exp\left(-\frac{t^2 / 2}{\sigma^2 + M t / 3}\right).$$

## Remarks

Proof idea: for $\lambda > 0$, the bound $\abs{X_i} \le M$ gives $E[e^{\lambda X_i}] \le \exp\left(E[X_i^2] \, \frac{e^{\lambda M} - 1 - \lambda M}{M^2}\right)$.
The [[Chernoff bound|014PEBGDYF]] with the best $\lambda$ then gives Bennett's inequality, and an elementary estimate of the resulting function gives the bound above.
Applying the bound to $-X_i$ gives the same bound for the lower tail, so $P(\abs{\sum_i X_i} \ge t)$ is at most twice the right-hand side.
For small $t$, the exponent is about $-t^2 / (2 \sigma^2)$, as for a normal variable with the true variance $\sigma^2$.
For large $t$, it is about $-3t / (2M)$, an exponential tail.
Example: take $n$ independent centred variables with $\abs{X_i} \le 1$ and $E[X_i^2] = p$ small.
[[Hoeffding's inequality|0WPDNXGSSN]] gives the exponent $-t^2 / (2n)$.
Bernstein gives about $-t^2 / (2np)$ when $t$ is much smaller than $np$, which is much better.
The name covers a family of related bounds; other versions replace the bound $\abs{X_i} \le M$ by moment conditions $E[\abs{X_i}^k] \le \frac{k!}{2} E[X_i^2] M^{k-2}$ for all $k \ge 2$.

## Explanations

### L1 human

A sum of many small independent random pieces is rarely far from its mean.
Bernstein's inequality says how rare, and it uses the actual variance of the pieces, not only their range.
When the pieces have small variance, it gives much stronger bounds than inequalities that see only the range.

### L2 human

Bernstein's inequality sits between [[Hoeffding's inequality|0WPDNXGSSN]], which uses only bounds on the range, and the [[central limit theorem|0EPZC4Z1FC]], which describes the normal approximation but gives no bound for finite $n$.
Its tail is Gaussian for moderate deviations and exponential for large ones, which matches the behaviour of sub-exponential variables.
It is a standard tool in statistics, randomized algorithms and the theory of empirical processes.
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# Principal symbol

## Definition

Let $\Omega$ be an open set in $\R^n$.
Let $P = \sum_{\abs{\alpha} \le m} a_\alpha(x) \, \partial^\alpha$ be a [[linear differential operator|6DZETZ5T9G]] of order $m$ on $\Omega$, with [[multi-indices|0PJXBR6QA3]] $\alpha$.
The *principal symbol* of $P$ is the following function of $x \in \Omega$ and $\xi \in \R^n$.
$$\sigma_m(P)(x, \xi) = \sum_{\abs{\alpha} = m} a_\alpha(x) \, \xi^\alpha.$$
Here $\xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n}$.
For each fixed $x$, it is a [[homogeneous polynomial|2C3PAQ7H50]] of degree $m$ in $\xi$.

## Remarks

Example: for $P = \partial_x^2 - \partial_y^2 + 5 \partial_x + 1$ on $\R^2$, the order is $2$ and $\sigma_2(P)(x, \xi) = \xi_1^2 - \xi_2^2$; the terms $5 \partial_x$ and $1$ do not contribute.
Non-example: the full polynomial $\xi_1^2 - \xi_2^2 + 5 \xi_1 + 1$ is not the principal symbol, because it keeps the lower-order terms.
Many texts write $P = \sum a_\alpha(x) D^\alpha$ with $D_j = -i \partial_j$ and define the principal symbol as $\sum_{\abs{\alpha} = m} a_\alpha(x) \xi^\alpha$ in those coefficients; this differs from the definition above by the factor $i^m$.
The vanishing of the principal symbol, and so the [[characteristic variety|CQJ2E2ARRF]], is the same in all conventions.
On a manifold, the principal symbol is a well-defined function on the cotangent bundle.

## Explanations

### L1 human

The principal symbol is what remains of a differential operator when only its top-order terms are kept and each derivative is replaced by a variable.
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<document source="https://tamnd.github.io/math2/node/MXEZM0QJ1K.md" tag="MXEZM0QJ1K" kind="concept">
# Anderson localization

## Summary

Anderson localization is the absence of wave transport in a disordered medium.
Mathematically, a random [[Schrödinger operator|89AA6XJY5K]] has, in some energy range, only eigenvalues, with eigenfunctions that decay exponentially away from a centre.
It was proposed by the physicist P. W. Anderson in 1958 for electrons in a random lattice.
It is proved for the one-dimensional Anderson model at all energies and in every dimension at large disorder.

## Explanations

### L1 human

Imagine a wave moving through a crystal whose atoms have random energies.
Each atom scatters the wave a little, and the scattered pieces interfere.
In a strongly disordered medium the interference is destructive on average, and the wave stays trapped near where it started.
For an electron this means the material does not conduct: the states are localized.
Mathematically, localized states are eigenvectors of the energy operator that are concentrated near one point and very small far away.
</document>

<document source="https://tamnd.github.io/math2/node/MXHTKVKB09.md" tag="MXHTKVKB09" kind="concept">
# Dowker space

## Summary

A Dowker space is a [[normal|0NSR9V8TED]] Hausdorff space that is not countably paracompact.
Equivalently, it is a normal Hausdorff space $X$ whose product $X \times [0, 1]$ with the unit interval is not normal.
Such spaces show that normality is not preserved by products, even with a compact metric factor.
Mary Ellen Rudin constructed a Dowker space in ZFC.

## Explanations

### L1 human

Normality is a separation property: two disjoint closed sets can be put inside disjoint open sets.
One might expect that multiplying a normal space by the plain interval $[0, 1]$ keeps it normal.
A Dowker space is a normal space for which this fails.
Every metrizable space and every [[paracompact|8R29G6NG9Q]] Hausdorff space is normal and countably paracompact, so none of them is a Dowker space.
Known Dowker spaces are hard to build, which is why they count as exotic.
</document>

<document source="https://tamnd.github.io/math2/node/MXNDW6AFGZ.md" tag="MXNDW6AFGZ" kind="definition">
# Kendall rank correlation coefficient

## Definition

Let $(x_1, y_1), \dots, (x_n, y_n)$ be $n \ge 2$ pairs of real numbers, with $x_1, \dots, x_n$ distinct and $y_1, \dots, y_n$ distinct.
A pair of indices $i < j$ is *concordant* if $(x_i - x_j)(y_i - y_j) > 0$ and *discordant* if $(x_i - x_j)(y_i - y_j) < 0$.
Let $C$ and $D$ be the numbers of concordant and discordant pairs.
The *Kendall rank correlation coefficient* is
$$\tau = \frac{C - D}{n(n-1)/2}.$$

## Remarks

Since every pair is concordant or discordant, $C + D = n(n-1)/2$ and $-1 \le \tau \le 1$.
For a pair $(X, Y)$ of [[random variables|0DKRR22AEQ]] with a continuous joint law and an independent copy $(X', Y')$, the population version is $P\big((X - X')(Y - Y') > 0\big) - P\big((X - X')(Y - Y') < 0\big)$.

Example: $x = (1, 2, 3)$ and $y = (1, 3, 2)$.
The pairs $(1, 2)$ and $(1, 3)$ are concordant, and the pair $(2, 3)$ is discordant because $(2 - 3)(3 - 2) < 0$.
So $\tau = (2 - 1)/3 = 1/3$.

Non-example: with ties, for example $y = (1, 1, 2)$, the pair $(1, 2)$ is neither concordant nor discordant, and the clause on distinct values fails.
Variants such as $\tau_b$ adjust the denominator for ties.

Under independence of $X$ and $Y$, the sample $\tau$ has mean $0$ and variance $2(2n + 5)/(9n(n-1))$, which gives a test of independence.
</document>

<document source="https://tamnd.github.io/math2/node/MXPTXV340K.md" tag="MXPTXV340K" kind="concept">
# Siegel modular form

## Summary

A Siegel modular form is a holomorphic function on the [[Siegel upper half-space|5PTYE97PF6]].
Under the Siegel modular group $Sp(2g, \Z)$ it transforms by a power of a determinant factor.
For $g = 1$ these are the classical [[modular forms|GWKN5VG78A]].
Siegel modular forms are sections of line bundles on the moduli space of principally polarized abelian varieties.
Their Fourier coefficients often encode arithmetic data, such as numbers of representations of one quadratic form by another.

## Explanations

### L1 human

A classical modular form is a function on the upper half-plane that changes in a prescribed way under the substitutions $z \mapsto (az + b)/(cz + d)$.
A Siegel modular form does the same in several variables: its argument is a symmetric matrix with positive definite imaginary part, and the substitutions are given by symplectic integer matrices.
Theta series of quadratic forms are typical examples.
As in one variable, the space of forms of a fixed weight is finite-dimensional.
</document>

<document source="https://tamnd.github.io/math2/node/MXRC891TQS.md" tag="MXRC891TQS" kind="concept">
# Complex Monge–Ampère operator

## Summary

The complex Monge–Ampère operator sends a function $u$ on an open subset of $\C^n$ to the determinant of its complex Hessian.
The complex Hessian is the matrix $\left(\partial^2 u / \partial z_j \partial \bar z_k\right)$, and the determinant is viewed as a measure.
In the language of forms it is the $n$-th exterior power $(dd^c u)^n$.
For $n = 1$ it is the Laplacian, so it is the nonlinear analogue of the Laplacian in pluripotential theory.
It is the operator of the complex [[Monge–Ampère equation|14AKAC0S0T]], which appears in the construction of Kähler–Einstein metrics and in the study of [[plurisubharmonic functions|FF2P7ER3QT]].

## Explanations

### L1 human

For a smooth function of several complex variables, one can form the matrix of its mixed second derivatives in $z_j$ and $\bar z_k$.
The complex Monge–Ampère operator takes the determinant of this matrix.
For a plurisubharmonic function the matrix is positive semidefinite, so the determinant is nonnegative and can be read as a density of mass.
For example, for $u(z) = \abs{z_1}^2 + \cdots + \abs{z_n}^2$ the matrix is the identity and the determinant is $1$ everywhere.
For $u(z) = \abs{z_1}^2$ in $\C^2$ the determinant is $0$, because $u$ does not depend on $z_2$.
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<document source="https://tamnd.github.io/math2/node/MXRCPAH09H.md" tag="MXRCPAH09H" kind="statement">
# Mordell–Weil theorem

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] and let $A$ be an [[abelian variety|R7W03TXRS9]] over $K$.
Then the group $A(K)$ of $K$-[[rational points|SWJYG87QQH]] of $A$ is a [[finitely generated|082DWWXMRJ]] [[abelian group|082GHNPJ53]].

## Remarks

The proof has two parts.
The weak Mordell–Weil theorem says that $A(K)/mA(K)$ is finite for an integer $m \ge 2$.
It is proved with Kummer theory and the finiteness of the class group and of the unit group modulo $m$-th powers.
Then a [[height function|F1FF505PWH]] on $A(K)$ has finitely many points of bounded height and satisfies $h(mP) \approx m^2 h(P)$.
A descent with it shows that representatives of $A(K)/mA(K)$ and the points of small height generate $A(K)$.
Mordell proved the case of an elliptic curve over $\Q$, which is [[Mordell's theorem|BZHGKN6Q3K]]; Weil proved the general case.
So $A(K) \cong \Z^r \oplus T$ with $T$ finite; the integer $r$ is the rank of $A$ over $K$.
The hypothesis on $K$ matters.
Over $\overline{\Q}$ the group $E(\overline{\Q})$ of an [[elliptic curve|427N6S6QQW]] contains points of every finite order, so its torsion subgroup is infinite and it is not finitely generated.
Over $\R$ the group $E(\R)$ is uncountable.

## Explanations

### L1 human

An abelian variety, such as an elliptic curve, has a group law on its points.
The theorem says that over a number field all rational points are built from finitely many of them with this group law.
So an infinite set of solutions to a system of equations has a finite description.

### L2 human

The theorem gives the rank and the torsion subgroup as the basic invariants of $A(K)$.
The rank is hard to compute: the descent gives only an upper bound through the Selmer group, and the gap is measured by the Tate–Shafarevich group.
The [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]] predicts the rank from an $L$-function.
Let $C$ be a curve of genus at least $2$ embedded in its Jacobian.
The theorem places $C(K)$ inside a finitely generated group, and [[Faltings's theorem|H0DB07WM20]] shows that $C(K)$ is finite.
</document>

<document source="https://tamnd.github.io/math2/node/MXX766RYDX.md" tag="MXX766RYDX" kind="definition">
# Complex Monge–Ampère operator

## Definition

Let $\Omega \subseteq \C^n$ be open and let $u \colon \Omega \to \R$ be of class $C^2$.
Let $d^c = i(\bar\partial - \partial)$, so that $dd^c = 2i \partial\bar\partial$.
The *complex Monge–Ampère operator* applied to $u$ is the $(n, n)$-form
$$(dd^c u)^n = dd^c u \wedge \cdots \wedge dd^c u \quad (n \text{ factors}).$$
In coordinates,
$$(dd^c u)^n = 4^n \, n! \, \det\left( \frac{\partial^2 u}{\partial z_j \partial \bar z_k} \right)_{j,k=1}^n dV,$$
where $dV$ is Lebesgue measure on $\C^n = \R^{2n}$.

## Remarks

The constant $4^n n!$ depends on the normalisation of $d^c$; many authors use $d^c = \frac{i}{4\pi}(\bar\partial - \partial)$ or write simply $\det\left(\partial^2 u / \partial z_j \partial \bar z_k\right)$.
If $u$ is [[plurisubharmonic|FF2P7ER3QT]], the complex Hessian is positive semidefinite, so $(dd^c u)^n$ is a positive measure.
Example: for $u(z) = \abs{z}^2 = \sum_j \abs{z_j}^2$ the complex Hessian is the identity, so $(dd^c u)^n = 4^n n! \, dV$.
For $n = 1$, $dd^c u = 4 \, \frac{\partial^2 u}{\partial z \partial \bar z} \, dV = \Delta u \, dV$, the Laplacian.
Non-example of positivity: for $u = \abs{z_1}^2 - \abs{z_2}^2$ on $\C^2$ the determinant is $-1$, so $(dd^c u)^2$ is a negative measure; this $u$ is not plurisubharmonic.
The operator is nonlinear for $n \ge 2$.
The extension of $(dd^c u)^n$ to locally bounded plurisubharmonic functions, where $u$ need not be differentiable, is the [[Bedford–Taylor theory|EEHQSWFHSB]].
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# Semisimple ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with $1$.
The ring $R$ is *(left) semisimple* if $R$, viewed as a left [[module|0NZ8JA25WK]] over itself, is a [[semisimple module|N3MRHDKDGC]].
That is, $R$ is a direct sum of minimal nonzero left ideals.

## Remarks

Example: for a field $F$, the ring $R = M_2(F)$ is semisimple.
The left ideals $L_1$ and $L_2$ of matrices that vanish outside the first and the second column are minimal, and $R = L_1 \oplus L_2$.

Non-example: $\Z$ is not semisimple.
A minimal nonzero ideal of $\Z$ would be some $n\Z$ with $n \ne 0$, but $2n\Z$ lies strictly inside it, so $\Z$ has no minimal nonzero ideal.

A ring is left semisimple if and only if it is right semisimple, so one simply says semisimple.
Equivalent conditions: every left $R$-module is semisimple; every left $R$-module is projective; $R$ is left [[Artinian|20QEAM8AX2]] and its [[Jacobson radical|XA0DCNH7WY]] is $0$.
A semisimple ring is a finite direct sum of minimal left ideals, because $1$ lies in a finite partial sum.
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<document source="https://tamnd.github.io/math2/node/MXZHMC2P99.md" tag="MXZHMC2P99" kind="definition">
# Perimeter of a set

## Definition

Let $n \ge 1$, let $\Omega \subseteq \R^n$ be an open set, and let $E \subseteq \R^n$ be a [[Lebesgue measurable|2Z2QVWZ262]] set.
Its *perimeter in $\Omega$* is the supremum below.
$$P(E, \Omega) = \sup \set{ \int_E \operatorname{div} \varphi \, dx : \varphi \in C^1_c(\Omega; \R^n), \ \abs{\varphi(x)} \le 1 \text{ for all } x \in \Omega }.$$
Here $\operatorname{div}$ is the [[divergence|0MWV6ZTK6S]].
The set $C^1_c(\Omega; \R^n)$ is the space of continuously differentiable vector fields with compact support in $\Omega$.
The value $P(E, \Omega)$ lies in $[0, \infty]$, and $P(E) = P(E, \R^n)$ is the *perimeter* of $E$.

## Remarks

Example: let $E$ be a bounded open set with $C^1$ boundary and outer unit normal $\nu$.
By the divergence theorem, $\int_E \operatorname{div} \varphi \, dx = \int_{\partial E} \varphi \cdot \nu \, d\mathcal{H}^{n-1} \le \mathcal{H}^{n-1}(\partial E)$.
Extending $\nu$ to a continuous field of norm at most $1$ and smoothing it shows that the supremum equals $\mathcal{H}^{n-1}(\partial E)$, the area of the boundary.
So the unit disc in $\R^2$ has perimeter $2\pi$.

The perimeter is not the size of the topological boundary.
The set $E = \Q^2 \cap [0, 1]^2$ has topological boundary $[0, 1]^2$, of positive area.
But $E$ has Lebesgue measure zero, so every integral above vanishes and $P(E) = 0$.

Equivalently, $P(E, \Omega)$ is the total variation $\abs{D \chi_E}(\Omega)$ of the distributional gradient of the indicator function $\chi_E$.
It is finite exactly when $\chi_E$ has bounded variation in $\Omega$.
Changing $E$ on a set of measure zero does not change $P(E, \Omega)$.
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<document source="https://tamnd.github.io/math2/node/MY2XG7XTS8.md" tag="MY2XG7XTS8" kind="definition">
# Octahedral axiom

## Definition

Let $\mathcal{T}$ be an [[additive category|B2VP96NJSK]] with an [[additive|KAN07YC9T2]] automorphism $\Sigma$.
Let a class of distinguished triangles $X \to Y \to Z \to \Sigma X$ be given that satisfies the axioms TR1, TR2 and TR3 of a triangulated category.
The class satisfies the *octahedral axiom* (TR4) if the following holds.
Let $f \colon X \to Y$ and $g \colon Y \to Z$ be morphisms, and let
$$X \xrightarrow{f} Y \xrightarrow{i} Z' \xrightarrow{i'} \Sigma X, \qquad Y \xrightarrow{g} Z \xrightarrow{j} X' \xrightarrow{j'} \Sigma Y, \qquad X \xrightarrow{g f} Z \xrightarrow{k} Y' \xrightarrow{k'} \Sigma X$$
be distinguished triangles.
Then there are morphisms $a \colon Z' \to Y'$ and $b \colon Y' \to X'$ such that

1. $Z' \xrightarrow{a} Y' \xrightarrow{b} X' \xrightarrow{(\Sigma i) j'} \Sigma Z'$ is a distinguished triangle;
2. $a i = k g$ and $k' a = i'$;
3. $b k = j$ and $j' b = (\Sigma f) k'$.

## Remarks

Here $Z'$, $X'$ and $Y'$ are cones of $f$, $g$ and $g f$.
Example: take $f = \id_X$, so that $Z' = 0$ with the triangle $X \xrightarrow{\id} X \to 0 \to \Sigma X$, and take the second and third triangles equal.
Then $a = 0$ and $b = \id_{X'}$ work: $0 \to X' \xrightarrow{\id} X' \to 0$ is distinguished, being a rotation of $X' \xrightarrow{\id} X' \to 0 \to \Sigma X'$; $a i = 0 = k g$ because $k g$ is a composite of consecutive maps of a distinguished triangle; and $k' a = 0 = i'$, $b k = k = j$, $j' b = j' = k' = (\Sigma f) k'$.
In the homotopy category of chain complexes the axiom holds, with $Y'$ and the maps $a$ and $b$ built from mapping cones.
The axiom is a form of the isomorphism $(Z/X)/(Y/X) \cong Z/Y$ for $X \subseteq Y \subseteq Z$: read $Z'$ as $Y/X$, $Y'$ as $Z/X$ and $X'$ as $Z/Y$.
</document>

<document source="https://tamnd.github.io/math2/node/MY3CBHARYQ.md" tag="MY3CBHARYQ" kind="example">
# Wallpaper group

## Example

A wallpaper group is a [[crystallographic group|7KTJKFQT2N]] of dimension $2$: a discrete group of isometries of the plane $\R^2$ with compact quotient.
A concrete one, called $p4$, is the group $\Gamma$ of all maps
$$v \mapsto R^k v + t, \qquad k \in \set{0, 1, 2, 3}, \ t \in \Z^2,$$
where $R(x, y) = (-y, x)$ is the rotation by a quarter turn about the origin.
The set $\Gamma$ is a group, because $R$ maps $\Z^2$ onto itself, so composites and inverses of such maps have the same form.
It is discrete, because its elements have only four linear parts and their translation parts lie in $\Z^2$.
The quotient is compact, because every orbit meets the unit square $[0, 1]^2$, already under the translations alone.
The translation subgroup is $\Z^2$, of index $4$, as part 1 of the [[Bieberbach theorems|YE87R5CDJY]] predicts.

## Remarks

Up to isomorphism, and also up to affine change of coordinates, there are exactly $17$ wallpaper groups.

## Explanations

### L1 human

A wallpaper group is the symmetry group of a pattern that repeats in two directions across the whole plane, like tiles on a bathroom floor.
The pattern $p4$ is a grid of square tiles each marked with a pinwheel, so it can be turned by a quarter turn but not reflected.
Although such patterns come in endless designs, only $17$ different symmetry groups occur.
</document>

<document source="https://tamnd.github.io/math2/node/MY491MR2PJ.md" tag="MY491MR2PJ" kind="statement">
# Hermite–Genocchi formula

## Statement

Let $n \ge 1$, let $I \subseteq \R$ be an interval, and let $f \colon I \to \R$ be $n$ times continuously differentiable.
Let $x_0, \dots, x_n \in I$ be pairwise distinct.
Let $\Sigma_n = \set{(t_1, \dots, t_n) \in \R^n : t_i \ge 0, \ t_1 + \cdots + t_n \le 1}$ and set $t_0 = 1 - t_1 - \cdots - t_n$.
Then the [[divided difference|N5ZHV4HWDC]] of $f$ is
$$f[x_0, \dots, x_n] = \int_{\Sigma_n} f^{(n)}(t_0 x_0 + t_1 x_1 + \cdots + t_n x_n) \, dt_1 \cdots dt_n,$$
where the integral is with respect to [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$.

## Remarks

The point $t_0 x_0 + \cdots + t_n x_n$ runs over the convex hull of the nodes, so the integrand is defined.
For $n = 1$ the formula reads $\frac{f(x_1) - f(x_0)}{x_1 - x_0} = \int_0^1 f'(x_0 + t(x_1 - x_0)) \, dt$, which is the fundamental theorem of calculus.
The general case follows by induction on $n$: integrating over the last variable and using the case $n - 1$ reproduces the recursion that defines divided differences.

Since $\Sigma_n$ is a [[simplex|YX1DJK2QAG]] of volume $1/n!$, the formula gives $f[x_0, \dots, x_n] = f^{(n)}(\xi)/n!$ for some $\xi$ between the smallest and largest node.
This is a form of the [[mean value theorem for divided differences|BQJ9C1CD5X]].
The right side is a continuous function of $(x_0, \dots, x_n)$ even when nodes coincide.
So it serves as the definition of divided differences with repeated nodes, which is what [[Hermite interpolation|9Z50M2WPTP]] uses.

## Explanations

### L1 human

A divided difference is a ratio of differences of values.
The formula says that it is also an average of a high derivative of $f$, taken over all weighted mixtures of the nodes.
So divided differences are smoothed derivatives.

### L2 human

The formula shows at once that $f[x_0, \dots, x_n]$ is symmetric in the nodes and depends continuously on them.
Letting all nodes tend to one point $x$ gives $f[x, \dots, x] = f^{(n)}(x)/n!$, the Taylor coefficient.
It is the bridge between [[Newton interpolation|9W77FPTM9E]] and [[Taylor polynomials|0HFSFVMWQB]].
</document>

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# Perfect field

## Definition

Let $K$ be a field.
The field $K$ is *perfect* if one of the following holds:

1. $K$ has [[characteristic|0P7CH1CND5]] $0$;
2. $K$ has characteristic $p > 0$ and the [[Frobenius endomorphism|0GHDDVAH8P]] $x \mapsto x^p$ of $K$ is surjective, that is, $K^p = K$.

## Remarks

A field $K$ is perfect if and only if every [[irreducible polynomial|00X2A9G9W3]] in $K[x]$ is [[separable|03J8BS7FMV]], if and only if every algebraic extension of $K$ is separable.

Example: a [[finite field|08RTDHX1SK]] $\F_q$ of characteristic $p$ is perfect.
Check: the Frobenius map is an injective ring homomorphism, and an injective map from a finite set to itself is surjective.

Non-example: $K = \F_p(t)$ is not perfect.
The element $t$ is not a $p$-th power.
If $(f/g)^p = t$ with polynomials $f, g$, then $f^p = t g^p$.
The degree of the left side is divisible by $p$, while the degree of the right side is not.
Accordingly, $x^p - t$ is irreducible over $K$ but not separable.
</document>

<document source="https://tamnd.github.io/math2/node/MY8ZZP84NP.md" tag="MY8ZZP84NP" kind="concept">
# Censoring

## Summary

Censoring is the situation in which the value of a measurement is only partly known.
One learns that it lies in some interval, but not its exact value.
The most common case is right censoring of event times, where one only knows that the event had not happened by the end of follow-up.
Ignoring censored observations, or treating them as exact, biases the estimates.
Methods of [[survival analysis|443ZWDX88P]], such as the [[Kaplan–Meier estimator|B7AJ8RWZRC]], are built to use censored data correctly.

## Explanations

### L1 human

A study of how long marriages last runs for ten years.
For couples still married at the end, we do not know the length of the marriage, only that it is more than their time in the study.
Such a record is censored on the right.
If a scale only reads up to $150$ kg, a reading of "over $150$" is censored in the same way.
If an infection is only detected at a check-up, we know it happened between two visits: that is interval censoring.
</document>

<document source="https://tamnd.github.io/math2/node/MYGWT305AB.md" tag="MYGWT305AB" kind="concept">
# Unit and counit

## Summary

The unit and the counit of an adjunction $F \dashv G$ are the two [[natural transformations|KYCPCC57FT]] $\eta \colon \mathrm{Id} \Rightarrow G F$ and $\varepsilon \colon F G \Rightarrow \mathrm{Id}$ that it determines.
They satisfy the two triangle identities, and conversely they determine the [[adjunction|TZHF6ZM92S]].
The unit and the counit give the [[monad|FF6XXA6D1A]] and the comonad of the adjunction.

## Explanations

### L1 human

For the free group and the forgetful functor, the unit sends each element of a set $X$ to the corresponding generator of the free group on $X$.
The counit takes the free group on the elements of a group $H$ and evaluates each word by multiplying in $H$.
So the unit says how to put things in, and the counit says how to compute the result.
The triangle identities say that putting in and then computing gives back what you started with.
</document>

<document source="https://tamnd.github.io/math2/node/MYQN2FX9SB.md" tag="MYQN2FX9SB" kind="definition">
# Morse function

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $f \colon M \to \R$ be a [[smooth function|0V33BNKZ9S]].
A point $p \in M$ is a [[critical point|0YTCMFB4HC]] of $f$ if $df_p = 0$.
A critical point $p$ is *nondegenerate* if, in some chart $(x_1, \dots, x_n)$ around $p$, the [[Hessian matrix|0693Y2METT]] $\big(\partial^2 f / \partial x_i \partial x_j(p)\big)$ is invertible.
The function $f$ is a *Morse function* if every critical point of $f$ is nondegenerate.
The *index* of a nondegenerate critical point is the number of negative eigenvalues of its Hessian.

## Remarks

At a critical point the Hessian transforms under a change of chart by $H \mapsto A^T H A$ with $A$ invertible, so nondegeneracy and the index do not depend on the chart.
Example: $f \colon \R^2 \to \R$, $f(x, y) = x^2 - y^2$, is a Morse function.
Its only critical point is the origin, where the Hessian is $\begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix}$, which is invertible; the index is $1$.
Non-example: $g \colon \R \to \R$, $g(x) = x^3$, is not a Morse function.
Its critical point $0$ has Hessian $g''(0) = 0$, which is not invertible.
Nondegenerate critical points are isolated, so on a compact manifold a Morse function has only finitely many critical points.
On a compact manifold the Morse functions form an open dense subset of $C^\infty(M, \R)$.
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<document source="https://tamnd.github.io/math2/node/MYV657YRDW.md" tag="MYV657YRDW" kind="concept">
# Spectral method

## Summary

A spectral method approximates the solution of a differential equation by a finite sum of smooth global functions, such as trigonometric functions or [[Chebyshev polynomials|M6EAR1JHPP]].
The coefficients are fixed by a [[Galerkin|Y5B0PEBKE8]] condition, a tau condition or collocation at chosen points.
For smooth solutions the error decreases faster than any power of the number of terms, often exponentially.
So they are the method of choice for smooth problems on simple domains.
Examples are turbulence in periodic boxes and weather models on the sphere.
See the [[definition|QFJEFGR171]].

## Explanations

### L1 human

A finite element method builds the solution from many small local pieces.
A spectral method instead uses a few waves that each stretch over the whole domain, as in a [[Fourier series|G12KXDHPPS]].
Derivatives of waves are easy: differentiating $e^{ikx}$ just multiplies it by $ik$.
For a smooth solution, the wave amplitudes drop very fast, so a few dozen waves can give many correct digits.
</document>

<document source="https://tamnd.github.io/math2/node/MZ02BKPHJA.md" tag="MZ02BKPHJA" kind="concept">
# Local ring

## Summary

A local ring is a [[commutative ring|0310REJJY8]] with exactly one [[maximal ideal|0PV74TR5FP]].
Equivalently, its non-[[units|0XVEZDE9J8]] form an ideal.
Local rings arise as [[localizations at prime ideals|X6JD4AG973]] and describe the behaviour of a ring, or of a geometric space, near one point.
Many theorems of commutative algebra are first proved for local rings, where [[Nakayama's lemma|15JHBYR6KH]] is a strong tool.

## Explanations

### L1 human

A local ring is a ring that "sees only one point".
A typical example is the ring of rational numbers whose denominators are odd: an element is invertible exactly when its numerator is odd too.
The elements that are not invertible, those with even numerator, form one ideal, the maximal ideal.
Another example is the ring of formal power series $k[\![x]\!]$, where a series is invertible exactly when its constant term is nonzero.
The [[definition|WJS9TPSGVM]] gives the condition and its equivalent forms.
</document>

<document source="https://tamnd.github.io/math2/node/MZ6XDATQKR.md" tag="MZ6XDATQKR" kind="statement">
# Chevalley's structure theorem

## Statement

Let $k$ be a perfect field.
Let $G$ be a smooth connected [[algebraic group|GH81SYAPPN]] over $k$.
Then there is a unique smooth connected [[normal subgroup|0CB0DX5BVV]] $H$ of $G$ such that:

1. $H$ is a [[linear algebraic group|GA81VVQHEP]];
2. the [[quotient|0KYX2N69J0]] $G / H$ is an [[abelian variety|R7W03TXRS9]].

## Remarks

Uniqueness is the easy part.
If $H$ and $H'$ both satisfy the conclusion, the image of $H$ in $G / H'$ is a connected linear algebraic group and also a closed subvariety of an abelian variety.
So it is both affine and complete, hence a point, and $H \subseteq H'$.
By symmetry $H = H'$.
Existence is the deep part of the theorem.

Example: let $E$ be an elliptic curve over $k$ and $G = E \times \mathbb{G}_m$.
Then $H = \set{0} \times \mathbb{G}_m$ is linear, and $G / H \cong E$ is an abelian variety.

Over imperfect fields the statement can fail, so the hypothesis that $k$ is perfect cannot simply be dropped.

## Explanations

### L1 human

Every connected algebraic group is assembled from two kinds of groups that are opposite in nature.
One kind is a group of matrices, which is affine.
The other kind is an abelian variety, which is complete.
The theorem says the matrix part sits inside as a normal subgroup, with an abelian variety as quotient.

### L2 human

The theorem reduces many questions about algebraic groups to the two separate theories of linear algebraic groups and abelian varieties.
For example, a connected algebraic group that is affine has trivial abelian quotient, and one that is complete has trivial linear part.
A standard family of examples are the generalized Jacobians of singular curves.
Such a group is an extension of the Jacobian of the normalized curve by a linear group.
</document>

<document source="https://tamnd.github.io/math2/node/MZHSEZDK3E.md" tag="MZHSEZDK3E" kind="definition">
# Integral equation of the third kind

## Definition

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ be a [[kernel|GSMBXT7FPK]], let $f \colon [a, b] \to \C$ be given, and let $\lambda \in \C$.
Let $a_0 \colon [a, b] \to \C$ be a [[continuous function|0X3WGX95FE]] that has at least one zero in $[a, b]$ and is not identically zero.
An *integral equation of the third kind* is a [[linear integral equation|AYZCKG34QK]] of the form
$$a_0(x) \varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds \quad \text{for all } x \in [a, b],$$
for the unknown function $\varphi \colon [a, b] \to \C$.

## Remarks

Example: $x \varphi(x) = x + \int_{-1}^{1} s \, \varphi(s) \, ds$ on $[-1, 1]$, with $a_0(x) = x$, $f(x) = x$, $K(x, s) = s$ and $\lambda = 1$.
The coefficient $a_0$ is continuous, vanishes at $0$, and is not identically zero.
The function $\varphi = 1$ is a solution, since $\int_{-1}^{1} s \, ds = 0$.

Non-example: $(2 + x) \varphi(x) = f(x) + \int_{-1}^{1} \varphi(s) \, ds$ on $[-1, 1]$ is not of the third kind, since $2 + x$ has no zero there.
Dividing by $2 + x$ turns it into an equation of the second kind.

Some authors also allow the upper limit $x$ in place of $b$.
</document>

<document source="https://tamnd.github.io/math2/node/MZM1AFXY82.md" tag="MZM1AFXY82" kind="statement">
# Interchange of summation order

## Statement

Let $(a_{m,n})_{m, n \in \N}$ be a [[double sequence|WK5V372TG3]] of complex numbers with
$$\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \abs{a_{m,n}} < \infty.$$
Then all the series below converge [[absolutely|0W3K3QPEWT]], and
$$\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} a_{m,n} = \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} a_{m,n}.$$

## Remarks

By the nonnegative case of [[Fubini's theorem for series|WW53KG71KH]], the hypothesis says that the family $(a_{m,n})$ over $\N \times \N$ is summable.
The same theorem, applied once with rows first and once with columns first, shows that both iterated sums equal the sum of the family.
The hypothesis cannot be dropped.
Let $a_{m,m} = 1$, $a_{m,m+1} = -1$ for all $m$, and $a_{m,n} = 0$ otherwise.
Each row sums to $0$, so $\sum_m \sum_n a_{m,n} = 0$.
Column $0$ sums to $1$ and every other column sums to $1 - 1 = 0$, so $\sum_n \sum_m a_{m,n} = 1$.
Example of use: with the [[Riemann zeta function|1XWBY4N0VT]], $\sum_{k \ge 2} (\zeta(k) - 1) = \sum_{k \ge 2} \sum_{n \ge 2} n^{-k}$.
Swapping the nonnegative sums gives $\sum_{n \ge 2} \sum_{k \ge 2} n^{-k} = \sum_{n \ge 2} \frac{1}{n(n-1)} = 1$.

## Explanations

### L1 human

For an infinite table of numbers, summing rows first or columns first gives the same total when the absolute values have a finite total.
Without this condition, the two orders can give different answers.

### L2 human

The result is used to swap two sums, which is one of the most common steps in analysis and number theory.
To apply it, bound the double sum of absolute values; for nonnegative terms the swap is always allowed, possibly with value $+\infty$.
The counterexample above shows why conditionally convergent double sums need care.
</document>

<document source="https://tamnd.github.io/math2/node/MZNFPJJMY9.md" tag="MZNFPJJMY9" kind="definition">
# Semimartingale

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_t)_{t \ge 0}$ that is right-continuous and complete.
A function $a \colon [0, \infty) \to \R$ has finite variation if $\sup \sum_{k} \abs{a(t_k) - a(t_{k-1})} < \infty$ on each interval $[0, t]$, the supremum being over finite partitions $0 = t_0 < \dots < t_m = t$.
A real [[adapted|YRNFE016FF]] process $(X_t)_{t \ge 0}$ with càdlàg paths is a *semimartingale* if it can be written as
$$X_t = X_0 + M_t + A_t, \qquad t \ge 0,$$
where:

1. $(M_t)$ is a [[local martingale|ZWJ9G89W3T]] with $M_0 = 0$.
2. $(A_t)$ is an adapted càdlàg process with $A_0 = 0$ whose paths have finite variation almost surely.

## Remarks

Brownian motion with drift, $X_t = W_t + \mu t$ with $W$ a Wiener process and $\mu \in \R$, is a semimartingale: $M = W$ is a martingale and $A_t = \mu t$ has finite variation.
[[Fractional Brownian motion|VC6R7XSKRQ]] with Hurst index $H \ne 1/2$ is not a semimartingale for its own filtration, so it admits no such decomposition.
The decomposition is not unique in general; it is unique when $A$ is required to be predictable and such a decomposition exists, as for continuous semimartingales.
By the Bichteler–Dellacherie theorem, semimartingales are exactly the good integrators: the adapted càdlàg processes for which the stochastic integral of simple predictable processes extends continuously.
Every semimartingale has a [[quadratic variation|ZBYC3MFW89]], and [[Itô's lemma|2J1RVQB0CM]] holds for $C^2$ functions of semimartingales.
</document>

<document source="https://tamnd.github.io/math2/node/MZT3Y1X53W.md" tag="MZT3Y1X53W" kind="concept">
# Analytic continuation in several variables

## Summary

Analytic continuation in several variables extends a [[holomorphic function|507C34W0ZT]] on an open set of $\C^n$ step by step through overlapping domains.
It generalises [[analytic continuation|0M3YQJ81NB]] in one variable, and by the identity theorem each step is unique.
As in one variable, continuation along different paths can lead to different values.
So the result lives on a domain spread over $\C^n$ rather than on a subset of $\C^n$.
Unlike in one variable, for $n \ge 2$ every holomorphic function on certain domains continues to a strictly larger domain.

## Explanations

### L1 human

Start with a holomorphic function on a small region and try to push it further, like extending a road piece by piece.
Each new piece is forced: there is only one holomorphic way to extend on an overlap.
In several variables something new happens: for some regions, such as a ball with a hole, every function can be extended across the hole.
For example, a holomorphic function on $\set{z \in \C^2 : 1 < \abs{z} < 2}$ always continues to the whole ball $\abs{z} < 2$.
</document>

<document source="https://tamnd.github.io/math2/node/MZT9D34B0A.md" tag="MZT9D34B0A" kind="concept">
# Analytic torsion

## Summary

Analytic torsion, or Ray–Singer torsion, is a spectral invariant of a closed Riemannian manifold with an orthogonal or unitary representation of its fundamental group.
It is an alternating product of [[zeta-regularized determinants|YT3ZQ2F7FG]] of the Laplacians on forms with values in the associated flat bundle.
Ray and Singer introduced it as an analytic counterpart of Reidemeister torsion, a combinatorial invariant that can distinguish homotopy equivalent lens spaces.
The [[Cheeger–Müller theorem|C9DT4Y63CY]] says that the two torsions agree.

## Explanations

### L1 human

Homology counts holes, and when all homology with twisted coefficients vanishes, a finer invariant called torsion still remains.
Reidemeister torsion computes it from a triangulation.
Analytic torsion computes it instead from the vibration frequencies of the manifold, the eigenvalues of Laplacians on forms.
On a circle with a twist by the unit complex number $e^{2\pi i a}$, both give $2\sin(\pi a)$.
</document>

<document source="https://tamnd.github.io/math2/node/MZW52JVTBX.md" tag="MZW52JVTBX" kind="definition">
# Infinitary operation

## Definition

Let $A$ be a set and let $I$ be an infinite set.
Write $A^I$ for the [[Cartesian product|09CEDJYBKP]] of copies of $A$ indexed by $I$, the set of families $(a_i)_{i \in I}$ of elements of $A$.
An *infinitary operation* on $A$ with index set $I$ is a [[function|0EPAT51DXQ]] $f \colon A^I \to A$.
Its [[arity|MSXX2P86CY]] is the cardinal $\abs{I}$.

## Remarks

Example: on $[0, 1]$, the map $s \colon [0,1]^{\N} \to [0, 1]$, $s((x_n)_{n \in \N}) = \sup_n x_n$, is an infinitary operation of arity $\aleph_0$.
Each sequence in $[0, 1]$ is bounded above by $1$, so its supremum exists and lies in $[0, 1]$.

Non-example: the map $(x_n)_{n \in \N} \mapsto \sum_{n} x_n$ is not an infinitary operation on $\R$, because it is not defined on the whole of $\R^{\N}$: the series $\sum_n 1$ diverges.

Some authors index the arguments by an infinite ordinal $\alpha$ instead of an arbitrary set $I$, and call $\alpha$ the arity.
</document>

<document source="https://tamnd.github.io/math2/node/MZXHSGFBRF.md" tag="MZXHSGFBRF" kind="statement">
# Inverse Mellin transform as inverse Fourier transform

## Statement

Let $c \in \R$ and let $\varphi$ be a complex function on the line $\operatorname{Re} s = c$ such that $y \mapsto \varphi(c + iy)$ is [[integrable|X7GTF3D98F]] on $\R$.
Define $\psi \colon \R \to \C$ by $\psi(t) = \varphi(c + 2\pi i t)$.
Then for every $x > 0$,
$$\mathcal{M}^{-1}_c \varphi(x) = x^{-c} \, \check{\psi}(-\log x),$$
where $\mathcal{M}^{-1}_c$ is the [[inverse Mellin transform|CHESR1W1CY]] along $\operatorname{Re} s = c$, and $\check{\psi}(u) = \int_{\R} \psi(t) e^{2\pi i u t} \, dt$ is the [[inverse Fourier transform|JB8QMT8DCH]].

## Remarks

Substitute $y = 2\pi t$ in the definition:
$$\mathcal{M}^{-1}_c \varphi(x) = \frac{1}{2\pi} \int_{\R} x^{-c - iy} \varphi(c + iy) \, dy = x^{-c} \int_{\R} e^{-2\pi i t \log x} \, \psi(t) \, dt,$$
and the last integral is $\check{\psi}(-\log x)$.
The function $\psi$ is integrable because $\int \abs{\psi(t)} \, dt = \frac{1}{2\pi} \int \abs{\varphi(c + iy)} \, dy$.
Example: $\varphi = \Gamma$ and $c = 1$ give $x^{-1} \check{\psi}(-\log x) = e^{-x}$, that is, $\check{\psi}(u) = e^{-u} e^{-e^{-u}}$ for $\psi(t) = \Gamma(1 + 2\pi i t)$.
Combined with the statement that the Mellin transform is a Fourier transform in logarithmic variables, this reduces the Mellin inversion theorem to the [[Fourier inversion theorem|T4Z8YQZKZV]].

## Explanations

### L1 human

The inverse Mellin transform is the inverse Fourier transform in disguise.
One rescales the vertical line so that it becomes the real line, takes the inverse Fourier transform, reads it at the point $-\log x$, and multiplies by the weight $x^{-c}$.

### L2 human

This identity is the inverse half of the dictionary between Mellin and Fourier analysis on $\R$.
It turns questions about the inverse Mellin integral, such as continuity in $x$, decay or behaviour under smoothing, into the corresponding questions for the inverse Fourier transform.
For instance, it shows at once that $x \mapsto x^{c} \mathcal{M}^{-1}_c \varphi(x)$ is bounded and continuous on $(0, \infty)$.
</document>

<document source="https://tamnd.github.io/math2/node/MZZ56W92DY.md" tag="MZZ56W92DY" kind="definition">
# Mittag-Leffler function

## Definition

Let $\alpha, \beta \in \C$ with $\operatorname{Re} \alpha > 0$, and let $\Gamma$ be the [[gamma function|0H5SYBSED9]], with the convention $1/\Gamma(-m) = 0$ for $m = 0, 1, 2, \ldots$.
The *two-parameter Mittag-Leffler function* is the [[power series|0MZB06ST2B]]
$$E_{\alpha, \beta}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}, \qquad z \in \C.$$
The *Mittag-Leffler function* is $E_\alpha = E_{\alpha, 1}$.

## Remarks

The series has infinite radius of convergence, because $\Gamma(\alpha k + \beta)$ grows faster than any power of $k$ when $\operatorname{Re} \alpha > 0$, so $E_{\alpha, \beta}$ is an [[entire function|0G4F0XB9X2]].

Example: for $\alpha = \beta = 1$, $\Gamma(k + 1) = k!$, so $E_{1,1}(z) = e^z$.
Also $E_2(z) = \sum_k \frac{z^k}{(2k)!} = \cosh \sqrt{z}$, and $E_{1/2}(z) = e^{z^2} \operatorname{erfc}(-z)$.

Non-example: for $\alpha = 0$ and $\beta = 1$ the series is $\sum_k z^k$, which is not entire; it converges only for $\abs{z} < 1$, so the condition $\operatorname{Re} \alpha > 0$ cannot be dropped.

Laplace transform: for $\operatorname{Re} s$ large enough,
$$\int_0^\infty e^{-st} \, t^{\beta - 1} E_{\alpha, \beta}(\lambda t^\alpha) \, dt = \frac{s^{\alpha - \beta}}{s^\alpha - \lambda},$$
which follows by integrating the series term by term.
</document>

<document source="https://tamnd.github.io/math2/node/N029V3JX60.md" tag="N029V3JX60" kind="concept">
# Proportional hazards model

## Summary

A proportional hazards model describes how covariates act on the [[hazard function|Y959NKQ3XW]] of an event time.
Each covariate multiplies the hazard by a factor that does not change with time.
In the Cox model, introduced by David Cox, the hazard is a baseline hazard times $e^{\beta^T x}$, and the baseline is left completely unspecified.
The coefficients $\beta$ can be estimated by maximising the partial likelihood, which does not involve the baseline hazard.
It is the most used regression model of [[survival analysis|443ZWDX88P]].

## Explanations

### L1 human

Suppose smokers have, at every age, twice the risk of a heart attack of non-smokers.
The risk itself may rise with age in a complicated way, but the ratio stays $2$.
A proportional hazards model says exactly this: each factor scales the whole hazard curve up or down by a fixed amount.
The doubling corresponds to a coefficient $\beta = \log 2$ for the smoking indicator.
The method can estimate this ratio without ever estimating the shape of the age curve.
</document>

<document source="https://tamnd.github.io/math2/node/N05N7W1BAB.md" tag="N05N7W1BAB" kind="definition">
# Jackson integral

## Definition

Let $0 < q < 1$, let $a, b \in \R$, and let $f$ be a real or complex function defined at the points $a q^k$ and $b q^k$ for all integers $k \ge 0$.
The *Jackson integral* of $f$ from $0$ to $a$ is
$$\int_0^a f(x) \, d_q x = (1 - q)\, a \sum_{k=0}^{\infty} q^k f(a q^k),$$
when the series converges.
The Jackson integral from $a$ to $b$ is
$$\int_a^b f(x) \, d_q x = \int_0^b f(x) \, d_q x - \int_0^a f(x) \, d_q x.$$

## Remarks

Example: for an integer $n \ge 0$,
$$\int_0^1 x^n \, d_q x = (1 - q) \sum_{k=0}^{\infty} q^{k(n+1)} = \frac{1 - q}{1 - q^{n+1}} = \frac{1}{[n+1]_q},$$
the [[q-analog|GFSDMJHHVA]] of $\int_0^1 x^n \, dx = \frac{1}{n+1}$.

Non-example: $f(x) = 1/x$ on $(0, 1]$ has no Jackson integral from $0$ to $1$, since the terms $q^k f(q^k) = 1$ do not tend to $0$.

If $f$ is continuous on $[0, a]$, the sum is a Riemann sum for the partition by the points $a q^k$, and it tends to $\int_0^a f(x)\, dx$ as $q \to 1^-$.
If $f$ is bounded near $0$, then $F(x) = \int_0^x f(t) \, d_q t$ satisfies $D_q F = f$ for the [[q-derivative|PMJT3FZE0Y]] $D_q F(x) = \frac{F(qx) - F(x)}{(q - 1) x}$, $x \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/N05WC9ZCWA.md" tag="N05WC9ZCWA" kind="concept">
# Displacement convexity

## Summary

A functional on probability measures is displacement convex if it is convex along the paths of optimal transport, rather than along straight-line mixtures of measures.
Robert McCann introduced the notion in 1997, and used it to prove uniqueness of minimizers for energies that are not convex in the usual sense.
In the language of the [[Wasserstein distance|VEY29CS26X]], displacement convexity is convexity along geodesics.
Key examples are potential energies with a convex potential, interaction energies with a convex kernel, and the entropy $\int \rho \log \rho$.

## Explanations

### L1 human

There are two ways to go from one cloud of sand to another.
One can fade the first cloud out while fading the second in, or one can move each grain along a straight line to its destination.
Displacement convexity asks for convexity along the second kind of path.
For example, measure the height of each grain by a bowl-shaped function and take the average.
This average is convex along such paths, since each grain moves on a straight line inside the bowl.
</document>

<document source="https://tamnd.github.io/math2/node/N087STXZKG.md" tag="N087STXZKG" kind="concept">
# Crisis

## Summary

A crisis is a bifurcation in which a chaotic [[attractor|7FWX24X5MW]] suddenly changes size or disappears as a parameter varies.
It happens when the attractor collides with an unstable periodic orbit or with its stable manifold.
In a boundary crisis the attractor touches the boundary of its basin and is destroyed; in an interior crisis it suddenly grows.
The notion was introduced by Grebogi, Ott and Yorke.

## Explanations

### L1 human

A chaotic system can wander inside a bounded region for ever, but the region itself depends on the parameters.
At a crisis the region abruptly bumps into an unstable orbit, a kind of trapdoor.
If the trapdoor leads out of the basin, orbits that used to stay chaotic forever now escape after a long chaotic transient.
If it leads to another part of space, the chaotic region suddenly becomes much larger.
</document>

<document source="https://tamnd.github.io/math2/node/N09YAJWA7E.md" tag="N09YAJWA7E" kind="definition">
# Torsion of a curve

## Definition

Let $I \subseteq \R$ be an interval and let $\gamma \colon I \to \R^3$ be a curve of class $C^3$ parametrised by [[arc length|00R1JDDV5X]] $s$, with [[curvature|7JQS6VKFEZ]] $\kappa(s) > 0$ for all $s \in I$.
Its unit tangent is $T = \gamma'$, its unit normal is $N = T' / \kappa$, and its binormal is $B = T \times N$, where $\times$ is the [[cross product|0SEDGXG9S3]].
The vector $B'(s)$ is a multiple of $N(s)$.
The *torsion* of $\gamma$ at $s$ is the number $\tau(s)$ with
$$B'(s) = -\tau(s) \, N(s).$$

## Remarks

For a regular $C^3$ curve with any parameter $t$ and $\gamma' \times \gamma'' \ne 0$, the torsion is
$$\tau = \frac{\det(\gamma', \gamma'', \gamma''')}{\norm{\gamma' \times \gamma''}^2}.$$
Example: take the helix $\gamma(t) = (a \cos t, a \sin t, bt)$ with $a > 0$ and $b \in \R$.
Then $\gamma' \times \gamma'' = (ab \sin t, -ab \cos t, a^2)$ and $\det(\gamma', \gamma'', \gamma''') = a^2 b$, so $\tau = b / (a^2 + b^2)$.
Non-example: on the curve $t \mapsto (t, t^3, 0)$ the curvature vanishes at $t = 0$, so $N$ and $B$ are not defined there and the torsion is not defined at that point.
Some books use the opposite sign convention $B' = \tau N$.
A curve with $\kappa > 0$ lies in a plane if and only if $\tau = 0$ everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/N0B51FAA9S.md" tag="N0B51FAA9S" kind="concept">
# Haar system

## Summary

The Haar system is an orthonormal basis of $L^2([0, 1])$.
It consists of the constant function $1$ and the rescaled, shifted copies of one step function.
This step function is $1$ on the left half and $-1$ on the right half of $[0, 1)$.
Alfréd Haar introduced it in 1910, as an orthonormal system in which every continuous function has a uniformly convergent expansion.
It is the simplest example of a [[wavelet|4PZ35C0GV3]] basis, and it is a [[Schauder basis|528G11SDWR]] of $L^p([0, 1])$ for $1 \le p < \infty$.

## Explanations

### L1 human

Start with the average of a function over $[0, 1]$.
Then record how the average over the left half differs from the average over the right half.
Then do the same inside each half, inside each quarter, and so on.
Each step uses one function that is $+1$ on one half of a dyadic interval and $-1$ on the other half, scaled to have length $1$ in $L^2$.
These numbers rebuild the function step by step, with finer and finer detail.
</document>

<document source="https://tamnd.github.io/math2/node/N0K0YWPQN6.md" tag="N0K0YWPQN6" kind="statement">
# Vitali–Hahn–Saks theorem

## Statement

Let $(X, \Sigma, \lambda)$ be a measure space with $\lambda$ a [[finite measure|63Z1Q4FS23]].
Let $(\mu_n)_{n \ge 1}$ be a sequence of [[complex measures|NY6TYHRGMS]] on $\Sigma$, each [[absolutely continuous|P0P21KDV6E]] with respect to $\lambda$.
Assume that for every $E \in \Sigma$ the limit $\mu(E) = \lim_{n \to \infty} \mu_n(E)$ exists in $\C$.
Then:

1. The $\mu_n$ are uniformly absolutely continuous with respect to $\lambda$: for every $\varepsilon > 0$ there is $\delta > 0$ such that $\lambda(E) < \delta$ implies $\abs{\mu_n(E)} < \varepsilon$ for all $n$.
2. The set function $\mu$ is a complex measure on $\Sigma$, and it is absolutely continuous with respect to $\lambda$.

## Remarks

A standard proof uses the Baire category theorem.
The measurable sets modulo $\lambda$-null sets form a complete metric space with distance $d(E, F) = \lambda(E \,\triangle\, F)$.
Each $\mu_n$ is continuous on this space, and a Baire argument on the sets where $\abs{\mu_n(E) - \mu_m(E)} \le \varepsilon$ for all $m, n \ge N$ gives the uniform bound.

Example: let $f_n \in L^1(\lambda)$ with $\int_E f_n \, d\lambda$ convergent for every $E$.
The theorem applied to $\mu_n = f_n \lambda$ shows that $(f_n)$ is uniformly integrable, which is a key step in the Dunford–Pettis theorem.

The hypothesis asks for convergence on every set of $\Sigma$.
Convergence on a generating family of sets, such as intervals, is not enough in general.

## Explanations

### L1 human

If a sequence of measures converges on every single set, then the limit is again a measure.
Moreover, the measures cannot concentrate mass on smaller and smaller sets at different speeds: they are controlled uniformly by the reference measure.

### L2 human

The theorem is the measure version of the uniform boundedness principle: pointwise convergence on a complete space forces a uniform estimate.
It is used to characterise weakly convergent and weakly compact sets in $L^1$.
The Nikodym convergence theorem is a version that needs no reference measure $\lambda$.
</document>

<document source="https://tamnd.github.io/math2/node/N0MAGPW8AP.md" tag="N0MAGPW8AP" kind="concept">
# Pluripolar set

## Summary

A pluripolar set is a subset of $\C^n$ that lies, at least locally, in the set where a [[plurisubharmonic function|FF2P7ER3QT]] takes the value $-\infty$.
Pluripolar sets are the negligible sets of pluripotential theory, in the same way as [[polar sets|C95QJ7YH5T]] are the negligible sets of classical potential theory.
They have Lebesgue measure zero, and plurisubharmonic functions bounded above extend across closed pluripolar sets.
By [[Josefson's theorem|V6XSDT5C21]], a set that is pluripolar locally is pluripolar for one plurisubharmonic function on all of $\C^n$.

## Explanations

### L1 human

The function $\log\abs{f}$ of a holomorphic function $f$ equals $-\infty$ exactly where $f$ vanishes.
So the zero set of a holomorphic function, such as the complex line $z_1 = 0$ in $\C^2$, is pluripolar.
Pluripolar sets are small sets: they cannot contain an open set, and they have volume zero.
A countable union of pluripolar sets is again pluripolar, so the union of countably many complex lines is pluripolar.
The real line $\R^n$ inside $\C^n$ is not pluripolar, although it has volume zero.
</document>

<document source="https://tamnd.github.io/math2/node/N0N7CMRCMH.md" tag="N0N7CMRCMH" kind="concept">
# Integrable system

## Summary

An integrable system is a [[Hamiltonian system|0EDKGA2FAK]] with as many independent conserved quantities in involution as it has degrees of freedom.
Such a system can be solved, in principle, by quadratures, and its motions on compact regular level sets are quasi-periodic motions on invariant tori.
The word is also used for infinite-dimensional systems, such as the Korteweg–De Vries equation.
These have infinitely many conserved quantities and can be solved by inverse scattering.
Integrable systems are the starting point of perturbation theory, where the question is what survives when the system is slightly changed.

## Explanations

### L1 human

Most mechanical systems cannot be solved by a formula, but integrable ones can.
They have enough conserved quantities, like energy and angular momentum, to pin down each motion.
Picture each motion as winding around the surface of a doughnut, at fixed speeds in each direction.
Two independent harmonic oscillators are integrable: the energy of each oscillator is conserved, and each one simply turns around a circle.
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<document source="https://tamnd.github.io/math2/node/N0QW1RGPR8.md" tag="N0QW1RGPR8" kind="definition">
# Resurgent function

## Definition

Let $\Omega$ be a closed discrete subset of $\C$.
A power series $\hat\varphi(\zeta) = \sum_{n \ge 0} c_n \zeta^n$ with positive radius of convergence is *$\Omega$-continuable* if it satisfies both conditions:

1. Its sum is a [[holomorphic function|0DG0Z72CQD]] on a disc $D$ centred at $0$ with $D \cap \Omega \subseteq \set{0}$.
2. It has an [[analytic continuation|0M3YQJ81NB]] along every path in $\C \setminus \Omega$ that starts in $D \setminus \set{0}$.

A formal series $\tilde\varphi(z) = \sum_{n \ge 0} a_n z^{-n-1}$ in the variable $z$ near $\infty$ is an *$\Omega$-resurgent series* if its formal Borel transform
$$\hat\varphi(\zeta) = \sum_{n \ge 0} \frac{a_n}{n!} \zeta^n$$
is $\Omega$-continuable.
The functions $\hat\varphi$ obtained in this way are the *resurgent functions* in the Borel plane.
The series $\tilde\varphi$ is a *resurgent series* if it is $\Omega$-resurgent for some closed discrete $\Omega$.

## Remarks

Example: $\tilde\varphi(z) = \sum_{n \ge 0} (-1)^n n! \, z^{-n-1}$ has Borel transform $\sum_{n \ge 0} (-1)^n \zeta^n = \frac{1}{1 + \zeta}$.
This converges on the unit disc and extends to a holomorphic function on $\C \setminus \set{-1}$, so $\tilde\varphi$ is $\set{-1}$-resurgent.
Non-example: $\sum_{n \ge 0} (n!)^2 z^{-n-1}$ is not resurgent, because its Borel transform $\sum_{n \ge 0} n! \, \zeta^n$ has radius of convergence $0$, so clause 1 fails.
Every convergent series $\sum a_n z^{-n-1}$ is resurgent, since then $\hat\varphi$ is entire.
Écalle's general notion allows the singular set to depend on the path of continuation; it is called endless continuability, and the $\Omega$-continuable germs are a special case.
The Borel transform of the formal product of two resurgent series is the convolution of their Borel transforms, and Écalle showed that resurgence is preserved under this product.
</document>

<document source="https://tamnd.github.io/math2/node/N155BSJ2DN.md" tag="N155BSJ2DN" kind="concept">
# Operator monotone function

## Summary

A real function $f$ on an interval is operator monotone if it keeps the order between [[Hermitian matrices|0P17NGQQMQ]]: from $A \le B$ it follows that $f(A) \le f(B)$.
Here $A \le B$ is the [[Loewner order|9G9S92T1VJ]], and $f(A)$ is defined through the eigenvalues of $A$.
Every operator monotone function is a [[monotone function|0A7KV2WB7A]], but the converse fails, so operator monotonicity is a much stronger condition.
[[Löwner's theorem|PA7ZQ6C6SP]] describes these functions through complex analysis.

## Explanations

### L1 human

For numbers, an increasing function keeps the order: if $a \le b$ then $f(a) \le f(b)$.
Hermitian matrices can also be compared: $A \le B$ means that $B - A$ has no negative eigenvalues.
An operator monotone function is one that keeps this matrix order, for matrices of every size.
The square root $\sqrt{t}$ and the logarithm $\log t$ on $(0, \infty)$ do this, while the square $t^2$ does not, even though it is increasing on $[0, \infty)$.
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<document source="https://tamnd.github.io/math2/node/N15YK7XG1H.md" tag="N15YK7XG1H" kind="concept">
# Large sieve

## Summary

The large sieve is a method of analytic number theory.
It applies when a set of integers is sifted by many residue classes modulo each prime.
Its core is an inequality for exponential sums: a trigonometric polynomial cannot be large at many well-spaced points at once.
In arithmetic form it bounds the size of a set of integers that misses many residue classes modulo many primes.
It is a key tool in the proof of the [[Bombieri–Vinogradov theorem|B3JMY54GJF]].

## Explanations

### L1 human

Classical sieves remove a few residue classes modulo each prime, for example the class $0$.
The large sieve handles the case where a large fraction of the classes is removed.
For example, the perfect squares fall into only $(p+1)/2$ of the $p$ residue classes modulo an odd prime $p$.
Its engine is a kind of uncertainty principle: a sum of waves $e^{2\pi i n \alpha}$ with $n$ in a short interval cannot be big at many points $\alpha$ that are far apart.
A set of integers that avoids many residue classes would make such a sum big at many fractions $a/q$, so such a set must be small.
</document>

<document source="https://tamnd.github.io/math2/node/N1H5NV5X0Z.md" tag="N1H5NV5X0Z" kind="definition">
# Uniserial module

## Definition

Let $R$ be a ring and let $M$ be a right $R$-[[module|0NZ8JA25WK]].
The module $M$ is *uniserial* if its set of [[submodules|05WDSN597C]] is [[totally ordered|0FRXBVVR03]] by inclusion.
That is, for any two submodules $N$ and $L$ of $M$, either $N \subseteq L$ or $L \subseteq N$.
Left uniserial modules are defined in the same way.

## Remarks

Example: for a prime $p$ and $n \ge 1$, the $\Z$-module $\Z/p^n\Z$ is uniserial.
Its submodules are the subgroups $p^i\Z/p^n\Z$ for $0 \le i \le n$, and $p^i\Z/p^n\Z \supseteq p^j\Z/p^n\Z$ when $i \le j$.
In the same way, $k[x]/(x^n)$ is uniserial as a module over itself, for a field $k$.

Non-example: the $\Z$-module $\Z$ is not uniserial.
Its submodules $2\Z$ and $3\Z$ are not comparable, since $2 \notin 3\Z$ and $3 \notin 2\Z$.

A nonzero uniserial module is indecomposable, since two nonzero submodules always meet in the smaller one, which is nonzero.
A uniserial module of finite length has exactly one composition series.
</document>

<document source="https://tamnd.github.io/math2/node/N1N5N2QJ86.md" tag="N1N5N2QJ86" kind="definition">
# Formalism

## Definition

*Formalism* is a position in the [[foundations of mathematics|EFFRJFW67Y]].
It holds that mathematical statements are strings of symbols in a formal system.
A statement is established when it is derived from the axioms of the system by its rules of inference.
The derivation is the whole content of a [[proof|0B94TKFQJX]], and a statement need not refer to abstract objects.

## Remarks

In a formal system for arithmetic, the string $S(0) + S(0) = S(S(0))$ is established by a derivation from the axioms; a formalist accepts it for that reason alone.
Some statements are true in the standard natural numbers but have no derivation in the system.
An example is a Gödel sentence of the [[incompleteness theorems|X9WYNYE29C]], and formalism does not count it as established.
Hilbert's version of formalism kept a contentful core of finitary mathematics.
It treated the rest as a formal system whose consistency should be proved by finitary means; this is [[Hilbert's program|P1RW6JWX73]].
</document>

<document source="https://tamnd.github.io/math2/node/N1NA5MEFWX.md" tag="N1NA5MEFWX" kind="definition">
# Empirical distribution function

## Definition

Let $X_1, \dots, X_n$ be real random variables, typically [[i.i.d.|0AJN7WHBJZ]] with [[cumulative distribution function|0MXMS3AE13]] $F$.
The *empirical distribution function* of the sample is the random function $\hat{F}_n \colon \R \to [0, 1]$ given by
$$\hat{F}_n(t) = \frac{1}{n} \sum_{i=1}^{n} \mathbf{1}\set{X_i \le t},$$
where $\mathbf{1}\set{X_i \le t}$ is $1$ if $X_i \le t$ and $0$ otherwise.

## Remarks

For each $\omega$, $\hat{F}_n$ is the distribution function of the empirical measure $\frac{1}{n}\sum_{i} \delta_{X_i(\omega)}$, so it is non-decreasing, right-continuous, and has limits $0$ and $1$ at $-\infty$ and $+\infty$.
Example: for the data $1, 2, 2, 5$, one has $\hat{F}_4(2) = 3/4$, because three of the four values are at most $2$.
Non-example: the function $t \mapsto \frac{1}{n}\#\set{i : X_i < t}$ uses strict inequality; it is left-continuous and is not a distribution function in the usual convention.
For fixed $t$ and an i.i.d. sample, $n \hat{F}_n(t)$ has the binomial distribution with parameters $n$ and $F(t)$.
So $\hat{F}_n(t)$ is an unbiased estimator of $F(t)$ with variance $F(t)(1 - F(t))/n$.
</document>

<document source="https://tamnd.github.io/math2/node/N1Q0BJAEFX.md" tag="N1Q0BJAEFX" kind="concept">
# Eulerian poset

## Summary

An Eulerian poset is a finite [[graded poset|1C7TKGTB2F]] with a least and a greatest element.
In it, every interval of positive length has as many elements of even rank as of odd rank.
Equivalently, its [[Möbius function|2Y75S8NG5B]] is $\mu(x, y) = (-1)^{\rho(y) - \rho(x)}$, where $\rho$ is the rank.
Face lattices of convex polytopes are Eulerian, and the condition encodes the Euler–Poincaré relation for their face numbers.

## Explanations

### L1 human

Count the elements in an interval of the poset, sorted by their level.
In an Eulerian poset the levels with even numbers and the levels with odd numbers always hold the same total.
The subsets of a finite set form an Eulerian poset: a set with $k \ge 1$ elements has as many subsets of even size as of odd size.
This is the same balance as $V - E + F = 2$ for a convex polyhedron.
</document>

<document source="https://tamnd.github.io/math2/node/N1QJQEGBFC.md" tag="N1QJQEGBFC" kind="concept">
# Cohen–Macaulay ring

## Summary

A Cohen–Macaulay ring is a Noetherian ring whose local rings have [[depth|M2RADBFW99]] equal to their [[Krull dimension|MVGBS3J1P7]].
This class contains [[regular local rings|63XXTF6B5M]], [[complete intersection rings|EPM2HKGMZS]] and [[Gorenstein rings|T6HCX959PJ]].
Cohen–Macaulay rings are the setting where dimension theory works best, since systems of parameters are regular sequences.
They also have a good duality theory through the [[canonical module|W5ECXPM6FW]].

## Explanations

### L1 human

The dimension of a ring counts its independent directions, and the depth counts how many independent equations can be imposed one after the other without hitting a zero divisor.
In a Cohen–Macaulay ring these two counts agree at every point.
Polynomial rings over a field are Cohen–Macaulay, and so is the ring of the two coordinate axes $K[x,y]/(xy)$.
The ring $K[x,y]/(x^2, xy)$, a line with an extra "fuzzy" point at the origin, is not.
</document>

<document source="https://tamnd.github.io/math2/node/N1V3YEYHNP.md" tag="N1V3YEYHNP" kind="statement">
# Hilbert's irreducibility theorem

## Statement

Let $r, s \ge 1$ and let $f_1, \dots, f_k$ be [[irreducible|00X2A9G9W3]] elements of the [[polynomial ring|0CES91M79A]] $\Q[t_1, \dots, t_r, x_1, \dots, x_s]$, each of positive degree in the variables $x_1, \dots, x_s$.
Then there is $a = (a_1, \dots, a_r) \in \Q^r$ such that each $f_i(a, x_1, \dots, x_s)$ is irreducible in $\Q[x_1, \dots, x_s]$.

## Remarks

The set of good $a$ is large: one may take $a \in \Z^r$, and there are infinitely many such $a$.
Classical proofs expand the roots of $f$ as series in $t$ near infinity and show that rational specialisations giving a factor are rare.
Example: $x^2 - t$ is irreducible in $\Q[t, x]$; the value $t = 2$ gives the irreducible $x^2 - 2$, while $t = 4$ gives $(x - 2)(x + 2)$.
So irreducibility can fail at some values, and the theorem only says that it survives at many.
A stronger form preserves the Galois group: if an irreducible $f \in \Q(t)[x]$ has Galois group $G$ over $\Q(t)$, then $f(a, x)$ has Galois group $G$ over $\Q$ for infinitely many $a \in \Q$.
The analogue fails over $\R$: $x^2 - t^2 - 1$ is irreducible in $\R[t, x]$, but for every $a \in \R$ the polynomial $x^2 - a^2 - 1$ has the real roots $\pm \sqrt{a^2 + 1}$.

## Explanations

### L1 human

A polynomial in several letters that cannot be factored usually stays unfactorable after you replace some letters by numbers.
Hilbert's theorem says that, over the rationals, you can always choose such numbers.

### L2 human

Fields with this property are called [[Hilbertian fields|SFPKB3NC0E]]; number fields are Hilbertian, while finite fields and algebraically closed fields are not.
The theorem is the main tool to pass from Galois groups over $\Q(t)$ to Galois groups over $\Q$.
For example, the generic polynomial with Galois group the full symmetric group $S_n$ over $\Q(t_1, \dots, t_n)$ specialises to polynomials over $\Q$ with Galois group $S_n$.
This is the starting point of many attacks on the [[inverse Galois problem|YJS6X8JN7D]].
</document>

<document source="https://tamnd.github.io/math2/node/N1WJR10WX0.md" tag="N1WJR10WX0" kind="statement">
# Killing–Hopf theorem

## Statement

Let $n \ge 2$, let $K$ be a real number, and let $X_K$ be the model space of dimension $n$.
It is the sphere of radius $1/\sqrt{K}$ if $K > 0$, Euclidean space $\R^n$ if $K = 0$, and hyperbolic space of curvature $K$ if $K < 0$.
Let $(M, g)$ be a [[space form|1VM1AQT956]] of dimension $n$ and curvature $K$.
Then:

1. If $M$ is simply connected, then $M$ is isometric to $X_K$.
2. In general, $M$ is isometric to a quotient $X_K / \Gamma$ by a group $\Gamma$ of [[isometries|R9YWQS15AA]] of $X_K$ that acts freely and properly discontinuously.
The group $\Gamma$ is isomorphic to the fundamental group of $M$.

## Remarks

The idea of the proof: a theorem of Cartan says that two Riemannian manifolds of the same dimension and the same constant [[sectional curvature|1X42JAA349]] are locally isometric.
For a complete simply connected $M$, a local isometry from a ball of $X_K$ extends along geodesics to a global isometry; this gives part 1.
For general $M$, the [[universal cover|Y1YPE6PM52]] with the pulled-back metric is complete, simply connected and of curvature $K$, so it is $X_K$ by part 1.
The deck transformations of the [[covering|3XZRRE165N]] are then isometries.

Example: the real projective space $\R P^n = S^n / \set{\pm \id}$ with the round metric is a space form of curvature $1$ with $\Gamma$ of order $2$.
Flat tori $\R^n / \Z^n$ are quotients of $\R^n$ by a lattice of translations.

Completeness is needed: an open ball in $\R^n$ is flat and simply connected, but it is not isometric to $\R^n$.

## Explanations

### L1 human

The theorem lists all spaces that curve by the same amount everywhere.
Each of them is a sphere, a flat space or a hyperbolic space, possibly folded up by a group of symmetries.
So to know them all, one only has to know which symmetry groups can be used for folding.

### L2 human

The theorem turns the classification of space forms into group theory: find the discrete groups of isometries of $S^n$, $\R^n$ or hyperbolic space that act freely.
For $K > 0$ these are finite subgroups of the orthogonal group acting freely on the sphere, which gives for example the lens spaces in dimension $3$.
For $K = 0$ and $M$ compact one gets the Bieberbach groups.
For $K < 0$ one gets the torsion-free discrete subgroups of the isometry group of hyperbolic space, a very rich family.
</document>

<document source="https://tamnd.github.io/math2/node/N219CDB623.md" tag="N219CDB623" kind="concept">
# Fitting subgroup

## Summary

The Fitting subgroup $F(G)$ of a finite group $G$ is the largest [[nilpotent|0Q7MMQHC1P]] [[normal subgroup|0CB0DX5BVV]] of $G$.
It exists because the product of two nilpotent normal subgroups is again nilpotent, a theorem of Hans Fitting.
In a [[solvable group|0FXJ7YFMMX]] $G$, the Fitting subgroup contains its own [[centraliser|02YYV36BFM]], so it controls much of the structure of $G$.
It is a basic tool in the theory of finite solvable groups.

## Explanations

### L1 human

Nilpotent groups are the groups that are closest to being abelian.
The Fitting subgroup is the biggest nilpotent piece that sits inside a group as a normal subgroup.
For the group $S_4$ of all permutations of four objects, it is the Klein four-group formed by the identity and the three products of two disjoint swaps.
For a nonabelian simple group, such as $A_5$, the Fitting subgroup is trivial.
</document>

<document source="https://tamnd.github.io/math2/node/N26TSVRH2S.md" tag="N26TSVRH2S" kind="concept">
# Risk function

## Summary

The risk function of a decision rule gives, for each value of the unknown parameter, the expected [[loss|3BTG6VVCGZ]] of the rule.
The data are drawn from the model with that parameter.
It turns a random loss into a deterministic function of the parameter.
Decision rules are compared through their risk functions, which leads to the notions of admissible, minimax and Bayes rules.

## Explanations

### L1 human

A rule that turns data into a decision is sometimes lucky and sometimes unlucky.
The risk is its average loss over many repetitions of the experiment, for a fixed true parameter.
Because the true parameter is unknown, one looks at the whole curve $\theta \mapsto R(\theta, \delta)$.
For example, estimating a normal mean $\theta$ by the mean of $n$ observations with variance $1$ has risk $1/n$ under squared error, the same for every $\theta$.
</document>

<document source="https://tamnd.github.io/math2/node/N298DRYE0W.md" tag="N298DRYE0W" kind="conjecture">
# Andrews–Curtis conjecture

## Statement

A [[presentation|0JGEQXQ4AC]] $\langle x_1, \ldots, x_n \mid r_1, \ldots, r_n \rangle$ is *balanced* if it has as many relators as generators.
The *Andrews–Curtis moves* on such a presentation, with $r_i$ elements of the [[free group|0KTW26C80J]] $F$ on $x_1, \ldots, x_n$, are:

1. replace some $r_i$ by $r_i^{-1}$;
2. replace some $r_i$ by $r_i r_j$ for some $j \ne i$;
3. replace some $r_i$ by $g r_i g^{-1}$ for some $g \in F$.

The conjecture: every balanced presentation of the trivial group can be turned into the trivial presentation $\langle x_1, \ldots, x_n \mid x_1, \ldots, x_n \rangle$ by a finite sequence of Andrews–Curtis moves.

## Remarks

Each move keeps the normal closure of the relators, so it keeps the group.
A presentation of the trivial group with $n$ generators has at least $n$ relators, since the abelianization $\Z^n$ must become trivial; so the balanced ones are the extreme case.
A weaker stable form of the conjecture also allows adding a generator $x_{n+1}$ with relator $x_{n+1}$, and removing such a pair.
Andrews and Curtis stated the conjecture in 1965, in connection with the topology of $2$-dimensional complexes and $4$-dimensional manifolds.
It is open; many experts expect it to be false, and several families of balanced presentations of the trivial group are studied as potential counterexamples.

## Explanations

### L1 human

Write the trivial group with $n$ generators and $n$ rules in a complicated way.
The conjecture says that one can always simplify the rules to the obvious ones, $x_1 = 1, \ldots, x_n = 1$.
Only three simple moves are allowed: invert a rule, multiply a rule by another, or conjugate a rule.
For example, $\langle x, y \mid x y^{-1}, y \rangle$ becomes trivial: multiply the first rule by the second to get $x$.
The difficulty is that the needed sequence of moves may be extremely long or may not exist at all.
</document>

<document source="https://tamnd.github.io/math2/node/N2JFN9VHD9.md" tag="N2JFN9VHD9" kind="statement">
# Laplace transform tends to zero at infinity

## Statement

Let $f \colon [0, \infty) \to \C$ be measurable, and assume that $\int_0^\infty \abs{f(t)} e^{-\sigma_0 t}\, dt < \infty$ for some real $\sigma_0$.
Let $F(s) = \int_0^\infty f(t) e^{-st}\, dt$ be its [[Laplace transform|05XT5AVJBK]], defined for $\operatorname{Re} s \ge \sigma_0$.
Then
$$\sup_{\operatorname{Re} s = \sigma} \abs{F(s)} \to 0 \quad (\sigma \to +\infty).$$
That is, $F(s) \to 0$ as $\operatorname{Re} s \to +\infty$, uniformly in $\operatorname{Im} s$.

## Remarks

For $\operatorname{Re} s = \sigma \ge \sigma_0$, $\abs{F(s)} \le \int_0^\infty \abs{f(t)} e^{-\sigma t}\, dt$.
The integrand decreases to $0$ for each $t > 0$ as $\sigma \to \infty$ and is bounded by $\abs{f(t)} e^{-\sigma_0 t}$, so the right side tends to $0$ by [[dominated convergence|BYNV84PF86]].
The hypothesis says that $\sigma_0$ lies to the right of the [[abscissa of absolute convergence|YCSH2WK69M]], or on it.
Example: $\mathcal{L}\{e^{at}\}(s) = 1/(s - a)$ tends to $0$ as $\operatorname{Re} s \to \infty$.
Consequence: functions such as $1$, $s$, $e^{s}$ or a rational function $P/Q$ with $P \ne 0$ and $\deg P \ge \deg Q$ are not Laplace transforms of any such $f$.
The constant $1$ is the transform of the Dirac delta, which is a measure and not a function.

## Explanations

### L1 human

As $\operatorname{Re} s$ grows, the weight $e^{-st}$ crushes everything except what happens at times very close to $0$.
An ordinary function has no mass concentrated at a single instant, so the transform dies out.
This gives a quick test: if a candidate $F(s)$ does not go to zero, it cannot be the transform of a function.

### L2 human

The test is used in partial fraction inversion: only proper rational functions, with numerator degree less than denominator degree, are inverted to ordinary functions.
Improper parts correspond to the Dirac delta and its derivatives, which belong to the Laplace transform of distributions.
The initial value theorem refines the statement: if $f(t) \to c$ as $t \to 0^+$, then $sF(s) \to c$, so $F(s)$ behaves like $c/s$.
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# Subdivision surface

## Summary

A subdivision surface is a smooth surface defined as the limit of repeatedly refining a coarse polygon mesh by fixed local averaging rules.
Each refinement step adds new vertices and moves old ones to weighted averages of their neighbors.
Subdivision surfaces handle meshes of any topology, generalize spline surfaces, and are standard in computer graphics and animation.
The [[Catmull–Clark|PF5Z57DDG2]] scheme is the most widely used example.

## Explanations

### L1 human

Start with a rough cage made of flat polygons, like a crude box model of a head.
Cut every face into smaller faces and nudge each corner toward the average of its neighbors.
Repeat this a few times, and the jagged cage turns into a smooth rounded shape.
The surface that these steps approach is the subdivision surface.
A curve version, Chaikin's corner cutting, replaces each edge of a polygon by two points at a quarter and three quarters of its length.
Its limit is a smooth curve.
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<document source="https://tamnd.github.io/math2/node/N2Q4RS43FD.md" tag="N2Q4RS43FD" kind="definition">
# Complete lattice

## Definition

Let $(L, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
It is a *complete lattice* if every subset $S \subseteq L$ has a [[supremum|0R3H5PANRQ]] $\bigvee S$ and an [[infimum|0QF1MZ9FRG]] $\bigwedge S$ in $L$.

## Remarks

It is enough to ask that every subset has a supremum: then $\bigwedge S$ is the supremum of the set of lower bounds of $S$.
Taking $S = \varnothing$ shows that a complete lattice has a least element $\bigvee \varnothing$ and a greatest element $\bigwedge \varnothing$; in particular it is nonempty.
Every complete lattice is a [[lattice|WH4SFAACAS]], and every finite nonempty lattice is complete.
Example: for a set $X$, the power set of $X$ ordered by inclusion is complete, with $\bigvee S = \bigcup S$ and $\bigwedge S = \bigcap S$, where $\bigcap \varnothing = X$.
Non-example: $\Q$ with its usual order is a lattice but not complete, because $\set{q \in \Q : q^2 < 2}$ has no supremum in $\Q$.
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<document source="https://tamnd.github.io/math2/node/N2SVK69S53.md" tag="N2SVK69S53" kind="concept">
# Circle map

## Summary

The circle map, or standard circle map, is the two-parameter family $\theta \mapsto \theta + \Omega - \frac{K}{2\pi} \sin(2\pi\theta)$ of maps of the [[circle|060W5CBZP6]] $\R/\Z$.
It models a rotation by $\Omega$ disturbed by a nonlinear kick of strength $K$.
It is the basic model of mode locking: its [[rotation number|DP4WWQBJJF]] locks to rational values on the [[Arnold tongues|WQF5ASXZTH]].
For $0 \le K < 1$ it is a family of circle [[diffeomorphisms|0KZZ70ZHXP]], and at $K = 1$ it has a critical point.

## Explanations

### L1 human

Picture a point moving around a circle in steps.
Without the kick ($K = 0$), each step is the same turn by $\Omega$.
The kick speeds the point up in some places and slows it down in others.
A pendulum clock pushed by a periodic force, or a heart driven by a pacemaker, can be modelled this way.
The question is whether the motion locks to the rhythm of the drive.
</document>

<document source="https://tamnd.github.io/math2/node/N2TSHRCJ1Y.md" tag="N2TSHRCJ1Y" kind="definition">
# Bootstrap

## Definition

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] with an unknown distribution $F$, and let $\hat\theta = T(X_1, \dots, X_n)$ be a [[statistic|036756NFD6]].
Let $\hat F_n$ be the [[empirical distribution|A05WR6TEK5]] of the observed values $x_1, \dots, x_n$, which puts mass $1/n$ at each $x_i$.
A *bootstrap sample* is a sample $X_1^*, \dots, X_n^*$ i.i.d. from $\hat F_n$, that is, $n$ draws with replacement from $x_1, \dots, x_n$.
The *bootstrap distribution* of $\hat\theta$ is the law of $\hat\theta^* = T(X_1^*, \dots, X_n^*)$ given the data.
The *bootstrap* estimates a feature of the [[sampling distribution|656Y8FA3RY]] of $\hat\theta$ under $F$ by the same feature of the bootstrap distribution.
An example of such a feature is the [[standard deviation|0VTN5RKNC5]].

## Remarks

In practice one draws $B$ independent bootstrap samples, computes $\hat\theta^*_1, \dots, \hat\theta^*_B$, and estimates the standard error of $\hat\theta$ by
$$\widehat{\mathrm{se}}_B = \Big( \frac{1}{B - 1} \sum_{b=1}^{B} \big(\hat\theta^*_b - \bar\theta^*\big)^2 \Big)^{1/2}, \qquad \bar\theta^* = \frac{1}{B} \sum_{b=1}^{B} \hat\theta^*_b.$$
The percentile interval takes the $\alpha/2$ and $1 - \alpha/2$ empirical quantiles of the $\hat\theta^*_b$ as a confidence interval of level about $1 - \alpha$.

Example: data $(1, 3)$ and $\hat\theta = \bar X$.
The four equally likely bootstrap samples $(1,1), (1,3), (3,1), (3,3)$ have means $1, 2, 2, 3$.
Their variance is $(1 + 0 + 0 + 1)/4 = 1/2$, which equals $\hat\sigma^2 / n$ with the plug-in variance $\hat\sigma^2 = 1$.

Non-example: for $X_i$ uniform on $[0, \theta]$ and $\hat\theta = \max_i X_i$, the bootstrap fails.
A bootstrap sample repeats the observed maximum with probability $1 - (1 - 1/n)^n \to 1 - e^{-1}$, so the bootstrap law of $n(\hat\theta - \hat\theta^*)$ has an atom at $0$, while the true law of $n(\theta - \hat\theta)$ is continuous.

The [[jackknife|WQ8TQKDKYT]] is an older resampling method that leaves out one observation at a time.
</document>

<document source="https://tamnd.github.io/math2/node/N32Z68F0D3.md" tag="N32Z68F0D3" kind="concept">
# Projective geometry

## Summary

Projective geometry studies the properties of figures that survive projection, such as incidence and the [[cross-ratio|09ASMRMD67]].
Length and angle are not among them.
Its spaces are [[projective spaces|ZT9A6JK8QF]], where any two lines of a plane meet.
It can be built either from linear algebra or from incidence axioms.
In it, points and hyperplanes play symmetric roles, which gives the principle of [[duality|DCMP1A0138]].
It underlies perspective drawing, algebraic geometry and computer vision.

## Explanations

### L1 human

When a lamp casts the shadow of a shape on a wall, lengths and angles change, but straight lines stay straight.
Projective geometry studies exactly what such shadows and perspective views keep.
In this geometry parallel lines are said to meet at a point at infinity, so any two lines in a plane meet.
For example, all ellipses, parabolas and hyperbolas become the same kind of curve, a conic seen from different viewpoints.
</document>

<document source="https://tamnd.github.io/math2/node/N3457SJ63D.md" tag="N3457SJ63D" kind="definition">
# Preadditive category

## Definition

A *preadditive category* is a [[category|A2DE66937D]] $\mathcal{C}$ with an [[abelian group|082GHNPJ53]] structure, written additively, on each set $\Hom(X, Y)$.
Composition must be bilinear:
$$h \circ (f + g) = h \circ f + h \circ g, \qquad (f + g) \circ k = f \circ k + g \circ k$$
for all $f, g \colon X \to Y$, $h \colon Y \to Z$ and $k \colon W \to X$.

## Remarks

Equivalently, a preadditive category is a category [[enriched|BPH9SJ30Y0]] over the monoidal category of abelian groups with the tensor product.
The zero element of $\Hom(X, Y)$ is the zero morphism $0_{XY}$, and bilinearity gives $h \circ 0 = 0$ and $0 \circ k = 0$.
Some authors call these categories $\mathbf{Ab}$-categories.

Example: for a [[ring|0B1261B5ED]] $R$, the category of left [[$R$-modules|0NZ8JA25WK]] with pointwise addition $(f + g)(x) = f(x) + g(x)$ is preadditive.
The sum of homomorphisms is a homomorphism, and $h(f(x) + g(x)) = h(f(x)) + h(g(x))$ because $h$ is additive, which gives the first identity; the second holds by definition of the sum.

Example: a ring $R$ is a preadditive category with one object $\ast$, with $\Hom(\ast, \ast) = R$ and composition the multiplication of $R$.

Non-example: the category of sets is not preadditive, since $\Hom(\set{0}, \emptyset)$ is empty and an abelian group is never empty.
</document>

<document source="https://tamnd.github.io/math2/node/N353JF8RBK.md" tag="N353JF8RBK" kind="concept">
# Free functor

## Summary

A free functor is a left adjoint to a [[forgetful functor|WV3BPYCHVG]].
It sends a set $X$ to the [[free algebra|GTS4DRRMZR]] on $X$ in a variety, together with the inclusion of $X$.
It sends a map of sets to the homomorphism that extends it.
The universal property of free algebras is exactly the statement that this assignment is left adjoint to forgetting.
The composite of the forgetful functor with the free functor is the monad whose algebras are the algebras of the variety.

## Explanations

### L1 human

Given a set of letters, the free group on them consists of all reduced words in the letters and their inverses.
A function between two sets of letters renames letters, and so it turns words into words; this is a group homomorphism.
The free functor is this construction, done for all sets and maps at once.
It is "free" because the words satisfy no relation beyond the group axioms, so every map from the letters into a group extends uniquely to a homomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/N35QM9G83G.md" tag="N35QM9G83G" kind="statement">
# Curtis–Hedlund–Lyndon theorem

## Statement

Let $A$ be a finite set and let $d \ge 1$.
Give $A^{\Z^d}$ the [[product topology|0Z57EC6YX3]] of the discrete topologies.
For $v \in \Z^d$ let $\sigma_v \colon A^{\Z^d} \to A^{\Z^d}$ be the shift $\sigma_v(x)_u = x_{u + v}$.
Let $F \colon A^{\Z^d} \to A^{\Z^d}$ be a map.
Then the following are equivalent:

1. $F$ is [[continuous|0P27W9RW0G]] and $F \circ \sigma_v = \sigma_v \circ F$ for all $v \in \Z^d$.
2. There are a finite set $N \subseteq \Z^d$ and a map $f \colon A^N \to A$ such that $F(x)_u = f\big((x_{u + n})_{n \in N}\big)$ for all $x$ and all $u \in \Z^d$.

## Remarks

Condition 2 says that $F$ is a [[cellular automaton|5QXNC6NTVN]] with neighbourhood $N$ and local rule $f$.
For 2 implies 1, the value $F(x)_u$ depends only on finitely many coordinates of $x$, which gives continuity.
The same rule is used at every site, which gives commutation with shifts.
For 1 implies 2, the space $A^{\Z^d}$ is compact, so $F$ is uniformly continuous.
Hence $F(x)_0$ is determined by the coordinates $x_n$ with $n$ in some finite set $N$, and this defines $f$.
Commutation with the shifts then gives the formula at every site $u$.
Compactness is essential: over an infinite alphabet there are continuous shift-commuting maps that are not given by a finite rule.
The same statement holds for maps between [[shift spaces|A1XE59ZHNM]], where condition 2 says that $F$ is a sliding block code.

## Explanations

### L1 human

A cellular automaton updates every cell of a grid by looking at a few neighbours and applying a fixed rule.
Such a map treats all positions alike and changes only a little when the input changes far away.
The theorem says that these two properties characterise cellular automata.
So one can define cellular automata either by rules or by topology.

### L2 human

The theorem lets one treat a cellular automaton as an endomorphism of the dynamical system $(A^{\Z^d}, \sigma)$.
It shows, for instance, that the inverse of a bijective cellular automaton is again a cellular automaton.
A continuous bijection of a compact space onto a Hausdorff space has a continuous inverse, and that inverse commutes with the shifts.
In one dimension the theorem identifies the conjugacies between shift spaces with invertible sliding block codes.
It is a basic tool in the classification of [[subshifts of finite type|624XDXX8FN]].
</document>

<document source="https://tamnd.github.io/math2/node/N35SYWJMA9.md" tag="N35SYWJMA9" kind="definition">
# Whitney sum

## Definition

Let $B$ be a topological space and let $p \colon E \to B$ and $p' \colon E' \to B$ be real (or complex) vector bundles over $B$, of ranks $n$ and $n'$.
The *Whitney sum* of $E$ and $E'$ is the space
$$E \oplus E' = \set{(v, v') \in E \times E' : p(v) = p'(v')},$$
with the [[subspace topology|0WS9P12FQR]] of the [[product|0GBG8B49AA]] $E \times E'$, together with the map $q \colon E \oplus E' \to B$, $q(v, v') = p(v)$.
The fiber $q^{-1}(b) = E_b \times E'_b$ is given the [[vector space|000AYPA9AJ]] structure of the direct sum $E_b \oplus E'_b$.

## Remarks

The map $q$ is a vector bundle of rank $n + n'$: local trivializations $p^{-1}(U) \cong U \times \R^n$ and $p'^{-1}(U) \cong U \times \R^{n'}$ combine into $q^{-1}(U) \cong U \times \R^{n + n'}$.
Equivalently, $E \oplus E'$ is the pullback of the product bundle $E \times E' \to B \times B$ along the diagonal $B \to B \times B$.
The Whitney sum is associative and commutative up to isomorphism, and the trivial bundle of rank $0$ is a unit.

Example: let $TS^2$ be the tangent bundle of $S^2 \subset \R^3$ and $\nu = \set{(x, t x) : x \in S^2, t \in \R}$ the normal line bundle.
At each $x \in S^2$, $T_x S^2 \oplus \R x = \R^3$, and so $TS^2 \oplus \nu$ is isomorphic to the trivial bundle $S^2 \times \R^3$, although $TS^2$ itself is not trivial.

Non-example: for bundles $E \to B$ and $E' \to B'$ over different bases, the product $E \times E' \to B \times B'$ is a vector bundle of rank $n + n'$.
It is not a Whitney sum, which needs a common base; it is the external direct sum.
</document>

<document source="https://tamnd.github.io/math2/node/N36F259F1B.md" tag="N36F259F1B" kind="definition">
# Decomposition group

## Definition

Let $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]] of [[algebraic number fields|MWBDFS2KJ5]] with [[Galois group|0XTY86BZKY]] $G$, and let $\mathcal{O}_K \subseteq \mathcal{O}_L$ be the rings of integers.
Let $\mathfrak{P}$ be a nonzero [[prime ideal|00MHHZ9SNE]] of $\mathcal{O}_L$.
The *decomposition group* of $\mathfrak{P}$ is the [[subgroup|0X1JXT47GA]]
$$D_{\mathfrak{P}} = \set{\sigma \in G : \sigma(\mathfrak{P}) = \mathfrak{P}}.$$
The *inertia group* of $\mathfrak{P}$ is
$$I_{\mathfrak{P}} = \set{\sigma \in D_{\mathfrak{P}} : \sigma(x) \equiv x \pmod{\mathfrak{P}} \text{ for all } x \in \mathcal{O}_L}.$$

## Remarks

Let $\mathfrak{p} = \mathfrak{P} \cap \mathcal{O}_K$, with ramification index $e$, residue degree $f$ and $g$ primes over it, as in [[splitting of prime ideals|B7KNZCBFYT]].
Then $D_{\mathfrak{P}}$ has order $ef$ and index $g$ in $G$.
Each $\sigma \in D_{\mathfrak{P}}$ induces an automorphism of the [[finite field|0ZXZ82FTX2]] $\mathcal{O}_L/\mathfrak{P}$ over $\mathcal{O}_K/\mathfrak{p}$; this gives a surjective homomorphism onto the Galois group of the residue extension, with kernel $I_{\mathfrak{P}}$, so $\abs{I_{\mathfrak{P}}} = e$.
For $\tau \in G$, one has $D_{\tau(\mathfrak{P})} = \tau D_{\mathfrak{P}} \tau^{-1}$.

Example: in $\Q(i)/\Q$, with $G = \set{1, c}$ where $c$ is complex conjugation, take $\mathfrak{P} = (3)$.
Then $c(\mathfrak{P}) = \mathfrak{P}$, so $D_{\mathfrak{P}} = G$; and $c(i) = -i \not\equiv i \pmod 3$, so $I_{\mathfrak{P}}$ is trivial, matching $e = 1$, $f = 2$.

Non-example: for $\mathfrak{P} = (2 + i)$, the element $c$ is not in $D_{\mathfrak{P}}$, because $c(\mathfrak{P}) = (2 - i) \ne \mathfrak{P}$ (the quotient $(2+i)/(2-i)$ is not in $\Z[i]$).
So $D_{\mathfrak{P}}$ is trivial, matching $e = f = 1$ and $g = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/N37SPDWYGS.md" tag="N37SPDWYGS" kind="definition">
# Localization at a prime ideal

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $\mathfrak p$ be a [[prime ideal|00MHHZ9SNE]] of $R$.
The set $S = R \setminus \mathfrak p$ is a [[multiplicative set|AN9SKEZ3VY]].
The *localization of $R$ at $\mathfrak p$* is the [[localization|4C1AE84N5N]]
$$R_{\mathfrak p} = S^{-1}R = \set{a/s : a \in R, \ s \in R \setminus \mathfrak p}.$$

## Remarks

The ring $R_{\mathfrak p}$ is a [[local ring|MZ02BKPHJA]].
Its unique maximal ideal is $\mathfrak p R_{\mathfrak p} = \set{a/s : a \in \mathfrak p, \ s \notin \mathfrak p}$.
Indeed, an element $a/s$ with $a \notin \mathfrak p$ has the inverse $s/a$, so every element outside $\mathfrak p R_{\mathfrak p}$ is a unit.
The [[residue field|NJ3APQ8BHH]] $R_{\mathfrak p} / \mathfrak p R_{\mathfrak p}$ is called the residue field of $R$ at $\mathfrak p$.

Example: for $R = \Z$ and $\mathfrak p = p\Z$ with $p$ prime, $\Z_{(p)}$ is the ring of rationals $a/b$ in lowest terms with $p \nmid b$.
Its maximal ideal is $p \Z_{(p)}$ and its residue field is $\Z/p\Z$.
For $\mathfrak p = (0)$ in an [[integral domain|0QVD7WBFGN]] $R$, the localization $R_{(0)}$ is the [[field of fractions|0F2PTGX1A2]] of $R$.

Non-example: the construction needs $\mathfrak p$ to be prime.
For the ideal $6\Z$ of $\Z$, the complement $\Z \setminus 6\Z$ contains $2$ and $3$ but not $6$, so it is not a multiplicative set.

Notation: for an element $f \in R$, the ring $R_f$ is the localization at the powers of $f$, which differs from $R_{\mathfrak p}$.
</document>

<document source="https://tamnd.github.io/math2/node/N383CX7WKS.md" tag="N383CX7WKS" kind="concept">
# Regularity theory

## Summary

Regularity theory is the part of the theory of partial differential equations that shows that solutions are smoother than first assumed.
Typically one first obtains a [[weak solution|SJZX81A783]] in a [[Sobolev space|M4CEZ5K9HP]].
Then one proves that it has more derivatives, in Sobolev or [[Hölder spaces|P101AX1ZSC]], when the coefficients and data are smooth.
For elliptic equations the classical results are [[elliptic regularity|2PVQY9PCT4]], the [[Schauder estimates|HSEKZMAH1V]], the [[Calderón–Zygmund estimates|RYQJH4RYB4]] and the [[De Giorgi–Nash–Moser theorem|32C3D8FG9V]].
An important source of the subject is [[Hilbert's nineteenth problem|QPMM2ZPWBW]] on the analyticity of minimisers.

## Explanations

### L1 human

Existence of a solution is often easiest to prove in a weak sense, where the function may not even have classical derivatives.
Regularity theory then asks how nice this solution really is.
For the Laplace equation $\Delta u = 0$ the answer is striking: any locally integrable weak solution is smooth, and even real analytic.
In general, an elliptic operator $L$ of order $m$ with smooth coefficients gains $m$ derivatives.
If $Lu$ has $k$ derivatives in $L^2$, then $u$ has $k + m$ derivatives in $L^2$ inside the domain.
</document>

<document source="https://tamnd.github.io/math2/node/N3CC5DRGT1.md" tag="N3CC5DRGT1" kind="concept">
# Hankel functions

## Summary

The Hankel functions $H^{(1)}_\nu$ and $H^{(2)}_\nu$ are the Bessel functions of the third kind.
They are the complex combinations $J_\nu \pm i Y_\nu$ of the [[Bessel functions of the first kind|CZCMRDWDYW]] and of the [[second kind|GRSNDJG363]].
They play the role of $e^{\pm i x}$ next to the roles of $\cos x$ and $\sin x$ played by $J_\nu$ and $Y_\nu$.
They describe outgoing and incoming cylindrical waves.

## Explanations

### L1 human

For the equation $y'' + y = 0$, one can use the real solutions $\cos x$ and $\sin x$ or the complex solutions $e^{ix}$ and $e^{-ix}$.
Bessel's equation has the same choice.
The pair $J_\nu$, $Y_\nu$ is the real one, and the Hankel functions are the complex one.
For large $x > 0$, $H^{(1)}_\nu(x)$ is close to $\sqrt{\frac{2}{\pi x}}\, e^{i(x - \nu\pi/2 - \pi/4)}$, a wave travelling outward whose size shrinks.
</document>

<document source="https://tamnd.github.io/math2/node/N3MRHDKDGC.md" tag="N3MRHDKDGC" kind="concept">
# Semisimple module

## Summary

A semisimple module is a [[module|0NZ8JA25WK]] that is a direct sum of [[simple modules|G00C22AEDH]].
Equivalently, every submodule of it has a complement, so the module splits as cleanly as a vector space.
Semisimple modules are the modules for which linear algebra works best.
The rings all of whose modules are semisimple are described completely by the Artin–Wedderburn theorem.
By Maschke's theorem, every representation of a finite group over a field of characteristic zero is semisimple.

## Explanations

### L1 human

A semisimple module is built from simple pieces placed side by side, with no gluing between them.
Every vector space is semisimple: it is a direct sum of lines, and each line is simple.
The group $\Z/6\Z$ is semisimple as a $\Z$-module, since it is $\Z/2\Z \oplus \Z/3\Z$.
The group $\Z/4\Z$ is not: its subgroup $\set{0, 2}$ has no complement.
</document>

<document source="https://tamnd.github.io/math2/node/N3PGYJJQ7W.md" tag="N3PGYJJQ7W" kind="concept">
# Ideal sheaf

## Summary

An ideal sheaf on a ringed space $X$ is a subsheaf of the structure sheaf $\mathcal{O}_X$ whose sections over each open set form an ideal.
It is the sheaf version of an ideal of a ring.
On a scheme, quasi-coherent ideal sheaves correspond exactly to closed subschemes, the subscheme being cut out by the functions in the ideal.

## Explanations

### L1 human

An ideal of a ring of functions is a set of functions closed under addition and under multiplication by any function.
An ideal sheaf picks such an ideal on every open set, in a way compatible with restriction.
The typical example is the sheaf of functions that vanish on a fixed closed subset, such as a point or a curve.
The ideal sheaf then remembers that subset, and even multiplicities, such as functions vanishing to order two at a point.
</document>

<document source="https://tamnd.github.io/math2/node/N3QA80M44M.md" tag="N3QA80M44M" kind="concept">
# Varifold

## Summary

An $m$-dimensional varifold in an open set $U \subseteq \R^n$ is a [[Radon measure|HZNEQQBCW4]] on a product space.
The product is $U$ times the space of $m$-dimensional linear subspaces of $\R^n$.
It records where a generalised surface lies and which tangent planes it has there, but not its orientation.
L. C. Young introduced them in 1951 as generalised surfaces.
Almgren gave them the name varifold in the 1960s, and Allard developed them as a setting for minimal surfaces with singularities.
Their compactness properties make them a main tool in the calculus of variations for area.

## Explanations

### L1 human

Think of a surface as a cloud of tiny flat pieces.
Each piece has a position and a tilt, given by its tangent plane.
A varifold is a mass distribution over all pairs of position and tilt.
A smooth surface gives such a distribution: at each point of the surface, its area sits on the tangent plane at that point.
Unlike a current, a varifold has no orientation, so two sheets on top of each other add up instead of cancelling.
</document>

<document source="https://tamnd.github.io/math2/node/N3QPQATSRN.md" tag="N3QPQATSRN" kind="statement">
# Serre duality

## Statement

Let $k$ be a field and let $X$ be a [[projective variety|A4H4CZ2V35]] over $k$ of dimension $n$.
Assume that $X$ is [[smooth|EXKGG9X6J0]] and geometrically connected, with [[canonical bundle|W05VYY4HBR]] $\omega_X$.
Then there is a trace isomorphism $H^n(X, \omega_X) \cong k$.
Let $\mathcal{E}$ be a [[vector bundle|P4DZM975BV]] on $X$ with dual $\mathcal{E}^\vee$, and let $0 \le i \le n$.
Then the cup product followed by the trace,
$$H^i(X, \mathcal{E}) \times H^{n-i}(X, \mathcal{E}^\vee \otimes \omega_X) \to H^n(X, \omega_X) \cong k,$$
is a perfect pairing of finite-dimensional vector spaces.
So $H^i(X, \mathcal{E}) \cong H^{n-i}(X, \mathcal{E}^\vee \otimes \omega_X)^\vee$, where $H^i$ is [[sheaf cohomology|PQ6Q143ARJ]].

## Remarks

The idea of the proof: first treat $\mathbb{P}^N_k$, where the cohomology of the sheaves $\mathcal{O}(d)$ is computed explicitly by Čech cohomology and the pairing is visible on monomials.
For $X \subseteq \mathbb{P}^N_k$ of codimension $r$, the dualizing sheaf of $X$ is $\mathcal{E}xt^r(\mathcal{O}_X, \omega_{\mathbb{P}^N})$, and for smooth $X$ this sheaf is $\omega_X$.

Example: for a smooth projective curve $X$ of genus $g$ and $\mathcal{E} = \mathcal{O}_X$, duality gives $h^1(X, \mathcal{O}_X) = h^0(X, \omega_X) = g$.
Combined with Riemann–Roch it gives $\deg \omega_X = 2g - 2$.

Properness is necessary: for $X = \mathbb{A}^1_k$, $H^0(X, \mathcal{O}_X) = k[t]$ is infinite-dimensional while $H^1(X, \omega_X) = 0$ by affine vanishing.
More generally the statement holds for projective Cohen–Macaulay schemes of pure dimension $n$, with $\omega_X$ replaced by the dualizing sheaf.

## Explanations

### L1 human

Serre duality says that the cohomology of a vector bundle in degree $i$ and the cohomology of a twisted dual bundle in the complementary degree $n - i$ carry the same information.
The twist is by the canonical bundle.
It halves the work in many computations and links the top cohomology to global sections, which are easier to understand.

### L2 human

Serre duality is the algebraic analogue of Poincaré duality for compact oriented manifolds.
It is used constantly with the Riemann–Roch theorem: on a curve, $h^1(L) = h^0(K - L)$, which turns Riemann–Roch into a statement about global sections only.
Its general form, valid for proper morphisms and complexes of sheaves, is Grothendieck duality.
</document>

<document source="https://tamnd.github.io/math2/node/N3XB4SYQ6B.md" tag="N3XB4SYQ6B" kind="definition">
# Pochhammer symbol

## Definition

Let $a$ be an element of a commutative ring, for example a complex number, and let $n$ be a natural number.
The *Pochhammer symbol* (or *rising factorial*) of $a$ of order $n$ is
$$(a)_n = a (a + 1) (a + 2) \cdots (a + n - 1) = \prod_{i=0}^{n-1} (a + i).$$
For $n = 0$ the product is empty, and $(a)_0 = 1$.

## Remarks

Example: $(1)_n = 1 \cdot 2 \cdots n = n!$, the [[factorial|0XWG0JGDFW]]; and $(2)_3 = 2 \cdot 3 \cdot 4 = 24$.
For a negative integer $a = -m$ and $n > m$, one factor is $0$, so $(-m)_n = 0$; this is why hypergeometric series with such a parameter terminate.
Other notations are $a^{\overline{n}}$ and $a^{(n)}$.
The rising factorial of $a$ is $(-1)^n$ times the [[falling factorial|0SJNCJCCQG]] of $-a$: $(a)_n = (-1)^n (-a)(-a-1)\cdots(-a-n+1)$.
If $a \in \C$ is not in $\set{0, -1, -2, \ldots}$, then $(a)_n = \Gamma(a + n)/\Gamma(a)$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
In Mathlib the rising factorial as a polynomial in $a$ is `ascPochhammer`.

## Explanations

### L1 human

The Pochhammer symbol is a factorial that may start at any number and goes up in steps of $1$.
For example $(5)_3 = 5 \cdot 6 \cdot 7 = 210$.
</document>

<document source="https://tamnd.github.io/math2/node/N3XH42KVV3.md" tag="N3XH42KVV3" kind="statement">
# Cohen structure theorem

## Statement

Let $(R, \mathfrak m)$ be a [[complete Noetherian local ring|G9Q61B8A7C]] with [[residue field|NJ3APQ8BHH]] $k = R/\mathfrak m$.

1. Suppose that $R$ contains a field. Then $R$ contains a coefficient field, a subfield $K$ such that $K \to R \to k$ is an isomorphism. Moreover $R \cong k[\![X_1, \dots, X_n]\!]/J$, a quotient of a [[formal power series ring|3E8ZG3NJYZ]], for some $n$ and some ideal $J$.
2. If $R$ contains a field and $R$ is a [[regular local ring|63XXTF6B5M]] of [[Krull dimension|MVGBS3J1P7]] $d$, then $R \cong k[\![X_1, \dots, X_d]\!]$.
3. In general, let $C = k$ if $k$ has [[characteristic|0F6ES3BRYP]] $0$. If $k$ has characteristic $p > 0$, let $C$ be a complete [[discrete valuation ring|MGM3CNJW9V]] of characteristic $0$ with maximal ideal $pC$ and residue field $k$. Such a ring exists. Then $R$ is isomorphic to a quotient $C[\![X_1, \dots, X_n]\!]/J$ for some $n$ and some ideal $J$.

## Remarks

The hard part is the existence of a coefficient field, or of a coefficient ring in the general case.
Once it exists, choose generators $x_1, \dots, x_n$ of $\mathfrak m$ and send $X_i$ to $x_i$.
This map from the power series ring is surjective because $R$ is complete and $\mathfrak m$ is finitely generated.
In clause 2 the number of generators can be taken equal to $d$.
Then the kernel is $0$, since a proper quotient of the domain $k[\![X_1, \dots, X_d]\!]$ has dimension less than $d$.
A complete local ring $R$ contains a field if and only if $R$ and $k$ have the same characteristic.
When $k$ is perfect of characteristic $p > 0$, the ring $C$ can be taken to be the ring of [[Witt vectors|6N3X2QX4W9]] $W(k)$; for $k = \F_p$ this is $\Z_p$.
Example: the completion of $k[x, y]_{(x, y)}/(y^2 - x^3)$ at its maximal ideal is $k[\![X, Y]\!]/(Y^2 - X^3)$.

## Explanations

### L1 human

Complete local rings look complicated in general, but the theorem says that each one is a quotient of a power series ring.
So, up to taking a quotient, the only complete local rings are rings of power series in several variables over a field or over a ring like the $p$-adic integers.
For example, the completed local ring at a smooth point of a $d$-dimensional variety over a field $k$ is the power series ring $k[\![X_1, \dots, X_d]\!]$.

### L2 human

Together with the faithful flatness of [[completion|NPF42DJMBX]], the theorem lets one reduce questions about Noetherian local rings to quotients of power series rings, where explicit computation is possible.
It implies, for example, that complete Noetherian local rings are excellent and that they are quotients of regular local rings.
</document>

<document source="https://tamnd.github.io/math2/node/N3YMNMWANM.md" tag="N3YMNMWANM" kind="statement">
# Gauss–Markov theorem

## Statement

Let $X$ be a fixed real $n \times p$ matrix of rank $p$, and let $\beta \in \R^p$ be an unknown parameter.
Let $\varepsilon$ be a random vector in $\R^n$ and
$$y = X\beta + \varepsilon.$$
Assume the following.

1. $E[\varepsilon] = 0$.
2. $\operatorname{Cov}(\varepsilon) = \sigma^2 I_n$ for some unknown $\sigma^2 > 0$, so the errors are uncorrelated and [[homoscedastic|D8VX6XQ3TT]].

Let $\hat\beta = (X^T X)^{-1} X^T y$ be the [[ordinary least squares|YM708KHV2W]] estimator.
Let $\tilde\beta = C y$ be any estimator with $C$ a fixed $p \times n$ matrix such that $E[\tilde\beta] = \beta$ for every $\beta \in \R^p$.
Then $\hat\beta$ is [[unbiased|0BWGTN92FG]], and the matrix $\operatorname{Cov}(\tilde\beta) - \operatorname{Cov}(\hat\beta)$ is [[positive semidefinite|01MMVW8CKC]].
In particular, $\operatorname{Var}(c^T \tilde\beta) \ge \operatorname{Var}(c^T \hat\beta)$ for every $c \in \R^p$.

## Remarks

Write $C = (X^T X)^{-1} X^T + D$.
Unbiasedness for every $\beta$ forces $D X = 0$.
Then $\operatorname{Cov}(\tilde\beta) = \sigma^2 C C^T = \sigma^2 (X^T X)^{-1} + \sigma^2 D D^T$, because the cross terms contain $D X$.
The matrix $D D^T$ is positive semidefinite, which gives the claim.

For the model $y_i = \mu + \varepsilon_i$, the estimator $\hat\beta$ is the sample mean.
Here the theorem says that the sample mean has the smallest variance among all weighted averages $\sum_i c_i y_i$ with $\sum_i c_i = 1$.

The errors need not be normal or independent.
If the [[covariance matrix|0V7G6WMFK0]] of $\varepsilon$ is not a multiple of $I_n$, then ordinary least squares is still unbiased but in general no longer best.
The theorem compares only linear unbiased estimators: a biased estimator, such as a ridge estimator, can have a smaller mean squared error.

## Explanations

### L1 human

In a linear model with well-behaved noise, the least squares fit is the most precise estimator among those that are linear in the data and correct on average.
"Best linear unbiased estimator", or BLUE, is the short name for this property.
It is the main reason why least squares is the default method for fitting linear models.

### L2 human

The hypotheses are only about the first two moments of the errors: mean zero, equal variances and no correlation.
When the errors are also normal, a stronger result holds: least squares is then the minimum-variance unbiased estimator among all unbiased estimators, linear or not.
When the variances differ or the errors are correlated with a known covariance structure, one transforms the model to restore the hypotheses.
That leads to generalized and weighted least squares.
The theorem says nothing about biased estimators, and shrinkage methods trade a small bias for a large drop in variance.
</document>

<document source="https://tamnd.github.io/math2/node/N3Z69FTQ7J.md" tag="N3Z69FTQ7J" kind="definition">
# Semianalytic set

## Definition

Let $M$ be a [[real analytic manifold|Z7H98E1NHP]], for example an open subset of $\R^n$.
A subset $X \subseteq M$ is *semianalytic* if every point $p \in M$ has an open [[neighbourhood|0ZCY0Z80B6]] $U$ with the following property.
The set $X \cap U$ is a finite union of sets of the form
$$\set{x \in U : f(x) = 0,\ g_1(x) > 0, \dots, g_r(x) > 0},$$
where $r \ge 0$ and $f, g_1, \dots, g_r$ are [[real analytic functions|081WKEDK6C]] on $U$.

## Remarks

The condition is required at every point of $M$, not only at points of $X$.
Equivalently, $X \cap U$ belongs to the smallest family of subsets of $U$ that contains the sets $\set{g > 0}$ for real analytic $g$ on $U$.
Here the family is required to be closed under finite unions, finite intersections and complements.
Several equations $f_1 = \cdots = f_s = 0$ can be merged into the single equation $f_1^2 + \cdots + f_s^2 = 0$, and $f = 0$ gives no equation at all.
Example: $X = \set{(x, y) \in \R^2 : y > 0,\ x^2 + y^2 < 1}$ is semianalytic, with $U = \R^2$, $f = 0$, $g_1 = y$ and $g_2 = 1 - x^2 - y^2$.
Non-example: $X = \set{0} \cup \set{1/n : n \ge 1}$ is not semianalytic in $\R$.
Let $U$ be any open interval about $0$, and $f, g_i$ real analytic on $U$.
If $f$ vanishes at infinitely many of the points $1/n$, then $f = 0$ on $U$ by the identity theorem; otherwise $\set{f = 0}$ is finite near $0$.
So each set $\set{f = 0, g_i > 0}$ is, near $0$, either finite or a union of finitely many open intervals, and a finite union of such sets cannot equal $X$ near $0$.
Every semialgebraic subset of $\R^n$ and every real analytic subset of $M$ is semianalytic.
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<document source="https://tamnd.github.io/math2/node/N41N5QHDAN.md" tag="N41N5QHDAN" kind="statement">
# Fefferman duality theorem

## Statement

Let $n \ge 1$, let $H^1(\R^n)$ be the [[real Hardy space|75WGJ014GR]], and let $\mathrm{BMO}(\R^n)$ be the space of functions of [[bounded mean oscillation|QJYTXFT8Y4]], modulo constants.
Let $H^1_0$ be the dense subspace of $H^1(\R^n)$ of finite linear combinations of atoms.

1. For every $b \in \mathrm{BMO}(\R^n)$ the linear functional $\ell_b(f) = \int_{\R^n} f b \, dx$ on $H^1_0$ extends to a bounded linear functional on $H^1(\R^n)$, with $\norm{\ell_b} \le C \norm{b}_{\mathrm{BMO}}$.
2. Every bounded linear functional $\ell$ on $H^1(\R^n)$ is of the form $\ell_b$ for a function $b \in \mathrm{BMO}(\R^n)$, unique up to an additive constant, with $\norm{b}_{\mathrm{BMO}} \le C \norm{\ell}$.

Here $C$ depends only on $n$.
So the [[continuous dual space|4A82S5FVRT]] of $H^1(\R^n)$ is $\mathrm{BMO}(\R^n)$.

## Remarks

The theorem is due to Charles Fefferman, and a full account appeared in the work of Fefferman and Stein on $H^p$ spaces.

With the [[atomic decomposition|AQ15060AAA]], part 1 is short.
For an atom $a$ on a ball $B$, the cancellation $\int a = 0$ gives $\abs{\int a b} = \abs{\int_B a (b - b_B)} \le \frac{1}{\abs{B}} \int_B \abs{b - b_B} \le C \norm{b}_{\mathrm{BMO}}$, and the bound passes to sums of atoms.
For part 2, the definition of $H^1$ through the [[Riesz transforms|FRNDFJ9WQ5]] embeds $H^1$ isometrically into a closed subspace of $L^1 \times \cdots \times L^1$ ($n + 1$ factors).
By the Hahn–Banach theorem, $\ell$ is given by bounded functions $b_0, \dots, b_n$, so $b = b_0 - \sum_j R_j b_j$, and each $R_j$ maps $L^\infty$ into BMO.

The pairing $\int f b$ need not converge absolutely for general $f \in H^1$ and $b \in \mathrm{BMO}$, which is why it is first defined on $H^1_0$.
An analogous duality on the circle identifies the dual of the real $H^1$ of the circle with BMO of the circle.

## Explanations

### L1 human

Every continuous linear way of measuring functions in the Hardy space $H^1$ is given by integrating against a function of bounded mean oscillation, and every BMO function gives one.
This is like the familiar fact that the dual of $L^1$ is $L^\infty$, with $L^1$ replaced by the smaller space $H^1$ and $L^\infty$ by the larger space BMO.

### L2 human

The theorem explains why BMO appears so often as the endpoint of $L^p$ theory at $p = \infty$.
Since Calderón–Zygmund operators map $H^1$ into $L^1$, their adjoints map $L^\infty$ into BMO by duality.
It is also the basis for interpolation between $H^1$ and $L^p$, or between $L^p$ and BMO.
Combined with the Carleson measure characterization of BMO, it links the real-variable theory with harmonic extensions to the half-space.
</document>

<document source="https://tamnd.github.io/math2/node/N42JDBT0VA.md" tag="N42JDBT0VA" kind="statement">
# Hardy–Littlewood Tauberian theorem

## Statement

Let $(a_n)_{n \ge 0}$ be real numbers with $a_n \ge 0$ for all $n$.
Assume that the [[power series|0VP56B4EMX]] $\sum_{n \ge 0} a_n x^n$ converges for $0 \le x < 1$ and that
$$\sum_{n \ge 0} a_n x^n \sim \frac{1}{1 - x} \quad (x \to 1^-),$$
where $\sim$ denotes [[asymptotic equivalence|0QR92XBV6H]].
Then
$$\sum_{k=0}^{n} a_k \sim n \quad (n \to \infty).$$

## Remarks

Karamata's proof shows first that $(1 - x) \sum_{n} a_n x^n P(x^n) \to \int_0^1 P(t) \, dt$ for every polynomial $P$, by checking it on monomials $P(t) = t^m$.
Then it approximates the function $g$ with $g(t) = 1/t$ on $[e^{-1}, 1]$ and $g(t) = 0$ on $[0, e^{-1})$ from above and below by polynomials with close integrals.
Positivity of $a_n$ makes these approximations compatible with the sums.
Taking $x = e^{-1/n}$, one has $x^k g(x^k) = 1$ for $k \le n$ and $0$ for $k > n$, which isolates the partial sum $\sum_{k \le n} a_k$.

Example: for $a_n = 1 + (-1)^n$, one has $\sum a_n x^n = \frac{1}{1-x} + \frac{1}{1+x} \sim \frac{1}{1-x}$, and indeed $\sum_{k \le n} a_k$ is $n + 2$ or $n + 1$.

Positivity is needed.
For $a_n = 1 + (-1)^n n$, one has $\sum a_n x^n = \frac{1}{1-x} - \frac{x}{(1+x)^2} \sim \frac{1}{1-x}$.
But $\sum_{k=0}^{n} (-1)^k k$ is $\frac{n}{2}$ for even $n$ and $-\frac{n+1}{2}$ for odd $n$, so the partial sums $\sum_{k \le n} a_k$ are close to $\frac{3n}{2}$ or to $\frac{n}{2}$ and are not $\sim n$.

## Explanations

### L1 human

Knowing how a power series with nonnegative coefficients blows up as $x \to 1$ tells you how fast its coefficients add up.
If the power series behaves like $\frac{1}{1-x}$, then the coefficients behave on average like $1$, so their first $n$ add up to about $n$.
The statement is false without the sign condition, since cancellations can hide in the power series.

### L2 human

This theorem is the prototype of Tauberian theorems with positivity conditions, where no condition on the size of single terms is needed.
In terms of $x = e^{-s}$ it says that $\sum a_n e^{-sn} \sim \frac1s$ as $s \to 0^+$ implies $\sum_{k \le n} a_k \sim n$; [[Karamata's Tauberian theorem|DBEGF38V9H]] extends it to Laplace–Stieltjes transforms with regularly varying growth.
The same method of proof gives [[Littlewood's Tauberian theorem|Z35TD4ZZRD]], and the theorem applies to counting problems in number theory where generating functions are easier to estimate than partial sums.
</document>

<document source="https://tamnd.github.io/math2/node/N43G50YHG2.md" tag="N43G50YHG2" kind="statement">
# Dedekind's lemma on independence of characters

## Statement

Let $G$ be a group and let $L$ be a field, with multiplicative [[group of units|05THZKR9AN]] $L^\times$.
Let $\chi_1, \ldots, \chi_n \colon G \to L^\times$ be pairwise distinct [[group homomorphisms|0HV9Y7WG5R]].
Then $\chi_1, \ldots, \chi_n$ are [[linearly independent|0EVHJFTG9Z]] over $L$ as functions $G \to L$: if $a_1, \ldots, a_n \in L$ and $a_1 \chi_1(g) + \cdots + a_n \chi_n(g) = 0$ for all $g \in G$, then $a_1 = \cdots = a_n = 0$.

## Remarks

Suppose there is a nontrivial relation, and take one $\sum a_i \chi_i = 0$ with the fewest nonzero coefficients; then at least two coefficients, say $a_1$ and $a_2$, are nonzero.
Choose $h \in G$ with $\chi_1(h) \ne \chi_2(h)$.
Evaluating the relation at $hg$ and subtracting $\chi_1(h)$ times the relation at $g$ gives $\sum_i a_i (\chi_i(h) - \chi_1(h)) \chi_i = 0$.
This relation has fewer nonzero coefficients but a nonzero coefficient at $\chi_2$, a contradiction.

The main application concerns [[field homomorphisms|EZVN87C88J]].
If $\sigma_1, \ldots, \sigma_n \colon K \to L$ are distinct field homomorphisms, their restrictions to the group $K^\times$ are distinct characters, so $\sigma_1, \ldots, \sigma_n$ are linearly independent over $L$.
For example, the identity and complex conjugation on $\C$ are independent over $\C$: if $a z + b \bar z = 0$ for all $z$, then $z = 1$ gives $a + b = 0$ and $z = i$ gives $a - b = 0$, so $a = b = 0$.

The same proof works when $G$ is only a monoid and the $\chi_i$ are monoid homomorphisms into $L^\times$.

## Explanations

### L1 human

A character is a map from a group to the nonzero numbers of a field that turns products into products.
Dedekind's lemma says that different characters are genuinely different as functions: no one of them is a combination of the others.
It is the field-theoretic cousin of the fact that the functions $e^{a_1 x}, \ldots, e^{a_n x}$ with distinct $a_i$ are linearly independent.

### L2 human

In Galois theory the lemma is used to show that a finite extension $L/K$ has at most $[L:K]$ automorphisms over $K$.
It is the key step in the proofs of [[Artin's theorem on fixed fields|X7R15C00ET]], of [[Hilbert's Theorem 90|YFF8B5A0QG]] and of the [[normal basis theorem|PXY1WYNRD4]].
It also shows that the [[field trace|WPE5Z53QYJ]] of a finite separable extension is not identically zero.
</document>

<document source="https://tamnd.github.io/math2/node/N44162Y9XS.md" tag="N44162Y9XS" kind="concept">
# Approximate point spectrum

## Summary

The approximate point spectrum of an operator $T$ is the set of numbers $\lambda$ that are eigenvalues up to an arbitrarily small error.
This means that there are unit vectors $x$ with $Tx - \lambda x$ as small as one likes.
It contains every eigenvalue and the whole continuous spectrum, and also the boundary of the spectrum.
For self-adjoint and normal operators on Hilbert spaces it is the whole spectrum.

## Explanations

### L1 human

An eigenvector is a vector $x$ with $Tx = \lambda x$ exactly.
An approximate eigenvector only needs $Tx$ to be very close to $\lambda x$, compared with the size of $x$.
For the operator that divides the $n$-th entry of a sequence by $n$, the $n$-th unit vector is sent to a vector of length $1/n$.
So $0$ is an approximate eigenvalue, although it is not an eigenvalue.
</document>

<document source="https://tamnd.github.io/math2/node/N469E47V75.md" tag="N469E47V75" kind="concept">
# Pseudoholomorphic curve

## Summary

A pseudoholomorphic curve, or $J$-holomorphic curve, is a map from a [[Riemann surface|5K0N8QMV6Q]] into an [[almost complex manifold|9SYWRGVTRS]] whose differential is complex linear.
On a symplectic manifold with a compatible almost complex structure, the moduli spaces of such curves carry topological information.
Gromov introduced them in 1985, and they are the basis of Gromov–Witten invariants, [[Floer homology|NCKFPAGHXF]] and [[Fukaya categories|4HS60JABC5]].

## Explanations

### L1 human

A holomorphic function of one complex variable bends the plane in a way that keeps angles.
A pseudoholomorphic curve is the same idea for maps of a surface into a bigger space that only has a pointwise notion of multiplication by $i$ on tangent vectors.
On a symplectic manifold these curves behave like soap films: they minimize area in their class.
For example, in $\C^2$ with the usual complex structure, the map $z \mapsto (z, z^2)$ is a pseudoholomorphic curve.
</document>

<document source="https://tamnd.github.io/math2/node/N470GNCK2Y.md" tag="N470GNCK2Y" kind="concept">
# Mean-periodic function

## Summary

A mean-periodic function is a continuous function on the real line that satisfies a convolution equation $f * \mu = 0$ for some nonzero compactly supported measure $\mu$.
A [[periodic function|0ATF6Q0633]] with period $p$ is the case $\mu = \delta_0 - \delta_p$, and exponentials $x^k e^{\lambda x}$ are also mean-periodic.
Equivalently, the translates of $f$ do not span a dense subspace of the continuous functions.
Laurent Schwartz proved that every mean-periodic function on $\R$ is a limit of exponential polynomials taken from the closed span of its translates.

## Explanations

### L1 human

A periodic function equals its own shift: $f(x) - f(x - p) = 0$.
A mean-periodic function satisfies a weaker rule: some fixed weighted average of its shifted copies is always zero.
For example, $e^{\lambda x}$ satisfies $f(x) - e^{\lambda} f(x - 1) = 0$, although it is not periodic for real $\lambda \ne 0$.
Such functions behave like sums of exponential waves, which explains the name.
</document>

<document source="https://tamnd.github.io/math2/node/N470JF7ZJX.md" tag="N470JF7ZJX" kind="example">
# Ackermann function

## Example

The Ackermann function, in the two-argument form of Rózsa Péter, is the function $A \colon \N \times \N \to \N$ given by
$$A(0, n) = n + 1, \qquad A(m + 1, 0) = A(m, 1), \qquad A(m + 1, n + 1) = A(m, A(m + 1, n)).$$
It is a total [[computable function|KZSB5E471K]].
Each right-hand side calls $A$ at a pair that is smaller in the lexicographic order on $\N \times \N$.
This order is a well-order, so the recursion stops on every input and gives an algorithm for $A$.

## Remarks

By induction on $n$, one gets $A(1, n) = n + 2$, $A(2, n) = 2n + 3$ and $A(3, n) = 2^{n+3} - 3$.
Then $A(4, 0) = A(3, 1) = 13$ and $A(4, 1) = A(3, 13) = 2^{16} - 3 = 65533$.
The function $A$ is not [[primitive recursive|5NVA2JRHPV]], see [[Ackermann function is not primitive recursive|97QF3KHSBY]].

## Explanations

### L1 human

The Ackermann function is a simple rule that produces enormous numbers very quickly.
Its first argument chooses the operation: adding, then something like multiplying, then something like taking powers, and so on.
The values $A(4, n)$ are already huge: $A(4, n) + 3$ is a tower of $n + 3$ twos, such as $A(4, 1) + 3 = 2^{2^{2^2}}$.
It was built to show that some computable functions cannot be defined with bounded loops alone.
</document>

<document source="https://tamnd.github.io/math2/node/N4EDJNMTEF.md" tag="N4EDJNMTEF" kind="concept">
# Proof theory

## Summary

Proof theory is the branch of mathematical logic that treats [[proofs|0B94TKFQJX]] as mathematical objects.
It studies formal proofs in systems of [[predicate logic|086QKPF0A6]] and of arithmetic by combinatorial means.
It began with Hilbert's aim to prove the consistency of mathematics by finitary methods.
Its main tools include normal forms of proofs, cut elimination and ordinal analysis.

## Explanations

### L1 human

In most of mathematics, a proof is a tool to reach a theorem.
In proof theory, the proof itself is the object of study, like a number or a graph.
A formal proof is a finite tree of formulas built by fixed rules.
So one can count its steps, transform it, and ask what it can never reach.
For example, one can show that a system never proves $0 = 1$.
One way is to put every proof in a simple normal form, and to check that no proof in normal form ends with $0 = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/N4FACX7QSW.md" tag="N4FACX7QSW" kind="definition">
# Holonomic function

## Definition

Let $K$ be a field of characteristic $0$, for example $\C$.
A [[formal power series|033WGN00BF]] $f \in K[\![x]\!]$ is *holonomic* (or *D-finite*) if there are an integer $r \ge 0$ and polynomials $p_0, p_1, \ldots, p_r \in K[x]$ with $p_r \ne 0$ such that
$$p_r(x) f^{(r)}(x) + \cdots + p_1(x) f'(x) + p_0(x) f(x) = 0.$$
Here $f^{(k)}$ is the $k$-th formal derivative of $f$ and $K[x]$ is the [[polynomial ring|0SDC934R52]].
An analytic function on an open set of $\C$ is holonomic if it satisfies such an equation with $K = \C$.

## Remarks

Example: $f(x) = \frac{1}{1 - x} = \sum_{n \ge 0} x^n$ satisfies $(1 - x) f'(x) - f(x) = 0$, since $f'(x) = \frac{1}{(1-x)^2}$; here $r = 1$, $p_1 = 1 - x$ and $p_0 = -1$.
Non-example: $\tan x$ is not holonomic.
An analytic holonomic function extends along paths avoiding the finitely many zeros of $p_r$, so it has only finitely many singularities, while $\tan$ has poles at all points $\frac{\pi}{2} + k\pi$.
Equivalently, $f$ is holonomic if and only if the derivatives $f, f', f'', \ldots$ span a finite-dimensional vector space over the field of [[rational functions|0XDK9CGYKJ]] $K(x)$.
Another equivalent form: $f = \sum_n a_n x^n$ is holonomic if and only if its coefficients satisfy a [[recurrence|0FYX44GP92]] $q_d(n) a_{n+d} + \cdots + q_0(n) a_n = 0$ for all $n$, with polynomials $q_i$ not all zero; such sequences are called P-recursive.

## Explanations

### L1 human

A holonomic function is one that obeys a linear differential equation with polynomial coefficients.
The exponential $e^x$ is the simplest example: it satisfies $f' - f = 0$.
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<document source="https://tamnd.github.io/math2/node/N4KDT83V3S.md" tag="N4KDT83V3S" kind="concept">
# Measurable space

## Summary

A measurable space is a set together with a [[σ-algebra|0SV8K0TDGY]] of its subsets.
The sets in the σ-algebra are the measurable sets.
Measurable spaces are the setting in which measures, measurable functions and random variables are defined.

## Explanations

### L1 human

A measurable space is a set with a chosen list of subsets that one is allowed to measure.
The list must contain the whole set and be stable under complements and countable unions.
Think of a dartboard: one wants to speak of the probability of hitting a region, so one fixes which regions count.
For example, the real line with the Borel sets, generated by the open intervals, is a measurable space.
No measure is chosen yet; the measurable space only fixes which sets will get a size.
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<document source="https://tamnd.github.io/math2/node/N4NTJ6W38N.md" tag="N4NTJ6W38N" kind="definition">
# Nonlinear integral equation

## Definition

Let $a < b$.
A *nonlinear integral equation* on $[a, b]$ is an [[integral equation|R3SG1SN7JQ]] for an unknown function $\varphi \colon [a, b] \to \C$ that is not a [[linear integral equation|AYZCKG34QK]].
The most common form is the *Urysohn equation*
$$\varphi(x) = f(x) + \int_{I_x} H\left(x, s, \varphi(s)\right) ds \quad \text{for all } x \in [a, b].$$
Here $I_x$ is $[a, b]$ or $[a, x]$, $f$ is given, and $H \colon [a, b] \times [a, b] \times \C \to \C$ is a given function that is not affine in its third variable.

## Remarks

Example: $\varphi(x) = x + \int_0^1 \varphi(s)^2 \, ds$ on $[0, 1]$ is nonlinear, with $H(x, s, u) = u^2$.
A solution has the form $\varphi(x) = x + c$, where $c = \int_0^1 (s + c)^2 \, ds = 1/3 + c + c^2$.
This gives $c^2 = -1/3$, so the equation has the two complex solutions $\varphi(x) = x \pm i/\sqrt{3}$ and no real one.

Non-example: $\varphi(x) = x + \int_0^1 (x s \, \varphi(s) + s^2) \, ds$ is linear, since $H(x, s, u) = x s u + s^2$ is affine in $u$.

When $H(x, s, u) = K(x, s) \, g(s, u)$ the equation is called a *Hammerstein equation*.
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<document source="https://tamnd.github.io/math2/node/N4QZS912MT.md" tag="N4QZS912MT" kind="statement">
# Minkowski inequality

## Statement

Let $(S, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $p \in [1, \infty)$.
Let $f, g \colon S \to \R$ be [[measurable functions|YYNCJSR0PJ]] with $\int_S \abs{f}^p \, d\mu < \infty$ and $\int_S \abs{g}^p \, d\mu < \infty$.
Then
$$\Big( \int_S \abs{f + g}^p \, d\mu \Big)^{1/p} \le \Big( \int_S \abs{f}^p \, d\mu \Big)^{1/p} + \Big( \int_S \abs{g}^p \, d\mu \Big)^{1/p}.$$

## Remarks

Proof idea: the case $p = 1$ follows from $\abs{f + g} \le \abs{f} + \abs{g}$.
For $p > 1$, first $\abs{f + g}^p \le 2^{p-1}(\abs{f}^p + \abs{g}^p)$ shows that the left side is finite.
Then write $\abs{f + g}^p \le \abs{f} \abs{f + g}^{p-1} + \abs{g} \abs{f + g}^{p-1}$ and apply [[Hölder's inequality|JQ9WXC729D]] to each term with $q = p / (p - 1)$.

With the counting measure on $\set{1, \dots, n}$ it shows that the [[p-norm on R^n|0KNJWEVSMG]] satisfies the triangle inequality.
In general it is the triangle inequality for the [[norm|08NYZ2FS6B]] of the [[Lp space|SASP14MA1Z]] $L^p(\mu)$.

The hypothesis $p \ge 1$ is needed.
For $p = 1/2$ on $\R^2$ with the counting measure, take $f = (1, 0)$ and $g = (0, 1)$: the left side is $(1 + 1)^2 = 4$ and the right side is $1 + 1 = 2$.

## Explanations

### L1 human

The Minkowski inequality is the triangle inequality for the $p$-size of functions or vectors.
The size of a sum is at most the sum of the sizes.
For $p = 2$ and vectors in the plane, it says that one side of a triangle is at most the sum of the other two.

### L2 human

This inequality is what makes $\norm{f}_p = (\int \abs{f}^p \, d\mu)^{1/p}$ a norm for $p \ge 1$, so that $L^p$ is a normed space.
Its proof is a short application of Hölder's inequality.
For $0 < p < 1$ the inequality fails, and the quantity $\norm{f}_p$ is then not a norm.
</document>

<document source="https://tamnd.github.io/math2/node/N528CFDN46.md" tag="N528CFDN46" kind="definition">
# Nest algebra

## Definition

Let $H$ be a Hilbert space and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.
A *nest* on $H$ is a family $\mathcal{N}$ of closed linear subspaces of $H$ such that:

1. $\set{0} \in \mathcal{N}$ and $H \in \mathcal{N}$;
2. $\mathcal{N}$ is a [[chain|0S8FE7ZMFM]] under inclusion: for $M, N \in \mathcal{N}$, $M \subseteq N$ or $N \subseteq M$;
3. $\mathcal{N}$ is complete: for every subfamily, the intersection and the closed linear span of its members belong to $\mathcal{N}$.

The *nest algebra* of $\mathcal{N}$ is
$$\operatorname{Alg} \mathcal{N} = \set{T \in B(H) : T(N) \subseteq N \text{ for all } N \in \mathcal{N}},$$
the set of operators for which every member of $\mathcal{N}$ is an [[invariant subspace|0ACX5QT8TE]].

## Remarks

Example: let $H = \C^n$ with standard basis $e_1, \ldots, e_n$, and let $\mathcal{N}$ consist of $N_0 = \set{0}$ and $N_k = \operatorname{span}(e_1, \ldots, e_k)$ for $1 \le k \le n$.
A matrix $T$ maps each $N_k$ into itself exactly when $T e_j \in N_j$ for all $j$, that is, when $T$ is upper triangular, so $\operatorname{Alg} \mathcal{N}$ is the algebra of upper triangular matrices.
Non-example: in $\C^2$, the family $\set{0}, \C e_1, \C e_2, \C^2$ satisfies clauses 1 and 3 but not clause 2.
Indeed $\C e_1$ and $\C e_2$ are not comparable.
The operators leaving this family invariant are the diagonal matrices.
The diagonal algebra is not a nest algebra at all, since the nests on $\C^2$ give only $M_2(\C)$ and algebras of dimension $3$.
Every nest algebra is weakly closed, contains the identity, and is reflexive.
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<document source="https://tamnd.github.io/math2/node/N56DHWH283.md" tag="N56DHWH283" kind="definition">
# Ordered geometry

## Definition

Let $\Pi$ be a set of points with a family of subsets called [[lines|0MP9BCF6DP]].
Let $[ABC]$ be a ternary relation on $\Pi$, read "$B$ lies [[between|CE8GX6ADZS]] $A$ and $C$".
Then $\Pi$ is an *ordered plane* if:

1. Two distinct points lie on exactly one line, each line has at least two points, and there are three points not on one line.
2. If $[ABC]$, then $A$, $B$, $C$ are three distinct points of one line, and $[CBA]$.
3. For distinct points $A$ and $C$, there is a point $B$ with $[ACB]$.
4. Of three distinct points of a line, at most one lies between the other two.
5. [[Pasch's axiom|KJT98G16Y4]] holds.

*Ordered geometry* is the study of the consequences of these axioms, and of their analogues in higher dimensions.

## Remarks

Example: the plane $F^2$ over an ordered field $F$, with the usual lines and with $[ABC]$ meaning $B = A + t(C - A)$ for some $t \in F$ with $0 < t < 1$, is an ordered plane.
Non-example: take $\R^2$ with the usual lines, and let $[ABC]$ mean that $A$, $B$, $C$ are three distinct points of one line.
This fails clause 4, since each of three collinear points then lies between the other two.
One proves from the axioms that, of three distinct collinear points, exactly one lies between the other two.
The Sylvester–Gallai theorem holds in every ordered plane.
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<document source="https://tamnd.github.io/math2/node/N587ECTQ4X.md" tag="N587ECTQ4X" kind="concept">
# Order polytope

## Summary

The order polytope of a finite poset $P$ is the set of functions $f \colon P \to [0, 1]$ that are [[order-preserving|0A7KV2WB7A]], seen as points of $\R^P$.
It is a convex polytope with integer vertices, namely the indicator functions of the upper sets of $P$.
Its volume is the number of [[linear extensions|W8DZ5KGGHN]] of $P$ divided by $\abs{P}!$, and its lattice point counts are given by the [[order polynomial|VDBDJZWPEA]].
It was introduced by Richard Stanley, together with the chain polytope of the poset.

## Explanations

### L1 human

Give each element of a poset a real number between $0$ and $1$, so that larger elements get numbers at least as large.
All such assignments fill a convex region in space, one coordinate per element.
For two unrelated elements you get the whole unit square.
For a chain $x < y$ you get the triangle where $0 \le f(x) \le f(y) \le 1$, which is half of the square.
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<document source="https://tamnd.github.io/math2/node/N59BW1QFD1.md" tag="N59BW1QFD1" kind="concept">
# Space of functions vanishing at infinity

## Summary

For a locally compact Hausdorff space $X$, the space $C_0(X)$ consists of the continuous functions that become small outside compact sets.
It is a closed subspace of the [[space of bounded continuous functions|MG3X8Y1VT4]], so it is a Banach space with the supremum norm.
By the Riesz–Markov–Kakutani theorem, its dual is the space of finite signed regular Borel measures on $X$, or complex ones for complex scalars.
It is also the basic example of a commutative C*-algebra without unit.

## Explanations

### L1 human

A function vanishes at infinity when it gets as small as you like once you go far enough away.
On $\R$, the bell curve $e^{-x^2}$ vanishes at infinity, and so does $1/(1 + x^2)$.
The constant function $1$ does not.
On the set $\N$ of natural numbers, these functions are just the sequences that tend to $0$.
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<document source="https://tamnd.github.io/math2/node/N5JCKN50NE.md" tag="N5JCKN50NE" kind="definition">
# Degenerate kernel

## Definition

Let $a < b$.
A *degenerate kernel* on $[a, b]$ is a [[kernel|GSMBXT7FPK]] $K \colon [a, b] \times [a, b] \to \C$ for which there are an integer $n \ge 1$ and functions $a_1, \ldots, a_n, b_1, \ldots, b_n \colon [a, b] \to \C$ such that
$$K(x, s) = \sum_{i=1}^{n} a_i(x) \, b_i(s) \quad \text{for all } x, s \in [a, b].$$

## Remarks

Example: $K(x, s) = \cos(x - s) = \cos x \cos s + \sin x \sin s$, with $n = 2$, $a_1 = b_1 = \cos$ and $a_2 = b_2 = \sin$.

Non-example: $K(x, s) = e^{x s}$ on $[0, 1]$.
If $e^{x s} = \sum_{i=1}^{n} a_i(x) b_i(s)$, then each function $x \mapsto e^{x s}$ would lie in the span of $a_1, \ldots, a_n$.
But the functions $e^{s_1 x}, \ldots, e^{s_{n+1} x}$ for distinct $s_j$ are [[linearly independent|0CRMGPAH83]], which gives $n + 1$ independent functions in a space of dimension at most $n$.

The names *separable kernel* and *kernel of finite rank* are also used.
One usually asks that $a_i$ and $b_i$ be continuous, or square-integrable, and one may take $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ linearly independent.
For $\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds$ with a degenerate $K$, every solution has the form $\varphi = f + \lambda \sum_i c_i a_i$ with $c_i = \int_a^b b_i \varphi \, ds$, and the $c_i$ solve an $n \times n$ linear system.
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# Polyharmonic function

## Definition

Let $n \ge 1$, let $m \ge 1$ be an integer, and let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]].
Let $\Delta$ be the [[Laplacian|0E0B6RA8K0]], and let $\Delta^m$ be $\Delta$ applied $m$ times.
A function $u \colon \Omega \to \R$ is *polyharmonic of order $m$* if it is of class $C^{2m}$ and
$$\Delta^m u = 0 \quad \text{on } \Omega.$$

## Remarks

For $m = 1$ this is the definition of a [[harmonic function|0DEB0JMN0W]], and for $m = 2$ that of a solution of the [[biharmonic equation|AS8A9F5SKE]].
A function that is polyharmonic of order $m$ is polyharmonic of every order $m' \ge m$.
Example: $u(x) = \abs{x}^4$ on $\R^n$ is polyharmonic of order $3$: $\Delta \abs{x}^4 = 4(n+2)\abs{x}^2$, $\Delta^2 \abs{x}^4 = 8n(n+2)$ and $\Delta^3 \abs{x}^4 = 0$.
Non-example: the same $u$ is not polyharmonic of order $2$, since $\Delta^2 u = 8n(n+2) \ne 0$.
Non-example: $u(x) = e^{x_1}$ is polyharmonic of no order, since $\Delta^m u = e^{x_1} \ne 0$ for all $m$.
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<document source="https://tamnd.github.io/math2/node/N5KE7FH6CX.md" tag="N5KE7FH6CX" kind="concept">
# Quotient algebra

## Summary

A quotient algebra is the algebra of classes of a [[congruence relation|DCSXXT4SER]], with the operations computed on representatives.
It generalizes the [[quotient group|0KYX2N69J0]] and the [[quotient ring|0076HWHGHQ]].
Every homomorphic image of an algebra is isomorphic to one of its quotients, by the [[homomorphism theorem|N783XNJNBV]].

## Explanations

### L1 human

Start with an algebra and a congruence, which says which elements to treat as equal.
Glue each class of equal elements into a single point.
To apply an operation to some classes, pick one element from each class, apply the operation, and take the class of the result.
The congruence guarantees that the choice does not matter.
For the integers with congruence modulo $5$, this gives the five-element ring $\Z/5\Z$.
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<document source="https://tamnd.github.io/math2/node/N5MRMYN61N.md" tag="N5MRMYN61N" kind="definition">
# Elliptic hypergeometric series

## Definition

Let $\sum_{n \ge 0} c_n$ be a series of nonzero complex numbers.
View the term ratio $c_{n+1} / c_n$ as a function of $n$ extended to complex values.
The series is an *elliptic hypergeometric series* if this ratio is an [[elliptic function|FF9ZPCZ59K]] of $n$.

In the usual explicit form, fix complex numbers $p, q$ with $\abs{p} < 1$ and $q \ne 0$.
The *modified theta function* is $\theta(x; p) = (x; p)_\infty \, (p/x; p)_\infty$ for $x \ne 0$, where $(\cdot\,; p)_\infty$ is the infinite [[q-Pochhammer symbol|Y1X8ZZQ7VM]].
The *elliptic shifted factorial* is $(a; q, p)_n = \prod_{k=0}^{n-1} \theta(a q^k; p)$.
For nonzero parameters $a_1, \ldots, a_{r+1}$, $b_1, \ldots, b_r$ and $z$, the series
$${}_{r+1}E_r(a_1, \ldots, a_{r+1}; b_1, \ldots, b_r; q, p; z) = \sum_{n \ge 0} \frac{(a_1; q, p)_n \cdots (a_{r+1}; q, p)_n}{(q; q, p)_n \, (b_1; q, p)_n \cdots (b_r; q, p)_n} z^n$$
is an elliptic hypergeometric series when the balancing condition $a_1 a_2 \cdots a_{r+1} = q \, b_1 b_2 \cdots b_r$ holds.

## Remarks

Why the balancing condition: write $q = e^{2\pi i \sigma}$ and $p = e^{2\pi i \tau}$, and view the term ratio as a function of a complex variable $n$.
The shift $n \mapsto n + 1/\sigma$ leaves $q^n$ unchanged.
The shift $n \mapsto n + \tau/\sigma$ replaces $q^n$ by $p q^n$, and the rule $\theta(px; p) = -x^{-1}\theta(x; p)$ shows that it multiplies the ratio by $q b_1 \cdots b_r / (a_1 \cdots a_{r+1})$.
So the ratio is elliptic with periods $1/\sigma$ and $\tau/\sigma$ exactly when this factor is $1$.

Example: for $p = 0$ one has $\theta(x; 0) = 1 - x$ and $(a; q, 0)_n = (a; q)_n$, so the series becomes the [[basic hypergeometric series|EB4N9RQR51]] ${}_{r+1}\phi_r$ with the same parameters.

Non-example: $\sum_n n! \, z^n$ has term ratio $(n + 1) z$, a nonconstant polynomial in the complex variable $n$.
A nonconstant elliptic function must have poles, so the series is not elliptic hypergeometric.
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<document source="https://tamnd.github.io/math2/node/N5NTPAR2KT.md" tag="N5NTPAR2KT" kind="example">
# The cosets of a subgroup that is not normal do not form a group

## Example

In the [[symmetric group|0515A6D2Y1]] $S_3$, let $H = \{e, (1\,2)\}$. Compose permutations from right to left.
Two of the [[left cosets|05FD317V2W]] are $(1\,3)H = \{(1\,3), (1\,2\,3)\}$ and $(2\,3)H = \{(2\,3), (1\,3\,2)\}$.
Try to set $(aH)(bH) = (ab)H$. With $a = (1\,3)$ and $b = (2\,3)$, we get $ab = (1\,3\,2)$, which is in $(2\,3)H$.
With $a = (1\,2\,3)$ from the same coset, we get $ab = (1\,2)$, which is in $H$.
The two results are in different cosets. Thus this product is not defined, and the cosets of $H$ do not form a [[quotient group|0YYWCNTPWR]].

## Remarks

$H$ is not [[normal|0W77CVBCHQ]] in $S_3$. For a normal subgroup the product of cosets is always defined.
</document>

<document source="https://tamnd.github.io/math2/node/N5PM2NCPWF.md" tag="N5PM2NCPWF" kind="concept">
# Liouvillian function

## Summary

A Liouvillian function is built from rational functions in finitely many steps by taking algebraic functions, integrals and exponentials of integrals.
They make precise the idea of a solution "in closed form" for differential equations.
[[Differential Galois theory|EEZS4H8QPH]] decides whether a linear differential equation has Liouvillian solutions.
See the [[definition|R80FA7993W]].

## Explanations

### L1 human

Start with fractions of polynomials in $x$.
Allow three operations again and again: solving polynomial equations, taking antiderivatives, and taking exponentials of antiderivatives.
Everything one can reach is Liouvillian; for example $e^{x^2}$ and the error function $\int e^{-x^2} dx$ are.
Elementary functions such as $\sin x$, $\ln x$ and $\sqrt{1 + x^3}$ are all Liouvillian.
The solutions of the [[Airy equation|PNESAD6R5T]] $y'' = x y$ are not.
</document>

<document source="https://tamnd.github.io/math2/node/N5PVGRJE1T.md" tag="N5PVGRJE1T" kind="concept">
# Mellin transform

## Summary

The Mellin transform sends a function $f$ on $(0, \infty)$ to the function $\mathcal{M}f(s) = \int_0^\infty x^{s-1} f(x) \, dx$ of a complex variable $s$.
It is the integral transform adapted to the multiplicative group $(0, \infty)$, in the way the [[Fourier transform|DPVM34ADZV]] is adapted to the additive group $\R$.
It usually converges on a vertical strip of the complex plane, and it turns dilations $x \mapsto ax$ into multiplication by $a^{-s}$.
It links power series and Dirichlet series with the [[gamma function|0H5SYBSED9]] and the [[Riemann zeta function|1XWBY4N0VT]], and it is a main tool for asymptotic expansions.

## Explanations

### L1 human

The Mellin transform measures how a function on the positive reals behaves against the powers $x^{s-1}$.
Changing the variable to $x = e^{u}$ turns it into a Laplace or Fourier transform in $u$, so it is the Fourier transform of a function viewed on a logarithmic scale.
Its basic example is $f(x) = e^{-x}$, whose Mellin transform is $\Gamma(s)$ for $\operatorname{Re} s > 0$.
Scaling the input, $x \mapsto 2x$, only multiplies the transform by $2^{-s}$.
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<document source="https://tamnd.github.io/math2/node/N5R1WAEPFB.md" tag="N5R1WAEPFB" kind="concept">
# Geometric morphism

## Summary

A geometric morphism $f \colon \mathcal{E} \to \mathcal{F}$ between [[toposes|6837RS7N82]] is a pair of [[adjoint functors|TZHF6ZM92S]] $f^* \dashv f_*$, with $f_* \colon \mathcal{E} \to \mathcal{F}$, whose left adjoint $f^* \colon \mathcal{F} \to \mathcal{E}$ preserves finite limits.
It is the notion of map between toposes that matches continuous maps between spaces.
A continuous map $g \colon X \to Y$ gives a geometric morphism $\mathrm{Sh}(X) \to \mathrm{Sh}(Y)$, with $g_*$ the direct image and $g^*$ the inverse image of sheaves.
Points of a topos, and the classifying toposes of geometric theories, are described through geometric morphisms.

## Explanations

### L1 human

A continuous map of spaces lets one push sheaves forward and pull sheaves back, and these two operations are adjoint.
Pulling back keeps finite limits, such as products and equalities of sections.
A geometric morphism is any such adjoint pair between toposes, so it acts as a continuous map between generalized spaces.
For example, a point $x$ of a space $X$ gives a geometric morphism $\mathbf{Set} \to \mathrm{Sh}(X)$ whose inverse image sends a sheaf to its stalk at $x$.
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<document source="https://tamnd.github.io/math2/node/N5ZHV4HWDC.md" tag="N5ZHV4HWDC" kind="concept">
# Divided differences

## Summary

Divided differences are quotients built recursively from the values of a function at a list of points.
First comes the slope between two points, then the difference of two slopes divided by the spread of three points, and so on.
The divided difference $f[x_0, \dots, x_n]$ is the leading coefficient of the polynomial of degree at most $n$ that interpolates $f$ at the points.
They are the coefficients of the [[Newton form|9W77FPTM9E]] of the interpolation polynomial.
For smooth functions they behave like scaled derivatives: by the [[mean value theorem for divided differences|BQJ9C1CD5X]], $f[x_0, \dots, x_n] = f^{(n)}(\xi)/n!$ for some $\xi$.

## Explanations

### L1 human

The first divided difference $\frac{f(x_1) - f(x_0)}{x_1 - x_0}$ is the slope of the chord between two points of a graph.
The second one compares two neighbouring slopes and so measures how the graph bends.
For $f(x) = x^2$ and the points $0, 1, 3$, the slopes are $1$ and $4$, and the second divided difference is $\frac{4 - 1}{3 - 0} = 1$, the leading coefficient of $x^2$.
Higher divided differences continue the same pattern.
The [[definition|T942SHWSSF]] gives the recursion.
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<document source="https://tamnd.github.io/math2/node/N602W0WAVT.md" tag="N602W0WAVT" kind="concept">
# Hyperplane arrangement

## Summary

A hyperplane arrangement is a finite set of hyperplanes in a vector space or affine space.
Over the real numbers the hyperplanes cut the space into convex pieces called regions.
One studies how many regions there are and how they fit together.
The combinatorics of an arrangement is captured by its intersection poset, the [[partially ordered|0QV2JS9RJ3]] set of all nonempty intersections of its hyperplanes.
Arrangements connect polytopes, [[matroids|PX31ZNYS3Z]], root systems and the topology of complements of hypersurfaces.

## Explanations

### L1 human

Draw a few straight lines on a sheet of paper.
They cut the sheet into pieces, and the question is how many pieces there are.
Three lines in general position give $7$ pieces, while three parallel lines give only $4$.
A hyperplane arrangement is the same picture in any dimension, with flat walls instead of lines.
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<document source="https://tamnd.github.io/math2/node/N63Z6EA6YS.md" tag="N63Z6EA6YS" kind="example">
# Smale horseshoe

## Example

Let $Q = [0,1]^2$, and let $H_0 = [0,1] \times [0, \tfrac13]$ and $H_1 = [0,1] \times [\tfrac23, 1]$ be two horizontal strips.
Consider a diffeomorphism $f$ of the plane that bends $Q$ into a horseshoe shape and is affine on the strips:
$$f(x, y) = \left(\tfrac{x}{3}, \; 3y\right) \text{ on } H_0, \qquad f(x, y) = \left(1 - \tfrac{x}{3}, \; 3 - 3y\right) \text{ on } H_1.$$
Then $f(H_0) = [0, \tfrac13] \times [0,1]$ and $f(H_1) = [\tfrac23, 1] \times [0, 1]$, and the middle strip of $Q$ is mapped outside $Q$.
Let $\Lambda = \bigcap_{n \in \Z} f^n(Q)$ be the set of points whose whole orbit stays in $Q$.

Check of the structure.
Consider the maps $y \mapsto 3y$ on $[0, \tfrac13]$ and $y \mapsto 3 - 3y$ on $[\tfrac23, 1]$.
A point stays in $Q$ for all forward times exactly when its $y$-coordinate never leaves $[0, \tfrac13] \cup [\tfrac23, 1]$ under these maps.
This means that $y$ lies in the middle-thirds [[Cantor set|0BXNDZFQME]] $C$.
By the same argument for $f^{-1}$, a point stays in $Q$ for all backward times exactly when $x \in C$.
So $\Lambda = C \times C$, a Cantor set.
The map $h \colon \Lambda \to \set{0,1}^{\Z}$, with $h(p)_n = i$ when $f^n(p) \in H_i$, is a homeomorphism with $h \circ f = \sigma \circ h$, so $f|_\Lambda$ is [[topologically conjugate|HNXXAPQKG3]] to the full shift on two symbols.
On $\Lambda$, $Df$ equals $\operatorname{diag}(\tfrac13, 3)$ or $\operatorname{diag}(-\tfrac13, -3)$.
So the horizontal direction is contracted and the vertical one expanded by the factor $3$, and $\Lambda$ is a [[hyperbolic set|MPFD24TYW2]] with $C = 1$ and $\lambda = \tfrac13$.

## Remarks

Through the conjugacy, $f|_\Lambda$ has $2^n$ points of period dividing $n$, a dense orbit, and topological entropy $\log 2$.
Hyperbolicity makes the horseshoe robust: every diffeomorphism $C^1$-close to $f$ has an invariant Cantor set on which it is conjugate to the same shift.
Smale showed that a transverse homoclinic point of a diffeomorphism produces a horseshoe for some iterate, which explains why such points imply chaotic behaviour.

## Explanations

### L1 human

Take a square, stretch it to a long thin strip, bend the strip into a horseshoe, and lay it back across the square.
Some points leave the square, and the points that stay forever form a dust of points, a Cantor set.
Each such point is described by an infinite list of $0$s and $1$s, telling whether it is in the lower or the upper part of the square at each time.
Every list occurs, so the motion is as unpredictable as tossing a coin, yet it comes from a simple smooth map.
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<document source="https://tamnd.github.io/math2/node/N67R59ZVC4.md" tag="N67R59ZVC4" kind="statement">
# Schur's lemma in abelian categories

## Statement

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]].
An object $S$ of $\mathcal{A}$ is *simple* if $S$ is not a zero object and its only [[subobjects|FS1FKAE89H]] are $0$ and $S$.
Let $S$ and $T$ be simple objects and let $f \colon S \to T$ be a morphism.
Then $f$ is either the [[zero morphism|YF9F9EG6M3]] or an isomorphism.
In particular, the endomorphism ring $\End(S)$ is a [[division ring|0EA5BTA0CX]].

## Remarks

The [[kernel|S417YNEJ6M]] of $f$ is a subobject of $S$, so it is $0$ or $S$.
The image of $f$ is a subobject of $T$, so it is $0$ or $T$.
If $f \ne 0$, then the kernel is not $S$ and the image is not $0$.
So $f$ is a [[monomorphism|Y2NP1JFR86]] and an [[epimorphism|RVXDSY04KG]], and in an abelian category such a morphism is an isomorphism.
For $\End(S)$, every nonzero element is invertible, and $1 \ne 0$ because $S$ is not zero.
In the category of modules over a ring, the simple objects are the simple modules, and one gets the classical Schur's lemma.
The simplicity hypothesis is needed: multiplication by $2$ on the $\Z$-module $\Z$ is nonzero and not an isomorphism, and $\Z$ is not simple since $2\Z$ is a proper nonzero submodule.

## Explanations

### L1 human

A simple object is one that cannot be cut into smaller nonzero pieces.
Between two such objects, a map either kills everything or matches them up perfectly.
There is nothing in between, because the kernel and the image of the map must each be all or nothing.

### L2 human

The lemma is the first step in studying objects of finite length, which are built from simple objects by extensions as in the Jordan–Hölder theorem.
It shows that $\Hom(S, T) = 0$ for non-isomorphic simple objects and that $\End(S)$ is a division ring.
In representation theory over an algebraically closed field $k$, a further argument with eigenvalues gives $\End(S) = k$ for finite-dimensional simple representations; that sharper form needs the extra hypotheses.
</document>

<document source="https://tamnd.github.io/math2/node/N685RXHDZX.md" tag="N685RXHDZX" kind="statement">
# Thom isomorphism

## Statement

Let $\pi \colon E \to B$ be a real [[vector bundle|P4DZM975BV]] of rank $n$ over a topological space $B$.
Assume it is oriented: each fibre $F_b = \pi^{-1}(b)$ has an [[orientation|0QJJAQ6ZYZ]], given locally by trivializations.
Let $E_0 \subseteq E$ be the complement of the zero section, and $F_{b,0} = F_b \setminus \set{0}$.
Then, in [[singular cohomology|6N7BC70CX0]] with integer coefficients:

1. $H^i(E, E_0) = 0$ for $i < n$.
2. There is a unique class $u \in H^n(E, E_0)$, the *Thom class*, whose restriction to $H^n(F_b, F_{b,0}) \cong \Z$ is the generator given by the orientation of $F_b$, for every $b \in B$.
3. For every $k$, the map $H^k(B) \to H^{k+n}(E, E_0)$, $x \mapsto \pi^*(x) \smile u$, is an isomorphism.

## Remarks

The relative group $H^{k+n}(E, E_0)$ is the cohomology analogue of a [[relative homology|4BVWH0BN28]] group, and $\smile$ is the relative [[cup product|JX47MS1R1H]].
The proof treats a trivial bundle $B \times \R^n$ first, where the result is the Künneth formula with $H^\bullet(\R^n, \R^n \setminus \set{0})$, which is $\Z$ in degree $n$ only.
It then glues local pieces by Mayer–Vietoris arguments and passes to general $B$ by a limit argument.
Without an orientation, the same statement holds with $\Z/2$ coefficients.
For paracompact $B$ one may choose a metric, and then $H^\bullet(E, E_0)$ is the reduced cohomology of the Thom space of the bundle.

Example: for the trivial bundle $B \times \R^n$, $u$ is the pullback of the generator of $H^n(\R^n, \R^n \setminus \set{0})$, and the isomorphism is $H^k(B) \cong H^{k+n}(B \times (\R^n, \R^n \setminus \set{0}))$.

## Explanations

### L1 human

A vector bundle attaches a copy of $\R^n$ to each point of a base space.
Looking at the cohomology of the bundle relative to the part away from the zero section, one finds the cohomology of the base again, shifted up by $n$ degrees.
The shift is given by one special class, the Thom class, which looks like the standard orientation class on each fibre.

### L2 human

The Thom isomorphism is the source of the Euler class: pull the Thom class back along the zero section to get $e \in H^n(B)$.
It leads to the Gysin sequence of a sphere bundle and to Poincaré duality type results for normal bundles.
Thom used it to compute the cohomology of Thom spaces and from that the cobordism ring.
Versions exist for any multiplicative cohomology theory, given a suitable orientation of the bundle.
</document>

<document source="https://tamnd.github.io/math2/node/N6AFVYBY2Z.md" tag="N6AFVYBY2Z" kind="definition">
# Rounding

## Definition

Let $F \subseteq \R$ be unbounded above and below, with only finitely many elements in each bounded interval.
An example is the set $\Z$ of [[integers|0C4JS9RF5M]].
For $x \in \R$ let $x^- = \max \set{f \in F : f \le x}$ and $x^+ = \min \set{f \in F : f \ge x}$.
A *rounding* to $F$ is a map $\mathrm{rd} \colon \R \to F$ with $\mathrm{rd}(x) \in \set{x^-, x^+}$ for every $x$.
The standard roundings are:

1. *rounding down*: $\mathrm{rd}(x) = x^-$;
2. *rounding up*: $\mathrm{rd}(x) = x^+$;
3. *rounding toward zero*: $\mathrm{rd}(x) = x^-$ if $x \ge 0$ and $x^+$ if $x < 0$;
4. *rounding to nearest*: $\mathrm{rd}(x)$ is the element of $\set{x^-, x^+}$ closest to $x$. When $x$ is exactly halfway, a tie rule decides, such as *ties to even* or *ties away from zero*.

## Remarks

Example: for $F = \Z$, rounding down is the [[floor function|08GR54CKEA]], rounding up is the ceiling function, and rounding to nearest with ties to even sends $2.5 \mapsto 2$, $3.5 \mapsto 4$ and $2.7 \mapsto 3$.
Check for $2.7$: $x^- = 2$, $x^+ = 3$, and $\abs{2.7 - 3} = 0.3 < 0.7 = \abs{2.7 - 2}$.
Non-example: the map $x \mapsto \floor{x} - 1$ is not a rounding to $\Z$, since $\floor{2.7} - 1 = 1 \notin \set{2, 3}$.
Each of the four standard roundings is monotone: $x \le y$ implies $\mathrm{rd}(x) \le \mathrm{rd}(y)$.
An arbitrary rounding need not be monotone, since it may choose $x^+$ for one point and $x^-$ for a larger point of the same gap.
For floating-point numbers, $F$ is extended by the subnormal numbers and by $\pm \infty$ to cover overflow.
</document>

<document source="https://tamnd.github.io/math2/node/N6BDTP0AZR.md" tag="N6BDTP0AZR" kind="definition">
# Algebraic statistics

## Definition

Let $\Delta_{m-1} = \set{p \in \R^m : p_i \ge 0, \ \sum_{i=1}^m p_i = 1}$ be the probability simplex of distributions on $m$ outcomes.
An *algebraic statistical model* is a [[statistical model|K3AMMSY8DD]] $\mathcal M \subseteq \Delta_{m-1}$ of one of the following forms:

1. the image $\mathcal M = \varphi(\Theta)$ of an open parameter set $\Theta \subseteq \R^d$ under a polynomial or rational map $\varphi \colon \Theta \to \Delta_{m-1}$, or
2. the set $\mathcal M = \set{p \in \Delta_{m-1} : f(p) = 0 \text{ for all } f \in F}$ for a set $F$ of polynomials in the [[polynomial ring|0SDC934R52]] $\R[p_1, \dots, p_m]$.

*Algebraic statistics* is the study of statistical models and methods through the [[ideals|096Z9MP2ME]] of polynomials vanishing on such models and the [[algebraic varieties|HV6YBS0YDH]] they define.

## Remarks

Example: the independence model for two binary variables is the image of $(a, b) \in (0, 1)^2$ under $\varphi(a, b) = (ab, a(1-b), (1-a)b, (1-a)(1-b))$.
It equals the set of $p = (p_{11}, p_{12}, p_{21}, p_{22}) \in \Delta_3$ with all $p_{ij} > 0$ and $p_{11} p_{22} - p_{12} p_{21} = 0$.
One inclusion: $ab \cdot (1-a)(1-b) - a(1-b) \cdot (1-a)b = 0$.
Conversely, such a $p$ is $\varphi(a, b)$ with $a = p_{11} + p_{12}$ and $b = p_{11} + p_{21}$, since $ab = p_{11}(p_{11} + p_{12} + p_{21}) + p_{12} p_{21} = p_{11}(p_{11} + p_{12} + p_{21} + p_{22}) = p_{11}$, and similarly for the other coordinates.

Non-example: the Poisson family $p_k = e^{-\lambda} \lambda^k / k!$, $k = 0, 1, 2, \dots$, is not an algebraic statistical model in this sense.
It lives on infinitely many outcomes, so it is not a subset of a simplex $\Delta_{m-1}$, and its coordinates are not polynomial or rational in $\lambda$.
</document>

<document source="https://tamnd.github.io/math2/node/N6DQ3Q0HBT.md" tag="N6DQ3Q0HBT" kind="concept">
# Adjunction space

## Summary

An adjunction space is obtained by gluing a space $Y$ onto a space $X$.
The gluing uses a continuous map $f$ from a subspace $A$ of $Y$ into $X$.
Each point $a$ of $A$ is identified with its image $f(a)$, and the result carries the [[quotient topology|0NQ1H2X495]].
Attaching cells to build CW complexes and the mapping cylinder are special cases.

## Explanations

### L1 human

Think of $X$ as a base and $Y$ as a piece you want to stick onto it.
A part $A$ of $Y$ is the glue strip, and the map $f$ says where on $X$ each glue point lands.
After gluing, nothing else changes: the rest of $Y$ stays attached only through $A$.
For example, glue a closed disc onto a circle $X$ by sending the boundary circle of the disc onto $X$ by the identity.
The result is again a closed disc.
If instead the boundary wraps twice around $X$, the result is the real projective plane.
</document>

<document source="https://tamnd.github.io/math2/node/N6DZXEGD9D.md" tag="N6DZXEGD9D" kind="definition">
# Boltzmann equation

## Definition

Let $B \colon \R^3 \times S^2 \to [0, \infty)$ be a given function, the *collision kernel*, where $S^2$ is the unit sphere in $\R^3$.
For $v, v_* \in \R^3$ and $\sigma \in S^2$ put
$$v' = \frac{v + v_*}{2} + \frac{\abs{v - v_*}}{2} \sigma, \qquad v'_* = \frac{v + v_*}{2} - \frac{\abs{v - v_*}}{2} \sigma.$$
For a function $g \colon \R^3 \to \R$ the *collision operator* is
$$Q(g, g)(v) = \int_{\R^3} \int_{S^2} B(v - v_*, \sigma) \big( g(v') g(v'_*) - g(v) g(v_*) \big) \, d\sigma \, dv_*.$$
The *Boltzmann equation* is the [[partial differential equation|GC5BDZ6QBC]]
$$\partial_t f + v \cdot \nabla_x f = Q(f, f)$$
for an unknown function $f \colon [0, \infty) \times \R^3 \times \R^3 \to [0, \infty)$, $(t, x, v) \mapsto f(t, x, v)$.
The operator $Q$ acts on the variable $v$, and $\nabla_x$ is the [[gradient|0ABK9V9Y64]] in $x$.

## Remarks

The pair $(v', v'_*)$ runs over the velocities after an elastic collision of two particles with velocities $v$ and $v_*$: one has $v' + v'_* = v + v_*$ and $\abs{v'}^2 + \abs{v'_*}^2 = \abs{v}^2 + \abs{v_*}^2$.

Example: for $\rho, T > 0$ and $w \in \R^3$, the Maxwellian $M(v) = \rho (2\pi T)^{-3/2} e^{-\abs{v - w}^2/(2T)}$, constant in $t$ and $x$, is a solution.
The left side vanishes, and $M(v') M(v'_*) = M(v) M(v_*)$ by the two conservation laws above, so $Q(M, M) = 0$.

Non-example: $f(t, x, v) = t M(v)$ is not a solution, because $\partial_t f = M \ne 0$ while $v \cdot \nabla_x f = 0$ and $Q(f, f) = t^2 Q(M, M) = 0$.

For hard spheres the kernel is $B(z, \sigma) = c \abs{z}$ for a constant $c > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/N6JC6KNXJJ.md" tag="N6JC6KNXJJ" kind="statement">
# Complete reducibility of representations of compact groups

## Statement

Let $G$ be a [[compact group|EVHCE5VR6Y]], let $V$ be a finite-dimensional complex vector space, and let $\rho \colon G \to GL(V)$ be a continuous [[representation|WYGRXCG2RN]].
Then:

1. there is an [[inner product|0NB0AWGJBW]] on $V$ such that $\ip{\rho(g)u}{\rho(g)v} = \ip{u}{v}$ for all $g \in G$ and $u, v \in V$;
2. $V$ is a direct sum $V = V_1 \oplus \cdots \oplus V_k$ of [[irreducible|FX3NEFYFFN]] subrepresentations.

## Remarks

For part 1, take any inner product $\ip{\cdot}{\cdot}_0$ on $V$ and average it with the normalised [[Haar measure|VY0HXD3GC4]]: $\ip{u}{v} = \int_G \ip{\rho(g)u}{\rho(g)v}_0 \, dg$.
Invariance of the Haar measure makes this inner product $G$-invariant, and it is positive definite because the integrand is positive when $u = v \ne 0$.
For part 2, the orthogonal complement of an [[invariant subspace|0ACX5QT8TE]] for an invariant inner product is again invariant, and induction on $\dim V$ finishes the proof.
For a finite group the Haar integral is the average over the group, and the theorem is Maschke's theorem over $\C$.
Compactness is needed: the representation of $\R$ on $\C^2$ given by $t \mapsto \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}$ has the invariant line $\C e_1$.
It has no invariant complement, since for $t \ne 0$ the only eigenvectors of the matrix are the multiples of $e_1$.
The same proof works for real representations, with a real inner product.

## Explanations

### L1 human

A representation of a compact group always breaks into irreducible pieces that do not interact.
The reason is that one can average over the group, which is possible because the group is compact.
Averaging gives a way of measuring lengths and angles that the group does not change, and then perpendicular pieces stay perpendicular.

### L2 human

The theorem reduces the study of finite-dimensional representations of compact groups to irreducible ones, which are then classified by characters through the [[Schur orthogonality relations|ETNDTECY97]].
For compact connected Lie groups, the irreducible representations are described by highest weights and the [[Weyl character formula|230F5FXPMQ]].
[[Weyl's unitary trick|DDQ2WBQ9GB]] transfers the theorem to complex semisimple Lie groups and Lie algebras, which are not compact.
The [[Peter–Weyl theorem|6VC9QE39XH]] extends the result to unitary representations on infinite-dimensional Hilbert spaces.
</document>

<document source="https://tamnd.github.io/math2/node/N6JX9BCKQP.md" tag="N6JX9BCKQP" kind="concept">
# Basic open set

## Summary

A basic open set of the [[spectrum|FB45BHHA2J]] of a [[commutative ring|0310REJJY8]] $R$ is a set $D(f)$ of [[prime ideals|00MHHZ9SNE]] not containing a fixed element $f$.
These sets form a [[basis|0NA0732YAG]] of the [[Zariski topology|8SQK7RTB63]].
The set $D(f)$ is in natural bijection with the spectrum of the [[localization|4C1AE84N5N]] $R_f$, so basic open sets are again spectra of rings.

## Explanations

### L1 human

Think of $f$ as a function on the space of prime ideals.
The basic open set $D(f)$ is the set of points where $f$ does not vanish.
On $\operatorname{Spec} \Z$, the set $D(6)$ consists of all points except $(2)$ and $(3)$.
Every open set is a union of such sets, just as every open set of the plane is a union of discs.
The [[definition|PS8DVS9WTA]] states this.
</document>

<document source="https://tamnd.github.io/math2/node/N6NX1VT837.md" tag="N6NX1VT837" kind="concept">
# Fractional Sobolev space

## Summary

A fractional Sobolev space $W^{s,p}$ measures smoothness of a non-integer order $s$ in the $L^p$ sense.
For $0 < s < 1$ it is defined by the Gagliardo seminorm, a double integral of difference quotients.
For $p = 2$ on $\R^n$ it agrees with the space $H^s$ defined by the weight $(1 + \abs{\xi}^2)^{s}$ on the [[Fourier transform|DPVM34ADZV]].
These spaces fill the gaps between the integer [[Sobolev spaces|M4CEZ5K9HP]].
They describe the boundary values of Sobolev functions and are the natural energy spaces for nonlocal operators such as the fractional Laplacian.

## Explanations

### L1 human

An ordinary derivative measures how fast a function changes at a point.
A fractional derivative of order $s$ between $0$ and $1$ asks for something weaker: the difference $u(x) - u(y)$ must be small compared with $\abs{x - y}^s$, on average over all pairs of points.
So $W^{s,p}$ is an averaged version of the Hölder condition with exponent $s$.
For example, on an interval, a function with a jump lies in $W^{s,2}$ exactly when $s < 1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/N6TE6WJ4EZ.md" tag="N6TE6WJ4EZ" kind="concept">
# Perfect map

## Summary

A perfect map is a [[continuous|0VGFHTM75Y]] [[closed map|00FSJTXPJM]] onto its target whose fibres are all [[compact|0TVJZYB6SH]].
Perfect maps carry many properties, such as compactness, local compactness and paracompactness, in both directions between the source and the target.
They are a central tool of general topology for comparing spaces.

## Explanations

### L1 human

Under a perfect map, each point of the target comes from a compact piece of the source, and the space is not torn.
Closedness means that the image of a closed set has no missing limit points.
The projection of the cylinder $S^1 \times \R$ onto the line $\R$ is perfect: each fibre is a circle, which is compact.
The projection of the plane onto a line is not, since each fibre is a whole line.
</document>

<document source="https://tamnd.github.io/math2/node/N6TH42TD7F.md" tag="N6TH42TD7F" kind="statement">
# Sunflower lemma

## Statement

A *sunflower* with $p$ petals is a [[family|0C21Q694AG]] of $p$ distinct sets $S_1, \ldots, S_p$ whose pairwise [[intersections|0HBRW4E6G1]] are all equal.
That is, there is a set $Y$, the core, with $S_i \cap S_j = Y$ for all $i \ne j$.
Let $k \ge 1$ and $p \ge 1$ be integers.
Let $\mathcal{F}$ be a family of sets, each with exactly $k$ elements, such that $|\mathcal{F}| > k! \, (p - 1)^k$.
Then $\mathcal{F}$ contains a sunflower with $p$ petals.

## Remarks

The proof is by induction on $k$.
Take a maximal subfamily of pairwise disjoint sets.
If it has at least $p$ members, they form a sunflower with empty core.
Otherwise their union $U$ has at most $(p - 1) k$ elements and meets every set of $\mathcal{F}$, so some $x \in U$ lies in more than $(k-1)! \, (p-1)^{k-1}$ sets.
Remove $x$ from these sets and apply the induction hypothesis to get a sunflower, then add $x$ back to its core.

Example: for $k = 1$ and $p = 3$, a family of $3$ distinct one-element sets is a sunflower with empty core, and $3 > 1! \cdot 2^1$.
The result is due to Erdős and Rado, who conjectured that $k! \, (p-1)^k$ can be replaced by $C_p^k$ for a constant $C_p$ that depends only on $p$; this is open.
In 2019 Alweiss, Lovett, Wu and Zhang improved the bound to $(C p \log k \log \log k)^k$ for an absolute constant $C$, and later work lowered it to $(C p \log k)^k$.

## Explanations

### L1 human

A sunflower is a group of sets that all overlap in exactly the same core, like petals around the centre of a flower.
The lemma says that any large enough collection of sets of the same size must contain a sunflower with as many petals as you like.
So size alone forces this very regular pattern.

### L2 human

The sunflower lemma is a basic tool in extremal set theory and in theoretical computer science, for example in lower bounds for monotone circuits.
Its proof is a typical Ramsey-type argument: either many sets are disjoint, or many share an element, and then one recurses.
It belongs to the circle of results of [[Ramsey theory|G5TWXV565S]], where a large structure always contains an orderly part.
</document>

<document source="https://tamnd.github.io/math2/node/N6YZFG66YP.md" tag="N6YZFG66YP" kind="concept">
# Linearly disjoint extensions

## Summary

Let $E$ and $F$ be extensions of a field $K$ inside a common field.
They are linearly disjoint over $K$ when elements of $E$ that are [[linearly independent|0CRMGPAH83]] over $K$ stay independent over $F$.
Equivalently, the natural map from the [[tensor product|ANH16GQJZQ]] $E \otimes_K F$ to the common field is injective.
For finite extensions this means that the degree of the [[compositum|T1S29NN8PE]] is as large as possible: $[EF:F] = [E:K]$.
The notion is the field version of independence, and it is used to compute degrees and Galois groups of composita.

## Explanations

### L1 human

Two extensions are linearly disjoint when they do not overlap in a hidden way.
A basis of one stays a basis after you extend the scalars by the other.
For example, $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are linearly disjoint over $\Q$: together they generate a field of degree $2 \cdot 2 = 4$.
Having only $\Q$ in common is not enough in general.
The fields $\Q(\sqrt[3]{2})$ and $\Q(\omega\sqrt[3]{2})$, with $\omega = e^{2\pi i/3}$, meet only in $\Q$, but together they generate a field of degree $6$, not $9$.
</document>

<document source="https://tamnd.github.io/math2/node/N6Z14G81VF.md" tag="N6Z14G81VF" kind="concept">
# Linear and stable summation methods

## Summary

A [[summation method|8X25Q4XYMN]] is linear when it respects sums and scalar multiples of series.
It is stable when deleting or adding a first term changes the sum by that term.
These two properties, together with [[regularity|29C7C41A52]], are the ones usually expected of a reasonable method.
They are strong enough to force values: any linear and stable method that sums $1 - 1 + 1 - \cdots$ must give it the value $\frac12$.

## Explanations

### L1 human

Ordinary sums obey two familiar rules: the sum of two series is the sum of their sums, and $a_0 + a_1 + a_2 + \cdots$ equals $a_0$ plus the sum of the rest.
Linear and stable methods keep these rules for the new, generalized sums.
For $x = 1 - 1 + 1 - \cdots$ these rules give $x = 1 - x$, so $x = \frac12$.
For $1 + 1 + 1 + \cdots$ they would give $x = 1 + x$, which is impossible, so no such method sums this series to a finite number.
</document>

<document source="https://tamnd.github.io/math2/node/N730JMCM7N.md" tag="N730JMCM7N" kind="definition">
# Minimal prime ideal

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
A *minimal prime ideal* of $R$ is a [[prime ideal|06HC6HK0Z8]] $P$ of $R$ such that no prime ideal $Q$ of $R$ satisfies $Q \subsetneq P$.
Let $I$ be an [[ideal|0MYSA6MYNA]] of $R$.
A *minimal prime over $I$* is a prime ideal $P \supseteq I$ such that no prime ideal $Q$ satisfies $I \subseteq Q \subsetneq P$.

## Remarks

Example: in $R = k[x, y]/(xy)$ with $k$ a field, the ideal $(x)$ is a minimal prime.
It is prime because $R/(x) \cong k[y]$ is a domain.
A prime $Q \subseteq (x)$ contains $xy = 0$, so it contains $x$ or $y$, and $y \notin (x)$ forces $x \in Q$.
Non-example: $2\Z$ is prime in $\Z$, but it is not minimal, since the prime ideal $(0)$ is strictly smaller.
By Zorn's lemma, every prime ideal containing $I$ contains a minimal prime over $I$; the key point is that the intersection of a chain of prime ideals is prime.
The minimal primes over $I$ are the minimal primes of $R/I$, pulled back to $R$.
</document>

<document source="https://tamnd.github.io/math2/node/N76R7JWMNJ.md" tag="N76R7JWMNJ" kind="statement">
# Gradient estimates for harmonic functions

## Statement

Let $n \ge 1$ and let $\omega_n$ be the volume of the unit ball of $\R^n$.
Let $u$ be [[harmonic|0DEB0JMN0W]] on an open set containing the closed ball $\overline{B}(x_0, r)$.

1. The [[gradient|0ABK9V9Y64]] of $u$ at the centre satisfies
$$\abs{\nabla u(x_0)} \le \frac{n}{r} \, \max_{\overline{B}(x_0, r)} \abs{u}.$$
2. For every multi-index $\alpha$ of order $k = \abs{\alpha}$,
$$\abs{\partial^\alpha u(x_0)} \le \frac{C_k}{r^{n+k}} \int_{B(x_0, r)} \abs{u(y)} \, dy, \qquad C_0 = \frac{1}{\omega_n}, \quad C_k = \frac{(2^{n+1} n k)^k}{\omega_n} \ (k \ge 1).$$

## Remarks

Proof idea for part 1: each partial derivative $\partial_i u$ is harmonic, so by the [[mean value property|0HYPR64JM2]] over the ball and the [[divergence theorem|0JX86PSSA9]],
$$\partial_i u(x_0) = \frac{1}{\omega_n r^n} \int_{B(x_0, r)} \partial_i u \, dy = \frac{1}{\omega_n r^n} \int_{\partial B(x_0, r)} u \, \nu_i \, dS.$$
The last integral is at most $\max \abs{u}$ times the area $n \omega_n r^{n-1}$ of the sphere, which gives $\abs{\partial_i u(x_0)} \le \frac{n}{r} \max \abs{u}$.
The same argument for the derivative in the direction of $\nabla u(x_0)$ gives the bound for $\abs{\nabla u(x_0)}$.
Part 2 follows by induction on $k$, applying the same argument on balls of radius $r/k$.
Example: $u(x, y) = x$ on the unit disc has $\abs{\nabla u} = 1$ and $\max \abs{u} = 1$, within the bound $2$ of part 1.
The estimates are interior: they control derivatives at the centre by the size of $u$ on the ball, and they fail near the boundary of the domain.
For instance $u_m(x, y) = \operatorname{Re}(x + iy)^m$ on the unit disc has $\abs{u_m} \le 1$ but $\abs{\nabla u_m(1, 0)} = m$.

## Explanations

### L1 human

A harmonic function cannot change quickly unless it is large.
If it stays between $-M$ and $M$ on a ball of radius $r$, its slope at the centre is at most $n M / r$.
Big balls therefore force small slopes, and this is why bounded harmonic functions on the whole space are constant.

### L2 human

These estimates give Liouville's theorem, the real analyticity of harmonic functions, and the compactness of bounded families of harmonic functions through the Arzelà–Ascoli theorem.
They are the model for interior Schauder and Sobolev estimates for general elliptic equations.
For nonnegative harmonic functions there is a sharper form, $\abs{\nabla u(x_0)} \le \frac{n}{r} u(x_0)$, which leads to Harnack's inequality.
</document>

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# Homomorphism theorem

## Statement

Let $\mathbf{A}$ and $\mathbf{B}$ be [[algebras|QJGPT2W0FT]] of the same signature and let $\alpha \colon \mathbf{A} \to \mathbf{B}$ be a [[homomorphism|KBXSZC6BV1]].
Let $\ker \alpha = \set{(a, b) \in A \times A : \alpha(a) = \alpha(b)}$.
Let $\theta$ be a [[congruence|DCSXXT4SER]] on $\mathbf{A}$ with $\theta \subseteq \ker \alpha$, and let $\nu \colon \mathbf{A} \to \mathbf{A}/\theta$ be the natural map onto the [[quotient algebra|N5KE7FH6CX]].

1. The relation $\ker \alpha$ is a congruence on $\mathbf{A}$.
2. There is a unique homomorphism $\beta \colon \mathbf{A}/\theta \to \mathbf{B}$ with $\beta \circ \nu = \alpha$.
3. The map $\beta$ is injective if and only if $\theta = \ker \alpha$.
4. In particular, $a/\ker \alpha \mapsto \alpha(a)$ is an isomorphism from $\mathbf{A}/\ker \alpha$ onto the subalgebra $\alpha(\mathbf{A})$ of $\mathbf{B}$.

## Remarks

Proof idea.
Compatibility of $\ker \alpha$ follows from $\alpha(f(a_1, \dots, a_n)) = f(\alpha(a_1), \dots, \alpha(a_n))$.
Put $\beta(a/\theta) = \alpha(a)$; this is well defined because $\theta \subseteq \ker \alpha$, and it is a homomorphism because $\alpha$ and $\nu$ are.
Uniqueness holds since $\nu$ is surjective, and $\beta(a/\theta) = \beta(b/\theta)$ means $(a, b) \in \ker \alpha$.

Example: the ring homomorphism $\Z \to \Z/n\Z$ has kernel congruence modulo $n$, so the quotient of $\Z$ by this congruence is isomorphic to $\Z/n\Z$.
For groups, part 4 is the [[first isomorphism theorem|0CXXJNG5B6]].

## Explanations

### L1 human

A homomorphism may send several elements to the same place.
If you glue together exactly the elements with the same image, what is left is a copy of the image.
So every homomorphic image of an algebra is, up to isomorphism, a quotient of it.

### L2 human

Part 2 is the universal property of the quotient: maps out of $\mathbf{A}/\theta$ are the maps out of $\mathbf{A}$ that do not separate $\theta$-related elements.
Part 1, together with $\ker \nu = \theta$, shows that congruences are exactly the kernels of homomorphisms.
With the [[correspondence theorem|JG7WB75BHC]] it yields the other isomorphism theorems.
It is basic in the proof of [[Birkhoff's HSP theorem|5083DHSK82]], since it identifies the class operator $H$ with taking quotients.
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# Brouwer fixed-point theorem

## Statement

Let $n \ge 1$ and let $D^n = \set{x \in \R^n : \norm{x} \le 1}$ be the [[closed unit ball|0Q0H13G8ZB]].
Every [[continuous map|0VGFHTM75Y]] $f \colon D^n \to D^n$ has a fixed point: there is $x \in D^n$ with $f(x) = x$.

## Remarks

The idea of the proof: suppose $f(x) \neq x$ for all $x$.
Send $x$ to the point where the ray from $f(x)$ through $x$ leaves the ball; this is a continuous [[retraction|68PFSGP39B]] of $D^n$ onto the sphere $S^{n-1}$ that fixes $S^{n-1}$.
No such retraction exists by the [[no-retraction theorem|Z6JR1GN2RE]], because $H_{n-1}(S^{n-1}) \neq 0$ for $n \ge 2$ while $D^n$ is contractible; for $n = 1$ the [[intermediate value theorem|00KAFBR32E]] applied to $f(x) - x$ suffices.
The theorem is also the case of a closed ball in the [[Lefschetz fixed-point theorem|G44NQXCWT9]], since $L(f) = 1$ for every self-map of a contractible compact polyhedron.
In the language of the [[fixed-point property|JQDHHFXBF3]], it says that $D^n$ has this property.
So does every space homeomorphic to $D^n$, such as every compact [[convex set|02AS45T3H9]] of $\R^n$ with nonempty interior.
Each hypothesis matters: the translation $x \mapsto x + 1$ of the non-compact space $\R$ has no fixed point.
The rotation $z \mapsto -z$ of the circle $S^1$, which is compact but has a hole, has none either.

## Explanations

### L1 human

If you continuously deform a disk and put it back on top of itself, at least one point ends up exactly where it started.
Stirring a cup of coffee gently, without breaking the surface, always leaves some point of the surface in its original place.
The result says nothing about which point it is; it only guarantees that one exists.

### L2 human

The theorem is the basic existence tool behind many other fixed-point results.
Schauder's theorem extends it to compact convex sets in normed spaces, and Kakutani's theorem to set-valued maps, which is how Nash proved the existence of equilibria in games.
Its proofs connect several subjects: degree theory, homology, the no-retraction theorem, and combinatorial arguments through Sperner's lemma.
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# Shadowing lemma

## Statement

Let $M$ be a [[Riemannian manifold|7CBT4SM83Y]] with distance $d$, let $f \colon M \to M$ be a $C^1$ [[diffeomorphism|0KZZ70ZHXP]], and let $\Lambda \subseteq M$ be a compact [[hyperbolic set|MPFD24TYW2]] for $f$.
For $\varepsilon > 0$, an *$\varepsilon$-pseudo-orbit* is a sequence $(x_n)_{n \in \Z}$ in $M$ with $d(f(x_n), x_{n+1}) < \varepsilon$ for all $n \in \Z$.
A point $x \in M$ *$\delta$-shadows* it if $d(f^n(x), x_n) < \delta$ for all $n \in \Z$.
Then there is an open neighbourhood $U$ of $\Lambda$ with the property that follows.
For every $\delta > 0$ there is $\varepsilon > 0$ such that every $\varepsilon$-pseudo-orbit contained in $U$ is $\delta$-shadowed by some $x \in M$.
Moreover, there is $\delta_0 > 0$ such that for $\delta < \delta_0$ the shadowing point $x$ is unique.

## Remarks

The idea of the proof: a pseudo-orbit is a near-solution of the equation $x_{n+1} = f(x_n)$ in a space of sequences.
Hyperbolicity makes the linearised equation invertible with a bounded inverse, contracting forward along stable directions and backward along unstable ones, and a fixed-point argument gives a true solution nearby.
Uniqueness comes from [[expansiveness|TAFJV2YWKR]] near $\Lambda$: two orbits that stay $\delta$-close for all times coincide.
For an Anosov diffeomorphism of a compact manifold, $\Lambda = M$, so every pseudo-orbit with small enough errors is shadowed.
Hyperbolicity is needed: consider the rotation of the circle by an irrational angle $\alpha$.
The pseudo-orbit that rotates by $\alpha + \varepsilon/2$ at each step drifts away from every true orbit at rate $\varepsilon/2$ per step.

## Explanations

### L1 human

When a computer follows an orbit, it makes a small rounding error at each step, so it computes a pseudo-orbit rather than a true one.
For chaotic hyperbolic systems the errors grow fast, so the computed orbit soon differs from the true orbit with the same start.
The shadowing lemma says that it still stays close to some other true orbit for all times.
So the pictures a computer draws of such systems are faithful in this sense: each computed orbit is close to some real orbit.

### L2 human

Shadowing is the main tool for building orbits with prescribed behaviour in hyperbolic dynamics.
One glues pieces of orbits into a pseudo-orbit and shadows it.
This gives the Anosov closing lemma, which finds a periodic orbit near any orbit segment that nearly closes up.
It also gives the Markov partitions used to code the dynamics by subshifts of finite type.
It also gives structural stability of Anosov diffeomorphisms, since a perturbed orbit is a pseudo-orbit of the original map.
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# Cap product

## Summary

The cap product combines a homology class of degree $n$ and a cohomology class of degree $k \le n$ into a homology class of degree $n - k$.
It makes [[singular homology|M0Q5CFJ7ZH]] a module over the [[cohomology ring|RDT4785M4D]].
Its main use is Poincaré duality, where capping with the fundamental class of a closed oriented manifold turns cohomology into homology.

## Explanations

### L1 human

A cohomology class is a test on simplices; a homology class is built from simplices.
The cap product uses the test on the front part of each simplex and keeps the back part as a smaller simplex.
So a degree $k$ test eats $k$ dimensions of an $n$-dimensional chain and leaves an $(n - k)$-dimensional chain.
When $k = n$, nothing but a point is left, and the cap product just records the value of the test on the chain.
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# Double complex

## Summary

A double complex is a grid of objects $C_{p,q}$ with a horizontal and a vertical differential, each squaring to zero, that anticommute with each other.
Summing along the diagonals $p + q = n$ gives the total complex, a single [[chain complex|WQ82C0M3QB]].
Double complexes arise from tensor products of complexes and from resolving complexes, and their total homology is computed by [[spectral sequences|GG8AYV847V]].

## Explanations

### L1 human

A chain complex is a line of objects with maps going one way.
A double complex is a whole grid of objects, with maps going left and maps going down, each family forming chain complexes.
Going left then down equals minus going down then left, and this sign lets one combine both into one map on the diagonals.
For example, the tensor product of two chain complexes is naturally a double complex.
Its total complex computes things like the homology of a product of spaces.
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<document source="https://tamnd.github.io/math2/node/N815PEPPFA.md" tag="N815PEPPFA" kind="definition">
# Combinatorics on words

## Definition

Let $A$ be a finite set, the *alphabet*, whose elements are *letters*.
A finite *word* over $A$ is a finite sequence $w = a_1 a_2 \cdots a_n$ of letters, also called a string; an infinite word is a sequence $(a_i)_{i \ge 1}$.
The finite words form the free [[monoid|0R47MAD57S]] $A^*$ under concatenation, with the empty word as identity.
A *factor* of $w$ is a word $u$ with $w = x u y$ for some words $x, y$.
*Combinatorics on words* is the branch of [[combinatorics|0JKW7W7XFR]] that studies the structure of words in $A^*$ and of infinite words over $A$.
Its typical questions concern factors, repetitions, periods and morphisms of $A^*$.

## Remarks

Example: a *square* is a nonempty word of the form $u u$.
Every binary word of length at least $4$ contains a square factor.
Indeed, a square-free binary word cannot contain $aa$ or $bb$, so it alternates, and $abab$ or $baba$ is a square.

Example: the [[Thue–Morse sequence|7JZVT385PC]] is an infinite binary word with no factor of the form $a x a x a$ with $a$ a letter, so it is overlap-free.

Non-example: the word $ac$ is not a factor of $abc$.
It is only a [[subsequence|0F1PR74BRX]] of $abc$, since a factor must consist of consecutive letters.

Results of the field include the Fine–Wilf theorem on words with two periods and the existence of infinite square-free words over three letters.
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# Lawvere's fixed-point theorem

## Statement

Let $\mathcal{C}$ be a [[Cartesian closed category|BFBNQ36QS8]] with [[terminal object|EK85VSSRMR]] $1$.
A *point* of an object $X$ is a morphism $1 \to X$.
Let $A$ and $B$ be objects, and let $B^A$ be the [[exponential object|RHWD8DY41R]].
Let $\phi \colon A \to B^A$ be point-surjective: for every point $q \colon 1 \to B^A$ there is a point $p \colon 1 \to A$ with $\phi \circ p = q$.
Then every morphism $f \colon B \to B$ has a fixed point: there is a point $s \colon 1 \to B$ with $f \circ s = s$.

## Remarks

Proof idea: let $\mathrm{ev} \colon B^A \times A \to B$ be evaluation and $\Delta \colon A \to A \times A$ the diagonal, built from the [[product|S5744QZV4N]].
Put $g = f \circ \mathrm{ev} \circ (\phi \times \id_A) \circ \Delta \colon A \to B$, so that informally $g(a) = f(\phi(a)(a))$.
The transpose of $g$ is a point of $B^A$, so it equals $\phi \circ p$ for some point $p$ of $A$.
Then $s = g \circ p$ satisfies $s = f(\phi(p)(p)) = f(g(p)) = f \circ s$.

The contrapositive is the form used most.
In $\mathbf{Set}$, take $B = \set{0, 1}$ and $f$ the swap of $0$ and $1$, which has no fixed point.
So no map $A \to \set{0,1}^A$ is surjective, which is [[Cantor's theorem|0WTEMGYV2C]] that no set maps onto its power set.
The proof is an abstract form of [[Cantor's diagonal argument|0XCH8QVDCQ]].

## Explanations

### L1 human

Suppose the elements of a set $A$ can serve as codes for all functions from $A$ to $B$.
Then every map from $B$ to itself must leave some value unchanged.
Flipping a bit leaves no value unchanged, so no set can code all bit-valued functions on itself.
This one short argument is behind Cantor's theorem and Russell's paradox.

### L2 human

The theorem needs only finite products and exponentials, so it applies in any Cartesian closed category, including toposes and models of the typed lambda calculus.
Read positively, it explains fixed-point combinators: if $A$ maps point-surjectively onto $A^A$, then every endomorphism of $A$ has a fixed point.
Read negatively, it unifies diagonal arguments: Cantor's theorem, Russell's paradox and Tarski's undefinability of truth all come from an $f$ without fixed point that forbids a point-surjection.
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# Depth

## Definition

Let $R$ be a commutative [[Noetherian ring|0670T59HGE]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module with $IM \ne M$.
The *$I$-depth* of $M$, written $\operatorname{depth}_I M$ or $\operatorname{grade}(I, M)$, is the length of a maximal [[$M$-regular sequence|69R5R2FCS5]] whose elements lie in $I$.
If $(R, \mathfrak{m})$ is a [[local ring|MZ02BKPHJA]], the *depth* of $M$ is $\operatorname{depth} M = \operatorname{depth}_{\mathfrak{m}} M$.

## Remarks

The definition makes sense because all maximal $M$-regular sequences in $I$ have the same length, a theorem of Rees.
That length is also $\min \set{i : \operatorname{Ext}^i_R(R/I, M) \ne 0}$.
If $IM = M$, one sets $\operatorname{depth}_I M = \infty$.
Example: for $R = K[x_1, \ldots, x_n]$ localized at $(x_1, \ldots, x_n)$, one has $\operatorname{depth} R = n$, with the maximal regular sequence $x_1, \ldots, x_n$.
Example: for $R = K[x,y]/(x^2, xy)$ localized at $\mathfrak{m} = (x, y)$, one has $\operatorname{depth} R = 0$, since $x \ne 0$ and $x \mathfrak{m} = 0$, so no element of $\mathfrak{m}$ is a non-zero-divisor.
For a nonzero finitely generated module over a Noetherian local ring, $\operatorname{depth} M \le \dim M$.
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# Geometric algebra

## Summary

Geometric algebra is the [[Clifford algebra|A38B1QFK20]] of a real vector space with a nondegenerate quadratic form, used as a language for geometry.
Its geometric product of two vectors combines the inner product and the outer product: $uv = u \cdot v + u \wedge v$.
Its elements, the multivectors, represent scalars, vectors, oriented plane areas, oriented volumes and so on, all in one algebra.
Rotations act by conjugation, $x \mapsto R x R^{-1}$, and the reflection in the hyperplane orthogonal to a vector $n$ acts by $x \mapsto -n x n^{-1}$, in every dimension.
This makes it popular in physics, computer graphics and robotics.

## Explanations

### L1 human

Geometric algebra lets you multiply vectors.
The product of two vectors has a number part, which measures how much they point the same way.
It also has a plane part, which measures the oriented area of the parallelogram they span.
In the plane, the product of the two unit axis vectors squares to $-1$, so the complex numbers appear inside the algebra by themselves.
Rotations then become products, just as multiplying by $e^{i\theta}$ rotates the complex plane.
</document>

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# Disintegration theorem

## Statement

Let $Y$ and $X$ be [[Polish spaces|584GXXY73M]] with their Borel σ-algebras.
Let $\mu$ be a Borel [[probability measure|059TJN0NZ1]] on $Y$, let $\pi \colon Y \to X$ be a Borel [[measurable map|YYNCJSR0PJ]], and let $\nu = \pi_* \mu$ be the [[pushforward measure|3N66TM3TSQ]].
Then there is a family $(\mu_x)_{x \in X}$ of Borel probability measures on $Y$ with the following properties.

1. For every Borel set $B \subseteq Y$, the map $x \mapsto \mu_x(B)$ is Borel measurable.
2. For $\nu$-almost every $x \in X$, $\mu_x$ is concentrated on the fibre: $\mu_x(Y \setminus \pi^{-1}(x)) = 0$.
3. For every Borel function $f \colon Y \to [0, \infty]$,
$$\int_Y f \, d\mu = \int_X \int_{\pi^{-1}(x)} f(y) \, d\mu_x(y) \, d\nu(x).$$

The family is unique up to $\nu$-null sets: two such families agree for $\nu$-almost every $x$.

## Remarks

One way to prove it is through [[regular conditional probabilities|PHHT5W3BEJ]].
For a Borel set $B$, the measure $A \mapsto \mu(B \cap \pi^{-1}(A))$ is absolutely continuous with respect to $\nu$, and the [[Radon–Nikodym theorem|1ZSZS5G1Q4]] gives a density $x \mapsto \mu_x(B)$.
The Polish assumption lets one choose these densities for a countable generating algebra of Borel sets at once and extend them to measures.

Example: let $\mu$ be Lebesgue measure on $Y = [0, 1]^2$ and $\pi(x, y) = x$.
Then $\nu$ is Lebesgue measure on $[0, 1]$, and $\mu_x$ is Lebesgue measure on the vertical segment $\set{x} \times [0, 1]$.
Property 3 is then Fubini's theorem.

The theorem is often stated for Radon spaces, which include Polish spaces.
Without such a regularity assumption a disintegration may fail to exist.

## Explanations

### L1 human

A measure on a big space can be cut into slices along the fibres of a map.
Each slice carries its own probability measure, and the total measure is recovered by averaging the slices according to how much weight each fibre gets.
It is the measure-theoretic form of computing a probability by first conditioning on the value of $\pi$.

### L2 human

Disintegration makes conditioning on events of probability zero rigorous: $\mu_x$ is the conditional law of the point $y$ given $\pi(y) = x$.
It is used in optimal transport to decompose transport plans, in ergodic theory for the ergodic decomposition, and in geometric measure theory together with the coarea formula.
The measures $\mu_x$ are only defined for almost every $x$.
So a conditional law given a single null event depends on the choice of the map $\pi$, as the [[Borel–Kolmogorov paradox|9RMS0ZJ618]] shows.
</document>

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# Adjoint of an unbounded operator

## Definition

Let $H$ and $K$ be [[Hilbert spaces|HM2HHE3J51]] over $\C$, with inner products linear in the first argument.
Let $T$ be a [[densely defined operator|BCJHNHKQFG]] from $H$ to $K$ with domain $D(T)$.
The *adjoint* of $T$ is the [[operator|KRJX5RA0PW]] $T^*$ from $K$ to $H$ given as follows.

1. Its domain $D(T^*)$ is the set of $y \in K$ for which there is $z \in H$ with $\ip{Tx}{y} = \ip{x}{z}$ for all $x \in D(T)$.
2. For $y \in D(T^*)$, $T^* y$ is this vector $z$.

## Remarks

The vector $z$ is unique because $D(T)$ is dense: two such vectors differ by a vector orthogonal to $D(T)$, hence to $H$.
Equivalently, $y \in D(T^*)$ if and only if the functional $x \mapsto \ip{Tx}{y}$ is bounded on $D(T)$.
If $T$ is bounded with $D(T) = H$, then $T^*$ is the [[Hilbert space adjoint|6HSJBJ8TTQ]].

The adjoint $T^*$ is always [[closed|VM7B38PA87]].
It is densely defined if and only if $T$ is [[closable|3TWMBFQJJ8]], and then $T^{**}$ is the [[closure|TRC892PJNN]] of $T$.
If $S \subseteq T$, then $T^* \subseteq S^*$.

Example: on $\ell^2$, let $D(T) = \set{x : \sum n^2 \abs{x_n}^2 < \infty}$ and $T(x_n) = (n x_n)$.
For $y \in D(T^*)$ with $T^* y = z$, testing with the unit vector $x = e_n$ gives $n \overline{y_n} = \overline{z_n}$, so $z = (n y_n) \in \ell^2$ and $y \in D(T)$.
Conversely each $y \in D(T)$ lies in $D(T^*)$ with $T^* y = Ty$, so $T^* = T$.

Non-example: if $D(T)$ is not dense, there is a nonzero $w \perp D(T)$, and then $z$ and $z + w$ both satisfy clause 1, so clause 2 does not define a map.
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# Fréchet space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$.
A *Fréchet space* over $\mathbb{K}$ is a topological vector space $E$ over $\mathbb{K}$ that is both [[locally convex|B6GY2KJMWW]] and an [[F-space|86BYEV9AX8]].

## Remarks

Equivalently, the topology of $E$ is generated by a countable family of [[seminorms|FP5GVCN8GV]] $p_1 \le p_2 \le \cdots$ that separates points, and $E$ is complete.
From such seminorms one gets a translation-invariant metric
$$d(x, y) = \sum_{n = 1}^{\infty} 2^{-n} \frac{p_n(x - y)}{1 + p_n(x - y)}$$
that induces the topology.
Example: the space $C(\R)$ of continuous real functions with the seminorms $p_n(f) = \sup_{\abs{t} \le n} \abs{f(t)}$ is a Fréchet space.
A sequence that is Cauchy for all $p_n$ converges uniformly on each $[-n, n]$, and the limit is continuous.
Every Banach space is a Fréchet space.
Non-example: $L^p([0, 1])$ with $0 < p < 1$ is an F-space but is not locally convex, so it is not a Fréchet space.
Some older texts use the name Fréchet space for what is here called an F-space.
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# Hopf module

## Summary

A Hopf module over a [[Hopf algebra|7XFYP0N84H]] $H$ is a vector space that is both an $H$-[[module|0NZ8JA25WK]] and an $H$-[[comodule|7WEM2E6RTV]].
The coaction must be compatible with the action.
The fundamental theorem of Hopf modules, due to Larson and Sweedler, says that every Hopf module is free.
It has the form $V \otimes H$, where $V$ is its space of coinvariants.
This theorem is the main tool behind the uniqueness of [[integrals|RJA5026RGX]] and the Frobenius property of finite-dimensional Hopf algebras.

## Explanations

### L1 human

The Hopf algebra $H$ itself can be multiplied by elements of $H$ and also split by its comultiplication, and the two operations fit together.
A Hopf module is any vector space with these two structures fitting together in the same way.
The simplest examples are copies of $H$ with a multiplicity space attached, $V \otimes H$.
The surprising theorem is that there are no others: every Hopf module looks like $V \otimes H$.
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# Stanley–Reisner ring

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $n \ge 1$.
Let $\Delta$ be a simplicial complex on the vertex set $[n] = \set{1, \ldots, n}$.
So $\Delta$ is a family of subsets of $[n]$ such that every subset of a member is a member.
For $F \subseteq [n]$, let $x_F = \prod_{i \in F} x_i$ in the [[polynomial ring|0SDC934R52]] $S = k[x_1, \ldots, x_n]$.
The *Stanley–Reisner ideal* of $\Delta$ is
$$I_\Delta = (x_F : F \subseteq [n], \; F \notin \Delta).$$
The *Stanley–Reisner ring*, or face ring, of $\Delta$ is the [[quotient ring|0076HWHGHQ]] $k[\Delta] = S/I_\Delta$.

## Remarks

The ideal $I_\Delta$ is already generated by the $x_F$ for the minimal non-faces $F$.
It is a squarefree [[monomial ideal|4G25S7PJ22]], and $\Delta \mapsto I_\Delta$ is a bijection from simplicial complexes on $[n]$ to squarefree monomial ideals of $S$.
The [[Krull dimension|MVGBS3J1P7]] of $k[\Delta]$ is $\dim \Delta + 1$, the largest size of a face.

Example: let $\Delta$ be the boundary of a triangle, with faces all subsets of $\set{1, 2, 3}$ except $\set{1, 2, 3}$.
The only non-face is $\set{1, 2, 3}$, so $I_\Delta = (x_1 x_2 x_3)$ and $k[\Delta] = k[x_1, x_2, x_3]/(x_1 x_2 x_3)$, of Krull dimension $2$.

Non-example: the ideal $(x_1^2)$ of $k[x_1, x_2]$ is a monomial ideal, but it is not of the form $I_\Delta$, because it is not generated by squarefree monomials.
Its quotient ring has the nonzero nilpotent element $x_1$, while every $k[\Delta]$ is reduced.
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# Cesàro summation

## Summary

Cesàro summation is the [[summability method|8X25Q4XYMN]] that assigns to a series the limit of the averages of its partial sums, when this limit exists.
These averages are the [[Cesàro means|08R2V9GZF7]] of the sequence of partial sums.
Every convergent series is Cesàro summable to its sum, and some divergent series, such as $1 - 1 + 1 - \cdots$, become summable.
Weighted versions of the averaging give the Cesàro methods $(C, \alpha)$ of higher order.
The method of order $1$ is used for Fourier series in [[Fejér's theorem|TPJZCSXSVN]].

## Explanations

### L1 human

Take the running totals of a series and then average them.
For $1 - 1 + 1 - 1 + \cdots$, the running totals are $1, 0, 1, 0, \dots$, which never settle down.
Their averages are $1, 1/2, 2/3, 1/2, 3/5, \dots$, and these tend to $1/2$.
So Cesàro summation gives the value $1/2$ to this series.
Averaging smooths out the oscillation while leaving the limit of a convergent series unchanged.
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# Sober space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A subset $C \subseteq X$ is irreducible when $C \ne \varnothing$ and the following holds.
If $C = A \cup B$ with $A$ and $B$ [[closed|0ZCK8DQYN1]] in the subspace $C$, then $A = C$ or $B = C$.
A point $x$ is a generic point of a closed set $C$ when the [[closure|0QEAJB91GZ]] $\overline{\set{x}}$ equals $C$.
$X$ is *sober* when every irreducible closed subset of $X$ has exactly one generic point.

## Remarks

For each point $x$, the closure $\overline{\set{x}}$ is irreducible and closed, with generic point $x$.
So the uniqueness part says that $\overline{\set{x}} = \overline{\set{y}}$ implies $x = y$, and a sober space is [[T0|07SG12TQH3]].
Every [[Hausdorff space|0S1FJ8VEX9]] is sober, because its irreducible subsets are single points.

Example: for a commutative ring $R$, the spectrum $\operatorname{Spec} R$ with the Zariski topology is sober.
An irreducible closed set has the form $V(\mathfrak p) = \set{\mathfrak q : \mathfrak q \supseteq \mathfrak p}$ for a unique [[prime ideal|00MHHZ9SNE]] $\mathfrak p$, and $\mathfrak p$ is its generic point.

Non-example: an infinite set $X$ with the [[cofinite topology|0B906ZBTMZ]] is not sober.
$X$ is irreducible, since two proper closed sets are finite and their union is not $X$.
But each point is closed, so $\overline{\set{x}} = \set{x} \ne X$, and $X$ has no generic point.
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# Fefferman's theorem on extension of biholomorphisms

## Statement

Let $\Omega_1, \Omega_2 \subset \C^n$ be bounded [[strongly pseudoconvex domains|Z7RVXBA2KB]] with $C^\infty$ boundaries.
Let $F \colon \Omega_1 \to \Omega_2$ be a [[biholomorphism|FX92PP3HZR]].
Then $F$ extends to a $C^\infty$ [[diffeomorphism|0KZZ70ZHXP]] from $\overline{\Omega_1}$ onto $\overline{\Omega_2}$.

## Remarks

Fefferman's proof studies the geodesics of the [[Bergman metric|BB7S4NTZ6V]], which a biholomorphism maps to geodesics, and uses [[his expansion of the Bergman kernel|GPQ9JNSDPA]] to control them near the boundary.
Later, Bell and Ligocka gave a shorter proof based on the regularity of the [[Bergman projection|6J1X63Z7QZ]] up to the boundary.

Consequence: the restriction of $F$ to $\partial\Omega_1$ is a smooth diffeomorphism onto $\partial\Omega_2$ that preserves the CR structures of the boundaries.
So every invariant of the CR structure of the boundary, such as those of the [[Chern–Moser normal form|FHG1N4EKP4]], is a biholomorphic invariant of the domain.
A generic small perturbation of the sphere has a CR structure that is not locally equivalent to that of the sphere.
So a generic smooth strongly pseudoconvex domain near the ball is not biholomorphic to the ball.

In one variable the analogous fact is classical: a conformal map between two bounded domains with $C^\infty$ boundary extends smoothly to the closures.

## Explanations

### L1 human

Two domains can be the same for complex analysis, through a biholomorphic map, while their boundaries look different.
Fefferman's theorem says that for smooth strongly pseudoconvex domains this cannot happen: the map between the domains also matches their boundaries smoothly.
So the shape of the boundary becomes a tool to tell such domains apart.

### L2 human

In several variables, the Riemann mapping theorem fails: by [[Poincaré's theorem|1NF8T07H9X]] the ball and the polydisc are not biholomorphic, and most domains are not biholomorphic to the ball.
Fefferman's theorem turns the classification of strongly pseudoconvex domains up to biholomorphism into the classification of their boundaries up to CR diffeomorphism.
The proof via the Bergman projection shows the role of boundary regularity of canonical operators, a theme that extends to many weakly pseudoconvex domains.
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# Finite morphism

## Summary

A finite morphism $X \to Y$ of schemes is affine over $Y$.
Over each affine piece $\operatorname{Spec} B$ of $Y$, it comes from a $B$-algebra that is a finitely generated $B$-module.
It is the scheme version of a [[finite morphism of rings|1NBACAEC2V]].
Finite morphisms have finite fibers and are closed maps; branched covers of curves are typical examples.

## Explanations

### L1 human

A finite morphism is like a map where each point has finitely many points above it, and nothing escapes to infinity.
The map $t \mapsto t^2$ from the complex line to the complex line is finite: above each nonzero point there are two points, and above $0$ there is one.
The inclusion of the punctured line into the line is not finite, because points near the removed origin escape.
Closed immersions are finite as well.
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# Heteroclinic orbit

## Summary

A heteroclinic orbit is a [[trajectory|07TZ2Q4C1M]] of a dynamical system that joins two different [[equilibrium points|0N8NGKDZDZ]].
It tends to the first one in the past and to the second one in the future.
It is also called a heteroclinic connection.
Heteroclinic orbits bound regions of different behaviour in the [[phase portrait|0T3GK0XCBC]].
Their breaking under perturbation is a source of complicated dynamics.
See the [[definition|F05PSBZP8Z]].

## Explanations

### L1 human

Picture a landscape with two mountain passes and a ridge between them.
A ball that leaves the first pass and rolls exactly along the ridge to the second pass follows a heteroclinic orbit.
For the equation $x' = 1 - x^2$ on the line, the solution $x(t) = \tanh t$ is near $-1$ in the far past and near $1$ in the far future.
Its orbit, the open interval $(-1, 1)$, is a heteroclinic orbit from the equilibrium $-1$ to the equilibrium $1$.
When the starting and the final equilibrium are the same, the orbit is called homoclinic instead.
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# Term operation

## Summary

A term operation of an algebra is an operation on the algebra obtained by [[evaluating|DH30WTA26T]] a [[term|6D3WA7N2W4]] in it.
The term operations are exactly the operations that can be built from the basic operations and the projections by composition.
They form the clone of the algebra, and two algebras with the same term operations behave alike for most questions of universal algebra.

## Explanations

### L1 human

The basic operations of an algebra are the ones it comes with, such as $+$ and $\cdot$ in a ring.
Combining them gives new operations, such as $(x, y) \mapsto x \cdot x + y$, and these are the term operations.
In a group, $(x, y) \mapsto x y x^{-1}$ is a term operation, built from multiplication and inversion.
Fixed elements of the algebra may not be used as constants unless the signature names them.
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# Chebyshev set

## Definition

Let $X$ be a [[normed vector space|03J5RAKB30]] and let $K \subseteq X$ be nonempty.
The set $K$ is a *Chebyshev set* if every $x \in X$ has exactly one [[best approximation|7N4988HN3P]] from $K$, that is, exactly one $k \in K$ with $\norm{x - k} = \inf_{y \in K} \norm{x - y}$.
The map $P_K \colon X \to K$ sending $x$ to this point is the *metric projection* onto $K$.

## Remarks

Example: in $\R^2$ with the Euclidean norm, the closed unit disc $D$ is a Chebyshev set.
For $\norm{x} \le 1$ the nearest point is $x$ itself.
For $\norm{x} > 1$ it is $x / \norm{x}$ and no other point of $D$, since the triangle inequality is strict off the ray through $x$.

Non-example: the unit circle in $\R^2$ is not a Chebyshev set, because the origin has every point of the circle as a nearest point.

A Chebyshev set is closed, since a point of the closure not in $K$ would have distance $0$ but no nearest point.
Every nonempty closed convex set in a Hilbert space is a Chebyshev set, by the [[Hilbert projection theorem|J5YNTHQMFX]].
In $\R^n$ with the Euclidean norm the converse holds: every Chebyshev set is convex.
Whether every Chebyshev set in an infinite-dimensional Hilbert space is convex is an open problem.
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# Anderson–Darling test

## Summary

The Anderson–Darling test is a goodness-of-fit [[hypothesis test|087P03NEND]] for whether a sample comes from a given distribution.
Its statistic is an integrated squared difference between the [[empirical distribution function|A05WR6TEK5]] and the hypothesised distribution function, weighted so that the tails count more.
It is a weighted form of the [[Cramér–von Mises criterion|G14XJXPKFS]], and it is often more sensitive than the [[Kolmogorov–Smirnov test|YZYHWC6C5F]] to departures in the tails.

## Explanations

### L1 human

Near the far ends of a distribution the staircase of the data and the expected curve are both close to $0$ or to $1$.
So a gap there looks small, even if it means that the data have too many or too few extreme values.
The Anderson–Darling test divides each squared gap by $F_0(1 - F_0)$, which is small in the tails, so that tail gaps count fully.
This makes it good at detecting heavy or light tails.
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# Lebesgue constant

## Definition

Let $a < b$, let $n \ge 0$, and let $x_0 < x_1 < \cdots < x_n$ be points of $[a, b]$.
Let $\ell_0, \dots, \ell_n$ be the [[Lagrange basis polynomials|DA3CQJ5MJ6]] of these nodes, so that $\ell_j(x_i) = 1$ if $i = j$ and $0$ otherwise.
The *Lebesgue function* of the nodes is $\lambda_n(x) = \sum_{j=0}^{n} \abs{\ell_j(x)}$.
The *Lebesgue constant* of the nodes is
$$\Lambda_n = \max_{a \le x \le b} \lambda_n(x).$$

## Remarks

Let $I_n \colon C[a, b] \to C[a, b]$ send $f$ to its [[Lagrange interpolation|237KR6FRCK]] polynomial $\sum_j f(x_j) \ell_j$.
With the uniform norm, $\Lambda_n$ is the [[operator norm|06EKW1AKXF]] of $I_n$.

Example: nodes $-1, 0, 1$ in $[-1, 1]$.
Here $\ell_0 = x(x - 1)/2$, $\ell_1 = 1 - x^2$ and $\ell_2 = x(x + 1)/2$.
For $0 \le x \le 1$ this gives $\lambda_2(x) = 1 + x - x^2$, and by symmetry $\Lambda_2 = \lambda_2(1/2) = 5/4$.

Non-example: the number $\max_x \abs{\sum_j \ell_j(x)}$ is not the Lebesgue constant; it always equals $1$, because $\sum_j \ell_j = 1$.
The absolute values must be taken term by term.

Always $\Lambda_n \ge 1$.
For equally spaced nodes $\Lambda_n$ grows like $2^{n+1} / (e \, n \log n)$.
For Chebyshev nodes $\Lambda_n = \frac{2}{\pi} \log n + O(1)$, and no choice of nodes does better than $\frac{2}{\pi} \log n - C$ for an absolute constant $C$.
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# Lacunary series

## Summary

A lacunary series is a [[trigonometric series|FZK2M8KGXQ]] or a [[power series|0D6KXBHKC0]] whose nonzero terms have frequencies that grow at least geometrically, such as $1, 2, 4, 8, \ldots$.
The large gaps make the terms behave almost like independent random variables.
As a result, lacunary series have rigid properties.
By the [[Hadamard gap theorem|DA8M8VZZHC]], a power series with such gaps cannot be continued past its circle of convergence.
A lacunary Fourier series of a bounded function converges absolutely.

## Explanations

### L1 human

Most of the terms of a lacunary series are zero, and the remaining ones are spaced further and further apart.
For example, $\sum_{k \ge 0} 2^{-k} \cos(2^k x)$ uses only the frequencies $1, 2, 4, 8, \ldots$.
Waves with such widely spaced frequencies hardly interfere with each other.
This makes lacunary series a source of both clean theorems and striking examples, such as the nowhere differentiable Weierstrass function.
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# Idempotent semiring

## Definition

Let $(S, +, \cdot, 0, 1)$ be a [[semiring|Q07RA7AMDE]].
The semiring $S$ is *idempotent* if $a + a = a$ for all $a \in S$.

## Remarks

In an idempotent semiring, the relation $a \le b \iff a + b = b$ is a [[partial order|011C29Q1S1]].
For this order, $a + b$ is the least upper bound of $a$ and $b$, the element $0$ is the least element, and multiplication is monotone on both sides.
Example: the Boolean semiring $\set{0, 1}$ with $a + b = \max(a, b)$ and $a \cdot b = \min(a, b)$ is idempotent, since $\max(a, a) = a$.
Example: the subsets of the [[free monoid|27ZPTJP9PP]] $\Sigma^*$ on an alphabet $\Sigma$, with union as addition and concatenation as multiplication, form an idempotent semiring, since $L \cup L = L$.
Non-example: $\N$ with the usual operations is a semiring but is not idempotent, since $1 + 1 = 2 \ne 1$.
A ring is idempotent only if it is the zero ring, since $a + a = a$ gives $a = 0$.
Idempotent semirings are also called *dioids*.
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# Minimal harmonic function

## Summary

A minimal harmonic function is a [[positive harmonic function|EYMQ82Z8XJ]] that has no positive harmonic minorants other than its own constant multiples.
Minimal functions play the role of extreme points in the convex cone of positive harmonic functions on a domain.
By Martin's theory, every positive harmonic function is an integral of minimal ones.
The minimal ones correspond to the points of the minimal Martin boundary.

## Explanations

### L1 human

Positive harmonic functions can be added and scaled by positive numbers.
A minimal one cannot be split into two genuinely different positive harmonic pieces.
On the unit disc, the Poisson kernel with its pole at one boundary point is minimal.
It is the response to a unit mass at a single point of the circle.
The constant function $1$ on the disc is not minimal, since it is the average of the Poisson kernels over the circle.
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# Weighted least squares

## Definition

Let $X$ be a real $n \times p$ matrix with rows $x_1^T, \dots, x_n^T$ and rank $p$, let $y \in \R^n$, and let $w_1, \dots, w_n > 0$ be weights.
Let $W$ be the diagonal matrix with diagonal entries $w_1, \dots, w_n$.
The *weighted least squares estimator* is the minimiser of
$$\sum_{i=1}^n w_i (y_i - x_i^T b)^2$$
over $b \in \R^p$, which is
$$\hat\beta_{\mathrm{WLS}} = (X^T W X)^{-1} X^T W y.$$

## Remarks

Example: take $p = 1$ and $X$ the column of ones, so that $x_i^T b = b$.
The function $\sum_i w_i (y_i - b)^2$ has derivative $-2 \sum_i w_i (y_i - b)$, which vanishes at $b = \sum_i w_i y_i / \sum_i w_i$, the weighted mean.
The formula agrees: $X^T W X = \sum_i w_i$ and $X^T W y = \sum_i w_i y_i$.

Non-example: if some weight is negative, the objective need not be bounded below.
With $n = 1$, $p = 1$, $x_1 = 1$ and $w_1 = -1$, the function $-(y_1 - b)^2$ has no minimum.

In a model $y = X\beta + \varepsilon$ with uncorrelated errors and $\operatorname{Var}(\varepsilon_i) = \sigma_i^2$, the choice $w_i = 1/\sigma_i^2$ gives the [[generalized least squares|P142WXJCF3]] estimator with $\Omega = \operatorname{diag}(\sigma_1^2, \dots, \sigma_n^2)$.
This is the usual remedy for known [[heteroscedasticity|D8VX6XQ3TT]].
With all weights equal it is [[ordinary least squares|YM708KHV2W]].
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# Positive element

## Summary

A positive element of a C*-algebra is a self-adjoint element whose [[spectrum|EH0CZ19RKP]] lies in $[0, \infty)$.
Equivalently, it has the form $b^* b$.
Positive elements generalize nonnegative functions and [[positive semidefinite matrices|01MMVW8CKC]], and they define the order $a \le b$ on self-adjoint elements.
They are the basis of the notions of [[states|F7MB2WREBJ]] and [[completely positive maps|CGY299QMC3]].

## Explanations

### L1 human

In the algebra of continuous functions, a positive element is a function that never takes a negative value.
In the algebra of matrices, it is a Hermitian matrix with no negative eigenvalue, like $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$, whose eigenvalues are $1$ and $3$.
Every element of the form $b^* b$ is positive, just as $\abs{z}^2 = \overline{z} z \ge 0$ for a complex number $z$.
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# Minimax approximation

## Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a [[continuous function|0X3WGX95FE]] on the [[interval|0TK8Y3J5RW]] $[a, b]$.
Let $n \ge 0$ be an integer and let $\mathcal{P}_n$ be the set of real polynomials of [[degree|0DPF6WRZW6]] at most $n$.
A *minimax approximation* of degree $n$ to $f$ is a polynomial $p^* \in \mathcal{P}_n$ such that
$$\max_{a \le x \le b} \abs{f(x) - p^*(x)} = \inf_{p \in \mathcal{P}_n} \max_{a \le x \le b} \abs{f(x) - p(x)}.$$
The common value is the *minimax error* $E_n(f)$.

## Remarks

Example: for $f(x) = x^2$ on $[-1, 1]$ and $n = 1$, the polynomial $p^*(x) = 1/2$ is the minimax approximation, with error $1/2$.
Check: $x^2 - 1/2$ takes the values $1/2, -1/2, 1/2$ at $x = -1, 0, 1$.
If a line $p$ had $\abs{x^2 - p(x)} < 1/2$ everywhere, then $p(-1) > 1/2$, $p(0) < 1/2$ and $p(1) > 1/2$, which no linear function can do, since $p(0)$ is the mean of $p(-1)$ and $p(1)$.

Non-example: the Taylor polynomial $1 + x$ of $e^x$ at $0$ is not the minimax approximation of degree $1$ on $[-1, 1]$.
Its error $e^x - 1 - x$ is never negative, so lowering the line a little makes the largest error smaller.

A minimax approximation exists and is unique for every continuous $f$.
It is characterised by equioscillation: the error $f - p^*$ attains its largest absolute value at $n + 2$ points with alternating signs.
In terms of norms, $p^*$ minimises $\norm{f - p}_\infty$ over $\mathcal{P}_n$, where $\norm{g}_\infty = \max_{[a, b]} \abs{g}$.
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# Trajectory optimization

## Definition

Let $U \subseteq \R^m$ be nonempty, and let $f \colon \R^n \times U \times \R \to \R^n$, $L \colon \R^n \times U \times \R \to \R$, $\phi \colon \R \times \R^n \times \R \times \R^n \to \R$ be continuous.
Let $c \colon \R^n \times U \times \R \to \R^q$ give path constraints and $b \colon \R \times \R^n \times \R \times \R^n \to \R^r$ give boundary conditions.
A *trajectory optimization problem* asks for times $t_0 < t_f$, a measurable control $u \colon [t_0, t_f] \to U$ and an absolutely continuous state $x \colon [t_0, t_f] \to \R^n$ that minimise
$$\phi(t_0, x(t_0), t_f, x(t_f)) + \int_{t_0}^{t_f} L(x(t), u(t), t) \, dt$$
subject to:

1. the [[control system|ZPDTVZQN5K]] $x'(t) = f(x(t), u(t), t)$ for almost every $t$;
2. the path constraints $c(x(t), u(t), t) \le 0$ componentwise for almost every $t$;
3. the boundary conditions $b(t_0, x(t_0), t_f, x(t_f)) = 0$.

*Trajectory optimization* is the numerical solution of such problems, which yields an open-loop control and its trajectory.

## Remarks

The problem is an [[optimal control|T3V6ZNAMDN]] problem in Bolza form with constraints; the word stresses the computation of one optimal trajectory.
Indirect methods solve the conditions of [[Pontryagin's maximum principle|7RPV2T29CF]]; direct methods discretise $x$ and $u$ and solve a finite-dimensional nonlinear program.

Example: $n = 2$, $x = (x_1, x_2)$, $f = (x_2, u)$, $U = \R$, $L = u^2$, $\phi = 0$, no path constraints, $t_0 = 0$, $t_f = 1$, and boundary conditions $x(0) = (0, 0)$, $x(1) = (1, 0)$.
The control $u(t) = 6 - 12t$ gives $x_2(t) = 6t - 6t^2$ and $x_1(t) = 3t^2 - 2t^3$, which meet the boundary conditions, with cost $\int_0^1 (6 - 12t)^2 \, dt = 12$.
The maximum principle forces the optimal $u$ to be affine in $t$, and this is the only affine control that meets the boundary conditions, so it is the solution.

Non-example: finding a feedback law $u = \kappa(x)$ that is good for all initial states is a synthesis problem, not a trajectory optimization problem in this sense.

## Explanations

### L1 human

Trajectory optimization finds one best motion of a system, together with the controls that produce it, from given start and end conditions.
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# Infinitesimal generator of a Markov process

## Definition

Let $E$ be a locally compact Hausdorff space with a countable base.
Let $C_0(E)$ be the [[Banach space|06GZ956657]] of continuous functions on $E$ that vanish at infinity, with the sup norm.
Let $(X_t)_{t \ge 0}$ be a [[Feller process|4HS3C12472]] on $E$ with semigroup $T_t f(x) = \mathbb{E}_x[f(X_t)]$.
The *infinitesimal generator* of $X$ is the linear operator $A$ with domain
$$D(A) = \set{f \in C_0(E) : \lim_{t \downarrow 0} \frac{T_t f - f}{t} \text{ exists in } C_0(E)},$$
given by
$$Af = \lim_{t \downarrow 0} \frac{T_t f - f}{t} \quad \text{for } f \in D(A),$$
where the limit is in the sup norm.

## Remarks

Example: for a continuous-time Markov chain on a finite set $S$ with rate matrix $Q$, one has $T_t = e^{tQ}$, so $(T_t f - f)/t \to Qf$ for every $f$, and $A = Q$ with $D(A)$ the whole space.
Example: for Brownian motion on $\R^d$, every twice continuously differentiable $f$ with $f$, its first and its second derivatives in $C_0(\R^d)$ lies in $D(A)$, and $Af = \tfrac12 \Delta f$.
Non-example: for Brownian motion on $\R$, the function $f(x) = e^{-x^2} \abs{x}$ is in $C_0(\R)$ but not in $D(A)$, since it is not differentiable at $0$.
For $f \in D(A)$, Dynkin's formula says that $f(X_t) - f(X_0) - \int_0^t Af(X_s)\, ds$ is a martingale.
The semigroup satisfies $\frac{d}{dt} T_t f = A T_t f = T_t A f$ for $f \in D(A)$, the [[backward|KQMJ7M57ER]] and forward equations.
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# Quaternion-Kähler manifold

## Summary

A quaternion-Kähler manifold is a Riemannian manifold of dimension $4n$ whose holonomy group lies in $Sp(n) \cdot Sp(1)$.
Locally it carries three almost complex structures that multiply like the quaternion units.
Parallel transport preserves the bundle they span, but not each of them.
For $n \ge 2$ such manifolds are Einstein, and despite the name they are in general not Kähler.
Quaternionic projective space $\mathbb{H}P^n$ is the basic example.

## Explanations

### L1 human

A [[hyperkähler manifold|87AWY7NQS0]] has three fixed ways to multiply by quaternion units.
In a quaternion-Kähler manifold the three ways may rotate among themselves as one moves around the manifold.
Only the three-dimensional family they span is well defined.
The quaternionic projective space, the quaternion analogue of the complex projective space, is the model example.
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# Hunt's extension of Carleson's theorem

## Statement

Let $1 < p < \infty$.
Let $f \colon \R \to \C$ be a $2\pi$-periodic function with $\int_{-\pi}^{\pi} \abs{f}^p < \infty$, that is, $f \in L^p(\mathbb{T})$ (see [[Lp space|SASP14MA1Z]]).
Let $S_N f(x) = \sum_{k=-N}^{N} \hat f(k) e^{ikx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]], where $\hat f(k)$ are its [[Fourier coefficients|5NN2YZZR3V]].
Then $S_N f(x) \to f(x)$ as $N \to \infty$ for [[almost every|RARTMA0ZEC]] $x \in \R$.

## Remarks

Hunt proved this in 1968 by extending the method of [[Carleson's theorem|FBH6KG484Z]], which is the case $p = 2$.
The key estimate is that the maximal operator $f \mapsto \sup_N \abs{S_N f}$ is bounded on $L^p(\mathbb{T})$ for $1 < p < \infty$.
The theorem is new only for $1 < p < 2$, since $L^p(\mathbb{T}) \subset L^2(\mathbb{T})$ for $p \ge 2$.
The case $p = 1$ fails: [[Kolmogorov's example|25V2MHPHZ7]] is an integrable function whose Fourier series diverges everywhere.

## Explanations

### L1 human

Carleson showed that Fourier series of square-integrable functions converge at almost every point.
Hunt showed the same for the larger classes $L^p$ with $p > 1$.
The boundary case $p = 1$ really is different, because there the convergence can fail everywhere.

### L2 human

The proof shows that the Carleson maximal operator is bounded on $L^p$ for $1 < p < \infty$.
With the maximal bound, a.e. convergence follows from convergence on the dense class of trigonometric polynomials.
The exact borderline between $L^1$ and the spaces $L^p$ with $p > 1$ for almost everywhere convergence is still studied.
</document>

<document source="https://tamnd.github.io/math2/node/NB057FEAAE.md" tag="NB057FEAAE" kind="statement">
# Brauer's main theorems

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a prime, and fix a $p$-modular system with residue field $k$ of characteristic $p$.
The $p$-blocks of a finite group $H$ are the blocks of the [[group algebra|MNP2HBWB5Q]] $k[H]$.
Each irreducible complex [[character|RMWMYBY8TC]] and each irreducible [[Brauer character|R2N3GBDBVQ]] of $H$ lies in exactly one block.
Each block $b$ has a defect group, a [[$p$-subgroup|06QYRTX9KQ]] of $H$ determined up to conjugacy.
Each block $b$ of $H$ has a central character $\lambda_b \colon Z(k[H]) \to k$.
For a subset $X$ of $G$, let $\hat{X}$ be the sum of the elements of $X$ in $k[G]$.
Let $H \le G$ and let $b$ be a block of $H$.
Suppose there is a block $B$ of $G$ with $\lambda_B(\hat{C}) = \lambda_b(\widehat{C \cap H})$ for each [[conjugacy class|0DEBXAJ5SA]] $C$ of $G$.
Then $B$ is unique, one says that $b^G$ is defined, and $b^G = B$ is the Brauer correspondent of $b$.
The principal block is the block containing the trivial character.

1. First main theorem. Let $D$ be a $p$-subgroup of $G$ with [[normaliser|0YGN3VAA0D]] $N_G(D)$. The map $b \mapsto b^G$ is a bijection from the blocks of $N_G(D)$ with defect group $D$ onto the blocks of $G$ with defect group $D$.
2. Second main theorem. Let $x \in G$ have order a power of $p$, and let $H = C_G(x)$ be its [[centraliser|02YYV36BFM]]. Let $\chi$ be an irreducible complex character of $G$. There are unique complex numbers $d^x_{\chi\varphi}$, for $\varphi$ in the set of irreducible Brauer characters of $H$, such that $\chi(xy) = \sum_{\varphi} d^x_{\chi\varphi} \, \varphi(y)$ for every $p$-regular $y \in H$. Let $\chi$ lie in the block $B$ and $\varphi$ in the block $b$ of $H$. If $d^x_{\chi\varphi} \ne 0$, then $b^G = B$.
3. Third main theorem. Let $H \le G$ and let $b$ be a block of $H$ with defect group $D$ such that $C_G(D) \le H$. Then $b^G$ is defined, and $b^G$ is the principal block of $G$ if and only if $b$ is the principal block of $H$.

## Remarks

The numbers $d^x_{\chi\varphi}$ are the generalised decomposition numbers; for $x = 1$ they are the ordinary decomposition numbers.
The first main theorem reduces the count of blocks with a given defect group to a question about the normaliser of a $p$-subgroup, a local subgroup.
For example, if $D$ is a normal Sylow $p$-subgroup of $G$, then $N_G(D) = G$ and the theorem is a tautology; the content appears when $N_G(D)$ is much smaller than $G$.
The second main theorem says that the values of characters of a block at elements with $p$-part $x$ are controlled by blocks of $C_G(x)$ that correspond to it.
The third main theorem identifies the principal block of $G$ through principal blocks of local subgroups, which is used in the local analysis of finite simple groups.

## Explanations

### L1 human

Brauer's three main theorems relate the blocks of a finite group, the natural pieces of its representation theory modulo $p$, to blocks of small subgroups built from $p$-subgroups.
They make precise the idea that much of the $p$-modular representation theory of a group is controlled locally, near its $p$-subgroups.

### L2 human

These results are the foundation of block theory.
The first gives the Brauer correspondence with normalisers, the second computes character values through centralisers of $p$-elements, and the third tracks the principal block.
Together they underlie local-global conjectures such as the [[McKay conjecture|WQQTFRZFBD]], Alperin's weight conjecture and [[Brauer's height zero conjecture|5M0HYY3RYV]].
</document>

<document source="https://tamnd.github.io/math2/node/NB0HE0TE1R.md" tag="NB0HE0TE1R" kind="definition">
# Distribution function of a measurable function

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
Let $f \colon X \to \C$ or $f \colon X \to [-\infty, \infty]$ be a [[measurable function|YYNCJSR0PJ]].
The *distribution function* of $f$ is the function $d_f \colon [0, \infty) \to [0, \infty]$ given by
$$d_f(t) = \mu(\set{x \in X : \abs{f(x)} > t}).$$

## Remarks

Example: let $X = [0, 1]$ with Lebesgue measure and $f(x) = x^2$.
For $0 \le t < 1$ the set $\set{x : x^2 > t}$ is $(\sqrt{t}, 1]$, so $d_f(t) = 1 - \sqrt{t}$.
For $t \ge 1$ the set is empty, so $d_f(t) = 0$.

Non-example: the cumulative distribution function $F(t) = \mu(\set{f \le t})$ of probability theory is a different object.
It measures where $f$ is small, uses $f$ and not $\abs{f}$, and is increasing rather than decreasing.

The function $d_f$ is non-increasing and right-continuous.
Two functions with the same distribution function are called *equimeasurable*, and they have the same $L^p$ norms.
Some authors write $\lambda_f$ or $\mu_f$ for $d_f$.
</document>

<document source="https://tamnd.github.io/math2/node/NB0NAQSKJZ.md" tag="NB0NAQSKJZ" kind="concept">
# Spheroidal wave function

## Summary

Spheroidal wave functions are solutions of the equations obtained by [[separation of variables|47NFRVB564]] in the [[Helmholtz equation|MNXKTXJ35G]] in prolate or oblate spheroidal coordinates.
The angular spheroidal wave functions $S_{mn}(c, x)$ are the solutions bounded on $[-1, 1]$, and they exist only for special eigenvalues $\lambda_{mn}(c)$.
For $c = 0$ they reduce to the [[associated Legendre polynomials|VPJCSVRY6S]], so they deform the spherical case to the spheroidal one.
The prolate functions with $m = 0$ are also central in signal processing.
There they are the functions of a given bandwidth that are most concentrated in a time interval.

## Explanations

### L1 human

A spheroid is a sphere stretched or squashed along one axis, like a rugby ball or a lentil.
Waves around a sphere are described with Legendre functions; waves around a spheroid need a deformed family, the spheroidal wave functions.
The parameter $c$ measures how far the problem is from the spherical one, and for $c = 0$ the deformation disappears.
As with Legendre polynomials, well-behaved solutions exist only for a discrete list of separation constants.
</document>

<document source="https://tamnd.github.io/math2/node/NBFNXA2WFM.md" tag="NBFNXA2WFM" kind="statement">
# Osgood's lemma

## Statement

Let $n \ge 1$, let $U \subseteq \C^n$ be an open set, and let $f \colon U \to \C$ be continuous.
Assume that $f$ is holomorphic in each variable separately: for each $j$ and each fixed value of the other variables, $z_j \mapsto f(z_1, \dots, z_n)$ is [[holomorphic|0DG0Z72CQD]].
Then $f$ is holomorphic on $U$.
That is, near each point $c \in U$, $f$ is the sum of a convergent [[power series|0VP56B4EMX]] in $z_1 - c_1, \dots, z_n - c_n$.

## Remarks

Proof idea: on a closed polydisc $\prod_j \set{\abs{z_j - c_j} \le r_j}$ in $U$, apply the one-variable [[Cauchy integral formula|0DG10TVYPN]] in each variable in turn.
Continuity makes the iterated integral an integral over the torus $\prod_j \set{\abs{\zeta_j - c_j} = r_j}$.
Expanding the kernel $\prod_j (\zeta_j - z_j)^{-1}$ as a geometric series gives the power series.
Hartogs's theorem on separate analyticity shows that the continuity hypothesis can be dropped, but its proof is much harder.
For real variables the analogue fails: $xy/(x^2 + y^2)$, set to $0$ at the origin, is real-analytic in each variable but not continuous.

## Explanations

### L1 human

In several complex variables a function can be tested one variable at a time.
If it is continuous and holomorphic in each variable when the others are frozen, it is holomorphic in all of them together.
This makes it easy to check that many functions of several variables are holomorphic.

### L2 human

Osgood's lemma is often the first step in a course on several complex variables.
With it, the Cauchy formula on polydiscs gives power series expansions, Cauchy estimates and Weierstrass convergence theorems in $\C^n$.
It also lets one define holomorphic functions of several variables either by power series or by separate holomorphy plus continuity.
</document>

<document source="https://tamnd.github.io/math2/node/NBK6G2VB7A.md" tag="NBK6G2VB7A" kind="concept">
# Sign test

## Summary

The sign test is a [[hypothesis test|087P03NEND]] about the [[median|07Y2E7P9NP]] of a distribution, or about the median difference within matched pairs.
It uses only whether each observation lies above or below the hypothesised median.
So under the null hypothesis the count of observations above has a [[binomial distribution|0WCWA1KT3M]] with success probability $1/2$.
It needs almost no assumptions on the shape of the distribution, which makes it one of the simplest tests of [[nonparametric statistics|VFAJ1EG3X5]].

## Explanations

### L1 human

Suppose a claim says that half of all values lie above $10$.
Look at a sample and count how many values are above $10$.
If the claim is true, each value is above $10$ with chance one half, like a fair coin toss.
Ten values out of ten above $10$ would be like ten heads in a row, which has chance $1/1024$, so the claim looks doubtful.
</document>

<document source="https://tamnd.github.io/math2/node/NBR81PR7E5.md" tag="NBR81PR7E5" kind="concept">
# Tomita–Takesaki theory

## Summary

Tomita–Takesaki theory starts from a [[von Neumann algebra|PSFWQVZ0FS]] with a cyclic and separating vector.
It attaches to it a positive self-adjoint modular operator $\Delta$ and an antiunitary modular conjugation $J$.
Its main theorem says that $J$ turns the algebra into its [[commutant|1VC7EJ1ZBZ]].
It also says that the unitaries $\Delta^{it}$ give a one-parameter group of automorphisms, the modular automorphism group.
So every faithful normal state comes with a natural time evolution.
The theory is the basis of Connes' classification of type $\mathrm{III}$ factors and of the use of KMS states in quantum statistical mechanics.

## Explanations

### L1 human

For a trace, the order of a product does not matter: $\tau(xy) = \tau(yx)$.
Most states are not traces, and Tomita–Takesaki theory measures exactly how far a state is from being one.
The answer is a flow $\sigma_t$ of automorphisms of the algebra that repairs the order of a product.
For a matrix algebra with the state $\varphi(x) = \operatorname{tr}(\rho x)$ given by an invertible density matrix $\rho$, this flow is $\sigma_t(x) = \rho^{it} x \rho^{-it}$.
Then $\varphi(xy) = \varphi(y \, \rho x \rho^{-1})$, so moving $x$ to the other side costs the twist $x \mapsto \rho x \rho^{-1}$.
For a trace, $\rho$ is a multiple of the identity and the flow does nothing.
</document>

<document source="https://tamnd.github.io/math2/node/NBX8FZY516.md" tag="NBX8FZY516" kind="concept">
# Independent component analysis

## Summary

Independent component analysis recovers hidden source signals from observed linear mixtures of them, assuming the sources are statistically independent.
Unlike [[principal component analysis|ABXD13M340]], which only makes the new variables uncorrelated, it asks for full [[independence|0GQ938GDE0]].
This is possible only when at most one source is Gaussian.
It is a standard tool for blind source separation in signal processing.

## Explanations

### L1 human

At a party, two microphones each record a different mix of two people talking.
Independent component analysis tries to unmix the recordings into the two separate voices, without knowing how they were mixed.
The key assumption is that what one person says does not depend on what the other says.
The voices come back only up to order and volume: the method cannot tell which voice is "first" or how loud it really was.
</document>

<document source="https://tamnd.github.io/math2/node/NC1C7EKV1V.md" tag="NC1C7EKV1V" kind="definition">
# Positive solutions of integral equations

## Definition

Let $(\Omega, \mu)$ be a measure space.
Let $X$ be a space of real-valued functions on $\Omega$, such as $C(\Omega)$ for compact $\Omega$ or $L^p(\Omega, \mu)$.
Its [[positive cone|Z344RY5XX1]] is $X_+ = \set{\varphi \in X : \varphi \ge 0}$, with the inequality almost everywhere in the case of $L^p$.
Consider an [[integral equation|R3SG1SN7JQ]] $\varphi = T\varphi$ on $X$, for example $T\varphi = f + \int_\Omega k(\cdot, y) \varphi(y) \, d\mu(y)$, or a nonlinear one such as a [[Hammerstein equation|T7E2EW2T0G]] $T\varphi = \int_\Omega k(\cdot, y) \, g(y, \varphi(y)) \, d\mu(y)$.

1. A *positive solution* is a solution $\varphi \in X_+$ with $\varphi \ne 0$.
2. A *strictly positive solution* is a solution with $\varphi(x) > 0$ for all $x \in \Omega$ (for almost all $x$, in the case of $L^p$).

## Remarks

Example: on $X = C[0, 1]$ take $\varphi(x) = 1 + \frac12 \int_0^1 \varphi(t) \, dt$.
The right side is constant, so $\varphi = c$ with $c = 1 + \frac{c}{2}$, that is $c = 2$, a strictly positive solution.
More generally, if $k \ge 0$, $f \ge 0$, and the integral operator $K$ has [[spectral radius|9V7YQA85PC]] $r(K) < 1$, the [[Liouville–Neumann series|H1HWSWME6B]] $\varphi = \sum_{n \ge 0} K^n f$ is a sum of nonnegative terms, so the solution is nonnegative.

Non-example: $\varphi(x) = 1 + 2 \int_0^1 \varphi(t) \, dt$ on $[0, 1]$ gives $c = 1 + 2c$, so its only solution is $\varphi = -1$, which is not in $X_+$.
Here $r(K) = 2 > 1$, and the Neumann series argument fails.

For the homogeneous eigenvalue problem $\lambda\varphi = K\varphi$, the [[Krein–Rutman theorem|VADXDK3TKM]] and [[Jentzsch's theorem|P8CAK0FHGJ]] give a positive eigenfunction for the eigenvalue $r(K)$ when $K$ is compact with nonnegative kernel and $r(K) > 0$.
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<document source="https://tamnd.github.io/math2/node/NC26CE0Y8B.md" tag="NC26CE0Y8B" kind="concept">
# Sweedler notation

## Summary

Sweedler notation is a shorthand for the comultiplication of a [[coalgebra|A06RA8KB0T]]: one writes $\Delta(c) = \sum c_{(1)} \otimes c_{(2)}$, or even $c_{(1)} \otimes c_{(2)}$, without naming the terms of the sum.
Coassociativity lets one write iterated coproducts as $c_{(1)} \otimes c_{(2)} \otimes c_{(3)}$ without brackets.
It is named after Moss Sweedler and is the standard notation for computations with [[Hopf algebras|7XFYP0N84H]] and [[comodules|7WEM2E6RTV]].

## Explanations

### L1 human

The coproduct of an element is a sum of pairs, but the pairs are rarely worth naming one by one.
Sweedler notation writes a typical pair as $c_{(1)} \otimes c_{(2)}$ and lets the summation stay implicit.
With it, the counit law reads $\varepsilon(c_{(1)}) c_{(2)} = c$, and the antipode law reads $S(c_{(1)}) c_{(2)} = \varepsilon(c) 1$.
The symbols $c_{(1)}$ and $c_{(2)}$ are not elements on their own; they only make sense inside expressions that are linear in each of them.
</document>

<document source="https://tamnd.github.io/math2/node/NC4B8KC479.md" tag="NC4B8KC479" kind="definition">
# Tate module

## Definition

Let $G$ be an [[abelian group|082GHNPJ53]], written additively, and let $\ell$ be a prime number.
For $n \ge 1$, let $G[\ell^n] = \set{x \in G : \ell^n x = 0}$, a subgroup of the [[torsion subgroup|0DBWXQRXQJ]].
Multiplication by $\ell$ maps $G[\ell^{n+1}]$ to $G[\ell^n]$.
The *$\ell$-adic Tate module* of $G$ is the inverse limit
$$T_\ell(G) = \varprojlim_n G[\ell^n]$$
along these maps.
It is a module over the [[$\ell$-adic integers|V6TJAFZ4MD]] $\Z_\ell$.

Let $A$ be an [[abelian variety|R7W03TXRS9]] over a field $k$, and let $k^{s}$ be a separable closure of $k$.
The Tate module of $A$ is $T_\ell(A) = T_\ell(A(k^{s}))$.

## Remarks

If $A$ has dimension $g$ and $\ell$ is not the characteristic of $k$, then $T_\ell(A)$ is a free $\Z_\ell$-module of rank $2g$.
The [[absolute Galois group|PGKWBJDTKX]] $\Gal(k^{s}/k)$ acts on each $A[\ell^n]$, and so continuously on $T_\ell(A)$.

Example: let $G = \mu_{\ell^\infty}$ be the group of roots of unity of $\ell$-power order in $\C^{*}$.
Then $G[\ell^n]$ is cyclic of order $\ell^n$, and the maps $\zeta \mapsto \zeta^\ell$ are onto.
So $T_\ell(G) \cong \Z_\ell$, and this module is written $\Z_\ell(1)$.

Degenerate example: $G = \Z$ has no nonzero torsion, so every $G[\ell^n]$ is $0$ and $T_\ell(\Z) = 0$.
The Tate module only sees the $\ell$-power torsion of $G$.
</document>

<document source="https://tamnd.github.io/math2/node/NC768TWZ3D.md" tag="NC768TWZ3D" kind="statement">
# Wiener's lemma

## Statement

Let $\mathbb{T} = \set{z \in \C : \abs{z} = 1}$ be the [[circle group|04EAFVFTF7]].
Let $f \colon \mathbb{T} \to \C$ be a function with an [[absolutely convergent Fourier series|HHBGB5R5S2]], that is, $f(z) = \sum_{n \in \Z} c_n z^n$ with $\sum_{n \in \Z} \abs{c_n} < \infty$.
If $f(z) \ne 0$ for every $z \in \mathbb{T}$, then $1/f$ also has an absolutely convergent Fourier series.

## Remarks

The result is also called Wiener's $1/f$ theorem.
Proof idea in Gelfand theory: the functions with absolutely convergent Fourier series form the Wiener algebra, isomorphic to the Banach algebra $\ell^1(\Z)$ with convolution.
Its [[characters|M7JVFJ3MMX]] are the evaluations at points of $\mathbb{T}$, and its [[Gelfand transform|VWNSHKJT92]] is $f$ itself.
An element of a commutative unital Banach algebra is invertible if and only if no character vanishes on it.
Since $f$ has no zero on $\mathbb{T}$, $f$ is invertible in the Wiener algebra, and its inverse is $1/f$.

Example: $f(z) = 2 + z$ has no zero on $\mathbb{T}$, and indeed $1/f(z) = \tfrac{1}{2} \sum_{n \ge 0} (-z/2)^n$ has absolutely summable coefficients.

The hypothesis is needed: $f(z) = 1 + z$ has absolutely convergent Fourier series but vanishes at $z = -1$, and $1/f$ is unbounded near $-1$, so it has no absolutely convergent Fourier series.

## Explanations

### L1 human

Some periodic functions are so nice that the sum of the sizes of their Fourier coefficients is finite.
Wiener's lemma says that if such a function never takes the value $0$, then its reciprocal is just as nice.
This is surprising at first: nothing obvious controls the Fourier coefficients of $1/f$.

### L2 human

Wiener's original proof was by direct estimates; Gelfand's proof with characters made the theory of commutative Banach algebras famous.
The same argument proves analogues for other Banach algebras of functions, and the lemma is a key step in Wiener's Tauberian theorem.
A function with an absolutely convergent Fourier series is continuous.
The lemma says that the invertible elements of the Wiener algebra are exactly its functions without zeros on the circle.
</document>

<document source="https://tamnd.github.io/math2/node/NC79GZ12QZ.md" tag="NC79GZ12QZ" kind="definition">
# Euler class

## Definition

Let $\pi \colon E \to B$ be an oriented real vector bundle of rank $n$ over a paracompact space $B$.
Let $s \colon B \to E$ be the zero section and let $E_0 \subset E$ be its complement.
Let $u \in H^n(E, E_0; \Z)$ be the Thom class of $E$.
It is the unique relative class whose restriction to $H^n(E_b, E_b \setminus \set{0}; \Z) \cong \Z$ is the oriented generator, for every $b \in B$.
Its existence and uniqueness is part of the [[Thom isomorphism|N685RXHDZX]] theorem.
The *Euler class* of $E$ is
$$e(E) = s^* j^* u \in H^n(B; \Z),$$
where $j^* \colon H^n(E, E_0; \Z) \to H^n(E; \Z)$ is restriction to absolute [[cohomology|6N7BC70CX0]].

## Remarks

The Euler class is natural under pullback, so it is a [[characteristic class|7HACJC0JE2]] of oriented bundles.
Reversing the orientation changes the sign of $e(E)$.
If $E$ has a nowhere-zero section $\sigma$, then $s$ is homotopic to $\sigma$ as maps into $E$ and $\sigma$ factors through $E_0$, where $j^* u$ restricts to $0$; so $e(E) = 0$.
If $n$ is odd, then $2 e(E) = 0$, because the map $v \mapsto -v$ reverses the orientation.
The reduction of $e(E)$ modulo $2$ is the top [[Stiefel–Whitney class|JQCK7A210W]] $w_n(E)$.

Example: for a closed oriented manifold $M$, $\langle e(TM), [M] \rangle = \chi(M)$.
For $M = S^2$ this gives $e(TS^2) = 2$ times the generator of $H^2(S^2; \Z)$.

Non-example: the Möbius line bundle over $S^1$ is not orientable, so it has no Euler class with integer coefficients in this sense; only its mod $2$ analogue $w_1$ is defined.
</document>

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# Operator system

## Definition

Let $A$ be a unital C*-algebra with unit $1$ and involution $a \mapsto a^*$.
An *operator system* in $A$ is a linear subspace $S \subseteq A$ such that:

1. $1 \in S$;
2. $a^* \in S$ for every $a \in S$.

## Remarks

Example: in the C*-algebra $C(\mathbb{T})$ of continuous functions on the unit circle $\mathbb{T} = \set{z \in \C : \abs{z} = 1}$, let $S$ be the span of $1, z, \bar z$.
It contains $1$, and $(a + b z + c \bar z)^* = \bar a + \bar c z + \bar b \bar z$ lies in $S$, so $S$ is an operator system.
Non-example: the span of $1$ and $z$ in $C(\mathbb{T})$ contains $1$ but not $\bar z$.
Indeed, if $\bar z = a + b z$ on $\mathbb{T}$, then $z = 1$ and $z = -1$ give $a = 0$ and $b = 1$, but then $z = i$ gives $-i = i$.
Every self-adjoint element $h$ of an operator system $S$ satisfies $-\norm{h} 1 \le h \le \norm{h} 1$, so $S$ is spanned by its positive elements.
Many authors take $A = B(H)$ for a Hilbert space $H$, and some also require $S$ to be norm closed.
</document>

<document source="https://tamnd.github.io/math2/node/NCCGNTBEYV.md" tag="NCCGNTBEYV" kind="statement">
# Equidistribution theorem

## Statement

Let $\alpha$ be an [[irrational number|0DZXAXEJQ9]].
Then the sequence $(n\alpha)_{n \ge 1}$ is [[equidistributed modulo one|XM6DH57BWY]].
That is, for all $0 \le a < b \le 1$, the proportion of $n \le N$ with $n\alpha - \floor{n\alpha} \in [a, b)$ tends to $b - a$ as $N \to \infty$.

## Remarks

A short proof uses Weyl's criterion.
For a nonzero integer $h$, the number $e^{2\pi i h \alpha}$ is not $1$, since $h\alpha \notin \Z$.
The sum $\sum_{n=1}^{N} e^{2\pi i h n \alpha}$ is a geometric sum, so its absolute value is at most $2 / \abs{1 - e^{2\pi i h \alpha}}$, a bound independent of $N$.
Dividing by $N$ gives a limit $0$, which is Weyl's condition.
The hypothesis that $\alpha$ is irrational is necessary: if $\alpha = p/q$ in lowest terms, then $n\alpha$ takes only $q$ values modulo one.
Example: $\log_{10} 2$ is irrational, so the fractional parts of $n \log_{10} 2$ are equidistributed.
Hence the leading digit of $2^n$ is $k$ for a proportion $\log_{10}(1 + 1/k)$ of the exponents $n$; for instance the digit $1$ occurs about $30.1\%$ of the time.

## Explanations

### L1 human

Walk around a circle of circumference $1$ in steps of an irrational length $\alpha$.
You never return to a point you have visited, and in the long run you visit every arc equally often.
This is a precise version of the idea that irrational rotations mix the circle well.

### L2 human

The theorem is the simplest case of Weyl's results on polynomial sequences.
If a real polynomial $p$ has at least one irrational coefficient other than the constant term, then $(p(n))$ is equidistributed modulo one.
Its proof shows the general pattern: equidistribution follows from cancellation in [[exponential sums|97PBMQCPPM]].
It strengthens Kronecker's theorem that $(n\alpha)$ is dense modulo one, and it expresses the unique ergodicity of the irrational rotation of the circle.
</document>

<document source="https://tamnd.github.io/math2/node/NCDZ969V7X.md" tag="NCDZ969V7X" kind="concept">
# Substitution

## Summary

A substitution replaces each variable of a [[term|6D3WA7N2W4]] by another term, all at the same time.
It is the syntactic counterpart of plugging values into a formula.
Substitution is one of the rules of equational logic, and it is the basic operation of unification and term rewriting.

## Explanations

### L1 human

Take the term $x + y$ and replace $x$ by $z \cdot z$ and $y$ by $1$.
The result is the term $z \cdot z + 1$.
All variables are replaced at once, so substituting $x \mapsto y$ and $y \mapsto x$ in $x + y$ gives $y + x$, not $x + x$.
A law such as $x + y \approx y + x$ stays true after any substitution, which is why laws can be applied to arbitrary expressions.
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<document source="https://tamnd.github.io/math2/node/NCKFPAGHXF.md" tag="NCKFPAGHXF" kind="concept">
# Floer homology

## Summary

Floer homology is a family of homology theories built like [[Morse homology|94BWTKKGJM]] on an infinite-dimensional space.
Its chain complex is generated by critical points of an action functional.
These are, for example, periodic orbits of a Hamiltonian or intersection points of two Lagrangian submanifolds.
Its differential counts solutions of a Cauchy–Riemann type equation that connect them.
Andreas Floer introduced it in the late 1980s to prove the Arnold conjecture on fixed points of Hamiltonian diffeomorphisms.
Versions of it are now central in symplectic topology and in the topology of three- and four-manifolds.

## Explanations

### L1 human

In Morse theory one studies a function on a space through its critical points and the gradient lines that run between them.
Counting these lines gives a homology that recovers the topology of the space.
Floer applied this idea to a functional on loops in a symplectic manifold, whose critical points are periodic orbits of a Hamiltonian system.
The gradient lines become cylinders that solve a perturbed [[pseudoholomorphic curve|N469E47V75]] equation.
Take a closed symplectic manifold with $\omega = 0$ on spheres and a nondegenerate Hamiltonian diffeomorphism of it.
The resulting homology shows that the number of fixed points is at least the sum of the Betti numbers of the manifold with coefficients in $\F_2$.
</document>

<document source="https://tamnd.github.io/math2/node/NCRPWS9W5R.md" tag="NCRPWS9W5R" kind="concept">
# Quasitriangular Hopf algebra

## Summary

A quasitriangular Hopf algebra is a [[Hopf algebra|7XFYP0N84H]] $H$ together with an invertible element $R \in H \otimes H$.
This element, the universal R-matrix, conjugates the coproduct into the opposite coproduct.
The element $R$ satisfies the [[Yang–Baxter equation|663BP8ZHS5]].
For two modules $V, W$ over $H$, the map $v \otimes w \mapsto \tau(R \cdot (v \otimes w))$, with $\tau$ the flip, is an isomorphism $V \otimes W \to W \otimes V$.
These isomorphisms make the category of $H$-modules a [[braided monoidal category|VT8SFWT0XW]], which is why such algebras feed knot invariants and [[quantum groups|AP2ZDXANKC]].

## Explanations

### L1 human

For a group, the representations $V \otimes W$ and $W \otimes V$ are the same up to swapping the factors.
For a general Hopf algebra the plain swap is no longer a map of modules.
A quasitriangular structure supplies a corrected swap: first act by a fixed element $R$, then swap.
In the simplest case, the group algebra of a group with $R = 1 \otimes 1$, the corrected swap is the ordinary swap.
The interesting examples, such as the Drinfeld double of a finite-dimensional Hopf algebra, can have a corrected swap that applied twice is not the identity.
</document>

<document source="https://tamnd.github.io/math2/node/NCWYTR6PH0.md" tag="NCWYTR6PH0" kind="definition">
# Clarke generalized gradient

## Definition

Let $X$ be a real [[Banach space|06GZ956657]] with [[continuous dual space|4A82S5FVRT]] $X^*$, and write $\ip{\zeta}{v} = \zeta(v)$.
Let $f \colon X \to \R$ be [[Lipschitz|0AW1E6YTJY]] on a neighbourhood of a point $x \in X$.
The *Clarke generalized directional derivative* of $f$ at $x$ in the direction $v \in X$ is
$$f^\circ(x; v) = \limsup_{y \to x, \ t \downarrow 0} \frac{f(y + t v) - f(y)}{t},$$
a [[limit superior|0T3RZQXDYD]] over $y \in X$ and $t > 0$.
The *Clarke generalized gradient* of $f$ at $x$ is the set
$$\partial_C f(x) = \set{\zeta \in X^* : \ip{\zeta}{v} \le f^\circ(x; v) \text{ for all } v \in X}.$$

## Remarks

If $K$ is a Lipschitz constant of $f$ near $x$, then $\abs{f^\circ(x; v)} \le K \norm{v}$, and $v \mapsto f^\circ(x; v)$ is sublinear.
So $\partial_C f(x)$ is convex, nonempty by the Hahn–Banach theorem, and weak-* compact by the Banach–Alaoglu theorem, since it is weak-* closed and bounded by $K$.
If $f$ is continuously differentiable near $x$, then $\partial_C f(x) = \set{f'(x)}$.
If $f$ is convex, then $\partial_C f(x)$ is the subdifferential of convex analysis.
For $X = \R^n$, Clarke's formula describes it through gradients, which exist almost everywhere by Rademacher's theorem.
It is the convex hull of the limits $\lim_k \nabla f(x_k)$ over sequences $x_k \to x$ at which $f$ is differentiable.
This formula still holds when the $x_k$ must avoid a given set of measure zero.

Example: $f(x) = -\abs{x}$ on $\R$ at $x = 0$.
For $y > 0$ small the difference quotient in direction $v$ is $-v$, and for $y < 0$ small it is $v$, so $f^\circ(0; v) = \abs{v}$.
Hence $\partial_C f(0) = \set{\zeta : \zeta v \le \abs{v} \text{ for all } v} = [-1, 1]$.

Non-example: $f(x) = \sqrt{\abs{x}}$ on $\R$ is not Lipschitz near $0$.
The quotient $(f(t) - f(0))/t = t^{-1/2}$ is unbounded as $t \downarrow 0$, so $f^\circ(0; 1) = +\infty$, and the construction gives no useful information.
</document>

<document source="https://tamnd.github.io/math2/node/NCZP5MPR9Z.md" tag="NCZP5MPR9Z" kind="definition">
# Desarguesian plane

## Definition

A [[projective plane|DANFNN8C4Y]] is *Desarguesian* if the conclusion of [[Desargues's theorem|5Z19Y02MME]] holds in it.
That is, let $ABC$ and $A'B'C'$ be triangles with $A \ne A'$, $B \ne B'$, $C \ne C'$ and distinct corresponding sides.
If $AA'$, $BB'$, $CC'$ pass through a common point, then $AB \cap A'B'$, $BC \cap B'C'$, $CA \cap C'A'$ are [[collinear|0AFNNTRTJE]].

## Remarks

A projective plane is Desarguesian exactly when it is isomorphic to $\mathbb{P}^2(D)$ for some division ring $D$, and then $D$ is determined up to isomorphism.

Example: $\mathbb{P}^2(\Q)$ is Desarguesian.
Check: it is $\mathbb{P}^2(k)$ for the field $k = \Q$, where Desargues's theorem holds.

Non-example: the projective completion of the [[Moulton plane|3V9GYYXM1H]] is not Desarguesian, since it contains two triangles in perspective from a point whose corresponding sides meet in three non-collinear points.

By Wedderburn's little theorem every finite division ring is a field, so the finite Desarguesian planes are the planes $\mathbb{P}^2(\F_q)$, of order $q$ a prime power.
Every Desarguesian plane is a [[Moufang plane|7F4HSNJJH8]].
</document>

<document source="https://tamnd.github.io/math2/node/NCZZ90R5Y6.md" tag="NCZZ90R5Y6" kind="definition">
# Fourier–Motzkin elimination

## Definition

Let a system $S$ of $m$ linear inequalities in real variables $x_1, \dots, x_n$ be given by
$$\sum_{j=1}^{n} a_{ij} x_j \le b_i, \qquad i = 1, \dots, m,$$
with $a_{ij}, b_i \in \R$.
Split the indices into $I_+ = \set{i : a_{in} > 0}$, $I_- = \set{i : a_{in} < 0}$ and $I_0 = \set{i : a_{in} = 0}$.
The *Fourier–Motzkin elimination* of $x_n$ from $S$ is the system $S'$ in $x_1, \dots, x_{n-1}$ consisting of:

1. the inequalities of $S$ with index in $I_0$;
2. for each pair $(i, k) \in I_+ \times I_-$, the inequality
$$\sum_{j=1}^{n-1} \left(a_{in} a_{kj} - a_{kn} a_{ij}\right) x_j \le a_{in} b_k - a_{kn} b_i,$$
obtained by adding $a_{in}$ times inequality $k$ to $-a_{kn}$ times inequality $i$, so that $x_n$ cancels.

## Remarks

The main property: $(x_1, \dots, x_{n-1})$ satisfies $S'$ if and only if there is $x_n \in \R$ such that $(x_1, \dots, x_n)$ satisfies $S$.
Indeed, inequalities in $I_+$ give upper bounds on $x_n$ and those in $I_-$ give lower bounds, and $S'$ says exactly that each lower bound is at most each upper bound.
So the solution set of $S'$ is the [[projection|033DA1WE8Y]] of the solution set of $S$ forgetting $x_n$.

Example: for $-x \le 0$, $-y \le 0$, $x + y \le 1$, eliminating $y$ pairs $x + y \le 1$ (in $I_+$) with $-y \le 0$ (in $I_-$) and gives $x \le 1$.
With $-x \le 0$ from $I_0$, the result is $0 \le x \le 1$, the projection of the triangle onto the $x$-axis.

Non-example: the method does not work for integer solutions.
The system $x - 2y \le 0$, $-x + 2y \le 0$ with $x = 1$ has the real solution $y = 1/2$, but no integer solution $y$, while elimination of $y$ gives no condition that excludes $x = 1$.

The new system can have up to $\abs{I_0} + \abs{I_+} \cdot \abs{I_-}$ inequalities, at most $m^2/4$ when $I_0$ is empty, so repeated elimination can blow up quickly.
</document>

<document source="https://tamnd.github.io/math2/node/ND1CBJS6V2.md" tag="ND1CBJS6V2" kind="definition">
# Holonomic sequence

## Definition

Let $K$ be a field.
A [[sequence|0G9EKM0E15]] $u \colon \N \to K$ is *holonomic* (or *P-recursive*) if it satisfies some [[linear recurrence with polynomial coefficients|K6QV4WSTXX]].
That is, there are an integer $r \ge 0$ and polynomials $p_0, \dots, p_r \in K[n]$, not all zero, with
$$p_r(n)\, u_{n+r} + \cdots + p_0(n)\, u_n = 0 \quad \text{for all } n \in \N.$$

## Remarks

Example: the factorials $u_n = n!$ form a holonomic sequence over $\Q$, with $r = 1$, $p_1 = 1$ and $p_0 = -(n + 1)$, since $(n+1)! - (n+1) \cdot n! = 0$.
Non-example: $u_n = 2^{n^2}$ over $\Q$ is not holonomic.
Indeed, take a recurrence as above with $p_r \ne 0$ and the other $p_i$ of degree at most $f$.
It gives $\abs{u_{n+r}} \le C n^{f} \max_{i < r} \abs{u_{n+i}}$ for large $n$ and some constant $C$.
So $\log \abs{u_n}$ grows at most like a constant times $n \log n$, while $\log 2^{n^2} = n^2 \log 2$.
Over a field of characteristic $0$, $u$ is holonomic if and only if its generating function $\sum u_n x^n$ is a [[holonomic function|7H44JD9NQD]].
Every solution of a linear recurrence with constant coefficients is holonomic.
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<document source="https://tamnd.github.io/math2/node/ND6XEY68P4.md" tag="ND6XEY68P4" kind="concept">
# Noetherian scheme

## Summary

A Noetherian scheme is a [[scheme|5A35TYZD02]] that is covered by finitely many affine pieces, each the spectrum of a [[Noetherian ring|0670T59HGE]].
It is the finiteness condition under which most of the standard theory of schemes works well.
All varieties over a field and all schemes of finite type over $\Z$ are Noetherian.

## Explanations

### L1 human

A Noetherian ring is one where every ideal has finitely many generators, so every system of polynomial equations is equivalent to a finite one.
A Noetherian scheme is built from finitely many pieces of this kind.
The finiteness has geometric consequences, for example the scheme splits into finitely many irreducible components.
The spectrum of a polynomial ring in infinitely many variables is a typical scheme that is not Noetherian.
</document>

<document source="https://tamnd.github.io/math2/node/ND7KXXEDJ1.md" tag="ND7KXXEDJ1" kind="definition">
# Lindelöf summation

## Definition

Let $\sum_{n \ge 0} a_n$ be a [[series|0JH4MN4K1C]] of complex numbers.
For $h > 0$ put
$$L(h) = a_0 + \sum_{n \ge 1} n^{-hn} a_n$$
whenever this series converges.
The series $\sum a_n$ is *Lindelöf summable* to $s$ if $L(h)$ is defined for every $h > 0$ and the [[one-sided limit|0909PWZR62]] $\lim_{h \to 0^+} L(h)$ equals $s$.

## Remarks

Example: let $\sum a_n$ converge to $s$.
For fixed $h > 0$ the factors $n^{-hn} = e^{-hn \ln n}$ decrease in $n \ge 1$ and lie in $(0, 1]$.
By Abel's partial summation, $L(h)$ converges, and the tails of $L(h)$ are bounded by the tails of $\sum a_n$ uniformly in $h$.
Since each factor tends to $1$ as $h \to 0^+$, one gets $L(h) \to s$, so the series is Lindelöf summable to its sum.

Non-example: for $1 + 1 + 1 + \cdots$, $L(h) = 1 + \sum_{n \ge 1} n^{-hn}$ is defined for every $h > 0$.
For each $N$, $L(h) \ge 1 + \sum_{n=1}^{N} n^{-hn} \to N + 1$ as $h \to 0^+$.
So $L(h) \to +\infty$, and the second condition fails.
</document>

<document source="https://tamnd.github.io/math2/node/NDRRH2YD2C.md" tag="NDRRH2YD2C" kind="conjecture">
# Lehmer's conjecture

## Statement

For a nonzero polynomial $f \in \Z[x]$, let $M(f)$ be its [[Mahler measure|QVPYBPRHRC]].
There is a constant $c > 1$ such that $M(f) \ge c$ for every $f \in \Z[x]$ with $M(f) > 1$.

## Remarks

The condition $M(f) > 1$ excludes exactly the polynomials $\pm x^k \Phi_{n_1} \cdots \Phi_{n_r}$ built from [[cyclotomic polynomials|08KEYFAH66]], which have $M(f) = 1$.
The problem goes back to D. H. Lehmer, who asked it in 1933.
The conjecture is open.
The smallest known value of $M(f) > 1$ is that of Lehmer's polynomial $x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$, about $1.17628$.
Smyth proved the conjecture for nonreciprocal polynomials.
If $f \in \Z[x]$ is irreducible of degree $d$, $f \ne \pm x$, and $x^d f(1/x) \ne \pm f(x)$, then $M(f) \ge M(x^3 - x - 1) \approx 1.3247$.
Dobrowolski proved a lower bound of the form $M(f) \ge 1 + c \left( \frac{\log \log d}{\log d} \right)^3$ for an irreducible noncyclotomic $f$ of degree $d$, with an absolute constant $c > 0$.

## Explanations

### L1 human

The Mahler measure of an integer polynomial is at least $1$, and it equals $1$ only for products of cyclotomic polynomials and powers of $x$.
The conjecture says that there is a gap: the next possible values cannot come arbitrarily close to $1$.
The best candidate for the smallest value above $1$ has been known for almost a century.
</document>

<document source="https://tamnd.github.io/math2/node/NDS0H727BA.md" tag="NDS0H727BA" kind="definition">
# Unit in the last place

## Definition

Consider a [[floating-point|RTKRSC3KPH]] system with base $\beta$, precision $p$ and minimal exponent $e_{\min}$.
Let $x \ne 0$ be a floating-point number written as $x = \pm (d_0 . d_1 \ldots d_{p-1})_\beta \times \beta^{e}$, with $d_0 \ne 0$ if $e > e_{\min}$.
The *unit in the last place* of $x$ is
$$\mathrm{ulp}(x) = \beta^{\,e - p + 1},$$
the value of the digit position $d_{p-1}$.

## Remarks

Example: with $\beta = 10$ and $p = 3$, the number $x = 3.14 = 3.14 \times 10^{0}$ has $e = 0$, so $\mathrm{ulp}(x) = 10^{-2}$; its neighbours are $3.13$ and $3.15$.
Non-example: in the same system, $0.001$ is not the ulp of $3.14$, since $3.141$ is not a floating-point number of precision $3$.
The ulp of $1$ is $\beta^{1-p}$, the [[machine epsilon|JA3QXVF2B8]].
Several variants of the definition exist; some define $\mathrm{ulp}(x)$ for every real $x$ as the gap between the two floating-point numbers closest to $x$.
They agree away from powers of $\beta$.
Rounding to nearest gives an error of at most $\tfrac12 \mathrm{ulp}$ of the result.
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<document source="https://tamnd.github.io/math2/node/NDVJ9VWA71.md" tag="NDVJ9VWA71" kind="example">
# Wilkinson's polynomial

## Example

*Wilkinson's polynomial* is the [[polynomial|1G0B73WPX8]]
$$w(x) = \prod_{i=1}^{20} (x - i) = (x - 1)(x - 2) \cdots (x - 20) = x^{20} - 210 x^{19} + \cdots + 20!.$$
Its [[roots|019MA1VTR2]] $1, 2, \ldots, 20$ are simple and well separated, yet they are extremely sensitive to the coefficients.
James H. Wilkinson changed the coefficient $-210$ of $x^{19}$ to $-210 - 2^{-23}$, a relative change of about $6 \times 10^{-10}$.
The perturbed polynomial has only $10$ real roots, and the roots near $10, \ldots, 19$ turn into five complex conjugate pairs.
For example, the roots $16$ and $17$ become approximately $16.73 \pm 2.81\,i$.
Check of the sensitivity: if $a$ is the coefficient of $x^{19}$ and $r$ a simple root, differentiating $w(r) = 0$ with respect to $a$ gives
$$\frac{\partial r}{\partial a} = -\frac{r^{19}}{w'(r)}.$$
For $r = 20$, $w'(20) = 19!$, so $\abs{\partial r/\partial a} = 20^{19}/19! \approx 4.3 \times 10^{7}$: the root-finding problem has a huge [[condition number|AB3S6MHJ8X]] with respect to this coefficient.

## Remarks

The difficulty lies in the problem, not in the algorithm: any method that works with the coefficients in floating-point arithmetic, where they are rounded, inherits this sensitivity.
The same roots are perfectly well conditioned as functions of the factors $(x - i)$, which shows that the choice of representation of a polynomial matters.

## Explanations

### L1 human

The polynomial with roots $1, 2, \ldots, 20$ looks harmless, since its roots are evenly spaced whole numbers.
But if one changes one coefficient by about one part in a billion, several roots move far away and even become complex numbers.
This shows that finding roots from the coefficients of a polynomial can be a badly conditioned problem.
</document>

<document source="https://tamnd.github.io/math2/node/NDYA85381W.md" tag="NDYA85381W" kind="statement">
# Dyson conjecture

## Statement

Let $n \ge 1$ and let $a_1, \dots, a_n$ be nonnegative integers.
In the [[ring of Laurent polynomials|1R5P656GS6]] $\Q[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$, the constant term of
$$\prod_{1 \le i \ne j \le n} \left( 1 - \frac{x_i}{x_j} \right)^{a_i}$$
is the multinomial coefficient
$$\frac{(a_1 + a_2 + \cdots + a_n)!}{a_1!\, a_2! \cdots a_n!}.$$

## Remarks

Freeman Dyson stated this as a conjecture in 1962, in work on the statistical theory of energy levels.
It was proved the same year by Gunson and by Wilson, and Good gave a short proof in 1970.
Good's proof shows that both sides satisfy the recurrence $F(a) = \sum_{i} F(a - e_i)$ when all $a_i \ge 1$, and that both reduce to the case of $n - 1$ variables when some $a_i = 0$.
The recurrence for the left side comes from the identity $\sum_{i=1}^n \prod_{j \ne i} \left(1 - \frac{x_j}{x_i}\right)^{-1} = 1$, a case of Lagrange interpolation.
Example: $n = 2$.
The constant term of $(1 - x_1/x_2)^{a_1} (1 - x_2/x_1)^{a_2}$ collects the products of the terms $(x_1/x_2)^k$ and $(x_2/x_1)^k$, and equals $\sum_{k} \binom{a_1}{k}\binom{a_2}{k} = \binom{a_1 + a_2}{a_1}$ by [[Vandermonde's identity|01JADB9VW5]].

## Explanations

### L1 human

Multiply out a product of many factors of the form $1 - x_i/x_j$ and keep only the terms with no variable left.
The Dyson conjecture, now a theorem, says that the sum of these terms is a multinomial coefficient, the number of ways to arrange $a_1 + \cdots + a_n$ objects of $n$ colours.

### L2 human

The case where all $a_i$ equal $k$ gives $\frac{(nk)!}{(k!)^n}$, which computes the normalizing constant of the circular ensembles of [[random matrices|1HS4RT7HK5]] through an integral over the torus.
Andrews proposed a $q$-analogue, proved by Zeilberger and Bressoud.
The equal-parameter case is the type $A$ case of the [[Macdonald constant term conjecture|5J25W265JA]].
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# Petersson inner product

## Definition

Let $k$ be an integer, and let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]] with coordinates $\tau = x + iy$.
Let
$$\mathcal{F} = \set{\tau \in \mathbb{H} : \abs{\tau} \ge 1, \ \abs{x} \le 1/2}$$
be the standard fundamental domain of the [[modular group|VEKV1EM77B]].
Let $f$ and $g$ be [[modular forms|GWKN5VG78A]] of weight $k$ for $\mathrm{SL}_2(\Z)$.
Assume that $f$ or $g$ is a [[cusp form|A50Y012RBP]].
Their *Petersson inner product* is
$$\langle f, g \rangle = \int_{\mathcal{F}} f(\tau)\, \overline{g(\tau)}\, y^k \, \frac{dx\, dy}{y^2}.$$

## Remarks

The function $f \overline{g} y^k$ and the measure $dx\, dy / y^2$ are invariant under $\mathrm{SL}_2(\Z)$, so any fundamental domain gives the same value.
A cusp form decays like $e^{-2\pi y}$ as $y \to \infty$, so the integral converges.
On the space $S_k$ of cusp forms, $\langle \cdot, \cdot \rangle$ is a positive definite Hermitian [[inner product|0NB0AWGJBW]].
Example: $\langle \Delta, \Delta \rangle > 0$, since $\Delta$ has no zeros on $\mathbb{H}$, so the integrand $\abs{\Delta}^2 y^{12} \cdot y^{-2}$ is positive on $\mathcal{F}$.
Non-example: for $f = g = E_4$ the integral diverges, because $\abs{E_4}^2 y^4 \cdot y^{-2}$ tends to infinity like $y^2$ as $y \to \infty$; this is why one form must be a cusp form.
For a congruence subgroup $\Gamma$, the integral is taken over a fundamental domain of $\Gamma$, often divided by the index of $\Gamma$ in $\mathrm{SL}_2(\Z)$.
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<document source="https://tamnd.github.io/math2/node/NE09EDK19V.md" tag="NE09EDK19V" kind="example">
# Surface of revolution

## Example

Let $s \mapsto (r(s), z(s))$, for $s$ in an open interval $I$, be a smooth regular curve in the $xz$-plane.
Assume that it is parametrized by [[arc length|0TH78S7GBZ]] and that $r(s) > 0$ for all $s$.
Rotating it about the $z$-axis gives the surface of revolution
$$x(s, \theta) = (r(s) \cos\theta, \; r(s) \sin\theta, \; z(s)), \qquad s \in I, \ \theta \in \R.$$
The partial derivatives are $x_s = (r' \cos\theta, r' \sin\theta, z')$ and $x_\theta = (-r \sin\theta, r \cos\theta, 0)$.
They are orthogonal, $\abs{x_s} = 1$ because $r'^2 + z'^2 = 1$, and $\abs{x_\theta} = r > 0$.
So $x$ is an immersion, and if the profile curve is a homeomorphism onto its image then the image of $x$ is a [[regular surface|6080X3FAD4]].
The [[first fundamental form|MAJ3DH7MZS]] is $ds^2 + r(s)^2 \, d\theta^2$.
The meridians ($\theta$ constant) and the parallels ($s$ constant) are lines of curvature, and the [[Gaussian curvature|SQVQ5DQT6W]] is
$$K = -\frac{r''(s)}{r(s)}.$$

## Remarks

For $r(s) = \cos s$ and $z(s) = \sin s$ with $s \in (-\pi/2, \pi/2)$, one gets the unit sphere without its poles, and $K = \cos s / \cos s = 1$.
Rotating a circle that does not meet the axis gives a torus.
Rotating a catenary gives the [[catenoid|6NN1HP54GB]], and rotating a tractrix gives the [[pseudosphere|44WEBQSKEY]].

## Explanations

### L1 human

Take a curve drawn in a half-plane and spin it around the edge line of the half-plane.
It sweeps out a surface, like a vase on a potter's wheel.
Spheres, cylinders, cones and doughnuts all arise this way.
Because of the symmetry, the curvature of such a surface depends only on the profile curve.
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<document source="https://tamnd.github.io/math2/node/NE0BW8Q2PX.md" tag="NE0BW8Q2PX" kind="definition">
# Exponential family

## Definition

Let $(\mathcal{X}, \mathcal{F})$ be a measurable space with a [[σ-finite measure|WEBBT18D4X]] $\mu$, and let $\set{P_\theta : \theta \in \Theta}$ be a [[parametric family|MSC0MR8VMH]] of probability measures on it.
The family is a *$k$-parameter exponential family* if there are

1. a measurable function $h \colon \mathcal{X} \to [0, \infty)$,
2. measurable functions $T_1, \dots, T_k \colon \mathcal{X} \to \R$,
3. functions $\eta_1, \dots, \eta_k, A \colon \Theta \to \R$,

such that each $P_\theta$ has a [[density|1NTKG64F7N]] with respect to $\mu$ of the form
$$p_\theta(x) = h(x) \exp\Big( \sum_{i=1}^{k} \eta_i(\theta) T_i(x) - A(\theta) \Big), \qquad x \in \mathcal{X}.$$

## Remarks

The function $A$ is fixed by normalization: $A(\theta) = \log \int h(x) \exp\big(\sum_i \eta_i(\theta) T_i(x)\big) \, \mu(dx)$.
The numbers $\eta_i = \eta_i(\theta)$ are the *natural parameters*, and writing the density in terms of $\eta$ gives the *canonical form*.

Example: the [[Bernoulli distribution|0386Y0Z7Y7]] with parameter $p \in (0, 1)$, with $\mu$ the counting measure on $\set{0, 1}$.
Its density is $p^x (1-p)^{1-x} = \exp\big(x \log \tfrac{p}{1-p} + \log(1-p)\big)$.
So it is a $1$-parameter exponential family with $h = 1$, $T(x) = x$, $\eta(p) = \log \frac{p}{1-p}$ and $A(p) = -\log(1-p)$.

Non-example: the uniform distributions on $[0, \theta]$ for $\theta > 0$.
In an exponential family the set $\set{x : p_\theta(x) > 0} = \set{x : h(x) > 0}$ does not depend on $\theta$, up to a $\mu$-null set, because the exponential factor is positive.
For the uniform family this set is $[0, \theta]$, which moves with $\theta$, so no choice of $h$, $T_i$, $\eta_i$, $A$ works.
</document>

<document source="https://tamnd.github.io/math2/node/NE0XB0AEDR.md" tag="NE0XB0AEDR" kind="definition">
# Algebraic cycle

## Definition

Let $X$ be an [[algebraic variety|HV6YBS0YDH]] over a field, or more generally a scheme of finite type over a field.
Let $r \ge 0$ be an integer.
An *$r$-cycle* on $X$ is a finite formal sum $Z = \sum_i n_i [V_i]$.
Here the $n_i$ are integers and the $V_i$ are irreducible closed subvarieties of $X$ of [[dimension|36X2WTHYCB]] $r$.
The $r$-cycles form the [[free abelian group|0YMSEZE1VH]] $Z_r(X)$ on the set of such subvarieties.
An *algebraic cycle* is an element of $Z_*(X) = \bigoplus_{r \ge 0} Z_r(X)$.
If $X$ is equidimensional of dimension $n$, one also writes $Z^j(X) = Z_{n-j}(X)$ for the cycles of codimension $j$.

## Remarks

A cycle is *effective* if all $n_i \ge 0$.
Let $W \subset X$ be a closed subscheme whose [[irreducible components|SY2MWC434Z]] $W_1, \ldots, W_s$ all have dimension $r$.
Its *fundamental cycle* is $[W] = \sum_i m_i [W_i]$, where $m_i$ is the [[length|V6JBKT168T]] of the local ring of $W$ at the generic point of $W_i$.
Example: in $X = \mathbb{P}^2$, with $L$ a line and $C$ a conic, $2[L] - [C]$ is a $1$-cycle.
The subscheme defined by $x^2 = 0$ is the double line, with fundamental cycle $2[L]$ for $L = \set{x = 0}$.
Non-example: the union $L_1 \cup L_2$ of two distinct lines is not one of the generators, since it is not irreducible; its fundamental cycle is $[L_1] + [L_2]$.
</document>

<document source="https://tamnd.github.io/math2/node/NE75TQGG79.md" tag="NE75TQGG79" kind="definition">
# Incidence algebra

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]] that is locally finite: for all $x \le y$ in $P$, the interval $[x, y] = \set{z \in P : x \le z \le y}$ is finite.
Let $K$ be a [[commutative ring|0310REJJY8]].
The *incidence algebra* $I(P, K)$ is the set of all functions $f \colon \set{(x, y) \in P \times P : x \le y} \to K$, with pointwise addition and multiplication by scalars, and with the product
$$(f * g)(x, y) = \sum_{z \in [x, y]} f(x, z) \, g(z, y).$$

## Remarks

The product is associative, and the function $\delta$ with $\delta(x, x) = 1$ and $\delta(x, y) = 0$ for $x < y$ is the identity.
The zeta function is $\zeta(x, y) = 1$ for all $x \le y$.
An element $f$ is invertible if and only if $f(x, x)$ is a unit of $K$ for every $x$.
The inverse of $\zeta$ is the Möbius function $\mu$ of $P$, defined by $\mu(x, x) = 1$ and $\mu(x, y) = -\sum_{x \le z < y} \mu(x, z)$ for $x < y$.

Example: let $P = \set{1, \dots, n}$ with the usual order.
Every interval is finite, and $f \mapsto (f(i, j))_{i \le j}$ identifies $I(P, K)$ with the algebra of upper triangular $n \times n$ matrices over $K$; the product $*$ becomes matrix multiplication.

Non-example: $\Q$ with its usual order is not locally finite, since $[0, 1]$ is infinite.
The sum defining $(\zeta * \zeta)(0, 1)$ would have infinitely many terms equal to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/NEAE07CSM4.md" tag="NEAE07CSM4" kind="definition">
# Logarithmically convex domain

## Definition

Let $D \subseteq \C^n$ be a [[Reinhardt domain|WA0SE56FVH]] with centre $0$.
Let $D^* = \set{z \in D : z_j \ne 0 \text{ for } j = 1, \dots, n}$, and let $\lambda \colon D^* \to \R^n$ be the map $\lambda(z) = (\log\abs{z_1}, \dots, \log\abs{z_n})$.
The domain $D$ is *logarithmically convex* if $\lambda(D^*)$ is a [[convex set|02AS45T3H9]] in $\R^n$.

## Remarks

Example: for the unit ball $\mathbb{B}^2$, $\lambda(\mathbb{B}^{2*}) = \set{x \in \R^2 : e^{2x_1} + e^{2x_2} < 1}$, a sublevel set of the convex function $e^{2x_1} + e^{2x_2}$, hence convex.
Every polydisc with centre $0$ is logarithmically convex, since its image is a product of half-lines $(-\infty, \log r_j)$.
Non-example: let $D = \set{\abs{z_1} < 1, \abs{z_2} < e^{-2}} \cup \set{\abs{z_1} < e^{-2}, \abs{z_2} < 1}$, a Reinhardt domain.
The points $(e^{-1}, e^{-3})$ and $(e^{-3}, e^{-1})$ lie in $D$, with images $(-1, -3)$ and $(-3, -1)$.
The midpoint $(-2, -2)$ of these images is the image of $(e^{-2}, e^{-2})$, which is not in $D$, so $\lambda(D^*)$ is not convex.
For a complete Reinhardt domain, logarithmic convexity is equivalent to being a domain of holomorphy.
</document>

<document source="https://tamnd.github.io/math2/node/NEB29GSM22.md" tag="NEB29GSM22" kind="definition">
# Influence function

## Definition

Let $\mathcal{D}$ be a convex set of [[probability distributions|0VGMEBVW1M]] on $\R$ that contains every [[point mass|099SXAG014]] $\delta_x$, and let $T \colon \mathcal{D} \to \R$ be a functional.
Let $F \in \mathcal{D}$ and $x \in \R$.
The *influence function* of $T$ at $F$ is
$$\mathrm{IF}(x; T, F) = \lim_{\varepsilon \downarrow 0} \frac{T\big((1 - \varepsilon) F + \varepsilon \delta_x\big) - T(F)}{\varepsilon},$$
defined at those $x$ where the limit exists.

## Remarks

It is the one-sided [[Gateaux derivative|PDXR52YGPD]] of $T$ at $F$ in the direction $\delta_x - F$.
The gross-error sensitivity is $\gamma^*(T, F) = \sup_x \abs{\mathrm{IF}(x; T, F)}$, and $T$ is called B-robust at $F$ when $\gamma^*$ is finite.
Under regularity conditions, $\sqrt{n}\big(T(F_n) - T(F)\big)$ is asymptotically normal with mean $0$ and variance $E_F\big[\mathrm{IF}(X; T, F)^2\big]$, where $F_n$ is the empirical distribution.

Example: the mean $T(F) = \int t \, dF(t)$, on distributions with finite mean.
Then $T((1-\varepsilon)F + \varepsilon\delta_x) = (1-\varepsilon)\mu + \varepsilon x$ with $\mu = T(F)$, so $\mathrm{IF}(x; T, F) = x - \mu$, which is unbounded.
For the median at a distribution $F$ with median $m$ and a density $f$ that is continuous and positive at $m$, $\mathrm{IF}(x; T, F) = \sgn(x - m) / (2 f(m))$ for $x \ne m$, which is bounded.

Non-example: the mean is not defined at a distribution $F$ without a finite [[expectation|0DCT0QZVQ1]], such as the Cauchy distribution.
Then $F$ lies outside $\mathcal{D}$, and the mean has no influence function at $F$.
</document>

<document source="https://tamnd.github.io/math2/node/NEC48RHNN0.md" tag="NEC48RHNN0" kind="statement">
# Śleszyński–Pringsheim theorem

## Statement

Let $a_1, a_2, \dots$ and $b_1, b_2, \dots$ be complex numbers such that $\abs{b_n} \ge \abs{a_n} + 1$ for every $n \ge 1$.
Then the [[continued fraction|0BK6JBK58W]]
$$f = \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}}}$$
converges, and its value satisfies $\abs{f} \le 1$.

## Remarks

Proof idea: with the [[fundamental recurrence formulas|6WYYMPH21Q]] ($b_0 = 0$), the denominators satisfy $\abs{B_n} \ge (\abs{a_n} + 1)\abs{B_{n-1}} - \abs{a_n}\abs{B_{n-2}}$.
By induction $\abs{B_n} - \abs{B_{n-1}} \ge \abs{a_1 \cdots a_n}$, so $1 = \abs{B_0} \le \abs{B_1} \le \cdots$ and no $B_n$ vanishes.
Then $\abs{f_n - f_{n-1}} = \frac{\abs{a_1 \cdots a_n}}{\abs{B_n}\abs{B_{n-1}}} \le \frac{1}{\abs{B_{n-1}}} - \frac{1}{\abs{B_n}}$, and this telescopes: the series $\sum (f_n - f_{n-1})$ converges absolutely with total at most $1$, where $f_0 = 0$.
Example: $a_n = 1$ and $b_n = 2$ satisfy the condition, and the value is $\sqrt{2} - 1$.
The condition cannot simply be dropped: for $a_n = -1$ and $b_n = 1$ one has $\abs{b_n} < \abs{a_n} + 1$, and the convergents repeat the cycle $-1, \infty, 0$, so the continued fraction diverges.

## Explanations

### L1 human

A continued fraction converges when each partial denominator is larger than the partial numerator by at least one in absolute value.
The value then lies in the closed unit disc.

### L2 human

This is a basic sufficient condition for convergence, in the same spirit as the comparison test for series.
Each map $w \mapsto \frac{a_n}{b_n + w}$ sends the closed unit disc into itself under the hypothesis, which explains the bound $\abs{f} \le 1$.
The [[Van Vleck theorem|KCCFZTP3MM]] is another classical criterion, for partial denominators in a sector.
</document>

<document source="https://tamnd.github.io/math2/node/NECFJN169C.md" tag="NECFJN169C" kind="concept">
# Hyperfunction

## Summary

A hyperfunction on an open set of the real line is a formal difference of boundary values of holomorphic functions.
One function lives above the real axis and the other below it.
Hyperfunctions were introduced by Mikio Sato and generalize distributions.
They form a flabby sheaf, so local hyperfunctions can always be glued and extended.
They are the basic objects of algebraic analysis of linear partial differential equations.

## Explanations

### L1 human

Take a holomorphic function $F$ above the real axis and another one below it.
Their "jump" across the real axis, $F(x + i0) - G(x - i0)$, is a hyperfunction, even when neither limit exists as a function.
Adding the same holomorphic function on both sides does not change the jump, so such pairs count as the same hyperfunction.
For example, the jump of $-1/(2\pi i z)$ across the real axis is the Dirac delta.
</document>

<document source="https://tamnd.github.io/math2/node/NEGHXN8RE1.md" tag="NEGHXN8RE1" kind="concept">
# Burgers' equation

## Summary

Burgers' equation $u_t + u\, u_x = \nu\, u_{xx}$ is the simplest equation that combines nonlinear transport with diffusion.
For $\nu = 0$ it is the inviscid Burgers' equation, the model scalar [[conservation law|8RA7AMWSAF]], whose smooth solutions break down and form [[shock waves|JSDMGGTG7G]].
For $\nu > 0$ the [[Cole–Hopf transformation|VBMQ9SJ3DH]] turns it into the heat equation, so it can be solved exactly.
It is a standard test case for theory and for numerical methods in fluid dynamics.
See the [[definition|P4RF8A1562]].

## Explanations

### L1 human

In Burgers' equation each value of $u$ moves with speed equal to itself.
So high parts of a wave move faster than low parts, and the front of a wave gets steeper and steeper.
Without viscosity the profile eventually becomes vertical and a jump, a shock, appears.
For example, starting from $u(x, 0) = -x$, the solution is $u(x, t) = x/(t - 1)$, which becomes infinitely steep at time $t = 1$.
With a little viscosity the jump is smoothed out into a thin but steep layer.
</document>

<document source="https://tamnd.github.io/math2/node/NES4B9CWW4.md" tag="NES4B9CWW4" kind="concept">
# Categorical logic

## Summary

Categorical logic is the branch of mathematics that studies logic with the tools of [[category theory|Q1RZT7DV0F]].
A logical theory is matched with a category built from its syntax.
A model of the theory is then a structure-preserving [[functor|28NQ0KRHD3]] from that category to another category, such as the category of sets.
Typical correspondences match the simply typed [[lambda calculus|BD756KR8FM]] with [[Cartesian closed categories|BFBNQ36QS8]], and higher-order [[intuitionistic logic|Q11N8X99FK]] with [[toposes|6837RS7N82]].
These correspondences let one transfer results between logic, algebra, geometry and the semantics of programming languages.

## Explanations

### L1 human

In ordinary logic one writes formulas and proofs, and then interprets them in sets.
Categorical logic allows the interpretation to take place in any suitable category, not only in sets.
Types become objects, terms become morphisms, and logical connectives become operations such as products and exponentials.
For example, a proof that $A$ and $B$ together give $C$ becomes a morphism $A \times B \to C$, and the rule "from $A$ and $B$ infer $A$" becomes the projection $A \times B \to A$.
</document>

<document source="https://tamnd.github.io/math2/node/NEXBTGYYRG.md" tag="NEXBTGYYRG" kind="concept">
# Bézier curve

## Summary

A Bézier curve is a polynomial curve determined by a finite list of control points, given by the [[Bernstein basis polynomials|0JT92VK307]].
It starts at the first control point, ends at the last one, and lies in the [[convex hull|0FQSDC2EQ2]] of all of them.
Bézier curves are the basic curves of vector graphics, fonts and computer-aided design.

## Explanations

### L1 human

Pick a start point, an end point, and a few extra points in between that the curve is attracted to but does not have to pass through.
The Bézier curve leaves the start point heading toward the second point and arrives at the end point coming from the second to last point.
Moving one control point changes the curve smoothly everywhere.
For three points $P_0, P_1, P_2$ one gets the parabola arc $(1-t)^2 P_0 + 2t(1-t) P_1 + t^2 P_2$.
</document>

<document source="https://tamnd.github.io/math2/node/NEYHC1XZ14.md" tag="NEYHC1XZ14" kind="statement">
# Dulac's theorem on finiteness of limit cycles

## Statement

Let $P$ and $Q$ be real polynomials in two variables.
Then the planar [[autonomous system|0673D85QKZ]]
$$x' = P(x, y), \qquad y' = Q(x, y)$$
has only finitely many [[limit cycles|0AAQ73WRFB]].

## Remarks

Henri Dulac published a proof in 1923, but a gap was found in it much later.
Complete proofs were given independently by Yulij Ilyashenko and by Jean Écalle around 1991–1992.
The hard part is to exclude limit cycles that accumulate on a polycycle, a closed chain of equilibria joined by orbits.
Limit cycles cannot accumulate on a periodic orbit of an analytic field, because the Poincaré return map near that orbit is analytic.

The theorem bounds the number of limit cycles of each single system.
It gives no bound in terms of the degree of $P$ and $Q$.
Whether such a bound exists, even for degree $2$, is the second part of Hilbert's sixteenth problem, and it is open.
For example, the van der Pol system $x' = y - (x^3/3 - x)$, $y' = -x$ has exactly one limit cycle.

## Explanations

### L1 human

A limit cycle is an isolated closed loop of motion in the plane.
The theorem says that a system given by polynomial formulas can only have finitely many of them.
It does not say how many, and nobody knows a general bound in terms of the degrees of the polynomials.

### L2 human

The theorem is the finiteness part of the study of Hilbert's sixteenth problem on limit cycles.
Near a periodic orbit, finiteness follows from the analyticity of the return map.
The difficulty lies at polycycles, where the return map has an asymptotic expansion that is not convergent.
Ilyashenko and Écalle overcame this with a theory of such expansions.
</document>

<document source="https://tamnd.github.io/math2/node/NEZYP9HJB9.md" tag="NEZYP9HJB9" kind="definition">
# Completely positive map

## Definition

Let $A$ and $B$ be C*-algebras, and let $\varphi \colon A \to B$ be a linear map.
For $n \ge 1$, let $M_n(A)$ be the [[matrix ring|0FHVPRAE60]] of $n \times n$ matrices with entries in $A$, which is again a C\*-algebra.
Let $\varphi_n \colon M_n(A) \to M_n(B)$ be the map $[a_{ij}] \mapsto [\varphi(a_{ij})]$.
The map $\varphi$ is *$n$-positive* if $\varphi_n$ sends [[positive elements|NA8DF5X3M4]] of $M_n(A)$ to positive elements of $M_n(B)$.
It is *completely positive* if it is $n$-positive for every $n \ge 1$.

## Remarks

Equivalently, $\varphi \otimes \id_{M_n}$ is positive for every $n$.
Every $*$-homomorphism $\pi$ is completely positive, since $\pi_n$ is a $*$-homomorphism and maps $x^* x$ to $\pi_n(x)^* \pi_n(x)$.

Example: for a fixed $m \times k$ complex matrix $V$, the map $\varphi \colon M_m(\C) \to M_k(\C)$, $\varphi(a) = V^* a V$, is completely positive.
Indeed $\varphi_n(x) = W^* x W$ with $W = \operatorname{diag}(V, \ldots, V)$, and $W^* y^* y W = (y W)^* (y W) \ge 0$.

Non-example: the transpose $\theta(a) = a^T$ on $M_2(\C)$ is positive, since $a$ and $a^T$ have the same eigenvalues.
It is not $2$-positive: the matrix $x = [E_{ij}]_{i,j=1}^2 \in M_2(M_2(\C))$, with $E_{ij}$ the matrix units, equals $v v^*$ with $v = e_1 \otimes e_1 + e_2 \otimes e_2$, so it is positive.
But $\theta_2(x) = [E_{ji}]_{i,j}$ is the swap matrix, which has the eigenvalue $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/NF0F8M86W4.md" tag="NF0F8M86W4" kind="statement">
# Monotone operator method for Hammerstein equations

## Statement

Let $H$ be a real [[Hilbert space|HM2HHE3J51]] with inner product $\ip{\cdot}{\cdot}$.
Let $K \colon H \to H$ be a bounded linear [[self-adjoint operator|0NEGN5CRSC]] with $\ip{K v}{v} \ge 0$ for all $v \in H$.
Let $F \colon H \to H$ be [[monotone|Y1M03JQKKR]], that is $\ip{F(u) - F(v)}{u - v} \ge 0$ for all $u, v \in H$, and [[hemicontinuous|XNMKX7H4FD]], that is $t \mapsto \ip{F(u + t v)}{w}$ is continuous on $[0, 1]$ for all $u, v, w \in H$.
Then for every $f \in H$ the abstract [[Hammerstein equation|T7E2EW2T0G]]
$$u + K F(u) = f$$
has exactly one solution $u \in H$.

## Remarks

Let $S$ be the positive [[square root|3JY5HNFRMF]] of $K$, so $K = S^2$ with $S$ self-adjoint.
Look for $u$ of the form $u = f - S v$, and put $G(v) = v - S F(f - S v)$.
For $a = f - S v$ and $b = f - S w$ one finds $\ip{G(v) - G(w)}{v - w} = \norm{v - w}^2 + \ip{F(a) - F(b)}{a - b} \ge \norm{v - w}^2$.
So $G$ is strongly monotone, hemicontinuous and coercive, and Minty's theorem in Hilbert space, the case of the [[Browder–Minty theorem|NWK9Z80BT2]], gives a unique $v$ with $G(v) = 0$.
Then $u = f - S v$ solves the equation, and each solution $u$ gives a zero $v = S F(u)$ of $G$, which yields uniqueness.

Application: let $\Omega \subseteq \R^m$ be measurable with finite measure and $H = L^2(\Omega)$.
Let $k$ be a symmetric square-integrable kernel with $\int\!\!\int k(x, t)\, v(x) v(t)\, dx\, dt \ge 0$ for all $v$, and let $g(x, u)$ be a Carathéodory function that is nondecreasing in $u$ with $\abs{g(x, u)} \le a(x) + b \abs{u}$, $a \in L^2(\Omega)$.
Then the [[Nemytskii operator|W8G3AZ0DNG]] $F = N_g$ is continuous and monotone on $L^2(\Omega)$, and $\varphi(x) + \int_\Omega k(x, t)\, g(t, \varphi(t))\, dt = f(x)$ has exactly one solution in $L^2(\Omega)$ for each $f \in L^2(\Omega)$.

The signs matter.
In $H = \R$ with $K = 1$ and $F(u) = -u$, which is not monotone, the equation reads $0 = f$ and has no solution for $f \ne 0$.

## Explanations

### L1 human

A Hammerstein equation mixes a linear smoothing step with a pointwise nonlinear rule.
If the linear part is "positive" and the nonlinear rule is "increasing", the two effects push in the same direction.
Then the equation behaves like a strictly increasing function on the real line: it takes every value exactly once.

### L2 human

The theorem is one of the main uses of [[monotone operator|Y1M03JQKKR]] theory outside partial differential equations.
The trick with the square root $S$ moves the problem to one where the operator is strongly monotone, which Browder–Minty theory handles directly.
Positive kernels arise as Green's functions of self-adjoint boundary value problems, so the theorem gives existence and uniqueness for $-u'' + g(x, u) = f$ with Dirichlet conditions when $g$ is nondecreasing in $u$.
Versions without symmetry of $K$ and in Banach spaces exist, where $K$ maps $X^*$ to $X$ and $F$ maps $X$ to $X^*$.
</document>

<document source="https://tamnd.github.io/math2/node/NF2GB0TFEW.md" tag="NF2GB0TFEW" kind="definition">
# Premeasure

## Definition

Let $X$ be a set and let $\mathcal{R}$ be a [[ring of sets|KWE1DBMN9B]] on $X$.
A *premeasure* on $\mathcal{R}$ is a map $\mu_0 \colon \mathcal{R} \to [0, \infty]$, with values in the [[extended real line|0GSZ7B0CGR]], such that:

1. $\mu_0(\varnothing) = 0$.
2. For every sequence $A_1, A_2, \ldots$ of pairwise disjoint sets of $\mathcal{R}$ whose union $\bigcup_n A_n$ lies in $\mathcal{R}$,
$$\mu_0\Big(\bigcup_{n=1}^{\infty} A_n\Big) = \sum_{n=1}^{\infty} \mu_0(A_n).$$

## Remarks

Example: on the ring of finite unions of half-open intervals $(a, b]$ in $\R$, the total length is a premeasure; countable additivity is checked with a compactness argument on closed subintervals.
Non-example: on the ring of finite and cofinite subsets of $\N$, put $\mu_0(A) = 0$ for finite $A$ and $\mu_0(A) = 1$ for cofinite $A$.
It is finitely additive, but $\N = \bigcup_n \set{n}$ gives $1 \ne \sum_n 0$, so clause 2 fails.
A premeasure is monotone on $\mathcal{R}$, and countably subadditive for sequences in $\mathcal{R}$ whose union lies in $\mathcal{R}$.
The same definition is used for a semiring of sets instead of a ring.
A measure is a premeasure whose domain is a $\sigma$-algebra.
</document>

<document source="https://tamnd.github.io/math2/node/NF61KYXFXF.md" tag="NF61KYXFXF" kind="definition">
# Y transform

## Definition

Fix a real number $\nu$, and let $Y_\nu$ be the [[Bessel function of the second kind|GRSNDJG363]] of order $\nu$.
Let $f \colon (0, \infty) \to \C$ be such that the integral below converges for $y$ in a set $Y \subseteq (0, \infty)$.
The *Y transform* of order $\nu$ of $f$ is the function
$$(\mathcal{Y}_\nu f)(y) = \int_0^\infty f(x)\, Y_\nu(xy)\, \sqrt{xy}\, dx, \qquad y \in Y.$$

## Remarks

Example: $Y_{1/2}(z) = -\sqrt{2/(\pi z)}\, \cos z$, because $Y_{1/2} = -J_{-1/2}$ and $J_{-1/2}(z) = \sqrt{2/(\pi z)}\, \cos z$.
So $\sqrt{xy}\, Y_{1/2}(xy) = -\sqrt{2/\pi}\, \cos(xy)$, and $\mathcal{Y}_{1/2} f(y) = -\sqrt{2/\pi} \int_0^\infty f(x) \cos(xy)\, dx$, the negative of the Fourier cosine transform.

Non-example: for $\nu > 0$, $Y_\nu(z)$ behaves like $-\frac{\Gamma(\nu)}{\pi} (2/z)^{\nu}$ as $z \to 0^+$.
For $\nu = 2$ and $f$ equal to $1$ on $(0, 1]$ and $0$ elsewhere, the integrand behaves like a constant times $x^{-3/2}$ near $0$, so the integral diverges for every $y > 0$.

The [[Hankel transform|2P9YWDGKC4]] is defined in the same way with $J_\nu$ in place of $Y_\nu$.
</document>

<document source="https://tamnd.github.io/math2/node/NFCQYZ3GRM.md" tag="NFCQYZ3GRM" kind="statement">
# Orlicz–Pettis theorem

## Statement

Let $X$ be a [[Banach space|0BDTTM24DP]] over $\R$ or $\C$, and let $X'$ be the space of continuous linear functionals on $X$.
Let $(x_n)_{n \ge 1}$ be a sequence in $X$.
Assume that for every strictly increasing sequence $n_1 < n_2 < \cdots$ of indices there is $y \in X$ with
$$\varphi(y) = \sum_{k=1}^{\infty} \varphi(x_{n_k}) \quad \text{for all } \varphi \in X'.$$
Then for every such sequence of indices the series $\sum_{k} x_{n_k}$ converges in norm.
In particular, the series $\sum_n x_n$ is [[unconditionally convergent|2ME9M61TJK]] in norm.

## Remarks

The hypothesis says that every subseries converges in the weak topology; the conclusion says that every subseries converges in norm.
The theorem needs all subseries: in $\ell^2$, the series $e_1 + (e_2 - e_1) + (e_3 - e_2) + \cdots$ has partial sums $e_n$, which tend to $0$ weakly but not in norm.
The theorem also holds in Hausdorff locally convex spaces, with the weak topology given by the continuous dual.

## Explanations

### L1 human

In an infinite-dimensional space, weak convergence is in general much weaker than convergence in norm.
For series, the theorem says that this gap closes as soon as one asks for every subseries.
If each subseries converges when tested by every continuous linear functional, then each subseries already converges in norm.

### L2 human

The theorem is a basic tool for [[series in Banach spaces|WN6M8B3ZKB]] and for vector measures.
A countably additive vector measure is defined by convergence of $\sum \mu(A_n)$ for disjoint sets $A_n$, and every subfamily of disjoint sets gives a subseries.
So the theorem shows that a weakly countably additive vector measure is countably additive in norm.
It is also used in the Bochner and Pettis theories of integration of vector-valued functions.
</document>

<document source="https://tamnd.github.io/math2/node/NFEEGWJFCK.md" tag="NFEEGWJFCK" kind="concept">
# Atomless Boolean algebra

## Summary

An atomless Boolean algebra is a [[Boolean algebra|0XADJKM699]] with no [[atoms|TTVCTW6TNC]]: below every nonzero element there is a strictly smaller nonzero element.
One example is the algebra of clopen subsets of the Cantor space.
Others are the [[free Boolean algebra|1MF0TVHDZD]] on infinitely many generators and the [[regular open algebra|17S8AM75AD]] of $\R$.
Any two countable atomless Boolean algebras with $0 \ne 1$ are isomorphic, so there is essentially one such algebra.
Atomless algebras are the opposite extreme to the [[complete atomic Boolean algebras|P5MA4CDRNA]], which are power sets.

## Explanations

### L1 human

In the subsets of a set, every nonempty set contains a single point, and one cannot split a point further.
In an atomless Boolean algebra there are no such smallest pieces: every nonzero element can be split again into two nonzero parts.
Think of the finite unions of intervals in $[0, 1)$: every such nonempty set can be cut into two smaller nonempty ones.
</document>

<document source="https://tamnd.github.io/math2/node/NFJTRCRB9W.md" tag="NFJTRCRB9W" kind="concept">
# Three-term recurrence relation

## Summary

A three-term recurrence relation is a linear recurrence of order two that links three consecutive terms $u_{n+1}$, $u_n$ and $u_{n-1}$.
It is the simplest recurrence beyond first order and appears throughout analysis.
[[Orthogonal polynomials|8TXASARQN1]] satisfy one, and so do the numerators and denominators of [[continued fractions|04P2RJ4WPS]] and many special functions.
Its solutions form a two-dimensional space, which leads to the study of minimal and dominant solutions.

## Explanations

### L1 human

Each new term is built from the two terms just before it, with coefficients that may change with $n$.
The Fibonacci rule $u_{n+1} = u_n + u_{n-1}$ is the simplest example.
The Chebyshev polynomials obey $T_{n+1}(x) = 2x\, T_n(x) - T_{n-1}(x)$, which lets you compute them one after another from $T_0 = 1$ and $T_1 = x$.
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<document source="https://tamnd.github.io/math2/node/NFN044536P.md" tag="NFN044536P" kind="concept">
# Krein space

## Summary

A Krein space is a complex vector space with an inner product that may take negative values on some vectors.
It splits as an orthogonal sum of a positive part and a negative part.
The positive part is a Hilbert space for the form, and the negative part for its negative.
Krein spaces are the natural setting for operators that are self-adjoint for an indefinite form.
Such operators occur in relativistic quantum field theory.
When the negative part has finite dimension, the space is a Pontryagin space.

## Explanations

### L1 human

In ordinary geometry every nonzero vector has positive squared length.
In a Krein space some vectors have negative or zero "squared length", as in the space-time of special relativity.
The space still breaks into a part where lengths are positive and a part where they are negative, and each part looks like a Hilbert space.
The simplest example is $\C^2$ with $[x, y] = x_1 \overline{y_1} - x_2 \overline{y_2}$.
</document>

<document source="https://tamnd.github.io/math2/node/NFQ9FVCP31.md" tag="NFQ9FVCP31" kind="concept">
# Axiom of regularity

## Summary

The axiom of regularity, also called the axiom of foundation, is an axiom of [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].
It says that every nonempty [[set|0BSJ7RQ9BW]] has an element that shares no element with it.
In other words, the membership relation $\in$ is [[well-founded|0DQPNJNPYK]].
It rules out sets such as $x = \set{x}$, and it implies that every set appears at some stage of the cumulative hierarchy $V_\alpha$.

## Explanations

### L1 human

Think of building sets in stages: first the empty set, then sets of what you have, and so on.
Regularity says that every set is built this way, from the bottom up.
So no set can contain itself, and there is no endless chain $x_1 \ni x_2 \ni x_3 \ni \cdots$ going down.
For example, in the set $\set{\emptyset, \set{\emptyset}}$, the element $\emptyset$ has no element in common with the set.
The axiom is almost never used in ordinary mathematics, but it makes the universe of sets much easier to describe.
</document>

<document source="https://tamnd.github.io/math2/node/NFTS5A3NXE.md" tag="NFTS5A3NXE" kind="definition">
# Intersection form

## Definition

Let $M$ be a closed, connected, [[oriented|913WEQERBE]] [[topological manifold|QQQ9B1DK4P]] of dimension $4$, with [[fundamental class|ZEA0PEM3B2]] $[M] \in H_4(M; \Z)$.
Let $H^2$ be [[singular cohomology|6N7BC70CX0]] and $\smile$ the [[cup product|JX47MS1R1H]].
Let $\langle \cdot, \cdot \rangle$ be the Kronecker pairing of $H^4(M; \Z)$ with $H_4(M; \Z)$.
The *intersection form* of $M$ is the map
$$Q_M \colon H^2(M; \Z) \times H^2(M; \Z) \to \Z, \qquad Q_M(a, b) = \langle a \smile b, [M] \rangle.$$

## Remarks

$Q_M$ is a [[symmetric bilinear form|0AXZNE5XNH]], since $a \smile b = b \smile a$ for classes of even degree.
It vanishes if $a$ or $b$ is a torsion class, so it induces a form on the free abelian group $H^2(M; \Z)/\text{torsion}$.
By [[Poincaré duality|9VY90HJVZX]] this induced form is unimodular: its matrix in any basis is a [[unimodular matrix|R6RBQHW191]], with determinant $\pm 1$.
Its rank is $b_2(M)$, and the [[signature|07EZA80NEH]] of its real extension is the signature of $M$.

Example: for the [[complex projective plane|7GJ4STCNNM]] $\C P^2$ with its complex orientation, $H^2 \cong \Z$ is generated by $x$ with $\langle x^2, [\C P^2] \rangle = 1$, so $Q = (1)$.
For $S^2 \times S^2$ the form has matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ in the basis given by the two factors.

Non-example: on a closed oriented surface $\Sigma$, the analogous form $H^1(\Sigma) \times H^1(\Sigma) \to \Z$ is skew-symmetric, not symmetric.
The reason is that $a \smile b = -\,b \smile a$ in degree $1$.
A symmetric form needs dimension $4$, or more generally $4k$ with middle degree $2k$.
</document>

<document source="https://tamnd.github.io/math2/node/NFXKNDJX9C.md" tag="NFXKNDJX9C" kind="statement">
# Functional equation of the Riemann zeta function

## Statement

Let $\zeta$ be the [[Riemann zeta function|1XWBY4N0VT]] and $\Gamma$ the [[gamma function|0H5SYBSED9]].
Then, as an identity of [[meromorphic functions|0ZZ7ZYDR8W]] on $\C$,
$$\zeta(s) = 2^s \pi^{s-1} \sin\!\left(\frac{\pi s}{2}\right) \Gamma(1 - s) \, \zeta(1 - s).$$
Equivalently, the completed zeta function $\xi_0(s) = \pi^{-s/2} \, \Gamma(s/2) \, \zeta(s)$ satisfies $\xi_0(s) = \xi_0(1 - s)$ for all $s \in \C \setminus \set{0, 1}$.

## Remarks

One proof uses the theta function $\theta(t) = \sum_{n \in \Z} e^{-\pi n^2 t}$ for $t > 0$.
Poisson summation gives $\theta(1/t) = \sqrt{t} \, \theta(t)$.
A Mellin transform expresses $\xi_0(s)$ as an integral of $(\theta(t) - 1)/2$ against $t^{s/2 - 1}$.
Splitting the integral at $t = 1$ and using the symmetry of $\theta$ gives a formula that is visibly invariant under $s \mapsto 1 - s$.
The same formula gives the meromorphic continuation of $\zeta$ to $\C$.
Example: at $s = -1$ the right side is $\frac{1}{2} \cdot \frac{1}{\pi^2} \cdot (-1) \cdot \Gamma(2) \cdot \zeta(2) = -\frac{1}{12}$, using $\zeta(2) = \pi^2/6$, so $\zeta(-1) = -1/12$.
At $s = -2, -4, -6, \dots$ the sine factor vanishes while $\Gamma(1 - s) \zeta(1 - s)$ is finite and nonzero, so these are zeros of $\zeta$, the trivial zeros.

## Explanations

### L1 human

The zeta function is first defined only for $\operatorname{Re} s > 1$.
The functional equation links its value at $s$ to its value at $1 - s$, a reflection in the point $1/2$.
So knowing $\zeta$ on the right half of the plane determines it on the left half.

### L2 human

The equation shows that the zeros of $\zeta$ with $\operatorname{Re} s < 0$ are the trivial zeros $-2, -4, \dots$.
The other zeros lie in the critical strip $0 \le \operatorname{Re} s \le 1$, placed symmetrically about the line $\operatorname{Re} s = 1/2$.
The [[Riemann hypothesis|VCGF9YR36X]] says that they lie on this line.
[[Dirichlet L-functions|9D2Z543EJQ]] of primitive characters and many other L-functions satisfy functional equations of the same shape.
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<document source="https://tamnd.github.io/math2/node/NG1F0HYC5G.md" tag="NG1F0HYC5G" kind="definition">
# Discriminant of a number field

## Definition

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] of degree $n = [K : \Q]$, and let $\mathcal{O}_K$ be its ring of integers.
Let $\omega_1, \ldots, \omega_n$ be an integral basis of $K$, that is, a basis of $\mathcal{O}_K$ as a free $\Z$-module.
Let $\sigma_1, \ldots, \sigma_n$ be the $n$ field embeddings $K \to \C$.
The *discriminant* of $K$ is the number
$$d_K = \Big(\det\big(\sigma_i(\omega_j)\big)_{1 \le i, j \le n}\Big)^2,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The value does not depend on the integral basis or on the order of the embeddings.
Two integral bases differ by a matrix in $\mathrm{GL}_n(\Z)$, whose determinant is $\pm 1$.
Reordering the rows changes only the sign of the determinant.
An equivalent form is $d_K = \det\big(\operatorname{Tr}_{K/\Q}(\omega_i \omega_j)\big)$, where the field trace $\operatorname{Tr}_{K/\Q}(\alpha)$ is the [[trace|0KZDNGA561]] of the $\Q$-linear map $x \mapsto \alpha x$ on $K$.
The field trace of an algebraic integer is an integer, so $d_K$ is an integer.
It is never $0$, and its sign is $(-1)^{r_2}$, where $r_2$ is the number of pairs of complex conjugate embeddings.

Example: for $K = \Q(i)$ the basis $1, i$ of $\Z[i]$ gives $\det \begin{pmatrix} 1 & i \\ 1 & -i \end{pmatrix} = -2i$, so $d_K = -4$.
More generally, for a squarefree integer $d \ne 1$, the discriminant of $\Q(\sqrt{d})$ is $d$ if $d \equiv 1 \pmod 4$ and $4d$ otherwise.

Non-example: the basis $1, 2i$ gives $\det \begin{pmatrix} 1 & 2i \\ 1 & -2i \end{pmatrix}^2 = (-4i)^2 = -16$.
This is not $d_{\Q(i)}$, because $1, 2i$ is a basis of the subring $\Z[2i]$ and not of the whole ring of integers $\Z[i]$.
</document>

<document source="https://tamnd.github.io/math2/node/NG1FPJQHA1.md" tag="NG1FPJQHA1" kind="statement">
# Steinhaus theorem on regular methods

## Statement

Let $A = (a_{nk})_{n, k \ge 0}$ be an infinite complex matrix whose [[matrix method|RKFB6AHPBE]] is [[regular|BXVVXJQV8M]].
Then there is a sequence $s$ with all terms in $\set{0, 1}$ that is not $A$-summable.
In particular, no regular matrix method sums every [[bounded sequence|05FQ3VGAR9]].

## Remarks

Proof idea: a "gliding hump" argument.
By the [[Silverman–Toeplitz theorem|S4J7SRPXMZ]], each fixed column of $A$ fades as $n$ grows, while row sums tend to $1$ and the absolute row sums stay bounded.
So one can pick rows $n_1 < n_2 < \cdots$ and disjoint blocks of columns such that row $n_j$ has almost all its weight on block $j$.
Put $s_k = 1$ on the blocks with odd $j$ and $s_k = 0$ elsewhere.
Then $(As)_{n_j}$ is close to $1$ for odd $j$ and close to $0$ for even $j$, so $(As)_n$ does not converge.
Example: for Cesàro summation, the sequence equal to $1$ for $2^{2j} \le k < 2^{2j+1}$ and $0$ otherwise has Cesàro means that oscillate between values near $\frac13$ and near $\frac23$.

## Explanations

### L1 human

Every reasonable averaging method of matrix type fails on some sequence of zeros and ones.
So there is no single regular matrix method that gives a limit to all bounded sequences.

### L2 human

The theorem shows that the bounded sequences are never all summable by one regular matrix.
Extensions of the limit to all bounded sequences do exist, such as [[Banach limits|SYF2ND6J97]], but they are not given by matrices, and their existence uses the Hahn–Banach theorem.
</document>

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# Survival function

## Definition

Let $T$ be a [[random variable|0PNRKMRC55]] with values in $[0, \infty)$, and let $F$ be its [[cumulative distribution function|0H8H7ZTHW2]].
The *survival function* of $T$ is the function $S \colon [0, \infty) \to [0, 1]$ given by
$$S(t) = P(T > t) = 1 - F(t).$$

## Remarks

The function $S$ is nonincreasing and right-continuous, and $S(t) \to 0$ as $t \to \infty$.
If $T$ has a density $f$, then $S(t) = \int_t^\infty f(u) \, du$ and $f = -S'$ wherever $f$ is continuous.
The mean of $T$ is $E[T] = \int_0^\infty S(t) \, dt$.
Some authors use $P(T \ge t)$ instead; the two agree at every $t$ with $P(T = t) = 0$.

Example: if $T$ has the [[exponential distribution|0E99F51GVB]] with rate $\lambda > 0$, then $S(t) = e^{-\lambda t}$.
Indeed $P(T > t) = \int_t^\infty \lambda e^{-\lambda u} \, du = e^{-\lambda t}$, and $\int_0^\infty e^{-\lambda t} \, dt = 1/\lambda$ is the mean.

Non-example: the function $G(t) = e^{-t} \cos^2 t$ takes values in $[0, 1]$, has $G(0) = 1$ and tends to $0$, but it is not a survival function.
It satisfies $G(\pi/2) = 0 < e^{-\pi} = G(\pi)$, so it is not nonincreasing, while $P(T > t)$ can only decrease as $t$ grows.
</document>

<document source="https://tamnd.github.io/math2/node/NG35MGYDZR.md" tag="NG35MGYDZR" kind="concept">
# Stopped process

## Summary

The stopped process $X^\tau$ of a [[stochastic process|0QH5ENKZM1]] $X$ at a [[stopping time|WND87FBAKC]] $\tau$ follows $X$ up to time $\tau$.
After time $\tau$ it stays frozen at the value $X_\tau$.
Stopping preserves the [[martingale|0D0GBNDKDH]] property, which is the heart of the [[optional stopping theorem|DC9HZT2HVJ]].
It is also the basic operation of localisation, used to define [[local martingales|XC2YPC2BSW]] and stochastic integrals.

## Explanations

### L1 human

Picture a gambler's fortune changing over time, and a rule telling the gambler when to leave the table.
After leaving, the fortune no longer changes.
The stopped process is this frozen record: equal to the fortune until the gambler leaves, and constant afterwards.
For a random walk stopped when it first reaches $5$ or $-5$, the stopped walk stays between $-5$ and $5$ for ever.
</document>

<document source="https://tamnd.github.io/math2/node/NG943XNQWH.md" tag="NG943XNQWH" kind="concept">
# Heyting algebra

## Summary

A Heyting algebra is a bounded lattice with an implication operation $a \to b$, the largest element whose meet with $a$ lies below $b$.
Every Heyting algebra is a [[distributive lattice|0HM0HQCK6N]], and every [[Boolean algebra|0XADJKM699]] is a Heyting algebra.
Heyting algebras are the algebraic models of [[intuitionistic logic|Q11N8X99FK]], in the same way that Boolean algebras are the models of classical logic.
The open sets of any topological space form a Heyting algebra.

## Explanations

### L1 human

In a Heyting algebra one can form "and", "or" and "implies", but "not $a$" need not satisfy "$a$ or not $a$".
The main picture is the family of open sets of the plane or the line.
"And" is intersection, "or" is union, and "not $U$" is the interior of the complement of $U$.
On the real line, the open set $U = \R \setminus \set{0}$ has "not $U$" equal to the empty set, so "$U$ or not $U$" is $\R \setminus \set{0}$, not the whole line.
This failure of the law of excluded middle is exactly what intuitionistic logic allows.
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<document source="https://tamnd.github.io/math2/node/NG9P9QC1NG.md" tag="NG9P9QC1NG" kind="definition">
# Subalgebra

## Definition

Let $K$ be a commutative ring and let $A$ be an [[associative algebra|1TZSV19D85]] over $K$ with unit element $1$.
A *subalgebra* of $A$ is a subset $B \subseteq A$ such that:

1. $1 \in B$.
2. If $a, b \in B$, then $a + b \in B$ and $ab \in B$.
3. If $k \in K$ and $b \in B$, then $kb \in B$.

## Remarks

Clause 3 with $k = -1$ gives $-b \in B$, so $B$ is a [[subring|0859D58WPD]] of $A$ and a [[submodule|0RGZ10DFPG]] of the $K$-module $A$.
With the operations of $A$, the set $B$ is then a $K$-algebra.
Let $\eta \colon K \to A$, $k \mapsto k1$, be the structure map.
A subset $B$ is a subalgebra exactly when it is a subring of $A$ that contains $\eta(K)$, because $kb = \eta(k)b$.

Example: in the $\R$-algebra $M_2(\R)$ of real $2 \times 2$ matrices, the set of upper triangular matrices is a subalgebra.
The identity matrix is upper triangular, and sums and real multiples of upper triangular matrices are upper triangular.
For upper triangular $P$ and $Q$, the $(2,1)$ entry of $PQ$ is $p_{21} q_{11} + p_{22} q_{21} = 0$, since $p_{21} = q_{21} = 0$.

Non-example: in the $\R$-algebra $\R[x]$, the set $\Z[x]$ of polynomials with integer coefficients satisfies clauses 1 and 2 but not clause 3.
For $k = 1/2$ and $b = 1$ the product $kb = 1/2$ is not in $\Z[x]$.
So $\Z[x]$ is a subring of $\R[x]$ but not an $\R$-subalgebra.

Some authors work with algebras without a unit element and drop clause 1.
For a field $K$, a subalgebra is a linear subspace of $A$ that contains $1$ and is closed under multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/NGGC4TTYNR.md" tag="NGGC4TTYNR" kind="concept">
# Inverse tangent integral

## Summary

The inverse tangent integral $\operatorname{Ti}_2$ is the function $x \mapsto \int_0^x \frac{\arctan t}{t} \, dt$.
It is the analogue of the [[dilogarithm|G0X0NWSDB7]] built from the [[arctangent|0CE768XNVP]] instead of the logarithm.
For real $x$ it equals the imaginary part of $\operatorname{Li}_2(ix)$.
Its value at $1$ is Catalan's constant.

## Explanations

### L1 human

The dilogarithm can be written as an integral of $-\log(1 - t)/t$.
Replace the logarithm by the arctangent and you get the inverse tangent integral.
Its power series is $x - \frac{x^3}{9} + \frac{x^5}{25} - \cdots$, with odd powers and squared odd numbers in the denominators.
At $x = 1$ this gives $1 - \frac19 + \frac1{25} - \cdots = G \approx 0.91597$, Catalan's constant.
</document>

<document source="https://tamnd.github.io/math2/node/NGHWJGCXS9.md" tag="NGHWJGCXS9" kind="definition">
# Umbilical point

## Definition

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]], let $p \in S$, and choose a smooth unit normal field near $p$.
The point $p$ is an *umbilical point* of $S$ if the two [[principal curvatures|FA8QE1SGP4]] of $S$ at $p$ are equal.

## Remarks

Equivalently, the [[shape operator|DQCD11X1VM]] is a multiple of the identity, $S_p = \lambda \id$.
Another equivalent form is $I\!I_p = \lambda I_p$, with $I\!I_p$ the [[second fundamental form|9KHQR16P1B]] and $I_p$ the [[first fundamental form|MAJ3DH7MZS]].
The condition does not depend on the choice of the normal.
Example: on the sphere of radius $r$, $S_p = \pm \frac{1}{r} \id$ at every point, so every point is umbilical; on a plane $S_p = 0$.
Non-example: no point of a cylinder of radius $r$ is umbilical, since its principal curvatures are $1/r$ and $0$, up to sign.
A connected regular surface all of whose points are umbilical is contained in a plane or in a sphere.
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# Kripke semantics

## Summary

Kripke semantics interprets [[modal logic|C3RZY20ETK]] in a set of possible worlds linked by an accessibility relation.
A formula $\square\varphi$ is true at a world when $\varphi$ is true at every world accessible from it.
Many modal logics are exactly the sets of formulas valid on a class of such frames.
A variant of the semantics also serves [[intuitionistic logic|Q11N8X99FK]].

## Explanations

### L1 human

Picture a set of points, the possible worlds, with arrows between some of them.
At each world, each basic statement is either true or false.
"Necessarily $p$" holds at a world when $p$ holds at every world that its arrows reach.
For example, with two worlds $u \to v$, where $p$ is true only at $v$, the formula $\square p$ holds at $u$, but $p$ itself does not.
</document>

<document source="https://tamnd.github.io/math2/node/NGQMPVNZDP.md" tag="NGQMPVNZDP" kind="example">
# Swallowtail catastrophe

## Example

The *swallowtail catastrophe* is the family of functions of one variable $x \in \R$
$$V(x; a, b, c) = x^5 + a x^3 + b x^2 + c x,$$
with three control parameters $(a, b, c) \in \R^3$.
For $a = b = c = 0$ the function is $x^5$, whose [[critical point|0YTCMFB4HC]] at $0$ is degenerate: the first four derivatives vanish there.

The critical points solve $V'(x) = 5x^4 + 3a x^2 + 2b x + c = 0$, and such a point is degenerate when also $V''(x) = 20x^3 + 6a x + 2b = 0$.
Solving these two equations for $b$ and $c$ in terms of $x$ and $a$ gives
$$b = -10x^3 - 3a x, \qquad c = 15x^4 + 3a x^2.$$
So the [[bifurcation set|JV45S0KQE6]] is the surface in $\R^3$ swept out by these curves as $a$ varies.

Fix $a$ and follow the curve $x \mapsto (b, c)$.
Its velocity is $(-30x^2 - 3a, \; 60x^3 + 6a x) = -3(10x^2 + a)\,(1, -2x)$.
For $a > 0$ the velocity never vanishes, and the slice is a smooth curve.
For $a < 0$ the velocity vanishes at $x = \pm\sqrt{-a/10}$, so the slice has two cusps, and between them the curve crosses itself.
This slice, with its two cusps and its crossing point, is the shape that gives the swallowtail its name.

## Explanations

### L1 human

The swallowtail is the third of Thom's seven elementary catastrophes, after the fold and the cusp.
Its potential depends on one state variable and three control knobs.
The set of knob settings where the potential has a degenerate critical point is a surface in space.
Cut it by a plane with $a < 0$ and you see a curve with two sharp points and a loop, like the tail of a swallow.
Crossing this surface changes the number of critical points of the potential by two.
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<document source="https://tamnd.github.io/math2/node/NGT5YCVAMR.md" tag="NGT5YCVAMR" kind="concept">
# Coercive functional

## Summary

A functional on a normed space is coercive if its values tend to $+\infty$ as the norm of the argument tends to infinity.
Then every minimising sequence is bounded, which is the compactness step of the [[direct method|TE379HBTY0]].
In reflexive spaces bounded sequences have weakly convergent subsequences, so coercivity and weak lower semicontinuity together give a minimiser.
The word is related to, but different from, a [[coercive bilinear form|86KRHS6X22]].

## Explanations

### L1 human

A coercive function grows without bound in every direction far from the origin.
So its low values all lie in a bounded region, and a search for the minimum cannot escape to infinity.
On the real line, $f(x) = x^2$ is coercive, while $f(x) = e^x$ is not, since it stays small as $x \to -\infty$.
The function $e^x$ indeed has no minimum: its infimum $0$ is approached only far away.
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# Krylov–Bogolyubov averaging

## Definition

Let $D \subseteq \R^n$ be open, let $T > 0$, and let $f \colon D \times \R \to \R^n$ be [[continuously differentiable|0VW90BBSJ4]] and [[periodic|0ATF6Q0633]] of period $T$ in its last variable.
For a small parameter $\varepsilon > 0$, consider the [[differential equation|0NT64JV96A]] in standard form
$$x' = \varepsilon f(x, t).$$
The *averaged function* is
$$\bar f(y) = \frac{1}{T} \int_0^T f(y, t)\, dt, \qquad y \in D,$$
and the *averaged equation* is $y' = \varepsilon \bar f(y)$.
*Krylov–Bogolyubov averaging* is the method of approximating the solutions of the equation in standard form by the solutions of the averaged equation.

## Remarks

The first averaging theorem justifies the method.
Let $x$ and $y$ solve the two equations with $x(0) = y(0)$, and let $L > 0$ be such that $y(t)$ stays in a compact subset of $D$ for $0 \le t \le L/\varepsilon$.
Then there is a constant $C$, independent of $\varepsilon$, with $\norm{x(t) - y(t)} \le C \varepsilon$ for $0 \le t \le L/\varepsilon$ and all small $\varepsilon$.

Example: $f(x, t) = x \sin^2 t$ on $\R$ with $T = \pi$.
Then $\bar f(y) = \frac{1}{\pi} \int_0^\pi y \sin^2 t\, dt = \frac{1}{2} y$.
The solutions are $x(t) = x_0 e^{\varepsilon (t/2 - \sin(2t)/4)}$ and $y(t) = x_0 e^{\varepsilon t/2}$.
Their ratio is $e^{-\varepsilon \sin(2t)/4} = 1 + O(\varepsilon)$, so the difference is $O(\varepsilon)$ on intervals of length of order $1/\varepsilon$.

Non-example: $x' = \varepsilon x t$ is not periodic in $t$, so it is not in the standard form above, and there is no period over which to average.

Weakly nonlinear oscillators $u'' + u = \varepsilon h(u, u')$ are put in standard form by writing $u = r \cos(t + \phi)$, $u' = -r \sin(t + \phi)$ and using $(r, \phi)$ as the new unknowns.
</document>

<document source="https://tamnd.github.io/math2/node/NH25G6GC07.md" tag="NH25G6GC07" kind="concept">
# Well-quasi-ordering

## Summary

A well-quasi-ordering is a [[preorder|0VABYWJWH9]] in which every infinite sequence $x_1, x_2, \ldots$ has indices $i < j$ with $x_i \le x_j$.
Equivalently, it is well-founded and has no infinite [[antichain|03HF498P1J]].
Well-quasi-orders give finiteness results: every upward closed set is the upward closure of finitely many of its elements.
Basic examples are $\N^k$ with the product order ([[Dickson's lemma|NQD0Q9S9M7]]), finite words under subword embedding ([[Higman's lemma|GHK66XE1S8]]) and finite trees under embedding ([[Kruskal's tree theorem|XVHKCX4F5Y]]).

## Explanations

### L1 human

In a well-quasi-order, any endless list of elements must contain an element that is below a later element.
You cannot keep writing down new elements that avoid all earlier ones.
For pairs of natural numbers ordered coordinatewise, the list $(5, 0), (4, 1), (3, 2), (2, 3), (1, 4), (0, 5)$ has no pair below a later one, but no such list can go on forever.
The positive integers under divisibility fail this: the primes $2, 3, 5, 7, \ldots$ never divide one another.
</document>

<document source="https://tamnd.github.io/math2/node/NH59FNJQG1.md" tag="NH59FNJQG1" kind="definition">
# Radial basis function interpolation

## Definition

Let $d \ge 1$, let $x_1, \dots, x_N \in \R^d$ be pairwise distinct, let $y_1, \dots, y_N \in \R$, and let $\varphi \colon [0, \infty) \to \R$ define a [[radial basis function|ZBSXSTQM2J]].
Let $A$ be the $N \times N$ matrix with entries $A_{ij} = \varphi(\norm{x_i - x_j})$.
If $A$ is invertible, the *radial basis function interpolant* of the data is
$$s(x) = \sum_{j=1}^{N} \lambda_j \, \varphi(\norm{x - x_j}),$$
where $\lambda = (\lambda_1, \dots, \lambda_N)$ is the solution of the [[linear system|0GT4WWGWH2]] $A \lambda = y$.

## Remarks

By construction $s(x_i) = (A\lambda)_i = y_i$, so $s$ is a [[multivariate interpolant|BM703GV34G]] of the data.
For the Gaussian $\varphi(r) = e^{-(\varepsilon r)^2}$ the matrix $A$ is a [[positive definite matrix|0KX66CZMBB]], hence invertible, for all distinct points in every dimension.
For conditionally positive definite functions such as the thin plate spline, a low-degree polynomial is added to $s$, with matching side conditions on $\lambda$, to get a uniquely solvable system.

Example: $d = 1$, $\varphi(r) = r$, nodes $0, 1$, values $y_1 = 0$, $y_2 = 1$.
Then $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $\lambda = (1, 0)$, so $s(x) = \abs{x}$, and indeed $s(0) = 0$, $s(1) = 1$.

Non-example: take $\varphi(r) = r^2$ on $\R$ and the four nodes $0, 1, 2, 3$.
Each translate $(x - x_j)^2$ lies in the space of polynomials of degree at most $2$, which has dimension $3$.
So the four columns of $A$, the values of these translates at the nodes, are linearly dependent, and $A$ is not invertible.
Then the interpolant is not defined.
</document>

<document source="https://tamnd.github.io/math2/node/NH5HWBPWS9.md" tag="NH5HWBPWS9" kind="concept">
# Pfaffian

## Summary

The Pfaffian is a polynomial in the entries of a [[skew-symmetric matrix|0PBY4E9GQF]] of even size whose square is the [[determinant|00KDZRW4C0]].
It is a sum over the [[perfect matchings|0NHK9MFFW9]] of the index set, with signs.
It appears in the study of alternating forms and symplectic groups, and in the counting of perfect matchings of planar graphs.

## Explanations

### L1 human

The determinant of a skew-symmetric matrix of even size is always a perfect square of a polynomial in its entries.
The Pfaffian is that square root, with a fixed choice of sign.
Each term of the Pfaffian pairs up the indices $1, \dots, 2n$ and multiplies the matching entries.
For $A = \begin{pmatrix} 0 & a \\ -a & 0 \end{pmatrix}$, the Pfaffian is $a$, and $\det A = a^2$.
</document>

<document source="https://tamnd.github.io/math2/node/NH739XXVXK.md" tag="NH739XXVXK" kind="concept">
# Partition of unity

## Summary

A partition of unity is a family of nonnegative functions on a space that add up to $1$ at every point.
Usually each function is required to live inside a chosen open set.
It is the standard tool for gluing local constructions into global ones.
On a smooth manifold one builds an object in each coordinate chart, multiplies it by the matching function, and adds the results.
Smooth partitions of unity exist on every smooth manifold, which is why Riemannian metrics and many other structures exist on all of them.

## Explanations

### L1 human

Imagine several lamps lighting a long hall, each lamp covering only part of it.
You want the total light to be exactly $1$ everywhere.
So you dim each lamp near the edges of its region, in such a way that the overlaps add up correctly.
The brightness profiles of the lamps form a partition of unity.
On the line, two smooth bumps $\rho_1$ and $\rho_2 = 1 - \rho_1$, where $\rho_1$ is $1$ far to the left and $0$ far to the right, give a simple example.
</document>

<document source="https://tamnd.github.io/math2/node/NH7J4YQMQC.md" tag="NH7J4YQMQC" kind="statement">
# Liouville's theorem on conformal mappings

## Statement

Let $n \ge 3$, let $U \subseteq \R^n$ be a connected open set, and let $f \colon U \to \R^n$ be a smooth [[conformal map|0YYT36R4YD]].
This means that at each $x \in U$ the derivative is $Df(x) = \lambda(x) A(x)$ with $\lambda(x) > 0$ and $A(x)$ an orthogonal matrix.
Then $f$ is the restriction to $U$ of a Möbius transformation of $\R^n \cup \set{\infty}$.
That is, $f$ is a composition of translations, orthogonal linear maps, dilations $x \mapsto c x$ with $c > 0$, and at most one [[inversion|YXST7NGMTC]] $x \mapsto x_0 + r^2 (x - x_0) / \abs{x - x_0}^2$ in a sphere.

## Remarks

Proof idea: the condition $Df^T Df = \lambda^2 I$ is a system of partial differential equations for $f$.
Differentiating it and using $n \ge 3$ forces the function $1/\lambda$ to be either constant or of the form $a \abs{x - x_0}^2$ with $a > 0$.
In the first case $f$ is a similarity; in the second, composing with an inversion centred at $x_0$ reduces to the first case.
The hypothesis $n \ge 3$ is necessary: in $\R^2 = \C$, every holomorphic function with nonzero derivative is conformal, for example $z \mapsto e^z$, and it is not a Möbius transformation since it is not injective.
The smoothness hypothesis can be weakened a great deal.
By results of Gehring and of Reshetnyak, a map of Sobolev class $W^{1,n}_{\mathrm{loc}}$ whose derivative is conformal almost everywhere is constant or a restricted Möbius transformation.

## Explanations

### L1 human

In the plane there are very many maps that keep angles, such as all the maps used by complex analysis.
In space of three or more dimensions, Liouville's theorem says there are very few.
Every angle-preserving map of a region of space is built from moves, rotations, scalings and reflections in spheres.
So in three dimensions there is no freedom to make conformal maps of arbitrary shapes, unlike in the plane.

### L2 human

The theorem explains why conformal geometry in dimension $n \ge 3$ is rigid: the conformal maps form the finite-dimensional group of Möbius transformations, of dimension $(n+1)(n+2)/2$.
In dimension $2$, by contrast, the Riemann mapping theorem gives a conformal map between any two simply connected proper open subsets of the plane.
The result is the local form of the fact that the conformal group of the round sphere $S^n$, $n \ge 3$, is the Möbius group.
It is a starting point for the theory of quasiconformal maps in higher dimensions.
</document>

<document source="https://tamnd.github.io/math2/node/NH7S9CHHBN.md" tag="NH7S9CHHBN" kind="definition">
# Finite geometry

## Definition

A *finite geometry* is an [[incidence structure|FGKNAFNHA3]] $(P, L, I)$ with a finite set $P$ of points that satisfies the axioms of a given type of geometry.
The usual types are [[projective planes|DANFNN8C4Y]], [[affine planes|EC7XT2ZM5S]], and projective and affine spaces of higher dimension.

## Remarks

Example: the [[projective space|ZT9A6JK8QF]] $PG(2, 2)$ over the field with two elements, the [[Fano plane|QQ8GW1ASZM]].
It has $7$ points and $7$ lines.
Any two points lie on exactly one line, and any two lines meet in exactly one point.
The points $[1 : 0 : 0]$, $[0 : 1 : 0]$, $[0 : 0 : 1]$, $[1 : 1 : 1]$ have no three on a line, so it is a finite projective plane.

Non-example: the real projective plane satisfies the axioms of a projective plane, but it has infinitely many points, so it is not a finite geometry.

The term also names the field of mathematics that studies these structures.
Finite projective spaces of dimension at least $3$ all come from finite fields, by the [[Veblen–Young theorem|NPNNXWMVCC]] and Wedderburn's little theorem; finite planes need not.
</document>

<document source="https://tamnd.github.io/math2/node/NHB3ZMN2BN.md" tag="NHB3ZMN2BN" kind="definition">
# Essential spectrum

## Definition

Let $X$ be a complex Banach space with identity map $I$, and let $T \colon X \to X$ be a bounded operator.
The *essential spectrum* of $T$ is the set
$$\sigma_{ess}(T) = \set{\lambda \in \C : T - \lambda I \text{ is not a Fredholm operator}}.$$
Here a bounded operator is [[Fredholm|8HWGKWD6ZM]] when its kernel is finite-dimensional and its image is closed and of finite codimension.

## Remarks

Since every invertible operator is Fredholm, $\sigma_{ess}(T)$ is a subset of the [[spectrum|BMKATJ802A]] $\sigma(T)$; it is compact.
It does not change when a [[compact operator|M2NEP1K5AE]] is added to $T$.
On a Hilbert space, it is the spectrum of the image of $T$ in the quotient algebra of bounded operators by compact operators, by Atkinson's theorem.

Example: if $X$ is infinite-dimensional, then $1 \in \sigma_{ess}(I)$, since $I - I = 0$ has an infinite-dimensional kernel.

Non-example: for $D(x_n) = (x_n / n)$ on $\ell^2$, $1 \in \sigma(D)$ but $1 \notin \sigma_{ess}(D)$.
The kernel of $D - I$ is spanned by $e_1$, and its image is the closed subspace $\set{x : x_1 = 0}$ of codimension $1$, so $D - I$ is Fredholm.
In fact $\sigma_{ess}(D) = \set{0}$.

In Banach spaces several non-equivalent variants of the essential spectrum are in use.
For a self-adjoint operator on a Hilbert space all the usual variants agree.
Then $\lambda \in \sigma_{ess}(T)$ if and only if $\lambda$ is in $\sigma(T)$ and is not an isolated eigenvalue of finite multiplicity.
</document>

<document source="https://tamnd.github.io/math2/node/NHERCK7C43.md" tag="NHERCK7C43" kind="example">
# Poincaré homology sphere

## Example

Identify the $3$-sphere $S^3$ with the group [[$SU(2)$|E5P8AG6CK7]] of unit quaternions.
The double cover $SU(2) \to SO(3)$ maps onto the rotation group of a regular icosahedron, a group of order $60$ isomorphic to $A_5$.
Its preimage $I^*$ in $SU(2)$ is the binary icosahedral group, of order $120$.
The Poincaré homology sphere is the quotient $P = S^3 / I^*$ of $S^3$ by the action of $I^*$ by left multiplication.

This action is free, so $P$ is a closed orientable [[3-manifold|R20TB2KEQQ]] with [[fundamental group|S1XK3HXHFX]] $I^*$.
The group $I^*$ is perfect, so $H_1(P; \Z)$, the abelianization of $I^*$, is $0$.
By [[Poincaré duality|9VY90HJVZX]], $H_2(P; \Z) \cong H^1(P; \Z) = \Hom(H_1(P; \Z), \Z) = 0$, and $H_3(P; \Z) = \Z$.
So $P$ has the homology of $S^3$ and is a [[homology sphere|HK8NS02NNE]].
It is not homeomorphic to $S^3$, because its fundamental group is not trivial.

## Remarks

The same manifold is obtained from a solid regular dodecahedron by gluing each face to the opposite face after a turn of $1/10$ of a full turn.
It is also the link of the singularity of $x^2 + y^3 + z^5 = 0$ in $\C^3$, often written $\Sigma(2, 3, 5)$.
Poincaré found it while testing whether homology characterises the $3$-sphere, and this example led him to the [[Poincaré conjecture|3R064KFF68]], which uses the fundamental group instead.

## Explanations

### L1 human

The Poincaré homology sphere looks like the $3$-sphere to homology: it has no holes of dimension $1$ or $2$.
Still, it contains loops that cannot be shrunk to a point.
It shows that counting holes with homology is not enough to recognise the sphere.
</document>

<document source="https://tamnd.github.io/math2/node/NHMJQBS04M.md" tag="NHMJQBS04M" kind="concept">
# Gosper's algorithm

## Summary

Gosper's algorithm decides, for a hypergeometric term $t_k$, whether its partial sums $\sum_{j < k} t_j$ are a hypergeometric term plus a constant.
When they are, it finds that term.
This is the discrete analogue of deciding whether an integral can be done in closed form.
It reduces the question to finding a polynomial solution of a linear recurrence with polynomial coefficients.
It is a building block of [[Zeilberger's algorithm|VA7MH05Z3B]] and of the search for [[WZ pairs|WC0QNABPZ4]].

## Explanations

### L1 human

Some sums have a neat closed form, such as $\sum_{j=0}^{k-1} j\cdot j! = k! - 1$.
Others, such as the partial sums $1 + \frac12 + \cdots + \frac1k$ of the harmonic series, have none of the same kind.
Gosper's algorithm looks for an answer $T_k$ that is the term $t_k$ times a rational function of $k$, with $T_{k+1} - T_k = t_k$.
It either finds such a $T_k$ or proves that none exists.
</document>

<document source="https://tamnd.github.io/math2/node/NHR4CYEQCE.md" tag="NHR4CYEQCE" kind="example">
# Variety of lattices

## Example

Take the signature with two binary operations $\vee$ and $\wedge$.
The *variety of lattices* is the class of algebras $(L, \vee, \wedge)$ with the [[identities|PKD440G2XW]]
$$x \vee y = y \vee x, \qquad (x \vee y) \vee z = x \vee (y \vee z), \qquad x \vee (x \wedge y) = x,$$
together with the same three identities with $\vee$ and $\wedge$ exchanged.
These algebras are exactly the [[lattices|WH4SFAACAS]].
The order is recovered by $x \le y$ if and only if $x \vee y = y$.
In this order, $x \vee y$ and $x \wedge y$ are the join and the meet.
The class is a [[variety|AJNZJD8442]], since it is defined by identities.

## Remarks

The idempotent laws $x \vee x = x$ and $x \wedge x = x$ follow from the absorption laws: $x \vee x = x \vee (x \wedge (x \vee x)) = x$.
The [[distributive lattices|0HM0HQCK6N]] and the [[modular lattices|QX7WZ92QAK]] form subvarieties, each defined by one more identity.
The class of complete lattices is not a variety in this signature.
For example, the rational numbers in $[0, 1]$ form a sublattice of the complete lattice $[0, 1]$, but this sublattice is not complete.

## Explanations

### L1 human

A lattice is usually defined as an ordered set in which any two elements have a least upper bound and a greatest lower bound.
The same structures can be described by a short list of equations for the two operations join and meet.
So lattices form a variety, and the tools of universal algebra apply to them.
</document>

<document source="https://tamnd.github.io/math2/node/NHRWCQQKTH.md" tag="NHRWCQQKTH" kind="concept">
# Flag variety

## Summary

The flag variety of a vector space $V$ parametrizes chains of subspaces $V_1 \subset V_2 \subset \cdots$ of prescribed dimensions.
It is a smooth projective [[homogeneous space|85TWR6E75P]] for $\mathrm{GL}(V)$; [[projective spaces|ZT9A6JK8QF]] and [[Grassmannians|CYPA882RB9]] are special cases.
In general, a flag variety is a quotient $G / P$ of a reductive group by a parabolic subgroup.
Flag varieties are where representation theory, combinatorics and geometry meet, through [[Schubert varieties|Z014Q8KPDD]] and the Borel–Weil theorem.

## Explanations

### L1 human

A flag in three-dimensional space is a line through the origin together with a plane through the origin that contains the line.
Think of a flagpole and a flag attached to it.
The set of all such flags is a smooth, compact space of dimension $3$, the full flag variety of $\C^3$.
In general a flag is a nested chain of subspaces, and the flag variety is the space of all of them.
</document>

<document source="https://tamnd.github.io/math2/node/NHS23SW780.md" tag="NHS23SW780" kind="definition">
# Heegner point

## Definition

Let $N \ge 1$ be an integer.
Let $X_0(N)$ be the modular curve of level $N$.
Its non-cuspidal points over $\C$ correspond to isomorphism classes of pairs $(E, C)$.
Here $E$ is an [[elliptic curve|427N6S6QQW]] over $\C$ and $C \subset E$ is a cyclic subgroup of order $N$.
Let $K$ be an imaginary [[quadratic field|7SV4WA92SM]] with ring of integers $\mathcal{O}_K$.
A *Heegner point* of $X_0(N)$ for $K$ is a point given by a pair $(E, C)$ such that $\End(E) \cong \End(E/C) \cong \mathcal{O}_K$.
So both $E$ and $E/C$ have [[complex multiplication|D3HEJTVY7G]] by $\mathcal{O}_K$.

Let $E$ be an elliptic curve over $\Q$ of conductor $N$ with a modular parametrisation $\varphi \colon X_0(N) \to E$.
The images $\varphi(x)$ of Heegner points $x$ are the Heegner points of $E$.

## Remarks

Heegner points for $K$ exist exactly when $\mathcal{O}_K$ has an ideal $\mathfrak{n}$ with $\mathcal{O}_K/\mathfrak{n} \cong \Z/N\Z$; then $(\C/\mathcal{O}_K, \mathfrak{n}^{-1}/\mathcal{O}_K)$ is one.
This holds, for example, under the Heegner hypothesis that every prime dividing $N$ splits in $K$.
By the theory of complex multiplication, Heegner points are defined over the [[Hilbert class field|WWCJ7CR2R3]] $H$ of $K$, and the trace $y_K = \sum_{\sigma \in \Gal(H/K)} \varphi(x)^\sigma$ lies in $E(K)$.

Example: for $N = 1$, $X_0(1)$ is the $j$-line and the condition is only that $E$ has CM by $\mathcal{O}_K$.
For $K = \Q(i)$, the curve $\C/\Z[i]$ has endomorphism ring $\Z[i] = \mathcal{O}_K$, so the point $j = 1728$ is a Heegner point for $K$.

Non-example: the curve $\C/(\Z + 2i\Z)$ has endomorphism ring $\Z[2i]$, an order of conductor $2$ in $\Q(i)$.
This is not the maximal order $\Z[i]$, so its point $j(2i)$ of $X_0(1)$ is not a Heegner point for $\Q(i)$ in this sense.

Some authors also allow pairs with CM by a non-maximal order of $K$, and speak of Heegner points of conductor $c$.
</document>

<document source="https://tamnd.github.io/math2/node/NHVQ6G3Z85.md" tag="NHVQ6G3Z85" kind="definition">
# Regular polytope

## Definition

Let $P \subseteq \R^d$ be a [[convex polytope|300K6CB1KH]] of dimension $d$.
A *flag* of $P$ is a sequence $(F_0, F_1, \dots, F_{d-1})$ of [[faces|KY7SB854VH]] of $P$ with $F_0 \subset F_1 \subset \cdots \subset F_{d-1}$ and $\dim F_i = i$ for each $i$.
The [[symmetry group|0ZG3Z5JZ85]] of $P$ is the group of isometries of $\R^d$ that map $P$ onto itself.
It acts on the set of flags of $P$.
The polytope $P$ is *regular* if this action is [[transitive|02GK6HZYBT]].
That is, for any two flags $\Phi$ and $\Psi$ of $P$ there is a symmetry $g$ of $P$ with $g(\Phi) = \Psi$.

## Remarks

Example: the cube $[0,1]^3$ has $8$ vertices, each vertex lies on $3$ edges, and each edge lies on $2$ square faces, so it has $8 \cdot 3 \cdot 2 = 48$ flags.
Its symmetry group has order $48$.
It can move any vertex to any vertex and then permute the three edges at a vertex freely, so it acts transitively on flags.
So the cube is regular.

Non-example: the rectangle $[0,2] \times [0,1]$ has $8$ flags but only $4$ symmetries.
No symmetry maps a flag whose edge has length $2$ to a flag whose edge has length $1$, so the action is not transitive.

For a convex polytope, a symmetry that fixes one flag is the identity, so the action of the symmetry group on the flags of a regular polytope is simply transitive.
In dimension $2$ the regular polytopes are the [[regular polygons|01KJJ9A400]], and in dimension $3$ they are the [[Platonic solids|017E5PRS1Q]].
The same flag condition defines regular star polytopes, which are not convex.
It also defines regular abstract polytopes, where the symmetry group is replaced by the automorphism group of a partially ordered set.
</document>

<document source="https://tamnd.github.io/math2/node/NHY3HE7RCN.md" tag="NHY3HE7RCN" kind="definition">
# Surgery theory

## Definition

Let $M$ be a smooth manifold of dimension $n$, and let $p, q \ge 0$ be integers with $p + q = n$.
Let $\varphi \colon S^p \times D^q \to M$ be a smooth embedding, where $S^p$ is the $p$-[[sphere|T62369RFH6]] and $D^q$ the closed $q$-disc.
The *surgery* on $M$ along $\varphi$ is the following manifold:
$$M' = \bigl(M \setminus \varphi(S^p \times \operatorname{int} D^q)\bigr) \cup_{\varphi} \bigl(D^{p+1} \times S^{q-1}\bigr).$$
Here the two pieces are glued along their common boundary $S^p \times S^{q-1}$ by $\varphi$, and corners are smoothed.
The number $p$ is the index of the surgery.
*Surgery theory* is the study of manifolds through such operations.
Its central problem is to turn a degree one normal map into a homotopy equivalence by a sequence of surgeries.

## Remarks

The *trace* of the surgery is $W = (M \times [0, 1]) \cup_{\varphi \times \set{1}} (D^{p+1} \times D^q)$, a $(p+1)$-handle attached to $M \times [0,1]$.
It is a [[cobordism|T8BVGPPSZP]] from $M$ to $M'$, and every compact cobordism is obtained by a finite sequence of such traces.
Example: let $M = S^1 \times S^1$ and $\varphi$ an embedding of $S^1 \times D^1$ as an annulus around the meridian circle.
Removing the open annulus leaves a cylinder $S^1 \times [0,1]$, and gluing in two discs $D^2 \times S^0$ caps it to $S^2$.
So surgery of index $1$ on a torus along a meridian gives the sphere.
Non-example: let $C$ be the core circle of a Möbius band inside $\R P^2$.
Every tubular neighbourhood of $C$ is a Möbius band and not an annulus, so $C$ has no neighbourhood of the form $S^1 \times D^1$.
So there is no embedding $\varphi$ as in the definition with core $C$, and no surgery along $C$ is defined.

## Explanations

### L1 human

To do surgery, find a sphere with a product neighbourhood inside a manifold and cut that neighbourhood out.
Then fill the hole with the other natural piece that has the same boundary.
Surgery along a non-separating circle in a surface removes a handle; surgery along two points adds one.
The operation changes the topology in a controlled way, and the trace records it as a cobordism.
</document>

<document source="https://tamnd.github.io/math2/node/NJ1FJSNK6A.md" tag="NJ1FJSNK6A" kind="concept">
# Lyapunov exponent

## Summary

A Lyapunov exponent measures the exponential rate at which a solution of a differential equation grows or decays as time goes to infinity.
For a linear system it is the upper limit of $\frac{1}{t} \ln \norm{x(t)}$ along a nonzero solution $x$.
For a nonlinear system one applies it to the linearised equation along a trajectory, and it then measures how fast nearby trajectories separate.
A negative exponent signals contraction, related to [[Lyapunov stability|0F2SYCKDT2]], and a positive one signals sensitive dependence on initial conditions, a mark of chaos.
See the [[definition|QREYWGEH41]].

## Explanations

### L1 human

Watch how the size of a solution changes over a long time and ask for the best fitting exponential rate $e^{\lambda t}$.
That rate $\lambda$ is the Lyapunov exponent.
For the scalar equation $x' = a x$, the solution $x(t) = e^{a t} x(0)$ has exponent exactly $a$.
If two nearby starting points drift apart like $e^{\lambda t}$ with $\lambda > 0$, small errors in the start blow up quickly.
This is why weather is hard to forecast far ahead.
</document>

<document source="https://tamnd.github.io/math2/node/NJ1PDJ3S4R.md" tag="NJ1PDJ3S4R" kind="statement">
# Oseledets theorem

## Statement

Let $(X, \mathcal{B}, \mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \colon X \to X$ be an [[ergodic|AZRPHS478B]] [[measure-preserving transformation|556WBDAHNP]].
Let $A \colon X \to GL(d, \R)$ be a measurable map, and let $\norm{\cdot}$ be the [[operator norm|06EKW1AKXF]].
Assume that $\log^+ \norm{A}$ and $\log^+ \norm{A^{-1}}$ are integrable, where $\log^+ t = \max(\log t, 0)$.
For $n \ge 1$ let $A^{(n)}(x) = A(T^{n-1}x) \cdots A(Tx) \, A(x)$.
Then there are an integer $k$ with $1 \le k \le d$ and real numbers $\lambda_1 > \lambda_2 > \cdots > \lambda_k$ with the following property.
For [[almost every|RARTMA0ZEC]] $x \in X$ there is a chain of linear subspaces
$$\R^d = V_1(x) \supsetneq V_2(x) \supsetneq \cdots \supsetneq V_k(x) \supsetneq V_{k+1}(x) = \set{0}$$
such that, for each $i = 1, \dots, k$:

1. $A(x) V_i(x) = V_i(Tx)$.
2. For every $v \in V_i(x) \setminus V_{i+1}(x)$,
$$\lim_{n \to \infty} \frac{1}{n} \log \norm{A^{(n)}(x) v} = \lambda_i.$$
3. The dimension $\dim V_i(x) - \dim V_{i+1}(x)$ does not depend on $x$.

## Remarks

The numbers $\lambda_i$ are the [[Lyapunov exponents|NJ1FJSNK6A]] of the cocycle $A$, and the dimensions in clause 3 are their multiplicities.
The largest one is $\lambda_1 = \lim_{n} \frac{1}{n} \log \norm{A^{(n)}(x)}$ for almost every $x$, which already follows from [[Kingman's subadditive ergodic theorem|833XVEEZCJ]], since $\log \norm{A^{(n)}}$ is subadditive along orbits.
Applying the same argument to exterior powers gives the sums $\lambda_1 + \cdots + \lambda_j$ with multiplicity, and linear algebra then builds the subspaces.
Without ergodicity the theorem still holds, but $k$, the $\lambda_i$ and the multiplicities become $T$-invariant functions of $x$.
Example: if $A$ is a constant diagonal matrix $\mathrm{diag}(e^{a_1}, \dots, e^{a_d})$ with $a_1 > \cdots > a_d$, then $\lambda_i = a_i$ and $V_i$ is spanned by the last $d - i + 1$ coordinate vectors.
If $T$ is invertible, the filtration refines to a splitting $\R^d = E_1(x) \oplus \cdots \oplus E_k(x)$ into subspaces on which the growth rate is $\lambda_i$ both forward and backward in time.

## Explanations

### L1 human

Multiply many random matrices together, each chosen by a stationary random process.
The theorem says that the product grows at a few well-defined exponential rates, the Lyapunov exponents.
Each vector grows at one of these rates, and which rate it gets depends only on which of a nested family of subspaces it belongs to.
This is the law of large numbers for products of matrices.

### L2 human

The main use is in smooth ergodic theory: take $A(x) = Df(x)$, the derivative of a diffeomorphism $f$ preserving a measure.
Then the Lyapunov exponents measure how fast nearby orbits separate in each direction, and positive exponents signal chaos.
Pesin theory builds stable and unstable manifolds from the Oseledets subspaces, and Ruelle's inequality bounds the entropy by the sum of the positive exponents.
For $d = 1$ the theorem reduces to [[Birkhoff's ergodic theorem|J6NMBX3E55]] applied to $\log \abs{A}$.
</document>

<document source="https://tamnd.github.io/math2/node/NJ21PN5ZGJ.md" tag="NJ21PN5ZGJ" kind="concept">
# Connection on a principal bundle

## Summary

A connection on a [[principal bundle|8E37E74XVP]] with structure group $G$ is an [[Ehresmann connection|TD6VEP3SCE]] that is invariant under the action of $G$.
It is usually given by a connection form, a $1$-form on the total space with values in the Lie algebra of $G$.
A principal connection induces connections on all associated vector bundles, so it is the universal form of a connection.
In physics these connections are the gauge fields, and their curvature is the field strength.

## Explanations

### L1 human

A principal bundle has, above each point of the base, a copy of a symmetry group whose origin has not been fixed.
A connection tells how to move from one fibre to a nearby one, and it must respect the symmetry.
The connection form does this by reading off, from each tangent vector, how fast it moves along the group direction.
For the trivial bundle $M \times G$ there is a flat choice: moving in $M$ while staying at the same group element.
</document>

<document source="https://tamnd.github.io/math2/node/NJ3APQ8BHH.md" tag="NJ3APQ8BHH" kind="concept">
# Residue field

## Summary

The residue field of a [[local ring|MZ02BKPHJA]] $R$ with maximal ideal $\mathfrak m$ is the [[quotient ring|0076HWHGHQ]] $R/\mathfrak m$, which is a [[field|04C5ZBG1WT]].
More generally, the residue field of any [[commutative ring|0310REJJY8]] at a [[prime ideal|00MHHZ9SNE]] $\mathfrak p$ is the residue field of the [[localization|X6JD4AG973]] $R_{\mathfrak p}$.
It is the field of "values at a point", and many invariants of local rings, such as the [[cotangent space|Z1CGB483J6]], are vector spaces over it.

## Explanations

### L1 human

Reducing modulo the maximal ideal throws away everything except the "value at the point".
For the ring $\Z_{(p)}$ of fractions with denominators prime to $p$, reducing modulo $p$ gives the field $\Z/p\Z$ with $p$ elements.
For the power series ring $k[\![x]\!]$, it keeps only the constant term, so the residue field is $k$.
The [[definition|8ST7CTZ2DE]] gives the construction.
</document>

<document source="https://tamnd.github.io/math2/node/NJ787GZS1N.md" tag="NJ787GZS1N" kind="concept">
# Genus

## Summary

The genus of a closed orientable [[surface|VGJJVBZ1MQ]] is its number of handles.
It is the largest number of disjoint simple closed curves along which the surface can be cut without falling apart.
By the [[classification of surfaces|2SFM73YET0]], the genus determines a closed orientable surface up to homeomorphism, and it is related to the [[Euler characteristic|RNDDWDAFJ5]] by $\chi = 2 - 2g$.
The same number appears as the [[genus of an algebraic curve|6W8TDV9VVN]] and of a compact Riemann surface.

## Explanations

### L1 human

The genus counts holes in the sense of a doughnut, not of a cup.
A sphere has genus $0$, a doughnut has genus $1$, and a pretzel with two holes has genus $2$.
On a doughnut you can cut along one circle around the tube and still have one piece, a cylinder.
Any second disjoint cut then splits it, so the genus is $1$.
</document>

<document source="https://tamnd.github.io/math2/node/NJEPVG39EZ.md" tag="NJEPVG39EZ" kind="definition">
# Itô calculus

## Definition

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, P)$ be a filtered probability space with a [[filtration|HNK1Y62G7V]] that satisfies the usual conditions.
Let $(B_t)_{t \ge 0}$ be a [[Brownian motion|22735FGJV6]] for this filtration and let $T > 0$.

1. A *simple process* is a process $H_t = \sum_{i=0}^{n-1} \xi_i \mathbf{1}_{(t_i, t_{i+1}]}(t)$, where $0 = t_0 < t_1 < \cdots < t_n = T$ and each $\xi_i$ is a bounded $\mathcal{F}_{t_i}$-measurable random variable.
Its Itô integral is $\int_0^T H_t \, dB_t = \sum_{i=0}^{n-1} \xi_i (B_{t_{i+1}} - B_{t_i})$.
2. Let $H$ be a progressively measurable process with $E\left[\int_0^T H_t^2 \, dt\right] < \infty$.
There are simple processes $H^{(n)}$ with $E\left[\int_0^T (H^{(n)}_t - H_t)^2 \, dt\right] \to 0$.
The *Itô integral* $\int_0^T H_t \, dB_t$ is the limit in $L^2(P)$ of $\int_0^T H^{(n)}_t \, dB_t$.
The limit exists and does not depend on the choice of the $H^{(n)}$.

*Itô calculus* is the study of these integrals, of their extensions to integrators that are semimartingales, and of the processes they define.

## Remarks

Example: for $H = \mathbf{1}_{(0, T]}$, the integral is $B_T - B_0$.
For $H_t = B_t$, the left-point sums converge to $\int_0^T B_t \, dB_t = \tfrac12 (B_T^2 - T)$ when $B_0 = 0$.
Non-example: the process $H_t = B_T$ for all $t < T$ is not [[adapted|YRNFE016FF]], so clause 2 does not apply to it.
The limit in clause 2 exists because of the [[Itô isometry|5MDK7HSYRM]], which makes the map $H \mapsto \int H \, dB$ an isometry on simple processes.
The process $t \mapsto \int_0^t H_s \, dB_s$ has a continuous version that is a square-integrable martingale.
The chain rule of Itô calculus is [[Itô's lemma|2J1RVQB0CM]].
</document>

<document source="https://tamnd.github.io/math2/node/NJP0AZ8R4K.md" tag="NJP0AZ8R4K" kind="problem">
# Minkowski problem

## Problem

Let $n \ge 2$, and let $S^{n-1}$ be the unit [[sphere|T62369RFH6]] of $\R^n$.
Let $\mathcal{H}^{n-1}$ be the $(n-1)$-dimensional [[Hausdorff measure|Q1TGZK8RN6]].
Let $K \subseteq \R^n$ be a [[convex body|437FSG3419]] and $\omega \subseteq S^{n-1}$ a Borel set.
Let $N_K(\omega)$ be the set of boundary points of $K$ with an outer unit normal in $\omega$.
The *surface area measure* of $K$ is the Borel measure $S_K(\omega) = \mathcal{H}^{n-1}(N_K(\omega))$ on $S^{n-1}$.
Let $\mu$ be a finite Borel measure on $S^{n-1}$.
Find a convex body $K$ with $S_K = \mu$, and decide when such a body exists and whether it is unique.

## Remarks

The problem is solved.
A convex body $K$ with $S_K = \mu$ exists if and only if $\mu$ is not concentrated on a great subsphere of $S^{n-1}$ and $\int_{S^{n-1}} u \, d\mu(u) = 0$.
The body is then unique up to translation.
Minkowski solved the case of polytopes.
Let $u_1, \ldots, u_m$ be distinct unit vectors spanning $\R^n$, and let $a_1, \ldots, a_m > 0$ with $\sum_i a_i u_i = 0$.
Then there is a polytope, unique up to translation, whose facets have outer normals $u_i$ and areas $a_i$.
The general case is due to Minkowski, Alexandrov, and Fenchel and Jessen.
When $\mu$ has a positive density $f$, the problem asks for a body whose [[Gaussian curvature|SQVQ5DQT6W]] at the point with outer normal $u$ is $1 / f(u)$.
This is a [[Monge–Ampère equation|14AKAC0S0T]] on the sphere.
The regularity of its solutions was studied by Pogorelov and by Cheng and Yau.
Example: the uniform measure $\mu = c \, \sigma$ on $S^{n-1}$, with $\sigma$ the surface measure and $c > 0$, is the surface area measure of a ball of radius $c^{1/(n-1)}$.

## Explanations

### L1 human

Suppose you know, for each direction, how much of the surface of a convex shape faces that direction.
Can you rebuild the shape?
Minkowski's answer is yes, up to sliding it around, provided the data are balanced: the outward pulls of all the pieces of surface must cancel.
For a polyhedron, the data are the directions and areas of its faces.
</document>

<document source="https://tamnd.github.io/math2/node/NJQ196M1WR.md" tag="NJQ196M1WR" kind="concept">
# Continuum

## Summary

A continuum is a nonempty [[compact|0TVJZYB6SH]] [[connected|0V032VWTHD]] metric space.
Continua are the objects of continuum theory, a part of topology that studies curves and their wild relatives.
Arcs, circles, discs and spheres are continua, and so are strange sets such as the [[pseudo-arc|VGCTDCGS6V]] and the [[Warsaw circle|17VG475BRV]].

## Explanations

### L1 human

A continuum is a space in one piece that is also closed and bounded, when it sits in a Euclidean space.
The closed interval $[0, 1]$, a circle and a filled square are continua.
Two disjoint closed intervals are not a continuum, since they are not in one piece.
The open interval $(0, 1)$ is not one either, since it is not compact.
Continua can be very strange: some cannot be split into two smaller continua at all.
</document>

<document source="https://tamnd.github.io/math2/node/NJQMWVK3QK.md" tag="NJQMWVK3QK" kind="concept">
# Reduced homology

## Summary

Reduced homology is a small change of [[singular homology|M0Q5CFJ7ZH]] that makes the homology of a point vanish in every degree.
It differs from ordinary homology only in degree $0$, where one copy of $\Z$ is removed.
Many statements, such as the homology of spheres, the suspension isomorphism and Alexander duality, take their cleanest form in reduced homology.

## Explanations

### L1 human

Ordinary homology always sees at least one copy of $\Z$ in degree $0$ for a nonempty space, because the space has at least one piece.
Reduced homology throws this copy away, so that it only records how far the space is from being a single point.
For a space made of three separate points, $H_0$ is $\Z^3$, while the reduced group $\tilde H_0$ is $\Z^2$.
For the sphere $S^n$, the reduced homology is $\Z$ in degree $n$ and zero in all other degrees, with no special case for $n = 0$.
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<document source="https://tamnd.github.io/math2/node/NJQRG3DVPP.md" tag="NJQRG3DVPP" kind="statement">
# Tonelli's existence theorem

## Statement

Let $\Omega \subseteq \R^n$ be a bounded open set with Lipschitz boundary, and let $1 < p < \infty$.
Let $L \colon \overline{\Omega} \times \R \times \R^n \to \R$, written $L(x, u, \xi)$, be continuous.
Assume:

1. $L$ is a [[convex integrand|X0Q69ZS09T]]: $\xi \mapsto L(x, u, \xi)$ is convex for all $(x, u)$;
2. there are constants $a > 0$ and $b \in \R$ with $L(x, u, \xi) \ge a \abs{\xi}^p + b$ for all $(x, u, \xi)$.

Let $u_0$ be in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\Omega)$ and let $J(u) = \int_\Omega L(x, u, \nabla u) \, dx$, an [[integral functional|JRQZ803Q5Y]].
If $J(u_0) < +\infty$, then $J$ attains its minimum on $u_0 + W^{1,p}_0(\Omega)$.

## Remarks

The proof is the [[direct method|TE379HBTY0]].
The growth bound and the Poincaré inequality make $J$ [[coercive|NGT5YCVAMR]] on $u_0 + W^{1,p}_0(\Omega)$, so a minimising sequence is bounded in $W^{1,p}(\Omega)$.
Since $1 < p < \infty$, a subsequence converges weakly in $W^{1,p}(\Omega)$, and strongly in $L^p(\Omega)$ by the Rellich–Kondrachov theorem.
Convexity in $\xi$ makes $J$ [[weakly lower semicontinuous|H6GTJRG26G]], so the weak limit is a minimiser.

Example: for $L = \frac{1}{2} \abs{\xi}^2 + \frac{1}{4} u^4$ and $p = 2$, the theorem gives a minimiser with boundary values $u_0$ whenever $J(u_0) < +\infty$; it is a weak solution of $-\Delta u + u^3 = 0$.

Each hypothesis matters.
For $L = (\xi^2 - 1)^2 + u^2$ with $n = 1$ and $u_0 = 0$, convexity fails and there is no minimiser.
For $L = \sqrt{1 + \abs{\xi}^2}$, which has growth $p = 1$, minimisers of the area of a graph need not exist in $W^{1,1}$ for some boundary data.

## Explanations

### L1 human

The theorem says that an energy that is convex in the slope and grows fast enough has a lowest state with given boundary values.
Fast growth keeps minimising sequences from running away.
Convexity prevents them from lowering the energy by oscillating.
Together they force a minimising sequence to settle on a true minimiser.

### L2 human

This is the model existence theorem of the calculus of variations, and it is the template for the direct method.
The minimiser is a priori only a Sobolev function; whether it is smooth is the separate question of regularity, raised in Hilbert's nineteenth problem.
For vector-valued $u$, convexity can be weakened to Morrey quasiconvexity, and in nonlinear elasticity to polyconvexity, as in Ball's existence theorem.
Without convexity one replaces the integrand by its convex envelope; this is relaxation, and it computes the limits of minimising sequences.
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<document source="https://tamnd.github.io/math2/node/NJY30DQG3H.md" tag="NJY30DQG3H" kind="definition">
# Ordinal sum of posets

## Definition

Let $(P, \le_P)$ and $(Q, \le_Q)$ be [[partially ordered sets|0QV2JS9RJ3]] with $P \cap Q = \varnothing$.
The *ordinal sum* $P \oplus Q$ is the set $P \cup Q$ with the relation $\le$ given by $x \le y$ if and only if one of the following holds:

1. $x, y \in P$ and $x \le_P y$;
2. $x, y \in Q$ and $x \le_Q y$;
3. $x \in P$ and $y \in Q$.

## Remarks

If $P$ and $Q$ are not disjoint, first replace them by disjoint copies.
Example: with $P = \set{a}$ and $Q = \set{b, c}$ an antichain, $P \oplus Q$ has $a < b$ and $a < c$ by clause 3, and $b$, $c$ stay incomparable.
Non-example: in $Q \oplus P$ the element $a$ is above $b$ and $c$, so $Q \oplus P$ differs from $P \oplus Q$; the ordinal sum is not commutative.
The ordinal sum is associative, and the ordinal sum of two total orders is a total order.
For well-orders, it gives the sum of [[ordinal numbers|0YSCSX5YZC]], so $\omega \oplus \set{x}$ has order type $\omega + 1$.
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<document source="https://tamnd.github.io/math2/node/NK3CZTVEJD.md" tag="NK3CZTVEJD" kind="concept">
# Boolean model

## Summary

The Boolean model is a random set in $\R^d$.
It is the union of independent random shapes, the grains, placed at the points of a [[Poisson point process|0MTNR28K60]], the germs.
It is the basic germ-grain model of [[stochastic geometry|2J9GB49FHA]].
Many of its mean values have closed formulas.
For example, a given point is covered with probability $1 - e^{-\lambda E[V_d(K)]}$, where $\lambda$ is the intensity and $E[V_d(K)]$ the mean grain volume.
It is used to model porous media, coverage of wireless networks and continuum [[percolation|BH7YPXCYDK]].

## Explanations

### L1 human

Throw points at random over the plane, with no point influencing any other.
Around each point draw a shape, such as a disc whose radius is also chosen at random.
The Boolean model is the union of all these shapes, a random pattern of overlapping blobs.
If all discs have radius $r$ and there are on average $\lambda$ centres per unit area, then each fixed point of the plane is uncovered with probability $e^{-\lambda \pi r^2}$.
With $\lambda \pi r^2 = 1$, about $63\%$ of the plane is covered.
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<document source="https://tamnd.github.io/math2/node/NK9ATH4WB6.md" tag="NK9ATH4WB6" kind="concept">
# Moduli of algebraic curves

## Summary

The moduli space $M_g$ is a geometric space whose points are the isomorphism classes of smooth projective [[algebraic curves|XJM2MT4S3Y]] of genus $g$.
Over $\C$ these are the compact [[Riemann surfaces|5K0N8QMV6Q]] of genus $g$, and $M_g$ is the quotient of [[Teichmüller space|ZZGC6BZNJ5]] by the [[mapping class group|JV5NYG913V]].
For $g \ge 2$, $M_g$ is an irreducible quasi-projective variety of dimension $3g - 3$, and it has a compactification $\overline{M}_g$ by stable curves due to Deligne and Mumford.
It is a central object of algebraic geometry, linking geometry, topology and number theory.

## Explanations

### L1 human

Every compact Riemann surface has a genus, the number of holes.
Surfaces of the same genus can still differ in shape, and the moduli space is the space of all these shapes.
For genus $0$ there is only one shape, the Riemann sphere, so $M_0$ is a point.
For genus $1$ the shapes are the tori $\C/\Lambda$, and the $j$-invariant identifies $M_1$ with the complex line.
For genus $g \ge 2$ one needs $3g - 3$ complex numbers to describe the shape, as Riemann already counted.
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# Alexander subbase theorem

## Statement

Let $X$ be a [[topological space|0CQHFPGJ45]], and let $\mathcal S$ be a [[subbasis|09S8DV34Y0]] of its topology.
Suppose that every [[cover|049PXYHW1T]] of $X$ by members of $\mathcal S$ has a finite subcover.
Then $X$ is [[compact|0TVJZYB6SH]].

## Remarks

Suppose $X$ is not compact.
By [[Zorn's lemma|04RXTQA5ZA]], there is an open cover $\mathcal C$ without finite subcover that is maximal for inclusion among such covers.
For $x \in U \in \mathcal C$, choose $S_1, \ldots, S_n \in \mathcal S$ with $x \in S_1 \cap \cdots \cap S_n \subseteq U$.
If no $S_i$ were in $\mathcal C$, maximality would give finite sets $\mathcal C_i \subseteq \mathcal C$ with $S_i \cup \bigcup \mathcal C_i = X$, and then $U$ with all $\mathcal C_i$ would be a finite subcover.
So $\mathcal C \cap \mathcal S$ covers $X$, and by hypothesis it has a finite subcover, a contradiction.

Example: the sets $[0, c)$ and $(c, 1]$ for $0 < c < 1$ form a subbasis of $[0, 1]$.
A cover by such sets contains some $[0, c)$ and some $(c', 1]$ with $c' < c$, by a supremum argument, so $[0, 1]$ is compact.
The theorem also gives a direct proof of [[Tychonoff's theorem|0JTRSJKCFB]].

The subbasis hypothesis is needed.
On $\R$, the family $\set{\R}$ has the finite subcover property, but it is not a subbasis of the usual topology, and $\R$ is not compact.

## Explanations

### L1 human

To show that a space is compact, one must handle every open cover.
The theorem says that it is enough to handle covers made of the simplest open sets, those of a chosen subbasis.
This cuts down the work a lot when the subbasis is easy to describe.

### L2 human

The product topology has a natural subbasis: the preimages of open sets under single projections.
With the theorem, compactness of a product of compact spaces reduces to covers by such sets, which is a short argument; this is one standard proof of Tychonoff's theorem.
The proof uses Zorn's lemma, and the theorem is equivalent to the ultrafilter lemma over the axioms of set theory without choice.
</document>

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# Ergodic hypothesis

## Summary

The ergodic hypothesis is an assumption of statistical mechanics about the long-run behaviour of a mechanical system.
It says that the system spends time in each region of its energy surface in proportion to the volume of that region.
In other words, time averages of an observable along a trajectory equal its average over the energy surface.
It justifies replacing hard time averages by phase-space averages, which are the averages that equilibrium statistical mechanics computes.
In mathematical terms it says that the [[Hamiltonian|0EDKGA2FAK]] flow restricted to an energy surface is [[ergodic|AZRPHS478B]] for the Liouville measure.

## Explanations

### L1 human

Think of a gas in a box: one cannot follow every molecule, but one can measure averages over a long time.
The ergodic hypothesis says that one long history of the system already visits all the states of the same energy, in a fair way.
So the average of a quantity over one long history is the same as its average over all possible states of that energy.
A simple model is a point moving around a circle at constant speed: the fraction of time it spends in an arc equals the length of the arc.
For real mechanical systems the hypothesis is hard to prove, and for many systems it is false.
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<document source="https://tamnd.github.io/math2/node/NKP5TPQ79X.md" tag="NKP5TPQ79X" kind="concept">
# Subobject classifier

## Summary

A subobject classifier in a [[category|A2DE66937D]] with finite limits is an object $\Omega$ with a morphism $\top \colon 1 \to \Omega$.
Every [[subobject|FS1FKAE89H]] of every object $X$ must be the pullback of $\top$ along exactly one morphism $X \to \Omega$.
In the category of sets, $\Omega = \set{0, 1}$ and the morphism $X \to \Omega$ of a subset is its [[characteristic function|0V5B6D6BQH]].
It is one of the defining features of a [[topos|6837RS7N82]], and $\Omega$ plays the role of the object of truth values.
In a sheaf topos, $\Omega$ need not have two elements, and its internal logic is in general [[intuitionistic|Q11N8X99FK]].

## Explanations

### L1 human

In sets, a subset $A \subseteq X$ is the same as a yes/no question on $X$: "is $x$ in $A$?".
A subobject classifier is an object $\Omega$ of answers that makes this work in another category.
For example, in sheaves on a space $X$, the answers over an open set $U$ are the open subsets of $U$.
The truth value of "$s = t$" for two sections is the open set where they agree.
So truth values come in many shades, not just true and false.
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<document source="https://tamnd.github.io/math2/node/NKTAN9XBZA.md" tag="NKTAN9XBZA" kind="statement">
# Douglas' lemma

## Statement

Let $H_1$, $H_2$ and $H$ be [[Hilbert spaces|HM2HHE3J51]], and let $A \colon H_1 \to H$ and $B \colon H_2 \to H$ be bounded linear operators.
The following are equivalent.

1. The range of $A$ is contained in the range of $B$.
2. There is $\lambda \ge 0$ with $AA^* \le \lambda^2 BB^*$, that is, $\norm{A^* x} \le \lambda \norm{B^* x}$ for all $x \in H$.
3. There is a bounded linear operator $C \colon H_1 \to H_2$ with $A = BC$.

Here $A^*$ and $B^*$ are the [[Hilbert space adjoints|6HSJBJ8TTQ]], and $\le$ is the order given by [[positive operators|77020CY5ZW]].
Moreover, if these hold, there is a unique $C$ with $A = BC$ whose range lies in the closure of the range of $B^*$.
It satisfies $\ker C = \ker A$ and $\norm{C}^2 = \inf \set{\mu \ge 0 : AA^* \le \mu BB^*}$.

## Remarks

Proof idea: (3) implies (1) at once, and (3) implies (2) because $AA^* = BCC^*B^* \le \norm{C}^2 BB^*$.
For (2) implies (3), the map $B^* x \mapsto A^* x$ is well defined and bounded by $\lambda$ on the range of $B^*$.
Extend it to the closure, set it to $0$ on the orthogonal complement, and let $C$ be the adjoint of the result.
For (1) implies (3), define $Cx$ as the unique $y \perp \ker B$ with $By = Ax$, and use the closed graph theorem to see that $C$ is bounded.

Example: on $\ell^2$ let $B$ be the diagonal operator with entries $1/n$ and $A$ the diagonal operator with entries $1/n^2$.
Then $A = BC$ with $C$ diagonal with entries $1/n$, and indeed $AA^* \le BB^*$ since $1/n^4 \le 1/n^2$.

## Explanations

### L1 human

The lemma relates three ways in which one operator can be smaller than another.
The range of $A$ fits inside the range of $B$ exactly when $A$ factors through $B$, and exactly when $AA^*$ is dominated by a multiple of $BB^*$.
It turns a question about ranges, which are often not closed, into an inequality between positive operators.

### L2 human

Douglas' lemma is a standard tool for factorization and for range inclusion questions.
For example, with $B = (AA^*)^{1/2}$, the [[square root|3JY5HNFRMF]] of $AA^*$, condition 2 holds in both directions, so $A$ and $(AA^*)^{1/2}$ have the same range.
It is used in dilation theory, in the theory of de Branges spaces, and in the study of operator equations $A = BX$.
</document>

<document source="https://tamnd.github.io/math2/node/NM1BCK3ZX9.md" tag="NM1BCK3ZX9" kind="concept">
# Absorbing set

## Summary

A subset of a real or complex vector space is absorbing if every vector lies in a large enough multiple of it.
Every neighbourhood of $0$ in a [[topological vector space|T32W553W95]] is absorbing.
The [[Minkowski functional|PAX9SHDHDW]] of an absorbing set is finite everywhere, so absorbing sets are the starting point for turning convex sets into [[seminorms|FP5GVCN8GV]].

## Explanations

### L1 human

An absorbing set reaches out a little in every direction from the origin.
Blow it up enough, and it swallows any given vector.
The open unit disc in $\R^2$ is absorbing: the point $(10, 0)$ lies in $11$ times the disc.
A line through the origin in $\R^2$ is not absorbing: no multiple of it contains a point off the line.
</document>

<document source="https://tamnd.github.io/math2/node/NM1KPS6E7Y.md" tag="NM1KPS6E7Y" kind="concept">
# Givens rotation

## Summary

A Givens rotation is an $n \times n$ [[orthogonal matrix|0W02QESPJQ]] that [[rotates|0HFREYRSZP]] the plane of two coordinate axes and leaves all other coordinates fixed.
With the angle chosen well, it turns one chosen entry of a vector or matrix into zero.
Because it touches only two rows, it is the tool of choice for zeroing single entries.
This is used in the [[QR decomposition|0ERWG4NXQ9]] of sparse or structured matrices and in the QR algorithm for eigenvalues.

## Explanations

### L1 human

Pick two coordinates, say the first and the third, and turn the vector inside the plane they span, keeping every other coordinate as it is.
Turning by a suitable angle makes one of the two coordinates zero, and moves all the length into the other one.
For example, rotating $(3, 4)$ clockwise by the angle whose cosine is $3/5$ gives $(5, 0)$.
Doing this entry by entry clears a matrix below its diagonal.
</document>

<document source="https://tamnd.github.io/math2/node/NM28RE8CHQ.md" tag="NM28RE8CHQ" kind="definition">
# Hermitian kernel

## Definition

Let $a < b$.
A *Hermitian kernel* on $[a, b]$ is a [[kernel|GSMBXT7FPK]] $K \colon [a, b] \times [a, b] \to \C$ such that
$$K(x, s) = \overline{K(s, x)} \quad \text{for all } x, s \in [a, b],$$
where the bar is the [[complex conjugate|0FTH2E9TAS]].

## Remarks

Example: $K(x, s) = e^{i (x - s)}$.
The check: $\overline{K(s, x)} = \overline{e^{i (s - x)}} = e^{-i (s - x)} = e^{i (x - s)} = K(x, s)$.

Non-example: $K(x, s) = i$, the constant kernel.
Here $\overline{K(s, x)} = -i \ne i$.

On the diagonal a Hermitian kernel is real, since $K(x, x) = \overline{K(x, x)}$.
A real-valued kernel is Hermitian exactly when it is symmetric.
For square-integrable kernels the condition is often required only for almost all $(x, s)$.
Then the integral operator $K$ on $L^2(a, b)$ satisfies $\ip{K u}{v} = \ip{u}{K v}$ for the inner product $\ip{u}{v} = \int_a^b u \bar v \, dx$.
</document>

<document source="https://tamnd.github.io/math2/node/NM8XCPS1Q3.md" tag="NM8XCPS1Q3" kind="concept">
# Lie subgroup

## Summary

A Lie subgroup of a [[Lie group|7JBRGCVJRB]] $G$ is a [[subgroup|0X1JXT47GA]] with its own Lie group structure.
The inclusion into $G$ must be a smooth [[immersion|6RW68JGBHC]].
Its topology may be finer than the subspace topology, so a Lie subgroup need not be closed or embedded.
The connected Lie subgroups of $G$ correspond exactly to the Lie subalgebras of the Lie algebra of $G$.

## Explanations

### L1 human

A Lie subgroup is a smaller Lie group sitting inside a bigger one.
It can be a nice closed piece, like the rotations about one axis inside all rotations of space.
It can also wind around in a complicated way.
In a torus, a line with irrational slope wraps around forever and comes close to every point, yet it is a Lie subgroup, a copy of $\R$.
Its own smooth structure is that of the line, not the one it would inherit as a subset of the torus.
</document>

<document source="https://tamnd.github.io/math2/node/NN6QM9JMKB.md" tag="NN6QM9JMKB" kind="definition">
# Complete theory

## Definition

Let $L$ be a signature and let $T$ be an $L$-[[theory|92SWXST1ZM]].
For an $L$-[[sentence|DKFGQTRYZE]] $\varphi$, write $T \models \varphi$ when every [[model|MVCH6N9SNW]] of $T$ satisfies $\varphi$.
The theory $T$ is complete when the following hold.

1. $T$ has a model.
2. For every $L$-sentence $\varphi$, $T \models \varphi$ or $T \models \neg\varphi$.

## Remarks

Example: for any $L$-structure $\mathcal{M}$, the theory $\mathrm{Th}(\mathcal{M})$ of all sentences true in $\mathcal{M}$ is complete.
It has the model $\mathcal{M}$, and for each sentence $\varphi$ either $\varphi$ or $\neg\varphi$ is true in $\mathcal{M}$, hence lies in $\mathrm{Th}(\mathcal{M})$.

Non-example: the theory of groups has a model but is not complete.
For $\varphi = \forall x \, \forall y \, (x \cdot y = y \cdot x)$, the group $\Z$ satisfies $\varphi$ and the symmetric group $S_3$ satisfies $\neg\varphi$.

A satisfiable theory $T$ is complete if and only if all its models are [[elementarily equivalent|P3B1V182N4]].
By the completeness theorem, $T \models \varphi$ can be replaced by $T \vdash \varphi$, provability from $T$.
Some authors do not require clause 1.

## Explanations

### L1 human

A complete theory answers every question that its language can ask.
Each structure has a complete theory, its full list of true sentences.
A natural list of axioms is complete only in special cases, such as the axioms of dense linear orders without endpoints, or of algebraically closed fields of a fixed characteristic.
</document>

<document source="https://tamnd.github.io/math2/node/NN77VYSGXR.md" tag="NN77VYSGXR" kind="concept">
# Cyclotomic field

## Summary

The $n$-th cyclotomic field is the field $\Q(\zeta_n)$ obtained by adjoining to $\Q$ a [[primitive $n$-th root of unity|0EHM5ZP5QH]] $\zeta_n$.
It is a [[Galois extension|0QMZAQ6QQ4]] of $\Q$ of degree $\varphi(n)$ whose Galois group is the [[group of units of $\Z/n\Z$|0F8X8ZRQ53]].
By the [[Kronecker–Weber theorem|AW00HPFB54]], every finite abelian extension of $\Q$ lies in some cyclotomic field.
Cyclotomic fields were studied by Kummer in his work on Fermat's Last Theorem, which led to the notion of a [[regular prime|B4VK33PCXZ]].

## Explanations

### L1 human

The $n$-th roots of unity are the $n$ corners of a regular $n$-gon inscribed in the unit circle, with one corner at $1$.
The cyclotomic field is what you get by adjoining these corners to the rational numbers.
For $n = 4$ the corners are $1, i, -1, -i$, and the field is $\Q(i)$.
For $n = 3$ the field is $\Q(\sqrt{-3})$, since $e^{2\pi i/3} = (-1 + \sqrt{-3})/2$.
The symmetries of the field are the maps $\zeta_n \mapsto \zeta_n^a$ with $a$ coprime to $n$.
</document>

<document source="https://tamnd.github.io/math2/node/NN8RWPC8WG.md" tag="NN8RWPC8WG" kind="concept">
# Reflexive space

## Summary

A normed space is reflexive when the [[canonical embedding into its bidual|QKR3D65TTH]] is onto.
Reflexive spaces are always [[Banach spaces|06GZ956657]].
They are the Banach spaces whose closed unit ball is weakly compact.
So bounded sequences have weakly convergent subsequences, which makes them the natural setting for existence proofs by minimization.
Hilbert spaces, $L^p$ spaces for $1 < p < \infty$ and finite-dimensional spaces are reflexive.

## Explanations

### L1 human

Every normed space sits inside its double dual, the dual of the dual.
In a reflexive space the double dual has nothing more: going to the dual twice brings you back to where you started.
For example, the dual of $\ell^2$ is $\ell^2$ again, so $\ell^2$ is reflexive.
The space $c_0$ of null sequences is not reflexive: its double dual is the bigger space $\ell^\infty$ of all bounded sequences.
</document>

<document source="https://tamnd.github.io/math2/node/NN8XH5RJYB.md" tag="NN8XH5RJYB" kind="concept">
# Inner function

## Summary

An inner function is a bounded [[holomorphic function|0DG0Z72CQD]] on the [[unit disc|8G5ABXRK85]] whose boundary values have modulus $1$ almost everywhere on the circle.
Inner functions are one half of the [[inner-outer factorization|X6ZQPAQF76]] of functions in [[Hardy spaces|PNNNFJVF6W]].
They describe the closed subspaces of $H^2$ that are invariant under the shift, by [[Beurling's theorem|2V1DXXM9FA]].

## Explanations

### L1 human

Think of an inner function as a map of the disc into itself that pushes almost all of the boundary circle onto the boundary circle.
The simplest example is $f(z) = z$.
Every [[Blaschke product|9N6R5ADJXT]] is inner, and its zeros can be placed almost anywhere in the disc.
There are also inner functions without zeros, such as $\exp\left(-\frac{1 + z}{1 - z}\right)$.
An inner function carries the "zeros and singular part" of a function, while its modulus on the circle carries no information.
</document>

<document source="https://tamnd.github.io/math2/node/NNAR1HN9BQ.md" tag="NNAR1HN9BQ" kind="definition">
# Γ-convergence

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]].
Let $F_n \colon X \to [-\infty, +\infty]$ for $n \in \N$ and $F \colon X \to [-\infty, +\infty]$ be functions with values in the [[extended real line|0GSZ7B0CGR]].
The [[sequence|0G9EKM0E15]] $(F_n)$ *Γ-converges* to $F$ if the following two conditions hold for every $x \in X$.

1. (Lower bound.) For every sequence $(x_n)$ in $X$ with $x_n \to x$,
$$F(x) \le \liminf_{n \to \infty} F_n(x_n).$$
2. (Recovery sequence.) There is a sequence $(x_n)$ in $X$ with $x_n \to x$ and
$$F(x) \ge \limsup_{n \to \infty} F_n(x_n).$$

The function $F$ is then called the *Γ-limit* of $(F_n)$, written $F = \Gamma\text{-}\lim_n F_n$.
Here $\liminf$ is the [[limit inferior|033RZ32T4Q]] and $\limsup$ is the [[limit superior|0T3RZQXDYD]].

## Remarks

The two conditions together give $F(x) = \lim_n F_n(x_n)$ along the recovery sequence.
A Γ-limit is unique when it exists, and it is always lower semicontinuous.
For a family $(F_\varepsilon)_{\varepsilon > 0}$, one says that $F_\varepsilon$ Γ-converges to $F$ as $\varepsilon \to 0$ if $F_{\varepsilon_k}$ Γ-converges to $F$ for every sequence $\varepsilon_k \to 0$.
The same definition with neighbourhoods instead of sequences works in any topological space, and it agrees with this one in metric spaces.

Example: $F_n(x) = \sin(nx)$ on $\R$ Γ-converges to the constant $F = -1$.
The lower bound holds because $\sin(n x_n) \ge -1$ for every $n$.
For a recovery sequence, pick $x_n$ with $\abs{x_n - x} \le 2\pi/n$ and $\sin(n x_n) = -1$; then $x_n \to x$ and $F_n(x_n) = -1$.

Non-example: let $F(0) = 1$ and $F(x) = 0$ for $x \ne 0$, and let $F_n = F$ for all $n$.
The constant sequence does not Γ-converge to $F$: for $x_n = 1/n \to 0$ one has $\liminf_n F_n(x_n) = 0 < 1 = F(0)$, so the lower bound fails at $0$.
Its Γ-limit is the function $0$, the lower semicontinuous envelope of $F$.
</document>

<document source="https://tamnd.github.io/math2/node/NNCBVR7Z2X.md" tag="NNCBVR7Z2X" kind="statement">
# Eilenberg–Moore spectral sequence

## Statement

Let $k$ be a field.
Let $\pi \colon E \to B$ be a [[Serre fibration|PB9JXQ2KC3]] and let $f \colon X \to B$ be a continuous map, where $X$, $E$ and $B$ are path-connected and $B$ is simply connected.
Assume that $H^n(X; k)$, $H^n(E; k)$ and $H^n(B; k)$ are finite-dimensional for every $n$.
Let $E_f = X \times_B E$ be the pullback fibration over $X$.
Then there is a second-quadrant [[spectral sequence|GG8AYV847V]] that converges strongly to $H^*(E_f; k)$.
Its second page is
$$E_2^{-p, q} = \operatorname{Tor}^{-p, q}_{H^*(B; k)}\big(H^*(X; k), H^*(E; k)\big),$$
the bigraded [[Tor|DDQRERAQCV]] over the [[cohomology ring|RDT4785M4D]] $H^*(B; k)$, where $p$ is the homological degree and $q$ the internal degree.

## Remarks

The proof compares the cochains of $E_f$ with the derived tensor product of the cochains of $X$ and $E$ over the cochains of $B$.
The spectral sequence comes from filtering a bar construction by bar length.
Some condition on $\pi_1(B)$ is needed for convergence; for general non-simply-connected $B$ the spectral sequence may fail to converge to the cohomology of the pullback.

Taking $X$ a point gives a spectral sequence $\operatorname{Tor}_{H^*(B)}(k, H^*(E)) \Rightarrow H^*(F)$ for the fibre $F$.
Taking $X$ a point and $E$ the contractible path space gives $\operatorname{Tor}_{H^*(B)}(k, k) \Rightarrow H^*(\Omega B)$, the cohomology of the [[loop space|2JFF5F3EGF]].

Example: for $B = S^n$ with $n \ge 2$, $H^*(S^n; k)$ is an exterior algebra on a class $x$ of degree $n$.
Then $\operatorname{Tor}_{H^*(S^n)}(k, k)$ has one copy of $k$ in each bidegree $(-j, jn)$, of total degree $j(n-1)$, and the spectral sequence collapses for degree reasons.
This gives $H^i(\Omega S^n; k) \cong k$ when $n - 1$ divides $i$, and $0$ otherwise.

## Explanations

### L1 human

The Eilenberg–Moore spectral sequence computes the cohomology of a pullback, such as a fibre or a loop space, from the cohomology of the spaces it is built from.
Where the Serre spectral sequence goes from base and fibre to total space, this one goes from base and total space to the fibre.
Its starting page is an algebraic Tor group over the cohomology ring of the base.

### L2 human

The spectral sequence is a key tool for computing the cohomology of loop spaces and of homogeneous spaces $G/H$ of Lie groups.
It shows, for instance, that the cohomology of $\Omega B$ depends only on the cochain algebra of $B$ up to quasi-isomorphism when $B$ is simply connected.
It is dual in spirit to the Serre spectral sequence, and the two are often used together.
</document>

<document source="https://tamnd.github.io/math2/node/NNHM06RCNV.md" tag="NNHM06RCNV" kind="definition">
# Approximation theory

## Definition

Let $X$ be a [[normed vector space|03J5RAKB30]] of functions, and let $A_0 \subseteq A_1 \subseteq A_2 \subseteq \cdots$ be nonempty subsets of $X$, the approximating classes.
*Approximation theory* is the study of the following questions for $f \in X$.

1. Existence: does $f$ have a [[best approximation|7N4988HN3P]] from $A_n$?
2. Uniqueness and characterization: is it unique, and which conditions single it out?
3. Rates: how fast does the [[error of best approximation|YDN0T4MEX0]] $E(f, A_n)$ tend to $0$ as $n \to \infty$, and how does this rate depend on $f$?
4. Construction: which explicit methods give good [[approximations|VNWH2KADN7]] of $f$ from $A_n$?

## Remarks

Example: $X = C[a, b]$ with the supremum norm and $A_n$ the polynomials of degree at most $n$.
Here a best approximation exists and is unique, it is characterized by the equioscillation theorem, and $E(f, A_n) \to 0$ for every $f$ by the Weierstrass approximation theorem.

Non-example: a single numerical estimate, such as $\abs{e^x - (1 + x)} \le 1$ on $[0, 1]$, is an approximation result but does not by itself pose these questions.
The theory studies whole families $A_n$ and all $f$ in a class.

Other common choices are trigonometric polynomials, splines with $n$ knots, rational functions of degree $n$, and $L^p$ norms instead of the supremum norm.
</document>

<document source="https://tamnd.github.io/math2/node/NNPHTWF645.md" tag="NNPHTWF645" kind="concept">
# Resultant

## Summary

The resultant of two polynomials is a polynomial expression in their coefficients that, for nonzero leading coefficients, vanishes exactly when they have a common root.
It is the [[determinant|00KDZRW4C0]] of the Sylvester matrix built from the coefficients.
It lets one decide whether two polynomial equations have a common solution without solving them.
The [[discriminant|0VVK64Q4JK]] of $f$ is, up to a simple factor, the resultant of $f$ and its derivative.

## Explanations

### L1 human

Two polynomials either share a root or they do not.
The resultant is a single number, computed from the coefficients alone, that is zero in the first case and nonzero in the second.
For $x - a$ and $x - b$ the resultant is $a - b$, which is zero exactly when the two roots agree.
The [[definition|8D277VG29T]] gives the matrix.
</document>

<document source="https://tamnd.github.io/math2/node/NNXSP5KKAC.md" tag="NNXSP5KKAC" kind="concept">
# Degree of a central simple algebra

## Summary

The dimension of a [[central simple algebra|5HRKJBAY2V]] over its center is always a perfect square $n^2$.
The number $n$ is the degree of the algebra.
It is the size of the matrices that the algebra becomes over an algebraically closed field.
It bounds the index and the period of the algebra.

## Explanations

### L1 human

The algebra of $n \times n$ matrices over a field has dimension $n^2$, and its degree is $n$.
Every central simple algebra becomes such a matrix algebra over a large enough field, so its dimension is also a square.
Hamilton's quaternions have dimension $4$ over $\R$, so their degree is $2$; over $\C$ they become the $2 \times 2$ complex matrices.
</document>

<document source="https://tamnd.github.io/math2/node/NP948D6FR0.md" tag="NP948D6FR0" kind="problem">
# Bernstein's problem

## Problem

Let $m \ge 1$ and let $u \colon \R^m \to \R$ be a twice continuously differentiable function that solves the [[minimal surface equation|3GV2E2YTZX]]
$$\operatorname{div} \left( \frac{\nabla u}{\sqrt{1 + \abs{\nabla u}^2}} \right) = 0$$
on all of $\R^m$.
Must $u$ be an affine function, so that its graph is a hyperplane in $\R^{m+1}$?

## Remarks

The answer is yes for $m \le 7$ and no for $m \ge 8$.
Sergei Bernstein proved the case $m = 2$: an entire [[minimal surface|41X0YW2R9N]] graph over the plane is a plane.
Wendell Fleming gave a new proof for $m = 2$ based on area-minimizing cones.
This approach led to the cases $m = 3$ by Ennio De Giorgi, $m = 4$ by Frederick Almgren and $m \le 7$ by James Simons.
Enrico Bombieri, Ennio De Giorgi and Enrico Giusti found a non-affine entire solution for $m = 8$, and so for every $m \ge 8$.
The dividing line comes from Simons' cone $\set{x \in \R^8 : x_1^2 + x_2^2 + x_3^2 + x_4^2 = x_5^2 + x_6^2 + x_7^2 + x_8^2}$, which is area-minimizing.
For $m = 1$ the statement is elementary: the equation says that $u'/\sqrt{1 + u'^2}$ is constant, so $u'$ is constant.

## Explanations

### L1 human

A soap film that spreads over the whole plane as the graph of a function must be flat.
Bernstein proved this, and the question was whether the same holds in higher dimensions.
It does up to dimension seven, but in dimension eight and beyond there are curved minimal graphs over the whole space.
The change at dimension eight is one of the surprising facts of geometric analysis.
</document>

<document source="https://tamnd.github.io/math2/node/NPBWWBGV37.md" tag="NPBWWBGV37" kind="definition">
# Hilbert–Schmidt operator

## Definition

Let $H$ and $K$ be [[Hilbert spaces|HM2HHE3J51]] and let $A \colon H \to K$ be a [[bounded linear operator|03TM135G0N]].
The operator $A$ is a *Hilbert–Schmidt operator* if there is an [[orthonormal basis|0S9GGTR0EX]] $(e_i)_{i \in I}$ of $H$ such that
$$\sum_{i \in I} \norm{A e_i}^2 < \infty.$$
The *Hilbert–Schmidt norm* of $A$ is $\norm{A}_{\mathrm{HS}} = \big( \sum_{i \in I} \norm{A e_i}^2 \big)^{1/2}$.

## Remarks

The condition and the value of the sum do not depend on the orthonormal basis.
Indeed, for an orthonormal basis $(f_j)_{j \in J}$ of $K$, Parseval's identity gives $\sum_i \norm{A e_i}^2 = \sum_i \sum_j \abs{\ip{A e_i}{f_j}}^2 = \sum_j \norm{A^* f_j}^2$, and the right side does not involve $(e_i)$.

Every Hilbert–Schmidt operator is [[compact|M2NEP1K5AE]] and satisfies $\norm{A} \le \norm{A}_{\mathrm{HS}}$.
A compact operator is Hilbert–Schmidt if and only if its [[singular values|QSJKPKPM7H]] $\sigma_n$ satisfy $\sum_n \sigma_n^2 < \infty$, and then $\norm{A}_{\mathrm{HS}}^2 = \sum_n \sigma_n^2$.
The Hilbert–Schmidt operators form the Schatten class $S_2$; every [[trace class operator|C37R11WREF]] is Hilbert–Schmidt.
On $L^2$ spaces of $\sigma$-finite measure spaces, they are the [[integral operators with square-integrable kernels|CB5G212Q03]], and $\norm{A}_{\mathrm{HS}}$ is the $L^2$ norm of the kernel.

Example: on $\ell^2$, the operator $A(x_1, x_2, \dots) = (x_1, x_2/2, x_3/3, \dots)$ is Hilbert–Schmidt.
For the standard basis, $\norm{A e_n} = 1/n$ and $\sum_n 1/n^2 = \pi^2/6$.

Non-example: on $\ell^2$, the operator $B(x_1, x_2, \dots) = (x_1, x_2/\sqrt{2}, x_3/\sqrt{3}, \dots)$ is compact but not Hilbert–Schmidt, since $\sum_n \norm{B e_n}^2 = \sum_n 1/n$ diverges.
The identity of an infinite-dimensional Hilbert space is not Hilbert–Schmidt either.

## Explanations

### L1 human

An operator is Hilbert–Schmidt when the total squared size of the images of the basis vectors is finite.
</document>

<document source="https://tamnd.github.io/math2/node/NPBYYK63FE.md" tag="NPBYYK63FE" kind="counterexample">
# Cantor function

## Counterexample

Let $C \subseteq [0, 1]$ be the [[Cantor set|0BXNDZFQME]].
For $x \in [0, 1]$, write $x$ in base $3$ as $x = 0.a_1 a_2 a_3 \ldots$ with $a_k \in \set{0, 1, 2}$.
If some digit equals $1$, delete all digits after the first $1$.
Then replace each digit $2$ by $1$, and read the result $0.b_1 b_2 b_3 \ldots$ in base $2$.
The value is $c(x) = \sum_k b_k 2^{-k}$, and it does not depend on the choice of the base-$3$ expansion.

The function $c \colon [0, 1] \to [0, 1]$ is [[continuous|0P27W9RW0G]] and nondecreasing, with $c(0) = 0$ and $c(1) = 1$.
It is constant on each open interval removed in the construction of $C$, for example $c(x) = 1/2$ on $(1/3, 2/3)$.
So $c'(x) = 0$ at every point of $[0, 1] \setminus C$.
The Cantor set is a [[null set|VQQM4Y550T]], so $c' = 0$ [[almost everywhere|RARTMA0ZEC]].
But $c$ is not constant.

## Remarks

The function refutes the claim that a continuous function with derivative $0$ almost everywhere is constant.
It also shows that the [[fundamental theorem of calculus|0NHGY0N0JB]] in the form $\int_0^1 f' = f(1) - f(0)$ can fail for a continuous [[monotone function|0WFF6AZPBP]] differentiable almost everywhere.
Here $\int_0^1 c'(x) \, dx = 0$, while $c(1) - c(0) = 1$.
So $c$ is not [[absolutely continuous|ZT6F4C2YHK]]; it is the standard example of a [[singular function|P5AWRD2793]].
It maps the null set $C$ onto all of $[0, 1]$.
Its graph is often called the devil's staircase.

## Explanations

### L1 human

The Cantor function climbs from $0$ to $1$ without jumps, yet it is flat almost everywhere.
All its growth happens on the Cantor set, a dust of points with total length zero.
On the middle third it sits at height $1/2$, on the next two middle thirds at heights $1/4$ and $3/4$, and so on.
So knowing that the derivative is zero almost everywhere does not tell you the function is constant.
</document>

<document source="https://tamnd.github.io/math2/node/NPEBABQ49M.md" tag="NPEBABQ49M" kind="concept">
# Group of Lie type

## Summary

A finite group of Lie type is the group of points of a reductive algebraic group that are fixed by a Frobenius-type map.
An example is $\mathrm{GL}_n(\F_q)$ inside $\mathrm{GL}_n$ over the algebraic closure of $\F_q$.
These groups are finite analogues of Lie groups, and their simple quotients form most of the families in the classification of finite simple groups.
They include the classical groups over finite fields, the Chevalley groups of exceptional type, and the twisted groups of Steinberg, Suzuki and Ree.

## Explanations

### L1 human

Matrix groups such as invertible matrices or orthogonal matrices make sense over any field, including a finite field with $q$ elements.
Over a finite field these groups are finite.
For example, $\mathrm{SL}_2(\F_2)$ is the group of $2 \times 2$ matrices with entries in $\{0, 1\}$, arithmetic modulo $2$ and determinant $1$; it has $6$ elements and is isomorphic to $S_3$.
Groups built this way, and their twisted relatives, are the groups of Lie type.
Their simple quotients, such as $\mathrm{PSL}_n(\F_q)$, are the main source of finite simple groups.
</document>

<document source="https://tamnd.github.io/math2/node/NPF2KSXK81.md" tag="NPF2KSXK81" kind="definition">
# Completion of a normed space

## Definition

Let $X$ be a [[normed vector space|03J5RAKB30]] over $\R$ or $\C$.
A *completion* of $X$ is a pair $(\widehat{X}, \iota)$ such that:

1. $\widehat{X}$ is a [[Banach space|06GZ956657]] over the same field.
2. $\iota \colon X \to \widehat{X}$ is a [[linear isometry|07SJXTWVQW]], that is, $\iota$ is linear and $\norm{\iota(x)} = \norm{x}$ for all $x \in X$.
3. The image $\iota(X)$ is [[dense|0BSJSX7ME2]] in $\widehat{X}$.

## Remarks

Every normed space has a completion: take the Cauchy sequences in $X$ modulo the sequences that tend to $0$, with $\norm{[(x_n)]} = \lim_n \norm{x_n}$, and $\iota(x)$ the constant sequence.
Any two completions $(\widehat{X}, \iota)$ and $(\widetilde{X}, \kappa)$ are related by a unique isometric isomorphism $\phi$ with $\phi \circ \iota = \kappa$.
Example: let $P$ be the space of real polynomial functions on $[0, 1]$ with the sup norm, and let $\iota$ be the inclusion of $P$ in $C[0, 1]$.
Clauses 1 and 2 hold, and clause 3 holds by the [[Weierstrass approximation theorem|0JQE130DHW]], so $C[0, 1]$ is the completion of $P$.
Non-example: the inclusion of the finitely supported sequences $c_{00}$, with the sup norm, into $\ell^\infty$ satisfies clauses 1 and 2 but not 3.
The reason is that the closure of $c_{00}$ in $\ell^\infty$ is the space $c_0$ of null sequences.
That closure, $c_0$, is the completion of $c_{00}$.
As a metric space, $\widehat{X}$ is the [[completion of the metric space|06RW4RKWH7]] $X$, with the vector operations extended by continuity.
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# Completion is flat

## Statement

Let $R$ be a commutative [[Noetherian ring|0670T59HGE]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $\hat R$ be the [[$I$-adic completion|NS84D2920R]] of $R$.

1. For every [[finitely generated|05QY1VEBYD]] $R$-module $M$, the natural map $\hat R \otimes_R M \to \hat M$ is an isomorphism, where $\hat M = \varprojlim_n M/I^n M$.
2. $\hat R$ is a [[flat|73RYYEVDKA]] $R$-module.
3. If $I$ is contained in the [[Jacobson radical|XA0DCNH7WY]] of $R$, then $\hat R$ is a [[faithfully flat|C31PYYPRGC]] $R$-module; this applies when $R$ is local and $I \neq R$.

## Remarks

By the [[Artin–Rees lemma|30H6B0G2FT]], completion $M \mapsto \hat M$ is an exact functor on finitely generated $R$-modules.
Both $M \mapsto \hat R \otimes_R M$ and $M \mapsto \hat M$ are right exact and agree on finite free modules, so they agree on finitely presented modules, which gives clause 1.
Then for an ideal $J$ of $R$ the map $\hat R \otimes_R J \to \hat R$ is the injective map $\hat J \to \hat R$, which gives clause 2 by the ideal criterion for flatness.
For clause 3, each maximal ideal $\mathfrak n$ contains $I$, so $\hat R/\mathfrak n \hat R \cong R/\mathfrak n \neq 0$.
Clause 3 can fail without its hypothesis.
The $p$-adic completion $\Z_p$ of $\Z$ is flat over $\Z$, but $\Z_p \otimes_\Z \Z/q\Z = 0$ for a prime $q \neq p$, because $q$ is a unit in $\Z_p$.
So $\Z_p$ is not faithfully flat over $\Z$.

## Explanations

### L1 human

Completing a ring adds limits of approximations, like passing from integers to $p$-adic integers.
Flatness means that this operation does not destroy injectivity of maps between modules.
So information about a Noetherian ring and its modules is kept faithfully when one passes to the completion, at least near the ideal used for completing.

### L2 human

Flatness lets one move properties between a Noetherian local ring $R$ and its completion $\hat R$.
For instance $\dim \hat R = \dim R$, and $R$ is regular, or Cohen–Macaulay, if and only if $\hat R$ is.
Combined with the [[Cohen structure theorem|N3XH42KVV3]], this reduces many questions about local rings to quotients of power series rings.
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# Derived functor

## Summary

The derived functors of a left exact functor $F$ are functors $R^n F$, for $n \ge 0$, with $R^0 F = F$.
They are computed by applying $F$ to an [[injective resolution|ZW3W6FX4PW]] and taking cohomology.
They turn a short exact sequence into a long exact sequence, and so measure how far $F$ is from being exact.
[[Ext|4P7GBWJ4RR]], [[Tor|DDQRERAQCV]], [[group cohomology|YXJXA0Q52R]] and sheaf cohomology are all derived functors.

## Explanations

### L1 human

Many natural functors keep injective maps injective but can destroy surjectivity, or the other way round.
The derived functors are correction terms that record exactly what is lost.
One computes them by replacing an object with a resolution by simple objects, applying the functor, and taking cohomology.
For abelian groups, the functor "elements killed by $2$" sends the surjection $\Z \to \Z/2\Z$ to the non-surjective map $0 \to \Z/2\Z$.
Its first derived functor is $\operatorname{Ext}^1(\Z/2\Z, -)$, which repairs this failure.
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<document source="https://tamnd.github.io/math2/node/NPNNXWMVCC.md" tag="NPNNXWMVCC" kind="statement">
# Veblen–Young theorem

## Statement

Let $S$ be a set of points with a family of subsets of $S$, called lines, such that:

1. any two distinct points lie on exactly one line;
2. every line has at least three points;
3. (Veblen's axiom) for distinct points $a, b, c, d$, if the lines $ab$ and $cd$ meet, then the lines $ac$ and $bd$ meet.

A subspace of $S$ is a set of points that contains the whole line through any two of its distinct points.
Suppose that the longest chains $S_0 \subsetneq S_1 \subsetneq \cdots \subsetneq S_n = S$ of nonempty subspaces have finite length $n$, and that $n \ge 3$.
Then there is a [[division ring|0EA5BTA0CX]] $D$ such that $S$ is isomorphic to the [[projective space|ZT9A6JK8QF]] $PG(n, D)$.
Its points are the $1$-dimensional subspaces of the [[vector space|000AYPA9AJ]] $D^{n+1}$, and its lines are the $2$-dimensional subspaces.
A point lies on a line when it is contained in it.
The division ring $D$ is unique up to isomorphism.

## Remarks

The key step is that Desargues's theorem holds in every projective space of dimension at least $3$.
Given two triangles in perspective in a plane of $S$, one lifts the figure out of the plane.
There the theorem follows by intersecting planes, and a projection brings it back.
Desargues's theorem then lets one define an addition and a multiplication on the points of a line minus one point, which form a division ring, and coordinates follow.

The hypothesis $n \ge 3$ is necessary.
For $n = 2$ the axioms describe [[projective planes|DANFNN8C4Y]], and there are planes such as the [[Moulton plane|3V9GYYXM1H]] that are not of the form $PG(2, D)$ for any division ring $D$.
If $S$ is finite, $D$ is a finite division ring, hence a field by [[Wedderburn's little theorem|0RRNVX1VKA]], so $S \cong PG(n, q)$ for a prime power $q$.
The division ring $D$ is commutative if and only if Pappus's theorem holds in the planes of $S$.

## Explanations

### L1 human

One can describe projective geometry by a few rules about points and lines, with no numbers at all.
The theorem says that as soon as the space has at least three dimensions, those rules force the space to come from linear algebra.
The points are the lines through the origin of a vector space, perhaps over a skew field such as the quaternions.
In two dimensions this fails, and there are many exotic planes.

### L2 human

The theorem is the reason why the synthetic and the analytic approach to projective geometry agree in dimension at least $3$.
It combines with the [[fundamental theorem of projective geometry|GKV4RVXBRE]], which describes the isomorphisms between such spaces through semilinear maps.
It explains why the theory of non-classical geometries concentrates on planes, where coordinatization needs [[planar ternary rings|Q0NXXFADT9]].
It is also a model for later results, such as the classification of polar spaces and buildings of higher rank by Tits.
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# Pullback

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $f \colon X \to Z$ and $g \colon Y \to Z$ be [[morphisms|859F0TFVA3]] of $\mathcal{C}$.
A *pullback* of $f$ and $g$ is an object $P$ with morphisms $p_1 \colon P \to X$ and $p_2 \colon P \to Y$ such that:

1. $f \circ p_1 = g \circ p_2$;
2. for every object $W$ and all morphisms $q_1 \colon W \to X$ and $q_2 \colon W \to Y$ with $f \circ q_1 = g \circ q_2$, there is exactly one morphism $u \colon W \to P$ with $p_1 \circ u = q_1$ and $p_2 \circ u = q_2$.

The object $P$ is written $X \times_Z Y$ and called the *fibre product* of $X$ and $Y$ over $Z$.

## Remarks

A pullback is the [[limit|CHS5AFJGWE]] of the diagram $X \xrightarrow{f} Z \xleftarrow{g} Y$, so it is unique up to unique isomorphism.
If the [[product|S5744QZV4N]] $X \times Y$ exists, the pullback is the [[equalizer|CTJ5AD8ST5]] of $f \circ \pi_1$ and $g \circ \pi_2 \colon X \times Y \to Z$.

Example: in the category of sets, let $P = \set{(x, y) \in X \times Y : f(x) = g(y)}$ with the two coordinate maps.
Clause 1 holds by the choice of $P$.
Given $q_1, q_2$ with $f \circ q_1 = g \circ q_2$, the map $u(w) = (q_1(w), q_2(w))$ lands in $P$ and is the only map with the required coordinates.
When $g$ is the inclusion of a subset $Y \subseteq Z$, the pullback is the [[preimage|01AHG4G9RT]] $f^{-1}(Y)$ with its inclusion into $X$.

Non-example: in the category of sets, let $X = Y = \{0, 1\}$, $Z = \{*\}$ and let $f, g$ be the constant maps.
The set $\{0, 1\}$ with $p_1 = p_2 = \id$ satisfies clause 1, but the pair $q_1 = \id$, $q_2 = (0 \mapsto 1, 1 \mapsto 0)$ on $W = \{0, 1\}$ admits no $u$ with $p_1 \circ u = q_1$ and $p_2 \circ u = q_2$, so clause 2 fails.
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# Topology generated by a family of seminorms

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, let $V$ be a [[vector space|05EGHEQZAW]] over $\mathbb{K}$, and let $\mathcal{P}$ be a family of [[seminorms|FP5GVCN8GV]] on $V$.
For $x \in V$, finitely many $p_1, \ldots, p_n \in \mathcal{P}$ and $\varepsilon > 0$, put
$$B(x; p_1, \ldots, p_n; \varepsilon) = \set{y \in V : p_i(y - x) < \varepsilon \text{ for } i = 1, \ldots, n}.$$
The *topology generated by* $\mathcal{P}$ is the [[topology|0CQHFPGJ45]] on $V$ defined as follows.
A set $U \subseteq V$ is open if for every $x \in U$ there are $p_1, \ldots, p_n \in \mathcal{P}$ and $\varepsilon > 0$ with $B(x; p_1, \ldots, p_n; \varepsilon) \subseteq U$.

## Remarks

With this topology $V$ is a [[topological vector space|T32W553W95]], each $p \in \mathcal{P}$ is continuous, and it is the coarsest topology on $V$ with these two properties.
A net $(x_i)$ converges to $x$ if and only if $p(x_i - x) \to 0$ for every $p \in \mathcal{P}$.
The topology is Hausdorff if and only if $\mathcal{P}$ separates points: for each $x \ne 0$ some $p \in \mathcal{P}$ has $p(x) \ne 0$.
Example: on $\R^X$, the space of all real functions on a set $X$, the seminorms $p_x(f) = \abs{f(x)}$ for $x \in X$ generate the topology of pointwise convergence, which is the product topology.
The check: a basic set $B(f; p_{x_1}, \ldots, p_{x_n}; \varepsilon)$ restricts the values at $x_1, \ldots, x_n$ only, as a basic open set of the product topology does.
Non-example: for $0 < p < 1$, the usual metric topology of $L^p([0, 1])$ is not generated by any family of seminorms.
Indeed, its only continuous linear form is $0$.
A nonzero continuous seminorm would give a nonzero continuous linear form by the Hahn–Banach theorem.
The topologies generated by families of seminorms are exactly the topologies of [[locally convex spaces|B6GY2KJMWW]].
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# Topological mixing

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
The map $f$ is *topologically mixing* if for all nonempty [[open sets|02BH3MCBQB]] $U, V \subseteq X$ there is $N \ge 1$ with $f^n(U) \cap V \neq \emptyset$ for every $n \ge N$.

## Remarks

Every topologically mixing map is [[topologically transitive|FR49Y9GQHR]].
Example: the doubling map $f(x) = 2x \bmod 1$ on the circle $\R / \Z$.
A nonempty open set $U$ contains an open arc of length $\ell > 0$; choose $k$ with $2^k \ell > 1$.
The map $f^n$ multiplies lengths of small arcs by $2^n$, so for $n \ge k$ the image $f^n(U)$ is the whole circle and meets every $V$.
Non-example: the rotation $f(x) = x + \alpha \bmod 1$ with $\alpha$ irrational, and $U = V = (-1/8, 1/8)$.
Here $f^n(U) \cap U \neq \emptyset$ only when $n\alpha$ is within $1/4$ of an integer.
By Weyl's equidistribution theorem, about half of all $n$ fail this, so there is no $N$ as required.

## Explanations

### L1 human

A mixing map brings each open region into each other open region and keeps it meeting that region at all large times.
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# Infinite Galois theory

## Definition

Let $L/K$ be a [[Galois extension|0DS2F7GK05]], that is, an [[algebraic extension|09EYMKMSB9]] that is normal and separable, of possibly infinite degree.
*Infinite Galois theory* is the study of $L/K$ through its [[Galois group|0XVR72M51M]] $G = \Gal(L/K)$ with the [[Krull topology|F3GTF5NT4M]].
Its central statement is the Galois correspondence:

1. the maps $H \mapsto L^H$, with $L^H$ the [[fixed field|0KMJ6919WS]] of $H$, and $E \mapsto \Gal(L/E)$ are inverse bijections that reverse inclusion.
   They match the closed subgroups $H$ of $G$ with the [[intermediate fields|02NBKVQFRQ]] $E$ of $L/K$;
2. a closed subgroup $H$ is open if and only if $L^H/K$ is finite;
3. a closed subgroup $H$ is normal if and only if $L^H/K$ is Galois, and then $\Gal(L^H/K) \cong G/H$ as topological groups.

## Remarks

Example: let $K = \F_p$ and $L$ an algebraic closure of $\F_p$.
The intermediate fields are the fields $\F_{p^n}$ and the infinite unions of such fields, and $\F_{p^n}$ corresponds to the open subgroup generated topologically by $\phi^n$, where $\phi(x) = x^p$.
Non-example of the correspondence for all subgroups: the subgroup $H = \set{\phi^k : k \in \Z}$ is not closed in $\Gal(L/\F_p)$.
Its fixed field is $\F_p$, the same as for the whole group, so $H \mapsto L^H$ is not injective on all subgroups.
When $L/K$ is finite, the Krull topology is discrete, every subgroup is closed, and the theory reduces to the [[fundamental theorem of Galois theory|0T1Q8D0FW8]].
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# Szemerédi–Trotter theorem

## Statement

There is an absolute constant $C > 0$ with the following property.
Let $P$ be a finite set of points and $L$ a finite set of lines in the [[Euclidean plane|0XW7V9ZZ05]] $\R^2$.
Let $I(P, L)$ be the number of pairs $(p, \ell) \in P \times L$ with $p \in \ell$.
Then
$$I(P, L) \le C \left( \abs{P}^{2/3} \abs{L}^{2/3} + \abs{P} + \abs{L} \right).$$

## Remarks

Székely gave a short proof with the crossing number inequality.
Draw a graph whose vertices are the points and whose edges join consecutive points on each line.
It has at least $I - \abs{L}$ edges and at most $\abs{L}^2$ crossings, and the crossing number inequality gives the bound.
The bound is sharp up to the constant.
Take the grid $\set{1, \dots, k} \times \set{1, \dots, 2k^2}$ and the lines $y = ax + b$ with $1 \le a \le k$ and $1 \le b \le k^2$.
Each such line meets the grid in $k$ points, so there are $k^4$ incidences, with $2k^3$ points and $k^3$ lines.
The theorem fails over finite fields.
In $\F_q^2$, the $q^2$ points and the $q^2 + q$ lines have $q^3 + q^2$ incidences, which is more than a constant times $q^{8/3}$ for large $q$.

## Explanations

### L1 human

Draw some points and some straight lines in the plane, and count how many times a point lies on a line.
The theorem gives the largest possible count, up to a constant factor.
With $n$ points and $n$ lines, there are at most about $n^{4/3}$ such incidences, far fewer than the trivial $n^2$.

### L2 human

The Szemerédi–Trotter theorem is the basic result of incidence geometry.
A corollary: $n$ points determine at most $O(n^2/k^3 + n/k)$ lines that contain at least $k$ of the points.
It has applications to the [[Erdős distinct distances problem|4H31EM5T0C]] and to sum-product estimates in [[additive combinatorics|3NM362A2JG]].
Its proof uses the topology of the real plane, which explains why it fails over finite fields.
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# Bernstein's theorem on absolutely convergent Fourier series

## Statement

Let $0 < \alpha \le 1$ and let $f \colon \R \to \C$ be a $2\pi$-periodic function that is [[Hölder continuous|012PBA0FMD]] with exponent $\alpha$.
So there is $K \ge 0$ with $\abs{f(x) - f(y)} \le K \abs{x - y}^\alpha$ for all $x, y \in \R$.
Let $\hat f(n)$, $n \in \Z$, be its [[Fourier coefficients|5NN2YZZR3V]].
If $\alpha > 1/2$, then $\sum_{n \in \Z} \abs{\hat f(n)} < \infty$, so the Fourier series of $f$ is [[absolutely convergent|HHBGB5R5S2]].

## Remarks

Proof idea: for $h > 0$ the function $x \mapsto f(x + h) - f(x - h)$ has Fourier coefficients $2i \sin(nh) \hat f(n)$.
By [[Parseval's identity|047RFCNVSK]] and the Hölder bound, $\sum_n 4 \sin^2(nh) \abs{\hat f(n)}^2 \le K^2 (2h)^{2\alpha}$.
Take $h = \pi / (3 \cdot 2^j)$; for $2^j \le \abs{n} < 2^{j+1}$, $\sin^2(nh) \ge 3/4$, so the sum of $\abs{\hat f(n)}^2$ over this block is $O(2^{-2j\alpha})$.
The block has fewer than $2^{j+2}$ terms, so by Cauchy–Schwarz the sum of $\abs{\hat f(n)}$ over it is $O(2^{j(1/2 - \alpha)})$.
These bounds form a convergent geometric series when $\alpha > 1/2$.

Example: every continuously differentiable periodic function is Hölder continuous with exponent $1$, so its Fourier series converges absolutely.

The bound $\alpha > 1/2$ is sharp: there are Hölder continuous functions with exponent $1/2$ whose Fourier series does not converge absolutely.
Mere continuity is far from enough, since there are continuous functions whose Fourier series even diverges at a point.

## Explanations

### L1 human

The smoother a periodic function is, the faster its Fourier coefficients decay.
Bernstein's theorem gives a precise threshold: Hölder continuity with an exponent above one half already makes the coefficients absolutely summable.
Then the Fourier series converges uniformly and without any delicate cancellation.

### L2 human

The proof is a model of a standard technique: estimate the coefficients in dyadic blocks with Parseval's identity, then pass from $\ell^2$ to $\ell^1$ with Cauchy–Schwarz.
The loss of a factor $2^{j/2}$ in the last step explains the threshold $1/2$.
The theorem gives a large class of functions in the [[Wiener algebra|5H8XESYCHQ]].
A related criterion of Zygmund says that a function of bounded variation that is Hölder continuous with some positive exponent also has an absolutely convergent Fourier series.
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# GCD domain

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]].
$R$ is a *GCD domain* if for all $a, b \in R$ there is an element $d \in R$ such that:

1. $d \mid a$ and $d \mid b$;
2. for every $c \in R$ with $c \mid a$ and $c \mid b$, one has $c \mid d$.

Such a $d$ is a [[greatest common divisor|0F7KNJ1ZYN]] of $a$ and $b$.

## Remarks

Here $x \mid y$ means [[divisibility|04GNK7K4VP]] in $R$; a greatest common divisor is unique up to multiplication by a unit.
Example: $\Z$ is a GCD domain, and so is every [[unique factorisation domain|06YMXJMXYS]], where a gcd is read off from the factorizations.
Non-example: in $\Z[\sqrt{-5}]$, with the norm $N(x + y\sqrt{-5}) = x^2 + 5y^2$, the elements $6$ and $2(1 + \sqrt{-5})$ have no gcd.
Indeed $2$ and $1 + \sqrt{-5}$ are common divisors, so the norm of a gcd $d$ would be divisible by $N(2) = 4$ and by $N(1 + \sqrt{-5}) = 6$.
Also $N(d)$ would divide $N(6) = 36$ and $N(2 + 2\sqrt{-5}) = 24$, so $N(d) = 12$.
But $x^2 + 5y^2 = 12$ has no integer solution, so clause 2 fails for every common divisor $d$.
Equivalent form: any two elements have a least common multiple.
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# Dickson's lemma

## Statement

Let $k \ge 1$ and order $\N^k$ by the [[product order|K0AWJK6NZH]]: $(a_1, \ldots, a_k) \le (b_1, \ldots, b_k)$ when $a_i \le b_i$ for every $i$.
Let $S$ be a subset of $\N^k$.
Then $S$ has only finitely many minimal elements, and every element of $S$ is greater than or equal to one of them.

## Remarks

Equivalently, $\N^k$ with the product order is a [[well-quasi-ordering|NH25G6GC07]].
The proof is by induction on $k$.
From any infinite sequence in $\N^k$, extract a subsequence that is non-decreasing in the first coordinate.
Then extract one that is also non-decreasing in the second coordinate, and so on.
After $k$ steps the subsequence is non-decreasing in the product order, so no infinite antichain exists, and $\N^k$ is well-founded.

Example: for $S = \set{(a, b) \in \N^2 : ab \ge 6}$, the minimal elements are $(1, 6), (2, 3), (3, 2), (6, 1)$.

The main application is to [[monomial ideals|4G25S7PJ22]].
Identify the monomial $x_1^{a_1} \cdots x_k^{a_k}$ with $(a_1, \ldots, a_k)$; divisibility becomes the product order.
So every monomial ideal in $F[x_1, \ldots, x_k]$ is generated by finitely many monomials.

## Explanations

### L1 human

Take any collection of points with natural number coordinates in $k$ dimensions.
Some of them are "corner" points, with no other point of the collection below and to the left in every coordinate.
Dickson's lemma says there are only finitely many corner points, and every point lies above one of them.

### L2 human

Dickson's lemma is one of the first examples of a well-quasi-order beyond $\N$ itself: a finite product of copies of $\N$.
It is the finiteness step behind Hilbert's basis theorem in the proof with Gröbner bases, and it shows that the Buchberger algorithm terminates.
Higman's lemma and Kruskal's tree theorem are its analogues for words and trees.
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# Period-doubling bifurcation

## Summary

A period-doubling bifurcation, or flip bifurcation, happens in a family of maps when the derivative at a fixed point crosses $-1$.
The fixed point changes stability, and an orbit of period $2$ is born from it.
For flows, the same happens to a periodic orbit whose Floquet multiplier crosses $-1$: a new periodic orbit of about twice the period appears.
A sequence of such bifurcations, at parameters that accumulate geometrically, is one of the main routes to chaos, governed by the [[Feigenbaum constants|F1EHT2CPHC]].

## Explanations

### L1 human

Picture a population that settles to the same size each year.
When a parameter such as the growth rate is increased, this steady state can lose stability by overshooting: too large one year, too small the next.
The population then settles into an alternating pattern, high and low, of period two.
This is a period-doubling bifurcation; in the logistic map $x \mapsto r x (1 - x)$ it happens at $r = 3$.
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# Infinitely divisible distribution

## Definition

Let $\mu$ be a [[probability distribution|0DVDCMA2DK]] on $\R$.
$\mu$ is *infinitely divisible* if for every integer $n \ge 1$ there are [[independent and identically distributed|00MRN0FARV]] random variables $Y_{n,1}, \dots, Y_{n,n}$ such that $Y_{n,1} + \dots + Y_{n,n}$ has law $\mu$.
A real random variable is infinitely divisible if its law is.

## Remarks

Equivalently, for every $n \ge 1$ there is a probability measure $\mu_n$ whose $n$-fold [[convolution|0F8GDKM0A7]] power is $\mu$; or, in terms of [[characteristic functions|0V5B6D6BQH]], $\varphi = \varphi_n^n$ for some characteristic function $\varphi_n$.
The same definition applies to laws on $\R^d$.
Example: the Poisson law with mean $\lambda$ is the law of a sum of $n$ independent Poisson variables with mean $\lambda / n$, since $e^{\lambda(e^{it} - 1)} = \left(e^{(\lambda/n)(e^{it} - 1)}\right)^n$.
Example: the normal law with mean $m$ and variance $\sigma^2$ is the law of a sum of $n$ independent normal variables with mean $m/n$ and variance $\sigma^2/n$.
Non-example: a [[Bernoulli|0386Y0Z7Y7]] law with parameter $p \in (0, 1)$.
Suppose $Y_1 + Y_2$ has this law, with $Y_1, Y_2$ i.i.d.
If $s < s'$ were two points of the support of $Y_1$, then $2s < s + s' < 2s'$ would be three values in $\set{0, 1}$.
So $Y_1$ is constant, and then $Y_1 + Y_2$ is constant, a contradiction.
The characteristic function of an infinitely divisible law has no zeros.
</document>

<document source="https://tamnd.github.io/math2/node/NQQBESGPA4.md" tag="NQQBESGPA4" kind="concept">
# Conway polynomial

## Summary

A Conway polynomial $C_{p,n}$ is a fixed choice of [[primitive polynomial|HQ7KMZYPSQ]] of degree $n$ over $\F_p$.
The choices for different $n$ are compatible, so the fields $\F_{p^n}$ built from them fit together as subfields.
Computer algebra systems use Conway polynomials as a standard model of [[finite fields|0ZXZ82FTX2]], so that results from different programs agree.

## Explanations

### L1 human

There are many ways to build the field with $p^n$ elements, one for each irreducible polynomial of degree $n$.
A Conway polynomial picks one of them by a fixed rule: the first in a dictionary order among the primitive polynomials that fit with the smaller choices.
For $p = 2$, the first ones are $x + 1$, $x^2 + x + 1$, $x^3 + x + 1$ and $x^4 + x + 1$.
</document>

<document source="https://tamnd.github.io/math2/node/NQVQFMXMGE.md" tag="NQVQFMXMGE" kind="definition">
# p-harmonic function

## Definition

Let $\Omega \subseteq \R^n$ be an open set and let $1 < p < \infty$.
Let $u$ lie in the local [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}_{\mathrm{loc}}(\Omega)$.
It is a *weak solution* of the [[p-Laplace equation|XJMYGY0ZZF]] $\operatorname{div}(\abs{\nabla u}^{p-2} \nabla u) = 0$ in $\Omega$ if
$$\int_\Omega \abs{\nabla u}^{p-2}\, \nabla u \cdot \nabla \varphi \, dx = 0 \quad \text{for every } \varphi \in C_c^\infty(\Omega),$$
where $\nabla u$ is the [[weak gradient|C3EC148VZD]].
A *$p$-harmonic function* in $\Omega$ is a continuous weak solution of this equation.

## Remarks

Every weak solution agrees almost everywhere with a continuous one, which is even of class $C^{1,\alpha}_{\mathrm{loc}}$ for some $\alpha > 0$, so the continuity requirement only picks a representative.
For $p = 2$ the condition is $\int \nabla u \cdot \nabla \varphi = 0$, and by Weyl's lemma the $2$-harmonic functions are the classical [[harmonic functions|0DEB0JMN0W]].
Example: for $p \ne n$, the function $u(x) = \abs{x}^{(p - n)/(p - 1)}$ is $p$-harmonic in $\R^n \setminus \set{0}$.
Indeed, for a radial function $u(r)$ the equation reads $\big(r^{n-1} \abs{u'}^{p-2} u'\big)' = 0$, and here $\abs{u'}^{p-2} u'$ is a constant multiple of $r^{1-n}$.
For $p = n$ the function $\log \abs{x}$ plays the same role.
Non-example: $u(x) = x_1^2$ on $\R^n$ is not $p$-harmonic.
Here $\abs{\nabla u}^{p-2} \nabla u = 2^{p-1} \abs{x_1}^{p-2} x_1\, e_1$, whose divergence $2^{p-1}(p - 1)\abs{x_1}^{p-2}$ is positive for $x_1 \ne 0$.
Equivalently, $u$ is $p$-harmonic in $\Omega$ when it is continuous and minimizes $\int_D \abs{\nabla v}^p \, dx$ among all $v$ with $v - u \in W^{1,p}_0(D)$, for every open $D$ with compact closure in $\Omega$.
</document>

<document source="https://tamnd.github.io/math2/node/NR6MGAB5V0.md" tag="NR6MGAB5V0" kind="definition">
# Ancillary statistic

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a [[statistical model|JZ6R7J36K5]] and let $V = V(X)$ be a [[statistic|036756NFD6]] with values in a measurable space $(\mathcal{V}, \mathcal{B})$.
The statistic $V$ is *ancillary* if its distribution does not depend on $\theta$: for every $B \in \mathcal{B}$, the probability $P_\theta(V \in B)$ is the same for all $\theta \in \Theta$.

## Remarks

Example: let $X_1, \dots, X_n$ be i.i.d. $\mathcal{N}(\mu, 1)$ with $\mu \in \R$ and $n \ge 2$.
Then $X_i - \mu$ are i.i.d. $\mathcal{N}(0, 1)$, and the sample variance $S^2 = \frac{1}{n-1}\sum_i (X_i - \bar{X})^2$ is a function of them alone, because $X_i - \bar{X} = (X_i - \mu) - (\bar{X} - \mu)$; so $S^2$ is ancillary for $\mu$.
Non-example: in the same model, $\bar{X} \sim \mathcal{N}(\mu, 1/n)$, whose law changes with $\mu$, so $\bar{X}$ is not ancillary.
In a location model $X_i = \mu + \varepsilon_i$ with a fixed law of the $\varepsilon_i$, every function of the differences $X_i - X_j$ is ancillary.
In a scale model $X_i = \sigma \varepsilon_i$, every function of the ratios $X_i / X_j$ is ancillary, when these are defined.
</document>

<document source="https://tamnd.github.io/math2/node/NR70GHZ37D.md" tag="NR70GHZ37D" kind="problem">
# Hilbert's fifth problem

## Problem

In its standard modern reading, the problem asks the following.
Let $G$ be a [[topological group|92700QAB5Y]] whose underlying space is a [[topological manifold|QQQ9B1DK4P]].
Is $G$ necessarily a [[Lie group|7JBRGCVJRB]]?
That is, does $G$ admit a smooth structure, compatible with its topology, for which multiplication and inversion are smooth?

## Remarks

The answer is yes.
It was obtained in 1952 by Andrew Gleason and by Deane Montgomery and Leo Zippin.
Hidehiko Yamabe later extended their methods to all locally compact groups, giving the [[Gleason–Yamabe theorem|XJF5V9WHKZ]].
A key step is the notion of a group with [[no small subgroups|HPMAJV2RDY]]: a locally compact group with this property is a Lie group.
Hilbert posed the problem in his 1900 list of problems, in terms of continuous transformation groups and without assuming differentiability.
The version for groups acting on manifolds, rather than groups that are manifolds, is the [[Hilbert–Smith conjecture|DS2AS9TSZP]], which is still open.

## Explanations

### L1 human

A Lie group is a group whose elements form a smooth space, like the rotations of the plane or of space.
Smoothness lets one use calculus on the group.
Hilbert asked whether smoothness is automatic: if a group is only known to look locally like ordinary space, with continuous operations, must it already be smooth?
The answer turned out to be yes, so for such groups continuity forces smoothness.
</document>

<document source="https://tamnd.github.io/math2/node/NRG03YP2PW.md" tag="NRG03YP2PW" kind="concept">
# Fourier transform method for convolution equations

## Summary

The Fourier transform method solves a [[convolution integral equation|EY75YD988Z]] on the whole line or on $\R^n$ by passing to the frequency side.
The [[convolution theorem|QY8ETVDA1Z]] turns the convolution with the kernel into multiplication by the transform of the kernel.
So the integral equation becomes an algebraic equation for the transform of the unknown.
One solves it by division and returns with the [[inverse Fourier transform|JB8QMT8DCH]].
The method also shows when the equation has exactly one solution in $L^2$ for every right side in $L^2$: the symbol $1 - \lambda \hat k(\xi)$ must not vanish.

## Explanations

### L1 human

A convolution mixes the values of a function along the whole line, which makes the equation hard to solve point by point.
The [[Fourier transform|DPVM34ADZV]] splits a function into pure waves $e^{i x \xi}$, and a convolution acts on each wave by a simple factor.
So in the wave picture the equation $\varphi - k * \varphi = f$ becomes $(1 - \hat k(\xi)) \hat\varphi(\xi) = \hat f(\xi)$, one equation for each frequency $\xi$.
For example, with $k(s) = \frac{1}{2} e^{-\abs{s}}$ one has $\hat k(\xi) = \frac{1}{1 + \xi^2}$, and dividing by $1 - \hat k(\xi) = \frac{\xi^2}{1 + \xi^2}$ is a problem only at $\xi = 0$.
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<document source="https://tamnd.github.io/math2/node/NRJQP7FXFM.md" tag="NRJQP7FXFM" kind="statement">
# Riemann's existence theorem

## Statement

Let $X$ be a compact connected [[Riemann surface|5K0N8QMV6Q]] and let $B \subset X$ be a finite set.

1. Let $p^\circ \colon Y^\circ \to X \setminus B$ be a finite topological covering map with $Y^\circ$ connected.
Then there is a compact connected Riemann surface $Y$ that contains $Y^\circ$ as the complement of a finite set.
There is a holomorphic [[branched covering|YARENZRXTD]] $p \colon Y \to X$ that extends $p^\circ$.
The pair $(Y, p)$ is unique up to a unique isomorphism over $X$ that is the identity on $Y^\circ$.
2. As a consequence, take $X = \hat{\C}$ and $B = \set{b_1, \ldots, b_n}$, and let $d \ge 1$.
Consider the connected holomorphic branched coverings $Y \to \hat{\C}$ of degree $d$ that are unbranched outside $B$, up to isomorphism over $\hat{\C}$.
Consider the tuples $(\sigma_1, \ldots, \sigma_n)$ in the [[symmetric group|0515A6D2Y1]] $S_d$ with $\sigma_1 \cdots \sigma_n = 1$ that generate a [[transitive|02GK6HZYBT]] subgroup.
Call two such tuples equivalent when they are conjugate by one element of $S_d$.
The coverings correspond one to one to the equivalence classes of such tuples.

## Remarks

For part 1, the complex structure of $X$ pulls back to $Y^\circ$ through the local homeomorphism $p^\circ$.
Near a point $b \in B$, each component of the preimage of a small punctured disc is a punctured disc that covers it like $z \mapsto z^e$, and one fills in the puncture.
Part 2 follows by applying part 1 to the coverings of $\hat{\C} \setminus B$, which correspond to transitive actions of the [[fundamental group|S1XK3HXHFX]].
That group is generated by loops $\gamma_1, \ldots, \gamma_n$ around the points $b_i$ with the single relation $\gamma_1 \cdots \gamma_n = 1$, and $\sigma_i$ is the permutation of the $d$ sheets given by $\gamma_i$.

Example: for $n = 2$, $B = \set{0, \infty}$ and $\sigma_1 = (1\,2\,\cdots\,d)$, $\sigma_2 = \sigma_1^{-1}$, the covering is $z \mapsto z^d$.

Since every compact Riemann surface carries enough meromorphic functions to be an algebraic curve, every such datum is realized by a polynomial equation $F(z, w) = 0$.

## Explanations

### L1 human

The theorem says that one can build Riemann surfaces from purely combinatorial data.
To describe a surface lying over the sphere with $d$ sheets, one only needs to say how the sheets are permuted when one walks around each branch point.
Any choice of permutations that multiply to the identity and connect all sheets gives a surface, and only one.

### L2 human

This is the link between the topology of coverings and the algebra of function fields.
Compact Riemann surfaces are the same as smooth projective complex curves.
So the theorem says that the finite extensions of $\C(z)$ unramified outside $B$ correspond to conjugacy classes of finite-index subgroups of the free group $\pi_1(\hat{\C} \setminus B)$.
It is used to count coverings, as in Hurwitz numbers, and in inverse Galois theory to show that every finite group is a Galois group over $\C(z)$.
Together with the [[uniformization theorem|5EJ7SW1RCV]] it is one of the main existence results for Riemann surfaces.
</document>

<document source="https://tamnd.github.io/math2/node/NRP019D9AE.md" tag="NRP019D9AE" kind="definition">
# Uniform module

## Definition

Let $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].
The module $M$ is *uniform* if:

1. $M \ne 0$;
2. $N_1 \cap N_2 \ne 0$ for all nonzero [[submodules|0RGZ10DFPG]] $N_1, N_2$ of $M$.

## Remarks

Equivalently, $M \ne 0$ and every nonzero submodule of $M$ is [[essential|8MS51FASMX]] in $M$.
Example: the $\Z$-module $\Z$ is uniform.
The check: nonzero submodules $a\Z$ and $b\Z$ both contain $ab \ne 0$.
Example: $\Z/p^n\Z$, for $p$ prime and $n \ge 1$, is uniform, since its subgroups form a chain.
Non-example: $\Z \oplus \Z$ fails clause 2, since $\Z \oplus 0$ and $0 \oplus \Z$ meet only in $0$.
Every uniform module is [[indecomposable|E2P29GCD9T]].
The converse fails: let $K$ be a field and $A = K[x, y]/(x^2, xy, y^2)$.
As a module over itself, $A$ is indecomposable, because $A$ is a commutative local ring and so has no idempotents other than $0$ and $1$.
It is not uniform, since its submodules $Ax = Kx$ and $Ay = Ky$ meet only in $0$.
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<document source="https://tamnd.github.io/math2/node/NRPQP49N9F.md" tag="NRPQP49N9F" kind="statement">
# Dvoretzky's theorem

## Statement

Let $k \ge 1$ be an integer and let $\varepsilon > 0$.
Then there is an integer $N = N(k, \varepsilon)$ with the following property.
Let $X$ be a real [[normed space|03J5RAKB30]] of dimension at least $N$.
Then $X$ has a $k$-dimensional [[linear subspace|00YC1D652K]] $E$ with a linear map $T \colon \R^k \to E$ such that
$$\abs{x} \le \norm{T x} \le (1 + \varepsilon) \abs{x} \quad \text{for all } x \in \R^k,$$
where $\abs{x}$ is the Euclidean norm of $x$.

## Remarks

The theorem was proved by Dvoretzky.
V. Milman gave a proof based on concentration of measure on the high-dimensional sphere.
His proof shows that one can take $k$ of order $c(\varepsilon) \log n$ in a space of dimension $n$.
The idea: a norm, seen as a Lipschitz function on the Euclidean sphere, is almost constant on most of the sphere.
A random $k$-dimensional subspace then sees only this almost constant part.

The bound $\log n$ cannot be improved in general.
For $\ell_\infty^n$, that is $\R^n$ with the max norm, the largest $k$ with a subspace at distance at most $2$ from Euclidean is of order $\log n$.
For $\ell_1^n$, in contrast, there are almost Euclidean subspaces of dimension proportional to $n$.

## Explanations

### L1 human

Every convex body that is symmetric about the origin and lives in very many dimensions has a slice through the center that is almost a round ball.
The slice can be large if the dimension is large.
So round balls, that is Euclidean geometry, appear inside every normed space of high dimension.

### L2 human

In the language of the local theory of Banach spaces, the theorem says that $\ell_2$ is finitely representable in every infinite-dimensional Banach space.
It is a starting point of asymptotic geometric analysis, together with the notions of [[type and cotype|BF9XR5SVPV]].
The distance used here is the Banach–Mazur distance of the [[Banach–Mazur compactum|RH1WTN38RQ]].
</document>

<document source="https://tamnd.github.io/math2/node/NRZC1SY4S3.md" tag="NRZC1SY4S3" kind="definition">
# Saccheri quadrilateral

## Definition

Let $A$, $B$, $C$, $D$ be four points of an [[absolute plane|VKFPCSRNDH]].
Assume that $C$ and $D$ lie on the same side of the line $AB$.
The quadrilateral $ABCD$ is a *Saccheri quadrilateral* with base $AB$ if:

1. the [[angles|0GJAJC6TNZ]] $\angle DAB$ and $\angle ABC$ are right angles;
2. the legs $AD$ and $BC$ are congruent.

The side $CD$ is the summit, and the angles $\angle BCD$ and $\angle CDA$ are the summit angles.

## Remarks

Example: in $\R^2$, the points $A = (0,0)$, $B = (2,0)$, $C = (2,1)$, $D = (0,1)$ form a Saccheri quadrilateral, since both base angles are right and both legs have length $1$.
Non-example: $A = (0,0)$, $B = (2,0)$, $C = (2,1)$, $D = (0,3)$ has right base angles but legs of lengths $1$ and $3$, so clause 2 fails.
In every absolute plane the summit angles are congruent, and the segment joining the midpoints of base and summit is perpendicular to both.
The summit angles are right in the Euclidean plane and acute in the hyperbolic plane.
</document>

<document source="https://tamnd.github.io/math2/node/NS1HWGTD8E.md" tag="NS1HWGTD8E" kind="concept">
# Hyperarithmetical theory

## Summary

Hyperarithmetical theory studies the hyperarithmetical sets of natural numbers, those computable from an iterate of the [[Turing jump|60P2XWY5QX]] along a computable ordinal.
The computable ordinals are those below the Church–Kleene ordinal $\omega_1^{CK}$, the least [[ordinal|0YSCSX5YZC]] that is not the order type of a computable well-ordering of $\N$.
By a theorem of Kleene, the hyperarithmetical sets are exactly the $\Delta^1_1$ sets of the [[analytical hierarchy|M6JGF4TPM6]].
The theory connects [[computability theory|X14Y880PN5]], descriptive set theory and the study of subsystems of second-order arithmetic.

## Explanations

### L1 human

The [[arithmetical hierarchy|7CVJCQKY41]] uses the jumps $\emptyset', \emptyset'', \ldots$, one for each natural number.
One can go further: join all of them into one set $\emptyset^{(\omega)}$, take its jump, and continue.
Doing this along every ordinal that a computer can describe gives the hyperarithmetical sets.
The set $\emptyset^{(\omega)} = \set{(n, x) : x \in \emptyset^{(n)}}$ is hyperarithmetical but not arithmetical.
These are the sets that can be defined by a single existential, and also by a single universal, quantifier over sets of numbers.
</document>

<document source="https://tamnd.github.io/math2/node/NS2057MFWP.md" tag="NS2057MFWP" kind="concept">
# Lattice in a locally compact group

## Summary

A lattice in a [[locally compact group|AHV05ZBCYW]] $G$ is a [[discrete subgroup|5M7D1BSHVZ]] $\Gamma$ of finite covolume.
That is, the quotient $G/\Gamma$ has finite volume for a $G$-invariant measure.
It is cocompact, or uniform, when $G/\Gamma$ is compact.
Lattices are discrete approximations of $G$; the integer lattice $\Z^n$ in $\R^n$ and the modular group $SL_2(\Z)$ in $SL_2(\R)$ are the guiding examples.
They link group theory with number theory, geometry and ergodic theory.

## Explanations

### L1 human

The integer points $\Z^2$ form a grid in the plane, and the grid squares tile the plane by shifts.
Each tile has area $1$, so the plane "folded up" by the grid, which is a torus, has finite area.
A lattice in a general group is the same idea: a discrete subgroup whose translates of one region of finite volume cover the group.
The group $SL_2(\Z)$ of integer matrices with determinant $1$ sits inside $SL_2(\R)$ in this way, even though the folded-up space is not compact.
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<document source="https://tamnd.github.io/math2/node/NS60VV786W.md" tag="NS60VV786W" kind="concept">
# Torus knot

## Summary

A torus knot is a [[knot|QYEAZTYGPT]] that lies on the surface of a standard unknotted [[torus|03NM7DZXXC]] in space.
For coprime integers $p$ and $q$, the torus knot $T(p, q)$ winds $p$ times around the axis of the torus and $q$ times around its tube.
Torus knots form one of the three classes in Thurston's division of knots into torus, [[satellite|A1E8TDEVJ9]] and [[hyperbolic|K47G2Q24DT]] knots.
Many invariants of torus knots can be computed exactly.

## Explanations

### L1 human

Picture a doughnut and wrap a string around its surface without crossing itself.
Go around the hole $p$ times while going through the hole $q$ times, then close the string up.
When $p$ and $q$ have no common factor you get a single loop, the torus knot $T(p, q)$.
The [[trefoil knot|VYTYSVHN4A]] is $T(2, 3)$: twice around the hole and three times through it.
If $p$ or $q$ is $\pm 1$, the loop can be slid off the doughnut and untied.
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<document source="https://tamnd.github.io/math2/node/NS84D2920R.md" tag="NS84D2920R" kind="definition">
# Completion of a ring

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $I$ be an [[ideal|096Z9MP2ME]] of $R$.
For $m \ge n$ let $\pi_{mn} \colon R/I^m \to R/I^n$ be the natural map of [[quotient rings|0076HWHGHQ]].
The *$I$-adic completion* of $R$ is the [[inverse limit|DZB378CNCK]]
$$\hat R = \varprojlim_n R/I^n = \set{(x_n)_{n \ge 1} \in \prod_{n \ge 1} R/I^n : \pi_{mn}(x_m) = x_n \text{ for all } m \ge n},$$
with componentwise addition and multiplication.

## Remarks

The map $R \to \hat R$, $r \mapsto (r + I^n)_n$, is a ring homomorphism with kernel $\bigcap_n I^n$.
Equivalently, $\hat R$ is the ring of Cauchy sequences for the [[$I$-adic topology|2VF2TSJ3PY]] modulo the sequences that tend to $0$.
For an $R$-module $M$ one defines $\hat M = \varprojlim_n M/I^n M$ in the same way.
Example: for $R = k[x]$, with $k$ a field, and $I = (x)$, a compatible sequence is the list of truncations of a formal power series, and $\hat R \cong k[\![x]\!]$.
Non-example: for $R = \Z$ and $I = 2\Z$, the sequence starting with $x_1 = 0 + 2\Z$ and $x_2 = 1 + 4\Z$ does not give an element of $\hat R$, since $\pi_{21}(x_2) = 1 + 2\Z \neq x_1$.
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<document source="https://tamnd.github.io/math2/node/NS852SSQ00.md" tag="NS852SSQ00" kind="definition">
# Symmetric function

## Definition

Let $x_1, x_2, \ldots$ be countably many commuting variables, and consider [[formal power series|033WGN00BF]] in them with integer coefficients.
A *symmetric function* is such a series $f$ with these properties:

1. $f$ has bounded degree: for some integer $d$, every monomial of $f$ has total degree at most $d$;
2. $f$ is invariant under every permutation of the variables.
   Thus $x_{i_1}^{a_1} \cdots x_{i_r}^{a_r}$ and $x_{j_1}^{a_1} \cdots x_{j_r}^{a_r}$ have equal coefficients when the $i_s$ are distinct and the $j_s$ are distinct.

The symmetric functions form a ring, written $\Lambda$.

## Remarks

The ring $\Lambda$ is graded, $\Lambda = \bigoplus_{k \ge 0} \Lambda^k$, where $\Lambda^k$ consists of the homogeneous symmetric functions of degree $k$.
For a partition $\lambda = (\lambda_1, \ldots, \lambda_r)$, the monomial symmetric function $m_\lambda$ is the sum of all distinct monomials $x_{i_1}^{\lambda_1} \cdots x_{i_r}^{\lambda_r}$ with distinct indices.
The $m_\lambda$ with $\lambda$ a partition of $k$ form a $\Z$-basis of $\Lambda^k$.
Setting $x_{n+1} = x_{n+2} = \cdots = 0$ maps $\Lambda$ onto the ring of [[symmetric polynomials|0VHSHB9B6J]] in $x_1, \ldots, x_n$.
Equivalently, $\Lambda^k$ is the inverse limit of the degree-$k$ symmetric polynomials in $n$ variables, as $n \to \infty$.

Example: the power sum $p_2 = x_1^2 + x_2^2 + x_3^2 + \cdots$ is a symmetric function of degree $2$.
Both clauses hold: all monomials have degree $2$, and a permutation of the variables only permutes the terms.

Non-example: the product $\prod_{i \ge 1} (1 + x_i) = 1 + e_1 + e_2 + \cdots$ is invariant under all permutations of the variables, but it fails clause 1.
It contains the monomial $x_1 x_2 \cdots x_k$ of degree $k$ for every $k$.

The elementary symmetric functions $e_k = \sum_{i_1 < \cdots < i_k} x_{i_1} \cdots x_{i_k}$ are algebraically independent, and $\Lambda = \Z[e_1, e_2, \ldots]$.
The same holds for the complete homogeneous symmetric functions $h_k$, the sum of all monomials of degree $k$.
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<document source="https://tamnd.github.io/math2/node/NS98MJFMYR.md" tag="NS98MJFMYR" kind="definition">
# Functor of points

## Definition

Let $\mathrm{Sch}$ be the category of [[schemes|5A35TYZD02]] and [[morphisms of schemes|JXRKH6P1Z4]], and let $X$ be a scheme.
The *functor of points* of $X$ is the [[functor|28NQ0KRHD3]]
$$h_X \colon \mathrm{Sch}^{\mathrm{op}} \to \mathrm{Set}, \quad h_X(T) = \Hom_{\mathrm{Sch}}(T, X),$$
on the [[opposite category|DAMK8XE1FR]] of $\mathrm{Sch}$.
For a morphism $u \colon T' \to T$, the map $h_X(u)$ sends $a \colon T \to X$ to $a \circ u$.
An element of $h_X(T)$ is a *$T$-point* of $X$; for a ring $A$, one writes $X(A) = h_X(\operatorname{Spec} A)$.

## Remarks

By the [[Yoneda lemma|D6DF1ARPVD]], the rule $X \mapsto h_X$ is fully faithful, so a scheme is determined up to unique isomorphism by its functor of points.
It suffices to know $X(A)$ for all commutative rings $A$.
Example: for $X = \mathbb{A}^n_\Z = \operatorname{Spec} \Z[x_1, \ldots, x_n]$, a morphism $\operatorname{Spec} A \to X$ is a ring map $\Z[x_1, \ldots, x_n] \to A$.
Such a map is given by the images of the $x_i$, so $X(A) = A^n$.
Non-example: the constant functor $F$ with $F(T) = \set{0, 1}$ for all $T$ is not of the form $h_X$.
For the empty scheme $\emptyset$, the set $h_X(\emptyset)$ has exactly one element for every $X$, while $F(\emptyset)$ has two.
A functor isomorphic to some $h_X$ is called [[representable|KCCHH87H6E]], and $X$ then represents it.
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# Kuratowski closure axioms

## Summary

The Kuratowski closure axioms are four rules for an operation that sends each subset of a set to a larger subset.
An operation that obeys them is a closure operator, and the closure operators on a set correspond exactly to the topologies on that set.
So a [[topological space|0CQHFPGJ45]] can be described by its [[closure|0QEAJB91GZ]] operation instead of by its [[open sets|02BH3MCBQB]].

## Explanations

### L1 human

The closure of a set adds to it all the points that are "infinitely close" to it.
Kuratowski picked out four simple rules that this operation always obeys.
Nothing comes from nothing, and a set lies in its closure.
Closing twice is the same as closing once, and the closure of a union is the union of the closures.
Any operation with these four rules comes from one and only one topology.
For example, on $\R$ the closure of the open interval $(0, 1)$ is $[0, 1]$, and closing $[0, 1]$ again gives nothing new.
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<document source="https://tamnd.github.io/math2/node/NSBYE143KS.md" tag="NSBYE143KS" kind="concept">
# Bayes factor

## Summary

The Bayes factor of two models is the ratio of the probabilities that each model gives to the observed data.
In each model the probability is averaged over the prior of its parameters.
It is the factor by which the data change the odds between the two models.
It is the Bayesian tool to compare hypotheses and models, an alternative to [[p-values|0GGKJGZF6D]] and likelihood-ratio tests.

## Explanations

### L1 human

Two theories compete to explain the same data.
Each theory predicts the data with some probability, averaged over what it leaves uncertain.
The Bayes factor is the ratio of these two predictions: a value of $10$ means that the data were $10$ times more expected under the first theory.
Whatever odds you gave the theories before, you multiply them by the Bayes factor to get your odds after seeing the data.
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<document source="https://tamnd.github.io/math2/node/NSCEBXM7ZM.md" tag="NSCEBXM7ZM" kind="concept">
# Hilbert transform

## Summary

The Hilbert transform is the linear operator on functions of a real variable given by convolution with $\frac{1}{\pi x}$ in the [[principal value|0SYR10HD4R]] sense.
Let $u + iv$ be holomorphic on the upper half-plane and decay at infinity.
Then the transform sends the boundary values of $u$ to those of $v$, so it is the operator of [[harmonic conjugation|050F2FMTKF]] on the line.
On the Fourier side it multiplies by $-i \sgn(\xi)$.
It is the model example of a singular integral operator, and it appears in signal processing and in the [[Kramers–Kronig relations|GFS95PZ5YK]].

## Explanations

### L1 human

The Hilbert transform shifts the phase of each frequency component of a signal by a quarter turn.
It sends $\cos$ to $\sin$ and $\sin$ to $-\cos$.
If $u + iv$ is holomorphic in the upper half-plane and decays at infinity, then on the real line $v$ is the Hilbert transform of $u$.
For example, $\frac{i}{x + i} = \frac{1}{1 + x^2} + i \frac{x}{1 + x^2}$, so the transform of $\frac{1}{1 + x^2}$ is $\frac{x}{1 + x^2}$.
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<document source="https://tamnd.github.io/math2/node/NSK7247MKX.md" tag="NSK7247MKX" kind="concept">
# Supermartingale

## Summary

A supermartingale is an adapted, integrable [[stochastic process|0QH5ENKZM1]] whose [[conditional expectation|0TTDGTEFMN]] of a future value, given the present information, is at most the present value.
It models the fortune of a player in a game that is unfavourable or fair.
A process is a supermartingale exactly when its negative is a [[submartingale|6M852B4RA8]].
Non-negative supermartingales converge almost surely, a fact used throughout probability and potential theory.

## Explanations

### L1 human

A supermartingale tends to go down, or at least not up, on average.
It is the right model for your money in a casino, where every game is slightly in favour of the house.
Given what you know now, the expected value of your fortune later is at most what you hold now.
For example, bet one unit on each toss of a coin that lands heads with probability $0.45$, and win the unit on heads or lose it on tails.
Your fortune after each toss is a supermartingale, since each bet loses $0.1$ on average.
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<document source="https://tamnd.github.io/math2/node/NSS3XW4BYZ.md" tag="NSS3XW4BYZ" kind="statement">
# Cantor's uniqueness theorem for trigonometric series

## Statement

Let $(c_n)_{n \in \Z}$ be complex numbers, and consider the [[trigonometric series|FZK2M8KGXQ]] $\sum_{n \in \Z} c_n e^{inx}$.
Assume that for every $x \in \R$ its symmetric partial sums converge to $0$:
$$\lim_{N \to \infty} \sum_{n=-N}^{N} c_n e^{inx} = 0.$$
Then $c_n = 0$ for every $n \in \Z$.

## Remarks

Equivalently, if two trigonometric series converge at every point to the same sum, then they have the same coefficients.
So a function has at most one representation by an everywhere convergent trigonometric series.

Proof idea, following Riemann's method.
First, the terms $c_n e^{inx} + c_{-n} e^{-inx}$ tend to $0$ for every $x$, so $c_n \to 0$ as $\abs{n} \to \infty$ by the [[Cantor–Lebesgue theorem|8Z55VMJSEX]].
Then the function $F(x) = \frac{c_0 x^2}{2} - \sum_{n \ne 0} \frac{c_n}{n^2} e^{inx}$ is continuous, and Riemann's first lemma shows that its second symmetric derivative at $x$ equals the sum of the series at $x$, which is $0$.
A lemma of Schwarz says that a continuous function with vanishing second symmetric derivative is linear, so $F(x) = ax + b$.
Comparing with the periodic part gives $c_0 = 0$, $a = 0$, and then $c_n = 0$ for all $n \ne 0$ by uniqueness of the coefficients of a uniformly convergent series.

Cantor proved the theorem in 1870 and soon allowed finitely many exceptional points.
In the language of [[sets of uniqueness|K7CNWXF9SN]], the theorem says that the empty set is a set of uniqueness.
The hypothesis of convergence at all points cannot be replaced by convergence almost everywhere.
Menshov constructed a nonzero trigonometric series that converges to $0$ outside a closed set of measure zero.

## Explanations

### L1 human

If a sum of waves is equal to $0$ at every point, every single wave in it must be $0$.
For finite sums this is easy linear algebra, but for infinite sums that only converge, it is a deep fact.
It means that a function can be written as a convergent trigonometric series in at most one way.

### L2 human

The theorem is about arbitrary trigonometric series, not only Fourier series, so no integral formula for the coefficients is available.
Riemann's idea of integrating the series twice turns the problem into one about continuous functions.
Trying to allow larger exceptional sets, Cantor was led to derived sets, transfinite ordinals and set theory.
The modern theory of sets of uniqueness continues this question.
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<document source="https://tamnd.github.io/math2/node/NSTD00PJPE.md" tag="NSTD00PJPE" kind="statement">
# Fenchel–Young inequality

## Statement

Let $X$ be a real [[normed space|03J5RAKB30]] with [[continuous dual space|4A82S5FVRT]] $X^*$.
Let $f \colon X \to (-\infty, +\infty]$ be a function that is not identically $+\infty$, and let $f^*$ be its [[Fenchel conjugate|T7V2VPY61S]].
Then for all $x \in X$ and $y \in X^*$,
$$f(x) + f^*(y) \ge y(x).$$
Moreover, let $x$ be a point with $f(x)$ finite.
Then equality holds if and only if $y$ is a subgradient of $f$ at $x$, that is, $f(z) \ge f(x) + y(z - x)$ for all $z \in X$.

## Remarks

Since $f$ is finite somewhere, $f^*$ never takes the value $-\infty$, so the left side is a well-defined element of $(-\infty, +\infty]$.
The proof is one line: $f^*(y) \ge y(x) - f(x)$ by the definition of $f^*$ as a supremum over $x$.
Equality means that $x$ attains this supremum, which is the subgradient inequality rearranged.
The set of subgradients is the [[subdifferential|9C5WXYYQEK]] $\partial f(x)$.

Example: for $p > 1$ and $q = p/(p - 1)$, the function $f(x) = \abs{x}^p / p$ on $\R$ has $f^*(y) = \abs{y}^q / q$.
The inequality becomes Young's inequality $xy \le \abs{x}^p/p + \abs{y}^q/q$, with equality exactly when $y = \sgn(x) \abs{x}^{p-1}$.
For $p = 2$ it is $xy \le \tfrac12 x^2 + \tfrac12 y^2$, which is $(x - y)^2 \ge 0$.

## Explanations

### L1 human

The conjugate $f^*$ is built so that this inequality holds: it is the smallest function with $f(x) + f^*(y) \ge xy$ for all $x$.
It generalizes the inequality $xy \le \tfrac12 x^2 + \tfrac12 y^2$ and Young's inequality for products.
Equality picks out the pairs where $y$ is a slope of $f$ at $x$.

### L2 human

The inequality gives weak duality in convex optimization: the value of a dual problem never exceeds the value of the primal problem.
Its equality case links the conjugate with the subdifferential: $y \in \partial f(x)$ if and only if $f(x) + f^*(y) = y(x)$.
For proper convex lower semicontinuous $f$, the Fenchel–Moreau theorem then gives the symmetric statement $x \in \partial f^*(y)$, so $\partial f^*$ is the inverse of $\partial f$.
It is the source of many classical inequalities obtained by choosing $f$.
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<document source="https://tamnd.github.io/math2/node/NSVEH6VMX8.md" tag="NSVEH6VMX8" kind="concept">
# Equicontinuous system

## Summary

An equicontinuous system is a topological dynamical system whose family of iterates $\set{f^n : n \ge 0}$ is [[equicontinuous|0XN66MZN0Y]].
Then points that start close stay close for all future times, uniformly in time.
Isometries are the typical examples, such as rotations of the circle or translations on a compact group.
Equicontinuity is the opposite of sensitive dependence on initial conditions, so equicontinuous systems are the most orderly ones in [[topological dynamics|5X6E97EMFR]].

## Explanations

### L1 human

In an equicontinuous system, if two starting points are close enough, their orbits remain close forever, and the needed closeness does not depend on how long one waits.
Rotating a circle is the model: it keeps all distances exactly.
The doubling map $x \mapsto 2x \bmod 1$ is not equicontinuous: two points at distance $\delta$ move to distance $2\delta$, then $4\delta$, until they are far apart.
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<document source="https://tamnd.github.io/math2/node/NSXT1MYFY8.md" tag="NSXT1MYFY8" kind="definition">
# Littlewood–Paley decomposition

## Definition

Let $n \ge 1$, and let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi} \, dx$ be the [[Fourier transform|DPVM34ADZV]], extended to [[tempered distributions|AES2T5JPFP]].
Fix a smooth function $\chi \colon \R^n \to [0, 1]$ with $\chi(\xi) = 1$ for $\abs{\xi} \le 1$ and $\chi(\xi) = 0$ for $\abs{\xi} \ge 2$.
Let $\psi(\xi) = \chi(\xi) - \chi(2\xi)$.
For $j \in \Z$ and $f$ a tempered distribution, the *Littlewood–Paley piece* $\Delta_j f$ is defined by
$$\widehat{\Delta_j f}(\xi) = \psi(2^{-j}\xi) \, \hat f(\xi).$$
The *Littlewood–Paley decomposition* of $f$ is the series $\sum_{j \in \Z} \Delta_j f$.

## Remarks

The function $\psi$ vanishes for $\abs{\xi} \le 1/2$ and for $\abs{\xi} \ge 2$, so $\widehat{\Delta_j f}$ is supported in $\set{2^{j-1} \le \abs{\xi} \le 2^{j+1}}$.
For $\xi \ne 0$ the sum telescopes: $\sum_{j = -N}^{M} \psi(2^{-j}\xi) = \chi(2^{-M}\xi) - \chi(2^{N+1}\xi)$, which tends to $1$ as $M, N \to \infty$.
Hence $f = \sum_{j \in \Z} \Delta_j f$ with convergence in $L^2(\R^n)$ for $f \in L^2(\R^n)$, and in the [[Lp space|SASP14MA1Z]] $L^p(\R^n)$ for $1 < p < \infty$.
Each $\Delta_j f = f * \check\psi_j$ is a convolution with the Schwartz function $\check\psi_j(x) = 2^{jn} \check\psi(2^j x)$, where $\check\psi$ is the inverse Fourier transform of $\psi$.
An inhomogeneous version keeps $\Delta_j$ for $j \ge 1$ and replaces all $j \le 0$ by the single low-frequency piece with multiplier $\chi$.

Example: if $\hat f$ vanishes outside $\set{2^{k-1} \le \abs{\xi} \le 2^{k+1}}$ for some $k \in \Z$, then $\Delta_j f = 0$ unless $j \in \set{k - 1, k, k + 1}$.
Indeed $\psi(2^{-j}\xi) \ne 0$ needs $2^{j-1} < \abs{\xi} < 2^{j+1}$, and this meets the support of $\hat f$ only when $k - 2 < j < k + 2$.

Non-example: the cut-offs $\mathbf{1}_{\set{2^j \le \abs{\xi} < 2^{j+1}}}$ also sum to $1$ on $\xi \ne 0$, but they are not smooth, so they do not fit this definition.
For $n \ge 2$ such sharp cut-offs are badly behaved on $L^p(\R^n)$ for $p \ne 2$, as Fefferman's ball multiplier theorem shows for the indicator of a ball.
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<document source="https://tamnd.github.io/math2/node/NSYMQMHPDN.md" tag="NSYMQMHPDN" kind="concept">
# Geodesic

## Summary

A geodesic on a [[Riemannian manifold|7CBT4SM83Y]] is a curve whose velocity is parallel along itself for the [[Levi-Civita connection|CAZYGE1T4E]].
So it has zero acceleration within the manifold.
Geodesics generalise straight lines: short pieces of a geodesic are the shortest curves between their endpoints.
They are solutions of a second-order ordinary differential equation, so a geodesic is determined by its starting point and starting velocity.
Geodesics are the basic tool for studying distance, curvature and topology of Riemannian manifolds.

## Explanations

### L1 human

A geodesic is the path of someone who walks straight ahead on a curved surface without ever turning left or right.
On a plane, geodesics are straight lines traversed at constant speed.
On a round sphere, geodesics are great circles, such as the equator or the meridians.
Airline routes follow great circles because, on the sphere, short arcs of great circles are the shortest paths.
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<document source="https://tamnd.github.io/math2/node/NT44831EFE.md" tag="NT44831EFE" kind="concept">
# Successor order

## Summary

A successor order is a partial order with a successor map.
This map sends each non-maximal element to the least element strictly above it.
It abstracts the [[successor function|0AK9TGSNXY]] $n \mapsto n + 1$ on the natural numbers.
The natural numbers, the integers, and every well-ordered set carry a successor order, while the rationals do not.

## Explanations

### L1 human

In the integers every number has a next number: after $n$ comes $n + 1$, and nothing lies strictly between them.
A successor order is an order with such a "next element" map.
In the rationals there is no next number after $0$, since between $0$ and any positive rational there is another rational.
So the rationals have no successor order.
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<document source="https://tamnd.github.io/math2/node/NT4797Z9H8.md" tag="NT4797Z9H8" kind="concept">
# Hermitian symmetric space

## Summary

A Hermitian symmetric space is a [[symmetric space|PAA93R4WYD]] with a compatible complex structure, for which every geodesic symmetry is holomorphic.
Examples are the complex projective spaces, the Grassmannians, the unit disc and the Siegel upper half space.
Such a space is automatically a Kähler manifold.
The irreducible ones of noncompact type are bounded symmetric domains in $\C^n$, and they are the home of automorphic forms in several variables.

## Explanations

### L1 human

Take a symmetric space, where each point has a mirror symmetry through it.
Suppose that the space also has complex coordinates, and that all these symmetries respect them.
Then it is a Hermitian symmetric space.
The unit disc with its hyperbolic metric is the main example: the symmetry through $0$ is $z \mapsto -z$, which is holomorphic.
The upper half plane, where modular forms live, is the same space in other coordinates.
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<document source="https://tamnd.github.io/math2/node/NT4SKSYKG9.md" tag="NT4SKSYKG9" kind="statement">
# Minkowski–Carathéodory theorem

## Statement

Let $n \ge 1$ and let $K \subseteq \R^n$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].
Let $\operatorname{ext} K$ be the set of [[extreme points|3MSGFJM2NX]] of $K$.
Then $K$ is the [[convex hull|0FQSDC2EQ2]] of $\operatorname{ext} K$:
$$K = \operatorname{conv}(\operatorname{ext} K).$$
Moreover, every point of $K$ is a [[convex combination|21SXYJYC59]] of at most $n + 1$ extreme points of $K$.

## Remarks

Proof idea: induct on the dimension of $K$.
If $x$ lies on the relative boundary of $K$, the [[supporting hyperplane theorem|DR9BXFWMQB]] gives a hyperplane $H$ through $x$.
Then $K \cap H$ is a [[face|KY7SB854VH]] of lower dimension whose extreme points are extreme points of $K$.
If $x$ lies in the relative interior, a line through $x$ meets $K$ in a segment with both endpoints on the relative boundary.
Then $x$ is a convex combination of these endpoints.
The bound $n + 1$ then follows from [[Carathéodory's theorem|0201Q4P2B5]].

Example: a filled triangle is the convex hull of its three vertices, and a closed disc is the convex hull of its boundary circle.

Both hypotheses on $K$ are needed.
The closed half-plane $\set{(x, y) : y \ge 0}$ is closed and convex but has no extreme points.
The open disc is bounded and convex but has no extreme points either.

## Explanations

### L1 human

A compact convex shape, that is, a closed and bounded one, is filled in by its corners.
Here a corner means an extreme point, a point not in the middle of any segment of the shape.
For a polygon the corners are the vertices; for a disc every boundary point counts.
In $\R^n$, every point of the shape is a weighted average of at most $n + 1$ corners, for example at most three in the plane.

### L2 human

This is the finite-dimensional case of the [[Krein–Milman theorem|RSRAY0Q68S]].
In $\R^n$ no closure is needed: the convex hull of $\operatorname{ext} K$ is already all of $K$, although $\operatorname{ext} K$ itself need not be closed when $n \ge 3$.
It explains why a linear function on a compact convex set attains its maximum at an extreme point, which is the basis of the simplex method in linear programming.
Applied to polytopes, it says that a bounded intersection of finitely many half-spaces is the convex hull of its vertices.
</document>

<document source="https://tamnd.github.io/math2/node/NT564Y315M.md" tag="NT564Y315M" kind="statement">
# Commutative Gelfand–Naimark theorem

## Statement

Let $A$ be a commutative C*-algebra, and let $\Delta(A)$ be its [[character space|M7JVFJ3MMX]].
Let $C_0(\Delta(A))$ be the C\*-algebra of [[continuous functions vanishing at infinity|N59BW1QFD1]] on $\Delta(A)$, with the sup norm and complex conjugation.
Then the [[Gelfand transform|VWNSHKJT92]] $\Gamma \colon A \to C_0(\Delta(A))$, $\Gamma(a)(\varphi) = \varphi(a)$, is an isometric $*$-isomorphism of $A$ onto $C_0(\Delta(A))$.

## Remarks

If $A$ has a unit, $\Delta(A)$ is compact and $A \cong C(\Delta(A))$.
Proof idea: a character takes real values on self-adjoint elements, so $\Gamma(a^*) = \overline{\Gamma(a)}$.
For self-adjoint $a$ the C\* identity gives $\norm{a^{2^k}} = \norm{a}^{2^k}$, so by [[Gelfand's spectral radius formula|T4RFYT8HDN]] $\norm{\Gamma(a)}_\infty = \norm{a}$; applying this to $a^* a$ shows that $\Gamma$ is isometric for all $a$.
So the range of $\Gamma$ is a closed $*$-subalgebra that separates points and vanishes at no point, and by the [[Stone–Weierstrass theorem|1P8D3MK79H]] it is all of $C_0(\Delta(A))$.

Example: for $A = C(X)$ with $X$ compact Hausdorff, $\Delta(A)$ is homeomorphic to $X$ through the evaluations, and $\Gamma$ is the identity map up to this homeomorphism.

In the unital case, the theorem gives a contravariant equivalence between commutative unital C\*-algebras with unital $*$-homomorphisms and compact Hausdorff spaces with continuous maps.

## Explanations

### L1 human

Every commutative C\*-algebra is an algebra of continuous functions on some space, with nothing lost.
The space is recovered from the algebra as the set of its characters.
So studying commutative C\*-algebras is the same as studying locally compact Hausdorff spaces.

### L2 human

This theorem is the starting point of "noncommutative topology": a general C\*-algebra is thought of as the functions on a space that does not exist.
It gives the [[continuous functional calculus|JB7S6WPZ77]].
For a normal element $a$, the C\*-algebra generated by $a$ and $1$ is commutative, and its character space is the spectrum of $a$.
So it is isomorphic to $C(\sigma(a))$.
The noncommutative Gelfand–Naimark theorem says that every C\*-algebra is isometrically $*$-isomorphic to a closed $*$-subalgebra of the bounded operators on a Hilbert space.
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<document source="https://tamnd.github.io/math2/node/NT65WHM5VV.md" tag="NT65WHM5VV" kind="example">
# Tetration

## Example

Let $a > 0$ and let $\exp_a(x) = a^x$ be [[real exponentiation|091AV9YHZY]] to base $a$.
For integers $n \ge 1$, the *tetration* ${}^{n}a$ is defined by
$${}^{1}a = a, \qquad {}^{n+1}a = a^{\left({}^{n}a\right)}.$$
So ${}^{n}a$ is a tower of $n$ copies of $a$, evaluated from the top down.
With ${}^{0}a = 1$, one has ${}^{n}a = \exp_a^{\,n}(1)$, the $n$-th [[iterate|AYVTR2ES5B]] of $\exp_a$ applied to $1$.

Check for $a = 2$: ${}^{1}2 = 2$, ${}^{2}2 = 2^2 = 4$, ${}^{3}2 = 2^4 = 16$ and ${}^{4}2 = 2^{16} = 65536$.
And $\exp_2(1) = 2$, $\exp_2(2) = 4$, $\exp_2(4) = 16$, $\exp_2(16) = 65536$, as claimed.

## Remarks

Tetration is the next operation after addition, multiplication and exponentiation: repeated exponentiation, as exponentiation is repeated multiplication.
It grows very fast: ${}^{5}2 = 2^{65536}$ has $19729$ decimal digits.
Defining ${}^{x}a$ for real heights $x$ is a problem of [[fractional iteration|5M3THA1S6E]] of $\exp_a$.
One asks for a function $F$ with $F(0) = 1$ and $F(x + 1) = a^{F(x)}$; its inverse is a solution of the [[Abel equation|68SHB7Z721]] for $\exp_a$.
These conditions alone do not determine $F$, so one adds regularity conditions such as real analyticity.
The infinite tower $a^{a^{a^{\cdots}}}$ is the limit of ${}^{n}a$ as $n \to \infty$; see [[iterated exponentiation|RDC61V812J]].

## Explanations

### L1 human

Multiplication repeats addition, and powers repeat multiplication.
Tetration repeats powers: ${}^{3}2$ means $2^{2^2} = 16$.
The values explode: ${}^{4}2 = 65536$, and ${}^{5}2$ is a number with almost twenty thousand digits.
Each step applies the map $x \mapsto 2^x$ once more, so tetration is a picture of iterating a function.
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# Fibred category

## Summary

A fibred category is a functor $p \colon \mathcal{E} \to \mathcal{B}$ along which objects can be pulled back.
For every object $y$ of $\mathcal{E}$ and every morphism $f$ into $p(y)$, there is a universal, or cartesian, morphism over $f$ ending at $y$.
Fibred categories describe families of categories indexed by the objects of $\mathcal{B}$, such as sheaves or vector bundles over varying spaces.
They are the setting of Grothendieck's descent theory and of stacks.

## Explanations

### L1 human

Think of $\mathcal{B}$ as a category of spaces and of the objects of $\mathcal{E}$ over a space $b$ as structures on $b$, for example vector bundles on $b$.
Given a map of spaces $f \colon b' \to b$ and a bundle on $b$, one can pull the bundle back to $b'$.
A fibred category is the abstract form of this pullback operation.
The pulled-back object is determined up to unique isomorphism by a universal property, not chosen once and for all.
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<document source="https://tamnd.github.io/math2/node/NT73Y95RY5.md" tag="NT73Y95RY5" kind="concept">
# Shallow water equations

## Summary

The shallow water equations describe a layer of fluid whose depth is small compared with the horizontal length scale of its motion.
Their unknowns are the depth of the layer and the horizontal velocity, both depending on horizontal position and time.
They form a system of [[hyperbolic conservation laws|XN081NCPF5]] for mass and momentum.
They model tides, tsunamis, floods and large-scale flows in the atmosphere and the ocean.

## Explanations

### L1 human

When water is shallow compared with the size of its waves, the water in a vertical column moves together.
So one only needs to know how deep the water is and how fast it moves horizontally at each place.
Water flows from where the surface is high to where it is low, and the amount of water is conserved.
Small waves travel at speed $\sqrt{g h}$, where $h$ is the depth and $g$ the gravitational acceleration, so a tsunami in a deep ocean is very fast.
Like other conservation laws, the equations can develop jumps, which model hydraulic jumps and bores.
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<document source="https://tamnd.github.io/math2/node/NT8W7R2KDR.md" tag="NT8W7R2KDR" kind="statement">
# Reiterman's theorem

## Statement

Let $\sigma$ be a finite algebraic signature and let $\mathcal{F}$ be the class of all finite $\sigma$-algebras.
For $n \ge 1$, an *$n$-ary implicit operation* $\pi$ assigns to each $A \in \mathcal{F}$ a map $\pi_A \colon A^n \to A$, subject to one condition.
For every [[homomorphism|KBXSZC6BV1]] $f \colon A \to B$ in $\mathcal{F}$ and all $a_1, \ldots, a_n \in A$, it requires $f(\pi_A(a_1, \ldots, a_n)) = \pi_B(f(a_1), \ldots, f(a_n))$.
A *pseudoidentity* is a pair $\pi \approx \rho$ of $n$-ary implicit operations, and $A \in \mathcal{F}$ satisfies it if $\pi_A = \rho_A$.
Let $V \subseteq \mathcal{F}$.
Then $V$ is a [[pseudovariety|XR4DPKX7T1]] if and only if there is a set $\Sigma$ of pseudoidentities with the following property.
An algebra $A \in \mathcal{F}$ is in $V$ exactly when it satisfies every member of $\Sigma$.

## Remarks

Each term gives an implicit operation, so every [[identity|PKD440G2XW]] is a pseudoidentity, but there are many more.
The $n$-ary implicit operations are the elements of the free profinite $\sigma$-algebra on $n$ generators, the inverse limit of all $n$-generated finite $\sigma$-algebras.
The easy direction holds because satisfaction of a pseudoidentity passes to quotients, subalgebras and finite products.
For the converse, one takes as $\Sigma$ all pseudoidentities true in $V$.
A finite algebra $A$ satisfying them is a continuous image of a free pro-$V$ algebra.
By compactness this map factors through a finite quotient of it that lies in $V$, so $A \in V$.

Example: in finite semigroups, each element $x$ has a unique idempotent power, written $x^\omega$; it is the limit of $x^{n!}$, and $x \mapsto x^\omega$ is a unary implicit operation.
The pseudovariety of finite groups is defined by $x^\omega y \approx y$ and $y x^\omega \approx y$.
The pseudovariety of finite aperiodic semigroups is defined by $x^\omega \approx x^{\omega + 1}$.
Neither class is the class of finite members of a variety of semigroups, so ordinary identities do not suffice.

## Explanations

### L1 human

Birkhoff's theorem says that classes of algebras closed under the usual constructions are exactly those defined by equations.
For finite algebras the equations are not enough.
Reiterman showed that they become enough once one allows "limit" operations, such as taking the idempotent power of an element, which make sense in every finite algebra.

### L2 human

The theorem is the finite counterpart of [[Birkhoff's HSP theorem|5083DHSK82]].
It is a basic tool in the algebraic theory of regular languages: through Eilenberg's correspondence, a class of regular languages is described by pseudoidentities of its syntactic monoids.
Decidability questions for pseudovarieties then become questions about the free profinite semigroup or monoid.
An earlier description, due to Eilenberg and Schützenberger, defines each pseudovariety by a sequence of identities that is eventually satisfied.
</document>

<document source="https://tamnd.github.io/math2/node/NT917KFR57.md" tag="NT917KFR57" kind="concept">
# Kummer's equation

## Summary

Kummer's equation is the second order linear differential equation $z w'' + (b - z) w' - a w = 0$, with complex parameters $a$ and $b$.
It has a [[regular singular point|H96HXG6CA0]] at $0$ and an [[irregular singularity|AT1W4MMTYR]] at $\infty$.
Its solutions are the confluent hypergeometric functions, which include Laguerre polynomials, Hermite functions, the error function and the incomplete gamma function.

## Explanations

### L1 human

Kummer's equation arises when two of the three singular points of the hypergeometric equation are pushed together.
One replaces $z$ by $z / \beta$ in the hypergeometric equation with parameters $a, \beta, b$, and lets $\beta \to \infty$; the singular points $\beta$ and $\infty$ merge into one.
The word "confluent" refers to this merging.
A small example: for $a = b$, the function $w = e^z$ solves the equation, since $z e^z + (b - z) e^z - b e^z = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/NTB5JQGPGC.md" tag="NTB5JQGPGC" kind="definition">
# Resonance

## Definition

Let $p(r) = r^n + a_{n-1} r^{n-1} + \cdots + a_0$ be a polynomial with complex coefficients, and write $D = \frac{d}{dt}$.
Consider the [[linear equation|0QQRP17Z7S]] with constant coefficients and exponential forcing
$$p(D)\, y = e^{\lambda t}, \qquad \lambda \in \C.$$
The equation is *resonant*, and $\lambda$ is a *resonant frequency*, if $p(\lambda) = 0$, that is, if $\lambda$ is a root of the [[characteristic equation|0GZWXM2DKE]] of $p(D) y = 0$.

## Remarks

If $p(\lambda) \ne 0$, then $y = \frac{e^{\lambda t}}{p(\lambda)}$ is a particular solution.
If $\lambda$ is a root of multiplicity $m \ge 1$, then $y = \frac{t^m e^{\lambda t}}{p^{(m)}(\lambda)}$ is a particular solution.
This follows from $p(D)(t^m e^{\lambda t}) = \frac{\partial^m}{\partial \lambda^m} \left( p(\lambda) e^{\lambda t} \right)$ and $p(\lambda) = \cdots = p^{(m-1)}(\lambda) = 0$.

Example: $y'' + \omega_0^2 y = e^{i \omega_0 t}$ with $\omega_0 > 0$ is resonant, since $p(r) = r^2 + \omega_0^2$ vanishes at $i \omega_0$.
Taking real parts, $y'' + \omega_0^2 y = \cos(\omega_0 t)$ has the solution $y = \frac{t \sin(\omega_0 t)}{2 \omega_0}$.
Indeed $y'' = \cos(\omega_0 t) - \frac{\omega_0 t \sin(\omega_0 t)}{2}$, so $y'' + \omega_0^2 y = \cos(\omega_0 t)$; this solution is unbounded.

Non-example: $y'' + y = e^{2 i t}$ is not resonant, since $p(2i) = -3 \ne 0$; the particular solution $-\frac{1}{3} e^{2 i t}$ is bounded.

For a damped oscillator, $p(r) = r^2 + c r + \omega_0^2$ with $c > 0$, the roots of $p$ have negative real part, so a forcing $e^{i \omega t}$ with real $\omega$ is never resonant in this sense.
When the damping is small, the amplitude $\frac{1}{\abs{p(i \omega)}}$ still has a sharp peak near the natural frequency; this is also called resonance.
</document>

<document source="https://tamnd.github.io/math2/node/NTCD3724BB.md" tag="NTCD3724BB" kind="example">
# Sierpiński space

## Example

Let $S = \set{0, 1}$ with the family of [[open sets|02BH3MCBQB]] $\set{\emptyset, \set{1}, \set{0, 1}}$.
This family contains $\emptyset$ and $S$, and it is closed under unions and intersections, as one checks on its three members.
So $S$ is a [[topological space|0CQHFPGJ45]], the *Sierpiński space*.

- $S$ is a [[T0 space|07SG12TQH3]]: the open set $\set{1}$ contains $1$ but not $0$.
- $S$ is not a [[T1 space|0NW6CH38YR]]: the only open set that contains $0$ is $S$, and it also contains $1$.
- The [[closed sets|0ZCK8DQYN1]] are $\emptyset$, $\set{0}$ and $S$, so the closure of $\set{1}$ is $S$.
- $S$ is [[connected|0V032VWTHD]], since its only clopen sets are $\emptyset$ and $S$.

For any topological space $X$, a map $f \colon X \to S$ is [[continuous|0VGFHTM75Y]] if and only if $f^{-1}(\set{1})$ is open in $X$.
So continuous maps $X \to S$ correspond one to one to open subsets of $X$.

## Explanations

### L1 human

The Sierpiński space is the smallest topological space that is neither discrete nor trivial.
It has two points, and only one of them, the point $1$, forms an open set by itself.
One can read $1$ as "yes, observed" and $0$ as "not yet observed": an open set of any space is the same as a continuous yes-or-no test on it.
This small space is a basic counterexample, since it separates points in one direction only.
</document>

<document source="https://tamnd.github.io/math2/node/NTE5C260S1.md" tag="NTE5C260S1" kind="definition">
# Cofinality

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
A subset $A \subseteq P$ is cofinal in $P$ if for each $p \in P$ there is $a \in A$ with $p \le a$.
Let $\alpha$ be an [[ordinal|0RDJK7EQYY]], ordered by $\in$.
The cofinality $\operatorname{cf}(\alpha)$ of $\alpha$ is the least ordinal $\beta$ such that $\alpha$ has a cofinal subset of order type $\beta$.

## Remarks

Equivalently, $\operatorname{cf}(\alpha)$ is the least $\beta$ for which there is a [[function|0XHDTE5YMZ]] $f \colon \beta \to \alpha$ whose range is cofinal in $\alpha$.
Example: $\operatorname{cf}(\aleph_\omega) = \omega$.
Check: the set $\set{\aleph_n : n < \omega}$ has order type $\omega$ and is cofinal in $\aleph_\omega$, and no finite subset of a limit ordinal is cofinal.
Non-example of a cofinal subset: a finite subset $A$ of $\omega$ is not cofinal, since $\max A + 1$ has no element of $A$ above it.
For a successor ordinal $\alpha + 1$, the set $\set{\alpha}$ is cofinal, so $\operatorname{cf}(\alpha + 1) = 1$, and $\operatorname{cf}(0) = 0$.
Always $\operatorname{cf}(\operatorname{cf}(\alpha)) = \operatorname{cf}(\alpha)$, and the cofinality of a limit ordinal is a regular cardinal.
In ZFC, König's theorem gives $\operatorname{cf}(2^\kappa) > \kappa$ for each infinite cardinal $\kappa$.
</document>

<document source="https://tamnd.github.io/math2/node/NTEFMRR3HR.md" tag="NTEFMRR3HR" kind="concept">
# Condenser capacity

## Summary

A condenser is a pair $(K, \Omega)$ of a compact set $K$ inside an open set $\Omega \subseteq \R^n$.
Its capacity is the least Dirichlet energy $\int_\Omega \abs{\nabla u}^2 \, dx$ of a function that is at least $1$ on $K$ and vanishes near the boundary of $\Omega$.
It is the case $p = 2$ of the [[variational capacity|A1KWT817JX]].
Condenser capacity works in every dimension, including the plane.
It is a basic tool in the theory of quasiconformal maps through the conformal modulus of ring domains.

## Explanations

### L1 human

A physical capacitor has two conductors: an inner plate $K$ and an outer shell, the boundary of $\Omega$.
Hold $K$ at voltage $1$ and the shell at voltage $0$.
The capacity is the charge needed to keep this voltage difference, and it equals the energy of the electric field between the plates.
For two concentric spheres of radii $r < R$ in space, the capacity is $4 \pi r R / (R - r)$, which grows without bound as the two spheres come close.
As $R$ tends to infinity it tends to $4 \pi r$, the [[capacity|4YZVFGJVSZ]] of a single ball.
</document>

<document source="https://tamnd.github.io/math2/node/NTFM7X5555.md" tag="NTFM7X5555" kind="concept">
# Specht module

## Summary

For each [[partition|0JV30XN73Z]] $\lambda$ of $n$, the Specht module $S^\lambda$ is a representation of the [[symmetric group|0515A6D2Y1]] $S_n$ built from [[Young tableaux|Q6TH9PH2R7]] of shape $\lambda$.
Over a field of characteristic $0$, the Specht modules are irreducible and give every irreducible representation of $S_n$ exactly once.
They are defined over any field, even over the integers.
The dimension of $S^\lambda$ is the number of standard Young tableaux of shape $\lambda$, given by the [[hook length formula|VCMS1CP91N]].

## Explanations

### L1 human

A Specht module turns a picture, a [[Young diagram|096AHMB3P4]], into a representation of the symmetric group.
One fills the diagram with the numbers $1$ to $n$ and forms a signed sum that is symmetric along rows and antisymmetric down columns.
The span of all such sums is the module.
A single row gives the trivial representation, and a single column gives the sign representation.
</document>

<document source="https://tamnd.github.io/math2/node/NTFND9FRDN.md" tag="NTFND9FRDN" kind="definition">
# Stopped process

## Definition

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in T}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]], where $T \subseteq [0, \infty)$.
Let $X = (X_t)_{t \in T}$ be a [[stochastic process|0QH5ENKZM1]] with values in a measurable space $S$, and let $\tau$ be a [[stopping time|WND87FBAKC]].
The *stopped process* $X^\tau = (X^\tau_t)_{t \in T}$ is defined by
$$X^\tau_t(\omega) = X_{\min(t, \tau(\omega))}(\omega) \quad \text{for } t \in T \text{ and } \omega \in \Omega.$$

## Remarks

The notation $t \wedge \tau = \min(t, \tau)$ is standard, so $X^\tau_t = X_{t \wedge \tau}$.
On the event $\set{\tau = \infty}$, the stopped process equals $X$.

In discrete time, if $X$ is [[adapted|YRNFE016FF]], then so is $X^\tau$, because
$$X^\tau_n = X_0 + \sum_{k=1}^{n} \mathbf{1}_{\set{\tau \ge k}} (X_k - X_{k-1})$$
for real-valued $X$, and $\set{\tau \ge k} = \Omega \setminus \set{\tau \le k - 1} \in \mathcal{F}_{k-1}$.
The same formula shows that a stopped [[martingale|0D0GBNDKDH]] is a martingale.
In continuous time, the stopped process of a progressively measurable process is progressively measurable.

Example: let $S$ be a simple symmetric [[random walk|04VZE3FZXF]] started at $0$ and let $\tau$ be the first time with $\abs{S_n} = 5$.
Then $\abs{S^\tau_n} \le 5$ for all $n$, so $S^\tau$ is a bounded martingale.

Non-example: if $\tau$ is not a stopping time, the formula still defines a process, but it need not be adapted.
Take the walk above and let $\sigma$ be the last visit to $0$ before time $10$.
The value $S^\sigma_1$ equals $S_1$ if the walk returns to $0$ between times $1$ and $10$, and $0$ otherwise.
This is not determined by $\mathcal{F}_1$, so $S^\sigma$ is not adapted.
</document>

<document source="https://tamnd.github.io/math2/node/NTFY23YDEJ.md" tag="NTFY23YDEJ" kind="concept">
# Balayage

## Summary

Balayage, French for "sweeping out", replaces a measure by a measure on a given set.
The new measure has the same potential as the old one outside that set, and on it up to small exceptional sets.
Poincaré introduced it to solve the Dirichlet problem: the charge of a measure in a domain is swept to the boundary through [[harmonic measure|CFD7BPM51W]].
In modern potential theory the balayage of a measure onto any set is defined through the [[reduced function|909B1WYQ4Q]] of its potential.
It is one of the basic operations of classical and axiomatic potential theory.

## Explanations

### L1 human

Think of a point charge inside a hollow metal ball.
Seen from outside the ball, the charge cannot be told apart from a suitable charge spread over the sphere.
Balayage is the rule that produces this spread-out charge.
For a unit charge at the centre of a ball in space, the swept charge is the uniform distribution of total mass $1$ on the sphere.
Its potential agrees with that of the point charge outside the ball and is smaller inside.
</document>

<document source="https://tamnd.github.io/math2/node/NTJ94H2CNS.md" tag="NTJ94H2CNS" kind="concept">
# Lawvere theory

## Summary

A Lawvere theory is a category with finite products whose objects are the natural numbers, where $n$ is the $n$-th power of $1$.
Its morphisms $n \to 1$ play the role of the $n$-ary operations of an algebraic theory.
They are closed under composition and identified modulo the equations of the theory.
Models are finite-product-preserving functors to $\mathbf{Set}$, and the models of the theory of a [[variety|AJNZJD8442]] form a category equivalent to the variety.
Lawvere theories give a presentation-free description of varieties, so that different choices of operations and axioms for the same variety give the same theory.

## Explanations

### L1 human

The theory of groups can be written with multiplication, inverse and identity, or with division alone, and the two descriptions look different.
A Lawvere theory removes this choice.
It collects all operations that can be built from the basic ones, such as $(x, y) \mapsto x y^{-1} x$, and identifies two of them when they agree in every group.
These operations, with their composition, form a category, and a group is then a way to interpret this category by sets and functions.
</document>

<document source="https://tamnd.github.io/math2/node/NTK1TVA50A.md" tag="NTK1TVA50A" kind="statement">
# Švarc–Milnor lemma

## Statement

Let $X$ be a [[proper metric space|0RN765S7DH]] that is geodesic: any two points $x, y$ are joined by a path of length $d(x, y)$.
Let a group $G$ [[act|0WCY42EV9V]] on $X$ by [[isometries|02HPP5VCZH]].
Assume that the action is proper: for every compact set $K \subseteq X$, the set $\set{g \in G : gK \cap K \neq \emptyset}$ is finite.
Assume also that the action is cocompact: there is a compact set $K \subseteq X$ with $GK = X$.
Then:

1. $G$ is [[finitely generated|082DWWXMRJ]].
2. Let $S$ be a finite generating set of $G$ and $x_0 \in X$. Then the orbit map $g \mapsto g x_0$ is a [[quasi-isometry|6FKYSH729T]] from $G$, with the [[word metric|VCHJVR9NGH]] of $S$, to $X$.

## Remarks

Idea of the proof.
Choose a closed ball $B$ around $x_0$ whose translates cover $X$, and let $S$ be the finite set of $g$ with $gB \cap B \neq \emptyset$.
A geodesic from $x_0$ to $g x_0$ can be cut into short pieces, and consecutive pieces lie in translates of $B$ that meet.
This writes $g$ as a word in $S$ of length at most a linear function of $d(x_0, g x_0)$.
The other inequality holds because each generator moves $x_0$ by a bounded distance.

Example: $\Z^n$ acts on $\R^n$ by translations, properly and cocompactly, so $\Z^n$ with its standard word metric is quasi-isometric to $\R^n$.
Example: the fundamental group of a closed Riemannian manifold acts on the universal cover in this way, so the group and the universal cover are quasi-isometric.

Cocompactness is necessary.
The group $\Z$ acts properly on $\R^2$ by $n \cdot (a, b) = (a + n, b)$.
But the orbit of a point lies on a line, which is not within bounded distance of every point of the plane, so the orbit map is not a quasi-isometry.

## Explanations

### L1 human

Suppose a group moves the points of a nice space around, without piling up, and the moved copies of one bounded region cover the space.
Then the group and the space have the same large-scale shape.
So one can study a group by looking at a space it acts on, and conversely.
For instance, the grid $\Z^2$ and the plane look the same from far away.

### L2 human

The lemma is the bridge between groups and geometry in [[geometric group theory|2T3218RXTD]].
It shows that the quasi-isometry class of a finitely generated group does not depend on the generating set.
It also equals the class of any space on which the group acts properly and cocompactly.
For instance, the fundamental group of a closed hyperbolic surface is quasi-isometric to the hyperbolic plane, and so it is a [[hyperbolic group|JWJ8YPR7EB]].
Large-scale invariants of spaces, such as [[growth|4CMRM3975W]], hyperbolicity or the number of ends, thus become invariants of groups.
</document>

<document source="https://tamnd.github.io/math2/node/NTKTY1A2TH.md" tag="NTKTY1A2TH" kind="definition">
# Jeffreys prior

## Definition

Let $\set{f(\cdot \mid \theta) : \theta \in \Theta}$ be a [[parametric model|Q0DXKAQXE1]] with $\Theta \subseteq \R^k$ open, regular enough that its [[Fisher information|G1Z3WSTE8Q]] matrix $I(\theta)$ exists for every $\theta \in \Theta$.
The *Jeffreys prior* is the [[prior|00NBBSXRQS]] on $\Theta$ with density with respect to Lebesgue measure
$$\pi(\theta) \propto \sqrt{\det I(\theta)},$$
where $\det$ is the [[determinant|0C3CFHDN6R]].
If $\int_\Theta \sqrt{\det I(\theta)} \, d\theta = \infty$, it is an improper prior, used only when the resulting posterior is proper.

## Remarks

Example: for one Bernoulli observation with chance $p \in (0, 1)$, $I(p) = 1/(p(1 - p))$.
So the Jeffreys prior is proportional to $p^{-1/2}(1 - p)^{-1/2}$, which is the proper distribution $\mathrm{Beta}(1/2, 1/2)$.
For the mean $\mu$ of $\mathcal{N}(\mu, \sigma^2)$ with $\sigma$ known, $I(\mu) = 1/\sigma^2$ is constant, so the Jeffreys prior is the improper flat prior on $\R$.
For the scale $\sigma > 0$ of $\mathcal{N}(0, \sigma^2)$, $I(\sigma) = 2/\sigma^2$, so the Jeffreys prior is proportional to $1/\sigma$.
Invariance: let $\varphi = h(\theta)$ for a smooth bijection $h$ with smooth inverse.
Then the Jeffreys prior computed in the parameter $\varphi$ is the image of the Jeffreys prior in $\theta$.
The reason is that $\det I$ is multiplied by the square of the Jacobian determinant of the change of variables.
Non-example: the flat prior on $p \in (0, 1)$ is not the Jeffreys prior of the Bernoulli model, since its density $1$ is not proportional to $p^{-1/2}(1 - p)^{-1/2}$.
</document>

<document source="https://tamnd.github.io/math2/node/NTQEXR0WDN.md" tag="NTQEXR0WDN" kind="definition">
# Fundamental group

## Definition

Let $(X, x_0)$ be a [[pointed space|T9DG8DDE75]] and let $I = [0, 1]$.
A *loop* at $x_0$ is a continuous map $\gamma \colon I \to X$ with $\gamma(0) = \gamma(1) = x_0$.
For loops $\alpha$ and $\beta$ at $x_0$, the *concatenation* $\alpha \cdot \beta$ is the loop with $(\alpha \cdot \beta)(s) = \alpha(2s)$ for $0 \le s \le 1/2$ and $(\alpha \cdot \beta)(s) = \beta(2s - 1)$ for $1/2 \le s \le 1$.
The *fundamental group* $\pi_1(X, x_0)$ is the set of [[path homotopy|CCNKA8C711]] classes $[\gamma]$ of loops at $x_0$, with the product $[\alpha][\beta] = [\alpha \cdot \beta]$.

## Remarks

The product is well defined, and it makes $\pi_1(X, x_0)$ a [[group|G5S1M9]].
The identity is the class of the constant loop, and the inverse of $[\gamma]$ is the class of the reversed loop $s \mapsto \gamma(1 - s)$.
A pointed map $f \colon (X, x_0) \to (Y, y_0)$ induces a homomorphism $f_* \colon \pi_1(X, x_0) \to \pi_1(Y, y_0)$, $[\gamma] \mapsto [f \circ \gamma]$.
If $X$ is [[path-connected|0WB6DH499T]], the group does not depend on $x_0$ up to isomorphism.
Example: for a convex set $C \subseteq \R^n$ and $x_0 \in C$, every loop is path homotopic to the constant loop by the straight-line homotopy, so $\pi_1(C, x_0)$ is trivial.
Non-example: loops themselves, without passing to classes, do not form a group under concatenation.
Indeed $(\alpha \cdot \beta) \cdot \gamma$ and $\alpha \cdot (\beta \cdot \gamma)$ are in general different maps, which agree only up to path homotopy.
A path-connected space with trivial fundamental group is *simply connected*.
</document>

<document source="https://tamnd.github.io/math2/node/NTVQB1DC56.md" tag="NTVQB1DC56" kind="concept">
# U-statistic

## Summary

A U-statistic is the average of a symmetric function, the kernel, over all subsets of a fixed size of the observations.
It is an unbiased estimator of the mean of the kernel.
The sample mean, the sample variance, Kendall's $\tau$ and the Mann–Whitney statistic are U-statistics or close relatives of them.
Under mild conditions, Hoeffding's theory gives their asymptotic normality in one stroke.

## Explanations

### L1 human

Suppose you want to estimate the average squared distance between two random members of a population.
Take every pair of points in your sample, compute half their squared distance, and average over all pairs.
This average is a U-statistic, and it turns out to be exactly the usual sample variance.
Averaging over all subsets, instead of using only some of them, uses the data as fully as possible.
</document>

<document source="https://tamnd.github.io/math2/node/NV0ZAWS3ES.md" tag="NV0ZAWS3ES" kind="statement">
# Steinitz theorem on convex hull interiors

## Statement

Let $d \ge 1$, let $S \subseteq \R^d$, and let $x$ be a point in the [[interior|0RZH6W98HK]] of the [[convex hull|0FQSDC2EQ2]] of $S$.
Then there is a set $T \subseteq S$ with at most $2d$ points such that $x$ lies in the interior of $\operatorname{conv}(T)$.

## Remarks

It is an interior analogue of [[Carathéodory's theorem|0201Q4P2B5]], where $d + 1$ points suffice for membership in the hull but $2d$ points are needed for the interior.

The bound $2d$ is sharp.
Let $S = \set{\pm e_1, \dots, \pm e_d}$ be the vertices of the cross-polytope and let $x = 0$.
If $T$ omits, say, $e_i$, then every point of $\operatorname{conv}(T)$ has $i$-th coordinate at most $0$, so $0$ is not interior to $\operatorname{conv}(T)$.

Example: in the plane, a point inside the convex hull of a finite set lies inside some quadrilateral, or some triangle, with vertices in the set.

The interior is taken in $\R^d$, so the hypothesis already forces $\operatorname{conv}(S)$ to have dimension $d$.

## Explanations

### L1 human

If a point sits strictly inside the convex hull of many points, a few of them already surround it.
In the plane four points always suffice, and in $d$ dimensions $2d$ points suffice.
The four points $(\pm 1, 0)$, $(0, \pm 1)$ around the origin show that three would not be enough.

### L2 human

The theorem was proved by Ernst Steinitz, who also extended Carathéodory's theorem from compact to arbitrary sets.
It is a standard companion of Carathéodory's and Helly's theorems in combinatorial convexity.
It is used for positively spanning sets.
A finite set positively spans $\R^d$ exactly when $0$ is in the interior of its convex hull.
By the theorem, a subset of at most $2d$ of its vectors then already positively spans $\R^d$.
</document>

<document source="https://tamnd.github.io/math2/node/NV54MWKT6T.md" tag="NV54MWKT6T" kind="statement">
# Tauberian theorems for integral transforms

## Statement

Let $f \colon [0, \infty) \to \R$ be locally integrable with $f(t) \ge 0$ for all $t$.
Assume that its [[Laplace transform|05XT5AVJBK]] $F(s) = \int_0^\infty f(t) e^{-st}\, dt$ is finite for every $s > 0$.
Let $c \ge 0$.
If $s F(s) \to c$ as $s \to 0^+$, then
$$\frac{1}{T} \int_0^T f(t)\, dt \to c \quad (T \to \infty).$$

## Remarks

This is the Hardy–Littlewood Tauberian theorem for the Laplace transform; the condition $f \ge 0$ is the [[Tauberian condition|6ZDP261K7R]].
For $c > 0$ it is the case $\rho = 1$ of [[Karamata's Tauberian theorem|DBEGF38V9H]], applied to $U(x) = \int_0^x f(t)\, dt$.
The proof follows Karamata.
One shows $s\int_0^\infty f(t) e^{-st} P(e^{-st})\, dt \to c \int_0^1 P(x)\, dx$ for polynomials $P$.
Then one approximates an indicator function by polynomials from above and below, using $f \ge 0$.
The Tauberian condition cannot be dropped.
Let $a_n = 1 + (-1)^n n$ and let $f = a_n$ on $[n, n + 1)$; then $sF(s) = (1 - x)\sum_n a_n x^n$ with $x = e^{-s}$, which tends to $1$ as $s \to 0^+$, as in the [[Hardy–Littlewood Tauberian theorem|N42JDBT0VA]] for series.
But $\frac{1}{N} \int_0^N f = \frac{1}{N} \sum_{k < N} a_k$ is close to $3/2$ for odd $N$ and to $1/2$ for even $N$, so it has no limit.
Converse direction: the Abelian theorem gives $sF(s) \to c$ from the convergence of the averages, with no sign condition (see [[Abelian theorems for integral transforms|16W9PP8ET0]]).

## Explanations

### L1 human

The Laplace transform smooths a function out, so knowing how the transform behaves does not, in general, tell how the function behaves.
A Tauberian theorem says that it does, provided the function cannot oscillate wildly.
Here the extra condition is that $f$ is never negative: then a limit of $sF(s)$ at $0$ forces the long-run average of $f$ to have the same limit.

### L2 human

Tauberian theorems are partial converses of Abelian theorems, and each needs a side condition such as positivity, monotonicity or a growth bound.
Karamata's theorem extends the statement to regularly varying growth, with a factor $\Gamma(1 + \rho)$.
Wiener's Tauberian theorem treats general convolution kernels.
The Wiener–Ikehara theorem, a Tauberian theorem for the Laplace transform with a condition on the boundary line, gives a short proof of the prime number theorem.
In probability, this theorem relates the behaviour of a Laplace transform near $0$ to the tail of a distribution.
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<document source="https://tamnd.github.io/math2/node/NVBD7Y86ES.md" tag="NVBD7Y86ES" kind="concept">
# Screw theory

## Summary

Screw theory describes the velocities of a rigid body and the forces on it by six-dimensional vectors called screws.
Twists describe motions, and wrenches describe forces and moments.
Its geometric basis is Chasles' theorem.
It says that every orientation-preserving rigid motion of space is a screw motion.
That is, it is a rotation about a line combined with a translation along the same line.
A screw has an axis, a line in space, and a pitch, the ratio of translation to rotation.
It is used in the kinematics and statics of mechanisms and robots.
There the power of a wrench on a twist is given by a simple pairing.

## Explanations

### L1 human

Turn a screw with a screwdriver: it rotates and at the same time moves forward along its axis.
Screw theory says that every motion of a rigid object is like this at each instant: a turn about some axis plus a slide along that axis.
The axis and the amount of slide per turn, the pitch, describe the motion completely, up to its speed.
A pure rotation is a screw of pitch $0$, and a pure translation is a screw of infinite pitch.
Forces work the same way: any system of forces on a body acts like one force along a line together with a twisting moment about that line.
</document>

<document source="https://tamnd.github.io/math2/node/NVD7R0T835.md" tag="NVD7R0T835" kind="concept">
# Splitting field of a central simple algebra

## Summary

Let $A$ be a [[central simple algebra|5HRKJBAY2V]] over a field $K$.
A field extension $L$ of $K$ splits $A$ if $A$ becomes a full matrix algebra after extending scalars to $L$.
Every central simple algebra has a splitting field of finite degree over $K$, and even one that is a finite Galois extension of $K$.
Splitting fields link central simple algebras with Galois cohomology: the algebras split by $L$ give the relative Brauer group $\mathrm{Br}(L/K)$.
The word is unrelated to the [[splitting field|0KXQKX9P63]] of a polynomial, apart from the idea of making something decompose.

## Explanations

### L1 human

A central simple algebra may be a division algebra over a small field, yet look like an ordinary matrix algebra over a larger one.
A splitting field is a larger field that makes this happen.
For example, Hamilton's quaternions are a division algebra over $\R$, but over $\C$ they become the algebra of $2 \times 2$ complex matrices, so $\C$ splits them.
</document>

<document source="https://tamnd.github.io/math2/node/NVD9GWQB6M.md" tag="NVD9GWQB6M" kind="definition">
# Kirillov orbit method

## Definition

Let $G$ be a connected and simply connected [[nilpotent Lie group|DK87YN458D]] with Lie algebra $\mathfrak g$ and [[exponential map|H3QX1ZFC8K]] $\exp \colon \mathfrak g \to G$.
Let $\mathfrak g^*$ be the [[dual space|07JBSZE0V4]] of $\mathfrak g$, with the coadjoint action $(g \cdot \ell)(X) = \ell(\mathrm{Ad}(g^{-1}) X)$, where $\mathrm{Ad}$ is the [[adjoint representation|P6ZP0TE07M]].
For $\ell \in \mathfrak g^*$, a *polarization* at $\ell$ is a Lie subalgebra $\mathfrak h \subseteq \mathfrak g$ such that $\ell([\mathfrak h, \mathfrak h]) = 0$ and $\mathfrak h$ is a maximal subspace with this property.
For such $\mathfrak h$, let $H = \exp \mathfrak h$ and let $\chi_\ell$ be the unitary character $\chi_\ell(\exp Y) = e^{i \ell(Y)}$ of $H$.
The *Kirillov orbit method* assigns to $\ell$ the [[induced representation|6Q7AHAZ6Q4]]
$$\pi_\ell = \mathrm{Ind}_H^G \chi_\ell.$$
By Kirillov's theorem, $\pi_\ell$ is irreducible, and its class depends only on the [[orbit|0YQPZB5Q1G]] $G \cdot \ell$, not on the choice of $\mathfrak h$.
The map $G \cdot \ell \mapsto \pi_\ell$ is a bijection from the set of coadjoint orbits onto the [[unitary dual|95Q8WZMN0W]] of $G$.

## Remarks

A polarization always exists for nilpotent $\mathfrak g$, and its dimension is $\tfrac12 (\dim \mathfrak g + \dim \mathfrak g_\ell)$, where $\mathfrak g_\ell = \set{X : \ell([X, \mathfrak g]) = 0}$.

Example: let $\mathfrak g$ have basis $X, Y, Z$ with $[X, Y] = Z$ and $Z$ central, the Lie algebra of the Heisenberg group.
For $\ell$ with $\ell(Z) = \lambda \ne 0$, the coadjoint orbit is the plane $\set{\ell' : \ell'(Z) = \lambda}$.
The subalgebra $\mathfrak h = \operatorname{span}(Y, Z)$ is abelian, so $\ell([\mathfrak h, \mathfrak h]) = 0$, and it is maximal with this property since $\ell([X, Y]) = \lambda \ne 0$.
The induced representation $\pi_\ell$ is the Schrödinger representation with central character $e^{i \lambda z}$.
For $\ell(Z) = 0$ the orbit is the single point $\ell$, and $\pi_\ell$ is the one-dimensional character $\exp(aX + bY + cZ) \mapsto e^{i(a \ell(X) + b \ell(Y))}$.

Non-example: for $\ell(Z) = \lambda \ne 0$, the center $\mathfrak h = \operatorname{span}(Z)$ satisfies $\ell([\mathfrak h, \mathfrak h]) = 0$ but is not maximal.
The induced representation $\mathrm{Ind}_H^G \chi_\ell$ acts on $L^2(\R^2)$ and is a direct sum of infinitely many copies of the Schrödinger representation, so it is reducible.
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<document source="https://tamnd.github.io/math2/node/NVFF58NEKV.md" tag="NVFF58NEKV" kind="definition">
# Crossed product

## Definition

Let $A$ be a unital C*-algebra, and let $G$ be a discrete group.
Let $\alpha \colon G \to \Aut(A)$ be a [[group action|0WCY42EV9V]] of $G$ on $A$ by $*$-[[automorphisms|DSQ17TNW1C]].
Let $C_c(G, A)$ be the space of finitely supported functions $f \colon G \to A$, with the product and involution
$$(f * g)(s) = \sum_{t \in G} f(t) \, \alpha_t\big(g(t^{-1} s)\big), \qquad f^*(s) = \alpha_s\big(f(s^{-1})^*\big).$$
Let $H$ be a [[Hilbert space|HM2HHE3J51]].
A *covariant representation* of $(A, G, \alpha)$ on $H$ is a pair $(\pi, u)$ with two parts.
The first, $\pi$, is a unital $*$-representation of $A$ on $H$; the second, $u$, is a unitary representation of $G$ on $H$.
The pair must satisfy $u_s \pi(a) u_s^* = \pi(\alpha_s(a))$ for all $s \in G$ and $a \in A$.
It gives the $*$-representation $(\pi \rtimes u)(f) = \sum_{s \in G} \pi(f(s)) u_s$ of $C_c(G, A)$.
The *crossed product* $A \rtimes_\alpha G$ is the completion of $C_c(G, A)$ in the norm
$$\norm{f} = \sup \set{\norm{(\pi \rtimes u)(f)} : (\pi, u) \text{ a covariant representation}}.$$

## Remarks

The element $a \delta_s$ that takes the value $a$ at $s$ and $0$ elsewhere is written $a u_s$; then $u_s a u_s^* = \alpha_s(a)$ in $A \rtimes_\alpha G$.
The *reduced crossed product* $A \rtimes_{\alpha, r} G$ uses only the regular covariant representations on $\ell^2(G, H)$; it equals the full one when $G$ is amenable.
For locally compact groups, $C_c(G, A)$ is replaced by continuous compactly supported functions and sums by integrals against Haar measure.

Example: take $A = \C$ and the trivial action.
Then $C_c(G, \C)$ is the group algebra of $G$ with convolution, and the covariant representations are the unitary representations of $G$.
So $\C \rtimes G$ is the full [[group C*-algebra|JKANN4VBYH]] $C^*(G)$.

Non-example: let $\Z$ act on $A = C(\mathbb{T})$ by an irrational rotation.
If $\alpha$ is replaced by the identity in the formulas for the product and the involution, the result is the crossed product for the trivial action.
That algebra is $C(\mathbb{T}) \otimes C^*(\Z) \cong C(\mathbb{T}^2)$, which is commutative.
The twisted product gives instead the noncommutative irrational rotation algebra.
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<document source="https://tamnd.github.io/math2/node/NVH3F8FTYS.md" tag="NVH3F8FTYS" kind="concept">
# Numerical differentiation

## Summary

Numerical differentiation approximates the [[derivative|0V1EJCT8G6]] of a function from its values at finitely many points.
The basic tools are difference quotients such as $\frac{f(x+h) - f(x-h)}{2h}$, whose error is controlled by [[Taylor's theorem|03QHBVW4TT]].
A small step $h$ reduces the truncation error but magnifies the [[round-off error|CPN6SD3BTV]], so the step must be chosen with care.
Systematic formulas are given by [[finite difference coefficients|752Z7DRT7P]], and [[automatic differentiation|BW7FEZ8WM1]] is an alternative that avoids truncation error altogether.

## Explanations

### L1 human

The derivative is the slope of a graph at a point.
Without a formula, one can still estimate it by taking two nearby points on the graph and computing the slope of the line through them.
For $f(x) = x^2$ at $x = 1$ with points $0.9$ and $1.1$, the slope is $\frac{1.21 - 0.81}{0.2} = 2$, which here is exact.
Moving the points closer usually helps, but on a computer, points that are too close cause rounding errors to dominate.
</document>

<document source="https://tamnd.github.io/math2/node/NVMCZWHTZE.md" tag="NVMCZWHTZE" kind="definition">
# Brown–Peterson cohomology

## Definition

Let $p$ be a [[prime number|01826KMG2H]], and let $\Z_{(p)}$ be the ring of rational numbers whose denominators are prime to $p$.
Let $MU_{(p)}$ be the $p$-localization of the [[complex cobordism|5CZHFNY9F3]] spectrum.
Quillen's idempotent is a map of [[ring spectra|BAW0T68GWK]] $e \colon MU_{(p)} \to MU_{(p)}$ with $e \circ e = e$.
On coefficients it sends the [[formal group law|TNHP7XXYZT]] of $MU_{(p)}$ to a $p$-typical one.
A formal group law is $p$-typical if its logarithm over $\Q$ has the form $\sum_{i \ge 0} \ell_i x^{p^i}$ with $\ell_0 = 1$.
The *Brown–Peterson spectrum* $BP$ (at $p$) is the image of $e$, that is, the homotopy colimit of $MU_{(p)} \xrightarrow{e} MU_{(p)} \xrightarrow{e} \cdots$.
*Brown–Peterson cohomology* is the [[generalized cohomology theory|H28PSZR7XG]] $BP^*(-)$ represented by $BP$.

## Remarks

The coefficient ring is $\pi_* BP \cong \Z_{(p)}[v_1, v_2, \ldots]$ with $\abs{v_i} = 2(p^i - 1)$.
With Hazewinkel's generators, the logarithm coefficients satisfy $p \ell_n = \sum_{i=0}^{n-1} \ell_i v_{n-i}^{p^i}$.
For example, $p \ell_1 = v_1$, so $\ell_1 = v_1 / p$.
The formal group law of $BP$ is the universal $p$-typical formal group law over a $\Z_{(p)}$-algebra.

Example: $MU_{(p)}$ is equivalent to a wedge $\bigvee_j \Sigma^{2 d_j} BP$, so $BP$ is a retract of $MU_{(p)}$, as an image of an idempotent should be.

Non-example: $MU_{(p)}$ itself is not $BP$, because $\pi_2 MU_{(p)} \cong \Z_{(p)}$ is nonzero for every $p$, while $\pi_2 BP = 0$ for $p$ odd since the first generator $v_1$ has degree $2(p - 1) \ge 4$.

Brown and Peterson first built $BP$ in 1966 by a different construction; Quillen's idempotent gives the description above.
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<document source="https://tamnd.github.io/math2/node/NVQ0N7QTCR.md" tag="NVQ0N7QTCR" kind="definition">
# Transcendence degree

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
The *transcendence degree* of $L/K$, written $\operatorname{trdeg}_K L$, is the [[cardinality|0JTGRV4WT4]] of a [[transcendence basis|E45PA267S4]] of $L/K$.

## Remarks

The definition makes sense because [[a transcendence basis exists|W635VBXEXG]] and any two transcendence bases have the same cardinality; the proof uses an exchange argument like the [[Steinitz exchange lemma|08ZDRN6H41]].
Example: $\operatorname{trdeg}_K K(x_1, \dots, x_n) = n$, with transcendence basis $\set{x_1, \dots, x_n}$.
Example: $\operatorname{trdeg}_\Q \R$ is the cardinality of the continuum, since $\R$ is uncountable and the algebraic closure of a countable field is countable.
Non-example: for $L = \Q(\pi, \pi^2)$, the transcendence degree over $\Q$ is $1$, not $2$, because $\set{\pi, \pi^2}$ is dependent and $\set{\pi}$ is a transcendence basis.
It is additive in towers: $\operatorname{trdeg}_K M = \operatorname{trdeg}_L M + \operatorname{trdeg}_K L$ for $K \subseteq L \subseteq M$.
The transcendence degree is $0$ if and only if the extension is [[algebraic|09EYMKMSB9]].
</document>

<document source="https://tamnd.github.io/math2/node/NVQJWDJG2T.md" tag="NVQJWDJG2T" kind="concept">
# Finitely generated variety

## Summary

A finitely generated variety is a [[variety|AJNZJD8442]] generated by finitely many finite algebras, or equivalently by one finite algebra.
It is the smallest variety that contains the given finite algebras.
Every finitely generated variety is a [[locally finite variety|8QPW65PZVW]], and such varieties are the main objects of the structure theory of finite algebras.

## Explanations

### L1 human

Start with one finite algebra, such as the two-element Boolean algebra.
Take all its subalgebras, all its direct powers, and all images of those under homomorphisms.
The class obtained is the variety generated by the algebra.
For the two-element Boolean algebra, this class is the variety of all Boolean algebras.
</document>

<document source="https://tamnd.github.io/math2/node/NVQN7HR9Y0.md" tag="NVQN7HR9Y0" kind="definition">
# Korteweg–De Vries equation

## Definition

The *Korteweg–De Vries equation* is the [[partial differential equation|GC5BDZ6QBC]]
$$u_t + u_{xxx} + 6 u u_x = 0$$
for an unknown real function $u(x, t)$ of $x \in \R$ and $t \in \R$.
Here subscripts denote [[partial derivatives|09EGC6F8PT]], for example $u_{xxx} = \partial_x^3 u$.

## Remarks

Example: for $c > 0$, the function $u(x,t) = f(x - ct)$ with $f(\xi) = \tfrac{c}{2} \mathrm{sech}^2\!\big(\tfrac{\sqrt{c}}{2}\xi\big)$ is a solution.
Substituting $u = f(x - ct)$ gives $-c f' + f''' + 6 f f' = 0$, and after one integration $f'' = c f - 3 f^2$.
With $b = \sqrt{c}/2$ and $s = \mathrm{sech}(b\xi)$, one computes $f'' = \tfrac{c}{2} b^2 (4 s^2 - 6 s^4) = \tfrac{c^2}{2} s^2 - \tfrac{3c^2}{4} s^4$, which equals $c f - 3 f^2$.

Non-example: $u(x,t) = x$ is not a solution, because $u_t + u_{xxx} + 6 u u_x = 6x \ne 0$.

Other normalisations are common, such as $u_t + u_{xxx} - 6 u u_x = 0$, obtained by replacing $u$ with $-u$.
Any equation $u_t + \alpha u_{xxx} + \beta u u_x = 0$ with $\alpha, \beta \ne 0$ reduces to this form by scaling $x$, $t$ and $u$.
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<document source="https://tamnd.github.io/math2/node/NVRR2TXQSE.md" tag="NVRR2TXQSE" kind="statement">
# Addition theorem for spherical harmonics

## Statement

Let $d \ge 3$ and $\ell \ge 0$ be integers.
Let $\mathcal{H}_\ell$ be the space of [[spherical harmonics|3F32Y21HP4]] of degree $\ell$ on the unit sphere $S^{d-1} \subset \R^d$, with dimension $N$.
Let $\sigma$ be the surface measure, $\omega = \sigma(S^{d-1})$, and $\lambda = \frac{d - 2}{2}$.
Let $Y_1, \dots, Y_N$ be an [[orthonormal basis|0S9GGTR0EX]] of $\mathcal{H}_\ell$ for the inner product $\ip{f, g} = \int_{S^{d-1}} f \, \overline{g} \, d\sigma$.
Then for all $x, y \in S^{d-1}$,
$$\sum_{j=1}^{N} Y_j(x) \, \overline{Y_j(y)} = \frac{N}{\omega} \, \frac{C_\ell^{(\lambda)}(x \cdot y)}{C_\ell^{(\lambda)}(1)},$$
where $C_\ell^{(\lambda)}$ is the [[Gegenbauer polynomial|5TRFA6F79B]].
In particular, for $d = 3$ and the standard basis $Y_\ell^m$, $-\ell \le m \le \ell$,
$$P_\ell(x \cdot y) = \frac{4\pi}{2\ell + 1} \sum_{m=-\ell}^{\ell} Y_\ell^m(x) \, \overline{Y_\ell^m(y)},$$
where $P_\ell$ is the [[Legendre polynomial|QFYGJVA996]].

## Remarks

Proof idea: the left side $K(x, y)$ is the [[zonal spherical harmonic|Q5D54R3RGE]] $Z_y^{(\ell)}(x)$, so it does not depend on the choice of basis.
Since rotations act unitarily on $\mathcal{H}_\ell$, $K(gx, gy) = K(x, y)$ for every rotation $g$, so $K$ depends only on $x \cdot y$.
A harmonic of degree $\ell$ invariant under the rotations fixing $y$ is a multiple of $C_\ell^{(\lambda)}(x \cdot y)$.
The constant comes from $K(x, x)$ being constant and $\int_{S^{d-1}} K(x, x) \, d\sigma(x) = \sum_j \norm{Y_j}^2 = N$, so $K(x, x) = N / \omega$.
Example: for $d = 3$ and $\ell = 1$, the basis $\sqrt{3/(4\pi)} \, x_i$, $i = 1, 2, 3$, gives $\frac{3}{4\pi} \, x \cdot y = \frac{3}{4\pi} P_1(x \cdot y)$.
Setting $x = y$ gives Unsöld's theorem: $\sum_{m=-\ell}^{\ell} \abs{Y_\ell^m(x)}^2 = \frac{2\ell + 1}{4\pi}$ at every point of $S^2$.

## Explanations

### L1 human

Pick two points on a sphere.
Summing the products of all basic patterns of degree $\ell$ at the two points gives a number that depends only on the angle between the points.
That number is a Legendre polynomial of the cosine of the angle, up to a fixed factor.
So a complicated sum over many functions collapses to a single simple function.

### L2 human

The theorem identifies the reproducing kernel of $\mathcal{H}_\ell$ with a Gegenbauer polynomial in $x \cdot y$.
It gives the projection of a function $f$ onto $\mathcal{H}_\ell$ as the convolution $\frac{N}{\omega} \int_{S^{d-1}} f(y) \frac{C_\ell^{(\lambda)}(x \cdot y)}{C_\ell^{(\lambda)}(1)} \, d\sigma(y)$.
Combined with the generating function of the Legendre polynomials, it expands $1/\norm{x - y}$ in spherical harmonics, the standard multipole expansion of potential theory.
It also gives the bound $\abs{Y(x)} \le \sqrt{N / \omega} \, \norm{Y}_{L^2}$ for every $Y \in \mathcal{H}_\ell$.
</document>

<document source="https://tamnd.github.io/math2/node/NVVGA3KEWE.md" tag="NVVGA3KEWE" kind="concept">
# Associated bundle

## Summary

Let $P \to B$ be a [[principal bundle|8E37E74XVP]] with group $G$, and let $G$ act on a space $F$.
The associated bundle $P \times_G F$ is a [[fiber bundle|CX4M8K61AB]] over $B$ with fiber $F$.
It is built by gluing copies of $F$ along $B$ with the same twisting as $P$.
In this way one principal bundle produces many bundles: vector bundles, sphere bundles, bundles of frames and so on.
For example, the tangent bundle of a manifold is associated to its frame bundle through the action of $GL_n(\R)$ on $\R^n$.

## Explanations

### L1 human

A principal bundle records only how the fibers are twisted, by elements of a group, as one moves around the base.
An associated bundle uses this twisting recipe on another fiber, any space on which the group acts.
For example, the circle double cover $S^1 \to S^1$, $z \mapsto z^2$, records a single flip as one goes around once.
Applying this flip to a line, by $t \mapsto -t$, gives the Möbius band.
</document>

<document source="https://tamnd.github.io/math2/node/NVXMVBS1ES.md" tag="NVXMVBS1ES" kind="definition">
# Weak equivalence

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
A *class of weak equivalences* in $\mathcal{C}$ is a class $W$ of morphisms of $\mathcal{C}$ such that:

1. every isomorphism of $\mathcal{C}$ is in $W$;
2. (two out of three) if $f \colon X \to Y$ and $g \colon Y \to Z$ are morphisms and two of $f$, $g$, $g \circ f$ are in $W$, then so is the third.

A pair $(\mathcal{C}, W)$ is a *category with weak equivalences*, and the members of $W$ are its *weak equivalences*.
The weak equivalences of a model category are the members of the class $W$ of its model structure, which satisfies these two clauses.

## Remarks

Example: in the category of topological spaces, the [[weak homotopy equivalences|4Q3WY1RX91]] form a class of weak equivalences.
Homeomorphisms induce isomorphisms on all homotopy groups, and two out of three follows from functoriality of $\pi_0$ and of $\pi_n(-, x)$ at all base points.
Example: in the category of chain complexes of modules over a ring, the [[quasi-isomorphisms|C9JEC2WQKV]] form a class of weak equivalences, by functoriality of homology.

Non-example: in the category of sets, the class of injective maps is closed under composition and contains the isomorphisms, but it fails two out of three.
With $f \colon \set{0} \to \set{0, 1}$ the inclusion and $g \colon \set{0, 1} \to \set{0}$, both $f$ and $g \circ f = \id$ are injective, but $g$ is not.

Some authors, for homotopical categories, ask the stronger two out of six property.
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<document source="https://tamnd.github.io/math2/node/NW1XQ95R5P.md" tag="NW1XQ95R5P" kind="statement">
# Inverse quadratic interpolation

## Statement

Let $f$ be a real function that is three times continuously differentiable on an open interval $I$, and let $r \in I$ satisfy $f(r) = 0$ and $f'(r) \ne 0$.
Given three points $x_{n-2}, x_{n-1}, x_n$ whose values $f_k = f(x_k)$ are pairwise distinct, *inverse quadratic interpolation* defines
$$x_{n+1} = \frac{f_{n-1} f_n \, x_{n-2}}{(f_{n-2} - f_{n-1})(f_{n-2} - f_n)} + \frac{f_{n-2} f_n \, x_{n-1}}{(f_{n-1} - f_{n-2})(f_{n-1} - f_n)} + \frac{f_{n-2} f_{n-1} \, x_n}{(f_n - f_{n-2})(f_n - f_{n-1})}.$$
Then there is $\delta > 0$ with the following property.
For all starting points $x_0, x_1, x_2$ within $\delta$ of $r$ for which the iteration is defined, the iterates [[converge|YWZQC5E54B]] to $r$ with order $q \approx 1.839$.
Here $q$ is the real root of $q^3 = q^2 + q + 1$.

## Remarks

The new point is the value at $y = 0$ of the [[Lagrange interpolation|237KR6FRCK]] polynomial of degree at most $2$ through the points $(f_k, x_k)$.
This polynomial approximates the [[inverse function|0J2WAR83WP]] of $f$ near $r$, so its value at $0$ approximates $r = f^{-1}(0)$.
The error analysis gives $e_{n+1} \approx C e_n e_{n-1} e_{n-2}$ for the errors $e_k = x_k - r$, which leads to the stated order.
Interpolating $x$ as a function of $y$ avoids solving a quadratic equation, unlike Muller's method.
The method breaks down when two of the values $f_k$ coincide, and it can diverge from poor starting points.
For this reason it is used inside safeguarded methods such as Brent's method.
With two points instead of three, the same idea gives the [[secant method|9ZC177PVPC]].

## Explanations

### L1 human

Swap the roles of $x$ and $y$: think of $x$ as a function of $y = f(x)$.
Fit a parabola in this swapped picture through three known points, and read off where it meets $y = 0$.
That reading is the next guess for the root.

### L2 human

The method needs no derivatives and only one new function value per step.
Its order is a little less than $2$ but better than the secant method's $1.618$.
It is fast but fragile, so libraries combine it with bisection to keep a bracket around the root.
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# Zero morphism

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
A *system of zero morphisms* on $\mathcal{C}$ is a choice of a [[morphism|859F0TFVA3]] $0_{XY} \colon X \to Y$ for all objects $X$ and $Y$, with the following property.
For all objects $W, Z$ and all morphisms $g \colon W \to X$ and $f \colon Y \to Z$:
$$f \circ 0_{XY} = 0_{XZ} \quad \text{and} \quad 0_{XY} \circ g = 0_{WY}.$$
A category with such a choice is a *category with zero morphisms*, and the morphisms $0_{XY}$ are its *zero morphisms*.

## Remarks

A category has at most one system of zero morphisms: if $0$ and $0'$ are two, then $0_{XY} = 0'_{YY} \circ 0_{XY} = 0'_{XY}$.
If $\mathcal{C}$ has a [[zero object|AQET9YHH6F]] $0$, setting $0_{XY}$ equal to the composite of the unique morphisms $X \to 0$ and $0 \to Y$ gives a system of zero morphisms.

Example: in the category of groups, let $0_{GH}$ send every element of $G$ to the identity $e_H$ of $H$.
For a [[group homomorphism|0JYTG5K9Y4]] $f \colon H \to K$, $f(e_H) = e_K$, so $f \circ 0_{GH} = 0_{GK}$; and $0_{GH} \circ g$ is again constant at $e_H$, so $0_{GH} \circ g = 0_{G'H}$ for $g \colon G' \to G$.

Non-example: the category of sets has no system of zero morphisms.
There is no function at all from a non-empty set $X$ to the empty set, so $0_{X \emptyset}$ cannot be chosen.
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# Inaccessible cardinal

## Summary

An inaccessible cardinal is an uncountable [[cardinal|0EBACQFAZW]] $\kappa$ that cannot be reached from smaller cardinals.
Power sets of smaller sets, and unions of fewer than $\kappa$ smaller sets, stay smaller than $\kappa$.
It is the smallest kind of [[large cardinal|X1FG7PSCTY]].
If $\kappa$ is inaccessible, the sets of rank less than $\kappa$ form a model of ZFC.
So ZFC cannot prove that inaccessible cardinals exist, if ZFC is consistent.
In category theory, inaccessible cardinals give [[Grothendieck universes|SMMK9E0MTM]].

## Explanations

### L1 human

The natural numbers cannot be reached from finite numbers: sums, products and powers of finite numbers stay finite.
An inaccessible cardinal is an uncountable size with the same property.
Taking power sets of smaller sets, or unions of fewer than $\kappa$ sets each smaller than $\kappa$, never gives a set of size $\kappa$.
So everything built from below stays below, and the sets below $\kappa$ form a whole universe of their own.
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# Pure weights

## Summary

A highest weight $\mu$ for $G_E$ is a family $(\mu_\tau)$ indexed by the set $S_\infty$ of infinite embeddings of the base field $E$. Each $\mu_\tau$ is a dominant weight of the diagonal torus of $\mathrm{GL}_3$, written as a triple $(m^+_\tau, m^-_\tau, v_\tau)$.

The weight $\mu$ is **pure** when $v_\tau$ is the same for all $\tau$ in $S_\infty$, and $m^+_\tau = m^-_\tau$ for every $\tau$.

The source paper uses this condition when it studies divisibility of automorphic periods for the base change of $\mathrm{GL}_3$ to a real quadratic field.
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<document source="https://tamnd.github.io/math2/node/NW57FTPF3E.md" tag="NW57FTPF3E" kind="concept">
# Monge point

## Summary

The Monge point of a [[tetrahedron|JFX8ASQS3E]] is the common point of six planes.
Each of them passes through the midpoint of an edge and is perpendicular to the opposite edge.
It is the reflection of the circumcentre in the [[centroid|0X7EEWES5G]].
It plays the role in a tetrahedron that the [[orthocentre|0WB619CA0V]] plays in a triangle, because the altitudes of a tetrahedron need not meet.
The definition extends to every [[simplex|YX1DJK2QAG]] of dimension at least $2$, and for a triangle it gives the orthocentre.

## Explanations

### L1 human

In a triangle, the three altitudes always meet at one point, the orthocentre.
In a tetrahedron, the four altitudes usually miss each other.
Gaspard Monge found a replacement: for each edge, take the plane through its midpoint at right angles to the opposite edge.
The six planes always meet at one point, the Monge point.
It lies on the line through the circumcentre and the centroid, on the other side of the centroid at the same distance.
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# Rational point

## Definition

Let $K$ be a [[field|04C5ZBG1WT]] and let $X$ be a [[scheme|5A35TYZD02]] over $K$, for example an [[algebraic variety|HV6YBS0YDH]] over $K$.
A *$K$-rational point* of $X$ is a morphism of $K$-schemes $\operatorname{Spec} K \to X$.
The set of $K$-rational points is written $X(K)$.

## Remarks

If $X \subseteq \mathbb{A}^n_K$ is the affine variety defined by polynomials $f_1, \dots, f_m \in K[x_1, \dots, x_n]$, then $X(K)$ is the set of $a \in K^n$ with $f_1(a) = \cdots = f_m(a) = 0$.
If $X \subseteq \mathbb{P}^n_K$ is defined by homogeneous polynomials, then $X(K)$ is the set of points $[a_0 : \cdots : a_n]$ with all $a_i \in K$, not all zero, at which the polynomials vanish.
Equivalently, when $X$ is of finite type over $K$, a $K$-rational point is a closed point of $X$ whose residue field is $K$.

Example: $(3/5, 4/5) \in X(\Q)$ for $X = V(x^2 + y^2 - 1) \subseteq \mathbb{A}^2_\Q$, since $9/25 + 16/25 = 1$.

Non-example: $(1, \sqrt{2})$ satisfies $x^2 + y^2 = 3$, but it is not a $\Q$-rational point, because $\sqrt{2} \notin \Q$.
In fact $x^2 + y^2 = 3$ has no $\Q$-rational point: clearing denominators gives $a^2 + b^2 = 3c^2$ in integers with $\gcd(a, b, c) = 1$, and reducing modulo $3$ forces $3$ to divide $a$, $b$ and then $c$.
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<document source="https://tamnd.github.io/math2/node/NW9MM370A5.md" tag="NW9MM370A5" kind="concept">
# Γ-convergence

## Summary

Γ-convergence is a notion of convergence for functionals that is designed for minimization problems.
A sequence $F_n$ Γ-converges to $F$ when two things hold.
A lower bound inequality holds along every convergent sequence of points, and the bound is reached along some sequence.
Under a compactness condition, minimum values and minimizers of $F_n$ then converge to those of $F$.
It was introduced by Ennio De Giorgi.
It is the standard tool to pass from a family of variational problems to a limit problem, as in homogenization and phase transitions.
The [[definition|NNAR1HN9BQ]] gives the precise conditions.

## Explanations

### L1 human

Pointwise convergence of functions does not control where their minima go.
Γ-convergence asks instead for two things.
The limit $F$ is never larger than what the $F_n$ give near a point.
And every value of $F$ can be approached by values of $F_n$ at nearby points.
Think of a landscape with finer and finer ripples: the ripples have no pointwise limit, but the lowest points of the landscape settle down.
For example, $F_n(x) = \sin(nx)$ on $\R$ has no pointwise limit, yet it Γ-converges to the constant $-1$, because near every point there are points where $\sin(nx) = -1$.
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# Frobenius element

## Definition

Let $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]] of [[algebraic number fields|MWBDFS2KJ5]] with [[Galois group|0XTY86BZKY]] $G$, and let $\mathcal{O}_K \subseteq \mathcal{O}_L$ be the rings of integers.
Let $\mathfrak{p}$ be a nonzero prime ideal of $\mathcal{O}_K$ that is [[unramified|H5RC67SD24]] in $L$, and let $\mathfrak{P}$ be a prime ideal of $\mathcal{O}_L$ over $\mathfrak{p}$.
Let $q = N(\mathfrak{p}) = \abs{\mathcal{O}_K/\mathfrak{p}}$ be the [[norm|M54ADM0CVF]] of $\mathfrak{p}$.
The *Frobenius element* of $\mathfrak{P}$ is the unique $\sigma \in G$ such that
$$\sigma(x) \equiv x^{q} \pmod{\mathfrak{P}} \quad \text{for all } x \in \mathcal{O}_L.$$
It is written $\mathrm{Frob}_{\mathfrak{P}}$ or $\left(\frac{L/K}{\mathfrak{P}}\right)$.

## Remarks

Existence and uniqueness: such a $\sigma$ fixes $\mathfrak{P}$, so it lies in the [[decomposition group|7X92G8CQ33]] $D_{\mathfrak{P}}$.
The map from $D_{\mathfrak{P}}$ to the Galois group of the residue extension is onto, with kernel the inertia group, which is trivial because $\mathfrak{p}$ is unramified.
So exactly one element of $D_{\mathfrak{P}}$ maps to the [[Frobenius automorphism|0CGCPX8Z70]] $y \mapsto y^q$ of the residue field.
$\mathrm{Frob}_{\mathfrak{P}}$ generates $D_{\mathfrak{P}}$, and its order is the residue degree $f$.

For $\tau \in G$, $\mathrm{Frob}_{\tau(\mathfrak{P})} = \tau \, \mathrm{Frob}_{\mathfrak{P}} \, \tau^{-1}$.
So the Frobenius elements of the primes over $\mathfrak{p}$ form one [[conjugacy class|0DEBXAJ5SA]] $\mathrm{Frob}_{\mathfrak{p}}$ of $G$; if $G$ is abelian, it is a single element, the *Artin symbol* of $\mathfrak{p}$.

Example: $K = \Q$, $L = \Q(\zeta_n)$ with $\zeta_n$ a primitive $n$-th root of unity, and $p$ a prime not dividing $n$.
The automorphism $\sigma_p$ with $\sigma_p(\zeta_n) = \zeta_n^p$ satisfies $\sigma_p(x) \equiv x^p$ modulo every prime over $p$, since $(a + b)^p \equiv a^p + b^p$ modulo $p$; so $\mathrm{Frob}_p = \sigma_p$.

Non-example: in $\Q(i)/\Q$, the prime $2$ is ramified.
Both elements $\sigma$ of $G$ satisfy $\sigma(x) \equiv x^2 \pmod{(1 + i)}$, because the residue field is $\F_2$ and $i \equiv -i \equiv 1 \pmod{(1+i)}$; so there is no unique Frobenius element.
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# Enumerative combinatorics

## Definition

Let $(S_n)_{n \ge 0}$ be a family of [[finite sets|0BBTH3YSZA]] given by a combinatorial rule, such as the subsets of $\set{1, \ldots, n}$ or the trees with $n$ vertices.
*Enumerative combinatorics* is the branch of [[combinatorics|0JKW7W7XFR]] that determines the counting function $n \mapsto \abs{S_n}$.
An answer is one of the following:

1. an explicit formula for $\abs{S_n}$;
2. a [[recurrence relation|0FYX44GP92]] that computes $\abs{S_n}$ from earlier values;
3. a closed form for a [[generating function|0PHAC833JF]] of the sequence $\abs{S_n}$;
4. an asymptotic estimate of $\abs{S_n}$ as $n \to \infty$.

## Remarks

Example: let $S_n$ be the set of subsets of $\set{1, \ldots, n}$ with no two consecutive elements.
A set in $S_n$ either avoids $n$, or contains $n$ and avoids $n - 1$.
This gives the recurrence $\abs{S_n} = \abs{S_{n-1}} + \abs{S_{n-2}}$, with $\abs{S_0} = 1$ and $\abs{S_1} = 2$, so $\abs{S_n}$ is the Fibonacci number $F_{n+2}$.

Non-example: the question whether a finite projective plane of order $10$ exists is a question of existence, not of counting.
It belongs to the theory of [[combinatorial designs|PAJTMQ604P]].

The parameter $n$ may be replaced by several parameters, as in the [[binomial coefficient|0GBZM0MT0F]] $\binom{n}{k}$.
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# Browder–Minty theorem

## Statement

Let $X$ be a real separable [[reflexive|NN8RWPC8WG]] [[Banach space|06GZ956657]] with norm $\norm{\cdot}$, and let $X^*$ be its [[continuous dual space|4A82S5FVRT]].
Write $\langle g, u \rangle$ for the value of $g \in X^*$ at $u \in X$.
Let $T \colon X \to X^*$ be a map with these three properties.

1. $T$ is [[monotone|Y1M03JQKKR]]: $\langle T(u) - T(v), u - v \rangle \ge 0$ for all $u, v \in X$.
2. $T$ is hemicontinuous: for all $u, v, w \in X$, the map $t \mapsto \langle T(u + t v), w \rangle$ is continuous on $[0, 1]$.
3. $T$ is coercive: $\langle T(u), u \rangle / \norm{u} \to +\infty$ as $\norm{u} \to \infty$.

Then for every $g \in X^*$ the equation $T(u) = g$ has a solution $u \in X$.
If moreover $\langle T(u) - T(v), u - v \rangle > 0$ for all $u \ne v$, then the solution is unique.

## Remarks

A standard proof uses the Galerkin method.
On an increasing sequence of finite-dimensional subspaces whose union is dense, the Brouwer fixed point theorem gives approximate solutions.
Coercivity bounds them, reflexivity gives a weakly convergent subsequence, and Minty's trick passes to the limit.
The trick: for a monotone hemicontinuous $T$, if $\langle g - T(v), u - v \rangle \ge 0$ for all $v \in X$, then $T(u) = g$.
For $X = \R$ the theorem says that a nondecreasing continuous function $T$ with $T(u) \to \pm\infty$ as $u \to \pm\infty$ takes every real value.
Coercivity cannot be dropped: $T(u) = \arctan u$ on $\R$ is monotone and continuous but never takes the value $2$.

## Explanations

### L1 human

The theorem solves equations $T(u) = g$ where $T$ is a nonlinear operator that "never goes backwards" and grows fast enough at infinity.
On the real line this is the intermediate value theorem for an increasing function that runs from $-\infty$ to $+\infty$.
The theorem carries this simple picture to infinite-dimensional spaces of functions.

### L2 human

The theorem is the nonlinear analogue of the Lax–Milgram theorem.
A bounded linear operator given by a coercive bilinear form is a special case of a coercive monotone operator.
Its main use is in quasilinear elliptic equations such as $-\operatorname{div}(\abs{\nabla u}^{p-2} \nabla u) = f$, whose weak form defines a monotone coercive operator from the Sobolev space $W_0^{1,p}$ to its dual for $1 < p < \infty$.
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# Holonomic sequence

## Summary

A holonomic sequence, also called a P-recursive sequence, is a sequence that satisfies a [[linear recurrence with polynomial coefficients|K6QV4WSTXX]].
A sequence is holonomic exactly when its generating function is a [[holonomic function|7H44JD9NQD]].
The class is closed under sums, termwise products and many other operations, which makes it the basis of automatic proofs of identities for sums.
It contains the solutions of linear recurrences with constant coefficients, the factorials, the Catalan numbers and the harmonic numbers.

## Explanations

### L1 human

Some sequences grow in complicated ways, but still obey a simple linear rule whose coefficients are polynomials in the position $n$.
The factorials obey $u_{n+1} = (n+1) u_n$, so they are holonomic.
Such a rule together with a few starting values is a complete, finite description of the sequence.
A computer can store it, add two such sequences, or check an identity by comparing rules.
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<document source="https://tamnd.github.io/math2/node/NWMYW2WK9S.md" tag="NWMYW2WK9S" kind="concept">
# Montel space

## Summary

A Montel space is a [[barrelled space|ZE6DXEJB8H]] in which every closed [[bounded set|3BAJ8ZTVPX]] is compact.
This is a Heine–Borel property, which fails in every infinite-dimensional normed space.
The name comes from [[Montel's theorem|ZVFVBVJMCB]]: the space of holomorphic functions on an open subset of $\C$ is a Montel space.
Spaces of smooth functions and of test functions used in distribution theory are Montel spaces too, and Montel spaces are reflexive.

## Explanations

### L1 human

In $\R^n$, a closed and bounded set is compact: every sequence in it has a convergent subsequence.
In an infinite-dimensional Banach space this fails, already for the closed unit ball.
Montel spaces are infinite-dimensional spaces where it holds again.
For holomorphic functions, a family that is bounded on each compact set has a subsequence that converges uniformly on compact sets.
That is the content of Montel's theorem, and it is why these spaces carry Montel's name.
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# Lemniscate elliptic functions

## Definition

Let $\varpi = 2\int_0^1 dt/\sqrt{1 - t^4}$ be the [[lemniscate constant|V5A2S8QYQQ]].
The map $x \mapsto \int_0^x dt/\sqrt{1 - t^4}$ is a strictly increasing bijection from $[-1, 1]$ onto $[-\varpi/2, \varpi/2]$.
The *lemniscate sine* $\operatorname{sl}$ is the inverse map on $[-\varpi/2, \varpi/2]$, extended to $\R$ by $\operatorname{sl}(x + \varpi) = -\operatorname{sl}(x)$.
It extends further to a meromorphic function on $\C$ by [[analytic continuation|0M3YQJ81NB]].
The *lemniscate cosine* is $\operatorname{cl}(z) = \operatorname{sl}(\varpi/2 - z)$.

## Remarks

The function $\operatorname{sl}$ is odd, satisfies $\operatorname{sl}(0) = 0$, $\operatorname{sl}(\varpi/2) = 1$, and $(\operatorname{sl}')^2 = 1 - \operatorname{sl}^4$.
It is an [[elliptic function|FF9ZPCZ59K]] whose periods form the lattice spanned by $(1 + i)\varpi$ and $(1 - i)\varpi$; in particular $2\varpi$ and $2i\varpi$ are periods.
It satisfies $\operatorname{sl}(iz) = i\operatorname{sl}(z)$.
The two functions are linked by $\operatorname{sl}^2 + \operatorname{cl}^2 + \operatorname{sl}^2\operatorname{cl}^2 = 1$.
Example: at $z = \varpi/2$ one has $\operatorname{sl} = 1$ and $\operatorname{cl} = \operatorname{sl}(0) = 0$, and indeed $1 + 0 + 0 = 1$.
Non-example: the inverse of $\int_0^x dt/\sqrt{1 - t^2}$ is $\sin$, which is not the lemniscate sine: it has the single period $2\pi$ and satisfies $(\sin')^2 = 1 - \sin^2$ instead of $1 - \operatorname{sl}^4$.
In terms of the [[Jacobi elliptic functions|P071B692JF]], $\operatorname{sl}(z) = \operatorname{sn}(z; k)$ with $k^2 = -1$, since this $\operatorname{sn}$ solves the same equation $(y')^2 = (1 - y^2)(1 + y^2)$ with $y(0) = 0$, $y'(0) = 1$.
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# Osculating circle

## Summary

The osculating circle of a curve at a point is the circle that fits the curve best at that point.
It touches the curve there, with the same tangent line and the same [[curvature|7JQS6VKFEZ]].
Its radius is the radius of curvature $1/\kappa$, and its centre is the centre of curvature.

## Explanations

### L1 human

Stand at a point of a curved road and try to draw a circle that follows the road as long as possible near you.
On a gentle bend the best circle is huge, and on a sharp bend it is small.
That best circle is the osculating circle; the word comes from the Latin for "kissing".
For the parabola $y = x^2$ at the origin, the osculating circle has radius $1/2$ and centre $(0, 1/2)$.
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# Lefschetz duality

## Statement

Let $M$ be a compact [[manifold with boundary|7DS6B2BFK8]] of dimension $n$ with boundary $\partial M$, and let $R$ be a commutative ring with $1$.
Assume that the interior of $M$ is $R$-[[oriented|913WEQERBE]], and let $[M] \in H_n(M, \partial M; R)$ be the corresponding relative [[fundamental class|ZEA0PEM3B2]].
Let $H^k$ be [[singular cohomology|6N7BC70CX0]], and use [[relative homology|4BVWH0BN28]] and relative cohomology for the pair $(M, \partial M)$.
Then for every $k$, the [[cap product|N7SXBQ7WK0]] with $[M]$ gives isomorphisms
$$H^k(M; R) \to H_{n-k}(M, \partial M; R) \qquad\text{and}\qquad H^k(M, \partial M; R) \to H_{n-k}(M; R).$$

## Remarks

The relative fundamental class is the class whose restriction to $H_n(M, M \setminus \set{x})$ is the local orientation at each interior point $x$.
The proof uses a collar of the boundary to compare $M$ with its interior.
It then applies the duality theorem behind [[Poincaré duality|9VY90HJVZX]] to the interior, a noncompact manifold without boundary.
When $\partial M = \emptyset$, both isomorphisms reduce to Poincaré duality.

Example: for the disc $M = D^n$ with $\partial M = S^{n-1}$, $H^0(D^n) \cong \Z$ and $H_n(D^n, S^{n-1}) \cong \Z$.
Also $H^n(D^n, S^{n-1}) \cong \Z$ matches $H_0(D^n) \cong \Z$, and all other groups vanish on both sides.

## Explanations

### L1 human

Poincaré duality pairs holes of complementary dimensions on a closed manifold.
For a manifold with an edge, the same pairing works if one side of the pairing is measured relative to the edge.
For a disc, the single point-like piece matches the whole disc counted modulo its boundary circle.

### L2 human

Lefschetz duality is the form of Poincaré duality used for compact manifolds with boundary, such as knot exteriors and cobordisms.
Take a compact oriented $(2m+1)$-manifold $M$ and a field $F$.
Lefschetz duality and the long exact sequence of $(M, \partial M)$ show that the kernel of $H_m(\partial M; F) \to H_m(M; F)$ has half the dimension of $H_m(\partial M; F)$.
It is the basic tool in the study of manifolds with boundary and of bordism.
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# Almost complex manifold

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] with [[tangent bundle|WY6K3P4YCG]] $TM$.
An *almost complex structure* on $M$ is a smooth [[tensor field|WX302T93TW]] $J$ of type $(1, 1)$ such that $J^2 = -\id$.
Thus $J \colon TM \to TM$ is a smooth map that is linear on each tangent space $T_p M$.
An *almost complex manifold* is a pair $(M, J)$ of a smooth manifold and an almost complex structure on it.

## Remarks

Example: on $\R^{2n}$ with coordinates $(x_1, \ldots, x_n, y_1, \ldots, y_n)$, put $J \partial_{x_k} = \partial_{y_k}$ and $J \partial_{y_k} = -\partial_{x_k}$.
Then $J^2 \partial_{x_k} = J \partial_{y_k} = -\partial_{x_k}$ and likewise for $\partial_{y_k}$, so $J^2 = -\id$.
This is multiplication by $i$ under $\R^{2n} \cong \C^n$, and every complex manifold carries the structure obtained in this way from its charts.

Non-example: no manifold of odd dimension $m$ has an almost complex structure.
A real linear map $J$ of $T_p M$ with $J^2 = -\id$ gives $\det(J)^2 = \det(-\id) = (-1)^m = -1$, which is impossible.

The *Nijenhuis tensor* of $J$ is
$$N_J(X, Y) = [JX, JY] - J[JX, Y] - J[X, JY] - [X, Y]$$
for vector fields $X, Y$, where $[\cdot, \cdot]$ is the [[Lie bracket of vector fields|68FGQEBD6J]].
The structure $J$ is called *integrable* when $N_J = 0$.
The sphere $S^6$ has an almost complex structure coming from the octonions, while $S^4$ has none.
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# Morrey–Campanato space

## Summary

Morrey spaces and Campanato spaces measure the size of a function on small balls, uniformly over all centres.
A Morrey space asks that the $L^p$ mass of $f$ on a ball of radius $r$ is at most a constant times $r^\lambda$.
A Campanato space asks the same of the oscillation of $f$ around its mean on the ball.
Special cases are [[Lebesgue spaces|SASP14MA1Z]], [[bounded mean oscillation|QJYTXFT8Y4]] and [[Hölder continuous|012PBA0FMD]] functions.
They are a basic tool for the regularity of solutions of elliptic equations.

## Explanations

### L1 human

Take a function and look at it through a small round window of radius $r$.
Measure how much of the function you see, or how much it varies, inside the window.
If this amount shrinks like a power $r^\lambda$ as the window gets smaller, wherever you put it, the function lies in the Morrey or Campanato space with exponent $\lambda$.
A larger $\lambda$ means a tamer function.
For instance, a function with a mild spike such as $\abs{x}^{-a}$ is still allowed for small enough $\lambda$.
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# Thin set

## Definition

Let $n \ge 2$, let $E \subseteq \R^n$ and let $x \in \R^n$.
The set $E$ is *thin at* $x$ if one of the following holds.

1. The point $x$ is not a [[limit point|07EGCNHAVW]] of $E$.
2. The point $x$ is a limit point of $E$, and there is a [[superharmonic function|X4V2S56YQQ]] $u$ on an open neighbourhood of $x$ such that
$$\liminf_{y \to x,\ y \in E \setminus \set{x}} u(y) > u(x).$$

## Remarks

The left side of clause 2 is a [[limit inferior|033RZ32T4Q]] along points of $E$ other than $x$.
Example: in $\R^3$ let $x_k = 2^{-k} e_1$ for $k \ge 1$ and $E = \set{x_k : k \ge 1}$, with $x = 0$.
Put $u(y) = \sum_{k \ge 1} 4^{-k} \abs{y - x_k}^{-1}$.
This is a series of positive Newtonian potentials with $u(0) = \sum_k 4^{-k} 2^{k} = 1$, so $u$ is superharmonic on $\R^3$.
Since $u(x_k) = +\infty$ for every $k$, clause 2 holds, and $E$ is thin at $0$.
Non-example: a closed ball $B$ is not thin at a point $x \in \partial B$.
If $u$ were as in clause 2, then $u(x)$ is finite and $u \ge u(x) + \varepsilon$ on $B$ near $x$ for some $\varepsilon > 0$, while $u \ge u(x) - \delta$ near $x$ for any $\delta > 0$ by lower semicontinuity.
A small sphere about $x$ has nearly half of its area in $B$, so its mean value of $u$ exceeds $u(x)$, against the super-mean value inequality.
For $n \ge 3$, Wiener's criterion says that $E$ is thin at a limit point $x$ if and only if $\sum_{k} 2^{k(n-2)} c(E_k) < \infty$, where $E_k = \set{y \in E : 2^{-k-1} \le \abs{y - x} \le 2^{-k}}$ and $c$ is Newtonian capacity.
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<document source="https://tamnd.github.io/math2/node/NX7WCTNYWT.md" tag="NX7WCTNYWT" kind="definition">
# Groupoid

## Definition

A *groupoid* is a [[category|A2DE66937D]] $\mathcal{G}$ in which every morphism is an isomorphism.
That is, for every morphism $f \colon x \to y$ of $\mathcal{G}$ there is a morphism $g \colon y \to x$ with $g \circ f = \id_x$ and $f \circ g = \id_y$.

## Remarks

A groupoid is *small* if its objects form a set.
For each object $x$, the morphisms $x \to x$ form a [[group|G5S1M9]], the vertex group at $x$.

Example: let $\sim$ be an [[equivalence relation|0G8VEZ3WTK]] on a set $X$.
Take $X$ as the set of objects, with exactly one morphism $x \to y$ when $x \sim y$ and none otherwise.
Composition exists by transitivity, identities by reflexivity, and the inverse of $x \to y$ is the morphism $y \to x$, which exists by symmetry.

Non-example: the category with objects $0, 1$ and one non-identity arrow $0 \to 1$, the poset $0 \le 1$.
There is no morphism $1 \to 0$, so the arrow $0 \to 1$ has no inverse.

A group $G$ gives a groupoid with one object whose morphisms are the elements of $G$, and every one-object groupoid arises in this way.
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# Grothendieck inequality

## Statement

There is a constant $K$ with the following property.
Let $m, n \ge 1$ and let $(a_{ij})$ be a real $m \times n$ matrix.
Assume that $\abs{\sum_{i,j} a_{ij} s_i t_j} \le 1$ for all $s_1, \dots, s_m, t_1, \dots, t_n \in \set{-1, 1}$.
Let $H$ be a real [[Hilbert space|HM2HHE3J51]] and let $u_1, \dots, u_m, v_1, \dots, v_n$ be vectors of norm at most $1$ in $H$.
Then
$$\Big| \sum_{i=1}^m \sum_{j=1}^n a_{ij} \ip{u_i}{v_j} \Big| \le K.$$

## Remarks

The constant $K$ does not depend on $m$, $n$ or $H$.
The smallest such constant is the real Grothendieck constant $K_G$.
Its exact value is unknown.
Krivine proved $K_G \le \pi / (2 \log(1 + \sqrt{2})) \approx 1.78$.

One proof uses random Gaussian vectors $g$.
For unit vectors $u$ and $v$, the average of $\sgn \ip{g}{u} \sgn \ip{g}{v}$ is $\frac{2}{\pi} \arcsin \ip{u}{v}$.
So signs drawn at random from the vectors reproduce a function of the [[inner products|0NB0AWGJBW]].
Krivine first transforms the vectors so that this function becomes a fixed multiple of the inner product.

The inequality has a complex version, with a different constant.
In computer science it shows that a semidefinite relaxation approximates the cut norm of a matrix within a constant factor.

## Explanations

### L1 human

Replace numbers $\pm 1$ by unit vectors in some Euclidean space, and products by inner products.
A sum built from a matrix can only grow by a bounded factor under this replacement, whatever the size of the matrix.
This links a hard combinatorial maximum over signs with an easy geometric maximum over vectors.

### L2 human

Grothendieck stated the result in his work on tensor products of Banach spaces.
In that language it says that every bounded linear map from $L^1$ to $L^2$ is absolutely summing.
The inequality is a key tool in the study of the [[projective tensor product|685ZV73JJ8]] and of operator ideals.
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<document source="https://tamnd.github.io/math2/node/NX8RE9B72A.md" tag="NX8RE9B72A" kind="concept">
# Total variation diminishing scheme

## Summary

A total variation diminishing (TVD) scheme is a numerical scheme for a scalar [[conservation law|8RA7AMWSAF]] in one space dimension.
Its discrete [[total variation|8QA5DEWNYD]] never increases from one time step to the next.
This copies a property of the exact entropy solutions, and it rules out the creation of new local extrema.
So the scheme gives no spurious oscillations near shocks.
The notion was introduced by Harten in 1983, and it is the design principle of the high-resolution schemes built with flux limiters.

## Explanations

### L1 human

The total variation of a list of numbers is the sum of the sizes of the jumps between neighbours: it measures how much the profile goes up and down.
Wiggles near a shock increase this sum.
A TVD scheme guarantees that the sum never grows, so wiggles cannot appear.
For example, a step from $1$ to $0$ has total variation $1$, and a TVD scheme can smear the step but never turn it into an overshoot such as $1, 1.1, 0.4, 0$.
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<document source="https://tamnd.github.io/math2/node/NX9X1A5EKY.md" tag="NX9X1A5EKY" kind="definition">
# Universal class

## Definition

Let $\sigma$ be a first-order [[signature|WDVVEYD6AT]].
A *universal sentence* of $\sigma$ is a [[sentence|DKFGQTRYZE]] of the form $\forall x_1 \cdots \forall x_n \, \varphi$, with $n \ge 0$ [[universal quantifiers|0GHS47PDAQ]].
Here $\varphi$ is a quantifier-free formula whose free variables are among $x_1, \ldots, x_n$.
Let $K$ be a class of [[$\sigma$-structures|983GZBQYT4]].
Then $K$ is a *universal class* if there is a set $\Sigma$ of universal sentences of $\sigma$ whose models are exactly the members of $K$.

## Remarks

Example: simple graphs, in the signature with one binary relation $E$, form a universal class with the axioms $\forall x \, \neg E(x, x)$ and $\forall x \forall y \, (E(x, y) \rightarrow E(y, x))$.

Non-example: fields, in the signature $(+, -, \cdot, 0, 1)$, do not form a universal class.
A universal sentence true in a structure is true in each [[substructure|676H7EGTKJ]], because it only speaks about tuples of elements and the operations on them.
The ring $\Z$ is a substructure of the field $\Q$ in this signature, and it is not a field.

Every universal class is an [[elementary class|MEC3ZMC4F1]].
The [[Łoś–Tarski preservation theorem|9RN0KYXNT1]] gives the converse characterization: an elementary class is universal exactly when it is closed under substructures.
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# Caristi fixed-point theorem

## Statement

Let $(X, d)$ be a nonempty [[complete metric space|0BY52JW6NK]].
Let $\varphi \colon X \to [0, \infty)$ be a lower semicontinuous function.
Let $T \colon X \to X$ be any map, not necessarily continuous, such that
$$d(x, T(x)) \le \varphi(x) - \varphi(T(x)) \quad \text{for all } x \in X.$$
Then $T$ has a fixed point: there is $x \in X$ with $T(x) = x$.

## Remarks

The theorem follows from [[Ekeland's variational principle|MJGPW5EVB3]].
With $\varepsilon = 1$, it gives a point $x_0$ with $\varphi(x_0) < \varphi(x) + d(x, x_0)$ for every $x \ne x_0$.
If $T(x_0) \ne x_0$, then taking $x = T(x_0)$ gives $\varphi(x_0) < \varphi(T(x_0)) + d(T(x_0), x_0) \le \varphi(x_0)$, a contradiction.

Example: the [[Banach fixed point theorem|0VK6VEH345]] is a special case.
If $d(T(x), T(y)) \le k\, d(x, y)$ with $0 \le k < 1$, put $\varphi(x) = d(x, T(x))/(1 - k)$, which is continuous.
The Caristi condition becomes $d(T(x), T(T(x))) \le k\, d(x, T(x))$, which holds.

Completeness is needed: on $X = (0, 1]$ with the usual metric, $T(x) = x/2$ and $\varphi(x) = x$ satisfy $d(x, T(x)) = x/2 = \varphi(x) - \varphi(T(x))$, but $T$ has no fixed point in $X$.

## Explanations

### L1 human

Think of $\varphi$ as a supply of fuel and of $T$ as a move.
Each move costs at least as much fuel as the distance travelled.
Since fuel cannot go below zero, the total distance of a run of moves is finite.
In a complete space the run settles down to a point that $T$ does not move.

### L2 human

The theorem does not ask $T$ to be continuous, which makes it more flexible than the Banach fixed point theorem.
It is equivalent in strength to Ekeland's variational principle, and each can be derived from the other.
It is used in nonlinear analysis for maps that are only known to decrease a lower semicontinuous potential, such as some descent schemes in optimisation.
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# Oka manifold

## Summary

An Oka manifold is a complex manifold $Y$ that receives many holomorphic maps from Euclidean spaces.
Holomorphic maps from neighbourhoods of compact convex sets in $\C^n$ to $Y$ can be approximated by entire maps $\C^n \to Y$.
For such targets the Oka principle holds: every continuous map from a Stein manifold to $Y$ can be deformed to a holomorphic map.
Oka manifolds are the opposite of Kobayashi hyperbolic manifolds, which receive no nonconstant entire maps from $\C$.

## Explanations

### L1 human

Some complex manifolds are "flexible": they contain many holomorphic images of $\C$ and of $\C^n$, as $\C^n$ itself does.
Others are "rigid": every holomorphic map from $\C$ into them is constant, as for the unit disc, by Liouville's theorem.
Oka manifolds are the flexible ones, made precise by an approximation property.
Examples are $\C^n$, the complex projective spaces, and complex Lie groups such as $\C^* = \C \setminus \set{0}$.
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# Carathéodory metric

## Definition

Let $\mathbb{D} = \set{\zeta \in \C : \abs{\zeta} < 1}$ and let $\rho(a, b) = \tanh^{-1} \abs{(a - b)/(1 - \bar a b)}$ be the Poincaré distance on $\mathbb{D}$.
Let $X$ be a [[complex manifold|JSGWAN2RJV]] and $p, q \in X$.
The *Carathéodory pseudodistance* of $X$ is
$$c_X(p, q) = \sup\set{\rho(f(p), f(q)) : f \colon X \to \mathbb{D} \text{ holomorphic}} .$$
The *Carathéodory–Reiffen pseudometric* is $\gamma_X(p, v) = \sup\set{\abs{df_p(v)} : f \colon X \to \mathbb{D} \text{ holomorphic},\ f(p) = 0}$ for $v$ tangent to $X$ at $p$.

## Remarks

$c_X$ is a [[pseudodistance|0A3HJX3KTF]], as a supremum of pseudodistances $(p, q) \mapsto \rho(f(p), f(q))$.
If $g \colon X \to Y$ is a [[holomorphic map|3MQTWKBBBT]], then $c_Y(g(p), g(q)) \le c_X(p, q)$, since $f \circ g$ is admissible for $X$ whenever $f$ is admissible for $Y$.
Example: $c_{\mathbb{D}} = \rho$; the identity gives $c_{\mathbb{D}} \ge \rho$, and the [[Schwarz–Pick theorem|V1QYA9RW73]] gives $\rho(f(a), f(b)) \le \rho(a, b)$ for every $f$.
Example: if $X$ is a bounded domain in $\C^n$, then $c_X$ is a distance, since a suitable linear function, rescaled, maps $X$ into $\mathbb{D}$ and separates any two given points.
Non-example: $c_{\C^n} \equiv 0$, since every bounded holomorphic function on $\C^n$ is constant.
For every connected $X$, $c_X \le k_X$, where $k_X$ is the [[Kobayashi pseudodistance|DY8QQ5GTTX]].
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# Elliptic integral

## Summary

An elliptic integral is an integral of a rational function of $t$ and $\sqrt{P(t)}$.
Here $P$ is a polynomial of degree $3$ or $4$ without repeated roots.
Such integrals arose from the arc length of the [[ellipse|0VKCCEXBJ1]] and in general cannot be written with elementary functions.
Legendre showed that every elliptic integral reduces to elementary functions and three standard integrals, of the first, second and third kind.
Inverting the integral of the first kind leads to the [[Jacobi elliptic functions|P071B692JF]].

## Explanations

### L1 human

Integrals with $\sqrt{1 - t^2}$ give the arcsine and the length of a circle arc.
Replacing the square root of a quadratic by the square root of a cubic or quartic gives an elliptic integral.
The simplest case is the length of an arc of an ellipse, which is the origin of the name.
For example, $\int_0^{x} dt / \sqrt{(1 - t^2)(1 - k^2 t^2)}$ with $0 < k < 1$ is an elliptic integral; for $k = 0$ it collapses to $\arcsin x$.
These integrals define new functions, which tables and computers evaluate like the logarithm or the sine.
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# Fourier basis of L2 of the circle

## Statement

Let $\mathbb{T} = \R / 2\pi\Z$, and let $L^2(\mathbb{T})$ be the [[Lebesgue space|SASP14MA1Z]] of square integrable functions on $\mathbb{T}$, with the inner product
$$\langle f, g \rangle = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \overline{g(x)} \, dx.$$
For $n \in \Z$ let $e_n(x) = e^{inx}$ be the [[Fourier monomial|TZYSW0975T]] of index $n$.
Then the family $(e_n)_{n \in \Z}$ is an [[orthonormal basis|0S9GGTR0EX]] of the [[Hilbert space|HM2HHE3J51]] $L^2(\mathbb{T})$.

## Remarks

Orthonormality is a direct computation: $\langle e_n, e_m \rangle = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{i(n-m)x} \, dx$ is $1$ if $n = m$ and $0$ otherwise.
Completeness means that the linear span of the $e_n$ is dense in $L^2(\mathbb{T})$.
This span is the space of trigonometric polynomials; it is dense in $C(\mathbb{T})$ for the supremum norm by the [[density of trigonometric polynomials|KDRZKC9T1V]], and $C(\mathbb{T})$ is dense in $L^2(\mathbb{T})$.
Since $\norm{g}_2 \le \sup \abs{g}$, uniform approximation gives approximation in $L^2$.
The coordinates of $f$ in this basis are its Fourier coefficients $\hat f(n) = \langle f, e_n \rangle$.
Example: $\cos x = \frac{1}{2} e_1 + \frac{1}{2} e_{-1}$, and indeed $\norm{\cos}_2^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} \cos^2 x \, dx = \frac{1}{2} = \frac{1}{4} + \frac{1}{4}$.
With the normalised measure $dx / 2\pi$ the $e_n$ have norm $1$; with the measure $dx$ one uses $e_n / \sqrt{2\pi}$ instead.

## Explanations

### L1 human

The pure waves $e^{inx}$ act like perpendicular unit axes in an infinite-dimensional space of functions.
Every square integrable periodic function has coordinates along these axes, namely its Fourier coefficients.
Completeness says that no direction is missing: a function perpendicular to every wave is zero.

### L2 human

The theorem puts Fourier series inside Hilbert space theory.
General facts about orthonormal bases then give at once the convergence of Fourier series in $L^2$, Parseval's identity and the Riesz–Fischer theorem.
The map $f \mapsto (\hat f(n))_{n \in \Z}$ is a unitary isomorphism from $L^2(\mathbb{T})$ onto $\ell^2(\Z)$.
The same holds on the torus $\mathbb{T}^d$ with the monomials $e^{i k \cdot x}$, $k \in \Z^d$, and, by the Peter–Weyl theorem, on any compact abelian group with its characters.
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# Gaussian measure

## Definition

Let $\gamma$ be a Borel [[probability measure|059TJN0NZ1]] on $\R$.
It is *Gaussian* in two cases.
The first case is $\gamma = \delta_m$, the [[Dirac measure|0ACRGYNHSR]] at some $m \in \R$.
The second case is that there are $m \in \R$ and $\sigma > 0$ with
$$\gamma(A) = \frac{1}{\sqrt{2 \pi \sigma^2}} \int_A e^{-(x - m)^2 / (2 \sigma^2)} \, dx$$
for every Borel set $A \subseteq \R$, where $dx$ is [[Lebesgue measure|2Z2QVWZ262]].
Let $E$ be a real [[separable|0Z2P41SVDP]] [[Banach space|06GZ956657]].
A Borel probability measure $\gamma$ on $E$ is *Gaussian* if for every [[continuous linear form|03TM135G0N]] $\ell \colon E \to \R$ the [[pushforward|3N66TM3TSQ]] $\ell_* \gamma$ is a Gaussian measure on $\R$.
It is *centered* if every $\ell_* \gamma$ has mean $0$.

## Remarks

In the second case on $\R$, $\gamma$ is the [[normal distribution|05AQE0QR0D]] $N(m, \sigma^2)$; the Dirac case is the degenerate normal distribution $N(m, 0)$.
The *standard* Gaussian measure on $\R^n$ has density $(2\pi)^{-n/2} e^{-\abs{x}^2/2}$.
Some authors call $\gamma$ Gaussian only when every $\ell_* \gamma$ with $\ell \ne 0$ is non-degenerate.

Example: on $E = \R^2$, the standard Gaussian measure is Gaussian.
For $\ell(x) = a x_1 + b x_2$, the law $\ell_* \gamma$ is $N(0, a^2 + b^2)$, since a linear combination of independent standard normal variables is normal.

Non-example: the uniform probability measure on $[-1, 1]$ is not Gaussian.
Its pushforward under the identity is not a Dirac measure and has no Gaussian density, since it vanishes outside $[-1, 1]$.

Wiener measure on the Banach space of continuous functions $\omega$ on $[0, 1]$ with $\omega(0) = 0$ is a centered Gaussian measure.

## Explanations

### L1 human

A measure on a vector space is Gaussian when every linear view of it is a bell curve, possibly squeezed to a point.
This definition works in any dimension, including infinite dimension.
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# Brody hyperbolicity

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]].
An *entire curve* in $X$ is a nonconstant [[holomorphic map|3MQTWKBBBT]] $f \colon \C \to X$.
The manifold $X$ is *Brody hyperbolic* if there is no entire curve in $X$, that is, if every holomorphic map $\C \to X$ is constant.

## Remarks

Example: the unit disc $\mathbb{D}$ is Brody hyperbolic, since a holomorphic map $\C \to \mathbb{D}$ is a bounded entire function, hence constant by [[Liouville's theorem|0EHKVRF4J6]].
Example: a compact Riemann surface of genus $g \ge 2$ is Brody hyperbolic, because an entire curve would lift to a holomorphic map from $\C$ to its universal covering, the unit disc.
Non-example: $\C^n$ and the complex projective space $\mathbb{P}^n$ contain lines $t \mapsto (t, 0, \dots, 0)$ and $t \mapsto [1 : t : 0 : \cdots : 0]$, so they are not Brody hyperbolic.
Every [[Kobayashi hyperbolic manifold|HRPAX8V7HA]] is Brody hyperbolic, since an entire curve $f$ gives $k_X(f(a), f(b)) \le k_\C(a, b) = 0$.
For compact $X$ the two notions agree by [[Brody's lemma|VEV5E2077C]]; for noncompact $X$ they can differ.
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# Real spectrum of self-adjoint operators

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $T \colon H \to H$ be a bounded linear operator with $T^* = T$, where $T^*$ is the [[Hilbert space adjoint|6HSJBJ8TTQ]].
Then the [[spectrum|BMKATJ802A]] $\sigma(T)$ is a subset of $\R$.
More precisely, with $m = \inf_{\norm{x} = 1} \ip{Tx}{x}$ and $M = \sup_{\norm{x} = 1} \ip{Tx}{x}$, one has $\sigma(T) \subseteq [m, M]$, and both $m$ and $M$ belong to $\sigma(T)$.

## Remarks

Proof idea: let $\lambda = a + ib$ with $a, b$ real and $b \ne 0$.
Because $T - a I$ is self-adjoint, $\norm{(T - \lambda I)x}^2 = \norm{(T - aI)x}^2 + b^2 \norm{x}^2 \ge b^2 \norm{x}^2$.
So $T - \lambda I$ is injective with closed range, and the orthogonal complement of its range is $\ker(T - \bar\lambda I) = \set{0}$ by the same estimate; hence $T - \lambda I$ is invertible.

Example: the multiplication operator $(Tf)(x) = x f(x)$ on $L^2[0,1]$ is self-adjoint, and its spectrum is $[0, 1]$, a real interval, although $T$ has no eigenvalues.

Self-adjointness is needed: the unilateral shift on $\ell^2$ is not self-adjoint, and its spectrum is the closed unit disc.
The finite-dimensional case is the fact that [[eigenvalues of a self-adjoint operator are real|0C4NT4GT3X]].
The same conclusion holds for unbounded self-adjoint operators, and for self-adjoint elements of a C*-algebra.

## Explanations

### L1 human

A self-adjoint operator is the infinite-dimensional analogue of a Hermitian matrix.
Its spectrum, the set of numbers $\lambda$ for which $T - \lambda I$ cannot be inverted, lies on the real line.
This is why self-adjoint operators model measurable quantities in quantum mechanics.

### L2 human

In infinite dimensions the spectrum can contain points that are not eigenvalues, so the finite-dimensional argument with an eigenvector does not suffice; the proof uses a lower bound instead.
The result is the first step towards the spectral theorem for self-adjoint operators, which represents $T$ as an integral of $\lambda$ over $\sigma(T) \subseteq \R$.
The numbers $m$ and $M$ are the ends of the closure of the [[numerical range|XWXZ0DGW3Z]], and $\norm{T} = \max(\abs{m}, \abs{M})$.
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# Kelvin transform

## Definition

Let $n \ge 2$ and, for $x \in \R^n \setminus \set{0}$, let $x^* = x / \abs{x}^2$ be its inversion in the unit sphere.
Let $\Omega \subseteq \R^n \setminus \set{0}$ be an [[open set|02BH3MCBQB]] and let $\Omega^* = \set{x^* : x \in \Omega}$.
The *Kelvin transform* of a function $u \colon \Omega \to \R$ is the function $K[u] \colon \Omega^* \to \R$ given by
$$K[u](x) = \abs{x}^{2-n} \, u(x^*).$$

## Remarks

The map $x \mapsto x^*$ is an involution, so $(\Omega^*)^* = \Omega$ and $K[K[u]] = u$.
For $n = 2$, $K[u] = u \circ {}^*$ is the composition with the [[inversion in the unit circle|YXST7NGMTC]], $z \mapsto 1/\bar z$.
For $u$ twice continuously differentiable, $\Delta(K[u]) = K[\abs{x}^4 \Delta u]$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]]; so $u$ is [[harmonic|0DEB0JMN0W]] on $\Omega$ if and only if $K[u]$ is harmonic on $\Omega^*$.
Example: in $\R^3$, $K[x_1](x) = \abs{x}^{-1} \cdot x_1 \abs{x}^{-2} = x_1 / \abs{x}^3 = -\partial_1(1/\abs{x})$, which is harmonic on $\R^3 \setminus \set{0}$, as a derivative of a harmonic function.
Non-example: without the weight, $u(x^*) = x_1 / \abs{x}^2$ is not harmonic on $\R^3 \setminus \set{0}$; a direct computation gives $\Delta(x_1 \abs{x}^{-2}) = -2 x_1 \abs{x}^{-4}$.
The transform turns the behaviour of $u$ near infinity into the behaviour of $K[u]$ near $0$.
For instance, let $n \ge 3$ and let $u$ be harmonic outside a ball; then $u(x) \to 0$ at infinity if and only if $K[u]$ has a removable singularity at $0$.
Inversion in a sphere of centre $a$ and radius $\rho$ gives the analogous transform $\left(\rho / \abs{x - a}\right)^{n-2} u\left(a + \rho^2 (x - a)/\abs{x - a}^2\right)$.
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# Dolbeault operator

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of complex dimension $n$.
For integers $p, q \ge 0$, let $\Omega^{p,q}(X)$ be the space of smooth complex [[differential forms|0E3G0GRPZ6]] of type $(p, q)$.
In each holomorphic chart $(z_1, \ldots, z_n)$ they are the sums $\sum_{\abs{I} = p, \abs{J} = q} f_{IJ} \, dz_I \wedge d\bar z_J$ with smooth functions $f_{IJ}$.
The [[exterior derivative|0ZHSTAG8ZY]] maps $\Omega^{p,q}(X)$ into $\Omega^{p+1,q}(X) \oplus \Omega^{p,q+1}(X)$.
The *Dolbeault operator* $\bar\partial \colon \Omega^{p,q}(X) \to \Omega^{p,q+1}(X)$ is $d$ followed by the projection onto $\Omega^{p,q+1}(X)$.

## Remarks

In a chart, with the [[Wirtinger derivatives|WMXDVF36ZT]] $\partial / \partial \bar z_k = \frac{1}{2}(\partial / \partial x_k + i \, \partial / \partial y_k)$ for $z_k = x_k + i y_k$,
$$\bar\partial \Bigl(\sum_{I, J} f_{IJ} \, dz_I \wedge d\bar z_J\Bigr) = \sum_{I, J} \sum_{k=1}^n \frac{\partial f_{IJ}}{\partial \bar z_k} \, d\bar z_k \wedge dz_I \wedge d\bar z_J.$$
The other component of $d$ is written $\partial$, so $d = \partial + \bar\partial$, and $d^2 = 0$ gives $\bar\partial^2 = 0$, $\partial^2 = 0$ and $\partial \bar\partial + \bar\partial \partial = 0$.
The splitting of $d$ into two components uses that the complex structure comes from holomorphic charts.
The cohomology of $\bar\partial$ on $\Omega^{p, \bullet}(X)$ is the Dolbeault cohomology $H^{p,q}_{\bar\partial}(X)$, and the same construction works for forms with values in a holomorphic vector bundle.

Example: on $X = \C$ with coordinate $z$, let $f(z) = z^2$.
Then $\partial f / \partial \bar z = 0$, so $\bar\partial f = 0$, as for every [[holomorphic function|0NAHXR1FRH]].

Non-example: the function $g(z) = \bar z$ on $\C$ is not holomorphic, and indeed $\bar\partial g = \frac{\partial \bar z}{\partial \bar z} \, d\bar z = d\bar z \ne 0$.

## Explanations

### L1 human

The Dolbeault operator takes the derivative of a form in the antiholomorphic directions only, and holomorphic functions are exactly the functions it kills.
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# Semisimple ring

## Summary

A semisimple ring is a ring that is a [[semisimple module|N3MRHDKDGC]] over itself: it is a direct sum of minimal left ideals.
Every module over a semisimple ring is semisimple, so every submodule has a complement and every module is projective and injective.
The Artin–Wedderburn theorem describes them completely as finite products of matrix rings over division rings.
Group algebras of finite groups over fields of characteristic zero are the main examples, by Maschke's theorem.

## Explanations

### L1 human

In linear algebra over a field, every subspace has a complementary subspace.
A semisimple ring is a ring whose modules behave in the same way: each submodule splits off as a direct summand.
So modules over a semisimple ring break into simple pieces, like vector spaces break into lines.
The ring $\R \times \R$ and the ring of $2 \times 2$ complex matrices are semisimple, while $\Z$ is not, since $2\Z$ has no complement in $\Z$.
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# Projective hierarchy

## Definition

Let $\mathcal{N} = \N^\N$ be the [[Baire space|P2WS16HKD7]].
For each [[Polish space|584GXXY73M]] $X$, define classes of subsets of $X$ by recursion on $n \ge 1$:

1. $\Sigma^1_1$ is the class of [[analytic subsets|3RWWZYPAFV]] of $X$.
2. $\Pi^1_n$ is the class of complements $X \setminus A$ of sets $A$ in $\Sigma^1_n$.
3. $\Sigma^1_{n+1}$ is the class of projections $\set{x \in X : \exists y \in \mathcal{N} \, (x, y) \in C}$ of sets $C \subseteq X \times \mathcal{N}$ in the class $\Pi^1_n$ of the Polish space $X \times \mathcal{N}$.
4. $\Delta^1_n = \Sigma^1_n \cap \Pi^1_n$.

A subset of $X$ is *projective* if it lies in $\Sigma^1_n$ for some $n$.
This system of classes is the *projective hierarchy*.

## Remarks

The coanalytic sets form $\Pi^1_1$, and by Suslin's theorem $\Delta^1_1$ is the class of Borel sets.
Example: the set of $x \in \set{0, 1}^{\N \times \N}$ that code a well-order of $\N$ is in $\Pi^1_1$ but not in $\Sigma^1_1$.
Non-example by counting: there are only $2^{\aleph_0}$ projective subsets of $\R$, and $2^{2^{\aleph_0}}$ subsets in all, so some subset of $\R$ is not projective.
Each class $\Sigma^1_n$ and $\Pi^1_n$ is closed under countable unions, countable intersections and preimages by continuous maps, and $\Sigma^1_n \cup \Pi^1_n \subseteq \Delta^1_{n+1}$.
If $X$ is uncountable, then $\Sigma^1_n \neq \Pi^1_n$ for every $n$.
The lightface version, defined with recursive codes instead of arbitrary parameters, is part of the [[analytical hierarchy|M6JGF4TPM6]].
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# Chebyshev system

## Summary

A Chebyshev system, or Haar system, is a family of $n$ continuous functions whose nonzero linear combinations have at most $n - 1$ zeros.
Polynomials of degree less than $n$ are the model case.
Their span is called a Haar subspace.
For Chebyshev systems, interpolation at any $n$ distinct points is uniquely solvable.
By Haar's theorem, they are also the families for which best uniform approximation is always unique.

## Explanations

### L1 human

A polynomial of degree at most $n - 1$ that is not zero crosses the axis at most $n - 1$ times.
A Chebyshev system is any family of functions that shares this zero-counting property.
Because of it, you can always find exactly one combination that passes through $n$ prescribed points with different abscissas.
Examples are $1, x, \dots, x^{n-1}$ on an interval, and the functions $1, \cos x, \sin x, \dots, \cos mx, \sin mx$ on the circle, with $x$ taken modulo $2\pi$.
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<document source="https://tamnd.github.io/math2/node/NY6FVCX61P.md" tag="NY6FVCX61P" kind="definition">
# Closed operator

## Definition

Let $X$ and $Y$ be normed spaces, and let $T$ be an [[unbounded operator|KRJX5RA0PW]] from $X$ to $Y$ with domain $D(T)$.
The operator $T$ is *closed* if its [[graph|3RCAY97W1E]] $\Gamma(T)$ is a [[closed subset|0ZCK8DQYN1]] of $X \times Y$.

## Remarks

Equivalently, $T$ is closed when, for every sequence $(x_n)$ in $D(T)$ with $x_n \to x$ in $X$ and $T x_n \to y$ in $Y$, one has $x \in D(T)$ and $Tx = y$.

Example: on $X = Y = C[0, 1]$ with the supremum norm, let $D(T) = C^1[0, 1]$ and $Tf = f'$.
If $f_n \to f$ and $f_n' \to g$ uniformly, then $f_n(x) = f_n(0) + \int_0^x f_n'$ gives $f(x) = f(0) + \int_0^x g$, so $f \in C^1[0, 1]$ and $f' = g$.
Hence $T$ is closed, although it is not bounded.

Non-example: the same formula with $D(S)$ the space of polynomials is not closed.
The Taylor polynomials $p_n$ of $e^x$ converge uniformly to $e^x$ and $p_n' = p_{n-1}$ converges to $e^x$, but $e^x$ is not a polynomial, so it is not in $D(S)$.

Every bounded operator defined on all of $X$ is closed.
If $X$ and $Y$ are Banach spaces and $T$ is closed with $D(T) = X$, then $T$ is bounded, by the [[closed graph theorem|GXM8BX8YM7]].
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<document source="https://tamnd.github.io/math2/node/NY6TYHRGMS.md" tag="NY6TYHRGMS" kind="concept">
# Complex measure

## Summary

A complex measure is a countably additive function from a σ-algebra to the complex numbers.
Its real and imaginary parts are finite [[signed measures|VG2P7R91Z7]].
Complex measures form a Banach space under the total variation norm.
On a locally compact Hausdorff space, the Riesz–Markov–Kakutani theorem identifies the complex Radon measures with the dual of the continuous functions vanishing at infinity.
They are the natural measures for Fourier analysis, where one integrates complex exponentials.

## Explanations

### L1 human

A complex measure assigns a complex number to each measurable set, and adds up correctly over countably many disjoint pieces.
One can think of it as two signed measures side by side, one for the real part and one for the imaginary part.
For example, on $[0, 2\pi]$ the rule $\nu(E) = \int_E e^{i t} \, dt$ is a complex measure, and $\nu([0, 2\pi]) = 0$.
Unlike a signed measure, a complex measure never takes infinite values.
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<document source="https://tamnd.github.io/math2/node/NY77DQ2AX8.md" tag="NY77DQ2AX8" kind="statement">
# Kodaira vanishing theorem

## Statement

Let $X$ be a [[smooth|EXKGG9X6J0]] [[projective variety|A4H4CZ2V35]] of dimension $n$ over a field of characteristic $0$, with [[canonical bundle|W05VYY4HBR]] $\omega_X$.
Let $L$ be an [[ample line bundle|RRK8P6TXG2]] on $X$.
Then
$$H^i(X, \omega_X \otimes L) = 0 \quad \text{for all } i > 0,$$
where $H^i$ is [[sheaf cohomology|PQ6Q143ARJ]].
Equivalently, by Serre duality, $H^i(X, L^{-1}) = 0$ for all $i < n$.

## Remarks

Kodaira's proof over $\C$ uses a Hermitian metric on $L$ with positive curvature and the Bochner–Kodaira–Nakano identity, which shows that harmonic forms representing the classes must vanish.
Deligne and Illusie gave an algebraic proof by reduction to positive characteristic, using a lifting of $X$ modulo $p^2$.

Example: on $X = \mathbb{P}^n_\C$ with $L = \mathcal{O}(d)$, $d > 0$, one has $\omega_X \otimes L = \mathcal{O}(d - n - 1)$, and indeed $H^i(\mathbb{P}^n, \mathcal{O}(m)) = 0$ for $0 < i < n$ and all $m$, and for $i = n$ when $m > -n-1$.

The ampleness hypothesis cannot be dropped: for a curve $X$ of genus $g \ge 1$ and $L = \mathcal{O}_X$, $H^1(X, \omega_X) \cong \C \ne 0$.
The characteristic hypothesis is also needed: Raynaud gave counterexamples in positive characteristic.

## Explanations

### L1 human

The Kodaira vanishing theorem says that certain cohomology groups of a positive line bundle, twisted by the canonical bundle, are all zero.
When cohomology vanishes, global sections are easy to count, because the Euler characteristic, which Riemann–Roch computes, equals the dimension of the space of sections.
So the theorem turns an existence question into a numerical computation.

### L2 human

The theorem is a key input of the Kodaira embedding theorem and of the theory of adjoint linear systems $\abs{K_X + L}$.
Its generalizations, the Nakano vanishing theorem for higher differential forms and the Kawamata–Viehweg vanishing theorem for nef and big bundles, are basic tools of the minimal model program.
It fails in positive characteristic, which makes it a test case for methods there.
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<document source="https://tamnd.github.io/math2/node/NY851TZ24X.md" tag="NY851TZ24X" kind="concept">
# Non-uniform discrete Fourier transform

## Summary

The non-uniform discrete Fourier transform (NUDFT) is a variant of the [[discrete Fourier transform|TST0GH66A5]].
In it the sample points, the frequencies, or both are not equally spaced.
It evaluates finite sums $\sum_n x_n e^{-2\pi i p_n f_k}$ with arbitrary real $p_n$ and $f_k$.
Such data arise in medical imaging, astronomy and spectral methods, where measurements are not taken on a regular grid.
Fast approximate algorithms for it are called non-uniform FFTs.

## Explanations

### L1 human

The ordinary discrete Fourier transform assumes the signal was sampled at evenly spaced times.
Real measurements often come at irregular times or positions, for instance along spiral paths in an MRI scanner.
The non-uniform transform uses the same kind of sum, but with the actual sample positions in the exponent.
For example, samples at times $0$, $0.3$ and $0.7$ give the coefficient $X_k = x_0 + x_1 e^{-0.6\pi i k} + x_2 e^{-1.4\pi i k}$ at frequency $k$.
The loss of regular spacing means the clever factorizations of the FFT no longer apply directly.
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<document source="https://tamnd.github.io/math2/node/NY91GA6WVT.md" tag="NY91GA6WVT" kind="definition">
# Periodic solution of a difference equation

## Definition

Let $k \ge 1$, let $S$ be a set, and let $F \colon S^k \to S$ be a map.
Consider the autonomous difference equation of order $k$
$$x_{n+k} = F(x_n, x_{n+1}, \dots, x_{n+k-1}), \qquad n \ge 0.$$
Let $p \ge 1$ be an integer.
A solution $(x_n)_{n \ge 0}$ is *periodic with period $p$* if it is a [[periodic sequence|Q3B6K9HYRG]] with period $p$, that is, $x_{n+p} = x_n$ for all $n \ge 0$.
Its *prime period* is its least period.

## Remarks

Example: for $x_{n+2} = x_{n+1} / x_n$ on $S = (0, \infty)$ with $x_0 = a$ and $x_1 = b$, the solution is $a, b, b/a, 1/a, 1/b, a/b, a, b, \dots$, so every solution is periodic with period $6$.

Non-example: for $x_{n+1} = 2 x_n$ on $\R$ with $x_0 = 1$, the solution $x_n = 2^n$ is strictly increasing, so $x_{n+p} \ne x_n$ for all $p \ge 1$ and it is not periodic.

For $k = 1$ and $F = f$, the solution starting at $x_0$ is periodic with period $p$ if and only if $f^p(x_0) = x_0$, that is, $x_0$ is a [[periodic point|T75CZ5H47N]] of $f$.
A periodic solution of period $1$ is an equilibrium.
For a non-autonomous equation, one asks for periodic solutions when the equation itself is periodic in $n$.
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# Hyperbolic set

## Definition

Let $M$ be a [[Riemannian manifold|5HDG31YEPW]] with norm $\norm{\cdot}$ on each tangent space, and let $f \colon M \to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]] with [[differential|565B0Y76XE]] $Df$.
A set $\Lambda \subseteq M$ is a *hyperbolic set* of $f$ if the following hold.

1. $\Lambda$ is [[compact|0QPHW703PV]] and $f(\Lambda) = \Lambda$.
2. For each $x \in \Lambda$ there is a splitting $T_x M = E^s_x \oplus E^u_x$ with $Df_x(E^s_x) = E^s_{f(x)}$ and $Df_x(E^u_x) = E^u_{f(x)}$.
3. There are constants $C > 0$ and $0 < \lambda < 1$ such that, for all $x \in \Lambda$ and $n \ge 0$, $\norm{Df^n_x v} \le C \lambda^n \norm{v}$ for $v \in E^s_x$ and $\norm{Df^{-n}_x v} \le C \lambda^n \norm{v}$ for $v \in E^u_x$.

## Remarks

The subspaces $E^s_x$ and $E^u_x$ are then unique and depend continuously on $x$.
Changing the Riemannian metric changes $C$ but not the property, because $\Lambda$ is compact; an adapted metric allows $C = 1$.
Example: a fixed point $p$ of $f$ is a hyperbolic set exactly when no eigenvalue of $Df_p$ has modulus $1$.
For $f(x, y) = (x/2, 2y)$ on $\R^2$ and $\Lambda = \set{0}$, take $E^s$ the $x$-axis, $E^u$ the $y$-axis, $C = 1$ and $\lambda = 1/2$.
Non-example: for the rotation $f(z) = e^{i\theta} z$ of $\R^2 = \C$, the set $\set{0}$ is compact and invariant, but $Df_0$ preserves lengths, so no splitting satisfies clause 3.
A flow version replaces clause 2 by $T_x M = E^s_x \oplus E^0_x \oplus E^u_x$, where $E^0_x$ is spanned by the vector field, as in the definition of an [[Anosov flow|EN4A9MM7RP]].
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# Site

## Summary

A site is a [[category|A2DE66937D]] together with a [[Grothendieck topology|GDRFSJ8E2B]] on it.
It is the setting in which [[sheaves|RT8A8FJ566]], sheafification and sheaf cohomology make sense beyond topological spaces.
The open sets of a topological space form a site, and so do the étale maps into a scheme, which give étale cohomology.
The category of sheaves on a small site is a [[Grothendieck topos|S3YVE24AQK]].

## Explanations

### L1 human

A site is a "generalized space" described by its pieces and by the rule saying which families of pieces cover a piece.
The pieces need not be subsets; they can be maps from other objects.
For example, the open sets of the real line, with the usual covers, form a site, and its sheaves are the usual sheaves on $\R$.
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<document source="https://tamnd.github.io/math2/node/NYHNTP9AND.md" tag="NYHNTP9AND" kind="concept">
# de Branges space

## Summary

A de Branges space is a Hilbert space of [[entire functions|0419794R0T]] built from one entire function $E$.
The function $E$ must be larger at each point of the upper half-plane than at the mirror point in the lower half-plane.
Its elements are the entire functions $F$ such that $F/E$ and $F^\#/E$ lie in the Hardy space of the [[upper half-plane|X6XGGH5FPW]].
The norm of $F$ is the $L^2$ norm of $F/E$ on the real line.
It is a [[reproducing kernel Hilbert space|4CK2GSRAGD]] with an explicit kernel, and it generalises the Paley–Wiener spaces.
De Branges used these spaces in the spectral theory of canonical systems.

## Explanations

### L1 human

Choose an entire function $E$ with $\abs{E(\bar z)} < \abs{E(z)}$ whenever $z$ is in the upper half-plane.
The space $\mathcal{H}(E)$ collects entire functions that are "no larger than $E$" in a precise square-integrable sense.
For $E(z) = e^{-iaz}$ with $a > 0$, one gets the Paley–Wiener space of square-integrable band-limited functions, whose Fourier transforms live on $[-a, a]$.
Each point evaluation $F \mapsto F(w)$ is a bounded functional, given by an inner product with an explicit kernel.
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<document source="https://tamnd.github.io/math2/node/NYJFYZ0B4M.md" tag="NYJFYZ0B4M" kind="definition">
# Adele ring

## Definition

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]].
For each [[place|BT5V3TSNY6]] $v$ of $K$, let $K_v$ be the completion of $K$ at $v$.
For each finite place $v$, let $\mathcal{O}_v = \set{x \in K_v : \abs{x}_v \le 1}$ be its valuation ring.
The *adele ring* of $K$ is
$$\mathbb{A}_K = \Big\{ (x_v)_v \in \prod_v K_v : x_v \in \mathcal{O}_v \text{ for all but finitely many finite } v \Big\},$$
with componentwise addition and multiplication, so it is a [[ring|0B1261B5ED]].
Its topology is the restricted product topology.
A basis of open sets is formed by the sets $\prod_{v \in S} U_v \times \prod_{v \notin S} \mathcal{O}_v$.
Here $S$ is a finite set of places containing all infinite places, and each $U_v \subseteq K_v$ is open.

## Remarks

With this topology, $\mathbb{A}_K$ is a [[locally compact|0FE2SB8WVJ]] topological ring.
The topology is finer than the one induced by the [[product topology|0GBG8B49AA]] on $\prod_v K_v$.
The diagonal map $K \to \mathbb{A}_K$, $x \mapsto (x)_v$, is well defined because an element of $K$ lies in $\mathcal{O}_v$ for all but finitely many $v$; its image is discrete and $\mathbb{A}_K / K$ is compact.

Example: for $K = \Q$, the places are $\infty$ and the primes $p$, with $\Q_\infty = \R$, $\Q_p$ the [[p-adic numbers|BT6T702C28]] and $\mathcal{O}_p = \Z_p$ the [[p-adic integers|V6TJAFZ4MD]].
The family $x = (x_v)$ with $x_\infty = \pi$, $x_2 = 1/2$ and $x_p = 1$ for $p \ge 3$ is an adele, since only one $p$-adic entry lies outside $\Z_p$.

Non-example: the family with $x_\infty = 0$ and $x_p = 1/p$ for every prime $p$ lies in $\prod_v \Q_v$ but not in $\mathbb{A}_{\Q}$, since $1/p \notin \Z_p$ for infinitely many $p$.
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# Post's theorem

## Statement

Let $\emptyset^{(0)} = \emptyset$, and let $\emptyset^{(n+1)}$ be the [[Turing jump|60P2XWY5QX]] of $\emptyset^{(n)}$.
Let $A \subseteq \N$ and $n \in \N$.

1. $A$ is $\Sigma^0_{n+1}$ in the [[arithmetical hierarchy|7CVJCQKY41]] if and only if $A$ is [[computably enumerable|TN9G24VABN]] relative to the oracle $\emptyset^{(n)}$.
2. If $n \ge 1$, the set $\emptyset^{(n)}$ is $\Sigma^0_n$-complete: it is $\Sigma^0_n$, and every $\Sigma^0_n$ set is [[many-one reducible|JTAH37JTHD]] to it.
3. $A$ is $\Delta^0_{n+1}$ if and only if $A$ is [[Turing reducible|PA4PYQ1PZ0]] to $\emptyset^{(n)}$.

## Remarks

The proof is by induction on $n$.
For $n = 0$, part 1 says that $\Sigma^0_1$ sets are the computably enumerable sets, which follows from Kleene's normal form.
For the inductive step, an existential quantifier over a set computable from $\emptyset^{(n)}$ is a search with that oracle.
A $\Pi^0_n$ condition can be answered by one question to $\emptyset^{(n)}$, by the completeness in part 2.

Example: the set of programs that compute total functions is $\Pi^0_2$, so it is computable from $\emptyset''$.

## Explanations

### L1 human

The Turing jump of a set is the halting problem for machines that may consult that set.
Post's theorem says that each extra quantifier alternation in a definition corresponds to one more jump.
So the logical complexity of a definition matches the computational complexity of deciding the set.

### L2 human

The theorem is the bridge between definability in arithmetic and relative computability.
It shows that the arithmetical hierarchy is proper, because $\emptyset^{(n+1)}$ is not computable from $\emptyset^{(n)}$.
In computability arguments, one often bounds the quantifier complexity of a set to know which oracle can decide it.
Iterating the jump through computable ordinals beyond $\omega$ leads to the [[hyperarithmetical sets|NS1HWGTD8E]].
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# Recurrence and capacity of graphs

## Statement

Let $(G, c)$ be an infinite [[electrical network|T7KRE7J3B4]] with vertex set $V$, and let $c(x) = \sum_y c(x, y)$.
Let $(X_n)$ be the [[random walk|04VZE3FZXF]] on $V$ with transition probabilities $p(x, y) = c(x, y)/c(x)$.
Fix $a \in V$ and finite sets $V_1 \subseteq V_2 \subseteq \cdots$ with union $V$ and $a \in V_1$.
Let $G_n$ be the finite network obtained from $G$ by gluing all vertices outside $V_n$ into one vertex $z_n$.
Edges with both ends outside $V_n$ are deleted, and the edge from $x \in V_n$ to $z_n$ gets the sum of the conductances $c(x, y)$ over $y \notin V_n$.
Then the [[effective resistances|XAE5YJ3H68]] $R_{\mathrm{eff}}(a, z_n)$ in $G_n$ increase to a limit $R(a \leftrightarrow \infty) \in (0, \infty]$ that does not depend on the choice of the sets $V_n$.
Moreover
$$\mathbb{P}_a\big[X_n \ne a \text{ for all } n \ge 1\big] = \frac{1}{c(a)\, R(a \leftrightarrow \infty)},$$
with $1/\infty = 0$.
In particular the walk is [[recurrent|PBZTBWZ4GS]] if and only if $R(a \leftrightarrow \infty) = \infty$, that is, if and only if the capacity $1/R(a \leftrightarrow \infty)$ of $a$ is $0$.

## Remarks

For a finite network, the probability that the walk from $a$ reaches $b$ before returning to $a$ is $1/\big(c(a) R_{\mathrm{eff}}(a, b)\big)$.
This follows from the fact that $x \mapsto \mathbb{P}_x[\text{hit } b \text{ before } a]$ is harmonic off $\set{a, b}$, so it is $1 - v$ for the voltage $v$ with $v(a) = 1$ and $v(b) = 0$.
The statement follows by applying this to $G_n$ and letting $n \to \infty$.
The limit increases because $G_{n+1}$ arises from $G_n$ by un-gluing vertices, which raises resistance by the [[Rayleigh monotonicity law|KX4152NVW8]].
Example: on $\Z^2$ with unit conductances, the edges between the boundary of the box $[-k, k]^2$ and the next box form $4(2k + 1)$ disjoint cutsets.
The Nash-Williams inequality gives $R(0 \leftrightarrow \infty) \ge \sum_{k \ge 0} \frac{1}{4(2k + 1)} = \infty$, so the simple random walk on $\Z^2$ is recurrent.
On $\Z^3$ the resistance to infinity is finite, which gives the transience part of [[Pólya's recurrence theorem|BEQH6AZPYM]].

## Explanations

### L1 human

Imagine the graph as an infinite network of wires and connect one pole of a battery to a vertex $a$ and the other pole to infinity.
If the resistance between $a$ and infinity is infinite, no current escapes, and a random walker started at $a$ comes back to $a$ with probability $1$.
If the resistance is finite, current escapes, and the walker has a positive chance never to return.
This turns a question about random paths into a question about circuits.

### L2 human

The theorem lets one decide recurrence with the tools of electrical networks.
Lower bounds on resistance come from gluing vertices, as in the Nash-Williams inequality; upper bounds come from flows of finite energy, by [[Thomson's principle|8BPMHA5MT4]].
Recurrence is a property of the graph and the conductances, not of the starting vertex.
Indeed $R(a \leftrightarrow \infty)$ is finite for one vertex exactly when it is finite for all vertices of the connected graph.
Comparisons between graphs, for example under bounded changes of conductances, preserve recurrence and transience.
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# Coulomb wave function

## Definition

Let $L$ be a nonnegative integer, let $\eta$ be a real number, and let $\rho > 0$.
The *Coulomb wave equation* is the [[linear differential equation|0QQRP17Z7S]]
$$\frac{d^2 u}{d\rho^2} + \left( 1 - \frac{2\eta}{\rho} - \frac{L(L + 1)}{\rho^2} \right) u = 0.$$
Let $M(a, b, z)$ be [[Kummer's function|6AZ4ZBN1M8]] and $\Gamma$ the [[gamma function|0H5SYBSED9]].
The *regular Coulomb wave function* is
$$F_L(\eta, \rho) = C_L(\eta)\, \rho^{L + 1} e^{-i\rho}\, M(L + 1 - i\eta,\ 2L + 2,\ 2i\rho),$$
where the *Coulomb normalising constant* is
$$C_L(\eta) = \frac{2^L e^{-\pi\eta/2} \abs{\Gamma(L + 1 + i\eta)}}{\Gamma(2L + 2)}.$$

## Remarks

The function $F_L(\eta, \rho)$ is real, solves the Coulomb wave equation, and satisfies $F_L(\eta, \rho) \sim C_L(\eta) \rho^{L + 1}$ as $\rho \to 0^+$.
The constant $C_L(\eta)$ is chosen so that $F_L(\eta, \rho) = \sin \theta_L(\eta, \rho) + o(1)$ as $\rho \to \infty$, with phase $\theta_L = \rho - \eta \log 2\rho - \frac{L\pi}{2} + \arg \Gamma(L + 1 + i\eta)$.
The *irregular Coulomb wave function* $G_L(\eta, \rho)$ is the solution with $G_L(\eta, \rho) = \cos \theta_L(\eta, \rho) + o(1)$ as $\rho \to \infty$.
Example: for $\eta = 0$ and $L = 0$, one has $C_0(0) = 1$ and $M(1, 2, 2i\rho) = \frac{e^{2i\rho} - 1}{2i\rho}$, so
$$F_0(0, \rho) = \rho\, e^{-i\rho} \frac{e^{2i\rho} - 1}{2i\rho} = \frac{e^{i\rho} - e^{-i\rho}}{2i} = \sin \rho.$$
Non-example: $\cos \rho$ also solves the equation for $\eta = 0$ and $L = 0$, but it is not a multiple of $F_0(0, \rho)$, since it does not vanish at $\rho = 0$; it is $G_0(0, \rho)$.
For $\eta = 0$, $F_L(0, \rho) = \rho\, j_L(\rho)$, where $j_L$ is the spherical Bessel function of the first kind.
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# Krein–Smulian theorem

## Statement

Let $X$ be a [[Banach space|06GZ956657]] over $\R$ or $\C$, and let $X^*$ be its [[continuous dual space|4A82S5FVRT]].
For $r > 0$ let $B_r = \set{f \in X^* : \norm{f} \le r}$.
Let $K \subseteq X^*$ be a [[convex set|02AS45T3H9]].
If $K \cap B_r$ is [[closed|0ZCK8DQYN1]] in the weak* topology of $X^*$ for every $r > 0$, then $K$ is closed in the weak* topology.

## Remarks

The weak* topology is the [[weak topology|P6KHVSTPZV]] on $X^*$ given by the evaluations $f \mapsto f(x)$ for $x \in X$.
The proof introduces the bounded weak* topology, the finest topology that agrees with the weak* topology on each ball $B_r$.
It shows that this topology is locally convex and that its continuous linear functionals are again the evaluations at points of $X$; this step uses completeness of $X$.
A convex set closed in the bounded weak* topology is then weak*-closed by the [[Hahn–Banach separation theorem|6GWMGFGFJJ]].

A main corollary, the Banach–Dieudonné theorem, says that a linear subspace $M \subseteq X^*$ is weak*-closed when $M \cap B_1$ is weak*-closed.
Convexity is needed: the theorem is a statement about convex sets, and a nonconvex set can meet each ball in a weak*-closed set without being weak*-closed.
For instance, in $\ell^2$ (identified with its own dual), the set $\set{\sqrt{n}\, e_n : n \ge 1}$ meets each ball in a finite set but has $0$ in its weak closure.
The name Krein–Smulian theorem is also used for a second result: in a Banach space, the closed convex hull of a weakly compact set is weakly compact.

## Explanations

### L1 human

The weak* topology on a dual space is hard to test directly, because its closed sets are not described by sequences.
The theorem says that for convex sets it is enough to look at one ball at a time.
If every bounded piece of a convex set is weak*-closed, the whole set is weak*-closed.

### L2 human

The theorem is used to check weak* closedness of subspaces and convex sets, for example to show that a subspace of a dual space is itself a dual space.
When $X$ is separable, each ball $B_r$ with the weak* topology is metrizable, so the test reduces to weak* convergent sequences in bounded sets.
It is a companion of the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], which gives weak* compactness of the balls $B_r$.
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# Relatively free algebra

## Summary

A relatively free algebra is an algebra that is free over some of its generating sets in the variety that it generates.
The word "relatively" contrasts it with absolutely free algebras: it is free relative to the identities that it satisfies.
Free groups, free abelian groups, free nilpotent groups of a given class and polynomial rings are relatively free.
Relatively free algebras are exactly the free algebras of all varieties.

## Explanations

### L1 human

An algebra is relatively free when some generators can be sent anywhere inside the algebra itself and the map always extends to an endomorphism.
For example, in $\Z^2$ the generators $(1, 0)$ and $(0, 1)$ can be sent to any two vectors, and a $2 \times 2$ integer matrix does the rest.
So $\Z^2$ is relatively free: it is the free algebra on two generators of the variety of abelian groups.
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# Nil ideal

## Summary

A nil ideal of a [[ring|0B1261B5ED]] is a one-sided or two-sided ideal all of whose elements are [[nilpotent|0ZB2JG1M9C]].
Nil ideals are a main object of radical theory, the study of the parts of a ring that are discarded to reach simpler rings.
In a commutative ring the largest nil ideal is the [[nilradical|QQDN6EYQNJ]].
Whether every nil one-sided ideal lies in a nil two-sided ideal is the open [[Köthe conjecture|RT2D9JEVRR]].

## Explanations

### L1 human

An element is nilpotent when some power of it is zero.
A nil ideal is an ideal in which every element has this property, though the power needed may depend on the element.
In the ring $\Z/8\Z$, the ideal of even residues is nil: every even residue $a$ satisfies $a^3 = 0$.
In $\Z$, the ideal $2\Z$ is not nil, since no power of $2$ is zero.
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# Hyperfinite type II1 factor

## Description

The hyperfinite type $\mathrm{II}_1$ factor $R$ is the [[von Neumann algebra|PSFWQVZ0FS]] obtained as the closure of an increasing union of matrix algebras.
Embed $M_{2^n}(\C)$ into $M_{2^{n+1}}(\C)$ by $x \mapsto I_2 \otimes x = \begin{pmatrix} x & 0 \\ 0 & x \end{pmatrix}$.
These embeddings preserve the normalized trace $\tau(x) = 2^{-n} \operatorname{tr}(x)$, so $\tau$ is well defined on the union $A_\infty = \bigcup_n M_{2^n}(\C)$.
Let $H$ be the [[Hilbert space|HM2HHE3J51]] completion of $A_\infty$ for the inner product $\langle x, y \rangle = \tau(y^* x)$, and let $A_\infty$ act on $H$ by left multiplication.
Then $R$ is the closure of $A_\infty$ in the [[weak operator topology|4XM1VN1C1N]] on $H$.

## Remarks

The algebra $R$ is a factor: its center consists of scalars.
The trace $\tau$ extends to a faithful normal tracial state on $R$, and the values of $\tau$ on projections fill $[0, 1]$, so $R$ has type $\mathrm{II}_1$.
It is hyperfinite: it contains an increasing sequence of finite-dimensional subalgebras whose union is weakly dense.
Murray and von Neumann proved that a hyperfinite $\mathrm{II}_1$ factor with separable predual is unique up to isomorphism.
Connes proved that, among $\mathrm{II}_1$ factors with separable predual, $R$ is also the unique injective one.
Other constructions give the same factor, for example the group von Neumann algebra of the group of finitely supported permutations of $\N$.
Every type $\mathrm{II}_1$ factor contains a von Neumann subalgebra isomorphic to $R$.

## Explanations

### L1 human

Matrices of size $2 \times 2$, $4 \times 4$, $8 \times 8$, and so on, sit inside each other by repeating a matrix twice along the diagonal.
Measured with the normalized trace, which gives the identity size $1$, the projections in these algebras have sizes $k / 2^n$.
Taking a suitable limit of all of them gives the factor $R$, where projections can have any size between $0$ and $1$.
It is the simplest and most important example of an operator algebra with this "continuous dimension".
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# Friedrichs extension

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] with inner product $\ip{\cdot}{\cdot}$.
Let $T$ be a [[densely defined|BCJHNHKQFG]] [[symmetric operator|H971WGMV6C]] on $H$ with domain $D(T)$.
Assume that $T$ is bounded below: there is a real number $c$ with $\ip{Tx}{x} \ge c \norm{x}^2$ for all $x \in D(T)$.
On $D(T)$, put
$$\ip{x}{y}_T = \ip{Tx}{y} + (1 - c)\ip{x}{y}.$$
This is an inner product on $D(T)$, and its norm satisfies $\norm{x}_T \ge \norm{x}$.
Let $H_T$ be the completion of $D(T)$ for $\norm{\cdot}_T$.
The inclusion of $D(T)$ into $H$ extends to an injective map from $H_T$ into $H$, so $H_T$ is a subspace of $H$.
The *Friedrichs extension* of $T$ is the operator $T_F$ with domain
$$D(T_F) = \set{x \in H_T : \text{there is } z \in H \text{ with } \ip{x}{y}_T = \ip{z + (1 - c)x}{y} \text{ for all } y \in H_T},$$
given by $T_F x = z$.

## Remarks

The operator $T_F$ is a [[self-adjoint|0NEGN5CRSC]] extension of $T$, and $\ip{T_F x}{x} \ge c \norm{x}^2$ for all $x \in D(T_F)$.
An equivalent description is that $T_F$ is the restriction of the [[adjoint|86B7HF79PQ]] $T^*$ to $D(T^*) \cap H_T$.
The space $H_T$ is the form domain of $T_F$, and the [[sesquilinear form|054YEQ0Y8M]] $(x, y) \mapsto \ip{Tx}{y}$ extends to it.

Example: on $H = L^2(0, 1)$, let $T f = -f''$ on smooth functions with compact support in $(0, 1)$.
Integration by parts gives $\ip{Tf}{f} = \int_0^1 \abs{f'}^2 \, dx \ge 0$, so $c = 0$ works.
The space $H_T$ is the Sobolev space $H^1_0(0, 1)$, and $T_F f = -f''$ on $D(T_F) = H^2(0, 1) \cap H^1_0(0, 1)$, the Dirichlet Laplacian.

Non-example: on $L^2(0, \infty)$, let $S f = -i f'$ on smooth functions with compact support in $(0, \infty)$.
It is symmetric, but $\ip{Sf}{f}$ is not bounded below by any $c \norm{f}^2$, so the construction does not apply.
In fact the [[deficiency indices|G4GMSNAX8X]] of $S$ are unequal, so $S$ has no self-adjoint extension at all.
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# Flux limiter

## Summary

A flux limiter is a function used in high-resolution schemes for [[conservation laws|8RA7AMWSAF]] to blend a low-order, non-oscillatory numerical flux with a high-order one.
It looks at the ratio of neighbouring slopes of the solution.
Where the solution is smooth it lets the high-order flux act, and near jumps or extrema it switches back to the low-order flux.
In this way the scheme is second-order accurate in smooth regions and still free of spurious oscillations.
This gets around the limits that [[Godunov's theorem|559N53ZTMV]] sets for linear schemes.
Standard examples are the minmod, superbee and van Leer limiters.

## Explanations

### L1 human

A careful but blurry method and a sharp but wiggly method can be mixed.
The flux limiter decides, at every grid point and every step, how much of the sharp method to use.
Where the solution is smooth, the ratio of consecutive slopes is close to $1$ and the sharp method is used in full.
Near a jump or a peak the ratio is far from $1$ or negative, and the limiter falls back to the careful method.
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# Versal unfolding

## Definition

Let $f \colon (\R^n, 0) \to \R$ be a [[germ|7SWSRGDT4A]] of a [[smooth function|0V33BNKZ9S]].
An *$r$-parameter unfolding* of $f$ is a smooth germ $F \colon (\R^n \times \R^r, 0) \to \R$ with $F(x, 0) = f(x)$.
An $s$-parameter unfolding $G$ of $f$ is *induced* from $F$ if there are smooth germs $\Phi \colon (\R^n \times \R^s, 0) \to (\R^n, 0)$, $\alpha \colon (\R^s, 0) \to (\R^r, 0)$ and $c \colon (\R^s, 0) \to \R$ with $\Phi(x, 0) = x$ and
$$G(x, v) = F(\Phi(x, v), \alpha(v)) + c(v).$$
The unfolding $F$ is *versal* if every unfolding of $f$, with any number of parameters, is induced from $F$.
A versal unfolding with the least possible number $r$ of parameters is *universal*, or *miniversal*.

## Remarks

Let $\mathcal{E}_n$ be the ring of smooth germs at $0$ on $\R^n$ and $J(f)$ the ideal generated by the partial derivatives of $f$.
The versality theorem says that $F$ is versal if and only if
$$\mathcal{E}_n = J(f) + \R \cdot 1 + \R \cdot \frac{\partial F}{\partial u_1}(\cdot, 0) + \cdots + \R \cdot \frac{\partial F}{\partial u_r}(\cdot, 0),$$
where $u_1, \ldots, u_r$ are the parameters.
Example: for $f = x^3$, $J(f) = (x^2)$ and $\mathcal{E}_1 / J(f)$ has basis the classes of $1$ and $x$; for $F(x, u) = x^3 + u x$ one has $\partial F / \partial u (x, 0) = x$, so $F$ is versal, and universal, since one parameter is needed.
Non-example: $G(x, u) = x^3 + u x^2$ is an unfolding of $x^3$, but $\partial G / \partial u (x, 0) = x^2$ lies in $J(f)$, so the class of $x$ is not reached and $G$ is not versal.
Indeed, every function $G(\cdot, u)$ has a critical point near $0$, while $x^3 + u x$ has none for $u > 0$.
A germ has a versal unfolding with finitely many parameters if and only if $\mathcal{E}_n / J(f)$ is finite-dimensional.
If moreover $f$ has a critical point at $0$, a universal unfolding has $\dim_\R \mathcal{E}_n / J(f) - 1$ parameters.
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# Synchronization of chaos

## Definition

Let $f \colon \R^n \to \R^n$ be a locally Lipschitz vector field whose [[autonomous system|0673D85QKZ]] $x' = f(x)$ has [[chaotic|DBKXCGDKTD]] solutions.
Let $\varepsilon > 0$ and let $\Gamma$ be an $n \times n$ matrix.
Consider the coupled system
$$x' = f(x) + \varepsilon \Gamma (y - x), \qquad y' = f(y) + \varepsilon \Gamma (x - y).$$
The diagonal $\set{(x, y) : x = y}$ is invariant under this system, and on it each component follows the chaotic system $x' = f(x)$.
The coupled system shows *complete synchronization* from a set $U \subseteq \R^n \times \R^n$ of initial states if every solution with $(x(0), y(0)) \in U$ satisfies
$$\lim_{t \to \infty} \norm{x(t) - y(t)} = 0.$$
Synchronization of chaos refers to such behaviour, with $U$ a neighbourhood of a chaotic attractor on the diagonal, and to its weaker forms.

## Remarks

The difference $e = x - y$ satisfies, to first order, $e' = \big(Df(x) - 2 \varepsilon \Gamma\big) e$.
Synchronization is stable when the largest Lyapunov exponent of this linear equation along the synchronized orbit, the transverse Lyapunov exponent, is negative.

Example: let $f(x) = a x$ be scalar with $a > 0$, which is not chaotic but shows the mechanism, and $\Gamma = 1$.
Then $e' = (a - 2\varepsilon) e$ exactly, so $e(t) \to 0$ for every initial state exactly when $\varepsilon > a/2$.

Non-example: for $\varepsilon = 0$ the two copies are uncoupled, and for a chaotic $f$ nearby initial states separate, so $\norm{x(t) - y(t)}$ does not tend to $0$ for most initial states.

Weaker forms are phase synchronization, where only the phases of oscillations lock, and generalized synchronization, where $y(t) - \psi(x(t)) \to 0$ for some function $\psi$.
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# Matrix congruence

## Definition

Let $F$ be a [[field|04C5ZBG1WT]] and let $n \ge 1$.
Let $A$ and $B$ be $n \times n$ matrices over $F$.
Then $A$ and $B$ are *congruent* if there is an [[invertible|0EB9N0S33C]] $n \times n$ matrix $P$ over $F$ with
$$B = P^{T} A P.$$
Here $P^{T}$ is the [[transpose|09DXWYRD43]] of $P$.

## Remarks

Example: over $\R$, let $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $P = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$.
Then $P$ is invertible, since $\det P = -2$, and $P^{T} A P = \begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix}$, so these two matrices are congruent.
Non-example: over $\R$, the identity $I_2$ and $-I_2$ are not congruent, because $x^{T} P^{T} I_2 P x = \norm{Px}^2 \ge 0$ for all $x \in \R^2$, while $x^{T} (-I_2) x < 0$ for $x \ne 0$.
Congruence is an equivalence relation on $n \times n$ matrices.
Let $A$ be the matrix of a [[bilinear form|0SQZMXB6GP]] in one basis, and let $P$ be the [[change of basis matrix|0MKRHZB3WS]] to a second basis.
Then $P^{T} A P$ is the matrix of the same form in the second basis; see [[change of basis for a bilinear form|0SPYAGE7T0]].
Congruence keeps symmetry and the rank, and it multiplies the determinant by the nonzero square $(\det P)^2$.
Over $\C$, the complex analogue often uses the [[conjugate transpose|0DXKE10PHF]] $P^{*}$ in place of $P^{T}$; this is called $*$-congruence.
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# Symplectic integrator

## Definition

Let $n \ge 1$ and let $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$, where $I_n$ is the $n \times n$ identity matrix.
For a smooth function $H \colon \R^{2n} \to \R$, consider the [[Hamiltonian system|0EDKGA2FAK]] $y' = J \nabla H(y)$ with $y = (q, p)$, that is, $q' = \nabla_p H$ and $p' = -\nabla_q H$.
A one-step method assigns to each such $H$ and each small step size $h > 0$ a map $\Phi_h$, and computes $y_{k+1} = \Phi_h(y_k)$.
The method is a *symplectic integrator* if $\Phi_h$ is a symplectic map whenever $H$ is smooth and $\Phi_h$ is defined and smooth.
This means
$$D\Phi_h(y)^T \, J \, D\Phi_h(y) = J \quad \text{for all } y.$$

## Remarks

For $n = 1$, a map of the plane is symplectic exactly when its Jacobian determinant is $1$, that is, when it preserves area.
Example: for $H = \frac12 (p^2 + q^2)$, the symplectic Euler method $p_{k+1} = p_k - h q_k$, $q_{k+1} = q_k + h p_{k+1}$ gives $q_{k+1} = (1 - h^2) q_k + h p_k$, with Jacobian determinant $(1 - h^2) \cdot 1 - h \cdot (-h) = 1$.
In general the symplectic Euler method is $p_{k+1} = p_k - h \nabla_q H(q_k, p_{k+1})$, $q_{k+1} = q_k + h \nabla_p H(q_k, p_{k+1})$, and it is symplectic for every $H$.
Non-example: the explicit [[Euler method|0JARTXX4XP]] for the same $H$ is $q_{k+1} = q_k + h p_k$, $p_{k+1} = p_k - h q_k$, with Jacobian determinant $1 + h^2 \ne 1$, so it is not symplectic.
A Runge–Kutta method with coefficients $a_{ij}$, $b_i$ is symplectic when $b_i a_{ij} + b_j a_{ji} = b_i b_j$ for all $i, j$; the implicit midpoint rule is an example.
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# Coupling from the past

## Summary

Coupling from the past is an algorithm due to Propp and Wilson.
It produces an exact sample from the [[stationary distribution|0KRH5157WP]] of an ergodic finite [[Markov chain|0G48PYM96Q]].
It runs copies of the chain from all states, starting further and further in the past with the same randomness.
It stops when all copies agree at time $0$.
Unlike ordinary [[Markov chain Monte Carlo|ZGNBR2SPZ0]], its output has exactly the stationary law, with no bias from stopping too early.

## Explanations

### L1 human

Imagine the chain has been running since the infinite past, so that today it is exactly in equilibrium.
If all possible starting states at some past time lead, with the same random choices, to the same state today, then that state is what the eternal chain shows today.
The algorithm looks further back in time until this happens.
The trick is to reuse the same random numbers for each past time step whenever one restarts from further back.
For chains with an order that the update preserves, it suffices to follow only the top and the bottom states.
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# Unique ergodicity

## Summary

A continuous map of a compact metric space is uniquely ergodic when it has exactly one invariant Borel probability measure.
That measure is then automatically [[ergodic|AZRPHS478B]].
For a uniquely ergodic map, time averages of every continuous function converge to the space average at every point, not only at almost every point.
Unique ergodicity is the dynamical form of equidistribution, as in [[Weyl's equidistribution theorem|NCCGNTBEYV]].

## Explanations

### L1 human

Every continuous map of a compact space has at least one invariant probability measure, a way of spreading mass that the map leaves unchanged.
Usually there are many: each fixed point carries its own point mass, for instance.
A uniquely ergodic map leaves no choice at all, so there is only one invariant way to spread the mass.
A rotation of the circle by an irrational angle is uniquely ergodic, and its only invariant measure is arc length.
A rotation by a quarter turn is not, because putting mass $1/4$ on each point of any orbit of four points gives an invariant measure.
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# Abelian differential

## Summary

An Abelian differential is a meromorphic $1$-form on a [[Riemann surface|5K0N8QMV6Q]], written locally as $f(z)\,dz$ with $f$ meromorphic.
The holomorphic ones are the differentials of the first kind.
Their integrals along paths give the abelian integrals, the periods, and the [[Abel–Jacobi map|HE8F7KBF42]].
On a compact Riemann surface of genus $g$, the holomorphic Abelian differentials form a complex vector space of dimension $g$.

## Explanations

### L1 human

A function on a Riemann surface has a value at each point, but a differential is the thing you integrate along a path.
In a local coordinate $z$ it looks like $f(z)\,dz$, and changing the coordinate changes $f$ by the derivative of the change.
On the torus $\C / \Lambda$, the form $dz$ is a holomorphic differential that never vanishes.
Integrating it along the two basic loops of the torus gives the two generators of the lattice $\Lambda$.
On the Riemann sphere there is no nonzero holomorphic differential, which matches genus $0$.
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# Jacobi elliptic functions

## Summary

The Jacobi elliptic functions $\operatorname{sn}$, $\operatorname{cn}$ and $\operatorname{dn}$ are [[elliptic functions|FF9ZPCZ59K]] that depend on a modulus $k$ and generalise the sine and cosine.
They come from inverting the incomplete [[elliptic integral|NXMCYGCB59]] of the first kind, through the [[Jacobi amplitude|WDHHJHVP7N]].
They satisfy $\operatorname{sn}^2 + \operatorname{cn}^2 = 1$ and simple differential equations, and they solve the pendulum equation exactly.
Nine further functions such as $\operatorname{sd} = \operatorname{sn}/\operatorname{dn}$ are formed as quotients.

## Explanations

### L1 human

The sine is the inverse of the integral $\int_0^x dt/\sqrt{1 - t^2}$.
If one puts a second factor under the root and inverts $\int_0^x dt/\sqrt{(1 - t^2)(1 - k^2t^2)}$, one gets $\operatorname{sn}$.
For $k = 0$ the functions reduce to $\sin$, $\cos$ and the constant $1$; for $k = 1$ they become $\tanh$, $\operatorname{sech}$ and $\operatorname{sech}$.
In between they look like sine waves that have been flattened at the tops.
Unlike the sine, they also repeat in an imaginary direction, so they are doubly periodic in the complex plane.
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# Gaussian measure

## Summary

A Gaussian measure is a [[probability measure|059TJN0NZ1]] whose one-dimensional projections are all [[normal distributions|05AQE0QR0D]].
On $\R^n$ these are the laws of Gaussian random vectors.
On infinite-dimensional spaces, such as spaces of continuous paths, Gaussian measures are the main examples of well-behaved measures, and Wiener measure is one of them.
There is no translation-invariant analogue of Lebesgue measure in infinite dimensions, so Gaussian measures often take its place.

## Explanations

### L1 human

The standard Gaussian measure on the line is the bell curve, read as a measure: the mass of an interval is the area under the bell curve over it.
On the plane, the standard Gaussian measure is the product of two bell curves, and it looks like a round hill.
In general, a measure is Gaussian if every straight-line view of it, through a linear functional, is a bell curve or a single point.
For example, the law of the Brownian path on $[0, 1]$ is Gaussian, because each value of the path, and each linear combination of finitely many values, is normally distributed.
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# Absolutely continuous measure

## Summary

A measure $\nu$ is absolutely continuous with respect to a measure $\mu$, written $\nu \ll \mu$, when every $\mu$-null set is also $\nu$-null.
By the [[Radon–Nikodym theorem|1ZSZS5G1Q4]], for σ-finite measures this holds exactly when $\nu$ has a density with respect to $\mu$.
It is the measure-theoretic counterpart of the [[absolute continuity|ZT6F4C2YHK]] of functions.
A nondecreasing function on a closed bounded interval is absolutely continuous exactly when its Lebesgue–Stieltjes measure is absolutely continuous with respect to Lebesgue measure.
In probability, random variables with a density are those whose law is absolutely continuous with respect to Lebesgue measure.

## Explanations

### L1 human

Absolute continuity of $\nu$ with respect to $\mu$ means that $\nu$ never sees anything that $\mu$ ignores.
If a set is negligible for $\mu$, it is negligible for $\nu$ too.
The normal distribution is absolutely continuous with respect to Lebesgue measure on $\R$, since it is given by a density.
A point mass at $0$ is not: the single point $\set{0}$ has length $0$ but carries mass $1$.
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# Schwarz reflection principle

## Statement

Let $\Omega \subseteq \C$ be an open set that is symmetric with respect to the real axis: $\bar z \in \Omega$ for each $z \in \Omega$.
Let $\Omega^+ = \set{z \in \Omega : \operatorname{Im} z > 0}$ and $\sigma = \Omega \cap \R$.
Let $f$ be continuous on $\Omega^+ \cup \sigma$, [[holomorphic|0DG0Z72CQD]] on $\Omega^+$, and real-valued on $\sigma$.
Define $F(z) = f(z)$ for $z \in \Omega^+ \cup \sigma$ and $F(z) = \overline{f(\bar z)}$ for $z \in \Omega$ with $\operatorname{Im} z < 0$.
Then $F$ is holomorphic on $\Omega$.

## Remarks

Proof idea: $F$ is holomorphic on $\Omega \setminus \R$, since $z \mapsto \overline{f(\bar z)}$ satisfies the Cauchy–Riemann equations when $f$ does.
$F$ is continuous on $\Omega$, because $f$ is real on $\sigma$, so the two definitions agree there.
The integral of $F$ over each small triangle in $\Omega$ is $0$: for triangles meeting $\R$, approximate by triangles slightly off the axis.
[[Morera's theorem|0VDD7W0EKZ]] then gives that $F$ is holomorphic.
Example: $f(z) = e^{z}$ on the upper half-plane is real on $\R$, and its reflection is $\overline{e^{\bar z}} = e^{z}$, as expected.
The hypothesis that $f$ is real on $\sigma$ is needed: $f(z) = i$ is continuous up to $\R$ but its reflection is $-i$ below the axis, which is not even continuous.
By a Möbius transformation the same principle holds for reflection across a circle, with $\bar z$ replaced by the inverse point.

## Explanations

### L1 human

If a holomorphic function on the upper half-plane takes real values on the real axis, it extends to the lower half-plane by mirror symmetry.
The value at the mirror point $\bar z$ is the complex conjugate of the value at $z$.
The two halves fit together into one holomorphic function.

### L2 human

The reflection principle is a basic tool of [[analytic continuation|0M3YQJ81NB]].
It is used to extend conformal maps whose boundary goes to a line or a circle, as for the Schwarz–Christoffel formula.
It also shows that a conformal map between domains bounded by analytic arcs extends across those arcs.
There is a version for harmonic functions that vanish on a line, which is the main step in some proofs.
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<document source="https://tamnd.github.io/math2/node/P0SYXV6RTK.md" tag="P0SYXV6RTK" kind="definition">
# Coalgebra

## Definition

Let $K$ be a [[field|04C5ZBG1WT]].
A *coalgebra* over $K$ is a [[vector space|000AYPA9AJ]] $C$ over $K$ with [[linear maps|0NRYHJPQG2]] $\Delta \colon C \to C \otimes C$, the comultiplication, and $\varepsilon \colon C \to K$, the counit, such that:

1. $(\Delta \otimes \id_C) \circ \Delta = (\id_C \otimes \Delta) \circ \Delta$ as maps $C \to C \otimes C \otimes C$ (coassociativity).
2. $(\varepsilon \otimes \id_C) \circ \Delta = \id_C = (\id_C \otimes \varepsilon) \circ \Delta$, where $K \otimes C$ and $C \otimes K$ are identified with $C$ (counit law).

Here $\otimes$ is the [[tensor product|0BF2DE2SYK]] over $K$.

## Remarks

Example: for a set $S$, let $C = K[S]$ be the vector space with basis $S$, and put $\Delta(s) = s \otimes s$ and $\varepsilon(s) = 1$ for $s \in S$.
Both sides of clause 1 send $s$ to $s \otimes s \otimes s$, and both maps in clause 2 send $s$ to $1 \cdot s = s$.
Non-example: the same $\Delta$ with $\varepsilon(s) = 0$ for all $s$ fails clause 2, since $(\varepsilon \otimes \id)\Delta(s) = 0 \ne s$.
Another example is $K[x]$ with $\Delta(x^n) = \sum_{k=0}^{n} \binom{n}{k} x^k \otimes x^{n-k}$ and $\varepsilon(x^n) = 1$ if $n = 0$ and $0$ otherwise.
The definition works in the same way over a commutative ring in place of $K$.
Over a field, every element of a coalgebra lies in a finite-dimensional subcoalgebra.
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<document source="https://tamnd.github.io/math2/node/P0XFYRE9P2.md" tag="P0XFYRE9P2" kind="definition">
# Indirect method by shooting

## Definition

Let $T > 0$, $x_0, x_1 \in \R^n$, let $U \subseteq \R^m$ be nonempty, and let $f \colon \R^n \times U \to \R^n$ and $L \colon \R^n \times U \to \R$ be continuously differentiable in $x$.
Consider the problem of minimising $\int_0^T L(x, u) \, dt$ subject to $x' = f(x, u)$, $x(0) = x_0$, $x(T) = x_1$.
Let $H(x, u, p) = p \cdot f(x, u) - L(x, u)$ be the [[Hamiltonian|RACVWG5QTK]] in the normal case of [[Pontryagin's maximum principle|7RPV2T29CF]].
Assume that for all $(x, p)$ the function $u \mapsto H(x, u, p)$ has a unique maximiser $u^*(x, p)$ in $U$, and that $u^*$ is continuously differentiable.
For $p_0 \in \R^n$, let $(x(\cdot\,; p_0), p(\cdot\,; p_0))$ be the solution on $[0, T]$, if it exists, of
$$x' = f(x, u^*(x, p)), \qquad p' = -\frac{\partial H}{\partial x}(x, u^*(x, p), p), \qquad x(0) = x_0, \qquad p(0) = p_0.$$
The *shooting function* is $S(p_0) = x(T; p_0) - x_1$.
The *indirect method by shooting* computes a zero $p_0^*$ of $S$, for example by [[Newton's method|07EXVNF9VP]], and returns the control $u(t) = u^*(x(t; p_0^*), p(t; p_0^*))$.

## Remarks

The second equation is the [[costate equation|WS9MRAZMFW]].
Every normal extremal of the maximum principle with these endpoints comes from a zero of $S$, so the method computes candidates for optimality, not proven minimisers.
The method is a special case of the [[shooting method|ECT6RNF865]], with unknowns $p_0$ instead of an initial slope.
Free final time or other boundary conditions add unknowns and equations to $S$, from the transversality conditions.
Multiple shooting splits $[0, T]$ into pieces to reduce the sensitivity of $S$.

Example: $n = m = 1$, $T = 1$, $f = u$, $L = u^2 / 2$, $U = \R$, $x_0 = 0$ and $x_1 = 1$.
Here $H = p u - u^2 / 2$, so $u^* = p$, and $p' = 0$.
Hence $p(t) = p_0$, $x(t) = p_0 t$ and $S(p_0) = p_0 - 1$.
The zero is $p_0^* = 1$, which gives the optimal control $u = 1$.

Non-example: if $U = [-1, 1]$ and $L = 0$ with $f = u$, the maximiser of $H = p u$ is not unique when $p = 0$ and is not differentiable in $p$.
Then the map $u^*$ of the definition does not exist, and the method must be adapted to the bang-bang structure.

## Explanations

### L1 human

The method guesses the initial costate, integrates the optimality conditions forward, and corrects the guess until the trajectory ends where it must.
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<document source="https://tamnd.github.io/math2/node/P101AX1ZSC.md" tag="P101AX1ZSC" kind="concept">
# Hölder space

## Summary

The Hölder space $C^{k,\alpha}$ consists of the functions whose derivatives up to order $k$ are bounded.
In addition, the derivatives of order $k$ must be [[Hölder continuous|012PBA0FMD]] with exponent $\alpha$, where $0 < \alpha \le 1$.
With the natural norm it is a [[Banach space|06GZ956657]].
Hölder spaces are the setting of the classical regularity theory of elliptic and parabolic equations, as in the [[Schauder estimates|HSEKZMAH1V]].
They fill the gaps between the spaces $C^k$ of $k$ times continuously differentiable functions.

## Explanations

### L1 human

A Hölder continuous function cannot change faster than a fixed power of the distance: $\abs{u(x) - u(y)} \le C \abs{x - y}^\alpha$.
The exponent $\alpha$ measures roughness: $\alpha = 1$ means Lipschitz, and small $\alpha$ allows sharp spikes like $\sqrt{\abs{x}}$.
The space $C^{k,\alpha}$ asks this of the $k$-th derivatives.
For example, $x \abs{x}$ is in $C^{1,1}$ on $(-1, 1)$, because its derivative $2 \abs{x}$ is Lipschitz, but it is not twice differentiable at $0$.
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<document source="https://tamnd.github.io/math2/node/P142WXJCF3.md" tag="P142WXJCF3" kind="concept">
# Generalized least squares

## Summary

Generalized least squares estimates the coefficients of a linear model whose errors have a known covariance structure that is not a multiple of the identity.
It minimises a squared distance weighted by the inverse of the error covariance matrix.
Under the analogue of the [[Gauss–Markov theorem|N3YMNMWANM]], due to Aitken, it is the best linear unbiased estimator in that setting.
[[Weighted least squares|H5CGZHBR58]] is the special case of uncorrelated errors with unequal variances.

## Explanations

### L1 human

[[Ordinary least squares|YM708KHV2W]] treats all observations as equally reliable and unrelated.
When some observations are noisier, or when errors are correlated, as in data over time, that is wasteful.
Generalized least squares first changes coordinates so that the errors become uncorrelated with equal variances, and then applies ordinary least squares.
For example, if each observation is the mean of a different number of measurements, the means of many measurements get more weight.
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<document source="https://tamnd.github.io/math2/node/P15K8RKHDV.md" tag="P15K8RKHDV" kind="statement">
# Harnack's principle

## Statement

Let $D \subseteq \R^n$ be a [[domain|093CM69AGS]], that is, a connected open set.
Let $(u_k)$ be a sequence of [[harmonic functions|0W5NJQ8MKY]] on $D$ with $u_1 \le u_2 \le u_3 \le \cdots$ at each point of $D$.
Then one of the following holds.

1. $u_k \to \infty$ uniformly on each compact subset of $D$.
2. $(u_k)$ converges uniformly on each compact subset of $D$ to a harmonic function $u$ on $D$.

## Remarks

Proof idea: the functions $u_k - u_1$ are nonnegative and harmonic.
[[Harnack's inequality|YNH832D50W]] compares the values of a nonnegative harmonic function at two points of a compact set by constants that do not depend on the function.
So the increasing limit is either infinite everywhere or finite everywhere, and in the finite case the convergence is uniform on compact sets.
A uniform limit of harmonic functions is harmonic, by the mean value property.
Connectedness is needed: on two disjoint discs, take $u_k = 0$ on one disc and $u_k = k$ on the other.

## Explanations

### L1 human

An increasing sequence of harmonic functions can do only two things.
Either it goes to infinity everywhere at once, or it converges nicely everywhere to a harmonic function.
It cannot converge at one point and blow up at another.

### L2 human

The principle is the harmonic analogue of the monotone convergence theorem, and it follows from Harnack's inequality.
It is used in Perron's method for the Dirichlet problem, where the solution is built as a limit of subharmonic functions.
It also appears in the construction of Green's functions and harmonic measures by exhaustion of a domain.
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<document source="https://tamnd.github.io/math2/node/P15VQS5DJ9.md" tag="P15VQS5DJ9" kind="concept">
# Measure space

## Summary

A measure space is a set together with a σ-algebra of subsets and a measure on that σ-algebra.
It is the basic object of measure theory and of the Lebesgue integral.
A probability space is a measure space of total measure $1$.

## Explanations

### L1 human

A measure space has three parts: a set of points, a list of subsets that one may measure, and a rule that gives each of them a size.
The real line with the Borel sets and length is the standard picture.
A fair die gives a small example: the six faces, all subsets of faces, and the size of a subset is its number of faces divided by $6$.
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<document source="https://tamnd.github.io/math2/node/P16AFZT3K0.md" tag="P16AFZT3K0" kind="concept">
# Character table

## Summary

The character table of a finite group lists the values of its irreducible complex [[characters|RMWMYBY8TC]] on its [[conjugacy classes|0DEBXAJ5SA]].
It is a square matrix, because the number of irreducible characters equals the number of conjugacy classes.
Much information about the group can be read from it, such as its order, its normal subgroups and whether it is simple or abelian.

## Explanations

### L1 human

A character table is a compact summary of all the ways a finite group can act by matrices.
Each row belongs to one irreducible representation, each column to one class of conjugate elements, and each entry is a trace.
For the cyclic group of order $2$ the table has rows $(1, 1)$ and $(1, -1)$.
Chemists use character tables of symmetry groups of molecules to classify vibrations and orbitals.
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<document source="https://tamnd.github.io/math2/node/P1ABGYRQXH.md" tag="P1ABGYRQXH" kind="concept">
# Higher-order logic

## Summary

Higher-order logic extends [[first-order logic|086QKPF0A6]] with variables, and quantifiers, for relations on elements, relations on such relations, and so on at every finite level.
Each variable carries a type that says at which level it lives.
It is the logic behind simple type theory and several proof assistants, such as Isabelle/HOL.

## Explanations

### L1 human

First-order logic talks about objects.
Second-order logic also talks about sets of objects.
Third-order logic adds sets of sets of objects, and higher-order logic allows every finite level of this hierarchy.
For example, "every collection of sets of numbers has a union" quantifies over collections of sets, so it is a third-order statement.
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<document source="https://tamnd.github.io/math2/node/P1ECPTAG8F.md" tag="P1ECPTAG8F" kind="statement">
# LOBPCG

## Statement

Let $A, B \in \R^{n \times n}$ be [[symmetric|0VR671PHE5]] with $B$ [[positive definite|0KX66CZMBB]], and let $\lambda_1 \le \cdots \le \lambda_n$ be the [[eigenvalues|0BQAA2BK3Q]] of the pencil $A x = \lambda B x$.
Let $T \in \R^{n \times n}$ be symmetric positive definite (the [[preconditioner|KE98SW3SBD]]), and let $\rho(x) = x^T A x / x^T B x$ be the [[Rayleigh quotient|06WHP8SE98]] for $x \ne 0$.
Consider the single-vector *locally optimal block preconditioned conjugate gradient method* (LOBPCG).
It starts from $x_0 \ne 0$, and for $k \ge 0$ it sets
$$r_k = A x_k - \rho(x_k) B x_k, \qquad w_k = T r_k, \qquad S_k = \mathrm{span}\set{x_k, w_k, x_{k-1}}$$
with $x_{-1}$ omitted for $k = 0$.
The next iterate $x_{k+1}$ is a vector that minimizes $\rho$ over $S_k \setminus \set{0}$, computed by the Rayleigh–Ritz method on $S_k$.
Then for every $k$:

1. $\lambda_1 \le \rho(x_{k+1}) \le \rho(x_k)$;
2. $\rho(x_{k+1}) \le \min_{t \in \R} \rho(x_k + t w_k)$, so each step is at least as good as one step of preconditioned steepest descent with the optimal step length;
3. if $r_k = 0$, then $x_k$ is an eigenvector and $\rho(x_k)$ is an eigenvalue of the pencil.

## Remarks

Parts 1 and 2 hold because $x_k$ and every $x_k + t w_k$ lie in $S_k$, over which $x_{k+1}$ minimizes $\rho$; the lower bound $\lambda_1$ is the [[min-max theorem|G11NE3AT6E]] for the pencil.
The Rayleigh–Ritz step reduces to a generalized eigenvalue problem of size at most $3$, the projections of $A$ and $B$ onto $S_k$.
The original method uses a direction $p_k$, a combination of $w_{k-1}$ and $p_{k-1}$, in place of $x_{k-1}$; the span $S_k$ is the same, and $p_k$ gives a better conditioned basis.

The block version iterates a block of $m$ vectors and approximates the $m$ smallest eigenvalues, with Rayleigh–Ritz on a subspace of dimension at most $3m$.
Since LOBPCG improves on preconditioned steepest descent, the convergence bounds of preconditioned inverse iteration (Knyazev and Neymeyr) apply to it.
Example: with $T = I$, $B = I$ and $x_{-1}$ dropped, the method becomes steepest descent for the Rayleigh quotient, since $r_k$ is a multiple of the gradient of $\rho$ at $x_k$.

## Explanations

### L1 human

LOBPCG looks for the smallest eigenvalue of a large symmetric problem by minimizing the Rayleigh quotient.
At each step it searches for the best vector in a three-dimensional space built from the current guess, the preconditioned residual, and the previous guess.
Because the search space contains the steepest descent step, the method can never do worse than steepest descent, and in practice it behaves like a conjugate gradient method.

### L2 human

LOBPCG was introduced by Andrew Knyazev in 2001.
It needs only products with $A$, $B$ and $T$, so it suits large sparse problems such as discretized PDE eigenproblems.
There a good preconditioner, for example [[multigrid|AC9PY6FBYF]], makes the convergence nearly independent of the mesh size.
Compared with the [[Lanczos algorithm|7RRB4PCRJB]], it uses a preconditioner directly and keeps a fixed small memory, at the price of solving a small dense eigenproblem at each step.
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<document source="https://tamnd.github.io/math2/node/P1G0M3S8QV.md" tag="P1G0M3S8QV" kind="concept">
# Bicubic interpolation

## Summary

Bicubic interpolation builds a smooth surface over a rectangular grid from values and derivatives at the grid points.
On each cell it uses a polynomial $\sum_{i,j=0}^{3} a_{ij} x^i y^j$, fixed by the values, the two first partial derivatives and the mixed derivative at the four corners.
The patches join with continuous first derivatives, giving smoother results than [[bilinear interpolation|5QJYCNRYMQ]], which is why it is common in image resampling.

## Explanations

### L1 human

Bilinear interpolation produces visible creases along grid lines because its slopes jump there.
Bicubic interpolation also matches slopes at the grid points, so neighboring patches meet smoothly.
Each patch is a cubic in each direction, with sixteen coefficients determined by sixteen pieces of corner data.
When the derivatives are unknown, as in image processing, they are estimated from neighboring values by finite differences.
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<document source="https://tamnd.github.io/math2/node/P1JVA39H5E.md" tag="P1JVA39H5E" kind="concept">
# Maurer–Cartan form

## Summary

The Maurer–Cartan form of a [[Lie group|7JBRGCVJRB]] $G$ is a [[differential form|0E3G0GRPZ6]] of degree one on $G$ with values in its Lie algebra $\mathfrak{g}$.
It translates each tangent vector of $G$ back to the identity by a left translation.
It is left-invariant and satisfies the Maurer–Cartan equation $d\omega + \tfrac12 [\omega \wedge \omega] = 0$, which encodes the Lie bracket.
It is the model for connection forms and for Cartan's method of moving frames.

## Explanations

### L1 human

At each point $g$ of a Lie group there is a tangent space, and left translation by $g^{-1}$ moves it to the tangent space at the identity.
The Maurer–Cartan form does exactly this: it reads every velocity of the group as a vector in one fixed space, the Lie algebra.
For a group of matrices, a velocity $V$ at $g$ is read as $g^{-1} V$, so the form is written $g^{-1}\,dg$.
For the circle group $e^{i\theta}$, it is $e^{-i\theta}\, d(e^{i\theta}) = i\, d\theta$.
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<document source="https://tamnd.github.io/math2/node/P1MKTF7MWQ.md" tag="P1MKTF7MWQ" kind="concept">
# Projective transformation

## Summary

A projective transformation, or homography, is a map between [[projective spaces|ZT9A6JK8QF]] induced by an invertible linear map of the underlying vector spaces.
In [[homogeneous coordinates|FS9FRJ17D2]] it is $[x] \mapsto [Ax]$ for an invertible matrix $A$, defined up to a nonzero factor.
The projective transformations of $\mathbb{P}^n(k)$ form the group $\mathrm{PGL}(n + 1, k)$.
They are the symmetries of [[projective geometry|N32Z68F0D3]], as Euclidean motions are those of Euclidean geometry.

## Explanations

### L1 human

Take a photograph of a flat floor from an angle: a square tile becomes a general quadrilateral.
The map from the floor to the photo is a projective transformation.
It keeps straight lines straight, but not lengths, angles or parallelism.
On the projective line over $\C$, the Möbius maps $z \mapsto (az + b)/(cz + d)$ with $ad - bc \ne 0$ are exactly the projective transformations.
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# Hilbert's program

## Problem

Write all of classical mathematics as a formal system with finitely describable axioms and rules of inference.
Then prove, by finitary means only, that this formal system is consistent.
Here a proof is finitary when it reasons only about finite strings of symbols in a concrete, checkable way.
Consistency means that no formula $\varphi$ is derivable together with its negation $\neg \varphi$.

## Remarks

The program is the main form of [[formalism|46JCQ6KDA4]].
It also asked for a proof of completeness: every statement of the system should be provable or refutable.
Gödel's [[incompleteness theorems|X9WYNYE29C]] of 1931 showed that the program cannot be carried out as first intended.
A consistent, effectively axiomatized system that contains enough arithmetic is incomplete.
Such a system also cannot prove its own consistency.
In 1936 Gerhard Gentzen proved the consistency of the [[Peano axioms|0QFT7Z76N5]] of first-order arithmetic, see [[Gentzen's consistency proof|QPV0HNP22S]].
His proof uses transfinite induction up to the ordinal $\varepsilon_0$, which goes beyond the finitary means of the original program.
Partial versions of the program are studied in [[proof theory|N4EDJNMTEF]] and in [[reverse mathematics|7SQS69T17R]].

## Explanations

### L1 human

Hilbert wanted to put all of mathematics on a safe base.
First, write every theorem as the result of fixed rules applied to fixed axioms, like moves in a game.
Second, show with simple, finite reasoning that the game can never produce a contradiction such as $0 = 1$.
Gödel showed that the second step is impossible in the strong form that Hilbert hoped for.
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# Sinc function

## Summary

The sinc function is $\operatorname{sinc} x = \frac{\sin x}{x}$, with the value $1$ at $x = 0$.
Signal processing often uses the normalised version $\frac{\sin \pi x}{\pi x}$, whose zeros are the non-zero integers.
The normalised sinc function is the [[Fourier transform|DPVM34ADZV]] of the indicator function of an interval, and it is the interpolation kernel in the sampling theorem.
Its integral over $(0, \infty)$ is the [[Dirichlet integral|0GZNSHMG4Q]], equal to $\frac{\pi}{2}$.

## Explanations

### L1 human

The sine wave $\sin x$ goes up and down forever with the same height.
Dividing by $x$ makes the waves smaller and smaller far from the origin.
Near $0$ both $\sin x$ and $x$ are small, and their quotient is close to $1$.
So the graph has a tall central hump of height $1$ and decaying ripples on both sides, crossing zero at $\pm\pi, \pm 2\pi, \ldots$.
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# G2 manifold

## Summary

A $G_2$ manifold is a Riemannian manifold of dimension $7$ whose holonomy group lies in the exceptional Lie group $G_2$.
The structure is encoded in a single $3$-form that is parallel, and the metric is determined by this form.
$G_2$ manifolds are Ricci-flat and form one of the two exceptional cases of Berger's list of holonomy groups.
Compact examples are hard to build and are studied in geometry and in M-theory.

## Explanations

### L1 human

The group $G_2$ is the symmetry group of the octonions, an eight-dimensional number system.
Its imaginary part is seven-dimensional, and $G_2$ preserves a special way of multiplying vectors there, a seven-dimensional cross product.
A $G_2$ manifold is a seven-dimensional curved space in which this cross product exists on every tangent space and does not change under parallel transport.
The flat space $\R^7$ and the flat torus $T^7$ are the simplest examples.
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# Fractional integration

## Summary

Fractional integration on $\R^n$ is the operator $I_\alpha$, for $0 < \alpha < n$, that convolves a function with the kernel $\abs{x}^{\alpha - n}$, suitably normalized.
It is also called the Riesz potential.
On the [[Fourier transform|DPVM34ADZV]] side it multiplies by a constant times $\abs{\xi}^{-\alpha}$, so it is the inverse of the [[fractional Laplacian|F8RSNQBMBE]] $(-\Delta)^{\alpha/2}$.
It is the Euclidean analogue of the [[Riemann–Liouville integral|JTKCZ44NN3]], and it is the main tool behind the Hardy–Littlewood–Sobolev and Sobolev inequalities.

## Explanations

### L1 human

Integrating a function once makes it smoother by one order; fractional integration makes it smoother by a fractional order $\alpha$.
In $\R^n$ this is done by averaging $f$ against the weight $\abs{x - y}^{\alpha - n}$, which is large near $x$ and decays far away.
For $n = 3$ and $\alpha = 2$ the result is the Newtonian potential $\frac{1}{4\pi}\int \frac{f(y)}{\abs{x - y}} \, dy$, the gravitational potential of a mass density $f$.
Doing it twice, of orders $\alpha$ and $\beta$, is the same as doing it once of order $\alpha + \beta$, as long as $\alpha + \beta < n$.
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# Harmonic functions on Riemann surfaces

## Summary

A real function on a Riemann surface is harmonic when it is harmonic in every holomorphic coordinate chart.
This makes sense because holomorphic changes of coordinates preserve harmonicity, even though the surface carries no preferred Laplacian.
Such functions are the starting point of the potential theory of surfaces and of the classification of open surfaces.
They also give the classical existence proofs for holomorphic functions and differentials.

## Explanations

### L1 human

A Riemann surface looks locally like a piece of the complex plane, but the local pictures are glued by holomorphic maps.
A function is harmonic on the surface if it looks harmonic in each local picture.
For example, on the punctured plane $\C \setminus \set{0}$ the function $\log\abs{z}$ is harmonic.
On a compact Riemann surface such as the Riemann sphere, every harmonic function is constant, because it attains a maximum.
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# Admissibility condition for wavelets

## Definition

Let $\psi \in L^2(\R)$ (see [[Lp space|SASP14MA1Z]]), and let $\hat\psi(\xi) = \int_\R \psi(t) e^{-2\pi i t \xi} \, dt$ be its [[Fourier transform|DPVM34ADZV]].
The function $\psi$ satisfies the *admissibility condition*, and is called an *admissible wavelet*, if
$$C_\psi = \int_\R \frac{\abs{\hat\psi(\xi)}^2}{\abs{\xi}} \, d\xi < \infty.$$

## Remarks

If $\psi$ is also in $L^1(\R)$, then $\hat\psi$ is continuous, and admissibility implies $\hat\psi(0) = \int_\R \psi(t) \, dt = 0$.
Conversely, if $\int_\R (1 + \abs{t}) \abs{\psi(t)} \, dt < \infty$ and $\int_\R \psi = 0$, then $\hat\psi$ is Lipschitz with $\hat\psi(0) = 0$, and $\psi$ is admissible.
The constant $C_\psi$ is the factor in Calderón's identity for the continuous wavelet transform.

Example: the Mexican hat $\psi(t) = (1 - t^2) e^{-t^2/2}$ is admissible.
It decays fast and $\int_\R \psi = \int_\R e^{-t^2/2} \, dt - \int_\R t^2 e^{-t^2/2} \, dt = \sqrt{2\pi} - \sqrt{2\pi} = 0$, so the converse criterion applies.

Non-example: the Gaussian $\psi(t) = e^{-\pi t^2}$ has $\hat\psi(\xi) = e^{-\pi \xi^2}$.
Near $\xi = 0$ the integrand behaves like $1/\abs{\xi}$, which is not integrable, so $C_\psi = \infty$.
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# Additive combinatorics

## Definition

Additive combinatorics is the part of mathematics that studies finite subsets $A, B$ of an [[abelian group|082GHNPJ53]] $G$ through their additive structure.
Its main objects are the [[sumsets|XNBN31HQWB]] $A + B$, the difference sets $A - B$, and the solutions of linear equations with all variables in $A$.
Its typical results are of three kinds:

1. Direct results bound the size of $A + B$ from below in terms of $\abs{A}$ and $\abs{B}$.
2. Inverse results describe the structure of $A$ when $\abs{A + A}$ is small compared with $\abs{A}$.
3. Density results show that large subsets of $G$ contain solutions of given linear equations.

## Remarks

The [[Cauchy–Davenport theorem|F7K7XFACSG]] is a direct result: in $\Z/p\Z$ it gives $\abs{A + B} \ge \min(p, \abs{A} + \abs{B} - 1)$.
[[Freiman's theorem|KEPFYE6WYZ]] is an inverse result: sets of integers with small doubling lie in small generalized arithmetic progressions.
[[Szemerédi's theorem|B23B0QEMN5]] is a density result about arithmetic progressions.
A statement about sets without any addition, such as a bound on chains in a poset, is not a result of additive combinatorics.
The field is also called arithmetic combinatorics.
</document>

<document source="https://tamnd.github.io/math2/node/P2E8BNYC7B.md" tag="P2E8BNYC7B" kind="concept">
# Linearly ordered group

## Summary

A linearly ordered group is an [[ordered group|KRDR04RQ7N]] whose order is a [[total order|0FRXBVVR03]].
Any two of its elements are comparable, and the order is preserved by multiplication on both sides.
The additive groups of $\Z$, $\Q$ and $\R$ are the basic examples.
Linearly ordered abelian groups are the value groups of valuations, and they are described by Hölder's theorem and the Hahn embedding theorem.

## Explanations

### L1 human

A linearly ordered group is a group whose elements stand in one line, from smaller to larger.
Multiplying by a fixed element on either side shifts the whole line without changing the order of its points.
The integers under addition are an example: adding $5$ to every integer keeps them in the same order.
The pairs of integers with the dictionary order are another example, where $(0, 1000) < (1, 0)$.
</document>

<document source="https://tamnd.github.io/math2/node/P2F66BS71T.md" tag="P2F66BS71T" kind="definition">
# Nerve of a cover

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $\mathcal{U} = (U_i)_{i \in I}$ be a family of subsets of $X$, usually an [[open cover|049PXYHW1T]].
The *nerve* $N(\mathcal{U})$ is the abstract [[simplicial complex|V8GZ0W1ZZ5]] with vertex set $\set{i \in I : U_i \ne \emptyset}$ whose faces are the finite nonempty subsets $J \subseteq I$ such that
$$\bigcap_{j \in J} U_j \ne \emptyset.$$

## Remarks

If $J$ is a face and $J' \subseteq J$ is nonempty, then $\bigcap_{j \in J'} U_j \supseteq \bigcap_{j \in J} U_j \ne \emptyset$, so $J'$ is a face; this is why $N(\mathcal{U})$ is a simplicial complex.

Example: cover the circle $S^1$ by three open arcs $U_0$, $U_1$, $U_2$, each a bit longer than one third of the circle.
Choose them so that consecutive arcs overlap and no point lies in all three.
Each pair $\set{i, j}$ is a face, and $\set{0, 1, 2}$ is not, so $N(\mathcal{U})$ is the boundary of a triangle.

Non-example: the complex with vertices $0, 1, 2$ and all subsets as faces is not the nerve of this cover, since it contains $\set{0, 1, 2}$ although $U_0 \cap U_1 \cap U_2 = \emptyset$.

The same definition applies to any family of sets; the nerve of a closed cover or of a family of convex sets is often used in combinatorics.
</document>

<document source="https://tamnd.github.io/math2/node/P2HKS3K7T2.md" tag="P2HKS3K7T2" kind="definition">
# Splitting field of a central simple algebra

## Definition

Let $K$ be a field and let $A$ be a [[central simple algebra|5HRKJBAY2V]] over $K$ of [[degree|NNXSP5KKAC]] $n$.
Let $L$ be a [[field extension|0EQWVHF3W5]] of $K$.
The field $L$ is a *splitting field* of $A$, or *splits* $A$, if there is an isomorphism of $L$-algebras
$$A \otimes_K L \cong M_n(L).$$
Here $\otimes_K$ is the [[tensor product of algebras|1735VGRT2P]] and $M_n(L)$ is the [[matrix ring|0FHVPRAE60]] of $n \times n$ matrices over $L$.

## Remarks

Example: $\C$ splits $\mathbb{H}$ over $\R$.
Indeed, $\mathbb{H} \otimes_\R \C$ is central simple of dimension $4$ over $\C$, and the only such algebra is $M_2(\C)$, since $\C$ is algebraically closed.

Non-example: $\R$ itself does not split $\mathbb{H}$, since $\mathbb{H} \otimes_\R \R \cong \mathbb{H}$ is a division algebra and $M_2(\R)$ has zero divisors.

$A$ is split over $K$ itself exactly when $A \cong M_n(K)$.
An algebraic closure of $K$ splits every central simple algebra over $K$, and so does a suitable finite Galois extension of $K$.
Every maximal subfield of a central division algebra $D$ is a splitting field of $D$.
</document>

<document source="https://tamnd.github.io/math2/node/P2NS0V36TH.md" tag="P2NS0V36TH" kind="definition">
# Operator monotone function

## Definition

Let $I \subseteq \R$ be an [[interval|0Q1D85B8G6]] and let $f \colon I \to \R$.
Let $A$ be an $n \times n$ [[Hermitian matrix|0P17NGQQMQ]] with all eigenvalues in $I$.
By the [[spectral theorem|0DGZBQWZWN]], $A = U \operatorname{diag}(\lambda_1, \dots, \lambda_n) U^*$ with $U$ unitary.
The [[matrix function|W7MPTRPXYD]] $f(A)$ is $U \operatorname{diag}(f(\lambda_1), \dots, f(\lambda_n)) U^*$; it does not depend on the choice of $U$.
For Hermitian matrices $A$ and $B$ of the same size, $A \le B$ in the [[Loewner order|9G9S92T1VJ]] means that $B - A$ is [[positive semidefinite|01MMVW8CKC]].
The function $f$ is *operator monotone* on $I$ if the following holds for every $n \ge 1$.
For all $n \times n$ Hermitian matrices $A$ and $B$ with eigenvalues in $I$, if $A \le B$, then $f(A) \le f(B)$.

## Remarks

For $n = 1$ the condition says that $f$ is nondecreasing, so an operator monotone function is a [[monotone function|0A7KV2WB7A]].
Example: $f(t) = -1/t$ on $(0, \infty)$.
If $0 < A \le B$, then $B^{-1/2} A B^{-1/2} \le I$, so its eigenvalues lie in $(0, 1]$, so $B^{1/2} A^{-1} B^{1/2} \ge I$, which gives $A^{-1} \ge B^{-1}$, that is, $-A^{-1} \le -B^{-1}$.
Non-example: $f(t) = t^2$ on $[0, \infty)$.
Take $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$, both positive semidefinite, with $B - A = \operatorname{diag}(1, 0) \ge 0$.
Then $B^2 - A^2 = \begin{pmatrix} 3 & 1 \\ 1 & 0 \end{pmatrix}$ has determinant $-1$, so it has a negative eigenvalue and $A^2 \le B^2$ fails.
The same definition with self-adjoint operators on a Hilbert space gives the same class of functions.

## Explanations

### L1 human

The definition asks that $f$ keep the order of Hermitian matrices of all sizes, not only the order of numbers.
The size-two example with $t^2$ shows that this is a real restriction.
</document>

<document source="https://tamnd.github.io/math2/node/P2QJ7NPW3Y.md" tag="P2QJ7NPW3Y" kind="statement">
# Propagation of singularities

## Statement

Let $X \subseteq \R^n$ be open and let $P = \sum_{\abs{\alpha} \le m} a_\alpha(x) D^\alpha$ be a linear differential operator of order $m$ with $C^\infty$ coefficients on $X$, where $D^\alpha = (-i\partial)^\alpha$.
Assume that its [[principal symbol|F31T650DKY]] $p(x, \xi) = \sum_{\abs{\alpha} = m} a_\alpha(x)\xi^\alpha$ is real-valued.
Let $u$ be a distribution on $X$ with $P u \in C^\infty(X)$, and let $\operatorname{WF}(u)$ be its [[wave front set|3T5Q9WVQX8]].
Then:

1. $\operatorname{WF}(u) \subseteq \set{(x, \xi) \in X \times (\R^n \setminus \set{0}) : p(x, \xi) = 0}$;
2. $\operatorname{WF}(u)$ is invariant under the flow of the [[Hamiltonian vector field|C5EH3B4BEP]] $H_p = \sum_{j} \big( \partial_{\xi_j} p\, \partial_{x_j} - \partial_{x_j} p\, \partial_{\xi_j} \big)$ on this set. That is, if $(x_0, \xi_0) \in \operatorname{WF}(u)$, then $\operatorname{WF}(u)$ contains the null bicharacteristic through $(x_0, \xi_0)$, as long as it stays in $X \times (\R^n \setminus \set{0})$.

## Remarks

Part 1 is microlocal elliptic regularity: where $p(x, \xi) \ne 0$, the operator $P$ has a microlocal parametrix, so $u$ is smooth in that direction.
Part 2 is due to Hörmander and to Duistermaat and Hörmander.
One proof uses a positive commutator argument: a pseudo-differential operator whose symbol grows along $H_p$ gives energy estimates that carry smoothness from one point of a bicharacteristic to the next.
Example: for the wave operator $P = \partial_t^2 - \Delta$ on $\R^{n+1}$, the symbol is $p = -\tau^2 + \abs{\xi}^2$ in the dual variables $(\tau, \xi)$ of $(t, x)$, and the null bicharacteristics project to the light rays $x = x_0 \pm t\,\xi_0/\abs{\xi_0}$.
So a singularity of a solution of the wave equation travels along a straight line at speed $1$ in its own direction.
Part 1 holds even without the assumption that $p$ is real.
Let $P$ be elliptic, such as the Cauchy–Riemann operator $\frac{1}{2}(\partial_x + i\partial_y)$ on $\R^2$.
Then its principal symbol has no zero with $\xi \ne 0$.
Microlocal elliptic regularity gives $\operatorname{WF}(u) = \emptyset$, so $u$ is smooth and there is nothing to propagate.

## Explanations

### L1 human

A singularity of a solution, such as a sharp edge or a jump, does not appear or disappear at random.
It moves along special curves, the rays of the equation, in a definite direction.
For the wave equation the rays are the straight lines along which light travels, so a sharp flash stays sharp and moves at the speed of light.
This is the precise form of the idea that singularities of waves travel like particles.

### L2 human

The theorem is a central result of microlocal analysis.
It locates singularities not only in space but also in direction, via the wave front set, and shows that they move along the bicharacteristics of the principal symbol.
For hyperbolic equations it gives the mathematical foundation of geometric optics: singularities follow the rays obtained from the eikonal equation.
It is used in inverse problems, in scattering theory and in the study of the wave equation on manifolds, where the rays are geodesics.
</document>

<document source="https://tamnd.github.io/math2/node/P2V0XS4F63.md" tag="P2V0XS4F63" kind="statement">
# Feit–Higman theorem

## Statement

Let $\mathcal{S}$ be a finite thick [[generalized $n$-gon|PCT2HY9AAZ]] with $n \ge 2$.
Then
$$n \in \set{2, 3, 4, 6, 8}.$$

## Remarks

Walter Feit and Graham Higman proved the theorem in 1964.
The proof is algebraic.
Counting paths in the incidence graph gives a matrix identity for the point-line incidence matrix, from which its [[eigenvalues|0BQAA2BK3Q]] can be computed in terms of the order $(s, t)$ and $n$.
The multiplicities of these eigenvalues must be nonnegative integers, and this fails for all values of $n$ outside the list.

Each value in the list occurs.
For $n = 3$ there are the [[projective planes|DANFNN8C4Y]] $PG(2, q)$, and for $n = 4$ the [[generalized quadrangles|GQJCPFSYVK]] of symplectic forms.
For $n = 6$ there are the split Cayley hexagons attached to the groups $G_2(q)$.
For $n = 8$ there are the Ree–Tits octagons attached to the groups ${}^2F_4(2^{2m+1})$.

Both hypotheses are needed.
Without thickness, take $n$ points and $n$ lines arranged in a cycle, the ordinary $n$-gon.
Its incidence graph is a cycle of length $2n$, so it is a finite generalized $n$-gon for every $n$.
Without finiteness, there are thick generalized $n$-gons for every $n \ge 3$, built by free constructions.

## Explanations

### L1 human

Generalized polygons are point-line geometries that look locally like the polygons of school geometry, but with many points on each line.
Consider finite ones with at least three points per line and three lines per point.
The theorem says that the underlying polygon can only be a digon, triangle, quadrangle, hexagon or octagon.
So, for example, there is no finite thick generalized pentagon.

### L2 human

Generalized polygons are the [[buildings|6BQJ50NEJ0]] of rank $2$, and finite buildings are tied to finite groups of Lie type.
The theorem explains why the labels of the Coxeter diagram of a finite thick building lie in $\set{2, 3, 4, 6, 8}$: its rank $2$ residues are finite thick generalized polygons.
The method, integrality of eigenvalue multiplicities, is a standard tool of algebraic combinatorics, and it also gives conditions on the order $(s, t)$.
</document>

<document source="https://tamnd.github.io/math2/node/P2WS16HKD7.md" tag="P2WS16HKD7" kind="object">
# Baire space of sequences

## Description

The *Baire space* $\N^{\N}$ is the set of all infinite sequences of [[natural numbers|0MWKJ44V5A]].
Its topology is the [[product topology|0GBG8B49AA]] of countably many copies of the discrete space $\N$.
Its basic open sets are the sets of all sequences that begin with a given finite sequence.
The topology comes from the metric $d(x, y) = 2^{-n}$, where $n$ is the first index with $x_n \neq y_n$, and $d(x, x) = 0$.
With this metric the space is [[complete|0BY52JW6NK]], and it is [[separable|0Z2P41SVDP]], since the eventually zero sequences are dense.
In descriptive set theory it is written $\mathcal{N}$ or $\omega^\omega$.

## Remarks

The Baire space is homeomorphic to the set of [[irrational numbers|0DZXAXEJQ9]] with the topology of the real line; one homeomorphism uses continued fraction expansions.
It is not locally compact: no basic open set is compact, since the next term of the sequence can take infinitely many values.
It has a base of clopen sets.
Every nonempty Polish space, that is a separable completely metrizable space, is a continuous image of $\N^{\N}$.
As a complete metric space it is a [[Baire space|0ZH5AJ2ED7]] in the sense of Baire category; the two uses of the name are related but different.

## Explanations

### L1 human

A point of the Baire space is an endless list of natural numbers.
Two points are close when their lists agree for a long time.
It is the basic space of descriptive set theory, where sets of reals are studied through the sequences that code them.
</document>

<document source="https://tamnd.github.io/math2/node/P32FQE1H0D.md" tag="P32FQE1H0D" kind="definition">
# Pseudo algebraically closed field

## Definition

Let $K$ be a [[field|04C5ZBG1WT]] with [[algebraic closure|0X6AB5P3G1]] $\bar K$.
An affine [[variety|HV6YBS0YDH]] $V \subseteq \bar K^n$ defined over $K$ is the set of common zeros in $\bar K^n$ of a set of polynomials in $K[X_1, \dots, X_n]$.
It is *absolutely irreducible* if it is irreducible as a variety over $\bar K$.
This means that it is not the union of two proper closed subsets defined over $\bar K$.
A point of $V$ is $K$-rational if all its coordinates lie in $K$.
The field $K$ is *pseudo algebraically closed* (PAC) if every nonempty absolutely irreducible affine variety defined over $K$ has a $K$-rational point.

## Remarks

It is equivalent to ask the condition only for curves, that is, for varieties of dimension $1$.

Example: every [[algebraically closed field|0PF6TM8TWT]] $K$ is PAC, since then $\bar K = K$ and every point of a nonempty variety is $K$-rational.
Every infinite algebraic extension of a [[finite field|0ZXZ82FTX2]] is PAC.

Non-example: $\Q$ and $\R$ are not PAC.
The curve $V$ defined by $x^2 + y^2 + 1 = 0$ is absolutely irreducible, since $x^2 + y^2 + 1$ is irreducible over $\C$, and $V$ is nonempty, since it contains $(i, 0)$.
But it has no point with real coordinates, because $x^2 + y^2 + 1 > 0$ for real $x, y$.

Non-example: a finite field $\F_q$ is not PAC.
For instance, the curve $y^q - y = x^{q+1} - x^2 + 1$ is absolutely irreducible, because the degree $q + 1$ of the right side is prime to the characteristic of $\F_q$.
It has no $\F_q$-rational point, because $y^q - y = 0$ for $y \in \F_q$, while $x^{q+1} - x^2 + 1 = 1$ for $x \in \F_q$.
</document>

<document source="https://tamnd.github.io/math2/node/P33CMYHJW9.md" tag="P33CMYHJW9" kind="concept">
# Strict total order

## Summary

A strict total order is a [[strict partial order|SWB1ZXFVKY]] in which any two distinct elements are comparable.
It is the strict form of a [[total order|0316AYH1SC]], in the same way that $<$ is the strict form of $\le$ on the real numbers.
For any two elements $a$ and $b$, exactly one of $a < b$, $a = b$, $b < a$ holds.

## Explanations

### L1 human

A strict total order puts all elements on one line and says which of two different elements comes first.
The relation $<$ on the integers is the standard example: for $3$ and $5$, one has $3 < 5$.
Alphabetical order of distinct words is another example.
</document>

<document source="https://tamnd.github.io/math2/node/P3A4EAA3RA.md" tag="P3A4EAA3RA" kind="definition">
# Least squares approximation

## Definition

Let $H$ be a real or complex [[inner product space|0QVNCJXRY3]] with inner product $\ip{\cdot}{\cdot}$ and [[induced norm|0KG4D8AP6G]] $\norm{\cdot}$.
Let $V \subseteq H$ be a finite-dimensional subspace and let $f \in H$.
A *least squares approximation* of $f$ from $V$ is an element $v^* \in V$ such that
$$\norm{f - v^*} = \min_{v \in V} \norm{f - v}.$$
In the main case, $w$ is a positive weight on $[a, b]$ and $H$ is the space $L^2_w$ of functions with $\int_a^b w \abs{f}^2 < \infty$.
Its inner product is $\ip{f}{g} = \int_a^b w(x) f(x) \overline{g(x)} \, dx$, and $V$ is the space of polynomials of degree at most $n$.

## Remarks

The least squares approximation exists, is unique, and is the [[orthogonal projection|0B8KZRT28M]] of $f$ onto $V$.
It is characterised by $\ip{f - v^*}{v} = 0$ for all $v \in V$.
If $\varphi_0, \dots, \varphi_n$ is an orthogonal basis of $V$, then $v^* = \sum_{k=0}^{n} \frac{\ip{f}{\varphi_k}}{\ip{\varphi_k}{\varphi_k}} \varphi_k$.
For a general basis, the coefficients solve the normal equations, whose matrix is the Gram matrix $(\ip{\varphi_j}{\varphi_k})$.

Example: $f(x) = x^2$ on $[-1, 1]$ with $w = 1$ and $V$ the polynomials of degree at most $1$.
With the orthogonal basis $1, x$, one gets $\ip{f}{1} / \ip{1}{1} = (2/3)/2 = 1/3$ and $\ip{f}{x} = 0$, so $v^* = 1/3$.
Check: $\int_{-1}^{1} (x^2 - 1/3) \, dx = 0$ and $\int_{-1}^{1} (x^2 - 1/3) x \, dx = 0$.

Non-example: the uniform best line for $x^2$ on $[-1, 1]$ is $1/2$, not $1/3$.
It minimises the largest error, not the integral of the squared error, so it fails the defining condition.

A discrete version replaces the integral by a weighted sum $\sum_i w_i \abs{f(x_i) - v(x_i)}^2$ over data points; this is curve fitting.
</document>

<document source="https://tamnd.github.io/math2/node/P3A4XF59FX.md" tag="P3A4XF59FX" kind="definition">
# Resolvent kernel

## Definition

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]], let $M = \max \abs{K}$, and let $K_n$ be its [[iterated kernels|9QNBFERMHD]].
For $\lambda \in \C$ with $\abs{\lambda} M (b - a) < 1$, the *resolvent kernel* of $K$ is
$$R(x, s; \lambda) = \sum_{n=1}^{\infty} \lambda^{n-1} K_n(x, s), \quad x, s \in [a, b].$$
The series converges absolutely and [[uniformly|08VZ57PT26]] on $[a, b] \times [a, b]$, because $\abs{\lambda^{n-1} K_n} \le M (\abs{\lambda} M (b - a))^{n-1}$.

## Remarks

Example: $K(x, s) = x s$ on $[0, 1]$, so $M = 1$ and $K_n = x s / 3^{n-1}$.
For $\abs{\lambda} < 1$ the series is $x s \sum_{n \ge 1} (\lambda/3)^{n-1} = 3 x s / (3 - \lambda)$.

Non-example: for $K = 1$ on $[0, 1]$ and $\lambda = 1$, all $K_n = 1$ and the series $\sum 1$ diverges.
This $\lambda$ violates the condition $\abs{\lambda} M (b - a) < 1$, and indeed the homogeneous equation $\varphi = \int_0^1 \varphi$ has the nonzero solution $\varphi = 1$.

The solution of the [[integral equation of the second kind|8KJCN49V64]] $\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds$ is then $\varphi(x) = f(x) + \lambda \int_a^b R(x, s; \lambda) f(s) \, ds$.
In operator terms, $R = K (I - \lambda K)^{-1}$.
By Fredholm theory, $R$ extends to a meromorphic function of $\lambda$ on $\C$, the quotient of two entire functions.
The resolvent kernel is then defined at every $\lambda$ that is not a pole.
The example shows that the extension can go beyond the disc of the definition: $3 x s / (3 - \lambda)$ has its only pole at $\lambda = 3$.
</document>

<document source="https://tamnd.github.io/math2/node/P3B1V182N4.md" tag="P3B1V182N4" kind="concept">
# Elementary equivalence

## Summary

Two [[structures|983GZBQYT4]] for the same signature are elementarily equivalent when they satisfy the same first-order sentences.
Isomorphic structures are elementarily equivalent, but the converse fails.
Elementary equivalence is the notion of "sameness" that first-order logic can see.

## Explanations

### L1 human

Imagine that you may only ask yes-or-no questions written as first-order sentences.
Two structures are elementarily equivalent when every such question gets the same answer in both.
The ordered sets $(\Q, <)$ and $(\R, <)$ are elementarily equivalent, although one is countable and the other is not.
The ordered sets $(\Z, <)$ and $(\Q, <)$ are not, because "between any two elements there is a third" is true in $\Q$ and false in $\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/P3H4MMC4ZH.md" tag="P3H4MMC4ZH" kind="concept">
# Discrete wavelet transform

## Summary

The discrete wavelet transform splits a sequence into a coarse part and a detail part.
It uses two filters followed by downsampling by $2$, and then repeats the split on the coarse part.
For filters from an orthonormal [[multiresolution analysis|1GHQMBR7DE]], each step is an orthogonal change of coordinates.
So the transform preserves energy and is easy to invert.
The outputs are the coefficients of a function in a [[wavelet|4PZ35C0GV3]] basis.
The [[fast wavelet transform|PDVYQAW887]] computes them in a number of steps proportional to the length of the signal.
It is the basis of wavelet compression, for example in the JPEG 2000 image standard.

## Explanations

### L1 human

Take a list of numbers and group them in pairs.
For each pair, keep a scaled sum, which records the average, and a scaled difference, which records the detail.
Then do the same with the list of sums, and again, until one number is left.
For a smooth signal most details are tiny, so one can drop them and still rebuild a good approximation.
This is the Haar case; other wavelets use longer filters in the same pattern.
</document>

<document source="https://tamnd.github.io/math2/node/P3PJGESDSF.md" tag="P3PJGESDSF" kind="definition">
# (p,q)-form

## Definition

Let $U \subseteq \C^n$ be an open set, with coordinates $z_j = x_j + i y_j$.
Consider complex-valued [[differential forms|0E3G0GRPZ6]] on $U$, and put $dz_j = dx_j + i\, dy_j$ and $d\bar z_j = dx_j - i\, dy_j$.
For increasing tuples $I = (i_1 < \cdots < i_p)$ and $J = (j_1 < \cdots < j_q)$ of elements of $\set{1, \dots, n}$, write $dz_I = dz_{i_1} \wedge \cdots \wedge dz_{i_p}$ and $d\bar z_J = d\bar z_{j_1} \wedge \cdots \wedge d\bar z_{j_q}$.
Let $p, q \ge 0$ be integers.
A *$(p, q)$-form* on $U$ is a differential form
$$\omega = \sum_{\abs{I} = p, \ \abs{J} = q} \omega_{IJ} \, dz_I \wedge d\bar z_J,$$
where the sum runs over increasing tuples and each $\omega_{IJ} \colon U \to \C$ is a [[smooth function|0V33BNKZ9S]].

## Remarks

The forms $dz_I \wedge d\bar z_J$ with $\abs{I} + \abs{J} = k$ form a basis of the complex $k$-th [[exterior power|0MDMZ965VE]] of the cotangent space at each point.
So every complex $k$-form is, in a unique way, a sum of $(p, q)$-forms with $p + q = k$.
Example: $\omega = \bar z_1 \, dz_1 \wedge d\bar z_2$ on $\C^2$ is a $(1, 1)$-form, with $\omega_{(1)(2)} = \bar z_1$ and all other coefficients zero.
Non-example: $dx_1 = \tfrac12 dz_1 + \tfrac12 d\bar z_1$ is a $1$-form whose $(1, 0)$ and $(0, 1)$ parts are both nonzero, so it is not a $(p, q)$-form for any $(p, q)$.
A holomorphic change of coordinates $w = F(z)$ gives $dw_k = \sum_j \frac{\partial F_k}{\partial z_j} dz_j$ with no $d\bar z_j$ terms, so the type $(p, q)$ is preserved; thus $(p, q)$-forms are defined on complex manifolds.
The space of $(p, q)$-forms on $U$ is often written $\mathcal{E}^{p, q}(U)$.
</document>

<document source="https://tamnd.github.io/math2/node/P3PZ73SMPZ.md" tag="P3PZ73SMPZ" kind="counterexample">
# Moments of the log-normal distribution

## Counterexample

Let $f$ be the density of the standard [[log-normal distribution|4W40NHWPQT]]:
$$f(x) = \frac{1}{x \sqrt{2\pi}} \exp\left(-\frac{(\ln x)^2}{2}\right) \quad (x > 0), \qquad f(x) = 0 \quad (x \le 0).$$
Its [[moments|01XAHPADBA]] are $\int_0^\infty x^n f(x)\, dx = e^{n^2/2}$ for every integer $n \ge 0$.
For a real number $a$ with $-1 \le a \le 1$, put
$$f_a(x) = f(x)\bigl(1 + a \sin(2\pi \ln x)\bigr) \quad (x > 0), \qquad f_a(x) = 0 \quad (x \le 0).$$
Each $f_a$ is a [[probability density|04YG690N5V]], because $f_a \ge 0$ and it has the same total mass as $f$ by the computation below.
Check that $f_a$ has the same moments as $f$.
It suffices to show $\int_0^\infty x^n f(x) \sin(2\pi \ln x)\, dx = 0$ for every integer $n \ge 0$.
The substitution $x = e^y$ turns this integral into $\int_\R e^{ny} \varphi(y) \sin(2\pi y)\, dy$, where $\varphi(y) = e^{-y^2/2}/\sqrt{2\pi}$ is the [[standard normal|05AQE0QR0D]] density.
Since $e^{ny} \varphi(y) = e^{n^2/2} \varphi(y - n)$, the substitution $y = u + n$ gives
$$e^{n^2/2} \int_\R \varphi(u) \sin(2\pi u + 2\pi n)\, du = e^{n^2/2} \int_\R \varphi(u) \sin(2\pi u)\, du = 0,$$
because $n$ is an integer and the last integrand is odd.
So the densities $f_a$, $-1 \le a \le 1$, are pairwise different and all have the moments $e^{n^2/2}$.
Hence the moment problem for this moment sequence is not [[determinate|H0GY2J14CA]]: it is [[indeterminate|GCTN4BZ1ET]].

## Remarks

The log-normal law has all moments finite, yet its moments do not determine it.
The moments grow like $e^{n^2/2}$, so fast that [[Carleman's condition|RVZ831CNYW]] fails.
The normal distribution, whose moments grow much more slowly, is determined by its moments.

## Explanations

### L1 human

One might hope that knowing all the averages $E[X], E[X^2], E[X^3], \dots$ pins down a random quantity.
The log-normal law shows that this can fail.
Adding a small wiggle $a \sin(2\pi \ln x)$ to its density changes the shape, but every moment stays exactly the same.
The wiggles cancel out in each moment because, after taking logarithms, they become an odd function against a bell curve.
</document>

<document source="https://tamnd.github.io/math2/node/P3T104G564.md" tag="P3T104G564" kind="concept">
# Band-limited function

## Summary

A band-limited function is a function on $\R$ whose [[Fourier transform|DPVM34ADZV]] vanishes outside a bounded interval of frequencies $[-B, B]$.
Such functions are very rigid: by the [[Paley–Wiener theorem|THY12GPZ8C]] they extend to entire functions, so they are determined by their values on any interval.
They can be recovered from samples taken at a rate of $2B$ per unit time.
This is the content of the sampling theorem.
See the [[definition|B4TYSQQC38]].

## Explanations

### L1 human

A band-limited signal contains no frequencies above some limit $B$.
Such a signal cannot change too fast, so it is pinned down by its values at evenly spaced times, $2B$ of them per second.
The function $\sin(\pi x)/(\pi x)$ is the basic example: its spectrum is a box on $[-\frac12, \frac12]$.
Audio recordings are filtered to be nearly band-limited before they are sampled.
</document>

<document source="https://tamnd.github.io/math2/node/P3VXR8FW17.md" tag="P3VXR8FW17" kind="concept">
# Strict hyperbolicity

## Summary

Strict hyperbolicity is a condition on a system of evolution equations that is stronger than [[hyperbolicity|92C3S03AQZ]].
For a first-order system $u_t + \sum_j A_j u_{x_j} = 0$ it asks that, in each direction, the matrix $\sum_j \xi_j A_j$ has real and distinct eigenvalues.
The eigenvalues are the speeds of the different waves, and distinct speeds keep the waves apart.
Strictly hyperbolic systems have a well-posed Cauchy problem, and they are the setting of the classical theory of [[hyperbolic conservation laws|XN081NCPF5]].

## Explanations

### L1 human

A hyperbolic system carries several kinds of waves, each with its own speed.
Strict hyperbolicity says that these speeds are real numbers and that no two of them are equal.
For example, the one-dimensional wave equation, written as a system, has the two speeds $c$ and $-c$, which are different, so it is strictly hyperbolic.
When two speeds coincide, waves can interact in delicate ways and some estimates break down.
</document>

<document source="https://tamnd.github.io/math2/node/P3YM2DDGX9.md" tag="P3YM2DDGX9" kind="definition">
# Autoregressive conditional heteroskedasticity

## Definition

Let $q \ge 1$ be an integer.
A real [[time series|EA3MF4VTEH]] $(\varepsilon_t)_{t \in \Z}$ is an *ARCH($q$) process* if it satisfies the following conditions.

1. $\varepsilon_t = \sigma_t z_t$ for all $t$, where $(z_t)_{t \in \Z}$ are [[independent and identically distributed|00MRN0FARV]] random variables with mean $0$ and variance $1$.
2. $\sigma_t^2 = \alpha_0 + \sum_{i=1}^{q} \alpha_i \varepsilon_{t-i}^2$ for all $t$, where $\alpha_0 > 0$ and $\alpha_1, \dots, \alpha_q \ge 0$ are constants and $\sigma_t \ge 0$.
3. $z_t$ is independent of $(\varepsilon_s)_{s < t}$ for all $t$.

Then $\sigma_t^2$ is the [[conditional variance|0VGPX25NXQ]] of $\varepsilon_t$ given the past $(\varepsilon_s)_{s < t}$.

## Remarks

The GARCH($p$, $q$) model replaces clause 2 by $\sigma_t^2 = \alpha_0 + \sum_{i=1}^{q} \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^{p} \beta_j \sigma_{t-j}^2$ with $\beta_j \ge 0$.

Example: take $q = 1$, $\alpha_0 = 1$ and $\alpha_1 = 1/2$.
If the process is weakly stationary with variance $v$, then taking expectations in clause 2 gives $v = 1 + v/2$, so $v = 2$.
In general a weakly stationary ARCH($q$) process exists if and only if $\alpha_1 + \cdots + \alpha_q < 1$, and then its variance is $\alpha_0 / (1 - \alpha_1 - \cdots - \alpha_q)$.

Non-example: the rule $\sigma_t^2 = 1 - \frac12 \varepsilon_{t-1}^2$ fails clause 2 because $\alpha_1 < 0$.
It gives a negative "variance" as soon as $\abs{\varepsilon_{t-1}} > \sqrt{2}$.

An ARCH process is uncorrelated in time, but it is not independent: the squares $\varepsilon_t^2$ follow an autoregressive pattern.
</document>

<document source="https://tamnd.github.io/math2/node/P43WKB4372.md" tag="P43WKB4372" kind="statement">
# Lie's theorems on transformation groups

## Statement

Let $M$ be a smooth manifold.
Let $G$ be a [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{g}$ and [[exponential map|H3QX1ZFC8K]] $\exp$.
Let $G$ have a smooth left [[action|96B4E2GPKJ]] on $M$.
The *fundamental vector field* of $X \in \mathfrak{g}$ is the [[vector field|0299VCW1TH]] $X_M$ on $M$ with $X_M(p) = \frac{d}{dt}\big|_{t=0} \exp(tX) \cdot p$.

1. The map $X \mapsto X_M$ is linear and satisfies $[X_M, Y_M] = -[X, Y]_M$, with the [[Lie bracket of vector fields|68FGQEBD6J]] on the left.
2. Each finite-dimensional Lie algebra $L$ of complete vector fields on $M$ is the set of fundamental vector fields of some simply connected Lie group action.
3. Every finite-dimensional real Lie algebra is the Lie algebra of a Lie group.

## Remarks

By part 1, the fundamental vector fields form a Lie algebra of vector fields of dimension at most $\dim G$.
Part 1 follows by differentiating the identity $\exp(sX)\exp(tY)\exp(-sX) = \exp(t\operatorname{Ad}(\exp sX)Y)$ along the action; the minus sign comes from using a left action together with the bracket of left-invariant vector fields.
Lie's original theorems were local: they concerned local groups of transformations and the vector fields that generate them.
The global form of part 2 is due to Palais.
Completeness of the vector fields is needed, since an incomplete field such as $x^2 \frac{d}{dx}$ on $\R$ generates no global action of $\R$.
Part 3 is [[Lie's third theorem|P6QNEK44RX]], which in the modern form is proved by Ado's theorem or by Cartan's method.
Example: the rotation action of $SO(3)$ on $\R^3$ has as fundamental vector fields the three infinitesimal rotations $y\partial_z - z\partial_y$, $z\partial_x - x\partial_z$, $x\partial_y - y\partial_x$ and their linear combinations, a Lie algebra isomorphic to $\mathfrak{so}(3)$.

## Explanations

### L1 human

A group of transformations of a space, such as the rotations of space, can be studied through its infinitesimal transformations, the velocity fields of motions that start at the identity.
Lie's theorems say that these velocity fields form a finite-dimensional space closed under the bracket of vector fields.
Conversely, such a space of vector fields generates a group of transformations.
So the study of continuous symmetry reduces to the study of finitely many vector fields and their brackets.

### L2 human

These results are the origin of Lie theory: Lie used them to study symmetries of differential equations.
Part 1 turns a group action into a Lie algebra homomorphism into vector fields, up to a sign, which is the basic tool for computing symmetries.
Part 2 integrates infinitesimal symmetries to actual group actions; without completeness one only gets a local action.
Part 3 shows that no Lie algebra is missed, so the classification of finite-dimensional Lie algebras gives all the local types of transformation groups.
</document>

<document source="https://tamnd.github.io/math2/node/P4B2Z6GS0R.md" tag="P4B2Z6GS0R" kind="concept">
# Spherical mean transform

## Summary

The spherical mean transform sends a function $f$ on $\R^n$ to a function $M f(x, r)$.
Its value is the average of $f$ over the [[sphere|P9CM4GQ2T0]] of center $x$ and radius $r$.
A continuous function is [[harmonic|0DEB0JMN0W]] exactly when all its spherical means equal its value at the center.
For a function of class $C^2$ the means satisfy the Darboux equation.
This reduces the [[wave equation|5EHD2XBQDG]] in $\R^n$ to an equation in one space variable.
Recovering $f$ from its means over spheres with centers on a fixed surface is the mathematical core of thermoacoustic and photoacoustic tomography.

## Explanations

### L1 human

Stand at a point $x$, draw a sphere of radius $r$ around it, and average the values of $f$ on that sphere.
Doing this for every center and every radius gives the spherical mean transform.
For example, for $f(x) = \abs{x}^2$ the mean over the sphere of center $x$ and radius $r$ is $\abs{x}^2 + r^2$.
For radius $0$ the mean is just the value $f(x)$, and for a harmonic function it stays the same for all radii.
</document>

<document source="https://tamnd.github.io/math2/node/P4DFMJTD09.md" tag="P4DFMJTD09" kind="definition">
# Empirical risk minimization

## Definition

Let $\mathcal X$ and $\mathcal Y$ be sets, let $L \colon \mathcal Y \times \mathcal Y \to [0, \infty)$ be a [[loss function|3BTG6VVCGZ]], and let $\mathcal F$ be a class of functions $f \colon \mathcal X \to \mathcal Y$.
Let $(x_1, y_1), \dots, (x_n, y_n) \in \mathcal X \times \mathcal Y$ be data.
The *empirical risk* of $f \in \mathcal F$ is
$$R_n(f) = \frac1n \sum_{i=1}^{n} L(f(x_i), y_i).$$
*Empirical risk minimization* over $\mathcal F$ selects a function $\hat f \in \mathcal F$ with $R_n(\hat f) = \inf_{f \in \mathcal F} R_n(f)$, when such a function exists.

## Remarks

When the data are a random sample from a distribution $P$, $R_n(f)$ is an unbiased estimate of the [[risk|N26TSVRH2S]] $R(f) = E[L(f(X), Y)]$ for each fixed $f$.
For the selected $\hat f$, however, $R_n(\hat f)$ tends to underestimate $R(\hat f)$, because $\hat f$ was chosen to fit the same data.

Example: with $\mathcal X = \R$, $\mathcal Y = \R$, the squared loss $L(a, y) = (a - y)^2$ and $\mathcal F = \set{x \mapsto a + bx : a, b \in \R}$, empirical risk minimization is fitting a line by [[least squares|0BYTQ4X1HC]].

Non-example: ridge regression minimises $R_n(f) + \lambda \norm{\beta}^2$ with $\lambda > 0$ over linear functions $f(x) = \beta^T x$.
Its minimiser is in general not a minimiser of $R_n$ alone, so it is regularized empirical risk minimization, not empirical risk minimization.
</document>

<document source="https://tamnd.github.io/math2/node/P4DZM975BV.md" tag="P4DZM975BV" kind="concept">
# Vector bundle

## Summary

In algebraic geometry, a vector bundle of rank $n$ on a scheme $X$ is usually identified with a locally free sheaf of rank $n$.
This is a sheaf of modules locally isomorphic to $\mathcal{O}_X^{\oplus n}$.
Such a sheaf is the sheaf of sections of a geometric vector bundle.
This is a scheme over $X$ whose fibres are affine spaces with a linear structure.
Vector bundles of rank one are the invertible sheaves, and tangent and cotangent bundles of smooth varieties are the main examples of higher rank.

## Explanations

### L1 human

A vector bundle attaches a vector space of fixed dimension to each point of a space, varying continuously.
Locally it is a product of the space with a vector space, but globally it may twist.
In algebraic geometry one usually works with the sheaf of its sections, which is locally a direct sum of copies of the sheaf of functions.
The tangent bundle of a smooth variety is an example: at each point, it gives the tangent space.
</document>

<document source="https://tamnd.github.io/math2/node/P4EKQS312D.md" tag="P4EKQS312D" kind="concept">
# Hankel operator

## Summary

A Hankel operator is a bounded operator on $\ell^2$ whose matrix $(a_{j+k})_{j,k \ge 0}$ is constant along each anti-diagonal, an infinite [[Hankel matrix|PQ1D8FQB3E]].
Up to a unitary identification of the spaces, the same operators arise in a function form.
They are the maps $f \mapsto (I - P)(\varphi f)$ from the [[Hardy space|PNNNFJVF6W]] $H^2$ to $L^2 \ominus H^2$ on the circle.
Here $\varphi$ is a bounded symbol and $P$ is the projection onto $H^2$.
Hankel operators are close relatives of [[Toeplitz operators|GQ0X3R3WM2]] and appear in rational approximation, control theory and prediction theory.
Nehari's theorem describes the bounded ones, and Kronecker's theorem the ones of finite rank.

## Explanations

### L1 human

A Hankel matrix has the same entry along every line that runs from the lower left to the upper right.
So the whole matrix is built from one sequence $a_0, a_1, a_2, \ldots$, with $a_{j+k}$ in row $j$ and column $k$.
The Hilbert matrix, with entries $1/(j + k + 1)$, is the classical example.
The operator theory question is which sequences give a bounded operator on square-summable sequences.
</document>

<document source="https://tamnd.github.io/math2/node/P4J5JZDW0B.md" tag="P4J5JZDW0B" kind="concept">
# Tensor algebra

## Summary

The tensor algebra of a vector space $V$ is the direct sum of all [[tensor powers|0BF2DE2SYK]] of $V$, with multiplication given by the tensor product.
It is the most general associative algebra generated by $V$.
Every linear map from $V$ to an associative unital algebra extends uniquely to a unital algebra homomorphism.
The [[symmetric algebra|PBGF9TXMB7]], the [[exterior algebra|0MDMZ965VE]] and the [[Clifford algebra|A38B1QFK20]] are quotients of it.

## Explanations

### L1 human

Take the elements of $V$ as letters and form words by writing letters side by side with the tensor product.
The tensor algebra contains all linear combinations of such words of every length, including the empty word $1$.
Multiplication puts one word after another, and the letters never commute.
If $V$ has a basis $x, y$, then $x \otimes y$ and $y \otimes x$ are different elements.
If $V$ has dimension $1$ with basis $x$, the tensor algebra is just the polynomial ring $F[x]$.
</document>

<document source="https://tamnd.github.io/math2/node/P4NVFFR2CH.md" tag="P4NVFFR2CH" kind="definition">
# Surface

## Definition

A *surface* is a topological space $S$ such that:

1. $S$ is [[Hausdorff|0S1FJ8VEX9]];
2. $S$ is second countable, that is, its topology has a countable base;
3. every point of $S$ has an open neighbourhood [[homeomorphic|0QT00PB5G2]] to an open subset of $\R^2$.

Equivalently, a surface is a [[topological manifold|QQQ9B1DK4P]] of dimension $2$.
A *surface with boundary* is defined in the same way, with open subsets of the closed half-plane $\set{(x, y) \in \R^2 : y \ge 0}$ in clause 3.
A *closed surface* is a compact connected surface without boundary.

## Remarks

Example: the unit sphere $S^2 \subset \R^3$ is a surface.
It is Hausdorff and second countable as a subspace of $\R^3$.
Stereographic projections from the north pole and from the south pole are homeomorphisms of two open subsets of $S^2$ onto $\R^2$, and these subsets cover $S^2$.

Non-example: let $X$ be the union of the planes $z = 0$ and $y = 0$ in $\R^3$.
Clauses 1 and 2 hold, but clause 3 fails at points of the $x$-axis.
A small sphere around a point $P$ of this axis meets $X$ in two great circles crossing at two points, a graph $G$ with $2$ vertices and $4$ edges.
So $P$ has a neighbourhood that is a cone on $G$, and the local homology group $H_2(X, X \setminus \set{P})$ is $\tilde H_1(G) \cong \Z^3$.
At a point of an open subset of $\R^2$ this group is $\tilde H_1(S^1) \cong \Z$, and local homology is preserved by homeomorphisms of neighbourhoods.

## Explanations

### L1 human

The three clauses say: points can be separated, the space is not too big, and near each point the space is a patch of the plane.
The last clause is the heart of the matter.
The first two rule out pathological spaces, such as the plane with two origins, which fails clause 1.
</document>

<document source="https://tamnd.github.io/math2/node/P4RF8A1562.md" tag="P4RF8A1562" kind="definition">
# Burgers' equation

## Definition

Let $\nu \ge 0$ be a constant, the *viscosity*.
*Burgers' equation* with viscosity $\nu$ is the equation
$$\partial_t u + u\, \partial_x u = \nu\, \partial_x^2 u$$
for an unknown function $u \colon \R \times (0, \infty) \to \R$, where $\partial_t$ and $\partial_x$ are [[partial derivatives|09EGC6F8PT]].
For $\nu > 0$ it is the *viscous* Burgers' equation.
For $\nu = 0$ it is the *inviscid* Burgers' equation, usually written in the conservation form
$$\partial_t u + \partial_x \left( \tfrac12 u^2 \right) = 0.$$

## Remarks

The inviscid equation is the scalar [[conservation law|8RA7AMWSAF]] with flux $F(u) = u^2/2$, which is uniformly convex.
For smooth $u$ the two forms agree, but for discontinuous weak solutions only the conservation form is meaningful, and it fixes the shock speed $(u_l + u_r)/2$.
By the method of characteristics, a smooth solution with initial value $g$ satisfies $u = g(x - ut)$; if $g$ is somewhere decreasing, the solution ceases to be smooth at time $1/\max(-g')$.

Example: $u(x, t) = x/(t + 1)$ solves the inviscid equation, and also the viscous one for every $\nu$.
Check: $\partial_t u = -x/(t + 1)^2$, $u\, \partial_x u = x/(t + 1)^2$, and $\partial_x^2 u = 0$.

Non-example: $u(x, t) = e^{-t} \sin x$ solves the linear heat equation $u_t = u_{xx}$ but not Burgers' equation with $\nu = 1$.
Indeed for it $\partial_t u - \partial_x^2 u = 0$, while $u\, \partial_x u = e^{-2t} \sin x \cos x$ is not identically zero.
</document>

<document source="https://tamnd.github.io/math2/node/P4X779XNEA.md" tag="P4X779XNEA" kind="definition">
# Hilbert manifold

## Definition

Let $k \ge 1$ be an integer or $\infty$.
A *Hilbert manifold of class $C^k$* is a [[Banach manifold|VB95VVVDM8]] of class $C^k$ with an atlas of charts with values in real [[Hilbert spaces|PYYXEXMARY]].
It is *modelled on* the Hilbert space $H$ if all charts can be taken with values in $H$.
Many authors also require the underlying space to be Hausdorff and second countable, or the model $H$ to be separable.

## Remarks

Example: let $H$ be a real Hilbert space and $S = \set{x \in H : \norm{x} = 1}$ its unit sphere.
For $p \in S$, the map $x \mapsto x - \ip{x}{p} p$ restricted to $\set{x \in S : \ip{x}{p} > 0}$ is a chart onto the open unit ball of the closed hyperplane $p^\perp$, with inverse $y \mapsto y + \sqrt{1 - \norm{y}^2}\, p$.
Both maps are smooth, $p^\perp$ is a Hilbert space, and these charts cover $S$, so $S$ is a smooth Hilbert manifold.

Non-example: the [[Banach space|0BDTTM24DP]] $\ell^1$ with its identity chart is a Banach manifold, but no compatible chart takes values in a Hilbert space.
A compatible chart $\varphi$ into a Hilbert space $H$ would have a [[derivative|06EH97V2A7]] at a point that is a linear isomorphism of Banach spaces from $\ell^1$ onto $H$.
But $\ell^1$ is not isomorphic to a Hilbert space.


Every finite-dimensional smooth manifold is a Hilbert manifold modelled on $\R^n$.
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<document source="https://tamnd.github.io/math2/node/P4YXE08ZXF.md" tag="P4YXE08ZXF" kind="concept">
# Kervaire invariant

## Summary

The Kervaire invariant is an invariant in $\Z/2$ of a closed framed smooth manifold of dimension $4k + 2$.
It is the Arf invariant of a quadratic refinement, built from the framing, of the mod $2$ intersection form on the middle cohomology.
Michel Kervaire used it in 1960 to give a closed topological manifold of dimension $10$ with no smooth structure.
Through the Pontryagin–Thom construction it becomes a homomorphism on the [[stable homotopy groups of spheres|HGZAYPQBXW]].
The question of where it is nonzero is the Kervaire invariant problem.

## Explanations

### L1 human

A framing of a manifold is a consistent choice of a basis of directions at every point.
On a framed manifold of dimension $4k + 2$, the framing lets one assign to each middle-dimensional mod $2$ class a value $0$ or $1$.
These values form a quadratic form over the field with two elements, and such a form has a single invariant, its Arf invariant.
That invariant is the Kervaire invariant.
For example, the torus $S^1 \times S^1$ with the framing that comes from its structure as a Lie group has Kervaire invariant $1$.
</document>

<document source="https://tamnd.github.io/math2/node/P52TCSHP7R.md" tag="P52TCSHP7R" kind="definition">
# Zassenhaus neighbourhood

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] with identity $e$.
A *Zassenhaus neighbourhood* of $G$ is an open [[neighbourhood|0ZCY0Z80B6]] $U$ of $e$ with the following property.
For every [[discrete subgroup|5M7D1BSHVZ]] $\Gamma$ of $G$, the subgroup generated by $\Gamma \cap U$ is [[nilpotent|0Q7MMQHC1P]].

## Remarks

Every Lie group has a Zassenhaus neighbourhood, by a theorem of Zassenhaus.
One can moreover choose $U$ so that $\Gamma \cap U$ lies in a connected nilpotent Lie subgroup of $G$.

Example: in $G = \R^n$, every open neighbourhood of $0$ is a Zassenhaus neighbourhood.
Indeed every subgroup of $\R^n$ is abelian, hence nilpotent.

Non-example: $U = \mathrm{SO}(3)$ in $G = \mathrm{SO}(3)$.
The rotation group of the icosahedron is a finite, hence discrete, subgroup isomorphic to the alternating group $A_5$.
It lies in $U$ and is not nilpotent, since $A_5$ is simple and non-abelian.
</document>

<document source="https://tamnd.github.io/math2/node/P546TSFBK2.md" tag="P546TSFBK2" kind="concept">
# Mahalanobis distance

## Summary

The Mahalanobis distance measures how far a point lies from the centre of a multivariate distribution.
Its units are adapted to the spread and the correlations of the distribution.
It is the [[Euclidean distance|0NPPGHQNXQ]] after the data are transformed so that the [[covariance matrix|0V7G6WMFK0]] becomes the identity.
It is used to detect outliers and in [[linear discriminant analysis|ZATS7NV9WD]].
The density of a [[multivariate normal distribution|0XN3K7RSZY]] is constant on the sets of constant Mahalanobis distance.

## Explanations

### L1 human

A point that is two units from the mean is unusual in a direction where the data hardly vary, but ordinary in a direction where they vary a lot.
The Mahalanobis distance takes this into account by dividing each direction by its own spread.
Its level sets are ellipses around the mean instead of circles.
Let the first coordinate have variance $4$ and the second variance $1$, with no correlation.
Then the points $(2, 0)$ and $(0, 1)$ are both at Mahalanobis distance $1$ from the origin.
</document>

<document source="https://tamnd.github.io/math2/node/P5AWRD2793.md" tag="P5AWRD2793" kind="concept">
# Singular function

## Summary

A singular function is a continuous, non-constant function on an interval whose derivative exists and equals zero almost everywhere.
Such a function cannot be recovered by integrating its derivative.
So it shows that the [[fundamental theorem of calculus|RX7JSHMP8Z]] needs [[absolute continuity|ZT6F4C2YHK]] in the Lebesgue setting.
The [[Cantor function|NPBYYK63FE]] and the [[Minkowski question-mark function|4D9HP5TS6W]] are the standard examples.
Every function of [[bounded variation|A8251BHCMP]] is a sum of three parts.
They are an absolutely continuous function, a jump function and a continuous function with derivative $0$ almost everywhere.

## Explanations

### L1 human

Picture a staircase with infinitely many tiny steps, where the flat parts of the stairs fill almost all of the interval.
The function climbs from one value to another, yet at almost every point its slope is zero.
All the growth happens on a set of length zero.
The Cantor function is such a staircase: it is flat on every interval removed in the construction of the Cantor set, and it still climbs from $0$ to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/P5F7ZGGRW8.md" tag="P5F7ZGGRW8" kind="definition">
# Mellin convolution

## Definition

Let $f, g \colon (0, \infty) \to \C$ be measurable.
Let $x > 0$ be such that $y \mapsto f(x/y) g(y) / y$ is [[integrable|X7GTF3D98F]] on $(0, \infty)$.
The *Mellin convolution* of $f$ and $g$ at $x$ is the [[Lebesgue integral|DWJF7ABPR1]]
$$(f \star g)(x) = \int_0^\infty f\left(\frac{x}{y}\right) g(y) \, \frac{dy}{y}.$$

## Remarks

The substitution $y \mapsto x/y$ shows that $f \star g = g \star f$.
Assume that $f$ and $g$ are both Mellin convergent at $s$.
Then $f \star g$ is defined at almost every $x > 0$, it is Mellin convergent at $s$, and $\mathcal{M}(f \star g)(s) = \mathcal{M}f(s) \, \mathcal{M}g(s)$.
The proof is Fubini's theorem and the substitution $x = yu$, which splits the double integral.
Example: $f = g = \mathbf{1}_{(0, 1]}$.
The integrand $\mathbf{1}_{(0,1]}(x/y) \mathbf{1}_{(0,1]}(y) / y$ is $1/y$ for $x \le y \le 1$ and $0$ otherwise, which is integrable for each $x > 0$.
So $(f \star g)(x) = \int_x^1 dy/y = -\log x$ for $0 < x \le 1$ and $0$ for $x > 1$.
As a check, $\int_0^1 x^{s-1} (-\log x) \, dx = 1/s^2 = (1/s)^2$ for $\operatorname{Re} s > 0$.
Non-example: $f = g = 1$ on $(0, \infty)$; the integrand $1/y$ is not integrable on $(0, \infty)$, so $(f \star g)(x)$ is defined at no $x$.
Some authors use the other normalization $\int_0^\infty f(xy) g(y) \, dy$, whose Mellin transform is $\mathcal{M}f(s) \mathcal{M}g(1 - s)$.
</document>

<document source="https://tamnd.github.io/math2/node/P5HCD49REV.md" tag="P5HCD49REV" kind="definition">
# Moufang plane

## Definition

A [[projective plane|DANFNN8C4Y]] is a *Moufang plane* if it satisfies the little Desargues property for every line $\ell$ and every point $O$ on $\ell$:

1. let $ABC$ and $A'B'C'$ be triangles with no vertex on $\ell$, with $A \ne A'$, $B \ne B'$, $C \ne C'$, and with $AA'$, $BB'$, $CC'$ through $O$;
2. suppose the points $AB \cap A'B'$ and $AC \cap A'C'$ lie on $\ell$;
3. then the point $BC \cap B'C'$ also lies on $\ell$.

## Remarks

So three [[collinear|0AFNNTRTJE]] intersection points are required only when the centre $O$ lies on the axis $\ell$.
Equivalently, every line of the plane is a translation line, that is, the group of elations with axis $\ell$ acts transitively on the points off $\ell$.
A projective plane is a Moufang plane exactly when it can be coordinatized by an alternative division ring.

Example: $\mathbb{P}^2(k)$ over a field $k$ is a Moufang plane.
Check: Desargues's theorem holds in $\mathbb{P}^2(k)$ for all triangles in perspective, in particular when the centre lies on the axis.

By the Artin–Zorn theorem every finite alternative division ring is a field, so every finite Moufang plane is Desarguesian.

Non-example: the Hall plane of order $9$ is a finite [[non-Desarguesian plane|HYE5FT4M34]].
So by the previous sentence it is not a Moufang plane: the little Desargues property fails in it for some line $\ell$ and point $O$.
</document>

<document source="https://tamnd.github.io/math2/node/P5MA4CDRNA.md" tag="P5MA4CDRNA" kind="concept">
# Complete atomic Boolean algebra

## Summary

A complete atomic Boolean algebra is a [[complete Boolean algebra|75BPV95AAS]] in which every nonzero element lies above an [[atom|TTVCTW6TNC]].
Every power set algebra is of this kind.
Conversely, every complete atomic Boolean algebra is isomorphic to the power set of its set of atoms.
So these algebras are the same as sets, seen algebraically.
With homomorphisms that preserve all suprema, their category is dual to the category of sets.

## Explanations

### L1 human

An atom is a smallest nonzero element, like a single point among the subsets of a set.
In a complete atomic Boolean algebra, every element is built from the atoms below it, and every collection of atoms builds an element.
So the algebra is just the family of all subsets of its atoms.
For example, the subsets of $\set{1, 2, 3}$ form such an algebra with the three atoms $\set{1}$, $\set{2}$, $\set{3}$.
</document>

<document source="https://tamnd.github.io/math2/node/P5N2ND3VCG.md" tag="P5N2ND3VCG" kind="concept">
# Turing degree

## Summary

The Turing degree of a set of natural numbers is the class of all sets that are Turing equivalent to it.
Two sets have the same degree when each is computable from the other, so the degree measures how uncomputable a set is.
The degrees are partially ordered by [[Turing reducibility|PA4PYQ1PZ0]], and the study of this order is a central part of computability theory.

## Explanations

### L1 human

Group sets of numbers by how hard they are to compute, where two sets are equally hard if each can be computed with the help of the other.
Each group is a Turing degree.
The lowest degree, written $\mathbf{0}$, contains all computable sets, such as the even numbers or the primes.
The set of halting programs lies in a higher degree, written $\mathbf{0}'$.
The degrees form a large and intricate ordering, with uncountably many degrees and many incomparable pairs.
</document>

<document source="https://tamnd.github.io/math2/node/P64V0HTE78.md" tag="P64V0HTE78" kind="concept">
# Queueing theory

## Summary

Queueing theory is the mathematical study of waiting lines.
Customers arrive at random times, wait, are served by one or more servers, and leave.
The theory computes the laws of the queue length and of the waiting time, and it asks when the queue stays stable.
It is used to design telephone networks, computer systems, call centres and traffic systems.

## Explanations

### L1 human

Think of a post office with one clerk.
People arrive at random and each needs a random amount of time at the desk.
If people arrive faster than the clerk can serve them, the line grows without bound.
If they arrive more slowly, the line goes up and down but stays of moderate size, and one can compute its average length.
The [[M/M/1 queue|GTFRPF5J2K]] is the simplest model of this kind, and [[Little's law|JN2YME6F4A]] links the average length with the average wait.
</document>

<document source="https://tamnd.github.io/math2/node/P65JDBW4FH.md" tag="P65JDBW4FH" kind="definition">
# Topological group

## Definition

A *topological group* is a [[group|G5S1M9]] $G$ together with a [[topology|0CQHFPGJ45]] on $G$ such that:

1. the multiplication $G \times G \to G$, $(x, y) \mapsto xy$, is a [[continuous multiplication|ZZK2GM1V15]], where $G \times G$ carries the product topology;
2. the inversion $G \to G$, $x \mapsto x^{-1}$, is [[continuous|0VGFHTM75Y]].

## Remarks

Example: $(\R, +)$ with the usual topology.
Addition $(x, y) \mapsto x + y$ is continuous on $\R^2$, and $x \mapsto -x$ is continuous.
Non-example: $(\R, +)$ with the lower limit topology, generated by the intervals $[a, b)$.
Addition is still continuous, but inversion fails clause 2: the preimage of the open set $[0, 1)$ under $x \mapsto -x$ is $(-1, 0]$, which is not open.
The two clauses together are equivalent to the single condition that $(x, y) \mapsto x y^{-1}$ is continuous from $G \times G$ to $G$.
Every group with the discrete topology, and every group with the indiscrete topology, is a topological group.
Many authors also require the topology to be Hausdorff.
</document>

<document source="https://tamnd.github.io/math2/node/P68G28E4AD.md" tag="P68G28E4AD" kind="concept">
# Slice knot

## Summary

A slice knot is a [[knot|QYEAZTYGPT]] in the $3$-sphere that bounds a smoothly embedded disk in the $4$-ball.
The notion links knot theory with the topology of [[4-manifolds|8H58Q6YJCH]].
Slice knots are exactly the knots concordant to the [[unknot|1TQ06R4Q66]], and they form the zero element of the knot concordance group.
There is also a topological version, with locally flat disks, and the two versions differ.

## Explanations

### L1 human

View ordinary space as the boundary of a four-dimensional ball, as a circle is the boundary of a disk.
A knot is slice if, with the help of the extra dimension, it bounds a disk without self-intersections.
In three dimensions only the unknot bounds such a disk, but in four dimensions many knots do.
The sum of a knot and its reversed mirror image is always slice.
The [[trefoil knot|VYTYSVHN4A]] is not slice.
</document>

<document source="https://tamnd.github.io/math2/node/P6AJ840RV4.md" tag="P6AJ840RV4" kind="definition">
# Chain complex

## Definition

Let $\mathcal{A}$ be an [[additive category|B2VP96NJSK]], for example the category of [[modules|0NZ8JA25WK]] over a ring.
A *chain complex* $C_\bullet$ in $\mathcal{A}$ consists of objects $C_n$ and morphisms $d_n \colon C_n \to C_{n-1}$, for $n \in \Z$, with
$$d_{n-1} \circ d_n = 0 \quad \text{for all } n \in \Z,$$
where $0$ is the [[zero morphism|YF9F9EG6M3]] $C_n \to C_{n-2}$.

## Remarks

A *cochain complex* has objects $C^n$ and differentials $d^n \colon C^n \to C^{n+1}$ with $d^{n+1} \circ d^n = 0$; setting $C_n = C^{-n}$ turns it into a chain complex.
A complex is *bounded* if only finitely many $C_n$ are nonzero.
One often writes $d$ for every $d_n$, so that the condition reads $d^2 = 0$.

Example: let $C_n = \Z/4\Z$ and let $d_n$ be multiplication by $2$ for all $n$.
Then $d_{n-1} \circ d_n$ is multiplication by $4$, which is $0$ on $\Z/4\Z$.

Non-example: let $C_2 = C_1 = C_0 = \Z$, with $d_2$ and $d_1$ the identity of $\Z$ and all other objects $0$.
Then $d_1 \circ d_2 = 1_{\Z} \ne 0$, so this is not a chain complex.
</document>

<document source="https://tamnd.github.io/math2/node/P6E9RZNRBZ.md" tag="P6E9RZNRBZ" kind="concept">
# Local solvability

## Summary

A linear differential operator $P$ is locally solvable at a point if $P u = f$ has a solution near that point for every smooth $f$.
Operators with constant coefficients are always locally solvable, but Hans Lewy found a first-order operator with polynomial coefficients that is locally solvable at no point.
This discovery started the study of necessary and sufficient conditions for local solvability.
See the [[definition|DQ0TSVK7ZT]].

## Explanations

### L1 human

For an ordinary differential equation such as $y' = f(x)$ one can always find a solution near any point, by integrating.
It was long expected that every reasonable linear partial differential equation also has local solutions.
Lewy's example shows that this is false: there is a simple first-order equation $L u = f$ in three variables, with smooth $f$, that has no solution in any open set.
Local solvability is the property that rules this out.
</document>

<document source="https://tamnd.github.io/math2/node/P6EQTZBBNT.md" tag="P6EQTZBBNT" kind="definition">
# Equicontinuous system

## Definition

Let $(X, d)$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
The system $(X, f)$ is *equicontinuous* if the family $\set{f^n : n \ge 0}$ is [[equicontinuous|0XN66MZN0Y]].
Since $X$ is compact, this means uniform equicontinuity: for every $\varepsilon > 0$ there is $\delta > 0$ with $d(f^n(x), f^n(y)) < \varepsilon$ for all $n \ge 0$ and all $x, y \in X$ with $d(x, y) < \delta$.

## Remarks

For a homeomorphism one often asks the same of the family $\set{f^n : n \in \Z}$.
Example: every [[isometry|02HPP5VCZH]] $f$ is equicontinuous with $\delta = \varepsilon$, since $d(f^n(x), f^n(y)) = d(x, y)$; for instance the rotation $x \mapsto x + \alpha \bmod 1$ of the circle.
Non-example: the doubling map $f(x) = 2x \bmod 1$ on the circle $\R / \Z$.
Given $\delta > 0$, take $x = 0$ and $y$ with $0 < y < \min(\delta, 1/4)$, and let $n$ be the first integer with $2^n y \ge 1/4$.
Then $f^n(y) = 2^n y$ lies in $[1/4, 1/2)$, at distance at least $1/4$ from $f^n(0) = 0$, so the condition fails for $\varepsilon = 1/4$.

## Explanations

### L1 human

A system is equicontinuous when nearby points stay nearby under all iterates, with one closeness threshold that works for all times.
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<document source="https://tamnd.github.io/math2/node/P6GR8ZQCK3.md" tag="P6GR8ZQCK3" kind="statement">
# Wirtinger presentation

## Statement

Let $K$ be an oriented tame [[knot|QYEAZTYGPT]] in $\R^3$, with a diagram that has $n \ge 1$ crossings.
The diagram breaks into $n$ arcs $a_1, \ldots, a_n$, each running from one undercrossing to the next.
Take a base point above the plane of the diagram.
For each arc $a_i$, let $x_i$ be the class of a loop from the base point that passes once under $a_i$, by the right-hand rule.
At a crossing, let $a_k$ be the over-arc, let $a_i$ be the incoming under-arc and let $a_j$ be the outgoing under-arc.
Let $\varepsilon = \pm 1$ be the sign of the crossing.
Write $r$ for the relation $x_j = x_k^{\varepsilon} x_i x_k^{-\varepsilon}$ at this crossing.
Then the [[knot group|AQVHM3FP0D]] of $K$ has the [[presentation|0JGEQXQ4AC]]
$$\pi_1(\R^3 \setminus K) \cong \langle x_1, \ldots, x_n \mid r_1, \ldots, r_n \rangle,$$
with one relation $r_c$ for each crossing $c$.
Any one of the relations may be omitted.

## Remarks

The proof applies the Seifert–van Kampen theorem to the complement, cut into the half-space above the diagram plane and small pieces near the crossings.
The half-space above, with tunnels under the arcs, gives a free group on the $x_i$, and each crossing adds one relation.
The sign convention for $\varepsilon$ depends on the orientation chosen for the loops $x_i$; with the opposite choice, $\varepsilon$ becomes $-\varepsilon$.
For the trefoil with arcs $a_1, a_2, a_3$, the relations read $x_3 = x_1 x_2 x_1^{-1}$, $x_1 = x_2 x_3 x_2^{-1}$ and $x_2 = x_3 x_1 x_3^{-1}$ for one orientation.
Eliminating $x_3$ and dropping the last relation gives $\langle x_1, x_2 \mid x_1 x_2 x_1 = x_2 x_1 x_2 \rangle$.
All generators are conjugate, because each relation conjugates one generator to another and the arcs are connected by the knot.
The hypothesis $n \ge 1$ is needed: a diagram of the unknot with no crossings has one arc and no crossing, and its group is $\Z$.

## Explanations

### L1 human

The knot group describes loops that wind around a knot.
The Wirtinger presentation reads this group off a picture of the knot: one generator for each strand of the picture, and one rule for each crossing.
The rule says that passing under one side of the crossing equals passing under the other side, conjugated by the strand on top.

### L2 human

It is the standard way to compute knot groups and their invariants.
Representations of the knot group into a finite group can be found from it by assigning elements to arcs.
[[Tricolorability|KNAGK0W65G]] is the case of the symmetric group on three letters.
The Alexander polynomial can be computed from it by Fox calculus.
The presentation has deficiency one, with one more generator than essential relations.
</document>

<document source="https://tamnd.github.io/math2/node/P6JYNDX5DR.md" tag="P6JYNDX5DR" kind="concept">
# Racah polynomials

## Summary

The Racah polynomials are a finite family of orthogonal polynomials, given by terminating balanced ${}_4F_3$ hypergeometric series.
They are polynomials in the quadratic variable $\lambda(x) = x(x + \gamma + \delta + 1)$, orthogonal for a weight on the points $x = 0, 1, \dots, N$.
They sit at the top of the [[Askey scheme|RR2S05C09B]] together with the [[Wilson polynomials|1CPF7SWQ3S]].
They are named after Giulio Racah because the [[6-j symbols|D17XX5N9VF]] of quantum angular momentum can be written in terms of them.

## Explanations

### L1 human

In the Askey scheme, families of orthogonal polynomials are arranged by how many parameters they have.
The Racah polynomials are the discrete family with the most parameters, four of them.
The other discrete families, such as the Hahn, Krawtchouk and Charlier polynomials, arise from them by letting parameters go to limits.
In physics they appear when three quantum angular momenta are added in two different orders.
</document>

<document source="https://tamnd.github.io/math2/node/P6KHVSTPZV.md" tag="P6KHVSTPZV" kind="concept">
# Weak topology

## Summary

Let $(X, Y)$ be a [[dual pair|R68HNB5SVR]].
The weak topology $\sigma(X, Y)$ is the coarsest topology on $X$ making every map $x \mapsto \langle x, y \rangle$ continuous.
For a topological vector space $E$, the weak topology is $\sigma(E, E')$, defined by its [[continuous linear forms|4A82S5FVRT]], and the weak-* topology on $E'$ is $\sigma(E', E)$.
Weak topologies have fewer open sets than norm topologies, so more sets are compact; this is the content of the [[Banach–Alaoglu theorem|1BXEJXQ2YT]].

## Explanations

### L1 human

In the weak topology, a sequence converges when every continuous linear measurement of it converges.
This is much easier to satisfy than convergence in norm.
In $\ell^2$, the unit vectors $e_1, e_2, e_3, \ldots$ converge weakly to $0$: for each fixed $y \in \ell^2$, the numbers $\langle e_n, y \rangle = y_n$ tend to $0$.
Yet $\norm{e_n} = 1$ for all $n$, so they do not converge to $0$ in norm.
</document>

<document source="https://tamnd.github.io/math2/node/P6MCQD23YR.md" tag="P6MCQD23YR" kind="definition">
# Algebraic K-theory of schemes

## Definition

Let $X$ be a [[scheme|5A35TYZD02]], and let $\mathcal{P}(X)$ be the category of [[vector bundles|P4DZM975BV]] on $X$, that is, locally free $\mathcal{O}_X$-modules of finite rank.
With the short exact sequences of $\mathcal{O}_X$-modules, $\mathcal{P}(X)$ is an [[exact category|TFADSC9YPZ]].
For $n \ge 0$, the *$n$-th algebraic K-group* of $X$ is $K_n(X) = \pi_{n+1}\big(B Q \mathcal{P}(X), 0\big)$.
This is a homotopy group of the classifying space of Quillen's Q-construction on $\mathcal{P}(X)$, at the base point given by the zero bundle.

## Remarks

The group $K_0(X)$ is the Grothendieck group of vector bundles.
It is the quotient of the [[free abelian group|0YMSEZE1VH]] on isomorphism classes $[E]$ of vector bundles by the following relations.
For every short exact sequence $0 \to E' \to E \to E'' \to 0$, one sets $[E] = [E'] + [E'']$.
The tensor product makes $K_0(X)$ a commutative ring.
Example: for $X = \operatorname{Spec} k$ with $k$ a field, every exact sequence of finite-dimensional vector spaces splits, and $K_0(X) \cong \Z$ through the dimension; also $K_1(X) \cong k^{\times}$.
For $X = \mathbb{P}^n_k$, $K_0(X)$ is free of rank $n + 1$ with basis $[\mathcal{O}], [\mathcal{O}(-1)], \ldots, [\mathcal{O}(-n)]$.
Non-example: using all coherent sheaves instead of vector bundles gives the groups $G_n(X)$, which differ in general.
For a regular separated Noetherian scheme the natural maps $K_n(X) \to G_n(X)$ are isomorphisms, but for a singular one they need not be.
For general schemes, Thomason and Trobaugh gave a better-behaved version using perfect complexes, which agrees with this one when $X$ has an ample family of line bundles.
</document>

<document source="https://tamnd.github.io/math2/node/P6QD6TKAGS.md" tag="P6QD6TKAGS" kind="statement">
# Grothendieck–Lefschetz trace formula

## Statement

Let $q$ be a power of a prime $p$, and let $\ell \ne p$ be another prime.
Let $X$ be a separated [[algebraic variety|HV6YBS0YDH]] over the finite field $\F_q$.
Let $\bar{X} = X \times_{\F_q} \bar{\F}_q$, where $\bar{\F}_q$ is an algebraic closure of $\F_q$.
Let $H^i_c(\bar{X}, \Q_\ell)$ be the [[ℓ-adic cohomology|E1DVHN06CR]] of $\bar{X}$ with compact support.
Let $F$ be the endomorphism of it induced by the [[Frobenius|0T3WVTD675]] morphism of $X$, which raises coordinates to the power $q$.
Then the number of [[rational points|SWJYG87QQH]] of $X$ is
$$\# X(\F_q) = \sum_{i=0}^{2 \dim X} (-1)^i \, \tr\big(F \mid H^i_c(\bar{X}, \Q_\ell)\big),$$
where $\tr$ is the [[trace|0RPDTWY2JS]] of a linear map.

## Remarks

The points of $X(\F_q)$ are the fixed points of the Frobenius morphism of $\bar{X}$, so the formula is a Lefschetz fixed point formula, with the [[Lefschetz number|ZSYCA4EGEZ]] on the right.
Grothendieck proved it in a more general form, for a constructible $\ell$-adic sheaf $\mathcal{F}$ on $X$.
Then the sum over $x \in X(\F_q)$ of the traces of Frobenius on the stalks $\mathcal{F}_{\bar{x}}$ equals the alternating sum of the traces of $F$ on $H^i_c(\bar{X}, \mathcal{F})$.
Example: for $X = \mathbb{A}^1$, the only nonzero group is $H^2_c(\bar{X}, \Q_\ell)$, of dimension $1$, on which $F$ acts by $q$.
The formula gives $\# \mathbb{A}^1(\F_q) = q$.
For $X = \mathbb{P}^1$, the groups $H^0$ and $H^2$ contribute $1$ and $q$, and the count is $q + 1$.
Compact supports are needed when $X$ is not proper: the ordinary cohomology of $\mathbb{A}^1_{\bar{\F}_q}$ is $\Q_\ell$ in degree $0$ only, which would give the wrong count $1$.
Applied to $X \times \F_{q^m}$ for all $m$, the formula expresses the zeta function of $X$ as an alternating product of characteristic polynomials of $F$, which proves its rationality.

## Explanations

### L1 human

Counting the solutions of polynomial equations over a finite field looks like pure arithmetic.
The formula says that this count can be read off from a geometric object: the cohomology of the variety, together with the action of the Frobenius map on it.
The points over $\F_q$ are exactly the points that the Frobenius map does not move, so the count is a count of fixed points, as in topology.

### L2 human

This is the main bridge between point counts over finite fields and $\ell$-adic cohomology.
It gives the rationality and the cohomological form of the zeta function.
Deligne's bounds on the eigenvalues of $F$ then give estimates such as $\abs{\# C(\F_q) - q - 1} \le 2g\sqrt{q}$ for a smooth projective curve $C$ of genus $g$.
In the sheaf version it turns sums of characters, such as Gauss and Kloosterman sums, into traces on cohomology, which is how they are bounded.
</document>

<document source="https://tamnd.github.io/math2/node/P6QNEK44RX.md" tag="P6QNEK44RX" kind="statement">
# Lie's third theorem

## Statement

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over $\R$.
Then there is a connected and simply connected real [[Lie group|7JBRGCVJRB]] $G$ whose Lie algebra is isomorphic to $\mathfrak{g}$.
Here the Lie algebra of $G$ is the algebra of [[left-invariant vector fields|SNA3DVPK0C]] on $G$.
Such a group $G$ is unique up to isomorphism of Lie groups.

## Remarks

One proof uses Ado's theorem: $\mathfrak{g}$ is isomorphic to a Lie subalgebra of $\mathfrak{gl}_n(\R)$ for some $n$.
The connected Lie subgroup of $GL_n(\R)$ with Lie algebra $\mathfrak{g}$ is then a Lie group with Lie algebra $\mathfrak{g}$, and its [[universal covering group|SWW2993CVJ]] is simply connected.
Uniqueness follows because a Lie algebra isomorphism between Lie algebras of simply connected groups integrates to a Lie group isomorphism.
Example: the Lie algebra $\R$ with zero bracket gives $G = \R$; the circle group has the same Lie algebra but is not simply connected, since its [[fundamental group|S1XK3HXHFX]] is $\Z$.
The group $G$ need not be a group of matrices: the universal cover of $SL_2(\R)$ has no faithful finite-dimensional representation.
In infinite dimension the analogue fails: there are Banach Lie algebras that are not the Lie algebra of any Banach Lie group.

## Explanations

### L1 human

A Lie algebra is a vector space with a bracket, the infinitesimal shadow of a Lie group.
The theorem says that every such shadow, in finite dimension, really comes from a group.
Among all groups with the given shadow there is a best one, the simply connected one, and it is unique.

### L2 human

With the correspondence between homomorphisms, this theorem shows that taking the Lie algebra is an equivalence between connected simply connected Lie groups and finite-dimensional real Lie algebras.
Every other connected Lie group with Lie algebra $\mathfrak{g}$ is a quotient $G / \Gamma$ of the simply connected one by a discrete central subgroup $\Gamma$.
So classifying connected Lie groups reduces to classifying Lie algebras and the discrete subgroups of the centres of the simply connected groups.
The name comes from Lie's work, where the third of his fundamental theorems was a local result about infinitesimal transformations.
</document>

<document source="https://tamnd.github.io/math2/node/P6WSG7DT4Q.md" tag="P6WSG7DT4Q" kind="statement">
# Margulis lemma

## Statement

For every $n \ge 1$ there is a constant $\varepsilon_n > 0$ with the following property.
Let $X$ be a complete, simply connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with [[sectional curvature|1X42JAA349]] between $-1$ and $0$.
Let $\Gamma$ be a discrete group of [[isometries|02HPP5VCZH]] of $X$, and let $x \in X$.
Let $\Gamma_x$ be the subgroup generated by the elements $\gamma \in \Gamma$ with $d(x, \gamma x) < \varepsilon_n$.
Then $\Gamma_x$ is virtually nilpotent: it has a [[nilpotent|0Q7MMQHC1P]] subgroup of finite [[index|0DTXBQPA2S]].

## Remarks

The proof combines a [[Zassenhaus neighbourhood|FZ798EZ6M0]] in a suitable Lie group with a compactness argument.
The curvature bounds give uniform control of the isometries that move a point only a little, which is why $\varepsilon_n$ depends only on $n$.

Example: in the upper half-plane model of $\mathbb{H}^2$, let $\Gamma$ be generated by the parabolic isometry $\gamma(z) = z + 1$.
At $x = i y$, the distance $d(x, \gamma x) = 2 \operatorname{arsinh}(1/(2y))$ tends to $0$ as $y \to \infty$.
So for large $y$ the group $\Gamma_x$ is all of $\Gamma \cong \Z$, which is abelian, as the lemma allows.

The lower curvature bound matters.
Multiplying the metric of $\mathbb{H}^2$ by $1/c$ gives curvature $-c$ and divides all displacements by $\sqrt{c}$.
For $c$ large, the generators of the fundamental group of a closed hyperbolic surface then move a point less than $\varepsilon_2$.
This group is not virtually nilpotent, so the conclusion fails without the bound $-1$.

## Explanations

### L1 human

Consider a group of symmetries of a negatively curved space, such as the hyperbolic plane, that acts without accumulating.
Look only at the symmetries that move a given point by a very small amount.
The lemma says that these symmetries generate a group that is almost commutative, namely virtually nilpotent.
The threshold for small depends only on the dimension.

### L2 human

The main application is the thick-thin decomposition of a hyperbolic manifold $M = \mathbb{H}^n/\Gamma$.
The thin part, where the injectivity radius is less than $\varepsilon_n / 2$, consists of tubes around short closed geodesics and of cusps.
This structure underlies the study of volumes of hyperbolic manifolds and of their convergence.
A Lie group form, with lattices in place of $\Gamma$, is closely related to the [[Kazhdan–Margulis theorem|C34RZMKKA4]].
</document>

<document source="https://tamnd.github.io/math2/node/P6ZP0TE07M.md" tag="P6ZP0TE07M" kind="concept">
# Adjoint representation of a Lie group

## Summary

The adjoint representation of a [[Lie group|7JBRGCVJRB]] $G$ is an action of $G$ on its own Lie algebra $\mathfrak{g}$.
It comes from differentiating conjugation at the identity.
It is a [[representation|WYGRXCG2RN]] $\operatorname{Ad} \colon G \to GL(\mathfrak{g})$ that every Lie group carries.
Its derivative is the map $X \mapsto [X, \cdot\,]$ on the Lie algebra, and its kernel, for a connected group, is the centre of $G$.
It is the starting point for root systems, the Killing form and the classification of semisimple groups.

## Explanations

### L1 human

Each element $g$ of a group acts on the group by conjugation, $h \mapsto g h g^{-1}$, and this map fixes the identity.
Its effect on tangent vectors at the identity is a linear map of the Lie algebra, called $\operatorname{Ad}(g)$.
For a group of matrices this is simply $X \mapsto g X g^{-1}$, a change of basis applied to the matrix $X$.
For a commutative group, conjugation does nothing, so every $\operatorname{Ad}(g)$ is the identity.
</document>

<document source="https://tamnd.github.io/math2/node/P70VGYNFS5.md" tag="P70VGYNFS5" kind="conjecture">
# Smooth four-dimensional Poincaré conjecture

## Statement

Let $M$ be a closed [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]] that is [[homotopy equivalent|XZG6MTSWCN]] to the [[$4$-sphere|T62369RFH6]] $S^4$.
Then $M$ is [[diffeomorphic|0KZZ70ZHXP]] to $S^4$ with its standard [[smooth structure|R2TQ82PCNB]].

## Remarks

The conjecture is open.
Its topological version is true: Michael Freedman proved in 1982 that such an $M$ is homeomorphic to $S^4$.
So the conjecture is equivalent to the statement that $S^4$ has no exotic smooth structure.
In dimensions $1$, $2$, $3$, $5$ and $6$ the smooth version holds, while in dimension $7$ it fails, by the exotic spheres of John Milnor.
Some proposed counterexamples, among them an infinite family of Cappell–Shaneson homotopy spheres, have been shown to be standard.
Gauge-theoretic invariants such as the [[Seiberg–Witten invariants|PRDDMSTR4X]] need $b_2^+ \ge 1$, so they say nothing about homotopy $4$-spheres.

## Explanations

### L1 human

A homotopy sphere is a space that cannot be told apart from a sphere by the methods of algebraic topology.
In dimension four, every such space is known to be a sphere up to continuous deformation.
The open question is whether it is always a sphere in the smooth sense, where one also keeps track of calculus.
</document>

<document source="https://tamnd.github.io/math2/node/P75CXH2Q2B.md" tag="P75CXH2Q2B" kind="statement">
# Characterizations of Galois extensions

## Statement

Let $L/K$ be a [[finite extension|0NRYPM9Y27]], and let $\Aut(L/K)$ be the group of [[automorphisms|0K1SCPBNKJ]] of $L$ that fix $K$ pointwise.
The following are equivalent:

1. $L/K$ is [[normal|0CBRFQ7DVZ]] and [[separable|0N019F6RE9]];
2. $L$ is the [[splitting field|02NZZ1J0FH]] over $K$ of a [[separable polynomial|03J8BS7FMV]] in $K[x]$;
3. $\abs{\Aut(L/K)} = [L:K]$, where $[L:K]$ is the [[degree|0GS74FGFEK]];
4. the [[fixed field|0KMJ6919WS]] of $\Aut(L/K)$ is $K$;
5. $K$ is the fixed field of some finite group of automorphisms of $L$.

An extension with these properties is a Galois extension.

## Remarks

The implication from 1 to 2 takes the product of the distinct minimal polynomials of a basis.
From 2 to 3, one counts the extensions of the identity of $K$ to automorphisms of $L$ step by step, using that the roots are distinct.
In general $\abs{\Aut(L/K)} \le [L:K]$, and [[Artin's theorem on fixed fields|X7R15C00ET]] gives $[L:L^{G}] = \abs{G}$ for $G = \Aut(L/K)$, which links 3, 4 and 5.
From 5 to 1, every element of $L$ is a root of the separable split polynomial $\prod (x - \beta)$ over its orbit.

Example: $\Q(\sqrt[3]{2})/\Q$ is not Galois.
Its only automorphism is the identity, since the other roots of $x^3 - 2$ are not real; so condition 3 fails ($1 \ne 3$) and condition 4 fails (the fixed field is $\Q(\sqrt[3]{2})$).

Finiteness is needed for condition 3 to make sense; for infinite algebraic extensions, conditions 1 and 4 remain equivalent.

## Explanations

### L1 human

A Galois extension is a field extension with as much symmetry as possible.
This theorem lists several tests for it.
One test: each irreducible polynomial with one root in $L$ has all its roots in $L$, with no repeated roots.
Another test: count the symmetries and compare with the degree.
A third test: only the elements of $K$ are fixed by all symmetries.

### L2 human

In practice one shows that an extension is Galois by exhibiting it as a splitting field of a separable polynomial, condition 2.
One uses condition 3 or 4 to compute with the Galois group, for example to show that $\abs{\Gal(L/K)} = [L:K]$.
Condition 5 is how Galois extensions arise from group actions.
These equivalences are the input for the [[fundamental theorem of Galois theory|0T1Q8D0FW8]].
</document>

<document source="https://tamnd.github.io/math2/node/P7TZ694FCM.md" tag="P7TZ694FCM" kind="statement">
# Higman embedding theorem

## Statement

Let $G$ be a [[finitely generated group|082DWWXMRJ]].
Call $G$ *recursively presented* if it has a [[presentation|0JGEQXQ4AC]] $\langle S \mid R \rangle$ with $S$ finite and $R$ [[computably enumerable|TN9G24VABN]].
Here $R$ is a set of words in the elements of $S$ and their inverses.
Then $G$ is isomorphic to a [[subgroup|0X1JXT47GA]] of a [[finitely presented group|TJKANTKASZ]] if and only if $G$ is recursively presented.

## Remarks

One direction is easy.
Let $G$ be generated by finitely many elements of a finitely presented group $P$.
A program can list all consequences of the finitely many relators of $P$.
So it can list all words in the generators of $G$ that equal $1$ in $P$.
The other direction is the hard part of Higman's 1961 proof.
It builds the finitely presented group by a long chain of [[HNN extensions|YEYERRY37K]] and amalgamated free products that encode the enumeration of $R$.

Example: the wreath product $\Z \wr \Z$ has the presentation $\langle a, t \mid [a, t^n a t^{-n}] = 1 \text{ for } n \ge 1 \rangle$.
The set of relators is computably enumerable, so $\Z \wr \Z$ embeds in a finitely presented group, although it is not finitely presented itself.

There are uncountably many $2$-generated groups up to isomorphism, but only countably many finitely presented groups, each with countably many finitely generated subgroups.
So the hypothesis cannot be dropped: some finitely generated groups do not embed in any finitely presented group.

## Explanations

### L1 human

A finitely presented group is described by finitely many generators and finitely many rules.
The theorem says exactly which finitely generated groups live inside such groups.
They are the groups whose rules can be listed one by one by a computer program, even if the list never ends.
It is a bridge between group theory and the theory of computation.

### L2 human

The theorem shows that there is a finitely presented group that contains every finitely presented group as a subgroup.
It also gives another route to finitely presented groups with unsolvable [[word problem|PBGTTAVMMT]].
Take a computably enumerable set of integers that is not computable.
Encode it into the relators of a recursively presented group with unsolvable word problem, and embed this group in a finitely presented group.
The word problem of the big group cannot be solvable, since it would solve the word problem of the subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/P7V1BQJPSS.md" tag="P7V1BQJPSS" kind="concept">
# Group law on an elliptic curve

## Summary

The points of an [[elliptic curve|427N6S6QQW]] $E$ over a field $K$, together with the point at infinity $O$, form an [[abelian group|082GHNPJ53]] $E(K)$.
The operation is the chord-and-tangent law: three points add up to $O$ exactly when they lie on a line.
This group structure is what makes elliptic curves so useful, from [[Mordell's theorem|BZHGKN6Q3K]] to elliptic curve cryptography.

## Explanations

### L1 human

To add two points $P$ and $Q$ on the curve, draw the line through them.
It meets the cubic curve in a third point; reflect that point in the $x$-axis to get $P + Q$.
To double a point, use the tangent line at that point instead.
The point at infinity plays the role of zero, and the reflection of a point is its negative.
For example, on $y^2 = x^3 + 1$ the tangent at $(2, 3)$ has slope $2$, and doubling gives $2 \cdot (2, 3) = (0, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/P7XYNN27AM.md" tag="P7XYNN27AM" kind="definition">
# Isotonic regression

## Definition

Let $y_1, \dots, y_n \in \R$ be data observed at points $x_1 < x_2 < \dots < x_n$, and let $w_1, \dots, w_n > 0$ be weights.
The *isotonic regression* of $y$ is the vector $\hat y \in \R^n$ that minimises
$$\sum_{i=1}^{n} w_i (y_i - z_i)^2$$
over all $z \in \R^n$ with $z_1 \le z_2 \le \dots \le z_n$.

## Remarks

The set of non-decreasing vectors is a closed [[convex|0K06X9QXDX]] cone, so $\hat y$ is the projection of $y$ onto it for the weighted inner product, and it exists and is unique.
It has the min-max formula $\hat y_i = \max_{s \le i} \min_{t \ge i} \mathrm{Av}(s, t)$, where $\mathrm{Av}(s, t) = \sum_{k=s}^{t} w_k y_k / \sum_{k=s}^{t} w_k$.
The pool adjacent violators algorithm computes $\hat y$ in $O(n)$ steps: it merges adjacent blocks whose averages are out of order and replaces them by their weighted average.
The same definition works for any partial order on the index set, with $z_i \le z_j$ required whenever $i$ precedes $j$.

Example: $y = (1, 3, 2)$ with equal weights.
The pair $3, 2$ violates the order, and pooling gives $\hat y = (1, 2.5, 2.5)$, which is non-decreasing.
Its squared error is $0 + 0.25 + 0.25 = 0.5$.

Non-example: $z = (1, 3, 2)$ itself has error $0$ but it is not non-decreasing, so it is not admissible; the [[monotone|0WFF6AZPBP]] constraint is what makes $\hat y$ differ from $y$.
</document>

<document source="https://tamnd.github.io/math2/node/P7YZFF8851.md" tag="P7YZFF8851" kind="definition">
# Parallel transport

## Definition

Let $M$ be a smooth manifold, let $E \to M$ be a smooth vector bundle with a [[connection|5ZYRRVE8BW]] $\nabla$, and let $\gamma : [a, b] \to M$ be a piecewise smooth curve.
Let $D_t$ be the [[covariant derivative along $\gamma$|5FZDQQHQKM]].
A section $V$ of $E$ along $\gamma$ is *parallel* if $D_t V = 0$.
For each $v \in E_{\gamma(a)}$ there is a unique parallel section $V$ along $\gamma$ with $V(a) = v$.
The *parallel transport* along $\gamma$ is the map
$$P_\gamma : E_{\gamma(a)} \to E_{\gamma(b)}, \quad v \mapsto V(b).$$

## Remarks

Existence and uniqueness hold because, in a local frame, $D_t V = 0$ is a linear system of ordinary differential equations, whose solutions exist on the whole interval.
$P_\gamma$ is a linear isomorphism, its inverse is the transport along the reversed curve, and the transport along a concatenation is the composition.
Example: for the flat connection on $T\R^n$, $D_t V = \dot V$, so parallel sections are constant and $P_\gamma$ is the identity of $\R^n$ for every curve $\gamma$.
Non-example of path independence: take the unit sphere $S^2$ with its Levi-Civita connection.
Transport around the boundary of an octant, a geodesic triangle with three right angles, rotates tangent vectors by $\pi/2$, the area of the octant.
For the Levi-Civita connection of a Riemannian metric, $P_\gamma$ is an isometry between tangent spaces.
The connection is recovered by $\nabla_{\gamma'(0)} s = \frac{d}{dt}\big|_{t = 0} P_{\gamma|_{[0, t]}}^{-1} s(\gamma(t))$.
</document>

<document source="https://tamnd.github.io/math2/node/P8055074WH.md" tag="P8055074WH" kind="definition">
# Train track

## Definition

Let $S$ be a smooth surface.
A *train track* in $S$ is a finite [[graph|00C0BR7D7H]] $\tau$ embedded in $S$, with vertices called *switches* and edges called *branches*, such that:

1. each branch is a smoothly embedded arc, and at each switch all branches that end there are tangent to one common line;
2. at each switch, at least one branch arrives from each of the two directions along this line;
3. no component of $S \setminus \tau$ is an open disc with $0$, $1$ or $2$ cusps, or an open annulus with no cusps.

Here a *cusp* of a complementary region is a corner of its boundary at a switch where two branches arriving from the same direction meet.

## Remarks

Clauses 1 and 2 let a smooth path run along $\tau$ through each switch without turning back.
Clause 3 rules out regions that would let curves carried by $\tau$ be shortened or be parallel copies of each other.
A *weight* on $\tau$ assigns a number $w(b) \ge 0$ to each branch $b$.
At each switch, the total weight arriving from one side must equal the total arriving from the other side.
Integer weights correspond to multicurves carried by $\tau$.

Example: let $S$ be a closed orientable surface of genus $2$ and $c$ a non-separating simple closed curve.
Put one switch on $c$; the rest of $c$ is one branch leaving the switch in one direction and coming back from the other, so clauses 1 and 2 hold.
The complement $S \setminus c$ is a surface of genus $1$ with two boundary circles, neither a disc nor an annulus, so clause 3 holds.

Non-example: two disjoint parallel copies of $c$, each with one switch, bound an annulus with no cusps between them, so clause 3 fails.

## Explanations

### L1 human

A train track is a smooth railway network on a surface.
The rules say that trains can pass through every junction smoothly, and that the network has no useless loops or parallel duplicate lines.
Numbers on the tracks, balanced at each switch, describe how a family of curves runs along the network.
</document>

<document source="https://tamnd.github.io/math2/node/P84GBDM0QR.md" tag="P84GBDM0QR" kind="definition">
# Cartan–Eilenberg resolution

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $C_\bullet$ be a [[chain complex|WQ82C0M3QB]] in $\mathcal{A}$.
A *Cartan–Eilenberg resolution* of $C_\bullet$ is a [[double complex|N80ABTQHEQ]] $P_{\bullet,\bullet}$ of [[projective objects|YXMXAGVC1Q]] with $P_{p,q} = 0$ for $q < 0$.
Its horizontal differential is $d^h \colon P_{p,q} \to P_{p-1,q}$ and its vertical differential is $d^v \colon P_{p,q} \to P_{p,q-1}$.
It comes with a [[chain map|HTK9WQZJ1G]] $\varepsilon \colon P_{\bullet,0} \to C_\bullet$ with $\varepsilon \circ d^v = 0$ on $P_{\bullet,1}$.
These data must satisfy, for every $p$:

1. if $C_p = 0$, then $P_{p,q} = 0$ for all $q$;
2. the maps induced by $\varepsilon$ on boundaries, $B_p(P_{\bullet,\bullet}, d^h) \to B_p(C)$, form a [[projective resolution|HK2AA1YSQ2]] of $B_p(C)$;
3. the maps induced by $\varepsilon$ on [[homology|2J4JJJYP22]], $H_p(P_{\bullet,\bullet}, d^h) \to H_p(C)$, form a projective resolution of $H_p(C)$.

Here $B_p(P_{\bullet,\bullet}, d^h)$ and $H_p(P_{\bullet,\bullet}, d^h)$ are the complexes, with vertical differential, of horizontal boundaries and horizontal homology in column $p$.

## Remarks

It follows that each column $P_{p,\bullet} \to C_p$ and the cycles $Z_p(P_{\bullet,\bullet}, d^h) \to Z_p(C)$ are projective resolutions too.
If $\mathcal{A}$ has [[enough projectives|J2TX4TDPCX]], every chain complex has a Cartan–Eilenberg resolution.
It is built with the [[horseshoe lemma|VNMBSYX9PY]] applied to $0 \to B_p \to Z_p \to H_p \to 0$ and $0 \to Z_p \to C_p \to B_{p-1} \to 0$.
Example: let $C$ be a module $M$ in degree $0$, and place a projective resolution $Q_\bullet \to M$ in column $0$.
This is a Cartan–Eilenberg resolution: the boundaries are $0$, resolved by $0$.
The horizontal homology of column $0$ is $Q_\bullet$, which resolves $H_0(C) = M$.
Non-example: over $\Z$, let $C$ be $\Z \xrightarrow{\cdot 2} \Z$ in degrees $1$ and $0$, and let $P = C$ placed in row $0$.
Clause 3 fails in degree $0$: the horizontal homology there is $\Z/2\Z$ alone.
It is not projective, so it is not a projective resolution of $H_0(C) = \Z/2\Z$.
Injective Cartan–Eilenberg resolutions of cochain complexes are defined dually.
</document>

<document source="https://tamnd.github.io/math2/node/P88X34MSVV.md" tag="P88X34MSVV" kind="concept">
# Pitchfork bifurcation

## Summary

A pitchfork bifurcation is a change in a family of dynamical systems with a symmetry, at which one [[equilibrium point|0N8NGKDZDZ]] splits into three.
It typically occurs when the system is unchanged under $x \mapsto -x$, so equilibria come in symmetric pairs.
In the supercritical case, a stable equilibrium becomes unstable and two new stable equilibria appear on either side.
See the [[definition|TRFDEJRPAZ]].

## Explanations

### L1 human

The model is $x' = \mu x - x^3$.
For $\mu \le 0$ the origin is the only equilibrium, and it attracts nearby solutions.
For $\mu > 0$ the origin repels, and two new stable equilibria appear at $\pm\sqrt{\mu}$.
Plotting the equilibria against $\mu$ gives a picture shaped like a pitchfork.
A ruler pressed at both ends that buckles to the left or to the right is a physical picture of this.
</document>

<document source="https://tamnd.github.io/math2/node/P89SJ7JWJE.md" tag="P89SJ7JWJE" kind="concept">
# Uniform structure of a topological group

## Summary

Every [[topological group|92700QAB5Y]] carries natural [[uniform structures|A54T86KS63]] that induce its topology.
The left uniformity declares $x$ and $y$ close when $x^{-1} y$ is close to the identity, and the right uniformity uses $y x^{-1}$ instead.
These uniformities give meaning to Cauchy nets, uniform continuity and completeness without any metric.
For abelian groups the two uniformities agree.

## Explanations

### L1 human

In a metric space you can compare distances at different places: two points here are as close as two points there.
A general topological group has no metric, but it has a group law, and that is enough.
Two elements are declared close when one is obtained from the other by multiplying by an element near the identity.
For the real line under addition, this says that $x$ and $y$ are close when $y - x$ is small, which is the usual notion.
</document>

<document source="https://tamnd.github.io/math2/node/P8CAK0FHGJ.md" tag="P8CAK0FHGJ" kind="statement">
# Jentzsch's theorem on positive kernels

## Statement

Let $a < b$ and let $K \colon [a, b] \times [a, b] \to \R$ be a [[continuous kernel|ZM5G144PZG]] with $K(x, y) > 0$ for all $x, y \in [a, b]$.
Let $T_K$ be the integral operator $(T_K \varphi)(x) = \int_a^b K(x, y) \varphi(y) \, dy$, and let $r$ be its [[spectral radius|9V7YQA85PC]].
Then:

1. $r > 0$, and $r$ is an eigenvalue of $T_K$ with an [[eigenfunction|4NSACSWRJZ]] $u$ that is continuous and satisfies $u(x) > 0$ for all $x \in [a, b]$;
2. the eigenspace of $r$ is spanned by $u$;
3. every other eigenvalue $\mu$ of $T_K$, real or complex, satisfies $\abs{\mu} < r$.

## Remarks

The operator $T_K$ is compact on $C[a, b]$ by the [[compactness of integral operators|MF77N00SK8]], and it maps the cone of nonnegative functions into itself.
Since $K$ has a positive minimum on the square, $T_K$ sends every nonnegative function that is not identically $0$ to a strictly positive function.
So it maps the cone minus $0$ into the interior of the cone.
The strong form of the [[Krein–Rutman theorem|VADXDK3TKM]] then gives all three parts.

Example: for $K(x, y) = 1 + xy$ on $[0, 1]$, $T_K$ has rank $2$ and maps into the span of $1$ and $x$.
Its matrix in this basis is $\begin{pmatrix} 1 & 1/2 \\ 1/2 & 1/3 \end{pmatrix}$, with eigenvalues $\frac{4 \pm \sqrt{13}}{6}$.
The larger one, $r = \frac{4 + \sqrt{13}}{6}$, has the eigenfunction $u(x) = 1 + 2(r - 1) x = 1 + \frac{\sqrt{13} - 2}{3} x$, which is positive on $[0, 1]$.
The other eigenvalues, $\frac{4 - \sqrt{13}}{6}$ and $0$, have modulus less than $r$.

Strict positivity matters.
Take $K(x, y) = 1$ when $y \le x$ and $0$ otherwise, which is nonnegative but neither continuous nor positive.
Its operator is the Volterra operator, whose spectral radius is $0$ and which has no eigenvalue at all.

## Explanations

### L1 human

If every point influences every other point positively, the integral operator has a dominant mode.
This mode is a positive function, it is unique up to scaling, and it grows faster under repeated application than every other mode.
This is the integral version of the [[Perron–Frobenius theorem|62NS7PV1M9]] for matrices with positive entries.

### L2 human

The theorem explains the long-time behaviour of models with a positive integral operator, such as population models structured by size.
After many steps, the state lines up with the positive eigenfunction $u$ and grows by the factor $r$ per step.
It is a special case of the Krein–Rutman theorem for strongly positive compact operators on ordered Banach spaces.
</document>

<document source="https://tamnd.github.io/math2/node/P8D6T5A1QD.md" tag="P8D6T5A1QD" kind="conjecture">
# Littlewood conjecture

## Statement

For a real number $x$, let $\norm{x}$ denote the distance from $x$ to the nearest integer.
For every pair of real numbers $\alpha$ and $\beta$,
$$\liminf_{n \to \infty} \, n \, \norm{n\alpha} \, \norm{n\beta} = 0,$$
where $n$ runs over the positive integers.

## Remarks

The problem is open; Littlewood posed it around 1930.
It holds if $\alpha$ is not badly approximable, that is, if $\liminf_n n \norm{n\alpha} = 0$.
Indeed $\norm{n\beta} \le 1/2$ gives $n \norm{n\alpha} \norm{n\beta} \le n \norm{n\alpha} / 2$.
Rational numbers are not badly approximable, and the same holds with the roles of $\alpha$ and $\beta$ exchanged.
By the theory of [[continued fractions|04P2RJ4WPS]], a number is badly approximable exactly when its partial quotients are bounded, so the open case is a pair of such [[irrational numbers|0DZXAXEJQ9]].
Cassels and Swinnerton-Dyer proved it in 1955 when $1, \alpha, \beta$ span a totally real cubic number field.
Einsiedler, Katok and Lindenstrauss proved in 2006 that the set of pairs $(\alpha, \beta)$ for which it fails has Hausdorff dimension $0$.
Their proof uses the action of the diagonal group on $\mathrm{SL}_3(\R)/\mathrm{SL}_3(\Z)$ and the rigidity of measures invariant under it.

## Explanations

### L1 human

Every real number can be approximated by fractions, and some numbers, like the golden ratio, resist this as much as possible.
The conjecture says that two such numbers cannot resist at the same time: some multiple $n\alpha$ and $n\beta$ are both unusually close to integers.
Here unusually close means that the product of the two errors beats $1/n$ by any factor you like.
It is a simple-looking statement about pairs of numbers that has been open for nearly a century.
</document>

<document source="https://tamnd.github.io/math2/node/P8JGMNC23R.md" tag="P8JGMNC23R" kind="definition">
# Differential graded algebra

## Definition

Let $K$ be a [[commutative ring|0GYGHKX834]].
Let $A = \bigoplus_{n \in \Z} A^n$ be a [[graded|CAPQBQDRA7]] [[associative algebra|3CTW1P4HBP]] over $K$, so that $A^m A^n \subseteq A^{m+n}$; for $a \in A^n$ write $\abs{a} = n$.
A *differential graded algebra* is such an $A$ with a $K$-linear map $d \colon A \to A$ such that:

1. $d(A^n) \subseteq A^{n+1}$ for all $n$.
2. $d \circ d = 0$.
3. $d(ab) = d(a)\, b + (-1)^{\abs{a}} a\, d(b)$ for all homogeneous $a \in A$ and all $b \in A$.

## Remarks

This is the cohomological convention; in the homological convention $d$ lowers the degree by one.
The cohomology $H(A) = \ker d / \operatorname{im} d$ is a graded algebra, because clause 3 makes $\ker d$ a subalgebra and $\operatorname{im} d$ a two-sided ideal of it.
Example: let $K$ be a field, $A^0 = K[x]$, $A^1 = K[x]\, dx$, and $A^n = 0$ otherwise, with $(dx)^2 = 0$.
Put $d(f) = f'\, dx$ on $A^0$ and $d = 0$ on $A^1$.
Clauses 1 and 2 are clear, and clause 3 reduces to the product rule $(fg)' = f'g + fg'$; for $a = f$, $b = g\,dx$ both sides are $0$ since $(dx)^2 = 0$.
Non-example: in the same algebra put $d(f) = f\, dx$ instead.
Clauses 1 and 2 hold, but clause 3 fails: $d(1 \cdot 1) = dx$ while $d(1)\cdot 1 + 1 \cdot d(1) = 2\, dx$.
Every graded algebra with $d = 0$ is a differential graded algebra.
</document>

<document source="https://tamnd.github.io/math2/node/P8PFMNCB8Y.md" tag="P8PFMNCB8Y" kind="concept">
# Octahedral axiom

## Summary

The octahedral axiom is the fourth axiom, TR4, in the definition of a [[triangulated category|T2V7GJJ2EN]].
It relates the cones of two composable morphisms $f$ and $g$ to the cone of their composite $g f$ by a further distinguished triangle.
It is the triangulated analogue of the third isomorphism theorem $(C/A)/(B/A) \cong C/B$.
Its name comes from drawing the four triangles involved as the faces of an octahedron.

## Explanations

### L1 human

If $X \subseteq Y \subseteq Z$ are abelian groups, then $Y/X$ sits inside $Z/X$ with quotient $Z/Y$.
The octahedral axiom says the same thing for cones in a triangulated category.
The cone of $X \to Y$ maps to the cone of $X \to Z$, and the "quotient" of that map is the cone of $Y \to Z$.
The four triangles that appear can be drawn as four of the eight faces of an octahedron.
</document>

<document source="https://tamnd.github.io/math2/node/P8TMYNPSYD.md" tag="P8TMYNPSYD" kind="statement">
# Lorentz's characterization of almost convergence

## Statement

Let $x = (x_n)_{n \ge 1}$ be a [[bounded sequence|0A8JNBGT2Z]] of real numbers and let $L$ be a real number.
Then $x$ is [[almost convergent|8HTHB1QPZY]] to $L$ if and only if
$$\lim_{p \to \infty} \frac{1}{p} \sum_{k = n}^{n + p - 1} x_k = L \quad \text{uniformly in } n \ge 1.$$

## Remarks

Proof idea for "if": let $\varphi$ be a [[Banach limit|SYF2ND6J97]] and $\varepsilon > 0$.
By shift invariance, $\varphi(x)$ equals $\varphi$ of the sequence of block averages $y_n = \frac{1}{p} \sum_{k=n}^{n+p-1} x_k$.
For $p$ large, $\abs{y_n - L} \le \varepsilon$ for all $n$, and positivity of $\varphi$ then gives $\abs{\varphi(x) - L} \le \varepsilon$.
For "only if", one shows with the [[Hahn–Banach theorem|QB5AE3VYZZ]] that Banach limits take every value between the lower and upper limits of the uniform block averages.

Example: for $x = (1, 0, 1, 0, \dots)$, any block of $p$ consecutive terms has sum $\floor{p/2}$ or $\ceil{p/2}$, so the averages tend to $1/2$ uniformly in $n$.

The uniformity in $n$ is necessary.
Let $x_n = 1$ when $4^k \le n < 4^k + k$ for some integer $k \ge 1$, and $x_n = 0$ otherwise.
Among $x_1, \dots, x_N$ there are at most about $(\log_4 N)^2$ ones, so the averages starting at $n = 1$ tend to $0$.
But the block of length $k$ starting at $4^k$ has average $1$, so the convergence is not uniform in $n$ and $x$ is not almost convergent.

## Explanations

### L1 human

A bounded sequence is almost convergent to $L$ exactly when the average of any long run of consecutive terms is close to $L$, no matter where the run starts.
This turns a definition about abstract functionals into a concrete test.

### L2 human

The definition of almost convergence quantifies over all Banach limits, which exist only by the Hahn–Banach theorem and cannot be written down.
Lorentz's theorem replaces this by a computable condition on block averages.
It also shows that almost convergence is a summability method in its own right.
For bounded sequences it lies between ordinary convergence and Cesàro summability, because the case $n = 1$ of the condition is Cesàro convergence.
</document>

<document source="https://tamnd.github.io/math2/node/P8YTMGEN9R.md" tag="P8YTMGEN9R" kind="concept">
# Local homomorphism

## Summary

A local homomorphism is a [[ring homomorphism|0VTH8KPCG2]] between [[local rings|MZ02BKPHJA]] that sends the maximal ideal into the maximal ideal.
These are the natural maps between local rings, and they induce maps of [[residue fields|NJ3APQ8BHH]].
In geometry, the map of local rings induced by a morphism of spaces at a point is local.

## Explanations

### L1 human

The maximal ideal of a local ring consists of the "functions that vanish at the point".
A local homomorphism must send functions vanishing at the point to functions vanishing at the image point.
For example, reducing the ring $\Z_{(p)}$ modulo $p$ to the field $\Z/p\Z$ is local.
But including $\Z_{(p)}$ into $\Q$ is not, since $p$ is not invertible in $\Z_{(p)}$ but becomes invertible in $\Q$.
The [[definition|RM0FMT0X8W]] states the condition.
</document>

<document source="https://tamnd.github.io/math2/node/P923DHV74J.md" tag="P923DHV74J" kind="concept">
# Hilbert metric

## Summary

The Hilbert metric is a distance on any bounded open [[convex set|02AS45T3H9]] of $\R^n$.
It is the logarithm of a [[cross-ratio|09ASMRMD67]] of the two points and the two boundary points on the line through them.
It turns the convex set into a [[metric space|0VKD1TP7F3]] in which straight segments are shortest paths.
When the convex set is a ball, it is the [[Beltrami–Klein model|1WN9B18H86]] of hyperbolic space, so Hilbert geometries generalize the hyperbolic [[Cayley–Klein metric|J7ZE3EGRYX]].

## Explanations

### L1 human

Take a convex region, such as a disc or a triangle.
To measure the distance between two points, extend the segment between them until it hits the boundary on both sides.
The distance compares how close each point is to each boundary point, through a logarithm.
Near the boundary, distances blow up, so the boundary is infinitely far away.
In a disc this gives a picture of the hyperbolic plane in which lines look straight.
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<document source="https://tamnd.github.io/math2/node/P93R13XP0P.md" tag="P93R13XP0P" kind="concept">
# Kadomtsev–Petviashvili equation

## Summary

The Kadomtsev–Petviashvili equation, or KP equation, is a nonlinear wave equation in two space variables $x$ and $y$.
It extends the [[Korteweg–De Vries equation|KTT75QC72Z]] to waves that move mainly in the $x$ direction but vary slowly in the $y$ direction.
It comes in two versions, KP-I and KP-II, which differ by the sign of the $u_{yy}$ term.
Like the KdV equation it is completely integrable.

## Explanations

### L1 human

The KdV equation describes waves in a narrow canal, where the water moves along one line.
On a wide sea surface a wave crest is a long line, and it can bend a little.
The KP equation adds a small term that accounts for this sideways bending.
Every KdV wave that does not depend on $y$, such as a soliton with a straight crest, is also a KP wave.
The two sign choices model water with weak surface tension (KP-II) and with strong surface tension (KP-I).
</document>

<document source="https://tamnd.github.io/math2/node/P93WWQ6BBB.md" tag="P93WWQ6BBB" kind="definition">
# Positive operator

## Definition

Let $H$ be a [[Hilbert space|HM2HHE3J51]] over $\R$ or $\C$ and let $A \colon H \to H$ be a bounded linear operator.
The operator $A$ is *positive* when

1. $A$ is self-adjoint, that is, $A^* = A$, where $A^*$ is the [[adjoint|6HSJBJ8TTQ]] of $A$;
2. $\ip{Ax}{x} \ge 0$ for all $x \in H$.

One writes $A \ge 0$, and $A \le B$ for self-adjoint $A$ and $B$ when $B - A \ge 0$.

## Remarks

Example: for every bounded operator $T$ on $H$, the operator $A = T^*T$ is positive.
It is self-adjoint since $(T^*T)^* = T^*T$, and $\ip{T^*Tx}{x} = \ip{Tx}{Tx} = \norm{Tx}^2 \ge 0$.

Non-example: the diagonal operator $A = \mathrm{diag}(1, -1)$ on $\C^2$ is self-adjoint, but $\ip{A e_2}{e_2} = -1 < 0$ for $e_2 = (0, 1)$, so condition 2 fails.

Over $\C$, condition 2 alone implies condition 1, because a bounded operator with $\ip{Ax}{x}$ real for all $x$ is self-adjoint by polarization.
Over $\R$, condition 1 is needed: a rotation by $90$ degrees of $\R^2$ has $\ip{Ax}{x} = 0$ for all $x$ but is not self-adjoint.
Equivalent forms: $A$ is positive if and only if $A$ is self-adjoint with spectrum in $[0, \infty)$, and if and only if $A = B^*B$ for some bounded operator $B$.
Some authors call these operators positive semidefinite or nonnegative.
</document>

<document source="https://tamnd.github.io/math2/node/P944X6S117.md" tag="P944X6S117" kind="concept">
# Bayesian hierarchical model

## Summary

A Bayesian hierarchical model specifies a [[prior|00NBBSXRQS]] in several stages.
The parameters of the data model have a prior, and that prior has unknown hyperparameters with their own hyperprior.
It describes data in related groups, whose parameters are different but drawn from a common population.
Its [[posterior|0KJ7ZS8AJA]] lets groups borrow strength from each other, and it is a fully Bayesian counterpart of the [[empirical Bayes method|8GG2Z2E61R]].

## Explanations

### L1 human

Think of test scores in many schools.
Each school has its own average level, but the schools are alike in many ways.
A hierarchical model says: each school's level comes from a common distribution of levels, and the shape of that distribution is itself unknown.
A small school with few scores then gets an estimate pulled towards the overall average, while a large school mostly keeps its own average.
</document>

<document source="https://tamnd.github.io/math2/node/P971S9AX0E.md" tag="P971S9AX0E" kind="definition">
# Nilpotent operator

## Definition

Let $X$ be a [[Banach space|06GZ956657]] and let $T \colon X \to X$ be a bounded [[linear operator|03TM135G0N]].
The operator $T$ is *nilpotent* when there is an integer $n \ge 1$ with
$$T^n = 0,$$
where $T^n$ is the $n$-fold composition of $T$ with itself.
The least such $n$ is the *index* of nilpotency of $T$.

## Remarks

Example: on $X = \ell^2 \oplus \ell^2$ let $T(x, y) = (0, x)$.
Then $T^2(x, y) = T(0, x) = (0, 0)$, so $T$ is nilpotent of index $2$, although its range is infinite-dimensional.

Non-example: the Volterra operator $(Vf)(x) = \int_0^x f(t) \, dt$ on $L^2[0, 1]$ has spectrum $\set{0}$, but $(V^n 1)(x) = x^n / n!$ is not the zero function for any $n$, so $V$ is not nilpotent.

If $T$ is nilpotent and $X \ne \set{0}$, then $\sigma(T) = \set{0}$: for $\lambda \ne 0$ the series $\lambda^{-1} \sum_{k=0}^{n-1} (T/\lambda)^k$ is a finite sum that inverts $\lambda I - T$.
</document>

<document source="https://tamnd.github.io/math2/node/P976WYEFMB.md" tag="P976WYEFMB" kind="concept">
# Homotopy group

## Summary

The $n$-th homotopy group $\pi_n(X, x_0)$ of a pointed space consists of the homotopy classes of based maps from the $n$-sphere to $(X, x_0)$.
For $n = 1$ it is the [[fundamental group|S1XK3HXHFX]], and for $n \ge 2$ it is abelian.
Homotopy groups are invariants of homotopy type, and on CW complexes they detect homotopy equivalences, by the [[Whitehead theorem|6P1E2FJPSD]].
They are easy to define and very hard to compute, even for spheres.

## Explanations

### L1 human

The group $\pi_n(X)$ records the ways an $n$-dimensional sphere can be placed in a space, where two placements count as the same if one can be deformed into the other.
A balloon inside a solid ball can always be shrunk to a point, so $\pi_2$ of a ball is trivial.
A balloon wrapped once around the surface of a sphere cannot be shrunk on that surface, and $\pi_2(S^2) \cong \Z$ counts how many times it wraps.
</document>

<document source="https://tamnd.github.io/math2/node/P9CM4GQ2T0.md" tag="P9CM4GQ2T0" kind="concept">
# Sphere

## Summary

A sphere is the set of points at a fixed positive distance, the radius, from a fixed point, the centre.
In three-dimensional [[Euclidean space|0XPR1E85W0]] it is the surface of a ball, and in the plane it is a [[circle|060W5CBZP6]].
The same definition makes sense in any [[metric space|0VKD1TP7F3]].
Spheres are among the most symmetric shapes, and many problems of geometry, physics and analysis are posed on them.

## Explanations

### L1 human

Think of the skin of a ball: every point on it is the same distance from the middle.
That common distance is the radius, and the middle point is the centre.
For example, the points $(x, y, z)$ with $x^2 + y^2 + z^2 = 1$ form the sphere of radius $1$ about the origin.
In the plane the same recipe draws a circle.
</document>

<document source="https://tamnd.github.io/math2/node/P9PYM71GMM.md" tag="P9PYM71GMM" kind="example">
# Hypercube

## Example

For $d \ge 1$, the $d$-dimensional *hypercube* is
$$C_d = [0,1]^d = \set{x \in \R^d : 0 \le x_i \le 1 \text{ for } i = 1, \dots, d}.$$
It is a [[convex polytope|300K6CB1KH]]: it is the convex hull of the $2^d$ points with all coordinates in $\set{0, 1}$, and these are its [[vertices|R7V3S9V891]].
Its [[facets|DPDXN2FJC4]] are the $2d$ sets where one coordinate is fixed to $0$ or to $1$.

A face of $C_d$ is obtained by fixing some coordinates to $0$ or $1$ and leaving the others free.
A $k$-dimensional face leaves $k$ coordinates free, so the number of $k$-faces is
$$f_k(C_d) = 2^{d-k} \binom{d}{k}.$$
For $d = 3$ this gives the [[f-vector|872BAKDQQK]] $(8, 12, 6)$ of the ordinary cube.

$C_d$ is a [[simple polytope|7E6FXTGH2Y]].
At a vertex $v$ the edges change exactly one coordinate of $v$, and there are $d$ coordinates, so every vertex lies in $d$ edges.

## Remarks

The graph of $C_d$ is the hypercube graph: its vertices are the $0/1$ strings of length $d$, joined when they differ in one place.
Many authors use $[-1, 1]^d$ instead; it has the same combinatorics and is centrally symmetric about $0$.
For $d = 4$ the hypercube is also called the tesseract, with $f$-vector $(16, 32, 24, 8)$.

## Explanations

### L1 human

The hypercube is the cube in any dimension.
A segment is a $1$-cube, a square is a $2$-cube and the ordinary cube is a $3$-cube.
To get the next one, slide the previous cube one unit in a new direction and fill in the space it sweeps out.
Each step doubles the number of corners, so the $d$-cube has $2^d$ corners.
</document>

<document source="https://tamnd.github.io/math2/node/P9R3777ZJF.md" tag="P9R3777ZJF" kind="concept">
# Thom–Mather stratified space

## Summary

A Thom–Mather stratified space is a space cut into smooth manifolds, the strata, together with control data near each stratum.
The control data are a tubular neighbourhood, a retraction onto the stratum and a distance function to it.
The control data must be compatible between strata.
Mather showed that every closed [[Whitney stratified|3F84AMVJNV]] subset of a smooth manifold carries such data.
The control data are what makes the [[Thom–Mather isotopy lemmas|7G1PCS2RPG]] work, so these spaces are locally topologically trivial along each stratum.

## Explanations

### L1 human

Think of a cone: it has a tip and a smooth part, two pieces of different dimension.
Near the tip one can measure the distance to the tip and project nearby points to it.
A Thom–Mather structure asks for such a projection and such a distance near every piece, and asks the projections and distances of different pieces to agree with each other.
With these data one can build vector fields that respect all the pieces, and flow along them.
This is how one proves that the space looks the same at all points of a stratum.
</document>

<document source="https://tamnd.github.io/math2/node/P9RN33VDFS.md" tag="P9RN33VDFS" kind="definition">
# Group homology

## Definition

Let $G$ be a [[group|G5S1M9]] and let $\Z[G]$ be its [[group ring|4NMTSX0XDS]].
A $G$-module is a left [[module|0NZ8JA25WK]] over $\Z[G]$.
Let $\Z$ be the right $\Z[G]$-module with the trivial action $n \cdot g = n$.
Let $P_\bullet \to \Z$ be a [[projective resolution|HK2AA1YSQ2]] of $\Z$ by right $\Z[G]$-modules.
Let $M$ be a $G$-module and let $n \ge 0$.
The $n$-th *homology group* of $G$ with coefficients in $M$ is
$$H_n(G, M) = H_n(P_\bullet \otimes_{\Z[G]} M) = \mathrm{Tor}^{\Z[G]}_n(\Z, M),$$
the homology of the [[chain complex|WQ82C0M3QB]] obtained by the [[tensor product|F0HQXQ0HCD]] with $M$.

## Remarks

The groups do not depend on the choice of projective resolution, up to canonical isomorphism.
In degree $0$, $H_0(G, M) = M_G$ is the module of coinvariants, the quotient of $M$ by the subgroup generated by all $gm - m$.
In degree $1$, $H_1(G, \Z) \cong G / [G, G]$.

Example: $G = \Z$ written multiplicatively with generator $t$, so that $\Z[G] = \Z[t, t^{-1}]$.
The sequence $0 \to \Z[G] \xrightarrow{\ t - 1\ } \Z[G] \to \Z \to 0$ is a free resolution of $\Z$.
Tensoring with the trivial module $\Z$ gives the complex $0 \to \Z \xrightarrow{0} \Z \to 0$.
So $H_0(\Z, \Z) = \Z$, $H_1(\Z, \Z) = \Z$, and $H_n(\Z, \Z) = 0$ for $n \ge 2$.

Non-example: the groups $\mathrm{Tor}^{\Z}_n(\Z, M)$, taken over $\Z$ instead of $\Z[G]$, forget the action.
They vanish for $n \ge 1$, while $H_1(\Z, \Z) = \Z$.
</document>

<document source="https://tamnd.github.io/math2/node/P9ZPH7EC8K.md" tag="P9ZPH7EC8K" kind="concept">
# Thin plate spline

## Summary

A thin plate spline is a smooth function of two variables that interpolates scattered data with the least bending energy.
It is a sum of translates of the radial function $r^2 \log r$ and an affine function, the case $d = 2$, $m = 2$ of a [[polyharmonic spline|XJ0JGPBC8R]].
The name comes from the physical picture of a thin metal sheet bent to pass through given heights.
Thin plate splines are widely used for surface fitting and for smooth image warping.

## Explanations

### L1 human

Pin a thin flexible sheet at a few points at given heights; it settles in the shape that bends the least.
The thin plate spline is the mathematical version of that shape.
Away from the pins it satisfies the [[biharmonic equation|AS8A9F5SKE]], the equation of a bent elastic plate with no load on it.
With three pins that are not on a line, the sheet is simply the tilted plane through them, because a plane needs no bending at all.
</document>

<document source="https://tamnd.github.io/math2/node/PA15QP0BP5.md" tag="PA15QP0BP5" kind="concept">
# S-finite measure

## Summary

An s-finite measure is a measure that is a countable sum of finite measures.
Every σ-finite measure is s-finite, but not conversely.
The class of s-finite measures is used in probability theory, for example for kernels and point processes.

## Explanations

### L1 human

An s-finite measure is built by stacking countably many finite measures on top of each other and adding them.
Each layer has finite total size, but the sum may be infinite everywhere it is not zero.
Infinitely many copies of length on the interval $[0, 1]$, added together, give a measure that is s-finite.
Every subset of $[0, 1]$ of positive length then has infinite size, and this measure is not σ-finite.
</document>

<document source="https://tamnd.github.io/math2/node/PA1B7DJ0C4.md" tag="PA1B7DJ0C4" kind="definition">
# Round-off error

## Definition

Let $F$ be a set of [[floating-point numbers|RTKRSC3KPH]] and let $\mathrm{fl} \colon \R \to F$ be a [[rounding|5NPVX29Y0C]].
The *round-off error* of representing $x \in \R$ is $\mathrm{fl}(x) - x$, and for $x \ne 0$ its *relative round-off error* is $(\mathrm{fl}(x) - x)/x$.
Let an algorithm compute $y = f(x)$ in exact real arithmetic, and let it return $\hat y$ when each operation is done in floating-point arithmetic.
Then the round-off error of the computation is $\hat y - y$.

## Remarks

Example: with base $10$, precision $3$ and rounding to nearest, $\mathrm{fl}(2/3) = 0.667$, so the round-off error is $0.667 - 2/3 = 1/3000$ and the relative round-off error is $1/2000 = 0.0005$.
This is within the unit roundoff $u = \tfrac12 \cdot 10^{-2}$ of the system.
Non-example: replacing $e = \sum_{k \ge 0} 1/k!$ by $\sum_{k=0}^{5} 1/k!$ in exact arithmetic gives an error, but it is a truncation error, not a round-off error.
If $x$ lies in the range of normalized numbers and $\mathrm{fl}$ rounds to nearest, then $\abs{\mathrm{fl}(x) - x} \le u \abs{x}$, where $u = \tfrac12 \beta^{1-p}$ is the unit roundoff, see [[machine epsilon|JA3QXVF2B8]].
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<document source="https://tamnd.github.io/math2/node/PA4HJDEP28.md" tag="PA4HJDEP28" kind="definition">
# Developable surface

## Definition

Let $S \subset \R^3$ be a [[regular surface|6080X3FAD4]] with [[Gaussian curvature|SQVQ5DQT6W]] $K \colon S \to \R$.
The surface $S$ is *developable* if $K(p) = 0$ for every point $p \in S$.

## Remarks

The cylinder $x^2 + y^2 = 1$ is developable.
Its principal curvatures are $1$ along the horizontal circles and $0$ along the vertical lines, so $K = 1 \cdot 0 = 0$.
The sphere of radius $r$ is not developable, since $K = 1/r^2 \neq 0$ at every point.

By the [[Theorema Egregium|YKSA8SJ81V]], $K$ depends only on the first fundamental form.
A surface with $K = 0$ is locally isometric to the plane, and conversely.
A developable surface without planar points is a ruled surface.
Away from a small exceptional set, it is locally a piece of a cylinder, a cone or a tangent developable.
Some authors define a developable surface as a ruled surface with $K = 0$.

## Explanations

### L1 human

A developable surface is one with zero Gaussian curvature everywhere.
At each point one of the two principal curvatures is zero, so the surface bends in only one direction at a time.
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<document source="https://tamnd.github.io/math2/node/PA4PYQ1PZ0.md" tag="PA4PYQ1PZ0" kind="concept">
# Turing reduction

## Summary

A set $A$ is Turing reducible to a set $B$ if some [[oracle machine|Z5CCVBME2V]] decides membership in $A$ by asking questions about $B$.
It is the most general notion of "$A$ is computable from $B$" in computability theory.
Turing reducibility is the basis of Turing degrees, which measure the amount of uncomputability of a set.

## Explanations

### L1 human

Imagine a computer that has a magic helper: whenever it asks "is this number in $B$?", the helper answers at once.
If such a computer can always decide whether a number lies in $A$, then $A$ is Turing reducible to $B$.
The computer may ask many questions, and later questions may depend on earlier answers.
For example, the complement of a set $B$ is Turing reducible to $B$: ask whether $x \in B$ and flip the answer.
This is more freedom than a [[many-one reduction|JTAH37JTHD]], which asks only one question and must copy its answer.
</document>

<document source="https://tamnd.github.io/math2/node/PA78NFEPCF.md" tag="PA78NFEPCF" kind="concept">
# Mean curvature

## Summary

The mean curvature of a surface at a point is the average of its two [[principal curvatures|FA8QE1SGP4]].
Unlike the Gaussian curvature, it depends on how the surface sits in space, and its sign depends on the choice of unit normal.
Surfaces with mean curvature zero are the minimal surfaces, such as soap films, and surfaces of constant mean curvature model soap bubbles.

## Explanations

### L1 human

At a point of a surface, the bending can be different in different directions.
The mean curvature averages the strongest and the weakest bending.
A soap film spanning a wire loop has mean curvature zero: at each point it bends up in one direction exactly as much as it bends down in the other.
A soap bubble, a sphere of radius $r$, has the same mean curvature $1/r$ at every point, measured with the inner normal.
</document>

<document source="https://tamnd.github.io/math2/node/PA7SWNRRPY.md" tag="PA7SWNRRPY" kind="definition">
# Inverse Mellin transform

## Definition

Let $c \in \R$ and let $\varphi$ be a complex function defined on the line $\operatorname{Re} s = c$.
Assume that $y \mapsto \varphi(c + iy)$ is [[integrable|X7GTF3D98F]] on $\R$.
For $x > 0$ and $s \in \C$, let $x^{-s} = e^{-s \log x}$ be the [[complex power|0RNGDMHT7M]].
The *inverse Mellin transform* of $\varphi$ along the line $\operatorname{Re} s = c$ is the function on $(0, \infty)$ given by
$$\mathcal{M}^{-1}_c \varphi(x) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} x^{-s} \varphi(s) \, ds = \frac{1}{2\pi} \int_{-\infty}^{\infty} x^{-c - iy} \varphi(c + iy) \, dy.$$

## Remarks

The [[Lebesgue integral|DWJF7ABPR1]] converges absolutely for every $x > 0$, because $\abs{x^{-c - iy}} = x^{-c}$.
The result usually depends on $c$.
When $\varphi$ is holomorphic on a vertical strip and decays fast enough, Cauchy's theorem shows that it does not change as $c$ moves inside the strip.
Example: $\varphi = \Gamma$ and $c > 0$.
Stirling's formula gives $\abs{\Gamma(c + iy)} = O(\abs{y}^{c - 1/2} e^{-\pi \abs{y}/2})$, so $\varphi$ is integrable on the line, and $\mathcal{M}^{-1}_c \Gamma(x) = e^{-x}$.
Non-example: $\varphi(s) = 1/s$ and $c > 0$.
Then $\abs{\varphi(c + iy)} = (c^2 + y^2)^{-1/2}$ is not integrable on $\R$, so the definition does not apply; the integral exists only as a limit of integrals over $[c - iT, c + iT]$.
It is common to drop the index $c$ and write $\mathcal{M}^{-1} \varphi$.
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<document source="https://tamnd.github.io/math2/node/PA7ZQ6C6SP.md" tag="PA7ZQ6C6SP" kind="statement">
# Löwner's theorem

## Statement

Let $a < b$ be real numbers, possibly $a = -\infty$ or $b = \infty$, and let $f \colon (a, b) \to \R$.
Let $\mathbb{H} = \set{z \in \C : \operatorname{Im} z > 0}$ be the upper half-plane.
Then $f$ is [[operator monotone|N155BSJ2DN]] on $(a, b)$ if and only if there is a holomorphic function $F$ on $\C \setminus \bigl( (-\infty, a] \cup [b, \infty) \bigr)$ such that:

1. $F(t) = f(t)$ for all $t \in (a, b)$;
2. $F(\bar z) = \overline{F(z)}$ for all $z$ in the domain of $F$;
3. $\operatorname{Im} F(z) \ge 0$ for all $z \in \mathbb{H}$.

## Remarks

So an operator monotone function is real analytic, and it extends to a holomorphic map from $\mathbb{H}$ into the closed upper half-plane, a so-called Pick function.
Example: for $0 < p < 1$, the function $t^p$ on $(0, \infty)$ extends to $z^p$ with the principal branch.
It maps $\mathbb{H}$ into the sector of arguments between $0$ and $p\pi$, which lies in $\mathbb{H}$, so $t^p$ is operator monotone.
In the same way $\log t$ is operator monotone on $(0, \infty)$, since $\log z$ maps $\mathbb{H}$ into the strip $0 < \operatorname{Im} w < \pi$.
In contrast, $z^2$ maps $-1 + i \in \mathbb{H}$ to $-2i$, so $t^2$ is not operator monotone on $(0, \infty)$.
One route to the proof goes through divided differences.
A function $f$ of class $C^1$ is operator monotone if and only if the following holds for all points $t_1, \dots, t_n$ of $(a, b)$.
The matrix with entries $(f(t_i) - f(t_j)) / (t_i - t_j)$ off the diagonal and $f'(t_i)$ on the diagonal is [[positive semidefinite|01MMVW8CKC]].

## Explanations

### L1 human

Keeping the order of Hermitian matrices of every size is a very strong demand on a function.
Löwner's theorem says that exactly the functions that extend to the complex upper half-plane and send it into itself meet this demand.
This turns a question about matrices into a question about holomorphic functions.

### L2 human

The theorem gives integral representations of operator monotone functions, through the Nevanlinna representation of Pick functions.
It explains at once why $t^p$ is operator monotone on $(0, \infty)$ exactly for $0 \le p \le 1$.
Operator monotone functions appear in matrix inequalities, in the theory of operator means and in quantum information.
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<document source="https://tamnd.github.io/math2/node/PAA93R4WYD.md" tag="PAA93R4WYD" kind="concept">
# Symmetric space

## Summary

A symmetric space is a Riemannian manifold that has, at each point, an isometry reversing all geodesics through that point.
Euclidean spaces, round spheres, hyperbolic spaces and projective spaces are symmetric spaces.
Each symmetric space is a quotient $G/K$ of a Lie group by a closed subgroup, so its geometry reduces to Lie theory.
[[Cartan's classification|9P8VK3H4YZ]] lists all of them, which makes them a large and well understood family of model spaces.

## Explanations

### L1 human

Stand at a point of the space and look in some direction.
In a symmetric space there is a symmetry of the whole space that keeps you fixed and swaps every direction with its opposite.
On a round sphere, this is the rotation by a half turn about the axis through you and the antipodal point.
So the space looks the same from every point and in every pair of opposite directions.
</document>

<document source="https://tamnd.github.io/math2/node/PAD37GKDPD.md" tag="PAD37GKDPD" kind="concept">
# Chern class

## Summary

Chern classes are [[characteristic classes|7HACJC0JE2]] of complex vector bundles, with values in cohomology.
In algebraic geometry they are defined for algebraic vector bundles, with values in the [[Chow ring|4HV4WTAX9P]].
A vector bundle $E$ of rank $r$ has classes $c_0(E) = 1, c_1(E), \ldots, c_r(E)$, with $c_i$ in degree $i$.
They measure how far $E$ is from being trivial, and they enter the Riemann–Roch theorems through the [[Todd class|2YDB5089FR]].
For a line bundle, $c_1$ is the class of the corresponding divisor.

## Explanations

### L1 human

A vector bundle attaches a vector space to each point of a space, and it may twist as one moves around.
Chern classes are invariants that record this twisting in a way that can be computed and added.
For the line bundle $\mathcal{O}(d)$ on the projective line, the first Chern class is $d$ times the class of a point: its sections vanish at $d$ points.
For the tangent bundle of a compact complex curve of genus $g$, the first Chern class has degree $2 - 2g$, the Euler characteristic.
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<document source="https://tamnd.github.io/math2/node/PADZ7D9KBT.md" tag="PADZ7D9KBT" kind="statement">
# Tucker's lemma

## Statement

Let $n \ge 1$ and let $T$ be a triangulation of the [[closed ball|0Q0H13G8ZB]] $B^n \subset \R^n$ by a finite [[simplicial complex|V8GZ0W1ZZ5]].
Assume that $T$ is antipodally symmetric on the boundary.
This means that $-\sigma$ is a simplex of $T$ for every simplex $\sigma$ of $T$ contained in the boundary sphere.
Let $\lambda$ be a labelling of the vertices of $T$ by elements of $\set{\pm 1, \pm 2, \ldots, \pm n}$ such that $\lambda(-v) = -\lambda(v)$ for every vertex $v$ on the boundary.
Then $T$ has a *complementary edge*: an edge $\set{u, w}$ of $T$ with $\lambda(u) = -\lambda(w)$.

## Remarks

Tucker's lemma is a combinatorial form of the [[Borsuk–Ulam theorem|QHJ0GBQE2E]].
To derive the lemma from Borsuk–Ulam, send a vertex with label $\pm i$ to $\pm e_i$ and extend linearly on simplices to a map $f \colon B^n \to \R^n$.
With no complementary edge, the image of $f$ avoids $0$.
Then $f / \norm{f}$ restricts to an odd map from the boundary sphere $S^{n-1}$ to $S^{n-1}$ that extends over the ball, which is impossible.
Conversely, applying the lemma to finer and finer triangulations proves the Borsuk–Ulam theorem.
There are also constructive path-following proofs, which give algorithms to find a complementary edge.

Example: for $n = 1$, subdivide $[-1, 1]$ into intervals and label the vertices by $\pm 1$ with $\lambda(-1) = -\lambda(1)$.
Moving from $-1$ to $1$ the label changes sign, so some interval of the subdivision has endpoints labelled $1$ and $-1$.

The antipodal condition on the boundary is needed: if every vertex gets the label $1$, no edge is complementary.

## Explanations

### L1 human

Cut a disc into small triangles and write one of the labels $\pm 1, \pm 2$ at each corner, so that opposite points on the rim get opposite labels.
Then somewhere two neighbouring corners carry opposite labels, such as $2$ and $-2$.

### L2 human

Tucker's lemma is to the Borsuk–Ulam theorem what [[Sperner's lemma|HQG3V5Z6MR]] is to the Brouwer fixed-point theorem.
Because it is finite and combinatorial, it is the usual route to applications of Borsuk–Ulam in combinatorics, such as necklace splitting and Kneser's conjecture, and to algorithms for them.
</document>

<document source="https://tamnd.github.io/math2/node/PAG39H901E.md" tag="PAG39H901E" kind="statement">
# Grauert's theorem on real analytic manifolds

## Statement

Let $M$ be a [[real analytic manifold|Z7H98E1NHP]] of dimension $n$ that is Hausdorff and second countable.
Then:

1. $M$ is a closed real analytic [[submanifold|VKFW27WXEZ]] of some [[Stein manifold|HMCD0JBFST]] $X$ of complex dimension $n$. Moreover $M$ is totally real in $X$: $T_pM \cap i\,T_pM = 0$ for $p \in M$.
2. Let $Y$ be a [[complex manifold|JSGWAN2RJV]] that contains $M$ as a closed totally real real analytic submanifold of real dimension $n$. Then $M$ has a basis of open neighbourhoods in $Y$ that are Stein manifolds.
3. $M$ admits a real analytic embedding onto a closed real analytic submanifold of $\R^N$ for some $N$.

## Remarks

The idea of the proof is as follows.
A complexification of $M$ exists by gluing the holomorphic extensions of the real analytic transition maps of $M$.
Near $M$, the square of a suitable distance to $M$ gives a strictly plurisubharmonic function whose sublevel sets shrink to $M$.
Grauert's solution of the Levi problem for manifolds shows that these sublevel sets are Stein manifolds.
A Stein manifold $X$ embeds holomorphically and properly in $\C^N$, and restricting such an embedding to $M$ and taking real and imaginary parts gives the real analytic embedding into $\R^{2N}$.
Example: the circle $M = \R/\Z$ sits in the cylinder $X = \C/\Z$ as the image of $\R$, and thin annuli $\set{\abs{\operatorname{Im} z} < \varepsilon}$ are Stein neighbourhoods of it.
The countability hypothesis is needed for an embedding into a Euclidean space, since every subset of $\R^N$ is second countable.

## Explanations

### L1 human

A real analytic manifold is a space that looks locally like $\R^n$, glued by maps given by convergent power series.
The theorem says that any such space is a real slice of a very well behaved complex manifold, in the way $\R^n$ is the real slice of $\C^n$.
As a consequence, every such space can be drawn inside some $\R^N$ by maps given by power series.

### L2 human

The result is the real analytic analogue of the Whitney embedding theorem for smooth manifolds.
Its proof shows how several complex variables helps with real problems: real analytic functions on $M$ are restrictions of holomorphic functions on a Stein neighbourhood.
Cartan's theorems A and B on that neighbourhood then give many global real analytic functions on $M$.
The Stein neighbourhoods are often called Grauert tubes.
</document>

<document source="https://tamnd.github.io/math2/node/PAGN759BNR.md" tag="PAGN759BNR" kind="definition">
# Laplace–Beltrami operator

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] with [[Levi-Civita connection|CAZYGE1T4E]] $\nabla$.
For a smooth function $f$ on $M$, the gradient $\operatorname{grad} f$ is the vector field with $g(\operatorname{grad} f, X) = df(X)$ for every vector field $X$.
For a vector field $Y$, the divergence $\operatorname{div} Y$ is the function $\tr(X \mapsto \nabla_X Y)$.
The *Laplace–Beltrami operator* is the operator $\Delta$ on smooth functions given by
$$\Delta f = \operatorname{div}(\operatorname{grad} f).$$

## Remarks

In local coordinates $x^1, \dots, x^n$, with $g_{ij}$ the metric, $g^{ij}$ its inverse and $\abs{g} = \det(g_{ij})$,
$$\Delta f = \frac{1}{\sqrt{\abs{g}}} \sum_{i,j} \partial_i \bigl(\sqrt{\abs{g}} \, g^{ij} \partial_j f\bigr).$$
Equivalently $\Delta f = \sum_{i,j} g^{ij} \bigl(\partial_i \partial_j f - \sum_k \Gamma^k_{ij} \partial_k f\bigr)$, with $\Gamma^k_{ij}$ the [[Christoffel symbols|FZBA1HA9ZM]].

Example: on the unit sphere $S^2$ with coordinates $(\theta, \varphi)$ and metric $d\theta^2 + \sin^2\theta \, d\varphi^2$, one has $\sqrt{\abs{g}} = \sin\theta$ and
$$\Delta f = \frac{1}{\sin\theta} \partial_\theta(\sin\theta \, \partial_\theta f) + \frac{1}{\sin^2\theta} \partial_\varphi^2 f.$$
For $f = \cos\theta$ this gives $\frac{1}{\sin\theta} \partial_\theta(-\sin^2\theta) = -2\cos\theta$.

Non-example: on $\R^2 \setminus \set{0}$ in polar coordinates $(r, \theta)$, the operator $\partial_r^2 + \partial_\theta^2$ is not the Laplace–Beltrami operator of the Euclidean metric $dr^2 + r^2 d\theta^2$.
The formula above gives $\Delta f = \partial_r^2 f + \frac{1}{r} \partial_r f + \frac{1}{r^2} \partial_\theta^2 f$.

With this sign convention $\Delta$ is the usual Laplacian $\sum_i \partial_i^2$ on $\R^n$ and $-\Delta$ has nonnegative spectrum on a compact manifold.
Many geometers use the opposite sign, $\Delta = -\operatorname{div} \operatorname{grad}$.
On functions, $-\Delta$ equals the Hodge Laplacian $\delta d$.
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<document source="https://tamnd.github.io/math2/node/PAJEHWXW6M.md" tag="PAJEHWXW6M" kind="concept">
# Symmetric function

## Summary

A symmetric function is a formal power series of bounded degree in infinitely many variables $x_1, x_2, \ldots$, see the [[definition|NS852SSQ00]].
It does not change under any permutation of the variables.
The symmetric functions form the ring $\Lambda$, which collects the [[symmetric polynomials|0VHSHB9B6J]] in all numbers of variables at once.
The ring has several bases indexed by [[integer partitions|0JV30XN73Z]]: monomial, [[elementary|0X5QVCF06N]], complete homogeneous and [[Schur functions|43VFMJCDYJ]].
Over $\Q$, the [[power sum|0ZNXZB4HNB]] symmetric functions give one more basis.
It is a central object of [[algebraic combinatorics|VPAQZ0XW74]] and of the representation theory of symmetric and general linear groups.

## Explanations

### L1 human

A symmetric polynomial, such as $x_1^2 + x_2^2 + x_3^2$, stays the same when the variables are shuffled.
Identities between symmetric polynomials usually do not depend on the number of variables.
So one works with infinitely many variables at once, as in $x_1^2 + x_2^2 + x_3^2 + \cdots$.
These infinite sums of bounded degree are the symmetric functions.
Setting all but finitely many variables to $0$ gives back the symmetric polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/PAJTMQ604P.md" tag="PAJTMQ604P" kind="concept">
# Combinatorial design

## Summary

A combinatorial design is a finite set together with a family of subsets, called blocks, that satisfies a balance condition.
A typical condition is that every pair of points lies in the same number of blocks.
Design theory studies when such systems exist and how to build them.
The main families include [[block designs|4JCSS2Z1BA]], [[Steiner systems|7BPSC4T4M3]], [[finite projective planes|2HWB7P1DZN]] and, in a wider sense, [[Latin squares|CDVRMVED2V]].
Designs are used in the planning of experiments, in coding theory and in finite geometry.

## Explanations

### L1 human

Suppose you want to test $7$ products in groups of $3$ so that every two products are compared in exactly one group.
This is possible with $7$ groups: $\set{1,2,4}$, $\set{2,3,5}$, $\set{3,4,6}$, $\set{4,5,7}$, $\set{5,6,1}$, $\set{6,7,2}$, $\set{7,1,3}$.
Such a fair arrangement is a combinatorial design; this one is the Fano plane.
</document>

<document source="https://tamnd.github.io/math2/node/PANSMHE0SW.md" tag="PANSMHE0SW" kind="concept">
# Contingency table

## Summary

A contingency table counts how many observations fall into each combination of categories of two or more categorical variables.
It is the basic form of data for studying the association between categorical variables.
The chi-squared test and [[Fisher's exact test|J6JP5CJNPX]] test independence of the variables from such a table.
Larger tables are analysed with log-linear models.

## Explanations

### L1 human

Ask $100$ people whether they smoke and whether they exercise.
Write the answers in a table with two rows, smoker and non-smoker, and two columns, exercises and does not.
Each cell holds the number of people with that pair of answers, and the four numbers add up to $100$.
Comparing the rows shows whether smoking and exercise seem related.
</document>

<document source="https://tamnd.github.io/math2/node/PAP76H0ND9.md" tag="PAP76H0ND9" kind="statement">
# Frobenius reciprocity

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
For a finite group $K$ and functions $\alpha, \beta \colon K \to \C$ write $\ip{\alpha}{\beta}_K = \frac{1}{\abs{K}} \sum_{k \in K} \alpha(k) \overline{\beta(k)}$.
Let $\theta$ be a class function on $H$, let $\psi$ be a class function on $G$, and let $\operatorname{Ind}_H^G \theta$ be the [[induced|JJP4JFH3C6]] class function.
Then
$$\ip{\operatorname{Ind}_H^G \theta}{\psi}_G = \ip{\theta}{\psi|_H}_H,$$
where $\psi|_H$ is the restriction of $\psi$ to $H$.

## Remarks

The proof is a direct computation: insert the formula $\operatorname{Ind}_H^G \theta(g) = \frac{1}{\abs{H}}\sum_{x \in G,\ x^{-1}gx \in H} \theta(x^{-1}gx)$, substitute $k = x^{-1} g x$, and use that $\psi$ is constant on [[conjugacy classes|0DEBXAJ5SA]].
The module form holds over any field $F$: for an $F[H]$-module $W$ and an $F[G]$-module $V$, $\Hom_G(\operatorname{Ind}_H^G W, V) \cong \Hom_H(W, V|_H)$ as vector spaces.
Example: for irreducible [[characters|RMWMYBY8TC]] $\theta$ of $H$ and $\psi$ of $G$, the multiplicity of $\psi$ in $\operatorname{Ind}_H^G \theta$ equals the multiplicity of $\theta$ in $\psi|_H$.
For $H = \set{1}$ and $\theta$ trivial this gives that each irreducible $\psi$ occurs $\psi(1)$ times in the regular representation.

## Explanations

### L1 human

Inducing goes up from a subgroup to the group, and restricting goes down.
Frobenius reciprocity says that these two moves are mirror images: counting how often one piece appears after going up gives the same number as counting after going down.
This turns hard questions about the big group into easier questions about a subgroup.

### L2 human

In the language of categories, induction is left adjoint to restriction.
The formula is used constantly to decompose induced characters, for instance in the proofs of [[Brauer's theorem on induced characters|V8JVWE87TE]] and [[Artin's induction theorem|2CABBY4EEB]].
It also gives Mackey-type computations and is the basis of [[Clifford theory|SAH4PD2J9T]].
</document>

<document source="https://tamnd.github.io/math2/node/PAQ4ZX2TX1.md" tag="PAQ4ZX2TX1" kind="definition">
# Natural deduction

## Definition

Consider formulas of [[propositional logic|09K9MWD2QQ]] with the [[connectives|QS0XVAN9BG]] $\land$, $\lor$, $\to$ and the constant $\bot$ for falsity; $\neg \varphi$ abbreviates $\varphi \to \bot$.
A *natural deduction* derivation is a finite tree of formulas.
Each leaf is an assumption, and each inner node follows from its children by one of these rules:

1. $\land$-introduction: from $\varphi$ and $\psi$, infer $\varphi \land \psi$. $\land$-elimination: from $\varphi \land \psi$, infer $\varphi$, and also $\psi$.
2. $\lor$-introduction: from $\varphi$, infer $\varphi \lor \psi$ and $\psi \lor \varphi$.
3. $\lor$-elimination: from $\varphi \lor \psi$, a derivation of $\chi$ from $\varphi$, and one of $\chi$ from $\psi$, infer $\chi$, discharging $\varphi$ and $\psi$.
4. $\to$-introduction: from a derivation of $\psi$ from the assumption $\varphi$, infer $\varphi \to \psi$, discharging $\varphi$.
5. $\to$-elimination, or modus ponens: from $\varphi$ and $\varphi \to \psi$, infer $\psi$.
6. $\bot$-elimination: from $\bot$, infer any formula $\varphi$.
7. Reductio ad absurdum: from a derivation of $\bot$ from the assumption $\neg \varphi$, infer $\varphi$, discharging $\neg \varphi$.

A discharged assumption is no longer open.
One writes $\Gamma \vdash \varphi$ if some derivation has root $\varphi$ and all its open assumptions in $\Gamma$.

## Remarks

A derivation of $\vdash \varphi \land \psi \to \psi \land \varphi$: assume $\varphi \land \psi$.
By $\land$-elimination get $\psi$ and $\varphi$, and by $\land$-introduction get $\psi \land \varphi$.
Then $\to$-introduction discharges the assumption, so no open assumption remains.
The tree with the single node $p$ is a derivation of $p$, but only from the open assumption $p$; it does not show $\vdash p$.
Without rule 7, the system is natural deduction for [[intuitionistic logic|Q11N8X99FK]]; with it, the system is classical.
Quantifier rules extend the system to [[predicate logic|086QKPF0A6]].
</document>

<document source="https://tamnd.github.io/math2/node/PAX9SHDHDW.md" tag="PAX9SHDHDW" kind="concept">
# Minkowski functional

## Summary

The Minkowski functional, or gauge, of an [[absorbing set|NM1BCK3ZX9]] $K$ in a vector space is a function on the space.
Its value at $x$ is the smallest factor by which $K$ must be scaled to contain $x$.
For convex sets it is sublinear, and for convex [[balanced|2TDSHJJYS4]] sets it is a [[seminorm|FP5GVCN8GV]].
It is the bridge between convex neighbourhoods of $0$ and seminorms, and so it is the main tool for describing [[locally convex spaces|B6GY2KJMWW]] by seminorms.

## Explanations

### L1 human

Take a convex shape around the origin, for example a disc or a square.
To measure a point $x$, blow up or shrink the shape until its boundary just reaches $x$; the scaling factor is the gauge of $x$.
If the shape is the unit disc, the gauge is the ordinary length.
If the shape is the square $[-1, 1]^2$, the gauge of $(x, y)$ is $\max(\abs{x}, \abs{y})$.
So every symmetric convex shape gives its own way of measuring size.
</document>

<document source="https://tamnd.github.io/math2/node/PAXCH4FFSV.md" tag="PAXCH4FFSV" kind="concept">
# Double series

## Summary

A double series $\sum_{m,n} a_{m,n}$ adds the terms of a [[double sequence|B70G8FYPG0]].
Its partial sums are sums over rectangles $\set{0, \ldots, M} \times \set{0, \ldots, N}$, and it converges when these tend to a limit as $M$ and $N$ both grow.
Double series arise from products of series, from power series in two variables, and from sums over lattices.
When the terms are absolutely summable, the double sum equals each iterated sum, by [[Fubini's theorem for series|WW53KG71KH]].
The exact definition is in [[double series|CWP26Y1TCZ]].

## Explanations

### L1 human

A double series adds up all the numbers of an infinite table.
One adds the entries in a growing rectangle in the top-left corner and asks whether these sums settle down.
For example, the table with entries $1/(2^m 3^n)$ has rectangle sums that approach $2 \cdot \tfrac32 = 3$.
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<document source="https://tamnd.github.io/math2/node/PAXJGJYDE3.md" tag="PAXJGJYDE3" kind="definition">
# Intrinsic volume

## Definition

Let $n \ge 1$ and let $K$ be a nonempty compact convex subset of $\R^n$ (a [[convex body|437FSG3419]]).
For $m \ge 0$ let $\kappa_m$ be the volume of the closed unit ball of $\R^m$, so $\kappa_0 = 1$, $\kappa_1 = 2$, $\kappa_2 = \pi$.
For $j = 0, 1, \dots, n$, the $j$-th *intrinsic volume* of $K$ is
$$V_j(K) = \binom{n}{j} \frac{W_{n-j}(K)}{\kappa_{n-j}},$$
where $W_{n-j}$ is the $(n-j)$-th [[quermassintegral|H1K36H9ZWB]].

## Remarks

Equivalently, with $B$ the closed unit ball, the Steiner formula reads
$$\operatorname{vol}(K + tB) = \sum_{j=0}^n \kappa_{n-j} V_j(K)\, t^{n-j} \quad (t \ge 0).$$
Special values: $V_n(K) = \operatorname{vol}(K)$, $V_{n-1}(K)$ is half the surface area, and $V_0(K) = 1$.
If $K$ lies in a $k$-dimensional affine subspace, then $V_j(K)$ is the same whether computed in $\R^n$ or in that subspace, and $V_j(K) = 0$ for $j > k$.

Example: the unit square $K = [0,1]^2$ in $\R^2$.
Its parallel body has area $1 + 4t + \pi t^2$, and the formula above gives $V_2(K) + 2 V_1(K)\, t + \pi V_0(K)\, t^2$.
So $V_2(K) = 1$, $V_1(K) = 2$ (half the perimeter) and $V_0(K) = 1$.

Non-example: $K = \set{0, 1} \subseteq \R$ is compact but not convex.
The length of $K + tB$ is $4t$ for $t \le \frac12$ and $1 + 2t$ for $t \ge \frac12$, which is not one polynomial, so there are no intrinsic volumes of $K$ in this sense.
</document>

<document source="https://tamnd.github.io/math2/node/PB0DSH9XCG.md" tag="PB0DSH9XCG" kind="concept">
# Motzkin number

## Summary

The Motzkin number $M_n$ counts certain [[lattice paths|040WY0TT2G]] from $(0, 0)$ to $(n, 0)$, see the [[definition|PHA1S6GQGV]].
The paths use up, down and flat steps and never go below the horizontal axis.
It also counts the ways to draw non-crossing chords between $n$ points on a circle.
The sequence starts $1, 1, 2, 4, 9, 21, 51, 127$.
The Motzkin numbers are a close relative of the [[Catalan numbers|0YWZJEABB2]], which count the same paths without flat steps.

## Explanations

### L1 human

Imagine a walker who takes $n$ steps to the right, and each step goes up, down or stays level.
The walker starts and ends at height $0$ and never goes underground.
For $n = 3$ the possible walks are level-level-level, up-down-level, level-up-down and up-level-down.
So $M_3 = 4$.
</document>

<document source="https://tamnd.github.io/math2/node/PB9JXQ2KC3.md" tag="PB9JXQ2KC3" kind="concept">
# Serre fibration

## Summary

A Serre fibration is a continuous map $p \colon E \to B$ that has the [[homotopy lifting property|G4QDM01P6J]] with respect to all cubes $I^n$.
It is the weak form of a fibration used most often in homotopy theory.
Every [[Hurewicz fibration|Y773B7EQA7]] and every [[fiber bundle|CX4M8K61AB]] is a Serre fibration.
For a Serre fibration the homotopy groups of the fiber, the total space and the base fit into the [[long exact sequence of homotopy groups|E0RV5D1GQ0]].

## Explanations

### L1 human

Think of $E$ as lying above $B$, with $p$ pressing each point of $E$ down to $B$.
In a Serre fibration, every deformation in $B$ of a cube-shaped family of points can be followed by a deformation in $E$.
One only needs to choose the starting position in $E$.
The projection of the product $B \times F$ onto $B$ is the simplest example: one keeps the $F$ coordinate fixed and moves the $B$ coordinate.
Covering maps, such as $\R \to S^1$, $t \mapsto e^{2\pi i t}$, are also Serre fibrations.
</document>

<document source="https://tamnd.github.io/math2/node/PBE65H0506.md" tag="PBE65H0506" kind="definition">
# Pseudo-element

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $A$ be an object of $\mathcal{A}$.
Consider the [[morphisms|859F0TFVA3]] with codomain $A$.
Two such morphisms $x \colon X \to A$ and $y \colon Y \to A$ are *equivalent*, written $x \sim y$, if there are an object $Z$ and [[epimorphisms|RVXDSY04KG]] $u \colon Z \to X$ and $v \colon Z \to Y$ with $x \circ u = y \circ v$.
This is an equivalence relation.
A *pseudo-element* of $A$ is an equivalence class for $\sim$; one writes $x \in_p A$.
For a morphism $f \colon A \to B$ and $x \in_p A$, the pseudo-element $f(x) \in_p B$ is the class of $f \circ x$.
The *zero pseudo-element* of $A$ is the class of a [[zero morphism|YF9F9EG6M3]] $0 \to A$.

## Remarks

Transitivity of $\sim$ uses that pullbacks of epimorphisms are epimorphisms in an abelian category.
A morphism $x$ is equivalent to a zero morphism exactly when $x = 0$.
The basic rules are these.
A morphism $f \colon A \to B$ is a monomorphism if and only if $f(x) = 0$ implies $x = 0$.
It is an epimorphism if and only if every $y \in_p B$ is $f(x)$ for some $x \in_p A$.
A sequence $A \xrightarrow{f} B \xrightarrow{g} C$ is exact if and only if $g \circ f = 0$ and every $y \in_p B$ with $g(y) = 0$ is $f(x)$ for some $x \in_p A$.

Example: in the category of abelian groups, let $x, y \colon \Z \to \Z$ be $x(n) = n$ and $y(n) = -n$.
Take $Z = \Z$, $u = \id$ and $v = y$, both epimorphisms; then $x \circ u = y \circ v$, so $x \sim y$.
More generally, two morphisms into an abelian group $A$ are equivalent exactly when they have the same image, so the pseudo-elements of $A$ correspond to the subgroups of $A$.

Non-example: the morphisms $x(n) = n$ and $w(n) = 2n$ from $\Z$ to $\Z$ are not equivalent.
If $x \circ u = w \circ v$ with $u, v$ epimorphisms, then $x$ and $w$ have the same image, but these images are $\Z$ and $2\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/PBEBXNHCPN.md" tag="PBEBXNHCPN" kind="definition">
# Haar system

## Definition

Let $\psi \colon \R \to \R$ be the [[step function|0AJXYH4DDQ]] with $\psi = 1$ on $[0, \tfrac12)$, $\psi = -1$ on $[\tfrac12, 1)$ and $\psi = 0$ elsewhere.
For integers $n \ge 0$ and $0 \le k < 2^n$, let
$$h_{n,k}(t) = 2^{n/2} \, \psi(2^n t - k), \qquad t \in [0, 1].$$
The *Haar system* on $[0, 1]$ is the family consisting of the constant function $1$ and all the functions $h_{n,k}$.

## Remarks

The function $h_{n,k}$ is supported on the dyadic interval $[k 2^{-n}, (k+1) 2^{-n}]$, where it equals $2^{n/2}$ on the left half and $-2^{n/2}$ on the right half.
The Haar system is an [[orthonormal basis|0S9GGTR0EX]] of $L^2([0, 1])$ (see [[Lp space|SASP14MA1Z]]).
On the whole line, the functions $2^{j/2}\psi(2^j t - k)$ with $j, k \in \Z$ form an orthonormal basis of $L^2(\R)$.

Example: $h_{1,0}$ and $h_{1,1}$ are orthogonal, since their supports $[0, \tfrac12]$ and $[\tfrac12, 1]$ meet in one point.
Also $h_{0,0}$ and $h_{1,0}$ are orthogonal: on $[0, \tfrac12]$ the function $h_{0,0}$ is constant $1$, and $h_{1,0}$ has integral $0$ there.
And $\norm{h_{1,0}}^2 = 2 \cdot \tfrac12 = 1$.

Non-example: the family of all $h_{n,k}$ without the constant function $1$ is orthonormal but not a basis of $L^2([0, 1])$.
Indeed, the constant $1$ is orthogonal to every $h_{n,k}$, since each $h_{n,k}$ has integral $0$.
</document>

<document source="https://tamnd.github.io/math2/node/PBGF9TXMB7.md" tag="PBGF9TXMB7" kind="concept">
# Symmetric algebra

## Summary

The symmetric algebra of a vector space $V$ is the [[tensor algebra|P4J5JZDW0B]] of $V$ made commutative.
It is the most general commutative algebra generated by $V$.
If $V$ has a finite basis $x_1, \ldots, x_n$, it is the [[polynomial ring|0SDC934R52]] $F[x_1, \ldots, x_n]$, so it gives a coordinate-free description of polynomials.

## Explanations

### L1 human

In the tensor algebra the order of the factors matters, so $x \otimes y$ and $y \otimes x$ are different.
The symmetric algebra forgets the order: products of vectors behave like products of numbers.
With a basis $x, y$ of $V$, the elements of degree $2$ are the combinations of $x^2$, $xy$ and $y^2$.
So the symmetric algebra is the ring of polynomials in the basis vectors.
</document>

<document source="https://tamnd.github.io/math2/node/PBGTTAVMMT.md" tag="PBGTTAVMMT" kind="problem">
# Word problem for groups

## Problem

Let $G = \langle S \mid R \rangle$ be a group given by a finite [[presentation|0JGEQXQ4AC]], with a finite set $S$ of generators and a finite set $R$ of relators.
A *word* is a finite sequence of elements of $S \cup S^{-1}$, that is, an element of the [[free group|0KTW26C80J]] $F(S)$.
The word problem for $G$ asks for an algorithm that decides, for every word $w$, whether $w$ represents the identity element of $G$.
Equivalently: is the set of words that represent the identity a [[computable set|PPGD5KW8JC]]?

## Remarks

The answer depends on the group.
The word problem is solvable for free groups, by free reduction, and for finite groups.
Magnus showed that it is solvable for every group with one defining relator.
Dehn gave an algorithm for the fundamental groups of closed orientable surfaces of genus at least $2$.
Novikov (1955) and Boone (1958) independently constructed finitely presented groups whose word problem is unsolvable.
The set of words that represent the identity is always computably enumerable, so the word problem asks only whether its complement is too.
Whether the word problem is solvable does not depend on the chosen finite presentation of $G$.

## Explanations

### L1 human

A group can be described by a few generators and a few rules saying which products equal the identity.
The question is whether there is a mechanical way to tell, for any product of generators, whether it equals the identity.
For many familiar groups there is, but there are groups, described by finitely many rules, where no computer program can answer the question.
</document>

<document source="https://tamnd.github.io/math2/node/PBK2XY4M0A.md" tag="PBK2XY4M0A" kind="concept">
# Fractional factorial design

## Summary

A fractional factorial design runs only a chosen fraction of the combinations of a [[factorial experiment|EHJ3R589ZX]].
An example is $2^{k-p}$ of the $2^k$ runs of a two-level design.
The fraction is chosen by a defining relation, so that the main effects and low-order interactions can still be estimated.
The price is that they are aliased with higher-order interactions.
The resolution of the design measures how severe the aliasing is.
Such designs make screening experiments with many factors affordable, on the assumption that high-order interactions are negligible.

## Explanations

### L1 human

With seven factors at two levels, a full factorial needs $128$ runs, often far too many.
A well chosen eighth of it, $16$ runs, still measures each factor's main effect.
The price is that some effects become indistinguishable: in the data, a main effect looks exactly like a certain interaction of other factors.
If those interactions are small, nothing important is lost.
</document>

<document source="https://tamnd.github.io/math2/node/PBPFVY20WT.md" tag="PBPFVY20WT" kind="statement">
# Muirhead's inequality

## Statement

Let $a = (a_1, \dots, a_n)$ and $b = (b_1, \dots, b_n)$ be vectors of nonnegative real numbers.
For $x_1, \dots, x_n > 0$, write
$$[a](x) = \frac{1}{n!} \sum_{\sigma \in S_n} x_{\sigma(1)}^{a_1} x_{\sigma(2)}^{a_2} \cdots x_{\sigma(n)}^{a_n},$$
where $S_n$ is the set of [[permutations|0BQMWCNS8S]] of $\set{1, \dots, n}$.
Then $[b](x) \le [a](x)$ for all $x_1, \dots, x_n > 0$ if and only if $a$ [[majorizes|5K7VT2VKZT]] $b$.

## Remarks

Proof idea: suppose $a$ majorizes $b$.
Then $b$ can be reached from $a$ by finitely many steps, each of which moves two entries closer together while keeping their sum.
For one such step it suffices to compare two variables, where the claim reduces to $x^{s} y^{t} + x^{t} y^{s} \ge x^{s'} y^{t'} + x^{t'} y^{s'}$ for $s + t = s' + t'$ and $\abs{s - t} \ge \abs{s' - t'}$, which follows by factoring.
Conversely, letting all $x_i$ but some tend to $0$ or $\infty$ shows that the partial sum conditions of majorization are needed.

Example: $a = (2, 0)$ majorizes $b = (1, 1)$, which gives $\frac{1}{2}(x^2 + y^2) \ge x y$.
With $a = (n, 0, \dots, 0)$ and $b = (1, 1, \dots, 1)$ one gets $\frac{1}{n}\sum_i x_i^n \ge x_1 x_2 \cdots x_n$, the [[AM–GM inequality|0DRF0F3XC9]] for $x_1^n, \dots, x_n^n$.

## Explanations

### L1 human

Muirhead's inequality compares symmetric sums of monomials in positive numbers.
Each such sum is labelled by its exponent vector.
The sum whose exponents are more spread out is never the smaller one, and this order is exactly majorization.

### L2 human

Muirhead's inequality settles many symmetric polynomial inequalities at once, and it is a standard tool in olympiad problems.
To use it, write both sides as combinations of symmetric sums $[a]$ and check majorization of the exponent vectors.
It does not apply directly when the two sides have different total degree, or when some coefficients are negative; then one combines it with other tools such as [[Schur's inequality|958TMJZ51Y]].
</document>

<document source="https://tamnd.github.io/math2/node/PBSMWBFFZH.md" tag="PBSMWBFFZH" kind="definition">
# Discrete spectrum

## Definition

Let $X$ be a complex Banach space, and let $T \colon X \to X$ be a bounded operator with [[spectrum|BMKATJ802A]] $\sigma(T)$.
A point $\lambda \in \sigma(T)$ is a *normal eigenvalue* of $T$ if:

1. $\lambda$ is an isolated point of $\sigma(T)$, and
2. the [[Riesz projector|HNW4WRSEEX]] $P_{\set{\lambda}}$ of $T$ for $\set{\lambda}$ has finite rank.

The *discrete spectrum* $\sigma_{disc}(T)$ is the set of normal eigenvalues of $T$.

## Remarks

A normal eigenvalue is an eigenvalue, and its algebraic multiplicity, the rank of $P_{\set{\lambda}}$, is finite.

Example: if $T$ is a [[compact operator|M2NEP1K5AE]], then every nonzero point of $\sigma(T)$ is a normal eigenvalue.
For $D(x_n) = (x_n / n)$ on $\ell^2$, each $1/n$ is isolated in $\sigma(D)$, and $P_{\set{1/n}}$ is the projection onto the span of $e_n$, of rank $1$.

Non-example: if $X$ is infinite-dimensional, then $\sigma(I) = \set{1}$ and $1$ is isolated, but $P_{\set{1}} = I$ has infinite rank, so clause 2 fails.
For $D$ above, $0$ fails clause 1, since $1/n \to 0$.

For a self-adjoint operator $T$ on a Hilbert space, $\sigma_{disc}(T)$ is the set of isolated eigenvalues of finite multiplicity.
Then $\sigma(T)$ is the disjoint union of $\sigma_{disc}(T)$ and the [[essential spectrum|FRZG0E15SK]].
</document>

<document source="https://tamnd.github.io/math2/node/PBT8VRJADK.md" tag="PBT8VRJADK" kind="concept">
# F-distribution

## Summary

The F-distribution is the distribution of a ratio of two independent chi-squared variables, each divided by its degrees of freedom.
It has two parameters, the numerator and the denominator degrees of freedom.
It is the reference distribution of the F-test, of analysis of variance and of tests that compare two variances in normal models.

## Explanations

### L1 human

Take two independent estimates of the same variance, each based on its own number of normal observations.
Their ratio should be near $1$, but it fluctuates by chance, and the F-distribution describes exactly how much.
A ratio far above $1$ is evidence that the two variances are not equal after all.
For example, the square of a Student $t$ variable with $\nu$ degrees of freedom has the F-distribution with $1$ and $\nu$ degrees of freedom.
</document>

<document source="https://tamnd.github.io/math2/node/PBYCBVG5CG.md" tag="PBYCBVG5CG" kind="definition">
# Simple polytope

## Definition

Let $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d \ge 1$.
An edge of $P$ is a face of dimension $1$.
$P$ is *simple* if every [[vertex|R7V3S9V891]] of $P$ lies in exactly $d$ edges of $P$.

## Remarks

Equivalently, every vertex of $P$ lies in exactly $d$ [[facets|DPDXN2FJC4]] of $P$.
Every vertex of a $d$-polytope lies in at least $d$ edges, so simple polytopes are those where this lower bound is always reached.

Example: the cube $[0,1]^3$ is simple.
At the vertex $0$ the edges are the three segments from $0$ to $e_1$, $e_2$ and $e_3$, and by symmetry every vertex lies in $3$ edges.

Non-example: the octahedron $\operatorname{conv}\set{\pm e_1, \pm e_2, \pm e_3}$ is not simple, since the vertex $e_1$ lies in the $4$ edges joining it to $\pm e_2$ and $\pm e_3$.

If $P$ contains the origin in its interior, then $P$ is simple if and only if its polar polytope is simplicial.
Every polygon is simple.
</document>

<document source="https://tamnd.github.io/math2/node/PBZDR2E342.md" tag="PBZDR2E342" kind="definition">
# Choquet theory

## Definition

Let $E$ be a real Hausdorff [[locally convex space|B6GY2KJMWW]], and let $K \subseteq E$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].
Let $\mu$ be a Borel [[probability measure|059TJN0NZ1]] on $K$, and let $x \in K$.
The measure $\mu$ *represents* $x$, and $x$ is the *barycenter* of $\mu$, if
$$\ell(x) = \int_K \ell \, d\mu$$
for every [[continuous linear form|03TM135G0N]] $\ell \colon E \to \R$.
*Choquet theory* is the study of representing measures of points of $K$ that are concentrated on the set $\operatorname{ext} K$ of [[extreme points|3MSGFJM2NX]] of $K$.

## Remarks

Every point of $K$ is represented by the [[Dirac measure|0ACRGYNHSR]] at that point; the theory asks for measures on the extreme points.
Choquet's theorem: if $K$ is metrizable, then $\operatorname{ext} K$ is a $G_\delta$ set, and every $x \in K$ is represented by a probability measure $\mu$ with $\mu(\operatorname{ext} K) = 1$.
The Bishop–de Leeuw theorem extends this to non-metrizable $K$ in a weaker form.
For metrizable $K$, the measure on $\operatorname{ext} K$ is unique for every $x$ exactly when $K$ is a Choquet simplex.

Example: let $K = [0, 1] \subseteq \R$, with $\operatorname{ext} K = \set{0, 1}$.
The measure $\mu = (1 - x) \delta_0 + x \delta_1$ represents $x$.
The check: for $\ell(t) = c t$, one gets $\int \ell \, d\mu = c x = \ell(x)$.

Non-example: for $x = 1/2$, the measure $\delta_0$ is concentrated on $\operatorname{ext} K$ but does not represent $x$, since its barycenter is $0$.

For the square $K = [0, 1]^2$, the centre is represented by $\frac12(\delta_{(0,0)} + \delta_{(1,1)})$ and by $\frac12(\delta_{(1,0)} + \delta_{(0,1)})$, so uniqueness fails.

## Explanations

### L1 human

A representing measure writes a point as an average of other points of the set.
Choquet theory asks for averages of extreme points only, the corners of the set.
It is the infinite-dimensional version of barycentric coordinates.
</document>

<document source="https://tamnd.github.io/math2/node/PBZK2A09X4.md" tag="PBZK2A09X4" kind="definition">
# Well-quasi-ordering

## Definition

Let $X$ be a set and let $\le$ be a [[preorder|0VABYWJWH9]] on $X$.
The preorder is a *well-quasi-ordering* when for every infinite [[sequence|0G9EKM0E15]] $x_1, x_2, x_3, \ldots$ of elements of $X$ there are indices $i < j$ with $x_i \le x_j$.

## Remarks

Example: every finite set with any preorder is well-quasi-ordered.
Check: an infinite sequence in a finite set has $x_i = x_j$ for some $i < j$, and then $x_i \le x_j$ by reflexivity.

Non-example: the positive integers ordered by divisibility are well-founded but not well-quasi-ordered.
The sequence of primes $2, 3, 5, 7, \ldots$ has no $i < j$ with $p_i \mid p_j$.

Equivalent form: every infinite sequence has an infinite subsequence $x_{i_1} \le x_{i_2} \le \cdots$ with $i_1 < i_2 < \cdots$.
Another equivalent form: the strict part of $\le$ is well-founded and every [[antichain|03HF498P1J]] is finite.
A sequence with no $i < j$ such that $x_i \le x_j$ is called bad, so a well-quasi-ordering is a preorder with no bad sequences.
The abbreviation wqo is common.
</document>

<document source="https://tamnd.github.io/math2/node/PBZTBWZ4GS.md" tag="PBZTBWZ4GS" kind="concept">
# Recurrence and transience of states

## Summary

A state of a [[Markov chain|0G48PYM96Q]] is recurrent if the chain, started there, returns to it with probability one, and transient otherwise.
A recurrent state is visited infinitely often, while a transient state is visited only finitely often, almost surely.
Recurrent states are further split into positive recurrent and null recurrent, according to whether the mean return time is finite.
In an irreducible chain all states have the same type, which decides whether a [[stationary distribution|0KRH5157WP]] exists.

## Explanations

### L1 human

Imagine a walker moving at random between rooms and ask whether she surely comes back to the room where she started.
If yes, the room is recurrent, and she will in fact come back again and again forever.
If there is a positive chance that she never returns, the room is transient, and after some last visit she leaves for good.
The fair coin walk on the integers comes back to $0$ for sure, while a walk that steps right with probability $2/3$ drifts away and may never return.
</document>

<document source="https://tamnd.github.io/math2/node/PC6HG1JEGM.md" tag="PC6HG1JEGM" kind="concept">
# Comma category

## Summary

Let $S \colon \mathcal{A} \to \mathcal{C}$ and $T \colon \mathcal{B} \to \mathcal{C}$ be [[functors|28NQ0KRHD3]].
The comma category $(S \downarrow T)$ has as objects the morphisms $S(A) \to T(B)$ of $\mathcal{C}$.
Its morphisms are the pairs of morphisms that make the evident square commute.
It turns morphisms of one category into objects of a new category.
[[Slice categories|1TC7W7BQT3]], arrow categories and [[categories of elements|S9EJKKRHW1]] are special comma categories.
[[Universal properties|EN4DD2FAJH]] and [[adjoint functors|TZHF6ZM92S]] can be expressed as initial or terminal objects of comma categories.

## Explanations

### L1 human

A comma category collects arrows of a category $\mathcal{C}$ that go from one kind of object to another, and treats each arrow as a new object.
An arrow between two such objects is a pair of maps at the two ends that fits with them in a commuting square.
For example, taking all arrows of $\mathcal{C}$ at both ends gives the arrow category, whose objects are morphisms $h \colon X \to Y$ of $\mathcal{C}$.
In the arrow category of sets, the map $\R \to \R$, $x \mapsto x^2$, is one object, and the inclusion $\Z \to \R$ is another.
</document>

<document source="https://tamnd.github.io/math2/node/PCE6XQTT6R.md" tag="PCE6XQTT6R" kind="statement">
# Bessel's integral

## Statement

Let $n$ be an integer and let $J_n$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $n$.
Then for every $z \in \C$,
$$J_n(z) = \frac{1}{\pi} \int_0^\pi \cos(n\tau - z \sin \tau)\, d\tau.$$

## Remarks

The generating function $\exp\left( \frac{z}{2} (t - t^{-1}) \right) = \sum_{k \in \Z} J_k(z)\, t^k$ with $t = e^{i\tau}$ gives $e^{iz \sin \tau} = \sum_{k} J_k(z)\, e^{ik\tau}$, the [[Jacobi–Anger expansion|KFHYQM8T7E]].
So $J_n(z)$ is the $n$-th Fourier coefficient $\frac{1}{2\pi} \int_{-\pi}^{\pi} e^{i(z \sin \tau - n\tau)}\, d\tau$.
Writing $e^{i(z \sin \tau - n\tau)} = \cos(n\tau - z\sin\tau) - i \sin(n\tau - z\sin\tau)$, the sine term is odd in $\tau$ and the cosine term is even, which gives the formula on $[0, \pi]$.
Example: for $n = 0$ and $z = 0$ the integral is $\frac{1}{\pi} \int_0^\pi 1\, d\tau = 1 = J_0(0)$.
The hypothesis that $n$ is an integer is needed: for non-integer $\nu$ the same integral defines the Anger function, which differs from $J_\nu$.
For $\Re z > 0$ and any $\nu$, Schläfli's extension reads $J_\nu(z) = \frac{1}{\pi} \int_0^\pi \cos(\nu\tau - z \sin\tau)\, d\tau - \frac{\sin(\nu\pi)}{\pi} \int_0^\infty e^{-z \sinh t - \nu t}\, dt$.

## Explanations

### L1 human

This formula writes a Bessel function as an average of a cosine over half a turn.
It was Bessel's own definition, coming from a problem in planetary motion.
It shows at once that $\abs{J_n(x)} \le 1$ for real $x$, since the integrand lies between $-1$ and $1$.

### L2 human

The integral turns many questions about $J_n$ into questions about an oscillatory integral.
Applying stationary phase to it gives the large-$x$ asymptotics of $J_n(x)$.
Differentiating under the integral sign checks Bessel's equation directly.
It is also the bridge between Bessel functions and Fourier series of $e^{iz\sin\tau}$, which is how they enter Kepler's equation and frequency modulation.
</document>

<document source="https://tamnd.github.io/math2/node/PCEVB6XV6J.md" tag="PCEVB6XV6J" kind="definition">
# Derived category

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $\operatorname{Ch}(\mathcal{A})$ be the category of [[chain complexes|WQ82C0M3QB]] in $\mathcal{A}$ and [[chain maps|HTK9WQZJ1G]].
The *derived category* of $\mathcal{A}$ is the [[localization|DNFR3YP0XF]] of $\operatorname{Ch}(\mathcal{A})$ at the class of [[quasi-isomorphisms|C9JEC2WQKV]].
That is, it is a category $D(\mathcal{A})$ with a functor $Q \colon \operatorname{Ch}(\mathcal{A}) \to D(\mathcal{A})$ such that:

1. $Q(f)$ is an isomorphism for every quasi-isomorphism $f$;
2. every functor $F \colon \operatorname{Ch}(\mathcal{A}) \to \mathcal{C}$ that sends quasi-isomorphisms to isomorphisms factors as $F = G \circ Q$ for a unique functor $G \colon D(\mathcal{A}) \to \mathcal{C}$.

## Remarks

The localization can be computed in two steps: first pass to the [[homotopy category|9Y5Y92R4D5]] $K(\mathcal{A})$, then invert quasi-isomorphisms there by a calculus of fractions.
A morphism $X \to Y$ in $D(\mathcal{A})$ is then represented by a roof $X \xleftarrow{s} X' \to Y$ with $s$ a quasi-isomorphism.
For general $\mathcal{A}$ the morphisms between two objects might not form a set.
They do form a set for module categories, and for bounded below complexes when $\mathcal{A}$ has enough injectives.
Example: over $\Z$, let $C$ be $\Z \xrightarrow{\cdot 2} \Z$ in degrees $1$ and $0$, and let $\Z/2\Z$ sit in degree $0$.
The quotient map $C_0 = \Z \to \Z/2\Z$ is a chain map, and it is a quasi-isomorphism since $H_0(C) \cong \Z/2\Z$ and $H_1(C) = 0$; so $C \cong \Z/2\Z$ in $D(\mathcal{A})$.
Non-example: $C$ and $\Z/2\Z$ are not isomorphic in $K(\mathcal{A})$, since every chain map $\Z/2\Z \to C$ is zero, because $\Hom(\Z/2\Z, \Z) = 0$; so $K(\mathcal{A})$ is not the derived category.
The bounded variants $D^b$, $D^+$ and $D^-$ are defined in the same way from bounded complexes.
</document>

<document source="https://tamnd.github.io/math2/node/PCH3G8BJ9Y.md" tag="PCH3G8BJ9Y" kind="definition">
# Multiplicative set

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] with identity $1$.
A subset $S \subseteq R$ is a *multiplicative set* (or *multiplicatively closed set*) if:

1. $1 \in S$;
2. $st \in S$ for all $s, t \in S$.

## Remarks

Example: let $\mathfrak p$ be a [[prime ideal|00MHHZ9SNE]] of $R$ and $S = R \setminus \mathfrak p$.
Then $1 \in S$ because $\mathfrak p$ is a proper ideal.
If $s, t \notin \mathfrak p$, then $st \notin \mathfrak p$ by the definition of a prime ideal, so $st \in S$.

Other examples are the powers $\set{1, f, f^2, \ldots}$ of an element $f \in R$, the [[units|0XVEZDE9J8]] of $R$, and the set of elements of $R$ that are not [[zero divisors|0NNRDW10RZ]].

Non-example: in $\Z$, the set $S = R \setminus 6\Z$ is not multiplicative.
It contains $2$ and $3$ but not $2 \cdot 3 = 6$.
The ideal $6\Z$ is not prime.

Some authors also require $0 \notin S$.
If $0 \in S$, the [[localization|4C1AE84N5N]] $S^{-1}R$ is the zero ring.
</document>

<document source="https://tamnd.github.io/math2/node/PCKK4SFZEK.md" tag="PCKK4SFZEK" kind="concept">
# Skew group ring

## Summary

The skew group ring $R * G$ of a group $G$ acting on a [[ring|0B1261B5ED]] $R$ by automorphisms is like the [[group ring|4NMTSX0XDS]] $R[G]$.
The difference is that group elements and coefficients do not commute: $g r = g(r) g$.
It is the algebraic counterpart of the C*-algebra [[crossed product|8HG8TS46A6]].
It turns a [[group action|0WCY42EV9V]] on a ring into a single ring, and its modules are the $R$-modules with a compatible action of $G$.
It is used in invariant theory, Galois theory of rings and the representation theory of finite groups.

## Explanations

### L1 human

Start with a ring $R$ and a group $G$ that moves the elements of $R$ around by symmetries.
Form sums $\sum r_g g$ as in the group ring.
When a group element passes a coefficient from left to right, it acts on it: $g r = g(r) g$.
If every $g \in G$ acts as the identity map of $R$, then $g r = r g$ for all $g$ and $r$, and this is the group ring.
For the complex numbers with complex conjugation, the skew group ring is the ring of $2 \times 2$ real matrices.
</document>

<document source="https://tamnd.github.io/math2/node/PCM3H6J3EE.md" tag="PCM3H6J3EE" kind="example">
# Sierpiński carpet

## Example

Let $S_0 = [0, 1]^2$.
Divide each square of side $3^{-k}$ in $S_k$ into $9$ equal squares of side $3^{-k-1}$.
Remove the interior of the middle one; the result is $S_{k+1}$.
The *Sierpiński carpet* is $S = \bigcap_{k \ge 0} S_k$.

The set $S$ is a [[continuum|NJQ196M1WR]].
Each $S_k$ is a finite union of closed squares, so it is [[compact|0TVJZYB6SH]].
It is [[connected|0V032VWTHD]], since the $8$ kept squares in each divided square form a ring in which neighbours share an edge.
The intersection of a decreasing sequence of nonempty compact connected sets in the plane is nonempty, compact and connected.

## Remarks

The set $S_k$ consists of $8^k$ squares of area $9^{-k}$, so $S$ has area $\lim_k (8/9)^k = 0$.
Its Hausdorff dimension is $\log 8 / \log 3 \approx 1.893$.
Its [[Lebesgue covering dimension|A3W7RGXXK9]] is $1$.
Sierpiński showed that every compact subset of the plane of dimension at most $1$ embeds in the carpet, so it is a universal plane curve.

## Explanations

### L1 human

Take a square, cut it into a three by three grid, and throw away the middle piece.
Do the same with each of the eight pieces left, and keep going forever.
What remains is a lacy set with holes at every scale.
It has no area, yet it is all in one piece, and it contains a copy of every curve that can be drawn in the plane.
</document>

<document source="https://tamnd.github.io/math2/node/PCMW7KN5P4.md" tag="PCMW7KN5P4" kind="concept">
# Inverse problem

## Summary

An inverse problem asks for the causes of observed effects: to recover the parameters of a model from measured data that the model produces.
The forward problem goes the other way and computes the data from known parameters.
Inverse problems are often ill-posed in the sense of Hadamard: small errors in the data can cause large errors in the answer.
So they need regularization.
Examples are medical imaging, seismic exploration of the Earth, and the [[Calderón problem|R8Y8DDTWBM]].

## Explanations

### L1 human

A doctor sees a CT scan, but the scanner only measures how much X-ray light passes through the body along many lines.
Turning those line measurements back into a picture of the inside is an inverse problem.
The forward direction, from body to measurements, is easy and stable.
The inverse direction can magnify noise: two very different pictures may give almost the same measurements.
A small example: knowing a function $y$, find $x = y'$; a tiny wiggle in $y$ can produce a huge wiggle in its derivative.
</document>

<document source="https://tamnd.github.io/math2/node/PCNPQ7YBDE.md" tag="PCNPQ7YBDE" kind="definition">
# Levi-Civita connection

## Definition

Let $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]], and write $\mathfrak{X}(M)$ for the space of smooth vector fields on $M$.
A *Levi-Civita connection* of $(M, g)$ is an [[affine connection|JX65KSVHSJ]] $\nabla$ on $M$ such that:

1. $\nabla$ is torsion-free: $\nabla_X Y - \nabla_Y X = [X, Y]$ for all $X, Y \in \mathfrak{X}(M)$, where $[X, Y]$ is the [[Lie bracket|68FGQEBD6J]];
2. $\nabla$ is compatible with $g$: $X\bigl(g(Y, Z)\bigr) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$ for all $X, Y, Z \in \mathfrak{X}(M)$.

## Remarks

Example: on $\R^n$ with $g = \sum_i dx_i^2$, let $\nabla_X Y$ be the vector field whose components are $X(Y^1), \ldots, X(Y^n)$.
Then $\nabla_X Y - \nabla_Y X$ has components $X(Y^i) - Y(X^i)$, which is $[X, Y]$, and the product rule gives clause 2 for $g(Y, Z) = \sum_i Y^i Z^i$.

Non-example: on $\R^3$ with the Euclidean metric, let $\nabla_X Y = D_X Y + X \times Y$, where $D$ is the flat connection above and $\times$ is the cross product.
This is an affine connection, and clause 2 holds because $\ip{X \times Y}{Z} + \ip{Y}{X \times Z} = 0$.
Its torsion is $\nabla_X Y - \nabla_Y X - [X, Y] = 2 \, X \times Y$, which is not zero, so clause 1 fails.

The two clauses determine $\nabla$; this is the [[fundamental theorem of Riemannian geometry|78JRW8KTH8]].
In coordinates, $\nabla_{\partial_i} \partial_j = \sum_k \Gamma^k_{ij} \partial_k$ with the [[Christoffel symbols|FZBA1HA9ZM]]
$$\Gamma^k_{ij} = \frac{1}{2} \sum_l g^{kl} \left( \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij} \right).$$
</document>

<document source="https://tamnd.github.io/math2/node/PCT2HY9AAZ.md" tag="PCT2HY9AAZ" kind="concept">
# Generalized polygon

## Summary

A generalized $n$-gon is an incidence structure of points and lines whose incidence graph has diameter $n$ and girth $2n$.
Ordinary $n$-gons are the trivial examples, [[projective planes|DANFNN8C4Y]] are the generalized $3$-gons, and the [[generalized quadrangles|GQJCPFSYVK]] are the generalized $4$-gons.
Jacques Tits introduced them as the [[buildings|6BQJ50NEJ0]] of rank $2$, and the [[Feit–Higman theorem|P2V0XS4F63]] limits the finite thick ones to $n \in \set{2, 3, 4, 6, 8}$.

## Explanations

### L1 human

Draw a triangle: three points, three sides, each side with two points and each point on two sides.
Make a graph whose vertices are the points and the sides, with an edge when a point lies on a side.
This graph is a cycle of length $6$: any two vertices are at distance at most $3$, and the shortest cycle has length $6 = 2 \cdot 3$.
A generalized $3$-gon keeps these two numbers but allows many more points on each line and many more lines through each point.
That is exactly a projective plane: two points always share a line, and two lines always share a point.
</document>

<document source="https://tamnd.github.io/math2/node/PCT8RW4MY4.md" tag="PCT8RW4MY4" kind="definition">
# Digamma function

## Definition

Let $\Gamma$ be the [[gamma function|0H5SYBSED9]], viewed as a [[meromorphic function|0ZZ7ZYDR8W]] on $\C$ with poles at $0, -1, -2, \ldots$ and no zeros.
The *digamma function* is the meromorphic function
$$\psi(z) = \frac{\Gamma'(z)}{\Gamma(z)} = \frac{d}{dz} \log \Gamma(z), \qquad z \in \C \setminus \set{0, -1, -2, \ldots}.$$

## Remarks

Example: from $\Gamma(z+1) = z\,\Gamma(z)$, taking logarithmic derivatives gives $\psi(z+1) = \psi(z) + \frac1z$.
With $\psi(1) = \Gamma'(1) = -\gamma$, where $\gamma$ is the [[Euler–Mascheroni constant|00C5P9JZBK]], this gives $\psi(n+1) = H_n - \gamma$ for every integer $n \ge 1$, where $H_n = 1 + \frac12 + \cdots + \frac1n$ is the [[harmonic number|0NANDX7V02]].
Another value is $\psi(\frac12) = -\gamma - 2\ln 2$.

Non-example of a valid argument: $z = 0$ is a pole of $\Gamma$, so $\psi$ is not defined there.
In fact $\psi$ has a simple pole with residue $-1$ at each of $0, -1, -2, \ldots$.

The function $\psi$ has no other poles, because $\Gamma$ has no zeros.
</document>

<document source="https://tamnd.github.io/math2/node/PCX8R81H96.md" tag="PCX8R81H96" kind="definition">
# State on a C*-algebra

## Definition

Let $A$ be a C*-algebra.
A linear functional $\varphi \colon A \to \C$ is *positive* if $\varphi(a) \ge 0$ for every [[positive element|NA8DF5X3M4]] $a \in A$.
A *state* on $A$ is a positive linear functional $\varphi$ with $\norm{\varphi} = 1$.

## Remarks

A positive linear functional is automatically bounded, so it lies in the [[dual space|4A82S5FVRT]] $A^*$.
If $A$ has a unit, then $\norm{\varphi} = \varphi(1)$ for positive $\varphi$, so a state is a positive linear functional with $\varphi(1) = 1$.
The set $S(A)$ of states of a unital $A$ is convex and weak-* compact; its [[extreme points|3MSGFJM2NX]] are the *pure states*.

Example: let $H$ be a Hilbert space, $A$ the bounded operators on $H$, and $\xi \in H$ a unit vector.
The vector state $\varphi(T) = \ip{T \xi}{\xi}$ is linear, satisfies $\varphi(T^* T) = \norm{T \xi}^2 \ge 0$, and $\varphi(I) = \norm{\xi}^2 = 1$; so it is a state.
On $M_n(\C)$, the normalized trace $\tfrac{1}{n} \tr$ is a state.

Non-example: the trace $\tr$ on $M_2(\C)$ is positive, but $\tr(I) = 2$, so its norm is $2$ and it is not a state.
</document>

<document source="https://tamnd.github.io/math2/node/PCYRWA7SYB.md" tag="PCYRWA7SYB" kind="concept">
# Convergence of an infinite product

## Summary

An [[infinite product|JYV1Y8XX16]] converges when, after finitely many zero factors are set aside, its partial products tend to a nonzero limit.
The nonzero condition makes the theory parallel to series: a product of positive terms converges exactly when the series of logarithms converges.
A convergent product has factors tending to $1$, and its value is $0$ only when some factor is $0$.
The exact definition is in [[convergence of an infinite product|H1Z2TKY3EQ]].

## Explanations

### L1 human

An infinite product converges when multiplying more and more factors brings you closer and closer to a fixed nonzero number.
Products whose partial products shrink to $0$ are said to diverge to $0$, just as a sum can diverge to infinity.
For example, $\tfrac12 \cdot \tfrac23 \cdot \tfrac34 \cdots$ has partial products $1/(N+1)$, so it diverges to $0$.
</document>

<document source="https://tamnd.github.io/math2/node/PCZBE7EX01.md" tag="PCZBE7EX01" kind="statement">
# Poisson integral formula

## Statement

Let $n \ge 2$ and $R > 0$, let $B = \set{x \in \R^n : \abs{x} < R}$ be the [[open ball|0YE5868HZB]], and let $S = \partial B$ be its boundary sphere with surface measure $dS$.
Let $\sigma_{n-1}$ be the [[surface area|0WQE3G7B88]] of the unit sphere in $\R^n$, and for $x \in B$ and $y \in S$ put
$$P(x, y) = \frac{R^2 - \abs{x}^2}{\sigma_{n-1} R \, \abs{x - y}^n}.$$

1. If $u$ is [[continuous|0X3WGX95FE]] on $\overline{B}$ and [[harmonic|0DEB0JMN0W]] on $B$, then $u(x) = \int_S P(x, y) \, u(y) \, dS(y)$ for every $x \in B$.
2. If $g$ is continuous on $S$, then $u(x) = \int_S P(x, y) \, g(y) \, dS(y)$ is harmonic on $B$, and $u(x) \to g(y_0)$ as $x \to y_0$ for every $y_0 \in S$.

## Remarks

Proof idea: $P(\cdot, y)$ is harmonic in $B$, positive, and has integral $1$ over $S$.
It is minus the outward normal derivative, in the second variable, of the Green's function of the ball.
Part 1 follows from Green's representation formula on smaller balls and a limit.
Part 2 holds because $P(x, \cdot)$ concentrates near $y_0$ as $x \to y_0$, so the integral is an approximate identity.
For $n = 2$ and $R = 1$ the formula is $u(re^{i\theta}) = \frac{1}{2\pi}\int_0^{2\pi} \frac{1 - r^2}{1 - 2r\cos(\theta - t) + r^2} \, u(e^{it}) \, dt$, the classical [[Poisson kernel|TCYSWCTYYM]] of the disc.
At $x = 0$ the formula gives the mean value property: $u(0)$ is the average of $u$ over $S$.
Together, parts 1 and 2 solve the [[Dirichlet problem|8D56GQ4AG8]] on a ball, with a unique solution.

## Explanations

### L1 human

If you know a harmonic function on the surface of a ball, you know it inside.
The value at an inside point is a weighted average of the boundary values.
Boundary points close to the inside point get large weights, and far points get small ones.

### L2 human

The Poisson formula is the main tool for local results on harmonic functions.
From it one gets [[Harnack's inequality|YNH832D50W]], gradient estimates, the real analyticity of harmonic functions, and the fact that uniform limits of harmonic functions are harmonic.
It also yields the converse of the mean value property.
It is the basic step in Perron's method, where a function is replaced by its Poisson integral on a small ball.
</document>

<document source="https://tamnd.github.io/math2/node/PD4XXFJSW0.md" tag="PD4XXFJSW0" kind="definition">
# Commutative algebra over a ring

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
A *commutative $R$-algebra* is a pair $(A, f)$ where $A$ is a commutative ring and $f \colon R \to A$ is a [[ring homomorphism|0GGE0Z0JQ1]].
The map $f$ is the *structure map* of the algebra.
For $r \in R$ and $a \in A$ one writes $r a = f(r) a$.
If $(A, f)$ and $(B, g)$ are commutative $R$-algebras, an *$R$-algebra homomorphism* is a ring homomorphism $h \colon A \to B$ with $h \circ f = g$.

## Remarks

Example: for $R = \Z$ and $A = \Z/n\Z$, the reduction map $\Z \to \Z/n\Z$ is a ring homomorphism, so $\Z/n\Z$ is a $\Z$-algebra.
Example: the polynomial ring $R[x_1, \ldots, x_n]$ with the inclusion of constants is an $R$-algebra.
Non-example: the ring $M_2(\R)$ of $2 \times 2$ real matrices receives the ring homomorphism $c \mapsto c I$ from $\R$.
But $M_2(\R)$ is not commutative, so it is an associative $\R$-algebra and not a commutative one.
Equivalent form: a commutative $R$-algebra is a commutative ring $A$ with an $R$-[[module|09VDS2W1E8]] structure such that $r(ab) = (ra)b$ for all $r \in R$ and $a, b \in A$; the structure map is then $r \mapsto r \cdot 1$.
The notion of a [[finite type algebra|AC5TFFQT9M]] and the [[tensor product of algebras|1735VGRT2P]] are built on this definition.
</document>

<document source="https://tamnd.github.io/math2/node/PD7R2CJSEP.md" tag="PD7R2CJSEP" kind="definition">
# Polydisc

## Definition

Let $n \ge 1$, let $a = (a_1, \dots, a_n) \in \C^n$ and let $r = (r_1, \dots, r_n)$ with each $r_j > 0$.
The *open polydisc* with centre $a$ and polyradius $r$ is
$$P(a, r) = \set{z \in \C^n : \abs{z_j - a_j} < r_j \text{ for } j = 1, \dots, n}.$$
The *closed polydisc* $\overline{P}(a, r)$ is defined in the same way with $\le$ in place of $<$.

## Remarks

So $P(a, r) = D(a_1, r_1) \times \cdots \times D(a_n, r_n)$, a product of open discs in $\C$.
The *unit polydisc* is $\Delta^n = P(0, (1, \dots, 1))$, the $n$-fold product of the [[unit disc|8G5ABXRK85]].
Example: $(1/2, i/2) \in \Delta^2$, since both coordinates have modulus $1/2 < 1$.
Non-example: $(0, 1) \notin \Delta^2$, since the second coordinate has modulus $1$, which fails the strict inequality.
The open polydisc is the open ball of radius $1$ about $a$ for the norm $\max_j \abs{z_j - a_j}/r_j$.
For $n = 1$ a polydisc is an ordinary disc.
</document>

<document source="https://tamnd.github.io/math2/node/PDDAJEYDE9.md" tag="PDDAJEYDE9" kind="definition">
# Residual set

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $R \subseteq X$.
The set $R$ is *residual* in $X$ if its complement $X \setminus R$ is [[meagre|0XGCZ4GDZ6]] in $X$.

## Remarks

Equivalently, $R$ contains a set $\bigcap_{n \ge 1} U_n$, where each $U_n$ is open and [[dense|0BSJSX7ME2]] in $X$.
Residual sets are also called comeagre sets.

Example: the set of irrational numbers is residual in $\R$.
Its complement $\Q$ is countable, so it is a countable union of single points, and each single point is [[nowhere dense|0VQYCCQVJ5]].

Non-example: $\Q$ is not residual in $\R$.
Otherwise $\R$ would be the union of the meagre sets $\Q$ and $\R \setminus \Q$.
This contradicts the [[Baire category theorem|0A7KGAYHGV]], since $\R$ is a complete metric space.
</document>

<document source="https://tamnd.github.io/math2/node/PDFC0MRYR8.md" tag="PDFC0MRYR8" kind="concept">
# Shelf

## Summary

A shelf is a set with a left [[self-distributive operation|VEFE5WZMS3]], that is, a [[magma|9689SFPVE9]] in which each left translation is an endomorphism.
A [[rack|X74VY86TSX]] is a shelf in which each left translation is a bijection.
A quandle is a rack in which $x \triangleright x = x$ for every element $x$.
Shelves that are not racks include the [[Laver tables|9H65YDXMGJ]], which come from set theory, and free shelves, whose study is linked to the braid groups.

## Explanations

### L1 human

In a shelf, every element $x$ acts on the other elements by $y \mapsto x \triangleright y$.
The only rule is that this action respects the operation itself: acting with $x$ on $y \triangleright z$ is the same as letting $x$ act on $y$ and on $z$ first.
Conjugation in a group is the model: conjugating a conjugate is the conjugate of the conjugates.
A simple example is any set with $x \triangleright y = y$, where every element acts as the identity.
Unlike in a rack, the action of an element in a shelf may collapse different elements together.
</document>

<document source="https://tamnd.github.io/math2/node/PDFK69ZMST.md" tag="PDFK69ZMST" kind="statement">
# CW approximation

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]].

1. There are a [[CW complex|5W95QFHXPR]] $Z$ and a [[weak homotopy equivalence|4Q3WY1RX91]] $f \colon Z \to X$.
2. Let $f' \colon Z' \to X$ be another weak homotopy equivalence from a CW complex $Z'$.
   Then there is a [[homotopy equivalence|XZG6MTSWCN]] $h \colon Z \to Z'$ such that $f' \circ h$ is homotopic to $f$.

The pair $(Z, f)$ is called a *CW approximation* of $X$.

## Remarks

Part 1 is proved by building $Z$ one dimension at a time.
For each path component one starts with a point.
At stage $n$ one attaches cells to make the map surjective on $\pi_n$ and further cells to kill the kernel on $\pi_n$.
Part 2 uses the fact that a weak homotopy equivalence $f' \colon Z' \to X$ induces a bijection $[Z, Z'] \to [Z, X]$ for every CW complex $Z$, together with the [[Whitehead theorem|6P1E2FJPSD]].

Example: the Warsaw circle $W$ has all homotopy groups trivial, so the inclusion of a point is a weak homotopy equivalence and a point is a CW approximation of $W$.
The space $W$ itself is not contractible, so a weak equivalence need not be a homotopy equivalence when the source or target is not a CW complex.

## Explanations

### L1 human

Many spaces are not CW complexes, but homotopy groups and singular homology cannot tell the difference.
The theorem says every space has a CW complex shadow with the same homotopy groups, mapped into it.
That shadow is unique up to homotopy equivalence.

### L2 human

CW approximation lets one prove results about singular homology and homotopy groups for CW complexes only and then extend them to all spaces.
It also shows that the homotopy category of CW complexes is equivalent to the category of all spaces with weak homotopy equivalences inverted.
Functorial versions are given by the geometric realization of the singular simplicial set, $\abs{\operatorname{Sing} X} \to X$.
</document>

<document source="https://tamnd.github.io/math2/node/PDG1HMR8GK.md" tag="PDG1HMR8GK" kind="definition">
# Laguerre–Pólya class

## Definition

An [[entire function|0419794R0T]] $f$ belongs to the *Laguerre–Pólya class* if there is a sequence $(P_k)$ of polynomials such that:

1. each $P_k$ has real coefficients;
2. all [[roots|00EHX6FWC0]] of each $P_k$ are real;
3. $P_k \to f$ [[uniformly|08FPABDQ4C]] on every compact subset of $\C$.

The class is written $\mathcal{L}\text{-}\mathcal{P}$.

## Remarks

A function $f$ that is not identically zero belongs to the class if and only if
$$f(z) = c\, z^m e^{-a z^2 + b z} \prod_{n} \left(1 - \frac{z}{x_n}\right) e^{z/x_n}.$$
Here $c, b \in \R$ with $c \ne 0$, $a \ge 0$, and $m \ge 0$ is an integer.
The $x_n$ are nonzero real numbers with $\sum_n x_n^{-2} < \infty$, and the product is a [[Weierstrass product|ACSF58HYDH]].
Example: $e^{-z^2}$ is the limit of the polynomials $(1 - z^2/k)^k$, which have real coefficients and only the real roots $\pm\sqrt{k}$.
Example: $\sin z = z \prod_{n \ge 1} (1 - z^2/(n^2\pi^2))$ is in the class.
Non-example: $1 + z^2$ has real coefficients but the roots $\pm i$.
It is not a limit of real-rooted real polynomials, because by Hurwitz's theorem such limits, when not identically zero, have only real zeros.
Non-example: $e^{z^2}$ has no zeros, but its product form would need $a = -1 < 0$.
The class is closed under differentiation.
</document>

<document source="https://tamnd.github.io/math2/node/PDP7CHN4B4.md" tag="PDP7CHN4B4" kind="definition">
# Displacement convexity

## Definition

Let $\mathcal{P}_2^{\mathrm{ac}}(\R^n)$ be the set of Borel probability measures on $\R^n$ with finite second moment that are [[absolutely continuous|P0P21KDV6E]] with respect to Lebesgue measure.
For $\mu_0, \mu_1 \in \mathcal{P}_2^{\mathrm{ac}}(\R^n)$, let $\nabla \varphi$ be the optimal map from $\mu_0$ to $\mu_1$ given by [[Brenier's theorem|CVQVBMHH4X]].
The *displacement interpolation* between them is the curve
$$\mu_t = \left( (1 - t) \id + t \nabla \varphi \right)_\# \mu_0, \qquad t \in [0, 1],$$
where $_\#$ denotes the [[pushforward|3N66TM3TSQ]].
A functional $F \colon \mathcal{P}_2^{\mathrm{ac}}(\R^n) \to (-\infty, +\infty]$ is *displacement convex* if, for all $\mu_0, \mu_1 \in \mathcal{P}_2^{\mathrm{ac}}(\R^n)$, the function $t \mapsto F(\mu_t)$ is convex on $[0, 1]$.

## Remarks

Example: let $V \colon \R^n \to \R$ be [[convex|07Q7CNM5WF]] and $F(\mu) = \int V \, d\mu$.
Then $F(\mu_t) = \int V\big((1 - t) x + t \nabla \varphi(x)\big) \, d\mu_0(x)$, and the integrand is convex in $t$ for each $x$, so $F$ is displacement convex.

Non-example: on $\R$, let $V(x) = -x^2$, let $\mu_0$ be uniform on $[0, 1]$ and $\mu_1$ uniform on $[2, 3]$.
The optimal map is $x \mapsto x + 2$, so $\mu_t$ is uniform on $[2t, 2t + 1]$ and $F(\mu_t) = -\int_0^1 (x + 2t)^2 \, dx = -\left(\tfrac13 + 2t + 4t^2\right)$, which is strictly concave in $t$.

The interpolation $\mu_t$ is a constant-speed geodesic for the Wasserstein distance $W_2$, so displacement convexity is convexity along $W_2$-geodesics.
The functional $F$ is *$\lambda$-displacement convex*, for a real $\lambda$, if $t \mapsto F(\mu_t) - \frac{\lambda}{2} t^2 W_2(\mu_0, \mu_1)^2$ is convex.
McCann showed that the entropy $\int \rho \log \rho \, dx$ of a density $\rho$ is displacement convex on $\R^n$.
The interaction energy $\iint W(x - y) \, d\mu(x) \, d\mu(y)$ with $W$ convex is also displacement convex.
It need not be convex in the usual sense: for $W(z) = \abs{z}^2$ it equals twice the variance of $\mu$, which is concave under mixtures $(1 - t)\mu_0 + t\mu_1$.
</document>

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# Unit ball in C^n

## Description

Let $n \ge 1$.
The *unit ball* of $\C^n$ is
$$\mathbb{B}^n = \set{z \in \C^n : \abs{z_1}^2 + \cdots + \abs{z_n}^2 < 1}.$$
It is the [[open ball|0YE5868HZB]] of radius $1$ about $0$ for the norm $\abs{z} = (\sum_j \abs{z_j}^2)^{1/2}$.
Writing $z_j = x_j + i y_j$, the condition becomes $\sum_j (x_j^2 + y_j^2) < 1$, so $\mathbb{B}^n$ is the Euclidean unit ball of $\R^{2n}$.

## Remarks

Its boundary is the unit sphere $S^{2n-1} = \set{z : \abs{z} = 1}$.
It is a [[Reinhardt domain|WA0SE56FVH]]: if $z \in \mathbb{B}^n$ and $\theta_j \in \R$, then $(e^{i\theta_1} z_1, \dots, e^{i\theta_n} z_n)$ has the same norm, so it lies in $\mathbb{B}^n$.
Every unitary map $U$ of $\C^n$ preserves the norm, so it maps $\mathbb{B}^n$ onto itself.
For $n = 1$ it is the unit disc, and for $n \ge 2$ it is not biholomorphic to the unit [[polydisc|DJBWP7D3M8]], by [[Poincaré's theorem|1NF8T07H9X]].

## Explanations

### L1 human

The unit ball of $\C^n$ is the set of points at distance less than $1$ from the origin.
As a set it is just the round ball of $\R^{2n}$; what is new is that one studies the holomorphic functions on it.
In $\C^2$, the point $(1/2, 1/2)$ is in the ball, since $1/4 + 1/4 < 1$, while $(0.9, 0.9)$ is not.
The ball is very symmetric: its holomorphic automorphisms can move any point of it to the origin.
</document>

<document source="https://tamnd.github.io/math2/node/PDVJTYCK8F.md" tag="PDVJTYCK8F" kind="statement">
# Bishop–Phelps theorem

## Statement

Let $X$ be a real [[Banach space|06GZ956657]] with [[continuous dual space|4A82S5FVRT]] $X^*$, and let $C \subseteq X$ be a nonempty, closed, bounded [[convex set|02AS45T3H9]].
Say that $f \in X^*$ attains its supremum on $C$ when $f(c) = \sup_{x \in C} f(x)$ for some $c \in C$.
Then the set of such functionals is [[dense|0BSJSX7ME2]] in $X^*$ for the norm topology.
In particular, taking $C$ to be the closed unit ball, the functionals $f \in X^*$ with $f(x) = \norm{f}$ for some $x$ with $\norm{x} \le 1$ are dense in $X^*$.

## Remarks

Proof idea: given $f \in X^*$ and $\varepsilon > 0$, one considers a cone determined by $f$ and $\varepsilon$.
A maximality argument, in the spirit of Zorn's lemma or of Ekeland's principle, gives $c \in C$ such that the cone translated to $c$ meets $C$ only at $c$.
The Hahn–Banach separation theorem then gives a functional $g$ that attains its supremum on $C$ at $c$ and satisfies $\norm{f - g} \le \varepsilon$.

Example: let $X = c_0$ be the space of real sequences tending to $0$, with the supremum norm.
The functional $f(x) = \sum_{n \ge 1} 2^{-n} x_n$ does not attain its norm $1$ on the unit ball, since that would need $x_n = 1$ for all $n$.
The truncations $g_N(x) = \sum_{n \le N} 2^{-n} x_n$ attain their norm at $x = (1, \ldots, 1, 0, 0, \ldots)$ and converge to $f$ in norm, as the theorem predicts.

The density cannot be improved to equality in general: by James' theorem, every functional attains its norm on the unit ball only when $X$ is reflexive.
The analogue for bounded operators between Banach spaces, in place of functionals, fails in general.

## Explanations

### L1 human

A functional attains its norm if some vector of length at most $1$ realizes its largest value.
In infinite dimensions this can fail, because the unit ball is not compact.
The theorem says that such failures are rare: every functional can be approximated as closely as one likes by functionals that do attain their norm.

### L2 human

Norm-attaining functionals correspond to supporting hyperplanes of the unit ball, so the theorem says that support points of closed bounded convex sets are plentiful.
It is used in the geometry of Banach spaces, for instance in the study of smoothness and of subdifferentials of convex functions.
The complex case follows from the real one, by viewing a complex Banach space as a real one.
</document>

<document source="https://tamnd.github.io/math2/node/PDVYQAW887.md" tag="PDVYQAW887" kind="concept">
# Fast wavelet transform

## Summary

The fast wavelet transform is the recursive filter-bank algorithm that computes the [[discrete wavelet transform|P3H4MMC4ZH]] of a signal.
At each level it convolves the current approximation with a low-pass and a high-pass filter and keeps every second output.
For a signal of length $N$ and filters of fixed length, it uses $O(N)$ operations, which is even faster than the fast Fourier transform.
It is also known as Mallat's algorithm, and it rests on the structure of a [[multiresolution analysis|1GHQMBR7DE]].

## Explanations

### L1 human

Start with a list of numbers.
Replace neighbouring pairs by a smoothed value and a difference value, which halves the length of each list.
Keep the differences, and repeat the same step on the smoothed list.
For the Haar wavelet, $(4, 2, 5, 5)$ becomes averages $(6, 10)/\sqrt{2}$ and differences $(2, 0)/\sqrt{2}$, and then the averages are split again.
Each level costs half as much as the one before, so the total work is about twice the cost of the first level.
</document>

<document source="https://tamnd.github.io/math2/node/PDXR52YGPD.md" tag="PDXR52YGPD" kind="definition">
# Gateaux derivative

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $X$ and $Y$ be [[normed vector spaces|03J5RAKB30]] over $\mathbb{K}$.
Let $U \subseteq X$ be open, let $f \colon U \to Y$, let $x \in U$ and let $h \in X$.
The *Gateaux differential* of $f$ at $x$ in the direction $h$ is the limit
$$df(x; h) = \lim_{t \to 0} \frac{f(x + t h) - f(x)}{t},$$
taken over $t \in \mathbb{K} \setminus \set{0}$, when this limit exists in $Y$.
The map $f$ is *Gateaux differentiable* at $x$ if $df(x; h)$ exists for every $h \in X$ and $h \mapsto df(x; h)$ is a [[continuous linear map|03TM135G0N]] $X \to Y$.
This map is the *Gateaux derivative* of $f$ at $x$.

## Remarks

For $X = \R^n$ and $Y = \R$, $df(x; h)$ is the [[directional derivative|06KWGHDN2J]] of $f$ along $h$.
Some authors do not ask $h \mapsto df(x; h)$ to be linear or continuous.
The definition works without change for locally convex spaces.
If $f$ has a [[Fréchet derivative|06EH97V2A7]] $Df(x)$, then $f$ is Gateaux differentiable at $x$ with $df(x; h) = Df(x) h$.

Example: let $H$ be a real Hilbert space and $f(x) = \norm{x}^2$.
Then $f(x + t h) = \norm{x}^2 + 2t \ip{x}{h} + t^2 \norm{h}^2$, so $df(x; h) = 2\ip{x}{h}$, which is continuous and linear in $h$.

Non-example: on $\R^2$, let $f(x, y) = x^3 / (x^2 + y^2)$ and $f(0, 0) = 0$.
For $h = (a, b) \neq 0$, $f(t a, t b) / t = a^3 / (a^2 + b^2)$, so $df(0; h)$ exists for every $h$.
But $h \mapsto a^3 / (a^2 + b^2)$ is not linear, so $f$ is not Gateaux differentiable at $0$.

A Gateaux differentiable map need not be Fréchet differentiable.
On $\R^2$, let $g(x, y) = 1$ when $y = x^2$ and $(x, y) \neq 0$, and $g = 0$ elsewhere.
Each line through $0$ meets the parabola in at most one point other than $0$, so $dg(0; h) = 0$ for all $h$.
But $g$ is not continuous at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/PDYF9AV5PN.md" tag="PDYF9AV5PN" kind="example">
# Particular values of the gamma function

## Example

Here are some values of the [[gamma function|0V3FZMVB4K]].

| $z$ | $\Gamma(z)$ |
|---|---|
| $n \in \set{1, 2, 3, \ldots}$ | $(n - 1)!$ |
| $\tfrac12$ | $\sqrt{\pi}$ |
| $\tfrac32$ | $\tfrac12\sqrt{\pi}$ |
| $\tfrac52$ | $\tfrac34\sqrt{\pi}$ |
| $n + \tfrac12$, $n \ge 0$ | $\dfrac{(2n)!}{4^n \, n!}\sqrt{\pi}$ |
| $-\tfrac12$ | $-2\sqrt{\pi}$ |
| $-\tfrac32$ | $\tfrac43\sqrt{\pi}$ |

Checks.
The first row is [[the gamma function extends the factorial|0FDCW77H44]].
The value at $\tfrac12$ is [[Gamma of one half|WM5FNBB7K0]].
The [[functional equation|ZCB806EV8A]] $\Gamma(z + 1) = z\Gamma(z)$ gives the others: $\Gamma(\tfrac52) = \tfrac32 \cdot \tfrac12 \sqrt{\pi} = \tfrac34\sqrt\pi$, and $\Gamma(-\tfrac12) = \Gamma(\tfrac12)/(-\tfrac12) = -2\sqrt\pi$, and $\Gamma(-\tfrac32) = \Gamma(-\tfrac12)/(-\tfrac32) = \tfrac43\sqrt\pi$.
For $n = 2$ the general half-integer formula gives $\frac{24}{16 \cdot 2}\sqrt{\pi} = \tfrac34\sqrt{\pi}$, in agreement.

Values at other rational points are less explicit, but the [[reflection formula|WF7KQAQTS2]] gives products:
$$\Gamma\!\left(\tfrac13\right)\Gamma\!\left(\tfrac23\right) = \frac{\pi}{\sin(\pi/3)} = \frac{2\pi}{\sqrt3}, \qquad \Gamma\!\left(\tfrac14\right)\Gamma\!\left(\tfrac34\right) = \frac{\pi}{\sin(\pi/4)} = \pi\sqrt2.$$

## Remarks

The values $\Gamma(\tfrac13)$ and $\Gamma(\tfrac14)$ are known to be [[transcendental numbers|0VTQXTCR2G]].
The value $\Gamma(\tfrac14)$ appears in the arc length of the lemniscate and in elliptic integrals.

## Explanations

### L1 human

At whole numbers the gamma function gives factorials, shifted by one.
At halves it gives rational multiples of $\sqrt{\pi}$, and the sign alternates for negative halves.
At thirds and quarters the single values are new constants, but some of their products are simple.
</document>

<document source="https://tamnd.github.io/math2/node/PE207P2DZD.md" tag="PE207P2DZD" kind="counterexample">
# Hedetniemi's conjecture

## Counterexample

Let $G$ and $H$ be finite [[graphs|0JD5C82QMB]].
Their tensor product $G \times H$ has vertex set $V(G) \times V(H)$.
In it, $(g, h)$ is adjacent to $(g', h')$ when $gg'$ is an edge of $G$ and $hh'$ is an edge of $H$.
Hedetniemi's conjecture asserted that $\chi(G \times H) = \min(\chi(G), \chi(H))$ for all finite graphs $G$ and $H$, where $\chi$ is the [[chromatic number|0C5M7AM6FH]].
The conjecture is false: in 2019 Yaroslav Shitov constructed finite graphs $G$ and $H$ with
$$\chi(G \times H) < \min(\chi(G), \chi(H)).$$

## Remarks

The inequality $\chi(G \times H) \le \min(\chi(G), \chi(H))$ always holds.
Indeed, the projection $(g, h) \mapsto g$ is a [[graph homomorphism|06SCB3R6BN]] to $G$, so a proper colouring of $G$ pulls back to one of $G \times H$.
The same holds for $H$.
Shitov's counterexample uses an exponential graph.
For a graph $G$ and $c \ge 1$, the vertices of $K_c^G$ are all maps $f \colon V(G) \to \set{1, \dots, c}$.
Two maps $f$ and $f'$ are adjacent when $f(u) \ne f'(v)$ for every edge $uv$ of $G$, in both orders of $u$ and $v$.
The map $(v, f) \mapsto f(v)$ is a proper colouring of $G \times K_c^G$ with $c$ colours.
Indeed, if $(u, f)$ and $(v, f')$ are adjacent, then $uv$ is an edge and $f$ is adjacent to $f'$, so $f(u) \ne f'(v)$.
So any $G$ with $\chi(G) > c$ and $\chi(K_c^G) > c$ is a counterexample, and Shitov showed that such $G$ exist for large $c$.
The conjecture holds when $\min(\chi(G), \chi(H)) \le 4$; the case $4$ is a theorem of El-Zahar and Sauer.

## Explanations

### L1 human

In the tensor product of two graphs, two pairs are joined when both coordinates are joined.
Colouring either factor gives a colouring of the product, so the product never needs more colours than the easier factor.
For over fifty years people believed it never needs fewer, but Shitov found very large graphs where it does.
</document>

<document source="https://tamnd.github.io/math2/node/PE2HN8MY2J.md" tag="PE2HN8MY2J" kind="definition">
# Projection body

## Definition

Let $n \ge 2$, and let $K \subseteq \R^n$ be a [[convex body|437FSG3419]].
For a nonzero vector $u$, let $K | u^\perp$ be the [[orthogonal projection|0Q8HZTJC7G]] of $K$ onto the hyperplane $u^\perp = \set{x : x \cdot u = 0}$.
Let $\mathrm{vol}_{n-1}$ be the $(n-1)$-dimensional volume in $u^\perp$.
The *projection body* of $K$ is the convex body $\Pi K$ whose [[support function|1H92CYA4ZC]] is
$$h_{\Pi K}(x) = \abs{x} \, \mathrm{vol}_{n-1}\bigl(K | x^\perp\bigr) \quad \text{for } x \ne 0, \qquad h_{\Pi K}(0) = 0.$$

## Remarks

That this function is a support function follows from Cauchy's projection formula $\mathrm{vol}_{n-1}(K | u^\perp) = \frac{1}{2} \int_{S^{n-1}} \abs{u \cdot v} \, dS_K(v)$, where $S_K$ is the surface area measure of $K$.
The formula also shows that $\Pi K$ is symmetric about the origin and is a [[zonoid|ZMZZ2QRFTZ]].
Example: let $C = [0, 1]^n$.
For a unit vector $u$, the shadow of $C$ on $u^\perp$ has volume $\abs{u_1} + \cdots + \abs{u_n}$.
This is the support function of $[-1, 1]^n$ at $u$, so $\Pi C = [-1, 1]^n$.
Non-example: for $n \ge 3$, every projection of a segment $K$ has $(n-1)$-volume $0$.
So the formula gives $h = 0$ and the set $\set{0}$, which is not a convex body; this is why $K$ must have nonempty interior.
For the unit ball $B$, every shadow is an $(n-1)$-ball, so $\Pi B = \omega_{n-1} B$, with $\omega_{n-1}$ the volume of the unit ball of $\R^{n-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/PE3Z7CB1RY.md" tag="PE3Z7CB1RY" kind="definition">
# Absolute summability

## Definition

Let $A = (a_{nk})_{n, k \ge 0}$ be an infinite matrix of real or complex numbers, viewed as a [[matrix summability method|M6311DWJ1E]].
Let $\sum_{k \ge 0} a_k$ be a series with [[partial sums|0NB643K60Q]] $s_k = a_0 + \dots + a_k$.
Assume that for every $n$ the series $t_n = \sum_{k \ge 0} a_{nk} s_k$ converges.
The series $\sum a_k$ is *absolutely $A$-summable* (or $\abs{A}$-summable) if
$$\sum_{n \ge 1} \abs{t_n - t_{n-1}} < \infty.$$

## Remarks

If $\sum_{n \ge 1} \abs{t_n - t_{n-1}}$ is finite, then $(t_n)$ is a Cauchy sequence, so it converges.
Thus an absolutely $A$-summable series is $A$-summable.

Example: for the identity matrix, $t_n = s_n$ and $t_n - t_{n-1} = a_n$.
So absolute summability by the identity method is [[absolute convergence|0YBDSJARWN]].

Non-example: Grandi's series $1 - 1 + 1 - \cdots$ is [[Cesàro summable|N8ZGG7EW6J]] to $1/2$, but not absolutely Cesàro summable.
Its partial sums are $1, 0, 1, 0, \dots$, so the Cesàro means are $t_{2m} = (m + 1)/(2m + 1)$ and $t_{2m+1} = 1/2$.
Then $\abs{t_{2m} - t_{2m-1}} = 1/(2(2m + 1))$, and the sum of these numbers diverges.

For the Cesàro method the notation is $\abs{C, 1}$.
For Abel summation one says that $\sum a_n$ is absolutely Abel summable when $x \mapsto \sum a_n x^n$ has bounded variation on $[0, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/PE5H892XVS.md" tag="PE5H892XVS" kind="statement">
# Chebyshev iteration

## Statement

Let $A$ be a real symmetric [[positive definite|0KX66CZMBB]] $n \times n$ matrix, and let $b \in \R^n$.
Let $0 < \alpha < \beta$ be numbers such that every [[eigenvalue|0BQAA2BK3Q]] of $A$ lies in $[\alpha, \beta]$, and let $\kappa = \beta / \alpha$.
Let $T_k$ be the [[Chebyshev polynomial|M6EAR1JHPP]] of the first kind of degree $k$, and set $\theta = (\beta + \alpha) / (\beta - \alpha)$.
The *Chebyshev iteration* produces iterates $x^{(k)}$ whose errors $e_k = x^{(k)} - A^{-1} b$ satisfy $e_k = p_k(A) e_0$, where
$$p_k(t) = \frac{T_k\!\left( \frac{\beta + \alpha - 2t}{\beta - \alpha} \right)}{T_k(\theta)}.$$
The iterates can be computed by a three-term recurrence that uses one product with $A$ per step and no inner products.
For every $k \ge 0$,
$$\norm{e_k}_2 \le \frac{\norm{e_0}_2}{T_k(\theta)} \le 2 \left( \frac{\sqrt{\kappa} - 1}{\sqrt{\kappa} + 1} \right)^k \norm{e_0}_2.$$

## Remarks

Since $A$ is symmetric, $\norm{p_k(A)}_2 = \max_\lambda \abs{p_k(\lambda)}$ over its eigenvalues, and on $[\alpha, \beta]$ the numerator of $p_k$ is at most $1$ in absolute value.
The bound $T_k(\theta) \ge \frac12 \sigma^{-k}$ with $\sigma = (\sqrt{\kappa} - 1)/(\sqrt{\kappa} + 1)$ follows from $T_k(x) = \frac12 \big( (x + \sqrt{x^2 - 1})^k + (x - \sqrt{x^2 - 1})^k \big)$ for $x \ge 1$.
Among all polynomials of degree $k$ with value $1$ at $0$, the scaled Chebyshev polynomial $p_k$ has the least maximum on $[\alpha, \beta]$.
So no method of the form $e_k = q(A) e_0$ with such $q$ can do better in the worst case over matrices with spectrum in $[\alpha, \beta]$.
For comparison, the optimal [[Richardson iteration|7CESYMHQEA]] contracts by $(\kappa - 1)/(\kappa + 1)$ per step.
Example: for $\kappa = 100$ the Chebyshev factor is $9/11 \approx 0.82$, while the Richardson factor is $99/101 \approx 0.98$.
If the interval does not contain the whole spectrum, the bound fails and the method can diverge.

## Explanations

### L1 human

Chebyshev iteration is Richardson's method with a cleverly varying step size.
The steps are chosen so that the error is damped evenly over the whole range of eigenvalues.
The price is that one must know bounds for the smallest and largest eigenvalues in advance.

### L2 human

The error factor depends on $\sqrt{\kappa}$ instead of $\kappa$, like the conjugate gradient method, where $\kappa$ is the [[condition number|AB3S6MHJ8X]].
Unlike conjugate gradients it needs no inner products, which helps on parallel machines where global sums are costly.
Its iterates lie in the [[Krylov subspace|QB45PCGQPV]] shifted by $x^{(0)}$, and the conjugate gradient method finds an iterate there that is at least as good in the energy norm.
Chebyshev polynomials are also used as smoothers in multigrid and as polynomial preconditioners.
</document>

<document source="https://tamnd.github.io/math2/node/PEB9DN0AHA.md" tag="PEB9DN0AHA" kind="concept">
# Moving-average model

## Summary

A moving-average model describes each value of a [[time series|EA3MF4VTEH]] as its mean plus a weighted sum of current and recent random shocks.
With $q$ past shocks the model is written MA($q$).
Its autocorrelation vanishes beyond lag $q$, so it suits series whose memory is short.
Together with the autoregressive model it forms the ARMA family.

## Explanations

### L1 human

Imagine a pond where each raindrop makes ripples that last exactly two seconds.
The water level now is the sum of the effects of the drops of the last two seconds, and older drops no longer matter.
In the MA($1$) model $X_t = \varepsilon_t + 0.5 \varepsilon_{t-1}$, each shock acts fully today and half as strongly tomorrow, and then disappears.
So values two or more steps apart are uncorrelated.
</document>

<document source="https://tamnd.github.io/math2/node/PEDBPRSV9V.md" tag="PEDBPRSV9V" kind="conjecture">
# Dürer's conjecture

## Statement

Every three-dimensional [[convex polytope|300K6CB1KH]] in $\R^3$ has a [[net|FXJ66Z6PQY]].
That is, some spanning tree of its edge graph gives an edge unfolding without overlap.

## Remarks

The conjecture is open.
It is named after Albrecht Dürer, whose book on geometry drew nets of many polyhedra, but it was first stated as a question by Shephard.
Cuts may also be allowed to run across faces and not only along edges.
Then every convex polyhedron can be unfolded without overlap, for example by the source unfolding or the star unfolding.
Convexity matters: there are non-convex polyhedra whose edge unfoldings all overlap.
For a given convex polyhedron, some spanning trees may give overlapping unfoldings, so the question is whether at least one good tree always exists.

## Explanations

### L1 human

Can every convex solid with flat faces be cut along some of its edges and flattened into a single piece of paper with no part lying on top of another?
For the cube and all the familiar solids the answer is yes.
Nobody knows whether it is true for every convex solid, and no counterexample has been found.
</document>

<document source="https://tamnd.github.io/math2/node/PEDD2CP5PF.md" tag="PEDD2CP5PF" kind="statement">
# Noether's second theorem

## Statement

Let $\Omega \subseteq \R^n$ be open, and consider maps $u \colon \Omega \to \R^m$.
Let $J(u) = \int_\Omega L(x, u, \partial u, \dots, \partial^k u) \, dx$ be an [[integral functional|JRQZ803Q5Y]] with smooth Lagrangian $L$.
Let $E_1(u), \dots, E_m(u)$ be its [[Euler–Lagrange expressions|PS0VV149ET]], so that the Euler–Lagrange equations read $E(u) = 0$.
Assume that $J$ has a symmetry depending on $r$ arbitrary smooth functions $g_1, \dots, g_r$ of $x$.
Its infinitesimal variations are
$$\delta u_\alpha = \sum_{j=1}^{r} \mathcal{D}_{\alpha j} \, g_j, \quad \alpha = 1, \dots, m,$$
where each $\mathcal{D}_{\alpha j}$ is a linear differential operator whose coefficients depend on $x$, $u$ and derivatives of $u$.
Assume that each such variation leaves $L$ invariant up to a total divergence.
Then the Euler–Lagrange expressions satisfy $r$ identities
$$\sum_{\alpha=1}^{m} \mathcal{D}_{\alpha j}^{*} \, E_\alpha(u) = 0, \quad j = 1, \dots, r,$$
for every smooth $u$, where $\mathcal{D}_{\alpha j}^{*}$ is the formal adjoint of $\mathcal{D}_{\alpha j}$.

## Remarks

Invariance up to a divergence gives $\sum_\alpha E_\alpha(u) \, \delta u_\alpha = \operatorname{div} S$ for some expression $S$ that is linear in the $g_j$ and their derivatives.
Integrating by parts moves the derivatives from $g_j$ onto $E_\alpha$, so $\sum_j g_j \sum_\alpha \mathcal{D}^*_{\alpha j} E_\alpha$ is a divergence too.
An expression of the form $\sum_j g_j F_j$ that is a divergence for all compactly supported $g_j$ must have $F_j = 0$.

Example: the length $J(x, y) = \int_a^b \sqrt{\dot x^2 + \dot y^2} \, dt$ of a regular plane curve, one with $(\dot x, \dot y) \ne 0$ everywhere, is invariant under reparametrisations $t \mapsto t + \varepsilon g(t)$.
The variations are $\delta x = \dot x \, g$ and $\delta y = \dot y \, g$, up to sign, so here $\mathcal{D}$ is multiplication by $\dot x$ and $\dot y$.
The Euler–Lagrange expressions are $E_x = -\frac{d}{dt} T_1$ and $E_y = -\frac{d}{dt} T_2$, with $T = (\dot x, \dot y) / \abs{(\dot x, \dot y)}$.
The identity is $\dot x \, E_x + \dot y \, E_y = 0$ for every regular curve; it holds because $T \cdot \dot T = 0$.

Unlike the first [[Noether's theorem|XHPDM5T2JG]], the result gives identities that hold for all $u$, not conservation laws along solutions.
So the Euler–Lagrange equations are not independent, and their solutions are not determined by initial data alone.

## Explanations

### L1 human

Some symmetries are not a single motion but a whole family that can vary from point to point, like a change of time scale that may differ at each moment.
Noether's second theorem says that such a large symmetry forces relations among the equations of motion themselves.
These relations hold for every function, not only for solutions.
They show that the equations do not fix the solution uniquely, because the symmetry can still move it.

### L2 human

This is the theorem behind gauge theories and general relativity.
In electromagnetism, gauge invariance gives the identity that the divergence of the Maxwell equations vanishes identically, which is linked to charge conservation.
In general relativity, invariance under diffeomorphisms gives the contracted Bianchi identities.
The identities explain why such equations are underdetermined and need gauge fixing before one can solve an initial value problem.
</document>

<document source="https://tamnd.github.io/math2/node/PEMNN32A43.md" tag="PEMNN32A43" kind="concept">
# Torsion-free group

## Summary

A torsion-free group is a [[group|G5S1M9]] in which the identity is the only element of finite [[order|0WE2NHZJ7B]].
It is the opposite extreme to a [[torsion group|JZNQTTMG58]].
For an abelian group $A$, the quotient of $A$ by its [[torsion subgroup|0DBWXQRXQJ]] is torsion-free.
A finitely generated torsion-free abelian group is a [[free abelian group|0WJX8MGG37]] $\Z^n$.

## Explanations

### L1 human

In a torsion-free group, no element other than the identity ever comes back to the identity when you repeat it.
The integers are the basic example: starting from $3$ and adding $3$ again and again, you get $6, 9, 12, \dots$ and never $0$.
A quarter turn of a circle comes back to the start after four steps.
So a group that contains a quarter turn is not torsion-free.
Torsion-free groups are the groups with no hidden clocks inside.
</document>

<document source="https://tamnd.github.io/math2/node/PETEQQCS0B.md" tag="PETEQQCS0B" kind="concept">
# Levi pseudoconvex domain

## Summary

A domain in $\C^n$ with $C^2$ boundary is Levi pseudoconvex when the [[Levi form|80MTNXWW80]] of its boundary is positive semidefinite at every boundary point.
This is the differential form of [[pseudoconvexity|AHYZMSBBNH]]: for domains with $C^2$ boundary the two notions agree.
It gives a test of pseudoconvexity that can be checked by computing second derivatives.

## Explanations

### L1 human

Convexity asks that the boundary never bends inwards in any real direction.
Levi pseudoconvexity asks this only for the complex directions tangent to the boundary, and only for the part of the bending that holomorphic maps see.
So it is a much weaker condition than convexity, but it is the right one for holomorphic functions.
For example, the unit ball in $\C^n$ is Levi pseudoconvex, and so is every convex domain with $C^2$ boundary.
The shell $\set{1 < \abs{z} < 2}$ in $\C^2$ is not: along the inner sphere its boundary bends the wrong way in the complex tangent directions.
</document>

<document source="https://tamnd.github.io/math2/node/PEXTM3MJ5Z.md" tag="PEXTM3MJ5Z" kind="concept">
# Axiom of determinacy

## Summary

The axiom of determinacy, written AD, says that every infinite game of natural numbers with perfect information is [[determined|QXECDPAYTN]].
Jan Mycielski and Hugo Steinhaus proposed it in 1962 as an alternative to the [[axiom of choice|0HTGB13FE2]], which it contradicts.
In ZF together with AD, every set of real numbers is Lebesgue measurable, has the [[Baire property|FZ7237VY65]] and has the [[perfect set property|H47ZGRR13V]].
AD is studied in models such as $L(\R)$, where it can hold under large cardinal hypotheses.

## Explanations

### L1 human

Pick any rule that sorts infinite sequences of natural numbers into wins and losses for the first player.
The axiom says that, whatever the rule, one of the two players has a strategy that always wins.
With the axiom of choice one can write down a strange rule for which no player has such a strategy, so AD and choice cannot both hold.
In exchange, AD removes the pathological sets of reals that choice creates, such as non-measurable sets.
</document>

<document source="https://tamnd.github.io/math2/node/PF3G0J5JD6.md" tag="PF3G0J5JD6" kind="definition">
# Wave front set

## Definition

Let $\Omega$ be an open subset of $\R^n$ and let $u$ be a distribution on $\Omega$.
A *cone* is a set $\Gamma \subseteq \R^n \setminus \set{0}$ such that $t \xi \in \Gamma$ for all $\xi \in \Gamma$ and all $t > 0$.
A point $(x_0, \xi_0) \in \Omega \times (\R^n \setminus \set{0})$ is *microlocally regular* for $u$ if there exist:

1. a test function $\varphi \in C_c^\infty(\Omega)$ with $\varphi(x_0) \ne 0$,
2. an open cone $\Gamma$ with $\xi_0 \in \Gamma$,
3. for each integer $N \ge 0$ a constant $C_N$,

such that the [[Fourier transform|DPVM34ADZV]] of the compactly supported distribution $\varphi u$ satisfies $\abs{\widehat{\varphi u}(\xi)} \le C_N (1 + \abs{\xi})^{-N}$ for all $\xi \in \Gamma$ and all $N$.
The *wave front set* $\operatorname{WF}(u)$ is the set of points of $\Omega \times (\R^n \setminus \set{0})$ that are not microlocally regular for $u$.

## Remarks

Since $\varphi u$ has compact support, $\widehat{\varphi u}$ is a [[smooth function|0V33BNKZ9S]], so the decay condition makes sense pointwise.
By construction $\operatorname{WF}(u)$ is closed, and it is conic: $(x, \xi) \in \operatorname{WF}(u)$ implies $(x, t\xi) \in \operatorname{WF}(u)$ for $t > 0$.
Its projection to $\Omega$ is the singular support of $u$, the complement of the largest open set on which $u$ is smooth.
In particular $\operatorname{WF}(u) = \emptyset$ if and only if $u$ is smooth.

Example: for the [[Dirac delta|0KPHENZ5S7]] $\delta$ at $0$ in $\R^n$, one has $\operatorname{WF}(\delta) = \set{0} \times (\R^n \setminus \set{0})$.
Check: at a point $x_0 \ne 0$ one may take $\varphi$ vanishing near $0$, so $\varphi \delta = 0$.
At $x_0 = 0$, $\widehat{\varphi \delta}(\xi) = \varphi(0)$ is a nonzero constant, which decays in no direction.

Non-example: let $u(x) = H(x_1)$ on $\R^2$, where $H$ is the [[Heaviside step function|09C4JP7Y0M]], and look at the point $(0, (0, 1))$.
Take $\varphi(x) = \chi(x_1) \chi(x_2)$ with $\chi \in C_c^\infty(\R)$ and $\chi(0) \ne 0$.
Then $\widehat{\varphi u}(\xi) = a(\xi_1)\, \hat\chi(\xi_2)$, where $a$ is the Fourier transform of $\chi H$ and is bounded.
In the cone $\abs{\xi_1} < \abs{\xi_2}$ around $(0, 1)$ one has $\abs{\xi} < 2 \abs{\xi_2}$, and $\hat\chi(\xi_2)$ decays faster than any power of $\abs{\xi_2}$.
So the point is microlocally regular, and it is not in $\operatorname{WF}(u)$.
In fact $\operatorname{WF}(u) = \set{x_1 = 0} \times \set{(\xi_1, 0) : \xi_1 \ne 0}$.

The wave front set transforms well under changes of coordinates, as a subset of the cotangent bundle minus its zero section, so the notion extends to distributions on manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/PF4RHTZWXH.md" tag="PF4RHTZWXH" kind="definition">
# Complete statistic

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a [[statistical model|JZ6R7J36K5]] and let $T = T(X)$ be a [[statistic|036756NFD6]] with values in a measurable space $\mathcal{T}$.
The statistic $T$ is *complete* if the following holds for every measurable $g \colon \mathcal{T} \to \R$ such that $g(T)$ is integrable under every $P_\theta$.
The condition
$$E_\theta[g(T)] = 0 \quad \text{for all } \theta \in \Theta$$
implies $P_\theta(g(T) = 0) = 1$ for all $\theta \in \Theta$.
Here $E_\theta$ is the [[expected value|0DCT0QZVQ1]] under $P_\theta$.
The statistic is *boundedly complete* if this implication holds for every bounded measurable $g$.

## Remarks

Every complete statistic is boundedly complete.
Example: let $T$ have the binomial law with parameters $n$ and $p \in (0, 1)$.
If $E_p[g(T)] = (1 - p)^n \sum_{t=0}^{n} g(t)\binom{n}{t} r^t = 0$ with $r = p/(1 - p)$ for all $p$, the polynomial in $r$ vanishes on $(0, \infty)$, so each $g(t) = 0$; thus $T$ is complete.
Non-example: let $X_1, X_2$ be i.i.d. $\mathcal{N}(\theta, 1)$ and $T = (X_1, X_2)$.
The function $g(T) = X_1 - X_2$ has $E_\theta[g(T)] = 0$ for all $\theta$, but $g(T) \ne 0$ with probability $1$; so $T$ is not complete.
Completeness is a property of the family of laws $\set{P_\theta \circ T^{-1} : \theta \in \Theta}$; enlarging $\Theta$ can only help it.
</document>

<document source="https://tamnd.github.io/math2/node/PF5HS26Q0H.md" tag="PF5HS26Q0H" kind="definition">
# Lattice polytope

## Definition

A *lattice polytope* in $\R^n$ is a [[convex polytope|300K6CB1KH]] $P \subseteq \R^n$ all of whose [[vertices|R7V3S9V891]] lie in $\Z^n$.
Equivalently, $P$ is the convex hull of a finite subset of $\Z^n$.

## Remarks

Example: the triangle with vertices $(0,0)$, $(3,0)$, $(0,2)$ is a lattice polygon, since all three vertices have integer coordinates.

Non-example: the triangle with vertices $(0,0)$, $(1,0)$, $(0, 1/2)$ is not a lattice polygon, since the vertex $(0, 1/2)$ is not in $\Z^2$.
It is a rational polytope: its vertices lie in $\Q^2$.

More generally, for a [[lattice|WN9V7WSJKF]] $\Lambda \subseteq \R^n$, a lattice polytope with respect to $\Lambda$ is a convex polytope with all vertices in $\Lambda$.
A linear change of coordinates turns $\Lambda$ into $\Z^n$, so the case $\Lambda = \Z^n$ is the standard one.
Two lattice polytopes are usually regarded as the same when an affine map $x \mapsto Ax + b$ with $A \in GL_n(\Z)$ and $b \in \Z^n$ carries one onto the other.
</document>

<document source="https://tamnd.github.io/math2/node/PF5Z57DDG2.md" tag="PF5Z57DDG2" kind="statement">
# Catmull–Clark subdivision surface

## Statement

Let $M$ be a closed polygon mesh in $\R^3$: finitely many vertices, edges and polygonal faces, where every edge lies on exactly two faces.
The *valence* of a vertex is the number of edges at it; on such a mesh it equals the number of faces at it.
One step of *Catmull–Clark subdivision* builds a new mesh $M'$ as follows.

1. For each face $f$, the face point $F_f$ is the average of the vertices of $f$.
2. For each edge $e$ with endpoints $P_1, P_2$ and adjacent faces $f_1, f_2$, the edge point is $E_e = \frac{1}{4}(P_1 + P_2 + F_{f_1} + F_{f_2})$.
3. Each old vertex $P$ of valence $n$ moves to $P' = \frac{1}{n}\big(Q + 2R + (n - 3) P\big)$.
   Here $Q$ is the average of the face points of the $n$ faces at $P$.
   Also, $R$ is the average of the midpoints of the $n$ edges at $P$.
4. Each face $f$ with $k$ vertices is replaced by $k$ quadrilaterals.
   The one at a vertex $P$ of $f$ has as vertices $P'$, the edge points of the two edges of $f$ at $P$, and $F_f$.

Call a vertex *extraordinary* if its valence is not $4$.
Then the following hold.

1. Every face of $M'$ is a quadrilateral, and from the second step on, the number of extraordinary vertices does not change.
2. Consider a part of the mesh where all faces are quadrilaterals and all vertices have valence $4$.
   There the step is the refinement rule of the uniform bicubic [[B-spline|62AFX32JGT]] surface with the mesh as control net.
   So the repeated meshes converge there to that surface, which is twice continuously differentiable.
3. The repeated meshes $M, M', M'', \ldots$ converge to a limit surface, which has a continuous tangent plane also at the limits of the extraordinary vertices.

## Remarks

Clause 1 is a count of valences.
Face points get the valence of the old face, edge points get valence $4$, and moved old vertices keep their valence.
So after one step only the old extraordinary vertices and the face points of non-quadrilateral faces are extraordinary, and later steps create no new ones.
Clause 2 is checked by writing a bicubic B-spline patch on a uniform grid and halving the knot spacing.
The new control points are then given by the three averaging rules above with $n = 4$, where the vertex rule becomes $P' = \frac14(Q + 2R + P)$.
Clause 3 needs a separate analysis of the subdivision matrix around an extraordinary vertex, through its eigenvalues; the surface is in general not twice differentiable there.
Example: start from a cube; after one step it has $24$ quadrilateral faces, and its $8$ corners stay extraordinary of valence $3$.
The limit surface is a smooth rounded cube.

## Explanations

### L1 human

The method turns a coarse box-like model into a smooth surface by repeated refinement.
At each step every face is cut into small four-sided pieces, and every point is moved toward a weighted average of its neighbours.
The corners get rounded a little more each time, and the shapes settle down to a smooth limit.
A cube becomes a rounded, pebble-like shape.

### L2 human

Catmull–Clark subdivision is the best known example of a [[subdivision surface|N2JTWEJ1F9]].
It generalises bicubic B-spline surfaces from regular grids to meshes of any topology, and it is widely used in computer graphics and animation.
The price is a loss of smoothness at the few extraordinary points, which is studied through the spectrum of the local subdivision matrix.
Other schemes, such as Loop subdivision for triangle meshes, follow the same pattern.
</document>

<document source="https://tamnd.github.io/math2/node/PF83EHMFEG.md" tag="PF83EHMFEG" kind="definition">
# Dbar operator

## Definition

Let $U \subseteq \C^n$ be an open set, with coordinates $z_1, \dots, z_n$.
For a smooth function $f \colon U \to \C$, put
$$\bar\partial f = \sum_{k=1}^n \frac{\partial f}{\partial \bar z_k} \, d\bar z_k,$$
where $\partial / \partial \bar z_k$ is the [[Wirtinger derivative|WMXDVF36ZT]] in the variable $z_k$.
For a [[$(p, q)$-form|DC81TCR98Y]] $\omega = \sum_{I, J} \omega_{IJ} \, dz_I \wedge d\bar z_J$ on $U$, the *$\bar\partial$ operator* gives the $(p, q+1)$-form
$$\bar\partial \omega = \sum_{I, J} \bar\partial \omega_{IJ} \wedge dz_I \wedge d\bar z_J = \sum_{I, J} \sum_{k=1}^n \frac{\partial \omega_{IJ}}{\partial \bar z_k} \, d\bar z_k \wedge dz_I \wedge d\bar z_J.$$
It is extended to all complex forms by additivity over the types.

## Remarks

The operator $\partial$ is defined in the same way with $\partial / \partial z_k$ and $dz_k$; it sends $(p, q)$-forms to $(p+1, q)$-forms.
On forms on $U$, the [[exterior derivative|0ZHSTAG8ZY]] is $d = \partial + \bar\partial$, and $\bar\partial \circ \bar\partial = 0$.
It obeys the Leibniz rule $\bar\partial(\alpha \wedge \beta) = \bar\partial \alpha \wedge \beta + (-1)^{k} \alpha \wedge \bar\partial \beta$ for $\alpha$ of degree $k$.
Example: for $f(z_1, z_2) = \abs{z_1}^2 + z_2$, $\partial f / \partial \bar z_1 = z_1$ and $\partial f / \partial \bar z_2 = 0$, so $\bar\partial f = z_1 \, d\bar z_1$.
Non-example: $\bar\partial$ is not the exterior derivative, since $\bar\partial z_1 = 0$ while $d z_1 = dz_1 \ne 0$; the $(1, 0)$ part of $d$ is dropped.
A $C^1$ function $f$ satisfies $\bar\partial f = 0$ on $U$ if and only if it is holomorphic on $U$.
Because holomorphic changes of coordinates preserve types, $\bar\partial$ is well defined on complex manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/PFF8SV6ENB.md" tag="PFF8SV6ENB" kind="concept">
# Content

## Summary

A content is a size function on a family of sets that is only finitely additive, not countably additive.
It is the same idea as a [[finitely additive measure|KFWQ24D9M6]], usually on a [[ring of sets|KWE1DBMN9B]] or a [[semiring of sets|PYH82FMXQ2]].
Contents are the first step toward measures: a content that is also countably additive is a [[premeasure|PV0XXSWFC7]].
Jordan measure is a classical content that is not a measure.

## Explanations

### L1 human

A content gives a size to some sets, so that the size of two disjoint pieces together is the sum of their sizes.
It says nothing about cutting a set into infinitely many pieces.
For example, the area of finite unions of rectangles in the plane is a content.
The [[Jordan measure|0KTGJW1ZTH]] of a bounded region, defined by inner and outer approximation by rectangles, is another one.
</document>

<document source="https://tamnd.github.io/math2/node/PFJFB459MT.md" tag="PFJFB459MT" kind="definition">
# Horvitz–Thompson estimator

## Definition

Let $U = \set{1, \dots, N}$ be a finite population, and let $y_1, \dots, y_N$ be fixed real values attached to the units.
Let $S \subseteq U$ be a random sample drawn by a sampling design, and let $\pi_i = P(i \in S)$ be the inclusion probability of unit $i$.
Assume $\pi_i > 0$ for every $i \in U$.
The *Horvitz–Thompson estimator* of the population total $Y = \sum_{i=1}^N y_i$ is the [[estimator|0TTNRP5PFQ]]
$$\hat{Y}_{HT} = \sum_{i \in S} \frac{y_i}{\pi_i}.$$
The Horvitz–Thompson estimator of the population mean $Y/N$ is $\hat{Y}_{HT}/N$.

## Remarks

The estimator is unbiased: write $\hat{Y}_{HT} = \sum_{i=1}^N \frac{y_i}{\pi_i} \mathbf{1}[i \in S]$, and take the [[expected value|0DCT0QZVQ1]], using $E(\mathbf{1}[i \in S]) = \pi_i$, to get $E(\hat{Y}_{HT}) = Y$.

Example: in a [[simple random sample|Y0PTCJBPEA]] of size $n$ drawn without replacement, every unit has $\pi_i = n/N$.
Then $\hat{Y}_{HT} = \frac{N}{n}\sum_{i \in S} y_i = N \bar{y}$, where $\bar{y}$ is the sample mean.

Non-example: let some unit $j$ have $\pi_j = 0$.
Then it never appears in the sample, so no estimator built from the sample can be unbiased for every choice of $y_j$.
The condition $\pi_i > 0$ fails, and the definition does not apply.

With the joint inclusion probabilities $\pi_{ij} = P(i \in S, j \in S)$ and $\pi_{ii} = \pi_i$, the variance is
$$\operatorname{Var}(\hat{Y}_{HT}) = \sum_{i=1}^N \sum_{j=1}^N (\pi_{ij} - \pi_i \pi_j) \frac{y_i}{\pi_i} \frac{y_j}{\pi_j}.$$
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<document source="https://tamnd.github.io/math2/node/PFJP8T03C4.md" tag="PFJP8T03C4" kind="concept">
# Caloric function

## Summary

A caloric function is a solution of the [[heat equation|VP2HN86VAG]] $\partial_t u = \Delta u$ on an open set of space-time.
Caloric functions play for the heat equation the role that [[harmonic functions|0DEB0JMN0W]] play for the Laplace equation.
Much of classical potential theory, such as mean value formulas, maximum principles and the Perron method, has a parallel version for caloric functions.
This parallel theory is often called parabolic or caloric potential theory.

## Explanations

### L1 human

Think of $u(x, t)$ as the temperature at the place $x$ at the time $t$ in a body where heat simply spreads out.
A caloric function is any temperature profile that obeys the law of heat flow.
A harmonic function that does not depend on time is caloric, since it describes a steady temperature.
A simple example in one space variable is $u(x, t) = x^2 + 2t$: here $\partial_t u = 2$ and $\partial_x^2 u = 2$.
Another is $u(x, t) = e^{-t} \sin x$, a wave of heat that fades as time goes on.
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<document source="https://tamnd.github.io/math2/node/PFMGZ7988V.md" tag="PFMGZ7988V" kind="concept">
# Algebra of sets

## Summary

An algebra of sets on a set $X$ is a family of subsets of $X$ that contains $X$.
It is closed under complements and finite unions.
It is also called a field of sets.
Algebras of sets are the natural domain of finitely additive measures.
They are also the starting point of Carathéodory's construction of measures.

## Explanations

### L1 human

An algebra of sets is a family of subsets that is stable under the basic set operations done finitely many times.
If two sets are in the family, then so are their union, their intersection and their complements.
For example, on the natural numbers, the finite sets together with the sets whose complement is finite form an algebra.
A σ-algebra asks more: it must also be stable under countable unions.
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<document source="https://tamnd.github.io/math2/node/PFQJTXX9YG.md" tag="PFQJTXX9YG" kind="definition">
# Contraction semigroup

## Definition

Let $X$ be a [[Banach space|06GZ956657]].
A *contraction semigroup* on $X$ is a [[strongly continuous semigroup|EVF180QJXY]] $(T(t))_{t \ge 0}$ of bounded operators on $X$ such that
$$\norm{T(t)} \le 1 \quad \text{for all } t \ge 0,$$
where $\norm{\cdot}$ is the [[operator norm|06EKW1AKXF]].

## Remarks

Example: on $L^p(\R)$ with $1 \le p < \infty$, let $(T(t)f)(s) = f(s + t)$.
Each $T(t)$ preserves the $L^p$ norm, so $\norm{T(t)} = 1$, and $t \mapsto T(t)f$ is continuous for every $f$, so this is a contraction semigroup.

Non-example: on $\C$, the semigroup $T(t)z = e^{t}z$ is strongly continuous, but $\norm{T(t)} = e^{t} > 1$ for $t > 0$.

The [[generator|60HSWZBTK9]] of a contraction semigroup is [[dissipative|3S7HJ5GQRF]].
Some authors write $C_0$ contraction semigroup to stress strong continuity.
</document>

<document source="https://tamnd.github.io/math2/node/PFT115EGPE.md" tag="PFT115EGPE" kind="definition">
# Modular function

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] and let $\mu$ be a left [[Haar measure|VY0HXD3GC4]] on $G$.
For $g \in G$, the measure $\mu_g(A) = \mu(Ag)$ on Borel sets $A$ is again a left Haar measure.
By the uniqueness part of the [[existence and uniqueness of Haar measure|KB792QTG4F]], there is a unique real number $\Delta(g) > 0$ such that
$$\mu(Ag) = \Delta(g) \, \mu(A) \quad \text{for every Borel set } A \subseteq G.$$
The map $\Delta \colon G \to (0, \infty)$ is the *modular function* of $G$.

## Remarks

The number $\Delta(g)$ does not depend on the choice of $\mu$, since any other left Haar measure is $c\mu$ with $c > 0$.
The map $\Delta$ is a continuous homomorphism into the multiplicative group $(0, \infty)$.
In integral form, $\int_G f(xg^{-1}) \, d\mu(x) = \Delta(g) \int_G f(x) \, d\mu(x)$ for compactly supported continuous $f$.

Example: let $G$ be the group of pairs $(a, b)$ with $a > 0$ and $b \in \R$, with product $(a, b)(c, d) = (ac, ad + b)$, the group of maps $x \mapsto ax + b$.
The measure $d\mu = da \, db / a^2$ is a left Haar measure.
Right multiplication by $(c, d)$ sends $(a, b)$ to $(ac, ad + b)$, a map with Jacobian determinant $c$.
So $\mu(A(c, d)) = \int_A c \, (ac)^{-2} a^2 \, d\mu = c^{-1} \mu(A)$, and $\Delta(c, d) = 1/c$.

For this group the measure $da \, db / a$ is a right Haar measure but not a left one, so the left and right Haar measures differ.

Some authors use the reciprocal convention, defining the modular function by $\mu(Ag) = \Delta(g)^{-1} \mu(A)$ or through right Haar measure, so formulas must be checked against the source.
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<document source="https://tamnd.github.io/math2/node/PG1CVVN7KA.md" tag="PG1CVVN7KA" kind="statement">
# Bruhat decomposition of a Lie group

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center, with an [[Iwasawa decomposition|4YDR9Q36DS]] $G = KAN$ and Lie algebra $\mathfrak{a}$ of $A$.
Let $M = Z_K(\mathfrak{a})$ and $M' = N_K(\mathfrak{a})$ be the centralizer and the normalizer of $\mathfrak{a}$ in $K$, and let $W = M'/M$ be the restricted Weyl group.
Let $P = MAN$ be the minimal [[parabolic subgroup|Z2FEBW5MQM]].
For $w \in W$, the double coset $P \tilde w P$ does not depend on the representative $\tilde w \in M'$ of $w$, and it is written $PwP$.
Then $G$ is the disjoint union
$$G = \bigsqcup_{w \in W} PwP.$$

## Remarks

The coset $P \tilde w P$ does not depend on $\tilde w$ because two representatives differ by an element of $M \subseteq P$.
A standard proof follows the algebraic case, with the minimal parabolic subgroup in place of a Borel subgroup.
Disjointness uses that $W$ acts simply transitively on the positive systems of restricted roots.

Example: $G = \mathrm{SL}_2(\R)$, where $P$ is the group of upper triangular matrices and $W = \set{1, s}$ with $s$ represented by $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.
A matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ lies in $P$ when $c = 0$ and in $PsP$ when $c \ne 0$.
So $\mathrm{SL}_2(\R) = P \sqcup PsP$.

The [[Bruhat decomposition for reductive algebraic groups|T91E86D0RH]] over an algebraically closed field is the analogue with a Borel subgroup in place of $P$.
For a real group, $W$ is the Weyl group of the restricted root system, which can be smaller than the Weyl group of the complexified group.

## Explanations

### L1 human

The group is cut into finitely many pieces, one for each element of a finite group $W$.
Each piece is a set of the form $PwP$, where $P$ is the minimal parabolic subgroup.
For $2 \times 2$ real matrices of determinant $1$, there are two pieces: the upper triangular matrices, and all the others.

### L2 human

Passing to the flag manifold $G/P$, the decomposition says that $P$ has finitely many orbits on $G/P$, one for each $w \in W$.
The orbit of the longest element of $W$ is open and dense, and its complement has lower dimension.
This open orbit is used to realize principal series representations on functions on $N^- = \theta(N)$.
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# Antipode

## Summary

The antipode of a [[bialgebra|Q9HHVN606G]] is a linear map $S$ that is a two-sided inverse of the identity map for the convolution product.
A bialgebra with an antipode is a [[Hopf algebra|7XFYP0N84H]].
The antipode generalizes the inverse map $g \mapsto g^{-1}$ of a group, and it is used to define dual representations.
When it exists it is unique, and it reverses the order of products and of coproducts.

## Explanations

### L1 human

In a group, multiplying an element by its inverse gives the identity.
In a bialgebra one first splits an element into pairs, applies $S$ to the left part of each pair, and multiplies the parts back together.
The antipode is the map for which this always gives the counit of the element times $1$.
For a group algebra, the antipode sends $g$ to $g^{-1}$.
For polynomials $K[x]$ with $x$ split as $x \otimes 1 + 1 \otimes x$, the antipode sends $x$ to $-x$.
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# Homotopy group

## Definition

Let $(X, x_0)$ be a [[pointed space|T9DG8DDE75]], and let $n \ge 1$ be an integer.
Let $I^n = [0, 1]^n$ be the $n$-cube, with boundary $\partial I^n$.
The *$n$-th homotopy group* $\pi_n(X, x_0)$ is the set of [[homotopy|9YA6S6HD9V]] classes of continuous maps $f \colon I^n \to X$ with $f(\partial I^n) = \set{x_0}$.
Here the homotopies $F \colon I^n \times [0, 1] \to X$ must satisfy $F(s, t) = x_0$ for all $s \in \partial I^n$ and all $t$.
The group operation is $[f] + [g] = [f + g]$, where
$$(f + g)(s_1, s_2, \ldots, s_n) = \begin{cases} f(2 s_1, s_2, \ldots, s_n) & \text{if } s_1 \le \tfrac12, \\ g(2 s_1 - 1, s_2, \ldots, s_n) & \text{if } s_1 \ge \tfrac12. \end{cases}$$

## Remarks

Since $I^n / \partial I^n$ is homeomorphic to $S^n$, the elements of $\pi_n(X, x_0)$ are also the pointed homotopy classes of pointed maps $(S^n, s_0) \to (X, x_0)$.
For $n = 1$ the definition gives the [[fundamental group|S1XK3HXHFX]].
The set $\pi_0(X, x_0)$ is defined as the set of path components of $X$, with the component of $x_0$ as base point; it is not a group in general.

Example: $\pi_n(\R^m, 0)$ is trivial for all $n \ge 1$.
For $f$ as above, $F(s, t) = (1 - t) f(s)$ is a homotopy that keeps $\partial I^n$ at $0$ and ends at the constant map.

Non-example: a map $f \colon I^n \to X$ that does not send all of $\partial I^n$ to $x_0$ does not define an element of $\pi_n(X, x_0)$.
Likewise, free homotopy classes of maps $S^1 \to X$ form in general a different set: they correspond to conjugacy classes of $\pi_1(X, x_0)$ when $X$ is path-connected.
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# Amenable group

## Definition

Let $G$ be a [[group|G5S1M9]], with no topology, and let $\mathcal{P}(G)$ be the set of all subsets of $G$.
The group $G$ is *amenable* if there is a function $\mu \colon \mathcal{P}(G) \to [0, 1]$ such that:

1. $\mu(G) = 1$.
2. $\mu(A \cup B) = \mu(A) + \mu(B)$ for all disjoint $A, B \subseteq G$.
3. $\mu(gA) = \mu(A)$ for all $g \in G$ and $A \subseteq G$.

Such a $\mu$ is a left-invariant finitely additive probability measure on $G$.

## Remarks

Example: a finite group $G$ is amenable, with $\mu(A) = \abs{A} / \abs{G}$; left translation is a bijection, so $\abs{gA} = \abs{A}$.

Non-example: the [[free group|0KTW26C80J]] $F_2$ on $a, b$ is not amenable.
Let $W(x)$ be the set of reduced words that begin with the letter $x$.
Then $F_2 = W(a) \sqcup a W(a^{-1})$ and $F_2 = W(b) \sqcup b W(b^{-1})$.
An invariant $\mu$ would give $\mu(W(a)) + \mu(W(a^{-1})) = 1$ and $\mu(W(b)) + \mu(W(b^{-1})) = 1$, but these four disjoint sets lie in $F_2$, so their total is at most $1$, a contradiction.

Equivalently, there is a left-invariant mean on $\ell^\infty(G)$, a positive linear functional $m$ with $m(1) = 1$ and $m(g \cdot f) = m(f)$.
For a locally compact group, one asks instead for a left-invariant mean on $L^\infty(G)$; for discrete groups the two notions agree.
For a countable group, amenability is equivalent to the [[Følner condition|82R5XGCART]].
Subgroups, quotients and extensions of amenable groups are amenable, and abelian and solvable groups are amenable.
</document>

<document source="https://tamnd.github.io/math2/node/PG8ZH3ZQ88.md" tag="PG8ZH3ZQ88" kind="statement">
# Kneser's theorem on solution sets

## Statement

Let $t_0 \in \R$, $x_0 \in \R^n$ and $a, b > 0$, and let $f$ be continuous on $R = [t_0, t_0 + a] \times \set{x : \norm{x - x_0} \le b}$.
Let $M = \max_R \norm{f} > 0$ and $\alpha = \min(a, b/M)$.
For $c \in [t_0, t_0 + \alpha]$, let $S_c$ be the set of values $x(c)$ of all solutions $x : [t_0, t_0 + \alpha] \to \R^n$ of $x' = f(t, x)$ with $x(t_0) = x_0$.
Then $S_c$ is a [[continuum|HG4S5HNWXF]]: a nonempty, compact and connected subset of $\R^n$.

## Remarks

$S_c$ is nonempty by the [[Peano existence theorem|098YP831QY]].
Compactness follows from the Arzelà–Ascoli theorem, since all solutions are bounded and $M$-Lipschitz.
Connectedness is the heart of the proof: one approximates $f$ by functions for which solutions are unique.
For these the value at time $c$ depends continuously on parameters, and one passes to the limit.
Example: $x' = 2\sqrt{\abs{x}}$, $x(0) = 0$ has the solutions $x = 0$ and, for each $s \ge 0$, $x(t) = 0$ for $t \le s$ and $x(t) = (t - s)^2$ for $t \ge s$.
So $S_c = [0, c^2]$, an interval, as the theorem predicts.
In particular, the [[initial value problem|075VBY4K4X]] has either exactly one solution on $[t_0, t_0 + \alpha]$, or some $S_c$ has more than one point and is then uncountable.

## Explanations

### L1 human

When an initial value problem has many solutions, their positions at a later time do not scatter in separate pieces.
They fill a connected, closed and bounded region, the funnel of solutions.
So non-uniqueness, when it happens, comes as a whole continuum of solutions.

### L2 human

The theorem refines the [[Peano existence theorem|098YP831QY]]: it describes the whole set of solutions, not just one.
A stronger form says that the set of solutions itself, as a subset of the space of continuous functions, is compact and connected.
These results are used in the theory of differential inclusions and in the study of non-unique flows.
</document>

<document source="https://tamnd.github.io/math2/node/PGCESD48XJ.md" tag="PGCESD48XJ" kind="definition">
# Quasicircle

## Definition

Let $K \ge 1$ and let $\mathbb{T} = \set{z \in \C : \abs{z} = 1}$ be the unit [[circle|060W5CBZP6]].
A subset $\Gamma$ of $\C$ is a *$K$-quasicircle* if $\Gamma = f(\mathbb{T})$ for some $K$-[[quasiconformal mapping|A9J2KK4CEJ]] $f \colon \C \to \C$.
It is a *quasicircle* if it is a $K$-quasicircle for some $K \ge 1$.

## Remarks

Every quasicircle is a bounded Jordan curve, since $f$ is a homeomorphism.
Example: an ellipse with semi-axes $a \ge b > 0$ is the image of $\mathbb{T}$ under the real-linear map $z \mapsto \frac{a+b}{2} z + \frac{a-b}{2}\bar z$, which is $K$-quasiconformal with $K = a/b$; so an ellipse is a quasicircle.
Ahlfors' three-point condition gives an equivalent form.
A bounded Jordan curve $\Gamma$ is a quasicircle if and only if there is a constant $C$ with the following property.
For any two points $z_1, z_2 \in \Gamma$, the arc of $\Gamma$ of smaller diameter between them has diameter at most $C \abs{z_1 - z_2}$.
Non-example: the cardioid $r = 1 - \cos\theta$ has a cusp at $0$.
Take the two points of the cardioid at angles $\pm\theta$ near the cusp.
Their distance is of order $\theta^3$, while the arc between them through the cusp has diameter of order $\theta^2$.
So the three-point condition fails, and the cardioid is not a quasicircle.
Every $C^1$ Jordan curve and the Koch snowflake are quasicircles.
</document>

<document source="https://tamnd.github.io/math2/node/PGJNYGAN75.md" tag="PGJNYGAN75" kind="definition">
# Mahlo cardinal

## Definition

Let $\kappa$ be an [[inaccessible cardinal|NW4DREQ5DQ]].
The cardinal $\kappa$ is *Mahlo* if the set
$$\set{\lambda < \kappa : \lambda \text{ is a regular cardinal}}$$
is a [[stationary subset|YE6WZR46RQ]] of $\kappa$, that is, it meets every closed unbounded subset of $\kappa$.

A weakly inaccessible cardinal $\kappa$ with the same property is *weakly Mahlo*.

## Remarks

For inaccessible $\kappa$, the uncountable strong limit cardinals below $\kappa$ form a closed unbounded set.
So $\kappa$ is Mahlo if and only if the inaccessible cardinals below $\kappa$ form a stationary set.
The least inaccessible cardinal $\kappa$ is not Mahlo.
Indeed, a regular cardinal among the uncountable strong limit cardinals below $\kappa$ would be an inaccessible cardinal below $\kappa$.
So the set of regular cardinals below $\kappa$ misses this closed unbounded set.
If $\kappa$ is Mahlo, then $\kappa$ is the $\kappa$-th inaccessible cardinal.
Every weakly compact cardinal is Mahlo.
</document>

<document source="https://tamnd.github.io/math2/node/PGKWBJDTKX.md" tag="PGKWBJDTKX" kind="concept">
# Absolute Galois group

## Summary

The absolute Galois group of a field $K$ is the Galois group of a [[separable closure|JJ91A1V8NQ]] of $K$ over $K$.
It contains, as quotients, the Galois groups of all finite Galois extensions of $K$, so it records the whole Galois theory of $K$ at once.
With the [[Krull topology|F3GTF5NT4M]] it is a [[profinite group|KA0YDB6QXE]].

## Explanations

### L1 human

Instead of studying one extension of a field at a time, take the largest separable algebraic extension and look at all its symmetries.
Every finite Galois extension is a piece of it, and its group is a quotient of the absolute Galois group.
For the real numbers, the separable closure is $\C$, and the absolute Galois group has two elements: the identity and complex conjugation.
For $\Q$, the absolute Galois group is a huge and still mysterious object.
</document>

<document source="https://tamnd.github.io/math2/node/PGMNNRN5YN.md" tag="PGMNNRN5YN" kind="statement">
# Nörlund–Rice integral

## Statement

Let $n \ge 1$ and $0 \le \alpha \le n$ be integers.
Let $\gamma$ be a positively oriented simple closed curve in $\C$ that passes through none of the integers $0, \dots, n$.
Assume that its interior contains $\alpha, \alpha + 1, \dots, n$ and none of $0, 1, \dots, \alpha - 1$.
Let $f$ be holomorphic on an open set that contains $\gamma$ and its interior.
Then
$$\sum_{k=\alpha}^{n} \binom{n}{k} (-1)^k f(k) = \frac{(-1)^n}{2\pi i} \oint_\gamma f(z)\, \frac{n!}{z(z-1)(z-2)\cdots(z-n)}\, dz,$$
where the integral is a [[contour integral|00FWNP9TXY]] and $\binom{n}{k}$ is the [[binomial coefficient|092GEF38YD]].

## Remarks

The proof is the [[residue theorem|032HKYS329]].
Inside $\gamma$ the integrand has simple poles only at $z = k$ with $\alpha \le k \le n$, and the residue there is $f(k)\, n! / \prod_{j \ne k} (k - j) = (-1)^{n-k} \binom{n}{k} f(k)$.
Multiplying by $(-1)^n$ gives the terms $(-1)^k \binom{n}{k} f(k)$.
For $\alpha = 0$ the left side is $(-1)^n \Delta^n f(0)$, an [[$n$-th forward difference|E38801KVJ1]], so the formula turns a high-order difference into a contour integral.
Example: for $f = 1$ and $\alpha = 0$, the left side is $(1 - 1)^n = 0$.
On the right, the integrand is $O(\abs{z}^{-2})$, and the integral over a circle does not depend on its radius once $\gamma$ encloses $0, \dots, n$.
So it equals its limit as the radius grows, which is $0$.

## Explanations

### L1 human

Alternating sums with binomial coefficients, such as $\sum_k \binom{n}{k} (-1)^k f(k)$, involve huge cancellations and are hard to estimate term by term.
The Nörlund–Rice integral rewrites such a sum as one complex integral.
The integral can then be estimated by moving the contour, which often gives precise asymptotics as $n$ grows.

### L2 human

The integrand is $f(z)$ times the kernel $n!/(z(z-1)\cdots(z-n)) = \Gamma(n+1)\Gamma(z-n)/\Gamma(z+1)$, so its size away from the integers is controlled by the gamma function.
To get asymptotics one extends $f$ meromorphically, enlarges the contour, and collects the residues at the poles of $f$ that are crossed.
This method is a standard tool in the analysis of algorithms, for example for digital trees and related recursive structures.
It is closely related to the Mellin transform method for harmonic sums.
</document>

<document source="https://tamnd.github.io/math2/node/PGXZJ2YWVJ.md" tag="PGXZJ2YWVJ" kind="concept">
# MacCormack method

## Summary

The MacCormack method is an explicit second-order [[finite difference|X7HYEBA5SA]] scheme for hyperbolic [[conservation laws|8RA7AMWSAF]].
It makes a predictor step with forward differences and then a corrector step with backward differences.
For linear equations it coincides with the [[Lax–Wendroff method|YHZ9BCNDRZ]].
For nonlinear systems it is simpler to program, because it needs no Jacobian of the flux.
It was introduced for problems of computational fluid dynamics, such as compressible flow.

## Explanations

### L1 human

The method first makes a rough prediction of the next time level, looking one grid point ahead.
It then corrects this prediction, looking one grid point back, and averages the result with the old value.
Alternating the two directions cancels the leading errors of each one-sided difference.
The result is second-order accurate, like a centred scheme, while each step uses only simple one-sided differences.
</document>

<document source="https://tamnd.github.io/math2/node/PGZSNWS5VN.md" tag="PGZSNWS5VN" kind="definition">
# Universal Horn class

## Definition

Let $\sigma$ be a first-order [[signature|WDVVEYD6AT]].
A *universal Horn sentence* of $\sigma$ is a [[sentence|DKFGQTRYZE]] $\forall x_1 \cdots \forall x_m \, \varphi$ in which $\varphi$ is a [[Horn clause|QR92EZDTKF]].
The atoms of $\varphi$ are atomic $\sigma$-formulas, possibly equations, in the variables $x_1, \ldots, x_m$.
Let $K$ be a class of [[$\sigma$-structures|983GZBQYT4]].
Then $K$ is a *universal Horn class* if there is a set $\Sigma$ of universal Horn sentences of $\sigma$ with the following property.
A $\sigma$-structure is in $K$ exactly when it satisfies every sentence of $\Sigma$.

## Remarks

Equivalently, each axiom has one of the forms $\forall \bar x \, (\alpha_1 \wedge \cdots \wedge \alpha_n \rightarrow \beta)$ or $\forall \bar x \, \neg (\alpha_1 \wedge \cdots \wedge \alpha_n)$ with atomic $\alpha_i$ and $\beta$.
Universal Horn sentences are preserved by substructures and by direct products of nonempty families, so every universal Horn class is closed under these constructions.

Example: [[partial orders|0QV2JS9RJ3]], in the signature with one binary relation $\le$, form a universal Horn class.
Its axioms $\forall x \, (x \le x)$, $\forall x \forall y \, (x \le y \wedge y \le x \rightarrow x = y)$ and $\forall x \forall y \forall z \, (x \le y \wedge y \le z \rightarrow x \le z)$ are universal Horn sentences.

Non-example: [[total orders|0FRXBVVR03]] do not form a universal Horn class.
The product of the 2-element chain $\set{0 < 1}$ with itself, ordered coordinatewise, is a partial order in which $(0, 1)$ and $(1, 0)$ are incomparable, so the class is not closed under products.

When $\sigma$ has only function symbols, the universal Horn sentences of the first form are exactly the [[quasi-identities|94GVY3APPQ]], so the corresponding classes are the [[quasivarieties|WRXEKT4NQS]].
</document>

<document source="https://tamnd.github.io/math2/node/PH4Z47SCK6.md" tag="PH4Z47SCK6" kind="statement">
# Christodoulou–Klainerman stability of Minkowski space

## Statement

Consider the Einstein vacuum equations $\operatorname{Ric}(g) = 0$ for a [[Lorentzian metric|BY4QYYS788]] $g$ on a $4$-dimensional manifold, the vacuum case of the [[Einstein field equations|3MCZ97SW6V]].
Let $(\Sigma, \bar g, k)$ be an initial data set for these equations, with $\Sigma$ diffeomorphic to $\R^3$.
Here $\bar g$ is a Riemannian metric on $\Sigma$, $k$ is a symmetric $2$-tensor on $\Sigma$, and the constraint equations hold.
Assume that the data are strongly asymptotically flat and maximal ($\operatorname{tr}_{\bar g} k = 0$).
Assume also that they are close enough to the data $(\R^3, \delta, 0)$ of Minkowski space in a suitable weighted Sobolev norm.
Then the maximal globally hyperbolic development of the data is a geodesically complete solution of the vacuum equations.
Moreover, it approaches Minkowski space in every direction as one goes to infinity, with precise decay rates for the curvature.

## Remarks

The result was proved by Demetrios Christodoulou and Sergiu Klainerman, and published as a monograph in 1993.
In a suitable gauge the vacuum equations form a system of nonlinear wave equations, so the theorem is a global existence result for small data of a [[hyperbolic|92C3S03AQZ]] system.
The nonlinearity is critical in three space dimensions.
For general quadratic nonlinearities small data can blow up in finite time, so global existence uses a special structure of the Einstein equations.
The proof uses a foliation of spacetime by maximal hypersurfaces and by null cones, and energy estimates for the Bel–Robinson tensor contracted with approximate symmetries of the solution.
Lindblad and Rodnianski later gave a different proof in harmonic (wave) gauge, under somewhat different assumptions on the data.

## Explanations

### L1 human

In general relativity, empty flat spacetime is the simplest solution of Einstein's equations.
The theorem says that this solution is stable: if one disturbs it a little, for instance by weak gravitational waves, the disturbance spreads out and dies away.
The spacetime does not collapse into a black hole or develop other singularities.
Far away and late in time it looks again like flat spacetime.

### L2 human

Proving stability means proving global existence and decay for a quasilinear hyperbolic system with small data.
Linear waves in three space dimensions decay like $t^{-1}$.
This is just barely too slow to control a general quadratic nonlinearity, as examples of John show for some nonlinear wave equations.
In harmonic gauge the Einstein equations do not satisfy the classical null condition of Klainerman, but their nonlinear terms still have enough structure to prevent blow-up.
The techniques of the proof, the vector field method and geometric energy estimates, became standard tools for nonlinear wave equations.
They are also used in later stability problems, such as the stability of black holes.
</document>

<document source="https://tamnd.github.io/math2/node/PH61T3B7W7.md" tag="PH61T3B7W7" kind="statement">
# Inverse Z-transform

## Statement

Let $x = (x_n)_{n \in \Z}$ be a sequence of complex numbers whose [[bilateral Z-transform|3DBAMXA80B]] $X(z) = \sum_{n \in \Z} x_n z^{-n}$ has a [[region of convergence|1P8WX3EGHF]] that contains a circle $\abs{z} = \rho$ with $\rho > 0$.
Then for every $n \in \Z$,
$$x_n = \frac{1}{2\pi i} \oint_{\abs{z} = \rho} X(z) \, z^{n-1} \, dz,$$
where the [[contour integral|06ZGPYTWK8]] runs once counterclockwise around the circle.
In particular, $x$ is determined by the function $X$ on any one such circle.

## Remarks

On the circle the series $\sum_k x_k z^{-k}$ converges absolutely and uniformly, so it can be integrated term by term after multiplication by $z^{n-1}$.
The integral of $z^{n-1-k}$ around the circle is $2\pi i$ when $k = n$ and $0$ otherwise, which leaves $x_n$.
This is the coefficient formula for the [[Laurent series|0GXMPDNYX3]] of $X$ on its annulus.
Example: $X(z) = z/(z - a)$ on $\abs{z} > \abs{a}$; for $n \ge 0$ the residue theorem gives $x_n = a^n$, and for $n < 0$ the integrand $z^n/(z - a)$ has total residue $0$ inside a large circle, so $x_n = 0$.
The circle must lie in the region of convergence: the same rational function $z/(z - a)$ on $\abs{z} < \abs{a}$ gives $x_n = -a^n$ for $n \le -1$ and $x_n = 0$ for $n \ge 0$.

## Explanations

### L1 human

The Z-transform packs a sequence into a function, and this formula unpacks it.
Each term of the sequence is recovered by an average of the function around a circle.
The circle must be chosen inside the region where the transform converges, since that region tells which sequence one started from.

### L2 human

In practice, $X$ is often a rational function, and one inverts it by partial fractions and a table of pairs such as $a^n \leftrightarrow z/(z - a)$.
The contour formula justifies this method and shows that the result is unique once the region of convergence is fixed.
With $z = e^{i\omega}$ on the unit circle, the formula becomes the inverse of the discrete-time Fourier transform.
It is the discrete analogue of the Bromwich inversion integral for the Laplace transform.
</document>

<document source="https://tamnd.github.io/math2/node/PHA1S6GQGV.md" tag="PHA1S6GQGV" kind="definition">
# Motzkin number

## Definition

Let $n \ge 0$ be an integer.
A *Motzkin path* of length $n$ is a [[lattice path|040WY0TT2G]] $(p_0, p_1, \ldots, p_n)$ in $\Z^2$ with these properties:

1. $p_0 = (0, 0)$ and $p_n = (n, 0)$;
2. each step $p_i - p_{i-1}$ is one of $U = (1, 1)$, $D = (1, -1)$ and $H = (1, 0)$;
3. each point $p_i$ has second coordinate at least $0$.

The *Motzkin number* $M_n$ is the number of Motzkin paths of length $n$.

## Remarks

Example: the Motzkin paths of length $3$ are $HHH$, $UDH$, $HUD$ and $UHD$, so $M_3 = 4$.
Non-example: the path $HDU$ ends at $(3, 0)$ but fails clause 3, because after two steps it is at height $-1$.

The first values are $M_0, \ldots, M_7 = 1, 1, 2, 4, 9, 21, 51, 127$.
Splitting a path at its first step gives the recurrence
$$M_{n+1} = M_n + \sum_{k=0}^{n-1} M_k M_{n-1-k}.$$
The [[generating function|0PHAC833JF]] $M(z) = \sum_n M_n z^n$ satisfies $M = 1 + z M + z^2 M^2$, hence
$$M(z) = \frac{1 - z - \sqrt{1 - 2z - 3z^2}}{2 z^2}.$$
Choosing the positions of the $H$ steps gives $M_n = \sum_{k} \binom{n}{2k} C_k$, where $C_k$ is the [[Catalan number|0YWZJEABB2]].
The Motzkin paths without $H$ steps are the [[Dyck words|0YTJ3RN3GE]].
</document>

<document source="https://tamnd.github.io/math2/node/PHEN4E1JPT.md" tag="PHEN4E1JPT" kind="concept">
# Hopfian group

## Summary

A [[group|G5S1M9]] is Hopfian if every surjective homomorphism from the group to itself is an isomorphism.
Equivalently, the group is not isomorphic to any of its proper quotients.
Finite groups, $\Z$ and every finitely generated [[residually finite group|HEE922EHQ8]] are Hopfian.
The [[Baumslag–Solitar group|429JJ5ZNBY]] $BS(2, 3)$ is a finitely presented group that is not Hopfian.

## Explanations

### L1 human

A homomorphism can collapse parts of a group.
A Hopfian group cannot collapse a part and still look like the whole group afterwards.
For finite groups this is clear by counting: a map from a finite set onto itself is a bijection.
An infinite group can fail.
Take the group of all sequences $(x_1, x_2, \ldots)$ of elements of $\Z/2$ that are $0$ from some point on.
Deleting the first term is onto, but it kills the sequence $(1, 0, 0, \ldots)$.
</document>

<document source="https://tamnd.github.io/math2/node/PHHT5W3BEJ.md" tag="PHHT5W3BEJ" kind="concept">
# Regular conditional probability

## Summary

A regular conditional probability is a version of conditional probability given a σ-algebra that is a genuine probability measure for each outcome.
Conditional probabilities $P(A \mid \mathcal{G})$ are defined one event at a time and only up to null sets.
So in general they need not fit together into a measure.
A regular version is a [[Markov kernel|2AVNDP8BJP]] that does fit them together, and it lets one compute conditional expectations as integrals.
It exists when the underlying space is nice enough, for example a [[Polish space|584GXXY73M]] with its Borel σ-algebra.

## Explanations

### L1 human

Given some partial information, one wants a whole new probability law that describes the remaining uncertainty.
For a point $(x, y)$ chosen uniformly in the unit square, knowing $x$ leaves $y$ uniform on $[0, 1]$, and that uniform law is the regular conditional probability.
The difficulty is that the information may be an event of probability zero, like "$x = 0.3$".
A regular conditional probability fixes one law for each value of the information at once, so that all of them together reproduce the original probabilities.
The [[Borel–Kolmogorov paradox|9RMS0ZJ618]] shows why the family must be chosen together, not one null event at a time.
</document>

<document source="https://tamnd.github.io/math2/node/PHJ8DD2W0E.md" tag="PHJ8DD2W0E" kind="definition">
# Bergman projection

## Definition

Let $\Omega \subseteq \C^n$ be a domain, and let $L^2(\Omega)$ be the [[Lebesgue space|SASP14MA1Z]] of square-integrable functions for Lebesgue measure $dV$.
Let $A^2(\Omega) \subseteq L^2(\Omega)$ be the [[Bergman space|2D619WTPYC]] of holomorphic functions in $L^2(\Omega)$, which is a closed subspace.
The *Bergman projection* of $\Omega$ is the [[orthogonal projection|0B8KZRT28M]]
$$P \colon L^2(\Omega) \to A^2(\Omega).$$
It is given by
$$P f(z) = \int_\Omega K(z, w) \, f(w) \, dV(w),$$
where $K$ is the [[Bergman kernel|8RWDHW0CXC]] of $\Omega$.

## Remarks

The integral formula follows from the reproducing property: for $f \in L^2(\Omega)$, $Pf(z) = \ip{Pf}{K(\cdot, z)} = \ip{f}{K(\cdot, z)}$, since $K(\cdot, z) \in A^2(\Omega)$ and $f - Pf$ is orthogonal to $A^2(\Omega)$.
Example: on the unit disc, the functions $z^k$, $k \ge 0$, form an orthogonal basis of $A^2$, and $\ip{\bar z}{z^k} = \int_{\mathbb{D}} \bar z \, \bar z^k \, dV = 0$ for every $k$, so $P \bar z = 0$.
Similarly $P \abs{z}^2 = \frac{1}{2}$, since $\ip{\abs{z}^2}{1} = \int_{\mathbb{D}} \abs{z}^2 dV = \frac{\pi}{2}$ and $\norm{1}^2 = \pi$, and $\abs{z}^2$ is orthogonal to $z^k$ for $k \ge 1$.
Non-example: the orthogonal projection of $L^2(\Omega)$ onto the harmonic functions in $L^2(\Omega)$ is a different operator, the harmonic Bergman projection; on the unit disc it fixes $\bar z$.
$P$ is self-adjoint, $P^2 = P$, and $\norm{P} = 1$ when $A^2(\Omega) \ne \set{0}$.
On smoothly bounded strongly pseudoconvex domains, $P$ maps $C^\infty(\bar\Omega)$ into itself.
</document>

<document source="https://tamnd.github.io/math2/node/PHT3TJCS90.md" tag="PHT3TJCS90" kind="statement">
# Ax–Kochen theorem

## Statement

For each positive integer $d$ there is a finite set $E(d)$ of [[prime numbers|0SN0MAEKYX]] with the following property.
Let $p$ be a prime with $p \notin E(d)$, and let $\Q_p$ be the field of [[p-adic numbers|BT6T702C28]].
Let $F$ be a [[homogeneous polynomial|2C3PAQ7H50]] of degree $d$ in $n$ variables with coefficients in $\Q_p$.
If $n > d^2$, then there is $x \in \Q_p^n$ with $x \ne 0$ and $F(x) = 0$.

## Remarks

The proof is model-theoretic.
Let $\varphi$ be a first-order sentence in the language of rings.
Then $\varphi$ holds in $\Q_p$ for almost all primes $p$ if and only if it holds in the [[Laurent series field|YZWYF7DXTW]] $\F_p((t))$ for almost all $p$.
Here "almost all" means all but finitely many.
This transfer comes from [[ultraproducts|AX9Q86GB8M]] over a nonprincipal ultrafilter on the primes.
The ultraproducts of the fields $\Q_p$ and of the fields $\F_p((t))$ are [[elementarily equivalent|P3B1V182N4]] valued fields.
Both are Henselian with residue field of characteristic $0$, and they have the same residue field and the same value group.
For a fixed degree $d$, the property "every form of degree $d$ in $d^2 + 1$ variables has a nontrivial zero" is one first-order sentence.
A theorem of Lang shows that it holds in every $\F_p((t))$.
The finite exceptional set cannot be dropped in general: Terjanian found a form of degree $4$ in $18$ variables over $\Q_2$ with no nontrivial zero, while $18 > 4^2$.

## Explanations

### L1 human

Artin asked whether every homogeneous polynomial of degree $d$ in more than $d^2$ variables has a nonzero root over the $p$-adic numbers.
The answer is no in general, but Ax and Kochen proved that it is yes for all but finitely many primes $p$, the finitely many depending on $d$.
The proof compares $p$-adic numbers with power series over the finite field $\F_p$, where the answer is known, and uses logic to carry it over.

### L2 human

The theorem is a celebrated application of model theory to number theory.
The key principle is now called the Ax–Kochen–Ershov principle.
It says that the first-order theory of a Henselian valued field of residue characteristic $0$ is determined by the theories of its residue field and value group.
The original proof gives no explicit bound for the exceptional primes.
For $d = 2$ and $d = 3$ the conclusion is known to hold for all primes.
</document>

<document source="https://tamnd.github.io/math2/node/PHT84D91B7.md" tag="PHT84D91B7" kind="definition">
# Measurable function

## Definition

Let $(X, \Sigma)$ and $(Y, \mathcal{T})$ be [[measurable spaces|N4KDT83V3S]].
A [[function|0EPAT51DXQ]] $f \colon X \to Y$ is *measurable* if $f^{-1}(B) \in \Sigma$ for every $B \in \mathcal{T}$.
A function $f \colon X \to \R$ is measurable if it is measurable for the [[Borel sigma-algebra|0WQXY32Q8D]] on $\R$.

## Remarks

A function $f \colon X \to \R$ is measurable if and only if $\set{x \in X : f(x) > a} \in \Sigma$ for every $a \in \R$.
It suffices to check preimages of a family that generates $\mathcal{T}$.
Example: every continuous $f \colon \R \to \R$ is Borel measurable, since preimages of open sets are open and the open sets generate the Borel sets.
Example: the indicator function $\mathbf{1}_A$ is measurable if and only if $A \in \Sigma$.
Non-example: if $V$ is a non-measurable subset of $\R$, then $\mathbf{1}_V$ is not Lebesgue measurable, because $\set{\mathbf{1}_V > 1/2} = V$.
For Lebesgue measure on $\R$, a measurable function usually means one with $\Sigma$ the Lebesgue measurable sets and $\mathcal{T}$ the Borel sets.
A composition of measurable functions is measurable.
</document>

<document source="https://tamnd.github.io/math2/node/PHTRQKMAC8.md" tag="PHTRQKMAC8" kind="definition">
# Electrical network

## Definition

An *electrical network* is a pair $(G, c)$ of a connected [[graph|00C0BR7D7H]] $G = (V, E)$, in which every vertex has finitely many neighbours, and a function $c \colon E \to (0, \infty)$.
The number $c(e)$ is the *conductance* of the edge $e$, and $r(e) = 1/c(e)$ is its *resistance*.
For $x, y \in V$, write $c(x, y) = c(\set{x, y})$ if $\set{x, y} \in E$ and $c(x, y) = 0$ otherwise.
A *voltage* is a function $v \colon V \to \R$.
The *current* of $v$ is the function on ordered pairs of neighbours given by Ohm's law: $i(x, y) = c(x, y)\big(v(x) - v(y)\big)$.

## Remarks

The current is antisymmetric: $i(y, x) = -i(x, y)$.
The net current out of $x$ is $\sum_y i(x, y) = \sum_y c(x, y)\big(v(x) - v(y)\big)$.
Kirchhoff's node law at $x$ says that this net current is $0$, which holds exactly when $v$ is [[harmonic|WRY0M7SW59]] at $x$.
Example: the path $a - m - b$ with two edges of conductance $1$ is a network.
The voltage with $v(a) = 1$, $v(m) = \tfrac12$, $v(b) = 0$ is harmonic at $m$, and it sends the current $\tfrac12$ along each edge from $a$ to $b$.
Non-example: giving an edge conductance $0$ is not allowed; such an edge carries no current and should be removed from the graph instead.
Kirchhoff's cycle law, that voltage drops add to $0$ around each cycle, is automatic here because the current comes from a voltage.
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<document source="https://tamnd.github.io/math2/node/PJ4DQ435XK.md" tag="PJ4DQ435XK" kind="statement">
# Parametric transversality theorem

## Statement

Let $X$, $S$ and $Y$ be [[smooth manifolds|F45VQ47GPV]] without boundary, and let $Z \subseteq Y$ be an embedded [[submanifold|VKFW27WXEZ]].
Let $F \colon X \times S \to Y$ be a smooth map that is [[transverse|A2PDZ8GA9C]] to $Z$.
For $s \in S$, let $f_s \colon X \to Y$ be the map $f_s(x) = F(x, s)$.
Then $f_s$ is transverse to $Z$ for all $s$ outside a [[set of measure zero|VQQM4Y550T]] in $S$.

## Remarks

Since $F$ is transverse to $Z$, the preimage $W = F^{-1}(Z)$ is a submanifold of $X \times S$.
Let $\pi \colon W \to S$ be the restriction of the projection.
A short linear algebra argument shows that $f_s$ is transverse to $Z$ exactly when $s$ is a regular value of $\pi$.
By [[Sard's theorem|Z7R8J71SJN]], the critical values of $\pi$ form a set of measure zero, which gives the theorem.

Example: let $X$ be a submanifold of $Y = \R^n$, let $S = \R^n$ and let $F(x, s) = x + s$.
For each $x$, the map $s \mapsto x + s$ is a submersion, so $F$ is transverse to every submanifold $Z$.
The theorem says that for almost every translation vector $s$, the translate $X + s$ meets $Z$ transversally.

The transversality of $F$ is needed: if $F(x, s) = f(x)$ does not depend on $s$ and $f$ is not transverse to $Z$, then no $f_s$ is transverse to $Z$.

## Explanations

### L1 human

Two curves in the plane may touch tangentially, but a tiny shift separates them or makes them cross cleanly.
The theorem makes this precise: if a family of maps moves in enough directions, then almost every member of the family meets a given submanifold cleanly, that is, transversally.

### L2 human

The theorem is the main step in proving that transversality is generic.
In the Thom transversality theorem one builds, locally, a family of perturbations of a given map by adding polynomials, checks that the family is transverse, and applies the parametric theorem.
It also gives the transversality homotopy theorem: every smooth map into a manifold can be perturbed by an arbitrarily small homotopy to a map transverse to a given submanifold.
In infinite dimension the same argument works with the Sard–Smale theorem in place of Sard's theorem.
This is how moduli spaces in gauge theory and symplectic topology are made smooth for generic data.
</document>

<document source="https://tamnd.github.io/math2/node/PJ61DXYG2C.md" tag="PJ61DXYG2C" kind="concept">
# Absorbing element

## Summary

An absorbing element, or zero element, for a [[binary operation|0M418FE7BB]] is an element that turns every product it enters into itself.
The number $0$ is absorbing for multiplication of numbers, and the empty set is absorbing for intersection.
A binary operation has at most one two-sided absorbing element.
Absorbing elements are central in semigroup theory and in rings, where $0$ is absorbing for multiplication.

## Explanations

### L1 human

An absorbing element swallows everything it meets.
Multiply any number by $0$ and you get $0$.
Intersect any set with the empty set and you get the empty set.
For the operation $\max$ on the interval $[0, 1]$, the number $1$ is absorbing, since $\max(x, 1) = 1$.
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<document source="https://tamnd.github.io/math2/node/PJA5H9ZW4S.md" tag="PJA5H9ZW4S" kind="definition">
# Lifting idempotents

## Definition

Let $R$ be a [[ring|0894J980RK]] and $I$ a [[two-sided ideal|0DDA6CY7GJ]] of $R$.
An [[idempotent|0X4JKZXJ86]] $x + I$ of $R/I$ *lifts* to $R$ if there is $e \in R$ with $e^2 = e$ and $e - x \in I$.
*Idempotents lift modulo $I$* if every idempotent of $R/I$ lifts, that is, for every $x \in R$ with $x^2 - x \in I$ there is $e \in R$ with $e^2 = e$ and $e - x \in I$.

## Remarks

Example: idempotents lift modulo $I = 4\Z/8\Z$ in $R = \Z/8\Z$.
The idempotents of $R/I \cong \Z/4\Z$ are $0$ and $1$, and they lift to the idempotents $0$ and $1$ of $R$.
Non-example: idempotents do not lift modulo $6\Z$ in $\Z$.
The class of $3$ is idempotent in $\Z/6\Z$, since $9 - 3 = 6$, but the idempotents of $\Z$ are $0$ and $1$, and neither $3 - 0$ nor $3 - 1$ lies in $6\Z$.
Idempotents lift modulo every nil ideal, and modulo every ideal of a ring that is complete with respect to the $I$-adic topology.
A semilocal ring in which idempotents lift modulo the Jacobson radical is called [[semiperfect|ZRC2S3ECFK]].
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<document source="https://tamnd.github.io/math2/node/PJFH51Z48M.md" tag="PJFH51Z48M" kind="statement">
# M. Riesz theorem on the Hilbert transform

## Statement

Let $1 < p < \infty$, and let $H$ be the [[Hilbert transform|NSCEBXM7ZM]] on $\R$.
Then there is a constant $C_p$, depending only on $p$, such that
$$\norm{Hf}_{L^p(\R)} \le C_p \norm{f}_{L^p(\R)} \quad \text{for all } f \in L^p(\R).$$
The same holds for the [[conjugate function|A1H48DDW49]] on the circle: $\norm{\tilde f}_{L^p} \le C_p \norm{f}_{L^p}$ for all $f \in L^p(\R/2\pi\Z)$.

## Remarks

Marcel Riesz's proof for $p = 2k$ an even integer uses complex analysis.
If $f$ is real and $F = f + iHf$ extends holomorphically to the upper half-plane, then $\int F^{2k} = 0$, and taking real parts bounds $\norm{Hf}_{2k}$ by $\norm{f}_{2k}$.
Interpolation then gives $2 \le p < \infty$, and duality gives $1 < p \le 2$, since $H$ is skew-adjoint.
A real-variable proof combines the bound on $L^2$ with a weak type $(1, 1)$ bound, through the Marcinkiewicz interpolation theorem.
The best constant is $C_p = \tan\frac{\pi}{2p}$ for $1 < p \le 2$ and $C_p = \cot\frac{\pi}{2p}$ for $2 \le p < \infty$, a result of Pichorides.

Example: for $p = 2$, $H$ is an isometry, since its multiplier $-i\sgn\xi$ has modulus $1$, and $C_2 = \tan(\pi/4) = 1$.

The theorem fails for $p = 1$ and $p = \infty$.
For $f = \mathbf{1}_{[0, 1]}$, one has $Hf(x) = \frac{1}{\pi} \log\abs{\frac{x}{x - 1}}$.
This is unbounded near $x = 0$ and $x = 1$, and it behaves like $\frac{1}{\pi x}$ at infinity, so it is not integrable.

## Explanations

### L1 human

The Hilbert transform turns each cosine wave into a sine wave, and so turns the real part of a holomorphic function into its imaginary part.
This theorem says it does not make functions much bigger, measured in any $L^p$ norm with $1 < p < \infty$.
At the two ends, $p = 1$ and $p = \infty$, a jump in the function creates a logarithmic blow-up, and the bound fails.

### L2 human

The theorem is equivalent to the convergence of Fourier series in $L^p(\R/2\pi\Z)$ for $1 < p < \infty$.
Indeed, the partial sum operators are combinations of modulated conjugate function operators, so the bound for $\tilde f$ gives uniform bounds for the partial sums.
It is the first case of the theory of singular integrals, which extends it to the [[Riesz transforms|FRNDFJ9WQ5]] and to [[Calderón–Zygmund operators|3SCXCW4AH3]].
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<document source="https://tamnd.github.io/math2/node/PJN54XYB5T.md" tag="PJN54XYB5T" kind="definition">
# Fractional Fourier transform

## Definition

Let $\psi_n(x) = (2^n n! \sqrt{\pi})^{-1/2} H_n(x) e^{-x^2/2}$, $n \ge 0$, be the normalized [[Hermite functions|52VS07H6GK]]; they form an orthonormal basis of $L^2(\R)$, the [[$L^2$ space|SASP14MA1Z]].
Let $\alpha \in \R$.
The *fractional Fourier transform* of angle $\alpha$ is the [[unitary operator|MM0EJ2GDE4]] $\mathcal{F}_\alpha$ on $L^2(\R)$ given by
$$\mathcal{F}_\alpha \Big( \sum_{n \ge 0} c_n \psi_n \Big) = \sum_{n \ge 0} e^{-i n \alpha}\, c_n \psi_n$$
for every square-summable sequence $(c_n)$.

## Remarks

Let $\mathcal{F} f(\xi) = \frac{1}{\sqrt{2\pi}} \int_\R f(x) e^{-i x \xi}\, dx$ be the unitary [[Fourier transform|DPVM34ADZV]].
Each $\psi_n$ satisfies $\mathcal{F} \psi_n = (-i)^n \psi_n$, so $\mathcal{F}_{\pi/2} = \mathcal{F}$.
Also $\mathcal{F}_0 = \id$, $\mathcal{F}_\pi f(x) = f(-x)$, $\mathcal{F}_\alpha \mathcal{F}_\beta = \mathcal{F}_{\alpha + \beta}$, and $\mathcal{F}_\alpha$ depends only on $\alpha$ modulo $2\pi$.
For $0 < \alpha < \pi$ it is an integral operator: for $f$ integrable and square-integrable,
$$\mathcal{F}_\alpha f(\xi) = \sqrt{\frac{1 - i \cot\alpha}{2\pi}} \int_\R \exp\Big( \frac{i}{2} \cot\alpha\, (x^2 + \xi^2) - \frac{i\, x \xi}{\sin\alpha} \Big) f(x)\, dx,$$
with the principal branch of the square root; at $\alpha = \pi/2$ this is the formula for $\mathcal{F}$.
Some authors use the order $a = 2\alpha/\pi$ instead of the angle, so that $\mathcal{F}$ has order $1$.

Example: $f = \psi_0 = \pi^{-1/4} e^{-x^2/2}$.
Its only coefficient is $c_0 = 1$, and $e^{-i \cdot 0 \cdot \alpha} = 1$, so $\mathcal{F}_\alpha \psi_0 = \psi_0$ for every $\alpha$.

Non-example: the transform $f \mapsto \int_\R f(x) e^{-i x \xi}\, dx$ without the factor $\frac{1}{\sqrt{2\pi}}$ is not $\mathcal{F}_{\pi/2}$.
It sends $\psi_n$ to $\sqrt{2\pi}\, (-i)^n \psi_n$, so it multiplies the coefficient $c_n$ by a number of modulus $\sqrt{2\pi} \ne 1$, and it is not unitary.
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<document source="https://tamnd.github.io/math2/node/PJNV4KR00F.md" tag="PJNV4KR00F" kind="definition">
# Short-time Fourier transform

## Definition

Let $g \in L^2(\R)$ be a nonzero function, the *window*, and let $f \in L^2(\R)$ (see [[Lp space|SASP14MA1Z]]).
The *short-time Fourier transform* of $f$ with respect to $g$ is the function $V_g f \colon \R^2 \to \C$,
$$V_g f(x, \omega) = \int_{\R} f(t) \, \overline{g(t - x)} \, e^{-2\pi i t \omega} \, dt,$$
for $x \in \R$ (time) and $\omega \in \R$ (frequency).

## Remarks

The integral converges absolutely, since $f \cdot \overline{g(\cdot - x)}$ is integrable by the Cauchy–Schwarz inequality.
So $V_g f(x, \cdot)$ is the [[Fourier transform|DPVM34ADZV]] of $f \cdot \overline{g(\cdot - x)}$.
In inner product form, $V_g f(x, \omega) = \ip{f}{M_\omega T_x g}$, where $T_x g(t) = g(t - x)$ and $M_\omega g(t) = e^{2\pi i \omega t} g(t)$.
Moyal's formula states $\norm{V_g f}_{L^2(\R^2)} = \norm{f}_2 \norm{g}_2$.
The squared modulus $\abs{V_g f}^2$ is the *spectrogram*.
In engineering texts the variable $\omega$ often denotes angular frequency, and the kernel is $e^{-i \omega t}$.

Example: let $f = g = 1_{[0, 1]}$.
For $0 \le x \le 1$, the product $f \cdot g(\cdot - x)$ is the indicator of $[x, 1]$, so $V_g f(x, 0) = 1 - x$.
For $\abs{x} > 1$ the supports are disjoint and $V_g f(x, \omega) = 0$.

Non-example: with the constant window $g = 1$, which is not in $L^2(\R)$, the formula gives $\hat f(\omega)$ for every $x$.
It carries no time information, and this choice is excluded by the definition.
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<document source="https://tamnd.github.io/math2/node/PJWK0PKN6Q.md" tag="PJWK0PKN6Q" kind="concept">
# Lie derivative

## Summary

The Lie derivative measures how a [[tensor field|WX302T93TW]] changes along the flow of a vector field $X$.
It is defined without any extra structure such as a metric or a connection.
On functions it is the directional derivative, and on vector fields it is the [[Lie bracket|68FGQEBD6J]].
A tensor field is invariant under the flow of $X$ exactly when its Lie derivative along $X$ is zero.

## Explanations

### L1 human

Imagine a fluid flowing on a surface, and some quantity painted on the surface, such as a field of arrows.
Carry the painted field back along the flow for a short time and compare it with the field that is already there.
The rate of change of this comparison is the Lie derivative.
For the rotation field $X = -y \, \partial_x + x \, \partial_y$ in the plane and the function $f = x^2 + y^2$, one gets $X f = 0$.
This says that $f$ does not change under rotations about the origin.
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<document source="https://tamnd.github.io/math2/node/PJWY3JJQQ7.md" tag="PJWY3JJQQ7" kind="concept">
# Discrete spectrum

## Summary

The discrete spectrum of an operator consists of its normal eigenvalues: isolated points of the spectrum whose Riesz projector has finite rank.
Such an eigenvalue behaves like an eigenvalue of a matrix, with finite algebraic multiplicity and a finite-dimensional generalised eigenspace.
For a self-adjoint operator on a Hilbert space, the spectrum is the disjoint union of the discrete spectrum and the essential spectrum.

## Explanations

### L1 human

Some points of the spectrum look exactly like matrix eigenvalues: they are isolated, and only finitely many directions belong to them.
These points form the discrete spectrum.
In quantum mechanics, the discrete spectrum of the energy operator gives the bound states, such as the energy levels of the hydrogen atom.
For the identity on an infinite-dimensional space, $1$ is isolated, but infinitely many directions belong to it, so it is not in the discrete spectrum.
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<document source="https://tamnd.github.io/math2/node/PK2KNMWNW3.md" tag="PK2KNMWNW3" kind="concept">
# Harmonic functions on Riemannian manifolds

## Summary

A harmonic function on a [[Riemannian manifold|7CBT4SM83Y]] is a function killed by the [[Laplace–Beltrami operator|F4NWZNTRMD]] of the metric.
This extends the classical [[harmonic functions|0DEB0JMN0W]] of Euclidean space to curved spaces.
Their behaviour depends on the geometry: on a connected closed manifold they are constant.
On a complete manifold with nonnegative Ricci curvature, the positive ones are constant.
They link analysis with geometry, through gradient estimates, Liouville theorems and the study of the manifold at infinity.

## Explanations

### L1 human

On a curved surface, one can still ask for functions whose value at each point is the average of nearby values, in a sense suited to the curvature.
The Laplace–Beltrami operator makes this precise, and harmonic functions are the functions it sends to $0$.
On a closed surface such as a sphere, only constant functions are harmonic, because a nonconstant function would have a maximum where it cannot be an average.
On the hyperbolic plane there are many bounded harmonic functions, which reflects the large size of the space at infinity.
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<document source="https://tamnd.github.io/math2/node/PK3RGWY032.md" tag="PK3RGWY032" kind="concept">
# Hausdorff summation

## Summary

Hausdorff summation is a family of [[matrix summability methods|M6311DWJ1E]], one for each sequence $(\mu_n)$ of numbers.
The matrix is built from $(\mu_n)$ and its iterated differences with binomial weights.
The family contains the [[Cesàro|N8ZGG7EW6J]] methods and the [[Euler|81Q4ERP594]] methods as special cases.
Hausdorff characterized the regular methods of the family by a moment condition on $(\mu_n)$, which links summability with the Hausdorff moment problem.

## Explanations

### L1 human

Many classical summation methods turn a sequence into weighted averages with binomial coefficients in the weights.
Hausdorff noticed that they all come from one recipe fed with a sequence $(\mu_n)$.
Feeding $\mu_n = \frac{1}{n+1}$ into the recipe gives the plain running averages of Cesàro.
Feeding $\mu_n = 2^{-n}$ gives Euler's averages with the weights of Pascal's triangle.
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<document source="https://tamnd.github.io/math2/node/PK3RPS6EF4.md" tag="PK3RPS6EF4" kind="definition">
# Uniform polytope

## Definition

Convex *uniform polytopes* of dimension $d \ge 2$ are defined by induction on $d$.

1. A convex polygon is uniform if it is a [[regular polygon|01KJJ9A400]].
2. For $d \ge 3$, a [[convex polytope|300K6CB1KH]] $P \subseteq \R^d$ of dimension $d$ is uniform if both of the following hold:
   - the [[symmetry group|0ZG3Z5JZ85]] of $P$ acts [[transitively|02GK6HZYBT]] on the set of [[vertices|R7V3S9V891]] of $P$;
   - every [[facet|DPDXN2FJC4]] of $P$ is a uniform polytope of dimension $d-1$.

## Remarks

Example: the triangular prism whose bases are equilateral triangles of side $1$ and whose sides are unit squares is uniform.
Its facets are regular polygons, hence uniform.
The symmetries of the prism, namely rotations about its axis, the swap of the two bases, and reflections, carry any vertex to any other.

Non-example: the square pyramid with equilateral triangles and a unit square base has regular facets, but it is not uniform.
Its apex lies on $4$ edges and each base vertex on $3$, so no symmetry maps the apex to a base vertex, and the first condition fails.
Second non-example: a box $[0,1] \times [0,1] \times [0,2]$ is vertex-transitive, but its rectangular facets are not regular polygons, so the second condition fails.

Every [[regular polytope|F60PD912D5]] is uniform.
In dimension $3$ the convex uniform polytopes are the five Platonic solids, the thirteen Archimedean solids, and the prisms and antiprisms over regular polygons.
The definition extends to non-convex star polytopes and to uniform tilings, by allowing star polygon faces or unbounded arrangements.
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<document source="https://tamnd.github.io/math2/node/PK77W5K15V.md" tag="PK77W5K15V" kind="definition">
# Similarity transformation

## Definition

Let $n \ge 1$ and consider the [[Euclidean space|0XPR1E85W0]] $\R^n$ with the [[Euclidean distance|0NPPGHQNXQ]] $\norm{x - y}$.
A *similarity transformation* of $\R^n$ is a bijection $f \colon \R^n \to \R^n$ for which there is a real number $r > 0$ such that
$$\norm{f(x) - f(y)} = r \norm{x - y} \quad \text{for all } x, y \in \R^n.$$
The number $r$ is the *ratio* of $f$.

## Remarks

A similarity of ratio $1$ is an [[isometry|02HPP5VCZH]].
Every similarity of ratio $r$ has the form $f(x) = rAx + b$ with $A$ in the [[orthogonal group|0TKYAEJ8N7]] $\mathrm{O}(n)$ and $b \in \R^n$.
So every similarity is a [[homothety|0ZG1PNEP35]] followed by an isometry, and the similarities of $\R^n$ form a group.
Similarities preserve angles and ratios of distances.

Example: $f(x) = 3x + b$ on $\R^n$ is a similarity of ratio $3$.
Check: $\norm{f(x) - f(y)} = \norm{3x - 3y} = 3 \norm{x - y}$.

Non-example: $g(x_1, x_2) = (2x_1, x_2)$ on $\R^2$ is a bijection.
It doubles the distance between $(0, 0)$ and $(1, 0)$ but keeps the distance between $(0, 0)$ and $(0, 1)$, so no single ratio $r$ works.
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# Identity

## Summary

In universal algebra, an identity is an equation between two [[terms|6D3WA7N2W4]] that is required to hold for every value of the variables.
Laws such as associativity $x(yz) \approx (xy)z$ and commutativity $xy \approx yx$ are identities.
Classes of algebras defined by identities are called varieties, and much of universal algebra studies which identities hold in which algebras.

## Explanations

### L1 human

An identity is a rule that holds no matter which elements you plug in.
In the integers, $x + y \approx y + x$ is an identity: swap any two numbers in a sum and the result does not change.
The equation $x + x \approx x$ is not an identity of the integers, since it fails for $x = 1$, even though it holds for $x = 0$.
In a Boolean algebra, however, $x \vee x \approx x$ is an identity.
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<document source="https://tamnd.github.io/math2/node/PKFPHK189T.md" tag="PKFPHK189T" kind="definition">
# Linear regression

## Definition

Let $n \ge p \ge 1$.
Let $X$ be a known $n \times p$ real matrix, the *design matrix*, whose $i$-th row $x_i^T$ holds the explanatory variables of unit $i$.
The *linear regression model* for a response vector $y = (y_1, \ldots, y_n)^T$ is
$$y = X \beta + \varepsilon, \qquad \text{that is,} \qquad y_i = x_i^T \beta + \varepsilon_i,$$
where:

1. $\beta \in \R^p$ is an unknown vector of coefficients;
2. $\varepsilon = (\varepsilon_1, \ldots, \varepsilon_n)^T$ is a random error vector with [[mean|0WQNRCNY3M]] $\mathbb{E}\varepsilon = 0$;
3. the errors are uncorrelated with a common unknown [[variance|0477R9PYAE]] $\sigma^2 > 0$, that is, $\operatorname{Cov}(\varepsilon) = \sigma^2 I_n$.

Usually $X$ is assumed to have full column [[rank|0ZC9YTXZRY]] $p$, and often the errors are assumed normal.

## Remarks

Example: simple linear regression $y_i = \beta_0 + \beta_1 t_i + \varepsilon_i$ is the case $p = 2$ with rows $x_i^T = (1, t_i)$.
The model $y_i = \beta_0 + \beta_1 t_i + \beta_2 t_i^2 + \varepsilon_i$ is also a linear regression, with rows $(1, t_i, t_i^2)$, because it is linear in $\beta$.
Non-example: $y_i = \beta_0 e^{\beta_1 t_i} + \varepsilon_i$ is not linear in $(\beta_0, \beta_1)$, so it is a nonlinear regression model.
The conditions 2 and 3 are the Gauss–Markov conditions; when they hold and $X$ has full column rank, ordinary least squares is the best linear unbiased estimator of $\beta$.
Weaker forms allow correlated or heteroscedastic errors, which leads to generalised least squares.
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# Mercer's theorem

## Statement

Let $a < b$ and let $K \colon [a, b] \times [a, b] \to \R$ be a [[continuous kernel|ZM5G144PZG]] that is [[symmetric|ADTYF8FG6C]], $K(x, y) = K(y, x)$, and positive semidefinite: for every $m \ge 1$, all $x_1, \dots, x_m \in [a, b]$ and all $c_1, \dots, c_m \in \R$,
$$\sum_{i=1}^{m} \sum_{j=1}^{m} c_i c_j K(x_i, x_j) \ge 0.$$
Let $T_K$ be the integral operator $(T_K \varphi)(x) = \int_a^b K(x, y) \varphi(y) \, dy$ on $L^2[a, b]$.
Then there are an orthonormal family $(e_n)_{n \in N}$ in $L^2[a, b]$, finite or countable, of continuous [[eigenfunctions|4NSACSWRJZ]] of $T_K$, and eigenvalues $\mu_n > 0$, such that:

1. the $\mu_n$ are all the nonzero eigenvalues of $T_K$, each repeated as often as the dimension of its eigenspace;
2. for all $x, y \in [a, b]$,
$$K(x, y) = \sum_{n \in N} \mu_n \, e_n(x) \, e_n(y),$$
where the series converges absolutely and [[uniformly|08VZ57PT26]] on $[a, b] \times [a, b]$;
3. $\sum_{n \in N} \mu_n = \int_a^b K(x, x) \, dx$.

## Remarks

Positivity of the kernel gives $\ip{T_K \varphi}{\varphi} \ge 0$ for all $\varphi$, so all eigenvalues are nonnegative.
Positivity of the remainder kernel $K(x, y) - \sum_{n \le m} \mu_n e_n(x) e_n(y)$ gives $\sum_{n \le m} \mu_n e_n(x)^2 \le K(x, x)$ for every $m$.
So the diagonal series converges pointwise, and one shows its sum is $K(x, x)$, a continuous function.
[[Dini's theorem|0XRRYF1HPE]] then makes the convergence on the diagonal uniform, and the Cauchy–Schwarz inequality transfers this to the whole square.
Part 3 is part 2 on the diagonal, integrated.

Example: $K(x, y) = \min(x, y)$ on $[0, 1]$ is the covariance of Brownian motion, hence positive semidefinite.
Its expansion is $\min(x, y) = \sum_{n \ge 1} \frac{2 \sin\left((n - \tfrac12)\pi x\right) \sin\left((n - \tfrac12)\pi y\right)}{(n - \tfrac12)^2 \pi^2}$, uniformly on the square.
On the diagonal, part 3 gives $\sum_{n \ge 1} \frac{1}{(n - \frac12)^2 \pi^2} = \int_0^1 x \, dx = \frac12$.

Without positivity, the conclusions can fail: for a general continuous symmetric kernel the expansion is only guaranteed in $L^2$, and $\sum \abs{\mu_n}$ may diverge.

## Explanations

### L1 human

A positive kernel is the infinite analogue of a positive semidefinite symmetric matrix.
Such a matrix is a sum of rank-one pieces built from its eigenvectors, with nonnegative weights.
Mercer's theorem says the same for continuous positive kernels, and the sum converges as well as one could hope, uniformly at every pair of points.

### L2 human

The theorem upgrades the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]], which gives only $L^2$ convergence, to uniform convergence of the kernel itself.
It is the basis of the Karhunen–Loève expansion of a stochastic process with continuous covariance.
In machine learning it explains why a positive kernel is an inner product of feature vectors $(\sqrt{\mu_n} e_n(x))_n$, which leads to reproducing kernel Hilbert spaces.
</document>

<document source="https://tamnd.github.io/math2/node/PKKD695HD2.md" tag="PKKD695HD2" kind="definition">
# Semihereditary ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
The ring $R$ is *left semihereditary* if every finitely generated [[left ideal|5N2CJZ47D4]] of $R$ is a [[projective module|BFFA03GE3V]] as a left $R$-module.
*Right semihereditary* is defined in the same way with right ideals, and $R$ is *semihereditary* if it is both.

## Remarks

Example: every [[hereditary ring|89VBQH8HNS]] is semihereditary, since the condition is asked only of some of the left ideals.
A non-hereditary example is the ring $\overline{\Z}$ of all algebraic integers.
It is a [[Bézout domain|HGEX40P0Z0]], so every finitely generated ideal is principal, hence free of rank $0$ or $1$, hence projective.
It is not Noetherian, because $2^{1/2}, 2^{1/4}, 2^{1/8}, \dots$ generate a strictly increasing chain of principal ideals; and a hereditary domain is a Dedekind domain, which is Noetherian.
Non-example: the ring $\Z/4\Z$ is not semihereditary, because its ideal $2\Z/4\Z$ is finitely generated and not projective.
Equivalent form: $R$ is left semihereditary if and only if every finitely generated submodule of a projective left $R$-module is projective.
A commutative integral domain is semihereditary if and only if it is a [[Prüfer domain|2RDAA4PCK8]].
</document>

<document source="https://tamnd.github.io/math2/node/PKP7RVAZK6.md" tag="PKP7RVAZK6" kind="statement">
# Nielsen–Thurston classification

## Statement

Let $S$ be a closed orientable [[surface|VGJJVBZ1MQ]], and let $f \colon S \to S$ be an orientation-preserving [[homeomorphism|0QT00PB5G2]].
Then $f$ is isotopic to a homeomorphism $g$ of at least one of the following three kinds:

1. $g$ is periodic: $g^n = \id_S$ for some $n \ge 1$.
2. $g$ is reducible: $g$ maps to itself a nonempty finite union $C$ of pairwise disjoint simple closed curves on $S$. No curve of $C$ bounds a disc, and no two are isotopic.
3. $g$ is [[pseudo-Anosov|GYW7GYABD9]].

A pseudo-Anosov homeomorphism is neither periodic nor reducible, and it is not isotopic to a homeomorphism of the first two kinds.

## Remarks

The statement is about the elements of the [[mapping class group|JV5NYG913V]] of $S$: each mapping class is periodic, reducible or pseudo-Anosov, and the third kind excludes the first two.
A mapping class can be both periodic and reducible.
The theorem also holds for compact surfaces with boundary and for surfaces with finitely many punctures.

Example: on the torus, mapping classes correspond to matrices $A$ in the [[special linear group|0V9PMN0RBH]] $SL(2, \Z)$.
If $\abs{\tr A} < 2$, then $A$ has finite order and the class is periodic.
If $\abs{\tr A} = 2$, then $A$ has an integer eigenvector, and the class preserves the corresponding simple closed curve.
For $A \neq \pm I$ it is a power of a [[Dehn twist|W890NRHGS4]] up to sign and is reducible.
If $\abs{\tr A} > 2$, then $A$ has real eigenvalues $\lambda^{\pm 1}$ with $\abs{\lambda} > 1$, and the class is Anosov, a pseudo-Anosov class without singularities.

Thurston's proof considers the action of $f$ on a compactification of the [[Teichmüller space|ZZGC6BZNJ5]] of $S$ by projective measured foliations.
This compactification is a closed ball, and the proof uses the Brouwer fixed-point theorem.
A fixed point inside the ball gives a periodic class.
A fixed point on the boundary gives either a reducing curve system or the stable and unstable foliations of a pseudo-Anosov map.
Bers gave another proof by minimizing a quasiconformal dilatation over the isotopy class.

## Explanations

### L1 human

Any way of deforming a surface back onto itself can be straightened out into one of three basic types.
It may repeat itself after finitely many steps, like a rotation.
It may keep some curves in place, and so split the surface into simpler pieces.
Or it stretches and squeezes the whole surface everywhere, mixing it thoroughly.
The third type, pseudo-Anosov, is the generic and most interesting one.

### L2 human

The theorem is the surface analogue of the classification of $2 \times 2$ integer matrices into elliptic, parabolic and hyperbolic ones.
By cutting along the reducing curves and repeating, every mapping class decomposes into periodic and pseudo-Anosov pieces on subsurfaces.
This decomposition controls the dynamics: the least topological entropy in an isotopy class is the largest $\log \lambda$ of its pseudo-Anosov pieces.
By Thurston's hyperbolization theorem for fibered 3-manifolds, if $S$ has genus at least $2$, the mapping torus of $f$ has a hyperbolic structure exactly when $f$ is pseudo-Anosov.
</document>

<document source="https://tamnd.github.io/math2/node/PKQ7ABPVSG.md" tag="PKQ7ABPVSG" kind="concept">
# Non-measurable set

## Summary

A non-measurable set is a set to which a given measure cannot assign a size, because it lies outside the $\sigma$-algebra of measurable sets.
The term most often refers to subsets of $\R^n$ that are not measurable for [[Lebesgue measure|2Z2QVWZ262]].
Their existence shows that length, area and volume cannot be defined for all sets in a way that is countably additive and invariant under translations.
Their construction needs some form of the [[axiom of choice|0HTGB13FE2]].
Solovay showed that ZF with only dependent choice cannot prove that they exist, if an inaccessible cardinal is consistent with ZFC.

## Explanations

### L1 human

Length works well for intervals and for all sets one usually meets.
But with the axiom of choice one can build sets of real numbers so scattered that no length fits them.
The classical example is a [[Vitali set|WYHGXY471C]], which picks one number from each class of numbers that differ by a fraction.
Shifted copies of it would have to have equal lengths that add up to a finite positive number, which no single length can do.
</document>

<document source="https://tamnd.github.io/math2/node/PKRGY8HD0A.md" tag="PKRGY8HD0A" kind="definition">
# Bromwich justification of the Heaviside calculus

## Definition

Let $c \ge 0$, and let $\varphi$ be a holomorphic function on the half-plane $\operatorname{Re} s > c$.
Let $\gamma > c$, and suppose that for each $t > 0$ the integral below converges.
Here the integral is the limit, as $R \to \infty$, of the integrals over the segment from $\gamma - iR$ to $\gamma + iR$.
The *Bromwich interpretation* of the Heaviside expression $\varphi(p) H$ is the function
$$x(t) = \frac{1}{2\pi i} \int_{\gamma - i\infty}^{\gamma + i\infty} \frac{\varphi(s)}{s} \, e^{st} \, ds, \qquad t > 0,$$
a [[Bromwich integral|5KNAXNHRY6]], with $x(t) = 0$ for $t < 0$.

## Remarks

Under suitable growth conditions on $\varphi$, the [[Laplace transform|0V49JEZW06]] of $x$ is $\varphi(s)/s$, so this interpretation agrees with the Laplace transform reading of the [[Heaviside operational calculus|QQ98C2AXZP]].
By Cauchy's theorem, the value does not depend on $\gamma > c$.
When $\varphi$ is meromorphic and the line can be closed to the left, $x(t)$ is the sum of the [[residues|0PHN07KVQT]] of $\varphi(s) e^{st} / s$.

Example: $\varphi(p) = p^{-n}$ with $n \ge 0$.
For $t > 0$, the line can be closed to the left, and the only pole of $e^{st}/s^{n+1}$ is at $0$, with residue $t^n / n!$.
So $x(t) = t^n/n!$, which is Heaviside's rule $p^{-n} H = t^n H / n!$.

Non-example: $\varphi(s) = e^{-s^2}$ is entire, but on the line $s = \gamma + iy$ one has $\abs{\varphi(s)} = e^{y^2 - \gamma^2}$.
So the integrand grows without bound, the integral diverges for every $t$, and $\varphi(p) H$ has no Bromwich interpretation.
</document>

<document source="https://tamnd.github.io/math2/node/PKS8JE1PW1.md" tag="PKS8JE1PW1" kind="statement">
# Atiyah–Bott fixed-point theorem

## Statement

Let $X$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary.
Let
$$0 \to \Gamma(E_0) \xrightarrow{d_0} \Gamma(E_1) \xrightarrow{d_1} \cdots \xrightarrow{d_{n-1}} \Gamma(E_n) \to 0$$
be an [[elliptic complex|M8GMH8GQR6]] of smooth sections of [[vector bundles|P4DZM975BV]] $E_i$ over $X$, with cohomology groups $H^i$.
Let $f \colon X \to X$ be a smooth map whose fixed points are all simple, that is, $\det(1 - df_x) \ne 0$ at each fixed point $x$.
For each $i$ let $\varphi_i \colon f^* E_i \to E_i$ be a smooth bundle map, and let $T_i \colon \Gamma(E_i) \to \Gamma(E_i)$ be $T_i s = \varphi_i(s \circ f)$.
Assume $d_i T_i = T_{i+1} d_i$ for all $i$, so that the $T_i$ induce linear maps $H^i(T) \colon H^i \to H^i$.
Then the [[Lefschetz number|ZSYCA4EGEZ]] $L(T) = \sum_i (-1)^i \tr H^i(T)$ satisfies
$$L(T) = \sum_{x \,:\, f(x) = x} \frac{\sum_i (-1)^i \tr \varphi_{i,x}}{\abs{\det(1 - df_x)}},$$
where $\varphi_{i,x} \colon (E_i)_{f(x)} = (E_i)_x \to (E_i)_x$ is the map on the fibre at the fixed point $x$.

## Remarks

The cohomology groups are finite-dimensional because the complex is elliptic, and the fixed points are finitely many because simple fixed points are isolated and $X$ is compact.
The idea of the proof is to write $L(T)$ as an alternating sum of traces of operators of the form $T_i e^{-t \Delta_i}$, where $\Delta_i$ are the Laplacians of the complex.
These traces are integrals of a kernel over the graph of $f$ restricted to the diagonal, and the transversality condition $\det(1 - df_x) \ne 0$ localizes the integral at the fixed points as $t \to 0$.

For the de Rham complex with $\varphi_i = \Lambda^i (df)^*$, the numerator at $x$ is $\det(1 - df_x)$, so each fixed point contributes $\sgn \det(1 - df_x) = \pm 1$.
This recovers the [[Lefschetz fixed-point theorem|G44NQXCWT9]] for maps with simple fixed points.
For the Dolbeault complex of a compact complex manifold and a holomorphic map $f$, it gives the holomorphic Lefschetz formula
$$\sum_q (-1)^q \tr\big(f^* \mid H^{0,q}(X)\big) = \sum_{f(x) = x} \frac{1}{\det_\C(1 - \partial f_x)}.$$

The hypothesis that the fixed points are simple cannot be dropped from the formula as stated.
For the identity map every point is fixed, and the right side is not defined.

## Explanations

### L1 human

The classical Lefschetz theorem counts fixed points of a map with signs and says the count equals a trace computed from cohomology.
Atiyah and Bott replaced the de Rham cohomology by the cohomology of any elliptic complex.
The global side is still a trace on cohomology.
The local side is a sum over the fixed points, where each point contributes a number read off from the derivative of the map at that point.

### L2 human

The theorem is a fixed-point counterpart of the Atiyah–Singer index theorem: for $f = \id$ and no fixed-point condition, $L(T)$ would be the index of the complex.
Applied to the Dolbeault complex, it computes traces of group elements on holomorphic cohomology, which leads to the Weyl character formula for compact Lie groups acting on flag manifolds.
Applied to the spin complex and the signature complex, it gives fixed-point formulas for the $\hat{A}$-genus and the signature.
</document>

<document source="https://tamnd.github.io/math2/node/PKYQY68PCB.md" tag="PKYQY68PCB" kind="definition">
# Lauricella hypergeometric series

## Definition

Let $n \ge 1$ be an integer, let $x = (x_1, \dots, x_n) \in \C^n$, and let $a, a_i, b, b_i, c, c_i$ be complex numbers with every lower parameter $c, c_i$ outside $\set{0, -1, -2, \dots}$.
Write $(x)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]], and for a multi-index $m = (m_1, \dots, m_n) \in \N^n$ write $\abs{m} = m_1 + \cdots + m_n$, $m! = m_1! \cdots m_n!$ and $x^m = x_1^{m_1} \cdots x_n^{m_n}$.
The four *Lauricella series* are:

1. $F_A^{(n)}(a; b_1, \dots, b_n; c_1, \dots, c_n; x) = \sum_{m} \frac{(a)_{\abs{m}} (b_1)_{m_1} \cdots (b_n)_{m_n}}{(c_1)_{m_1} \cdots (c_n)_{m_n}} \frac{x^m}{m!}$, convergent for $\abs{x_1} + \cdots + \abs{x_n} < 1$;
2. $F_B^{(n)}(a_1, \dots, a_n; b_1, \dots, b_n; c; x) = \sum_{m} \frac{(a_1)_{m_1} \cdots (a_n)_{m_n} (b_1)_{m_1} \cdots (b_n)_{m_n}}{(c)_{\abs{m}}} \frac{x^m}{m!}$, convergent for $\abs{x_i} < 1$ for all $i$;
3. $F_C^{(n)}(a; b; c_1, \dots, c_n; x) = \sum_{m} \frac{(a)_{\abs{m}} (b)_{\abs{m}}}{(c_1)_{m_1} \cdots (c_n)_{m_n}} \frac{x^m}{m!}$, convergent for $\sqrt{\abs{x_1}} + \cdots + \sqrt{\abs{x_n}} < 1$;
4. $F_D^{(n)}(a; b_1, \dots, b_n; c; x) = \sum_{m} \frac{(a)_{\abs{m}} (b_1)_{m_1} \cdots (b_n)_{m_n}}{(c)_{\abs{m}}} \frac{x^m}{m!}$, convergent for $\abs{x_i} < 1$ for all $i$.

Each sum runs over all $m \in \N^n$.

## Remarks

For $n = 2$ the four series are the [[Appell series|QG9VDK1HN9]]: $F_A = F_2$, $F_B = F_3$, $F_C = F_4$ and $F_D = F_1$.
Example: for $n = 1$ all four series reduce to $\sum_{k} \frac{(a)_k (b)_k}{(c)_k} \frac{x^k}{k!} = {}_2F_1(a, b; c; x)$, with $b = b_1$ and $c = c_1$ as needed; the check is a direct comparison of coefficients.
Non-example: the point $x_1 = x_2 = \frac12$ is outside the region of convergence of $F_A^{(2)}$; with $a = 1$ and $b_i = c_i$ the series is $\sum_{N \ge 0} (x_1 + x_2)^N$, which diverges there.
For $\operatorname{Re} c > \operatorname{Re} a > 0$, with $\Gamma$ the [[gamma function|0H5SYBSED9]], the series $F_D$ has the integral form
$$F_D^{(n)}(a; b_1, \dots, b_n; c; x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(c - a)} \int_0^1 t^{a - 1} (1 - t)^{c - a - 1} \prod_{i = 1}^{n} (1 - x_i t)^{-b_i}\, dt.$$
</document>

<document source="https://tamnd.github.io/math2/node/PM5PQX6TVP.md" tag="PM5PQX6TVP" kind="concept">
# Optimal experimental design

## Summary

Optimal experimental design chooses the conditions of an experiment so that the parameters of a given model are estimated as precisely as possible.
Precision is measured by the information matrix of the design, and an optimality criterion turns that matrix into a number.
D-optimality maximises its determinant, A-optimality minimises the trace of its inverse, and E-optimality maximises its smallest eigenvalue.
The approximate theory, where a design is a probability measure on the design region, makes these problems convex.
The [[Kiefer–Wolfowitz equivalence theorem|71XBFB49HE]] shows that D-optimal and G-optimal designs coincide in that theory.

## Explanations

### L1 human

To estimate the slope of a straight line over an interval, where should one measure?
Spreading the points evenly feels natural, but putting half of them at each end gives a more precise slope.
Optimal design makes such choices systematic: it picks the measuring points that make the estimates as sharp as possible.
Different goals, such as precise coefficients or precise predictions, lead to different criteria and sometimes to different designs.
</document>

<document source="https://tamnd.github.io/math2/node/PM71JAZV31.md" tag="PM71JAZV31" kind="definition">
# Valuation on convex bodies

## Definition

Let $n \ge 1$ and let $\mathcal{K}^n$ be the set of nonempty compact convex subsets of $\R^n$ (the [[convex bodies|437FSG3419]] in the broad sense).
Let $G$ be an [[abelian group|082GHNPJ53]], for example $\R$.
A *valuation* on $\mathcal{K}^n$ is a map $\varphi \colon \mathcal{K}^n \to G$ such that
$$\varphi(K \cup L) + \varphi(K \cap L) = \varphi(K) + \varphi(L)$$
for all $K, L \in \mathcal{K}^n$ with $K \cup L \in \mathcal{K}^n$.

## Remarks

If $K \cup L$ is convex, then $K \cap L$ is nonempty, since a convex set is connected and cannot be the union of two disjoint nonempty closed sets.
So $K \cap L \in \mathcal{K}^n$ and the condition makes sense.
A valuation is *continuous* if it is continuous for the [[Hausdorff distance|F79WYYC07C]], and *invariant under rigid motions* if $\varphi(gK) = \varphi(K)$ for every rigid motion $g$ of $\R^n$.

Example: $\varphi = \operatorname{vol}$, Lebesgue measure on $\R^n$.
For measurable sets, $\operatorname{vol}(K \cup L) + \operatorname{vol}(K \cap L) = \operatorname{vol}(K) + \operatorname{vol}(L)$ always holds, so volume is a valuation.
The constant function $\chi(K) = 1$ is a valuation as well; it is the Euler characteristic.

Non-example: $\varphi(K) = \operatorname{vol}(K)^2$ on $\mathcal{K}^1$.
For $K = [0, 2]$ and $L = [1, 3]$ one gets $\varphi(K \cup L) + \varphi(K \cap L) = 9 + 1 = 10$, but $\varphi(K) + \varphi(L) = 4 + 4 = 8$.

Many authors include the empty set in $\mathcal{K}^n$ and require $\varphi(\emptyset) = 0$.
Then the Euler characteristic takes the value $0$ at $\emptyset$.
</document>

<document source="https://tamnd.github.io/math2/node/PMDXYNPE7D.md" tag="PMDXYNPE7D" kind="definition">
# Sub-Gaussian distribution

## Definition

Let $X$ be a real [[random variable|0PNRKMRC55]].
$X$ is *sub-Gaussian* if there is a constant $K > 0$ such that
$$P(\abs{X} \ge t) \le 2 e^{-t^2 / K^2} \qquad \text{for all } t \ge 0.$$
A distribution on $\R$ is sub-Gaussian if a random variable with this distribution is sub-Gaussian.

## Remarks

Up to changing $K$ by a constant factor, the condition is equivalent to each of the following: $E[e^{X^2/K^2}] \le 2$; and $(E[\abs{X}^p])^{1/p} \le K \sqrt{p}$ for all $p \ge 1$.
If $E[X] = 0$, it is also equivalent to a bound on the [[moment-generating function|0WQVXP7PDB]]: there is $\sigma > 0$ with $E[e^{\lambda X}] \le e^{\lambda^2 \sigma^2 / 2}$ for all real $\lambda$.
In this last form one says that $X$ is sub-Gaussian with variance proxy $\sigma^2$.
Example: if $X$ is [[normal|05AQE0QR0D]] with mean $0$ and variance $\sigma^2$, then $E[e^{\lambda X}] = e^{\lambda^2 \sigma^2 / 2}$, so $X$ is sub-Gaussian with variance proxy $\sigma^2$.
Example: by Hoeffding's lemma, if $a \le X \le b$ and $E[X] = 0$, then $E[e^{\lambda X}] \le e^{\lambda^2 (b - a)^2 / 8}$, so $X$ is sub-Gaussian with variance proxy $(b - a)^2 / 4$.
Non-example: an [[exponential|0E99F51GVB]] variable with rate $1$ has $P(X \ge t) = e^{-t}$, and $e^{-t} \le 2 e^{-t^2/K^2}$ fails for large $t$, whatever $K$ is.
If $X_1, \dots, X_n$ are independent and centred with variance proxies $\sigma_i^2$, then $X_1 + \dots + X_n$ has variance proxy $\sigma_1^2 + \dots + \sigma_n^2$.
</document>

<document source="https://tamnd.github.io/math2/node/PMGTV643X1.md" tag="PMGTV643X1" kind="statement">
# Petersen's theorem

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]] that is cubic, that is, [[$3$-regular|0KA4S8R39M]], and that has no [[bridge|0D81KWH9ZM]].
Then $G$ has a [[perfect matching|05ZPH8KF9K]].

## Remarks

The proof uses [[Tutte's theorem|08J7R35B6Q]].
It is enough to show that, for each set $S$ of vertices, the graph $G - S$ has at most $\abs{S}$ components with an odd number of vertices.
If $C$ is such an odd component, the degrees in $C$ add up to $3\abs{C}$, which is odd, so an odd number of edges joins $C$ to $S$.
This number is not $1$, since $G$ has no bridge, so it is at least $3$.
Since at most $3\abs{S}$ edges end in $S$, there are at most $\abs{S}$ odd components.
For example, the [[Petersen graph|0NR95CTFCZ]] is cubic and bridgeless, and its five spokes form a perfect matching.
The hypothesis on bridges is needed: there are cubic graphs with bridges that have no perfect matching.
The theorem also holds for cubic multigraphs without bridges, with the same proof.

## Explanations

### L1 human

In a cubic graph each vertex has exactly three edges.
If no single edge disconnects the graph, then the vertices can be paired off along edges, with every vertex in exactly one pair.
This is one of the oldest results of graph theory.

### L2 human

Petersen's theorem is a standard application of Tutte's theorem, by a counting argument on edges leaving odd components.
Removing a perfect matching from a bridgeless cubic graph leaves a $2$-factor, a disjoint union of cycles.
This is a starting point for edge colouring of cubic graphs.
A cubic graph has chromatic index $3$ exactly when it has a perfect matching whose complement consists of even cycles.
</document>

<document source="https://tamnd.github.io/math2/node/PMJT3FZE0Y.md" tag="PMJT3FZE0Y" kind="concept">
# q-derivative

## Summary

The $q$-derivative of a function $f$ is the quotient $\frac{f(qx) - f(x)}{qx - x}$, where $q \ne 1$ is a fixed number.
It replaces the limit in the ordinary [[derivative|0V1EJCT8G6]] by a comparison of $f$ at $x$ and at the scaled point $qx$.
It is the basic operator of $q$-calculus and of the theory of [[q-analogs|GFSDMJHHVA]], and it tends to the ordinary derivative as $q \to 1$.
It is also called the Jackson derivative.

## Explanations

### L1 human

The ordinary derivative compares $f(x)$ with $f(x + h)$ for tiny $h$ and takes a limit.
The $q$-derivative compares $f(x)$ with $f(qx)$, a point that is a fixed factor away, and takes no limit at all.
For $f(x) = x^2$ it gives $\frac{q^2 x^2 - x^2}{qx - x} = (1 + q) x$, which tends to the usual $2x$ as $q \to 1$.
The number $1 + q$ in place of $2$ is a first example of a $q$-number.
</document>

<document source="https://tamnd.github.io/math2/node/PMMXG4Q8JR.md" tag="PMMXG4Q8JR" kind="statement">
# Littlewood subordination theorem

## Statement

Let $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$.
Let $\omega \colon \mathbb{D} \to \mathbb{D}$ be a [[holomorphic function|0NAHXR1FRH]] with $\omega(0) = 0$, and let $F$ be a holomorphic function on $\mathbb{D}$.
Then for every $p$ with $0 < p < \infty$ and every $r$ with $0 < r < 1$,
$$\int_0^{2\pi} \abs{F(\omega(r e^{i\theta}))}^p \, d\theta \le \int_0^{2\pi} \abs{F(r e^{i\theta})}^p \, d\theta.$$

## Remarks

The function $\abs{F}^p$ is [[subharmonic|Z1MHMGFNFW]].
Let $u$ be the harmonic function on the disc $\abs{z} < r$ with boundary values $\abs{F}^p$ on $\abs{z} = r$, given by the [[Poisson kernel|TCYSWCTYYM]]; then $\abs{F}^p \le u$ on $\abs{z} \le r$.
By the [[Schwarz lemma|0MB48XDBY3]], $\abs{\omega(z)} \le \abs{z}$, so $\omega$ maps the closed disc of radius $r$ into itself and $\abs{F \circ \omega}^p \le u \circ \omega$ there.
Since $u \circ \omega$ is harmonic, its mean over $\abs{z} = r$ is $u(\omega(0)) = u(0)$, which is the mean of $\abs{F}^p$ over $\abs{z} = r$.
The hypothesis $\omega(0) = 0$ is needed.
For the constant map $\omega \equiv c$ with $0 < c < 1$ and $F(z) = z$, the left side is $2\pi c^p$ and the right side is $2\pi r^p$.
So the inequality fails for $r < c$.

## Explanations

### L1 human

Composing a function on the disc with a self-map of the disc that fixes the centre can only make the function smaller on average along circles.
The self-map pulls each circle inside a smaller region, and averages of nice functions over circles grow with the radius.
So the theorem compares the size of $F \circ \omega$ with the size of $F$ itself.

### L2 human

Taking the supremum over $r$ shows that the composition operator $F \mapsto F \circ \omega$ is a contraction on each [[Hardy space|PNNNFJVF6W]] $H^p$ when $\omega(0) = 0$.
For a general holomorphic self-map $\omega$ one composes with a disc automorphism moving $\omega(0)$ to $0$, which shows that every composition operator on $H^p$ is bounded.
The proof only uses that $\abs{F}^p$ is subharmonic, so the same inequality holds for any subharmonic function in place of $\abs{F}^p$.
This is the starting point of the theory of composition operators on spaces of analytic functions.
</document>

<document source="https://tamnd.github.io/math2/node/PMRGCQTFZC.md" tag="PMRGCQTFZC" kind="definition">
# Crossed product algebra

## Definition

Let $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]] with [[Galois group|0XTY86BZKY]] $G$ and degree $n = [L : K]$.
A *factor set* (or $2$-cocycle) is a map $f \colon G \times G \to L^\times$ such that
$$f(\sigma, \tau) \, f(\sigma\tau, \rho) = \sigma\big(f(\tau, \rho)\big) \, f(\sigma, \tau\rho) \quad \text{for all } \sigma, \tau, \rho \in G.$$
The *crossed product algebra* $(L/K, f)$ is the $L$-vector space with basis $\set{u_\sigma : \sigma \in G}$, with the multiplication determined by
$$u_\sigma \, x = \sigma(x) \, u_\sigma \quad (x \in L), \qquad u_\sigma \, u_\tau = f(\sigma, \tau) \, u_{\sigma\tau}.$$

## Remarks

The cocycle condition is exactly what makes the multiplication associative.
The algebra $(L/K, f)$ is a [[central simple algebra|5HRKJBAY2V]] over $K$ of degree $n$, it contains $L$ as a maximal subfield, and $L$ splits it.
Two factor sets give isomorphic algebras if and only if they differ by a coboundary, so the classes correspond to $H^2(G, L^\times)$ in [[Galois cohomology|9MPCZ3EAJM]].
This gives $\mathrm{Br}(L/K) \cong H^2(G, L^\times)$.

Example: take $L = \C$, $K = \R$, $G = \set{1, \sigma}$ with $\sigma$ complex conjugation, and $f(\sigma, \sigma) = -1$, $f = 1$ otherwise.
With $j = u_\sigma$ one gets $j z = \bar{z} j$ and $j^2 = -1$, which are the relations of $\mathbb{H} = \C + \C j$.
With $f(\sigma, \sigma) = 1$ instead one gets an algebra isomorphic to $M_2(\R)$.

Non-example: for $L = \C$, $K = \R$ as above, the map with $f(\sigma, \sigma) = i$ and $f = 1$ otherwise is not a factor set.
The condition for $\sigma, \sigma, \sigma$ would give $i = \sigma(i) = -i$, and indeed $(u_\sigma u_\sigma) u_\sigma = i u_\sigma$ while $u_\sigma (u_\sigma u_\sigma) = \sigma(i) u_\sigma = -i u_\sigma$, so the product is not associative.

A central simple algebra is called a crossed product if it is isomorphic to some $(L/K, f)$; Amitsur showed that some central division algebras are not crossed products.
</document>

<document source="https://tamnd.github.io/math2/node/PMTK0C47E0.md" tag="PMTK0C47E0" kind="definition">
# Quasisymmetric map

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]].
Let $\eta \colon [0, \infty) \to [0, \infty)$ be a [[homeomorphism|05GZ11PWRY]].
An injective map $f \colon X \to Y$ is *$\eta$-quasisymmetric* if
$$\frac{d_Y(f(x), f(a))}{d_Y(f(x), f(b))} \le \eta\!\left( \frac{d_X(x, a)}{d_X(x, b)} \right)$$
for all $x, a, b \in X$ with $b \ne x$.
The map $f$ is *quasisymmetric* if it is $\eta$-quasisymmetric for some such $\eta$.

## Remarks

Example: an $L$-bi-[[Lipschitz|0TMZ21WKCN]] map, with $L^{-1} d_X(x, y) \le d_Y(f(x), f(y)) \le L\, d_X(x, y)$, is $\eta$-quasisymmetric with $\eta(t) = L^2 t$.
Indeed the numerator is at most $L\, d_X(x, a)$ and the denominator is at least $L^{-1} d_X(x, b)$.
A similarity is quasisymmetric with $\eta(t) = t$.

Non-example: $f(x) = e^x$ on $\R$ is not quasisymmetric.
For $s > 0$ take $a = x + s$ and $b = x - s$; the ratio on the right is $1$, while the ratio on the left is $\frac{e^s - 1}{1 - e^{-s}} = e^s$.
No single value $\eta(1)$ can bound $e^s$ for all $s$.

A quasisymmetric map is a topological embedding, and its inverse on $f(X)$ is quasisymmetric too.
An increasing homeomorphism $h$ of $\R$ is quasisymmetric if and only if there is $M \ge 1$ with $\frac{1}{M} \le \frac{h(x + t) - h(x)}{h(x) - h(x - t)} \le M$ for all $x \in \R$ and $t > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/PMYDHW00KF.md" tag="PMYDHW00KF" kind="statement">
# Pentagonal number theorem

## Statement

In the ring of [[formal power series|033WGN00BF]] $\Z[\![x]\!]$,
$$\prod_{n=1}^{\infty} (1 - x^n) = \sum_{k=-\infty}^{\infty} (-1)^k x^{k(3k-1)/2} = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + \cdots.$$
The same identity holds for complex $x$ with $\abs{x} < 1$, both sides converging absolutely.

## Remarks

The exponents $k(3k-1)/2$ for $k = 0, \pm 1, \pm 2, \ldots$ are the generalized pentagonal numbers $0, 1, 2, 5, 7, 12, 15, 22, 26, \ldots$.
Let $E(m)$ and $O(m)$ be the numbers of partitions of $m$ into an even and an odd number of distinct parts.
The coefficient of $x^m$ on the left is $E(m) - O(m)$.
Franklin's bijection pairs these two kinds of partitions, and the only unpaired partitions occur when $m$ is a generalized pentagonal number, with sign $(-1)^k$.
Example: for $m = 3$, the distinct-part partitions are $3$ (one part) and $2 + 1$ (two parts), so the coefficient is $1 - 1 = 0$; indeed $3$ is not a generalized pentagonal number.
The [[generating function for partitions|0578PA53VA]] is $\prod_n (1 - x^n)^{-1} = \sum_m p(m) x^m$.
Multiplying the theorem by it gives the recurrence
$$p(m) = p(m-1) + p(m-2) - p(m-5) - p(m-7) + p(m-12) + p(m-15) - \cdots,$$
with $p(0) = 1$ and $p(j) = 0$ for $j < 0$.

## Explanations

### L1 human

Multiply out $(1 - x)(1 - x^2)(1 - x^3)\cdots$.
One expects a mess, but almost everything cancels: only the powers $0, 1, 2, 5, 7, 12, 15, \ldots$ survive, each with coefficient $+1$ or $-1$.
These exponents are the pentagonal numbers, which count dots in nested pentagons, together with a second family of the same shape.

### L2 human

Euler found the identity, and it gives a fast recurrence for the [[partition function|001CX45SWQ]] $p(m)$.
It is a special case of the Jacobi triple product identity.
The left side is, up to the factor $x^{1/24}$, the [[Dedekind eta function|MX3CVYMNDV]], so the theorem is also a statement about a modular form.
</document>

<document source="https://tamnd.github.io/math2/node/PN55VEF0J7.md" tag="PN55VEF0J7" kind="concept">
# Normal complex space

## Summary

A normal complex space is a [[complex analytic space|HXSZA44J5J]] whose local rings are integrally closed domains.
Normal spaces are [[reduced|831N2Q4GCB]], locally irreducible, and their singular locus has codimension at least $2$.
On a normal space, the Riemann extension theorem holds: a function that is holomorphic off the singular locus and locally bounded is holomorphic.
Every reduced complex space has a normalization, so normal spaces are a standard first step in the study of singularities.

## Explanations

### L1 human

Singular points come in different degrees of badness.
At a normal point, the space does not cross itself and is not pinched like the cusp $z_2^2 = z_1^3$.
For example, the cone $\set{z_1 z_2 = z_3^2}$ in $\C^3$ is normal, even though its vertex is a singular point.
A normal curve is the same as a smooth curve, but in higher dimension normal spaces can have singularities.
</document>

<document source="https://tamnd.github.io/math2/node/PNERH1096H.md" tag="PNERH1096H" kind="definition">
# Noetherian module

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $M$ be a left [[module|09VDS2W1E8]] over $R$.
The module $M$ is *Noetherian* if the set of its [[submodules|0RGZ10DFPG]], ordered by inclusion, satisfies the [[ascending chain condition|0ZCAXY9563]].
That is, for every chain $N_1 \subseteq N_2 \subseteq \cdots$ of submodules of $M$, there is an index $k$ with $N_n = N_k$ for all $n \ge k$.

## Remarks

The following conditions are equivalent: $M$ is Noetherian; every submodule of $M$ is [[finitely generated|0XHM30WBVB]]; every nonempty set of submodules of $M$ has a maximal element for inclusion.
Example: $\Z$ as a $\Z$-module is Noetherian, because each submodule is $n\Z$ for some $n$, generated by one element.
Non-example: $\Q$ as a $\Z$-module is not Noetherian, because $\Z \subsetneq \frac{1}{2}\Z \subsetneq \frac{1}{4}\Z \subsetneq \cdots$ is a chain of submodules that never stops.
Submodules and quotients of Noetherian modules are Noetherian, and an extension of two Noetherian modules is Noetherian.
Over a [[Noetherian ring|0MVH4Y1CRZ]], every finitely generated module is Noetherian.

## Explanations

### L1 human

A Noetherian module is a module with no endless increasing chain of submodules.
Equivalently, each of its submodules can be described by finitely many generators.
</document>

<document source="https://tamnd.github.io/math2/node/PNESAD6R5T.md" tag="PNESAD6R5T" kind="example">
# Airy equation

## Example

The *Airy equation* is
$$y'' - z\, y = 0.$$
Its coefficients are polynomials, so every point of $\C$ is an [[ordinary point|QK2J038VB4]], and all solutions are entire functions.
A [[power series solution|08RT9GDDPQ]] $y = \sum_{k \ge 0} a_k z^k$ gives $2 a_2 = 0$ and
$$(k + 2)(k + 1)\, a_{k+2} = a_{k-1} \quad \text{for } k \ge 1.$$
With $a_0 = 1$ and $a_1 = 0$ one gets $y_1 = 1 + \frac{z^3}{6} + \frac{z^6}{180} + \cdots$.
With $a_0 = 0$ and $a_1 = 1$ one gets $y_2 = z + \frac{z^4}{12} + \frac{z^7}{504} + \cdots$.
They are linearly independent, since their Wronskian at $0$ is $y_1(0) y_2'(0) - y_1'(0) y_2(0) = 1$.

## Remarks

The standard solutions are the Airy functions $\operatorname{Ai}$ and $\operatorname{Bi}$, which are fixed linear combinations of $y_1$ and $y_2$.
For real $x \to +\infty$, $\operatorname{Ai}(x)$ decays like $\exp\left( -\frac{2}{3} x^{3/2} \right)$ up to a power of $x$, and $\operatorname{Bi}(x)$ grows like $\exp\left( \frac{2}{3} x^{3/2} \right)$.
For $x \to -\infty$, both oscillate with an amplitude that decays like $\abs{x}^{-1/4}$.
The point $\infty$ is an irregular singular point, and $\operatorname{Ai}$ is the classical example of the [[Stokes phenomenon|TTGF8KCA8C]].

## Explanations

### L1 human

The Airy equation is the simplest equation whose solutions switch from oscillating to growing or decaying.
For $x < 0$ it behaves like $y'' = -\abs{x} y$, a spring, so solutions wiggle.
For $x > 0$ it behaves like $y'' = x y$, so solutions grow or decay exponentially.
It describes light near a rainbow caustic and a quantum particle near a turning point.
</document>

<document source="https://tamnd.github.io/math2/node/PNF97STBAW.md" tag="PNF97STBAW" kind="definition">
# k-means clustering

## Definition

Let $x_1, \ldots, x_n \in \R^d$ and let $1 \le k \le n$.
For a [[partition|08QMKTWF9T]] of $\set{1, \ldots, n}$ into non-empty sets $S_1, \ldots, S_k$, let $m_j = \frac{1}{\abs{S_j}} \sum_{i \in S_j} x_i$ be the mean of cluster $j$.
The *$k$-means problem* is to find a partition that minimises the within-cluster sum of squares
$$W(S_1, \ldots, S_k) = \sum_{j=1}^k \sum_{i \in S_j} \abs{x_i - m_j}^2,$$
where $\abs{\cdot}$ is the [[Euclidean norm|0NPPGHQNXQ]].
*$k$-means clustering* is the [[clustering|D9Y3A9RH6J]] given by a solution, or by an approximate solution, of this problem.

## Remarks

Lloyd's algorithm starts from $k$ centres and repeats two steps: assign each $x_i$ to a nearest centre, then replace each centre by the mean of its assigned points.
Neither step increases $W$, and there are finitely many partitions, so the algorithm stops after finitely many steps, at a local minimum that need not be global.
Example: for $0, 1, 10, 11 \in \R$ and $k = 2$, the partition $\set{0, 1}, \set{10, 11}$ has means $0.5$ and $10.5$ and $W = 4 \cdot 0.25 = 1$.
Non-example: the partition $\set{0}, \set{1, 10, 11}$ has means $0$ and $22/3$ and $W = 546/9 \approx 60.7$, so it is not a solution.
The mean is used because, for fixed clusters, $m_j$ is the point $c$ that minimises $\sum_{i \in S_j} \abs{x_i - c}^2$.
</document>

<document source="https://tamnd.github.io/math2/node/PNFNGBXRM3.md" tag="PNFNGBXRM3" kind="example">
# Hopf surface

## Example

A *Hopf surface* is a compact complex surface whose [[universal cover|Y1YPE6PM52]] is biholomorphic to $\C^2 \setminus \set{0}$.
It is *primary* if its fundamental group is infinite cyclic.

Example: let $0 < \abs{\lambda} < 1$ and $X = (\C^2 \setminus \set{0}) / \langle g \rangle$ with $g(z, w) = (\lambda z, \lambda w)$, the [[Hopf manifold|Z8R2D33WSM]] of dimension $2$.
Since $\C^2 \setminus \set{0}$ is homotopy equivalent to $S^3$, it is simply connected, so it is the universal cover of $X$, and $\pi_1(X) \cong \Z$.

Check that $X$ is a [[class VII surface|Y4NSSJT4RC]].
Its first [[Betti number|ST71ZV3FES]] is $1$, since $X$ is diffeomorphic to $S^1 \times S^3$.
Its [[Kodaira dimension|ZW00Q0SAGR]] is $-\infty$.
A section of $K_X^{\otimes m}$ with $m \ge 1$ pulls back to $h(z, w)\,(dz \wedge dw)^{\otimes m}$ with $h$ holomorphic on $\C^2 \setminus \set{0}$, hence, by Hartogs's extension theorem, on $\C^2$.
Invariance under $g$ gives $h(\lambda z, \lambda w)\, \lambda^{2m} = h(z, w)$, so each Taylor coefficient $c_{jk}$ of $h$ satisfies $c_{jk} \lambda^{j + k + 2m} = c_{jk}$; since $\abs{\lambda} < 1$, all $c_{jk}$ vanish and $h = 0$.

## Remarks

Kodaira showed that every primary Hopf surface is the quotient of $\C^2 \setminus \set{0}$ by the infinite cyclic group generated by a holomorphic contraction that fixes $0$.
Every Hopf surface has $b_1 = 1$ and $b_2 = 0$, so it is not Kähler.
The function $z/w$ on the example above is invariant under $g$, so it gives a nonconstant meromorphic function on $X$.
It also gives a map to $\mathbb{CP}^1$ whose fibres are elliptic curves, and this $X$ has [[algebraic dimension|F0GF0JEZTG]] $1$.

## Explanations

### L1 human

A Hopf surface is a compact complex surface obtained from $\C^2$ minus the origin by gluing each point to its images under a contraction toward the origin.
Topologically it is usually the product of a circle and a three-dimensional sphere.
It is the first and simplest example of a compact complex surface that is not Kähler, and so not algebraic.
</document>

<document source="https://tamnd.github.io/math2/node/PNGY60EBDJ.md" tag="PNGY60EBDJ" kind="concept">
# Strongly minimal theory

## Summary

A strongly minimal theory is a complete first-order [[theory|92SWXST1ZM]] with infinite models.
In each of its models, every definable subset of the line is finite or cofinite.
These theories are the simplest infinite structures from the point of view of definability.
In a strongly minimal theory, algebraic closure behaves like linear span, so each model has a well-defined dimension.
They are basic building blocks of [[stable theories|T7JW5QY6KR]] and are central to the study of [[categorical theories|1JJCGNX240]].

## Explanations

### L1 human

Think of a structure where a formula in one variable can only cut out a few points, or everything except a few points.
Nothing in between, like a half-line or an infinite set with an infinite complement, can be described.
The theory of an infinite set with no structure at all is the simplest case: a formula can only pick out finitely many named elements or their complement.
Another case is the theory of [[algebraically closed fields|MT8TDVPBXB]]: a nonzero polynomial in one variable has finitely many roots.
So in $\C$, every definable set of numbers is finite or cofinite.
</document>

<document source="https://tamnd.github.io/math2/node/PNMT15BB91.md" tag="PNMT15BB91" kind="concept">
# Stratified sampling

## Summary

Stratified sampling divides a finite population into disjoint groups, the strata, and draws a separate sample from each stratum.
Population quantities are then estimated by combining the stratum estimates with weights equal to the stratum shares.
When the strata are internally homogeneous, this gives more precise estimates than a [[simple random sample|Y0PTCJBPEA]] of the same total size.

## Explanations

### L1 human

To estimate the average income in a country, one can first split the country into regions and then survey some people in each region.
This guarantees that every region is represented, which a purely random draw might fail to do by bad luck.
The regional averages are then combined, each weighted by the share of the population living there.
If people within a region are similar, the combined estimate is much more accurate.
</document>

<document source="https://tamnd.github.io/math2/node/PNNNFJVF6W.md" tag="PNNNFJVF6W" kind="concept">
# Hardy space

## Summary

Let $0 < p \le \infty$.
The Hardy space $H^p$ of the unit disc is the space of [[holomorphic functions|0NAHXR1FRH]] whose $L^p$ means over the circles $\abs{z} = r$ stay bounded as $r \to 1$.
For $p \ge 1$ it is a [[Banach space|06GZ956657]], and $H^2$ is the Hilbert space of power series $\sum a_n z^n$ with $\sum \abs{a_n}^2 < \infty$.
By [[Fatou's theorem|Q2N917JQ6A]], functions in $H^p$ have boundary values almost everywhere on the circle.
So $H^p$ can be seen as a subspace of $L^p$ of the circle.
Hardy spaces connect complex analysis with Fourier analysis and operator theory, through [[inner functions|NN8XH5RJYB]], [[Blaschke products|9N6R5ADJXT]] and [[Toeplitz operators|GQ0X3R3WM2]].

## Explanations

### L1 human

Take a holomorphic function on the unit disc and look at its size along circles of radius $r$ around $0$.
The function is in a Hardy space if these average sizes stay bounded as the circles approach the rim.
So the function may blow up at some rim points, but not too strongly on average.
For example, $\log(1 - z)$ is unbounded near $z = 1$, yet it lies in $H^2$ because its Taylor coefficients $-1/n$ are square-summable.
</document>

<document source="https://tamnd.github.io/math2/node/PNQCW85HTJ.md" tag="PNQCW85HTJ" kind="statement">
# Liouville–Arnold theorem

## Statement

Let $(M, \omega)$ be a [[symplectic manifold|CAN490EPRT]] of dimension $2n$.
Let $F_1, \ldots, F_n \colon M \to \R$ be smooth functions with [[Poisson brackets|2S0MXDT4YH]] $\{F_i, F_j\} = 0$ for all $i, j$, and let $H = F_1$.
Let $F = (F_1, \ldots, F_n) \colon M \to \R^n$, let $c \in \R^n$ be a regular value of $F$, and let $M_c = F^{-1}(c)$.
Then:

1. $M_c$ is a smooth [[Lagrangian submanifold|H8ACPTMX08]] of dimension $n$. It is invariant under the flows of the [[Hamiltonian vector fields|C5EH3B4BEP]] $X_{F_1}, \ldots, X_{F_n}$.
2. Each compact connected component of $M_c$ is diffeomorphic to the [[torus|03NM7DZXXC]] $\mathbb{T}^n = \R^n / 2\pi \Z^n$.
3. On such a component there are angle coordinates $\theta \in \mathbb{T}^n$ in which the flow of $X_H$ is linear: $\theta(t) = \theta(0) + t \, \nu(c)$ for a vector $\nu(c) \in \R^n$.
4. A neighbourhood of such a component has [[action-angle coordinates|M3T81YNSKC]] $(I, \theta)$ in which $H$ is a function of $I$ only.

## Remarks

Since the $F_i$ Poisson-commute, the flows of $X_{F_1}, \ldots, X_{F_n}$ commute and preserve $M_c$.
They give an action of $\R^n$ on each component of $M_c$, which is locally free because the $dF_i$ are independent there.
A compact connected manifold with a transitive locally free $\R^n$-action is $\R^n / \Lambda$ for a lattice $\Lambda$, that is, a torus.
The actions are built from the integrals $\oint p \, dq$ over the basic cycles of the tori.

Example: for two uncoupled harmonic oscillators on $\R^4$, with $F_1 = \frac{1}{2}(p_1^2 + q_1^2) + \frac{1}{2}(p_2^2 + q_2^2)$ and $F_2 = \frac{1}{2}(p_2^2 + q_2^2)$, the level sets with $c_1 > c_2 > 0$ are products of two circles, that is, $2$-tori.

Compactness is necessary for the torus conclusion.
For a free particle on $\R$ with $H = p^2/2$, the level set $H = c > 0$ is two lines, not circles, and the motion is not periodic.

## Explanations

### L1 human

If a mechanical system with $n$ degrees of freedom has $n$ independent conserved quantities that are compatible with each other, then its motion is very simple.
Each bounded motion lives on a doughnut-shaped surface, an $n$-dimensional torus.
On the torus, the motion is a straight line wound around at constant speeds.
This is one precise meaning of "the system can be solved".

### L2 human

The theorem is the structure theorem for [[integrable systems|N0N7CMRCMH]].
It turns the algebraic condition of commuting integrals into a geometric picture: a fibration of phase space by invariant Lagrangian tori, with linear flow on each torus.
The frequency vector $\nu$ decides the type of motion.
When its entries are rationally independent, each orbit is dense in its torus.
When they satisfy an integer relation, the torus is filled by invariant tori of lower dimension, and no orbit is dense in it.
This picture is the starting point of [[KAM theory|3ZZSXKM16Z]], which asks which tori survive a small perturbation of $H$.
</document>

<document source="https://tamnd.github.io/math2/node/PNRMHP5A0C.md" tag="PNRMHP5A0C" kind="concept">
# Simson line

## Summary

Let $P$ be a point on the [[circumcircle|03VZJE6C8C]] of a [[triangle|0CTNFGYMM2]].
Its Simson line is the line through the feet of the perpendiculars from $P$ to the three side lines.
These three feet are [[collinear|0AFNNTRTJE]] exactly when $P$ lies on the circumcircle.
The Simson line is a basic object of triangle geometry and appears in many angle chasing arguments.

## Explanations

### L1 human

Pick a point on the circle through the three corners of a triangle.
Drop a perpendicular from it to each of the three sides, extended if needed.
The three feet of these perpendiculars always lie on one straight line.
For example, if the point is a corner $A$, two of the feet are $A$ itself, and the line is the altitude from $A$.
</document>

<document source="https://tamnd.github.io/math2/node/PNT397N3HW.md" tag="PNT397N3HW" kind="definition">
# Faithful module

## Definition

Let $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].
The module $M$ is *faithful* if its [[annihilator|VQ31D68MDD]]
$$\operatorname{ann}_R(M) = \set{r \in R : rm = 0 \text{ for all } m \in M}$$
is $\set{0}$.

## Remarks

Equivalently, the ring homomorphism $R \to \End_\Z(M)$ that sends $r$ to the map $m \mapsto rm$ is injective.
Example: for a field $K$, the column space $K^n$ is a faithful $M_n(K)$-module.
The check: if $A e_j = 0$ for every standard basis vector $e_j$, then every column of $A$ is zero, so $A = 0$.
Non-example: for $n \ge 2$, the $\Z$-module $\Z/n\Z$ is not faithful, since $n \ne 0$ lies in its annihilator.
Every module $M$ is faithful as a module over $R / \operatorname{ann}_R(M)$.
</document>

<document source="https://tamnd.github.io/math2/node/PNTCSE6DWA.md" tag="PNTCSE6DWA" kind="definition">
# Fredholm equation of the second kind

## Definition

Let $a < b$ be real numbers.
Let $K \colon [a, b] \times [a, b] \to \C$ be a [[kernel|GSMBXT7FPK]], for example continuous or square-integrable.
Let $f \colon [a, b] \to \C$ be a given function, and let $\lambda \in \C$ be a parameter.
Let $\varphi$ denote an unknown function on $[a, b]$.
The *Fredholm equation of the second kind* with kernel $K$, parameter $\lambda$ and right-hand side $f$ is
$$\varphi(x) - \lambda \int_a^b K(x, y) \, \varphi(y) \, dy = f(x), \qquad x \in [a, b].$$
It is *homogeneous* when $f = 0$.

## Remarks

Example: on $[0, 1]$, take $K(x, y) = xy$, $\lambda = 1$ and $f(x) = x$.
Then $\varphi(x) = 3x/2$ is a solution, since $\int_0^1 x y \cdot \tfrac{3y}{2} \, dy = \tfrac{3x}{2} \cdot \tfrac{1}{3} = \tfrac{x}{2}$ and $\tfrac{3x}{2} - \tfrac{x}{2} = x$.

Non-example: $\int_a^b K(x, y) \varphi(y) \, dy = f(x)$ is not of the second kind, because $\varphi$ does not appear outside the integral; it is a [[Fredholm equation of the first kind|2RMWAVR1HV]].

Some books write the equation as $\varphi - \lambda T_K \varphi = f$, where $(T_K \varphi)(x) = \int_a^b K(x, y) \varphi(y) \, dy$.
Others put $\lambda$ on the other side, as $\lambda \varphi - T_K \varphi = f$; for $\lambda \ne 0$ the two forms are equivalent after dividing by $\lambda$.
</document>

<document source="https://tamnd.github.io/math2/node/PNWQZD5GRQ.md" tag="PNWQZD5GRQ" kind="definition">
# Slice knot

## Definition

Let $B^4$ be the closed unit ball in $\R^4$, with boundary the $3$-sphere $S^3$.
A tame [[knot|QYEAZTYGPT]] $K \subset S^3$ is *smoothly slice* if there is a smoothly embedded $2$-disk $D \subset B^4$ with $\partial D = D \cap S^3 = K$.
It is *topologically slice* if there is such a disk $D$ that is only locally flat.
Without qualification, *slice* means smoothly slice.

## Remarks

The disk $D$ must be properly embedded, meeting $S^3$ only in its boundary.
Example: let $K^*$ be the mirror image of $K$ with reversed orientation.
Then $K \# K^*$ bounds a ribbon disk, made by spinning a half of $K$ in $B^4$, so it is slice; the square knot is an instance.
Non-example: the [[trefoil knot|VYTYSVHN4A]] is not slice, because its signature is $\pm 2$ and the signature of a slice knot is $0$.
By Fox and Milnor, if $K$ is slice then $\Delta_K(t) = f(t) f(t^{-1})$ for some $f \in \Z[t, t^{-1}]$, where $\Delta_K$ is the [[Alexander polynomial|JKA25KDPQP]].
Freedman proved that every knot with $\Delta_K = 1$ is topologically slice.
Some of these knots are not smoothly slice, which shows that the two notions differ.
</document>

<document source="https://tamnd.github.io/math2/node/PNZT35FFN2.md" tag="PNZT35FFN2" kind="statement">
# Positive definiteness of the energy

## Statement

Let $n \ge 2$ and let $\mu$ and $\nu$ be finite positive [[Borel measures|DYR1QFNGNJ]] on $\R^n$ with compact supports.
Use the kernel $\abs{x}^{2-n}$ when $n \ge 3$ and $\log(1/\abs{x})$ when $n = 2$, and assume that the [[energies|SFSRQCM3EG]] $I(\mu)$ and $I(\nu)$ are finite.
When $n = 2$, assume also that $\mu(\R^2) = \nu(\R^2)$.
Then the [[mutual energy|6VR8GASZJY]] $I(\mu, \nu)$ is finite and
$$I(\mu - \nu) = I(\mu) - 2 I(\mu, \nu) + I(\nu) \ge 0,$$
with equality if and only if $\mu = \nu$.

## Remarks

Proof idea for $n \ge 3$: the [[Fourier transform|DPVM34ADZV]] of $\abs{x}^{2-n}$, as a tempered distribution, is $c_n \abs{\xi}^{-2}$ with a constant $c_n > 0$.
For $\lambda = \mu - \nu$ this gives
$$I(\lambda) = c\, \int_{\R^n} \frac{\abs{\hat{\lambda}(\xi)}^2}{\abs{\xi}^2}\, d\xi$$
with a constant $c > 0$, which is nonnegative and vanishes only when $\hat\lambda = 0$, that is, $\mu = \nu$.
In the plane the logarithmic kernel has a similar representation on measures of total mass $0$, which is why the masses must agree.

The mass condition is necessary in the plane.
Let $\mu$ be the uniform probability measure on the circle $\abs{z} = R$ with $R > 1$, and $\nu = 0$.
Then $I(\mu - \nu) = I(\mu) = \log(1/R) < 0$.

For $n \ge 3$ the theorem implies the [[Cauchy–Schwarz inequality|0RFWW56Y22]] $I(\mu, \nu)^2 \le I(\mu) I(\nu)$, since $I(\mu - t\nu) = I(\mu) - 2t\, I(\mu, \nu) + t^2 I(\nu) \ge 0$ for every $t > 0$ and $I(\mu, \nu) \ge 0$; take $t = I(\mu, \nu)/I(\nu)$ when $I(\nu) > 0$.

## Explanations

### L1 human

The energy of a charge distribution with both positive and negative charges is never negative, as long as the total charge is balanced in the plane.
It is zero only when there is no charge at all.
So energy behaves like the square of a length on the space of charge distributions.

### L2 human

This is the key fact of the energy method in potential theory.
It makes $\sqrt{I(\mu - \nu)}$ a distance on measures of finite energy, and a theorem of Cartan says that positive measures of finite energy are complete for this distance when $n \ge 3$.
Uniqueness of the equilibrium measure follows: if $\mu$ and $\nu$ both minimise the energy among probability measures on a compact set, then the energy of $\frac12(\mu + \nu)$ forces $I(\mu - \nu) = 0$, so $\mu = \nu$.
</document>

<document source="https://tamnd.github.io/math2/node/PP11DV3ZMQ.md" tag="PP11DV3ZMQ" kind="concept">
# Krull dimension of a noncommutative ring

## Summary

The Krull dimension of a module over a possibly noncommutative ring, in the sense of Gabriel and Rentschler, is an ordinal.
It measures how far the lattice of submodules is from satisfying the descending chain condition.
It is defined by transfinite recursion: dimension $0$ means nonzero [[Artinian|YQNZTQGP1X]].
Dimension at most $\alpha$ means that every descending chain of submodules has, from some point on, factors of dimension less than $\alpha$.
The right Krull dimension of a ring is the Krull dimension of the ring as a right module over itself.
For commutative [[Noetherian rings|0670T59HGE]] it agrees with the classical [[Krull dimension|MVGBS3J1P7]] defined by chains of prime ideals.

## Explanations

### L1 human

In a commutative ring, dimension counts the longest chain of prime ideals, but in noncommutative rings prime ideals can be scarce.
The Gabriel–Rentschler dimension looks instead at descending chains of submodules.
Artinian modules, where every descending chain stops, get dimension $0$.
The integers get dimension $1$: a descending chain of ideals $\Z \supseteq a_1\Z \supseteq a_2\Z \supseteq \cdots$ can go on forever, but its factors are finite groups, which have dimension $0$.
A polynomial ring in two variables gets dimension $2$, and so on, matching the usual dimension.
</document>

<document source="https://tamnd.github.io/math2/node/PP1A0JB47Q.md" tag="PP1A0JB47Q" kind="concept">
# Bayes estimator

## Summary

A Bayes estimator is an estimator that minimizes the [[Bayes risk|VEBDNCJJTN]], the risk averaged over a prior distribution on the parameter.
In practice it minimizes, for each observed data set, the expected loss under the [[posterior distribution|0KJ7ZS8AJA]].
Under squared error loss it is the posterior mean.
Bayes estimators link [[Bayesian inference|0JWRJX2RMP]] with frequentist decision theory, since they are central in the study of admissible and minimax rules.

## Explanations

### L1 human

Before seeing the data, a Bayesian describes beliefs about the unknown quantity by a prior distribution.
After seeing the data, the beliefs are updated to a posterior distribution.
The Bayes estimator is the single number that is best on average according to these updated beliefs.
For a coin with a uniform prior on its heads probability, $x$ heads in $n$ tosses give the Bayes estimate $(x + 1)/(n + 2)$ under squared error.
</document>

<document source="https://tamnd.github.io/math2/node/PP3XTZWHNH.md" tag="PP3XTZWHNH" kind="definition">
# Butterfly effect

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \colon X \to X$ be a [[continuous map|0X3WGX95FE]], with [[iterates|AYVTR2ES5B]] $f^n$.
The map $f$ has *sensitive dependence on initial conditions*, the butterfly effect, if there is a constant $\delta > 0$ with the following property.
For every $x \in X$ and every $\varepsilon > 0$, there are a point $y \in X$ with $d(x, y) < \varepsilon$ and an integer $n \ge 0$ with $d(f^n(x), f^n(y)) > \delta$.

## Remarks

The constant $\delta$ is the same for all $x$; the point $y$ and the time $n$ depend on $x$ and $\varepsilon$.
For a flow $(\varphi_t)$ one asks instead for a time $t \ge 0$ with $d(\varphi_t(x), \varphi_t(y)) > \delta$.
Sensitive dependence is one of the conditions in Devaney's definition of chaos.

Example: the doubling map $f(x) = 2x \bmod 1$ on the circle $\R / \Z$, with the distance $d$ to the nearest integer of the difference.
Take $\delta = 1/8$.
Given $x$ and $\varepsilon > 0$, choose $n$ with $2^{-n} < \varepsilon$ and let $y = x + 2^{-n-2}$.
Then $d(x, y) < \varepsilon$, and $f^n(y) - f^n(x) = 1/4 \bmod 1$, so $d(f^n(x), f^n(y)) = 1/4 > \delta$.

Non-example: the rotation $f(x) = x + \alpha \bmod 1$ on $\R / \Z$ is an isometry, so $d(f^n(x), f^n(y)) = d(x, y)$ for all $n$.
Points closer than $\delta$ stay closer than $\delta$ forever, so no $\delta > 0$ works.
</document>

<document source="https://tamnd.github.io/math2/node/PPEWPSQB9K.md" tag="PPEWPSQB9K" kind="definition">
# Horocycle

## Definition

Let $\mathbb{H}^2$ be the [[hyperbolic plane|GN4N94GQ0Z]] with distance $d$, and let $\omega$ be an ideal point, a point at infinity of $\mathbb{H}^2$.
Let $\gamma \colon [0, \infty) \to \mathbb{H}^2$ be a [[geodesic|NSYMQMHPDN]] ray parametrized by arc length with $\gamma(t) \to \omega$.
The *Busemann function* of $\gamma$ is
$$b(x) = \lim_{t \to \infty} \bigl( d(x, \gamma(t)) - t \bigr), \qquad x \in \mathbb{H}^2.$$
A *horocycle centred at* $\omega$ is a level set $\set{x \in \mathbb{H}^2 : b(x) = c}$ with $c \in \R$.

## Remarks

Another ray to $\omega$ changes $b$ by an additive constant, so the set of horocycles centred at $\omega$ depends only on $\omega$.
Example: in the [[Poincaré half-plane model|RHKSMPBSQ1]], take $\omega = \infty$ and $\gamma(t) = i e^t$.
For $x = u + iy$, the distance formula $\cosh d(z, w) = 1 + \abs{z - w}^2 / (2 \operatorname{Im} z \operatorname{Im} w)$ gives $d(x, i e^t) = t - \ln y + o(1)$ as $t \to \infty$.
So $b(x) = -\ln y$, and the horocycles centred at $\infty$ are the horizontal lines $y = \text{const}$.
Their normal geodesics are the vertical half-lines, which all end at $\infty$.
Horocycles centred at a real point $\omega$ are Euclidean circles tangent to the real axis at $\omega$, without $\omega$.
Non-example: the ray $\set{r e^{i\theta} : r > 0}$ with $0 < \theta < \pi/2$ is not a horocycle.
Its normal geodesics are the semicircles centred at $0$, and they end at different real points, so no single ideal point $\omega$ works.
This ray is a curve at constant distance from the imaginary axis, called a hypercycle.
</document>

<document source="https://tamnd.github.io/math2/node/PPGD5KW8JC.md" tag="PPGD5KW8JC" kind="concept">
# Computable set

## Summary

A computable set, also called a recursive or decidable set, is a set of natural numbers for which an algorithm decides membership.
The algorithm must halt on every input and answer yes or no correctly.
Computable sets are exactly the sets whose characteristic function is a total [[computable function|KZSB5E471K]].
They are the bottom level of the hierarchies of complexity of sets in [[computability theory|X14Y880PN5]].

## Explanations

### L1 human

A set of numbers is computable if a program can always answer the question "is $n$ in the set?" in finitely many steps.
The even numbers and the prime numbers are computable sets.
The set of numbers of programs that halt is not computable, although a program can confirm membership when it holds.
Each finite set is computable, since a program can simply contain the list.
</document>

<document source="https://tamnd.github.io/math2/node/PPM7V7WWHQ.md" tag="PPM7V7WWHQ" kind="definition">
# Cole–Hopf transformation

## Definition

Let $\nu > 0$, let $I \subseteq \R$ be an open interval, and let $T > 0$.
Let $\varphi \colon I \times (0, T) \to (0, \infty)$ be a positive function with continuous [[partial derivatives|09EGC6F8PT]] $\varphi_t$, $\varphi_x$, $\varphi_{xx}$.
The *Cole–Hopf transformation* of $\varphi$ is the function
$$u = -2\nu\, \frac{\varphi_x}{\varphi} = -2\nu\, \partial_x \log \varphi.$$
Its basic property: if $\varphi$ solves the [[heat equation|VP2HN86VAG]] $\varphi_t = \nu \varphi_{xx}$, then $u$ solves the viscous [[Burgers' equation|NEGHXN8RE1]] $u_t + u u_x = \nu u_{xx}$.

## Remarks

Proof of the basic property: put $\psi = \log \varphi$, so $u = -2\nu \psi_x$.
The heat equation for $\varphi$ becomes $\psi_t = \nu \psi_{xx} + \nu \psi_x^2$.
Differentiating in $x$ and multiplying by $-2\nu$ gives $u_t = \nu u_{xx} - u u_x$.

Inverse direction: given initial values $u_0$, take $\varphi_0(x) = \exp\left(-\frac{1}{2\nu} \int_0^x u_0(y)\, dy\right)$.
Solving the heat equation with data $\varphi_0$ and transforming gives the solution of Burgers' equation with data $u_0$, for bounded continuous $u_0$ on $I = \R$.

Example: $\varphi(x, t) = 1 + e^{-k(x - \nu k t)}$ with $k > 0$ solves $\varphi_t = \nu \varphi_{xx}$, since both sides equal $\nu k^2 e^{-k(x - \nu k t)}$.
Its transformation is $u = \nu k \left(1 - \tanh\left(\tfrac{k}{2}(x - \nu k t)\right)\right)$.
This is a [[traveling wave|M37M411KP7]] of speed $\nu k$ from $2\nu k$ to $0$.

Non-example: $\varphi(x, t) = \sin x$ is not allowed on $\R$, since it is not positive.
Its transformation $-2\nu \cot x$ is undefined at the zeros of $\sin x$.

## Explanations

### L1 human

The transformation takes the logarithmic derivative of a solution of the heat equation.
This one formula converts a linear equation into a nonlinear one.
Read backwards, it solves a nonlinear equation with the tools of a linear one.
</document>

<document source="https://tamnd.github.io/math2/node/PPS8T94AWS.md" tag="PPS8T94AWS" kind="definition">
# Calderón–Zygmund kernel

## Definition

Let $n \ge 1$ and let $\Delta = \set{(x, x) : x \in \R^n}$ be the diagonal of $\R^n \times \R^n$.
A *Calderón–Zygmund kernel*, or standard kernel, is a function $K \colon (\R^n \times \R^n) \setminus \Delta \to \C$ for which there are constants $A > 0$ and $0 < \delta \le 1$ such that:

1. $\abs{K(x, y)} \le A \abs{x - y}^{-n}$ for all $x \ne y$;
2. $\abs{K(x, y) - K(x', y)} \le A \dfrac{\abs{x - x'}^\delta}{\abs{x - y}^{n + \delta}}$ whenever $\abs{x - x'} \le \frac{1}{2}\abs{x - y}$;
3. $\abs{K(x, y) - K(x, y')} \le A \dfrac{\abs{y - y'}^\delta}{\abs{x - y}^{n + \delta}}$ whenever $\abs{y - y'} \le \frac{1}{2}\abs{x - y}$.

## Remarks

Example: $n = 1$ and $K(x, y) = \frac{1}{\pi(x - y)}$, the kernel of the Hilbert transform.
Clause 1 holds with $A = 1/\pi$.
For clause 2, if $\abs{x - x'} \le \frac{1}{2}\abs{x - y}$ then $\abs{x' - y} \ge \frac{1}{2}\abs{x - y}$, and
$$\abs{K(x, y) - K(x', y)} = \frac{\abs{x - x'}}{\pi \abs{x - y}\abs{x' - y}} \le \frac{2}{\pi}\frac{\abs{x - x'}}{\abs{x - y}^2},$$
so clause 2 holds with $\delta = 1$; clause 3 is the same by symmetry.

Non-example: $K(x, y) = \abs{x - y}^{-n - 1/2}$ fails clause 1, because it is too large near the diagonal.

A convolution kernel $K(x, y) = k(x - y)$ is a Calderón–Zygmund kernel when $\abs{k(x)} \le A\abs{x}^{-n}$ and $\abs{\nabla k(x)} \le A\abs{x}^{-n-1}$ for $x \ne 0$, by the mean value theorem.
A weaker smoothness condition often used for convolution kernels is the Hörmander condition $\int_{\abs{x} > 2\abs{y}} \abs{k(x - y) - k(x)} \, dx \le A$ for all $y \ne 0$.
The size and smoothness conditions alone do not give an $L^2$ bound; one also needs cancellation, as for the [[singular integral operators|HTMCEARGDD]] with mean-zero angular part.
</document>

<document source="https://tamnd.github.io/math2/node/PPSSTHTEDE.md" tag="PPSSTHTEDE" kind="concept">
# Cauchy–Riemann operator

## Summary

The Cauchy–Riemann operator is the first-order differential operator $\partial / \partial \bar z = \tfrac12 (\partial_x + i \partial_y)$ on complex-valued functions of $z = x + iy$.
A continuously differentiable function $f$ is [[holomorphic|0NAHXR1FRH]] exactly when $\partial f / \partial \bar z = 0$, which is a compact form of the [[Cauchy–Riemann equations|0T71SSYZ39]].
It is the simplest elliptic operator, with fundamental solution $1/(\pi z)$.
Its several-variable version, the $\bar\partial$ operator, is a main tool of complex analysis and complex geometry.

## Explanations

### L1 human

A function of $z = x + iy$ can depend on $x$ and $y$ in any way, for example through $\bar z = x - iy$.
The Cauchy–Riemann operator measures how much a function depends on $\bar z$.
Holomorphic functions, such as polynomials in $z$, do not depend on $\bar z$ at all, so the operator gives zero on them.
For example, it gives $0$ on $z^2$ and $1$ on $\bar z$.
</document>

<document source="https://tamnd.github.io/math2/node/PPT1ZSSHBN.md" tag="PPT1ZSSHBN" kind="definition">
# Reproducing kernel Hilbert space

## Definition

Let $X$ be a nonempty set and let $\mathbb{K}$ be $\R$ or $\C$.
Let $H$ be a vector space of functions $X \to \mathbb{K}$, with pointwise operations, that is a [[Hilbert space|HM2HHE3J51]] for some inner product.
The space $H$ is a *reproducing kernel Hilbert space* if the following holds.
For every $x \in X$, the evaluation map $f \mapsto f(x)$ is a [[continuous linear map|03TM135G0N]] from $H$ to $\mathbb{K}$.

## Remarks

For each $x$ there is then a unique $k_x \in H$ with $f(x) = \ip{f}{k_x}$ for all $f \in H$.
The function $K(x, y) = k_y(x) = \ip{k_y}{k_x}$ on $X \times X$ is the *reproducing kernel* of $H$.

Example: $H = \ell^2(\N)$, seen as functions on $X = \N$.
Evaluation at $n$ is $f \mapsto f(n)$, and $\abs{f(n)} \le \norm{f}_2$, so it is continuous.
Here $k_n$ is the unit vector $e_n$ and $K(m, n) = 1$ if $m = n$ and $0$ otherwise.

Non-example: $L^2([0, 1])$ is not a reproducing kernel Hilbert space.
Its elements are classes of functions equal almost everywhere, so $f \mapsto f(x)$ is not even defined on it.
The continuous functions on $[0, 1]$ with the $L^2$ inner product are not one either, because this space is not complete.

The Hardy space of the unit disc is a reproducing kernel Hilbert space with kernel $K(z, w) = 1/(1 - z \bar{w})$.
</document>

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# Quadratic transformations of hypergeometric functions

## Statement

Let ${}_2F_1$ be the [[hypergeometric function|463X5V31D9]], with principal powers, and let $a, b \in \C$.

1. If $a + b + \frac12 \notin \set{0, -1, -2, \ldots}$, then for all $z \in \C$ with $\operatorname{Re} z < \frac12$,
$${}_2F_1\left(2a, 2b; a + b + \tfrac12; z\right) = {}_2F_1\left(a, b; a + b + \tfrac12; 4z(1 - z)\right).$$
2. If $2b \notin \set{0, -1, -2, \ldots}$, then for all $z \in \C$ with $\abs{z} < 1$,
$${}_2F_1(a, b; 2b; z) = (1 - z)^{-a/2} \, {}_2F_1\left(\tfrac{a}{2}, b - \tfrac{a}{2}; b + \tfrac12; \frac{z^2}{4z - 4}\right).$$
3. If $1 + a - b \notin \set{0, -1, -2, \ldots}$, then for all $z \in \C$ with $\abs{z} < 1$,
$${}_2F_1(a, b; 1 + a - b; z) = (1 - z)^{-a} \, {}_2F_1\left(\tfrac{a}{2}, \tfrac{1 + a}{2} - b; 1 + a - b; \frac{-4z}{(1 - z)^2}\right).$$

## Remarks

In each formula both sides solve the same second-order hypergeometric equation in $z$.
Indeed, the substitution of a quadratic rational function of $z$ into a hypergeometric equation with special parameters gives back a hypergeometric equation.
Both sides are holomorphic near $z = 0$ with value $1$ there and the exponent $0$ there, which singles out one solution; analytic continuation then extends the identity to the stated domains.
For instance in part 1, $4z(1 - z) = 1 - (1 - 2z)^2$ maps the half-plane $\operatorname{Re} z < \frac12$ into $\C \setminus [1, \infty)$.

Such transformations exist only for special parameters.
Consider the exponent differences $1 - c$, $c - a - b$ and $a - b$ of the equation.
A quadratic transformation exists if two of them are equal up to sign or one of them is $\pm\frac12$.
All three parts fit this pattern: in part 1, the right side has $c - a - b = \frac12$; in part 2, $c = 2b$ gives $c - a - b = -(a - b)$; in part 3, $c = 1 + a - b$ gives $1 - c = -(a - b)$.

Example: with $a = b = \frac12$ in part 3, $1 + a - b = 1$ and the formula expresses ${}_2F_1(\frac12, \frac12; 1; z)$, which is $\frac{2}{\pi}$ times the complete elliptic integral of the first kind with parameter $z$, through $(1 - z)^{-1/2} \, {}_2F_1(\frac14, \frac14; 1; \frac{-4z}{(1-z)^2})$.

## Explanations

### L1 human

Some hypergeometric functions can be rewritten in terms of another hypergeometric function of a squared or quadratic expression in $z$, such as $4z(1 - z)$.
This only works when the three parameters satisfy a special relation.
These formulas connect hypergeometric functions with elliptic integrals and the arithmetic–geometric mean.

### L2 human

Linear transformations such as Pfaff and Euler exist for all parameters and come from the Möbius symmetries of $\set{0, 1, \infty}$.
Quadratic transformations come instead from rational maps of degree $2$ of the sphere that send singular points to singular points, which is possible only for special exponents.
Kummer and Goursat studied them systematically; for more special parameters there are also transformations of higher degree.
</document>

<document source="https://tamnd.github.io/math2/node/PPTX47481A.md" tag="PPTX47481A" kind="definition">
# Field of extremals

## Definition

Let $L(x, y, p)$ be a $C^2$ function on $\R \times \R \times \R$, with [[integral functional|JRQZ803Q5Y]] $J(y) = \int L(x, y, y') \, dx$.
Let $D \subseteq \R^2$ be a simply connected open set.
A *field of extremals* of $J$ on $D$ is a family of functions $y = \varphi(x, c)$, with $c$ in an open interval $I$, such that:

1. for each $c \in I$, the function $x \mapsto \varphi(x, c)$ solves the [[Euler–Lagrange equation|PS0VV149ET]] of $J$;
2. through each point $(x, y) \in D$ passes exactly one curve $y = \varphi(x, c)$ of the family;
3. the parameter $c(x, y)$ of that curve is a [[continuously differentiable|0VW90BBSJ4]] function on $D$.

The *slope function* of the field is $p(x, y) = \partial_x \varphi(x, c(x, y))$.
An extremal $y_0$ is *embedded* in the field if $y_0 = \varphi(\cdot, c_0)$ for some $c_0 \in I$.

## Remarks

Example: for $L = \sqrt{1 + p^2}$ the extremals are straight lines.
The family $\varphi(x, c) = x + c$, $c \in \R$, is a field on $D = \R^2$, with $c(x, y) = y - x$ and slope function $p = 1$.

Non-example: the family $\varphi(x, c) = c x$, $c \in \R$, of lines through the origin is not a field on $\R^2$.
All its curves pass through $(0, 0)$, and no curve passes through the points $(0, y)$ with $y \ne 0$.
On the half-plane $x > 0$ it is a field, with $c = y / x$; it is then a *central field*, whose curves all start at one point on the boundary.

Let an extremal $y_0$ on $[a, b]$ satisfy $L_{pp} > 0$ along it, with no point conjugate to $a$ in $(a, b]$.
Then it can be embedded in a field on a neighbourhood of its graph.
</document>

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# Dixon elliptic functions

## Summary

The Dixon elliptic functions $\operatorname{sm}$ and $\operatorname{cm}$ are two [[elliptic functions|FF9ZPCZ59K]] that parametrise the Fermat cubic $x^3 + y^3 = 1$, as $\sin$ and $\cos$ parametrise the circle $x^2 + y^2 = 1$.
They satisfy $\operatorname{sm}' = \operatorname{cm}^2$ and $\operatorname{cm}' = -\operatorname{sm}^2$.
Their period lattice has a threefold symmetry: rotation by a primitive cube root of unity $\omega$ maps it to itself.
They are named after Alfred Cardew Dixon.

## Explanations

### L1 human

The point $(\cos t, \sin t)$ runs around the circle $x^2 + y^2 = 1$.
The point $(\operatorname{cm} t, \operatorname{sm} t)$ runs along the cubic curve $x^3 + y^3 = 1$ in the same way.
Near $t = 0$, $\operatorname{sm} t \approx t$ and $\operatorname{cm} t \approx 1$, just as $\sin t \approx t$ and $\cos t \approx 1$.
Unlike sine and cosine, they have poles, because the real cubic curve goes off to infinity.
In the complex plane they repeat along a lattice of triangles.
</document>

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# Real algebraic geometry

## Summary

Real algebraic geometry studies the sets of real solutions of systems of polynomial equations and inequalities, and the polynomials that are nonnegative on them.
Its basic objects are real algebraic sets and [[semialgebraic sets|BX5WSK8ZCT]] over the [[real numbers|08J2M1JYAG]] or over a [[real closed field|DDY3Q3PJ6W]].
Its main tools are the [[Tarski–Seidenberg theorem|3W5JGDZAN0]], orderings of fields, and certificates of positivity such as the [[Positivstellensatz|FN734RQYM6]].
It has applications to optimization, robotics and the topology of real curves and surfaces.

## Explanations

### L1 human

Over the complex numbers every nonconstant polynomial in one variable has a root, and algebraic geometry is clean.
Over the real numbers, $x^2 + 1$ has no root, and inequalities like $x^2 + y^2 \le 1$ matter as much as equations.
Real algebraic geometry studies such real shapes: discs, ovals, regions bounded by polynomial curves.
A typical question asks how many pieces such a shape can have.
Another asks how to prove that a polynomial never takes negative values, for instance by writing it as a sum of squares.
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<document source="https://tamnd.github.io/math2/node/PPX0AK81QV.md" tag="PPX0AK81QV" kind="concept">
# Fredholm determinant

## Summary

The Fredholm determinant of a continuous kernel $K$ on $[a, b]$ is the entire function $D(\lambda) = \sum_{n \ge 0} \frac{(-\lambda)^n}{n!} \int \cdots \int \det[K(x_i, x_j)] \, dx_1 \cdots dx_n$.
It plays for the integral operator $\varphi \mapsto \int K(\cdot, y) \varphi(y) \, dy$ the role that $\det(I - \lambda A)$ plays for a square matrix $A$.
Its zeros are the [[characteristic values|DC7HMHVREP]] of the kernel.
Together with the [[Fredholm minors|HP5VJ9EBST]] it gives an explicit formula for the solution of the [[Fredholm equation of the second kind|DEYYNVCV9J]].
When the integral operator $T$ of $K$ is a [[trace class operator|Z2CP530WDE]], $D(\lambda)$ equals the operator determinant $\det(I - \lambda T)$.

## Explanations

### L1 human

Replace the integral by a sum over $n$ equally spaced points, and the integral operator becomes an $n \times n$ matrix.
For that matrix one can compute $\det(I - \lambda A)$ by expanding it into sums of principal minors.
Letting the number of points grow turns these sums into multiple integrals, and the result is the Fredholm determinant.
For the kernel $K(x, y) = xy$ on $[0, 1]$ only the first two terms survive, and $D(\lambda) = 1 - \lambda/3$.
It vanishes at $\lambda = 3$, which is exactly where the equation $\varphi = \lambda \int_0^1 x y \varphi(y) \, dy$ has the nonzero solution $\varphi(x) = x$.
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<document source="https://tamnd.github.io/math2/node/PQ1D8FQB3E.md" tag="PQ1D8FQB3E" kind="concept">
# Hankel matrix

## Summary

A Hankel matrix is a matrix that is constant along each anti-diagonal, the lines that go from lower left to upper right.
Its entry in row $i$ and column $j$ depends only on the sum $i + j$.
Hankel matrices appear for moment sequences, for the realisation of linear systems from their impulse responses, and in the theory of orthogonal polynomials.
Reversing the order of the rows turns a square Hankel matrix into a [[Toeplitz matrix|GKY2WSHE26]].

## Explanations

### L1 human

Take a sequence of numbers $h_2, h_3, h_4, \dots$ and write it into a square so that it runs along the anti-diagonals.
For example, $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{pmatrix}$ is Hankel, built from $1, 2, 3, 4, 5$.
Each row is the row above it shifted one step to the left, with a new number at the end.
A square Hankel matrix is always symmetric.
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<document source="https://tamnd.github.io/math2/node/PQ25NHV0KS.md" tag="PQ25NHV0KS" kind="concept">
# Calculus of variations

## Summary

The calculus of variations studies the extreme values of functionals, that is, of maps that send functions to numbers.
Examples of functionals are length, area and energy.
A typical functional is an integral $\int_\Omega L(x, u, \nabla u) \, dx$, minimised among functions $u$ with given boundary values.
Its smooth critical points solve a partial differential equation, the [[Euler–Lagrange equation|PS0VV149ET]].
Conversely, many equations of physics and geometry are Euler–Lagrange equations, and the calculus of variations gives existence through the [[direct method|TE379HBTY0]].

## Explanations

### L1 human

Ordinary calculus finds the point where a function of a number is smallest.
The calculus of variations finds the curve, surface or function that makes a quantity smallest.
Examples are the shortest path between two points, the soap film of least area spanning a wire, and the shape of a hanging chain of least potential energy.
For the shortest path in the plane, the functional is the length $\int_a^b \sqrt{1 + y'(x)^2} \, dx$, and the answer is the straight line.
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<document source="https://tamnd.github.io/math2/node/PQ6Q143ARJ.md" tag="PQ6Q143ARJ" kind="concept">
# Sheaf cohomology

## Summary

Sheaf cohomology assigns to a sheaf $\mathcal{F}$ on a space $X$ a sequence of groups $H^i(X, \mathcal{F})$, the right derived functors of the global sections functor.
In algebraic geometry it is applied to sheaves of modules on schemes, above all to [[quasi-coherent|KPKND0K7W6]] and [[coherent sheaves|56RXEJ8Z6S]].
For these the groups are modules, and in good cases finite-dimensional vector spaces.
It measures the failure of local sections to glue and of surjections of sheaves to be surjective on global sections.
The topological side of the theory, with constant and locally constant coefficients, is described in [[sheaf cohomology in topology|KB76923A1M]].

## Explanations

### L1 human

Global sections of a sheaf are the objects you can define on the whole space at once, such as functions or sections of a bundle.
When a short exact sequence of sheaves is given, the global sections need not form an exact sequence: something can exist locally but not globally.
Sheaf cohomology measures exactly this defect, in degree $1$, $2$, and so on.
For example, on the projective line over a field, $H^0$ of the structure sheaf is the field itself, the constant functions.
The bundle $\mathcal{O}(-2)$ has no sections, but its first cohomology group is one-dimensional.
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<document source="https://tamnd.github.io/math2/node/PQ7JJY0VPH.md" tag="PQ7JJY0VPH" kind="concept">
# Symplectic vector space

## Summary

A symplectic vector space is a vector space with a nondegenerate [[alternating bilinear form|0DGDS5XY84]], called the symplectic form.
A finite-dimensional symplectic vector space has even dimension and a standard basis in which the form has one fixed shape.
The linear maps that keep the form make up the [[symplectic group|X12FS2JAP0]].
Symplectic vector spaces are the linear model of phase space in classical mechanics and of symplectic manifolds.

## Explanations

### L1 human

An inner product measures lengths and angles; a symplectic form measures signed areas instead.
On $\R^2$, the form $\omega((x_1, y_1), (x_2, y_2)) = x_1 y_2 - x_2 y_1$ gives the signed area of the parallelogram spanned by two vectors.
Every vector has area zero with itself, but each nonzero vector has nonzero area with some other vector.
In higher dimensions, the space splits into planes like this one, so the dimension is always even.
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<document source="https://tamnd.github.io/math2/node/PQB48WPK5C.md" tag="PQB48WPK5C" kind="definition">
# Classical group

## Definition

Let $F$ be a [[field|04C5ZBG1WT]] and let $V$ be a [[finite-dimensional vector space|018W3V3TCT]] over $F$.
A *classical group* over $F$ is one of the following groups:

1. the [[general linear group|0FCHE902D7]] $\mathrm{GL}(V)$ of invertible linear maps $V \to V$, or the [[special linear group|0V9PMN0RBH]] $\mathrm{SL}(V)$ of those of determinant $1$;
2. the [[orthogonal group|0TKYAEJ8N7]] $\mathrm{O}(V, B) = \set{g \in \mathrm{GL}(V) : B(gu, gv) = B(u, v) \text{ for all } u, v}$ of a nondegenerate symmetric [[bilinear form|0SQZMXB6GP]] $B$ on $V$, where $F$ has characteristic not $2$;
3. the symplectic group $\mathrm{Sp}(V, \omega)$ of all $g \in \mathrm{GL}(V)$ preserving a nondegenerate alternating bilinear form $\omega$ on $V$;
4. the [[unitary group|0R9R0Y1HDK]] $\mathrm{U}(V, h)$ of all $g \in \mathrm{GL}(V)$ preserving $h$. Here $h$ is a nondegenerate Hermitian [[sesquilinear form|054YEQ0Y8M]] for an automorphism $\sigma$ of $F$ of order $2$;

together with their subgroups of elements of determinant $1$, such as $\mathrm{SO}(V, B)$ and $\mathrm{SU}(V, h)$.

## Remarks

The term is a name for this list rather than a single axiom, and authors vary.
Many include the analogous groups over the quaternions or over other division rings, and the projective quotients such as $\mathrm{PSL}(V)$, which give finite simple groups when $F$ is finite.

Example: $\mathrm{SO}(3)$, the special orthogonal group of the standard form $B(u, v) = u_1 v_1 + u_2 v_2 + u_3 v_3$ on $\R^3$, is classical; $B$ is symmetric and nondegenerate, since its matrix is the identity.

Non-example: the group of $g \in \mathrm{GL}_2(\R)$ preserving $B(u, v) = u_1 v_1$ is not given by clause 2, since $B$ is degenerate: $B(u, (0, 1)) = 0$ for all $u$.
This group consists of the matrices $\begin{pmatrix} \pm 1 & 0 \\ c & d \end{pmatrix}$ with $d \neq 0$.

When $F = \C$ and $\sigma$ is complex conjugation, clause 4 gives $\mathrm{U}(n)$ for the standard positive definite form; a nondegenerate alternating form exists only when $\dim V$ is even.
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<document source="https://tamnd.github.io/math2/node/PQBW43X5RE.md" tag="PQBW43X5RE" kind="concept">
# Kullback–Leibler divergence

## Summary

The Kullback–Leibler divergence $D(P \,\|\, Q)$, also called relative entropy, measures how much a probability distribution $P$ differs from a reference distribution $Q$.
It is non-negative and equals $0$ exactly when $P = Q$, but it is not symmetric and is not a metric.
It is the rate function in [[Sanov's theorem|QWEF2WAXJM]] and the expected log-likelihood ratio in statistics.
In information theory, it is the extra code length when one codes for $Q$ but the data follow $P$.

## Explanations

### L1 human

Suppose you believe that data come from a distribution $Q$, but they really come from $P$.
The Kullback–Leibler divergence measures, on average, how surprised you are by this mistake.
For a coin that really lands heads with probability $1/2$, the divergence from a model that says $1/4$ is $\frac{1}{2} \log \frac{4}{3}$, about $0.14$ in natural units.
It is zero only when the model is right.
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<document source="https://tamnd.github.io/math2/node/PQEPV7F2RV.md" tag="PQEPV7F2RV" kind="statement">
# Equivalence of Boolean rings and Boolean algebras

## Statement

1. Let $(B, \wedge, \vee, \neg, 0, 1)$ be a [[Boolean algebra|0XADJKM699]].
Put $a + b = (a \wedge \neg b) \vee (\neg a \wedge b)$ and $a \cdot b = a \wedge b$.
Then $(B, +, \cdot, 0, 1)$ is a [[Boolean ring|0RCV1ZFP8G]].
2. Let $(R, +, \cdot, 0, 1)$ be a Boolean ring.
Put $a \wedge b = ab$, $a \vee b = a + b + ab$ and $\neg a = 1 + a$.
Then $(R, \wedge, \vee, \neg, 0, 1)$ is a Boolean algebra.
3. The two constructions are inverse to each other.
4. A map between Boolean algebras is a homomorphism if and only if it is a unital ring homomorphism of the matching rings.

## Remarks

The proof is a direct check of axioms.
In a Boolean ring, $(a + b)^2 = a + b$ gives $ab + ba = 0$, and $a = -a$ follows from $(a + a)^2 = a + a$, so the ring is commutative of characteristic $2$; this makes the lattice laws routine.
In the other direction, the key point is that $+$ is associative, which reduces to the distributive laws and [[De Morgan's laws|0XYVDWGQX2]].
For the power set of a set $X$, the ring addition is the [[symmetric difference|0ME21QN125]] $A \mathbin{\triangle} B$ and the product is $A \cap B$.
In categorical terms, the theorem gives an isomorphism between the category of Boolean algebras and the category of Boolean rings.

## Explanations

### L1 human

Boolean algebras use "and", "or" and "not"; Boolean rings use "exclusive or" and "and" as plus and times.
The theorem says that these are two notations for the same thing.
One can always translate between them without losing anything.

### L2 human

The ring view brings tools of commutative algebra to Boolean algebras.
Ideals of the ring are exactly the ideals of the Boolean algebra, and prime ideals are maximal, with quotient the field $\F_2$.
So ultrafilters, which are complements of maximal ideals, correspond to ring homomorphisms onto $\F_2$, and the Stone space of the algebra is the prime spectrum of the ring.
This is the bridge used in one standard proof of [[Stone's representation theorem|S4QJ395PA5]].
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<document source="https://tamnd.github.io/math2/node/PQFEA12AHA.md" tag="PQFEA12AHA" kind="concept">
# Natural boundary

## Summary

A holomorphic function on a domain has the boundary as natural boundary if it cannot be continued across any boundary point.
See the [[definition|WER1M071EK]].
It is the extreme case of the obstruction to [[analytic continuation|0M3YQJ81NB]].
The typical example is a [[power series|0D6KXBHKC0]] with large gaps, whose circle of convergence is a natural boundary by the [[Hadamard gap theorem|DA8M8VZZHC]].

## Explanations

### L1 human

Many holomorphic functions given on a disc can be extended past the circle, except at a few bad points.
For $1/(1 - z)$, only the point $1$ is bad.
For some functions, every point of the circle is bad, so the disc is the end of the world for the function.
Then the circle is called a natural boundary.
An example is $z + z^2 + z^4 + z^8 + \cdots$, which blows up near each root of unity whose order is a power of $2$, and these points are dense on the circle.
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<document source="https://tamnd.github.io/math2/node/PQG7VAZ1R2.md" tag="PQG7VAZ1R2" kind="concept">
# Infinitary logic

## Summary

An infinitary logic is an extension of [[first-order logic|086QKPF0A6]] that allows formulas of infinite length.
Examples are conjunctions of infinitely many formulas and infinite blocks of quantifiers.
The logics $L_{\kappa\lambda}$ are the standard family, and $L_{\omega_1\omega}$, with countable conjunctions and finite quantifier blocks, is the most studied one.
These logics can express properties that first-order logic cannot, but they lose the compactness theorem.

## Explanations

### L1 human

In ordinary logic a formula is a finite string of symbols.
Infinitary logic allows a formula to say "$A_1$ or $A_2$ or $A_3$ or ..." without end.
For example, "every element $x$ of the group has some finite order" is the infinite disjunction "$x = e$ or $x^2 = e$ or $x^3 = e$ or ...".
No single first-order sentence says this about all groups.
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<document source="https://tamnd.github.io/math2/node/PQPDQCCD02.md" tag="PQPDQCCD02" kind="statement">
# von Neumann mean ergodic theorem

## Statement

Let $H$ be a [[Hilbert space|HM2HHE3J51]] and let $U \colon H \to H$ be a bounded linear operator with $\norm{U} \le 1$, for example a [[unitary operator|MM0EJ2GDE4]].
Let $P$ be the [[orthogonal projection|0B8KZRT28M]] of $H$ onto the closed subspace $\set{v \in H : Uv = v}$.
Then for every $v \in H$,
$$\lim_{n \to \infty} \norm{\frac{1}{n} \sum_{k=0}^{n-1} U^k v - P v} = 0.$$

## Remarks

Proof idea: let $F = \set{v : Uv = v}$ and $B = \set{w - Uw : w \in H}$.
For $\norm{U} \le 1$ one shows that $Uv = v$ if and only if $U^* v = v$, and this gives $H = F \oplus \overline{B}$, an [[orthogonal decomposition|0XJXTBCB2T]].
On $F$ the averages are constantly $v$.
On $B$ they telescope: $\frac{1}{n} \sum_{k=0}^{n-1} U^k (w - Uw) = \frac{1}{n}(w - U^n w)$, of norm at most $2\norm{w}/n$.
Since all averaging operators have norm at most $1$, the convergence to $0$ extends from $B$ to its closure.

Main application: let $T$ be a [[measure-preserving transformation|556WBDAHNP]] of a probability space $(X, \mathcal{A}, \mu)$, and let $U f = f \circ T$ on [[$L^2(\mu)$|SASP14MA1Z]].
Then $U$ is an isometry, and the theorem says that the Birkhoff averages of every $f \in L^2(\mu)$ converge in $L^2$ to the projection of $f$ onto the $T$-invariant functions.
If $T$ is ergodic, the invariant functions are the constants, and the limit is the constant $\int_X f \, d\mu$.

Example: for $U = -\id$ on $H = \C$, the fixed space is $\set{0}$.
The averages of $v$ equal $v/n$ for odd $n$ and $0$ for even $n$, so they tend to $0 = Pv$.

## Explanations

### L1 human

Apply a norm-non-increasing operator again and again to a vector and average the results.
The averages always settle down, and the limit is the part of the vector that the operator does not move.
The parts that move get averaged out.

### L2 human

This is the Hilbert space form of the ergodic theorem, proved by von Neumann in 1932, and it gives convergence in mean rather than at almost every point.
Through the composition operator $f \mapsto f \circ T$, it turns questions about a measure-preserving map into questions about a linear operator, which is the starting point of the spectral theory of dynamical systems.
Birkhoff's theorem gives almost-everywhere convergence and needs a different, harder argument based on a maximal inequality.
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<document source="https://tamnd.github.io/math2/node/PQRJ30QTYD.md" tag="PQRJ30QTYD" kind="definition">
# Pro-object

## Definition

Let $\mathcal{C}$ be a category.
A small category $\mathcal{I}$ is *cofiltered* if its [[opposite category|DAMK8XE1FR]] $\mathcal{I}^{\mathrm{op}}$ is [[filtered|975PMY4FG8]].
A *pro-object* in $\mathcal{C}$ is a functor $X \colon \mathcal{I} \to \mathcal{C}$ with $\mathcal{I}$ small and cofiltered, written $(X_i)_{i \in \mathcal{I}}$.
For pro-objects $X \colon \mathcal{I} \to \mathcal{C}$ and $Y \colon \mathcal{J} \to \mathcal{C}$ the morphisms are
$$\mathrm{Pro}(\mathcal{C})(X, Y) = \lim_{j} \operatorname{colim}_{i} \mathcal{C}(X_i, Y_j).$$
The pro-objects with these morphisms form the category $\mathrm{Pro}(\mathcal{C})$.

## Remarks

There is an equivalence $\mathrm{Pro}(\mathcal{C}) \simeq \mathrm{Ind}(\mathcal{C}^{\mathrm{op}})^{\mathrm{op}}$ with the [[Ind-completion|1DKFWGWHH5]] of the opposite category.
An object $c$ gives the constant pro-object indexed by the one-point category, and this gives a fully faithful functor $\mathcal{C} \to \mathrm{Pro}(\mathcal{C})$.
Every pro-object is the [[limit|CHS5AFJGWE]] in $\mathrm{Pro}(\mathcal{C})$ of its stages.

Example: in the category of finite groups, the groups $\Z / p^n \Z$ for $n \ge 1$, with the reduction maps, form a pro-object.
Its index category is the cofiltered poset $\N$ with an arrow $m \to n$ whenever $m \ge n$.
More generally, every [[profinite group|KA0YDB6QXE]] is the limit of the pro-object of its finite quotients.

Non-example: a functor from the discrete category with two objects to $\mathcal{C}$ is not a pro-object.
The discrete category with two objects is not cofiltered, since its two objects have no common object mapping to both.
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<document source="https://tamnd.github.io/math2/node/PQV6DNE0CW.md" tag="PQV6DNE0CW" kind="definition">
# Binary quadratic form

## Definition

A *binary quadratic form* (with integer coefficients) is a polynomial
$$f(x, y) = ax^2 + bxy + cy^2$$
in two variables $x, y$, with coefficients $a, b, c$ in the [[integers|0C4JS9RF5M]].
Its *discriminant* is $D = b^2 - 4ac$.
The form is *primitive* if the [[greatest common divisor|0C8WYW4KYQ]] of $a$, $b$, $c$ is $1$.
Two forms $f$ and $g$ are *properly equivalent* if $g(x, y) = f(\alpha x + \beta y, \gamma x + \delta y)$ for some matrix $\begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}$ in the [[special linear group|0V9PMN0RBH]] $\mathrm{SL}_2(\Z)$.

## Remarks

Properly equivalent forms have the same discriminant and represent the same integers.
The discriminant always satisfies $D \equiv 0$ or $1 \pmod 4$.
The form is positive definite if and only if $D < 0$ and $a > 0$.
Some authors write the form as $ax^2 + 2bxy + cy^2$ and require $a, b, c$ to be integers; this is Gauss's convention.

Example: $x^2 + y^2$ has $a = 1$, $b = 0$, $c = 1$, so it is a primitive binary quadratic form of discriminant $-4$.

Non-example: $x^2 + y^2 + z^2$ is a quadratic form with integer coefficients, but it has three variables, so it is not binary.
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<document source="https://tamnd.github.io/math2/node/PR11BZQQPG.md" tag="PR11BZQQPG" kind="concept">
# Measure

## Summary

A measure on a measurable space is a function that gives each measurable set a size in $[0, \infty]$.
It gives the empty set size $0$ and is countably additive: the size of a countable disjoint union is the sum of the sizes.
Measures generalize length, area, volume, counting and probability, and they are the basis of the Lebesgue integral.

## Explanations

### L1 human

A measure is a consistent way to give sizes to sets.
If a set is cut into countably many separate pieces, its size is the sum of the sizes of the pieces.
Length on the real line is a measure: the interval $[0, 2]$ has size $2$, the union of $[0, 1)$ and $[1, 2]$.
Counting the elements of a set is also a measure, and so is probability.
Sizes may be infinite, as for the length of the whole line.
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<document source="https://tamnd.github.io/math2/node/PR2DV03M4H.md" tag="PR2DV03M4H" kind="definition">
# Separation of variables

## Definition

*Separation of variables* is the following method, in two standard settings.

1. Ordinary equations. Let $g$ be continuous on an open interval $I$ and $h$ continuous on an open interval $J$, and consider the [[separable equation|0V4E623THY]] $y' = g(x)\, h(y)$. Let $H$ be an [[antiderivative|0BW9N5CDSB]] of $1/h$ and $G$ an antiderivative of $g$. On an interval where $h(y(x)) \ne 0$, $y$ is a solution if and only if $H(y(x)) = G(x) + C$ for a constant $C$. The constant functions $y = c$ with $h(c) = 0$ are further solutions.
2. Linear partial equations. For a linear homogeneous equation in $u(x, t)$, one looks for solutions of product form $u(x, t) = X(x)\, T(t)$. Sometimes the equation can then be rewritten with one side depending only on $x$ and the other only on $t$. Then both sides equal a common constant $\lambda$, which gives an ordinary differential equation for $X$ and one for $T$.

## Remarks

Example for 1: $y' = x y$ gives $\ln\abs{y} = x^2/2 + C$, so $y = A e^{x^2/2}$ with $A \ne 0$, and $y = 0$ comes from the zero of $h(y) = y$.
Check: $(A e^{x^2/2})' = x \cdot A e^{x^2/2}$.
Example for 2: for the heat equation $u_t = k u_{xx}$ with $k > 0$, the product $X T$ gives $X(x) T'(t) = k X''(x) T(t)$, so $T'/(kT) = X''/X = \lambda$.
With $\lambda = -1$ one finds $X = \sin x$ and $T = e^{-kt}$, and $u = e^{-kt} \sin x$ indeed satisfies $u_t = -k e^{-kt}\sin x = k u_{xx}$.
Non-example: $y' = x + y$ is not separable, since $x + y = g(x) h(y)$ for all $x, y$ is impossible.
Indeed, $x = 0$ gives $y = g(0) h(y)$ for all $y$, so $h(0) = 0$; then $y = 0$ gives $x = g(x) h(0) = 0$ for all $x$, which is false.
In the partial differential setting, superposition of product solutions and boundary conditions lead to eigenvalue problems and Fourier series.
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<document source="https://tamnd.github.io/math2/node/PR7Q3QMEHZ.md" tag="PR7Q3QMEHZ" kind="statement">
# Playfair's axiom

## Statement

Let $\ell$ be a [[line|0MP9BCF6DP]] in a [[plane|0V3XAMF3C1]] and let $P$ be a point of the plane not on $\ell$.
Then there is at most one line in the plane through $P$ that does not meet $\ell$.

## Remarks

In [[absolute geometry|VKFPCSRNDH]], at least one such line always exists: drop the perpendicular from $P$ to $\ell$ and take the perpendicular to it at $P$.
So, over those axioms, Playfair's axiom says that the parallel through $P$ exists and is unique.
It is the parallel axiom in Hilbert's system, and it is equivalent to Euclid's fifth postulate over the other axioms.
In the hyperbolic plane it fails: through $P$ there are infinitely many lines that do not meet $\ell$.
In the real plane $\R^2$ with $\ell = \set{(x, 0) : x \in \R}$ and $P = (0, 1)$, the only line through $P$ missing $\ell$ is $y = 1$.

## Explanations

### L1 human

Through a point off a line, you can draw exactly one line that never meets it.
This is the parallel axiom in its most common form.
It is named after John Playfair, who used it in his edition of Euclid.

### L2 human

Playfair's axiom is the form of the parallel postulate that one uses in practice.
It speaks only of incidence, and not of angles, so it also makes sense in an affine plane, where it is one of the defining axioms.
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<document source="https://tamnd.github.io/math2/node/PR8KWEA3MH.md" tag="PR8KWEA3MH" kind="concept">
# Riesz measure of a subharmonic function

## Summary

The Riesz measure of a [[subharmonic function|Z1MHMGFNFW]] is its [[Laplacian|0E0B6RA8K0]] taken in the sense of distributions, which is always a positive [[Radon measure|HZNEQQBCW4]].
It records where and how strongly the function fails to be [[harmonic|0DEB0JMN0W]]: it vanishes exactly on the open sets where the function is harmonic.
Through the [[Riesz decomposition theorem|CBZ20SS8RQ]], a subharmonic function is locally the potential of its Riesz measure plus a harmonic function.
For $\log\abs{f}$ with $f$ holomorphic, the Riesz measure counts the zeros of $f$.

## Explanations

### L1 human

Think of a subharmonic function as being shaped by a distribution of mass, possibly smeared out.
The Riesz measure is that mass distribution: it is the Laplacian of the function, which is never negative.
For $\log\abs{z}$ in the plane the mass is a single point mass $2\pi$ at the origin.
For $\abs{x}^2$ in $\R^n$ the mass is spread evenly everywhere, with density $2n$.
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<document source="https://tamnd.github.io/math2/node/PR8SESTZAN.md" tag="PR8SESTZAN" kind="definition">
# U-statistic

## Definition

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] random variables with values in a measurable space $E$.
Let $r \le n$ be a positive integer, and let $h \colon E^r \to \R$ be a measurable function that is symmetric in its $r$ arguments, the *kernel*.
The *U-statistic* with kernel $h$ is
$$U_n = \binom{n}{r}^{-1} \sum_{1 \le i_1 < i_2 < \dots < i_r \le n} h(X_{i_1}, \dots, X_{i_r}).$$
The integer $r$ is the degree of the kernel.

## Remarks

If $E\abs{h(X_1, \dots, X_r)} < \infty$, then $U_n$ is an unbiased estimator of $\theta = E[h(X_1, \dots, X_r)]$, by linearity of [[expectation|0DCT0QZVQ1]].
Let $h_1(x) = E[h(x, X_2, \dots, X_r)]$ and $\zeta_1 = \operatorname{Var}(h_1(X_1))$.
If $E[h^2] < \infty$, Hoeffding's theorem gives $\sqrt{n}(U_n - \theta) \to N(0, r^2 \zeta_1)$ in distribution.

Example: $r = 2$ and $h(x, y) = (x - y)^2 / 2$.
The identity $\sum_{i < j} (x_i - x_j)^2 = n \sum_i (x_i - \bar x)^2$ gives $U_n = \frac{2}{n(n-1)} \cdot \frac{n}{2} \sum_i (X_i - \bar X)^2 = \frac{1}{n-1} \sum_i (X_i - \bar X)^2$, the sample variance.
Here $\theta = \operatorname{Var}(X_1)$.
With $r = 1$ and $h(x) = x$, $U_n$ is the sample mean.

Non-example: the average $\frac{1}{n^2} \sum_{i, j} h(X_i, X_j)$ over all ordered pairs, including $i = j$, is a V-statistic, not a U-statistic.
For $h(x, y) = (x - y)^2/2$ it equals $\frac{1}{n} \sum_i (X_i - \bar X)^2$, which is biased.

Two-sample U-statistics average a kernel over one subset from each of two samples; the [[Mann–Whitney statistic|CKF9KDXBP7]] is one.
</document>

<document source="https://tamnd.github.io/math2/node/PR9A84ZPT1.md" tag="PR9A84ZPT1" kind="definition">
# Biharmonic equation

## Definition

Let $n \ge 1$ and let $\Omega \subseteq \R^n$ be an open set.
Let $u \colon \Omega \to \R$ be four times continuously differentiable.
The *biharmonic equation* on $\Omega$ is the [[partial differential equation|GC5BDZ6QBC]]
$$\Delta^2 u = 0,$$
where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $\Delta^2 u = \Delta(\Delta u)$.
A solution is called a *biharmonic function*.

## Remarks

In coordinates, $\Delta^2 u = \sum_{i,j=1}^n \partial_i^2 \partial_j^2 u$; in the plane, $\Delta^2 u = u_{xxxx} + 2 u_{xxyy} + u_{yyyy}$.
Example: on $\R^n$, $u(x) = \abs{x}^2$ is biharmonic, since $\Delta \abs{x}^2 = 2n$ is constant and so $\Delta^2 u = 0$.
Non-example: $u(x) = \abs{x}^4$ is not biharmonic, since $\Delta \abs{x}^4 = 4(n + 2) \abs{x}^2$ and then $\Delta^2 u = 8n(n + 2) \ne 0$.
Every [[harmonic function|0DEB0JMN0W]] is biharmonic.
The inhomogeneous equation $\Delta^2 u = f$ is the plate equation; boundary conditions usually prescribe two quantities, such as $u$ and its normal derivative.
</document>

<document source="https://tamnd.github.io/math2/node/PRC4WH5WYJ.md" tag="PRC4WH5WYJ" kind="statement">
# Tian–Todorov theorem

## Statement

Let $X$ be a compact [[Kähler manifold|X60JG9AEQJ]] of dimension $n$ whose [[canonical bundle|W05VYY4HBR]] $K_X$ is trivial.
Let $\Theta_X$ be the sheaf of holomorphic vector fields on $X$.
Then the deformations of $X$ are unobstructed: the base of the [[Kuranishi family|R0QABVGPRX]] of $X$ is smooth, of dimension $\dim_\C H^1(X, \Theta_X) = h^{n-1,1}(X)$.

## Remarks

The theorem is also called the Bogomolov–Tian–Todorov theorem: Bogomolov proved the case of hyperkähler manifolds, and Tian and Todorov proved the general case independently.
A nowhere vanishing holomorphic $n$-form $\Omega$ gives an isomorphism $\Theta_X \cong \Omega^{n-1}_X$, $v \mapsto \iota_v \Omega$, which explains the equality of dimensions.
Under this isomorphism the bracket of two vector-valued forms becomes $\partial$-exact (the Tian–Todorov lemma).
The $\partial\bar\partial$-lemma of compact Kähler manifolds then lets one solve the integrability equation $\bar\partial\varphi + \frac12[\varphi, \varphi] = 0$ order by order, so all obstructions vanish.

Example: a K3 surface $X$ has $n = 2$, $\Theta_X \cong \Omega^1_X$ and $h^{1,1}(X) = 20$, so its Kuranishi space is a smooth germ of dimension $20$.

The Kähler hypothesis is used through the $\partial\bar\partial$-lemma.
The Iwasawa manifold is a compact complex manifold with trivial canonical bundle that is not Kähler, and its deformations are obstructed.

## Explanations

### L1 human

Calabi–Yau manifolds are compact Kähler manifolds with a nowhere zero holomorphic volume form.
The theorem says that every first-order deformation of such a manifold extends to an actual family.
So their local moduli spaces are smooth, with a dimension computed from Hodge numbers.

### L2 human

In general the Kuranishi space is cut out by obstructions with values in $H^2(X, \Theta_X)$, and for some manifolds these obstructions are nonzero.
For Calabi–Yau manifolds $H^2(X, \Theta_X) \cong H^{n-1,2}(X)$ is usually nonzero, so the earlier existence theorem of Kodaira, Nirenberg and Spencer does not apply.
The theorem shows that the obstructions vanish anyway.
It is the starting point for the study of moduli of Calabi–Yau threefolds, of their period maps, and of mirror symmetry.
</document>

<document source="https://tamnd.github.io/math2/node/PRCQG82QTK.md" tag="PRCQG82QTK" kind="concept">
# Stochastic partial differential equation

## Summary

A stochastic partial differential equation, or SPDE, is a [[partial differential equation|GC5BDZ6QBC]] driven by random noise.
Its unknown is a random field $u(t, x)$, and the noise is often space-time white noise, a random distribution that is very irregular.
SPDEs extend [[stochastic differential equations|B219G9SHJA]] from finitely many to infinitely many variables.
They model systems with random forcing in physics, biology and finance, such as growing interfaces described by the [[Kardar–Parisi–Zhang equation|RQ196EHXKC]].

## Explanations

### L1 human

Take the [[heat equation|VP2HN86VAG]], which describes how temperature spreads in a rod.
Now let the rod be heated by tiny random kicks, independent at every point and every instant.
The temperature then becomes a random function, and the equation that governs it is an SPDE.
Because the kicks are so rough, the solution is rough too, and even the meaning of the equation needs care.
In one space dimension the solution is still a continuous function, but in two or more dimensions it is only a distribution.
</document>

<document source="https://tamnd.github.io/math2/node/PRDDMSTR4X.md" tag="PRDDMSTR4X" kind="concept">
# Seiberg–Witten invariants

## Summary

The Seiberg–Witten invariants are integer invariants of smooth closed oriented [[4-manifolds|8H58Q6YJCH]], one for each spin$^c$ structure.
They count solutions of the Seiberg–Witten equations, which couple a $U(1)$-connection with a spinor field through a Dirac operator.
They were introduced by Edward Witten in 1994, following work of Nathan Seiberg and Witten in physics.
They recover many results of [[Donaldson theory|ZQW6M0S6M0]] with much simpler analysis, because the moduli spaces are compact.

## Explanations

### L1 human

The Seiberg–Witten equations are a system of differential equations that one can write on any smooth four-dimensional space with a little extra structure.
One counts their solutions, with signs, and the count does not depend on the choices made along the way.
This count is a number attached to the smooth space, and spaces that look the same to topology can have different counts.
For example, take a K3 surface blown up at one point, and the connected sum of $3$ complex projective planes with $20$ reversed ones.
These two spaces are homeomorphic, but their counts differ, so they are not diffeomorphic.
</document>

<document source="https://tamnd.github.io/math2/node/PRFRD5RQ1W.md" tag="PRFRD5RQ1W" kind="definition">
# Subobject classifier

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] with finite limits, and let $1$ be its [[terminal object|EK85VSSRMR]].
A *subobject classifier* is an object $\Omega$ of $\mathcal{C}$ with a morphism $\top \colon 1 \to \Omega$ with the following property.
For every [[monomorphism|Y2NP1JFR86]] $m \colon S \to X$, there is exactly one morphism $\chi_m \colon X \to \Omega$ for which the square
$$\begin{array}{ccc} S & \longrightarrow & 1 \\ {\scriptstyle m}\downarrow & & \downarrow{\scriptstyle \top} \\ X & \xrightarrow{\chi_m} & \Omega \end{array}$$
is a [[pullback|Q05K04QJ71]].
The morphism $\chi_m$ is the *characteristic morphism* of $m$.

## Remarks

Equivalently, when $\mathcal{C}$ is locally small, there is a natural bijection between the [[subobjects|FS1FKAE89H]] of $X$ and $\Hom(X, \Omega)$, and $\Omega$ represents the subobject functor.
Example: in $\mathbf{Set}$, take $\Omega = \set{0, 1}$ and $\top(*) = 1$; for $S \subseteq X$, the unique $\chi$ is the characteristic function, since the pullback of $\set{1}$ along $\chi$ is $\chi^{-1}(1) = S$.
Non-example: the category of [[groups|G5S1M9]] has no subobject classifier.
Its terminal object is the trivial group, so every pullback of $\top \colon 1 \to \Omega$ is the kernel of a homomorphism, which is a normal subgroup.
A subgroup of order $2$ in $S_3$ is not normal, so it is not classified.
In a presheaf category $\mathbf{Set}^{\mathcal{C}^{\mathrm{op}}}$, the object $\Omega(c)$ is the set of [[sieves|CNCXNWKC6R]] on $c$, and $\top$ picks the maximal sieves.
</document>

<document source="https://tamnd.github.io/math2/node/PRJCH788AM.md" tag="PRJCH788AM" kind="concept">
# Eta invariant

## Summary

The eta invariant of a self-adjoint elliptic operator on a closed manifold measures the asymmetry of its spectrum between positive and negative eigenvalues.
It is the value at $s = 0$ of the eta function $\sum \operatorname{sign}(\lambda) \abs{\lambda}^{-s}$, defined by analytic continuation.
Atiyah, Patodi and Singer introduced it as the boundary correction in their [[index theorem for manifolds with boundary|GGFAH1M80J]].

## Explanations

### L1 human

A self-adjoint operator has real eigenvalues, some positive and some negative, usually infinitely many of each.
One would like to count the positive ones minus the negative ones, but both counts are infinite.
The eta invariant is a regularized value of this difference, obtained by weighting each eigenvalue by $\abs{\lambda}^{-s}$ and continuing to $s = 0$.
If the spectrum is symmetric about $0$, the eta invariant is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/PRQKKB7HNX.md" tag="PRQKKB7HNX" kind="definition">
# Paratopological group

## Definition

A *paratopological group* is a [[group|G5S1M9]] $G$ together with a [[topology|0CQHFPGJ45]] on $G$ such that the multiplication $G \times G \to G$, $(x, y) \mapsto xy$, is a [[continuous multiplication|ZZK2GM1V15]].
No condition is placed on the inversion $x \mapsto x^{-1}$.

## Remarks

Example: $(\R, +)$ with the topology of the [[Sorgenfrey line|Y4T3GVK8GV]], generated by the intervals $[a, b)$.
If $x + y \in [c, d)$, choose $\varepsilon > 0$ with $x + y + 2\varepsilon < d$; then $[x, x + \varepsilon) + [y, y + \varepsilon) \subseteq [x + y, x + y + 2\varepsilon) \subseteq [c, d)$, so addition is continuous.
Inversion is not continuous, since the preimage of $[0, 1)$ under $x \mapsto -x$ is $(-1, 0]$, which is not open.
So this is a paratopological group that is not a [[topological group|92700QAB5Y]].
Non-example: $(\Z, +)$ with the cofinite topology is a group, but addition is not jointly continuous, because every nonempty open set $U \times V$ of $\Z \times \Z$ contains a pair $(x, -x)$.
A paratopological group is the same as a topological monoid whose underlying monoid is a group.
</document>

<document source="https://tamnd.github.io/math2/node/PRWP1JF486.md" tag="PRWP1JF486" kind="statement">
# Alexander's theorem on closed braids

## Statement

Let $L$ be a tame oriented [[link|1A5WRCNTM3]] in $\R^3$.
Then there are an integer $n \ge 1$ and a braid $\beta$ in the [[braid group|BNR4H5GCTP]] $B_n$ with this property.
The link $L$ is ambient isotopic, as an oriented link, to the closure $\hat\beta$ of $\beta$.
Here $\hat\beta$ is obtained from a picture of $\beta$ in a cylinder in the following way.
The top end of each strand is joined to the bottom end in the same position, outside the cylinder and without new crossings.
All strands are oriented downward.

## Remarks

The idea of the proof is to choose an axis and to make the link wind around it in one direction.
Take a polygonal diagram of $L$ and a point $O$ of the plane off the diagram.
Each edge that runs clockwise around $O$ can be replaced by two edges that pass on the other side of $O$, so that they run counterclockwise.
After finitely many such moves every edge runs counterclockwise around $O$, and cutting along a ray from $O$ shows the link as a closed braid.
For example, the trefoil knot is the closure of $\sigma_1^3$ in $B_2$, and the Hopf link is the closure of $\sigma_1^2$ in $B_2$.
The braid $\beta$ is far from unique; the [[Markov theorem|7F3QQQ63WR]] says exactly when two braids have the same closure.

## Explanations

### L1 human

Every knot or link can be drawn as a braid whose ends are joined around the back.
Imagine a pole through the middle of the knot: you can move the string so that it always goes around the pole in the same direction.
Then the string looks like a braid that has been wrapped into a ring.

### L2 human

The theorem turns questions about links into questions about the braid groups $B_n$, which are groups with simple presentations.
Together with the [[Markov theorem|7F3QQQ63WR]], it shows that a link invariant is the same as a function on all braid groups that is unchanged by the two Markov moves.
This is how Jones first found the [[Jones polynomial|4SPC17FPXE]], from traces on representations of braid groups.
The least $n$ for which $L$ is a closed $n$-braid is called the braid index of $L$.
</document>

<document source="https://tamnd.github.io/math2/node/PS0VV149ET.md" tag="PS0VV149ET" kind="concept">
# Euler–Lagrange equation

## Summary

The Euler–Lagrange equation of an integral functional $\int L(x, u, \nabla u) \, dx$ is the differential equation satisfied by its smooth critical points, in particular by its smooth minimisers.
It is the analogue, for functionals, of the condition that the derivative vanishes at a minimum.
Many equations of physics and geometry arise this way.
Examples are Laplace's equation from the Dirichlet energy, the minimal surface equation from area, and Newton's equations from the action.
It is the basic necessary condition of the [[calculus of variations|PQ25NHV0KS]].

## Explanations

### L1 human

If a function makes an energy as small as possible, then small changes of the function cannot lower the energy, to first order.
Writing out this condition and integrating by parts gives a differential equation, the Euler–Lagrange equation.
For the length $\int \sqrt{1 + y'^2} \, dx$ of a graph, the equation says $y'' = 0$, so shortest graphs are straight lines.
For the energy $\int \frac{1}{2} \abs{\nabla u}^2 \, dx$, the equation is Laplace's equation $\Delta u = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/PS28BSAJKJ.md" tag="PS28BSAJKJ" kind="concept">
# Vector measure

## Summary

A vector measure is an additive set function whose values lie in a [[Banach space|06GZ956657]] instead of the real or complex numbers.
The basic notion is finite additivity; countable additivity is an extra property, with convergence of series in norm.
Vector measures arise from spectral theory, where a self-adjoint operator gives a projection-valued measure, and from integration of vector-valued functions.
Their theory, including the Bartle–Dunford–Schwartz integral and Lyapunov's convexity theorem, studies how far scalar measure theory extends to infinite dimensions.

## Explanations

### L1 human

An ordinary measure gives each set a number.
A vector measure gives each set a vector, for example a function in some function space, and it adds up over disjoint sets.
A simple example sends a set $E \subseteq [0, 1]$ to its indicator function $\mathbf{1}_E$, viewed as an element of $L^1([0, 1])$.
New phenomena appear in infinite dimensions: the same rule with values in $L^\infty([0, 1])$ adds up over finitely many sets but not over countably many.
</document>

<document source="https://tamnd.github.io/math2/node/PS2MR7P299.md" tag="PS2MR7P299" kind="concept">
# Noncommutative geometry

## Summary

Noncommutative geometry studies algebras that are not commutative as if they were algebras of functions on a space.
Its starting point is the [[commutative Gelfand–Naimark theorem|NT564Y315M]].
By this theorem, a locally compact Hausdorff space is determined by its C*-algebra of continuous functions that vanish at infinity.
A noncommutative C*-algebra is then read as a "noncommutative space", and tools of topology and geometry are carried over to it.
The theory gives a geometric language for spaces of orbits, foliations, quantum physics and number theory, where the usual point sets behave badly.

## Explanations

### L1 human

On an ordinary space, functions can be multiplied in any order: $fg = gf$.
Much of geometry can be recovered from these functions alone, without looking at the points.
Noncommutative geometry keeps the algebra and drops the rule $fg = gf$.
A standard example is the [[noncommutative torus|PZ97S3QQF1]], made of two unitaries $u$ and $v$ with $vu = e^{2\pi i \theta} uv$.
For $\theta = 0$ it is the algebra of continuous functions on the ordinary torus, and for other $\theta$ it behaves like a "quantum" torus.
Geometric ideas such as K-theory, distance and dimension still make sense for such algebras.
</document>

<document source="https://tamnd.github.io/math2/node/PS8DVS9WTA.md" tag="PS8DVS9WTA" kind="definition">
# Basic open set

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $f \in R$.
The *basic open set* (or *principal open set*) of $f$ is the subset
$$D(f) = \set{\mathfrak p \in \operatorname{Spec} R : f \notin \mathfrak p}$$
of the [[spectrum|FB45BHHA2J]] of $R$.
It is the complement of the closed set $V(\set{f})$ of the [[Zariski topology|8SQK7RTB63]], hence open.

## Remarks

The basic open sets have these properties, for $f, g \in R$:

1. $D(f) \cap D(g) = D(fg)$, since a [[prime ideal|00MHHZ9SNE]] contains $fg$ exactly when it contains $f$ or $g$;
2. each open set $X \setminus V(E)$ equals $\bigcup_{f \in E} D(f)$, so the sets $D(f)$ form a [[basis|0NA0732YAG]] of the topology;
3. $D(f) = \emptyset$ exactly when $f$ is [[nilpotent|0ZB2JG1M9C]], and $D(f) = \operatorname{Spec} R$ exactly when $f$ is a [[unit|0XVEZDE9J8]];
4. the map $\operatorname{Spec} R_f \to \operatorname{Spec} R$ induced by $R \to R_f$ is a homeomorphism onto $D(f)$; see [[prime ideals of a localization|W8QNZZ4QJS]].

Example: in $\operatorname{Spec} \Z$, the set $D(6)$ consists of $(0)$ and all $p\Z$ with $p \ne 2, 3$.

Non-example: not every open set is basic.
Let $k$ be a field and $R = k[x, y]$.
The set $U = \operatorname{Spec} R \setminus \set{(x, y)}$ is open, because $\set{(x,y)} = V(\set{x, y})$.
If $U = D(f)$, then $f \ne 0$ and $f$ is not a unit, since $U$ is neither empty nor everything.
Then an irreducible factor $g$ of $f$ generates a prime ideal $(g)$ that contains $f$, and $(g) \ne (x, y)$ because $(x,y)$ is not principal.
So $(g) \notin D(f)$ but $(g) \in U$, a contradiction.
</document>

<document source="https://tamnd.github.io/math2/node/PSAX8W3ZYV.md" tag="PSAX8W3ZYV" kind="example">
# Rudin–Shapiro polynomials

## Example

Define polynomials $P_m$ and $Q_m$ by $P_0(z) = Q_0(z) = 1$ and, for $m \ge 0$,
$$P_{m+1}(z) = P_m(z) + z^{2^m} Q_m(z), \qquad Q_{m+1}(z) = P_m(z) - z^{2^m} Q_m(z).$$
These are the *Rudin–Shapiro polynomials*.
By induction, $P_m$ and $Q_m$ have degree $2^m - 1$, and all $2^m$ of their coefficients are $+1$ or $-1$.
Indeed $P_m$ fills the coefficients of $1, \ldots, z^{2^m - 1}$ and $z^{2^m} Q_m$ fills those of $z^{2^m}, \ldots, z^{2^{m+1} - 1}$.
So they are [[Littlewood polynomials|5A16A9W335]].
The first ones are $P_1(z) = 1 + z$, $Q_1(z) = 1 - z$ and $P_2(z) = 1 + z + z^2 - z^3$.

For $\abs{z} = 1$, the parallelogram law $\abs{u + v}^2 + \abs{u - v}^2 = 2\abs{u}^2 + 2\abs{v}^2$ with $\abs{z^{2^m}} = 1$ gives
$$\abs{P_{m+1}(z)}^2 + \abs{Q_{m+1}(z)}^2 = 2 \left( \abs{P_m(z)}^2 + \abs{Q_m(z)}^2 \right).$$
Hence $\abs{P_m(z)}^2 + \abs{Q_m(z)}^2 = 2^{m+1}$, and
$$\max_{\abs{z} = 1} \abs{P_m(z)} \le \sqrt{2} \cdot \sqrt{2^m}.$$

## Remarks

Since the mean square of $\abs{P_m}$ on the circle is $2^m$, the maximum is at most $\sqrt 2$ times the root mean square.
For a polynomial with $N$ coefficients $\pm 1$, the maximum on the circle is always at least $\sqrt N$, so this is optimal up to the factor $\sqrt 2$.
A random choice of signs typically gives a maximum of order $\sqrt{N \log N}$, so the Rudin–Shapiro signs are much better than random.
The coefficients of the $P_m$ form the Rudin–Shapiro sequence, an automatic sequence.

## Explanations

### L1 human

The Rudin–Shapiro polynomials are built by a doubling rule.
Glue the old polynomial to a shifted copy of its partner, once with a plus sign and once with a minus sign.
All coefficients stay $\pm 1$.
The rule keeps the sum of the squared sizes of the pair exactly doubling at every point of the circle.
So neither polynomial can ever be much larger than its average size, which makes them almost flat from above.
</document>

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# Janko groups

## Description

The Janko groups $J_1$, $J_2$, $J_3$ and $J_4$ are four [[sporadic groups|PZ49Z4CBFZ]], found by Zvonimir Janko.
Their [[orders|0B85S2V2DR]] are:

1. $\abs{J_1} = 175560 = 2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19$;
2. $\abs{J_2} = 604800 = 2^7 \cdot 3^3 \cdot 5^2 \cdot 7$;
3. $\abs{J_3} = 50232960 = 2^7 \cdot 3^5 \cdot 5 \cdot 17 \cdot 19$;
4. $\abs{J_4} = 86775571046077562880 = 2^{21} \cdot 3^3 \cdot 5 \cdot 7 \cdot 11^3 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 43$.

Each of them is a finite [[simple group|0YMYMNQ0QZ]].

## Remarks

$J_1$ was the first sporadic group found after the [[Mathieu groups|T1KR4KPX8P]], about a century later.
Janko characterised it by the [[centraliser|02YYV36BFM]] of an involution, which is $C_2 \times A_5$.
$J_1$ is a subgroup of the Chevalley group $G_2(\F_{11})$.
$J_2$ is also called the Hall–Janko group.
$J_1$, $J_3$ and $J_4$ are [[pariah groups|ZYZ5FYG7QC]], since they are not subquotients of the [[Monster group|C9VY20TAPT]], while $J_2$ is a subquotient of the Monster.

## Explanations

### L1 human

The Janko groups are four of the exceptional finite simple groups.
The first of them showed, after a century without new examples, that the Mathieu groups were not the only exceptions.
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# Von Neumann algebra

## Summary

A von Neumann algebra is an algebra of bounded operators on a Hilbert space that contains the identity and is closed under taking adjoints.
It must also be closed in the [[weak operator topology|4XM1VN1C1N]].
It is a special kind of C*-algebra, one that is rich in projections and admits a measurable functional calculus.
Commutative von Neumann algebras are the algebras $L^\infty$ of measure spaces, so the theory is often called noncommutative measure theory.
John von Neumann introduced them and studied them with Francis Murray.
They are basic objects in operator algebras, quantum physics and representation theory.

## Explanations

### L1 human

Take some operators on a Hilbert space and keep adding sums, products, adjoints and limits.
If the limits are taken in the norm, one gets a C*-algebra.
If they are taken in the much weaker sense of matrix coefficients, one gets a von Neumann algebra.
The weaker limits add many projections, the operator analogues of indicator functions of sets.
For example, the continuous functions on $[0,1]$, acting by multiplication on $L^2[0,1]$, do not form a von Neumann algebra.
Adding the weak limits gives all bounded measurable functions, which do.
The matrix algebras $M_n(\C)$ are the simplest examples.
</document>

<document source="https://tamnd.github.io/math2/node/PSK6VJQR9B.md" tag="PSK6VJQR9B" kind="concept">
# Amoeba

## Summary

The amoeba of a complex polynomial in $n$ variables is the image of its zero set in $(\C^\times)^n$ under a logarithmic map.
This map takes the logarithm of the absolute value of each coordinate.
The name, given by Gelfand, Kapranov and Zelevinsky, comes from its shape: a body with thin tentacles going off to infinity.
Each connected component of the complement is convex, and the number of these components is bounded by the [[Newton polytope|CYTTX8KHC8]].
Amoebas link complex algebraic geometry with convex geometry and degenerate to tropical varieties, see [[tropical geometry|7YTC3BHK60]].

## Explanations

### L1 human

Take the complex solutions of $1 + z + w = 0$ and record only the sizes $\abs{z}$ and $\abs{w}$, on a logarithmic scale.
The resulting region in the plane looks like a creature with three long arms.
The arms reach to infinity in three directions, where one of $z$, $w$ is tiny or both are huge.
Between the arms lie three convex holes.
Zooming out, the region shrinks to three rays from a point, the tropical line.
</document>

<document source="https://tamnd.github.io/math2/node/PSNW0ZQ0E9.md" tag="PSNW0ZQ0E9" kind="definition">
# IEEE 754 standard

## Definition

The *IEEE 754 standard* is the technical standard for [[floating-point arithmetic|RTKRSC3KPH]] of the Institute of Electrical and Electronics Engineers.
Its main requirements are the following.

1. Formats: a format has a base $\beta \in \set{2, 10}$, a precision $p$ and an exponent range $[e_{\min}, e_{\max}]$ with $e_{\min} = 1 - e_{\max}$. The formats binary32 and binary64 have $(p, e_{\max}) = (24, 127)$ and $(53, 1023)$.
2. Special values: each format contains $+0$ and $-0$, $+\infty$ and $-\infty$, NaN ("not a number") values, and [[subnormal numbers|DHXXSN12JC]].
3. Encoding: a binary64 number has $1$ sign bit, $11$ bits of biased exponent (bias $1023$) and $52$ bits of fraction. The leading significand bit of a normalized number is implicit.
4. Correct rounding: the operations $+$, $-$, $\times$, $/$, square root and fused multiply-add must return the exact result [[rounded|5NPVX29Y0C]] by the current rounding mode.
5. Rounding modes: the default is round to nearest with ties to even. The other modes are ties away from zero and directed rounding toward $0$, $+\infty$ and $-\infty$.

## Remarks

Example: the decimal number $1$ in binary64 has sign bit $0$, biased exponent $0 + 1023 = 1023$ and fraction bits all $0$.
Non-example: an arithmetic that always computes $x + y$ by chopping the extra digits does not meet clause 4, since in the default mode the result must be rounded to nearest.
In binary64 the largest finite number is $(2 - 2^{-52}) \times 2^{1023} \approx 1.8 \times 10^{308}$, and the smallest positive normalized number is $2^{-1022}$.
An overflowing result is rounded to $\pm\infty$ under round to nearest, and an invalid operation such as $0/0$ or $\sqrt{-1}$ returns NaN.
</document>

<document source="https://tamnd.github.io/math2/node/PSR7JAN6HZ.md" tag="PSR7JAN6HZ" kind="definition">
# Beurling–Ahlfors transform

## Definition

Let $f \colon \C \to \C$ be a smooth function with compact support.
Its *Beurling–Ahlfors transform* is the function $\mathcal{B} f \colon \C \to \C$ given by the [[principal value|0V3RDQV3ZX]] integral
$$\mathcal{B} f(z) = -\frac{1}{\pi} \lim_{\varepsilon \to 0} \int_{\abs{w - z} > \varepsilon} \frac{f(w)}{(z - w)^2} \, dA(w),$$
where $dA$ is the Lebesgue area measure on $\C$.
The operator $\mathcal{B}$ extends uniquely to a bounded linear operator on $L^p(\C)$ for each $p$ with $1 < p < \infty$, also called the Beurling–Ahlfors transform.

## Remarks

It is a singular integral operator of Calderón–Zygmund type, with the kernel $-\frac{1}{\pi}(z - w)^{-2}$.
The principal value is needed: the kernel has absolute value $\frac{1}{\pi}\abs{z - w}^{-2}$, which is not integrable near $w = z$ in the plane, but its integral over each circle around $z$ is $0$.
With the [[Wirtinger derivatives|WMXDVF36ZT]] $\partial$ and $\bar{\partial}$, it satisfies $\mathcal{B}(\bar{\partial}\varphi) = \partial\varphi$ for every smooth compactly supported $\varphi$.
Example: for $\varphi(z) = e^{-\abs{z}^2}$, $\bar{\partial}\varphi = -z e^{-\abs{z}^2}$ and $\partial\varphi = -\bar{z} e^{-\abs{z}^2}$, so $\mathcal{B}(z e^{-\abs{z}^2}) = \bar{z} e^{-\abs{z}^2}$.
On the [[Fourier transform|DPVM34ADZV]] side, $\mathcal{B}$ is multiplication by $\bar{\xi}/\xi$, where $\xi = \xi_1 + i\xi_2$ is the frequency variable; as $\abs{\bar{\xi}/\xi} = 1$, it is an isometry of $L^2(\C)$ onto itself.
The norm of $\mathcal{B}$ on $L^p(\C)$ is conjectured by Iwaniec to be $\max(p, p/(p-1)) - 1$; this is an open problem.
</document>

<document source="https://tamnd.github.io/math2/node/PSX7N8WAVB.md" tag="PSX7N8WAVB" kind="definition">
# Riemannian distance

## Definition

Let $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]] with [[Riemannian metric|Q61VFHCH2J]] $g$.
For a piecewise smooth curve $\gamma \colon [a, b] \to M$, its [[length|0TH78S7GBZ]] is
$$L(\gamma) = \int_a^b \sqrt{g_{\gamma(t)}(\gamma'(t), \gamma'(t))} \, dt.$$
The *Riemannian distance* between points $p, q \in M$ is the [[infimum|0QF1MZ9FRG]]
$$d_g(p, q) = \inf \set{ L(\gamma) : \gamma \text{ is a piecewise smooth curve from } p \text{ to } q }.$$

## Remarks

Example: on $\R^n$ with the Euclidean metric $g = \sum_i dx_i^2$, the straight segment from $p$ to $q$ has length $\abs{q - p}$, and no curve is shorter, so $d_g(p, q) = \abs{q - p}$.

Non-example: if $M$ is not connected, two points in different components are joined by no curve.
Then the infimum is over the empty set, and $d_g$ is not a finite distance.
Some authors then set $d_g(p, q) = \infty$.

The function $d_g$ is a [[metric|0VKD1TP7F3]] on $M$, and the topology it induces is the given topology of $M$.
Positivity $d_g(p, q) > 0$ for $p \ne q$ is the nontrivial part; it follows by comparing $g$ with a Euclidean metric in a chart.
The infimum need not be attained: in $\R^2 \setminus \set{0}$ with the Euclidean metric, $d_g((-1, 0), (1, 0)) = 2$, but every curve between these points avoiding $0$ is longer than $2$.
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<document source="https://tamnd.github.io/math2/node/PT1P6Z6GEY.md" tag="PT1P6Z6GEY" kind="definition">
# Complex manifold

## Definition

Let $n \ge 0$ and let $M$ be a [[Hausdorff|0S1FJ8VEX9]], [[second-countable|03PPNCM9GG]] topological space.
A *complex chart* on $M$ is a pair $(U, \varphi)$ with $U \subset M$ open and $\varphi$ a [[homeomorphism|0QT00PB5G2]] of $U$ onto an open subset of $\C^n$.
A *holomorphic atlas* is a family of complex charts $(U_\alpha, \varphi_\alpha)$ such that:

1. the sets $U_\alpha$ cover $M$;
2. for all $\alpha, \beta$, the transition map $\varphi_\beta \circ \varphi_\alpha^{-1} \colon \varphi_\alpha(U_\alpha \cap U_\beta) \to \varphi_\beta(U_\alpha \cap U_\beta)$ is [[holomorphic|3MQTWKBBBT]].

Two holomorphic atlases are equivalent if their union is a holomorphic atlas.
A *complex manifold of dimension $n$* is such a space $M$ together with an equivalence class of holomorphic [[atlases|J7P32JQTVE]], called a *complex structure*.
A function $f \colon V \to \C$ on an open set $V \subset M$ is holomorphic if $f \circ \varphi^{-1}$ is holomorphic for every chart $(U, \varphi)$ of the structure.

## Remarks

Example: the [[Riemann sphere|0J78KBPFZB]] $\hat\C = \C \cup \set{\infty}$ with the charts $\varphi_0(z) = z$ on $\C$ and $\varphi_\infty(z) = 1/z$ on $\hat\C \setminus \set{0}$, with $\varphi_\infty(\infty) = 0$.
On the overlap, $\varphi_0(\C \setminus \set{0}) = \C \setminus \set{0}$, and the transition map $w \mapsto 1/w$ is holomorphic there; so $\hat\C$ is a complex manifold of dimension $1$.

Non-example: on $\C$, the two charts $\varphi_1(z) = z$ and $\varphi_2(z) = \bar z$ do not form a holomorphic atlas.
Each is a homeomorphism onto $\C$, but the transition map $w \mapsto \bar w$ is not holomorphic, so clause 2 fails.

Every complex manifold of dimension $n$ is a smooth manifold of dimension $2n$, since holomorphic maps are smooth.
It is canonically oriented, because the real Jacobian determinant of a holomorphic map is $\abs{\det J_\C}^2 \ge 0$, where $J_\C$ is its complex Jacobian matrix.
Further examples are open subsets of $\C^n$, [[complex projective space|7GJ4STCNNM]], [[complex tori|Y7MA6KBE9M]] and [[Hopf manifolds|Z8R2D33WSM]].
A complex structure induces an [[almost complex structure|9SYWRGVTRS]]; by the [[Newlander–Nirenberg theorem|VCT848WV1T]], the almost complex structures that arise this way are exactly the integrable ones.
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<document source="https://tamnd.github.io/math2/node/PT67C5ANSD.md" tag="PT67C5ANSD" kind="statement">
# Descartes's theorem on total angular defect

## Statement

Let $P \subseteq \R^3$ be a three-dimensional [[convex polytope|300K6CB1KH]].
Let $v$ be a [[vertex|R7V3S9V891]] of $P$.
For a face $F$ of $P$ that contains $v$, let $\alpha_F(v)$ be the interior angle of the polygon $F$ at $v$.
The *angular defect* at $v$ is
$$\delta(v) = 2\pi - \sum_F \alpha_F(v),$$
where the sum runs over the faces $F$ of $P$ that contain $v$.
Then
$$\sum_{v} \delta(v) = 4\pi,$$
where the sum runs over all vertices of $P$.

## Remarks

Let $P$ have $V$ vertices, $E$ edges and $F$ faces.
A face with $k$ sides has angle sum $(k - 2)\pi$, and the numbers of sides of all faces add up to $2E$.
So all face angles of $P$ add up to $2\pi E - 2\pi F$.
Hence $\sum_v \delta(v) = 2\pi V - 2\pi E + 2\pi F = 2\pi (V - E + F)$, and [[Euler's polyhedron formula|09HT3P35YP]] $V - E + F = 2$ gives $4\pi$.
The same count shows that for a polyhedral surface of [[Euler characteristic|RNDDWDAFJ5]] $\chi$ the total defect is $2\pi\chi$.

Example: the cube has $8$ vertices, each with three right angles, so each defect is $2\pi - 3\pi/2 = \pi/2$, and the total is $8 \cdot \pi/2 = 4\pi$.
The regular tetrahedron has $4$ vertices with three angles of $\pi/3$ each, so each defect is $\pi$, and the total is again $4\pi$.

Convexity enters only through Euler's formula: the statement holds for every polyhedron whose surface is a topological sphere, where a defect may be negative at a non-convex vertex.

## Explanations

### L1 human

At each corner of a convex solid, the face angles add up to less than a full turn; the missing amount is the defect.
If you add the missing amounts over all corners, you always get $720^\circ$, two full turns, whatever the solid.
A cube has eight corners each missing $90^\circ$, and $8 \cdot 90^\circ = 720^\circ$.

### L2 human

The theorem is the discrete form of the [[Gauss–Bonnet theorem|ESGEHKPD60]] for the sphere: the defect at a vertex plays the role of the total [[Gaussian curvature|SQVQ5DQT6W]] concentrated there.
It is equivalent to Euler's formula, given the angle count in the proof.
It shows that a convex polyhedron has at least one vertex with defect at least $4\pi/V$, and it constrains the possible vertex configurations of Platonic and Archimedean solids.
It also explains why the total of the cone angles is fixed in [[Alexandrov's theorem on polyhedra|ZCDWXVMDFN]].
</document>

<document source="https://tamnd.github.io/math2/node/PT6B8TPA8H.md" tag="PT6B8TPA8H" kind="definition">
# Fractional Brownian motion

## Definition

Let $H \in (0, 1)$.
A *fractional Brownian motion* with Hurst index $H$ is a real stochastic process $(B^H_t)_{t \ge 0}$ such that:

1. $(B^H_t)$ is a [[Gaussian process|22WEM6RTW0]] with $E[B^H_t] = 0$ for all $t \ge 0$.
2. Its [[covariance|0KBFAWYRTE]] is
$$E[B^H_s B^H_t] = \tfrac{1}{2}\bigl(s^{2H} + t^{2H} - \abs{t - s}^{2H}\bigr) \quad \text{for all } s, t \ge 0.$$
3. Almost surely, the path $t \mapsto B^H_t$ is continuous.

## Remarks

For $H = 1/2$ the covariance is $\frac{1}{2}(s + t - \abs{t - s}) = \min(s, t)$, so $B^{1/2}$ is a [[Wiener process|552AWX3QXN]].
Clause 2 gives $B^H_0 = 0$ and $E[(B^H_t - B^H_s)^2] = \abs{t - s}^{2H}$, so the increments are stationary and the process is [[self-similar|KS3KDRRJ58]] with index $H$.
The value $H = 1$ is excluded: the formula then gives covariance $st$, realised by the degenerate process $t \xi$ with $\xi$ standard normal, which has no randomness in its shape.
Disjoint increments are positively correlated for $H > 1/2$ and negatively correlated for $H < 1/2$.
Almost every path is Hölder continuous of every order less than $H$, and for $H \ne 1/2$ the process is not a [[semimartingale|MZNFPJJMY9]].
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<document source="https://tamnd.github.io/math2/node/PT8PN4CS1H.md" tag="PT8PN4CS1H" kind="definition">
# Quasi-coherent sheaf

## Definition

Let $A$ be a ring and $M$ an $A$-[[module|0NZ8JA25WK]].
For $f \in A$, let $D(f)$ be the basic open set of $\operatorname{Spec} A$ where $f$ does not vanish.
The sheaf $\widetilde{M}$ on $\operatorname{Spec} A$ is the unique sheaf of modules with $\widetilde{M}(D(f)) = M_f$ for every $f \in A$, with restriction maps given by localization.
Let $X$ be a [[scheme|5A35TYZD02]].
A [[sheaf of $\mathcal{O}_X$-modules|S33B2H3MYD]] $\mathcal{F}$ is *quasi-coherent* if $X$ has a cover by open sets $U$ with the following property.
Each $U$ is an [[affine scheme|B3V3XJ2W8G]] $\operatorname{Spec} A$, and $\mathcal{F}|_U \cong \widetilde{M}$ for some $A$-module $M$.

## Remarks

On $X = \operatorname{Spec} A$, the sheaf $\widetilde{I}$ of an ideal $I \subseteq A$ is quasi-coherent, with the cover by $X$ itself.
Let $R$ be a discrete valuation ring with fraction field $K$, and $X = \operatorname{Spec} R = \set{\eta, s}$, with $\eta$ the generic point.
The sheaf $\mathcal{F}$ with $\mathcal{F}(X) = 0$ and $\mathcal{F}(\set{\eta}) = K$ is an $\mathcal{O}_X$-module but is not quasi-coherent.
The only open set containing $s$ is $X$, so $\mathcal{F} \cong \widetilde{M}$ would force $M = \mathcal{F}(X) = 0$ and then $\mathcal{F}(\set{\eta}) = 0$.
If $\mathcal{F}$ is quasi-coherent, then $\mathcal{F}|_U \cong \widetilde{\mathcal{F}(U)}$ for every affine open $U$, not only for those of the chosen cover.
On a general ringed space, $\mathcal{F}$ is called quasi-coherent if it is locally the cokernel of a map between free modules; on schemes the two definitions agree.
</document>

<document source="https://tamnd.github.io/math2/node/PT9FSNP9T5.md" tag="PT9FSNP9T5" kind="definition">
# Killing vector field

## Definition

Let $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]].
Let $X$ be a smooth vector field on $M$, that is, a smooth section of the [[tangent bundle|WY6K3P4YCG]].
It is a *Killing vector field* if the [[Lie derivative|PJWK0PKN6Q]] of $g$ along $X$ vanishes:
$$\mathcal{L}_X g = 0.$$

## Remarks

Equivalently, with the [[Levi-Civita connection|CAZYGE1T4E]] $\nabla$, the field $X$ is Killing if and only if $g(\nabla_Y X, Z) + g(Y, \nabla_Z X) = 0$ for all vector fields $Y, Z$.
In coordinates this is Killing's equation $\nabla_i X_j + \nabla_j X_i = 0$.
Another equivalent form: the local flow of $X$ consists of isometries.

Example: on $\R^2$ with the Euclidean metric, the rotation field $X = -y \, \partial_x + x \, \partial_y$ has $\nabla X$ given by the constant matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, which is antisymmetric, so Killing's equation holds.

Non-example: on $\R^2$, the field $X = x \, \partial_x + y \, \partial_y$ has $\nabla X$ equal to the identity matrix, which is not antisymmetric; indeed $\mathcal{L}_X g = 2g \ne 0$.

The Killing fields of $(M, g)$ form a Lie algebra under the Lie bracket, of dimension at most $n(n + 1)/2$ on a connected manifold of dimension $n$.
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<document source="https://tamnd.github.io/math2/node/PT9V59MJZZ.md" tag="PT9V59MJZZ" kind="concept">
# Gauss–Kronrod quadrature formula

## Summary

A Gauss–Kronrod quadrature formula extends an $n$-point [[Gauss–Legendre rule|RGGXDRCJVE]] by $n + 1$ new nodes into a rule with $2n + 1$ nodes.
The old nodes are kept, and the new nodes and all weights are chosen to give the highest possible [[degree of exactness|ZYT1JNT1Y0]].
The difference between the two results is a cheap estimate of the error of the integral.
For this reason Gauss–Kronrod pairs are the core of many adaptive integration programs.

## Explanations

### L1 human

To know how good an approximate integral is, one usually computes it twice with two rules and compares.
Two unrelated Gauss rules share almost no sample points, so the second computation costs almost as much as starting over.
The Kronrod idea reuses every sample of the first rule and only adds a few new ones in between.
For example, the popular pair "G7–K15" uses a $7$-point Gauss rule and a $15$-point Kronrod rule that needs only $8$ extra function values.
</document>

<document source="https://tamnd.github.io/math2/node/PTK8XCQ0GK.md" tag="PTK8XCQ0GK" kind="definition">
# Bayesian hierarchical model

## Definition

A *Bayesian hierarchical model* (with two levels of prior) consists of the following.

1. A [[parametric model|Q0DXKAQXE1]] for the data: given a parameter $\theta \in \Theta$, the data $x$ have density $f(x \mid \theta)$.
2. A [[prior|00NBBSXRQS]] for $\theta$ that depends on a hyperparameter $\varphi \in \Phi$: given $\varphi$, the parameter $\theta$ has density $\pi(\theta \mid \varphi)$.
3. A hyperprior: a probability distribution on $\Phi$ with density $\pi(\varphi)$.

The joint density of $(x, \theta, \varphi)$ is $f(x \mid \theta)\,\pi(\theta \mid \varphi)\,\pi(\varphi)$, and the [[posterior|0KJ7ZS8AJA]] of $(\theta, \varphi)$ given $x$ is proportional to it, by [[Bayes' theorem|0NM6E9VW04]].

## Remarks

Models with more levels add further hyperpriors in the same way.
Example: data in $J$ groups with $X_{ij} \mid \theta_j \sim \mathcal{N}(\theta_j, \sigma^2)$, group means $\theta_j \mid \mu, \tau \sim \mathcal{N}(\mu, \tau^2)$ i.i.d., and a hyperprior on $(\mu, \tau)$.
All three clauses hold, with $\theta = (\theta_1, \dots, \theta_J)$ and $\varphi = (\mu, \tau)$.
Given $\mu$ and $\tau$, the posterior mean of each $\theta_j$ lies between the sample mean of group $j$ and the overall level $\mu$.
Non-example: the same model with $\mu$ and $\tau$ fixed numbers has no hyperprior, so clause 3 fails; it is an ordinary one-level Bayesian model.
Integrating out $\theta$ gives the marginal $\int f(x \mid \theta)\pi(\theta \mid \varphi)\,d\theta$; estimating $\varphi$ from it instead of using a hyperprior is the empirical Bayes approach.
</document>

<document source="https://tamnd.github.io/math2/node/PTKR703G6A.md" tag="PTKR703G6A" kind="concept">
# Clone

## Summary

A clone on a set $A$ is a set of finitary operations on $A$ that contains all [[projections|769B35T6WG]] and is closed under composition.
The term operations of any algebra with universe $A$ form a clone, and every clone arises in this way.
Clones let one compare algebras by what they can express, regardless of which basic operations were chosen.
Two algebras on the same set with the same clone of term operations have the same subalgebras, congruences and automorphisms.

## Explanations

### L1 human

Start with some operations on a set and allow yourself to plug operations into each other, repeat variables and ignore variables.
Everything you can build in this way forms a clone.
For example, on $\set{0, 1}$ the operations $\wedge$ and $\neg$ generate a clone that contains $x \vee y = \neg(\neg x \wedge \neg y)$.
In fact this clone contains every operation on $\set{0, 1}$.
A clone records the expressive power of a family of operations, not the particular family.
</document>

<document source="https://tamnd.github.io/math2/node/PTRKN3DFAA.md" tag="PTRKN3DFAA" kind="definition">
# Bifurcation theory

## Definition

Let $\Lambda \subseteq \R^k$ be an open set of parameters.
For each $\mu \in \Lambda$, let $\dot x = F(x, \mu)$ be an [[ordinary differential equation|0NT64JV96A]] on an open set of $\R^n$, with $F$ smooth in $(x, \mu)$.
Two such systems are *topologically equivalent* if a homeomorphism maps the orbits of one onto the orbits of the other.
The homeomorphism must preserve the direction of time.
A parameter value $\mu_0 \in \Lambda$ is a *bifurcation value* if the following holds.
Every neighbourhood of $\mu_0$ contains some $\mu$ whose system is not topologically equivalent to the system at $\mu_0$.
*Bifurcation theory* studies the bifurcation values of such families and the changes of the [[phase portrait|0T3GK0XCBC]] that occur at them.

## Remarks

The same definitions apply to families of maps $x \mapsto f_\mu(x)$, with topological conjugacy in place of equivalence.
Example: for $\dot x = \mu - x^2$ on $\R$, the value $\mu_0 = 0$ is a bifurcation value.
For $\mu < 0$ there is no [[equilibrium|0N8NGKDZDZ]], for $\mu = 0$ there is one, and for $\mu > 0$ there are two, at $\pm\sqrt{\mu}$.
This is a [[saddle-node bifurcation|MAP5J6HK0T]].
Non-example: $\mu_0 = 1$ is not a bifurcation value of this family.
For all $\mu$ near $1$ there are two equilibria, $\sqrt{\mu}$ attracting and $-\sqrt{\mu}$ repelling, and the phase portraits are equivalent.
A system whose topological type is unchanged by every small perturbation of the vector field, not only by changes of $\mu$, is [[structurally stable|D4MGFAHVMT]].
Local bifurcations concern a neighbourhood of an equilibrium or periodic orbit; global ones, such as homoclinic bifurcations, involve whole orbits.
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<document source="https://tamnd.github.io/math2/node/PV0XXSWFC7.md" tag="PV0XXSWFC7" kind="concept">
# Premeasure

## Summary

A premeasure is a countably additive size function defined only on a [[ring of sets|KWE1DBMN9B]].
Such a ring is closed under finite unions and differences, but not under countable unions.
It is the raw material for building a [[measure|PR11BZQQPG]].
By [[Carathéodory's extension theorem|Q8EXG7EKYW]] every premeasure extends to a measure on the generated $\sigma$-algebra.

## Explanations

### L1 human

Often one knows the size of simple sets, such as finite unions of intervals, before one knows the size of complicated sets.
A premeasure records these sizes and asks that they add up correctly, even when a simple set is cut into countably many simple pieces.
For example, length on finite unions of half-open intervals $(a, b]$ is a premeasure.
If $(0, 1]$ is cut into the pieces $(1/2, 1]$, $(1/4, 1/2]$, $(1/8, 1/4]$, and so on, the lengths $1/2 + 1/4 + 1/8 + \cdots$ add up to $1$.
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<document source="https://tamnd.github.io/math2/node/PV187HG1K6.md" tag="PV187HG1K6" kind="definition">
# L-estimator

## Definition

Let $X_1, \dots, X_n$ be real random variables, and let $X_{(1)} \le X_{(2)} \le \dots \le X_{(n)}$ be their [[order statistics|58NHGZDSRE]].
An *L-estimator* is an [[estimator|0TTNRP5PFQ]] of the form
$$T_n = \sum_{i=1}^{n} c_{n,i} \, X_{(i)},$$
where the coefficients $c_{n,1}, \dots, c_{n,n} \in \R$ are fixed numbers that do not depend on the data.

## Remarks

Common examples:

1. the sample mean, with $c_{n,i} = 1/n$;
2. the sample median, with $c_{n,(n+1)/2} = 1$ for odd $n$, and $c_{n,n/2} = c_{n,n/2+1} = 1/2$ for even $n$;
3. the $k$-trimmed mean, with $c_{n,i} = 1/(n - 2k)$ for $k < i \le n - k$ and $c_{n,i} = 0$ otherwise;
4. the midrange $(X_{(1)} + X_{(n)})/2$ and the range $X_{(n)} - X_{(1)}$.

Example: data $(4, 1, 100, 2, 3)$, sorted $(1, 2, 3, 4, 100)$.
The $1$-trimmed mean drops $1$ and $100$ and gives $(2 + 3 + 4)/3 = 3$, while the mean is $22$.

Non-example: the sample variance $\frac{1}{n-1}\sum_i (X_i - \bar X)^2$ is a quadratic function of the order statistics, not a linear one, so it is not an L-estimator.

Often the weights come from a function $J$ on $[0, 1]$ through $c_{n,i} = \frac1n J\big(\frac{i}{n+1}\big)$; then $T_n$ estimates $\int_0^1 J(u) F^{-1}(u) \, du$, where $F^{-1}$ is the quantile function.
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<document source="https://tamnd.github.io/math2/node/PV2DGD23AY.md" tag="PV2DGD23AY" kind="definition">
# Spectral theory of dynamical systems

## Definition

Let $(X, \mathcal{B}, \mu)$ and $(Y, \mathcal{C}, \nu)$ be [[probability spaces|0NFPJVJJHX]], and let $T \colon X \to X$ and $S \colon Y \to Y$ be invertible [[measure-preserving transformations|556WBDAHNP]] with measurable measure-preserving inverses.
Let $U_T$ and $U_S$ be their [[Koopman operators|37TRG5DBQC]] on $L^2(\mu)$ and $L^2(\nu)$.

1. $T$ and $S$ are *spectrally isomorphic* if there is a [[unitary operator|MM0EJ2GDE4]] $W \colon L^2(\mu) \to L^2(\nu)$ with $W U_T = U_S W$.
2. A *spectral invariant* is a property of such transformations that holds for $S$ whenever it holds for $T$ and $T$, $S$ are spectrally isomorphic.
3. $T$ has *discrete spectrum* if $L^2(\mu)$ has an orthonormal basis of [[eigenvectors|0BQAA2BK3Q]] of $U_T$.

The spectral theory of dynamical systems is the study of transformations up to spectral isomorphism, through the spectrum and the spectral measures of $U_T$.

## Remarks

Isomorphic transformations are spectrally isomorphic: an isomorphism $\varphi$ with $\varphi \circ T = S \circ \varphi$ gives $W f = f \circ \varphi^{-1}$.

Example of a spectral invariant: ergodicity, since $T$ is ergodic if and only if the eigenvalue $1$ of $U_T$ has multiplicity $1$.
Weak mixing is another: it holds if and only if $1$ is a simple eigenvalue and $U_T$ has no other eigenvalue.
An irrational rotation $x \mapsto x + \alpha \bmod 1$ has discrete spectrum, with basis $e^{2\pi i k x}$ and eigenvalues $e^{2\pi i k \alpha}$, $k \in \Z$.

Non-example: the [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] is not a spectral invariant.
Two Bernoulli shifts with different entropies, such as the $(1/2, 1/2)$ and the $(1/3, 1/3, 1/3)$ shifts, are spectrally isomorphic.
Both Koopman operators have countable Lebesgue spectrum on the orthogonal complement of the constants.
</document>

<document source="https://tamnd.github.io/math2/node/PV82BVXE1A.md" tag="PV82BVXE1A" kind="conjecture">
# Lindelöf hypothesis

## Statement

Let $\zeta$ be the [[Riemann zeta function|1XWBY4N0VT]].
For every $\varepsilon > 0$ there is a constant $C_\varepsilon$ such that
$$\abs{\zeta(\tfrac12 + it)} \le C_\varepsilon \abs{t}^{\varepsilon}$$
for all real $t$ with $\abs{t} \ge 1$.

## Remarks

The conjecture is open.
The [[Riemann hypothesis|VCGF9YR36X]] implies it.
The functional equation and the Phragmén–Lindelöf principle give the convexity bound $\zeta(\tfrac12 + it) = O(\abs{t}^{1/4 + \varepsilon})$.
Every improvement of the exponent $1/4$ is called a subconvexity bound; Hardy and Littlewood obtained $1/6 + \varepsilon$ with Weyl's method, and Bourgain obtained $13/84 + \varepsilon$.
An equivalent form: for every integer $k \ge 1$ and every $\varepsilon > 0$,
$$\frac{1}{T} \int_{0}^{T} \abs{\zeta(\tfrac12 + it)}^{2k} \, dt = O(T^{\varepsilon}).$$

## Explanations

### L1 human

On the critical line $\operatorname{Re} s = 1/2$, the zeta function oscillates and its size grows slowly with the height $t$.
The Lindelöf hypothesis says that it grows more slowly than any power of $t$, however small.
It is weaker than the Riemann hypothesis but still out of reach.
It would imply strong bounds on the moments of zeta and on the number of zeros off the critical line.
</document>

<document source="https://tamnd.github.io/math2/node/PV8DQ41Q55.md" tag="PV8DQ41Q55" kind="definition">
# Partial differential equation

## Definition

Let $\Omega$ be an [[open set|02BH3MCBQB]] in $\R^n$ with $n \ge 2$, and let $k \ge 1$ be an integer.
For a function $u \colon \Omega \to \R$ of class $C^k$, write $D^j u(x)$ for the collection of all [[partial derivatives|09EGC6F8PT]] $\partial^\alpha u(x)$ of order $\abs{\alpha} = j$, in [[multi-index notation|0PJXBR6QA3]].
Let $F$ be a given function of these derivatives and of $x$.
A *partial differential equation* of order at most $k$ in $\Omega$ is an equation of the following form.
$$F\left(D^k u(x), D^{k-1} u(x), \dots, D u(x), u(x), x\right) = 0 \quad \text{for } x \in \Omega.$$
The function $u$ is the unknown.
A *classical solution* is a function $u \in C^k(\Omega)$ that satisfies the equation at every point of $\Omega$.
A *system* of partial differential equations is defined in the same way with $u$ and $F$ taking values in $\R^N$ and $\R^M$.

## Remarks

Example: with $n = 2$ and variables $(x, y)$, the equation $\partial_x^2 u + \partial_y^2 u = 0$ is a partial differential equation.
The function $u(x, y) = x^2 - y^2$ is a classical solution on $\R^2$, since $\partial_x^2 u = 2$ and $\partial_y^2 u = -2$.
Non-example: $u'(x) = u(x)$ for $u \colon \R \to \R$ involves derivatives with respect to one variable only, so it is an [[ordinary differential equation|0NT64JV96A]].
Complex-valued unknowns are treated in the same way.
Many PDEs also have solutions in a weaker sense, such as [[weak solutions|SJZX81A783]], which need not be of class $C^k$.

## Explanations

### L1 human

A partial differential equation is a condition on a function of several variables and its partial derivatives, which must hold at every point.
The unknown is a whole function, not a number.
</document>

<document source="https://tamnd.github.io/math2/node/PV90EW00XK.md" tag="PV90EW00XK" kind="concept">
# Sublattice

## Summary

A sublattice of a [[lattice|WH4SFAACAS]] $L$ is a nonempty subset of $L$ that is closed under the [[join and meet|J2T75ATRV5]] of $L$.
It is a lattice in its own right, with the operations of $L$.
Sublattices play the role that subgroups play for groups, and forbidden sublattices characterise classes of lattices, such as the distributive ones.

## Explanations

### L1 human

Take a lattice and keep only some of its elements.
The kept part is a sublattice when the join and the meet of any two kept elements, computed in the big lattice, are again kept.
For example, in the subsets of $\set{1, 2, 3}$, the four sets $\emptyset$, $\set{1}$, $\set{2}$, $\set{1, 2}$ form a sublattice.
A subset can be a lattice in its own order without being a sublattice, because its joins may differ from those of the big lattice.
</document>

<document source="https://tamnd.github.io/math2/node/PVA1QJVS2E.md" tag="PVA1QJVS2E" kind="statement">
# Grauert's solution of the Levi problem for manifolds

## Statement

Let $X$ be a [[complex manifold|JSGWAN2RJV]] that is second countable.
Then $X$ is a [[Stein manifold|HMCD0JBFST]] if and only if $X$ has a smooth strictly [[plurisubharmonic exhaustion function|FGSKR6ZS3Q]].

## Remarks

The direction from Stein to the exhaustion function is easy.
A Stein manifold has a proper holomorphic embedding $F \colon X \to \C^N$, and then $\varphi = \abs{F}^2$ is smooth, strictly plurisubharmonic, and an exhaustion because $F$ is proper.
Grauert proved the converse in 1958.
His proof enlarges the sublevel sets $\set{\varphi < c}$ step by step and uses a finiteness theorem for cohomology on strongly pseudoconvex domains.
A later proof by Hörmander uses $L^2$ estimates for $\bar\partial$.
Example: the unit ball $B \subseteq \C^n$ is Stein, since $-\log(1 - \abs{z}^2)$ is a smooth strictly plurisubharmonic exhaustion function.
Strictness is needed: on a compact complex manifold $X$ of positive dimension, every constant is a smooth plurisubharmonic exhaustion function, yet $X$ is not Stein.

## Explanations

### L1 human

A Stein manifold is a complex manifold with very many global holomorphic functions.
The theorem says that this is the same as having a function that behaves like $\abs{z}^2$ on $\C^n$.
Such a function grows to infinity at the ends of the manifold and is strictly convex in the complex sense.
So a question about the existence of holomorphic functions is reduced to the existence of one real function with a convexity property.

### L2 human

For domains in $\C^n$ the theorem gives back [[Oka's solution of the Levi problem|6BJB149MDK]], after one smooths a continuous plurisubharmonic exhaustion and adds $\abs{z}^2$ to make it strict.
The function $\varphi$ is a Morse function after a small perturbation, and the strict plurisubharmonicity bounds the indices of its critical points by $\dim_\C X$.
This gives the result of Andreotti and Frankel that a Stein manifold of complex dimension $n$ has the homotopy type of a CW complex of real dimension at most $n$.
The Andreotti–Grauert theorem generalizes the theorem to $q$-convex manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/PVD7VZJZ9B.md" tag="PVD7VZJZ9B" kind="concept">
# Transport equation

## Summary

The transport equation $u_t + b \cdot \nabla u = 0$ says that a quantity $u$ is carried along by a flow of velocity $b$ without changing.
It is the simplest first-order [[partial differential equation|GC5BDZ6QBC]], also called the advection equation.
Its solutions are the initial profile moved along the flow.
It is the model problem for the [[method of characteristics|7XAPBCJK3B]] and for numerical schemes of hyperbolic equations.
See the [[definition|R3X5WFHV8P]].

## Explanations

### L1 human

Think of dye in a river that flows at constant speed.
If the dye does not spread out, the picture of the dye at a later time is the first picture shifted downstream.
The transport equation is the formula for this.
In one dimension, with speed $c$, it reads $u_t + c\, u_x = 0$, and its solutions are $u(x, t) = g(x - ct)$.
For example, a bump centred at $x = 0$ at time $0$ is centred at $x = 3c$ at time $3$.
</document>

<document source="https://tamnd.github.io/math2/node/PVR5EX71PD.md" tag="PVR5EX71PD" kind="example">
# Doubling on the integers is not an isomorphism

## Example

Let $G = (\mathbb{Z}, +)$ be the [[integers|0C4JS9RF5M]] with addition. Define $d : G \to G$ by $d(u) = 2u$.
Then $d(u + v) = 2u + 2v = d(u) + d(v)$, so $d$ is a homomorphism.
If $2u = 0$, then $u = 0$. Thus the kernel is $\{0\}$, and $d$ is [[injective|0PVG1CRAJN]].
The image of $d$ is the set of even integers. Thus $1$ is not in the image, and $d$ is not [[surjective|0XKH881BZ7]].
Thus $d$ is not an [[isomorphism|085PND4KNT]].

## Remarks

The image $2\mathbb{Z}$ is isomorphic to $\mathbb{Z}$. Thus $d$ is an isomorphism from $\mathbb{Z}$ onto $2\mathbb{Z}$, but not onto $\mathbb{Z}$.
</document>

<document source="https://tamnd.github.io/math2/node/PVTA9RQKWM.md" tag="PVTA9RQKWM" kind="concept">
# Faber polynomials

## Summary

The Faber polynomials of a compact set $K$ in the plane are polynomials $F_0, F_1, F_2, \dots$, with $F_n$ of degree $n$.
They are built from the [[conformal map|0VHNKP1N3K]] of the complement of $K$ onto the exterior of the unit disc.
They play for $K$ the role that the powers $z^n$ play for a disc.
Functions holomorphic on a neighbourhood of $K$ can be expanded in Faber series $\sum_n b_n F_n(z)$.
They are a standard tool of polynomial approximation and of geometric function theory.

## Explanations

### L1 human

On a disc centred at $0$, the natural building blocks for holomorphic functions are the powers $z^n$.
For another shape, such as an ellipse or a segment, one first straightens its outside into the outside of a circle by a conformal map $\Phi$.
The $n$-th Faber polynomial is the polynomial part of $\Phi(z)^n$ near infinity.
For the closed unit disc, $\Phi(z) = z$ and $F_n(z) = z^n$.
For the segment $[-1, 1]$ one gets twice the Chebyshev polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/PVZCDPKF31.md" tag="PVZCDPKF31" kind="definition">
# Matroid

## Definition

Let $E$ be a [[finite set|070R982G4B]], called the ground set.
A *matroid* on $E$ is a pair $M = (E, \mathcal{I})$, where $\mathcal{I}$ is a family of subsets of $E$, called independent sets, such that:

1. $\emptyset \in \mathcal{I}$.
2. If $A \in \mathcal{I}$ and $B \subseteq A$, then $B \in \mathcal{I}$.
3. If $A, B \in \mathcal{I}$ and $\abs{A} > \abs{B}$, then there is $x \in A \setminus B$ with $B \cup \set{x} \in \mathcal{I}$.

A subset of $E$ that is not in $\mathcal{I}$ is *dependent*.

## Remarks

Example: let $E$ be the set of columns of a matrix over a field, and let $\mathcal{I}$ be the sets of columns that are [[linearly independent|0CRMGPAH83]].
Clauses 1 and 2 are clear.
For clause 3, if $\abs{A} > \abs{B}$, then $A$ spans a space of larger dimension than $B$.
So some $x \in A$ lies outside the span of $B$, and $B \cup \set{x}$ is independent.
Similarly, the edge sets of the forests of a graph form a matroid, the graphic matroid.
The uniform matroid $U_{k,n}$ has ground set $\set{1, \dots, n}$ and independent sets all subsets with at most $k$ elements.

Non-example: on $E = \set{1, 2, 3}$, let $\mathcal{I} = \set{\emptyset, \set{1}, \set{2}, \set{3}, \set{1, 2}}$.
Clauses 1 and 2 hold, but clause 3 fails for $A = \set{1, 2}$ and $B = \set{3}$: neither $\set{1, 3}$ nor $\set{2, 3}$ is independent.

Clause 3 is called the augmentation or exchange axiom.
Equivalent definitions use the [[bases|QTSQR7130G]], the [[rank function|SYRCTH853H]] or the circuits (minimal dependent sets).
Some authors allow an infinite ground set with extra conditions.
</document>

<document source="https://tamnd.github.io/math2/node/PW2NKAPXWA.md" tag="PW2NKAPXWA" kind="concept">
# Weak ordering

## Summary

A weak ordering ranks the elements of a set where ties are allowed.
In its strict form it is a [[strict partial order|SWB1ZXFVKY]] in which incomparability is transitive.
Then the elements fall into classes of tied elements, and these classes are totally ordered.
Weak orderings model rankings with ties, such as the results of a race with shared places.

## Explanations

### L1 human

Think of the final ranking of a competition where several people can share a place.
Each person is strictly ahead of, strictly behind, or tied with each other person.
If Ann and Bob are tied and Bob and Cem are tied, then Ann and Cem are tied too.
For example, sorting people by age in whole years gives a weak ordering: "is younger than" in whole years, with people of the same age tied.
</document>

<document source="https://tamnd.github.io/math2/node/PW2TVRD47Y.md" tag="PW2TVRD47Y" kind="definition">
# Small cancellation theory

## Definition

Let $F(S)$ be the [[free group|0KTW26C80J]] on a set $S$, with elements written as freely reduced words.
Let $\abs{w}$ be the length of a word $w$.
A set $R$ of cyclically reduced words in $F(S)$ is *symmetrized* if it is closed under taking inverses and cyclic permutations.
Let $R$ be symmetrized.
A *piece* relative to $R$ is a nonempty word $u$ such that two distinct elements $r_1 \ne r_2$ of $R$ both begin with $u$.
That is, $r_1 = u v_1$ and $r_2 = u v_2$ with no cancellation.

1. Let $\lambda$ be a real number with $0 < \lambda < 1$. Then $R$ satisfies the *condition $C'(\lambda)$* if $\abs{u} < \lambda \abs{r}$ whenever $r \in R$ and $r = u v$ with $u$ a piece.
2. For an integer $p \ge 2$, $R$ satisfies the *condition $C(p)$* if no element of $R$ is a product of fewer than $p$ pieces.

A [[presentation|0JGEQXQ4AC]] $\langle S \mid R_0 \rangle$ is a *small cancellation presentation* of type $C'(\lambda)$ or $C(p)$ if the symmetrized closure of $R_0$ satisfies the condition.
*Small cancellation theory* is the study of groups with such presentations.

## Remarks

The condition $C'(\lambda)$ implies $C(p)$ for $p$ with $\lambda \le 1/(p-1)$; in particular $C'(1/6)$ implies $C(7)$.
Greendlinger's lemma: if $R$ satisfies $C'(1/6)$ and a nontrivial reduced word $w$ equals $1$ in $\langle S \mid R \rangle$, then $w$ contains a subword of some $r \in R$ of length greater than $\abs{r}/2$.

Example: let $S = \set{a, b, c, d}$ and $r = a b a^{-1} b^{-1} c d c^{-1} d^{-1}$.
In $r$ and its cyclic permutations each two-letter subword occurs once, and no two-letter subword of $r^{-1}$ or of its cyclic permutations occurs in $r$.
So all pieces have length $1$, and $1 < \frac{1}{6} \cdot 8$, so the presentation satisfies $C'(1/6)$.

Non-example: $\langle a, b \mid a b a^{-1} b^{-1} \rangle$, a presentation of $\Z^2$, does not satisfy $C'(1/6)$.
Each letter is a piece, but the relator has length $4$ and $1 \ge \frac{1}{6} \cdot 4$.
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<document source="https://tamnd.github.io/math2/node/PW313WB7HE.md" tag="PW313WB7HE" kind="definition">
# Wilf–Zeilberger pair

## Definition

A function $F(n, k)$ of two integer variables is a *hypergeometric term* if the ratios $F(n + 1, k)/F(n, k)$ and $F(n, k + 1)/F(n, k)$ are [[rational functions|0XDK9CGYKJ]] of $n$ and $k$.
A *Wilf–Zeilberger pair* (WZ pair) is a pair of hypergeometric terms $F$ and $G$ such that

1. $F(n + 1, k) - F(n, k) = G(n, k + 1) - G(n, k)$ for all integers $n, k$;
2. $G(n, k) = R(n, k)\,F(n, k)$ for a rational function $R$, called the *WZ certificate*.

## Remarks

Consequence: suppose that for each $n$ the sum $\sum_k F(n, k)$ converges and $G(n, k) \to 0$ as $k \to \pm\infty$.
Then summing clause 1 over $k$ gives a [[telescoping sum|0GXKM1R7E4]] on the right.
So $\sum_k F(n + 1, k) = \sum_k F(n, k)$, and the sum is constant in $n$.

Example: let $F(n, k) = \binom{n}{k}/2^n$ and $G(n, k) = -\binom{n}{k - 1}/2^{n + 1}$, with [[binomial coefficients|0GBZM0MT0F]] that vanish outside $0 \le k \le n$.
Pascal's rule gives $F(n + 1, k) - F(n, k) = \big(\binom{n}{k - 1} - \binom{n}{k}\big)/2^{n + 1}$, which is $G(n, k + 1) - G(n, k)$.
The certificate is $R(n, k) = -k/(2(n - k + 1))$.
So $\sum_k \binom{n}{k}/2^n$ is constant, and its value $1$ at $n = 0$ proves $\sum_k \binom{n}{k} = 2^n$.

Non-example: the same $F$ with $G = 0$ is not a WZ pair, since clause 1 would need $F(n + 1, k) = F(n, k)$, which fails for $n = 0$, $k = 0$: $F(1, 0) = 1/2$ but $F(0, 0) = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/PW5SKE8D5R.md" tag="PW5SKE8D5R" kind="example">
# Sawtooth wave

## Example

Let $f$ be the $2\pi$-periodic function with $f(x) = \frac{\pi - x}{2}$ for $0 < x < 2\pi$ and $f(0) = 0$.
Its graph is a sawtooth: it falls linearly from $\frac{\pi}{2}$ to $-\frac{\pi}{2}$ on each period and jumps back up at the multiples of $2\pi$.
It is odd, so its cosine coefficients vanish, and integration by parts gives for $n \ge 1$
$$b_n = \frac{1}{\pi} \int_0^{2\pi} \frac{\pi - x}{2} \sin nx \, dx = \frac{1}{\pi} \cdot \frac{1}{2} \cdot \frac{2\pi}{n} = \frac{1}{n}.$$
So its [[Fourier series|G12KXDHPPS]] is
$$\sum_{n=1}^{\infty} \frac{\sin nx}{n}.$$
The function has bounded variation, so by the [[Dirichlet conditions|75ZQW12WY0]] the series converges to $\frac{\pi - x}{2}$ for $0 < x < 2\pi$, and to $0$ at the [[jumps|004FH91BN5]] $x \in 2\pi\Z$.

## Remarks

At $x = \pi/2$ the series gives $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \cdots$, the Leibniz formula.
The coefficients $\frac{1}{n}$ are square summable but not summable, as expected for a function with a jump.
Integrating the series term by term from $0$ to $x$ gives $\sum_{n \ge 1} \frac{1 - \cos nx}{n^2} = \frac{\pi x}{2} - \frac{x^2}{4}$ for $0 \le x \le 2\pi$, and at $x = \pi$ this yields $\sum_{n \text{ odd}} \frac{2}{n^2} = \frac{\pi^2}{4}$.

## Explanations

### L1 human

The sawtooth wave ramps down steadily and then snaps back, like the sweep of an old television beam.
It contains every harmonic, the $n$-th one with amplitude $\frac{1}{n}$.
Because of the snap back, the amplitudes shrink slowly, and many harmonics are needed for a good picture.
</document>

<document source="https://tamnd.github.io/math2/node/PW8XX59WNH.md" tag="PW8XX59WNH" kind="definition">
# Laplace transform of distributions

## Definition

Let $T$ be a distribution on $\R$ with support in $[0, \infty)$.
Let $\sigma_0 \in \R$ be such that $e^{-\sigma_0 t} T$ is a [[tempered distribution|AES2T5JPFP]].
Fix a smooth function $\chi \colon \R \to [0, 1]$ with $\chi = 1$ on $[-1/2, \infty)$ and $\chi = 0$ on $(-\infty, -1]$.
The *Laplace transform* of $T$ is the function defined for $\operatorname{Re} s > \sigma_0$ by
$$(\mathcal{L} T)(s) = \left\langle e^{-\sigma_0 t} T, \ \chi(t) \, e^{-(s - \sigma_0) t} \right\rangle.$$

## Remarks

For $\operatorname{Re} s > \sigma_0$, the function $\chi(t) e^{-(s - \sigma_0) t}$ lies in the [[Schwartz space|CG9JEATW3Q]], so the pairing makes sense.
The value does not depend on the choice of $\chi$ or of $\sigma_0$, because $T$ is supported in $[0, \infty)$; formally it is $\langle T, e^{-st} \rangle$.
$\mathcal{L} T$ is holomorphic on $\operatorname{Re} s > \sigma_0$.
For a locally integrable function $f$ of exponential order vanishing on $(-\infty, 0)$, it agrees with the ordinary [[Laplace transform|0V49JEZW06]] $\int_0^\infty e^{-st} f(t) \, dt$.
For all such $T$, $\mathcal{L}(T') = s \, \mathcal{L} T$, and the transform of a convolution is the product of the transforms.

Example: the [[Dirac delta|0KPHENZ5S7]] $\delta$ has support $\set{0}$ and is tempered, so $\sigma_0 = 0$ works, and $\mathcal{L}\delta(s) = \chi(0) e^{0} = 1$.
Likewise $\mathcal{L}\delta'(s) = -\frac{d}{dt}\left(\chi(t) e^{-st}\right)\big|_{t=0} = s$.
With the [[Heaviside step function|09C4JP7Y0M]] $H$, $\mathcal{L} H = 1/s$ and $H' = \delta$, consistent with $\mathcal{L}(H') = s \cdot \frac{1}{s} = 1$.

Non-example: $T = H(t) e^{t^2}$ is a distribution supported in $[0, \infty)$, but the positive function $e^{-\sigma_0 t} e^{t^2}$ grows faster than any polynomial. Hence $e^{-\sigma_0 t} T$ is tempered for no $\sigma_0$, and $T$ has no Laplace transform.
</document>

<document source="https://tamnd.github.io/math2/node/PW9D9QKAD4.md" tag="PW9D9QKAD4" kind="concept">
# Unknotting number

## Summary

The unknotting number of a [[knot|QYEAZTYGPT]] is the least number of crossing changes needed to turn it into the [[unknot|1TQ06R4Q66]].
It is a [[knot invariant|GZQ5ZE3X6G]] that measures how far a knot is from being trivial.
It is easy to define but often hard to compute, and it is still unknown for some knots with only ten crossings.

## Explanations

### L1 human

Think of a knot as a tangled loop of rope, drawn on paper with over- and under-crossings.
A crossing change lets one strand pass through the other at one crossing, so the over-strand becomes the under-strand.
The unknotting number counts how many such passes are needed, at best, to get a plain loop.
You may move the rope around freely between the passes.
The [[trefoil knot|VYTYSVHN4A]] has unknotting number $1$: one crossing change in its usual picture undoes it.
</document>

<document source="https://tamnd.github.io/math2/node/PWDQ80EKZ8.md" tag="PWDQ80EKZ8" kind="definition">
# Antipode

## Definition

Let $K$ be a field and let $(H, m, \eta, \Delta, \varepsilon)$ be a [[bialgebra|Q9HHVN606G]] over $K$.
An *antipode* for $H$ is a [[linear map|0NRYHJPQG2]] $S \colon H \to H$ such that
$$m \circ (S \otimes \id_H) \circ \Delta = \eta \circ \varepsilon = m \circ (\id_H \otimes S) \circ \Delta.$$

## Remarks

The *convolution* of linear maps $f, g \colon H \to H$ is $f * g = m \circ (f \otimes g) \circ \Delta$.
It makes $\End_K(H)$ an associative algebra with identity $\eta \circ \varepsilon$, and the condition says $S * \id_H = \eta \circ \varepsilon = \id_H * S$.
So $S$ is a two-sided inverse of $\id_H$, and an antipode is unique when it exists.
Example: $H = K[x]$ with $\Delta(x) = x \otimes 1 + 1 \otimes x$ and $\varepsilon(x) = 0$ has the antipode $S(x^n) = (-x)^n$.
On $x$ the check is $S(x) \cdot 1 + S(1) \cdot x = -x + x = 0 = \varepsilon(x) 1$.
Non-example: the identity map of $K[x]$ is not an antipode, since $m(\id \otimes \id)\Delta(x) = 2x \ne 0$ when the characteristic of $K$ is not $2$.
An antipode is an algebra anti-homomorphism, $S(ab) = S(b)S(a)$, and a coalgebra anti-homomorphism.
If $H$ is commutative or cocommutative, then $S \circ S = \id_H$.
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<document source="https://tamnd.github.io/math2/node/PWG3GKZG80.md" tag="PWG3GKZG80" kind="concept">
# Hodge Laplacian

## Summary

The Hodge Laplacian is the operator $\Delta = d\delta + \delta d$ on [[differential forms|0E3G0GRPZ6]] of a Riemannian manifold, built from the [[exterior derivative|0ZHSTAG8ZY]] $d$ and its adjoint $\delta$.
On functions it is the [[Laplace–Beltrami operator|F4NWZNTRMD]], up to sign convention.
It is elliptic, self-adjoint and nonnegative, and on a closed manifold its kernel, the harmonic forms, represents the de Rham cohomology.
This is the content of [[Hodge theory|32YJXG30MF]], which links analysis on a manifold with its topology.

## Explanations

### L1 human

The ordinary Laplacian measures how much a function differs from its average nearby.
The Hodge Laplacian does the same for differential forms, such as vector fields written as $1$-forms.
On flat $\R^n$ it simply applies the negative of the usual Laplacian $-\sum_j \partial_j^2$ to each coefficient of the form.
On a curved manifold extra curvature terms appear, which is why it carries geometric information.
A form killed by it is called harmonic, and on a closed manifold each cohomology class contains exactly one harmonic form.
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<document source="https://tamnd.github.io/math2/node/PWHJZYBNEY.md" tag="PWHJZYBNEY" kind="definition">
# Signature

## Definition

A *signature* $\sigma$ consists of:

1. a set $F$ of function symbols;
2. a set $R$ of relation symbols, disjoint from $F$;
3. an arity [[function|0EPAT51DXQ]] $\operatorname{ar} \colon F \cup R \to \N$ that gives the number of arguments of each symbol.

A function symbol of arity $0$ is a *constant symbol*.
The symbols of $\sigma$ are the non-logical symbols of a first-order language; the variables, connectives, quantifiers and equality are not part of $\sigma$.

## Remarks

The signature of rings is $\set{+, \cdot, -, 0, 1}$ with arities $2, 2, 1, 0, 0$; it has no relation symbols.
The signature of ordered sets has one relation symbol $<$ of arity $2$ and no function symbols.
The list "a binary operation $\cdot$ that is associative" is not a signature: associativity is an axiom, and a signature contains only symbols and their arities.
Some books list the constant symbols as a third set instead of treating them as functions of arity $0$.
In Mathlib a signature is `FirstOrder.Language`.
</document>

<document source="https://tamnd.github.io/math2/node/PWHVYMN5T3.md" tag="PWHVYMN5T3" kind="definition">
# Grothendieck universe

## Definition

A *Grothendieck universe* is a set $U$ such that:

1. If $x \in U$ and $y \in x$, then $y \in U$.
2. If $x, y \in U$, then $\set{x, y} \in U$.
3. If $x \in U$, then the [[power set|00E0YZCERH]] $\mathcal{P}(x)$ is in $U$.
4. If $I \in U$ and $(x_i)_{i \in I}$ is a family with $x_i \in U$ for each $i \in I$, then $\bigcup_{i \in I} x_i \in U$.

## Remarks

The set $V_\omega$ of hereditarily finite sets is a Grothendieck universe.
Each clause holds because elements, pairs, power sets and finite unions of hereditarily finite sets are again hereditarily finite, and an index set $I \in V_\omega$ is finite.
The set $V_{\omega + \omega}$ is a non-example: it contains $\omega$ and each ordinal $\omega + n$.
The union of the family $(\omega + n)_{n \in \omega}$ is $\omega + \omega$, which is not in $V_{\omega + \omega}$, so clause 4 fails.
In ZFC the Grothendieck universes are exactly the empty set, $V_\omega$, and the sets $V_\kappa$ for [[inaccessible cardinals|NW4DREQ5DQ]] $\kappa$.
Many authors add the clause $\omega \in U$, which leaves only the universes $V_\kappa$ with $\kappa$ inaccessible.
The axiom of universes states that every set is an element of some Grothendieck universe; it is equivalent to the existence of arbitrarily large inaccessible cardinals.
</document>

<document source="https://tamnd.github.io/math2/node/PWKN6ANA0G.md" tag="PWKN6ANA0G" kind="statement">
# Levy–Desplanques theorem

## Statement

Let $A = (a_{ij})$ be an $n \times n$ complex matrix that is strictly [[diagonally dominant|89P5GTKW87]] by rows:
$$\abs{a_{ii}} > \sum_{j \ne i} \abs{a_{ij}} \quad \text{for } i = 1, \dots, n.$$
Then $A$ is [[invertible|0EB9N0S33C]], that is, $\det A \ne 0$.

## Remarks

Suppose $Ax = 0$ with $x \ne 0$, and choose $i$ with $\abs{x_i}$ maximal, so $\abs{x_i} > 0$.
Row $i$ gives $a_{ii} x_i = -\sum_{j \ne i} a_{ij} x_j$, so $\abs{a_{ii}} \abs{x_i} \le \sum_{j \ne i} \abs{a_{ij}} \abs{x_i}$.
This contradicts strict dominance in row $i$.
Equivalently, $0$ lies in no disc of the [[Gershgorin circle theorem|0ETGP0XVCJ]].

Example: $\begin{pmatrix} 3 & 1 & 1 \\ 0 & 2 & 1 \\ 1 & -1 & 4 \end{pmatrix}$ is strictly diagonally dominant, so it is invertible without any computation.

Strictness is necessary: $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ satisfies $\abs{a_{ii}} \ge \sum_{j \ne i} \abs{a_{ij}}$ with equality, and it is singular.
The theorem also holds for strict dominance by columns, by applying it to $A^T$.

## Explanations

### L1 human

If each diagonal entry is larger in absolute value than all other entries of its row together, the matrix cannot send a nonzero vector to zero.
The diagonal wins against the rest of the row, so it can never be cancelled.

### L2 human

The theorem gives a cheap test for invertibility, used for the matrices that arise when differential equations are discretised.
For strictly diagonally dominant matrices, the Jacobi and Gauss–Seidel iterations for $Ax = b$ converge.
It is the special case "$0$ is not an eigenvalue" of the Gershgorin circle theorem.
</document>

<document source="https://tamnd.github.io/math2/node/PWNZQVQ175.md" tag="PWNZQVQ175" kind="concept">
# Perfect closure

## Summary

The perfect closure of a field $K$ of characteristic $p > 0$ is the smallest [[perfect field|VWBVKT794W]] that contains $K$ inside an algebraic closure.
It is obtained by adjoining all $p^n$-th roots of all elements of $K$, so it is a [[purely inseparable extension|966GNVGB8A]] of $K$.
It is the purely inseparable counterpart of the [[separable closure|JJ91A1V8NQ]].
For a perfect field, and for any field of characteristic $0$, the perfect closure is the field itself.

## Explanations

### L1 human

A field of characteristic $p$ is perfect when every element has a $p$-th root.
If some roots are missing, add them, then add their $p$-th roots, and so on forever.
The union of all these steps is the perfect closure.
For $\F_p(t)$ one adds $t^{1/p}$, then $t^{1/p^2}$, and so on.
The result is the union of the fields $\F_p(t^{1/p^n})$ for $n \ge 0$.
</document>

<document source="https://tamnd.github.io/math2/node/PWQ031W07T.md" tag="PWQ031W07T" kind="definition">
# Cyclic extension

## Definition

Let $L/K$ be a [[Galois extension|0DS2F7GK05]] with [[Galois group|0XVR72M51M]] $\Gal(L/K)$.
The extension $L/K$ is *cyclic* if $\Gal(L/K)$ is a [[cyclic group|0B85Q41RP6]].
It is *cyclic of degree $n$* if moreover $\Gal(L/K)$ has order $n$.

## Remarks

Example: let $p$ be a prime, $q$ a power of $p$, and $n \ge 1$.
The extension $\F_{q^n}/\F_q$ of finite fields is cyclic of degree $n$, generated by the Frobenius map $x \mapsto x^q$.
Non-example: $\Q(\sqrt{2}, \sqrt{3})/\Q$ is Galois with group $\Z/2 \times \Z/2$.
Every element of this group has order at most $2$, while the group has order $4$, so it is not cyclic.
Every cyclic extension is an [[abelian extension|EQDEXBWGJH]].
</document>

<document source="https://tamnd.github.io/math2/node/PWQ5ZR5PK5.md" tag="PWQ5ZR5PK5" kind="definition">
# Atlas

## Definition

Let $M$ be a [[topological space|0CQHFPGJ45]] and let $n \ge 0$ be an integer.
A *chart* of dimension $n$ on $M$ is a pair $(U, \varphi)$ with $U \subseteq M$ [[open|02BH3MCBQB]].
The map $\varphi$ must be a [[homeomorphism|0QT00PB5G2]] from $U$ onto an open subset $\varphi(U)$ of [[Euclidean space|0XPR1E85W0]] $\R^n$.
An *atlas* of dimension $n$ on $M$ is a family $\set{(U_i, \varphi_i)}_{i \in I}$ of charts of dimension $n$ on $M$ such that $\bigcup_{i \in I} U_i = M$.

For two charts $(U_i, \varphi_i)$ and $(U_j, \varphi_j)$ of an atlas, the *transition map* is the homeomorphism
$$\varphi_j \circ \varphi_i^{-1} \colon \varphi_i(U_i \cap U_j) \to \varphi_j(U_i \cap U_j)$$
between open subsets of $\R^n$.

## Remarks

On the circle $S^1 \subset \R^2$, let $N = (0, 1)$ and $S = (0, -1)$.
The stereographic projections $\varphi_N(x, y) = x / (1 - y)$ on $S^1 \setminus \set{N}$ and $\varphi_S(x, y) = x / (1 + y)$ on $S^1 \setminus \set{S}$ are homeomorphisms onto $\R$, and their domains cover $S^1$.
So they form an atlas of dimension $1$; the transition map is $t \mapsto 1/t$ on $\R \setminus \set{0}$.

The single chart $S^1 \setminus \set{N}$ is not an atlas of $S^1$, because its domain misses the point $N$.

A Hausdorff, second-countable space has an atlas of dimension $n$ exactly when it is a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.
An atlas is called smooth when all its transition maps are smooth, and piecewise linear when they are piecewise linear.
Two smooth atlases are equivalent when their union is again a smooth atlas, and a smooth structure is an equivalence class of smooth atlases.
In Mathlib, a space with an atlas modelled on a space $H$ is a `ChartedSpace H M`.
</document>

<document source="https://tamnd.github.io/math2/node/PWV2JFX6RR.md" tag="PWV2JFX6RR" kind="conjecture">
# Erdős–Hajnal conjecture

## Statement

For every finite [[graph|00C0BR7D7H]] $H$ there is a constant $\varepsilon(H) > 0$ with the following property.
Let $G$ be a graph with $n$ vertices that has no [[induced subgraph|0EREPBVG2G]] isomorphic to $H$.
Then $G$ has a [[clique|0BYX1D7C2E]] or an [[independent set|08ANQHHG3P]] with at least $n^{\varepsilon(H)}$ vertices.

## Remarks

Without the hypothesis on $H$, only a logarithmic bound holds.
Every graph on $n$ vertices has a clique or an independent set of size at least $\tfrac12 \log_2 n$, by [[Ramsey's theorem|00TQV3F5S4]].
Random graphs show that this order $\log n$ cannot be improved in general.
Erdős and Hajnal proved that excluding $H$ gives a bound of the form $e^{c\sqrt{\log n}}$ with $c > 0$ depending on $H$.
The conjecture is known for all graphs $H$ with at most four vertices.
For $H$ the path on four vertices it is easy: such graphs are perfect, so the product of the clique number and the independence number is at least $n$.
The conjecture is open in general; the path on five vertices was settled only in 2023.

## Explanations

### L1 human

Ramsey theory says that every large graph has a fairly large set of vertices that are all joined or all unjoined.
In general this set may be tiny, about the logarithm of the number of vertices.
The conjecture says that forbidding any one fixed pattern as an induced subgraph forces a much larger such set, a power of the number of vertices.
</document>

<document source="https://tamnd.github.io/math2/node/PWZAR115MC.md" tag="PWZAR115MC" kind="definition">
# Countable additivity

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]] and let $\mathcal{F}$ be a family of subsets of $X$.
Let $\mu \colon \mathcal{F} \to [0, \infty]$ be a function with values in the [[extended real number line|0GSZ7B0CGR]].
The function $\mu$ is *countably additive*, or *σ-additive*, if the following holds.
For every sequence $A_1, A_2, \ldots$ of pairwise disjoint sets in $\mathcal{F}$ whose [[union|07KVB4JSC7]] is in $\mathcal{F}$,
$$\mu\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} \mu(A_n).$$

## Remarks

On the power set of $\N$, the function $\mu(A) = \sum_{n \in A} 2^{-n}$ is countably additive.
The [[series|0NM626FH6Q]] has non-negative terms, so its sum does not depend on how the terms are grouped.
On the family $\mathcal{F}$ of subsets $A \subseteq \N$ such that $A$ or $\N \setminus A$ is finite, let $\nu(A) = 0$ for finite $A$ and $\nu(A) = 1$ otherwise.
This $\nu$ is finitely additive but not countably additive.
Indeed $\N = \bigcup_{n} \set{n}$ is a disjoint union of sets in $\mathcal{F}$, but $\nu(\N) = 1$ and $\sum_n \nu(\set{n}) = 0$.
If $\varnothing \in \mathcal{F}$ and $\mu$ takes at least one finite value, then countable additivity forces $\mu(\varnothing) = 0$: take $A_1 = A$ with $\mu(A) < \infty$ and $A_n = \varnothing$ for $n \ge 2$.
The same definition is used for functions with values in $\R$, $\C$ or a Banach space; then the series must converge.
</document>

<document source="https://tamnd.github.io/math2/node/PX31ZNYS3Z.md" tag="PX31ZNYS3Z" kind="concept">
# Matroid

## Summary

A matroid is a finite set together with a notion of independence for its subsets that behaves like [[linear independence|0CRMGPAH83]] of vectors.
Matroids abstract both linear independence in vector spaces and acyclic sets of edges ([[forests|0GGXWDNKBR]]) in graphs.
They have many equivalent descriptions, by independent sets, [[bases|QTSQR7130G]], [[rank functions|SYRCTH853H]], circuits or closure, and every matroid has a [[dual|6QF83CFK2G]].
Matroids explain why the greedy algorithm works and carry invariants such as the [[Tutte polynomial|C81RAXD6DV]].

## Explanations

### L1 human

Take some vectors, and call a set of them independent if none is a combination of the others.
Or take the edges of a graph, and call a set of edges independent if it contains no cycle.
In both cases, small independent sets can always be enlarged using elements of bigger ones.
A matroid keeps only this rule and forgets where the elements came from.
</document>

<document source="https://tamnd.github.io/math2/node/PX7SS1X16K.md" tag="PX7SS1X16K" kind="problem">
# Happy ending problem

## Problem

A finite set of points in the plane is in general position if no three of its points lie on a line.
For an integer $n \ge 3$, let $N(n)$ be the least integer $N$ with the following property.
Every set of $N$ points in the plane in general position contains $n$ points that are the vertices of a [[convex|02AS45T3H9]] polygon.
The problem asks for the value of $N(n)$.
Erdős and Szekeres conjectured that $N(n) = 2^{n-2} + 1$ for every $n \ge 3$.

## Remarks

Erdős and Szekeres proved that $N(n)$ exists, using [[Ramsey's theorem|00TQV3F5S4]] in one of their proofs.
They also proved the lower bound $N(n) \ge 2^{n-2} + 1$ by a construction.
The known values are $N(3) = 3$, $N(4) = 5$, $N(5) = 9$ and $N(6) = 17$, all equal to $2^{n-2} + 1$.
The value $N(6) = 17$ was established with a computer search by Szekeres and Peters.
Suk proved in 2017 that $N(n) \le 2^{n + o(n)}$.
The conjecture $N(n) = 2^{n-2} + 1$ is open for $n \ge 7$.

## History

The case $n = 4$ was posed by Esther Klein.
The name comes from the marriage of Klein and George Szekeres, who met through this problem.

## Explanations

### L1 human

Put five points on a sheet of paper so that no three are on one straight line.
Then four of them always form a convex quadrilateral, a shape with no dent.
The problem asks how many points you need to be sure of finding a convex shape with $n$ corners.
The guess is $2^{n-2} + 1$ points, and it is proved only for small $n$.
</document>

<document source="https://tamnd.github.io/math2/node/PXF8731MBK.md" tag="PXF8731MBK" kind="definition">
# Uniformly convex space

## Definition

Let $X$ be a [[normed vector space|03J5RAKB30]] over $\R$ or $\C$.
The space $X$ is *uniformly convex* if the following holds.
For every $\varepsilon$ with $0 < \varepsilon \le 2$ there is $\delta > 0$ such that for all $x, y \in X$ with $\norm{x} \le 1$, $\norm{y} \le 1$ and $\norm{x - y} \ge \varepsilon$,
$$\left\| \frac{x + y}{2} \right\| \le 1 - \delta.$$

## Remarks

Example: an inner product space is uniformly convex.
By the parallelogram law, for $\norm{x}, \norm{y} \le 1$ and $\norm{x - y} \ge \varepsilon$,
$$\left\| \frac{x + y}{2} \right\|^2 = \frac{\norm{x}^2 + \norm{y}^2}{2} - \left\| \frac{x - y}{2} \right\|^2 \le 1 - \frac{\varepsilon^2}{4},$$
so $\delta = 1 - \sqrt{1 - \varepsilon^2/4}$ works.
Non-example: $\R^2$ with the norm $\norm{(a, b)}_\infty = \max(\abs{a}, \abs{b})$.
For $x = (1, 1)$ and $y = (1, -1)$ one has $\norm{x - y}_\infty = 2$ but $\norm{(x + y)/2}_\infty = \norm{(1, 0)}_\infty = 1$, so no $\delta > 0$ works for $\varepsilon = 2$.
The spaces $\ell^p$ and $L^p$ for $1 < p < \infty$ are uniformly convex by [[Clarkson's inequalities|1V11QM5C07]]; $\ell^1$ and $\ell^\infty$ are not.
Uniform convexity depends on the norm, not only on the topology: $\R^2$ is uniformly convex with the Euclidean norm and not with the norm $\norm{\cdot}_\infty$.
</document>

<document source="https://tamnd.github.io/math2/node/PXFYDT3M52.md" tag="PXFYDT3M52" kind="concept">
# Pivotal quantity

## Summary

A pivotal quantity is a function of the data and the unknown parameter whose law does not depend on the parameter.
It is the standard tool to build a [[confidence interval|0S6T9KQNC0]]: one fixes an interval of probability $1 - \alpha$ for the pivot and solves for the parameter.
Many classical intervals, such as the Student t interval for a normal mean, come from a pivot.

## Explanations

### L1 human

A statistic cannot contain the unknown parameter, but a pivot may.
The point is that the pivot has a known law whatever the true parameter is.
For example, if $X$ is normal with unknown mean $\mu$ and variance $1$, then $X - \mu$ is standard normal for every $\mu$.
So $X - \mu$ lies in $[-1.96, 1.96]$ with probability about $0.95$, and this gives the interval $[X - 1.96, X + 1.96]$ for $\mu$.
</document>

<document source="https://tamnd.github.io/math2/node/PXPC6E40ZW.md" tag="PXPC6E40ZW" kind="statement">
# Dahlberg's theorem on harmonic measure

## Statement

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a bounded [[Lipschitz domain|1GNKQ7NBAG]] that is connected.
Let $\sigma$ be the surface measure on $\partial\Omega$, that is, the $(n-1)$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] restricted to $\partial\Omega$.
Let $x \in \Omega$ and let $\omega^x$ be the [[harmonic measure|CFD7BPM51W]] of $\Omega$ at $x$.
Then $\omega^x$ and $\sigma$ are mutually [[absolutely continuous|P0P21KDV6E]], and the density $k^x = d\omega^x / d\sigma$ belongs to [[Lp space|SASP14MA1Z]] $L^2(\partial\Omega, \sigma)$.

## Remarks

More precisely, $k^x$ satisfies a reverse Hölder inequality with exponent $2$ on surface balls, and by Gehring's lemma it lies in $L^{2 + \varepsilon}(\sigma)$ for some $\varepsilon > 0$ depending on $\Omega$.
As a consequence, the Dirichlet problem on $\Omega$ is solvable for boundary data in $L^p(\sigma)$ with $p \ge 2$, with nontangential convergence to the data and a nontangential maximal function estimate.
Idea of the proof: the [[boundary Harnack principle|PYJ7T889KC]] compares $\omega^x$ of a surface ball with the Green function at a nearby interior point.
One proof then uses a Rellich-type integral identity for harmonic functions on Lipschitz domains to get the $L^2$ bound.
Example: for a ball, $k^x$ is the Poisson kernel, which is bounded, so $k^x \in L^p$ for every $p$.
The exponent $2$ cannot be replaced by a fixed larger one for all Lipschitz domains.
In the plane, at the vertex of a sector of opening $\theta \in (\pi, 2\pi)$, the density behaves like $t^{\pi/\theta - 1}$ at distance $t$ from the vertex.
It lies in $L^p$ near the vertex only for $p < 1 / (1 - \pi/\theta)$, and this bound tends to $2$ as $\theta \to 2\pi$.

## Explanations

### L1 human

Harmonic measure tells where a random path started at $x$ first hits the boundary.
On a domain with corners but no cusps, this hitting distribution has a density with respect to surface area, and the density is square-integrable.
So a set of zero area is never hit, and a set that is hit with probability zero has zero area.

### L2 human

Dahlberg's theorem made Lipschitz domains the natural setting for harmonic analysis on rough domains.
It is the basis of the $L^p$ theory of the Dirichlet and Neumann problems and of layer potentials on Lipschitz boundaries.
It shows that harmonic measure is an $A_\infty$ weight with respect to surface measure, a property later studied on much more general domains.
In the plane, for a Jordan domain with rectifiable boundary, the [[F. and M. Riesz theorem|X9V1E3V145]] already gives mutual absolute continuity with arc length.
</document>

<document source="https://tamnd.github.io/math2/node/PXTF18X433.md" tag="PXTF18X433" kind="definition">
# Normal function

## Definition

Let $f$ be a function from the class of [[ordinals|0RDJK7EQYY]] to the class of ordinals.
The function $f$ is normal if it satisfies both conditions:

1. $f$ is strictly increasing: $\alpha < \beta$ implies $f(\alpha) < f(\beta)$.
2. $f$ is continuous: for each nonzero limit ordinal $\lambda$, $f(\lambda) = \sup \set{f(\beta) : \beta < \lambda}$, the [[supremum|0BTMQ04556]] of the earlier values.

## Remarks

Example: with [[ordinal arithmetic|2C7MBG93AZ]], $f(\alpha) = \omega + \alpha$ is normal.
It is strictly increasing, and $\omega + \lambda = \sup_{\beta < \lambda} (\omega + \beta)$ holds by the definition of addition at limits.
Non-example: $g(\alpha) = \alpha + 1$ is strictly increasing but not continuous.
Indeed $\sup_{n < \omega} g(n) = \omega$, while $g(\omega) = \omega + 1$.
Other normal functions are $\alpha \mapsto \omega^\alpha$ and $\alpha \mapsto \aleph_\alpha$.
A strictly increasing $f$ always satisfies $f(\alpha) \ge \alpha$.
Fixed-point lemma: for each normal $f$ and each ordinal $\gamma$, there is an ordinal $\alpha \ge \gamma$ with $f(\alpha) = \alpha$.
The fixed points form a closed unbounded class, and the function that lists them in increasing order is again normal.
The same definition is used for functions from a limit ordinal to itself, or from a regular cardinal to itself.
</document>

<document source="https://tamnd.github.io/math2/node/PXX4P2BHJW.md" tag="PXX4P2BHJW" kind="definition">
# Morley rank

## Definition

Let $M$ be an $L$-structure that is $\aleph_0$-[[saturated|RMTZ5J64FX]], and let $\varphi(\bar{x})$ be an $L$-[[formula|HYJ10AF6RT]] with parameters from $M$.
Write $\varphi(M)$ for the [[definable set|CY1TXHF6KQ]] of tuples of $M$ that satisfy $\varphi$.
For each [[ordinal|0YSCSX5YZC]] $\alpha$, the relation $\mathrm{RM}(\varphi) \ge \alpha$ is defined by induction on $\alpha$:

1. $\mathrm{RM}(\varphi) \ge 0$ if $\varphi(M)$ is nonempty;
2. $\mathrm{RM}(\varphi) \ge \alpha + 1$ if there are formulas $\psi_0, \psi_1, \dots$ with parameters from $M$ whose sets $\psi_i(M)$ are pairwise disjoint subsets of $\varphi(M)$ with $\mathrm{RM}(\psi_i) \ge \alpha$;
3. for a limit ordinal $\alpha$, $\mathrm{RM}(\varphi) \ge \alpha$ if $\mathrm{RM}(\varphi) \ge \beta$ for every $\beta < \alpha$.

The *Morley rank* $\mathrm{RM}(\varphi)$ is $-1$ if $\varphi(M)$ is empty.
Otherwise it is the largest ordinal $\alpha$ with $\mathrm{RM}(\varphi) \ge \alpha$ if one exists, and $\infty$ if $\mathrm{RM}(\varphi) \ge \alpha$ for every ordinal $\alpha$.
For an arbitrary $L$-structure $N$ and a formula with parameters from $N$, the Morley rank is computed in an $\aleph_0$-saturated [[elementary extension|CGH8C8EA9N]] of $N$.
It does not depend on the choice of the extension.

## Remarks

Example: in an [[algebraically closed field|MT8TDVPBXB]] $K$, the formula $x = x$ in one variable has Morley rank $1$.
The singletons $\set{a}$ for distinct $a \in K$ are infinitely many disjoint sets of rank $0$, so the rank is at least $1$.
By quantifier elimination every definable subset of $K$ is finite or cofinite, so no two disjoint definable subsets are both infinite.
A set of rank at least $1$ is infinite, so the rank of $x = x$ is not at least $2$.
More generally, in $K^n$ the Morley rank of an irreducible variety is its Krull dimension.

Non-example: in the theory of dense linear orders without endpoints, the formula $x = x$ has Morley rank $\infty$.
The rank of an open interval is at least $\alpha$ for every $\alpha$, by induction, since an interval contains infinitely many pairwise disjoint open intervals.

A complete theory $T$ in a countable signature is $\omega$-stable, a strong form of [[stability|T7JW5QY6KR]], if and only if every formula has Morley rank less than $\infty$.
</document>

<document source="https://tamnd.github.io/math2/node/PXY1WYNRD4.md" tag="PXY1WYNRD4" kind="statement">
# Normal basis theorem

## Statement

Let $L/K$ be a finite [[Galois extension|0DS2F7GK05]] with [[Galois group|0XVR72M51M]] $G = \Gal(L/K)$.
Then there is an element $\alpha \in L$ such that the family $(\sigma(\alpha))_{\sigma \in G}$ is a [[basis|0VTQTPQGAV]] of $L$ as a $K$-vector space.

## Remarks

Such a basis is called a normal basis, and $\alpha$ is a normal basis generator.
Equivalently, $L$ is isomorphic to the group algebra $K[G]$ as a module over $K[G]$.

When $K$ is infinite, write $G = \set{\sigma_1, \ldots, \sigma_n}$ and fix a basis $b_1, \ldots, b_n$ of $L$.
The family $(\sigma_i(\alpha))$ is a basis exactly when $\det\left(\sigma_i^{-1}\sigma_j(\alpha)\right) \ne 0$; writing $\alpha = \sum x_k b_k$ makes this determinant a polynomial in $x_1, \ldots, x_n$.
An argument in the spirit of [[Dedekind's lemma|N43G50YHG2]], using that $K$ is infinite, shows that this polynomial is not zero.
A nonzero polynomial over an infinite field has a non-root in $K^n$.
When $K$ is finite, $G$ is cyclic, generated by the Frobenius $\varphi$.
Dedekind's lemma shows that the minimal polynomial of the $K$-linear map $\varphi$ is $x^n - 1$, and the theory of cyclic modules then gives the normal basis.

Example: $L = \Q(i)$ over $K = \Q$, with $G = \set{\id, \sigma}$ and $\sigma(i) = -i$.
For $\alpha = 1 + i$, the family $1 + i, 1 - i$ is a basis of $\Q(i)$ over $\Q$.
The choices $\alpha = 1$ and $\alpha = i$ fail: they give $1, 1$ and $i, -i$, which are linearly dependent.

## Explanations

### L1 human

A Galois extension has a group of symmetries.
The normal basis theorem says that one well-chosen element, together with all its images under the symmetries, already forms a basis.
So the whole field is spanned by the conjugates of a single element.

### L2 human

In coordinates of a normal basis, the Galois group acts by permuting the coordinates, which makes many computations simple.
Over finite fields, normal bases make the Frobenius map $x \mapsto x^q$ a cyclic shift of coordinates, which is used in computer arithmetic.
The theorem says that $L$ is a free $K[G]$-module of rank $1$.
Whether the ring of integers of a number field is free over $\Z[G]$ is a much subtler question of the theory of Galois modules.
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<document source="https://tamnd.github.io/math2/node/PY430TXMG0.md" tag="PY430TXMG0" kind="concept">
# Monoidal functor

## Summary

A monoidal functor is a [[functor|28NQ0KRHD3]] between [[monoidal categories|M3MSQ3NKAE]] that is compatible with the tensor products and units.
It comes with maps $F(A) \otimes F(B) \to F(A \otimes B)$ and $I \to F(I)$ that respect the associators and unitors.
When these maps are isomorphisms the functor is called strong, and when they are identities it is called strict.
Monoidal functors send monoid objects to monoid objects, and they are the morphisms of the category of monoidal categories.

## Explanations

### L1 human

A monoidal functor translates one "side-by-side" world into another.
It gives a way to combine the images of $A$ and $B$ into the image of $A \otimes B$, in a way that ignores how the brackets are placed.
For example, the free vector space functor sends a product of sets $X \times Y$ to a space isomorphic to $k[X] \otimes k[Y]$.
The forgetful functor from abelian groups to sets is monoidal in the weaker sense: there is a map $(a, b) \mapsto a \otimes b$ from $A \times B$ to $A \otimes B$, but it is not a bijection.
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<document source="https://tamnd.github.io/math2/node/PY4EWJ8BQA.md" tag="PY4EWJ8BQA" kind="statement">
# Dold–Kan correspondence

## Statement

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]].
Let $s\mathcal{A}$ be the category of [[simplicial objects|WK0A32TZHB]] in $\mathcal{A}$, and let $\mathrm{Ch}_{\ge 0}(\mathcal{A})$ be the category of [[chain complexes|WQ82C0M3QB]] in $\mathcal{A}$ concentrated in degrees $n \ge 0$.
For $A \in s\mathcal{A}$ with face maps $d_i \colon A_n \to A_{n-1}$, let the normalized chain complex $N(A)$ be
$$N(A)_n = \bigcap_{i=1}^{n} \ker\left(d_i \colon A_n \to A_{n-1}\right), \qquad \partial_n = d_0 \colon N(A)_n \to N(A)_{n-1},$$
with $N(A)_0 = A_0$, where $\ker$ denotes the [[kernel|S417YNEJ6M]].
Then $N \colon s\mathcal{A} \to \mathrm{Ch}_{\ge 0}(\mathcal{A})$ is an [[equivalence of categories|K7A04ZBD92]].

## Remarks

A quasi-inverse $\Gamma$ is given by
$$\Gamma(C)_n = \bigoplus_{[n] \twoheadrightarrow [k]} C_k.$$
The sum runs over all surjective order-preserving maps from $\set{0, \ldots, n}$ onto $\set{0, \ldots, k}$.
The face and degeneracy maps are defined from the simplicial identities and the differential of $C$.
To see that $N$ is an equivalence, one shows that every $A_n$ splits as the direct sum of the images of $N(A)_k$ under the iterated degeneracies, indexed by these surjections.

The equivalence carries more structure.
For a simplicial abelian group $A$, the homotopy groups $\pi_n(A)$ are the [[homology|2J4JJJYP22]] groups $H_n(N(A))$.
Simplicial homotopies correspond to [[chain homotopies|4EBHAWJJ8C]].
The normalized complex $N(A)$ is a direct summand of the unnormalized complex with differential $\sum_i (-1)^i d_i$, and the inclusion is a chain homotopy equivalence.

Example: for an abelian group $M$ and $n \ge 0$, the chain complex $M[n]$ with $M$ in degree $n$ and $0$ elsewhere corresponds to a simplicial abelian group $K(M, n) = \Gamma(M[n])$.
Its geometric realization is an Eilenberg–MacLane space for $M$ in degree $n$.

## Explanations

### L1 human

Simplicial abelian groups are combinatorial models of spaces that carry an addition.
Chain complexes are the basic objects of homological algebra.
The Dold–Kan correspondence says that the two are the same thing, once one discards the redundant degenerate simplices.
This lets one move freely between homotopy theory and homological algebra.

### L2 human

The correspondence turns homotopy groups of simplicial abelian groups into homology of chain complexes.
It is used to build Eilenberg–MacLane spaces, to define derived functors of non-additive functors, as Dold and Puppe did, and to compare simplicial and differential graded algebra.
The functors $N$ and $\Gamma$ are not monoidal equivalences; the Eilenberg–Zilber and Alexander–Whitney maps relate the tensor products only up to homotopy.
The statement extends to any idempotent complete additive category, and it has an $\infty$-categorical version for stable $\infty$-categories.
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<document source="https://tamnd.github.io/math2/node/PY6K9AWR33.md" tag="PY6K9AWR33" kind="concept">
# Reciprocal gamma function

## Summary

The reciprocal gamma function is the function $z \mapsto 1/\Gamma(z)$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
Because $\Gamma$ has no zeros and only simple poles at $0, -1, -2, \ldots$, the reciprocal is an [[entire function|0G4F0XB9X2]] with simple zeros exactly at these points.
It is often more convenient than $\Gamma$ itself: it has an everywhere convergent [[Weierstrass product|53MZ2DGDTQ]] and a [[Hankel contour integral|2V0TG9Z3ZW]] valid for all $z$.

## Explanations

### L1 human

The gamma function has infinitely many points where it blows up.
Turning it upside down, $1/\Gamma$, changes these points into ordinary zeros, and the result is defined and smooth on the whole complex plane.
For example $1/\Gamma(1) = 1$, $1/\Gamma(\tfrac12) = 1/\sqrt{\pi}$, and $1/\Gamma(0) = 0$.
On the positive real axis it starts at $0$, rises to a maximum between $1$ and $2$, and then decays very fast, like one over a factorial.
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<document source="https://tamnd.github.io/math2/node/PYABBA1QQE.md" tag="PYABBA1QQE" kind="definition">
# Universally catenary ring

## Definition

Let $R$ be a commutative [[Noetherian ring|0670T59HGE]].
The ring $R$ is *universally catenary* if every [[finitely generated|AC5TFFQT9M]] commutative $R$-algebra is [[catenary|B2CYR292J6]].

## Remarks

An equivalent condition is that the [[polynomial ring|0SDC934R52]] $R[X_1, \dots, X_n]$ is catenary for every $n \ge 0$.
This holds because every finitely generated $R$-algebra is a quotient of such a ring, and a quotient of a catenary ring is catenary.
Every Cohen–Macaulay ring is universally catenary; for example a field $k$ is, so $k[x, y]/(xy)$ is catenary.
Nagata gave a Noetherian local domain that is catenary but not universally catenary: it satisfies the condition for $n = 0$ and fails it for some polynomial ring over it.
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<document source="https://tamnd.github.io/math2/node/PYEN6SFS9P.md" tag="PYEN6SFS9P" kind="statement">
# Nekhoroshev estimates

## Statement

Let $n \ge 2$, let $D \subseteq \R^n$ be a bounded open set and let $\mathbb{T}^n = \R^n / 2\pi \Z^n$.
Use [[action-angle coordinates|M3T81YNSKC]] $(I, \theta)$ on $D \times \mathbb{T}^n$.
Consider the [[Hamiltonian system|0EDKGA2FAK]] with Hamiltonian
$$H_\varepsilon(I, \theta) = h(I) + \varepsilon f(I, \theta).$$
Assume that $h$ and $f$ are real-analytic on a complex neighbourhood of $\overline{D} \times \mathbb{T}^n$.
Assume that $h$ is steep, for example strictly convex.
Then there are constants $a, b, C, \varepsilon_0 > 0$ with the following property.
Let $0 < \varepsilon < \varepsilon_0$, and take a solution with $I(0)$ in $D$, away from the boundary of $D$.
Then
$$\abs{I(t) - I(0)} \le C \varepsilon^b \quad \text{for} \quad \abs{t} \le C \exp\left(\varepsilon^{-a}\right).$$

## Remarks

The proof has two parts.
The analytic part builds, near each resonance, a normal form in which the angle dependence that does not belong to the resonance is exponentially small in $1/\varepsilon$.
The geometric part uses steepness to show that an orbit cannot move far along the directions allowed by a resonance before it leaves the resonance zone.

For strictly convex $h$ one can take $a = b = \frac{1}{2n}$.

Unlike the [[KAM theorem|3ZZSXKM16Z]], the estimate holds for all initial conditions, not only for most of them, but only for a finite, exponentially long time.
Together with the existence of [[Arnold diffusion|ZVY046R4XB]], it shows that instability can occur in nearly integrable systems but only on exponentially long time scales.

Some steepness condition is needed.
For $n = 2$, $h(I) = I_1^2 - I_2^2$, which is not steep, and $f = \sin(\theta_1 + \theta_2)$, the equations give $(\theta_1 + \theta_2)' = 2(I_1 - I_2)$ and $I_1' = I_2' = -\varepsilon \cos(\theta_1 + \theta_2)$.
The solution with $I_1(0) = I_2(0)$ and $\theta_1(0) + \theta_2(0) = 0$ has $I_1(t) = I_1(0) - \varepsilon t$ and $I_2(t) = I_2(0) - \varepsilon t$, so the actions drift by order one in time of order $1/\varepsilon$.

## Explanations

### L1 human

Take an [[integrable system|N0N7CMRCMH]], where the actions never change, and add a small perturbation of size $\varepsilon$.
The theorem says that the actions then change only by a small amount for an extremely long time, longer than any power of $1/\varepsilon$.
So even when a system is not stable forever, it can be stable for times much longer than the age of the universe.

### L2 human

The Nekhoroshev estimates complement KAM theory.
KAM gives stability forever on a large set of invariant tori, but says nothing about the orbits between them.
Nekhoroshev gives stability for all orbits, over times that are exponentially long in a power of $1/\varepsilon$.
They are used in celestial mechanics, for example to discuss the long-time stability of asteroids near the Lagrange points.
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<document source="https://tamnd.github.io/math2/node/PYH82FMXQ2.md" tag="PYH82FMXQ2" kind="concept">
# Semiring of sets

## Summary

A semiring of sets is a family of sets that contains the empty set and is closed under intersections of two members.
In it, the difference of two members is a finite disjoint union of members.
The half-open intervals of the real line form the main example.
Measures such as Lebesgue measure are often first defined on a semiring and then extended.

## Explanations

### L1 human

A semiring of sets is a family of simple building blocks.
Two blocks overlap in a block, and cutting one block out of another leaves finitely many separate blocks.
Half-open intervals $[a, b)$ are the picture to keep in mind: $[0, 3) \setminus [1, 2)$ is the two blocks $[0, 1)$ and $[2, 3)$.
Length is easy to define on such blocks, and measure theory extends it to much more complicated sets.
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<document source="https://tamnd.github.io/math2/node/PYJ7T889KC.md" tag="PYJ7T889KC" kind="statement">
# Boundary Harnack principle

## Statement

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a bounded [[Lipschitz domain|1GNKQ7NBAG]].
Then there are constants $r_0 > 0$ and $C \ge 1$, depending only on $\Omega$, with the following property.
Let $\zeta \in \partial\Omega$ and $0 < r < r_0$, and let $u, v$ be positive [[harmonic functions|0DEB0JMN0W]] on $\Omega \cap B(\zeta, 2r)$ that extend continuously by $0$ to $\partial\Omega \cap B(\zeta, 2r)$.
Then
$$\frac{u(x)}{v(x)} \le C\, \frac{u(y)}{v(y)} \qquad \text{for all } x, y \in \Omega \cap B(\zeta, r).$$

## Remarks

The constant $C$ does not depend on $\zeta$, $r$, $u$ or $v$; it depends only on $n$ and on the Lipschitz character of $\Omega$.
Inside $\Omega$, the ordinary [[Harnack inequality|YNH832D50W]] compares the values of one positive harmonic function at nearby points.
The boundary version compares two functions up to the boundary, where both tend to $0$: their ratio stays bounded above and below.
Idea of the proof: a Carleson-type estimate bounds $u$ near the boundary by its value at an interior point at distance about $r$ from $\zeta$.
Comparison with harmonic measure, using that the Lipschitz graph satisfies an exterior cone condition at every scale, then gives matching lower bounds.
Example, a local check in the upper half-plane near $\zeta = 0$: take $u(x, y) = y$ and $v(x, y) = y / ((x - 3)^2 + y^2)$, a constant multiple of the Poisson kernel with pole $3$.
On $B(0, 1)$ the ratio $v / u = 1 / ((x - 3)^2 + y^2)$ lies between $1/16$ and $1/4$, so the conclusion holds with $C = 4$.
The vanishing hypothesis is needed: with $u = y$ and $v = 1$, the ratio $u / v = y$ tends to $0$ at the boundary, and no constant works.

## Explanations

### L1 human

Two positive harmonic functions that both vanish on a piece of the boundary vanish there at the same rate.
So near that piece, one is roughly a constant multiple of the other.
This holds even when the boundary has corners, as long as it is a Lipschitz graph.

### L2 human

The principle is the key estimate in the boundary theory of harmonic functions on rough domains.
It implies that ratios of positive harmonic functions vanishing on a boundary portion are Hölder continuous up to the boundary.
From this one finds the Martin boundary of a Lipschitz domain: it equals the Euclidean boundary, and every boundary point is minimal.
It also underlies the comparison of harmonic measure with the Green function and the doubling property of harmonic measure on Lipschitz domains.
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<document source="https://tamnd.github.io/math2/node/PYN08Z14ZP.md" tag="PYN08Z14ZP" kind="definition">
# Total variation distance between probability measures

## Definition

Let $P$ and $Q$ be [[probability measures|059TJN0NZ1]] on a measurable space $(\mathcal{X}, \mathcal{F})$.
The *total variation distance* between $P$ and $Q$ is
$$\delta(P, Q) = \sup_{A \in \mathcal{F}} \abs{P(A) - Q(A)}.$$

## Remarks

It lies in $[0, 1]$, and it is a [[metric|0VKD1TP7F3]] on the probability measures on $(\mathcal{X}, \mathcal{F})$.
If $\lambda$ is a measure with densities $p = dP/d\lambda$ and $q = dQ/d\lambda$, then $\delta(P, Q) = \frac{1}{2} \int \abs{p - q} \, d\lambda$, and the supremum is attained at $A = \set{p > q}$.
Equivalently, $\delta(P, Q) = \frac{1}{2} \norm{P - Q}$, where $\norm{\cdot}$ is the [[total variation|EPBND51CR5]] norm of the signed measure $P - Q$.
Some authors call $\norm{P - Q}$ itself the total variation distance, which differs by the factor $2$.

Example: let $P$ and $Q$ be Bernoulli distributions with parameters $p$ and $q$.
The four events are $\emptyset$, $\set{0}$, $\set{1}$ and $\set{0, 1}$, with differences $0$, $\abs{q - p}$, $\abs{p - q}$ and $0$.
So $\delta(P, Q) = \abs{p - q}$.

Non-example: the largest difference over single points, $\sup_x \abs{P(\set{x}) - Q(\set{x})}$, is not the total variation distance.
For the [[normal distributions|05AQE0QR0D]] with means $0$ and $1$ and variance $1$, every point has probability $0$, so this quantity is $0$.
Yet $\delta(P, Q) > 0$, since the two measures are different.
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<document source="https://tamnd.github.io/math2/node/PYRPXHNQTB.md" tag="PYRPXHNQTB" kind="statement">
# Positive mass theorem

## Statement

Let $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $3$ that is asymptotically flat in the following sense.
There is a compact set $C \subset M$ and a diffeomorphism from $M \setminus C$ to $\R^3$ minus a closed ball.
In the coordinates $x = (x^1, x^2, x^3)$ it gives, for some $q > 1/2$,
$$g_{ij} - \delta_{ij} = O(\abs{x}^{-q}), \quad \partial_k g_{ij} = O(\abs{x}^{-q-1}), \quad \partial_k \partial_l g_{ij} = O(\abs{x}^{-q-2}),$$
and the [[scalar curvature|ZHX2VYQDZ3]] $R$ of $g$ is integrable on $M$.
Let the ADM mass of $(M, g)$ be
$$m = \frac{1}{16\pi} \lim_{r \to \infty} \int_{\abs{x} = r} \sum_{i,j=1}^{3} \bigl(\partial_i g_{ij} - \partial_j g_{ii}\bigr) \frac{x^j}{r} \, dA.$$
If $R \ge 0$ on $M$, then $m \ge 0$.
Moreover, $m = 0$ if and only if $(M, g)$ is [[isometric|R9YWQS15AA]] to Euclidean space $\R^3$.

## Remarks

The theorem was proved by Schoen and Yau in 1979 using minimal surfaces, and by Witten in 1981 using spinors and a Weitzenböck formula for the Dirac operator.
In Schoen and Yau's argument, if $m < 0$ one builds a complete stable minimal surface that is asymptotically a plane.
The stability inequality, together with $R \ge 0$ and the Gauss–Bonnet theorem, gives a contradiction.
Both methods extend to higher dimensions under extra hypotheses: dimension at most $7$ for minimal hypersurfaces, and spin manifolds for Witten's method.

Example: the Riemannian Schwarzschild metric $g = \bigl(1 + \frac{m}{2\abs{x}}\bigr)^4 \delta$ on $\R^3 \setminus \set{0}$, with $m > 0$, has scalar curvature $0$.
It has two asymptotically flat ends, as $\abs{x} \to \infty$ and as $\abs{x} \to 0$, and each has ADM mass $m$.
The theorem extends to manifolds with several such ends: each end has nonnegative mass.

Completeness is needed: for $m < 0$ the same formula on $\set{\abs{x} > -m/2}$ is scalar-flat and asymptotically flat with mass $m < 0$.
This metric is not complete, because the conformal factor vanishes at $\abs{x} = -m/2$, which lies at finite distance.

## Explanations

### L1 human

In general relativity, a space that has nonnegative energy density everywhere has nonnegative total mass, measured from far away.
The only way to have total mass zero is to have no gravity at all: flat space.

### L2 human

The Riemannian version above is the time-symmetric case of the positive energy theorem for initial data sets of the Einstein equations, where the condition $R \ge 0$ is the dominant energy condition.
The ADM mass is read off from the decay of the metric at infinity and does not depend on the chosen asymptotic coordinates.
The theorem is a basic tool in conformal geometry: Schoen used it to finish the solution of the Yamabe problem.
The Riemannian Penrose inequality is a sharper version that bounds the mass below by the area of a horizon.
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<document source="https://tamnd.github.io/math2/node/PYSBNWCHCF.md" tag="PYSBNWCHCF" kind="concept">
# Finitely complete category

## Summary

A finitely complete category is a [[category|A2DE66937D]] in which every finite diagram has a [[limit|CHS5AFJGWE]].
Equivalently, it has a terminal object and all [[pullbacks|Q05K04QJ71]].
Finite limits are enough for much of category theory, such as internal groups, regular categories and elementary toposes.
Every [[complete category|FWGPKXCPB7]] is finitely complete.

## Explanations

### L1 human

A finite diagram is a picture with finitely many objects and arrows.
A category is finitely complete when every such picture has a limit, such as a product of two objects or a solution set of an equation.
The category of finite sets is finitely complete, since finite products and subsets of finite sets are finite.
It is not complete, because infinite products of finite sets are not finite.
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<document source="https://tamnd.github.io/math2/node/PYYQQKAZ0C.md" tag="PYYQQKAZ0C" kind="concept">
# Spinor bundle

## Summary

The spinor bundle of a spin manifold is the complex [[vector bundle|P4DZM975BV]] associated with its [[spin structure|RSPZ9TCFQ8]] and the spin representation of the [[spin group|JJSXXJQGEH]].
Its sections are the spinor fields.
It carries a Hermitian metric, a connection induced by the Levi-Civita connection and a Clifford multiplication by tangent vectors, and these define the Dirac operator.

## Explanations

### L1 human

On a spin manifold, each point has a small complex vector space of spinors, and these spaces fit together into a bundle.
A tangent vector acts on spinors, and acting twice by a unit vector gives minus the identity.
On flat space $\R^4$ the spinor bundle is the trivial bundle with fibre $\C^4$, matching the four complex components of a spinor in Dirac's equation for the electron.
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<document source="https://tamnd.github.io/math2/node/PYYXEXMARY.md" tag="PYYXEXMARY" kind="definition">
# Hilbert space

## Definition

Let $H$ be an [[inner product space|0QVNCJXRY3]] over $\R$ or $\C$ with inner product $\ip{\cdot}{\cdot}$.
Let $\norm{x} = \sqrt{\ip{x}{x}}$ be the [[induced norm|0MFJJV97RJ]] and $d(x, y) = \norm{x - y}$ the induced metric.
The space $H$ is a *Hilbert space* if the metric space $(H, d)$ is [[complete|0BY52JW6NK]].

## Remarks

Example: $\ell^2$, the space of real sequences $x$ with $\sum_n x_n^2 < \infty$ and $\ip{x}{y} = \sum_n x_n y_n$.
The series converges by the Cauchy–Schwarz inequality.
It is complete: a Cauchy sequence converges in each coordinate, and the coordinatewise limit is in $\ell^2$ and is the limit in norm.

Non-example: $C([0, 1])$ with $\ip{f}{g} = \int_0^1 f g \, dx$ is an inner product space but not complete.
The continuous functions $f_k$ equal to $0$ on $[0, 1/2]$, to $1$ on $[1/2 + 1/k, 1]$ and linear in between form a Cauchy sequence.
Their only possible limit is the indicator function of $(1/2, 1]$, which is not continuous.

Every finite-dimensional inner product space is a Hilbert space.
An inner product space that is not complete is often called a pre-Hilbert space, and its completion is a Hilbert space.
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# Projective algebra

## Summary

An algebra in a class of algebras is projective when every homomorphism from it into a quotient lifts through the quotient map.
It is the universal-algebra form of a projective object, with surjective homomorphisms in the role of epimorphisms.
In a variety, the projective algebras are exactly the retracts of free algebras.
For modules they are the [[projective modules|BFFA03GE3V]].

## Explanations

### L1 human

A projective algebra can always be lifted back along an onto map.
If its image lands in a quotient, one can choose preimages for everything in a way that respects the operations.
Free algebras are projective, because one only needs to choose a preimage for each free generator.
The group $\Z/2\Z$ is not projective among abelian groups: there is no way to lift the identity of $\Z/2\Z$ through $\Z \to \Z/2\Z$, since $\Z$ has no element of order $2$.
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# Sporadic group

## Summary

A sporadic group is a finite simple group that belongs to none of the infinite families of the classification.
These families are the cyclic groups of prime order, the alternating groups, and the groups of Lie type.
By the classification of finite simple groups there are exactly $26$ of them.
The smallest is the Mathieu group $M_{11}$, of order $7920$, and the largest is the Monster group.

## Explanations

### L1 human

Almost all finite simple groups come in infinite families with a regular pattern.
The sporadic groups are the $26$ exceptions that fit no pattern.
They include the five Mathieu groups, symmetry groups of special codes and designs, the Conway groups, symmetry groups of the Leech lattice, and the Monster.
Twenty of them live inside the Monster as quotients of its subgroups, and the other six are called the pariahs.
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<document source="https://tamnd.github.io/math2/node/PZ8KRWR8RT.md" tag="PZ8KRWR8RT" kind="definition">
# Convex independence

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]], let $I$ be a set, and let $(x_i)_{i \in I}$ be a family of points of $V$.
The family is *convexly independent* if, for every subset $J \subseteq I$ and every $i \in I$,
$$x_i \in \operatorname{conv}\set{x_j : j \in J} \implies i \in J,$$
where $\operatorname{conv}$ denotes the [[convex hull|0FQSDC2EQ2]].
A subset $S \subseteq V$ is *in convex position* if the family $(x)_{x \in S}$ is convexly independent, that is, if no $x \in S$ lies in $\operatorname{conv}(S \setminus \set{x})$.

## Remarks

Example: the points $(0,0)$, $(1,0)$, $(0,1)$, $(1,1)$ of $\R^2$ are in convex position.
Each of them is a vertex of the unit square, and no vertex is a convex combination of the other three.
For instance, the other three points satisfy $x + y \le 1$, so all their convex combinations do, while $(1,1)$ does not.
The other vertices are handled by symmetry.

Non-example: add the point $(\frac12, \frac12)$.
It equals $\frac12 (1,0) + \frac12 (0,1)$, so it lies in the convex hull of the other points.

A convexly independent family is injective, because $x_i = x_j$ with $i \neq j$ puts $x_i$ in the hull of $\set{x_j}$.
A finite set $S$ is in convex position if and only if every point of $S$ is an [[extreme point|3MSGFJM2NX]] of $\operatorname{conv}(S)$, that is, a vertex of the polytope $\operatorname{conv}(S)$.
Affinely independent families are convexly independent, but not conversely: the four corners of a square are affinely dependent.
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# Noncommutative torus

## Example

Let $\theta \in \R$.
The *noncommutative torus* $A_\theta$ is the universal C*-algebra generated by two unitaries $U$ and $V$ with
$$V U = e^{2 \pi i \theta} U V.$$

Concrete check: on $L^2(\mathbb{T})$, with $\mathbb{T}$ the unit circle and its arc length measure, let $(U f)(z) = z f(z)$ and $(V f)(z) = f(e^{2 \pi i \theta} z)$.
Both are unitary.
Then $(V U f)(z) = e^{2 \pi i \theta} z f(e^{2 \pi i \theta} z)$ and $(U V f)(z) = z f(e^{2 \pi i \theta} z)$, so $V U = e^{2 \pi i \theta} U V$.
For irrational $\theta$, the C\*-algebra generated by these $U$ and $V$ is isomorphic to $A_\theta$.

## Remarks

For $\theta = 0$, $U$ and $V$ commute, and $A_0 \cong C(\mathbb{T}^2)$, the continuous functions on the ordinary torus; this explains the name.
For irrational $\theta$, the algebra $A_\theta$ is simple and has a unique tracial state; it is also called the irrational rotation algebra.
It is the [[crossed product|8HG8TS46A6]] $C(\mathbb{T}) \rtimes \Z$ for the action of $\Z$ on $\mathbb{T}$ by rotation through the angle $2 \pi \theta$.
For irrational $\theta$ and $\theta'$, $A_\theta \cong A_{\theta'}$ if and only if $\theta' \equiv \pm \theta \pmod{1}$.

## Explanations

### L1 human

Functions on the ordinary torus are generated by the two coordinate functions $z_1$ and $z_2$, which commute.
The noncommutative torus keeps two generators but lets them commute only up to a fixed phase $e^{2 \pi i \theta}$.
It is a basic example of a "noncommutative space" in noncommutative geometry.
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# One-relator group

## Definition

Let $G$ be a [[group|G5S1M9]].
$G$ is a *one-relator group* if it has a [[presentation|0JGEQXQ4AC]] $\langle S \mid r \rangle$ with a single relator $r$.
Here $r$ is an element of the [[free group|0KTW26C80J]] $F(S)$ on the set $S$.
That is, $G$ is isomorphic to the quotient of $F(S)$ by the normal subgroup generated by $r$.

## Remarks

The relator may be taken cyclically reduced, since a cyclic permutation of $r$ is a conjugate of $r$ and generates the same normal subgroup.
A one-relator group has an element of finite order other than $1$ if and only if $r$ is a proper power $s^n$ in $F(S)$ with $n \ge 2$.
In that case, if $s$ is chosen not to be a proper power itself, then $s$ has order exactly $n$ in $G$.

Example: $\Z^2$ is a one-relator group.
The map $\langle a, b \mid a b a^{-1} b^{-1} \rangle \to \Z^2$ with $a \mapsto (1, 0)$ and $b \mapsto (0, 1)$ is an isomorphism, since the relator says exactly that $a$ and $b$ commute.

Non-example: $\Z/2 \times \Z/2$ is not a one-relator group.
If $\langle S \mid r \rangle$ with $S$ finite of size $k$ defined it, its abelianization would be $\Z^k$ modulo the subgroup generated by one vector, which is $\Z^{k-1} \oplus \Z/d$ for some integer $d \ge 0$.
The group $\Z/2 \times \Z/2$ is abelian, so it equals its abelianization.
It is finite, so $k - 1 = 0$, and then it would be cyclic, which it is not.
A presentation with $S$ infinite gives an abelianization with infinitely many free factors $\Z$, so it fails as well.
</document>

<document source="https://tamnd.github.io/math2/node/PZCJV40M4K.md" tag="PZCJV40M4K" kind="concept">
# Systolic geometry

## Summary

Systolic geometry studies the systole of a Riemannian manifold, the length of its shortest closed curve that cannot be shrunk to a point.
Its main questions compare the systole with the volume of the manifold.
Typical results say that a manifold with a long systole must have a large volume, as in [[Loewner's torus inequality|1Y9X1YW6GH]] and [[Gromov's systolic inequality|HB2K72PSZ6]].
The field links Riemannian geometry with topology and with the geometry of numbers.

## Explanations

### L1 human

On a doughnut surface some loops cannot be pulled tight to a point, such as a loop around the hole.
The systole is the length of the shortest such loop.
If all these loops are long, the surface must be large: it cannot have small area.
For a flat square torus made from a unit square, the systole is $1$ and the area is $1$.
Systolic geometry asks how small the area can be once the systole is fixed.
</document>

<document source="https://tamnd.github.io/math2/node/PZES1MKY8M.md" tag="PZES1MKY8M" kind="definition">
# Relational clone

## Definition

Let $A$ be a nonempty set.
Let $\Gamma$ be a set of finitary relations on $A$.
A *primitive positive formula* over $\Gamma$ is a formula of the form
$$\exists y_1 \cdots \exists y_k \, \bigl(\varphi_1 \wedge \cdots \wedge \varphi_r\bigr).$$
Here each $\varphi_j$ is an equation $z_1 = z_2$ or an atomic formula $R(z_1, \dots, z_m)$ with $R \in \Gamma$ of arity $m$.
The $z_i$ are variables.
A relation $S \subseteq A^p$ is *primitive positive definable* from $\Gamma$ if $S = \set{(a_1, \dots, a_p) \in A^p : \psi(a_1, \dots, a_p)}$.
Here $\psi(x_1, \dots, x_p)$ is a primitive positive formula over $\Gamma$ with free variables $x_1, \dots, x_p$.
A *relational clone* on $A$ is a set $\Gamma$ of finitary relations on $A$ such that:

1. $\Gamma$ contains the binary equality relation $\set{(a, a) : a \in A}$.
2. $\Gamma$ contains the empty relation of each arity.
3. Every relation that is primitive positive definable from $\Gamma$ belongs to $\Gamma$.

## Remarks

Example: for any set $F$ of operations on $A$, the set $\operatorname{Inv}(F)$ of all relations preserved by every $f \in F$ is a relational clone.
Check: equality and empty relations are preserved by every operation.
If the relations in a primitive positive formula are preserved by $f$, so is the defined relation, because $f$ can be applied to the witnesses $y_i$ too.
Non-example: on $\set{0, 1}$, the set consisting of equality, the empty relations and the single relation $R = \set{(0, 1)}$ is not a relational clone.
The formula $\exists y \, R(x, y)$ defines the unary relation $\set{0}$, which is not in this set.
Conventions differ slightly between texts on whether the empty relation is included.
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<document source="https://tamnd.github.io/math2/node/PZGNRA2DAY.md" tag="PZGNRA2DAY" kind="definition">
# Power of a test

## Definition

Let $X$ be data whose law $P_\theta$ depends on $\theta \in \Theta$, and consider a [[hypothesis test|087P03NEND]] of $H_0 \colon \theta \in \Theta_0$ against $H_1 \colon \theta \in \Theta_1$.
Let the test reject $H_0$ when $X$ lies in a rejection region $R$.
The *power function* of the test is $\pi(\theta) = P_\theta(X \in R)$ for $\theta \in \Theta$.
The *power* of the test at $\theta \in \Theta_1$ is $\pi(\theta)$.

## Remarks

For $\theta \in \Theta_1$, the power is $1 - \beta(\theta)$, where $\beta(\theta)$ is the probability of a [[type II error|0R41PVFA39]].
For $\theta \in \Theta_0$, the value $\pi(\theta)$ is the probability of a type I error, and $\sup_{\theta \in \Theta_0} \pi(\theta)$ is the size of the test.
For a randomized test $\varphi$ with values in $[0, 1]$, the power function is $\pi(\theta) = E_\theta[\varphi(X)]$.
Example: let $X \sim \mathcal{N}(\theta, 1)$, $H_0 \colon \theta = 0$, $H_1 \colon \theta > 0$, and $R = \set{x > 1.645}$.
Then $\pi(\theta) = 1 - \Phi(1.645 - \theta)$, where $\Phi$ is the standard normal distribution function; so $\pi(0) \approx 0.05$ and $\pi(2) \approx 0.64$.
Non-example: $\pi(0) \approx 0.05$ is not a power in this example, since $0 \notin \Theta_1$; it is the size.
</document>

<document source="https://tamnd.github.io/math2/node/PZHDNJ2J73.md" tag="PZHDNJ2J73" kind="concept">
# Locally presentable category

## Summary

A locally presentable category is an [[accessible category|FC8C0V4TFZ]] that has all small colimits.
Every object is a highly filtered colimit of objects from a fixed small set of presentable objects.
Sets, groups, rings, modules, small categories and presheaf categories are locally presentable, while topological spaces are not.
These categories are also complete, and the adjoint functor theorem takes a simple form for them.

## Explanations

### L1 human

A locally presentable category is a large category built from a small set of building blocks in a controlled way.
In the category of groups, every group is a filtered colimit of finitely presented groups.
Because of this, questions about the whole category can often be reduced to questions about the small building blocks.
</document>

<document source="https://tamnd.github.io/math2/node/PZKBP4F5HB.md" tag="PZKBP4F5HB" kind="definition">
# Epimorphism

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $f \colon A \to B$ be a [[morphism|859F0TFVA3]] of $\mathcal{C}$.
The morphism $f$ is an *epimorphism* if, for every object $Y$ and all morphisms $g_1, g_2 \colon B \to Y$, the equality $g_1 \circ f = g_2 \circ f$ implies $g_1 = g_2$.

## Remarks

Equivalently, for every object $Y$ the map $\Hom_{\mathcal{C}}(B, Y) \to \Hom_{\mathcal{C}}(A, Y)$, $g \mapsto g \circ f$, is injective.
One writes $f \colon A \twoheadrightarrow B$, or says that $f$ is *epic*.
Example: in the category of sets, every [[surjective function|01RVCMSN5N]] $f$ is an epimorphism.
If $g_1 \circ f = g_2 \circ f$ and $b \in B$, write $b = f(a)$; then $g_1(b) = g_1(f(a)) = g_2(f(a)) = g_2(b)$.
Non-example: the inclusion $f \colon \set{0} \to \set{0, 1}$ is not an epimorphism of sets.
The identity of $\set{0, 1}$ and the constant map with value $0$ agree after $f$ but are different.
In the category of rings with unit, the inclusion $\Z \to \Q$ is an epimorphism that is not surjective.
Two [[ring homomorphisms|0VTH8KPCG2]] $g_1, g_2 \colon \Q \to R$ that agree on $\Z$ agree on $m/n$, since $g_i(m/n) = g_i(m) g_i(n)^{-1}$.
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<document source="https://tamnd.github.io/math2/node/PZMGQMDN25.md" tag="PZMGQMDN25" kind="concept">
# Isomorphism

## Summary

An isomorphism between two [[structures|983GZBQYT4]] of the same signature is a bijection that preserves and reflects all operations, constants and relations.
Two structures are isomorphic when an isomorphism between them exists; then they have the same properties expressible in their signature.
[[Group isomorphisms|08MG7H8JW5]], [[ring isomorphisms|0KN2S81K72]], [[linear isomorphisms|0B8T10CE7N]] and [[order isomorphisms|0D3WA36K2Q]] are special cases.
An isomorphism of a structure with itself is an [[automorphism|DSQ17TNW1C]].

## Explanations

### L1 human

Two structures are isomorphic when they are the same up to renaming of their elements.
The even integers with addition and the integers with addition are isomorphic: the map $x \mapsto 2x$ renames each integer as an even integer and turns sums into sums.
The integers and the rational numbers with addition are not isomorphic, although both are infinite: in $\Q$ every element is twice another, and in $\Z$ the element $1$ is not.
</document>

<document source="https://tamnd.github.io/math2/node/PZR6RC79ED.md" tag="PZR6RC79ED" kind="statement">
# Quintuple product identity

## Statement

Let $s, t \in \C$ with $\abs{s} < 1$ and $t \ne 0$.
Then
$$\prod_{n=1}^{\infty} \left(1 - s^n\right) \left(1 - s^n t\right) \left(1 - s^{n-1} t^{-1}\right) \left(1 - s^{2n-1} t^2\right) \left(1 - s^{2n-1} t^{-2}\right) = \sum_{n=-\infty}^{\infty} s^{(3n^2 + n)/2} \left( t^{3n} - t^{-3n-1} \right).$$

## Remarks

Both sides are holomorphic in $t$ on $\C \setminus \set{0}$, and the [[infinite product|RZ0Z2BTMT4]] converges because $\abs{s} < 1$.
Proof idea: as for the [[Jacobi triple product|JFP2DGMNXT]], one checks that both sides $F$ satisfy $F(st) = s^{-2} t^{-3} F(t)$.
This relation links the Laurent coefficients of index $k$ and $k + 3$, so it fixes them from three of them, and these are compared directly.
Example: at $t = 1$ the factor $1 - s^0 t^{-1}$ of index $n = 1$ vanishes, so the left side is $0$.
On the right the terms $s^{(3n^2+n)/2}(1 - 1)$ also vanish, so both sides agree.
The exponents $(3n^2 + n)/2$ are the generalized pentagonal numbers, as in [[Euler's pentagonal number theorem|R587SR9GNV]].

## Explanations

### L1 human

Like the triple product identity, this identity turns an infinite product into a sum, but now each step of the product has five factors.
The sum on the right is very sparse: only exponents of the form $(3n^2 + n)/2$ appear.
It is used to prove identities for partitions and for theta functions.

### L2 human

The identity has been rediscovered many times, and there are many equivalent forms obtained by changes of variables.
It can be derived from the [[Jacobi triple product|JFP2DGMNXT]], and it is a source of identities for theta functions and for the [[Dedekind eta function|MX3CVYMNDV]].
</document>

<document source="https://tamnd.github.io/math2/node/PZRK2EPH1Y.md" tag="PZRK2EPH1Y" kind="definition">
# Hörmander's condition

## Definition

Let $\Omega \subseteq \R^n$ be an open set and let $X_0, X_1, \ldots, X_m$ be smooth [[vector fields|0299VCW1TH]] on $\Omega$.
Let $[\cdot, \cdot]$ be the [[Lie bracket of vector fields|68FGQEBD6J]], and let the indices $j_1, j_2, \ldots$ range over $\set{0, 1, \ldots, m}$.
The family satisfies *Hörmander's condition* if, at every point $x \in \Omega$, the vectors
$$X_{j_1}(x), \quad [X_{j_1}, X_{j_2}](x), \quad [X_{j_1}, [X_{j_2}, X_{j_3}]](x), \quad \ldots$$
span $\R^n$.
Equivalently, the Lie algebra generated by $X_0, \ldots, X_m$ spans the tangent space at every point.

## Remarks

Hörmander's theorem: under this condition, for every smooth function $c$, the operator $P = \sum_{j=1}^m X_j^2 + X_0 + c$ is [[hypoelliptic|K9DSR21DQC]], where each vector field acts as a first-order differential operator.
Example: on $\R^2$, let $X_1 = \partial_x$ and $X_0 = x \, \partial_y$.
Then $[X_1, X_0] = \partial_y$, so $X_1$ and $[X_1, X_0]$ span $\R^2$ at every point, and the Kolmogorov operator $\partial_x^2 + x \partial_y$ is hypoelliptic.
Non-example: on $\R^2$, the single field $X_1 = \partial_x$ fails the condition, since all its brackets vanish and $\partial_x$ alone spans a line.
Accordingly, $\partial_x^2$ is not hypoelliptic: $u(x, y) = \abs{y}$ satisfies $\partial_x^2 u = 0$ but is not smooth.
In stochastic analysis one often uses a stronger parabolic version, in which $X_0$ itself is left out of the first level of the spanning family.
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<document source="https://tamnd.github.io/math2/node/PZX9J3VEMT.md" tag="PZX9J3VEMT" kind="definition">
# Sequent calculus

## Definition

Fix a language of [[predicate logic|086QKPF0A6]].
A *sequent* is an expression $\Gamma \vdash \Delta$, where $\Gamma$ and $\Delta$ are finite lists of formulas.
The *sequent calculus* LK has the following parts.

1. Axioms: $A \vdash A$ for each formula $A$.
2. Structural rules: weakening, contraction and exchange on each side, and the cut rule. The cut rule infers $\Gamma, \Pi \vdash \Delta, \Lambda$ from $\Gamma \vdash \Delta, A$ and $A, \Pi \vdash \Lambda$.
3. Logical rules: for each connective and quantifier, rules that introduce it on the left and on the right.

The logical rules for [[conjunction|0M28ASWG8T]], [[disjunction|0H5GKDTB01]], [[implication|0ZNX6DQ3EJ]], [[negation|0XR7A7RSXB]] and the quantifiers are these.

- $\wedge$R: from $\Gamma \vdash \Delta, A$ and $\Gamma \vdash \Delta, B$ infer $\Gamma \vdash \Delta, A \wedge B$.
- $\wedge$L: from $A, \Gamma \vdash \Delta$ infer $A \wedge B, \Gamma \vdash \Delta$ and $B \wedge A, \Gamma \vdash \Delta$.
- $\vee$L: from $A, \Gamma \vdash \Delta$ and $B, \Gamma \vdash \Delta$ infer $A \vee B, \Gamma \vdash \Delta$.
- $\vee$R: from $\Gamma \vdash \Delta, A$ infer $\Gamma \vdash \Delta, A \vee B$ and $\Gamma \vdash \Delta, B \vee A$.
- $\to$R: from $A, \Gamma \vdash \Delta, B$ infer $\Gamma \vdash \Delta, A \to B$.
- $\to$L: from $\Gamma \vdash \Delta, A$ and $B, \Pi \vdash \Lambda$ infer $A \to B, \Gamma, \Pi \vdash \Delta, \Lambda$.
- $\neg$R: from $A, \Gamma \vdash \Delta$ infer $\Gamma \vdash \Delta, \neg A$.
- $\neg$L: from $\Gamma \vdash \Delta, A$ infer $\neg A, \Gamma \vdash \Delta$.
- $\forall$R: from $\Gamma \vdash \Delta, A(y)$ infer $\Gamma \vdash \Delta, \forall x\, A(x)$, where the variable $y$ is not free in $\Gamma, \Delta, \forall x\, A(x)$.
- $\forall$L: from $A(t), \Gamma \vdash \Delta$ infer $\forall x\, A(x), \Gamma \vdash \Delta$, where $t$ is a term.
- $\exists$L: from $A(y), \Gamma \vdash \Delta$ infer $\exists x\, A(x), \Gamma \vdash \Delta$, where $y$ is not free in $\Gamma, \Delta, \exists x\, A(x)$.
- $\exists$R: from $\Gamma \vdash \Delta, A(t)$ infer $\Gamma \vdash \Delta, \exists x\, A(x)$, where $t$ is a term.

A *derivation* is a finite tree of sequents.
Each leaf is an axiom, and each other node follows from its children by a rule.
The calculus LJ for intuitionistic logic is LK with the restriction that each right side has at most one formula.

## Remarks

Example: the [[law of excluded middle|0RYYK8PG4S]] has an LK derivation.
Start from $A \vdash A$ and get $\vdash A, \neg A$ by $\neg$R.
Then $\vee$R twice gives $\vdash A \vee \neg A, A \vee \neg A$, and contraction gives $\vdash A \vee \neg A$.
Non-example: this is not an LJ derivation, because $\vdash A, \neg A$ has two formulas on the right.
In fact $\vdash A \vee \neg A$ has no LJ derivation at all.
A sequent $A_1, \dots, A_m \vdash B_1, \dots, B_n$ means the same as the formula $(A_1 \wedge \dots \wedge A_m) \to (B_1 \vee \dots \vee B_n)$.
Some books use multisets or sets in place of lists, and then drop the exchange rule.
</document>

<document source="https://tamnd.github.io/math2/node/PZZ4X908Y8.md" tag="PZZ4X908Y8" kind="definition">
# Lagrangian submanifold

## Definition

Let $(M, \omega)$ be a [[symplectic manifold|CAN490EPRT]] of dimension $2n$.
Let $L \subseteq M$ be a submanifold with inclusion map $\iota \colon L \to M$.
The submanifold $L$ is *Lagrangian* if:

1. $\dim L = n$, and
2. $\iota^* \omega = 0$, that is $\omega_p(v, w) = 0$ for all $p \in L$ and $v, w \in T_p L$.

## Remarks

Example: in $\R^4$ with $\omega_0 = dx_1 \wedge dy_1 + dx_2 \wedge dy_2$, the plane $L = \set{y_1 = y_2 = 0}$.
It has dimension $2$, and on it $dy_1 = dy_2 = 0$, so $\iota^* \omega_0 = 0$.

Non-example: the plane $P = \set{x_2 = y_2 = 0}$ in the same space has dimension $2$, but $\omega_0$ restricts to $dx_1 \wedge dy_1 \ne 0$ on it, so clause 2 fails.
Such a plane is called symplectic.

Clause 2 alone forces $\dim L \le n$, so Lagrangian submanifolds are the isotropic submanifolds of largest dimension.
In a cotangent bundle $T^*Q$ with its canonical form, the graph of a $1$-form $\alpha$ on $Q$ is Lagrangian exactly when $\alpha$ is [[closed|0ZV14XRNF0]].
A diffeomorphism $\phi \colon M \to M$ is a [[symplectomorphism|A8TS2SNX9Z]] exactly when its graph is Lagrangian in $M \times M$ with the form $\mathrm{pr}_1^* \omega - \mathrm{pr}_2^* \omega$.
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<document source="https://tamnd.github.io/math2/node/PZZNV9SEF1.md" tag="PZZNV9SEF1" kind="definition">
# Holomorphic fiber bundle

## Definition

Let $E$, $B$ and $F$ be [[complex manifolds|JSGWAN2RJV]].
A *holomorphic fiber bundle* over $B$ with fibre $F$ and total space $E$ is a surjective [[holomorphic map|3MQTWKBBBT]] $\pi \colon E \to B$ with the following property.
Each point of $B$ has an open neighbourhood $U$ and a [[biholomorphism|FX92PP3HZR]] $\varphi \colon \pi^{-1}(U) \to U \times F$ with $\mathrm{pr}_1 \circ \varphi = \pi$ on $\pi^{-1}(U)$.

## Remarks

The underlying smooth map is a [[fiber bundle|CX4M8K61AB]] in the usual sense.
On overlaps the trivializations differ by maps $(b, f) \mapsto (b, g_{ij}(b)(f))$, where each $g_{ij}(b)$ is a biholomorphism of $F$ that depends holomorphically on $b$.
When $F = \C^r$ and each $g_{ij}(b)$ is linear, the bundle is a holomorphic vector bundle.

Example: the map $\pi \colon \C^2 \setminus \set{0} \to \mathbb{P}^1$, $(z_0, z_1) \mapsto [z_0 : z_1]$, with fibre $F = \C^*$.
Over $U_0 = \set{z_0 \ne 0}$ the map $(z_0, z_1) \mapsto ([z_0 : z_1], z_0)$ is a biholomorphism $\pi^{-1}(U_0) \to U_0 \times \C^*$, with inverse $([1 : w], c) \mapsto (c, c w)$, and similarly over $U_1$.

Non-example: $\pi \colon \C^2 \to \C$, $(z, w) \mapsto zw$.
Each fibre over $t \ne 0$ is biholomorphic to $\C^*$, but the fibre over $0$ is the union of the two axes, which is not a manifold.
So no neighbourhood of $0$ has a trivialization.

By a theorem of Fischer and Grauert, a proper holomorphic submersion between connected complex manifolds whose fibres are all biholomorphic to one another is a holomorphic fiber bundle.
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<document source="https://tamnd.github.io/math2/node/Q05K04QJ71.md" tag="Q05K04QJ71" kind="concept">
# Pullback

## Summary

The pullback, or fibre product, of two morphisms $f \colon X \to Z$ and $g \colon Y \to Z$ with a common target is a universal commutative square over them.
It is the [[limit|CHS5AFJGWE]] of the diagram $X \to Z \leftarrow Y$.
In sets it is the set of pairs $(x, y)$ with $f(x) = g(y)$, and it includes preimages and intersections as special cases.
Pullbacks are used to restrict, intersect and change the base of objects throughout geometry and topology.

## Explanations

### L1 human

Two maps land in the same place $Z$.
The pullback collects the pairs of points, one from $X$ and one from $Y$, that land on the same point of $Z$.
If $Y$ is a subset of $Z$ and $g$ is the inclusion, the pairs are just the points $x$ with $f(x) \in Y$, so the pullback is the preimage $f^{-1}(Y)$.
If both $X$ and $Y$ are subsets of $Z$ with their inclusions, the pullback is the intersection $X \cap Y$.
</document>

<document source="https://tamnd.github.io/math2/node/Q07RA7AMDE.md" tag="Q07RA7AMDE" kind="definition">
# Semiring

## Definition

Let $S$ be a set with two binary operations $+$ and $\cdot$, and two elements $0$ and $1$ of $S$.
The tuple $(S, +, \cdot, 0, 1)$ is a *semiring* if the following hold.

1. $(S, +, 0)$ is a commutative [[monoid|0N6R902FG1]].
2. $(S, \cdot, 1)$ is a monoid.
3. Multiplication distributes over addition on both sides: $a \cdot (b + c) = a \cdot b + a \cdot c$ and $(a + b) \cdot c = a \cdot c + b \cdot c$ for all $a, b, c \in S$.
4. Zero is absorbing: $0 \cdot a = a \cdot 0 = 0$ for all $a \in S$.

The semiring is *commutative* if its multiplication is commutative.

## Remarks

The natural numbers $\N = \set{0, 1, 2, \dots}$ with the usual addition and multiplication form a commutative semiring.
The four clauses are the familiar rules of arithmetic on $\N$.
The tropical semiring is $\R \cup \set{\infty}$ with $\min$ as addition, ordinary $+$ as multiplication, $\infty$ as zero and $0$ as one.
For instance, $a + \min(b, c) = \min(a + b, a + c)$ gives distributivity, and $a + \infty = \infty$ gives the absorbing clause.
The positive integers $\set{1, 2, 3, \dots}$ with the usual operations are not a semiring: there is no additive identity, so clause 1 fails.
A [[ring|0B1261B5ED]] is exactly a semiring in which $(S, +)$ is a group; clause 4 then follows from the others.
Some authors omit $0$ or $1$ from the definition, and some do not require clause 4.
</document>

<document source="https://tamnd.github.io/math2/node/Q0C1H111EN.md" tag="Q0C1H111EN" kind="statement">
# Schwartz impossibility theorem

## Statement

Let $\mathcal{D}'(\R)$ be the space of distributions on $\R$, the dual of the space of test functions.
Let $C(\R)$ be the space of continuous functions, viewed as distributions.
There is no [[associative algebra|3CTW1P4HBP]] $(\mathcal{A}, \circ)$ over $\R$ with these properties:

1. $\mathcal{D}'(\R)$ is a linear subspace of $\mathcal{A}$, and the constant function $1$ is the unit of $\mathcal{A}$.
2. There is a linear map $D \colon \mathcal{A} \to \mathcal{A}$ with $D(u \circ v) = (Du) \circ v + u \circ (Dv)$ for all $u, v \in \mathcal{A}$.
3. The restriction of $D$ to $\mathcal{D}'(\R)$ is the derivative of distributions.
4. The restriction of $\circ$ to $C(\R) \times C(\R)$ is the pointwise product of functions.

## Remarks

The proof computes inside $\mathcal{A}$ with continuous functions such as $x$, $x \log\abs{x} - x$ and their derivatives in $\mathcal{D}'(\R)$.
The Leibniz rule and the pointwise product give identities for products like $x \circ \delta$, where $\delta$ is the [[Dirac delta|0KPHENZ5S7]].
Associativity then leads to a contradiction such as $\delta = 0$.

So distributions cannot be multiplied in a way that keeps all the rules of calculus.
[[Colombeau algebras|XGQT4WEVGG]] keep clauses 1, 2 and 3 and weaken clause 4: they keep the pointwise product of smooth functions only.

## Explanations

### L1 human

Distributions such as the delta function are very useful, but one cannot multiply them freely.
This theorem shows that the problem is not a lack of cleverness.
Any product that extends the product of continuous functions and obeys the product rule for derivatives must break down.

### L2 human

Nonlinear equations with distributional data, for example in shock waves or in quantum field theory, need products of distributions.
The theorem shows which rule must be given up.
In the Colombeau algebra the product agrees with the usual one on smooth functions, and on continuous functions only up to association, a weak equality tested against test functions.
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<document source="https://tamnd.github.io/math2/node/Q0DXKAQXE1.md" tag="Q0DXKAQXE1" kind="definition">
# Parametric model

## Definition

Let $\mathcal{P} = \set{P_\theta : \theta \in \Theta}$ be a [[statistical model|JZ6R7J36K5]] on a measurable space $(S, \mathcal{S})$.
The model is a *parametric model* if its parameter space $\Theta$ is a subset of $\R^k$ for some finite $k \ge 1$.
The number $k$ is the dimension of the parameter.

## Remarks

Example: the normal model on $S = \R$ with $P_{(\mu, \sigma^2)} = \mathcal{N}(\mu, \sigma^2)$ and $\Theta = \R \times (0, \infty) \subseteq \R^2$ is parametric with $k = 2$.
Non-example: the model of all probability measures on $\R$ with finite variance is not given by a natural finite-dimensional parameter, and it is called nonparametric.
In practice one also asks that the map $\theta \mapsto P_\theta$ is regular, for example that the densities depend smoothly on $\theta$.
Without such a condition the definition is weak, since any set of at most continuum many laws can be indexed by a subset of $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/Q0GBCB9RA4.md" tag="Q0GBCB9RA4" kind="example">
# SL(2,R)

## Example

The group $SL(2, \R)$ is the [[special linear group|0V9PMN0RBH]] of real $2 \times 2$ matrices with determinant $1$:
$$SL(2, \R) = \set{\begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \R, \ ad - bc = 1}.$$
It is a closed subgroup of $GL_2(\R)$ cut out by one equation, so it is a [[Lie group|7JBRGCVJRB]] of dimension $3$.

It acts on the [[upper half-plane|X6XGGH5FPW]] $\mathbb{H} = \set{z \in \C : \operatorname{Im} z > 0}$ by [[Möbius transformations|0SG74WBFA2]]
$$z \mapsto \frac{az + b}{cz + d}.$$
Check that $\mathbb{H}$ is preserved: multiplying numerator and denominator by $c\bar{z} + d$ gives
$$\operatorname{Im} \frac{az + b}{cz + d} = \frac{(ad - bc) \operatorname{Im} z}{\abs{cz + d}^2} = \frac{\operatorname{Im} z}{\abs{cz + d}^2} > 0.$$

## Remarks

The matrices $I$ and $-I$ act as the identity map, and the quotient $PSL(2, \R) = SL(2, \R)/\set{\pm I}$ is the group of orientation-preserving isometries of the hyperbolic plane in the upper half-plane model.
$SL(2, \R)$ is connected but not compact.
Its maximal compact subgroup is the rotation group [[SO(2)|0CGYNDCWRR]], and $SL(2, \R)$ is homeomorphic to $SO(2) \times \R^2$.
So its fundamental group is $\Z$, and it is not simply connected.
Its universal cover is a Lie group with no faithful finite-dimensional representation.
An element $g \ne \pm I$ is called elliptic, parabolic or hyperbolic when $\abs{\tr g}$ is less than, equal to, or greater than $2$.
Its Lie algebra $\mathfrak{sl}(2, \R)$ of trace-zero real $2 \times 2$ matrices is simple, so $SL(2, \R)$ is a simple Lie group.

## Explanations

### L1 human

$SL(2, \R)$ is the group of linear maps of the plane that keep area and orientation.
Rotations, shears such as $\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}$, and squeezes such as $\begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}$ all belong to it.
The same group also moves the points of the upper half-plane by fractional linear maps, and these are the symmetries of the hyperbolic plane.
It is among the smallest noncompact simple Lie groups, and its study is the entry point to the representation theory of such groups.
</document>

<document source="https://tamnd.github.io/math2/node/Q0K15AWV8H.md" tag="Q0K15AWV8H" kind="concept">
# Linear congruential generator

## Summary

A linear congruential generator is a [[pseudorandom number generator|C5FG6ARWVV]] whose state is an integer modulo $m$, updated by an affine map $x \mapsto (ax + c) \bmod m$.
It is one of the oldest and simplest generators, and its period and lattice structure can be analysed completely with elementary number theory.
Its weaknesses, such as short periods of the low-order bits and points lying on few hyperplanes, made it a classic lesson in generator design.

## Explanations

### L1 human

Take a number, multiply it by a fixed number, add another fixed number, and keep only the remainder after division by a modulus $m$.
Repeat, and divide each result by $m$ to get numbers between $0$ and $1$.
With $m = 8$, multiplier $5$ and increment $1$, starting from $0$, the states are $0, 1, 6, 7, 4, 5, 2, 3$ and then the cycle repeats.
In practice $m$ is huge, such as $2^{64}$, so that the cycle is far longer than any computation.
The same simplicity that makes the generator fast also makes its patterns easy to detect, so it is no longer the default choice in serious simulations.
</document>

<document source="https://tamnd.github.io/math2/node/Q0KB9QDH8V.md" tag="Q0KB9QDH8V" kind="definition">
# Category theory

## Definition

*Category theory* is the branch of mathematics whose objects of study are:

1. [[categories|A2DE66937D]], made of objects, morphisms between objects, and an associative composition of morphisms with identities;
2. [[functors|28NQ0KRHD3]], the maps between categories that preserve composition and identities;
3. [[natural transformations|KYCPCC57FT]], the maps between two functors with the same source and target.

## Remarks

The three levels are tied together: small categories and functors form a category, and functors from one category to another with natural transformations between them form a category again.
Example: the statement "a functor sends isomorphisms to isomorphisms" belongs to category theory, since it only uses composition and identities.
Indeed, if $g \circ f$ and $f \circ g$ are identities, then so are $F(g) \circ F(f)$ and $F(f) \circ F(g)$.
Non-example: the statement "every finite group of prime order is cyclic" is not a categorical statement, since it refers to the elements of a group.
Its main constructions are limits and colimits, adjoint functors, and the Yoneda lemma.
</document>

<document source="https://tamnd.github.io/math2/node/Q0M7PT9VN2.md" tag="Q0M7PT9VN2" kind="statement">
# Error bound for the trapezoidal rule

## Statement

Let $a < b$, let $f \colon [a, b] \to \R$ be twice continuously differentiable, let $N \ge 1$, $h = (b - a)/N$, and let $T_N(f)$ be the composite [[trapezoidal rule|FH0538XCZR]] with $N$ subintervals.
Then there is $\xi \in [a, b]$ with
$$\int_a^b f(x)\, dx - T_N(f) = -\frac{(b - a)\, h^2}{12}\, f''(\xi).$$
In particular,
$$\left| \int_a^b f(x)\, dx - T_N(f) \right| \le \frac{(b - a)\, h^2}{12} \max_{a \le x \le b} \abs{f''(x)}.$$

## Remarks

On one subinterval $[x_{i-1}, x_i]$, the [[interpolation error formula|HASA2XSG66]] gives $f(x) - L(x) = \tfrac{1}{2} f''(\eta_x)(x - x_{i-1})(x - x_i)$ for the linear interpolant $L$.
Since $(x - x_{i-1})(x - x_i) \le 0$ there, the [[mean value theorem for integrals|0N91N98EG5]] gives the local error $-\tfrac{h^3}{12} f''(\xi_i)$.
Summing over the $N$ subintervals and using the intermediate value theorem for $f''$ gives the global formula with $N h^3 = (b - a) h^2$.
Example: $f(x) = x^2$ on $[0, 1]$ with $N = 2$: the error is $\tfrac{1}{3} - \tfrac{3}{8} = -\tfrac{1}{24}$, and the formula gives $-\tfrac{1 \cdot 1/4}{12} \cdot 2 = -\tfrac{1}{24}$.
Without the smoothness hypothesis the rate $h^2$ can fail: for $f(x) = \sqrt{x}$ on $[0, 1]$, whose second derivative is unbounded near $0$, the error is only of order $h^{3/2}$.
Mathlib proves the bound in the form `trapezoidal_error_le_of_c2`.

## Explanations

### L1 human

The trapezoidal rule replaces the curve by straight chords, and the mistake on each piece depends on how much the curve bends there.
The theorem says the total mistake is at most a constant times the largest bending times the square of the piece width.
So using twice as many pieces makes the error about four times smaller.
The sign also tells something: for a convex function, with $f'' \ge 0$, the trapezoidal rule overestimates the integral.

### L2 human

This bound is the model for error analysis of composite [[Newton–Cotes formulas|BQPQAY1FDJ]]: a local error of order $h^{d+2} f^{(d+1)}$ summed over $N$ pieces gives a global error of order $h^{d+1}$.
It also tells how many subintervals are needed for a given tolerance, once a bound on $\abs{f''}$ is known.
The Euler–Maclaurin formula refines it into an expansion of the error in even powers of $h$, which is the basis of Richardson extrapolation and [[Romberg's method|VP40T7MHGM]].
For smooth periodic functions all the correction terms vanish, which explains the very fast convergence of the trapezoidal rule in that case.
</document>

<document source="https://tamnd.github.io/math2/node/Q0NXXFADT9.md" tag="Q0NXXFADT9" kind="concept">
# Planar ternary ring

## Summary

A planar ternary ring is a set with a [[ternary operation|WC47MFR6GW]] $T(x, m, b)$ that plays the role of $x m + b$ in the equation $y = x m + b$ of a line.
Marshall Hall showed that every [[projective plane|DANFNN8C4Y]] can be coordinatized by a planar ternary ring.
Conversely, every planar ternary ring gives a projective plane.
Algebraic properties of the ternary ring match geometric properties of the plane.
For instance, the plane is a [[Desarguesian plane|HPX85PA8NX]] when the ring is a division ring with $T(x, m, b) = x m + b$.
It is the main tool for constructing and classifying [[non-Desarguesian planes|HYE5FT4M34]].

## Explanations

### L1 human

In the usual plane, a non-vertical line is the set of points with $y = x m + b$.
An abstract projective plane may have no multiplication and no addition.
Still, a rule tells which $y$ belongs to $x$ on the line with slope $m$ and intercept $b$.
That rule is the ternary operation $T(x, m, b)$.
Its axioms say exactly what is needed for two points to determine one line and two lines to meet once.
</document>

<document source="https://tamnd.github.io/math2/node/Q0PAZ5QG79.md" tag="Q0PAZ5QG79" kind="definition">
# Pointed space

## Definition

A *pointed space* is a pair $(X, x_0)$ where $X$ is a [[topological space|0CQHFPGJ45]] and $x_0 \in X$.
The point $x_0$ is the *base point*.
A *pointed map* $f \colon (X, x_0) \to (Y, y_0)$ is a [[continuous map|0VGFHTM75Y]] $f \colon X \to Y$ with $f(x_0) = y_0$.

## Remarks

Pointed spaces and pointed maps form a category, often written $\mathbf{Top}_*$.
A pointed homotopy between pointed maps is a homotopy relative to the base point, and $[X, Y]_*$ denotes the set of pointed homotopy classes.
Example: $(S^n, e_1)$ with $e_1 = (1, 0, \ldots, 0)$ is a pointed space.
Non-example: the empty space has no point, so it carries no structure of a pointed space.
Every space $X$ gives a pointed space $X_+ = X \sqcup \set{*}$ with a new base point $*$.
</document>

<document source="https://tamnd.github.io/math2/node/Q0RW86P85F.md" tag="Q0RW86P85F" kind="concept">
# Discrete geometry

## Summary

Discrete geometry is the part of geometry that studies finite or discrete configurations of simple objects, such as points, lines, circles, polygons and convex bodies.
Its questions are combinatorial: how many incidences, distances or intersections such configurations can have.
Typical topics are [[tilings|GKJK293DXS]], [[sphere packings|HF32CBN5MA]], [[polytopes|QP7GEKDTYX]], incidence problems and geometric graphs.
It is closely tied to convex geometry and to computational geometry.

## Explanations

### L1 human

Discrete geometry asks counting questions about shapes placed in the plane or in space.
How many pairs among $n$ points can be at distance exactly $1$?
How many lines through at least two of the points must there be?
How densely can equal coins be packed on a table?
Such questions are easy to state, but many of them are still open.
</document>

<document source="https://tamnd.github.io/math2/node/Q0TY362K0V.md" tag="Q0TY362K0V" kind="concept">
# Complemented subspace

## Summary

A closed subspace of a Banach space is complemented when it has a closed [[complementary subspace|0M83V17KHX]].
Equivalently, it is complemented when it is the range of a bounded linear projection.
In linear algebra every subspace has a complement, but in Banach spaces a closed complement may fail to exist.
Complemented subspaces are the pieces into which a Banach space splits as a direct sum.
Much of the isomorphic theory of Banach spaces studies them.

## Explanations

### L1 human

In the plane, a line through the origin can be paired with another line so that every vector is a unique sum of one vector from each line.
In an infinite-dimensional Banach space one wants the second piece to be closed as well, so that the splitting is continuous.
In a Hilbert space this always works, using the orthogonal complement.
In other Banach spaces it can fail: the space $c_0$ of null sequences sits inside the space $\ell^\infty$ of bounded sequences with no closed complement.
</document>

<document source="https://tamnd.github.io/math2/node/Q0XKJ01SY6.md" tag="Q0XKJ01SY6" kind="definition">
# Spinor bundle

## Definition

Let $(M, g)$ be an oriented Riemannian manifold of dimension $n \ge 2$ with a [[spin structure|RSPZ9TCFQ8]] $(P, \xi)$, where $P$ is a principal $\operatorname{Spin}(n)$-bundle.
Let $\Sigma_n$ be a complex vector space of dimension $2^{\floor{n/2}}$ that is an irreducible module over the complexified [[Clifford algebra|A38B1QFK20]] $\mathbb{C}\ell_n$.
Let $\kappa \colon \operatorname{Spin}(n) \to GL(\Sigma_n)$ be the restriction of this module structure to $\operatorname{Spin}(n) \subset \mathbb{C}\ell_n$.
It is the complex spin representation of the [[spin group|JJSXXJQGEH]].
The *spinor bundle* of $(M, g, P, \xi)$ is the associated complex [[vector bundle|P4DZM975BV]]
$$S = P \times_\kappa \Sigma_n = (P \times \Sigma_n) / \sim, \qquad (p \cdot a, v) \sim (p, \kappa(a) v) \text{ for } a \in \operatorname{Spin}(n).$$

## Remarks

The bundle $S$ has rank $2^{\floor{n/2}}$.
Clifford multiplication of $\mathbb{C}\ell_n$ on $\Sigma_n$ gives a bundle map $TM \otimes S \to S$, $X \otimes \psi \mapsto X \cdot \psi$, with $X \cdot X \cdot \psi = -g(X, X) \psi$.
The Levi-Civita connection lifts to a connection on $S$, and the Dirac operator is $D\psi = \sum_i e_i \cdot \nabla_{e_i} \psi$ for a local orthonormal frame $e_1, \dots, e_n$.
If $n$ is even, $S = S^+ \oplus S^-$ splits into half-spinor bundles of rank $2^{n/2 - 1}$.

Example: on $\R^n$ with its unique spin structure $P = \R^n \times \operatorname{Spin}(n)$, the spinor bundle is the trivial bundle $\R^n \times \Sigma_n$, and spinor fields are maps $\R^n \to \C^{2^{\floor{n/2}}}$.

Non-example: the bundle $P \times_{\kappa'} \C^n$ for the representation $\kappa'$ of $\operatorname{Spin}(n)$ through $\rho \colon \operatorname{Spin}(n) \to SO(n) \subset GL(\C^n)$ is the complexified tangent bundle $TM \otimes \C$, not the spinor bundle.
The element $-1 \in \operatorname{Spin}(n)$ acts as the identity in $\kappa'$ but as $-\id$ in the spin representation $\kappa$.
</document>

<document source="https://tamnd.github.io/math2/node/Q0YPJZH60H.md" tag="Q0YPJZH60H" kind="definition">
# Tridiagonal matrix

## Definition

Let $R$ be a [[ring|0B1261B5ED]] and let $A = (a_{ij})$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $A$ is *tridiagonal* when $a_{ij} = 0$ for all $i, j$ with $\abs{i - j} > 1$.

## Remarks

Example: $\begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}$ is tridiagonal, because the only entries with $\abs{i - j} > 1$ are $a_{13}$ and $a_{31}$, and both are $0$.
Non-example: $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ is not tridiagonal, because $a_{13} = 1$ and $\abs{1 - 3} = 2$.
A tridiagonal matrix is often written with diagonal $a_1, \dots, a_n$, superdiagonal $b_1, \dots, b_{n-1}$ and subdiagonal $c_1, \dots, c_{n-1}$.
Then the leading principal minors $f_k$ satisfy $f_k = a_k f_{k-1} - b_{k-1} c_{k-1} f_{k-2}$, with $f_0 = 1$ and $f_{-1} = 0$.
Equivalently, $A$ is tridiagonal when it is both upper and lower [[Hessenberg|YCEEWJTX56]].
</document>

<document source="https://tamnd.github.io/math2/node/Q104C67P26.md" tag="Q104C67P26" kind="statement">
# Stationary-action principle

## Statement

Let a physical system be described by generalized coordinates $q = (q_1, \dots, q_n)$ in an open set $Q \subseteq \R^n$, and by a Lagrangian $L \colon [t_0, t_1] \times Q \times \R^n \to \R$ of class $C^2$.
For a $C^2$ path $q \colon [t_0, t_1] \to Q$, its *action* is the [[functional|1EQM0DJ4NC]]
$$S[q] = \int_{t_0}^{t_1} L\big(t, q(t), \dot q(t)\big) \, dt.$$
Fix endpoints $q(t_0) = a$ and $q(t_1) = b$.
The principle states that the motions of the system between them are exactly the $C^2$ paths $q$ with these endpoints that satisfy one condition.
The [[first variation|5B8NFYB2XQ]] of $S$ at $q$ vanishes: $\frac{d}{d\varepsilon} S[q + \varepsilon h] \big|_{\varepsilon = 0} = 0$ for every $C^2$ map $h$ with $h(t_0) = h(t_1) = 0$.
Equivalently, the motions are the solutions of the [[Euler–Lagrange equations|PS0VV149ET]]
$$\frac{d}{dt} \frac{\partial L}{\partial \dot q_i} = \frac{\partial L}{\partial q_i}, \qquad i = 1, \dots, n.$$

## Remarks

The equivalence of the two forms is a theorem of the calculus of variations.
Differentiating under the integral and integrating by parts gives $\frac{d}{d\varepsilon} S[q + \varepsilon h] \big|_{\varepsilon = 0} = \int_{t_0}^{t_1} \sum_i \left( \frac{\partial L}{\partial q_i} - \frac{d}{dt} \frac{\partial L}{\partial \dot q_i} \right) h_i \, dt$.
By the fundamental lemma of the calculus of variations, this vanishes for all $h$ exactly when the Euler–Lagrange equations hold.
The physical content is the choice of $L$; for a particle of mass $m$ in a potential $V$, the choice $L = \frac12 m \abs{\dot q}^2 - V(q)$ gives Newton's law $m \ddot q = -\nabla V(q)$.
The principle asks for a stationary action, not a least one.
For the harmonic oscillator $L = \frac12 (\dot q^2 - \omega^2 q^2)$ with $\omega > 0$, the solution is a minimum of $S$ when $t_1 - t_0 < \pi / \omega$, but not when $t_1 - t_0 > \pi / \omega$: then $h(t) = \sin(\pi (t - t_0)/(t_1 - t_0))$ makes the second variation $\int (\dot h^2 - \omega^2 h^2) \, dt$ negative.
The same principle, with an action given by the integral of a Lagrangian density over space-time, gives the field equations of electromagnetism and of general relativity.
It is also the setting of Noether's theorem on symmetries and conservation laws.
Hamilton's principle in mechanics and Maupertuis's principle are particular forms of it.

## Explanations

### L1 human

Nature seems to choose its motions by an economy rule.
Consider all the ways a system could go from one state to another in a given time.
It takes one for which a single number, the action, does not change to first order when the path is bent slightly.
From this one rule, with the right choice of Lagrangian, come the equations of mechanics, optics, electromagnetism and gravitation.

### L2 human

In practice the principle is a recipe: write the kinetic and potential energies in any convenient coordinates, form $L$, and read off the equations of motion from the Euler–Lagrange equations.
This avoids computing constraint forces, and it makes symmetries visible, since a symmetry of $L$ gives a conserved quantity by Noether's theorem.
The Legendre transform of $L$ leads to Hamilton's equations, and the action itself, as a function of the endpoints, solves the Hamilton–Jacobi equation.
</document>

<document source="https://tamnd.github.io/math2/node/Q11N8X99FK.md" tag="Q11N8X99FK" kind="concept">
# Intuitionistic logic

## Summary

Intuitionistic logic is the logic of constructive reasoning, in which a statement counts as true only when it has a proof.
It is classical logic without the [[law of excluded middle|0RYYK8PG4S]] $\varphi \lor \neg\varphi$ and without double negation elimination $\neg\neg\varphi \to \varphi$.
Its semantics include [[Heyting algebras|NG943XNQWH]], [[Kripke models|NGHWZ9Y7ZV]] and the [[Brouwer–Heyting–Kolmogorov interpretation|7B53ZGXXEQ]].
Through the [[Curry–Howard correspondence|A77FER1Q5H]], its proofs correspond to typed programs.

## Explanations

### L1 human

In intuitionistic logic, to prove "$A$ or $B$" you must prove $A$ or prove $B$, and to prove "there exists $x$" you must exhibit one.
So "$A$ or not $A$" is not a law, because you may have a proof of neither.
For example, a classical proof that some pair of irrational numbers $a, b$ has $a^b$ rational, without telling which pair, is not an intuitionistic proof.
Everything provable intuitionistically is provable classically, but not conversely.
</document>

<document source="https://tamnd.github.io/math2/node/Q12AB1MJ64.md" tag="Q12AB1MJ64" kind="concept">
# Energy functional on paths

## Summary

The energy of a path in a [[Riemannian manifold|7CBT4SM83Y]] is half the integral of the squared speed.
Among paths with fixed ends, its critical points are exactly the [[geodesics|NSYMQMHPDN]] with constant speed.
In variational work it is preferred to the length.
It is not invariant under reparametrization, so its critical points are better behaved and its Hessian is easier to compute.
It is the main example of a function to which Morse theory on infinite-dimensional manifolds applies.

## Explanations

### L1 human

The length of a path does not care how fast it is travelled; the energy does.
The energy is smallest, for a given length and travel time, when the speed is constant.
So minimizing energy finds the shortest path and also a uniform way to travel along it.
For example, in the plane the path $t \mapsto (t, 0)$ for $0 \le t \le 1$ has energy $\frac{1}{2}$, while $t \mapsto (t^2, 0)$ covers the same segment with energy $\frac{2}{3}$.
</document>

<document source="https://tamnd.github.io/math2/node/Q134D9EMAZ.md" tag="Q134D9EMAZ" kind="statement">
# Shilnikov theorem

## Statement

Let $\dot x = F(x)$ be a smooth vector field on $\R^3$.
Let $x_0$ be a [[hyperbolic equilibrium|01TPMXM6NM]] whose Jacobian matrix $DF(x_0)$ has [[eigenvalues|0BQAA2BK3Q]] $\lambda > 0$ and $-\rho \pm i \omega$ with $\rho > 0$ and $\omega \ne 0$.
Let $\Gamma$ be a [[homoclinic orbit|6DBSQKZK4W]] to $x_0$.
Assume the Shilnikov condition $\rho < \lambda$.
Then every neighbourhood of $\Gamma \cup \set{x_0}$ contains infinitely many [[periodic orbits|0AA97XA70K]] of saddle type.

## Remarks

Such an equilibrium is called a saddle-focus: orbits near it spiral in along a plane and leave along a line.
The proof studies the return map to a small section transverse to $\Gamma$.
Near $x_0$ the flow is close to linear, and the passage time near the equilibrium makes the returning image a spiral that winds infinitely often.
When $\rho < \lambda$ this spiral is stretched rather than contracted, and it crosses the section so as to form infinitely many [[Smale horseshoes|N63Z6EA6YS]].
Each horseshoe carries infinitely many periodic orbits, which gives the conclusion.
The condition $\rho < \lambda$ is needed: if $\rho > \lambda$, a sufficiently small neighbourhood of $\Gamma$ contains at most one periodic orbit.
The theorem is due to Leonid Shilnikov.

## Explanations

### L1 human

Imagine a flow in space with a point that pulls nearby orbits inward along a spiralling disc and pushes them out along a line.
If an orbit leaves along the line and comes back spiralling into the point, the theorem says that complicated behaviour is forced near this loop.
When the push outwards is stronger than the pull inwards, infinitely many closed orbits appear near the loop.
This is a simple test for chaos that can be checked from the eigenvalues at one point.

### L2 human

The theorem is a main source of chaos in three-dimensional flows, because it reduces chaos to the existence of a single homoclinic orbit plus an inequality on eigenvalues.
A [[homoclinic bifurcation|JQYCTY3Q8V]] to a saddle-focus that satisfies the Shilnikov condition is therefore a route to chaos in families of flows.
In the plane, by contrast, the breaking of a homoclinic loop creates at most one limit cycle.
</document>

<document source="https://tamnd.github.io/math2/node/Q19EMZ4CP7.md" tag="Q19EMZ4CP7" kind="definition">
# Wiener amalgam space

## Definition

Let $d \ge 1$, let $1 \le p, q \le \infty$ and let $Q = [0, 1)^d$.
For a measurable function $f \colon \R^d \to \C$ and $k \in \Z^d$, let $a_k(f) = \norm{f \cdot 1_{k + Q}}_{L^p(\R^d)}$, the norm in the [[Lebesgue space|SASP14MA1Z]] $L^p$ of the restriction of $f$ to the cube $k + Q$.
The *Wiener amalgam space* $W(L^p, \ell^q)(\R^d)$ is the set of measurable $f$ with
$$\norm{f}_{W(L^p, \ell^q)} = \Big( \sum_{k \in \Z^d} a_k(f)^q \Big)^{1/q} < \infty,$$
with the supremum over $k$ in place of the sum when $q = \infty$.
Functions that agree almost everywhere are identified.

## Remarks

$L^p$ is the local component and $\ell^q$ the global component.
In the general theory of Feichtinger, $L^p$ is replaced by a Banach space $B$ of local behaviour.
The space $\ell^q$ is replaced by a solid sequence or function space $C$ of global behaviour, and the sharp cut-off $1_{k + Q}$ by a smooth partition of unity.
Two facts follow from the definition:

1. $W(L^p, \ell^p) = L^p(\R^d)$ with equal norms.
2. If $p_1 \ge p_2$ and $q_1 \le q_2$, then $W(L^{p_1}, \ell^{q_1}) \subseteq W(L^{p_2}, \ell^{q_2})$, since the cube $Q$ has measure $1$.

Example: let $d = 1$ and $f(x) = 1/(1 + \abs{x})$, with $p = \infty$.
On $[k, k+1)$ the supremum of $f$ is $1/(1 + \min(\abs{k}, \abs{k + 1}))$.
These numbers behave like $1/\abs{k}$, so their $q$-th powers have a finite sum for every $q > 1$.
Hence $f \in W(L^\infty, \ell^q)(\R)$ for $q > 1$.

Non-example: the same $f$ is not in $W(L^\infty, \ell^1)(\R)$, because the sum of these numbers is a harmonic series and diverges.
</document>

<document source="https://tamnd.github.io/math2/node/Q1H6BCKTBE.md" tag="Q1H6BCKTBE" kind="statement">
# Kissing number problem in three dimensions

## Statement

The [[kissing number|B408FZYXF9]] in dimension $3$ is $\tau(3) = 12$.
That is, at most $12$ unit balls in $\R^3$ with pairwise disjoint interiors can touch a given unit ball, and $12$ is attained.

## Remarks

The lower bound comes from the face-centred cubic packing: the $12$ vectors $(\pm 1, \pm 1, 0)$, $(\pm 1, 0, \pm 1)$, $(0, \pm 1, \pm 1)$, scaled to norm $2$, are pairwise at distance at least $2$.
The configuration of the $12$ vertices of a regular icosahedron also works, and in it no two outer balls touch, so they can move.
The question goes back to a discussion between Isaac Newton, who held that the answer is $12$, and David Gregory, who thought $13$ might fit.
Kurt Schütte and B. L. van der Waerden gave the first complete proof in 1953.
Later proofs bound the number of points on the sphere with pairwise angular distance at least $60^\circ$, for example with a refined form of the [[Delsarte linear programming bound|7RPEDHAT7R]].

## Explanations

### L1 human

Hold one ball and try to make as many equal balls as possible touch it.
Twelve fit, with some space left over, so it seems a thirteenth might squeeze in after shuffling.
The theorem says it never does.

### L2 human

The plain linear programming bound gives only $\tau(3) \le 13$, so the three-dimensional case needs extra geometric arguments.
In dimensions $8$ and $24$ the same linear programming method is exactly sharp and gives $240$ and $196560$.
Dimension $4$ was settled by Oleg Musin, who showed $\tau(4) = 24$.
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# Information-based complexity

## Summary

Information-based complexity studies the minimal cost of solving continuous problems, such as integration or approximation of functions, to a given accuracy.
An algorithm may learn the input only through finitely many measurements, for example function values, so the information it has is partial.
The central quantities are the smallest error reachable with $n$ measurements and the smallest $n$ that reaches a given error $\varepsilon$.
It explains, for example, when [[Monte Carlo|HWP11G3HTN]] or [[quasi-Monte Carlo|ABSRB2KXA6]] methods can beat the curse of dimensionality.

## Explanations

### L1 human

A computer cannot see a whole function; it can only ask for its values at finitely many points.
Different functions can give the same answers, so any method must guess, and its error is limited by this uncertainty.
Information-based complexity asks how many questions are needed, at least, to reach a given accuracy.
Take the functions on $[0,1]$ whose slope is at most $1$ in size.
To integrate them to within $\varepsilon$, one needs about $1/(4\varepsilon)$ function values, and the midpoint rule achieves this.
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# Category theory

## Summary

Category theory is the part of mathematics that studies [[categories|A2DE66937D]], [[functors|28NQ0KRHD3]] between them and [[natural transformations|KYCPCC57FT]] between functors.
It describes mathematical structures through the maps between them rather than through their elements.
It was introduced by Samuel Eilenberg and Saunders Mac Lane in the 1940s, first to make precise the word "natural" in algebraic topology.
Today it is the common language of algebraic topology, algebraic geometry, homological algebra and parts of logic and computer science.

## Explanations

### L1 human

Many kinds of objects come with their own kind of maps: sets with functions, groups with homomorphisms, spaces with continuous maps.
Category theory looks only at the objects, the arrows between them, and how arrows compose.
From this one picture it defines notions that work in every field at once, such as products, isomorphisms and universal properties.
For example, the product of two sets, of two groups and of two spaces are all instances of one categorical notion of product.
A theorem proved at this level holds in every field that fits the picture.
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# Hausdorff measure

## Summary

The $s$-dimensional Hausdorff measure of a subset of a [[metric space|0VKD1TP7F3]] measures its $s$-dimensional size, for any real $s \ge 0$.
It is built from covers by small sets, each counted by its [[diameter|0KW3PN547K]] to the power $s$.
For whole numbers $s = n$ it extends length, area and volume to rough sets, and in $\R^n$ it agrees with [[Lebesgue measure|2Z2QVWZ262]] up to a constant.
It defines Hausdorff dimension and is the basic measure of fractal geometry.

## Explanations

### L1 human

To measure the length of a curve, cover it by many tiny pieces and add up their sizes.
For area one adds the squares of the sizes, and for volume the cubes.
Hausdorff measure allows any power $s$, also non-integer powers such as $0.63$.
For a given set, there is one critical power: below it the measure is infinite, above it the measure is zero.
For example, a line segment of length $1$ has $1$-dimensional measure $1$ and $2$-dimensional measure $0$.
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# Khinchin integral

## Summary

The Khinchin integral is an integral on an interval defined through primitives rather than through sums.
It is also called the Denjoy–Khinchin integral or the wide Denjoy integral.
A function is Khinchin integrable when it is, almost everywhere, the [[approximate derivative|4W31MKE01V]] of a continuous function that is generalized [[absolutely continuous|ZT6F4C2YHK]].
It extends the [[Henstock–Kurzweil integral|FCGHXDGVV9]], and hence the [[Lebesgue integral|DWJF7ABPR1]] and the [[Perron integral|M077AXJACA]].

## Explanations

### L1 human

In Lebesgue's theory, the indefinite integrals are exactly the absolutely continuous functions, and each one has the integrand as its derivative almost everywhere.
The Khinchin integral relaxes this in two ways.
The primitive only has to be absolutely continuous on each of countably many pieces of the interval.
It only has to have the function as an approximate derivative, a derivative that may ignore thin sets of points.
The integral of $f$ is then the change of the primitive from one end of the interval to the other.
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# Dual of Lp

## Statement

Let $(X, \Sigma, \mu)$ be a measure space, let $1 \le p < \infty$, and let $q \in (1, \infty]$ be the conjugate exponent, with $1/p + 1/q = 1$.
If $p = 1$, assume moreover that $\mu$ is [[σ-finite|WEBBT18D4X]].
For $g \in L^q(\mu)$, let $\Lambda_g$ be the functional on [[$L^p(\mu)$|SASP14MA1Z]] given by
$$\Lambda_g(f) = \int_X f g \, d\mu.$$
Then $g \mapsto \Lambda_g$ is an [[isometric isomorphism|0ZXSF2YQV2]] of $L^q(\mu)$ onto the [[dual space|07JBSZE0V4]] $L^p(\mu)^*$.

## Remarks

Hölder's inequality gives $\abs{\Lambda_g(f)} \le \norm{f}_p \norm{g}_q$, so $\norm{\Lambda_g} \le \norm{g}_q$.
For $1 < p < \infty$, the choice $f = \abs{g}^{q-1} \overline{\sgn g}$ gives equality, so the map is an isometry.
Surjectivity is the main point.
Let $\mu$ be $\sigma$-finite and let $\Lambda$ be a bounded functional.
Then $E \mapsto \Lambda(\mathbf{1}_E)$, on sets of finite measure, is a set function absolutely continuous with respect to $\mu$.
The [[Radon–Nikodym theorem|1ZSZS5G1Q4]] gives $g$.
For $1 < p < \infty$ and a general measure space, one reduces to the $\sigma$-finite case, or uses the uniform convexity of $L^p$.

Example: for counting measure on $\N$, the theorem says that $(\ell^p)^* = \ell^q$ for $1 \le p < \infty$.

The $\sigma$-finiteness for $p = 1$ cannot be dropped.
Let $X = \set{x}$ with $\mu(\set{x}) = \infty$.
Then $L^1(\mu) = \set{0}$, whose dual is $\set{0}$, but $L^\infty(\mu)$ is the one-dimensional space of constants.

The case $p = \infty$ is excluded.
The map $g \mapsto \Lambda_g$ from $L^1$ into $(L^\infty)^*$ is an isometry, but in general it is not onto.
For example, take the functional on convergent sequences that gives the limit.
By the Hahn–Banach theorem it extends to a bounded functional on $\ell^\infty$, and this extension is not given by any element of $\ell^1$.

## Explanations

### L1 human

Every continuous linear way of turning a function in $L^p$ into a number is integration against a fixed function $g$.
The function $g$ lives in the partner space $L^q$, where $1/p + 1/q = 1$.
So $L^2$ is its own partner, $L^1$ is paired with $L^\infty$, and $L^{3/2}$ is paired with $L^3$.

### L2 human

The theorem identifies $(L^p)^* = L^q$ for $1 \le p < \infty$ under the stated hypotheses.
For $1 < p < \infty$ it follows that $L^p$ is reflexive, since $(L^p)^{**} = (L^q)^* = L^p$.
Weak convergence in $L^p$ then means convergence of $\int f_n g \, d\mu$ for each $g \in L^q$, and bounded sequences have weakly convergent subsequences.
Neither $L^1$ nor $L^\infty$ is reflexive in general, which is the source of many technical difficulties.
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# Axiom of constructibility

## Definition

Work in [[ZF|0HE5WDQ9FJ]], and let $L = \bigcup_\alpha L_\alpha$ be the [[constructible universe|6TF2JBTPPJ]], where $\alpha$ runs over the [[ordinals|0YSCSX5YZC]].
The *axiom of constructibility* is the sentence
$$\forall x \, \exists \alpha \, (\alpha \text{ is an ordinal} \wedge x \in L_\alpha).$$
It is written $V = L$, where $V$ is the class of all sets.

## Remarks

The sentence holds inside $L$ itself: the construction of $L_\alpha$ is absolute for transitive models of ZF that contain $\alpha$, so the class $L$ computed inside $L$ is again $L$.
This gives the relative consistency of $V = L$ with ZF.
For a non-example, take a forcing extension $L[c]$ of $L$ by a Cohen real $c$; the real $c$ is not in $L$, so $V = L$ fails in $L[c]$.
Under $V = L$ there is a well-order of the reals that is $\Delta^1_2$ in the [[projective hierarchy|8DZTNVTGY2]].
Scott proved that $V = L$ implies that there is no [[measurable cardinal|AKBWT323RQ]].
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# Direct limit of algebras

## Definition

Let $(I, \le)$ be a nonempty [[directed set|D9RY4667K6]].
A *directed system* of algebras over $I$ consists of [[algebras|QJGPT2W0FT]] $\mathbf{A}_i$ of one signature, for $i \in I$, and [[homomorphisms|KBXSZC6BV1]] $f_{ij} \colon \mathbf{A}_i \to \mathbf{A}_j$ for $i \le j$, such that:

1. $f_{ii}$ is the identity of $\mathbf{A}_i$ for each $i$;
2. $f_{jk} \circ f_{ij} = f_{ik}$ whenever $i \le j \le k$.

On the disjoint union of the universes, let $a \sim b$ for $a \in A_i$ and $b \in A_j$ when $f_{ik}(a) = f_{jk}(b)$ for some $k \ge i, j$.
The *direct limit* $\varinjlim \mathbf{A}_i$ has as universe the set of classes $[a]$ of this [[equivalence relation|0G8VEZ3WTK]].
For an $n$-ary operation $f$ and $a_m \in A_{i_m}$, choose $k \ge i_1, \dots, i_n$ and put $f([a_1], \dots, [a_n]) = [f^{\mathbf{A}_k}(f_{i_1 k}(a_1), \dots, f_{i_n k}(a_n))]$.

## Remarks

The relation $\sim$ is transitive and the operations do not depend on the choices of $k$ and representatives, because $I$ is directed and the $f_{ij}$ are homomorphisms.
For a nullary operation $c$, take any $k \in I$ and put $c = [c^{\mathbf{A}_k}]$.
The maps $a \mapsto [a]$ are homomorphisms $\mathbf{A}_i \to \varinjlim \mathbf{A}_i$, and they make the direct limit a colimit of the system.

Example: let $\mathbf{A}_n = \Z/p^n\Z$ for $n \ge 1$, with $f_{nm}(x) = p^{m-n} x$ for $n \le m$.
These maps are injective homomorphisms satisfying clauses 1 and 2, and the direct limit is the Prüfer $p$-group.

Non-example: over $I = \N$, take $\mathbf{A}_n = \Z$ for all $n$ with $f_{nm}(x) = 2x$ for all $n < m$ and $f_{nn} = \id$.
Clause 2 fails, since $f_{12} \circ f_{01}(x) = 4x$ while $f_{02}(x) = 2x$.
Setting instead $f_{nm}(x) = 2^{m-n} x$ gives a directed system with direct limit $\Z[1/2]$.
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# Foundations of mathematics

## Definition

The *foundations of mathematics* is the part of mathematical logic and philosophy that studies the basic framework of mathematics.
It studies:

1. the formal languages in which mathematical statements are written, such as [[predicate logic|086QKPF0A6]];
2. the axiom systems from which mathematical theorems are derived, such as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]];
3. the notion of [[proof|0B94TKFQJX]], and the properties of the whole framework, such as its consistency and its completeness.

## Remarks

For example, the [[Peano axioms|0QFT7Z76N5]] can be interpreted in set theory.
Each natural number becomes a set, and each axiom becomes a theorem of set theory.
A theorem about prime numbers alone, such as the infinitude of primes, is not a foundational result: it is about objects inside the framework, not about the framework.
The main schools of the early twentieth century were [[logicism|5PGSQYB5Y1]], [[formalism|46JCQ6KDA4]] and [[intuitionism|XQHF63BBYF]].
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# Semi-local ring

## Definition

A [[commutative ring|0310REJJY8]] $R$ is *semi-local* if it has only finitely many [[maximal ideals|0PV74TR5FP]].

## Remarks

If $\mathfrak m_1, \ldots, \mathfrak m_n$ are the maximal ideals, then the Chinese remainder theorem gives
$$R / J(R) \cong R/\mathfrak m_1 \times \cdots \times R/\mathfrak m_n,$$
where $J(R) = \mathfrak m_1 \cap \cdots \cap \mathfrak m_n$ is the [[Jacobson radical|XA0DCNH7WY]].
For a commutative ring $R$, being semi-local is equivalent to $R/J(R)$ being a finite product of fields.
For noncommutative rings, the usual definition asks that $R/J(R)$ be semisimple.

Example: let $S = \Z \setminus (2\Z \cup 3\Z)$, the set of integers prime to $6$.
It is a multiplicative set, and by [[prime ideals of a localization|W8QNZZ4QJS]] the prime ideals of $S^{-1}\Z$ correspond to the primes of $\Z$ disjoint from $S$.
These are $(0)$, $2\Z$ and $3\Z$, so $S^{-1}\Z$ has exactly the two maximal ideals generated by $2$ and $3$.
Every [[local ring|MZ02BKPHJA]] is semi-local, and so is $\Z/n\Z$ for $n \ge 2$, with one maximal ideal for each prime divisor of $n$.

Non-example: $\Z$ is not semi-local, because $p\Z$ is a maximal ideal for each of the infinitely many primes $p$.
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# Asymptotic analysis

## Summary

Asymptotic analysis describes the behaviour of functions near a limit point, such as $x \to \infty$ or a small parameter $\varepsilon \to 0$.
Its basic tools are [[big O|0TGV80E6A4]] and [[little o|00T5EPFQB9]] notation, [[asymptotic equivalence|0QR92XBV6H]] and asymptotic expansions.
An asymptotic expansion need not converge, yet a few of its terms often give an excellent approximation.
In differential equations, it gives approximate solutions when exact ones are out of reach, as in perturbation methods.

## Explanations

### L1 human

Asymptotic analysis asks what a function looks like when its argument becomes very large or very small.
It keeps the main terms and controls the size of what is left out.
For example, $x^2 + 3x + 7$ behaves like $x^2$ for large $x$, since the ratio tends to $1$.
A finer statement is $x^2 + 3x + 7 = x^2 + 3x + O(1)$.
Stirling's formula $n! \sim \sqrt{2 \pi n} \, (n / e)^n$ is a famous result of this kind.
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# Duhamel's principle

## Statement

Let $I$ be an open interval, and let $A \colon I \to \R^{n \times n}$ and $f \colon I \to \R^n$ be continuous.
Let $\Phi$ be a [[fundamental matrix|0H1GN6P9R5]] of the homogeneous [[linear system|037ZNY7NAG]] $x' = A(t) x$ on $I$, and put $\Phi(t, s) = \Phi(t) \Phi(s)^{-1}$ for $t, s \in I$.
Let $t_0 \in I$ and $x_0 \in \R^n$.
Then the [[initial value problem|0FXDW992DS]]
$$x' = A(t) x + f(t), \qquad x(t_0) = x_0$$
has exactly one solution on $I$, and it is
$$x(t) = \Phi(t, t_0)\, x_0 + \int_{t_0}^{t} \Phi(t, s)\, f(s)\, ds.$$
If $A$ is constant, then $\Phi(t, s) = e^{(t - s) A}$, the [[matrix exponential|020M04FN1H]].

## Remarks

Differentiating under the integral sign gives $x'(t) = A(t) \Phi(t, t_0) x_0 + \Phi(t, t) f(t) + \int_{t_0}^t A(t) \Phi(t, s) f(s)\, ds = A(t) x(t) + f(t)$, since $\Phi(t, t)$ is the identity.
Uniqueness holds because the difference of two solutions solves the homogeneous system with initial value $0$, so it is zero.
The formula is the matrix form of [[variation of parameters|0Y6TH5PQVS]].

Example: for $x' = -x + 1$ and $x(0) = 0$, one has $\Phi(t, s) = e^{-(t - s)}$.
The formula gives $x(t) = \int_0^t e^{-(t - s)}\, ds = 1 - e^{-t}$, and indeed $x' = e^{-t} = -x + 1$.

## Explanations

### L1 human

The principle says how to solve a linear equation with an outside force once one can solve it without the force.
At each instant $s$, the force gives the system a small kick $f(s)\, ds$.
That kick then evolves by the unforced equation from time $s$ to time $t$.
The solution is the free evolution of the initial state plus the sum of all these evolved kicks.

### L2 human

The function $\Phi(t, s)$ is often called the state transition matrix or propagator.
The same idea works for evolution equations such as the heat equation and the wave equation, where $\Phi(t, s)$ becomes a solution operator.
In the theory of $C_0$-semigroups, the formula defines the mild solution of an inhomogeneous abstract Cauchy problem.
For a boundary value problem, the analogous object is the [[Green's function|SH6JY9RN2E]].
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# de Bruijn–Erdős theorem

## Statement

Let $P$ be a finite set with $n$ elements and let $L$ be a family of subsets of $P$, called lines, such that:

1. every line has at least two points;
2. any two distinct points lie on exactly one line;
3. no line contains all points of $P$.

Then $\abs{L} \ge n$.
Moreover $\abs{L} = n$ if and only if $(P, L)$ is either a [[projective plane|DANFNN8C4Y]] or a near-pencil.
A near-pencil consists of one line with $n - 1$ points and the $n - 1$ lines of two points that join the remaining point to them.

## Remarks

For a point $p$ let $r_p$ be the number of lines through $p$, and for a line $\ell$ let $k_\ell$ be its number of points.
If $p$ is not on $\ell$, the lines joining $p$ to the points of $\ell$ are distinct, so $r_p \ge k_\ell$.
Write $b = \abs{L}$ and sum $1/(n(b - r_p))$ and $1/(b(n - k_\ell))$ over all pairs $(p, \ell)$ with $p \notin \ell$: each sum equals $1$.
If $b < n$, then $n r_p > b k_\ell$ for each such pair, so each term of the first sum is larger than the matching term of the second, a contradiction.
A closer look at the case $b = n$ gives the description of equality.

Hypothesis 3 is needed: a single line through all $n \ge 3$ points satisfies clauses 1 and 2 and has only one line.

A consequence: $n$ points of the real plane, not all on one line, determine at least $n$ distinct lines through two or more of them.
The theorem was proved by Nicolaas Govert de Bruijn and Paul Erdős.

## Explanations

### L1 human

Take a finite set of points in which every two points lie on a unique line, and not all points are on one line.
The theorem says there are at least as many lines as points.
There are two ways to reach equality.
One is a projective plane, such as the Fano plane with seven points and seven lines.
The other is the near-pencil: a line with all points but one, plus the lines through the last point.

### L2 human

The statement lives in a [[linear space|CX07CWPCDX]], a partial linear space in which any two points are joined.
It is close in spirit to Fisher's inequality for block designs.
Applied to points in the real plane, it gives a lower bound for the number of lines they determine, a question related to the Sylvester–Gallai theorem.
The equality case shows how rigid projective planes are among all finite linear spaces.
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# Legendrian submanifold

## Summary

A Legendrian submanifold of a [[contact manifold|HBQ8BZ7XB5]] of dimension $2n + 1$ is a submanifold of dimension $n$ that is everywhere tangent to the contact hyperplanes.
Legendrian submanifolds play in contact geometry the role that [[Lagrangian submanifolds|H8ACPTMX08]] play in symplectic geometry.
In a contact $3$-manifold they are Legendrian knots, whose classification up to Legendrian isotopy is a rich subject with invariants such as the Thurston–Bennequin number.

## Explanations

### L1 human

A contact structure gives a plane at each point of a three-dimensional space, and these planes twist so much that no surface can follow them.
Curves can still follow them: a Legendrian curve is a curve whose velocity always lies in the given plane.
In $\R^3$ with the planes $\ker(dz - y\,dx)$, the curve $t \mapsto (t, f'(t), f(t))$ is Legendrian for every smooth function $f$, because $dz - y\,dx$ vanishes on its velocity.
Legendrian knots appear in geometric optics as the fronts of propagating waves, lifted with their direction of travel.
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# Fatou's theorem

## Statement

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].
For $\theta \in \R$ and $\alpha > 1$, let $\Gamma_\alpha(\theta) = \set{z \in \mathbb{D} : \abs{z - e^{i\theta}} < \alpha (1 - \abs{z})}$ be the Stolz angle at $e^{i\theta}$.
Say that a function $u$ on $\mathbb{D}$ has nontangential limit $L$ at $e^{i\theta}$ if $u(z) \to L$ as $z \to e^{i\theta}$ inside $\Gamma_\alpha(\theta)$, for every $\alpha > 1$.

1. Let $\varphi$ be a Lebesgue integrable function on the circle ([[Lp space|SASP14MA1Z]] with $p = 1$), and let $u$ be its [[Poisson integral|TCYSWCTYYM]],
$$u(r e^{i\theta}) = \frac{1}{2\pi} \int_0^{2\pi} \frac{1 - r^2}{1 - 2r \cos(\theta - t) + r^2} \, \varphi(e^{it}) \, dt.$$
Then for [[almost every|RARTMA0ZEC]] $\theta$, $u$ has nontangential limit $\varphi(e^{i\theta})$ at $e^{i\theta}$.
2. Let $f$ be a bounded [[holomorphic function|0DG0Z72CQD]] on $\mathbb{D}$.
Then for almost every $\theta$, $f$ has a nontangential limit at $e^{i\theta}$.
In particular the radial limit $\lim_{r \to 1^-} f(r e^{i\theta})$ exists for almost every $\theta$.

## Remarks

Proof idea: the nontangential maximal function of $u$ is bounded by a constant times the Hardy–Littlewood maximal function of $\varphi$, which is finite almost everywhere.
The limit holds for continuous $\varphi$, and the maximal estimate extends it to all integrable $\varphi$.
For part 2, the functions $f(r_k e^{it})$ have a weak-star limit $\varphi$ in $L^\infty$ along some sequence $r_k \to 1$, and $f$ is the Poisson integral of $\varphi$, so part 1 applies.

Example: the singular inner function $\exp\left(-\frac{1 + z}{1 - z}\right)$ is bounded and has radial limit of modulus $1$ at every point $e^{it} \ne 1$, while its radial limit at $1$ is $0$.

Boundedness cannot simply be dropped: there are holomorphic functions on $\mathbb{D}$ with a finite radial limit at almost no point.
Using the [[inner-outer factorization|X6ZQPAQF76]], part 2 extends to every function in the [[Hardy space|PNNNFJVF6W]] $H^p$, $0 < p \le \infty$.

## Explanations

### L1 human

A bounded holomorphic function on the disc may oscillate near the boundary, but not too wildly.
Fatou's theorem says that, approaching almost any point of the circle, the function settles down to a limit.
So the function has well defined boundary values, up to a set of length zero.
The same is true for the harmonic extension of any integrable function on the circle, and the limit is the function you started with.

### L2 human

Fatou's theorem is the reason Hardy spaces can be viewed as subspaces of $L^p$ of the circle.
It is used to define [[inner functions|NN8XH5RJYB]] and [[outer functions|KGJD4A4ATY]] through their boundary values.
The approach must be nontangential: Littlewood showed that limits along curves tangent to the circle can fail at almost every point.
The proof is a model case of the maximal function method in harmonic analysis.
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# Domain theory

## Summary

Domain theory studies special partially ordered sets, called domains, whose elements are read as pieces of partial information.
Its basic objects are [[directed-complete partial orders|7TSXP95QN6]], the [[way-below relation|RDM7J5J9HQ]], [[Scott-continuous maps|C6GAHS1Z2P]] and the [[Scott topology|ZR8A03FV67]].
Dana Scott created it in the late 1960s to give a mathematical semantics for programming languages and for the [[lambda calculus|BD756KR8FM]].
Recursive definitions are interpreted as [[least fixed points|DBD64TH52D]] of continuous maps.

## Explanations

### L1 human

A program that has not finished yet has produced only part of its answer.
Domain theory orders such partial answers by "carries less information than", and treats a full answer as the limit of partial ones.
For example, partial functions from $\N$ to $\N$ are ordered by extension.
A recursive definition of a function is then the limit of the finite approximations produced by unfolding the recursion.
The theory sits between order theory, topology and computer science.
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<document source="https://tamnd.github.io/math2/node/Q2QFXPP9EE.md" tag="Q2QFXPP9EE" kind="concept">
# Configuration

## Summary

A configuration is a finite system of points and lines in which two points share at most one line.
Every point lies on the same number of lines, and every line contains the same number of points.
A configuration with $p$ points on $\gamma$ lines each and $\ell$ lines with $\pi$ points each is written $(p_\gamma\ \ell_\pi)$, and $(p_k)$ when $p = \ell$ and $\gamma = \pi = k$.
Many classical theorems of projective geometry, such as those of Desargues and Pappus, are statements that certain configurations close up.

## Explanations

### L1 human

Draw some points and some lines so that the picture is completely regular: each point is on $3$ of the lines and each line passes through $3$ of the points.
Such a balanced picture is a configuration.
Counting the pairs (point, line through it) in two ways gives $p \gamma = \ell \pi$.
The Fano plane, with $7$ points and $7$ lines, is a $(7_3)$ configuration, but it cannot be drawn with straight lines in the real plane.
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<document source="https://tamnd.github.io/math2/node/Q2V8JVPB5E.md" tag="Q2V8JVPB5E" kind="definition">
# Homogeneous coordinates

## Definition

Let $k$ be a field and let $n \ge 0$.
A point $p$ of the [[projective space|ZT9A6JK8QF]] $\mathbb{P}^n(k)$ is an [[equivalence class|0W95W33CMP]] of nonzero vectors of $k^{n+1}$.
Two vectors are equivalent when one is a nonzero scalar multiple of the other.
A tuple $(x_0, \dots, x_n) \in k^{n+1} \setminus \set{0}$ in this class is a tuple of *homogeneous coordinates* of $p$, and one writes $p = [x_0 : \dots : x_n]$.

## Remarks

Two tuples are homogeneous coordinates of the same point exactly when they differ by a nonzero factor $\lambda \in k$.

Example: in $\mathbb{P}^2(\R)$, $[1 : 2 : 3] = [-2 : -4 : -6]$.
Check: $(-2, -4, -6) = -2 \cdot (1, 2, 3)$.

Non-example: $(0, 0, 0)$ is not a tuple of homogeneous coordinates of any point, since the zero vector is excluded.

The affine point $(x_1, \dots, x_n)$ of $k^n$ corresponds to $[x_1 : \dots : x_n : 1]$; the points with last coordinate $0$ form the hyperplane at infinity.
For a homogeneous polynomial $F$, the condition $F(x_0, \dots, x_n) = 0$ does not depend on the chosen coordinates of a point, since $F(\lambda x) = \lambda^d F(x)$ for $F$ of degree $d$.
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<document source="https://tamnd.github.io/math2/node/Q2W26ZY10B.md" tag="Q2W26ZY10B" kind="concept">
# Unary operation

## Summary

A unary operation on a set $A$ is a [[finitary operation|EJ8YJVM9BW]] of [[arity|MSXX2P86CY]] $1$, that is, a function from $A$ to $A$.
Negation of numbers, the inverse in a [[group|G5S1M9]] and the complement of a set are unary operations.
An algebra whose operations are all unary is a [[unary algebra|XD10P9CJMD]].

## Explanations

### L1 human

A unary operation takes one element and gives back one element of the same set.
On the integers, $x \mapsto -x$ and $x \mapsto x + 1$ are unary operations.
On the subsets of a fixed set $X$, taking the complement $Y \mapsto X \setminus Y$ is a unary operation.
The square root is not a unary operation on the real numbers, because $\sqrt{-1}$ is not a real number.
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<document source="https://tamnd.github.io/math2/node/Q2WXAR0PXW.md" tag="Q2WXAR0PXW" kind="definition">
# Topology of uniform convergence

## Definition

Let $X$ be a set, let $(Y, d)$ be a [[metric space|0VKD1TP7F3]], and let $Y^X$ be the set of all maps from $X$ to $Y$.
The uniform metric on $Y^X$ is
$$\bar d(f, g) = \sup_{x \in X} \min\bigl(d(f(x), g(x)), 1\bigr).$$
The *topology of uniform convergence* on $Y^X$ is the [[metric topology|0YVKNB7V5F]] of $\bar d$.
On a subset of $Y^X$, it is the [[subspace topology|0WS9P12FQR]].

## Remarks

The cut-off at $1$ makes $\bar d$ finite even for unbounded maps; it does not change which sequences converge.
A sequence $(f_n)$ converges to $f$ in this topology if and only if $f_n \to f$ [[uniformly|08VZ57PT26]] on $X$.
On bounded maps, the sup metric $\sup_x d(f(x), g(x))$ gives the same topology.
If $(Y, d)$ is complete, then $(Y^X, \bar d)$ is complete.
If $X$ is a topological space, the set $C(X, Y)$ of continuous maps is closed in $Y^X$, since a uniform limit of continuous maps is continuous.

Example: on $\R^{[0, 1]}$, let $f_n(x) = x / n$.
Then $\bar d(f_n, 0) \le 1/n$, so $f_n \to 0$ in this topology.

Non-example: let $f_n(x) = x^n$ on $X = [0, 1)$ and $Y = \R$.
Then $f_n \to 0$ pointwise, but $\bar d(f_n, 0) = 1$ for every $n$, since $x^n$ takes values close to $1$ near $x = 1$.
So $(f_n)$ converges in the topology of pointwise convergence, but not in the topology of uniform convergence.
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<document source="https://tamnd.github.io/math2/node/Q2XBHJ96GV.md" tag="Q2XBHJ96GV" kind="statement">
# Incomparability theorem

## Statement

Let $A \subseteq B$ be commutative rings with $B$ [[integral over|B7X0KFY1S1]] $A$.
Let $\mathfrak{q} \subseteq \mathfrak{q}'$ be [[prime ideals|00MHHZ9SNE]] of $B$ with $\mathfrak{q} \cap A = \mathfrak{q}' \cap A$.
Then $\mathfrak{q} = \mathfrak{q}'$.

## Remarks

Let $\mathfrak{p} = \mathfrak{q} \cap A$ and localize at $\mathfrak{p}$.
Then $B_{\mathfrak{p}}/\mathfrak{q} B_{\mathfrak{p}}$ is a domain integral over the field $A_{\mathfrak{p}}/\mathfrak{p} A_{\mathfrak{p}}$, so it is a field by [[integral extensions of fields|5GCANV7337]].
Hence $\mathfrak{q} B_{\mathfrak{p}}$ is maximal, and the larger prime $\mathfrak{q}' B_{\mathfrak{p}}$ must equal it, which gives $\mathfrak{q} = \mathfrak{q}'$.

A consequence: a prime $\mathfrak{q}$ of $B$ is maximal if and only if $\mathfrak{q} \cap A$ is maximal in $A$.

Example: in $\Z \subseteq \Z[i]$, the primes $0 \subsetneq (2 + i)$ contract to $0$ and $5\Z$, which differ, as the theorem demands.

Integrality is needed: for $k[x] \subseteq k[x, y]$, the primes $0 \subsetneq (y)$ both contract to $0$.

## Explanations

### L1 human

Over a fixed prime of the small ring there may be several primes of the big ring, but none of them lies inside another.
In geometry: the fibres of an integral map have dimension zero.

### L2 human

Incomparability says that chains of primes cannot get longer when one passes to an integral extension.
With [[going-up|GHKYR5PBQC]] it gives $\dim B = \dim A$.
It is also the reason that the maximal ideals of $B$ are exactly the primes over maximal ideals of $A$.
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<document source="https://tamnd.github.io/math2/node/Q390KATZ87.md" tag="Q390KATZ87" kind="statement">
# Householder's method

## Statement

Let $d \ge 1$ be an integer.
Let $f$ be a real function that is $d + 1$ times continuously differentiable on an open interval $I$.
Let $r \in I$ be a simple zero of $f$, that is, $f(r) = 0$ and $f'(r) \ne 0$.
*Householder's method of order $d + 1$* is the iteration
$$x_{n+1} = x_n + d \, \frac{(1/f)^{(d-1)}(x_n)}{(1/f)^{(d)}(x_n)},$$
where $(1/f)^{(k)}$ is the $k$-th [[derivative|0F1WR9C83K]] of the function $1/f$.
Then there is $\delta > 0$ such that for every $x_0$ with $\abs{x_0 - r} < \delta$ the iterates are defined and [[converge|YWZQC5E54B]] to $r$ with order $d + 1$.

## Remarks

For $d = 1$ the formula gives $x_{n+1} = x_n - f(x_n)/f'(x_n)$, which is [[Newton's method|07EXVNF9VP]].
For $d = 2$ it gives [[Halley's method|RPME2B5MS2]].
The idea of the proof: near $r$, the function $1/f$ has a simple pole at $r$, so it equals $c/(x - r)$ plus a function that is smoother at $r$.
For the pole part alone, the ratio of consecutive derivatives gives exactly $-(x - r)/d$, so the step lands on $r$.
The remainder adds an error of size $O(\abs{x - r}^{d+1})$.
The hypothesis $f'(r) \ne 0$ is needed: at a multiple zero the pole of $1/f$ has higher order and the convergence becomes linear.

## Explanations

### L1 human

Householder's methods form a ladder of root-finding methods.
The rung $d$ uses derivatives of $f$ up to order $d$, and near a simple root it multiplies the number of correct digits by about $d + 1$ at each step.
The first rung is Newton's method and the second is Halley's method.

### L2 human

Higher order does not always mean less work, because each step needs more derivatives.
In practice $d = 1$ or $d = 2$ is used, unless the derivatives come almost for free.
The formula is a convenient way to see all these methods as one family built on the reciprocal $1/f$.
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# Periodic sequence

## Summary

A periodic sequence is a [[sequence|0G9EKM0E15]] that repeats the same finite block of terms forever.
The length of the shortest repeating block is its least period.
Periodic sequences are the discrete analogue of [[periodic functions|0ATF6Q0633]].
They appear as periodic solutions of difference equations, as decimal expansions of rational numbers, and as Fibonacci numbers modulo a fixed integer.

## Explanations

### L1 human

A periodic sequence is a pattern that repeats like a necklace with a fixed set of beads.
The sequence $1, 2, 3, 1, 2, 3, \dots$ repeats after $3$ terms, so it has period $3$.
The digits after the decimal point of $1/7 = 0.142857142857\ldots$ form a periodic sequence of period $6$.
A constant sequence is periodic with period $1$.
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<document source="https://tamnd.github.io/math2/node/Q3C2ZGRHSG.md" tag="Q3C2ZGRHSG" kind="definition">
# Integro-partial differential equation

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be open.
An *integro-partial differential equation* on $\Omega$ is an equation for an unknown function $u \colon \Omega \to \R$ (or $\C$, or $\R^m$) such that:

1. at least one partial derivative of $u$ appears in it, and
2. at least one integral of $u$, or of a derivative of $u$, with respect to one or more of the variables appears in it.

A typical linear form is
$$\sum_{\abs{\alpha} \le k} a_\alpha(x) \, \partial^\alpha u(x) + \int_{\Omega} K(x, y) \, u(y) \, dy = f(x), \quad x \in \Omega,$$
with given coefficients $a_\alpha$, kernel $K$ and right side $f$.

## Remarks

Example: let $J \colon \R \to [0, \infty)$ be even and integrable, and consider $\partial_t u(x, t) = \int_{\R} J(x - y) \big(u(y, t) - u(x, t)\big) \, dy$ on $\R \times (0, \infty)$.
Write $\hat J(\omega) = \int_{\R} J(z) \cos(\omega z) \, dz$.
Then $u(x, t) = e^{(\hat J(\omega) - \hat J(0)) t} \cos(\omega x)$ is a solution, because $\int_{\R} J(x - y) \cos(\omega y) \, dy = \hat J(\omega) \cos(\omega x)$ for even $J$.

Non-example: the heat equation $\partial_t u = \partial_x^2 u$ contains no integral of $u$, so it fails clause 2 and is a [[partial differential equation|GC5BDZ6QBC]].
The equation $x'(t) = \int_0^t x(s) \, ds$ has an unknown of one variable only, so it is an [[integro-differential equation|YRKW2KX2CW]] but not a partial one.
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<document source="https://tamnd.github.io/math2/node/Q3EAVBDA8J.md" tag="Q3EAVBDA8J" kind="definition">
# Linear extension

## Definition

Let $X$ be a set and let $\le$ be a [[partial order|0QV2JS9RJ3]] on $X$.
A *linear extension* of $\le$ is a [[total order|0FRXBVVR03]] $\preceq$ on $X$ such that $x \le y$ implies $x \preceq y$ for all $x, y \in X$.

## Remarks

Example: on $\set{1, 2, 3, 6}$ ordered by divisibility, the usual order $1 < 2 < 3 < 6$ is a linear extension, because each divisor of a positive integer is at most that integer.
Non-example: the order $1 \prec 3 \prec 6 \prec 2$ is total but not a linear extension, because $2 \mid 6$ and yet $6 \prec 2$.
If $X$ is finite with $n$ elements, a linear extension is the same as a bijection $f \colon X \to \set{1, \ldots, n}$ with $f(x) < f(y)$ whenever $x < y$.
The number of linear extensions of a finite poset $P$ is often written $e(P)$.
A total order has exactly one linear extension, itself, and an antichain with $n$ elements has $n!$.
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# Graphical model

## Summary

A probabilistic graphical model is a family of joint distributions of random variables whose [[conditional independences|0THJ7416DX]] are read off a [[graph|00C0BR7D7H]].
The vertices are the variables, and missing edges mean conditional independence.
Directed acyclic graphs give [[Bayesian networks|21JT983G60]], and undirected graphs give [[Markov random fields|6W69H3A3G5]].
The graph makes large models easier to describe, to fit and to compute with.

## Explanations

### L1 human

A joint distribution of many variables is too large to write down directly.
A graphical model draws one dot for each variable and joins two dots when the variables influence each other directly.
Variables that are not joined interact only through the others.
For example, the weather affects whether the grass is wet both directly and through the sprinkler, and the graph shows exactly these routes.
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<document source="https://tamnd.github.io/math2/node/Q3JWBZ7EFY.md" tag="Q3JWBZ7EFY" kind="concept">
# Vandermonde matrix

## Summary

A Vandermonde matrix has rows of the form $1, x, x^2, \dots, x^{n-1}$, one row for each of the numbers $x_1, \dots, x_m$.
Multiplying it by the coefficient vector of a polynomial gives the values of the polynomial at $x_1, \dots, x_m$.
Its determinant, the [[Vandermonde determinant|0P84B554AS]], is nonzero exactly when the $x_i$ are distinct.
This is why a polynomial of degree less than $n$ is fixed by its values at $n$ distinct points.

## Explanations

### L1 human

Take a few numbers, say $1, 2, 3$, and write each one's powers in a row: $1, x, x^2$.
This gives the square table $\begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{pmatrix}$.
To find a parabola through three given points, one solves a linear system with this matrix.
Because the numbers are distinct, the matrix is invertible, so there is exactly one such parabola.
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<document source="https://tamnd.github.io/math2/node/Q3Q4FF23VG.md" tag="Q3Q4FF23VG" kind="concept">
# Quadric

## Summary

A quadric is the set of points of an affine or projective space where a polynomial of degree $2$ vanishes.
Conics in the plane and quadric surfaces such as spheres, ellipsoids and hyperboloids in space are quadrics.
In projective geometry, a quadric is given by a [[quadratic form|0RBQS1JJCR]].
The points, lines and subspaces lying on a non-degenerate quadric form a [[polar space|Z7T0V2VAWJ]].

## Explanations

### L1 human

A quadric is the next simplest shape after a flat one: flat sets are cut out by equations of degree $1$, quadrics by equations of degree $2$.
The sphere $x^2 + y^2 + z^2 = 1$ is a quadric in space.
So is the saddle $z = xy$, which contains two families of straight lines.
In the projective setting all equations are written homogeneously, for example $x_0 x_1 + x_2 x_3 = 0$.
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<document source="https://tamnd.github.io/math2/node/Q3TVJ4A3D4.md" tag="Q3TVJ4A3D4" kind="definition">
# Grothendieck topology

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
For a [[sieve|CNCXNWKC6R]] $S$ on $c$ and a morphism $h \colon d \to c$, let $h^*S = \set{g : h \circ g \in S}$ be the pullback sieve on $d$.
A *Grothendieck topology* $J$ on $\mathcal{C}$ assigns to each object $c$ a set $J(c)$ of sieves on $c$, the *covering sieves*, such that:

1. (maximal sieve) the sieve of all morphisms with codomain $c$ is in $J(c)$;
2. (stability) if $S \in J(c)$ and $h \colon d \to c$ is a morphism, then $h^*S \in J(d)$;
3. (transitivity) if $S \in J(c)$ and $R$ is a sieve on $c$ such that $h^*R \in J(d)$ for every $h \colon d \to c$ in $S$, then $R \in J(c)$.

## Remarks

Example: let $X$ be a topological space and $\mathcal{O}(X)$ the poset of open sets as a category.
Let $J(U)$ be the set of sieves $S$ on $U$ whose members have union $U$.
Clause 1 holds because $U \subseteq U$ is in the maximal sieve.
Clause 2 holds because the sets $V \cap W$ with $V \in S$ cover $W$ when $S$ covers $U \supseteq W$.
Clause 3 holds because covers of the members of a cover together give a cover.
Non-example: on $\mathcal{O}(X)$ with $X$ nonempty, the rule $J(U) = \emptyset$ for all $U$ fails clause 1.
On every category, the rule in which only maximal sieves cover is a Grothendieck topology, and so is the rule in which every sieve covers.
Many authors start from a *pretopology*, a choice of covering families of morphisms; it generates a Grothendieck topology by taking the sieves that contain a covering family.
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# K3 surface

## Summary

A K3 surface is a smooth projective [[algebraic surface|KFTECAATKG]], or more generally a compact complex surface, with trivial canonical bundle and no holomorphic $1$-forms.
K3 surfaces are the two-dimensional [[Calabi–Yau manifolds|STYAB4PFDW]], the surface analogue of elliptic curves.
All complex K3 surfaces are diffeomorphic to each other and simply connected, with second Betti number $22$.
They are central in the classification of surfaces, in arithmetic geometry and in string theory.

## Explanations

### L1 human

The simplest K3 surface is a smooth surface of degree $4$ in projective three-space, such as $x^4 + y^4 + z^4 + w^4 = 0$.
Its canonical bundle is trivial, which means that it carries a nowhere vanishing holomorphic $2$-form, a kind of volume form.
This makes K3 surfaces flat in an averaged sense, between the positively curved rational surfaces and the negatively curved surfaces of general type.
</document>

<document source="https://tamnd.github.io/math2/node/Q3X1CNXASD.md" tag="Q3X1CNXASD" kind="statement">
# Peetre theorem on local operators

## Statement

Let $M$ be a [[smooth manifold|F45VQ47GPV]] and let $E$ and $F$ be smooth real [[vector bundles|P4DZM975BV]] over $M$.
Let $\Gamma(E)$ and $\Gamma(F)$ be their spaces of smooth [[sections|WWX7H5W1RP]].
Let $D \colon \Gamma(E) \to \Gamma(F)$ be an $\R$-linear map that is local, that is,
$$\operatorname{supp}(Ds) \subseteq \operatorname{supp}(s) \quad \text{for all } s \in \Gamma(E).$$
Then every point $p \in M$ has an open neighbourhood $U$ with an integer $k \ge 0$ such that the following holds.
Over $U$, $D$ is a [[linear differential operator|CGBX1N9V8H]] of order at most $k$ with smooth coefficients.
In local coordinates and trivialisations over $U$, this means $(Ds)|_U = \sum_{\abs{\alpha} \le k} a_\alpha \, \partial^\alpha (s|_U)$ for every $s \in \Gamma(E)$, with smooth matrix-valued functions $a_\alpha$.

## Remarks

No continuity of $D$ is assumed; it is a consequence.
The idea of the proof is to show first that, near each point, $(Ds)(x)$ depends only on the $k$-jet of $s$ at $x$ for one fixed $k$.
If this failed, one could build sections with small disjoint supports accumulating at a point and add them up, and locality would force a contradiction.
Linearity then shows that $D$ is given by a linear map on $k$-jets, which is exactly a differential operator of order $k$.

Example: the exterior derivative $d \colon \Omega^j(M) \to \Omega^{j+1}(M)$ is local, and it is indeed a differential operator of order $1$.

Locality is needed.
On $M = \R$ the operator $(Df)(x) = \int_0^x f(t)\, dt$ is linear but not local, and it is not a differential operator.

The order need not be bounded on all of $M$ when $M$ is not compact.
On $\R$, let $\varphi_n$ be a smooth function with support in $(n, n + 1)$ and $\varphi_n \ne 0$.
Then $Df = \sum_{n \ge 1} \varphi_n f^{(n)}$ is a locally finite sum, so it is a local linear operator, but its order is unbounded.
If $M$ is compact, finitely many sets $U$ cover $M$, so $D$ is a differential operator of finite order.

## Explanations

### L1 human

A linear operator on functions is local if the output near a point depends only on the input near that point.
Derivatives have this property, integrals do not.
Peetre's theorem says that derivatives are the only examples: a linear local operator is a differential operator, at least near each point.

### L2 human

The theorem gives an algebraic characterisation of differential operators by their support property.
It is used to show that natural operators in differential geometry, such as those that commute with diffeomorphisms, are differential operators of finite order.
Together with invariance arguments, it leads to classification results for natural operators on bundles of tensors.
It also explains why locality is the key property of the exterior derivative, the Lie derivative and covariant derivatives.
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<document source="https://tamnd.github.io/math2/node/Q3Y1HBD8NN.md" tag="Q3Y1HBD8NN" kind="statement">
# Dunford–Pettis theorem

## Statement

Let $(X, \Sigma, \mu)$ be a measure space with $\mu$ a [[finite measure|63Z1Q4FS23]], and let $K$ be a subset of [[$L^1(\mu)$|SASP14MA1Z]].
Then the following are equivalent.

1. $K$ is relatively compact in the [[weak topology|P6KHVSTPZV]] of $L^1(\mu)$.
2. $K$ is [[uniformly integrable|VRVVKMXDGV]]: $\lim_{M \to \infty} \sup_{f \in K} \int_{\set{\abs{f} > M}} \abs{f} \, d\mu = 0$.

## Remarks

When $\mu$ is finite, condition 2 implies that $K$ is bounded in $L^1$.
One proof of 2 $\Rightarrow$ 1 views $K$ inside the bidual $L^\infty(\mu)^*$ and takes a weak-$*$ cluster point.
Uniform integrability makes the set functions $E \mapsto \int_E f \, d\mu$, $f \in K$, uniformly absolutely continuous, so the cluster point is a countably additive measure absolutely continuous with respect to $\mu$.
The [[Radon–Nikodym theorem|1ZSZS5G1Q4]] then gives it as an element of $L^1(\mu)$.
The converse uses the Eberlein–Šmulian theorem to reduce to sequences.

Example: on $[0, 1]$ with Lebesgue measure, a set bounded in $L^2$ is uniformly integrable, because $\int_{\set{\abs{f} > M}} \abs{f} \le \norm{f}_2^2 / M$.
So bounded subsets of $L^2$ are weakly relatively compact in $L^1$.

Boundedness in $L^1$ alone is not enough.
The functions $f_n = n \mathbf{1}_{[0, 1/n]}$ on $[0, 1]$ all have norm $1$, but $\int_{\set{f_n > M}} f_n = 1$ for $n > M$.
No subsequence converges weakly in $L^1$: as measures, $f_n \, dx$ concentrate at the point $0$.

## Explanations

### L1 human

In $L^1$, bounded sequences need not have limits even in a weak sense, because the mass of the functions can pile up on smaller and smaller sets.
The theorem says that this concentration is the only obstruction.
If the functions cannot put a fixed amount of mass where they are very large, then every sequence has a weakly convergent subsequence.

### L2 human

The theorem is the $L^1$ substitute for the weak compactness of bounded sets in reflexive spaces such as $L^p$, $1 < p < \infty$.
In probability it is the analytic side of the fact that uniformly integrable families of random variables behave well under limits, as in the [[Vitali convergence theorem|C3CKZAZ57Z]].
In the calculus of variations it gives weak limits for sequences with superlinear integral bounds, by the de la Vallée-Poussin criterion.
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# Krein's condition

## Statement

Let $\mu$ be a measure on $\R$ of the form $d\mu(x) = f(x) \, dx$, where $f \ge 0$ is Lebesgue integrable.
Suppose that every [[moment|01XAHPADBA]] $m_n = \int x^n f(x) \, dx$ is finite.
If
$$\int_{-\infty}^{\infty} \frac{-\ln f(x)}{1 + x^2} \, dx < \infty,$$
then the [[Hamburger moment problem|TBNCFXTMMW]] for $\mu$ is [[indeterminate|GCTN4BZ1ET]]: there is a positive measure $\nu \ne \mu$ on $\R$ with the same moments.

## Remarks

The integral is well defined in $(-\infty, \infty]$, since $\ln f \le f$ where $f \ge 1$ and $f$ is integrable.
The result is a corollary of a density theorem of Krein: the integral is infinite if and only if the exponential sums $\sum_k a_k e^{i \lambda_k x}$ with $\lambda_k \ge 0$ are dense in $L^2(\mu)$.
Example: $f(x) = c \, e^{-\abs{x}^{\alpha}}$ with $0 < \alpha < 1$ and $c > 0$ has $-\ln f(x) = \abs{x}^{\alpha} - \ln c$, and $\int \abs{x}^{\alpha}/(1 + x^2) \, dx < \infty$ for $\alpha < 1$, so this density is not determined by its moments.
For $\alpha \ge 1$ the integral diverges, and in fact Carleman's condition shows that the moment problem is determinate.
The condition is sufficient for indeterminacy but not necessary.

## Explanations

### L1 human

A density whose tails are very heavy, but not so heavy that the moments are infinite, can share all its moments with another distribution.
Krein's condition measures heaviness of the tails by how slowly $-\ln f$ grows.
If $-\ln f(x)$ grows more slowly than $\abs{x}$, in the averaged sense of the integral, the moments do not determine the distribution.

### L2 human

It complements [[Carleman's condition|RVZ831CNYW]], which gives determinacy when the moments grow slowly.
The two meet at densities like $e^{-\abs{x}^{\alpha}}$: determinate for $\alpha \ge 1$ and indeterminate for $\alpha < 1$.
Versions for the Stieltjes problem on $[0, \infty)$ and converses under regularity hypotheses on $f$ are known.
</document>

<document source="https://tamnd.github.io/math2/node/Q43RD1GK8G.md" tag="Q43RD1GK8G" kind="statement">
# Morrey's lower semicontinuity theorem

## Statement

Let $\Omega \subseteq \R^n$ be a bounded open set, let $m \ge 1$, and let $f \colon \R^{m \times n} \to \R$ be continuous.
Let $J(u) = \int_\Omega f(\nabla u) \, dx$, an [[integral functional|JRQZ803Q5Y]] on maps $u \colon \Omega \to \R^m$.

1. $J$ is sequentially weak-$*$ lower semicontinuous on the [[Sobolev space|M4CEZ5K9HP]] $W^{1,\infty}(\Omega; \R^m)$ if and only if $f$ is [[quasiconvex|DVJ17KVWXP]].
2. Let $1 < p < \infty$ and assume $0 \le f(\xi) \le C (1 + \abs{\xi}^p)$ for a constant $C$. Then $J$ is sequentially [[weakly lower semicontinuous|H6GTJRG26G]] on $W^{1,p}(\Omega; \R^m)$ if and only if $f$ is quasiconvex.

## Remarks

For necessity, take $u_k(x) = \xi x + k^{-1} \varphi(kx)$ with $\varphi$ periodic on the unit cube and zero on its boundary.
Then $u_k$ converges weak-$*$ to the affine map $\xi x$, and $J(u_k)$ tends to $\abs{\Omega}$ times the average of $f(\xi + \nabla \varphi)$ over the cube.
Lower semicontinuity then gives the quasiconvexity inequality.
For sufficiency, one localises: on small cubes $\nabla u$ is nearly constant, and quasiconvexity controls the energy of the oscillations of $u_k$ around it.

Part 1 is Morrey's theorem; part 2 under these growth conditions is due to Acerbi and Fusco, after work of Meyers.

Example: $f(\xi) = \abs{\xi}^2 + (\det \xi)^2$ for $2 \times 2$ matrices is polyconvex, hence quasiconvex, so $J$ is weakly lower semicontinuous on $W^{1,4}(\Omega; \R^2)$.

When $m = 1$ or $n = 1$, quasiconvexity is convexity, and the theorem recovers the classical role of convexity in Tonelli's existence theorem.

## Explanations

### L1 human

The theorem answers exactly when the energy of a map can drop in a weak limit.
It cannot drop precisely when the integrand is quasiconvex, that is, when wiggling an affine map never lowers its energy.
So quasiconvexity is not just a convenient assumption; it is the right condition.

### L2 human

Combined with coercivity, the theorem gives existence of minimisers for vectorial problems by the direct method.
Its drawback is that quasiconvexity is hard to verify, which is why one often works with polyconvexity, a sufficient condition, and rank-one convexity, a necessary one.
When the integrand is not quasiconvex, the lower semicontinuous envelope of $J$ is the integral of the quasiconvex envelope of $f$, under suitable growth conditions.
This is the basis of the relaxation theory of microstructure in materials.
</document>

<document source="https://tamnd.github.io/math2/node/Q468P6FP4A.md" tag="Q468P6FP4A" kind="definition">
# Dehn surgery

## Definition

Let $M$ be a closed orientable [[3-manifold|R20TB2KEQQ]] and let $K \subset M$ be a smooth [[knot|QYEAZTYGPT]], that is, a smoothly embedded circle.
Let $N(K)$ be a closed tubular neighbourhood of $K$, a solid torus, and let $E = M \setminus \operatorname{int} N(K)$ be the knot exterior, with boundary torus $\partial E = \partial N(K)$.
Let $\alpha \subset \partial E$ be an essential simple closed curve, called the *slope*.
Let $V = D^2 \times S^1$ and let $h \colon \partial V \to \partial E$ be a [[homeomorphism|0QT00PB5G2]] with $h(\partial D^2 \times \set{1}) = \alpha$.
The *Dehn surgery* on $K$ with slope $\alpha$ is the manifold
$$M_K(\alpha) = E \cup_h V.$$
Its homeomorphism type depends only on the isotopy class of $\alpha$ in $\partial E$, not on its orientation.
For a [[link|1A5WRCNTM3]] one does this on each component, with one slope for each.

## Remarks

For a knot $K$ in $S^3$, let $\mu$ be a meridian, bounding a disc in $N(K)$, and $\lambda$ a longitude that is null-homologous in $E$, both oriented.
Each slope is isotopic to a curve representing $p\mu + q\lambda$ in $H_1(\partial E; \Z)$ with $p, q$ coprime, and the slope is labelled by $p/q \in \Q \cup \set{\infty}$.
The slope $\infty = 1/0$ gives back $S^3$; a slope with $q = \pm 1$ is *integral*.

Example: let $U$ be the unknot in $S^3$.
Its exterior is a solid torus in which $\lambda$ bounds a disc.
Surgery with slope $0$, that is $\alpha = \lambda$, glues two solid tori so that their meridians match, and the result is $S^2 \times S^1$.
Surgery on $U$ with slope $p/q$, $p \ne 0$, gives a lens space $L(p, q)$, up to the orientation convention.

Non-example: a curve $\alpha$ that bounds a disc in $\partial E$ is not a slope.
No homeomorphism $h$ can send $\partial D^2 \times \set{1}$ to it, because a homeomorphism of tori sends essential curves to essential curves.

## Explanations

### L1 human

Drill out a thin tube around a knot and fill the hole with a solid tube glued in a new way.
The only choice is which loop on the boundary torus becomes the boundary of a disc of the new tube.
This choice is a fraction, and different fractions usually give different spaces.
</document>

<document source="https://tamnd.github.io/math2/node/Q48KGV33AH.md" tag="Q48KGV33AH" kind="concept">
# Type theory

## Summary

A type theory is a formal system in which every term has a type.
Its rules say which terms can be formed in which types.
Russell introduced a theory of types to avoid [[Russell's paradox|0HESMMSYEG]], by forbidding a collection to contain objects of its own level.
Church's simple type theory of 1940 and Martin-Löf's intuitionistic type theory are the main later systems.
Type theories serve as foundations of mathematics alternative to set theory, and as the basis of programming languages and proof assistants.

## Explanations

### L1 human

In a type theory, everything comes with a label, its type, such as "natural number" or "function from numbers to numbers".
You may only apply a function to an argument of the right type.
So the expression $f(f)$, which feeds a function to itself, is not allowed for a function $f \colon \N \to \N$.
This simple discipline blocks the paradoxes of naive [[set|0BSJ7RQ9BW]] theory.
It also catches many errors in computer programs before they run.
</document>

<document source="https://tamnd.github.io/math2/node/Q490XBH1Q2.md" tag="Q490XBH1Q2" kind="statement">
# Littlewood–Richardson rule

## Statement

Let $\lambda$ and $\mu$ be [[partitions|0JV30XN73Z]], and let $s_\lambda$ denote the [[Schur polynomial|43VFMJCDYJ]] (in any number of variables, or the Schur function).
Then
$$s_\lambda \, s_\mu = \sum_{\nu} c^{\nu}_{\lambda \mu} \, s_\nu,$$
where $c^{\nu}_{\lambda\mu}$ is the number of Littlewood–Richardson tableaux of shape $\nu / \lambda$ and content $\mu$.
Such a tableau is a filling $T$ of the skew shape $\nu / \lambda$ with these properties.
Here $\nu / \lambda$ is the set of cells of the [[Young diagram|096AHMB3P4]] of $\nu$ that are not in the diagram of $\lambda$.

1. The entries weakly increase along rows and strictly increase down columns.
2. For each $k$, exactly $\mu_k$ entries are equal to $k$.
3. Read the rows from right to left, from the top row down. In every initial segment of this word, for each $k \ge 1$, the entry $k$ occurs at least as often as $k+1$.

## Remarks

In particular $c^{\nu}_{\lambda\mu} = 0$ unless the diagram of $\lambda$ lies in the diagram of $\nu$ and $\abs{\nu} = \abs{\lambda} + \abs{\mu}$.
The coefficients are symmetric: $c^{\nu}_{\lambda\mu} = c^{\nu}_{\mu\lambda}$.
Modern proofs use the [[Robinson–Schensted–Knuth correspondence|8JW8F4VVED]] or Schützenberger's jeu de taquin, which turns each tableau of shape $\nu/\lambda$ into a straight-shape [[Young tableau|Q6TH9PH2R7]].

Example: take $\lambda = (1)$ and $\mu = (1, 1)$.
For $\nu = (2, 1)$ the two cells of $\nu / \lambda$ are in different rows and columns.
The reading word must start with $1$, so the only tableau has $1$ in the first row and $2$ in the second row.
For $\nu = (1, 1, 1)$ the two cells form a column filled with $1$ above $2$, and the reading word $1\,2$ is allowed.
For $\nu = (3)$ the cells form a row $1\,2$ whose reading word $2\,1$ fails clause 3.
Hence $s_{(1)} s_{(1,1)} = s_{(2,1)} + s_{(1,1,1)}$.

## Explanations

### L1 human

Schur polynomials are building blocks, and the product of two of them is again a sum of them.
The rule says how many times each building block appears: count certain fillings of a skew shape with numbers.
The reading condition makes sure that, at every point while reading, the $1$s are ahead of the $2$s, the $2$s ahead of the $3$s, and so on.

### L2 human

The numbers $c^\nu_{\lambda\mu}$ have several meanings at once.
They are the multiplicities in the tensor product of two irreducible polynomial representations of the general linear group.
They are the multiplicities of the irreducible representations of $S_{\abs\nu}$ in the representation induced from the irreducible representation of $S_{\abs\lambda} \times S_{\abs\mu}$ indexed by the pair $(\lambda, \mu)$.
They are also the structure constants of the cohomology ring of a Grassmannian in the basis of Schubert classes.
When $\mu = (k)$ is a single row, the rule reduces to Pieri's rule.
It says that $s_\lambda s_{(k)}$ is the sum of $s_\nu$ over the $\nu$ obtained by adding $k$ cells to $\lambda$, no two in the same column.
</document>

<document source="https://tamnd.github.io/math2/node/Q4HGMRA9JQ.md" tag="Q4HGMRA9JQ" kind="concept">
# Dehn function

## Summary

The Dehn function of a [[finitely presented group|TJKANTKASZ]] measures how many relators are needed to prove that a word of length $n$ equals the identity.
Up to a standard equivalence it does not depend on the presentation, and it is a [[quasi-isometry|6FKYSH729T]] invariant.
It is a geometric measure of the difficulty of the word problem.
The word problem is solvable exactly when the Dehn function is bounded by a computable function.
A group is [[hyperbolic|JWJ8YPR7EB]] if and only if its Dehn function is linear.

## Explanations

### L1 human

A word in the generators that equals the identity can be reduced to the empty word by inserting relators, one at a time, and cancelling.
The area of the word is the least number of relators needed.
The Dehn function gives the largest area of such a word of length at most $n$.
In $\Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle$, the word $a^n b^n a^{-n} b^{-n}$ traces the boundary of an $n \times n$ square and needs $n^2$ relators, one per unit square.
So the Dehn function of $\Z^2$ grows like $n^2$, like the area enclosed by a loop in the plane.
</document>

<document source="https://tamnd.github.io/math2/node/Q4HR7AF2R9.md" tag="Q4HR7AF2R9" kind="definition">
# Sine and cosine transforms

## Definition

Let $f \colon [0, \infty) \to \C$ be a Lebesgue [[integrable function|X7GTF3D98F]].
The *cosine transform* of $f$ is the function on $[0, \infty)$ given by
$$\mathcal{F}_c f(\xi) = \int_0^\infty f(x) \cos(2\pi x \xi)\, dx.$$
The *sine transform* of $f$ is the function on $[0, \infty)$ given by
$$\mathcal{F}_s f(\xi) = \int_0^\infty f(x) \sin(2\pi x \xi)\, dx.$$

## Remarks

Both integrals converge absolutely because $\abs{\cos}$ and $\abs{\sin}$ are at most $1$.
Example: for $f(x) = e^{-x}$, two integrations by parts give $\mathcal{F}_c f(\xi) = \frac{1}{1 + a^2}$ and $\mathcal{F}_s f(\xi) = \frac{a}{1 + a^2}$ with $a = 2\pi \xi$.
Non-example: the constant function $f = 1$ is not integrable on $[0, \infty)$, and $\int_0^\infty \cos(2\pi x \xi)\, dx$ diverges for every $\xi$.

Let $f_e$ and $f_o$ be the even and the odd extensions of $f$ to $\R$, and let $\hat g$ denote the [[Fourier transform|DPVM34ADZV]] $\int_\R g(x) e^{-2\pi i x \xi}\, dx$.
Then $\hat f_e(\xi) = 2 \mathcal{F}_c f(\xi)$ and $\hat f_o(\xi) = -2i\, \mathcal{F}_s f(\xi)$.
Consequently, if $f$ is continuous and $\mathcal{F}_c f$ is integrable, then $f(x) = 4 \int_0^\infty \mathcal{F}_c f(\xi) \cos(2\pi x \xi)\, d\xi$ for $x \ge 0$.

Other conventions use $\cos(x \xi)$ and $\sin(x \xi)$, often with a factor $\sqrt{2/\pi}$ that makes the transforms their own inverses on $L^2(0, \infty)$.
</document>

<document source="https://tamnd.github.io/math2/node/Q4KF0MAGSC.md" tag="Q4KF0MAGSC" kind="definition">
# Lacunary series

## Definition

A strictly increasing sequence $(n_k)_{k \ge 1}$ of positive integers is *lacunary* (in the sense of Hadamard) if there is a number $q > 1$ with
$$\frac{n_{k+1}}{n_k} \ge q \quad \text{for all } k \ge 1.$$
A *lacunary trigonometric series* is a [[trigonometric series|FZK2M8KGXQ]] of the form
$$\sum_{k \ge 1} \left( a_k \cos n_k x + b_k \sin n_k x \right),$$
with real or complex coefficients $a_k, b_k$ and a lacunary sequence $(n_k)$.
A *lacunary power series* is a [[power series|0D6KXBHKC0]] $\sum_{k \ge 1} c_k z^{n_k}$ with a lacunary sequence $(n_k)$.

## Remarks

Example: $\sum_{k \ge 0} 2^{-k} \cos(2^k x)$ is lacunary with $q = 2$, since $2^{k+1}/2^k = 2$.

Non-example: $\sum_{k \ge 1} \frac{\cos(k^2 x)}{k^2}$ has frequencies $n_k = k^2$ with $n_{k+1}/n_k = (1 + 1/k)^2 \to 1$.
So no $q > 1$ is a lower bound for all ratios, and the series is not lacunary in the sense of Hadamard.

A theorem of Sidon says that if a lacunary trigonometric series is the Fourier series of a bounded function, then $\sum_k (\abs{a_k} + \abs{b_k}) < \infty$.
For power series the [[Hadamard gap theorem|DA8M8VZZHC]] applies.
Weaker gap conditions, such as $n_k / k \to \infty$ in the [[Fabry gap theorem|GE0HX2KMMZ]], are also studied.
</document>

<document source="https://tamnd.github.io/math2/node/Q4P4R2GDNG.md" tag="Q4P4R2GDNG" kind="problem">
# dbar-Neumann problem

## Problem

Let $\Omega \subseteq \C^n$ be a bounded open set with smooth boundary, and let $1 \le q \le n$.
Let $L^2_{(0,q)}(\Omega)$ be the Hilbert space of [[(0,q)-forms|DC81TCR98Y]] on $\Omega$ with square integrable coefficients.
Regard $\bar\partial$ as a closed, densely defined operator from $L^2_{(0,q)}(\Omega)$ to $L^2_{(0,q+1)}(\Omega)$.
It acts in the sense of distributions on the forms for which the result is square integrable.
Let $\bar\partial^*$ be its [[Hilbert space adjoint|86B7HF79PQ]], and let
$$\square = \bar\partial \bar\partial^* + \bar\partial^* \bar\partial$$
on the set of $u \in L^2_{(0,q)}(\Omega)$ with $u \in \operatorname{Dom}(\bar\partial) \cap \operatorname{Dom}(\bar\partial^*)$, $\bar\partial u \in \operatorname{Dom}(\bar\partial^*)$ and $\bar\partial^* u \in \operatorname{Dom}(\bar\partial)$.
Given $\alpha \in L^2_{(0,q)}(\Omega)$, find $u$ in this domain with $\square u = \alpha$, and study the regularity of $u$ up to the boundary.

## Remarks

For forms smooth up to the boundary, membership in the domain of $\square$ means two boundary conditions, the $\bar\partial$-Neumann conditions.
If $\rho$ is a defining function of $\Omega$, they say that the contraction of $u$ with $\bar\partial \rho$ and the contraction of $\bar\partial u$ with $\bar\partial \rho$ both vanish on $\partial\Omega$.
These conditions are not elliptic, so the standard theory of elliptic boundary value problems does not apply.
Spencer proposed the problem in the 1950s as a way to extend Hodge theory to manifolds with boundary.
Kohn solved it for strongly pseudoconvex domains in 1963: $\square$ has a bounded inverse $N$, the Neumann operator.
Moreover $N$ maps forms that are smooth up to the boundary to forms that are smooth up to the boundary.
For a bounded [[pseudoconvex|AHYZMSBBNH]] domain, Hörmander's $L^2$ estimates show that $\square$ is invertible on $L^2_{(0,q)}(\Omega)$ for $q \ge 1$.
When $\bar\partial \alpha = 0$, the form $\bar\partial^* N \alpha$ solves $\bar\partial v = \alpha$ and is the solution of least $L^2$ norm.
Regularity up to the boundary may fail for smooth bounded pseudoconvex domains: Christ showed that $N$ does not preserve $C^\infty(\overline{\Omega})$ on worm domains.

## Explanations

### L1 human

The problem asks to invert a Laplace-type operator built from $\bar\partial$, with boundary conditions that come from the adjoint of $\bar\partial$ in a Hilbert space.
Once it is inverted, one gets a canonical solution of the equation $\bar\partial v = \alpha$ with good estimates.
The difficulty is that the boundary conditions are weaker than in classical problems, so the behaviour at the boundary depends on the shape of the domain.
</document>

<document source="https://tamnd.github.io/math2/node/Q4TKABC1N2.md" tag="Q4TKABC1N2" kind="definition">
# Integral transform

## Definition

Let $I$ and $J$ be intervals of $\R$, with $-\infty \le t_1 < t_2 \le \infty$ the endpoints of $I$.
Let $K \colon I \times J \to \C$ be a measurable function.
Let $f \colon I \to \C$ be a [[function|0XHDTE5YMZ]].
The *integral transform* of $f$ with kernel $K$ is the function $Tf$ given by
$$(Tf)(u) = \int_{t_1}^{t_2} f(t)\, K(t, u)\, dt$$
at each $u \in J$ for which the integral converges.
The integral is a [[Lebesgue integral|XVEWR93EMV]] or an improper Riemann integral, depending on the setting.

## Remarks

Example: take $I = [0, \infty)$, $J = (0, \infty)$ and $K(t, u) = e^{-ut}$.
This is the Laplace transform; for $f = 1$ the integral $\int_0^\infty e^{-ut}\, dt = 1/u$ converges for every $u \in J$.
Non-example: the map $f \mapsto f'$ is linear but is not an integral transform with a function kernel.
The value $f'(u)$ depends only on the values of $f$ near $u$, and no integrable kernel produces this for all smooth $f$.
The map $f \mapsto Tf$ is linear wherever both sides converge.
The same definition works with $I$ and $J$ replaced by measure spaces, for example subsets of $\R^n$.
</document>

<document source="https://tamnd.github.io/math2/node/Q4YAST023H.md" tag="Q4YAST023H" kind="statement">
# Newton–Gregory forward formula

## Statement

Let $x_0 \in \R$, let $h > 0$, and let $x_i = x_0 + i h$ for $i = 0, 1, \dots, n$.
Let $f$ be a real function defined at these points.
Define the forward differences by $\Delta^0 f(x) = f(x)$ and $\Delta^{k+1} f(x) = \Delta^k f(x + h) - \Delta^k f(x)$.
Let $p$ be the interpolating polynomial of degree at most $n$ with $p(x_i) = f(x_i)$ for $i = 0, \dots, n$.
Then for every real $s$,
$$p(x_0 + s h) = \sum_{k=0}^{n} \binom{s}{k} \Delta^k f(x_0), \qquad \binom{s}{k} = \frac{s (s - 1) \cdots (s - k + 1)}{k!}.$$

## Remarks

Proof idea: the [[Newton polynomial|9W77FPTM9E]] gives $p(x) = \sum_{k=0}^{n} f[x_0, \dots, x_k] \prod_{j<k} (x - x_j)$.
For equally spaced nodes, induction on $k$ shows that the [[divided difference|N5ZHV4HWDC]] is $f[x_0, \dots, x_k] = \Delta^k f(x_0) / (k!\, h^k)$.
With $x = x_0 + s h$ one has $\prod_{j<k} (x - x_j) = h^k s (s - 1) \cdots (s - k + 1)$, and the two formulas combine.

Example: $f(x) = x^2$, $x_0 = 0$, $h = 1$, $n = 2$.
The values are $0, 1, 4$, so $\Delta f(0) = 1$ and $\Delta^2 f(0) = 2$.
The formula gives $p(s) = 0 + s + \binom{s}{2} \cdot 2 = s + s(s - 1) = s^2$, which is $f$ itself.

The formula needs equal spacing; for unequal nodes the general Newton form with divided differences must be used.

## Explanations

### L1 human

Given equally spaced values of a function, the formula builds the polynomial through them from a table of differences.
The first differences measure slope, the second differences measure curvature, and so on.
The weights are binomial coefficients in the position $s$, measured in steps of $h$.
It is the discrete analogue of the Taylor series, with differences in place of derivatives and $\binom{s}{k}$ in place of $s^k / k!$.

### L2 human

The formula is the basis of classical table interpolation and of many finite difference formulas.
Differentiating or integrating it term by term gives numerical differentiation rules and the Newton–Cotes formulas.
The error is that of [[polynomial interpolation|SS1YMS2NPC]]: if $f$ is $n + 1$ times differentiable, then $f(x_0 + s h) - p(x_0 + s h) = h^{n+1} \binom{s}{n+1} f^{(n+1)}(\xi)$ for some $\xi$ in the smallest interval containing $x_0, x_n$ and $x_0 + s h$.
There is a matching backward formula, based at $x_n$ with backward differences.
</document>

<document source="https://tamnd.github.io/math2/node/Q5B5YGY7D6.md" tag="Q5B5YGY7D6" kind="definition">
# Fredholm operator

## Definition

Let $X$ and $Y$ be [[Banach spaces|06GZ956657]], and let $T \colon X \to Y$ be a [[bounded linear map|03TM135G0N]].
The map $T$ is a *Fredholm operator* if both of the following hold.

1. The [[kernel|0S23E6YE83]] $\ker T$ is finite-dimensional.
2. The cokernel $\operatorname{coker} T = Y / T(X)$, the [[quotient|0BR3FY5Q8M]] of $Y$ by the [[image|0WSFKKT6FF]] of $T$, is finite-dimensional.

## Remarks

If $T$ is Fredholm, then its image $T(X)$ is automatically closed in $Y$.
Example: let $S$ be the right shift on $\ell^2$, $S(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$.
Its kernel is $\set{0}$, and its image is $\set{x \in \ell^2 : x_1 = 0}$, of codimension $1$, so $S$ is Fredholm.
Non-example: the diagonal operator $D(x_n) = (x_n / n)$ on $\ell^2$ has kernel $\set{0}$.
Its image contains every finitely supported sequence, so it is dense, but it misses $(1/n)$, so it is not closed.
By the previous remark, the image of a bounded operator with finite-dimensional cokernel is closed, so clause 2 fails.
If $K \colon X \to X$ is a [[compact operator|M2NEP1K5AE]], then $I - K$ is Fredholm.
Every linear map between finite-dimensional spaces is Fredholm.
</document>

<document source="https://tamnd.github.io/math2/node/Q5C1E10CSQ.md" tag="Q5C1E10CSQ" kind="concept">
# H-cone

## Summary

An H-cone is an ordered convex cone whose axioms abstract the properties of the cone of positive [[superharmonic functions|X4V2S56YQQ]] on a domain.
The theory of H-cones was developed by Nicu Boboc, Gheorghe Bucur and Aurel Cornea.
It treats potential theory as the study of an ordered cone, with no underlying space of points given in advance.
Under further conditions an H-cone can be represented as a cone of functions on a suitable space.

## Explanations

### L1 human

Positive superharmonic functions can be added, multiplied by positive numbers, and compared, and they form a cone with a rich order structure.
Many results of potential theory, such as balayage and the decomposition of a function into a potential and a harmonic part, depend mainly on this structure.
An H-cone is a cone with these order properties, taken as axioms.
In this way one theory applies at once to harmonic spaces, to Markov processes and to other settings.
</document>

<document source="https://tamnd.github.io/math2/node/Q5D54R3RGE.md" tag="Q5D54R3RGE" kind="concept">
# Zonal spherical harmonics

## Summary

Fix a point $y$ of the sphere and a degree $\ell$.
The zonal spherical harmonic $Z_y^{(\ell)}$ is the spherical harmonic of degree $\ell$ that reproduces the value at $y$ of every harmonic of degree $\ell$.
It is invariant under the rotations that fix $y$, so it depends only on the angle between $x$ and $y$.
It is a multiple of a [[Gegenbauer polynomial|5TRFA6F79B]] in $x \cdot y$, and of the [[Legendre polynomial|QFYGJVA996]] $P_\ell(x \cdot y)$ on the sphere in $\R^3$.
It is the building block of expansions in [[spherical harmonics|3F32Y21HP4]] and of the Poisson kernel of the ball.

## Explanations

### L1 human

Pick a point $y$ on a globe, as if it were the north pole.
A zonal function is one that is constant along each circle of latitude around $y$, so it only depends on the distance from $y$.
Among the spherical harmonics of a given degree there is, up to a factor, exactly one zonal one.
With the right factor, integrating any harmonic of that degree against it returns the value of that harmonic at $y$.
On the ordinary sphere it is $\frac{2\ell + 1}{4\pi} P_\ell(\cos \theta)$, where $\theta$ is the angle from $y$.
</document>

<document source="https://tamnd.github.io/math2/node/Q5EWVTA2RW.md" tag="Q5EWVTA2RW" kind="statement">
# Riemannian Penrose inequality

## Statement

Let $(M, g)$ be a complete, asymptotically flat, three-dimensional [[Riemannian manifold|7CBT4SM83Y]] with nonnegative [[scalar curvature|ZHX2VYQDZ3]].
Assume that the boundary $\partial M$ is an outermost [[minimal surface|41X0YW2R9N]], possibly with several components.
This means that $\partial M$ is a closed minimal surface and that no other closed minimal surface in $M$ encloses it.
Let $A$ be the total area of $\partial M$ and let $m$ be the ADM mass of $(M, g)$.
Then
$$m \ge \sqrt{\frac{A}{16\pi}}.$$
Equality holds if and only if $(M, g)$ is isometric to the spatial [[Schwarzschild metric|6Y50T68WKY]] of mass $m$ outside its horizon.

## Remarks

Huisken and Ilmanen proved the inequality in 2001 when $A$ is the area of a largest component of $\partial M$, using a weak form of [[inverse mean curvature flow|5XVEF5DKAG]].
Along this flow the Hawking mass of the evolving surface does not decrease, a monotonicity first observed by Geroch.
The Hawking mass equals $\sqrt{A/16\pi}$ on the horizon and tends to at most $m$ at infinity.
Bray proved the version with total area in 2001 by a conformal flow of metrics.
For the spatial Schwarzschild metric of mass $m$, the horizon is the sphere $r = 2m$ of area $16\pi m^2$, so both sides equal $m$.
Nonnegative scalar curvature is needed: without it the mass can even be negative.

## Explanations

### L1 human

In general relativity, a black hole has a horizon with a certain area.
The inequality says that the total mass of the space is at least a fixed multiple of the square root of that area.
So black holes cannot carry more horizon than their mass allows.
The Schwarzschild black hole is the extreme case where the two sides are equal.

### L2 human

The statement is the time-symmetric case of a conjecture of Penrose; the scalar curvature condition encodes nonnegative energy density.
It strengthens the [[positive mass theorem|PYRPXHNQTB]], which gives $m \ge 0$ when there is no horizon.
The two proofs use different flows: inverse mean curvature flow of surfaces, and a flow of conformal metrics.
The general case of the Penrose conjecture, without time symmetry, is still open.
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<document source="https://tamnd.github.io/math2/node/Q5H9VYFK98.md" tag="Q5H9VYFK98" kind="concept">
# Method of steepest descent

## Summary

The method of steepest descent, also called the saddle point method, finds the asymptotic behaviour of contour integrals $\int_C g(z)\, e^{\lambda f(z)}\,dz$ as the parameter $\lambda$ grows.
One deforms the contour so that it passes through [[saddle points|0JW6MDHAZD]] of the holomorphic function $f$ along paths on which $\operatorname{Im} f$ is constant.
On these paths the integrand does not oscillate, and [[Laplace's method|9G7K8DBT9Y]] gives the leading terms.
It yields the asymptotics of many special functions, such as the Airy and Bessel functions, and of coefficients of generating functions.

## Explanations

### L1 human

Picture $\operatorname{Re} f$ as a landscape over the complex plane, with mountains and valleys.
A holomorphic function has no peaks or pits, only passes, the saddle points.
For large $\lambda$ the integrand is tiny everywhere except where $\operatorname{Re} f$ is largest along the path.
The trick is to walk from one valley to the next over the lowest pass, along the steepest route.
Then the integrand is concentrated at the pass and does not oscillate.
For example, rotating the line of integration of $\int_{\R} e^{i\lambda x^2}\,dx$ by $45$ degrees turns it into a real Gaussian integral and gives $\sqrt{\pi/\lambda}\, e^{i\pi/4}$.
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<document source="https://tamnd.github.io/math2/node/Q5JHEPRYQ6.md" tag="Q5JHEPRYQ6" kind="concept">
# Galois connection

## Summary

A Galois connection is a pair of maps $f \colon P \to Q$ and $g \colon Q \to P$ between [[partially ordered sets|011C29Q1S1]] with $f(x) \le y$ exactly when $x \le g(y)$.
It is a weak form of an order isomorphism, and it is the order-theoretic case of an adjunction between functors.
The [[correspondence between subgroups and subfields|R3625YKQC0]] in Galois theory is the classical example, which gives the notion its name.
Every Galois connection produces a [[closure operator|17FVAN6HXX]] $g \circ f$ on $P$.

## Explanations

### L1 human

A Galois connection pairs two ordered sets so that each map is the best approximation to an inverse of the other.
The map from integers to reals and the floor function form an example.
For an integer $n$ and a real $x$, the statement "$n \le x$" is the same as "$n \le \lfloor x \rfloor$".
So the floor of $x$ is the best integer approximation of $x$ from below.
</document>

<document source="https://tamnd.github.io/math2/node/Q5N7E7P6F6.md" tag="Q5N7E7P6F6" kind="definition">
# Hamilton–Jacobi–Isaacs equation

## Definition

Let $A \subseteq \R^{k}$ and $B \subseteq \R^{l}$ be nonempty compact sets and let $T > 0$.
Let $f \colon \R^n \times A \times B \to \R^n$, $h \colon \R^n \times A \times B \to \R$ and $g \colon \R^n \to \R$ be continuous.
Consider the two-player zero-sum [[differential game|THSHGNJNPS]] with dynamics $x' = f(x, a, b)$ and payoff $\int_t^T h(x(s), a(s), b(s)) \, ds + g(x(T))$.
The player with control $a(s) \in A$ minimises the payoff, and the player with control $b(s) \in B$ maximises it.
The *upper* and *lower Hamiltonians* are, for $x, p \in \R^n$,
$$H^+(x, p) = \min_{a \in A} \max_{b \in B} \left\{ f(x, a, b) \cdot p + h(x, a, b) \right\}, \qquad H^-(x, p) = \max_{b \in B} \min_{a \in A} \left\{ f(x, a, b) \cdot p + h(x, a, b) \right\}.$$
The *Hamilton–Jacobi–Isaacs equations* are the terminal value problems
$$V_t + H^\pm(x, \nabla_x V) = 0 \quad \text{in } (0, T) \times \R^n, \qquad V(T, x) = g(x),$$
for an unknown function $V \colon [0, T] \times \R^n \to \R$, one with $H^+$ and one with $H^-$.
The game satisfies *Isaacs' condition* if $H^+ = H^-$, and then the two equations coincide.

## Remarks

Always $H^- \le H^+$.
If $B$ has one point, both equal $\min_{a} \{ f \cdot p + h \}$ and the equation is the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]], a special [[Hamilton–Jacobi equation|5TRXQKP7DY]].
Assume also that $f$, $h$ and $g$ are bounded and Lipschitz in $x$.
Then the upper and lower values of the game, defined with nonanticipating strategies, are the [[viscosity solutions|H1ERJ3T75T]] of these equations.
Under Isaacs' condition the two values are equal, so the game has a value.

Example: if $f(x, a, b) = f_1(x, a) + f_2(x, b)$ and $h(x, a, b) = h_1(x, a) + h_2(x, b)$, then the expression splits into a part that depends only on $a$ and a part that depends only on $b$.
So $\min_a \max_b$ and $\max_b \min_a$ are both the sum of $\min_a$ of the first part and $\max_b$ of the second, and Isaacs' condition holds.

Non-example: $n = 1$, $A = B = [-1, 1]$, $f(x, a, b) = (a - b)^2$ and $h = 0$.
For $p = 1$, $\max_b (a - b)^2 = (1 + \abs{a})^2$, so $H^+(x, 1) = 1$, while $\min_a (a - b)^2 = 0$, so $H^-(x, 1) = 0$.
Isaacs' condition fails, and the two equations are different.

## Explanations

### L1 human

The equation says how the value of the game changes in time when both players play their best against each other at each instant.
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<document source="https://tamnd.github.io/math2/node/Q5QFYKD004.md" tag="Q5QFYKD004" kind="statement">
# Hahn–Mazurkiewicz theorem

## Statement

Let $X$ be a nonempty [[Hausdorff space|0S1FJ8VEX9]].
Then there is a [[continuous|0X3WGX95FE]] [[surjection|01RVCMSN5N]] $[0, 1] \to X$ if and only if $X$ is [[compact|0TVJZYB6SH]], [[connected|0V032VWTHD]], [[locally connected|FDVAHRJKF3]] and [[second-countable|03PPNCM9GG]].

## Remarks

A continuous image of $[0, 1]$ in a Hausdorff space is compact, connected and metrizable, and the map is a quotient map, which carries local connectedness over to $X$.
For the converse, $X$ is metrizable by the [[Urysohn metrization theorem|HBFHDH4EJX]], so it is a locally connected [[continuum|NJQ196M1WR]].
The [[Cantor set|0BXNDZFQME]] maps continuously onto every compact metric space.
Local connectedness lets one join the images of the ends of each removed interval by small arcs, which extends the map to all of $[0, 1]$.
For example, the square $[0, 1]^2$ satisfies the conditions, so there is a [[space-filling curve|QPRHC4DZ20]] onto it.
The [[Warsaw circle|17VG475BRV]] is compact, connected and metrizable but not locally connected, so it is not a continuous image of $[0, 1]$.

## Explanations

### L1 human

The theorem says exactly which spaces can be traced by a single continuous curve that passes through every point.
They are the compact spaces in one piece that are also connected at small scales near every point.
So squares, cubes and even the Hilbert cube can be filled by a curve.

### L2 human

The metric spaces in the theorem are the Peano continua, the locally connected continua.
The theorem turns the surprise of the Peano curve into a general rule.
It shows that a continuous image of an interval can have any finite dimension.
A related fact is that every Peano continuum is arcwise connected.
</document>

<document source="https://tamnd.github.io/math2/node/Q5RN2FNX6Z.md" tag="Q5RN2FNX6Z" kind="example">
# Meyer wavelet

## Example

Use the [[Fourier transform|DPVM34ADZV]] $\hat f(\omega) = \int_{\R} f(x) e^{-i\omega x} \, dx$.
Fix an infinitely differentiable function $\nu \colon \R \to [0,1]$ with $\nu(x) = 0$ for $x \le 0$, $\nu(x) = 1$ for $x \ge 1$, and $\nu(x) + \nu(1-x) = 1$ for all $x$.
The *Meyer wavelet* $\psi$ is the function in $L^2(\R)$ whose Fourier transform is
$$\hat\psi(\omega) = \begin{cases} e^{i\omega/2} \sin\left(\frac{\pi}{2} \nu\left(\frac{3\abs{\omega}}{2\pi} - 1\right)\right) & \text{if } \frac{2\pi}{3} \le \abs{\omega} \le \frac{4\pi}{3}, \\ e^{i\omega/2} \cos\left(\frac{\pi}{2} \nu\left(\frac{3\abs{\omega}}{4\pi} - 1\right)\right) & \text{if } \frac{4\pi}{3} \le \abs{\omega} \le \frac{8\pi}{3}, \\ 0 & \text{otherwise.} \end{cases}$$
The functions $\psi_{j,k}(x) = 2^{j/2} \psi(2^j x - k)$, for $j, k \in \Z$, form an orthonormal basis of $L^2(\R)$.
So $\psi$ is an orthonormal wavelet.
It comes from a [[multiresolution analysis|1GHQMBR7DE]] whose scaling function $\varphi$ has $\hat\varphi(\omega) = 1$ for $\abs{\omega} \le 2\pi/3$ and $\hat\varphi(\omega) = 0$ for $\abs{\omega} \ge 4\pi/3$.

## Remarks

The two formulas agree at $\abs{\omega} = 4\pi/3$, since there the sine has argument $\pi/2$ and the cosine has argument $0$.
The condition $\nu(x) + \nu(1-x) = 1$ makes the values of $\abs{\hat\psi}^2$ on overlapping bands add up to $1$, which is the key to orthonormality.
Since $\hat\psi$ vanishes near $\omega = 0$, every moment $\int x^n \psi(x) \, dx$ is zero.
Since $\hat\psi$ has compact support, $\psi$ is band-limited and infinitely differentiable.
Since $\nu$ is $C^\infty$, $\hat\psi$ is $C^\infty$ with compact support, so $\psi$ decays faster than any power of $\abs{x}$.
The support of $\psi$ itself is not compact.
The Meyer wavelet can be seen as a smoothed version of the [[Shannon wavelet|6C1S8HD38Z]], whose Fourier transform is a sharp indicator function.

## Explanations

### L1 human

The Meyer wavelet is built in frequency space rather than in time.
Its spectrum lives in two bands, one on each side of the origin, and the edges of the bands are rounded off smoothly instead of being cut sharply.
Because the edges are smooth, the wavelet is concentrated in time as well: it is a smooth wave that dies out quickly away from the origin.
Shifting and dyadically stretching it gives an orthonormal basis for all finite-energy signals on the line.
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<document source="https://tamnd.github.io/math2/node/Q5T05ET56S.md" tag="Q5T05ET56S" kind="concept">
# Direct limit of rings

## Summary

The direct limit of rings glues a directed family of [[rings|0B1261B5ED]] $R_i$, connected by ring homomorphisms $R_i \to R_j$ for $i \le j$, into one ring.
This ring receives a map from each $R_i$.
It is the [[direct limit|X9HEQ29Q49]] taken in the category of rings, and it is computed on the underlying sets.
Every element of the limit comes from some $R_i$, and two elements are equal if they become equal in some later $R_k$.
Polynomial rings in infinitely many variables and many infinite-dimensional algebras arise this way.

## Explanations

### L1 human

Imagine a chain of rings, each sitting inside the next one, such as $k[x_1] \subseteq k[x_1, x_2] \subseteq k[x_1, x_2, x_3] \subseteq \cdots$.
The direct limit is the union of the chain, here the polynomial ring in infinitely many variables.
When the maps are not inclusions, elements of different rings are identified as soon as they become equal further along.
Each computation in the limit takes place in one of the rings of the family, since only finitely many elements are involved.
</document>

<document source="https://tamnd.github.io/math2/node/Q61VFHCH2J.md" tag="Q61VFHCH2J" kind="concept">
# Riemannian metric

## Summary

A Riemannian metric on a smooth manifold is a smoothly varying [[inner product|0NB0AWGJBW]] on each tangent space.
It lets one measure lengths of tangent vectors and angles between them, and from these the lengths of curves, distances, areas and volumes.
It is a symmetric positive definite [[tensor field|WX302T93TW]] of type $(0, 2)$.
Every smooth manifold admits Riemannian metrics, and the choice of one is the starting point of Riemannian geometry.

## Explanations

### L1 human

On a curved surface, a small piece looks almost flat, and on a flat piece you can use a ruler and a protractor.
A Riemannian metric gives such a ruler at every point, possibly different from point to point.
To measure a curve, add up the lengths of its tiny pieces with the local ruler.
In the upper half-plane, the metric $(dx^2 + dy^2)/y^2$ makes rulers shrink as you approach the $x$-axis, so the axis is infinitely far away.
This is the hyperbolic plane.
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<document source="https://tamnd.github.io/math2/node/Q630X0VHAE.md" tag="Q630X0VHAE" kind="statement">
# Property P conjecture

## Statement

Let $K \subset S^3$ be a tame [[knot|QYEAZTYGPT]] that is not the unknot.
Let $r = p/q \in \mathbb{Q}$ be a slope other than $\infty = 1/0$, and let $S^3_r(K)$ be the result of [[Dehn surgery|ZA405B0Z1S]] on $K$ with slope $r$.
Then $S^3_r(K)$ is not simply connected.

## Remarks

The homology of $S^3_{p/q}(K)$ is $\Z / p$, or $\Z$ when $p = 0$, so it is nontrivial unless $p = \pm 1$.
The content of the theorem is therefore about the slopes $r = \pm 1/q$.
Kronheimer and Mrowka proved it in 2004 using instanton Floer homology and work on taut foliations and contact structures.
With the Poincaré conjecture, proved by Perelman, it also follows from the Gordon–Luecke result that a nontrivial surgery on a nontrivial knot never gives $S^3$.
The hypothesis that $K$ is knotted is needed: $1/q$ surgery on the unknot gives $S^3$.
Example: $+1$ surgery on the right-handed trefoil gives the [[Poincaré homology sphere|NHERCK7C43]], up to orientation, which has the same homology as $S^3$ but a fundamental group of order $120$.

## Explanations

### L1 human

Dehn surgery cuts out a thickened knot from space and glues it back in a twisted way, making a new three-dimensional space.
For the unknot, some twists give back the $3$-sphere.
The theorem says that for a genuine knot, every twisted regluing gives a space with a loop that cannot be shrunk.
So surgery on a real knot always leaves a trace in the fundamental group.

### L2 human

By the [[Lickorish–Wallace theorem|1YP4RF1QXV]] every closed orientable $3$-manifold arises by surgery on a link, so it is natural to ask what surgery on a single knot can produce.
Property P says that, apart from the slope $\infty$ which gives back $S^3$, it never produces a fake or real $3$-sphere.
It implies the key step of the [[Gordon–Luecke theorem|EBASBW5TYS]], that nontrivial surgery on a nontrivial knot never gives $S^3$.
Conversely, that step together with the Poincaré conjecture implies Property P.
The proof by Kronheimer and Mrowka shows that the fundamental group of $S^3_{\pm 1/q}(K)$ has a nontrivial representation into $SU(2)$.
</document>

<document source="https://tamnd.github.io/math2/node/Q699KKMVJD.md" tag="Q699KKMVJD" kind="concept">
# Pushout

## Summary

A pushout is the [[colimit|BJS1CGTKQ5]] of a diagram made of two morphisms with a common source.
It glues two objects together along the images of a third object.
Pushouts give gluing of spaces, amalgamated free products of groups and tensor products of commutative rings.
It is the dual notion of a [[pullback|Q05K04QJ71]].

## Explanations

### L1 human

Take two objects $X$ and $Y$ and a common piece $Z$ that maps into both.
The pushout is the smallest object in which $X$ and $Y$ sit side by side and the two copies of $Z$ are glued into one.
For sets, take the disjoint union of $X$ and $Y$ and identify $f(z)$ with $g(z)$ for each $z$ in $Z$.
For example, gluing two closed intervals at one end point gives a longer interval.
In the same way, gluing two discs along their boundary circles gives a sphere.
</document>

<document source="https://tamnd.github.io/math2/node/Q6KCF4688J.md" tag="Q6KCF4688J" kind="statement">
# Korovkin's theorem

## Statement

Let $a < b$ and let $C[a, b]$ be the space of continuous functions $[a, b] \to \R$ with the supremum norm.
Let $L_n \colon C[a, b] \to C[a, b]$, $n \ge 1$, be [[positive linear operators|AQDXWH4812]]: each $L_n$ is linear, and $L_n f \ge 0$ whenever $f \ge 0$.
Let $e_0(x) = 1$, $e_1(x) = x$ and $e_2(x) = x^2$.
If $L_n e_i \to e_i$ [[uniformly|08FPABDQ4C]] on $[a, b]$ for $i = 0, 1, 2$, then $L_n f \to f$ uniformly on $[a, b]$ for every $f \in C[a, b]$.

## Remarks

The idea of the proof: fix $f$ and $\varepsilon > 0$.
By [[uniform continuity|0Q4DR37DVE]] there is $\delta > 0$ with $\abs{f(t) - f(x)} \le \varepsilon + 2 \norm{f}_\infty (t - x)^2 / \delta^2$ for all $t, x \in [a, b]$.
Apply the positive operator $L_n$ in the variable $t$ and evaluate at $x$.
This bounds $\abs{L_n f(x) - f(x)}$ by $\varepsilon$ plus terms in $L_n e_0 - e_0$, $L_n e_1 - e_1$ and $L_n e_2 - e_2$, which tend to $0$ uniformly.

Example: the Bernstein operators $B_n$ on $[0, 1]$ satisfy $B_n e_0 = e_0$, $B_n e_1 = e_1$ and $B_n e_2 = e_2 + x(1 - x)/n$.
So Korovkin's theorem gives $B_n f \to f$ uniformly for every continuous $f$.

Positivity is needed.
In the periodic setting the test functions are $1, \cos x, \sin x$.
The partial sums of Fourier series reproduce these test functions for $n \ge 1$, but they are not positive operators.
And there are continuous periodic functions whose Fourier series do not converge uniformly.

## Explanations

### L1 human

Many approximation schemes are positive linear operators: they average values of $f$ with nonnegative weights.
Korovkin's theorem says that to check such a scheme works for every continuous function, it is enough to check three functions: $1$, $x$ and $x^2$.
This turns a hard question into a short computation.

### L2 human

The theorem is the standard tool to prove convergence of the Bernstein, Szász–Mirakyan, Baskakov and Kantorovich operators and of Fejér means.
The periodic version uses the test functions $1, \cos x, \sin x$, and it proves Fejér's theorem on Cesàro means of Fourier series.
Quantitative forms bound $\norm{L_n f - f}_\infty$ by the modulus of continuity of $f$ and the errors on the test functions.
The theorem also shows a rigidity: a single positive linear operator on $C[a, b]$ that fixes $1$, $x$ and $x^2$ must be the identity.
</document>

<document source="https://tamnd.github.io/math2/node/Q6KKBP3RRP.md" tag="Q6KKBP3RRP" kind="concept">
# Polygon triangulation

## Summary

A polygon triangulation cuts a simple polygon into triangles by drawing diagonals between its vertices.
Every simple polygon with $n$ vertices has one, and each triangulation has exactly $n - 2$ triangles and $n - 3$ diagonals.
Triangulations reduce questions about a complicated polygon to questions about triangles.
They are a basic tool of [[computational geometry|KAT7K2NE9Z]], used for area computation, guarding problems such as the [[art gallery theorem|G4WSPQJ6W1]], and graphics.

## Explanations

### L1 human

Take a polygon drawn on paper whose sides do not cross.
Draw straight cuts from corner to corner, staying inside the polygon, until no more cuts fit without crossing.
The polygon is then split into triangles whose corners are corners of the polygon.
For a convex hexagon, the three cuts from one corner to the three non-adjacent corners give $4$ triangles.
It is a planar cousin of the [[triangulation|YH170WBQ6M]] of a space in topology.
</document>

<document source="https://tamnd.github.io/math2/node/Q6T1TTJJZG.md" tag="Q6T1TTJJZG" kind="definition">
# Vanishing moments of a wavelet

## Definition

Let $\psi \colon \R \to \C$ be a [[wavelet|4PZ35C0GV3]] and let $p \ge 1$ be an integer.
Suppose that $x \mapsto x^k \psi(x)$ is [[Lebesgue integrable|DWJF7ABPR1]] on $\R$ for $k = 0, \dots, p - 1$.
The wavelet $\psi$ has *$p$ vanishing moments* if its [[moments|01XAHPADBA]] of order less than $p$ vanish:
$$\int_\R x^k \, \psi(x) \, dx = 0 \qquad \text{for } k = 0, 1, \dots, p - 1.$$

## Remarks

Equivalently, $\int_\R P(x) \psi(x) \, dx = 0$ for every polynomial $P$ of degree less than $p$.
In terms of the Fourier transform, under the same integrability, this says that $\hat\psi$ has a zero of order at least $p$ at $\xi = 0$.
The *number* of vanishing moments of $\psi$ is the largest such $p$.

Example: the [[Haar wavelet|JW5PA9XA7F]] $\psi = 1_{[0, 1/2)} - 1_{[1/2, 1)}$ has $\int \psi = \tfrac12 - \tfrac12 = 0$, so it has $1$ vanishing moment.

Non-example: the Haar wavelet does not have $2$ vanishing moments, since $\int_\R x \psi(x) \, dx = \tfrac18 - \tfrac38 = -\tfrac14 \ne 0$.
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# Young tableau

## Summary

A Young tableau is a [[Young diagram|096AHMB3P4]] whose boxes are filled with positive integers, with the entries increasing along rows and down columns.
In a standard tableau the entries are $1, \dots, n$, each used once; in a semistandard tableau repetitions are allowed along rows.
Tableaux index bases of the irreducible representations of the symmetric group and of the general linear group.
They are the combinatorial heart of [[Schur polynomials|43VFMJCDYJ]], the [[hook length formula|VCMS1CP91N]] and the [[Robinson–Schensted correspondence|1SCF8X47G1]].

## Explanations

### L1 human

Draw the rows of boxes of a [[partition|0JV30XN73Z]], longest row on top.
Now write a number in each box so that numbers grow when you move right and when you move down.
For the shape with rows of lengths $2$ and $1$, the fillings of $1, 2, 3$ that work are $\begin{smallmatrix} 1 & 2 \\ 3 & \end{smallmatrix}$ and $\begin{smallmatrix} 1 & 3 \\ 2 & \end{smallmatrix}$.
Counting such fillings answers questions in algebra, probability and the study of permutations.
</document>

<document source="https://tamnd.github.io/math2/node/Q6W7PDWN4V.md" tag="Q6W7PDWN4V" kind="definition">
# Cartesian closed category

## Definition

A [[category|A2DE66937D]] $\mathcal{C}$ is *Cartesian closed* if:

1. $\mathcal{C}$ has a [[terminal object|EK85VSSRMR]].
2. Any two objects $X$ and $Y$ of $\mathcal{C}$ have a [[product|S5744QZV4N]] $X \times Y$.
3. Any two objects $Y$ and $Z$ of $\mathcal{C}$ have an [[exponential object|RHWD8DY41R]] $Z^Y$.

## Remarks

Clauses 1 and 2 say that $\mathcal{C}$ has finite products.
Clause 3 then says that each functor $- \times Y$ has a right adjoint $(-)^Y$.
So there are bijections $\Hom(X \times Y, Z) \cong \Hom(X, Z^Y)$, natural in $X$ and $Z$.

Example: the category of sets is Cartesian closed.
A one-point set is terminal, $X \times Y$ is the Cartesian product, and $Z^Y$ is the set of maps $Y \to Z$ with evaluation.

Non-example: the category of abelian groups is not Cartesian closed, although clauses 1 and 2 hold.
In a Cartesian closed category, $- \times Y$ is a left adjoint, so it preserves the initial object $0$, and $0 \times Y \cong 0$.
In abelian groups $0 \times \Z \cong \Z$, which is not $0$, so clause 3 fails.
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<document source="https://tamnd.github.io/math2/node/Q6WPMDYWN2.md" tag="Q6WPMDYWN2" kind="definition">
# Equational logic

## Definition

Let $\sigma$ be a signature with only function symbols and let $X$ be a set of variables.
Let $\Sigma$ be a set of [[identities|PKD440G2XW]] over $\sigma$ and $X$.
The identities *derivable* from $\Sigma$ in *equational logic*, written $\Sigma \vdash s \approx t$, form the smallest set of identities that satisfies the following rules, for all [[terms|6D3WA7N2W4]] $s, t, u, s_i, t_i$:

1. (axioms) every identity in $\Sigma$ is derivable;
2. (reflexivity) $t \approx t$ is derivable;
3. (symmetry) if $s \approx t$ is derivable, so is $t \approx s$;
4. (transitivity) if $s \approx t$ and $t \approx u$ are derivable, so is $s \approx u$;
5. (replacement) if $s_i \approx t_i$ is derivable for $i = 1, \ldots, n$ and $f$ is a function symbol of arity $n$, then $f(s_1, \ldots, s_n) \approx f(t_1, \ldots, t_n)$ is derivable;
6. (substitution) if $s \approx t$ is derivable and $\theta$ is a [[substitution|NCDZ969V7X]], then $s\theta \approx t\theta$ is derivable.

## Remarks

A derivation is a finite list of identities, each one in $\Sigma$ or obtained from earlier ones by a rule.
Each rule preserves truth in every algebra, so every derivable identity holds in every algebra that satisfies $\Sigma$; Birkhoff's completeness theorem gives the converse.

Example: let $\Sigma = \set{x + 0 \approx x, \ x + y \approx y + x}$.
Rule 6 applied to the second axiom with $\theta = \set{x \mapsto 0, \ y \mapsto x}$ gives $0 + x \approx x + 0$, and rule 4 with the first axiom gives $\Sigma \vdash 0 + x \approx x$.

Non-example: $x + y \approx y$ is not derivable from $\Sigma$.
The group $(\Z, +, 0)$ satisfies $\Sigma$, and every derivable identity holds in it, but $1 + 0 \ne 0$.
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# Hermitian metric on a vector bundle

## Summary

A Hermitian metric on a complex [[vector bundle|P4DZM975BV]] is a choice of [[Hermitian inner product|0NB0AWGJBW]] on each fibre that varies smoothly with the base point.
It lets one measure the length of sections and define the Chern connection and its curvature.
Every complex vector bundle on a paracompact manifold has one, by a partition of unity.
A Hermitian metric on the holomorphic tangent bundle of a complex manifold is what makes it a [[Hermitian manifold|CK1DCPC3D3]].

## Explanations

### L1 human

Each fibre of a complex vector bundle is a complex vector space, but it comes with no notion of length.
A Hermitian metric gives each fibre an inner product, as on $\C^r$, and asks that it change smoothly from fibre to fibre.
Then each vector in the bundle has a length, and each section has a length function on the base.
On the trivial line bundle over $\C$, the rule $\abs{v}_z^2 = e^{-\abs{z}^2} \abs{v}^2$ is such a metric: it is the usual length, rescaled by a positive smooth factor at each point.
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<document source="https://tamnd.github.io/math2/node/Q6XNC8JAZD.md" tag="Q6XNC8JAZD" kind="concept">
# Compactification

## Summary

A compactification of a [[topological space|0CQHFPGJ45]] $X$ is a [[compact space|0TVJZYB6SH]] that contains a copy of $X$ as a [[dense|0BSJSX7ME2]] subspace.
It adds "points at infinity" so that sequences or nets that escape $X$ get limits.
Standard examples are the [[one-point compactification|0GWN3CMRK8]] and the [[Stone–Čech compactification|D9MWP5C85W]].
A space has a Hausdorff compactification exactly when it is a [[Tychonoff space|C5N2J2K6Z3]].

## Explanations

### L1 human

The open interval $(0, 1)$ is not compact, because points can run toward the missing ends.
Adding the two end points gives $[0, 1]$, a compactification of $(0, 1)$.
Bending the real line into a circle and adding one point at the top gives another compactification, of $\R$.
Different compactifications of the same space add different sets of ideal points.
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<document source="https://tamnd.github.io/math2/node/Q702WCDQQ3.md" tag="Q702WCDQQ3" kind="concept">
# Large deviations theory

## Summary

Large deviations theory studies the exponential decay of the probabilities of rare events.
Its central notion is the large deviation principle, with a [[rate function|140N2PQSTE]] $I$.
A family of probability measures $\mu_n$ satisfies it when $\mu_n(A)$ behaves like $\exp(-n \inf_{x \in A} I(x))$.
Basic results are [[Cramér's theorem|RZZK51CXMN]] for sample means, [[Sanov's theorem|QWEF2WAXJM]] for empirical measures and [[Varadhan's lemma|BD1CNVV5P0]] for exponential integrals.
The theory is used in statistical mechanics, information theory, queueing and risk analysis.

## Explanations

### L1 human

The [[law of large numbers|0ZVEAYVRC3]] says that the average of many coin tosses is close to $1/2$.
Large deviations theory asks how unlikely it is that the average is far from $1/2$, say above $0.6$.
The answer is that such a probability shrinks exponentially in the number of tosses, with a precise exponent.
For fair coins, the probability that the average of $n$ tosses is at least $0.6$ is about $e^{-0.020 n}$, up to factors that grow more slowly than any exponential.
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# Spectral mapping theorem for polynomials

## Statement

Let $A$ be a complex [[Banach algebra|9KYWN94QDN]] with unit $1 \ne 0$, let $a \in A$, and let $p \in \C[z]$ be a [[polynomial|0SDC934R52]].
Let $\sigma(b)$ denote the [[spectrum|EH0CZ19RKP]] of $b \in A$.
Then
$$\sigma(p(a)) = p(\sigma(a)) = \set{p(\lambda) : \lambda \in \sigma(a)}.$$

## Remarks

Proof idea: if $p$ is constant equal to $c$, both sides are $\set{c}$, using the [[nonemptiness of the spectrum|QWQC5SV0SD]].
Otherwise fix $\mu \in \C$ and factor $p(z) - \mu = c (z - \lambda_1) \cdots (z - \lambda_k)$ with $c \ne 0$.
Then $p(a) - \mu 1 = c (a - \lambda_1 1) \cdots (a - \lambda_k 1)$, a product of commuting factors.
A product of commuting elements is invertible if and only if each factor is invertible.
So $\mu \in \sigma(p(a))$ if and only if some root $\lambda_i$ of $p - \mu$ lies in $\sigma(a)$, that is, if and only if $\mu = p(\lambda)$ for some $\lambda \in \sigma(a)$.

Example: if $a$ is a $2 \times 2$ complex matrix with eigenvalues $1$ and $-1$, then $a^2$ has the single eigenvalue $1$, matching $p(\sigma(a)) = \set{1}$ for $p(z) = z^2$.

## Explanations

### L1 human

If you know the spectrum of $a$, you know the spectrum of every polynomial in $a$: just apply the polynomial to each point.
For matrices this is the familiar fact that $p(A)$ has eigenvalues $p(\lambda)$ for the eigenvalues $\lambda$ of $A$.

### L2 human

The theorem gives $\sigma(a^n) = \set{\lambda^n : \lambda \in \sigma(a)}$, which is the easy half of [[Gelfand's spectral radius formula|T4RFYT8HDN]].
It extends to holomorphic functions in the holomorphic functional calculus, and to continuous functions of normal elements in the [[continuous functional calculus|JB7S6WPZ77]] of C*-algebras.
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<document source="https://tamnd.github.io/math2/node/Q72NHN7ATC.md" tag="Q72NHN7ATC" kind="concept">
# Completeness in order theory

## Summary

In order theory, a completeness property of a partially ordered set says that every subset of a certain kind has a supremum, or an infimum.
All subsets give [[complete lattices|D7A0ZGZT5S]], and nonempty bounded subsets give [[conditionally complete lattices|CZPBQY67ZJ]].
Directed subsets give [[directed-complete partial orders|7TSXP95QN6]].
These notions organize much of lattice theory and domain theory.

## Explanations

### L1 human

An ordered set is "complete" when some collections of its elements are sure to have a least upper bound.
Which collections are required depends on the context.
Asking it for all finite nonempty sets gives a join-semilattice; asking it for every set gives a complete lattice.
The real numbers are complete for nonempty sets that are bounded above, and this is the form of completeness used in calculus.
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<document source="https://tamnd.github.io/math2/node/Q75YRG25JK.md" tag="Q75YRG25JK" kind="concept">
# Polynomial greatest common divisor

## Summary

The greatest common divisor of two polynomials over a field is the monic common divisor that every common divisor divides.
It is the [[greatest common divisor|0ZAF9KHDDF]] in the ring $K[x]$, made unique by asking it to be monic.
The [[Euclidean algorithm|023VY493K0]] computes it with repeated [[long division|D6D0SR1FK3]].
It detects common roots, and $\gcd(f, f')$ detects [[repeated roots|067F8YVYMH]].

## Explanations

### L1 human

Two polynomials may share factors, as two integers may share prime factors.
The greatest common divisor is the product of all the shared factors, counted as often as both contain them.
For example, $x^2 - 1 = (x - 1)(x + 1)$ and $x^2 - 3x + 2 = (x - 1)(x - 2)$ share only $x - 1$, which is their greatest common divisor.
The [[definition|99J1WXG3F7]] gives the conditions.
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# Modular lattice

## Definition

Let $(L, \vee, \wedge)$ be a [[lattice|WH4SFAACAS]] with [[join and meet|J2T75ATRV5]] $\vee$ and $\wedge$ and order $\le$.
The lattice $L$ is *modular* if for all $a, b, x \in L$ with $a \le b$,
$$a \vee (x \wedge b) = (a \vee x) \wedge b.$$

## Remarks

The inequality $a \vee (x \wedge b) \le (a \vee x) \wedge b$ holds in every lattice when $a \le b$, so only the reverse inequality is a condition.
Equivalently, $L$ satisfies the identity $((a \wedge b) \vee x) \wedge b = (a \wedge b) \vee (x \wedge b)$ for all $a, b, x \in L$, so the modular lattices form a variety.

Example: the lattice of subspaces of a vector space $V$ is modular.
If $A \subseteq B$ and $X$ are subspaces, and $b = a + x \in (A + X) \cap B$ with $a \in A$, $x \in X$, then $x = b - a \in B$, so $b \in A + (X \cap B)$.

Non-example: the [[pentagon lattice N5|DHC8TZXGYS]] with $0 < a < c < 1$ and $b$ incomparable to $a$ and $c$.
Here $a \le c$, but $a \vee (b \wedge c) = a \vee 0 = a$ while $(a \vee b) \wedge c = 1 \wedge c = c$.

Every [[distributive lattice|0HM0HQCK6N]] is modular.
The [[diamond lattice M3|JQ6FJH2TP8]] is modular but not distributive.
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# Surgery exact sequence

## Statement

Let $X$ be a closed connected [[topological manifold|QQQ9B1DK4P]] of dimension $n \ge 5$, with [[fundamental group|S1XK3HXHFX]] $\pi$ and orientation character $w \colon \pi \to \set{\pm 1}$.
Consider [[simple homotopy equivalences|AR8SP73SXX]] $f \colon M \to X$ from closed topological manifolds $M$.
Identify $f$ with $f' \colon M' \to X$ when there is a homeomorphism $h \colon M \to M'$ with $f' \circ h$ homotopic to $f$.
The *structure set* $\mathcal{S}(X)$ is the set of classes.
Let $\mathcal{N}(X)$ be the set of normal bordism classes of degree one normal maps to $X$.
Let $\mathcal{N}(X \times [0,1], \partial)$ be the same set for $X \times [0,1]$ relative to the boundary.
Let $L_m(\Z[\pi], w)$ be the simple surgery obstruction groups of Wall.
Then there is an exact sequence of pointed sets:
$$\mathcal{N}(X \times [0,1], \partial) \xrightarrow{\ \theta\ } L_{n+1}(\Z[\pi], w) \xrightarrow{\ \partial\ } \mathcal{S}(X) \xrightarrow{\ \eta\ } \mathcal{N}(X) \xrightarrow{\ \theta\ } L_n(\Z[\pi], w).$$
Here $\eta$ sends a homotopy equivalence to its normal invariant and $\theta$ is the surgery obstruction.
The map $\partial$ comes from an action of $L_{n+1}(\Z[\pi], w)$ on $\mathcal{S}(X)$.
Exactness at $\mathcal{S}(X)$ means the following.
Two elements have the same image under $\eta$ if and only if they lie in the same orbit of this action.

## Remarks

Idea of the proof: surgeries below the middle dimension make a degree one normal map to $X$ highly connected.
By Wall's theorem, the remaining obstruction to reach a homotopy equivalence is $\theta$.
The action of $L_{n+1}$ realises an element as the obstruction of a normal bordism from a given homotopy equivalence; the [[s-cobordism theorem|E1E5ABP89W]] turns trivial obstructions into homeomorphisms.
In the topological category $\mathcal{N}(X)$ is in bijection with the set $[X, G/\mathrm{TOP}]$ of homotopy classes of maps from $X$ to the space $G/\mathrm{TOP}$.

The sequence at work: for $X = S^n$ with $n \ge 5$, the structure set is a single point, which is the topological generalized Poincaré conjecture.
For simply connected $X$ the groups $L_n(\Z)$ are $\Z, 0, \Z/2, 0$ for $n \equiv 0, 1, 2, 3 \pmod 4$, and the obstruction in $L_{4k}(\Z) = \Z$ is a multiple of a difference of signatures.

## Explanations

### L1 human

Suppose a space looks like a manifold to homotopy theory.
How many genuinely different manifolds have this homotopy type?
The surgery exact sequence answers this by fitting the set of such manifolds between two computable pieces.
One piece is homotopy theory of a classifying space, the other is algebra of quadratic forms over the group ring of the fundamental group.

### L2 human

The sequence is the main tool of the Browder–Novikov–Sullivan–Wall surgery theory.
To classify manifolds homotopy equivalent to $X$, one computes the normal invariants, the surgery obstruction map and the action of $L_{n+1}$.
It is the setting of the [[Borel conjecture|X057MN9YY9]], which says that for aspherical $X$ the structure set is a single point.
The algebraic theory of these groups is L-theory.
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# Denjoy theorem on rotation number

## Statement

Let $f$ be an orientation-preserving $C^1$ [[diffeomorphism|0KZZ70ZHXP]] of the circle $S^1 = \R / \Z$ such that $\log f'$ has [[bounded variation|A8251BHCMP]]; for example, let $f$ be of class $C^2$.
Assume that the [[rotation number|DP4WWQBJJF]] $\rho$ of $f$ is irrational.
Then $f$ is [[topologically conjugate|HNXXAPQKG3]] to the rotation $R_\rho(x) = x + \rho$: there is a homeomorphism $h$ of $S^1$ with $h \circ f = R_\rho \circ h$.

## Remarks

By Poincaré's classification, $f$ is semiconjugate to $R_\rho$, and it is conjugate unless $f$ has a wandering interval, an open interval $J$ whose images $f^n(J)$, $n \in \Z$, are pairwise disjoint.
The proof rules out wandering intervals.
Let $q_n$ be the denominators of the continued fraction of $\rho$.
A combinatorial argument finds intervals whose images under $f^{q_n}$ overlap little, and the Denjoy inequality bounds $\abs{\log (f^{q_n})'}$ by the total variation of $\log f'$.
For a wandering interval $J$ this gives a constant $c > 0$ with $\abs{f^{q_n}(J)} + \abs{f^{-q_n}(J)} \ge c \abs{J}$ for all $n$.
But the images of $J$ are pairwise disjoint, so their lengths tend to $0$, which is a contradiction.

The smoothness hypothesis cannot be dropped.
Denjoy also built $C^1$ diffeomorphisms with irrational rotation number that have a wandering interval.
For them the omega-limit set of every point is a [[Cantor set|0BXNDZFQME]] and $f$ is not conjugate to a rotation.
Such examples exist even of class $C^{1 + \alpha}$ for every $\alpha < 1$.

The conjugacy $h$ is only a homeomorphism in general; whether it is smooth depends on arithmetic properties of $\rho$, a question studied by Arnold, Herman and Yoccoz.

## Explanations

### L1 human

A smooth map of the circle that never brings a point back to itself turns out to be a rigid irrational rotation in disguise.
One can change coordinates on the circle, continuously, so that the map becomes a plain rotation.
For maps that are only once differentiable this can fail: some intervals may wander around the circle forever without overlapping.

### L2 human

The theorem completes Poincaré's classification for smooth circle diffeomorphisms: up to topological conjugacy, a $C^2$ diffeomorphism with irrational rotation number is determined by that number.
It implies that every orbit of such a map is dense and that the map is minimal and [[uniquely ergodic|NZZE9KCAD8]], since the [[irrational rotation|2Q1J7RZ4TK]] has these properties.
The key tool, control of the distortion of $(f^n)'$ through bounded variation of $\log f'$, reappears in many places in one-dimensional dynamics.
Denjoy published the theorem in 1932.
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# Biorthogonal wavelet

## Summary

A biorthogonal wavelet is a pair of wavelets $\psi$ and $\tilde\psi$ whose dilated and translated copies are dual to each other.
It replaces a single wavelet whose copies are orthonormal.
One family is used to compute the coefficients of a function and the other to rebuild it.
Giving up orthogonality allows wavelets that are both compactly supported and symmetric.
Among real orthonormal wavelets, only the [[Haar wavelet|JW5PA9XA7F]] has both properties.
Biorthogonal wavelets are used in image compression, for example in the JPEG 2000 standard.

## Explanations

### L1 human

With an orthonormal wavelet, the same function is used to measure a signal and to rebuild it.
A biorthogonal wavelet splits these two jobs between two different functions, an analysis wavelet and a synthesis wavelet.
The two families fit together like a basis and its dual basis in linear algebra: each analysis function sees exactly one synthesis function and is blind to all the others.
This extra freedom lets designers make both functions symmetric and short, which avoids visible distortions in compressed images.
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<document source="https://tamnd.github.io/math2/node/Q7QK3PF806.md" tag="Q7QK3PF806" kind="statement">
# Lagrange multipliers on Banach spaces

## Statement

Let $X$ and $Y$ be real [[Banach spaces|06GZ956657]], let $U \subseteq X$ be open, and let $f \colon U \to \R$ and $g \colon U \to Y$ be continuously [[Fréchet differentiable|06EH97V2A7]].
Let $x_0 \in U$ with $g(x_0) = 0$, and suppose that $x_0$ is a [[local extremum|0WFBH1Q3WC]] of the restriction of $f$ to $M = \set{x \in U : g(x) = 0}$.
Suppose that the derivative $Dg(x_0) \colon X \to Y$ is surjective.
Then there is a unique continuous linear functional $\lambda \in Y^*$, an element of the [[dual space|07JBSZE0V4]] of $Y$, such that
$$Df(x_0) = \lambda \circ Dg(x_0).$$

## Remarks

Let $h \in \ker Dg(x_0)$.
By [[Lyusternik's theorem|3PGXH0T1K6]] there is a curve $t \mapsto x_0 + th + r(t)$ in $M$ with $r(t) = o(t)$, and $t \mapsto f(x_0 + th + r(t))$ has a local extremum at $t = 0$, so $Df(x_0) h = 0$.
Thus $Df(x_0)$ vanishes on $\ker Dg(x_0)$, and $\lambda(Dg(x_0) h) = Df(x_0) h$ is a well defined linear map on $Y$.
It is continuous by the [[open mapping theorem|03YEM7JEDN]], and it is unique because $Dg(x_0)$ is onto.

For $X = \R^n$ and $Y = \R^m$, the functional $\lambda$ is a vector $(\lambda_1, \dots, \lambda_m)$, and the result is the [[Lagrange multiplier theorem|0VKKP8X6Q6]].

Example: an isoperimetric problem minimises $J(y) = \int_a^b L(x, y, y') \, dx$ subject to $K(y) = \int_a^b G(x, y, y') \, dx = \ell$ and fixed ends.
Here $Y = \R$, and the result gives a number $\lambda$ with $\delta J(y_0; \eta) = \lambda \, \delta K(y_0; \eta)$ for all admissible variations $\eta$.

Surjectivity is needed.
For $X = \R^2$, $Y = \R$, $f(x_1, x_2) = x_1$ and $g(x_1, x_2) = x_2^2 - x_1^3$, the point $0$ minimises $f$ on $M$, since $x_1 \ge 0$ on $M$.
But $Dg(0) = 0$, so $\lambda \circ Dg(0) = 0 \ne Df(0)$ for every $\lambda$.

## Explanations

### L1 human

At a constrained extremum, the derivative of the objective is a combination of the derivatives of the constraints.
In infinite dimensions there may be infinitely many constraints, so the coefficients form a continuous linear functional on the constraint space instead of a list of numbers.

### L2 human

This is the abstract form of the multiplier rule used for constrained problems in the calculus of variations and in optimal control.
With $Y = \R$ it gives Euler's multiplier rule for isoperimetric problems, and with $Y$ a function space it handles pointwise constraints.
The onto condition on $Dg(x_0)$ is the infinite-dimensional constraint qualification; without it the multiplier may not exist, as the example shows.
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# Suspension

## Summary

The suspension $SX$ of a topological space $X$ is the cylinder $X \times [0, 1]$ with each end collapsed to its own point.
It looks like two cones on $X$ glued along their bases.
The suspension of the $n$-sphere is the $(n+1)$-sphere, and suspension shifts reduced homology up by one degree.
It is a basic construction in algebraic topology and stable homotopy theory.

## Explanations

### L1 human

Take the space $X$ and stretch it into a tube $X \times [0, 1]$.
Then pinch the whole bottom end to one point and the whole top end to another point.
The space now hangs between a south pole and a north pole.
For example, the suspension of two points is a circle: each point becomes a path from the south pole to the north pole.
The suspension of a circle is a sphere, like a globe with its meridians.
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# Feigenbaum–Cvitanović functional equation

## Summary

The Feigenbaum–Cvitanović functional equation is $g(x) = -\alpha \, g(g(x/\alpha))$, for an even function $g$ with $g(0) = 1$ and a constant $\alpha$.
Its solution is the fixed point of the renormalization operator that doubles the period of a unimodal map and rescales space.
The constant $\alpha$ that comes with the solution is the second of the [[Feigenbaum constants|F1EHT2CPHC]], $\alpha \approx 2.5029$.
The equation explains why the [[period-doubling cascade|6J921PEDN2]] looks the same at every scale, for a large class of maps.

## Explanations

### L1 human

Take a map with one hump, such as $x \mapsto 1 - \mu x^2$, and look at its second iterate near the top of the hump.
After shrinking the picture by a fixed factor and flipping it upside down, it looks almost like the original map.
The equation describes a map that reproduces itself exactly under this operation.
The shrinking factor of that map, about $2.5$, then shows up in the period-doubling route to chaos of many different systems.
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# Polynomial identity ring

## Definition

Let $k$ be a [[commutative ring|0310REJJY8]] with identity, and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$.
Let $k\langle x_1, \ldots, x_n \rangle$ be the [[free algebra|2P3Y2FRCSZ]] on $n$ variables.
A polynomial $f \in k\langle x_1, \ldots, x_n \rangle$ is a *polynomial identity* of $A$ if $f(a_1, \ldots, a_n) = 0$ for all $a_1, \ldots, a_n \in A$.
The polynomial $f$ is *monic* if some monomial of $f$ of highest degree has coefficient $1$.
The algebra $A$ is a *PI algebra* if it satisfies some monic polynomial identity.
A [[ring|0B1261B5ED]] is a *polynomial identity ring*, or *PI ring*, if it is a PI algebra over $k = \Z$.

## Remarks

Over a field $k$, every nonzero polynomial is a scalar multiple of a monic one, so a $k$-algebra is PI when it satisfies some nonzero polynomial identity.
Monic polynomials are used for rings because, for example, every ring of characteristic $2$ satisfies the identity $2 x_1 = 0$, which says nothing about its multiplication.

Example: a commutative ring satisfies the monic identity $x_1 x_2 - x_2 x_1$.

Example: the ring $M_2(k)$ of $2 \times 2$ matrices over a commutative ring $k$ satisfies the monic identity $[[x_1, x_2]^2, x_3]$, where $[a, b] = ab - ba$.
Indeed, $c = [a_1, a_2]$ has trace $0$, so the Cayley–Hamilton theorem gives $c^2 = -\det(c) \, I$, a scalar matrix that commutes with every $a_3$.

Non-example: let $k$ be a field and $A = k\langle x, y \rangle$.
If $f(x_1, \ldots, x_n)$ is nonzero, then $f(x, xy, x y^2, \ldots, x y^{n-1}) \ne 0$, because distinct words in the $x y^i$ are distinct words in $x$ and $y$.
So the free algebra on two letters satisfies no nonzero identity and is not PI.

Every subring and every homomorphic image of a PI ring is a PI ring.
The [[exterior algebra|0MDMZ965VE]] of an infinite-dimensional vector space over a field of characteristic $0$ is PI, since it satisfies $[[x_1, x_2], x_3]$, but it satisfies no standard identity.
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# Lévy–Khintchine representation

## Statement

Let $\mu$ be a [[probability measure|059TJN0NZ1]] on $\R$ with [[characteristic function|0V5B6D6BQH]] $\varphi(u) = \int_\R e^{iux} \, \mu(dx)$.
Then $\mu$ is [[infinitely divisible|BX3V3NYM1Y]] if and only if there are $b \in \R$, $\sigma \ge 0$ and a measure $\nu$ on $\R \setminus \set{0}$ with the following properties.
The measure satisfies $\int_{\R \setminus \set{0}} \min(1, x^2) \, \nu(dx) < \infty$, and $\varphi(u) = e^{\psi(u)}$ for all $u \in \R$, where
$$\psi(u) = i b u - \frac{\sigma^2 u^2}{2} + \int_{\R \setminus \set{0}} \left(e^{iux} - 1 - iux \, \mathbf{1}_{\set{\abs{x} < 1}}\right) \nu(dx).$$
In that case the triple $(b, \sigma^2, \nu)$ is uniquely determined by $\mu$.

## Remarks

The measure $\nu$ is the Lévy measure and $(b, \sigma^2, \nu)$ is the characteristic triple; the value of $b$ depends on the choice of the cut-off $\mathbf{1}_{\set{\abs{x} < 1}}$.
The three parts are a drift, a Gaussian part with variance $\sigma^2$, and a jump part.
Proof idea for "only if": write $\mu$ as the $n$-fold convolution power of some $\mu_n$.
Then $\exp(n(\varphi_n - 1))$ is the characteristic function of a compound Poisson law, and it tends to $\varphi$.
The triple is obtained as a limit of the jump measures $n \mu_n$.
Example: the [[normal|05AQE0QR0D]] law with mean $m$ and variance $\sigma^2$ has the triple $(m, \sigma^2, 0)$.
Example: the [[Poisson|0XB9GQTN9G]] law with mean $\lambda$ has the triple $(0, 0, \lambda \delta_1)$, since the cut-off term vanishes at $x = 1$ and $\psi(u) = \lambda(e^{iu} - 1)$.
For a [[Lévy process|8NT5CFG2ZR]] $(X_t)_{t \ge 0}$, the law of $X_1$ is infinitely divisible with exponent $\psi$, and $E[e^{iuX_t}] = e^{t \psi(u)}$ for all $t \ge 0$.
The same theorem holds on $\R^d$, with $\sigma^2$ replaced by a positive semidefinite matrix and $ux$ by a dot product.

## Explanations

### L1 human

Every law that can be split into any number of equal independent parts is built from three ingredients: a steady drift, a normal fluctuation, and random jumps of various sizes.
The theorem says this precisely through the characteristic function, and the three ingredients are unique.
The measure $\nu$ tells how often jumps of each size happen.

### L2 human

The theorem classifies [[infinitely divisible|BX3V3NYM1Y]] laws, and through them all [[Lévy processes|8NT5CFG2ZR]].
The Lévy–Itô decomposition writes a Lévy process as a Brownian motion with drift plus an independent jump process.
The jump process is built from a Poisson random measure with intensity $\nu$.
[[Stable laws|F3N6E1Y2AY]] with index $\alpha \in (0, 2)$ are the case $\sigma = 0$ and $\nu(dx) = (c_+ \mathbf{1}_{\set{x > 0}} + c_- \mathbf{1}_{\set{x < 0}}) \abs{x}^{-1-\alpha} \, dx$ with $c_+, c_- \ge 0$ not both $0$.
The condition $\int \min(1, x^2) \, d\nu < \infty$ allows infinitely many small jumps, as long as they are square-summable.
</document>

<document source="https://tamnd.github.io/math2/node/Q85MMH5DEY.md" tag="Q85MMH5DEY" kind="definition">
# Sentence

## Definition

Let $\sigma$ be a signature.
The set $\operatorname{FV}(\varphi)$ of *free variables* of a [[formula|HYJ10AF6RT]] $\varphi$ over $\sigma$ is defined by recursion on $\varphi$:

1. for an atomic formula, $\operatorname{FV}(\varphi)$ is the set of variables that occur in its [[terms|6D3WA7N2W4]];
2. $\operatorname{FV}(\neg \varphi) = \operatorname{FV}(\varphi)$, and $\operatorname{FV}(\varphi \ast \psi) = \operatorname{FV}(\varphi) \cup \operatorname{FV}(\psi)$ for each binary connective $\ast$;
3. $\operatorname{FV}(\forall x \, \varphi) = \operatorname{FV}(\exists x \, \varphi) = \operatorname{FV}(\varphi) \setminus \set{x}$.

A *sentence* over $\sigma$ is a formula $\varphi$ with $\operatorname{FV}(\varphi) = \emptyset$.

## Remarks

The formula $\forall x \, \exists y \, (x < y)$ is a sentence.
Indeed $\operatorname{FV}(x < y) = \set{x, y}$, removing $y$ gives $\set{x}$, and removing $x$ gives $\emptyset$.
The formula $\exists y \, (x < y)$ is not a sentence, since $x$ remains free.
Sentences are also called closed formulas.
In Mathlib sentences are `FirstOrder.Language.Sentence`.
</document>

<document source="https://tamnd.github.io/math2/node/Q8AYGD93CW.md" tag="Q8AYGD93CW" kind="statement">
# Snake lemma

## Statement

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], such as the category of modules over a ring.
Consider a [[commutative diagram|WV898P4X16]]
$$\begin{array}{ccccccc} & A & \xrightarrow{f} & B & \xrightarrow{g} & C & \to 0 \\ & \downarrow{\scriptstyle a} & & \downarrow{\scriptstyle b} & & \downarrow{\scriptstyle c} & \\ 0 \to & A' & \xrightarrow{f'} & B' & \xrightarrow{g'} & C' & \end{array}$$
in $\mathcal{A}$ whose rows are [[exact|JB7ZBFBA9C]].
Then there is a morphism $\partial \colon \ker c \to \operatorname{coker} a$ such that the sequence
$$\ker a \to \ker b \to \ker c \xrightarrow{\partial} \operatorname{coker} a \to \operatorname{coker} b \to \operatorname{coker} c$$
is exact, where the unlabeled morphisms are induced by $f, g, f', g'$ on [[kernels|S417YNEJ6M]] and [[cokernels|H8W38933NC]].
Moreover, if $f$ is a [[monomorphism|Y2NP1JFR86]], then $\ker a \to \ker b$ is a monomorphism, and if $g'$ is an [[epimorphism|RVXDSY04KG]], then $\operatorname{coker} b \to \operatorname{coker} c$ is an epimorphism.

## Remarks

The morphism $\partial$ is called the *connecting morphism*.
For modules, $\partial$ is defined by a chase.
Given $z \in \ker c$, choose $y \in B$ with $g(y) = z$.
Then $g'(b(y)) = c(z) = 0$, so $b(y) = f'(x')$ for a unique $x' \in A'$, and $\partial(z)$ is the class of $x'$ in $\operatorname{coker} a$.
One checks that this class does not depend on the choice of $y$, and then checks exactness at each place by similar chases.
In a general abelian category one uses [[pseudo-elements|1WV2THA6NY]] or an embedding theorem.

Example: map $0 \to \Z \xrightarrow{2} \Z \to \Z/2\Z \to 0$ to itself by multiplication by $2$ in each place, which is $0$ on $\Z/2\Z$.
The kernels are $0, 0, \Z/2\Z$ and the cokernels are $\Z/2\Z, \Z/2\Z, \Z/2\Z$.
Exactness forces $\partial \colon \Z/2\Z \to \Z/2\Z$ to be an isomorphism, the map $\operatorname{coker} a \to \operatorname{coker} b$ to be $0$, and $\operatorname{coker} b \to \operatorname{coker} c$ to be an isomorphism, which one can confirm directly.

## Explanations

### L1 human

The lemma starts from two exact rows joined by three vertical maps.
It produces one long exact sequence that runs through the three kernels and then the three cokernels.
The surprising link is the connecting map from the last kernel to the first cokernel; drawn in the diagram, the whole sequence winds through it like a snake.

### L2 human

The snake lemma is the engine behind most long exact sequences.
Applied to a short exact sequence of chain complexes, it gives the long exact sequence in homology.
It also gives the long exact sequences of derived functors and the nine lemma.
In practice, it is how one computes a kernel or cokernel that is hard to see directly, from ones that are easy to see.
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<document source="https://tamnd.github.io/math2/node/Q8CTDCZH01.md" tag="Q8CTDCZH01" kind="concept">
# Minimal model program

## Summary

The minimal model program, or MMP, is a strategy for finding a simplest variety in each birational class of projective varieties.
It modifies a variety step by step, by contractions and flips, so as to make the [[canonical bundle|W05VYY4HBR]] [[nef|RZ66X0NAM5]].
It ends either with a minimal model or with a Mori fibre space, a fibration whose general fibres are Fano varieties.
For surfaces it is the classical process of contracting $(-1)$-curves; in dimension $3$ it was carried out by Mori and others.

## Explanations

### L1 human

Many different varieties have the same function field, because one can always blow up points to make a variety more complicated.
The program runs this backwards: it looks for curves on which the canonical bundle is negative and removes them.
For a surface, it repeatedly contracts curves that look like a projective line with self-intersection $-1$.
When no such curves are left, either the canonical bundle is nef and the surface is minimal, or the surface is $\mathbb{P}^2$ or fibred in projective lines over a curve.
</document>

<document source="https://tamnd.github.io/math2/node/Q8DWXQ2WG0.md" tag="Q8DWXQ2WG0" kind="definition">
# Bicubic interpolation

## Definition

Let $U = [0, 1]^2$, and for each corner $c \in \set{0, 1}^2$ let real numbers $f(c)$, $f_x(c)$, $f_y(c)$ and $f_{xy}(c)$ be given.
The *bicubic interpolant* on $U$ is the unique polynomial
$$p(x, y) = \sum_{i=0}^{3} \sum_{j=0}^{3} a_{ij} x^i y^j$$
such that, at each corner $c$, $p = f(c)$, $\partial_x p = f_x(c)$, $\partial_y p = f_y(c)$ and $\partial_x \partial_y p = f_{xy}(c)$.
On a rectangular grid, *bicubic interpolation* applies this on each cell after an affine change of variables to $U$.
The derivative data are scaled by the cell sides.

## Remarks

The sixteen conditions determine the sixteen coefficients.
Indeed, $p$ is the tensor product of one-dimensional [[cubic Hermite interpolation|HCQWPAHYAT]] in $x$ and in $y$, so it exists and is unique.
Example: for $f(x, y) = x^3 y^3$, the data are $f = f_x = f_y = 0$ at every corner except $(1, 1)$, where $f = 1$, $f_x = f_y = 3$ and $f_{xy} = 9$; the interpolant is $x^3 y^3$, since it has the required form.
Non-example: the function $\sin(\pi x) \sin(\pi y)$ has zero value at all corners, but it is not a polynomial of the required form, so it is not a bicubic interpolant of any data.
On a grid, along a shared edge the value and the normal derivative are cubics determined by data at the two common corners only.
So adjacent patches join with continuous value and first derivatives.
The variant used for images replaces the derivatives by differences of neighboring pixel values, or uses a cubic convolution kernel on a $4 \times 4$ neighborhood.

## Explanations

### L1 human

On each square of the grid, bicubic interpolation fits a polynomial that is cubic in $x$ and cubic in $y$.
It matches the value, the slopes and the twist at the four corners, sixteen numbers in all, and this makes the whole surface smooth across cell edges.
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<document source="https://tamnd.github.io/math2/node/Q8EXG7EKYW.md" tag="Q8EXG7EKYW" kind="statement">
# Carathéodory's extension theorem

## Statement

Let $X$ be a set and let $\mathcal{S}$ be a [[semiring of sets|PYH82FMXQ2]] on $X$.
Let $\mu_0 \colon \mathcal{S} \to [0, \infty]$ be a [[premeasure|PV0XXSWFC7]], that is, $\mu_0(\varnothing) = 0$ and $\mu_0\big(\bigcup_n A_n\big) = \sum_n \mu_0(A_n)$ for every sequence of pairwise disjoint sets $A_n \in \mathcal{S}$ whose union lies in $\mathcal{S}$.
Let $\sigma(\mathcal{S})$ be the [[sigma-algebra|0SV8K0TDGY]] generated by $\mathcal{S}$.

1. There is a [[measure|PR11BZQQPG]] $\mu$ on $\sigma(\mathcal{S})$ with $\mu(A) = \mu_0(A)$ for all $A \in \mathcal{S}$.
2. If moreover $X = \bigcup_{n} X_n$ for a sequence of sets $X_n \in \mathcal{S}$ with $\mu_0(X_n) < \infty$, then this measure $\mu$ is unique.

## Remarks

For existence, define $\mu^*(B) = \inf \set{ \sum_n \mu_0(A_n) : A_n \in \mathcal{S}, \ B \subseteq \bigcup_n A_n }$.
This is an outer measure, and every set of $\mathcal{S}$ satisfies the Carathéodory splitting condition.
Countable additivity of $\mu_0$ gives $\mu^* = \mu_0$ on $\mathcal{S}$.
So $\mu^*$ restricted to the Carathéodory-measurable sets is the required extension.
Uniqueness follows from Dynkin's $\pi$-$\lambda$ theorem, since $\mathcal{S}$ is closed under finite intersections.
Example: the half-open intervals $(a, b]$ form a semiring on $\R$, and $\mu_0((a, b]) = b - a$ is a premeasure; the theorem gives Lebesgue measure on the Borel sets, and it is unique.
Uniqueness can fail without the finiteness hypothesis in clause 2.
On $\Q$, take the semiring of sets $(a, b] \cap \Q$ and the premeasure that is $\infty$ on nonempty sets and $0$ on $\varnothing$.
It extends both to counting measure and to $2$ times counting measure on the power set of $\Q$.

## Explanations

### L1 human

It is often easy to say how big simple sets are, such as intervals or boxes.
The theorem says that if this size is countably additive on the simple sets, it can be extended to a measure on every set built from them by countable operations.
If the whole space is covered by countably many simple sets of finite size, there is only one way to do it.

### L2 human

This is the main tool to construct measures: Lebesgue measure, Lebesgue–Stieltjes measures, product measures and measures on sequence spaces all come from it.
The only work left in each case is to check that the given set function is countably additive on the semiring.
A version for an algebra of sets, with the extension given by an explicit outer measure, is the [[Hahn–Kolmogorov theorem|FN2WRRF6V9]].
The proof goes through an [[outer measure|BJ314H17HC]] and [[Carathéodory's criterion|JDGQ135YK2]].
</document>

<document source="https://tamnd.github.io/math2/node/Q8FDSM86X6.md" tag="Q8FDSM86X6" kind="definition">
# Mutually singular measures

## Definition

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]] and let $\mu$ and $\nu$ be [[measures|PR11BZQQPG]] on $\Sigma$.
The measures $\mu$ and $\nu$ are *mutually singular* if there are sets $A, B \in \Sigma$ such that:

1. $A \cap B = \emptyset$ and $A \cup B = X$.
2. $\mu(B) = 0$.
3. $\nu(A) = 0$.

This is written $\mu \perp \nu$.

## Remarks

Example: on the Borel sets of $\R$, let $\lambda$ be Lebesgue measure and let $\delta_0$ be the Dirac measure at $0$.
Take $A = \R \setminus \{0\}$ and $B = \{0\}$.
Then $\lambda(B) = 0$ and $\delta_0(A) = 0$, so $\lambda \perp \delta_0$.

Non-example: let $\mu$ be Lebesgue measure restricted to $[0, 1]$ and $\nu$ Lebesgue measure restricted to $[0, 2]$.
If $\mu(B) = 0$, then $B \cap [0, 1]$ has length $0$, so $A$ contains almost all of $[0, 1]$ and $\nu(A) \ge 1$.
So clause 3 fails for every choice, and $\mu$ and $\nu$ are not mutually singular.

The relation is symmetric.
For signed or complex measures, $\mu \perp \nu$ means that the total variations $\abs{\mu}$ and $\abs{\nu}$ are mutually singular.
A measure $\nu$ is called singular with respect to $\mu$ when $\nu \perp \mu$.
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<document source="https://tamnd.github.io/math2/node/Q8FERXTZ8B.md" tag="Q8FERXTZ8B" kind="statement">
# Banach–Stone theorem

## Statement

Let $X$ and $Y$ be [[compact|0TVJZYB6SH]] [[Hausdorff spaces|0S1FJ8VEX9]].
Let $C(X)$ and $C(Y)$ be the spaces of continuous real functions with the [[supremum norm|0PARCKE8WP]].
Let $T \colon C(X) \to C(Y)$ be a surjective [[linear isometry|07SJXTWVQW]].
Then there are a [[homeomorphism|0QT00PB5G2]] $\varphi \colon Y \to X$ and a continuous function $h \colon Y \to \set{-1, 1}$ such that
$$(T f)(y) = h(y) \, f(\varphi(y)) \quad \text{for all } f \in C(X), \ y \in Y.$$

## Remarks

The idea of the proof: the extreme points of the unit ball of the dual of $C(X)$ are the functionals $\pm \delta_x$, with $\delta_x(f) = f(x)$.
The adjoint of $T$ is an isometry of the duals and maps extreme points to extreme points.
So it sends each $\delta_y$ to $h(y) \delta_{\varphi(y)}$, and the weak-* topology makes $\varphi$ and $h$ continuous.

Example: on $X = Y = [0, 1]$, the map $(T f)(t) = -f(1 - t)$ is a surjective linear isometry.
Here $h = -1$ and $\varphi(t) = 1 - t$.

For complex scalars the same holds with $h$ a continuous function of absolute value $1$.
As a consequence, $X$ and $Y$ are homeomorphic if and only if $C(X)$ and $C(Y)$ are isometrically isomorphic.
Isometry is needed.
Let $S = \set{0} \cup \set{1/n : n \ge 1}$, a convergent sequence with its limit.
Then $C(S)$ and $C(S \times \set{1, 2})$ are linearly isomorphic, but $S$ has one limit point and $S \times \set{1, 2}$ has two.

## Explanations

### L1 human

The norm of $C(X)$ remembers the space $X$.
Every way to match $C(X)$ with $C(Y)$ that keeps distances comes from matching the points of $Y$ with the points of $X$, up to a change of sign.

### L2 human

The theorem shows that the Banach space $C(X)$, with its norm, determines the compact space $X$ up to homeomorphism.
Gelfand–Naimark theory gives a similar result for $C(X)$ as an algebra, and the Banach–Stone theorem gets it from the metric structure alone.
Without the isometry condition the result fails badly: by a theorem of Milutin, $C(X)$ is isomorphic to $C([0, 1])$ for every uncountable compact metric space $X$.
</document>

<document source="https://tamnd.github.io/math2/node/Q8K5MWEXB9.md" tag="Q8K5MWEXB9" kind="concept">
# Bloch–Wigner function

## Summary

The Bloch–Wigner function $D$ is a real-valued function on the complex plane built from the imaginary part of the [[dilogarithm|G0X0NWSDB7]] and a logarithm correction.
Unlike the dilogarithm, it is single-valued and continuous on all of $\C$.
It satisfies a clean five-term relation and gives the volume of an ideal tetrahedron in hyperbolic space.
This makes it central in hyperbolic geometry and in the theory of the Bloch group.

## Explanations

### L1 human

The dilogarithm is a multivalued function: going around the point $1$ changes its value.
The Bloch–Wigner function takes the imaginary part and adds just the right correction so that these jumps cancel.
The result is a smooth real function on the plane with the points $0$ and $1$ removed, and it extends continuously to them with value $0$.
It is zero on the real line and changes sign under complex conjugation.
Its largest value, about $1.0149$, is reached at $z = e^{i\pi/3}$, and this number is the volume of the regular ideal tetrahedron in hyperbolic space.
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<document source="https://tamnd.github.io/math2/node/Q8MTV1Z4AQ.md" tag="Q8MTV1Z4AQ" kind="definition">
# Elastic net

## Definition

Let $X$ be a real $n \times p$ matrix, let $y \in \R^n$, and let $\lambda_1 \ge 0$ and $\lambda_2 \ge 0$.
For $b \in \R^p$ write $\norm{b}_1 = \sum_{j=1}^p \abs{b_j}$ and let $\norm{\cdot}_2$ be the Euclidean norm.
An *elastic net estimate* with parameters $\lambda_1, \lambda_2$ is a minimiser over $b \in \R^p$ of
$$\norm{y - X b}_2^2 + \lambda_1 \norm{b}_1 + \lambda_2 \norm{b}_2^2.$$

## Remarks

If $\lambda_2 > 0$, the objective is strictly [[convex|07Q7CNM5WF]] and tends to infinity with $\norm{b}_2$, so the estimate exists and is unique.
With $\lambda_2 = 0$ it is a [[lasso|KZ226116HW]] estimate, and with $\lambda_1 = 0$ it is the [[ridge regression|3KZW2930V2]] estimate.
A common parametrisation uses one level $\lambda$ and a mixing weight $\alpha \in [0, 1]$, with penalty $\lambda \big( \alpha \norm{b}_1 + \frac{1 - \alpha}{2} \norm{b}_2^2 \big)$.
Some authors multiply the estimate defined above by $1 + \lambda_2$ to undo part of the double shrinkage.

Example: assume $X^T X = I_p$ and put $z = X^T y$.
Then $\norm{y - Xb}_2^2 = \norm{z - b}_2^2 + \text{const}$, and the problem splits into coordinates.
Minimising $(z_j - b_j)^2 + \lambda_1 \abs{b_j} + \lambda_2 b_j^2$ gives
$$\hat\beta_j = \frac{\sgn(z_j) \max(\abs{z_j} - \lambda_1/2, \, 0)}{1 + \lambda_2}.$$
So coefficients with $\abs{z_j} \le \lambda_1 / 2$ are exactly zero, and the others are shrunk by both penalties.

Non-example: if $\lambda_1 = 0$ and $\lambda_2 > 0$, no coefficient is set to zero unless $z_j = 0$, so no selection takes place; that is ridge regression.
</document>

<document source="https://tamnd.github.io/math2/node/Q9176JRJ5M.md" tag="Q9176JRJ5M" kind="statement">
# Gödel's consistency of CH and AC

## Statement

Let $L$ be the [[constructible universe|6TF2JBTPPJ]].

1. The axioms of [[ZF|0HE5WDQ9FJ]] and the [[axiom of constructibility|4QKRPRCSXH]] $V = L$ hold in $L$.
2. ZF together with $V = L$ proves the [[axiom of choice|0HTGB13FE2]] and the generalized [[continuum hypothesis|0MN2MER9WN]] $2^{\aleph_\alpha} = \aleph_{\alpha + 1}$ for every ordinal $\alpha$.
3. Hence, if ZF is consistent, then ZF together with the axiom of choice and the generalized continuum hypothesis is consistent.

## Remarks

The construction of $L$ can be carried out inside any model of ZF, and it is absolute, so $L$ computed inside $L$ is $L$ again; this gives part 1.
The axiom of choice holds in $L$ because $L$ has a definable well-order.
A set enters at a least stage and is defined there by a formula with parameters, and these data can be ordered.
For the continuum hypothesis, assume $V = L$; the condensation lemma shows that every subset of $\omega_\alpha$ lies in $L_{\omega_{\alpha + 1}}$, a set of size $\aleph_{\alpha + 1}$.
The result is a relative consistency proof: by [[Gödel's second incompleteness theorem|X9WYNYE29C]], ZF cannot prove its own consistency, so the hypothesis "ZF is consistent" cannot be removed.
Together with [[Cohen's theorem|EMGBZ46GCD]], it shows that the continuum hypothesis is independent of ZFC.

## Explanations

### L1 human

The continuum hypothesis says that there is no size of infinity strictly between the natural numbers and the real numbers.
Gödel found a world of sets, the constructible universe, in which all the usual axioms hold and the continuum hypothesis is true as well.
So the usual axioms can never disprove the continuum hypothesis.
The same world also shows that the axiom of choice cannot be disproved from the other axioms.

### L2 human

The theorem is the first half of the independence of the continuum hypothesis; the second half is Cohen's forcing argument.
The method is the method of inner models: one finds a definable class in which the axioms hold together with the desired statement.
Inner models alone cannot show that the continuum hypothesis can fail: if $V = L$, then $L$ is the only inner model, and the continuum hypothesis holds in it.
To show that the continuum hypothesis can fail, one needs a way to enlarge the universe, and this is what forcing does.
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<document source="https://tamnd.github.io/math2/node/Q93QR7EJKY.md" tag="Q93QR7EJKY" kind="definition">
# Near-ring

## Definition

A *(right) near-ring* is a set $N$ with two binary operations $+$ and $\cdot$ such that:

1. $(N, +)$ is a [[group|G5S1M9]], not necessarily abelian, with identity element $0$.
2. $(N, \cdot)$ is a [[semigroup|0V3CFG1KK9]]: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in N$.
3. $(a + b) \cdot c = a \cdot c + b \cdot c$ for all $a, b, c \in N$.

## Remarks

A *left near-ring* asks instead $c \cdot (a + b) = c \cdot a + c \cdot b$; the two notions are mirror images.
Every ring is a near-ring.
Example: for a group $(G, +)$, let $M(G)$ be the set of all maps $G \to G$, with $(f + g)(x) = f(x) + g(x)$ and $f \cdot g = f \circ g$.
Clause 1 holds pointwise, clause 2 because composition is associative, and clause 3 because $((f + g) \circ h)(x) = f(h(x)) + g(h(x))$.
The other distributive law fails in $M(\Z)$: for the constant map $f = 1$, $f \circ (g + h) = 1$ but $f \circ g + f \circ h = 2$.
Non-example: $\N = \set{0, 1, 2, \ldots}$ with the usual operations satisfies clauses 2 and 3, but fails clause 1, since $1$ has no additive inverse.
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<document source="https://tamnd.github.io/math2/node/Q94Z7TB8B5.md" tag="Q94Z7TB8B5" kind="concept">
# Strong amalgamation property

## Summary

A class of algebras has the strong amalgamation property when any two members over a common part can be joined in a third member.
The join must identify only the common part.
It strengthens the [[amalgamation property|XSCSX39GEZ]], which allows the amalgam to identify more.
Groups and lattices have it, while fields have the amalgamation property but not the strong one.

## Explanations

### L1 human

Take two structures that share a common piece.
Amalgamation asks for a bigger structure containing both, with the two copies of the shared piece glued together.
Strong amalgamation asks in addition that nothing else gets glued: the two structures overlap exactly in the shared piece.
For finite graphs, gluing two graphs along a common subgraph and adding nothing else is such a strong amalgam.
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<document source="https://tamnd.github.io/math2/node/Q958DANJNA.md" tag="Q958DANJNA" kind="example">
# Kempner series

## Example

The *Kempner series* is the sum of $1/n$ over the positive integers $n$ whose decimal expansion has no digit $9$:
$$\sum_{\substack{n \ge 1 \\ \text{no digit } 9}} \frac{1}{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{8} + \frac{1}{10} + \cdots + \frac{1}{18} + \frac{1}{20} + \cdots.$$
It is a [[convergent series|0AF7X08PCR]], and its sum is less than $80$.

Check: a positive integer with exactly $k$ digits and no digit $9$ has a first digit in $\set{1, \ldots, 8}$ and $k - 1$ further digits in $\set{0, \ldots, 8}$.
So there are $8 \cdot 9^{k-1}$ such integers, and each is at least $10^{k-1}$.
Their reciprocals add up to at most $8 \cdot 9^{k-1} / 10^{k-1} = 8 \cdot (9/10)^{k-1}$.
Summing over $k \ge 1$ gives the bound $8 / (1 - 9/10) = 80$.
The terms are positive, so the partial sums are increasing and bounded, and the series converges.

## Remarks

The true sum is about $22.92$.
This contrasts with the [[harmonic series|0CXTMAY5PX]], which [[diverges|0E9M24KS9J]].
The same argument works for any digit and any base: removing every integer that contains a fixed digit string leaves a convergent series.
The series is named after A. J. Kempner, who proved its convergence in 1914.
Most large integers contain a $9$: the proportion of $k$-digit integers without one is $(8/9)(9/10)^{k-1}$, which tends to $0$.

## Explanations

### L1 human

The harmonic series $1 + 1/2 + 1/3 + \cdots$ grows without bound.
Throw away every term whose denominator has a $9$ somewhere in it.
This looks like a small change, but almost all large numbers contain a $9$, so most terms disappear, and what remains adds up to a finite number, about $22.92$.
</document>

<document source="https://tamnd.github.io/math2/node/Q95N549AE3.md" tag="Q95N549AE3" kind="definition">
# Periodicity equation

## Definition

Let $p$ be a nonzero real number.
The *periodicity equation* with period $p$ is the [[functional equation|SRAN4EKWS0]]
$$f(x + p) = f(x) \quad \text{for all } x \in \R,$$
for an unknown [[function|0EPAT51DXQ]] $f \colon \R \to Y$, where $Y$ is a set.
The solutions are the [[periodic functions|0ATF6Q0633]] with period $p$.

## Remarks

Example: $f(x) = \cos(\pi x)$ is a solution for $p = 2$, since $\cos(\pi x + 2\pi) = \cos(\pi x)$.
Non-example: $f(x) = x^2$ is not a solution for any $p \ne 0$, since $f(0 + p) = p^2 \ne 0 = f(0)$.
With $q = \abs{p}$, the general solution is $f(x) = h\big(x - q \floor{x/q}\big)$, where $h \colon [0, q) \to Y$ is arbitrary.
Every solution for $p$ is also a solution for each nonzero integer multiple $np$.
The analogous equation $f(x + p) = -f(x)$ is called the antiperiodicity equation; its solutions have period $2p$.
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<document source="https://tamnd.github.io/math2/node/Q9B74BYZ8C.md" tag="Q9B74BYZ8C" kind="concept">
# Germ of an analytic set

## Summary

The germ of an analytic set at a point $a$ records the shape of a [[complex analytic variety|GB7M582M5Y]] arbitrarily close to $a$.
It forgets the rest of the set.
Two analytic sets have the same germ at $a$ when they coincide on some neighbourhood of $a$.
Germs of analytic sets at $a$ correspond to radical ideals of the ring of [[germs of holomorphic functions|7PJ8KVZZCQ]] at $a$.
This makes local analytic geometry a branch of commutative algebra.

## Explanations

### L1 human

Zoom in on an analytic set at one point and keep only what is visible at every scale.
For instance, the curve $\set{z_2^2 = z_1^2(1 + z_1)}$ in $\C^2$ is connected and irreducible as a whole, but its germ at the origin looks like two smooth branches crossing.
Far away points, such as other components of the set, do not affect the germ.
</document>

<document source="https://tamnd.github.io/math2/node/Q9D5TDRBEF.md" tag="Q9D5TDRBEF" kind="statement">
# Coifman–Rochberg–Weiss commutator theorem

## Statement

Let $n \ge 1$ and $1 < p < \infty$.
For a locally integrable function $b$ on $\R^n$ and an operator $T$, let $[b, T]$ be the commutator $[b, T] f = b \, Tf - T(bf)$, defined for smooth $f$ with compact support.

Let $T$ be a [[singular integral operator|HTMCEARGDD]] with kernel $\Omega(x/\abs{x})/\abs{x}^n$.
Here $\Omega$ is infinitely differentiable on the unit sphere and has mean zero.
Let $R_1, \dots, R_n$ be the [[Riesz transforms|FRNDFJ9WQ5]], and write $\norm{\cdot}$ for the operator norm on the [[Lp space|SASP14MA1Z]] $L^p(\R^n)$.

1. If $b$ has [[bounded mean oscillation|QJYTXFT8Y4]], then $[b, T]$ extends to a bounded operator on $L^p(\R^n)$.
Moreover $\norm{[b, T]} \le C \norm{b}_{\mathrm{BMO}}$, where $C$ depends only on $n$, $p$ and $\Omega$.
2. If $[b, R_j]$ is bounded on $L^p(\R^n)$ for every $j = 1, \dots, n$, then $b$ is in BMO.
Moreover $\norm{b}_{\mathrm{BMO}} \le C \sum_{j=1}^n \norm{[b, R_j]}$, where $C$ depends only on $n$ and $p$.

## Remarks

One proof of part 1 uses a conjugation trick.
For complex $z$, the operator $T_z f = e^{zb} T(e^{-zb} f)$ satisfies $\frac{d}{dz} T_z \big\vert_{z = 0} = [b, T]$.
By the John–Nirenberg inequality, $e^{p \, \mathrm{Re}(z) b}$ is a Muckenhoupt $A_p$ weight when $\abs{z} \norm{b}_{\mathrm{BMO}}$ is small.
So, by weighted theory, $T_z$ is bounded on $L^p$ uniformly for small $\abs{z}$.
The Cauchy integral formula for the derivative then gives the bound on $[b, T]$.

For $n = 1$ the only Riesz transform is the [[Hilbert transform|NSCEBXM7ZM]] $H$, and the theorem says that $[b, H]$ is bounded on $L^p(\R)$ exactly when $b \in \mathrm{BMO}(\R)$.
Example: $b(x) = \log \abs{x}$ is unbounded, so the multiplication operator by $b$ is unbounded on $L^p$, yet $[b, H]$ is bounded because $b \in \mathrm{BMO}$.

## Explanations

### L1 human

A commutator $[b, T]$ measures how much the order of two operations matters: first apply $T$ and then multiply by $b$, or first multiply by $b$ and then apply $T$.
For a singular integral $T$, the theorem says that this difference is a bounded operator exactly when $b$ has bounded mean oscillation.
So BMO is precisely the class of multipliers that almost commute with singular integrals.

### L2 human

The theorem gives a new characterization of BMO through operators, complementing its definition by mean oscillations and Fefferman's duality with $H^1$.
By duality it is equivalent to a factorization theorem for $H^1$: every $f \in H^1(\R^n)$ can be written as $\sum_k \sum_j \left( g_{k,j} R_j h_{k,j} + h_{k,j} R_j g_{k,j} \right)$ with $\sum_k \sum_j \norm{g_{k,j}}_{p'} \norm{h_{k,j}}_p < \infty$.
Commutators with BMO functions appear in the study of div-curl lemmas in compensated compactness and of elliptic equations with VMO coefficients.
The commutator $[b, T]$ is compact on $L^p$ exactly when $b$ is in the closure of smooth compactly supported functions in BMO, a result of Uchiyama.
</document>

<document source="https://tamnd.github.io/math2/node/Q9HHVN606G.md" tag="Q9HHVN606G" kind="concept">
# Bialgebra

## Summary

A bialgebra is a vector space that is both an [[associative algebra|3CTW1P4HBP]] and a [[coalgebra|A06RA8KB0T]] in a compatible way.
The compatibility is that the comultiplication and the counit are algebra homomorphisms.
Group algebras, monoid algebras, polynomial rings and universal enveloping algebras of Lie algebras are bialgebras.
The compatibility lets one form tensor products of modules, which is the key feature of [[Hopf algebras|7XFYP0N84H]] and quantum groups.

## Explanations

### L1 human

In a bialgebra one can multiply two elements into one and split one element into pairs.
The rule linking the two says that splitting a product is the same as splitting each factor and multiplying the pieces.
For a group algebra, each group element $g$ splits as $g \otimes g$, and indeed $gh$ splits as $gh \otimes gh = (g \otimes g)(h \otimes h)$.
This structure is what makes it possible to let a group act on a tensor product of two representations at once.
</document>

<document source="https://tamnd.github.io/math2/node/Q9J0VHGTSD.md" tag="Q9J0VHGTSD" kind="definition">
# Quantum invariant of 3-manifolds

## Definition

Let $\mathcal{C}$ be a modular tensor category.
An example is the category of tilting representations of the [[quantum group|AP2ZDXANKC]] $U_q(\mathfrak{sl}_2)$ at a root of unity $q$, modulo negligible morphisms.
The Reshetikhin–Turaev functor of $\mathcal{C}$ assigns a number $\langle L \rangle$ to each framed link $L$ in $S^3$ whose components are colored by objects of $\mathcal{C}$.

The *quantum invariant* $\tau_{\mathcal{C}}(M)$ of a closed oriented [[3-manifold|R20TB2KEQQ]] $M$ is obtained as follows.

1. Present $M$ by integral [[Dehn surgery|ZA405B0Z1S]] on a framed link $L \subset S^3$ with $n$ components.
2. Color each component of $L$ by the Kirby color $\omega = \sum_i \dim(V_i)\, V_i$, and evaluate $\langle L \rangle$ by linearity.
   Here $\dim$ is the quantum dimension, and the sum runs over the isomorphism classes of simple objects $V_i$.
3. Multiply by a normalising factor that depends only on $\mathcal{C}$, on $n$ and on the signature of the linking matrix of $L$.

The normalising factor is chosen so that the result does not change under the [[Kirby moves|AVQ9WN1MDV]], and hence depends only on $M$.

## Remarks

Let $\mathcal{C}$ come from $U_q(\mathfrak{sl}_2)$ with $q = e^{2\pi i / r}$.
Then the evaluation in step 2 is a combination of colored [[Jones polynomials|4SPC17FPXE]] of $L$, and $\tau_{\mathcal{C}}$ is the $SU(2)$ Witten–Reshetikhin–Turaev invariant at level $r - 2$.
Suitably normalised, the invariant is multiplicative under connected sum.
Step 3 is needed: the links $\emptyset$ and the $(\pm 1)$-framed unknot both present $S^3$, while their colored evaluations differ in general.
Normalisation conventions differ between authors.
</document>

<document source="https://tamnd.github.io/math2/node/Q9KPGBD49F.md" tag="Q9KPGBD49F" kind="definition">
# Order (ring theory)

## Definition

Let $R$ be a Noetherian [[integral domain|0QVD7WBFGN]] with [[field of fractions|0F2PTGX1A2]] $K$, and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $K$.
An *$R$-order* in $A$ is a [[subring|0GNCRPYXYV]] $\Lambda$ of $A$, with the same identity, such that:

1. $\Lambda$ is a [[finitely generated|05QY1VEBYD]] $R$-module.
2. $\Lambda$ contains a $K$-basis of $A$, that is $K \Lambda = A$.

## Remarks

Example: $\Lambda = \Z[i]$ in $A = \Q(i)$, with $R = \Z$.
It is a subring containing $1$, it is generated as a $\Z$-module by $1$ and $i$, and $1, i$ is a $\Q$-basis of $\Q(i)$.
Other examples are $M_n(\Z)$ in $M_n(\Q)$ and the group ring $\Z G$ of a finite group $G$ in $\Q G$.
Non-example: $\Z[\tfrac{1}{2}]$ is a subring of $\Q$ that spans $\Q$, but it is not a finitely generated $\Z$-module, so clause 1 fails.
Non-example: $\Z$ inside $\Q(i)$ is finitely generated but fails clause 2, since $\Q \cdot \Z = \Q \ne \Q(i)$.
When $R$ is a principal ideal domain, clause 1 makes $\Lambda$ a free $R$-module, and clause 2 then says its rank is $\dim_K A$.
Every element of an order is integral over $R$.
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<document source="https://tamnd.github.io/math2/node/Q9VG3EGT74.md" tag="Q9VG3EGT74" kind="definition">
# Morse theory on Hilbert manifolds

## Definition

Let $M$ be a $C^\infty$ [[Hilbert manifold|R8DMMW0E9M]] with a complete Riemannian metric $g$, and let $f \colon M \to \R$ be a $C^2$ function.
The gradient $\nabla f$ is defined by $g(\nabla f, v) = df(v)$.
The function $f$ satisfies *condition (C)* if every sequence $(p_n)$ in $M$ on which $f$ is bounded and $\norm{\nabla f(p_n)} \to 0$ has a convergent subsequence.
This is the [[Palais–Smale condition|4724KRNHAT]] on $M$.
Through $g$, the Hessian of $f$ at a critical point $p$ is a bounded self-adjoint operator on $T_p M$.
The point $p$ is *nondegenerate* if this operator is invertible.
Its *index* is the supremum of the dimensions of subspaces of $T_p M$ on which the Hessian is negative definite.

*Morse theory on Hilbert manifolds* is the [[critical point theory|9ZQC7ARWTT]] of $C^2$ functions $f$ on such $M$ with three properties.
The function is bounded below, it satisfies condition (C), and its critical points are [[nondegenerate|MGPPCDFQDG]] of finite [[index|MAX2ZYG6H9]].

## Remarks

For such $f$ the two basic theorems of Morse theory hold.
If $a < b$ and $f^{-1}[a, b]$ contains no critical point, then the sublevel set $M^a = \set{f \le a}$ is a deformation retract of $M^b$.
If $f^{-1}[a, b]$ contains exactly one critical point, of index $\lambda$, in its interior, then $M^b$ has the homotopy type of $M^a$ with a $\lambda$-cell attached.

Example: let $(N, h)$ be a compact Riemannian manifold, let $p, q \in N$, and let $M$ be the Hilbert manifold of paths $\omega \colon [0, 1] \to N$ of Sobolev class $H^1$ with $\omega(0) = p$ and $\omega(1) = q$.
The [[energy|Q12AB1MJ64]] $E(\omega) = \frac{1}{2} \int_0^1 \abs{\omega'}^2 \, dt$ is bounded below and satisfies condition (C).
Its critical points are the geodesics from $p$ to $q$, of finite index, and they are nondegenerate when $q$ is not conjugate to $p$ along any geodesic.

Non-example: on $M = \R$ the function $f(x) = \arctan x$ is bounded below and has no critical points, but condition (C) fails along $x_n = -n$, since $f(x_n)$ is bounded and $f'(x_n) \to 0$.
Accordingly $f$ does not attain its infimum $-\pi/2$, while a function that is bounded below and satisfies condition (C) attains its infimum at a critical point.
</document>

<document source="https://tamnd.github.io/math2/node/Q9Y4CJE1X5.md" tag="Q9Y4CJE1X5" kind="concept">
# Perfectoid space

## Summary

A perfectoid space is an analytic space over a non-archimedean field of residue characteristic $p$.
It is so ramified that the Frobenius map is surjective modulo $p$.
Peter Scholze introduced perfectoid spaces in 2012.
Their key feature is tilting: perfectoid spaces in mixed characteristic correspond to perfectoid spaces in characteristic $p$.
This lets one transfer problems between the two worlds.
They have become a basic tool of $p$-adic geometry and the arithmetic of [[Galois representations|JQEYEQJY2G]].

## Explanations

### L1 human

The $p$-adic numbers and the field of Laurent series over $\F_p$ look very different, since one has characteristic $0$ and the other characteristic $p$.
Adjoin all $p$-power roots of $p$ to the first, all $p$-power roots of the variable to the second, and complete both.
The two resulting fields have equivalent Galois theories.
Perfectoid spaces extend this surprising match from fields to geometry.
A hard question in characteristic $0$ can then be moved to characteristic $p$, where the Frobenius map gives extra tools.
</document>

<document source="https://tamnd.github.io/math2/node/QA9YR4GBNB.md" tag="QA9YR4GBNB" kind="definition">
# Geometric algebra

## Definition

Let $V$ be a finite-dimensional real [[vector space|000AYPA9AJ]] with a nondegenerate symmetric [[bilinear form|0SQZMXB6GP]] $u \cdot v$.
The *geometric algebra* $\mathcal{G}(V)$ is the [[Clifford algebra|A38B1QFK20]] of $V$ for the quadratic form $Q(v) = v \cdot v$.
So it is the associative unital real algebra generated by $V$ subject to $v v = v \cdot v$ for all $v \in V$.
Its product is the *geometric product*.
For vectors $u, v \in V$, the *outer product* is $u \wedge v = \tfrac{1}{2}(uv - vu)$, and then $uv = u \cdot v + u \wedge v$.
An element of $\mathcal{G}(V)$ is a *multivector*.
A product of $k$ pairwise orthogonal vectors is a $k$-blade, and the span of the $k$-blades is the grade $k$ part $\mathcal{G}^k(V)$.

## Remarks

Polarizing $v v = v \cdot v$ gives $uv + vu = 2\, u \cdot v$, which is why $u \cdot v = \tfrac{1}{2}(uv + vu)$.
As a graded vector space, $\mathcal{G}(V) = \bigoplus_{k=0}^{n} \mathcal{G}^k(V)$ is isomorphic to the [[exterior algebra|0MDMZ965VE]] $\Lambda V$, of dimension $2^n$ for $n = \dim V$.
Example: $V = \R^2$ with the dot product and orthonormal basis $e_1, e_2$.
Then $e_1 e_2 = -e_2 e_1$, and $I = e_1 e_2$ satisfies $I^2 = e_1 e_2 e_1 e_2 = -e_1 e_1 e_2 e_2 = -1$, so $\set{a + bI}$ is a copy of $\C$.
With $R = \cos(\theta/2) - I \sin(\theta/2)$ and $\tilde R = \cos(\theta/2) + I \sin(\theta/2)$, one checks $R e_1 \tilde R = e_1 (\cos \theta + I \sin \theta) = \cos\theta \, e_1 + \sin\theta \, e_2$, a rotation by $\theta$.
Non-example: the exterior algebra $\Lambda V$ with the wedge product alone is not a geometric algebra in this sense, because there $v \wedge v = 0$ for every $v$, while $v v = v \cdot v \ne 0$ for a non-null vector.
The notation $\mathcal{G}(p, q)$ or $\mathrm{Cl}_{p,q}(\R)$ is used when the form has $p$ positive and $q$ negative squares.
</document>

<document source="https://tamnd.github.io/math2/node/QAAV27WAAP.md" tag="QAAV27WAAP" kind="definition">
# Alexandrov space

## Definition

Let $k$ be a real number.
Let $M_k$ and $D_k$ be the model surface of constant curvature $k$ and its diameter bound, as for [[CAT(k) spaces|1FAF2VNA4G]].
Let $(X, d)$ be a [[metric space|0VKD1TP7F3]].
The space $X$ is a *length space* if each distance $d(x, y)$ is the infimum of the lengths of continuous curves from $x$ to $y$.
Geodesic triangles and comparison triangles in $M_k$ are as for CAT($k$) spaces.

The space $X$ is an *Alexandrov space of curvature $\ge k$* if:

1. $X$ is a [[complete|0BY52JW6NK]], [[locally compact|0FE2SB8WVJ]] length space, and
2. each point of $X$ has a neighbourhood $U$ with the following property.
   Take any geodesic triangle in $U$ of perimeter less than $2 D_k$, any vertex $x$ and any point $p$ on the opposite side.
   Let $\bar x, \bar p$ be the comparison points in a comparison triangle in $M_k$.
   Then $d(x, p) \ge d_{M_k}(\bar x, \bar p)$.

Many authors also require $X$ to have finite Hausdorff dimension; that dimension is then an integer.

## Remarks

Example: a complete Riemannian manifold with [[sectional curvature|1X42JAA349]] $\ge k$ is an Alexandrov space of curvature $\ge k$, by [[Toponogov's theorem|KX0B3AWD5H]].
In particular $\R^n$ has curvature $\ge 0$; its triangles equal their comparison triangles.

Non-example: the hyperbolic plane of curvature $-1$ does not have curvature $\ge 0$.
Take a geodesic triangle in it and the midpoint of one side.
The distance from the opposite vertex to this midpoint is strictly less than in the flat comparison triangle, so clause 2 fails for $k = 0$.

For an integer $m \ge 2$, let $C_m$ be the quotient of $\R^2$ by the group of rotations about $0$ through multiples of $2\pi/m$.
This cone is an Alexandrov space of curvature $\ge 0$, and it is not a smooth Riemannian manifold at its tip.
</document>

<document source="https://tamnd.github.io/math2/node/QAC7BYK9KX.md" tag="QAC7BYK9KX" kind="concept">
# Holomorphic line bundle

## Summary

A holomorphic line bundle is a [[holomorphic vector bundle|TKA958Z4KZ]] of rank one.
Its transition functions are nowhere vanishing holomorphic functions.
So the isomorphism classes of holomorphic line bundles on a complex manifold $X$ form the [[Picard group|3Z9DCWYXQZ]] $H^1(X, \mathcal{O}_X^*)$ under tensor product.
Line bundles encode divisors, maps to projective space and positivity, as with the [[canonical bundle|W05VYY4HBR]] and the bundles $\mathcal{O}(k)$ on $\mathbb{P}^n$.

## Explanations

### L1 human

A holomorphic line bundle attaches a complex line to each point of a complex manifold, in a holomorphic way.
Locally it is just $U \times \C$, and two local pictures are related by multiplying by a nowhere zero holomorphic function.
A section is then like a holomorphic function, except that its values are only defined up to these gluing factors.
For example, the homogeneous polynomials of degree $k$ in $z_0, z_1$ are not functions on the Riemann sphere, but they are exactly the holomorphic sections of the line bundle $\mathcal{O}(k)$.
</document>

<document source="https://tamnd.github.io/math2/node/QADCFWETJC.md" tag="QADCFWETJC" kind="statement">
# Rubio de Francia extrapolation theorem

## Statement

Let $n \ge 1$ and $1 \le p_0 < \infty$.
For a weight $w$ on $\R^n$ and $1 \le p < \infty$, write $\norm{g}_{L^p(w)} = \left( \int_{\R^n} \abs{g}^p \, w \, dx \right)^{1/p}$, and let $[w]_{A_p}$ be the constant of the [[Muckenhoupt class|T86RSJ8M86]] $A_p$.
Let $\mathcal{F}$ be a family of pairs $(f, g)$ of nonnegative measurable functions on $\R^n$.
Assume that there is an increasing function $N \colon [1, \infty) \to [0, \infty)$ such that
$$\norm{f}_{L^{p_0}(w)} \le N([w]_{A_{p_0}}) \, \norm{g}_{L^{p_0}(w)}$$
for every $w \in A_{p_0}$ and every $(f, g) \in \mathcal{F}$ with $\norm{f}_{L^{p_0}(w)} < \infty$.
Then let $1 < p < \infty$ and $w \in A_p$.
There is a constant $C$, depending only on $n$, $p$, $p_0$, $N$ and $[w]_{A_p}$, such that
$$\norm{f}_{L^p(w)} \le C \norm{g}_{L^p(w)}$$
for every $(f, g) \in \mathcal{F}$ with $\norm{f}_{L^p(w)} < \infty$.

## Remarks

In the typical use, the pairs are $(\abs{Tf}, \abs{f})$ for an operator $T$.
Assume $T$ is bounded on $L^{p_0}(w)$ for all $w \in A_{p_0}$, with a bound depending only on $[w]_{A_{p_0}}$.
Then $T$ is bounded on $L^p(w)$ for all $1 < p < \infty$ and all $w \in A_p$.
In particular, taking $w = 1$, one exponent $p_0$ gives unweighted bounds for all $1 < p < \infty$.

The proof uses the Rubio de Francia algorithm.
Let $v$ be a weight and $q > 1$ such that the [[maximal operator|VX1A56P5KQ]] $M$ is bounded on $L^q(v)$, with norm $\norm{M}$.
For $h \ge 0$ in $L^q(v)$, set $Rh = \sum_{k \ge 0} \frac{M^k h}{(2\norm{M})^k}$, where $M^k$ is the $k$-th iterate.
Then $h \le Rh$, $\norm{Rh}_{L^q(v)} \le 2 \norm{h}_{L^q(v)}$, and $M(Rh) \le 2 \norm{M} \, Rh$, so $Rh$ is an $A_1$ weight.
These $A_1$ weights are combined into an $A_{p_0}$ weight to which the hypothesis applies, and [[Muckenhoupt's theorem|S9BKNBRV4H]] controls the constants.

## Explanations

### L1 human

Proving an inequality for one exponent $p$ is often much easier than for all of them, for example $p = 2$ where Fourier methods work.
Extrapolation says that if the inequality holds for one exponent and for all Muckenhoupt weights at that exponent, then it holds for all exponents $1 < p < \infty$ and all corresponding weights.
So weights act as a bridge between different exponents.

### L2 human

The theorem is a main reason to prove weighted inequalities.
A single weighted $L^2$ bound for a Calderón–Zygmund operator, a commutator or a square function yields all the $L^p$ bounds, weighted and unweighted.
It also gives vector-valued inequalities almost for free, since a sum $\left( \sum_j \abs{T f_j}^q \right)^{1/q}$ can be treated by applying the pair formulation to suitable functions.
It relies on the theory of $A_p$ weights, in particular on Muckenhoupt's theorem and on the factorization of $A_p$ weights as $w = w_1 w_2^{1 - p}$ with $w_1, w_2 \in A_1$.
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<document source="https://tamnd.github.io/math2/node/QAGW4V0S62.md" tag="QAGW4V0S62" kind="definition">
# Tor functor

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $M$ and $N$ be [[$R$-modules|0NZ8JA25WK]].
Choose a [[projective resolution|HK2AA1YSQ2]] $\cdots \to P_1 \xrightarrow{d_1} P_0 \to M \to 0$ of $M$.
[[Tensoring|F0HQXQ0HCD]] the complex $P_\bullet$ with $N$ gives a chain complex
$$\cdots \to P_2 \otimes_R N \xrightarrow{d_2 \otimes \id_N} P_1 \otimes_R N \xrightarrow{d_1 \otimes \id_N} P_0 \otimes_R N \to 0.$$
For $n \ge 0$, the $n$-th *Tor module* is its homology
$$\operatorname{Tor}_n^R(M, N) = \ker(d_n \otimes \id_N) / \im(d_{n+1} \otimes \id_N),$$
with $d_0 \otimes \id_N = 0$.

## Remarks

Up to a canonical isomorphism, $\operatorname{Tor}_n^R(M, N)$ does not depend on the resolution, and resolving $N$ instead of $M$ gives the same modules.
One has $\operatorname{Tor}_0^R(M, N) \cong M \otimes_R N$ and $\operatorname{Tor}_n^R(M, N) \cong \operatorname{Tor}_n^R(N, M)$.
Example: for integers $m, n \ge 1$, tensoring $0 \to \Z \xrightarrow{\cdot m} \Z \to \Z/m\Z \to 0$ with $\Z/n\Z$ gives $\Z/n\Z \xrightarrow{\cdot m} \Z/n\Z$, whose kernel is cyclic of order $\gcd(m, n)$; so $\operatorname{Tor}_1^\Z(\Z/m\Z, \Z/n\Z) \cong \Z/\gcd(m,n)\Z$.
Non-example: the exact sequence $0 \to \Z/m\Z \xrightarrow{\id} \Z/m\Z \to 0$ is not a projective resolution over $\Z$ for $m \ge 2$, and using it would wrongly give $\operatorname{Tor}_1 = 0$.
A module $M$ is [[flat|73RYYEVDKA]] if and only if $\operatorname{Tor}_1^R(M, N) = 0$ for all $R$-modules $N$.
For a noncommutative ring $R$, the same construction works for a right module $M$ and a left module $N$.
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<document source="https://tamnd.github.io/math2/node/QAP2SBSK2G.md" tag="QAP2SBSK2G" kind="definition">
# Riesz mean

## Definition

Let $(\lambda_n)_{n \ge 0}$ be a strictly increasing sequence of real numbers with $\lambda_0 \ge 0$ and $\lambda_n \to \infty$, and let $\delta > 0$.
Let $\sum_{n \ge 0} a_n$ be a [[series|0JH4MN4K1C]] of complex numbers.
For real $\omega > 0$, the *Riesz mean* of type $\lambda$ and order $\delta$ is the finite sum
$$R^{\delta}_{\lambda}(\omega) = \sum_{n \,:\, \lambda_n < \omega} \left(1 - \frac{\lambda_n}{\omega}\right)^{\delta} a_n.$$
The series is $(R, \lambda, \delta)$ summable to $s$ if $R^{\delta}_{\lambda}(\omega) \to s$ as $\omega \to \infty$.

## Remarks

Example: take $\lambda_n = n$, $\delta = 1$ and an integer cut-off $\omega = N + 1$.
Then
$$R^1_{\lambda}(N + 1) = \sum_{n=0}^{N} \frac{N + 1 - n}{N + 1} a_n = \frac{s_0 + s_1 + \cdots + s_N}{N + 1},$$
where $s_j$ are the [[partial sums|0NB643K60Q]], because $a_n$ occurs in exactly the $N + 1 - n$ partial sums $s_n, \dots, s_N$.
So these Riesz means are the Cesàro means of the partial sums.

Non-example: if $\lambda_n = 1 - 2^{-n}$, the sequence is increasing but bounded, so the condition $\lambda_n \to \infty$ fails.
Then for $\omega > 1$ every term enters $R^{\delta}_{\lambda}(\omega)$, which is not a finite sum, and the definition does not apply.

For Dirichlet series $\sum a_n n^{-s}$ one usually takes $\lambda_n = \ln n$ for $n \ge 1$.
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<document source="https://tamnd.github.io/math2/node/QAQAS3JWXW.md" tag="QAQAS3JWXW" kind="concept">
# Congruence-modular variety

## Summary

A congruence-modular variety is a [[variety|AJNZJD8442]] in which the [[congruence lattice|REE07CSXHR]] of every algebra is a [[modular lattice|QX7WZ92QAK]].
It includes every congruence-permutable variety, such as groups, rings and modules, and every congruence-distributive variety, such as lattices.
Congruence-modular varieties are the natural setting of commutator theory, which brings ideas of group theory to general algebras.

## Explanations

### L1 human

The modular law is a weak form of the distributive law, and it holds for the normal subgroups of every group.
A variety is congruence modular when all its algebras have congruence lattices that obey this law.
Groups and lattices both qualify, though for different reasons.
Sets with no operations do not: the equivalence relations on a four-element set do not form a modular lattice.
</document>

<document source="https://tamnd.github.io/math2/node/QARH15D2PN.md" tag="QARH15D2PN" kind="concept">
# Lusternik–Schnirelmann theory of critical points

## Summary

Lusternik–Schnirelmann theory gives lower bounds for the number of [[critical points|0YTCMFB4HC]] of a function from the topology of its domain, without any nondegeneracy assumption.
Its main result is that a $C^1$ function on a closed manifold has at least $\operatorname{cat}(M) + 1$ critical points, where $\operatorname{cat}(M)$ is the normalized [[Lusternik–Schnirelmann category|W5BHHXCYXA]].
The critical values are obtained by minimax over closed sets of increasing category.
It is a branch of [[critical point theory|9ZQC7ARWTT]], complementary to Morse theory, and it extends to infinite dimensions under the [[Palais–Smale condition|4724KRNHAT]].

## Explanations

### L1 human

Morse theory counts critical points of a function with only clean critical points, and gives bounds from the homology of the space.
Lusternik and Schnirelmann asked what can be said for any smooth function, however degenerate.
Their answer is a number measuring how many contractible pieces are needed to cover the space.
For example, a torus needs three such pieces, so every smooth function on a torus has at least three critical points.
On a sphere the bound is two: a maximum and a minimum.
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<document source="https://tamnd.github.io/math2/node/QATW7046PD.md" tag="QATW7046PD" kind="concept">
# Weierstrass zeta and sigma functions

## Summary

For a lattice $\Lambda$ in $\C$, the Weierstrass sigma function $\sigma$ is an [[entire function|0G4F0XB9X2]] with a simple zero at each lattice point.
The Weierstrass zeta function is its logarithmic derivative $\zeta = \sigma'/\sigma$.
They are companions of the [[Weierstrass elliptic function|361NAK68QK]] $\wp$, since $\zeta' = -\wp$.
Neither is elliptic: shifting by a period adds a constant to $\zeta$ and multiplies $\sigma$ by an exponential factor.
They are the building blocks for writing any elliptic function from its zeros and poles.

## Explanations

### L1 human

A polynomial can be written as a product over its roots.
The sigma function plays this role for the lattice: it is a product over all lattice points, with exponential correction factors so that the product converges.
Taking its logarithmic derivative gives the zeta function, a sum of simple poles, one at each lattice point.
Differentiating once more and changing sign gives $\wp$, which has double poles and is truly periodic.
So $\sigma$, $\zeta$, $\wp$ form a chain, and only the last one is elliptic.
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<document source="https://tamnd.github.io/math2/node/QAWDC7SWKC.md" tag="QAWDC7SWKC" kind="statement">
# Murray–von Neumann classification of factors

## Statement

Let $M$ be a [[von Neumann algebra|PSFWQVZ0FS]] on a separable complex Hilbert space, with [[commutant|1VC7EJ1ZBZ]] $M'$.
Assume that $M$ is a factor, that is, its center $M \cap M'$ is $\C I$.
For [[orthogonal projections|0B8KZRT28M]] $p, q \in M$, write $p \sim q$ if there is a [[partial isometry|GGDSF85JPD]] $v \in M$ with $v^*v = p$ and $vv^* = q$.
Write $p \precsim q$ if $p \sim q_0$ for some projection $q_0 \in M$ with $q_0 \le q$.
Call $p$ finite if $p \sim q_0 \le p$ implies $q_0 = p$.
Then:

1. for all projections $p, q \in M$, $p \precsim q$ or $q \precsim p$;
2. there is a dimension function $D$ from the projections of $M$ to $[0, \infty]$. It is unique up to a positive scalar factor. It satisfies: $p \precsim q$ if and only if $D(p) \le D(q)$. Also $D(p + q) = D(p) + D(q)$ when $pq = 0$. And $D(p) < \infty$ if and only if $p$ is finite;
3. after normalization, the range of $D$ is exactly one of the following sets, which defines the type of $M$. The set $\set{0, 1, \dots, n}$ gives type $\mathrm{I}_n$, for $n \ge 1$. The set $\set{0, 1, 2, \dots, \infty}$ gives type $\mathrm{I}_\infty$. The interval $[0, 1]$ gives type $\mathrm{II}_1$. The interval $[0, \infty]$ gives type $\mathrm{II}_\infty$. The set $\set{0, \infty}$ gives type $\mathrm{III}$;
4. $M$ has type $\mathrm{I}_n$ or $\mathrm{I}_\infty$ if and only if $M \cong B(K)$ for a [[Hilbert space|HM2HHE3J51]] $K$ of dimension $n$ or $\aleph_0$, respectively. Here $B(K)$ is the [[algebra of bounded operators|BMDG4JQPEH]] on $K$.

## Remarks

The comparison in part 1 comes from a maximality argument.
Take a maximal family of pairwise orthogonal equivalent pieces of $p$ and $q$.
It exhausts $p$ or $q$, since otherwise, in a factor, the two remainders would contain further equivalent nonzero pieces.
The dimension function $D$ is built by cutting projections into many equivalent pieces and counting, much as a length is measured with smaller and smaller rulers.
In type $\mathrm{II}_1$, $D$ is the restriction of the unique faithful normal tracial state to the projections.

Examples: $M_n(\C)$ has type $\mathrm{I}_n$, with $D(p) = \rank p$.
The [[hyperfinite type II1 factor|NZ4NY8SK1C]] has type $\mathrm{II}_1$.
The tensor product of a type $\mathrm{II}_1$ factor with $B(\ell^2)$ has type $\mathrm{II}_\infty$.
Factors of type $\mathrm{III}$ also exist; they admit no nonzero normal semifinite trace.

The hypothesis that $M$ is a factor is needed: for the non-factor $M = \C \oplus \C$ acting diagonally on $\C^2$, the two projections $(1, 0)$ and $(0, 1)$ are not comparable, so part 1 fails.

## Explanations

### L1 human

In a factor one can compare the "sizes" of projections, like comparing the dimensions of subspaces.
Murray and von Neumann showed that this size takes values in only five kinds of ranges.
For matrices the sizes are whole numbers, as dimensions should be.
In type $\mathrm{II}_1$ factors the sizes fill the whole interval $[0,1]$, giving "continuous dimension".
In type $\mathrm{III}$ factors every nonzero projection is infinite.

### L2 human

Every von Neumann algebra on a separable Hilbert space is a direct integral of factors, so factors are the building blocks of the theory.
The classification by types is the first step; the next steps distinguish factors within one type.
Type $\mathrm{I}$ factors are completely understood by part 4.
Type $\mathrm{III}$ factors are further divided by Connes into types $\mathrm{III}_\lambda$, $0 \le \lambda \le 1$, using [[Tomita–Takesaki theory|NBR81PR7E5]].
Type $\mathrm{II}_1$ factors carry a trace and are studied through subfactors and free probability.
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<document source="https://tamnd.github.io/math2/node/QB05EN6HEQ.md" tag="QB05EN6HEQ" kind="definition">
# Permuting congruences

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$, and let $\theta$ and $\psi$ be [[congruences|DCSXXT4SER]] on $\mathbf{A}$.
Their relational product is
$$\theta \circ \psi = \set{(a, c) \in A \times A : (a, b) \in \theta \text{ and } (b, c) \in \psi \text{ for some } b \in A}.$$
The congruences $\theta$ and $\psi$ *permute* when $\theta \circ \psi = \psi \circ \theta$.
The algebra $\mathbf{A}$ is *congruence-permutable* when every two congruences on $\mathbf{A}$ permute.

## Remarks

The congruences $\theta$ and $\psi$ permute if and only if $\theta \circ \psi$ is a congruence.
In that case $\theta \circ \psi = \theta \vee \psi$, the join in the [[congruence lattice|REE07CSXHR]].
A congruence-permutable algebra has a modular congruence lattice.

Example: let $G$ be a group and $N$, $M$ normal subgroups, with congruences $a \mathrel{\theta} b \iff a^{-1} b \in N$ and $a \mathrel{\psi} b \iff a^{-1} b \in M$.
Then $(a, c) \in \theta \circ \psi$ if and only if $a^{-1} c \in NM$, and $(a, c) \in \psi \circ \theta$ if and only if $a^{-1} c \in MN$.
Since $NM = MN$ for normal subgroups, $\theta$ and $\psi$ permute.

Non-example: on the set $\set{1, 2, 3}$ with no operations, let $\theta$ have classes $\set{1, 2}, \set{3}$ and $\psi$ have classes $\set{1}, \set{2, 3}$.
Then $(1, 3) \in \theta \circ \psi$ through $b = 2$.
But $(1, 3) \notin \psi \circ \theta$, since the only $b$ with $(1, b) \in \psi$ is $1$, and $(1, 3) \notin \theta$.
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<document source="https://tamnd.github.io/math2/node/QB3Q5FBJ08.md" tag="QB3Q5FBJ08" kind="concept">
# Mixed volume

## Summary

Let $K_1, \dots, K_r$ be nonempty compact convex sets of $\R^n$ and $\lambda_i \ge 0$.
The volume of the [[Minkowski sum|SXS6D87PHE]] $\lambda_1 K_1 + \cdots + \lambda_r K_r$ is a homogeneous polynomial of degree $n$ in the $\lambda_i$.
Its coefficients are the mixed volumes $V(K_{i_1}, \dots, K_{i_n})$.
Mixed volumes unify volume, surface area, mean width and many other measurements of convex bodies.
They satisfy deep inequalities, above all the [[Alexandrov–Fenchel inequality|D3AVHVYQW9]].

## Explanations

### L1 human

Blow up two convex shapes by factors $\lambda$ and $\mu$ and add them.
The area of the result is $A(K) \lambda^2 + 2 V(K, L) \lambda \mu + A(L) \mu^2$, a quadratic expression.
The middle coefficient $V(K, L)$ is the mixed area of $K$ and $L$: it measures how the two shapes interact.
When $L$ is the unit disk, $V(K, L)$ is half the perimeter of $K$.
So perimeter, and in higher dimensions surface area, are special mixed volumes.
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<document source="https://tamnd.github.io/math2/node/QB45PCGQPV.md" tag="QB45PCGQPV" kind="concept">
# Krylov subspace

## Summary

The Krylov subspace of order $r$ generated by a square matrix $A$ and a vector $b$ is the [[linear span|0DTY8B9HMS]] of $b, A b, \dots, A^{r-1} b$.
Its elements are the vectors $p(A) b$ for polynomials $p$ of degree less than $r$.
Many iterative methods for large linear systems and eigenvalue problems search for approximations in Krylov subspaces.
These subspaces can be built with matrix-vector products alone.

## Explanations

### L1 human

Start with a vector $b$ and keep applying the matrix: $b$, $A b$, $A^2 b$, and so on.
The Krylov subspace is everything one can build from the first few of these vectors by adding and scaling.
A method that can only multiply by $A$ sees exactly these vectors, so it is natural to search for the answer there.
For example, if $A$ is diagonal with entries $1$ and $2$ and $b = (1, 1)$, then $A b = (1, 2)$, and the subspace of order $2$ is all of $\R^2$.
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<document source="https://tamnd.github.io/math2/node/QB5AE3VYZZ.md" tag="QB5AE3VYZZ" kind="statement">
# Hahn–Banach theorem

## Statement

Let $X$ be a [[vector space|000AYPA9AJ]] over $\R$, and let $p \colon X \to \R$ be a [[sublinear function|K3356N0SAX]].
Let $M$ be a [[linear subspace|00YC1D652K]] of $X$, and let $f \colon M \to \R$ be a [[linear form|07YQ1C78G0]] with $f(m) \le p(m)$ for all $m \in M$.
Then there is a linear form $F \colon X \to \R$ such that $F(m) = f(m)$ for all $m \in M$ and $F(x) \le p(x)$ for all $x \in X$.

## Remarks

The proof first extends $f$ to $M + \R x_0$ for one vector $x_0 \notin M$.
The value $F(x_0)$ must be a number $c$ with $\sup_{m \in M} (f(m) - p(m - x_0)) \le c \le \inf_{m \in M} (p(m + x_0) - f(m))$, and subadditivity of $p$ shows that such a $c$ exists.
[[Zorn's lemma|04RXTQA5ZA]] applied to the partial extensions dominated by $p$ then gives an extension to all of $X$.

The most used case takes $p(x) = \norm{f}\,\norm{x}$ on a real [[normed vector space|03J5RAKB30]].
It shows that every bounded linear functional on a subspace extends to the whole space with the same [[operator norm|06EKW1AKXF]].
The complex case of this norm version follows by applying the real theorem to the real part of $f$.
A consequence is that for each $x \ne 0$ in a normed space there is a functional $F$ of norm $1$ with $F(x) = \norm{x}$, so the [[continuous dual space|4A82S5FVRT]] separates points.

## Explanations

### L1 human

The theorem says that a linear rule defined on part of a space can be extended to the whole space without breaking a given bound.
The bound is a sublinear function, such as a norm.
It is the basic tool that guarantees there are enough continuous linear functionals on a normed space to tell its points apart.

### L2 human

In a course on functional analysis this is the first of the main theorems, next to the [[uniform boundedness principle|YG0042V8CJ]] and the [[bounded inverse theorem|MR5FJXR348]].
Its norm form gives the isometric embedding of a space into its [[bidual|QKR3D65TTH]].
Its geometric form is the [[Hahn–Banach separation theorem|6GWMGFGFJJ]], which separates disjoint convex sets by a closed hyperplane.
The two forms are linked by the [[Minkowski functional|PAX9SHDHDW]] of a convex neighbourhood of $0$.
In a general real vector space the proof needs Zorn's lemma or a weaker choice principle.
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<document source="https://tamnd.github.io/math2/node/QB6EZ3HP73.md" tag="QB6EZ3HP73" kind="definition">
# Simple function

## Definition

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]].
A *simple function* on $X$ is a [[measurable function|YYNCJSR0PJ]] $s \colon X \to \R$ whose image $s(X)$ is a finite set.

## Remarks

Equivalently, $s = \sum_{k=1}^{n} a_k \mathbf{1}_{A_k}$ for some $n \ge 0$, real numbers $a_k$ and sets $A_k \in \Sigma$, where $\mathbf{1}_{A_k}$ is the [[indicator function|0RCQZ36MME]] of $A_k$.
The sets $A_k$ can be chosen pairwise disjoint, with distinct values $a_k$.
For a measure $\mu$ and $a_k \ge 0$, the integral is $\int s \, d\mu = \sum_k a_k \mu(A_k)$, with $0 \cdot \infty = 0$.
Example: on $\R$ with the Borel sets, $s = 3 \cdot \mathbf{1}_{[0, 1]} - \mathbf{1}_{\Q}$ is simple; its values are $-1$, $0$, $2$ and $3$.
Non-example: $x \mapsto x$ on $[0, 1]$ is measurable but has infinitely many values.
Every measurable $f \colon X \to [0, \infty]$ is the pointwise limit of a non-decreasing sequence of nonnegative simple functions.
Some authors drop the measurability condition, or allow complex values.
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<document source="https://tamnd.github.io/math2/node/QBDT694BAP.md" tag="QBDT694BAP" kind="concept">
# Free object

## Summary

A free object on a set $X$ is an object generated by $X$ with no relations other than those forced by the axioms.
It is defined by a [[universal property|EN4DD2FAJH]]: every map from $X$ into the underlying set of an object extends uniquely to a morphism.
[[Free groups|0KTW26C80J]], [[free modules|0DZ9DKAKZR]] and [[free monoids|CV013KHVJG]] are the main examples.
When free objects exist on every set, they form a left [[adjoint|TZHF6ZM92S]] of the forgetful functor.

## Explanations

### L1 human

A free object is the most general object you can build from a set of letters.
In the free group on $\set{a, b}$, the elements are words such as $a b a^{-1} b^{-1}$, and no two different reduced words are equal.
Because nothing extra holds, you can send $a$ and $b$ to any two elements of any group, and this gives exactly one homomorphism.
A vector space with basis $X$ is the free vector space on $X$ in the same sense.
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<document source="https://tamnd.github.io/math2/node/QBHNB4CP22.md" tag="QBHNB4CP22" kind="definition">
# Finitely generated variety

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras of a fixed signature.
The variety $\mathcal{V}$ is *finitely generated* if $\mathcal{V} = \mathbf{HSP}(\mathcal{K})$ for some finite set $\mathcal{K}$ of finite algebras.
Here $\mathbf{H}$, $\mathbf{S}$ and $\mathbf{P}$ are the [[class operators|S7MJVSP79N]] of homomorphic images, subalgebras and direct products.

## Remarks

If $\mathcal{K} = \set{A_1, \dots, A_k}$, then $\mathbf{HSP}(\mathcal{K}) = \mathbf{HSP}(A_1 \times \cdots \times A_k)$, so one finite algebra suffices.
Example: the variety of abelian groups that satisfy the identity $6x = 0$ is generated by the finite group $\Z/6\Z$.
Each such group is a direct sum of copies of $\Z/2\Z$ and $\Z/3\Z$, and both of these are subgroups of $\Z/6\Z$.
A direct sum is a subgroup of the direct product, so each such group lies in $\mathbf{SP}(\Z/6\Z)$.
Non-example: the variety of all abelian groups is not finitely generated.
If $A$ is a finite abelian group with $m$ elements, then every group in $\mathbf{HSP}(A)$ satisfies the identity $mx = 0$, while $\Z$ does not.
Every finitely generated variety is a [[locally finite variety|8QPW65PZVW]].

## Explanations

### L1 human

A variety is finitely generated when one finite algebra already produces the whole variety by taking subalgebras, products and images.
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# Siciak extremal function

## Summary

The Siciak extremal function $V_E$ of $E \subseteq \C^n$ is the supremum of all functions of the [[Lelong class|X4ZHQG9995]] that are at most $0$ on $E$.
For compact $E$ it can also be computed from polynomials.
It is the supremum of $\frac{1}{\deg p} \log\abs{p}$ over polynomials $p$ of degree at least $1$ with $\abs{p} \le 1$ on $E$.
It is the higher-dimensional version of the Green function of a planar compact set with pole at infinity.
It is also called the pluricomplex Green function of $E$ with pole at infinity.
It controls polynomial approximation and interpolation on $E$, and for compact $E$ its upper semicontinuous regularisation is identically $+\infty$ exactly when $E$ is [[pluripolar|N0MAGPW8AP]].

## Explanations

### L1 human

Suppose a polynomial is at most $1$ in size on a set $E$.
How large can it be at a point $z$ outside $E$, given its degree?
The Siciak extremal function answers this: the best bound is $\abs{p(z)} \le e^{d \, V_E(z)}$ for polynomials of degree $d$.
For the closed unit ball in $\C^n$, the answer is $V_E(z) = \log\abs{z}$ outside the ball.
So a polynomial of degree $d$ bounded by $1$ on the ball is bounded by $\abs{z}^d$ outside it.
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# Double complex

## Definition

Let $\mathcal{A}$ be an [[additive category|B2VP96NJSK]], for example the category of [[modules|0NZ8JA25WK]] over a ring.
A *double complex* in $\mathcal{A}$ consists of objects $C_{p,q}$ for all integers $p, q$ and morphisms
$$d^h_{p,q} \colon C_{p,q} \to C_{p-1,q}, \qquad d^v_{p,q} \colon C_{p,q} \to C_{p,q-1},$$
called the horizontal and vertical differentials, such that:

1. $d^h \circ d^h = 0$;
2. $d^v \circ d^v = 0$;
3. $d^h \circ d^v + d^v \circ d^h = 0$.

Suppose that the needed direct sums exist, for example for modules.
Then the *total complex* is the [[chain complex|WQ82C0M3QB]] with
$$\mathrm{Tot}(C)_n = \bigoplus_{p + q = n} C_{p,q}, \qquad d = d^h + d^v.$$

## Remarks

The three conditions give $d^2 = (d^h)^2 + d^h d^v + d^v d^h + (d^v)^2 = 0$, so the total complex is a complex.
The sums are finite, and exist in any additive category, when only finitely many $C_{p,q}$ with $p + q = n$ are nonzero for each $n$.
Some texts ask instead that the squares commute, $d^h d^v = d^v d^h$, and put the sign into the total complex as $d = d^h + (-1)^p d^v$; the two conventions are equivalent.
Using products instead of direct sums gives a second total complex, which differs when infinitely many terms are nonzero on a diagonal.

Example: for chain complexes $(P_\bullet, d^P)$ and $(Q_\bullet, d^Q)$ of modules over a commutative ring, set $C_{p,q} = P_p \otimes Q_q$, $d^h = d^P \otimes 1$ and $d^v = (-1)^p \, 1 \otimes d^Q$ on $C_{p,q}$.
For $x \in P_p$ and $y \in Q_q$, $d^h d^v (x \otimes y) = (-1)^p\, d^P x \otimes d^Q y$ and $d^v d^h (x \otimes y) = (-1)^{p-1}\, d^P x \otimes d^Q y$, so condition 3 holds.

Non-example: the same data without the sign, $d^v = 1 \otimes d^Q$, fails condition 3 in general, since then $d^h d^v + d^v d^h = 2\, d^P \otimes d^Q$; over $\Z$ with $P = Q = (\Z \xrightarrow{1} \Z)$ this is $2 \ne 0$.
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# Stability of Volterra equations

## Definition

Let $n \ge 1$, let $\R_+ = [0, \infty)$, and let $k \in L^1_{\mathrm{loc}}(\R_+; \C^{n \times n})$ be a locally integrable matrix-valued kernel.
Write $(k * x)(t) = \int_0^t k(t - s) \, x(s) \, ds$.
Consider the linear [[Volterra equation|DND8CBEVXT]] of convolution type
$$x(t) + (k * x)(t) = f(t), \qquad t \ge 0.$$
Its resolvent is the unique $r \in L^1_{\mathrm{loc}}(\R_+; \C^{n \times n})$ with $r + k * r = k$ and $r + r * k = k$.
For every $f \in L^1_{\mathrm{loc}}(\R_+; \C^n)$ the unique locally integrable solution is $x = f - r * f$.
The equation is called *stable*, or is said to have an *integrable resolvent*, if $r \in L^1(\R_+; \C^{n \times n})$.

## Remarks

If the equation is stable and $1 \le p \le \infty$, then $f \in L^p(\R_+)$ gives $x \in L^p(\R_+)$ with $\norm{x}_p \le (1 + \norm{r}_1) \norm{f}_p$, by Young's inequality in the [[Lp spaces|SASP14MA1Z]].
If moreover $f$ is bounded and $f(t) \to f_\infty$ as $t \to \infty$, then $x(t) \to \big(I - \int_0^\infty r(s) \, ds\big) f_\infty$, by dominated convergence.

Example: $n = 1$ and $k(t) = a e^{-t}$ with $\operatorname{Re} a > -1$.
The resolvent is $r(t) = a e^{-(1 + a) t}$.
Check: $(k * r)(t) = a^2 e^{-t} \int_0^t e^{-a s} \, ds = a e^{-t} - a e^{-(1 + a) t}$ for $a \ne 0$, so $r + k * r = k$, and the case $a = 0$ is trivial.
Since $\operatorname{Re}(1 + a) > 0$, $r$ is integrable and the equation is stable.

Non-example: $n = 1$ and $k(t) = -1$.
Then $r(t) = -e^t$, since $-e^t + \int_0^t e^s \, ds = -1$, and $r$ is not integrable.
For $f = 1$ the solution is $x(t) = 1 + (e^t - 1) = e^t$, which is unbounded although $f$ is bounded.

For nonlinear Volterra equations one speaks instead of stability of a particular solution, in the sense of Lyapunov.

## Explanations

### L1 human

The resolvent turns the data $f$ into the solution $x$.
When the resolvent has finite total size, it cannot turn bounded data into unbounded solutions.
That is what stability means here.
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# HOMFLY polynomial

## Summary

The HOMFLY polynomial is an invariant of oriented [[links|1A5WRCNTM3]] with values in Laurent polynomials in two variables.
It is defined by a skein relation and contains both the [[Alexander polynomial|JKA25KDPQP]] and the [[Jones polynomial|4SPC17FPXE]] as special cases.
Its name comes from the initials of Hoste, Ocneanu, Millett, Freyd, Lickorish and Yetter, who found it soon after the Jones polynomial.
Przytycki and Traczyk found it independently, so it is also called the HOMFLYPT polynomial.

## Explanations

### L1 human

The HOMFLY polynomial is a two-variable polynomial attached to a knot or link.
Like the Jones polynomial, you compute it by undoing crossings one at a time with a fixed rule.
Setting the two variables to special values gives back older invariants.
So one calculation gives both the Alexander and the Jones polynomial at once.
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# Annihilator method

## Definition

Let $D = d/dt$ be the [[differentiation operator|BB15T3J891]], and let $L = a_n D^n + \dots + a_1 D + a_0$ with $a_0, \dots, a_n \in \C$ and $a_n \ne 0$.
Let $f$ be a function on $\R$, and let $A(D) = b_m D^m + \dots + b_0$ be a nonzero polynomial in $D$ with constant coefficients.
If $A(D) f = 0$, then $A$ is an *annihilator* of $f$.
The *annihilator method* for $L y = f$ consists of these steps.

1. Find an annihilator $A$ of $f$.
2. Solve the homogeneous equation $A(D) L(D) y = 0$ by the roots of the polynomial $A(r) L(r)$. A root $r$ of multiplicity $k$ gives $e^{rt}, t e^{rt}, \dots, t^{k-1} e^{rt}$.
3. Remove from this solution the terms that already solve $L y = 0$, and keep the remaining terms with unknown coefficients as a trial function $y_p$.
4. Substitute $y_p$ into $L y = f$ and solve the resulting linear equations for the coefficients.

## Remarks

The functions with an annihilator are exactly the finite sums of terms $t^k e^{\alpha t}$, $t^k e^{\alpha t} \cos \beta t$ and $t^k e^{\alpha t} \sin \beta t$.
For instance, $t^k e^{\alpha t}$ is annihilated by $(D - \alpha)^{k+1}$, and $e^{\alpha t} \cos \beta t$ by $(D - \alpha)^2 + \beta^2$.
Step 4 always has a solution, which justifies the trial functions of the [[method of undetermined coefficients|0XRB469Z32]].

Example: $y'' - y = e^t$.
Here $L = D^2 - 1$ and $A = D - 1$, and $A L = (D - 1)^2 (D + 1)$ gives the solutions $e^t, t e^t, e^{-t}$.
Removing $e^t$ and $e^{-t}$, which solve $L y = 0$, leaves $y_p = c\, t e^t$.
Then $y_p'' - y_p = c (t + 2) e^t - c t e^t = 2c \, e^t$, so $c = 1/2$ and $y_p = \frac{1}{2} t e^t$.

Non-example: $f(t) = 1/t$ on $(0, \infty)$ is not a finite sum of the terms above, so it has no annihilator.
So the method does not apply to $y'' - y = 1/t$, and [[variation of parameters|0Y6TH5PQVS]] is used instead.
</document>

<document source="https://tamnd.github.io/math2/node/QBTK36P203.md" tag="QBTK36P203" kind="definition">
# Boundary layer

## Definition

Let $a < b$ and let $(y_\varepsilon)_{0 < \varepsilon < \varepsilon_0}$ be a family of functions $[a, b] \to \R^n$, for example the solutions of a [[singularly perturbed|G6J21TGRMM]] boundary value problem.
Assume there is a continuous function $y_0 \colon (a, b] \to \R^n$, the *outer limit*, such that $y_\varepsilon \to y_0$ uniformly on $[a + \eta, b]$ for every $\eta \in (0, b - a)$.
The family has a *boundary layer* at $a$ if the convergence is not uniform on $(a, b]$, or $y_\varepsilon(a)$ does not tend to $\lim_{x \to a^+} y_0(x)$.
Let $\delta(\varepsilon) > 0$ with $\delta(\varepsilon) \to 0$, and consider the rescaled functions $X \mapsto y_\varepsilon(a + \delta(\varepsilon) X)$.
If they converge to a nonconstant function, uniformly on compact subsets of $[0, \infty)$, then $\delta$ is a *thickness* of the layer.
A boundary layer at $b$ is defined in the same way.

## Remarks

Example: $y_\varepsilon(x) = (1 - e^{-x/\varepsilon}) / (1 - e^{-1/\varepsilon})$ on $[0, 1]$, the solution of $\varepsilon y'' + y' = 0$, $y(0) = 0$, $y(1) = 1$.
The outer limit is $y_0 = 1$, and on $[\eta, 1]$ the error is at most $e^{-\eta/\varepsilon} / (1 - e^{-1/\varepsilon}) \to 0$.
But $y_\varepsilon(0) = 0$, so there is a boundary layer at $0$.
With $\delta(\varepsilon) = \varepsilon$, $y_\varepsilon(\varepsilon X) \to 1 - e^{-X}$, so the thickness is $\varepsilon$.

Non-example: $y_\varepsilon(x) = \varepsilon + (1 - \varepsilon) e^{-x}$ converges uniformly on all of $[0, 1]$, so it has no boundary layer.

Layers can also sit inside the interval; these are called *interior layers*, or *shocks* when the solution jumps.

## Explanations

### L1 human

A boundary layer is where the simple approximation of the solution stops working.
It is a strip near the boundary whose width shrinks with the small parameter.
</document>

<document source="https://tamnd.github.io/math2/node/QBXW04EC4V.md" tag="QBXW04EC4V" kind="definition">
# Hilbertian field

## Definition

Let $K$ be a [[field|04C5ZBG1WT]], and let $K[T, X]$ be the [[polynomial ring|07GNX8GCYP]] in two variables over $K$.
Consider [[irreducible polynomials|03FKX7Z5DC]] $f_1, \dots, f_m \in K[T, X]$, each of positive degree in $X$ and separable in $X$, and a nonzero $g \in K[T]$.
The field $K$ is *Hilbertian* if, for all such data, there is $a \in K$ such that:

1. $g(a) \ne 0$;
2. for each $i$, the polynomial $f_i(a, X) \in K[X]$ is irreducible over $K$ and has the same degree in $X$ as $f_i$.

## Remarks

If $K$ is Hilbertian, then for such data there are infinitely many $a \in K$ with these properties, since one can add to $g$ factors $T - a$ for the values already found.
A Hilbertian field is infinite, and the same condition then holds with $T$ replaced by several variables $T_1, \dots, T_r$.
In characteristic $0$, the separability condition is automatic, and $K$ is Hilbertian exactly when $K$ itself is not a thin subset of the affine line over $K$.

Example: $\Q$ is Hilbertian by [[Hilbert's irreducibility theorem|N1V3YEYHNP]], and so is every number field.
For any field $k$, the rational function field $k(t)$ is Hilbertian.

Non-example: $\R$ is not Hilbertian.
The polynomial $f = X^2 - T^2 - 1$ is irreducible in $\R[T, X]$, because $T^2 + 1$ is not a square in $\R(T)$.
But for every $a \in \R$ one has $a^2 + 1 > 0$, so $f(a, X) = (X - \sqrt{a^2 + 1})(X + \sqrt{a^2 + 1})$ factors over $\R$.
An algebraically closed field is not Hilbertian either, because every polynomial in $X$ of degree at least $2$ factors over it.
</document>

<document source="https://tamnd.github.io/math2/node/QC4TP5D6HE.md" tag="QC4TP5D6HE" kind="statement">
# Dixon's identity

## Statement

1. For every nonnegative integer $a$, the [[binomial coefficients|0GBZM0MT0F]] satisfy
$$\sum_{k = -a}^{a} (-1)^k \binom{2a}{a + k}^3 = \frac{(3a)!}{(a!)^3}.$$
2. For nonnegative integers $a, b, c$, with $k$ running over all integers and $\binom{n}{j}$ read as $0$ for $j < 0$ or $j > n$,
$$\sum_{k} (-1)^k \binom{a + b}{a + k} \binom{b + c}{b + k} \binom{c + a}{c + k} = \frac{(a + b + c)!}{a!\, b!\, c!}.$$
3. Let $a, b, c$ be complex numbers such that $1 + a - b$, $1 + a - c$ and $1 + \frac{a}{2}$ are not in $\set{0, -1, -2, \dots}$ and $\operatorname{Re}\left( \frac{a}{2} - b - c \right) > -1$.
Then, with $\Gamma$ the [[gamma function|0H5SYBSED9]] and ${}_3F_2$ the [[generalized hypergeometric series|VDBAW9PN88]],
$${}_3F_2\left( \begin{matrix} a,\ b,\ c \\ 1 + a - b,\ 1 + a - c \end{matrix} ; 1 \right) = \frac{\Gamma(1 + \frac{a}{2})\, \Gamma(1 + a - b)\, \Gamma(1 + a - c)\, \Gamma(1 + \frac{a}{2} - b - c)}{\Gamma(1 + a)\, \Gamma(1 + \frac{a}{2} - b)\, \Gamma(1 + \frac{a}{2} - c)\, \Gamma(1 + a - b - c)}.$$

## Remarks

Part 1 is the case $a = b = c$ of part 2.
The ${}_3F_2$ in part 3 is *well-poised*: each upper parameter plus the matching lower parameter gives $1 + a$.
Part 3 is the case $d = \frac{a}{2}$ of the summation of a well-poised ${}_5F_4$.
That series has upper parameters $a, 1 + \frac{a}{2}, b, c, d$ and lower parameters $\frac{a}{2}, 1 + a - b, 1 + a - c, 1 + a - d$.
For $d = \frac{a}{2}$, two upper and two lower parameters cancel.
The condition on $\operatorname{Re}\left( \frac{a}{2} - b - c \right)$ is exactly the condition for the series at $1$ to converge.
For $a = 1$ in part 1, the sum is $-1 + 8 - 1 = 6 = 3!/(1!)^3$.

## Explanations

### L1 human

Take row $2a$ of Pascal's triangle, cube each entry, and add the cubes with alternating signs.
Dixon's identity says the result is the multinomial coefficient $\frac{(3a)!}{(a!)^3}$, the number of ways to split $3a$ objects into three groups of $a$.
For $a = 1$ the row is $1, 2, 1$, and $-1 + 8 - 1 = 6$.

### L2 human

The binomial forms are the terminating cases of the hypergeometric form in part 3.
Part 3 is one of the classical closed-form evaluations of ${}_3F_2(1)$, next to the Pfaff–Saalschütz theorem for balanced series and Watson's and Whipple's sums.
Dixon's identity also has a short proof by the Wilf–Zeilberger method of certificate proofs.
</document>

<document source="https://tamnd.github.io/math2/node/QC5MB5WM8F.md" tag="QC5MB5WM8F" kind="statement">
# Banach–Mazur theorem

## Statement

Let $X$ be a [[separable Banach space|AH45GS6RCR]] over $\R$.
Let $C([0,1])$ be the Banach space of continuous real functions on $[0,1]$ with the [[supremum norm|0PARCKE8WP]].
Then there is a [[linear isometry|07SJXTWVQW]] $T \colon X \to C([0,1])$.
In particular, $X$ is isometrically isomorphic to the closed subspace $T(X)$ of $C([0,1])$.

## Remarks

The idea of the proof: let $B$ be the closed unit ball of the dual space $X^*$ with the weak-* topology.
Since $X$ is separable, $B$ is a compact metrizable space, so it is a continuous image of the Cantor set $\Delta$ under some map $\varphi$.
The map $x \mapsto (t \mapsto \varphi(t)(x))$ is a linear isometry of $X$ into $C(\Delta)$, by the Hahn–Banach theorem.
Each function in $C(\Delta)$ extends to $[0,1]$ by linear interpolation on the open intervals removed from $[0,1]$, and this extension is a linear isometry from $C(\Delta)$ into $C([0,1])$.

For example, the Hilbert space $\ell^2$ and every space $L^p([0,1])$ with $1 \le p < \infty$ are isometric to subspaces of $C([0,1])$.
Separability is necessary: $C([0,1])$ is separable, so each of its subspaces is separable, and the non-separable space $\ell^\infty$ does not embed in it.

## Explanations

### L1 human

The space of continuous functions on an interval is big enough to hold a copy of every separable Banach space.
The copy keeps all distances exactly.
So questions about separable Banach spaces can be asked inside one single space.

### L2 human

The theorem says that $C([0,1])$ is universal for the class of separable Banach spaces.
It is the linear analogue of the fact that every compact metric space is a continuous image of the Cantor set, and the proof uses exactly that fact.
The weak-* compactness of the dual ball comes from the Banach–Alaoglu theorem.
Universality makes $C([0,1])$ itself very complicated: every property that fails in some separable space fails in some subspace of $C([0,1])$.
</document>

<document source="https://tamnd.github.io/math2/node/QC9M3Q8B8Q.md" tag="QC9M3Q8B8Q" kind="concept">
# Lindenbaum–Tarski algebra

## Summary

The Lindenbaum–Tarski algebra of a logical theory is the set of its formulas taken up to provable equivalence, with the operations induced by the connectives.
For a theory of classical [[propositional logic|09K9MWD2QQ]] it is a [[Boolean algebra|0XADJKM699]].
It turns statements about provability into statements about an algebra, and it is the basic tool in algebraic proofs of completeness theorems.

## Explanations

### L1 human

Two formulas such as $p \wedge q$ and $q \wedge p$ look different but say the same thing.
The Lindenbaum–Tarski algebra glues together all formulas that the theory proves to be equivalent.
Each glued class becomes one element, and "and", "or" and "not" act on these classes.
With one propositional variable $p$ and no extra axioms, there are exactly four classes, those of $p$, $\neg p$, $p \vee \neg p$ and $p \wedge \neg p$.
</document>

<document source="https://tamnd.github.io/math2/node/QCBYAWBYWM.md" tag="QCBYAWBYWM" kind="definition">
# Separably generated extension

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
A *separating transcendence basis* of $L/K$ is a [[transcendence basis|E45PA267S4]] $B$ of $L/K$ such that $L$ is a [[separable|0N019F6RE9]] algebraic extension of $K(B)$.
The extension $L/K$ is *separably generated* if it has a separating transcendence basis.

## Remarks

An algebraic extension is separably generated if and only if it is separable, since then $B = \emptyset$.
In characteristic $0$ every extension is separably generated.
Every finitely generated extension of a [[perfect field|VWBVKT794W]] is separably generated.
Example: $K(t)/K$ is separably generated, with separating transcendence basis $\set{t}$.
Non-example: let $K = \F_p(t)$ and $L = K(t^{1/p})$; then $L/K$ is algebraic and [[purely inseparable|966GNVGB8A]] of degree $p$, so its only transcendence basis is $\emptyset$ and $L/K$ is not separable.
Not every transcendence basis of a separably generated extension is separating: $\set{t^p}$ is a transcendence basis of $\F_p(t)/\F_p$, but $\F_p(t)/\F_p(t^p)$ is purely inseparable.
</document>

<document source="https://tamnd.github.io/math2/node/QCCD2JNFBT.md" tag="QCCD2JNFBT" kind="statement">
# Hopf–Rinow theorem

## Statement

Let $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]] with [[Riemannian distance|B4QR306D8G]] $d$.
The following are equivalent:

1. the metric space $(M, d)$ is [[complete|0BY52JW6NK]];
2. every [[closed|0ZCK8DQYN1]] and [[bounded|09QSZVVM26]] subset of $M$ is [[compact|0TVJZYB6SH]];
3. there is a point $p \in M$ such that the [[exponential map|Y7Z0AGY1A7]] $\exp_p$ is defined on all of $T_pM$;
4. for every point $p \in M$, $\exp_p$ is defined on all of $T_pM$;
5. $M$ is geodesically complete: every [[geodesic|NSYMQMHPDN]] $\gamma$ can be extended to a geodesic defined on all of $\R$.

If these conditions hold, then any two points $p, q \in M$ are joined by a geodesic of length $d(p, q)$.

## Remarks

The key step is to show that if $\exp_p$ is defined on all of $T_pM$, then every $q$ is reached by a minimizing geodesic from $p$.
Choose a point $x$ nearest to $q$ on a small sphere around $p$, and follow the geodesic from $p$ through $x$.
One shows that it decreases the distance to $q$ at unit rate until it reaches $q$.
From this, bounded closed sets are closed subsets of images of compact balls under $\exp_p$, which gives the other conditions.
Example: $\R^n$ and every compact Riemannian manifold satisfy the conditions.
The open disc $\set{x \in \R^2 : \abs{x} < 1}$ with the Euclidean metric fails them: a radial geodesic leaves the disc in finite time, and the sequence $(1 - 1/k, 0)$ is Cauchy with no limit.
The last conclusion can hold without completeness, as in the open disc, so it is not equivalent to the conditions.

## Explanations

### L1 human

For a curved space, several natural meanings of "having no holes or missing edges" turn out to be the same.
One can never fall off the space by walking straight; Cauchy sequences converge; closed bounded regions are compact.
And when this holds, any two points are joined by a shortest path that is a geodesic.

### L2 human

The theorem is used constantly in global Riemannian geometry.
Completeness is a standing hypothesis in the theorems of Myers, Cartan–Hadamard and others, and one can check it in whichever form is easiest.
For example, compact manifolds and homogeneous Riemannian manifolds are complete.
It also shows that $\exp_p$ is surjective for complete $M$, which is the starting point for the study of the [[cut locus|X5WKGC5Q6T]].
The theorem fails for pseudo-Riemannian manifolds, where geodesic completeness and connection by geodesics behave differently.
</document>

<document source="https://tamnd.github.io/math2/node/QCDA5QM2BZ.md" tag="QCDA5QM2BZ" kind="problem">
# Kuratowski's closure-complement problem

## Problem

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $A \subseteq X$.
Apply to $A$, again and again and in any order, the two operations of taking the [[closure|0QEAJB91GZ]] and taking the [[complement|0RKF1ZT5JJ]] in $X$.
How many distinct subsets of $X$ can arise in this way, at most?

## Remarks

The problem is solved: at most $14$ distinct sets arise, counting $A$ itself, and the bound $14$ is attained in $\R$.
The [[interior|0RZH6W98HK]] of $A$ is the complement of the closure of the complement of $A$.
With $k$ for closure and $i$ for interior, every set that arises is one of $A, kA, iA, ikA, kiA, kikA, ikiA$ or the complement of one of them.
The key identity is $kikik A = kik A$, which stops the list.

In $\R$, the set
$$A = (0, 1) \cup (1, 2) \cup \set{3} \cup ([4, 5] \cap \Q)$$
gives $14$ distinct sets.
For it, $kA = [0, 2] \cup \set{3} \cup [4, 5]$, $iA = (0, 1) \cup (1, 2)$, $ikA = (0, 2) \cup (4, 5)$, $kiA = [0, 2]$, $kikA = [0, 2] \cup [4, 5]$ and $ikiA = (0, 2)$.
These seven sets are distinct and bounded, and their seven complements are unbounded, so all $14$ are distinct.

## Explanations

### L1 human

Start with a set and keep pressing two buttons: "close it" and "take what is outside".
One might expect to get infinitely many different sets this way.
In fact the process always cycles back after a short while, and one can never get more than $14$ different sets.
A clever subset of the real line, made of intervals, a single point and some rational numbers, shows that all $14$ can really appear.
</document>

<document source="https://tamnd.github.io/math2/node/QCJN05GN64.md" tag="QCJN05GN64" kind="concept">
# Injective hull

## Summary

The injective hull of a module $M$ is the smallest [[injective module|85H7PRHDPZ]] that contains $M$.
Every module has one, and it is unique up to an isomorphism that fixes $M$.
Injective hulls are the building blocks of minimal injective resolutions.
The injective hull of the residue field of a local ring is the dualizing object of [[Matlis duality|2V9BR5XX6A]].

## Explanations

### L1 human

An injective module is one into which maps can always be extended.
The injective hull is the most economical way to enlarge $M$ into such a module.
Every nonzero piece of the hull touches $M$ itself, so nothing superfluous is added.
For example, the injective hull of the integers $\Z$, as an abelian group, is the group of rational numbers $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/QCYZRQX7QG.md" tag="QCYZRQX7QG" kind="statement">
# Generic smoothness

## Statement

Let $k$ be an [[algebraically closed field|02SJG1HC32]] of characteristic $0$.
Let $f \colon X \to Y$ be a morphism of [[varieties|HV6YBS0YDH]] over $k$, and assume that $X$ is nonsingular, that is, a [[regular scheme|ESPG0BJERZ]].
Then there is a nonempty open subset $V \subseteq Y$ such that the restriction $f^{-1}(V) \to V$ is a [[smooth morphism|EXKGG9X6J0]].

## Remarks

The proof reduces to the case where $f$ is dominant and then compares tangent spaces.
In characteristic $0$, the field extension $k(Y) \subseteq k(X)$ is separable, so the map on differentials is generically surjective.
Then $\Omega_{X/Y}$ is locally free of the right rank over a dense open set, which gives smoothness there.
For example, the map $\mathbb{A}^2_k \to \mathbb{A}^1_k$, $(x, y) \mapsto x^2 + y^2$, is smooth over $V = \mathbb{A}^1_k \setminus \set{0}$.
Its fibre over $0$ is the pair of lines $x = \pm i y$, which is singular at the origin.
The characteristic hypothesis is necessary.
In characteristic $p > 0$, the Frobenius map $\mathbb{A}^1_k \to \mathbb{A}^1_k$, $t \mapsto t^p$, is a morphism of smooth curves.
Its fibre over $a$ is $\operatorname{Spec} k[t]/(t^p - a) = \operatorname{Spec} k[t]/((t - b)^p)$ with $b^p = a$, a non-reduced point, so it is smooth over no nonempty open set.

## Explanations

### L1 human

Take a smooth variety and a map from it to another variety.
Some fibres of the map may be singular, but generic smoothness says that this only happens over a small closed set.
Over almost every point of the target, the fibre is a smooth variety.
In positive characteristic this can fail, because a map like $t \mapsto t^p$ has only thickened points as fibres.

### L2 human

Generic smoothness is the algebraic counterpart of Sard's theorem on critical values of smooth maps.
It is the main tool behind Bertini's theorem: a general hyperplane section of a smooth projective variety in characteristic $0$ is smooth.
It also shows that, in characteristic $0$, in a family with smooth total space over a curve, only finitely many fibres are singular.
Its proof uses that every finitely generated field extension in characteristic $0$ is separably generated.
</document>

<document source="https://tamnd.github.io/math2/node/QD02B3DEMR.md" tag="QD02B3DEMR" kind="concept">
# Martin boundary

## Summary

The Martin boundary of a domain is an ideal boundary.
It is built from the limits of the [[Martin kernel|1DBC9FYF7M]] as the pole tends to the edge of the domain.
Each of its points is a positive harmonic function, and every positive harmonic function on the domain is an integral of these over the boundary.
R. S. Martin introduced it in 1941 to extend the Poisson representation from the ball to arbitrary domains.
For smooth or Lipschitz domains it coincides with the Euclidean boundary, but for domains with slits or cusps it can be quite different.
In probability it is the boundary seen by Brownian motion and by Markov chains through Doob's $h$-transforms.

## Explanations

### L1 human

The geometric boundary of a domain may not reflect how harmonic functions see it.
For a disc with a slit cut along a radius, the two sides of the slit behave like different boundary points.
The Martin boundary separates such points by asking which positive harmonic functions arise as limits of normalised Green's functions.
For the unit ball, the Martin boundary is just the unit sphere.
</document>

<document source="https://tamnd.github.io/math2/node/QD1MCREE5A.md" tag="QD1MCREE5A" kind="concept">
# Integral transforms of tempered distributions

## Summary

Many [[integral transforms|YCK709QB06]] extend from functions to [[tempered distributions|8TK1JNG87X]] by duality.
One moves the transform onto the [[Schwartz test functions|CG9JEATW3Q]].
The transform $Tu$ of a distribution $u$ acts on a test function $\varphi$ as $u$ acts on $T^t\varphi$.
Here $T^t$ is the transform with the transposed kernel.
This works when $T^t$ maps the Schwartz space continuously into itself.
The model case is the [[Fourier transform of tempered distributions|5PVSQXCTCT]]; the [[Laplace transform of distributions|6Z3XJVYT2Q]] uses a related construction.

## Explanations

### L1 human

Objects like the Dirac delta or the constant function $1$ on $\R$ have no Fourier transform as an ordinary integral.
The trick is never to integrate them directly.
Instead one asks how the transform of the object would act on nice test functions, and uses the rule $\int (Tf)\varphi = \int f\, (T^t\varphi)$, which holds for ordinary functions.
The right-hand side makes sense for distributions, so it is taken as the definition.
In this way the delta at $0$ gets the Fourier transform $1$.
</document>

<document source="https://tamnd.github.io/math2/node/QD23PJR6RP.md" tag="QD23PJR6RP" kind="statement">
# Lidskii's theorem

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]], and let $T$ be a [[trace class operator|Z2CP530WDE]] on $H$ with trace $\operatorname{tr} T$ and trace norm $\norm{T}_1$.
Let $(\lambda_n)$ be the nonzero [[eigenvalues|0BQAA2BK3Q]] of $T$, each repeated according to its algebraic multiplicity $\dim \bigcup_{k \ge 1} \ker (T - \lambda_n I)^k$.
Then $\sum_n \abs{\lambda_n} \le \norm{T}_1$, and
$$\operatorname{tr} T = \sum_n \lambda_n.$$

## Remarks

A trace class operator is [[compact|M2NEP1K5AE]], so its nonzero eigenvalues form a finite or null sequence, and each has finite algebraic multiplicity.
The inequality $\sum \abs{\lambda_n} \le \norm{T}_1$ is a consequence of Weyl's inequality between eigenvalues and singular values.
For the identity, one writes $T$ in upper triangular form with respect to a Schur basis of the span of its generalized eigenvectors.
This reduces the theorem to the case of a quasinilpotent trace class operator, whose trace must be shown to be $0$.
This last step is the hard part; one proof uses the Fredholm determinant $\det(I + zT)$, an entire function of $z$ whose zeros are the numbers $-1/\lambda_n$.
For a matrix, the theorem is the fact that the [[trace|0XC1RDBY7A]] is the sum of the eigenvalues.
Example: a trace class operator may have no nonzero eigenvalue at all; then the theorem says its trace is $0$.
The hypothesis matters: for a merely compact operator, the sum $\sum_i \ip{T e_i}{e_i}$ can diverge or depend on the basis.

## Explanations

### L1 human

For a square matrix, adding the diagonal entries gives the same number as adding the eigenvalues.
Lidskii's theorem says that this remains true for trace class operators on infinite-dimensional spaces.
This is surprising, since such an operator may have very few eigenvalues, or none.

### L2 human

The theorem links two different definitions of the trace: a basis-dependent sum of diagonal entries and a spectral sum.
It was proved by Lidskii in 1959; the case of self-adjoint or normal operators is much easier and follows from the spectral theorem.
It is used in the theory of Fredholm determinants, in random matrix theory, and in the study of trace formulas.
</document>

<document source="https://tamnd.github.io/math2/node/QD9H1WHHSW.md" tag="QD9H1WHHSW" kind="definition">
# Preservation of limits

## Definition

Let $G \colon \mathcal{C} \to \mathcal{D}$ be a [[functor|28NQ0KRHD3]], and let $F \colon J \to \mathcal{C}$ be a [[diagram|HJK75HDWFE]].
For a [[cone|7TPBQPPN97]] $(L, (\lambda_j)_{j \in J})$ over $F$, the image $(G L, (G \lambda_j)_{j \in J})$ is a cone over $G \circ F$.
$G$ *preserves the limit* of $F$ if for every [[limit|CHS5AFJGWE]] cone $(L, \lambda)$ of $F$, the cone $(G L, G \lambda)$ is a limit cone of $G \circ F$.
$G$ *preserves limits* if it preserves the limit of every diagram $F \colon J \to \mathcal{C}$ with $J$ a small category.

## Remarks

A functor that preserves limits is also called *continuous*.
Dually, $G$ preserves colimits if it sends colimit cones to colimit cones.
Since limits are unique up to isomorphism, it suffices to check one limit cone of each diagram.

Example: the forgetful functor $U$ from groups to sets preserves products.
The product of groups $H \times K$ has underlying set $U H \times U K$, and the projections are the projections of sets.

Non-example: the functor $X \mapsto X \sqcup \set{*}$ from sets to sets does not preserve the terminal object.
The terminal object is a one-point set, which is the limit of the empty diagram, and its image has two points.

A related notion goes the other way.
$G$ *reflects* limits if every cone over $F$ whose image under $G$ is a limit cone is itself a limit cone.
</document>

<document source="https://tamnd.github.io/math2/node/QDCD33CS21.md" tag="QDCD33CS21" kind="concept">
# Topological index

## Summary

The topological index of an elliptic operator on a closed manifold is an integer built only from topological data.
These data are the K-theory class of the principal symbol and the characteristic classes of the manifold.
Atiyah and Singer defined it by pushing the symbol class forward along an embedding of the manifold into a Euclidean space.
Their index theorem states that it equals the [[analytic index|44GBFZ16SG]].
In cohomology it is an integral of the Chern character of the symbol times the [[Todd class|2YDB5089FR]].

## Explanations

### L1 human

The analytic index is computed by solving equations, which is hard.
The topological index is computed from the symbol alone, the top-order part of the operator, read as a family of matrices over the cotangent bundle.
A small change of the operator does not change the topology of its symbol, so the topological index cannot jump.
For the operator $d + d^*$ from even to odd forms, the topological index is the Euler characteristic of the manifold.
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# Standard probability space

## Summary

A standard probability space is a [[probability space|0NFPJVJJHX]] that looks like a simple model after null sets are removed.
The model is an interval with Lebesgue measure, a countable set of atoms, or a combination of both.
The theory goes back to von Neumann in 1932 and was shaped by Rokhlin, who called these spaces Lebesgue spaces.
Most probability spaces of analysis are standard, and on standard spaces conditional probabilities and disintegrations behave well.
For this reason ergodic theory and probability often assume their spaces to be standard.

## Explanations

### L1 human

A standard probability space is one that, after ignoring events of probability zero, looks like a simple model.
The model is a piece of the unit interval where probability is length, plus some single points that carry positive probability.
Picking a random point of a square, or flipping infinitely many coins, gives a standard space.
Strange spaces built with the axiom of choice can fail to be standard, and they are the exception.
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# Frobenius theorem on real division algebras

## Statement

Let $D$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over the field $\R$ of [[real numbers|08J2M1JYAG]].
Suppose $D$ is a [[division ring|0WZXK05ME7]]: it has a unit $1 \ne 0$ and every nonzero element has a two-sided inverse.
Then $D$ is isomorphic, as an $\R$-algebra, to exactly one of $\R$, the [[complex numbers|09FTAET5Z4]] $\C$, and the [[quaternions|0GE4DVQKR8]] $\mathbb{H}$.

## Remarks

Proof idea: each $a \in D$ lies in the commutative subalgebra $\R[a]$, which is a finite field extension of $\R$, hence $\R$ or $\C$.
Using this, one shows that the set $V$ of elements $u$ with $u^2 \in \R$ and $u^2 \le 0$ is a real subspace with $D = \R \oplus V$.
The rule $\ip{u}{w} = -(uw + wu)/2$ is a positive definite inner product on $V$.
An orthonormal basis of $V$ satisfies the multiplication rules of $\C$ or $\mathbb{H}$, and a fourth basis vector would lead to a contradiction.
The dimensions are $1$, $2$ and $4$.
Associativity is needed: the octonions form an $8$-dimensional real algebra, not associative, in which every nonzero element has an inverse.
Finite dimension is needed: the field $\R(t)$ of rational functions is a commutative division algebra over $\R$ of infinite dimension.

## Explanations

### L1 human

Consider number systems of finite dimension over the reals, where one can divide by every nonzero number.
If the order of brackets in a product does not matter, there are only three of them.
They are the real numbers, the complex numbers and Hamilton's quaternions.
The quaternions are the only one of the three in which $ab$ and $ba$ can differ.

### L2 human

In the language of the [[Brauer group|BEW48TA9WZ]], the only central division algebras over $\R$ are $\R$ and $\mathbb{H}$.
So the Brauer group of $\R$ has order $2$.
This agrees with [[Galois cohomology|9MPCZ3EAJM]]: $\C$ is the [[algebraic closure|0X6AB5P3G1]] of $\R$, and $H^2(\Gal(\C/\R), \C^\times) \cong \Z/2\Z$.
Without associativity, a finite-dimensional real division algebra has dimension $1$, $2$, $4$ or $8$, by work of Kervaire and of Bott and Milnor.
Hurwitz's theorem shows that the normed ones are $\R$, $\C$, $\mathbb{H}$ and the octonions.
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# Ring of sets

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]].
A *ring of sets* on $X$ is a nonempty family $\mathcal{R}$ of subsets of $X$ such that, for all $A, B \in \mathcal{R}$:

1. the [[union|07KVB4JSC7]] $A \cup B$ is in $\mathcal{R}$;
2. the difference $A \setminus B$ is in $\mathcal{R}$.

## Remarks

The finite subsets of $\N$ form a ring of sets: the union and the difference of two finite sets are finite.
This ring does not contain $\N$, so it is not an algebra of sets.
The bounded intervals of $\R$ do not form a ring of sets, since clause 2 fails: $[0, 3] \setminus [1, 2] = [0, 1) \cup (2, 3]$ is not an interval.
A ring contains $\varnothing = A \setminus A$ and the [[intersection|0HBRW4E6G1]] $A \cap B = A \setminus (A \setminus B)$.
It also contains the symmetric difference $A \mathbin{\triangle} B = (A \setminus B) \cup (B \setminus A)$.
With $\triangle$ as addition and $\cap$ as multiplication, a ring of sets is a commutative ring in the sense of algebra, possibly without identity.
Every ring of sets is a [[semiring of sets|PYH82FMXQ2]].
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# Voronoi diagram

## Summary

The Voronoi diagram of a finite set of points, the sites, divides space into cells, one for each site.
The cell of a site consists of the points that are at least as close to it as to any other site.
Voronoi diagrams are a basic structure of computational geometry, with uses in nearest-neighbour search, mesh generation and the geometry of numbers.

## Explanations

### L1 human

Imagine a few post offices in a town, and assign every house to its nearest post office.
The regions that you get are the cells of the Voronoi diagram.
For two sites the dividing line is the perpendicular bisector of the segment between them.
For more sites each cell is a convex polygon, possibly unbounded, bounded by pieces of such bisectors.
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# Hammerstein integral equation

## Definition

Let $\Omega$ be a measurable subset of $\R^m$, for example a closed interval $[a, b]$.
Let $k \colon \Omega \times \Omega \to \R$ be a [[kernel|GSMBXT7FPK]], let $g \colon \Omega \times \R \to \R$ be a function, and let $f \colon \Omega \to \R$ be given.
A *Hammerstein integral equation* for an unknown function $\varphi \colon \Omega \to \R$ is
$$\varphi(x) = f(x) + \int_\Omega k(x, t)\, g\bigl(t, \varphi(t)\bigr)\, dt, \quad x \in \Omega.$$
With the linear operator $(K v)(x) = \int_\Omega k(x, t)\, v(t)\, dt$ and the [[Nemytskii operator|W8G3AZ0DNG]] $(N_g \varphi)(t) = g(t, \varphi(t))$, it reads $\varphi = f + K N_g \varphi$.

## Remarks

It is the [[Urysohn equation|WZ4Z26XSVC]] with the product kernel $K(x, t, u) = k(x, t)\, g(t, u)$.
Many authors write it as $\varphi + K N_g \varphi = f$, which only changes the sign of $g$.

Example: on $[0, 1]$, take $f = 0$, $k(x, t) = x t$ and $g(t, u) = u^2$.
Try $\varphi(x) = c x$; the right side is $x \int_0^1 t \cdot c^2 t^2\, dt = \frac{c^2}{4} x$, so $c = c^2/4$, that is $c = 0$ or $c = 4$.
So $\varphi = 0$ and $\varphi(x) = 4x$ are two solutions, and the equation has no uniqueness.

Non-example: $\varphi(x) = \int_0^1 \sin\bigl(x\, \varphi(t)\bigr)\, dt$ is a Urysohn equation but not a Hammerstein one with a single product kernel, because $\sin(xu)$ is not of the form $k(x, t)\, g(t, u)$.
Indeed, the functions $u \mapsto \sin(x u)$ for different $x$ are not all multiples of one function.

Example of origin: on $[0, 1]$ the problem $-u'' = h(x, u)$, $u(0) = u(1) = 0$, with continuous $h$, is equivalent to $u(x) = \int_0^1 G(x, t)\, h(t, u(t))\, dt$, where $G(x, t) = \min(x, t)\bigl(1 - \max(x, t)\bigr)$ is the Green's function.
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# Structure theorem for compactly generated LCA groups

## Statement

Let $G$ be a [[locally compact abelian group|ZR09S9R3AW]].
Assume that $G$ is compactly generated, that is, there is a compact subset $C \subseteq G$ such that $G$ is the smallest subgroup of $G$ containing $C$.
Then there are integers $n, m \ge 0$ and a compact abelian group $K$ such that $G$ is isomorphic, as a topological group, to $\R^n \times \Z^m \times K$.

## Remarks

For a discrete group, compactly generated means finitely generated, and the theorem gives the [[structure theorem for finitely generated abelian groups|0RY0B72XAS]] up to the finite part, which goes into $K$.
Example: $G = \R \times \Z/2\Z$ is compactly generated by $[0, 1] \times \Z/2\Z$, and it already has the form $\R^1 \times \Z^0 \times K$ with $K = \Z/2\Z$.
The hypothesis is needed: the field $\Q_p$ of $p$-adic numbers is locally compact abelian but not compactly generated.
It is not of the form $\R^n \times \Z^m \times K$.
Indeed each element of $\Q_p$ lies in a compact subgroup $p^{-k}\Z_p$, which would force $n = m = 0$, but $\Q_p$ is not compact.
Every locally compact abelian group has an open compactly generated subgroup, so it has an open subgroup isomorphic to $\R^n \times K$ with $K$ compact.

## Explanations

### L1 human

Any locally compact abelian group that is generated by a compact piece is built from three simple kinds of blocks.
There are copies of the real line, copies of the integers, and one compact group.
This reduces many questions about such groups to these three cases.

### L2 human

Some proofs of [[Pontryagin duality|QTVC6F2P9D]] use the theorem: one checks duality on $\R$, $\Z$ and compact groups, and then passes to open subgroups and quotients.
It also shows that a connected locally compact abelian group is isomorphic to $\R^n \times K$ with $K$ compact and connected.
For non-compactly generated groups, such as $\Q_p$ or the adeles, the theorem applies only to open subgroups, and the global structure can be much richer.
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# Van der Corput lemma

## Statement

Let $k \ge 1$ be an integer.
There is a constant $c_k$, depending only on $k$, with the following property.
Let $a < b$ be real numbers, let $\phi \colon [a, b] \to \R$ be a [[smooth function|0V33BNKZ9S]], and assume that $\abs{\phi^{(k)}(x)} \ge 1$ for all $x \in [a, b]$.
If $k = 1$, assume also that $\phi'$ is monotonic on $[a, b]$.
Then, for every $\lambda > 0$,
$$\Big| \int_a^b e^{i\lambda\phi(x)} \, dx \Big| \le c_k \lambda^{-1/k}.$$

## Remarks

The point is that the bound does not depend on $a$, $b$ or $\phi$.
For $k = 1$, write $e^{i\lambda\phi} = (i\lambda\phi')^{-1} \frac{d}{dx} e^{i\lambda\phi}$ and integrate by parts; monotonicity of $\phi'$ makes the integral of $\abs{(1/\phi')'}$ telescope, and one gets $c_1 = 3$.
For $k \ge 2$, use induction on $k$.
On the set where $\abs{\phi^{(k-1)}} \ge \lambda^{-1/k}$, apply the case $k - 1$ after rescaling.
This set misses at most an interval of length about $\lambda^{-1/k}$, where the integrand is bounded by $1$.

Example: for $\phi(x) = x^2/2$ and $k = 2$, the lemma gives $\abs{\int_a^b e^{i\lambda x^2/2} \, dx} \le c_2 \lambda^{-1/2}$ uniformly in $a, b$; the Fresnel integral $\int_{-\infty}^{\infty} e^{i\lambda x^2/2} \, dx = \sqrt{2\pi/\lambda} \, e^{i\pi/4}$ shows that the power $\lambda^{-1/2}$ cannot be improved.

For $k = 1$ the monotonicity of $\phi'$ is needed.
Without it, no bound $c_1\lambda^{-1}$ holds uniformly in $\phi$ and in the length of the interval.

A corollary adds an amplitude: if $\psi$ is smooth on $[a, b]$, then $\abs{\int_a^b e^{i\lambda\phi}\psi} \le c_k\lambda^{-1/k} \big(\abs{\psi(b)} + \int_a^b \abs{\psi'}\big)$.

## Explanations

### L1 human

An integral of $e^{i\lambda\phi}$ is small when the phase $\phi$ changes fast, because the integrand spins around the circle and cancels itself.
The lemma measures this: if some derivative of $\phi$ of order $k$ is at least $1$, the integral is at most a constant times $\lambda^{-1/k}$.
Higher $k$ means a flatter possible phase and a weaker decay.

### L2 human

This is the basic one-dimensional tool for [[oscillatory integrals|D5HZKE8E1A]].
It gives decay rates for Fourier transforms of measures on curves, such as $\abs{\hat\sigma(\xi)} \lesssim \abs{\xi}^{-1/2}$ for arc length on a curve with nonzero curvature.
In higher dimensions there is no equally clean analogue, and one uses stationary phase or sublevel set estimates instead.
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# Green's identities

## Statement

Let $\Omega \subseteq \R^n$ be a bounded open set with $C^1$ boundary $\partial\Omega$, outward unit normal $\nu$ and surface measure $dS$.
For a function $w$ of class $C^1$ near $\partial\Omega$, write $\partial w / \partial \nu = \nabla w \cdot \nu$ for its normal [[derivative|06KWGHDN2J]].
Let $u$ and $v$ be twice continuously differentiable on an open set that contains the closure $\overline{\Omega}$.

1. (First identity.) $\displaystyle \int_\Omega \left( u \, \Delta v + \nabla u \cdot \nabla v \right) dx = \int_{\partial\Omega} u \, \frac{\partial v}{\partial \nu} \, dS.$
2. (Second identity.) $\displaystyle \int_\Omega \left( u \, \Delta v - v \, \Delta u \right) dx = \int_{\partial\Omega} \left( u \, \frac{\partial v}{\partial \nu} - v \, \frac{\partial u}{\partial \nu} \right) dS.$
3. (Third identity.) For every $x \in \Omega$,
$$u(x) = \int_{\partial\Omega} \left( \Phi(y - x) \, \frac{\partial u}{\partial \nu}(y) - u(y) \, \frac{\partial \Phi}{\partial \nu_y}(y - x) \right) dS(y) - \int_\Omega \Phi(y - x) \, \Delta u(y) \, dy,$$
where $\Phi$ is the [[fundamental solution of the Laplacian|BSJ8X2XRN9]], normalised by $-\Delta\Phi = \delta$.

Here $\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $\nabla$ is the [[gradient|0ABK9V9Y64]].

## Remarks

The first identity is the [[divergence theorem|0JX86PSSA9]] applied to the vector field $u \nabla v$, whose divergence is $u \Delta v + \nabla u \cdot \nabla v$.
The second identity follows by writing the first one also with $u$ and $v$ exchanged and subtracting.
The third identity applies the second one with $v(y) = \Phi(y - x)$ on $\Omega$ minus a small ball $B(x, \varepsilon)$, and lets $\varepsilon \to 0$; the small sphere contributes $u(x)$ in the limit.
For the first identity it is enough that $u$ is of class $C^1$.
Example: with $u = 1$, the first identity gives $\int_\Omega \Delta v \, dx = \int_{\partial\Omega} \partial v / \partial \nu \, dS$; so a harmonic function has zero total flux through $\partial\Omega$.
Example: with $u = v$ harmonic, it gives $\int_\Omega \abs{\nabla u}^2 dx = \int_{\partial\Omega} u \, \partial u / \partial \nu \, dS$, which proves uniqueness for the Dirichlet problem.

## Explanations

### L1 human

Green's identities are the integration by parts formulas for the Laplacian.
They trade an integral over a region for an integral over its boundary.
In one dimension the first identity is just $\int_a^b (u v'' + u' v') \, dx = [u v']_a^b$.

### L2 human

These identities are the basic tool for boundary value problems of Laplace and Poisson type.
The first one gives energy estimates and uniqueness of solutions, and it underlies the weak formulation of elliptic equations.
The second one shows that the Laplacian is symmetric for functions that vanish on the boundary, which leads to the symmetry of Green's functions.
The third one is the representation formula: a function is determined inside $\Omega$ by its Laplacian, its boundary values and its normal derivatives.
It leads to the Poisson integral formula once a Green's function is known.
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# Allard regularity theorem

## Statement

Let $1 \le m < n$ be integers and let $p > m$.
Then there are constants $\varepsilon > 0$ and $\gamma \in (0, 1)$, depending only on $n$, $m$ and $p$, with the following property.
Let $\rho > 0$, let $U \supseteq B_\rho(0)$ be an open subset of $\R^n$, and let $V$ be an $m$-dimensional rectifiable [[varifold|N3QA80M44M]] in $U$ with weight measure $\norm{V}$.
Suppose that:

1. the density of $V$ is at least $1$ at $\norm{V}$-almost every point;
2. $V$ has generalized [[mean curvature|PA78NFEPCF]] $H$ in $U$, with $H \in L^p(\norm{V})$;
3. $0$ lies in the support of $\norm{V}$;
4. $\dfrac{\norm{V}(B_\rho(0))}{\omega_m \rho^m} \le 1 + \varepsilon$, where $\omega_m$ is the volume of the unit ball of $\R^m$;
5. $\rho^{1 - m/p} \left( \int_{B_\rho(0)} \abs{H}^p \, d\norm{V} \right)^{1/p} \le \varepsilon$.

Let $\alpha = 1 - m/p$.
Then there are an $m$-dimensional linear subspace $T \subseteq \R^n$ and a $C^{1, \alpha}$ map $u$ from an open subset of $T$ to $T^\perp$ with this property.
The support of $\norm{V}$ inside $B_{\gamma\rho}(0)$ is the graph of $u$ intersected with $B_{\gamma\rho}(0)$.

## Remarks

The theorem is due to William Allard (1972).
Hypothesis 4 says that, at scale $\rho$, $V$ has nearly the area of a single flat $m$-disc; hypothesis 5 says that its mean curvature is small at that scale.
The proof shows that $V$ is close to a plane in a scale-invariant tilt-excess sense.
It then approximates $V$ by the graph of a Lipschitz function and shows that this function is nearly harmonic.
An iteration over smaller and smaller balls then gives a [[Hölder continuous|012PBA0FMD]] tangent plane, and the graph is $C^{1,\alpha}$.
The density bound in hypothesis 4 cannot be dropped.
Two planes crossing at the origin, or one plane with multiplicity $2$, are stationary varifolds with density $2$ at $0$.
Neither is the graph of a single function near $0$.
A main corollary concerns a stationary $V$ with density at least $1$.
The set of points near which its support is a smooth embedded submanifold is open and dense in the support.

## Explanations

### L1 human

A varifold is a very general kind of surface, possibly with folds, crossings and corners.
Suppose such a surface is almost balanced, in the sense of small mean curvature, and looks almost like a single flat sheet in a ball.
Allard's theorem says that it is then a smooth sheet in a smaller ball.
So roughness cannot hide inside a region where the surface has nearly the area of a flat disc.

### L2 human

The theorem is the basic $\varepsilon$-regularity result for stationary and almost stationary varifolds.
It plays the role for varifolds that De Giorgi's regularity theorem plays for perimeter minimizers.
Combined with the monotonicity formula, it gives regularity at every point where the density is close to $1$.
It does not decide what happens at points of higher density, which is where branch points and crossings can occur.
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# Finite difference coefficients

## Definition

Let $m \ge 0$ and $N \ge m + 1$ be integers, and let $s_1, \ldots, s_N$ be distinct real numbers, the *stencil*.
The *finite difference coefficients* for the $m$-th derivative on this stencil are the real numbers $a_1, \ldots, a_N$ that solve
$$\sum_{j=1}^{N} a_j \, s_j^{\,i} = \begin{cases} m! & \text{if } i = m, \\ 0 & \text{if } i \ne m, \end{cases} \qquad i = 0, 1, \ldots, N - 1.$$
The associated formula is $f^{(m)}(x) \approx h^{-m} \sum_{j=1}^{N} a_j f(x + s_j h)$ for a step $h > 0$.

## Remarks

The matrix of the system is a [[Vandermonde matrix|Q3JWBZ7EFY]] in the distinct numbers $s_j$, so the coefficients exist and are unique.
Expanding $f(x + s_j h)$ by [[Taylor's theorem|03QHBVW4TT]] up to order $N - 1$ shows that for $f \in C^N$ the formula has error $O(h^{N - m})$; for symmetric stencils the order is often one higher.
Example: $m = 1$ and the forward stencil $0, 1, 2$ give $a = (-\tfrac32, 2, -\tfrac12)$.
Check: $\sum a_j = 0$, $\sum a_j s_j = 2 - 1 = 1 = 1!$, and $\sum a_j s_j^2 = 2 - 2 = 0$.
So $f'(x) \approx \frac{-3 f(x) + 4 f(x+h) - f(x+2h)}{2h}$, with error $O(h^2)$.
Non-example: for $m = 2$ the weights $(1, -2, 1)$ on the stencil $0, 1, 3$ are not the coefficients, since $\sum a_j s_j = -2 + 3 = 1 \ne 0$; the correct ones there are $(\tfrac23, -1, \tfrac13)$.

## Explanations

### L1 human

The weighted sum of the samples cancels every term of the Taylor expansion except the wanted derivative.
It does so up to the degree that the number of points allows.
More points give more cancellations and a more accurate formula.
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# Scaling rule for the Laplace transform

## Statement

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$, and let $F$ be its [[Laplace transform|05XT5AVJBK]].
Let $c > 0$ be real, and let $s$ be a complex number such that the Laplace integral of $f$ converges at $s / c$.
Then the Laplace integral of $t \mapsto f(c t)$ converges at $s$, and
$$\mathcal{L}\{f(c t)\}(s) = \frac{1}{c} F\!\left(\frac{s}{c}\right).$$

## Remarks

The substitution $\tau = c t$ turns $\int_0^\infty f(c t) e^{-s t} \, dt$ into $\frac{1}{c} \int_0^\infty f(\tau) e^{-(s / c) \tau} \, d\tau$.
Example: from $\mathcal{L}\{\sin t\}(s) = 1 / (s^2 + 1)$ one gets $\mathcal{L}\{\sin \omega t\}(s) = \frac{1}{\omega} \cdot \frac{1}{(s / \omega)^2 + 1} = \frac{\omega}{s^2 + \omega^2}$ for $\omega > 0$.
The condition $c > 0$ is needed: for $c < 0$ the function $f(c t)$ uses values of $f$ at negative times.

## Explanations

### L1 human

Speeding up time by a factor $c$ stretches the transform by the same factor and scales its height by $1/c$.
Fast signals have transforms that are spread out in $s$.

### L2 human

The rule lets one derive a whole family of transforms from one transform, for example all $\sin \omega t$ from $\sin t$.
It also explains how the abscissa of convergence changes: it is multiplied by $c$.
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# Least harmonic majorant

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]], and let $u \colon \Omega \to [-\infty, \infty)$ be a function.
A *least harmonic majorant* of $u$ is a [[harmonic majorant|ZDHPQWWCQW]] $h_0$ of $u$ such that $h_0 \le h$ on $\Omega$ for every harmonic majorant $h$ of $u$.

## Remarks

A least harmonic majorant is unique when it exists, since two of them are each at most the other.
If $\Omega$ is connected and $u$ is [[subharmonic|Z1MHMGFNFW]] with at least one harmonic majorant, then $u$ has a least harmonic majorant.
The *greatest harmonic minorant* of a function $v$ is defined in the same way; it is $-h_0$, where $h_0$ is the least harmonic majorant of $-v$.

Example: on the unit disc $\mathbb{D} \subseteq \C$, the least harmonic majorant of $u(z) = \abs{z}^2$ is the constant $1$.
It is a harmonic majorant.
Let $h$ be any harmonic majorant, and let $\abs{a} < r < 1$.
Since $h \ge r^2$ on the circle of radius $r$, the [[Poisson integral formula|PCZBE7EX01]] on that circle gives $h(a) \ge r^2$.
Letting $r \to 1$ gives $h(a) \ge 1$.

Non-example: the constant $2$ is a harmonic majorant of $\abs{z}^2$ on $\mathbb{D}$, but it is not least, because $1$ is also a harmonic majorant and $2 \le 1$ fails.
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# Taylor term

## Summary

A Taylor term is an idempotent term with one identity in two variables $x, y$ for each argument place.
The identity for a place has $x$ in that place on one side and $y$ in that place on the other side.
Having a Taylor term is one of the weakest nontrivial [[Mal'cev conditions|6R0CA4JWWY]].
For an idempotent variety, having a Taylor term means that no nontrivial algebra in the variety has only projections as term operations.
It is central in the [[algebraic approach to constraint satisfaction|3EH4F5H7FT]].
There, a finite core structure has a polynomial-time problem when its idempotent polymorphisms include a Taylor operation.
Otherwise its problem is NP-complete.

## Explanations

### L1 human

A projection, such as $(x, y) \mapsto x$, simply copies one of its inputs and carries no real information.
A Taylor term is an operation that is provably not a projection: for every input place, some equation forces the output not to depend on that place alone.
For example, a binary operation $x \cdot y$ with $x \cdot x = x$ and $x \cdot y = y \cdot x$ is a Taylor term.
Commutativity rules out both projections, because $(x, y) \mapsto x$ is not commutative.
Algebras with such an operation have enough structure for strong algorithms and structure theorems.
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# Horvitz–Thompson estimator

## Summary

The Horvitz–Thompson estimator estimates the total of a finite population from a random sample in which units may have unequal chances of selection.
Each sampled value is divided by its inclusion probability, the probability that its unit enters the sample.
It is unbiased for every sampling design in which every unit has a positive inclusion probability.
It is the basic estimator of survey sampling and the model for inverse probability weighting.

## Explanations

### L1 human

A unit that had a small chance of being sampled stands for many units like it that were missed.
So its value is weighted up by the inverse of its chance of selection.
For example, if a farm had a chance of $1/10$ of being sampled and it grows $50$ tonnes of wheat, it contributes $500$ tonnes to the estimated total.
Adding these weighted values over the sample gives an estimate of the whole population total that is right on average.
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<document source="https://tamnd.github.io/math2/node/QEEZEDBZ50.md" tag="QEEZEDBZ50" kind="statement">
# Narasimhan lemma

## Statement

Let $\Omega \subseteq \C^n$ be a domain with $C^2$ boundary, and let $p \in \partial\Omega$ be a [[strongly pseudoconvex|Z7RVXBA2KB]] point.
Then there is a [[biholomorphic map|FX92PP3HZR]] $\Phi$ from an open neighbourhood $U$ of $p$ onto an open subset of $\C^n$ with this property.
The set $\Phi(U \cap \Omega)$ is strictly convex.
That is, the hypersurface $\Phi(U \cap \partial\Omega)$ has a defining function whose real Hessian is positive definite on the real tangent space at each of its points.

## Remarks

The proof is a computation with the second-order Taylor expansion of a defining function $\rho$ at $p$.
After a complex-linear change of coordinates, $p = 0$ and $\rho(z) = 2\operatorname{Re}\big(z_n + \sum_{j,k} a_{jk} z_j z_k\big) + \sum_{j,k} \rho_{j\bar k}\, z_j \bar z_k + o(\abs{z}^2)$, where $(\rho_{j \bar k})$ is the [[Levi form|80MTNXWW80]] matrix.
The holomorphic quadratic change of coordinates $z_n' = z_n + \sum_{j,k} a_{jk} z_j z_k$, $z_j' = z_j$ for $j < n$, removes the pure terms $z_j z_k$, so the new real Hessian at $0$ is given by the Levi form matrix.
Strong pseudoconvexity makes it positive definite on the complex tangent space $\set{z_n = 0}$.
The real tangent space at $0$ also contains the direction of $\operatorname{Im} z_n$.
A further change $z_n' = z_n + c z_n^2 + \sum_{j < n} b_j z_j z_n$, with suitable constants $c$ and $b_j$, makes the Hessian positive definite on the whole real tangent space.
Positive definiteness at $p$ then persists at nearby boundary points, by continuity.
Example of use: local holomorphic peak functions at $p$ are obtained from linear functions in the convex coordinates.
This is the first step towards the fact that every boundary point of a strongly pseudoconvex domain is a peak point.
Without strong pseudoconvexity the lemma fails.
If the boundary contains a complex disc, its image under a biholomorphic map is a complex disc in the new boundary, and a strictly convex hypersurface contains no such disc.

## Explanations

### L1 human

Strong pseudoconvexity is a complex form of strict convexity, and it is preserved by holomorphic changes of coordinates, whereas convexity is not.
The lemma says that the two notions agree locally: near each strongly pseudoconvex boundary point, one can bend the coordinates holomorphically so that the domain becomes strictly convex there.
So local problems at such points can be solved with the tools of convex geometry.

### L2 human

The lemma is used to build local [[peak functions|SKE8QTW3YM]] and local holomorphic support functions, and to construct integral kernels, such as the Henkin–Ramírez kernels, near strongly pseudoconvex boundaries.
It does not extend to weakly pseudoconvex points.
Kohn and Nirenberg gave a pseudoconvex domain in $\C^2$ with a boundary point near which no holomorphic coordinate change makes the domain convex.
Globally the lemma also fails: a strongly pseudoconvex domain need not be biholomorphic to a convex domain, for instance because it need not be contractible, while convex domains are.
</document>

<document source="https://tamnd.github.io/math2/node/QEFSJXTCZA.md" tag="QEFSJXTCZA" kind="definition">
# Lommel function

## Definition

Let $\mu, \nu \in \C$ be such that neither $\mu + \nu$ nor $\mu - \nu$ is a negative odd integer, and let $z \in \C \setminus (-\infty, 0]$ with the principal branch of powers.
The *Lommel function* $s_{\mu,\nu}$ is
$$s_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{(-1)^k\, z^{\mu + 1 + 2k}}{\prod_{j=0}^{k} \left( (\mu + 2j + 1)^2 - \nu^2 \right)}.$$

## Remarks

The hypothesis on $\mu \pm \nu$ says exactly that no factor $(\mu + 2j + 1)^2 - \nu^2$ vanishes.
The function $s_{\mu,\nu}$ is a particular solution of
$$z^2 w'' + z w' + (z^2 - \nu^2)\, w = z^{\mu + 1}.$$
Check: the operator $w \mapsto z^2 w'' + z w' - \nu^2 w$ multiplies $z^p$ by $p^2 - \nu^2$, and $z^2 w$ shifts each term up by two powers.
So the coefficients $c_k$ of $z^{\mu + 1 + 2k}$ must satisfy $\left( (\mu + 1)^2 - \nu^2 \right) c_0 = 1$ and $\left( (\mu + 2k + 1)^2 - \nu^2 \right) c_k = -c_{k-1}$, which the series does.
Example: for $\mu = 1$ and $\nu = 0$ the first factor is $4$, so $s_{1,0}$ starts with $\frac{z^2}{4}$.
In fact $s_{1,0}$ is $1 - J_0(z)$, where $J_0$ is the [[Bessel function of the first kind|CZCMRDWDYW]].
A non-example: for $\mu = -2$ and $\nu = 1$ one has $\mu + \nu = -1$, so the factor with $j = 0$ is $(-2 + 1)^2 - 1 = 0$ and $s_{-2,1}$ is not defined by the series.
The second Lommel function $S_{\mu,\nu}$ is obtained from $s_{\mu,\nu}$ by adding a suitable combination of $J_\nu$ and $Y_\nu$ so that it has a simple asymptotic expansion at infinity.
</document>

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# Reductive Lie group

## Definition

Let $G$ be a real or complex [[Lie group|7JBRGCVJRB]], and let $\mathfrak{g}$ be its Lie algebra, with centre $\mathfrak{z}(\mathfrak{g})$ and derived algebra $[\mathfrak{g}, \mathfrak{g}]$.
The group $G$ is a *reductive Lie group* if $\mathfrak{g}$ is a reductive Lie algebra, that is:
$$\mathfrak{g} = \mathfrak{z}(\mathfrak{g}) \oplus [\mathfrak{g}, \mathfrak{g}] \quad \text{and} \quad [\mathfrak{g}, \mathfrak{g}] \text{ is semisimple}.$$

## Remarks

Over a field of characteristic $0$, $\mathfrak{g}$ is reductive exactly when its adjoint representation is completely reducible.

Example: $G = GL_n(\R)$ with $n \ge 1$.
Here $\mathfrak{g} = \mathfrak{gl}_n(\R)$, whose centre is the scalar matrices $\R I_n$ and whose derived algebra is $\mathfrak{sl}_n(\R)$, the trace-zero matrices.
Every matrix $X$ is uniquely $\frac{\tr X}{n} I_n + \left(X - \frac{\tr X}{n} I_n\right)$, so $\mathfrak{g} = \R I_n \oplus \mathfrak{sl}_n(\R)$, and $\mathfrak{sl}_n(\R)$ is $0$ or simple, hence semisimple.

Non-example: the group $B$ of invertible real upper triangular $2 \times 2$ matrices.
Its Lie algebra $\mathfrak{b}$ of upper triangular matrices has centre $\R I_2$ and derived algebra $\mathfrak{n}$ of strictly upper triangular matrices.
Then $\mathfrak{z}(\mathfrak{b}) \oplus \mathfrak{n}$ has dimension $2$, not $3$, and $\mathfrak{n}$ is abelian and nonzero, so not semisimple; both conditions fail.

Many authors add further conditions, for example that $G$ has finitely many connected components, or that $G$ is the group of real points of a reductive algebraic group.
Under the definition above, connected semisimple Lie groups and compact Lie groups are reductive.
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<document source="https://tamnd.github.io/math2/node/QEKVW55VV7.md" tag="QEKVW55VV7" kind="definition">
# Operator theory

## Definition

Let $X$ and $Y$ be [[normed spaces|03J5RAKB30]], and let $D(T)$ be a linear subspace of $X$.
An *operator* from $X$ to $Y$ is a [[linear map|0NRYHJPQG2]] $T \colon D(T) \to Y$.
*Operator theory* is the branch of mathematics that studies operators.
Its main objects are the [[continuous linear maps|03TM135G0N]] between [[Banach spaces|06GZ956657]] and the operators on a [[Hilbert space|HM2HHE3J51]].
Its central questions concern invertibility, the spectrum, adjoints, invariant subspaces and the classification of operators.

## Remarks

Example: the right shift $S(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$ on the space $\ell^2$ of square-summable sequences is an operator.
It preserves norms, so it is bounded with norm $1$.
It is injective but not surjective, since every vector in its image has first coordinate $0$.

Non-example: the map $x \mapsto \norm{x}^2 x$ on $\ell^2$ is not an operator in this sense, since it is not linear.
Such maps belong to nonlinear functional analysis.

On a finite-dimensional space, operator theory reduces to the linear algebra of matrices.

## Explanations

### L1 human

Operator theory is linear algebra for spaces of infinite dimension.
It studies maps that respect sums and scalar multiples, and that act on sequences or functions instead of finite lists of numbers.
</document>

<document source="https://tamnd.github.io/math2/node/QETCC3GZXC.md" tag="QETCC3GZXC" kind="definition">
# Ahlfors theory

## Definition

Ahlfors theory studies a [[meromorphic function|0NSD1TPHR0]] $f$ on $\C$ through the covering surfaces $f(\set{\abs{z} < r})$ over the [[Riemann sphere|0CDMXGPSES]].
These surfaces are measured in the spherical metric $\abs{dw}/(1 + \abs{w}^2)$, for which the whole sphere has area $\pi$.
Its basic quantities, for $r > 0$, are:

1. the *mean sheet number*
$$S(r) = \frac{1}{\pi} \iint_{\abs{z} < r} \left(\frac{\abs{f'(z)}}{1 + \abs{f(z)}^2}\right)^2 dx \, dy,$$
the spherical area of the image counted with multiplicity, divided by the area of the sphere;
2. the *boundary length*
$$L(r) = \int_{\abs{z} = r} \frac{\abs{f'(z)}}{1 + \abs{f(z)}^2} \, \abs{dz},$$
the spherical length of the image of the circle $\abs{z} = r$.

## Remarks

Example: for $f(z) = z$, $S(r) = \frac{1}{\pi}\int_0^r \frac{2\pi t}{(1+t^2)^2} dt = \frac{r^2}{1+r^2}$, which tends to $1$, the sheet number of a map that covers the sphere once.
Here $L(r) = 2\pi r/(1 + r^2)$ tends to $0$, so the boundary becomes negligible.
The Ahlfors–Shimizu characteristic $T_0(r) = \int_0^r S(t)/t \, dt$ differs from the [[Nevanlinna characteristic|YT6Z0GWS0W]] $T(r, f)$ by a bounded function.
The main results compare, for disjoint regions $D_1, \dots, D_q$ of the sphere, the number of times the surface covers each $D_j$ with $S(r)$, with an error proportional to $L(r)$.
For a transcendental meromorphic function on $\C$ the ratio $L(r)/S(r)$ tends to $0$ along a sequence of radii.
This leads to the Ahlfors five islands theorem.
Let $f$ be a transcendental meromorphic function on $\C$, and let $D_1, \dots, D_5$ be Jordan domains on the sphere with pairwise disjoint closures.
Then $f$ maps some bounded simply connected domain in $\C$ conformally onto one of the $D_j$.
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# Rademacher's theorem

## Statement

Let $n, m \ge 1$, let $U \subseteq \R^n$ be an open set, and let $f \colon U \to \R^m$ be [[Lipschitz continuous|0QNVQ4VD9T]].
Then $f$ is [[Fréchet differentiable|08EVNYZF42]] at [[almost every|RARTMA0ZEC]] point of $U$ with respect to [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$.

## Remarks

For $n = 1$ the theorem follows from Lebesgue's theorem: a Lipschitz function of one variable is [[absolutely continuous|ZT6F4C2YHK]], hence of bounded variation, hence differentiable almost everywhere.
In general one works with each coordinate of $f$, so one may take $m = 1$.
Applying the case $n = 1$ on lines and Fubini's theorem, the directional derivative of $f$ in each fixed direction exists almost everywhere.
One then shows that almost everywhere it depends linearly on the direction, first for a countable dense set of directions, and the Lipschitz bound turns this into full differentiability.
For example, $f(x) = \norm{x}$ is Lipschitz on $\R^n$ and differentiable everywhere except at $0$.
Continuity alone is not enough: the Weierstrass function is continuous on $\R$ and differentiable nowhere.

## Explanations

### L1 human

A Lipschitz function is one whose graph is never steeper than some fixed slope.
Such a function may have corners, but the theorem says the corners are rare: at almost every point the function has a tangent plane.

### L2 human

The theorem is the starting point of analysis on rough sets and maps.
It lets one use the derivative of a Lipschitz map in integrals, for example in the area and coarea formulas of geometric measure theory.
It also shows that Lipschitz functions belong to the Sobolev space $W^{1,\infty}$ locally, with weak gradient equal to the almost everywhere gradient.
The one-dimensional case is closely tied to [[absolute continuity|ZT6F4C2YHK]] and to [[Lebesgue's theorem on monotone functions|1Y4N1DDQNR]].
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# Vanishing cycle

## Definition

Let $U$ be an open subset of $\C^{n+1}$ and let $f \colon U \to \C$ be a [[holomorphic function|507C34W0ZT]].
Let $p \in U$ be a nondegenerate [[critical point|0YTCMFB4HC]] of $f$ with $f(p) = 0$.
This means that the partial derivatives of $f$ vanish at $p$ and the Hessian matrix $\left( \partial^2 f / \partial z_j \partial z_k (p) \right)$ is invertible.
By the holomorphic Morse lemma there are holomorphic coordinates $w_0, \dots, w_n$ centred at $p$, on a ball $B$ about $p$, in which $f = w_0^2 + \cdots + w_n^2$.
For small real $t > 0$, the set
$$\Sigma_t = \set{w \in B : w_0, \dots, w_n \in \R, \ w_0^2 + \cdots + w_n^2 = t}$$
is an $n$-dimensional [[sphere|P9CM4GQ2T0]] in the fibre $f^{-1}(t) \cap B$.
The *vanishing cycle* at $p$ is the class $\delta = [\Sigma_t]$ in the [[homology group|M0Q5CFJ7ZH]] $H_n(f^{-1}(t) \cap B; \Z)$, defined up to the sign fixed by an orientation of $\Sigma_t$.

## Remarks

When $B$ is a ball in the coordinates $w$, the class $\delta$ generates the reduced homology $\tilde H_n(f^{-1}(t) \cap B; \Z) \cong \Z$, and $\Sigma_t$ shrinks to the point $p$ as $t \to 0$.
Let $f^{-1}(c)$ be a fibre further away, joined to $t$ by a path $\gamma$ of regular values.
Transporting $\delta$ along $\gamma$ gives a vanishing cycle in $H_n(f^{-1}(c); \Z)$, which depends on the homotopy class of $\gamma$.

Example: $n = 0$ and $f(w) = w^2$.
The fibre $f^{-1}(t)$ is the two points $\pm\sqrt{t}$, and $\delta = [\sqrt{t}] - [-\sqrt{t}]$ in $H_0$.

Example: $n = 1$ and $f(w) = w_0^2 + w_1^2$.
The local fibre is an annulus, and $\delta$ is the class of its core circle.

Non-example: the critical point $0$ of $f(w) = w^3$ is degenerate, since the Hessian is $0$.
Its local fibre is three points, with reduced $H_0$ of rank $2$, so it is not described by one vanishing cycle.
After a small perturbation $w^3 - sw$ with $s \ne 0$, it splits into two nondegenerate critical points, each with its own vanishing cycle.
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# Double-negation translation

## Definition

Fix a first-order language, and read $\neg A$ as $A \to \bot$.
The *Gödel–Gentzen negative translation* is the map $A \mapsto A^N$ on formulas defined by these clauses.

1. $\bot^N = \bot$, and $P^N = \neg \neg P$ for every other atomic formula $P$.
2. $(A \wedge B)^N = A^N \wedge B^N$.
3. $(A \vee B)^N = \neg (\neg A^N \wedge \neg B^N)$.
4. $(A \to B)^N = A^N \to B^N$.
5. $(\forall x\, A)^N = \forall x\, A^N$.
6. $(\exists x\, A)^N = \neg \forall x\, \neg A^N$.

For a set of formulas $\Gamma$, let $\Gamma^N = \set{A^N : A \in \Gamma}$.
Its main property is this: $\Gamma \vdash A$ in classical logic if and only if $\Gamma^N \vdash A^N$ in [[intuitionistic logic|Q11N8X99FK]].
A *double-negation translation* is any map with this property obtained by inserting double [[negations|0XR7A7RSXB]], such as the translations of Kolmogorov and Kuroda.

## Remarks

Example: for the [[law of excluded middle|0RYYK8PG4S]] with $P$ atomic, $(P \vee \neg P)^N = \neg (\neg \neg \neg P \wedge \neg \neg \neg \neg P)$.
Since $\neg \neg \neg P$ is equivalent to $\neg P$ in intuitionistic logic, this is equivalent to $\neg (\neg P \wedge \neg \neg P)$, which is intuitionistically provable.
Non-example: the identity map $A \mapsto A$ is not a double-negation translation, because $P \vee \neg P$ is classically provable but not intuitionistically provable.
For arithmetic, [[Peano arithmetic|0QFT7Z76N5]] proves $A$ if and only if [[Heyting arithmetic|RXHJPVMPJH]] proves $A^N$.
In particular, if HA is consistent, then PA is consistent, because $\bot^N = \bot$.
In arithmetic, atomic formulas are equations, which are decidable in HA, so one may take $P^N = P$.
</document>

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# Blakers–Massey theorem

## Statement

Let $X$ be a [[CW complex|5W95QFHXPR]] that is the union of two subcomplexes $A$ and $B$ whose intersection $C = A \cap B$ is nonempty and connected.
Let $m, n \ge 0$, and assume that the pair $(A, C)$ is $m$-connected and the pair $(B, C)$ is $n$-connected.
Then, for a base point in $C$, the map $\pi_i(A, C) \to \pi_i(X, B)$ of [[relative homotopy groups|ZHHMDZWA7F]] induced by inclusion has the following properties:

1. it is an isomorphism for $i < m + n$;
2. it is surjective for $i = m + n$.

## Remarks

Here a pair $(Y, Z)$ is $k$-connected if every path component of $Y$ meets $Z$ and $\pi_i(Y, Z, z) = 0$ for $1 \le i \le k$ and all $z \in Z$, as for [[n-connected spaces|1A943BWBQT]].
The theorem is also called the homotopy excision theorem.
For homology, excision gives $H_i(A, C) \cong H_i(X, B)$ in all degrees; for homotopy groups excision holds only in the range above.
The proof reduces to the case where $A$ and $B$ are obtained from $C$ by attaching cells.
It then works cell by cell with a general position argument in the cubes $I^i$.
Applied to the two cones $A$ and $B$ that form the suspension of an $(n-1)$-connected complex, it gives the [[Freudenthal suspension theorem|7DM2KSS2DB]].

## Explanations

### L1 human

In homology, gluing two pieces together behaves predictably: what happens near the seam can be cut out.
For homotopy groups this fails in general, and new homotopy classes can appear after gluing.
The theorem says that the failure only starts in high dimensions.
The more connected the pieces are, the higher it starts.

### L2 human

The theorem is the reason homotopy theory becomes homology-like in the stable range.
It implies Freudenthal suspension, and it compares a cofibre sequence with a fibre sequence in a range of degrees.
A version for cubical diagrams of spaces, due to Goodwillie, is a basic tool in the calculus of functors.
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# Goldie ring

## Summary

A right Goldie ring is a [[ring|0B1261B5ED]] with two finiteness conditions.
It contains no infinite direct sum of nonzero right ideals, and it has the ascending chain condition on right [[annihilators|HB4XDKJTW8]].
Every right [[Noetherian ring|0670T59HGE]] is right Goldie, and so is every commutative integral domain.
These are the conditions in [[Goldie's theorem|MDS8655N0Z]], which describes the semiprime right Goldie rings as the rings with a semisimple Artinian [[classical ring of quotients|C5971PGKPH]].

## Explanations

### L1 human

The Goldie conditions are much weaker than the Noetherian condition, but they are strong enough to make the ring behave like a ring with finitely many "directions".
The first condition says that one cannot place infinitely many independent right ideals side by side.
The second condition limits only the special right ideals that arise as annihilators.
For example, an infinite product of copies of a field is not Goldie, because the copies of the field in each coordinate give infinitely many independent ideals.
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# Graph minor

## Definition

Let $G$ and $H$ be finite [[graphs|0JD5C82QMB]].
Let $uv$ be an edge of $G$.
The contraction $G / uv$ is obtained from $G$ by removing $u$ and $v$ and adding a new vertex $w$.
The new vertex $w$ is adjacent to each other vertex that was adjacent to $u$ or to $v$.
The graph $H$ is a minor of $G$ if finitely many edge contractions turn a [[subgraph|0P11X5HKAX]] of $G$ into a graph [[isomorphic|00CT0PTH74]] to $H$.

## Remarks

Equivalently, there are pairwise disjoint nonempty sets $B_x \subseteq V(G)$, one for each vertex $x$ of $H$, with two properties.
Each $B_x$ induces a [[connected|0FFAYNMAJZ]] subgraph of $G$.
For each edge $xy$ of $H$, some edge of $G$ joins $B_x$ to $B_y$.
The sets $B_x$ are called branch sets.
Example: $K_5$ is a minor of the [[Petersen graph|0NR95CTFCZ]].
Take as branch sets the five spokes, each made of an outer vertex and its inner neighbour.
Then the outer $5$-cycle and the inner pentagram join every two branch sets.
Non-example: the triangle $K_3$ is not a minor of any tree, because deleting and contracting edges of a forest always gives a forest.
Every subgraph is a minor, but a minor need not be a subgraph.
For example, $K_3$ is a minor of every cycle $C_n$ with $n \ge 3$ but a subgraph only of $C_3$.
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# Endomorphism monoid

## Summary

The endomorphism monoid of a [[structure|983GZBQYT4]] is the set of all [[homomorphisms|KBXSZC6BV1]] from the structure to itself, under composition.
Its identity element is the identity map.
It is a [[monoid|0N6R902FG1]], usually written $\End(\mathbf{A})$, and its group of invertible elements is the [[automorphism group|0K1SCPBNKJ]].
For a vector space it is the monoid underlying the [[algebra of endomorphisms|0E3X2G1ZQB]], whose elements are the [[endomorphisms|0J7H0YBFDJ]] of the space.
Every monoid is isomorphic to the endomorphism monoid of some structure.

## Explanations

### L1 human

An endomorphism is a map from a structure to itself that respects its operations.
Doing one endomorphism after another gives again an endomorphism, and doing nothing is an endomorphism.
So the endomorphisms form a monoid, like the maps of a set to itself, but smaller.
For the integers with addition, the endomorphisms are the maps $x \mapsto nx$, and composing them multiplies the factors $n$.
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<document source="https://tamnd.github.io/math2/node/QFF87AXWN2.md" tag="QFF87AXWN2" kind="statement">
# Bohr–Mollerup theorem

## Statement

Let $f \colon (0, \infty) \to (0, \infty)$ be a function such that:

1. $f(1) = 1$;
2. $f(x + 1) = x f(x)$ for every $x > 0$;
3. $\ln f$ is [[convex|0VV0HSW1ZS]] on $(0, \infty)$.

Then $f(x) = \Gamma(x)$ for every $x > 0$, where $\Gamma$ is the [[gamma function|0V3FZMVB4K]].

## Remarks

The gamma function satisfies the three conditions; its [[log-convexity|8FRWATNNMX]] follows from Hölder's inequality applied to $\Gamma(x) = \int_0^\infty t^{x-1} e^{-t}\, dt$.
For uniqueness, conditions 1 and 2 give $f(n + 1) = n!$, so the values at integers are fixed.
Fix $0 < x \le 1$ and an integer $n \ge 2$.
Convexity of $\ln f$ compares its slope on $[n, n + x]$ with its slopes on $[n - 1, n]$ and $[n, n + 1]$.
This squeezes $f(n + x)$ between two explicit bounds, and condition 2 turns them into bounds for $f(x)$.
Letting $n \to \infty$ yields $f(x) = \lim_{n \to \infty} \frac{n!\, n^x}{x (x + 1) \cdots (x + n)}$, the same limit for every such $f$.
Without condition 3 uniqueness fails: $f(x) = \Gamma(x)\,(1 + \frac12 \sin(2\pi x))$ satisfies conditions 1 and 2, takes positive values, and differs from $\Gamma$ at $x = \frac14$.

## Explanations

### L1 human

Many functions interpolate the factorials, taking the value $(n-1)!$ at each positive integer $n$.
The theorem says that the gamma function is the only one that obeys the rule $f(x + 1) = x f(x)$ and whose logarithm bends upward everywhere.
So the gamma function is not an arbitrary choice: it is the natural extension of the factorial.

### L2 human

The theorem is a convenient way to prove identities for $\Gamma$: show that the other side satisfies the three conditions.
For example, it gives the formula $B(x, y) = \Gamma(x)\Gamma(y)/\Gamma(x + y)$ for the [[beta function|QHNWVSC6AQ]], by applying the theorem to $x \mapsto B(x, y)\,\Gamma(x + y)/\Gamma(y)$ for fixed $y$.
It also gives the Gauss product formula for $\Gamma$ and the Legendre duplication formula.
It links the theory of [[convex functions|0VV0HSW1ZS]] with special functions.
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# Spectral method

## Definition

Let $L$ be a linear differential operator on an interval or a box $\Omega$, and consider $L u = f$ with boundary or periodicity conditions.
Fix a family of smooth functions $\varphi_k$ on all of $\Omega$, each nonzero on most of $\Omega$.
Typical choices are $e^{ikx}$ for periodic problems, [[Chebyshev polynomials|M6EAR1JHPP]] or [[Legendre polynomials|QFYGJVA996]] otherwise.
Let $I_N$ be a finite set of $n$ indices.
A *spectral method* seeks $u_N = \sum_{k \in I_N} a_k \varphi_k$ and fixes the $a_k$ by one of three conditions:

1. Galerkin: $\ip{L u_N - f}{\varphi_j} = 0$ for all $j \in I_N$, with the $\varphi_k$ satisfying the boundary conditions;
2. tau: the same equations for all but $m$ indices $j$, plus the $m$ boundary conditions written as equations on the $a_k$;
3. collocation: $L u_N(x_j) = f(x_j)$ at chosen points $x_j$, together with the boundary conditions, with $n$ equations in all.

Here $\ip{\cdot}{\cdot}$ is the $L^2$ inner product, with the weight of the chosen polynomial family.

## Remarks

Example: on $[0, 2\pi]$ with periodic conditions, solve $-u'' + u = f$ with the Fourier basis $e^{ikx}$, $\abs{k} \le N$.
The Galerkin condition gives $(k^2 + 1) a_k = \hat f_k$, where $\hat f_k$ are the [[Fourier coefficients|G12KXDHPPS]] of $f$.
So $u_N$ is the truncation of the exact Fourier series of $u$.
If $f$ is analytic, $\hat f_k$ decays like $e^{-c \abs{k}}$, and so does the error of $u_N$.

Non-example: the finite element method with hat functions uses basis functions that vanish outside two mesh cells.
These are not global functions, so it is not a spectral method; its error for smooth $u$ decreases only like a fixed power of $1/N$.
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<document source="https://tamnd.github.io/math2/node/QFPFF15PXC.md" tag="QFPFF15PXC" kind="statement">
# Poles of the gamma function

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\C$.
The [[poles|066E7PJCYE]] of $\Gamma$ are exactly the points $z = -n$ with $n \in \set{0, 1, 2, \ldots}$.
Each of them is a simple pole, and the [[residue|0PHN07KVQT]] of $\Gamma$ at $-n$ is
$$\operatorname{Res}_{z = -n} \Gamma(z) = \frac{(-1)^n}{n!}.$$

## Remarks

Iterating the functional equation gives $\Gamma(z) = \dfrac{\Gamma(z + n + 1)}{z (z + 1) \cdots (z + n)}$ near $z = -n$.
The numerator is holomorphic at $-n$ with value $\Gamma(1) = 1$, and only the factor $z + n$ of the denominator vanishes there, to first order.
So the pole is simple, and the residue is $1/\big((-n)(-n + 1) \cdots (-1)\big) = (-1)^n/n!$.
Away from these points $\Gamma$ is holomorphic, see [[holomorphy of the gamma function|ZKJF80DTNB]].
For example, near $z = 0$ one has $\Gamma(z) = \frac{1}{z} - \gamma + O(z)$, where $\gamma$ is the [[Euler–Mascheroni constant|00C5P9JZBK]].

## Explanations

### L1 human

The gamma function blows up at $0$ and at every negative integer.
Near each such point it behaves like a constant divided by the distance to the point, and the constants $1, -1, \tfrac12, -\tfrac16, \ldots$ are $(-1)^n/n!$.

### L2 human

The residues explain contour integral formulas such as the Mellin–Barnes integrals, where closing the contour to the left picks up the poles of $\Gamma$ and produces power series.
For instance, summing the residues of $\Gamma(s) x^{-s}$ at $s = -n$ gives $\sum_n (-1)^n x^n / n! = e^{-x}$.
The fact that the poles are simple and $\Gamma$ has no zeros makes $1/\Gamma$ an entire function with simple zeros.
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<document source="https://tamnd.github.io/math2/node/QFQAB091MQ.md" tag="QFQAB091MQ" kind="problem">
# Signorini problem

## Problem

Let $\Omega \subseteq \R^3$ be a bounded open set with Lipschitz boundary, the reference shape of a linearly elastic body.
Split the boundary into disjoint parts $\Gamma_D$, $\Gamma_N$ and $\Gamma_C$, with $\Gamma_D$ of positive surface measure, and let $n$ be the outward unit normal.
For a displacement $u \colon \Omega \to \R^3$ let $\varepsilon(u) = \tfrac12 (\nabla u + \nabla u^T)$ be the strain and $\sigma(u) = \mathcal{C} \varepsilon(u)$ the stress.
Here $\mathcal{C}$ is a bounded elasticity tensor with the usual symmetries that is uniformly positive definite on symmetric matrices.
Let $f$ be a body force and $g$ a surface force on $\Gamma_N$.
Write $u_n = u \cdot n$, $\sigma_n = (\sigma(u) n) \cdot n$ and $\sigma_T = \sigma(u) n - \sigma_n n$.
Find $u$ such that:

1. $-\operatorname{div} \sigma(u) = f$ in $\Omega$;
2. $u = 0$ on $\Gamma_D$;
3. $\sigma(u) n = g$ on $\Gamma_N$;
4. on $\Gamma_C$: $u_n \le 0$, $\sigma_n \le 0$, $\sigma_n u_n = 0$ and $\sigma_T = 0$.

Clause 4 describes frictionless contact with a rigid support.
The body cannot enter the support, the support can only push, and it pushes only where there is contact.

## Remarks

The difficulty is that it is not known in advance where on $\Gamma_C$ the body touches the support, so the boundary condition there is ambiguous.
The weak form is a [[variational inequality|6QS7X5GVMY]].
Let $V = \set{v \in H^1(\Omega; \R^3) : v = 0 \text{ on } \Gamma_D}$, using the [[Sobolev space|M4CEZ5K9HP]] $H^1$, and $K = \set{v \in V : v \cdot n \le 0 \text{ on } \Gamma_C}$.
Let $a(u, v) = \int_\Omega \sigma(u) : \varepsilon(v) \, dx$ and $\ell(v) = \int_\Omega f \cdot v \, dx + \int_{\Gamma_N} g \cdot v \, dS$.
Find $u \in K$ with $a(u, v - u) \ge \ell(v - u)$ for all $v \in K$.
The set $K$ is closed and convex, and Korn's inequality makes $a$ coercive on $V$, so the [[Lions–Stampacchia theorem|H80M01T0H8]] gives exactly one weak solution.
Without the contact part, the problem is the mixed boundary value problem of linear elasticity, governed by the [[Lamé equations|GQ3K59MRJP]] in the isotropic case.

Status: solved.
Signorini posed the problem in a course in 1959.
Gaetano Fichera proved existence and uniqueness of the weak solution in the early 1960s, and this work started the theory of variational inequalities.

## Explanations

### L1 human

Put an elastic block on a rigid table and push or pull it.
Some parts of its bottom face stay on the table, others lift off, and nobody tells you which.
The Signorini problem asks for the deformation of the block together with the unknown contact zone.
The table may push the block but cannot pull it, and the block may not sink into the table.
These one-sided conditions turn the usual equations of elasticity into a variational inequality.
</document>

<document source="https://tamnd.github.io/math2/node/QFY1JR1TK4.md" tag="QFY1JR1TK4" kind="concept">
# Dual cone

## Summary

The dual cone of a set $C$ in $\R^n$ is the set of vectors that make a nonnegative [[dot product|0S52TNNTH9]] with every vector of $C$.
It is always a closed [[convex cone|6VE4KA12VD]], and for a closed convex cone $C$ the dual of the dual is $C$ again.
Dual cones are the geometric side of duality in linear and conic optimisation, and they appear in Farkas-type theorems of the alternative.

## Explanations

### L1 human

For each vector $x$ of $C$, the vectors $y$ with $x \cdot y \ge 0$ form a closed half-space whose boundary passes through the origin.
The dual cone is the intersection of all these half-spaces.
So it collects the directions that make an angle of at most $90°$ with every vector of $C$.
For the closed first quadrant of the plane, the dual cone is the first quadrant itself.
For a single ray in direction $(1, 0)$, it is the right half-plane $\set{(x, y) : x \ge 0}$.
</document>

<document source="https://tamnd.github.io/math2/node/QFYGJVA996.md" tag="QFYGJVA996" kind="concept">
# Legendre polynomials

## Summary

The Legendre polynomials $P_0, P_1, P_2, \dots$ are the polynomials obtained by making $1, x, x^2, \dots$ orthogonal on $[-1, 1]$ with the constant weight, normalised by $P_n(1) = 1$.
They are the simplest [[classical orthogonal polynomials|47WJ34NC9K]], the Jacobi case $\alpha = \beta = 0$.
They solve [[Legendre's equation|W6VE30V10Y]] and appear in the Laplace equation in spherical coordinates, in multipole expansions, and in Gauss–Legendre quadrature.

## Explanations

### L1 human

On the interval from $-1$ to $1$, two functions are called orthogonal when the integral of their product is zero.
Start with $1$, $x$, $x^2$, and so on, and adjust each one so that it is orthogonal to all the earlier ones.
The results, scaled to equal $1$ at $x = 1$, are $P_0 = 1$, $P_1 = x$, $P_2 = \frac{3x^2 - 1}{2}$, and so on.
They play the role on $[-1, 1]$ that sines and cosines play on a circle.
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<document source="https://tamnd.github.io/math2/node/QFZ3WKEFH6.md" tag="QFZ3WKEFH6" kind="concept">
# Radical of a module

## Summary

The radical of a module is the intersection of all its maximal submodules.
It is the dual notion of the [[socle|S2KXCAH5MM]], which is a sum of minimal submodules.
For the ring viewed as a module over itself, the radical is the [[Jacobson radical|XA0DCNH7WY]].
It is used to pass from a module to the quotient $M/\operatorname{rad}(M)$ in many arguments, such as Nakayama's lemma.

## Explanations

### L1 human

A maximal submodule is as large as possible without being the whole module.
The radical is what all these largest proper pieces have in common.
Elements of the radical are the "small" or "negligible" part of the module.
For the integers as a $\Z$-module, the maximal submodules are $p\Z$ for primes $p$, and no nonzero integer is divisible by every prime, so the radical is $0$.
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<document source="https://tamnd.github.io/math2/node/QFZMEFZH76.md" tag="QFZMEFZH76" kind="definition">
# Path space fibration

## Definition

Let $(X, x_0)$ be a [[pointed space|T9DG8DDE75]] and let $I = [0, 1]$.
The *path space* of $(X, x_0)$ is
$$PX = \set{\gamma \colon I \to X \text{ continuous} : \gamma(0) = x_0},$$
the set of [[paths|03NAMV1DHJ]] starting at $x_0$, with the [[subspace topology|0WS9P12FQR]] of the compact-open topology.
The *path space fibration* is the map
$$p \colon PX \to X, \quad p(\gamma) = \gamma(1).$$

## Remarks

The map $p$ is a Hurewicz fibration, and its fiber $p^{-1}(x_0)$ is the [[loop space|2JFF5F3EGF]] $\Omega X$.
The space $PX$ is contractible: $h_s(\gamma)(t) = \gamma((1 - s) t)$ deforms every path to the constant path at $x_0$.
So the long exact sequence of homotopy groups gives $\pi_n(\Omega X) \cong \pi_{n+1}(X, x_0)$ for $n \ge 0$.

Example: for $X = \R$ and $x_0 = 0$, the fiber over $a \in \R$ is the space of paths from $0$ to $a$, and $p$ is surjective, since the path $t \mapsto t a$ ends at $a$.

Non-example: if $X$ is not path-connected, $p$ is not surjective.
For $X = \set{0, 1}$ with the discrete topology and $x_0 = 0$, every path from $0$ is constant, so $PX$ is a point and $p^{-1}(1)$ is empty.

The fiber of $p$ over $y \in X$ is the [[homotopy fiber|4F3KWXYXBA]] over $y$ of the inclusion $\set{x_0} \to X$.
So $p$ replaces this inclusion by a fibration with a contractible total space.
</document>

<document source="https://tamnd.github.io/math2/node/QG1ZXRC8EH.md" tag="QG1ZXRC8EH" kind="concept">
# Algebra over a monad

## Summary

An algebra over a [[monad|FF6XXA6D1A]] $T$ on a category $\mathcal{C}$ is an object $A$ with a structure map $a \colon T(A) \to A$.
The structure map must be compatible with the unit and the multiplication of $T$.
These algebras, with the maps that respect the structure maps, form the Eilenberg–Moore category of $T$.
Many algebraic structures are algebras over a monad on sets.
Examples are monoids for the list monad, complete join-semilattices for the power set monad, and modules for the free module monad.

## Explanations

### L1 human

A monad describes formal expressions, and an algebra is a way to evaluate them.
The structure map $a$ takes a formal expression in elements of $A$ and returns an element of $A$.
The first law says that evaluating a single element gives back that element.
The second law says that evaluating in stages gives the same result as flattening first and evaluating once.
For the list monad, a monoid $M$ is an algebra: a list $(m_1, \dots, m_n)$ evaluates to the product $m_1 \cdots m_n$.
</document>

<document source="https://tamnd.github.io/math2/node/QG2JHK1VXX.md" tag="QG2JHK1VXX" kind="definition">
# Parabolic cylinder function

## Definition

Let $\nu$ be a complex number.
*Weber's equation* with parameter $\nu$ is the [[linear differential equation|0QQRP17Z7S]]
$$\frac{d^2 y}{dz^2} + \left( \nu + \frac12 - \frac{z^2}{4} \right) y = 0.$$
Let $M(a, b, x)$ be [[Kummer's function|6AZ4ZBN1M8]] and $\Gamma$ the [[gamma function|0H5SYBSED9]], with $1/\Gamma$ read as $0$ at the poles of $\Gamma$.
The *parabolic cylinder function* $D_\nu$ is the entire function
$$D_\nu(z) = 2^{\nu/2} e^{-z^2/4} \left( \frac{\sqrt{\pi}}{\Gamma\left( \frac{1 - \nu}{2} \right)} M\left( -\frac{\nu}{2}, \frac12, \frac{z^2}{2} \right) - \frac{\sqrt{2\pi}\, z}{\Gamma\left( -\frac{\nu}{2} \right)} M\left( \frac{1 - \nu}{2}, \frac32, \frac{z^2}{2} \right) \right).$$

## Remarks

The function $D_\nu$ solves Weber's equation, and it is the solution with $D_\nu(z) \sim z^\nu e^{-z^2/4}$ as $z \to +\infty$.
The functions $D_\nu(z)$ and $D_\nu(-z)$ are solutions too, and they are linearly independent unless $\nu$ is a nonnegative integer.
Another common notation is $U(a, z) = D_{-a - \frac12}(z)$.
Example: for $\nu = 0$, the first coefficient is $\sqrt{\pi}/\Gamma(\frac12) = 1$ and the second is $0$, since $1/\Gamma(0) = 0$; so $D_0(z) = e^{-z^2/4}$.
Check: $y = e^{-z^2/4}$ gives $y'' = \left( \frac{z^2}{4} - \frac12 \right) y$, so $y'' + \left( \frac12 - \frac{z^2}{4} \right) y = 0$.
More generally, for $\nu = n \in \N$, one has $D_n(z) = 2^{-n/2} e^{-z^2/4} H_n(z / \sqrt{2})$ with $H_n$ the [[Hermite polynomials|MGACY98E9H]].
Non-example: $y = e^{z^2/4}$ solves Weber's equation for $\nu = -1$, since $y'' = \left( \frac12 + \frac{z^2}{4} \right) y$, but it is not $D_{-1}$: it grows as $z \to +\infty$ instead of decaying.
</document>

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# G2 manifold

## Definition

On $\R^7$ with coordinates $x_1, \ldots, x_7$, write $dx_{ijk} = dx_i \wedge dx_j \wedge dx_k$ and let
$$\varphi_0 = dx_{123} + dx_{145} + dx_{167} + dx_{246} - dx_{257} - dx_{347} - dx_{356}.$$
The group $G_2$ is the group of linear maps $A \in GL(7, \R)$ with $A^* \varphi_0 = \varphi_0$.
It is a compact Lie group of dimension $14$ contained in $SO(7)$.

Let $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $7$.
Identify the tangent space at a point with $\R^7$ by a suitable linear isomorphism.
The manifold $M$ is a *$G_2$ manifold* if its [[holonomy|WHDZ3MK56B]] group is then contained in $G_2$.

## Remarks

Equivalently, $M$ carries a [[$3$-form|0E3G0GRPZ6]] $\varphi$ that agrees with $\varphi_0$ at each point in suitable linear coordinates, that determines $g$, and that is parallel for the [[Levi-Civita connection|CAZYGE1T4E]].
Such a $\varphi$ is a [[$G_2$-structure|VYNQVZ8XVF]], and it is parallel exactly when $d\varphi = 0$ and $d{*\varphi} = 0$, where $*$ is the [[Hodge star|1Y4WX4XQWJ]] of $g$.

Example: $\R^7$ with the Euclidean metric and $\varphi = \varphi_0$; the form has constant coefficients, so it is parallel and the holonomy is trivial.
The flat torus $\R^7 / \Z^7$ is a compact example with the same form.

Non-example: the round sphere $S^7$ is not a $G_2$ manifold.
A metric with holonomy in $G_2$ is Ricci-flat, while the round metric has positive Ricci curvature.

Compact manifolds with holonomy exactly $G_2$ were first constructed by Joyce, by resolving singularities of quotients of $T^7$ by finite groups.
</document>

<document source="https://tamnd.github.io/math2/node/QG9QRZAFTG.md" tag="QG9QRZAFTG" kind="concept">
# Convergence along an ultrafilter

## Summary

Convergence along an [[ultrafilter|J64Y3JC0RE]] is the [[filter limit|F6QDHKE6J3]] of a function along an ultrafilter.
Its key feature is that every function into a [[compact|0TVJZYB6SH]] space has such a limit, and the limit is unique when the space is [[Hausdorff|0S1FJ8VEX9]].
So every bounded real sequence has a limit along a fixed non-principal ultrafilter on $\N$, and this limit is linear and multiplicative.
Ultrafilter limits are used to build ultraproducts, Banach-limit-like functionals and the Stone–Čech compactification.

## Explanations

### L1 human

An ultrafilter on $\N$ decides, for every set of indices, whether it is "large" or "small", and exactly one of a set and its complement is large.
A sequence converges to $x$ along the ultrafilter if, for each small region around $x$, the indices whose terms land there form a large set.
Because the ultrafilter takes a side on every set, a bounded sequence always has such a limit.
For the sequence $1, 0, 1, 0, \dots$, indexed from $1$, the limit is $1$ if the set of odd indices is large and $0$ if the set of even indices is large.
</document>

<document source="https://tamnd.github.io/math2/node/QG9VDK1HN9.md" tag="QG9VDK1HN9" kind="concept">
# Appell series

## Summary

The Appell series $F_1, F_2, F_3, F_4$ are four double power series in two variables $x$ and $y$ that generalise the Gauss [[hypergeometric series|428M453CR2]].
Their coefficients are ratios of [[Pochhammer symbols|YZHM3HSM0F]] in the two summation indices.
Each of them reduces to a Gauss hypergeometric function when one of the variables is $0$.
They appear in integrals of products of powers, in the theory of algebraic curves and in physics.

## Explanations

### L1 human

The Gauss hypergeometric series is a power series in one variable whose coefficients are built from rising factorials.
To get a series in two variables, one lets the rising factorials depend on $m$, on $n$, or on $m + n$, where $x^m y^n$ is the monomial.
There are four natural ways to do this that give genuinely new functions, and these are Appell's four series.
For example $F_1(a; b_1, b_2; c; x, x) = {}_2F_1(a, b_1 + b_2; c; x)$, so on the diagonal $F_1$ collapses back to one variable.
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<document source="https://tamnd.github.io/math2/node/QGA9ABJEXQ.md" tag="QGA9ABJEXQ" kind="concept">
# Vanishing mean oscillation

## Summary

A function has vanishing mean oscillation if it has [[bounded mean oscillation|QJYTXFT8Y4]] and its mean oscillation over cubes tends to zero as the cubes shrink.
The space VMO plays, inside BMO, the role that continuous functions play inside bounded functions.
It contains all uniformly continuous functions in BMO, and also some unbounded functions.
It was introduced by Sarason, and it appears in the theory of Toeplitz and Hankel operators, of commutators, and of elliptic equations with discontinuous coefficients.

## Explanations

### L1 human

A BMO function may oscillate at every scale, as long as the oscillation stays bounded.
A VMO function must become calm at small scales: on tiny cubes it is nearly equal to its average.
Uniformly continuous functions with bounded oscillation have this property, but so do some functions with very mild singularities.
The function $\log \abs{x}$ is in BMO but not in VMO, because zooming in at $0$ shows the same oscillation at every scale.
</document>

<document source="https://tamnd.github.io/math2/node/QGFRT7QXWA.md" tag="QGFRT7QXWA" kind="definition">
# Linear algebraic group

## Definition

Let $k$ be a field.
A *linear algebraic group* over $k$ is an [[algebraic group|GH81SYAPPN]] $G$ over $k$ whose underlying scheme is [[affine|B3V3XJ2W8G]].

## Remarks

Every linear algebraic group over $k$ is isomorphic to a closed subgroup scheme of $\mathrm{GL}_n$ for some $n$.
So an equivalent definition is: a closed subgroup scheme of some [[general linear group|0FCHE902D7]] $\mathrm{GL}_n$.
In the classical language, with $k$ algebraically closed, it is a subgroup of $\mathrm{GL}_n(k)$ that is the zero set of a set of polynomials in the matrix entries.
Many authors also require $G$ to be smooth.

Example: $\mathrm{SL}_n = \operatorname{Spec} k[x_{ij}] / (\det - 1)$.
It is a closed subgroup of $\mathrm{GL}_n$, cut out by $\det = 1$, so it is affine.

Non-example: an elliptic curve $E$ over $k$ is an algebraic group, but it is projective of positive dimension and so not affine.
</document>

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# Monoidal functor

## Definition

Let $(\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)$ and $(\mathcal{D}, \otimes', I', \alpha', \lambda', \rho')$ be [[monoidal categories|M3MSQ3NKAE]].
A (lax) *monoidal functor* from $\mathcal{C}$ to $\mathcal{D}$ is a [[functor|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathcal{D}$ together with:

1. a morphism $\varepsilon \colon I' \to F(I)$;
2. a [[natural transformation|KYCPCC57FT]] $\mu$ with components $\mu_{A,B} \colon F(A) \otimes' F(B) \to F(A \otimes B)$;

such that for all objects $A, B, C$ of $\mathcal{C}$:

1. associativity: $F(\alpha_{A,B,C}) \circ \mu_{A \otimes B, C} \circ (\mu_{A,B} \otimes' \id_{F(C)}) = \mu_{A, B \otimes C} \circ (\id_{F(A)} \otimes' \mu_{B,C}) \circ \alpha'_{F(A),F(B),F(C)}$;
2. left unitality: $F(\lambda_A) \circ \mu_{I,A} \circ (\varepsilon \otimes' \id_{F(A)}) = \lambda'_{F(A)}$;
3. right unitality: $F(\rho_A) \circ \mu_{A,I} \circ (\id_{F(A)} \otimes' \varepsilon) = \rho'_{F(A)}$.

The functor is *strong monoidal* if $\varepsilon$ and all $\mu_{A,B}$ are isomorphisms, and *strict monoidal* if they are identities.

## Remarks

Example: the forgetful functor $U \colon (\mathbf{Ab}, \otimes, \Z) \to (\mathbf{Set}, \times, \set{*})$ with $\mu_{A,B}(a, b) = a \otimes b$ and $\varepsilon(*) = 1$.
Both sides of the associativity axiom send $((a, b), c)$ to $a \otimes (b \otimes c)$, and the left unit axiom reads $1 \otimes a \mapsto a$.
It is lax but not strong, since $A \times B \to A \otimes B$ is not a bijection in general.
Non-example: the forgetful functor from $\R$-vector spaces with $\otimes$ to $(\mathbf{Set}, \times)$, with $\mu(u, v) = u \otimes v$ but with $\varepsilon(*) = 0$.
The left unit axiom fails: it would send $(*, v)$ to $0 \otimes v = 0$, while $\lambda'$ sends it to $v$.
Some authors say "monoidal functor" for strong monoidal functors only; the lax version is then called lax monoidal.
</document>

<document source="https://tamnd.github.io/math2/node/QGGMQNFHJM.md" tag="QGGMQNFHJM" kind="example">
# Hankel transform of radial functions in the plane

## Example

Let $f_0 \colon (0, \infty) \to \C$ satisfy $\int_0^\infty \abs{f_0(r)}\, r\, dr < \infty$, and let $f(x) = f_0(\abs{x})$ on $\R^2$, an integrable radial function.
With the [[Fourier transform|DPVM34ADZV]] $\hat f(\xi) = \int_{\R^2} f(x) e^{-2\pi i x \cdot \xi}\, dx$, one has for $\rho = \abs{\xi}$
$$\hat f(\xi) = 2\pi \int_0^\infty f_0(r)\, J_0(2\pi r \rho)\, r\, dr,$$
which is $2\pi$ times the [[Hankel transform|2P9YWDGKC4]] of order $0$ of $f_0$ at $k = 2\pi\rho$.
Check: rotate so that $\xi$ lies along the first axis.
In polar coordinates $x = (r\cos\theta, r\sin\theta)$ one gets
$$\hat f(\xi) = \int_0^\infty f_0(r) \Big( \int_0^{2\pi} e^{-2\pi i r \rho \cos\theta}\, d\theta \Big) r\, dr.$$
The inner integral is $2\pi J_0(2\pi r\rho)$ by Bessel's integral $J_0(z) = \frac{1}{2\pi} \int_0^{2\pi} e^{-iz\cos\theta}\, d\theta$, where $J_0$ is the [[Bessel function|CZCMRDWDYW]] of order $0$.
This is the case $n = 2$ of the [[Fourier transform of radial functions|3CSEGRAD75]].

For the Gaussian $f_0(r) = e^{-\pi r^2}$, the known integral $\int_0^\infty e^{-a r^2} J_0(kr)\, r\, dr = \frac{1}{2a} e^{-k^2/(4a)}$ with $a = \pi$ and $k = 2\pi\rho$ gives $\hat f(\xi) = 2\pi \cdot \frac{1}{2\pi} e^{-\pi \rho^2} = e^{-\pi\abs{\xi}^2}$, as expected for $e^{-\pi \abs{x}^2}$.

## Explanations

### L1 human

A function in the plane that only depends on the distance to the origin has a Fourier transform with the same symmetry.
Its two-dimensional Fourier transform then reduces to a one-dimensional integral over the radius, with the Bessel function $J_0$ as kernel.
That integral is the Hankel transform of order $0$, which is why Hankel transforms appear in every problem with circular symmetry.
</document>

<document source="https://tamnd.github.io/math2/node/QGNCCJ125J.md" tag="QGNCCJ125J" kind="definition">
# Generic filter

## Definition

Let $(P, \le)$ be a [[partial order|0QV2JS9RJ3]], and let $M$ be a set or class.
A subset $D \subseteq P$ is *dense* if for every $p \in P$ there is $q \in D$ with $q \le p$.
A subset $G \subseteq P$ is a *filter* if:

1. $G$ is nonempty.
2. If $p \in G$ and $p \le q$, then $q \in G$.
3. If $p, q \in G$, then there is $r \in G$ with $r \le p$ and $r \le q$.

A filter $G$ is *generic over $M$*, or *$M$-generic*, if $G \cap D \neq \emptyset$ for every dense set $D \subseteq P$ with $D \in M$.

## Remarks

If $P$ has a least element $0$, then $G = P$ is generic over every $M$: each dense set $D$ contains some $q \le 0$, so $0 \in D$.
Non-example: let $P$ be the set of finite partial functions from $\omega$ to $\set{0, 1}$, ordered by reverse inclusion.
Let $M$ be a transitive model of ZFC with $P \in M$, and let $G \in M$ be a filter.
The set $D = P \setminus G$ is in $M$, and it is dense.
Given $p$, pick $n \notin \operatorname{dom} p$, and extend $p$ at $n$ by a value that disagrees with $\bigcup G$ at $n$, or by any value if $\bigcup G$ is undefined at $n$.
So $G \cap D = \emptyset$, and $G$ is not generic over $M$.
If $M$ is countable, the Rasiowa–Sikorski lemma gives, for each $p \in P$, a generic filter over $M$ that contains $p$.
Some authors also allow open dense sets or maximal antichains in the definition; these give the same notion of genericity.
</document>

<document source="https://tamnd.github.io/math2/node/QGPAB6AS9F.md" tag="QGPAB6AS9F" kind="statement">
# Tannaka–Krein duality

## Statement

Let $G$ be a compact Hausdorff topological [[group|G5S1M9]].
Let $\mathrm{Rep}(G)$ be the category of finite-dimensional continuous complex [[representations|WYGRXCG2RN]] of $G$, and let $\omega \colon \mathrm{Rep}(G) \to \mathbf{Vect}_\C$ be the forgetful functor.
Let $T(G)$ be the group of families $u = (u_V)_{V \in \mathrm{Rep}(G)}$ of linear automorphisms $u_V \colon \omega(V) \to \omega(V)$ such that:

1. $u$ is a [[natural transformation|KYCPCC57FT]] $\omega \to \omega$;
2. $u_{V \otimes W} = u_V \otimes u_W$ for all $V, W$, and $u$ is the identity on the trivial representation $\C$;
3. $u_{\overline{V}} = \overline{u_V}$ for every $V$, where $\overline{V}$ is the complex conjugate representation.

Then the map $G \to T(G)$ that sends $g$ to the family of operators $(\pi_V(g))_V$ is an isomorphism of groups.

## Remarks

This is Tannaka's theorem: a compact group is determined by its category of representations together with the forgetful functor.
Proof idea: by the Peter–Weyl theorem, the matrix coefficients of the representations span a dense subalgebra $\mathcal{R}(G)$ of the algebra $C(G)$ of continuous functions.
These functions also separate the points of $G$.
An element of $T(G)$ defines an algebra homomorphism $\mathcal{R}(G) \to \C$ that commutes with complex conjugation.
One shows that every such homomorphism is evaluation at a unique point of $G$.

Without clause 3 the group is larger.
For $G = U(1)$, the representations are the $\Z$-graded spaces, and families satisfying clauses 1 and 2 are given by $t \in \C^\times$ acting by $t^n$ in degree $n$; clause 3 forces $\abs{t} = 1$.
For a compact Lie group, clauses 1 and 2 alone give its complexification.
Krein's part of the duality describes which categories with such a functor arise from compact groups.
The algebraic version concerns a neutral Tannakian category over a field $k$.
It says that such a category is equivalent, as a rigid symmetric monoidal category, to the category of finite-dimensional representations of an affine group scheme over $k$.

## Explanations

### L1 human

A compact group can be rebuilt from the way it acts on vector spaces.
One records all its finite-dimensional representations, all the maps between them, and how tensor products of representations behave.
The group is then recovered as the set of consistent ways to act on all these spaces at once.
This generalizes Pontryagin duality, which rebuilds a compact abelian group from its characters.

### L2 human

For a compact abelian group, every irreducible representation is one-dimensional, and the theorem reduces to recovering $G$ from its character group, as in Pontryagin duality.
For non-abelian groups the representations do not form a group, but they do form a [[rigid|A10WW8GEKR]] [[symmetric monoidal category|5S7ER95XAC]], and the forgetful functor is a strong [[monoidal functor|PY430TXMG0]].
The group appears as the automorphisms of this functor that respect the tensor product.
This idea is central to the theory of Tannakian categories, motivic Galois groups, and the reconstruction of quantum groups from their representation categories.
</document>

<document source="https://tamnd.github.io/math2/node/QGPHXKCNNX.md" tag="QGPHXKCNNX" kind="definition">
# Ehrhart polynomial

## Definition

Let $P \subseteq \R^n$ be a [[lattice polytope|F39N9NBFPZ]] of dimension $d$.
For each integer $t \ge 1$, let
$$L_P(t) = \#\left(tP \cap \Z^n\right), \qquad tP = \set{tx : x \in P},$$
be the number of integer points in the dilate $tP$.
By Ehrhart's theorem, there is a unique [[polynomial|1G0B73WPX8]] $E_P(t)$ with rational coefficients such that $E_P(t) = L_P(t)$ for all integers $t \ge 1$.
This polynomial $E_P$ is the *Ehrhart polynomial* of $P$.

## Remarks

The polynomial $E_P$ has degree $d$, and $E_P(0) = 1$.
If $d = n$, the leading coefficient of $E_P$ is the volume of $P$.
In general the leading coefficient is the volume of $P$ measured inside its affine hull, normalised by the lattice of integer points of that affine hull.

Example: for the unit square $P = [0,1]^2$, the dilate $tP = [0,t]^2$ contains $(t+1)^2$ integer points, so $E_P(t) = t^2 + 2t + 1$.
For the triangle with vertices $(0,0)$, $(1,0)$, $(0,1)$, the dilate contains the points $(a,b) \in \Z^2$ with $a, b \ge 0$ and $a + b \le t$, so $E_P(t) = (t+1)(t+2)/2$.
In both cases the leading coefficient is the area, $1$ and $1/2$.

Non-example: the segment $P = [0, 1/2]$ is a rational polytope but not a lattice polytope.
Here $L_P(t) = \floor{t/2} + 1$, which is not a polynomial in $t$; it is only a quasi-polynomial of period $2$.
So the hypothesis that the vertices are integer points is needed.

The generating function $\sum_{t \ge 0} E_P(t) z^t$ is a rational function of the form $h^*(z)/(1-z)^{d+1}$, where $h^*$ is a polynomial with nonnegative integer coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/QGQ2PZHZ5R.md" tag="QGQ2PZHZ5R" kind="definition">
# Singular point of an algebraic variety

## Definition

Let $X$ be an [[algebraic variety|HV6YBS0YDH]] over an algebraically closed field $k$ and let $p \in X$.
The point $p$ is *nonsingular* (or smooth) if $\dim_k T_p X = \dim X$, where $T_p X$ is the [[Zariski tangent space|E7A5DHFT01]] and $\dim X$ is the [[dimension|36X2WTHYCB]] of $X$.
Otherwise $\dim_k T_p X > \dim X$, and $p$ is a *singular point* of $X$.
The variety $X$ is *nonsingular* if it has no singular point.

## Remarks

Jacobian criterion: let $X \subseteq k^n$ be an [[affine variety|JERVQT4C12]] of dimension $d$ with $I(X) = (f_1, \dots, f_r)$.
Then $p$ is nonsingular if and only if the [[Jacobian matrix|047NDTEPCV]] $\left(\frac{\partial f_j}{\partial x_i}(p)\right)$ has rank $n - d$.
Algebraic form: $p$ is nonsingular if and only if the [[local ring|MZ02BKPHJA]] of $X$ at $p$ is a [[regular local ring|63XXTF6B5M]].
The singular points of $X$ form a closed subset different from $X$.

Example: for $X = V(f) \subseteq k^2$ with $f = y^2 - x^2(x + 1)$, a curve of dimension $1$, the [[partial derivatives|09EGC6F8PT]] are $\partial f / \partial x = -3x^2 - 2x$ and $\partial f / \partial y = 2y$.
Assume $\operatorname{char} k \ne 2$.
Both vanish together with $f$ only at $(0, 0)$, so the origin is the only singular point.

Non-example: on the circle $V(x^2 + y^2 - 1)$ with $\operatorname{char} k \ne 2$, the gradient $(2x, 2y)$ vanishes only at $(0, 0)$, which is not on the circle, so the circle has no singular point.
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<document source="https://tamnd.github.io/math2/node/QGQPHRANWZ.md" tag="QGQPHRANWZ" kind="concept">
# Analytic capacity

## Summary

The analytic capacity of a compact set $K \subset \C$ measures how large a bounded holomorphic function on $\C \setminus K$ can be near infinity, see the [[definition|X252NY7CS1]].
It is zero exactly when $K$ is removable for bounded holomorphic functions.
The search for a geometric description of the sets of analytic capacity zero is the Painlevé problem.

## Explanations

### L1 human

Remove a compact set $K$ from the plane and look at holomorphic functions on what is left that stay bounded by $1$ and vanish at infinity.
If $K$ is a single point, the only such function is $0$, so the point is too small to matter.
If $K$ is a disc of radius $r$, the function $r/z$ works, and the capacity is $r$.
Analytic capacity is a number that tells how big $K$ is from the point of view of such functions.
</document>

<document source="https://tamnd.github.io/math2/node/QGQTYT6YZR.md" tag="QGQTYT6YZR" kind="definition">
# Lagrange interpolation

## Definition

Let $K$ be a [[field|0QA8CHH8MG]], let $n \ge 0$, let $x_0, \dots, x_n \in K$ be pairwise distinct, and let $y_0, \dots, y_n \in K$.
For $j = 0, \dots, n$, the Lagrange basis polynomial is
$$\ell_j(x) = \prod_{k \ne j} \frac{x - x_k}{x_j - x_k} \in K[x].$$
The Lagrange interpolation polynomial of the data is
$$L(x) = \sum_{j=0}^{n} y_j \, \ell_j(x).$$

## Remarks

Each $\ell_j$ has degree $n$, with $\ell_j(x_j) = 1$ and $\ell_j(x_k) = 0$ for $k \ne j$.
So $L$ has degree at most $n$ and $L(x_i) = y_i$ for all $i$: it is the unique [[interpolation polynomial|SS1YMS2NPC]] of the data.
Example: for the data $(0, 1)$, $(1, 3)$, $(2, 7)$, the basis polynomials are $\ell_0 = \frac{(x-1)(x-2)}{2}$, $\ell_1 = -x(x-2)$ and $\ell_2 = \frac{x(x-1)}{2}$.
Then $L = \ell_0 + 3 \ell_1 + 7 \ell_2 = x^2 + x + 1$, and indeed $L(0) = 1$, $L(1) = 3$, $L(2) = 7$.
Non-example: if two nodes coincide, $x_j = x_k$ with $j \ne k$, the denominator $x_j - x_k$ is $0$ and the formula has no meaning.

## Explanations

### L1 human

Each basis polynomial is a switch that is on at one node and off at all others.
Weighting the switches by the wanted values gives a polynomial through all the points.
</document>

<document source="https://tamnd.github.io/math2/node/QGRBR04JJN.md" tag="QGRBR04JJN" kind="concept">
# Leray–Schauder method

## Summary

The Leray–Schauder method proves that a nonlinear equation $u = T(u)$ with a compact map $T$ has a solution.
It deforms the equation into one that is easy to solve.
One embeds the problem in a family $u = T(\lambda, u)$, $0 \le \lambda \le 1$, and proves an a priori bound: all solutions for all $\lambda$ lie in a fixed ball.
The [[Leray–Schauder degree|3VJC5Q9MYA]] is then the same for every $\lambda$, and a nonzero degree at $\lambda = 0$ forces a solution at $\lambda = 1$.
It is a main tool for nonlinear integral equations and for quasilinear elliptic boundary value problems.

## Explanations

### L1 human

Imagine slowly turning a knob that changes an easy equation into the hard one you care about.
Solutions move as the knob turns, but they cannot vanish into thin air unless they escape to infinity.
If one can show that no solution ever gets too large, then the solution of the easy equation must survive until the end.
The degree is the count of solutions, with signs, that stays fixed along the way.
</document>

<document source="https://tamnd.github.io/math2/node/QH76Y4JD89.md" tag="QH76Y4JD89" kind="definition">
# Elliptic geometry

## Definition

Let $n \ge 2$, let $S^n \subset \R^{n+1}$ be the unit [[sphere|T62369RFH6]], and let $x \cdot y$ be the [[dot product|0S52TNNTH9]].
*Elliptic $n$-space* is the [[real projective space|1DDYB6VVSR]] $\mathbb{RP}^n = S^n / \set{x \sim -x}$ with these data:

1. A point is a pair $\set{x, -x}$ of antipodal points of $S^n$, written $[x]$.
2. A line is the image of a great circle of $S^n$, the intersection of $S^n$ with a 2-dimensional linear subspace.
3. The distance is $d([x], [y]) = \arccos \abs{x \cdot y} \in [0, \pi/2]$.

*Elliptic geometry* is the geometry of these spaces; for $n = 2$ it is the geometry of the elliptic plane.

## Remarks

The distance is well defined, since replacing $x$ by $-x$ does not change $\abs{x \cdot y}$.
Example: two distinct great circles of $S^2$ meet in exactly one antipodal pair $\set{x, -x}$, so two distinct lines of the elliptic plane meet in exactly one point.
Two distinct points $[x] \ne [y]$ span a 2-dimensional subspace, so they lie on exactly one line.
Non-example: the sphere $S^2$ with great circles as lines, that is [[spherical geometry|SQ9RQVANMG]], does not satisfy the fact that two distinct points lie on exactly one line.
The antipodal points $(0, 0, 1)$ and $(0, 0, -1)$ lie on infinitely many great circles.
For this reason spherical geometry is sometimes called double elliptic geometry, and elliptic geometry single elliptic geometry.
The elliptic plane has constant curvature $1$, and the double cover $S^2 \to \mathbb{RP}^2$ is a local isometry.
</document>

<document source="https://tamnd.github.io/math2/node/QH84JHCQMY.md" tag="QH84JHCQMY" kind="definition">
# Picard group of a ring

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]].
An $R$-module $P$ is *invertible* if $P$ is a [[finitely generated|05QY1VEBYD]] [[projective module|BFFA03GE3V]] of rank one.
This means that for every [[maximal ideal|0PV74TR5FP]] $\mathfrak{m}$ of $R$, the [[localization|X6JD4AG973]] $P_{\mathfrak{m}}$ is isomorphic to $R_{\mathfrak{m}}$.
The *Picard group* $\operatorname{Pic}(R)$ is the set of isomorphism classes $[P]$ of invertible $R$-modules, with the operation
$$[P] \cdot [Q] = [P \otimes_R Q],$$
where $\otimes_R$ is the [[tensor product|F0HQXQ0HCD]].
The identity element is $[R]$, and the inverse of $[P]$ is $[\Hom_R(P, R)]$.

## Remarks

The operation is well defined, associative and commutative because the tensor product is so up to natural isomorphism.
For invertible $P$, the evaluation map $P \otimes_R \Hom_R(P, R) \to R$ is an isomorphism, since it is one after localization at every maximal ideal.
An $R$-module $P$ is invertible exactly when there is a module $Q$ with $P \otimes_R Q \cong R$.
If $R$ is an integral domain, $\operatorname{Pic}(R)$ is isomorphic to the group of [[invertible fractional ideals|E190TAV2S6]] modulo the principal fractional ideals.

Example: for $R = \Z[\sqrt{-5}]$, the ideal $P = (2, 1 + \sqrt{-5})$ is an invertible module that is not free, while $P \otimes_R P \cong P^2 = (2) \cong R$; in fact $\operatorname{Pic}(R)$ has order $2$.
If $R$ is a [[principal ideal domain|0CMD0BCH8T]] or a [[local ring|MZ02BKPHJA]], then $\operatorname{Pic}(R)$ is trivial.

Non-example: the module $R^2$ is finitely generated and projective, but $(R^2)_{\mathfrak{m}} \cong R_{\mathfrak{m}}^2$ is not isomorphic to $R_{\mathfrak{m}}$ for $R \ne 0$; so $R^2$ is not invertible and has no class in $\operatorname{Pic}(R)$.
</document>

<document source="https://tamnd.github.io/math2/node/QHD4KJYFSA.md" tag="QHD4KJYFSA" kind="statement">
# Levitzki's theorem

## Statement

Let $R$ be a right [[Noetherian ring|0670T59HGE]].
Then every [[nil|NZ1G9WRWFB]] one-sided ideal of $R$ is [[nilpotent|C2H26BYWHE]].
That is, if $I$ is a right or [[left ideal|5N2CJZ47D4]] of $R$ in which every element is [[nilpotent|0ZB2JG1M9C]], then $I^n = 0$ for some $n \ge 1$.

## Remarks

The sum of all nilpotent ideals of a right Noetherian ring is itself a nilpotent ideal $N$, by the ascending chain condition.
The ring $R/N$ has no nonzero nilpotent ideals, and it suffices to show that it has no nonzero nil one-sided ideals.
In a nonzero nil one-sided ideal, choose a nonzero element $a$ whose right [[annihilator|HB4XDKJTW8]] is maximal, which is possible by the chain condition.
A short computation with this maximality gives $aRa = 0$, so $aR$ is a nonzero right ideal with $(aR)^2 = 0$, a contradiction.

Example: in the ring of upper triangular $n \times n$ matrices over a field, the strictly upper triangular matrices form a nil ideal $N$, and indeed $N^n = 0$.

The Noetherian hypothesis is needed.
Let $k$ be a field and $R = k[x_1, x_2, \ldots]/(x_1^2, x_2^3, x_3^4, \ldots)$.
The ideal $I$ generated by the images of all $x_i$ is nil, because $R$ is commutative and $I$ is generated by nilpotent elements.
But $I$ is not nilpotent, because $x_n^n \ne 0$ in $R$, so $I^n \ne 0$ for every $n$.

## Explanations

### L1 human

An ideal is nil if each of its elements becomes zero after raising it to some power.
It is nilpotent if one fixed power kills all products of its elements at once.
Nilpotent implies nil, but in general not the other way round.
Levitzki's theorem says that in a ring with the ascending chain condition on right ideals the two notions agree, even for one-sided ideals.

### L2 human

The theorem extends the classical fact that nil ideals of Artinian rings are nilpotent, since Artinian rings are Noetherian by the [[Hopkins–Levitzki theorem|W29AA0266Z]].
A typical consequence: the prime radical of a right Noetherian ring is nilpotent, and it is the largest nil ideal.
So for right Noetherian rings the various nil radicals all coincide.
</document>

<document source="https://tamnd.github.io/math2/node/QHH8RY9XCZ.md" tag="QHH8RY9XCZ" kind="concept">
# Schnirelmann density

## Summary

The Schnirelmann density of a set $A$ of positive integers is the infimum over $n \ge 1$ of the proportion of $\set{1, \dots, n}$ that lies in $A$.
Unlike the [[natural density|RVG9Z8NYA6]], it always exists and it is very sensitive to small elements: it is $0$ whenever $1 \notin A$.
It behaves well under sums of sets, which makes it a tool for additive problems.
Schnirelmann used it to show that every integer greater than $1$ is a sum of a bounded number of [[primes|01826KMG2H]].

## Explanations

### L1 human

Look at the first $n$ positive integers and ask what fraction of them belongs to $A$.
The Schnirelmann density is the worst such fraction over all $n$.
For the odd numbers the fraction is $1$ at $n = 1$ and $1/2$ at every even $n$, so the density is $1/2$.
For the even numbers it is already $0$ at $n = 1$, so the density is $0$.
A set with positive Schnirelmann density has to start early and keep up its pace everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/QHJ0GBQE2E.md" tag="QHJ0GBQE2E" kind="statement">
# Borsuk–Ulam theorem

## Statement

Let $n \ge 1$ and let $f \colon S^n \to \R^n$ be a [[continuous map|0VGFHTM75Y]] on the [[$n$-sphere|T62369RFH6]] $S^n \subset \R^{n+1}$.
Then there is a point $x \in S^n$ with $f(x) = f(-x)$.

## Remarks

An equivalent form says that there is no continuous odd map $g \colon S^n \to S^{n-1}$, that is, no map with $g(-x) = -g(x)$ for all $x$.
Indeed, if $f(x) \neq f(-x)$ for all $x$, then $g(x) = (f(x) - f(-x)) / \norm{f(x) - f(-x)}$ is such a map.
One proof uses the fact that an odd map $S^{n-1} \to S^{n-1}$ has odd [[degree|TMFG2P912R]].
The restriction of an odd map $S^n \to S^{n-1}$ to the equator is an odd map that extends over a hemisphere.
So it is null-homotopic and has degree $0$, a contradiction.

Example: for $n = 1$, let $h(x) = f(x) - f(-x)$ on the circle.
Then $h(-x) = -h(x)$, so $h$ takes values of both signs or vanishes, and by the intermediate value theorem it has a zero.

Example of use: at any moment there are two antipodal points on the Earth with the same temperature and the same air pressure, if both are continuous functions on $S^2$.
The theorem fails for maps $S^n \to \R^{n+1}$: the inclusion $S^n \subset \R^{n+1}$ takes different values at $x$ and $-x$.

## Explanations

### L1 human

If you squash a sphere flat into a space of one dimension less, two opposite points of the sphere always land on the same spot.
For a balloon pressed flat onto a table, some pair of opposite points ends up one on top of the other.

### L2 human

The theorem is the starting point of the use of topology in combinatorics and geometry.
It implies the [[ham sandwich theorem|GDXX60SNG5]], the Lusternik–Schnirelmann covering theorem, and Lovász's computation of the chromatic number of Kneser graphs.
Its combinatorial counterpart is [[Tucker's lemma|PADZ7D9KBT]], from which it can also be proved.
</document>

<document source="https://tamnd.github.io/math2/node/QHN67MSH8S.md" tag="QHN67MSH8S" kind="concept">
# Inverse distance weighting

## Summary

Inverse distance weighting is a simple method of [[multivariate interpolation|BM703GV34G]] for scattered data.
The value at a new point is a weighted average of the data values.
The weight of each data point is a negative power of its distance to the new point.
At a data point the method returns the data value, and near it the weight of that point dominates, so the result is continuous.
It is widely used in geography and engineering because it needs no linear system and its values never leave the range of the data.

## Explanations

### L1 human

To guess the rainfall at a village from nearby rain gauges, trust the closest gauges most.
Inverse distance weighting does exactly this: each gauge gets weight $1/\text{distance}^p$, and the guess is the weighted average.
With two gauges at distances $1$ and $2$ reading $10$ and $40$, and $p = 2$, the weights are $1$ and $1/4$, so the guess is $\frac{10 + 10}{1 + 1/4} = 16$.
At a gauge the distance is $0$, and the method returns the gauge reading itself.
The [[definition|55V8WAJXXJ]] gives the formula.
</document>

<document source="https://tamnd.github.io/math2/node/QHNWVSC6AQ.md" tag="QHNWVSC6AQ" kind="concept">
# Beta function

## Summary

The beta function $B(x, y)$ is the integral $\int_0^1 t^{x-1}(1 - t)^{y-1}\, dt$ for positive $x$ and $y$.
It is expressed through the [[gamma function|0V3FZMVB4K]] by $B(x, y) = \Gamma(x)\Gamma(y)/\Gamma(x + y)$.
It turns many integrals over intervals and over angles into products of gamma values.
It also gives the normalising constants of the beta distributions in probability.
At positive integers it is the reciprocal of a multiple of a binomial coefficient.

## Explanations

### L1 human

The beta function measures the area under the curve $t^{x-1}(1-t)^{y-1}$ between $0$ and $1$.
Large $x$ pushes the mass of this curve toward $t = 1$, and large $y$ pushes it toward $t = 0$.
For $x = y = 1$ the curve is the constant $1$ and the area is $1$.
For $x = 2$, $y = 1$ the curve is $t$ and the area is $\frac12$.
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# Steiner system

## Definition

Let $t, k, n$ be positive integers with $t < k < n$.
A Steiner system $S(t, k, n)$ is a pair $(X, \mathcal{B})$ where $X$ is a [[finite set|0BBTH3YSZA]] with $n$ elements.
The family $\mathcal{B}$ is a set of $k$-element subsets of $X$, called blocks.
The condition is that each $t$-element subset of $X$ is contained in exactly one block.

## Remarks

Equivalently, a Steiner system $S(t, k, n)$ is a $t$-$(n, k, 1)$ [[block design|TGZATGBHTH]].
Counting $t$-sets gives the number of blocks: $\binom{n}{t} / \binom{k}{t}$.
Example: on the $9$ points of the grid $(\Z/3\Z)^2$, the $12$ lines of the affine plane form an $S(2, 3, 9)$.
Two distinct points $p, q$ lie on exactly one line, namely $\set{p, q, -p - q}$.
Non-example: there is no $S(2, 3, 8)$.
In an $S(2, 3, n)$, the blocks through a point split the other $n - 1$ points into pairs, so $n - 1$ must be even, and $8 - 1 = 7$ is odd.
An $S(2, 3, n)$ is a Steiner triple system; for $n \ge 7$, one exists if and only if $n \equiv 1$ or $3 \pmod 6$.
A [[finite projective plane|2HWB7P1DZN]] of order $q$ is an $S(2, q + 1, q^2 + q + 1)$.
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<document source="https://tamnd.github.io/math2/node/QHPXB39SN7.md" tag="QHPXB39SN7" kind="statement">
# Rogers' bound

## Statement

Let $n \ge 1$.
Let $T$ be a regular simplex in $\R^n$ with edge length $2$.
Let $\sigma_n$ be the fraction of the volume of $T$ covered by the $n + 1$ unit balls centred at the vertices of $T$.
Then every [[sphere packing|HF32CBN5MA]] of unit balls in $\R^n$ has upper [[packing density|4P5PSWADJD]] at most $\sigma_n$.

## Remarks

C. A. Rogers proved the bound in 1958.
The proof splits space into the Voronoi cells of the centres, and splits each cell further into simplices.
A local volume argument shows that the balls fill at most the fraction $\sigma_n$ of each Voronoi cell.
For $n = 2$, the triangle of side $2$ has area $\sqrt 3$, and the three discs cover three sectors of angle $\pi/3$, of total area $\pi/2$.
So $\sigma_2 = \pi/\sqrt{12}$, and the bound is sharp by [[Thue's theorem|3P246RHXMF]].
For $n = 3$ one gets $\sigma_3 \approx 0.7797$, a little above the true optimum $\pi/\sqrt{18} \approx 0.7405$ of the [[Kepler conjecture|AFMBQT06ZV]].

## Explanations

### L1 human

In the plane, the best packing of coins puts every three neighbouring coins at the corners of an equilateral triangle.
Rogers' bound says that in any dimension, a packing of balls cannot be denser than the share of a regular simplex filled by balls at its corners.
In dimension $3$ this simplex is a regular tetrahedron, and the bound is close to, but above, the true answer.

### L2 human

Rogers' bound was the best general upper bound on sphere packing density for about twenty years.
For large $n$ it is roughly $\tfrac{n}{e} 2^{-n/2}$, and the Kabatiansky–Levenshtein bound improved it exponentially in 1978 by a method based on linear programming.
In dimensions $4$ to $36$ the Cohn–Elkies linear programming bound is better, and it is sharp in dimensions $8$ and $24$.
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<document source="https://tamnd.github.io/math2/node/QHTAEPTKEB.md" tag="QHTAEPTKEB" kind="object">
# Hurwitz quaternions

## Description

Let $\mathbb{H}$ be the [[quaternions|0GE4DVQKR8]], with basis $1, i, j, k$ over $\R$ and $i^2 = j^2 = k^2 = ijk = -1$.
The *Hurwitz quaternions* are the elements
$$a + bi + cj + dk \in \mathbb{H}$$
such that either $a, b, c, d$ are all in $\Z$ or all in $\Z + \tfrac{1}{2}$.
They form a ring, written $\mathcal{H}$ here.
Equivalently, $\mathcal{H}$ is the $\Z$-span of $i$, $j$, $k$ and $\omega = \tfrac{1}{2}(1 + i + j + k)$.

## Remarks

The ring $\mathcal{H}$ is a maximal [[order|1ZBX4TKVZ9]] in the rational [[quaternion algebra|ZWXW0T3EAC]] $(-1, -1)_{\Q}$.
It contains the Lipschitz quaternions, those with $a, b, c, d \in \Z$, as a subring of index $2$; the Lipschitz quaternions are an order but not a maximal one.
The reduced norm $a^2 + b^2 + c^2 + d^2$ takes integer values on $\mathcal{H}$; for example $\omega$ has norm $4 \cdot \tfrac14 = 1$.
The units of $\mathcal{H}$ are the $24$ elements $\pm 1, \pm i, \pm j, \pm k$ and $\tfrac12(\pm 1 \pm i \pm j \pm k)$; they form the binary tetrahedral group.
Hurwitz quaternions admit division with remainder on each side, so every one-sided ideal of $\mathcal{H}$ is principal.
This is a standard route to [[Lagrange's four-square theorem|0KYYSMEVHH]].

## Explanations

### L1 human

The Hurwitz quaternions are the quaternion analogue of the integers.
One allows all four coordinates to be integers, or all four to be halves of odd integers.
Adding the half-integer points makes the set a better ring than the plain integer quaternions: one can divide with a small remainder, as for whole numbers.
The quaternion $\tfrac12(1 + i + j + k)$ is a typical new element, and it is even a unit, with norm $1$.
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<document source="https://tamnd.github.io/math2/node/QJ0P26J9BV.md" tag="QJ0P26J9BV" kind="concept">
# Overdetermined system

## Summary

An overdetermined system is a [[system of linear equations|0GT4WWGWH2]] with more equations than unknowns.
Such a system usually has no solution; a system with no solution is called [[inconsistent|0X6V9B1V10]].
Overdetermined systems arise whenever many measurements are fitted with a model that has few parameters.
The method of [[least squares|0BYTQ4X1HC]] then finds the vector that comes closest to solving all equations.

## Explanations

### L1 human

Each equation is a condition, and each unknown is a freedom.
With more conditions than freedoms, the conditions usually contradict each other.
Picture three lines in the plane: two lines usually meet in a point, but a third line usually misses that point.
For example, $x + y = 2$, $x - y = 0$ and $x + 2y = 4$ have no common solution, because the first two force $x = y = 1$ and then $x + 2y = 3$.
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<document source="https://tamnd.github.io/math2/node/QJAXM2SF1Z.md" tag="QJAXM2SF1Z" kind="definition">
# Spectrum of a ring

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]].
The *spectrum* of $R$ is the set
$$\operatorname{Spec} R = \set{\mathfrak p : \mathfrak p \text{ is a prime ideal of } R},$$
equipped with the [[Zariski topology|8SQK7RTB63]], whose closed sets are the sets $V(E) = \set{\mathfrak p \in \operatorname{Spec} R : E \subseteq \mathfrak p}$ for subsets $E \subseteq R$.

## Remarks

A [[ring homomorphism|0VTH8KPCG2]] $f \colon R \to A$ induces the map $\operatorname{Spec} A \to \operatorname{Spec} R$, $\mathfrak q \mapsto f^{-1}(\mathfrak q)$.
It is continuous, since the preimage of $V(E)$ is $V(f(E))$.

Example: $\operatorname{Spec} \Z$ consists of $(0)$ and the ideals $p\Z$ for the prime numbers $p$.
The spectrum of a field $k$ is a single point $(0)$.
The spectrum of $\Z/n\Z$ for $n \ge 2$ is the finite set of ideals $p\Z/n\Z$ for the primes $p$ dividing $n$.

Non-example: the ring $R$ itself is not a point of $\operatorname{Spec} R$, because a [[prime ideal|00MHHZ9SNE]] is by definition a proper ideal.
So the zero ring has empty spectrum.

Every nonzero ring has a [[maximal ideal|0PV74TR5FP]], which is prime, so $\operatorname{Spec} R \ne \emptyset$ when $R \ne 0$.
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<document source="https://tamnd.github.io/math2/node/QJGPT2W0FT.md" tag="QJGPT2W0FT" kind="concept">
# Algebraic structure

## Summary

An algebraic structure, or algebra, is a set together with a family of [[finitary operations|EJ8YJVM9BW]] on it.
The names and arities of the operations form a [[signature|WDVVEYD6AT]] with no relation symbols, so an algebraic structure is a [[structure|983GZBQYT4]] of a special kind.
Groups, rings, fields, lattices and vector spaces are the main examples.
A kind of algebraic structure is usually fixed by its signature and by a list of axioms, often identities, that the operations satisfy.

## Explanations

### L1 human

An algebraic structure is a set with ways to combine its elements.
The integers with addition, negation and the number $0$ form one: $+$ combines two integers, negation changes one integer, and $0$ is a fixed element.
The axioms say how these operations behave, for example $a + 0 = a$.
A [[group|G5S1M9]] is an algebraic structure with one binary operation, one unary operation and one constant that satisfy the group axioms.
</document>

<document source="https://tamnd.github.io/math2/node/QJK35TCFCJ.md" tag="QJK35TCFCJ" kind="definition">
# Muckenhoupt weights

## Definition

Let $n \ge 1$.
A *weight* on $\R^n$ is a locally [[integrable|X7GTF3D98F]] function $w \colon \R^n \to [0, \infty)$ with $w > 0$ [[almost everywhere|RARTMA0ZEC]].
For a cube $Q \subseteq \R^n$ with sides parallel to the axes, let $\abs{Q}$ be its [[Lebesgue measure|2Z2QVWZ262]].

1. For $1 < p < \infty$, the weight $w$ is in the *Muckenhoupt class* $A_p$ if
$$[w]_{A_p} = \sup_Q \left( \frac{1}{\abs{Q}} \int_Q w \, dx \right) \left( \frac{1}{\abs{Q}} \int_Q w^{-1/(p - 1)} \, dx \right)^{p - 1} < \infty,$$
where the supremum is over all such cubes $Q$.
2. The weight $w$ is in $A_1$ if there is a constant $C$ with $Mw(x) \le C w(x)$ for almost every $x$, where $M$ is the [[Hardy–Littlewood maximal function|VX1A56P5KQ]].

A weight in some class $A_p$ is called a *Muckenhoupt weight*.

## Remarks

By Hölder's inequality $[w]_{A_p} \ge 1$, and $A_1 \subseteq A_p \subseteq A_q$ for $1 < p < q$.
The union $A_\infty = \bigcup_{p \ge 1} A_p$ has several equivalent descriptions, for example through a reverse Hölder inequality.
For $p = 2$ the condition reads $\sup_Q \left( \frac{1}{\abs{Q}}\int_Q w \right) \left( \frac{1}{\abs{Q}}\int_Q w^{-1} \right) < \infty$.

Example: $n = 1$, $p = 2$, $w(x) = \abs{x}^a$ with $-1 < a < 1$.
On $Q = [0, L]$ the two averages are $L^a/(1 + a)$ and $L^{-a}/(1 - a)$, whose product is $1/(1 - a^2)$ for every $L$.
Intervals symmetric about $0$ give the same averages.
On intervals far from $0$ compared with their length, $w$ is almost constant, so the product is close to $1$.
The remaining intervals lie in a symmetric interval of comparable length, and this case analysis shows $w \in A_2$.
More generally $\abs{x}^a \in A_p(\R^n)$ exactly when $-n < a < n(p - 1)$.

Non-example: $n = 1$, $p = 2$, $w(x) = \abs{x}$.
It is a weight, but on $Q = [0, 1]$ the average of $w^{-1} = 1/x$ is infinite, so $w \notin A_2$.
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<document source="https://tamnd.github.io/math2/node/QJNS6RJ743.md" tag="QJNS6RJ743" kind="definition">
# Left-orderable group

## Definition

Let $G$ be a [[group|G5S1M9]].
A *left order* on $G$ is a [[total order|0316AYH1SC]] $\le$ on $G$ such that for all $a, b, c \in G$,
$$a \le b \implies ca \le cb.$$
The group $G$ is *left-orderable* if it has a left order.

## Remarks

Every [[orderable group|ZZFFBXDTWN]] is left-orderable.
Example: the Klein bottle group $K = \langle a, b \mid b a b^{-1} = a^{-1} \rangle$ is left-orderable but not orderable.
Every element of $K$ is $b^n a^m$ for unique integers $n, m$.
Set $b^n a^m \le b^{n'} a^{m'}$ when $n < n'$, or $n = n'$ and $m \le m'$.
Left multiplication by $b$ adds $1$ to $n$, and left multiplication by $a$ sends $b^n a^m$ to $b^n a^{m + (-1)^n}$.
Both keep this order, so it is a left order.
In a two-sided order, $a > e$ would give $a^{-1} = b a b^{-1} > e$, which contradicts $a > e$.
The case $a < e$ is the same, so $K$ is not orderable.
Non-example: a group with an element $g \ne e$ of finite order $n$ is not left-orderable, since $e < g$ gives $e < g < g^2 < \cdots < g^n = e$.
If $g < e$, the same argument applies to $g^{-1}$.
So a left-orderable group is a [[torsion-free group|VQ4SSKS53D]].
Equivalently, $G$ is left-orderable if and only if it has a subsemigroup $P$ with $G = P \sqcup \set{e} \sqcup P^{-1}$.
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# Gysin sequence

## Statement

Let $\pi \colon E \to B$ be an oriented real [[vector bundle|P4DZM975BV]] of rank $n \ge 1$, let $E_0$ be the complement of its zero section, and let $\pi_0 = \pi|_{E_0} \colon E_0 \to B$.
Let $e \in H^n(B; \Z)$ be its [[Euler class|BJX6QS0ANT]].
Then there is a long [[exact sequence|JB7ZBFBA9C]] in integral [[singular cohomology|6N7BC70CX0]]
$$\cdots \to H^i(B) \xrightarrow{\ \smile e\ } H^{i+n}(B) \xrightarrow{\ \pi_0^*\ } H^{i+n}(E_0) \to H^{i+1}(B) \xrightarrow{\ \smile e\ } H^{i+n+1}(B) \to \cdots,$$
where $\smile e$ is the [[cup product|JX47MS1R1H]] with $e$.

## Remarks

The space $E_0$ deformation retracts onto the unit [[sphere bundle|3VQ2V5HH3R]] $S(E)$ of any metric on $E$, so the sequence is often written with $S(E)$, a bundle with fibre $S^{n-1}$.
The proof writes the long exact sequence of the pair $(E, E_0)$, replaces $H^\bullet(E)$ by $H^\bullet(B)$ since $E$ deformation retracts onto $B$, and replaces $H^{\bullet}(E, E_0)$ by $H^{\bullet - n}(B)$ with the [[Thom isomorphism|N685RXHDZX]].
The composite $H^{i}(B) \to H^{i+n}(E, E_0) \to H^{i+n}(E) \cong H^{i+n}(B)$ is then the cup product with $e$.
With $\Z/2$ coefficients the sequence holds without the orientation hypothesis, with the top Stiefel–Whitney class in place of $e$.

Example: the tautological complex line bundle over [[complex projective space|7GJ4STCNNM]] $\C P^m$ has rank $2$ as a real bundle, and its unit sphere bundle is the Hopf map $S^{2m+1} \to \C P^m$.
Since $H^j(S^{2m+1}) = 0$ for $0 < j < 2m+1$, the sequence shows that $\smile e \colon H^i(\C P^m) \to H^{i+2}(\C P^m)$ is an isomorphism for $0 \le i \le 2m - 2$.
With $H^0(\C P^m) = \Z$, this gives $H^\bullet(\C P^m; \Z) \cong \Z[e]/(e^{m+1})$.

## Explanations

### L1 human

A sphere bundle is a space made of spheres, one over each point of a base space.
The Gysin sequence computes the cohomology of the total space from the cohomology of the base and one class, the Euler class, which measures how twisted the bundle is.
If the bundle is a product, the Euler class is zero and the sequence splits into short pieces.

### L2 human

The Gysin sequence is the special case of the Serre spectral sequence for a sphere bundle, where only two rows are nonzero.
It is a standard way to compute cohomology rings of projective spaces, of unit tangent bundles and of lens spaces.
It also shows that an oriented bundle with a nowhere zero section has Euler class zero, because then $\pi_0^*$ is injective in the relevant degree.
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<document source="https://tamnd.github.io/math2/node/QJWCFAK4AK.md" tag="QJWCFAK4AK" kind="statement">
# First main theorem of Nevanlinna theory

## Statement

Let $f$ be a [[meromorphic function|0NSD1TPHR0]] on $\C$ that is not constant, and let $a \in \C$.
For $r > 0$, let $m(r, a)$ and $N(r, a)$ be the proximity function and the counting function of $f$ at the value $a$, as in [[Nevanlinna theory|1EHH63VNAD]].
Let $T(r, f)$ be the [[Nevanlinna characteristic|YT6Z0GWS0W]] of $f$.
Then
$$m(r, a) + N(r, a) = T(r, f) + O(1) \qquad (r \to \infty),$$
where the bounded term depends on $f$ and $a$ but not on $r$.
More precisely, if $f(0) \ne a$ and $0$ is not a pole of $f$, then
$$\abs{\, m(r, a) + N(r, a) - T(r, f) + \log \abs{f(0) - a} \,} \le \log^+ \abs{a} + \log 2 \quad \text{for all } r > 0.$$

## Remarks

Apply [[Jensen's formula|497K99BQ4Y]] to $g = f - a$.
In Nevanlinna's notation it reads $T(r, g) - T(r, 1/g) = \log \abs{g(0)}$, and $T(r, 1/g) = m(r, a) + N(r, a)$.
The inequalities $\log^+ \abs{u - a} \le \log^+ \abs{u} + \log^+ \abs{a} + \log 2$ and its symmetric form show that $T(r, f - a)$ and $T(r, f)$ differ by at most $\log^+ \abs{a} + \log 2$.
For example, for $f(z) = e^z$ and $a = 0$ there are no solutions of $f(z) = 0$, so $N(r, 0) = 0$, and $m(r, 0) = r/\pi = T(r, f)$.

## Explanations

### L1 human

For each value $a$, the quantity $m(r, a) + N(r, a)$ measures how much $f$ is attracted to $a$ on the disc of radius $r$.
The counting part $N$ comes from the points where $f$ takes the value $a$.
The proximity part $m$ comes from the places on the circle where $f$ is very close to $a$.
The theorem says that the total is essentially the same for every value $a$, namely $T(r, f)$.

### L2 human

The first main theorem is a reformulation of Jensen's formula, and it is easy compared with the [[second main theorem|GWYE6745JK]].
It gives the upper bound $N(r, a) \le T(r, f) + O(1)$, which says that no value is taken too often.
It also justifies the definition of the defect $\delta(a) = 1 - \limsup_{r \to \infty} N(r, a)/T(r, f)$, which lies between $0$ and $1$.
The second main theorem gives the much deeper lower bound for most values.
</document>

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# Bounded mean oscillation

## Summary

A locally integrable function has bounded mean oscillation if its average deviation from its own mean, over every cube, is bounded by one constant.
The space BMO of such functions contains $L^\infty$ and also some unbounded functions such as $\log \abs{x}$.
It is the dual of the real Hardy space $H^1$, by Fefferman's theorem.
It is the natural replacement for $L^\infty$ in the theory of singular integrals.

## Explanations

### L1 human

Take any cube, compute the average of $f$ on it, and measure how far $f$ is from that average on the cube, on average.
If this number stays below a fixed bound for all cubes, large and small, then $f$ is in BMO.
A bounded function is always in BMO, because it can never be far from its averages.
The function $\log \abs{x}$ is unbounded, but rescaling only adds a constant to it, so its oscillation looks the same at every scale and stays bounded.
The function $f(x) = x$ on $\R$ is not in BMO: on $[0, L]$ its mean deviation is $L/4$, which grows without bound.
</document>

<document source="https://tamnd.github.io/math2/node/QK03YC87QC.md" tag="QK03YC87QC" kind="definition">
# Randomization

## Definition

Let $T$ be a finite set of treatments, let $U$ be a set of $n$ experimental units, and let $r_t \ge 1$ be integers with $\sum_{t \in T} r_t = n$.
Let $\mathcal{D}$ be the set of [[experimental designs|2TE42ZWJ3E]] $d \colon U \to T$ with $\abs{d^{-1}(t)} = r_t$ for every $t \in T$.
*Complete randomization* is the choice of a design $D$ at random from $\mathcal{D}$ with the uniform [[distribution|0QB913WM5E]], that is,
$$P(D = d) = \frac{1}{\abs{\mathcal{D}}} = \frac{\prod_{t \in T} r_t!}{n!} \quad \text{for every } d \in \mathcal{D}.$$
More generally, a *randomized design* is a design chosen at random from a set of designs by a known probability distribution.
In a restricted randomization the set is smaller, for example the designs that give every treatment once in each block.

## Remarks

Example: $T = \set{A, B}$, $U = \set{1, 2, 3, 4}$ and $r_A = r_B = 2$.
Then $\abs{\mathcal{D}} = 4!/(2!\,2!) = 6$, and each of the six splits, such as $\set{1, 2}$ to $A$ and $\set{3, 4}$ to $B$, has probability $1/6$.
Each unit receives $A$ with probability $1/2$.

Non-example: giving $A$ to the first two units to arrive and $B$ to the next two is a fixed rule, not a chance mechanism.
If early arrivals differ from late ones, the comparison of $A$ and $B$ is biased.

Fix the observed responses and compute a test statistic for every design in $\mathcal{D}$.
Under the hypothesis of no treatment effect, these values give the exact null distribution of a randomization test.
</document>

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# Liouville's theorem for harmonic functions

## Statement

Let $n \ge 1$ and let $u \colon \R^n \to \R$ be [[harmonic|0DEB0JMN0W]] on all of $\R^n$.
If $u$ is bounded above or bounded below on $\R^n$, then $u$ is constant.

## Remarks

Proof idea for a bounded function, $\abs{u} \le M$: by the [[gradient estimate|N76R7JWMNJ]] on the ball $B(x, r)$, $\abs{\nabla u(x)} \le n M / r$ for every $r > 0$.
Letting $r \to \infty$ gives $\nabla u = 0$ everywhere, so $u$ is constant.
If $u$ is only bounded below, say $u \ge m$, apply [[Harnack's inequality|YNH832D50W]] to the nonnegative harmonic function $u - m$ on larger and larger balls.
It shows that $u - m$ is bounded, and the first case applies.
The theorem fails on proper subsets: on the half-space $\set{x_n > 0}$, $u = x_n$ is bounded below and not constant.
Some one-sided condition is needed: $u(x, y) = x$ is harmonic on $\R^2$ and not constant.
A harmonic function on $\R^n$ with $\abs{u(x)} \le C(1 + \abs{x})^k$ is a harmonic polynomial of degree at most $k$.

## Explanations

### L1 human

A harmonic function defined on the whole space cannot stay below a ceiling, or above a floor, unless it is flat.
Every nonconstant harmonic function on the whole space takes arbitrarily large and arbitrarily small values.

### L2 human

For $n = 2$ this is the real counterpart of Liouville's theorem for entire functions, and the two imply each other through harmonic conjugates.
The theorem gives uniqueness of solutions of Poisson's equation on $\R^n$ that tend to $0$ at infinity.
It is the prototype of Liouville theorems for elliptic equations and for harmonic maps.
On $\R^n$ minus a point, or on other spaces, the analogous question leads to the distinction between parabolic and hyperbolic spaces.
</document>

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# Ordinary point and singular point

## Summary

For a linear differential equation written with leading coefficient $1$, a point is ordinary when all the coefficients are analytic there, and singular otherwise.
At an ordinary point the solutions are themselves analytic, and they can be found as [[power series|08RT9GDDPQ]].
At a singular point the solutions may blow up, branch or oscillate wildly.
The singular points are split further into [[regular|H96HXG6CA0]] and [[irregular|AT1W4MMTYR]] ones.
See the [[definition|T01779NV3Z]].

## Explanations

### L1 human

Divide the equation by the coefficient of the highest derivative.
Where the new coefficients are nice power series, nothing special happens: the point is ordinary.
Where one of them blows up, the point is singular.
For $x y'' + y = 0$, dividing by $x$ gives the coefficient $\frac{1}{x}$, so $x = 0$ is a singular point and every other point is ordinary.
</document>

<document source="https://tamnd.github.io/math2/node/QK55379SXW.md" tag="QK55379SXW" kind="statement">
# Coarea formula

## Statement

Let $n \ge m \ge 1$ be integers and let $u \colon \R^n \to \R^m$ be [[Lipschitz|0AW1E6YTJY]].
By Rademacher's theorem, $u$ is differentiable at [[Lebesgue|2Z2QVWZ262]]-almost every $x$, with [[Jacobian matrix|047NDTEPCV]] $Du(x)$ of size $m \times n$.
Let $J_m u(x) = \sqrt{\det\big(Du(x) \, Du(x)^T\big)}$ at those points.
Let $\mathcal{H}^{n-m}$ be the $(n-m)$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] on $\R^n$.
Then for every Lebesgue [[measurable|YYNCJSR0PJ]] $g \colon \R^n \to [0, \infty]$,
$$\int_{\R^n} g(x) \, J_m u(x) \, dx = \int_{\R^m} \left( \int_{u^{-1}(y)} g \, d\mathcal{H}^{n-m} \right) dy.$$
The same holds for every Lebesgue integrable $g \colon \R^n \to \R$, the inner integral being defined for almost every $y$.

## Remarks

The [[level sets|0B2RZ0778N]] $u^{-1}(y)$ slice $\R^n$, and the formula says that integrating over $\R^n$ is the same as integrating over each slice and then over the parameter $y$.
The weight $J_m u$ measures how much $u$ squeezes volume across the slices.
For a linear $u$ of full rank, the formula is a computation with Fubini's theorem; the general case uses approximation of a Lipschitz map by $C^1$ maps on large sets.

Example: with $m = 1$ and $u(x) = \abs{x}$, one has $J_1 u = \abs{\nabla u} = 1$ for $x \ne 0$, and the level sets are spheres.
The formula becomes integration in polar coordinates: $\int_{\R^n} g \, dx = \int_0^\infty \int_{\set{\abs{x} = r}} g \, d\mathcal{H}^{n-1} \, dr$.

For $m = 1$ and $g = 1$ on a set $A$, it reads $\int_A \abs{\nabla u} \, dx = \int_{\R} \mathcal{H}^{n-1}(A \cap \set{u = t}) \, dt$.
For $m = n$, $\mathcal{H}^0$ is counting measure and the formula is the area formula for maps $\R^n \to \R^n$.

## Explanations

### L1 human

Cut a solid into thin layers along the level sets of a function, like slicing an onion.
The volume of the solid equals the sum of the areas of the layers, weighted by how far apart neighbouring layers are.
Where the function changes fast, the layers are close together, and the weight $\abs{\nabla u}$ corrects for that.

### L2 human

The coarea formula generalises Fubini's theorem from straight slices to curved ones.
For $m = 1$ and $g = 1$ it links the total variation of a function to the perimeters of its level sets.
This is the starting point of the Fleming–Rishel formula for functions of bounded variation.
It is used to prove isoperimetric and Sobolev inequalities and to compute integrals over level sets in probability and geometry.
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<document source="https://tamnd.github.io/math2/node/QKAFRDPGE4.md" tag="QKAFRDPGE4" kind="concept">
# Spectral geometry

## Summary

Spectral geometry studies how the geometry of a Riemannian manifold relates to the spectra of natural differential operators on it.
The main such operator is the [[Laplace–Beltrami operator|F4NWZNTRMD]].
Direct problems bound or compute eigenvalues from geometric data such as curvature, diameter and volume.
Inverse problems ask how much of the geometry is determined by the [[spectrum|B8TYKV7AA1]], as in the question of [[hearing the shape of a drum|X3VH8H1XV3]].
Its tools include the heat kernel, [[Weyl's law|2JK9YMZV2B]] and trace formulas.

## Explanations

### L1 human

A drum, a bell or a guitar string sounds at certain frequencies fixed by its shape.
Spectral geometry studies this link for curved spaces: the frequencies are the eigenvalues of the Laplacian.
From the frequencies one can hear the dimension, the volume and the total curvature.
But not everything can be heard: there exist [[isospectral manifolds|18MQEGN13R]] that sound the same and have different shapes.
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<document source="https://tamnd.github.io/math2/node/QKEE8AT2NR.md" tag="QKEE8AT2NR" kind="concept">
# Whitehead product

## Summary

The Whitehead product is an operation $[\alpha, \beta] \in \pi_{p+q-1}(X)$ on [[homotopy groups|P976WYEFMB]], defined for $\alpha \in \pi_p(X)$ and $\beta \in \pi_q(X)$.
It comes from the attaching map of the top cell of the product $S^p \times S^q$.
For $p = q = 1$ it is the commutator in the fundamental group.
With it, the homotopy groups of a space form a graded Lie algebra, up to a shift of degree and signs.

## Explanations

### L1 human

The product of two spheres $S^p \times S^q$ is built from the wedge $S^p \vee S^q$ by gluing in one more cell.
The gluing map is a sphere of dimension $p + q - 1$ sitting in the wedge.
Given two spheres in a space $X$, sending that gluing map into $X$ gives a new element of a higher homotopy group: their Whitehead product.
For two loops on a torus the product is trivial, which reflects that the two loops commute there.
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<document source="https://tamnd.github.io/math2/node/QKGW3Z9BHW.md" tag="QKGW3Z9BHW" kind="concept">
# Fully invariant congruence

## Summary

A fully invariant congruence of an [[algebra|QJGPT2W0FT]] is a [[congruence|DCSXXT4SER]] that every endomorphism of the algebra maps into itself.
On a [[term algebra|3YYKGWJJND]], the fully invariant congruences are exactly the [[equational theories|F1M51B4MDG]].
These are the sets of identities closed under the rules of equational logic.
This is how equational theories become lattice-theoretic objects in universal algebra.

## Explanations

### L1 human

A congruence says which elements count as equal.
It is fully invariant when this equality survives every way of mapping the algebra into itself by a homomorphism.
In a term algebra, an endomorphism is the same as a substitution of terms for variables.
So a fully invariant congruence there is a set of equations between terms that stays true after any substitution, which is exactly how identities like $x \cdot y = y \cdot x$ behave.
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<document source="https://tamnd.github.io/math2/node/QKJ743GE3J.md" tag="QKJ743GE3J" kind="definition">
# Cardinal function

## Definition

A *cardinal function* is a rule $\varphi$ that assigns to each [[topological space|0CQHFPGJ45]] $X$ a [[cardinal number|0EBACQFAZW]] $\varphi(X)$.
It is required that $\varphi(X) = \varphi(Y)$ whenever $X$ and $Y$ are [[homeomorphic|0QT00PB5G2]].

## Remarks

Common examples, for a space $X$:

1. the weight $w(X)$, the least [[cardinality|0G0ASKNA6Z]] of a [[base|0NA0732YAG]] of the topology;
2. the density $d(X)$, the least cardinality of a [[dense|0BSJSX7ME2]] subset;
3. the character $\chi(X)$, the least cardinal $\kappa$ such that every point has a neighbourhood base of cardinality at most $\kappa$;
4. the cellularity $c(X)$, the supremum of the cardinalities of families of pairwise disjoint nonempty open sets.

Many authors add $\aleph_0$ to each value, so that these functions are infinite.
Two basic inequalities follow.
First, $d(X) \le w(X)$: choose one point in each nonempty member of a base.
Second, $\abs{X} \le 2^{w(X)}$ for a [[T0 space|07SG12TQH3]] $X$, since a point is determined by the base members that contain it.

Example: for $X = \R$, one has $w(X) = d(X) = \aleph_0$, by the base of intervals with rational endpoints and the dense set $\Q$.

Non-example: the rule $X \mapsto$ the number of points of $X$ that lie in $[0, 1]$, defined only for subspaces of $\R$, is not a cardinal function.
It is not invariant under homeomorphism: $[0, 1]$ and $[2, 3]$ are homeomorphic but give $\mathfrak c$ and $0$.
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<document source="https://tamnd.github.io/math2/node/QKJPM71RMG.md" tag="QKJPM71RMG" kind="definition">
# Model category

## Definition

A *model category* is a [[category|A2DE66937D]] $\mathcal{M}$ with all small [[limits|CHS5AFJGWE]] and [[colimits|BJS1CGTKQ5]], together with three classes of morphisms.
They are the [[weak equivalences|E2870TEDG7]] $W$, the *fibrations* $\mathrm{Fib}$ and the *cofibrations* $\mathrm{Cof}$.
Each class contains the identities and is closed under composition.
A morphism in $\mathrm{Fib} \cap W$ is a *trivial fibration*, and a morphism in $\mathrm{Cof} \cap W$ is a *trivial cofibration*.
The classes satisfy the following axioms.

1. (two out of three) If $f$ and $g$ are composable and two of $f$, $g$, $g \circ f$ are in $W$, then so is the third.
2. (retracts) Each of $W$, $\mathrm{Fib}$ and $\mathrm{Cof}$ is closed under [[retracts|DXJH3DGXBC]] in the category of arrows of $\mathcal{M}$.
3. (lifting) Cofibrations have the left [[lifting property|AZK4RD8XV0]] with respect to trivial fibrations, and trivial cofibrations have the left lifting property with respect to fibrations.
4. (factorization) Every morphism $f$ factors as $f = p \circ i$ with $i$ a trivial cofibration and $p$ a fibration. It also factors as $f = q \circ j$ with $j$ a cofibration and $q$ a trivial fibration.

## Remarks

Some authors ask only for finite limits and colimits, as Quillen did, and some ask the factorizations in axiom 4 to be functorial.

Example: let $\mathcal{C}$ be any category with all small limits and colimits, let $W$ be the isomorphisms, and let $\mathrm{Fib}$ and $\mathrm{Cof}$ be all morphisms.
Isomorphisms satisfy two out of three and are closed under retracts.
Every square lifts against an isomorphism, which settles axiom 3, and $f = f \circ \id = \id \circ f$ gives axiom 4.

Non-example: in the category of sets, let all three classes be all maps.
Then the map $i \colon \set{0, 1} \to \set{0}$ is a cofibration and a trivial fibration.
Consider the commutative square with $i$ on both vertical sides, $\id$ on $\set{0, 1}$ on top and $\id$ on $\set{0}$ below.
A lift would be a map $h \colon \set{0} \to \set{0, 1}$ with $h \circ i = \id$, which does not exist, so axiom 3 fails.

The main example is the Quillen model structure on topological spaces: weak equivalences are weak homotopy equivalences, fibrations are Serre fibrations, and cofibrations are retracts of relative cell complexes.
An object $X$ is *fibrant* if $X \to \ast$ is a fibration and *cofibrant* if $\emptyset \to X$ is a cofibration.
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<document source="https://tamnd.github.io/math2/node/QKPMR9MC9W.md" tag="QKPMR9MC9W" kind="concept">
# Lagrangian singularity

## Summary

Take a [[Lagrangian submanifold|H8ACPTMX08]] of a cotangent bundle and project it to the base manifold.
A Lagrangian singularity is a point where this projection fails to be a local diffeomorphism.
The projection of a Lagrangian submanifold is called a Lagrangian map, and its singular points are the Lagrangian singularities.
Their images form the [[caustic|GMRYG4S3G0]], which in optics is where rays focus.
Arnold's theory classifies the generic Lagrangian singularities in low dimensions by the simple singularity types $A_k$, $D_k$ and $E_k$.

## Explanations

### L1 human

Think of a sheet lying in phase space, where each point records a position and a momentum.
Look at the sheet from above, so that you only see positions.
Where the sheet folds over, several of its points sit above the same position, and at the crease the view from above is singular.
Such a crease point is a Lagrangian singularity.
For example, the parabola $q = p^2$ in the $(q, p)$-plane, seen on the $q$-axis, folds at $p = 0$: positions $q > 0$ are hit twice and positions $q < 0$ not at all.
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<document source="https://tamnd.github.io/math2/node/QKR3D65TTH.md" tag="QKR3D65TTH" kind="concept">
# Bidual and canonical embedding

## Summary

The bidual $X^{**}$ of a normed space $X$ is the dual of its [[continuous dual space|4A82S5FVRT]] $X^*$.
The canonical embedding $J \colon X \to X^{**}$ sends a vector $x$ to the functional "evaluate at $x$".
By the [[Hahn–Banach theorem|QB5AE3VYZZ]], $J$ is a linear isometry, so every normed space sits inside its bidual.
Spaces for which $J$ is onto are the [[reflexive spaces|NN8RWPC8WG]].

## Explanations

### L1 human

A vector $x$ can be turned into a machine that eats a functional $f$ and returns the number $f(x)$.
This machine is linear and bounded, so it is an element of the bidual.
In finite dimensions every element of the bidual arises this way, which is the familiar fact that a space is naturally the dual of its dual.
In infinite dimensions the bidual can be bigger: the bidual of the space $c_0$ of null sequences is the space $\ell^\infty$ of bounded sequences.
</document>

<document source="https://tamnd.github.io/math2/node/QKWVXJWHP6.md" tag="QKWVXJWHP6" kind="concept">
# Prime ideal of a lattice

## Summary

A prime ideal of a [[lattice|WH4SFAACAS]] is a proper [[ideal|5GW2T2AM1A]] that contains one of $x$ or $y$ whenever it contains their meet $x \wedge y$.
It is the lattice analogue of a [[prime ideal|00MHHZ9SNE]] of a commutative ring, with the meet in the role of the product.
The complement of a prime ideal is a prime [[filter|Z15FXM9WM1]].
Prime ideals are the points of the spaces used to represent distributive lattices and Boolean algebras.

## Explanations

### L1 human

An ideal is prime when it cannot catch the common part of two elements without catching one of the two elements.
Fix a point $p$ of a set $X$ and look at all subsets of $X$ that do not contain $p$.
They form a prime ideal of the subsets of $X$: if $A \cap B$ misses $p$, then $A$ or $B$ misses $p$.
Splitting the lattice into this ideal and its complement is like asking the single yes-or-no question "does the set contain $p$?".
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<document source="https://tamnd.github.io/math2/node/QM0CGSQ4TF.md" tag="QM0CGSQ4TF" kind="concept">
# Fermat point

## Summary

The Fermat point of a [[triangle|0CTNFGYMM2]] is the point of the plane whose total distance to the three vertices is as small as possible.
When every angle of the triangle is less than $120^\circ$, it is the interior point that sees each side under $120^\circ$.
It is a model case of a geometric minimisation problem and of a [[triangle center|WRKFH25NCX]].

## Explanations

### L1 human

Three villages want to build one meeting point and join it to each village by a straight road.
The Fermat point is the place that makes the total length of road as short as possible.
At this point the three roads meet at equal angles of $120^\circ$, unless one corner of the triangle is already very blunt.
For an equilateral triangle the Fermat point is simply the centre.
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<document source="https://tamnd.github.io/math2/node/QM2EBW5CD9.md" tag="QM2EBW5CD9" kind="statement">
# Ky Fan minimax inequality

## Statement

Let $E$ be a real Hausdorff [[topological vector space|T32W553W95]], and let $C \subseteq E$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].
Let $f \colon C \times C \to \R$ be a function such that:

1. for every $y \in C$, the function $x \mapsto f(x, y)$ is lower [[semicontinuous|RN4G8SGP6P]] on $C$;
2. for every $x \in C$, the function $y \mapsto f(x, y)$ is quasiconcave on $C$, that is, $y \mapsto -f(x, y)$ is [[quasiconvex|S4RMH7JQB8]].

Then there is $\bar x \in C$ with
$$\sup_{y \in C} f(\bar x, y) \le \sup_{y \in C} f(y, y).$$
In particular, if $f(y, y) \le 0$ for all $y \in C$, then there is $\bar x \in C$ with $f(\bar x, y) \le 0$ for all $y \in C$.

## Remarks

It suffices to prove the second form, applied to $f - c$ with $c = \sup_y f(y, y)$ when $c$ is finite; the first form is trivial when $c = +\infty$.
For each $y$ the set $S(y) = \set{x \in C : f(x, y) \le 0}$ is closed by clause 1.
If a point $x$ of the convex hull of $y_1, \dots, y_k$ lay outside every $S(y_i)$, clause 2 would give $f(x, x) \ge \min_i f(x, y_i) > 0$, which is impossible.
So the sets $S(y)$ satisfy the hypothesis of the Knaster–Kuratowski–Mazurkiewicz lemma in Fan's form, and by compactness of $C$ they have a common point $\bar x$.

Example: let $\Delta_k$ be the set of probability vectors in $\R^k$, and let $A$ be a real $m \times n$ matrix.
The payoff of the matrix game is $g(p, q) = p^T A q$ for $p \in \Delta_m$ and $q \in \Delta_n$, as in [[von Neumann's minimax theorem|GJ8VTBFNP2]].
Let $C = \Delta_m \times \Delta_n$, and for $z = (p, q)$ and $w = (p', q')$ in $C$ let $f(z, w) = g(p', q) - g(p, q')$.
This $f$ is continuous, affine in $w$, and $f(z, z) = 0$.
The point $\bar z = (\bar p, \bar q)$ given by the theorem satisfies $g(p', \bar q) \le g(\bar p, q')$ for all $p', q'$, so $(\bar p, \bar q)$ is a pair of optimal strategies.

Compactness cannot be dropped: on $C = [0, \infty) \subseteq \R$, the function $f(x, y) = y - x$ satisfies both clauses and $f(y, y) = 0$, but for every $x$ there is $y > x$ with $f(x, y) > 0$.

## Explanations

### L1 human

The inequality gives a point $\bar x$ that no other point $y$ can beat, as measured by $f(\bar x, y)$.
It asks only that comparing a point with itself gives no gain, and that $f$ has some mild continuity and convexity.
Many existence results for equilibria have this form, so they all follow from this one statement.

### L2 human

Ky Fan proved the inequality in 1972 as a common source of minimax theorems, fixed-point theorems and existence theorems for equilibria.
From it one derives the minimax theorems of von Neumann and Sion, Nash equilibria of games with convex strategy sets, and solutions of [[variational inequalities|6QS7X5GVMY]] on compact convex sets.
Its proof rests on the KKM lemma, which is equivalent to the Brouwer fixed-point theorem.
</document>

<document source="https://tamnd.github.io/math2/node/QM5ZCBJYP1.md" tag="QM5ZCBJYP1" kind="definition">
# Finite type algebra

## Definition

Let $A$ be a commutative ring and let $B$ be a commutative [[$A$-algebra|DPBJ04JMH8]] with structure map $f \colon A \to B$.
The algebra $B$ is *of finite type* over $A$, or *finitely generated* as an $A$-algebra, if it has a finite generating set.
That means there are $b_1, \ldots, b_n \in B$ such that every element of $B$ is $p(b_1, \ldots, b_n)$ for a polynomial $p$ with coefficients in $f(A)$.
Equivalently, there is a surjective $A$-algebra homomorphism from the [[polynomial ring|0SDC934R52]] $A[x_1, \ldots, x_n]$ onto $B$ for some $n \ge 0$.

## Remarks

Example: $k[x, x^{-1}]$ is of finite type over a field $k$.
It is generated by $x$ and $x^{-1}$, and it is isomorphic to $k[x, y]/(xy - 1)$.

Non-example: $\Q$ is not of finite type over $\Z$.
Suppose $\Q = \Z[q_1, \ldots, q_n]$, and let $d$ be the product of the denominators of the $q_i$.
Then every element of $\Q$ would have a power of $d$ as a denominator, which fails for $1/p$ with $p$ a prime not dividing $d$.

A [[finite|1NBACAEC2V]] algebra is of finite type, but $k[x]$ over $k$ shows that the converse fails.
A finitely generated algebra over a Noetherian ring is Noetherian, by [[Hilbert's basis theorem|0YKA9J3XEE]].
</document>

<document source="https://tamnd.github.io/math2/node/QMH530N27M.md" tag="QMH530N27M" kind="statement">
# Weyl integration formula

## Statement

Let $G$ be a connected [[compact Lie group|9TFC6ADC1C]] with Lie algebra $\mathfrak{g}$.
Let $T$ be a [[maximal torus|3JVGZVVTB0]] with Lie algebra $\mathfrak{t}$, and let $W$ be the [[Weyl group|EZGTEK3HQN]].
Give $G$, $T$ and $G/T$ their invariant probability measures $dg$, $dt$ and $d\dot g$ (normalised [[Haar measures|VY0HXD3GC4]]).
For $t \in T$ let
$$\delta(t) = \det\left(\id - \operatorname{Ad}(t^{-1})\big|_{\mathfrak{g}/\mathfrak{t}}\right),$$
where $\operatorname{Ad}$ is the [[adjoint representation|P6ZP0TE07M]].
Then for every continuous function $f \colon G \to \C$,
$$\int_G f(g) \, dg = \frac{1}{\abs{W}} \int_T \delta(t) \int_{G/T} f(g t g^{-1}) \, d\dot g \, dt.$$
In particular, if $f$ is a class function, then $\int_G f(g) \, dg = \frac{1}{\abs{W}} \int_T f(t)\,\delta(t) \, dt$.

## Remarks

The map $G/T \times T \to G$, $(gT, t) \mapsto gtg^{-1}$, is surjective by the conjugacy of maximal tori, and over the dense set of regular elements it is a covering of degree $\abs{W}$.
Its Jacobian at $(gT, t)$ is $\delta(t)$, which gives the formula by change of variables.
In terms of roots, $\delta(t) = \prod_{\alpha} (1 - t^{-\alpha})$, the product over all roots $\alpha$, which equals $\abs{\prod_{\alpha > 0} (t^{\alpha/2} - t^{-\alpha/2})}^2$ when this makes sense, so $\delta \ge 0$.
Example: for $G = SU(2)$ and $t = \operatorname{diag}(e^{i\theta}, e^{-i\theta})$, $\delta(t) = \abs{e^{i\theta} - e^{-i\theta}}^2 = 4\sin^2\theta$ and $\abs{W} = 2$.
So for a class function $\int_{SU(2)} f \, dg = \frac{1}{\pi}\int_0^{2\pi} f(t_\theta) \sin^2\theta \, d\theta$; for $f = 1$ this gives $\frac{1}{\pi} \cdot \pi = 1$.

## Explanations

### L1 human

To integrate a function over a compact Lie group, one can first average over conjugates and then integrate over a maximal torus.
The torus is much smaller, but its points must be weighted: points where many conjugates pile up get more weight.
The weight is an explicit function $\delta$, and the formula divides by the number of times the torus is covered.

### L2 human

The formula is the main tool for computing integrals of characters.
With the [[Schur orthogonality relations|ETNDTECY97]] it shows that an irreducible character times the Weyl denominator is an alternating sum of characters of $T$.
This is the core of the proof of the [[Weyl character formula|230F5FXPMQ]].
In random matrix theory, the case $G = U(n)$ gives the joint density of eigenvalues of a Haar-random unitary matrix, proportional to $\prod_{j < k} \abs{z_j - z_k}^2$.
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# Graphon

## Summary

A graphon is a symmetric measurable function $W \colon [0, 1]^2 \to [0, 1]$.
Graphons are the limit objects of sequences of dense [[graphs|00C0BR7D7H]].
A sequence converges when the density of each fixed small graph in it converges, and then the limit is a graphon.
They also give the most general exchangeable model of a [[random graph|TNT3DG0HRJ]].
Graphons turn questions of [[extremal graph theory|SC5QB5F4TE]], such as [[Sidorenko's conjecture|KN2PXSCT9J]], into questions of analysis.

## Explanations

### L1 human

Draw the adjacency matrix of a large graph as a black and white picture in a unit square, black for an edge and white for no edge.
When the graph grows, the pictures can blur into a grey picture with shades between $0$ and $1$.
That grey picture is a graphon: the shade at a point $(x, y)$ is the probability of an edge between "vertices" $x$ and $y$.
For example, large random graphs with edge probability $1/2$ converge to the constant graphon $W = 1/2$.
</document>

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# Mather–Yau theorem

## Statement

Let $\mathcal{O}_{n+1} = \C\{z_0, \dots, z_n\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]].
For $f \in \mathcal{O}_{n+1}$ with $f(0) = 0$, let the *Tjurina algebra* be
$$A(f) = \mathcal{O}_{n+1} \Big/ \left( f, \frac{\partial f}{\partial z_0}, \dots, \frac{\partial f}{\partial z_n} \right),$$
the quotient by the [[ideal|096Z9MP2ME]] generated by $f$ and its partial derivatives.
Let $f, g \in \mathcal{O}_{n+1}$ have [[isolated singularities|VTGYEES3VS]] at $0$.
Then the following are equivalent:

1. the hypersurface germs $(f^{-1}(0), 0)$ and $(g^{-1}(0), 0)$ are biholomorphic, that is, there are a biholomorphic germ $\phi$ of $(\C^{n+1}, 0)$ and a unit $u \in \mathcal{O}_{n+1}$ with $g = u \cdot (f \circ \phi)$;
2. $A(f)$ and $A(g)$ are isomorphic as $\C$-algebras.

## Remarks

John Mather and Stephen Yau proved the theorem in 1982.
The direction from 1 to 2 is a direct computation, since the ideal $(f, \partial f / \partial z_j)$ is carried to $(g, \partial g / \partial z_j)$ by $\phi$ and $u$.
The converse is the hard part.
Its proof uses the finite determinacy of isolated singularities and a study of the action of the group of coordinate changes on jets.

Example: for $f = x^2 + y^{k+1}$ with $k \ge 1$, the ideal is $(x, y^k)$, so $A(f) \cong \C[y] / (y^k)$.
These algebras have different dimensions for different $k$, so the singularities $A_k$ are pairwise not biholomorphic.

The dimension $\tau(f) = \dim_\C A(f)$ is the *Tjurina number*.
It satisfies $\tau(f) \le \mu(f)$, and K. Saito proved that $\tau(f) = \mu(f)$ if and only if $f$ is right equivalent to a weighted homogeneous polynomial.

## Explanations

### L1 human

Two hypersurfaces with an isolated singular point may look different only because of the choice of coordinates.
The theorem gives a finite test for whether they are really the same: compute one finite-dimensional algebra for each and compare.
So a geometric classification problem becomes a problem about finite-dimensional commutative algebras.

### L2 human

The Tjurina algebra is also the space of first-order deformations of the hypersurface germ: its dimension $\tau$ is the number of parameters of the miniversal deformation.
The theorem says that this algebra remembers the whole singularity.
Together with the [[Milnor number|1ZE51P7GRQ]] and the [[classification of simple singularities|YE5F548WMS]], it is a basic tool of the analytic classification of isolated hypersurface singularities.
</document>

<document source="https://tamnd.github.io/math2/node/QMN3W21C9M.md" tag="QMN3W21C9M" kind="concept">
# Box spline

## Summary

A box spline is a piecewise polynomial function of several variables obtained by projecting a unit cube of higher dimension onto $\R^d$.
Its value at a point is the volume of the slice of the cube that lands at that point.
Box splines generalize the uniform [[B-splines|62AFX32JGT]] to several variables, and their integer translates give smooth spline spaces on regular grids and triangulations.

## Explanations

### L1 human

Shine light through a cube onto a lower dimensional screen and record how thick the cube is above each point of the screen.
That thickness profile is a box spline.
Mapping the square $[0,1]^2$ to the line by $(u_1, u_2) \mapsto u_1 + u_2$ gives a tent, the hat function on $[0, 2]$.
More directions give smoother bumps, and in the plane one gets smooth bumps adapted to grids of squares or triangles.
</document>

<document source="https://tamnd.github.io/math2/node/QMRAVX14BJ.md" tag="QMRAVX14BJ" kind="concept">
# Dirac operator on a spin manifold

## Summary

The Dirac operator of a Riemannian spin manifold, also called the Atiyah–Singer operator, acts on sections of the [[spinor bundle|PYYQQKAZ0C]].
It combines the spin connection, induced by the [[Levi-Civita connection|CAZYGE1T4E]], with Clifford multiplication by tangent vectors.
It is a first-order elliptic and formally self-adjoint operator whose square is the rough Laplacian plus a quarter of the scalar curvature.
On a closed spin manifold of dimension divisible by $4$, the index of its restriction to positive spinors is the $\hat{A}$-genus.
This ties the existence of metrics of positive scalar curvature to topology.
It is the main example of a [[Dirac operator|3DXGJTEQZH]].

## Explanations

### L1 human

On flat $\R^n$, Dirac found a first-order operator whose square is minus the Laplacian, using matrices that anticommute.
To do the same on a curved manifold one needs a bundle on which these matrices can act consistently from point to point.
A spin structure is exactly the extra data that makes this possible, and the spinor bundle is the resulting bundle.
The Dirac operator differentiates a spinor field along each direction of an orthonormal frame and multiplies by the matching Clifford matrix, then adds the results.
</document>

<document source="https://tamnd.github.io/math2/node/QMW1MWQH6J.md" tag="QMW1MWQH6J" kind="concept">
# Abel transform

## Summary

The Abel transform of a function $f$ of a radius $r \ge 0$ is $F(y) = 2 \int_y^\infty \frac{f(r)\, r}{\sqrt{r^2 - y^2}}\, dr$.
It gives the integral of a circularly symmetric function of the plane along a line at distance $y$ from the centre.
So it is the [[Radon transform|VJNZJ352PC]] of a radial function.
It can be inverted by a formula of the same type, which solves an integral equation first studied by Abel.
It is used to recover radial profiles of emission or density from line-of-sight measurements, for example in plasma physics and astronomy.

## Explanations

### L1 human

Picture a round, glowing cloud that is brighter near the centre.
Looking at it from the side, each line of sight passes through the cloud and adds up its brightness along the way.
The Abel transform turns the brightness at each radius into the total seen along each line.
For a uniform disc of radius $R$, the line at distance $y < R$ meets it in a chord of length $2\sqrt{R^2 - y^2}$, and this is the Abel transform.
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<document source="https://tamnd.github.io/math2/node/QMY5JTZZB8.md" tag="QMY5JTZZB8" kind="concept">
# Laurent polynomial ring

## Summary

The Laurent polynomial ring $R[x, x^{-1}]$ over a [[ring|0B1261B5ED]] $R$ consists of finite sums $\sum a_i x^i$ in which the exponents $i$ may be negative.
It is the [[monoid ring|1HM7ZYMPD7]] of the additive group $\Z$ over $R$.
In it the variable $x$ is a [[unit|0XVEZDE9J8]] with inverse $x^{-1}$.
It is the group ring of the infinite cyclic group.
Over a field it is the coordinate ring of the line with the origin removed.

## Explanations

### L1 human

A Laurent polynomial is a polynomial that may also contain negative powers of the variable, like $3x^{-2} + 1 + 5x$.
There are still only finitely many terms.
Adding and multiplying works as for polynomials, with the rule $x^i x^j = x^{i+j}$ for all integers $i, j$.
For example, $(x + x^{-1})^2 = x^2 + 2 + x^{-2}$.
The point of the construction is that $x$ now has an inverse, which it does not have among ordinary polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/QMYZ798EAR.md" tag="QMYZ798EAR" kind="definition">
# Identity component

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] with identity element $e$.
The *identity component* of $G$, written $G_0$ or $G^\circ$, is the [[connected component|0JG9W3KA6B]] of $G$ that contains $e$.

## Remarks

$G_0$ is a closed [[normal subgroup|0CB0DX5BVV]] of $G$.
It is a subgroup because $G_0 G_0^{-1}$ is the image of the connected set $G_0 \times G_0$ under $(x, y) \mapsto xy^{-1}$ and contains $e$, so it lies in $G_0$.
It is normal because each inner automorphism $x \mapsto gxg^{-1}$ is a homeomorphism fixing $e$.
The [[quotient|417ZRDCHY6]] $G/G_0$ is a [[totally disconnected group|AW3J9AQSA0]].
Example: for $G = \mathrm{GL}_n(\R)$, the identity component is the subgroup of matrices with positive determinant.
For $G = (\Q, +)$ with the topology from $\R$, the identity component is $\set{0}$, since $\Q$ is totally disconnected.
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<document source="https://tamnd.github.io/math2/node/QN2210XCST.md" tag="QN2210XCST" kind="statement">
# Moore–Osgood theorem

## Statement

Let $(X, d)$ be a [[complete metric space|08EWT7J0SC]], for example $\R$ or $\C$, and let $(a_{m,n})_{m, n \ge 1}$ be a double sequence in $X$.
Assume:

1. for each $m$, the limit $b_m = \lim_{n \to \infty} a_{m,n}$ exists;
2. there is a sequence $(c_n)$ such that $a_{m,n} \to c_n$ as $m \to \infty$, [[uniformly|08FPABDQ4C]] in $n$, that is, $\sup_n d(a_{m,n}, c_n) \to 0$.

Then the limits $\lim_{m \to \infty} b_m$ and $\lim_{n \to \infty} c_n$ exist, the [[Pringsheim double limit|4FK55CYMVZ]] $\lim_{m,n \to \infty} a_{m,n}$ exists, and all three are equal.

## Remarks

Proof idea: uniform convergence gives an $M$ such that $d(a_{m,n}, a_{m',n}) < 2\varepsilon$ for all $n$ when $m, m' \ge M$.
Letting $n \to \infty$ gives $d(b_m, b_{m'}) \le 2\varepsilon$, so $(b_m)$ is a Cauchy sequence and has a limit $L$ by completeness.
The triangle inequality $d(c_n, L) \le d(c_n, a_{M,n}) + d(a_{M,n}, b_M) + d(b_M, L)$ then shows $c_n \to L$, and the same splitting gives the double limit.
Example: for $a_{m,n} = \frac{1}{m} + \frac{1}{n}$ the convergence in $m$ is uniform, and all limits equal $0$.
Without uniformity the conclusion fails: $a_{m,n} = \frac{m}{m+n}$ gives $\lim_m \lim_n a_{m,n} = 0$ but $\lim_n \lim_m a_{m,n} = 1$.
Without completeness it also fails: in $\mathbb{Q}$, take rationals $x_k \to \sqrt{2}$ and $a_{m,n} = x_{\min(m,n)}$; both hypotheses hold, but $b_m = x_m$ has no limit in $\mathbb{Q}$.

## Explanations

### L1 human

Two limits taken one after the other may give different answers, depending on the order.
The theorem says that if one of the inner limits is reached uniformly, the order does not matter, and the double limit exists too.

### L2 human

This is the abstract principle behind many interchange results in analysis.
The theorem that a uniform limit of continuous functions is continuous, and the theorem that uniform limits commute with integration, follow the same pattern.
For [[double series|PAXCH4FFSV]], it gives conditions under which summing by rows, by columns, or over squares gives the same value.
</document>

<document source="https://tamnd.github.io/math2/node/QN7M1JCGT9.md" tag="QN7M1JCGT9" kind="concept">
# Satisfiability

## Summary

A formula is satisfiable when some interpretation makes it true.
In [[propositional logic|09K9MWD2QQ]] an interpretation is an assignment of truth values to the variables.
In [[predicate logic|086QKPF0A6]] it is a [[structure|983GZBQYT4]] with an assignment of its elements to the free variables.
Satisfiability is dual to validity: a formula is valid exactly when its negation is not satisfiable.

## Explanations

### L1 human

A formula is satisfiable when it is possible to make it true.
For example, $p \land \neg q$ is satisfiable: take $p$ true and $q$ false.
The formula $p \land \neg p$ is not satisfiable, since $p$ cannot be true and false at once.
Deciding satisfiability of propositional formulas is the SAT problem, a central problem of computer science.
</document>

<document source="https://tamnd.github.io/math2/node/QN9X9BNKTX.md" tag="QN9X9BNKTX" kind="example">
# Logistic map

## Example

For a parameter $r \in [0, 4]$, the *logistic map* is
$$f_r \colon [0, 1] \to [0, 1], \qquad f_r(x) = r x (1 - x).$$
It is a [[discrete-time dynamical system|5AA9AJEARJ]] on $[0, 1]$: the maximum of $f_r$ on $[0, 1]$ is $f_r(1/2) = r/4 \le 1$ and $f_r \ge 0$ there, so $f_r$ maps $[0, 1]$ into itself.

Check of the first bifurcations.
The [[fixed points|R6QVS6P21N]] are $0$ and, for $r > 1$, $x_r = 1 - 1/r$.
Since $f_r'(x) = r(1 - 2x)$, one gets $f_r'(0) = r$ and $f_r'(x_r) = 2 - r$.
So for $1 < r < 3$ the fixed point $x_r$ attracts nearby points.
At $r = 3$ its derivative is $-1$, and the map has a [[period-doubling bifurcation|NQDTTZ59X8]].
For $r$ slightly above $3$ an attracting orbit of period $2$ appears.

## Remarks

As $r$ grows further, attracting orbits of periods $4, 8, 16, \dots$ appear at parameters that accumulate at $r_\infty \approx 3.5699$.
For $r = 4$ the map is [[topologically conjugate|HNXXAPQKG3]] to the full [[tent map|D0A22MEB1F]] $T(y) = 1 - \abs{1 - 2y}$ by $h(y) = \sin^2(\pi y / 2)$.
Indeed $f_4(h(y)) = 4 \sin^2(\pi y/2) \cos^2(\pi y/2) = \sin^2(\pi y)$, and $h(T(y)) = \sin^2(\pi y)$ for $y \le 1/2$ and $\sin^2(\pi - \pi y) = \sin^2(\pi y)$ for $y > 1/2$.
For $r = 4$ the measure with density $\frac{1}{\pi \sqrt{x(1 - x)}}$ on $(0, 1)$ is invariant.
The map is the discrete-time analogue of the [[logistic equation|0CC4VC7KR3]] of population growth.

## Explanations

### L1 human

The logistic map is a toy model of a population from one year to the next: it grows when small and is held back when large.
One step takes just two multiplications and a subtraction.
Yet as $r$ increases the long-run behaviour goes from a steady state to a cycle of $2$ years, then $4$, $8$, and then to chaos.
It became famous as the simplest example of how chaos can arise from a very simple rule.
</document>

<document source="https://tamnd.github.io/math2/node/QNGGCTXK9S.md" tag="QNGGCTXK9S" kind="statement">
# Whipple's transformation

## Statement

Let $n$ be a nonnegative integer and let $a, b, c, d, e$ be complex numbers.
Assume that none of $\frac{a}{2}, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a + n$ and $d + e - a - n$ lies in $\set{0, -1, \dots, 1 - n}$.
Write $(x)_k = x (x + 1) \cdots (x + k - 1)$ for the [[Pochhammer symbol|YZHM3HSM0F]].
Then the terminating [[generalized hypergeometric series|VDBAW9PN88]] satisfy
$${}_7F_6\left( \begin{matrix} a,\ 1 + \frac{a}{2},\ b,\ c,\ d,\ e,\ -n \\ \frac{a}{2},\ 1 + a - b,\ 1 + a - c,\ 1 + a - d,\ 1 + a - e,\ 1 + a + n \end{matrix} ; 1 \right)$$
$$= \frac{(1 + a)_n\, (1 + a - d - e)_n}{(1 + a - d)_n\, (1 + a - e)_n}\; {}_4F_3\left( \begin{matrix} -n,\ d,\ e,\ 1 + a - b - c \\ 1 + a - b,\ 1 + a - c,\ d + e - a - n \end{matrix} ; 1 \right).$$

## Remarks

The left side is a very well-poised ${}_7F_6$: each upper parameter plus the matching lower parameter gives $1 + a$.
The ${}_4F_3$ on the right is balanced: its lower parameters sum to $1$ more than its upper parameters.
So the transformation trades well-poised structure for balanced structure, and the number of free parameters stays the same.
A standard proof uses the Pfaff–Saalschütz summation together with an exchange of the order of summation.
For $n = 0$ both sides equal $1$.
If moreover $1 + 2a = b + c + d + e - n$, then $d + e - a - n = 1 + a - b - c$, the ${}_4F_3$ reduces to a balanced ${}_3F_2$, and the Pfaff–Saalschütz theorem sums it; this proves Dougall's formula.

## Explanations

### L1 human

Whipple's transformation rewrites one long finite hypergeometric sum as a shorter one with a simple factor in front.
The two sums look very different, but they have the same value.
Identities of this kind let one move a hard sum into a form where a known summation formula applies.

### L2 human

The theorem connects the two main families of hypergeometric series at $1$: the well-poised ones and the balanced ones.
Combined with the Pfaff–Saalschütz theorem it gives Dougall's ${}_7F_6$ summation.
Its $q$-analogue is Watson's transformation between a very well-poised ${}_8\phi_7$ and a balanced ${}_4\phi_3$.
The name Whipple is attached to several other transformations of ${}_3F_2$ and ${}_7F_6$ series as well.
</document>

<document source="https://tamnd.github.io/math2/node/QNHYXDFJJY.md" tag="QNHYXDFJJY" kind="concept">
# Prolate spheroidal wave functions

## Summary

The prolate spheroidal wave functions are the bounded solutions on $[-1, 1]$ of a second-order differential equation that arises from separating variables in prolate spheroidal coordinates.
D. Slepian and H. O. Pollak found that the same functions are eigenfunctions of a time and band limiting operator.
This operator first restricts a function to $[-1, 1]$ and then removes all frequencies outside a fixed band.
So they are the [[band-limited functions|P3T104G564]] that are most concentrated on an interval.
They answer the question of how much of its energy a band-limited signal can keep in a given time window.
They are used in signal processing and spectral estimation.
See the [[definition|SDYEK7HEZQ]].

## Explanations

### L1 human

No nonzero signal can be both limited in time and limited in frequency.
One can still ask which band-limited signal puts as large a share of its energy as possible inside a given time interval.
The answer is the first prolate spheroidal wave function; the next ones give the best signals that are orthogonal to the previous ones.
When the product of the time interval and the frequency band is large, many of them keep almost all their energy inside the interval.
Their number is roughly proportional to that product.
</document>

<document source="https://tamnd.github.io/math2/node/QNMR5F525C.md" tag="QNMR5F525C" kind="definition">
# Sufficient statistic

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a [[statistical model|JZ6R7J36K5]] on a measurable space $(S, \mathcal{S})$, and let $X$ be the identity random element of $S$.
Let $T = T(X)$ be a [[statistic|036756NFD6]] with values in a measurable space.
The statistic $T$ is *sufficient* for $\theta$ if the following holds for every $A \in \mathcal{S}$.
There is a version of the conditional probability $P_\theta(X \in A \mid T)$ that is the same for all $\theta \in \Theta$.
That is, there is a function $k_A(T)$ with $k_A(T) = P_\theta(X \in A \mid T)$ $P_\theta$-almost surely, for every $\theta$, as a [[conditional expectation|0YAMJRK04E]] of $\mathbf{1}\set{X \in A}$ given $T$.

## Remarks

Example: let $X = (X_1, \dots, X_n)$ be i.i.d. Bernoulli($p$), $p \in (0, 1)$, and $T = X_1 + \dots + X_n$.
For $x \in \set{0, 1}^n$ with $x_1 + \dots + x_n = t$, one has $P_p(X = x \mid T = t) = p^t(1 - p)^{n - t} / \left(\binom{n}{t} p^t (1 - p)^{n - t}\right) = 1/\binom{n}{t}$, which does not depend on $p$; so $T$ is sufficient.
Non-example: for $n \ge 2$, the statistic $X_1$ is not sufficient, since given $X_1$ the variable $X_2$ is still Bernoulli($p$), and its law depends on $p$.
The whole sample $X$ is always sufficient, and any one-to-one measurable function of a sufficient statistic with measurable inverse is sufficient.
</document>

<document source="https://tamnd.github.io/math2/node/QNMSHKBBSG.md" tag="QNMSHKBBSG" kind="definition">
# Fox H-function transforms

## Definition

Fix integers $0 \le m \le q$ and $0 \le n \le p$, complex numbers $a_1, \dots, a_p$, $b_1, \dots, b_q$, and positive reals $A_1, \dots, A_p$, $B_1, \dots, B_q$.
Let $H(z)$ be the [[Fox H-function|EAZMKENA4N]] $H^{m,n}_{p,q}$ with the parameter pairs $(a_j, A_j)$ and $(b_j, B_j)$.
Let $f \colon (0, \infty) \to \C$ be such that the integral below converges for $y$ in a set $Y \subseteq (0, \infty)$.
The *H-transform* of $f$ with these parameters is the function
$$g(y) = \int_0^\infty H(xy)\, f(x)\, dx, \qquad y \in Y.$$

## Remarks

The [[Mellin transform|N5PVGRJE1T]] of the kernel is
$$\mathcal{M} H(s) = \frac{\prod_{j=1}^{m} \Gamma(b_j + B_j s) \prod_{j=1}^{n} \Gamma(1 - a_j - A_j s)}{\prod_{j=m+1}^{q} \Gamma(1 - b_j - B_j s) \prod_{j=n+1}^{p} \Gamma(a_j + A_j s)},$$
and formally $\mathcal{M} g(s) = \mathcal{M} H(s)\, \mathcal{M} f(1 - s)$.

Example: with all $A_j = B_j = 1$, the kernel is a Meijer G-function and the H-transform is a [[Meijer G-function transform|C0BCHX106R]].
In particular $m = q = 1$, $n = p = 0$, $b_1 = 0$, $B_1 = 1$ gives the kernel $e^{-z}$, whose Mellin transform is $\Gamma(s)$, and the H-transform is the [[Laplace transform|0V49JEZW06]].

Example with a scale different from $1$: $m = q = 1$, $n = p = 0$, $b_1 = 0$, $B_1 = 1/2$ gives the kernel with Mellin transform $\Gamma(s/2)$, which is $2 e^{-z^2}$.
The check is $\int_0^\infty 2 e^{-z^2} z^{s-1}\, dz = \Gamma(s/2)$, by the substitution $u = z^2$.

Non-example: if some $A_j$ or $B_j$ is $0$ or negative, the parameters are outside the definition of the H-function.
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<document source="https://tamnd.github.io/math2/node/QNMWK3GCF2.md" tag="QNMWK3GCF2" kind="definition">
# Pseudofunctor

## Definition

Let $\mathcal{B}$ and $\mathcal{B}'$ be [[bicategories|TAHV24ED74]] with associators $\alpha, \alpha'$ and unitors $\lambda, \rho, \lambda', \rho'$.
A *pseudofunctor* $F \colon \mathcal{B} \to \mathcal{B}'$ consists of:

1. for each object $a$ of $\mathcal{B}$, an object $F(a)$ of $\mathcal{B}'$;
2. for each pair $a, b$, a [[functor|28NQ0KRHD3]] $F_{a,b} \colon \mathcal{B}(a, b) \to \mathcal{B}'(F(a), F(b))$;
3. for composable $f \colon a \to b$ and $g \colon b \to c$, invertible 2-morphisms $\phi_{g,f} \colon F(g) \circ F(f) \Rightarrow F(g \circ f)$, [[natural|KYCPCC57FT]] in $f$ and $g$;
4. for each object $a$, an invertible 2-morphism $\phi_a \colon 1_{F(a)} \Rightarrow F(1_a)$.

These data satisfy, for all $f \colon a \to b$, $g \colon b \to c$, $h \colon c \to d$:

1. $F(\alpha_{h,g,f}) \cdot \phi_{h \circ g, f} \cdot (\phi_{h,g} \circ 1_{F(f)}) = \phi_{h, g \circ f} \cdot (1_{F(h)} \circ \phi_{g,f}) \cdot \alpha'_{F(h),F(g),F(f)}$;
2. $F(\lambda_f) \cdot \phi_{1_b, f} \cdot (\phi_b \circ 1_{F(f)}) = \lambda'_{F(f)}$;
3. $F(\rho_f) \cdot \phi_{f, 1_a} \cdot (1_{F(f)} \circ \phi_a) = \rho'_{F(f)}$.

Here $\cdot$ is vertical composition of 2-morphisms.

## Remarks

If the $\phi_{g,f}$ and $\phi_a$ are only required to be 2-morphisms, not invertible, $F$ is called a lax functor; if they are identities, $F$ is a strict 2-functor.
Example: regard the category of rings as a [[2-category|CQACXG10JZ]] with only identity 2-morphisms.
Send a ring $R$ to its category of left modules, and a ring map $u \colon R \to S$ to extension of scalars $S \otimes_R -$.
The canonical isomorphisms $T \otimes_S (S \otimes_R M) \cong T \otimes_R M$ and $R \otimes_R M \cong M$ give $\phi$, and the axioms hold because these isomorphisms are the standard ones on elementary tensors.
Non-example: a monoidal category is a one-object bicategory, and a lax [[monoidal functor|PY430TXMG0]] gives a lax functor.
The forgetful functor $(\mathbf{Ab}, \otimes) \to (\mathbf{Set}, \times)$ with $\phi(a, b) = a \otimes b$ is a lax functor but not a pseudofunctor, since $A \times B \to A \otimes B$ is not invertible.
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<document source="https://tamnd.github.io/math2/node/QNS8WFYFXJ.md" tag="QNS8WFYFXJ" kind="definition">
# Cauchy–Riemann equations in several variables

## Definition

Let $D \subseteq \C^n$ be an open set, write $z_j = x_j + i y_j$ with $x_j, y_j$ real, and let $f = u + iv \colon D \to \C$ with $u, v$ real-valued.
Suppose that all first [[partial derivatives|09EGC6F8PT]] of $u$ and $v$ exist on $D$.
The function $f$ satisfies the *Cauchy–Riemann equations* on $D$ if for every $j = 1, \dots, n$, at every point of $D$,
$$\frac{\partial u}{\partial x_j} = \frac{\partial v}{\partial y_j}, \qquad \frac{\partial u}{\partial y_j} = -\frac{\partial v}{\partial x_j}.$$

## Remarks

With the [[Wirtinger derivatives|WMXDVF36ZT]] $\frac{\partial}{\partial \bar z_j} = \frac12 \left( \frac{\partial}{\partial x_j} + i \frac{\partial}{\partial y_j} \right)$, the $2n$ real equations are the $n$ complex equations $\partial f / \partial \bar z_j = 0$, $j = 1, \dots, n$.
For $n = 1$ they are the classical [[Cauchy–Riemann equations|0T71SSYZ39]].
A function $f$ of class $C^1$ on $D$ is [[holomorphic|507C34W0ZT]] if and only if it satisfies the Cauchy–Riemann equations on $D$.
Example: $f(z_1, z_2) = z_1 z_2$ satisfies them, since $\partial f / \partial \bar z_1 = 0$ and $\partial f / \partial \bar z_2 = 0$.
Non-example: $f(z_1, z_2) = z_1 \bar z_2$ satisfies the pair for $j = 1$ but not the pair for $j = 2$, since $\partial f / \partial \bar z_2 = z_1$, which is not zero when $z_1 \ne 0$.
</document>

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# Congruence generated by a relation

## Summary

The congruence generated by a relation $R$ on an [[algebra|QJGPT2W0FT]] is the smallest [[congruence|DCSXXT4SER]] that contains $R$.
It exists because any intersection of congruences is a congruence.
It is the tool for forcing given pairs of elements to become equal while keeping the operations meaningful, as in presentations by generators and relations.

## Explanations

### L1 human

Suppose you want some pairs of elements of an algebra to count as equal.
Once $a$ counts as equal to $b$, compatibility forces many other pairs to be equal too, such as $a + c$ and $b + c$.
The congruence generated by your pairs collects exactly what is forced and nothing more.
In the group of integers under addition, asking for $0$ to equal $6$ forces exactly congruence modulo $6$.
</document>

<document source="https://tamnd.github.io/math2/node/QP0MY1E9RK.md" tag="QP0MY1E9RK" kind="definition">
# Symplectomorphism

## Definition

Let $(M, \omega)$ and $(N, \eta)$ be [[symplectic manifolds|CAN490EPRT]].
A *symplectomorphism* from $M$ to $N$ is a [[diffeomorphism|0KZZ70ZHXP]] $\phi \colon M \to N$ such that $\phi^* \eta = \omega$, that is
$$\eta_{\phi(p)}\big(d\phi_p(v), d\phi_p(w)\big) = \omega_p(v, w)$$
for all $p \in M$ and $v, w \in T_p M$.

## Remarks

Example: on $\R^2$ with $\omega = dx \wedge dy$, let $\phi(x, y) = (x, y + f(x))$ for a smooth function $f$.
It is a diffeomorphism with inverse $(x, y) \mapsto (x, y - f(x))$, and $\phi^* \omega = dx \wedge (dy + f'(x)\, dx) = dx \wedge dy$.

Non-example: $\psi(x, y) = (2x, y)$ on the same plane is a diffeomorphism, but $\psi^* \omega = 2\, dx \wedge dy \ne \omega$.

A linear map of $(\R^{2n}, \omega_0)$ is a symplectomorphism exactly when it lies in the [[symplectic group|X12FS2JAP0]] $Sp(2n, \R)$.
The flow of a [[Hamiltonian vector field|C5EH3B4BEP]] consists of symplectomorphisms.
A symplectomorphism preserves the volume form $\omega^n / n!$, so it preserves volume; in dimension $2$ the converse holds for orientation-preserving diffeomorphisms.
</document>

<document source="https://tamnd.github.io/math2/node/QP7GEKDTYX.md" tag="QP7GEKDTYX" kind="concept">
# Polytope

## Summary

A polytope is a geometric figure with flat sides: a polygon in the plane, a [[polyhedron|0325SGSJAR]] in space, and their analogues in any dimension.
The word is used with several meanings.
In convex geometry and combinatorics it usually means a [[convex polytope|300K6CB1KH]].
In the wider sense used here, a polytope is a finite union of convex polytopes, so it may be non-convex.
Further generalizations include star polytopes, whose faces may cross each other, and abstract polytopes, which keep only the incidence pattern of vertices, edges and faces.

## Explanations

### L1 human

Polytopes are the shapes built from flat pieces: triangles and squares, cubes and pyramids, and the four-dimensional hypercube.
A polytope need not be convex: an L-shaped region made of three unit squares is a polytope.
Curved shapes such as discs and balls are not polytopes.
</document>

<document source="https://tamnd.github.io/math2/node/QP8B74828C.md" tag="QP8B74828C" kind="statement">
# Cartan's magic formula

## Statement

Let $M$ be a [[smooth manifold|F45VQ47GPV]], let $X$ be a smooth [[vector field|0299VCW1TH]] on $M$ and let $\omega$ be a smooth [[differential form|0E3G0GRPZ6]] on $M$.
Let $\mathcal{L}_X$ be the [[Lie derivative|PJWK0PKN6Q]] along $X$, and let $d$ be the [[exterior derivative|0ZHSTAG8ZY]].
Let $\iota_X$ be the interior product, given on a $k$-form by $(\iota_X \omega)(Y_1, \dots, Y_{k-1}) = \omega(X, Y_1, \dots, Y_{k-1})$.
Then
$$\mathcal{L}_X \omega = d(\iota_X \omega) + \iota_X(d\omega).$$

## Remarks

The idea of the proof: both sides are derivations of degree $0$ of the algebra of forms, and both commute with $d$.
On a function $f$ both sides give $X f$, since $\iota_X f = 0$ and $\iota_X df = Xf$.
On an exact $1$-form $df$ both sides give $d(Xf)$.
Locally every form is a sum of products $f\, dg_1 \wedge \dots \wedge dg_k$, so the two sides agree everywhere.

Example: on $\R^2$, let $X = -y\, \partial_x + x\, \partial_y$, the infinitesimal rotation, and $\omega = dx \wedge dy$.
Then $d\omega = 0$ and $\iota_X \omega = -y\, dy - x\, dx$, so $d(\iota_X \omega) = 0$.
Hence $\mathcal{L}_X \omega = 0$: rotations preserve area.

The formula has two quick consequences.
First, $\mathcal{L}_X$ commutes with $d$, since $d \mathcal{L}_X = d \iota_X d = \mathcal{L}_X d$ by $d^2 = 0$.
Second, if $\omega$ is closed, then $\mathcal{L}_X \omega = d(\iota_X \omega)$ is exact.

## Explanations

### L1 human

The Lie derivative measures how a form changes when it is dragged along the flow of a vector field.
Cartan's formula computes this change from two simple operations: plugging the vector field into the form, and taking the exterior derivative.
It turns a question about flows into a short algebraic computation.

### L2 human

The second consequence above is the heart of the homotopy invariance of de Rham cohomology.
Integrating $\mathcal{L}_X$ along a flow shows that the pullbacks of a closed form by homotopic maps differ by an exact form.
This gives the Poincaré lemma for star-shaped domains.
In symplectic geometry the formula shows that a vector field $X$ preserves a symplectic form $\omega$ exactly when $\iota_X \omega$ is closed, which leads to Hamiltonian vector fields.
</document>

<document source="https://tamnd.github.io/math2/node/QP92G4ZGJE.md" tag="QP92G4ZGJE" kind="concept">
# Deformation of algebras

## Summary

A deformation of an [[associative algebra|3CTW1P4HBP]] $A$ is a family of associative products on $A$ depending on a parameter $t$.
At $t = 0$ it gives back the original product.
In the formal theory of Murray Gerstenhaber, the parameter is a formal variable and the products are power series in $t$.
The first-order terms are classified by the second Hochschild cohomology of $A$, and the obstructions to extending them lie in the third.
Deformations of algebras are the algebraic core of [[deformation quantization|3MKSJ1J04K]].

## Explanations

### L1 human

Imagine changing the multiplication of an algebra a little bit, in a way controlled by a small parameter $t$.
At $t = 0$ one has the old algebra, and for small $t$ a nearby algebra.
For example, the commutative polynomial ring in $x$ and $y$ can be bent into an algebra where $yx = xy - t$, which is a Weyl algebra.
Some algebras cannot be bent at all except in trivial ways; they are called rigid.
The theory asks which bendings exist and when two of them are really the same after a change of coordinates.
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<document source="https://tamnd.github.io/math2/node/QP9DY5K8QW.md" tag="QP9DY5K8QW" kind="concept">
# Total curvature

## Summary

The total curvature of a curve is the integral of its [[curvature|7JQS6VKFEZ]] with respect to arc length.
It measures how much the direction of the curve turns in all, from start to end.
For closed curves it is bounded below by $2\pi$, by Fenchel's theorem, and knotted curves need more than $4\pi$, by the Fáry–Milnor theorem.

## Explanations

### L1 human

Walk once around a closed path and add up all the turning you do, ignoring left and right.
That sum is the total curvature.
Walking once around a circle you turn by a full turn, so the total curvature is $2\pi$.
A wiggly path makes you turn back and forth, and the total grows.
</document>

<document source="https://tamnd.github.io/math2/node/QPH28QH036.md" tag="QPH28QH036" kind="concept">
# Bargmann transform

## Summary

The Bargmann transform is a unitary map from $L^2(\R)$ onto the Segal–Bargmann space, the Hilbert space of [[entire functions|0G4F0XB9X2]] $F$ on $\C$ with $\int_\C \abs{F(z)}^2 e^{-\abs{z}^2}\, dA(z) < \infty$.
It sends the [[Hermite functions|52VS07H6GK]] to the normalized monomials $z^n / \sqrt{n!}$.
It turns the creation and annihilation operators of the harmonic oscillator into multiplication by $z$ and differentiation in $z$.
This makes the oscillator very simple.
It is closely related to the short-time Fourier transform with a Gaussian window.

## Explanations

### L1 human

A function on the real line is turned into a holomorphic function on the complex plane.
The real part of $z$ plays the role of position and the imaginary part the role of momentum, so the transform shows a quantum state in phase space.
The Gaussian $e^{-x^2/2}$, the ground state of the harmonic oscillator, becomes the constant function $1$.
Higher energy states become the powers $z^n$, up to a factor.
</document>

<document source="https://tamnd.github.io/math2/node/QPKKTE1ZY0.md" tag="QPKKTE1ZY0" kind="concept">
# Interval arithmetic

## Summary

Interval arithmetic computes with closed bounded [[intervals|0Q1D85B8G6]] of real numbers instead of single numbers.
The result of an operation on two intervals is the smallest interval that contains every result of the operation on numbers taken from them.
When a computer rounds the lower end down and the upper end up, the computed interval is sure to contain the exact answer.
This gives rigorous error bounds and is the base of verified numerical computation and of some computer-assisted proofs.

## Explanations

### L1 human

If you know only that a length lies between $1$ and $2$ and another lies between $3$ and $5$, then their sum lies between $4$ and $7$.
Interval arithmetic makes this kind of reasoning into a set of rules for adding, subtracting, multiplying and dividing ranges.
Each number with an uncertainty becomes an interval, and the final interval shows how large the uncertainty of the answer can be.
A catch is that the same quantity used twice is treated as two independent quantities, so the bounds can be wider than needed: $[0, 1] - [0, 1]$ is $[-1, 1]$, not $\set{0}$.
</document>

<document source="https://tamnd.github.io/math2/node/QPMM2ZPWBW.md" tag="QPMM2ZPWBW" kind="problem">
# Hilbert's nineteenth problem

## Problem

Let $\Omega \subseteq \R^n$ be an open set and let $L \colon \R^n \to \R$, written $L(p)$, be a real [[analytic function|081WKEDK6C]].
Assume $L$ is uniformly convex: for constants $0 < \lambda \le \Lambda$, the eigenvalues of the Hessian matrix $D^2 L(p)$ lie in $[\lambda, \Lambda]$ for all $p$.
Consider the functional $J(u) = \int_\Omega L(\nabla u) \, dx$ of the [[calculus of variations|PQ25NHV0KS]].
Call $u \in W^{1,2}_{\mathrm{loc}}(\Omega)$ a local minimiser of $J$ if $\int_{S} L(\nabla u) \, dx \le \int_{S} L(\nabla u + \nabla \varphi) \, dx$ for every $\varphi \in W^{1,2}(\Omega)$ that vanishes outside a compact set $S \subseteq \Omega$.
Decide whether every local minimiser of $J$ is real analytic in $\Omega$.
In Hilbert's words, the question is whether the solutions of regular problems of the calculus of variations are always analytic.

## Remarks

Hilbert asked the question in a more general form, for analytic Lagrangians $L(x, u, \nabla u)$ that satisfy the Legendre condition; the form above is the model case, with the same answer.
The answer is yes for scalar functions $u$.
Bernstein proved in 1904 that $C^3$ solutions of analytic nonlinear elliptic equations in two variables are analytic.
The general case was settled by De Giorgi in 1957 and, independently, by Nash in 1958.
They proved that weak solutions of linear elliptic equations in divergence form with bounded measurable coefficients are Hölder continuous.
The derivatives of a minimiser solve such an equation, so $u \in C^{1,\alpha}$.
Then Schauder theory gives $u \in C^\infty$, and the analyticity of smooth solutions of analytic elliptic equations gives that $u$ is analytic.
For vector-valued $u$ the answer is no: De Giorgi gave a counterexample in 1968 of a non-smooth minimiser for an elliptic system.

## Explanations

### L1 human

Hilbert noticed that the minimisers of natural energies, such as the area of a soap film, tend to be not just smooth but analytic, given by convergent power series.
He asked whether this is always so for well-behaved energies.
The answer, found more than fifty years later, is yes for a single unknown function and no for systems of unknown functions.
The work on this problem created much of modern regularity theory for elliptic equations.
</document>

<document source="https://tamnd.github.io/math2/node/QPR7DCHXPG.md" tag="QPR7DCHXPG" kind="concept">
# Cartan connection

## Summary

A Cartan connection describes the geometry of a manifold as a curved version of a model homogeneous space $G/H$.
At each point the manifold is identified to first order with the model, and the connection says how these identifications fit together.
Its curvature measures how far the manifold is from being locally the model, and it vanishes exactly when the manifold is locally isomorphic to $G/H$.
Riemannian, conformal and projective geometry all fit in this frame, with models Euclidean space, the conformal sphere and projective space.

## Explanations

### L1 human

Think of rolling a ball without slipping over a curved surface.
At each moment the ball touches the surface at one point, and rolling compares the surface with the ball's round shape.
A Cartan connection works the same way, with any model space in place of the ball.
For a flat plane rolled on itself nothing changes, and the curvature is zero.
For a curved surface rolled on the flat plane, going around a loop leaves the plane slightly turned, and this is the curvature.
</document>

<document source="https://tamnd.github.io/math2/node/QPRHC4DZ20.md" tag="QPRHC4DZ20" kind="example">
# Space-filling curve

## Example

A *space-filling curve* is a [[continuous|0X3WGX95FE]] [[surjection|01RVCMSN5N]] from $[0, 1]$ onto the square $[0, 1]^2$.
Here is one, due to Lebesgue.
Each point $x$ of the [[Cantor set|0BXNDZFQME]] $C$ has a unique expansion $x = \sum_{n \ge 1} 2 a_n 3^{-n}$ with digits $a_n \in \set{0, 1}$.
Set
$$f(x) = \left( \sum_{n \ge 1} a_{2n-1} 2^{-n}, \ \sum_{n \ge 1} a_{2n} 2^{-n} \right).$$
On each open interval removed in the construction of $C$, extend $f$ linearly between its values at the two ends.

The check has three steps.
The map $f$ is onto.
Take binary expansions of both coordinates of a point of $[0, 1]^2$; interleaving their digits gives a point of $C$ that $f$ sends to it.
The map $f$ is continuous on $C$.
Two points of $C$ less than $3^{-2m}$ apart share their first $2m$ digits, so their images differ by at most $2^{-m}$ in each coordinate.
The two ends of a removed interval of length $3^{-k}$ share their first $k - 1$ digits.
So short removed intervals carry small linear pieces, and the extended map is continuous.

## Remarks

No space-filling curve is injective.
A continuous bijection $[0, 1] \to [0, 1]^2$ would be a homeomorphism, since $[0, 1]$ is compact and the square is Hausdorff.
That is impossible: removing an inner point disconnects $[0, 1]$, but removing a point does not disconnect the square.
The first space-filling curve was found by Peano, and Hilbert gave another one.

## Explanations

### L1 human

A space-filling curve is a pencil line drawn without lifting the pencil, in a finite time, that passes through every point of a filled square.
It comes as the limit of ever finer zigzag paths.
The curve must cross itself: it passes through some points more than once.
</document>

<document source="https://tamnd.github.io/math2/node/QPS8VSJGMJ.md" tag="QPS8VSJGMJ" kind="concept">
# Dynamic programming in continuous time

## Summary

Dynamic programming in continuous time solves an optimal control problem by studying its [[value function|K6G0DRQ3CP]] for all initial times and states at once.
Its basis is Bellman's principle of optimality: the tail of an optimal trajectory is optimal for the problem that starts where the tail starts.
In infinitesimal form this principle becomes the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]].
It gives optimal controls in feedback form, which complements the open-loop necessary conditions of [[Pontryagin's maximum principle|7RPV2T29CF]].

## Explanations

### L1 human

To plan the cheapest trip, one can work backward from the destination.
If you know the cheapest remaining cost from every place at a later time, the best move now is easy to find.
Dynamic programming in continuous time applies this idea at every instant.
The result is a rule that tells you what to do in every situation, not just along one planned path.
</document>

<document source="https://tamnd.github.io/math2/node/QPT5H6MM99.md" tag="QPT5H6MM99" kind="definition">
# Abstract data type as an algebra

## Definition

Let $S$ be a set of sorts and let $\Sigma$ be an $S$-sorted signature.
An *algebraic specification* is a pair $(\Sigma, E)$, where $E$ is a set of [[equations|PKD440G2XW]] between $\Sigma$-terms with variables, each side of the same sort.
A *model* of $(\Sigma, E)$ is a [[many-sorted $\Sigma$-algebra|WAQJJ5GX68]] that satisfies every equation of $E$.
A model $I$ is *initial* if for every model $A$ there is exactly one [[homomorphism|KBXSZC6BV1]] $I \to A$.
The *abstract data type* specified by $(\Sigma, E)$ is the class of initial models of $(\Sigma, E)$.

## Remarks

Initial models exist, and any two of them are isomorphic, so the abstract data type is one isomorphism class of algebras; this is the [[initial algebra semantics|HVF403PQ4H]] of the specification.
An initial model has two features.
Every element is the value of a ground term ("no junk"), and two ground terms are equal in it only if $E$ proves them equal ("no confusion").

Example: sorts $\mathrm{nat}$ and $\mathrm{bool}$; constants $\mathrm{true}, \mathrm{false} \colon \mathrm{bool}$ and $0 \colon \mathrm{nat}$; operations $s \colon \mathrm{nat} \to \mathrm{nat}$ and $\mathrm{iszero} \colon \mathrm{nat} \to \mathrm{bool}$; equations $\mathrm{iszero}(0) = \mathrm{true}$ and $\mathrm{iszero}(s(x)) = \mathrm{false}$.
The model with carriers $\N$ and $\set{\mathrm{true}, \mathrm{false}}$ is initial.
Each element is the value of a ground term, and two ground terms with the same value are equal by the equations.

Non-example: the model with one element in each sort satisfies $E$ but is not initial.
A homomorphism from it to the model above would send its single Boolean element to both $\mathrm{true}$ and $\mathrm{false}$, which is impossible.
</document>

<document source="https://tamnd.github.io/math2/node/QPV0HNP22S.md" tag="QPV0HNP22S" kind="statement">
# Gentzen's consistency proof

## Statement

Let PA be first-order [[Peano arithmetic|0QFT7Z76N5]], and let PRA be primitive recursive arithmetic.
Fix a standard [[primitive recursive|5NVA2JRHPV]] notation system for the ordinals below [[epsilon nought|KT50SMH3EC]], ordered by a primitive recursive relation $\prec$.
Then the statement that PA is consistent is provable in PRA extended by the rule of [[transfinite induction|0ZHQMQ2C1N]] along $\prec$ for quantifier-free formulas.

## Remarks

Gentzen published the proof in 1936, and a second proof in 1938.
In the second proof, one assigns to each derivation of a sequent in a sequent calculus for PA an ordinal below $\varepsilon_0$.
One then gives a primitive recursive reduction step that turns any derivation of the empty sequent into another derivation of the empty sequent with a smaller ordinal.
Transfinite induction along $\prec$ shows that no derivation of the empty sequent exists.

By [[Gödel's second incompleteness theorem|X9WYNYE29C]], PA does not prove its own consistency.
So PA does not prove the transfinite induction used here, although every part of the proof other than this induction is finitary.
Gentzen later showed directly that PA proves transfinite induction along $\prec$ up to each ordinal below $\varepsilon_0$, but not up to $\varepsilon_0$.
Thus $\varepsilon_0$ is exactly the length of induction that PA cannot reach.

## Explanations

### L1 human

Gödel showed that arithmetic cannot prove its own freedom from contradiction.
Gentzen found what extra principle is enough to prove it.
The principle is induction along an ordering of length $\varepsilon_0$, which is easy to describe with finite symbols.
Every other step of the proof uses only simple, finite reasoning about formulas.

### L2 human

The proof assigns ordinals to proofs and shows that reduction steps make the ordinal go down.
Since the ordinals below $\varepsilon_0$ are well-ordered, the reduction must stop, and so no proof of a contradiction can exist.
This is the first ordinal analysis: it shows that $\varepsilon_0$ is the proof-theoretic ordinal of PA.
The method extends to stronger theories, which get larger ordinals.
</document>

<document source="https://tamnd.github.io/math2/node/QQ1YT0MV2X.md" tag="QQ1YT0MV2X" kind="concept">
# Method of matched asymptotic expansions

## Summary

The method of matched asymptotic expansions builds approximate solutions of [[singularly perturbed|G6J21TGRMM]] problems.
It uses one expansion away from the thin layers where the solution changes fast, and another expansion inside each layer, in a stretched variable.
The two are joined by requiring that they agree in an intermediate region.
It is a standard tool for [[boundary layers|YCFPD0GE5G]] in fluid dynamics and for many problems in applied mathematics.

## Explanations

### L1 human

Imagine a solution that is smooth almost everywhere but has a steep cliff near one end.
One formula describes the flat part well, and a zoomed-in formula describes the cliff well.
Matching makes the two formulas agree where the cliff meets the flat part.
Adding them and removing the shared part gives one formula that is good everywhere.
For $\varepsilon y'' + y' + y = 0$, $y(0) = 0$, $y(1) = 1$, this gives $y \approx e^{1 - x} - e^{1 - x/\varepsilon}$.
</document>

<document source="https://tamnd.github.io/math2/node/QQ3PBZS9J8.md" tag="QQ3PBZS9J8" kind="definition">
# Mesh generation

## Definition

Let $d \in \set{2, 3}$ and let $\Omega \subset \R^d$ be a bounded open set whose closure $\overline{\Omega}$ is a polygon ($d = 2$) or a polyhedron ($d = 3$).
A *conforming mesh* of $\overline{\Omega}$ is a finite set $\mathcal{T}$ of closed cells, each a simplex or each a convex quadrilateral or hexahedron, such that:

1. the union of the cells of $\mathcal{T}$ is $\overline{\Omega}$;
2. the interiors of two distinct cells are disjoint;
3. the intersection of two distinct cells is empty or a common face of both, that is, a common vertex, edge or facet.

*Mesh generation* is the task of computing a conforming mesh of $\overline{\Omega}$ from a description of its boundary.
Usually it must meet conditions on the size $h_K = \operatorname{diam} K$ and the shape of each cell $K$.
A typical shape condition is an upper bound on $h_K / \rho_K$, where $\rho_K$ is the radius of the largest ball contained in $K$.

## Remarks

When all cells are simplices, a conforming mesh is a [[triangulation|YH170WBQ6M]] of $\overline{\Omega}$, the underlying space of a [[simplicial complex|V8GZ0W1ZZ5]].
Example: the unit square $[0,1]^2$ with the two triangles of vertices $(0,0), (1,0), (1,1)$ and $(0,0), (1,1), (0,1)$ is a conforming mesh.
The triangles cover the square, their interiors are disjoint, and they meet along the common edge from $(0,0)$ to $(1,1)$.
Non-example: cover $[0,2] \times [0,1]$ by the squares $[0,1] \times [0,1]$, $[1,2] \times [0, \tfrac12]$ and $[1,2] \times [\tfrac12, 1]$.
Clauses 1 and 2 hold, but clause 3 fails: the first and second squares meet in the segment $\set{1} \times [0, \tfrac12]$.
This segment is an edge of the second square but not of the first.
The point $(1, \tfrac12)$ is called a hanging node.
A family of meshes with $\max_K h_K \to 0$ and $h_K / \rho_K$ bounded uniformly is called shape regular.
Finite element error estimates are usually proved for such families.

## Explanations

### L1 human

A mesh cuts a region into simple pieces that fit together edge to edge.
There are no gaps, no overlaps, and no corner of one piece in the middle of a side of another.
Mesh generation is the job of producing such a cutting, with pieces that are neither too big nor too thin.
</document>

<document source="https://tamnd.github.io/math2/node/QQ5F8B8DYC.md" tag="QQ5F8B8DYC" kind="definition">
# Differential operator between vector bundles

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $\mathbb{K}$ be $\R$ or $\C$.
Let $E$ and $F$ be smooth [[vector bundles|P4DZM975BV]] over $M$ with fibres $\mathbb{K}^r$ and $\mathbb{K}^s$.
Write $\Gamma(E)$ for the space of smooth [[sections|WWX7H5W1RP]] of $E$.
Let $m \ge 0$ be an integer.
Consider a linear map $P \colon \Gamma(E) \to \Gamma(F)$.
It is a *differential operator of order at most $m$* if the following holds.
Let $(U, x_1, \ldots, x_n)$ be any chart of $M$ over which $E$ and $F$ are trivial.
In the trivialisations, there are smooth $A_\alpha \colon U \to \mathbb{K}^{s \times r}$ for $\abs{\alpha} \le m$ with
$$(P u)(x) = \sum_{\abs{\alpha} \le m} A_\alpha(x) \, \partial^\alpha u(x) \quad \text{for all } u \in \Gamma(E) \text{ and } x \in U,$$
where $u$ is written on $U$ as a function $U \to \mathbb{K}^r$.
The operator has *order $m$* if moreover some $A_\alpha$ with $\abs{\alpha} = m$ is not identically zero.

## Remarks

The condition does not depend on the charts and trivialisations.
A change of coordinates or of trivialisation turns a sum of this form into another one of the same order.
A differential operator is local: the value of $Pu$ near a point depends only on $u$ near that point.
By the Peetre theorem, linear local operators are, locally, differential operators.

Example: the [[exterior derivative|0ZHSTAG8ZY]] $d \colon \Gamma(\Lambda^k T^* M) \to \Gamma(\Lambda^{k+1} T^* M)$ is a differential operator of order $1$.
In a chart, $d(\sum_I f_I \, dx_I) = \sum_I \sum_{j} \partial_j f_I \, dx_j \wedge dx_I$, whose coefficients are constant matrices.

Non-example: on the circle $S^1 = \R / 2\pi\Z$, the map $u \mapsto \bigl(\int_0^{2\pi} u(t) \, dt\bigr) \cdot 1$ on functions is linear but not a differential operator.
It is not local: changing $u$ on a small interval changes $Pu$ everywhere.

## Explanations

### L1 human

A differential operator between bundles is, in every coordinate patch, a matrix of partial derivatives with smooth coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/QQ62R5ZAXJ.md" tag="QQ62R5ZAXJ" kind="concept">
# Boundary integral method for the Dirichlet problem

## Summary

The boundary integral method solves the [[Dirichlet problem|8D56GQ4AG8]] for the Laplace equation with a [[double layer potential|EG8Z3FWV9A]].
The unknown harmonic function is written as that potential, with an unknown density on the boundary.
The [[jump relations|J58YA8QNNX]] turn the boundary condition into a Fredholm integral equation of the second kind for the density.
The [[Fredholm alternative|4HH8C9V8JQ]] then reduces existence of a solution to uniqueness.
Its discretisation is the boundary element method of numerical analysis.

## Explanations

### L1 human

Instead of looking for a function everywhere inside a region, look only for a function on its boundary.
Spread unknown dipoles over the boundary; whatever their strengths, they produce a harmonic function inside.
Then choose the strengths so that this function has the prescribed values on the boundary, which is one equation on the boundary only.
For the unit ball in space and boundary value $1$, the constant density $-1$ does the job, and the potential it produces is $1$ throughout the ball.
</document>

<document source="https://tamnd.github.io/math2/node/QQ6DA5QKXB.md" tag="QQ6DA5QKXB" kind="definition">
# Funk transform

## Definition

Let $S^2 = \set{x \in \R^3 : \abs{x} = 1}$ be the unit [[sphere|P9CM4GQ2T0]].
For $\theta \in S^2$, let $C_\theta = \set{x \in S^2 : x \cdot \theta = 0}$ be the great circle with pole $\theta$, with its arc length measure $ds$.
Let $f \colon S^2 \to \C$ be continuous.
The *Funk transform* of $f$ is the function on $S^2$ given by
$$(F f)(\theta) = \int_{C_\theta} f(x)\, ds(x).$$

## Remarks

Since $C_{-\theta} = C_\theta$, $F f$ is even.
If $f$ is odd, $f(-x) = -f(x)$, then $F f = 0$, because $C_\theta$ is symmetric under $x \mapsto -x$.
On a [[spherical harmonic|3F32Y21HP4]] $Y_k$ of degree $k$, $F Y_k = 2\pi P_k(0)\, Y_k$, where $P_k$ is the Legendre polynomial; $P_k(0) = 0$ for odd $k$ and $P_k(0) \ne 0$ for even $k$.
Hence $F$ is injective on even continuous functions.

Example: $f = 1$ gives $F f = 2\pi$, the length of a great circle; this is the case $k = 0$, with $P_0(0) = 1$.

Example: $f(x) = x_3$ is odd, so $F f = 0$, although $f \ne 0$.

Non-example: the integral of $f$ over a small circle $\set{x \in S^2 : x \cdot \theta = c}$ with $0 < \abs{c} < 1$ is not a value of the Funk transform, which uses only circles through the center of the sphere.

The same definition works on $S^{n-1}$ with the great subspheres $S^{n-1} \cap \theta^\perp$.
</document>

<document source="https://tamnd.github.io/math2/node/QQ717VSFNN.md" tag="QQ717VSFNN" kind="conjecture">
# Vojta's conjecture

## Statement

Let $F$ be an [[algebraic number field|MWBDFS2KJ5]] and let $X$ be a smooth projective variety over $F$.
Let $D$ be an effective reduced [[divisor|Z9DN23DAE2]] on $X$ with at most normal crossings.
Let $K_X$ be a [[canonical divisor|W05VYY4HBR]] and let $A$ be an [[ample|RRK8P6TXG2]] divisor on $X$.
Let $S$ be a finite set of places of $F$.
Let $h_{K_X}$ and $h_A$ be Weil [[heights|F1FF505PWH]] for $K_X$ and $A$, and let $\lambda_{D,v}$ be a local height function for $D$ at the place $v$.
Then for every $\varepsilon > 0$ there are a proper Zariski closed subset $Z \subsetneq X$ and a constant $C$ such that
$$\sum_{v \in S} \lambda_{D,v}(P) + h_{K_X}(P) \le \varepsilon \, h_A(P) + C$$
for all $P \in X(F)$ with $P \notin Z$.

## Remarks

Vojta formulated the conjecture in 1987 by analogy with the second main theorem of Nevanlinna theory, through a dictionary between Diophantine approximation and value distribution.
For $X = \mathbb{P}^1$ and $D$ a sum of distinct points, the conjecture is Roth's theorem, which is known.
For $X = \mathbb{P}^n$ and $D$ a sum of hyperplanes in general position, it is Schmidt's subspace theorem, which is also known.
Let $X$ be of general type and $D = 0$.
Outside a proper closed subset, the canonical height is at least $\delta h_A - C'$ for some $\delta > 0$.
Taking $\varepsilon < \delta$ then forces $X(F)$ into a proper closed subset, which is the [[Bombieri–Lang conjecture|TQ1QDAK2BR]].
Generalized forms for algebraic points, with truncated local heights, are higher-dimensional analogues of the [[abc conjecture|G7JH2NXPC0]].
The conjecture is open in general.

## Explanations

### L1 human

Heights measure how complicated a rational point is.
Vojta's conjecture is a single inequality between heights that predicts how close rational points can come to a fixed divisor, given the curvature of the variety.
It unifies many famous results and conjectures.
Roth's theorem on approximating algebraic numbers, Faltings's theorem on curves and the Bombieri–Lang conjecture appear as special cases.
Its generalizations are related to the abc conjecture.
</document>

<document source="https://tamnd.github.io/math2/node/QQ8GW1ASZM.md" tag="QQ8GW1ASZM" kind="object">
# Fano plane

## Description

The Fano plane has the $7$ points $1, 2, \dots, 7$ and the $7$ lines
$$\set{1,2,3},\ \set{1,4,5},\ \set{1,6,7},\ \set{2,4,6},\ \set{2,5,7},\ \set{3,4,7},\ \set{3,5,6}.$$
Check: each line has $3$ points, so the lines cover $7 \cdot 3 = 21$ pairs of points.
These $21$ pairs are distinct, and there are $\binom{7}{2} = 21$ pairs in all, so every two points lie on exactly one line.
Every two lines meet in exactly one point, as one sees from the list.
So the Fano plane is a [[finite projective plane|2HWB7P1DZN]] of order $2$ and a [[Steiner system|7BPSC4T4M3]] $S(2, 3, 7)$.

## Remarks

Write each point $i$ in binary as a nonzero vector of $\F_2^3$.
Then the lines are the sets $\set{a, b, a + b}$ with $a \ne b$, for example $1 + 6 = 7$ in binary addition.
So the Fano plane is $PG(2, 2)$, the projective plane over the field with two elements.
Its automorphism group is $GL(3, 2)$, a simple group of order $168$.
It is the smallest projective plane.

## Explanations

### L1 human

The Fano plane is a tiny geometry with $7$ points and $7$ lines, each line with $3$ points.
It is usually drawn as a triangle with its three medians and its inscribed circle; the circle counts as a line.
Any two points share one line and any two lines share one point.
</document>

<document source="https://tamnd.github.io/math2/node/QQ8M0T04Q9.md" tag="QQ8M0T04Q9" kind="concept">
# Lie group method for PDEs

## Summary

The Lie group method studies a differential equation through its symmetries: continuous groups of changes of variables that map solutions to solutions.
Sophus Lie showed that such symmetries can be found by solving a linear system of determining equations for their infinitesimal generators.
Once found, they produce new solutions from old ones and reduce the equation to one with fewer variables.
For ordinary differential equations it is the [[Lie symmetry method|X3S9XHX83H]].
[[Similarity solutions|579J5K9K5Z]] and [[traveling waves|M37M411KP7]] are the invariant solutions of scaling and translation symmetries.
See the [[definition|SYXNSN5JX0]].

## Explanations

### L1 human

Moving a solution of the heat equation to the right, or starting it later, again gives a solution.
Stretching space by $2$ and time by $4$ also gives a solution.
These are symmetries of the equation, and each comes in a family with a continuous parameter.
Lie's method finds all such families by a mechanical computation.
A solution that is unchanged by one family depends on fewer variables, so it solves a simpler equation.
</document>

<document source="https://tamnd.github.io/math2/node/QQ98C2AXZP.md" tag="QQ98C2AXZP" kind="concept">
# Heaviside operational calculus

## Summary

The Heaviside operational calculus is the formal method of Oliver Heaviside for linear differential equations, developed for electrical circuits.
The derivative $d/dt$ is written as a symbol $p$.
The response of a system to a unit step input is written as a function of $p$ applied to the step.
This expression is then expanded and interpreted by fixed rules.
The method gave correct answers long before it was justified; the [[Laplace transform|0V49JEZW06]] and the [[Bromwich integral|5KNAXNHRY6]] later explained why.

## Explanations

### L1 human

Heaviside treated "differentiate" as a letter $p$ that can be multiplied, divided and expanded like a number.
Dividing by $p$ means integrating from time $0$.
A circuit switched on at time $0$ is then described by a fraction in $p$, and expanding that fraction gives the current or voltage as a function of time.
For example, the expression $1/(1 - a/p)$ applied to the unit step expands to $1 + a t + a^2 t^2/2 + \cdots = e^{at}$.
</document>

<document source="https://tamnd.github.io/math2/node/QQAWHC9X60.md" tag="QQAWHC9X60" kind="statement">
# Schreier refinement theorem

## Statement

Let $G$ be a [[group|G5S1M9]].
A refinement of a [[subnormal series|0ZZR8D7Y1Z]] of $G$ is a subnormal series of $G$ that contains every term of the given series.
Two subnormal series of $G$ are equivalent when they have the same length and their factors agree up to a permutation and isomorphism.
Then any two subnormal series of $G$ have refinements that are equivalent.

## Remarks

Let the two series be $(G_i)_{0 \le i \le n}$ and $(H_j)_{0 \le j \le m}$.
The proof inserts the groups $G_{i,j} = G_{i-1}(G_i \cap H_j)$ between $G_{i-1}$ and $G_i$, and the groups $H_{j,i} = H_{j-1}(H_j \cap G_i)$ between $H_{j-1}$ and $H_j$.
Both refined series have length $nm$, and the [[Zassenhaus lemma|21JJ523V95]] gives $G_{i,j}/G_{i,j-1} \cong H_{j,i}/H_{j,i-1}$ for each pair $(i, j)$.
For example, in $\Z/12\Z$ the series $0 \le 6\Z/12\Z \le \Z/12\Z$ and $0 \le 4\Z/12\Z \le \Z/12\Z$ refine to $0 \le 6\Z/12\Z \le 2\Z/12\Z \le \Z/12\Z$ and $0 \le 4\Z/12\Z \le 2\Z/12\Z \le \Z/12\Z$.
Both refinements have factors of orders $2, 3, 2$ in some order, namely $\Z/2\Z$, $\Z/3\Z$, $\Z/2\Z$ and $\Z/3\Z$, $\Z/2\Z$, $\Z/2\Z$.

## Explanations

### L1 human

A subnormal series cuts a group into layers.
Two different ways of cutting may look unrelated.
The theorem says that you can always cut each of them further so that both give the same list of layers, up to order.

### L2 human

The theorem is the main step in the proof of the [[Jordan–Hölder theorem|0E088ZP533]].
A [[composition series|0XV2A49X6Q]] has no proper refinement other than itself with repeated terms, so two composition series are already equivalent once the repeated terms are removed.
The same argument works for groups with operators and for modules, where it gives the uniqueness of composition factors of modules of finite length.
</document>

<document source="https://tamnd.github.io/math2/node/QQCWHB37ZC.md" tag="QQCWHB37ZC" kind="definition">
# Discrete subgroup

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] and let $\Gamma$ be a [[subgroup|0X1JXT47GA]] of $G$.
The subgroup $\Gamma$ is *discrete* if the subspace topology on $\Gamma$ is the [[discrete topology|0S2TEHX4E1]].

## Remarks

Equivalently, there is an open neighbourhood $U$ of the identity $e$ in $G$ with $U \cap \Gamma = \set{e}$.
Indeed, then $\gamma U$ is an open set with $\gamma U \cap \Gamma = \set{\gamma}$ for each $\gamma \in \Gamma$.

Example: $\Z$ is a discrete subgroup of $\R$, since $U = (-\tfrac12, \tfrac12)$ meets $\Z$ only in $0$.
Likewise $\mathrm{SL}_n(\Z)$ is a discrete subgroup of $\mathrm{SL}_n(\R)$, since two distinct integer matrices differ by at least $1$ in some entry.

Non-example: $\Gamma = \set{a + b\sqrt2 : a, b \in \Z}$ is a subgroup of $\R$ that is not discrete.
The numbers $(\sqrt2 - 1)^k$ lie in $\Gamma$, are nonzero, and tend to $0$, so every neighbourhood of $0$ meets $\Gamma$ in more than one point.

In a Hausdorff topological group, every discrete subgroup is closed.
</document>

<document source="https://tamnd.github.io/math2/node/QQDN6EYQNJ.md" tag="QQDN6EYQNJ" kind="concept">
# Nilradical

## Summary

The nilradical of a commutative ring is the set of its [[nilpotent elements|0N9R5NN6TW]].
It is an ideal, the [[radical|437YRCNYRJ]] of the zero ideal.
It equals the [[intersection of all prime ideals|3CZ8PXR9JY]] of the ring, and a ring with zero nilradical is called [[reduced|SZHY975FFX]].

## Explanations

### L1 human

An element is nilpotent when some power of it is zero.
The nilradical collects all such elements.
In $\Z/12\Z$, the nilpotent elements are $0$ and $6$, since $6^2 = 36 = 0$ in $\Z/12\Z$, while for instance no power of $2$ or $4$ is divisible by $3$.
In $\Z$ or in any field the nilradical is $\set{0}$.
Geometrically, nilpotent functions vanish at every point, so the nilradical is invisible to the points of the [[spectrum|FB45BHHA2J]].
</document>

<document source="https://tamnd.github.io/math2/node/QQEASX79RW.md" tag="QQEASX79RW" kind="counterexample">
# Alexander horned sphere

## Counterexample

The Alexander horned sphere is a subset $\Sigma \subset \R^3$ built by an infinite process.
It is the image of a [[topological embedding|00ZG9Y9TR4]] of the $2$-[[sphere|T62369RFH6]] $S^2$.
Start with a ball and pull out two horns that grow towards each other.
Near their tips, let each horn split into a pair of thinner horns.
Let the two pairs hook through each other like two links of a chain, without touching.
Repeat this splitting inside each pair, at smaller and smaller scales, infinitely often.
The limit set of the surfaces is $\Sigma$; the tips shrink to a Cantor set of points, and $\Sigma$ is still homeomorphic to $S^2$.

The closure of the bounded component of $\R^3 \setminus \Sigma$ is homeomorphic to the closed ball $D^3$.
The unbounded component is not simply connected.
A loop around the base of one horn cannot be shrunk to a point in the complement.
Each attempt to slide it off the horns is blocked by the next pair of hooked horns.
So the closure of the unbounded component is not homeomorphic to the closure of the complement of a round ball, whose [[fundamental group|S1XK3HXHFX]] is trivial.

## Remarks

The example shows that the [[Schoenflies theorem|3VX06XRVZG]] fails in dimension $3$ without its local flatness hypothesis.
The sphere $\Sigma$ is not locally flat at the points of the Cantor set.
J. W. Alexander described it in 1924.
Gluing two copies of the closed bounded side along $\Sigma$ gives a space homeomorphic to $S^3$, a result of R. H. Bing.

## Explanations

### L1 human

In the plane, every loop bounds a disk on the inside and the rest of the plane on the outside, both as nice as possible.
In space one would expect the same for a sphere, but the horned sphere breaks this.
Its horns branch and interlock forever, so a rubber band around a horn on the outside can never be pulled free.
The surface is still a perfect topological sphere; the problem is only how it sits in space.
</document>

<document source="https://tamnd.github.io/math2/node/QQFYC988SJ.md" tag="QQFYC988SJ" kind="statement">
# Variation of parameters for difference equations

## Statement

Let $K$ be a field and let $m \ge 1$.
Let $A(0), A(1), \dots$ be [[invertible|0EB9N0S33C]] $m \times m$ matrices over $K$, and let $b(0), b(1), \dots$ be vectors in $K^m$.
Let $\Phi(0)$ be an invertible $m \times m$ matrix and let $\Phi(n+1) = A(n)\, \Phi(n)$ for $n \ge 0$.
Then each $\Phi(n)$ is invertible, and for each $x(0) \in K^m$ the solution of the [[matrix difference equation|9VA6P5KQQT]]
$$x(n+1) = A(n)\, x(n) + b(n) \qquad (n \ge 0)$$
is
$$x(n) = \Phi(n)\, \Phi(0)^{-1} x(0) + \sum_{k=0}^{n-1} \Phi(n)\, \Phi(k+1)^{-1} b(k) \qquad (n \ge 0).$$

## Remarks

The idea is to replace the constant vector $c$ in the homogeneous solution $\Phi(n) c$ by a varying vector $c(n)$.
Putting $x(n) = \Phi(n) c(n)$ into the equation gives $\Phi(n+1) c(n+1) = \Phi(n+1) c(n) + b(n)$, so $c(n+1) = c(n) + \Phi(n+1)^{-1} b(n)$.
Summing this telescoping relation from $0$ to $n - 1$ gives the formula.
Example: for $m = 1$ and a constant $A(n) = a \ne 0$, take $\Phi(n) = a^n$; the formula reads $x(n) = a^n x(0) + \sum_{k=0}^{n-1} a^{n-1-k} b(k)$.
Invertibility of the $A(n)$ is used to invert $\Phi(k+1)$.
Without it, the formula still holds if $\Phi(n)\, \Phi(0)^{-1}$ is replaced by the product $A(n-1) \cdots A(0)$.
Likewise $\Phi(n)\, \Phi(k+1)^{-1}$ is replaced by $A(n-1) \cdots A(k+1)$, which is the identity matrix when $k = n - 1$.
A scalar recurrence of order $m$ becomes such a system for the vector $(u_n, \dots, u_{n+m-1})$.
For $u_{n+2} + p_n u_{n+1} + q_n u_n = g_n$ with $q_n \ne 0$ and two independent homogeneous solutions $y_1, y_2$, this gives the particular solution with $u_0 = u_1 = 0$:
$$u_n = \sum_{k=0}^{n-1} g_k\, \frac{y_1(k+1)\, y_2(n) - y_1(n)\, y_2(k+1)}{C(k+1)},$$
where $C(j) = y_1(j) y_2(j+1) - y_2(j) y_1(j+1)$ is the Casoratian, which never vanishes.

## Explanations

### L1 human

The homogeneous equation is easy once you know a fundamental matrix: its solutions are $\Phi(n) c$ for constant vectors $c$.
To handle the extra input $b(n)$, let the constant $c$ change with $n$.
Each input $b(k)$ is then carried forward from time $k+1$ to time $n$ by the homogeneous dynamics, and the output is the sum of all these contributions.

### L2 human

This is the discrete form of the [[variation of parameters|0Y6TH5PQVS]] method for linear differential equations, where the sum becomes an integral.
It is also the discrete Duhamel principle: the response to the inputs is a sum of free responses started at each time.
With constant $A$, it reads $x(n) = A^n x(0) + \sum_{k=0}^{n-1} A^{n-1-k} b(k)$, which is the standard formula for linear time-invariant systems in discrete time.
Together with [[superposition|7W7VV861JW]], it describes all solutions of an inhomogeneous linear recurrence.
</document>

<document source="https://tamnd.github.io/math2/node/QQMJZWDS1F.md" tag="QQMJZWDS1F" kind="example">
# Counting measure

## Example

Let $X$ be a set and let $\Sigma$ be its power set.
For $A \subseteq X$, let $\mu(A)$ be the number of elements of $A$ if $A$ is finite, and $\mu(A) = \infty$ if $A$ is infinite.
Then $\mu$ is a [[measure|PR11BZQQPG]] on $(X, \Sigma)$, the counting measure.

For the check, $\mu(\varnothing) = 0$.
Let $A_1, A_2, \ldots$ be pairwise disjoint subsets of $X$ with union $A$.
Suppose all $A_n$ are finite and only finitely many are nonempty.
Then $A$ is finite, and its number of elements is the sum of those of the $A_n$.
Otherwise $A$ is infinite, and the sum $\sum_n \mu(A_n)$ is $\infty$ too, since it has a term $\infty$ or infinitely many terms at least $1$.

## Remarks

Counting measure is [[σ-finite|WEBBT18D4X]] exactly when $X$ is countable.
On $X = \N$, the integral of a non-negative function $f$ with respect to counting measure is the sum $\sum_{n} f(n)$.
So series of non-negative terms are a special case of Lebesgue integrals.

## Explanations

### L1 human

Counting measure gives each set its number of elements.
The set $\set{2, 5, 7}$ has measure $3$, and the set of even numbers has measure $\infty$.
It is the simplest measure, and it turns sums into integrals.
</document>

<document source="https://tamnd.github.io/math2/node/QQMQGXS0XE.md" tag="QQMQGXS0XE" kind="definition">
# Saddle point of a Lagrangian

## Definition

Let $C \subseteq \R^n$ be a set, and let $f, g_1, \dots, g_m \colon C \to \R$ be functions.
Consider the problem of minimizing $f(x)$ over $x \in C$ subject to $g_i(x) \le 0$ for $i = 1, \dots, m$.
Its *Lagrangian* is the function $L \colon C \times [0, \infty)^m \to \R$ given by
$$L(x, \lambda) = f(x) + \sum_{i=1}^m \lambda_i g_i(x).$$
A pair $(x^*, \lambda^*) \in C \times [0, \infty)^m$ is a *saddle point of the Lagrangian* if
$$L(x^*, \lambda) \le L(x^*, \lambda^*) \le L(x, \lambda^*) \quad \text{for all } x \in C \text{ and } \lambda \in [0, \infty)^m.$$

## Remarks

The function $L$ is minimal in the first variable and maximal in the second at $(x^*, \lambda^*)$, as for a saddle point of any function on a product.
The numbers $\lambda^*_i$ play the role of [[Lagrange multipliers|0VKKP8X6Q6]].
If $(x^*, \lambda^*)$ is a saddle point, the left inequality forces $g_i(x^*) \le 0$ for all $i$ and $\lambda^*_i g_i(x^*) = 0$ for all $i$.
The right inequality then gives $f(x^*) = L(x^*, \lambda^*) \le L(x, \lambda^*) \le f(x)$ for every feasible $x$, so $x^*$ is optimal.
This needs no convexity; the converse needs convexity of $f$, $g_i$ and $C$ and a constraint qualification such as Slater's condition.

Example: minimize $x^2$ over $C = \R$ subject to $g(x) = 1 - x \le 0$, with $L(x, \lambda) = x^2 + \lambda (1 - x)$.
The pair $(1, 2)$ is a saddle point.
For every $\lambda \ge 0$, $L(1, \lambda) = 1 = L(1, 2)$.
For every $x$, $L(x, 2) = (x - 1)^2 + 1 \ge 1$.

Non-example: the pair $(1, 1)$ for the same problem.
The right inequality fails, since $L(\tfrac12, 1) = \tfrac34 < 1 = L(1, 1)$.
</document>

<document source="https://tamnd.github.io/math2/node/QQPC9ASQHR.md" tag="QQPC9ASQHR" kind="concept">
# Wigner D-matrix

## Summary

The Wigner D-matrix $D^j(\alpha, \beta, \gamma)$ is the matrix of a rotation, given by its Euler angles, in the irreducible representation of spin $j$ of [[SU(2)|E5P8AG6CK7]].
Its entries are functions of the three angles, and they factor as $e^{-i m' \alpha} d^j_{m' m}(\beta) e^{-i m \gamma}$.
The entries for all $j$ form an orthogonal basis of the square-integrable functions on $\mathrm{SU}(2)$.
Those with integer $j$ form such a basis for the rotation group $\mathrm{SO}(3)$.
They contain the [[spherical harmonics|3F32Y21HP4]] and the [[Legendre polynomials|QFYGJVA996]] as special cases, and they are the standard tool for rotations in quantum mechanics.

## Explanations

### L1 human

A rotation of space acts on functions on the sphere, and on states of a quantum system with angular momentum $j$.
In a fixed basis of $2j + 1$ states this action is a square matrix, and this matrix is the Wigner D-matrix.
For spin $j = 1/2$ it is a $2 \times 2$ unitary matrix; a rotation by $\beta$ about the $y$-axis gives the rows $(\cos \frac{\beta}{2}, -\sin \frac{\beta}{2})$ and $(\sin \frac{\beta}{2}, \cos \frac{\beta}{2})$.
The half angles show that a full turn by $2\pi$ gives $-1$ and not $1$, a famous feature of spin.
</document>

<document source="https://tamnd.github.io/math2/node/QQQ9B1DK4P.md" tag="QQQ9B1DK4P" kind="concept">
# Topological manifold

## Summary

A topological manifold of dimension $n$ is a [[topological space|0CQHFPGJ45]] that looks like the Euclidean space $\R^n$ near each of its points.
It is also required to be Hausdorff and second-countable.
It is the basic object of geometric topology.
Curves, surfaces and the space-times of physics are manifolds, and smooth and piecewise linear manifolds are topological manifolds with extra structure.

## Explanations

### L1 human

Imagine a tiny ant on a surface: around it, the world looks like a flat plane, even though the whole surface may be a sphere or a doughnut.
A manifold is a space with this property in some fixed dimension $n$.
The circle is a $1$-dimensional manifold, because a short arc of a circle looks like a short open interval.
The surface of the Earth is a $2$-dimensional manifold, which is why flat maps of small regions work.
A figure eight is not a manifold, because near the crossing point four arcs meet, which no open interval can imitate.
</document>

<document source="https://tamnd.github.io/math2/node/QQRD38DF4X.md" tag="QQRD38DF4X" kind="definition">
# Haefliger structure

## Definition

Let $q \ge 0$ be an integer.
Let $\Gamma_q$ be the [[groupoid|FH3C0T643F]] whose objects are the points of $\R^q$.
Its arrows from $a$ to $b$ are the [[germs|VZXF7RNFKS]] at $a$ of local diffeomorphisms $\phi$ of $\R^q$ with $\phi(a) = b$.
Give the set of arrows the sheaf topology, in which the germs at nearby points of one local diffeomorphism form an open set.
Let $X$ be a [[topological space|0CQHFPGJ45]].
A *$\Gamma_q$-cocycle* on $X$ is given by an open cover $(U_i)_{i \in I}$ of $X$ and continuous maps $\gamma_{ij} \colon U_i \cap U_j \to \Gamma_q$ such that
$$\gamma_{ik}(x) = \gamma_{ij}(x) \, \gamma_{jk}(x) \quad \text{for all } x \in U_i \cap U_j \cap U_k.$$
Two $\Gamma_q$-cocycles are equivalent if their union, over the disjoint union of the two index sets, is again a $\Gamma_q$-cocycle.
A *Haefliger structure* of codimension $q$ on $X$, or $\Gamma_q$-structure, is an equivalence class of $\Gamma_q$-cocycles.

## Remarks

The cocycle condition with $i = j = k$ shows that $\gamma_{ii}(x)$ is the germ of the identity at a point $f_i(x) \in \R^q$, and this defines continuous maps $f_i \colon U_i \to \R^q$.
The germ $\gamma_{ij}(x)$ goes from $f_j(x)$ to $f_i(x)$, so near $x$ the local maps satisfy $f_i = \phi \circ f_j$ for a representative $\phi$.
The same definition works with any topological groupoid in place of $\Gamma_q$.

Example: let $\mathcal{F}$ be a [[foliation|98HTB7MSPX]] of codimension $q$ of a manifold $X$, given by submersions $f_i \colon U_i \to \R^q$ with $f_i = g_{ij} \circ f_j$ on overlaps for local diffeomorphisms $g_{ij}$.
Taking $\gamma_{ij}(x)$ to be the germ of $g_{ij}$ at $f_j(x)$ gives a $\Gamma_q$-cocycle, since the transition maps compose as $g_{ik} = g_{ij} \circ g_{jk}$ near each point.
Another example: any continuous map $f \colon X \to \R^q$ gives a $\Gamma_q$-structure with one open set $U_1 = X$ and $\gamma_{11}(x)$ the germ of the identity at $f(x)$.

Non-example: let $X$ be a point, $U_1 = U_2 = X$, $\gamma_{11} = \gamma_{22} = \gamma_{21}$ the germ of the identity at $0$ and $\gamma_{12}$ the germ of $t \mapsto 2t$ at $0$.
The cocycle condition fails for $(i, j, k) = (1, 2, 1)$, since $\gamma_{12} \gamma_{21}$ is the germ of $t \mapsto 2t$ and not that of the identity.

Haefliger structures pull back along continuous maps.
For a CW complex $X$, homotopy classes of $\Gamma_q$-structures on $X$ correspond to homotopy classes of maps from $X$ into a [[classifying space|WSZ40H5E4T]] $B\Gamma_q$.

## Explanations

### L1 human

A Haefliger structure is the gluing data of a foliation, kept even where the local maps fail to be submersions.
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<document source="https://tamnd.github.io/math2/node/QQSNZX1QMM.md" tag="QQSNZX1QMM" kind="concept">
# Dunford–Pettis property

## Summary

A [[Banach space|06GZ956657]] has the Dunford–Pettis property when the following holds.
Pairing a weakly null sequence of vectors with a weakly null sequence of functionals always gives a null sequence of numbers.
The spaces $L^1(\mu)$ and $C(K)$ have it, by results of Dunford and Pettis and of Grothendieck.
No infinite-dimensional [[reflexive space|NN8RWPC8WG]] has it.
The property is a tool to show that some operators on these classical spaces are compact.

## Explanations

### L1 human

In infinite dimensions a sequence can tend to zero in a weak sense while its points stay far from zero.
The same happens to sequences of functionals.
In many spaces, two such sequences can still meet: the values $f_n(x_n)$ need not go to zero.
In $\ell^2$, the unit vectors $e_n$ and the functionals "take the $n$-th coordinate" are both weakly null, yet each value is $1$.
Spaces with the Dunford–Pettis property forbid this.
</document>

<document source="https://tamnd.github.io/math2/node/QQSPWH4NG9.md" tag="QQSPWH4NG9" kind="concept">
# Absolutely monotonic function

## Summary

An absolutely monotonic function on an interval is a [[smooth function|0V33BNKZ9S]] whose derivatives of every order, including order $0$, are nonnegative there.
The mirror notion is a completely monotone function, whose derivatives alternate in sign.
On the half-lines $(-\infty, 0)$ and $(0, \infty)$ respectively, both classes are described by [[Bernstein's theorem on monotone functions|ZBS1JA62QP]] as mixtures of exponential functions.

## Explanations

### L1 human

The function is nonnegative, increasing, convex, and so on for every higher derivative.
The exponential function $e^x$ is the main example: all its derivatives equal $e^x > 0$.
A polynomial with nonnegative coefficients is absolutely monotonic on $[0, \infty)$.
A power series with nonnegative coefficients is absolutely monotonic on $[0, r)$, where $r$ is its radius of convergence.
</document>

<document source="https://tamnd.github.io/math2/node/QQTPD2TZ0Z.md" tag="QQTPD2TZ0Z" kind="definition">
# Triangulation

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A *triangulation* of $X$ is a pair $(K, h)$ with $K$ a [[simplicial complex|V8GZ0W1ZZ5]].
The map $h \colon \abs{K} \to X$ must be a [[homeomorphism|0QT00PB5G2]] from the geometric realization $\abs{K}$ of $K$ onto $X$.
$X$ is *triangulable* if it has a triangulation.

## Remarks

Let $K$ be the simplicial complex of all proper faces of a $3$-[[simplex|YX1DJK2QAG]] in $\R^3$, with $4$ vertices, $6$ edges and $4$ triangles.
Radial projection from the barycentre onto the sphere centred there is a homeomorphism $\abs{K} \to S^2$, so $K$ with this map is a triangulation of $S^2$.

The CW structure of $S^2$ with one $0$-cell and one $2$-cell is not a triangulation.
Its $2$-cell is glued to the point along its whole boundary circle, while in a simplicial complex each triangle has three distinct vertices and three edges.

Some authors allow only finite or locally finite complexes $K$, and some require $h$ to be piecewise linear in a given structure.
Every smooth manifold is triangulable, while some topological $4$-manifolds, such as the $E_8$ manifold, are not.
</document>

<document source="https://tamnd.github.io/math2/node/QQW1AWRFZH.md" tag="QQW1AWRFZH" kind="definition">
# Condition number

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, let $n \ge 1$, and let $\norm{\cdot}$ be a submultiplicative [[matrix norm|54T6Y6DSY4]] on $\mathbb{K}^{n \times n}$.
Let $A$ be an [[invertible|0EB9N0S33C]] $n \times n$ matrix over $\mathbb{K}$.
The *condition number* of $A$ with respect to $\norm{\cdot}$ is
$$\kappa(A) = \norm{A} \, \norm{A^{-1}}.$$

## Remarks

By convention, $\kappa(A) = \infty$ when $A$ is not invertible.
If $\norm{\cdot}$ is an [[operator norm|06EKW1AKXF]], then $\kappa(A) \ge \norm{A A^{-1}} = \norm{I} = 1$.
For the operator norm of the Euclidean norm, $\kappa_2(A) = \sigma_{\max} / \sigma_{\min}$, the ratio of the largest and smallest [[singular values|09VMDSRXWJ]].
If $Ax = b$ with $b \ne 0$ and $A(x + \delta x) = b + \delta b$, then $\norm{\delta x} / \norm{x} \le \kappa(A) \, \norm{\delta b} / \norm{b}$ for the operator norm of a vector norm.
Example: $A = \begin{pmatrix} 1 & 0 \\ 0 & 10^{-3} \end{pmatrix}$ has $\norm{A}_2 = 1$ and $\norm{A^{-1}}_2 = 10^{3}$, so $\kappa_2(A) = 10^3$.
Non-example: $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is not invertible, so the formula does not apply.
</document>

<document source="https://tamnd.github.io/math2/node/QQZPWDWS29.md" tag="QQZPWDWS29" kind="statement">
# Lehmann–Scheffé theorem

## Statement

Let $\set{P_\theta : \theta \in \Theta}$ be a [[parametric model|MSC0MR8VMH]] for an observation $X$, and let $g \colon \Theta \to \R$.
Let $T = T(X)$ be a [[sufficient statistic|WP4TGXJQJG]] that is [[complete|DED6JFAV1K]].
Let $h$ be a measurable function such that $h(T)$ is [[unbiased|0BWGTN92FG]] for $g(\theta)$ and $\operatorname{Var}_\theta(h(T)) < \infty$ for every $\theta \in \Theta$.
Then $h(T)$ is a [[uniformly minimum-variance unbiased estimator|ZCJX7THBNJ]] of $g(\theta)$: for every unbiased estimator $\delta$ of $g(\theta)$ and every $\theta \in \Theta$,
$$\operatorname{Var}_\theta(h(T)) \le \operatorname{Var}_\theta(\delta).$$
Moreover, $h(T)$ is the only such estimator, up to equality with $P_\theta$-probability $1$ for every $\theta$.

## Remarks

Completeness gives uniqueness: if $h_1(T)$ and $h_2(T)$ are both unbiased for $g(\theta)$, then $E_\theta[h_1(T) - h_2(T)] = 0$ for all $\theta$, so $h_1(T) = h_2(T)$ almost surely.
Given any unbiased $\delta$ with finite [[variance|0477R9PYAE]], the [[Rao–Blackwell theorem|Y3TX4J41TP]] gives the unbiased estimator $E[\delta \mid T]$, a function of $T$, with variance at most $\operatorname{Var}_\theta(\delta)$.
By uniqueness this function is $h(T)$, which proves the claim.

Example: let $X_1, \dots, X_n$ be independent Poisson variables with mean $\lambda > 0$.
The sum $T = X_1 + \cdots + X_n$ is complete and sufficient, and $T/n$ is unbiased for $\lambda$.
So the sample mean is the unique minimum-variance unbiased estimator of $\lambda$.

Completeness cannot be dropped from the uniqueness argument.
If $T$ is not complete, two different functions of $T$ can be unbiased for the same $g(\theta)$.
Then the Rao–Blackwell step alone does not single out a best one.

## Explanations

### L1 human

If a summary of the data is sufficient and complete, then any unbiased estimate built only from that summary is automatically the best unbiased estimate.
"Best" means smallest variance, for every value of the unknown parameter at once.

### L2 human

This is the standard route to minimum-variance unbiased estimators.
First find a complete sufficient statistic, often the natural statistic of an exponential family of full rank.
Then find any function of it that is unbiased, for example by solving $E_\theta[h(T)] = g(\theta)$ or by Rao–Blackwellizing a simple unbiased estimator.
The theorem then certifies optimality without comparing with other estimators one by one.
</document>

<document source="https://tamnd.github.io/math2/node/QR83860XAZ.md" tag="QR83860XAZ" kind="definition">
# Gaussian process

## Definition

Let $T$ be a set and let $X = (X_t)_{t \in T}$ be a real-valued [[stochastic process|0QH5ENKZM1]].
The process $X$ is a *Gaussian process* if, for every $k \ge 1$ and all $t_1, \dots, t_k \in T$, the random vector $(X_{t_1}, \dots, X_{t_k})$ has a [[multivariate normal distribution|0XN3K7RSZY]], possibly degenerate.

## Remarks

Equivalently, every finite linear combination $\sum_{i=1}^{k} c_i X_{t_i}$ with real $c_i$ is normal or constant.

The *mean function* is $m(t) = E[X_t]$ and the *covariance function* is $K(s, t) = \operatorname{Cov}(X_s, X_t)$.
They determine the finite-dimensional distributions, hence the law of $X$.
The function $K$ is positive semidefinite: the matrix $(K(t_i, t_j))_{i, j}$ is positive semidefinite for all $t_1, \dots, t_k$.
Conversely, by the [[Kolmogorov extension theorem|WR4X7MMTH2]], for every function $m$ and every symmetric positive semidefinite $K$ there is a Gaussian process with mean $m$ and covariance $K$.

Example: the [[Wiener process|22735FGJV6]] is a Gaussian process on $[0, \infty)$ with $m = 0$ and $K(s, t) = \min(s, t)$.
Check: $(W_{t_1}, \dots, W_{t_k})$ for $t_1 < \cdots < t_k$ is a linear image of the independent normal increments $W_{t_1}, W_{t_2} - W_{t_1}, \dots$, hence multivariate normal.

Non-example: let $Z$ be standard normal and let $\varepsilon$ be independent of $Z$ with $P(\varepsilon = 1) = P(\varepsilon = -1) = 1/2$.
Let $X_1 = Z$ and $X_2 = \varepsilon Z$ on $T = \set{1, 2}$.
Each $X_t$ is standard normal, but $X_1 + X_2 = (1 + \varepsilon) Z$ equals $0$ with probability $1/2$ and is not constant, so it is neither normal nor constant.
Thus $(X_1, X_2)$ is not multivariate normal, and $X$ is not a Gaussian process.
</document>

<document source="https://tamnd.github.io/math2/node/QR92EZDTKF.md" tag="QR92EZDTKF" kind="concept">
# Horn clause

## Summary

A Horn clause is a disjunction of literals in which at most one literal is unnegated.
Such a clause can be read as a rule: "if all of these hold, then that holds".
With no unnegated literal, it is a constraint: "these do not all hold".
Satisfiability of a [[conjunctive normal form|0B3D4C8YF2]] made of Horn clauses can be decided in polynomial time, unlike general [[satisfiability|QN7M1JCGT9]].
Horn clauses are the basis of logic programming, and their first-order universal forms axiomatize [[universal Horn classes|9A4CYWXPEA]] and [[quasivarieties|WRXEKT4NQS]].

## Explanations

### L1 human

The clause $\neg p \vee \neg q \vee r$ says the same as "if $p$ and $q$, then $r$".
Rules of this shape can be chained: from known facts one keeps deriving new facts until nothing changes.
This is why Horn formulas are easy to solve, while clauses such as $p \vee q$, which allow a choice, make satisfiability hard in general.
A clause with no positive literal, such as $\neg p \vee \neg q$, says "not both $p$ and $q$".
</document>

<document source="https://tamnd.github.io/math2/node/QRAPTPVH8C.md" tag="QRAPTPVH8C" kind="concept">
# Uniformly most powerful test

## Summary

A uniformly most powerful test of level $\alpha$ is a [[hypothesis test|087P03NEND]] of level $\alpha$ with the largest possible [[power|4RJSN0QARD]].
Its power beats that of every other test of level $\alpha$, at every parameter value of the alternative.
For simple hypotheses the [[Neyman–Pearson lemma|0EE19CFEEF]] gives such a test.
For one-sided hypotheses in families with a [[monotone likelihood ratio|4Z73AFN7WT]], the [[Karlin–Rubin theorem|RCASN31FFS]] gives one.
For many other problems no such test exists.

## Explanations

### L1 human

Among all tests with the same false-alarm rate, one would like the test that best detects every possible alternative at once.
Such a test is called uniformly most powerful.
For a normal mean with known variance, the test that rejects when the sample mean is large is uniformly most powerful against all larger means.
Against "the mean is different", no single test is best everywhere: a test tuned for larger means loses against smaller means.
</document>

<document source="https://tamnd.github.io/math2/node/QREAJQH24A.md" tag="QREAJQH24A" kind="example">
# Tamari lattice

## Example

Let $n \ge 1$, and consider the ways to fully parenthesize a product $x_0 x_1 \cdots x_n$ of $n + 1$ factors.
There are $C_n$ of them, where $C_n$ is the $n$-th [[Catalan number|0YWZJEABB2]].
A rotation replaces a subexpression $(A B) C$ by $A (B C)$, where $A$, $B$, $C$ are subexpressions.
The Tamari order $T_n$ sets $p \le q$ when $q$ can be reached from $p$ by a finite sequence of rotations.

For $n = 3$ there are $C_3 = 5$ bracketings of $abcd$.
From the least element $((ab)c)d$ to the greatest element $a(b(cd))$ there are two routes:
$$((ab)c)d \to (ab)(cd) \to a(b(cd)),$$
$$((ab)c)d \to (a(bc))d \to a((bc)d) \to a(b(cd)).$$
These are all the relations, so $T_3$ is the pentagon $N_5$, with a route of two steps beside a route of three steps.

## Remarks

Each $T_n$ is a [[lattice|WH4SFAACAS]].
Since $T_3$ is the [[pentagon lattice|DHC8TZXGYS]], the Tamari lattices are not [[modular|QX7WZ92QAK]] in general.
Bracketings of $n + 1$ factors correspond to [[binary trees|0D7EZ10V2R]] with $n$ internal nodes, and a rotation is the tree rotation used in balanced search trees.
The Hasse diagram of $T_n$ is the graph of the associahedron.

## Explanations

### L1 human

There are several ways to put brackets into a long product, and moving brackets to the right is a single step.
The Tamari lattice records which bracketings can be reached from which by such steps.
With four factors there are five bracketings, arranged like a pentagon.
</document>

<document source="https://tamnd.github.io/math2/node/QREYWGEH41.md" tag="QREYWGEH41" kind="definition">
# Lyapunov exponent

## Definition

Let $I = [t_0, \infty)$ and let $A \colon I \to \R^{n \times n}$ be continuous.
Consider the [[linear system|037ZNY7NAG]] $x' = A(t) x$, and fix a [[norm|0V2WP9B1MF]] $\norm{\cdot}$ on $\R^n$.
Let $x$ be a nonzero [[solution|07VX34QXAV]] of this system on $I$.
The *Lyapunov exponent* (or characteristic exponent) of $x$ is
$$\lambda(x) = \limsup_{t \to \infty} \frac{1}{t} \ln \norm{x(t)} \in [-\infty, \infty],$$
where $\limsup$ is the [[limit superior|0T3RZQXDYD]].

## Remarks

The value does not depend on the choice of norm, because any two norms on $\R^n$ are within constant factors of each other.
A nonzero solution never vanishes, by uniqueness of solutions, so the logarithm is defined for all $t$.
If $A$ is bounded on $I$, then every exponent is finite.

Example: for constant $A$ with an eigenvalue $\mu$ and eigenvector $v$, the solution $x(t) = e^{\mu t} v$ satisfies $\norm{x(t)} = e^{(\operatorname{Re} \mu) t} \norm{v}$.
So $\frac{1}{t} \ln \norm{x(t)} = \operatorname{Re} \mu + \frac{1}{t} \ln \norm{v} \to \operatorname{Re} \mu$, and $\lambda(x) = \operatorname{Re} \mu$.

Non-example: the zero solution has no Lyapunov exponent in this sense, since $\ln \norm{0}$ is undefined; some authors set it to $-\infty$.

For a nonlinear system $x' = f(x)$ with $f$ of class $C^1$, the exponents along a trajectory are those of the variational equation $y' = Df(x(t))\, y$.
The largest of them measures the rate of separation of nearby trajectories.
</document>

<document source="https://tamnd.github.io/math2/node/QRGSRA0CYV.md" tag="QRGSRA0CYV" kind="definition">
# Fractional differential equation

## Definition

Let $\alpha > 0$ be a real number that is not an integer, let $n = \ceil{\alpha}$, let $a < b$ be real numbers, and let $f \colon [a, b] \times \R \to \R$ be a function.
Write ${}^C D_a^\alpha$ for the [[Caputo fractional derivative|6VMGQYF8RW]] of order $\alpha$ with base point $a$.
The *fractional differential equation* of Caputo type with right side $f$ is
$$ {}^C D_a^\alpha y(t) = f(t, y(t)), \qquad t \in (a, b],$$
usually with initial conditions $y^{(k)}(a) = y_k$ for $k = 0, 1, \ldots, n - 1$.
More generally, a fractional differential equation relates an unknown function to some of its derivatives in the sense of [[fractional calculus|T73XK15SA2]].
At least one of these derivatives has non-integer order.

## Remarks

Let $f$ be continuous.
A continuous function $y$ solves the initial value problem if and only if it solves the Volterra integral equation
$$y(t) = \sum_{k=0}^{n-1} \frac{y_k}{k!} (t - a)^k + \frac{1}{\Gamma(\alpha)} \int_a^t (t - s)^{\alpha - 1} f(s, y(s)) \, ds.$$
Here $\Gamma$ is the [[gamma function|0H5SYBSED9]], and the integral is the [[Riemann–Liouville integral|JTKCZ44NN3]] of order $\alpha$.
Example: for $0 < \alpha < 1$, $a = 0$, $f \equiv 1$ and $y(0) = 0$, the integral equation gives $y(t) = t^\alpha / \Gamma(\alpha + 1)$, and indeed ${}^C D_0^\alpha t^\alpha = \Gamma(\alpha + 1)$.
For $0 < \alpha < 1$, the equation ${}^C D_0^\alpha y = \lambda y$ with $y(0) = 1$ has the solution $y(t) = \sum_{k \ge 0} \lambda^k t^{\alpha k} / \Gamma(\alpha k + 1)$, a [[Mittag-Leffler function|12G9M694YG]] of $\lambda t^\alpha$; it reduces to $e^{\lambda t}$ when $\alpha = 1$.
Non-example: the delay equation $y'(t) = y(t - 1)$ is not a fractional differential equation, since it contains only a derivative of integer order.
Equations with the Riemann–Liouville derivative are also common; their natural initial conditions are on fractional integrals of $y$ at $a$ rather than on $y^{(k)}(a)$.
</document>

<document source="https://tamnd.github.io/math2/node/QS0XVAN9BG.md" tag="QS0XVAN9BG" kind="concept">
# Logical connective

## Summary

A logical connective is a symbol that builds a new formula from one or more given formulas.
The usual connectives are [[negation|029JBBF693]] $\neg$, [[conjunction|025YZN3VJW]] $\land$, [[disjunction|0F72FK8EKB]] $\lor$, [[implication|0NMG7W7Q11]] $\to$ and the [[biconditional|0S6GMNJ62Y]] $\leftrightarrow$.
In classical logic, each connective is given by a truth function, which a [[truth table|05RSK4Q657]] displays.
Connectives are the glue of [[propositional logic|09K9MWD2QQ]] and of [[predicate logic|086QKPF0A6]].

## Explanations

### L1 human

Connectives are the words "not", "and", "or", "if ... then" and "if and only if", written as symbols.
They join simple statements into compound ones.
For example, from $p$: "it rains" and $q$: "it is cold", the connective $\land$ builds $p \land q$: "it rains and it is cold".
In classical logic, the truth of the compound statement depends only on the truth of its parts.
</document>

<document source="https://tamnd.github.io/math2/node/QS6TMVTTHW.md" tag="QS6TMVTTHW" kind="definition">
# Haken manifold

## Definition

Let $M$ be a compact connected [[orientable|702SNFQG0W]] [[3-manifold|R20TB2KEQQ]].
The manifold $M$ is *irreducible* if every embedded $2$-sphere in $M$ bounds an embedded $3$-ball in $M$.
The manifold $M$ is a *Haken manifold* if:

1. $M$ is irreducible, and
2. $M$ contains a properly embedded two-sided [[incompressible surface|6N9GW98MR6]].

## Remarks

Example: the $3$-torus $T^3 = T^2 \times S^1$ is irreducible, since its universal cover $\R^3$ is irreducible by Alexander's theorem.
The torus $T^2 \times \set{1}$ is two-sided, and the inclusion $\Z^2 \to \Z^3$ on fundamental groups is injective, so it is incompressible.
Hence $T^3$ is Haken.

Non-example: the $3$-sphere $S^3$ is irreducible by Alexander's theorem, but it is not Haken.
A closed surface $S \subseteq S^3$ of positive genus has $\pi_1(S) \ne 1 = \pi_1(S^3)$, so by the loop theorem it is compressible.

Every closed orientable irreducible $3$-manifold with infinite first homology group is Haken.
For non-orientable manifolds, irreducible is replaced by $P^2$-irreducible, which also forbids embedded two-sided projective planes.
</document>

<document source="https://tamnd.github.io/math2/node/QS9PG4MNTR.md" tag="QS9PG4MNTR" kind="statement">
# Jordan curve theorem for polygons

## Statement

Let $C \subset \R^2$ be a simple closed polygonal curve.
That is, $C$ is a closed polygonal chain that meets itself only where its last side ends at the start of its first side.
Then $\R^2 \setminus C$ has exactly two connected components.
One component is bounded and the other is unbounded, and $C$ is the boundary of each of them.

## Remarks

The proof uses parity of crossings.
For a point $x \notin C$, take a ray from $x$ in a direction that avoids the vertices of $C$, and count how many times it crosses $C$.
The parity of this count does not depend on the ray, and it is constant along paths that avoid $C$.
So points of even and odd parity lie in different [[components|0V032VWTHD]].
Conversely, two points of the same parity can be joined by a path that runs close to $C$, so there are exactly two components.
Points far away have even parity and form the unbounded one.

This crossing count is also the standard test of whether a point lies inside a polygon.

## Explanations

### L1 human

A closed loop made of straight pieces that never crosses itself splits the plane into an inside and an outside.
To go from inside to outside, a path has to cross the loop.
A point is inside when a ray from it crosses the loop an odd number of times.

### L2 human

This is the special case of the [[Jordan curve theorem|687RGCHBTP]] for polygonal curves, and its proof is elementary.
The general theorem for arbitrary [[closed curves|045YFKRW0P]] without self-intersections is usually proved by approximating the curve by polygons or by homology.
The polygonal case is what is used for simple polygons in [[polygon triangulation|Q6KKBP3RRP]] and other parts of computational geometry.
</document>

<document source="https://tamnd.github.io/math2/node/QSH01NSVXN.md" tag="QSH01NSVXN" kind="definition">
# Packing measure

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]], let $s \ge 0$, and let $A \subseteq X$.
For $\delta > 0$, a *$\delta$-packing* of $A$ is a countable family of pairwise disjoint closed balls $\bar B(x_i, r_i)$ with centres $x_i \in A$ and radii $0 < r_i \le \delta$.
The construction has three steps.

1. $P^s_\delta(A) = \sup \sum_i (2 r_i)^s$, the [[supremum|0R3H5PANRQ]] over all $\delta$-packings of $A$, and $P^s_0(A) = \lim_{\delta \to 0} P^s_\delta(A)$.
2. The *$s$-dimensional packing measure* of $A$ is
$$\mathcal{P}^s(A) = \inf \set{ \sum_{i=1}^\infty P^s_0(A_i) : A \subseteq \bigcup_{i=1}^\infty A_i }.$$
3. The *packing dimension* of $A$ is $\dim_P A = \inf \set{ s \ge 0 : \mathcal{P}^s(A) = 0 }$.

## Remarks

The limit in step 1 exists since $P^s_\delta(A)$ decreases as $\delta$ decreases.
The map $\mathcal{P}^s$ is a [[metric outer measure|3T6WG315GC]], so it is a measure on the Borel sets.
Some authors use the diameters of the balls instead of $2 r_i$; in $\R^n$ the two agree.

Example: on $\R$ with $s = 1$, a $\delta$-packing of $[a, b]$ by intervals has total length at most $b - a + 2\delta$, and packings with total length close to $b - a$ exist.
So $P^1_0([a, b]) = b - a$, and in fact $\mathcal{P}^1$ is Lebesgue measure on $\R$.

Step 2 is needed: $P^s_0$ alone is not countably subadditive.
The set $\Q \cap [0, 1]$ has $P^1_0 = 1$, because centres in a dense set can approximate any packing of $[0, 1]$.
But it is a countable union of points, each with $P^1_0 = 0$, so $\mathcal{P}^1(\Q \cap [0, 1]) = 0$.

For $A \subseteq \R^n$, $\dim_H A \le \dim_P A$, where $\dim_H$ is defined by [[Hausdorff measure|Q1TGZK8RN6]].

## Explanations

### L1 human

Packing measure fills a set with disjoint small balls centred in it and adds the $s$-th powers of their diameters.
A second step, through countable covers, turns this into a true measure.
</document>

<document source="https://tamnd.github.io/math2/node/QSJKPKPM7H.md" tag="QSJKPKPM7H" kind="concept">
# Singular values of a compact operator

## Summary

The singular values of a compact operator $T$ between Hilbert spaces are the eigenvalues of the positive operator $\abs{T} = (T^* T)^{1/2}$, listed in decreasing order.
They extend the singular values of matrices to infinite dimensions and form a sequence that tends to $0$.
They measure how well $T$ can be approximated by finite-rank operators.
Their rate of decay defines the Schatten classes, among them the trace class and the Hilbert–Schmidt operators.

## Explanations

### L1 human

A compact operator stretches the unit ball into an infinite-dimensional ellipsoid whose axes shrink to zero.
The singular values are the lengths of these semi-axes, from the longest to the shortest.
For example, the operator on square-summable sequences that multiplies the $n$-th coordinate by $(-1)^n / n$ has singular values $1, 1/2, 1/3, \ldots$.
The faster the singular values decrease, the closer the operator is to a matrix of small size.
</document>

<document source="https://tamnd.github.io/math2/node/QSK2J5HEER.md" tag="QSK2J5HEER" kind="statement">
# Lenagan's theorem

## Statement

Let $R$ be a [[ring|0B1261B5ED]] with identity.
For a right $R$-[[module|0NZ8JA25WK]] $M$, the [[Krull dimension|PP11DV3ZMQ]] $\operatorname{Kdim} M$ (in the sense of Gabriel and Rentschler) is defined by transfinite recursion on ordinals.
One sets $\operatorname{Kdim} 0 = -1$.
Let $\alpha \ge 0$ be an ordinal, and consider descending chains $M_0 \supseteq M_1 \supseteq \cdots$ of submodules of $M$.
Then $\operatorname{Kdim} M \le \alpha$ means: in every such chain, all but finitely many factors $M_i / M_{i+1}$ have Krull dimension less than $\alpha$.
If such ordinals exist, $\operatorname{Kdim} M$ is the least of them; otherwise $M$ has no Krull dimension.
The ring $R$ has right Krull dimension if $\operatorname{Kdim} R_R \le \alpha$ for some ordinal $\alpha$.

Assume that $R$ has right Krull dimension.
Then the [[upper nilradical|3G2JQEYBEK]] of $R$ is a [[nilpotent ideal|C2H26BYWHE]].
Equivalently, every [[nil ideal|NZ1G9WRWFB]] of $R$, that is, every [[two-sided ideal|0CD87SFM2N]] whose elements are all nilpotent, is nilpotent.

## Remarks

The nonzero modules of Krull dimension $0$ are exactly the nonzero [[Artinian modules|YQNZTQGP1X]].
Every right Noetherian ring has right Krull dimension.
So the theorem contains the two-sided case of [[Levitzki's theorem|QHD4KJYFSA]]: in a right Noetherian ring every nil ideal is nilpotent.
It also covers right Artinian rings, where the nil ideals lie in the [[Jacobson radical|7HWZ06SRMF]], which is nilpotent.

An outline of the proof: one first shows that the [[prime radical|9AVE43G8RT]] $N$ of $R$ is nilpotent.
The quotient $R/N$ is semiprime and still has right Krull dimension, which makes it a right [[Goldie ring|4AY3RMCTA3]].
A semiprime right Goldie ring has no nonzero nil ideals, so every nil ideal of $R$ lies in $N$.

The hypothesis cannot be dropped.
Let $k$ be a field and $R = k[x_1, x_2, \dots] / (x_1, x_2^2, x_3^3, \dots)$, with $\bar x_i$ the image of $x_i$.
The ideal $I$ generated by all $\bar x_i$ is nil, since each element of $I$ lies in an ideal generated by finitely many nilpotent elements of a commutative ring.
It is not nilpotent, since $\bar x_n^{\,n-1} \ne 0$ lies in $I^{n-1}$ for every $n$.
This ring has no Krull dimension.

## Explanations

### L1 human

An element is nilpotent when some power of it is zero.
An ideal can consist of nilpotent elements and still have no single power that kills all of its elements at once.
Lenagan's theorem says that this cannot happen in rings that satisfy a mild finiteness condition, the existence of Krull dimension.
In such rings, "every element dies at some power" upgrades to "every product of a fixed number of elements of the ideal is zero".

### L2 human

Krull dimension of a module measures how far the module is from being Artinian, by a deviation of its lattice of submodules.
Rings with right Krull dimension include all right Noetherian and all right Artinian rings, and many rings that are neither.
The theorem extends the classical facts that nil ideals are nilpotent in right Artinian rings and, by Levitzki's theorem, in right Noetherian rings.
It is a typical use of Krull dimension: a statement proved for Noetherian rings by an ascending chain argument is carried over by induction on the deviation.
The commutative example $k[x_1, x_2, \dots] / (x_1, x_2^2, x_3^3, \dots)$ shows that some finiteness condition is needed even without noncommutativity.
</document>

<document source="https://tamnd.github.io/math2/node/QSMVQ4EJ3D.md" tag="QSMVQ4EJ3D" kind="example">
# The symmetric group S3 is not abelian

## Example

Let $S_3$ be the [[symmetric group|0515A6D2Y1]] on $\{1, 2, 3\}$. Compose permutations from right to left.
Take the [[transpositions|02DTRYKGE8]] $(1\,2)$ and $(1\,3)$.
Then $(1\,2)(1\,3) = (1\,3\,2)$ and $(1\,3)(1\,2) = (1\,2\,3)$.
These two products are different. Thus $S_3$ is not [[abelian|071DDFVG4J]].

## Remarks

$S_3$ has $6$ elements. It is the smallest group that is not abelian.
</document>

<document source="https://tamnd.github.io/math2/node/QSTN8NMSK0.md" tag="QSTN8NMSK0" kind="definition">
# Join

## Definition

Let $X$ and $Y$ be topological spaces and let $I = [0, 1]$.
The *join* of $X$ and $Y$ is the quotient space
$$X * Y = (X \times Y \times I) / \sim,$$
where $X \times Y \times I$ has the [[product topology|0GBG8B49AA]], and $X * Y$ carries the [[quotient topology|0NQ1H2X495]].
The relation $\sim$ is the equivalence relation generated by two rules.
First, $(x, y, 0) \sim (x, y', 0)$ for all $x \in X$ and $y, y' \in Y$.
Second, $(x, y, 1) \sim (x', y, 1)$ for all $x, x' \in X$ and $y \in Y$.

## Remarks

The class of $(x, y, t)$ is often written $(1 - t) x + t y$, since it lies on the segment from $x$ to $y$.
The images of $t = 0$ and $t = 1$ are copies of $X$ and of $Y$ in $X * Y$.
Example: if $Y$ is one point, then $X * Y$ is the cone on $X$.
Example: if $Y = S^0$ is two points, then $X * S^0$ is the unreduced [[suspension|Q7RFNJH76Q]] of $X$.
Example: $S^0 * S^0$ is homeomorphic to the circle $S^1$, a square with four vertices.
Non-example: the product $X \times Y \times I$ with no identification is not the join; for $X = Y = S^0$ it is four disjoint segments, while $S^0 * S^0$ is a circle.
</document>

<document source="https://tamnd.github.io/math2/node/QSWSVYRCS1.md" tag="QSWSVYRCS1" kind="object">
# Mandelbrot set

## Description

For $c \in \C$ let $f_c(z) = z^2 + c$, the [[quadratic family|WDZFV2ACXR]].
The *Mandelbrot set* is
$$M = \set{c \in \C : \text{the sequence } f_c^n(0),\ n \ge 0, \text{ is bounded}}.$$
Here $0$ is the only critical point of $f_c$.
Check of a few points:

1. $c = 0$: the orbit of $0$ is $0, 0, 0, \ldots$, so $0 \in M$.
2. $c = -1$: the orbit is $0, -1, 0, -1, \ldots$, so $-1 \in M$.
3. $c = i$: the orbit is $0, i, -1 + i, -i, -1 + i, -i, \ldots$, so $i \in M$.
4. $c = 1$: the orbit is $0, 1, 2, 5, 26, \ldots$, which is unbounded, so $1 \notin M$.

If some $\abs{f_c^n(0)} > 2$, the orbit tends to $\infty$.
So $M$ is the set of $c$ with $\abs{f_c^n(0)} \le 2$ for all $n$, and $M$ is a compact subset of the disc $\abs{c} \le 2$.

## Remarks

A parameter $c$ lies in $M$ exactly when the [[Julia set|5M4S7P04GR]] of $f_c$ is connected.
For $c \notin M$ the Julia set is a Cantor set.
Douady and Hubbard proved that $M$ is connected.
Its intersection with the real line is the interval $[-2, 1/4]$.
Shishikura proved that the boundary of $M$ has [[Hausdorff dimension|SSNFTHZ6YJ]] $2$.
The main cardioid is the set of $c = \lambda/2 - \lambda^2/4$ with $\abs{\lambda} < 1$.
For these $c$, the map $f_c$ has a fixed point with multiplier $\lambda$, hence an attracting fixed point.
Whether $M$ is [[locally connected|FDVAHRJKF3]] is an open problem, known as the MLC conjecture.

## Explanations

### L1 human

For each complex number $c$, start at $0$ and apply $z \mapsto z^2 + c$ again and again.
Colour $c$ black if the numbers you get stay bounded, and leave it white if they run off to infinity.
The black region is the Mandelbrot set, a cardioid with discs and filaments attached, and with smaller copies of itself everywhere along its edge.
It is a map of the whole quadratic family: each point tells whether the Julia set of that map is one connected piece or dust.
</document>

<document source="https://tamnd.github.io/math2/node/QSX12D8RQM.md" tag="QSX12D8RQM" kind="concept">
# Valuation ring

## Summary

A valuation ring is an [[integral domain|0QVD7WBFGN]] $R$ with a simple property, see the [[definition|YVQ9CKGA0T]].
For each nonzero $x$ in its [[field of fractions|0F2PTGX1A2]], either $x$ or $1/x$ lies in $R$.
Valuation rings are exactly the rings of elements of nonnegative value for a [[valuation|B85T5VKR3M]].
They are [[local rings|MZ02BKPHJA]], and their ideals are totally ordered by inclusion.
[[Discrete valuation rings|MGM3CNJW9V]] are the Noetherian valuation rings that are not fields.

## Explanations

### L1 human

Think of the fractions whose denominator is not divisible by a prime $p$, such as $3/5$ or $7/2$ for $p = 3$.
They form a ring, and any fraction that is not in it, such as $5/3$, has its inverse $3/5$ in it.
So the ring takes "half" of the fractions in a precise sense: from each pair $x, 1/x$ at least one is in.
This half is decided by one rule, the size of the power of $p$, and in general by a valuation.
</document>

<document source="https://tamnd.github.io/math2/node/QT1DYNX2QA.md" tag="QT1DYNX2QA" kind="definition">
# Pincherle derivative

## Definition

Let $K$ be a field and let $K[x]$ be the [[polynomial ring|0SDC934R52]] in one variable over $K$.
Let $T \colon K[x] \to K[x]$ be a $K$-[[linear map|0NRYHJPQG2]].
The *Pincherle derivative* of $T$ is the linear map $T'$ on $K[x]$ given by
$$T'(f) = T(x f) - x \, T(f) \qquad (f \in K[x]).$$
Writing $x$ also for the operator of multiplication by $x$, this says $T' = T x - x T$.

## Remarks

The map $T \mapsto T'$ is linear and satisfies the product rule $(ST)' = S'T + ST'$, since $STx - xST = S(Tx - xT) + (Sx - xS)T$.

Example: for $D = d/dx$, $D'(f) = (xf)' - x f' = f$, so $D' = \id$.
For the shift $E^h f(x) = f(x + h)$ with $h \in K$, $(E^h)'(f)(x) = (x + h) f(x + h) - x f(x + h) = h f(x + h)$, so $(E^h)' = h E^h$.

Operators with Pincherle derivative zero: a multiplication operator $M_q f = q f$ with $q \in K[x]$ commutes with $x$, so $M_q' = 0$.
Conversely, $T' = 0$ forces $T(x^n) = x^n T(1)$ for all $n$, so $T = M_{T(1)}$.
</document>

<document source="https://tamnd.github.io/math2/node/QT3M720TCV.md" tag="QT3M720TCV" kind="concept">
# Multidimensional Laplace transform

## Summary

The multidimensional Laplace transform takes a function of $n$ variables on the orthant $[0, \infty)^n$ to a function of $n$ complex variables, by integrating against $e^{-s_1 x_1 - \cdots - s_n x_n}$.
It is the [[Laplace transform|0V49JEZW06]] applied in each variable in turn.
It turns partial derivatives into multiplication by the variables $s_j$, up to boundary terms.
So it is used for linear partial differential equations with constant coefficients.
In probability it gives the joint Laplace transform of nonnegative random vectors.

## Explanations

### L1 human

The ordinary Laplace transform weighs a function of time by a decaying exponential.
In several variables one uses one decay rate for each variable.
For example, for $f(x_1, x_2) = 1$ on the quarter plane the transform is $\int_0^\infty \int_0^\infty e^{-s_1 x_1 - s_2 x_2}\, dx_1\, dx_2 = \frac{1}{s_1 s_2}$ for $s_1, s_2 > 0$.
A product of functions of single variables becomes a product of one-variable Laplace transforms.
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<document source="https://tamnd.github.io/math2/node/QT4CF7ZKK5.md" tag="QT4CF7ZKK5" kind="concept">
# Shimura variety

## Summary

A Shimura variety is a quotient of a Hermitian [[symmetric space|PAA93R4WYD]] by a congruence subgroup of a [[reductive group|2XS84PWE3S]] over $\Q$.
It is viewed as an algebraic variety.
The family includes [[modular curves|T5W0CMS0DM]] and the Siegel moduli spaces of abelian varieties.
Shimura varieties are defined over number fields, and their cohomology carries both Galois actions and automorphic forms.
This makes them a main bridge in the Langlands program.

## Explanations

### L1 human

The upper half-plane, divided out by a group such as $\mathrm{SL}_2(\Z)$, gives a modular curve, which parametrizes elliptic curves.
Shimura varieties are the higher-dimensional versions of this picture.
They start from a symmetric space with a complex structure and divide it by an arithmetic group.
An example is the Siegel upper half-space of symmetric complex matrices with positive definite imaginary part.
The result is an algebraic variety that often parametrizes abelian varieties with extra structure.
Because it has a natural model over a number field, it links geometry, automorphic forms and Galois representations.
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<document source="https://tamnd.github.io/math2/node/QT4HP1H2YE.md" tag="QT4HP1H2YE" kind="definition">
# Seifert surface

## Definition

Let $L \subset S^3$ be an oriented tame [[link|1A5WRCNTM3]], for example an oriented [[knot|QYEAZTYGPT]].
A *Seifert surface* for $L$ is a subset $\Sigma \subset S^3$ such that:

1. $\Sigma$ is a compact connected [[surface|VGJJVBZ1MQ]] with boundary, embedded in $S^3$ in a tame way;
2. $\Sigma$ is [[orientable|702SNFQG0W]];
3. the boundary of $\Sigma$ is $L$, and the orientation of $L$ is the boundary orientation induced by an orientation of $\Sigma$.

## Remarks

The *genus* $g(K)$ of a knot $K$ is the least [[genus|NJ787GZS1N]] of a Seifert surface for $K$.
Example: the unknot $\set{(x, y, 0) : x^2 + y^2 = 1}$ in $\R^3 \subset S^3$ bounds the flat disk $\set{(x, y, 0) : x^2 + y^2 \le 1}$, which is compact, connected, orientable and has the unknot as boundary; so the unknot has genus $0$.
Conversely, a knot of genus $0$ bounds a disk and is the unknot.
Non-example: a band with one half-twist, a Möbius band, has the unknot as boundary, but it is not orientable, so it is not a Seifert surface.
A band with three half-twists is a Möbius band whose boundary is the trefoil, so it fails for the same reason; the trefoil has genus $1$ instead.
Seifert's algorithm builds a Seifert surface from any diagram.
One smooths each crossing according to the orientation, fills the resulting circles with disks, and joins the disks by twisted bands at the crossings.
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<document source="https://tamnd.github.io/math2/node/QTC57AJEVR.md" tag="QTC57AJEVR" kind="definition">
# Categorical theory

## Definition

Let $T$ be a [[theory|92SWXST1ZM]] in a first-order signature $L$, and let $\kappa$ be an infinite [[cardinal|0EBACQFAZW]].
The theory $T$ is *$\kappa$-categorical* if two conditions hold.

1. $T$ has a [[model|MVCH6N9SNW]] whose domain has [[cardinality|0G0ASKNA6Z]] $\kappa$.
2. Any two models of $T$ of cardinality $\kappa$ are isomorphic.

## Remarks

A theory is *uncountably categorical* if it is $\kappa$-categorical for every uncountable $\kappa$.
Some authors drop the existence of a model of size $\kappa$ from the definition.

Example: the theory $\mathrm{ACF}_0$ of [[algebraically closed fields|MT8TDVPBXB]] of characteristic $0$ is $\kappa$-categorical for every uncountable $\kappa$.
By [[Steinitz's theorem|CAEHMSX9RT]], such a field is determined up to isomorphism by its transcendence degree over $\Q$, and a field of uncountable size $\kappa$ has transcendence degree $\kappa$.

Non-example: $\mathrm{ACF}_0$ is not $\aleph_0$-categorical.
The fields $\overline{\Q}$ and the algebraic closure of $\Q(t)$ are both countable models, of transcendence degree $0$ and $1$, so they are not isomorphic.

The theory of dense linear orders without endpoints is $\aleph_0$-categorical, by Cantor's back-and-forth argument, but not $\kappa$-categorical for any uncountable $\kappa$.
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<document source="https://tamnd.github.io/math2/node/QTHPN6V7XM.md" tag="QTHPN6V7XM" kind="definition">
# Birkhoff min-max method

## Definition

Let $g$ be a smooth [[Riemannian metric|Q61VFHCH2J]] on the [[sphere|P9CM4GQ2T0]] $S^2$.
A *sweepout* of $S^2$ is a continuous map $c \colon [0, 1] \times S^1 \to S^2$ such that:

1. each curve $c_t = c(t, \cdot)$ is piecewise smooth, and $t \mapsto c_t$ is continuous for the $C^0$ topology and for the [[length|0TH78S7GBZ]];
2. the curves $c_0$ and $c_1$ are constant;
3. the map $S^2 \to S^2$ induced by $c$ on the quotient of $[0, 1] \times S^1$ that collapses $\set{0} \times S^1$ and $\set{1} \times S^1$ to points has nonzero [[degree|TMFG2P912R]].

The *width* of $(S^2, g)$ is
$$W(g) = \inf_{c} \ \max_{t \in [0, 1]} L(c_t),$$
the infimum over all sweepouts $c$ of the length of the longest curve.
The *Birkhoff min-max method* shows that $W(g) > 0$ and that $W(g)$ is the length of a [[closed geodesic|4MCCP8GG4J]].
The geodesic is a limit of longest curves of a minimizing sequence of sweepouts.
Before taking the limit, a curve-shortening process is applied to all curves of each sweepout at once.

## Remarks

The width is positive because curves of small length lie in small discs, so a sweepout by short curves would induce a map of degree $0$.
Birkhoff's curve-shortening process replaces each curve by a broken geodesic through finitely many of its points, and repeats.
It acts continuously on all curves of a sweepout and does not increase lengths.

Example: on the round unit sphere, the circles of latitude $c_t(s) = (\sin \pi t \cos s, \sin \pi t \sin s, -\cos \pi t)$ form a sweepout of degree $1$, whose longest curve is the equator, of length $2\pi$.
Here $W = 2\pi$, the length of a great circle.

Non-example: a family that starts and ends at constant curves but stays in the southern hemisphere, such as $c_t(s) = (\tfrac12 \sin \pi t \cos s, \tfrac12 \sin \pi t \sin s, -\sqrt{1 - \tfrac14 \sin^2 \pi t})$, is not a sweepout.
The induced map misses the north pole, so its degree is $0$, and clause 3 fails.
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<document source="https://tamnd.github.io/math2/node/QTM1HZM53Q.md" tag="QTM1HZM53Q" kind="concept">
# Survival function

## Summary

The survival function of a lifetime $T$ gives, for each time $t$, the probability that the event has not yet happened at time $t$.
It equals one minus the [[cumulative distribution function|0H8H7ZTHW2]] of $T$.
It is the main quantity estimated in [[survival analysis|443ZWDX88P]], and it determines the distribution of $T$.
In reliability engineering it is called the reliability function.

## Explanations

### L1 human

If $T$ is the lifetime of a light bulb, $S(1000)$ is the chance that the bulb still works after $1000$ hours.
The curve $S$ starts at $1$, because every bulb works at the start, and goes down as time passes.
It never goes up again: a broken bulb stays broken.
For a lifetime with constant risk of failure, the curve is the decaying exponential $e^{-\lambda t}$.
</document>

<document source="https://tamnd.github.io/math2/node/QTQN93B305.md" tag="QTQN93B305" kind="definition">
# Fixed-point property of posets

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
$P$ has the *fixed-point property* if every [[monotone|0A7KV2WB7A]] map $f \colon P \to P$ has a fixed point, that is, an element $x \in P$ with $f(x) = x$.

## Remarks

Example: every finite chain $c_1 < c_2 < \cdots < c_n$ has the fixed-point property.
For monotone $f$, $c_1 \le f(c_1)$, and induction gives $f^k(c_1) \le f^{k+1}(c_1)$; this increasing sequence in a finite set is eventually constant, at a fixed point.
Non-example: the two-element antichain $\set{a, b}$ fails it, since the map exchanging $a$ and $b$ is monotone and has no fixed point.
The chain $\Z$ also fails it, with the monotone map $x \mapsto x + 1$.
Every complete lattice has the fixed-point property, by the Knaster–Tarski theorem.
More generally, every finite poset with a least element has it, by the same iteration as for chains.
The property is preserved by order isomorphisms and passes to retracts.
</document>

<document source="https://tamnd.github.io/math2/node/QTRPNFDFJ8.md" tag="QTRPNFDFJ8" kind="concept">
# Natural boundary condition

## Summary

A natural boundary condition is a boundary condition that a minimiser satisfies automatically at an end where no value is prescribed.
For $J(y) = \int_a^b L(x, y, y') \, dx$ with $y(b)$ free, it reads $L_p = 0$ at $x = b$.
It comes from the [[first variation|5B8NFYB2XQ]], not from the statement of the problem.
For the Dirichlet energy in several variables, it is the [[Neumann boundary condition|GS3KT2H645]].

## Explanations

### L1 human

If one end of a curve is free to move up and down, the best curve has to be best also with respect to that freedom.
This gives an extra condition at the free end, which nobody imposed by hand.
Think of a hanging string with one end on a frictionless vertical rail.
The rail cannot push along the string, so the string settles with its end meeting the rail at a right angle.
</document>

<document source="https://tamnd.github.io/math2/node/QTSJT4JZXT.md" tag="QTSJT4JZXT" kind="statement">
# John–Nirenberg inequality

## Statement

Let $n \ge 1$.
There are constants $c_1, c_2 > 0$, depending only on $n$, with the following property.
Let $f$ be a function of [[bounded mean oscillation|QJYTXFT8Y4]] on $\R^n$ that is not constant, with norm $\norm{f}_{\mathrm{BMO}}$.
Let $Q \subseteq \R^n$ be a cube with sides parallel to the axes, let $\abs{E}$ denote [[Lebesgue measure|2Z2QVWZ262]], and let $f_Q = \frac{1}{\abs{Q}}\int_Q f$.
Then for every $\lambda > 0$,
$$\abs{\set{x \in Q : \abs{f(x) - f_Q} > \lambda}} \le c_1 \abs{Q} \exp\left( - \frac{c_2 \lambda}{\norm{f}_{\mathrm{BMO}}} \right).$$

## Remarks

The proof applies the [[Calderón–Zygmund lemma|K1YKXVPVDG]] to $f - f_Q$ on the cube $Q$, at a height of about $2\norm{f}_{\mathrm{BMO}}$.
On the selected subcubes $Q_k$ the mean $f_{Q_k}$ differs from $f_Q$ by at most a constant times $\norm{f}_{\mathrm{BMO}}$, and their total measure is at most half of $\abs{Q}$.
Repeating the argument inside each $Q_k$ shows that the set where $\abs{f - f_Q}$ exceeds $C m \norm{f}_{\mathrm{BMO}}$ has measure at most $2^{-m}\abs{Q}$, which is the exponential decay.

Consequences: for $c < c_2$ the function $\exp(c \abs{f - f_Q}/\norm{f}_{\mathrm{BMO}})$ has average over $Q$ bounded by a constant depending on $n$ and $c$.
Also, for every $1 \le p < \infty$, the quantity $\sup_Q \left( \frac{1}{\abs{Q}} \int_Q \abs{f - f_Q}^p \right)^{1/p}$ is a norm on BMO equivalent to $\norm{f}_{\mathrm{BMO}}$.

Example: $f(x) = \log \abs{x}$ on $\R$ and $Q = [-1, 1]$, where $f_Q = -1$.
The set $\set{\abs{f + 1} > \lambda}$ contains $\set{\abs{x} < e^{-1 - \lambda}}$, of measure $2 e^{-1 - \lambda}$.
So the exponential decay in $\lambda$ cannot be improved to a faster decay for all BMO functions.

## Explanations

### L1 human

A BMO function can be unbounded, but only in a mild way.
The inequality says that on any cube, the set where $f$ is far from its average is exponentially small in the distance.
So BMO functions behave like logarithms: they can blow up, but only very slowly.

### L2 human

The inequality shows that BMO functions are locally integrable to every power, and even exponentially integrable after subtracting means.
It is used to show that $\log w$ is in BMO for every Muckenhoupt weight $w$, and conversely that $e^{\delta b}$ is a Muckenhoupt weight for $b \in$ BMO and $\delta$ small.
It enters the proof of Fefferman's duality theorem and of the boundedness of commutators with BMO functions.
</document>

<document source="https://tamnd.github.io/math2/node/QTSQR7130G.md" tag="QTSQR7130G" kind="concept">
# Basis of a matroid

## Summary

A basis of a [[matroid|PX31ZNYS3Z]] is a maximal independent set.
All bases of a matroid have the same size, the [[rank|SYRCTH853H]] of the matroid, and they satisfy an exchange property.
The bases determine the matroid, and they generalize both [[bases|0KCQ9Y3QKA]] of a vector space and [[spanning trees|0S4ZEZM7KE]] of a connected graph.

## Explanations

### L1 human

Keep adding elements to an independent set as long as it stays independent; when you cannot add any more, you have a basis.
For vectors in a plane, any two non-parallel vectors form a basis.
For a connected graph with the matroid of forests, the bases are exactly the spanning trees.
Any two bases have the same number of elements.
</document>

<document source="https://tamnd.github.io/math2/node/QTTW954WCK.md" tag="QTTW954WCK" kind="definition">
# Lattice of equational theories

## Definition

Let $\sigma$ be a signature and let $X$ be a countably infinite set of variables.
An [[equational theory|F1M51B4MDG]] of signature $\sigma$ is a set $T$ of [[identities|PKD440G2XW]] over $X$ that contains every identity true in all models of $T$.
The *lattice of equational theories* of $\sigma$ is the set of all equational theories of signature $\sigma$, ordered by inclusion.
For an equational theory $T_0$, the lattice of equational theories *extending* $T_0$ is its subset of theories that contain $T_0$.

## Remarks

The meet of a family of theories is their intersection.
The join is the smallest theory containing their union, that is, the set of consequences of the union.
So the lattice is a [[complete lattice|D7A0ZGZT5S]], whose top is the theory containing every identity, including $x \approx y$.
Identify a theory $T$ with the set of pairs $(s, t)$ such that $s \approx t \in T$.
This makes $T$ a [[fully invariant congruence|QKGW3Z9BHW]] of the [[term algebra|3YYKGWJJND]] over $X$, and every fully invariant congruence arises this way.
Example: the identities true in all groups form a theory, and so do those true in all abelian groups; the first is contained in the second.
Non-example: the set $\set{xy \approx yx}$ alone is not a theory, because it does not contain its consequences $zw \approx wz$ and $x \approx x$.
The map $T \mapsto \operatorname{Mod}(T)$ is a dual isomorphism onto the lattice of varieties of signature $\sigma$.
</document>

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# Pontryagin duality theorem

## Statement

Let $G$ be a [[locally compact abelian group|ZR09S9R3AW]], let $\widehat{G}$ be its [[Pontryagin dual|FMW57WNPDB]], and let $\widehat{\widehat{G}}$ be the Pontryagin dual of $\widehat{G}$.
For $x \in G$, define $\alpha_G(x) \in \widehat{\widehat{G}}$ by $\alpha_G(x)(\chi) = \chi(x)$ for all [[characters|H1WNVN5V7F]] $\chi \in \widehat{G}$.
Then the map $\alpha_G \colon G \to \widehat{\widehat{G}}$ is an isomorphism of topological groups.

## Remarks

It is easy to check that $\alpha_G$ is a continuous homomorphism.
Injectivity means that continuous characters separate the points of $G$.
This is a hard step; one proof uses the Gelfand–Raikov theorem, or the Peter–Weyl theorem in the compact case.
Surjectivity and openness are usually proved with the Plancherel theorem, or by reducing to compactly generated groups of the form $\R^n \times \Z^m \times K$ with $K$ compact.
Example: $\widehat{\Z} \cong \mathbb{T}$ and $\widehat{\mathbb{T}} \cong \Z$, where $n \in \Z$ gives the character $z \mapsto z^n$ of $\mathbb{T}$, and the theorem says that $n \mapsto (z \mapsto z^n)$ is the natural isomorphism.
The map $\alpha_G$ is natural in $G$, so duality is a contravariant equivalence of the category of locally compact abelian groups with itself.
The theorem needs local compactness: for general abelian topological groups the dual may fail to be locally compact, and $\alpha_G$ may fail to be an isomorphism.

## Explanations

### L1 human

Every locally compact abelian group has a dual group made of its waves, its continuous characters.
The theorem says that taking the dual twice brings back the original group, in a natural way.
So the group and its dual carry the same information, seen from two sides.
The circle and the integers are dual to each other, and the real line is dual to itself.

### L2 human

Duality turns properties of $G$ into properties of $\widehat{G}$: compact groups have discrete duals, and discrete groups have compact duals.
A compact group is connected exactly when its discrete dual is torsion-free.
Closed subgroups $H \subseteq G$ correspond to closed subgroups of $\widehat{G}$ through annihilators, with $\widehat{G/H}$ equal to the annihilator of $H$.
The theorem is the background of the [[Fourier transform on LCA groups|VHN5VW5Q92]].
The transform turns functions on $G$ into functions on $\widehat{G}$, and duality lets the inverse transform go back from $\widehat{G}$ to $G$.
It also underlies the [[Plancherel theorem for abelian groups|V9K9TG0K3J]] and [[Bochner's theorem|ZRHAG4NXHC]].
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# Iterative refinement

## Statement

Let $A$ be an [[invertible|0EB9N0S33C]] $n \times n$ real or complex matrix, and let $b$ be a vector.
Let $x = A^{-1} b$ be the solution of the [[linear system|0GT4WWGWH2]] $Ax = b$.
Fix a vector norm and its [[operator norm|06EKW1AKXF]], and a number $\eta$ with $0 \le \eta < 1/2$.
*Iterative refinement* starts from an approximation $x_0$ and repeats, for $m = 0, 1, 2, \dots$:

1. compute the residual $r_m = b - A x_m$;
2. compute a correction $d_m$ with $(A + E_m) d_m = r_m$, for some matrix $E_m$ with $\norm{A^{-1}} \norm{E_m} \le \eta$;
3. set $x_{m+1} = x_m + d_m$.

Then, if the residuals and updates are computed exactly,
$$\norm{x_{m+1} - x} \le \frac{\eta}{1 - \eta} \norm{x_m - x}$$
for all $m$, so $x_m \to x$ linearly with rate at most $\eta / (1 - \eta) < 1$.

## Remarks

Let $e_m = x - x_m$, so that $r_m = A e_m$.
Then $e_{m+1} = e_m - (A + E_m)^{-1} A e_m = (A + E_m)^{-1} E_m e_m$.
Since $\norm{A^{-1} E_m} \le \eta < 1$, the matrix $A + E_m = A(I + A^{-1} E_m)$ is invertible with $\norm{(A + E_m)^{-1}} \le \norm{A^{-1}} / (1 - \eta)$, which gives the bound.
The matrix $E_m$ models the error of an inexact solver, such as a computed [[LU decomposition|0RKG5YW66Y]] reused at each step, or a solve in lower precision.
If the solver is too inaccurate, so that $\eta \ge 1/2$, the bound gives no contraction and the process can diverge.
This happens when the [[condition number|AB3S6MHJ8X]] of $A$ times the solver's relative accuracy is not small.
In floating-point practice, the residual is the critical step: computing $r_m$ in higher precision lets refinement reach a solution accurate to the working precision, even for fairly ill-conditioned $A$.
In exact arithmetic with $E_m = 0$, one step gives the exact solution.

## Explanations

### L1 human

Solve the system approximately, then check how far off you are by computing the residual $b - Ax$.
The residual is itself the right-hand side of a system for the error, so solve that system approximately too, and add the correction.
Each round removes most of the remaining error, as long as the solver is not too crude.

### L2 human

Iterative refinement turns a cheap or low-precision solver into an accurate one.
It reuses one factorization of $A$, so each extra round costs only $O(n^2)$ operations after the $O(n^3)$ factorization.
Modern mixed-precision solvers factor in low precision and refine in higher precision to get both speed and accuracy.
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<document source="https://tamnd.github.io/math2/node/QTX0GBZMT8.md" tag="QTX0GBZMT8" kind="statement">
# Removable sets for bounded harmonic functions

## Statement

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be an open set.

1. Let $E \subseteq \Omega$ be a closed subset of $\Omega$ that is a [[polar set|C95QJ7YH5T]].
   Let $h$ be a [[harmonic function|0DEB0JMN0W]] on $\Omega \setminus E$ such that each point of $E$ has a neighbourhood $V$ with $h$ bounded on $V \setminus E$.
   Then $h$ extends uniquely to a harmonic function on $\Omega$.
2. Conversely, let $\Omega$ be bounded and let $K \subseteq \Omega$ be a [[compact set|0R6EWFKRCF]] that is not polar.
   Then there is a bounded harmonic function on $\Omega \setminus K$ that does not extend to a harmonic function on $\Omega$.

## Remarks

Proof idea for part 1: the question is local, so work in a small ball $B$ with $\overline B \subseteq \Omega$.
Let $H$ be the Poisson integral on $B$ of $h$ on $\partial B$, which is bounded and defined outside $E \cap \partial B$, a set of surface measure zero.
Let $u$ be a subharmonic function, not identically $-\infty$, with $u \le 0$ on $B$ and $u = -\infty$ on $E$.
For each $\varepsilon > 0$ the function $h - H + \varepsilon u$ is subharmonic on $B \setminus E$ and tends to $-\infty$ at $E$, so the maximum principle gives $h - H + \varepsilon u \le 0$.
Letting $\varepsilon \to 0$ and repeating the argument with $-h$ gives $h = H$ off the polar set $\set{u = -\infty}$, hence on $B \setminus E$.
For part 2, the restriction to $\Omega \setminus K$ of the potential of the equilibrium measure of $K$ is bounded and harmonic, and it does not extend.

Example: in $\R^2$, a single point is polar, so a bounded harmonic function on a punctured disc extends harmonically across the puncture.
The function $\log \abs{x}$ on the punctured disc is harmonic but unbounded, and it does not extend, so boundedness is needed.

Example of part 2: in $\R^3$ let $K$ be the closed unit ball and let $\Omega$ be the open ball of radius $2$.
The function $h(x) = 1/\abs{x}$ is bounded and harmonic on $\Omega \setminus K$.
Suppose that it had a harmonic extension $H$ to $\Omega$.
The function equal to $H$ on $\Omega$ and to $1/\abs{x}$ outside $K$ would be harmonic and bounded on $\R^3$ and tend to $0$ at infinity.
By Liouville's theorem it would be $0$, which is false.

## Explanations

### L1 human

A harmonic function that stays bounded cannot notice a polar set: any hole of this kind can be filled in.
Among compact sets, the polar ones are exactly those with this property.
So capacity zero is the precise notion of "too small to matter" for bounded harmonic functions.

### L2 human

This theorem extends the classical removable singularity theorem for a single point.
In the plane it gives the removability of polar sets for bounded holomorphic functions as well, since their real and imaginary parts are harmonic.
For bounded holomorphic functions the exact class of removable compact sets is larger: it is described by analytic capacity, and a segment is not removable while some non-polar sets are.
The same proof scheme, with a subharmonic function equal to $-\infty$ on the exceptional set, is used throughout potential theory to show that polar sets can be ignored.
</document>

<document source="https://tamnd.github.io/math2/node/QTZWZR1KWQ.md" tag="QTZWZR1KWQ" kind="concept">
# Perspectivity

## Summary

A perspectivity maps the points of one line to the points of another line by central projection from a fixed point, the centre.
It is the basic map of [[projective geometry|N32Z68F0D3]] and models what an eye at the centre sees.
A composition of perspectivities is called a projectivity, and on the real projective line the projectivities are exactly the [[projective transformations|P1MKTF7MWQ]].
Perspectivities preserve the [[cross-ratio|09ASMRMD67]].

## Explanations

### L1 human

Stand at a point $O$ and look at points on a line $\ell$.
Each line of sight from $O$ through a point $X$ of $\ell$ hits a second line $m$ at a point $X'$.
The map $X \mapsto X'$ is the perspectivity from $\ell$ to $m$ with centre $O$.
Equal steps on $\ell$ usually become unequal steps on $m$, as in a perspective drawing of a fence.
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<document source="https://tamnd.github.io/math2/node/QV3V8HJW2F.md" tag="QV3V8HJW2F" kind="definition">
# Idempotence

## Definition

1. Let $\ast$ be a [[binary operation|0M418FE7BB]] on a set $S$.
An element $x \in S$ is *idempotent* for $\ast$ if $x \ast x = x$.
The operation $\ast$ is *idempotent* if every element of $S$ is idempotent for $\ast$.
2. Let $X$ be a set.
A [[function|0EPAT51DXQ]] $f \colon X \to X$ is *idempotent* if $f \circ f = f$, that is, $f(f(x)) = f(x)$ for all $x \in X$, where $\circ$ is [[composition|0THBK9V825]].

## Remarks

Example: union is an idempotent operation on the power set of a set $X$, because $Y \cup Y = Y$ for every $Y \subseteq X$.

Non-example: multiplication on $\Z$ is not idempotent, because $2 \cdot 2 = 4 \ne 2$; its only idempotent elements are $0$ and $1$.

The two notions agree.
A function $f \colon X \to X$ is idempotent in the sense of clause 2 if and only if it is idempotent for composition on the set of maps $X \to X$.
A function is idempotent if and only if it fixes every point of its image.
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<document source="https://tamnd.github.io/math2/node/QV75TT6131.md" tag="QV75TT6131" kind="statement">
# Cheeger–Gromoll splitting theorem

## Statement

Let $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] with nonnegative [[Ricci curvature|4GTNM0WHDH]].
Suppose $M$ contains a line, that is, a unit-speed [[geodesic|NSYMQMHPDN]] $\gamma \colon \R \to M$ with $d(\gamma(s), \gamma(t)) = \abs{s - t}$ for all $s, t \in \R$, where $d$ is the [[Riemannian distance|B4QR306D8G]].
Then $M$ is [[isometric|R9YWQS15AA]] to a Riemannian product $N \times \R$, where $N$ is a connected complete Riemannian manifold with nonnegative Ricci curvature.

## Remarks

The idea of the proof: let $b^{+}$ and $b^{-}$ be the Busemann functions of the two rays $t \mapsto \gamma(t)$ and $t \mapsto \gamma(-t)$, $t \ge 0$.
The Laplacian comparison theorem gives $\Delta b^{\pm} \le 0$ in a weak sense.
The triangle inequality gives $b^{+} + b^{-} \ge 0$, with equality on $\gamma$, so the strong maximum principle forces $b^{+} + b^{-} = 0$ and $b^{+}$ is harmonic and smooth.
The [[Bochner formula|DF7KYMAGTC]] applied to $b^{+}$, with $\abs{\nabla b^{+}} = 1$, gives that its Hessian vanishes, so $\nabla b^{+}$ is a parallel vector field and $M$ splits along its flow.

Example: the flat cylinder $S^1 \times \R$ contains the lines $\set{\theta} \times \R$ and splits with $N = S^1$.

The curvature hypothesis is needed: the hyperbolic plane contains lines but is not a product with a line.
The line is needed too: the paraboloid $z = x^2 + y^2$ has positive curvature and contains no line.

## Explanations

### L1 human

Suppose a space is nowhere negatively curved in the Ricci sense and contains one infinitely long shortest path.
Then the whole space is a product of a cross-section and that line, like a cylinder.

### L2 human

Toponogov proved the theorem earlier under the stronger hypothesis of nonnegative sectional curvature; Cheeger and Gromoll proved the Ricci version in 1971.
A main consequence: if $M$ is compact with $\operatorname{Ric} \ge 0$, then its universal cover is isometric to $N \times \R^k$ with $N$ compact.
It follows that the fundamental group of such $M$ contains a free abelian subgroup of finite index.
Versions of the theorem for limit spaces are central in the work of Cheeger and Colding.
</document>

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# Lebesgue–Stieltjes measure

## Summary

A Lebesgue–Stieltjes measure is a measure on the real line built from a non-decreasing, right-continuous function $F$, so that the interval $(a, b]$ gets measure $F(b) - F(a)$.
Every locally finite Borel measure on $\R$ arises this way.
For $F(x) = x$ it is Lebesgue measure, and for the distribution function of a random variable it is the law of that variable.
Integration against it is Lebesgue–Stieltjes integration, which extends the Riemann–Stieltjes integral.

## Explanations

### L1 human

Think of $F(x)$ as the total mass lying to the left of $x$ on a line.
Then the mass of the interval $(a, b]$ is $F(b) - F(a)$.
Where $F$ grows fast, there is a lot of mass, and where $F$ jumps, there is a point mass equal to the jump.
For example, if $F(x) = 0$ for $x < 0$ and $F(x) = 1$ for $x \ge 0$, the measure is a single unit mass at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/QVCKQ46TFM.md" tag="QVCKQ46TFM" kind="definition">
# Continuous q-Hermite polynomials

## Definition

Let $q \in \C$ with $q^k \ne 1$ for all integers $k \ge 1$, and let $x = \cos\theta$.
For an integer $n \ge 0$, the *continuous $q$-Hermite polynomial* is
$$H_n(x \mid q) = \sum_{k=0}^{n} \binom{n}{k}_q\, e^{i(n - 2k)\theta},$$
where $\binom{n}{k}_q$ is the [[Gaussian binomial coefficient|YS09Z1YNEK]].

## Remarks

The terms for $k$ and $n - k$ have the same coefficient, so they pair to $2\binom{n}{k}_q \cos((n - 2k)\theta)$, and $H_n$ is a polynomial of degree $n$ in $x$ with leading coefficient $2^n$.
They satisfy $H_0 = 1$, $H_1 = 2x$ and the recurrence $2x\, H_n(x \mid q) = H_{n+1}(x \mid q) + (1 - q^n)\, H_{n-1}(x \mid q)$ for $n \ge 1$.
For $\abs{q} < 1$ and $\abs{t} < 1$ the generating function is $\sum_{n \ge 0} \frac{H_n(x \mid q)}{(q; q)_n} t^n = \frac{1}{(t e^{i\theta}; q)_\infty\, (t e^{-i\theta}; q)_\infty}$, with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].

Example: $n = 2$.
With $\binom{2}{1}_q = 1 + q$, $H_2 = e^{2i\theta} + (1 + q) + e^{-2i\theta} = 2\cos 2\theta + 1 + q = 4x^2 - 1 + q$.
The recurrence gives the same: $2x \cdot 2x - (1 - q) \cdot 1 = 4x^2 - 1 + q$.

Non-example: the ordinary Hermite polynomial $4x^2 - 2$ is not $H_2(x \mid q)$ for any $q$: matching constants would need $q = -1$, which is excluded since $q^2 = 1$.

For $-1 < q < 1$ they are orthogonal: $\frac{1}{2\pi} \int_0^{\pi} H_m(\cos\theta \mid q)\, H_n(\cos\theta \mid q)\, (e^{2i\theta}, e^{-2i\theta}; q)_\infty \, d\theta = \delta_{mn} \frac{(q; q)_n}{(q; q)_\infty}$.
As $q \to 1^-$, $\left(\frac{2}{1 - q}\right)^{n/2} H_n\left( x \sqrt{\tfrac{1 - q}{2}} \,\middle|\, q \right)$ tends to the [[Hermite polynomial|MGACY98E9H]] $H_n(x)$.
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# Mahler measure

## Summary

The Mahler measure of a complex polynomial is the absolute value of its leading coefficient times a product over its roots.
Each root outside the unit circle contributes its absolute value, and the other roots contribute $1$.
It is multiplicative and, for integer polynomials, it is at least $1$.
It measures the size of a polynomial in a way adapted to its roots.
Small Mahler measures of integer polynomials are the subject of [[Lehmer's conjecture|NDRRH2YD2C]].

## Explanations

### L1 human

Factor the polynomial into its leading coefficient and linear factors.
Keep the leading coefficient and every root of absolute value bigger than $1$, and multiply their absolute values.
For $x^2 - 2$, the roots are $\pm \sqrt{2}$, so the Mahler measure is $\sqrt{2} \cdot \sqrt{2} = 2$.
The [[definition|1PJS951C8N]] states the formula.
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# Hypersingular integral equation

## Summary

A hypersingular integral equation is an integral equation with a very singular kernel.
Its integral exists neither as an ordinary integral nor as a [[Cauchy principal value|0SYR10HD4R]].
The typical kernel on an interval is $1/(t - x)^2$, and the integral is read as a [[Hadamard finite part|G7VE41JHHH]].
Such equations arise in crack problems of elasticity and in the scattering of waves by thin screens.
They also arise in boundary element methods that use the normal derivative of a double layer potential.

## Explanations

### L1 human

In a Cauchy singular equation the kernel behaves like $1/(t - x)$, and the two sides of the singularity cancel.
In a hypersingular equation the kernel behaves like $1/(t - x)^2$, which is positive on both sides, so nothing cancels.
One gives the integral a meaning by removing its divergent part.
A useful picture: the hypersingular operator is the derivative of a Cauchy singular operator.
For example, the finite part of $\frac{1}{\pi}\int_{-1}^{1} \frac{\sqrt{1 - t^2}}{(t - x)^2}\, dt$ equals $-1$ for every $x$ in $(-1, 1)$.
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<document source="https://tamnd.github.io/math2/node/QVWHWDJ5X1.md" tag="QVWHWDJ5X1" kind="definition">
# MV-algebra

## Definition

An *MV-algebra* is a structure $(A, \oplus, \neg, 0)$, where $\oplus$ is a binary operation, $\neg$ is a unary operation and $0 \in A$, such that for all $x, y \in A$:

1. $(A, \oplus, 0)$ is a commutative [[monoid|0N6R902FG1]].
2. $\neg \neg x = x$.
3. $x \oplus \neg 0 = \neg 0$.
4. $\neg (\neg x \oplus y) \oplus y = \neg (\neg y \oplus x) \oplus x$.

## Remarks

Example: the unit interval $[0, 1]$ with $x \oplus y = \min(1, x + y)$, $\neg x = 1 - x$ and $0$ is an MV-algebra.
Clauses 1 to 3 are direct, since $\neg 0 = 1$.
For clause 4, if $y \ge x$ the left side is $\neg 1 \oplus y = y$, and if $y < x$ it is $(x - y) \oplus y = x$; so both sides equal $\max(x, y)$.

Non-example: the integers $(\Z, +, -, 0)$ satisfy clauses 1 and 2, but fail clause 3, since $1 + (-0) = 1 \ne 0$.

Every [[Boolean algebra|0XADJKM699]] is an MV-algebra with $\oplus = \vee$ and $\neg$ the complement.
The constant $\neg 0$ is written $1$, and $x \odot y = \neg(\neg x \oplus \neg y)$ is the dual operation.
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<document source="https://tamnd.github.io/math2/node/QW17YJ6ZCT.md" tag="QW17YJ6ZCT" kind="definition">
# Galois representation

## Definition

Let $K$ be a [[field|04C5ZBG1WT]] with separable closure $K^{\mathrm{sep}}$, and let $G_K = \Gal(K^{\mathrm{sep}}/K)$ be its absolute [[Galois group|0XTY86BZKY]] with the Krull topology, a profinite group.
Let $A$ be a topological commutative ring and let $n \ge 1$.
Typical choices of $A$ are $\C$, a finite field, $\Z_\ell$, or a finite extension of $\Q_\ell$.
Give the [[general linear group|0FCHE902D7]] $\mathrm{GL}_n(A)$ the topology induced from $A^{n^2}$.
A *Galois representation* of $K$ of dimension $n$ over $A$ is a [[continuous|0X3WGX95FE]] [[group homomorphism|0JYTG5K9Y4]] $\rho \colon G_K \to \mathrm{GL}_n(A)$.

## Remarks

Equivalently, it is a continuous action of $G_K$ on a free $A$-module of rank $n$, by $A$-linear maps.
Two representations are isomorphic if they are conjugate by a matrix of $\mathrm{GL}_n(A)$.

Example: let $\ell$ be a prime different from the characteristic of $K$.
The $\ell$-adic cyclotomic character $\chi_\ell \colon G_K \to \Z_\ell^\times = \mathrm{GL}_1(\Z_\ell)$ is given by $\sigma(\zeta) = \zeta^{\chi_\ell(\sigma) \bmod \ell^k}$.
This holds for all $k$ and every root of unity $\zeta$ of order dividing $\ell^k$.
It is a homomorphism, and it is continuous because its reduction modulo $\ell^k$ factors through the finite quotient $\Gal(K(\mu_{\ell^k})/K)$.
Another example: the Tate module $T_\ell(E) = \varprojlim E[\ell^k]$ of an elliptic curve $E$ over $K$ gives a $2$-dimensional representation over $\Z_\ell$.

Non-example: the constant map $G_K \to \mathrm{GL}_n(A)$ with value $-I$, for $A$ of characteristic not $2$, is continuous but sends the identity to $-I \ne I$, so it is not a homomorphism.

With $A = \C$ and its usual topology, every continuous representation has finite image; these are the Artin representations.
For $K$ a number field, a Galois representation is unramified at a finite place $v$ if it is trivial on the inertia group at $v$.
Then the image of a Frobenius element at $v$ is defined up to conjugacy.
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# Lawson topology

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
For $x \in P$, write ${\uparrow} x = \set{y \in P : x \le y}$.
The *lower topology* on $P$ is the topology with [[subbasis|09S8DV34Y0]] the sets $P \setminus {\uparrow} x$ for $x \in P$.
The *Lawson topology* on $P$ is the smallest [[topology|0CQHFPGJ45]] on $P$ that contains both the [[Scott topology|ZR8A03FV67]] and the lower topology.
Equivalently, it is the topology generated by the union of these two topologies as a subbasis.

## Remarks

Example: on $P = [0, 1]$ with the usual order, the Scott open sets are $\emptyset$, $[0, 1]$ and the intervals $(a, 1]$ with $0 \le a < 1$.
The lower open sets are generated by the intervals $P \setminus {\uparrow} b = [0, b)$.
Finite intersections of these sets give all intervals $(a, b)$, $[0, b)$ and $(a, 1]$, so the Lawson topology is the usual topology of $[0, 1]$.

Non-example: the Scott topology of $[0, 1]$ is not its Lawson topology.
The set $[0, 1/2)$ is Lawson open, but it is not an upper set, so it is not Scott open.

The Lawson topology of a [[continuous lattice|RXBD58SGXZ]] is compact and Hausdorff.
Some authors define the Lawson topology only for directed-complete partial orders or for continuous posets, where it is best behaved.
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# Domain of finite type

## Definition

Let $\Omega \subseteq \C^n$ be a domain with $C^\infty$ boundary.
Let $\rho$ be a [[smooth|0V33BNKZ9S]] defining function of $\Omega$, as in the [[definition of the Levi form|5S30FAJNGM]].
Let $D \subseteq \C$ be the unit disc.
For a function $g$ of class $C^\infty$ near $0 \in \C$, let $\nu(g)$ be its order of vanishing at $0$.
This is the smallest $k$ such that some partial derivative of $g$ of order $k$ is nonzero at $0$, and $\nu(g) = \infty$ if there is none.
For $p \in \partial\Omega$, the *type* of $\partial\Omega$ at $p$ is
$$\Delta(p) = \sup_{\gamma} \frac{\nu(\rho \circ \gamma)}{\nu(\gamma - p)},$$
where $\gamma \colon D \to \C^n$ runs over the nonconstant [[holomorphic maps|3MQTWKBBBT]] with $\gamma(0) = p$.
Here $\nu(\gamma - p)$ is the least order of vanishing of the components of $\gamma - p$.
The domain $\Omega$ is *of finite type* if $\Delta(p) < \infty$ for every $p \in \partial\Omega$.

## Remarks

The number $\Delta(p)$ does not depend on the choice of $\rho$, since two defining functions differ by a smooth positive factor.
It is also called the D'Angelo type, or the $1$-type.
Example: at a strongly pseudoconvex point $\Delta(p) = 2$, so a bounded [[strongly pseudoconvex domain|Z7RVXBA2KB]] is of finite type.
For $\Omega = \set{\abs{z}^2 + \abs{w}^{2m} < 1}$ in $\C^2$, with $m \ge 1$ an integer, the disc $\gamma(t) = (1, t)$ gives $\rho \circ \gamma(t) = \abs{t}^{2m}$, so $\Delta((1, 0)) \ge 2m$.
Non-example: if $\partial\Omega$ contains a nonconstant holomorphic disc $\gamma$ through $p$, then $\rho \circ \gamma = 0$, so $\Delta(p) = \infty$.
When $n = 2$, the type agrees with the type of Kohn, which is defined by iterated brackets of complex tangent vector fields.
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# Sanov's theorem

## Statement

Let $\Sigma$ be a [[Polish space|584GXXY73M]], and let $M_1(\Sigma)$ be the set of [[probability measures|059TJN0NZ1]] on $\Sigma$ with the topology of [[weak convergence|MBW9JCGZC1]].
Let $X_1, X_2, \dots$ be [[i.i.d.|00MRN0FARV]] random elements of $\Sigma$ with law $\mu$.
Let $L_n = \frac{1}{n} \sum_{k=1}^{n} \delta_{X_k}$ be the empirical measure, where $\delta_x$ is the [[Dirac measure|0ACRGYNHSR]] at $x$.
Then the laws of $L_n$ satisfy the [[large deviation principle|MNPWBBNCAD]] on $M_1(\Sigma)$ with the good rate function $\nu \mapsto D(\nu \,\|\, \mu)$, the [[Kullback–Leibler divergence|PQBW43X5RE]] of $\nu$ from $\mu$.

## Remarks

For a finite set $\Sigma = \set{1, \dots, m}$, the proof is a counting argument.
The probability that $L_n$ equals a given type $\nu$, with $n \nu(i)$ an integer for each $i$, is a multinomial probability.
By Stirling's formula, it equals $e^{-n D(\nu \| \mu) + O(\log n)}$, and there are only polynomially many types.
The general case reduces to finite partitions of $\Sigma$, or uses Cramér's theorem in a space of measures.

Example: for fair coin tosses, $\mu$ is uniform on $\set{0, 1}$.
The event that the fraction of ones is at least $0.6$ has probability $e^{-n c + o(n)}$, where $c = D(\nu \| \mu)$ for $\nu = (0.4, 0.6)$, that is $c = 0.6 \log 1.2 + 0.4 \log 0.8 \approx 0.020$.

## Explanations

### L1 human

Draw many independent samples from a known distribution and make a histogram.
The histogram almost always resembles the true distribution.
Sanov's theorem says that the chance that it resembles some other distribution $\nu$ instead is about $e^{-n D(\nu \| \mu)}$, where $D$ is the Kullback–Leibler divergence.

### L2 human

Sanov's theorem gives the Kullback–Leibler divergence a probabilistic meaning and is a basic link between probability and information theory.
By the contraction principle, it implies Cramér's theorem for bounded continuous functions of the samples.
In statistics it yields error exponents of tests, such as Stein's lemma for hypothesis testing.
The conditional limit theorem, or Gibbs conditioning principle, also follows: given a rare constraint on $L_n$, the empirical measure concentrates near the minimiser of $D(\cdot \| \mu)$ under the constraint.
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# Cayley–Abels graph

## Summary

Let $G$ be a compactly generated totally disconnected [[locally compact group|AHV05ZBCYW]].
A Cayley–Abels graph of $G$ is a connected locally finite graph on which $G$ acts vertex-transitively with compact open vertex stabilisers.
It generalises the [[Cayley graph|0GYQK9NDJF]] of a finitely generated group, which is the case of a discrete group.
Every such $G$ has a Cayley–Abels graph, and any two of them are [[quasi-isometric|6FKYSH729T]].
So $G$ has a well-defined large-scale geometry, as in geometric group theory.

## Explanations

### L1 human

For a finitely generated group, the Cayley graph draws one vertex for each element and joins elements that differ by a generator.
For a group like $\mathrm{PGL}_2(\Q_p)$ this is not possible with finitely many generators.
Instead one draws one vertex for each coset of a compact open subgroup $U$, and joins cosets using finitely many extra elements.
For $\mathrm{PGL}_2(\Q_p)$ with $U = \mathrm{PGL}_2(\Z_p)$, one can obtain in this way the regular tree in which each vertex has $p + 1$ neighbours.
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# Lorentz group

## Example

Let $\eta = \operatorname{diag}(1, -1, -1, -1)$.
Give $\R^4$ with coordinates $(t, x, y, z)$ the quadratic form
$$Q(t, x, y, z) = t^2 - x^2 - y^2 - z^2,$$
which makes it Minkowski space (in units where the speed of light is $1$).
The *Lorentz group* is the group of Lorentz transformations, the linear maps of $\R^4$ preserving $Q$:
$$O(1, 3) = \set{\Lambda \in GL_4(\R) : \Lambda^T \eta \Lambda = \eta}.$$
It is the [[indefinite orthogonal group|M2900DMEW6]] of signature $(1, 3)$, a [[Lie group|7JBRGCVJRB]] of dimension $6$.

Check with a boost along the $x$-axis with rapidity $\phi \in \R$:
$$\Lambda = \begin{pmatrix} \cosh\phi & \sinh\phi & 0 & 0 \\ \sinh\phi & \cosh\phi & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.$$
It sends $(t, x, y, z)$ to $(t \cosh\phi + x \sinh\phi, \ t \sinh\phi + x \cosh\phi, \ y, \ z)$, and
$$(t \cosh\phi + x \sinh\phi)^2 - (t \sinh\phi + x \cosh\phi)^2 = (t^2 - x^2)(\cosh^2\phi - \sinh^2\phi) = t^2 - x^2,$$
so $Q$ is preserved.

## Remarks

The rotations of space, $\Lambda = \operatorname{diag}(1, R)$ with $R \in$ [[SO(3)|0CGYNDCWRR]], form a compact subgroup, and the boosts make the group noncompact.
$O(1, 3)$ has four connected components, distinguished by the sign of $\det \Lambda$ and the sign of the entry $\Lambda_{00}$.
The identity component $SO^+(1, 3)$ is the *proper orthochronous* Lorentz group: it keeps the orientation of space-time and the direction of time.
The other components are reached from it by space inversion, time reversal, or both.
$SO^+(1, 3)$ is isomorphic to $PSL(2, \C)$, and [[SL(2,C)|J58ED1GV6N]] is its universal cover, a double cover.

## Explanations

### L1 human

The Lorentz group collects the changes of reference frame of special relativity that fix the origin of space-time.
These are the rotations of space and the boosts, which pass to an observer moving at constant velocity.
Because they preserve $t^2 - x^2 - y^2 - z^2$, they map the light cone $t^2 = x^2 + y^2 + z^2$ to itself, so every observer measures the same speed of light.
A boost mixes time with one space coordinate in the way a rotation mixes two space coordinates, but with hyperbolic functions in place of sines and cosines.
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# Exponential functions on semigroups

## Summary

An exponential function on a [[semigroup|0XMEM406JM]] $S$ is a complex-valued function $m$ with $m(xy) = m(x) m(y)$ for all $x, y \in S$.
On the additive group $\R$, the continuous ones are the functions $x \mapsto e^{\lambda x}$ with $\lambda \in \C$, and the zero function.
Exponentials are the multiplicative solutions of Cauchy's equation on $S$, and they generalize the [[characters of abelian groups|H1WNVN5V7F]].
Many functional equations on groups and semigroups have solutions built from a few exponentials, such as $\frac{1}{2}(m(x) + m(x^{-1}))$ for d'Alembert's equation.

## Explanations

### L1 human

An exponential function turns the operation of the semigroup into multiplication of numbers.
On the natural numbers under addition, $m(n) = 2^n$ is an example, since $2^{a + b} = 2^a \cdot 2^b$.
On the positive integers under multiplication, $m(n) = n^3$ is an example.
These functions play the role that $e^x$ plays on the real line.
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# Nonemptiness of the spectrum

## Statement

Let $A$ be a [[Banach algebra|9KYWN94QDN]] over the [[complex numbers|09FTAET5Z4]] with unit $1 \ne 0$.
For $a \in A$, let $\sigma(a) = \set{\lambda \in \C : a - \lambda 1 \text{ is not invertible in } A}$ be the [[spectrum|EH0CZ19RKP]] of $a$.
Then $\sigma(a)$ is nonempty for every $a \in A$.

## Remarks

Proof idea: suppose $\sigma(a)$ is empty.
Then the resolvent $R(\lambda) = (a - \lambda 1)^{-1}$ is defined on all of $\C$.
For each continuous linear functional $\varphi$ on $A$, the function $\lambda \mapsto \varphi(R(\lambda))$ is entire and tends to $0$ as $\abs{\lambda} \to \infty$.
By [[Liouville's theorem|0EHKVRF4J6]] it is zero, and by the [[Hahn–Banach theorem|QB5AE3VYZZ]] this forces $R(0) = a^{-1} = 0$, which is impossible since $1 \ne 0$.
Together with the [[compactness of the spectrum|FA5V5KRWDF]], the spectrum of every element is a nonempty compact subset of $\C$.

Example: in the algebra $C([0,1])$ of continuous complex functions with the sup norm, the spectrum of $f$ is its range $f([0,1])$, which is nonempty.

The complex scalars are needed.
In the real Banach algebra of real $2 \times 2$ matrices, the rotation $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ minus $\lambda 1$ has determinant $\lambda^2 + 1 > 0$ for every real $\lambda$, so its real spectrum is empty.
The condition $1 \ne 0$ is also needed: in the zero algebra every element is invertible.

## Explanations

### L1 human

For a square complex matrix, the spectrum is the set of eigenvalues, and every such matrix has an eigenvalue.
The theorem says the same thing for every element of a complex Banach algebra: there is always at least one number $\lambda$ for which $a - \lambda 1$ cannot be inverted.
The proof uses complex analysis instead of polynomials.

### L2 human

This result is the first step of spectral theory in Banach algebras.
It gives the [[Gelfand–Mazur theorem|3D6XH41A3B]]: a complex Banach division algebra is $\C$.
It also makes the spectral radius $\max \set{\abs{\lambda} : \lambda \in \sigma(a)}$ well defined, which is the subject of [[Gelfand's spectral radius formula|T4RFYT8HDN]].
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# Quasi-primal algebra

## Summary

A quasi-primal algebra is a finite algebra in which the [[ternary discriminator|72PG6XCQ7D]] is a term operation.
Every primal algebra is quasi-primal, and so is every finite field.
The variety generated by a quasi-primal algebra is a [[discriminator variety|MC9MX40Z7S]], so it is arithmetical and well understood.
Quasi-primal algebras are a natural middle ground between primal algebras and arbitrary finite algebras.

## Explanations

### L1 human

In a quasi-primal algebra, the operations can test whether two elements are equal and act on the answer.
This is weaker than expressing every function, as in a primal algebra.
For example, the field with four elements can express the test, but it cannot express a constant that is moved by its Frobenius automorphism.
Such algebras still behave much like Boolean algebras in many structural questions.
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# Normal family

## Definition

Let $U \subseteq \C$ be an open set and let $\mathcal{F}$ be a family of [[holomorphic|0DG0Z72CQD]] functions $U \to \C$.
The family $\mathcal{F}$ is *normal* if every sequence $(f_n)$ in $\mathcal{F}$ has a [[subsequence|0R8J3ZQMJS]] that converges [[uniformly|08HJRTG5N7]] on each [[compact|0QPHW703PV]] subset of $U$.

## Remarks

Example: $\mathcal{F} = \set{z^n : n \ge 1}$ on the unit disc $\mathbb{D}$ is normal.
On a compact set $K \subset \mathbb{D}$ there is $r < 1$ with $\abs{z} \le r$ on $K$, so $\abs{z^n} \le r^n \to 0$ uniformly on $K$.
Non-example: $\set{n z : n \ge 1}$ on $\mathbb{D}$ is not normal.
Each subsequence tends to $\infty$ at each $z \neq 0$ but stays $0$ at $z = 0$, so it does not converge on a disc around $0$.
The limit of a convergent subsequence is holomorphic, by Weierstrass's theorem that [[uniform limits of holomorphic functions are holomorphic|0431RN0R1N]].
Many authors also accept a subsequence that tends to $\infty$ uniformly on compact subsets.
With this convention, and with the chordal metric of the [[Riemann sphere|0CDMXGPSES]], the definition extends to families of meromorphic functions.
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# Addition formulas for Jacobi elliptic functions

## Statement

Let $0 < k < 1$, and let $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ be the [[Jacobi elliptic functions|P071B692JF]] with modulus $k$.
For all complex $u, v$ such that both sides are defined, with
$$\Delta = 1 - k^2\operatorname{sn}^2 u \operatorname{sn}^2 v,$$
one has
$$\operatorname{sn}(u + v) = \frac{\operatorname{sn} u \operatorname{cn} v \operatorname{dn} v + \operatorname{sn} v \operatorname{cn} u \operatorname{dn} u}{\Delta},$$
$$\operatorname{cn}(u + v) = \frac{\operatorname{cn} u \operatorname{cn} v - \operatorname{sn} u \operatorname{sn} v \operatorname{dn} u \operatorname{dn} v}{\Delta},$$
$$\operatorname{dn}(u + v) = \frac{\operatorname{dn} u \operatorname{dn} v - k^2 \operatorname{sn} u \operatorname{sn} v \operatorname{cn} u \operatorname{cn} v}{\Delta}.$$

## Remarks

Proof idea: fix $c$ and set $v = c - u$; using the derivative rules $\operatorname{sn}' = \operatorname{cn}\operatorname{dn}$, $\operatorname{cn}' = -\operatorname{sn}\operatorname{dn}$, $\operatorname{dn}' = -k^2\operatorname{sn}\operatorname{cn}$ and the relations $\operatorname{sn}^2 + \operatorname{cn}^2 = 1$, $\operatorname{dn}^2 + k^2\operatorname{sn}^2 = 1$, one checks that each right side has derivative $0$ in $u$.
So each right side depends only on $u + v = c$, and putting $u = 0$ (where $\operatorname{sn} 0 = 0$, $\operatorname{cn} 0 = \operatorname{dn} 0 = 1$) shows it equals the left side.
By analytic continuation the identities hold wherever both sides are defined.
Example: for $k \to 0$ the formulas become $\sin(u + v) = \sin u\cos v + \sin v\cos u$ and the cosine addition formula, with $\operatorname{dn} = 1$ and $\Delta = 1$.
With $k = 1$, where $\operatorname{sn} = \tanh$ and $\operatorname{cn} = \operatorname{dn} = \operatorname{sech}$, the first formula becomes $\tanh(u + v) = (\tanh u + \tanh v)/(1 + \tanh u\tanh v)$, after factoring $1 - \tanh^2 u\tanh^2 v = (1 - \tanh u\tanh v)(1 + \tanh u\tanh v)$.
Example of use: with $v = K$, where $\operatorname{sn} K = 1$, $\operatorname{cn} K = 0$, $\operatorname{dn} K = k'$, one gets $\operatorname{sn}(u + K) = \operatorname{cn} u / \operatorname{dn} u$.

## Explanations

### L1 human

The sine and cosine of a sum can be written with the sines and cosines of the parts.
The Jacobi elliptic functions have the same property, but the formulas carry the denominator $1 - k^2\operatorname{sn}^2 u\operatorname{sn}^2 v$.
When the modulus $k$ is $0$, the denominator is $1$ and the familiar trigonometric formulas come back.

### L2 human

The addition formulas reflect the group law on the elliptic curve $y^2 = (1 - x^2)(1 - k^2x^2)$, parametrised by $x = \operatorname{sn} u$, $y = \operatorname{cn} u\operatorname{dn} u$.
They give the shifts by quarter and half periods, such as $\operatorname{sn}(u + 2K) = -\operatorname{sn} u$, from the special values at $K$ and $2K$.
They are also the algebraic addition theorem behind Euler's addition theorem for the integral $\int dt/\sqrt{(1 - t^2)(1 - k^2t^2)}$.
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# Solvability of x^n - a

## Statement

Let $K$ be a field, let $n \ge 1$ be an integer that is not divisible by the [[characteristic|1ZB13N78XW]] of $K$, and let $a \in K$.
Then the [[Galois group|0JZY3R908H]] of $x^n - a$ over $K$ is a [[solvable group|0P3KZK0FTV]].

## Remarks

If $a = 0$, the polynomial $x^n$ splits over $K$ and the group is trivial.
If $a \ne 0$, let $L$ be a splitting field and $\beta \in L$ a root; the roots are $\beta \zeta^k$, where $\zeta$ is a [[primitive $n$-th root of unity|0J817RTVG6]], so $\zeta \in L$.
Put $E = K(\zeta)$; $E/K$ is a [[cyclotomic extension|VD6VC3ZMMY]], and $\Gal(E/K)$ embeds in $(\Z/n\Z)^\times$, so it is abelian.
The map $\sigma \mapsto \sigma(\beta)/\beta$ embeds $\Gal(L/E)$ in the cyclic group of $n$-th roots of unity, so $L/E$ is a [[cyclic extension|VFE5B71E7Y]].
Then $\Gal(L/E)$ is a normal subgroup of $\Gal(L/K)$ with abelian quotient $\Gal(E/K)$, so $\Gal(L/K)$ is solvable.
Example: for $x^3 - 2$ over $\Q$, the subgroup is cyclic of order $3$ and the quotient has order $2$, giving $S_3$.

## Explanations

### L1 human

The equation $x^n = a$ is the simplest equation that needs an $n$-th root.
Its roots are one $n$-th root of $a$ times the $n$-th roots of unity.
The proposition says that its Galois group is solvable, as it should be, since its roots are written with radicals.

### L2 human

This is the base case of the easy direction of [[Galois' theorem on solvability|VHCWCFA3MX]].
The two-step tower $K \subseteq K(\zeta) \subseteq L$ is the model for every radical extension: first roots of unity, then pure roots.
The same argument shows that $\Gal(L/K)$ embeds in the group of maps $x \mapsto ux + v$ on $\Z/n\Z$ with $u$ a unit.
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# Frankel conjecture

## Statement

Let $X$ be a compact [[Kähler manifold|X60JG9AEQJ]] of complex dimension $n$, with Kähler metric $g$ and curvature tensor $R$.
Suppose that $g$ has positive holomorphic bisectional curvature: $R(u, \bar u, v, \bar v) > 0$ for every point $p \in X$ and all nonzero $u, v \in T^{1,0}_p X$.
Then $X$ is biholomorphic to complex projective space $\mathbb{CP}^n$.

## Remarks

Frankel stated the conjecture in 1961 for compact Kähler manifolds of positive sectional curvature; the version above uses the weaker hypothesis of positive bisectional curvature.
It was proved in 1979 by Mori, as a consequence of his proof of the Hartshorne conjecture that a smooth projective variety with ample tangent bundle is a projective space.
Independently, Siu and Yau gave a differential-geometric proof in 1980, using energy-minimizing harmonic maps from $\mathbb{CP}^1$ to produce rational curves.

Example: the Fubini–Study metric on [[complex projective space|7GJ4STCNNM]] has constant positive holomorphic sectional curvature and positive bisectional curvature, so $\mathbb{CP}^n$ satisfies the hypothesis.

Positive Ricci curvature is not enough.
The product $\mathbb{CP}^1 \times \mathbb{CP}^1$ with the product of Fubini–Study metrics has positive Ricci curvature, but its bisectional curvature vanishes for $u$ tangent to the first factor and $v$ to the second.
It is not biholomorphic to $\mathbb{CP}^2$, since its second Betti number is $2$, while that of $\mathbb{CP}^2$ is $1$.

## Explanations

### L1 human

Positive curvature makes a space round, like a sphere.
The Frankel conjecture, now a theorem, says that in Kähler geometry a strong form of positive curvature leaves only one possibility: the space is complex projective space.
It is a complex analogue of the sphere theorems of Riemannian geometry.

### L2 human

Holomorphic bisectional curvature compares two complex directions at a point; it is weaker than sectional curvature and stronger than Ricci curvature.
Mori's proof works in algebraic geometry: positivity makes the tangent bundle ample, and then one finds and bends rational curves to recognize $\mathbb{CP}^n$.
Mori's method of bending rational curves became a basis of the minimal model program.
The theorem was later extended by Mok to compact Kähler manifolds of nonnegative bisectional curvature, which are built from projective spaces, Hermitian symmetric spaces and flat tori.
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# Numerical stability

## Definition

Let $X$ and $Y$ be normed spaces with [[norms|0V2WP9B1MF]] $\norm{\cdot}$, and let $f \colon X \to Y$ be a problem.
An *algorithm* for $f$ is a family of maps $\tilde f \colon X \to Y$, one for each floating-point system, with unit roundoff $u$ as in [[machine epsilon|JA3QXVF2B8]].
All [[$O$ bounds|0TGV80E6A4]] below are as $u \to 0$, uniform in $x \ne 0$.

1. The algorithm is *backward stable* if for each $x$ there is $\tilde x$ with $\tilde f(x) = f(\tilde x)$ and $\norm{\tilde x - x}/\norm{x} = O(u)$.
2. The algorithm is *stable* if for each $x$ there is $\tilde x$ with $\norm{\tilde x - x}/\norm{x} = O(u)$ and $\norm{\tilde f(x) - f(\tilde x)}/\norm{f(\tilde x)} = O(u)$.
3. The algorithm is *forward stable* if $\norm{\tilde f(x) - f(x)}/\norm{f(x)} = O(\kappa(x) \, u)$, where $\kappa(x)$ is the relative condition number of $f$ at $x$.

## Remarks

Example: computing $x_1 + x_2$ as $\mathrm{fl}(x_1 + x_2) = (x_1 + x_2)(1 + \delta)$, $\abs{\delta} \le u$, is backward stable, with $\tilde x = (x_1(1 + \delta), x_2(1 + \delta))$; this is the [[backward error|CMXHRNWY35]] computation.
Non-example: computing $1 - \cos t$ by forming $\mathrm{fl}(\cos t)$ and subtracting it from $1$ is not stable for small $t$.
The result, of order $t^2/2$, carries an absolute error of order $u$, so its relative error is of order $u/t^2$.
This is not $O(u)$, and a relative perturbation of $t$ of order $u$ cannot explain it.
The formula $2 \sin^2(t/2)$ computes the same quantity stably.
Backward stability implies stability, and a backward stable algorithm has relative forward error $O(\kappa(x) u)$.
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# Modular lattice

## Summary

A modular lattice is a [[lattice|WH4SFAACAS]] that satisfies the modular law: if $a \le b$, then $a \vee (x \wedge b) = (a \vee x) \wedge b$ for every $x$.
The lattices of subspaces of a vector space, of submodules of a module and of normal subgroups of a group are modular.
Every [[distributive lattice|0HM0HQCK6N]] is modular, but not conversely.
Modularity is what makes dimension and length behave well, as in the [[Jordan–Hölder theorem for modular lattices|BCG5KM83AP]].

## Explanations

### L1 human

Take subspaces $A \subseteq B$ of a vector space and any subspace $X$.
One can add $X$ to $A$ and then cut down to $B$, or cut $X$ down to $B$ and then add $A$; the result is the same.
The modular law asks for this rule in an abstract lattice.
It is a weak form of the distributive law, which only has to hold when two of the elements are comparable.
The smallest lattice that breaks it is the [[pentagon lattice N5|DHC8TZXGYS]].
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# Topological semigroup

## Summary

A topological semigroup is a [[semigroup|0XMEM406JM]] with a topology in which the semigroup operation is jointly continuous.
It is a very general setting where algebra and topology meet through one associative operation.
Topological monoids and topological groups are special cases.
Compact topological semigroups have a rich structure theory with uses in topological dynamics.

## Explanations

### L1 human

A semigroup is a set with an associative way of combining two elements, without any demand for an identity or inverses.
A topological semigroup adds a notion of nearness, and asks that combining nearby elements gives nearby results.
For example, the interval $[0, 1]$ with ordinary multiplication is a topological semigroup.
The product of two numbers in $[0, 1]$ is again in $[0, 1]$, and the product depends continuously on both factors.
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# Missing data mechanisms

## Definition

Let $Y$ be the complete data, a [[random variable|0DKRR22AEQ]] with values in a product space, split as $Y = (Y_{\mathrm{obs}}, Y_{\mathrm{mis}})$ into its observed and its missing parts.
Let $R$ be the missingness indicator: $R_j = 1$ if entry $j$ of $Y$ is observed and $R_j = 0$ otherwise.
The *missing data mechanism* is the conditional [[distribution|0QB913WM5E]] of $R$ given $Y$, written $p(r \mid y; \phi)$ with a parameter $\phi$.

1. The data are *missing completely at random* (MCAR) if $p(r \mid y; \phi) = p(r; \phi)$ for all $y$, that is, $R$ does not depend on $Y$.
2. The data are *missing at random* (MAR) if $p(r \mid y_{\mathrm{obs}}, y_{\mathrm{mis}}; \phi) = p(r \mid y_{\mathrm{obs}}; \phi)$ for all $y_{\mathrm{mis}}$, that is, $R$ depends on $Y$ only through the observed part.
3. The data are *missing not at random* (MNAR) if they are not MAR.

## Remarks

MCAR implies MAR.

Example: let $Y = (Y_1, Y_2)$, where $Y_1$ (age) is always observed and $Y_2$ (income) is missing with probability $h(Y_1)$ for a function $h$ that is not constant.
Then $p(r \mid y)$ depends only on $y_1 = y_{\mathrm{obs}}$, so the data are MAR, but not MCAR.

Non-example: let $Y_2$ be missing with probability $h(Y_2)$ for a function $h$ that is not constant.
Then $p(r \mid y)$ depends on the missing value itself, so clause 2 fails and the data are MNAR.

Assume the data are MAR and the parameters $\theta$ of the data model and $\phi$ of the mechanism are distinct, that is, they vary in a product set.
Then the mechanism is called *ignorable*.
In that case, the [[likelihood|05FSTXM92W]] of $\theta$ based on $(y_{\mathrm{obs}}, r)$ is proportional to the likelihood based on $y_{\mathrm{obs}}$ alone.
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# Determinacy

## Summary

Determinacy is the property of an infinite two-player game of perfect information that one of the two players has a winning strategy.
In the games of descriptive set theory, the players alternately choose natural numbers forever.
The first player wins when the resulting [[sequence|0G9EKM0E15]] lies in a fixed payoff set.
Gale and Stewart proved that games with an open or a closed payoff set are determined.
The [[Borel determinacy theorem|8FVWYEZWTK]] extends this to all Borel payoff sets.
Stronger determinacy hypotheses, such as [[projective determinacy|2GY3131ZPS]] and the [[axiom of determinacy|PEXTM3MJ5Z]], imply strong regularity properties of sets of reals.

## Explanations

### L1 human

Two players take turns writing down numbers, and they never stop.
Before the game starts, a rule decides which infinite sequences count as a win for the first player.
A strategy is a recipe that tells a player which number to write in each position, and it is winning if it wins against every way the opponent can play.
In a finite game where every play ends with a win for one of the players, one of them always has a winning strategy.
For infinite games this is no longer automatic, and determinacy is the statement that some player has a winning strategy.
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# Holomorphically convex hull

## Definition

Let $D \subseteq \C^n$ be an [[open set|02BH3MCBQB]] and let $K \subseteq D$ be a [[compact set|0R6EWFKRCF]].
The *holomorphically convex hull* of $K$ in $D$ is
$$\hat K_D = \set{z \in D : \abs{f(z)} \le \sup_{w \in K} \abs{f(w)} \text{ for every holomorphic function } f \text{ on } D},$$
where holomorphic means [[holomorphic|507C34W0ZT]] in the sense of several variables.

## Remarks

Always $K \subseteq \hat K_D$, and $\hat K_D$ is closed in $D$.
Using $f(z) = z_j$ shows that $\hat K_D$ is bounded, and using $f = e^{\ell}$ for complex linear $\ell$, with $\abs{e^{\ell}} = e^{\operatorname{Re} \ell}$, shows that $\hat K_D$ lies in the convex hull of $K$.

Example: $D = \C$ and $K = \set{\abs{z} = 1}$.
By the maximum principle every entire $f$ satisfies $\abs{f(z)} \le \max_K \abs{f}$ for $\abs{z} \le 1$, and $f(z) = z$ excludes the points with $\abs{z} > 1$; so $\hat K_D$ is the closed unit disc.

With the same $K$ and $D = \C \setminus \set{0}$, the function $f(z) = 1/z$ is holomorphic on $D$ and has $\abs{f(z)} > 1 = \max_K \abs{f}$ for $0 < \abs{z} < 1$; so here $\hat K_D = K$.

When $D = \C^n$, the hull coincides with the polynomially convex hull, since entire functions are uniform limits of polynomials on compact sets.
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# Castelnuovo's rationality criterion

## Statement

Let $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field, with [[canonical divisor|W05VYY4HBR]] $K_X$.
Let $q = \dim H^1(X, \mathcal{O}_X)$ be its irregularity, in the sense of [[sheaf cohomology|PQ6Q143ARJ]], and let $P_2 = \dim H^0(X, \mathcal{O}_X(2K_X))$ be its second plurigenus.
If $q = 0$ and $P_2 = 0$, then $X$ is a [[rational surface|55A4A53TRF]].

## Remarks

The converse holds as well, since $q$ and $P_2$ are birational invariants and both vanish for $\mathbb{P}^2$.
Castelnuovo proved it over $\C$, and Zariski extended it to positive characteristic.
The proof passes to a minimal model and uses Riemann–Roch on the surface to show that $K_X$ is not nef.
A minimal surface with this property is $\mathbb{P}^2$ or a ruled surface, and $q = 0$ makes the base curve of the ruling rational.
The plurigenus $P_2$ cannot be replaced by the geometric genus $p_g = h^0(K_X)$: an [[Enriques surface|Y30JE2DC6Y]] has $q = p_g = 0$ but $P_2 = 1$, and it is not rational.
Example: over $\C$ a unirational surface $X$ is dominated by $\mathbb{P}^2$.
Pulling back forms shows that $q$ and $P_2$ of $X$ are at most those of $\mathbb{P}^2$, which vanish.
Hence every complex unirational surface is rational.

## Explanations

### L1 human

Deciding whether a surface can be parametrized by the plane seems hard, since one would have to find the parametrization.
Castelnuovo's criterion replaces this by two numbers that can be computed: the number of independent holomorphic $1$-forms and the number of independent sections of twice the canonical bundle.
If both are zero, the surface is rational.

### L2 human

The criterion is the key step in the classification of surfaces of Kodaira dimension $-\infty$.
It shows that, for surfaces over $\C$, the Lüroth problem has a positive answer: unirational implies rational.
This fails in dimension three, where there are unirational threefolds that are not rational, and it fails for surfaces in positive characteristic.
The Enriques surface shows that the weaker condition $q = p_g = 0$ is not enough.
</document>

<document source="https://tamnd.github.io/math2/node/QXK4Q72WCV.md" tag="QXK4Q72WCV" kind="definition">
# Conservative system

## Definition

Let $(X, \mathcal{A}, \mu)$ be a measure space with $\mu$ [[σ-finite|WEBBT18D4X]], and let $T \colon X \to X$ be a [[quasi-measure-preserving map|F608XX51KA]], with [[iterates|AYVTR2ES5B]] $T^n$.
The map $T$ is *conservative* if for every $A \in \mathcal{A}$ with $\mu(A) > 0$ there is an integer $n \ge 1$ with
$$\mu(A \cap T^{-n}(A)) > 0.$$

## Remarks

Equivalently, $T$ has no wandering set of positive measure, where $W \in \mathcal{A}$ is wandering if the sets $T^{-n}(W)$, $n \ge 0$, are pairwise disjoint.
For a conservative $T$, almost every point of each $A \in \mathcal{A}$ returns to $A$ infinitely often.
The Hopf decomposition splits $X$, up to null sets, into an invariant part on which $T$ is conservative and a part that is a union of wandering sets.

Example: every measure-preserving map of a finite measure space is conservative, by the [[Poincaré recurrence theorem|TVYJPBHD0S]].
For instance, an irrational rotation of the circle with Lebesgue measure is conservative.

Non-example: the translation $T(x) = x + 1$ on $\R$ with Lebesgue measure preserves the measure, but for $A = [0, 1)$ one has $T^{-n}(A) = [-n, 1 - n)$, which is disjoint from $A$ for every $n \ge 1$.
So $\mu(A \cap T^{-n}(A)) = 0$ for all $n \ge 1$, and $T$ is not conservative.
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<document source="https://tamnd.github.io/math2/node/QY8ETVDA1Z.md" tag="QY8ETVDA1Z" kind="statement">
# Convolution theorem

## Statement

Let $n \ge 1$ and let $f, g \in L^1(\R^n)$.
Let $f * g$ be their [[convolution|2Z29JN3VES]], $(f * g)(x) = \int_{\R^n} f(y)\, g(x - y)\, dy$.
Let $\hat h(\xi) = \int_{\R^n} h(x) e^{-2\pi i x \cdot \xi}\, dx$ denote the [[Fourier transform|DPVM34ADZV]] of $h \in L^1(\R^n)$.
Then $f * g$ lies in [[$L^1(\R^n)$|SASP14MA1Z]], and
$$\widehat{f * g}(\xi) = \hat f(\xi)\, \hat g(\xi) \quad \text{for every } \xi \in \R^n.$$

## Remarks

The proof is one application of [[Fubini's theorem|0R0T1VWH5X]].
Write $e^{-2\pi i x \cdot \xi} = e^{-2\pi i y \cdot \xi} e^{-2\pi i (x - y) \cdot \xi}$ inside the double integral of $f(y) g(x - y) e^{-2\pi i x \cdot \xi}$.
The double integral converges absolutely because $\int\int \abs{f(y)} \abs{g(x - y)}\, dy\, dx = \norm{f}_1 \norm{g}_1$.
Integrating first in $x$ and then in $y$ gives $\hat f(\xi) \hat g(\xi)$.

Example: on $\R$, let $f = g = \mathbf{1}_{[-1/2, 1/2]}$.
Then $\hat f(\xi) = \sin(\pi \xi)/(\pi \xi)$, and $f * g$ is the triangle function $\max(1 - \abs{x}, 0)$.
So the Fourier transform of the triangle function is $\sin^2(\pi \xi)/(\pi \xi)^2$.

There is a dual form: for $f, g$ in the Schwartz space, $\widehat{f g} = \hat f * \hat g$.
The same identity holds for Fourier series, where the convolution on the circle becomes the product of Fourier coefficients.

## Explanations

### L1 human

Convolution is a kind of moving weighted average of one function against another.
On the frequency side it becomes plain multiplication.
So to smooth a signal with a filter, one can multiply its spectrum by the spectrum of the filter.

### L2 human

The theorem is why the Fourier transform solves linear equations with constant coefficients: such equations are convolution equations, and the transform makes them algebraic.
It also shows that $L^1(\R^n)$ with convolution is a commutative algebra, and that $f \mapsto \hat f$ is an algebra homomorphism into bounded continuous functions with pointwise product.
Fast convolution of long sequences uses the discrete version of the theorem together with the fast Fourier transform.
</document>

<document source="https://tamnd.github.io/math2/node/QYCFH68RD1.md" tag="QYCFH68RD1" kind="concept">
# Open subgroup

## Summary

An open subgroup of a [[topological group|92700QAB5Y]] is a [[subgroup|0X1JXT47GA]] that is an open set.
[[Open subgroups are also closed|DKX3EAVQX6]], so they are unions of connected components.
They are central in the study of totally disconnected groups.
In a profinite group the open normal subgroups form a base of neighbourhoods of the identity.
A connected topological group has no open subgroup other than itself.

## Explanations

### L1 human

An open subgroup is a subgroup that contains a whole neighbourhood of each of its points.
Because the group can be shifted around, it is enough that it contains a neighbourhood of the identity.
For example, in the $p$-adic integers the multiples of $p^n$ form an open subgroup for each $n$.
In the real line under addition, the only open subgroup is the whole line.
</document>

<document source="https://tamnd.github.io/math2/node/QYDQ28X8MA.md" tag="QYDQ28X8MA" kind="counterexample">
# Wild knot

## Counterexample

A [[knot|QYEAZTYGPT]] is a [[topological embedding|00ZG9Y9TR4]] $K \colon S^1 \to \R^3$.
It is *tame* if there is a [[homeomorphism|0QT00PB5G2]] $h \colon \R^3 \to \R^3$ such that $h(K(S^1))$ is a closed polygon with finitely many sides; otherwise it is *wild*.

Let $p$ be a point of $\R^3$.
Take a sequence of small trefoil knots, of sizes tending to $0$, placed one after the other along an arc that ends at $p$.
Tie them all into one closed curve that passes through $p$, so that every neighbourhood of $p$ contains infinitely many of the small trefoils.
The result is a knot, and it is wild.
Every point of a polygonal knot has a small ball around it that meets the knot in one unknotted arc.
A homeomorphism of $\R^3$ carries this local picture along.
At $p$ the knot has no such ball, because every small ball around $p$ contains whole knotted trefoils of the sequence.
One shows that this local knotting cannot be undone.

## Remarks

So not every topological knot is tame, and knot theory usually restricts to tame knots, for example to smooth or polygonal ones.
The local flatness property at every point, which fails at $p$, characterizes tame knots among topological ones.
Wild knots and other wild embeddings, such as the Alexander horned sphere, are studied in the theory of wild embeddings.

## Explanations

### L1 human

Ordinary knots are tied in a rope with a finite number of crossings.
A wild knot is a closed curve with infinitely many knots tied in it, getting smaller and smaller and piling up at one point.
Near that point the curve is knotted at every scale, so it cannot be straightened into a nice polygon.
</document>

<document source="https://tamnd.github.io/math2/node/QYEAZTYGPT.md" tag="QYEAZTYGPT" kind="concept">
# Knot

## Summary

A knot is a closed curve in three-dimensional space that does not cross itself, considered up to continuous deformation of the surrounding space.
Formally it is an embedding of the [[circle|060W5CBZP6]] in $\R^3$ or in the $3$-sphere.
Knot theory asks when two knots are equivalent, and answers it with [[knot invariants|GZQ5ZE3X6G]] such as the [[knot group|AQVHM3FP0D]] and knot polynomials.
Knots are also the basic examples in the study of [[3-manifolds|R20TB2KEQQ]].

## Explanations

### L1 human

Take a piece of string, tie it in some way and then glue its two ends together.
The result is a knot: one can move it around in space, but one cannot pass the string through itself or cut it.
A round circle is the [[unknot|1TQ06R4Q66]], the simplest knot.
The [[trefoil|VYTYSVHN4A]], with three crossings, is the simplest knot that cannot be undone.
</document>

<document source="https://tamnd.github.io/math2/node/QYHKS4B7M0.md" tag="QYHKS4B7M0" kind="definition">
# Herman ring

## Definition

Let $f \colon \hat{\C} \to \hat{\C}$ be a [[rational function|0XDK9CGYKJ]] of degree at least $2$.
A *Herman ring* of $f$ is a component $U$ of the [[Fatou set|C0N0XGC6GK]] of $f$ with the following properties.

1. There is $p \ge 1$ with $f^p(U) = U$.
2. There are $r > 1$ and a [[conformal map|0YYT36R4YD]] $\varphi$ from $U$ onto the round [[annulus|0CW52NV5TC]] $A_r = \set{w \in \C : 1 < \abs{w} < r}$.
3. There is an [[irrational number|0DZXAXEJQ9]] $\theta$ with $\varphi(f^p(z)) = e^{2\pi i \theta} \varphi(z)$ for all $z \in U$.

## Remarks

The circles $\abs{w} = s$ with $1 < s < r$ pull back to invariant closed curves of $f^p$ in $U$.
Non-example: a polynomial $P$ has no Herman ring.
If $U$ were one, an invariant curve $\gamma$ in $U$ would bound a disc $D$, and by the maximum principle the iterates of $P$ would be bounded on $D$.
Then $D$ would lie in the Fatou set, so $U$ could not be a component of it with a hole.
Herman constructed the first examples of rational maps with Herman rings.
Shishikura later built them by quasiconformal surgery from maps with Siegel discs.
</document>

<document source="https://tamnd.github.io/math2/node/QYNEBRGMR6.md" tag="QYNEBRGMR6" kind="definition">
# Struve function

## Definition

Let $\nu \in \C$ with $\nu + \frac{3}{2}$ not a non-positive integer, let $\Gamma$ be the [[gamma function|0H5SYBSED9]], and let $z \in \C \setminus (-\infty, 0]$ with the principal branch of powers.
The *Struve function* of order $\nu$ is
$$\mathbf{H}_\nu(z) = \sum_{m=0}^{\infty} \frac{(-1)^m}{\Gamma\left( m + \frac{3}{2} \right) \Gamma\left( m + \nu + \frac{3}{2} \right)} \left( \frac{z}{2} \right)^{2m + \nu + 1}.$$

## Remarks

It is a particular solution of the inhomogeneous Bessel equation
$$z^2 w'' + z w' + (z^2 - \nu^2)\, w = \frac{4 \left( z/2 \right)^{\nu + 1}}{\sqrt{\pi}\, \Gamma\left( \nu + \frac{1}{2} \right)}.$$
The general solution adds to $\mathbf{H}_\nu$ any solution of Bessel's equation, such as a combination of the [[Bessel functions|CZCMRDWDYW]] $J_\nu$ and $Y_\nu$.
Example: for $\nu = \frac{1}{2}$ and $x > 0$, $\mathbf{H}_{1/2}(x) = \sqrt{\frac{2}{\pi x}}\,(1 - \cos x)$; the first term of the series is $\frac{(x/2)^{3/2}}{\Gamma(3/2)\Gamma(2)} = \frac{x^{3/2}}{\sqrt{2\pi}}$, which matches $\sqrt{\frac{2}{\pi x}} \cdot \frac{x^2}{2}$.
A non-example: $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$ is not a Struve function; it solves the equation with right-hand side $0$.
The modified Struve function is $\mathbf{L}_\nu(z) = -i e^{-i\nu\pi/2}\, \mathbf{H}_\nu(iz)$.
</document>

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# Feichtinger conjecture

## Statement

Let $H$ be a separable [[Hilbert space|HM2HHE3J51]], and let $(f_i)_{i \in \N}$ be a frame of $H$ with $\inf_{i} \norm{f_i} > 0$.
That is, there are constants $0 < A \le B < \infty$ with $A \norm{f}^2 \le \sum_{i} \abs{\ip{f}{f_i}}^2 \le B \norm{f}^2$ for all $f \in H$.
Then $\N$ is the union of finitely many sets $I_1, \dots, I_r$ such that each family $(f_i)_{i \in I_j}$ is a Riesz sequence.
Here a Riesz sequence is a family that is a [[Riesz basis|ZYDYD7FB1Z]] of its closed linear span.

## Remarks

The question is due to Hans Feichtinger, who came to it from time–frequency analysis.
In the 2000s Peter Casazza and coauthors showed that it is equivalent to the [[Kadison–Singer problem|C8TTRMVVS7]].
The positive solution of the Kadison–Singer problem by Marcus, Spielman and Srivastava therefore proves the statement.
The hypothesis $\inf_i \norm{f_i} > 0$ is necessary: a Riesz sequence has norms bounded below, so a frame containing vectors of norm tending to $0$ cannot be split into finitely many Riesz sequences.

Example: the union of two orthonormal bases $(e_n)$ and $(e'_n)$ of $H$ is a frame with $\norm{f_i} = 1$; it splits into the two Riesz sequences $(e_n)$ and $(e'_n)$.

## Explanations

### L1 human

A frame is a redundant spanning family in which every vector can be recovered stably.
A Riesz sequence is a family that is as good as an orthonormal basis of the space it spans, with no redundancy.
The statement says that any frame whose vectors do not shrink to zero can be cut into finitely many pieces, each of them free of redundancy in this strong sense.

### L2 human

The statement lets one replace a redundant frame by finitely many non-redundant pieces, each of which behaves like an orthonormal basis of its span.
The equivalence with the Kadison–Singer problem passes through Anderson's paving conjecture and Weaver's discrepancy conjecture.
The proof of Marcus, Spielman and Srivastava works with sums of random rank-one matrices and interlacing families of polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/QYTGCW1GTK.md" tag="QYTGCW1GTK" kind="concept">
# Pringsheim convergence

## Summary

Pringsheim convergence is the usual notion of convergence for a [[double sequence|B70G8FYPG0]] $(a_{m,n})$.
The double sequence converges to $L$ when $a_{m,n}$ is close to $L$ as soon as both indices $m$ and $n$ are large.
It is the basis for the theory of double series and for comparing double limits with iterated limits.

## Explanations

### L1 human

Think of the terms $a_{m,n}$ written in an infinite table with rows $m$ and columns $n$.
Pringsheim convergence to $L$ means that, outside a large enough corner square of the table, every entry is within any given tolerance of $L$.
For example, $a_{m,n} = \frac{1}{m} + \frac{1}{n}$ tends to $0$ in this sense.
On the other hand, $a_{m,n} = \frac{m}{m+n}$ has no double limit: on the diagonal it is $\frac12$, but far out along a row it is near $0$.
</document>

<document source="https://tamnd.github.io/math2/node/QYVY3DV9K4.md" tag="QYVY3DV9K4" kind="statement">
# Cartan's classification of bounded symmetric domains

## Statement

1. Every [[bounded symmetric domain|Y7Q2SNKG0C]] is [[biholomorphic|FX92PP3HZR]] to a product of irreducible bounded symmetric domains, and the factors are unique up to order and biholomorphism.
2. Every irreducible bounded symmetric domain is biholomorphic to one of the following domains.
   Here $Z^*$ is the conjugate transpose of $Z$, and $M > 0$ means that the Hermitian matrix $M$ is positive definite.
   - type $I_{m,n}$: $\set{Z \in M_{m \times n}(\C) : I_n - Z^* Z > 0}$;
   - type $II_n$: $\set{Z \in M_{n \times n}(\C) : Z^T = -Z,\ I_n - Z^* Z > 0}$;
   - type $III_n$: $\set{Z \in M_{n \times n}(\C) : Z^T = Z,\ I_n - Z^* Z > 0}$;
   - type $IV_n$: $\set{z \in \C^n : \abs{z^T z} < 1,\ 1 - 2 z^* z + \abs{z^T z}^2 > 0}$;
   - two exceptional domains, of complex dimensions $16$ and $27$.

## Remarks

Élie Cartan obtained the classification in 1935.
The proof passes to Lie theory.
A bounded symmetric domain is $G/K$, where $G$ is the identity component of its automorphism group, a semisimple Lie group, and $K$ is a maximal compact subgroup.
Irreducible domains correspond to simple $G$ for which $K$ has a nontrivial centre.
Reading these off from the [[classification of symmetric spaces|9P8VK3H4YZ]] gives the list: $G$ is $SU(m, n)$, $SO^*(2n)$, $Sp(2n, \R)$, $SO_0(n, 2)$, or real forms of $E_6$ and $E_7$.

Example: $I_{m,1}$ is the unit ball of $\C^m$, and $I_{1,1} = III_1$ is the unit disc.
The type $III_n$ domain is the bounded model of the [[Siegel upper half-space|5PTYE97PF6]] of degree $n$.

For small indices some types coincide, so the list is without repetitions only for suitable ranges of the indices.

## Explanations

### L1 human

Bounded symmetric domains are the most symmetric bounded regions in $\C^n$.
The theorem says that there are only four infinite families of them, made of matrices of small norm, plus two exceptional ones, and products of these.
So a seemingly general class of domains is described by an explicit list.

### L2 human

Through the list, analysis on bounded symmetric domains reduces to matrix computations.
The domains are the [[Hermitian symmetric spaces|NT4797Z9H8]] of noncompact type, and each has a compact dual, such as a Grassmannian for type $I$.
Quotients of these domains by arithmetic groups include the Shimura varieties and the moduli spaces of abelian varieties, so the list organises much of the theory of automorphic forms.
</document>

<document source="https://tamnd.github.io/math2/node/QYW79XKJTP.md" tag="QYW79XKJTP" kind="concept">
# Tate's thesis

## Summary

Tate's thesis is the 1950 doctoral thesis of John Tate, written under Emil Artin.
It proves the meromorphic continuation and the functional equation of the L-functions of [[Hecke characters|R65XKHN4D1]].
The tool is harmonic analysis on the [[adele ring|SEH68BA1QQ]] and on the idele group.
The L-function appears as a zeta integral of a test function against a character over the ideles.
The functional equation then follows from the Poisson summation formula for the discrete subgroup $K$ of $\mathbb{A}_K$.
The method made adeles a standard tool and is a model for the theory of automorphic L-functions.

## Explanations

### L1 human

The [[Riemann zeta function|1XWBY4N0VT]] has a symmetry that relates its values at $s$ and $1 - s$.
Tate found one uniform reason for this symmetry and for its analogues over every number field.
He split the zeta function into local pieces, one for each completion, and studied each piece with Fourier analysis on that completion.
The pieces are glued with the Poisson summation formula, and the symmetry drops out.
For the rational numbers, this gives a new proof of the functional equation of the Riemann zeta function.
</document>

<document source="https://tamnd.github.io/math2/node/QYXGVES1XM.md" tag="QYXGVES1XM" kind="statement">
# Tits alternative

## Statement

Let $F$ be a [[field|04C5ZBG1WT]], let $n \ge 1$, and let $G$ be a [[finitely generated|082DWWXMRJ]] [[subgroup|0X1JXT47GA]] of the [[general linear group|0FCHE902D7]] $\mathrm{GL}_n(F)$.
Then exactly one of the following holds:

1. $G$ contains a non-abelian [[free group|0KTW26C80J]].
2. $G$ is virtually [[solvable|0FXJ7YFMMX]]: it has a solvable subgroup of finite [[index|0DTXBQPA2S]].

If $F$ has characteristic $0$, the same conclusion holds for every subgroup $G$ of $\mathrm{GL}_n(F)$, finitely generated or not.

## Remarks

The two cases exclude each other, because subgroups of virtually solvable groups are virtually solvable and a non-abelian free group is not.
Suppose $G$ is not virtually solvable.
Tits embeds the relevant field into a suitable local field and finds two elements of $G$ that act on projective space with attracting and repelling points in general position.
High powers of these elements then satisfy the ping-pong lemma, and so generate a free group of rank $2$.

Example: $\mathrm{SL}_2(\Z)$ is not virtually solvable, and indeed the matrices $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$ generate a free group of rank $2$.
Example: the group of upper triangular invertible matrices is solvable, so it has no non-abelian free subgroup.

In positive characteristic, finite generation is needed.
Let $\bar{\F}_p$ be an algebraic closure of $\F_p$.
Every finitely generated subgroup of $\mathrm{GL}_2(\bar{\F}_p)$ is finite, so $\mathrm{GL}_2(\bar{\F}_p)$ has no free subgroup of rank $2$.
But it is not virtually solvable, since it has the infinite simple group $\mathrm{PSL}_2(\bar{\F}_p)$ as a quotient of a subgroup.

## Explanations

### L1 human

A group of invertible matrices is either tame or wild.
Tame means that, after passing to a finite-index piece, it is built from commutative layers.
Wild means it contains two matrices that satisfy no relation at all, like the generators of a free group.
The theorem says there is nothing in between for finitely generated matrix groups.

### L2 human

The theorem gives strong consequences for linear groups.
A finitely generated linear group is [[amenable|CEAT9BESKK]] if and only if it is virtually solvable, since groups with free subgroups are not amenable.
Combined with the Milnor–Wolf theorem on finitely generated solvable groups, it shows that the [[growth|4CMRM3975W]] of a finitely generated linear group is either polynomial or exponential.
Groups outside linear groups are said to satisfy the Tits alternative when the same dichotomy holds for their subgroups; examples include [[hyperbolic groups|JWJ8YPR7EB]] and [[Out(F_n)|7Z5ZS7KNXS]].
</document>

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# Wiener–Hopf equation

## Summary

A Wiener–Hopf equation is a [[convolution integral equation|EY75YD988Z]] on the half-line.
The kernel depends only on $x - t$, but the unknown lives on $[0, \infty)$ and the integral runs over $[0, \infty)$.
The cut at $0$ breaks the translation symmetry, so the [[Fourier transform|DPVM34ADZV]] alone does not solve it.
It is solved by the [[Wiener–Hopf method|H4NXEY6QTZ]].
This method factors the symbol of the equation into pieces that extend holomorphically to the two half-planes.
Such equations model radiative transfer in a half-space, diffraction by a half-plane, and optimal causal filtering.

## Explanations

### L1 human

Imagine a medium that fills half of space, such as a star's atmosphere seen as a slab of infinite depth.
Each layer interacts with the others through a rule that depends only on their distance, but there are no layers beyond the surface.
The equation for the state of the layers is then a convolution, cut off at the surface.
A small example: the function $\varphi(x) = 1 + x$ satisfies $\varphi(x) = \int_0^\infty \frac{1}{2} e^{-\abs{x - t}} \varphi(t)\, dt$ for all $x \ge 0$.
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# Incomplete Cholesky factorization

## Summary

An incomplete Cholesky factorization of a sparse symmetric positive definite matrix $A$ is a sparse lower triangular matrix $L$ with $L L^T \approx A$.
It is computed like the [[Cholesky decomposition|0WKG8CYY6A]], but every entry outside a prescribed sparsity pattern is dropped.
Its main use is as a [[preconditioner|KE98SW3SBD]] for the conjugate gradient method.

## Explanations

### L1 human

The exact Cholesky factor of a sparse matrix is often much denser than the matrix itself.
The incomplete factor keeps only the entries in allowed positions, for example the positions where $A$ itself is nonzero.
The result is cheap to store and to use, and $L L^T$ is close to $A$, though not equal.
Solving with $L L^T$ then serves as a fast rough inverse of $A$ inside an iterative method.
</document>

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# Spin structure

## Definition

Let $(M, g)$ be an [[oriented|913WEQERBE]] [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \ge 2$.
Let $P_{SO}(M)$ be its bundle of oriented orthonormal frames, a principal $SO(n)$-bundle (see [[frame bundle|F4HWTRNXAP]]).
Let $\rho \colon \operatorname{Spin}(n) \to SO(n)$ be the double covering homomorphism from the [[spin group|JJSXXJQGEH]].
A *spin structure* on $(M, g)$ is a pair $(P, \xi)$ such that:

1. $P$ is a [[principal|8E37E74XVP]] $\operatorname{Spin}(n)$-bundle over $M$.
2. $\xi \colon P \to P_{SO}(M)$ is a smooth map that commutes with the projections to $M$.
3. $\xi(p \cdot a) = \xi(p) \cdot \rho(a)$ for all $p \in P$ and $a \in \operatorname{Spin}(n)$.

Two spin structures $(P, \xi)$ and $(P', \xi')$ are equivalent if there is a $\operatorname{Spin}(n)$-equivariant bundle isomorphism $\phi \colon P \to P'$ with $\xi' \circ \phi = \xi$.

## Remarks

Clauses 1 to 3 imply that $\xi$ is a two-sheeted covering map.
An oriented manifold admits a spin structure if and only if its second [[Stiefel–Whitney class|JQCK7A210W]] $w_2(TM)$ vanishes.
In that case the equivalence classes of spin structures are in bijection with $H^1(M; \Z/2)$, after one class is fixed.
Existence and the set of classes do not depend on the choice of the metric $g$.

Example: the sphere $S^n$ with $n \ge 2$ has $w_2 = 0$, because its tangent bundle becomes trivial after adding a trivial line bundle.
Since $H^1(S^n; \Z/2) = 0$, it has exactly one spin structure up to equivalence.
For $S^n$ it is $\operatorname{Spin}(n+1) \to SO(n+1) = P_{SO}(S^n)$.

Non-example: the complex projective plane $\C P^2$ is oriented but $w_2(T\C P^2) \ne 0$, so no pair $(P, \xi)$ satisfies the clauses.
A manifold with a spin structure is called a *spin manifold*.
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# Spectral space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A subset of $X$ is *compact* if every cover of it by open sets of $X$ has a finite subcover; no Hausdorff condition is assumed.
The space $X$ is *spectral* if:

1. $X$ is [[compact|0TVJZYB6SH]];
2. the compact open subsets of $X$ form a [[basis|0NA0732YAG]] of the topology of $X$;
3. the intersection of any two compact open subsets of $X$ is compact;
4. $X$ is [[sober|4N4DNDR22Z]]: every irreducible closed subset of $X$ is the closure of exactly one point.

Here a closed set is *irreducible* if it is nonempty and is not the union of two closed proper subsets.

## Remarks

A sober space is [[T0|07SG12TQH3]], so every spectral space is T0.

Example: the prime spectrum $\operatorname{Spec} \Z$ with the Zariski topology.
Its open sets are $\emptyset$ and the complements of finite sets of nonzero primes, each of which is compact.
The irreducible closed sets are the points $(p)$ for primes $p$ and the whole space, which is the closure of $(0)$.

Non-example: the interval $[0, 1]$ with its usual topology.
It is compact and sober, but it is connected, so its only compact open subsets are $\emptyset$ and $[0, 1]$, and these do not form a basis; clause 2 fails.

[[Hochster's theorem|C82AXZK2BS]] says that the spectral spaces are exactly the spaces homeomorphic to the spectrum of a commutative ring.
A Hausdorff spectral space is the same as a [[Stone space|ZV38HA5PAQ]].
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# Locally presentable category

## Definition

Let $\lambda$ be a [[regular cardinal|13ZV2F8JAR]] and let $\mathcal{C}$ be a locally small category.
The category $\mathcal{C}$ is *locally $\lambda$-presentable* if both conditions hold.

1. $\mathcal{C}$ is $\lambda$-[[accessible|FC8C0V4TFZ]].
2. $\mathcal{C}$ has all small [[colimits|BJS1CGTKQ5]].

The category $\mathcal{C}$ is *locally presentable* if it is locally $\lambda$-presentable for some regular cardinal $\lambda$.

## Remarks

Equivalently, $\mathcal{C}$ is cocomplete and has a set $S$ of $\lambda$-presentable objects such that every object is a $\lambda$-filtered colimit of objects of $S$.
For $\lambda = \aleph_0$ one says locally finitely presentable.

Example: the category of sets is locally finitely presentable.
It is cocomplete, and it is finitely accessible because every set is the filtered colimit of its finite subsets, which are finitely presentable.

Non-example: the category of fields and field homomorphisms is accessible but not locally presentable.
It has no initial object, which is the colimit of the empty diagram.
An initial field would map to both $\Q$ and $\F_2$, but a field homomorphism preserves the characteristic.
So clause 2 fails.

A locally presentable category is also complete, well-powered and co-well-powered.
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# Wynn's epsilon algorithm

## Definition

Let $(A_n)_{n \ge 0}$ be a [[sequence|0QE8STGJH2]] of real or complex numbers.
*Wynn's epsilon algorithm* defines numbers $\varepsilon_k^{(n)}$ for $k \ge -1$ and $n \ge 0$ by
$$\varepsilon_{-1}^{(n)} = 0, \qquad \varepsilon_0^{(n)} = A_n, \qquad \varepsilon_{k+1}^{(n)} = \varepsilon_{k-1}^{(n+1)} + \frac{1}{\varepsilon_k^{(n+1)} - \varepsilon_k^{(n)}},$$
as long as no denominator is $0$.
The even entries are the values of the [[Shanks transformation|S6E3H24FGB]]: $\varepsilon_{2k}^{(n)} = e_k(A_{n+k})$.
The odd entries are auxiliary quantities.

## Remarks

Check for $k = 1$: $\varepsilon_1^{(n)} = \frac{1}{\Delta A_n}$, where $\Delta A_n = A_{n+1} - A_n$, and
$$\varepsilon_2^{(n)} = A_{n+1} + \frac{1}{\frac{1}{\Delta A_{n+1}} - \frac{1}{\Delta A_n}} = A_{n+1} - \frac{\Delta A_n \Delta A_{n+1}}{\Delta A_{n+1} - \Delta A_n},$$
which equals $S(A_{n+1}) = \frac{A_{n+2}A_n - A_{n+1}^2}{A_{n+2} - 2A_{n+1} + A_n}$.
Example: for $A_0 = 1$, $A_1 = \frac23$, $A_2 = \frac{13}{15}$ one gets $\varepsilon_1^{(0)} = -3$, $\varepsilon_1^{(1)} = 5$ and $\varepsilon_2^{(0)} = \frac23 + \frac18 = \frac{19}{24}$.
Non-example: for a constant sequence $A_n = c$ already $\varepsilon_1^{(n)} = \frac{1}{0}$ is undefined; in practice the algorithm stops when two neighbouring entries agree.

## Explanations

### L1 human

Each new entry comes from the entry two columns back and the difference of two entries one column back.
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# Cohomological dimension of a field

## Summary

The cohomological dimension of a field $K$ is defined through its [[Galois cohomology|9MPCZ3EAJM]], see the [[definition|XKMBKAN1MR]].
It is the largest degree in which this cohomology with torsion coefficients can be nonzero.
It is the [[cohomological dimension|EF4ZWACHD8]] of the [[absolute Galois group|PGKWBJDTKX]] of $K$, and it can be computed one prime at a time.
Separably closed fields have dimension $0$, finite fields have dimension $1$, and $p$-adic fields have dimension $2$.
It controls which Galois cohomology groups can appear in arithmetic questions over the field.

## Explanations

### L1 human

Galois cohomology attaches to a field a sequence of groups $H^0, H^1, H^2, \ldots$.
For many fields these groups vanish above some degree, and that degree is the cohomological dimension.
A field in which every polynomial splits has nothing above degree $0$.
A finite field has nothing above degree $1$, and the field of $p$-adic numbers has nothing above degree $2$.
For the real numbers the groups never stop, so the dimension is infinite.
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# Valuative criterion

## Statement

Let $f \colon X \to Y$ be a [[morphism|JXRKH6P1Z4]] [[of finite type|AHTE5DVX61]], where $X$ is a [[Noetherian scheme|ND6XEY68P4]].
Let $R$ be a [[valuation ring|QSX12D8RQM]] with [[field of fractions|0F2PTGX1A2]] $K$, and let $\operatorname{Spec}$ denote the [[spectrum|FB45BHHA2J]].
Let $i \colon \operatorname{Spec} K \to \operatorname{Spec} R$ be induced by the inclusion $R \subseteq K$.
Consider pairs of morphisms $u \colon \operatorname{Spec} K \to X$ and $v \colon \operatorname{Spec} R \to Y$ with $f \circ u = v \circ i$.
A *lift* of such a pair is a morphism $h \colon \operatorname{Spec} R \to X$ with $h \circ i = u$ and $f \circ h = v$.

1. $f$ is [[separated|X30G4V7VAJ]] if and only if, for all such $R$, every such pair has at most one lift.
2. $f$ is [[proper|9537RKBZPP]] if and only if, for all such $R$, every such pair has exactly one lift.

## Remarks

The space $\operatorname{Spec} R$ has a generic point and a closed point, so it is an algebraic picture of a small curve.
The space $\operatorname{Spec} K$ is that curve with its special point removed.
Existence of a lift says that a curve in $X$ that is defined except at one point extends over that point, so limits exist; uniqueness says that limits are unique.
For the proof, one realizes each specialization of points by a morphism from the spectrum of a valuation ring.
This turns the closedness of images into the existence of lifts.
For example, $\mathbb{A}^1_k \to \operatorname{Spec} k$ is not proper: with $R = k[t]_{(t)}$ and $u$ given by $x \mapsto 1/t$, there is no lift, since $1/t \notin R$.
In $\mathbb{P}^1_k$ the same point is $[1 : 1/t] = [t : 1]$, which does extend over $t = 0$.
If $Y$ is also Noetherian, then in both parts it suffices to use discrete valuation rings.

## Explanations

### L1 human

The criterion tests properness with small curves.
Take a curve in $X$ with one point punched out, lying over a full curve in $Y$.
The map is proper exactly when one can always fill in the missing point, and in exactly one way.
This is the algebraic form of the fact that a space is compact when every sequence has a limit point.

### L2 human

The definition of a proper morphism asks for universal closedness, which is hard to check directly.
The valuative criterion replaces it by a lifting problem that one can test on explicit rings.
It gives quick proofs that projective morphisms are proper and that properness is stable under composition and base change.
It also underlies the construction of compact moduli spaces, where one shows that a family over a punctured curve has a unique limit.
More general forms drop the Noetherian hypothesis and state the criterion for quasi-compact and quasi-separated morphisms.
</document>

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# Hilbert's syzygy theorem

## Statement

Let $K$ be a [[field|04C5ZBG1WT]], let $n \ge 0$, and let $S = K[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]], [[graded|8KB4SQGC0C]] by degree.
Let $M$ be a [[finitely generated|05QY1VEBYD]] graded $S$-module.
Then $M$ has a graded [[free resolution|HK2AA1YSQ2]]
$$0 \to F_m \to F_{m-1} \to \cdots \to F_0 \to M \to 0$$
by finitely generated graded [[free|0DZ9DKAKZR]] $S$-modules $F_i$, with homogeneous maps of degree $0$, of length $m \le n$.

## Remarks

One proof uses the [[Koszul complex|YZY11W08SD]] of $x_1, \ldots, x_n$, which is a free resolution of $K = S/(x_1, \ldots, x_n)$ of length $n$.
It shows that $\operatorname{Tor}_i^S(M, K) = 0$ for $i > n$, and the length of a minimal graded free resolution of $M$ is the largest $i$ with $\operatorname{Tor}_i^S(M, K) \ne 0$.
The bound $n$ is sharp: for $n = 2$, the resolution $0 \to S \xrightarrow{(-y,\, x)^T} S^2 \xrightarrow{(x \;\; y)} S \to K \to 0$ of $K = S/(x, y)$ has length $2$, and no shorter resolution exists.
The same method shows that the [[global dimension|GWA0T1WKCN]] of $K[x_1, \ldots, x_n]$ is $n$.

## Explanations

### L1 human

A syzygy is a relation among generators, and a second syzygy is a relation among relations.
Hilbert showed that for modules over a polynomial ring in $n$ variables this process stops after at most $n$ steps.
So the structure of such a module is described by finitely many finite lists of polynomials.

### L2 human

The theorem says that the polynomial ring $K[x_1, \ldots, x_n]$ has global dimension $n$.
It is closely related to [[Serre's theorem on regular local rings|93RQRG9R1P]], since the localizations of $K[x_1, \ldots, x_n]$ at maximal ideals are regular.
The minimal free resolution gives the graded [[Betti numbers|3B1743XS7S]] of $M$, and Gröbner bases make it computable.
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# Pincherle derivative

## Summary

The Pincherle derivative of a linear operator $T$ on polynomials is the commutator $T' = Tx - xT$ of $T$ with multiplication by $x$.
It behaves like a derivative: it is linear and satisfies the product rule $(ST)' = S'T + ST'$.
It turns the [[differentiation operator|BB15T3J891]] into the identity, and each shift operator $f(x) \mapsto f(x + h)$ into $h$ times itself.
It is a basic tool in umbral calculus.

## Explanations

### L1 human

Operators on polynomials can be "differentiated" too.
The Pincherle derivative measures how far an operator is from commuting with multiplication by $x$.
Operators that commute with multiplication by $x$, such as multiplication by a fixed polynomial, have derivative $0$, like constants.
Differentiation $d/dx$ has Pincherle derivative the identity, just as $x$ has derivative $1$.
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# Ising model

## Example

Let $G = (V, E)$ be a finite [[graph|00C0BR7D7H]], and let $J > 0$, $h \in \R$ and $\beta \ge 0$ be real numbers.
A spin configuration is a map $\sigma \colon V \to \set{-1, +1}$.
Its energy is
$$H(\sigma) = -J \sum_{\set{u, v} \in E} \sigma(u) \sigma(v) - h \sum_{v \in V} \sigma(v).$$
The *Ising model* on $G$ at inverse temperature $\beta$ is the [[probability measure|059TJN0NZ1]] on $\set{-1, +1}^V$ given by
$$\mu_\beta(\sigma) = \frac{e^{-\beta H(\sigma)}}{Z_\beta}, \qquad Z_\beta = \sum_{\tau \in \set{-1, +1}^V} e^{-\beta H(\tau)}.$$
Check: each weight $e^{-\beta H(\sigma)}$ is positive and there are finitely many configurations.
So $Z_\beta > 0$, and the values of $\mu_\beta$ are positive and sum to $1$.

## Remarks

With $J > 0$, neighbouring spins prefer to agree, and the measure favours aligned configurations more strongly as $\beta$ grows.
At $\beta = 0$ the spins are independent and uniform.
On the infinite lattice $\Z^d$, limits of these measures as $G$ grows are Gibbs measures.
For $h = 0$ and $d \ge 2$ there is a phase transition: there is more than one Gibbs measure for large $\beta$ and only one for small $\beta$.
For $d = 1$ there is no phase transition.
For $d = 2$ and $h = 0$, Onsager's solution gives the critical value $\beta_c$ by $\sinh(2 \beta_c J) = 1$, that is $\beta_c = \frac{\ln(1 + \sqrt 2)}{2J}$.
Its Glauber dynamics, in which single spins flip at random, is an [[interacting particle system|GR49AM3PSZ]] with $\mu_\beta$ as invariant measure.

## Explanations

### L1 human

Picture a magnet as a grid of tiny arrows, each pointing up or down.
Neighbouring arrows like to point the same way, and heat shakes them at random.
At high temperature the arrows are disordered; at low temperature, in two or more dimensions, most of them line up and the material is magnetised.
For two spins joined by one edge with $h = 0$, the aligned configurations each get weight $e^{\beta J}$ and the opposite ones $e^{-\beta J}$.
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# Kuranishi family

## Summary

The Kuranishi family of a compact [[complex manifold|JSGWAN2RJV]] $X$ is a semi-universal [[deformation|VKQSM3BCE4]] of $X$.
Every small deformation of $X$ is pulled back from it, and the pullback map is unique to first order.
Kuranishi constructed it in 1962 over a germ of [[complex analytic space|HXSZA44J5J]], the Kuranishi space.
That space is the zero set of a holomorphic obstruction map from a neighbourhood of $0$ in $H^1(X, \Theta_X)$ to $H^2(X, \Theta_X)$.
It is the local model for moduli of compact complex manifolds.

## Explanations

### L1 human

Every compact complex manifold has one "best" small family of deformations that contains all others.
It is best in the sense that any other small deformation of $X$ is obtained from it by a change of parameter.
The parameter space can be singular, because some first-order directions may fail to extend to actual deformations.
For an elliptic curve the Kuranishi family is a small disc of the family $\C / (\Z + \tau \Z)$; for projective space it is a single point, because projective space is rigid.
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# Smooth morphism

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].
The morphism $f$ is *smooth* if it satisfies the following three conditions.

1. $f$ is locally of finite presentation.
This means that, for all affine opens $\operatorname{Spec} B \subseteq X$ and $\operatorname{Spec} A \subseteq Y$ with $f(\operatorname{Spec} B) \subseteq \operatorname{Spec} A$, the $A$-algebra $B$ is [[finitely presented|764GZHTGV8]].
2. $f$ is [[flat|5M6KS8F47R]].
3. For every $y \in Y$ with residue field $\kappa(y)$, the [[fiber|963QWX2E2Z]] $X_y$ is geometrically regular over $\kappa(y)$.
This means that $X_y \times_{\kappa(y)} \operatorname{Spec} \overline{\kappa(y)}$ is a [[regular scheme|ESPG0BJERZ]], where $\overline{\kappa(y)}$ is an algebraic closure of $\kappa(y)$.

## Remarks

For a ring $A$, the projection $\mathbb{A}^n_A \to \operatorname{Spec} A$ is smooth.
It is of finite presentation and flat, and each fibre is affine space over a field, which stays regular over the algebraic closure.
For a field $k$, the morphism $X = \operatorname{Spec} k[x, y]/(y^2 - x^3) \to \operatorname{Spec} k$ is not smooth.
Clause 3 fails, since the local ring at the origin has maximal ideal $(x, y)$, which needs two generators while the curve has dimension one.
Equivalently, $f$ is smooth of relative dimension $n$ if and only if three conditions hold.
It is flat and locally of finite presentation, its nonempty fibres have pure dimension $n$, and $\Omega_{X/Y}$ is locally free of rank $n$.
The ring version of this notion is a [[smooth morphism of rings|286G0P1B69]].
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# Perron–Wiener–Brelot solution

## Summary

The Perron–Wiener–Brelot solution is the generalized solution of the [[Dirichlet problem|8D56GQ4AG8]] given by the [[Perron method|AEZ7PPCNQS]] with both upper and lower classes.
The upper solution is the infimum of [[superharmonic functions|X4V2S56YQQ]] lying above the boundary data.
The lower solution is the supremum of [[subharmonic functions|Z1MHMGFNFW]] lying below it.
When the two agree and are harmonic, the boundary function is called resolutive, and the common value is the PWB solution.
Wiener showed that every continuous boundary function is resolutive, and Brelot characterized resolutive functions as those integrable for harmonic measure.

## Explanations

### L1 human

Squeeze the unknown solution between functions that are too large at the boundary and functions that are too small.
If the best upper bound and the best lower bound meet, the meeting point is the solution.
For continuous boundary data on a bounded open set they always meet, even when the solution does not take the boundary values at every point.
For the punctured disc with data $0$ on the circle and $1$ at the centre, the PWB solution is $0$.
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# Reduced scheme

## Definition

A [[scheme|5A35TYZD02]] $X$ is *reduced* if for every open set $U \subseteq X$ the ring $\mathcal{O}_X(U)$ is a [[reduced ring|SZHY975FFX]].
That is, $\mathcal{O}_X(U)$ has no nonzero [[nilpotent elements|0ZB2JG1M9C]].

## Remarks

Equivalently, every [[stalk|JTAKAPTT8H]] $\mathcal{O}_{X,x}$ is a reduced ring.
An affine scheme $\operatorname{Spec} A$ is reduced if and only if $A$ is reduced.
Example: $\operatorname{Spec} k[x, y]/(xy)$ for a field $k$ is reduced, since $k[x,y]/(xy)$ has no nonzero nilpotents.
Indeed, a power of a polynomial lies in $(xy)$ only if the polynomial itself is divisible by both $x$ and $y$.
Non-example: $\operatorname{Spec} k[x]/(x^2)$ is not reduced, since the class of $x$ is nonzero but $x^2 = 0$.
Every scheme $X$ has a reduced scheme $X_{\mathrm{red}}$ with the same underlying space, obtained by dividing out the nilpotents.
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# Representation theory of the Lorentz group

## Example

Let $G = \mathrm{SO}^+(1, 3)$ be the restricted [[Lorentz group|QWK3BYVJ9W]]: the linear maps of $\R^4$ that preserve $t^2 - x^2 - y^2 - z^2$, have determinant $1$ and preserve the direction of time.
Its universal cover is the group [[SL(2, C)|J58ED1GV6N]] $\mathrm{SL}_2(\C)$, a double cover with kernel $\set{\pm I}$.

For half-integers $m, n \in \set{0, \frac12, 1, \frac32, \dots}$, let $V_{2m}$ be the space of homogeneous polynomials of degree $2m$ in two variables, with $\mathrm{SL}_2(\C)$ acting by $(\pi_{2m}(g) P)(z) = P(g^{-1} z)$.
Restricted to [[SU(2)|E5P8AG6CK7]], this is the irreducible representation of dimension $2m + 1$.
Put
$$\pi_{m,n}(g) = \pi_{2m}(g) \otimes \pi_{2n}(\bar g) \quad \text{on} \quad V_{2m} \otimes V_{2n},$$
where $\bar g$ is the matrix with complex conjugate entries.
This is a continuous homomorphism, so it is a [[representation|7JA7WH92K0]] of $\mathrm{SL}_2(\C)$ of dimension $(2m + 1)(2n + 1)$.

The classification: the $\pi_{m,n}$ are [[irreducible|FX3NEFYFFN]] and pairwise non-isomorphic.
View $\mathrm{SL}_2(\C)$ as a real Lie group.
Then each irreducible finite-dimensional complex representation of it is isomorphic to exactly one $\pi_{m,n}$.
The element $-I$ acts by $(-1)^{2m + 2n}$, so $\pi_{m,n}$ is a representation of $G$ exactly when $m + n$ is an integer.

Check on the Lie algebra: the complexification of $\mathfrak{so}(1, 3)$ is $\mathfrak{sl}_2(\C) \oplus \mathfrak{sl}_2(\C)$.
On it, $\pi_{m,n}$ becomes the tensor product of the irreducible representations of highest weights $2m$ and $2n$ of the two factors.

## Remarks

The named cases are used throughout physics.
The representation $(0, 0)$ is the scalar one, $(\frac12, 0)$ and $(0, \frac12)$ are the left- and right-handed Weyl spinors, and $(\frac12, \frac12)$ is the defining action on four-vectors, after complexification.
The sum $(1, 0) \oplus (0, 1)$ is the action on antisymmetric tensors such as the electromagnetic field tensor, and $(\frac12, 0) \oplus (0, \frac12)$ is the Dirac spinor.

None of the $\pi_{m,n}$ other than $(0, 0)$ is unitary, because $G$ is a noncompact simple Lie group.
The irreducible unitary representations of $\mathrm{SL}_2(\C)$ are infinite-dimensional, apart from the trivial one, and they were classified by Gelfand and Naimark.

## Explanations

### L1 human

The Lorentz group is the group of changes of reference frame in special relativity.
Physical quantities change in definite ways under it: numbers stay put, vectors rotate and boost, spinors change by a "square root" of this rule.
The finite-dimensional ways in which the group can act are labelled by two half-integers $(m, n)$.
Each label gives one kind of field or particle, from scalars to spinors to the electromagnetic field.
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# Derivation

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $R$.
A *derivation* of $A$ over $R$ is a map $D \colon A \to A$ such that:

1. $D$ is $R$-[[linear|0RJF668P1T]]: $D(a + b) = D(a) + D(b)$ and $D(ra) = r D(a)$ for all $a, b \in A$ and $r \in R$.
2. $D(ab) = D(a) b + a D(b)$ for all $a, b \in A$ (the Leibniz rule).

## Remarks

Example: let $k$ be a field and $A = k[X]$, and let $D$ be the formal derivative, $D(\sum_i c_i X^i) = \sum_i i c_i X^{i-1}$.
It is $k$-linear, and on monomials $D(X^i X^j) = (i + j) X^{i+j-1} = i X^{i-1} X^j + X^i \cdot j X^{j-1}$, which gives clause 2 by linearity.
Non-example: on $k[X]$ with $k$ of characteristic $0$, the second derivative $f \mapsto f''$ is linear but fails clause 2, since $(X \cdot X)'' = 2$ while $X'' \cdot X + X \cdot X'' = 0$.
Consequences: $D(1) = 0$, hence $D(r \cdot 1) = 0$ for all $r \in R$, and $D(a^n) = n a^{n-1} D(a)$ when $A$ is commutative.
If $D_1$ and $D_2$ are derivations, then $D_1 D_2 - D_2 D_1$ is a derivation.
More generally, for an $A$-bimodule $M$, a derivation $A \to M$ is an $R$-linear map with $D(ab) = D(a) b + a D(b)$.
A derivation of a ring $A$ without a chosen base ring is a derivation over $\Z$, that is, an additive map with the Leibniz rule.
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# Artin–Wedderburn theorem

## Statement

Let $R$ be a nonzero [[semisimple ring|NY25HXA478]].
Then there are integers $k \ge 1$ and $n_1, \dots, n_k \ge 1$ and [[division rings|0EA5BTA0CX]] $D_1, \dots, D_k$ such that
$$R \cong M_{n_1}(D_1) \times \cdots \times M_{n_k}(D_k),$$
where $M_n(D)$ is the [[matrix ring|0FHVPRAE60]] of $n \times n$ matrices over $D$.
The number $k$ is unique, and the pairs $(n_i, D_i)$ are unique up to order and up to isomorphism of the $D_i$.
Conversely, every such finite product is a semisimple ring.

## Remarks

Group the minimal left ideals of $R$ by isomorphism type; there are finitely many types $S_1, \dots, S_k$, and $R \cong S_1^{n_1} \oplus \cdots \oplus S_k^{n_k}$ as left modules.
Then $R^{\mathrm{op}} \cong \End_R(R)$ is the product of the rings $\End_R(S_i^{n_i}) \cong M_{n_i}(\End_R(S_i))$.
By Schur's lemma each $\End_R(S_i)$ is a division ring, and passing to opposite rings gives the result.
For example, the group algebra $\C[S_3]$ of the symmetric group is isomorphic to $\C \times \C \times M_2(\C)$.
If $R$ is a finite-dimensional semisimple algebra over an algebraically closed field $K$, every $D_i$ equals $K$.

## Explanations

### L1 human

Semisimple rings, which look abstract, are all built from matrices.
Each one is a product of a few matrix rings, and the entries of each matrix ring come from a division ring.
So knowing the division rings gives all semisimple rings.

### L2 human

The theorem reduces the study of semisimple rings, and so of the representations of finite groups in characteristic zero, to division rings and matrix sizes.
The factors $M_{n_i}(D_i)$ correspond to the isomorphism classes of simple modules, and $n_i$ is the multiplicity of the $i$-th simple module in $R$.
For $R = \C[G]$ with $G$ finite this gives $\abs{G} = n_1^2 + \cdots + n_k^2$, where $k$ is the number of irreducible representations.
For a simple Artinian ring there is only one factor, which is Wedderburn's theorem $R \cong M_n(D)$.
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# Urysohn universal space

## Description

The *Urysohn universal space* $\mathbb{U}$ is a [[complete|0BY52JW6NK]] [[separable|0Z2P41SVDP]] [[metric space|0VKD1TP7F3]] with two properties.

1. Universality: every separable metric space is [[isometric|02HPP5VCZH]] to a subset of $\mathbb{U}$.
2. Ultrahomogeneity: every isometry between two finite subsets of $\mathbb{U}$ extends to an isometry of $\mathbb{U}$ onto itself.

Up to isometry, there is exactly one complete separable metric space with both properties.

## Remarks

An equivalent characterisation uses a one-point extension property.
Let $A \subseteq \mathbb{U}$ be finite, and let $A \cup \set{x}$ carry a metric extending that of $A$.
Then there is a point $u \in \mathbb{U}$ with $d(u, a) = d(x, a)$ for all $a \in A$.
A complete separable metric space with this property is isometric to $\mathbb{U}$.

The space $C[0, 1]$ with the sup norm is also universal for separable metric spaces, but it is not ultrahomogeneous.

## Explanations

### L1 human

The Urysohn space is one separable metric space that contains a perfect copy of every other one.
It is also as symmetric as possible: any finite configuration of points can be moved onto any other congruent configuration by a symmetry of the whole space.
It is built step by step, adding at each stage every point that a finite metric extension asks for.
</document>

<document source="https://tamnd.github.io/math2/node/R1PHMVBJ9J.md" tag="R1PHMVBJ9J" kind="statement">
# Marcinkiewicz interpolation theorem

## Statement

Let $(X, \mu)$ and $(Y, \nu)$ be [[σ-finite|WEBBT18D4X]] measure spaces and let $1 \le p_0 < p_1 \le \infty$.
Let $T$ map $L^{p_0}(\mu) + L^{p_1}(\mu)$ into the measurable functions on $Y$.
Assume that $T$ is sublinear: for all $f, g$ and scalars $c$, almost everywhere,
$$\abs{T(f + g)} \le \abs{T f} + \abs{T g}, \qquad \abs{T(c f)} = \abs{c} \abs{T f}.$$
Assume that $T$ is of weak type $(p_0, p_0)$: there is $A_0$ with
$$\nu(\set{y : \abs{T f(y)} > t}) \le \left( \frac{A_0 \norm{f}_{p_0}}{t} \right)^{p_0} \quad \text{for all } f \in L^{p_0}(\mu) \text{ and } t > 0.$$
Assume that $T$ is of weak type $(p_1, p_1)$ in the same sense with a constant $A_1$, where for $p_1 = \infty$ this means $\norm{T f}_\infty \le A_1 \norm{f}_\infty$.
Then for every $p$ with $p_0 < p < p_1$ there is a constant $A$, depending only on $p$, $p_0$, $p_1$, $A_0$ and $A_1$, with
$$\norm{T f}_p \le A \norm{f}_p \quad \text{for all } f \in L^p(\mu).$$

## Remarks

The weak type condition says that $T$ maps $L^{p_i}$ boundedly into the [[weak Lp space|H4C888M5AZ]] $L^{p_i,\infty}$.
The proof writes $\norm{T f}_p^p = p \int_0^\infty t^{p-1} \nu(\set{\abs{T f} > t}) \, dt$ by the [[layer cake representation|H78RXJC1K1]].
For each $t$, it splits $f$ into the part where $\abs{f} > c t$, which lies in $L^{p_0}$, and the rest, which lies in $L^{p_1}$.
The two weak-type bounds and Fubini's theorem then give the bound.

Example: the [[Hardy–Littlewood maximal function|VX1A56P5KQ]] operator $M$ on $\R^n$ is of weak type $(1, 1)$, and $\norm{M f}_\infty \le \norm{f}_\infty$.
The theorem gives $\norm{M f}_p \le A_p \norm{f}_p$ for $1 < p < \infty$.

The endpoints are not included in the conclusion.
The maximal operator is not bounded on $L^1(\R^n)$: for $f$ the indicator function of the unit ball, $M f(x)$ is of order $\abs{x}^{-n}$ for large $\abs{x}$, which is not integrable.

## Explanations

### L1 human

Some important operators on functions are not bounded on $L^1$, but satisfy a weaker estimate there that controls the measure of the set where the output is large.
The theorem says that two such weak estimates, at two exponents, give a genuine $L^p$ bound for every exponent strictly between them.
It works even for operators that are not linear, such as maximal functions.

### L2 human

In harmonic analysis one usually proves a weak $(1, 1)$ bound by a covering or Calderón–Zygmund argument and an $L^2$ or $L^\infty$ bound by other means.
Marcinkiewicz interpolation then gives all the $L^p$ bounds in between.
Compared with the [[Riesz–Thorin theorem|AGRKRG1TMJ]], it needs less (weak type, sublinearity) and gives less (an unspecified constant).
There is also an off-diagonal version, from weak type $(p_i, q_i)$ to strong type $(p, q)$, which requires $p_0 \le q_0$, $p_1 \le q_1$, $q_0 \ne q_1$, and gives the conclusion for $p \le q$.
</document>

<document source="https://tamnd.github.io/math2/node/R1PTHQACWM.md" tag="R1PTHQACWM" kind="concept">
# Filtered algebra

## Summary

A filtered algebra is an [[associative algebra|3CTW1P4HBP]] with an increasing chain of subspaces $F_0 \subseteq F_1 \subseteq \cdots$ that exhausts it and is compatible with multiplication, $F_i F_j \subseteq F_{i+j}$.
It generalizes a [[graded ring|8KB4SQGC0C]], where elements have an exact degree, to a setting where elements only have a degree "at most $i$".
Each filtered algebra has an [[associated graded|8HFAEGFQT1]] algebra, which is often simpler, and many properties pass from the graded algebra back to the filtered one.
The [[universal enveloping algebra|SCZ8PRVY00]] of a Lie algebra and the [[Weyl algebra|98QE7KY55W]] are the classic examples.

## Explanations

### L1 human

In a polynomial ring every element has a degree, and the degree of a product is at most the sum of the degrees.
A filtered algebra keeps only this rule, measuring "size at most $i$" by the subspace $F_i$.
In a noncommutative algebra, such as the algebra of differential operators, there may be no exact degree, but there is still a sensible notion of order.
Passing to the associated graded algebra forgets the lower-order terms; for differential operators it leaves only the leading symbols, which commute.
</document>

<document source="https://tamnd.github.io/math2/node/R1QBNFTHFP.md" tag="R1QBNFTHFP" kind="statement">
# Rejection sampling

## Statement

Let $f$ and $g$ be [[probability density functions|0ZY9DX8FBF]] on $\R^d$, and let $M \ge 1$ be a constant with $f(x) \le M g(x)$ for all $x$.
Let $(Y_1, V_1), (Y_2, V_2), \ldots$ be independent pairs, where $Y_j$ has density $g$, $V_j$ has the [[uniform distribution|0AREFQ9Z65]] on $(0, 1)$, and $Y_j$ and $V_j$ are independent.
Call trial $j$ *accepted* if $g(Y_j) > 0$ and $V_j \le f(Y_j) / (M g(Y_j))$.
Let $\tau$ be the index of the first accepted trial and $X = Y_\tau$.
Then:

1. each trial is accepted with probability $1/M$;
2. the [[conditional distribution|0990W5PN13]] of $Y_j$ given that trial $j$ is accepted has density $f$;
3. $\tau$ is finite almost surely, has the [[geometric distribution|04P2SR252N]] with success probability $1/M$, so $\mathbb E\tau = M$, and $X$ has density $f$.

## Remarks

For a measurable set $A$, $P(Y_j \in A,\ \text{accepted}) = \int_A g(y) \frac{f(y)}{M g(y)}\, dy = \frac{1}{M} \int_A f(y)\, dy$, where the integral runs over the points with $g > 0$, which carry all the mass of $f$ since $f \le M g$.
Taking $A = \R^d$ gives clause 1, and dividing by $1/M$ gives clause 2.
Clause 3 follows because the trials are independent and $X$ is the value of the first success.
Example: to sample the density $f(x) = 2x$ on $[0, 1]$, take $g = 1$ on $[0, 1]$ and $M = 2$; a uniform $Y$ is accepted when $V \le Y$, which happens half the time.
The bound $f \le M g$ is necessary: if $f > M g$ on a set of positive measure, the accepted values follow the density proportional to $\min(f, M g)$, not $f$.
The method needs $f$ only up to a constant factor: if $f = c \tilde f$ with unknown $c$, one uses $\tilde f \le \tilde M g$ and the acceptance test $V \le \tilde f(Y)/(\tilde M g(Y))$.

## Explanations

### L1 human

To sample from a difficult distribution, sample from an easier one that sits above it after scaling, and throw away some of the samples.
Picture the graph of $f$ under the graph of $M g$: pick a random point under $M g$, and keep it only if it also lies under $f$.
The kept points are uniform under the graph of $f$, so their horizontal positions have density $f$.
On average $M$ tries are needed per kept sample, so the envelope $M g$ should fit $f$ closely.

### L2 human

Rejection sampling, due to von Neumann, is exact, needs no inverse distribution function, and works in any dimension.
Its weakness is the acceptance rate $1/M$, which typically decays exponentially with the dimension, so in high dimension Markov chain Monte Carlo is used instead.
Many fast samplers are refinements of it.
The ziggurat algorithm uses a cheap envelope made of rectangles.
Adaptive rejection sampling builds the envelope from tangent lines of $\log f$ when $f$ is log-concave.
The number of trials before acceptance is geometric, so the cost per sample is random but with all moments finite.
</document>

<document source="https://tamnd.github.io/math2/node/R1RQ3226WV.md" tag="R1RQ3226WV" kind="definition">
# Simplicial homology

## Definition

Let $K$ be a [[simplicial complex|V8GZ0W1ZZ5]] with a total order on its vertex set.
For $n \ge 0$ let $C_n(K)$ be the [[free abelian group|0WJX8MGG37]] with basis the $n$-simplices of $K$, and let $C_n(K) = 0$ for $n < 0$.
An $n$-simplex with vertices $v_0 < v_1 < \cdots < v_n$ is written $[v_0, \dots, v_n]$.
The *boundary map* $\partial_n \colon C_n(K) \to C_{n-1}(K)$ is the homomorphism given on basis elements by
$$\partial_n [v_0, \dots, v_n] = \sum_{i=0}^{n} (-1)^i [v_0, \dots, \widehat{v_i}, \dots, v_n],$$
where $\widehat{v_i}$ means that $v_i$ is omitted, and $\partial_0 = 0$.
Then $\partial_{n} \circ \partial_{n+1} = 0$, so $(C_\bullet(K), \partial)$ is a [[chain complex|WQ82C0M3QB]].
The *$n$-th simplicial homology group* of $K$ is
$$H_n(K) = \ker \partial_n / \operatorname{im} \partial_{n+1}.$$

## Remarks

Elements of $\ker \partial_n$ are $n$-cycles and elements of $\operatorname{im} \partial_{n+1}$ are $n$-boundaries; the [[kernel|0Z424HK8CQ]] contains the image because $\partial_n \circ \partial_{n+1} = 0$.
The groups do not depend on the vertex order up to isomorphism.
With coefficients in an abelian group $A$, one uses $C_n(K) \otimes A$.

Example: let $K$ be the boundary of a triangle, with vertices $0 < 1 < 2$ and edges $[0,1]$, $[0,2]$, $[1,2]$.
Then $\partial_1 [0,1] = [1] - [0]$, $\partial_1 [0,2] = [2] - [0]$, $\partial_1 [1,2] = [2] - [1]$.
The kernel of $\partial_1$ is generated by $[0,1] - [0,2] + [1,2]$, and there are no $2$-simplices, so $H_1(K) \cong \Z$; also $H_0(K) \cong \Z$.

Non-example: dropping the signs $(-1)^i$ does not give a chain complex over $\Z$.
For the $2$-simplex, the unsigned boundary of $[0,1,2]$ is $[1,2] + [0,2] + [0,1]$, whose unsigned boundary is $2([0] + [1] + [2]) \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/R1S7NT139R.md" tag="R1S7NT139R" kind="definition">
# Intersection body

## Definition

Let $n \ge 2$, and let $K \subseteq \R^n$ be a [[convex body|437FSG3419]] with the origin in its interior.
Let $S^{n-1}$ be the unit [[sphere|T62369RFH6]] of $\R^n$, and for $u \in S^{n-1}$ let $u^\perp = \set{x : x \cdot u = 0}$.
Write $\mathrm{vol}_{n-1}$ for the $(n-1)$-dimensional volume in the hyperplane $u^\perp$.
The *intersection body* of $K$ is the set
$$IK = \set{r u : u \in S^{n-1}, \ 0 \le r \le \mathrm{vol}_{n-1}(K \cap u^\perp)}.$$
So $IK$ is star-shaped about the origin, and its radial function is $\rho_{IK}(u) = \mathrm{vol}_{n-1}(K \cap u^\perp)$.

## Remarks

Example: let $B$ be the closed unit ball of $\R^n$ and $\omega_{n-1}$ the volume of the unit ball of $\R^{n-1}$.
Every section $B \cap u^\perp$ is a unit ball of dimension $n - 1$, so $\rho_{IB} = \omega_{n-1}$ and $IB = \omega_{n-1} B$.
Non-example: let the origin lie outside $K$.
Then some hyperplane $u^\perp$ misses $K$, and the radial function vanishes in the direction $u$.
So the construction no longer gives a body with the origin in its interior, which is why the definition requires the origin to be an interior point of $K$.
Busemann's theorem says that $IK$ is convex when $K$ is convex and symmetric about the origin.
The definition is often stated more generally for star bodies, with the same formula for the radial function.
</document>

<document source="https://tamnd.github.io/math2/node/R1W62VM8W6.md" tag="R1W62VM8W6" kind="problem">
# Poincaré problem

## Problem

Let $X$ be a connected [[complex manifold|JSGWAN2RJV]] and let $f$ be a [[meromorphic function|W6FTSF57T1]] on $X$.
Find [[holomorphic functions|507C34W0ZT]] $g$ and $h$ on $X$, with $h$ not identically zero, such that $f = g / h$ on $X$.
Also ask that the germs of $g$ and $h$ be coprime in the local ring $\mathcal{O}_{X,x}$ at every point $x \in X$.

## Remarks

Without the coprimality condition this is the weak Poincaré problem.
Henri Poincaré proved in 1883 that every meromorphic function on $\C^2$ is a quotient of two entire functions.
Locally, every meromorphic function is a quotient of coprime holomorphic germs, because the ring $\mathcal{O}_{X,x}$ is a unique factorization domain.
The local denominators form data for the [[second Cousin problem|D44KAF8PNK]]: a global solution $h$ of it gives $g = f h$ holomorphic, and $f = g / h$ with coprime germs.
So the Poincaré problem is solvable on every manifold where all second Cousin problems are solvable, for example on a Stein manifold $X$ with $H^2(X, \Z) = 0$, such as $\C^n$.
On every Stein manifold the weak Poincaré problem is solvable.

## Explanations

### L1 human

In one variable every meromorphic function on the plane is a quotient of two entire functions, by the Weierstrass factorization theorem.
The Poincaré problem asks the same in several variables, with the extra wish that numerator and denominator share no common factor at any point.
It reduces to the second Cousin problem, the problem of building a global function with prescribed zeros, and it has a positive answer on $\C^n$.
</document>

<document source="https://tamnd.github.io/math2/node/R1XGWTVY3N.md" tag="R1XGWTVY3N" kind="definition">
# Lambert W function

## Definition

Let $f \colon \R \to \R$ be $f(w) = w e^w$, where $e^w$ is the [[exponential function|0J0FCWV32D]].
1. The *principal branch* $W_0 \colon [-1/e, \infty) \to [-1, \infty)$ is the [[inverse|0J2WAR83WP]] of the restriction of $f$ to $[-1, \infty)$.
2. The *lower branch* $W_{-1} \colon [-1/e, 0) \to (-\infty, -1]$ is the inverse of the restriction of $f$ to $(-\infty, -1]$.

Thus $W(x)\, e^{W(x)} = x$ for both branches $W$ on their domains.

## Remarks

The two restrictions are bijections because $f'(w) = (1 + w) e^w$ is positive for $w > -1$ and negative for $w < -1$, $f(-1) = -1/e$, $f(w) \to 0^-$ as $w \to -\infty$, and $f(w) \to \infty$ as $w \to \infty$.

Example: $W_0(e) = 1$, because $1 \cdot e^1 = e$ and $1 \ge -1$.
Also $W_0(0) = 0$, $W_0(-1/e) = W_{-1}(-1/e) = -1$, and $W_0(1) = \Omega \approx 0.5671$, the omega constant.

Non-example: $x = -1$ is in the domain of neither branch, because $w e^w \ge -1/e > -1$ for every real $w$.

Over $\C$, the equation $w e^w = z$ has infinitely many solutions for $z \ne 0$, and they are grouped into branches $W_k$, $k \in \Z$, which extend $W_0$ and $W_{-1}$.
Implicit differentiation gives $W'(x) = \frac{W(x)}{x\,(1 + W(x))}$ for $x \ne 0, -1/e$.
</document>

<document source="https://tamnd.github.io/math2/node/R20TB2KEQQ.md" tag="R20TB2KEQQ" kind="concept">
# 3-manifold

## Summary

A $3$-manifold is a [[topological manifold|QQQ9B1DK4P]] of dimension $3$: a space that looks like ordinary space $\R^3$ near each of its points.
By Moise's theorem every $3$-manifold has an essentially unique triangulation.
It also has an essentially unique smooth structure, so the topological, piecewise linear and smooth theories agree.
Their study combines topology with geometry, through the [[geometrization conjecture|B27HB43K86]], proved by Perelman, which includes the [[Poincaré conjecture|3R064KFF68]].

## Explanations

### L1 human

A $3$-manifold is a possible shape for a universe: around every point, space looks like the space we live in.
The space $\R^3$, the $3$-sphere and the $3$-torus are examples.
The $3$-torus is a cube whose opposite faces are glued: leaving through the right face, one comes back through the left face.
Globally such spaces can be very different, even though they all look the same locally.
</document>

<document source="https://tamnd.github.io/math2/node/R2658B61VX.md" tag="R2658B61VX" kind="definition">
# Prevariety

## Definition

Let $\sigma$ be an algebraic signature.
A class $K$ of $\sigma$-algebras is a *prevariety* if it satisfies these three conditions.

1. If $A \in K$ and $B$ is [[isomorphic|PZMGQMDN25]] to $A$, then $B \in K$.
2. If $A \in K$ and $B$ is a [[subalgebra|7CHG6SD365]] of $A$, then $B \in K$.
3. If $(A_i)_{i \in I}$ is a family of members of $K$, then the [[direct product|WJ73XYBNE4]] $\prod_{i \in I} A_i$ is in $K$; for $I = \emptyset$ this product is the one-element algebra.

## Remarks

In operator notation, $K$ is a prevariety exactly when $\mathbf{ISP}(K) = K$, and $\mathbf{ISP}(K)$ is the prevariety generated by $K$.

Example: residually finite groups form a prevariety.
A subgroup of a residually finite group is residually finite, by restricting the finite quotients.
A product of residually finite groups is residually finite.
Indeed, a nontrivial element has a nontrivial coordinate.
Compose the projection to that coordinate with a finite quotient that separates this coordinate from $1$.
This prevariety is not a [[quasivariety|WRXEKT4NQS]], since it is not closed under ultraproducts.
A nonprincipal ultraproduct of the groups $\Z / p\Z$ over the primes $p$ is a nontrivial torsion-free divisible abelian group, which has no proper subgroup of finite index.

Non-example: the class of fields fails condition 3, since $\Q \times \Q$ has the zero divisor $(1, 0)$.
</document>

<document source="https://tamnd.github.io/math2/node/R2AV47AX0Y.md" tag="R2AV47AX0Y" kind="definition">
# Bernstein–Sato polynomial

## Definition

Let $f \in \C[x_1, \dots, x_n]$ be a nonzero [[polynomial|0SDC934R52]].
Let $D_n = \C[x_1, \dots, x_n]\langle \partial_1, \dots, \partial_n \rangle$ be the [[Weyl algebra|98QE7KY55W]] of polynomial differential operators.
Let $D_n[s]$ be the ring of polynomials in a new central variable $s$ with coefficients in $D_n$.
The operators of $D_n[s]$ act on expressions $g \, f^{s}$ with $g$ in $\C[x, s, 1/f]$ by the usual rules, with $\partial_i (f^s) = s \, (\partial_i f) \, f^{s-1}$.
The set of polynomials $b(s) \in \C[s]$ for which there is $P(s) \in D_n[s]$ with
$$P(s) \, f^{s+1} = b(s) \, f^s$$
is an ideal of $\C[s]$.
The *Bernstein–Sato polynomial* $b_f(s)$ is the monic generator of this ideal.

## Remarks

Bernstein proved that this ideal is nonzero, so $b_f$ exists.
For a nonconstant $f$, $s + 1$ divides $b_f(s)$, as one sees by setting $s = -1$.

Example: $f = x_1^2 + \cdots + x_n^2$ and $P = \Delta = \partial_1^2 + \cdots + \partial_n^2$.
One has $\partial_i (f^{s+1}) = 2 (s + 1) x_i f^s$, and then
$$\Delta (f^{s+1}) = 2(s+1) \left( n f^s + 2s \sum_i x_i^2 f^{s-1} \right) = 4 (s + 1) \left( s + \tfrac{n}{2} \right) f^s.$$
So $(s+1)(s + n/2)$ lies in the ideal; it is in fact $b_f(s)$.

Non-example: for $f = x_1$, the polynomial $(s + 1)(s + 2)$ lies in the ideal, with $P = (s + 2) \partial_1$, but it is not $b_f$.
It is not of least degree, since $\partial_1 (x_1^{s+1}) = (s+1) x_1^s$ gives $b_f(s) = s + 1$.

The same definition works for a germ of holomorphic function, with the ring of differential operators with convergent power series coefficients; this gives the local $b$-function at a point.
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<document source="https://tamnd.github.io/math2/node/R2EHQ06B98.md" tag="R2EHQ06B98" kind="definition">
# Principal component analysis

## Definition

Let $X$ be a random vector in $\R^p$ with finite second moments, mean $\mu$ and [[covariance matrix|0V7G6WMFK0]] $\Sigma$.
By the [[spectral theorem|06J4PXRG22]], $\Sigma$ has an [[orthonormal basis|0S9GGTR0EX]] of [[eigenvectors|0N6960W0BV]] $v_1, \ldots, v_p$ with [[eigenvalues|0BQAA2BK3Q]] $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_p \ge 0$.
The $k$-th *principal component* of $X$ is the random variable
$$Y_k = v_k^T (X - \mu), \qquad k = 1, \ldots, p.$$
The vector $v_k$ is the $k$-th principal direction, or loading vector.
*Principal component analysis* is the passage from $X$ to $(Y_1, \ldots, Y_p)$, usually followed by keeping only $Y_1, \ldots, Y_q$ for some $q < p$.
For a data matrix, the same construction is applied to the sample covariance matrix.

## Remarks

The components are uncorrelated and $\operatorname{Var}(Y_k) = \lambda_k$.
Equivalently, $v_1$ maximises the [[variance|0477R9PYAE]] $\operatorname{Var}(a^T X) = a^T \Sigma a$ over unit vectors $a$, and $v_k$ maximises it over unit vectors orthogonal to $v_1, \ldots, v_{k-1}$.
The principal directions are unique up to sign when the eigenvalues are distinct.

Example: let $\Sigma = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$.
Its eigenvalues are $3$ and $1$, with unit eigenvectors $v_1 = (1, 1)/\sqrt{2}$ and $v_2 = (1, -1)/\sqrt{2}$.
Check: $v_1^T \Sigma v_1 = (2 + 1 + 1 + 2)/2 = 3$.
Non-example: the direction $a = (1, 0)$ gives $a^T \Sigma a = 2 < 3$, so $a^T (X - \mu)$ is not the first principal component.

The fraction $(\lambda_1 + \cdots + \lambda_q)/(\lambda_1 + \cdots + \lambda_p)$ is the proportion of the total variance kept by the first $q$ components.
For a centred data matrix, the principal directions are the right singular vectors of its singular value decomposition.
</document>

<document source="https://tamnd.github.io/math2/node/R2KZVG9SQ6.md" tag="R2KZVG9SQ6" kind="definition">
# Volterra equations with completely monotone kernels

## Definition

A function $k \colon (0, \infty) \to \R$ is *completely monotone* if it is infinitely differentiable and $(-1)^j k^{(j)}(t) \ge 0$ for all integers $j \ge 0$ and all $t > 0$.
Let $k \colon (0, \infty) \to \R$ and let $f$ be a given function on $[0, \infty)$.
Consider the [[Volterra equation of convolution type|SNFYTC1TTA]]
$$x(t) + \int_0^t k(t - s) \, x(s) \, ds = f(t), \quad t \ge 0,$$
and its first-kind analogue $\int_0^t k(t - s) \, x(s) \, ds = f(t)$.
Each is a *Volterra equation with a completely monotone kernel* if:

1. $k$ is completely monotone on $(0, \infty)$;
2. $k$ is integrable on $(0, 1)$, so that it is locally integrable on $[0, \infty)$.

## Remarks

Example: $k(t) = t^{-1/2}$.
Its derivatives are $k^{(j)}(t) = (-1)^j \frac{1 \cdot 3 \cdots (2j - 1)}{2^j} t^{-1/2 - j}$, so $(-1)^j k^{(j)} \ge 0$, and $\int_0^1 t^{-1/2} \, dt = 2$.
So [[Abel's integral equation|G4GWBYTW7B]] is a first-kind equation with a completely monotone kernel.
Non-example: $k(t) = t$ is positive but increasing, so $(-1)^1 k'(t) = -1 < 0$ and clause 1 fails.
Non-example: $k(t) = 1/t$ is completely monotone but not integrable near $0$, so clause 2 fails.
By [[Bernstein's theorem|ZBS1JA62QP]], clause 1 holds exactly when $k(t) = \int_{[0, \infty)} e^{-st} \, d\mu(s)$ for a positive Borel measure $\mu$ for which the integral converges for $t > 0$.
For $k(t) = t^{-1/2}$ the measure is $d\mu(s) = \pi^{-1/2} s^{-1/2} \, ds$.
</document>

<document source="https://tamnd.github.io/math2/node/R2N3GBDBVQ.md" tag="R2N3GBDBVQ" kind="concept">
# Brauer character

## Summary

A Brauer character is the analogue of the [[character of a representation|RMWMYBY8TC]] for representations of a finite group over a field of prime characteristic $p$.
It is a complex-valued function on the elements of order prime to $p$.
Its values come from lifting the eigenvalues of each matrix to complex roots of unity.
Brauer characters determine the composition factors of a modular representation, which ordinary traces in characteristic $p$ cannot do.
They are the basic tool of modular representation theory.

## Explanations

### L1 human

In characteristic $p$ the trace of a matrix is only a number modulo $p$, and it loses information; for example the identity matrix of size $p$ has trace $0$.
Brauer's idea is to replace each eigenvalue, a root of unity in the finite field world, by a matching complex root of unity, and add those up.
This gives a genuine complex number that behaves like an ordinary character.
The price is that it is only defined on elements whose order is not divisible by $p$.
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<document source="https://tamnd.github.io/math2/node/R2PHM118NG.md" tag="R2PHM118NG" kind="statement">
# Picard's little theorem

## Statement

Let $f$ be an [[entire function|0419794R0T]] that is not constant.
Then the set $\C \setminus f(\C)$ has at most one point.
That is, $f$ takes every complex value with at most one exception.

## Remarks

One classical proof uses the [[modular lambda function|4Z1YXT957Z]], which gives a holomorphic covering map from the unit disc onto $\C \setminus \set{0, 1}$.
If $f$ omits two values, one may assume that they are $0$ and $1$.
Then $f$ lifts through this covering to an entire function with values in the unit disc, which is constant by [[Liouville's theorem|0EHKVRF4J6]], so $f$ is constant.
Another proof uses [[Bloch's theorem|KW31KDX7TB]].
The bound is sharp: $e^z$ is not constant and omits exactly the value $0$.
For polynomials no value is omitted, by the fundamental theorem of algebra.

## Explanations

### L1 human

Liouville's theorem says that a bounded entire function is constant, so a nonconstant entire function cannot avoid a whole disc of values.
Picard's little theorem is much stronger: it cannot avoid even two points.
The exponential function shows that avoiding one point is possible.

### L2 human

The theorem has a version for meromorphic functions: a nonconstant meromorphic function on $\C$ omits at most two values of the Riemann sphere, as $\tan z$ omits $i$ and $-i$.
[[Picard's great theorem|08QA8QTM71]] is a local version near an essential singularity and implies the little theorem: if $f$ is not a polynomial, then $f(1/z)$ has an essential singularity at $0$.
The [[second main theorem of Nevanlinna theory|GWYE6745JK]] gives a quantitative form: for three distinct values, the solutions of $f(z) = a$ must be numerous.
Through Bloch's principle, the theorem is linked to [[Montel's theorem|ZVFVBVJMCB]] on normal families.
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<document source="https://tamnd.github.io/math2/node/R2TQ82PCNB.md" tag="R2TQ82PCNB" kind="concept">
# Smooth structure

## Summary

A smooth structure on a [[topological manifold|QQQ9B1DK4P]] is a maximal atlas of charts whose transition maps are smooth.
It is the extra data that turns a topological manifold into a [[smooth manifold|F45VQ47GPV]], so that one can differentiate functions and maps on it.
A topological manifold may have no smooth structure, exactly one up to diffeomorphism, or many.
The study of these possibilities is a central theme of differential topology.

## Explanations

### L1 human

A topological manifold only tells you which points are near which.
A smooth structure also tells you which functions on it count as smooth, by fixing a family of coordinate systems that fit together without kinks.
For example, the coordinate $t$ on the line $\R$ gives the usual smooth structure.
The coordinate $t^3$ gives a different atlas, because $t \mapsto t^{1/3}$ is not smooth at $0$, but the map $t \mapsto t^{1/3}$ is a diffeomorphism between the two structures.
Some spaces are more surprising: the $7$-sphere carries smooth structures that are not diffeomorphic to the standard one.
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<document source="https://tamnd.github.io/math2/node/R32HEEN55K.md" tag="R32HEEN55K" kind="statement">
# Liouville's theorems on elliptic functions

## Statement

Let $\Lambda = \Z\omega_1 + \Z\omega_2$ be a lattice in $\C$, with $\omega_1, \omega_2$ linearly independent over $\R$.
Let $f$ be an [[elliptic function|FF9ZPCZ59K]] with respect to $\Lambda$.
Let $P = \set{a + s\omega_1 + t\omega_2 : 0 \le s, t < 1}$ be a fundamental parallelogram, for some $a \in \C$.

1. If $f$ has no poles, then $f$ is constant.
2. The sum of the [[residues|0JFGKKWWFK]] of $f$ at its poles in $P$ is $0$.
3. If $f$ is not constant and $c \in \C$, then $f$ has as many solutions of $f(z) = c$ in $P$ as [[poles|0WPQYW8MTE]] in $P$, both counted with multiplicity.

## Remarks

1. An elliptic function without poles is continuous on the closure of $P$, hence bounded on $\C$ by periodicity, hence constant by [[Liouville's theorem|0EHKVRF4J6]].
2. By the [[residue theorem|032HKYS329]], the sum of the residues is $\frac{1}{2\pi i}\int_{\partial P} f(z)\,dz$, and the integrals over opposite sides cancel by periodicity.
3. Apply part 2 to the elliptic function $f'/(f - c)$ and use the [[argument principle|0PN2BXRS8G]].

The common number in part 3 is the *order* of $f$.
Part 2 shows that the order is never $1$: a single simple pole would have a nonzero residue.
So a nonconstant elliptic function has order at least $2$; the [[Weierstrass function|361NAK68QK]] $\wp$ has order $2$, with one double pole in $P$.
A further result of the same kind says that the sum of the zeros minus the sum of the poles in $P$, with multiplicity, lies in $\Lambda$.
Each point of $\C$ is congruent modulo $\Lambda$ to exactly one point of $P$, so the counts in parts 2 and 3 do not depend on $a$.
The contour arguments use a choice of $a$ for which $f$ has no zeros, poles or $c$-points on the boundary of $P$.

## Explanations

### L1 human

These three facts constrain what a doubly periodic function can look like.
It must have poles unless it is constant.
Its residues inside one period cell add up to zero.
And it takes every value the same number of times in a period cell.

### L2 human

All three proofs use one idea: integrate around the boundary of a period parallelogram, where opposite sides cancel by periodicity.
The results show that the smallest possible order of a nonconstant elliptic function is $2$.
This is why the Weierstrass function has a double pole and the Jacobi functions have two simple poles per cell.
On the torus $\C/\Lambda$, they are the first instances of general facts on compact Riemann surfaces.
There a holomorphic function is constant, and a meromorphic function has as many zeros as poles.
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<document source="https://tamnd.github.io/math2/node/R3625YKQC0.md" tag="R3625YKQC0" kind="concept">
# Galois connection between subgroups and subfields

## Summary

Let $L/K$ be a field extension.
One map sends a group $H$ of automorphisms of $L$ over $K$ to its fixed field.
The other sends an intermediate field $E$ to the group of automorphisms that fix $E$.
Together they form an antitone [[Galois connection|Q5JHEPRYQ6]].
Both maps reverse inclusions, and applying one after the other can only enlarge a subgroup or an intermediate field.
For a finite [[Galois extension|0DS2F7GK05]] the two maps are inverse bijections; this is the [[fundamental theorem of Galois theory|0T1Q8D0FW8]].

## Explanations

### L1 human

Think of two lists side by side: the subgroups of the automorphism group, and the fields between $K$ and $L$.
Each subgroup points to the field of elements it leaves fixed, and each field points to the group of automorphisms that leave it fixed.
A bigger group fixes fewer elements, so the arrows turn the order upside down.
For $L = \Q(\sqrt{2})$ over $K = \Q$, the trivial group points to $L$, and the group generated by $\sqrt{2} \mapsto -\sqrt{2}$ points to $\Q$.
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<document source="https://tamnd.github.io/math2/node/R38W9GFMQX.md" tag="R38W9GFMQX" kind="example">
# Coiflet

## Example

Coiflets are a family of compactly supported orthonormal wavelets constructed by Ingrid Daubechies and named after Ronald Coifman.
For an even integer $L \ge 2$, the coiflet of order $L$ is a real orthonormal wavelet $\psi$ with [[scaling function|YH4BERB952]] $\varphi$ such that:

1. $\psi$ has $L$ [[vanishing moments|7QERXM0C52]]: $\int_{\R} x^l \psi(x) \, dx = 0$ for $l = 0, 1, \dots, L-1$.
2. $\int_{\R} \varphi(x) \, dx = 1$ and $\int_{\R} x^l \varphi(x) \, dx = 0$ for $l = 1, \dots, L-1$.

The low-pass filter of the coiflet of order $L$ has $3L$ coefficients, so $\varphi$ and $\psi$ are supported on intervals of length $3L - 1$.
For $L = 2$ the filter has $6$ coefficients.

## Remarks

The vanishing moments of the scaling function are the distinctive feature.
For a smooth function $f$, they make the scaling coefficient $\ip{f}{\varphi_{j,k}}$ close to $2^{-j/2} f(2^{-j} k)$, where $\varphi_{j,k}(x) = 2^{j/2}\varphi(2^j x - k)$, with an error that is small to high order in $2^{-j}$.
So samples of $f$ can be used directly as starting coefficients of the discrete wavelet transform.
To get these extra conditions, coiflets need longer filters than the [[Daubechies wavelets|90V8S5CXE6]] with the same number of vanishing moments of $\psi$.
Coiflets are also closer to symmetric than Daubechies wavelets.

## Explanations

### L1 human

Most wavelets only ask that the wavelet itself have zero average against low-degree polynomials.
Coiflets also ask this of the scaling function, apart from its total mass.
As a result, the scaling function acts almost like a point sample of the signal.
This makes them convenient when the input data are just samples of a smooth function.
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<document source="https://tamnd.github.io/math2/node/R3AR5JK8WW.md" tag="R3AR5JK8WW" kind="definition">
# Dedekind zeta function

## Definition

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\mathcal{O}_K$.
For a nonzero [[ideal|096Z9MP2ME]] $\mathfrak{a}$ of $\mathcal{O}_K$, let $N(\mathfrak{a}) = \abs{\mathcal{O}_K/\mathfrak{a}}$ be its [[norm|M54ADM0CVF]].
Let $s$ be a complex variable with $\operatorname{Re} s > 1$.
The *Dedekind zeta function* of $K$ is
$$\zeta_K(s) = \sum_{\mathfrak{a}} \frac{1}{N(\mathfrak{a})^{s}},$$
where $\mathfrak{a}$ runs over the nonzero ideals of $\mathcal{O}_K$.

## Remarks

The series converges absolutely for $\operatorname{Re} s > 1$.
Unique factorisation of ideals gives the Euler product $\zeta_K(s) = \prod_{\mathfrak{p}} \big(1 - N(\mathfrak{p})^{-s}\big)^{-1}$ over the nonzero prime ideals $\mathfrak{p}$.
The function has an [[analytic continuation|0M3YQJ81NB]] to a meromorphic function on $\C$ whose only pole is a simple pole at $s = 1$.
Let $K$ have $r_1$ real embeddings, $r_2$ pairs of complex embeddings, and $w_K$ roots of unity.
Let $h_K$ be its [[class number|KTKK9B1X41]], $R_K$ its [[regulator|RWK0HNF1NA]] and $d_K$ its [[discriminant|E0VZTNPJXA]].
The residue is
$$\operatorname{Res}_{s = 1} \zeta_K(s) = \frac{2^{r_1} (2\pi)^{r_2} h_K R_K}{w_K \sqrt{\abs{d_K}}},$$
the [[class number formula|55XX5A7XZV]].

Example: for $K = \Q$ every nonzero ideal is $n\Z$ for a unique $n \ge 1$, with norm $n$, so $\zeta_{\Q}$ is the [[Riemann zeta function|1XWBY4N0VT]].
For $K = \Q(i)$, comparing Euler factors prime by prime gives $\zeta_K(s) = \zeta(s) L(s, \chi_4)$, where $L(s, \chi_4)$ is the [[Dirichlet L-function|9D2Z543EJQ]] of the non-trivial character modulo $4$.

Non-example: the sum $\sum_{\alpha} \abs{N_{K/\Q}(\alpha)}^{-s}$ over nonzero elements $\alpha \in \mathcal{O}_K$ is not $\zeta_K(s)$.
It counts each principal ideal once for each unit, so for $K = \Q(\sqrt{2})$, which has infinitely many units, it diverges for every $s$.
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# Fréchet–Urysohn space

## Summary

A Fréchet–Urysohn space is a topological space with the following property.
Each point of the closure of a set is the limit of a sequence from that set.
So closures, and not only closed sets, can be computed with sequences.
Every [[first-countable space|0CP5NDD21C]] is Fréchet–Urysohn, and every Fréchet–Urysohn space is [[sequential|WMB2MAEDX4]].
The converse implications fail in general.

## Explanations

### L1 human

In a metric space, a point is in the closure of a set exactly when some sequence from the set runs to it.
A Fréchet–Urysohn space is a space where this familiar fact still holds.
For example, in $\R$ the point $\sqrt 2$ is in the closure of $\Q$, and the decimal approximations $1, 1.4, 1.41, \ldots$ form a sequence of rationals that converges to it.
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<document source="https://tamnd.github.io/math2/node/R3QTCPTJK7.md" tag="R3QTCPTJK7" kind="concept">
# Power series in several variables

## Summary

A power series in several variables is an infinite sum $\sum_\alpha c_\alpha (z - a)^\alpha$ of monomials in $z_1 - a_1, \dots, z_n - a_n$, indexed by [[multi-indices|0PJXBR6QA3]] $\alpha$.
It extends the one-variable [[power series|0MZB06ST2B]].
A function on an open set of $\C^n$ is [[holomorphic|507C34W0ZT]] exactly when it is locally the sum of a convergent power series.
The domain of convergence of such a series is the interior of the set where it converges.
Unlike in one variable, it is not always a polydisc; it is a [[logarithmically convex|EHWH6EF4QY]] complete [[Reinhardt domain|WA0SE56FVH]].

## Explanations

### L1 human

A polynomial in $z_1$ and $z_2$ is a finite sum of terms $c \, z_1^j z_2^k$.
A power series allows infinitely many such terms.
For example, $\sum_{j, k \ge 0} z_1^j z_2^k$ is the product of two geometric series, and it equals $\frac{1}{(1 - z_1)(1 - z_2)}$ when $\abs{z_1} < 1$ and $\abs{z_2} < 1$.
The terms have no natural order, so one asks the sum of their absolute values to be finite.
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<document source="https://tamnd.github.io/math2/node/R3SG1SN7JQ.md" tag="R3SG1SN7JQ" kind="concept">
# Integral equation

## Summary

An integral equation is an equation in which the unknown function appears under an integral sign.
The typical linear example is $\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds$, where $f$ and the [[kernel|GSMBXT7FPK]] $K$ are given and $\varphi$ is unknown.
Many problems for differential equations, such as [[initial value problems|0FXDW992DS]] and [[boundary value problems|TM8VQJWGHZ]], can be rewritten as integral equations.
The integral form often needs less smoothness and suits fixed point and operator methods.

## Explanations

### L1 human

In an ordinary equation the unknown is a number; here the unknown is a whole function.
The function appears once on its own and once inside an integral, which mixes its values over a range of points.
A picture: the value of $\varphi$ at $x$ is a known value $f(x)$ plus a weighted sum, an integral, of the values of $\varphi$ over a range.
For example, $\varphi(x) = 1 + \int_0^x \varphi(s) \, ds$ has the solution $\varphi(x) = e^x$, since $1 + \int_0^x e^s \, ds = e^x$.
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<document source="https://tamnd.github.io/math2/node/R3SHZX0NZR.md" tag="R3SHZX0NZR" kind="concept">
# Semilocally simply connected space

## Summary

A space is semilocally simply connected if every point has a neighbourhood whose loops can all be shrunk to a point inside the whole space.
The loops need not shrink inside the neighbourhood itself, which is why the condition is only "semi" local.
It is exactly the condition, together with local path-connectedness, under which a connected space has a [[universal cover|Y1YPE6PM52]].

## Explanations

### L1 human

The condition forbids arbitrarily small loops that cannot be undone.
Manifolds satisfy it, because every point has a neighbourhood that looks like a ball, and CW complexes satisfy it too.
The [[Hawaiian earring|73EBKQPP8Q]], a union of circles of radius $1/n$ that all touch at one point, fails it.
Every neighbourhood of the touching point contains a whole small circle, and that circle cannot be shrunk to a point in the earring.
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<document source="https://tamnd.github.io/math2/node/R3TMVHTP3A.md" tag="R3TMVHTP3A" kind="concept">
# Projective conic

## Summary

A projective conic is the set of points of a projective plane over a field where a quadratic form in three variables vanishes.
It is the projective version of the [[conic section|00HJCMC2CD]].
Ellipses, parabolas and hyperbolas become the same kind of curve once points at infinity are added.
Non-degenerate conics are the simplest curves after lines, and in finite planes they are the model examples of [[ovals|7B03TB64N9]].

## Explanations

### L1 human

An ellipse, a parabola and a hyperbola look different in the plane.
In the projective plane, a parabola gains one point at infinity and a hyperbola gains two, and all three become closed curves of the same type.
In homogeneous coordinates $[x : y : z]$ each one is given by an equation of degree $2$, such as $xz = y^2$.
This curve contains the points $[1 : t : t^2]$ for every $t$, and also the point $[0 : 0 : 1]$.
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# Hurwitz-stable matrix

## Definition

Let $A$ be an $n \times n$ [[square matrix|0JXZKX526R]] with real or complex entries.
The matrix $A$ is *Hurwitz-stable* if every [[eigenvalue|0BQAA2BK3Q]] $\lambda$ of $A$ satisfies $\operatorname{Re} \lambda < 0$.

## Remarks

Equivalently, $e^{tA} \to 0$ as $t \to \infty$, that is, the origin is asymptotically stable for $x' = Ax$.
Then there are $C \ge 1$ and $\alpha > 0$ with $\norm{e^{tA}} \le C e^{-\alpha t}$ for all $t \ge 0$.
A real matrix $A$ is Hurwitz-stable if and only if, for one (equivalently, every) positive definite matrix $Q$, the [[Lyapunov equation|92GVPZH9FT]] $A^T P + P A = -Q$ has a positive definite solution $P$.

Example: $A = \begin{pmatrix} -1 & 2 \\ 0 & -3 \end{pmatrix}$.
It is triangular, so its eigenvalues are its diagonal entries $-1$ and $-3$, both with negative real part.

Non-example: $A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.
Its eigenvalues are $\pm i$, with real part $0$, so $A$ is not Hurwitz-stable; the solutions of $x' = Ax$ are rotations that do not decay.

For a real matrix, the condition can be tested on the coefficients of the characteristic polynomial by the Routh–Hurwitz criterion.
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# Transport equation

## Definition

Let $n \ge 1$ and let $b \in \R^n$ be a fixed vector.
The *transport equation* with velocity $b$ is the [[partial differential equation|GC5BDZ6QBC]]
$$\partial_t u + b \cdot \nabla u = 0$$
for $u \colon \R^n \times (0, \infty) \to \R$.
Here $\nabla u$ is the [[gradient|0ABK9V9Y64]] of $u$ in the space variable $x$.
The *initial value problem* adds the condition $u(x, 0) = g(x)$ for a given function $g \colon \R^n \to \R$.

## Remarks

For $g \in C^1(\R^n)$ the initial value problem has the unique $C^1$ solution $u(x, t) = g(x - tb)$.
The reason is that $u$ is constant along each line $s \mapsto (x + sb, t + s)$, since the derivative of $u$ along it is $\partial_t u + b \cdot \nabla u = 0$.
The inhomogeneous equation $\partial_t u + b \cdot \nabla u = f$ has the solution $u(x, t) = g(x - tb) + \int_0^t f(x + (s - t) b, s)\, ds$.
With a variable velocity field $b(x, t)$ one also calls $\partial_t u + b \cdot \nabla u = 0$ a transport equation, and $\partial_t u + \nabla \cdot (u\, b) = 0$ the continuity equation; the two agree when $\nabla \cdot b = 0$.

Example: $n = 1$, $b = 2$, $g(x) = e^{-x^2}$, and $u(x, t) = e^{-(x - 2t)^2}$.
Check: $\partial_t u = 4(x - 2t) u$ and $\partial_x u = -2(x - 2t) u$, so $\partial_t u + 2 \partial_x u = 0$, and $u(x, 0) = g(x)$.

Non-example: $u(x, t) = e^{-t} \sin x$ with $n = 1$ is not a solution for any $b$.
Indeed $\partial_t u + b\, \partial_x u = e^{-t}(-\sin x + b \cos x)$, which is not identically zero.
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# Aitken's delta-squared process

## Definition

Let $(s_n)_{n \ge 0}$ be a [[sequence|0QE8STGJH2]] of real or complex numbers.
Write $\Delta s_n = s_{n+1} - s_n$ and $\Delta^2 s_n = s_{n+2} - 2 s_{n+1} + s_n$.
For each $n$ with $\Delta^2 s_n \ne 0$, *Aitken's delta-squared process* gives
$$A_n = s_n - \frac{(\Delta s_n)^2}{\Delta^2 s_n}.$$

## Remarks

Equivalently, $A_n = \dfrac{s_n s_{n+2} - s_{n+1}^2}{s_{n+2} - 2 s_{n+1} + s_n}$.
Example: if $s_n = s + c \lambda^n$ with $c \ne 0$ and $\lambda \ne 0, 1$, then $\Delta s_n = c\lambda^n(\lambda - 1)$ and $\Delta^2 s_n = c \lambda^n (\lambda - 1)^2$, so $A_n = s_n - c\lambda^n = s$ for every $n$.
Non-example: for an arithmetic sequence such as $s_n = n$ one has $\Delta^2 s_n = 0$, so $A_n$ is not defined.
A classical result: if $s_n \to s$, $s_n \ne s$, and $\frac{s_{n+1} - s}{s_n - s} \to \lambda$ with $\lambda \ne 1$, then $(A_n)$ [[accelerates|90ANMX55DP]] the convergence of $(s_n)$.

## Explanations

### L1 human

The formula subtracts from $s_n$ the correction that a perfectly geometric error would require.
It uses only the first and second differences of the sequence.
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# Bruck–Ryser–Chowla theorem

## Statement

Let $v, k, \lambda$ be integers with $0 < \lambda < k < v$.
Suppose that there is a symmetric $2$-$(v, k, \lambda)$ [[block design|TGZATGBHTH]], that is, one with exactly $v$ blocks.
Then:

1. If $v$ is even, then $k - \lambda$ is a perfect square.
2. If $v$ is odd, then the equation $x^2 = (k - \lambda) y^2 + (-1)^{(v-1)/2} \lambda z^2$ has a solution in integers $x, y, z$ that are not all zero.

## Remarks

Proof idea: the [[incidence matrix|9DNFN8M83X]] $N$ of the design is a square matrix with $N N^T = (k - \lambda) I + \lambda J$, where $J$ is the all-ones matrix.
For $v$ even, taking determinants gives $\det(N)^2 = k^2 (k - \lambda)^{v - 1}$, and since $v - 1$ is odd, $k - \lambda$ must be a square.
For $v$ odd, the identity says that two quadratic forms are equivalent over $\Q$; Lagrange's four-square theorem and a variable-by-variable reduction turn this into the stated equation.
For a [[finite projective plane|2HWB7P1DZN]] of order $n$, take $v = n^2 + n + 1$, $k = n + 1$ and $\lambda = 1$.
If $n \equiv 1$ or $2 \pmod 4$, then $(v-1)/2 = n(n+1)/2$ is odd, and the condition becomes: $n$ is a [[sum of two squares|04NPWJMA6H]].
Example: $6 \equiv 2 \pmod 4$ and $6$ is not a sum of two squares, so there is no projective plane of order $6$.
The theorem gives only a necessary condition: $10 = 1^2 + 3^2$ passes the test, yet there is no plane of order $10$.

## Explanations

### L1 human

The theorem says that some parameter sets for symmetric designs are impossible, by a test from number theory.
For projective planes it says, for instance, that there is no plane of order $6$, $14$ or $21$.

### L2 human

The theorem is the main known non-existence result for symmetric designs and projective planes.
It refines [[Fisher's inequality|M9XF82GGW8]], whose equality case defines symmetric designs, by looking at the incidence matrix over the rationals instead of only its rank.
The open question of whether every projective plane has prime power order lies beyond its reach.
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# Sturm's theorem

## Statement

Let $p \in \R[x]$ be a nonconstant polynomial without multiple roots.
Define the *Sturm sequence* by $p_0 = p$, $p_1 = p'$ (the [[derivative|0F1WR9C83K]]), and $p_{i+1} = -\operatorname{rem}(p_{i-1}, p_i)$.
Here $\operatorname{rem}$ is the remainder of [[polynomial division|D6D0SR1FK3]], and the sequence stops at the last nonzero term $p_m$.
For $\xi \in \R$, let $V(\xi)$ be the number of sign changes in the sequence $p_0(\xi), p_1(\xi), \dots, p_m(\xi)$, with zeros ignored.
Let $a < b$ be real numbers that are not [[roots|00EHX6FWC0]] of $p$.
Then the number of distinct real roots of $p$ in the interval $(a, b)$ is $V(a) - V(b)$.

## Remarks

The sequence is the Euclidean algorithm for $p$ and $p'$ with sign changes, so $p_m$ is a [[greatest common divisor|Q75YRG25JK]] of $p$ and $p'$, a nonzero constant here.
As $\xi$ increases, $V(\xi)$ can only change when $\xi$ passes a root of some $p_i$.
At a root of $p_i$ with $0 < i < m$, the relation $p_{i-1} = q_i p_i - p_{i+1}$ forces $p_{i-1}$ and $p_{i+1}$ to have opposite signs, so $V$ does not change.
At a root of $p$, the signs of $p$ and $p'$ go from opposite to equal, so $V$ drops by exactly one.
Example: for $p = x^2 - 2$, the sequence is $x^2 - 2$, $2x$, $2$; then $V(0) = 1$ and $V(2) = 0$, so there is one root in $(0, 2)$, namely $\sqrt 2$.
The theorem also holds without the hypothesis on multiple roots, still counting distinct roots, but the proof then divides the sequence by $p_m$.

## Explanations

### L1 human

Sturm's theorem counts the real roots of a polynomial in an interval exactly, with no guessing.
One builds a short chain of polynomials by repeated division, evaluates the chain at the two ends of the interval, and counts sign changes.
The difference of the two counts is the number of roots in between.

### L2 human

Unlike [[Descartes' rule of signs|T6DQQ36JZ5]], which only gives an upper bound, Sturm's theorem gives the exact number of distinct roots.
By bisection, it isolates every real root in an interval as small as one likes, which is the basis of exact real root isolation in computer algebra.
Taking $a$ below and $b$ above all roots, it counts all real roots of $p$, using the signs of leading coefficients only.
The same argument works over any real closed field, where the [[intermediate value theorem|00KAFBR32E]] holds for polynomials.
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# Kummer–Artin–Schreier–Witt theory

## Summary

Kummer–Artin–Schreier–Witt theory describes the [[abelian extensions|EQDEXBWGJH]] of bounded exponent of a [[field|04C5ZBG1WT]] by explicit equations.
For an exponent $m$ prime to the [[characteristic|1ZB13N78XW]], this is [[Kummer theory|XZQENYV15Y]].
It uses $m$-th roots and needs the $m$-th [[roots of unity|0YV2CKP2AV]] in the base field.
For an exponent $p^n$ in characteristic $p > 0$, this is Artin–Schreier–Witt theory.
It uses equations $F(x) - x = a$ in the ring of [[Witt vectors|6N3X2QX4W9]] of length $n$.
For $n = 1$ it reduces to [[Artin–Schreier theory|7YNMNRGH3T]].
Together the two parts describe all finite cyclic extensions of a field that contains enough roots of unity.

## Explanations

### L1 human

Kummer theory builds extensions by adjoining roots such as $\sqrt[m]{a}$.
In characteristic $p$ this fails for $m = p$, because $x^p - a$ has only one root, repeated.
Artin and Schreier replaced $x^p = a$ by $x^p - x = a$, which gives the cyclic extensions of degree $p$.
Witt found that to reach cyclic extensions of degree $p^2, p^3, \ldots$ one must solve the same kind of equation with Witt vectors in place of field elements.
For example, over the finite field $\F_p$ the Frobenius map fixes every Witt vector, because $c^p = c$ for each $c \in \F_p$.
From this the theory recovers that $\F_p$ has exactly one extension of each degree $p^n$, and that it is cyclic.
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# Torus integral

## Summary

The torus integral of a function $f$ on $\C^n$ is the iterated contour integral of $f(\zeta) \, d\zeta_1 \cdots d\zeta_n$ over a product of $n$ circles.
This product is the [[distinguished boundary|A6HN207Y2T]] of a [[polydisc|DJBWP7D3M8]].
It is computed by parametrising each circle by an angle and integrating over the cube $[0, 2\pi]^n$.
It appears in the [[Cauchy integral formula on a polydisc|FG5448NZ18]].
It also gives the coefficients of a [[power series in several variables|R3QTCPTJK7]].
For $n = 1$ it is the [[contour integral|06ZGPYTWK8]] over a circle.

## Explanations

### L1 human

In one variable, one integrates a function around a circle by walking around the circle once.
In $n$ variables, each coordinate $\zeta_j$ walks around its own circle, independently of the others.
The points visited form a torus, like the surface of a doughnut when $n = 2$.
For example, the torus integral of $1/(\zeta_1 \zeta_2)$ over the product of two unit circles is $(2\pi i)^2$, the product of two one-variable integrals equal to $2\pi i$.
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# Projective resolution

## Definition

Let $R$ be a [[ring|0B1261B5ED]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].
A *projective resolution* of $M$ is an [[exact sequence|JB7ZBFBA9C]] of $R$-modules
$$\cdots \to P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 \xrightarrow{\varepsilon} M \to 0$$
in which every $P_i$ is a [[projective module|BFFA03GE3V]].
It is a *free resolution* if every $P_i$ is a [[free module|0DZ9DKAKZR]].
It has *length* $n$ if $P_n \ne 0$ and $P_i = 0$ for all $i > n$.

## Remarks

Every module has a free resolution: take a free module $F_0$ mapping onto $M$, then a free module mapping onto the kernel, and repeat.
Example: over $\Z$, the sequence $0 \to \Z \xrightarrow{\cdot n} \Z \to \Z/n\Z \to 0$ is a free resolution of length $1$, since multiplication by $n \ne 0$ is injective and its image $n\Z$ is the kernel of the reduction map.
Over $R = \Z/4\Z$, the module $\Z/2\Z$ has the free resolution $\cdots \to R \xrightarrow{\cdot 2} R \xrightarrow{\cdot 2} R \to \Z/2\Z \to 0$, which never stops.
Non-example: $0 \to \Z \xrightarrow{\cdot 4} \Z \to \Z/2\Z \to 0$ is not a resolution, because it is not exact at the middle $\Z$.
The kernel of the reduction is $2\Z$, but the image of multiplication by $4$ is $4\Z$.
One often writes the resolution as the complex $P_\bullet$ together with the map $\varepsilon \colon P_0 \to M$; then $H_0(P_\bullet) \cong M$ and $H_i(P_\bullet) = 0$ for $i > 0$.
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# Neumann's addition theorem

## Statement

Let $n$ be an integer and let $J_k$ denote the [[Bessel function of the first kind|CZCMRDWDYW]] of integer order $k$.
For all $x, y \in \C$,
$$J_n(x + y) = \sum_{k=-\infty}^{\infty} J_k(x)\, J_{n-k}(y),$$
where the series converges absolutely.

## Remarks

The proof uses the [[generating function|0PHAC833JF]] $G(z, t) = \exp\left( \frac{z}{2}\left( t - \frac{1}{t} \right) \right) = \sum_{k \in \Z} J_k(z)\, t^k$.
Since $G(x + y, t) = G(x, t)\, G(y, t)$, multiplying the two Laurent series in $t$ and comparing the coefficients of $t^n$ gives the formula.
Example: with $n = 0$ and $y = -x$, one gets $1 = J_0(0) = \sum_{k} J_k(x) J_{-k}(-x) = \sum_k J_k(x)^2$, using $J_{-k}(-x) = J_k(x)$; so $J_0(x)^2 + 2\sum_{k \ge 1} J_k(x)^2 = 1$.
This identity shows $\abs{J_0(x)} \le 1$ and $\abs{J_k(x)} \le \frac{1}{\sqrt{2}}$ for $k \ge 1$ and real $x$.
The formula as written is for integer orders; for non-integer orders there are related addition theorems of Graf and Gegenbauer.

## Explanations

### L1 human

The value of a Bessel function at a sum $x + y$ is built from its values at $x$ and at $y$.
This is the Bessel analogue of the rule $e^{x+y} = e^x e^y$, and indeed it comes from that rule applied to the generating function.

### L2 human

The theorem is a convolution identity: the sequence $(J_k(x + y))_k$ is the convolution of $(J_k(x))_k$ and $(J_k(y))_k$.
In terms of Fourier series, it says that $e^{i(x+y)\sin\theta} = e^{ix\sin\theta} e^{iy\sin\theta}$, read coefficient by coefficient.
Its consequence $\sum_k J_k(x)^2 = 1$ gives uniform bounds on Bessel functions of integer order on the real line.
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# Gelfand–Levitan theory

## Summary

Gelfand–Levitan theory reconstructs the potential $q$ of a [[Sturm–Liouville|7WV6B2502K]] operator $-y'' + q y$ from its spectral measure.
The key step is a linear integral equation, the Gelfand–Levitan equation, whose kernel is built from the spectral data.
Its solution $K(x, t)$ is a transformation kernel that turns cosines into the solutions of the perturbed equation, and $q(x) = 2 \frac{d}{dx} K(x, x)$.
Israel Gelfand and Boris Levitan introduced it, and the related Marchenko equation is a basic tool of the inverse scattering transform.

## Explanations

### L1 human

Suppose you know the notes a string can play, together with some extra information, and you want to find the varying stiffness of the string.
Gelfand–Levitan theory gives a recipe for this inverse problem.
It compares the given spectral data with those of a plain string, writes the difference as a kernel, and solves one linear integral equation.
The answer $q$ then appears from the diagonal values of the solution.
When the spectral data are those of the plain string, the kernel is zero and the recipe returns $q = 0$.
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# Incomplete gamma function

## Definition

Let $s$ and $x$ be real numbers.
1. For $s > 0$ and $x \ge 0$, the *lower incomplete gamma function* is
$$\gamma(s, x) = \int_0^x t^{s-1} e^{-t}\, dt.$$
2. For $x > 0$ and any real $s$, the *upper incomplete gamma function* is the [[improper integral|0VCTBJMEJF]]
$$\Gamma(s, x) = \int_x^\infty t^{s-1} e^{-t}\, dt.$$

## Remarks

For $s > 0$ and $x > 0$, $\gamma(s, x) + \Gamma(s, x) = \Gamma(s)$, the [[gamma function|0H5SYBSED9]].

Example: for $s = 1$, $\gamma(1, x) = \int_0^x e^{-t}\, dt = 1 - e^{-x}$ and $\Gamma(1, x) = e^{-x}$, and their sum is $1 = \Gamma(1)$.
Integration by parts gives $\gamma(s+1, x) = s\,\gamma(s, x) - x^s e^{-x}$.

Non-example of a valid argument: for $s = 0$ the lower integral $\int_0^x t^{-1} e^{-t}\, dt$ diverges at $0$, so $\gamma(0, x)$ is not defined.
The upper function is still defined: $\Gamma(0, x) = E_1(x)$ is the [[exponential integral|EV6ZDQ0NDY]].

The quotient $P(s, x) = \gamma(s, x)/\Gamma(s)$ is the regularised lower incomplete gamma function, the distribution function of the [[gamma distribution|0B80PAQD3C]] with shape $s$ and scale $1$.
Both functions extend to complex arguments: $\gamma(s, x)$ for $\operatorname{Re} s > 0$, and $\Gamma(s, x)$ by analytic continuation.
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# Multidimensional scaling

## Definition

Let $D = (d_{ij})$ be an $n \times n$ [[symmetric matrix|0VR671PHE5]] with $d_{ii} = 0$ and $d_{ij} \ge 0$, the dissimilarities of $n$ objects, and let $1 \le k < n$.
*Multidimensional scaling* seeks points $x_1, \ldots, x_n \in \R^k$ whose distances $\abs{x_i - x_j}$ approximate the $d_{ij}$.
*Classical multidimensional scaling* computes them as follows:

1. Let $D^{(2)} = (d_{ij}^2)$ and $J = I_n - \tfrac{1}{n} \mathbf{1} \mathbf{1}^T$, where $\mathbf{1}$ is the vector of ones, and let $B = -\tfrac{1}{2} J D^{(2)} J$.
2. Let $\lambda_1 \ge \cdots \ge \lambda_k$ be the $k$ largest [[eigenvalues|0BQAA2BK3Q]] of $B$, assumed positive, with orthonormal [[eigenvectors|0N6960W0BV]] $e_1, \ldots, e_k$.
3. Let $x_i \in \R^k$ be the $i$-th row of the $n \times k$ matrix $\begin{pmatrix} \sqrt{\lambda_1} e_1 & \cdots & \sqrt{\lambda_k} e_k \end{pmatrix}$.

## Remarks

The matrix $D$ is the matrix of [[Euclidean distances|0NPPGHQNXQ]] of some points in $\R^k$ exactly when $B$ is [[positive semidefinite|01MMVW8CKC]] of rank at most $k$.
In that case classical scaling recovers the points up to a translation and an orthogonal transformation, since $B$ is then the Gram matrix of the centred points.
Example: three points on a line at distances $d_{12} = d_{23} = 1$, $d_{13} = 2$.
The centred configuration $-1, 0, 1$ has Gram matrix $B = \begin{pmatrix} 1 & 0 & -1 \\ 0 & 0 & 0 \\ -1 & 0 & 1 \end{pmatrix}$, with largest eigenvalue $2$ and eigenvector $(1, 0, -1)/\sqrt{2}$.
Step 3 gives the coordinates $1, 0, -1$, the original points after a reflection.
Non-example: if $d_{12} = d_{23} = 1$ and $d_{13} = 3$, the triangle inequality fails, so no points in any $\R^k$ realise $D$, and $B$ has a negative eigenvalue.
Metric and non-metric scaling instead minimise a stress function, such as $\sum_{i < j} (d_{ij} - \abs{x_i - x_j})^2$, by iteration.
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# Cauchy–Riemann operator

## Definition

Let $\Omega \subseteq \C$ be an open set, and write $z = x + iy$ with $x, y$ real.
For a function $f \colon \Omega \to \C$ that has [[partial derivatives|09EGC6F8PT]] $\partial_x f$ and $\partial_y f$, the *Cauchy–Riemann operator* applied to $f$ is
$$\frac{\partial f}{\partial \bar z} = \frac12 \left( \frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y} \right).$$
It is also written $\bar\partial f$.

## Remarks

Writing $f = u + iv$ with $u, v$ real, $\partial f / \partial \bar z = 0$ is equivalent to $u_x = v_y$ and $u_y = -v_x$, the [[Cauchy–Riemann equations|0T71SSYZ39]].
So a continuously differentiable function $f$ on $\Omega$ is [[holomorphic|0NAHXR1FRH]] if and only if $\partial f / \partial \bar z = 0$ on $\Omega$.
Example: $\partial z / \partial \bar z = \tfrac12 (1 + i \cdot i) = 0$, so $z$ is in the kernel.
Non-example: $\partial \bar z / \partial \bar z = \tfrac12 (1 + i \cdot (-i)) = 1 \ne 0$, so $\bar z$ is not holomorphic.
The companion operator $\partial / \partial z = \tfrac12 (\partial_x - i \partial_y)$ satisfies $4 \, \partial_z \partial_{\bar z} = \Delta$.
In distributions on $\C$, $\partial_{\bar z} \big( 1/(\pi z) \big) = \delta_0$, the Dirac measure at $0$.
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# Hadamard finite part

## Definition

Let $a < x < b$ and let $\varphi \colon [a, b] \to \R$ be integrable, differentiable near $x$, with $\varphi'$ [[Hölder continuous|012PBA0FMD]] near $x$.
The *Hadamard finite part* integral of $\varphi(t)/(t - x)^2$ is
$$\mathrm{f.p.}\!\int_a^b \frac{\varphi(t)}{(t - x)^2}\, dt = \lim_{\varepsilon \to 0^+} \left( \int_{\set{t \in [a, b] : \abs{t - x} > \varepsilon}} \frac{\varphi(t)}{(t - x)^2}\, dt - \frac{2\varphi(x)}{\varepsilon} \right).$$
Under these hypotheses the limit exists.

## Remarks

The general rule is the same.
Expand the integral over the region at distance more than $\varepsilon$ from the singularity in powers of $\varepsilon$ and $\log \varepsilon$.
Then drop the terms that diverge as $\varepsilon \to 0$ and keep the limit of the rest.
For example, $\int_\varepsilon^1 t^{-3/2}\, dt = 2\varepsilon^{-1/2} - 2$, so $\mathrm{f.p.}\!\int_0^1 t^{-3/2}\, dt = -2$.

Example: $\varphi = 1$ on $[-1, 1]$ and $x = 0$.
Then $\int_{\varepsilon < \abs{t} \le 1} t^{-2}\, dt = \frac{2}{\varepsilon} - 2$, so the finite part is $-2$.

Non-example: for $\varphi(t) = \abs{t}^{1/2}$ and $x = 0$, the derivative condition fails at $0$.
The integral over $\varepsilon < \abs{t} \le 1$ equals $4\varepsilon^{-1/2} - 4$, and subtracting $2\varphi(0)/\varepsilon = 0$ leaves a divergent quantity, so this limit does not exist.

For $\varphi$ as above, the finite part is the derivative of a principal value:
$$\mathrm{f.p.}\!\int_a^b \frac{\varphi(t)}{(t - x)^2}\, dt = \frac{d}{dx}\, \mathrm{PV}\!\int_a^b \frac{\varphi(t)}{t - x}\, dt,$$
where $\mathrm{PV}$ is the [[Cauchy principal value|0SYR10HD4R]].
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# Normal scheme

## Definition

A [[scheme|5A35TYZD02]] $X$ is *normal* if for every point $x \in X$ the [[stalk|JTAKAPTT8H]] $\mathcal{O}_{X,x}$ is an [[integrally closed domain|H20P600HQS]].

## Remarks

An affine scheme $\operatorname{Spec} A$ is normal if and only if $A_{\mathfrak p}$ is an integrally closed domain for every prime $\mathfrak{p}$, that is, $A$ is a [[normal ring|KR7SWBYHRQ]].
For an integral domain $A$, this holds if and only if $A$ is integrally closed.
Example: $\operatorname{Spec} \Z$ is normal, since $\Z$ is a unique factorization domain and hence integrally closed.
Non-example: let $A = k[x, y]/(y^2 - x^3)$ for a field $k$; it is a domain.
The element $t = y/x$ of its field of fractions satisfies $t^2 - x = 0$, which is monic over $A$, but $t \notin A$.
So $A$ is not integrally closed, and the cusp $\operatorname{Spec} A$ is not normal; the failure is at the origin.
A normal scheme is reduced, but it need not be irreducible: a disjoint union of two normal schemes is normal.
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# Permanent

## Definition

Let $K$ be a [[commutative ring|0310REJJY8]], and let $A = (a_{ij})$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $K$.
The *permanent* of $A$ is
$$\operatorname{perm}(A) = \sum_{\sigma \in S_n} \prod_{i=1}^{n} a_{i, \sigma(i)},$$
where $S_n$ is the [[symmetric group|0515A6D2Y1]] of permutations of $\set{1, \dots, n}$.

## Remarks

Example: for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, the two permutations give $\operatorname{perm}(A) = 1 \cdot 4 + 2 \cdot 3 = 10$, while $\det A = -2$.

The permanent is not multiplicative, unlike the [[determinant|00KDZRW4C0]].
For $J = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, $\operatorname{perm}(J) = 2$, but $J^2 = 2J$ has permanent $8 \ne 2 \cdot 2$.

The permanent is linear in each row and each column, and it does not change when rows or columns are permuted.
If $A$ is the $0/1$ biadjacency matrix of a [[bipartite graph|0ZVQKTRKVQ]] with $n$ vertices on each side, then $\operatorname{perm}(A)$ is the number of [[perfect matchings|0NHK9MFFW9]] of the graph.
The permanent of the $n \times n$ all-ones matrix is $n!$.

## Explanations

### L1 human

The permanent is computed like the determinant, but every term gets a plus sign.
This small change makes it lose most of the good algebraic properties of the determinant.
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# Euler's pentagonal number theorem

## Statement

In the ring of [[formal power series|033WGN00BF]] $\Z[\![x]\!]$,
$$\prod_{n=1}^{\infty} (1 - x^n) = \sum_{k=-\infty}^{\infty} (-1)^k x^{k(3k-1)/2} = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + \cdots.$$
The same identity holds for complex numbers $x$ with $\abs{x} < 1$, where both sides converge absolutely.

## Remarks

The exponents $k(3k - 1)/2$ for $k = 1, -1, 2, -2, 3, -3, \ldots$ are $1, 2, 5, 7, 12, 15, \ldots$, the generalised pentagonal numbers.
Franklin's bijective proof reads the identity as a statement on [[integer partitions|0JV30XN73Z]] of $m$ into distinct parts.
The partitions with an even number of parts and those with an odd number of parts are equinumerous, except when $m = k(3k \pm 1)/2$.
In that case the difference, even minus odd, is $(-1)^k$.
Franklin's involution moves the smallest part onto the largest parts, or back, and it changes the number of parts by one.

Example: for $m = 5$, the partitions into distinct parts are $5$, $4 + 1$ and $3 + 2$.
One has an odd number of parts and two have an even number, so the coefficient of $x^5$ is $2 - 1 = 1$, as in the series.

Multiplying by the [[generating function for partitions|0578PA53VA]] $\prod_n (1 - x^n)^{-1}$ gives a recurrence for the [[partition function|0CV69WDRXW]]:
$$p(m) = \sum_{k \ne 0} (-1)^{k+1}\, p\big(m - k(3k - 1)/2\big),$$
with $p(0) = 1$ and $p(j) = 0$ for $j < 0$.
For example, $p(5) = p(4) + p(3) - p(0) = 5 + 3 - 1 = 7$.
The theorem is a special case of the [[Jacobi triple product|JFP2DGMNXT]].

## Explanations

### L1 human

Multiply out $(1 - x)(1 - x^2)(1 - x^3)\cdots$ without end.
One expects a messy series, but almost everything cancels.
Only the powers $x^1, x^2, x^5, x^7, x^{12}, x^{15}, \ldots$ survive, with signs in the pattern $-, -, +, +, -, -, \ldots$, and every coefficient is $0$ or $\pm 1$.
The surviving exponents are the pentagonal numbers and their close relatives.

### L2 human

The product $\prod (1 - x^n)$ is the reciprocal of the partition generating function, so the theorem controls the partition numbers $p(m)$.
The recurrence above computes $p(m)$ with about $\sqrt{m}$ terms, which was the standard method for tables of $p(m)$.
The bijective proof by Franklin is a model of a sign-reversing involution.
The identity is the simplest of a family of product-sum identities that includes the Jacobi triple product and the [[Rogers–Ramanujan identities|M24H4GJPPW]].
</document>

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# Oriented matroid

## Summary

An oriented matroid is a [[matroid|PX31ZNYS3Z]] in which each minimal dependence also carries signs, recording which elements lie on which side.
It abstracts the combinatorics of real vector configurations, of point configurations, of [[hyperplane arrangements|N602W0WAVT]] and of directed graphs.
Faces of [[convex polytopes|300K6CB1KH]], regions of real arrangements and the results of linear programming can all be described in terms of oriented matroids.
Not every oriented matroid comes from vectors; by the topological representation theorem, oriented matroids correspond to arrangements of pseudospheres.

## Explanations

### L1 human

Take some vectors in the plane or in space.
A dependence like $v_1 + v_2 - v_3 = 0$ tells you that $v_1$ and $v_2$ lie on one side and $v_3$ on the other.
An oriented matroid keeps only these signs, not the numbers.
Remarkably, a lot of geometry, such as which points form the corners of their convex hull, can be recovered from the signs alone.
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# Stochastic ordering

## Definition

Let $X$ and $Y$ be real [[random variables|0PNRKMRC55]], not necessarily on the same probability space.
$X$ is *smaller than $Y$ in the usual stochastic order*, written $X \le_{\mathrm{st}} Y$, if
$$P(X > t) \le P(Y > t) \qquad \text{for all } t \in \R.$$

## Remarks

In terms of [[distribution functions|0H8H7ZTHW2]], $X \le_{\mathrm{st}} Y$ means $F_X(t) \ge F_Y(t)$ for all $t$.
It is equivalent to $E[u(X)] \le E[u(Y)]$ for every nondecreasing function $u \colon \R \to \R$ for which both [[expectations|0WQNRCNY3M]] exist.
It is also equivalent to the existence of a [[coupling|HTR7J0W66V]] $(X', Y')$ with $X'$ distributed as $X$, $Y'$ distributed as $Y$ and $X' \le Y'$ almost surely.
One can take $X' = F_X^{-1}(U)$ and $Y' = F_Y^{-1}(U)$ for $U$ uniform on $(0, 1)$, where $F^{-1}$ is the [[quantile function|0X60VE98HN]].
The relation depends only on the laws, and it is a partial order on the probability measures on $\R$.
If $X \le_{\mathrm{st}} Y$ and $E[X] = E[Y]$ is finite, then $X$ and $Y$ have the same law.
Example: if $X$ is exponential with rate $\lambda$ and $Y$ is exponential with rate $\lambda' \le \lambda$, then $P(X > t) = e^{-\lambda t} \le e^{-\lambda' t} = P(Y > t)$ for $t \ge 0$, and both sides are $1$ for $t < 0$; so $X \le_{\mathrm{st}} Y$.
Non-example: if $X$ is normal with mean $0$ and variance $1$ and $Y$ is normal with mean $0$ and variance $4$, then $P(X > 1) < P(Y > 1)$ but $P(X > -1) > P(Y > -1)$, so neither $X \le_{\mathrm{st}} Y$ nor $Y \le_{\mathrm{st}} X$.
The name stochastic ordering also covers other orders, such as the convex order ($E[u(X)] \le E[u(Y)]$ for convex $u$) and the hazard rate and likelihood ratio orders.
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# Contractible space

## Summary

A contractible space is a topological space that has the [[homotopy type|SK0ZJGWXDP]] of a point.
Equivalently, it is nonempty and its identity map can be deformed continuously into a constant map.
All homotopy groups and all reduced homology groups of a contractible space vanish.
Contractible spaces are the trivial objects of homotopy theory.

## Explanations

### L1 human

A space is contractible when it can be shrunk continuously to a single point inside itself.
A disk, a ball, a tree and the whole space $\R^n$ are contractible.
For $\R^n$, slide every point $x$ to the origin along the segment $(1 - t) x$.
A circle is not contractible: shrinking it to a point would have to break it.
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<document source="https://tamnd.github.io/math2/node/R65XKHN4D1.md" tag="R65XKHN4D1" kind="concept">
# Hecke character

## Summary

A Hecke character of a [[number field|MWBDFS2KJ5]] $K$ is a continuous homomorphism from the [[idele class group|9GHD9MXNYV]] $C_K$ to the nonzero complex numbers.
Hecke characters generalize [[Dirichlet characters|7XB5VJKDPG]], which are the Hecke characters of finite order of $\Q$.
Each Hecke character has an L-function with an Euler product, a meromorphic continuation and a functional equation, proved by Hecke and again by [[Tate's thesis|QYW79XKJTP]].

## Explanations

### L1 human

A Dirichlet character is a pattern of complex numbers attached to the residues modulo $m$, compatible with multiplication.
A Hecke character does the same for a number field, but it may also depend on the real and complex embeddings in a continuous way.
In the language of ideles it is just a continuous character of the idele group that ignores the elements of $K$.
A simple example is the idele norm $x \mapsto \prod_v \abs{x_v}_v$, which is $1$ on every element of $K$ by the product formula.
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<document source="https://tamnd.github.io/math2/node/R68D2P175Z.md" tag="R68D2P175Z" kind="example">
# The natural numbers under addition are not a group

## Example

Let $\mathbb{N} = \{0, 1, 2, \dots\}$ be the [[natural numbers|0MWKJ44V5A]] with addition.
Addition is associative, and $0$ is an identity element.
The element $1$ has no [[inverse|0BRHKKP333]], because $1 + m \ge 1$ for each $m \in \mathbb{N}$, so $1 + m \ne 0$.
Thus $(\mathbb{N}, +)$ is not a group. It is a commutative monoid.

## Remarks

If we add the negative integers, we get the group $(\mathbb{Z}, +)$.
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<document source="https://tamnd.github.io/math2/node/R68HNB5SVR.md" tag="R68HNB5SVR" kind="concept">
# Dual pair

## Summary

A dual pair, or dual system, consists of two vector spaces and a [[bilinear form|0SQZMXB6GP]] between them that separates points on both sides.
It is the abstract framework of duality theory for [[locally convex spaces|B6GY2KJMWW]].
The basic example is a Hausdorff locally convex space $E$ paired with its [[continuous dual|4A82S5FVRT]] $E'$ by $\langle x, f \rangle = f(x)$.
Topologies such as the [[weak topology|P6KHVSTPZV]] and the [[Mackey topology|W18EY4E5TH]] are defined from a dual pair alone.

## Explanations

### L1 human

In a dual pair, each side can be used to test the other.
A vector $x$ is known once one knows all the numbers $\langle x, y \rangle$, and likewise for $y$.
The simplest case is $\R^n$ paired with itself by the dot product.
In functional analysis one pairs a space of functions with a space of measures or of other functions, for example $\langle f, g \rangle = \int f g$.
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<document source="https://tamnd.github.io/math2/node/R6DXPZPVV7.md" tag="R6DXPZPVV7" kind="definition">
# n-skeleton

## Definition

Let $n \ge 0$ be an integer.

1. Let $X$ be a [[CW complex|5W95QFHXPR]]. The *$n$-skeleton* $X^{(n)}$ of $X$ is the union of all closed cells of $X$ of dimension at most $n$.
2. Let $K$ be a [[simplicial complex|V8GZ0W1ZZ5]]. The *$n$-skeleton* $K^{(n)}$ of $K$ is the subcomplex of all simplices of $K$ of dimension at most $n$.

The notation $X^n$ is also common.

## Remarks

The torus $T^2$ has a CW structure with one $0$-cell, two $1$-cells $a$ and $b$, and one $2$-cell attached along $a b a^{-1} b^{-1}$.
Its $0$-skeleton is a point, its $1$-skeleton is a wedge of two circles, and its $2$-skeleton is $T^2$.

The skeleton depends on the cell structure and not only on the space.
The sphere $S^2$ with one $0$-cell and one $2$-cell has a point as $1$-skeleton.
The sphere $S^2$ as the boundary of a tetrahedron has as $1$-skeleton a graph with $4$ vertices and $6$ edges.
So "the $1$-skeleton of $S^2$" has no meaning before a cell structure is chosen.

For a CW complex, $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching $n$-cells along maps from their boundary spheres, and $X$ is the union of its skeleta with the weak topology.
The inclusion $X^{(n)} \to X$ induces an isomorphism on [[homotopy groups|P976WYEFMB]] $\pi_k$ for $k < n$ and a surjection for $k = n$.
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<document source="https://tamnd.github.io/math2/node/R6PWCDFEVR.md" tag="R6PWCDFEVR" kind="definition">
# Tetrahedron

## Definition

Let $A$, $B$, $C$, $D$ be four points of the [[Euclidean space|0XPR1E85W0]] $\R^3$ that do not lie in one [[plane|0V3XAMF3C1]].
The *tetrahedron* $ABCD$ is the [[convex hull|0FQSDC2EQ2]] of $\set{A, B, C, D}$.
Its vertices are the four points, its edges are the six segments between them, and its faces are the four triangles on three of them.

## Remarks

The four points do not lie in one plane exactly when the vectors $B - A$, $C - A$, $D - A$ are linearly independent.
Then the volume of $ABCD$ is
$$V = \frac{1}{6} \abs{\det(B - A, C - A, D - A)}.$$
A tetrahedron is a 3-dimensional [[simplex|YX1DJK2QAG]], and its four vertices always lie on one [[circumsphere|MRC4WMVTAH]].
It is regular when all six edges have the same length; the regular tetrahedron is a [[Platonic solid|017E5PRS1Q]].

Example: $(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)$.
The determinant of the three standard basis vectors is $1$, so the points are not coplanar and the volume is $1/6$.

Non-example: $(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0)$ all lie in the plane $z = 0$.
Their convex hull is a flat unit square, not a tetrahedron.
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<document source="https://tamnd.github.io/math2/node/R6QVS6P21N.md" tag="R6QVS6P21N" kind="concept">
# Fixed point

## Summary

A fixed point of a map is a point that the map sends to itself.
In a dynamical system, a fixed point is a state that never changes, the discrete-time analogue of an [[equilibrium point|0N8NGKDZDZ]].
Many problems, such as solving an equation $g(x) = 0$, can be rewritten as finding a fixed point.
Theorems such as the [[Banach fixed point theorem|0VK6VEH345]] guarantee that one exists.
Near a fixed point, the behaviour of the iterates tells whether nearby states are attracted or repelled.

## Explanations

### L1 human

A fixed point is a point that a map leaves where it is.
For $f(x) = x^2$ on the real line, the fixed points are the solutions of $x^2 = x$, namely $0$ and $1$.
On a graph, the fixed points are where the graph of $f$ meets the diagonal line $y = x$.
Stir a cup of coffee and let it settle.
If the stirring is a continuous map of the disc to itself, at least one point of the coffee ends where it started.
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<document source="https://tamnd.github.io/math2/node/R6RBQHW191.md" tag="R6RBQHW191" kind="concept">
# Unimodular matrix

## Summary

A unimodular matrix is a square matrix with integer entries and [[determinant|00KDZRW4C0]] $+1$ or $-1$.
These are exactly the integer matrices whose inverse also has integer entries.
They form the group $\mathrm{GL}_n(\Z)$, the [[general linear group|0FCHE902D7]] over the integers.
Unimodular matrices describe the changes of basis of the lattice $\Z^n$, and they appear in the Smith and Hermite normal forms.

## Explanations

### L1 human

For real matrices, any nonzero determinant allows an inverse.
For integer matrices, an inverse with integer entries needs more: the determinant must be $1$ or $-1$.
For example, $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ has determinant $1$ and inverse $\begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$.
Such a matrix maps the grid of integer points of the plane onto itself, one to one.
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<document source="https://tamnd.github.io/math2/node/R72NAFZ722.md" tag="R72NAFZ722" kind="concept">
# Seeley–DeWitt coefficients

## Summary

The Seeley–DeWitt coefficients are the coefficients $a_k(x)$ of the small-time asymptotic expansion of the heat kernel of a Laplace-type operator on the diagonal.
They are local invariants: each is a universal polynomial in the curvature, the endomorphism part of the operator, and their covariant derivatives.
Their integrals are spectral invariants, and in index theory and quantum field theory they compute indices, anomalies and divergences.
For the scalar Laplacian they are the coefficients of the [[Minakshisundaram–Pleijel expansion|45P2SEGGR1]].

## Explanations

### L1 human

Heat spreading for a short time on a curved space behaves almost as in flat space.
The deviation is a series in powers of the time, and its coefficients are the Seeley–DeWitt coefficients.
The first one is $1$, the second involves the scalar curvature, and later ones involve more and more curvature terms.
Physicists use them to read off the short-distance behaviour of quantum fields on curved spacetimes.
</document>

<document source="https://tamnd.github.io/math2/node/R73WRFDXRY.md" tag="R73WRFDXRY" kind="concept">
# Clifford–Klein form

## Summary

Let $G/H$ be a [[homogeneous space|85TWR6E75P]] of a [[Lie group|7JBRGCVJRB]] $G$.
A Clifford–Klein form is a quotient $\Gamma \backslash G / H$ by a [[discrete subgroup|5M7D1BSHVZ]] $\Gamma$ that acts [[properly discontinuously|J5X5ED0B2Y]].
When $\Gamma$ is torsion-free, the quotient is a manifold that looks locally like $G/H$.
Compact [[space forms|1VM1AQT956]] such as flat tori and closed hyperbolic manifolds are the classical examples.
A central question is which homogeneous spaces admit compact Clifford–Klein forms.

## Explanations

### L1 human

Start with a very symmetric space, such as the plane or the hyperbolic plane.
Pick a discrete group of its symmetries that moves points far enough apart, and glue each point to all its images.
The glued space keeps the local geometry of the original space but can be compact.
For example, gluing the plane $\R^2$ along the translations by vectors of $\Z^2$ gives a flat torus.
Such glued spaces are called Clifford–Klein forms of the original space.
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<document source="https://tamnd.github.io/math2/node/R76EC56V9R.md" tag="R76EC56V9R" kind="concept">
# Initial algebra

## Summary

An initial algebra of an endofunctor $F$ is an [[F-algebra|4RB8834SEJ]] from which there is exactly one homomorphism to every other $F$-algebra.
It is unique up to unique isomorphism, and by [[Lambek's lemma|S9FVESJ0CS]] its structure map is an isomorphism, so it is a fixed point of $F$.
Initial algebras give a uniform account of inductive definitions and of definitions by recursion.
The unique homomorphism out of the initial algebra is the function defined by structural recursion.
For a signature, the initial algebra is the term algebra, which is the basis of [[initial algebra semantics|HVF403PQ4H]] for abstract data types.

## Explanations

### L1 human

The natural numbers are built from $0$ by repeatedly adding $1$, and nothing else.
That is why a function on them can be defined by giving its value at $0$ and a rule for passing from $n$ to $n + 1$.
Initiality is the abstract form of this fact.
Lists, finite trees and terms are initial algebras of other functors, and the same recursion principle holds for them.
</document>

<document source="https://tamnd.github.io/math2/node/R7E5A4W6FJ.md" tag="R7E5A4W6FJ" kind="statement">
# Ruzsa triangle inequality

## Statement

Let $G$ be an [[abelian group|082GHNPJ53]], and let $A$, $B$ and $C$ be finite subsets of $G$.
Then the difference sets satisfy
$$\abs{A} \, \abs{B - C} \le \abs{A - B} \, \abs{A - C}.$$

## Remarks

The proof is an injection from $A \times (B - C)$ into $(A - B) \times (A - C)$.
For each $d \in B - C$, fix one pair $(b_d, c_d)$ with $d = b_d - c_d$.
Send $(a, d)$ to $(a - b_d, \ a - c_d)$.
The difference of the two coordinates is $d$, which recovers $b_d$ and $c_d$, and then $a$.
The name comes from the distance $d(A, B) = \log \big( \abs{A - B} / \sqrt{\abs{A}\abs{B}} \big)$ on nonempty finite sets.
The inequality is exactly $d(B, C) \le d(B, A) + d(A, C)$.
This $d$ is not a metric, since $d(A, A) > 0$ unless $A$ is a coset of a finite subgroup.
For example, with $A = B = C = \set{0, 1}$ in $\Z$, the inequality reads $2 \cdot 3 \le 3 \cdot 3$.

## Explanations

### L1 human

The size of a difference set measures how well two sets fit together under subtraction.
The inequality says that if $A$ fits well with $B$ and with $C$, then $B$ and $C$ fit well with each other.
It behaves like the triangle inequality for distances.

### L2 human

The Ruzsa triangle inequality is one of the first tools of [[additive combinatorics|3NM362A2JG]].
Together with the [[Plünnecke–Ruzsa inequality|EBEVWSRD2E]], it lets one pass between bounds on $A + A$, $A - A$ and longer sums.
The same injection proves a version in nonabelian groups, with products and inverses in place of sums and differences.
</document>

<document source="https://tamnd.github.io/math2/node/R7FTZ813DA.md" tag="R7FTZ813DA" kind="definition">
# Borel subgroup

## Definition

Let $k$ be an algebraically closed field, and let $G$ be a [[linear algebraic group|GA81VVQHEP]] over $k$.
A *Borel subgroup* of $G$ is a subgroup $B \subseteq G$ such that:

1. $B$ is closed in the [[Zariski topology|8SQK7RTB63]];
2. $B$ is connected and [[solvable|0FXJ7YFMMX]];
3. $B$ is maximal among the subgroups with properties 1 and 2.

## Remarks

Over an arbitrary field $k$, a closed subgroup $B$ of $G$ is called a Borel subgroup if $B \times_k \bar{k}$ is a Borel subgroup of $G \times_k \bar{k}$.

Example: in $G = \mathrm{GL}_n$ the group $B$ of invertible upper triangular matrices is a Borel subgroup.
It is closed, cut out by $x_{ij} = 0$ for $i > j$.
It is connected and solvable, since its commutator subgroup consists of unipotent upper triangular matrices.
It is maximal: by the Lie–Kolchin theorem, a connected solvable subgroup $B' \supseteq B$ fixes some full flag, and the only full flag fixed by $B$ is the standard one, so $B' \subseteq B$.

Non-example: in $\mathrm{GL}_2$, the diagonal torus $T$ is closed, connected and solvable, but it fails clause 3, since $T \subsetneq B$ for the upper triangular group $B$.

All Borel subgroups of $G$ are conjugate, and $G / B$ is a projective variety.
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<document source="https://tamnd.github.io/math2/node/R7G4ZC0963.md" tag="R7G4ZC0963" kind="definition">
# Localization of a space

## Definition

Let $P$ be a set of [[prime numbers|01826KMG2H]].
Let $\Z_{(P)}$ be the [[localization|KET2GSPZFG]] of $\Z$ at $P$: the rational numbers whose denominators are divisible by no prime in $P$.
Let $X$ be a simply connected [[CW complex|5W95QFHXPR]].
The space $X$ is *$P$-local* if each [[homotopy group|P976WYEFMB]] $\pi_n(X)$, $n \ge 2$, is a $\Z_{(P)}$-module, that is, multiplication by every prime $q \notin P$ is bijective on it.
A *$P$-localization* of $X$ is a map $f \colon X \to X_P$ to a simply connected $P$-local space such that $f$ induces isomorphisms
$$\pi_n(X) \otimes \Z_{(P)} \to \pi_n(X_P) \quad \text{for all } n \ge 2.$$
For $P = \emptyset$ one has $\Z_{(P)} = \Q$, and $f$ is called a *rationalization*, written $X \to X_\Q$.

## Remarks

Every simply connected CW complex has a $P$-localization, unique up to homotopy equivalence under $X$.
The condition on homotopy groups is equivalent to: $f$ induces isomorphisms $\tilde H_n(X; \Z) \otimes \Z_{(P)} \to \tilde H_n(X_P; \Z)$ in reduced [[singular homology|M0Q5CFJ7ZH]] for all $n$.
The localization is universal: every map from $X$ to a $P$-local space factors through $f$, uniquely up to homotopy.
The theory extends to nilpotent spaces.

Example: for $n$ odd, the map $S^n \to K(\Q, n)$ that represents a generator of $H^n(S^n; \Q)$ is a rationalization.
Indeed, $\pi_k(S^n) \otimes \Q$ is $\Q$ for $k = n$ and $0$ otherwise, by [[Serre's finiteness theorem|Y40ZRJJNH0]], which matches the homotopy of $K(\Q, n)$.

Non-example: the identity of $S^2$ is not a rationalization, because $S^2$ is not $P$-local for $P = \emptyset$: $\pi_2(S^2) \cong \Z$ is not a $\Q$-vector space.
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# Unital magma

## Definition

Let $(M, \cdot)$ be a [[magma|9689SFPVE9]].
It is a *unital magma* if there is an element $e \in M$ with $e \cdot x = x$ and $x \cdot e = x$ for all $x \in M$.
Such an $e$ is an [[identity element|07WYQPB8E3]] of $M$.

## Remarks

The identity is unique: if $e$ and $e'$ are identities, then $e = e \cdot e' = e'$.

Example: the set $\set{e, a, b}$ with $e \cdot x = x \cdot e = x$ for all $x$ and $u \cdot v = e$ for $u, v \in \set{a, b}$.
Both identity equations hold by construction, so this is a unital magma.
It is not associative: $(a \cdot a) \cdot b = e \cdot b = b$, but $a \cdot (a \cdot b) = a \cdot e = a$.

Non-example: $(\Z, -)$ has no identity.
If $e$ were an identity, then $e - x = x$ for all $x \in \Z$, so $e = 2x$ for every $x$, which is impossible.

In Mathlib the class `MulOneClass` is a unital magma written multiplicatively.
</document>

<document source="https://tamnd.github.io/math2/node/R7RNMX78KZ.md" tag="R7RNMX78KZ" kind="concept">
# Kernel approximation in reproducing kernel Hilbert spaces

## Summary

Kernel approximation reconstructs a function from its values at finitely many points $x_j$.
It uses a combination $\sum_j c_j K(\cdot, x_j)$ of copies of a positive definite kernel $K$.
The natural setting is the [[reproducing kernel Hilbert space|4CK2GSRAGD]] $H_K$ of the kernel.
There the kernel interpolant is the element of smallest norm that matches the data.
Its pointwise error is bounded by a power function times the norm of the target.
It covers radial basis function interpolation and works in many variables and on scattered points.

## Explanations

### L1 human

Place a bump, given by the kernel, at each data point, and choose the heights of the bumps so that their sum passes through the data.
For the Gaussian kernel $K(x, y) = e^{-\abs{x - y}^2}$ and one data point $x_1 = 0$ with value $1$, the interpolant is just $e^{-x^2}$.
With more points one solves a linear system with the matrix of kernel values $K(x_i, x_j)$.
The reproducing kernel Hilbert space explains why this is a good choice.
Among all functions of the space that fit the data, this sum is the smoothest in the sense of the norm.
Adding a penalty to the fit gives kernel ridge regression and other methods of machine learning.
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<document source="https://tamnd.github.io/math2/node/R7SH3ECJRN.md" tag="R7SH3ECJRN" kind="definition">
# Ordered semigroup

## Definition

Let $(S, \cdot)$ be a [[semigroup|0V3CFG1KK9]] and let $\le$ be a [[partial order|011C29Q1S1]] on $S$.
The triple $(S, \cdot, \le)$ is an *ordered semigroup* if for all $a, b, c \in S$,
$$a \le b \implies ca \le cb \text{ and } ac \le bc.$$

## Remarks

Example: $(\N, +)$ with the usual order is an ordered semigroup, since $a \le b$ gives $c + a \le c + b$ and $a + c \le b + c$.
Non-example: $(\Z, \cdot)$ with the usual order is not an ordered semigroup.
Indeed $1 \le 2$, but $(-1) \cdot 1 = -1 > -2 = (-1) \cdot 2$.
An ordered semigroup with an identity element is an *ordered monoid*.
Some authors require the order to be total and call the general case a partially ordered semigroup.
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<document source="https://tamnd.github.io/math2/node/R7V3S9V891.md" tag="R7V3S9V891" kind="concept">
# Vertex of a polytope

## Summary

A vertex of a [[convex polytope|300K6CB1KH]] is a [[face|KY7SB854VH]] of dimension zero, that is, a corner point of the polytope.
Vertices are exactly the [[extreme points|3MSGFJM2NX]] of the polytope: the points that are not inside a segment between two other points of it.
A convex polytope is the [[convex hull|0FQSDC2EQ2]] of its vertices, and no smaller set of points has this hull.
So the vertices form the most economical description of a polytope, the V-description.

## Explanations

### L1 human

The vertices of a polytope are its corners.
A square has four, a cube has eight, a tetrahedron has four.
At a vertex you can tilt a flat plane so that it touches the solid at that point only.
If you know the corners, you know the whole solid: stretch a skin tightly around them and it gives the polytope back.
</document>

<document source="https://tamnd.github.io/math2/node/R7VKRV8AGQ.md" tag="R7VKRV8AGQ" kind="statement">
# Cartan's theorem B

## Statement

Let $X$ be a [[Stein space|Z4AMJS5KA6]] and let $\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.
Then the [[sheaf cohomology|PQ6Q143ARJ]] groups vanish in positive degree:
$$H^q(X, \mathcal{F}) = 0 \quad \text{for all } q \ge 1.$$

## Remarks

The converse holds: a complex space on which $H^1(X, \mathcal{F}) = 0$ for every coherent analytic sheaf $\mathcal{F}$ is Stein, so theorem B characterizes Stein spaces.
The proof first treats compact polydiscs and analytic polyhedra.
There, vanishing follows from Cousin-type gluing and from the local theory of coherent sheaves.
Then it passes to the limit over an exhaustion of $X$ by holomorphically convex compact sets, using approximation to control the inverse limit.
Example: on a Stein manifold, $H^1(X, \mathcal{O}) = 0$, so the first Cousin problem is always solvable.
Example: for an analytic subset $Y$ with ideal sheaf $\mathcal{I}$, the exact sequence $0 \to \mathcal{I} \to \mathcal{O}_X \to \mathcal{O}_X / \mathcal{I} \to 0$ and $H^1(X, \mathcal{I}) = 0$ show that every holomorphic function on $Y$ extends to a holomorphic function on $X$.
The Stein hypothesis is needed: on the projective line, $H^1(\mathbb{P}^1, \mathcal{O}(-2)) \cong \C$ is not zero; and on $\C^2 \setminus \set{0}$, the group $H^1(X, \mathcal{O})$ is infinite-dimensional.

## Explanations

### L1 human

Many problems in complex analysis ask to glue local holomorphic solutions into a global one.
The obstruction to such gluing lies in a cohomology group.
Theorem B says that on a Stein space these obstructions vanish, for all problems described by coherent sheaves.
So on Stein spaces, whatever can be solved locally can be solved globally.

### L2 human

Theorem B, with the exponential sequence, solves the Cousin problems: the additive one always, and the multiplicative one exactly when a topological obstruction in $H^2(X, \Z)$ vanishes.
It gives extension of holomorphic functions from closed analytic subsets, interpolation on discrete sets, and the embedding of Stein manifolds into $\C^N$.
It is the analytic analogue of Serre's vanishing theorem for quasi-coherent sheaves on affine schemes.
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# Abelian variety

## Summary

An abelian variety is a complete [[algebraic variety|HV6YBS0YDH]] that is also an [[algebraic group|GH81SYAPPN]], with group operations given by morphisms.
Its group law is automatically commutative, and the variety is automatically projective.
Abelian varieties of dimension one are the [[elliptic curves|427N6S6QQW]], and the [[Jacobian variety|ZQ84M9D9TC]] of a curve is a basic example in higher dimension.
They are central in arithmetic geometry, for instance in the [[Mordell–Weil theorem|MXRCPAH09H]].

## Explanations

### L1 human

An abelian variety is a shape cut out by polynomial equations whose points can be added like numbers.
Over the complex numbers it looks like a doughnut in several complex dimensions, a [[complex torus|Y7MA6KBE9M]].
The simplest example is an elliptic curve such as $y^2 = x^3 - x$, together with its point at infinity.
Two points on it are added by drawing the line through them and taking the third intersection point, reflected.
Because the shape is closed up with no missing points, the group law is very rigid, and this rigidity makes the theory rich.
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<document source="https://tamnd.github.io/math2/node/R7WFJGJBV4.md" tag="R7WFJGJBV4" kind="concept">
# Willmore energy

## Summary

The Willmore energy of a closed surface in space is the integral of the square of its mean curvature over the surface.
It measures how far the surface is from being round: it is at least $4\pi$, with equality only for round spheres.
It does not change under scaling, and more generally under Möbius transformations of space.
It appears in the study of elastic membranes, such as cell membranes, where bending costs energy.

## Explanations

### L1 human

Think of a closed surface as a thin elastic shell; bending it costs energy.
The Willmore energy adds up, over the whole surface, the square of how much it bends on average at each point.
A round ball is the most relaxed shape, with energy $4\pi$ whatever its size.
A long thin sausage or a wrinkled surface has much larger energy.
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<document source="https://tamnd.github.io/math2/node/R8071FWJRK.md" tag="R8071FWJRK" kind="concept">
# Convex hull algorithms

## Summary

Convex hull algorithms take a finite set of $n$ points in $\R^d$ and compute its [[convex hull|0FQSDC2EQ2]].
The output is usually the list of its vertices or of its facets.
In the plane, Graham scan and other methods run in $O(n \log n)$ time.
Gift wrapping runs in $O(n h)$ time, where $h$ is the number of hull vertices.
Chan's algorithm combines both ideas and runs in $O(n \log h)$ time.
Since sorting reduces to the planar convex hull problem, $\Omega(n \log n)$ operations are needed in the worst case in the algebraic decision tree model.
These algorithms are a basic building block of [[computational geometry|KAT7K2NE9Z]].

## Explanations

### L1 human

Hammer nails into a board at some points and stretch a rubber band around all of them; the band takes the shape of the convex hull.
A convex hull algorithm finds which nails the band touches.
Gift wrapping starts at the leftmost nail and repeatedly turns to the nail that keeps all others on one side, like wrapping a present.
For the points $(0,0)$, $(2,0)$, $(1,1)$, $(1,3)$, the hull is the triangle with corners $(0,0)$, $(2,0)$, $(1,3)$, and $(1,1)$ lies inside it.
</document>

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# Liouvillian function

## Definition

Let $K$ be a [[differential field|T43TQ817XC]] of characteristic $0$ with derivation $a \mapsto a'$.
Let $L \supseteq K$ be a differential field extension with the same field of constants as $K$.
It is *Liouvillian* if there is a tower of differential fields $K = K_0 \subseteq \dots \subseteq K_m = L$ with the following property.
For each $i$, $K_i = K_{i-1}(t_i)$, where $t_i$ satisfies one of these conditions:

1. $t_i$ is algebraic over $K_{i-1}$;
2. $t_i' \in K_{i-1}$, so $t_i$ is an integral of an element of $K_{i-1}$;
3. $t_i \ne 0$ and $t_i'/t_i \in K_{i-1}$, so $t_i$ is an exponential of an integral of an element of $K_{i-1}$.

Let $\C(x)$ be the field of rational functions with the derivation $d/dx$.
A *Liouvillian function* is an element of a Liouvillian extension of $\C(x)$, realized as a field of meromorphic functions on some domain.

## Remarks

Example: $t = e^{x^2}$ is Liouvillian over $\C(x)$, since $t'/t = 2x \in \C(x)$ (clause 3).
An antiderivative of $e^{-x^2}$ is Liouvillian too: adjoin $e^{-x^2}$ by clause 3 and then the integral by clause 2.
Also $\ln x$ is Liouvillian by clause 2, since $(\ln x)' = 1/x$, and $\sin x = (e^{ix} - e^{-ix})/(2i)$ is Liouvillian.
Non-example: no nonzero solution of the Airy equation $y'' = x y$ is Liouvillian; this is a result of differential Galois theory, since the differential Galois group of that equation is $\mathrm{SL}_2(\C)$.
Elementary functions are the special case in which clause 2 is restricted to logarithms and clause 3 to exponentials.
So every elementary function is Liouvillian, but $\int e^{-x^2} dx$ is Liouvillian and not elementary, by [[Liouville's theorem|8VRTPECBPZ]].
</document>

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# Cylinder set measure

## Definition

Let $E$ be a real [[topological vector space|T32W553W95]].
For $n \ge 0$, let $\mathcal{T}_n$ be the set of continuous [[linear maps|0NRYHJPQG2]] $T \colon E \to \R^n$ that are surjective, and let $\mathcal{T} = \bigcup_n \mathcal{T}_n$.
A *cylinder set measure* on $E$ is a family $(\mu_T)_{T \in \mathcal{T}}$ with these two properties.

1. For each $T \in \mathcal{T}_n$, $\mu_T$ is a [[probability measure|059TJN0NZ1]] on the [[Borel σ-algebra|0WQXY32Q8D]] of $\R^n$.
2. If $S \in \mathcal{T}_m$, $T \in \mathcal{T}_n$ and $\pi \colon \R^m \to \R^n$ is linear with $T = \pi \circ S$, then $\mu_T$ is the [[pushforward|3N66TM3TSQ]] $\pi_* \mu_S$.

## Remarks

A *cylinder set* is a set $T^{-1}(B)$ with $T \in \mathcal{T}_n$ and $B \subseteq \R^n$ Borel.
Condition 2 makes the rule $\mu(T^{-1}(B)) = \mu_T(B)$ well defined, and $\mu$ is a [[finitely additive measure|KFWQ24D9M6]] on the algebra of cylinder sets.

Example: a Borel probability measure $\nu$ on $E$ gives the cylinder set measure $\mu_T = T_* \nu$.
Condition 2 holds since $\pi_*(S_* \nu) = (\pi \circ S)_* \nu$.

Example: let $E = H$ be a real Hilbert space and let $\mu_T$ be the centered normal law on $\R^n$ with covariance matrix $T T^*$.
Condition 2 holds, because $\pi_*$ of the centered normal law with covariance $S S^*$ is centered normal with covariance $\pi S S^* \pi^*$.
This is the canonical Gaussian cylinder set measure on $H$.

Non-example: on $E = \R^2$, let every $\mu_T$ with $T \in \mathcal{T}_1$ be the standard [[normal distribution|05AQE0QR0D]] $N(0, 1)$.
For $T(x) = x_1$ and $T'(x) = 2 x_1$, one has $T' = \pi \circ T$ with $\pi(t) = 2t$, and $\pi_* N(0, 1) = N(0, 4) \ne N(0, 1)$, so condition 2 fails.

## Explanations

### L1 human

A cylinder set measure gives the law of every finite set of linear readings of a random point, in a consistent way.
It is a measure that is only known on sets described by finitely many coordinates.
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# Self-similar measure

## Summary

A self-similar measure is a [[probability measure|059TJN0NZ1]] on $\R^n$ that is a weighted average of its own images under finitely many contracting similarities.
For given similarities and weights there is exactly one such measure, by a theorem of Hutchinson.
Its support is the self-similar fractal, or attractor, of the similarities.
Self-similar measures are the basic examples of fractal measures, and their dimensions and Fourier decay are studied in detail.

## Explanations

### L1 human

Take a unit of sand on the line and copy it in two shrunken versions, each with half of the sand.
Put one copy in the left third and one in the right third, and repeat this forever.
The limit distribution of the sand is the Cantor measure, which lives on the middle-thirds Cantor set.
It is self-similar: its part on the left third and its part on the right third each look like the whole measure, shrunk by $3$ and holding half the mass.
</document>

<document source="https://tamnd.github.io/math2/node/R8DMMW0E9M.md" tag="R8DMMW0E9M" kind="concept">
# Hilbert manifold

## Summary

A Hilbert manifold is a [[Banach manifold|9K15AMGBV9]] whose charts take values in [[Hilbert spaces|HM2HHE3J51]].
Inner products give partitions of unity, orthogonal complements and Riemannian metrics, so much of finite-dimensional differential topology carries over.
Spaces of paths and loops of Sobolev class $H^1$ in a finite-dimensional manifold are Hilbert manifolds.
This makes them the natural setting for Morse theory in infinite dimensions.
Take an infinite-dimensional separable model and a separable metrizable manifold.
Then the [[open embedding theorem|JWCDDMJE3K]] shows that it is an open subset of its model space, up to diffeomorphism.

## Explanations

### L1 human

A Hilbert manifold is built from patches that look like pieces of a Hilbert space, glued by smooth maps.
A Hilbert space is a possibly infinite-dimensional space in which lengths and angles make sense, such as the space $\ell^2$ of square-summable sequences.
The unit sphere of $\ell^2$, the set of sequences with $\sum x_n^2 = 1$, is a Hilbert manifold.
Near each of its points it looks like the hyperplane orthogonal to that point.
</document>

<document source="https://tamnd.github.io/math2/node/R8GZAJ85YF.md" tag="R8GZAJ85YF" kind="definition">
# Complex analytic space

## Definition

A *local model* is built as follows.
Let $D \subseteq \C^n$ be open, let $\mathcal{O}_D$ be the [[sheaf|TB6TM6R4MD]] of [[holomorphic functions|507C34W0ZT]] on $D$, and let $f_1, \dots, f_k$ be holomorphic functions on $D$.
Let $\mathcal{I} \subseteq \mathcal{O}_D$ be the [[ideal sheaf|N3PGYJJQ7W]] generated by $f_1, \dots, f_k$, and let $Z = \set{z \in D : f_1(z) = \dots = f_k(z) = 0}$.
The local model is the ringed space $(Z, (\mathcal{O}_D / \mathcal{I})|_Z)$.

A *complex analytic space* is a pair $(X, \mathcal{O}_X)$ such that:

1. $X$ is a [[Hausdorff space|0S1FJ8VEX9]];
2. $\mathcal{O}_X$ is a sheaf of $\C$-algebras on $X$;
3. each point of $X$ has an open neighbourhood $U$ such that $(U, \mathcal{O}_X|_U)$ is isomorphic to a local model as a $\C$-ringed space.

## Remarks

Example: $\C^n$ with its sheaf of holomorphic functions is a complex space; take $D = \C^n$ and $k = 0$.
The double point is the local model with $D = \C$ and $f_1 = z^2$: the underlying space is $\set{0}$ and the ring of functions is $\C[z]/(z^2)$, which contains the nilpotent $z$.

Non-example: $\C$ with the sheaf of continuous complex functions satisfies clauses 1 and 2 but not 3.
The stalks of a local model are quotients of the [[ring of convergent power series|EVPZ7ZG1HB]], hence [[Noetherian|0670T59HGE]].
Let $\mathfrak{m}$ be the maximal ideal of the ring of germs of continuous functions at $0$.
Then $\mathfrak{m} = \mathfrak{m}^2$, since $f = \sqrt{\abs{f}} \cdot (f / \sqrt{\abs{f}})$.
By Nakayama's lemma $\mathfrak{m}$ is not finitely generated, so this ring is not Noetherian.

A complex space is a [[locally ringed space|B7FPT8JTXR]].
Morphisms of complex spaces, also called holomorphic maps, are morphisms of locally ringed spaces of $\C$-algebras.
Some authors drop the Hausdorff condition.
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# Hasse–Minkowski theorem

## Statement

Let $Q$ be a nondegenerate [[quadratic form|0RBQS1JJCR]] in $n$ variables with coefficients in $\Q$.
For a prime $p$, let $\Q_p$ be the field of [[p-adic numbers|BT6T702C28]].
Then $Q$ has a zero $x \in \Q^n$ with $x \ne 0$ if and only if both of these hold:

1. $Q$ has a nonzero zero in $\R^n$.
2. $Q$ has a nonzero zero in $\Q_p^n$ for every prime $p$.

## Remarks

The direction "only if" is clear; the content is that local zeros give a global zero.
The proof goes by the number of variables, with the [[Hilbert symbol|WFG4YMBYVQ]] as the main tool.
The case $n = 3$ is essentially Legendre's theorem on $ax^2 + by^2 + cz^2 = 0$; the case $n = 4$ uses [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]], and the case $n \ge 5$ reduces to $n = 4$.
The theorem holds over every number field, with its completions in place of $\R$ and the $\Q_p$.
Example: $x^2 + y^2 - 3z^2$ has the real zero $(\sqrt{3}, 0, 1)$, but no nonzero zero in $\Q_3^3$; hence it has no nonzero rational zero.
A consequence: a nonzero $a \in \Q$ is represented by $Q$ over $\Q$ if and only if it is represented over $\R$ and over every $\Q_p$.
Another: two quadratic forms over $\Q$ are equivalent over $\Q$ if and only if they are equivalent over $\R$ and over every $\Q_p$.
The restriction to quadratic forms is necessary: Selmer's cubic $3x^3 + 4y^3 + 5z^3$ has nonzero zeros over $\R$ and every $\Q_p$, but none over $\Q$.

## Explanations

### L1 human

To decide whether a quadratic equation such as $ax^2 + by^2 + cz^2 = 0$ has a solution in fractions, it is enough to check real numbers and the $p$-adic numbers for each prime.
Each such check is a finite computation, and only finitely many primes need real work.
So the theorem turns an infinite search into a finite test.

### L2 human

This is the main example of the [[Hasse principle|EMAR4X3W8D]].
In practice one computes Hilbert symbols $(a, b)_p$ at the primes dividing $2$ and the coefficients; at all other primes a form of rank at least $3$ is automatically isotropic.
The theorem is the basis of the classification of quadratic forms over $\Q$ by discriminant, signature and Hasse invariants.
For cubic and higher degree equations the principle can fail, and the study of the failure leads to the Tate–Shafarevich group of elliptic curves.
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<document source="https://tamnd.github.io/math2/node/R8MZAZ6GCR.md" tag="R8MZAZ6GCR" kind="statement">
# Frobenius theorem on Cesàro and Abel summability

## Statement

Let $\sum_{n \ge 0} a_n$ be a series of complex numbers that is [[Cesàro summable|N8ZGG7EW6J]] to $s$, that is, $\sigma_n = \frac{s_0 + \cdots + s_n}{n + 1} \to s$, where $s_n = a_0 + \cdots + a_n$.
Then the [[power series|0VP56B4EMX]] $f(x) = \sum_{n \ge 0} a_n x^n$ converges for $\abs{x} < 1$, and $f(x) \to s$ as $x \to 1^-$.
So the series is [[Abel summable|EZXPY4A19C]] to $s$.

## Remarks

The sequence $(\sigma_n)$ converges, so it is bounded, and then $s_n = (n+1)\sigma_n - n\sigma_{n-1}$ and $a_n = s_n - s_{n-1}$ are $O(n)$; hence the power series converges for $\abs{x} < 1$.
Multiplying by $\frac{1}{1-x}$ twice gives, for $\abs{x} < 1$,
$$f(x) = (1 - x)^2 \sum_{n \ge 0} (n + 1) \sigma_n x^n.$$
Since $(1 - x)^2 \sum_{n \ge 0} (n+1) x^n = 1$, one gets $f(x) - s = (1 - x)^2 \sum_{n \ge 0} (n + 1)(\sigma_n - s) x^n$.
Splitting the sum at a large $N$ where $\abs{\sigma_n - s} < \varepsilon$ for $n > N$ shows that this tends to $0$ as $x \to 1^-$.

Example: $1 - 1 + 1 - \cdots$ is Cesàro summable to $\frac12$, and indeed $\sum (-1)^n x^n = \frac{1}{1+x} \to \frac12$.

The converse fails: $1 - 2 + 3 - 4 + \cdots$ is Abel summable to $\frac14$, since $\sum (-1)^n (n+1) x^n = \frac{1}{(1+x)^2}$, but it is not Cesàro summable of order $1$.

## Explanations

### L1 human

Cesàro summation averages the partial sums; Abel summation damps the terms by powers of $x$ and lets $x \to 1$.
The theorem says that the second method is at least as strong as the first and gives the same answer.
Every series that averaging can sum, damping can sum too, to the same value.

### L2 human

This is an Abelian theorem in the sense of summability: it compares two methods and shows an inclusion with consistency.
It is a strengthening of [[Abel's theorem|0S4EMDQS2F]], because every convergent series is Cesàro summable to its sum.
The same argument shows that Cesàro summability of any order $k$ implies Abel summability to the same value.
</document>

<document source="https://tamnd.github.io/math2/node/R8N0CCDJZD.md" tag="R8N0CCDJZD" kind="definition">
# Likelihood-ratio test

## Definition

Let $X$ be data with [[likelihood function|05FSTXM92W]] $L(\theta \mid x)$ for $\theta$ in a parameter set $\Theta$, and let $\Theta_0 \subset \Theta$.
For data $x$ with $\sup_{\theta \in \Theta} L(\theta \mid x) > 0$, the *likelihood-ratio statistic* is
$$\lambda(x) = \frac{\sup_{\theta \in \Theta_0} L(\theta \mid x)}{\sup_{\theta \in \Theta} L(\theta \mid x)}.$$
The *likelihood-ratio test* of $H_0 \colon \theta \in \Theta_0$ against $H_1 \colon \theta \in \Theta \setminus \Theta_0$ rejects $H_0$ when $\lambda(x) \le c$.
The constant $c \in [0, 1]$ is chosen so that $\sup_{\theta \in \Theta_0} P_\theta(\lambda(X) \le c) \le \alpha$, for a given level $\alpha$.

## Remarks

Since $\Theta_0 \subset \Theta$, one has $0 \le \lambda(x) \le 1$.
The two suprema are the likelihoods at the restricted and the unrestricted [[maximum likelihood estimates|0FREPFKNS7]], when these exist.
One often uses $-2 \log \lambda(X)$, which is large when $\lambda$ is small.
Example: for $X_1, \dots, X_n$ i.i.d. $\mathcal{N}(\mu, 1)$ and $\Theta_0 = \set{0}$, one finds $-2 \log \lambda = n \bar X^2$.
So the test rejects when $\abs{\sqrt{n}\, \bar X}$ is large, which is the usual two-sided z-test.
For two simple hypotheses, the test is the most powerful test of the [[Neyman–Pearson lemma|0EE19CFEEF]].
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# Linking number

## Definition

Let $K_1$ and $K_2$ be disjoint oriented closed curves in $\R^3$, each the image of a smooth or piecewise-linear embedding of the circle.
The group $H_1(\R^3 \setminus K_2; \Z)$ of [[singular homology|M0Q5CFJ7ZH]] is infinite cyclic.
It has a preferred generator $\mu$, the class of a small circle that bounds a disc meeting $K_2$ once.
This circle is oriented by the right-hand rule with respect to the orientation of $K_2$.
The *linking number* $\operatorname{lk}(K_1, K_2)$ is the integer $m$ such that the class of $K_1$ in $H_1(\R^3 \setminus K_2; \Z)$ equals $m \mu$.

## Remarks

Equivalently, take a regular diagram of the [[link|1A5WRCNTM3]] $K_1 \cup K_2$ and give each crossing between $K_1$ and $K_2$ the sign $+1$ or $-1$ by the right-hand rule.
Then $\operatorname{lk}(K_1, K_2)$ is half the sum of these signs.
In this form one sees that $\operatorname{lk}(K_1, K_2) = \operatorname{lk}(K_2, K_1)$.

Example: in the Hopf link, the two circles cross twice in a standard diagram, and both crossings have the same sign.
So the linking number is $\pm 1$, the sign depending on the orientations.

Non-example of linking: in the unlink of two circles, the diagram has no crossings between the components, so the linking number is $0$.
The converse fails: the Whitehead link has linking number $0$, but its two components cannot be pulled apart.

For smooth curves $\gamma_1, \gamma_2$ parametrising $K_1, K_2$, the Gauss integral gives
$$\operatorname{lk}(K_1, K_2) = \frac{1}{4\pi} \oint_{\gamma_1} \oint_{\gamma_2} \frac{(\gamma_1 - \gamma_2) \cdot (d\gamma_1 \times d\gamma_2)}{\abs{\gamma_1 - \gamma_2}^3}.$$
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# Cardinal characteristics of the continuum

## Definition

Let $\omega^\omega$ be the set of [[functions|0XHDTE5YMZ]] from $\N$ to $\N$.
For $f, g \in \omega^\omega$, write $f \le^* g$ if $f(n) \le g(n)$ for all but finitely many $n$.
Let $\mathcal{M}$ be the family of [[meagre|0FQPYAY9PJ]] subsets of $\R$.
A cardinal characteristic of the continuum is an uncountable [[cardinal|0QKPNZKNEA]] $\kappa \le 2^{\aleph_0}$ of the following kind.
It is the least size of a family of reals, or of sets of reals, with a given property.
The standard examples include:

1. The bounding number $\mathfrak{b}$ is the least size of a set $F \subseteq \omega^\omega$ with no $g$ such that $f \le^* g$ for all $f \in F$.
2. The dominating number $\mathfrak{d}$ is the least size of a set $F \subseteq \omega^\omega$ such that each $g \in \omega^\omega$ has $g \le^* f$ for some $f \in F$.
3. The additivity $\operatorname{add}(\mathcal{M})$ is the least size of a subfamily of $\mathcal{M}$ whose union is not meagre.
4. The covering number $\operatorname{cov}(\mathcal{M})$ is the least size of a subfamily of $\mathcal{M}$ whose union is $\R$.
5. The uniformity $\operatorname{non}(\mathcal{M})$ is the least size of a subset of $\R$ that is not meagre.

## Remarks

Example: $\aleph_1 \le \mathfrak{b} \le \mathfrak{d} \le \mathfrak{c}$, where $\mathfrak{c} = 2^{\aleph_0}$ is the [[cardinality of the continuum|0WC28W9DNM]].
Check: for countably many $f_k$, the function $g(n) = \max_{k \le n} f_k(n)$ bounds them all, so $\mathfrak{b} > \aleph_0$.
Each dominating family is unbounded, which gives $\mathfrak{b} \le \mathfrak{d}$, and $\omega^\omega$ itself dominates, so $\mathfrak{d} \le \mathfrak{c}$.
Non-example: the least size of a family of meagre sets whose union is meagre is $1$.
It is defined in the same style, but it is not uncountable, so it is not a cardinal characteristic.
The same three definitions with the Lebesgue null sets in place of $\mathcal{M}$ give three more cardinals.
Together with $\mathfrak{b}$, $\mathfrak{d}$ and the cofinalities of the two ideals, they make up the ten cardinals of Cichoń's diagram.
Under the continuum hypothesis all these cardinals equal $\aleph_1$.
Martin's axiom implies that they all equal $\mathfrak{c}$.
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# Floating-point arithmetic

## Definition

Fix [[integers|0C4JS9RF5M]] $\beta \ge 2$ (the *base*), $p \ge 1$ (the *precision*), and $e_{\min} < e_{\max}$.
A *floating-point number* of this system is $0$ or a real number of the form
$$x = \pm\, (d_0 . d_1 d_2 \ldots d_{p-1})_\beta \times \beta^{e} = \pm \left( \sum_{i=0}^{p-1} d_i \beta^{-i} \right) \beta^{e}.$$
Here the digits $d_i$ are integers with $0 \le d_i \le \beta - 1$, and the exponent $e$ is an integer with $e_{\min} \le e \le e_{\max}$.
The number is *normalized* if $d_0 \ne 0$.
Let $F$ be the set of these numbers.
Let $\mathrm{fl}$ be a [[rounding|5NPVX29Y0C]] to $F$ of the real numbers between $\min F$ and $\max F$, usually rounding to nearest.
*Floating-point arithmetic* replaces each operation $\circ \in \set{+, -, \times, /}$ on $x, y \in F$, whenever $x \circ y$ lies in this range, by
$$x \circledast y = \mathrm{fl}(x \circ y),$$
that is, the exact result rounded to $F$.

## Remarks

Example: with $\beta = 10$ and $p = 3$, the numbers $x = 1.23$ and $y = 4.56 \times 10^{-2}$ lie in $F$, and $x \oplus y = \mathrm{fl}(1.2756) = 1.28$ under rounding to nearest.
Non-example: with $\beta = 10$ and $p = 3$, the number $1.234$ is not in $F$, since it needs four significant digits.
Floating-point addition is commutative but not associative: with $\beta = 10$ and $p = 3$, $(1.00 \oplus 0.004) \oplus 0.004 = 1.00$, while $1.00 \oplus (0.004 \oplus 0.004) = 1.01$.
In the range of normalized numbers and with rounding to nearest, $x \circledast y = (x \circ y)(1 + \delta)$ with $\abs{\delta} \le \tfrac12 \beta^{1-p}$.
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# Zolotarev polynomials

## Summary

Zolotarev polynomials solve an extremal problem close to the one solved by [[Chebyshev polynomials|M6EAR1JHPP]].
Among the real polynomials of degree $n$ whose two leading coefficients are fixed, they have the least maximum absolute value on $[-1, 1]$.
Yegor Zolotarev studied them in the nineteenth century, and in general they are expressed through elliptic functions.

## Explanations

### L1 human

The Chebyshev problem fixes only the leading coefficient and asks for the polynomial that stays smallest on $[-1, 1]$.
The Zolotarev problem fixes the first two coefficients and asks the same question.
When the second coefficient is $0$, the answer is still the scaled Chebyshev polynomial $2^{1-n} T_n$.
When the second coefficient is large, the optimal polynomial still wiggles between two equal bounds, but its shape is more complicated.
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# Weierstrass excess function

## Summary

The Weierstrass excess function of a Lagrangian $L(x, y, p)$ is $E(x, y, p, q) = L(x, y, q) - L(x, y, p) - (q - p) \cdot L_p(x, y, p)$.
It measures how far the graph of $q \mapsto L(x, y, q)$ lies above its tangent at $q = p$.
It gives the Weierstrass necessary condition for a strong minimum, and through Hilbert's invariant integral it gives the classical sufficient condition for a strong minimum.

## Explanations

### L1 human

Fix a point and a slope $p$ of a curve, and compare the integrand at another slope $q$ with its tangent-line prediction from $p$.
The excess is the gap between the true value and the prediction.
If the integrand is convex in the slope, the gap is never negative.
For $L = p^2$ the excess is $q^2 - p^2 - 2p(q - p) = (q - p)^2$.
Along a strong minimiser the excess must be nonnegative; otherwise a short sudden switch to the slope $q$ would lower the integral.
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# Calderón problem

## Problem

Let $\Omega \subset \R^n$, $n \ge 2$, be a bounded connected open set with smooth boundary.
A conductivity is a measurable function $\gamma$ on $\Omega$ with $c \le \gamma \le c^{-1}$ for some constant $c > 0$.
Let $\Lambda_\gamma$ be the [[Dirichlet-to-Neumann operator|W783KDGT8E]] of the equation $\nabla \cdot (\gamma \nabla u) = 0$ in $\Omega$.
The problem has two parts:

1. Uniqueness: if $\Lambda_{\gamma_1} = \Lambda_{\gamma_2}$, does it follow that $\gamma_1 = \gamma_2$?
2. Reconstruction: give a method that recovers $\gamma$ from $\Lambda_\gamma$.

## Remarks

This is the model [[inverse problem|PCMW7KN5P4]] behind electrical impedance tomography, where voltages are applied on the surface of a body and the resulting currents are measured.
Calderón posed it in his 1980 paper "On an inverse boundary value problem", where he proved uniqueness for the linearized problem near constant conductivities.
Sylvester and Uhlmann proved uniqueness for smooth conductivities in dimension $n \ge 3$ in 1987, using complex geometrical optics solutions.
Astala and Päivärinta proved uniqueness for all bounded measurable conductivities in dimension $n = 2$ in 2006.
In dimension $n \ge 3$ uniqueness for general bounded measurable conductivities is open.
The problem is severely ill-posed: the stability of $\gamma$ with respect to $\Lambda_\gamma$ is only logarithmic in general.

## Explanations

### L1 human

Can one see inside a body using only electrical measurements made on its skin?
Each choice of voltage pattern on the surface gives a pattern of currents flowing out, and the full list of these pairs is the data.
The Calderón problem asks whether these data determine the electrical conductivity at every point inside, and how to compute it.
Different tissues conduct differently, so a positive answer gives an imaging method that uses no radiation.
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# Univalent function

## Summary

A univalent function is a holomorphic function that is injective on its domain.
It is a conformal map of its domain onto its image, which is again an open set.
The theory of univalent functions studies the class $S$ of univalent functions on the unit disc with $f(0) = 0$ and $f'(0) = 1$.
Its main results are distortion, covering and coefficient theorems, such as the [[Koebe quarter theorem|WX0HMW4GT7]] and the [[Bieberbach conjecture|ZMRFSYVWQT]].

## Explanations

### L1 human

A univalent function never takes the same value twice.
So it maps its domain onto a region without folding or overlapping, like a sheet of rubber stretched without tearing or doubling.
The map $z \mapsto z^2$ is not univalent on the unit disc, since $z$ and $-z$ have the same square.
On the right half-plane, $z \mapsto z^2$ is univalent, and its image is the plane minus the negative real axis and $0$.
</document>

<document source="https://tamnd.github.io/math2/node/R997ZAD141.md" tag="R997ZAD141" kind="statement">
# Schur complement method

## Statement

Let $A$ be an invertible real matrix.
Write it in [[block form|0ASDFEQS6P]] as
$$A = \begin{pmatrix} A_{II} & A_{I\Gamma} \\ A_{\Gamma I} & A_{\Gamma\Gamma} \end{pmatrix}, \qquad A_{II} = \begin{pmatrix} A_{11} & & \\ & \ddots & \\ & & A_{NN} \end{pmatrix}.$$
Here the unknowns are split into interior unknowns $u_I = (u_1, \dots, u_N)$ of $N$ subdomains and interface unknowns $u_\Gamma$.
Each block $A_{jj}$ is invertible.
Let $S = A_{\Gamma\Gamma} - A_{\Gamma I} A_{II}^{-1} A_{I\Gamma}$ be the [[Schur complement|6HWFBXYPZP]] of $A_{II}$ in $A$.
Then $(u_I, u_\Gamma)$ solves $A u = f$ with $f = (f_I, f_\Gamma)$ if and only if
$$S u_\Gamma = f_\Gamma - A_{\Gamma I} A_{II}^{-1} f_I \quad \text{and} \quad u_j = A_{jj}^{-1} \bigl( f_j - (A_{I\Gamma} u_\Gamma)_j \bigr) \text{ for } j = 1, \dots, N.$$
If $A$ is symmetric [[positive definite|0KX66CZMBB]], then $S$ is symmetric positive definite.

## Remarks

The equivalence is block Gaussian elimination: the first block row gives $u_I = A_{II}^{-1}(f_I - A_{I\Gamma} u_\Gamma)$, and substituting into the second block row gives the interface equation.
Since $A_{II}$ is block diagonal, every application of $A_{II}^{-1}$ splits into independent solves on the subdomains.
If $A$ is symmetric positive definite, then $v^T S v = w^T A w$ for $w = (-A_{II}^{-1} A_{I\Gamma} v, v)$, which is positive for $v \ne 0$.

In practice one does not form $S$.
One solves the interface system by the [[conjugate gradient method|AFXZGQB4EN]], and each product $S v$ costs one solve with each $A_{jj}$.
Take a finite element discretization of an elliptic problem with a fixed set of subdomains.
Then the condition number of $S$ grows like $h^{-1}$ in the mesh size $h$, against $h^{-2}$ for $A$.
Preconditioners for $S$, such as [[balancing domain decomposition|X8MCVW56QP]], reduce this further.

## Explanations

### L1 human

Cut the domain of a problem into pieces that touch only along thin interfaces.
If the values on the interfaces were known, each piece could be solved on its own.
The method first finds the interface values from a smaller system, the Schur complement system, and then fills in each piece independently.

### L2 human

This is the non-overlapping branch of [[domain decomposition methods|AN054RHT1K]], also called iterative substructuring.
The Schur complement acts on interface functions like a discrete Steklov–Poincaré operator, mapping Dirichlet data on the interfaces to the jump of the normal fluxes.
Its solution by a preconditioned Krylov method is the starting point of the Neumann–Neumann, BDD and FETI families.
</document>

<document source="https://tamnd.github.io/math2/node/R9JJ44TMQB.md" tag="R9JJ44TMQB" kind="statement">
# Cauchy–Fantappiè formula

## Statement

Let $n \ge 1$ and let $D \subseteq \C^n$ be a bounded domain with $C^1$ boundary $\partial D$, oriented as the boundary of $D$.
Fix $z \in D$.
Let $\eta = (\eta_1, \dots, \eta_n)$ be a map of class $C^1$ from a neighbourhood of $\partial D$ to $\C^n$ such that
$$\sum_{j=1}^n \eta_j(\zeta) (\zeta_j - z_j) = 1 \quad \text{for all } \zeta \in \partial D.$$
Put
$$\omega_\eta(\zeta) = \frac{(n-1)!}{(2\pi i)^n} \sum_{j=1}^n \eta_j(\zeta) \, d\zeta_j \wedge \bigwedge_{k \ne j} \left( \bar\partial \eta_k \wedge d\zeta_k \right),$$
where $\bar\partial$ is the [[$\bar\partial$ operator|M1VKCW3RFW]] in $\zeta$.
Let $f$ be a [[holomorphic function|507C34W0ZT]] on a neighbourhood of $\overline{D}$.
Then
$$f(z) = \int_{\partial D} f(\zeta) \, \omega_\eta(\zeta).$$

## Remarks

The forms $\bar\partial \eta_k \wedge d\zeta_k$ have degree $2$, so the order of the factors in the product does not matter.
With $\eta_j(\zeta) = (\bar\zeta_j - \bar z_j)/\abs{\zeta - z}^2$, the form $\omega_\eta$ is the kernel of the [[Bochner–Martinelli formula|YMHYAEJQ80]], and the statement reduces to that formula for holomorphic $f$.
For a general $\eta$, a homotopy argument joining $\eta$ to the Bochner–Martinelli choice shows that the boundary integral does not depend on $\eta$.
For $n = 1$ the condition forces $\eta = 1/(\zeta - z)$, and the formula is the Cauchy integral formula.
On a convex domain with $C^2$ boundary, one can choose $\eta$ depending holomorphically on $z$, which gives integral formulas with kernels holomorphic in $z$.

## Explanations

### L1 human

There are many integral formulas that recover a holomorphic function inside a domain from its boundary values.
The Cauchy–Fantappiè formula produces one formula for each choice of an auxiliary map $\eta$ satisfying a single linear condition.
Different choices of $\eta$ adapt the formula to the shape of the domain.

### L2 human

The Bochner–Martinelli kernel is universal but not holomorphic in $z$, which limits its use.
The freedom in $\eta$ allows kernels that are holomorphic in $z$ on convex or strongly pseudoconvex domains.
Such kernels, built by Henkin, Ramírez and others, give solutions of the $\bar\partial$ equation with estimates and results on approximation by holomorphic functions.
</document>

<document source="https://tamnd.github.io/math2/node/R9JZA3G2NX.md" tag="R9JZA3G2NX" kind="statement">
# Thue–Siegel–Roth theorem

## Statement

Let $\alpha$ be a real [[algebraic number|0MNF0BYFQC]] that is [[irrational|0DZXAXEJQ9]], and let $\varepsilon > 0$.
Then there are only finitely many [[rational numbers|0NCJ5WXN6Z]] $p/q$, with $p, q \in \Z$ and $q \ge 1$, such that
$$\abs{\alpha - \frac{p}{q}} < \frac{1}{q^{2 + \varepsilon}}.$$

## Remarks

The proof builds an auxiliary polynomial in many variables with integer coefficients that vanishes to high order at $(\alpha, \dots, \alpha)$.
Suppose there were many very good approximations with fast growing denominators.
Then this polynomial would have to vanish to high order at a rational point too, and a nonvanishing lemma (Roth's lemma) rules this out.
The proof is not effective: it gives a bound for the number of exceptional fractions, but not for their size.
The exponent $2$ cannot be lowered, because [[Dirichlet's approximation theorem|04JAY9DGB7]] gives infinitely many $p/q$ with $\abs{\alpha - p/q} < 1/q^2$.
The hypothesis that $\alpha$ is algebraic is necessary: a [[Liouville number|0PYEZM6CN9]] has, for each exponent $\mu$, infinitely many approximations with error less than $1/q^{\mu}$.
For example, $\alpha = \sqrt[3]{2}$ has only finitely many approximations with error less than $1/q^{2.01}$.
The theorem sharpens [[Liouville's theorem on Diophantine approximation|0N3WFZEFPM]], where the exponent is the degree of $\alpha$, and the later results of Thue and Siegel.

## Explanations

### L1 human

An algebraic irrational number, such as $\sqrt{2}$ or $\sqrt[3]{2}$, cannot be approximated by fractions much better than any irrational can.
Every irrational has infinitely many fractions $p/q$ within $1/q^2$, but only finitely many within $1/q^{2.001}$ when the number is algebraic.
Numbers that are approximated extremely well, like Liouville numbers, are therefore transcendental.

### L2 human

Roth's theorem implies that a [[Thue equation|3Q6CFPQZDF]] $F(x, y) = m$, with $F$ an irreducible binary form of degree at least $3$, has only finitely many integer solutions.
Its higher-dimensional generalization is the [[Schmidt subspace theorem|9TPS3W497D]].
The theorem says that the irrationality exponent of each algebraic irrational is $2$, the same as for almost all real numbers.
</document>

<document source="https://tamnd.github.io/math2/node/R9YWQS15AA.md" tag="R9YWQS15AA" kind="concept">
# Isometry of Riemannian manifolds

## Summary

An isometry of Riemannian manifolds is a [[diffeomorphism|0KZZ70ZHXP]] that carries one [[Riemannian metric|Q61VFHCH2J]] to the other.
Two isometric manifolds are the same for Riemannian geometry: lengths, angles, geodesics and curvature correspond.
The isometries of a manifold onto itself form a group, which by the [[Myers–Steenrod theorem|Z3VRMW8ETQ]] is a Lie group.

## Explanations

### L1 human

An isometry moves a curved space onto another without stretching or shrinking anything, even infinitesimally.
Rotations of the round sphere are isometries of the sphere onto itself.
Rolling a flat sheet of paper into a cylinder is locally an isometry, because the paper is bent but not stretched.
A map from the sphere to the plane can never be an isometry, even on a small piece, because the sphere is curved and the plane is flat.
</document>

<document source="https://tamnd.github.io/math2/node/RA358M9QCN.md" tag="RA358M9QCN" kind="statement">
# Friedberg–Muchnik theorem

## Statement

There are [[computably enumerable sets|TN9G24VABN]] $A, B \subseteq \N$ such that $A \not\le_T B$ and $B \not\le_T A$, where $\le_T$ is [[Turing reducibility|PA4PYQ1PZ0]].

## Remarks

The sets are built in stages by the [[priority method|3BZFSC5QZ2]].
The requirements are $R_{2e}$: $A \ne \Phi_e^B$ and $R_{2e+1}$: $B \ne \Phi_e^A$, where $\Phi_e$ is the $e$-th oracle machine.
To meet $R_{2e}$, one picks a witness $x$ and puts $x$ into $A$ once $\Phi_e^B(x) = 0$ appears, then tries to keep $B$ unchanged below the use of that computation.
Actions for requirements of lower priority may be undone, but each requirement is injured only finitely often, so all of them are met.

As a consequence, the degrees $\deg(A)$ and $\deg(B)$ are incomparable, so neither is $\mathbf{0}$ or $\mathbf{0}'$.
So both lie strictly between $\mathbf{0}$ and $\mathbf{0}'$, which solves [[Post's problem|1RVF7DWG8C]].

## Explanations

### L1 human

Some sets of numbers can be listed by a computer but not decided by one.
The theorem says that two such sets can be built so that neither helps to compute the other.
So the hardness of listable sets is not a single ladder: there are incomparable levels.

### L2 human

This is the result that solved Post's problem, and its proof introduced the finite-injury [[priority method|3BZFSC5QZ2]].
The same method, refined, shows that the computably enumerable [[Turing degrees|P5N2ND3VCG]] are dense (Sacks) and that every countable partial order embeds in them.
The construction is also a model for many other constructions in computability theory, where infinitely many conflicting requirements must be met at once.
</document>

<document source="https://tamnd.github.io/math2/node/RA56GHB9JZ.md" tag="RA56GHB9JZ" kind="concept">
# Topological monoid

## Summary

A topological monoid is a [[monoid|0N6R902FG1]] with a topology in which the multiplication is jointly continuous.
It is a [[topological semigroup|QXB6JDG68F]] that has an identity element.
Topological monoids arise in algebraic topology, for example as Moore loop spaces, where loops of varying length are concatenated.
They also arise in the study of operator semigroups.

## Explanations

### L1 human

A monoid is a set with an associative operation and a neutral element, like the natural numbers under addition with neutral element $0$.
A topological monoid adds a topology so that the operation respects nearness.
An example is the set of real numbers under multiplication with the usual topology, with neutral element $1$.
It is not a group, since $0$ has no inverse, but it is a topological monoid.
</document>

<document source="https://tamnd.github.io/math2/node/RA5SQ908G1.md" tag="RA5SQ908G1" kind="concept">
# Width of a poset

## Summary

The width of a [[partially ordered set|0QV2JS9RJ3]] is the largest size of an [[antichain|03HF498P1J]] in it.
It measures how many elements can be pairwise incomparable at the same time.
By [[Dilworth's theorem|GT224XDA4E]], for a finite poset it is also the least number of chains that cover the poset.

## Explanations

### L1 human

Draw a poset with smaller elements lower down.
The width is the largest number of elements that sit side by side, with no two of them related.
A chain has width $1$.
In the subsets of $\set{1, 2, 3}$ ordered by inclusion, the three one-element subsets form an antichain, and no antichain is larger, so the width is $3$.
</document>

<document source="https://tamnd.github.io/math2/node/RABJ6X1BXC.md" tag="RABJ6X1BXC" kind="statement">
# Toeplitz index theorem

## Statement

Let $\mathbb{T}$ be the unit circle, let $H^2 \subseteq L^2(\mathbb{T})$ be the Hardy space, and let $\varphi \colon \mathbb{T} \to \C$ be a continuous function with no zeros.
Then the [[Toeplitz operator|GQ0X3R3WM2]] $T_\varphi$ on $H^2$ is a [[Fredholm operator|8HWGKWD6ZM]].
Its [[index|2V9Q4KHY1T]] is
$$\operatorname{ind} T_\varphi = -\operatorname{wind}(\varphi, 0).$$
Here $\operatorname{wind}(\varphi, 0)$ is the [[winding number|0NQSV2EWDR]] of the closed curve $\varphi$ around $0$.

## Remarks

Idea of the proof: for continuous $\varphi$ and $\psi$, the operator $T_\varphi T_\psi - T_{\varphi\psi}$ is compact.
So $T_{1/\varphi}$ is an inverse of $T_\varphi$ modulo compact operators, and $T_\varphi$ is Fredholm by Atkinson's theorem.
The index is constant along homotopies of nonvanishing symbols, and $\varphi$ is homotopic to $z^n$ with $n = \operatorname{wind}(\varphi, 0)$.
For $n \ge 0$, $T_{z^n}$ is the $n$-th power of the shift, which is injective with cokernel of dimension $n$, so its index is $-n$; for $n < 0$ one takes adjoints.
The hypothesis that $\varphi$ has no zeros is necessary: for continuous $\varphi$, the operator $T_\varphi$ is Fredholm if and only if $\varphi$ has no zeros on $\mathbb{T}$.

## Explanations

### L1 human

A Toeplitz operator multiplies by a function on the circle and then discards negative frequencies.
When the function never vanishes, the operator is nearly invertible, and the failure is measured by an integer.
That integer is minus the number of times the function winds around the origin.
For the simplest case, multiplication by $z$ shifts the coefficients of a power series one step and loses exactly one dimension.

### L2 human

The theorem is the one-dimensional model of the Atiyah–Singer index theorem: an analytic index equals a topological invariant of the symbol.
It is also a basic example of index theory for $C^*$-algebras, through the Toeplitz extension $0 \to \mathcal{K} \to \mathcal{T} \to C(\mathbb{T}) \to 0$, whose index map $K_1(C(\mathbb{T})) \to K_0(\mathcal{K}) = \Z$ is the winding number up to a sign convention.
Higher-dimensional versions for Toeplitz operators on strictly pseudoconvex domains are due to Boutet de Monvel.
</document>

<document source="https://tamnd.github.io/math2/node/RACSNQXDQV.md" tag="RACSNQXDQV" kind="statement">
# Weyl's inequality

## Statement

Let $A$ and $B$ be $n \times n$ [[Hermitian matrices|0P17NGQQMQ]].
For a Hermitian matrix $M$, let $\lambda_1(M) \ge \dots \ge \lambda_n(M)$ be its [[eigenvalues|0BQAA2BK3Q]], listed with multiplicity.
Then, for all $i, j \ge 1$ with $i + j - 1 \le n$,
$$\lambda_{i+j-1}(A + B) \le \lambda_i(A) + \lambda_j(B).$$

## Remarks

The proof uses the [[Courant–Fischer min-max theorem|CJK2HS2508]].
The span of the eigenvectors for $\lambda_i(A), \dots, \lambda_n(A)$ and the span of those for $\lambda_j(B), \dots, \lambda_n(B)$ meet in a subspace of dimension at least $n - i - j + 2$.
On this subspace the Rayleigh quotient of $A + B$ is at most $\lambda_i(A) + \lambda_j(B)$, and the min-max formula gives the bound.

Taking $j = 1$ and applying the result also to $(A + B) + (-B)$ gives
$$\lambda_k(A) + \lambda_n(B) \le \lambda_k(A + B) \le \lambda_k(A) + \lambda_1(B).$$
Hence $\abs{\lambda_k(A + B) - \lambda_k(A)} \le \norm{B}_2$, where $\norm{B}_2$ is the [[operator norm|06EKW1AKXF]] induced by the Euclidean norm.

Example: $A = \operatorname{diag}(1, 0)$ and $B = \operatorname{diag}(0, 1)$ give $A + B = I_2$, and $\lambda_2(A + B) = 1 \le \lambda_1(A) + \lambda_2(B) = 1$, with equality.

## Explanations

### L1 human

Adding two symmetric matrices cannot move the eigenvalues in an arbitrary way.
Each eigenvalue of the sum is bounded by sums of eigenvalues of the two parts.
In particular, a small symmetric perturbation moves each eigenvalue by at most the size of the perturbation.

### L2 human

The corollary $\abs{\lambda_k(A + B) - \lambda_k(A)} \le \norm{B}_2$ says that the eigenvalues of Hermitian matrices are stable under perturbation.
For non-Hermitian matrices this fails, and bounds such as the [[Bauer–Fike theorem|AK9J4RNZ51]] are needed instead.
Weyl's inequalities are among the inequalities in Horn's problem on the eigenvalues of a sum of Hermitian matrices.
</document>

<document source="https://tamnd.github.io/math2/node/RACVWG5QTK.md" tag="RACVWG5QTK" kind="concept">
# Hamiltonian of an optimal control problem

## Summary

The Hamiltonian of an optimal control problem is a function that combines the dynamics and the running cost with a vector of multipliers, the costate.
It has the form $H = p \cdot f - \lambda_0 L$, or $H = L + p \cdot f$ in the sign convention of the minimum principle.
[[Pontryagin's maximum principle|7RPV2T29CF]] says that an optimal control maximizes the Hamiltonian at almost every time.
It plays the role that the Hamiltonian of [[Hamiltonian systems|0EDKGA2FAK]] plays in mechanics.

## Explanations

### L1 human

The Hamiltonian is a bookkeeping device that weighs the immediate cost of a control against its effect on the future.
The costate $p$ acts as a price for each unit of change in the state.
A control is good at an instant if it gives a high value of "price times rate of change minus cost".
For example, for $x' = u$ with cost $\int u^2/2 \, dt$, the Hamiltonian is $H = p u - u^2/2$, which is largest at $u = p$.
</document>

<document source="https://tamnd.github.io/math2/node/RADE160VJN.md" tag="RADE160VJN" kind="concept">
# Minimal sufficient statistic

## Summary

A minimal sufficient statistic is a [[sufficient statistic|WP4TGXJQJG]] that reduces the data as much as possible without losing information about the parameter.
It is a function of every other sufficient statistic.
In regular models it exists and can be found from ratios of likelihoods.

## Explanations

### L1 human

The whole data set is always sufficient, but it is not a summary at all.
A minimal sufficient statistic is the most compressed summary that still keeps everything relevant about the parameter.
For coin tosses it is the number of heads; for normal data with unknown mean and variance it is the pair (sample mean, sample variance).
Any further compression would lose information.
</document>

<document source="https://tamnd.github.io/math2/node/RAGJSF4V9V.md" tag="RAGJSF4V9V" kind="concept">
# Monad in nonstandard topology

## Summary

Let $X$ be a topological space and ${}^*X$ its extension in [[nonstandard analysis|WM62GKCBWN]].
The monad of a point $x$ of $X$ is the set of points of ${}^*X$ that are infinitely close to $x$.
It is the intersection of the nonstandard extensions of all open neighbourhoods of $x$.
Monads turn topological notions into simple statements about infinitely close points.
For example, a space is Hausdorff exactly when distinct points have disjoint monads.

## Explanations

### L1 human

In the [[hyperreal numbers|421QMTFG9T]] there are infinitesimals, numbers smaller in size than every positive real number but not zero.
The monad of a real number $x$ is the cloud of hyperreal numbers $x + \varepsilon$ with $\varepsilon$ infinitesimal.
It is a picture of "all points that are infinitely near $x$".
For a general space, the open sets around $x$ take the place of distances, and the monad is what all of them have in common after the nonstandard extension.
Continuity then reads: points infinitely near $x$ go to points infinitely near $f(x)$.
</document>

<document source="https://tamnd.github.io/math2/node/RAGR38Q40B.md" tag="RAGR38Q40B" kind="concept">
# Lp multiplier

## Summary

An $L^p$ multiplier is a bounded function $m$ on $\R^n$ such that the operator $f \mapsto (m \hat f)^\vee$ is bounded on the [[$L^p$ space|SASP14MA1Z]] $L^p(\R^n)$.
It is a [[Fourier multiplier|ZQEFCY61T7]] that respects the size of functions measured in the $L^p$ norm.
For $p = 2$ every bounded function is an $L^p$ multiplier, but for $p \ne 2$ the question is subtle.
Many operators of analysis and PDE are multiplier operators, such as partial sums of Fourier integrals and Riesz transforms.
So criteria for $L^p$ multipliers are a central topic of harmonic analysis.

## Explanations

### L1 human

A multiplier operator changes a function by turning each of its frequencies up or down by a factor $m(\xi)$.
The question is whether this change can blow up the size of the function.
In the $L^2$ norm, the size is the total energy over all frequencies, so a bounded factor can never blow it up.
In other $L^p$ norms, sharp cuts in the frequency picture can make the function much larger, so not every bounded $m$ works.
For example, $m(\xi) = e^{2\pi i \xi h}$ only shifts a function by $h$, so it keeps every $L^p$ norm and is an $L^p$ multiplier for all $p$.
</document>

<document source="https://tamnd.github.io/math2/node/RARTMA0ZEC.md" tag="RARTMA0ZEC" kind="concept">
# Almost everywhere

## Summary

A property of the points of a measure space holds almost everywhere if the points where it fails lie in a set of measure zero.
Most statements about integrals and measurable functions only hold, and only need to hold, almost everywhere.
In probability theory the same notion is called almost surely.

## Explanations

### L1 human

"Almost everywhere" means "everywhere except on a negligible set".
The function that is $1$ at rational numbers and $0$ elsewhere is $0$ almost everywhere on the real line.
The rational numbers form a countable set, which has length zero.
Two functions that agree almost everywhere have the same integral, so measure theory treats them as the same.
</document>

<document source="https://tamnd.github.io/math2/node/RARXGWQ85S.md" tag="RARXGWQ85S" kind="counterexample">
# Volterra's function

## Counterexample

Let $S \subseteq [0, 1]$ be the [[Smith–Volterra–Cantor set|HZTRH10Z8Y]].
It is closed, contains no interval, and has positive [[Lebesgue measure|2Z2QVWZ262]].
Let $g(t) = t^2 \sin(1/t)$ for $t \ne 0$ and $g(0) = 0$.
Then $g$ is differentiable on $\R$, with $g'(t) = 2t \sin(1/t) - \cos(1/t)$ for $t \ne 0$ and $g'(0) = 0$.

The complement $[0, 1] \setminus S$ is a union of disjoint open intervals.
On each such interval $(a, b)$, let $t_0$ be the largest $t \in (0, (b - a)/2]$ with $g'(t) = 0$.
Define $V(x) = g(x - a)$ for $a < x \le a + t_0$, $V(x) = g(t_0)$ for $a + t_0 \le x \le b - t_0$, and $V(x) = g(b - x)$ for $b - t_0 \le x < b$.
Set $V(x) = 0$ for $x \in S$.

Then $V$ is differentiable at every point of $[0, 1]$, and its derivative $V'$ is bounded.
At each point of $S$, one has $V'(x) = 0$, but $V'$ is not continuous there.
So $V'$ is discontinuous on a set of positive measure, and by [[Lebesgue's criterion|0565EGK6XP]] it is not [[Riemann integrable|0N1JQXZXHC]].

## Remarks

The number $t_0$ exists because $g'$ has zeros at points of $(0, \varepsilon)$ for every $\varepsilon > 0$.
Near an endpoint $a$, the derivative $V'(x) = g'(x - a)$ keeps taking values close to $1$ and close to $-1$.
Every point of $S$ is a limit of such endpoints, which gives the discontinuity of $V'$ on $S$.
The example shows that a function can have a bounded [[derivative|0F1WR9C83K]] that has no Riemann integral.
So the [[fundamental theorem of calculus|0NHGY0N0JB]] in the Riemann form $\int_a^b f' = f(b) - f(a)$ needs the hypothesis that $f'$ is Riemann integrable.
The Lebesgue integral repairs this: a bounded derivative is Lebesgue integrable, and the formula holds.

## Explanations

### L1 human

Volterra's function is differentiable everywhere, and its slope stays bounded.
Yet the slope jumps around wildly on a fat Cantor set, a set with no intervals but with positive length.
Because the bad set is so large, the Riemann integral cannot handle the derivative.
The function shows that the Riemann integral is too weak to undo every differentiation.
</document>

<document source="https://tamnd.github.io/math2/node/RASAC44RTQ.md" tag="RASAC44RTQ" kind="definition">
# Unramified morphism of rings

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $A$ be a commutative [[$R$-algebra|DPBJ04JMH8]].
Then $A$ is *unramified* over $R$, and the structure map $R \to A$ is an *unramified morphism*, if both conditions hold:

1. $A$ is an $R$-algebra of [[finite type|AC5TFFQT9M]].
2. The module of [[Kähler differentials|7HCFARHG7D]] $\Omega_{A/R}$ is zero.

## Remarks

Some authors, following Grothendieck, require $A$ to be [[finitely presented|764GZHTGV8]] rather than only of finite type; the two agree when $R$ is Noetherian.
Condition 2 alone is called *formally unramified*.
An equivalent form, for $A$ of finite type, is a condition on each [[prime ideal|00MHHZ9SNE]] $\mathfrak q$ of $A$, with $\mathfrak p = \mathfrak q \cap R$.
One needs $\mathfrak p A_{\mathfrak q} = \mathfrak q A_{\mathfrak q}$ in the [[localization|4C1AE84N5N]].
One also needs the [[residue field|NJ3APQ8BHH]] of $\mathfrak q$ to be a finite [[separable extension|0WCX7NB4N7]] of that of $\mathfrak p$.

Example: let $R = \Z[\frac12]$ and $A = R[x]/(x^2 + 1) = \Z[\frac12][i]$.
It is of finite type, and $\Omega_{A/R} = A \, dx / (2x \, dx) \cong A/(2i) = 0$, since $2i$ is a unit in $A$.

Non-example for clause 2: for $A = \Z[i]$ over $\Z$, one gets $\Omega_{A/\Z} \cong \Z[i]/(2) \ne 0$; the prime $2$ ramifies.
Non-example for clause 1: the algebraic closure $\overline{\Q}$ of $\Q$ has $\Omega_{\overline{\Q}/\Q} = 0$, because every element is separable algebraic over $\Q$, but it is not of finite type over $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/RASDTDEKP7.md" tag="RASDTDEKP7" kind="concept">
# Self-similar process

## Summary

A self-similar process is a [[stochastic process|0QH5ENKZM1]] whose law is unchanged when time is scaled by $a$ and space by $a^H$.
The number $H$ is the Hurst index.
The [[Wiener process|22735FGJV6]] and [[fractional Brownian motion|VC6R7XSKRQ]] are the main examples.
Self-similar processes model random signals that look alike at every scale, in physics, hydrology, networks and finance.

## Explanations

### L1 human

Zoom into the graph of a self-similar process and rescale the vertical axis in the right way: the picture you get is statistically the same as the original.
The vertical rescaling is a power $a^H$ of the horizontal zoom $a$.
For Brownian motion $H = 1/2$: watch it for four times as long and its typical size doubles.
A straight line $t \mapsto t$ is self-similar with $H = 1$ in a trivial, non-random way.
</document>

<document source="https://tamnd.github.io/math2/node/RATW595XR9.md" tag="RATW595XR9" kind="concept">
# Huber loss

## Summary

The Huber loss is a [[loss function|3BTG6VVCGZ]] that is quadratic for small residuals and linear for large ones.
It combines the efficiency of [[least squares|0BYTQ4X1HC]] for well-behaved data with the resistance of absolute loss to outliers.
Peter Huber introduced it in 1964 to define a robust [[M-estimator|4503R8BVF8]] of location, and it is now also common in regression and machine learning.

## Explanations

### L1 human

Squared error punishes a residual of $10$ a hundred times as much as a residual of $1$, so a single outlier can dominate a fit.
Absolute error is gentler on outliers but has a sharp corner at zero.
The Huber loss uses squared error near zero and switches to a straight line beyond a cutoff $\delta$.
So small errors are treated as in least squares, while large errors count only in proportion to their size.
</document>

<document source="https://tamnd.github.io/math2/node/RB08381QYR.md" tag="RB08381QYR" kind="definition">
# Hilbert's invariant integral

## Definition

Let $L(x, y, p)$ be a $C^2$ function on $\R \times \R \times \R$, and let $J(y) = \int_a^b L(x, y, y') \, dx$.
Let $p(x, y)$ be the slope function of a [[field of extremals|K4DFT3H9BJ]] of $J$ on a simply connected open set $D \subseteq \R^2$.
For a piecewise $C^1$ curve $\gamma$ in $D$, *Hilbert's invariant integral* is the [[line integral|0H5MR9B6GT]]
$$I^*(\gamma) = \int_\gamma \Big( L(x, y, p) - p \, L_p(x, y, p) \Big) \, dx + L_p(x, y, p) \, dy,$$
where $p = p(x, y)$ inside the integrand.

## Remarks

The integrand is a [[closed differential form|0ZV14XRNF0]] on $D$ because the field consists of extremals.
Since $D$ is simply connected, the form is exact, so $I^*(\gamma)$ depends only on the end points of $\gamma$.

Along a curve of the field one has $dy = p \, dx$, so $I^*(\gamma) = \int L(x, y, p) \, dx = J$ on that curve.
For the graph of any piecewise $C^1$ function $y$ in $D$, subtracting $I^*$ from $J$ gives
$$J(y) - I^*(y) = \int_a^b E\big(x, y(x), p(x, y(x)), y'(x)\big) \, dx,$$
with $E$ the [[Weierstrass excess function|R8TWRPSPNV]].
If $y_0$ is a field curve and $y$ has the same end points, then $I^*(y) = J(y_0)$, which gives the Weierstrass formula $J(y) - J(y_0) = \int_a^b E \, dx$.

Example: for $L = p^2$ and the field $y = x + c$ with $p = 1$, the integrand is $(1 - 2) \, dx + 2 \, dy = d(2y - x)$, which is exact.

Non-example: with the slope function $p = 1$ and $L = p^2 + y^2$, the curves $y = x + c$ are not extremals, and the form $(y^2 - 1) \, dx + 2 \, dy$ is not closed, since $\partial_y (y^2 - 1) = 2y$ is not identically equal to $\partial_x 2 = 0$.
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<document source="https://tamnd.github.io/math2/node/RBDJBK63HX.md" tag="RBDJBK63HX" kind="statement">
# Dugundji extension theorem

## Statement

Let $X$ be a [[metric space|0VKD1TP7F3]], let $A \subseteq X$ be a [[closed|0ZCK8DQYN1]] subset, and let $L$ be a [[locally convex|B6GY2KJMWW]] topological vector space over $\R$.
Let $f \colon A \to L$ be a [[continuous|0VGFHTM75Y]] map.
Then there is a continuous map $F \colon X \to L$ with $F(a) = f(a)$ for all $a \in A$.
Moreover, $F$ can be chosen with $F(X)$ contained in the [[convex hull|0FQSDC2EQ2]] of $f(A)$.

## Remarks

The proof covers $X \setminus A$ by small open balls whose radii are comparable to the distance to $A$.
It then takes a locally finite open refinement, which exists because metric spaces are paracompact.
For each set $W$ of the refinement, pick a point $a_W \in A$ close to $W$, and let $(\varphi_W)$ be a partition of unity subordinate to the refinement.
Set $F(x) = \sum_W \varphi_W(x) f(a_W)$ for $x \notin A$ and $F = f$ on $A$.
Continuity at points of $A$ uses that $a_W$ is near $x$ when $W$ is near $x$, together with the local convexity of $L$.

Example: with $L = \R$, the theorem gives the [[Tietze extension theorem|0X52TNMTCJ]] for metric spaces, with the bound $\inf f \le F \le \sup f$.

The hypothesis that $A$ is closed is necessary.
On $X = [0, 1]$, the map $f(x) = \sin(1/x)$ is continuous on $A = (0, 1]$, but it has no continuous extension to $X$, because $f$ has no limit at $0$.

## Explanations

### L1 human

If a continuous function is known on a closed part of a metric space, one can always fill in its values on the rest of the space continuously.
This holds not only for real values, but for values in any space where one can form averages in a controlled way.
The filled-in values are averages of known values, so they stay inside the convex hull of the original values.

### L2 human

The theorem extends the Tietze extension theorem from real-valued to vector-valued maps, at the price of assuming that $X$ is metric.
Its main consequence is that every convex subset of a locally convex metrizable topological vector space is an [[absolute retract|DK3MT3HYX5]].
The proof is a model use of partitions of unity on paracompact spaces, and the same technique appears in the [[Michael selection theorem|3RJ1K36A9A]].
</document>

<document source="https://tamnd.github.io/math2/node/RBPR7BK5T5.md" tag="RBPR7BK5T5" kind="definition">
# Bimodule

## Definition

Let $R$ and $S$ be [[rings|0894J980RK]].
An *$(R, S)$-bimodule* is an abelian group $M$ with these structures:

1. a left [[$R$-module|09VDS2W1E8]] structure $(r, m) \mapsto rm$;
2. a right $S$-module structure $(m, s) \mapsto ms$;
3. the compatibility $(rm)s = r(ms)$ for all $r \in R$, $m \in M$ and $s \in S$.

## Remarks

An $(R, R)$-bimodule is also called an $R$-bimodule.
Example: for a field $K$, the space $M_{m \times n}(K)$ of $m \times n$ matrices is an $(M_m(K), M_n(K))$-bimodule under matrix multiplication.
Clauses 1 and 2 are the usual rules of matrix arithmetic, and clause 3 is associativity $(AX)B = A(XB)$.
Every ring $R$ is an $R$-bimodule, and its sub-bimodules are its two-sided ideals.
Non-example: let $R$ be a noncommutative ring and give $R$ the "right action" $m \cdot s = sm$.
This fails clause 2, since $(m \cdot s) \cdot t = tsm$ while $m \cdot (st) = stm$, and $ts \ne st$ for some $s, t$.
For a commutative ring $K$, every $K$-module is a $K$-bimodule with $mk = km$.
</document>

<document source="https://tamnd.github.io/math2/node/RBVCDV3GCY.md" tag="RBVCDV3GCY" kind="definition">
# Eigenvalue algorithm

## Definition

Let $n \ge 1$ and let $\mathbb{K}$ be $\R$ or $\C$.
An *eigenvalue algorithm* is an algorithm that takes as input a [[matrix|0VD9ZSWWJG]] $A \in \mathbb{K}^{n \times n}$ (and, for iterative algorithms, a tolerance $\varepsilon > 0$) and returns:

1. numbers $\tilde\lambda_1, \dots, \tilde\lambda_k \in \C$, with $1 \le k \le n$, that approximate [[eigenvalues|0BQAA2BK3Q]] $\lambda_1, \dots, \lambda_k$ of $A$, counted with algebraic multiplicity, to within the tolerance;
2. optionally, vectors $\tilde v_1, \dots, \tilde v_k$ that approximate corresponding [[eigenvectors|0N6960W0BV]] or bases of invariant subspaces.

It computes *all* eigenvalues when $k = n$, and *some* eigenvalues otherwise, for example those of largest modulus or those closest to a given shift.

## Remarks

Example: the [[QR algorithm|B6F8YWHYZ5]] reduces $A$ to Hessenberg form and applies shifted QR steps until the matrix is nearly upper triangular.
Its diagonal then approximates all $n$ eigenvalues, so it is an eigenvalue algorithm with $k = n$.
Non-example: an algorithm that only computes the coefficients of the [[characteristic polynomial|0VSSPAFM2C]] does not return eigenvalues, so clause 1 fails.
Finding the roots is a further step, and it is badly conditioned in general.
The eigenvalues of a general matrix with $n \ge 5$ are not expressible by radicals in its entries.
So every eigenvalue algorithm that works for all such matrices is iterative and stops at a tolerance.
Accuracy is usually measured by backward error: the output should be the exact eigenvalues of a matrix $A + E$ with $\norm{E}$ small compared with $\norm{A}$.

## Explanations

### L1 human

An eigenvalue algorithm is any recipe that, given a square matrix, outputs its eigenvalues (and maybe eigenvectors) up to a chosen accuracy.
Some recipes find all of them, others find only a few that one asks for.
</document>

<document source="https://tamnd.github.io/math2/node/RBZPZTSPZE.md" tag="RBZPZTSPZE" kind="definition">
# Hyperbolic geometry

## Definition

A *hyperbolic plane* is a model of the axioms of [[absolute geometry|VKFPCSRNDH]], including Dedekind's axiom of continuity, in which the hyperbolic parallel axiom holds.
That axiom says: for each line $\ell$ and each point $P$ not on $\ell$, at least two distinct lines through $P$ do not meet $\ell$.
*Hyperbolic geometry* is the study of the hyperbolic plane and of its higher-dimensional analogues.

## Remarks

Any two hyperbolic planes in this sense are isomorphic, so hyperbolic geometry is a [[non-Euclidean geometry|SPCYMFN5V2]] with a single model up to isomorphism.
In differential geometry, the hyperbolic $n$-space $\mathbb{H}^n$ is the complete simply connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with constant sectional curvature $-1$.
For $n = 2$ this is a surface of [[Gaussian curvature|SQVQ5DQT6W]] $-1$, and it is a hyperbolic plane in the axiomatic sense.
In it, a triangle with angles $\alpha, \beta, \gamma$ has area $\pi - \alpha - \beta - \gamma$.

Example: the [[Poincaré half-plane model|RHKSMPBSQ1]] $\set{(x, y) : y > 0}$ with the metric $(dx^2 + dy^2)/y^2$ satisfies the axioms.
Its lines are the vertical half-lines and the semicircles centred on the $x$-axis.

Non-example: the Euclidean plane satisfies the axioms of absolute geometry.
It fails the hyperbolic parallel axiom, because each point off a line lies on exactly one parallel.
</document>

<document source="https://tamnd.github.io/math2/node/RC22S41ZDB.md" tag="RC22S41ZDB" kind="definition">
# Wasserstein metric

## Definition

Let $(M, d)$ be a separable complete metric space, that is, a [[Polish space|584GXXY73M]], and let $p \in [1, \infty)$.
Let $\mathcal{P}_p(M)$ be the set of Borel [[probability measures|059TJN0NZ1]] $\mu$ on $M$ with $\int_M d(x, x_0)^p \, d\mu(x) < \infty$ for some point $x_0 \in M$.
For $\mu, \nu \in \mathcal{P}_p(M)$, a *coupling* of $\mu$ and $\nu$ is a Borel probability measure $\gamma$ on $M \times M$ whose [[marginals|0MXT6PRZZ6]] are $\mu$ and $\nu$.
That is, $\gamma(A \times M) = \mu(A)$ and $\gamma(M \times A) = \nu(A)$ for every Borel set $A \subseteq M$.
Let $\Gamma(\mu, \nu)$ be the set of couplings.
The *$p$-Wasserstein distance* is
$$W_p(\mu, \nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{M \times M} d(x, y)^p \, d\gamma(x, y) \right)^{1/p}.$$

## Remarks

The condition on $\mu$ does not depend on the choice of $x_0$, by the triangle inequality.
The product measure $\mu \otimes \nu$ is always a coupling, so $\Gamma(\mu, \nu)$ is not empty and $W_p(\mu, \nu) < \infty$.
The function $W_p$ is a metric on $\mathcal{P}_p(M)$, and the infimum is attained.

Example: let $\delta_x$ be the [[Dirac measure|0ACRGYNHSR]] at $x$.
The only coupling of $\delta_x$ and $\delta_y$ is $\delta_{(x, y)}$, so $W_p(\delta_x, \delta_y) = d(x, y)$.

Non-example: on $M = \R$, the standard Cauchy distribution has $\int \abs{x} \, d\mu = \infty$.
So it does not lie in $\mathcal{P}_1(\R)$, and $W_1$ is not defined for it.

On $\R$, with distribution functions $F_\mu$ and $F_\nu$, one has $W_1(\mu, \nu) = \int_{-\infty}^{\infty} \abs{F_\mu(t) - F_\nu(t)} \, dt$.
In $\mathcal{P}_p(M)$, $W_p(\mu_n, \mu) \to 0$ if and only if $\mu_n \to \mu$ weakly and the $p$-th moments converge.

## Explanations

### L1 human

A coupling is a plan for moving the mass of $\mu$ to the mass of $\nu$.
The distance is the cost of the cheapest plan, where moving mass by $r$ costs $r^p$ per unit.
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<document source="https://tamnd.github.io/math2/node/RC5A5T7YY2.md" tag="RC5A5T7YY2" kind="statement">
# Golod–Shafarevich theorem

## Statement

Let $k$ be a [[field|04C5ZBG1WT]], let $d \ge 1$, and let $F = k\langle x_1, \ldots, x_d \rangle$ be the [[free algebra|2P3Y2FRCSZ]], graded by degree.
Let $I$ be the [[two-sided ideal|0CD87SFM2N]] of $F$ generated by a set of homogeneous elements of degrees at least $2$.
For $i \ge 2$, let $r_i$ be the number of these generators of degree $i$, and assume each $r_i$ is finite.
Let $A = F/I$, which is a [[graded algebra|8KB4SQGC0C]] $A = \bigoplus_{n \ge 0} A_n$, and let $H_A(t) = \sum_{n \ge 0} (\dim_k A_n) t^n$ be its Hilbert series.
Let $R(t) = \sum_{i \ge 2} r_i t^i$.

1. Every coefficient of the [[formal power series|033WGN00BF]] $H_A(t) \big(1 - d t + R(t)\big) - 1$ is nonnegative.
2. If there is a real number $t_0$ with $0 < t_0 < 1$ such that $R(t_0)$ converges and $1 - d t_0 + R(t_0) \le 0$, then $A$ is infinite-dimensional.

## Remarks

Part 1 comes from a careful count of the degree $n$ part of $I$: it is spanned by products $a f b$ with $f$ a generator, and the count gives $\dim A_n \ge d \dim A_{n-1} - \sum_{i=2}^{n} r_i \dim A_{n-i}$.
For part 2, if $A$ were finite-dimensional, then $H_A$ would be a polynomial with $H_A(t_0) > 0$.
Evaluating the series of part 1 at $t_0$ would give $H_A(t_0)\big(1 - d t_0 + R(t_0)\big) \ge 1 > 0$, which contradicts $1 - d t_0 + R(t_0) \le 0$.

Example: take $d = 4$ and at most one relation in each degree $i \ge 2$.
Then $R(t) \le t^2/(1 - t)$, and at $t_0 = 1/2$ one gets $1 - 2 + R(1/2) \le -1 + 1/2 < 0$.
So every such quotient of $k\langle x_1, x_2, x_3, x_4 \rangle$ is infinite-dimensional.

Golod used this criterion to build finitely generated infinite-dimensional algebras in which every element is nilpotent.
A version for groups concerns a nontrivial finite group $G$ of order a power of a prime $p$, that is, a [[p-group|06QYRTX9KQ]].
Let $d = \dim H^1(G, \F_p)$, its minimal number of generators, and $r = \dim H^2(G, \F_p)$; then $r > d^2/4$.

## Explanations

### L1 human

Start with a free algebra on $d$ letters, which is infinite-dimensional, and impose relations.
Each relation cuts down the size, but only by a limited amount in each degree.
The theorem says: if there are few relations, especially in low degrees, the quotient is still infinite-dimensional.

### L2 human

The theorem is a tool for proving that objects are infinite.
Through the group version, it shows that some groups with few relations are infinite, which gave infinite finitely generated periodic groups.
In number theory, it shows that the class field tower of a number field can be infinite.
Some imaginary quadratic fields have no finite tower of [[Hilbert class fields|WWCJ7CR2R3]] ending in a field with class number $1$.
</document>

<document source="https://tamnd.github.io/math2/node/RC7HCZCRC5.md" tag="RC7HCZCRC5" kind="definition">
# Mori domain

## Definition

Let $A$ be an [[integral domain|09S3PFBEPM]] with [[field of fractions|0VF11ZTTMD]] $K$.
For a nonzero [[ideal|0MYSA6MYNA]] $I$ of $A$, let $I^{-1} = \set{x \in K : xI \subseteq A}$ and $I_v = \set{x \in K : x I^{-1} \subseteq A}$.
The ideal $I$ is *divisorial* if $I_v = I$.
$A$ is a *Mori domain* if every strictly ascending chain $I_1 \subsetneq I_2 \subsetneq \cdots$ of nonzero divisorial ideals of $A$ is finite.

## Remarks

Every nonzero principal ideal $aA$ is divisorial, since $(aA)^{-1} = a^{-1}A$ and then $(aA)_v = aA$.
Example: every Noetherian domain is a Mori domain, since it has no infinite strictly ascending chain of ideals at all.
Non-example: the ring $\overline{\Z}$ of all algebraic integers is not Mori; the principal ideals $(2^{1/2^n})$ for $n \ge 1$ form a strictly ascending chain, since $2^{1/2^{n+1}} / 2^{1/2^{n}} = 2^{-1/2^{n+1}}$ is not an algebraic integer.
Every [[Krull domain|CB9WJCN9BM]] is a Mori domain, and a Mori domain is Krull if and only if it is completely integrally closed.
</document>

<document source="https://tamnd.github.io/math2/node/RCASN31FFS.md" tag="RCASN31FFS" kind="statement">
# Karlin–Rubin theorem

## Statement

Let $\Theta \subseteq \R$, let $X$ have a density $f_\theta$ with $\theta \in \Theta$, and assume the family has a [[monotone likelihood ratio|4Z73AFN7WT]] in a [[statistic|036756NFD6]] $T(X)$.
Let $\theta_0 \in \Theta$ and $\alpha \in (0, 1)$.
For constants $c \in \R$ and $\gamma \in [0, 1]$, let $\varphi(x) = 1$ if $T(x) > c$, $\varphi(x) = \gamma$ if $T(x) = c$, and $\varphi(x) = 0$ if $T(x) < c$.
Then $c$ and $\gamma$ can be chosen so that $E_{\theta_0}[\varphi(X)] = \alpha$.
With this choice, $\varphi$ is a [[uniformly most powerful test|QRAPTPVH8C]] at level $\alpha$ of $H_0 \colon \theta \le \theta_0$ against $H_1 \colon \theta > \theta_0$.

## Remarks

Proof idea: fix $\theta_1 > \theta_0$.
By the monotone likelihood ratio, $\varphi$ rejects exactly when $f_{\theta_1}/f_{\theta_0}$ is large, so it is the most powerful test of $\theta_0$ against $\theta_1$ by the [[Neyman–Pearson lemma|0EE19CFEEF]].
This test does not depend on $\theta_1$.
One then shows that its [[power function|4RJSN0QARD]] is nondecreasing in $\theta$, so its type I error is at most $\alpha$ on all of $\set{\theta \le \theta_0}$.
Example: for $X$ binomial with parameters $n$ and $p$, the UMP test of $p \le p_0$ against $p > p_0$ rejects for a large number of successes, with randomization at the boundary value.
The theorem is about one-sided hypotheses; for $H_1 \colon \theta \ne \theta_0$ a UMP test usually does not exist.

## Explanations

### L1 human

When larger values of a statistic always point towards larger parameters, the best one-sided test is the obvious one.
It rejects "the parameter is at most $\theta_0$" when the statistic is large.
The theorem says that this simple test is the best possible against every larger parameter at once.

### L2 human

The Karlin–Rubin theorem extends the Neyman–Pearson lemma from a simple alternative to a one-sided composite alternative.
The key is that the most powerful test against one alternative is the same for all alternatives on one side.
It applies to one-parameter exponential families, such as binomial, Poisson and normal models with one unknown parameter.
For two-sided problems, one restricts to unbiased tests to get an optimal test.
For discrete data, the randomization $\gamma$ is needed to reach the exact level $\alpha$.
</document>

<document source="https://tamnd.github.io/math2/node/RCNY89A4NY.md" tag="RCNY89A4NY" kind="concept">
# Semilinear PDE

## Summary

A semilinear PDE is a [[partial differential equation|GC5BDZ6QBC]] whose highest-order part is linear with coefficients depending only on the variables.
Its lower-order terms may be nonlinear.
The principal part, which decides the type of the equation, is then the same as for a linear equation.
Many models of physics and biology have this form, such as reaction–diffusion equations and nonlinear wave equations.
See the [[definition|X3P4TQZ9X2]].

## Explanations

### L1 human

Think of a linear equation such as the heat equation, and add a nonlinear source term that depends on the unknown itself.
For example, $\partial_t u = \partial_x^2 u + u(1 - u)$ is [[Fisher's equation|AXV2PH7FEZ]].
The second derivatives enter linearly, but the term $u(1 - u)$ does not.
Because the highest derivatives behave as in a linear equation, much of the linear theory still applies.
</document>

<document source="https://tamnd.github.io/math2/node/RCQH8P6DPQ.md" tag="RCQH8P6DPQ" kind="concept">
# Anosov diffeomorphism

## Summary

An Anosov diffeomorphism is a diffeomorphism of a compact manifold for which the whole manifold is a [[hyperbolic set|MPFD24TYW2]].
At every point it contracts some directions and expands the complementary ones, uniformly.
Anosov diffeomorphisms are [[structurally stable|D4MGFAHVMT]], satisfy [[Axiom A|Z26Z4XZFVA]] and admit [[Markov partitions|4VNHVDHTNN]].
The basic examples are the hyperbolic toral automorphisms, such as [[Arnold's cat map|BSCZN8Y1AN]].

## Explanations

### L1 human

Take a torus and a map that stretches it in one direction and squeezes it in another, everywhere and by a fixed rate.
Such a map mixes points very fast, like dough being kneaded, but it never tears the surface.
Arnold's cat map, given by the matrix with rows $(2, 1)$ and $(1, 1)$ acting on the torus, is the standard example.
Small smooth changes of an Anosov diffeomorphism do not change its orbit structure.
See the [[definition|X1P48KQFXQ]].
</document>

<document source="https://tamnd.github.io/math2/node/RCRD8CAVB4.md" tag="RCRD8CAVB4" kind="problem">
# Tarski's high school algebra problem

## Problem

Consider the signature with a constant $1$ and binary operations $+$, $\cdot$ and exponentiation $x^y$, interpreted in the positive integers.
Let $\mathrm{HSI}$ be the following eleven [[identities|PKD440G2XW]]:

1. $x + y \approx y + x$ and $(x + y) + z \approx x + (y + z)$;
2. $x \cdot 1 \approx x$, $x \cdot y \approx y \cdot x$ and $(x \cdot y) \cdot z \approx x \cdot (y \cdot z)$;
3. $x \cdot (y + z) \approx x \cdot y + x \cdot z$;
4. $1^x \approx 1$, $x^1 \approx x$, $x^{y + z} \approx x^y \cdot x^z$, $(x \cdot y)^z \approx x^z \cdot y^z$ and $(x^y)^z \approx x^{y \cdot z}$.

The problem asks whether every identity in this signature that holds in the positive integers can be derived from $\mathrm{HSI}$ in [[equational logic|72S3YSQFZ7]].

## Remarks

The answer is no.
Alex Wilkie found, around 1980, an identity true in the positive integers that cannot be derived from $\mathrm{HSI}$.
Later, R. Gurevič showed that the identities true in the positive integers with $1$, $+$, $\cdot$ and exponentiation have no finite [[equational basis|9Q008GMZSW]] at all.
Every identity without exponentiation that holds in the positive integers is derivable from the identities in items 1 to 3.
Indeed, both sides expand to the same polynomial; this is why the question was natural.

## Explanations

### L1 human

School algebra teaches a short list of rules for adding, multiplying and taking powers.
Tarski asked whether these rules are enough to prove every true equation built from $1$, $+$, $\times$ and powers.
Surprisingly, they are not: there are true equations that the school rules cannot prove.
Even worse, no finite list of rules can do the job.
</document>

<document source="https://tamnd.github.io/math2/node/RCSXGDTTET.md" tag="RCSXGDTTET" kind="definition">
# Shape derivative

## Definition

Let $d \ge 1$ and let $W = W^{1,\infty}(\R^d; \R^d)$ be the Banach space of bounded Lipschitz vector fields $\theta \colon \R^d \to \R^d$, with norm $\norm{\theta}_W = \norm{\theta}_{L^\infty} + \norm{\nabla \theta}_{L^\infty}$.
Let $\mathcal{O}$ be a class of open subsets of $\R^d$, and let $J \colon \mathcal{O} \to \R$ be a shape [[functional|1EQM0DJ4NC]].
Let $\Omega \in \mathcal{O}$ be such that $(\mathrm{Id} + \theta)(\Omega) \in \mathcal{O}$ whenever $\norm{\theta}_W$ is small enough.
Then $J$ is *shape differentiable* at $\Omega$ if the map $\theta \mapsto J((\mathrm{Id} + \theta)(\Omega))$ is [[Fréchet differentiable|08EVNYZF42]] at $\theta = 0$.
Its Fréchet derivative, a bounded linear map $J'(\Omega) \colon W \to \R$, is the *shape derivative* of $J$ at $\Omega$.

## Remarks

For $\norm{\theta}_W < 1$ the map $\mathrm{Id} + \theta$ is a bi-Lipschitz homeomorphism of $\R^d$, so $(\mathrm{Id} + \theta)(\Omega)$ is again open.
Example: the volume $J(\Omega) = \abs{\Omega}$ of a bounded open set.
By the change of variables formula $J((\mathrm{Id} + \theta)(\Omega)) = \int_\Omega \det(I + \nabla \theta) \, dx$.
Since $\det(I + \nabla \theta) = 1 + \operatorname{div} \theta + O(\norm{\nabla\theta}^2)$, the shape derivative is $J'(\Omega)(\theta) = \int_\Omega \operatorname{div} \theta \, dx$.
If $\Omega$ has a $C^1$ boundary with outward unit [[normal vector|0XYP2KKSPC]] $n$, the divergence theorem gives $J'(\Omega)(\theta) = \int_{\partial \Omega} \theta \cdot n \, ds$.
Non-example: $J(\Omega) = \abs{\abs{\Omega} - 1}$ at a set $\Omega$ with $\abs{\Omega} = 1$ is not shape differentiable.
Take $\Omega$ bounded and a smooth function $\chi$ with compact support that equals $1$ on a neighbourhood of $\overline{\Omega}$.
Along $\theta_s(x) = s \chi(x) x$ one gets $(\mathrm{Id} + \theta_s)(\Omega) = (1 + s) \Omega$ and $J = \abs{(1 + s)^d - 1}$, which has a corner at $s = 0$.
For smooth $\Omega$ and smooth $J$, the Hadamard structure theorem gives $J'(\Omega)(\theta) = \int_{\partial \Omega} g \, (\theta \cdot n) \, ds$ for a function $g$ on $\partial \Omega$, the shape gradient.
</document>

<document source="https://tamnd.github.io/math2/node/RCWHDMWP1T.md" tag="RCWHDMWP1T" kind="concept">
# Order topology

## Summary

The order topology is the natural topology on a [[totally ordered set|0FRXBVVR03]], generated by the open rays $\set{x : x > a}$ and $\set{x : x < b}$.
The usual topology of the real line is the order topology of its usual order.
Ordinals, the long line and many classical counterexamples of general topology are built as order topologies.
Every order topology is Hausdorff, and in fact normal.

## Explanations

### L1 human

On a line of ordered points, call a set open when, around each of its points, it contains a whole open interval.
This is exactly how the open sets of the real line are described, but it uses only the order and not the distance.
So any totally ordered set, such as the rationals, the integers or a set of ordinals, gets a topology in the same way.
On the integers every point $n$ is the open interval $(n - 1, n + 1)$, so the order topology is discrete.
</document>

<document source="https://tamnd.github.io/math2/node/RD1ES06VHN.md" tag="RD1ES06VHN" kind="definition">
# Polar space

## Definition

Let $(P, L)$ be a [[partial linear space|CX07CWPCDX]] with point set $P$ and line set $L$.
Two points are *collinear* if they are equal or lie on a common line.
The space $(P, L)$ is a *polar space* (in the sense of Buekenhout and Shult) if:

1. every line has at least three points;
2. for every line $\ell$ and point $p$ not on $\ell$, the point $p$ is collinear with exactly one or with all points of $\ell$.

The polar space is *non-degenerate* if, in addition, no point is collinear with all points of $P$.

## Remarks

Example: let $F$ be a field and $Q(x) = x_0 x_1 + x_2 x_3$ on $F^4$, with polar form $B(x, y) = Q(x + y) - Q(x) - Q(y)$.
Let $P$ be the points of the [[quadric|Q3Q4FF23VG]] of $Q$ in $\mathbb{P}^3(F)$ and $L$ the projective lines contained in it.
For $p, q \in P$ one has $Q(\lambda p + \mu q) = \lambda \mu B(p, q)$, so $p$ and $q$ are collinear exactly when $B(p, q) = 0$.
For a line $\ell \in L$ and $p \in P$ off $\ell$, the points $q$ of $\ell$ with $B(p, q) = 0$ form a linear condition on $\ell$, so they are one point or all of $\ell$.
Every line has $\abs{F} + 1 \ge 3$ points.
The points collinear with $p$ lie in the plane $B(p, \cdot) = 0$, which does not contain all of $P$, so the polar space is non-degenerate.

Non-example: a projective plane with its lines satisfies clauses 1 and 2, since any two points are collinear.
It is degenerate, because every point is collinear with all points.

The rank of a non-degenerate polar space is the largest $n$ such that there is a chain of $n$ nonempty subspaces in which every two points are collinear.
The non-degenerate polar spaces of rank $2$ are exactly the generalized quadrangles with at least three points on each line.
Veldkamp and Tits showed that the non-degenerate polar spaces of rank at least $3$ come from forms on vector spaces, with some exceptions in rank $3$.
The forms needed here include pseudo-quadratic forms.
</document>

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# Post algebra

## Summary

Post algebras are the algebras of Post's $n$-valued logic, in the same way as Boolean algebras are the algebras of classical two-valued logic.
For each $n \ge 2$, the Post algebras of order $n$ form the variety generated by an $n$-element chain with a cyclic negation.
This generating algebra is [[primal|XD01Y7B6XW]], so Post algebras have a Boolean-like structure theory.
For $n = 2$ they are exactly the Boolean algebras.

## Explanations

### L1 human

Classical logic has two truth values, and Boolean algebras capture its laws.
Emil Post studied logics with $n$ truth values $0, 1, \dots, n - 1$, ordered like a ladder, with a negation that moves every value one step up and the top value back to $0$.
Post algebras collect all algebras that obey the same laws as this $n$-valued system.
With three values, for example, the negation sends $0 \mapsto 1 \mapsto 2 \mapsto 0$.
</document>

<document source="https://tamnd.github.io/math2/node/RD3K5HADYG.md" tag="RD3K5HADYG" kind="definition">
# Matching pursuit

## Definition

Let $H$ be a real or complex [[Hilbert space|HM2HHE3J51]] with inner product $\ip{\cdot}{\cdot}$ and norm $\norm{\cdot}$.
Let $\mathcal{D} \subseteq H$ be a dictionary of unit vectors, and let $f \in H$.
*Matching pursuit* produces residuals $r_k$, elements $g_k \in \mathcal{D}$ and approximants $f_k$ as follows.

1. Set $r_0 = f$ and $f_0 = 0$.
2. For $k \ge 1$, choose $g_k \in \mathcal{D}$ with $\abs{\ip{r_{k-1}}{g_k}} = \sup\set{\abs{\ip{r_{k-1}}{g}} : g \in \mathcal{D}}$.
3. Set $r_k = r_{k-1} - \ip{r_{k-1}}{g_k} g_k$ and $f_k = f_{k-1} + \ip{r_{k-1}}{g_k} g_k$.

## Remarks

By construction $f = f_k + r_k$, and $f_k$ is an [[n-term approximant|GEK0NP5WQM]] with $n = k$.
Since $r_k$ is orthogonal to $g_k$, $\norm{r_k}^2 = \norm{r_{k-1}}^2 - \abs{\ip{r_{k-1}}{g_k}}^2$, so the residual norms never increase.
When the supremum in clause 2 is not attained, one uses a weak version that only asks for a fixed fraction $t \in (0, 1]$ of it.
Example: in $\R^2$ with $\mathcal{D} = \set{e_1, e_2, u}$, $u = (1, 1)/\sqrt{2}$, and $f = (2, 1)$.
The correlations are $2$, $1$ and $3/\sqrt{2} \approx 2.12$, so $g_1 = u$, $f_1 = (3/2, 3/2)$ and $r_1 = (1/2, -1/2)$.
Then $\ip{r_1}{u} = 0$ and $\abs{\ip{r_1}{e_1}} = \abs{\ip{r_1}{e_2}} = 1/2$; choosing $g_2 = e_1$ gives $r_2 = (0, -1/2)$, and $g_3 = e_2$ gives $r_3 = 0$.
Non-example: choosing $g_1 = e_1$ at the first step fails clause 2, since $\abs{\ip{f}{e_1}} = 2 < 3/\sqrt{2}$.
*Orthogonal matching pursuit* replaces clause 3 by taking $f_k$ to be the orthogonal projection of $f$ onto the span of $g_1, \dots, g_k$ and $r_k = f - f_k$.
</document>

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# Morita equivalence

## Summary

Two [[rings|0B1261B5ED]] are Morita equivalent when their categories of [[modules|0NZ8JA25WK]] are equivalent.
Such rings need not be isomorphic, but every property that is defined through modules alone is the same for both.
A ring $R$ and the matrix ring $M_n(R)$ are the basic example.
Morita theory describes all such equivalences through [[progenerators|ZQ9ABBKF1N]] and their endomorphism rings.

## Explanations

### L1 human

Often one studies a ring through the modules over it, much as one studies a group through its actions.
Two different rings can have module theories that look exactly alike, with the same pattern of maps between modules.
Such rings are called Morita equivalent.
For example, the field $\R$ and the ring of $2 \times 2$ real matrices have the same module theory.
A module over the matrix ring is a sum of copies of $\R^2$, and these behave just like real vector spaces.
So any question about modules can be moved from one ring to the other.
</document>

<document source="https://tamnd.github.io/math2/node/RD7DVC2NJ6.md" tag="RD7DVC2NJ6" kind="definition">
# Weierstrass polynomial

## Definition

Let $n \ge 1$, write $z' = (z_1, \dots, z_{n-1})$, and let $\mathcal{O}_{n-1} = \C\{z'\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]] in $n - 1$ variables.
Let $d \ge 0$ be an integer.
A *Weierstrass polynomial* of degree $d$ in $z_n$ is a [[monic polynomial|0C3QBCR46M]]
$$P(z', z_n) = z_n^d + a_{d-1}(z')\, z_n^{d-1} + \cdots + a_1(z')\, z_n + a_0(z')$$
with coefficients $a_0, \dots, a_{d-1} \in \mathcal{O}_{n-1}$ such that $a_j(0) = 0$ for every $j$.

## Remarks

Equivalently, $P \in \mathcal{O}_{n-1}[z_n]$ is monic of degree $d$ and $P(0, z_n) = z_n^d$.
The only Weierstrass polynomial of degree $0$ is $P = 1$.
Example: $P = z_2^3 - z_1 z_2 + z_1^2$ is a Weierstrass polynomial of degree $3$ in $z_2$, since $a_2 = 0$, $a_1 = -z_1$ and $a_0 = z_1^2$ all vanish at $0$.
Non-example: $Q = z_2^2 + z_2 + z_1$ is monic in $z_2$, but its coefficient $a_1 = 1$ does not vanish at $0$; indeed $Q(0, z_2) = z_2^2 + z_2 \ne z_2^2$.
The same definition is used with $\mathcal{O}_{n-1}$ replaced by the ring of holomorphic functions on a neighbourhood of $0 \in \C^{n-1}$.
</document>

<document source="https://tamnd.github.io/math2/node/RDC61V812J.md" tag="RDC61V812J" kind="example">
# Iterated exponentiation

## Example

For $x > 0$, the infinite power tower $x^{x^{x^{\cdots}}}$ is defined as the limit of $t_1 = x$, $t_{n+1} = x^{t_n}$.
It is an [[infinite composition|B88PFYXWBK]] of the map $t \mapsto x^t$.
Take $x = \sqrt2$; the tower converges to $2$.

Check: $t_1 = \sqrt2 < 2$, and if $t_n < 2$ then $t_{n+1} = \sqrt2^{\,t_n} < \sqrt2^{\,2} = 2$.
Since $t \mapsto \sqrt2^{\,t}$ is increasing and $t_2 = \sqrt2^{\sqrt2} > \sqrt2 = t_1$, the sequence is increasing.
By the [[monotone convergence theorem|0FM6N4R0VB]] it has a limit $L \le 2$ with $\sqrt2^{\,L} = L$.
The equation $\sqrt2^{\,L} = L$ has the two solutions $L = 2$ and $L = 4$, and $L \le 2$ gives $L = 2$.

A classical result, going back to Euler, is that the tower converges exactly for $e^{-e} \le x \le e^{1/e}$.
For example, $x = 2 > e^{1/e} \approx 1.444$ gives $2, 4, 16, 65536, \dots$, which tends to infinity.

## Explanations

### L1 human

A power tower stacks the same number in the exponent again and again.
For $\sqrt2$ the values creep up towards $2$ and never pass it, because $\sqrt2$ raised to the power $2$ is again $2$.
For larger bases the tower explodes.
</document>

<document source="https://tamnd.github.io/math2/node/RDE658H8Z4.md" tag="RDE658H8Z4" kind="definition">
# Projection-valued measure

## Definition

Let $(\Omega, \Sigma)$ be a measurable space, so that $\Sigma$ is a [[sigma-algebra|0SV8K0TDGY]] of subsets of $\Omega$.
Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] with identity map $I$.
A *projection-valued measure* on $(\Omega, \Sigma)$ is a map $E$ from $\Sigma$ to the bounded operators on $H$ such that:

1. $E(B)$ is an [[orthogonal projection|0B8KZRT28M]] for every $B \in \Sigma$;
2. $E(\emptyset) = 0$ and $E(\Omega) = I$;
3. $E(B \cap C) = E(B) E(C)$ for all $B, C \in \Sigma$;
4. for all $x, y \in H$, the map $E_{x,y}(B) = \ip{E(B) x}{y}$ is a [[complex measure|NY6TYHRGMS]] on $\Sigma$.

## Remarks

Clause 4 is equivalent to the strong countable additivity: if $B = \bigcup_n B_n$ with $B_n \in \Sigma$ pairwise disjoint, then $E(B) x = \sum_n E(B_n) x$ for every $x \in H$.
By clause 3, the projections of disjoint sets have orthogonal images, and all $E(B)$ commute.
For each $x$, $E_{x,x}$ is a positive measure of total mass $\norm{x}^2$.
Example: let $H = \ell^2$ with orthonormal basis $(e_n)_{n \ge 1}$, let $\Omega = \R$ with its Borel sets, and let $(\lambda_n)$ be real numbers.
Let $E(B)$ be the orthogonal projection onto the closed span of $\set{e_n : \lambda_n \in B}$; clauses 1 to 4 follow from the fact that $E_{x,y}(B) = \sum_{n : \lambda_n \in B} x_n \overline{y_n}$.
Non-example: on $\Omega = \R$, let $E(B) = P$ if $0 \in B$ and $E(B) = 0$ otherwise, for an orthogonal projection $P \ne I$; then $E(\R) = P \ne I$, so clause 2 fails.
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<document source="https://tamnd.github.io/math2/node/RDFA0SYHEQ.md" tag="RDFA0SYHEQ" kind="definition">
# Self-avoiding walk

## Definition

Let $d \ge 1$ and $n \ge 0$ be integers, and let $\abs{\cdot}$ be the Euclidean norm on $\R^d$.
An $n$-step *self-avoiding walk* on $\Z^d$ starting at $0$ is a sequence $\omega = (\omega_0, \omega_1, \ldots, \omega_n)$ of points of $\Z^d$ such that:

1. $\omega_0 = 0$;
2. $\abs{\omega_i - \omega_{i-1}} = 1$ for $1 \le i \le n$;
3. $\omega_i \ne \omega_j$ for $0 \le i < j \le n$.

Let $c_n$ be the number of $n$-step self-avoiding walks.
The uniform probability measure on these $c_n$ walks is also called the self-avoiding walk with $n$ steps.

## Remarks

Example: in $\Z^2$, the walk $(0,0), (1,0), (1,1), (0,1)$ satisfies all three clauses, and it is one of the $c_3 = 36$ walks with three steps.
Non-example: $(0,0), (1,0), (0,0)$ satisfies clauses 1 and 2 but not clause 3, since it returns to $0$.
Cutting an $(n + m)$-step walk after $n$ steps gives $c_{n+m} \le c_n c_m$, so by Fekete's lemma the connective constant $\mu_d = \lim_{n \to \infty} c_n^{1/n}$ exists.
Walks that only move in positive coordinate directions are self-avoiding, and walks that never reverse their last step include all self-avoiding walks, so $d \le \mu_d \le 2d - 1$.
</document>

<document source="https://tamnd.github.io/math2/node/RDFRDH3E45.md" tag="RDFRDH3E45" kind="definition">
# Gelfand–Levitan–Marchenko equation

## Definition

Let $u \colon \R \to \R$ be a potential with $\int_{\R} (1 + \abs{x}) \abs{u(x)} \, dx < \infty$, and consider $-\psi'' + u \psi = k^2 \psi$ on $\R$.
Let its scattering data be the reflection coefficient $r(k)$ for $k \in \R$, the bound state energies $-\kappa_1^2, \ldots, -\kappa_N^2$ with $\kappa_j > 0$, and the norming constants $\beta_1, \ldots, \beta_N > 0$.
Put
$$F(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} r(k) \, e^{ikx} \, dk + \sum_{j=1}^{N} \beta_j e^{-\kappa_j x}.$$
The *Gelfand–Levitan–Marchenko equation* is the equation for an unknown function $K(x, y)$, $y \ge x$:
$$K(x, y) + F(x + y) + \int_x^\infty K(x, z) \, F(z + y) \, dz = 0, \quad y > x.$$
For each fixed $x$ it is a linear integral equation of the second kind on $(x, \infty)$, and the potential is recovered as $u(x) = -2 \frac{d}{dx} K(x, x)$.

## Remarks

Example: the reflectionless data $r = 0$, $N = 1$, give $F(x) = \beta e^{-\kappa x}$.
Try $K(x, y) = L(x) e^{-\kappa y}$; the equation becomes $L(x) + \beta e^{-\kappa x} + L(x) \frac{\beta}{2\kappa} e^{-2\kappa x} = 0$.
With $w(x) = \frac{\beta}{2\kappa} e^{-2\kappa x}$ this gives $K(x, x) = -\frac{2\kappa w}{1 + w}$, and since $w' = -2\kappa w$,
$$u(x) = -2 \frac{d}{dx} K(x, x) = -\frac{8 \kappa^2 w}{(1 + w)^2} = -2\kappa^2 \operatorname{sech}^2\!\big(\kappa (x - x_0)\big),$$
where $e^{2\kappa x_0} = \frac{\beta}{2\kappa}$.
This potential has the single bound state energy $-\kappa^2$ and reflects nothing.

Non-example: the equation $K(x, y) + F(x - y) + \int_x^\infty K(x, z) F(z - y) \, dz = 0$ has kernels $F(x - y)$ and $F(z - y)$ that depend on differences; in the Gelfand–Levitan–Marchenko equation they must depend on the sums $x + y$ and $z + y$.

Several conventions are in use, with other signs and with data on the left instead of the right.
The Gelfand–Levitan equation of [[Gelfand–Levitan theory|R4MRNP9EQT]] is a related equation, with an integral over $(0, x)$, that recovers a potential on a half-line from its spectral function.
</document>

<document source="https://tamnd.github.io/math2/node/RDFYHS2Q55.md" tag="RDFYHS2Q55" kind="example">
# Laplace transform of a power function

## Example

Let $\alpha > -1$ be real and $f(t) = t^\alpha$ for $t > 0$.
Its [[Laplace transform|05XT5AVJBK]] is
$$\mathcal{L}\{t^\alpha\}(s) = \frac{\Gamma(\alpha + 1)}{s^{\alpha + 1}} \quad (\operatorname{Re} s > 0),$$
where $\Gamma$ is the [[gamma function|0V3FZMVB4K]] and $s^{\alpha + 1}$ is the principal power.
For $\alpha = n \in \N$ this is $\mathcal{L}\{t^n\}(s) = n!/s^{n + 1}$, with $n!$ the [[factorial|0WFCGVMRRC]].
Check for real $s > 0$: the substitution $u = st$ gives
$$\int_0^\infty t^\alpha e^{-st}\, dt = \frac{1}{s^{\alpha + 1}} \int_0^\infty u^\alpha e^{-u}\, du = \frac{\Gamma(\alpha + 1)}{s^{\alpha + 1}}.$$
The integral converges at $0$ because $\alpha > -1$, and at $\infty$ because of the factor $e^{-st}$.
Both sides are holomorphic in $s$ on $\operatorname{Re} s > 0$ and agree for real $s > 0$, so they agree on the whole half-plane.
For example, $\mathcal{L}\{t^{-1/2}\}(s) = \sqrt{\pi / s}$, since $\Gamma(1/2) = \sqrt{\pi}$.

## Explanations

### L1 human

Powers of $t$ become negative powers of $s$: $1 \mapsto 1/s$, $t \mapsto 1/s^2$, $t^2 \mapsto 2/s^3$, and so on.
The factorials come from integrating by parts once for each power of $t$.
For powers that are not whole numbers the factorial is replaced by the gamma function.
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# Way-below relation

## Summary

The way-below relation $x \ll y$ on a partially ordered set says that $x$ is a necessary part of every directed approximation of $y$.
It refines the order $x \le y$ and formalizes the idea of "a finite piece of information about $y$".
It is used to define [[continuous lattices|RXBD58SGXZ]] and continuous domains, and an element is [[compact|9F7TYES4KE]] exactly when it is way below itself.

## Explanations

### L1 human

Suppose $y$ is approached by a directed family of elements, for instance an increasing sequence.
Then $x$ is way below $y$ if every such approach to $y$ passes above $x$ at some finite stage.
In the real interval $[0, 1]$, the number $0.4$ is way below $0.5$, because any increasing sequence with limit at least $0.5$ eventually exceeds $0.4$.
But $0.5$ is not way below itself, because the sequence $0.5 - 1/n$ approaches $0.5$ without reaching it.
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# Semiprimary ring

## Definition

Let $R$ be a [[ring|0894J980RK]] with identity and $J = J(R)$ its [[Jacobson radical|XA0DCNH7WY]].
$R$ is *semiprimary* if the following hold.

1. $R$ is [[semilocal|YH75M7XAM1]], that is, $R/J$ is a semisimple ring.
2. $J$ is a [[nilpotent ideal|C2H26BYWHE]]: $J^n = 0$ for some integer $n \ge 1$.

## Remarks

Example: the ring $R$ of upper triangular $2 \times 2$ matrices over a field $k$ is semiprimary: $J$ is the set of strictly upper triangular matrices, $J^2 = 0$, and $R/J \cong k \times k$ is semisimple.
Non-example: $\Z_{(p)}$ satisfies clause 1, since $\Z_{(p)}/p\Z_{(p)} \cong \mathbb{F}_p$, but not clause 2, since $p^n \ne 0$ for every $n$.
Every left Artinian ring is semiprimary, and a semiprimary ring is left Artinian if and only if it is left Noetherian.
The definition is left-right symmetric.
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# Lanczos sigma factor

## Definition

Let $m \ge 1$ be an integer.
Let $k$ be an integer with $\abs{k} < m$.
The *Lanczos sigma factor* of $k$ is $\sigma_{m,0} = 1$ for $k = 0$, and for $k \ne 0$ it is
$$\sigma_{m,k} = \frac{\sin(\pi k / m)}{\pi k / m}.$$
For a $2\pi$-periodic integrable function $f$ with [[Fourier coefficients|5NN2YZZR3V]] $\hat f(k)$, the *sigma approximation* of order $m$ of $f$ is
$$\Sigma_m f(x) = \sum_{\abs{k} < m} \sigma_{m,k} \, \hat f(k) e^{ikx}.$$

## Remarks

Let $S_{m-1} f(x) = \sum_{\abs{k} \le m-1} \hat f(k) e^{ikx}$ be the partial sum of the [[Fourier series|G12KXDHPPS]].
Since $\frac{m}{2\pi} \int_{x - \pi/m}^{x + \pi/m} e^{ikt} \, dt = \frac{\sin(\pi k/m)}{\pi k/m} e^{ikx}$ for $k \ne 0$,
$$\Sigma_m f(x) = \frac{m}{2\pi} \int_{x - \pi/m}^{x + \pi/m} S_{m-1} f(t) \, dt.$$
So the sigma approximation is the moving average of the partial sum over a window of length $2\pi/m$.
In the notation $\operatorname{sinc} u = \frac{\sin \pi u}{\pi u}$, the factors are $\sigma_{m,k} = \operatorname{sinc}(k/m)$.

Example: for $m = 2$ and $f(x) = \cos x$, the factor is $\sigma_{2, \pm 1} = \frac{\sin(\pi/2)}{\pi/2} = \frac{2}{\pi}$, so $\Sigma_2 f(x) = \frac{2}{\pi} \cos x$.
This agrees with the window average $\frac{1}{\pi} \int_{x - \pi/2}^{x + \pi/2} \cos t \, dt = \frac{2}{\pi} \cos x$.

Non-example: the [[Fejér weights|MA8X1G11DH]] $1 - \abs{k}/m$ also fade out high frequencies, but they are not the sigma factors; for $m = 2$ and $k = 1$ they give $1/2$, not $2/\pi$.
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# Relation algebra

## Definition

A *relation algebra* is a structure $(A, \vee, \wedge, -, 0, 1, ;, c, 1')$, where $;$ is a binary operation, $c$ is a unary operation and $1'$ is an element of $A$.
Write $\breve{x}$ for $c(x)$.
The following hold for all $x, y, z \in A$:

1. $(A, \vee, \wedge, -, 0, 1)$ is a [[Boolean algebra|0XADJKM699]], with order $\le$.
2. $(A, ;, 1')$ is a [[monoid|0N6R902FG1]].
3. $(x \vee y) ; z = (x ; z) \vee (y ; z)$.
4. $\breve{\breve{x}} = x$ and $\breve{(x \vee y)} = \breve{x} \vee \breve{y}$.
5. $\breve{(x ; y)} = \breve{y} ; \breve{x}$.
6. $\breve{x} ; -(x ; y) \le -y$.

## Remarks

Example: let $U$ be a set and let $A$ be the set of all [[binary relations|05AVARPXA2]] on $U$, with the set operations.
Let $;$ be the [[composition|0PYSR20HGF]] $R ; S = \set{(a, c) : (a, b) \in R \text{ and } (b, c) \in S \text{ for some } b}$.
Let $\breve{R}$ be the [[converse|054K3HNN5Z]] relation and $1'$ the identity relation.
Check of clause 6: if $(a, c) \in \breve{R} ; -(R ; S)$, there is $b$ with $(b, a) \in R$ and $(b, c) \notin R ; S$.
If $(a, c)$ were in $S$, then $(b, c) \in R ; S$; so $(a, c) \notin S$.

Non-example: on the relations of a set $U$ with at least two elements, keep the Boolean operations, $;$ and $1'$, but take $\breve{R} = R$.
Clause 5 then says $R ; S = S ; R$ for all relations, which fails for $U = \set{a, b}$, $R = \set{(a, b)}$ and $S = \set{(b, b)}$, since $R ; S = \set{(a, b)}$ and $S ; R = \emptyset$.
</document>

<document source="https://tamnd.github.io/math2/node/RDT4785M4D.md" tag="RDT4785M4D" kind="concept">
# Cohomology ring

## Summary

The cohomology ring of a space $X$ with coefficients in a commutative ring $R$ is the direct sum $H^\bullet(X; R) = \bigoplus_{k \ge 0} H^k(X; R)$ of its [[singular cohomology|6N7BC70CX0]] groups.
Its multiplication is the [[cup product|JX47MS1R1H]].
It is a graded commutative ring and a homotopy invariant of $X$.
It separates spaces that have isomorphic homology groups, and it is the setting for characteristic classes and cohomology operations.

## Explanations

### L1 human

The cohomology groups of a space can be added, and with the cup product they can also be multiplied.
Putting all degrees together gives one ring, much like a ring of polynomials where each variable has a degree.
For the complex projective plane, the ring is generated by one class $x$ of degree $2$ with $x^3 = 0$, while $x^2$ is not zero.
The space made by joining a $2$-sphere and a $4$-sphere at a point has the same groups, but there the square of the degree $2$ class is zero.
So the two spaces are not homotopy equivalent.
</document>

<document source="https://tamnd.github.io/math2/node/RDTESNZTKQ.md" tag="RDTESNZTKQ" kind="concept">
# Indirect method by shooting

## Summary

The indirect method by shooting solves an [[optimal control|T3V6ZNAMDN]] problem by first writing the necessary conditions of [[Pontryagin's maximum principle|7RPV2T29CF]] and then solving them numerically.
These conditions form a two-point [[boundary value problem|TM8VQJWGHZ]] for the state and the costate.
The unknown initial costate is found by the [[shooting method|ECT6RNF865]]: one integrates forward from a guess and corrects the guess until the final conditions hold.
The method is very accurate near a good guess, and it is the main alternative to direct methods such as [[direct collocation|T58W34D60G]].

## Explanations

### L1 human

The maximum principle tells you what an optimal trajectory looks like, except for a few unknown numbers at the start.
If you knew the initial costate, you could integrate the equations forward and get the whole optimal path.
So you guess it, integrate, measure how far you miss the required final state, and adjust the guess, as when aiming a cannon.
For instance, take $x' = u$ from $0$ to $1$ in unit time at least cost $\int u^2 / 2$.
The conditions say $u$ is a constant $p$, and shooting finds $p = 1$ after one Newton step.
The hard part in practice is that the miss can depend very sensitively on the guess.
</document>

<document source="https://tamnd.github.io/math2/node/RDV3C0PXTH.md" tag="RDV3C0PXTH" kind="definition">
# Local field

## Definition

Let $K$ be a [[field|04C5ZBG1WT]] with an [[absolute value|0FFXKTKKQC]] $\abs{\cdot}$ that is nontrivial, that is, $\abs{x} \neq 1$ for some $x \neq 0$.
Give $K$ the topology of the metric $d(x, y) = \abs{x - y}$.
Then $K$ is a *local field* if $K$ is a [[locally compact space|0FE2SB8WVJ]] for this topology.

## Remarks

A local field is Archimedean when it is $\R$ or $\C$, and non-Archimedean otherwise.
Many authors use "local field" only for the non-Archimedean case.
A non-Archimedean field is a local field exactly when it is complete for a discrete valuation and its [[residue field|NJ3APQ8BHH]] is finite.
The non-Archimedean local fields of characteristic $0$ are the finite extensions of the fields $\Q_p$.
Those of characteristic $p$ are the fields $\F_q((t))$ with $q$ a power of $p$.
Example: $\Q_p$ is a local field, because the ball $\Z_p$ is a compact neighbourhood of $0$.
Non-example: $\Q$ with $\abs{\cdot}_p$ is not a local field, because it is not complete, and a field with such an absolute value is locally compact only when complete.
Non-example: $\C((t))$ is complete for a discrete valuation, but its residue field $\C$ is infinite, so it is not locally compact.
</document>

<document source="https://tamnd.github.io/math2/node/RDX55HQ3XT.md" tag="RDX55HQ3XT" kind="statement">
# Gromov's non-squeezing theorem

## Statement

Let $n \ge 1$ and give $\R^{2n}$ the coordinates $(x_1, y_1, \ldots, x_n, y_n)$ and the standard symplectic form $\omega_0 = \sum_{i=1}^n dx_i \wedge dy_i$.
For $r > 0$, let $B^{2n}(r)$ be the open ball of radius $r$ centered at $0$.
For $R > 0$, let
$$Z^{2n}(R) = \set{(x, y) \in \R^{2n} : x_1^2 + y_1^2 < R^2}$$
be the open cylinder of radius $R$ over a disc in the $(x_1, y_1)$-plane.
If there is a symplectic embedding $\varphi \colon B^{2n}(r) \to Z^{2n}(R)$, that is, a smooth embedding with $\varphi^* \omega_0 = \omega_0$, then $r \le R$.

## Remarks

Gromov proved the theorem in 1985 with [[pseudoholomorphic curves|N469E47V75]].
Assume $r > R$ and an embedding exists; compactify the cylinder to $S^2 \times T^{2n-2}$ with $S^2$ of area slightly more than $\pi R^2$.
For a suitable almost complex structure, there is a pseudoholomorphic sphere in the class of $S^2$ through the image of the center of the ball.
Its area is less than $\pi R^2 + \varepsilon$.
Pulled back to the ball, it gives a proper minimal surface through the center with area less than $\pi r^2$.
This contradicts the monotonicity formula, which says that such a surface has area at least $\pi r^2$.

The hypothesis that the disc lies in a symplectic plane is needed.
The map $(x_1, y_1, x_2, y_2) \mapsto (\lambda x_1, \lambda^{-1} y_1, \lambda x_2, \lambda^{-1} y_2)$ with small $\lambda > 0$ is a linear symplectomorphism that squeezes any ball into the cylinder $\set{x_1^2 + x_2^2 < R^2}$ over the $(x_1, x_2)$-plane, which is Lagrangian.
For $n = 1$ the theorem is just the statement that area-preserving maps preserve area.

## Explanations

### L1 human

A symplectic map preserves volume, but in dimension four and higher it is much more rigid than that.
A ball cannot be squeezed by a symplectic map through a round hole whose radius is smaller than the ball's, even though the cylinder over the hole has infinite volume.
This is sometimes called the principle of the symplectic camel.

### L2 human

The theorem marks the birth of symplectic topology: it shows that symplectic maps are not just volume-preserving maps.
It leads to the notion of a [[symplectic capacity|8KCB5KJKNT]], a size of a symplectic set that is invariant under symplectic embeddings.
By Liouville's theorem, Hamiltonian flows preserve volume; non-squeezing gives a finer constraint on them, a classical analogue of the uncertainty principle.
</document>

<document source="https://tamnd.github.io/math2/node/RDZGJ21WRX.md" tag="RDZGJ21WRX" kind="statement">
# Jordan–Zassenhaus theorem

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $R$.
Let $A$ be a finite-dimensional [[semisimple|NY25HXA478]] algebra over $K$, and let $\Lambda$ be an $R$-[[order|1ZBX4TKVZ9]] in $A$.
A *$\Lambda$-lattice* is a [[finitely generated|05QY1VEBYD]] left $\Lambda$-module that is torsion-free as an $R$-module.
Then for every finitely generated left $A$-module $V$ there are only finitely many isomorphism classes of $\Lambda$-lattices $M$ with $K \otimes_R M \cong V$.

## Remarks

The proof combines a reduction theory for lattices with the finiteness of the class number.
In outline, one fixes a lattice $M_0$ with $K \otimes_R M_0 \cong V$ and finds a nonzero $c \in R$ such that every lattice $M$ with $K \otimes_R M \cong V$ is isomorphic to a lattice $N$ with $c M_0 \subseteq N \subseteq M_0$.
There are only finitely many such $N$, because $M_0 / c M_0$ is finite.
Example: take $A = K$, $\Lambda = R$ and $V = K$.
The $\Lambda$-lattices with $K M \cong K$ are the nonzero fractional ideals of $R$, up to isomorphism, so in this case the theorem is the finiteness of the [[class number|KTKK9B1X41]] of $K$.
Example: take $K = \Q$ and $\Lambda = \Z G$ for a finite group $G$; there are finitely many $\Z G$-lattices of each $\Z$-rank $n$ up to isomorphism.
Together with Minkowski's bound on the orders of finite subgroups of $GL_n(\Z)$, it follows that $GL_n(\Z)$ has only finitely many conjugacy classes of finite subgroups.

## Explanations

### L1 human

An order is a ring of integers inside an algebra, and a lattice over it is a module shaped like a grid.
Over a field, a module is determined by little more than its dimension, but grids can differ in subtle arithmetic ways.
The theorem says that only finitely many different grids, counted up to isomorphism, fill out a given module.
For the integers of a number field this is the classical fact that there are only finitely many ideal classes.

### L2 human

The theorem is the basic finiteness result of integral representation theory.
It lets one define the class number of an order, the number of isomorphism classes of lattices in a given genus.
This number can then be studied with adelic methods.
It extends to any Dedekind domain whose field of fractions is a global field, with $A$ a separable algebra.
Applied to integral group rings, it controls the integral representations of finite groups, and in crystallography it gives the finiteness of arithmetic classes of finite subgroups of $GL_n(\Z)$.
</document>

<document source="https://tamnd.github.io/math2/node/RE02MP1H45.md" tag="RE02MP1H45" kind="definition">
# Loss function

## Definition

Let $\Theta$ be a set of states of nature, for example the parameter set of a [[parametric model|MSC0MR8VMH]].
Let $\mathcal{A}$ be a set of actions.
A *loss function* is a function $L \colon \Theta \times \mathcal{A} \to [0, \infty)$.
The value $L(\theta, a)$ is the loss incurred by taking the action $a$ when the true state is $\theta$.

## Remarks

For estimating a real quantity $g(\theta)$, the action set is $\mathcal{A} = \R$ and standard choices are the following.
The *squared error loss* is $L(\theta, a) = (a - g(\theta))^2$.
The *absolute error loss* is $L(\theta, a) = \abs{a - g(\theta)}$.
In a test of $\theta \in \Theta_0$ against $\theta \notin \Theta_0$, with $\mathcal{A} = \set{0, 1}$, the *$0$-$1$ loss* is $1$ for a wrong decision and $0$ for a right one.
Some authors allow real-valued losses that are only bounded below; this changes little in the theory.

Example: $\Theta = \mathcal{A} = \R$ and $L(\theta, a) = (a - \theta)^2$.
It is a function of the pair $(\theta, a)$ with values in $[0, \infty)$, and it is $0$ exactly when $a = \theta$.

Non-example: $L(\theta, a) = a - \theta$ on $\R \times \R$.
It takes negative values, so it is not a loss function in the sense above.
It would also reward underestimating $\theta$ without limit.
</document>

<document source="https://tamnd.github.io/math2/node/RE5T59YZ46.md" tag="RE5T59YZ46" kind="definition">
# Injective module

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $E$ be an [[$R$-module|0NZ8JA25WK]].
The module $E$ is *injective* if it satisfies the following extension property.
Let $M$, $N$ be $R$-modules, $i \colon M \to N$ an injective $R$-[[linear map|0NRYHJPQG2]], and $f \colon M \to E$ an $R$-linear map.
Then there is an $R$-linear map $g \colon N \to E$ with $g \circ i = f$.

## Remarks

The following are equivalent for an $R$-module $E$:

1. $E$ is injective.
2. Every injective linear map $E \to N$ has a left inverse, that is, it splits.
3. The functor $\Hom_R(-, E)$ takes [[exact sequences|JB7ZBFBA9C]] to exact sequences.
4. ([[Baer's criterion|MR5HNV0Q07]]) Every linear map from an ideal $I$ of $R$ to $E$ extends to $R$.

Over a [[principal ideal domain|0CMD0BCH8T]], a module $E$ is injective exactly when it is divisible: $rE = E$ for every nonzero $r \in R$.
The same definition works over any ring, commutative or not.

Example: $\Q$ is an injective $\Z$-module.
By Baer's criterion it suffices to extend a map $f \colon n\Z \to \Q$ with $n \ne 0$; with $q = f(n)$, the map $g(x) = xq/n$ on $\Z$ satisfies $g(n) = q$.

Non-example: $\Z$ is not an injective $\Z$-module.
Take $i \colon \Z \to \Z$, $x \mapsto 2x$, and $f$ the identity of $\Z$.
An extension $g$ would satisfy $1 = f(1) = g(2) = 2g(1)$, which has no solution in $\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/RE5TFTDP4D.md" tag="RE5TFTDP4D" kind="definition">
# Quantifier elimination

## Definition

Let $L$ be a [[signature|WDVVEYD6AT]] and let $T$ be an $L$-[[theory|92SWXST1ZM]].
The theory $T$ *has quantifier elimination* if, for every $L$-[[formula|HYJ10AF6RT]] $\varphi(x_1, \dots, x_n)$, there is a quantifier-free $L$-formula $\psi(x_1, \dots, x_n)$ such that
$$T \models \forall x_1 \cdots \forall x_n \, \bigl(\varphi(x_1, \dots, x_n) \leftrightarrow \psi(x_1, \dots, x_n)\bigr).$$

## Remarks

For sentences ($n = 0$) one allows the quantifier-free sentences $\top$ and $\bot$ when $L$ has no constant symbols.
It suffices to treat formulas of the form $\exists y \, \chi(x_1, \dots, x_n, y)$ with $\chi$ a conjunction of atomic and negated atomic formulas.

Example: the theory of dense linear orders without endpoints, in the signature $\set{<}$, has quantifier elimination.
For instance, $\exists y \, (x_1 < y \wedge y < x_2)$ is equivalent to $x_1 < x_2$ by density, and $\exists y \, (x_1 < y)$ is equivalent to $x_1 = x_1$ since there is no greatest element.

Non-example: the theory of $(\R, +, \cdot, 0, 1)$, without the order symbol, does not have quantifier elimination.
The formula $\exists y \, (x = y \cdot y)$ defines $[0, \infty)$, which is infinite and co-infinite.
A quantifier-free formula in $x$ is a Boolean combination of polynomial equations $p(x) = 0$, so it defines a finite or cofinite subset of $\R$.
With the order symbol added, the theory has quantifier elimination; this is [[Tarski's theorem|FSJ9NGRRPA]].
</document>

<document source="https://tamnd.github.io/math2/node/RE9YFFS3TS.md" tag="RE9YFFS3TS" kind="concept">
# Quiver

## Summary

A quiver is a [[directed graph|0KFYGSHHF5]] in which several arrows between the same two vertices and loops at a vertex are allowed.
Quivers are the combinatorial input of the representation theory of algebras.
A [[representation of a quiver|TXVWFWP6VQ]] places a vector space at each vertex and a linear map along each arrow.
Every quiver gives a [[path algebra|2PYANBGXFD]], and many finite-dimensional algebras are quotients of path algebras.
In category theory, a quiver is the underlying data of a category without composition.

## Explanations

### L1 human

A quiver is a picture made of dots and arrows between them.
Two dots may be joined by several arrows, and an arrow may start and end at the same dot.
The word quiver, a case for arrows, was introduced by Pierre Gabriel.
The Kronecker quiver has two vertices and two parallel arrows from the first to the second.
Quivers are used as blueprints for systems of vector spaces and linear maps.
</document>

<document source="https://tamnd.github.io/math2/node/RECR96V97B.md" tag="RECR96V97B" kind="concept">
# Contravariant functor

## Summary

A contravariant functor is an assignment like a [[functor|28NQ0KRHD3]] that reverses the direction of [[morphisms|859F0TFVA3]]: it sends $f \colon A \to B$ to a morphism $F(B) \to F(A)$.
It is the same thing as an ordinary functor out of the [[opposite category|DAMK8XE1FR]].
The [[dual space|07JBSZE0V4]] construction, cohomology and [[presheaves|AND2WHZC8P]] are contravariant functors.
The hom functor $\Hom(-, X)$ is the basic example, and the Yoneda lemma is about such functors.

## Explanations

### L1 human

Some constructions turn arrows around.
A map of sets $f \colon A \to B$ pulls back each subset of $B$ to a subset of $A$, its preimage.
So "subsets of" sends $f$ to a map going backwards, from subsets of $B$ to subsets of $A$.
Such a backward rule that still respects identities and composition is a contravariant functor.
Composition flips order too: pulling back along $g \circ f$ is pulling back along $g$, then along $f$.
</document>

<document source="https://tamnd.github.io/math2/node/REE07CSXHR.md" tag="REE07CSXHR" kind="concept">
# Congruence lattice

## Summary

The congruence lattice of an [[algebra|QJGPT2W0FT]] is the set of all its [[congruences|DCSXXT4SER]], ordered by inclusion.
It is an [[algebraic lattice|3J0BTXSCVN]], and it records all the [[quotient algebras|N5KE7FH6CX]] of the algebra and how they relate.
Many properties of algebras and varieties, such as simplicity, subdirect irreducibility and congruence modularity, are properties of this lattice.
By a theorem of Grätzer and Schmidt, every algebraic lattice is isomorphic to the congruence lattice of some algebra.

## Explanations

### L1 human

Each congruence is one way of gluing elements of an algebra together.
Some gluings are finer than others, and comparing them by inclusion gives a lattice.
At the bottom is "glue nothing", and at the top is "glue everything".
For a group, this lattice is the same as the lattice of normal subgroups.
For the group $\Z/6\Z$ it has four elements, matching the subgroups $\set{0}$, $\set{0, 3}$, $\set{0, 2, 4}$ and the whole group.
</document>

<document source="https://tamnd.github.io/math2/node/REER3CG7DS.md" tag="REER3CG7DS" kind="definition">
# Fatou set

## Definition

Let $f \colon \hat{\C} \to \hat{\C}$ be a [[rational function|0XDK9CGYKJ]] of degree at least $2$ on the [[Riemann sphere|0J78KBPFZB]] $\hat{\C}$.
Let $f^n$ be its $n$-th [[iterate|AYVTR2ES5B]].
A point $z \in \hat{\C}$ belongs to the *Fatou set* $F(f)$ if the following holds.
There is an [[open|02BH3MCBQB]] neighbourhood $U$ of $z$ such that $\set{f^n|_U : n \ge 0}$ is a [[normal family|JTVWRXXAKM]].
Normality refers to maps into $\hat{\C}$ with the spherical metric.
So every sequence in the family has a subsequence that converges locally uniformly on $U$, possibly to the constant $\infty$.
A connected component of $F(f)$ is a *Fatou component*.

## Remarks

By definition $F(f)$ is open.
Example: for $f(z) = z^2$, the open unit disc lies in $F(f)$, since there $f^n(z) = z^{2^n} \to 0$ locally uniformly.
Non-example: $z = 1$ is not in $F(z^2)$; every neighbourhood of $1$ contains points tending to $0$ and points tending to $\infty$, so no subsequence of iterates has a continuous limit there.
The Fatou set is completely invariant: $z \in F(f)$ if and only if $f(z) \in F(f)$.
The same definition with $\C$ in place of $\hat{\C}$ is used for transcendental entire functions.
</document>

<document source="https://tamnd.github.io/math2/node/REH63MEBNC.md" tag="REH63MEBNC" kind="definition">
# Neumann boundary condition

## Definition

Let $a < b$, and consider a second-order ordinary differential equation for $y$ on $[a, b]$.
A *Neumann boundary condition* is a pair of conditions
$$y'(a) = \alpha, \qquad y'(b) = \beta,$$
with given real numbers $\alpha, \beta$.
More generally, let $\Omega \subseteq \R^d$ be a bounded domain, that is, a connected open set, whose [[boundary|05D5EGMFYV]] $\partial \Omega$ is of class $C^1$.
Let $\nu(x)$ be the outward unit [[normal vector|0WQJJCCTC8]] at $x \in \partial \Omega$, and let $g \colon \partial \Omega \to \R$ be a given function.
For an equation for $u$ on $\Omega$, the Neumann boundary condition is
$$\frac{\partial u}{\partial \nu}(x) = \nabla u(x) \cdot \nu(x) = g(x) \quad \text{for all } x \in \partial \Omega,$$
where $\frac{\partial u}{\partial \nu}$ is the [[directional derivative|06KWGHDN2J]] of $u$ along $\nu$.

## Remarks

On $[a, b]$ the outward normal derivatives are $-y'(a)$ and $y'(b)$, so the two forms agree up to the sign at $a$.

Example: $y'' = 2$ on $[0, 1]$ with $y'(0) = 0$ and $y'(1) = 2$.
The function $y = x^2$ has $y'' = 2$, $y'(0) = 0$ and $y'(1) = 2$, so it solves this Neumann problem; so does $x^2 + c$ for every constant $c$.

Non-example: $y(0) = 0$ together with $y'(1) = 0$ is not a Neumann condition, since the first condition prescribes a value.

For $\Delta u = f$ on $\Omega$ with Neumann data $g$, the divergence theorem gives the necessary condition $\int_\Omega f = \int_{\partial \Omega} g$.
Solutions, when they exist, are unique only up to an additive constant.
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<document source="https://tamnd.github.io/math2/node/REJDQ6BTQV.md" tag="REJDQ6BTQV" kind="statement">
# Fundamental identity for ramification and inertia

## Statement

Let $L/K$ be a finite extension of [[algebraic number fields|MWBDFS2KJ5]] of [[degree|0EAJ2XXADC]] $n$, with rings of integers $\mathcal{O}_K \subseteq \mathcal{O}_L$.
Let $\mathfrak{p}$ be a nonzero prime ideal of $\mathcal{O}_K$, with [[factorisation|B7KNZCBFYT]] $\mathfrak{p}\mathcal{O}_L = \mathfrak{P}_1^{e_1} \cdots \mathfrak{P}_g^{e_g}$ into distinct prime ideals.
Let $f_i = [\mathcal{O}_L/\mathfrak{P}_i : \mathcal{O}_K/\mathfrak{p}]$ be the residue degrees.
Then
$$\sum_{i=1}^{g} e_i f_i = n.$$
If moreover $L/K$ is a [[Galois extension|0QMZAQ6QQ4]], then $e_1 = \cdots = e_g = e$ and $f_1 = \cdots = f_g = f$, and $efg = n$.

## Remarks

Proof idea: count elements of $\mathcal{O}_L/\mathfrak{p}\mathcal{O}_L$.
It is a vector space over $\mathcal{O}_K/\mathfrak{p}$ of dimension $n$, by localising at $\mathfrak{p}$, where $\mathcal{O}_L$ becomes free of rank $n$.
By the Chinese remainder theorem it is the product of the rings $\mathcal{O}_L/\mathfrak{P}_i^{e_i}$, and each has dimension $e_i f_i$, using the filtration by powers of $\mathfrak{P}_i$.
In the Galois case, the Galois group acts transitively on the $\mathfrak{P}_i$, and conjugate primes have the same $e$ and $f$.

Example: in $\Q(\sqrt{-5})/\Q$, with $n = 2$: the prime $2$ has $e = 2, f = 1, g = 1$; the prime $3$ has $e = 1, f = 1, g = 2$; the prime $11$ is inert since $-5$ is not a square modulo $11$, with $e = 1, f = 2, g = 1$.
In each case $efg = 2$.

## Explanations

### L1 human

When a prime ideal of a small field breaks into primes of a larger field, the pieces must add up to the degree of the extension.
Each piece counts with its exponent times the size of its residue field extension.
So a prime in a quadratic field either splits into two, stays prime with a bigger residue field, or becomes a square.

### L2 human

The identity limits the possible splitting types and is used constantly in computations.
In a Galois extension, $e f$ is the order of the [[decomposition group|7X92G8CQ33]] of any prime over $\mathfrak{p}$, and $e$ is the order of its inertia group.
For a cyclic extension of prime degree $\ell$, the identity leaves only three cases: split completely ($g = \ell$), inert ($f = \ell$) or totally ramified ($e = \ell$).
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<document source="https://tamnd.github.io/math2/node/RERRK4RFZE.md" tag="RERRK4RFZE" kind="definition">
# Affine independence

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $P$ be an [[affine space|09XNAPTJQ7]] over $V$.
Let $p_0, \dots, p_k$ be points of $P$.
They are *affinely independent* if the vectors $p_1 - p_0, \dots, p_k - p_0$ of $V$ are [[linearly independent|0CRMGPAH83]].

## Remarks

Example: in $\R^2$, the points $(0,0)$, $(1,0)$, $(0,1)$ are affinely independent, since the vectors $(1,0)$ and $(0,1)$ are linearly independent.
Non-example: the points $(0,0)$, $(1,1)$, $(2,2)$ are not, since $(2,2) = 2 \cdot (1,1)$.
The condition does not depend on which point plays the role of $p_0$.
An equivalent form: for all $\lambda_0, \dots, \lambda_k \in F$ with $\sum_i \lambda_i = 0$ and $\sum_i \lambda_i (p_i - p_0) = 0$, all $\lambda_i$ are $0$.
Another equivalent form: no $p_j$ is an [[affine combination|0KJH6SRFZK]] of the other points.
In an affine space of dimension $n$, at most $n + 1$ points can be affinely independent.
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<document source="https://tamnd.github.io/math2/node/RF3Y3CF0YF.md" tag="RF3Y3CF0YF" kind="definition">
# Carleson measure

## Definition

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]] and $\mathbb{T}$ the unit circle.
For an arc $I \subseteq \mathbb{T}$ of length $\ell(I)$, the *Carleson box* over $I$ is
$$S(I) = \set{r \zeta : \zeta \in I, \ 1 - \ell(I) \le r < 1}.$$
A positive [[Borel measure|DYR1QFNGNJ]] $\mu$ on $\mathbb{D}$ is a *Carleson measure* if there is a constant $C$ such that
$$\mu(S(I)) \le C \, \ell(I) \quad \text{for every arc } I \subseteq \mathbb{T}.$$

## Remarks

Carleson's embedding theorem: for $0 < p < \infty$, $\mu$ is a Carleson measure if and only if there is $C_p$ with $\int_{\mathbb{D}} \abs{f}^p \, d\mu \le C_p \norm{f}_{H^p}^p$ for all $f$ in the [[Hardy space|PNNNFJVF6W]] $H^p$.

Example: area measure $A$ on $\mathbb{D}$.
In polar coordinates, $A(S(I)) \le \ell(I) \int_{\max(0, 1 - \ell(I))}^1 r \, dr \le \ell(I) \min(\ell(I), 1) \le \ell(I)$.
So $A$ is a Carleson measure with $C = 1$.

Non-example: let $z_n = 1 - 2^{-n}$ and $\mu = \sum_{n \ge 1} n^{-2} \delta_{z_n}$, a finite measure.
Let $I_k$ be the arc of length $2^{-k}$ centred at $1$.
Then $S(I_k)$ contains $z_n$ for all $n \ge k$, so $\mu(S(I_k)) \ge \sum_{n \ge k} n^{-2} > 1/k$.
The ratio $\mu(S(I_k)) / \ell(I_k) > 2^k / k$ is unbounded, so $\mu$ is not a Carleson measure.
</document>

<document source="https://tamnd.github.io/math2/node/RF66FF634A.md" tag="RF66FF634A" kind="definition">
# Tjurina number

## Definition

Let $\mathcal{O}_n = \C\{x_1, \ldots, x_n\}$ be the ring of [[germs|VZXF7RNFKS]] at $0$ of [[holomorphic functions|0NAHXR1FRH]] on $\C^n$, that is, convergent power series.
Let $f \in \mathcal{O}_n$ with $f(0) = 0$, and let $I$ be the [[ideal|096Z9MP2ME]] generated by $f$ and its partial derivatives.
The *Tjurina number* of $f$ is
$$\tau(f) = \dim_\C \mathcal{O}_n / I, \qquad I = \left( f, \frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n} \right).$$
It may be infinite.

## Remarks

The quotient is the Tjurina algebra of $f$.
The number $\tau(f)$ is finite if and only if the hypersurface $f = 0$ has an isolated singular point at $0$ or is smooth there.
Example: for $f = x^2 + y^3$ the ideal is $(x^2 + y^3, 2x, 3y^2) = (x, y^2)$, so $\mathcal{O}_2 / (x, y^2)$ has basis $1, y$ and $\tau(f) = 2$.
Non-example: for $f = x^2$ on $\C^2$ the ideal is $(x)$, and $\mathcal{O}_2 / (x) \cong \C\{y\}$ is infinite-dimensional; the hypersurface $x^2 = 0$ is singular along the whole line $x = 0$, so the singularity is not isolated.
The Milnor number is $\mu(f) = \dim_\C \mathcal{O}_n / (\partial f / \partial x_1, \ldots, \partial f / \partial x_n)$, so $\tau(f) \le \mu(f)$.
For an isolated singularity, $\tau(f) = \mu(f)$ if and only if $f$ is quasi-homogeneous in suitable coordinates, by a theorem of K. Saito.
The Tjurina number depends only on the hypersurface germ, not on the choice of $f$: if $u$ is a unit of $\mathcal{O}_n$, then $uf$ and $f$ give the same ideal.
</document>

<document source="https://tamnd.github.io/math2/node/RF8WD9TSE7.md" tag="RF8WD9TSE7" kind="example">
# Grassmannian as classifying space

## Example

Let $n \ge 1$.
For $k \ge n$, let $G_n(\R^k)$ be the [[Grassmannian|CYPA882RB9]] of $n$-dimensional linear subspaces of $\R^k$.
The inclusions $\R^k \subset \R^{k+1}$ give inclusions $G_n(\R^k) \subset G_n(\R^{k+1})$, and $G_n(\R^\infty) = \bigcup_k G_n(\R^k)$ carries the weak topology of this union.
The tautological bundle
$$\gamma^n = \set{(V, v) \in G_n(\R^\infty) \times \R^\infty : v \in V}$$
is a real vector bundle of rank $n$ over $G_n(\R^\infty)$.

Let $X$ be a paracompact space.
Then $[f] \mapsto f^* \gamma^n$ is a bijection from homotopy classes of maps $X \to G_n(\R^\infty)$ to isomorphism classes of real rank $n$ bundles over $X$.
Rank $n$ real vector bundles correspond to principal $O(n)$-bundles through their bundles of orthonormal frames.
So $G_n(\R^\infty)$ is a [[classifying space|WSZ40H5E4T]] $BO(n)$ of the orthogonal group $O(n)$.
The frame bundle of $\gamma^n$ is the universal bundle; its total space is the infinite [[Stiefel manifold|W7RCXS3CHT]] $V_n(\R^\infty)$, which is contractible.

## Remarks

For $n = 1$, $G_1(\R^\infty) = \R P^\infty$, and real line bundles over a CW complex $X$ correspond to classes in $H^1(X; \Z/2\Z)$ through the first [[Stiefel–Whitney class|JQCK7A210W]].
In the same way, the complex Grassmannian $G_n(\C^\infty)$ is $BU(n)$, and $\C P^\infty = BU(1)$ classifies complex line bundles.
The cohomology ring $H^*(G_n(\R^\infty); \Z/2\Z)$ is the polynomial ring over $\Z/2\Z$ on $w_1(\gamma^n), \ldots, w_n(\gamma^n)$, which is why every mod $2$ [[characteristic class|7HACJC0JE2]] of rank $n$ bundles is a polynomial in Stiefel–Whitney classes.

## Explanations

### L1 human

A rank $n$ vector bundle over $X$ attaches an $n$-dimensional space of vectors to each point of $X$.
Place every such space inside one huge space $\R^\infty$.
Then the bundle becomes a map that sends each point of $X$ to an $n$-dimensional subspace, that is, to a point of the Grassmannian.
Conversely, any map into the Grassmannian gives a bundle: over each point of $X$, take the subspace it is sent to.
So the infinite Grassmannian lists all vector bundles of rank $n$ at once.
</document>

<document source="https://tamnd.github.io/math2/node/RFA8HAEW9D.md" tag="RFA8HAEW9D" kind="statement">
# Gauss–Jacobi quadrature

## Statement

Let $\alpha, \beta > -1$ and let $\omega(x) = (1 - x)^{\alpha} (1 + x)^{\beta}$ on $(-1, 1)$.
Let $n \ge 1$ and let $x_1, \ldots, x_n$ be the zeros of the [[Jacobi polynomial|TRD7DSF2KR]] $P_n^{(\alpha, \beta)}$; they are real, simple and lie in $(-1, 1)$.
Let $\ell_1, \ldots, \ell_n$ be the [[Lagrange basis polynomials|Y7KSGE97GK]] of these nodes, and set $w_i = \int_{-1}^{1} \omega(x) \ell_i(x) \, dx$.
Then $w_i = \int_{-1}^{1} \omega(x) \ell_i(x)^2 \, dx > 0$ for each $i$, and for every polynomial $p$ of degree at most $2n - 1$,
$$\int_{-1}^{1} (1 - x)^{\alpha} (1 + x)^{\beta} p(x) \, dx = \sum_{i=1}^{n} w_i \, p(x_i).$$

## Remarks

The proof is the general argument for [[Gaussian quadrature|FP37BTP93K]].
Divide $p$ by $P_n^{(\alpha,\beta)}$ and use the orthogonality of $P_n^{(\alpha,\beta)}$ to all polynomials of lower degree for the weight $\omega$.
Then integrate the remainder, of degree at most $n - 1$, by interpolation.
Applying the exactness to $\ell_i^2$, of degree $2n - 2$, gives $w_i = \int \omega \ell_i^2 > 0$.
Example: $\alpha = \beta = 0$ gives Gauss–Legendre quadrature.
Example: $\alpha = \beta = -\tfrac12$ gives Gauss–Chebyshev quadrature, with nodes $x_i = \cos\frac{(2i - 1)\pi}{2n}$ and equal weights $w_i = \pi / n$; for $n = 1$ the rule is $\pi f(0)$, and indeed $\int_{-1}^1 (1 - x^2)^{-1/2} dx = \pi$.
The condition $\alpha, \beta > -1$ is needed for $\omega$ to be integrable; for $\alpha \le -1$ even $\int \omega$ diverges.

## Explanations

### L1 human

Some integrals on $[-1, 1]$ have a factor that blows up or vanishes at an endpoint, such as $1/\sqrt{1 - x}$.
Ordinary rules handle such endpoint behaviour badly.
Gauss–Jacobi quadrature builds the factor $(1-x)^\alpha (1+x)^\beta$ into the weights, so that only the smooth part of the integrand is sampled.
With $n$ points it is exact for that factor times any polynomial of degree up to $2n - 1$.

### L2 human

The family contains the Gauss–Legendre and Gauss–Chebyshev rules as special cases.
It is used for integrands with known algebraic endpoint singularities, for example in boundary integral methods and in spectral methods based on Jacobi polynomials.
The nodes and weights are computed in practice from the three-term recurrence of the Jacobi polynomials, as eigenvalues and eigenvector components of a symmetric tridiagonal matrix.
</document>

<document source="https://tamnd.github.io/math2/node/RFCQT6TKZF.md" tag="RFCQT6TKZF" kind="statement">
# Euler's theorem in geometry

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane.
Let $O$ and $R$ be the centre and radius of its [[circumcircle|03VZJE6C8C]], and $I$ and $r$ the centre and radius of its [[incircle|07DFG5JYPT]].
Then the distance $d = OI$ satisfies
$$d^2 = R(R - 2r).$$
In particular $R \ge 2r$, with equality if and only if the triangle is equilateral.

## Remarks

Let the line $AI$ meet the circumcircle again at $M$, the midpoint of the arc $BC$ not containing $A$.
An angle chase shows $MI = MB$, and the law of sines in the triangle $ABM$ gives $MB = 2R \sin \frac{A}{2}$.
Also $AI = r / \sin \frac{A}{2}$, so $AI \cdot IM = 2Rr$.
The power of the point $I$ with respect to the circumcircle is $R^2 - d^2 = AI \cdot IM$, and the formula follows.

The inequality $R \ge 2r$ is Euler's inequality.
Equality $d = 0$ means that the [[circumcentre|0F98FZK14A]] and the [[incentre|0M9MX8YW0A]] coincide, which happens only for an equilateral triangle.

Example: for the right triangle with sides $3$, $4$ and $5$, the circumradius is $R = \frac{5}{2}$ and the inradius is $r = 1$.
The formula gives $d^2 = \frac{5}{2} \cdot \frac{1}{2} = \frac{5}{4}$.
With $C = (0, 0)$, $A = (4, 0)$, $B = (0, 3)$ one has $O = (2, \frac{3}{2})$ and $I = (1, 1)$, and indeed $OI^2 = 1 + \frac{1}{4} = \frac{5}{4}$.

## Explanations

### L1 human

Every triangle has a circle through its corners and a circle inside touching its sides.
Euler's theorem gives the distance between their centres from the two radii alone.
As a consequence, the outer circle is always at least twice as large as the inner one.
The two centres meet only when the triangle is equilateral.

### L2 human

The theorem is a basic tool for inequalities in triangle geometry, through $R \ge 2r$.
It also enters the proof of Feuerbach's theorem via the distance between the incentre and the nine-point centre.
Its converse is a closure statement.
For two circles with $d^2 = R(R - 2r)$, every point of the outer circle is a vertex of a triangle inscribed in one and circumscribed about the other.
This is the case of triangles in [[Poncelet's closure theorem|XWYXSRM5YS]].
</document>

<document source="https://tamnd.github.io/math2/node/RFK1AVB9H7.md" tag="RFK1AVB9H7" kind="definition">
# Approximate derivative

## Definition

Let $I \subseteq \R$ be an open interval, let $f \colon I \to \R$ be a function, and let $x \in I$.
Write $\lambda$ for [[Lebesgue measure|2Z2QVWZ262]].
A measurable set $E \subseteq \R$ has *density $1$* at $x$ if
$$\lim_{h \to 0^+} \frac{\lambda(E \cap [x - h, x + h])}{2h} = 1.$$
The function $f$ is *approximately differentiable* at $x$, with *approximate derivative* $c \in \R$, if the following holds.
There is a measurable set $E \subseteq I$ with $x \in E$ and density $1$ at $x$ such that
$$\lim_{\substack{y \to x \\ y \in E \setminus \set{x}}} \frac{f(y) - f(x)}{y - x} = c.$$
The approximate derivative is written $f'_{\mathrm{ap}}(x)$.

## Remarks

Example: let $f(1/n) = 1$ for $n \ge 2$ and $f(y) = 0$ at all other $y \in (-1, 1)$.
At $x = 0$ the ordinary [[derivative|0F1WR9C83K]] does not exist, since the quotient at $y = 1/n$ equals $n$.
The set $E = (-1, 1) \setminus \set{1/n : n \ge 2}$ has density $1$ at $0$, because the removed set is countable, and on $E$ the quotient is $0$; so $f'_{\mathrm{ap}}(0) = 0$.

Non-example: let $f(y) = 1$ for $y \ge 0$ and $f(y) = 0$ for $y < 0$, at $x = 0$.
For $y < 0$ the quotient is $-1/y$, which tends to $+\infty$ as $y \to 0^-$.
Every set of density $1$ at $0$ contains points $y < 0$ arbitrarily close to $0$.
So $f$ has no approximate derivative at $0$.

If $f'(x)$ exists, then $f'_{\mathrm{ap}}(x)$ exists and equals it, by taking $E = I$.
The approximate derivative is unique when it exists, because two sets of density $1$ at $x$ meet in a set of density $1$ at $x$.
</document>

<document source="https://tamnd.github.io/math2/node/RFKHKD4KXY.md" tag="RFKHKD4KXY" kind="definition">
# Green's relations

## Definition

Let $S$ be a [[semigroup|0XMEM406JM]].
Let $S^1 = S$ if $S$ is a [[monoid|0N6R902FG1]], and otherwise let $S^1$ be $S$ with an identity element $1$ adjoined.
For $a, b \in S$, *Green's relations* are:

1. $a \mathrel{\mathcal{L}} b$ if $S^1 a = S^1 b$.
2. $a \mathrel{\mathcal{R}} b$ if $a S^1 = b S^1$.
3. $a \mathrel{\mathcal{J}} b$ if $S^1 a S^1 = S^1 b S^1$.
4. $a \mathrel{\mathcal{H}} b$ if $a \mathrel{\mathcal{L}} b$ and $a \mathrel{\mathcal{R}} b$.
5. $a \mathrel{\mathcal{D}} b$ if there is $c \in S$ with $a \mathrel{\mathcal{L}} c$ and $c \mathrel{\mathcal{R}} b$.

## Remarks

All five are [[equivalence relations|0G8VEZ3WTK]]; for $\mathcal{D}$ this uses the fact that $\mathcal{L}$ and $\mathcal{R}$ commute as relations.
Equivalently, $a \mathrel{\mathcal{L}} b$ if and only if $a = xb$ and $b = ya$ for some $x, y \in S^1$.
Always $\mathcal{H} \subseteq \mathcal{L}, \mathcal{R} \subseteq \mathcal{D} \subseteq \mathcal{J}$, and $\mathcal{D} = \mathcal{J}$ when $S$ is finite.

Example: let $T_X$ be the monoid of all maps of a finite set $X$ to itself, with $(fg)(x) = f(g(x))$.
If $f = gh$, then the image of $f$ is contained in the image of $g$.
Conversely, if the images are equal, choose $h(x)$ to be a preimage of $f(x)$ under $g$; then $f = gh$.
So $f \mathrel{\mathcal{R}} g$ if and only if $f$ and $g$ have the same image.
Similarly, $f \mathrel{\mathcal{L}} g$ if and only if $f$ and $g$ have the same fibres, and $f \mathrel{\mathcal{D}} g$ if and only if their images have the same size.

Non-example: in the monoid $(\N, +)$, the left ideal generated by $a$ is $a + \N$, so $a \mathrel{\mathcal{L}} b$ only when $a = b$.
Unlike in a group, the identity $0$ and the element $1$ are not $\mathcal{L}$-related.
</document>

<document source="https://tamnd.github.io/math2/node/RG15F0TWPW.md" tag="RG15F0TWPW" kind="concept">
# Compact-open topology

## Summary

The compact-open topology is a standard topology on the set $C(X, Y)$ of continuous maps between two topological spaces.
Its basic open sets ask that the maps send a given compact set into a given open set.
When $Y$ is a metric space, it is the topology of uniform convergence on compact sets.
It is the usual topology on mapping spaces in homotopy theory.

## Explanations

### L1 human

To say that two continuous maps are close, you can look at them on a compact piece of the source.
The compact-open topology says a map is near $f$ when it sends each chosen compact piece into an open target region that $f$ also sends it into.
For maps from $[0, 1]$ to $\R$, this is the same as being uniformly close on all of $[0, 1]$.
For maps from $\R$ to $\R$, it means being uniformly close on each bounded closed interval, but not necessarily on all of $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/RG1HFAYD98.md" tag="RG1HFAYD98" kind="concept">
# Stieltjes transform

## Summary

The Stieltjes transform of a finite measure $\mu$ on the real line is the function $S_\mu(z) = \int \frac{d\mu(t)}{t - z}$ of a complex variable $z$ off the real axis.
It is a [[holomorphic function|0NAHXR1FRH]] that encodes $\mu$ completely, and the measure can be read back from its boundary values on the real axis.
It is central in the [[moment problem|CS7TWHY7JM]], in spectral theory, and in random matrix theory, where limiting eigenvalue distributions are found through their Stieltjes transforms.
For a function $f$ on $(0, \infty)$ the classical form is $\int_0^\infty \frac{f(t)}{x + t}\, dt$, which is the [[Laplace transform|0V49JEZW06]] applied twice.

## Explanations

### L1 human

Think of $\mu$ as a distribution of electric charge along the real line.
The Stieltjes transform at a point $z$ of the plane adds up the contributions $1/(t - z)$ of all charges, a kind of complex potential.
When $z$ comes close to the real line at a point $x$, the imaginary part feels mostly the charge near $x$, so it reveals the density of $\mu$ there.
For one unit charge at $0$, the transform is $-1/z$.
</document>

<document source="https://tamnd.github.io/math2/node/RG52DZ1A8Y.md" tag="RG52DZ1A8Y" kind="concept">
# Strong dual space

## Summary

The strong dual space of a [[topological vector space|T32W553W95]] $E$ is its [[continuous dual|4A82S5FVRT]] $E'$ with a certain topology.
It is the topology of uniform convergence on the [[bounded subsets|3BAJ8ZTVPX]] of $E$.
For a normed space this is just the norm topology of the dual norm.
The strong dual is the default topology on $E'$ in the theory of reflexivity.
A locally convex space $E$ is reflexive when the canonical map from $E$ to the strong dual of its strong dual is a topological isomorphism.

## Explanations

### L1 human

There are several ways to say that two continuous linear forms are close.
The weakest asks that they agree nearly at each single vector.
The strong one asks that they agree nearly, uniformly, on every bounded set of vectors.
For a normed space, being close on the unit ball is the same as being close in the dual norm, so the strong dual is the usual normed dual.
</document>

<document source="https://tamnd.github.io/math2/node/RG6WFFW7K3.md" tag="RG6WFFW7K3" kind="definition">
# Kähler differentials

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $A$ be a commutative [[$R$-algebra|DPBJ04JMH8]].
For an $A$-[[module|0NZ8JA25WK]] $M$, an *$R$-derivation* $D \colon A \to M$ is an $R$-linear map with $D(ab) = a \, D(b) + b \, D(a)$ for all $a, b \in A$; see [[derivation|3J89REV9WQ]].
The *module of Kähler differentials* $\Omega_{A/R}$ is the $A$-module with one generator $da$ for each $a \in A$ and the following relations, for all $a, b \in A$ and $r \in R$:

1. $d(a + b) = da + db$.
2. $d(ab) = a \, db + b \, da$.
3. $d(r \cdot 1) = 0$.

The map $d \colon A \to \Omega_{A/R}$, $a \mapsto da$, is the *universal derivation*.

## Remarks

The map $d$ is an $R$-derivation, and it is universal: for each $R$-derivation $D \colon A \to M$ there is a unique $A$-linear map $\varphi \colon \Omega_{A/R} \to M$ with $D = \varphi \circ d$.
An equivalent form: if $I$ is the kernel of the multiplication map $A \otimes_R A \to A$ (see [[tensor product of algebras|1735VGRT2P]]), then $\Omega_{A/R} \cong I/I^2$, with $da$ corresponding to the class of $a \otimes 1 - 1 \otimes a$.

Example: for the [[polynomial ring|0SDC934R52]] $A = R[x_1, \ldots, x_n]$, the module $\Omega_{A/R}$ is free with basis $dx_1, \ldots, dx_n$, and $df = \sum_i \frac{\partial f}{\partial x_i} \, dx_i$, with [[formal partial derivatives|0EGN1E1JPB]].
For a quotient $A = R[x]/(f)$, the module $\Omega_{A/R}$ is $A \, dx / (f'(x) \, dx) \cong A/(f'(x))$.
So $\Omega_{\Q(\sqrt 2)/\Q} = 0$, because $f = x^2 - 2$ and $f'(\sqrt 2) = 2\sqrt 2$ is a unit.
For $A = \F_p(t^{1/p})$ over $R = \F_p(t)$, with $f = x^p - t$, one has $f' = 0$, so $\Omega_{A/R} \cong A \ne 0$.

Non-example: the identity map $A \to A$ is not a derivation when $A \ne 0$, because it gives $1 = D(1 \cdot 1) = 2 \cdot D(1) = 2$, which fails in $A$ unless $1 = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/RGCZ2P9QRM.md" tag="RGCZ2P9QRM" kind="statement">
# Bing metrization theorem

## Statement

Let $X$ be a [[topological space|0CQHFPGJ45]].
A family $\mathcal{F}$ of subsets of $X$ is *discrete* if each point of $X$ has a neighbourhood that meets at most one member of $\mathcal{F}$.
A family is *$\sigma$-discrete* if it is a union of countably many discrete families.
Then $X$ is [[metrizable|091GJ3FYD7]] if and only if $X$ is [[regular|03HMJASQBQ]], [[Hausdorff|0S1FJ8VEX9]] and has a $\sigma$-discrete base.

## Remarks

A discrete family is locally finite, so one direction follows from the [[Nagata–Smirnov metrization theorem|KB11T6389G]].
For the other direction, take a metric space.
The cover by balls of radius $1/n$ has a $\sigma$-discrete open refinement, built with a well-order of the balls.
The union over $n$ of these refinements is a $\sigma$-discrete base.
In the real line, the open intervals with rational endpoints form a countable base, and a countable base is $\sigma$-discrete.

## Explanations

### L1 human

The theorem is another exact test for when a topology comes from a distance.
It asks for basic open sets in countably many layers, where in each layer the sets are spread apart.
Near each point at most one set of the layer is present.

### L2 human

The theorems of Bing and of Nagata–Smirnov were found independently and give the same class of spaces.
Bing's form is often handy in proofs, because a discrete family behaves like a disjoint family of sets that are also kept apart from each other.
In the same work Bing proved that a collectionwise normal [[Moore space|68MA0RM5P6]] is metrizable.
</document>

<document source="https://tamnd.github.io/math2/node/RGGXDRCJVE.md" tag="RGGXDRCJVE" kind="concept">
# Gauss–Legendre quadrature

## Summary

Gauss–Legendre quadrature is [[Gaussian quadrature|FP37BTP93K]] for the constant weight $1$ on the interval $[-1, 1]$.
Its $n$ nodes are the zeros of the [[Legendre polynomial|QFYGJVA996]] $P_n$, and it integrates every polynomial of degree at most $2n - 1$ exactly.
An integral over any bounded interval $[a, b]$ is handled by a linear change of variable.
It is the default Gaussian rule for smooth functions on a bounded interval.

## Explanations

### L1 human

To integrate a function over $[-1, 1]$, sample it at a few special points and add the samples with special weights.
The points are not equally spaced: they crowd towards the two ends of the interval.
With two points, at $\pm 1/\sqrt{3}$, and weights $1$, the rule already gives the exact integral of every cubic.
For example, for $f(x) = x^2$ it gives $1/3 + 1/3 = 2/3$, which is $\int_{-1}^{1} x^2\,dx$.
</document>

<document source="https://tamnd.github.io/math2/node/RGT7WEZG8X.md" tag="RGT7WEZG8X" kind="definition">
# Spline interpolation

## Definition

Let $n \ge 1$, let $a = x_0 < x_1 < \cdots < x_k = b$ with $k \ge 1$, and let $y_0, \dots, y_k \in \R$.
A *spline interpolant of degree $n$* of the data $(x_i, y_i)$ is a [[spline|84TJ8KQ926]] $s$ of degree $n$ with knots $x_0, \dots, x_k$ such that $s(x_i) = y_i$ for $i = 0, \dots, k$.

## Remarks

The splines of degree $n$ with these knots form a space of dimension $n + k$, and the interpolation conditions are $k + 1$ equations.
So $n - 1$ further conditions are needed for uniqueness.
For $n = 1$ the interpolant is unique: it is the broken line through the data.
For $n = 3$ two end conditions are added.
Examples are $s''(a) = s''(b) = 0$ for the natural [[cubic spline|V42H0R3EKX]], and prescribed values of $s'(a)$ and $s'(b)$ for the clamped cubic spline.
Each of these choices gives a unique interpolant.

Example: data $(0, 0)$, $(1, 1)$, $(2, 0)$ with $n = 1$.
The broken line $s(x) = x$ on $[0, 1]$ and $s(x) = 2 - x$ on $[1, 2]$ is continuous, linear on each piece and takes the values $0, 1, 0$, so it is the spline interpolant of degree $1$.

Non-example: for the same data, $s(x) = 2x - x^2$ takes the values $0, 1, 0$.
But it is not a spline of degree $1$, because its pieces have degree $2$.
It is a [[polynomial interpolant|SS1YMS2NPC]], not a linear spline interpolant.
</document>

<document source="https://tamnd.github.io/math2/node/RGT9HH7SQQ.md" tag="RGT9HH7SQQ" kind="definition">
# Hardy–Littlewood circle method

## Definition

For real $\alpha$ write $e(\alpha) = e^{2\pi i \alpha}$.
Let $\mathcal{S}$ be a finite set of positive integers, and let $s \ge 1$ and $N \ge 1$ be integers.
Let $r(N)$ be the number of $s$-tuples $(x_1, \ldots, x_s) \in \mathcal{S}^s$ with $x_1 + \cdots + x_s = N$.
Let $f(\alpha) = \sum_{x \in \mathcal{S}} e(x\alpha)$, an [[exponential sum|97PBMQCPPM]].
The *circle method* consists of the following steps.

1. Write $r(N) = \int_0^1 f(\alpha)^s\, e(-N\alpha)\, d\alpha$.
2. Choose a parameter $Q$ and split the circle $[0, 1)$ into *major arcs* $\mathfrak{M}$ and *minor arcs* $\mathfrak{m} = [0, 1) \setminus \mathfrak{M}$. The major arcs are the points near a rational $a/q$ with $q \le Q$ and $\gcd(a, q) = 1$.
3. On $\mathfrak{M}$, approximate $f$ by explicit functions to get a main term, usually the product of a singular series and a singular integral.
4. On $\mathfrak{m}$, bound $\abs{f}$ uniformly, for example by Weyl's inequality or Vinogradov's estimates, to show that $\int_{\mathfrak{m}}$ is smaller than the main term.

## Remarks

Step 1 holds because $\int_0^1 e(m\alpha)\, d\alpha$ equals $1$ for $m = 0$ and $0$ for every other integer $m$; expanding $f(\alpha)^s$ gives one term $e((x_1 + \cdots + x_s - N)\alpha)$ for each tuple.
Example: for $\mathcal{S} = \set{1, 2}$, $s = 2$ and $N = 3$, $f(\alpha)^2 e(-3\alpha) = e(-\alpha) + 2 + e(\alpha)$, whose integral is $2$, matching the tuples $(1, 2)$ and $(2, 1)$.
Non-example: the interval of integration must be a full period; $\int_0^{1/2} e(\alpha)\, d\alpha = i/\pi \ne 0$, so integrating over $[0, 1/2)$ does not isolate the solutions.
For [[Waring's problem|7VWGA444K3]], $\mathcal{S}$ is the set of $k$-th powers up to $N$; for [[Vinogradov's theorem|C01Z9QRGZY]], one uses primes weighted by $\log p$.
Hardy and Ramanujan worked with power series $\sum a_n z^n$ and integrals over a circle $\abs{z} = \rho < 1$, which is the origin of the name.
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# Mild solution

## Definition

Let $X$ be a [[Banach space|06GZ956657]], and let $A$ be the [[generator|60HSWZBTK9]] of a [[C0-semigroup|EVF180QJXY]] $(T(t))_{t \ge 0}$ on $X$.
Let $\tau > 0$, $u_0 \in X$, and let $f \colon [0, \tau] \to X$ be Bochner integrable.
A *mild solution* of the initial value problem
$$u'(t) = A u(t) + f(t) \text{ for } 0 < t \le \tau, \qquad u(0) = u_0$$
is the continuous function $u \colon [0, \tau] \to X$ given by
$$u(t) = T(t) u_0 + \int_0^t T(t - s) f(s) \, ds \quad \text{for } 0 \le t \le \tau.$$

## Remarks

The formula is the variation of constants formula, the analogue of [[Duhamel's principle|Q28DM6ZCMF]] for linear systems of ordinary differential equations.
The mild solution always exists and is unique.
A classical solution, that is, a continuously differentiable $u$ with values in the domain of $A$ that satisfies the equation, is equal to the mild solution.
Conversely, if $u_0$ lies in the domain of $A$ and $f$ is continuously differentiable on $[0, \tau]$, the mild solution is a classical solution.

Example: let $X = \R$, $A = a$ for a real number $a \ne 0$, so $T(t) = e^{at}$, and let $f = 1$.
The mild solution is $u(t) = e^{at} u_0 + (e^{at} - 1)/a$, and it satisfies $u' = a e^{at} u_0 + e^{at} = a u + 1$ with $u(0) = u_0$.

A mild solution need not be classical.
Let $X = L^2(\R)$, $A = d/dx$ with domain the Sobolev space $H^1(\R)$, so $T(t)$ is the shift $(T(t) g)(x) = g(x + t)$, and let $f = 0$.
For $u_0$ the indicator function of $[0, 1]$, the mild solution $u(t) = u_0(\cdot + t)$ is not differentiable as a map into $L^2(\R)$, so the problem has no classical solution.

Non-example: for $X = \R$, $A = 0$ and $f = 0$, the function $u(t) = t$ is not a mild solution of $u' = 0$, $u(0) = 0$.
Here $T(t) = 1$, so the formula forces $u(t) = 0$ for all $t$.
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# Order embedding

## Summary

An order embedding is a map between [[partially ordered sets|011C29Q1S1]] that both preserves and reflects the order.
It places a copy of one poset inside another, with exactly the same comparisons.
An order embedding is automatically injective, and a surjective order embedding is an [[order isomorphism|0Y74KN99A5]].

## Explanations

### L1 human

An order embedding copies one ordered set into another without changing any comparison.
The map $n \mapsto 2n$ from the natural numbers to the natural numbers is an order embedding: $m \le n$ exactly when $2m \le 2n$.
The inclusion of the rationals into the reals is another one.
A map that only keeps "smaller" as "smaller", but can make incomparable elements comparable, is not an order embedding.
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# Banach–Mazur compactum

## Description

Let $n \ge 1$ be an integer.
For two real [[normed spaces|03J5RAKB30]] $X$ and $Y$ of dimension $n$, the Banach–Mazur distance is
$$d(X, Y) = \inf \set{\norm{T} \, \norm{T^{-1}} : T \colon X \to Y \text{ a linear isomorphism}}.$$
Here $\norm{T}$ is the [[operator norm|06EKW1AKXF]].
The Banach–Mazur compactum $\mathcal{Q}_n$ is the set of isometry classes of $n$-dimensional real normed spaces with the metric $\delta(X, Y) = \log d(X, Y)$.
It is a compact [[metric space|0VKD1TP7F3]].

## Remarks

The function $\delta$ is a metric on isometry classes.
It is symmetric because $\norm{T^{-1}} \, \norm{T}$ is the same number for $T$ and $T^{-1}$.
The triangle inequality comes from $\norm{ST} \le \norm{S} \, \norm{T}$, so $d(X, Z) \le d(X, Y) \, d(Y, Z)$.
If $d(X, Y) = 1$, then a compactness argument gives an isomorphism with $\norm{T} \, \norm{T^{-1}} = 1$, and a multiple of it is an isometry.

By John's theorem, $d(X, \ell_2^n) \le \sqrt{n}$ for every $n$-dimensional normed space $X$, where $\ell_2^n$ is $\R^n$ with the Euclidean norm.
So the diameter of $\mathcal{Q}_n$ in the multiplicative form is at most $n$.
Gluskin proved that this order is right: there are $n$-dimensional spaces at distance at least $c n$ from each other, for a constant $c > 0$ that does not depend on $n$.

## Explanations

### L1 human

Each point of the Banach–Mazur compactum is a way to measure length in $n$-dimensional space, up to a change of coordinates that keeps lengths.
Two such ways are close when a linear change of coordinates can squeeze one unit ball between the other unit ball and a slightly larger copy of it.
For $n = 2$, the square ball of the max norm and the round disc of the Euclidean norm are at distance $\log \sqrt{2}$.
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# Whitney extension theorem

## Statement

Let $n \ge 1$ and $m \ge 0$ be integers and let $A \subseteq \R^n$ be a [[closed set|0ZCK8DQYN1]].
For each [[multi-index|0PJXBR6QA3]] $\alpha$ with $\abs{\alpha} \le m$, let $f_\alpha \colon A \to \R$ be continuous.
For $x, y \in A$ and $\abs{\alpha} \le m$, set
$$R_\alpha(x, y) = f_\alpha(x) - \sum_{\abs{\beta} \le m - \abs{\alpha}} \frac{f_{\alpha + \beta}(y)}{\beta!} (x - y)^\beta.$$
Assume that for every compact set $K \subseteq A$ and every $\abs{\alpha} \le m$,
$$\frac{\abs{R_\alpha(x, y)}}{\norm{x - y}^{m - \abs{\alpha}}} \to 0 \quad \text{as } \norm{x - y} \to 0,$$
uniformly for $x, y \in K$ with $x \ne y$.
Then there is a function $F \colon \R^n \to \R$ of [[class $C^m$|0TCBZMEYGN]] such that $D^\alpha F = f_\alpha$ on $A$ for every $\abs{\alpha} \le m$.
Moreover, $F$ can be chosen of class $C^\infty$ on $\R^n \setminus A$.

## Remarks

The family $(f_\alpha)$ is a list of candidate derivatives of $F$ on $A$, called a jet.
The condition says that the candidate values obey [[Taylor's theorem|0J8T4K4YTE]] uniformly on compact parts of $A$.
It is necessary: if $F$ is of class $C^m$ on $\R^n$, then the restrictions of $D^\alpha F$ to $A$ satisfy it, by Taylor's theorem.
For $m = 0$ the condition holds automatically, because a continuous function is uniformly continuous on compact sets, and the theorem gives a continuous extension, as the Tietze extension theorem does.
The proof covers $\R^n \setminus A$ by cubes whose size is comparable to their distance to $A$.
It then glues Taylor polynomials taken at nearby points of $A$ with a partition of unity subordinate to these cubes.

## Explanations

### L1 human

Suppose the values of a function and of its derivatives are prescribed only on a closed set.
The question is whether a smooth function on all of space has exactly these values and derivatives there.
Whitney's theorem says yes, as soon as the prescribed values fit together the way Taylor expansions of a smooth function would.

### L2 human

The theorem turns questions about functions on arbitrary closed sets into questions about jets, which are easier to handle.
It is used to define smooth functions on closed subsets, and in the study of ideals of smooth functions.
For $n = 1$ and $m = 1$ the condition says that $f_0$ is differentiable along $A$ with derivative $f_1$, uniformly on compact sets.
The related [[Borel's lemma|KZHNA19FSS]] extends a jet given at a single point.
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# Representation of a quiver

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $Q = (Q_0, Q_1, s, t)$ be a [[quiver|RE9YFFS3TS]].
A *representation* $V$ of $Q$ over $k$ consists of:

1. a [[vector space|000AYPA9AJ]] $V_i$ over $k$ for each vertex $i \in Q_0$;
2. a [[linear map|0NRYHJPQG2]] $V_a \colon V_{s(a)} \to V_{t(a)}$ for each arrow $a \in Q_1$.

A *morphism* $\varphi \colon V \to W$ of representations is a family of linear maps $\varphi_i \colon V_i \to W_i$, $i \in Q_0$, such that $\varphi_{t(a)} \circ V_a = W_a \circ \varphi_{s(a)}$ for every arrow $a$.
If $Q_0$ is finite and each $V_i$ is finite-dimensional, $V$ is *finite-dimensional*, and its *dimension vector* is $(\dim V_i)_{i \in Q_0}$.

## Remarks

Example: for the quiver $1 \to 2$ with arrow $a$, the data $V_1 = V_2 = k$ and $V_a = \id_k$ form a representation with dimension vector $(1, 1)$.
Non-example: for the same quiver, a vector space at each vertex with a linear map $V_2 \to V_1$ is not a representation.
Clause 2 asks for a map from the source of $a$ to its target.
Direct sums are taken vertexwise, and a nonzero representation is indecomposable if it is not a direct sum of two nonzero representations.
For a finite quiver, representations of $Q$ correspond to left modules over the [[path algebra|2PYANBGXFD]] $kQ$, and this is an equivalence of categories.
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# Butcher tableau

## Summary

A Butcher tableau is the table of coefficients that specifies a [[Runge–Kutta method|7DBPJKYC6C]].
It lists the nodes $c_i$ in a column, the matrix $(a_{ij})$ beside them, and the weights $b_i$ in a bottom row.
Its shape shows at a glance whether the method is explicit, with $(a_{ij})$ strictly lower triangular, or implicit.
Order conditions, stability functions and the algebraic theory of the [[Butcher group|6MH1DHMCPF]] are all expressed in terms of these coefficients.

## Explanations

### L1 human

A Runge–Kutta method mixes several slope evaluations within each step.
The Butcher tableau is a compact recipe that says where in the step each slope is measured and how the slopes are combined.
For example, the tableau with nodes $0, 1$, the single entry $a_{21} = 1$, and weights $\frac12, \frac12$ is [[Heun's method|RJEK60G28F]].
Different tables give different methods, and catalogues of methods are usually written as lists of tableaux.
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# Structural rigidity

## Summary

Rigidity theory studies when a framework of rigid bars joined at flexible joints cannot be deformed except by moving it as a whole.
A framework is a [[graph|00C0BR7D7H]] together with positions of its vertices in [[Euclidean space|0XPR1E85W0]], with each edge a bar of fixed length.
The subject links geometry, combinatorics and linear algebra, and it is used in engineering, chemistry and robotics.
Classical results include Cauchy's rigidity theorem for convex polyhedra and Laman's characterisation of generically rigid graphs in the plane.

## Explanations

### L1 human

Build a shape out of sticks and hinges and ask whether it wobbles.
A square of four sticks wobbles: it can be pushed into a rhombus.
A triangle of three sticks does not, since three side lengths fix a triangle.
Adding one diagonal stick to the square splits it into two triangles and makes it rigid.
Rigidity theory turns this into precise mathematics and counts how many sticks a rigid structure needs.
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# Nontangential limit

## Summary

A function on the unit ball has a nontangential limit at a boundary point if it converges along every approach inside a cone.
The cones have their vertex at the boundary point.
Such cones, called Stolz angles in the plane, keep the approach away from tangential directions.
Nontangential limits are the right notion of boundary value for harmonic and holomorphic functions.
[[Fatou's theorem|Q2N917JQ6A]] says that Poisson integrals and bounded holomorphic functions have them almost everywhere.

## Explanations

### L1 human

Walk towards a point on the boundary circle of the unit disc.
If you come in at an angle, staying inside a fixed wedge with tip at the point, you approach nontangentially.
A function has a nontangential limit if it tends to one value along all such walks.
Walks that hug the circle, tangential walks, are allowed to give other values.
For example, $\exp\left(-\frac{1 + z}{1 - z}\right)$ tends to $0$ at $1$ nontangentially, but has modulus $1/e$ on a circle through $1$ tangent to the unit circle.
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# Steiner chain

## Example

Let $\alpha$ and $\beta$ be two [[circles|060W5CBZP6]] in the plane with $\beta$ inside $\alpha$ and no common point.
A *Steiner chain* for $\alpha$ and $\beta$ is a cyclic sequence of $n \ge 3$ circles $\gamma_1, \ldots, \gamma_n$ with two properties.
Each $\gamma_i$ is tangent to $\alpha$ from inside and to $\beta$ from outside.
Each $\gamma_i$ is tangent to $\gamma_{i+1}$, where $\gamma_{n+1} = \gamma_1$.

Take $\alpha$ of radius $3$ and $\beta$ of radius $1$, both centred at the origin.
Let $\gamma_k$ be the circle of radius $1$ centred at $2(\cos(k\pi/3), \sin(k\pi/3))$ for $k = 1, \ldots, 6$.
The centre of $\gamma_k$ is at distance $2 = 3 - 1$ from the origin, so $\gamma_k$ touches $\alpha$ from inside.
It is also at distance $2 = 1 + 1$, so $\gamma_k$ touches $\beta$ from outside.
Two consecutive centres are at distance $2 \cdot 2 \sin(\pi/6) = 2$, the sum of the radii, so $\gamma_k$ and $\gamma_{k+1}$ touch.
So these six circles form a Steiner chain.

## Remarks

For concentric circles of radii $R > r$, a chain of $n$ circles that goes once around $\beta$ exists exactly when $\sin(\pi/n) = (R - r)/(R + r)$.
Steiner's porism says that if one chain closes up for $\alpha$ and $\beta$, then a chain closes up starting from any first circle tangent to both.
The proof uses an [[inversion in a circle|YXST7NGMTC]] that sends $\alpha$ and $\beta$ to concentric circles, where rotation moves one chain to any other.

## Explanations

### L1 human

Put a small coin inside a large ring, and fill the space between them with coins that touch both and touch their neighbours.
For the right sizes, the coins close up into a full necklace.
Then the necklace can be rolled around: start it from any position and it still closes up.
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# Poincaré asymptotic series

## Definition

Let $D$ be a subset of $\C$ that is unbounded, for example a sector or a ray, and let $f \colon D \to \C$.
A formal series $\sum_{n \ge 0} a_n z^{-n}$ with complex coefficients is a *Poincaré asymptotic series* of $f$ as $z \to \infty$ in $D$ if for every $N \ge 0$
$$z^N \Big( f(z) - \sum_{n=0}^{N} a_n z^{-n} \Big) \to 0 \quad \text{as } z \to \infty \text{ in } D.$$
This is written $f(z) \sim \sum_{n \ge 0} a_n z^{-n}$.
It is the [[asymptotic expansion|BJQTEPEZ9G]] of $f$ in the [[asymptotic scale|XEGB8Y44GG]] $\varphi_n(z) = z^{-n}$.

## Remarks

An equivalent condition is that $f(z) - \sum_{n=0}^{N} a_n z^{-n} = O(z^{-N-1})$ for every $N$.
The coefficients are determined by $f$: $a_0 = \lim f(z)$ and $a_N = \lim z^N \big( f(z) - \sum_{n<N} a_n z^{-n} \big)$.
The same definition with $(z - z_0)^n$ in place of $z^{-n}$ gives Poincaré series at a finite point $z_0$.

Example: $f(z) = \frac{1}{z - 1}$ on $D = (2, \infty)$ has $f(z) \sim \sum_{n \ge 1} z^{-n}$.
Check: $f(z) - \sum_{n=1}^{N} z^{-n} = \frac{z^{-N}}{z - 1}$, and $z^N$ times this is $\frac{1}{z - 1} \to 0$.

Non-example: $f(z) = \sin z$ on $D = (0, \infty)$ has no Poincaré asymptotic series, because $a_0 = \lim_{z \to \infty} \sin z$ does not exist.

The function $e^{-z}$ on $(0, \infty)$ has the zero series, like the function $0$, so the series does not determine the function.
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# Poincaré half-plane model

## Example

The *Poincaré half-plane model* is the [[upper half-plane|X6XGGH5FPW]] $\mathbb{H} = \set{(x, y) \in \R^2 : y > 0}$ with the [[Riemannian metric|Q61VFHCH2J]]
$$ds^2 = \frac{dx^2 + dy^2}{y^2}.$$
Its [[geodesics|NSYMQMHPDN]], the lines of the geometry, are the vertical half-lines and the semicircles centred on the $x$-axis.
The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is given by
$$\cosh d = 1 + \frac{(x_2 - x_1)^2 + (y_2 - y_1)^2}{2 y_1 y_2}.$$

Check of the distance on the $y$-axis: the length of the segment from $(0, 1)$ to $(0, e)$ is $\int_1^e dy/y = 1$.
The formula gives $\cosh d = 1 + (e - 1)^2/(2e) = (e^2 + 1)/(2e) = \cosh 1$, so $d = 1$ as well.

Check of the hyperbolic parallel axiom: take the line $x = 0$ and the point $(1, 1)$.
The vertical line $x = 1$ does not meet it.
The semicircle centred at $(2, 0)$ through $(1, 1)$ has radius $\sqrt 2$, so it stays in $x \ge 2 - \sqrt 2 > 0$ and does not meet it either.
So $\mathbb{H}$ is a model of [[hyperbolic geometry|GN4N94GQ0Z]].

## Remarks

The metric has constant Gaussian curvature $-1$.
The orientation-preserving isometries are the [[Möbius transformations|0SG74WBFA2]] $z \mapsto (az + b)/(cz + d)$ with $a, b, c, d \in \R$ and $ad - bc = 1$, written with $z = x + iy$.
So the group of these isometries is $\mathrm{PSL}_2(\R)$.
The model is conformal: hyperbolic angles equal Euclidean angles.

## Explanations

### L1 human

The half-plane model draws the whole hyperbolic plane inside the region above a horizontal line.
Distances grow as you approach that line, so the line itself is infinitely far away.
The straight lines of the geometry look like vertical rays and half-circles standing on the boundary line.
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# Parabolic and hyperbolic Riemann surfaces

## Summary

An open Riemann surface is hyperbolic, in the sense of potential theory, when it carries a nonconstant negative subharmonic function, or equivalently a Green's function.
It is parabolic when it is noncompact and carries no such function.
This splits the open surfaces into those that are "large", like the disc, and those that are "small", like the plane.
On a parabolic surface Brownian motion is recurrent, while on a hyperbolic surface it is transient.

## Explanations

### L1 human

Think of a surface as a sheet with a heat source at one point and the far-away part held at temperature zero.
On a hyperbolic surface heat escapes to infinity well enough that a steady temperature profile exists; this profile is the Green's function.
On a parabolic surface, such as the plane, heat cannot escape fast enough and no such profile exists.
The unit disc is hyperbolic, and the complex plane is parabolic.
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# Graeffe's method

## Statement

Let $p(x) = a \prod_{i=1}^n (x - x_i)$ be a polynomial of degree $n \ge 1$ with [[complex|09FTAET5Z4]] coefficients and [[roots|019MA1VTR2]] $x_1, \dots, x_n$.

1. There is a unique polynomial $q$ with $q(x^2) = (-1)^n p(x) p(-x)$, and $q(y) = a^2 \prod_{i=1}^n (y - x_i^2)$. So the roots of $q$ are the squares of the roots of $p$.
2. Let $p_0 = p$, let $p_{k+1}$ be obtained from $p_k$ as in clause 1, and write $p_k(y) = \sum_{j=0}^n b_j^{(k)} y^j$. Suppose that $\abs{x_1} > \abs{x_2} > \cdots > \abs{x_n} > 0$. Then for each $m = 1, \dots, n$,
$$\abs{x_m} = \lim_{k \to \infty} \abs{\frac{b_{n-m}^{(k)}}{b_{n-m+1}^{(k)}}}^{1/2^k}.$$

## Remarks

Clause 1: $p(-x) = a \prod_i (-x - x_i) = (-1)^n a \prod_i (x + x_i)$, so $(-1)^n p(x) p(-x) = a^2 \prod_i (x^2 - x_i^2)$.
Clause 2: the roots of $p_k$ are $y_i = x_i^{2^k}$, and their moduli separate very fast.
By the relations between roots and coefficients, $b_{n-m}^{(k)} / b_{n-m+1}^{(k)}$ is a ratio of elementary symmetric functions of the $y_i$, and the ratio is dominated by $-y_m$ as $k \to \infty$.
For example, $p(x) = x^2 - 3x + 2$ has roots $2$ and $1$, and one step gives $q(y) = y^2 - 5y + 4$, with roots $4$ and $1$.
The hypothesis of distinct moduli is needed for clause 2.
For $p(x) = x^3 - 1$ the three roots have modulus $1$, but every $p_k$ equals $y^3 - 1$, so $b_2^{(k)} / b_3^{(k)} = 0$ and the formula gives $0$ instead of $1$ for $m = 1$.
The method gives only the moduli of the roots; the signs or arguments need extra work.

## Explanations

### L1 human

Squaring numbers that differ in size makes the gap between them much larger.
Graeffe's method repeatedly replaces a polynomial by one whose roots are the squares of the old roots.
After a few rounds one root dominates each coefficient ratio, and the sizes of the roots can be read off directly from the coefficients.

### L2 human

Each squaring step costs about $n^2$ operations and needs no starting guess.
The coefficients grow or shrink very fast, so a practical version must work with scaled numbers or logarithms.
The squaring step, also called the Dandelin–Graeffe step, reappears in modern root finders, for example to estimate root moduli in the splitting circle method.
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# Beeman's algorithm

## Statement

Let $a \colon \R^d \to \R^d$ be continuous, and consider the equation $x'' = a(x)$.
Fix a step size $h > 0$ and write $a_n = a(x_n)$.
Beeman's algorithm computes, from $x_n$, $v_n$, $a_n$ and $a_{n-1}$,
$$x_{n+1} = x_n + h v_n + \frac{h^2}{6} (4 a_n - a_{n-1}),$$
$$v_{n+1} = v_n + \frac{h}{6} (2 a_{n+1} + 5 a_n - a_{n-1}).$$
Then for all $n \ge 2$ for which the sequences are defined,
$$x_{n+1} = 2 x_n - x_{n-1} + h^2 a_n.$$
So Beeman's algorithm produces the same positions as the [[Störmer–Verlet method|ESWE8Y3JD4]], once both start from the same $x_1$ and $x_2$.

## Remarks

Subtract the position formula at step $n - 1$ from the one at step $n$:
$$x_{n+1} - 2x_n + x_{n-1} = h (v_n - v_{n-1}) + \frac{h^2}{6} (4 a_n - 5 a_{n-1} + a_{n-2}).$$
The velocity formula gives $h (v_n - v_{n-1}) = \frac{h^2}{6}(2 a_n + 5 a_{n-1} - a_{n-2})$, and the sum is $h^2 a_n$.
The difference from Verlet lies only in the velocities, which Beeman's formula computes more accurately than the central difference $(x_{n+1} - x_{n-1})/(2h)$; this improves the computed kinetic energy.
The velocity formula above is the implicit form; a common explicit variant predicts $v_{n+1}$ first when the force depends on the velocity.

## Explanations

### L1 human

Beeman's method is a molecular dynamics integrator that uses the forces at the current and previous steps.
Its positions turn out to be exactly those of the simpler Verlet method.
What it adds is a more accurate formula for the velocities, so that the energy is measured more accurately.

### L2 human

The identity shows that Beeman's method inherits the stability and the long-time behaviour of Verlet's positions.
It is a multistep method, so it needs a starting procedure to produce $a_{-1}$ or $x_1$.
In practice velocity Verlet is more common, because it gives similar accuracy with less storage.
</document>

<document source="https://tamnd.github.io/math2/node/RHWD8DY41R.md" tag="RHWD8DY41R" kind="concept">
# Exponential object

## Summary

An exponential object $Z^Y$ in a category with [[products|S5744QZV4N]] is an object of morphisms from $Y$ to $Z$, together with an evaluation morphism.
It is defined by a [[universal property|EN4DD2FAJH]] that expresses currying: maps $X \times Y \to Z$ match maps $X \to Z^Y$.
Exponential objects are the defining feature of [[Cartesian closed categories|BFBNQ36QS8]].
They give the semantics of function types in [[lambda calculus|BD756KR8FM]].

## Explanations

### L1 human

A function of two variables $g(x, y)$ can be seen as a function of $x$ whose values are functions of $y$.
This trick is called currying.
For sets, the exponential $Z^Y$ is the set of all functions from $Y$ to $Z$.
For example, if $Y$ has $2$ elements and $Z$ has $3$, then $Z^Y$ has $3^2 = 9$ elements.
An exponential object is the same idea in any category with products.
</document>

<document source="https://tamnd.github.io/math2/node/RHXP3HA64M.md" tag="RHXP3HA64M" kind="statement">
# Knaster–Tarski theorem

## Statement

Let $(L, \le)$ be a [[complete lattice|D7A0ZGZT5S]] and let $f \colon L \to L$ be a [[monotone|0A7KV2WB7A]] map, so that $x \le y$ implies $f(x) \le f(y)$.
Let $\operatorname{Fix}(f) = \set{x \in L : f(x) = x}$.
Then $\operatorname{Fix}(f)$, with the order of $L$, is a complete lattice.
In particular $f$ has a [[least fixed point|DBD64TH52D]] and a greatest fixed point.
They are given by
$$\operatorname{lfp}(f) = \inf \set{x \in L : f(x) \le x}, \qquad \operatorname{gfp}(f) = \sup \set{x \in L : x \le f(x)},$$
where $\inf$ and $\sup$ are the [[infimum|0QF1MZ9FRG]] and the [[supremum|0R3H5PANRQ]] in $L$.

## Remarks

Proof idea for the least fixed point: let $D = \set{x : f(x) \le x}$ and $m = \inf D$.
For $x \in D$, $m \le x$ gives $f(m) \le f(x) \le x$, so $f(m)$ is a lower bound of $D$ and $f(m) \le m$.
Then $f(f(m)) \le f(m)$, so $f(m) \in D$ and $m \le f(m)$; hence $f(m) = m$, and every fixed point lies in $D$, so $m$ is the least one.
Let $W$ be a set of fixed points with supremum $a$ in $L$.
The map $f$ sends the complete lattice $\set{x \in L : a \le x}$ into itself.
Its least fixed point there is the join of $W$ in $\operatorname{Fix}(f)$, and meets are handled dually.
The join of a set of fixed points in $\operatorname{Fix}(f)$ need not be its join in $L$.
Example: on the power set of $\N$, the map $f(S) = \set{0} \cup \set{n + 2 : n \in S}$ is monotone, and its least fixed point is the set of even numbers.
Monotonicity is needed: on the two-element lattice $\set{0, 1}$, the map that swaps $0$ and $1$ has no fixed point.
Completeness is needed too: on the lattice $\Z$ the monotone map $x \mapsto x + 1$ has no fixed point.

## Explanations

### L1 human

If a rule moves the points of a complete lattice and never reverses the order, then some points stay put.
There is a smallest such point and a largest one, and the fixed points even form a complete lattice themselves.
The smallest fixed point is the meet of all points that the rule pushes downward.
This is why inductive definitions, such as "the smallest set containing $0$ and closed under adding $2$", always make sense.

### L2 human

The theorem is the basic existence result for fixed points in order theory.
In logic and computer science it gives the semantics of inductive and coinductive definitions as least and greatest fixed points.
It also underlies the proof of the Cantor–Schröder–Bernstein theorem through a monotone map on a power set.
When $f$ is moreover Scott-continuous, the [[Kleene fixed-point theorem|DDAHG7QRY8]] computes the least fixed point as the supremum of the iterates $f^n(\bot)$.
A converse holds: a lattice in which every monotone self-map has a fixed point is complete.
</document>

<document source="https://tamnd.github.io/math2/node/RJ9767FFTF.md" tag="RJ9767FFTF" kind="definition">
# Black–Scholes equation

## Definition

Let $\sigma > 0$ and $r \ge 0$ be constants and let $T > 0$.
The *Black–Scholes equation* is the [[parabolic equation|1CCTH8X2FT]] for a function $V(S, t)$ with $S > 0$ and $0 \le t < T$:
$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0.$$
Here $S$ is the price of the underlying asset, modelled by [[geometric Brownian motion|D48DR46B97]] with volatility $\sigma$.
The constant $r$ is the risk-free interest rate, and $V$ is the price of a derivative security.
It is solved with a terminal condition $V(S, T) = \Phi(S)$ given by the payoff $\Phi$.

## Remarks

Example: $V(S, t) = S$ is a solution: $V_t = 0$, $V_{SS} = 0$, and $r S \cdot 1 - r S = 0$.
It is the value of holding one share.
Another solution is $V(S, t) = K e^{-r(T - t)}$, the value of a bond paying $K$ at time $T$: $V_t - r V = 0$.
Non-example: $V(S, t) = S^2$ fails, since $\frac{1}{2}\sigma^2 S^2 \cdot 2 + r S \cdot 2S - r S^2 = (\sigma^2 + r) S^2 \ne 0$.
With $x = \log S$ and $\tau = T - t$, the equation becomes a heat equation with constant drift and decay terms, so it is backward parabolic in $t$.
</document>

<document source="https://tamnd.github.io/math2/node/RJA5026RGX.md" tag="RJA5026RGX" kind="concept">
# Integral of a Hopf algebra

## Summary

A left integral in a [[Hopf algebra|7XFYP0N84H]] $H$ is an element $\Lambda$ that absorbs left multiplication through the counit: $h \Lambda = \varepsilon(h) \Lambda$ for all $h \in H$.
For a finite group algebra $KG$, the sum of all group elements is an integral.
Integrals generalize averaging over a finite group and the Haar measure of a compact group.
By the [[Larson–Sweedler theorem|BZPKCPCWTM]], the left integrals of a finite-dimensional Hopf algebra form a one-dimensional space.
Such an $H$ is semisimple exactly when $\varepsilon(\Lambda) \ne 0$ for a nonzero left integral $\Lambda$.

## Explanations

### L1 human

For a finite group, the sum $\sum_g g$ of all its elements does not change when it is multiplied by any element of the group, because multiplication only reorders the terms.
An integral is the Hopf algebra version of such an invariant sum.
Dividing it by the size of the group gives the averaging operator used to prove Maschke's theorem.
The criterion $\varepsilon(\Lambda) \ne 0$ is the Hopf algebra form of the condition that the size of the group is invertible in the field.
</document>

<document source="https://tamnd.github.io/math2/node/RJBDSA2ETP.md" tag="RJBDSA2ETP" kind="statement">
# Beth definability theorem

## Statement

Let $L$ be a signature, let $P$ be an $n$-ary relation symbol not in $L$, and let $T$ be a [[theory|92SWXST1ZM]] in the signature $L \cup \set{P}$.
Suppose that $P$ is implicitly defined by $T$.
That is, any two [[models|MVCH6N9SNW]] of $T$ with the same domain and the same interpretation of each symbol of $L$ interpret $P$ alike.
Then $P$ is explicitly defined by $T$.
That is, there is an $L$-[[formula|HYJ10AF6RT]] $\varphi(x_1, \dots, x_n)$ with
$$T \models \forall x_1 \cdots \forall x_n \, \big(P(x_1, \dots, x_n) \leftrightarrow \varphi(x_1, \dots, x_n)\big).$$

## Remarks

The converse is immediate: an explicit definition fixes $P$ in each [[structure|983GZBQYT4]] for $L$.
The standard proof uses the [[Craig interpolation theorem|E2CBFNAVZY]].
Take a copy $P'$ of $P$ and the copy $T'$ of $T$; implicit definability says $T \cup T' \models P(\bar x) \to P'(\bar x)$.
By compactness, finitely many sentences suffice, and an interpolant between the $P$-side and the $P'$-side is the required formula $\varphi$.

Example: let $L = \set{+, \cdot, 0, 1}$ and let $T$ be the theory of real closed ordered fields, with the order symbol $\le$ in the role of $P$.
In each model of $T$, the nonnegative elements are exactly the squares, so $\le$ is implicitly defined.
An explicit definition is $x \le y \leftrightarrow \exists z \, (y = x + z \cdot z)$.

## Explanations

### L1 human

Suppose some axioms pin down a new notion uniquely, once the old notions are fixed.
Beth's theorem says that the new notion can then be written as a formula in the old notions.
In first-order logic, a unique description always gives an explicit definition.

### L2 human

The theorem was proved by Evert Willem Beth in 1953.
It follows from Craig interpolation, and both results fail for many logics beyond first-order logic.
In abstract model theory, logics with the Beth property are studied as those in which implicit definitions can always be made explicit.
</document>

<document source="https://tamnd.github.io/math2/node/RJE1W5NRE3.md" tag="RJE1W5NRE3" kind="statement">
# Jensen's covering theorem

## Statement

Assume ZFC, and assume that [[zero sharp|HCTNBZZQ0W]] does not exist.
Then for every uncountable set $X$ of [[ordinals|0YSCSX5YZC]] there is a set $Y$ in the [[constructible universe|6TF2JBTPPJ]] $L$ such that $X \subseteq Y$ and $\abs{Y} = \abs{X}$.

## Remarks

The proof by Jensen uses the fine structure theory of $L$; one takes an elementary substructure that contains $X$ and analyses its transitive collapse.
A consequence: if $0^\sharp$ does not exist, then every singular cardinal $\kappa$ of $V$ is singular in $L$, and $(\kappa^+)^L = \kappa^+$.
It also follows that the singular cardinals hypothesis holds when $0^\sharp$ does not exist.
The hypothesis is necessary.
If $0^\sharp$ exists, then $\aleph_\omega$ is a regular cardinal in $L$.
Then the set $X = \omega_1 \cup \set{\aleph_n : n < \omega}$ of size $\aleph_1$ cannot be covered.
Indeed, a set $Y \in L$ with $\abs{Y} = \aleph_1$ has order type less than $\omega_2$.
So $Y \cap \aleph_\omega$ has $L$-cardinality less than $\aleph_\omega$, and it is bounded in $\aleph_\omega$.

## Explanations

### L1 human

The theorem says that, unless zero sharp exists, the constructible universe is a good approximation of the full universe.
Every uncountable set of ordinals sits inside a constructible set of the same size.
So the constructible universe cannot miss much: it has the same large singular cardinals and the same successors of them.
This gives a dichotomy: either the constructible universe is close to everything, or it is very thin.

### L2 human

The theorem is the model for many covering results for larger inner models, such as the Dodd–Jensen core model and the [[core model|A6ZJ8MSCAY]] in general.
It turns the nonexistence of a large cardinal object into strong combinatorial consequences, such as the singular cardinals hypothesis.
Read the other way, it gives lower bounds for consistency strength.
If the singular cardinals hypothesis fails, then $0^\sharp$ exists, and in fact much stronger large cardinal hypotheses hold in inner models.
</document>

<document source="https://tamnd.github.io/math2/node/RJEK60G28F.md" tag="RJEK60G28F" kind="statement">
# Heun's method

## Statement

Let $f \colon [t_0, T] \times \R^d \to \R^d$ be of class $C^2$, and consider the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$ with exact solution $y$.
*Heun's method* with step $h > 0$ computes a predictor $\tilde y_{n+1} = y_n + h f(t_n, y_n)$ and then
$$y_{n+1} = y_n + \frac{h}{2}\Big( f(t_n, y_n) + f(t_{n+1}, \tilde y_{n+1}) \Big), \qquad t_{n+1} = t_n + h.$$
Its [[local truncation error|T220HF0562]] is $O(h^3)$, so it is a Runge–Kutta method of order $2$.

## Remarks

Proof idea: by [[Taylor's theorem|03QHBVW4TT]], $f\big(t + h, y + h f\big) = f + h(f_t + f_y f) + O(h^2)$ at $(t, y)$, so the increment is $h f + \frac{h^2}{2}(f_t + f_y f) + O(h^3)$, which matches the Taylor expansion of $y(t + h)$ up to $h^2$.
In Butcher form it has $c = (0, 1)$, $a_{21} = 1$ and $b = (\tfrac12, \tfrac12)$.
Example: for $y' = y$, $y(0) = 1$ and $h = 0.1$, one step gives $\tilde y_1 = 1.1$ and $y_1 = 1 + 0.05 (1 + 1.1) = 1.105$, against $e^{0.1} \approx 1.10517$; the [[Euler method|0JARTXX4XP]] gives $1.1$.
If $f$ does not depend on $y$, the method is the trapezoidal rule for $\int f(t)\,dt$.

## Explanations

### L1 human

Heun's method first takes a trial Euler step to guess where the solution goes.
It then measures the slope at the guessed endpoint, averages it with the slope at the start, and redoes the step with this average slope.
The averaging cancels most of the error of the plain Euler step.

### L2 human

Heun's method is also called the improved Euler method or the explicit trapezoidal method.
It is the simplest predictor–corrector scheme: an explicit Euler predictor followed by one correction with the trapezoidal rule.
Like the [[midpoint method|WHK8D383YT]], it is a two-stage explicit [[Runge–Kutta method|7DBPJKYC6C]] of order $2$.
The difference between its result and the Euler predictor gives a cheap estimate of the local error for step size control.
</document>

<document source="https://tamnd.github.io/math2/node/RJJ9D0WF53.md" tag="RJJ9D0WF53" kind="concept">
# Unification

## Summary

Unification is the problem of finding a [[substitution|NCDZ969V7X]] that makes two terms equal.
Such a substitution is a unifier.
When two terms have a unifier, they have a most general one, from which every other unifier is obtained by a further substitution.
Robinson's unification algorithm computes it, and unification is at the core of resolution theorem proving, logic programming and type inference.

## Explanations

### L1 human

Two expressions with blanks are given, and the task is to fill in the blanks so that the two become the same expression.
For example, $f(x, b)$ and $f(a, y)$ become the same once $x$ is replaced by $a$ and $y$ by $b$.
The expressions $f(x)$ and $g(y)$ can never be made equal, because their outer symbols differ.
Neither can $x$ and $f(x)$, because $x$ would have to contain itself.
</document>

<document source="https://tamnd.github.io/math2/node/RJK6ZKNF6R.md" tag="RJK6ZKNF6R" kind="statement">
# Fundamental convergence theorem

## Statement

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]].
Let $(u_i)_{i \in I}$ be a nonempty family of [[superharmonic functions|X4V2S56YQQ]] on $\Omega$ that is locally uniformly bounded below.
Let $u = \inf_{i \in I} u_i$, and let $\hat u$ be its lower semicontinuous regularisation, $\hat u(x) = \min\left(u(x), \liminf_{y \to x} u(y)\right)$.
Then:

1. $\hat u$ is superharmonic on $\Omega$, and $\hat u \le u$.
2. The set $\set{x \in \Omega : \hat u(x) < u(x)}$ is a [[polar set|C95QJ7YH5T]].

## Remarks

The infimum $u$ satisfies the super-mean value inequality but need not be lower semicontinuous, so it need not be superharmonic.
Idea of the proof: Choquet's topological lemma gives a countable subfamily with the same regularised infimum.
Replacing it by its finite minima, which are superharmonic, reduces the problem to a decreasing sequence.
For a decreasing sequence, the limit keeps the super-mean value inequality by monotone convergence, and its regularisation is then shown to be superharmonic.
The exceptional set is shown to be polar by a capacity argument.
Example: in $\R^3$ let $u_k(y) = \min\left(1, k^{-1}\abs{y}^{-1}\right)$, which are positive superharmonic functions.
Their infimum is $0$ for $y \ne 0$ and $1$ at $y = 0$, while $\hat u = 0$; the exceptional set $\set{0}$ is polar.
The lower bound is needed: the constants $u_k = -k$ on $\R^n$ are superharmonic, but their infimum is $-\infty$ everywhere.

## Explanations

### L1 human

Take the lowest values of many superharmonic functions.
The result may fail to be superharmonic only because of a few bad points.
After a small repair at those points it is superharmonic again, and the bad points form a set that potential theory cannot see.

### L2 human

The theorem is the tool behind most limiting constructions in classical potential theory.
It shows that regularised reduced functions are superharmonic, which gives balayage and equilibrium potentials.
It also yields that the set where a reduced function differs from its regularisation is polar, and hence the quasi-everywhere statements of capacity theory.
</document>

<document source="https://tamnd.github.io/math2/node/RJYKTAPSBJ.md" tag="RJYKTAPSBJ" kind="example">
# Dihedral quandle

## Example

Let $n \ge 1$ and let $R_n = \Z/n\Z$ be the [[integers modulo $n$|03VFZ2WETT]], with
$$a \triangleright b = 2a - b.$$
Then $(R_n, \triangleright)$ is a [[quandle|X74VY86TSX]], the *dihedral quandle* of order $n$.

Check of the three axioms:

1. Self-distributivity: $a \triangleright (b \triangleright c) = 2a - 2b + c$ and $(a \triangleright b) \triangleright (a \triangleright c) = 2(2a - b) - (2a - c) = 2a - 2b + c$.
2. For fixed $a$ and $b$, the equation $a \triangleright c = b$ has the unique solution $c = 2a - b$.
3. Idempotence: $a \triangleright a = 2a - a = a$.

The name comes from the [[dihedral group|0JZ27DMJ5D]] $D_n$.
Let $s_a$ be the reflection $x \mapsto 2a - x$ of $\Z/n\Z$.
Then $s_a s_b s_a^{-1} = s_{2a - b}$, so the operation of $R_n$ is conjugation of reflections.
For odd $n$ the map $a \mapsto s_a$ is injective, and $R_n$ is the conjugacy class of reflections in $D_n$ with conjugation.

## Remarks

Colourings of a [[knot|QYEAZTYGPT]] diagram by $R_3$ are the Fox $3$-colourings: at each crossing, the three arcs carry either one colour or three different colours.
The trefoil has $9$ colourings by $R_3$, three of them constant, while the unknot has only the $3$ constant ones.
Since the number of colourings is a knot invariant, the trefoil is knotted.
In Mathlib this quandle is `Quandle.Dihedral`.

## Explanations

### L1 human

Place $n$ points evenly on a circle and reflect the circle in the line through point $a$ and the centre.
Point $b$ moves to point $2a - b$.
The dihedral quandle records exactly this rule.
These simple quandles already detect interesting knots, through the familiar game of colouring a knot diagram with three colours.
</document>

<document source="https://tamnd.github.io/math2/node/RJYSXGZTTE.md" tag="RJYSXGZTTE" kind="concept">
# Functional equations on groups

## Summary

A functional equation on a group is a [[functional equation|SRAN4EKWS0]] whose unknown functions are defined on a [[group|G5S1M9]].
The arguments of these functions are formed from the variables by the group operation and inversion.
The basic example is Cauchy's equation $f(xy) = f(x) f(y)$ for maps $f$ from $G$ to another group, whose solutions are exactly the [[group homomorphisms|0JYTG5K9Y4]].
Classical equations on $\R$, such as the d'Alembert, Jensen and quadratic equations, have analogues on arbitrary groups.
Their solutions are often described by homomorphisms, [[characters|H1WNVN5V7F]] and representations of the group, which links the subject to harmonic analysis.

## Explanations

### L1 human

Many functional equations on the real line only use addition and subtraction of the variables.
Any group has an operation and inverses, so the same equations make sense on any group.
For example, on the group of nonzero real numbers under multiplication, $f(xy) = f(x) + f(y)$ is solved by $f(x) = \ln \abs{x}$.
On non-commutative groups, such as groups of matrices, new phenomena appear, because $xy$ and $yx$ may differ.
</document>

<document source="https://tamnd.github.io/math2/node/RK2G9HN63E.md" tag="RK2G9HN63E" kind="concept">
# Fourier–Legendre series

## Summary

A Fourier–Legendre series expands a function on $[-1, 1]$ as a sum of [[Legendre polynomials|QFYGJVA996]].
It is the [[generalized Fourier series|SNJC87TBHD]] for the orthogonal system of Legendre polynomials in $L^2([-1, 1])$.
Since this system is complete, the series of every square-integrable function converges to it in the mean-square sense.
Such series appear when one solves the Laplace equation in spherical coordinates.

## Explanations

### L1 human

The Legendre polynomials $P_0, P_1, P_2, \dots$ play on the interval $[-1, 1]$ the role that sines and cosines play on a circle.
Each one is orthogonal to all the others, so one can read off how much of each $P_n$ a function contains by one integral.
Summing these pieces rebuilds the function.
For example, $x^2 = \tfrac13 P_0(x) + \tfrac23 P_2(x)$, where $P_0(x) = 1$ and $P_2(x) = (3x^2 - 1)/2$.
</document>

<document source="https://tamnd.github.io/math2/node/RK2N3QP4FQ.md" tag="RK2N3QP4FQ" kind="concept">
# Attainable null space

## Summary

For a continuous [[seminorm|FP5GVCN8GV]] $q$ on the [[Schwartz space|CG9JEATW3Q]] $\mathcal{S}$, the **attainable null space** $\mathcal{N}_q$ is a set of [[tempered distributions|AES2T5JPFP]].

A distribution $p$ in $\mathcal{S}'$ lies in $\mathcal{N}_q$ when some sequence $(\chi_k)$ in $\mathcal{S}$ converges to $p$ in $\mathcal{S}'$ and has $q(\chi_k)\to 0$.

Because $q$ is a seminorm, $\mathcal{N}_q$ is a [[linear subspace|00YC1D652K]] of $\mathcal{S}'$: sums and scalar multiples of such sequences are again such sequences.

So $\mathcal{N}_q$ collects the distributions that test functions of vanishing $q$-size can approximate.

For a positive-semi-definite operator $\mathrm{B}$ from $\mathcal{S}$ to $\mathcal{S}'$, the set $\mathcal{N}_{\mathrm{B}}$ is the case where $q$ is the seminorm induced by $\mathrm{B}$.
</document>

<document source="https://tamnd.github.io/math2/node/RK6DCJVJCR.md" tag="RK6DCJVJCR" kind="statement">
# Ramanujan's 1psi1 summation

## Statement

Let $a, b, q, z \in \C$ with $0 < \abs{q} < 1$, $a \ne 0$, $b \notin \set{q^{-m} : m \ge 0}$, and $\abs{b/a} < \abs{z} < 1$.
For an integer $n \ge 1$ put $(a; q)_{-n} = 1 / (a q^{-n}; q)_n$, so that $(a; q)_n$ is defined for all $n \in \Z$ through the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].
Then
$$\sum_{n=-\infty}^{\infty} \frac{(a; q)_n}{(b; q)_n} z^n = \frac{(q; q)_\infty\, (b/a; q)_\infty\, (az; q)_\infty\, (q/(az); q)_\infty}{(b; q)_\infty\, (q/a; q)_\infty\, (z; q)_\infty\, (b/(az); q)_\infty}.$$
In the notation of [[bilateral basic hypergeometric series|EB4N9RQR51]], the left side is ${}_1\psi_1(a; b; q, z)$.

## Remarks

The condition $\abs{z} < 1$ makes the part with $n \ge 0$ converge, and $\abs{b/a} < \abs{z}$ makes the part with $n < 0$ converge.
Proof idea: for $b = q$ the terms with $n < 0$ vanish and the identity is the [[q-binomial theorem|VVT27RB46B]]; then $b = q^{m+1}$ follows by a shift of the summation index.
Both sides are analytic in $b$ in a disc, and they agree on the points $q^{m}$, $m \ge 1$, which accumulate at $0$, so they agree everywhere by the identity theorem.
Example: with $b = 0$ (a limit) and $z$ replaced by $-z/a$, letting $a \to \infty$ turns the identity into the [[Jacobi triple product|JFP2DGMNXT]].
The identity is due to Ramanujan, who stated it without proof; Hahn and Jackson gave the first published proofs around 1950.

## Explanations

### L1 human

This formula sums a two-sided series, running over all integers $n$, whose terms are ratios of $q$-Pochhammer symbols.
The answer is a ratio of eight infinite products.
It is the two-sided version of the $q$-binomial theorem.

### L2 human

Many product formulas of the theory of partitions and of theta functions are special cases.
It is the starting point for the bilateral series ${}_r\psi_r$, of which Bailey's ${}_6\psi_6$ summation is the main generalization.
</document>

<document source="https://tamnd.github.io/math2/node/RK84ANP6BP.md" tag="RK84ANP6BP" kind="concept">
# Noncentral chi-squared distribution

## Summary

The noncentral chi-squared distribution is the distribution of a sum of squares of independent normal variables with unit variance and arbitrary means.
It has two parameters: the number $k$ of terms and the noncentrality $\lambda$, the sum of the squared means.
For $\lambda = 0$ it is the ordinary chi-squared distribution.
It describes the distribution of chi-squared test statistics under alternatives, so it is used to compute the power of these tests.

## Explanations

### L1 human

Throw a dart at a target in $k$ dimensions, with independent normal errors in each direction, but aim at a point at distance $\sqrt{\lambda}$ from the centre.
The squared distance of the dart from the centre has the noncentral chi-squared distribution.
When the aim is at the centre, this is the usual chi-squared distribution.
For example, if $X$ is normal with mean $2$ and variance $1$, then $X^2$ is noncentral chi-squared with $k = 1$ and $\lambda = 4$, and its mean is $1 + 4 = 5$.
</document>

<document source="https://tamnd.github.io/math2/node/RK8B7KEA3M.md" tag="RK8B7KEA3M" kind="definition">
# Adjoint integral equation

## Definition

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ be a [[kernel|GSMBXT7FPK]], and let $\lambda \in \C$.
Consider the [[integral equation of the second kind|8KJCN49V64]]
$$\varphi(x) - \lambda \int_a^b K(x, s) \varphi(s) \, ds = f(x), \quad x \in [a, b].$$
Its *adjoint kernel* is $K^*(x, s) = \overline{K(s, x)}$, where the bar is the [[complex conjugate|0FTH2E9TAS]].
For a given function $g \colon [a, b] \to \C$, the *adjoint integral equation* is
$$\psi(x) - \bar\lambda \int_a^b \overline{K(s, x)} \, \psi(s) \, ds = g(x), \quad x \in [a, b],$$
for the unknown function $\psi \colon [a, b] \to \C$.

## Remarks

Example: for $K(x, s) = i x$ on $[0, 1]$, the adjoint kernel is $K^*(x, s) = \overline{i s} = -i s$.
The check: put $x$ and $s$ in reverse order and conjugate.

Non-example: the equation with kernel $K(s, x)$ and parameter $\lambda$, without conjugation, is the adjoint only when $K$ is real and $\lambda$ is real.
For $K(x, s) = i x$ it gives the kernel $i s$, not $-i s$.

With the inner product $\ip{u}{v} = \int_a^b u(x) \overline{v(x)} \, dx$, the integral operators satisfy $\ip{K u}{v} = \ip{u}{K^* v}$ when $K$ is continuous or square-integrable.
For real kernels and real $\lambda$, the adjoint equation is often called the *transposed* or *associated* equation.
A kernel equal to its adjoint kernel is called Hermitian.
</document>

<document source="https://tamnd.github.io/math2/node/RK8RV2C998.md" tag="RK8RV2C998" kind="definition">
# Group representation

## Definition

Let $G$ be a [[group|G5S1M9]] and let $F$ be a field.
Let $V$ be a [[vector space|000AYPA9AJ]] over $F$, and let $GL(V)$ be the group of invertible [[linear maps|0NRYHJPQG2]] $V \to V$ under composition.
A *representation* of $G$ on $V$ is a [[group homomorphism|0JYTG5K9Y4]] $\rho \colon G \to GL(V)$.
That is, $\rho(gh) = \rho(g) \circ \rho(h)$ for all $g, h \in G$.
The dimension of $V$ is the degree of $\rho$.
Two representations $\rho \colon G \to GL(V)$ and $\sigma \colon G \to GL(W)$ are isomorphic if there is a linear isomorphism $T \colon V \to W$ with $T \circ \rho(g) = \sigma(g) \circ T$ for all $g \in G$.

## Remarks

Example: let $G = \Z / n\Z$ and $V = \C$.
The map $\rho(k) = e^{2\pi i k / n}$, viewed as multiplication on $\C$, is a representation of degree $1$, because $e^{2\pi i (k + l)/n} = e^{2\pi i k/n} e^{2\pi i l/n}$, and the value does not depend on the representative $k$ of its class.
Non-example: on $G = \Z / 2\Z$, the map sending $0$ to $\id$ and $1$ to $2 \cdot \id$ on $\C$ is not a representation, because $\rho(1 + 1) = \rho(0) = \id$ but $\rho(1)\rho(1) = 4 \cdot \id$.
When $V = F^n$, the group $GL(V)$ is the [[general linear group|0FCHE902D7]] $GL_n(F)$, and a representation is a matrix-valued homomorphism.
Equivalently, a representation is a linear action of $G$ on $V$, or a module over the [[group algebra|MNP2HBWB5Q]] $F[G]$.
</document>

<document source="https://tamnd.github.io/math2/node/RKCSWWVYTC.md" tag="RKCSWWVYTC" kind="definition">
# Cylindric algebra

## Definition

Let $\alpha$ be an [[ordinal|0YSCSX5YZC]].
A *cylindric algebra of dimension $\alpha$* is a structure $(A, +, \cdot, -, 0, 1, c_\kappa, d_{\kappa\lambda})_{\kappa, \lambda < \alpha}$ with the following properties.
The reduct $(A, +, \cdot, -, 0, 1)$ is a [[Boolean algebra|0XADJKM699]], each $c_\kappa$ is a unary operation on $A$, and each $d_{\kappa\lambda}$ is an element of $A$.
For all $x, y \in A$ and $\kappa, \lambda, \mu < \alpha$:

1. $c_\kappa 0 = 0$.
2. $x \le c_\kappa x$.
3. $c_\kappa (x \cdot c_\kappa y) = c_\kappa x \cdot c_\kappa y$.
4. $c_\kappa c_\lambda x = c_\lambda c_\kappa x$.
5. $d_{\kappa\kappa} = 1$.
6. If $\kappa \ne \lambda$ and $\kappa \ne \mu$, then $d_{\lambda\mu} = c_\kappa (d_{\lambda\kappa} \cdot d_{\kappa\mu})$.
7. If $\kappa \ne \lambda$, then $c_\kappa (d_{\kappa\lambda} \cdot x) \cdot c_\kappa (d_{\kappa\lambda} \cdot -x) = 0$.

Here $\le$ is the order of the Boolean algebra.

## Remarks

Example: let $U$ be a nonempty set and let $A$ be the set of all subsets of $U^\alpha$, with the set operations.
Put $c_\kappa X = \set{y \in U^\alpha : y \text{ agrees with some } x \in X \text{ outside the coordinate } \kappa}$ and $d_{\kappa\lambda} = \set{x \in U^\alpha : x_\kappa = x_\lambda}$.
For clause 7, a point $y$ in both sets would agree outside $\kappa$ with some $x \in X$ and some $x' \notin X$, both with $\kappa$-th coordinate equal to $y_\lambda$; then $x = x'$, a contradiction.
The other clauses are checked in the same way, and this is a cylindric set algebra.

Non-example: on a Boolean algebra with $0 \ne 1$, taking $c_\kappa x = 0$ for all $x$ fails clause 2 at $x = 1$.

A cylindric algebra is called representable if it is isomorphic to a subalgebra of a product of cylindric set algebras.
For dimension at least $3$, not every cylindric algebra is representable.
</document>

<document source="https://tamnd.github.io/math2/node/RKFB6AHPBE.md" tag="RKFB6AHPBE" kind="definition">
# Matrix summability method

## Definition

Let $A = (a_{nk})_{n, k \ge 0}$ be an infinite matrix of complex numbers.
For a complex [[sequence|0QE8STGJH2]] $s = (s_k)_{k \ge 0}$, the *$A$-transform* of $s$ is the sequence
$$(As)_n = \sum_{k=0}^{\infty} a_{nk} s_k \qquad (n \ge 0),$$
defined when each of these [[series|0JH4MN4K1C]] converges.
The sequence $s$ is *$A$-summable to $L$* if $(As)_n$ is defined for every $n$ and $(As)_n \to L$ as $n \to \infty$.
The resulting [[summability method|X80DZEXQDQ]], with $D_A$ the set of $A$-summable sequences and $A(s) = \lim_n (As)_n$, is the *matrix method* given by $A$.

## Remarks

Example: the Cesàro matrix has $a_{nk} = \frac{1}{n+1}$ for $k \le n$ and $a_{nk} = 0$ for $k > n$.
For $s = (1, 0, 1, 0, \dots)$ one has $(As)_n = \frac{\lceil (n+1)/2 \rceil}{n+1} \to \frac12$, so $s$ is $A$-summable to $\frac12$.
Non-example: for $a_{nk} = 2^{-k-1}$ and $s_k = 2^k$, the series $\sum_k a_{nk} s_k = \sum_k \frac12$ diverges, so $(As)_n$ is not defined and $s$ is not $A$-summable.
The identity matrix gives ordinary convergence.
Every matrix method is linear, because $A(\lambda s + \mu t) = \lambda As + \mu At$.

## Explanations

### L1 human

Multiply the infinite matrix by the sequence, viewed as an infinite column, and take the limit of the result.
</document>

<document source="https://tamnd.github.io/math2/node/RKJBNRKEDT.md" tag="RKJBNRKEDT" kind="example">
# Segre embedding

## Example

Let $k$ be an algebraically closed field and let $m, n \ge 0$.
Let $N = (m + 1)(n + 1) - 1$ and give $\mathbb{P}^N$ the homogeneous coordinates $z_{ij}$ for $0 \le i \le m$ and $0 \le j \le n$.
The *Segre embedding* is the map of [[projective spaces|ZT9A6JK8QF]]
$$\sigma \colon \mathbb{P}^m \times \mathbb{P}^n \to \mathbb{P}^N, \qquad ([x_0 : \dots : x_m], [y_0 : \dots : y_n]) \mapsto [x_i y_j]_{i, j}.$$

Check that it is well defined: scaling $x$ by $\lambda$ and $y$ by $\mu$ scales every $x_i y_j$ by $\lambda \mu \ne 0$, and some $x_i y_j$ is nonzero.

Check that it is injective: the matrix $(x_i y_j)$ is the outer product $x y^T$, of rank $1$.
Its nonzero rows are multiples of $y$ and its nonzero columns are multiples of $x$, so the point determines $[x]$ and $[y]$.

The image is the set of $[z]$ whose $(m+1) \times (n+1)$ matrix $(z_{ij})$ has rank $1$, that is, the zero set of the [[homogeneous polynomials|2C3PAQ7H50]]
$$z_{ij} z_{kl} - z_{il} z_{kj}.$$
This image is [[irreducible|SY2MWC434Z]], so it is a [[projective variety|A4H4CZ2V35]], and it is used to make $\mathbb{P}^m \times \mathbb{P}^n$ into a projective variety of dimension $m + n$.

For $m = n = 1$, the image is the quadric surface $z_{00} z_{11} - z_{01} z_{10} = 0$ in $\mathbb{P}^3$.
The fibres $\sigma(\set{a} \times \mathbb{P}^1)$ and $\sigma(\mathbb{P}^1 \times \set{b})$ are two families of lines on this surface.

## Explanations

### L1 human

A pair of points, one in each projective space, is turned into one point by multiplying all coordinates of the first with all coordinates of the second.
The result is a matrix of rank one, read as a single list of coordinates.
So the product of two projective spaces becomes a subset of a bigger projective space, cut out by the vanishing of all $2 \times 2$ minors.
For two projective lines, this gives the familiar saddle-shaped quadric surface, which is covered by two families of straight lines.
</document>

<document source="https://tamnd.github.io/math2/node/RKK3XHG4MR.md" tag="RKK3XHG4MR" kind="statement">
# Dirichlet's principle

## Statement

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]] with $C^1$ [[boundary|05D5EGMFYV]], and let $g$ be a [[continuous function|0X3WGX95FE]] on $\partial\Omega$.
Let $\mathcal{A}$ be the set of functions $w$ that are twice continuously differentiable on $\overline{\Omega}$ with $w = g$ on $\partial\Omega$.
Let $E[w] = \frac12 \int_\Omega \abs{\nabla w}^2 \, dx$ be the [[Dirichlet energy|YG3W8CBB3E]].
For $u \in \mathcal{A}$, the following are equivalent:

1. $u$ is [[harmonic|0DEB0JMN0W]] on $\Omega$.
2. $E[u] \le E[w]$ for every $w \in \mathcal{A}$.

## Remarks

Proof idea: for $w \in \mathcal{A}$ put $v = w - u$, which vanishes on $\partial\Omega$.
Then $E[w] = E[u] + \int_\Omega \nabla u \cdot \nabla v \, dx + E[v]$, and integration by parts gives $\int_\Omega \nabla u \cdot \nabla v \, dx = -\int_\Omega v \, \Delta u \, dx$.
If $u$ is harmonic, the middle term is $0$, so $E[w] \ge E[u]$.
If $u$ minimizes, the derivative of $t \mapsto E[u + tv]$ at $t = 0$ is $0$ for all such $v$, which forces $\Delta u = 0$.
The theorem does not say that a minimizer exists in $\mathcal{A}$; Weierstrass pointed out that this needs proof, and it was later justified by Hilbert.
Existence can fail even for continuous $g$: on the unit disc, $g(\theta) = \sum_{k \ge 1} k^{-2} \cos(2^k \theta)$ is continuous, but its harmonic extension $\sum_k k^{-2} r^{2^k} \cos(2^k\theta)$ has energy $\frac{\pi}{2}\sum_k 2^k k^{-4} = \infty$.

## Explanations

### L1 human

Among all ways to fill in a function with given boundary values, the harmonic one is the least wiggly.
It has the smallest Dirichlet energy.
This is why a soap film or a stretched membrane with a fixed rim takes the shape of a harmonic function, for small slopes.

### L2 human

Dirichlet's principle turns the [[Dirichlet problem|8D56GQ4AG8]] into a minimization problem.
In modern form, one minimizes the energy over the Sobolev space $H^1$ with given trace.
The minimizer exists by the Hilbert space projection theorem, and it is a weak solution of Laplace's equation.
Elliptic regularity then shows that the weak solution is a classical harmonic function.
The same variational idea gives existence for many other elliptic equations.
</document>

<document source="https://tamnd.github.io/math2/node/RKKWS7CDNF.md" tag="RKKWS7CDNF" kind="statement">
# Mackey–Arens theorem

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, let $(X, Y, \langle \cdot, \cdot \rangle)$ be a [[dual pair|R68HNB5SVR]] over $\mathbb{K}$, and let $\mathcal{T}$ be a [[locally convex|B6GY2KJMWW]] topology on $X$.
Say that $\mathcal{T}$ is *compatible* with the pair if the [[continuous dual|4A82S5FVRT]] of $(X, \mathcal{T})$ is exactly the set of linear forms $x \mapsto \langle x, y \rangle$ with $y \in Y$.
Then $\mathcal{T}$ is compatible with the pair if and only if
$$\sigma(X, Y) \subseteq \mathcal{T} \subseteq \tau(X, Y),$$
where $\sigma(X, Y)$ is the [[weak topology|P6KHVSTPZV]] and $\tau(X, Y)$ is the [[Mackey topology|W18EY4E5TH]].

## Remarks

The weak topology is compatible by construction, and it is the coarsest topology for which all the forms $\langle \cdot, y \rangle$ are continuous.
If $\mathcal{T}$ is compatible, each $\mathcal{T}$-neighbourhood of $0$ contains a closed convex balanced neighbourhood $U$ of $0$.
Its polar $U^\circ$ in $Y$ is convex, balanced and $\sigma(Y, X)$-compact by the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], and $U = U^{\circ\circ}$ by the [[bipolar theorem|FX5PWKCMBQ]].
So $U$ is a $\tau(X, Y)$-neighbourhood of $0$, which gives $\mathcal{T} \subseteq \tau(X, Y)$.
That $\tau(X, Y)$ itself is compatible is the harder half; it uses the [[bipolar theorem|FX5PWKCMBQ]] and a compactness argument in the algebraic dual of $Y$.
Example: for a normed space $E$, the weak topology $\sigma(E, E')$ and the norm topology are both compatible with the pair $(E, E')$, so they have the same continuous linear forms.
A consequence: all compatible topologies have the same closed convex sets.
Indeed, by the Hahn–Banach separation theorem, a closed convex set is an intersection of closed half-spaces defined by continuous forms.

## Explanations

### L1 human

Different topologies on a space can give the same continuous linear forms.
The theorem describes all of them at once: they are exactly the locally convex topologies between the weak topology and the Mackey topology.
So the set of continuous linear forms pins down a whole interval of topologies, with a smallest and a largest element.

### L2 human

In duality theory one often starts from a dual pair and asks which topologies on one side give the other side as dual.
The Mackey–Arens theorem answers this completely.
It explains why closed convex sets and bounded sets depend only on the dual pair: they are the same for all compatible topologies.
For bounded sets this is Mackey's theorem.
It also shows that the [[Mackey topology|W18EY4E5TH]] is natural: metrizable and barrelled locally convex spaces carry their Mackey topology.
The proof rests on the [[bipolar theorem|FX5PWKCMBQ]] and on [[Banach–Alaoglu|1BXEJXQ2YT]] compactness.
</document>

<document source="https://tamnd.github.io/math2/node/RKX9XJ0RVW.md" tag="RKX9XJ0RVW" kind="statement">
# Fermat's principle

## Statement

Let $\Omega \subseteq \R^3$ be a region filled with an isotropic optical medium of refractive index $n \colon \Omega \to (0, \infty)$.
For a piecewise smooth curve $\gamma \colon [a, b] \to \Omega$ define the *optical path length*
$$\mathcal{L}[\gamma] = \int_a^b n(\gamma(s))\, \abs{\gamma'(s)} \, ds.$$
Let $A$ and $B$ be two points of $\Omega$.
In geometric optics, the path of a light ray from $A$ to $B$ is a critical point of $\mathcal{L}$ among curves from $A$ to $B$.

## Remarks

Since light has speed $c/n$ in the medium, $\mathcal{L}[\gamma]/c$ is the travel time along $\gamma$, so the principle says the ray makes the travel time stationary.
The ray need not minimise the time: a ray reflected by a concave mirror can be a saddle point or a maximum.
When $n$ is smooth, the rays are the solutions of the [[Euler–Lagrange equation|PS0VV149ET]] of $\mathcal{L}$, which in arc length $\sigma$ reads $\frac{d}{d\sigma}\big(n\, \frac{dx}{d\sigma}\big) = \nabla n$.
Equivalently, the rays are the [[geodesics|NSYMQMHPDN]] of the metric $n^2 \abs{dx}^2$.
In a homogeneous medium ($n$ constant) the rays are straight lines.
Take two homogeneous media with indices $n_1, n_2$ separated by a plane.
Minimising $\mathcal{L}$ over broken lines gives Snell's law $n_1 \sin \theta_1 = n_2 \sin \theta_2$.
Here $\theta_i$ is the angle between the ray and the normal of the plane.
The principle is a statement of physics, not a theorem.
In wave optics it is recovered in the limit of short wavelength, where the phase solves the [[eikonal equation|HGMX1QGTAZ]] and the rays are its characteristic curves.

## Explanations

### L1 human

Light travelling from one point to another follows the path that takes the least time.
More exactly, its travel time does not change to first order under small changes of the path.
This explains why light goes straight in air, why it bends when it enters water, and why angles of incidence and reflection are equal at a mirror.

### L2 human

Fermat's principle is the model for all variational principles of physics.
The ray equations are the Euler–Lagrange equations of the optical length, and the rays are geodesics of the optical metric $n^2 \abs{dx}^2$.
The same formalism appears in mechanics through [[Maupertuis's principle|ZXY0AYKMXZ]], whose Jacobi metric plays the role of $n^2 \abs{dx}^2$.
</document>

<document source="https://tamnd.github.io/math2/node/RM0FMT0X8W.md" tag="RM0FMT0X8W" kind="definition">
# Local homomorphism

## Definition

Let $(R, \mathfrak m)$ and $(S, \mathfrak n)$ be [[local rings|MZ02BKPHJA]].
A [[ring homomorphism|0VTH8KPCG2]] $f \colon R \to S$ is a *local homomorphism* if $f(\mathfrak m) \subseteq \mathfrak n$.

## Remarks

Equivalent forms:

1. $f^{-1}(\mathfrak n) = \mathfrak m$;
2. for every $a \in R$, if $f(a)$ is a [[unit|0XVEZDE9J8]] of $S$, then $a$ is a unit of $R$.

Indeed $f^{-1}(\mathfrak n)$ is a proper ideal containing $\mathfrak m$ when $f(\mathfrak m) \subseteq \mathfrak n$, so it equals $\mathfrak m$.
A local homomorphism induces a field homomorphism $R/\mathfrak m \to S/\mathfrak n$ of [[residue fields|NJ3APQ8BHH]].

Example: the quotient map $\Z_{(p)} \to \Z/p\Z$ is local, since it sends $p\Z_{(p)}$ to $0$, the maximal ideal of the field $\Z/p\Z$.
For a prime ideal $\mathfrak q \subseteq \mathfrak p$ of a ring $R$, the natural map $R_{\mathfrak p} \to R_{\mathfrak q}$ is local only when $\mathfrak q = \mathfrak p$.

Non-example: the inclusion $\Z_{(p)} \to \Q$ is a ring homomorphism between local rings, but it is not local.
It sends $p \in p\Z_{(p)}$ to $p$, which is a unit of $\Q$ and so not in the maximal ideal $(0)$ of $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/RM54CPPE12.md" tag="RM54CPPE12" kind="concept">
# Complete elliptic integral of the first kind

## Summary

The complete elliptic integral of the first kind is the function $K(k) = \int_0^{\pi/2} d\theta / \sqrt{1 - k^2\sin^2\theta}$ of the modulus $k$.
It is a quarter of the real period of the [[Jacobi elliptic functions|P071B692JF]] $\operatorname{sn}$ and $\operatorname{cn}$.
It gives the exact period of a pendulum and can be computed very fast with the arithmetic–geometric mean.

## Explanations

### L1 human

For a pendulum with small swings, the period does not depend on the amplitude.
For large swings it does, and the correction is given by $K(k)$, where $k$ is the sine of half the largest angle.
At $k = 0$ the integrand is $1$ and $K(0) = \pi/2$.
As $k$ grows toward $1$ the integral grows without bound, because the pendulum takes longer and longer to swing near the top.
</document>

<document source="https://tamnd.github.io/math2/node/RM64J2CP1M.md" tag="RM64J2CP1M" kind="problem">
# Dedekind's problem

## Problem

For each integer $n \ge 0$, let $M(n)$ be the [[Dedekind number|481ZPQ0GC3]], the number of monotone Boolean functions of $n$ variables.
Find the value of $M(n)$ for every $n$, ideally by a closed formula.

## Remarks

No closed formula of a practical kind is known, and the problem is open in this form.
Exact values are known only for $n \le 9$; the value $M(9)$ was computed in 2023 with large computer searches.
Simple bounds come from antichains: every family of $\floor{n/2}$-element subsets of $\set{1, \ldots, n}$ is an antichain, so $M(n) \ge 2^{\binom{n}{\floor{n/2}}}$.
Kleitman proved that $\log_2 M(n) = (1 + o(1)) \binom{n}{\floor{n/2}}$ as $n \to \infty$, and finer asymptotic formulas are known.

## Explanations

### L1 human

The question is how many monotone yes/no rules on $n$ switches exist.
For a few switches one can count by hand: $2, 3, 6, 20, 168$ for $n = 0, 1, 2, 3, 4$.
The counts explode so quickly that even with computers only the first ten values are known.
A formula that gives them all is still missing.
</document>

<document source="https://tamnd.github.io/math2/node/RM94KBV8P7.md" tag="RM94KBV8P7" kind="concept">
# Turing pattern

## Summary

A Turing pattern is a stationary, spatially periodic pattern that forms in a [[reaction–diffusion system|TAAHWE917M]] from an instability caused by diffusion.
Alan Turing described the mechanism in 1952 in "The Chemical Basis of Morphogenesis".
The surprise is that diffusion, which usually smooths out differences, can destabilise a uniform state when two substances diffuse at different rates.
Turing patterns are proposed as explanations of spots and stripes in animal coats and other biological patterns.

## Explanations

### L1 human

Think of an activator that makes more of itself and of an inhibitor, and suppose the inhibitor spreads much faster.
A small local excess of activator grows, but the inhibitor it produces spreads around it and stops growth nearby.
The result is a regular arrangement of peaks with a typical spacing, such as spots or stripes.
Without diffusion the mixture would stay uniform; the difference in diffusion rates is what creates the pattern.
</document>

<document source="https://tamnd.github.io/math2/node/RMEF8T6C0E.md" tag="RMEF8T6C0E" kind="definition">
# Tomita–Takesaki theory

## Definition

Let $M$ be a [[von Neumann algebra|PSFWQVZ0FS]] on a complex [[Hilbert space|HM2HHE3J51]] $H$, with [[commutant|1VC7EJ1ZBZ]] $M'$.
Let $\Omega \in H$ be cyclic for $M$, that is, $M\Omega$ is dense in $H$.
Assume also that $\Omega$ is separating for $M$: if $x \in M$ and $x\Omega = 0$, then $x = 0$.

1. The operator $S_0$ with domain $M\Omega$ defined by $S_0(x\Omega) = x^*\Omega$ is a well-defined, densely defined, closable antilinear [[unbounded operator|KRJX5RA0PW]]; let $S$ be its closure.
2. The *modular operator* is the positive self-adjoint operator $\Delta = S^* S$, and the *modular conjugation* is the antiunitary operator $J$ in the polar decomposition $S = J \Delta^{1/2}$.
3. The *modular automorphism group* is $\sigma_t(x) = \Delta^{it} x \Delta^{-it}$ for $t \in \R$ and $x \in M$.

*Tomita–Takesaki theory* is the study of these objects.
Its main theorem states that $J M J = M'$ and $\Delta^{it} M \Delta^{-it} = M$ for all $t \in \R$.
So each $\sigma_t$ is an automorphism of $M$.

## Remarks

Example: let $M = M_n(\C)$ act by left multiplication on $H = M_n(\C)$ with inner product $\langle \xi, \eta \rangle = \operatorname{tr}(\eta^* \xi)$, and let $\Omega = \rho^{1/2}$ for an invertible density matrix $\rho$.
Then $\Omega$ is cyclic and separating, $\Delta \xi = \rho \xi \rho^{-1}$ and $J \xi = \xi^*$.
Check: $J \Delta^{1/2}(x \rho^{1/2}) = J(\rho^{1/2} x) = x^* \rho^{1/2} = S(x\rho^{1/2})$.
Here $JMJ$ is the algebra of right multiplications, which is $M'$, and $\sigma_t(x) = \rho^{it} x \rho^{-it}$.

Non-example: let $M = B(H)$ with $\dim H \ge 2$, let $\Omega$ be a unit vector, and let $\eta$ be a unit vector orthogonal to $\Omega$.
Every nonzero vector is cyclic for $B(H)$, but $\Omega$ is not separating.
Indeed, the operator $x = \langle \cdot, \eta \rangle \Omega$ is nonzero and $x\Omega = 0$.
Its adjoint is $x^* = \langle \cdot, \Omega \rangle \eta$, so $x^*\Omega = \eta \ne 0$.
Since $x\Omega = 0\Omega$ but $x^*\Omega \ne 0^*\Omega$, the formula for $S_0$ does not define a map.

The vector state $\varphi(x) = \langle x\Omega, \Omega \rangle$ is a KMS state for the modular automorphism group, and Takesaki showed that $\sigma$ is the only one-parameter automorphism group of $M$ with this property.
</document>

<document source="https://tamnd.github.io/math2/node/RMH4FBCFDA.md" tag="RMH4FBCFDA" kind="concept">
# Hilbert polynomial

## Summary

Let $M$ be a finitely generated graded module over a [[polynomial ring|07GNX8GCYP]] over a field.
Its Hilbert polynomial gives the dimension of the degree $n$ part of $M$ for all large $n$.
Its existence is the [[Hilbert–Serre theorem|HP56DJECMT]].
For the coordinate ring of a projective variety, the degree of the Hilbert polynomial is the dimension of the variety.
Its leading coefficient encodes the degree of the variety.

## Explanations

### L1 human

Count how many independent homogeneous polynomials of degree $n$ survive on some shape.
For large $n$ this count follows a polynomial formula in $n$.
For a plane curve given by one equation of degree $d$, the count is $dn - d(d-3)/2$ for large $n$.
The growth rate of the count tells the dimension of the shape.
The leading coefficient tells its degree, the number of points in which it meets a general linear space of complementary dimension.
For instance, a plane curve of degree $d$ meets a general line in $d$ points.
The [[definition|5FR6JX04RS]] makes this precise.
</document>

<document source="https://tamnd.github.io/math2/node/RMH7ZQ2F40.md" tag="RMH7ZQ2F40" kind="definition">
# Borel hierarchy

## Definition

Let $X$ be a [[Polish space|584GXXY73M]], and let $\omega_1$ be the first uncountable [[ordinal|0YSCSX5YZC]].
Define classes of subsets of $X$ by recursion on $\alpha$ with $1 \le \alpha < \omega_1$:

1. $\Sigma^0_1$ is the class of open subsets of $X$.
2. For $\alpha \ge 1$, $\Pi^0_\alpha$ is the class of complements $X \setminus A$ of sets $A$ in $\Sigma^0_\alpha$.
3. For $\alpha > 1$, $\Sigma^0_\alpha$ is the class of countable unions $\bigcup_{n} A_n$ with each $A_n$ in $\Pi^0_{\beta_n}$ for some $\beta_n < \alpha$.
4. For $\alpha \ge 1$, $\Delta^0_\alpha = \Sigma^0_\alpha \cap \Pi^0_\alpha$.

This system of classes is the *Borel hierarchy* of $X$.

## Remarks

The class $\Pi^0_1$ is the class of closed sets, $\Sigma^0_2$ is the class of $F_\sigma$ sets, and $\Pi^0_2$ is the class of $G_\delta$ sets.
In $X = \R$, the set $\Q$ is in $\Sigma^0_2$, as the countable union of the closed sets $\set{q}$.
It is not in $\Pi^0_2$, since by the Baire category theorem a dense $G_\delta$ subset of $\R$ is uncountable.
Every class satisfies $\Sigma^0_\alpha \cup \Pi^0_\alpha \subseteq \Delta^0_{\alpha + 1}$, and the union of all the classes $\Sigma^0_\alpha$ is the [[Borel sigma-algebra|0WQXY32Q8D]] of $X$.
A non-example: an analytic set that is not Borel lies in no class of the hierarchy.
If $X$ is uncountable, then $\Sigma^0_\alpha \neq \Pi^0_\alpha$ for every $\alpha$, so the hierarchy does not collapse.
The same definition works in any metrizable space; metrizability is used to show that closed sets are $G_\delta$ sets.
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<document source="https://tamnd.github.io/math2/node/RMTZ5J64FX.md" tag="RMTZ5J64FX" kind="concept">
# Saturated model

## Summary

A saturated model is a model that realizes as many [[types|9V71Q3SD1G]] as its size allows.
More precisely, $M$ is $\kappa$-saturated when every complete type over fewer than $\kappa$ parameters of $M$ is realized in $M$.
It is saturated when it is $\abs{M}$-saturated.
Saturated models of a complete theory are unique up to isomorphism in each cardinality.
Every model of the theory of at most the same size embeds elementarily into a saturated one.
They serve as large "universal" models in which most model-theoretic work is done.

## Explanations

### L1 human

A saturated model has no gaps that its own language can describe.
If a description of an element mentions only a few given elements and is consistent, such an element already exists in the model.
For example, the rational numbers with their order are saturated.
Given finitely many rationals, a consistent wish such as "between $1/3$ and $1/2$" or "larger than all of them" is met by some rational number.
The integers with their order are not: no integer is "larger than $0$ and with infinitely many integers between it and $0$".
</document>

<document source="https://tamnd.github.io/math2/node/RMWMYBY8TC.md" tag="RMWMYBY8TC" kind="concept">
# Character of a representation

## Summary

The character of a finite-dimensional [[group representation|WYGRXCG2RN]] $\rho$ is the function $g \mapsto \tr \rho(g)$ on the group.
It is constant on [[conjugacy classes|0DEBXAJ5SA]], because the trace does not change under conjugation.
For a finite group over the complex numbers, a representation is determined up to isomorphism by its character.
So characters turn the study of representations into the study of a finite list of numbers.

## Explanations

### L1 human

Replace each matrix of a representation by one number, its trace, the sum of its diagonal entries.
Surprisingly little is lost: for finite groups over $\C$ these numbers already tell which representation one has.
For example, let the symmetric group $S_3$ permute the coordinates of $\C^3$.
A permutation matrix has trace equal to the number of points it fixes.
So the character takes the value $3$ at the identity, $1$ at each transposition and $0$ at each $3$-cycle.
</document>

<document source="https://tamnd.github.io/math2/node/RMWW2X86TT.md" tag="RMWW2X86TT" kind="concept">
# Bartle–Dunford–Schwartz integral

## Summary

The Bartle–Dunford–Schwartz integral integrates a scalar function against a countably additive [[vector measure|PS28BSAJKJ]] with values in a [[Banach space|06GZ956657]].
The integral of a function is then a vector of the Banach space.
It is built from simple functions, as the Lebesgue integral is.
It is a standard tool for spectral measures and for operators on spaces of continuous functions.

## Explanations

### L1 human

An ordinary measure gives each set a number, and the integral of a function is a weighted sum of these numbers.
A vector measure gives each set a vector, so the weighted sum is a vector.
For a step function that takes the value $a_i$ on the set $E_i$, the integral is $\sum_i a_i \, \mu(E_i)$.
For general functions one approximates them by step functions and takes the limit of the vectors.
</document>

<document source="https://tamnd.github.io/math2/node/RMXR0FEF54.md" tag="RMXR0FEF54" kind="object">
# Rado graph

## Description

The Rado graph $R$ has vertex set $\N = \set{0, 1, 2, \dots}$.
For $i < j$, the vertices $i$ and $j$ are adjacent when the binary digit of $j$ at position $i$ is $1$.
Equivalently, $\floor{j / 2^i}$ is odd.

It has the extension property, stated as follows.
For any two disjoint finite sets $U, V \subseteq \N$, some vertex $z \notin U \cup V$ is adjacent to every vertex of $U$ and to no vertex of $V$.
Check: let $N$ be larger than every element of $U \cup V$ and put $z = 2^N + \sum_{u \in U} 2^u$.
Then $z > N$, so $z \notin U \cup V$, and every $w \in U \cup V$ satisfies $w < z$.
The binary digit of $z$ at position $w$ is $1$ if $w \in U$ and $0$ if $w \in V$.
So $z$ is adjacent to every vertex of $U$ and to no vertex of $V$.

## Remarks

Any two countable [[graphs|00C0BR7D7H]] with the extension property are [[isomorphic|0F7QKGJS6B]], by a back-and-forth argument.
So the Rado graph is the [[Fraïssé limit|HJVRBDQSGG]] of the class of finite graphs.
Every finite graph is an induced subgraph of it, and every isomorphism between finite induced subgraphs extends to an automorphism.
A countable [[random graph|TNT3DG0HRJ]] in which each edge is present independently with probability $1/2$ is isomorphic to $R$ with probability $1$.
For this reason $R$ is also called the countable random graph.
The first-order theory of $R$ is $\aleph_0$-categorical.

## Explanations

### L1 human

The Rado graph is an infinite graph that contains every finite and every countable graph as an induced subgraph.
Its key feature is that any finite wish list "be joined to these vertices and not to those" is met by some vertex.
If one draws a countable graph by flipping a fair coin for each pair of vertices, one gets this same graph with probability one.
</document>

<document source="https://tamnd.github.io/math2/node/RMXW3D9XZ3.md" tag="RMXW3D9XZ3" kind="statement">
# Analyticity of the bilateral Laplace transform of a law

## Statement

Let $\mu$ be a [[probability measure|059TJN0NZ1]] on $\R$.
Let $I$ be the set of real $\sigma$ with $\int_{\R} e^{-\sigma x} \, d\mu(x) < \infty$.
Then:

1. $I$ is an interval that contains $0$.
2. For every $s \in \C$ with $\operatorname{Re} s \in I$, the integral $\mathcal{B}\mu(s) = \int_{\R} e^{-sx} \, d\mu(x)$ converges absolutely.
3. On the open strip $S = \set{s \in \C : \operatorname{Re} s \in \operatorname{int} I}$, the function $\mathcal{B}\mu$ is [[holomorphic|0NAHXR1FRH]], and for every $n \ge 0$ and $s \in S$,
$$(\mathcal{B}\mu)^{(n)}(s) = \int_{\R} (-x)^n e^{-sx} \, d\mu(x),$$
where this integral converges absolutely.

Here $\operatorname{int} I$ is the interior of $I$, and $\mathcal{B}\mu$ is the [[two-sided Laplace transform|6J2AGD0BYY]] of $\mu$.

## Remarks

Part 1 holds because $\sigma \mapsto e^{-\sigma x}$ is convex, so $e^{-\sigma x} \le e^{-a x} + e^{-b x}$ for $a \le \sigma \le b$; and $0 \in I$ because $\mu$ has total mass $1$.
Part 2 holds because $\abs{e^{-sx}} = e^{-x \operatorname{Re} s}$.
For part 3, fix $a < \operatorname{Re} s < b$ in $I$; then $\abs{x}^n e^{-x \operatorname{Re} s} \le C_n (e^{-a x} + e^{-b x})$ for a constant $C_n$, so [[differentiation under the integral sign|0AR06W1E1B]] is allowed by dominated convergence.
On the real axis this says that the [[moment-generating function|0WQVXP7PDB]] $M(t) = \mathcal{B}\mu(-t)$ is real analytic on the interior of its domain, with $M^{(n)}(t) = \int x^n e^{tx} \, d\mu(x)$.
Example: for the exponential law with rate $\lambda$, $I = (-\lambda, \infty)$ and $\mathcal{B}\mu(s) = \lambda/(\lambda + s)$, which is holomorphic for $\operatorname{Re} s > -\lambda$ and has a pole at $s = -\lambda$.
The open strip is needed in part 3: the derivative formula uses room on both sides of $\operatorname{Re} s$, and at an end point of $I$ the integral $\int \abs{x} e^{-x \operatorname{Re} s} \, d\mu(x)$ may diverge.
If $I = \set{0}$, as for the Cauchy law, the strip is empty and the statement says nothing.

## Explanations

### L1 human

Averaging $e^{-sX}$ over a random quantity $X$ gives a function of $s$.
This theorem says that, wherever the average is finite with a little room to spare, the function is as smooth as possible.
It is given locally by a convergent power series.
Its derivatives are found by differentiating inside the average.

### L2 human

This is the reason why a moment-generating function that is finite near $0$ produces all the moments, as its derivatives at $0$.
It is also the starting point of large deviation theory, where the convex function $\log M$ and its derivatives are used.
The domain of finiteness is the [[strip of convergence|HYA7QQ1E38]] of the bilateral transform.
The same proof gives analyticity of the two-sided Laplace transform of any measure on its open strip.
On the imaginary axis the transform is the characteristic function, which is always defined but need not be analytic.
</document>

<document source="https://tamnd.github.io/math2/node/RN4G8SGP6P.md" tag="RN4G8SGP6P" kind="concept">
# Semicontinuity

## Summary

Semicontinuity is a one-sided form of [[continuity|0VGFHTM75Y]] for functions with values in the [[extended real line|0GSZ7B0CGR]].
A function is upper semicontinuous when its value at a point may lie above the nearby values but not below them.
It is lower semicontinuous in the opposite case.
A function is continuous exactly when it is both.
Lower semicontinuous functions attain their minimum on nonempty [[compact spaces|0TVJZYB6SH]], which makes them basic in optimisation and the calculus of variations.

## Explanations

### L1 human

Think of the graph of a function with a jump.
At an upper semicontinuous point, the value is at least as high as the nearby values tend to be.
The function may drop sharply as you move away from the point, but it cannot rise sharply.
The [[floor function|08GR54CKEA]] is upper semicontinuous: at an integer $n$ its value is $n$, and just to the left it is $n - 1$.
The [[indicator function|0RCQZ36MME]] of an open set is lower semicontinuous, and the indicator function of a closed set is upper semicontinuous.
</document>

<document source="https://tamnd.github.io/math2/node/RN4J24CD83.md" tag="RN4J24CD83" kind="counterexample">
# Tychonoff's non-uniqueness example

## Counterexample

There is a function $u \colon \R \times \R \to \R$, smooth and not identically zero, such that
$$\partial_t u = \partial_x^2 u \ \text{on } \R \times \R \quad \text{and} \quad u(x, t) = 0 \ \text{for all } t \le 0.$$
So the Cauchy problem for the [[heat equation|VP2HN86VAG]] on $\R \times (0, \infty)$ with initial value $0$ has a solution other than $u = 0$.
One construction: fix $\alpha > 1$, let $g(t) = \exp(-t^{-\alpha})$ for $t > 0$ and $g(t) = 0$ for $t \le 0$, and set
$$u(x, t) = \sum_{k=0}^{\infty} g^{(k)}(t) \, \frac{x^{2k}}{(2k)!}.$$
Term by term, $\partial_x^2$ of the $k$-th term is the $(k-1)$-th term with $g^{(k-1)}$ replaced by $g^{(k)}$, so formally $\partial_x^2 u = \sum_k g^{(k+1)}(t) x^{2k}/(2k)! = \partial_t u$.
There are constants $\theta, c > 0$ with $\abs{g^{(k)}(t)} \le k! \, (\theta t)^{-k} \exp(-c \, t^{-\alpha})$ for $t > 0$.
These bounds show that the series and its differentiated series converge locally uniformly, so $u$ is a genuine solution.
Since $g$ is not identically zero, neither is $u$.

## Remarks

The solution grows very fast in $x$: on no strip $\R \times (0, T)$ is it bounded by $A e^{a x^2}$ for constants $A$ and $a$.
Tychonoff showed that uniqueness does hold in the class of solutions with $\abs{u(x, t)} \le A e^{a \abs{x}^2}$.
So a growth condition at infinity is needed for uniqueness on the whole line; on a bounded interval with boundary values, the [[maximum principle|EMDVD4E83K]] gives uniqueness without one.
Nonnegative solutions are also unique, by Widder's theorem, so $u$ must change sign.

## Explanations

### L1 human

One would expect a rod that starts at temperature zero everywhere and receives no heat to stay at zero.
On an infinite rod this is false in a strange way: heat can arrive from infinity, through temperatures that grow enormously far away.
The example shows that the heat equation alone does not determine the future; one must also rule out such wild behaviour at infinity.
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<document source="https://tamnd.github.io/math2/node/RN58Z2NWAC.md" tag="RN58Z2NWAC" kind="statement">
# Gelfond–Schneider theorem

## Statement

Let $a$ and $b$ be complex [[algebraic numbers|0MNF0BYFQC]] with $a \ne 0$, $a \ne 1$, and $b \notin \Q$.
Let $\log a$ be any value of the [[complex logarithm|02QGGD5G87]] of $a$.
Then the number $a^b = e^{b \log a}$ is [[transcendental|0VTQXTCR2G]].

## Remarks

The proof builds an auxiliary function $F(z) = \sum_{i,j} c_{ij} e^{(i + j b) z \log a}$ with integer coefficients $c_{ij}$, not all zero, chosen so that $F$ vanishes at many integer points.
If $a^b$ were algebraic, the values of $F$ and its derivatives at integers would be algebraic numbers of controlled size.
An analytic estimate and a lower bound for nonzero algebraic numbers then force too many zeros, which is a contradiction.
Examples: $2^{\sqrt{2}}$ is transcendental, and so is $e^{\pi}$, since $e^{\pi} = (-1)^{-i}$ with $\log(-1) = i\pi$.
Another consequence: $\log 3 / \log 2$ is transcendental, since it is irrational and $2^{\log 3 / \log 2} = 3$.
The hypothesis $b \notin \Q$ is necessary: $2^{1/2} = \sqrt{2}$ is algebraic.
The theorem answers Hilbert's seventh problem.
[[Baker's theorem|WMBJSNG57F]] generalizes it to linear forms in several logarithms.

## Explanations

### L1 human

Raise an algebraic number to an irrational algebraic power, for example $2^{\sqrt{2}}$.
The theorem says that the result is never a root of a polynomial with integer coefficients.
So numbers such as $2^{\sqrt{2}}$ and $e^{\pi}$ are transcendental, like $\pi$ and $e$.

### L2 human

An equivalent form: if $\alpha$ and $\beta$ are nonzero algebraic numbers and $\log \alpha$ and $\log \beta$ are linearly independent over $\Q$, then they are linearly independent over the algebraic numbers.
This is the case of two logarithms in Baker's theorem.
Together with the [[Lindemann–Weierstrass theorem|0JNCBFMPXR]], it is one of the classical results of transcendence theory.
The question whether $e + \pi$ or $e \pi$ is transcendental is still open.
</document>

<document source="https://tamnd.github.io/math2/node/RN8RBM2A8A.md" tag="RN8RBM2A8A" kind="definition">
# Stereographic projection

## Definition

Let $S^2 = \set{(x_1, x_2, x_3) \in \R^3 : x_1^2 + x_2^2 + x_3^2 = 1}$ be the unit sphere and let $N = (0, 0, 1)$ be its north pole.
Identify the plane $x_3 = 0$ with $\C$ by $(x, y, 0) \mapsto x + iy$.
The *stereographic projection* is the map $\sigma \colon S^2 \setminus \set{N} \to \C$ defined as follows.
For $P \in S^2 \setminus \set{N}$, the point $\sigma(P)$ is where the line through $N$ and $P$ meets the plane $x_3 = 0$.
In coordinates,
$$\sigma(x_1, x_2, x_3) = \frac{x_1 + i x_2}{1 - x_3}.$$
Setting $\sigma(N) = \infty$ extends $\sigma$ to a bijection from $S^2$ onto the [[Riemann sphere|0CDMXGPSES]] $\C \cup \set{\infty}$.

## Remarks

The inverse map is
$$\sigma^{-1}(z) = \left( \frac{2 \operatorname{Re} z}{\abs{z}^2 + 1}, \frac{2 \operatorname{Im} z}{\abs{z}^2 + 1}, \frac{\abs{z}^2 - 1}{\abs{z}^2 + 1} \right).$$
Example: the south pole $(0, 0, -1)$ goes to $0/2 = 0$.
A point $(x_1, x_2, 0)$ of the equator goes to $x_1 + i x_2$, so the equator goes onto the unit circle.
Non-example: the projection from the center of the sphere onto a tangent plane is not stereographic.
It sends two antipodal points $P$ and $-P$ to the same point, so it is not injective.
The map $\sigma$ is a homeomorphism, it preserves angles, and it sends circles on $S^2$ to circles or lines in $\C$; the circles through $N$ go to lines.
Some authors project onto the plane tangent at the south pole, which multiplies $\sigma$ by $2$.
</document>

<document source="https://tamnd.github.io/math2/node/RN8Y1823FX.md" tag="RN8Y1823FX" kind="definition">
# Soliton

## Definition

Let $E$ be an evolution equation for $u(x, t)$ with $x, t \in \R$.
A *solitary wave* of $E$ is a [[traveling wave|M37M411KP7]] solution $u(x, t) = \varphi(x - ct)$ with $c \in \R$ and a profile $\varphi$ that tends to constant limits as $x \to \pm \infty$.
A *soliton* is a solitary wave of $E$ that survives interaction with other solitary waves of $E$.
That is, after each such interaction it emerges with the same shape and speed, shifted only in position.

## Remarks

The second clause is often checked by the existence of exact multi-soliton solutions, which split for $t \to \pm \infty$ into separated solitary waves with the same speeds.
There is no single formal definition used by all authors.

Example: for the [[Korteweg–De Vries equation|KTT75QC72Z]] $u_t + 6 u u_x + u_{xxx} = 0$ and $c > 0$, put $\varphi(\xi) = \frac{c}{2} \operatorname{sech}^2\big(\frac{\sqrt{c}}{2} \xi\big)$, using the [[hyperbolic secant|01PPDSDP2N]].
For $u = \varphi(x - ct)$ the equation becomes $-c\varphi' + 6\varphi\varphi' + \varphi''' = 0$, which integrates to $\varphi'' = c\varphi - 3\varphi^2$.
With $\varphi = A \operatorname{sech}^2(k\xi)$ one has $\varphi'' = 4k^2 A \operatorname{sech}^2(k\xi) - 6k^2 A \operatorname{sech}^4(k\xi)$, so the equation holds with $4k^2 = c$ and $A = 2k^2 = c/2$.
The KdV equation also has explicit two-soliton solutions, so this solitary wave is a soliton.

Non-example: the linear equation $u_t + u_{xxx} = 0$ has no solitary wave with $c \ne 0$ that decays at infinity.
Indeed, $\varphi''' = c\varphi'$ gives $\varphi'' = c\varphi$, whose decaying solutions on all of $\R$ are only $\varphi = 0$.

## Explanations

### L1 human

A solitary wave is a single hump that moves without changing shape.
It is a soliton when it also survives collisions with other such humps, keeping its size and speed.
</document>

<document source="https://tamnd.github.io/math2/node/RN9D96ESWN.md" tag="RN9D96ESWN" kind="concept">
# Kan extension

## Summary

A Kan extension extends a [[functor|28NQ0KRHD3]] $F \colon \mathcal{A} \to \mathcal{C}$ along another functor $K \colon \mathcal{A} \to \mathcal{B}$ to a functor on $\mathcal{B}$, in the best possible way.
There is a left Kan extension, which is universal among functors receiving a [[natural transformation|KYCPCC57FT]] from $F$, and a dual right Kan extension.
Limits, colimits and [[adjoint functors|TZHF6ZM92S]] are all special cases.
This is why Kan extensions are often called the most basic universal construction of category theory.

## Explanations

### L1 human

Suppose a functor is only known on a part of a category, or on a category that maps into a bigger one.
A Kan extension asks for the closest functor on the bigger category that is compatible with the given one.
The left Kan extension approximates from one side, the right Kan extension from the other side.
When the bigger category has just one object and one arrow, the left Kan extension is the colimit of $F$ and the right one is the limit.
For a subgroup $H$ of a group $G$, the left Kan extension of a representation of $H$ along the inclusion is the induced representation of $G$.
</document>

<document source="https://tamnd.github.io/math2/node/RNB6QX0GCA.md" tag="RNB6QX0GCA" kind="statement">
# Sokhotski–Plemelj theorem

## Statement

Let $a < 0 < b$ and let $f \colon [a, b] \to \C$ be [[Hölder continuous|0WZTM0EFR6]].
Then the two limits below exist, and
$$\lim_{\varepsilon \to 0^+} \int_a^b \frac{f(x)}{x + i\varepsilon} \, dx = -i\pi f(0) + \mathrm{p.v.} \int_a^b \frac{f(x)}{x} \, dx,$$
$$\lim_{\varepsilon \to 0^+} \int_a^b \frac{f(x)}{x - i\varepsilon} \, dx = i\pi f(0) + \mathrm{p.v.} \int_a^b \frac{f(x)}{x} \, dx,$$
where $\mathrm{p.v.}$ denotes the [[Cauchy principal value|0V3RDQV3ZX]].

## Remarks

Proof idea: write $\frac{1}{x \pm i\varepsilon} = \frac{x}{x^2 + \varepsilon^2} \mp i \frac{\varepsilon}{x^2 + \varepsilon^2}$.
The second kernel, divided by $\pi$, is an approximate identity, so its integral against $f$ tends to $\pi f(0)$.
The first kernel, integrated against $f$, tends to the principal value; the Hölder condition makes $(f(x) - f(-x))/x$ integrable near $0$.
In the language of distributions, $\frac{1}{x \pm i0} = \mathrm{p.v.} \frac{1}{x} \mp i\pi \delta$, where $\delta$ is the [[Dirac delta|0KPHENZ5S7]].
Example: take $f = 1$ on $[-1, 1]$ and let $\operatorname{Log}$ be the principal logarithm.
The first integral is $\operatorname{Log}(1 + i\varepsilon) - \operatorname{Log}(-1 + i\varepsilon)$, which tends to $0 - i\pi$, and the principal value is $0$.
There is a version for Cauchy-type integrals over a curve, the Sokhotski–Plemelj formulae, which give the jump of $\frac{1}{2\pi i} \int_C \frac{\varphi(\zeta)}{\zeta - z} \, d\zeta$ across $C$.

## Explanations

### L1 human

The function $1/x$ cannot be integrated across $0$ in the usual sense.
Moving the pole slightly off the real line, to $\mp i\varepsilon$, makes the integral finite.
As the pole moves back, the integral splits into a principal value plus a point contribution $\mp i\pi f(0)$.
The sign of that contribution records from which side the pole approached.

### L2 human

The theorem gives the boundary values of Cauchy integrals, which is the starting point of the theory of singular integral equations and of the Riemann–Hilbert problem.
In physics it appears whenever one adds $i\varepsilon$ to a denominator, as in Green's functions of quantum mechanics and in the [[Kramers–Kronig relations|GFS95PZ5YK]].
It is also the computation that relates the [[Hilbert transform|NSCEBXM7ZM]] to boundary values of holomorphic functions on a half-plane.
</document>

<document source="https://tamnd.github.io/math2/node/RNCFXWTPFK.md" tag="RNCFXWTPFK" kind="definition">
# Finitely complete category

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
$\mathcal{C}$ is *finitely complete* if every [[diagram|HJK75HDWFE]] $F \colon J \to \mathcal{C}$ with $J$ a finite category has a [[limit|CHS5AFJGWE]] in $\mathcal{C}$.
Here a category $J$ is finite when it has finitely many objects and finitely many morphisms.

## Remarks

The following are equivalent for a category $\mathcal{C}$:

1. $\mathcal{C}$ is finitely complete.
2. $\mathcal{C}$ has a terminal object, binary products and equalizers.
3. $\mathcal{C}$ has a terminal object and pullbacks.

Example: the category of finite sets is finitely complete.
Its terminal object is a one-point set, and the pullback of $f \colon X \to Z$ and $g \colon Y \to Z$ is the finite set $\set{(x, y) : f(x) = g(y)}$.

Non-example: the totally ordered set $(\Z, \le)$, seen as a category, has all pullbacks but no terminal object.
In a preorder the pullback of $x \to z \leftarrow y$ is $\min(x, y)$, and a terminal object would be a greatest element.
</document>

<document source="https://tamnd.github.io/math2/node/RNDDWDAFJ5.md" tag="RNDDWDAFJ5" kind="concept">
# Euler characteristic

## Summary

The Euler characteristic $\chi(X)$ of a space is the alternating sum $b_0 - b_1 + b_2 - \cdots$ of its [[Betti numbers|ST71ZV3FES]].
For a finite [[CW complex|5W95QFHXPR]] it equals the alternating count of cells, which recovers $V - E + F$ for polyhedra.
It is a homotopy invariant, and it is multiplicative for products and finite coverings.
It links topology with geometry through results such as the Gauss–Bonnet theorem.

## Explanations

### L1 human

Count the vertices, edges and faces of a convex polyhedron, and compute $V - E + F$.
The answer is always $2$, whatever the polyhedron, for example $8 - 12 + 6 = 2$ for a cube.
This number belongs to the sphere, not to the particular polyhedron.
For the surface of a doughnut the same count always gives $0$.
The Euler characteristic extends this count to any reasonable space, using homology in place of counting cells.
</document>

<document source="https://tamnd.github.io/math2/node/RNDKD74V17.md" tag="RNDKD74V17" kind="definition">
# Mosco convergence

## Definition

Let $X$ be a real [[normed space|03J5RAKB30]].
Write $x_n \rightharpoonup x$ for convergence in the [[weak topology|P6KHVSTPZV]] $\sigma(X, X^*)$ and $x_n \to x$ for convergence in norm, where $X^*$ is the [[continuous dual space|4A82S5FVRT]] of $X$.
Let $F_n \colon X \to [-\infty, +\infty]$ for $n \in \N$ and $F \colon X \to [-\infty, +\infty]$ be functions with values in the [[extended real line|0GSZ7B0CGR]].
The sequence $(F_n)$ *Mosco converges* to $F$ if the following two conditions hold for every $x \in X$.

1. (Weak lower bound.) For every sequence $(x_n)$ in $X$ with $x_n \rightharpoonup x$,
$$F(x) \le \liminf_{n \to \infty} F_n(x_n).$$
2. (Strong recovery sequence.) There is a sequence $(x_n)$ in $X$ with $x_n \to x$ and
$$F(x) \ge \limsup_{n \to \infty} F_n(x_n).$$

The indicator function of a set $K \subseteq X$ in convex analysis is $0$ on $K$ and $+\infty$ outside $K$.
Subsets $K_n \subseteq X$ Mosco converge to $K \subseteq X$ if their indicator functions do.

## Remarks

For sets the two conditions read as follows.
If $n_k$ is increasing, $x_k \in K_{n_k}$ and $x_k \rightharpoonup x$, then $x \in K$.
Every $x \in K$ is the norm limit of a sequence with $x_n \in K_n$.
The notation $F = \operatorname{M\text{-}lim}_n F_n$ is common.
If $X$ is finite-dimensional, weak and norm convergence agree, and Mosco convergence is [[Γ-convergence|NW9MM370A5]].

Example: let $F$ be convex and lower semicontinuous, and let $F_n = F$ for all $n$.
Then $(F_n)$ Mosco converges to $F$.
The recovery sequence $x_n = x$ gives clause 2.
For clause 1, a convex lower semicontinuous function is also weakly lower semicontinuous, because its closed sublevel sets are convex and hence weakly closed.

Non-example: in the sequence space $\ell^2$ with unit vectors $e_n$, let $K_n = \set{e_n}$ and $K = \set{0}$.
Clause 1 holds, since $e_n \rightharpoonup 0$.
Clause 2 fails at $0$: every choice $x_n \in K_n$ is $x_n = e_n$, and $\norm{e_n - 0} = 1$ for all $n$.
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# Hyperoval

## Summary

A hyperoval in a finite projective plane of order $n$ is a set of $n + 2$ points with no three on a line.
It is the largest possible size of such a set, and it can occur only when $n$ is even.
When $n$ is even, every [[oval|7B03TB64N9]] extends to a hyperoval by adding one point, the nucleus, where all tangents of the oval meet.
The classification of hyperovals in $PG(2, 2^h)$ is a long-standing problem of finite geometry.

## Explanations

### L1 human

In a finite projective plane, how many points can one choose with no three on a line?
The answer is at most $n + 2$, and when that many can be chosen, the set is a hyperoval.
Every line then meets the set in $0$ or $2$ points, never in exactly one.
In the Fano plane, the four points off any line form a hyperoval.
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<document source="https://tamnd.github.io/math2/node/RNS272QBGT.md" tag="RNS272QBGT" kind="concept">
# Brunn–Minkowski theory

## Summary

Brunn–Minkowski theory is the part of convex geometry that studies [[convex bodies|437FSG3419]] through the interplay of volume and [[Minkowski addition|SXS6D87PHE]].
Its central objects are the [[mixed volumes|QB3Q5FBJ08]], the coefficients of the polynomial that gives the volume of a Minkowski combination of convex bodies.
Its central results are inequalities between them, starting from the [[Brunn–Minkowski theorem|9B02RPN7XD]] and leading to the Alexandrov–Fenchel inequality.
Surface area, mean width, quermassintegrals and the isoperimetric inequality all fit into this framework.

## Explanations

### L1 human

Add two convex shapes by adding each point of one to each point of the other, and ask how the volume of the result depends on the pieces.
The volume of $s K + t L$ turns out to be a polynomial in $s$ and $t$, and its coefficients measure how $K$ and $L$ interact.
For a convex body $K$ and the unit ball $B$, the volume of $K + t B$ is a polynomial in $t$ whose coefficients include the volume and the surface area of $K$.
Brunn–Minkowski theory studies these coefficients and the inequalities between them.
</document>

<document source="https://tamnd.github.io/math2/node/RNTAAHTD0M.md" tag="RNTAAHTD0M" kind="definition">
# Semilattice

## Definition

Let $(S, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
$S$ is a *join-semilattice* if every two elements $x, y \in S$ have a [[join|J2T75ATRV5]] $x \vee y$ in $S$.
$S$ is a *meet-semilattice* if every two elements $x, y \in S$ have a meet $x \wedge y$ in $S$.

Algebraic form: a *semilattice* is a set $S$ with a binary operation $\cdot$ such that, for all $x, y, z \in S$:

1. $(x \cdot y) \cdot z = x \cdot (y \cdot z)$;
2. $x \cdot y = y \cdot x$;
3. $x \cdot x = x$.

## Remarks

The two forms are equivalent.
A join-semilattice gives the operation $x \cdot y = x \vee y$.
Conversely, a semilattice $(S, \cdot)$ becomes a join-semilattice for the order $x \le y \iff x \cdot y = y$, and a meet-semilattice for the order $x \le y \iff x \cdot y = x$.
So a semilattice is a commutative [[semigroup|0XMEM406JM]] in which every element is idempotent.
Example: the finite subsets of $\N$, ordered by $\subseteq$, form a join-semilattice, since the union of two finite sets is finite and is their join.
Example: $\set{a, b, 1}$ with $a < 1$ and $b < 1$ is a join-semilattice with $a \vee b = 1$.
It is not a meet-semilattice, since $a$ and $b$ have no common lower bound.
Non-example: a two-element antichain $\set{a, b}$ is not a join-semilattice, since $a$ and $b$ have no upper bound.
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<document source="https://tamnd.github.io/math2/node/RNV041SNCT.md" tag="RNV041SNCT" kind="definition">
# Risk function

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a [[statistical model|K3AMMSY8DD]] for an observation $X$ with values in a measurable space $\mathcal{X}$.
Let $\mathcal{A}$ be a measurable space of actions and let $L \colon \Theta \times \mathcal{A} \to [0, \infty)$ be a [[loss function|3BTG6VVCGZ]], measurable in $a$ for each $\theta$.
A (non-randomized) *decision rule* is a measurable map $\delta \colon \mathcal{X} \to \mathcal{A}$.
The *risk function* of $\delta$ is the map $R(\cdot, \delta) \colon \Theta \to [0, \infty]$ given by
$$R(\theta, \delta) = E_\theta\big[ L(\theta, \delta(X)) \big] = \int_{\mathcal{X}} L(\theta, \delta(x)) \, P_\theta(dx).$$

## Remarks

The risk is an [[expected value|0WQNRCNY3M]] over the data, for a fixed $\theta$.
Under the squared error loss $L(\theta, a) = (a - g(\theta))^2$, the risk of an estimator is its [[mean squared error|05NZ1HHQMV]].
For randomized rules, one also averages over the randomization.

Example: let $X_1, \dots, X_n$ be independent [[normal|05AQE0QR0D]] with mean $\theta$ and variance $1$, with squared error loss.
For $\delta = \bar X$, the risk is $R(\theta, \delta) = E_\theta[(\bar X - \theta)^2] = \operatorname{Var}_\theta(\bar X) = 1/n$ for every $\theta$.

Non-example: the realized loss $L(\theta, \delta(x))$ at the observed data $x$ is not the risk.
It depends on the data, while the risk is an average over all possible data.
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<document source="https://tamnd.github.io/math2/node/RNX150HS1P.md" tag="RNX150HS1P" kind="concept">
# Strictly singular operator

## Summary

A strictly singular operator is a bounded linear map between Banach spaces that is not an isomorphism on any infinite-dimensional closed subspace.
Every compact operator is strictly singular, but the converse fails in general Banach spaces.
Strictly singular operators share an important property with compact ones: adding one of them to a Fredholm operator does not change the Fredholm index.

## Explanations

### L1 human

An isomorphism onto its image keeps every vector at a length comparable to its original length.
A strictly singular operator cannot do that on any infinite-dimensional piece of the space: on every such piece some unit vectors are shrunk almost to zero.
For example, the inclusion of the sequence space $\ell^1$ into $\ell^2$ is strictly singular.
It is not compact, since it sends the unit vectors $e_n$ to unit vectors at distance $\sqrt{2}$ from each other.
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<document source="https://tamnd.github.io/math2/node/RNXPV3KC5J.md" tag="RNXPV3KC5J" kind="concept">
# Henselian ring

## Summary

A Henselian ring is a [[local ring|MZ02BKPHJA]] in which [[Hensel's lemma|010QVB71T2]] holds.
There, factorizations of polynomials over the [[residue field|NJ3APQ8BHH]] into coprime factors lift to the ring.
Every complete local ring, such as the $p$-adic integers or a formal power series ring, is Henselian.
Henselian rings behave like small neighbourhoods of a point in the étale topology.
Every local ring has a [[Henselization|917PGHVZPM]], a smallest Henselian ring over it.

## Explanations

### L1 human

Hensel's lemma says that a root of a polynomial modulo $p$ can be refined to a root modulo $p^2$, $p^3$, and so on, if the root is simple.
A Henselian ring is a local ring where this refinement always ends in an actual root inside the ring.
The $p$-adic integers have this property, but the fractions with denominators prime to $p$ do not.
For example, $x^2 + 1$ has a simple root modulo $5$, yet no square root of $-1$ is rational.
The [[definition|WSQ8GH8XWZ]] states the lifting condition.
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<document source="https://tamnd.github.io/math2/node/RPAVY0W0XS.md" tag="RPAVY0W0XS" kind="concept">
# Orbifold

## Summary

An orbifold is a space that looks near each point like the quotient of an open subset of $\R^n$ by a finite group.
It generalizes a [[smooth manifold|R2TQ82PCNB]], which is the case where all these groups are trivial.
Orbifolds arise as quotients of manifolds by properly discontinuous actions of discrete groups, which may have fixed points.
They are central in the geometry of surfaces and $3$-manifolds.

## Explanations

### L1 human

Fold a sheet of paper into a cone: near the tip the cone is not flat, but it still looks like a flat disk divided by a rotation.
An orbifold is a space built from such pieces, where each piece is a flat piece of $\R^n$ divided by a finite group of symmetries.
For example, rotate the plane $\R^2$ by a third of a turn around the origin.
Points that are moved into each other are glued, and the result is a cone with one special point, the image of the origin.
The cone point remembers that the group of order $3$ fixed it.
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<document source="https://tamnd.github.io/math2/node/RPBKZ7E9VE.md" tag="RPBKZ7E9VE" kind="concept">
# Constant-mean-curvature surface

## Summary

A constant-mean-curvature surface, or CMC surface, is a surface in space whose mean curvature takes the same value at every point.
Minimal surfaces are the case where this value is zero.
CMC surfaces are the critical points of area among surfaces that enclose a fixed volume.
So they model soap bubbles and other interfaces with a pressure difference.
Round spheres and circular cylinders are the first examples.

## Explanations

### L1 human

A soap bubble has the same pressure difference across its wall everywhere, so its mean curvature is the same everywhere.
That is why a free bubble is round.
A cylinder of radius $r$ is also a CMC surface: it bends with curvature $1/r$ around and not at all along, so the average is $1/(2r)$ at every point.
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<document source="https://tamnd.github.io/math2/node/RPFX0V6BYF.md" tag="RPFX0V6BYF" kind="definition">
# Littlewood polynomial

## Definition

Let $n \ge 0$ be an integer.
A *Littlewood polynomial* of degree $n$ is a polynomial
$$p(z) = \sum_{k=0}^{n} a_k z^k$$
in one complex variable $z$ whose coefficients satisfy $a_k \in \set{-1, 1}$ for $k = 0, \ldots, n$.

## Remarks

On the unit circle, $\theta \mapsto p(e^{i\theta})$ is a [[trigonometric polynomial|J3NJQ5FZJ5]] with coefficients $a_k$.
By orthogonality of the functions $e^{ik\theta}$,
$$\frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{p(e^{i\theta})}^2 \, d\theta = \sum_{k=0}^{n} a_k^2 = n + 1.$$
So $\max_{\abs{z} = 1} \abs{p(z)} \ge \sqrt{n+1}$ and $\min_{\abs{z} = 1} \abs{p(z)} \le \sqrt{n+1}$.
Littlewood asked whether there are constants $0 < A \le B$ and Littlewood polynomials of every degree with $A \sqrt{n+1} \le \abs{p(z)} \le B \sqrt{n+1}$ on $\abs{z} = 1$.
Balister, Booth, Bollobás, Morris and Tiba answered this positively in 2019.

Example: $p(z) = 1 + z - z^2$ has degree $2$ and coefficients $1, 1, -1$, all in $\set{-1, 1}$.
At $z = 1$ its value is $1$, and at $z = -1$ it is $-1$.

Non-example: $q(z) = 1 + z^2 = 1 + 0 \cdot z + z^2$ is not a Littlewood polynomial, since its coefficient of $z$ is $0$.
Polynomials with coefficients in $\set{-1, 0, 1}$ form a different, larger class.
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# Halley's method

## Statement

Let $f$ be a real function that is three times continuously differentiable on an open interval $I$.
Let $r \in I$ be a simple zero of $f$, that is, $f(r) = 0$ and $f'(r) \ne 0$.
*Halley's method* is the iteration
$$x_{n+1} = x_n - \frac{2 f(x_n) f'(x_n)}{2 f'(x_n)^2 - f(x_n) f''(x_n)}.$$
Then there is $\delta > 0$ such that for every $x_0$ with $\abs{x_0 - r} < \delta$ the iterates are defined and converge to $r$.
The [[convergence|YWZQC5E54B]] has order at least $3$:
$$\abs{x_{n+1} - r} \le C \abs{x_n - r}^3$$
for some constant $C$ and all $n$.

## Remarks

The step is [[Newton's method|07EXVNF9VP]] applied to $g = f / \sqrt{\abs{f'}}$.
At a simple zero, $g$ has the zero $r$ and satisfies $g''(r) = 0$, and Newton's method on such a function converges cubically.
The same formula can be written $x_{n+1} = x_n - \dfrac{f/f'}{1 - f f''/(2 f'^2)}$, evaluated at $x_n$.
Here $f'$ is the [[derivative|0F1WR9C83K]] of $f$.
For $f(x) = x^2 - a$ with $a > 0$, the method gives $x_{n+1} = x_n \dfrac{x_n^2 + 3a}{3x_n^2 + a}$, which roughly triples the number of correct digits of $\sqrt{a}$ per step near the root.
At a multiple zero the order of convergence drops to $1$.

## Explanations

### L1 human

Newton's method replaces the graph by a tangent line.
Halley's method also uses the second derivative, so it follows the bending of the graph.
Near a simple root, each step roughly triples the number of correct digits, against doubling for Newton's method.

### L2 human

Each step costs one evaluation of $f$, $f'$ and $f''$.
So it pays off when the second derivative is cheap, as for polynomials.
It is the case $d = 2$ of the Householder family of methods, whose case $d = 1$ is Newton's method.
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<document source="https://tamnd.github.io/math2/node/RPN59B2XSR.md" tag="RPN59B2XSR" kind="concept">
# c-cyclical monotonicity

## Summary

Let $c$ be a cost function.
A set of pairs $(x, y)$ is $c$-cyclically monotone if no cyclic reassignment of finitely many of its pairs lowers the total cost.
It is the condition that an optimal transport plan must satisfy on its support: one cannot improve the plan by swapping destinations along a cycle.
For the quadratic cost on $\R^n$ it is the cyclical monotonicity of Rockafellar.
That notion characterizes the subsets of the graph of the subdifferential of a convex function.
Under mild hypotheses, a [[coupling|HTR7J0W66V]] with finite cost is optimal exactly when it is concentrated on a $c$-cyclically monotone set.

## Explanations

### L1 human

Suppose bakeries send bread to cafés, and each pair (bakery, café) on the delivery list has a cost.
The list is $c$-cyclically monotone if no group of bakeries can save money by passing their deliveries around in a circle.
On the real line with cost $(x - y)^2$, sending $0 \to 0$ and $1 \to 1$ passes the test.
Sending $0 \to 1$ and $1 \to 0$ fails, since swapping the two destinations lowers the cost from $2$ to $0$.
An optimal transport plan can never contain such a wasteful cycle.
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<document source="https://tamnd.github.io/math2/node/RPQNSM5HKC.md" tag="RPQNSM5HKC" kind="statement">
# Mellin transform of a power times a function

## Statement

Let $f \colon (0, \infty) \to \C$ be measurable, let $a \in \C$, and let $g(x) = x^{a} f(x)$ for $x > 0$, where $x^{a} = e^{a \log x}$ is the [[complex power|0RNGDMHT7M]].
Let $s \in \C$.
Then $g$ is [[Mellin convergent|SGZZ4Y0D36]] at $s$ if and only if $f$ is Mellin convergent at $s + a$.
In that case
$$\mathcal{M}g(s) = \mathcal{M}f(s + a),$$
where $\mathcal{M}$ is the [[Mellin transform|N5PVGRJE1T]].

## Remarks

The proof is the identity $x^{s-1} \cdot x^{a} = x^{(s + a) - 1}$ for $x > 0$, which holds because $\log x$ is real.
Taking absolute values gives $\abs{x^{s-1} g(x)} = x^{\operatorname{Re}(s + a) - 1} \abs{f(x)}$, hence the equivalence of convergence.
So the strip of convergence is shifted by $-\operatorname{Re} a$.
Example: $f(x) = e^{-x}$ and $a = 1$ give $\mathcal{M}[x e^{-x}](s) = \Gamma(s + 1)$ for $\operatorname{Re} s > -1$.

## Explanations

### L1 human

Multiplying a function by a power $x^{a}$ just shifts its Mellin transform by $a$ in the variable $s$.
This is the Mellin counterpart of the rule that multiplying by $e^{i\omega x}$ shifts a Fourier transform.

### L2 human

The rule shows that the choice of the exponent $s - 1$ rather than $s$ in the Mellin kernel is a convention: changing it only shifts $s$.
With the dilation rule and the power substitution rule, it gives the Mellin transforms of all functions $x^{b} f(c x^{a})$ from that of $f$.
It is also how a weight $x^{\sigma}$ is moved in and out of Plancherel-type identities for the Mellin transform.
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<document source="https://tamnd.github.io/math2/node/RPSYWTKB4V.md" tag="RPSYWTKB4V" kind="definition">
# Singularity of a smooth map

## Definition

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] of dimensions $m$ and $n$, and let $f \colon M \to N$ be a smooth map.
For $p \in M$, let $df_p \colon T_p M \to T_{f(p)} N$ be the [[differential|6P9PXQDSEN]] of $f$ at $p$.
The point $p$ is a *singular point*, or *singularity*, of $f$ if the [[rank|0ZZXSBTR4J]] of $df_p$ is less than $\min(m, n)$.
Otherwise $p$ is a *regular point* of $f$.
The set of singular points of $f$ is denoted $\Sigma(f)$.

## Remarks

Example: $f \colon \R^2 \to \R^2$, $f(x, y) = (x, y^2)$, has $df_{(x,y)} = \begin{pmatrix} 1 & 0 \\ 0 & 2y \end{pmatrix}$, of rank $1 < 2$ exactly when $y = 0$.
So $\Sigma(f)$ is the line $y = 0$, along which $f$ folds the plane.
Non-example: for $f \colon \R \to \R^2$, $f(t) = (t, t^2)$, the differential $df_t$ has matrix $(1, 2t)^T$ of rank $1 = \min(1, 2)$ at every $t$, so $f$ has no singular points.
A point is regular if and only if $f$ is an [[immersion|6RW68JGBHC]] near it (when $m \le n$) or a [[submersion|BMY55EW3RF]] near it (when $m \ge n$).
For $N = \R$ the singular points are the [[critical points|0YTCMFB4HC]] of $f$, where $df_p = 0$.
Some authors call every point with $df_p$ not surjective a critical point; this agrees with the definition above when $m \ge n$.
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# Kardar–Parisi–Zhang equation

## Summary

The Kardar–Parisi–Zhang (KPZ) equation is the [[stochastic partial differential equation|PRCQG82QTK]] $h_t = \nu h_{xx} + \frac{\lambda}{2} (h_x)^2 + \xi$, where $\xi$ is space-time white noise.
It was proposed by Kardar, Parisi and Zhang in 1986 as a model for a randomly growing interface.
Its solutions are so rough that the square $(h_x)^2$ has no classical meaning, and giving the equation a meaning is a hard problem.
It is the centre of the KPZ universality class, which contains many growth and particle models.
The large-time fluctuations in this class are described by distributions from random matrix theory, such as the Tracy–Widom distribution.

## Explanations

### L1 human

Picture the edge of a burning sheet of paper or a growing bacterial colony.
The height $h$ of the edge grows faster where it is tilted, is smoothed by the second derivative, and is kicked by random noise.
The fluctuations of such an interface grow like $t^{1/3}$, not like $t^{1/4}$ as for the linear equation without the square term.
The [[Cole–Hopf transformation|VBMQ9SJ3DH]] $Z = e^{\lambda h/(2\nu)}$ turns the equation formally into a linear heat equation with random multiplicative noise.
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# Spectral theory

## Definition

*Spectral theory* is the part of [[operator theory|YG9NR68ZH3]] that studies a linear operator $T$ on a Banach or Hilbert space through its [[spectrum|BMKATJ802A]] $\sigma(T)$.
This is the set of complex numbers $\lambda$ for which $T - \lambda I$ has no bounded inverse.
It extends the theory of [[eigenvalues|0BQAA2BK3Q]] and [[eigenvectors|0N6960W0BV]] of matrices.
Its main topics are the location and the parts of the spectrum, the resolvent and functional calculus.
It also contains the spectral theorems, which represent operators as multiplication operators.

## Remarks

Example: for the diagonal operator $D(x_n) = (x_n / n)$ on $\ell^2$, spectral theory gives $\sigma(D) = \set{1/n : n \ge 1} \cup \set{0}$.
Each $1/n$ is an eigenvalue with eigenvector the $n$-th unit vector, and $0$ is in the spectrum without being an eigenvalue.

Non-example: the study of the solutions of a nonlinear equation such as $u'' = u^3$ is not spectral theory in this sense, since no linear operator and no spectrum are involved.
Only its linearisation can be studied spectrally.

The word "spectrum" was used by David Hilbert before quantum mechanics.
Only later did it turn out that the spectral lines of atoms correspond to differences of eigenvalues of a quantum Hamiltonian.

## Explanations

### L1 human

Spectral theory is the study of the "eigenvalues" of operators in infinite dimensions.
It explains what replaces eigenvalues and eigenvectors when an operator has too few of them.
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<document source="https://tamnd.github.io/math2/node/RQ3JKGWD03.md" tag="RQ3JKGWD03" kind="concept">
# Volterra series

## Summary

A Volterra series writes the output of a nonlinear, time-invariant system with memory as a sum of multiple integrals of the input.
The first term is a [[convolution|2Z29JN3VES]] with an impulse response, as for a linear system.
The next terms involve products of two, three or more past values of the input, weighted by kernels of several variables.
It plays for nonlinear operators the role that the [[Taylor series|0Z49YH6ZWY]] plays for functions.
It is used to model and identify nonlinear systems in engineering and biology.

## Explanations

### L1 human

A linear system remembers its past inputs and adds them up with fixed weights.
A nonlinear system may also multiply past inputs with each other, for example the input a second ago times the input two seconds ago.
The Volterra series lists all these effects: first the single inputs, then the pairs, then the triples, each with its own weights.
For example, squaring the output of a simple linear filter gives a system whose series has only a second order term.
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<document source="https://tamnd.github.io/math2/node/RQ44FKS5QM.md" tag="RQ44FKS5QM" kind="concept">
# Dodgson condensation

## Summary

Dodgson condensation is a method to compute the [[determinant|00KDZRW4C0]] of a square matrix by repeatedly replacing it with a smaller matrix of connected $2 \times 2$ minors.
Each step divides by interior entries of the matrix from two steps before.
It rests on the Desnanot–Jacobi identity, a case of [[Sylvester's determinant identity|7SSKBQYFW1]].
It was proposed by Charles Dodgson, better known as Lewis Carroll.

## Explanations

### L1 human

Start with an $n \times n$ matrix and replace each $2 \times 2$ block of neighbouring entries by its determinant.
This gives an $(n-1) \times (n-1)$ matrix.
Repeat, but now divide each new entry by the matching inner entry of the matrix two steps back.
After $n - 1$ steps a single number is left, and it is the determinant.
For $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{pmatrix}$, the first step gives $\begin{pmatrix} -3 & -3 \\ -3 & 2 \end{pmatrix}$, and the second gives $(-6 - 9)/5 = -3 = \det A$.
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<document source="https://tamnd.github.io/math2/node/RQ4X0EH6NH.md" tag="RQ4X0EH6NH" kind="concept">
# Archimedean equivalence classes

## Summary

Let $G$ be an abelian [[linearly ordered group|P2E8BNYC7B]].
Two nonzero elements of $G$ are Archimedean equivalent when each is bounded by a multiple of the other, in absolute value.
The equivalence classes measure the size of elements up to finite factors, in the way orders of magnitude do.
They form a totally ordered set, and the group is Archimedean exactly when it has at most one class of nonzero elements.
These classes index the coordinates in the [[Hahn embedding theorem|FVPXXFNVQY]].

## Explanations

### L1 human

Compare the sizes of things up to any finite factor.
A millimetre and a kilometre are in the same class, since a million millimetres make a kilometre.
In a group with infinitely large elements, some element is larger than every multiple of another one, and then they lie in different classes.
In $\Z^2$ with the dictionary order, $(0, 1)$ and $(0, 5)$ are in the same class, but $(1, 0)$ is infinitely larger than both.
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# Bang–bang control

## Definition

Let $x' = f(x, u)$ be a [[control system|ZPDTVZQN5K]] whose control set $U \subseteq \R^m$ is a compact [[convex set|02AS45T3H9]], for example a box $\prod_{i=1}^m [a_i, b_i]$ with $a_i < b_i$.
An admissible control $u \colon [0, T] \to U$ is *bang–bang* if $u(t)$ is an extreme point of $U$ for almost every $t \in [0, T]$.
For a box, this means $u_i(t) \in \set{a_i, b_i}$ for each $i$ and almost every $t$.
The times at which $u$ jumps between extreme points are its switching times.

## Remarks

Example: for $U = [-1, 1]$, the control $u(t) = 1$ on $[0, 1)$ and $u(t) = -1$ on $[1, 2]$ is bang–bang, with one switching time $t = 1$.
For the double integrator $x_1' = x_2$, $x_2' = u$ starting at rest at $x_1 = -1$, it brings the state to rest at $x_1 = 0$ at time $2$, and this is the minimum time with $\abs{u} \le 1$.
Non-example: $u(t) = \sin t$ on $[0, \pi]$ with $U = [-1, 1]$ is not bang–bang, since $\abs{u(t)} < 1$ except at $t = \pi / 2$.
Consider a linear system $x' = A x + B u$ whose control set $U$ is a compact convex polytope.
The bang–bang principle says that every state reachable at time $T$ is also reachable at time $T$ by a bang–bang control.
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<document source="https://tamnd.github.io/math2/node/RQDZX250B2.md" tag="RQDZX250B2" kind="definition">
# Strictly convex space

## Definition

Let $X$ be a [[normed vector space|03J5RAKB30]] over $\R$ or $\C$.
The space $X$ is *strictly convex* if for all $x, y \in X$ with $\norm{x} = \norm{y} = 1$ and $x \ne y$,
$$\left\| \frac{x + y}{2} \right\| < 1.$$

## Remarks

Equivalent forms: the unit sphere contains no nondegenerate line segment; and $\norm{x + y} = \norm{x} + \norm{y}$ with $x, y \ne 0$ implies $y = c x$ for some real $c > 0$.
Example: $\R^2$ with the Euclidean norm.
If $\norm{x} = \norm{y} = 1$ and $x \ne y$, the parallelogram law gives $\norm{(x + y)/2}^2 = 1 - \norm{(x - y)/2}^2 < 1$.
Non-example: $\R^2$ with $\norm{(a, b)}_1 = \abs{a} + \abs{b}$.
For $x = (1, 0)$ and $y = (0, 1)$ the midpoint $(1/2, 1/2)$ has norm $1$.
Every [[uniformly convex space|BEAMC5011M]] is strictly convex, but not conversely.
For instance, $\ell^1$ with the norm $\norm{x}_1 + \norm{x}_2$ is strictly convex.
Indeed, equality in the triangle inequality forces equality for $\norm{\cdot}_2$.
But this space is not reflexive, so it is not uniformly convex by the [[Milman–Pettis theorem|AJ5RG9XQYZ]].
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<document source="https://tamnd.github.io/math2/node/RQHPBZ9682.md" tag="RQHPBZ9682" kind="definition">
# Mean-periodic function

## Definition

Let $f \colon \R \to \C$ be a [[continuous function|0X3WGX95FE]].
Let $\mu$ range over the nonzero complex [[Radon measures|HZNEQQBCW4]] on $\R$ with compact [[support|FQTXV6BQZG]].
The function $f$ is *mean-periodic* if the [[convolution|2Z29JN3VES]] of $f$ with some such $\mu$ vanishes:
$$(f * \mu)(x) = \int_{\R} f(x - y) \, d\mu(y) = 0 \quad \text{for all } x \in \R.$$

## Remarks

Give $C(\R)$ the topology of uniform convergence on compact sets.
Its dual space is the space of compactly supported complex Radon measures.
By the Hahn–Banach theorem, $f$ is mean-periodic if and only if the closed linear span of the translates $x \mapsto f(x - a)$, $a \in \R$, is not all of $C(\R)$.
Some authors allow $\mu$ to be a compactly supported distribution instead of a measure.

Example: if $f$ is continuous with period $p > 0$, take $\mu = \delta_0 - \delta_p$.
Then $(f * \mu)(x) = f(x) - f(x - p) = 0$, so $f$ is mean-periodic.
Similarly $f(x) = e^{\lambda x}$ with $\lambda \in \C$ satisfies $f * (\delta_0 - e^{\lambda} \delta_1) = 0$.

Non-example: $f(x) = e^{x^2}$ is not mean-periodic.
Suppose $f * \mu = 0$ with $\mu$ compactly supported, and let $\nu$ be the measure $e^{y^2} d\mu(y)$.
Then $(f * \mu)(x) = e^{x^2} \int e^{-2xy} \, d\nu(y)$, so $F(z) = \int e^{-2zy} \, d\nu(y)$ vanishes for real $z$.
Since $\nu$ has compact support, $F$ is an entire function, so $F = 0$ on $\C$.
On the imaginary axis $F$ is the Fourier transform of $\nu$, so $\nu = 0$ and $\mu = 0$.
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<document source="https://tamnd.github.io/math2/node/RQJC67S0QJ.md" tag="RQJC67S0QJ" kind="statement">
# Law of the iterated logarithm

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|00MRN0FARV]] real random variables with $E[X_1] = 0$ and $E[X_1^2] = 1$.
Let $S_n = X_1 + \cdots + X_n$.
Then, [[almost surely|0ST21PGDAP]],
$$\limsup_{n \to \infty} \frac{S_n}{\sqrt{2 n \log \log n}} = 1 \quad \text{and} \quad \liminf_{n \to \infty} \frac{S_n}{\sqrt{2 n \log \log n}} = -1,$$
where $\log$ is the [[natural logarithm|04DXAFN2ES]] and $\limsup$ is the [[limit superior|0T3RZQXDYD]].

## Remarks

This version under a finite second moment is the Hartman–Wintner theorem; Khinchin proved the first case, for coin tossing.
One proof first treats the [[Wiener process|22735FGJV6]] $W$, for which $\limsup_{t \to \infty} W_t / \sqrt{2 t \log \log t} = 1$ almost surely.
The upper bound uses the Gaussian tail and the [[Borel–Cantelli lemma|0T6WS7TT77]] along a geometric sequence of times $t_k = q^k$.
The lower bound uses the [[second Borel–Cantelli lemma|0NWXRAMEE7]] for the independent increments between these times.
The case of general $X_k$ then follows by a strong approximation of the walk by Brownian motion, such as [[Skorokhod's embedding theorem|XHTYZ7ER4Z]].

Example: take a simple symmetric [[random walk|04VZE3FZXF]], with steps $\pm 1$, and fix $\varepsilon > 0$.
Almost surely, the walk exceeds $(1 - \varepsilon)\sqrt{2 n \log \log n}$ for infinitely many $n$.
Almost surely, it exceeds $(1 + \varepsilon)\sqrt{2 n \log \log n}$ for only finitely many $n$.

The theorem sits between two other limit theorems.
The [[strong law of large numbers|0DFWHQ839T]] gives $S_n / n \to 0$ almost surely.
The [[central limit theorem|0EPZC4Z1FC]] gives that $S_n / \sqrt{n}$ converges in distribution, and it implies that $\limsup_n S_n / \sqrt{n} = \infty$ almost surely.

## Explanations

### L1 human

A random walk with fair steps of size about $1$ drifts away from $0$ by about $\sqrt{n}$ after $n$ steps.
Over an infinite time, it reaches slightly further now and then.
The law of the iterated logarithm gives the exact size of its record excursions: $\sqrt{2 n \log \log n}$.

### L2 human

The result describes the almost sure size of the fluctuations of sums, which neither the strong law nor the central limit theorem determines.
The normalisation $\sqrt{2 n \log \log n}$ is the boundary between the scale $\sqrt{n}$ of the central limit theorem and larger scales that the walk exceeds only finitely often.
Strassen's functional form describes the whole set of limit points of the rescaled path.
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<document source="https://tamnd.github.io/math2/node/RQP2HJ4WQ1.md" tag="RQP2HJ4WQ1" kind="definition">
# Ring of invariants

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $G$ be a group that [[acts|0WCY42EV9V]] on $R$ by ring [[automorphisms|DSQ17TNW1C]], written $r \mapsto g \cdot r$.
The *ring of invariants* of the action is
$$R^G = \set{ r \in R : g \cdot r = r \text{ for all } g \in G }.$$

## Remarks

$R^G$ is a [[subring|0GNCRPYXYV]] of $R$, because each $g$ acts by a ring homomorphism and so fixes $0$, $1$, sums and products of fixed elements.
The main case is a group $G$ acting linearly on a finite-dimensional vector space $V$ over a field $k$.
Then $G$ acts on the [[polynomial ring|0SDC934R52]] $k[V]$ of polynomial functions by $(g \cdot f)(v) = f(g^{-1} v)$.
Emmy Noether proved that for a finite group $G$ acting by $k$-algebra automorphisms on a finitely generated commutative $k$-algebra $R$, the ring $R^G$ is finitely generated.

Example: the symmetric group $S_n$ acts on $k[x_1, \ldots, x_n]$ by permuting the variables.
Its ring of invariants is the ring of [[symmetric polynomials|0VHSHB9B6J]], which is a polynomial ring in the elementary symmetric polynomials by the [[fundamental theorem of symmetric polynomials|0GW1ZZH0R4]].

Example: let $G = \set{1, g}$ act on $k[x, y]$ by $g \cdot x = -x$ and $g \cdot y = -y$, where the characteristic of $k$ is not $2$.
A monomial $x^a y^b$ is fixed exactly when $a + b$ is even, so $k[x, y]^G = k[x^2, xy, y^2]$.

Non-example: in the last example, $x \notin k[x, y]^G$, because $g \cdot x = -x \ne x$.
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# Stable vector bundle

## Summary

A stable vector bundle is a [[vector bundle|P4DZM975BV]] whose proper nonzero subbundles all have smaller slope than the bundle itself.
The slope of a bundle is its degree divided by its rank.
Stability is the condition that makes vector bundles on a curve form a good [[moduli space|FJZRW9P0K3]].
It was introduced by David Mumford in the setting of [[geometric invariant theory|9EG3A21ET1]].
Over a compact Riemann surface, the Narasimhan–Seshadri theorem identifies stable bundles of degree $0$ with irreducible unitary representations of the fundamental group.

## Explanations

### L1 human

A vector bundle on a curve has two numbers: its rank $r$ and its degree $d$, which measures how twisted it is.
The slope $d/r$ is the twist per dimension.
The bundle is stable when no smaller piece inside it is relatively more twisted than the whole.
A direct sum $L_1 \oplus L_2$ of line bundles with $\deg L_1 > \deg L_2$ is not stable, because $L_1$ is a piece with larger slope.
Stable bundles have only scalar automorphisms, which is why they can be classified by a well-behaved space.
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<document source="https://tamnd.github.io/math2/node/RQS6YHAR8N.md" tag="RQS6YHAR8N" kind="statement">
# Mazur's torsion theorem

## Statement

Let $E$ be an [[elliptic curve|427N6S6QQW]] over $\Q$, and let $E(\Q)_{\mathrm{tors}}$ be the [[torsion subgroup|0DBWXQRXQJ]] of the [[group of rational points|P7V1BQJPSS]].
Then $E(\Q)_{\mathrm{tors}}$ is isomorphic to one of the following $15$ groups:

1. the [[cyclic group|0KVGH0J5B3]] $\Z/N\Z$ with $1 \le N \le 10$ or $N = 12$;
2. $\Z/2\Z \times \Z/2N\Z$ with $1 \le N \le 4$.

## Remarks

Each of the $15$ groups occurs for infinitely many curves with different $j$-invariants; this part was known before Mazur, from explicit families.
The hard part is that no other group occurs.
A rational point of order $N$ on $E$ gives a non-cuspidal rational point on the modular curve $X_1(N)$.
Mazur proved in 1977, with his study of the Eisenstein ideal, that for primes $N > 7$ the curve $X_1(N)$ has no such points.
Together with earlier results on the composite orders, this gives the list.
Example: on $y^2 = x^3 + 1$ the torsion subgroup is cyclic of order $6$.
Over larger number fields, the list of possible torsion groups is different, but it is still finite for each degree, by a theorem of Merel.

## Explanations

### L1 human

The points of finite order on an elliptic curve over the rationals form a small finite group.
Mazur's theorem gives the complete list of what that group can be: never more than $16$ points, and only $15$ shapes.

### L2 human

Together with the [[Nagell–Lutz theorem|G38B7E4ESW]], it makes the torsion part of [[Mordell's theorem|BZHGKN6Q3K]] fully explicit.
The method, through rational points on modular curves, is an early deep use of modular curves in arithmetic.
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# Injective tensor product

## Summary

Let $X$ and $Y$ be Banach spaces.
Their injective tensor product is the completion of the algebraic [[tensor product|0BF2DE2SYK]] $X \otimes Y$ for the smallest reasonable norm on it.
This norm sees a tensor only through its values on pairs of functionals of norm at most $1$.
It describes spaces of continuous vector-valued functions: $C(K)$ with the injective tensor product by $X$ is the space of continuous functions from $K$ to $X$.
With the [[projective tensor product|685ZV73JJ8]] it is one of the two extreme tensor norms of Grothendieck's theory.

## Explanations

### L1 human

A tensor $u = \sum_i x_i \otimes y_i$ gives a number $\sum_i f(x_i) g(y_i)$ for each pair of functionals $f$ and $g$.
The injective norm is the largest such number when $f$ and $g$ have norm at most $1$.
If $X$ and $Y$ are spaces of continuous functions on compact spaces $K$ and $L$, the tensor becomes the function $(s, t) \mapsto \sum_i x_i(s) y_i(t)$ on $K \times L$.
Then the injective norm is just the maximum of its absolute value.
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<document source="https://tamnd.github.io/math2/node/RQYX6E42D0.md" tag="RQYX6E42D0" kind="definition">
# Borel summation

## Definition

Let $\sum_{k \ge 0} a_k$ be a [[series|0JH4MN4K1C]] of complex numbers.
Its *Borel transform* is the [[power series|0VP56B4EMX]]
$$\mathcal{B}(t) = \sum_{k \ge 0} \frac{a_k}{k!} t^k.$$
The series $\sum_k a_k$ is *Borel summable* if:

1. $\mathcal{B}$ has a positive [[radius of convergence|0660QX1VPR]];
2. $\mathcal{B}$ has an [[analytic continuation|0M3YQJ81NB]] to an open set that contains the half-line $[0, \infty)$, still written $\mathcal{B}$;
3. the improper integral $\int_0^\infty e^{-t} \mathcal{B}(t) \, dt$ converges.

The value of this integral is the *Borel sum* of the series.

## Remarks

Example: for $a_k = z^k$ with a complex number $z$, $\mathcal{B}(t) = e^{zt}$, and the integral converges exactly when $\operatorname{Re} z < 1$, with value $1/(1 - z)$.
So the geometric series is Borel summable to $1/(1-z)$ on the half-plane $\operatorname{Re} z < 1$, also where it diverges, such as at $z = -1$, where the sum is $1/2$.
Non-example: for $a_k = (k!)^2$, the Borel transform $\sum_k k! \, t^k$ has radius of convergence $0$, so clause 1 fails.
The method is regular: a convergent series is Borel summable, and its Borel sum is its ordinary sum.
For a power series $\sum_k a_k z^k$, the Borel sum is $\int_0^\infty e^{-t} \mathcal{B}(zt) \, dt$, which is a [[Laplace transform|05XT5AVJBK]] of $\mathcal{B}$ up to a change of variable.
A weaker variant, the weak Borel method, uses the limit of $e^{-t} \sum_{n \ge 0} s_n t^n/n!$ as $t \to \infty$, where $s_n$ are the partial sums.
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<document source="https://tamnd.github.io/math2/node/RR10R8JN02.md" tag="RR10R8JN02" kind="definition">
# Intuitionism

## Definition

*Intuitionism* is the position in the [[foundations of mathematics|EFFRJFW67Y]] that consists of these theses:

1. mathematical objects are mental constructions, and an object exists only if it can be constructed;
2. a mathematical statement is true only if it has a [[constructive proof|01XM7JEKHY]], so truth means provability by construction;
3. as a consequence, the law of excluded middle $P \lor \neg P$ is not accepted for arbitrary statements $P$.

## Remarks

The statement "there exist irrational numbers $a$ and $b$ with $a^b$ rational" has an intuitionistic proof.
Take $a = \sqrt 2$ and $b = 2 \log_2 3$; then $a^b = 2^{\log_2 3} = 3$.
Both numbers are shown to be irrational directly.
The classical proof considers the two cases "$\sqrt 2^{\sqrt 2}$ is rational" and "it is irrational".
This proof is not intuitionistic: it uses excluded middle and does not say which case holds.
The logic of intuitionism was formalized by Arend Heyting as [[intuitionistic logic|Q11N8X99FK]], and its meaning of the connectives is the [[Brouwer–Heyting–Kolmogorov interpretation|7B53ZGXXEQ]].
</document>

<document source="https://tamnd.github.io/math2/node/RR2S05C09B.md" tag="RR2S05C09B" kind="concept">
# Askey scheme

## Summary

The Askey scheme is a chart that organises the hypergeometric orthogonal polynomials into levels connected by limit relations.
At the top are the [[Wilson|1CPF7SWQ3S]] and [[Racah polynomials|P6JYNDX5DR]], given by ${}_4F_3$ series; at the bottom are the [[Hermite polynomials|MGACY98E9H]].
Each arrow says that one family is a limit of another as a parameter tends to a special value or to infinity.
It shows the classical families of Jacobi, Laguerre and Hermite as degenerations of a few very general ones.

## Explanations

### L1 human

There are many named families of orthogonal polynomials, and at first they look unrelated.
The Askey scheme is a family tree for them.
Families with more parameters are higher up, and moving down an arrow means fixing or removing a parameter by a limit.
For example, the [[Krawtchouk polynomials|CJGKW80KBR]] for $N$ coin tosses with probability $a/N$ become the [[Charlier polynomials|6KH8KVYS3R]] as $N \to \infty$, just as the binomial distribution becomes the Poisson distribution.
</document>

<document source="https://tamnd.github.io/math2/node/RR5FCG92PN.md" tag="RR5FCG92PN" kind="concept">
# Stiff equation

## Summary

A stiff equation is an ordinary differential equation on which explicit numerical methods must take extremely small steps to stay stable.
This happens even where the solution is smooth, for example with the [[Euler method|0JARTXX4XP]].
Stiffness typically comes from processes that decay on very different time scales.
Stiff equations are solved with implicit methods, which remain stable with large steps.
See the [[definition|AXND5JRQAZ]].

## Explanations

### L1 human

Imagine a system with one part that settles in a millisecond and another part that changes over hours.
After the first millisecond the fast part is at rest, and the solution looks slow and smooth.
Still, a simple step-by-step method must keep tiny steps, or its errors in the fast part blow up.
For $y' = -1000\,(y - \cos t)$ the solution quickly follows $\cos t$, yet the explicit Euler method needs steps below $0.002$.
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# Baire property

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $A \subseteq X$.
The set $A$ has the *Baire property* if there is an [[open set|02BH3MCBQB]] $U \subseteq X$ such that the [[symmetric difference|0ME21QN125]] $A \mathbin{\triangle} U$ is [[meagre|0XGCZ4GDZ6]] in $X$.

## Remarks

Example: every closed set $F \subseteq X$ has the Baire property.
Take $U$ to be the [[interior|0RZH6W98HK]] of $F$.
Then $F \mathbin{\triangle} U = F \setminus U$ is closed and has empty interior, so it is [[nowhere dense|0VQYCCQVJ5]] and hence meagre.

Non-example: a Vitali set $V \subseteq \R$ contains exactly one point of each coset of $\Q$ in $\R$.
Such a set does not have the Baire property; its construction uses the axiom of choice.

The sets with the Baire property form a sigma-algebra that contains the open sets, so every Borel set has the Baire property.
Equivalently, $A$ has the Baire property if $A = G \cup M$ for a $G_\delta$ set $G$ and a meagre set $M$.
</document>

<document source="https://tamnd.github.io/math2/node/RREV63H509.md" tag="RREV63H509" kind="definition">
# Moduli of algebraic curves

## Definition

Let $g \ge 0$ be an integer.
Let $T$ be a scheme over $\C$.
A *family of curves of [[genus|NJ787GZS1N]] $g$* over $T$ is a smooth proper morphism $\pi \colon C \to T$.
Its fibres must be connected [[algebraic curves|XJM2MT4S3Y]] of genus $g$.
Let $F_g$ be the functor that sends $T$ to the set of isomorphism classes of families of curves of genus $g$ over $T$.
It sends a morphism $T' \to T$ to the pullback of families.
A *coarse [[moduli space|FJZRW9P0K3]] of curves of genus $g$* is a scheme $M_g$ over $\C$ with a natural transformation $\Phi \colon F_g \to \Hom(-, M_g)$ such that:

1. $\Phi$ is a bijection from $F_g(\operatorname{Spec} \C)$ onto the set of $\C$-points of $M_g$, and
2. for every scheme $N$ and natural transformation $\Psi \colon F_g \to \Hom(-, N)$ there is a unique morphism $u \colon M_g \to N$ with $\Psi = u \circ \Phi$.

## Remarks

Here $F_g(\operatorname{Spec} \C)$ is the set of isomorphism classes of smooth projective connected curves of genus $g$.
A coarse moduli space is unique up to unique isomorphism, and it exists for every $g$.
It is not a fine moduli space, that is, $\Phi$ is not an isomorphism of functors, because some curves have nontrivial automorphisms.
The analytic space underlying $M_g$ is the quotient of the [[Teichmüller space|ZZGC6BZNJ5]] $T_g$ by the mapping class group.

Example: $g = 1$.
Each curve of genus $1$ over $\C$ is an elliptic curve, and the $j$-invariant gives a bijection from isomorphism classes to $\C$.
This bijection comes from a natural transformation as in the definition, and $M_1$ is the affine line $\mathbb{A}^1$.

Non-example: the bare set $F_g(\operatorname{Spec} \C)$ has no scheme structure.
So it is not a coarse moduli space by itself, although it is in bijection with the points of $M_g$.
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# Normal coordinates

## Summary

Normal coordinates at a point $p$ of a [[Riemannian manifold|7CBT4SM83Y]] are local coordinates centred at $p$.
They identify a neighbourhood of $p$ with a ball in the tangent space through the [[exponential map|Y7Z0AGY1A7]].
In them, the metric equals the Euclidean metric at $p$ and its first derivatives vanish at $p$, so the [[Christoffel symbols|FZBA1HA9ZM]] vanish at $p$.
They make many local computations, such as those of curvature identities, much simpler.

## Explanations

### L1 human

Near a point of a curved space, normal coordinates are the best possible imitation of flat coordinates.
Each nearby point gets the coordinates of the direction and distance one walks along a geodesic to reach it from $p$.
Straight lines through the origin in these coordinates are exactly the geodesics through $p$.
On the Earth, this is like the azimuthal equidistant map centred at one city, which gets distances and directions from that city right.
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<document source="https://tamnd.github.io/math2/node/RRK8P6TXG2.md" tag="RRK8P6TXG2" kind="concept">
# Ample line bundle

## Summary

A line bundle is ample when its high tensor powers have enough global sections to embed the scheme into projective space.
Let $X$ be a scheme of finite type over a Noetherian ring.
A line bundle $L$ on $X$ is ample exactly when some power $L^{\otimes m}$ is very ample.
Very ample means that it is the pullback of $\mathcal{O}(1)$ under an immersion into projective space.
Ampleness is the algebraic form of positivity: it plays the role of positive curvature in complex geometry.
Many vanishing, finiteness and embedding theorems assume an ample line bundle.

## Explanations

### L1 human

A line bundle attaches a line to each point of a space, and its sections are like functions with values in these lines.
A line bundle is ample if, after taking a high enough power, its sections can tell all points apart and all tangent directions apart.
Then the sections give coordinates that place the space inside a projective space without folding it.
On the projective line, the bundle $\mathcal{O}(1)$, whose sections are the linear forms, is ample.
Its dual $\mathcal{O}(-1)$ has no nonzero sections in any positive power, so it is not ample.
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<document source="https://tamnd.github.io/math2/node/RRZGEFJS7H.md" tag="RRZGEFJS7H" kind="concept">
# Symmetric and self-reciprocal kernels

## Summary

A [[kernel|9M0NNNFJGM]] $K(x, y)$ of an [[integral transform|YCK709QB06]] is symmetric when it does not change if $x$ and $y$ are swapped.
It is self-reciprocal (a Fourier kernel) when the transform is its own inverse: applying it twice gives back the original function.
The Fourier cosine and sine kernels and the [[Hankel transform|2P9YWDGKC4]] kernel $\sqrt{xy}\, J_\nu(xy)$ are both symmetric and self-reciprocal.
For such kernels the transform and its [[inverse transform|GC96MQC0PP]] are given by the same formula, which makes the theory and the tables symmetric.

## Explanations

### L1 human

A symmetric kernel is like a symmetric matrix.
A self-reciprocal kernel is like a matrix whose square is the identity, such as a reflection.
For the cosine kernel $\sqrt{2/\pi}\cos(xy)$ on $(0, \infty)$, if $g$ is the transform of $f$, then $f$ is the transform of $g$.
So a table of cosine transforms can be read from left to right or from right to left.
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# Normal number

## Definition

Let $b \ge 2$ be an integer, and let $x$ be a real number.
Write the fractional part $x - \floor{x}$ in base $b$ as $0.d_1 d_2 d_3 \ldots$ with digits $d_i \in \set{0, 1, \dots, b-1}$.
When there are two such expansions, take the one that does not end in an infinite run of the digit $b - 1$.
For a finite string $w$ of $k$ digits and an integer $n \ge 1$, let $N(w, n)$ be the number of indices $i$ with $1 \le i \le n - k + 1$ such that $d_i d_{i+1} \cdots d_{i+k-1} = w$.
The number $x$ is *normal in base $b$* if
$$\lim_{n \to \infty} \frac{N(w, n)}{n} = b^{-k}$$
for every $k \ge 1$ and every string $w$ of $k$ digits.
The number $x$ is *absolutely normal* if it is normal in base $b$ for every integer $b \ge 2$.

## Remarks

Example: Champernowne's number $0.1234567891011121314\ldots$, formed by the decimal digits of the positive integers in order, is normal in base $10$.
Non-example: $x = 1/3 = 0.333\ldots$ in base $10$ is not normal, since for $w = 1$ one has $N(w, n) = 0$ for all $n$, so the limit is $0 \ne 1/10$.
In the same way, no [[rational number|0NCJ5WXN6Z]] is normal in any base, because its expansion is eventually periodic.
A number is *simply normal* in base $b$ if the condition holds for strings of length $k = 1$.
Normality in base $b$ is equivalent to the [[equidistribution modulo one|XM6DH57BWY]] of the sequence $(b^n x)_{n \ge 0}$.
Here $\floor{x}$ is the [[floor|08GR54CKEA]] of $x$.
</document>

<document source="https://tamnd.github.io/math2/node/RS8Z48E784.md" tag="RS8Z48E784" kind="concept">
# Riemann's differential equation

## Summary

Riemann's differential equation is the general second-order linear [[Fuchsian equation|ZBVH2YE50B]] with exactly three [[regular singular points|H96HXG6CA0]] on the Riemann sphere.
It is determined by the positions of the three points and the two exponents at each.
The six exponents must add up to $1$.
The [[hypergeometric differential equation|W8VE25R5A7]] is the case where the points are $0$, $1$ and $\infty$.
Every Riemann equation reduces to it by a [[Möbius transformation|0SG74WBFA2]] and a power factor.

## Explanations

### L1 human

A second-order linear equation whose solutions behave like powers near three special points, and nowhere else misbehave, is completely fixed by those three points and the six powers.
Riemann wrote down this equation explicitly.
Moving the three points to $0$, $1$, $\infty$ by a Möbius map, and multiplying the unknown by suitable powers, turns it into the hypergeometric equation.
So the theory of the hypergeometric function covers every such equation.
</document>

<document source="https://tamnd.github.io/math2/node/RS9VHTWR4K.md" tag="RS9VHTWR4K" kind="definition">
# Second variation

## Definition

Let $X$ be a real vector space, let $U \subseteq X$, and let $J \colon U \to \R$ be a [[functional|1EQM0DJ4NC]].
Let $y_0 \in U$ and $h \in X$ be such that $y_0 + \varepsilon h \in U$ for all small $\abs{\varepsilon}$.
The *second variation* of $J$ at $y_0$ in the direction $h$ is
$$\delta^2 J(y_0)[h] = \frac{d^2}{d\varepsilon^2} J(y_0 + \varepsilon h) \Big|_{\varepsilon = 0},$$
when this derivative exists.

Let $J(y) = \int_a^b L(x, y, y') \, dx$ be an [[integral functional|JRQZ803Q5Y]] with $L(x, y, p)$ of class $C^2$.
For $y_0, h \in C^1([a, b])$ the second variation is
$$\delta^2 J(y_0)[h] = \int_a^b \big( L_{pp} \, h'^2 + 2 L_{yp} \, h h' + L_{yy} \, h^2 \big) \, dx,$$
where the second partial derivatives of $L$ are taken at $(x, y_0(x), y_0'(x))$.

## Remarks

Some books put a factor $\frac{1}{2}$ in front, so that $J(y_0 + \varepsilon h) = J(y_0) + \varepsilon \, \delta J(y_0)[h] + \varepsilon^2 \, \delta^2 J(y_0)[h] + o(\varepsilon^2)$, where $\delta J$ is the [[first variation|5B8NFYB2XQ]].

Example: let $J(y) = \int_0^\pi (y'^2 - y^2) \, dx$ and $y_0 = 0$.
Then $J(\varepsilon h) = \varepsilon^2 \int_0^\pi (h'^2 - h^2) \, dx$, so $\delta^2 J(0)[h] = 2 \int_0^\pi (h'^2 - h^2) \, dx$.
This agrees with the formula, since $L_{pp} = 2$, $L_{yp} = 0$ and $L_{yy} = -2$.

Non-example: let $J(y) = \int_0^1 \abs{y'} \, dx$ on $C^1([0, 1])$ and $y_0 = 0$.
Then $J(\varepsilon h) = \abs{\varepsilon} \int_0^1 \abs{h'} \, dx$, which is not differentiable at $\varepsilon = 0$ when $h$ is not constant.
So the second variation at $0$ does not exist in such a direction $h$; the integrand $\abs{p}$ is not of class $C^2$.

If $h(a) = h(b) = 0$ and $x \mapsto L_{yp}(x, y_0(x), y_0'(x))$ is $C^1$, an integration by parts gives $\delta^2 J(y_0)[h] = \int_a^b \big( P h'^2 + Q h^2 \big) \, dx$ with $P = L_{pp}$ and $Q = L_{yy} - \frac{d}{dx} L_{yp}$.
</document>

<document source="https://tamnd.github.io/math2/node/RSAK8YHMXF.md" tag="RSAK8YHMXF" kind="definition">
# Integrable function

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
Let $f \colon X \to \R$ or $f \colon X \to \C$ be a [[measurable function|YYNCJSR0PJ]].
The function $f$ is *integrable*, or *Lebesgue integrable*, if
$$\int_X \abs{f} \, d\mu < \infty,$$
where the left side is the [[integral of the non-negative function|DG76G5Z6XW]] $\abs{f}$.

## Remarks

Example: on $\R$ with Lebesgue measure, $f(x) = e^{-\abs{x}}$ is measurable since it is continuous.
Its integral is $\int_\R e^{-\abs{x}} \, dx = 2 < \infty$, so $f$ is integrable.

Non-example: $g(x) = \sin(x) / x$ on $[1, \infty)$ with Lebesgue measure is measurable but not integrable.
On each interval $[k\pi, (k+1)\pi]$ the integral of $\abs{g}$ is at least $2 / ((k+1)\pi)$, and these bounds have an infinite sum.
Its improper Riemann integral still converges, so improper integrals and Lebesgue integrals differ here.

A real function $f$ is integrable if and only if both $f^+ = \max(f, 0)$ and $f^- = \max(-f, 0)$ have finite integral.
A function with values in $[-\infty, \infty]$ is called integrable under the same condition, and it is then finite almost everywhere.
The integrable functions form a vector space, often written $\mathcal{L}^1(\mu)$.
</document>

<document source="https://tamnd.github.io/math2/node/RSBFHHPTWX.md" tag="RSBFHHPTWX" kind="statement">
# Gromov's theorem on groups of polynomial growth

## Statement

Let $G$ be a [[finitely generated group|082DWWXMRJ]].
Then $G$ has polynomial [[growth|4CMRM3975W]] if and only if $G$ is virtually nilpotent, that is, $G$ has a [[nilpotent|0Q7MMQHC1P]] [[subgroup|0X1JXT47GA]] of finite [[index|0DTXBQPA2S]].

## Remarks

The direction "virtually nilpotent implies polynomial growth" was known before Gromov.
For a finitely generated nilpotent group the degree of growth is the integer $\sum_{k \ge 1} k \, r_k$.
Here $r_k$ is the torsion-free rank of the quotient $\gamma_k(G) / \gamma_{k+1}(G)$ of consecutive terms of the lower central series.
Gromov proved the hard direction.
His idea is to rescale the word metric of $G$ by $1/n$ and pass to a limit space as $n \to \infty$.
Polynomial growth makes this limit a finite-dimensional, locally compact space.
The solution of Hilbert's fifth problem turns its isometry group into a Lie group, on which $G$ acts.
From this action one extracts the nilpotent subgroup by induction on the degree.
Kleiner later gave a different proof based on harmonic functions on the Cayley graph.

Example: the integer Heisenberg group of upper unitriangular $3 \times 3$ integer matrices is nilpotent, with $r_1 = 2$ and $r_2 = 1$, so its growth is polynomial of degree $2 + 2 = 4$.

A consequence is a gap: the growth of a finitely generated group is either polynomial of an integer degree, or faster than every polynomial.
Groups of intermediate growth exist, such as the [[Grigorchuk group|M0WAT1VQZH]], so "faster than every polynomial" does not mean exponential.

## Explanations

### L1 human

Count the elements of a group that can be reached in at most $n$ steps from the identity.
If this count grows only like a power of $n$, the theorem says the group is very close to being commutative: up to a finite index, it is nilpotent.
So a purely geometric counting condition forces a strong algebraic structure.

### L2 human

The theorem shows that being virtually nilpotent is a [[quasi-isometry|6FKYSH729T]] invariant of finitely generated groups, because growth is.
For finitely generated linear groups the gap is sharper: by the [[Tits alternative|QYXGVES1XM]] and the Milnor–Wolf theorem for solvable groups, their growth is either polynomial or exponential.
It is also used in the study of random walks and of manifolds with almost nonnegative curvature.
</document>

<document source="https://tamnd.github.io/math2/node/RSC5KTVNZD.md" tag="RSC5KTVNZD" kind="definition">
# Scheme

## Definition

A *scheme* is a [[locally ringed space|B7FPT8JTXR]] $(X, \mathcal{O}_X)$ such that every point $x \in X$ has an open neighbourhood $U$ for which $(U, \mathcal{O}_X|_U)$ is an [[affine scheme|B3V3XJ2W8G]].

## Remarks

Example: every affine scheme $\operatorname{Spec} A$ is a scheme, with $U = X$ for every point.
Example: the projective line $\mathbb{P}^1_k$ over a field $k$ is a scheme: it is covered by two open sets, each isomorphic to $\operatorname{Spec} k[t]$.
It is not affine, since its ring of global functions is $k$, while $\operatorname{Spec} k$ is a single point.
Non-example: the one-point space with sheaf $\Z$ is a ringed space but not a scheme.
Its stalk $\Z$ is not a local ring, so it is not even locally ringed.
An open subset $U$ of a scheme, with the restricted sheaf $\mathcal{O}_X|_U$, is again a scheme, called an open subscheme.
</document>

<document source="https://tamnd.github.io/math2/node/RSFYN75DCS.md" tag="RSFYN75DCS" kind="concept">
# Eilenberg swindle

## Summary

The Eilenberg swindle is an argument with infinite direct sums that cancels an object by regrouping an infinite sum.
Its basic form shows that every [[projective module|BFFA03GE3V]] $P$ becomes free after adding a large enough [[free module|0DZ9DKAKZR]]: $P \oplus F \cong F$ for some free $F$.
It explains why the Grothendieck group is built only from finitely generated modules: with infinite sums allowed, every class would be zero.
The same trick, used by Mazur in topology, shows that no nontrivial knot has an inverse under connected sum.

## Explanations

### L1 human

The swindle is the old joke $1 = 1 + (-1 + 1) + (-1 + 1) + \cdots = (1 - 1) + (1 - 1) + \cdots = 0$, made correct.
With numbers the regrouping is not allowed, but with infinite direct sums of modules it is.
If $P \oplus Q$ is free, then adding infinitely many copies of $P \oplus Q$ to $P$ gives the same as infinitely many copies of $Q \oplus P$.
So $P$ disappears into an infinite free module.
</document>

<document source="https://tamnd.github.io/math2/node/RSNZNG0WA1.md" tag="RSNZNG0WA1" kind="concept">
# Askey–Wilson polynomials

## Summary

The Askey–Wilson polynomials $p_n(x; a, b, c, d \mid q)$ are a family of orthogonal polynomials in one variable with four parameters and a base $q$.
They are defined by a terminating [[basic hypergeometric series|EB4N9RQR51]] ${}_4\phi_3$ in the variable $x = \cos\theta$.
They sit at the top of the $q$-Askey scheme.
The [[continuous q-Hermite polynomials|XFKVZJ55Q5]], the [[Rogers polynomials|7H7SACRYEF]] and many other $q$-orthogonal polynomials are special or limiting cases.
As $q \to 1$ they give the [[Wilson polynomials|1CPF7SWQ3S]].
Askey and Wilson introduced them in 1985.

## Explanations

### L1 human

Classical orthogonal polynomials, like the Legendre or Chebyshev polynomials, are each defined by a formula with few parameters.
The Askey–Wilson polynomials are one formula with five parameters $a, b, c, d, q$ that contains a large number of them as special cases or limits.
They are orthogonal on $[-1, 1]$ for a weight built from infinite products.
For example, the polynomial of degree $1$ is $2(1 - abcd)\, x + (abc + abd + acd + bcd) - (a + b + c + d)$, which is symmetric in the four parameters.
</document>

<document source="https://tamnd.github.io/math2/node/RSPZ9TCFQ8.md" tag="RSPZ9TCFQ8" kind="concept">
# Spin structure

## Summary

A spin structure on an oriented [[Riemannian manifold|7CBT4SM83Y]] lifts its bundle of oriented orthonormal frames to a principal bundle for the [[spin group|JJSXXJQGEH]].
The spin group is the double cover of the special orthogonal group.
It is the extra structure needed to define spinors and the Dirac operator on a manifold.
An oriented manifold admits a spin structure exactly when its second Stiefel–Whitney class vanishes.

## Explanations

### L1 human

Rotating an object a full turn brings it back, but some quantities, called spinors, change sign after one turn and return only after two turns.
To have spinors on a curved space, one must choose consistently, over the whole space, how to "unwind" rotations twice.
Such a consistent choice is a spin structure.
Every sphere of dimension at least $2$ has exactly one spin structure, while the complex projective plane has none.
</document>

<document source="https://tamnd.github.io/math2/node/RSRAY0Q68S.md" tag="RSRAY0Q68S" kind="statement">
# Krein–Milman theorem

## Statement

Let $X$ be a Hausdorff [[locally convex space|B6GY2KJMWW]] over $\R$ or $\C$, and let $K \subseteq X$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].
Then $K$ has at least one [[extreme point|3MSGFJM2NX]].
Moreover, $K$ is the [[closure|0QEAJB91GZ]] of the [[convex hull|0FQSDC2EQ2]] of the set $\operatorname{ext} K$ of its extreme points:
$$K = \overline{\operatorname{conv}}(\operatorname{ext} K).$$

## Remarks

Call a nonempty closed subset $F \subseteq K$ a face if every open segment of $K$ that meets $F$ has both endpoints in $F$.
By [[Zorn's lemma|04RXTQA5ZA]] and compactness, every face contains a minimal face.
A minimal face is a single point, since otherwise a continuous functional that separates two of its points would cut out a smaller face; that point is extreme.
If $\overline{\operatorname{conv}}(\operatorname{ext} K)$ were smaller than $K$, the [[Hahn–Banach separation theorem|6GWMGFGFJJ]] would give a continuous functional whose real part has its maximum on $K$ only on a face disjoint from $\operatorname{ext} K$.
But that face contains a minimal face, which is an extreme point of $K$, a contradiction.

For the square $[0, 1]^2$ in $\R^2$ the theorem says that the square is the convex hull of its four corners.
A standard application: the closed unit ball of a dual Banach space is weak*-compact by the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], so it has extreme points.
The unit ball of $c_0$ has no extreme points, so $c_0$ is not isometrically isomorphic to the dual of any Banach space.
Compactness is needed: the closed half-plane $\set{(x, y) : y \ge 0}$ in $\R^2$ is closed and convex but has no extreme points.

## Explanations

### L1 human

A compact convex set is determined by its corners.
Taking all mixtures of the corners, and then all limits of such mixtures, gives back the whole set.
For a polygon this is clear; the theorem says the same in infinite-dimensional spaces.

### L2 human

The theorem is used to prove that certain Banach spaces are not dual spaces, by showing that their unit balls lack extreme points.
In measure theory and probability, the extreme points of compact convex sets of measures or states are often the "pure" or "ergodic" objects.
The theorem shows that they generate everything.
Choquet theory refines the closed convex hull into an integral representation over the extreme points.
The proof combines Zorn's lemma with the [[Hahn–Banach separation theorem|6GWMGFGFJJ]].
</document>

<document source="https://tamnd.github.io/math2/node/RSW29AVKGM.md" tag="RSW29AVKGM" kind="statement">
# Clifford's theorem on special divisors

## Statement

Let $X$ be a smooth projective [[algebraic curve|XJM2MT4S3Y]] of genus $g$ over an algebraically closed field, with [[canonical divisor|W05VYY4HBR]] $K$.
For a [[divisor|Z9DN23DAE2]] $D$, let $\ell(D)$ be the dimension of the space of rational functions $f$ with $\operatorname{div}(f) + D \ge 0$.
Then the [[complete linear system|GHRPE8CXJ9]] $\abs{D}$ has dimension $\ell(D) - 1$.
Let $D$ be an effective divisor that is special, meaning $\ell(K - D) > 0$.
Then
$$\ell(D) - 1 \le \frac{\deg D}{2}.$$
Equality holds if and only if one of the following holds:

1. $D = 0$;
2. $D$ is linearly equivalent to $K$;
3. $X$ is [[hyperelliptic|5SB8EKSQRN]], and $D$ is linearly equivalent to a multiple of a fibre of the degree-$2$ map $X \to \mathbb{P}^1$.

## Remarks

The inequality comes from adding linear systems.
Since $D$ and $K - D$ are both effective up to linear equivalence, the addition map $\abs{D} \times \abs{K - D} \to \abs{K}$ is finite-to-one, so $\dim \abs{D} + \dim \abs{K - D} \le \dim \abs{K} = g - 1$.
Adding the [[Riemann–Roch theorem|WMVAT9MBZ1]] $\ell(D) - \ell(K - D) = \deg D - g + 1$ gives $2(\ell(D) - 1) \le \deg D$.
The hypothesis that $D$ is special is needed: for $\deg D > 2g - 2$, Riemann–Roch gives $\ell(D) - 1 = \deg D - g$, which exceeds $\deg D / 2$ once $\deg D > 2g$.
Example: on a hyperelliptic curve with degree-$2$ map divisor $g^1_2$, the divisor $D = r \cdot g^1_2$ with $r \le g - 1$ has $\ell(D) = r + 1$ and $\deg D = 2r$, so equality holds.

## Explanations

### L1 human

A special divisor is one that is still small compared to the canonical divisor.
Clifford's theorem bounds how many functions can have poles bounded by such a divisor: the dimension of the linear system is at most half the degree.
So small divisors cannot move in large families.
The bound is reached only in trivial cases and on hyperelliptic curves.

### L2 human

Riemann–Roch computes $\ell(D)$ exactly for nonspecial divisors, and Clifford's theorem is the first general bound in the special range $0 \le \deg D \le 2g - 2$.
It is used to show that a non-hyperelliptic curve has a canonical embedding, and that a divisor of degree $2$ moving in a pencil forces the curve to be hyperelliptic.
Brill–Noether theory refines the bound for a general curve, where the true dimensions are much smaller.
</document>

<document source="https://tamnd.github.io/math2/node/RSZS9SFFZ7.md" tag="RSZS9SFFZ7" kind="example">
# Fuller's phenomenon

## Example

*Fuller's problem* is the optimal control problem
$$\text{minimize } \int_0^\infty x_1(t)^2 \, dt \quad \text{subject to } x_1' = x_2, \; x_2' = u, \; \abs{u(t)} \le 1, \; x(0) = \xi,$$
where $\xi \in \R^2$ is a given nonzero initial state and $u$ is a measurable control.
For every $\xi \ne 0$ the optimal control is [[bang–bang|GK1Z3Q1WRZ]] with $u = \pm 1$.
It switches infinitely many times, and the switching times accumulate at a finite time $t^*$, at which the state reaches the origin.
After $t^*$ the optimal control is $u = 0$ and the state stays at the origin.
So the optimal control is a [[chattering control|6K8Q2YERTF]].
This behaviour is called *Fuller's phenomenon*.

## Remarks

A scaling argument explains the pattern.
If $(x_1(t), x_2(t), u(t))$ is admissible, then so is $(\lambda^2 x_1(t/\lambda), \lambda x_2(t/\lambda), u(t/\lambda))$ for every $\lambda > 0$, and its cost is multiplied by $\lambda^5$.
So the optimal feedback is invariant under this scaling, and the time between two switches shrinks by a fixed factor at each switch.
The intervals then form a geometric series with finite sum, which gives the finite time $t^*$.
The optimal feedback has the form $u = -\sgn(x_1 + C x_2 \abs{x_2})$ for a certain constant $C > 0$.
Necessary conditions of [[Pontryagin's maximum principle|7RPV2T29CF]] together with this scaling are the main tools of the analysis.

## Explanations

### L1 human

A cart on a rail is pushed by a force that can be at most one in either direction.
The goal is to keep the cart's position close to zero in total, measured by the integral of the squared position.
The best strategy pushes at full force, alternating direction, with shorter and shorter pushes.
Infinitely many switches happen before the cart comes to rest at zero, after a finite time.
This shows that optimal controls can be surprisingly wild even for very simple systems.
</document>

<document source="https://tamnd.github.io/math2/node/RT2D9JEVRR.md" tag="RT2D9JEVRR" kind="conjecture">
# Köthe conjecture

## Statement

Let $R$ be a [[ring|0894J980RK]].
A one-sided ideal of $R$ is *nil* if each of its elements is nilpotent.
The conjecture states that if $R$ has no nonzero [[nil ideal|NZ1G9WRWFB]] (two-sided), then $R$ has no nonzero nil [[left ideal|5N2CJZ47D4]].

## Remarks

The conjecture is open.
Equivalent forms: in every ring the sum of two nil left ideals is nil; in every ring every nil left ideal is contained in the [[upper nilradical|3G2JQEYBEK]].
Krempa showed that it is equivalent to the statement that for every nil ring $R$ the polynomial ring $R[x]$ is equal to its Jacobson radical.
It holds for left Noetherian rings, where every nil one-sided ideal is nilpotent by [[Levitzki's theorem|QHD4KJYFSA]].
Smoktunowicz constructed a nil ring $R$ such that $R[x]$ is not nil, which rules out a stronger form that had been suggested.

## Explanations

### L1 human

An element is nilpotent when some power of it is zero.
In a noncommutative ring there are several ways to collect nilpotent elements: in two-sided ideals or in one-sided ideals.
The conjecture says that one-sided ideals of nilpotent elements can only exist if a two-sided one does.
It is one of the oldest open problems of ring theory.
</document>

<document source="https://tamnd.github.io/math2/node/RT5MJT6CTB.md" tag="RT5MJT6CTB" kind="statement">
# Apéry's theorem

## Statement

The number
$$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3}$$
is [[irrational|0DZXAXEJQ9]].
Here $\zeta$ is the [[Riemann zeta function|1XWBY4N0VT]].

## Remarks

The proof constructs two sequences of rational numbers $a_n$ and $b_n$, where the $b_n$ and the $2 d_n^3 a_n$ are integers, with $d_n$ the [[least common multiple|007NPQX34K]] of $1, 2, \dots, n$.
The differences $b_n \zeta(3) - a_n$ are nonzero and tend to $0$ so fast that $2 d_n^3 (b_n \zeta(3) - a_n)$ still tends to $0$.
If $\zeta(3) = p/q$, then $2 q d_n^3 (b_n \zeta(3) - a_n)$ would be a nonzero integer tending to $0$, which is impossible.
A useful series in the proof is
$$\zeta(3) = \frac{5}{2} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^3 \binom{2n}{n}}.$$
For even arguments much more is known: $\zeta(2k)$ is a rational multiple of $\pi^{2k}$, so it is transcendental, as in the [[Basel problem|0DMY75D6FV]].
It is not known whether $\zeta(5)$ is irrational, and it is not known whether $\zeta(3)$ is transcendental.

## Explanations

### L1 human

The sum $1 + 1/8 + 1/27 + 1/64 + \cdots$ of the reciprocals of the cubes is about $1.202$.
Apéry's theorem says that this number is not a fraction.
Unlike the sum of the reciprocals of the squares, which equals $\pi^2/6$, no closed form for it is known.

### L2 human

Apéry's sequences satisfy a three-term recurrence with polynomial coefficients, and their growth controls the speed of approximation.
Later work gave other proofs, for example with Beukers' integrals over the unit cube.
Rivoal proved that infinitely many of the values $\zeta(2k+1)$ are irrational, and Zudilin proved that at least one of $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational.
</document>

<document source="https://tamnd.github.io/math2/node/RT62ZAN3JB.md" tag="RT62ZAN3JB" kind="definition">
# Equidistribution modulo one

## Definition

For a real number $x$, let $\{x\} = x - \floor{x}$ be its fractional part, where $\floor{x}$ is the [[floor|08GR54CKEA]] of $x$.
A sequence $(x_n)_{n \ge 1}$ of real numbers is *equidistributed modulo one* if for all real $a, b$ with $0 \le a < b \le 1$,
$$\lim_{N \to \infty} \frac{\#\set{n : 1 \le n \le N, \ \{x_n\} \in [a, b)}}{N} = b - a.$$

## Remarks

Example: for irrational $\alpha$, the sequence $x_n = n\alpha$ is equidistributed modulo one; this is the [[equidistribution theorem|NCCGNTBEYV]].
Non-example: $x_n = n/2$ has $\{x_n\} \in \set{0, 1/2}$, so for $[a, b) = [0.1, 0.4)$ the count is $0$ for every $N$, and the limit $0$ differs from $0.3$.
Weyl's criterion gives an equivalent form: the sequence is equidistributed modulo one if and only if
$$\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} e^{2\pi i h x_n} = 0$$
for every nonzero integer $h$.
Another equivalent form: $\frac{1}{N} \sum_{n=1}^{N} f(\{x_n\}) \to \int_0^1 f(t)\,dt$ for every continuous function $f$ on $[0, 1]$ with $f(0) = f(1)$.
Some authors also say *uniformly distributed modulo one*.
</document>

<document source="https://tamnd.github.io/math2/node/RT8A8FJ566.md" tag="RT8A8FJ566" kind="concept">
# Sheaf on a site

## Summary

A sheaf on a [[site|NYE0Z94MME]] $(\mathcal{C}, J)$ is a [[presheaf|AND2WHZC8P]] on $\mathcal{C}$ with a gluing property.
Compatible local data on every covering [[sieve|CNCXNWKC6R]] glue to a unique global datum.
It generalizes a [[sheaf on a topological space|TB6TM6R4MD]], which is the case of the site of open sets.
Sheaves on a small site form a [[Grothendieck topos|S3YVE24AQK]], and every presheaf has a best approximation by a sheaf, its [[sheafification|2HYK3KBQNA]].
Sheaves on the étale site of a scheme give étale cohomology.

## Explanations

### L1 human

A presheaf gives, for each piece of a space, a set of data, and a way to restrict data to smaller pieces.
It is a sheaf when data can be checked and built locally.
If pieces of data on a cover agree where they overlap, there is exactly one datum on the whole that restricts to them.
For example, continuous real functions on the open sets of a space form a sheaf, since continuity is a local property.
Constant functions do not: a function equal to $0$ on one component and $1$ on another is locally constant but not constant.
</document>

<document source="https://tamnd.github.io/math2/node/RT9EYEQ6S2.md" tag="RT9EYEQ6S2" kind="definition">
# Harmonic functions on Riemann surfaces

## Definition

Let $X$ be a [[Riemann surface|5K0N8QMV6Q]] and let $U \subseteq X$ be an open set.
A function $u \colon U \to \R$ is *harmonic* if for every holomorphic chart $\phi \colon V \to \phi(V) \subseteq \C$ of $X$, the function $u \circ \phi^{-1}$ is [[harmonic|0DEB0JMN0W]] on the open set $\phi(U \cap V)$.

## Remarks

It suffices to check the condition for the charts of one atlas.
Indeed, transition maps are holomorphic, and [[harmonicity is preserved by holomorphic maps|1VGB63K5HS]].
Equivalently, $u$ is harmonic if each point of $U$ has a neighbourhood on which $u = \operatorname{Re} f$ for a [[holomorphic|0NAHXR1FRH]] function $f$.
Example: on $X = \C \setminus \set{0}$ with the chart $z \mapsto z$, the function $u(z) = \log\abs{z}$ is harmonic, since $\Delta \log\abs{z} = 0$ there.
Non-example: $u(z) = \abs{z}^2$ on $X = \C$ is not harmonic, since in the chart $z \mapsto z$ its Laplacian is $4$.
On the [[Riemann sphere|0J78KBPFZB]], or on any compact Riemann surface, the maximum principle forces each harmonic function on the whole surface to be constant.
The Laplacian of $u$ depends on the chart, but the condition $\Delta(u \circ \phi^{-1}) = 0$ does not; in coordinate-free form it reads $\partial \bar\partial u = 0$.
Subharmonic and superharmonic functions on $X$ are defined in the same way, chart by chart.
</document>

<document source="https://tamnd.github.io/math2/node/RTA8BG53KZ.md" tag="RTA8BG53KZ" kind="definition">
# Chebyshev polynomials of the second kind

## Definition

The *Chebyshev polynomials of the second kind* are the polynomials $U_n \in \Z[x]$, $n \ge 0$, in the [[polynomial ring|0SDC934R52]] $\Z[x]$ given by
$$U_0 = 1, \qquad U_1 = 2x, \qquad U_{n+1} = 2x \, U_n - U_{n-1} \quad (n \ge 1).$$

## Remarks

Equivalently, $U_n$ is the unique polynomial with $U_n(\cos \theta) \sin \theta = \sin((n + 1)\theta)$ for all real $\theta$.
Example: $U_2 = 2x \cdot 2x - 1 = 4x^2 - 1$, and indeed $\sin(3\theta) = \sin \theta \, (3 - 4 \sin^2 \theta) = \sin \theta \, (4 \cos^2 \theta - 1)$.
Non-example: the [[Chebyshev polynomials|M6EAR1JHPP]] of the first kind $T_n$ satisfy the same recurrence but start with $T_1 = x$, so they fail the clause $U_1 = 2x$.
The polynomial $U_n$ has degree $n$, leading coefficient $2^n$, and roots $\cos \frac{k\pi}{n + 1}$ for $k = 1, \dots, n$.
They are orthogonal: $\int_{-1}^{1} U_m(x) U_n(x) \sqrt{1 - x^2} \, dx = \frac{\pi}{2} \delta_{mn}$.
They are linked to the first kind by $T_n' = n \, U_{n-1}$ for $n \ge 1$.
</document>

<document source="https://tamnd.github.io/math2/node/RTAVY44P4Z.md" tag="RTAVY44P4Z" kind="definition">
# Legendrian submanifold

## Definition

Let $(M, \xi)$ be a [[contact manifold|HBQ8BZ7XB5]] of dimension $2n + 1$.
A *Legendrian submanifold* of $(M, \xi)$ is a submanifold $L \subset M$ of dimension $n$ such that $T_x L \subset \xi_x$ for every $x \in L$.
A *Legendrian knot* is a Legendrian submanifold of a contact $3$-manifold that is diffeomorphic to a circle.

## Remarks

If $\xi = \ker \alpha$, the condition is that $\alpha$ restricts to zero on $L$.
Then $d\alpha$ also restricts to zero on $L$, so each $T_x L$ is an isotropic subspace of the symplectic vector space $(\xi_x, d\alpha)$, of dimension $2n$.
Hence a submanifold tangent to $\xi$ has dimension at most $n$, and Legendrian submanifolds are the tangent submanifolds of maximal dimension.

Example: in $\R^3$ with $\alpha = dz - y\,dx$, the $x$-axis $\set{y = z = 0}$ is Legendrian, since along it $dz = 0$ and $y = 0$.
More generally, for a smooth $f \colon \R \to \R$, the curve $\set{(x, f'(x), f(x))}$ is Legendrian, since $dz - y\,dx = f'(x)\,dx - f'(x)\,dx = 0$ on it.

Non-example: the $z$-axis in the same space is not Legendrian, since $\alpha(\partial_z) = 1 \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/RTBJM23ETS.md" tag="RTBJM23ETS" kind="statement">
# Petrov–Galerkin method

## Statement

Let $U$ and $V$ be real [[Hilbert spaces|HM2HHE3J51]].
Let $a \colon U \times V \to \R$ be a [[bilinear form|0SQZMXB6GP]] with $\abs{a(w, v)} \le M \norm{w}_U \norm{v}_V$ for all $w \in U$ and $v \in V$, and let $f$ be in the [[continuous dual space|4A82S5FVRT]] $V'$.
Let $u \in U$ satisfy $a(u, v) = f(v)$ for all $v \in V$.
Let $U_h \subseteq U$ (trial space) and $V_h \subseteq V$ (test space) be finite-dimensional subspaces of the same dimension, with
$$\beta_h = \inf_{w \in U_h \setminus \set{0}} \ \sup_{v \in V_h \setminus \set{0}} \frac{a(w, v)}{\norm{w}_U \norm{v}_V} > 0.$$
Then there is a unique $u_h \in U_h$ with $a(u_h, v) = f(v)$ for all $v \in V_h$, and
$$\norm{u - u_h}_U \le \Big( 1 + \frac{M}{\beta_h} \Big) \inf_{w \in U_h} \norm{u - w}_U.$$

## Remarks

In bases of $U_h$ and $V_h$ the discrete problem is a square linear system.
The condition $\beta_h > 0$ says that $a(w, v) = 0$ for all $v \in V_h$ forces $w = 0$, so the system matrix is injective and therefore invertible.
Subtracting the equations gives $a(u - u_h, v) = 0$ for all $v \in V_h$.
For $w \in U_h$ this gives $\beta_h \norm{u_h - w} \le \sup_{v} a(u_h - w, v) / \norm{v} = \sup_{v} a(u - w, v) / \norm{v} \le M \norm{u - w}$.
The triangle inequality $\norm{u - u_h} \le \norm{u - w} + \norm{w - u_h}$ then gives the bound.

The case $U = V$ and $U_h = V_h$ is the [[Galerkin method|Y5B0PEBKE8]]; there, coercivity with constant $\alpha$ gives $\beta_h \ge \alpha$.
Example: in the streamline upwind Petrov–Galerkin method for $-\varepsilon u'' + u' = f$, each test function is a trial hat function plus a multiple of its derivative, which adds stabilisation along the flow.

A positive $\beta_h$ is needed: with $\beta_h = 0$ the discrete system is singular, and with $\beta_h \to 0$ along a family of meshes the bound degenerates.

## Explanations

### L1 human

In the Galerkin method one searches and tests in the same space of functions.
The Petrov–Galerkin method allows two different spaces: one where the approximate solution lives, and one that is used to test the equation.
The theorem says that the method works, and gives a nearly best approximation, as long as the two spaces fit together well, as measured by the number $\beta_h$.

### L2 human

The theorem is the discrete half of the Babuška theory of problems that are well posed but not coercive.
It explains why different test spaces can stabilise convection-dominated problems and why some choices of spaces fail.
Its constant $1 + M / \beta_h$ can be improved to $M / \beta_h$ in Hilbert spaces, a result of Xu and Zikatanov.
For saddle-point problems, the analogous condition is the [[Ladyzhenskaya–Babuška–Brezzi condition|SC5FV3GHV5]].
</document>

<document source="https://tamnd.github.io/math2/node/RTG32790CW.md" tag="RTG32790CW" kind="definition">
# Devaney chaos

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
The map $f$ is *chaotic in the sense of Devaney* if:

1. $f$ is [[topologically transitive|FR49Y9GQHR]].
2. The set of [[periodic points|T75CZ5H47N]] of $f$ is [[dense|0BSJSX7ME2]] in $X$.
3. $f$ has *sensitive dependence on initial conditions*: there is $\delta > 0$ such that each $x \in X$ and $\eta > 0$ admit $y \in X$ and $n \ge 0$ as follows.
   They satisfy $d(x, y) < \eta$ and $d(f^n(x), f^n(y)) > \delta$.

## Remarks

Banks, Brooks, Cairns, Davis and Stacey showed that if $X$ is infinite, then conditions 1 and 2 imply condition 3.
Example: the doubling map $f(x) = 2x \bmod 1$ on the circle $\R / \Z$.
It is transitive, since $f^n$ maps any arc of length greater than $2^{-n}$ onto the whole circle.
Its periodic points are the points $k / (2^n - 1)$, which are dense.
It is sensitive with $\delta = 1/8$: for $y \neq x$ close to $x$, the distance between $f^n(x)$ and $f^n(y)$ doubles at each step while it is below $1/4$, so it eventually exceeds $1/8$.
Non-example: the rotation $x \mapsto x + \alpha \bmod 1$ with $\alpha$ irrational is transitive but has no periodic points, so condition 2 fails; it is also not sensitive, being an isometry.

## Explanations

### L1 human

A map is chaotic in the sense of Devaney if it links all regions, has repeating orbits near every point, and amplifies small differences in starting points.
</document>

<document source="https://tamnd.github.io/math2/node/RTHHCX84HN.md" tag="RTHHCX84HN" kind="concept">
# Set theory

## Summary

Set theory is the branch of mathematical logic that studies [[sets|0BSJ7RQ9BW]], collections of objects called their [[elements|0EZ758XE73]].
Cantor created it in the 1870s, with his study of infinite sets and their sizes.
In its axiomatic form, most often [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with choice, it serves as the standard foundation of mathematics.
Its own subjects include ordinals and cardinals, independence results, large cardinals and descriptive set theory.

## Explanations

### L1 human

Almost every mathematical object can be built from sets.
For example, the number $2$ can be taken to be the set $\set{\emptyset, \set{\emptyset}}$, and a function is a set of ordered pairs.
Set theory writes down the basic rules for sets, and then studies what follows from them.
A famous discovery is that infinite sets come in different sizes: there are more real numbers than natural numbers.
Another is that some natural questions, such as the continuum hypothesis, cannot be settled by the usual rules.
</document>

<document source="https://tamnd.github.io/math2/node/RTJ6Y1AH7S.md" tag="RTJ6Y1AH7S" kind="definition">
# Coupled map lattice

## Definition

Let $N \ge 3$, let $f \colon \R \to \R$ be a map, and let $\varepsilon \in [0, 1]$ be a coupling strength.
Index the sites by $i \in \Z / N\Z$.
The *diffusively coupled map lattice* with local map $f$ is the discrete-time [[dynamical system|BBZP1R872N]] on $\R^{N}$ given by $x_{n+1} = F(x_n)$, where
$$F(x)_i = (1 - \varepsilon) f(x_i) + \frac{\varepsilon}{2} \big( f(x_{i-1}) + f(x_{i+1}) \big), \qquad i \in \Z / N\Z.$$

## Remarks

The sites form a ring, which is the usual periodic boundary condition; the same rule makes sense on $\Z$ or on higher-dimensional lattices, with the average over all nearest neighbours.
More general coupled map lattices allow other local maps and other coupling rules.

Example: if $\varepsilon = 0$, then $F(x)_i = f(x_i)$, so each site evolves by [[iterating|AYVTR2ES5B]] $f$ on its own.
If all sites start equal, $x_i = s$ for all $i$, then $F(x)_i = (1 - \varepsilon) f(s) + \varepsilon f(s) = f(s)$ for all $i$.
So the spatially uniform states form an invariant set on which the lattice reduces to the single map $f$.

Non-example: a cellular automaton has the same discrete space and time.
But its states take values in a finite set, not in $\R$, so it is not a coupled map lattice.

A common choice is the [[logistic map|QN9X9BNKTX]] $f(x) = r x (1 - x)$ or the quadratic map $f(x) = 1 - a x^2$, with $\varepsilon$ and the parameter of $f$ as the two control parameters.
</document>

<document source="https://tamnd.github.io/math2/node/RTJQPZC0VY.md" tag="RTJQPZC0VY" kind="definition">
# Atomic domain

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]].
$R$ is *atomic* if every nonzero element $r$ of $R$ that is not a [[unit|019NFGFPR5]] has a factorization into irreducibles.
That is, $r = a_1 a_2 \cdots a_n$ for some $n \ge 1$ and some [[irreducible elements|052SDEPZH7]] $a_1, \ldots, a_n$ of $R$.

## Remarks

Example: $\Z[\sqrt{-5}]$ is atomic, because it is Noetherian.
In a Noetherian domain, a nonzero nonunit without a factorization would give an infinite strictly ascending chain of principal ideals.
Non-example: the ring $\overline{\Z}$ of all algebraic integers is not atomic.
For a nonzero nonunit $\alpha$, a square root $\beta$ of $\alpha$ is an algebraic integer and a nonunit, and $\alpha = \beta^2$.
So no element of $\overline{\Z}$ is irreducible, and $2$ has no factorization.
A domain that satisfies the ascending chain condition on principal ideals is atomic; the converse is false.
The factorization is not required to be unique; uniqueness up to order and units is what makes $R$ a [[unique factorisation domain|06YMXJMXYS]].
</document>

<document source="https://tamnd.github.io/math2/node/RTKRSC3KPH.md" tag="RTKRSC3KPH" kind="concept">
# Floating-point arithmetic

## Summary

Floating-point arithmetic is the way computers represent real numbers and compute with them.
A floating-point number is a fixed number of significant digits times a power of a base, such as $1.0110 \times 2^{-3}$.
The result of each operation is [[rounded|5NPVX29Y0C]] to a nearby floating-point number, so most computations carry a small [[round-off error|CPN6SD3BTV]].
Its relative precision is measured by the [[machine epsilon|JA3QXVF2B8]], and the [[IEEE 754 standard|GNDWPYYFW4]] fixes the formats used in practice.

## Explanations

### L1 human

Floating point is scientific notation for a computer.
The number $6.02 \times 10^{23}$ keeps three significant digits and an exponent, so it can describe huge and tiny numbers with the same relative accuracy.
A computer does the same in base $2$, with a fixed number of binary digits.
Because only finitely many numbers fit, most results must be rounded.
For example, $0.1$ has no finite binary expansion, so in a computer $0.1 + 0.2$ is not exactly equal to $0.3$.
</document>

<document source="https://tamnd.github.io/math2/node/RTMYXRDGWR.md" tag="RTMYXRDGWR" kind="example">
# Semiautomaton as a unary algebra

## Example

A *semiautomaton* is a triple $(Q, \Sigma, \delta)$ with a set of states $Q$ and a set of input letters $\Sigma$.
The map $\delta \colon Q \times \Sigma \to Q$ is the transition function.
For each letter $a \in \Sigma$ put $\delta_a(q) = \delta(q, a)$.
Then each $\delta_a$ maps $Q$ to $Q$, so $(Q, (\delta_a)_{a \in \Sigma})$ is a [[unary algebra|XD10P9CJMD]] with index set $\Sigma$.
Conversely, a unary algebra $(Q, (f_a)_{a \in \Sigma})$ gives the semiautomaton with $\delta(q, a) = f_a(q)$, so the two notions are the same data.

Concretely, let $Q = \set{\mathrm{even}, \mathrm{odd}}$ and $\Sigma = \set{0, 1}$, with $\delta_0$ the identity of $Q$ and $\delta_1$ the map that swaps $\mathrm{even}$ and $\mathrm{odd}$.
Reading a binary word moves the state according to the parity of the number of letters $1$ read so far.

Under this translation, a homomorphism of unary algebras is a map of states that commutes with every transition.
A subalgebra is a set of states closed under all transitions.
A congruence is an equivalence of states compatible with all transitions.
A deterministic finite automaton is a finite semiautomaton with a start state and a set of accepting states added.

## Explanations

### L1 human

A semiautomaton is a machine that changes its state each time it reads a letter.
For each letter there is a rule that sends every state to a new state.
So each letter is a map from the set of states to itself, and the machine is a set with a family of such maps.
That is exactly a unary algebra, so the tools of universal algebra, such as quotients and homomorphisms, apply directly to machines.
</document>

<document source="https://tamnd.github.io/math2/node/RTSAFF4KFJ.md" tag="RTSAFF4KFJ" kind="concept">
# Mesh generation

## Summary

Mesh generation is the construction of a mesh: a subdivision of a geometric domain into many small simple cells.
Typical cells are triangles, tetrahedra, quadrilaterals or hexahedra.
Meshes are the discrete setting of the [[finite element method|98GZ9DVHTA]], the [[finite volume method|HGKN8E09RS]] and many other numerical methods for partial differential equations.
The quality of a mesh, the size and shape of its cells, controls both the accuracy and the cost of the computation.
Standard tools include [[Delaunay triangulations|259Y10BYJC]], advancing front methods, and [[adaptive mesh refinement|X001F42KH8]].

## Explanations

### L1 human

To compute heat flow in an engine part, a computer cannot handle the solid shape directly.
It first cuts the shape into many small pieces, like a mosaic of tiny triangles or bricks, and computes a simple approximation on each piece.
Making these pieces is mesh generation.
A square, for example, can be cut along a diagonal into two triangles.
Each triangle can be cut again into four smaller ones by joining the midpoints of its sides.
Good meshes use small pieces where the solution changes fast and avoid very thin, flat pieces.
</document>

<document source="https://tamnd.github.io/math2/node/RV0MKJ3CQ6.md" tag="RV0MKJ3CQ6" kind="concept">
# Nerve of a category

## Summary

The nerve of a small [[category|A2DE66937D]] $\mathcal{C}$ is the [[simplicial set|WW81ARB8K9]] whose $n$-simplices are the strings of $n$ composable morphisms of $\mathcal{C}$.
The construction is a fully faithful [[functor|28NQ0KRHD3]] from small categories to simplicial sets, so no information about $\mathcal{C}$ is lost.
Its geometric realization is the classifying space of $\mathcal{C}$; for a group $G$ viewed as a one-object category it gives the classifying space $BG$.
Nerves are the basic examples of [[quasi-categories|9328WEVJEK]] and, for groupoids, of [[Kan complexes|FDFYCM3YYC]].

## Explanations

### L1 human

Draw a point for each object of the category and an edge for each arrow.
Fill in a triangle for each pair of arrows that can be composed, with the composite as the third side.
Then fill in a tetrahedron for each string of three composable arrows, and so on.
The resulting shape is the nerve.
For example, the category $0 \to 1$ with one non-identity arrow has as nerve a single edge, the standard $1$-simplex $\Delta^1$, plus its degenerate simplices.
</document>

<document source="https://tamnd.github.io/math2/node/RV67AGM6EJ.md" tag="RV67AGM6EJ" kind="definition">
# Multidimensional Mellin transform

## Definition

Let $n \ge 1$ and let $f \colon (0, \infty)^n \to \C$ be locally integrable.
For $s = (s_1, \dots, s_n) \in \C^n$ and $x \in (0, \infty)^n$ write $x^{s - 1} = x_1^{s_1 - 1} \cdots x_n^{s_n - 1}$.
The *$n$-dimensional Mellin transform* of $f$ is the function
$$(\mathcal{M} f)(s) = \int_{(0, \infty)^n} x^{s - 1} f(x)\, dx,$$
defined at those $s \in \C^n$ where the integral converges absolutely.

## Remarks

For $n = 1$ this is the [[Mellin transform|N5PVGRJE1T]].
Absolute convergence at $s$ depends only on $(\operatorname{Re} s_1, \dots, \operatorname{Re} s_n)$, and the set of such real parts is convex.

Example: $f(x) = e^{-x_1 - \cdots - x_n}$.
The integral splits into a product, and $(\mathcal{M} f)(s) = \Gamma(s_1) \cdots \Gamma(s_n)$ for $\operatorname{Re} s_j > 0$, by the [[gamma function|0H5SYBSED9]] integral.

Non-example: $f = 1$ has no Mellin transform.
For every $s$, the factor $\int_0^\infty x_1^{\operatorname{Re} s_1 - 1}\, dx_1$ diverges, at $0$ or at $\infty$.
</document>

<document source="https://tamnd.github.io/math2/node/RV6PWJBH5P.md" tag="RV6PWJBH5P" kind="statement">
# Almgren's regularity theorem

## Statement

Let $m \ge 2$ and $n \ge 1$ be integers.
Let $M$ be a smooth [[Riemannian manifold|7CBT4SM83Y]] of dimension $m + n$, and let $T$ be an $m$-dimensional [[integral current|EG4AJVEFT2]] in $M$ that is [[area-minimizing|4NKP73KVVW]].
Let $\operatorname{sing} T$ be the set of points of the support of $T$, outside the support of its boundary, with this property.
Near the point, the support of $T$ is not a smooth embedded submanifold.
Then the [[Hausdorff dimension|SSNFTHZ6YJ]] of $\operatorname{sing} T$ is at most $m - 2$.

## Remarks

Frederick Almgren proved the theorem in a very long work that was published in 2000, after his death.
Camillo De Lellis and Emanuele Spadaro later gave a shorter proof.
The proof studies blow-ups at points where the tangent cone is a plane with integer multiplicity $Q \ge 2$.
Near such points the current is approximated by the graph of a $Q$-valued function that minimizes a Dirichlet energy.
A frequency function controls the rate at which the current departs from the plane.
The bound $m - 2$ is sharp in higher codimension.
By Wirtinger's inequality, complex algebraic curves in $\C^2 = \R^4$ are area-minimizing, and the curve $\set{(z, w) : z^2 = w^3}$ has a singular point at the origin, a set of dimension $0 = 2 - 2$.
In codimension $n = 1$ the singular set is much smaller: its dimension is at most $m - 7$, by the work of Federer and Simons.

## Explanations

### L1 human

An area-minimizing surface of any dimension, in a space of any dimension, can have singular points where it is not smooth.
Almgren's theorem says that these singular points form a set that is small: two dimensions smaller than the surface itself.
For two-dimensional surfaces, the singular points are isolated, as for complex curves crossing themselves.

### L2 human

In codimension one, regularity follows from De Giorgi's excess-decay method, because the tangent planes have multiplicity one.
In higher codimension a tangent plane can carry multiplicity, so branching phenomena occur, and De Giorgi's method no longer applies.
Almgren's theorem handles this with $Q$-valued functions and the frequency function, tools that have since spread to free boundary problems.
</document>

<document source="https://tamnd.github.io/math2/node/RVEFNEJSHF.md" tag="RVEFNEJSHF" kind="definition">
# Global field

## Definition

A *global field* is a [[field|04C5ZBG1WT]] $K$ of one of the following two kinds.

1. $K$ is a [[finite extension|0C1Y04SZ1W]] of the [[rational numbers|0NCJ5WXN6Z]] $\Q$; then $K$ is a [[number field|MWBDFS2KJ5]].
2. $K$ is a finite extension of the field $\F_q(t)$ of rational functions in one variable $t$ over a [[finite field|0ZXZ82FTX2]] $\F_q$. Then $K$ is a global function field.

## Remarks

Example: $\Q(\sqrt{2})$ is a global field, because it has degree $2$ over $\Q$.
Example: $\F_3(t)$ itself is a global function field, of degree $1$ over $\F_3(t)$.
Non-example: $\R$ is not a global field, because it has infinite degree over $\Q$ and has characteristic $0$.
Non-example: $\C(t)$ is not a global field, because its field of constants $\C$ is not finite.
The global function fields are exactly the function fields of smooth projective curves over finite fields.
</document>

<document source="https://tamnd.github.io/math2/node/RVG9Z8NYA6.md" tag="RVG9Z8NYA6" kind="concept">
# Natural density

## Summary

The natural density, or asymptotic density, of a set $A$ of positive integers is a limit as $n \to \infty$.
It is the limit of the proportion of $\set{1, \dots, n}$ that lies in $A$.
It makes precise statements such as "half of all integers are even" or "almost all integers are composite".
Not every set has a natural density, but every set has an upper and a lower density.
Many results of number theory, such as the [[Chebotarev density theorem|X8VKW5Y1CZ]], are statements about densities.

## Explanations

### L1 human

Count how many of the numbers $1, 2, \dots, n$ lie in $A$, divide by $n$, and let $n$ grow.
If the ratio settles down, its limit is the density of $A$.
The multiples of $3$ have density $1/3$, and the perfect squares have density $0$, because there are only about $\sqrt{n}$ of them up to $n$.
The set of numbers whose first decimal digit is $1$ has no density: the ratio keeps swinging between about $1/9$ and about $5/9$.
</document>

<document source="https://tamnd.github.io/math2/node/RVKPM94MDF.md" tag="RVKPM94MDF" kind="concept">
# Embedded prime

## Summary

An embedded prime of an ideal $I$ is an [[associated prime|ERFZ6S8X0X]] of $I$ that strictly contains another associated prime of $I$.
The associated primes that are not embedded are the minimal ones, and they are the [[minimal primes|S2D8823P8B]] over $I$.
In geometry, the minimal primes give the irreducible components of the zero set of $I$.
The embedded primes give smaller sets inside them that carry extra structure.

## Explanations

### L1 human

In the plane, the ideal $(x^2, xy)$ of $k[x, y]$ has zero set the line $x = 0$.
Its associated primes are $(x)$, for the line, and $(x, y)$, for the origin.
The origin lies inside the line, so $(x, y)$ is an embedded prime: it records a "thickening" of the line at one point that the zero set alone does not see.
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# Hessenberg matrix

## Definition

Let $R$ be a [[ring|0B1261B5ED]] and let $A = (a_{ij})$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $A$ is *upper Hessenberg* when $a_{ij} = 0$ for all $i > j + 1$.
It is *lower Hessenberg* when $a_{ij} = 0$ for all $j > i + 1$.

## Remarks

Example: $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 0 & 7 & 8 \end{pmatrix}$ is upper Hessenberg, because its only entry with $i > j + 1$ is $a_{31} = 0$.
Non-example: $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ is not upper Hessenberg, because $a_{31} = 7 \ne 0$.
Every [[upper triangular matrix|0N4XY7Z3NN]] is upper Hessenberg.
A matrix that is both upper and lower Hessenberg is a [[tridiagonal matrix|DARX1KZRHZ]].
An upper Hessenberg matrix is *unreduced* when all its subdiagonal entries $a_{i+1, i}$ are nonzero.
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# Infinite Galois theory

## Summary

Infinite Galois theory extends [[Galois theory|WNQDKQYZ2R]] to [[Galois extensions|0DS2F7GK05]] of infinite degree, such as the algebraic closure of $\Q$ over $\Q$.
The Galois group then carries the [[Krull topology|F3GTF5NT4M]], which makes it a [[profinite group|KA0YDB6QXE]].
The fundamental theorem survives with one change: intermediate fields correspond to the closed subgroups, not to all subgroups.

## Explanations

### L1 human

An infinite Galois extension is the union of its finite Galois pieces.
Each symmetry of the whole extension is a compatible choice of symmetries of all these pieces.
Two symmetries are close in the Krull topology when they agree on a large finite piece.
For example, the symmetries of the algebraic closure of $\F_p$ include the Frobenius map $x \mapsto x^p$, and the closed subgroup it generates is the whole group.
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<document source="https://tamnd.github.io/math2/node/RVVRYKVGXK.md" tag="RVVRYKVGXK" kind="definition">
# Partial correlation

## Definition

Let $X$ and $Y$ be real random variables and let $Z$ be a random vector in $\R^k$, all with finite second moments.
Assume that the [[covariance matrix|0V7G6WMFK0]] $\Sigma_{ZZ}$ of $Z$ is [[invertible|0EB9N0S33C]].
Write $\Sigma_{XZ}$ for the row vector of [[covariances|0KBFAWYRTE]] $\operatorname{Cov}(X, Z_j)$, and similarly $\Sigma_{YZ}$.
The residuals of the best linear predictions of $X$ and $Y$ from $Z$ are
$$e_X = X - \mathbb{E}X - \Sigma_{XZ} \Sigma_{ZZ}^{-1} (Z - \mathbb{E}Z), \qquad e_Y = Y - \mathbb{E}Y - \Sigma_{YZ} \Sigma_{ZZ}^{-1} (Z - \mathbb{E}Z).$$
If $\operatorname{Var}(e_X) > 0$ and $\operatorname{Var}(e_Y) > 0$, the *partial correlation* of $X$ and $Y$ given $Z$ is the [[correlation coefficient|0DNN9B6Q2F]]
$$\rho_{XY \cdot Z} = \operatorname{Corr}(e_X, e_Y).$$

## Remarks

Example: let $Z, U, V$ be uncorrelated with variance $1$ and mean $0$, and let $X = Z + U$, $Y = Z + V$.
Then $\Sigma_{XZ} = \Sigma_{YZ} = 1$ and $\Sigma_{ZZ} = 1$, so $e_X = U$ and $e_Y = V$, and $\rho_{XY \cdot Z} = 0$.
Yet $\operatorname{Corr}(X, Y) = 1/2$.
Non-example: if $X = Z$, then $e_X = 0$, so $\operatorname{Var}(e_X) = 0$ and the partial correlation is not defined.

For a single control variable $Z$, with $\rho_{XY}$, $\rho_{XZ}$, $\rho_{YZ}$ the pairwise correlations,
$$\rho_{XY \cdot Z} = \frac{\rho_{XY} - \rho_{XZ} \rho_{YZ}}{\sqrt{(1 - \rho_{XZ}^2)(1 - \rho_{YZ}^2)}}.$$
If $P = (p_{ij})$ is the inverse of the covariance matrix of $(X, Y, Z)$, then $\rho_{XY \cdot Z} = -p_{12} / \sqrt{p_{11} p_{22}}$.
The sample partial correlation uses the sample covariances, or equivalently the residuals of least squares fits on the data.
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# Rational variety

## Definition

Let $k$ be a field and let $X$ be an integral [[algebraic variety|HV6YBS0YDH]] over $k$ of dimension $n$.
The variety $X$ is *rational* over $k$ if it is birationally equivalent to the [[projective space|ZT9A6JK8QF]] $\mathbb{P}^n_k$.
This means that there is a [[birational map|KA5890R9YM]] $\mathbb{P}^n_k \dashrightarrow X$.

## Remarks

Equivalently, the [[function field|2D5SS6WM14]] $k(X)$ is purely transcendental over $k$: it is isomorphic to $k(t_1, \ldots, t_n)$ for elements $t_1, \ldots, t_n$ of [[transcendence degree|D7TR0YWR9H]] $n$.
Example: the conic $C \colon x^2 + y^2 = z^2$ in $\mathbb{P}^2_\Q$ is rational.
The map $[s : t] \mapsto [s^2 - t^2 : 2st : s^2 + t^2]$ sends $\mathbb{P}^1$ into $C$, since $(s^2 - t^2)^2 + (2st)^2 = (s^2 + t^2)^2$.
Its inverse is the projection $[x : y : z] \mapsto [y : z - x]$ from the point $[1 : 0 : 1]$ of $C$.
Non-example: the conic $x^2 + y^2 + z^2 = 0$ in $\mathbb{P}^2_\R$ is not rational over $\R$.
It has no real points, while a variety birational to $\mathbb{P}^1_\R$ has a dense set of real points.
Over $\C$ the same conic is rational.
So rationality depends on the field $k$.
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# Epimorphism

## Summary

An epimorphism is a [[morphism|859F0TFVA3]] $f$ that can be cancelled on the right: $g_1 \circ f = g_2 \circ f$ implies $g_1 = g_2$.
It is the categorical version of a [[surjective function|01RVCMSN5N]], and the dual notion of a [[monomorphism|Y2NP1JFR86]].
In the categories of sets and of groups, the epimorphisms are exactly the surjective homomorphisms.
In other categories an epimorphism need not be surjective: the inclusion of the integers into the rationals is an epimorphism of rings.

## Explanations

### L1 human

A surjective map reaches every point of its target.
An epimorphism is defined without points: two maps out of the target that agree after $f$ must already be equal.
If $f$ missed a point $b$, two maps that differ only at $b$ would agree after $f$; so for sets an epimorphism must be surjective.
For rings the situation is different.
A ring homomorphism out of $\Q$ is fixed by its values on $\Z$, because $1/n$ must go to the inverse of the image of $n$.
So $\Z \to \Q$ is an epimorphism of rings although it is far from surjective.
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# Carleman's condition

## Statement

1. Let $\mu$ be a positive Borel measure on $\R$ with finite [[moments|01XAHPADBA]] $m_n = \int x^n \, d\mu(x)$ and with $m_{2n} > 0$ for all $n \ge 1$. If
$$\sum_{n=1}^{\infty} m_{2n}^{-1/(2n)} = \infty,$$
then the [[Hamburger moment problem|TBNCFXTMMW]] for $(m_n)$ is [[determinate|H0GY2J14CA]]: $\mu$ is the only positive measure on $\R$ with these moments.
2. Let $\mu$ be a positive Borel measure on $[0, \infty)$ with finite moments $m_n$ and with $m_n > 0$ for all $n \ge 1$. If
$$\sum_{n=1}^{\infty} m_{n}^{-1/(2n)} = \infty,$$
then the [[Stieltjes moment problem|Y9KH8P9JC0]] for $(m_n)$ is determinate: $\mu$ is the only positive measure on $[0, \infty)$ with these moments.

## Remarks

The proof goes through quasi-analytic functions.
The condition puts the Fourier–Stieltjes transform of the difference of two solutions in a quasi-analytic class.
That transform has all its derivatives zero at $0$, so it vanishes identically.
Example: for the standard normal distribution, $m_{2n} = (2n - 1)!! \le (2n)^n$, so $m_{2n}^{-1/(2n)} \ge (2n)^{-1/2}$ and the series diverges; the normal law is determined by its moments.
Example: if $\mu$ has bounded support in $[-R, R]$, then $m_{2n} \le m_0 R^{2n}$, so every term is at least $m_0^{-1/(2n)}/R$, and the series diverges.
The condition is sufficient but not necessary for determinacy.
The hypothesis $m_{2n} > 0$ only excludes the measures concentrated at $0$, which are determinate anyway.

## Explanations

### L1 human

The moments of a distribution may fail to identify it, but only if they grow very fast.
Carleman's condition is a precise way of saying that they do not grow too fast.
When it holds, two distributions with the same moments are the same distribution.

### L2 human

It is the test used in practice to check that a law is determined by its moments, for instance before applying the method of moments to prove convergence in distribution.
A simpler sufficient condition is that the moment generating function is finite near $0$, and that condition implies Carleman's.
In the other direction, Krein's condition gives indeterminacy for densities with very heavy tails, such as the log-normal density.
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# Euler equations

## Summary

The Euler equations describe the motion of an ideal fluid, that is, a fluid without viscosity.
They express conservation of mass and Newton's second law for each parcel of fluid, with pressure as the only internal force.
The incompressible Euler equations are the case $\nu = 0$ of the [[Navier–Stokes equations|18KXZHVF4M]].
They were among the first partial differential equations written down, and many questions about them are still open.

## Explanations

### L1 human

Imagine a fluid with no internal friction at all, so that layers can slide past each other freely.
Each small parcel of such a fluid is pushed only by differences in pressure around it.
The Euler equations state this, together with the rule that fluid is neither created nor destroyed.
A simple example is water spinning as a rigid body in a bucket: the velocity is $(-y, x, 0)$ and the pressure grows like $(x^2 + y^2)/2$ away from the axis.
That higher pressure at the rim is what bends the water surface into a curve.
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<document source="https://tamnd.github.io/math2/node/RW1ZF04CRF.md" tag="RW1ZF04CRF" kind="proof">
# Proof of the uniqueness of inverses

## Proof

1. Let $a$ be an element of a [[group|G5S1M9]] $G$ with identity $e$. Let $b$ and $c$ both be [[inverses|0BRHKKP333]] of $a$, so $ba = e$ and $ac = e$.
2. Then $b = be = b(ac)$.
3. By associativity, $b(ac) = (ba)c$.
4. Since $ba = e$, $(ba)c = ec = c$.
5. Thus $b = c$, and $a$ has only one inverse.

## Idea

Multiply one inverse by the element and then by the other inverse. Associativity lets the product collapse in two ways.
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# Jordan–Dedekind chain condition

## Summary

A partially ordered set satisfies the Jordan–Dedekind chain condition when, between any two comparable elements, all [[maximal chains|3AX549FFFM]] have the same finite length.
It is the property behind the Jordan–Hölder theorem, which says that composition series of a group have the same length.
For a finite poset with a least element, it is the same as being a [[graded poset|1C7TKGTB2F]].

## Explanations

### L1 human

Think of climbing from one element to a higher one by steps that cannot be split further.
The condition says that every such climb takes the same number of steps, whatever route is chosen.
In the divisors of $12$, every climb from $1$ to $12$ by single prime factors takes $3$ steps, such as $1, 2, 4, 12$ or $1, 3, 6, 12$.
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# Group action on a variety

## Summary

An action of an [[algebraic group|GH81SYAPPN]] $G$ on a [[variety|HV6YBS0YDH]] $X$ is a [[group action|0WCY42EV9V]] given by a morphism $G \times X \to X$.
Orbits, stabilizers and fixed points of such actions are again algebraic objects.
Quotients by such actions are the subject of geometric invariant theory.
Many important varieties, such as projective spaces, Grassmannians and toric varieties, are best understood through a group action.

## Explanations

### L1 human

A group acts on a space when each group element moves the points of the space, compatibly with the group law.
In algebraic geometry one asks that this motion be given by polynomial formulas in the group element and the point together.
For example, invertible matrices act on vectors by matrix multiplication, and the formula is polynomial in the entries.
The nonzero numbers act on the plane by scaling, and the orbits are the lines through the origin with the origin removed, together with the origin itself.
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# Differential Galois theory

## Definition

Let $(K, \partial)$ be a [[differential field|T43TQ817XC]].
Let $(L, \partial)$ be a differential field that is a [[field extension|0WQ1A8SHTZ]] of $K$, with derivation equal to $\partial$ on $K$.
The *differential Galois group* $\Gal_\partial(L/K)$ is the group, under composition, of the field automorphisms $\sigma \colon L \to L$ such that:

1. $\sigma(a) = a$ for all $a \in K$.
2. $\sigma(\partial b) = \partial(\sigma b)$ for all $b \in L$.

Differential Galois theory is the study of these groups.
The main case is a [[Picard–Vessiot extension|TB53Y5NMJD]] $L$ of $K$ for a linear differential equation.
There the field of constants of $K$ is assumed algebraically closed of characteristic $0$.

## Remarks

Example: let $K = \C(x)$ with $\partial = d/dx$ and $L = K(e^x)$, the Picard–Vessiot extension of $y' = y$.
An element $\sigma$ of $\Gal_\partial(L/K)$ satisfies $\partial(\sigma e^x) = \sigma(e^x)$, so $\sigma(e^x)/e^x$ has derivative $0$ and is a nonzero constant $c \in \C$.
Each $c \in \C^\times$ gives such a $\sigma$, because $e^x$ is transcendental over $K$, so $\Gal_\partial(L/K) \cong \C^\times$.
Non-example: let $K = \C$ with $\partial = 0$ and $L = \C(x)$ with $\partial = d/dx$; the field automorphism $\sigma$ of $L$ over $\C$ with $\sigma(x) = 2x$ satisfies clause 1 but fails clause 2, since $\partial(\sigma x) = 2$ while $\sigma(\partial x) = 1$.
The elements of $\Gal_\partial(\C(x)/\C)$ are the maps $x \mapsto x + c$ with $c \in \C$, so this group is isomorphic to $(\C, +)$.
For a Picard–Vessiot extension for an equation of order $n$, the group acts on the $n$-dimensional space of solutions over the constants $C$ and is a Zariski-closed subgroup of $\mathrm{GL}_n(C)$.
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# Singular simplicial set

## Summary

The singular [[simplicial set|WW81ARB8K9]] $\operatorname{Sing} X$ of a topological space $X$ is a simplicial set.
Its $n$-simplices are the continuous maps from the standard topological $n$-simplex into $X$.
It turns a space into combinatorial data without losing its weak homotopy type.
Its chains give singular homology, and it is a Kan complex, so its simplicial homotopy groups are the homotopy groups of $X$.

## Explanations

### L1 human

Look at all the ways of drawing a point, a path, a triangle, a tetrahedron, and so on, inside a space $X$.
The drawings may be crumpled or collapsed; they only have to be continuous.
Restricting a drawn triangle to one of its edges gives a drawn path, and this is the face operation.
The collection of all these drawings, with the face and degeneracy operations, is $\operatorname{Sing} X$.
When $X$ is a single point, there is exactly one drawing in each dimension.
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# Regulator of a number field

## Summary

The regulator of an [[algebraic number field|MWBDFS2KJ5]] $K$ is a positive real number $R_K$ that measures the size of the unit group.
By [[Dirichlet's unit theorem|1D99CSKX6N]], the logarithms of the units form a lattice.
Up to normalisation, $R_K$ is the volume of a fundamental domain of that lattice.
It appears with the [[class number|KTKK9B1X41]] in the [[class number formula|55XX5A7XZV]] for the residue of the [[Dedekind zeta function|KXS5DQDFYD]].

## Explanations

### L1 human

The units of a number field are generated by a few fundamental units, up to roots of unity.
If the fundamental units are large numbers, the regulator is large.
For a real quadratic field with fundamental unit $\varepsilon > 1$, the regulator is $\log \varepsilon$.
For $\Q(\sqrt{2})$ the fundamental unit is $1 + \sqrt{2}$, so the regulator is $\log(1 + \sqrt{2}) \approx 0.881$.
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# Lebesgue constants of the Dirichlet kernel

## Statement

For an integer $N \ge 1$, let $D_N$ be the [[Dirichlet kernel|3JY4BVE4DW]] of order $N$ and let
$$L_N = \frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{D_N(x)} \, dx$$
be its $N$-th *Lebesgue constant*.
Then
$$L_N = \frac{4}{\pi^2} \log N + O(1) \qquad \text{as } N \to \infty.$$
In particular $L_N \to \infty$.
Let $C(\mathbb{T})$ be the space of [[continuous|0X3WGX95FE]] $2\pi$-periodic functions with the supremum norm.
Then $L_N$ is the operator norm of the partial sum map $f \mapsto S_N f$ on $C(\mathbb{T})$.
It is also the norm of the linear functional $f \mapsto S_N f(0)$ on $C(\mathbb{T})$.

## Remarks

For the estimate, use $\abs{\sin(x/2)} \le \abs{x}/2$ and compare $\abs{\sin((N + \frac{1}{2}) x)}$ on each interval between consecutive zeros with its mean value $\frac{2}{\pi}$.
Summing over about $N$ intervals gives a harmonic sum, hence the $\log N$.
The norm of $f \mapsto S_N f(0) = \frac{1}{2\pi} \int f(-t) D_N(t) \, dt$ is at most $L_N$, and it equals $L_N$ by testing on continuous functions that approximate $\operatorname{sgn} D_N$.
Example: $L_0 = 1$ and $L_N > 1$ for $N \ge 1$; the values grow very slowly, like $0.405 \log N$.

## Explanations

### L1 human

The Dirichlet kernel has integral $1$, but the integral of its absolute value is much larger and grows like a constant times $\log N$.
This number measures how much a partial Fourier sum can amplify a bounded function.
Since the amplification is unbounded, partial Fourier sums of bounded continuous functions can become unbounded.

### L2 human

By the uniform boundedness principle, the unboundedness of $L_N$ implies that some continuous function has a Fourier series that diverges at $0$.
It also implies that $S_N f$ does not converge to $f$ in $L^1(\mathbb{T})$ for every $f \in L^1(\mathbb{T})$, because the norm of $S_N$ on $L^1$ is also $L_N$.
On the positive side, $\abs{S_N f - f} \le (1 + L_N) E_N(f)$, where $E_N(f)$ is the distance from $f$ to trigonometric polynomials of degree at most $N$; combined with Jackson's theorem this gives the Dini–Lipschitz criterion.
The Fejér kernel, which has $L^1$ norm $1$, avoids the problem and gives Fejér's theorem.
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# Zeilberger's algorithm

## Definition

Let $F(n, k)$ be a hypergeometric term in both variables, as in the definition of a [[WZ pair|WC0QNABPZ4]].
*Zeilberger's algorithm* searches, for $J = 0, 1, 2, \ldots$ in turn, for polynomials $a_0(n), \ldots, a_J(n)$, not all zero, and a hypergeometric term $G(n, k)$ with $G/F$ rational, such that
$$\sum_{j=0}^{J} a_j(n)\,F(n + j, k) = G(n, k + 1) - G(n, k).$$
For each $J$ it applies [[Gosper's algorithm|NHMJQBS04M]] to the term $t_k = \sum_{j} a_j(n) F(n + j, k)$ in $k$, with unknown $a_j$.
The polynomial equation of Gosper's method is then linear in the $a_j$ and in the unknown coefficients of $x$.
It stops at the first $J$ for which a solution exists and returns the $a_j$ and $G$.

## Remarks

If $F(n, k)$ vanishes for $k$ outside a finite range depending on $n$, summing the relation over all $k$ makes the right side a [[telescoping sum|0GXKM1R7E4]] equal to $0$.
So $f(n) = \sum_k F(n, k)$ satisfies the [[recurrence|0FYX44GP92]] $\sum_{j=0}^{J} a_j(n) f(n + j) = 0$.
The search terminates when $F$ is a proper hypergeometric term.
Such a term is a polynomial, times a product of factorials of integer-linear forms in $n$ and $k$, times $x^k$.
This termination result is part of the fundamental theorem of Wilf and Zeilberger.
Example: for $F(n, k) = \binom{n}{k}$ the algorithm stops at $J = 1$ with $a_0 = -2$, $a_1 = 1$ and $G(n, k) = -\binom{n}{k - 1}$.
Check by Pascal's rule: $\binom{n + 1}{k} - 2\binom{n}{k} = \binom{n}{k - 1} - \binom{n}{k} = G(n, k + 1) - G(n, k)$, which gives $f(n + 1) = 2f(n)$.
Non-example: for $J = 0$ no solution exists here, since that would need $\sum_{j<k}\binom{n}{j}$ to be a hypergeometric term in $k$, and Gosper's algorithm shows it is not.
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# Church–Turing thesis

## Summary

The Church–Turing thesis is a claim about the informal notion of an algorithm.
It says that a function on the natural numbers can be calculated by an effective method exactly when a [[Turing machine|758EYCN27B]] computes it.
It is not a theorem, because "effective method" is an informal notion and cannot be the subject of a proof.
It is supported by the fact that all proposed formal models of computation define the same class of functions.
These models include [[general recursive functions|HBK5QHC1GC]], the [[lambda calculus|BD756KR8FM]] and Turing machines.
The thesis lets one prove that a problem has no algorithm at all by proving that no Turing machine solves it.

## Explanations

### L1 human

People computed with pencil and paper long before there were precise definitions of computation.
In the 1930s Alonzo Church and Alan Turing each proposed a precise definition, and these turned out to describe the same functions.
The thesis claims that these definitions capture everything that can be computed by following exact rules.
For example, when one shows that no Turing machine decides whether a program halts, the thesis lets one conclude that no algorithm of any kind does.
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# Noncommutative geometry

## Definition

A *noncommutative space* is a C*-algebra $A$, possibly not commutative, often with extra structure such as a dense $*$-subalgebra or a spectral triple.
*Noncommutative geometry* is the study of such algebras with the methods of topology and geometry, through the following dictionary.

1. A locally compact Hausdorff space $X$ corresponds to the commutative C*-algebra $C_0(X)$ of continuous complex functions on $X$ that vanish at infinity.
2. By the [[commutative Gelfand–Naimark theorem|NT564Y315M]], every commutative C*-algebra is isomorphic to $C_0(X)$ for a locally compact Hausdorff space $X$, unique up to homeomorphism.
3. A notion of the geometry of $X$ that can be expressed through $C_0(X)$ alone is then defined for every C*-algebra by the same formula.

## Remarks

Example: the space $X$ is compact if and only if $C_0(X)$ has a unit.
So a unital C*-algebra is read as a "compact noncommutative space".
Another example: vector bundles over a compact space $X$ correspond to finitely generated projective $C(X)$-modules.
So these modules play the role of vector bundles for any unital C*-algebra, and this leads to operator K-theory.

Non-example: the notion "a point of the space" does not carry over.
A point $x$ of $X$ gives the character $f \mapsto f(x)$ of $C_0(X)$, that is, a nonzero multiplicative linear functional.
The matrix algebra $M_2(\C)$ has no such functional.
Indeed, if $\varphi$ is one and $E_{ij}$ are the matrix units, then $\varphi(E_{11}) = \varphi(E_{12})\varphi(E_{21}) = \varphi(E_{22})$ and $\varphi(E_{11})\varphi(E_{22}) = \varphi(E_{11}E_{22}) = 0$, so $\varphi(I) = \varphi(E_{11}) + \varphi(E_{22}) = 0$.
Then $\varphi(a) = \varphi(I)\varphi(a) = 0$ for every $a$, a contradiction.

The approach of Alain Connes adds a [[spectral triple|FTQMQR3YS7]] to $A$ to describe metric and differential structure.
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# Serre's modularity conjecture

## Statement

Let $p$ be a prime, and let $\overline{\F}_p$ be an algebraic closure of the [[finite field|0ZXZ82FTX2]] $\F_p$.
Let $\bar\rho \colon \Gal(\overline{\Q}/\Q) \to \mathrm{GL}_2(\overline{\F}_p)$ be a continuous [[Galois representation|JQEYEQJY2G]] that is irreducible and odd, that is, $\det \bar\rho(c) = -1$ for a complex conjugation $c$.
Then $\bar\rho$ is modular, in the following sense.
There is a normalised [[cusp form|A50Y012RBP]] $f = \sum_{n \ge 1} a_n q^n$ of some weight $k \ge 2$ and level $N \ge 1$ that is an eigenform of all [[Hecke operators|S4SV3QZ8B1]].
Let $\mathrm{Frob}_\ell$ denote a [[Frobenius element|H5145REFMW]] at a prime $\ell$.
There is also a ring homomorphism $a \mapsto \bar{a}$ from the ring generated by the $a_n$ to $\overline{\F}_p$ such that, for every prime $\ell \nmid pN$,
$$\tr \bar\rho(\mathrm{Frob}_\ell) = \bar{a}_\ell.$$

## Remarks

Serre stated the conjecture in 1987 in a strong form.
In it, $N$ is the Artin conductor of $\bar\rho$ away from $p$, and $k$ is an explicit weight $k(\bar\rho)$ read off from the restriction of $\bar\rho$ to inertia at $p$.
Ribet's level-lowering theorem and later work showed that the weak form above implies the strong form.
Khare and Wintenberger proved the conjecture, with a step by Kisin, in work published in 2009.
Their proof is an induction on the prime $p$ and the conductor, using modularity lifting theorems and the existence of compatible families of Galois representations.
Oddness is necessary: for every modular eigenform the attached representation is odd.
Example: for an elliptic curve $E$ over $\Q$, the action on the $p$-torsion points $E[p]$ gives an odd $\bar\rho$, since $\det \bar\rho$ is the cyclotomic character modulo $p$.
When this $\bar\rho$ is irreducible, its modularity also follows from the [[modularity theorem|CS1NVJBZMH]].

## Explanations

### L1 human

Serre's conjecture says that every two-dimensional Galois representation modulo $p$ of the right kind comes from a modular form.
The "right kind" is just irreducible and odd, two conditions that are easy to check.
So a large class of arithmetic objects is matched with analytic ones.

### L2 human

The strong form gives a route to Fermat's Last Theorem.
A putative solution would give a representation of level $2$ and weight $2$, and there are no nonzero cusp forms of that level and weight.
The theorem is a mod $p$ case of the [[Langlands program|J9T2M084W5]] for $\mathrm{GL}_2$ over $\Q$.
It also implies that the Artin conjecture holds for odd two-dimensional complex Galois representations of $\Q$.
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<document source="https://tamnd.github.io/math2/node/RWZ5KMPD1R.md" tag="RWZ5KMPD1R" kind="concept">
# Random ordinary differential equation

## Summary

A random ordinary differential equation is an [[ordinary differential equation|0NT64JV96A]] whose right side depends on a [[stochastic process|0QH5ENKZM1]] with regular sample paths.
For each fixed outcome of the randomness it is an ordinary, non-autonomous differential equation, and it is solved path by path with classical calculus.
Such equations generate [[random dynamical systems|DMTFJY6CS0]], and many [[stochastic differential equations|B219G9SHJA]] can be turned into them by a change of variables.

## Explanations

### L1 human

Think of a boat whose speed depends on a wind that changes in a random but smooth way.
Once the weather of a given day is known, the motion of the boat is an ordinary differential equation.
A random ordinary differential equation collects all these equations, one for each possible weather.
For example, $x' = -x + \eta_t$ with a random continuous signal $\eta$ has, for each path of $\eta$, the solution $x(t) = e^{-t} x(0) + \int_0^t e^{-(t - s)} \eta_s \, ds$.
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<document source="https://tamnd.github.io/math2/node/RX1NWE1WMD.md" tag="RX1NWE1WMD" kind="definition">
# Dynkin quiver

## Definition

Let $Q$ be a finite [[quiver|RE9YFFS3TS]].
The *underlying graph* of $Q$ is the graph with vertex set $Q_0$ and one undirected edge between $s(a)$ and $t(a)$ for each arrow $a \in Q_1$.
The quiver $Q$ is a *Dynkin quiver* if its underlying graph is one of the simply laced Dynkin diagrams:

1. $A_n$ for $n \ge 1$, a path with $n$ vertices;
2. $D_n$ for $n \ge 4$, a path with $n - 2$ vertices and two extra vertices joined to one end vertex;
3. $E_6$, $E_7$ or $E_8$, a path with $n - 1$ vertices and one extra vertex joined to the third vertex from one end, for $n = 6, 7, 8$.

## Remarks

Example: $1 \to 2 \leftarrow 3$ is a Dynkin quiver, since its underlying graph is the path with $3$ vertices, of type $A_3$.
Non-example: the Kronecker quiver, with two arrows $1 \to 2$, is not Dynkin, since its underlying graph has a double edge.
Non-example: a quiver with a loop, or whose underlying graph is a cycle, is not Dynkin.
A Dynkin quiver is connected and has no oriented cycles, and every orientation of a Dynkin diagram is a Dynkin quiver.
</document>

<document source="https://tamnd.github.io/math2/node/RX4CVSSRRJ.md" tag="RX4CVSSRRJ" kind="example">
# Dirichlet distribution

## Example

Let $K \ge 2$ and $\alpha_1, \dots, \alpha_K > 0$, and let $\alpha_0 = \alpha_1 + \dots + \alpha_K$.
The *Dirichlet distribution* $\operatorname{Dir}(\alpha_1, \dots, \alpha_K)$ is the law of a random vector $(X_1, \dots, X_K)$ in the open [[simplex|YX1DJK2QAG]] $\set{x \in \R^K : x_i > 0, \ x_1 + \dots + x_K = 1}$.
The vector $(X_1, \dots, X_{K-1})$ has the density
$$f(x_1, \dots, x_{K-1}) = \frac{\Gamma(\alpha_0)}{\Gamma(\alpha_1) \cdots \Gamma(\alpha_K)} \prod_{i=1}^K x_i^{\alpha_i - 1}$$
on the set where all $x_i > 0$ and $x_1 + \dots + x_{K-1} < 1$, with $x_K = 1 - x_1 - \dots - x_{K-1}$; here $\Gamma$ is the [[gamma function|0H5SYBSED9]].
A construction: if $Y_1, \dots, Y_K$ are independent and $Y_i$ has the [[gamma distribution|0B80PAQD3C]] with shape $\alpha_i$ and scale $1$, then $(Y_1, \dots, Y_K) / (Y_1 + \dots + Y_K)$ has law $\operatorname{Dir}(\alpha_1, \dots, \alpha_K)$.
Check for $K = 2$: the density becomes $\frac{\Gamma(\alpha_1 + \alpha_2)}{\Gamma(\alpha_1)\Gamma(\alpha_2)} x^{\alpha_1 - 1} (1 - x)^{\alpha_2 - 1}$ on $(0, 1)$, so $X_1$ has the [[beta distribution|0EW824Z7G2]] with parameters $\alpha_1, \alpha_2$.
The means are $E[X_i] = \alpha_i / \alpha_0$ and the variances are $\operatorname{Var}(X_i) = \frac{\alpha_i (\alpha_0 - \alpha_i)}{\alpha_0^2 (\alpha_0 + 1)}$.

## Remarks

If all $\alpha_i = 1$, the density is constant, so the law is uniform on the simplex.
The Dirichlet law is the [[conjugate prior|EHY450Q5CW]] of the multinomial law: with prior $\operatorname{Dir}(\alpha_1, \dots, \alpha_K)$ for the cell probabilities and observed counts $n_1, \dots, n_K$, the posterior is $\operatorname{Dir}(\alpha_1 + n_1, \dots, \alpha_K + n_K)$.
It is the finite-dimensional building block of the [[Dirichlet process|XT4SWG6J7N]].

## Explanations

### L1 human

The Dirichlet distribution is a law on random probability vectors: lists of $K$ positive numbers that add up to $1$.
It is used to model an unknown die with $K$ faces, before the die has been thrown.
Large parameters concentrate the vector near the mean $(\alpha_1, \dots, \alpha_K)/\alpha_0$, and small parameters push it towards the corners of the simplex.
For two categories it is the beta distribution.
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<document source="https://tamnd.github.io/math2/node/RX5MYWE73J.md" tag="RX5MYWE73J" kind="concept">
# Pfaffian system

## Summary

A Pfaffian system is an [[exterior differential system|901CCKQHVY]] generated by [[one-forms|02Z5B2HA3F]].
Equivalently, it is given by a field of linear conditions on tangent vectors, that is, by a distribution, the common kernel of the generating forms.
Its integral manifolds are the submanifolds tangent to this distribution.
The [[Frobenius theorem|5Q4BHBFXJY]] tells exactly when the integral manifolds fill out the manifold as the leaves of a foliation.

## Explanations

### L1 human

A one-form at each point says which directions are allowed: those on which it vanishes.
A Pfaffian system is a list of such restrictions.
For example, the form $dz$ on $\R^3$ allows only horizontal directions, and the horizontal planes are surfaces that obey it everywhere.
The form $dz - y\, dx$ also allows a plane of directions at each point, but these planes twist, and no surface is tangent to them everywhere.
Pfaffian systems describe rolling without slipping, parking a car and many other constrained motions.
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<document source="https://tamnd.github.io/math2/node/RX7JSHMP8Z.md" tag="RX7JSHMP8Z" kind="statement">
# Fundamental theorem of calculus for Lebesgue integrals

## Statement

Let $a < b$ be real numbers, and let integrals on $[a, b]$ be taken with respect to [[Lebesgue measure|2Z2QVWZ262]].

1. Let $g \colon [a, b] \to \R$ be [[integrable|X7GTF3D98F]] and let $F(x) = \int_a^x g(t) \, dt$ for $x \in [a, b]$. Then $F$ is [[absolutely continuous|ZT6F4C2YHK]], and $F$ is [[differentiable|0V1EJCT8G6]] with $F'(x) = g(x)$ for [[almost every|RARTMA0ZEC]] $x \in [a, b]$.
2. Let $f \colon [a, b] \to \R$ be absolutely continuous. Then $f$ is differentiable at almost every point of $[a, b]$, the derivative $f'$ is integrable on $[a, b]$, and $f(x) = f(a) + \int_a^x f'(t) \, dt$ for every $x \in [a, b]$.

## Remarks

Part 1 follows from the Lebesgue differentiation theorem in dimension one; absolute continuity of $F$ comes from the absolute continuity of the integral.
For part 2, an absolutely continuous function has bounded variation, so it is a difference of two increasing functions and is differentiable almost everywhere with $f' \in L^1$.
The function $h = f - f(a) - \int_a^x f'$ is then absolutely continuous with $h' = 0$ almost everywhere, and an absolutely continuous function with zero derivative almost everywhere is constant.

Example: $f(x) = \abs{x}$ on $[-1, 1]$ is Lipschitz, hence absolutely continuous.
Its derivative is $-1$ on $(-1, 0)$ and $1$ on $(0, 1)$, undefined at $0$, and $\abs{x} = 1 + \int_{-1}^{x} f'(t) \, dt$ for every $x$.

Absolute continuity cannot be weakened to continuity in part 2.
The [[Cantor function|NPBYYK63FE]] $c$ is continuous and increasing on $[0, 1]$ with $c' = 0$ almost everywhere, but $c(1) - c(0) = 1 \ne 0 = \int_0^1 c'(t) \, dt$.

## Explanations

### L1 human

Integration and differentiation undo each other also for the Lebesgue integral, provided the functions are of the right kind.
Integrating an integrable function and then differentiating gives back the function, except on a set of length zero.
Differentiating and then integrating gives back the function exactly when the function is absolutely continuous.

### L2 human

The theorem shows that the absolutely continuous functions on $[a, b]$ are exactly the indefinite Lebesgue integrals.
It is the one-dimensional prototype of Sobolev spaces: absolutely continuous functions are the functions in $W^{1,1}(a, b)$, up to the choice of representative.
In the language of measures, it says that the Lebesgue–Stieltjes measure $df$ of an absolutely continuous $f$ has density $f'$ with respect to Lebesgue measure.
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<document source="https://tamnd.github.io/math2/node/RXBD58SGXZ.md" tag="RXBD58SGXZ" kind="concept">
# Continuous lattice

## Summary

A continuous lattice is a complete lattice in which every element is the supremum of the elements [[way below|RDM7J5J9HQ]] it.
Continuous lattices were introduced by Dana Scott as models for the semantics of programming languages, and they connect order theory with topology.
Every [[algebraic lattice|3J0BTXSCVN]] is continuous, and so is the unit interval $[0, 1]$.

## Explanations

### L1 human

An element $x$ is way below $y$ when $x$ is reached by every increasing approach to $y$ before the approach is finished.
In $[0, 1]$, the number $x$ is way below $y$ when $x < y$, or $x = 0$.
A continuous lattice is one where every element can be approximated from below by elements way below it.
So $[0, 1]$ is continuous: each $y > 0$ is the supremum of the numbers $x < y$.
In an algebraic lattice the approximating elements can be taken compact, that is, "finite".
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<document source="https://tamnd.github.io/math2/node/RXFJ47VYGX.md" tag="RXFJ47VYGX" kind="concept">
# Discrete-time Fourier transform

## Summary

The discrete-time Fourier transform turns a sequence $(x_n)_{n \in \Z}$ into a $2\pi$-periodic function $X(\omega) = \sum_n x_n e^{-i \omega n}$ of a continuous frequency $\omega$.
It is the Fourier transform of the group $\Z$, whose dual group is the circle.
Read backwards, it is a [[Fourier series|G12KXDHPPS]]: up to the sign of the index, the sequence consists of the Fourier coefficients of $X$.
It is the basic tool for analysing digital filters and sampled signals.
Sampling the frequency variable at $N$ points leads to the [[discrete Fourier transform|TST0GH66A5]].
See the [[definition|65QTEF1FTN]].

## Explanations

### L1 human

A digital signal is a list of numbers indexed by time steps, going on in both directions.
Its discrete-time Fourier transform says how much of each frequency between $-\pi$ and $\pi$ radians per step it contains.
Frequencies outside this range look the same as frequencies inside it, because time is sampled; this is why the transform is periodic.
For the signal that is $1$ at time $0$ and $0$ elsewhere, the transform is the constant $1$.
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<document source="https://tamnd.github.io/math2/node/RXHJPVMPJH.md" tag="RXHJPVMPJH" kind="definition">
# Heyting arithmetic

## Definition

*Heyting arithmetic*, written HA, is the first-order theory in the language $0, S, +, \cdot, =$ with these parts.

1. Its logic is first-order [[intuitionistic logic|Q11N8X99FK]] with equality.
2. Its axioms are those of first-order [[Peano arithmetic|0QFT7Z76N5]]. They are $S x \ne 0$, $S x = S y \to x = y$, and $x + 0 = x$, $x + S y = S(x + y)$, $x \cdot 0 = 0$, $x \cdot S y = x \cdot y + x$. They also include the induction scheme $A(0) \wedge \forall x\, (A(x) \to A(S x)) \to \forall x\, A(x)$ for every formula $A$.

Thus HA has the same axioms as Peano arithmetic PA.
The theory PA is HA together with the [[law of excluded middle|0RYYK8PG4S]] $A \vee \neg A$ for all formulas $A$.

## Remarks

Example: HA proves $\forall x\, \forall y\, (x = y \vee \neg x = y)$, by induction on $x$ and $y$.
So excluded middle holds in HA for equations, and, by induction on formulas, for all formulas without unbounded quantifiers.
Non-example: HA does not prove $\forall x\, (\exists y\, T(x, x, y) \vee \neg \exists y\, T(x, x, y))$, where $T$ is Kleene's predicate for "program $x$ on input $x$ halts within $y$ steps".
This formula is provable in PA, so the two theories differ.
HA has the disjunction property: if HA proves a closed formula $A \vee B$, then it proves $A$ or it proves $B$.
It has the existence property: if HA proves a closed formula $\exists x\, A(x)$, then it proves $A(\overline{n})$ for some numeral $\overline{n}$.
</document>

<document source="https://tamnd.github.io/math2/node/RXWA8TF2RM.md" tag="RXWA8TF2RM" kind="concept">
# Locally cyclic group

## Summary

A group is locally cyclic if every finitely generated subgroup is cyclic. Every cyclic group is locally cyclic, and every finitely-generated locally cyclic group is cyclic. Every locally cyclic group is abelian. Every subgroup, every quotient group and every homomorphic image of a locally cyclic group is locally cyclic.
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<document source="https://tamnd.github.io/math2/node/RY2RAH3802.md" tag="RY2RAH3802" kind="definition">
# Phase space

## Definition

Let $U \subseteq \R^n$ be an open set and let $f : U \to \R^n$ be a continuous map.
The *phase space* of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ is the set $U$.
Its points are called states, and the curve $t \mapsto x(t)$ of a solution in $U$ is its [[trajectory|0BVT6VSB3S]].
For an [[ordinary differential equation|0P97V20RW7]] $y^{(n)} = g(y, y', \dots, y^{(n-1)})$ of order $n$, the phase space is the phase space of the equivalent first-order system for the state $(y, y', \dots, y^{(n-1)})$.

## Remarks

Example: the equation $x'' = -x$ becomes the system $x' = v$, $v' = -x$ on $U = \R^2$.
Its phase space is the plane with coordinates $(x, v)$.
The solution $x = \cos t$ has the trajectory $(\cos t, -\sin t)$, which runs on the unit circle, since $\cos^2 t + \sin^2 t = 1$.
Non-example: for a pendulum, the set of angles alone is not the phase space.
The angle at one time does not determine the motion; the angular velocity is also needed.
For a non-autonomous system $x' = f(t, x)$ on an open set $D \subseteq \R \times \R^n$, the set $D$ is called the extended phase space.
For a [[Hamiltonian system|0C2AKPDCD9]] with $n$ degrees of freedom, the phase space has the $2n$ coordinates $(q_1, \dots, q_n, p_1, \dots, p_n)$ of positions and momenta.
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<document source="https://tamnd.github.io/math2/node/RY3QGW87HX.md" tag="RY3QGW87HX" kind="statement">
# Bluestein's FFT algorithm

## Statement

Let $N \ge 1$, let $x \in \C^N$ and let $X$ be its [[discrete Fourier transform|TST0GH66A5]], $X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i nk/N}$.
For integers $m$ let $b_m = e^{\pi i m^2/N}$, and for $0 \le n < N$ let $a_n = x_n e^{-\pi i n^2/N}$.
Then for $0 \le k < N$,
$$X_k = \overline{b_k} \sum_{n=0}^{N-1} a_n\, b_{k-n}.$$
Let $M \ge 2N - 1$, let $\alpha \in \C^M$ be $a$ followed by zeros, and let $\beta \in \C^M$ have $\beta_j = b_j$ for $0 \le j < N$, $\beta_{M-j} = b_{-j}$ for $1 \le j < N$, and zeros elsewhere.
Then $X_k = \overline{b_k}\, (\alpha \circledast \beta)_k$ for $0 \le k < N$, where $(\alpha \circledast \beta)_k = \sum_{j=0}^{M-1} \alpha_j \beta_{(k-j) \bmod M}$ is the cyclic [[convolution|2Z29JN3VES]] of length $M$.
Take $M$ a power of $2$ and compute the cyclic convolution with three [[radix-2 FFTs|9N8D9E5TB2]] of length $M$.
Then the DFT of any length $N$ costs $O(N \log N)$ operations.

## Remarks

The key identity is $nk = \frac12 \bigl(n^2 + k^2 - (k-n)^2\bigr)$, which gives $e^{-2\pi i nk/N} = e^{-\pi i n^2/N} e^{-\pi i k^2/N} e^{\pi i (k-n)^2/N}$.
So the DFT is a pointwise product with a chirp, a convolution with the conjugate chirp, and another pointwise product.
The index $k - n$ runs from $-(N-1)$ to $N-1$, and the zero padding to $M \ge 2N-1$ makes the cyclic convolution agree with this linear convolution on $0 \le k < N$.
The cyclic convolution of length $M$ equals the inverse DFT of the product of the DFTs of $\alpha$ and $\beta$.
Since one can choose $M < 4N$, the constant in $O(N \log N)$ does not depend on the arithmetic of $N$; in particular prime $N$ is allowed.
The same identity with $e^{-2\pi i/N}$ replaced by any complex $z$ evaluates the chirp z-transform $\sum_n x_n z^{-nk}$, which is the subject of the source article.

## Explanations

### L1 human

The fast Fourier transform is easiest when the length is a power of two.
Bluestein's trick rewrites a transform of any length, even a prime length, as a convolution.
A convolution can be padded with zeros to a power-of-two length and computed with power-of-two FFTs.
So every length gets the speed $N \log N$, at the price of a larger constant.

### L2 human

The algorithm is the standard fallback in FFT libraries for lengths with large prime factors.
Compared with Rader's algorithm, it does not need a primitive root and works for every $N$, not only primes.
It needs about three FFTs of length $M \approx 2N$ to $4N$, so it is slower than a direct Cooley–Tukey FFT of a highly composite length.
The chirp $e^{\pi i m^2/N}$ is the same object as in the fractional Fourier transform and in radar signal processing.
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<document source="https://tamnd.github.io/math2/node/RY8K5RQJSR.md" tag="RY8K5RQJSR" kind="statement">
# Kurosh subgroup theorem

## Statement

Let $(A_i)_{i \in I}$ be a family of groups, and let $G = \mathop{*}_{i \in I} A_i$ be their [[free product|TM7QS7P7X9]].
Let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then there are a [[free group|0KTW26C80J]] $F$, groups $B_j$ for $j \in J$, and elements $g_j \in G$ with the following property.
Each $B_j$ is a subgroup of some factor $A_{i(j)}$, and
$$H = F * \mathop{*}_{j \in J} g_j B_j g_j^{-1}.$$
In words, $H$ is the free product of a free group and of [[conjugates|0DXD4QSZ8A]] of subgroups of the factors.

## Remarks

One proof uses [[Bass–Serre theory|6M731X66RD]].
The free product $G$ acts on a tree with trivial edge stabilisers, and the vertex stabilisers are the conjugates of the factors $A_i$.
The subgroup $H$ acts on the same tree, so it is the fundamental group of a [[graph of groups|DC1CQMD3WK]] with trivial edge groups.
Such a group is the free product of its vertex groups and of a free group, the fundamental group of the underlying graph.
A topological proof uses covering spaces of a wedge of spaces whose fundamental groups are the $A_i$.

Example: if every $A_i$ is infinite cyclic, then $G$ is free.
Each subgroup of $\Z$ is trivial or infinite cyclic, so each $g_j B_j g_j^{-1}$ is free, and $H$ is free.
This recovers the [[Nielsen–Schreier theorem|W97X7WX5G8]].
Another example: in $\Z/2 * \Z/3$, which is isomorphic to $PSL(2, \Z)$, every subgroup without elements of finite order is free.

## Explanations

### L1 human

A free product glues groups together with no relations between them.
The theorem says that every subgroup of such a product is again built in the same simple way.
It is a free product of pieces taken from the original factors, moved around by conjugation, together with a free group.

### L2 human

The theorem is the main structure result for subgroups of free products.
It generalises the Nielsen–Schreier theorem, which says that subgroups of free groups are free.
It implies that a subgroup that meets every conjugate of every factor only in the identity is free.
In Bass–Serre theory it is the special case of trees with trivial edge stabilisers; the general case describes subgroups of amalgamated products and [[HNN extensions|YEYERRY37K]].
</document>

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# Riemann–Hilbert problem

## Problem

Let $\Gamma \subset \C$ be a smooth simple closed curve, oriented counterclockwise.
Let $D^+$ be its interior domain and $D^-$ its exterior domain.
Let $G \colon \Gamma \to \C \setminus \set{0}$ and $g \colon \Gamma \to \C$ be [[Hölder continuous|0WZTM0EFR6]] functions.
Find a function $\Phi$ that is [[holomorphic|0NAHXR1FRH]] on $D^+$ and on $D^-$ and vanishes at $\infty$.
It must have continuous boundary values $\Phi^+$ from $D^+$ and $\Phi^-$ from $D^-$ on $\Gamma$ with
$$\Phi^+(t) = G(t)\, \Phi^-(t) + g(t) \quad \text{for } t \in \Gamma.$$
In the matrix form of the problem, $\Phi$ takes values in $\C^n$.
Then $G$ is a Hölder continuous map from $\Gamma$ to the invertible $n \times n$ complex matrices.

## Remarks

The scalar problem is solved explicitly.
Let $\kappa$ be the [[winding number|0X7SBS8E0F]] of the curve $G(\Gamma)$ about $0$, called the index of the problem.
Assume $0 \in D^+$; taking a branch of $\log$ of $t^{-\kappa} G(t)$, which has winding number $0$, turns the problem into a jump problem $\Psi^+ - \Psi^- = h$.
The Cauchy integral $\frac{1}{2\pi i} \int_\Gamma \frac{h(\tau)}{\tau - z}\, d\tau$ solves it by the [[Sokhotski–Plemelj theorem|RNB6QX0GCA]].
If $\kappa \ge 0$, the problem is solvable for every $g$, and the solutions form an affine space of dimension $\kappa$.
If $\kappa < 0$, it is solvable if and only if $g$ satisfies $-\kappa$ linear conditions, and then the solution is unique.

Example: for $G = 1$ the index is $0$ and the unique solution is the Cauchy integral of $g$.

The matrix problem cannot be solved by logarithms, since matrices do not commute.
It is closely tied to [[Hilbert's twenty-first problem|7RHW0KA5RY]] on linear differential equations with prescribed monodromy, and it underlies the Riemann–Hilbert method for integrable systems and random matrices.

## Explanations

### L1 human

One looks for a function that is holomorphic on both sides of a curve and jumps across the curve in a prescribed way.
The jump says that the value from inside equals a given function times the value from outside, plus another given function.
For scalar functions this is solved completely, and the answer depends on how many times the given function $G$ winds around $0$.
For matrix functions the problem is much harder and connects to differential equations and mathematical physics.
</document>

<document source="https://tamnd.github.io/math2/node/RYA0R6P73C.md" tag="RYA0R6P73C" kind="definition">
# Height function

## Definition

Let $\overline{\Q}$ be an algebraic closure of $\Q$ and let $n \ge 1$.
Let $P = [x_0 : \cdots : x_n] \in \mathbb{P}^n(\overline{\Q})$ be a point of [[projective space|ZT9A6JK8QF]], and let $K$ be an [[algebraic number field|MWBDFS2KJ5]] that contains $x_0, \ldots, x_n$.
For each [[place|BT5V3TSNY6]] $v$ of $K$, let $\abs{\cdot}_v$ be the absolute value of $v$ that extends the usual or a $p$-adic absolute value of $\Q$.
Let $n_v = [K_v : \Q_v]$ be the local degree.
The *absolute logarithmic height* of $P$ is
$$h(P) = \frac{1}{[K : \Q]} \sum_{v} n_v \log \max\left(\abs{x_0}_v, \ldots, \abs{x_n}_v\right).$$
It does not depend on the choice of $K$ or of homogeneous coordinates.

More generally, let $V$ be a [[projective variety|A4H4CZ2V35]] over $\overline{\Q}$ and let $\phi \colon V \to \mathbb{P}^n$ be a morphism.
The *height function* relative to $\phi$ is $h_\phi = h \circ \phi \colon V(\overline{\Q}) \to \R$.
The height of an [[algebraic number|0MNF0BYFQC]] $\alpha$ is $h(\alpha) = h([1 : \alpha])$.

## Remarks

Independence of the coordinates follows from the product formula $\prod_v \abs{x}_v^{n_v} = 1$ for $x \in K^\times$.

Example: for $P = [a_0 : \cdots : a_n] \in \mathbb{P}^n(\Q)$ with $a_i \in \Z$ and $\gcd(a_0, \ldots, a_n) = 1$, take $K = \Q$.
Each $p$-adic term is $\log 1 = 0$, since some $a_i$ is not divisible by $p$, so $h(P) = \log \max_i \abs{a_i}$.
For the rational number $2/3$, that is $[3 : 2]$, this gives $h = \log 3$.

Non-example: the rule $P \mapsto \log \max_i \abs{a_i}$ for arbitrary integer coordinates, without the coprimality condition, is not a function on $\mathbb{P}^n(\Q)$.
The coordinates $[1 : 1]$ and $[2 : 2]$ give the same point but the values $\log 1$ and $\log 2$.

Northcott's theorem: for all $B$ and $d$, the set of $P \in \mathbb{P}^n(\overline{\Q})$ with $h(P) \le B$ and $[\Q(P) : \Q] \le d$ is finite.
The multiplicative height is $H = e^h$.
On an elliptic curve, the Néron–Tate height $\hat{h}(P) = \lim_{m \to \infty} 4^{-m} h_x(2^m P)$, with $h_x$ the height relative to the $x$-coordinate, is a positive semi-definite quadratic form on $E(\overline{\Q})$; some authors include a factor $\frac{1}{2}$.
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<document source="https://tamnd.github.io/math2/node/RYB4DQTAK0.md" tag="RYB4DQTAK0" kind="statement">
# Euler's theorem on normal curvatures

## Statement

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]] with a smooth unit normal field near a point $p \in S$.
Let $k_1, k_2$ be the [[principal curvatures|FA8QE1SGP4]] of $S$ at $p$, with orthonormal principal directions $e_1, e_2 \in T_p S$.
For a unit vector $w \in T_p S$, let $k_n(w) = I\!I_p(w)$ be the normal curvature in the direction $w$, where $I\!I_p$ is the [[second fundamental form|9KHQR16P1B]].
If $w = \cos \theta \, e_1 + \sin \theta \, e_2$, then
$$k_n(w) = k_1 \cos^2 \theta + k_2 \sin^2 \theta.$$

## Remarks

The proof is one line: $I\!I_p(w) = \ip{S_p w}{w}$ with the [[shape operator|DQCD11X1VM]] $S_p$, and $S_p e_i = k_i e_i$, so the cross terms vanish.
It follows that $k_1$ and $k_2$ are the maximum and the minimum of the normal curvature over all directions.
Example: on a cylinder of radius $r$ with the inner normal, $k_1 = 1/r$ around the cylinder and $k_2 = 0$ along it.
A direction at angle $\theta$ to the circular sections has normal curvature $\cos^2 \theta / r$.
The normal curvature $k_n(w)$ equals the signed curvature of the normal section, the curve cut out of $S$ by the plane through $p$ spanned by $w$ and the normal.

## Explanations

### L1 human

At a point of a surface, the bending changes with the direction you look in.
Euler's theorem says that this change follows a very simple rule.
Once you know the strongest and the weakest bending, and their directions, you know the bending in every other direction.

### L2 human

The theorem is the diagonal form of the quadratic form $I\!I_p$ in an orthonormal eigenbasis of the self-adjoint map $S_p$.
It explains why the Gaussian curvature $k_1 k_2$ and the mean curvature $(k_1 + k_2)/2$ capture the whole second-order shape, up to the choice of principal directions.
The sign of $k_1 k_2$ decides whether the normal curvature keeps one sign (elliptic point), changes sign (hyperbolic point), or vanishes in one direction (parabolic point).
The directions with $k_n(w) = 0$ at a hyperbolic point are the asymptotic directions, at angles $\theta$ with $\tan^2 \theta = -k_1 / k_2$.
</document>

<document source="https://tamnd.github.io/math2/node/RYB6F7JFGV.md" tag="RYB6F7JFGV" kind="statement">
# Adjunction formula

## Statement

Let $k$ be a field and let $X$ be a variety that is [[smooth|EXKGG9X6J0]] over $k$.
Let $Y \subseteq X$ be a closed subvariety that is smooth over $k$ and of codimension $1$.
Then $Y$ is an effective [[Cartier divisor|FHV05ZK99F]] on $X$, with [[invertible sheaf|7JCF2TE3V5]] $\mathcal{O}_X(Y)$.
The [[canonical bundles|W05VYY4HBR]] satisfy
$$\omega_Y \cong \left(\omega_X \otimes \mathcal{O}_X(Y)\right)|_Y.$$
In terms of divisors, $K_Y = (K_X + Y)|_Y$.

## Remarks

The idea of the proof: for $Y$ smooth with ideal sheaf $\mathcal{I}$, the conormal sequence $0 \to \mathcal{I}/\mathcal{I}^2 \to \Omega_X|_Y \to \Omega_Y \to 0$ is an exact sequence of vector bundles.
Taking top exterior powers gives $\omega_X|_Y \cong \omega_Y \otimes \mathcal{I}/\mathcal{I}^2$.
Since $\mathcal{I} = \mathcal{O}_X(-Y)$, the conormal bundle is $\mathcal{I}/\mathcal{I}^2 \cong \mathcal{O}_X(-Y)|_Y$, and the formula follows.

Example: let $C \subseteq \mathbb{P}^2_k$ be a smooth plane curve of degree $d$.
Then $K_{\mathbb{P}^2} = -3H$ for a line $H$ and $C \sim dH$, so $K_C = (d-3)H|_C$ has degree $d(d-3)$.
Since $\deg K_C = 2g - 2$, this gives the genus $g = (d-1)(d-2)/2$.

For a smooth closed subvariety $Y$ of any codimension $r$, the same argument gives $\omega_Y \cong \omega_X|_Y \otimes \det N_{Y/X}$, where $N_{Y/X}$ is the normal bundle.

## Explanations

### L1 human

The adjunction formula computes the canonical bundle of a hypersurface from that of the ambient space.
One only needs to add the hypersurface itself, viewed as a divisor, and restrict.
This turns questions about a subvariety into questions about the bigger space, where computations are often easy.

### L2 human

The formula is the standard way to compute the genus of curves on surfaces and the canonical class of complete intersections.
For a smooth hypersurface of degree $d$ in $\mathbb{P}^n$ it gives $\omega_Y \cong \mathcal{O}_Y(d - n - 1)$, which decides whether $Y$ is Fano, Calabi–Yau or of general type.
On a surface $X$ it gives the genus formula $2g - 2 = C \cdot (C + K_X)$ for a smooth curve $C$.
</document>

<document source="https://tamnd.github.io/math2/node/RYQJH4RYB4.md" tag="RYQJH4RYB4" kind="statement">
# Calderón–Zygmund estimates

## Statement

Let $n \ge 1$ be an integer and let $1 < p < \infty$.
Let $\Delta$ be the [[Laplacian|0E0B6RA8K0]] and let $\norm{\cdot}_{L^p}$ be the norm of the [[Lebesgue space|SASP14MA1Z]] $L^p(\R^n)$.
There is $C = C(n, p)$ such that each $u \in C_c^\infty(\R^n)$ satisfies
$$\norm{\partial_i \partial_j u}_{L^p(\R^n)} \le C \norm{\Delta u}_{L^p(\R^n)} \quad \text{for all } i, j \in \set{1, \ldots, n}.$$

## Remarks

On the Fourier side, $\widehat{\partial_i \partial_j u}(\xi) = \frac{\xi_i \xi_j}{\abs{\xi}^2} \widehat{\Delta u}(\xi)$.
So $\partial_i \partial_j u = R_i R_j \Delta u$, up to sign, where $R_i$ is the Riesz transform with multiplier $-i \xi_i / \abs{\xi}$.
Each $R_i R_j$ is a [[singular integral operator|HTMCEARGDD]], and Calderón–Zygmund theory shows that such operators are bounded on $L^p$ for $1 < p < \infty$.
The proof of that bound uses the [[Calderón–Zygmund lemma|K1YKXVPVDG]] to get a weak type $(1,1)$ bound, then interpolation with the $L^2$ bound from Plancherel, then duality.
The estimate fails for $p = 1$ and for $p = \infty$: a function with bounded Laplacian need not have bounded second derivatives.
Combined with cut-off functions, the estimate gives the local $W^{2,p}$ estimates for solutions of $\Delta u = f$, and by freezing coefficients for elliptic equations with continuous leading coefficients.

## Explanations

### L1 human

If you know how large the Laplacian of a function is, on average in the $L^p$ sense, then you also know how large all its second derivatives are.
The Laplacian is only one combination of second derivatives, so this is a surprising gain.
It holds for every exponent strictly between $1$ and $\infty$, but not at the two ends.

### L2 human

These estimates are the $L^p$ counterpart of the Schauder estimates in Hölder spaces.
They are the main tool to show that a weak solution of an elliptic equation with an $L^p$ right-hand side has second derivatives in $L^p$.
They are also a model case of the general theory of singular integrals, which treats operators like the Riesz and Hilbert transforms.
</document>

<document source="https://tamnd.github.io/math2/node/RZ0Z2BTMT4.md" tag="RZ0Z2BTMT4" kind="definition">
# Infinite product

## Definition

Let $(a_n)_{n \ge 1}$ be a [[sequence|0QE8STGJH2]] of [[complex numbers|09FTAET5Z4]].
The infinite product with factors $a_n$ is the sequence $(p_N)_{N \ge 1}$ of partial products
$$p_N = a_1 a_2 \cdots a_N = \prod_{n=1}^{N} a_n.$$
It is written $\prod_{n=1}^{\infty} a_n$.

## Remarks

The symbol $\prod_{n=1}^{\infty} a_n$ also denotes the value of the product when the product [[converges|PCYRWA7SYB]].
Example: for $a_n = 1 - 1/(n+1)^2$ the partial products are $p_N = \prod_{n=2}^{N+1} \frac{(n-1)(n+1)}{n^2} = \frac{N+2}{2(N+1)}$, by telescoping.
So $p_N \to \tfrac12$ and the product converges with value $\tfrac12$.
Non-example of convergence: for $a_n = 1 - 1/(n+1)$ the partial products are $p_N = 1/(N+1) \to 0$.
By convention this product does not converge but diverges to $0$, since a limit $0$ reached by nonzero factors is excluded in the definition of convergence.
A product can start at another index, such as $\prod_{n=2}^{\infty} (1 - 1/n^2)$.

## Explanations

### L1 human

An infinite product is the list of products of the first $N$ factors, for $N = 1, 2, 3, \ldots$.
</document>

<document source="https://tamnd.github.io/math2/node/RZ66X0NAM5.md" tag="RZ66X0NAM5" kind="concept">
# Nef line bundle

## Summary

A line bundle on a proper variety is nef, short for numerically effective or numerically eventually free, when its degree on every curve is nonnegative.
Nef line bundles are the limits of [[ample line bundles|RRK8P6TXG2]]: on a projective variety, the nef cone is the closure of the ample cone.
Nefness is a numerical condition, so it depends only on intersection numbers with curves.
It is central in the [[minimal model program|Q8CTDCZH01]], where a minimal model is a variety with mild (terminal) singularities whose canonical divisor is nef.

## Explanations

### L1 human

Each line bundle on a variety has a degree on each curve inside the variety, an integer that counts zeros minus poles of a section restricted to the curve.
An ample bundle has positive degree on every curve.
A nef bundle is allowed degree zero on some curves, but never negative degree.
On the product $\mathbb{P}^1 \times \mathbb{P}^1$, pulling back $\mathcal{O}(1)$ from the first factor gives a nef bundle.
It has degree $1$ on the curves $\mathbb{P}^1 \times \set{q}$ and degree $0$ on the curves $\set{p} \times \mathbb{P}^1$, so it is nef but not ample.
</document>

<document source="https://tamnd.github.io/math2/node/RZHB962EMF.md" tag="RZHB962EMF" kind="definition">
# Contingency table

## Definition

Let $X$ and $Y$ be categorical [[variables|0PNRKMRC55]] with values in $\set{1, \ldots, r}$ and $\set{1, \ldots, c}$, observed on $n$ units.
The $r \times c$ *contingency table* of the sample is the matrix $N = (n_{ij})$, where $n_{ij}$ is the number of units with $X = i$ and $Y = j$.
The row totals are $n_{i+} = \sum_j n_{ij}$, the column totals are $n_{+j} = \sum_i n_{ij}$, and $\sum_{i,j} n_{ij} = n$.
A contingency table of $m \ge 3$ variables is defined in the same way as an $m$-way array of counts.

## Remarks

The cells correspond to a [[partition|08QMKTWF9T]] of the sample: each unit is counted in exactly one cell.
Example: among $100$ people, $10$ smoke and exercise, $20$ smoke and do not exercise, $40$ do not smoke and exercise, and $30$ do neither.
The table is $\begin{pmatrix} 10 & 20 \\ 40 & 30 \end{pmatrix}$, with row totals $30, 70$, column totals $50, 50$, and grand total $100$.
Non-example: in a survey where people may tick several sports, a table of persons by sport counts some people more than once.
Then the cells do not partition the sample, and the entries need not add up to $n$.
To test independence, the expected counts are $E_{ij} = n_{i+} n_{+j} / n$, and Pearson's statistic is $\sum_{i,j} (n_{ij} - E_{ij})^2 / E_{ij}$.
Under independence and for large $n$, it is approximately chi-squared with $(r - 1)(c - 1)$ degrees of freedom.
</document>

<document source="https://tamnd.github.io/math2/node/RZKC8B00HZ.md" tag="RZKC8B00HZ" kind="statement">
# An infinite product converges iff its series of logarithms converges

## Statement

Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers.
Then the infinite product $\prod_{n=1}^{\infty} a_n$ [[converges|H1Z2TKY3EQ]] if and only if the series $\sum_{n=1}^{\infty} \ln a_n$ [[converges|0CVVKY7GY2]], where $\ln$ is the [[natural logarithm|0044CPHJBV]].
In that case
$$\prod_{n=1}^{\infty} a_n = \exp\left( \sum_{n=1}^{\infty} \ln a_n \right).$$

## Remarks

The partial products and the partial sums are linked by $\prod_{n \le N} a_n = \exp\left(\sum_{n \le N} \ln a_n\right)$.
The map $\exp$ is a continuous bijection from $\R$ onto $(0, \infty)$, with continuous inverse $\ln$.
So the partial products tend to a positive limit exactly when the partial sums tend to a real limit.
The nonzero condition in the definition of convergence is what makes this work: partial sums tending to $-\infty$ correspond to products diverging to $0$.
A consequence: if $u_n \ge 0$, then $\prod (1 + u_n)$ converges if and only if $\sum u_n$ converges.
Indeed, if $u_n \to 0$ then $\ln(1 + u_n) / u_n \to 1$, and the [[limit comparison test|0HW7BNHHGF]] applies; if $u_n \not\to 0$, both diverge.
Example: $\prod (1 + 1/n^2)$ converges and $\prod (1 + 1/n)$ diverges.

## Explanations

### L1 human

Taking logarithms turns products into sums.
So an infinite product of positive numbers behaves exactly like the infinite series of their logarithms.
This lets the convergence tests for series decide convergence of products.

### L2 human

The statement reduces the theory of products of positive numbers to the theory of series.
Its most used form is the criterion for $\prod (1 + u_n)$ with $u_n \ge 0$, which compares the product with $\sum u_n$.
For complex factors one uses a branch of the logarithm near $1$, and absolute convergence of $\sum (a_n - 1)$ gives convergence of the product.
</document>

<document source="https://tamnd.github.io/math2/node/RZT4WMH4SG.md" tag="RZT4WMH4SG" kind="definition">
# Schauder basis

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $X$ be a [[Banach space|0BDTTM24DP]] over $\mathbb{K}$ with norm $\norm{\cdot}$.
A sequence $(e_n)_{n \ge 1}$ in $X$ is a *Schauder basis* of $X$ if the following holds.
For every $x \in X$ there is a unique sequence $(\alpha_n)_{n \ge 1}$ in $\mathbb{K}$ such that
$$\lim_{N \to \infty} \norm{x - \sum_{n=1}^{N} \alpha_n e_n} = 0,$$
that is, the [[series|WN6M8B3ZKB]] $\sum_n \alpha_n e_n$ converges to $x$ in $X$.

## Remarks

Example: in $\ell^p$ with $1 \le p < \infty$, see [[spaces ℓ^p|08P6T2WH6R]], the unit vectors $e_n$ (with $1$ in place $n$ and $0$ elsewhere) form a Schauder basis.
For $x \in \ell^p$ and $\alpha_n = x_n$, one has $\norm{x - \sum_{n \le N} x_n e_n}_p = \left( \sum_{n > N} \abs{x_n}^p \right)^{1/p} \to 0$.
The coefficients are unique, because a norm-convergent series converges coordinatewise, so the $n$-th coordinate of $\sum_k \alpha_k e_k$ is $\alpha_n$.

Non-example: in $\ell^\infty$ the unit vectors are not a Schauder basis.
For $x = (1, 1, 1, \dots)$ the only possible coefficients are $\alpha_n = 1$, and $\norm{x - \sum_{n \le N} e_n}_\infty = 1$ for every $N$.
In fact $\ell^\infty$ has no Schauder basis at all, because it is not [[separable|0Z2P41SVDP]].

The order of the basis matters, since the series need not converge after a rearrangement.
A Schauder basis of an infinite-dimensional Banach space is never a [[Hamel basis|0VTQTPQGAV]].
The coefficient functionals $x \mapsto \alpha_n$ are always continuous.
</document>

<document source="https://tamnd.github.io/math2/node/RZZ78GXP5F.md" tag="RZZ78GXP5F" kind="concept">
# Abel–Poisson summation of Fourier series

## Summary

Abel–Poisson summation applies [[Abel summation|EZXPY4A19C]] to the [[Fourier series|G12KXDHPPS]] of a periodic function.
The $k$-th term is damped by the factor $r^{\abs{k}}$, and $r$ tends to $1$ from below.
The damped series equals the convolution of the function with the [[Poisson kernel|TCYSWCTYYM]].
As a function of the point $r e^{ix}$ of the unit disc, the damped sum is a [[harmonic function|0DEB0JMN0W]].
Its boundary values, in the limit $r \to 1^-$, are the given function.
For a continuous function the Abel–Poisson means converge uniformly to it.

## Explanations

### L1 human

Write a periodic function as a sum of waves $e^{ikx}$.
Multiply the wave of frequency $k$ by $r^{\abs{k}}$ for some $r < 1$, which tames the high frequencies.
The result is a smooth function, and it is the steady temperature at radius $r$ in a round plate whose rim is held at the given function.
As $r$ goes to $1$, the smoothed function approaches the original one.
</document>

<document source="https://tamnd.github.io/math2/node/RZZK51CXMN.md" tag="RZZK51CXMN" kind="statement">
# Cramér's theorem

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|00MRN0FARV]] real random variables.
Assume that the logarithmic [[moment-generating function|0WQVXP7PDB]] $\Lambda(\lambda) = \log E[e^{\lambda X_1}]$ is finite for every $\lambda \in \R$.
Let $\Lambda^*(x) = \sup_{\lambda \in \R} (\lambda x - \Lambda(\lambda))$ be its convex conjugate, and let $\bar X_n = (X_1 + \cdots + X_n)/n$.
Then the laws of $\bar X_n$ satisfy the [[large deviation principle|MNPWBBNCAD]] on $\R$ with the good [[rate function|140N2PQSTE]] $\Lambda^*$.
In particular, for every $x > E[X_1]$,
$$\lim_{n \to \infty} \frac{1}{n} \log P(\bar X_n \ge x) = -\Lambda^*(x).$$

## Remarks

The upper bound is the Chernoff bound.
For $\lambda \ge 0$, Markov's inequality gives $P(\bar X_n \ge x) \le e^{-n \lambda x} E[e^{\lambda S_n}] = e^{-n(\lambda x - \Lambda(\lambda))}$, with $S_n = n \bar X_n$, and one optimises over $\lambda$.
The lower bound uses a change of measure, the tilted law $e^{\lambda y - \Lambda(\lambda)} \, P(X_1 \in dy)$ with $\lambda$ chosen so that its mean is $x$.
Under the tilted law the event $\bar X_n \approx x$ is typical, and the cost of the change of measure is $\Lambda^*(x)$.

The function $\Lambda^*$ is convex, non-negative, and equal to $0$ exactly at the mean $E[X_1]$.

Example: for standard normal $X_k$, $\Lambda(\lambda) = \lambda^2/2$ and $\Lambda^*(x) = x^2/2$.
Example: for fair coin tosses with values $0$ and $1$, $\Lambda^*(x) = x \log x + (1 - x) \log(1 - x) + \log 2$ on $[0, 1]$, and $\Lambda^* = \infty$ outside $[0, 1]$.

On $\R$ the large deviation principle holds for every i.i.d. sequence, with rate function $\Lambda^*$, but without the finiteness assumption $\Lambda^*$ need not be good.

## Explanations

### L1 human

The average of many independent copies of a random number is very likely close to the mean.
Cramér's theorem says how unlikely a given deviation is: the probability falls exponentially in the number of copies.
The exponent is computed from the moment-generating function by a Legendre transform.

### L2 human

The theorem is the first large deviation principle and the model for the others.
Its proof introduces the two main tools of the subject: exponential Chebyshev bounds for the upper bound and exponential tilting for the lower bound.
In $\R^d$ the same result holds with $\lambda x$ replaced by an inner product, and the Gärtner–Ellis theorem extends it to dependent sequences.
[[Sanov's theorem|QWEF2WAXJM]] is the corresponding result for empirical measures; Cramér's theorem for finitely valued variables can be deduced from it by the contraction principle.
</document>

<document source="https://tamnd.github.io/math2/node/S026HQSGY9.md" tag="S026HQSGY9" kind="concept">
# Zak transform

## Summary

The Zak transform turns a function on $\R$ into a function of two variables by periodizing it with phase factors.
It is a unitary map from $L^2(\R)$ onto $L^2([0, 1)^2)$, and its values are quasi-periodic.
It diagonalizes Gabor systems at critical density, which makes it the main tool behind the [[Balian–Low theorem|XVVHKCYF2Z]].
It has been discovered several times; Joshua Zak introduced it in solid-state physics.

## Explanations

### L1 human

Take a function on the line and cut it into pieces of length $1$.
Stack the pieces on top of each other, and add them up with phase factors $e^{-2\pi i k \omega}$ that depend on a second variable $\omega$.
The result depends on a position $x$ within one piece and on $\omega$.
It is a Fourier series in the index $k$ of the pieces, for each fixed $x$.
For the box function $1_{[0, 1)}$ there is just one piece, and the Zak transform is constant $1$ on the unit square.
</document>

<document source="https://tamnd.github.io/math2/node/S06VTCGCHN.md" tag="S06VTCGCHN" kind="definition">
# Nilpotent Lie group

## Definition

Let $G$ be a connected real or complex [[Lie group|7JBRGCVJRB]].
Its [[lower central series|0K98R7A4VC]] is $C^1 G = G$ and $C^{k+1} G = [G, C^k G]$ for $k \ge 1$, where $[G, H]$ is the subgroup generated by the commutators $g h g^{-1} h^{-1}$ with $g \in G$ and $h \in H$.
The group $G$ is a *nilpotent Lie group* if $C^{k} G = \set{e}$ for some $k \ge 1$, that is, if $G$ is a [[nilpotent group|0Q7MMQHC1P]].

## Remarks

A connected Lie group is nilpotent if and only if its Lie algebra $\mathfrak{g}$ is a nilpotent Lie algebra, that is, the series $\mathfrak{g} \supseteq [\mathfrak{g}, \mathfrak{g}] \supseteq [\mathfrak{g}, [\mathfrak{g}, \mathfrak{g}]] \supseteq \cdots$ reaches $0$.

Example: the group $N$ of real upper triangular $3 \times 3$ matrices with $1$ on the diagonal.
Writing $[a, b, c]$ for the matrix with entries $a$, $b$ above the diagonal and $c$ in the corner, a direct computation shows that the commutator of $[a, b, c]$ and $[a', b', c']$ is $[0, 0, ab' - a'b]$.
So $C^2 N = \set{[0, 0, c]}$, which is the centre of $N$, and $C^3 N = \set{e}$; $N$ is nilpotent.

Non-example: the group $G$ of affine maps $x \mapsto a x + b$ of $\R$ with $a > 0$, written $(a, b)$.
The commutator of $(a, 0)$ and $(1, t)$ is $(1, (a - 1)t)$, so $C^2 G$ is the group of all translations $(1, b)$, and then $C^3 G = [G, C^2 G]$ again contains every translation.
So $C^k G$ is the translation group for all $k \ge 2$ and never trivial: $G$ is solvable but not nilpotent.
</document>

<document source="https://tamnd.github.io/math2/node/S0DAGR1WN7.md" tag="S0DAGR1WN7" kind="example">
# Median algebra as a ternary algebra

## Example

A median algebra is a set $A$ with one [[ternary operation|WC47MFR6GW]] $m$, written $\langle x, y, z \rangle$, such that for all $x, y, z, w \in A$:

1. $\langle x, y, y \rangle = y$;
2. $\langle x, y, z \rangle = \langle z, x, y \rangle$;
3. $\langle x, y, z \rangle = \langle x, z, y \rangle$;
4. $\langle \langle x, w, y \rangle, w, z \rangle = \langle x, w, \langle y, w, z \rangle \rangle$.

So it is an algebra with a single operation of arity $3$, defined by identities.

The basic instance is a [[distributive lattice|0HM0HQCK6N]] $(L, \wedge, \vee)$ with
$$\langle x, y, z \rangle = (x \wedge y) \vee (y \wedge z) \vee (z \wedge x).$$
On a [[total order|0FRXBVVR03]] this is the middle one of the three elements.
For instance on $\Z$ with the usual order, $\langle 1, 5, 3 \rangle = 3$ and $\langle 4, 4, 9 \rangle = 4$.
On $\set{0, 1}$ it is the majority function, and on $\set{0, 1}^n$ it is the majority taken in each coordinate.
Laws 1 to 3 are clear for the majority function, and law 4 can be checked on $\set{0, 1}$ by going through the cases.

## Remarks

In a [[tree|062D9KXJB1]], the median of three vertices is the unique vertex that lies on all three paths between them, and this gives a median algebra on the vertex set.
By laws 1 to 3, $\langle x, x, y \rangle = \langle x, y, x \rangle = \langle y, x, x \rangle = x$.
So $m$ is a [[majority term|9879ZMW5AA]] for the variety of median algebras.
The operation $x - y + z$ on $\Z$ is a non-example: it fails law 1, since it gives $x - y + y = x$, which differs from $y$ when $x \ne y$.

## Explanations

### L1 human

Given three numbers, the median picks the one in the middle.
A median algebra keeps only this operation and the rules it obeys.
If two of the three inputs agree, the answer is that common value, and the order of the inputs does not matter.
The same rules hold for the majority vote of three bits, and for the meeting point of three towns joined by roads that form a tree.
</document>

<document source="https://tamnd.github.io/math2/node/S0QKEJQ1WM.md" tag="S0QKEJQ1WM" kind="definition">
# Mean-field game

## Definition

Let $\mathbb{T}^d = \R^d / \Z^d$ be the torus, let $T > 0$, and let $\nu > 0$ be a constant.
Let $\mathcal{P}$ be the set of probability densities on $\mathbb{T}^d$.
Let $H \colon \mathbb{T}^d \times \R^d \to \R$ be a smooth function, convex in its second variable, the Hamiltonian.
Let $F, G \colon \mathbb{T}^d \times \mathcal{P} \to \R$ be given coupling functions, and let $m_0 \in \mathcal{P}$ be the initial distribution.
The *mean-field game system* is the system
$$\begin{aligned} -\partial_t u - \nu \Delta u + H(x, \nabla u) &= F(x, m(t)), \\ \partial_t m - \nu \Delta m - \nabla \cdot \big( m \, \partial_p H(x, \nabla u) \big) &= 0, \\ m(0, x) = m_0(x), \qquad u(T, x) &= G(x, m(T)), \end{aligned}$$
for unknown functions $u, m \colon [0, T] \times \mathbb{T}^d \to \R$.
Here $\Delta$ is the [[Laplacian|0E0B6RA8K0]], $\nabla$ the [[gradient|0ABK9V9Y64]] and $\nabla \cdot$ the [[divergence|0MWV6ZTK6S]] in $x$, and $\partial_p H$ is the gradient of $H$ in its second variable.
A solution $(u, m)$ is a *mean-field game equilibrium*.
In it, $u$ is the value function of a typical player and $m(t)$ is the distribution of the players at time $t$.

## Remarks

The first equation is a [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] solved backward from time $T$.
The second is a [[Fokker–Planck equation|FVQWR7NS00]] solved forward from time $0$, for players who use the optimal feedback $-\partial_p H(x, \nabla u)$ with noise of intensity $\nu$.

Example: let $H(x, p) = \tfrac{1}{2} \abs{p}^2$ and $F = G = 0$.
Then $u = 0$ and $m$ equal to the solution of the heat equation $\partial_t m = \nu \Delta m$ with $m(0) = m_0$ form a solution.
The first equation and the terminal condition hold since $u = 0$.
The second equation reduces to the heat equation because $\partial_p H(x, 0) = 0$.

Non-example: in the same setting, if $m_0$ is not constant, then $u = 0$ and $m(t, x) = m_0(x)$ for all $t$ is not a solution, because $\partial_t m - \nu \Delta m = -\nu \Delta m_0 \ne 0$.

Lasry and Lions proved existence of solutions under regularity assumptions, and uniqueness when $F$ and $G$ are monotone.
</document>

<document source="https://tamnd.github.io/math2/node/S0TGRX3ARS.md" tag="S0TGRX3ARS" kind="concept">
# Dehn invariant

## Summary

The Dehn invariant of a [[polyhedron|0325SGSJAR]] combines the lengths of its edges with their [[dihedral angles|05HN08ZSRJ]] in a tensor product of abelian groups.
It does not change when a polyhedron is cut into pieces and the pieces are reassembled.
Max Dehn used it in 1900 to solve [[Hilbert's third problem|81035XQ5HE]].
A cube and a regular tetrahedron of the same volume have different Dehn invariants, so neither can be cut into the other.

## Explanations

### L1 human

When you cut a solid into pieces, new edges appear.
Around a new edge inside the solid, the angles of the pieces add up to a full turn or a half turn.
The Dehn invariant adds up "edge length times angle" over all edges, but counts angles only up to rational multiples of $\pi$.
With this rule the new edges contribute nothing.
A cube has only right angles, which are rational multiples of $\pi$, so its invariant is $0$.
A regular tetrahedron has the angle $\arccos(1/3)$, which is not, so its invariant is not $0$.
</document>

<document source="https://tamnd.github.io/math2/node/S17DY98VF2.md" tag="S17DY98VF2" kind="statement">
# Seifert–van Kampen theorem

## Statement

Let $X$ be a topological space with $X = U \cup V$, where $U$ and $V$ are open in $X$.
Assume that $U$, $V$ and $U \cap V$ are [[path-connected|0WB6DH499T]], and let $x_0 \in U \cap V$.
Write $\pi_1(-)$ for $\pi_1(-, x_0)$.
Let $i_U \colon \pi_1(U \cap V) \to \pi_1(U)$, $i_V \colon \pi_1(U \cap V) \to \pi_1(V)$, $j_U \colon \pi_1(U) \to \pi_1(X)$ and $j_V \colon \pi_1(V) \to \pi_1(X)$ be maps induced by inclusions.
They are [[homomorphisms|0JYTG5K9Y4]] of [[fundamental groups|S1XK3HXHFX]].
Let $\Phi \colon \pi_1(U, x_0) * \pi_1(V, x_0) \to \pi_1(X, x_0)$ be the homomorphism from the [[free product|TM7QS7P7X9]] that restricts to $j_U$ and $j_V$.
Then $\Phi$ is surjective, and its kernel is the [[normal subgroup|0CB0DX5BVV]] generated by the elements $i_U(\omega)\, i_V(\omega)^{-1}$ for $\omega \in \pi_1(U \cap V, x_0)$.
Hence
$$\pi_1(X, x_0) \cong \pi_1(U, x_0) *_{\pi_1(U \cap V, x_0)} \pi_1(V, x_0),$$
the free product with amalgamation.

## Remarks

Surjectivity comes from cutting a loop in $X$ into pieces that each lie in $U$ or in $V$, by a Lebesgue number argument.
The cut points are then joined to $x_0$ by paths inside $U \cap V$.
The description of the kernel comes from cutting a path homotopy in $X$ into small squares in the same way.
Example: for the figure eight $S^1 \vee S^1$, let $U$ be the wedge minus a point of the first circle, and $V$ the wedge minus a point of the second circle.
Each of them deformation retracts onto a circle, and $U \cap V$ is contractible.
So $\pi_1(S^1 \vee S^1)$ is the free group on two generators.
Example: for $n \ge 2$, cover $S^n$ by the complements of the two poles; both sets are contractible and their intersection is path-connected, so $S^n$ is simply connected.
The hypothesis that $U \cap V$ is path-connected is needed: for $S^1$ covered by two open arcs, the intersection has two components, both arcs are contractible, and yet $\pi_1(S^1) \cong \Z$ is not trivial.

## Explanations

### L1 human

The theorem computes the loops of a space glued from two overlapping pieces.
Each loop in the whole space is made of loops in the two pieces.
The only new relations come from loops that live in the overlap, since such a loop can be seen in either piece.

### L2 human

Together with the [[fundamental group of the circle|6SQXPE0JPN]], the theorem computes the fundamental group of every connected CW complex from its $2$-skeleton.
Attaching a $2$-cell adds one relation.
The result is a [[presentation|0JGEQXQ4AC]] with one generator for each $1$-cell outside a maximal tree and one relation for each $2$-cell.
It gives the fundamental groups of closed surfaces and of knot complements.
A version with the [[fundamental groupoid|ZBHGXJF24E]] and a set of base points removes the condition that $U \cap V$ is path-connected.
</document>

<document source="https://tamnd.github.io/math2/node/S1BHE358VX.md" tag="S1BHE358VX" kind="statement">
# Erdős–Rado theorem

## Statement

For [[cardinals|0QKPNZKNEA]] $\kappa$, $\lambda$, $\mu$ and an integer $n \ge 1$, write $\kappa \to (\lambda)^n_\mu$ for the following property.
Let $X$ be a set of size $\kappa$, and let $c$ map the $n$-element subsets of $X$ to $\mu$.
Then some $H \subseteq X$ of size $\lambda$ has all its $n$-element subsets of the same colour under $c$.
For an infinite cardinal $\kappa$, let $\beth_0(\kappa) = \kappa$ and $\beth_{m+1}(\kappa) = 2^{\beth_m(\kappa)}$, as for the [[beth numbers|ZC4BHVHWE9]].
Then, for each infinite cardinal $\kappa$ and each integer $n \ge 0$,
$$\beth_n(\kappa)^+ \to (\kappa^+)^{n+1}_\kappa.$$

## Remarks

The case $n = 0$ is the pigeonhole principle: a map from $\kappa^+$ points to $\kappa$ colours is constant on $\kappa^+$ points.
The case $n = 1$ reads $(2^\kappa)^+ \to (\kappa^+)^2_\kappa$, and for $\kappa = \aleph_0$ it gives $(2^{\aleph_0})^+ \to (\aleph_1)^2_{\aleph_0}$.
The proof goes by induction on $n$.
It builds a long sequence in which the colour of an $(n+1)$-element set depends only on its $n$ smallest elements, and then applies the case $n - 1$.
The bound is sharp for $n = 1$: by a theorem of Sierpiński, $2^{\aleph_0} \not\to (\aleph_1)^2_2$, so $2^{\aleph_0}$ points do not suffice even for two colours.
The theorem extends the infinite [[Ramsey theorem|00TQV3F5S4]] $\aleph_0 \to (\aleph_0)^n_k$ to uncountable homogeneous sets.

## Explanations

### L1 human

Colour the pairs of points of a set with a small number of colours.
Ramsey's theorem says that an infinite set always contains an infinite set whose pairs all have one colour.
For uncountable sets this is no longer free: to get an uncountable one-coloured set, the starting set must be large.
The Erdős–Rado theorem says how large is enough: more than continuum many points always suffice for pairs.

### L2 human

The notation $\kappa \to (\lambda)^n_\mu$ is the arrow notation of the partition calculus of Erdős and Rado.
The theorem is the basic positive result of uncountable Ramsey theory, and Sierpiński's colouring shows that it cannot be improved for pairs.
It is used in model theory, for example to find indiscernible sequences in large models, and in the study of large cardinals defined by partition properties.
</document>

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# Oka principle

## Statement

Let $X$ be a [[Stein manifold|HMCD0JBFST]].
Then two [[holomorphic line bundles|QAC7BYK9KX]] on $X$ are holomorphically isomorphic if and only if they are isomorphic as topological complex line bundles.
Moreover, every topological complex line bundle on $X$ is isomorphic to a holomorphic one.
Equivalently, the first Chern class map $c_1 \colon H^1(X, \mathcal{O}^*) \to H^2(X, \Z)$ is a group isomorphism.

## Remarks

The proof uses the exponential sequence $0 \to \Z \to \mathcal{O} \to \mathcal{O}^* \to 0$, where the second map is $f \mapsto e^{2\pi i f}$.
Its long exact [[cohomology|PQ6Q143ARJ]] sequence contains $H^1(X, \mathcal{O}) \to H^1(X, \mathcal{O}^*) \to H^2(X, \Z) \to H^2(X, \mathcal{O})$.
By [[Cartan's theorem B|R7VKRV8AGQ]], the two outer groups are zero, so the middle map, which is $c_1$, is an isomorphism.
Topological complex line bundles are classified by $H^2(X, \Z)$ through their Chern class, which gives the first two sentences.
Oka proved this result in 1939, in the language of the second Cousin problem.
In that form it says that on a domain of holomorphy, a second Cousin problem is solvable holomorphically if and only if it is solvable continuously.
The name Oka principle is also used for a general heuristic.
On Stein spaces, holomorphic problems with only topological obstructions should be solvable holomorphically as soon as they are solvable continuously.
The Stein hypothesis is needed.
On a compact complex torus of dimension $1$, the holomorphic line bundles of degree $0$ are all topologically trivial, but they form a family parametrized by the torus itself.

## Explanations

### L1 human

On a Stein manifold, a holomorphic line bundle is determined, up to isomorphism, by its topology alone.
So a question about holomorphic objects, which looks hard, is the same as a question about continuous objects, which is a question of algebraic topology.
This surprising rigidity is the original Oka principle.

### L2 human

The result solves the second Cousin problem on Stein manifolds: it is solvable exactly when the associated line bundle has Chern class zero, in particular always when $H^2(X, \Z) = 0$.
Grauert extended the principle from line bundles to vector bundles and principal bundles with a complex Lie group as structure group.
Gromov and Forstnerič extended it further to sections of holomorphic submersions and to maps into Oka manifolds.
</document>

<document source="https://tamnd.github.io/math2/node/S1FB2K14NV.md" tag="S1FB2K14NV" kind="statement">
# Uncertainty principle in Fourier analysis

## Statement

Let $n \ge 1$ and let $f \in L^1(\R^n)$.
Let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\, dx$ be its [[Fourier transform|DPVM34ADZV]].
Suppose that $f$ vanishes almost everywhere outside a bounded set and that $\hat f$ vanishes outside a bounded set.
Then $f = 0$ almost everywhere.

## Remarks

If $f$ vanishes outside a ball of radius $R$, then the integral $\int f(x) e^{-2\pi i x \cdot z}\, dx$ makes sense for every $z \in \C^n$.
It defines an [[entire function|0G4F0XB9X2]] of $z$ that extends $\hat f$.
An entire function that vanishes on a nonempty open subset of $\R^n$ vanishes identically, so $\hat f = 0$, and then $f = 0$ by injectivity of the transform.

The statement is the simplest of a family of results that say a nonzero function and its Fourier transform cannot both be sharply concentrated.
The Heisenberg inequality bounds the product of the variances of $\abs{f}^2$ and $\abs{\hat f}^2$ from below.
Hardy's uncertainty principle is about Gaussian decay, and the entropic uncertainty principle measures spread by entropy.
Benedicks' theorem replaces bounded sets by sets of finite measure.

Example: $\operatorname{rect}$, the indicator of $[-\frac12, \frac12]$, has compact support, and its transform $\sin(\pi\xi)/(\pi\xi)$ has zeros only at the nonzero integers, so it does not vanish outside any bounded set.

## Explanations

### L1 human

A signal cannot be both short in time and limited to a finite band of frequencies.
If it lasts only a finite time, it must contain arbitrarily high frequencies, and if all its frequencies lie in a bounded band, it must go on forever.

### L2 human

The proof uses the Paley–Wiener idea that compact support in $x$ makes $\hat f$ analytic, indeed entire of exponential type.
Analyticity is very rigid, and this rigidity is the source of all qualitative uncertainty principles.
In signal processing the statement says that no nonzero band-limited signal has finite duration, so every real filter or sampling scheme is a compromise between the two.
</document>

<document source="https://tamnd.github.io/math2/node/S1G61TDW0K.md" tag="S1G61TDW0K" kind="concept">
# Exact functor

## Summary

An exact functor is an [[additive functor|KAN07YC9T2]] between [[abelian categories|Y9CZ3MTNF1]] that sends [[short exact sequences|KJKNJMGKZZ]] to short exact sequences.
Many natural functors, such as $\Hom(M, -)$ and $M \otimes -$, preserve only part of the exactness; they are called left exact or right exact.
Measuring how far such a functor is from being exact is the purpose of derived functors such as [[Ext|4P7GBWJ4RR]] and [[Tor|DDQRERAQCV]].

## Explanations

### L1 human

An exact sequence records how an object is built from a subobject and a quotient.
An exact functor carries this information over without loss: the image of a subobject stays a subobject, and the image of the quotient stays the quotient.
For example, tensoring abelian groups with $\Q$ is exact.
Tensoring with $\Z/2\Z$ is not: it turns the injective map "multiply by $2$" on $\Z$ into the zero map on $\Z/2\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/S1PG1N2VEX.md" tag="S1PG1N2VEX" kind="statement">
# Kirchhoff's matrix tree theorem

## Statement

Let $G$ be a finite [[multigraph|0C62PZXX02]] without loops, with vertices $v_1, \dots, v_n$ and $n \ge 2$.
Let $D$ be the diagonal matrix of the vertex degrees.
Let $A$ be the matrix with $A_{ii} = 0$ and $A_{ij}$ the number of edges between $v_i$ and $v_j$ for $i \ne j$.
Let $L = D - A$ be the [[Laplacian matrix|TN6M6DVZRP]] of $G$.
For each $i$, let $L_i$ be the matrix obtained from $L$ by deleting row $i$ and column $i$.
Then the number of [[spanning trees|0S4ZEZM7KE]] of $G$ equals $\det L_i$, for every $i \in \set{1, \dots, n}$.

## Remarks

Orient each edge and let $B$ be the $n \times m$ incidence matrix, with entries $+1$ and $-1$ at the head and tail of each edge; then $L = B B^T$.
Delete row $i$ of $B$ to get $B_i$, so that $L_i = B_i B_i^T$.
By the [[Cauchy–Binet formula|08WE2W7DYS]], $\det L_i$ is the sum of $(\det N)^2$ over the $(n-1) \times (n-1)$ submatrices $N$ of $B_i$.
Such a minor is $\pm 1$ if its $n - 1$ edges form a spanning tree and $0$ otherwise, which gives the count.

Example: for the triangle $K_3$, every vertex has degree $2$ and
$$L_3 = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}, \qquad \det L_3 = 3,$$
which matches the $3$ spanning trees obtained by removing one edge.
For the complete graph $K_n$ the theorem gives $n^{n-2}$, which is [[Cayley's formula|0M2SSGSYPE]].
If $G$ is not connected, both sides are $0$.

## Explanations

### L1 human

Counting spanning trees by hand is hard even for small graphs.
The theorem gives a recipe.
Write the degree of each vertex on the diagonal and $-1$ for each edge off the diagonal.
Cross out one row and the matching column, and take the [[determinant|00KDZRW4C0]].
The number you get is the number of spanning trees, whichever row you crossed out.

### L2 human

An equivalent form uses the eigenvalues $0 = \mu_1 \le \mu_2 \le \dots \le \mu_n$ of $L$: the number of spanning trees is $\frac{1}{n} \mu_2 \cdots \mu_n$.
This links spectral graph theory to enumeration.
The theorem goes back to Kirchhoff's study of electrical networks.
It has a weighted version, where each edge carries a variable and the determinant sums the weights of the spanning trees, and a version for directed graphs that counts arborescences.
</document>

<document source="https://tamnd.github.io/math2/node/S1WS18QEGJ.md" tag="S1WS18QEGJ" kind="example">
# Quintic equation

## Example

A quintic equation is an equation $ax^5 + bx^4 + cx^3 + dx^2 + ex + g = 0$ with $a \ne 0$.
Consider the quintic $f(x) = x^5 - 4x + 2$ over $\Q$.

1. $f$ is irreducible over $\Q$ by [[Eisenstein's criterion|02BPKD7TMD]] at $2$, as $2$ divides $-4$ and $2$ but not $1$, and $4$ does not divide $2$.
2. $f(-2) = -22$, $f(0) = 2$, $f(1) = -1$ and $f(2) = 26$, so by the [[intermediate value theorem|00KAFBR32E]] $f$ has at least three real roots.
3. $f'(x) = 5x^4 - 4$ has exactly two real zeros, so by [[Rolle's theorem|0194DY35BG]] $f$ has at most three real roots.
4. Hence $f$ has exactly three real roots and two nonreal roots.

By the [[theorem on prime degree polynomials with two nonreal roots|4CMPPTSJNP]], the Galois group of $f$ over $\Q$ is $S_5$.
Since [[$S_5$ is not solvable|0J53NE674D]], [[Galois' theorem on solvability|VHCWCFA3MX]] shows that $f$ is not [[solvable by radicals|J7YS0DA3HQ]].
By contrast, $x^5 - 2$ is solvable by radicals: its roots are $\sqrt[5]{2}\, \zeta^k$, with $\zeta = e^{2\pi i/5}$ and $k = 0, \dots, 4$.

## Explanations

### L1 human

The equation $x^5 - 4x + 2 = 0$ has five complex solutions, and they can be computed to any accuracy.
But none of them can be written using only whole numbers, arithmetic and $n$-th roots.
This concrete equation shows that the [[Abel–Ruffini theorem|163A2WQQJX]] is about specific equations, not only about a general formula.
</document>

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# Fundamental group

## Summary

The fundamental group $\pi_1(X, x_0)$ of a [[pointed space|T9DG8DDE75]] is the [[group|G5S1M9]] of loops at $x_0$ up to [[path homotopy|CCNKA8C711]], with concatenation of loops as product.
It is the first invariant of algebraic topology that detects one-dimensional holes.
Homotopy equivalent path-connected spaces have isomorphic fundamental groups.
For example, the plane has trivial fundamental group, while the [[circle|060W5CBZP6]] has fundamental group $\Z$.

## Explanations

### L1 human

Stand at a point of a space and walk along loops that come back to where you started.
Two loops count as the same if one can be deformed into the other while the start stays fixed.
Doing one loop after another is the product.
On a circle, a loop is classified by how many times it goes around, counted with sign, so the group is the integers.
</document>

<document source="https://tamnd.github.io/math2/node/S26GYRVDR5.md" tag="S26GYRVDR5" kind="example">
# Octonionic projective plane

## Example

Let $\mathbb{O}$ be the [[octonions|1HRMT3DSX2]], the $8$-dimensional real division algebra that is alternative but not associative.
The *octonionic projective plane* $\mathbb{O}P^2$, also called the Cayley plane, has the following points and lines.

1. Affine points $(x, y)$ with $x, y \in \mathbb{O}$, points $(m)$ with $m \in \mathbb{O}$, and one more point $(\infty)$.
2. Lines $[m, b] = \set{(x, y) : y = x m + b} \cup \set{(m)}$ for $m, b \in \mathbb{O}$, lines $[c] = \set{(c, y) : y \in \mathbb{O}} \cup \set{(\infty)}$ for $c \in \mathbb{O}$, and the line at infinity $\set{(m) : m \in \mathbb{O}} \cup \set{(\infty)}$.

This is a [[projective plane|DANFNN8C4Y]].
The key check concerns two affine points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \ne x_2$.
They lie on a line $[m, b]$ exactly when $(x_1 - x_2) m = y_1 - y_2$ and $b = y_1 - x_1 m$.
In an alternative division algebra, $a m = c$ with $a \ne 0$ has the unique solution $m = a^{-1} c$, since $a (a^{-1} c) = c$.

It is a [[Moufang plane|7F4HSNJJH8]], since it is coordinatized by the alternative division ring $\mathbb{O}$.
It is not a [[Desarguesian plane|HPX85PA8NX]]: a Desarguesian plane is coordinatized by an associative division ring, and $\mathbb{O}$ is not associative.

## Remarks

As a topological space $\mathbb{O}P^2$ is a compact manifold of dimension $16$.
For its natural Riemannian metric, its group of isometries is the compact Lie group $F_4$.
Its group of continuous collineations is a real form of $E_6$.
There is no octonionic projective space of dimension $3$ or more.
Indeed, [[Desargues's theorem|5Z19Y02MME]] holds in every projective space of dimension at least $3$, which would force the coordinates to be associative.

## Explanations

### L1 human

The real, complex and quaternionic numbers each give a projective plane, built from lines through the origin.
The octonions are a larger number system whose multiplication is not associative, so the usual construction with lines through the origin breaks down.
Still, one can write down points and lines with octonion coordinates, and they obey the rules of a projective plane.
This plane is special: it satisfies a weak form of Desargues's theorem but not the full theorem, and there is no octonionic projective space of dimension $3$ or more.
</document>

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# Tessellation

## Definition

Let $n \ge 1$ and let $\R^n$ be [[Euclidean space|0XPR1E85W0]].
A *tessellation* or *tiling* of $\R^n$ is a countable family $\mathcal{T} = \set{T_1, T_2, \dots}$ of subsets of $\R^n$, called *tiles*, such that:

1. each tile $T_i$ is a [[closed set|0ZCK8DQYN1]] with nonempty [[interior|0RZH6W98HK]];
2. the union of all tiles is $\R^n$;
3. the interiors of two distinct tiles are disjoint.

## Remarks

Example: the squares $T_{a,b} = [a, a+1] \times [b, b+1]$ with $(a, b) \in \Z^2$ form a tiling of $\R^2$.
Each square is closed with nonempty interior, every point $(x, y)$ lies in $T_{\floor{x}, \floor{y}}$, and two distinct open squares $(a, a+1) \times (b, b+1)$ do not meet.

Non-example: the closed unit disks centred at the points of $\Z^2$ cover $\R^2$, but the disks centred at $(0, 0)$ and $(1, 0)$ have overlapping interiors, so clause 3 fails.

Many authors also require each tile to be homeomorphic to a closed ball, so that tiles are "topological disks" in the plane.
A *prototile* set of a tiling is a set of shapes such that each tile is congruent to one of them.
In dimension $3$ and higher, tilings by polyhedra are also called [[honeycombs|HTJ03RFKFH]].
</document>

<document source="https://tamnd.github.io/math2/node/S2BBZPAFDR.md" tag="S2BBZPAFDR" kind="concept">
# Optimal control of partial differential equations

## Summary

Optimal control of partial differential equations studies how to choose a control, such as a heat source or a boundary temperature.
The aim is that the solution of a [[partial differential equation|GC5BDZ6QBC]] is as good as possible for a given cost.
The state lives in an infinite-dimensional function space, such as a [[Sobolev space|M4CEZ5K9HP]].
It is the infinite-dimensional branch of [[optimal control|T3V6ZNAMDN]] and is also called PDE-constrained optimization.
Its main tools are existence theory in function spaces, optimality systems with adjoint equations, and numerical methods such as the [[adjoint state method|YD8SA6WS01]].

## Explanations

### L1 human

Think of a metal plate heated by adjustable burners underneath.
You want the plate to reach a given temperature profile, while spending little energy on the burners.
The temperature obeys the heat equation, a partial differential equation, and the burner settings are the control.
Choosing the best burner settings is an optimal control problem for a partial differential equation.
</document>

<document source="https://tamnd.github.io/math2/node/S2CV71FCY2.md" tag="S2CV71FCY2" kind="definition">
# Suspension flow

## Definition

Let $X$ be a topological space, let $T \colon X \to X$ be a [[homeomorphism|0QT00PB5G2]], and let $r \colon X \to (0, \infty)$ be a [[continuous function|0X3WGX95FE]], the *roof function*, with $\inf_X r > 0$.
On $X \times \R$, let $\sim$ be the equivalence relation generated by
$$(x, s + r(x)) \sim (T(x), s) \quad \text{for all } x \in X,\ s \in \R.$$
Let $X_r = (X \times \R) / \sim$ with the [[quotient topology|0NQ1H2X495]], and write $[x, s]$ for the class of $(x, s)$.
The *suspension flow* of $T$ under $r$ is the flow $(\varphi_t)_{t \in \R}$ on $X_r$ given by
$$\varphi_t([x, s]) = [x, s + t].$$

## Remarks

The flow is well defined because the generating relation commutes with the translations $(x, s) \mapsto (x, s + t)$.
Every class has a representative $(x, s)$ with $0 \le s < r(x)$.
So $X_r$ can be pictured as the region under the graph of $r$, with the top point $(x, r(x))$ glued to the bottom point $(T(x), 0)$.
For $0 \le s < r(x)$ and $t \ge 0$, the point $\varphi_t([x, s])$ equals $[T^n(x), s + t - \sum_{k=0}^{n-1} r(T^k(x))]$, where $T^k$ is the $k$-th [[iterate|AYVTR2ES5B]] and $n \ge 0$ is the number of times the orbit has hit the roof.
The first-return map of the flow to the base $\set{[x, 0] : x \in X}$ is $T$, and the return time of $[x, 0]$ is $r(x)$.
If $\mu$ is a $T$-invariant measure, the product of $\mu$ with Lebesgue measure on the region under the graph of $r$ is invariant under the flow.

Example: let $X = \R / \Z$, $T(x) = x + \alpha \bmod 1$ and $r = 1$.
The map $[x, s] \mapsto (x + \alpha s, s) \bmod \Z^2$ is well defined, since $(x, s + 1)$ and $(x + \alpha, s)$ both go to $(x + \alpha s + \alpha, s) \bmod \Z^2$.
It identifies $X_r$ with the torus $\R^2 / \Z^2$, and it carries the suspension flow to the linear flow $(u, v) \mapsto (u + \alpha t, v + t)$ in the direction $(\alpha, 1)$.

The condition $\inf_X r > 0$ makes the sums $\sum_{k=0}^{n-1} r(T^k(x))$ and $\sum_{k=1}^{n} r(T^{-k}(x))$ tend to infinity, which is what gives every class a representative under the roof.
Non-example: the doubling map $x \mapsto 2x \bmod 1$ is not a homeomorphism, so the construction above does not apply to it; for non-invertible maps one uses a suspension semiflow, defined only for $t \ge 0$.
With $r = 1$, the space $X_r$ is the mapping torus of $T$.
</document>

<document source="https://tamnd.github.io/math2/node/S2D8823P8B.md" tag="S2D8823P8B" kind="concept">
# Minimal prime ideal

## Summary

A minimal prime ideal of a commutative ring is a [[prime ideal|06HC6HK0Z8]] that contains no smaller prime ideal.
Every prime ideal contains a minimal prime, and the [[nilradical|QQDN6EYQNJ]] is the intersection of the minimal primes.
Geometrically, the minimal primes correspond to the irreducible components of the [[spectrum|FB45BHHA2J]].
A Noetherian ring has only finitely many minimal primes.

## Explanations

### L1 human

Prime ideals can be ordered by inclusion, and the minimal ones sit at the bottom.
In an integral domain such as $\Z$, the zero ideal is prime, so it is the only minimal prime.
In the ring $k[x, y]/(xy)$, which describes the union of the two coordinate axes, there are two minimal primes, generated by $x$ and by $y$.
They correspond to the two axes, the two irreducible pieces of the picture.
</document>

<document source="https://tamnd.github.io/math2/node/S2GBXQ26QM.md" tag="S2GBXQ26QM" kind="concept">
# Karoubi envelope

## Summary

The Karoubi envelope of a category $\mathcal{C}$ is the category obtained by formally adding an image for every [[idempotent|0N5NJZKKN3]] morphism.
Its objects are pairs of an object and an idempotent endomorphism of it.
It is the smallest category containing $\mathcal{C}$ in which every idempotent splits, and it is also called the idempotent completion.
For example, the Karoubi envelope of the category of finitely generated free modules over a ring is the category of finitely generated projective modules.

## Explanations

### L1 human

An idempotent map $e$ satisfies $e \circ e = e$, like a projection onto a subspace.
In vector spaces such a projection has an image, and $e$ factors through it.
In some categories the image is missing, and the Karoubi envelope adds it as the formal object $(A, e)$.
In this way projective modules appear as images of projections on free modules.
</document>

<document source="https://tamnd.github.io/math2/node/S2J9CBPJ0V.md" tag="S2J9CBPJ0V" kind="statement">
# Minlos theorem

## Statement

Let $E$ be a real [[nuclear space|XC6XPMC2FQ]] that is also a Fréchet space, and let $E'$ be its topological [[dual space|07JBSZE0V4]].
Give $E'$ the σ-algebra generated by the maps $x \mapsto x(f)$ for $f \in E$.
Let $\varphi \colon E \to \C$ be a function.
Then there is a [[probability measure|059TJN0NZ1]] $\mu$ on $E'$ with
$$\varphi(f) = \int_{E'} e^{i x(f)} \, d\mu(x) \quad \text{for all } f \in E$$
if and only if these three conditions hold:

1. $\varphi$ is continuous.
2. $\varphi$ is [[positive-definite|F73VWGF637]]: $\sum_{j,k=1}^n c_j \overline{c_k} \, \varphi(f_j - f_k) \ge 0$ for all $n$, all $f_1, \dots, f_n \in E$ and all $c_1, \dots, c_n \in \C$.
3. $\varphi(0) = 1$.

When $\mu$ exists, it is unique.

## Remarks

The function $\varphi$ is the characteristic functional of $\mu$, the infinite-dimensional analogue of a [[characteristic function|0V5B6D6BQH]].
Necessity of the three conditions is direct.
For sufficiency, [[Bochner's theorem|ZRHAG4NXHC]] on each finite-dimensional quotient gives a [[cylinder set measure|G95YDNFXQN]] on $E'$.
Nuclearity of $E$ is what makes this cylinder set measure countably additive.

Example: on the Schwartz space $E = \mathcal{S}(\R^n)$, the function $\varphi(f) = \exp\left(-\frac12 \int_{\R^n} f(t)^2 \, dt\right)$ satisfies the three conditions.
The measure it gives on the space of tempered distributions $\mathcal{S}'(\R^n)$ is white noise.

Nuclearity cannot be dropped.
On an infinite-dimensional real Hilbert space $H$, the function $\exp(-\frac12 \norm{f}^2)$ is continuous, positive-definite and equal to $1$ at $0$.
Yet there is no probability measure on $H' \cong H$ with this characteristic functional.

## Explanations

### L1 human

A probability measure on $\R^n$ is determined by its characteristic function, and Bochner's theorem says which functions occur.
Minlos's theorem gives the same answer in infinite dimensions, for the dual of a nuclear space.
So to build a random distribution, it suffices to write down a suitable characteristic functional.

### L2 human

The theorem is the standard way to construct Gaussian and other random fields as random elements of $\mathcal{S}'(\R^n)$.
White noise, the Gaussian free field and Euclidean quantum fields are built this way.
It contrasts with the Hilbert space case, where by Sazonov's theorem continuity in a stronger topology is needed.
</document>

<document source="https://tamnd.github.io/math2/node/S2JE8M3FMP.md" tag="S2JE8M3FMP" kind="concept">
# Schlegel diagram

## Summary

A Schlegel diagram is a picture of a $d$-dimensional [[convex polytope|300K6CB1KH]] inside one of its [[facets|DPDXN2FJC4]], one dimension lower.
It is obtained by looking at the polytope from a point just outside that facet and projecting the rest of the boundary onto it.
The result is a subdivision of the facet into convex polytopes, one for each other facet, that keeps the full face structure.
Schlegel diagrams make $4$-dimensional polytopes visible in space and give a quick proof that graphs of $3$-polytopes are [[planar|0NC9R64TXP]].

## Explanations

### L1 human

Put your eye very close to one face of a cube and look through it.
You see the far face as a small square inside the near one, joined to it by the four side faces seen as trapezoids.
That drawing, a square inside a square with the corners joined, is the Schlegel diagram of the cube.
The same trick shows a four-dimensional cube as a small cube inside a big cube, with six squashed cubes in between.
</document>

<document source="https://tamnd.github.io/math2/node/S2KXCAH5MM.md" tag="S2KXCAH5MM" kind="concept">
# Socle of a module

## Summary

The socle of a module is the sum of all its [[simple submodules|G00C22AEDH]].
It is the largest [[semisimple submodule|N3MRHDKDGC]] of the module.
It measures the part of a module that is built directly from simple pieces.
It is a basic tool in the study of Artinian rings and injective modules.

## Explanations

### L1 human

Simple modules are the atoms of module theory: they have no proper nonzero submodules.
The socle collects every atom that sits at the bottom of a module.
Picture a module as a building: the socle is its ground floor.
For example, in the $\Z$-module $\Z/12\Z$ the simple submodules are $\set{0, 6}$ and $\set{0, 4, 8}$, and their sum is the socle, the subgroup generated by $2$, which has $6$ elements.
</document>

<document source="https://tamnd.github.io/math2/node/S2NAGK9YHF.md" tag="S2NAGK9YHF" kind="example">
# Affine plane over a finite field

## Example

Let $q$ be a prime power and let $\F_q$ be the [[finite field|0ZXZ82FTX2]] with $q$ elements.
The plane $\mathrm{AG}(2, q)$ has as points the pairs $(x, y) \in \F_q^2$.
Its lines are the sets $\set{(x, y) : ax + by = c}$ with $a, b, c \in \F_q$ and $(a, b) \ne (0, 0)$.

Check that it is an [[affine plane|EC7XT2ZM5S]].
Two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ lie on exactly one line.
Indeed, the two linear equations in $(a, b, c)$ that say both points lie on the line have a one-dimensional solution space.
Let $ax + by = c$ be a line and $(x_0, y_0)$ a point not on it.
The line $ax + by = a x_0 + b y_0$ passes through the point and is disjoint from the given line.
Every other line through the point has a different direction, so it meets the given line.
The points $(0, 0)$, $(1, 0)$ and $(0, 1)$ are not on a common line.

It has $q^2$ points and $q^2 + q$ lines, each line has $q$ points, and the lines fall into $q + 1$ parallel classes.
For $q = 2$ it has $4$ points and $6$ lines, and every pair of points is a line.
Adding one point at infinity for each parallel class and a line at infinity gives the [[projective plane $\mathrm{PG}(2, q)$|CSMB49BDPT]].

## Explanations

### L1 human

Take the usual coordinate plane, but let the coordinates run over a finite field instead of the real numbers.
Lines are still the solution sets of equations $ax + by = c$, and two points still determine a line.
Parallel lines still exist and never meet.
This gives finite planes with $q^2$ points, one for each prime power $q$.
</document>

<document source="https://tamnd.github.io/math2/node/S2T8X7QYNM.md" tag="S2T8X7QYNM" kind="definition">
# Hardy spaces of harmonic functions

## Definition

Let $n \ge 2$, let $B = \set{x \in \R^n : \abs{x} < 1}$ be the open unit ball, and let $S = \partial B$ be the unit sphere.
Let $\sigma$ be the rotation-invariant Borel probability measure on $S$.
Let $1 \le p < \infty$.
The *harmonic Hardy space* $h^p(B)$ is the set of [[harmonic functions|0DEB0JMN0W]] $u \colon B \to \R$ such that
$$\norm{u}_{h^p} = \sup_{0 < r < 1} \left( \int_S \abs{u(r\zeta)}^p \, d\sigma(\zeta) \right)^{1/p} < \infty.$$
The space $h^\infty(B)$ is the set of bounded harmonic functions on $B$, with $\norm{u}_{h^\infty} = \sup_B \abs{u}$.

## Remarks

Each $h^p$ with $1 \le p \le \infty$ is a Banach space for the norm above.
Write $P(x, \zeta) = (1 - \abs{x}^2) / \abs{x - \zeta}^n$ for the [[Poisson kernel|TCYSWCTYYM]] of $B$.
For $1 < p \le \infty$, $u \in h^p$ if and only if $u(x) = \int_S P(x, \zeta) f(\zeta)\, d\sigma(\zeta)$ for some $f$ in the [[Lp space|SASP14MA1Z]] $L^p(\sigma)$, and then $\norm{u}_{h^p} = \norm{f}_{L^p}$.
For $p = 1$, $u \in h^1$ if and only if $u$ is the Poisson integral of a finite signed Borel measure on $S$.
Example: fix $\eta \in S$ and let $u = P(\cdot, \eta)$.
It is positive and harmonic, and by the mean value property its mean over each sphere $\abs{x} = r$ is $u(0) = 1$, so $u \in h^1$ with norm $1$.
It is not in $h^p$ for $p > 1$, since it is the Poisson integral of the point mass at $\eta$, which has no $L^p$ density.
Non-example: in the plane, let $u = \operatorname{Re}\left( \frac{1 + z}{1 - z} \right)^2$ on the unit disc.
For $e^{i\theta} \ne 1$, $u(r e^{i\theta}) \to -\cot^2(\theta / 2)$ as $r \to 1$, and this function is not integrable on the circle.
By Fatou's lemma the means of $\abs{u}$ over circles are unbounded, so $u \notin h^1$.
In the plane, if $1 < p < \infty$ and $f = u + iv$ is holomorphic with $v(0) = 0$, then $u \in h^p$ if and only if $f$ is in the holomorphic [[Hardy space|PNNNFJVF6W]] $H^p$, by the [[M. Riesz theorem|PJFH51Z48M]].
This fails for $p = 1$ and for $p = \infty$.
</document>

<document source="https://tamnd.github.io/math2/node/S31M1ZYJRK.md" tag="S31M1ZYJRK" kind="statement">
# Hopf's formula

## Statement

Let $G$ be a group, let $F$ be a [[free group|0KTW26C80J]], and let $R$ be a [[normal subgroup|0CB0DX5BVV]] of $F$ with $G \cong F/R$.
Write $[A, B]$ for the subgroup generated by the [[commutators|0PXW0BN7DW]] $aba^{-1}b^{-1}$ with $a \in A$ and $b \in B$.
Then the second integral [[homology group|CFAG3VRXEA]] of $G$ is
$$H_2(G, \Z) \cong \frac{R \cap [F, F]}{[F, R]}.$$

## Remarks

Idea of the proof: the short exact sequence $1 \to R \to F \to G \to 1$ gives the exact sequence of low-degree terms
$$H_2(F, \Z) \to H_2(G, \Z) \to R / [F, R] \to F / [F, F] \to G / [G, G] \to 0.$$
Since $F$ is free, $H_2(F, \Z) = 0$, so $H_2(G, \Z)$ is the kernel of $R / [F, R] \to F / [F, F]$, which is $(R \cap [F, F]) / [F, R]$.

In particular, the right side does not depend on the chosen [[presentation|0JGEQXQ4AC]] of $G$.

Example: $G = \Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle$, with $F$ free on $a, b$.
Here $R = [F, F]$, because the normal closure of $aba^{-1}b^{-1}$ in $F$ is the commutator subgroup $[F, F]$.
So $H_2(\Z^2, \Z) \cong [F, F] / [F, [F, F]]$, which is infinite cyclic, generated by the image of $aba^{-1}b^{-1}$.
This matches the homology of the torus.

Example: $G = \Z / m\Z$, with $F = \Z$ and $R = m\Z$.
Then $[F, F]$ is trivial, so $H_2(\Z/m\Z, \Z) = 0$.

## Explanations

### L1 human

Hopf's formula computes a subtle invariant of a group from any description of the group by generators and relations.
It looks at the relations that are products of commutators, and it divides out the ones that every presentation has automatically.
The answer is the same for every presentation, which is surprising at first sight.

### L2 human

For a finite group $G$, the group $H_2(G, \Z)$ is the Schur multiplier, which controls the central extensions of $G$ and its projective representations.
Hopf's formula gives a purely algebraic way to compute it.
The exact sequence in the proof is the five-term sequence of the [[Lyndon–Hochschild–Serre spectral sequence|VC3QNG6TKR]].
</document>

<document source="https://tamnd.github.io/math2/node/S31RTTJPDC.md" tag="S31RTTJPDC" kind="definition">
# Axiom A

## Definition

Let $M$ be a compact smooth manifold and let $f \colon M \to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]].
Let $\Omega(f)$ be the [[non-wandering set|EY2W34JZA4]] of $f$.
Then $f$ satisfies *Axiom A* if both of the following hold.

1. $\Omega(f)$ is a [[hyperbolic set|NY9MCZXY6N]] of $f$.
2. The set of [[periodic points|T75CZ5H47N]] of $f$ is [[dense|0D198GPMTJ]] in $\Omega(f)$.

## Remarks

Every periodic point is non-wandering, so clause 2 says that $\Omega(f)$ is the closure of the set of periodic points.
Example: the north-south map of the circle, the time-one map of $\theta' = \sin\theta$, has $\Omega(f) = \set{0, \pi}$.
Both fixed points are hyperbolic, with derivatives $e$ and $e^{-1}$, so both clauses hold.
Non-example: an irrational rotation of the circle has $\Omega(f)$ equal to the whole circle, but it has no periodic points, and the derivative $1$ is not hyperbolic; both clauses fail.
Another non-example is the time-one map of $\theta' = 1 - \cos\theta$: its only non-wandering point $0$ is fixed, but the derivative there is $1$, so clause 1 fails.
For flows, Axiom A asks that the non-wandering set be hyperbolic in the flow sense and that closed orbits be dense in it.
</document>

<document source="https://tamnd.github.io/math2/node/S33B2H3MYD.md" tag="S33B2H3MYD" kind="concept">
# Sheaf of modules

## Summary

Let $(X, \mathcal{O}_X)$ be a ringed space.
A sheaf of modules on it is a sheaf of abelian groups $\mathcal{F}$ in which each $\mathcal{F}(U)$ is a module over $\mathcal{O}_X(U)$, compatibly with restriction.
It is the sheaf-theoretic version of a module over a ring.
Vector bundles, ideal sheaves, quasi-coherent and coherent sheaves are all sheaves of modules.

## Explanations

### L1 human

On a space, the functions on each open set form a ring.
A sheaf of modules attaches to each open set a module over that ring, in a way that is compatible with shrinking the open set.
The simplest example is the sheaf of functions itself.
Another example is the sheaf of vector fields on a manifold: one can multiply a vector field by a function and get a vector field.
</document>

<document source="https://tamnd.github.io/math2/node/S34G7679AK.md" tag="S34G7679AK" kind="definition">
# Polyconvexity

## Definition

Let $m, n \ge 1$ and let $T(\xi) \in \R^{\tau}$ denote the vector of all [[minors|0BT9BYWVRY]] of all orders $1, \dots, \min(m, n)$ of a matrix $\xi \in \R^{m \times n}$.
A function $f \colon \R^{m \times n} \to (-\infty, +\infty]$ is *polyconvex* if there is a [[convex function|07GR3TNW4H]] $g \colon \R^{\tau} \to (-\infty, +\infty]$ with
$$f(\xi) = g\big(T(\xi)\big) \quad \text{for all } \xi \in \R^{m \times n}.$$

## Remarks

For $m = n = 2$, $T(\xi) = (\xi, \det \xi) \in \R^5$.
For $m = n = 3$, $T(\xi) = (\xi, \operatorname{cof} \xi, \det \xi) \in \R^{19}$, with $\operatorname{cof} \xi$ the cofactor matrix.

Example: for $m = n = 2$, $f(\xi) = \det \xi$ is polyconvex, with $g$ linear, although $f$ is not convex.

Example: $f(\xi) = \abs{\xi}^2 + h(\det \xi)$ for $3 \times 3$ matrices, with $h(d) = d^{-1}$ for $d > 0$ and $h(d) = +\infty$ for $d \le 0$.
It is polyconvex, since $h$ is convex on $\R$ with values in $(-\infty, +\infty]$.

Non-example: for $m = n = 2$, $f(\xi) = -(\det \xi)^2$ is not polyconvex.
Along $\xi(t) = I + t \, e_1 \otimes e_1$ it equals $-(1 + t)^2$, which is not convex.
So $f$ is not [[rank-one convex|GK4P8G0PB2]], while every finite polyconvex function is.

For real-valued $f$, polyconvexity implies [[quasiconvexity|DVJ17KVWXP]], and the converse fails when $m, n \ge 2$.
</document>

<document source="https://tamnd.github.io/math2/node/S357810Z25.md" tag="S357810Z25" kind="definition">
# Localization of a module

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]], let $S \subseteq R$ be a [[multiplicative set|AN9SKEZ3VY]], and let $M$ be an $R$-[[module|0NZ8JA25WK]].
On $M \times S$, let $(m, s) \sim (n, t)$ if there is $u \in S$ with $u(tm - sn) = 0$.
This is an [[equivalence relation|0G8VEZ3WTK]]; write $m/s$ for the class of $(m, s)$.
The *localization* of $M$ at $S$ is the set $S^{-1}M$ of these classes with the operations
$$\frac{m}{s} + \frac{n}{t} = \frac{tm + sn}{st}, \qquad \frac{a}{t} \cdot \frac{m}{s} = \frac{am}{ts}$$
for $a \in R$.
These operations are well defined and make $S^{-1}M$ a module over the [[localization|4C1AE84N5N]] $S^{-1}R$.
For a [[prime ideal|00MHHZ9SNE]] $\mathfrak p$ of $R$, one writes $M_{\mathfrak p}$ for $S^{-1}M$ with $S = R \setminus \mathfrak p$.

## Remarks

There is a natural isomorphism of $S^{-1}R$-modules $S^{-1}R \otimes_R M \cong S^{-1}M$, given by $(a/s) \otimes m \mapsto am/s$; see [[tensor product of modules|F0HQXQ0HCD]].
The functor $M \mapsto S^{-1}M$ is exact, so $S^{-1}R$ is a [[flat|73RYYEVDKA]] $R$-module.

Example: let $R = \Z$, $M = \Z/3\Z$ and $\mathfrak p = 2\Z$.
Then $3 \in S = \Z \setminus 2\Z$, and $m/s = 3m/(3s) = 0$ for every $m$, so $M_{(2)} = 0$.
For $\mathfrak p = 3\Z$, the elements of $S$ act invertibly on $\Z/3\Z$, and $M_{(3)} \cong \Z/3\Z$.

Non-example: the relation "$tm = sn$" without the factor $u$ is in general not transitive, as for the [[localization of a ring|4C1AE84N5N]], which is the case $M = R$.
</document>

<document source="https://tamnd.github.io/math2/node/S360JBEYKD.md" tag="S360JBEYKD" kind="concept">
# Circle packing

## Summary

A circle packing is an arrangement of circles in the plane, or on a surface, whose discs do not overlap.
One studies dense packings of equal circles, where the hexagonal packing is optimal by [[Thue's theorem|3P246RHXMF]].
One also studies packings with prescribed tangencies, where the [[circle packing theorem|GR1DDT1B6B]] realizes every planar graph.
Circle packings give a discrete analogue of [[conformal maps|0YYT36R4YD]], and fractal examples such as the [[Apollonian gasket|8SETG842RX]].

## Explanations

### L1 human

Lay coins on a table so that no two coins overlap; that is a circle packing.
The coins may all have the same size, as in a tray of identical cans, or different sizes.
If the coins are equal, the honeycomb arrangement, where each coin touches six others, wastes the least space.
</document>

<document source="https://tamnd.github.io/math2/node/S36T7TR7AG.md" tag="S36T7TR7AG" kind="definition">
# Hereditary ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
The ring $R$ is *left hereditary* if every [[left ideal|5N2CJZ47D4]] of $R$ is a [[projective module|BFFA03GE3V]] as a left $R$-module.
It is *right hereditary* if every right ideal is projective as a right $R$-module.
It is *hereditary* if it is both left and right hereditary.

## Remarks

Example: the ring $\Z$ is hereditary.
Each ideal is $n\Z$ for some $n \ge 0$; it is $0$ or isomorphic to $\Z$ through $x \mapsto x/n$, so it is free and hence projective.
Non-example: the ring $\Z/4\Z$ is not hereditary.
It is a local ring, so its projective modules are free.
The ideal $2\Z/4\Z$ has $2$ elements, while a nonzero free $\Z/4\Z$-module has at least $4$, so this ideal is not projective.
Equivalent forms: $R$ is left hereditary if and only if every submodule of a projective left $R$-module is projective.
This holds if and only if the left [[global dimension|GWA0T1WKCN]] of $R$ is at most $1$.
For a ring that is Noetherian on both sides, left and right hereditary are equivalent.
An integral domain is hereditary if and only if it is a [[Dedekind domain|7NPY2CN5KB]].
The path algebra of a finite quiver without oriented cycles over a field is hereditary.
</document>

<document source="https://tamnd.github.io/math2/node/S36XX2PVNH.md" tag="S36XX2PVNH" kind="definition">
# Fourier series

## Definition

Let $f \colon \R \to \C$ be a [[periodic function|089KWVYD69]] of period $2\pi$ whose real and imaginary parts are [[Riemann integrable|0N1JQXZXHC]] on $[-\pi, \pi]$.
For each integer $n$, the $n$-th *Fourier coefficient* of $f$ is
$$\hat f(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx,$$
where $e^{-inx}$ is the [[complex exponential|07GJV36896]].
The *Fourier series* of $f$ is the formal [[series|0NM626FH6Q]]
$$\sum_{n = -\infty}^{\infty} \hat f(n) e^{inx},$$
whose $N$-th partial sum is $S_N f(x) = \sum_{n=-N}^{N} \hat f(n) e^{inx}$.

## Remarks

For real $f$ the series is often written $\frac{a_0}{2} + \sum_{n \ge 1} (a_n \cos nx + b_n \sin nx)$, with $a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos nx \, dx$ and $b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin nx \, dx$.
Example: let $f(x) = x$ for $-\pi < x \le \pi$, extended with period $2\pi$.
Then $a_n = 0$ since $f$ is odd, and integration by parts gives $b_n = \frac{2}{\pi} \int_0^\pi x \sin nx \, dx = \frac{2 (-1)^{n+1}}{n}$.
So the Fourier series is $2 \sum_{n \ge 1} \frac{(-1)^{n+1}}{n} \sin nx$; at $x = \pi/2$ the series converges to $\pi/2$, which is the Leibniz formula $1 - \frac{1}{3} + \frac{1}{5} - \cdots = \frac{\pi}{4}$.
Non-example: the $2\pi$-periodic function equal to $1/\abs{x}$ for $0 < \abs{x} \le \pi$ is not integrable on $[-\pi, \pi]$, so its Fourier coefficients are not defined.
The series is formal: the definition says nothing about convergence, and there are continuous functions whose Fourier series diverge at some points.
In Lebesgue theory the same formulas define the Fourier series of every integrable $2\pi$-periodic function.
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<document source="https://tamnd.github.io/math2/node/S37DQGW3TZ.md" tag="S37DQGW3TZ" kind="statement">
# Zero divisors are the union of associated primes

## Statement

Let $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $M$ be an $R$-[[module|09VDS2W1E8]].
An element $r \in R$ is a zero divisor on $M$ if $rm = 0$ for some nonzero $m \in M$.
Then the set of zero divisors on $M$ is the union of the [[associated primes|1RJJ33KKSS]] of $M$.

## Remarks

If $P = \operatorname{Ann}(m)$ is associated, then each $r \in P$ kills $m \ne 0$, so $r$ is a zero divisor.
Conversely, if $rm = 0$ with $m \ne 0$, the set of [[annihilators|HB4XDKJTW8]] of nonzero elements that contain $\operatorname{Ann}(m)$ has a maximal member, since $R$ is Noetherian.
A maximal annihilator $\operatorname{Ann}(m')$ is prime: if $ab \in \operatorname{Ann}(m')$ and $b m' \ne 0$, then $\operatorname{Ann}(b m') \supseteq \operatorname{Ann}(m') + (a)$, so $a \in \operatorname{Ann}(m')$.
For example, an integer $r$ is a zero divisor on $\Z/12\Z$ if and only if $\gcd(r, 12) > 1$, that is, $r \in (2) \cup (3)$.
Taking $M = R$, the set of [[zero divisors|0K26YKNBSA]] of $R$, together with $0$, is a finite union of prime ideals.

## Explanations

### L1 human

The elements of the ring that kill some nonzero element of the module are exactly the elements of the associated primes.
So the zero divisors, which form a messy set in general, are a union of a few prime ideals.

### L2 human

Combine it with [[finiteness of associated primes|XR0WZJAM0R]] and [[prime avoidance|FMP5AWG6PR]].
This shows: if an ideal $I$ consists of zero divisors on a finitely generated module $M$, then $I$ kills a nonzero element of $M$.
This is the basic step in the theory of regular sequences and depth.
It is also used to show that $R$ has a non-zero-divisor in an ideal not contained in any associated prime of $R$.
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<document source="https://tamnd.github.io/math2/node/S3N3CHZV89.md" tag="S3N3CHZV89" kind="concept">
# Countably compact space

## Summary

A countably compact space is a topological space in which every countable open cover has a finite subcover.
It is a weakening of [[compactness|0TVJZYB6SH]] that is equivalent to the statement that every sequence has a cluster point.
Compact spaces and [[sequentially compact|0FRMJH3CY7]] spaces are countably compact.
For metric spaces, countable compactness, compactness and sequential compactness all coincide.

## Explanations

### L1 human

Compactness asks that every open cover can be cut down to finitely many sets.
Countable compactness only asks this for covers with countably many sets.
An equivalent picture: every sequence in the space keeps coming back near some point.
For example, the interval $[0, 1]$ is countably compact, but $(0, 1]$ is not: the cover by the sets $(1/n, 1]$ has no finite subcover.
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<document source="https://tamnd.github.io/math2/node/S3RJQ9CX2P.md" tag="S3RJQ9CX2P" kind="concept">
# Mixing

## Summary

A [[measure-preserving transformation|556WBDAHNP]] of a probability space is mixing, or strongly mixing, when any two events become independent in the long run.
The points of $B$ that land in $A$ after $n$ steps form a set whose measure tends to $\mu(A)\mu(B)$.
Mixing implies [[weak mixing|2MGZFRH5PX]], which implies [[ergodicity|AZRPHS478B]].
It is the basic way to say that a system forgets its initial state.

## Explanations

### L1 human

Pour a drop of dye into a glass of water and stir.
After a long time, any region of the glass holds a share of dye equal to its share of the volume.
Mixing says exactly this for a measure-preserving map: the image of any set spreads out evenly over the whole space.
The doubling map $x \mapsto 2x \bmod 1$ on $[0,1)$ is mixing.
A rotation of a circle is never mixing, because a small arc stays a small arc and just moves around.
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<document source="https://tamnd.github.io/math2/node/S3YF52GRJM.md" tag="S3YF52GRJM" kind="definition">
# Birkhoff sum and average

## Definition

Let $X$ be a set, let $T \colon X \to X$ be a map with [[iterates|AYVTR2ES5B]] $T^k$, and let $f \colon X \to \R$ be a [[function|0EPAT51DXQ]].
For an integer $n \ge 1$, the *Birkhoff sum* of $f$ is the function
$$S_n f(x) = \sum_{k=0}^{n-1} f(T^k(x)), \qquad x \in X,$$
and the *Birkhoff average*, or ergodic average, of $f$ is
$$A_n f(x) = \frac{1}{n} S_n f(x).$$

## Remarks

The same formulas make sense for $f$ with values in $\C$ or in any real vector space.
The sums satisfy the cocycle relation $S_{m+n} f = S_m f + (S_n f) \circ T^m$ for all $m, n \ge 1$.
If $\mu$ is a $T$-invariant probability measure and $f$ is integrable, then $\int A_n f \, d\mu = \int f \, d\mu$ for every $n$.

Example: let $T(x) = x + 1/2 \bmod 1$ on $[0, 1)$ and $f(x) = x$.
For $x \in [0, 1/2)$ the orbit alternates between $x$ and $x + 1/2$, so $S_{2m} f(x) = m(2x + 1/2)$ and $A_{2m} f(x) = x + 1/4$.

Non-example: the space average $\int_0^1 f(x) \, dx = 1/2$ is not a Birkhoff average, since it does not depend on a starting point; in this example $A_{2m} f(x) = x + 1/4$ differs from $1/2$ unless $x = 1/4$.
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<document source="https://tamnd.github.io/math2/node/S3YVE24AQK.md" tag="S3YVE24AQK" kind="concept">
# Grothendieck topos

## Summary

A Grothendieck topos is a category equivalent to the category of [[sheaves|RT8A8FJ566]] of sets on a small [[site|NYE0Z94MME]].
Grothendieck introduced these categories as generalized spaces.
A topological space is studied through its category of sheaves, and many sites that are not spaces still give such categories.
Every Grothendieck topos is an elementary [[topos|6837RS7N82]], and [[Giraud's theorem|57BXX32WF0]] characterizes Grothendieck toposes by intrinsic properties.
Different sites can give equivalent toposes, so the topos is the invariant object.

## Explanations

### L1 human

A topological space can be studied through the sheaves on it, rather than through its points.
A Grothendieck topos is any category that looks like "all sheaves on some space", where the space may be a generalized one given by a site.
The category of sets is the topos of sheaves on a one-point space.
For a group $G$, the category of sets with a $G$-action is the topos of presheaves on $G$ viewed as a one-object category, so it is also a Grothendieck topos.
</document>

<document source="https://tamnd.github.io/math2/node/S417YNEJ6M.md" tag="S417YNEJ6M" kind="concept">
# Kernel

## Summary

In a category with zero morphisms, the kernel of a morphism $f$ is the universal morphism $k$ with $f \circ k = 0$.
It is the [[equalizer|CTJ5AD8ST5]] of $f$ and the zero morphism.
In categories of groups, modules or vector spaces it is the inclusion of the usual [[kernel|0W88CEGQP1]], the set of elements sent to zero.
Kernels and their dual notion, cokernels, are the building blocks of abelian categories.

## Explanations

### L1 human

The kernel of a linear map is the set of vectors that it sends to zero.
Category theory cannot look at elements, so it describes this subspace by a property of its inclusion map.
The inclusion is killed by $f$, and every other map that is killed by $f$ passes through it in exactly one way.
For the map $\Z \to \Z/2$ that reduces mod $2$, the kernel is the inclusion of the even integers $2\Z \to \Z$.
</document>

<document source="https://tamnd.github.io/math2/node/S4543S2GBJ.md" tag="S4543S2GBJ" kind="definition">
# Submanifold

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $0 \le k \le n$.
A subset $S \subseteq M$ is an *embedded submanifold* of dimension $k$ if the following holds.
For every point $p \in S$ there is a chart $(U, \varphi)$ of the smooth [[atlas|J7P32JQTVE]] of $M$ with $p \in U$ such that
$$\varphi(U \cap S) = \varphi(U) \cap (\R^k \times \set{0}).$$
Such a chart is a *slice chart* for $S$.
The restrictions of the slice charts give $S$ a unique smooth structure for which the inclusion $S \to M$ is a smooth embedding.

## Remarks

Example: the unit circle $S^1 \subseteq \R^2$ is an embedded submanifold of dimension $1$.
Near a point with $y > 0$, the chart $(x, y) \mapsto (x, y - \sqrt{1 - x^2})$ on $\set{(x, y) : \abs{x} < 1, y > 0}$ sends the circle onto the axis $\R \times \set{0}$; the other points are handled in the same way.
Non-example: the cross $C = \set{(x, y) \in \R^2 : xy = 0}$ is not an embedded submanifold.
Removing the origin from a small connected neighbourhood of $0$ in $C$ leaves four components.
Removing a point from an open interval leaves two, so no slice chart exists at the origin.
The embedded submanifolds of $M$ are exactly the images of smooth embeddings into $M$.
An *immersed submanifold* is the image of an injective [[immersion|6RW68JGBHC]] $S \to M$, with the topology and smooth structure carried over from $S$; its topology may be finer than the subspace topology.
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<document source="https://tamnd.github.io/math2/node/S47R8TG0T3.md" tag="S47R8TG0T3" kind="definition">
# Comonad

## Definition

Let $\mathcal{C}$ be a category.
A *comonad* on $\mathcal{C}$ is a triple $(G, \varepsilon, \delta)$ made of these data.

1. $G \colon \mathcal{C} \to \mathcal{C}$ is a [[functor|28NQ0KRHD3]].
2. $\varepsilon \colon G \Rightarrow \mathrm{Id}_{\mathcal{C}}$ is a [[natural transformation|KYCPCC57FT]], the counit.
3. $\delta \colon G \Rightarrow G \circ G$ is a natural transformation, the comultiplication.

The data must satisfy two laws.
The coassociativity law is $G\delta \circ \delta = \delta G \circ \delta$ as natural transformations $G \Rightarrow G G G$.
The counit laws are $\varepsilon G \circ \delta = \id_G$ and $G\varepsilon \circ \delta = \id_G$.

## Remarks

A comonad on $\mathcal{C}$ is the same as a [[monad|FF6XXA6D1A]] on the opposite category $\mathcal{C}^{\mathrm{op}}$.

Example: fix a set $A$ and let $G(X) = A \times X$ on the category of sets, with $G(f) = \id_A \times f$.
Put $\varepsilon_X(a, x) = x$ and $\delta_X(a, x) = (a, (a, x))$.
Both sides of the coassociativity law send $(a, x)$ to $(a, (a, (a, x)))$.
The counit laws hold since $\varepsilon_{G(X)}(a, (a, x)) = (a, x)$ and $G(\varepsilon_X)(a, (a, x)) = (a, x)$.

Non-example: let $A$ have two elements $a_0 \ne a_1$, and keep $G$ and $\varepsilon$ but put $\delta_X(a, x) = (a_0, (a, x))$.
Then $G(\varepsilon_X)(\delta_X(a_1, x)) = (a_0, x) \ne (a_1, x)$, so a counit law fails.

If $F \colon \mathcal{C} \to \mathcal{D}$ is left adjoint to $U \colon \mathcal{D} \to \mathcal{C}$ with counit $\varepsilon$ and unit $\eta$, then $(F U, \varepsilon, F \eta U)$ is a comonad on $\mathcal{D}$.
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<document source="https://tamnd.github.io/math2/node/S4842D7MAE.md" tag="S4842D7MAE" kind="concept">
# Levi-flat hypersurface

## Summary

A Levi-flat hypersurface is a real hypersurface in a [[complex manifold|JSGWAN2RJV]] whose [[Levi form|80MTNXWW80]] vanishes at every point.
For a smooth hypersurface this means that it is filled, near each point, by complex hypersurfaces stacked in a one-parameter family.
Levi-flat hypersurfaces are the opposite extreme to strongly pseudoconvex ones, and CR functions on them need not extend holomorphically to either side.

## Explanations

### L1 human

A real hypersurface in $\C^n$ has one real dimension less than the space.
Most such hypersurfaces bend in the complex directions, and the Levi form measures this bending.
A Levi-flat hypersurface does not bend at all in these directions.
The model is $\set{z \in \C^2 : \operatorname{Im} z_1 = 0}$, which is the union of the complex lines $\set{z_1 = t}$ for real $t$.
So it looks like a stack of complex lines indexed by a real number, like the pages of a book.
</document>

<document source="https://tamnd.github.io/math2/node/S49XMZKGQC.md" tag="S49XMZKGQC" kind="example">
# Tsirelson space

## Example

Let $c_{00}$ be the space of real sequences with finitely many nonzero terms.
For finite sets $E, F \subset \N$, write $E < F$ when $\max E < \min F$.
For $x \in c_{00}$ and a finite set $E$, let $E x$ be the sequence equal to $x$ on $E$ and to $0$ off $E$.
Call finite sets $E_1, \dots, E_n$ admissible when $n \le \min E_1$ and $E_1 < E_2 < \cdots < E_n$.
There is exactly one norm on $c_{00}$ that satisfies
$$\norm{x} = \max \Big( \norm{x}_\infty, \ \frac{1}{2} \sup \sum_{j=1}^n \norm{E_j x} \Big),$$
where the supremum runs over all $n$ and all admissible $E_1, \dots, E_n$.
The completion $T$ of $c_{00}$ for this norm is the Tsirelson space.
It is a [[Banach space|06GZ956657]].

Check of two values.
For a unit vector $e_k$, at most one set $E_j$ meets $k$, so the sum is at most $\norm{e_k}$, and the equation gives $\norm{e_k} = 1$.
For $x = e_{n+1} + \cdots + e_{2n}$, the sets $\set{n+1}, \dots, \set{2n}$ are admissible, so $n/2 \le \norm{x} \le n$.

## Remarks

The unit vectors form an unconditional basis of $T$, and $T$ is [[reflexive|NN8RWPC8WG]].
The space $T$ contains no subspace isomorphic to $c_0$ or to any [[sequence space|08P6T2WH6R]] $\ell^p$ with $1 \le p < \infty$.
Tsirelson built the first space with this property in 1974.
The space above, described by Figiel and Johnson, is the dual of Tsirelson's original space.
The second check shows how the norm forces blocks far out in the sequence to behave like $\ell^1$ of their length.

## Explanations

### L1 human

For a long time it was open whether every Banach space contains a copy of one of the classical sequence spaces.
Tsirelson space shows that the answer is no.
Its norm is defined by an equation that refers to itself.
This rule makes the space look like $\ell^1$ on short blocks and stops it from containing any classical space as a whole.
</document>

<document source="https://tamnd.github.io/math2/node/S4BZ697DSH.md" tag="S4BZ697DSH" kind="concept">
# Torsion of a curve

## Summary

The torsion of a space curve measures how fast the curve twists out of its osculating plane.
This is the plane that best fits the curve at a point.
A plane curve has torsion $0$ everywhere.
With [[curvature|7JQS6VKFEZ]], torsion appears in the Frenet–Serret formulas and determines a space curve up to a rigid motion.

## Explanations

### L1 human

Curvature tells how a curve bends inside a plane, and torsion tells how that plane itself turns.
Picture the thread of a screw: it bends around the axis and also climbs, so it leaves every plane.
A circle lies flat in one plane and has torsion $0$.
The helix $t \mapsto (\cos t, \sin t, t)$ climbs at a steady rate and has constant torsion $1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/S4GQN5V54X.md" tag="S4GQN5V54X" kind="concept">
# Radical ideal

## Summary

A radical ideal of a commutative ring is an [[ideal|0MYSA6MYNA]] that equals its own [[radical|437YRCNYRJ]].
Equivalently, if some power of an element lies in the ideal, then the element itself lies in it.
Every [[prime ideal|06HC6HK0Z8]] is radical.
By [[Hilbert's Nullstellensatz|D85VZWS9QK]], the radical ideals of a polynomial ring over an algebraically closed field correspond to the algebraic sets.

## Explanations

### L1 human

A radical ideal has no "repeated factors".
In $\Z$, the ideal $n\Z$ with $n \ge 1$ is radical exactly when $n$ is square-free, such as $6$ or $30$.
The ideal $12\Z$ is not radical: $6^2 = 36$ lies in it, but $6$ does not.
The quotient by a radical ideal is a [[reduced ring|SZHY975FFX]], a ring without nonzero nilpotent elements.
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<document source="https://tamnd.github.io/math2/node/S4J7SRPXMZ.md" tag="S4J7SRPXMZ" kind="statement">
# Silverman–Toeplitz theorem

## Statement

Let $A = (a_{nk})_{n, k \ge 0}$ be an infinite complex matrix.
The [[matrix method|RKFB6AHPBE]] given by $A$ is [[regular|BXVVXJQV8M]], that is, every convergent sequence $s$ is $A$-summable to $\lim s_k$, if and only if:

1. $\sup_{n} \sum_{k=0}^{\infty} \abs{a_{nk}} < \infty$;
2. $\lim_{n \to \infty} a_{nk} = 0$ for every $k$;
3. $\lim_{n \to \infty} \sum_{k=0}^{\infty} a_{nk} = 1$.

## Remarks

Necessity of conditions 2 and 3: apply $A$ to the constant sequence $1$ and to the unit sequence $e_k$.
Here $e_k$ has $1$ in place $k$ and $0$ elsewhere, so it tends to $0$.
Necessity of condition 1 uses the uniform boundedness principle, or a direct "gliding hump" construction of a null sequence with unbounded transform.
Sufficiency: write $s_k = L + r_k$ with $r_k \to 0$; condition 3 handles $L$, and for $r$ one splits $\sum_k a_{nk} r_k$ into finitely many terms, small by condition 2, and a tail, small by condition 1.
Example: the Cesàro matrix $a_{nk} = \frac{1}{n+1}$ for $k \le n$ has row sums of absolute values equal to $1$, entries tending to $0$ and row sums $1$, so Cesàro summation is regular.
Non-example: the matrix with $a_{nk} = 1$ for $k \le n$ and $0$ otherwise violates condition 1, and indeed it sends the convergent sequence $s_k = \frac{1}{k+1}$ to the divergent harmonic partial sums.

## Explanations

### L1 human

A weighted-average method gives the right answer on every convergent sequence under three conditions.
The weights stay bounded in total size, each fixed weight fades away, and the weights in each row add up to nearly $1$.

### L2 human

The theorem reduces regularity, a statement about all convergent sequences, to three checks on the matrix.
It is the standard tool for showing that Cesàro, Nörlund, Riesz and Euler methods are regular.
The [[Kojima–Schur theorem|JB8GQBVW6Q]] characterizes the wider class of matrices that only send convergent sequences to convergent sequences, and the Silverman–Toeplitz theorem is its special case with limits $0$ and $1$.
The [[Steinhaus theorem|NG1FPJQHA1]] shows that no regular matrix sums every bounded sequence.
</document>

<document source="https://tamnd.github.io/math2/node/S4QJ395PA5.md" tag="S4QJ395PA5" kind="statement">
# Stone's representation theorem for Boolean algebras

## Statement

Let $B$ be a [[Boolean algebra|0XADJKM699]], and let $S(B)$ be the set of [[ultrafilters|CQM84EW10G]] of $B$.
Give $S(B)$ the topology with basis the sets $\hat a = \set{U \in S(B) : a \in U}$ for $a \in B$.
Then $S(B)$ is a [[Stone space|ZV38HA5PAQ]].
The map $a \mapsto \hat a$ is an isomorphism from $B$ onto the Boolean algebra of [[clopen|DQ4E55T3BW]] subsets of $S(B)$.
In particular, every Boolean algebra is isomorphic to a [[field of sets|PFMGZ7988V]].

## Remarks

The map $a \mapsto \hat a$ preserves the operations because an ultrafilter contains $a \wedge b$ exactly when it contains $a$ and $b$, and contains $\neg a$ exactly when it does not contain $a$.
It is injective by the [[ultrafilter lemma|3MKZST79MK]]: if $a \ne b$, then $a \wedge \neg b \ne 0$ or $\neg a \wedge b \ne 0$, and an ultrafilter containing that element separates $a$ and $b$.
Compactness of $S(B)$ also uses the ultrafilter lemma, and surjectivity onto the clopen sets follows from compactness, since a clopen set is a finite union of basic sets $\hat a$.
The ultrafilter lemma cannot be dropped: over ZF, the theorem implies it.

Example: let $B$ be the algebra of finite and cofinite subsets of $\N$.
Its ultrafilters are the principal ones and one more, made of the cofinite sets, and $S(B)$ is the one-point compactification of $\N$.
For a finite Boolean algebra the space is finite and discrete, which gives [[Stone's representation theorem for finite Boolean algebras|0DYS9C3HH5]].

## Explanations

### L1 human

Every Boolean algebra can be pictured as a family of subsets of some set, with union, intersection and complement as its operations.
The set to use is the set of ultrafilters, the consistent ways to declare each element true or false.
So the axioms of a Boolean algebra say nothing beyond what holds for sets.

### L2 human

The theorem is the object part of [[Stone duality|B9T33D5MJK]], which also matches homomorphisms with continuous maps.
It implies that an equation holds in all Boolean algebras as soon as it holds in the two-element algebra.
This is the completeness theorem for classical propositional logic in algebraic form.
The same idea, with prime filters instead of ultrafilters, gives [[Priestley duality|SA22SEC75K]] for distributive lattices.
</document>

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# Quasiconvex function

## Summary

A quasiconvex function is a real function on a convex set whose sublevel sets are all [[convex|02AS45T3H9]].
Every [[convex function|07Q7CNM5WF]] is quasiconvex, but many quasiconvex functions are not convex.
The notion is used in optimization and economics, where only the shape of the sublevel sets matters, not the exact values.
The [[definition|393PNMXBFJ]] gives the precise condition.

## Explanations

### L1 human

A function is quasiconvex if, on each segment, its value never rises above the larger of the two values at the ends.
Picture a valley whose sides may be steep or flat but never have a bump in the middle.
Every increasing function on the real line is quasiconvex, even if its graph bends downward.
For example $\sqrt{\abs{x}}$ is quasiconvex on $\R$ but not convex.
</document>

<document source="https://tamnd.github.io/math2/node/S4SV3QZ8B1.md" tag="S4SV3QZ8B1" kind="concept">
# Hecke operator

## Summary

The Hecke operators $T_n$, for $n \ge 1$, are linear maps on the space $M_k$ of [[modular forms|GWKN5VG78A]] of weight $k$.
They average a modular form over the sublattices of index $n$ of a lattice.
They commute with each other and preserve the [[cusp forms|A50Y012RBP]].
A common eigenform of all $T_n$, suitably normalized, has multiplicative Fourier coefficients and an L-function with an Euler product.

## Explanations

### L1 human

A modular form can be seen as a function of lattices in the plane.
The operator $T_n$ replaces the value at a lattice by the sum of the values at all its sublattices of index $n$.
This produces a new modular form of the same weight.
Since the space of forms is small, a form is often sent to a multiple of itself.
For $\Delta$ of weight $12$, $T_n \Delta = \tau(n) \Delta$, which explains why the [[Ramanujan tau function|6DNZAGCSQS]] is multiplicative.
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<document source="https://tamnd.github.io/math2/node/S4XMYJ3V42.md" tag="S4XMYJ3V42" kind="statement">
# Rankine–Hugoniot conditions

## Statement

Let $F \colon \R^m \to \R^m$ be a $C^1$ function and consider the system of [[conservation laws|8RA7AMWSAF]] $u_t + F(u)_x = 0$ in one space dimension.
Let $V \subseteq \R \times (0, \infty)$ be open, and let $C = \set{(s(t), t) : t \in I} \subseteq V$ be a curve with $s$ of class $C^1$.
Assume that $C$ splits $V$ into a left part $V_l$ and a right part $V_r$.
Let $u \colon V \to \R^m$ be bounded and of class $C^1$ on $V_l$ and on $V_r$.
Assume that $u$ has one-sided limits $u_l(t)$ from $V_l$ and $u_r(t)$ from $V_r$ at $(s(t), t)$, continuous in $t$.
Then $u$ is a [[weak solution|SJZX81A783]] in $V$ if and only if $u$ solves the equation classically in $V_l$ and in $V_r$, and
$$F(u_l(t)) - F(u_r(t)) = s'(t)\, \big( u_l(t) - u_r(t) \big) \qquad \text{for all } t \in I.$$

## Remarks

Write $[\,\cdot\,]$ for the jump across the curve; then the condition is $[F(u)] = \sigma [u]$ with shock speed $\sigma = s'$.
The proof tests the weak formulation with a test function supported near a point of $C$.
On each side $u$ is a classical solution, so integrating by parts on $V_l$ and on $V_r$ leaves only boundary integrals over $C$.
These combine to $\int_C \big( [F(u)] - s' [u] \big) \varphi \, dt = 0$ for all test functions, which gives the jump condition.

Example: for the scalar [[Burgers' equation|NEGHXN8RE1]], $F(u) = u^2/2$, the condition gives $s' = \dfrac{u_l^2/2 - u_r^2/2}{u_l - u_r} = \dfrac{u_l + u_r}{2}$.
So a jump from $u_l = 2$ to $u_r = 0$ moves at speed $1$.

The conditions are necessary but do not select a unique solution.
A jump from $0$ to $1$ for Burgers' equation also satisfies them and gives a weak solution, but not the physical one.
An entropy condition is needed in addition.
In several space dimensions the same argument gives $\sum_j [F_j(u)]\, \nu_{x_j} = - \nu_t [u]$, where $(\nu_x, \nu_t)$ is a normal to the surface of discontinuity.

## Explanations

### L1 human

When a conserved quantity jumps across a moving front, the front cannot move at any speed it likes.
The amount that piles up at the front per unit time must match the difference of the fluxes on the two sides.
This balance fixes the speed of the front: it equals the jump of the flux divided by the jump of the quantity.
In gas dynamics these are the jump relations for mass, momentum and energy across a shock.

### L2 human

The theorem shows that piecewise smooth weak solutions are exactly the piecewise classical solutions whose jumps obey the Rankine–Hugoniot relations.
It is used to construct explicit solutions, for example of the [[Riemann problem|GJWKX22VVE]], and to check proposed shock solutions.
Since it allows non-physical jumps, it is always combined with an entropy condition such as the Lax condition, which leads to the notion of [[entropy solution|1A01A3EJRX]].
For a system, the set of states $u_r$ that can be joined to a given $u_l$ is the Hugoniot locus.
For a strictly hyperbolic system, near $u_l$ it consists of $m$ curves through $u_l$.
</document>

<document source="https://tamnd.github.io/math2/node/S5744QZV4N.md" tag="S5744QZV4N" kind="concept">
# Product

## Summary

The product of a family of objects in a [[category|A2DE66937D]] is an object with a projection to each member of the family.
Any family of morphisms into the members factors uniquely through it.
It is the [[limit|CHS5AFJGWE]] of a diagram with no non-identity arrows.
It generalises the [[Cartesian product|09CEDJYBKP]] of sets, the [[direct product of groups|00HF4RJ5QN]] and the product of topological spaces.

## Explanations

### L1 human

A map into a product is the same thing as a list of maps, one into each factor.
For sets, a function $f \colon Z \to X \times Y$ is the same as the pair of functions $z \mapsto$ first coordinate and $z \mapsto$ second coordinate of $f(z)$.
The categorical product takes this property as the definition and forgets how the product is built.
In a partially ordered set seen as a category, the product of two elements is their greatest lower bound.
</document>

<document source="https://tamnd.github.io/math2/node/S5H153HFWC.md" tag="S5H153HFWC" kind="concept">
# Hochschild homology

## Summary

Hochschild homology is a homology theory for [[associative algebras|3CTW1P4HBP]], with coefficients in a [[bimodule|EM7FHAFD3N]].
It is computed from the Hochschild complex, whose $n$-th term is $M \otimes A^{\otimes n}$.
In degree $0$ it gives $A/[A, A]$, the quotient of the algebra by its commutators.
For smooth commutative algebras in characteristic $0$ it recovers [[Kähler differential forms|7HCFARHG7D]], so it is a noncommutative analogue of differential forms.

## Explanations

### L1 human

Hochschild homology measures how far an algebra is from being commutative, and in which ways.
In the lowest degree it keeps an element $a$ only up to adding commutators $bc - cb$.
For the matrix algebra $M_n(k)$ this quotient is a single copy of $k$, detected by the trace.
In higher degrees it records relations among such commutators, built from tensor products of elements of the algebra.
For the polynomial ring $k[x]$ it is $k[x]$ in degree $0$, the forms $f(x)\,dx$ in degree $1$, and zero above.
</document>

<document source="https://tamnd.github.io/math2/node/S5HVHEPSMA.md" tag="S5HVHEPSMA" kind="definition">
# Period-doubling bifurcation

## Definition

Let $I, J \subseteq \R$ be open intervals and let $(x, \mu) \mapsto f_\mu(x)$ be a map $I \times J \to \R$ of class $C^3$.
Write $f_x, f_{xx}, f_{xxx}$ for the partial derivatives in $x$.
The family $(f_\mu)$ has a *period-doubling bifurcation* (or *flip bifurcation*) at $(x_0, \mu_0) \in I \times J$ if the following hold:

1. $f_{\mu_0}(x_0) = x_0$, so $x_0$ is a [[fixed point|R6QVS6P21N]] of $f_{\mu_0}$.
2. $f_x(x_0, \mu_0) = -1$.
3. Let $x(\mu)$ be the branch of fixed points through $x_0$, given by the implicit function theorem. Then $\mu \mapsto f_x(x(\mu), \mu)$ has a nonzero derivative at $\mu_0$.
4. $\frac{1}{2} f_{xx}(x_0, \mu_0)^2 + \frac{1}{3} f_{xxx}(x_0, \mu_0) \neq 0$.

## Remarks

Under these conditions, for $\mu$ near $\mu_0$ on one side of $\mu_0$ there is exactly one orbit of [[period|T75CZ5H47N]] $2$ near $x_0$.
On the other side there is none, and the period-$2$ orbit tends to $x_0$ as $\mu \to \mu_0$.
The sign of the quantity in clause 4 decides whether the bifurcation is supercritical, with an attracting period-$2$ orbit, or subcritical, with a repelling one.

Example: the logistic family $f_r(x) = r x (1 - x)$ at $r_0 = 3$ and $x_0 = 2/3$.
Clause 1: $3 \cdot \frac{2}{3} \cdot \frac{1}{3} = \frac{2}{3}$.
Clause 2: $f_x = r(1 - 2x) = 3 \cdot (-\frac{1}{3}) = -1$.
Clause 3: on the branch $x(r) = 1 - 1/r$ one has $f_x = 2 - r$, with derivative $-1$ in $r$.
Clause 4: $f_{xx} = -2r = -6$ and $f_{xxx} = 0$, so the quantity is $18 \neq 0$.
For $r > 3$ the period-$2$ orbit is $\frac{r + 1 \pm \sqrt{(r - 3)(r + 1)}}{2r}$.

Non-example: the family $f_\mu(x) = \mu - x$ satisfies clauses 1 and 2 at $(x_0, \mu_0) = (0, 0)$, but $f_x = -1$ for every $\mu$, so clause 3 fails.
Here every point other than the fixed point $\mu/2$ has period $2$ for every $\mu$, and no isolated orbit is born.

A fixed point with derivative $+1$ instead of $-1$ gives a [[saddle-node bifurcation|MAP5J6HK0T]] or one of its degenerate relatives.
</document>

<document source="https://tamnd.github.io/math2/node/S5VTE2R86M.md" tag="S5VTE2R86M" kind="definition">
# Haar measure on a Lie group

## Definition

Let $G$ be a real [[Lie group|7JBRGCVJRB]] of dimension $n$, with identity $e$, and for $g \in G$ let $L_g(h) = gh$.
Fix a nonzero alternating $n$-linear form $\omega_e$ on the [[tangent space|E7A5DHFT01]] $T_eG$.
Let $\omega$ be the left-invariant $n$-form on $G$ given by $\omega_g = (L_{g^{-1}})^* \omega_e$, a nowhere-vanishing [[differential form|0E3G0GRPZ6]] of top degree.
The *left Haar measure* of $\omega$ is the Borel measure $\mu_\omega$ on $G$ such that
$$\int_G f \, d\mu_\omega = \int_G f \, \abs{\omega}$$
for every continuous function $f$ with compact support.
Here $\abs{\omega}$ is the density associated with $\omega$.

## Remarks

Since $L_h^* \omega = \omega$ for all $h \in G$, the measure $\mu_\omega$ is invariant under left translations, so it is a left [[Haar measure|VY0HXD3GC4]] of $G$; changing $\omega_e$ multiplies it by a positive constant.

Example: on $G = GL_n(\R)$, open in $M_n(\R) = \R^{n^2}$, the measure $\abs{\det X}^{-n} \, dX$, with $dX$ Lebesgue measure, is a left Haar measure.
Left multiplication $X \mapsto AX$ is linear with determinant $(\det A)^n$, which cancels the change of $\abs{\det X}^{-n}$.

Example and non-example: on $G = \set{(a, b) : a > 0, b \in \R}$ with $(a, b)(a', b') = (aa', ab' + b)$, the measure $a^{-2}\, da\, db$ is a left Haar measure.
The measure $a^{-1}\, da\, db$ is not left-invariant: left translation by $(c, 0)$ maps $(a, b)$ to $(ca, cb)$, with Jacobian $c^2$, and multiplies $a^{-1}\, da\, db$ by $c$.
It is a right Haar measure instead.

A right Haar measure arises in the same way from right-invariant forms.
The group is unimodular, meaning that left and right Haar measures coincide, if and only if $\abs{\det \operatorname{Ad}(g)} = 1$ for all $g \in G$, where $\operatorname{Ad}$ is the [[adjoint representation|P6ZP0TE07M]].
</document>

<document source="https://tamnd.github.io/math2/node/S5Z1XS6MZN.md" tag="S5Z1XS6MZN" kind="definition">
# Group of Lie type

## Definition

Let $p$ be a prime and let $k$ be an [[algebraic closure|0X6AB5P3G1]] of the [[finite field|0ZXZ82FTX2]] $\F_p$.
Let $\mathbf{G}$ be a connected [[reductive|2XS84PWE3S]] [[linear algebraic group|GH81SYAPPN]] over $k$.
A Steinberg endomorphism of $\mathbf{G}$ is an endomorphism $F$ of the algebraic group $\mathbf{G}$ with the following property.
Some power $F^m$ with $m \ge 1$ is the [[Frobenius endomorphism|0T3WVTD675]] of $\mathbf{G}$ for some structure of $\mathbf{G}$ over a finite subfield of $k$.
A *finite group of Lie type* is a group
$$\mathbf{G}^F = \set{g \in \mathbf{G} : F(g) = g}$$
for such $\mathbf{G}$ and $F$.

## Remarks

The group $\mathbf{G}^F$ is always finite.
Example: take $\mathbf{G} = \mathrm{GL}_n(k)$ and let $F$ raise every matrix entry to the power $q$, where $q$ is a power of $p$.
Then $F$ is the Frobenius endomorphism for the standard structure over $\F_q$, and $\mathbf{G}^F = \mathrm{GL}_n(\F_q)$, since the elements of $k$ fixed by $x \mapsto x^q$ are exactly the elements of $\F_q$.
Non-example: $\mathrm{GL}_n(\R)$ for $n \ge 1$ is not a finite group of Lie type, since it is infinite.
Many authors also call the simple groups obtained from $\mathbf{G}^F$, such as $\mathrm{PSL}_n(\F_q)$, groups of Lie type.
The [[Chevalley groups|YPCH4Q0V34]] over $\F_q$ come from the case where $\mathbf{G}$ is split over $\F_q$ and $F$ is the corresponding Frobenius endomorphism.
The twisted groups, such as the unitary groups, come from the other Steinberg endomorphisms.
For the [[Suzuki groups|36X3GET13B]] and the [[Ree groups|YB62XEQJAC]], the map $F$ is not a Frobenius endomorphism, but $F^2$ is one.

## Explanations

### L1 human

A finite group of Lie type is a matrix group over a finite field, or a twisted variant of one, described as the fixed points of a Frobenius-type map.
</document>

<document source="https://tamnd.github.io/math2/node/S60PA0RXRY.md" tag="S60PA0RXRY" kind="concept">
# Fox H-function transforms

## Summary

A Fox H-function transform, or H-transform, is an [[integral transform|YCK709QB06]] on $(0, \infty)$ whose kernel is a [[Fox H-function|EAZMKENA4N]] of the product $xy$.
The H-function generalizes the Meijer G-function by allowing positive scaling factors in front of the variable in each gamma factor of its Mellin–Barnes integral.
So the H-transforms contain the [[Meijer G-function transforms|C0BCHX106R]] and, through them, the Laplace, Stieltjes and Hankel-type transforms.
They are a common framework for fractional calculus and for transforms with kernels of Mittag-Leffler or Wright type.

## Explanations

### L1 human

The Fox H-function is an even larger family of special functions than the Meijer G-function.
Its definition is a contour integral of a ratio of gamma functions, and each gamma factor may carry its own scale.
An H-transform integrates a function against such a kernel.
If every scale equals $1$, the H-function is a G-function and the H-transform is a G-transform, for example the Laplace transform with kernel $e^{-xy}$.
</document>

<document source="https://tamnd.github.io/math2/node/S6C8KDBS04.md" tag="S6C8KDBS04" kind="concept">
# Skew polynomial ring

## Summary

A skew polynomial ring $R[x; \sigma]$ is a ring of polynomials over a [[ring|0B1261B5ED]] $R$ in which the variable twists the coefficients.
The rule is $x a = \sigma(a) x$ for a fixed endomorphism $\sigma$ of $R$.
It is the [[Ore extension|VF3PV13BA1]] with zero derivation.
Some authors use the name skew polynomial ring for every Ore extension $R[x; \sigma, \delta]$.
Skew polynomial rings over finite fields are used in coding theory, and quantum planes are skew polynomial rings.

## Explanations

### L1 human

Polynomials are added as usual, but multiplying $x$ past a coefficient $a$ replaces it with $\sigma(a)$.
Over the complex numbers with $\sigma$ the complex conjugation, $x \cdot i = -i \cdot x$.
So $x$ anticommutes with $i$ but commutes with real numbers.
The order of factors now matters, while every element is still a sum $a_0 + a_1 x + \cdots + a_n x^n$.
</document>

<document source="https://tamnd.github.io/math2/node/S6CCJBJKHR.md" tag="S6CCJBJKHR" kind="definition">
# Fiber product of schemes

## Definition

Let $S$, $X$ and $Y$ be [[schemes|5A35TYZD02]], and let $f \colon X \to S$ and $g \colon Y \to S$ be [[morphisms of schemes|JXRKH6P1Z4]].
A *fiber product* of $X$ and $Y$ over $S$ is a scheme $P$ with morphisms $p \colon P \to X$ and $q \colon P \to Y$ such that:

1. $f \circ p = g \circ q$.
2. For every scheme $T$ and morphisms $a \colon T \to X$ and $b \colon T \to Y$ with $f \circ a = g \circ b$, there is a unique morphism $h \colon T \to P$ with $p \circ h = a$ and $q \circ h = b$.

The fiber product exists for all such data and is unique up to unique isomorphism.
It is written $X \times_S Y$; it is the [[pullback|Q05K04QJ71]] in the category of schemes.

## Remarks

Example: for ring maps $A \to B$ and $A \to C$, the fiber product of $\operatorname{Spec} B$ and $\operatorname{Spec} C$ over $\operatorname{Spec} A$ is $\operatorname{Spec}(B \otimes_A C)$.
The projections come from $b \mapsto b \otimes 1$ and $c \mapsto 1 \otimes c$, and clause 2 follows from the universal property of the [[tensor product of algebras|1735VGRT2P]].
Non-example: the underlying set of $X \times_S Y$ is in general not the fiber product of the underlying sets.
For instance, $\operatorname{Spec} \C \times_{\operatorname{Spec} \R} \operatorname{Spec} \C = \operatorname{Spec}(\C \otimes_\R \C) \cong \operatorname{Spec}(\C \times \C)$ has two points, while the set product has one.
When $S = \operatorname{Spec} \Z$, one writes $X \times Y$ for $X \times_S Y$.
</document>

<document source="https://tamnd.github.io/math2/node/S6E3H24FGB.md" tag="S6E3H24FGB" kind="definition">
# Shanks transformation

## Definition

Let $(A_n)_{n \ge 0}$ be a [[sequence|0QE8STGJH2]] of real or complex numbers, and write $\Delta A_m = A_{m+1} - A_m$.
For $n \ge 1$, the *Shanks transformation* of $(A_n)$ is
$$S(A_n) = \frac{A_{n+1} A_{n-1} - A_n^2}{A_{n+1} - 2A_n + A_{n-1}},$$
defined when the denominator is not $0$.
More generally, let $k \ge 1$ and $n \ge k$.
The transformation of order $k$ is the quotient of two [[determinants|0C3CFHDN6R]]:
$$e_k(A_n) = \frac{\begin{vmatrix} A_{n-k} & \cdots & A_n \\ \Delta A_{n-k} & \cdots & \Delta A_n \\ \vdots & & \vdots \\ \Delta A_{n-1} & \cdots & \Delta A_{n+k-1} \end{vmatrix}}{\begin{vmatrix} 1 & \cdots & 1 \\ \Delta A_{n-k} & \cdots & \Delta A_n \\ \vdots & & \vdots \\ \Delta A_{n-1} & \cdots & \Delta A_{n+k-1} \end{vmatrix}}.$$
For $1 \le j \le k$, row $j$ of each determinant is $\Delta A_{n-k+j-1}, \dots, \Delta A_{n+j-1}$.
The denominator must not be $0$.
Thus $e_1(A_n) = S(A_n)$.

## Remarks

Check of $e_1 = S$: the numerator is $A_{n-1}\Delta A_n - A_n \Delta A_{n-1} = A_{n+1}A_{n-1} - A_n^2$ and the denominator is $\Delta A_n - \Delta A_{n-1}$.
Example: for the partial sums $A_0 = 1$, $A_1 = \frac23$, $A_2 = \frac{13}{15}$ of the Leibniz series,
$S(A_1) = \dfrac{\frac{13}{15} - \frac49}{\frac{13}{15} - \frac43 + 1} = \dfrac{19/45}{8/15} = \dfrac{19}{24}$.
Non-example: for an arithmetic sequence such as $A_n = n$ the denominator $A_{n+1} - 2A_n + A_{n-1}$ is $0$, so $S(A_n)$ is not defined.
Up to a shift of index, $S$ is [[Aitken's delta-squared process|R40BXAKSV4]]: $S(A_{n+1}) = A_n - \frac{(\Delta A_n)^2}{\Delta^2 A_n}$.
The transformation can be iterated, as in $S(S(A_n))$; this is different from $e_2$ in general.

## Explanations

### L1 human

The first-order formula uses three consecutive terms; the order-$k$ formula uses $2k + 1$ terms and fits a more flexible model of the error.
</document>

<document source="https://tamnd.github.io/math2/node/S6KGGNFKDF.md" tag="S6KGGNFKDF" kind="definition">
# Relaxation of a variational problem

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $J \colon X \to (-\infty, +\infty]$ be a [[functional|1EQM0DJ4NC]].
The *relaxed functional*, or lower semicontinuous envelope, of $J$ is
$$\bar J(u) = \sup \set{ G(u) : G \colon X \to [-\infty, +\infty] \text{ is lower semicontinuous and } G \le J }.$$
The *relaxed problem* of the problem $\inf_X J$ is the problem $\inf_X \bar J$.

## Remarks

The function $\bar J$ is itself [[lower semicontinuous|RN4G8SGP6P]], and $\bar J \le J$.
If $X$ is a metric space, then $\bar J(u) = \inf \set{ \liminf_k J(u_k) : u_k \to u }$.
In the calculus of variations one uses the sequential form of this formula for weak convergence in a Sobolev space.

The relaxed problem has the same infimum: $\inf_X \bar J = \inf_X J$.
If $X$ is a metric space, every limit of a subsequence of a minimising sequence of $J$ is a minimiser of $\bar J$.

Example: on $X = \R$, let $J(u) = u^2$ for $u \ne 0$ and $J(0) = 1$.
Then $\bar J(u) = u^2$ for all $u$, which attains the common infimum $0$ at $u = 0$, while $J$ does not attain it.

Example: let $J(u) = \int_0^1 \big( (u'^2 - 1)^2 + u^2 \big) \, dx$ on $W^{1,4}_0(0, 1)$ with weak convergence.
Its relaxation is $\bar J(u) = \int_0^1 \big( f^{**}(u') + u^2 \big) \, dx$, where $f^{**}(\xi) = \big(\max(\xi^2 - 1, 0)\big)^2$ is the convex envelope of $f(\xi) = (\xi^2 - 1)^2$.
The function $u = 0$ minimises $\bar J$, with $\bar J(0) = 0 = \inf J$, although $J$ has no minimiser.

Non-example: for maps $u \colon \Omega \to \R^m$ with $m, n \ge 2$, replacing the integrand by its convex envelope does not give $\bar J$ in general.
For $2 \times 2$ matrices, $\det$ is weakly continuous on $W^{1,p}$ for $p > 2$, so it needs no relaxation, but its convex envelope is $-\infty$.
The correct integrand is the quasiconvex envelope, under growth conditions.
</document>

<document source="https://tamnd.github.io/math2/node/S6PHT75MEK.md" tag="S6PHT75MEK" kind="statement">
# Unique factorization of ideals in a Dedekind domain

## Statement

Let $R$ be a [[Dedekind domain|7NPY2CN5KB]], that is, a [[Noetherian|0670T59HGE]] [[integrally closed domain|H20P600HQS]] in which every nonzero [[prime ideal|00MHHZ9SNE]] is [[maximal|0PV74TR5FP]].
Let $I$ be a nonzero [[ideal|096Z9MP2ME]] of $R$ with $I \ne R$.
Then there are distinct nonzero prime ideals $P_1, \ldots, P_r$ and integers $e_1, \ldots, e_r \ge 1$ with
$$I = P_1^{e_1} P_2^{e_2} \cdots P_r^{e_r}.$$
The primes $P_i$ and exponents $e_i$ are unique up to the order of the factors.

## Remarks

One proof shows first that every nonzero ideal contains a product of nonzero primes, using the Noetherian condition.
Then it shows that every nonzero prime $P$ is [[invertible|E190TAV2S6]]: there is a fractional ideal $P^{-1}$ with $P P^{-1} = R$; here integral closedness and dimension $1$ are used.
Dividing $I$ by a prime that contains it gives a strictly larger ideal, and the Noetherian condition ends the process.
Uniqueness follows by cancelling invertible primes, since $P \supseteq P_1 \cdots P_r$ forces $P = P_i$ for some $i$.

Example: in $R = \Z[\sqrt{-5}]$, the element $6$ factors in two ways as $2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$, so $R$ is not a [[UFD|0W4X9G6263]].
With $P = (2, 1 + \sqrt{-5})$, $Q = (3, 1 + \sqrt{-5})$ and $Q' = (3, 1 - \sqrt{-5})$, one has $P^2 = (2)$ and $Q Q' = (3)$, so $(6) = P^2 Q Q'$, and the two factorizations of $6$ come from regrouping these prime ideals.

Conversely, an integral domain in which every nonzero proper ideal is a product of prime ideals is a Dedekind domain.
The dimension hypothesis is needed: $k[x, y]$ over a field $k$ is Noetherian and integrally closed, but it has dimension $2$.
In it, the only prime ideal that contains $(x, y^2)$ is $(x, y)$, and no power of $(x, y)$ equals $(x, y^2)$, so $(x, y^2)$ is not a product of prime ideals.

## Explanations

### L1 human

In many rings of algebraic numbers, numbers do not factor uniquely into primes.
Dedekind's idea was to factor ideals instead of numbers.
In a Dedekind domain, every ideal splits into prime ideals in exactly one way, just as every integer splits into prime numbers.
This rescues unique factorization for all rings of integers of number fields.

### L2 human

The theorem shows that the nonzero fractional ideals of a Dedekind domain form a free abelian group with the nonzero prime ideals as a basis.
The quotient by the principal fractional ideals is the [[ideal class group|WQ46BD3ARY]], which measures how far $R$ is from being a [[principal ideal domain|0CMD0BCH8T]].
Locally, the exponent $e_i$ is the valuation of $I$ in the [[discrete valuation ring|MGM3CNJW9V]] $R_{P_i}$.
In number theory, the factorization of $pR$ for a prime number $p$ describes how $p$ splits in the number field.
</document>

<document source="https://tamnd.github.io/math2/node/S6VR0CVZ86.md" tag="S6VR0CVZ86" kind="definition">
# Survival analysis

## Definition

*Survival analysis* is the branch of [[statistical inference|JZ2S01BHJJ]] whose data are times to an event.
Its setting is a nonnegative random variable $T$, the event time, together with observations of $T$ that may be [[censored|MY8ZZP84NP]].
Its aims are to estimate the [[survival function|QTM1HZM53Q]] $S(t) = P(T > t)$ and the [[hazard function|Y959NKQ3XW]] of $T$.
It also compares them between groups and models how they depend on covariates.

## Remarks

The data of subject $i$ usually take the form $(\tilde t_i, \delta_i, x_i)$, where $x_i$ is a vector of covariates.
Here $\tilde t_i = \min(T_i, C_i)$ is the observed time, $C_i$ is a censoring time, and $\delta_i = 1$ if $T_i \le C_i$ and $0$ otherwise.

Example: estimating the fraction of light bulbs that still work after $1000$ hours, when the test is stopped at $1500$ hours, is a survival analysis problem.
The bulbs still working at $1500$ hours give right-censored times.

Non-example: estimating the mean weight of the bulbs is not survival analysis, since the quantity of interest is not a time to an event.
</document>

<document source="https://tamnd.github.io/math2/node/S70JBRSQZX.md" tag="S70JBRSQZX" kind="example">
# Arithmetic progression

## Example

Let $a$ and $d$ be real numbers.
The sequence $(a_n)_{n \ge 0}$ with $a_n = a + n d$ is the *arithmetic progression* with first term $a$ and common difference $d$.
It is a [[sequence|0G9EKM0E15]] in which each term minus the previous one is the same number: $a_{n+1} - a_n = d$ for every $n \ge 0$.
For example, $a = 3$ and $d = 4$ give $3, 7, 11, 15, \ldots$.

The sum of the first $n$ terms is
$$a_0 + a_1 + \cdots + a_{n-1} = n a + \frac{n(n-1)}{2} d = \frac{n (a_0 + a_{n-1})}{2}.$$
Check: $\sum_{k=0}^{n-1} (a + kd) = na + d \sum_{k=0}^{n-1} k$, and $\sum_{k=0}^{n-1} k = n(n-1)/2$ by the [[sum of the first natural numbers|01QZNHQWYD]].
For $3 + 7 + 11 + 15$ this gives $4 \cdot (3 + 15) / 2 = 36$.

## Remarks

The progression converges if and only if $d = 0$, since otherwise $\abs{a_n} \ge n \abs{d} - \abs{a}$ tends to $\infty$.
So the series $\sum_n a_n$ of an arithmetic progression diverges unless $a = d = 0$.
Each term is the average of its two neighbours: $a_n = (a_{n-1} + a_{n+1})/2$ for $n \ge 1$.

## Explanations

### L1 human

An arithmetic progression goes up or down in equal steps, like $2, 5, 8, 11$.
To add its terms, pair the first with the last, the second with the second to last, and so on.
Every pair has the same sum, so the total is the number of terms times the average of the first and the last term.
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# Hardy's inequality

## Statement

Let $p > 1$ be a real number.

1. (Discrete form.) Let $a_1, a_2, \dots$ be nonnegative real numbers. Then
$$\sum_{n=1}^{\infty} \Big( \frac{a_1 + a_2 + \dots + a_n}{n} \Big)^{p} \le \Big( \frac{p}{p - 1} \Big)^{p} \sum_{n=1}^{\infty} a_n^{p}.$$
2. (Integral form.) Let $f \colon (0, \infty) \to [0, \infty)$ be a [[measurable function|YYNCJSR0PJ]], and let $F(x) = \frac{1}{x} \int_0^x f(t) \, dt$ for $x > 0$. Then
$$\int_0^{\infty} F(x)^{p} \, dx \le \Big( \frac{p}{p - 1} \Big)^{p} \int_0^{\infty} f(x)^{p} \, dx.$$

Here the [[series|0CVVKY7GY2]] and [[integrals|DWJF7ABPR1]] take values in $[0, \infty]$.

## Remarks

Proof idea for the integral form: substitute $t = s x$ to get $F(x) = \int_0^1 f(s x) \, ds$, an average of the dilates of $f$.
By Minkowski's integral inequality, the continuous form of the [[Minkowski inequality|N4QZS912MT]], $\norm{F}_p \le \int_0^1 \norm{f(s \, \cdot)}_p \, ds$.
Since $\norm{f(s \, \cdot)}_p = s^{-1/p} \norm{f}_p$ and $\int_0^1 s^{-1/p} \, ds = \frac{p}{p-1}$, the result follows.

The constant $\big(\frac{p}{p-1}\big)^p$ is the best possible in both forms.

The hypothesis $p > 1$ is needed.
For $p = 1$ and $f$ the indicator function of $(0, 1]$, one gets $F(x) = 1/x$ for $x > 1$, so the left side is infinite while the right side is $1$.

## Explanations

### L1 human

Take a sequence of nonnegative numbers and replace each term by the average of the terms up to it.
Hardy's inequality says the new sequence is not much larger than the old one, when size is measured by the sum of $p$-th powers.
The loss is at most the factor $\big(\frac{p}{p-1}\big)^p$, which is $4$ for $p = 2$.

### L2 human

In operator language, the averaging operator $f \mapsto F$ is bounded on the [[Lp space|SASP14MA1Z]] $L^p(0, \infty)$ for $1 < p < \infty$, with norm $\frac{p}{p-1}$.
It is a basic example of a bounded integral operator, and it is used in Sobolev-type estimates and in the theory of weighted inequalities.
Letting $p \to \infty$ after replacing $a_n$ by $a_n^{1/p}$ leads to [[Carleman's inequality|B54HPY6C5Z]].
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<document source="https://tamnd.github.io/math2/node/S7MC88TW6W.md" tag="S7MC88TW6W" kind="statement">
# Bidiagonalization

## Statement

Let $A \in \R^{m \times n}$ with $m \ge n$.
Then there are [[orthogonal matrices|0W02QESPJQ]] $U \in \R^{m \times m}$ and $V \in \R^{n \times n}$ such that
$$U^T A V = \begin{pmatrix} B \\ 0 \end{pmatrix},$$
where $B \in \R^{n \times n}$ is upper bidiagonal: $b_{ij} = 0$ unless $j = i$ or $j = i + 1$.
Moreover $U$ can be taken as a product of at most $n$ [[Householder reflections|JDH0THTVGM]], and $V$ as a product of at most $n - 2$ of them.
This is the Golub–Kahan bidiagonalization.
The matrices $A$ and $B$ have the same [[singular values|09VMDSRXWJ]].

## Remarks

The construction alternates sides.
A reflection from the left zeros column $1$ below the diagonal.
A reflection from the right acting on columns $2, \dots, n$ then zeros row $1$ to the right of the superdiagonal.
The right reflection does not touch column $1$, so the zeros there survive.
Repeating on the trailing submatrix gives $B$ after $n$ left and $n - 2$ right reflections.
Singular values are preserved because multiplying by orthogonal matrices does not change $A^T A$ up to similarity: $(U^T A V)^T (U^T A V) = V^T A^T A V$.

The cost with Householder reflections is about $4 m n^2 - \tfrac{4}{3} n^3$ floating point operations.
The matrix $B^T B$ is symmetric tridiagonal.
So bidiagonalization is the first phase of the standard [[singular value decomposition|000NPTMX6J]] algorithm, followed by an implicit QR iteration or a divide-and-conquer method on $B$.
One cannot go further to a diagonal matrix in finitely many steps in general, because the singular values are roots of polynomials of arbitrary degree.
Example: let $n = 2$.
One left reflection maps the first column to a multiple of $e_1$, and one more acts on rows $2, \dots, m$ of the second column.
This already gives an upper bidiagonal $2 \times 2$ block, with $V = I$.
When $m \gg n$, it is cheaper to compute a QR factorization first and bidiagonalize the triangular factor.

## Explanations

### L1 human

Bidiagonalization uses rotations and reflections to turn any rectangular matrix into one whose only nonzero entries are on the diagonal and just above it.
These operations keep the singular values, which measure how much the matrix stretches in its principal directions.
The simpler shape makes computing those singular values much faster.

### L2 human

It plays for the singular value decomposition the role that reduction to tridiagonal or Hessenberg form plays for eigenvalues.
A Lanczos-type variant, Golub–Kahan–Lanczos bidiagonalization, builds $B$ one column at a time using only products with $A$ and $A^T$.
It is the basis of the LSQR method for sparse least squares and of methods for a few singular values.
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<document source="https://tamnd.github.io/math2/node/S7MJVSP79N.md" tag="S7MJVSP79N" kind="concept">
# Class operators H S P

## Summary

The class operators $H$, $S$ and $P$ act on classes of algebras of one signature.
They give the homomorphic images, the subalgebras and the direct products of the members of the class.
They are the three closure operations that appear in Birkhoff's characterisation of varieties.
Rules for composing them, such as $SH \le HS$ and $PS \le SP$, lead to Tarski's formula $V(K) = HSP(K)$ for the variety generated by a class $K$.

## Explanations

### L1 human

Start with some algebras, for example the single group $\Z$.
The operator $S$ adds all subgroups, such as $2\Z$.
The operator $H$ adds all homomorphic images, such as $\Z / 6\Z$.
The operator $P$ adds all direct products, such as $\Z \times \Z$.
Doing all three, in the order $P$, then $S$, then $H$, gives every abelian group.
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<document source="https://tamnd.github.io/math2/node/S7WJW2YES2.md" tag="S7WJW2YES2" kind="concept">
# Green's relations

## Summary

Green's relations are five [[equivalence relations|0G8VEZ3WTK]] $\mathcal{L}$, $\mathcal{R}$, $\mathcal{J}$, $\mathcal{H}$ and $\mathcal{D}$ on a [[semigroup|0XMEM406JM]].
Two elements are related when they generate the same left, right or two-sided ideal.
They sort the elements of a semigroup into classes that play the role of the subgroups and cosets of group theory.
In a group, any two elements are related under each of the five relations.
So the relations carry information only for semigroups that are not groups.
An $\mathcal{H}$-class that contains an [[idempotent|0N5NJZKKN3]] is a subgroup, by Green's theorem.

## Explanations

### L1 human

In a group, every element can be reached from every other by multiplying on the left.
In a semigroup this fails, and Green's relations record which elements can reach each other.
Two elements are $\mathcal{L}$-related if each is a left multiple of the other, and $\mathcal{R}$-related if each is a right multiple of the other.
For maps of a finite set, two maps are $\mathcal{R}$-related exactly when they have the same image.
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<document source="https://tamnd.github.io/math2/node/S805N6Z8G7.md" tag="S805N6Z8G7" kind="definition">
# Operator space

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.
A *concrete operator space* is a linear subspace $E \subseteq B(H)$ that is closed in the [[operator norm|06EKW1AKXF]].
For each $n \ge 1$, let $M_n(E)$ be the space of $n \times n$ matrices with entries in $E$.
A matrix $[x_{ij}] \in M_n(E)$ acts on the direct sum $H^n$ of $n$ copies of $H$, and its norm is the operator norm of this action:
$$[x_{ij}] (h_1, \ldots, h_n) = \Big( \sum_{j} x_{1j} h_j, \ldots, \sum_{j} x_{nj} h_j \Big).$$
The operator space is $E$ together with this sequence of norms $(\norm{\cdot}_n)_{n \ge 1}$.

## Remarks

Example: $B(H)$ itself, and every norm-closed subalgebra of it, are operator spaces.
Example: the set of matrices $\begin{pmatrix} a & b \\ 0 & a \end{pmatrix}$ with $a, b \in \C$ is a closed subspace of $M_2(\C) = B(\C^2)$, since every subspace of a finite-dimensional space is closed.
Non-example: if $H$ is infinite-dimensional, the finite-rank operators form a subspace of $B(H)$ that is not closed, so it is not an operator space.
Its closure, the space of compact operators, is one.
Abstractly, an operator space is a complex vector space $E$ with a complete norm on $E$ and norms $\norm{\cdot}_n$ on each $M_n(E)$ with two properties.
First, $\norm{x \oplus y}_{m+n} = \max(\norm{x}_m, \norm{y}_n)$ for $x \in M_m(E)$ and $y \in M_n(E)$.
Second, $\norm{\alpha x \beta}_n \le \norm{\alpha} \norm{x}_m \norm{\beta}$ for $x \in M_m(E)$ and scalar matrices $\alpha$ of size $n \times m$ and $\beta$ of size $m \times n$.
Ruan's theorem states that every such abstract operator space is completely isometric to a concrete one.
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# Riccati difference equation

## Definition

Let $a, b, c, d$ be real numbers with $c \ne 0$ and $ad - bc \ne 0$.
The *Riccati difference equation* with these coefficients is the [[rational difference equation|T8YDEASD1H]]
$$x_{n+1} = \frac{a x_n + b}{c x_n + d} \qquad (n \ge 0),$$
for real sequences $(x_n)_{n \ge 0}$ with $c x_n + d \ne 0$ for all $n$.

## Remarks

Example: $x_{n+1} = 1/(1 + x_n)$ has $a = 0$, $b = c = d = 1$, so $c \ne 0$ and $ad - bc = -1 \ne 0$.
Non-example: $x_{n+1} = (2 x_n + 1)/3$ has $c = 0$; it is an affine linear recurrence, not a Riccati equation.
The condition $ad - bc \ne 0$ excludes the degenerate case where the right side is a constant.
Linearization: if $(y_n)$ satisfies $y_{n+2} = (a + d)\, y_{n+1} - (ad - bc)\, y_n$ with all $y_n \ne 0$, then $x_n = (y_{n+1}/y_n - d)/c$ solves the equation.
To check, put $w_n = c x_n + d$; the equation becomes $w_{n+1} = (a + d) - (ad - bc)/w_n$, and $w_n = y_{n+1}/y_n$ turns this into the linear recurrence.
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# Toda bracket

## Definition

Let $W$, $X$, $Y$, $Z$ be [[pointed spaces|T9DG8DDE75]], and let $f \colon W \to X$, $g \colon X \to Y$, $h \colon Y \to Z$ be pointed maps such that $g \circ f$ and $h \circ g$ are [[null-homotopic|SMEV5BSQ5S]].
Let $CW = (W \times [0, 1]) / (W \times \set{1} \cup \set{w_0} \times [0, 1])$ be the reduced cone on $W$, with $W$ included as $W \times \set{0}$.

1. Choose a pointed null-homotopy of $g \circ f$, that is, a map $A \colon CW \to Y$ that restricts to $g \circ f$ on $W$.
2. Choose a pointed null-homotopy of $h \circ g$, that is, a map $B \colon CX \to Z$ that restricts to $h \circ g$ on $X$.
3. The maps $h \circ A \colon CW \to Z$ and $B \circ Cf \colon CW \to Z$ both restrict to $h \circ g \circ f$ on $W$.
   Gluing them along $W$ gives a map from two cones on $W$ glued along their bases, which is the reduced [[suspension|2VD7GR64JJ]] $\Sigma W$, to $Z$.

The *Toda bracket* $\langle h, g, f \rangle$ is the set of [[homotopy classes|F2F143KQ2M]] of all maps $\Sigma W \to Z$ obtained in this way, over all choices of $A$ and $B$.

## Remarks

Changing the choices of $A$ and $B$ changes the result by an element of $h_* [\Sigma W, Y] + (\Sigma f)^* [\Sigma X, Z]$.
This subgroup of $[\Sigma W, Z]$ is the *indeterminacy*, and $\langle h, g, f \rangle$ is a coset of it.
The bracket depends only on the homotopy classes of $f$, $g$, $h$.

Example: in the stable homotopy groups of spheres, let $2$ denote twice the identity and $\eta$ the stable Hopf map in degree $1$.
Then $2 \eta = 0$, so $\langle 2, \eta, 2 \rangle$ is defined, and it equals $\set{\eta^2}$ in degree $2$.

Non-example: for the identity $\iota$ of $S^n$, the composite $\iota \circ \iota$ is not null-homotopic, so $\langle \iota, \iota, \iota \rangle$ is not defined.

Some authors use the opposite order of the gluing, which changes the bracket by a sign.
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# Incidence structure

## Definition

An *incidence structure* is a triple $(P, L, I)$ in which $P$ and $L$ are disjoint sets and $I \subseteq P \times L$ is a [[binary relation|05AVARPXA2]] from $P$ to $L$.
The elements of $P$ are the *points*, the elements of $L$ are the *lines* (or *blocks*), and $I$ is the *incidence relation*.
A point $p$ and a line $\ell$ are *incident*, and one says that $p$ lies on $\ell$, when $(p, \ell) \in I$.

## Remarks

Example: let $P = \set{1, 2, 3}$, let $L = \set{a, b}$, and let $I = \set{(1, a), (2, a), (2, b), (3, b)}$.
Check: $P$ and $L$ are disjoint and $I$ is a subset of $P \times L$, so $(P, L, I)$ is an incidence structure, a path of two lines meeting at the point $2$.

Non-example: with the same $P$ and $L$, a set $I$ that contains the pair $(1, 2)$ is not an incidence relation, because $(1, 2)$ is not in $P \times L$.

The *dual* of $(P, L, I)$ is $(L, P, I^{\top})$, where $(\ell, p) \in I^{\top}$ exactly when $(p, \ell) \in I$.
An *isomorphism* from $(P, L, I)$ to $(P', L', I')$ is a pair of bijections $P \to P'$ and $L \to L'$ that preserves incidence in both directions.
Often lines are taken to be subsets of $P$, with incidence given by membership.
The general definition also allows two different lines with the same set of points.
A finite incidence structure is described by its [[incidence matrix|9DNFN8M83X]].
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# Principal symbol of an operator on a manifold

## Summary

The principal symbol of a [[differential operator between vector bundles|CGBX1N9V8H]] records the top-order part of the operator.
It is a bundle map that depends on a covector.
It is a function on the cotangent bundle with values in linear maps between the fibres.
It is homogeneous in the covector, of degree the order of the operator.
Unlike the full local expression of the operator, it does not depend on coordinates or trivialisations.
Ellipticity, characteristic directions and the index theorem are all phrased in terms of the principal symbol.
It is the manifold version of the [[principal symbol|F31T650DKY]] on $\R^n$.

## Explanations

### L1 human

In coordinates, a differential operator is a sum of partial derivatives with coefficients.
If one keeps only the derivatives of highest order and replaces each $\partial_j$ by a variable $\xi_j$, one gets a polynomial in $\xi$ with matrix coefficients.
The lower-order terms change when the coordinates change, but this top part does not, once $\xi$ is read as a covector.
For example, the exterior derivative $d$ has principal symbol "wedge with $\xi$": it sends a form $\omega$ to $\xi \wedge \omega$.
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<document source="https://tamnd.github.io/math2/node/S8RG6N1PRT.md" tag="S8RG6N1PRT" kind="concept">
# Laplace's equation

## Summary

Laplace's equation is the [[partial differential equation|GC5BDZ6QBC]] $\Delta u = 0$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]], the sum of the second partial derivatives.
Its solutions are the [[harmonic functions|0DEB0JMN0W]].
It is the model elliptic equation.
It describes steady states: equilibrium temperatures, electrostatic and gravitational potentials in empty space, and potential flows of an ideal fluid.
The [[Dirichlet problem|8D56GQ4AG8]] asks for a solution with given values on the boundary of a region.

## Explanations

### L1 human

Laplace's equation says that at every point the value of $u$ equals the average of its values nearby.
A stretched elastic membrane fixed along a wire frame takes a shape whose height satisfies it.
So does the temperature in a metal plate once it has stopped changing.
In the plane, $u(x, y) = x^2 - y^2$ is a solution, because $u_{xx} = 2$ and $u_{yy} = -2$ add up to zero.
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<document source="https://tamnd.github.io/math2/node/S8S72AA9C8.md" tag="S8S72AA9C8" kind="concept">
# Brown–Peterson cohomology

## Summary

Brown–Peterson cohomology is a [[generalized cohomology theory|H28PSZR7XG]] attached to a prime $p$, represented by a [[ring spectrum|BAW0T68GWK]] $BP$.
After [[complex cobordism|5CZHFNY9F3]] $MU$ is localized at $p$, it splits as a wedge of suspensions of copies of $BP$.
So $BP$ keeps all the $p$-local information of $MU$ in a much smaller object, with coefficient ring $\pi_* BP = \Z_{(p)}[v_1, v_2, \ldots]$ and $\abs{v_i} = 2(p^i - 1)$.
It is the basic tool of the Adams–Novikov spectral sequence and of chromatic homotopy theory.

## Explanations

### L1 human

Complex cobordism is a very powerful cohomology theory, but it is large and hard to compute with.
When one only cares about one prime $p$, most of it repeats the same pattern many times.
Brown–Peterson cohomology is one copy of this pattern.
Its coefficients are polynomials in variables $v_1, v_2, \ldots$, and for $p = 2$ these sit in degrees $2, 6, 14, 30, \ldots$, that is $2(2^i - 1)$.
The variable $v_n$ is the source of the "height $n$" layer studied in chromatic homotopy theory.
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<document source="https://tamnd.github.io/math2/node/S8SNBHVPSP.md" tag="S8SNBHVPSP" kind="concept">
# Persistent homology

## Summary

Persistent homology studies how the homology of a growing family of spaces changes, and records when each homology class is born and when it dies.
The input is a filtration of a space, often built from a finite point cloud at increasing scales.
The output is a barcode or persistence diagram, a multiset of intervals that summarises the shape of the data across all scales at once.
It is the central tool of topological data analysis.

## Explanations

### L1 human

Start with a finite cloud of points in the plane, and draw a disc of radius $r$ around each point.
As $r$ grows, discs merge, loops appear, and later loops fill in.
Persistent homology records for each component and each loop the radius at which it appears and the radius at which it disappears.
Features that last for a long range of radii are taken as real shape; short-lived ones are taken as noise.
For points spread evenly on a circle, one loop is born early and survives for a long time, which reveals the circle.
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<document source="https://tamnd.github.io/math2/node/S8TA91BAN8.md" tag="S8TA91BAN8" kind="definition">
# Foliation

## Definition

Let $M$ be a smooth manifold of dimension $n$, and let $p, q \ge 0$ be integers with $p + q = n$.
A *foliation of dimension $p$* (and *codimension $q$*) of $M$ is a maximal atlas of charts $\varphi_i \colon U_i \to \R^p \times \R^q$ with the following property.
The transition maps have the local form
$$\varphi_j \circ \varphi_i^{-1}(x, y) = \bigl(g_{ij}(x, y), h_{ij}(y)\bigr), \qquad x \in \R^p, \ y \in \R^q.$$
In such a chart, the sets $\varphi_i^{-1}(\R^p \times \set{y}) \cap U_i$ are the *plaques*.
Call two points of $M$ equivalent when they can be joined by a finite chain of plaques, each meeting the next.
The *leaves* of the foliation are the equivalence classes of this relation.

## Remarks

Each leaf is a connected injectively immersed submanifold of dimension $p$, and the leaves partition $M$.
The tangent spaces of the leaves form a subbundle of the [[tangent bundle|WY6K3P4YCG]] that is involutive; by the [[Frobenius theorem|5Q4BHBFXJY]], foliations of dimension $p$ correspond to involutive rank $p$ subbundles.

Example: on the torus $\R^2 / \Z^2$, the lines of slope $\alpha$ give a foliation of dimension $1$.
The charts are restrictions of the linear map $(s, t) \mapsto (s + \alpha t, t - \alpha s)$ composed with local inverses of the quotient map, and transition maps are translations, which have the required form.
If $\alpha$ is irrational, each leaf is dense in the torus.

Non-example: on $\R^3$ with coordinates $(x, y, z)$, the plane field spanned by $\partial_x$ and $\partial_y + x \, \partial_z$ is not involutive, because the bracket of these two fields is $\partial_z$.
So it is not the tangent field of any foliation by surfaces.

## Explanations

### L1 human

A foliation is a way to slice a manifold into thin sheets of one fixed dimension.
In small pieces the slicing looks like the horizontal slices of a stack.
The charts must respect the slices: when you change chart, slices go to slices.
A leaf is what you get by following one slice as far as it goes.
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# Selberg integral

## Statement

Let $n \ge 1$ be an integer, and let $\alpha, \beta, \gamma \in \C$ with $\operatorname{Re} \alpha > 0$, $\operatorname{Re} \beta > 0$ and
$$\operatorname{Re} \gamma > -\min\left( \frac{1}{n}, \frac{\operatorname{Re} \alpha}{n - 1}, \frac{\operatorname{Re} \beta}{n - 1} \right),$$
where the last two terms are omitted when $n = 1$.
Let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
Then
$$\int_{[0,1]^n} \prod_{i=1}^{n} t_i^{\alpha - 1} (1 - t_i)^{\beta - 1} \prod_{1 \le i < j \le n} \abs{t_i - t_j}^{2\gamma} \, dt_1 \cdots dt_n = \prod_{j=0}^{n-1} \frac{\Gamma(\alpha + j\gamma)\, \Gamma(\beta + j\gamma)\, \Gamma(1 + (j+1)\gamma)}{\Gamma(\alpha + \beta + (n + j - 1)\gamma)\, \Gamma(1 + \gamma)}.$$

## Remarks

For $n = 1$ the formula is Euler's [[beta integral|QHNWVSC6AQ]] $\int_0^1 t^{\alpha-1} (1-t)^{\beta-1} \, dt = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}$.
Selberg proved it in 1944.
One proof first treats integer $\gamma$, where the integrand is a polynomial times the beta weights.
It then extends to complex $\gamma$ by Carlson's theorem on entire functions of exponential type.
Another proof, due to Aomoto, derives a recurrence in $\alpha$ by integrating a derivative and using the symmetry of the integrand.
Example: $n = 2$, $\alpha = \beta = 1$, $\gamma = 1$ gives $\int_0^1 \int_0^1 (t_1 - t_2)^2 \, dt_1 dt_2 = \frac{1}{6}$, and the right side is $\frac{\Gamma(1)^2 \Gamma(2)}{\Gamma(3)\Gamma(2)} \cdot \frac{\Gamma(2)^2 \Gamma(3)}{\Gamma(4)\Gamma(2)} = \frac12 \cdot \frac{2}{6} = \frac16$.
The bound on $\operatorname{Re} \gamma$ is needed for convergence: the factor $\abs{t_i - t_j}^{2\gamma}$ is not integrable near the diagonal when $\gamma$ is too negative.

## Explanations

### L1 human

The Selberg integral is an $n$-dimensional version of the beta integral.
It averages a power of the product of all the pairwise distances between $n$ random points in $[0, 1]$, with beta weights at the endpoints.
The surprise is that the answer is an explicit product of gamma values.

### L2 human

The factor $\prod_{i<j} \abs{t_i - t_j}^{2\gamma}$ is the power of a [[Vandermonde determinant|0P84B554AS]], the same factor that governs the eigenvalue densities of [[random matrices|1HS4RT7HK5]].
So the Selberg integral computes the normalizing constants of the Jacobi ensembles, and by limits those of the Laguerre and Gaussian ensembles; the Gaussian case is the [[Mehta integral|4N53T1BZ9S]].
It also has $q$-analogues and analogues for other root systems, linked to the [[Macdonald constant term conjecture|5J25W265JA]].
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# Prime model

## Summary

A prime model of a complete theory is a model that embeds elementarily into every model of the theory.
It is the smallest model in a strong sense, and when it exists in a countable language it is unique up to isomorphism.
For a complete theory in a countable language, the prime models are exactly the countable atomic models, in which every tuple satisfies an isolated [[type|9V71Q3SD1G]].
The field of algebraic numbers is the prime model of the theory of algebraically closed fields of characteristic $0$.

## Explanations

### L1 human

Among all models of a theory, the prime model is the one that sits inside every other model, in a way that keeps all first-order truths.
It contains only the elements that the theory forces to exist.
For algebraically closed fields of characteristic $0$, these are the algebraic numbers: every such field contains a copy of them, and nothing smaller works.
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<document source="https://tamnd.github.io/math2/node/S8X2N1DW8P.md" tag="S8X2N1DW8P" kind="statement">
# Chebyshev's sum inequality

## Statement

Let $a_1 \ge a_2 \ge \dots \ge a_n$ and $b_1 \ge b_2 \ge \dots \ge b_n$ be real numbers.
Then
$$\frac{1}{n} \sum_{i=1}^{n} a_i b_i \ge \Big( \frac{1}{n} \sum_{i=1}^{n} a_i \Big) \Big( \frac{1}{n} \sum_{i=1}^{n} b_i \Big).$$

## Remarks

Proof idea: expand to get the identity
$$n \sum_{i=1}^{n} a_i b_i - \sum_{i=1}^{n} a_i \sum_{j=1}^{n} b_j = \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} (a_i - a_j)(b_i - b_j).$$
Each term on the right is $\ge 0$, because $a_i - a_j$ and $b_i - b_j$ have the same sign for [[monotone sequences|0KN22AAW5R]] ordered the same way.

If one sequence decreases and the other increases, the inequality is reversed.
The common ordering is needed: for $a = (1, 0)$ and $b = (0, 1)$, the left side is $0$ and the right side is $1/4$.

Example: taking $b = a$ gives $n \sum a_i^2 \ge (\sum a_i)^2$, which holds for every real sequence after sorting.

Mathlib states a form as `Monovary.sum_mul_sum_le_card_mul_sum`.

## Explanations

### L1 human

If two lists are sorted in the same order, the average of their products is at least the product of their averages.
Large values meet large values, so the products are pushed up.
For lists sorted in opposite orders the reverse holds.

### L2 human

The inequality follows from the [[rearrangement inequality|6E9TAF8HCY]] by summing over the $n$ cyclic shifts of the second sequence.
It has an integral form: if $f$ and $g$ are both increasing on $[0, 1]$, then $\int_0^1 f g \ge \int_0^1 f \int_0^1 g$.
In probability it says that two increasing functions of the same random variable have nonnegative covariance.
</document>

<document source="https://tamnd.github.io/math2/node/S8Y8RVZY79.md" tag="S8Y8RVZY79" kind="definition">
# Divisible group

## Definition

Let $A$ be an [[abelian group|082GHNPJ53]], written additively.
The group $A$ is *divisible* if for every $a \in A$ and every integer $n \ge 1$ there is $x \in A$ with $nx = a$.

## Remarks

Example: $\Q$ is divisible: given $a \in \Q$ and $n \ge 1$, take $x = a/n$.
Example: $\Q/\Z$ is divisible, as a quotient of the divisible group $\Q$.

Non-example: $\Z$ is not divisible, since there is no integer $x$ with $2x = 1$.
A nonzero finite abelian group is not divisible either: if $\abs{A} = m$ and $a \neq 0$, then $mx = 0 \neq a$ for every $x$.

An abelian group is divisible if and only if it is an [[injective|85H7PRHDPZ]] $\Z$-module.
It is enough to check the condition for $n$ prime.
Every divisible group is isomorphic to a direct sum of copies of $\Q$ and of the [[Prüfer groups|F0XAMM2PPS]] $\Z(p^\infty)$, for various primes $p$.
</document>

<document source="https://tamnd.github.io/math2/node/S95TR54TCG.md" tag="S95TR54TCG" kind="statement">
# Fatou's lemma

## Statement

Let $(X, \Sigma, \mu)$ be a measure space.
Let $f_1, f_2, \dots \colon X \to [0, \infty]$ be measurable functions.
Then $\liminf_{n} f_n$ is measurable, and
$$\int_X \liminf_{n \to \infty} f_n \, d\mu \le \liminf_{n \to \infty} \int_X f_n \, d\mu,$$
where $\liminf$ is the [[limit inferior|0H4R3RESGV]] and the integrals are [[integrals of non-negative functions|DG76G5Z6XW]].

## Remarks

Proof idea: let $g_k = \inf_{n \ge k} f_n$.
The functions $g_k$ increase to $\liminf_n f_n$, and $\int g_k \le \int f_n$ for every $n \ge k$.
The [[monotone convergence theorem|G2S6H789RG]] gives $\int \liminf_n f_n = \lim_k \int g_k \le \liminf_n \int f_n$.

The inequality can be strict.
For $f_n = n \mathbf{1}_{(0, 1/n)}$ on $\R$ with Lebesgue measure, $f_n \to 0$ pointwise but $\int f_n = 1$ for all $n$.

Non-negativity is needed.
For $f_n = -\mathbf{1}_{[n, n+1]}$ on $\R$, the left side is $0$ and the right side is $-1$.

## Explanations

### L1 human

When a sequence of non-negative functions converges, mass can escape, but it cannot appear from nowhere.
So the integral of the limit is at most the limit of the integrals.
A tall thin spike that keeps its area while it shrinks is the typical way mass escapes.

### L2 human

Fatou's lemma is the step between the monotone convergence theorem and the dominated convergence theorem.
It is often used to show that a limit function is integrable: if $\int f_n \le C$ for all $n$, then $\int \liminf f_n \le C$.
It also shows that the $L^p$ norm is lower semicontinuous under pointwise convergence almost everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/S966EZX3ZT.md" tag="S966EZX3ZT" kind="statement">
# Fast Fourier transform

## Statement

There is a constant $C$ with the following property.
For $x \in \C^N$ let $X$ be its [[discrete Fourier transform|TST0GH66A5]],
$$X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i nk/N}, \qquad 0 \le k < N.$$
For every integer $N \ge 2$ some algorithm computes $X$ from any $x \in \C^N$ and precomputed roots of unity.
It uses at most $C N \log_2 N$ complex additions and multiplications.
An algorithm of this kind is called a *fast Fourier transform* (FFT).

## Remarks

For $N = 2^m$ the radix-2 [[Cooley–Tukey algorithm|9N8D9E5TB2]] gives the bound: it uses $\frac{N}{2} \log_2 N$ complex multiplications and $N \log_2 N$ complex additions.
For general $N$, [[Bluestein's algorithm|RY3QGW87HX]] rewrites the transform as a cyclic convolution of a power-of-two length $M < 4N$, computed by three radix-2 FFTs plus $O(N)$ operations.
Evaluating the $N$ defining sums directly needs about $N^2$ multiplications, so the gain is a factor of order $N / \log N$.
The inverse transform has the same form with $e^{+2\pi i nk/N}$ and a factor $1/N$, so it is also computed by an FFT.
The bound counts arithmetic operations.
In floating-point arithmetic the FFT is also accurate: its normwise error bound grows like $\log_2 N$ times the unit roundoff, against about $N$ times for direct evaluation.
Whether $\Theta(N \log N)$ operations are necessary is not known in general; lower bounds are known only for restricted models of computation.

## Explanations

### L1 human

The discrete Fourier transform turns a list of $N$ samples into $N$ frequency coefficients.
Done naively, it needs $N^2$ multiplications, which is a trillion for a million samples.
A fast Fourier transform does the same job in a multiple of $N \log N$ steps, about twenty million for a million samples.
This speed-up is what makes digital audio, image compression and fast multiplication of large numbers practical.

### L2 human

"FFT" names a family of algorithms rather than one method: Cooley–Tukey for composite lengths, the [[prime-factor algorithm|BH3NNAYJT5]] for coprime factors, and [[Rader's|9HYM2ZMS31]] or Bluestein's algorithm for prime lengths.
All exploit the structure of the DFT matrix $(\omega^{nk})$, which factors into $O(\log N)$ sparse matrices.
Combined with the cyclic convolution property of the DFT, the FFT computes convolutions and polynomial products of length $N$ in $O(N \log N)$ operations.
Spectral methods for differential equations and fast polynomial interpolation at roots of unity rest on this.
</document>

<document source="https://tamnd.github.io/math2/node/S977PT8S3S.md" tag="S977PT8S3S" kind="definition">
# Wave equation

## Definition

Let $n \ge 1$, let $c > 0$, and let $\Omega \subseteq \R^n$ be open.
Let $I \subseteq \R$ be an interval.
The *wave equation* with speed $c$ is the [[hyperbolic partial differential equation|92C3S03AQZ]] for $u \colon \Omega \times I \to \R$:
$$\frac{\partial^2 u}{\partial t^2} = c^2 \Delta u,$$
where $\Delta = \sum_{i=1}^{n} \partial^2/\partial x_i^2$ is the [[Laplacian|0E0B6RA8K0]] in the space variables.
The *inhomogeneous wave equation* is $u_{tt} - c^2 \Delta u = f$ for a given function $f$.

## Remarks

Example: for $n = 1$ and any $F \in C^2(\R)$, the function $u(x, t) = F(x - ct)$ is a solution, since $u_{tt} = c^2 F''(x - ct)$ and $u_{xx} = F''(x - ct)$.
Non-example: $u(x, t) = x^2$ fails for $n = 1$, since $u_{tt} = 0$ but $c^2 u_{xx} = 2c^2 \ne 0$.
The operator $\partial_t^2 - c^2\Delta$ is called the d'Alembertian or wave operator.
The Cauchy problem prescribes $u(\cdot, 0) = g$ and $u_t(\cdot, 0) = h$.
</document>

<document source="https://tamnd.github.io/math2/node/S99HFQ9WPH.md" tag="S99HFQ9WPH" kind="definition">
# Conjugate prior

## Definition

Let $\set{f(\cdot \mid \theta) : \theta \in \Theta}$ be a [[parametric model|Q0DXKAQXE1]] given by densities or mass functions, with [[likelihood|05FSTXM92W]] $L(\theta \mid x) = f(x \mid \theta)$.
Let $\mathcal{C}$ be a family of probability distributions on $\Theta$.
The family $\mathcal{C}$ is *conjugate* for the model if the following holds for every [[prior|00NBBSXRQS]] $\pi \in \mathcal{C}$.
For every observation $x$ with $0 < \int_\Theta f(x \mid \theta) \, \pi(d\theta) < \infty$, the [[posterior|0KJ7ZS8AJA]] $\pi(\cdot \mid x)$ belongs to $\mathcal{C}$.
A *conjugate prior* is a member of such a family, usually a family indexed by finitely many hyperparameters.

## Remarks

Example: for $k$ successes in $n$ Bernoulli trials with chance $p \in (0, 1)$, the [[beta distributions|0PP5ZAJPP7]] form a conjugate family.
Indeed, with prior density proportional to $p^{a - 1}(1 - p)^{b - 1}$, the posterior density is proportional to $p^{a + k - 1}(1 - p)^{b + n - k - 1}$, so the posterior is $\mathrm{Beta}(a + k, b + n - k)$.
Non-example: for a Poisson observation $x$ with rate $\lambda > 0$, the log-normal family is not conjugate.
The posterior density is proportional to $\lambda^{x} e^{-\lambda}$ times a log-normal density, and the factor $e^{-\lambda}$ cannot be written in log-normal form.
The gamma family is conjugate for the Poisson rate: a $\mathrm{Gamma}(\alpha, \beta)$ prior, with rate $\beta$, gives the posterior $\mathrm{Gamma}(\alpha + x, \beta + 1)$.
The family of all distributions on $\Theta$ is conjugate, because every posterior is a distribution on $\Theta$; so the notion is useful only for small families.
</document>

<document source="https://tamnd.github.io/math2/node/S9BKNBRV4H.md" tag="S9BKNBRV4H" kind="statement">
# Muckenhoupt's theorem

## Statement

Let $n \ge 1$, let $w$ be a weight on $\R^n$, and let $M$ be the [[Hardy–Littlewood maximal function|VX1A56P5KQ]] operator.
For $1 \le p < \infty$ write $L^p(w)$ for the [[Lp space|SASP14MA1Z]] of the measure $w(x) \, dx$, and $w(E) = \int_E w \, dx$.

1. For $1 < p < \infty$, there is a constant $C$ with $\int_{\R^n} (Mf)^p \, w \, dx \le C \int_{\R^n} \abs{f}^p \, w \, dx$ for all $f \in L^p(w)$ if and only if $w$ is in the [[Muckenhoupt class|T86RSJ8M86]] $A_p$.
2. There is a constant $C$ with $w(\set{Mf > \lambda}) \le \frac{C}{\lambda} \int_{\R^n} \abs{f} \, w \, dx$ for all $f \in L^1(w)$ and $\lambda > 0$ if and only if $w \in A_1$.

## Remarks

Necessity in part 1 comes from testing on one cube.
Let $\sigma = w^{-1/(p - 1)}$ and $f = \sigma \mathbf{1}_Q$.
On $Q$ one has $Mf \ge \frac{1}{\abs{Q}} \int_Q \sigma$, and inserting this in the inequality gives the $A_p$ bound for $Q$, after a cut-off to make $f \in L^p(w)$.
For sufficiency, one first shows the [[weak-type|9534R6VMY0]] bound on $L^p(w)$ from the $A_p$ condition and the covering argument.
Then one uses that an $A_p$ weight is also in $A_{p - \varepsilon}$ for some $\varepsilon > 0$, which follows from a reverse Hölder inequality, and interpolates with the [[Marcinkiewicz interpolation theorem|R1PHMVBJ9J]].

Example: $w(x) = \abs{x}$ on $\R$ is not in $A_2$, and indeed the weighted inequality fails for $p = 2$.
For $0 < \delta < 1$ let $f = \abs{x}^{-1} \mathbf{1}_{[\delta, 1]}$; then $\int f^2 w = \log(1/\delta)$.
For $x \in [-1, 1]$ the interval of radius $2$ centred at $x$ contains $[\delta, 1]$, so $Mf(x) \ge \frac{1}{4} \log(1/\delta)$.
Hence $\int (Mf)^2 w \ge \frac{1}{16} \log^2(1/\delta)$, and the ratio of the two sides tends to infinity as $\delta \to 0$.

## Explanations

### L1 human

The maximal function is bounded on ordinary $L^p$ for $p > 1$.
The theorem says for which weights this stays true when the $L^p$ norm is weighted.
The answer is a simple test on cubes, the $A_p$ condition.

### L2 human

Muckenhoupt proved the theorem in 1972.
Shortly after, Hunt, Muckenhoupt and Wheeden showed that the same $A_p$ condition characterizes the weights for which the [[Hilbert transform|NSCEBXM7ZM]] is bounded on $L^p(w)$, $1 < p < \infty$.
Coifman and Fefferman extended the sufficiency to general Calderón–Zygmund operators.
The theorem is the starting point of weighted theory, including extrapolation and the sharp dependence of operator norms on $[w]_{A_p}$.
</document>

<document source="https://tamnd.github.io/math2/node/S9DK13K5PS.md" tag="S9DK13K5PS" kind="concept">
# Kleene's T predicate

## Summary

Kleene's T predicate is a [[primitive recursive|5NVA2JRHPV]] relation $T(e, x, c)$ about computations.
It says that $c$ codes a halting computation of the program with index $e$ on input $x$.
Together with a primitive recursive function $U$ that reads off the output, it gives Kleene's normal form theorem.
Every partial [[computable function|KZSB5E471K]] has the form $x \mapsto U(\mu c \, T(e, x, c))$ for some $e$.
So a single unbounded search is enough for any computation.

## Explanations

### L1 human

Checking that a given record of a computation is correct is easy: you go through it step by step.
The T predicate is exactly this check, done for every program and every input at once.
Finding a correct record may be impossible, because the program may never stop.
So every computation is "search for a valid record, then read the answer from it".
For instance, the halting set is the set of $e$ for which some $c$ satisfies $T(e, e, c)$.
</document>

<document source="https://tamnd.github.io/math2/node/S9EJKKRHW1.md" tag="S9EJKKRHW1" kind="concept">
# Category of elements

## Summary

The category of elements of a [[functor|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathbf{Set}$ has as objects the pairs $(A, x)$ with $x \in F(A)$.
Its morphisms are the morphisms of $\mathcal{C}$ that carry one chosen element to the other.
It turns a set-valued functor into a category lying over $\mathcal{C}$, and it is a special [[comma category|PC6HG1JEGM]].
It is used to write every [[presheaf|AND2WHZC8P]] as a colimit of representable functors, and it is the simplest case of the [[Grothendieck construction|272T16NWZ9]].

## Explanations

### L1 human

A set-valued functor gives a set $F(A)$ for each object $A$, and functions between these sets for each arrow.
The category of elements spreads all these sets out: each element of each set becomes a point.
There is an arrow from $(A, x)$ to $(B, y)$ for each arrow $f \colon A \to B$ whose function sends $x$ to $y$.
For a group acting on a set, this gives a picture with one point per element of the set and one arrow per group element moving one point to another.
</document>

<document source="https://tamnd.github.io/math2/node/S9FVESJ0CS.md" tag="S9FVESJ0CS" kind="statement">
# Lambek's lemma

## Statement

Let $F \colon \mathcal{C} \to \mathcal{C}$ be a [[functor|28NQ0KRHD3]].
An $F$-algebra is a pair $(A, a)$ of an object $A$ and a morphism $a \colon F(A) \to A$.
A morphism of $F$-algebras $h \colon (A, a) \to (B, b)$ is a morphism $h \colon A \to B$ with $h \circ a = b \circ F(h)$.
If $(I, i)$ is an [[initial object|EK85VSSRMR]] of the category of $F$-algebras, then $i \colon F(I) \to I$ is an isomorphism.

## Remarks

The pair $(F(I), F(i))$ is an $F$-algebra, so by initiality there is a unique algebra morphism $j \colon (I, i) \to (F(I), F(i))$.
Then $i \circ j$ is an algebra morphism from $(I, i)$ to itself, so it equals $\id_I$ by uniqueness.
Also $j \circ i = F(i) \circ F(j) = F(i \circ j) = \id_{F(I)}$, using that $j$ is an algebra morphism.
So $j$ is the inverse of $i$.

Example: on the category of sets let $F(X) = 1 + X$, where $1 = \set{*}$.
The initial $F$-algebra is the set of [[natural numbers|0MWKJ44V5A]] with the map sending $*$ to $0$ and $n$ to $n + 1$.
The lemma says that this map $1 + \N \to \N$ is a bijection: every natural number is either $0$ or a successor, in exactly one way.

Example of a consequence: the [[power set|00E0YZCERH]] functor on sets has no initial algebra.
An initial algebra would give a bijection between a set and its power set, which Cantor's theorem rules out.

The dual statement holds as well: the structure map of a terminal $F$-coalgebra is an isomorphism.

## Explanations

### L1 human

An initial algebra is the smallest structure built by a recipe $F$, like the natural numbers built from zero and successor.
The lemma says that such a structure is a fixed point of the recipe: applying the recipe once more gives nothing new.
For lists, it says that every list is either empty or an element followed by a list, in a unique way.

### L2 human

The lemma is the starting point of the theory of inductive and coinductive data types.
It shows that an initial algebra, when it exists, is a fixed point $F(I) \cong I$ of the functor.
It also gives quick non-existence results, as with the power set functor.
Its dual, for terminal coalgebras, says that infinite streams over $A$ satisfy $A \times S \cong S$.
</document>

<document source="https://tamnd.github.io/math2/node/S9KDHJF5YV.md" tag="S9KDHJF5YV" kind="definition">
# Herglotz–Riesz kernel

## Definition

Let $c \in \C$ and $R > 0$, and let $D = \set{z \in \C : \abs{z - c} < R}$.
For $\zeta$ on the circle $\abs{\zeta - c} = R$ and $z \in \C$ with $z \ne \zeta$, the *Herglotz–Riesz kernel* of $D$ is
$$H(\zeta, z) = \frac{(\zeta - c) + (z - c)}{(\zeta - c) - (z - c)} = \frac{\zeta + z - 2c}{\zeta - z}.$$

## Remarks

For fixed $\zeta$, the function $z \mapsto H(\zeta, z)$ is [[holomorphic|0NAHXR1FRH]] on $D$.
Its real part is the [[Poisson kernel|TCYSWCTYYM]]: with $a = \zeta - c$ and $b = z - c$,
$$\operatorname{Re} H(\zeta, z) = \operatorname{Re} \frac{(a + b)(\bar a - \bar b)}{\abs{a - b}^2} = \frac{R^2 - \abs{z - c}^2}{\abs{\zeta - z}^2},$$
because $b \bar a - a \bar b$ is purely imaginary.
Example: for the unit disc ($c = 0$, $R = 1$), $\zeta = 1$ and $z = 1/2$, one gets $H = 3$, and the Poisson kernel $(1 - 1/4)/(1/4)$ is also $3$.
Non-example: the Cauchy kernel $\zeta/(\zeta - z)$ is also holomorphic in $z$, but at $\zeta = 1$, $z = 1/2$ it equals $2$, so its real part is not the Poisson kernel.
For $c = 0$ the two kernels are related by $H(\zeta, z) = 2\zeta/(\zeta - z) - 1$.
If $u$ is harmonic on a neighbourhood of $\overline{D}$, the [[Schwarz integral formula|61WT6CE08W]] reads
$$u(z) + i v(z) = \frac{1}{2\pi} \int_0^{2\pi} H(c + R e^{i\theta}, z) \, u(c + R e^{i\theta}) \, d\theta,$$
where $v$ is the harmonic conjugate of $u$ on $D$ with $v(c) = 0$.
The same kernel appears in the [[Herglotz representation theorem|9ZBAX4ZPX0]].
Other names in use are Herglotz kernel and Schwarz kernel.
</document>

<document source="https://tamnd.github.io/math2/node/S9VTAYV1VJ.md" tag="S9VTAYV1VJ" kind="statement">
# Kummer's transformation

## Statement

Let $a$ and $b$ be complex numbers with $b \notin \set{0, -1, -2, \dots}$, and let $M$ be [[Kummer's confluent hypergeometric function|6AZ4ZBN1M8]].
Then for every $z \in \C$,
$$M(a, b, z) = e^{z}\, M(b - a, b, -z).$$

## Remarks

Proof idea: put $w(z) = e^z v(-z)$ into [[Kummer's equation|NT917KFR57]] with parameters $a, b$.
A direct computation shows that $w$ is a solution exactly when $v$ solves Kummer's equation with parameters $b - a, b$.
So $e^z M(b - a, b, -z)$ is a solution of the first equation that is analytic at $0$ with value $1$.
For $b \notin \set{0, -1, -2, \dots}$ there is only one such solution, namely $M(a, b, z)$.
Indeed, the other exponent at $0$ is $1 - b$, and a solution with that exponent cannot be analytic with value $1$ there.
Example: for $a = b$, the left side is $e^z$ and the right side is $e^z M(0, b, -z) = e^z \cdot 1$.
Example: for $a = 1$, $b = 2$, the right side is $e^z \frac{e^{-z} - 1}{-z} = \frac{e^z - 1}{z} = M(1, 2, z)$.

## Explanations

### L1 human

Kummer's transformation says that changing $z$ to $-z$ in Kummer's function costs only a factor $e^z$ and a change of the first parameter from $a$ to $b - a$.
It is the confluent analogue of the Pfaff and Euler transformations of the Gauss hypergeometric function.
For instance it turns the trivial fact $M(0, b, z) = 1$ into $M(b, b, z) = e^z$.

### L2 human

The identity is often used to move between large positive and large negative arguments, since one side may be much easier to estimate than the other.
It also shows that if $a$ or $b - a$ is a nonpositive integer, then $M(a, b, z)$ is a polynomial or $e^z$ times a polynomial.
The Tricomi function satisfies a companion identity, $U(a, b, z) = z^{1 - b} U(a - b + 1, 2 - b, z)$, often called Kummer's second transformation.
</document>

<document source="https://tamnd.github.io/math2/node/S9Z6PKTXAS.md" tag="S9Z6PKTXAS" kind="definition">
# Residuated lattice

## Definition

A *residuated lattice* is a structure $(L, \wedge, \vee, \cdot, e, \backslash, /)$ such that:

1. $(L, \wedge, \vee)$ is a [[lattice|WH4SFAACAS]], with order $\le$;
2. $(L, \cdot, e)$ is a [[monoid|0R47MAD57S]];
3. $\backslash$ and $/$ are binary operations on $L$ such that for all $x, y, z \in L$,
$$x \cdot y \le z \iff y \le x \backslash z \iff x \le z / y.$$

The operations $\backslash$ and $/$ are the *left residual* and the *right residual*.

## Remarks

Clause 3 says that $x \backslash z$ is the largest $y$ with $x \cdot y \le z$, and $z / y$ is the largest $x$ with $x \cdot y \le z$.
Example: a [[Heyting algebra|BND6A96S3D]] $H$ with $x \cdot y = x \wedge y$, $e = 1$ and $x \backslash z = z / x = x \to z$ is a residuated lattice.
Clause 3 is then the defining rule $x \wedge y \le z \iff y \le x \to z$ of the Heyting implication.
Example: the group $\Z$ with $\min$, $\max$, $x \cdot y = x + y$, $e = 0$ and $x \backslash z = z / x = z - x$ is a residuated lattice, since $x + y \le z \iff y \le z - x$.
Non-example: $\N$ with $\min$, $\max$, ordinary multiplication and $e = 1$ satisfies clauses 1 and 2, but has no left residual.
For $x = 0$ and $z = 0$, every $y \in \N$ satisfies $0 \cdot y \le 0$, and $\N$ has no largest element.
Many authors also require $L$ to be bounded, or the monoid to be commutative, in which case $x \backslash z = z / x$ is written $x \to z$.
Multiplication in a residuated lattice is monotone and distributes over all existing joins.
</document>

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# Priestley duality

## Statement

Let $L$ be a bounded [[distributive lattice|0HM0HQCK6N]].
A *prime filter* of $L$ is a [[filter|Z15FXM9WM1]] $F$ with $0 \notin F$ such that $a \vee b \in F$ implies $a \in F$ or $b \in F$.
Let $X_L$ be the set of prime filters of $L$, ordered by inclusion.
Give $X_L$ the topology generated by the sets $\set{F : a \in F}$ and $\set{F : a \notin F}$ for $a \in L$.
For a [[Priestley space|ZGBV8V3823]] $X$, let $\mathrm{ClopUp}(X)$ be the lattice of subsets of $X$ that are clopen [[upper sets|ETCE287E2R]].
Its meet and join are intersection and union.

1. $X_L$ is a Priestley space, and the map $a \mapsto \set{F \in X_L : a \in F}$ is an isomorphism of bounded lattices from $L$ onto $\mathrm{ClopUp}(X_L)$.
2. For every Priestley space $X$, the map $x \mapsto \set{U \in \mathrm{ClopUp}(X) : x \in U}$ is a homeomorphism and order isomorphism from $X$ onto $X_{\mathrm{ClopUp}(X)}$.
3. These constructions extend to a dual equivalence of categories.
On one side are the bounded distributive lattices with bounded lattice homomorphisms; on the other, the Priestley spaces with continuous order-preserving maps.
A homomorphism $h \colon L \to M$ corresponds to the map $X_M \to X_L$, $G \mapsto h^{-1}(G)$.

## Remarks

The key input is the [[prime ideal theorem for distributive lattices|ME8GM0VTWP]]: if $a \not\le b$, some prime filter contains $a$ and not $b$.
This gives the injectivity in part 1 and the separation axiom of $X_L$.
Compactness of $X_L$ comes from viewing $X_L$ as a closed subspace of the product space $\set{0, 1}^L$.

Example: if $L$ is finite, the prime filters are the sets ${\uparrow} j$ for $j$ join-irreducible, and $X_L$ is a finite poset with the discrete topology.
Then part 1 becomes [[Birkhoff's representation theorem|6RDM3069KC]] up to reversing the order.

When $L$ is a [[Boolean algebra|0XADJKM699]], every prime filter is an ultrafilter and the order on $X_L$ is trivial.
Then the duality restricts to Stone duality between Boolean algebras and [[Stone spaces|ZV38HA5PAQ]].

## Explanations

### L1 human

Every bounded distributive lattice can be drawn as a lattice of sets: the clopen upper sets of a special ordered space.
Conversely, from such a space one gets back the lattice, and from the lattice the space.
So questions about distributive lattices can be translated into questions about ordered compact spaces, and back.

### L2 human

Priestley duality is a widely used form of duality for distributive lattices.
Homomorphisms turn into continuous order-preserving maps in the opposite direction, so algebraic questions become geometric ones.
It extends Stone duality for Boolean algebras.
Under it, Heyting algebras correspond to [[Esakia spaces|996BWM6ZQ7]], a special class of Priestley spaces.
An equivalent version, due to Stone, uses [[spectral spaces|G3W2WXY2VN]] instead of Priestley spaces.
</document>

<document source="https://tamnd.github.io/math2/node/SA40H8EGJZ.md" tag="SA40H8EGJZ" kind="definition">
# Weak solution

## Definition

Let $U \subseteq \R^n$ be open, let $m \ge 1$, and let
$$P = \sum_{\abs{\alpha} \le m} a_\alpha(x) \, \partial^\alpha$$
be a linear differential operator on $U$ with coefficients $a_\alpha \in C^\infty(U)$, where $\alpha$ runs over multi-indices.
Its formal transpose is the operator $P^t \varphi = \sum_{\abs{\alpha} \le m} (-1)^{\abs{\alpha}} \partial^\alpha (a_\alpha \varphi)$.
Let $f$ be a locally integrable function on $U$.
A locally integrable function $u$ on $U$ is a *weak solution* of $P u = f$ if
$$\int_U u \, P^t \varphi \, dx = \int_U f \, \varphi \, dx \quad \text{for all } \varphi \in C_c^\infty(U),$$
where $C_c^\infty(U)$ is the space of test functions on $U$.

## Remarks

The identity is what integration by parts gives when $u$ is of class $C^m$ and $P u = f$ holds pointwise.
So every classical solution is a weak solution.
Conversely, a weak solution of class $C^m$ satisfies $P u = f$ almost everywhere, and everywhere if $f$ is continuous.
If $u$ has [[weak derivatives|C3EC148VZD]] up to order $m$, the condition says that $\sum a_\alpha D^\alpha u = f$ almost everywhere.

Example: on $\R^2$ with variables $(x, t)$, let $u(x, t) = 1$ for $x > t$ and $u(x, t) = 0$ for $x \le t$, and let $P = \partial_t + \partial_x$, so $P^t \varphi = -(\varphi_t + \varphi_x)$.
Put $\psi(y, t) = \varphi(y + t, t)$; then $\partial_t \psi = \varphi_x + \varphi_t$ at the point $(y + t, t)$.
Hence $\int\!\!\int u \, (\varphi_t + \varphi_x) \, dx \, dt = \int_0^\infty \int_{\R} \partial_t \psi(y, t) \, dt \, dy = 0$, because $\psi(y, \cdot)$ has compact support.
So $u$ is a weak solution of $u_t + u_x = 0$, though it is discontinuous.

Non-example: the step $w(x, t) = 1$ for $x > 0$ and $0$ for $x \le 0$ is not a weak solution of the same equation.
Here $\int\!\!\int w \, (\varphi_t + \varphi_x) \, dx \, dt = -\int_{\R} \varphi(0, t) \, dt$, which is not zero for a test function with $\varphi(0, t) > 0$ near some $t$.

For second-order equations in divergence form, such as $-\sum_{i,j} \partial_j (a_{ij} \partial_i u) = f$ with bounded coefficients, one often asks instead that $u$ lie in a Sobolev space $H^1_0(U)$ and that $\int_U \sum_{i,j} a_{ij} \, \partial_i u \, \partial_j v \, dx = \int_U f v \, dx$ for all $v \in H^1_0(U)$.
This is the variational form of the definition.
</document>

<document source="https://tamnd.github.io/math2/node/SA493EH7DT.md" tag="SA493EH7DT" kind="definition">
# Reduced homology

## Definition

Let $X$ be a nonempty topological space with singular [[chain complex|WQ82C0M3QB]] $C_\bullet(X)$.
The *augmentation* is the homomorphism $\varepsilon \colon C_0(X) \to \Z$ given by $\varepsilon\left(\sum_i n_i \sigma_i\right) = \sum_i n_i$.
The *augmented chain complex* is
$$\cdots \to C_2(X) \xrightarrow{\partial_2} C_1(X) \xrightarrow{\partial_1} C_0(X) \xrightarrow{\varepsilon} \Z \to 0,$$
with $\Z$ placed in degree $-1$; it is a chain complex because $\varepsilon \circ \partial_1 = 0$.
The *reduced [[homology|2J4JJJYP22]]* $\tilde H_n(X)$ is the homology of the augmented chain complex in degree $n$.

## Remarks

For $n \ge 1$, $\tilde H_n(X) = H_n(X)$, where $H_n$ is [[singular homology|M0Q5CFJ7ZH]].
In degree $0$, $H_0(X) \cong \tilde H_0(X) \oplus \Z$, and $\tilde H_{-1}(X) = 0$ because $\varepsilon$ is onto.
For any point $x_0 \in X$, $\tilde H_n(X) \cong H_n(X, \set{x_0})$ for all $n$, a [[relative homology|4BVWH0BN28]] group.

Example: for a one-point space $X$, $C_n(X) \cong \Z$ for every $n \ge 0$, and $\varepsilon$ is an isomorphism, so $\tilde H_n(X) = 0$ for all $n$.

Non-example: the hypothesis that $X$ is nonempty matters.
For $X = \emptyset$ the map $\varepsilon \colon 0 \to \Z$ is not onto, so the augmented complex gives $\tilde H_{-1}(\emptyset) = \Z$, and the formula $H_0 \cong \tilde H_0 \oplus \Z$ fails.
</document>

<document source="https://tamnd.github.io/math2/node/SA5ZDFBHNZ.md" tag="SA5ZDFBHNZ" kind="definition">
# Einstein–Hilbert action

## Definition

Let $M$ be a smooth oriented $4$-manifold, and let $g$ be a [[Lorentzian metric|BY4QYYS788]] on $M$ of signature $(-,+,+,+)$.
Let $R$ be the [[scalar curvature|ZHX2VYQDZ3]] of $g$, and let $\sqrt{-\det g}\; d^4x$ be its volume form in local coordinates $x^0, \dots, x^3$.
Let $G$ be Newton's gravitational constant, $c$ the speed of light, and $\kappa = 8\pi G / c^4$.
For a compact region $K \subseteq M$, the *Einstein–Hilbert action* of $g$ on $K$ is
$$S_{EH}[g] = \frac{1}{2\kappa} \int_K R \,\sqrt{-\det g}\; d^4x.$$

## Remarks

Let $g_t$ be a variation of the metric supported in the interior of $K$.
Write $\delta g^{\mu\nu} = \frac{d}{dt}\big|_{t=0} g_t^{\mu\nu}$.
Then
$$\frac{d}{dt}\Big|_{t=0} S_{EH}[g_t] = \frac{1}{2\kappa} \int_K \Big(R_{\mu\nu} - \tfrac12 R\, g_{\mu\nu}\Big)\, \delta g^{\mu\nu} \sqrt{-\det g}\; d^4x,$$
where $R_{\mu\nu}$ is the Ricci tensor; the term with the variation of $R_{\mu\nu}$ is a divergence and drops out.
So $g$ is a critical point of $S_{EH}$ for all such variations if and only if $R_{\mu\nu} - \tfrac12 R g_{\mu\nu} = 0$, the vacuum [[Einstein field equations|3MCZ97SW6V]].
Example: Minkowski space $\R^4$ with $g = -dt^2 + dx^2 + dy^2 + dz^2$ is flat, so its Ricci tensor vanishes, and it is a critical point.
Non-example: the Euclidean metric $dt^2 + dx^2 + dy^2 + dz^2$ is not Lorentzian, so the definition above does not apply to it; the analogous Riemannian functional $\int R \sqrt{\det g}\, d^4x$ is a different object.
With a cosmological constant $\Lambda$ one replaces $R$ by $R - 2\Lambda$.
Adding a matter action $S_M$ and setting $T_{\mu\nu} = -\frac{2}{\sqrt{-\det g}} \frac{\delta S_M}{\delta g^{\mu\nu}}$ gives $R_{\mu\nu} - \tfrac12 R g_{\mu\nu} = \kappa T_{\mu\nu}$.
On a region with boundary, one adds the Gibbons–Hawking–York boundary term to obtain a well-posed variational problem with the metric fixed on the boundary.
</document>

<document source="https://tamnd.github.io/math2/node/SADNTQAY3G.md" tag="SADNTQAY3G" kind="concept">
# Real form of a Lie group

## Summary

A real form of a complex Lie group $G_\C$ is a closed real Lie subgroup $G$.
Its Lie algebra must be a real form of the Lie algebra of $G_\C$.
So $G$ has half the real dimension of $G_\C$, and complexifying its Lie algebra gives back that of $G_\C$.
One complex group has several real forms, for example $\mathrm{SL}_n(\R)$ and $\mathrm{SU}(n)$ in $\mathrm{SL}_n(\C)$.
The classification of real semisimple groups is largely the classification of real forms of complex semisimple groups.

## Explanations

### L1 human

A complex Lie group can be cut down to a real group in several ways.
Inside the complex $n \times n$ matrices of determinant $1$, one can keep the matrices with real entries.
One can also keep the unitary ones, which preserve lengths in $\C^n$.
Both choices have half the dimension, and both "complexify" back to the whole group.
Such real subgroups are the real forms.
</document>

<document source="https://tamnd.github.io/math2/node/SAGNEFJGVP.md" tag="SAGNEFJGVP" kind="example">
# Mellin transform of a theta series and the zeta function

## Example

For $x > 0$ let $\psi(x) = \sum_{n=1}^{\infty} e^{-\pi n^2 x}$.
With the [[theta function|2F5CYQJVAF]] $\theta(x) = \sum_{n \in \Z} e^{-\pi n^2 x}$, one has $\psi(x) = (\theta(x) - 1)/2$.
For $\operatorname{Re} s > 1$, the [[Mellin transform|N5PVGRJE1T]] of $\psi$ at $s/2$ is
$$\int_0^\infty x^{s/2 - 1} \psi(x) \, dx = \pi^{-s/2} \, \Gamma\left(\frac{s}{2}\right) \zeta(s),$$
where $\Gamma$ is the [[gamma function|0H5SYBSED9]] and $\zeta$ is the [[Riemann zeta function|1XWBY4N0VT]].

Check: for each $n \ge 1$ the dilation rule gives
$$\int_0^\infty x^{s/2 - 1} e^{-\pi n^2 x} \, dx = (\pi n^2)^{-s/2} \, \Gamma\left(\frac{s}{2}\right) = \pi^{-s/2} \, \Gamma\left(\frac{s}{2}\right) n^{-s}.$$
Let $\sigma = \operatorname{Re} s > 1$.
The sum over $n$ of the integrals of the absolute values is $\pi^{-\sigma/2} \Gamma(\sigma/2) \sum_n n^{-\sigma} < \infty$.
So, by Fubini's theorem, the sum and the integral may be exchanged, and the sum of the right sides is $\pi^{-s/2} \Gamma(s/2) \zeta(s)$.

The series $\psi$ is Mellin convergent at $s/2$ exactly for $\operatorname{Re} s > 1$.
Near $\infty$, $\psi(x) \le C e^{-\pi x}$; near $0$, $\psi(x) \sim \frac{1}{2} x^{-1/2}$, which follows from the identity $\theta(1/x) = \sqrt{x}\, \theta(x)$.

## Explanations

### L1 human

The theta series adds up Gaussian bumps $e^{-\pi n^2 x}$, one for each whole number $n$.
Taking its Mellin transform turns each bump into a multiple of $n^{-s}$, so the whole series turns into the zeta function, times a gamma factor.
This is how Riemann connected the symmetry $\theta(1/x) = \sqrt{x}\, \theta(x)$ of the theta function with the [[functional equation of the zeta function|NFXKNDJX9C]].
Splitting the integral at $x = 1$ and using that symmetry gives the analytic continuation of $\zeta$ and the symmetry under $s \mapsto 1 - s$.
</document>

<document source="https://tamnd.github.io/math2/node/SAH0W43RMW.md" tag="SAH0W43RMW" kind="definition">
# Model predictive control

## Definition

Let $f \colon \R^n \times \R^m \to \R^n$ define the discrete-time [[control system|ZPDTVZQN5K]] $x_{k+1} = f(x_k, u_k)$.
Let $\mathbb{X} \subseteq \R^n$, $\mathbb{U} \subseteq \R^m$ and $\mathbb{X}_f \subseteq \mathbb{X}$ be constraint sets.
Let $\ell \colon \R^n \times \R^m \to \R$ be a stage cost, let $V_f \colon \R^n \to \R$ be a terminal cost, and let $N \ge 1$ be the horizon.
For $x \in \R^n$, let $P_N(x)$ be the problem: minimise
$$\sum_{i=0}^{N-1} \ell(z_i, v_i) + V_f(z_N)$$
over $v_0, \dots, v_{N-1} \in \mathbb{U}$, subject to $z_0 = x$, $z_{i+1} = f(z_i, v_i)$, $z_i \in \mathbb{X}$ for $i < N$, and $z_N \in \mathbb{X}_f$.
Assume that $P_N(x)$ has a minimiser $(v_0^*(x), \dots, v_{N-1}^*(x))$ for each $x$ in a set $\mathcal{X}_N$, and fix a choice of it.
The *model predictive control law* is $\kappa_N(x) = v_0^*(x)$ for $x \in \mathcal{X}_N$.
*Model predictive control* operates the system in closed loop by $u_k = \kappa_N(x_k)$: at each time $k$ it solves $P_N(x_k)$ and applies only the first control.

## Remarks

The problem $P_N(x)$ is a finite-horizon [[optimal control|T3V6ZNAMDN]] problem, and the horizon moves forward with time, hence the name receding-horizon control.
Stability of the closed loop is not automatic.
Standard sufficient conditions ask for a terminal set $\mathbb{X}_f$ that is invariant under some local controller.
They also ask for a terminal cost $V_f$ that decreases along this controller by at least the stage cost.
Then the optimal value of $P_N$ is a Lyapunov function of the closed loop.

Example: $n = m = 1$, $f(x, u) = x + u$, no constraints, $\ell(x, u) = x^2 + u^2$, $N = 1$ and $V_f(z) = z^2$.
Then $P_1(x)$ minimises $x^2 + v^2 + (x + v)^2$, whose minimiser is $v = -x/2$.
So $\kappa_1(x) = -x/2$, and the closed loop $x_{k+1} = x_k / 2$ converges to $0$.

Non-example of a stabilising choice: in the same setting with $V_f = 0$, the problem minimises $x^2 + v^2$, so $\kappa_1(x) = 0$.
The closed loop is $x_{k+1} = x_k$, which does not converge to $0$, although each step is optimal for the short horizon.

## Explanations

### L1 human

At each time the controller plans the best moves over the next $N$ steps, applies the first one, observes the new state, and plans again.
</document>

<document source="https://tamnd.github.io/math2/node/SAH4PD2J9T.md" tag="SAH4PD2J9T" kind="statement">
# Clifford theory

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$, and let $F$ be a field.
Let $V$ be an [[irreducible|FX3NEFYFFN]] [[representation|WYGRXCG2RN]] of $G$ over $F$ of finite dimension.
For a representation $W$ of $N$ and $g \in G$, the conjugate ${}^g W$ is the space $W$ with $n \in N$ acting as $g^{-1} n g$ acts on $W$.

1. The restriction $V|_N$ is a [[semisimple|N3MRHDKDGC]] representation of $N$. There are an irreducible representation $W$ of $N$, an integer $e \ge 1$ and elements $g_1, \ldots, g_t \in G$ with $V|_N \cong e \, ({}^{g_1}W \oplus \cdots \oplus {}^{g_t}W)$. The ${}^{g_i}W$ are pairwise non-isomorphic, and every $G$-conjugate of $W$ is isomorphic to one of them.
2. Let $T = \set{g \in G : {}^g W \cong W}$, the inertia group of $W$, and let $U$ be the sum of the subrepresentations of $V|_N$ isomorphic to $W$. Then $U$ is invariant under $T$, $U$ is an irreducible representation of $T$, $t = [G : T]$, and $V \cong \operatorname{Ind}_T^G U$.

## Remarks

For the proof of part 1, take an irreducible $N$-subrepresentation $W$ of $V$.
Each $gW$ is again $N$-invariant and irreducible, and isomorphic to a conjugate of $W$.
The sum $\sum_g gW$ is a nonzero $G$-invariant subspace, so it is $V$.
The elements of $G$ permute the isotypic components of $V|_N$ transitively, and $T$ is the [[stabiliser|0AYMP41THN]] of the component $U$.
Part 2 follows, since a representation that is a direct sum of transitively permuted subspaces is [[induced|JJP4JFH3C6]] from a stabiliser.
Example: let $G = S_3$, $N = A_3 \cong \Z/3$ and let $V$ be the irreducible complex representation of degree $2$.
Then $V|_N$ is the sum of the two non-trivial characters of $A_3$, which conjugation by a transposition swaps.
Here $e = 1$, $t = 2$, $T = N$, and $V = \operatorname{Ind}_{A_3}^{S_3} \omega$ for a non-trivial character $\omega$ of $A_3$.
The theorem also holds when $G$ is any group and $N$ has finite index in $G$.

## Explanations

### L1 human

Clifford theory explains how an irreducible representation of a group looks when one only watches a normal subgroup.
It breaks into equally many copies of a few irreducible pieces, and those pieces are all twists of one another by the big group.
Moreover the whole representation can be rebuilt by induction from the subgroup that fixes one piece.

### L2 human

This is the basic tool for building representations of a group from those of a normal subgroup and of the quotient.
It applies well to solvable groups, which have chains of normal subgroups.
In character language it states that the irreducible constituents of $\chi|_N$ form one $G$-orbit and all occur with the same multiplicity $e$.
The Clifford correspondence, part 2, gives a bijection between irreducible characters of $T$ lying over $W$ and irreducible characters of $G$ lying over $W$, by induction.
It is a standard step in inductive arguments in character theory, which pass from $G$ to a normal subgroup.
</document>

<document source="https://tamnd.github.io/math2/node/SAHAEZNCVD.md" tag="SAHAEZNCVD" kind="concept">
# Homological algebra

## Summary

Homological algebra is the part of algebra that studies [[chain complexes|WQ82C0M3QB]], their [[homology|2J4JJJYP22]], and the [[exact sequences|JB7ZBFBA9C]] and functors built from them.
It grew out of algebraic topology, where homology groups of spaces are computed from chain complexes, and out of the study of syzygies of modules.
Its main tools include [[abelian categories|Y9CZ3MTNF1]], [[projective|YXMXAGVC1Q]] and [[injective objects|5QRFTBF4PG]], resolutions, [[derived functors|NPMC3K6J2F]] such as [[Ext|4P7GBWJ4RR]] and [[Tor|DDQRERAQCV]], [[spectral sequences|GG8AYV847V]] and [[derived categories|5ST7NVJFSM]].
It provides a common language for algebraic topology, algebraic geometry, commutative algebra and representation theory.

## Explanations

### L1 human

Homological algebra studies sequences of objects joined by maps in which applying two maps in a row always gives zero.
The homology measures how far such a sequence is from being exact, that is, how much of what one map kills is not produced by the previous map.
In topology this measures holes: a loop that is not the boundary of any surface detects a hole.
In algebra the same machinery measures, for example, how far a module is from being free.
A small example is $0 \to \Z \xrightarrow{2} \Z \to 0$, whose homology at the second $\Z$ is $\Z/2\Z$; this records that multiplication by $2$ is not onto.
</document>

<document source="https://tamnd.github.io/math2/node/SAQ5EFQP7H.md" tag="SAQ5EFQP7H" kind="definition">
# Frame operator

## Definition

Let $H$ be a [[Hilbert space|HM2HHE3J51]] with [[inner product|0NB0AWGJBW]] $\ip{\cdot}{\cdot}$.
Let $(f_i)_{i \in I}$ be a *frame* of $H$: there are constants $0 < A \le B < \infty$ with $A \norm{f}^2 \le \sum_{i \in I} \abs{\ip{f}{f_i}}^2 \le B \norm{f}^2$ for all $f \in H$.
The *frame operator* of $(f_i)_{i \in I}$ is the map $S \colon H \to H$,
$$S f = \sum_{i \in I} \ip{f}{f_i} f_i,$$
where the series converges unconditionally in $H$.

## Remarks

$S$ is bounded, [[self-adjoint|0NEGN5CRSC]] and [[positive|77020CY5ZW]], since $\ip{Sf}{f} = \sum_i \abs{\ip{f}{f_i}}^2$.
The frame inequality reads $A I \le S \le B I$, so $S$ is [[invertible|CPAD5W1CGW]] with $\norm{S^{-1}} \le A^{-1}$.
The family $(S^{-1} f_i)_{i \in I}$ is the *canonical dual frame*, and $f = \sum_i \ip{f}{S^{-1} f_i} f_i = \sum_i \ip{f}{f_i} S^{-1} f_i$ for all $f$.
With the analysis operator $T f = (\ip{f}{f_i})_{i \in I} \in \ell^2(I)$, one has $S = T^* T$.
A frame is tight with bound $A$ exactly when $S = A I$.

Example: for an orthonormal basis $(e_i)$, $S f = \sum_i \ip{f}{e_i} e_i = f$, so $S = I$.

Example: in $\C^2$ with the frame $(e_1, e_1, e_2)$, $S(x, y) = 2x e_1 + y e_2$, so $S = \mathrm{diag}(2, 1)$, with spectrum $\set{1, 2}$ equal to the optimal frame bounds.

Non-example: in $\ell^2(\N)$ the family $(e_n / n)_{n \ge 1}$ satisfies only the upper bound.
The map $f \mapsto \sum_n \ip{f}{e_n/n} e_n/n$ is still defined, but it sends $e_n$ to $e_n/n^2$, so it has no bounded inverse; the family is not a frame and the reconstruction formula fails.
</document>

<document source="https://tamnd.github.io/math2/node/SAR7PBYF07.md" tag="SAR7PBYF07" kind="counterexample">
# Moore plane

## Counterexample

Let $\Gamma = \set{(x, y) \in \R^2 : y \ge 0}$ be the closed upper half-plane.
Give a point $(x, y)$ with $y > 0$ the usual Euclidean open discs around it, of radius less than $y$, as basic [[neighbourhoods|0ZCY0Z80B6]].
Give a point $(x, 0)$ the basic neighbourhoods $\set{(x, 0)} \cup D$, where $D$ is an open disc of radius $r > 0$ centred at $(x, r)$.
Such a disc $D$ is tangent to the $x$-axis at $(x, 0)$.
The space $\Gamma$ with this topology is the Moore plane, also called the Niemytzki plane.
It is a [[Tychonoff space|C5N2J2K6Z3]], hence [[Hausdorff|0S1FJ8VEX9]] and regular, but it is not [[normal|0NSR9V8TED]].

## Remarks

The $x$-axis $A = \set{(x, 0) : x \in \R}$ is [[closed|0ZCK8DQYN1]], since every point with $y > 0$ has a disc neighbourhood missing $A$.
The subspace $A$ is discrete, because a tangent disc neighbourhood of $(x, 0)$ meets $A$ only in $(x, 0)$.
So every subset of $A$ is closed in $\Gamma$.
The points with rational coordinates and $y > 0$ form a countable dense set, so $\Gamma$ is [[separable|0Z2P41SVDP]].
By Jones' lemma, a separable normal space has no closed discrete subset $A$ with $2^{\abs{A}} > 2^{\aleph_0}$.
Here $\abs{A} = 2^{\aleph_0}$, so $\Gamma$ is not normal.
For example, the points of $A$ with rational $x$ and the points of $A$ with irrational $x$ are disjoint closed sets that cannot be separated by disjoint open sets.
Since $\Gamma$ is Tychonoff, it is a subspace of a compact Hausdorff space, which is normal; so a subspace of a normal space need not be normal.

## Explanations

### L1 human

Away from the bottom edge, the Moore plane is the ordinary half-plane.
On the bottom edge, a point is only near the points of small discs that sit on top of it and touch the edge there.
So the points of the edge cannot see each other, and the edge breaks up into isolated points.
This makes it impossible to pull apart some disjoint closed pieces of the edge with open sets.
</document>

<document source="https://tamnd.github.io/math2/node/SARA216BGB.md" tag="SARA216BGB" kind="definition">
# Variety of algebras

## Definition

Let $\sigma$ be a signature of function symbols, and let $\Sigma$ be a set of [[identities|PKD440G2XW]] $s \approx t$ in the language of $\sigma$.
An algebra $A$ of signature $\sigma$ *satisfies* $s \approx t$ if $s$ and $t$ take the same value in $A$ for every assignment of the variables.
The class $\operatorname{Mod}(\Sigma)$ is the class of all algebras of signature $\sigma$ that satisfy every identity in $\Sigma$.
A *variety* (or *equational class*) of signature $\sigma$ is a class $\mathcal{V}$ of algebras of signature $\sigma$ with $\mathcal{V} = \operatorname{Mod}(\Sigma)$ for some $\Sigma$.

## Remarks

Example: abelian groups, in the signature $(+, -, 0)$, form a variety.
It is $\operatorname{Mod}(\Sigma)$ for $\Sigma$ the identities $(x + y) + z \approx x + (y + z)$, $x + 0 \approx x$, $x + (-x) \approx 0$ and $x + y \approx y + x$.
Non-example: fields do not form a variety of signature $(+, \cdot, -, 0, 1)$.
Every variety is closed under direct products, but the product $\F_2 \times \F_2$ of two fields is not a field, since $(1, 0) \cdot (0, 1) = (0, 0)$.
By Birkhoff's theorem, a class is a variety if and only if it is closed under homomorphic images, subalgebras and direct products.
Some authors write $\operatorname{Mod}(\Sigma)$ as $V(\Sigma)$.
</document>

<document source="https://tamnd.github.io/math2/node/SARNZPAKE5.md" tag="SARNZPAKE5" kind="definition">
# Kolmogorov–Smirnov test

## Definition

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with [[cumulative distribution function|0MXMS3AE13]] $F$, and let $F_n$ be their [[empirical distribution function|A05WR6TEK5]].
Let $F_0$ be a fixed continuous cumulative distribution function.
The *one-sample Kolmogorov–Smirnov statistic* is
$$D_n = \sup_{x \in \R} \abs{F_n(x) - F_0(x)}.$$
The *Kolmogorov–Smirnov test* of the null hypothesis $H_0 \colon F = F_0$ at level $\alpha$ rejects $H_0$ when $D_n > c_{n, \alpha}$, where $c_{n, \alpha}$ is chosen so that $P(D_n > c_{n, \alpha}) \le \alpha$ under $H_0$.

Let two independent i.i.d. samples of sizes $n$ and $m$ have empirical distribution functions $F_n$ and $G_m$.
The *two-sample statistic* is $D_{n,m} = \sup_{x} \abs{F_n(x) - G_m(x)}$.
It tests whether the two samples have the same continuous distribution.

## Remarks

Under $H_0$ the law of $D_n$ is the same for every continuous $F_0$, because $F_0(X_i)$ is uniform on $[0, 1]$.
As $n \to \infty$, $\sqrt{n} D_n$ converges in distribution to the Kolmogorov distribution, the law of $\sup_{0 \le t \le 1} \abs{B(t)}$ for a [[Brownian bridge|3MCVMBAGD1]] $B$:
$$\lim_{n \to \infty} P(\sqrt{n} D_n \le t) = 1 - 2 \sum_{k=1}^{\infty} (-1)^{k-1} e^{-2k^2 t^2}, \qquad t > 0.$$
For ordered data $x_{(1)} \le \dots \le x_{(n)}$ the supremum is a finite maximum: $D_n = \max_{i} \max\big(i/n - F_0(x_{(i)}), \, F_0(x_{(i)}) - (i - 1)/n\big)$.

Example: $n = 1$, $X_1 = 0.3$ and $F_0(x) = x$ on $[0, 1]$.
Then $F_1$ jumps from $0$ to $1$ at $0.3$, and $D_1 = \max(0.3, 1 - 0.3) = 0.7$.

Non-example: testing normality with the mean and variance estimated from the same data is not this test.
Then $F_0$ is not fixed in advance, and the null law of $D_n$ changes; the corrected procedure is the Lilliefors test.
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# Lp space

## Summary

For $1 \le p < \infty$, the space $L^p(\mu)$ consists of the measurable functions $f$ on a measure space with $\int \abs{f}^p \, d\mu < \infty$.
Two functions that agree almost everywhere count as the same element.
The norm is $\norm{f}_p = (\int \abs{f}^p \, d\mu)^{1/p}$, and with this norm $L^p(\mu)$ is a Banach space.
The case $p = 2$ is a Hilbert space.
These spaces are the basic function spaces of analysis, probability and partial differential equations.

## Explanations

### L1 human

The number $\norm{f}_p$ measures the size of a function by averaging the $p$-th power of its absolute value.
A larger $p$ gives more weight to the places where the function is large.
Two functions that differ only on a set of measure zero have the same size, so they are treated as the same element.
For example, $f(x) = x^{-1/3}$ on $(0, 1]$ lies in $L^2$, since $\int_0^1 x^{-2/3} \, dx = 3$, but not in $L^3$, since $\int_0^1 x^{-1} \, dx = \infty$.
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# Quaternionic analysis

## Summary

Quaternionic analysis studies functions of a quaternion variable with quaternion values, by analogy with complex analysis.
The naive analogue of the complex derivative is too restrictive.
So the central class is that of Fueter regular functions, the solutions of a quaternionic Cauchy–Riemann equation.
These functions have an integral formula like Cauchy's and are harmonic in $\R^4$.
A wider theory, [[Clifford analysis|DKVSAZQWF7]], extends these ideas to all dimensions.

## Explanations

### L1 human

The quaternions extend the complex numbers with three imaginary units $i, j, k$, but their multiplication is not commutative.
In complex analysis, a function is holomorphic when it has a derivative $\lim_{h \to 0} (f(z + h) - f(z))/h$.
For quaternions this limit exists only for affine functions, so this road leads nowhere.
Instead one copies the Cauchy–Riemann equations: one adds the four partial derivatives, each multiplied by $1, i, j$ or $k$, and asks the sum to be $0$.
Surprisingly, even $f(q) = q$ fails this test, while some functions of the real coordinates pass it.
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# Artin group

## Summary

An Artin group, or Artin–Tits group, is a [[group|G5S1M9]] defined by a Coxeter matrix.
Each pair of generators satisfies a braid relation of a given length, or no relation.
Adding the relations $s_i^2 = 1$ turns it into the corresponding [[Coxeter group|G6FND7X69J]].
The [[braid groups|BNR4H5GCTP]], the [[free groups|0KTW26C80J]], the free abelian groups and the [[right-angled Artin groups|6D1DV09D6P]] are Artin groups.
Many basic questions about general Artin groups, such as whether they are torsion-free, are still open.

## Explanations

### L1 human

A braid relation says that two generators can be pushed past each other in a fixed pattern.
For example, $aba = bab$ is the relation of two neighbouring crossings in a braid with three strands.
An Artin group chooses, for each pair of generators, how long the alternating words in such a relation are, or imposes no relation at all.
It is like a Coxeter group without the rule that each generator squares to the identity, so the generators have infinite order.
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# Lebesgue differentiation theorem

## Statement

Let $n \ge 1$, let $\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$, and let $f \colon \R^n \to \R$ be locally [[integrable|X7GTF3D98F]], that is, integrable on every bounded measurable set.
For $x \in \R^n$ and $r > 0$, let $B(x, r)$ be the [[open ball|0YE5868HZB]] of centre $x$ and radius $r$.
Then for [[almost every|RARTMA0ZEC]] $x \in \R^n$,
$$\lim_{r \to 0^+} \frac{1}{\lambda(B(x, r))} \int_{B(x, r)} \abs{f(y) - f(x)} \, d\lambda(y) = 0.$$
In particular, $\lim_{r \to 0^+} \frac{1}{\lambda(B(x, r))} \int_{B(x, r)} f \, d\lambda = f(x)$ for almost every $x$.

## Remarks

The conclusion says that almost every point is a [[Lebesgue point|19WRHM5NAT]] of $f$.
The proof first notes that the claim is clear for continuous $f$ with compact support, which are dense in $L^1$.
The error for general $f$ is controlled by the [[Hardy–Littlewood maximal function|VX1A56P5KQ]], which satisfies the weak type bound $\lambda(\set{Mf > t}) \le 3^n \norm{f}_1 / t$.
This bound comes from the [[Vitali covering lemma|FCA0J1X07K]].

Example: for $n = 1$ and $f$ integrable on $\R$, the theorem says $F(x) = \int_0^x f$ satisfies $F'(x) = f(x)$ for almost every $x$.

The balls can be replaced by other families that shrink nicely, such as cubes containing $x$.
They cannot be replaced by arbitrary rectangles containing $x$: in dimension $n \ge 2$, for rectangles of unbounded eccentricity the conclusion can fail for some integrable $f$.

## Explanations

### L1 human

Average an integrable function over smaller and smaller balls around a point.
For almost every point, these averages tend to the value of the function at that point.
So an integrable function, however rough, is determined by its local averages.

### L2 human

The theorem is the measure-theoretic form of the fundamental theorem of calculus.
It gives Lebesgue's density theorem when $f$ is an indicator function.
It extends to Radon measures on $\R^n$ through the Besicovitch covering theorem, and to doubling metric measure spaces.
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# Injective tensor product

## Definition

Let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\mathbb{K} = \R$ or $\C$, with [[continuous duals|4A82S5FVRT]] $X^*$ and $Y^*$.
Let $X \otimes Y$ be their algebraic [[tensor product|0BF2DE2SYK]].
For $u = \sum_{i=1}^n x_i \otimes y_i$ in $X \otimes Y$, let
$$\varepsilon(u) = \sup \Big\{ \Big| \sum_{i=1}^n f(x_i) \, g(y_i) \Big| : f \in X^*, \ g \in Y^*, \ \norm{f} \le 1, \ \norm{g} \le 1 \Big\}.$$
This number does not depend on the representation of $u$, and $\varepsilon$ is a norm on $X \otimes Y$, the *injective norm*.
The *injective tensor product* $X \check{\otimes}_\varepsilon Y$ is the completion of $X \otimes Y$ for $\varepsilon$.

## Remarks

The norm is a cross norm: $\varepsilon(x \otimes y) = \norm{x} \, \norm{y}$, by the Hahn–Banach theorem.
It is at most the [[projective norm|685ZV73JJ8]] $\pi$.
The bounds on $f$ and $g$ are needed: without them, scaling $f$ makes the supremum infinite for every $u \ne 0$.

Example: let $X = Y = \R^2$ with the Euclidean norm and $u = e_1 \otimes e_1 + e_2 \otimes e_2$.
Then $\varepsilon(u) = \sup \abs{\ip{f}{g}} = 1$, over unit vectors $f, g$.
The projective norm of $u$ is $2$, so the two norms differ.

For compact Hausdorff spaces $K$ and $L$, the map $f \otimes g \mapsto ((s, t) \mapsto f(s) g(t))$ extends to an isometry of $C(K) \check{\otimes}_\varepsilon C(L)$ onto $C(K \times L)$.
More generally, $C(K) \check{\otimes}_\varepsilon X$ is the space of continuous functions from $K$ to $X$ with the sup norm.
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# MUSCL scheme

## Definition

Let $f \colon \R \to \R$ be a $C^1$ function and consider the [[conservation law|8RA7AMWSAF]] $u_t + f(u)_x = 0$.
Take a uniform grid of cells $[x_{i-1/2}, x_{i+1/2}]$ of width $\Delta x$, and let $u_i(t)$ be the cell averages of a [[finite volume method|HGKN8E09RS]].
Let $H \colon \R^2 \to \R$ be a numerical flux, for example one given by a [[Riemann solver|ZY4GY93C4B]], with $H(v, v) = f(v)$ for all $v$.
Let $\phi \colon \R \to \R$ be a [[flux limiter|NZ8XFNCKQW]] with $\phi(r) = 0$ for $r \le 0$.
A *MUSCL scheme* is the semi-discrete scheme built in three steps.

1. Slope ratio: $r_i = \dfrac{u_i - u_{i-1}}{u_{i+1} - u_i}$, with $\phi(r_i)$ read as $0$ when $u_{i+1} = u_i$.
2. Limited reconstruction at each interface:
$$u^L_{i+1/2} = u_i + \tfrac{1}{2} \phi(r_i) (u_{i+1} - u_i), \qquad u^R_{i+1/2} = u_{i+1} - \tfrac{1}{2} \phi(r_{i+1}) (u_{i+2} - u_{i+1}).$$
3. Flux update:
$$\frac{d u_i}{dt} = -\frac{1}{\Delta x} \Big( H(u^L_{i+1/2}, u^R_{i+1/2}) - H(u^L_{i-1/2}, u^R_{i-1/2}) \Big).$$

The system of ordinary differential equations is then advanced in time, for example by a Runge–Kutta method.

## Remarks

Example: take the minmod limiter $\phi(r) = \max(0, \min(1, r))$.
For linear data $u_j = j \Delta x$ one gets $r_i = 1$ and $\phi(r_i) = 1$, so $u^L_{i+1/2} = u_i + \tfrac12 \Delta x$, the exact interface value.
For step data $u_{i-1} = u_i = 0$ and $u_{i+1} = u_{i+2} = 1$ one gets $r_i = 0$, so $u^L_{i+1/2} = u_i = 0$ and no overshoot appears.

Non-example: the choice $\phi \equiv 1$ fails the condition $\phi(r) = 0$ for $r \le 0$.
It gives unlimited central slopes, and the resulting scheme creates oscillations near discontinuities.

With $\phi \equiv 0$ the reconstruction is constant and the scheme reduces to a first-order scheme of Godunov type.
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# Modification of a complex space

## Summary

A modification of a [[reduced complex space|831N2Q4GCB]] $Y$ is a [[proper|15XEBGNGHC]] surjective holomorphic map $X \to Y$ that is biholomorphic outside thin analytic subsets.
It changes $Y$ only over a nowhere dense analytic set, where it may replace points by larger sets.
[[Blowing up|C4KGYAFPN7]], normalization and resolution of singularities give modifications.
Modifications are the analytic counterpart of proper birational morphisms in algebraic geometry.

## Explanations

### L1 human

A modification keeps a space as it is except along a small bad set, where it may replace points by curves or other larger pieces.
The basic example is the blow-up of the plane $\C^2$ at the origin.
It keeps every point except the origin, and it replaces the origin by a projective line that records the directions of lines through the origin.
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# Denjoy–Carleman–Ahlfors theorem

## Statement

Let $f$ be an [[entire function|0419794R0T]] of [[order|SHF8Z7GMFX]] $\rho$.
Call $a \in \C$ a finite asymptotic value of $f$ if there is a continuous curve $\gamma \colon [0, 1) \to \C$ with $\gamma(t) \to \infty$ and $f(\gamma(t)) \to a$ as $t \to 1$.
If $f$ has $n \ge 1$ distinct finite asymptotic values, then $n \le 2\rho$.

## Remarks

Between two curves along which $f$ tends to different finite limits, $f$ must be unbounded, so $f$ has at least $n$ separate regions near $\infty$ where $\abs{f}$ is large.
Ahlfors's proof bounds the growth of $f$ in such regions with an estimate for conformal maps of strip-like domains.
This estimate is now called the Ahlfors distortion theorem.
It is proved by the length–area method, which later grew into the theory of [[extremal length|HX72KT9D0G]].
For example, $e^z$ has order $1$ and the single finite asymptotic value $0$, reached along the negative real axis, so $1 \le 2$.
For $n = 1$ the theorem says that an entire function of order less than $1/2$ has no finite asymptotic value.

## Explanations

### L1 human

An asymptotic value is a limit that the function approaches along some road to infinity.
The theorem says that an entire function cannot have many different such limits unless it grows fast.
Each extra finite limit costs half a unit of order.

### L2 human

The order is defined by $\rho = \limsup_{r \to \infty} \log \log M(r)/\log r$, where $M(r)$ is the maximum of $\abs{f}$ on $\abs{z} = r$.
The case $n = 1$ is consistent with the $\cos \pi\rho$ theorem, which shows that functions of order less than $1/2$ are large on a sequence of whole circles.
The method of the proof, through extremal length and conformal invariants, became a central tool of geometric function theory.
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# Quasiconformal mapping

## Summary

A quasiconformal mapping is an orientation-preserving [[homeomorphism|0QT00PB5G2]] between plane domains that distorts infinitesimal circles into ellipses of bounded eccentricity.
The bound $K \ge 1$ on the ratio of the axes is the maximal dilatation, and $1$-quasiconformal maps are exactly the [[conformal maps|0YYT36R4YD]].
Quasiconformal maps are much more flexible than conformal maps, yet they keep many of their compactness and boundary properties.
They are a basic tool in Teichmüller theory, complex dynamics and the theory of Kleinian groups.

## Explanations

### L1 human

A conformal map sends tiny circles to tiny circles.
A quasiconformal map is allowed to squash tiny circles into tiny ellipses, as long as the ellipses never get too thin.
The number $K$ measures the worst ratio of the long axis to the short axis.
For example, the stretch $(x, y) \mapsto (2x, y)$ of the plane turns every circle into an ellipse with axis ratio $2$, so it is $2$-quasiconformal.
Such maps can bend shapes in ways that conformal maps cannot, which makes them easy to build by hand.
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# Weyl–von Neumann theorem

## Statement

Let $H$ be a [[separable|0Z2P41SVDP]] complex [[Hilbert space|HM2HHE3J51]], and let $\varepsilon > 0$.
Let $A$ be a bounded operator on $H$ with $A = A^*$, where $A^*$ is the [[adjoint|6HSJBJ8TTQ]] of $A$.
Then there are an [[orthonormal basis|0S9GGTR0EX]] $(e_n)$ of $H$, real numbers $(\lambda_n)$ and a [[compact operator|M2NEP1K5AE]] $K$ on $H$ such that:

1. $K = K^*$ and $\norm{K} < \varepsilon$;
2. $(A + K) e_n = \lambda_n e_n$ for every $n$.

## Remarks

In words, every bounded self-adjoint operator is the sum of a diagonal operator and a compact self-adjoint operator of arbitrarily small norm.
Weyl proved the version with $K$ compact, and von Neumann showed that $K$ can be chosen in the Hilbert–Schmidt class, the [[Schatten class|153N5DW55Z]] $S_2$, with Hilbert–Schmidt norm less than $\varepsilon$.
The idea of the proof is to use spectral projections of $A$ for small intervals.
They give finite-dimensional subspaces that are almost invariant under $A$, and $A$ is diagonalized up to a small error on each of them.
Example: the multiplication operator $(Af)(x) = x f(x)$ on $L^2([0,1])$ has no eigenvalues at all, yet by the theorem a compact perturbation of small norm has an orthonormal basis of eigenvectors.
The theorem does not extend to trace class perturbations.
By the Kato–Rosenblum theorem, if $A$ has a nonzero absolutely continuous part, then so does $A + K$ for every trace class self-adjoint $K$.
Then $A + K$ is not diagonal.

## Explanations

### L1 human

A self-adjoint matrix can be diagonalized exactly, but a self-adjoint operator on an infinite-dimensional space may have no eigenvectors at all.
The theorem says that this failure is small: after adding a tiny compact correction, there is a full orthonormal basis of eigenvectors.
So, up to compact errors, every self-adjoint operator is diagonal.

### L2 human

The theorem shows that the point spectrum and the continuous spectrum of a self-adjoint operator are not stable under compact perturbations, while the essential spectrum is, by Weyl's theorem.
It is a starting point of the theory of unitary equivalence modulo compact operators, which later led to the Brown–Douglas–Fillmore theory of extensions of C*-algebras.
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# Binary quadratic form

## Summary

A binary quadratic form is a [[quadratic form|0RBQS1JJCR]] in two variables, $ax^2 + bxy + cy^2$, usually with integer coefficients.
Its discriminant $b^2 - 4ac$ is unchanged by integral changes of variables of determinant $1$.
Gauss developed the theory of these forms: which integers they represent, how they fall into equivalence classes, and how the classes combine into a group.
This theory is equivalent to the arithmetic of ideals in [[quadratic fields|7SV4WA92SM]], and the number of classes is a [[class number|KTKK9B1X41]].

## Explanations

### L1 human

The forms $x^2 + y^2$ and $x^2 + 2y^2$ are binary quadratic forms.
A classical question is which numbers they produce when $x$ and $y$ run through the integers.
Fermat found that an odd prime is of the form $x^2 + y^2$ exactly when it leaves remainder $1$ on division by $4$.
Changing variables by an integer matrix of determinant $1$ does not change the set of values, so such forms are grouped into classes.
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# Koenigs function

## Definition

Let $U \subseteq \C$ be an open set containing $0$, and let $f \colon U \to \C$ be [[holomorphic|0NAHXR1FRH]] with $f(0) = 0$.
Let $\lambda = f'(0)$ and assume $0 < \abs{\lambda} < 1$.
Let $V$ be a neighbourhood of $0$ on which the iterates $f^n$ of $f$ are defined and tend to $0$.
The *Koenigs function* of $f$ is the function $\varphi \colon V \to \C$ given by
$$\varphi(z) = \lim_{n \to \infty} \frac{f^n(z)}{\lambda^n},$$
where $f^n$ is the $n$-th [[iterate|AYVTR2ES5B]] of $f$.

## Remarks

The limit exists, uniformly near $0$, and $\varphi$ is holomorphic with $\varphi(0) = 0$ and $\varphi'(0) = 1$.
It solves [[Schröder's equation|VTJV7JVJEA]] $\varphi \circ f = \lambda \varphi$, since replacing $z$ by $f(z)$ shifts the index $n$ by one.
It is the unique solution with these normalisations, by the [[Koenigs linearization theorem|G02NBNN6CJ]].

Example: $f(z) = z / (2 - z)$ on the unit disc, with $\lambda = f'(0) = 1/2$.
For $\varphi(z) = z / (1 - z)$ one has $\varphi(f(z)) = \frac{z/(2-z)}{(2 - 2z)/(2 - z)} = \frac{z}{2(1 - z)} = \frac{1}{2} \varphi(z)$.
So $\varphi(f^n(z)) = 2^{-n} \varphi(z)$, and since $f^n(z) \to 0$ and $\varphi(w) / w \to 1$ as $w \to 0$, the limit of $2^n f^n(z)$ is $\varphi(z)$.

Non-example: for $f(z) = z^2$ one has $\lambda = 0$, so the quotient $f^n(z) / \lambda^n$ is not defined, and $f$ has no Koenigs function.

When $f$ is a univalent holomorphic map of the unit disc into itself with $f(0) = 0$ and $0 < \abs{f'(0)} < 1$, the limit exists on the whole disc and $\varphi$ is univalent there.
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# Weyl tensor

## Summary

The Weyl tensor is defined on a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] of dimension at least $3$.
It is the part of the [[Riemann curvature tensor|JA29ETCM02]] left after removing all that the [[Ricci curvature|4GTNM0WHDH]] determines.
It is unchanged, as a $(1,3)$-tensor, when the metric is multiplied by a positive function, so it measures conformal geometry.
In dimension at least $4$, it vanishes exactly when the metric is locally conformally flat.
In general relativity it describes tidal forces and gravitational waves in vacuum.

## Explanations

### L1 human

Curvature in several dimensions has many components.
Some of them say how volumes change, and these are captured by the Ricci curvature.
The rest say how shapes are distorted without changing volume, and that rest is the Weyl tensor.
A round sphere has zero Weyl tensor, because it can be mapped to flat space by a map that keeps angles, the stereographic projection.
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# Marsden–Weinstein reduction

## Summary

Marsden–Weinstein reduction, also called symplectic reduction, builds a smaller [[symplectic manifold|CAN490EPRT]] from one with a Hamiltonian symmetry.
One takes a level set of the [[moment map|ZARM34FZ6Z]] and divides it by the symmetry group.
It is the geometric form of removing conserved quantities from a mechanical system.
It produces many important spaces, such as complex projective spaces and moduli spaces.

## Explanations

### L1 human

A symmetry of a mechanical system gives conserved quantities, and fixing their values cuts down the phase space.
The symmetry still moves points inside this smaller set, so one also identifies points that differ by a symmetry.
What is left is again a phase space, with two dimensions less for each dimension of the symmetry group.
For example, fixing the energy of $n + 1$ uncoupled harmonic oscillators with equal frequency gives a sphere $S^{2n+1}$, and dividing by the common phase rotation gives the complex projective space $\mathbb{CP}^n$.
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# Control system

## Definition

Let $n, m \ge 1$, and let $U \subseteq \R^m$ be a nonempty closed set, the control set.
Let $f \colon \R^n \times U \to \R^n$ be continuous and locally [[Lipschitz|0AW1E6YTJY]] in $x$, locally uniformly in $u$.
The *control system* given by $f$ is the equation
$$x'(t) = f(x(t), u(t)).$$
An *admissible control* on an interval $[0, T]$ is a measurable, essentially bounded function $u \colon [0, T] \to U$.
A *trajectory* for $u$ is an [[absolutely continuous|ZT6F4C2YHK]] function $x \colon [0, T] \to \R^n$ that satisfies the equation for almost every $t \in [0, T]$.
Often a control system also has an output $y(t) = h(x(t))$ with a map $h \colon \R^n \to \R^p$.

## Remarks

Fix an admissible control $u$ and an initial state $x(0) = x_0$.
Then the equation is an [[ordinary differential equation|0NT64JV96A]] in the sense of Carathéodory, and it has a unique trajectory on some interval $[0, \tau)$.
The most important case is the linear system $x' = A x + B u$ with matrices $A \in \R^{n \times n}$ and $B \in \R^{n \times m}$.
Example: the double integrator $x_1' = x_2$, $x_2' = u$ with $U = [-1, 1]$ is linear, with $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$; the control $u = 1$ on $[0, 1)$ and $u = -1$ on $[1, 2]$ is admissible.
Non-example: $u(t) = 1/t$ for $t \in (0, 1]$ is not an admissible control for $U = \R$ on $[0, 1]$, because it is not essentially bounded.
Some authors allow locally integrable controls, or controls of a smaller class such as piecewise continuous ones.
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# F-space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a [[topological vector space|T32W553W95]] over $\mathbb{K}$.
$E$ is an *F-space* if there is a [[metric|063W5JMY85]] $d$ on $E$ such that:

1. $d$ induces the topology of $E$.
2. $d$ is translation-invariant: $d(x + z, y + z) = d(x, y)$ for all $x, y, z \in E$.
3. The metric space $(E, d)$ is [[complete|08EWT7J0SC]].

## Remarks

Some authors use the name for a vector space with a metric satisfying 2 and 3 for which addition and scalar multiplication are continuous; this is the same notion.
Example: every Banach space is an F-space with $d(x, y) = \norm{x - y}$; this metric induces the norm topology, it is translation-invariant since $(x + z) - (y + z) = x - y$, and it is complete by definition of a Banach space.
Example: for $0 < p < 1$, $L^p([0, 1])$ with $d(f, g) = \int_0^1 \abs{f - g}^p$ is an F-space that is not locally convex.
Non-example: the space $c_{00}$ of finitely supported real sequences with $d(x, y) = \sup_n \abs{x_n - y_n}$ satisfies 1 and 2 but not 3: the sequences $x^{(k)} = (1, 1/2, \ldots, 1/k, 0, 0, \ldots)$ are Cauchy and have no limit in $c_{00}$.
Every metrizable topological vector space has a translation-invariant metric inducing its topology, so condition 2 matters only together with condition 3.
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# Supercompact cardinal

## Definition

Let $\kappa$ be a [[cardinal|0EBACQFAZW]], and let $\lambda \ge \kappa$ be an [[ordinal|0YSCSX5YZC]].
The cardinal $\kappa$ is *$\lambda$-supercompact* if there is an [[elementary embedding|CGH8C8EA9N]] $j \colon V \to M$, with $M$ a transitive class, such that:

1. The critical point of $j$ is $\kappa$, that is, $j(\alpha) = \alpha$ for $\alpha < \kappa$ and $j(\kappa) > \kappa$.
2. $j(\kappa) > \lambda$.
3. $M$ is closed under $\lambda$-sequences: every function from $\lambda$ to $M$ is an element of $M$.

The cardinal $\kappa$ is *supercompact* if it is $\lambda$-supercompact for every ordinal $\lambda \ge \kappa$.

## Remarks

A cardinal is $\kappa$-supercompact if and only if it is measurable: the ultrapower embedding by a $\kappa$-complete nonprincipal ultrafilter on $\kappa$ satisfies the three clauses with $\lambda = \kappa$.
So the least measurable cardinal is $\kappa$-supercompact, but it is not supercompact, since a supercompact cardinal has measurable cardinals below it.
An equivalent form uses ultrafilters on the set $P_\kappa(\lambda)$ of subsets of $\lambda$ of size less than $\kappa$.
The cardinal $\kappa$ is $\lambda$-supercompact if and only if there is a fine normal $\kappa$-complete ultrafilter on $P_\kappa(\lambda)$.
Every supercompact cardinal is a Woodin cardinal.
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# Ladyzhenskaya–Babuška–Brezzi condition

## Statement

Let $V$ and $Q$ be real [[Hilbert spaces|HM2HHE3J51]].
Let $a \colon V \times V \to \R$ and $b \colon V \times Q \to \R$ be bounded [[bilinear forms|0SQZMXB6GP]], and let $f \in V'$ and $g \in Q'$, where $'$ denotes the [[continuous dual space|4A82S5FVRT]].
Consider the saddle-point problem: find $(u, p) \in V \times Q$ with
$$a(u, v) + b(v, p) = f(v) \ \text{ for all } v \in V, \qquad b(u, q) = g(q) \ \text{ for all } q \in Q.$$
Let $V_h \subseteq V$ and $Q_h \subseteq Q$ be finite-dimensional subspaces, and let $Z_h = \set{v \in V_h : b(v, q) = 0 \text{ for all } q \in Q_h}$.
Assume that there are constants $\alpha > 0$ and $\beta > 0$, independent of $h$, such that:

1. $a$ is [[coercive|86KRHS6X22]] on $Z_h$: $a(v, v) \ge \alpha \norm{v}_V^2$ for all $v \in Z_h$;
2. the inf-sup condition, or LBB condition, holds:
$$\inf_{q \in Q_h \setminus \set{0}} \ \sup_{v \in V_h \setminus \set{0}} \frac{b(v, q)}{\norm{v}_V \norm{q}_Q} \ge \beta.$$

Then the discrete problem, posed with $V_h$ and $Q_h$ in place of $V$ and $Q$, has a unique solution $(u_h, p_h)$.
If the continuous problem has a solution $(u, p)$, then
$$\norm{u - u_h}_V + \norm{p - p_h}_Q \le C \Big( \inf_{v \in V_h} \norm{u - v}_V + \inf_{q \in Q_h} \norm{p - q}_Q \Big),$$
where $C$ depends only on $\alpha$, $\beta$ and the bounds of $a$ and $b$.

## Remarks

The same two conditions with $V$, $Q$ and the [[kernel|S417YNEJ6M]] $Z = \set{v \in V : b(v, q) = 0 \text{ for all } q \in Q}$ give existence and uniqueness for the continuous problem; this is Brezzi's theorem.
The inf-sup condition says that the operator $B \colon V \to Q'$ given by $b$ maps onto $Q'$ with a bounded right inverse.
So one first finds $u_g$ with $b(u_g, q) = g(q)$ and $\norm{u_g} \le \norm{g} / \beta$.
Then one solves for $u - u_g$ in $Z$ by the [[Lax–Milgram theorem|85KK79FCN5]].
Finally one recovers $p$ from the first equation, using the inf-sup condition again.
The point of the discrete theorem is that $\beta$ does not depend on $h$; otherwise the constant $C$ can blow up as the mesh is refined.

Example: for the Stokes equations, $V = H^1_0(\Omega)^d$, $Q = L^2_0(\Omega)$, $a(u, v) = \int_\Omega \nabla u : \nabla v$ and $b(v, q) = -\int_\Omega q \, \nabla \cdot v$.
The Taylor–Hood pair, continuous piecewise quadratic velocities with continuous piecewise linear pressures, satisfies the inf-sup condition uniformly on suitable meshes.

Necessity: the pair of continuous piecewise linear velocities and pressures on a triangle mesh violates the condition.
The discrete problem then has spurious pressure modes or a constant $\beta_h$ that tends to $0$, and the computed pressure oscillates.

## Explanations

### L1 human

Some problems have two unknowns that play different roles, such as the velocity and the pressure of a slow viscous flow.
When both are approximated by finite elements, the two spaces must be balanced.
If the pressure space is too rich compared with the velocity space, the pressure is not controlled and the computation fails.
The inf-sup condition is the exact test of this balance.

### L2 human

The condition is the key stability requirement of [[mixed finite element methods|87VMQ892VW]].
It replaces the coercivity of the Lax–Milgram setting, which a saddle-point problem never has on the whole space $V \times Q$.
Checking it for a given pair of spaces is often done with a Fortin operator: a bounded projection $\Pi_h \colon V \to V_h$ with $b(v - \Pi_h v, q) = 0$ for all $q \in Q_h$.
Stable pairs include Taylor–Hood elements for Stokes flow and [[Raviart–Thomas elements|DVB79BK06Y]] with piecewise constants for the mixed Poisson problem.
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# Extremal graph theory

## Summary

Extremal graph theory studies how large or small a [[graph|00C0BR7D7H]] can be when it must have, or must avoid, a given structure.
Its basic quantity is the extremal number $\mathrm{ex}(n, H)$, the largest number of edges of a graph on $n$ vertices with no [[subgraph|0WBFFC8XZ9]] isomorphic to $H$.
[[Turán's theorem|0JG1KZSPFZ]] computes it for complete graphs, and the [[Erdős–Stone theorem|XZF9BFJFXB]] gives it asymptotically for every non-bipartite $H$.
For bipartite $H$ the question leads to the [[Zarankiewicz problem|KT9WJFRQ3F]], which is still largely open.

## Explanations

### L1 human

Extremal graph theory asks questions of the form "how many edges can you draw before a certain pattern is forced to appear?".
For example, a graph on $n$ vertices without a triangle has at most $n^2/4$ edges.
The best graph puts the vertices in two equal groups and joins every vertex of one group to every vertex of the other.
Results of this kind show that density alone forces structure.
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# Preadditive category

## Summary

A preadditive category is a [[category|A2DE66937D]] in which morphisms with the same source and target can be added, and composition distributes over this addition.
Equivalently, it is a category [[enriched|BPH9SJ30Y0]] over abelian groups.
Modules over a ring, with their homomorphisms, are the basic example.
Preadditive categories are the first step towards additive and abelian categories, the setting of homological algebra.

## Explanations

### L1 human

Two linear maps between the same vector spaces can be added, and the sum is again linear.
Moreover $h \circ (f + g) = h \circ f + h \circ g$, so composition and addition get along.
A preadditive category asks for exactly this: every set of arrows from one object to another is an abelian group, and composing distributes over sums.
A ring is the same thing as a preadditive category with one object, whose arrows are the elements of the ring and whose composition is multiplication.
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# Periodic boundary conditions

## Definition

Let $a < b$ be real numbers and consider a [[linear differential equation|0QQRP17Z7S]] of order $2$ on $[a, b]$.
A function $y \in C^1([a, b])$ satisfies the periodic boundary conditions on $[a, b]$ if
$$y(a) = y(b) \quad \text{and} \quad y'(a) = y'(b).$$
For an equation of order $n$, the periodic boundary conditions are $y^{(k)}(a) = y^{(k)}(b)$ for $k = 0, 1, \ldots, n - 1$.

## Remarks

Example: on $[0, 2\pi]$, the function $y(x) = \sin x$ satisfies $y(0) = 0 = y(2\pi)$ and $y'(0) = 1 = y'(2\pi)$.
So it satisfies the periodic boundary conditions, and it solves $y'' + y = 0$.
Non-example: on $[0, \pi]$, the same function has $y(0) = y(\pi) = 0$ but $y'(0) = 1 \ne -1 = y'(\pi)$, so the second condition fails.
The eigenvalue problem $-y'' = \lambda y$ with periodic boundary conditions on $[0, 2\pi]$ has the eigenvalues $\lambda = k^2$ for $k = 0, 1, 2, \ldots$.
The eigenvalue $0$ has the eigenfunction $1$, and each $k^2$ with $k \ge 1$ has the two independent eigenfunctions $\cos kx$ and $\sin kx$.
Unlike separated conditions, periodic conditions couple the two endpoints, so eigenvalues of multiplicity $2$ can occur.
If the coefficients of the equation are $(b - a)$-periodic, a solution with these conditions extends to a $(b - a)$-periodic solution on $\R$.

## Explanations

### L1 human

Periodic boundary conditions say that the start and the end of the interval look the same to the solution.
The value and the slope must match at both ends.
Then copies of the solution glue together into a smooth repeating function on the whole line.
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# Elliptic operator

## Definition

Let $\Omega \subseteq \R^n$ be an open set and let $m \ge 1$ be an integer.
For a multi-index $\alpha = (\alpha_1, \ldots, \alpha_n)$ of nonnegative integers, write $\abs{\alpha} = \alpha_1 + \cdots + \alpha_n$, $\partial^\alpha = \partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n}$ for the [[partial derivatives|09EGC6F8PT]], and $\xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n}$ for $\xi \in \R^n$.
Let
$$L = \sum_{\abs{\alpha} \le m} a_\alpha(x) \partial^\alpha$$
be a linear differential operator of order $m$ on $\Omega$ with complex-valued continuous coefficients $a_\alpha$.
Its [[principal symbol|F31T650DKY]] is $\sigma_L(x, \xi) = \sum_{\abs{\alpha} = m} a_\alpha(x) \xi^\alpha$.
The operator $L$ is *elliptic* if
$$\sigma_L(x, \xi) \ne 0 \quad \text{for all } x \in \Omega \text{ and all } \xi \in \R^n \setminus \set{0}.$$

## Remarks

Many authors put a factor $i^m$ in the principal symbol; this does not change the condition.
For a second-order operator $L = \sum_{j,k} a_{jk}(x)\partial_j\partial_k + \sum_j b_j(x)\partial_j + c(x)$ with real symmetric $(a_{jk})$, ellipticity means that the matrix $(a_{jk}(x))$ is positive definite or negative definite at each $x$.
The operator is *uniformly elliptic* if there is $\theta > 0$ with $\sum_{j,k} a_{jk}(x)\xi_j\xi_k \ge \theta \abs{\xi}^2$ for all $x$ and $\xi$.
The definition extends to systems, where the symbol is a matrix that must be invertible, and to operators between vector bundles on manifolds.

Example: the [[Laplacian|0E0B6RA8K0]] $\Delta = \partial_1^2 + \cdots + \partial_n^2$ has symbol $\abs{\xi}^2$, which is nonzero for $\xi \ne 0$, so it is elliptic.
The Cauchy–Riemann operator $\partial_x + i\partial_y$ on $\R^2$ has symbol $\xi_1 + i\xi_2$, also nonzero for $\xi \ne 0$.

Non-example: the wave operator $\partial_1^2 - \partial_2^2$ on $\R^2$ has symbol $\xi_1^2 - \xi_2^2$, which vanishes at $\xi = (1, 1)$.
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# Accessible category

## Definition

Let $\lambda$ be a [[regular cardinal|13ZV2F8JAR]].
A small category $\mathcal{I}$ is *$\lambda$-filtered* if every subcategory of $\mathcal{I}$ with fewer than $\lambda$ morphisms has a cocone in $\mathcal{I}$.
Let $\mathcal{C}$ be a locally small [[category|A2DE66937D]].
An object $A$ of $\mathcal{C}$ is *$\lambda$-presentable* if the [[hom functor|VJXS8AK67E]] $\mathcal{C}(A, -)$ preserves [[colimits|BJS1CGTKQ5]] of $\lambda$-filtered diagrams.
The category $\mathcal{C}$ is *$\lambda$-accessible* if both conditions hold.

1. Every diagram $\mathcal{I} \to \mathcal{C}$ with $\mathcal{I}$ small and $\lambda$-filtered has a colimit in $\mathcal{C}$.
2. There is a set $S$ of $\lambda$-presentable objects such that every object of $\mathcal{C}$ is a colimit of a $\lambda$-filtered diagram of objects of $S$.

The category $\mathcal{C}$ is *accessible* if it is $\lambda$-accessible for some regular cardinal $\lambda$.

## Remarks

For $\lambda = \aleph_0$, the $\lambda$-filtered categories are the [[filtered categories|975PMY4FG8]], and one says finitely presentable and finitely accessible.

Example: the category of sets is finitely accessible.
It has all colimits, so clause 1 holds.
A finite set $A$ is finitely presentable, since a map from $A$ into a filtered colimit factors through some stage, essentially uniquely.
Every set is the directed union, hence the filtered colimit, of its finite subsets, and the finite sets $\set{1, \dots, n}$ form a set $S$ up to isomorphism.

Non-example: let $\mathcal{M}$ be the category with one object $*$ and morphisms $1$ and $e$, where $e \circ e = e$.
The identity diagram of $\mathcal{M}$ is $\lambda$-filtered for every $\lambda$, with the cocone $e$.
It has no colimit: the cocone $e$ is the only one, and both $1$ and $e$ factor it through itself.
So clause 1 fails for every $\lambda$, and $\mathcal{M}$ is not accessible.
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# Embedding of structures

## Summary

An embedding of one [[structure|983GZBQYT4]] into another is an injective [[homomorphism|KBXSZC6BV1]] that also reflects the relations.
It identifies the first structure with a [[substructure|676H7EGTKJ]] of the second, up to [[isomorphism|PZMGQMDN25]].
Embeddings let one say that a structure "sits inside" another, as $\Z$ sits inside $\Q$.
[[Elementary embeddings|CGH8C8EA9N]] are the embeddings that preserve all first-order formulas.

## Explanations

### L1 human

An embedding places a copy of one structure inside another without losing any information.
The map that sends an integer $n$ to the rational number $n/1$ is an embedding of the ring of integers into the field of rational numbers.
Different elements go to different elements, sums go to sums and products go to products.
For relational structures, such as graphs, an embedding must also keep non-edges as non-edges.
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<document source="https://tamnd.github.io/math2/node/SCYW8DVPKQ.md" tag="SCYW8DVPKQ" kind="definition">
# Design of experiments

## Definition

Let $T$ be a finite set of *treatments* and let $U$ be a finite set of $n$ *experimental units*.
An *experimental design* for $T$ on $U$ is a map $d \colon U \to T$, which assigns treatment $d(u)$ to unit $u$.
The number $r_t = \abs{d^{-1}(t)}$ is the *replication* of treatment $t$.
After the treatments are applied, a response $y_u$ is observed on each unit.
A [[statistical model|K3AMMSY8DD]] relates $y_u$ to the treatment $d(u)$ and to the unit $u$.
*Design of experiments* is the choice of $T$, $U$ and $d$, together with the plan of analysis, before the data are collected.

## Remarks

Example: two diets $T = \set{A, B}$ and four animals $U = \set{1, 2, 3, 4}$.
The map $d(1) = A$, $d(2) = B$, $d(3) = B$, $d(4) = A$ is a design with $r_A = r_B = 2$.

Non-example: in an observational study the units choose their own treatments, for example smokers and non-smokers.
The map $d$ is not chosen by the investigator, so this is not a designed experiment.
Differences in response may then come from other differences between the groups.

For experiments with a quantitative input $x$ in a region $\mathcal{X}$, a design is a choice of the input values $x_1, \dots, x_n \in \mathcal{X}$.
In the approximate theory, it is a probability measure on $\mathcal{X}$ with finite support.
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# Universal enveloping algebra

## Summary

The universal enveloping algebra $U(\mathfrak g)$ of a Lie algebra $\mathfrak g$ is the most general associative algebra containing $\mathfrak g$.
In it the Lie bracket becomes the commutator $xy - yx$.
Representations of $\mathfrak g$ are the same as modules over $U(\mathfrak g)$, so Lie theory can use the tools of ring theory.
By the [[Poincaré–Birkhoff–Witt theorem|HFEA4FKGW1]], $U(\mathfrak g)$ has a basis of ordered monomials in a basis of $\mathfrak g$.
It is also a [[Hopf algebra|7XFYP0N84H]], and its deformations give the quantum groups.

## Explanations

### L1 human

A Lie algebra has a bracket $[x, y]$ but no product $xy$.
The universal enveloping algebra adds a product, subject only to the rule $xy - yx = [x, y]$.
For an abelian Lie algebra, the rule says that the elements commute, and $U(\mathfrak g)$ is a polynomial ring.
For a Lie algebra with basis $x, y$ and $[x, y] = y$, it is the algebra of noncommutative polynomials in $x, y$ with $xy - yx = y$.
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# Homogeneous polynomial

## Definition

Let $R$ be a commutative ring and let $f \in R[x_1, \ldots, x_n]$ be a [[polynomial in several variables|0CES91M79A]].
The total degree of a monomial $x_1^{a_1} \cdots x_n^{a_n}$ is $a_1 + \cdots + a_n$.
Let $d \ge 0$ be an integer.
The polynomial $f$ is *homogeneous of degree $d$* if every monomial that appears in $f$ with a nonzero coefficient has total degree $d$.
The zero polynomial is homogeneous of every degree.

## Remarks

Example: $f = x^2 y + 3 y z^2 - z^3$ in $\Q[x, y, z]$ is homogeneous of degree $3$.
Check: the monomials $x^2 y$, $y z^2$ and $z^3$ have total degrees $2 + 1$, $1 + 2$ and $3$.

Non-example: $g = x^2 + y$ is not homogeneous, because its monomials have total degrees $2$ and $1$.

The polynomial $f$ is homogeneous of degree $d$ if and only if $f(\lambda x_1, \ldots, \lambda x_n) = \lambda^d f(x_1, \ldots, x_n)$ in $R[\lambda, x_1, \ldots, x_n]$.
Every polynomial is in a unique way a finite sum $f = f_0 + f_1 + \cdots$ where each $f_d$ is homogeneous of degree $d$; these are the homogeneous components of $f$.
A homogeneous polynomial is also called a form; forms of degree $1$, $2$ and $3$ are linear, quadratic and cubic forms.
In one variable, the nonzero homogeneous polynomials of degree $d$ are the $c x^d$ with $c \ne 0$, whose [[degree|0DPF6WRZW6]] is $d$.
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# Maximum and minimum

## Definition

Let $D$ be a nonempty set, let $f \colon D \to \R$ be a [[function|0XHDTE5YMZ]], and let $a \in D$.

1. The function $f$ attains its *maximum* at $a$ if $f(x) \le f(a)$ for all $x \in D$. Then $f(a)$ is the *maximum value* of $f$, written $\max_{x \in D} f(x)$.
2. The function $f$ attains its *minimum* at $a$ if $f(x) \ge f(a)$ for all $x \in D$. Then $f(a)$ is the *minimum value* of $f$, written $\min_{x \in D} f(x)$.
3. A point $a$ as in 1 or 2 is a *global extremum*, or *absolute extremum*, of $f$.

## Remarks

The maximum value of $f$ is the [[greatest element|0Z7HDDXX3X]] of the image $f(D)$, when it exists.
Then it equals the [[supremum|0BTMQ04556]] of $f(D)$, and the supremum is attained.
The maximum value is unique, but it can be attained at several points.
For $D \subseteq \R^n$, every global extremum is a [[local extremum|04CPG9K1C4]].

Example: $f(x) = x^2$ on $D = [-1, 2]$ attains its maximum at $a = 2$.
Indeed, for $x \in [-1, 2]$ one has $\abs{x} \le 2$, so $x^2 \le 4 = f(2)$.

Non-example: $f(x) = x$ on $D = (0, 1)$ has no maximum.
For each $a \in (0, 1)$, the point $x = (a + 1)/2$ lies in $D$ and $f(x) > f(a)$.
The supremum of $f(D)$ is $1$, but it is not a value of $f$.
The [[extreme value theorem|0WS1A8E2P9]] gives a maximum and a minimum for every continuous function on a closed bounded interval.
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<document source="https://tamnd.github.io/math2/node/SD89TZ1PHR.md" tag="SD89TZ1PHR" kind="definition">
# Fourier multiplier

## Definition

Let $n \ge 1$ and let $\mathcal{F} \colon L^2(\R^n) \to L^2(\R^n)$ be the [[Fourier transform|DPVM34ADZV]] on $L^2$, extended from $\mathcal{F} f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\, dx$ by the [[Plancherel theorem|HPZHRZ0502]].
Let $m \colon \R^n \to \C$ be a bounded measurable function.
The *Fourier multiplier* with symbol $m$ is the operator $T_m \colon L^2(\R^n) \to L^2(\R^n)$ given by
$$T_m f = \mathcal{F}^{-1}\big( m \cdot \mathcal{F} f \big).$$

## Remarks

By Plancherel, $T_m$ is a bounded linear operator on [[$L^2(\R^n)$|SASP14MA1Z]] with operator norm equal to $\norm{m}_{L^\infty}$.
It commutes with every translation, and conversely every bounded operator on $L^2(\R^n)$ that commutes with all translations is $T_m$ for a unique $m \in L^\infty(\R^n)$.

Example: on $\R$, the symbol $m(\xi) = -i \operatorname{sgn}(\xi)$ is bounded, and $T_m$ is the [[Hilbert transform|NSCEBXM7ZM]].
Example: for $t > 0$, the symbol $m(\xi) = e^{-4\pi^2 t \abs{\xi}^2}$ gives the heat semigroup: $u(t, \cdot) = T_m f$ solves $\partial_t u = \Delta u$ with $u(0, \cdot) = f$, because $\Delta$ acts on the Fourier side as multiplication by $-4\pi^2 \abs{\xi}^2$.

Non-example: the symbol $m(\xi) = 2\pi i \xi$ on $\R$ is not bounded; the corresponding operator is differentiation, which is not defined on all of $L^2(\R)$.
It is still a Fourier multiplier in the wider sense of an operator on the Schwartz space or on tempered distributions.

For the torus $\mathbb{T}^n$ the same definition uses a bounded sequence $(m(k))_{k \in \Z^n}$ that multiplies the Fourier coefficients.
The symbol $m$ is called an $L^p$ multiplier when $T_m$, restricted to $L^2 \cap L^p$, extends to a bounded operator on $L^p$.
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# Weierstrass polynomial

## Summary

A Weierstrass polynomial is a monic polynomial in one variable $z_n$ whose other coefficients are holomorphic germs in $z_1, \dots, z_{n-1}$ vanishing at the origin.
The [[Weierstrass preparation theorem|49JSD49P7X]] says that every holomorphic germ that does not vanish identically on the $z_n$-axis is a unit times such a polynomial.
This reduces many local questions about holomorphic functions of $n$ variables to questions about polynomials over the ring of germs in $n - 1$ variables.

## Explanations

### L1 human

Near the origin, the zeros of a Weierstrass polynomial of degree $d$ in $z_n$ behave like the $d$ roots of an ordinary polynomial.
For each fixed $z' = (z_1, \dots, z_{n-1})$ near $0$, there are exactly $d$ roots $z_n$, counted with multiplicity, and they all tend to $0$ as $z' \to 0$.
For example, $z_2^2 - z_1$ is a Weierstrass polynomial in $z_2$ of degree $2$, and its roots are the two square roots of $z_1$.
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# Spherical code

## Summary

A spherical code is a finite set of points on the unit sphere in $\R^n$, judged by how far apart its points are.
The main question asks for the largest number of points with a given minimal angle.
The dual question asks for the largest minimal angle for a given number of points.
Kissing configurations are spherical codes with minimal angle $60^\circ$, and the [[Tammes problem|9WD2TZ4ZB5]] is the case $n = 3$.
Upper bounds come mainly from the [[Delsarte linear programming bound|7RPEDHAT7R]].

## Explanations

### L1 human

Choose a number of directions in space, pointing from the centre of a ball to its surface, and try to keep any two directions far apart.
Such a choice is a spherical code.
In the plane the best choice is clear: space the points evenly around the circle.
On a sphere and in higher dimensions it is a hard problem, studied for its use in signal transmission and for its links with sphere packing.
</document>

<document source="https://tamnd.github.io/math2/node/SDHZRWMSTT.md" tag="SDHZRWMSTT" kind="statement">
# Shoelace formula

## Statement

Let $P$ be a simple [[polygon|8VEFWD3EJB]] in $\R^2$ with vertices $(x_1, y_1), \ldots, (x_n, y_n)$ listed in order along its boundary.
Put $(x_{n+1}, y_{n+1}) = (x_1, y_1)$.
Then the area $A$ of the region bounded by $P$ is
$$A = \frac{1}{2} \abs{\sum_{i=1}^{n} \left(x_i y_{i+1} - x_{i+1} y_i\right)}.$$

## Remarks

Without the absolute value, the sum gives the signed area, which is positive when the vertices run counterclockwise.
Each term $\frac{1}{2}(x_i y_{i+1} - x_{i+1} y_i)$ is the signed area of the triangle with vertices $0$, $P_i$, $P_{i+1}$, and these signed areas add up to the area of $P$.
Equivalently, the formula is [[Green's theorem|0HR3N02317]] applied to the line integral $\frac{1}{2}\oint (x \, dy - y \, dx)$ along the boundary.
Example: for the triangle $(0, 0), (4, 0), (0, 3)$ the sum is $0 + 12 + 0 = 12$, so $A = 6$.
Simplicity is needed: for a self-crossing figure eight the two lobes have opposite orientation, and the sum gives the difference of their areas.

## Explanations

### L1 human

The shoelace formula computes the area of a polygon from the coordinates of its corners alone.
Write the coordinates in two columns, multiply them crosswise like the laces of a shoe, and take half the difference of the two totals.

### L2 human

The formula is the standard way to compute polygon areas in computational geometry and surveying.
It is a discrete case of Green's theorem, and it is also the main tool in many proofs of [[Pick's theorem|K9SBQZMWB9]].
</document>

<document source="https://tamnd.github.io/math2/node/SDJ4ADZK8M.md" tag="SDJ4ADZK8M" kind="definition">
# Tensor product of fields

## Definition

Let $K$ be a field and let $E$ and $F$ be [[field extensions|0WQ1A8SHTZ]] of $K$.
The *tensor product* $E \otimes_K F$ is the [[tensor product|0G31VJ5YS5]] of the $K$-vector spaces $E$ and $F$, with the unique $K$-bilinear multiplication such that
$$(a \otimes b)(c \otimes d) = ac \otimes bd \quad \text{for } a, c \in E \text{ and } b, d \in F.$$
It is a [[commutative ring|0GYGHKX834]] with identity $1 \otimes 1$ and a $K$-algebra.

## Remarks

The maps $a \mapsto a \otimes 1$ and $b \mapsto 1 \otimes b$ embed $E$ and $F$ into $E \otimes_K F$.
If $E = K(\alpha)$ is finite and $f$ is the [[minimal polynomial|0GK1WB9R98]] of $\alpha$ over $K$, then $E \otimes_K F \cong F[x]/(f)$.
If moreover $f = g_1 \cdots g_r$ with distinct monic irreducible $g_i \in F[x]$, the Chinese remainder theorem gives $E \otimes_K F \cong \prod_{i=1}^r F[x]/(g_i)$, a product of fields.

Example: $\C \otimes_\R \C \cong \C \times \C$.
Check: $\C = \R(i)$ with $f = x^2 + 1$, and over $F = \C$ one has $x^2 + 1 = (x - i)(x + i)$, so $\C \otimes_\R \C \cong \C[x]/(x - i) \times \C[x]/(x + i) \cong \C \times \C$.

Non-example: $E \otimes_K F$ need not be a field, even when $E$ and $F$ are; $\C \otimes_\R \C$ has the zero divisors $(1, 0)$ and $(0, 1)$ in the product above.
</document>

<document source="https://tamnd.github.io/math2/node/SDQTNBDJ2H.md" tag="SDQTNBDJ2H" kind="concept">
# Identity component

## Summary

The identity component of a [[topological group|92700QAB5Y]] $G$ is the [[connected component|0JG9W3KA6B]] of $G$ that contains the identity element.
It is a closed normal subgroup of $G$, and the other connected components are its cosets.
The quotient of $G$ by its identity component is totally disconnected.
So every topological group is built from a connected group and a totally disconnected group.

## Explanations

### L1 human

Start at the identity of the group and collect every element you can reach without jumping across a gap.
The elements you collect form the identity component.
For example, in the group of invertible real $2 \times 2$ matrices, the identity component consists of the matrices with positive determinant.
The matrices with negative determinant form the other piece.
To reach them from the identity you would have to pass through a matrix of determinant zero, which is not in the group.
</document>

<document source="https://tamnd.github.io/math2/node/SDSGN4B6JD.md" tag="SDSGN4B6JD" kind="problem">
# Hilbert's eighteenth problem

## Problem

The problem consists of three questions about filling Euclidean space $\R^n$.

1. For each $n$, are there only finitely many [[crystallographic groups|7KTJKFQT2N]] of dimension $n$, up to a suitable notion of equivalence?
2. Is there a polyhedron that [[tiles|GKJK293DXS]] $\R^3$ by congruent copies, such that no such tiling has a symmetry group transitive on the tiles?
3. What is the densest way to pack congruent balls in $\R^3$, and more generally to pack copies of a given solid?

## Remarks

David Hilbert posed the problem in his list of 1900.
The first question was answered positively by Ludwig Bieberbach in 1910 and 1911; see the [[Bieberbach theorems|YE87R5CDJY]].
In dimension $2$ there are $17$ [[wallpaper groups|MY3CBHARYQ]], and in dimension $3$ there are $219$ space groups up to affine equivalence.
The second question was answered positively by Karl Reinhardt in 1928, who found such a polyhedron; Heinrich Heesch gave a plane example in 1935.
Tiles of this kind are called anisohedral.
The ball case of the third question is the [[Kepler conjecture|AFMBQT06ZV]], proved by Thomas Hales, who announced his proof in 1998.
The densest packing of balls in $\R^3$ has density $\pi/\sqrt{18}$, attained by the face-centred cubic packing.
See [[sphere packing|HF32CBN5MA]] for the general setting.

## Explanations

### L1 human

Hilbert asked three questions about filling space with copies of one shape.
How many kinds of symmetry can a crystal have?
Can a shape fill space even though its copies cannot all be placed in the same way relative to the pattern?
And how tightly can equal balls, like oranges in a crate, be packed?
All three have been answered: finitely many, yes, and the grocer's pyramid stacking is the best.
</document>

<document source="https://tamnd.github.io/math2/node/SDTZVZ6B4Q.md" tag="SDTZVZ6B4Q" kind="concept">
# Model category

## Summary

A model category is a [[category|A2DE66937D]] with three chosen classes of morphisms, called weak equivalences, fibrations and cofibrations, that satisfy axioms modelled on topological spaces.
Quillen introduced model categories in 1967 to do homotopy theory in settings such as simplicial sets and chain complexes.
The weak equivalences are the maps to be treated as isomorphisms, and the fibrations and cofibrations give the tools to compute with them.
The homotopy category of a model category, obtained by inverting the weak equivalences, has a concrete description in terms of homotopy classes of maps.

## Explanations

### L1 human

In topology one often wants to regard two spaces as the same when one can be deformed into the other.
Making such deformations into honest isomorphisms is easy to say but hard to compute with.
A model category adds two kinds of well-behaved maps, fibrations and cofibrations, which play the roles of nice projections and nice inclusions.
With them one can replace any object by a good one and compute maps up to homotopy as in topology.
Topological spaces, simplicial sets and chain complexes of modules all carry standard model structures.
</document>

<document source="https://tamnd.github.io/math2/node/SDV692Z915.md" tag="SDV692Z915" kind="statement">
# Shafarevich's theorem on solvable Galois groups

## Statement

Let $G$ be a finite [[solvable group|0P3KZK0FTV]].
Then there is a [[Galois extension|0DS2F7GK05]] $L/\Q$ whose [[Galois group|0XVR72M51M]] $\Gal(L/\Q)$ is isomorphic to $G$.

## Remarks

The proof is long and uses class field theory and embedding problems over number fields.
It builds the extension in steps, and each step solves an embedding problem.
Given a normal subgroup $N$ of $G$ and an extension with group $G/N$, one must find a larger extension with group $G$ that contains it.
Example: the symmetric group $S_4$ is solvable, so it is a Galois group over $\Q$; the splitting field of $x^4 - x - 1$ is one such extension.
The theorem answers the [[inverse Galois problem|YJS6X8JN7D]] for solvable groups only; nonsolvable groups need other methods.

## Explanations

### L1 human

The inverse Galois problem asks which finite groups arise as symmetry groups of rational polynomials.
Shafarevich's theorem says that all solvable groups do.
These are the groups that can be built from commutative pieces, the same groups that correspond to equations solvable by radicals.

### L2 human

Abelian groups are easy to realise inside cyclotomic fields, but solvable groups need many successive extensions.
The difficulty lies in the embedding problem: an extension with a given quotient group need not extend to one with the larger group.
The theorem holds over every number field, not only over $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/SDYEK7HEZQ.md" tag="SDYEK7HEZQ" kind="definition">
# Prolate spheroidal wave functions

## Definition

Let $c \ge 0$ be a real parameter, called the bandwidth parameter.
Consider the differential equation on $(-1, 1)$
$$\frac{d}{dx}\Big( (1 - x^2) \frac{d\psi}{dx} \Big) - c^2 x^2 \psi = -\chi \psi,$$
where $\chi$ is a real number.
There is an increasing sequence $\chi_0 < \chi_1 < \chi_2 < \cdots$ that tends to $\infty$ and has the following property.
The equation has a nonzero bounded solution on $(-1, 1)$ exactly when $\chi = \chi_m$ for some $m$.
For each $\chi_m$ this solution is unique up to a constant factor.
The *prolate spheroidal wave functions* with parameter $c$ are these solutions $\psi_0, \psi_1, \psi_2, \dots$, normalized by $\int_{-1}^1 \abs{\psi_m(x)}^2\, dx = 1$ and taken real.

## Remarks

Example: for $c = 0$ the equation is Legendre's equation $((1 - x^2)\psi')' = -\chi \psi$.
Its bounded solutions are the Legendre polynomials $P_m$ with $\chi_m = m(m + 1)$, so $\psi_m = \sqrt{(2m + 1)/2}\, P_m$; for instance $\psi_0 = 1/\sqrt{2}$ satisfies the equation with $\chi_0 = 0$.

Non-example: for $c = 0$ and $\chi = 0$, the function $\psi(x) = \frac12 \log \frac{1 + x}{1 - x}$ also solves the equation, since $(1 - x^2)\psi'(x) = 1$.
It is not bounded near $\pm 1$, so it is not a prolate spheroidal wave function.

For $c > 0$, the functions $\psi_m$ are also the [[eigenfunctions|0BQAA2BK3Q]] of the integral operator
$$Q_c \psi(x) = \int_{-1}^1 \frac{\sin c(x - y)}{\pi (x - y)}\, \psi(y)\, dy$$
on $L^2(-1, 1)$, which is [[self-adjoint|0NEGN5CRSC]], positive and [[compact|M2NEP1K5AE]].
Its eigenvalues satisfy $1 > \lambda_0 > \lambda_1 > \cdots > 0$.
The kernel of $Q_c$ is that of the projection onto functions whose Fourier transform, in the convention $\int f(t) e^{-i\omega t}\, dt$, vanishes outside $[-c, c]$.
Extending $\psi_m$ to $\R$ by the same integral, divided by $\lambda_m$, gives a [[band-limited function|P3T104G564]], and $\lambda_m$ is the fraction of its energy that lies in $[-1, 1]$.
</document>

<document source="https://tamnd.github.io/math2/node/SEAW0VNYJ6.md" tag="SEAW0VNYJ6" kind="definition">
# Weyl's unitary trick

## Definition

Let $\mathfrak{g}$ be a complex semisimple Lie algebra.
*Weyl's unitary trick* is the method of studying the finite-dimensional complex representations of $\mathfrak{g}$ through the following steps:

1. choose a compact real form $\mathfrak{k}$ of $\mathfrak{g}$, that is, a real Lie subalgebra with $\mathfrak{g} = \mathfrak{k} \oplus i\mathfrak{k}$ whose Killing form is negative definite;
2. restrict from $\mathfrak{g}$ to $\mathfrak{k}$, a bijection from complex representations of $\mathfrak{g}$ to representations of $\mathfrak{k}$ on complex spaces that keeps invariant subspaces;
3. pass to the simply connected Lie group $K$ with Lie algebra $\mathfrak{k}$, a [[compact Lie group|9TFC6ADC1C]] to whose [[representations|WYGRXCG2RN]] those of $\mathfrak{k}$ integrate;
4. prove the desired property for representations of the compact group $K$, for example by averaging over its Haar measure, and transfer it back to $\mathfrak{g}$.

## Remarks

Example: for $\mathfrak{g} = \mathfrak{sl}(2, \C)$, one takes $\mathfrak{k} = \mathfrak{su}(2)$ and $K = SU(2)$.
Complete reducibility of representations of $SU(2)$, proved by averaging an inner product, gives complete reducibility of finite-dimensional representations of $\mathfrak{sl}(2, \C)$.
Non-example: the real form $\mathfrak{sl}(2, \R)$ of $\mathfrak{sl}(2,\C)$ cannot replace $\mathfrak{k}$ in step 3, because every connected Lie group with Lie algebra $\mathfrak{sl}(2,\R)$, such as $SL(2,\R)$, is not compact.
The same method applies to a connected complex semisimple Lie group $G$ with a compact real form $K$, using holomorphic representations of $G$.
In step 3, the integrated representation of $K$ has the same invariant subspaces as the representation of $\mathfrak{k}$.
The compactness of $K$ in step 3 is a theorem of Weyl.
It says that a connected Lie group whose Lie algebra has a negative definite Killing form is compact, and so is its universal cover.
</document>

<document source="https://tamnd.github.io/math2/node/SEH68BA1QQ.md" tag="SEH68BA1QQ" kind="concept">
# Adele ring

## Summary

The adele ring $\mathbb{A}_K$ of a number field $K$ is a restricted product of the completions $K_v$ of $K$ at all its [[places|BT5V3TSNY6]].
The restriction is taken with respect to the rings of integers of the non-archimedean completions.
It is a locally compact topological ring in which $K$ sits diagonally as a discrete subring with compact quotient.
It allows all completions of $K$ to be handled at once.
It is the setting of Tate's thesis, of the [[idele class group|9GHD9MXNYV]] form of [[class field theory|BEKCEYHCHX]] and of the theory of automorphic forms.

## Explanations

### L1 human

A rational number can be seen inside the real numbers and inside each field of [[p-adic numbers|BT6T702C28]] $\Q_p$.
An adele of $\Q$ is a family with one real number and one $p$-adic number for each prime $p$.
All but finitely many of the $p$-adic entries must be $p$-adic integers.
For example, $1/6$ gives the adele whose entries are all $1/6$; it is allowed, since $1/6$ is a $p$-adic integer for every $p$ except $2$ and $3$.
Working with all these entries together turns many local statements into one global statement.
</document>

<document source="https://tamnd.github.io/math2/node/SEKY67RPX5.md" tag="SEKY67RPX5" kind="definition">
# Loop-erased random walk

## Definition

Let $G$ be a [[graph|00C0BR7D7H]] and let $\gamma = (\gamma_0, \ldots, \gamma_n)$ be a finite path in $G$.
Its *loop erasure* $\mathrm{LE}(\gamma) = (\sigma_0, \ldots, \sigma_m)$ is defined inductively.
Let $s_0 = \max \set{k : \gamma_k = \gamma_0}$ and $\sigma_0 = \gamma_0$.
If $s_i < n$, let $\sigma_{i+1} = \gamma_{s_i + 1}$ and $s_{i+1} = \max \set{k : \gamma_k = \sigma_{i+1}}$.
Stop at the first $i = m$ with $s_m = n$.

Let $G$ be connected, let $A$ be a nonempty vertex set, and let $X$ be a simple [[random walk|04VZE3FZXF]] on $G$ from a vertex $x$.
Assume that the hitting time $\tau_A = \min \set{k \ge 0 : X_k \in A}$, a [[stopping time|WND87FBAKC]], is finite almost surely.
The *loop-erased random walk* from $x$ to $A$ is the random path $\mathrm{LE}(X_0, X_1, \ldots, X_{\tau_A})$.

## Remarks

The loop erasure visits each vertex at most once, starts at $\gamma_0$ and ends at $\gamma_n$.
Example: for $\gamma = (0, 1, 2, 1, 2, 3)$ on $\Z$, one gets $s_0 = 0$, $\sigma_1 = 1$, $s_1 = 3$, $\sigma_2 = \gamma_4 = 2$, $s_2 = 4$, $\sigma_3 = \gamma_5 = 3$, $s_3 = 5$, so $\mathrm{LE}(\gamma) = (0, 1, 2, 3)$.
Non-example: the path $(0, 1, 2, 1, 2, 3)$ itself is not a loop erasure of anything, since it visits $1$ and $2$ twice.
On a transient graph, such as $\Z^d$ with $d \ge 3$, one also erases the loops of the infinite walk, since each vertex is visited only finitely often.
Wilson's algorithm builds a uniform spanning tree of a finite connected graph from successive loop-erased random walks.
</document>

<document source="https://tamnd.github.io/math2/node/SENTYNSKSF.md" tag="SENTYNSKSF" kind="concept">
# Puiseux series

## Summary

A Puiseux series is a formal Laurent series in a fractional power $X^{1/m}$ of the variable, for some positive integer $m$, see the [[definition|D5BDZZW9D4]].
Puiseux series form a field that contains the [[field of formal Laurent series|YZWYF7DXTW]] $k((X))$.
By the [[Newton–Puiseux theorem|ZWT5HGRFXX]], over an algebraically closed field of characteristic $0$ they give all roots of polynomial equations with Laurent series coefficients.
They describe the branches of a plane algebraic curve near a point.

## Explanations

### L1 human

The equation $Y^2 = X$ has no solution that is a power series in $X$, but it has the solution $Y = X^{1/2}$.
Puiseux series allow such fractional powers, as long as all exponents have a common denominator and only finitely many are negative.
An example is $X^{-1/3} + 2 + X^{1/3} + X^{2/3} + \cdots$.
Near the origin, the curve $Y^2 = X^3$ has the two branches $Y = X^{3/2}$ and $Y = -X^{3/2}$, which are Puiseux series.
</document>

<document source="https://tamnd.github.io/math2/node/SEVHW7TPPZ.md" tag="SEVHW7TPPZ" kind="definition">
# Krein space

## Definition

Let $K$ be a complex vector space and let $[\cdot, \cdot]$ be a [[Hermitian form|48P4A1CX85]] on $K$.
The pair $(K, [\cdot, \cdot])$ is a *Krein space* if there are subspaces $K_+$ and $K_-$ of $K$ such that:

1. $K = K_+ \oplus K_-$ as a direct sum of vector spaces.
2. $[x, y] = 0$ for all $x \in K_+$ and $y \in K_-$.
3. $(K_+, [\cdot, \cdot])$ is a [[Hilbert space|HM2HHE3J51]].
4. $(K_-, -[\cdot, \cdot])$ is a Hilbert space.

Such a splitting is a *fundamental decomposition* of $K$.

## Remarks

Let $P_\pm$ be the projections of $K$ onto $K_\pm$ along the decomposition, and let $J = P_+ - P_-$.
Then $\ip{x}{y}_J = [Jx, y]$ is a Hilbert space inner product on $K$, and $[x, y] = \ip{Jx}{y}_J$.
The map $J$ is the *fundamental symmetry*; the decomposition is not unique, but all the norms $\ip{x}{x}_J^{1/2}$ are equivalent.
If $K_-$ has finite dimension $\kappa$, the space is a *Pontryagin space* $\Pi_\kappa$.

Example: $K = \C^2$ with $[x, y] = x_1 \overline{y_1} - x_2 \overline{y_2}$, $K_+ = \C e_1$ and $K_- = \C e_2$.
Clause 2 holds since $[e_1, e_2] = 0$, and $[x, x] = \abs{x_1}^2$ on $K_+$ and $-[x, x] = \abs{x_2}^2$ on $K_-$ are positive definite.
The vector $e_1 + e_2$ satisfies $[e_1 + e_2, e_1 + e_2] = 0$, so the form is indefinite.

Non-example: $K = \C^2$ with $[x, y] = x_1 \overline{y_1}$.
The vector $e_2$ satisfies $[e_2, y] = 0$ for every $y$.
In a Krein space, write $e_2 = x_+ + x_-$; then $[x_+, x_+] = [e_2, x_+] = 0$ forces $x_+ = 0$, and in the same way $x_- = 0$.
So the form of a Krein space is nondegenerate, and this degenerate form admits no fundamental decomposition.
</document>

<document source="https://tamnd.github.io/math2/node/SF1VEDT29F.md" tag="SF1VEDT29F" kind="definition">
# Scorer's function

## Definition

For real $x$, *Scorer's functions* are
$$\operatorname{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left( \frac{t^3}{3} + x t \right) dt, \qquad \operatorname{Hi}(x) = \frac{1}{\pi} \int_0^\infty \exp\left( -\frac{t^3}{3} + x t \right) dt,$$
where the first integral is an improper Riemann integral.

## Remarks

They solve the inhomogeneous [[Airy equations|PNESAD6R5T]] $\operatorname{Gi}'' - x \operatorname{Gi} = -\frac{1}{\pi}$ and $\operatorname{Hi}'' - x \operatorname{Hi} = \frac{1}{\pi}$.
Check for $\operatorname{Hi}$: differentiating under the integral sign, $\operatorname{Hi}'' - x \operatorname{Hi} = \frac{1}{\pi} \int_0^\infty (t^2 - x)\, e^{-t^3/3 + xt}\, dt = -\frac{1}{\pi} \left[ e^{-t^3/3 + xt} \right]_0^\infty = \frac{1}{\pi}$.
By comparing with the integral that defines the [[Airy function|1N5D4PN92W]] $\operatorname{Bi}$, one has $\operatorname{Gi} + \operatorname{Hi} = \operatorname{Bi}$.
A non-example: $\operatorname{Bi}$ itself is not a Scorer function; it solves the homogeneous equation $y'' - x y = 0$, not one with right-hand side $\pm \frac{1}{\pi}$.
Both functions extend to entire functions of $z \in \C$.
</document>

<document source="https://tamnd.github.io/math2/node/SF3X43X837.md" tag="SF3X43X837" kind="statement">
# Hölder's theorem

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], as a holomorphic function on $\C \setminus \set{0, -1, -2, \ldots}$.
Let $n \ge 0$ be an integer and let $P \in \C[Y, X_0, X_1, \ldots, X_n]$ be a polynomial in $n + 2$ variables (an element of a [[polynomial ring|0SDC934R52]]).
If
$$P\big(z, \Gamma(z), \Gamma'(z), \ldots, \Gamma^{(n)}(z)\big) = 0$$
for all $z$ in the domain of $\Gamma$, then $P = 0$.

## Remarks

In words, $\Gamma$ satisfies no algebraic differential equation whose coefficients are polynomials in $z$; one says that $\Gamma$ is differentially transcendental.
One proof assumes a nonzero $P$ of least possible degree and uses the functional equation $\Gamma(z + 1) = z\Gamma(z)$ to build from it a nonzero polynomial relation of smaller degree, a contradiction.
In particular $\Gamma$ is not a [[holonomic function|7H44JD9NQD]], since a linear differential equation with polynomial coefficients is a special algebraic one.
By contrast, the exponential and many other [[special functions|HWY72ZK273]], such as Bessel functions, do satisfy algebraic differential equations.
The theorem is named after Otto Hölder; it is not the same as [[Hölder's theorem on Archimedean groups|CDM40MT3HZ]].

## Explanations

### L1 human

Many familiar functions obey a differential equation built from polynomials, like $f' = f$ for the exponential.
Hölder's theorem says the gamma function obeys no such equation, however complicated.
So it lives outside the world of functions described by polynomial differential equations.

### L2 human

The theorem shows that the gamma function is not a solution of any of the classical differential equations of mathematical physics.
This contrasts with the hypergeometric and Bessel functions, which are defined by linear differential equations, and explains why $\Gamma$ is usually studied through functional equations, integrals and products instead.
</document>

<document source="https://tamnd.github.io/math2/node/SF738NKWJ0.md" tag="SF738NKWJ0" kind="concept">
# Block system

## Summary

A block system for a group acting on a set is a partition of the set into parts that the group permutes among themselves.
Each part is a block: a set that every group element either maps onto itself or moves off itself entirely.
Block systems measure how far a [[transitive action|02GK6HZYBT]] is from being [[primitive|Y10S0751KG]].
For a transitive action, block systems correspond to subgroups between a point [[stabiliser|0AYMP41THN]] and the whole group.

## Explanations

### L1 human

Picture the four corners of a square and the eight symmetries of the square.
Every symmetry maps a diagonal to a diagonal: the pair of opposite corners stays a pair of opposite corners.
So the two diagonals form a block system: the symmetries may swap the two diagonals, but never break one apart.
The two trivial block systems, one part with everything or every point alone, exist for every action.
A group is primitive when these trivial ones are the only block systems.
</document>

<document source="https://tamnd.github.io/math2/node/SFGFWZ4F5N.md" tag="SFGFWZ4F5N" kind="definition">
# Transform pair

## Definition

Let $T$ be an [[integral transform|YCK709QB06]], defined on a class $A$ of functions.
A *transform pair* for $T$ is a pair $(f, F)$ with $f \in A$ and $F = Tf$.
It is written $f \leftrightarrow F$.

## Remarks

Example: for the Laplace transform $T$ and a real number $a$, the pair $(e^{at}, 1/(s - a))$ is a transform pair, because $\int_0^\infty e^{at} e^{-st}\, dt = 1/(s - a)$ for $s > a$.
Non-example: $(1, 1/s^2)$ is not a Laplace transform pair, because the Laplace transform of $1$ is $1/s$, not $1/s^2$.
When $T$ is injective on $A$, each $F$ in the image determines $f$, and the pair can be read in both directions.
</document>

<document source="https://tamnd.github.io/math2/node/SFJ329YB9F.md" tag="SFJ329YB9F" kind="concept">
# Linking number

## Summary

The linking number of two disjoint oriented closed curves in $\R^3$ is an integer that counts how many times one curve winds around the other.
It does not change when the curves move without crossing each other, so it is an invariant of two-component [[links|1A5WRCNTM3]].
It can be read off a diagram, computed by a homology class, or given by the Gauss double integral.

## Explanations

### L1 human

Take two closed loops of string in space that do not touch.
The linking number says how many times the first loop goes through the second, counted with a sign for the direction.
Two separate rings that can be pulled apart have linking number $0$.
Two rings hooked once, as in a chain, have linking number $1$ or $-1$, depending on the chosen directions.
You cannot unhook the rings without cutting one of them, and the number shows this.
</document>

<document source="https://tamnd.github.io/math2/node/SFMD8RKRSE.md" tag="SFMD8RKRSE" kind="concept">
# Relaxed control

## Summary

A relaxed control assigns to each time a probability measure on the set of control values, instead of a single control value.
The system then moves with the average of the velocities with respect to this measure.
Relaxed controls fill in the convex hulls of the velocity sets, so optimal control problems without a minimizer gain one in the relaxed sense.
They are the control theory counterpart of [[Young measures|6TQSVJQDEH]] and of the [[relaxation of a variational problem|DPS8QZBVB0]].

## Explanations

### L1 human

Suppose a car may only steer fully left or fully right, but you want it to drive straight.
By switching very fast between left and right, it moves almost straight.
A relaxed control describes the limit of such fast switching: at each moment it says "half left, half right".
In the limit the car goes straight, even though no ordinary control does that.
[[Filippov's existence theorem|XEX7QZV1BS]] needs convex velocity sets, and relaxed controls supply this convexity.
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<document source="https://tamnd.github.io/math2/node/SFPKB3NC0E.md" tag="SFPKB3NC0E" kind="concept">
# Hilbertian field

## Summary

A field is Hilbertian if irreducible polynomials in several variables stay irreducible after suitable values in the field are substituted for all but one variable.
[[Hilbert's irreducibility theorem|N1V3YEYHNP]] says that the field $\Q$ of rational numbers is Hilbertian.
Hilbertian fields are the natural setting for realizing finite groups as [[Galois groups|0XTY86BZKY]] by specialization, an approach to the [[inverse Galois problem|YJS6X8JN7D]].
In characteristic $0$, a field is Hilbertian exactly when its affine line is not a thin set in the sense of Serre.

## Explanations

### L1 human

Take a polynomial such as $X^2 - T$, which cannot be factored as a polynomial in $X$ and $T$.
If you put a number in place of $T$, the result may factor: $X^2 - 4 = (X - 2)(X + 2)$.
Over the rational numbers, most choices, such as $T = 2$ or $T = 3$, keep it irreducible.
A field is Hilbertian if this always happens: for any irreducible polynomial in $X$ and $T$ that involves $X$, some (in fact infinitely many) values of $T$ keep it irreducible.
The real numbers are not Hilbertian, since $X^2 - T^2 - 1$ factors for every real value of $T$.
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<document source="https://tamnd.github.io/math2/node/SFSRQCM3EG.md" tag="SFSRQCM3EG" kind="concept">
# Energy of a measure

## Summary

The energy of a positive measure is the double integral of a potential-theoretic kernel against the measure in both variables.
Typical kernels are $\abs{x - y}^{2-n}$ in $\R^n$ for $n \ge 3$ and $\log(1/\abs{x - y})$ in the plane.
Physically it is the electrostatic energy of the charge distribution described by the measure.
Minimising the energy over probability measures on a compact set defines its [[capacity|4YZVFGJVSZ]] and its equilibrium measure.
Sets that carry no nonzero measure of finite energy are the negligible sets of potential theory.

## Explanations

### L1 human

Spread a unit of electric charge over a body.
Every pair of tiny bits of charge repels with an energy that grows as they get closer, and the energy of the measure adds up all these pair energies.
A point charge has infinite energy, because the charge sits on top of itself.
Charge spread evenly over the unit sphere in space has energy $1$ for the kernel $1/\abs{x - y}$.
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<document source="https://tamnd.github.io/math2/node/SG5C5TVGRB.md" tag="SG5C5TVGRB" kind="concept">
# Quasi-arithmetic mean

## Summary

A quasi-arithmetic mean is the arithmetic mean taken after a change of scale.
Given a continuous strictly monotone function $\varphi$, the mean of $x_1, \dots, x_n$ is $\varphi^{-1}\big(\frac{1}{n} \sum_i \varphi(x_i)\big)$.
The arithmetic, geometric, harmonic and power means are all of this form.
These means are characterized by a few natural axioms, which is the [[Kolmogorov–Nagumo theorem|9Q5DPYQ5NP]].

## Explanations

### L1 human

To average numbers in a quasi-arithmetic way, first move them to a new scale, take the ordinary average there, and move the result back.
With the logarithm as the scale, the average of $2$ and $8$ is computed as the average of $\ln 2$ and $\ln 8$, which is $\ln 4$, so the mean is $4$.
This is the geometric mean.
With the scale $\varphi(x) = 1/x$, the same recipe gives the harmonic mean, used for averaging speeds over equal distances.
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<document source="https://tamnd.github.io/math2/node/SG5Q94QTQE.md" tag="SG5Q94QTQE" kind="definition">
# Fourier coefficient

## Definition

Let $f \colon \R \to \C$ be a [[periodic function|089KWVYD69]] of period $2\pi$ that is Lebesgue [[integrable|X7GTF3D98F]] on $[-\pi, \pi]$, and let $n \in \Z$.
The $n$-th *Fourier coefficient* of $f$ is the complex number
$$\hat f(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx,$$
where the integral is the [[Lebesgue integral|DWJF7ABPR1]].

## Remarks

In terms of the [[Fourier monomials|TZYSW0975T]] $e_n$, $\hat f(n) = \langle f, e_n \rangle$ for the inner product $\langle f, g \rangle = \frac{1}{2\pi} \int_{-\pi}^{\pi} f \overline{g}$.
By periodicity the integral may be taken over any interval of length $2\pi$.
The bound $\abs{\hat f(n)} \le \frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{f}$ holds for every $n$.
Example: for $f(x) = \cos x$, the formula $\cos x = \frac{1}{2}(e^{ix} + e^{-ix})$ and the orthogonality of the $e_n$ give $\hat f(1) = \hat f(-1) = \frac{1}{2}$ and $\hat f(n) = 0$ for $\abs{n} \ne 1$.
Non-example: the $2\pi$-periodic function equal to $1/x$ on $(0, 2\pi]$ is not integrable on a period, so its Fourier coefficients are not defined.
The real coefficients $a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos nx \, dx$ and $b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin nx \, dx$ satisfy $a_n = \hat f(n) + \hat f(-n)$ and $b_n = i(\hat f(n) - \hat f(-n))$.
Other notations are $c_n$ and $\mathcal{F}f(n)$; for period $T$ the coefficient is $\frac{1}{T} \int_0^T f(x) e^{-2\pi i n x / T} \, dx$.
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<document source="https://tamnd.github.io/math2/node/SG7AQ9T8HD.md" tag="SG7AQ9T8HD" kind="concept">
# Ore localization

## Summary

Ore localization builds a ring of fractions $R S^{-1}$ of a noncommutative [[ring|0B1261B5ED]] $R$.
In it the elements of a chosen set $S$ become units, and every element has the form $a s^{-1}$.
It exists when $S$ satisfies the [[Ore condition|TJP4ARC2X7]] together with a reversibility condition.
It extends the commutative [[localization|4C1AE84N5N]] and is the main tool to build division rings of fractions of noncommutative domains and [[classical rings of quotients|C5971PGKPH]].

## Explanations

### L1 human

From the integers one builds the rationals by allowing division by nonzero integers.
Ore localization does the same for a noncommutative ring, but only on one side: every new element is a fraction $a s^{-1}$ with the denominator on the right.
For this to give a ring, one must be able to rewrite $s^{-1} a$ as some $b t^{-1}$, and this is what the Ore condition guarantees.
For example, the polynomial ring $k[x]$ over a field localizes to the field $k(x)$ of rational functions, and the [[Weyl algebra|98QE7KY55W]] localizes to a division ring.
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<document source="https://tamnd.github.io/math2/node/SGGXZ3TBSA.md" tag="SGGXZ3TBSA" kind="object">
# Free loop space

## Description

Let $M$ be a smooth manifold and $S^1 = \R / \Z$ the [[circle|060W5CBZP6]].
The *free loop space* of $M$ is
$$L M = C^\infty(S^1, M),$$
the set of all smooth loops in $M$, with no condition on a base point.
Since $S^1$ is compact, it is a manifold of smooth mappings, which is a Fréchet manifold.
Its tangent space at a loop $\gamma$ is the space of smooth vector fields along $\gamma$, that is, [[sections|WWX7H5W1RP]] of $\gamma^* TM$.

## Remarks


The circle acts on $L M$ by rotating the parameter, $(\theta \cdot \gamma)(t) = \gamma(t + \theta)$, and the fixed points of this action are the constant loops, a copy of $M$.
Evaluation at $0$ is a map $L M \to M$, $\gamma \mapsto \gamma(0)$, whose fibre over $x$ is the smooth based [[loop space|2JFF5F3EGF]] $\Omega_x M$.
For $M = S^1$, a loop has a degree in $\Z$, and the connected components of $L S^1$ correspond to the integers.

In topology one also uses the space $\mathrm{Map}(S^1, X)$ of continuous loops in a space $X$ with the [[compact-open topology|RG15F0TWPW]].
For a manifold $M$, the inclusion of smooth loops into continuous loops is a homotopy equivalence.


## Explanations

### L1 human

A point of the free loop space is a whole closed curve in $M$, which may sit anywhere and need not pass through a chosen point.
Moving a little in this space means wiggling the curve.
Constant curves, which sit still at one point, form a copy of $M$ inside the loop space.
The free loop space is where one looks for closed geodesics, which are the critical points of the energy of a loop.
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<document source="https://tamnd.github.io/math2/node/SGH803FGY4.md" tag="SGH803FGY4" kind="definition">
# Delaunay triangulation

## Definition

Let $P$ be a finite set of points in $\R^2$, not all collinear.
For a triangle $\Delta$, its *circumdisk* is the open disk bounded by the [[circumcircle|03VZJE6C8C]] of $\Delta$.
A [[triangulation|V0B6C6TMWD]] $T$ of $P$ is a *Delaunay triangulation* if no point of $P$ lies in the circumdisk of any triangle of $T$.

## Remarks

Example: let $P = \set{(0,0), (2,0), (1,2)}$, a single triangle.
Its only triangulation is the triangle itself, and the open circumdisk contains no point of $P$, since the three points lie on the circle.

Non-example: let $P = \set{(-2, 0), (2, 0), (0, 1), (0, -1)}$ and take the triangulation by the diagonal from $(-2, 0)$ to $(2, 0)$.
The circumcircle of $(-2, 0), (2, 0), (0, 1)$ has centre $(0, -3/2)$ and radius $5/2$, and $(0, -1)$ is at distance $1/2$ from the centre, so it lies inside.
The other diagonal, from $(0, 1)$ to $(0, -1)$, gives the Delaunay triangulation.

A Delaunay triangulation of $P$ always exists, and it is unique when no four points of $P$ lie on a common circle.
Among all triangulations of $P$, a Delaunay triangulation maximizes the smallest angle of its triangles.
When no four points are concyclic, two points of $P$ are joined by an edge exactly when their cells in the [[Voronoi diagram|QDNEBBVZ84]] share an edge.
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<document source="https://tamnd.github.io/math2/node/SGHMSBHHA0.md" tag="SGHMSBHHA0" kind="statement">
# Excision theorem

## Statement

Let $X$ be a topological space and let $A \subseteq X$ be a subspace.
Let $Z \subseteq A$ be a subset whose [[closure|0QEAJB91GZ]] in $X$ is contained in the [[interior|0RZH6W98HK]] of $A$.
Then the inclusion of pairs $(X \setminus Z, A \setminus Z) \to (X, A)$ induces isomorphisms of [[relative homology|4BVWH0BN28]] groups
$$H_n(X \setminus Z, A \setminus Z) \to H_n(X, A)$$
for every $n \ge 0$.

## Remarks

An equivalent form: if $A, B \subseteq X$ are subspaces whose interiors cover $X$, then the inclusion $(B, A \cap B) \to (X, A)$ induces isomorphisms on all $H_n$.
The proof uses repeated [[barycentric subdivision|J5DZEYCEX2]] to show that every singular chain is homologous to a sum of small simplices, each lying in the interior of $A$ or in $X \setminus Z$.
The chains made of such small simplices compute the same homology as all chains, and on them the excision map is an isomorphism at chain level.

Example: if $A$ is a nonempty closed subspace that is a deformation retract of some open neighbourhood in $X$, then excision gives $H_n(X, A) \cong \tilde H_n(X / A)$, where $\tilde H$ is [[reduced homology|NJQMWVK3QK]].
For $(D^n, S^{n-1})$ this gives $H_n(D^n, S^{n-1}) \cong \tilde H_n(S^n)$.

The closure hypothesis is needed.
Take $X = [-1, 1]$, $A = [0, 1]$ and $Z = A$; the closure of $Z$ is not inside the interior $(0, 1]$ of $A$.
Then $H_0(X \setminus Z, A \setminus Z) = H_0([-1, 0)) \cong \Z$, while $H_0(X, A) = 0$ because $X$ is path-connected and $A$ is nonempty.

## Explanations

### L1 human

Relative homology of $X$ modulo $A$ only depends on what happens near and outside $A$.
So one can cut away a piece deep inside $A$ without changing it.
This turns a hard global computation into a local one, for example the homology near a single point.

### L2 human

Excision is one of the Eilenberg–Steenrod axioms, and it is the axiom that fails for homotopy groups.
With the long exact sequence of a pair it gives the Mayer–Vietoris sequence and the suspension isomorphism $\tilde H_n(X) \cong \tilde H_{n+1}(\Sigma X)$.
For manifolds it gives $H_n(M, M \setminus \set{x}) \cong H_n(\R^n, \R^n \setminus \set{0}) \cong \Z$, the starting point of orientations.
</document>

<document source="https://tamnd.github.io/math2/node/SGQQGP261M.md" tag="SGQQGP261M" kind="example">
# Duffing equation

## Example

Let $\delta, \alpha, \beta, \gamma, \omega$ be real constants.
The Duffing equation is the second-order [[ordinary differential equation|0NT64JV96A]]
$$x'' + \delta x' + \alpha x + \beta x^3 = \gamma \cos(\omega t).$$
Here $\delta$ is a damping coefficient, $\alpha x + \beta x^3$ is a restoring force with a cubic term, and $\gamma \cos(\omega t)$ is a periodic forcing.

Take $\delta = \gamma = 0$, $\alpha = -1$ and $\beta = 1$, so $x'' - x + x^3 = 0$.
This is a [[Hamiltonian system|0EDKGA2FAK]] with $H(x, y) = \frac{1}{2} y^2 - \frac{1}{2} x^2 + \frac{1}{4} x^4$ and $y = x'$.
Its equilibria are $(0, 0)$, a [[hyperbolic|0C5PREPZWG]] saddle, and $(\pm 1, 0)$, two centres.
The function $x(t) = \sqrt{2}\, \operatorname{sech} t$ is a solution.
Check: with $u = \operatorname{sech} t$ one has $u'' = u - 2u^3$, so $x'' - x + x^3 = \sqrt{2}(u - 2u^3) - \sqrt{2} u + 2\sqrt{2} u^3 = 0$.
Since $x(t) \to 0$ and $x'(t) \to 0$ as $t \to \pm\infty$, this solution is a [[homoclinic orbit|6DBSQKZK4W]] of the saddle.
By the symmetry $x \mapsto -x$, there is a second one in the half-plane $x < 0$.

## Remarks

Solutions of the unforced, undamped equation stay on level sets of $H$.
So each solution is an equilibrium, a periodic solution, or a homoclinic solution.
With small damping and small forcing, the two homoclinic orbits can break into a tangle of stable and unstable manifolds.
The [[Melnikov method|A755BDRP1A]] decides when this happens in terms of $\gamma$, $\delta$ and $\omega$.
In that regime, numerical solutions of the Duffing equation look chaotic.
For $\alpha > 0$ and small $\beta$, the equation is a perturbed harmonic oscillator whose frequency depends on the amplitude.

## Explanations

### L1 human

The Duffing equation is a spring whose force is not exactly proportional to the stretch, pushed back and forth periodically.
With $\alpha < 0 < \beta$ the spring has two stable rest positions, like a ball in a double well.
A forced ball can jump between the two wells in an irregular way, which is a classical example of chaos.
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<document source="https://tamnd.github.io/math2/node/SGVQ5ZMEGE.md" tag="SGVQ5ZMEGE" kind="definition">
# Completely monotone function

## Definition

Let $f \colon (0, \infty) \to \R$ be [[infinitely differentiable|0V33BNKZ9S]], and let $f^{(k)}$ be its [[derivative of order $k$|04TKTF8KDK]], with $f^{(0)} = f$.
The function $f$ is *completely monotone* if
$$(-1)^k f^{(k)}(x) \ge 0$$
for every integer $k \ge 0$ and every $x > 0$.

## Remarks

Example: $f(x) = 1/x$ is completely monotone, since $f^{(k)}(x) = (-1)^k k! \, x^{-k-1}$ and so $(-1)^k f^{(k)}(x) = k! \, x^{-k-1} > 0$.
Non-example: $g(x) = e^{-x^2}$ satisfies the conditions for $k = 0$ and $k = 1$, but $g''(x) = (4 x^2 - 2) e^{-x^2} < 0$ for $0 < x < 1/\sqrt{2}$, so the condition fails for $k = 2$.
Sums, products and positive multiples of completely monotone functions are completely monotone.
The same definition is used on other open intervals; on $[0, \infty)$ one also asks that $f$ be continuous at $0$.
The function $f$ is completely monotone on $(0, \infty)$ if and only if $x \mapsto f(-x)$ is [[absolutely monotonic|QQSPWH4NG9]] on $(-\infty, 0)$.
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<document source="https://tamnd.github.io/math2/node/SGXYY92GRS.md" tag="SGXYY92GRS" kind="definition">
# Walsh function

## Definition

For $k \ge 1$, let $r_k$ be the $k$-th [[Rademacher function|51GRFQ4WY6]] on $[0, 1)$.
Thus $r_k(t) = (-1)^{t_k}$, where $t_k$ is the $k$-th binary digit of $t$, taken from the expansion that does not end in an infinite run of $1$s.
Let $n \ge 0$ be an integer with binary expansion $n = \sum_{k \ge 0} b_k 2^k$, $b_k \in \set{0, 1}$.
The *Walsh function* of index $n$ (in Paley order) is
$$w_n = \prod_{k \ge 0} r_{k+1}^{\,b_k},$$
a finite product, with $w_0 = 1$.

## Remarks

The family $(w_n)_{n \ge 0}$ is the *Walsh system*.
It is an [[orthonormal basis|0S9GGTR0EX]] of $L^2([0, 1))$ (see [[Lp space|SASP14MA1Z]]).
Other orderings of the same functions are in use, such as the Walsh order by number of sign changes.

Example: $n = 3 = 1 + 2$, so $w_3 = r_1 r_2$.
Here $r_1 = 1$ on $[0, \tfrac12)$ and $-1$ on $[\tfrac12, 1)$, while $r_2 = 1$ on $[0, \tfrac14) \cup [\tfrac12, \tfrac34)$ and $-1$ elsewhere.
So $w_3$ is $1, -1, -1, 1$ on the four quarters of $[0, 1)$, and $\int_0^1 w_3 = 0$, as orthogonality to $w_0$ requires.

Non-example: the function $\cos(2\pi t)$ is orthogonal to many Walsh functions, but it is not a Walsh function, since it takes values other than $\pm 1$.
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<document source="https://tamnd.github.io/math2/node/SGZZ4Y0D36.md" tag="SGZZ4Y0D36" kind="concept">
# Mellin convergence

## Summary

A function $f$ on $(0, \infty)$ is Mellin convergent at a complex number $s$ when $x^{s-1} f(x)$ is integrable on $(0, \infty)$.
This is exactly the condition under which the [[Mellin transform|N5PVGRJE1T]] $\mathcal{M}f(s)$ is defined as an absolutely convergent integral.
It depends only on $\operatorname{Re} s$, and the set of real parts where it holds is an interval.
Its interior gives the vertical strip on which the Mellin transform is holomorphic.

## Explanations

### L1 human

The power $x^{s-1}$ is large near $0$ when $\operatorname{Re} s < 1$ and large near $\infty$ when $\operatorname{Re} s > 1$.
So Mellin convergence is a balance: $f$ must be small enough near $0$ to tame one end and small enough near $\infty$ to tame the other.
For example, $f(x) = 1/(1 + x)$ behaves like $1$ near $0$ and like $1/x$ near $\infty$.
It is Mellin convergent exactly when $0 < \operatorname{Re} s < 1$.
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<document source="https://tamnd.github.io/math2/node/SH3R5VTMZH.md" tag="SH3R5VTMZH" kind="statement">
# Cayley–Hamilton theorem for modules

## Statement

Let $R$ be a [[commutative ring|0310REJJY8]], let $I$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be an $R$-module [[generated|05QY1VEBYD]] by $n$ elements.
Let $\varphi \colon M \to M$ be an $R$-linear [[endomorphism|0J7H0YBFDJ]] with $\varphi(M) \subseteq IM$.
Then there are elements $a_1, \ldots, a_n \in R$ with $a_i \in I^i$ for each $i$ such that
$$\varphi^n + a_1 \varphi^{n-1} + \cdots + a_{n-1} \varphi + a_n \id_M = 0.$$

## Remarks

Let $x_1, \ldots, x_n$ generate $M$ and write $\varphi(x_i) = \sum_j a_{ij} x_j$ with $a_{ij} \in I$.
View $M$ as a module over $R[t]$ with $t$ acting as $\varphi$.
The matrix $B = t I_n - (a_{ij})$ kills the column vector $(x_1, \ldots, x_n)$; multiplying by the [[adjugate|07W9H8QF38]] of $B$ shows that $\det B$ kills every $x_j$, hence all of $M$.
Expanding $\det B$ gives the polynomial, and its coefficient of $t^{n-i}$ is a sum of products of $i$ entries in $I$.
For $I = R$ and $M = R^n$, with $\varphi$ given by a matrix $A$, one recovers the [[Cayley–Hamilton theorem|0F765KFC2J]] $p_A(A) = 0$.

## Explanations

### L1 human

Every endomorphism of a finite-dimensional vector space satisfies its characteristic polynomial.
This result says the same thing for any finitely generated module over a commutative ring.
It adds a bonus: if the map pushes $M$ into $IM$, the coefficients of the polynomial lie in powers of $I$.

### L2 human

This is the engine behind the [[determinant trick|A8HRT9H3XA]] and hence behind [[Nakayama's lemma|15JHBYR6KH]].
Taking $\varphi = \id_M$ when $IM = M$ gives an element $1 + a_1 + \cdots + a_n$ with $a_i \in I$ that annihilates $M$.
It is also the standard tool to prove that an element is integral.
If $x$ lies in a ring $S \supseteq R$ and $M$ is a faithful $R[x]$-module that is finitely generated over $R$, then $x$ satisfies a monic polynomial over $R$.
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# Green's function

## Summary

A Green's function of a linear differential operator $L$ with boundary conditions is a kernel $G(x, s)$ that turns the source term into the solution.
The function $y(x) = \int G(x, s) f(s)\, ds$ solves $L y = f$.
Formally it is the response of the system to a unit point source at $s$, that is, $L G(\cdot, s) = \delta_s$ with the [[Dirac delta|0KPHENZ5S7]].
It turns a differential equation into an integral formula, and it is basic in boundary value problems, potential theory and physics.
The [[Green's function for the Laplacian|M36SD56MWQ]] is the main case for partial differential equations.
See the [[definition|ZH4V0WHSXT]] for ordinary differential equations.

## Explanations

### L1 human

Think of a taut string fixed at both ends.
Pushing it at one point $s$ with a unit force bends it into a shape $G(x, s)$.
A general load $f$ is a sum of many point pushes, so the string takes the shape $\int G(x, s) f(s)\, ds$.
For $y'' = f$ on $[0, 1]$ with $y(0) = y(1) = 0$, the Green's function is $G(x, s) = x (s - 1)$ for $x \le s$ and $G(x, s) = s (x - 1)$ for $x \ge s$.
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# Singular homology

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]].
For $n \ge 0$ let $\Delta^n = \set{(t_0, \dots, t_n) \in \R^{n+1} : t_i \ge 0, \ \sum_i t_i = 1}$ be the standard $n$-simplex, and for $0 \le i \le n$ let $\delta^i \colon \Delta^{n-1} \to \Delta^n$ be the map that inserts a $0$ in position $i$.
A *singular $n$-simplex* in $X$ is a [[continuous map|0SY9ECQ19Q]] $\sigma \colon \Delta^n \to X$.
Let $S_n(X)$ be the [[free abelian group|0WJX8MGG37]] on the singular $n$-simplices, and $S_n(X) = 0$ for $n < 0$.
The boundary map $\partial_n \colon S_n(X) \to S_{n-1}(X)$ is given on generators by
$$\partial_n \sigma = \sum_{i=0}^{n} (-1)^i \, \sigma \circ \delta^i,$$
and $\partial_0 = 0$.
Then $\partial_n \circ \partial_{n+1} = 0$, so $S_\bullet(X)$ is a [[chain complex|WQ82C0M3QB]].
The *$n$-th singular homology group* of $X$ is
$$H_n(X) = \ker \partial_n / \operatorname{im} \partial_{n+1}.$$

## Remarks

A continuous map $f \colon X \to Y$ induces $f_* \colon H_n(X) \to H_n(Y)$ by $\sigma \mapsto f \circ \sigma$, and homotopic maps induce the same homomorphism.
Homology with coefficients in an abelian group $A$ uses $S_n(X) \otimes A$.

Example: let $X$ be a point.
There is exactly one singular $n$-simplex $\sigma_n$ for each $n$, and $\partial_n \sigma_n = \sum_{i=0}^n (-1)^i \sigma_{n-1}$, which is $\sigma_{n-1}$ for even $n \ge 2$ and $0$ for odd $n$.
So $\partial_n$ is an isomorphism for even $n \ge 2$ and zero for odd $n$, which gives $H_0(X) \cong \Z$ and $H_n(X) = 0$ for $n \ge 1$.

Non-example: using all functions $\Delta^n \to X$, continuous or not, in place of singular simplices still gives a chain complex, but not singular homology.
Any two points of $X$ are then joined by a $1$-simplex, so its $H_0$ is $\Z$ for every nonempty $X$.
For $X$ a discrete space with two points, singular homology gives $H_0(X) \cong \Z^2$.
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# Relaxed control

## Definition

Let $U$ be a nonempty [[compact set|0R6EWFKRCF]] in $\R^m$, and let $\mathcal{P}(U)$ be the set of Borel [[probability measures|059TJN0NZ1]] on $U$.
Let $T > 0$.
A *relaxed control* on $[0, T]$ is a family $(\mu_t)_{t \in [0, T]}$ with $\mu_t \in \mathcal{P}(U)$ for every $t$.
It must be measurable in this sense: for every continuous function $h \colon U \to \R$, the map $t \mapsto \int_U h \, d\mu_t$ is Lebesgue measurable on $[0, T]$.
For a [[control system|ZPDTVZQN5K]] $x' = f(t, x, u)$ with $f$ continuous, the trajectory of the relaxed control $(\mu_t)$ solves the relaxed equation
$$x'(t) = \int_U f(t, x(t), v) \, d\mu_t(v).$$

## Remarks

Every ordinary [[admissible control|8M31QKB0GW]] $u \colon [0, T] \to U$ gives the relaxed control $\mu_t = \delta_{u(t)}$, the Dirac measure at $u(t)$.
Its relaxed equation is $x' = f(t, x, u(t))$, the original one.
Example: $U = \set{-1, 1}$ and $\mu_t = \frac12 \delta_{-1} + \frac12 \delta_1$ for all $t$.
It is a relaxed control, since $\int_U h \, d\mu_t = \frac12 (h(-1) + h(1))$ is constant in $t$.
For $f(t, x, u) = u$ its trajectory has $x' = 0$, a velocity that no ordinary control with values in $\set{-1, 1}$ gives.
Non-example: $\mu_t = \delta_{-1} + \delta_1$ is not a relaxed control, since $\mu_t(U) = 2$, so $\mu_t$ is not a probability measure.
The velocities of relaxed controls fill the convex hull of $f(t, x, U)$, so the convexity hypothesis of [[Filippov's existence theorem|XEX7QZV1BS]] holds for the relaxed problem.
A relaxed control is a [[Young measure|6TQSVJQDEH]] on $[0, T]$ with values in $U$.
</document>

<document source="https://tamnd.github.io/math2/node/SH9NBM8JRR.md" tag="SH9NBM8JRR" kind="object">
# 24-cell

## Description

The *24-cell* is the convex polytope in $\R^4$ whose vertices are the $24$ points obtained from $(\pm 1, \pm 1, 0, 0)$ by permuting the coordinates.
It is a convex [[regular polytope|F60PD912D5]] with [[Schläfli symbol|MAE6BF4DFK]] $\set{3, 4, 3}$.
It has $24$ vertices, $96$ edges, $96$ triangular faces and $24$ cells, which are regular octahedra.

Check of the vertex and edge counts: there are $\binom{4}{2} = 6$ ways to place the two nonzero coordinates and $4$ choices of signs, so $24$ vertices.
All vertices lie at distance $\sqrt2$ from the origin.
The nearest neighbours of $(1,1,0,0)$ are the $8$ points $(1,0,\pm1,0)$, $(1,0,0,\pm1)$, $(0,1,\pm1,0)$, $(0,1,0,\pm1)$, at distance $\sqrt2$.
So each vertex lies on $8$ edges, and there are $24 \cdot 8 / 2 = 96$ edges.
The Euler characteristic check $24 - 96 + 96 - 24 = 0$ holds, as for every convex $4$-polytope.

## Remarks

The vertex figure is a cube, in agreement with the symbol: the last two entries $\set{4, 3}$ are the symbol of the cube.
The symbol reads the same backwards, and indeed the 24-cell is self-dual.
Apart from polygons and simplices, it is the only self-dual convex regular polytope, and it has no analogue in other dimensions.
Its symmetry group has order $1152$.
Another model: the $24$ units $\pm 1$, $\pm i$, $\pm j$, $\pm k$ and $\frac12(\pm 1 \pm i \pm j \pm k)$ of the [[Hurwitz quaternions|QHTAEPTKEB]] are the vertices of a 24-cell.

## Explanations

### L1 human

The 24-cell is a four-dimensional solid built from $24$ octahedra, glued so that each triangle is shared by two of them.
It is one of the six perfectly regular solids of four-dimensional space.
Unlike the other five, it is its own dual and it has no cousin among the regular solids of three or five dimensions.
</document>

<document source="https://tamnd.github.io/math2/node/SHEN158T3M.md" tag="SHEN158T3M" kind="concept">
# Local class field theory

## Summary

Local class field theory describes the abelian extensions of a non-Archimedean [[local field|J85FZEQG38]] $K$ in terms of the multiplicative group $K^\times$.
Its main result is the local reciprocity map $K^\times \to \Gal(K^{ab} / K)$, where $K^{ab}$ is the maximal abelian extension of $K$.
For each finite abelian extension $L / K$, it induces an isomorphism $K^\times / N_{L/K}(L^\times) \cong \Gal(L / K)$.
The finite abelian extensions of $K$ correspond to the open subgroups of finite index of $K^\times$, through their norm groups.

## Explanations

### L1 human

Class field theory says that the abelian extensions of a field are controlled by data inside the field itself.
Over a local field such as $\Q_p$ the data is simple: the group of nonzero elements $K^\times$.
Each abelian extension corresponds to a subgroup of $K^\times$ of finite index, namely the norms from the extension.
The Galois group of the extension is the quotient of $K^\times$ by this subgroup.
For example, take $K = \Q_p$ and its unramified extension of degree $n$.
It corresponds to the elements whose [[p-adic absolute value|WGN82TCEF2]] is an integer power of $p^n$.
It is the local half of [[class field theory|BEKCEYHCHX]]; the global theory glues the local ones together.
</document>

<document source="https://tamnd.github.io/math2/node/SHF8Z7GMFX.md" tag="SHF8Z7GMFX" kind="concept">
# Order of an entire function

## Summary

The order of an [[entire function|0419794R0T]] measures how fast its maximum modulus grows, on the scale of functions $\exp(r^\lambda)$.
It is the least exponent $\rho$ such that $\abs{f(z)}$ is at most about $\exp(\abs{z}^{\rho})$ for large $\abs{z}$.
Polynomials have order $0$, $e^z$ has order $1$, and $e^{z^2}$ has order $2$.
The order controls the number of zeros of the function and the shape of its factorization, as in the [[Hadamard factorization theorem|KYSR7NHNFQ]].

## Explanations

### L1 human

Entire functions can grow at very different speeds as $\abs{z}$ grows.
To compare them, one looks at the largest value $M(r)$ of $\abs{f}$ on the circle of radius $r$ and asks which power $r^\rho$ appears in the exponent.
For $e^z$, $M(r) = e^r$, so the order is $1$.
For $\cos \sqrt{z}$ the growth is like $e^{\sqrt{r}}$, so the order is $1/2$.
A function such as $e^{e^z}$ grows faster than every $\exp(r^\rho)$ and has infinite order.
</document>

<document source="https://tamnd.github.io/math2/node/SHFJH5JZ5P.md" tag="SHFJH5JZ5P" kind="definition">
# Siegel zero

## Definition

There is an absolute constant $c > 0$ with the following property.
Let $q \ge 3$ be an integer and $\chi$ a [[Dirichlet character|7XB5VJKDPG]] modulo $q$.
Then the [[Dirichlet L-function|9D2Z543EJQ]] $L(s, \chi)$ has no zero $s = \sigma + it$ in the region
$$\sigma \ge 1 - \frac{c}{\log\big(q(\abs{t} + 2)\big)}.$$
The one possible exception is a single zero when $\chi$ is a real nonprincipal character; such a zero is real and simple.
Fix such a constant $c$.
A *Siegel zero* (or *exceptional zero*) for the modulus $q$ is a real number $\beta$ with
$$1 - \frac{c}{\log q} < \beta < 1 \quad\text{and}\quad L(\beta, \chi) = 0$$
for some real nonprincipal character $\chi$ modulo $q$.

## Remarks

No Siegel zero is known, so there is no example to check.
If the [[generalized Riemann hypothesis|DNKA4WBKDS]] holds, every zero of $L(s, \chi)$ in the strip $0 < \operatorname{Re} s < 1$ has real part $1/2$, so for large $q$ no Siegel zero exists.
A non-example: a zero of $L(s, \chi)$ for a character $\chi$ that is not real is never a Siegel zero, because the zero-free region has no exception for such $\chi$.
Siegel's theorem bounds a possible exceptional zero: for every $\varepsilon > 0$ there is $c(\varepsilon) > 0$ with $\beta < 1 - c(\varepsilon) q^{-\varepsilon}$, but $c(\varepsilon)$ is not effectively computable.
The notion depends on the choice of $c$; statements about Siegel zeros are usually about a sequence of moduli $q$ with $(1 - \beta)\log q \to 0$.
</document>

<document source="https://tamnd.github.io/math2/node/SHMTFRH28E.md" tag="SHMTFRH28E" kind="concept">
# Gumm terms

## Summary

Gumm terms are a finite chain of ternary terms of a [[variety|AJNZJD8442]].
They combine a chain like [[Jónsson terms|3M8Y76E4AG]] with a final step like a [[Mal'cev term|Y203PPTS61]].
A variety has Gumm terms if and only if it is a [[congruence-modular variety|QAQAS3JWXW]].
This form of the Mal'cev condition for modularity shows that congruence modularity mixes the two basic cases, congruence distributivity and congruence permutability.

## Explanations

### L1 human

Lattices have a chain of formulas of one kind, and groups have one formula of another kind.
Gumm terms allow a chain of the lattice kind followed by one formula of the group kind.
A variety has such a chain exactly when all its congruence lattices satisfy the modular law.
So congruence modularity sits between the lattice-like and the group-like behaviour, and contains both.
</document>

<document source="https://tamnd.github.io/math2/node/SHRXX7EW76.md" tag="SHRXX7EW76" kind="definition">
# Chain homotopy

## Definition

Let $\mathcal{A}$ be an [[additive category|B2VP96NJSK]], let $(C_\bullet, d^C)$ and $(D_\bullet, d^D)$ be [[chain complexes|WQ82C0M3QB]] in $\mathcal{A}$, and let $f, g \colon C_\bullet \to D_\bullet$ be [[chain maps|HTK9WQZJ1G]].
A *chain homotopy* from $f$ to $g$ is a family of morphisms $h_n \colon C_n \to D_{n+1}$, one for each integer $n$, such that
$$f_n - g_n = d^D_{n+1} \circ h_n + h_{n-1} \circ d^C_n \quad \text{for all } n \in \Z.$$
The maps $f$ and $g$ are *chain homotopic*, written $f \simeq g$, if such a family exists.

## Remarks

Chain homotopy is an equivalence relation compatible with composition.
A chain map $f$ is a *chain homotopy equivalence* if there is a chain map $g \colon D_\bullet \to C_\bullet$ with $g \circ f \simeq 1_C$ and $f \circ g \simeq 1_D$.
A complex is *contractible* if its identity is chain homotopic to the zero map.
In an abelian category, chain homotopic maps induce the same morphisms on [[homology|2J4JJJYP22]]: a cycle $z$ is sent by $f - g$ to $d^D h(z)$, which is a boundary.

Example: let $C_\bullet = D_\bullet$ have $C_1 = C_0 = \Z$, $d_1 = 1_{\Z}$, and $C_n = 0$ otherwise; let $f$ be the identity and $g = 0$.
Take $h_0 = 1_{\Z} \colon C_0 \to D_1$ and all other $h_n = 0$.
In degree $0$, $d_1 h_0 + h_{-1} d_0 = 1 + 0 = 1 = f_0 - g_0$; in degree $1$, $d_2 h_1 + h_0 d_1 = 0 + 1 = 1 = f_1 - g_1$.

Non-example: let $C_\bullet = D_\bullet$ be $\Z$ in degree $0$ and $0$ elsewhere.
In degree $0$, the map $h_0$ lands in $D_1 = 0$ and $d^C_0$ lands in $C_{-1} = 0$, so $d_1 h_0 + h_{-1} d_0 = 0$, while $f_0 - g_0 = 1$.
So the identity is not chain homotopic to $0$; indeed it induces the identity of $H_0 = \Z$.
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<document source="https://tamnd.github.io/math2/node/SHSXQ98PHB.md" tag="SHSXQ98PHB" kind="definition">
# Julia set

## Definition

Let $f \colon \hat{\C} \to \hat{\C}$ be a [[rational function|0XDK9CGYKJ]] of degree at least $2$ on the [[Riemann sphere|0J78KBPFZB]] $\hat{\C}$.
Let $F(f)$ be the [[Fatou set|C0N0XGC6GK]] of $f$.
The *Julia set* of $f$ is the complement $J(f) = \hat{\C} \setminus F(f)$.
So a point $z$ lies in $J(f)$ if and only if the iterates $f^n$, $n \ge 0$, form a normal family on no open neighbourhood of $z$.

## Remarks

Example: for $f(z) = z^2$, $J(f)$ is the unit circle.
On $\abs{z} < 1$ the iterates $z^{2^n}$ tend to $0$ locally uniformly, and on $\abs{z} > 1$ they tend to $\infty$, so both regions lie in $F(f)$.
Every neighbourhood of a point with $\abs{z} = 1$ contains points whose orbits tend to $0$ and points whose orbits tend to $\infty$.
So a limit of a subsequence of iterates would be discontinuous there, and the point is in $J(f)$.
Non-example: $z = 1/2$ is not in $J(z^2)$, because the disc $\abs{z} < 3/4$ lies in the Fatou set.
For degree at least $2$, $J(f)$ is nonempty, compact and perfect, it satisfies $f^{-1}(J(f)) = J(f)$, and it is the closure of the set of repelling periodic points.
The same definition, with $\C$ in place of $\hat{\C}$, is used for transcendental entire functions.
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<document source="https://tamnd.github.io/math2/node/SHX9JPKJBE.md" tag="SHX9JPKJBE" kind="statement">
# Linearity of the Laplace transform

## Statement

Let $f, g \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$ with $T > 0$, and let $\alpha, \beta \in \C$.
Let $s \in \C$ be such that the [[Laplace integrals|05XT5AVJBK]] $F(s) = \int_0^\infty f(t) e^{-st}\, dt$ and $G(s) = \int_0^\infty g(t) e^{-st}\, dt$ both converge.
Then the Laplace integral of $\alpha f + \beta g$ converges at $s$, and
$$\mathcal{L}\{\alpha f + \beta g\}(s) = \alpha F(s) + \beta G(s).$$

## Remarks

By [[linearity of the integral|080RWVVSB6]], $\int_0^T (\alpha f + \beta g) e^{-st}\, dt = \alpha \int_0^T f e^{-st}\, dt + \beta \int_0^T g e^{-st}\, dt$ for each $T$, and the limit of a sum is the sum of the limits.
So the identity holds on the half-plane $\operatorname{Re} s > \max(\sigma_c(f), \sigma_c(g))$, where $\sigma_c$ is the [[abscissa of convergence|FTHFSJ30WJ]].
Example: $\cosh t = \frac{1}{2}(e^t + e^{-t})$ has transform $\frac{1}{2}\left(\frac{1}{s - 1} + \frac{1}{s + 1}\right) = \frac{s}{s^2 - 1}$ for $\operatorname{Re} s > 1$.
The transform of $\alpha f + \beta g$ may converge on a larger half-plane than those of $f$ and $g$: for $f(t) = 1 + e^t$ and $g(t) = -e^t$, both have abscissa $1$, while $f + g = 1$ has abscissa $0$.

## Explanations

### L1 human

The transform of a sum is the sum of the transforms, and constants can be pulled out.
This is what lets one use a table: split a function into simple pieces, transform each piece, and add.
It also works backwards, when a transform is split into partial fractions and each fraction is inverted separately.

### L2 human

Linearity is the reason the Laplace transform fits linear differential equations with constant coefficients: together with the rule for derivatives, it turns the equation into an algebraic equation for $F$.
In operator language, $\mathcal{L}$ is a linear map from a space of functions (for instance those of exponential order) to a space of holomorphic functions on half-planes.
Combined with Lerch's uniqueness theorem, linearity also makes the inverse transform linear.
</document>

<document source="https://tamnd.github.io/math2/node/SJ2BVG9F4W.md" tag="SJ2BVG9F4W" kind="definition">
# Elliptic curve

## Definition

Let $K$ be a [[field|04C5ZBG1WT]].
An *elliptic [[curve|XJM2MT4S3Y]] over $K$* is a pair $(E, O)$ with two properties.
$E$ is a smooth projective curve over $K$ of [[genus|6W8TDV9VVN]] $1$, and $O \in E(K)$ is a point of $E$ defined over $K$.

Equivalently, it is a curve in the [[projective plane|ZT9A6JK8QF]] given by a [[Weierstrass equation|Z3C8B6QTS6]] with $a_1, \ldots, a_6 \in K$:
$$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.$$
Its discriminant $\Delta$ must be nonzero, and $O = [0 : 1 : 0]$ is the point at infinity.
If the characteristic of $K$ is not $2$ or $3$, every elliptic curve has an equation $y^2 = x^3 + ax + b$ with $a, b \in K$, and the condition $\Delta \ne 0$ becomes $4a^3 + 27b^2 \ne 0$.

## Remarks

Example: $E \colon y^2 = x^3 - x$ over $\Q$.
Here $a = -1$, $b = 0$, and $4a^3 + 27b^2 = -4 \ne 0$, so the cubic $x^3 - x = x(x-1)(x+1)$ has three distinct roots and $E$ is smooth.

Non-example: $y^2 = x^3 + x^2$ has $\Delta = 0$.
Its projective closure has a node at $(0, 0)$, where both partial derivatives of $y^2 - x^3 - x^2$ vanish, so the smoothness condition fails.

The condition $4a^3 + 27b^2 \ne 0$ says that $x^3 + ax + b$ has no repeated root in an algebraic closure of $K$.
For $y^2 = x^3 + ax + b$ one has $\Delta = -16(4a^3 + 27b^2)$.
The set $E(K)$ of points over $K$, including $O$, is an abelian group with identity $O$ under the chord-and-tangent group law.
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<document source="https://tamnd.github.io/math2/node/SJ9AMSQA7P.md" tag="SJ9AMSQA7P" kind="definition">
# Homology sphere

## Definition

Let $n \ge 1$ be an integer.
A *homology $n$-sphere* is a closed (compact, without boundary) [[topological $n$-manifold|QQQ9B1DK4P]] $X$ whose integral [[singular homology|M0Q5CFJ7ZH]] is that of the [[$n$-sphere|T62369RFH6]]:
$$H_i(X; \Z) \cong \begin{cases} \Z & \text{if } i = 0 \text{ or } i = n, \\ 0 & \text{otherwise.} \end{cases}$$

## Remarks

The condition $H_0(X; \Z) \cong \Z$ says that $X$ is connected, and $H_n(X; \Z) \cong \Z$ says that $X$ is orientable.

Example: the Poincaré homology sphere $P = S^3 / I^\ast$, where $I^\ast \subset S^3$ is the binary icosahedral group of order $120$ acting by left multiplication.
Its fundamental group $I^\ast$ is perfect, so $H_1(P; \Z)$, the abelianisation of $I^\ast$, is $0$; Poincaré duality and the universal coefficient theorem then give $H_2(P; \Z) = 0$ and $H_3(P; \Z) \cong \Z$.

Non-example: the lens space $L(p, q)$ with $p \ge 2$ is a closed orientable $3$-manifold, but $H_1(L(p, q); \Z) \cong \Z/p \neq 0$.

A simply connected homology $n$-sphere is homotopy equivalent to $S^n$, by the Hurewicz and Whitehead theorems.
</document>

<document source="https://tamnd.github.io/math2/node/SJDZH6V0BC.md" tag="SJDZH6V0BC" kind="definition">
# f-vector

## Definition

Let $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d \ge 1$.
For $0 \le i \le d - 1$, let $f_i(P)$ be the number of [[faces|KY7SB854VH]] of $P$ of dimension $i$.
The *f-vector* of $P$ is $(f_0(P), f_1(P), \dots, f_{d-1}(P))$.

## Remarks

Some authors add $f_{-1} = 1$ for the empty face and $f_d = 1$ for $P$ itself.
For a finite [[simplicial complex|V8GZ0W1ZZ5]] $\Delta$, $f_i(\Delta)$ is the number of $i$-dimensional simplices of $\Delta$, defined in the same way.

Example: the cube $[0,1]^3$ has $8$ vertices, $12$ edges and $6$ square faces, so its f-vector is $(8, 12, 6)$.

Non-example: $(4, 6, 3)$ is not the f-vector of any $3$-dimensional polytope.
Every $3$-polytope satisfies $f_0 - f_1 + f_2 = 2$, while $4 - 6 + 3 = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/SJH7M2VCTE.md" tag="SJH7M2VCTE" kind="concept">
# Vlasov equation

## Summary

The Vlasov equation describes a large system of particles that interact through long-range forces.
Examples are charged particles in a plasma and stars in a galaxy.
It is a [[transport equation|PVD7VZJZ9B]] in position and velocity for the particle density $f(t, x, v)$, without a collision term.
The force is the mean field created by all particles together, so the equation is nonlinear.
Coupled with the Poisson equation it gives the Vlasov–Poisson system.

## Explanations

### L1 human

In a plasma each electron feels the electric pull and push of billions of other charges.
Close encounters between two particles matter little, but the combined field of all of them matters a lot.
The Vlasov equation follows the cloud of particles in the space of positions and velocities, as each particle moves under this averaged force.
For a single particle in the force $-x$ on a line, the orbits in the $(x, v)$ plane are circles.
Any density that is constant on these circles stays the same over time.
</document>

<document source="https://tamnd.github.io/math2/node/SJP9HW82VX.md" tag="SJP9HW82VX" kind="concept">
# Hamilton–Jacobi–Bellman equation

## Summary

The Hamilton–Jacobi–Bellman equation is a first-order partial differential equation for the [[value function|K6G0DRQ3CP]] of an [[optimal control|T3V6ZNAMDN]] problem.
It is the continuous-time form of the principle of [[dynamic programming|QPS8VSJGMJ]]: an optimal strategy stays optimal from every later state.
A solution gives the optimal cost from every starting point and, through a pointwise minimization, an optimal feedback control.
In general the value function is not differentiable, and the equation is understood in the sense of [[viscosity solutions|H1ERJ3T75T]].

## Explanations

### L1 human

Think of the cheapest cost to reach the end of a journey from each place and each time.
If you know this cost everywhere, the best move now is the one that leads to the place with the smallest remaining cost plus the cost of the move.
The Hamilton–Jacobi–Bellman equation writes this idea as an equation for the cost function.
It is the counterpart of [[Pontryagin's maximum principle|7RPV2T29CF]], which describes one optimal path instead of all optimal costs.
</document>

<document source="https://tamnd.github.io/math2/node/SJQDDY9FMX.md" tag="SJQDDY9FMX" kind="concept">
# Filtered back-projection

## Summary

Filtered back-projection is the standard method to reconstruct a function in the plane from its [[Radon transform|VJNZJ352PC]].
It has two steps.
First each projection is convolved with a filter whose Fourier transform is close to $\abs{\sigma}$.
Then the filtered projections are fed into the [[dual Radon transform|938SXZQS8G]].
With the exact ramp filter it is the [[Radon inversion formula|HN93ST5CBT]] in dimension $2$.
Practical filters cut off high frequencies, and the method then reconstructs a smoothed version of the function.
It was for a long time the main algorithm of computed tomography scanners.

## Explanations

### L1 human

Back-projection alone smears each measurement along its line and gives a blurred image.
The blur can be computed: it boosts low frequencies too much compared with high ones.
So before smearing back, each measured profile is filtered to undo this imbalance.
The filter sharpens the edges of each profile, and after back-projection the blur is gone.
</document>

<document source="https://tamnd.github.io/math2/node/SJVWPZZF54.md" tag="SJVWPZZF54" kind="definition">
# d'Alembert's functional equation

## Definition

*d'Alembert's functional equation* is the [[functional equation|SRAN4EKWS0]]
$$f(x + y) + f(x - y) = 2 f(x) f(y) \quad \text{for all } x, y \in \R,$$
for an unknown [[function|0EPAT51DXQ]] $f \colon \R \to \R$.
A solution is a function $f$ that satisfies this identity for all real $x$ and $y$.

## Remarks

Example: $f(x) = \cos(2x)$ is a solution, by the product formula $2 \cos u \cos v = \cos(u + v) + \cos(u - v)$.
Non-example: $f(x) = 2$ is not a solution, since the left side is $4$ and the right side is $8$.
Setting $y = 0$ gives $2 f(x) = 2 f(x) f(0)$, so either $f = 0$ or $f(0) = 1$.
If $f(0) = 1$, setting $x = 0$ gives $f(-y) = f(y)$, so every nonzero solution is even.
The continuous solutions are $f = 0$, $f(x) = \cos(ax)$ and $f(x) = \cosh(ax)$ with $a \in \R$.
The same equation is studied for complex-valued functions on groups, in the form $f(xy) + f(xy^{-1}) = 2 f(x) f(y)$.
</document>

<document source="https://tamnd.github.io/math2/node/SJZW2HN1SC.md" tag="SJZW2HN1SC" kind="statement">
# Krull's principal ideal theorem

## Statement

Let $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $x \in R$.
Let $P$ be a [[prime ideal|06HC6HK0Z8]] of $R$ that is [[minimal|S2D8823P8B]] among the prime ideals containing $x$.
Then the [[height|9BKCZPMQT6]] of $P$ is at most $1$.

## Remarks

After localizing at $P$, one may assume that $R$ is local with maximal ideal $P$, and then $R/xR$ is Artinian.
If $Q \subsetneq P$ is prime, the chain of ideals $xR + Q^{(n)}$ stabilizes, where $Q^{(n)}$ is the [[symbolic power|7VFPQVCNJ7]]; with Nakayama's lemma this gives $Q^{(n)} = Q^{(n+1)}$ for large $n$.
Then $Q^n R_Q = 0$ in $R_Q$, so $R_Q$ is Artinian, and $Q$ has height $0$.
For example, in $k[x, y]$ the prime $(x)$ is minimal over $x$ and has height $1$.
The Noetherian hypothesis is needed.
In a valuation domain whose value group is $\Z \times \Z$ with the lexicographic order, the maximal ideal has height $2$ but is minimal over a single element.
If $x$ is not a zero divisor and not a unit, every prime minimal over $x$ has height exactly $1$.

## Explanations

### L1 human

Cutting a space by one equation lowers its dimension by at most one.
In algebra: a prime ideal that is minimal over one element sits at most one step above a minimal prime.

### L2 human

The theorem, the Hauptidealsatz, is the base case of [[Krull's height theorem|TBRTZMHFTV]] for ideals generated by $n$ elements.
Through that generalisation, it shows that every prime ideal of a Noetherian ring has finite height.
It also gives the criterion that a Noetherian domain is a unique factorisation domain exactly when every height one prime is principal.
It is a basic tool in dimension theory, for example in the [[dimension theorem for local rings|KW3HY77V6D]].
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<document source="https://tamnd.github.io/math2/node/SJZX81A783.md" tag="SJZX81A783" kind="concept">
# Weak solution

## Summary

A weak solution of a [[partial differential equation|GC5BDZ6QBC]] is a function that satisfies the equation after multiplication by test functions and integration by parts.
In this form the derivatives fall on the test functions.
It need not be differentiable, and so it can describe shocks, corners and other rough behaviour.
Existence theory usually first produces a weak solution, often in a [[Sobolev space|M4CEZ5K9HP]].
Then regularity theory shows that it is smoother.

## Explanations

### L1 human

A classical solution must have all the derivatives that appear in the equation.
A weak solution only has to satisfy the equation "on average" against every smooth bump function.
For the transport equation $u_t + u_x = 0$, a moving step $u(x, t) = 1$ for $x > t$ and $0$ for $x < t$ is a weak solution.
It is not differentiable along the line $x = t$, but it describes a profile moving to the right with speed $1$.
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<document source="https://tamnd.github.io/math2/node/SK0ZJGWXDP.md" tag="SK0ZJGWXDP" kind="concept">
# Homotopy type

## Summary

Two topological spaces have the same homotopy type when there is a [[homotopy equivalence|XZG6MTSWCN]] between them.
The homotopy type of a space is its class under this equivalence relation.
Most invariants of algebraic topology, such as homotopy groups, homology and cohomology, depend only on the homotopy type.
Classifying spaces up to homotopy type is a central aim of homotopy theory.

## Explanations

### L1 human

The homotopy type of a space is what is left of it after you allow any continuous squashing that can be undone up to deformation.
A solid ball, a line and a point all have the same homotopy type.
A circle, an annulus and a Möbius band also share one homotopy type, different from that of a point.
The homotopy type remembers the holes of a space but forgets its size and dimension.
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<document source="https://tamnd.github.io/math2/node/SK224Y1HNG.md" tag="SK224Y1HNG" kind="definition">
# Monodromy

## Definition

Let $U \subseteq \C$ be a connected open set, and let $A \colon U \to \C^{n \times n}$ be [[holomorphic|0NAHXR1FRH]].
Consider the [[linear system|037ZNY7NAG]] $Y' = A(z)\, Y$, fix $z_0 \in U$, and let $\Phi$ be a [[fundamental matrix|0H1GN6P9R5]] of the system on a disc around $z_0$ in $U$.
Let $\gamma$ be a loop in $U$ based at $z_0$.
The [[analytic continuation|0M3YQJ81NB]] of $\Phi$ along $\gamma$ is again a fundamental matrix near $z_0$, so it equals $\Phi\, M_\gamma$ for a unique matrix $M_\gamma \in \mathrm{GL}_n(\C)$.
The matrix $M_\gamma$ is the *monodromy matrix* of $\gamma$.
It depends only on the homotopy class $[\gamma]$ of $\gamma$ in the fundamental group $\pi_1(U, z_0)$.
The map
$$\rho \colon \pi_1(U, z_0) \to \mathrm{GL}_n(\C), \qquad \rho([\gamma]) = M_\gamma^{-1},$$
is a group homomorphism, the *monodromy representation* of the system, and its image is the *monodromy group*.

## Remarks

Continuing $\Phi$ along $\gamma_1$ and then along $\gamma_2$ gives $\Phi M_{\gamma_2} M_{\gamma_1}$, so $\gamma \mapsto M_\gamma$ reverses products; taking inverses makes $\rho$ a [[representation|WYGRXCG2RN]] of $\pi_1$ in the [[general linear group|0FCHE902D7]].
Changing $\Phi$ to $\Phi C$ with $C$ invertible conjugates every $M_\gamma$ by $C$, so the representation is determined up to conjugation.
The same definition applies to a scalar equation of order $n$, using a fundamental system of solutions.

Example: for $y' = \frac{a}{z} y$ on $U = \C \setminus \set{0}$ with $a \in \C$, take $\Phi(z) = z^a = e^{a \log z}$ near $z_0 = 1$.
Continuation once counterclockwise around $0$ adds $2 \pi i$ to $\log z$, so $M_\gamma = e^{2 \pi i a}$.
The monodromy is trivial exactly when $a \in \Z$.

Non-example: for $y' = y$ on $\C$, the monodromy is trivial.
The solution $e^z$ is single-valued and $U = \C$ is simply connected, so every monodromy matrix is $1$.
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# Tracy–Widom distribution

## Definition

Let $\mathrm{Ai}$ be the Airy function.
Let $q \colon \R \to \R$ be the Hastings–McLeod solution of the Painlevé II equation
$$q''(s) = s\, q(s) + 2 q(s)^3,$$
that is, the unique solution with $q(s) \sim \mathrm{Ai}(s)$ as $s \to +\infty$.
The *Tracy–Widom distribution* (for $\beta = 2$) is the [[probability distribution|0DVDCMA2DK]] on $\R$ with distribution function
$$F_2(s) = \exp\Big( -\int_s^\infty (x - s)\, q(x)^2 \, dx \Big), \qquad s \in \R.$$
The laws for $\beta = 1$ and $\beta = 4$ are given by
$$F_1(s) = \exp\Big( -\frac12 \int_s^\infty q(x)\, dx \Big) F_2(s)^{1/2}, \qquad F_4\Big(\frac{s}{\sqrt 2}\Big) = \cosh\Big( \frac12 \int_s^\infty q(x)\, dx \Big) F_2(s)^{1/2}.$$

## Remarks

Example: let $H_n$ be an $n \times n$ matrix of the Gaussian unitary ensemble, Hermitian with independent standard real normal diagonal entries and independent complex normal entries above the diagonal with $\mathbb{E}\abs{H_{ij}}^2 = 1$.
If $\lambda_{\max}$ is its largest [[eigenvalue|0BQAA2BK3Q]], then $n^{1/6} (\lambda_{\max} - 2\sqrt n)$ [[converges in distribution|0SRYJGXKR4]] to $F_2$.
Equivalently, $F_2(s) = \det(I - K_{\mathrm{Ai}})$ on $L^2(s, \infty)$, the Fredholm determinant of the Airy kernel $K_{\mathrm{Ai}}(x, y) = \frac{\mathrm{Ai}(x) \mathrm{Ai}'(y) - \mathrm{Ai}'(x) \mathrm{Ai}(y)}{x - y}$.
Non-example: the standard normal law is not a Tracy–Widom law; $F_2$ has mean about $-1.77$ and is not symmetric about its mean.
Baik, Deift and Johansson showed that the length $\ell_n$ of the longest increasing subsequence of a uniform random permutation of $n$ letters satisfies $(\ell_n - 2\sqrt n)/n^{1/6} \to F_2$ in distribution.
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# Peak function

## Summary

A peak function at a boundary point $p$ of a domain is a holomorphic function, continuous up to the boundary, that equals $1$ at $p$.
At every other point of the closure its modulus is less than $1$.
A point with a peak function is a peak point.
Peak functions help to localize problems at the boundary and to show that a domain is a domain of holomorphy.
They are also used to study the boundary behaviour of holomorphic maps.

## Explanations

### L1 human

A peak function is a holomorphic function whose modulus has a single highest point on the closed domain, and that point is on the boundary.
Powers $f^k$ of a peak function $f$ are then close to $1$ near $p$ and close to $0$ away from $p$, so they act like holomorphic "spotlights" on $p$.
On the closed unit disc, $f(z) = (1 + z)/2$ peaks at $z = 1$.
On the bidisc, the boundary points $(1, b)$ with $\abs{b} < 1$ have no peak function, because the boundary contains a flat disc through them.
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<document source="https://tamnd.github.io/math2/node/SKJB3ANDF4.md" tag="SKJB3ANDF4" kind="definition">
# Martin boundary

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be a domain with a Green's function, and fix $x_0 \in \Omega$.
Let $M$ be the [[Martin kernel|1DBC9FYF7M]] of $\Omega$ with reference point $x_0$.
Call a sequence $(y_k)$ in $\Omega$ *fundamental* if it eventually leaves each compact subset of $\Omega$ and the functions $M(\cdot, y_k)$ converge locally uniformly on $\Omega$.
The *Martin boundary* $\Delta$ of $\Omega$ is the set of all limits
$$h = \lim_{k \to \infty} M(\cdot, y_k)$$
of fundamental sequences, and for $\xi = h \in \Delta$ one writes $M(\cdot, \xi) = h$.
The set $\Omega \cup \Delta$ is the *Martin compactification* of $\Omega$.

## Remarks

Each $M(\cdot, \xi)$ with $\xi \in \Delta$ is a positive [[harmonic function|0DEB0JMN0W]] on $\Omega$ with value $1$ at $x_0$.
The set $\Omega \cup \Delta$ carries a compact metrisable topology that induces the usual topology on $\Omega$.
In it, $\Omega$ is open and dense, and each fundamental sequence converges to its limit in $\Delta$.
So it is a [[compactification|Q6XNC8JAZD]] of $\Omega$, and for each fixed $x \in \Omega$ the function $y \mapsto M(x, y)$ extends continuously to $\Delta$ by $\xi \mapsto M(x, \xi)$.
It does not depend on the choice of $x_0$, up to a homeomorphism that is the identity on $\Omega$.
A point $\xi \in \Delta$ is *minimal* if $M(\cdot, \xi)$ is a [[minimal harmonic function|NA4BTZ6WRB]]; the minimal points form the *minimal Martin boundary* $\Delta_1$.
Example: let $B$ be the unit ball of $\R^n$ and $x_0 = 0$.
A sequence is fundamental if and only if it converges to a point $\zeta$ of the sphere, and the limit is the [[Poisson kernel|TCYSWCTYYM]] $P(\cdot, \zeta)$.
Distinct $\zeta$ give distinct kernels, so $\Delta$ is homeomorphic to the unit sphere.
By contrast, consider the slit disc $\set{\abs{z} < 1} \setminus [0, 1)$.
Sequences converging to $1/2$ from above the slit and from below it give different limits, so the Martin boundary has two points over $1/2$.
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<document source="https://tamnd.github.io/math2/node/SKMQ02FP9J.md" tag="SKMQ02FP9J" kind="definition">
# Unilateral Z-transform

## Definition

Let $(x_n)_{n \in \Z}$ or $(x_n)_{n \ge 0}$ be a [[sequence|0G9EKM0E15]] of complex numbers.
The *unilateral Z-transform* of $x$ is
$$X(z) = \sum_{n = 0}^{\infty} x_n z^{-n},$$
as a function of the nonzero complex numbers $z$ for which the series converges absolutely.

## Remarks

Let $r = \limsup_{n \to \infty} \abs{x_n}^{1/n}$.
The series converges absolutely for $\abs{z} > r$ and diverges for $0 < \abs{z} < r$, by the root test.
Example: $x_n = n$ gives $X(z) = \sum_{n \ge 0} n z^{-n} = z/(z - 1)^2$ for $\abs{z} > 1$, by differentiating the geometric series.
Non-example: $x_n = n!$ gives $r = \infty$, so the series converges for no $z$, and $x$ has no unilateral Z-transform as a function.
The unilateral transform of $x$ equals the [[Z-transform|5FY0JQGHYZ]] of the sequence that agrees with $x$ for $n \ge 0$ and is $0$ for $n < 0$.
Writing $w = z^{-1}$ turns it into the ordinary generating function $\sum_{n \ge 0} x_n w^n$.
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# Euler–Gauss limit formula

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\C$.
For every $z \in \C \setminus \set{0, -1, -2, \ldots}$,
$$\Gamma(z) = \lim_{n \to \infty} \frac{n! \; n^{z}}{z (z + 1) (z + 2) \cdots (z + n)},$$
where $n^z = e^{z \ln n}$.

## Remarks

For $\operatorname{Re} z > 0$, the quotient equals $\int_0^n t^{z-1} (1 - t/n)^n \, dt$, by repeated integration by parts.
Since $(1 - t/n)^n$ increases to $e^{-t}$, the [[dominated convergence theorem|BYNV84PF86]] gives the limit $\int_0^\infty t^{z-1} e^{-t}\,dt = \Gamma(z)$.
For other $z$, the functional equation $\Gamma(z + 1) = z\Gamma(z)$ moves the argument into the half-plane, and the quotients transform in the same way up to a factor that tends to $1$.
Example: at $z = 1$ the quotient is $\frac{n!\, n}{(n+1)!} = \frac{n}{n+1} \to 1 = \Gamma(1)$.
In Mathlib the statement is `Complex.GammaSeq_tendsto_Gamma`.

## Explanations

### L1 human

The formula computes the gamma function as a limit of simple fractions built from factorials and powers.
It was Euler's first way to extend the factorial to non-integer arguments, and Gauss used it as the definition.

### L2 human

The limit makes sense for every allowed complex $z$ at once, so it can serve as a definition of the meromorphic gamma function.
Taking reciprocals and rewriting $n^z = \exp\big(z(\ln n - 1 - \tfrac12 - \cdots - \tfrac1n)\big)\exp\big(z(1 + \cdots + \tfrac1n)\big)$ turns it into the [[Weierstrass product|53MZ2DGDTQ]].
It is also the limit that appears in the proof of the [[Bohr–Mollerup theorem|QFF87AXWN2]].
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# Hypoelliptic operator

## Definition

Let $\Omega \subseteq \R^n$ be an open set and let $P = \sum_{\abs{\alpha} \le m} a_\alpha(x) \partial^\alpha$ be a linear differential operator on $\Omega$ whose coefficients $a_\alpha$ are [[smooth functions|0V33BNKZ9S]].
A distribution on an open set $U$ is a continuous linear functional on the space of test functions on $U$.
The operator $P$ is *hypoelliptic* if the following holds for every open set $U \subseteq \Omega$ and every distribution $u$ on $U$.
If $Pu$ is a smooth function on $U$, then $u$ is a smooth function on $U$.

## Remarks

Equivalently, $\operatorname{sing\,supp} u = \operatorname{sing\,supp} Pu$ for every distribution $u$, where the singular support is the complement of the largest open set on which the distribution is smooth.
Example: every [[elliptic operator|XY8GEQSWPR]] with smooth coefficients is hypoelliptic; this is elliptic regularity.
The heat operator $\partial_t - \Delta$ on $\R^{n+1}$ is hypoelliptic but not elliptic.
Non-example: the wave operator $\partial_t^2 - \partial_x^2$ on $\R^2$ is not hypoelliptic.
For the Heaviside function $H$, the distribution $u(x, t) = H(x - t)$ satisfies $\partial_t^2 u - \partial_x^2 u = H''(x - t) - H''(x - t) = 0$, which is smooth, but $u$ is not smooth.
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# Lefschetz principle

## Statement

Let $\varphi$ be a first-order [[sentence|DKFGQTRYZE]] in the signature of rings $\set{+, -, \cdot, 0, 1}$.
The following are equivalent:

1. $\varphi$ holds in the field $\C$ of [[complex numbers|09FTAET5Z4]].
2. $\varphi$ holds in every [[algebraically closed field|0PF6TM8TWT]] of [[characteristic|1ZB13N78XW]] $0$.
3. $\varphi$ holds in some algebraically closed field of characteristic $0$.
4. There is an integer $m$ such that $\varphi$ holds in every algebraically closed field of characteristic $p$, for every prime $p > m$.
5. $\varphi$ holds in algebraically closed fields of characteristic $p$ for infinitely many primes $p$.

## Remarks

Clauses 1, 2 and 3 are equivalent because the theory $\mathrm{ACF}_0$ of [[algebraically closed fields|MT8TDVPBXB]] of characteristic $0$ is complete.
Suppose $\varphi$ holds in $\mathrm{ACF}_0$.
By the [[compactness theorem|1NR3HAWG9D]], $\varphi$ follows from $\mathrm{ACF}$ and finitely many of the axioms $1 + \cdots + 1 \ne 0$.
So it holds in characteristic $p$ for all large $p$, which is clause 4.
Clause 4 gives 5, since there are infinitely many primes.
If $\varphi$ fails in $\mathrm{ACF}_0$, then $\neg \varphi$ holds there, so $\neg \varphi$ holds in all large characteristics and 5 fails.

Example: fix $n$ and $d$.
The statement "every injective polynomial map $K^n \to K^n$ whose components have degree at most $d$ is surjective" is a first-order sentence about the field $K$.
It holds in the algebraic closure of each finite field $\F_p$, so by clause 5 it holds in $\C$.
This is the model-theoretic proof of the Ax–Grothendieck theorem.

The principle holds only for first-order sentences.
The statement "every element is algebraic over the prime field" holds in the algebraic closure of each $\F_p$ but fails in $\C$.
It is not expressible by a single first-order sentence.

## Explanations

### L1 human

Algebraically closed fields of the same characteristic cannot be told apart by first-order statements in the language of rings.
So a statement of this kind about the complex numbers is true exactly when it is true in every algebraically closed field of characteristic $0$.
It is also true exactly when it holds in algebraically closed fields of all large prime characteristics.

### L2 human

The principle lets one prove algebraic facts about $\C$ by working over finite fields, where counting arguments are available, or the other way around.
It comes from the completeness of $\mathrm{ACF}_p$ for each $p$, which itself follows from quantifier elimination or from uncountable categoricity and the Łoś–Vaught test.
Take the algebraic closures of the fields $\F_p$ and a non-principal ultrafilter on the primes.
Their ultraproduct is an algebraically closed field of characteristic $0$, which gives another view of clause 4.
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# Hellinger distance

## Summary

The Hellinger distance is a distance between two probability distributions.
It compares the square roots of their densities in the $L^2$ sense.
It takes values in $[0, 1]$ and behaves well under independent products.
It is a standard tool in asymptotic statistics, for example for rates of convergence of estimators and in testing.

## Explanations

### L1 human

Take the square roots of two density functions, so that each becomes a unit vector in a space of functions.
The Hellinger distance measures how far apart these two unit vectors are.
Two equal distributions have distance $0$, and two distributions that live on disjoint sets have distance $1$.
For instance, a fair coin and a coin that always shows heads have squared Hellinger distance $1 - 1/\sqrt{2} \approx 0.29$.
[[Total variation distance|3SZESKX3N5]] is another common distance, and the two are always comparable.
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# Golay code

## Description

The extended binary Golay code $G_{24}$ is a linear code of length $24$ over the field $\F_2$, of dimension $12$ and minimum distance $8$.
So it has $2^{12} = 4096$ codewords, and any two differ in at least $8$ of the $24$ positions; as an [[error-correcting code|X122TKFP9G]] it corrects $3$ errors.
Deleting one coordinate gives the perfect binary Golay code $G_{23}$, of length $23$, dimension $12$ and minimum distance $7$.
Check that $G_{23}$ is perfect: a Hamming ball of radius $3$ in $\F_2^{23}$ contains $1 + 23 + 253 + 1771 = 2048 = 2^{11}$ words.
The $2^{12}$ balls around the codewords are disjoint, since $7 \ge 2 \cdot 3 + 1$, and together they contain $2^{12} \cdot 2^{11} = 2^{23}$ words, which is all of $\F_2^{23}$.

## Remarks

The weights of the codewords of $G_{24}$ are $0, 8, 12, 16, 24$, with $1, 759, 2576, 759, 1$ codewords of each weight.
The $759$ codewords of weight $8$, seen as $8$-element subsets of the $24$ positions, are the blocks of the [[Steiner system|7BPSC4T4M3]] $S(5, 8, 24)$.
The automorphism group of $G_{24}$ is the Mathieu group $M_{24}$, one of the sporadic simple groups.
Besides the binary codes there is also a ternary Golay code.

## Explanations

### L1 human

The binary Golay code is a set of $4096$ strings of $24$ bits, chosen so that any two differ in at least $8$ bits.
If a string is damaged in up to $3$ bits, the original string can still be recovered.
Its shorter version of length $23$ is perfect: the spheres of radius $3$ around its codewords fill the whole space with no gap and no overlap.
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# Hom functor

## Definition

Let $\mathcal{C}$ be a locally small [[category|A2DE66937D]], so that for all objects $X, Y$ the [[morphisms|859F0TFVA3]] $X \to Y$ form a [[set|0BSJ7RQ9BW]] $\Hom(X, Y)$.
Let $\mathbf{Set}$ be the category of sets and functions, and let $A$ be an object of $\mathcal{C}$.

1. The *covariant hom functor* $\Hom(A, -) \colon \mathcal{C} \to \mathbf{Set}$ sends an object $X$ to $\Hom(A, X)$ and a morphism $f \colon X \to Y$ to the function $\Hom(A, f) \colon \Hom(A, X) \to \Hom(A, Y)$, $g \mapsto f \circ g$.
2. The *contravariant hom functor* $\Hom(-, A) \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ sends an object $X$ to $\Hom(X, A)$ and a morphism $f \colon X \to Y$ to the function $\Hom(f, A) \colon \Hom(Y, A) \to \Hom(X, A)$, $h \mapsto h \circ f$.

## Remarks

Associativity of composition gives $\Hom(A, f' \circ f) = \Hom(A, f') \circ \Hom(A, f)$, and $\Hom(A, \id_X)$ is the identity, so clause 1 defines a [[functor|28NQ0KRHD3]].
Clause 2 defines a contravariant functor, that is, a functor on the [[opposite category|DAMK8XE1FR]] $\mathcal{C}^{\mathrm{op}}$.
The two together form the bifunctor $\Hom(-, -) \colon \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathbf{Set}$ on the [[product category|AFNE9C4NH6]].
Other notations are $\mathcal{C}(A, X)$, $h^A = \Hom(A, -)$ and $h_A = \Hom(-, A)$.

Example: in the category of real vector spaces, take $A = \R$.
A linear map $\R \to V$ is determined by the image of $1$, so $\Hom(\R, V)$ is in bijection with $V$, and $\Hom(\R, f)$ corresponds to $f$ itself.

Non-example: the assignment $X \mapsto \Hom(A, X)$ with $f \mapsto (g \mapsto g \circ f)$ is not a covariant hom functor.
For $f \colon X \to Y$ and $g \colon A \to X$, the composite $g \circ f$ is in general not defined.
So it does not give a function $\Hom(A, X) \to \Hom(A, Y)$.
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# Siciak extremal function

## Definition

Let $E \subseteq \C^n$ be a nonempty set and let $\mathcal{L}$ be the [[Lelong class|X4ZHQG9995]] of $\C^n$.
The *Siciak extremal function* of $E$ is the function $V_E \colon \C^n \to [0, +\infty]$ given by
$$V_E(z) = \sup\set{u(z) : u \in \mathcal{L},\ u \le 0 \text{ on } E} .$$
Its upper semicontinuous regularisation is $V_E^*(z) = \limsup_{w \to z} V_E(w)$.

## Remarks

The value is at least $0$, since $u = 0$ is admissible.
For compact $E$, Siciak's theorem gives the polynomial formula
$$V_E(z) = \sup\set{\tfrac{1}{\deg p} \log\abs{p(z)} : p \text{ a polynomial},\ \deg p \ge 1,\ \sup_E \abs{p} \le 1} .$$
Example: for the closed unit ball $E = \set{\abs{z} \le 1}$, $V_E(z) = \max(\log\abs{z}, 0)$.
Check: this function lies in $\mathcal{L}$ and vanishes on $E$, so $V_E \ge \max(\log\abs{z}, 0)$.
Conversely, let $u \in \mathcal{L}$ with $u \le 0$ on $E$, and let $w$ be a unit vector.
The function $\lambda \mapsto u(\lambda w) - \log\abs{\lambda}$ is subharmonic and bounded above on $\abs{\lambda} > 1$, so it extends across $\infty$.
It is at most $0$ on $\abs{\lambda} = 1$, so by the maximum principle $u(\lambda w) \le \log\abs{\lambda}$ for $\abs{\lambda} > 1$.
Non-example: if $E$ is [[pluripolar|N0MAGPW8AP]], then $V_E^* \equiv +\infty$, so $V_E^*$ is not in $\mathcal{L}$; for compact $E$ the converse holds as well, and $V_E^* \in \mathcal{L}^+$ when $E$ is compact and not pluripolar.
For $n = 1$ and compact $E$ of positive logarithmic capacity, $V_E^*$ is the Green function of the unbounded component of $\C \setminus E$ with pole at infinity, extended by $0$.
When $V_E$ is continuous, $E$ is called $L$-regular.
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# Indefinite sum

## Summary

An indefinite sum, or antidifference, of a function $f$ is a function $F$ whose forward difference is $f$, that is $F(x + 1) - F(x) = f(x)$.
It plays the role for sums that an [[antiderivative|0BYWRWFT4F]] plays for integrals.
With an indefinite sum, every finite sum $\sum_{k=a}^{b-1} f(k)$ becomes the difference $F(b) - F(a)$.

## Explanations

### L1 human

Summing a sequence is the reverse of taking differences.
If you know a function $F$ whose steps are exactly the terms you want to add, then the total of the terms is just how far $F$ has moved.
For $f(x) = x$, the function $F(x) = x(x-1)/2$ works, since $F(x + 1) - F(x) = x$.
So $0 + 1 + \dots + (n-1) = F(n) - F(0) = n(n-1)/2$.
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# Group algebra

## Definition

Let $G$ be a [[group|G5S1M9]] and let $F$ be a [[field|04C5ZBG1WT]].
The *group algebra* $F[G]$ is the set of functions $a \colon G \to F$ with $a(g) \ne 0$ for only finitely many $g \in G$, written as finite formal sums $\sum_{g \in G} a_g \, g$.
It is a [[vector space|000AYPA9AJ]] over $F$ under termwise addition and scalar multiplication, with basis $G$.
Its multiplication is
$$\Big(\sum_{g} a_g \, g\Big)\Big(\sum_{h} b_h \, h\Big) = \sum_{k \in G} \Big(\sum_{gh = k} a_g b_h\Big) k.$$

## Remarks

With this product $F[G]$ is an associative [[ring|0B1261B5ED]] with identity $1_G$, and an algebra over $F$ of dimension $\abs{G}$.
It is commutative if and only if $G$ is abelian.
Example: for $G = \set{1, t}$ cyclic of order $2$ and $F = \Q$, the elements $e = \tfrac{1}{2}(1 + t)$ and $f = \tfrac{1}{2}(1 - t)$ satisfy $e^2 = e$, $f^2 = f$, $ef = 0$ and $e + f = 1$, since $t^2 = 1$.
So $\Q[G] \cong \Q \times \Q$ as rings.
Non-example: for infinite $G$, the set of all functions $G \to F$ is not the group algebra, since it fails the finiteness condition.
For $G = \Z$ the product formula then breaks down, as the inner sum $\sum_{gh = k} a_g b_h$ can be an infinite sum.
A [[representation|WYGRXCG2RN]] of $G$ on $V$ is the same as a left $F[G]$-module structure on $V$.
For a commutative ring $R$, the same definition gives the group ring $R[G]$.
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# Null-homotopic map

## Summary

A continuous map is null-homotopic when it is [[homotopic|F2F143KQ2M]] to a constant map.
Null-homotopic maps are the trivial maps of homotopy theory.
For example, a map $S^n \to Y$ is null-homotopic exactly when it extends over the disk $D^{n+1}$.

## Explanations

### L1 human

A map is null-homotopic when its image can be shrunk continuously to a single point inside the target.
Every map into a disk is null-homotopic.
Every loop in the plane is null-homotopic, but in the plane without the origin a loop that goes once around the origin is not.
Null-homotopic maps carry no information about the holes of the target.
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# Tensor product of modules

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $M$, $N$ be [[$R$-modules|0NZ8JA25WK]].
A map $\beta \colon M \times N \to P$ to an $R$-module $P$ is *$R$-bilinear* if it is $R$-[[linear|0NRYHJPQG2]] in each variable when the other is fixed.
A *tensor product* of $M$ and $N$ is an $R$-module $M \otimes_R N$ with an $R$-bilinear map $\otimes \colon M \times N \to M \otimes_R N$, written $(m, n) \mapsto m \otimes n$.
It must satisfy this universal property.
For every $R$-module $P$ and every $R$-bilinear map $\beta \colon M \times N \to P$ there is exactly one $R$-linear map $\varphi \colon M \otimes_R N \to P$ with $\varphi(m \otimes n) = \beta(m, n)$ for all $m \in M$ and $n \in N$.

## Remarks

A tensor product exists and is unique up to a unique isomorphism compatible with the maps $\otimes$.
One construction is the [[free module|0DZ9DKAKZR]] on the set $M \times N$ modulo the submodule generated by the elements $(m + m', n) - (m, n) - (m', n)$, $(m, n + n') - (m, n) - (m, n')$, $(rm, n) - r(m, n)$ and $(m, rn) - r(m, n)$.
The elements $m \otimes n$ generate $M \otimes_R N$, but a general element is a finite sum $\sum_i m_i \otimes n_i$.
Standard isomorphisms are $R \otimes_R N \cong N$, $M \otimes_R N \cong N \otimes_R M$, and $R^m \otimes_R R^n \cong R^{mn}$.

Example: for integers $a, b \ge 1$, $\Z/a\Z \otimes_\Z \Z/b\Z \cong \Z/d\Z$ with $d = \gcd(a, b)$.
The map $(x, y) \mapsto xy \bmod d$ is well defined and bilinear, so it gives a surjection onto $\Z/d\Z$; conversely $1 \otimes 1$ generates and is killed by $a$ and $b$, hence by $d$.

Non-example: the module $M \oplus N$ with the map $(m, n) \mapsto (m, n)$ is not a tensor product when $M$ and $N$ are nonzero, because this map is not bilinear.
For instance, with $M = N = \Z$, it sends $(2, 1)$ to $(2, 1)$, while bilinearity would require $2 \cdot (1, 1) = (2, 2)$.
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# Comparability graph

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
The *comparability graph* of $P$ is the [[graph|00C0BR7D7H]] with vertex set $P$ in which distinct $x, y \in P$ are adjacent if and only if $x \le y$ or $y \le x$.
A graph $G$ is a *comparability graph* if it is isomorphic to the comparability graph of some partially ordered set.

## Remarks

Equivalently, $G$ is a comparability graph when its edges can be oriented so that $x \to y$ and $y \to z$ always imply $x \to z$; such an orientation is called transitive.
Example: the complete bipartite graph $K_{m,n}$ is the comparability graph of the poset with $m$ minimal elements, $n$ maximal elements, and each minimal element below each maximal element.
Non-example: the cycle $C_5$ is not a comparability graph.
Its largest clique has $2$ vertices but it needs $3$ colours, while in a comparability graph the clique number equals the chromatic number, by [[Mirsky's theorem|5VPXR3RDF1]].
The cliques of the comparability graph are the chains of $P$, and its independent sets are the antichains of $P$.
Different partial orders, such as an order and its dual, can have the same comparability graph.
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# Grothendieck universe

## Summary

A Grothendieck universe is a set $U$ that is closed under the usual operations of set theory.
These are taking elements, pairs, [[power sets|00E0YZCERH]] and unions of families indexed by elements of $U$.
Inside a universe one can do ordinary mathematics without leaving $U$.
Category theorists use universes to speak of the category of all "small" sets or groups as a set-sized object.
Uncountable Grothendieck universes are exactly the sets $V_\kappa$ for [[inaccessible cardinals|NW4DREQ5DQ]] $\kappa$, so their existence goes beyond ZFC.

## Explanations

### L1 human

In category theory one wants to talk about "the category of all groups", but the collection of all groups is too big to be a set.
A universe is a large set that is closed under every construction one normally uses.
One then calls a set small when it lies in the universe, and the category of all small groups is a legitimate object.
The axiom that every set lies in some universe was used by Grothendieck in algebraic geometry.
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# Tensor rank

## Definition

Let $F$ be a field, let $d \ge 1$, and let $V_1, \ldots, V_d$ be finite-dimensional [[vector spaces|000AYPA9AJ]] over $F$.
A *simple tensor* in the [[tensor product|0BF2DE2SYK]] $V_1 \otimes \cdots \otimes V_d$ is a nonzero tensor of the form $v_1 \otimes \cdots \otimes v_d$ with $v_i \in V_i$.
The *rank* of a tensor $T \in V_1 \otimes \cdots \otimes V_d$ is the smallest integer $r \ge 0$ such that $T$ is a sum of $r$ simple tensors.
The zero tensor has rank $0$.

## Remarks

For $d = 2$ and $V_1 = F^m$, $V_2 = F^n$, a tensor is an $m \times n$ matrix, a simple tensor is a rank-one matrix $u v^T$, and the tensor rank is the [[rank of the matrix|0ZC9YTXZRY]].
Example: in $F^2 \otimes F^2 \otimes F^2$ with standard basis $e_1, e_2$, the tensor $T = e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_2 \otimes e_2$ has rank $2$.
The written sum shows rank at most $2$.
The tensor is not simple: flattening it to a $2 \times 4$ matrix gives rank $2$, while a simple tensor flattens to a matrix of rank $1$.
Non-example of rank $2$: over $\C$, the tensor $e_1 \otimes e_1 \otimes e_2 + e_1 \otimes e_2 \otimes e_1 + e_2 \otimes e_1 \otimes e_1$ has rank $3$, although it is a limit of tensors of rank $2$.
So the set of tensors of rank at most $r$ need not be closed, which leads to the notion of border rank.
The rank of a real tensor can be larger than its rank over $\C$.
Other names are canonical polyadic rank and CP rank.
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# Left-invariant vector field

## Summary

A left-invariant vector field on a [[Lie group|7JBRGCVJRB]] is a [[vector field|0299VCW1TH]] that is unchanged by every left translation.
Such a field is fixed by its value at the identity, so left-invariant vector fields correspond to tangent vectors at the identity.
They are closed under the [[Lie bracket of vector fields|68FGQEBD6J]].
This makes the tangent space at the identity a Lie algebra, the Lie algebra of the group.

## Explanations

### L1 human

Pick an arrow at the identity element of a Lie group.
Carry it to every other point $g$ by multiplying on the left by $g$; you get an arrow at every point, all fitting together smoothly.
On the plane $\R^2$ as an additive group, this gives the constant vector fields: the same arrow attached at every point.
On the circle group, it gives a field of arrows of the same length all pointing anticlockwise, or all clockwise.
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# Laplace's equation

## Definition

Let $n \ge 1$ and let $U \subseteq \R^n$ be open.
*Laplace's equation* on $U$ is the equation
$$\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2} = 0 \quad \text{in } U$$
for an unknown function $u \colon U \to \R$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
A classical solution is a function $u$ that is twice continuously differentiable on $U$ and satisfies the equation at every point of $U$.

## Remarks

The classical solutions are exactly the harmonic functions on $U$.
Every classical solution is in fact infinitely differentiable, and even real analytic.
The inhomogeneous equation $\Delta u = f$ is Poisson's equation.
Some authors write the Laplacian as $\nabla^2$ or $\nabla \cdot \nabla$, and in physics the equation is often written $\nabla^2 \phi = 0$.

Example: on $\R^3 \setminus \set{0}$, the function $u(x) = 1 / \abs{x}$ is a solution.
With $r = \abs{x}$, one has $\partial_i u = -x_i / r^3$ and $\partial_i^2 u = -1/r^3 + 3 x_i^2 / r^5$, and the sum over $i = 1, 2, 3$ is $-3/r^3 + 3 r^2 / r^5 = 0$.

Non-example: on $\R^2$, the function $u(x, y) = x^2 + y^2$ is not a solution, since $\Delta u = 2 + 2 = 4 \ne 0$.
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# Delta derivative on a time scale

## Definition

Let $\mathbb{T}$ be a time scale, that is, a nonempty [[closed subset|0G5AV9Q4RR]] of $\R$.
Let $\sigma(t) = \inf \set{s \in \mathbb{T} : s > t}$ for $t \in \mathbb{T}$, with $\sigma(t) = t$ when $t = \max \mathbb{T}$.
Let $\mathbb{T}^\kappa$ be $\mathbb{T}$ without $m = \max \mathbb{T}$ if this maximum exists and $\sup \set{s \in \mathbb{T} : s < m} < m$, and $\mathbb{T}^\kappa = \mathbb{T}$ otherwise.
Let $f \colon \mathbb{T} \to \R$ and $t \in \mathbb{T}^\kappa$.
A number $f^\Delta(t) \in \R$ is the *delta derivative* of $f$ at $t$ if the following holds.
For every $\varepsilon > 0$ there is $\delta > 0$ such that
$$\abs{f(\sigma(t)) - f(s) - f^\Delta(t)\,(\sigma(t) - s)} \le \varepsilon \abs{\sigma(t) - s}$$
for all $s \in \mathbb{T}$ with $\abs{t - s} < \delta$.
Then $f$ is called *delta differentiable* at $t$.

## Remarks

This is the derivative of [[time-scale calculus|C4B1FA1H0J]]; when it exists it is unique.
Example: on $\mathbb{T} = \Z$, $\sigma(t) = t + 1$; with $\delta = \tfrac{1}{2}$ the only admissible $s$ is $t$, and the condition holds exactly for $f^\Delta(t) = f(t+1) - f(t)$.
Non-example: on $\mathbb{T} = \R$, $\sigma(t) = t$ and the condition says that $f$ is differentiable at $t$ with $f^\Delta(t) = f'(t)$; so $f(t) = \abs{t}$ has no delta derivative at $0$.
If $\sigma(t) > t$ and $f$ is continuous at $t$, then $f^\Delta(t) = \frac{f(\sigma(t)) - f(t)}{\sigma(t) - t}$.
If $\sigma(t) = t$ and $t$ is not isolated from the right, then $f^\Delta(t) = \lim_{s \to t,\, s \in \mathbb{T}} \frac{f(t) - f(s)}{t - s}$ when this limit exists.
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# Expectation–maximization algorithm

## Definition

Let $p(x, z; \theta)$, for $\theta \in \Theta$, be a family of joint densities of observed data $x$ and unobserved (latent) data $z$.
The observed-data [[likelihood|05FSTXM92W]] is $L(\theta) = \int p(x, z; \theta)\, dz$, and $p(z \mid x; \theta) = p(x, z; \theta)/L(\theta)$ is the conditional density of $z$.
The *expectation–maximization algorithm* starts from $\theta^{(0)} \in \Theta$ and, for $t = 0, 1, 2, \dots$, repeats two steps:

1. (E-step) compute the function $Q(\theta \mid \theta^{(t)}) = E_{Z \sim p(\cdot \mid x; \theta^{(t)})}\big[\log p(x, Z; \theta)\big]$;
2. (M-step) choose $\theta^{(t+1)}$ that maximizes $\theta \mapsto Q(\theta \mid \theta^{(t)})$ over $\Theta$.

## Remarks

The key property is monotonicity: $L(\theta^{(t+1)}) \ge L(\theta^{(t)})$.
Indeed, $\log L(\theta) = Q(\theta \mid \theta^{(t)}) - E\big[\log p(Z \mid x; \theta)\big]$ with the expectation over $Z \sim p(\cdot \mid x; \theta^{(t)})$, and by [[Jensen's inequality|0RWMRTPC87]] the second term is smallest at $\theta = \theta^{(t)}$.
The E-step is a [[conditional expectation|0YAMJRK04E]] given the observed data.
The iterates need not converge to the [[maximum likelihood estimate|0FREPFKNS7]]: they may stop at a local maximum or a saddle point.

Example: a mixture of two normal densities $\pi\, \varphi_{\mu_1, 1} + (1 - \pi)\, \varphi_{\mu_2, 1}$ with latent labels $z_i \in \set{1, 2}$.
The E-step computes the weights $w_i = \pi \varphi_{\mu_1,1}(x_i) / \big(\pi \varphi_{\mu_1,1}(x_i) + (1-\pi)\varphi_{\mu_2,1}(x_i)\big)$ at the current parameters; the M-step sets $\pi = \frac{1}{n}\sum_i w_i$, $\mu_1 = \sum_i w_i x_i / \sum_i w_i$ and $\mu_2 = \sum_i (1 - w_i) x_i / \sum_i (1 - w_i)$.

Non-example: if the M-step only chooses some $\theta^{(t+1)}$ with $Q(\theta^{(t+1)} \mid \theta^{(t)}) \ge Q(\theta^{(t)} \mid \theta^{(t)})$, without maximizing, clause 2 fails; this variant is called a generalized EM algorithm, and it keeps the monotonicity property.
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# Volterra equation of convolution type

## Summary

A Volterra equation of convolution type is a [[Volterra integral equation|DND8CBEVXT]] whose kernel depends only on the difference $t - s$.
Its integral term is the convolution $\int_0^t k(t - s) \, x(s) \, ds$ on the half-line.
Such equations describe systems whose rules do not change in time, and the [[Laplace transform|0V49JEZW06]] turns them into algebraic equations.
Renewal equations, Abel's equation and many models with memory are of this type.

## Explanations

### L1 human

The weight given to a past value $x(s)$ depends only on how long ago it was, the lag $t - s$, and not on the date $s$ itself.
This is the integral version of a linear differential equation with constant coefficients.
For example, $x(t) = 1 + \int_0^t e^{-(t - s)} x(s) \, ds$ has memory that fades exponentially with the lag.
Its solution is $x(t) = 1 + t$, as one can check by computing the integral.
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# Generalized Fourier series

## Summary

A generalized Fourier series expands a vector of an [[inner product space|0QVNCJXRY3]] along an orthogonal family, with coefficients computed by inner products.
The classical [[Fourier series|G12KXDHPPS]] is the case of the exponentials $e^{inx}$ in $L^2$ of the circle.
Other cases are expansions in Legendre polynomials, Bessel functions or eigenfunctions of a Sturm–Liouville problem.
The partial sums are the best approximations from the span of the first terms, and the series recovers the vector when the family is complete.

## Explanations

### L1 human

In the plane, the coordinates of a vector along two perpendicular directions are found by dot products.
A generalized Fourier series does the same in a space of functions, with infinitely many perpendicular directions.
For example, on $[-1, 1]$ the function $x^2$ equals $\frac13 \cdot 1 + \frac23 \cdot \frac{3x^2 - 1}{2}$, a sum along the first and third Legendre polynomials.
Each coefficient tells how much of one basic shape is present in the function.
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# Order of a difference equation

## Summary

The order of a difference equation is the gap between the largest and the smallest index of the unknown sequence that occur in it.
When the equation can be solved for its highest term, the order is the number of initial values needed to fix a solution.
It plays the role that the order of a differential equation plays for derivatives.

## Explanations

### L1 human

A difference equation ties a term of a sequence to some terms before it.
The order counts how far back the equation looks.
The Fibonacci rule $u_{n+2} = u_{n+1} + u_n$ looks two steps back, so it has order $2$.
A rule like $u_{n+1} = 3 u_n$ looks only one step back and has order $1$.
To start a sequence of order $k$ you must choose its first $k$ terms by hand.
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# Whitney immersion theorem

## Statement

Let $n \ge 2$ and let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, Hausdorff and second countable.
Then there is a smooth [[immersion|6RW68JGBHC]] of $M$ into the [[Euclidean space|0XPR1E85W0]] $\R^{2n-1}$.

## Remarks

The weak version gives an immersion of any smooth $n$-manifold, for every $n \ge 1$, into $\R^{2n}$, and shows that any smooth map $M \to \R^{2n}$ can be approximated by immersions.
It follows from a general position argument: the maps whose differential drops rank form a set of codimension $2n - n + 1 = n + 1 > n$ in the space of $1$-jets, so a generic map avoids them.
The hypothesis $n \ge 2$ is needed: the circle $S^1$ admits no immersion into $\R^1$.
Such an immersion would be a local diffeomorphism, so its image would be open and compact in $\R$, which is impossible.
Example: the real projective plane $\R P^2$ immerses in $\R^3$, as Boy's surface shows.
For compact $M$, Ralph Cohen proved the sharper bound $\R^{2n - \alpha(n)}$, where $\alpha(n)$ is the number of ones in the binary expansion of $n$.

## Explanations

### L1 human

An immersion may cross itself, so it needs less room than an embedding.
Whitney showed that every $n$-dimensional manifold with $n \ge 2$ can be drawn without corners or folds, but possibly with self-crossings, in Euclidean space of dimension $2n - 1$.
For surfaces this means $\R^3$: even the Klein bottle and the projective plane can be drawn there with self-crossings.

### L2 human

Compare with the Whitney embedding theorem, which needs dimension $2n$ to avoid self-crossings.
The easy bound $2n$ comes from transversality of the $1$-jet of a map to the set of rank-deficient jets.
The step down to $2n - 1$ needs a finer study of the singularities of a generic map into $\R^{2n-1}$.
The Smale–Hirsch theorem later reduced the existence of immersions to a question about vector bundles.
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# Wang's finiteness theorem

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center and without compact factors.
Assume that no simple factor of $G$ is locally isomorphic to $\mathrm{SL}_2(\R)$ or to $\mathrm{SL}_2(\C)$.
Fix a [[Haar measure|VY0HXD3GC4]] on $G$, and for a [[lattice|BTYF5Y5PFY]] $\Gamma$ of $G$ let $\operatorname{covol}(\Gamma)$ be the measure of $G/\Gamma$ for the induced invariant measure.
Then for every $v > 0$ there are only finitely many lattices $\Gamma$ in $G$ with $\operatorname{covol}(\Gamma) \le v$, up to conjugation by elements of $G$.

## Remarks

The result is due to H. C. Wang.
The proof combines two facts.
The [[Kazhdan–Margulis theorem|C34RZMKKA4]] gives a fixed neighbourhood of the identity that every lattice avoids up to conjugation, which makes the set of lattices of bounded covolume compact up to conjugation.
[[Weil's local rigidity theorem|V2HTMG4KT1]] and its extension to non-uniform lattices make each lattice isolated up to conjugation, and a compact set of isolated points is finite.
The excluded factors are necessary.
Fix a genus $g \ge 2$.
In $\mathrm{PSL}_2(\R)$ the fundamental groups of closed hyperbolic surfaces of genus $g$ all have covolume $4\pi(g - 1)$, by the Gauss–Bonnet theorem.
Yet they form uncountably many conjugacy classes, parametrised by [[Teichmüller space|ZZGC6BZNJ5]].
In $\mathrm{PSL}_2(\C)$ hyperbolic Dehn filling gives infinitely many closed hyperbolic $3$-manifolds whose volumes are bounded by the volume of one cusped hyperbolic $3$-manifold.

## Explanations

### L1 human

A lattice gives a finite-volume space $G/\Gamma$, and one can measure its size.
The theorem says that, for most semisimple groups, only finitely many essentially different lattices have size below any given bound.
So lattices are sparse, unlike in the plane, where lattices can be deformed continuously.
For example, there are only finitely many closed hyperbolic manifolds of dimension at least $4$ with volume below any fixed number.

### L2 human

Wang's theorem is the global consequence of two local facts: lattices cannot contain elements too close to the identity in a uniform way, and lattices cannot be deformed.
It fails for factors of type $\mathrm{SL}_2(\R)$, where lattices deform continuously.
It also fails for $\mathrm{SL}_2(\C)$, where hyperbolic Dehn filling produces infinitely many lattices of bounded covolume.
It is a starting point for counting lattices of bounded covolume, and for the study of the smallest covolume of a lattice in a given group.
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# Delay differential equation

## Definition

Let $n, k \ge 1$, let $0 < \tau_1 < \cdots < \tau_k$ be constants, and let $f \colon \R \times (\R^n)^{k+1} \to \R^n$ be a function.
A *delay differential equation* with constant delays $\tau_1, \ldots, \tau_k$ is an equation
$$x'(t) = f\bigl(t, x(t), x(t - \tau_1), \ldots, x(t - \tau_k)\bigr)$$
for an unknown function $x$ with values in $\R^n$.
Let $t_0 \in \R$ and let $\phi \colon [-\tau_k, 0] \to \R^n$ be continuous, the *history*.
A *solution* of the initial value problem is a continuous $x \colon [t_0 - \tau_k, t_0 + T) \to \R^n$, for some $T > 0$, with these properties.
It equals the history: $x(t_0 + \theta) = \phi(\theta)$ for $\theta \in [-\tau_k, 0]$.
It is differentiable on $(t_0, t_0 + T)$, and the equation holds there.

## Remarks

Example: $x'(t) = -x(t - 1)$, with $n = k = 1$, $\tau_1 = 1$ and $f(t, u, v) = -v$.
Its characteristic equation, from $x = e^{\lambda t}$, is $\lambda = -e^{-\lambda}$, which has infinitely many complex roots.

Non-example: $x'(t) = x(t + 1)$ is not a delay equation, since the rate at time $t$ depends on a later time; such equations are called advanced.
An [[ordinary differential equation|0NT64JV96A]] $x' = f(t, x)$ is the degenerate case in which $f$ does not depend on the delayed arguments.

Delay equations are a special case of [[functional differential equations|ZMCFEEPEKT]] of retarded type, with $F(t, x_t) = f(t, x_t(0), x_t(-\tau_1), \ldots, x_t(-\tau_k))$.
Delays may also vary with time or with the state, as in $x'(t) = f(t, x(t), x(t - \tau(t)))$.

## Explanations

### L1 human

The derivative now depends on where the solution was some time ago.
So the start of the problem is a whole piece of past, not one point.
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# Hasse principle

## Definition

For a prime $p$, let $\Q_p$ be the field of [[p-adic numbers|BT6T702C28]], and let $\Q_\infty = \R$.
Let $\mathcal{C}$ be a class of systems of polynomial equations with coefficients in $\Q$.
For a system of homogeneous equations, a solution means a solution other than the zero solution.
Let $V$ be the set of all primes together with the symbol $\infty$.
The class $\mathcal{C}$ *satisfies the Hasse principle* if each system in $\mathcal{C}$ with a solution in $\Q_v$ for every $v \in V$ has a solution in $\Q$.

## Remarks

The converse always holds, since $\Q$ is a subfield of each $\Q_v$.
The same definition makes sense over any number field, with its completions in place of the $\Q_v$.

Example: the class of homogeneous quadratic equations $Q(x_1, \dots, x_n) = 0$, with $Q$ a quadratic form over $\Q$, satisfies the Hasse principle; this is the [[Hasse–Minkowski theorem|R8HCWA981S]].

Non-example: the class of homogeneous cubic equations in three variables does not.
Selmer's equation $3x^3 + 4y^3 + 5z^3 = 0$ has nonzero solutions in $\R$ and in every $\Q_p$, but it has no nonzero solution in $\Q$.
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# Non-Euclidean geometry

## Summary

A non-Euclidean geometry is a geometry that keeps the other basic assumptions of [[Euclidean geometry|KRZD96GSVC]] but drops the [[parallel postulate|2PC3XY0PYZ]].
The two classical ones are [[hyperbolic geometry|GN4N94GQ0Z]], where a line has many parallels through an outside point, and [[elliptic geometry|H5GB19W2H3]], where it has none.
Their discovery showed that the parallel postulate cannot be proved from the other axioms.
They also became models for curved space in differential geometry and physics.

## Explanations

### L1 human

In school geometry, through a point off a line there is exactly one line that never meets it.
For two thousand years people tried to prove this from simpler rules and failed.
Non-Euclidean geometries are consistent worlds where this rule is false.
On a sphere, where great circles play the role of lines, any two lines meet, and the angles of a triangle add up to more than $180^\circ$.
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<document source="https://tamnd.github.io/math2/node/SPFP3GHVYJ.md" tag="SPFP3GHVYJ" kind="statement">
# Segre's theorem

## Statement

Let $q$ be an odd prime power, let $\F_q$ be the [[finite field|0ZXZ82FTX2]] with $q$ elements, and let $\mathrm{PG}(2, q)$ be the [[projective plane over $\F_q$|CSMB49BDPT]].
An [[oval|7B03TB64N9]] of $\mathrm{PG}(2, q)$ is a set of $q + 1$ points with no three on a common line.
Then every oval of $\mathrm{PG}(2, q)$ is a non-degenerate [[conic|R3TMVHTP3A]].

## Remarks

The theorem was proved by Beniamino Segre.
The key step is the "lemma of tangents" for an oval in $\mathrm{PG}(2, q)$ with $q$ odd.
It says that any triangle of points of the oval and the triangle of tangent lines at these points are in perspective.
Its proof uses that the product of all nonzero elements of $\F_q$ is $-1$.
From this one shows that the oval agrees with the conic through five of its points.

The hypothesis that $q$ is odd is necessary.
Let $q = 2^h$ with $h \ge 3$, let $C$ be a conic and let $N$ be its nucleus, the common point of all its tangent lines.
Removing a point $P \ne N$ of $C$ from $C \cup \set{N}$ gives an oval.
It is not a conic: it shares $q \ge 8$ points with $C$, while two distinct conics share at most $4$ points.

## Explanations

### L1 human

In a finite plane over a field with an odd number of elements, take $q + 1$ points with no three on a line.
This is the largest such set possible, and conics are one way to get it.
Segre's theorem says that conics are the only way.
So a purely combinatorial condition forces an algebraic shape, a curve of degree two.

### L2 human

Segre's theorem is the starting point of Galois geometry, the study of point sets in projective spaces over finite fields.
It shows that in $\mathrm{PG}(2, q)$ with $q$ odd the ovals have a single type up to projective equivalence.
In even characteristic the picture is different: every oval extends to a hyperoval of $q + 2$ points, and many non-conic examples exist.
The same circle of ideas classifies ovoids of $\mathrm{PG}(3, q)$ for $q$ odd as elliptic quadrics.
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<document source="https://tamnd.github.io/math2/node/SPV5SRAS9H.md" tag="SPV5SRAS9H" kind="example">
# Carleman's singular integral equation

## Example

Let $a$ and $b$ be real constants with $b \ne 0$, and let $f$ be a given function on the interval $(-1, 1)$.
*Carleman's equation* asks for a function $\varphi$ on $(-1, 1)$ with
$$a\, \varphi(x) + \frac{b}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\varphi(t)}{t - x}\, dt = f(x), \quad -1 < x < 1,$$
where $\mathrm{PV}$ is the [[Cauchy principal value|0SYR10HD4R]].
It is a [[singular integral equation|E30EA6ETZ4]] of the second kind with constant coefficients on an arc, and its singular part is the [[finite Hilbert transform|5JYBXH9BP6]].

A check with an explicit solution uses two classical principal value integrals, valid for $-1 < x < 1$:
$$\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\sqrt{1 - t^2}}{t - x}\, dt = -x, \qquad \frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{dt}{\sqrt{1 - t^2}\,(t - x)} = 0.$$
Hence $\varphi(t) = \sqrt{1 - t^2}$ solves the equation with right side $f(x) = a\sqrt{1 - x^2} - b x$.
The function $(1 - t^2)^{-1/2}$ solves it with right side $f(x) = a (1 - x^2)^{-1/2}$.

## Remarks

Put $\Phi(z) = \frac{1}{2\pi i} \int_{-1}^{1} \frac{\varphi(t)}{t - z}\, dt$ for $z \notin [-1, 1]$.
By the [[Sokhotski–Plemelj theorem|RNB6QX0GCA]], $\Phi^+ - \Phi^- = \varphi$ and $\Phi^+ + \Phi^- = \frac{1}{\pi i}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\varphi(t)}{t - x}\, dt$ on the cut.
The equation becomes $(a + ib)\, \Phi^+ = (a - ib)\, \Phi^- + f$, a [[Riemann–Hilbert problem|RY8YQ9JZF9]] on the segment with constant coefficient $(a - ib)/(a + ib)$ of modulus $1$.
Solving it gives the solutions in closed form, with powers of $1 - x$ and $1 + x$ at the endpoints whose exponents are fixed by the argument of $a + ib$.
For $a = 0$ the equation is the [[airfoil equation|W6Z6CWRE25]].

## Explanations

### L1 human

This is the simplest singular integral equation on an interval: the unknown appears once by itself and once inside a Cauchy principal value integral.
The integral blows up where $t = x$, so it is taken as a principal value.
Because the coefficients are constants, the equation can be solved exactly by turning it into a jump problem for holomorphic functions.
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<document source="https://tamnd.github.io/math2/node/SQ6APWZBTR.md" tag="SQ6APWZBTR" kind="concept">
# Topological group structure of a vector space

## Summary

Every [[topological vector space|T32W553W95]] is, under addition, a commutative [[topological group|92700QAB5Y]].
This additive structure is the source of many basic facts about topological vector spaces.
Translations are homeomorphisms, so the whole topology is fixed by the neighbourhoods of $0$.
It also gives each topological vector space a canonical uniform structure, so Cauchy nets and completeness make sense without a metric.

## Explanations

### L1 human

In a topological vector space you may add vectors, and adding is a continuous operation.
So, if you forget scalar multiplication, what is left is a group with a topology that fits the group law.
Shifting the whole space by a fixed vector moves open sets to open sets.
Hence, to know what "near $x$" means, it is enough to know what "near $0$" means and then shift.
For example, in $\R^2$ the open discs around a point $x$ are exactly the open discs around $0$ moved by $x$.
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<document source="https://tamnd.github.io/math2/node/SQ9RQVANMG.md" tag="SQ9RQVANMG" kind="concept">
# Spherical geometry

## Summary

Spherical geometry is the geometry of the surface of a sphere, where the role of lines is played by great circles.
Any two great circles meet, so there are no parallel lines, and the angle sum of a triangle exceeds $\pi$.
It is the oldest [[non-Euclidean geometry|SPCYMFN5V2]] in practice, used for navigation and astronomy long before the axioms of geometry were questioned.
Identifying antipodal points turns it into [[elliptic geometry|H5GB19W2H3]].

## Explanations

### L1 human

On a globe, the shortest route between two cities follows a great circle, a circle whose centre is the centre of the Earth.
These great circles are the straight lines of spherical geometry.
The equator and any meridian are great circles; the circles of latitude other than the equator are not.
A triangle with one corner at the North Pole and two on the equator, a quarter turn apart, has three right angles.
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<document source="https://tamnd.github.io/math2/node/SQB74J5E8Q.md" tag="SQB74J5E8Q" kind="statement">
# Auslander–Buchsbaum formula

## Statement

Let $(R, \mathfrak{m})$ be a commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]].
Let $M$ be a nonzero [[finitely generated|05QY1VEBYD]] $R$-module of finite [[projective dimension|MR9QCH1PNA]].
Then
$$\operatorname{pd}_R M + \operatorname{depth} M = \operatorname{depth} R,$$
where $\operatorname{depth}$ is the [[depth|M2RADBFW99]] with respect to $\mathfrak{m}$.

## Remarks

One proof goes by induction on $\operatorname{depth} R$, using a minimal free resolution of $M$.
When the depths are positive, one divides by an element of $\mathfrak{m}$ that is a non-zero-divisor on both $R$ and $M$.
Example: for $R = K[\![x, y]\!]$, which has depth $2$, the residue field $K$ has projective dimension $2$ and depth $0$, and $R/(x)$ has projective dimension $1$ and depth $1$.
The hypothesis $\operatorname{pd}_R M < \infty$ is needed: for $R = K[x]/(x^2)$ and $M = K$, the depths are $0$ and $0$, but $\operatorname{pd}_R K = \infty$.
In particular, a finitely generated module of finite projective dimension has projective dimension at most $\operatorname{depth} R$.

## Explanations

### L1 human

For a finitely generated module of finite projective dimension over a local ring, two numbers add up to a constant.
The projective dimension counts how complicated the module is homologically.
The depth counts how much room the module has for regular sequences.
The more depth a module has, the shorter its resolution.

### L2 human

The formula is the main link between resolutions and regular sequences.
Over a [[regular local ring|63XXTF6B5M]], every finitely generated module has finite projective dimension, so the formula always applies there.
It shows that a module of finite projective dimension over a Cohen–Macaulay local ring is maximal Cohen–Macaulay exactly when it is free.
</document>

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# Remez algorithm

## Definition

Let $f \colon [a, b] \to \R$ be [[continuous|0X3WGX95FE]], let $n \ge 0$, and let $\mathcal{P}_n$ be the real polynomials of [[degree|0DPF6WRZW6]] at most $n$.
The *Remez algorithm* computes the [[minimax approximation|ASWF03X5AM]] of $f$ in $\mathcal{P}_n$ by the following iteration.

1. Choose a reference set of $n + 2$ points $a \le x_0 < x_1 < \cdots < x_{n+1} \le b$.
2. Solve the linear system $p(x_i) + (-1)^i E = f(x_i)$, $i = 0, \dots, n + 1$, for the $n + 1$ coefficients of $p \in \mathcal{P}_n$ and the number $E$.
3. Let $e = f - p$. Replace the reference set by $n + 2$ points $y_0 < \cdots < y_{n+1}$ in $[a, b]$ at which $e$ alternates in sign, $\abs{e(y_i)} \ge \abs{E}$ for all $i$, and $\abs{e(y_j)} = \max_{[a, b]} \abs{e}$ for some $j$.
4. Repeat steps 2 and 3 until $\max_{[a, b]} \abs{e} - \abs{E}$ is below a chosen tolerance.

## Remarks

The system in step 2 has a unique solution for any $n + 2$ distinct points.
After each step 2, the minimax error $E_n(f)$ satisfies $\abs{E} \le E_n(f) \le \max_{[a, b]} \abs{e}$, so the gap between the two bounds measures the progress.

Example: $f(x) = x^2$ on $[-1, 1]$, $n = 1$, reference set $-1, 0, 1$.
Step 2 for $p(x) = c_0 + c_1 x$ gives $c_0 - c_1 + E = 1$, $c_0 - E = 0$ and $c_0 + c_1 + E = 1$, so $c_1 = 0$ and $c_0 = E = 1/2$.
The error $x^2 - 1/2$ has maximum $1/2 = \abs{E}$, so the algorithm stops with the minimax line $p = 1/2$.

Non-example: solving $p(x_i) = f(x_i)$ at $n + 1$ points is plain interpolation, not a Remez step.
It lacks the levelled error term $(-1)^i E$, and its error need not alternate in sign with equal size.

For continuous $f$ the iterates converge to the minimax polynomial.
The method also applies to rational approximation and to other Chebyshev systems in place of $\mathcal{P}_n$.
</document>

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# Carleson measure

## Summary

A Carleson measure is a positive measure on the [[unit disc|8G5ABXRK85]] with a size bound near the boundary.
Its mass in each box touching the boundary is at most a constant times the length of the box.
Carleson showed that these are exactly the measures $\mu$ for which the [[Hardy space|PNNNFJVF6W]] $H^p$ embeds continuously into $L^p(\mu)$.
They appear in the [[corona theorem|WMGA92P2G4]], in interpolation problems and in the duality between $H^1$ and [[BMO|QJYTXFT8Y4]].

## Explanations

### L1 human

Picture a small arc $I$ on the circle and the "box" in the disc that sits just inside it, of depth equal to the length of $I$.
A measure is Carleson if each such box carries mass at most a constant times the length of $I$.
Area measure on the disc is Carleson, since a box of side $\ell$ has area about $\ell^2$, which is at most a constant times $\ell$.
A measure that piles too much mass close to one boundary point is not Carleson.
The condition is exactly what is needed to integrate $\abs{f}^p$ against $\mu$ when $f$ is in $H^p$.
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<document source="https://tamnd.github.io/math2/node/SQT6QCEXZ8.md" tag="SQT6QCEXZ8" kind="concept">
# Hadamard product

## Summary

The Hadamard product of two [[matrices|0VD9ZSWWJG]] of the same size is the matrix of the products of their corresponding entries.
It is also called the entrywise product or the Schur product.
Unlike ordinary [[matrix multiplication|032Q9FB8MT]], it is commutative.
The Schur product theorem says that it keeps positive semidefinite matrices positive semidefinite.

## Explanations

### L1 human

Lay two tables of numbers of the same shape on top of each other.
Multiply the two numbers in each cell, and write the result in that cell.
For example, the Hadamard product of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$ is $\begin{pmatrix} 5 & 12 \\ 21 & 32 \end{pmatrix}$.
This is different from the usual matrix product, which mixes rows with columns.
</document>

<document source="https://tamnd.github.io/math2/node/SQVQ5DQT6W.md" tag="SQVQ5DQT6W" kind="concept">
# Gaussian curvature

## Summary

The Gaussian curvature of a surface at a point is the product of its two [[principal curvatures|FA8QE1SGP4]].
It is positive at dome-like points, negative at saddle-like points, and zero on surfaces that can be unrolled flat.
By Gauss's Theorema Egregium it depends only on the first fundamental form, so it can be measured from inside the surface.

## Explanations

### L1 human

Try to flatten an orange peel on a table: it tears, because the sphere has positive Gaussian curvature.
A piece of a cylinder, like a rolled sheet of paper, flattens without stretching, because its Gaussian curvature is zero.
A saddle or a potato crisp has negative Gaussian curvature: it curves up one way and down the other.
A sphere of radius $r$ has Gaussian curvature $1/r^2$ at every point.
</document>

<document source="https://tamnd.github.io/math2/node/SR51EVK9H0.md" tag="SR51EVK9H0" kind="definition">
# Abelian extension

## Definition

Let $L/K$ be a [[Galois extension|0DS2F7GK05]] with [[Galois group|0XVR72M51M]] $\Gal(L/K)$.
The extension $L/K$ is *abelian* if $\Gal(L/K)$ is an [[abelian group|071DDFVG4J]].

## Remarks

Example: $\Q(\sqrt{2}, \sqrt{3})/\Q$ is Galois of degree $4$.
Its Galois group consists of the four maps $\sqrt{2} \mapsto \pm\sqrt{2}$, $\sqrt{3} \mapsto \pm\sqrt{3}$, which commute, so the extension is abelian.
Non-example: the splitting field $L = \Q(\sqrt[3]{2}, \omega)$ of $x^3 - 2$, with $\omega = e^{2\pi i/3}$, is Galois over $\Q$.
Its Galois group permutes the three roots of $x^3 - 2$ in all six ways, so it is isomorphic to $S_3$, which is not abelian.
Every subextension of a finite abelian extension is abelian, because every subgroup of an abelian group is normal and every quotient is abelian.
</document>

<document source="https://tamnd.github.io/math2/node/SR56KCFYYS.md" tag="SR56KCFYYS" kind="definition">
# Ultrafilter

## Definition

Let $X$ be a set.
An *ultrafilter* on $X$ is a [[filter|15PF73BKY6]] $\mathcal{U}$ on $X$ such that for every filter $\mathcal{F}$ on $X$ with $\mathcal{U} \subseteq \mathcal{F}$, one has $\mathcal{F} = \mathcal{U}$.

## Remarks

A filter $\mathcal{U}$ on $X$ is an ultrafilter if and only if, for every $A \subseteq X$, either $A \in \mathcal{U}$ or $X \setminus A \in \mathcal{U}$.

Example: for $a \in X$, the principal filter $\mathcal{U}_a = \set{A \subseteq X : a \in A}$ is an ultrafilter.
For each $A \subseteq X$, either $a \in A$ or $a \in X \setminus A$.

Non-example: on $\N$, the Fréchet filter of sets with finite complement is not an ultrafilter.
The set $E$ of even numbers and its complement are both infinite, so neither has finite complement.

By [[Zorn's lemma|04RXTQA5ZA]], every filter on $X$ is contained in an ultrafilter.
On an infinite set, an ultrafilter that contains the Fréchet filter is not principal, so non-principal ultrafilters exist.
</document>

<document source="https://tamnd.github.io/math2/node/SR8DQDKHQN.md" tag="SR8DQDKHQN" kind="definition">
# Hotelling's T-squared distribution

## Definition

Let $p$ and $m$ be positive integers with $m \ge p$, and let $\Sigma$ be a $p \times p$ [[positive definite matrix|0QQCG3JWJ4]].
Let $x$ be a random vector with the [[multivariate normal distribution|0G558GJQMW]] $\mathcal{N}_p(0, \Sigma)$.
Let $M$ be a random matrix with the [[Wishart distribution|8BEMNJW0B9]] $W_p(\Sigma, m)$, [[independent|0E9502NT92]] of $x$.
The distribution of
$$T^2 = m \, x^T M^{-1} x$$
does not depend on $\Sigma$.
It is *Hotelling's T-squared distribution* with dimension $p$ and $m$ degrees of freedom, written $T^2(p, m)$.

## Remarks

The condition $m \ge p$ makes $M$ invertible with probability $1$.
The distribution is a rescaled [[F-distribution|PBT8VRJADK]]:
$$\frac{m - p + 1}{p m} T^2 \sim F(p, m - p + 1).$$

Example: for $p = 1$, $x = \sigma Z$ with $Z$ standard normal and $M = \sigma^2 V$ with $V$ chi-squared with $m$ degrees of freedom.
Then $T^2 = Z^2/(V/m)$ is the square of a [[Student t|0NQ5798SB1]] variable with $m$ degrees of freedom, and the formula above gives $T^2 \sim F(1, m)$.

Example: let $Y_1, \dots, Y_n$ be i.i.d. $\mathcal{N}_p(\mu, \Sigma)$ with $n - 1 \ge p$, with sample mean $\bar{Y}$ and sample covariance matrix $\hat{\Sigma}$.
Then $\sqrt{n}(\bar{Y} - \mu) \sim \mathcal{N}_p(0, \Sigma)$ and $(n - 1)\hat{\Sigma} \sim W_p(\Sigma, n - 1)$ are independent, so $n (\bar{Y} - \mu)^T \hat{\Sigma}^{-1} (\bar{Y} - \mu) \sim T^2(p, n - 1)$.

Non-example: if $M$ and $x$ are dependent, for instance $p = 1$, $m = 1$ and $M = x^2$, then $T^2 = 1$ always, which is not $T^2(1, 1)$.
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# Functional equation

## Summary

A functional equation is an equation whose unknown is a [[function|0EPAT51DXQ]] and which must hold for all values of some variables.
Both sides are built by composing the unknown function with known operations.
Classic examples are [[Cauchy's functional equation|ZBMEVWN100]] $f(x + y) = f(x) + f(y)$ and the rule $\Gamma(x + 1) = x\,\Gamma(x)$ of the gamma function.
By convention, equations with derivatives or integrals of the unknown are called differential or integral equations instead.
Functional equations characterize many classical functions, such as the exponential, the logarithm and the power functions, by a few simple rules.

## Explanations

### L1 human

In school algebra the unknown is a number; in a functional equation the unknown is a whole function.
One looks for every function that obeys a rule for all inputs.
For example, the rule $f(x + y) = f(x) f(y)$ says that adding inputs multiplies outputs.
The continuous solutions on the real line are $f = 0$ and $f(x) = a^x$ with $a > 0$, so the rule pins down the exponential functions.
</document>

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# Weak homotopy equivalence

## Definition

Let $X$ and $Y$ be topological spaces, and let $f \colon X \to Y$ be a [[continuous map|0VGFHTM75Y]].
The map $f$ is a *weak homotopy equivalence* if both of the following hold:

1. $f$ induces a bijection $\pi_0(X) \to \pi_0(Y)$ between the sets of path components;
2. for every $x \in X$ and every $n \ge 1$, the induced map $f_* \colon \pi_n(X, x) \to \pi_n(Y, f(x))$ of [[homotopy groups|P976WYEFMB]] is an isomorphism.

## Remarks

Example: every homotopy equivalence $f$ with homotopy inverse $g$ is a weak homotopy equivalence, since $g_*$ inverts $f_*$ up to the change-of-base-point isomorphisms along the homotopies.

Example: all homotopy groups of the [[Warsaw circle|17VG475BRV]] are trivial, so the map from it to a point is a weak homotopy equivalence.
This map is not a homotopy equivalence.

Non-example: the inclusion $S^1 \hookrightarrow D^2$ of the boundary circle into the disk is a bijection on $\pi_0$.
It fails clause 2, because $\pi_1(S^1) \cong \Z$ while $\pi_1(D^2)$ is trivial.

Weak homotopy equivalences induce isomorphisms on singular homology and cohomology.
A weak homotopy equivalence is also called a weak equivalence.
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# Heaviside cover-up method

## Summary

The Heaviside cover-up method is a quick way to find coefficients in a [[partial fraction decomposition|030VFEREKG]].
Let $(s - a)^m$ be a factor of the denominator.
The coefficient of $1/(s - a)^m$ is found by deleting ("covering up") this factor and evaluating the rest of the fraction at $s = a$.
When all roots of the denominator are simple, it gives the whole decomposition, which is why it is a standard step in inverting [[Laplace transforms|0V49JEZW06]].

## Explanations

### L1 human

Take $\frac{3 s + 5}{(s + 1)(s + 2)} = \frac{A}{s + 1} + \frac{B}{s + 2}$.
To find $A$, cover the factor $(s + 1)$ with your finger and put $s = -1$ in what is left: $\frac{3(-1) + 5}{(-1) + 2} = 2$.
To find $B$, cover $(s + 2)$ and put $s = -2$: $\frac{3(-2) + 5}{(-2) + 1} = 1$.
So the fraction is $\frac{2}{s + 1} + \frac{1}{s + 2}$, which one can check by adding the two terms back.
</document>

<document source="https://tamnd.github.io/math2/node/SRRMJFCKVT.md" tag="SRRMJFCKVT" kind="definition">
# String diagram

## Definition

Let $(\mathcal{C}, \otimes, I)$ be a strict [[monoidal category|M3MSQ3NKAE]].
A *string diagram* in $\mathcal{C}$ is a picture in the strip $\R \times [0, 1]$ with the following parts.

1. Finitely many disjoint boxes, each labelled by a morphism $f \colon A_1 \otimes \cdots \otimes A_m \to B_1 \otimes \cdots \otimes B_n$ of $\mathcal{C}$.
2. Finitely many disjoint wires, each an arc on which the height coordinate is strictly increasing, each labelled by an object of $\mathcal{C}$.
3. Each wire starts at the bottom edge of the strip or at the top of a box. It ends at the top edge of the strip or at the bottom of a box.
4. The wires entering a box labelled $f$, read from left to right, carry $A_1, \ldots, A_m$, and the wires leaving it carry $B_1, \ldots, B_n$.

The *value* of the diagram is a morphism from the tensor product of the bottom labels to the tensor product of the top labels.
To compute it, cut the strip by horizontal lines so that each slice contains at most one box.
Each slice is then the tensor product of identities and one box.
The value is the composite of the slices from bottom to top.

## Remarks

This node reads diagrams from bottom to top; reading from top to bottom or from left to right is also common.
The Joyal–Street theorem says that the value does not depend on the slicing, and is unchanged by deformations of the plane that keep the wires increasing.
An empty region stands for the unit object $I$, so a box with no input wires is a morphism $I \to B$.
Example: a box labelled $f \colon A \to B$ with one wire $A$ below and one wire $B$ above has value $f$.
Placing a box $g \colon C \to D$ to its right gives $f \otimes g$, and stacking a box $h \colon B \to E$ on the wire $B$ above $f$ gives $h \circ f$.
Non-example: a wire labelled $A$ entering a box labelled $f \colon B \to C$ with $B \ne A$ breaks clause 4, so the picture has no value.
By Mac Lane's coherence theorem, every monoidal category is equivalent to a strict one, so string diagrams apply to all monoidal categories.
Braided and symmetric monoidal categories allow wires to cross, and categories with duals allow wires to bend.
</document>

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# Braid group

## Definition

Let $n \ge 1$ be an integer.
The *braid group on $n$ strands* is the [[group|G5S1M9]] $B_n$ with the [[presentation|0JGEQXQ4AC]] with generators $\sigma_1, \ldots, \sigma_{n-1}$ and the relations

1. $\sigma_i \sigma_j = \sigma_j \sigma_i$ for $1 \le i, j \le n - 1$ with $\abs{i - j} \ge 2$;
2. $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ for $1 \le i \le n - 2$.

## Remarks

Geometrically, $\sigma_i$ is the braid in which strand $i$ crosses over strand $i + 1$ and the other strands go straight down.
By a theorem of Artin, this presentation gives the group of isotopy classes of geometric braids on $n$ strands.
It is the [[Artin group|SAWBDBHE41]] of the Coxeter matrix of $S_n$.

Example: $B_1$ is trivial and $B_2 = \langle \sigma_1 \mid \ \rangle \cong \Z$.
The group $B_3 = \langle \sigma_1, \sigma_2 \mid \sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2 \rangle$ is not abelian.
The map $\sigma_i \mapsto (i \ i{+}1)$ respects both relations, so it gives a surjective homomorphism $B_n \to S_n$ onto the [[symmetric group|0515A6D2Y1]], and $S_3$ is not abelian.
The kernel of $B_n \to S_n$ is the pure braid group.

Non-example: the symmetric group $S_n$ with the transpositions $(i \ i{+}1)$ satisfies relations 1 and 2, but it is not $B_n$ for $n \ge 2$.
In $S_n$ the generators also satisfy $(i \ i{+}1)^2 = 1$, while in $B_n$ each $\sigma_i$ has infinite order, since the map $B_n \to \Z$ with $\sigma_i \mapsto 1$ respects the relations and sends $\sigma_i^k$ to $k$.
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<document source="https://tamnd.github.io/math2/node/SRW9PD2RSA.md" tag="SRW9PD2RSA" kind="statement">
# Equioscillation theorem

## Statement

Let $a < b$, let $f \colon [a, b] \to \R$ be continuous, and let $n \ge 0$.
Let $\mathcal{P}_n$ be the set of real [[polynomial functions|1G0B73WPX8]] of degree at most $n$, and let $\norm{g} = \max_{a \le x \le b} \abs{g(x)}$.
Then $p \in \mathcal{P}_n$ is a [[best approximation|7N4988HN3P]] of $f$ from $\mathcal{P}_n$ if and only if there are $n + 2$ points $a \le x_0 < x_1 < \cdots < x_{n+1} \le b$ and a sign $\sigma \in \set{-1, 1}$ with
$$f(x_i) - p(x_i) = \sigma (-1)^i \norm{f - p} \quad \text{for } i = 0, 1, \dots, n + 1.$$
Moreover, the best approximation from $\mathcal{P}_n$ exists and is unique.

## Remarks

The idea of the proof: if the error $e = f - p$ alternates at $n + 2$ points with full size $\norm{e}$, the [[de la Vallée Poussin lower bound|XNPVW5Z126]] gives $E_n(f) \ge \norm{e}$, so $p$ is best.
Conversely, suppose $e$ has fewer than $n + 2$ alternation points.
Then one can split $[a, b]$ into at most $n + 1$ pieces on which the extreme values of $e$ have one sign.
A polynomial $q$ of degree at most $n$ with the same sign pattern then makes $\norm{f - p - t q} < \norm{e}$ for small $t > 0$.
Uniqueness follows: if $p_1, p_2$ are best, then $(p_1 + p_2)/2$ is best, and at its $n + 2$ alternation points $p_1 = p_2$, so $p_1 - p_2$ has $n + 2$ zeros.

Example: for $f(x) = x^2$ on $[0, 1]$ and $n = 1$, the line $p(x) = x - 1/8$ is the best approximation.
The error $x^2 - x + 1/8$ takes the values $1/8, -1/8, 1/8$ at $0, 1/2, 1$, which is $n + 2 = 3$ alternating extreme values, and $\norm{f - p} = 1/8$, so the [[error of best approximation|YDN0T4MEX0]] is $E_1(f) = 1/8$.
Another example: the monic polynomial of degree $n + 1$ of least norm on $[-1, 1]$ is $2^{-n} T_{n+1}$, where $T_{n+1}$ is the [[Chebyshev polynomial|M6EAR1JHPP]], because $T_{n+1}$ attains $\pm 1$ alternately at $n + 2$ points.

## Explanations

### L1 human

The best polynomial approximation in the uniform sense is recognized by the shape of its error curve.
The error must reach its largest size $n + 2$ times, with alternating signs, like a wave of constant height.
If the error had fewer such peaks, you could tilt the polynomial a little and lower all the peaks at once.

### L2 human

The theorem, due to Chebyshev, is the main characterization of minimax approximation by polynomials.
It is the basis of the Remez algorithm, which moves a set of $n + 2$ reference points until the error equioscillates.
It holds in the same form when $\mathcal{P}_n$ is replaced by any Haar subspace of dimension $n + 1$ on $[a, b]$.
It also gives the extremal property of Chebyshev polynomials and many sharp constants in approximation theory.
</document>

<document source="https://tamnd.github.io/math2/node/SRY5M3HCKH.md" tag="SRY5M3HCKH" kind="definition">
# Convergence of Fourier series

## Definition

Let $f \colon \R \to \C$ be $2\pi$-periodic and integrable on $[-\pi, \pi]$, and let $S_N f(x) = \sum_{n=-N}^{N} \hat f(n) e^{inx}$ be the symmetric [[partial sums|09RJ859N7A]] of its [[Fourier series|G12KXDHPPS]].

1. The Fourier series of $f$ *converges at* $x \in \R$ if the limit $\lim_{N \to \infty} S_N f(x)$ exists in $\C$.
It *converges to $f$ at $x$* if this limit equals $f(x)$.
2. It *converges pointwise* on a set $E \subseteq \R$ if it converges at each $x \in E$ (see [[pointwise convergence|0GFGK40582]]).
It converges *almost everywhere* if it converges to $f$ at each $x$ outside a set of measure zero.
3. It *converges uniformly* to $f$ on $E$ if $\sup_{x \in E} \abs{S_N f(x) - f(x)} \to 0$ as $N \to \infty$ (see [[uniform convergence|08VZ57PT26]]).
4. For $1 \le p < \infty$ and $f \in L^p(\mathbb{T})$, it *converges in $L^p$* if $\int_{-\pi}^{\pi} \abs{S_N f - f}^p \, dx \to 0$ as $N \to \infty$, in the [[Lebesgue space|SASP14MA1Z]] $L^p(\mathbb{T})$.

## Remarks

The partial sums are taken symmetric in $n$; with this convention the real form $\frac{a_0}{2} + \sum_{n=1}^{N} (a_n \cos nx + b_n \sin nx)$ has the same partial sums.
Example: for $f(x) = \abs{x}$ on $[-\pi, \pi]$, extended periodically, $\abs{\hat f(n)} \le \frac{2}{\pi n^2}$, so the series converges uniformly to $f$ on $\R$.
Non-example: for the square wave equal to $\operatorname{sgn} x$ on $(-\pi, \pi)$, the series converges at every point but not uniformly on $\R$.
Indeed a uniform limit of continuous functions would be continuous.
Convergence of $S_N f(x)$ is a different question from summability, where averages such as the Cesàro means of $S_N f(x)$ are used.
</document>

<document source="https://tamnd.github.io/math2/node/SS1YMS2NPC.md" tag="SS1YMS2NPC" kind="concept">
# Polynomial interpolation

## Summary

Polynomial interpolation finds a polynomial whose graph passes through finitely many given points.
Through $n + 1$ points with distinct first coordinates there is exactly one polynomial of degree at most $n$.
[[Lagrange interpolation|237KR6FRCK]] gives an explicit formula for it.
It is a basic tool to approximate functions, to derive numerical integration rules and to reconstruct polynomials from their values.

## Explanations

### L1 human

Through two points there is one line, and through three points with distinct $x$-values there is one parabola or line.
In general, $n + 1$ points fix a polynomial of degree at most $n$.
For the points $(0, 1)$, $(1, 3)$ and $(2, 7)$ the polynomial is $x^2 + x + 1$.
The [[definition|EKDSDN89FR]] states the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/SS62J715AZ.md" tag="SS62J715AZ" kind="concept">
# Bäcklund transform

## Summary

A Bäcklund transform is a system of first-order partial differential equations that links two functions $u$ and $v$.
The system can be solved for $v$ exactly when $u$ solves one [[partial differential equation|GC5BDZ6QBC]].
Then $v$ solves a second equation.
When the two equations are the same, it is an auto-Bäcklund transform, and it turns known solutions into new ones.
It is one of the main tools of the theory of integrable equations and [[solitons|63D4AXYCT4]].

## Explanations

### L1 human

Think of a machine that takes one solution of an equation and outputs another solution.
The machine is itself a small system of equations of first order, which is easier to solve than the original equation.
The simplest example is the pair of Cauchy–Riemann equations $u_x = v_y$, $u_y = -v_x$.
If $u$ is harmonic, these equations give a harmonic conjugate $v$, so they carry one solution of Laplace's equation to another.
For the [[sine-Gordon equation|802QSF9NM3]], applying the machine to the zero solution gives a kink, and applying it again gives solutions with two kinks.
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<document source="https://tamnd.github.io/math2/node/SS6WMEN6KX.md" tag="SS6WMEN6KX" kind="concept">
# Beth number

## Summary

The beth numbers $\beth_\alpha$ are the infinite [[cardinal numbers|0EBACQFAZW]] obtained from $\aleph_0$ by taking power sets again and again, and suprema at limit stages.
So $\beth_0 = \aleph_0$, and $\beth_1 = 2^{\aleph_0}$ is the cardinality of the real numbers.
The [[precise definition|ZC4BHVHWE9]] is by transfinite recursion.
The generalized continuum hypothesis says exactly that $\beth_\alpha = \aleph_\alpha$ for every ordinal $\alpha$.

## Explanations

### L1 human

Start with the size of the natural numbers.
The set of all subsets of the natural numbers is strictly bigger, by [[Cantor's theorem|0WTEMGYV2C]]; its size is $\beth_1$, the size of the real line.
The set of all subsets of the real line is bigger still; its size is $\beth_2$, the number of all functions from $\R$ to $\R$.
The beth numbers continue this ladder of power sets forever.
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<document source="https://tamnd.github.io/math2/node/SS8T32TMEX.md" tag="SS8T32TMEX" kind="concept">
# Complete set of orthogonal idempotents

## Summary

A complete set of orthogonal idempotents in a [[ring|0B1261B5ED]] is a finite family of [[idempotent elements|0N5NJZKKN3]].
Their pairwise products are zero and their sum is the unit element.
Such a family cuts the ring into pieces.
As a left module over itself, the ring is the direct sum of the left ideals generated by the idempotents.
It is the algebraic form of splitting a space into complementary parts by projections.
When the idempotents are central, the ring itself is a direct product of rings.

## Explanations

### L1 human

Think of projections onto the coordinate axes of $\R^n$.
Each one, applied twice, gives the same result as applied once.
Applying one and then another gives zero, and together they add up to the identity.
In the ring of $2 \times 2$ matrices, the diagonal matrices $E_{11} = \mathrm{diag}(1, 0)$ and $E_{22} = \mathrm{diag}(0, 1)$ form such a family.
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<document source="https://tamnd.github.io/math2/node/SSCJWAQ7AX.md" tag="SSCJWAQ7AX" kind="definition">
# Seeley–DeWitt coefficients

## Definition

Let $(M, g)$ be a closed [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ and let $E$ be a Hermitian [[vector bundle|P4DZM975BV]] over $M$.
Let $L$ be a Laplace-type operator on $E$, that is, a second-order operator with principal symbol $\abs{\xi}^2 \id_E$.
Equivalently, $L = \nabla^* \nabla + F$ for a unique [[connection|5ZYRRVE8BW]] $\nabla$ on $E$ and a unique endomorphism field $F$.
Let $K_L(t, x, y)$ be the [[heat kernel|327NP6S7H5]] of $L$.
There are unique smooth sections $a_k(L)$ of $\End(E)$, $k = 0, 1, 2, \ldots$, such that for every $N$
$$K_L(t, x, x) = (4\pi t)^{-n/2} \left( \sum_{k = 0}^{N} a_k(L)(x) \, t^k + O(t^{N+1}) \right) \quad \text{as } t \to 0^+,$$
uniformly in $x$.
The $a_k(L)$ are the *Seeley–DeWitt coefficients* of $L$.

## Remarks

Existence follows by the same parametrix construction as for the [[Minakshisundaram–Pleijel expansion|45P2SEGGR1]], and uniqueness from the uniqueness of asymptotic expansions.
The first two are $a_0 = \id_E$ and $a_1 = \tfrac16 S \, \id_E - F$, where $S$ is the [[scalar curvature|ZHX2VYQDZ3]].
Some authors call the integrated traces $\int_M \operatorname{tr} a_k(L) \, dv$ the heat invariants, and on manifolds with boundary there are also half-integer powers $t^{k/2}$.

Example: take the scalar Laplacian on a flat torus, so $F = 0$.
Its heat kernel on the diagonal is $(4\pi t)^{-n/2}$ times $1$ plus exponentially small terms, so $a_0 = 1$ and $a_k = 0$ for $k \ge 1$.

Non-example: a Dirac operator $D$ is of first order, so it is not Laplace-type and has no Seeley–DeWitt coefficients in this sense.
Its square $D^2$ is Laplace-type, and the coefficients of $D^2$ are the ones used in index theory.
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<document source="https://tamnd.github.io/math2/node/SSFSDQX8RR.md" tag="SSFSDQX8RR" kind="statement">
# Pitman–Koopman–Darmois theorem

## Statement

Let $\set{f_\theta : \theta \in \Theta}$ be a family of [[probability densities|0ZY9DX8FBF]] on $\R$ with the following properties.

1. There is an open interval $I$, the same for all $\theta$, such that $f_\theta(x) > 0$ for $x \in I$ and $f_\theta(x) = 0$ for $x \notin I$.
2. Each $f_\theta$ is continuously differentiable on $I$.

Let $n \ge 2$ and let $X_1, \dots, X_n$ be [[independent and identically distributed|00MRN0FARV]] with density $f_\theta$.
Assume that there is a [[sufficient statistic|WP4TGXJQJG]] $T = (T_1, \dots, T_k) \colon I^n \to \R^k$ for $\theta$ that is continuously differentiable, with $k < n$.
Then the family is an [[exponential family|EP9VPZ0S6G]] with at most $k$ terms: there are functions $h$, $u_1, \dots, u_k$, $\eta_1, \dots, \eta_k$ and $A$ such that
$$f_\theta(x) = h(x) \exp\Big( \sum_{i=1}^{k} \eta_i(\theta) u_i(x) - A(\theta) \Big), \qquad x \in I.$$

## Remarks

The converse is easy.
For an exponential family, the sums $\sum_{j=1}^{n} u_i(X_j)$, $i = 1, \dots, k$, form a sufficient statistic of dimension $k$ for every sample size $n$, by the factorization criterion.
So, among families with a fixed support, exponential families are exactly those with a sufficient statistic whose dimension does not grow with $n$.

The idea of the proof is to write the factorization of the joint density and take logarithms.
Differentiating in the $x_j$ shows that $\partial_x \log f_\theta(x)$ lies, as a function of $\theta$, in a space of dimension at most $k$.
Integrating in $x$ gives the exponential form.

The fixed support is necessary.
For the uniform densities on $(0, \theta)$, the maximum $\max_j X_j$ is sufficient and one-dimensional for every $n$.
Yet the support moves with $\theta$, and this family is not an exponential family.

## Explanations

### L1 human

Sometimes a whole sample can be squeezed into a few numbers without losing information about the parameter, for any sample size.
The theorem says that, when the range of possible values does not depend on the parameter, this only happens for exponential families.
So exponential families are the natural home of data compression in statistics.

### L2 human

The theorem explains why exponential families appear everywhere in classical parametric statistics.
Sufficient statistics of fixed dimension give conjugate priors, complete statistics and explicit minimum-variance unbiased estimators.
The result is named after Pitman, Koopman and Darmois.
The precise regularity conditions differ from one textbook to another, but some smoothness of the sufficient statistic is always needed to exclude pathological maps from $\R^n$ to $\R^k$.
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<document source="https://tamnd.github.io/math2/node/SSFYQK86PK.md" tag="SSFYQK86PK" kind="definition">
# Inner regular measure

## Definition

Let $X$ be a topological space and let $\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.
A Borel set $A \subseteq X$ is *inner regular* for $\mu$ if
$$\mu(A) = \sup \set{\mu(K) : K \subseteq A, \ K \text{ compact and Borel}}.$$
The measure $\mu$ is *inner regular* if every Borel set is inner regular for $\mu$.

## Remarks

In a Hausdorff space every [[compact set|0R6EWFKRCF]] is closed, hence Borel, so the condition "Borel" on $K$ is automatic.

Example: counting measure on $\R$ is inner regular.
If $A$ is finite, take $K = A$; if $A$ is infinite, its finite subsets are compact and have arbitrarily large counting measure, so the supremum is $\infty = \mu(A)$.

Non-example: let $X$ be an uncountable set with the cocountable topology, whose open sets are $\emptyset$ and the complements of countable sets.
Its Borel sets are the countable sets and their complements; let $\mu$ be $0$ on countable sets and $1$ on the others.
Compact subsets of $X$ are finite, so they have measure $0$, while $\mu(X) = 1$; thus $\mu$ is not inner regular.

Variants: some authors approximate by closed sets instead of compact sets, and some ask inner regularity only for open sets.
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<document source="https://tamnd.github.io/math2/node/SSNFTHZ6YJ.md" tag="SSNFTHZ6YJ" kind="concept">
# Hausdorff dimension

## Summary

Let $A$ be a subset of a [[metric space|0VKD1TP7F3]].
Its Hausdorff dimension is the exponent $s$ at which the $s$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] of $A$ jumps from infinity to zero.
It is a real number, not always an integer.
It agrees with the usual dimension for smooth curves, surfaces and open sets, and it gives fractals such as the [[Cantor set|0BXNDZFQME]] non-integer dimensions.

## Explanations

### L1 human

A curve has length but no area, and a square has area but infinite length.
The Hausdorff dimension finds the exact exponent where a set switches from infinite size to zero size.
For a segment this happens at $1$, and for a filled square at $2$.
The middle-thirds Cantor set switches at $\log 2 / \log 3 \approx 0.63$: it is too thin to have length but too big to be counted as a few points.
</document>

<document source="https://tamnd.github.io/math2/node/SSP0CC967X.md" tag="SSP0CC967X" kind="definition">
# Monomorphism

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $f \colon A \to B$ be a [[morphism|859F0TFVA3]] of $\mathcal{C}$.
The morphism $f$ is a *monomorphism* if, for every object $X$ and all morphisms $g_1, g_2 \colon X \to A$, the equality $f \circ g_1 = f \circ g_2$ implies $g_1 = g_2$.

## Remarks

Equivalently, for every object $X$ the map $\Hom_{\mathcal{C}}(X, A) \to \Hom_{\mathcal{C}}(X, B)$, $g \mapsto f \circ g$, is injective.
One writes $f \colon A \rightarrowtail B$, or says that $f$ is *monic*.
Example: in the category of sets, every [[injective function|09NWETVTNY]] $f$ is a monomorphism.
If $f \circ g_1 = f \circ g_2$, then $f(g_1(x)) = f(g_2(x))$ for each $x \in X$, so $g_1(x) = g_2(x)$ by injectivity.
Non-example: the map $f \colon \set{1, 2} \to \set{0}$ is not a monomorphism.
The two maps $g_1, g_2 \colon \set{0} \to \set{1, 2}$ with $g_1(0) = 1$ and $g_2(0) = 2$ satisfy $f \circ g_1 = f \circ g_2$ but $g_1 \ne g_2$.
Every isomorphism is a monomorphism, and a composite of monomorphisms is a monomorphism.
If $g \circ f$ is a monomorphism, then $f$ is a monomorphism.
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<document source="https://tamnd.github.io/math2/node/SSWFTA4GP8.md" tag="SSWFTA4GP8" kind="statement">
# Whitney's theorem on folds and cusps

## Statement

Let $M$ be a smooth surface, and give $C^\infty(M, \R^2)$ the [[Whitney $C^\infty$ topology|7PN9MD1PQB]].
The smooth maps $f \colon M \to \R^2$ with the following property form a dense subset of $C^\infty(M, \R^2)$.
At every point $p \in M$, there are local coordinates centred at $p$ and at $f(p)$ in which $f$ has one of these forms:

1. $(x, y) \mapsto (x, y)$, a regular point;
2. $(x, y) \mapsto (x, y^2)$, a fold point;
3. $(x, y) \mapsto (x, y^3 + x y)$, a cusp point.

## Remarks

At a fold point the [[singular set|1JG3HFWX3T]] is a smooth curve and the kernel of the differential is transverse to it.
At a cusp point the kernel is tangent to the singular curve.
For the cusp map $f(x, y) = (x, y^3 + xy)$ the singular set is the parabola $x = -3y^2$, and its image is the cusp curve $(-3y^2, -2y^3)$, which satisfies $27 v^2 = -4 u^3$ in the target coordinates $(u, v)$.
The proof uses transversality of the jet of $f$ to the subsets of the jet space defined by these singularities.
Worse singularities have codimension greater than $2$, so generic maps of a surface avoid them.
Both the fold and the cusp are stable germs, so they cannot be removed by small perturbations.
The theorem is due to Whitney and was a starting point of the theory of [[stable maps|1991WETZF9]].

## Explanations

### L1 human

Project a smooth surface onto a plane, as when you look at a curved object and see its outline.
Whitney's theorem says that, after a tiny wiggle of the surface, only two kinds of special points appear.
Along fold curves the surface turns over like the edge of a folded sheet of paper.
At isolated cusp points a fold curve stops, and its outline forms a sharp spike.
Nothing more complicated is typical.

### L2 human

This is the first nontrivial case of the classification of generic singularities of maps between manifolds of equal dimension.
In the language of [[Thom–Boardman singularities|A9TYDPD5D7]], folds are the $\Sigma^{1,0}$ points and cusps are the $\Sigma^{1,1,0}$ points.
Higher-dimensional analogues are the [[Morin singularities|DF8CJ1A9MF]], and the fold and cusp reappear in [[catastrophe theory|6FKJ6SXP3G]] as the fold and cusp catastrophes.
</document>

<document source="https://tamnd.github.io/math2/node/SSYN7X11MB.md" tag="SSYN7X11MB" kind="concept">
# Initial value problem for a recurrence

## Summary

An initial value problem for a recurrence asks for a sequence that satisfies a given recurrence and has prescribed first terms.
For a recurrence of order $k$ in explicit form, prescribing $k$ initial values gives exactly one solution.
It is the discrete counterpart of the [[initial value problem|0FXDW992DS]] for ordinary differential equations.

## Explanations

### L1 human

A recurrence is a rule that builds each term of a sequence from the terms before it.
The rule alone does not tell you where to start.
An initial value problem fixes the starting terms, and then the rule produces everything else.
For example, $u_{n+1} = 2u_n$ with $u_0 = 3$ gives $3, 6, 12, 24, \dots$, that is $u_n = 3 \cdot 2^n$.
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<document source="https://tamnd.github.io/math2/node/SSYPTQC4C1.md" tag="SSYPTQC4C1" kind="definition">
# Elliptic boundary value problem

## Definition

Let $\Omega \subseteq \R^n$ be a bounded open set with smooth boundary, with $n \ge 2$, and let $m \ge 1$.
Let $A$ be a linear differential operator of order $2m$ with smooth coefficients on $\overline{\Omega}$.
Let $B_1, \ldots, B_m$ be linear differential operators with smooth coefficients on $\partial \Omega$, of orders $m_1, \ldots, m_m$ less than $2m$.
Consider the [[boundary value problem|TM8VQJWGHZ]]
$$A u = f \ \text{in } \Omega, \qquad B_j u = g_j \ \text{on } \partial \Omega \quad (j = 1, \ldots, m).$$
It is *elliptic* if:

1. $A$ is properly [[elliptic|XY8GEQSWPR]] on $\overline{\Omega}$;
2. $B_1, \ldots, B_m$ satisfy the [[Lopatinskii–Shapiro condition|48MEJEMEXM]] with respect to $A$ at every point of $\partial \Omega$.

## Remarks

Example: the Dirichlet problem $-\Delta u = f$ in $\Omega$, $u = g$ on $\partial \Omega$ has $m = 1$, the Laplacian is properly elliptic, and the Dirichlet condition satisfies the Lopatinskii–Shapiro condition.
Non-example: $-\Delta u = f$ with the two conditions $u = 0$ and $\partial_\nu u = 0$ on $\partial \Omega$ is not elliptic, since an operator of order $2$ takes exactly one boundary condition.
Agmon, Douglis and Nirenberg proved an a priori estimate for elliptic problems.
For each integer $s \ge 0$ there is $C > 0$ with
$$\norm{u}_{H^{2m+s}(\Omega)} \le C \Big( \norm{Au}_{H^s(\Omega)} + \sum_{j=1}^m \norm{B_j u}_{H^{2m+s-m_j-1/2}(\partial \Omega)} + \norm{u}_{L^2(\Omega)} \Big),$$
where $H^k$ denotes Sobolev spaces.
For a properly elliptic operator of order $2$, the Dirichlet condition always satisfies the Lopatinskii–Shapiro condition.
Indeed, the bounded solutions of the ordinary differential equation form a line spanned by a function that does not vanish at $0$.
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<document source="https://tamnd.github.io/math2/node/ST71ZV3FES.md" tag="ST71ZV3FES" kind="concept">
# Betti number

## Summary

The $k$-th Betti number $b_k(X)$ of a topological space $X$ is the rank of its $k$-th integral [[singular homology|M0Q5CFJ7ZH]] group.
Informally, $b_0$ counts the path components, $b_1$ counts independent loops that do not bound, and $b_2$ counts enclosed cavities.
Betti numbers are homotopy invariants, and their alternating sum is the [[Euler characteristic|RNDDWDAFJ5]].

## Explanations

### L1 human

Betti numbers count holes of each dimension.
A circle has one piece and one loop around a hole, so $b_0 = 1$ and $b_1 = 1$.
The surface of a doughnut has one piece, two independent loops (around the tube and around the central hole) and one enclosed cavity, so its Betti numbers are $1, 2, 1$.
A solid ball has $b_0 = 1$ and no holes at all, so its other Betti numbers are zero.
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<document source="https://tamnd.github.io/math2/node/ST8B3EV6YA.md" tag="ST8B3EV6YA" kind="definition">
# Tight frame

## Definition

Let $H$ be a [[Hilbert space|HM2HHE3J51]] with [[inner product|0NB0AWGJBW]] $\ip{\cdot}{\cdot}$, and let $(f_i)_{i \in I}$ be a family of vectors of $H$.
The family is a *frame* if there are constants $0 < A \le B < \infty$ with
$$A \norm{f}^2 \le \sum_{i \in I} \abs{\ip{f}{f_i}}^2 \le B \norm{f}^2 \quad \text{for all } f \in H.$$
The numbers $A$ and $B$ are *frame bounds*.
The frame is *tight* if one can take $A = B$, that is, if $\sum_{i \in I} \abs{\ip{f}{f_i}}^2 = A \norm{f}^2$ for all $f \in H$.
A tight frame with $A = B = 1$ is a *Parseval frame*.

## Remarks

For a tight frame with bound $A$, every $f \in H$ satisfies $f = A^{-1} \sum_{i \in I} \ip{f}{f_i} f_i$, with convergence in $H$.
Every orthonormal basis is a Parseval frame, and the union of two orthonormal bases is a tight frame with $A = 2$.

Example: in $\R^2$, let $u_1 = (0, 1)$, $u_2 = (-\tfrac{\sqrt3}{2}, -\tfrac12)$, $u_3 = (\tfrac{\sqrt3}{2}, -\tfrac12)$.
For $f = (x, y)$, $\sum_k \ip{f}{u_k}^2 = y^2 + \tfrac14(\sqrt3 x + y)^2 + \tfrac14(\sqrt3 x - y)^2 = \tfrac32 (x^2 + y^2)$.
So $(u_1, u_2, u_3)$ is a tight frame with bound $\tfrac32$.

Non-example: in $\R^2$, the family $(e_1, e_1, e_2)$ gives $\sum \ip{f}{f_i}^2 = 2x^2 + y^2$.
It is a frame with optimal bounds $A = 1$ and $B = 2$, but it is not tight.
</document>

<document source="https://tamnd.github.io/math2/node/STHTW16717.md" tag="STHTW16717" kind="concept">
# Poisson regression

## Summary

Poisson regression models a count response as [[Poisson distributed|0XB9GQTN9G]] with a mean whose logarithm is linear in the regressors.
It is the [[generalized linear model|K6C48KMRKC]] for Poisson responses with the canonical log link, and it is fitted by [[maximum likelihood|0FREPFKNS7]].
Each coefficient acts multiplicatively on the expected count, and an offset handles different exposure times or population sizes.
It is the basic model for counts of events and for log-linear models of [[contingency tables|PANSMHE0SW]].

## Explanations

### L1 human

Counts such as the number of accidents at a crossing per year cannot be negative, and their spread grows with their size.
Poisson regression says that the expected count is $e^{x^T \beta}$, which is always positive.
A coefficient of $0.1$ for a regressor means that one more unit of it multiplies the expected count by $e^{0.1} \approx 1.105$, an increase of about $10.5\%$.
A Poisson variable has variance equal to its mean, and when real counts vary more than that, analysts switch to a negative binomial or a quasi-Poisson model.
</document>

<document source="https://tamnd.github.io/math2/node/STJAXZG1K1.md" tag="STJAXZG1K1" kind="definition">
# Cauchy's power equation

## Definition

*Cauchy's power equation* is the [[functional equation|SRAN4EKWS0]]
$$f(xy) = f(x)\, f(y) \quad \text{for all } x, y > 0,$$
for an unknown [[function|0EPAT51DXQ]] $f \colon (0, \infty) \to \R$.
A solution is a function $f$ that satisfies this identity for all positive $x$ and $y$.

## Remarks

Example: $f(x) = \sqrt{x}$ is a solution, since $\sqrt{xy} = \sqrt{x} \sqrt{y}$.
Non-example: $f(x) = 2x$ is not a solution, since $f(1 \cdot 1) = 2$ while $f(1) f(1) = 4$.
The substitution $x = e^s$, $y = e^t$ turns the equation into [[Cauchy's exponential equation|CEGF8W64VQ]] for $g(s) = f(e^s)$.
So the continuous solutions are $f = 0$ and $f(x) = x^c$ with $c \in \R$.
</document>

<document source="https://tamnd.github.io/math2/node/STMT1K1865.md" tag="STMT1K1865" kind="statement">
# Wiener's criterion

## Statement

Let $n \ge 3$, let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]], and let $\xi \in \partial\Omega$ be a [[boundary|05D5EGMFYV]] point.
For each integer $k \ge 1$ let
$$E_k = \set{x \in \R^n \setminus \Omega : 2^{-k-1} \le \abs{x - \xi} \le 2^{-k}},$$
and let $\operatorname{cap}(E_k)$ be its Newtonian [[capacity|4YZVFGJVSZ]].
Then $\xi$ is a [[regular boundary point|B7948M365K]] of $\Omega$ if and only if
$$\sum_{k=1}^{\infty} 2^{k(n-2)} \operatorname{cap}(E_k) = \infty.$$

## Remarks

The factor $2^{k(n-2)}$ normalizes the capacity of $E_k$ by the capacity of a ball of radius $2^{-k}$, which is a constant times $2^{-k(n-2)}$.
So the criterion says that the complement of $\Omega$ is not too thin near $\xi$, measured shell by shell.
Any ratio $\lambda \in (0, 1)$ in place of $\frac12$ gives an equivalent criterion, and constant factors in the normalization of capacity do not matter.
Example: suppose the complement of $\Omega$ contains a cone with vertex $\xi$.
Then each $E_k$ contains a ball of radius comparable to $2^{-k}$, so every term of the series is bounded below and $\xi$ is regular.
Example: if $\xi$ is an isolated point of $\R^n \setminus \Omega$, then $E_k$ is empty for large $k$, the series converges, and $\xi$ is irregular.
In the plane there is an analogous criterion with logarithmic capacity.

## Explanations

### L1 human

Whether a boundary point is good for the Dirichlet problem depends on how much of the outside of the set lies near that point.
Wiener's criterion measures this outside part in rings around the point, using capacity, the electrical size of a set.
If the rescaled sizes of the rings add up to infinity, the point is regular; otherwise it is irregular.

### L2 human

Wiener's criterion is the complete answer to the question of regularity for the Laplacian.
It explains the [[exterior cone condition|B51J3BC610]], the irregularity of the [[Lebesgue spine|EFFC7688KR]], and the irregularity of single points for $n \ge 3$.
The logarithmic version gives the same in the plane.
It also shows that regularity depends only on the complement of $\Omega$ near $\xi$, and it extends to a large class of elliptic equations.
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<document source="https://tamnd.github.io/math2/node/STPBXBKSZP.md" tag="STPBXBKSZP" kind="statement">
# Boolean prime ideal theorem

## Statement

Let $B$ be a [[Boolean algebra|06GKBTAA5E]] with operations $\wedge$, $\vee$, $\neg$, least element $0$ and greatest element $1$.
An ideal of $B$ is a set $I \subseteq B$ such that $0 \in I$, $a \vee b \in I$ for all $a, b \in I$, and $a \in I$ whenever $a \le b$ and $b \in I$.
An ideal $I$ is prime if $1 \notin I$ and, for each $a \in B$, either $a \in I$ or $\neg a \in I$.
Then each ideal $I$ of $B$ with $1 \notin I$ is contained in a prime ideal of $B$.

## Remarks

In ZFC the proof applies [[Zorn's lemma|04RXTQA5ZA]] to the ideals that contain $I$ and omit $1$.
A maximal one $P$ is prime: if $a \notin P$ and $\neg a \notin P$, then the ideals generated by $P \cup \set{a}$ and by $P \cup \set{\neg a}$ both contain $1$, and this forces $1 \in P$.
The theorem is usually abbreviated BPI.
By duality, BPI is equivalent to the ultrafilter lemma: each filter on a set extends to an [[ultrafilter|SR56KCFYYS]].
Over ZF, BPI follows from the [[axiom of choice|0HTGB13FE2]] but does not imply it.
Over ZF, BPI is equivalent to the compactness theorem of first-order logic and to Stone's representation theorem for Boolean algebras.
Example: in the power set Boolean algebra of a set $X$, with $x \in X$, the family of subsets of $X$ that do not contain $x$ is a prime ideal.

## Explanations

### L1 human

A Boolean algebra is an abstract version of the algebra of subsets of a set, with "and", "or" and "not".
A prime ideal is a way to split the algebra into "small" elements and "large" elements, so that each element or its complement is small.
The theorem says that such a split always exists, extending any consistent choice of small elements.
It needs a weak form of the axiom of choice.

### L2 human

BPI is the choice principle behind many compactness arguments.
It gives the ultrafilter lemma, the compactness theorem of first-order logic, and the representation of every Boolean algebra as an algebra of clopen sets of a Stone space.
It also gives [[Tychonoff's theorem|0JTRSJKCFB]] for products of compact Hausdorff spaces, though not for arbitrary compact spaces, which needs full choice.
So it sits strictly between ZF and ZFC, and it is a common test of how much choice a theorem really needs.
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<document source="https://tamnd.github.io/math2/node/STSV9XY3R5.md" tag="STSV9XY3R5" kind="definition">
# Krull dimension of a poset

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
A finite [[chain|0S8FE7ZMFM]] $x_0 < x_1 < \cdots < x_n$ in $P$ has length $n$.
The *Krull dimension* $\dim P$ of a nonempty $P$ is the supremum of the lengths of the finite chains in $P$, an element of $\N \cup \set{\infty}$.
The empty poset has Krull dimension $-\infty$ by convention; some authors use $-1$ instead.

## Remarks

Example: the power set of $\set{1, \ldots, n}$ ordered by inclusion has Krull dimension $n$.
The chain $\varnothing \subset \set{1} \subset \cdots \subset \set{1, \ldots, n}$ has length $n$, and a longer chain would need two subsets of the same size.
Example: $\N$ with its usual order has Krull dimension $\infty$.
The dimension can be infinite even when every chain is finite: the disjoint union of chains of lengths $1, 2, 3, \ldots$ has no infinite chain but has dimension $\infty$.
For a commutative ring $R$, the [[Krull dimension of the ring|MVGBS3J1P7]] is the Krull dimension of its set of prime ideals ordered by inclusion.
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<document source="https://tamnd.github.io/math2/node/STW9S1Q66W.md" tag="STW9S1Q66W" kind="definition">
# Inductive dimension

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]], and write $\partial V$ for the [[boundary|05D5EGMFYV]] of a set $V \subseteq X$.
The *small inductive dimension* $\operatorname{ind} X$ is defined by induction on $n \ge -1$:

1. $\operatorname{ind} X = -1$ if and only if $X = \emptyset$.
2. For $n \ge 0$, $\operatorname{ind} X \le n$ if for every point $x \in X$ and every [[open set|02BH3MCBQB]] $U$ with $x \in U$ there is an open set $V$ with $x \in V \subseteq U$ and $\operatorname{ind} \partial V \le n - 1$.
3. $\operatorname{ind} X = n$ if $\operatorname{ind} X \le n$ and not $\operatorname{ind} X \le n - 1$; and $\operatorname{ind} X = \infty$ if $\operatorname{ind} X \le n$ holds for no $n$.

The *large inductive dimension* $\operatorname{Ind} X$ is defined in the same way, with a new clause 2.
For $n \ge 0$, $\operatorname{Ind} X \le n$ if for every [[closed set|0ZCK8DQYN1]] $A$ and every open set $U \supseteq A$ there is an open set $V$ with $A \subseteq V \subseteq U$ and $\operatorname{Ind} \partial V \le n - 1$.
Here the boundary $\partial V$ carries the subspace topology.

## Remarks

The rational numbers $\Q$ have $\operatorname{ind} \Q = 0$.
Given $x \in U$, choose irrational numbers $a < x < b$ with $(a, b) \cap \Q \subseteq U$; then $V = (a, b) \cap \Q$ is open and closed in $\Q$, so $\partial V = \emptyset$.
The real line has $\operatorname{ind} \R = 1$.
Small open intervals have a two-point boundary, which has dimension $0$, so $\operatorname{ind} \R \le 1$.
But $\operatorname{ind} \R \le 0$ fails: a nonempty open set $V \neq \R$ has nonempty boundary, because $\R$ is connected.
For separable metric spaces, $\operatorname{ind} X = \operatorname{Ind} X = \dim X$, where $\dim$ is the [[Lebesgue covering dimension|A3W7RGXXK9]].

## Explanations

### L1 human

A space has inductive dimension at most $n$ when small pieces of it can be cut out by walls of dimension at most $n - 1$.
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<document source="https://tamnd.github.io/math2/node/STYAB4PFDW.md" tag="STYAB4PFDW" kind="concept">
# Calabi–Yau manifold

## Summary

A Calabi–Yau manifold is a compact [[Kähler manifold|X60JG9AEQJ]] with trivial canonical bundle.
By Yau's proof of the [[Calabi conjecture|WJKA5YBTJ7]], each Kähler class on it contains a Ricci-flat Kähler metric.
Calabi–Yau manifolds sit between Fano varieties and varieties of general type in the classification of varieties.
They are central in string theory and in [[mirror symmetry|YK3DMN2NSK]].

## Explanations

### L1 human

The canonical bundle measures how a complex manifold curves on average, and on a Calabi–Yau manifold it is trivial, so the manifold is flat on average.
In complex dimension $1$ the Calabi–Yau manifolds are the elliptic curves, which are tori.
In dimension $2$ the simply connected ones are the K3 surfaces.
A smooth quintic hypersurface in $\mathbb{P}^4$ is a Calabi–Yau threefold, and it is the most studied example in physics.
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<document source="https://tamnd.github.io/math2/node/SV0816DJP9.md" tag="SV0816DJP9" kind="statement">
# Moise's theorem

## Statement

Let $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $3$.

1. $M$ has a [[triangulation|YH170WBQ6M]].
2. Any two triangulations $h_1 \colon \abs{K_1} \to M$ and $h_2 \colon \abs{K_2} \to M$ are related by a piecewise linear homeomorphism. That is, $K_1$ and $K_2$ have subdivisions that are isomorphic as simplicial complexes.

## Remarks

Equivalently, every topological $3$-manifold has a [[piecewise linear structure|15MKGRNM9K]], unique up to piecewise linear homeomorphism.
Combined with results comparing piecewise linear and smooth structures in low dimensions, it follows that every topological $3$-manifold has a smooth structure, unique up to diffeomorphism.
Edwin Moise proved the theorem in 1952; R. H. Bing later gave another proof.
A key step approximates a homeomorphism of $3$-manifolds by piecewise linear ones, which needs tameness results for surfaces in $3$-manifolds.
The statement fails in dimension $4$, where the $E_8$ manifold has no triangulation, and part 2 fails for manifolds in higher dimensions.

## Explanations

### L1 human

Every $3$-dimensional manifold can be cut into tetrahedra glued face to face, and in essentially only one way.
So in dimension three, the topological, the piecewise linear and the smooth points of view all describe the same manifolds.

### L2 human

The theorem lets $3$-manifold topology use triangulations, normal surfaces and smooth tools freely, without losing generality.
It is the dimension-three case of the Hauptvermutung and of the triangulation conjecture, both of which hold in dimensions up to three and fail in general.
</document>

<document source="https://tamnd.github.io/math2/node/SV4CK2XP8M.md" tag="SV4CK2XP8M" kind="concept">
# Surface of general type

## Summary

A surface of general type is an [[algebraic surface|KFTECAATKG]] of [[Kodaira dimension|ZW00Q0SAGR]] $2$, the largest possible value.
Its pluricanonical systems $\abs{nK}$ grow as fast as possible, quadratically in $n$.
Most surfaces are of this type, for example all smooth surfaces of degree at least $5$ in $\mathbb{P}^3$.
Unlike the other classes in the Enriques–Kodaira classification, they are not classified in detail.
Their study centres on inequalities between invariants and on their moduli spaces.

## Explanations

### L1 human

Surfaces of general type are the two-dimensional analogue of curves of genus at least two.
They are, in a loose sense, negatively curved and rigid, with many differential forms.
A surface given by a random polynomial equation of high degree in three-space is of general type.
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<document source="https://tamnd.github.io/math2/node/SV5EEBF861.md" tag="SV5EEBF861" kind="definition">
# Cusp form

## Definition

Let $k$ be an integer and let $f$ be a [[modular form|GWKN5VG78A]] of weight $k$ for the [[modular group|VEKV1EM77B]] $\mathrm{SL}_2(\Z)$, with $q$-expansion $f(\tau) = \sum_{n \ge 0} a_n q^n$, where $q = e^{2 \pi i \tau}$.
Then $f$ is a *cusp form* if $a_0 = 0$.
Equivalently, $f(\tau) \to 0$ as $\operatorname{Im} \tau \to \infty$.

## Remarks

For a [[congruence subgroup|1H6R856P3C]] $\Gamma$, a modular form $f$ for $\Gamma$ of weight $k$ is a cusp form if $(c\tau + d)^{-k} f(\gamma \tau) \to 0$ as $\operatorname{Im} \tau \to \infty$, for every $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\Z)$.
Example: $\Delta = (E_4^3 - E_6^2) / 1728$ is a cusp form of weight $12$; the constant terms of $E_4^3$ and $E_6^2$ are both $1$, so they cancel.
Non-example: the [[Eisenstein series|EYREVY31BN]] $E_4 = 1 + 240 q + \cdots$ is a modular form but not a cusp form, since its constant term is $1$.
For $\mathrm{SL}_2(\Z)$, $S_k = 0$ for $k < 12$ and $S_{12} = \C \Delta$.
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<document source="https://tamnd.github.io/math2/node/SV6A17MEYZ.md" tag="SV6A17MEYZ" kind="definition">
# Lelong class

## Definition

Let $n \ge 1$ and let $\abs{z}$ be the Euclidean norm on $\C^n$.
The *Lelong class* $\mathcal{L} = \mathcal{L}(\C^n)$ is the set of [[plurisubharmonic functions|FF2P7ER3QT]] $u$ on $\C^n$ for which there is a constant $C_u \in \R$ with
$$u(z) \le \log(1 + \abs{z}) + C_u \quad \text{for all } z \in \C^n .$$
The subclass $\mathcal{L}^+$ consists of the $u \in \mathcal{L}$ for which there is also a constant $c_u \in \R$ with $u(z) \ge \log(1 + \abs{z}) + c_u$ for all $z \in \C^n$.

## Remarks

Example: let $p$ be a polynomial on $\C^n$ of degree $d \ge 1$ and $u = \frac{1}{d} \log\abs{p}$.
Then $u$ is plurisubharmonic, and $\abs{p(z)} \le M (1 + \abs{z})^d$ for some $M > 0$, so $u(z) \le \log(1 + \abs{z}) + \frac{1}{d} \log M$; hence $u \in \mathcal{L}$.
It is not in $\mathcal{L}^+$ when $p$ has a zero, since $u = -\infty$ there.
The function $\log(1 + \abs{z}^2)/2$ is in $\mathcal{L}^+$.
Non-example: $u(z) = \abs{z_1}$ is plurisubharmonic, but $\abs{z_1} - \log(1 + \abs{z})$ is unbounded above, so $u \notin \mathcal{L}$.
For $n = 1$, the logarithmic potentials $\int \log\abs{z - w} \, d\mu(w)$ for compactly supported probability measures $\mu$ are in $\mathcal{L}$.
Some authors write $\log^+\abs{z} = \max(\log\abs{z}, 0)$ in place of $\log(1 + \abs{z})$; the class is the same, since the two functions differ by a bounded amount.
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<document source="https://tamnd.github.io/math2/node/SV96KN4WR9.md" tag="SV96KN4WR9" kind="concept">
# Non-atomic measure

## Summary

An atom of a measure is a measurable set of positive measure that cannot be split into two parts of positive measure.
A measure is non-atomic, or atomless, if it has no atoms.
Non-atomic measures can be divided finely: by Sierpiński's theorem, a set of finite measure $m$ has subsets of every measure between $0$ and $m$.

## Explanations

### L1 human

An atom is a lump of mass that cannot be split.
A point mass, such as the Dirac measure at a point, is one big atom.
Length on the real line has no atoms: any set of positive length can be cut into two pieces of positive length.
A non-atomic measure is spread out like sand, not concentrated in lumps.
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<document source="https://tamnd.github.io/math2/node/SVABG134WE.md" tag="SVABG134WE" kind="concept">
# Trigonometric interpolation

## Summary

Trigonometric interpolation fits a [[trigonometric polynomial|J3NJQ5FZJ5]] through prescribed values at given points of a period.
With $2n + 1$ distinct nodes in one period there is exactly one interpolating trigonometric polynomial of degree at most $n$.
It is the periodic counterpart of [[Lagrange interpolation|237KR6FRCK]] by ordinary polynomials.
For equally spaced nodes its coefficients are given by the [[discrete Fourier transform|TST0GH66A5]].
This makes it fast to compute and central in numerical analysis and signal processing.

## Explanations

### L1 human

Suppose you measure a periodic quantity, like temperature over a day, at a few moments.
Trigonometric interpolation finds a sum of a few waves that passes exactly through all the measurements.
For example, through the values $1, 0, 0$ at the angles $0, 2\pi/3, 4\pi/3$ passes the curve $\frac{1}{3} + \frac{2}{3} \cos x$.
With equally spaced measurements, the waves are found by the discrete Fourier transform.
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# Isomonodromic deformation

## Definition

Let $m \ge 1$ and let $T \subseteq \C^m$ be a simply connected domain of points $t = (t_1, \ldots, t_m)$ with pairwise distinct coordinates.
For each $t \in T$, consider the [[Fuchsian|ZBVH2YE50B]] [[linear system|037ZNY7NAG]]
$$\frac{dY}{dz} = \sum_{i=1}^{m} \frac{A_i(t)}{z - t_i}\, Y,$$
where $A_1, \ldots, A_m \colon T \to \C^{n \times n}$ are holomorphic.
Fix a base point $z_0$ with $z_0 \ne t_i$ for all $t \in T$ and all $i$.
The family is an *isomonodromic deformation* if it has a [[fundamental matrix|0H1GN6P9R5]] $Y(z, t)$ with the following property.
$Y$ is holomorphic in $(z, t)$ for $z$ near $z_0$ and $t \in T$.
Its [[monodromy|C8HD62AB6Q]] matrices along loops around the points $t_i$, moved continuously with $t$, do not depend on $t$.

## Remarks

The family is isomonodromic if the matrices $A_i$ satisfy the Schlesinger equations
$$\frac{\partial A_i}{\partial t_j} = \frac{[A_i, A_j]}{t_i - t_j} \ (j \ne i), \qquad \frac{\partial A_i}{\partial t_i} = - \sum_{j \ne i} \frac{[A_i, A_j]}{t_i - t_j},$$
where $[A, B] = A B - B A$.
For $2 \times 2$ systems with singular points $0$, $1$, $t$ and $\infty$, these equations reduce to the sixth Painlevé equation in the variable $t$.

Example: if all $A_i$ are constant and commute with each other, the Schlesinger equations hold, since all commutators vanish.
For $n = 1$, the system is $y' = \sum_i \frac{a_i}{z - t_i} y$ with solution $\prod_i (z - t_i)^{a_i}$, and the monodromy around $t_i$ is $e^{2 \pi i a_i}$; it does not depend on $t$ when the $a_i$ are constant.

Non-example: for $n = 1$ and $m = 1$, take $a_1(t) = t_1$ on a domain $T \subseteq \C$; the monodromy $e^{2 \pi i t_1}$ around $t_1$ changes with $t_1$, so the family is not isomonodromic.
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# Homogeneous and inhomogeneous linear recurrences

## Definition

Let $K$ be a field and let $d \ge 1$ be an integer.
Let $a_0, \dots, a_{d-1} \colon \N \to K$ and $g \colon \N \to K$ be [[sequences|0G9EKM0E15]].
A linear recurrence of order $d$ is the [[recurrence relation|07TZV87Y0T]]
$$u_{n+d} + a_{d-1}(n)\, u_{n+d-1} + \cdots + a_0(n)\, u_n = g(n) \qquad (n \in \N)$$
for a sequence $u \colon \N \to K$.
The sequence $g$ is its right-hand side.

1. The recurrence is *homogeneous* if $g(n) = 0$ for all $n$.
2. The recurrence is *inhomogeneous* if $g(n) \ne 0$ for some $n$.
3. The *associated homogeneous recurrence* is the one with the same $a_0, \dots, a_{d-1}$ and with $g$ replaced by $0$.

## Remarks

Example: $u_{n+1} - 2 u_n = 1$ is inhomogeneous of order $1$, with $a_0(n) = -2$ and $g(n) = 1$.
Its associated homogeneous recurrence is $u_{n+1} - 2 u_n = 0$, whose solutions are $c\, 2^n$.
Non-example: $u_{n+1} = u_n^2 + 1$ is not a linear recurrence at all, since the term $u_n^2$ is not linear in $u_n$; so it is neither homogeneous nor inhomogeneous in this sense.
When all $a_i$ are constant, the homogeneous case is a [[linear recurrence with constant coefficients|081THG13JZ]].
Other names are homogeneous and nonhomogeneous linear difference equations.
The difference of two solutions of the inhomogeneous recurrence solves the associated homogeneous one.
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# Kummer–Artin–Schreier–Witt theory

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $K^s$ be a [[separable closure|JJ91A1V8NQ]] of $K$.
*Kummer–Artin–Schreier–Witt theory* is the description of the finite [[abelian extensions|EQDEXBWGJH]] of $K$ inside $K^s$ of exponent dividing a given integer.
It has two parts.

The Kummer part is [[Kummer theory|XZQENYV15Y]].
It applies to an integer $m \ge 1$ not divisible by the [[characteristic|1ZB13N78XW]] of $K$, when $K$ contains $m$ distinct $m$-th roots of unity.
It describes the abelian extensions of exponent dividing $m$ by adjoining $m$-th roots.

The Artin–Schreier–Witt part concerns a field $K$ of characteristic $p > 0$ and an integer $n \ge 1$.
Let $W_n(K)$ be the ring of [[Witt vectors|6N3X2QX4W9]] of length $n$.
Its elements are sequences $(a_0, \ldots, a_{n-1})$ in $K$, with the first $n$ Witt sum and product polynomials as operations.
Let $F(a_0, \ldots, a_{n-1}) = (a_0^p, \ldots, a_{n-1}^p)$ be the [[Frobenius|0T3WVTD675]] endomorphism of $W_n(K)$, and let $\wp = F - \id$, an additive homomorphism.
For a group $\Delta$ with $\wp W_n(K) \subseteq \Delta \subseteq W_n(K)$, let $K_\Delta$ be the field generated over $K$ by the entries of all $x \in W_n(K^s)$ with $\wp(x) \in \Delta$.
The theory states:

1. the map $\wp \colon W_n(K^s) \to W_n(K^s)$ is surjective, and its kernel is $W_n(\F_p) \cong \Z/p^n\Z$;
2. the map $\Delta \mapsto K_\Delta$ is a bijection from the groups $\Delta$ as above with $\Delta / \wp W_n(K)$ finite onto a set of extensions.
   This set consists of the finite abelian extensions of $K$ inside $K^s$ whose [[Galois group|0XTY86BZKY]] has exponent dividing $p^n$;
3. for such $\Delta$, the pairing $\operatorname{Gal}(K_\Delta/K) \times \Delta/\wp W_n(K) \to \Z/p^n\Z$, $(\sigma, a) \mapsto \sigma(x) - x$ with $\wp(x) = a$, is well defined and perfect, so $\operatorname{Gal}(K_\Delta/K) \cong \Hom(\Delta/\wp W_n(K), \Z/p^n\Z)$.

## Remarks

For $n = 1$, $W_1(K) = K$ and $\wp(x) = x^p - x$, so the Artin–Schreier–Witt part is [[Artin–Schreier theory|7YNMNRGH3T]].
The cyclic extensions of degree $p^n$ correspond to the groups $\Delta$ generated by $\wp W_n(K)$ and one $a$ whose class has order $p^n$.

Example: take $K = \F_p$ and $n = 2$.
Here $F$ is the identity on $W_2(\F_p)$, because $c^p = c$ for $c \in \F_p$, so $\wp W_2(\F_p) = 0$.
Thus $W_2(\F_p)/\wp W_2(\F_p) \cong \Z/p^2\Z$ is cyclic of order $p^2$.
By clauses 2 and 3, $\F_p$ has exactly one extension in $K^s$ with Galois group of exponent dividing $p^2$ and of degree $p^2$, and it is cyclic: it is $\F_{p^{p^2}}$.

Non-example: the Kummer part needs the roots of unity in the base field.
Over $\Q$ with $m = 3$, the field $\Q(\sqrt[3]{2})$ is not Galois over $\Q$, since its other conjugates $\omega \sqrt[3]{2}$, with $\omega$ a primitive cube root of unity, are not real.
So adjoining $m$-th roots does not give abelian extensions when $K$ lacks the $m$-th roots of unity.
</document>

<document source="https://tamnd.github.io/math2/node/SVP75YY386.md" tag="SVP75YY386" kind="conjecture">
# Hadwiger covering conjecture

## Statement

Let $n \ge 1$, and let $K \subseteq \R^n$ be a [[convex body|437FSG3419]].
Then there are $\lambda \in (0, 1)$ and points $x_1, \ldots, x_{2^n} \in \R^n$ with
$$K \subseteq \bigcup_{i=1}^{2^n} (x_i + \lambda K),$$
that is, $K$ can be covered by $2^n$ smaller [[homothetic|0ZG1PNEP35]] copies of itself.
Moreover, $2^n$ copies are needed only when $K$ is a parallelotope.

## Remarks

The conjecture is open for $n \ge 3$; Levi proved the case $n = 2$.
The cube $[0, 1]^n$ shows that $2^n$ cannot be lowered.
A translate of $\lambda [0, 1]^n$ with $\lambda < 1$ has all side lengths below $1$, so it contains at most one of the $2^n$ vertices.
General upper bounds of exponential order, about $4^n$ times a polynomial factor in $n$, are known.
Boltyanski showed that the conjecture is equivalent to a statement about illuminating the boundary of $K$ by $2^n$ directions.

## Explanations

### L1 human

The conjecture says that any convex shape in $n$-dimensional space can be covered by $2^n$ slightly shrunk copies of itself.
In the plane four copies are enough, and a square really needs four, one for each corner.
In space it predicts eight copies, but this is still unproved.
</document>

<document source="https://tamnd.github.io/math2/node/SVSVMWA9NQ.md" tag="SVSVMWA9NQ" kind="statement">
# Pairwise summation

## Statement

Let $x_1, \ldots, x_n$ be [[floating-point|RTKRSC3KPH]] numbers in a system with unit roundoff $u$ (see [[machine epsilon|JA3QXVF2B8]]), with no overflow or underflow, and let $\gamma_k = ku/(1 - ku)$ for $ku < 1$.
*Pairwise summation* computes $\hat s$ recursively, with $\hat s = x_1$ if $n = 1$.
If $n \ge 2$, let $m = \floor{n/2}$; it computes $\hat s_1$ for $x_1, \ldots, x_m$ and $\hat s_2$ for $x_{m+1}, \ldots, x_n$ in the same way.
Then it sets $\hat s = \mathrm{fl}(\hat s_1 + \hat s_2)$.
If $\lceil \log_2 n \rceil u < 1$, then
$$\Bigl| \hat s - \sum_{i=1}^{n} x_i \Bigr| \le \gamma_{\lceil \log_2 n \rceil} \sum_{i=1}^{n} \abs{x_i}.$$

## Remarks

The recursion has depth $\lceil \log_2 n \rceil$, and each term $x_i$ takes part in at most $\lceil \log_2 n \rceil$ floating-point additions.
Each addition multiplies the term by a factor $1 + \delta$ with $\abs{\delta} \le u$, so $\hat s = \sum x_i (1 + \theta_i)$ with $\abs{\theta_i} \le \gamma_{\lceil \log_2 n \rceil}$, which gives the bound.
For $n = 10^6$ terms in double precision, the bound is about $20u \sum \abs{x_i}$, against about $10^6 u \sum \abs{x_i}$ for plain recursive summation.
The cost is the same $n - 1$ additions as recursive summation.
In practice the recursion stops at blocks of moderate size that are summed in a simple loop, which gives a bound of order $(B + \log_2 n) u \sum \abs{x_i}$ for blocks of size $B$.

## Explanations

### L1 human

When one adds a long list of numbers one by one, the running total gets large and each new small term is rounded against it.
Pairwise summation adds the numbers in pairs, then the pair sums in pairs, and so on, like a sports tournament bracket.
Each number then passes through only about $\log_2 n$ additions instead of up to $n$, so the rounding errors grow much more slowly.

### L2 human

The error bound grows like $u \log_2 n$, against $u n$ for recursive summation.
The [[Kahan summation algorithm|4JAEKN22Y5]] does better still, with a leading term $2u$ independent of $n$, at about four times the cost.
Pairwise summation is the default in several numerical libraries because it costs no extra arithmetic.
The same divide-and-conquer pattern appears in the fast Fourier transform, whose rounding errors also grow only logarithmically.
For random rounding errors the typical error is even smaller, of order $u \sqrt{\log n}$ in a statistical model.
</document>

<document source="https://tamnd.github.io/math2/node/SVTT5MWB2X.md" tag="SVTT5MWB2X" kind="definition">
# Large sieve

## Definition

For real $x$ write $e(x) = e^{2\pi i x}$ and let $\norm{x}$ be the distance from $x$ to the nearest integer.
Let $M$ be an integer, $N \ge 1$ an integer and $(a_n)_{M < n \le M+N}$ complex numbers, and let
$$S(\alpha) = \sum_{M < n \le M + N} a_n\, e(n\alpha),$$
an [[exponential sum|97PBMQCPPM]].
Let $\delta \in (0, 1]$, and let $\alpha_1, \ldots, \alpha_R$ be real numbers with $\norm{\alpha_r - \alpha_s} \ge \delta$ for all $r \ne s$.
The *large sieve inequality* is
$$\sum_{r=1}^{R} \abs{S(\alpha_r)}^2 \le \left(N - 1 + \delta^{-1}\right) \sum_{M < n \le M+N} \abs{a_n}^2.$$
The *large sieve* is the method that bounds sifted sets by this inequality.
Its arithmetic form reads as follows.
For each prime $p \le Q$, let $\omega(p)$ residue classes modulo $p$ be removed, with $0 \le \omega(p) < p$.
If $\mathcal{Z}$ is a set of integers in $(M, M+N]$ that meets none of the removed classes, then
$$\#\mathcal{Z} \le \frac{N + Q^2}{L}, \qquad L = \sum_{q \le Q} \mu(q)^2 \prod_{p \mid q} \frac{\omega(p)}{p - \omega(p)},$$
where $\mu$ is the [[Möbius function|0FANE7P0DY]].

## Remarks

Example: take $N = 1$, so $S(\alpha) = a_{M+1} e((M+1)\alpha)$ and $\abs{S(\alpha_r)}^2 = \abs{a_{M+1}}^2$ for each $r$.
The $\delta$-separated points in $[0, 1)$ number at most $\delta^{-1}$, so the left side is at most $\delta^{-1} \abs{a_{M+1}}^2$, which is the right side.
Non-example: if the separation fails, say $\alpha_1 = \cdots = \alpha_R$, the left side is $R \abs{S(\alpha_1)}^2$, which exceeds any fixed bound as $R$ grows.
The fractions $a/q$ with $q \le Q$ and $\gcd(a, q) = 1$ are $Q^{-2}$-separated modulo $1$, which gives the form
$$\sum_{q \le Q} \sum_{\substack{a = 1 \\ \gcd(a,q) = 1}}^{q} \abs{S(a/q)}^2 \le (N - 1 + Q^2) \sum_{n} \abs{a_n}^2.$$
The constant $N - 1 + \delta^{-1}$ is due to Montgomery–Vaughan and Selberg.
</document>

<document source="https://tamnd.github.io/math2/node/SVXBKKB3T4.md" tag="SVXBKKB3T4" kind="definition">
# Costate equation

## Definition

Let $U \subseteq \R^m$ be nonempty, $T > 0$, and let $f \colon [0, T] \times \R^n \times U \to \R^n$ and $L \colon [0, T] \times \R^n \times U \to \R$ be continuous with continuous partial derivatives in $x$.
Let $H(t, x, u, p, \lambda_0) = p \cdot f(t, x, u) - \lambda_0 L(t, x, u)$ be the [[Hamiltonian|RACVWG5QTK]], and fix $\lambda_0 \ge 0$.
Let $u \colon [0, T] \to U$ be a measurable bounded control with trajectory $x$ of the [[control system|ZPDTVZQN5K]] $x' = f(t, x, u)$.
The *costate equation* along $(x, u)$ is the linear differential equation
$$p'(t) = -\frac{\partial H}{\partial x}(t, x(t), u(t), p(t), \lambda_0) = -\Big(\frac{\partial f}{\partial x}(t, x(t), u(t))\Big)^{T} p(t) + \lambda_0 \nabla_x L(t, x(t), u(t))$$
for an [[absolutely continuous|ZT6F4C2YHK]] function $p \colon [0, T] \to \R^n$, required to hold for almost every $t \in [0, T]$.
A solution $p$ is a *costate* or *adjoint* vector along $(x, u)$.

## Remarks

Example: $x' = u$ in $\R$ with running cost $\int \frac12 (x^2 + u^2) \, dt$ and $\lambda_0 = 1$.
Then $H = p u - \frac12 (x^2 + u^2)$ and $\partial H / \partial x = -x$, so the costate equation is $p' = x$.
Non-example: $p' = +\partial H / \partial x$ is not the costate equation, because the sign is wrong.
With it, the quantity $p \cdot \delta x$ for a solution $\delta x$ of the linearized equation $\delta x' = (\partial f / \partial x) \delta x$ would not be constant when $\lambda_0 = 0$.
The costate equation is fixed by a boundary condition, the transversality condition.
For a free final state and an endpoint cost $\varphi(x(T))$ to be minimized, it is $p(T) = -\lambda_0 \nabla \varphi(x(T))$.
</document>

<document source="https://tamnd.github.io/math2/node/SVY882HKSJ.md" tag="SVY882HKSJ" kind="definition">
# Valuation

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $\Gamma$ be a [[totally ordered|P2E8BNYC7B]] [[abelian group|071DDFVG4J]], written additively.
Add to $\Gamma$ a symbol $\infty$ with $\gamma < \infty$ and $\gamma + \infty = \infty + \gamma = \infty + \infty = \infty$ for all $\gamma \in \Gamma$.
A *valuation* on $K$ with values in $\Gamma$ is a map $v \colon K \to \Gamma \cup \set{\infty}$ such that:

1. $v(x) = \infty$ if and only if $x = 0$.
2. $v(xy) = v(x) + v(y)$ for all $x, y \in K$.
3. $v(x + y) \ge \min(v(x), v(y))$ for all $x, y \in K$.

The subgroup $v(K^\times)$ of $\Gamma$ is the *value group* of $v$.
The pair $(K, v)$ is a *valued field*.

## Remarks

Example: for a prime $p$, the [[p-adic valuation|04VCPS30ZE]] $v_p$ on $\Q$ with values in $\Z$ is a valuation.
If $x = p^m a/b$ and $y = p^n c/d$ with $a, b, c, d$ prime to $p$ and $m \le n$, then $xy = p^{m+n} ac/(bd)$ gives clause 2, and $x + y = p^m (ad + p^{n-m} cb)/(bd)$ gives $v_p(x + y) \ge m$, which is clause 3.
Example: on the field $k(t)$ of rational functions over a field $k$, the map $f/g \mapsto \deg g - \deg f$ is a valuation with values in $\Z$.
Non-example: the map $v \colon \Q \to \Z \cup \set{\infty}$ with $v(0) = \infty$ and $v(x) = 1$ for $x \ne 0$ fails clause 2, since $v(1 \cdot 1) = 1 \ne 2 = v(1) + v(1)$.
Every field has the *trivial valuation*, with $v(x) = 0$ for all $x \ne 0$.
Clauses 1 and 2 give $v(1) = 0$, $v(-x) = v(x)$ and $v(x^{-1}) = -v(x)$ for $x \ne 0$.
The set $\set{x \in K : v(x) \ge 0}$ is a [[valuation ring|QSX12D8RQM]] of $K$.
Some texts write valuations multiplicatively, with the reverse order; for $\Gamma \subseteq \R$ the map $x \mapsto c^{v(x)}$ with $0 < c < 1$ is a [[non-Archimedean absolute value|1H5F1BP6XS]].
</document>

<document source="https://tamnd.github.io/math2/node/SW0SN1NEJE.md" tag="SW0SN1NEJE" kind="concept">
# Lindelöf space

## Summary

A Lindelöf space is a [[topological space|0CQHFPGJ45]] in which every [[open cover|049PXYHW1T]] has a countable subcover.
It is a weakening of [[compactness|0TVJZYB6SH]], where finite subcovers are replaced by countable ones.
Every [[second-countable space|03PPNCM9GG]] is Lindelöf, and every regular Lindelöf space is [[paracompact|8R29G6NG9Q]].
For [[metric spaces|0VKD1TP7F3]], being Lindelöf, [[separable|0Z2P41SVDP]] and second-countable are the same.

## Explanations

### L1 human

Compactness says that finitely many members of any open cover always suffice to cover the space.
The Lindelöf property says that countably many always suffice.
The real line is not compact, since the intervals $(-n, n)$ cover it and no finitely many of them do.
It is Lindelöf, since it has a countable base of open intervals with rational end points.
An uncountable set with the discrete topology is not Lindelöf, because the cover by single points has no countable subcover.
</document>

<document source="https://tamnd.github.io/math2/node/SWB1ZXFVKY.md" tag="SWB1ZXFVKY" kind="definition">
# Strict partial order

## Definition

Let $A$ be a set.
A *strict partial order* on $A$ is a [[binary relation|08EJC6Q25H]] $<$ on $A$ with these two properties.

1. Irreflexivity: $a < a$ holds for no $a \in A$.
2. [[Transitivity|0CYRECG2YQ]]: for all $a, b, c \in A$, if $a < b$ and $b < c$, then $a < c$.

## Remarks

The relation "is a proper subset of" on the [[power set|0BC7KX51Z8]] of a set is a strict partial order.
It is irreflexive since no set is a proper subset of itself, and it is transitive since $X \subsetneq Y \subsetneq Z$ gives $X \subsetneq Z$.
The relation $\ne$ on $\R$ is irreflexive but not transitive: $1 \ne 2$ and $2 \ne 1$, but $1 \ne 1$ fails.
A strict partial order is asymmetric: $a < b$ and $b < a$ together would give $a < a$.
From a [[partial order|011C29Q1S1]] $\le$ one gets a strict partial order by $a < b$ if and only if $a \le b$ and $a \ne b$.
Conversely, from a strict partial order one gets a partial order by $a \le b$ if and only if $a < b$ or $a = b$.
These two constructions are inverse to each other.

## Explanations

### L1 human

A strict partial order is the relation "strictly smaller than" in an order where some pairs may be incomparable.
The relation $<$ on numbers is the main example.
</document>

<document source="https://tamnd.github.io/math2/node/SWDN96J72T.md" tag="SWDN96J72T" kind="definition">
# Linearly disjoint extensions

## Definition

Let $\Omega$ be a field and let $K$, $E$ and $F$ be [[subfields|07QY5RPFBR]] of $\Omega$ with $K \subseteq E$ and $K \subseteq F$.
The fields $E$ and $F$ are *linearly disjoint over $K$* if the following holds.
Every finite family of elements of $E$ that is [[linearly independent|0EVHJFTG9Z]] over $K$ is linearly independent over $F$.

## Remarks

The definition is symmetric in $E$ and $F$.
It holds if and only if the $K$-linear map $E \otimes_K F \to \Omega$ with $x \otimes y \mapsto xy$ is injective, where $\otimes_K$ is the [[tensor product|0G31VJ5YS5]].
If $E/K$ is finite, it holds if and only if $[EF:F] = [E:K]$, where $EF$ is the [[compositum|X4ZZKC319Q]].
If $E/K$ is finite Galois, it holds if and only if $E \cap F = K$.

Example: $E = \Q(\sqrt{2})$ and $F = \Q(\sqrt{3})$ in $\C$ are linearly disjoint over $\Q$.
Check: $EF = \Q(\sqrt{2}, \sqrt{3})$ has degree $4$ over $\Q$, so $[EF:F] = 2 = [E:\Q]$.

Non-example: let $\omega = e^{2\pi i/3}$, $E = \Q(\sqrt[3]{2})$ and $F = \Q(\omega\sqrt[3]{2})$.
Then $E \cap F = \Q$, but $EF = \Q(\sqrt[3]{2}, \omega)$ has degree $6$, so $[EF:F] = 2 < 3 = [E:\Q]$.
Hence the $\Q$-basis $1, \sqrt[3]{2}, \sqrt[3]{4}$ of $E$ is linearly dependent over $F$.
</document>

<document source="https://tamnd.github.io/math2/node/SWFR1GD6QT.md" tag="SWFR1GD6QT" kind="concept">
# Deformation theory

## Summary

Deformation theory studies infinitesimal variations of a geometric or algebraic object: families over small base spaces whose special member is the given object.
In algebraic geometry the small base spaces are spectra of local Artinian rings, and the first step is the ring of dual numbers $k[\varepsilon]/(\varepsilon^2)$.
For a smooth variety, first-order deformations are classified by the first cohomology group of the tangent sheaf.
The obstructions to extending them lie in the second cohomology group.
It describes the local structure of [[moduli spaces|FJZRW9P0K3]] near a point.

## Explanations

### L1 human

Given a curve, one can ask in how many independent directions it can be bent while staying a curve of the same kind.
Deformation theory answers this question to first order, like a derivative, and then asks whether each first-order bending extends to an actual family.
The projective line cannot be deformed at all: every small deformation is again a projective line.
A smooth curve of genus $g \ge 2$ has $3g - 3$ independent directions of deformation, which is the dimension of the moduli space of such curves.
</document>

<document source="https://tamnd.github.io/math2/node/SWHVQD9YEB.md" tag="SWHVQD9YEB" kind="statement">
# Burkholder–Davis–Gundy inequality

## Statement

Let $p > 0$.
There are constants $0 < c_p \le C_p < \infty$, depending only on $p$, with the following property.
Let $(M_t)_{t \ge 0}$ be a continuous [[local martingale|ZWJ9G89W3T]] with $M_0 = 0$, with [[quadratic variation|ZBYC3MFW89]] $[M]$, and let $\tau$ be a [[stopping time|WND87FBAKC]].
Let $M^*_\tau = \sup_{0 \le s \le \tau} \abs{M_s}$.
Then
$$c_p\, E\bigl[[M]_\tau^{p/2}\bigr] \le E\bigl[(M^*_\tau)^p\bigr] \le C_p\, E\bigl[[M]_\tau^{p/2}\bigr].$$

## Remarks

For $p = 2$ the inequality follows from [[Doob's martingale inequality|4G53E7BRXF]] in its $L^2$ form and the identity $E[M_\tau^2] = E[[M]_\tau]$ for suitable $\tau$, with $c_2 = 1$ and $C_2 = 4$.
For other $p$, one standard proof applies Itô's formula to $\abs{M}^p$ or uses a good-$\lambda$ inequality comparing the laws of $M^*$ and $[M]^{1/2}$.
Example: for a Wiener process $W$ and a fixed time $t$, $[W]_t = t$, so $E[\sup_{s \le t} \abs{W_s}^p]$ is between $c_p t^{p/2}$ and $C_p t^{p/2}$.
For càdlàg local martingales the inequality holds for $p \ge 1$, with the same form.
The constants do not depend on $M$ or $\tau$, and this uniformity is what makes the inequality useful.

## Explanations

### L1 human

The size of the largest swing of a fair game is comparable to the square root of its accumulated squared fluctuations.
Both quantities measure how much the game has moved, and the inequality says they agree up to constant factors.
It holds in every $L^p$ norm.

### L2 human

The inequality lets one bound moments of a stochastic integral $\int_0^t H_s \, dW_s$ by moments of $\int_0^t H_s^2 \, ds$.
It is a key estimate in proving existence, uniqueness and moment bounds for solutions of stochastic differential equations.
It also shows that a continuous local martingale with $E[[M]_\infty^{1/2}] < \infty$ is a uniformly integrable martingale.
</document>

<document source="https://tamnd.github.io/math2/node/SWJYG87QQH.md" tag="SWJYG87QQH" kind="concept">
# Rational point

## Summary

A rational point of a variety defined over a field $K$ is a point whose coordinates all lie in $K$.
Over $K = \Q$ these are the rational solutions of the defining equations, so rational points are the geometric form of [[Diophantine equations|08B1RTW7VZ]].
Deciding whether rational points exist, and describing them when they do, is the central problem of [[arithmetic geometry|824JAXQ5M6]].

## Explanations

### L1 human

The circle $x^2 + y^2 = 1$ has many points, but only some have both coordinates rational, such as $(3/5, 4/5)$.
These are its rational points over $\Q$.
Each one corresponds to a Pythagorean triple, here $3^2 + 4^2 = 5^2$.
The circle $x^2 + y^2 = 3$ has real points but no rational points at all.
Which curves and surfaces have rational points, and how many, is a deep question.
</document>

<document source="https://tamnd.github.io/math2/node/SWJZ4QRF9W.md" tag="SWJZ4QRF9W" kind="concept">
# Crossing number

## Summary

The crossing number of a [[knot|QYEAZTYGPT]] is the least number of crossings in any diagram of the knot.
It is a [[knot invariant|GZQ5ZE3X6G]] with values in the natural numbers, and knot tables list knots by their crossing number.
It is easy to define but hard to compute, because one must rule out all diagrams with fewer crossings.

## Explanations

### L1 human

Draw a knot on paper, with a gap wherever one strand passes under another.
Different drawings of the same knot can have different numbers of crossings.
The crossing number is the smallest number that any drawing can achieve.
The unknot has crossing number $0$, the [[trefoil|VYTYSVHN4A]] has $3$, and the [[figure-eight knot|C01XBGAHC6]] has $4$.
</document>

<document source="https://tamnd.github.io/math2/node/SWW2993CVJ.md" tag="SWW2993CVJ" kind="concept">
# Universal covering group

## Summary

The universal covering group of a connected [[Lie group|7JBRGCVJRB]] $G$ is its [[universal cover|Y1YPE6PM52]] $\tilde G$ with a Lie group structure.
This structure makes the covering map $\tilde G \to G$ a homomorphism.
It is simply connected, and $G$ is the quotient of $\tilde G$ by a discrete central subgroup isomorphic to the [[fundamental group|S1XK3HXHFX]] of $G$.
All connected Lie groups with a given Lie algebra are quotients of the same universal covering group.

## Explanations

### L1 human

Some Lie groups have loops that cannot be shrunk, like the circle.
The universal covering group unwinds all such loops while keeping a group structure.
For the circle of unit complex numbers, it is the real line, mapped onto the circle by $t \mapsto e^{2\pi i t}$; adding real numbers matches multiplying points of the circle.
For the rotations of space $SO(3)$, it is the group $SU(2)$, and each rotation is covered by exactly two elements.
</document>

<document source="https://tamnd.github.io/math2/node/SWWW9X4A3N.md" tag="SWWW9X4A3N" kind="concept">
# Functor of points

## Summary

The functor of points of a [[scheme|5A35TYZD02]] $X$ sends each scheme $T$ to the set of morphisms $T \to X$, called the $T$-points of $X$.
By the [[Yoneda lemma|D6DF1ARPVD]], the scheme $X$ is determined by this [[functor|28NQ0KRHD3]].
This lets one define schemes by the data they parametrize, as is done for moduli spaces and Hilbert schemes.

## Explanations

### L1 human

Instead of asking what a space is made of, one can ask what all the maps into it look like.
For the affine plane over $\Z$, a map from $\operatorname{Spec} A$ into it is just a pair $(a, b)$ of elements of the ring $A$.
So the plane "is" the rule that sends each ring $A$ to the set $A^2$.
Solutions of equations work the same way: the scheme of $x^2 + y^2 = 1$ sends $A$ to the set of solutions of $x^2 + y^2 = 1$ in $A$.
</document>

<document source="https://tamnd.github.io/math2/node/SWX539TCG0.md" tag="SWX539TCG0" kind="concept">
# Tjurina number

## Summary

The Tjurina number of an isolated hypersurface singularity is the dimension of the space of its first-order deformations up to coordinate changes.
It is the number of parameters of a miniversal deformation of the hypersurface germ.
It is computed from the defining function and its partial derivatives, and it is at most the Milnor number.
K. Saito proved that the two numbers are equal exactly when the function is quasi-homogeneous in suitable coordinates.

## Explanations

### L1 human

The Tjurina number counts the essentially different ways to perturb a singular point of a hypersurface, such as the tip of a cone or the cusp of a curve.
For the cusp curve $x^2 + y^3 = 0$ it is $2$: the perturbations $x^2 + y^3 + a y + b$ already capture every nearby curve up to a change of coordinates.
It differs from the Milnor number because it allows multiplying the equation by a nonvanishing function, which changes the function but not the hypersurface.
</document>

<document source="https://tamnd.github.io/math2/node/SX5RBFJVYC.md" tag="SX5RBFJVYC" kind="definition">
# Alexander–Spanier cohomology

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $G$ be an [[abelian group|082GHNPJ53]].
For $q \ge 0$, let $C^q(X; G)$ be the group of all functions $f \colon X^{q+1} \to G$.
The coboundary $\delta \colon C^q(X; G) \to C^{q+1}(X; G)$ is
$$(\delta f)(x_0, \ldots, x_{q+1}) = \sum_{i=0}^{q+1} (-1)^i f(x_0, \ldots, \widehat{x_i}, \ldots, x_{q+1}),$$
where the hat means that $x_i$ is left out.
Then $\delta \circ \delta = 0$, so $C^\bullet(X; G)$ is a [[cochain complex|WQ82C0M3QB]].

A function $f \in C^q(X; G)$ is *locally zero* if there is an open cover $\mathcal{U}$ of $X$ with the following property.
If the points $x_0, \ldots, x_q$ all lie in one member of $\mathcal{U}$, then $f(x_0, \ldots, x_q) = 0$.
The locally zero functions form a subcomplex $C^\bullet_0(X; G)$, because $\delta$ maps locally zero functions to locally zero functions.
Let $\bar{C}^\bullet(X; G) = C^\bullet(X; G) / C^\bullet_0(X; G)$.
The *Alexander–Spanier cohomology* of $X$ with coefficients in $G$ is the cohomology of this cochain complex:
$$\bar{H}^q(X; G) = H^q(\bar{C}^\bullet(X; G)).$$

## Remarks

Example: let $X$ be a single point.
Then $C^q(X; G) \cong G$ for every $q$, and no nonzero function is locally zero.
The coboundary $C^q \to C^{q+1}$ is multiplication by $\sum_{i=0}^{q+1} (-1)^i$, which is $0$ for even $q$ and $1$ for odd $q$.
So $\bar{H}^0(X; G) \cong G$ and $\bar{H}^q(X; G) = 0$ for $q > 0$.

Non-example: a function on $X^{q+1}$ that is nonzero at some diagonal point $(x, \ldots, x)$ is not locally zero, since every cover has a member that contains $x$.

There is a relative version for a pair $(X, A)$ and a version with compact supports.
On paracompact Hausdorff spaces, $\bar{H}^q(X; G)$ is isomorphic to Čech cohomology and to the sheaf cohomology of $X$ with coefficients in the constant sheaf $G$.
</document>

<document source="https://tamnd.github.io/math2/node/SX7ZPP72EM.md" tag="SX7ZPP72EM" kind="statement">
# Doob–Meyer decomposition theorem

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_t)_{t \ge 0}$ that is right-continuous and complete.
Let $(X_t)_{t \ge 0}$ be a [[submartingale|EQ055QMACW]] with càdlàg paths, that is, right-continuous paths with left limits.
Assume $X$ is of class (D): the family of random variables $X_\tau$, where $\tau$ runs over the finite [[stopping times|WND87FBAKC]], is uniformly integrable.
Then there is a unique decomposition $X_t = M_t + A_t$ for $t \ge 0$ with these properties:

1. $(M_t)$ is a uniformly integrable càdlàg [[martingale|0CDVK9RQBH]].
2. $(A_t)$ is a predictable, non-decreasing càdlàg process with $A_0 = 0$.
3. $E[A_\infty] < \infty$, where $A_\infty = \lim_{t \to \infty} A_t$.

Uniqueness is up to indistinguishability.

## Remarks

Here a process is predictable if it is measurable for the σ-algebra on $[0, \infty) \times \Omega$ generated by the left-continuous [[adapted|YRNFE016FF]] processes.
One proof applies the discrete [[Doob decomposition theorem|YHSSQ67GDS]] on dyadic time grids and shows that the predictable parts converge, using class (D) for uniform integrability.
Example: for a Wiener process $W$ and $T > 0$, the stopped process $X_t = W_{t \wedge T}^2$ is a submartingale of class (D), and its decomposition is $M_t = W_{t \wedge T}^2 - t \wedge T$, $A_t = t \wedge T$.
Without class (D) a decomposition still exists locally, with $M$ a [[local martingale|ZWJ9G89W3T]]; this is used for $W_t^2 = (W_t^2 - t) + t$ on $[0, \infty)$.
Predictability is what makes $A$ unique.
For a Poisson process $N$ with rate $1$, both $N_t = 0 + N_t$ and $N_t = (N_t - t) + t$ split $N$ into a martingale and an increasing process.
Only $A_t = t$ is predictable.

## Explanations

### L1 human

A process that tends to rise splits into a fair game plus a rising trend.
The trend can be foreseen an instant before it happens, and the fair game holds all the surprise.
This is the continuous-time version of the Doob decomposition for sequences.

### L2 human

The theorem is the source of the predictable quadratic variation $\langle M \rangle$ of a square-integrable martingale $M$, the compensator $A$ of $M^2$.
It underlies the construction of the stochastic integral with respect to martingales and the theory of [[semimartingales|MZNFPJJMY9]].
Compensators of counting processes, such as $t$ for the Poisson process, come from the same theorem and are central in point process theory.
</document>

<document source="https://tamnd.github.io/math2/node/SX8V8GR5ZG.md" tag="SX8V8GR5ZG" kind="concept">
# Cartan–Eilenberg resolution

## Summary

A Cartan–Eilenberg resolution of a chain complex $C$ is a [[double complex|N80ABTQHEQ]] of projective objects.
It resolves not only each term of $C$ but also its boundaries, cycles and homology, all at once.
It is the tool for defining derived functors of complexes, called hyperderived functors.
It is also the main step in the proof of the [[Grothendieck spectral sequence|ZV7VGE2SQ2]].

## Explanations

### L1 human

A projective resolution replaces one object by a chain of simple objects.
For a whole chain complex one wants to do this term by term, but in a way that respects the structure of the complex.
A Cartan–Eilenberg resolution is a grid of projective objects, and each column resolves one term of the complex.
The rows are arranged so that the homology of the complex is resolved as well.
For a complex with a single nonzero term $M$, it is just a projective resolution of $M$ placed in one column.
</document>

<document source="https://tamnd.github.io/math2/node/SXD7X0WQH1.md" tag="SXD7X0WQH1" kind="concept">
# Determinantal ideal

## Summary

A determinantal ideal is the ideal generated by all minors of a fixed size of a matrix with entries in a commutative ring.
Take the generic matrix, whose entries are independent variables.
Then the zero set of the ideal is the determinantal variety of matrices of rank less than the size of the minors.
These ideals are among the best understood classes of ideals.
For the generic matrix they are prime, and their quotient rings are Cohen–Macaulay with a known height.
They appear in the study of resolutions, of rank conditions, and of secant varieties.

## Explanations

### L1 human

A matrix has rank less than $t$ exactly when all of its $t \times t$ minors vanish.
So the condition "rank less than $t$" is a system of polynomial equations in the entries.
The determinantal ideal is the ideal generated by those minors.
For a $2 \times 2$ matrix of variables $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $t = 2$, it is the principal ideal $(ad - bc)$, and its zero set is the set of singular $2 \times 2$ matrices.
</document>

<document source="https://tamnd.github.io/math2/node/SXE9KQ02YE.md" tag="SXE9KQ02YE" kind="problem">
# Secretary problem

## Problem

Let $n \ge 1$ applicants with distinct qualities be interviewed one at a time, in an order given by a uniformly random [[permutation|05RWFKVTJ7]].
After each interview, the only information is the relative rank of the current applicant among those seen so far.
After each interview the applicant must be accepted or rejected at once, and the decision cannot be changed.
The process ends at the first acceptance.
Find a rule that maximises the probability of accepting the best of the $n$ applicants, and find this maximal probability.
A rule is a [[stopping time|WND87FBAKC]] for the observed relative ranks.

## Remarks

The problem is solved, and it is a standard example of optimal stopping.
An optimal rule has a threshold form: reject the first $r - 1$ applicants, then accept the first applicant who is better than all earlier ones.
For $r \ge 2$ this rule picks the best applicant with probability
$$P_n(r) = \frac{r - 1}{n} \sum_{i=r}^{n} \frac{1}{i - 1},$$
and $P_n(1) = 1/n$.
The optimal $r$ satisfies $r/n \to 1/e$ and the maximal probability tends to $1/e \approx 0.368$ as $n \to \infty$.
For $n = 3$, the rule with $r = 2$ succeeds with probability $\frac{1}{3}(1 + \frac{1}{2}) = \frac{1}{2}$, which is optimal.

## Explanations

### L1 human

You must hire one of $n$ candidates who arrive in random order, and you must say yes or no to each one on the spot.
A good plan is to look at about $37\%$ of the candidates without hiring anyone, to learn the level of the field.
After that, hire the first candidate who beats everyone seen so far.
Surprisingly, this plan finds the very best candidate about $37\%$ of the time, however large $n$ is.
</document>

<document source="https://tamnd.github.io/math2/node/SXMPSYESTP.md" tag="SXMPSYESTP" kind="definition">
# Holonomy groupoid of a foliation

## Definition

Let $M$ be a smooth manifold of dimension $n$ with a [[foliation|98HTB7MSPX]] $\mathcal{F}$ of codimension $q$.
A *leafwise path* is a [[path|03NAMV1DHJ]] $\gamma \colon [0, 1] \to M$ whose image lies in a single leaf.
For each $x \in M$ fix a transversal $T_x$: a [[submanifold|VKFW27WXEZ]] of dimension $q$ through $x$ that meets the leaves transversally.
Let $\gamma$ be a leafwise path from $x$ to $y$.
Following the leaves near $\gamma$ from $T_x$ to $T_y$ defines a [[germ|VZXF7RNFKS]] at $x$ of a diffeomorphism $\operatorname{hol}(\gamma) \colon (T_x, x) \to (T_y, y)$, the *holonomy* of $\gamma$.
Two leafwise paths are *holonomy equivalent* if they have the same endpoints $x, y$ and the same holonomy germ.
The *holonomy groupoid* $\operatorname{Hol}(M, \mathcal{F})$ is the [[groupoid|FH3C0T643F]] with these data.

1. Its objects are the points of $M$.
2. Its arrows from $x$ to $y$ are the holonomy equivalence classes of leafwise paths from $x$ to $y$.
3. Composition is induced by concatenation of paths, and the inverse of the class of $\gamma$ is the class of the reversed path.

## Remarks

The holonomy germ does not depend on the choice of transversals up to the natural identifications, so the equivalence relation is well defined.
Homotopic leafwise paths with fixed endpoints have the same holonomy, so each arrow is a union of leafwise homotopy classes of paths.
The holonomy groupoid has a natural structure of a [[Lie groupoid|HNVW0E732J]] of dimension $2n - q$, whose manifold of arrows need not be Hausdorff.
Its source fibre over $x$ is the holonomy cover of the leaf through $x$.

Example: foliate $\R^2$ by the horizontal lines $y = c$.
All holonomy germs are identities, so there is one arrow from $(a, c)$ to $(b, c)$ for all $a, b, c$, and no arrow between points with different second coordinates.
The holonomy groupoid is the equivalence relation $\set{((b, c), (a, c))}$, a manifold of dimension $3 = 2 \cdot 2 - 1$.

Non-example: the monodromy groupoid, whose arrows are leafwise homotopy classes of leafwise paths, is in general different.
For the foliation of $S^1 \times \R$ by the circles $S^1 \times \set{c}$, a loop going once around a circle has trivial holonomy but is not null-homotopic in its leaf.
So it gives the identity arrow of the holonomy groupoid but a nontrivial arrow of the monodromy groupoid.
In the [[Reeb foliation|W58NYR1DGR]], the compact leaf has nontrivial holonomy, so some loops in it give nontrivial arrows in both.

## Explanations

### L1 human

The holonomy groupoid records all ways of walking from one point to another inside a leaf, counting two walks as equal when they drag nearby leaves in the same way.
</document>

<document source="https://tamnd.github.io/math2/node/SXNWHN2C01.md" tag="SXNWHN2C01" kind="statement">
# Resolvent as Laplace transform of a semigroup

## Statement

Let $X$ be a complex [[Banach space|0BDTTM24DP]], and let $(T(t))_{t \ge 0}$ be a [[$C_0$-semigroup|EVF180QJXY]] on $X$ with [[generator|60HSWZBTK9]] $A$.
Let $M \ge 1$ and $\omega \in \R$ be constants with $\norm{T(t)} \le M e^{\omega t}$ for all $t \ge 0$.
Then every complex $\lambda$ with $\operatorname{Re} \lambda > \omega$ lies in the [[resolvent set|X0243CZD9D]] of $A$, and for every $x \in X$,
$$(\lambda I - A)^{-1} x = \int_0^\infty e^{-\lambda t} \, T(t) x \, dt = \lim_{r \to \infty} \int_0^r e^{-\lambda t} \, T(t) x \, dt.$$
Moreover $\norm{(\lambda I - A)^{-1}} \le M / (\operatorname{Re} \lambda - \omega)$.

## Remarks

Constants $M$ and $\omega$ as above exist for every $C_0$-semigroup, by the uniform boundedness principle.
Let $R(\lambda) x$ denote the integral; it converges absolutely by the bound on $\norm{T(t)}$, which also gives the norm estimate.
A direct computation with the semigroup law shows $\frac{1}{h}(T(h) - I) R(\lambda) x \to \lambda R(\lambda) x - x$ as $h \to 0^+$, so $R(\lambda) x \in D(A)$ and $(\lambda I - A) R(\lambda) x = x$.
For $x \in D(A)$, $A$ commutes with the integral, so $R(\lambda)(\lambda I - A) x = x$.
Example: for a square matrix $A$ and $T(t) = e^{t A}$, the formula reads $(\lambda I - A)^{-1} = \int_0^\infty e^{-\lambda t} e^{t A} \, dt$ when $\operatorname{Re} \lambda$ exceeds the real parts of all eigenvalues of $A$.

## Explanations

### L1 human

For a number $a$, $\int_0^\infty e^{-\lambda t} e^{a t} \, dt = 1/(\lambda - a)$.
The theorem says that the same identity holds for operators: the [[Laplace transform|0V49JEZW06]] of the semigroup $e^{t A}$ is the inverse of $\lambda - A$.

### L2 human

The formula is the bridge between semigroups and their generators.
It gives the necessity part of the [[Hille–Yosida theorem|5YR4GJKPCV]], and the norm bound $M/(\operatorname{Re} \lambda - \omega)$ is exactly the resolvent estimate in that theorem.
Conversely, one recovers $T(t)$ from the resolvent by a complex inversion integral or by $T(t) x = \lim_{n \to \infty} \left(\frac{n}{t} R(\frac{n}{t})\right)^n x$, an operator form of [[Post's inversion formula|D6QKPBS2EN]].
</document>

<document source="https://tamnd.github.io/math2/node/SXQGHWHFRF.md" tag="SXQGHWHFRF" kind="concept">
# Spherical trigonometry

## Summary

Spherical trigonometry studies the relations between the sides and the angles of spherical triangles, the triangles on a sphere bounded by arcs of great circles.
Its main formulas are the spherical law of cosines, $\cos c = \cos a \cos b + \sin a \sin b \cos \gamma$, and the spherical law of sines, $\sin a / \sin \alpha = \sin b / \sin \beta = \sin c / \sin \gamma$.
It is the part of [[spherical geometry|SQ9RQVANMG]] used in astronomy, geodesy and navigation.
The area of a spherical triangle is given by [[Girard's theorem|Y9DM9B0RA3]].

## Explanations

### L1 human

On a globe, a triangle is made of three arcs of great circles, and its sides are measured as angles seen from the centre of the Earth.
Spherical trigonometry answers questions such as: given two sides and the angle between them, how long is the third side?
A pilot uses it to find the great-circle distance between two airports from their latitudes and longitudes.
For small triangles the formulas reduce to the plane ones, which is why plane trigonometry works on a field but not across an ocean.
</document>

<document source="https://tamnd.github.io/math2/node/SXR93CAP1R.md" tag="SXR93CAP1R" kind="concept">
# Sequential probability ratio test

## Summary

The sequential probability ratio test, or SPRT, is a sequential test of a simple null hypothesis against a simple alternative.
After each observation it updates the likelihood ratio of the data and stops as soon as this ratio leaves a fixed interval.
Abraham Wald introduced it, and it is the basic procedure of [[sequential analysis|9CWYNB9003]].
Among all tests with the same error probabilities, it needs the smallest expected number of observations under both hypotheses.

## Explanations

### L1 human

Two explanations of the data compete, and each new observation makes one of them a little more plausible than the other.
The test keeps a running score, the ratio of how likely the data are under the two explanations.
It stops and picks the alternative when the score climbs above an upper bar, and picks the null hypothesis when it falls below a lower bar.
For example, to test whether a coin shows heads with probability $1/2$ or $2/3$, each head multiplies the score by $4/3$ and each tail by $2/3$.
</document>

<document source="https://tamnd.github.io/math2/node/SXS38FWHEN.md" tag="SXS38FWHEN" kind="definition">
# Newton polytope

## Definition

Let $K$ be a [[field|04C5ZBG1WT]] and let $f = \sum_{a \in \Z^n} c_a x^a$ be a Laurent polynomial in $x_1, \dots, x_n$ over $K$, where $x^a = x_1^{a_1} \cdots x_n^{a_n}$ and only finitely many $c_a$ are nonzero.
The *support* of $f$ is $\operatorname{supp}(f) = \set{a \in \Z^n : c_a \ne 0}$.
The *Newton polytope* of $f$ is the [[convex hull|0FQSDC2EQ2]]
$$\operatorname{Newt}(f) = \operatorname{conv}(\operatorname{supp}(f)) \subseteq \R^n.$$

## Remarks

The Newton polytope is a lattice polytope: its vertices lie in $\Z^n$, and each vertex is the exponent of a monomial of $f$.
For nonzero $f, g$, one has $\operatorname{Newt}(fg) = \operatorname{Newt}(f) + \operatorname{Newt}(g)$, the Minkowski sum.

Example: for $f = 1 + x + y + xy$ the support is $\set{(0,0), (1,0), (0,1), (1,1)}$ and $\operatorname{Newt}(f)$ is the unit square.
Indeed $f = (1 + x)(1 + y)$, and the unit square is the Minkowski sum of the segments $[0,1] \times \set{0}$ and $\set{0} \times [0, 1]$.

Non-example: for $f = x^2 - (x - 1)(x + 1) = 1$, the support is $\set{0}$, not $\set{0, 1, 2}$.
The Newton polytope uses the monomials that remain after simplification, so $\operatorname{Newt}(f)$ is the single point $0$ and not the segment $[0, 2]$.

In one variable, the Newton polytope of $f$ is the segment from the lowest to the highest exponent.
</document>

<document source="https://tamnd.github.io/math2/node/SXS6D87PHE.md" tag="SXS6D87PHE" kind="concept">
# Minkowski addition

## Summary

The Minkowski sum of two subsets $A$ and $B$ of a vector space is the set $A + B$ of all sums $a + b$ with $a \in A$ and $b \in B$.
Together with scaling $\lambda A = \set{\lambda a : a \in A}$, it makes the convex bodies of $\R^n$ into a cone-like algebraic structure.
Volume behaves well under this addition, which leads to the Brunn–Minkowski inequality and the theory of mixed volumes.

## Explanations

### L1 human

To add two shapes, place a copy of the second shape at every point of the first and take everything covered.
Adding a disk of radius $r$ to a square gives a square with rounded corners, thickened by $r$ on every side.
Adding two segments in different directions gives a parallelogram.
Adding a shape to a single point just moves the shape.
</document>

<document source="https://tamnd.github.io/math2/node/SY0TJ4NCES.md" tag="SY0TJ4NCES" kind="statement">
# Commutativity of convolution

## Statement

Let $n \ge 1$, and let $f, g \colon \R^n \to \C$ be Lebesgue measurable.
Let $x \in \R^n$.
Then $\int_{\R^n} \abs{f(y) g(x - y)} \, dy < \infty$ if and only if $\int_{\R^n} \abs{g(y) f(x - y)} \, dy < \infty$.
In that case $(f * g)(x) = (g * f)(x)$, where $*$ is the [[convolution|2Z29JN3VES]].
In particular $f * g = g * f$ for all $f, g \in L^1(\R^n)$.

## Remarks

The proof is the substitution $y = x - z$.
[[Lebesgue measure|2Z2QVWZ262]] is invariant under the reflection $z \mapsto -z$ and under translations, so
$$\int_{\R^n} f(y) g(x - y) \, dy = \int_{\R^n} f(x - z) g(z) \, dz = (g * f)(x),$$
and the same computation with absolute values gives the equivalence of the two convergence conditions.

Example: on $\R$, with $f = \mathbf{1}_{[0, 1]}$ and $g = \mathbf{1}_{[0, 2]}$, both $f * g$ and $g * f$ give at $x$ the length of $[0, 1] \cap [x - 2, x]$ and of $[0, 2] \cap [x - 1, x]$, which are equal: both are $\min(x, 1)$ for $0 \le x \le 2$.

The statement uses that the group $\R^n$ is abelian.
On a non-abelian locally compact group, convolution $f * g$ with $(f * g)(x) = \int f(y) g(y^{-1} x) \, dy$ is in general not commutative.

## Explanations

### L1 human

When you convolve two functions, it does not matter which one you slide and which one stays put.
The answer is the same either way.
This is like the product of two polynomials, which does not depend on the order of the factors.

### L2 human

Commutativity, together with [[associativity|HZ5J8GGS16]] and bilinearity, makes $L^1(\R^n)$ a commutative algebra under convolution.
It is the reason why the [[convolution on the line|ERP5737VZ1]] can be written with either function shifted.
Under the Fourier transform it becomes the commutativity of the pointwise product $\hat f \hat g = \hat g \hat f$.
</document>

<document source="https://tamnd.github.io/math2/node/SY22C7XVCW.md" tag="SY22C7XVCW" kind="definition">
# Universal enveloping algebra

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $\mathfrak g$ be a Lie algebra over $k$ with bracket $[\cdot, \cdot]$.
Let $T(\mathfrak g)$ be the [[tensor algebra|P4J5JZDW0B]] of the vector space $\mathfrak g$.
Let $I$ be the [[two-sided ideal|0CD87SFM2N]] of $T(\mathfrak g)$ generated by the elements $x \otimes y - y \otimes x - [x, y]$ for $x, y \in \mathfrak g$.
The *universal enveloping algebra* of $\mathfrak g$ is $U(\mathfrak g) = T(\mathfrak g) / I$, together with the map $\iota \colon \mathfrak g \to U(\mathfrak g)$ induced by the inclusion of $\mathfrak g$ in $T(\mathfrak g)$.

## Remarks

The map $\iota$ is linear and satisfies $\iota([x, y]) = \iota(x)\iota(y) - \iota(y)\iota(x)$.
Universal property: for every [[associative algebra|3CTW1P4HBP]] $A$ over $k$ with identity and every linear map $\varphi \colon \mathfrak g \to A$ with $\varphi([x, y]) = \varphi(x)\varphi(y) - \varphi(y)\varphi(x)$, there is a unique algebra homomorphism $\tilde\varphi \colon U(\mathfrak g) \to A$ with $\tilde\varphi \circ \iota = \varphi$.
Hence representations of $\mathfrak g$ on a vector space $V$ are the same as left $U(\mathfrak g)$-module structures on $V$.
Example: if $\mathfrak g$ is abelian, all brackets are $0$, so $I$ is generated by the elements $x \otimes y - y \otimes x$.
Then $U(\mathfrak g)$ is the [[symmetric algebra|PBGF9TXMB7]] $S(\mathfrak g)$, a polynomial ring in a basis of $\mathfrak g$.
Non-example: for the Lie algebra $\mathfrak g$ with basis $x, y$ and $[x, y] = y$, the tensor algebra $T(\mathfrak g)$ itself does not satisfy $xy - yx = [x, y]$, since $x \otimes y - y \otimes x \ne y$ there; one must pass to the quotient.
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<document source="https://tamnd.github.io/math2/node/SY2MWC434Z.md" tag="SY2MWC434Z" kind="concept">
# Irreducible component

## Summary

A topological space is irreducible when it is nonempty and is not the union of two proper closed subsets.
An irreducible component of a space is a maximal irreducible subset.
In a Noetherian space, such as an algebraic set with its Zariski topology, there are only finitely many irreducible components.
The space is their union.
This splits every algebraic set into irreducible pieces, the varieties.

## Explanations

### L1 human

An algebraic set can be made of several pieces that cannot be split further.
The set of solutions of $xy = 0$ in the plane is the union of the two coordinate axes.
Each axis is a single piece that is not a union of two smaller algebraic sets, and the two axes are the irreducible components.
A finite set of two or more points is not irreducible: each point is a piece of its own.
</document>

<document source="https://tamnd.github.io/math2/node/SY38F954FE.md" tag="SY38F954FE" kind="concept">
# Quillen adjunction

## Summary

A Quillen adjunction is a pair of [[adjoint functors|TZHF6ZM92S]] between [[model categories|SDTZVZ6B4Q]] that respects the model structures.
The left adjoint preserves cofibrations and trivial cofibrations, or equivalently the right adjoint preserves fibrations and trivial fibrations.
Such a pair induces an adjunction between the homotopy categories, through the derived functors.
When this derived adjunction is an equivalence, the pair is a [[Quillen equivalence|KZCXHTEYK7]], and the two model categories present the same homotopy theory.

## Explanations

### L1 human

An adjunction is a precise way in which two constructions between categories mirror each other.
In homotopy theory one also wants the constructions to respect weak equivalences, and most do not do so on the nose.
A Quillen adjunction asks for less: the functors must respect the good maps of the model structures.
That is enough to give well-defined functors after one inverts the weak equivalences.
Geometric realization of simplicial sets and the singular simplicial set functor form the basic example.
</document>

<document source="https://tamnd.github.io/math2/node/SYF2ND6J97.md" tag="SYF2ND6J97" kind="concept">
# Banach limit

## Summary

A Banach limit is a positive, shift-invariant linear functional on the space $\ell^\infty$ of bounded real sequences.
It gives the value $1$ to the constant sequence $1$.
It extends the ordinary limit from convergent sequences to all bounded sequences.
Banach limits exist by the [[Hahn–Banach theorem|QB5AE3VYZZ]], but none can be written down explicitly, and there are many of them.
They define [[almost convergence|8HTHB1QPZY]] and are a basic example of an invariant mean.

## Explanations

### L1 human

The sequence $1, 0, 1, 0, \dots$ has no limit, but it is natural to say that its "average limit" is $1/2$.
A Banach limit is a rule that assigns such a value to every bounded sequence.
It must be linear, it must give a nonnegative value to a nonnegative sequence, and it must not change when you drop the first term.
These rules force the value $1/2$ for $1, 0, 1, 0, \dots$ and the usual limit for every convergent sequence.
</document>

<document source="https://tamnd.github.io/math2/node/SYNST4W3FR.md" tag="SYNST4W3FR" kind="concept">
# Carathéodory metric

## Summary

The Carathéodory metric, or Carathéodory pseudodistance, of a complex manifold $X$ uses holomorphic maps from $X$ to the unit disc.
The distance of two points is the largest Poincaré distance between their images under such maps.
It is the smallest pseudodistance for which all holomorphic maps from $X$ to the disc are distance-decreasing, and it is a biholomorphic invariant.
It is always at most the [[Kobayashi metric|DY8QQ5GTTX]], and on bounded convex domains the two agree by [[Lempert's theorem|2PANNSQ3G6]].
Constantin Carathéodory introduced it in the 1920s for domains in $\C^n$.

## Explanations

### L1 human

Bounded holomorphic functions separate the points of a bounded domain.
The Carathéodory distance uses them as rulers: map the domain into the unit disc by a holomorphic function and measure the hyperbolic distance of the two image points.
The largest value over all such maps is the Carathéodory distance.
On $\C$ every bounded holomorphic function is constant by Liouville's theorem, so the Carathéodory distance of $\C$ is $0$.
On the unit disc it is the hyperbolic distance.
</document>

<document source="https://tamnd.github.io/math2/node/SYP1ATSTTH.md" tag="SYP1ATSTTH" kind="statement">
# Decay of the Fourier transform of surface measure

## Statement

Let $n \ge 2$ and let $S \subseteq \R^n$ be a smooth hypersurface whose [[Gaussian curvature|SQVQ5DQT6W]] is nonzero at every point.
Let $\sigma$ be the surface measure on $S$, and let $\psi \colon \R^n \to \C$ be a [[smooth function|0V33BNKZ9S]] with compact support.
Let $\mu = \psi \, d\sigma$, and let its [[Fourier transform|DPVM34ADZV]] be
$$\hat\mu(\xi) = \int_S e^{-2\pi i x \cdot \xi} \psi(x) \, d\sigma(x).$$
Then there is a constant $C$ such that
$$\abs{\hat\mu(\xi)} \le C (1 + \abs{\xi})^{-(n-1)/2} \quad \text{for all } \xi \in \R^n.$$

## Remarks

By a partition of unity, one may assume that the support of $\psi$ meets $S$ in a small piece.
This piece is the graph of a smooth function $h$ over a ball in $\R^{n-1}$.
Then $\hat\mu(\xi)$ is an [[oscillatory integral|D5HZKE8E1A]] in $n - 1$ variables with phase $-2\pi (x' \cdot \xi' + h(x')\xi_n)$.
If $\xi$ is far from the normal directions to the piece, the phase has no critical point and the integral decays faster than any power.
If $\xi$ is close to a normal direction, the critical points are nondegenerate, because the curvature hypothesis makes the Hessian of $h$ invertible.
Then stationary phase gives the decay $\abs{\xi}^{-(n-1)/2}$.

Example: for the unit sphere $S^2 \subseteq \R^3$ and $\psi = 1$ near $S^2$, a computation in polar coordinates gives
$$\hat\sigma(\xi) = 2\pi \int_{-1}^{1} e^{-2\pi i \abs{\xi} t} \, dt = \frac{2 \sin(2\pi\abs{\xi})}{\abs{\xi}},$$
which decays like $\abs{\xi}^{-1} = \abs{\xi}^{-(3-1)/2}$ and no faster.

The curvature hypothesis is needed.
If $S$ is the hyperplane $\set{x_n = 0}$, which has curvature zero, then $e^{-2\pi i x \cdot \xi} = 1$ on $S$ when $\xi = (0, \ldots, 0, \xi_n)$.
So $\hat\mu(0, \ldots, 0, \xi_n) = \int_S \psi \, d\sigma$ for every $\xi_n$, and there is no decay when this integral is not $0$.

## Explanations

### L1 human

A measure spread on a curved surface, like the sphere, has a Fourier transform that tends to zero at infinity at a definite rate.
The more the surface curves in all directions, the more the waves cancel, and the rate is half the dimension of the surface.
A flat surface gives no decay in the direction perpendicular to it.

### L2 human

This decay is the starting point of the [[Stein–Tomas restriction theorem|ZZVWHNFY6F]], whose proof only uses the bound $\abs{\hat\mu(\xi)} \lesssim \abs{\xi}^{-(n-1)/2}$.
It also gives the dispersive decay of waves and the asymptotics of Bessel functions, since $\hat\sigma$ for the sphere is a Bessel function.
When some principal curvatures vanish, the decay is weaker and is measured by the number of nonvanishing curvatures.
</document>

<document source="https://tamnd.github.io/math2/node/SYP3FVM71M.md" tag="SYP3FVM71M" kind="definition">
# Euler characteristic

## Definition

Let $X$ be a topological space whose [[singular homology|M0Q5CFJ7ZH]] groups $H_k(X)$ are finitely generated for all $k$ and zero for all but finitely many $k$.
The *Euler characteristic* of $X$ is
$$\chi(X) = \sum_{k \ge 0} (-1)^k b_k(X),$$
where $b_k(X)$ is the $k$-th [[Betti number|ST71ZV3FES]] of $X$.

## Remarks

If $X$ is a finite [[CW complex|5W95QFHXPR]] with $c_k$ cells of dimension $k$, then $\chi(X) = \sum_k (-1)^k c_k$.
For the boundary of a convex polyhedron this is [[Euler's polyhedron formula|09HT3P35YP]] $V - E + F = 2$.
For any field $F$, $\chi(X) = \sum_k (-1)^k \dim_F H_k(X; F)$ under the same hypothesis.

Example: the torus has $b_0 = 1$, $b_1 = 2$, $b_2 = 1$, so $\chi(T^2) = 1 - 2 + 1 = 0$.
Its usual CW structure has one $0$-cell, two $1$-cells and one $2$-cell, and $1 - 2 + 1 = 0$ again.

Non-example: the wedge of countably many circles has $H_1$ free abelian of infinite rank, so it fails the finiteness hypothesis and has no Euler characteristic in this sense.
</document>

<document source="https://tamnd.github.io/math2/node/SYRCTH853H.md" tag="SYRCTH853H" kind="concept">
# Matroid rank

## Summary

The rank function of a [[matroid|PX31ZNYS3Z]] assigns to each subset of the ground set the size of its largest independent subsets.
The rank of the matroid is the rank of the whole ground set, the common size of its [[bases|QTSQR7130G]].
Rank generalizes the [[rank of a matrix|0ZC9YTXZRY]] and the size of a spanning forest of a graph, and its properties, notably submodularity, characterize matroids.

## Explanations

### L1 human

Given some of the elements of a matroid, ask how many of them you can keep while staying independent.
That number is their rank.
For a set of vectors it is the dimension of the space they span.
For a set of edges of a graph, keep all the vertices but only these edges.
The rank is the number of vertices minus the number of connected pieces.
In a triangle, the three edges together have rank $3 - 1 = 2$.
</document>

<document source="https://tamnd.github.io/math2/node/SYSVE5P1YN.md" tag="SYSVE5P1YN" kind="statement">
# Bogomolov–Miyaoka–Yau inequality

## Statement

Let $X$ be a smooth projective [[surface of general type|SV4CK2XP8M]] over $\C$.
Let $c_1^2 = K_X^2$ be the self-intersection of its [[canonical divisor|W05VYY4HBR]], and let $c_2$ be its second [[Chern number|PAD37GKDPD]], equal to the topological Euler characteristic of $X$.
Then
$$c_1^2 \le 3 c_2.$$

## Remarks

Miyaoka and Yau proved the inequality independently, Miyaoka by algebraic methods and Yau by constructing Kähler–Einstein metrics; Bogomolov had earlier proved the weaker bound $c_1^2 \le 4 c_2$.
Yau's proof shows that equality holds exactly when $X$ is a quotient of the unit ball in $\C^2$ by a discrete group acting freely.
Examples are the fake projective planes, with $c_1^2 = 9$ and $c_2 = 3$.
Example: a smooth quintic surface in $\mathbb{P}^3$ has $c_1^2 = 5$ and $c_2 = 55$, and $5 \le 165$.
The hypothesis on the Kodaira dimension is needed: for $X = C \times \mathbb{P}^1$ with $C$ of genus $2$, one has $c_1^2 = 8(1 - 2) = -8$ and $c_2 = 4(1 - 2) = -4$, so $c_1^2 > 3 c_2 = -12$.
The inequality fails in positive characteristic in general.

## Explanations

### L1 human

Two numbers measure a complex surface: the self-intersection of its canonical class and its Euler characteristic.
For surfaces of general type, the first is never more than three times the second.
So surfaces of general type cannot be too curved compared with their topology.

### L2 human

Together with Noether's inequality $c_1^2 \ge 2 \chi(\mathcal{O}_X) - 6$ for minimal surfaces, the inequality confines the Chern numbers of surfaces of general type to a wedge, the domain of the geography of surfaces.
It is the surface case of the Chern number inequalities for varieties with Kähler–Einstein metrics.
Yau used the equality case to prove that every compact complex surface homotopy equivalent to $\mathbb{P}^2$ is isomorphic to $\mathbb{P}^2$.
</document>

<document source="https://tamnd.github.io/math2/node/SYVAV7V0B3.md" tag="SYVAV7V0B3" kind="definition">
# Dual matroid

## Definition

Let $M$ be a [[matroid|PX31ZNYS3Z]] on a finite ground set $E$, and let $\mathcal{B}$ be its set of [[bases|QTSQR7130G]].
The *dual matroid* $M^*$ is the matroid on $E$ whose set of bases is
$$\mathcal{B}^* = \set{E \setminus B : B \in \mathcal{B}}.$$

## Remarks

It is a theorem that $\mathcal{B}^*$ satisfies the basis exchange property, so $M^*$ is indeed a matroid.
Its independent sets are the subsets of $E$ disjoint from some basis of $M$.
Its [[rank function|SYRCTH853H]] is $r^*(A) = \abs{A} - r(E) + r(E \setminus A)$, where $r$ is the rank function of $M$, and $(M^*)^* = M$.
The bases of $M^*$ are the cobases of $M$, and the circuits of $M^*$ are the cocircuits of $M$.

Example: for the uniform matroid $U_{k,n}$, whose bases are the $k$-element subsets of $\set{1, \dots, n}$, the dual has as bases the $(n-k)$-element subsets, so $U_{k,n}^* = U_{n-k,n}$.

Non-example: the complements of the independent sets of $M$ are in general not the independent sets of $M^*$.
For $M = U_{1,2}$ on $\set{1, 2}$, the complement of $\emptyset$ is $\set{1, 2}$, which is not independent in $M^* = U_{1,2}$.
</document>

<document source="https://tamnd.github.io/math2/node/SYVXQQK57V.md" tag="SYVXQQK57V" kind="statement">
# Lusternik–Schnirelmann theorem

## Statement

Let $n \ge 0$ and let $A_1, \ldots, A_{n+1}$ be closed subsets of the [[$n$-sphere|T62369RFH6]] $S^n \subset \R^{n+1}$ with $A_1 \cup \cdots \cup A_{n+1} = S^n$.
Then some $A_i$ contains a pair of antipodal points $x$ and $-x$.

## Remarks

The theorem follows from the [[Borsuk–Ulam theorem|QHJ0GBQE2E]].
We may assume that every $A_i$ is nonempty, by replacing an empty one with a copy of a nonempty one.
Let $f(x) = (d(x, A_1), \ldots, d(x, A_n)) \in \R^n$, where $d(x, A)$ is the [[distance from $x$ to $A$|0WQ19TJV24]], and choose $x$ with $f(x) = f(-x)$.
If some coordinate of $f(x)$ is $0$, then $x$ and $-x$ both lie in the closed set $A_i$; otherwise neither lies in $A_1 \cup \cdots \cup A_n$, so both lie in $A_{n+1}$.
The same conclusion holds when all the $A_i$ are open.

Example: for $n = 1$, if the circle is the union of two closed sets, one of them contains two opposite points.

The number $n + 1$ cannot be raised to $n + 2$.
Project the facets of a regular $(n+1)$-simplex centred at the origin radially onto $S^n$.
This covers $S^n$ by $n + 2$ closed sets, none of which contains an antipodal pair.

## Explanations

### L1 human

If you paint a sphere with $n + 1$ colours, where each colour region is closed, then some colour appears at two opposite points.
On an ordinary globe, painted with three colours, some colour is used at two antipodal places.

### L2 human

The theorem is one of several equivalent forms of the Borsuk–Ulam theorem, and it can be proved from it and conversely.
It gives a short proof of Kneser's conjecture on the chromatic number of Kneser graphs.
More generally, covering and coincidence statements of this kind are a main entry point of topological methods in combinatorics.
</document>

<document source="https://tamnd.github.io/math2/node/SYXNSN5JX0.md" tag="SYXNSN5JX0" kind="definition">
# Lie group method for PDEs

## Definition

Let $\Delta(x, u, \partial u, \dots, \partial^k u) = 0$ be a system of [[partial differential equations|GC5BDZ6QBC]] for $u \colon U \to \R^q$ with $U \subseteq \R^p$ open.
A *one-parameter group of point transformations* is a local [[flow|06738GGWWK]] $g_\varepsilon$ on an open subset of $\R^p \times \R^q$, generated by a smooth vector field
$$v = \sum_{i=1}^p \xi^i(x, u)\, \partial_{x^i} + \sum_{\alpha=1}^q \phi^\alpha(x, u)\, \partial_{u^\alpha}.$$
The group acts on functions by moving their graphs: $g_\varepsilon \cdot u$ is the function whose graph is $g_\varepsilon(\text{graph of } u)$, where this is again a graph.
It is a *Lie point symmetry* of the system if $g_\varepsilon \cdot u$ is a solution for every solution $u$ and every small $\varepsilon$.

The *Lie group method* consists of the following steps.

1. Write the condition that the $k$-th prolongation of $v$ annihilates $\Delta$ on the solution set.
2. Split this condition into a linear system of determining equations for $\xi^i$ and $\phi^\alpha$, and solve it.
3. Use the resulting symmetries to map known solutions to new ones.
   Find invariant solutions by reducing the system to fewer independent variables.

## Remarks

For systems of maximal rank, step 1 is necessary and sufficient for $v$ to generate symmetries.
The set of all such $v$ is a Lie algebra of vector fields.

Example: the heat equation $u_t = u_{xx}$ has the symmetries $u(x, t) \mapsto u(x - \varepsilon, t)$, $u(x, t) \mapsto u(e^{-\varepsilon} x, e^{-2\varepsilon} t)$ and $u \mapsto e^{\varepsilon} u$.
It also has the Galilean symmetry $u(x, t) \mapsto e^{-\varepsilon x + \varepsilon^2 t}\, u(x - 2\varepsilon t, t)$.
Check of the last one: if $w = e^{-\varepsilon x + \varepsilon^2 t} u(x - 2\varepsilon t, t)$, a direct computation gives $w_t - w_{xx} = e^{-\varepsilon x + \varepsilon^2 t} (u_t - u_{xx})$, which vanishes.
Solutions invariant under the scaling are the [[similarity solutions|579J5K9K5Z]] $u = v(x / \sqrt t)$, which solve the ordinary differential equation $v'' + \tfrac{\eta}{2} v' = 0$ with $\eta = x / \sqrt t$.

Non-example: the map $u(x, t) \mapsto w(x, t) = u(x/\lambda, t/\lambda)$ with $\lambda > 0$, $\lambda \ne 1$, is not a symmetry of the heat equation.
Indeed $w_t - w_{xx} = \lambda^{-1} u_t - \lambda^{-2} u_{xx}$ at $(x/\lambda, t/\lambda)$, which equals $(\lambda^{-1} - \lambda^{-2}) u_t$ there and is non-zero for most solutions.

## Explanations

### L1 human

The method first finds all continuous symmetries of an equation by solving linear equations.
Then it uses them to build new solutions and to reduce the equation to simpler ones.
</document>

<document source="https://tamnd.github.io/math2/node/SYXSZD8EFK.md" tag="SYXSZD8EFK" kind="statement">
# Kantorovich–Rubinstein duality

## Statement

Let $(M, d)$ be a [[Polish space|584GXXY73M]].
Let $\mu$ and $\nu$ be Borel [[probability measures|059TJN0NZ1]] on $M$ with finite first moments, that is, $\int_M d(x, x_0) \, d\mu(x) < \infty$ and $\int_M d(x, x_0) \, d\nu(x) < \infty$ for some $x_0 \in M$.
Let $W_1$ be the $1$-[[Wasserstein distance|VEY29CS26X]].
Then
$$W_1(\mu, \nu) = \sup \set{ \int_M f \, d\mu - \int_M f \, d\nu },$$
where the [[supremum|0R3H5PANRQ]] is over all functions $f \colon M \to \R$ with $\abs{f(x) - f(y)} \le d(x, y)$ for all $x, y \in M$.

## Remarks

The inequality $\ge$ is easy.
For a coupling $\gamma$ and a $1$-[[Lipschitz|0AW1E6YTJY]] $f$, one has $\int f \, d\mu - \int f \, d\nu = \int (f(x) - f(y)) \, d\gamma(x, y) \le \int d(x, y) \, d\gamma(x, y)$.
The reverse inequality is a case of Kantorovich duality in optimal transport, for the cost $c(x, y) = d(x, y)$.
For this cost the dual potentials can be taken $1$-Lipschitz, with the second potential equal to $-f$.

Example: for Dirac measures $\delta_x$ and $\delta_y$, the function $f = d(\cdot, y)$ is $1$-Lipschitz and gives $f(x) - f(y) = d(x, y)$.
This matches $W_1(\delta_x, \delta_y) = d(x, y)$.

The moment condition makes all the integrals finite, since a $1$-Lipschitz $f$ satisfies $\abs{f(x)} \le \abs{f(x_0)} + d(x, x_0)$.

## Explanations

### L1 human

The cheapest cost of moving one mass distribution onto another can be computed in a second way.
One looks for a price function that changes by at most the distance between two places.
The largest difference of average prices under the two distributions equals the transport cost.

### L2 human

The duality turns a minimum over couplings into a maximum over Lipschitz functions.
It shows that $W_1(\mu, \nu)$ depends only on the signed measure $\mu - \nu$, as a dual norm for the Lipschitz seminorm.
This is the form of $W_1$ used in Wasserstein generative adversarial networks, where the Lipschitz function is a trained critic.
It is often called the Kantorovich–Rubinstein theorem.
</document>

<document source="https://tamnd.github.io/math2/node/SZ1R8N1NR7.md" tag="SZ1R8N1NR7" kind="concept">
# Manifold with corners

## Summary

A smooth manifold with corners is a space modelled near each point on an open subset of the closed orthant $[0, \infty)^n$.
Its changes of chart are smooth.
It allows boundary points where several boundary faces meet at an angle, like the vertices of a square or a cube.
Products of manifolds with boundary are manifolds with corners.
This is one reason the notion is needed, for example in cobordism theory and in Stokes' theorem on cubes.

## Explanations

### L1 human

A closed square has an edge, like a closed disk, but its edge has four sharp corners.
As a topological space the square is just a disk, so the corners only matter when one wants to do calculus.
A manifold with corners keeps track of them: near a corner it looks like the corner of the quarter-plane $\set{x \ge 0, y \ge 0}$.
The solid cube is a $3$-dimensional example, with faces, edges and vertices as boundary strata.
</document>

<document source="https://tamnd.github.io/math2/node/SZ2C28JG5K.md" tag="SZ2C28JG5K" kind="definition">
# Caloric function

## Definition

Let $n \ge 1$ and let $\Omega$ be an open subset of $\R^n \times \R$, with points written $(x, t)$, $x \in \R^n$, $t \in \R$.
A *caloric function* on $\Omega$ is a function $u \colon \Omega \to \R$ such that:

1. $u$ is twice continuously differentiable in $x$ and once continuously differentiable in $t$ on $\Omega$.
2. $u$ satisfies the [[heat equation|VP2HN86VAG]]
$$\frac{\partial u}{\partial t}(x, t) = \Delta_x u(x, t) \quad \text{for all } (x, t) \in \Omega,$$
where $\Delta_x = \sum_{i=1}^n \partial^2 / \partial x_i^2$ is the [[Laplacian|0E0B6RA8K0]] in the space variables.

## Remarks

Example: on $\R \times \R$, the function $u(x, t) = x^2 + 2t$ is caloric, since $\partial_t u = 2$ and $\partial_x^2 u = 2$.
On $\R^n \times (0, \infty)$, the [[heat kernel|KD404MF042]] $\Phi(x, t) = (4 \pi t)^{-n/2} e^{-\abs{x}^2 / (4t)}$ is caloric.

Non-example: $u(x, t) = x^2 + t$ on $\R \times \R$ is smooth but not caloric, because $\partial_t u = 1$ while $\partial_x^2 u = 2$.

Some authors write the heat equation as $\partial_t u = k \Delta u$ with a constant $k > 0$; a change of the time scale reduces this to $k = 1$.
A function with $\partial_t u \le \Delta u$ is called subcaloric, and one with $\partial_t u \ge \Delta u$ is called supercaloric.
Caloric functions are also called temperatures.
</document>

<document source="https://tamnd.github.io/math2/node/SZ6MEVM89M.md" tag="SZ6MEVM89M" kind="counterexample">
# Hadamard's example of ill-posedness

## Counterexample

Consider the Cauchy problem for the [[Laplace equation|S8RG6N1PRT]] in the half-plane $y > 0$:
$$\partial_x^2 u + \partial_y^2 u = 0 \ \text{ for } y > 0, \qquad u(x, 0) = 0, \qquad \partial_y u(x, 0) = g(x).$$
For an integer $n \ge 1$, take the datum $g_n(x) = \frac{1}{n} \sin(n x)$.
The function
$$u_n(x, y) = \frac{1}{n^2} \sinh(n y) \sin(n x)$$
solves the problem.
Indeed $\partial_x^2 u_n = -n^2 u_n$ and $\partial_y^2 u_n = n^2 u_n$, so the Laplace equation holds.
Also $u_n(x, 0) = 0$, since $\sinh 0 = 0$, and $\partial_y u_n(x, 0) = \frac{1}{n} \cosh(0) \sin(n x) = g_n(x)$.
The data tend to $0$ uniformly: $\sup_x \abs{g_n(x)} = 1/n \to 0$.
For the zero datum $g = 0$ the solution is $u = 0$.
But for every fixed $y > 0$, $\sup_x \abs{u_n(x, y)} = \sinh(n y) / n^2 \to \infty$.
So the solution does not depend continuously on the data, and the problem is not [[well posed|W09B9NZ9B5]].

## Remarks

Uniqueness holds here, by [[Holmgren's uniqueness theorem|1VVKF0Y19B]], so the failure is only in the continuous dependence.
The same holds with any norm on the data that involves finitely many derivatives, since the $k$-th derivative of $g_n$ has size $n^{k-1}$, while $\sinh(n y)$ grows exponentially in $n$.
The example also explains why the [[Cauchy–Kovalevskaya theorem|97HMV9BT2W]], which does give a solution for analytic data, is not a well-posedness result.
The natural problems for the Laplace equation are boundary value problems such as the Dirichlet problem, which are well posed.

## Explanations

### L1 human

For the Laplace equation one may not prescribe both the value and the slope of the solution along a line.
Data that are tiny, but oscillate fast, produce solutions that become enormous a short distance away.
So arbitrarily small errors in the data can cause arbitrarily large errors in the solution.
</document>

<document source="https://tamnd.github.io/math2/node/SZ6XCSQA5H.md" tag="SZ6XCSQA5H" kind="definition">
# Gaussian quadrature

## Definition

Let $(a, b)$ be an interval, bounded or not, and let $w$ be a weight function on $(a, b)$.
This means that $w$ is integrable, $w > 0$ almost everywhere, and every moment $\int_a^b x^k w(x)\,dx$ is finite.
Let $n \ge 1$.
Consider a [[quadrature rule|ADSVMCY77A]]
$$\int_a^b f(x)\, w(x)\,dx \approx \sum_{i=1}^{n} w_i f(x_i)$$
with distinct nodes $x_1, \dots, x_n$ in $(a, b)$ and real weights $w_1, \dots, w_n$.
It is an $n$-point *Gaussian quadrature* rule for $w$ if it is an equality for every polynomial $f$ of degree at most $2n - 1$.

## Remarks

For each $w$ and $n$ such a rule exists and is unique.
Its nodes are the zeros of the degree $n$ [[orthogonal polynomial|8TXASARQN1]] $p_n$ for the inner product $\langle f, g \rangle = \int_a^b f g\, w\,dx$, and its weights are positive.
No $n$-point rule is exact for all polynomials of degree $2n$: the polynomial $\prod_{i}(x - x_i)^2$ has a positive integral but vanishes at every node.
So $2n - 1$ is the largest possible [[degree of exactness|ZYT1JNT1Y0]].

Example: $w = 1$ on $(-1, 1)$ and $n = 1$.
The rule $\int_{-1}^{1} f \approx 2 f(0)$ is exact for $f = 1$ (both sides are $2$) and for $f = x$ (both sides are $0$), so it is the one-point Gaussian rule, the midpoint rule.

Non-example: the trapezoidal rule $\int_{-1}^{1} f \approx f(-1) + f(1)$ has two nodes but fails for $f = x^2$, since the left side is $2/3$ and the right side is $2$.
So it is not exact up to degree $3$ and is not a Gaussian rule.

The rule for $w = 1$ on $(-1, 1)$ is [[Gauss–Legendre quadrature|RGGXDRCJVE]].
</document>

<document source="https://tamnd.github.io/math2/node/SZ7QZBDPYJ.md" tag="SZ7QZBDPYJ" kind="definition">
# Cross-validation

## Definition

Let $(x_1, y_1), \dots, (x_n, y_n)$ be data, and let a fitting rule map any subset $S$ of the data to a prediction function $\hat f_S$.
Let $\ell(y, \hat y) \ge 0$ be a [[loss function|3BTG6VVCGZ]].
Let $K$ be an integer with $2 \le K \le n$, and let $F_1, \dots, F_K$ be a partition of $\set{1, \dots, n}$ into nonempty folds.
For each $k$, let $\hat f^{(-k)}$ be the prediction function fitted on the data with indices outside $F_k$.
The *$K$-fold cross-validation estimate* of the prediction error is
$$\mathrm{CV} = \frac{1}{n} \sum_{k=1}^{K} \sum_{i \in F_k} \ell\big(y_i, \hat f^{(-k)}(x_i)\big).$$

## Remarks

The case $K = n$, with one observation per fold, is *leave-one-out* cross-validation.
To choose a tuning parameter, one computes $\mathrm{CV}$ for each candidate value and keeps a value with the smallest $\mathrm{CV}$.
Example: for least squares with $\ell(y, \hat y) = (y - \hat y)^2$, leave-one-out has a closed form.
It equals $\frac{1}{n} \sum_i \big((y_i - \hat y_i)/(1 - h_{ii})\big)^2$, where $\hat y_i$ is the full fit and $h_{ii}$ the $i$-th diagonal entry of the hat matrix.
Non-example: the training error $\frac{1}{n} \sum_i \ell(y_i, \hat f(x_i))$ with $\hat f$ fitted on all data is not cross-validation.
Each $y_i$ is predicted by a function that used it, so the error is biased downwards.
</document>

<document source="https://tamnd.github.io/math2/node/SZ8ARMDEMY.md" tag="SZ8ARMDEMY" kind="statement">
# Szegő's theorem

## Statement

Let $\mu$ be a probability measure on the unit circle with infinite support, and let $(\alpha_n)_{n \ge 0}$ be its [[Verblunsky coefficients|1RP3NVGNA6]].
Write the Lebesgue decomposition of $\mu$ as $d\mu = w(\theta) \, \frac{d\theta}{2\pi} + d\mu_s$, where $w \ge 0$ is integrable and $\mu_s$ is singular with respect to arc length.
Then
$$\prod_{n=0}^{\infty} \big( 1 - \abs{\alpha_n}^2 \big) = \exp\left( \int_0^{2\pi} \log w(\theta) \, \frac{d\theta}{2\pi} \right),$$
where both sides are $0$ when $\int_0^{2\pi} \log w(\theta) \, d\theta = -\infty$.
In particular, $\sum_{n} \abs{\alpha_n}^2 < \infty$ if and only if $\int_0^{2\pi} \log w(\theta) \, d\theta > -\infty$.

## Remarks

The integral of $\log w$ is well defined in $[-\infty, 0]$, because Jensen's inequality gives $\int \log w \, \frac{d\theta}{2\pi} \le \log \int w \, \frac{d\theta}{2\pi} \le 0$.
By the norm formula for the [[monic orthogonal polynomials|3F9G3DRQ62]], the left side equals $\lim_{n \to \infty} \norm{\Phi_n}^2$.
And $\norm{\Phi_n}^2 = \min \set{ \norm{p}^2 : p \text{ monic of degree } n }$, so the theorem computes the best mean-square prediction error of $z^n$ from the past.
A standard proof treats this minimum as an extremal problem in the Hardy space $H^2$ of the disk.
Along the way, the singular part $\mu_s$ turns out not to affect the limit.

Equivalently, in terms of the [[Toeplitz matrices|GKY2WSHE26]] $T_n = (c_{j-k})_{j,k=0}^{n}$ of the moments $c_k = \int e^{-ik\theta} \, d\mu$, one has $\det T_n / \det T_{n-1} = \norm{\Phi_n}^2$.
So the theorem also gives $\lim_{n \to \infty} (\det T_n)^{1/(n+1)} = \exp \int \log w \, \frac{d\theta}{2\pi}$, Szegő's first limit theorem.

Example: for $d\mu = d\theta/2\pi$ one has $w = 1$ and all $\alpha_n = 0$, and both sides equal $1$.
If $\mu$ is singular, $w = 0$ almost everywhere, so $\prod (1 - \abs{\alpha_n}^2) = 0$ and $\sum \abs{\alpha_n}^2 = \infty$.

## Explanations

### L1 human

A measure on the circle is encoded by its Verblunsky coefficients, a sequence of points in the unit disk.
Szegő's theorem links the size of these coefficients to how small the density of the measure can get.
The coefficients are square-summable exactly when the density is not too close to zero, in the sense that the integral of its logarithm is finite.

### L2 human

The condition $\int \log w > -\infty$ is the Szegő condition.
The same condition governs the factorization $w = \abs{D}^2$ with an outer function $D$ in the Hardy space.
It also governs prediction for stationary sequences: the future is not determined by the past exactly when it holds.
The theorem is a sum rule, relating the coefficients to the measure, and it is a starting point for finer results such as the [[Szegő limit theorems|WNJPRHZ680]] on Toeplitz determinants.
</document>

<document source="https://tamnd.github.io/math2/node/SZ8NCJ4TC5.md" tag="SZ8NCJ4TC5" kind="definition">
# Semiorder

## Definition

Let $X$ be a set and let $\prec$ be a [[strict partial order|60FNZ35H6R]] on $X$.
Say that $x$ and $y$ are incomparable when neither $x \prec y$ nor $y \prec x$.
The order $\prec$ is a *semiorder* if, for all $a, b, c, d \in X$:

1. if $a \prec b$ and $c \prec d$, then $a \prec d$ or $c \prec b$;
2. if $a \prec b$ and $b \prec c$, then $a \prec d$ or $d \prec c$.

## Remarks

Clause 1 forbids two incomparable two-element chains, and clause 2 forbids a three-element chain with an element incomparable to all three of its elements.
Example: on $\R$, let $x \prec y$ when $x + 1 < y$.
For clause 2, if $a + 1 < b$ and $b + 1 < c$ but $d \le a + 1$, then $d + 1 \le a + 2 < c$, so $d \prec c$; clause 1 has a similar check.
Non-example: the poset with $a \prec b \prec c$ and an element $d$ incomparable to $a$, $b$ and $c$ satisfies clause 1 but fails clause 2.
Every semiorder is an [[interval order|VSNQV29KFJ]].
By a theorem of Scott and Suppes, a finite strict partial order is a semiorder if and only if there is a map $u \colon X \to \R$ with $x \prec y \iff u(x) + 1 < u(y)$.
</document>

<document source="https://tamnd.github.io/math2/node/SZ9X10ATEK.md" tag="SZ9X10ATEK" kind="definition">
# Bounded poset

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
The poset $P$ is *bounded* when it has a [[greatest element|0Z7HDDXX3X]] $\top$ and a [[least element|0T2NTJCV83]] $\bot$, that is, $\bot \le x \le \top$ for all $x \in P$.

## Remarks

The positive divisors of $12$, ordered by divisibility, form a bounded poset with $\bot = 1$ and $\top = 12$, since $1 \mid d$ and $d \mid 12$ for each divisor $d$.
The poset $(\Z, \le)$ is not bounded: it has neither a greatest nor a least element.
The interval $(0, 1]$ in $\R$ has the top $1$ but no bottom, so it is not bounded.
For a subset of $P$, being bounded usually means something weaker: having an [[upper bound|0H5J02P8S9]] and a lower bound in $P$.
The elements $\top$ and $\bot$ are unique when they exist, by antisymmetry.
A bounded poset with one element has $\top = \bot$.

## Explanations

### L1 human

A bounded poset has a top and a bottom.
Every element lies between the two.
</document>

<document source="https://tamnd.github.io/math2/node/SZEYSY99AM.md" tag="SZEYSY99AM" kind="example">
# Fundamental domain of the modular group

## Example

Let the [[modular group|VEKV1EM77B]] $\Gamma = \mathrm{SL}_2(\Z)$ act on the [[upper half-plane|X6XGGH5FPW]] $\mathbb H$ by [[Möbius transformations|0SG74WBFA2]], generated by $T \tau = \tau + 1$ and $S \tau = -1/\tau$.
Let
$$\mathcal F = \set{\tau \in \mathbb H : \abs{\operatorname{Re} \tau} \le \tfrac12, \ \abs{\tau} \ge 1}.$$
Then $\mathcal F$ is a [[fundamental domain|KNJA2NEZ9F]] for $\Gamma$:

1. every [[orbit|0YQPZB5Q1G]] of $\Gamma$ in $\mathbb H$ meets $\mathcal F$;
2. two distinct points of $\mathcal F$ lie in the same orbit only if they lie on the boundary of $\mathcal F$.

Check of 1: for $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, one has $\operatorname{Im}(\gamma \tau) = \operatorname{Im} \tau / \abs{c\tau + d}^2$.
Only finitely many pairs $(c, d)$ of integers have $\abs{c\tau + d} \le 1$, so some $\gamma$ makes $\operatorname{Im}(\gamma \tau)$ maximal.
Apply a power of $T$ to move $\gamma \tau$ into the strip $\abs{\operatorname{Re} \tau} \le \tfrac12$; this does not change the imaginary part.
If the result $\tau'$ had $\abs{\tau'} < 1$, then $\operatorname{Im}(S \tau') = \operatorname{Im} \tau' / \abs{\tau'}^2 > \operatorname{Im} \tau'$, against maximality, so $\tau' \in \mathcal F$.

Check of 2: there are two boundary identifications.
The map $T$ sends the line $\operatorname{Re} \tau = -\tfrac12$ to the line $\operatorname{Re} \tau = \tfrac12$.
The map $S$ sends the arc $\abs{\tau} = 1$ to itself by $\tau \mapsto -\bar\tau$.
A short computation with $\abs{c\tau + d} \ge 1$ on $\mathcal F$ shows there are no other identifications.

## Remarks

The hyperbolic area of $\mathcal F$ is
$$\int_{-1/2}^{1/2} \int_{\sqrt{1 - x^2}}^{\infty} \frac{dy \, dx}{y^2} = \int_{-1/2}^{1/2} \frac{dx}{\sqrt{1 - x^2}} = 2 \arcsin \tfrac12 = \frac{\pi}{3}.$$
In $\mathcal F$, the points with nontrivial stabilizer in $\mathrm{PSL}_2(\Z)$ are the following.
The point $i$ is fixed by $S$ of order $2$, and the corners $e^{2\pi i/3}$ and $e^{\pi i/3}$ are fixed by elements of order $3$.
Gluing the sides of $\mathcal F$ by $T$ and $S$ gives the modular curve $\Gamma \backslash \mathbb H$, a sphere with one puncture at the cusp $i\infty$.

## Explanations

### L1 human

The modular group moves points of the upper half-plane around in many ways.
The region $\mathcal F$, a strip of width one above the unit circle, contains a copy of every point up to these moves, and only its edges are counted twice.
Its images under the group tile the whole half-plane by curved triangles.
A modular form is determined by its values on this one region.
</document>

<document source="https://tamnd.github.io/math2/node/SZHY975FFX.md" tag="SZHY975FFX" kind="concept">
# Reduced ring

## Summary

A reduced ring is a ring with no nonzero [[nilpotent elements|0N9R5NN6TW]].
For a commutative ring this means that its [[nilradical|QQDN6EYQNJ]] is zero.
Every [[integral domain|09S3PFBEPM]] is reduced, and the coordinate rings of algebraic varieties are reduced.
Dividing a commutative ring by its nilradical always gives a reduced ring.

## Explanations

### L1 human

In a reduced ring, if some power of an element is zero, then the element is already zero.
The ring $\Z/6\Z$ is reduced, while $\Z/4\Z$ is not, because $2 \ne 0$ but $2^2 = 4 = 0$ there.
A reduced ring may still have zero divisors: in $\Z/6\Z$, $2 \cdot 3 = 0$.
Geometrically, a reduced ring is a ring of functions in which a function that vanishes everywhere is zero.
</document>

<document source="https://tamnd.github.io/math2/node/SZV8Y806Y7.md" tag="SZV8Y806Y7" kind="concept">
# Admissible representation

## Summary

An admissible representation of a reductive group is one in which each irreducible representation of a maximal compact subgroup occurs only finitely many times.
The condition singles out the representations that can be studied by algebraic means, through their [[Harish-Chandra modules|2MKHH636C3]].
Every irreducible unitary representation of a real reductive group is admissible, a theorem of Harish-Chandra.
Over $p$-adic fields the analogous notion asks that the vectors fixed by each compact open subgroup form a finite-dimensional space.

## Explanations

### L1 human

Representations of non-compact groups such as $\mathrm{SL}_2(\R)$ usually act on infinite-dimensional spaces.
Restrict the action to the compact subgroup of rotations $\mathrm{SO}(2)$, and the space breaks into simple pieces, one for each "frequency" $e^{in\theta}$.
Admissible means that each frequency appears only a finite number of times.
So the representation is infinite, but it is infinite in a controlled way.
</document>

<document source="https://tamnd.github.io/math2/node/T01779NV3Z.md" tag="T01779NV3Z" kind="definition">
# Ordinary point and singular point

## Definition

Let $U \subseteq \C$ be open, $z_0 \in U$, and let $p_1, \ldots, p_n$ be [[meromorphic functions|0ZZ7ZYDR8W]] on $U$.
Consider the homogeneous [[linear equation|0QQRP17Z7S]]
$$y^{(n)} + p_1(z)\, y^{(n-1)} + \cdots + p_n(z)\, y = 0.$$
The point $z_0$ is an *ordinary point* of the equation if every $p_k$ is [[analytic|081WKEDK6C]] at $z_0$ (after removing removable singularities).
Otherwise $z_0$ is a *singular point*: some $p_k$ has a pole at $z_0$.

## Remarks

The point $\infty$ of the [[Riemann sphere|0J78KBPFZB]] is ordinary or singular according as $w = 0$ is ordinary or singular for the equation obtained by the substitution $z = 1/w$.
For real equations with real-analytic coefficients the same words are used, with $z$ replaced by a real variable $x$.
The definition applies to the normalised equation; for $a_0(z) y'' + a_1(z) y' + a_2(z) y = 0$ one first divides by $a_0$.

Example: for $y'' - 2 z y' + 4 y = 0$, the coefficients $-2z$ and $4$ are polynomials, so every $z_0 \in \C$ is an ordinary point.

Non-example: for $z^2 y'' + z y' + (z^2 - 1) y = 0$, the normalised coefficient $p_1(z) = \frac{1}{z}$ has a pole at $0$, so $0$ is not an ordinary point but a singular point.

At an ordinary point $z_0$, every solution is analytic, and its Taylor series at $z_0$ converges at least on the largest disc around $z_0$ on which all $p_k$ are analytic.
</document>

<document source="https://tamnd.github.io/math2/node/T046A1477B.md" tag="T046A1477B" kind="definition">
# Fourier–Legendre series

## Definition

Let $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$, for $n \ge 0$, normalized by $P_n(1) = 1$.
Let $f \colon [-1, 1] \to \C$ be a [[Lebesgue integrable|DWJF7ABPR1]] function.
The *Fourier–Legendre coefficients* of $f$ are
$$a_n = \frac{2n + 1}{2} \int_{-1}^{1} f(x) P_n(x) \, dx, \qquad n \ge 0.$$
The *Fourier–Legendre series* of $f$ is the formal series
$$\sum_{n=0}^{\infty} a_n P_n(x).$$

## Remarks

The factor $(2n+1)/2$ comes from the orthogonality relations $\int_{-1}^{1} P_m P_n \, dx = \frac{2}{2n+1} \delta_{mn}$.
For $f$ in the [[space|SASP14MA1Z]] $L^2([-1, 1])$, the series converges to $f$ in $L^2$, because the Legendre polynomials form a complete orthogonal system there.

Example: let $f(x) = x^2$.
Then $a_0 = \frac12 \int_{-1}^1 x^2 \, dx = \frac13$, $a_1 = \frac32 \int_{-1}^1 x^3 \, dx = 0$, and $a_2 = \frac52 \int_{-1}^1 x^2 \cdot \frac{3x^2 - 1}{2} \, dx = \frac52 \cdot \frac{4}{15} = \frac23$.
All $a_n$ with $n \ge 3$ vanish, since $x^2$ is a combination of $P_0$ and $P_2$ and is orthogonal to $P_n$ for $n \ge 3$.
So the series is $\frac13 P_0 + \frac23 P_2$, which equals $x^2$.

Non-example: $f(x) = 1/(1 - x)$ on $[-1, 1)$ is not integrable near $x = 1$.
Already $a_0 = \frac12 \int_{-1}^{1} \frac{dx}{1 - x}$ diverges, so $f$ has no Fourier–Legendre series.
</document>

<document source="https://tamnd.github.io/math2/node/T060JHSH05.md" tag="T060JHSH05" kind="counterexample">
# Simpson's paradox

## Counterexample

The claim refuted is this.
If treatment $A$ has a higher success rate than treatment $B$ in every subgroup, then $A$ has the higher rate in the whole population.

Consider two subgroups, mild and severe cases, summarised in a [[contingency table|PANSMHE0SW]] for each subgroup.

1. Mild cases: $A$ succeeds in $9$ of $10$ patients ($90\%$), $B$ in $80$ of $100$ patients ($80\%$).
2. Severe cases: $A$ succeeds in $30$ of $100$ patients ($30\%$), $B$ in $2$ of $10$ patients ($20\%$).
3. All cases: $A$ succeeds in $39$ of $110$ patients (about $35\%$), $B$ in $82$ of $110$ patients (about $75\%$).

So $A$ is better in each subgroup, yet $B$ is better overall.

## Remarks

The reason is that the overall rate is a weighted average of the subgroup rates, and the weights differ between the treatments.
Most patients given $A$ are severe cases, and most patients given $B$ are mild cases.
In terms of [[conditional probability|0990W5PN13]], $P(S \mid A, G) > P(S \mid B, G)$ for each subgroup $G$ does not imply $P(S \mid A) > P(S \mid B)$.
The same reversal can happen for correlations: a trend in each group can be reversed when the groups are pooled.
Which comparison is relevant depends on the causal structure, not on the numbers alone.

## Explanations

### L1 human

A treatment can look better in every group of patients and still look worse when all patients are counted together.
This happens when the treatments were given to very different mixes of patients.
Here the better treatment was mostly given to the hard cases, so its overall success rate is dragged down.
</document>

<document source="https://tamnd.github.io/math2/node/T06PG3RWC0.md" tag="T06PG3RWC0" kind="example">
# Permutohedron

## Example

Let $n \ge 1$.
The *permutohedron* $\Pi_n$ is the convex hull in $\R^n$ of the $n!$ points
$$(\sigma(1), \sigma(2), \dots, \sigma(n)),$$
where $\sigma$ runs over all [[permutations|05RWFKVTJ7]] of $\set{1, \dots, n}$.

All these points satisfy $x_1 + \cdots + x_n = n(n+1)/2$, so $\Pi_n$ lies in a hyperplane, and it has dimension $n - 1$.
Each of the $n!$ points is a [[vertex|R7V3S9V891]]: the vector $c = (\sigma(1), \dots, \sigma(n))$ is maximised over the points uniquely at that point, by the rearrangement inequality.
The [[facets|DPDXN2FJC4]] correspond to the nonempty proper subsets $S \subset \set{1, \dots, n}$, with facet inequality
$$\sum_{i \in S} x_i \ge \frac{\abs{S}(\abs{S}+1)}{2},$$
so there are $2^n - 2$ facets.
Two vertices are joined by an edge exactly when one is obtained from the other by swapping the values $k$ and $k+1$ for some $k$.
Hence every vertex lies in $n - 1$ edges and $\Pi_n$ is a [[simple polytope|7E6FXTGH2Y]].

$\Pi_n$ is a [[zonotope|Z5SA6A67KF]]: it is a translate of the Minkowski sum of the segments $[0, e_j - e_i]$ for $1 \le i < j \le n$.

## Remarks

For $n = 3$, $\Pi_3$ is a regular hexagon with vertices the six permutations of $(1,2,3)$.
For $n = 4$, $\Pi_4$ is the truncated octahedron, with $24$ vertices, $14$ facets ($6$ squares and $8$ hexagons) and $36$ edges.

## Explanations

### L1 human

Write the numbers $1$ to $n$ in every possible order and treat each order as a point in space.
The solid spanned by these points is the permutohedron.
Its corners are the orderings, and walking along an edge means swapping two neighbouring values.
For three numbers it is a hexagon, for four numbers a solid with $24$ corners built from squares and hexagons.
</document>

<document source="https://tamnd.github.io/math2/node/T08FF77H16.md" tag="T08FF77H16" kind="definition">
# Complemented lattice

## Definition

Let $L$ be a [[bounded lattice|72V1Z82M5F]] with least element $0$, greatest element $1$, and [[join and meet|J2T75ATRV5]] $\vee$ and $\wedge$.
A *complement* of an element $a \in L$ is an element $b \in L$ such that
$$a \wedge b = 0 \quad\text{and}\quad a \vee b = 1.$$
The lattice $L$ is *complemented* if every element of $L$ has at least one complement.

## Remarks

Example: the lattice of subspaces of a finite-dimensional vector space $V$.
For a subspace $U$, extend a basis of $U$ to a basis of $V$ and let $W$ be the span of the added vectors; then $U \cap W = \set{0}$ and $U + W = V$.

Non-example: the chain $0 < a < 1$.
The element $a$ has no complement: $a \wedge x = 0$ forces $x = 0$, and then $a \vee x = a \ne 1$.

In the [[diamond lattice M3|JQ6FJH2TP8]] each of the three middle elements has the other two as complements, so complements are not unique.
In a [[distributive lattice|0HM0HQCK6N]] an element has at most one complement, and a complemented distributive lattice is a [[Boolean algebra|0XADJKM699]].
</document>

<document source="https://tamnd.github.io/math2/node/T09DXZAT6R.md" tag="T09DXZAT6R" kind="definition">
# Morphism of finite type

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].
It is *locally of finite type* if there is a cover of $Y$ by open [[affine|B3V3XJ2W8G]] subschemes $V_i = \operatorname{Spec} B_i$ with the following property.
For each $i$, the open set $f^{-1}(V_i)$ is covered by open affine subschemes $U_{ij} = \operatorname{Spec} A_{ij}$, where each $A_{ij}$ is a [[finitely generated|AC5TFFQT9M]] $B_i$-algebra.
It is *of finite type* if moreover each $f^{-1}(V_i)$ can be covered by finitely many such $U_{ij}$.

## Remarks

Equivalently, $f$ is of finite type if and only if it is locally of finite type and [[quasi-compact|41009J4M5D]].
A morphism $\operatorname{Spec} A \to \operatorname{Spec} B$ is of finite type if and only if $A$ is a finitely generated $B$-algebra.
Example: for a field $k$, the morphism $\mathbb{A}^n_k \to \operatorname{Spec} k$ is of finite type, since $k[x_1, \ldots, x_n]$ is generated by the $x_i$.
Projective space $\mathbb{P}^n_k \to \operatorname{Spec} k$ is of finite type too, being covered by $n + 1$ such affine pieces.
Non-example: $\operatorname{Spec} k(x) \to \operatorname{Spec} k$ is not of finite type.
If the field $k(x)$ were a finitely generated $k$-algebra, then by [[Zariski's lemma|YBJ2VQHFDQ]] it would be a finite extension of $k$, which it is not.
</document>

<document source="https://tamnd.github.io/math2/node/T0AEF3Z8C9.md" tag="T0AEF3Z8C9" kind="concept">
# Frölicher space

## Summary

A Frölicher space is a set with a family of smooth curves into it and a family of smooth real functions on it.
Each of the two families determines the other.
A curve is smooth when every smooth function composed with it is smooth.
A function is smooth when it composes smoothly with every smooth curve.
Frölicher spaces generalize [[smooth manifolds|F45VQ47GPV]] and are used for infinite-dimensional spaces such as spaces of smooth maps.
They are closely related to [[diffeological spaces|7JS0JFR89E]] and form a cartesian closed category.

## Explanations

### L1 human

One can learn a lot about a space by testing it from two sides: with curves going into it and with functions going out of it.
A Frölicher space fixes both families so that they match perfectly.
A curve passes the test if every allowed function sees it as smooth, and a function passes if it sees every allowed curve as smooth.
For ordinary Euclidean space, the usual smooth curves and smooth functions match in this way.
</document>

<document source="https://tamnd.github.io/math2/node/T0BWYYR508.md" tag="T0BWYYR508" kind="statement">
# Equivalent definition with relaxed axioms

## Statement

The group axioms for identity and inverses may be "weakened" to assert only the existence of a left identity and left inverses. From these *one-sided axioms*, one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are not weaker.

In particular, assuming associativity and the existence of a left identity $e$ (that is, $e\cdot f=f$) and a left inverse $f^{-1}$ for each element $f$ (that is, $f^{-1}\cdot f=e$), it follows that every left inverse is also a right inverse of the same element as follows. Indeed, one has

$$\begin{aligned} f \cdot f^{-1} &=e \cdot (f \cdot f^{-1}) && \text{(left identity)}\\ &=((f^{-1})^{-1} \cdot f^{-1}) \cdot (f \cdot f^{-1}) && \text{(left inverse)}\\ &=(f^{-1})^{-1} \cdot ((f^{-1} \cdot f) \cdot f^{-1}) && \text{(associativity)}\\ &=(f^{-1})^{-1} \cdot (e \cdot f^{-1}) && \text{(left inverse)}\\ &=(f^{-1})^{-1} \cdot f^{-1} && \text{(left identity)}\\ &=e && \text{(left inverse)} \end{aligned}$$

Similarly, the left identity is also a right identity:

$$\begin{aligned} f\cdot e &= f \cdot ( f^{-1} \cdot f) && \text{(left inverse)}\\ &= (f \cdot f^{-1}) \cdot f && \text{(associativity)}\\ &= e \cdot f && \text{(right inverse)}\\ &= f && \text{(left identity)} \end{aligned}$$

These results do not hold if any of these axioms (associativity, existence of left identity and existence of left inverse) is removed. For a structure with a looser definition (like a [[semigroup|0XMEM406JM]]) one may have, for example, that a left identity is not necessarily a right identity.

The same result can be obtained by only assuming the existence of a right identity and a right inverse.

However, only assuming the existence of a *left* identity and a *right* inverse (or vice versa) is not sufficient to define a group. For example, consider the set $G = \{ e,f \}$ with the operator $\,\!\cdot$ satisfying $e \cdot e = f \cdot e = e$ and $e \cdot f = f \cdot f = f$. This structure does have a left identity (namely, $e$), and each element has a right inverse (which is $e$ for both elements). Furthermore, this operation is associative (since the product of any number of elements is always equal to the rightmost element in that product, regardless of the order in which these operations are applied). However, $( G , \cdot )$ is not a group, since it lacks a right identity.
</document>

<document source="https://tamnd.github.io/math2/node/T0JZN6WP5X.md" tag="T0JZN6WP5X" kind="concept">
# Moving least squares

## Summary

Moving least squares builds a smooth function from scattered data in $\R^d$.
At each point $x$ it fits a low-degree polynomial to the data by [[weighted least squares|H5CGZHBR58]], with weights that favour data points close to $x$.
The value of the reconstruction at $x$ is the value of this local fit at $x$.
As $x$ moves, the weights and so the local polynomial change continuously, which gives a smooth global approximation.
It is a basic tool of [[multivariate interpolation|BM703GV34G]] and approximation, of surface reconstruction from point clouds and of meshfree methods for differential equations.

## Explanations

### L1 human

To estimate a value at a point, look at the data near that point.
Fit a simple model such as a plane by least squares, and read off the model at that point.
Do this afresh at every point, always giving more weight to nearby data.
Because the weights change gradually from point to point, the estimates fit together into a smooth surface.
If the model is just a constant, the fit is a weighted average of the data, as in [[inverse distance weighting|QHN67MSH8S]].
The [[definition|9HWTPQHHZK]] gives the formula.
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# Subalgebra lattice

## Summary

The subalgebra lattice of an [[algebra|QJGPT2W0FT]] is the set of all its subuniverses, ordered by inclusion.
It is a [[complete lattice|D7A0ZGZT5S]], and in fact an [[algebraic lattice|3J0BTXSCVN]] whose [[compact elements|9F7TYES4KE]] are the finitely generated subuniverses.
It generalizes the [[lattice of subgroups|0B8BW4YWTF]] of a group.
By the [[Birkhoff–Frink theorem|YFT50501A0]], every algebraic lattice arises in this way.

## Explanations

### L1 human

Collect all the subsets of an algebra that are closed under its operations, and order them by inclusion.
Two of them always have a largest common lower bound, their intersection, and a smallest common upper bound, the subalgebra generated by their union.
So they form a lattice, which records how the algebra is built from its pieces.
For the group $\Z/4\Z$ the lattice is a chain of three subgroups: $\set{0}$, $\set{0, 2}$ and the whole group.
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# Minakshisundaram–Pleijel zeta function

## Summary

The Minakshisundaram–Pleijel zeta function of a closed Riemannian manifold is the Dirichlet series $\sum \lambda_j^{-s}$ over the nonzero eigenvalues of its Laplacian.
It has a pointwise version, with each term weighted by eigenfunctions.
It converges for $\operatorname{Re} s > n/2$ and continues to a meromorphic function on $\C$.
Its poles and special values carry geometric information, and its derivative at $0$ defines the [[zeta-regularized determinant|YT3ZQ2F7FG]] of the Laplacian.
It generalizes the [[Riemann zeta function|1XWBY4N0VT]], which arises from the circle.

## Explanations

### L1 human

The eigenvalues of the Laplacian on a closed manifold are the squared frequencies of its vibrations.
Summing their negative powers gives a function of $s$, built like the Riemann zeta function from the positive integers.
For the circle of length $2\pi$ the eigenvalues are $1, 1, 4, 4, 9, 9, \ldots$, and the zeta function is $2\zeta(2s)$.
In general one cannot write it down, but its analytic properties follow from the heat kernel.
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# Liouville's theorem for planar harmonic functions

## Statement

Let $u \colon \C \to \R$ be a [[harmonic function|0DEB0JMN0W]] on the whole plane.
Assume that $u$ is bounded above or bounded below.
Then $u$ is constant.

## Remarks

Proof idea: $\C$ is simply connected, so by [[the existence of harmonic conjugates|CVDGP7DKZ3]] there is an entire function $f$ with $\operatorname{Re} f = u$.
If $u \le M$ on $\C$, then $g = e^{f}$ is entire with $\abs{g} = e^{u} \le e^{M}$.
By [[Liouville's theorem|0EHKVRF4J6]], $g$ is constant, so $f' e^{f} = g' = 0$, hence $f' = 0$ and $u$ is constant.
If $u$ is bounded below, apply this to $-u$.
A one-sided bound is needed: $u(x + iy) = x$ is harmonic, nonconstant, and unbounded in both directions.
The theorem is the planar case of [[Liouville's theorem for harmonic functions|QK1KWYZNRD]] on $\R^n$.
In the plane even more is true: a [[subharmonic function|Z1MHMGFNFW]] on $\C$ that is bounded above is constant.
This stronger form fails in $\R^n$ for $n \ge 3$, where $-\min(1, \abs{x}^{2-n})$ is subharmonic, bounded above, and not constant.

## Explanations

### L1 human

A harmonic function on the whole plane cannot stay below a ceiling unless it is flat.
Think of a membrane stretched over the entire plane: if it never rises above some height, it must be level everywhere.

### L2 human

The proof shows a typical planar technique: pass to a holomorphic function with real part $u$ and use complex analysis.
The theorem gives quick uniqueness results, for example for harmonic functions on $\C$ with a given one-sided bound.
Its subharmonic form says that $\C$ is a parabolic surface in potential theory: it carries no nonconstant negative subharmonic function.
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# Band matrix

## Definition

Let $R$ be a [[ring|0B1261B5ED]], let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$, and let $p, q \ge 0$ be integers.
The matrix $A$ is a *band matrix* with *lower bandwidth* $p$ and *upper bandwidth* $q$ when
$$a_{ij} = 0 \quad \text{whenever } i > j + p \text{ or } j > i + q.$$
The *bandwidth* of $A$ is often taken as the least $k$ such that the condition holds with $p = q = k$.

## Remarks

Example: $\begin{pmatrix} 1 & 2 & 0 & 0 \\ 3 & 4 & 5 & 0 \\ 6 & 7 & 8 & 9 \\ 0 & 1 & 2 & 3 \end{pmatrix}$ has lower bandwidth $2$ and upper bandwidth $1$: the entries with $i > j + 2$ or $j > i + 1$ are all $0$.
Non-example: $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ has no upper bandwidth $q = 1$, because $a_{13} = 1$ and $3 > 1 + 1$.
With $p = q = 0$ one gets the diagonal matrices, and with $p = q = 1$ the [[tridiagonal matrices|DARX1KZRHZ]].
A square matrix with $p = 0$ is [[upper triangular|0N4XY7Z3NN]].
The term is mostly used when $p$ and $q$ are small compared with the size of the matrix.
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# Positive current

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $n$ and let $0 \le p \le n$.
A [[current|360H7MD9RD]] $T$ of bidegree $(p, p)$ on $X$ is *positive* if
$$T \wedge \left(i \alpha_1 \wedge \bar\alpha_1\right) \wedge \cdots \wedge \left(i \alpha_{n-p} \wedge \bar\alpha_{n-p}\right) \ge 0$$
for every choice of smooth [[$(1, 0)$-forms|DC81TCR98Y]] $\alpha_1, \dots, \alpha_{n-p}$ on $X$.
Here an $(n, n)$-current $S$ is identified with a distribution, and $S \ge 0$ means $S(\chi) \ge 0$ for every smooth compactly supported function $\chi \ge 0$.
Then $S$ is a positive measure.
A current is *closed* if $dT = 0$ in the sense of currents.

## Remarks

The condition is the same as $T(\varphi) \ge 0$ for all compactly supported smooth strongly positive $(n - p, n - p)$-forms $\varphi$, with the current acting on forms of complementary degree.
A positive current is real, and its coefficients in local coordinates are complex measures.
Example: for $X = \C^n$ and $p = 1$, the smooth form $\omega = i \partial\bar\partial \abs{z}^2 = i \sum_j dz_j \wedge d\bar z_j$ is positive, because its coefficient matrix is the identity matrix (see the criterion for $p = 1$ below).
Example: let $Z \subseteq X$ be a complex analytic subset of pure dimension $n - p$.
By Lelong's theorem, the current of integration $[Z](\varphi) = \int_{Z_{\mathrm{reg}}} \varphi$ is a closed positive current of bidegree $(p, p)$.
Example: for $u$ [[plurisubharmonic|FF2P7ER3QT]], $i \partial\bar\partial u$ is a closed positive $(1, 1)$-current.
Non-example: on $\C^2$, the form $i \, dz_1 \wedge d\bar z_1 - i \, dz_2 \wedge d\bar z_2$ is not positive; take $\alpha_1 = dz_1$, then the product is $-i\,dz_2 \wedge d\bar z_2 \wedge i \, dz_1 \wedge d\bar z_1$, which is $-4$ times Lebesgue measure.
For $p = 1$, a $(1,1)$-form $i \sum h_{jk} \, dz_j \wedge d\bar z_k$ is positive exactly when the Hermitian matrix $(h_{jk})$ is positive semidefinite at each point.
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# Holomorphic map between Riemann surfaces

## Summary

A holomorphic map between [[Riemann surfaces|5K0N8QMV6Q]] is a continuous map that is holomorphic when written in local coordinates on both sides.
These maps are the morphisms of the category of Riemann surfaces; the bijective ones are the biholomorphisms, which identify two Riemann surfaces.
Meromorphic functions on a Riemann surface are the same as holomorphic maps to the [[Riemann sphere|0CDMXGPSES]] that are not constantly $\infty$.

## Explanations

### L1 human

On a Riemann surface you cannot differentiate directly, but you can always look through a chart, which is a piece of the complex plane.
A map between two surfaces is holomorphic if every such view of it is an ordinary holomorphic function.
For instance $z \mapsto z^2$ on the Riemann sphere, with $\infty \mapsto \infty$, is holomorphic: near $\infty$ in the coordinate $w = 1/z$ it reads $w \mapsto w^2$.
Near any point where it is not constant, a holomorphic map looks like $z \mapsto z^k$ for some $k \ge 1$ in suitable coordinates.
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# Mathieu groups

## Description

The Mathieu groups are five finite [[simple groups|0YMYMNQ0QZ]], written $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$ and $M_{24}$.
Each $M_n$ is a permutation group on $n$ points.
Their orders are:

1. $\abs{M_{11}} = 7920 = 2^4 \cdot 3^2 \cdot 5 \cdot 11$,
2. $\abs{M_{12}} = 95040 = 2^6 \cdot 3^3 \cdot 5 \cdot 11$,
3. $\abs{M_{22}} = 443520 = 2^7 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11$,
4. $\abs{M_{23}} = 10200960 = 2^7 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 \cdot 23$,
5. $\abs{M_{24}} = 244823040 = 2^{10} \cdot 3^3 \cdot 5 \cdot 7 \cdot 11 \cdot 23$.

They are sporadic groups: they belong to none of the infinite families of finite simple groups.

## Remarks

The Mathieu groups are highly [[multiply transitive|BDJ1AX4XBB]].
$M_{11}$ is sharply $4$-transitive on $11$ points, and $M_{12}$ is sharply $5$-transitive on $12$ points.
$M_{22}$ is $3$-transitive on $22$ points, $M_{23}$ is $4$-transitive on $23$ points, and $M_{24}$ is $5$-transitive on $24$ points.
Apart from the symmetric and alternating groups, $M_{11}$, $M_{12}$, $M_{23}$ and $M_{24}$ are the only finite $4$-transitive groups.
The [[stabiliser|0AYMP41THN]] of a point in $M_{24}$ is $M_{23}$, and the stabiliser of a point in $M_{23}$ is $M_{22}$; likewise the stabiliser of a point in $M_{12}$ is $M_{11}$.
$M_{24}$ is the [[automorphism group|0K1SCPBNKJ]] of the Steiner system $S(5, 8, 24)$ and of the extended binary Golay code, and $M_{12}$ is the automorphism group of the Steiner system $S(5, 6, 12)$.
They were the first sporadic simple groups to be found, by Émile Mathieu in the nineteenth century.

## Explanations

### L1 human

Most finite simple groups come in infinite families, like the alternating groups.
The Mathieu groups are five exceptions that do not fit any family.
They are permutation groups on $11$, $12$, $22$, $23$ and $24$ points, and they are surprisingly symmetric.
For example, $M_{24}$ can move any $5$ of its $24$ points to any other $5$ points, in any order.
They are also the symmetry groups of remarkable combinatorial objects, such as the Golay code used in error correction.
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# Borel–Moore homology

## Summary

Borel–Moore homology is a homology theory for locally compact spaces in which cycles may be infinite, as long as they are locally finite.
It is also called homology with closed supports.
For an oriented $n$-manifold $M$ it is dual to ordinary cohomology, $H^{\mathrm{BM}}_i(M) \cong H^{n-i}(M)$, even when $M$ is not compact.
It is the natural home of fundamental classes of non-compact manifolds and of closed subvarieties of complex varieties.

## Explanations

### L1 human

In ordinary homology a cycle is a finite sum of small pieces.
In Borel–Moore homology a cycle may have infinitely many pieces, provided only finitely many of them crowd near any one point.
The whole real line, cut into the intervals $[k, k+1]$ for all integers $k$, is such a cycle.
It has no boundary, because each endpoint is shared by two intervals and cancels.
So the line has a fundamental class in Borel–Moore homology, though it has none in ordinary homology.
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# Compositum of fields

## Summary

The compositum of two subfields $E$ and $F$ of a field $\Omega$ is the smallest subfield of $\Omega$ that contains both, written $EF$.
It is the field [[generated|05NAS7AGMZ]] by $E \cup F$, and when $E = K(\alpha_1, \ldots, \alpha_n)$ it equals $F(\alpha_1, \ldots, \alpha_n)$.
Composita appear throughout Galois theory: the compositum of two Galois extensions is Galois, and its degree is controlled by [[linear disjointness|N6YZFG66YP]].
Without a common field $\Omega$ the compositum is not defined, and the [[tensor product of fields|ANH16GQJZQ]] describes all possible ways to form one.

## Explanations

### L1 human

Put two fields side by side inside one big field.
Their union is usually not a field, because sums like $\sqrt{2} + \sqrt{3}$ are missing.
The compositum is what you get by allowing all sums, products and quotients of elements from both.
For $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ inside $\C$, the compositum is $\Q(\sqrt{2}, \sqrt{3})$, a field of degree $4$ over $\Q$.
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# Partial autocorrelation function

## Definition

Let $(X_t)_{t \in \Z}$ be a real weakly [[stationary process|DT0QXJE705]] with mean $\mu$ and autocovariance $\gamma$, with $\gamma(0) > 0$.
Let $L^2$ be the space of square-integrable random variables.
For $t \in \Z$ and $h \ge 2$, let $P_{t,h} Y$ be the [[orthogonal projection|0B8KZRT28M]] in $L^2$ of $Y$ onto the span of $1, X_{t+1}, \dots, X_{t+h-1}$.
So $P_{t,h} Y$ is the best linear predictor of $Y$ from these variables.
The *partial autocorrelation function* $\alpha$ is defined for $h \ge 1$ by:

1. $\alpha(1) = \rho(1)$, where $\rho$ is the [[autocorrelation function|YCZHYY8VE5]].
2. For $h \ge 2$, $\alpha(h)$ is the [[correlation coefficient|0DNN9B6Q2F]] of $X_{t+h} - P_{t,h} X_{t+h}$ and $X_t - P_{t,h} X_t$.

By stationarity, $\alpha(h)$ does not depend on $t$.

## Remarks

Equivalently, $\alpha(h) = \phi_{hh}$, the last coefficient of the best linear predictor $\mu + \sum_{j=1}^{h} \phi_{hj} (X_{t+h-j} - \mu)$ of $X_{t+h}$ from $X_{t+h-1}, \dots, X_t$, provided the $h \times h$ matrix $[\gamma(i - j)]$ is nonsingular.
The sample partial autocorrelation is obtained by putting the sample autocovariances into these formulas, for instance by the Durbin–Levinson recursion.

Example: let $X_t = \phi X_{t-1} + \varepsilon_t$ be a stationary [[autoregressive model|2JC96FMGA1]] of order $1$ with $\abs{\phi} < 1$ and uncorrelated noise $\varepsilon_t$.
Then $\alpha(1) = \rho(1) = \phi$.
For $h \ge 2$, the best linear predictor of $X_{t+h}$ from $X_{t+1}, \dots, X_{t+h-1}$ is $\phi X_{t+h-1}$, and the residual $\varepsilon_{t+h}$ is uncorrelated with $X_t - P_{t,h} X_t$, so $\alpha(h) = 0$.

Non-example: a random walk is not weakly stationary, so clause 1 and clause 2 do not define a function of the lag alone.
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# Spectral element method

## Summary

The spectral element method is a [[finite element method|98GZ9DVHTA]] that uses high-degree polynomials on a few large elements.
Its nodes and quadrature come from a [[spectral method|MYV657YRDW]].
On each element, usually a quadrilateral or hexahedron, the basis consists of [[Lagrange polynomials|Y7KSGE97GK]] at the Gauss–Lobatto–Legendre points.
It combines the geometric flexibility of finite elements with the fast convergence of spectral methods for smooth solutions.
It was introduced by Anthony Patera for fluid dynamics, and is widely used in seismology and in simulations of turbulent flow.
See the [[definition|8PQ405CKA3]].

## Explanations

### L1 human

Cut the domain into a few blocks, and on each block use a polynomial of high degree, such as degree $8$.
Neighbouring blocks agree along their common edges, so the whole approximation is continuous.
The polynomial on each block is stored by its values at special points that cluster near the block edges.
Using the same points for the integrals makes the mass matrix diagonal, which makes time stepping cheap.
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# Local truncation error

## Summary

The local truncation error of a method for an [[initial value problem|0FXDW992DS]] is the error of one step started from exact values of the solution.
It measures how well the formula of the method matches the differential equation, and it defines the order of consistency of the method.
Together with a stability condition it controls the global error.
For a one-step method with local error $O(h^{p+1})$ per step, the global error after $O(1/h)$ steps is $O(h^p)$.
It is a special case of [[truncation error|4AKCQERM89]].

## Explanations

### L1 human

Imagine standing exactly on the true solution curve and taking one step with the method.
The local truncation error is how far from the curve that single step lands.
For the [[Euler method|0JARTXX4XP]] applied to $y' = y$ starting at $y = 1$, one step of length $h$ gives $1 + h$, while the truth is $e^h = 1 + h + \frac{h^2}{2} + \cdots$, so the local error is about $\frac{h^2}{2}$.
Over the whole interval many such small errors add up to the global error.
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# Associative quotient of a magma

## Summary

The associative quotient of a [[magma|9689SFPVE9]] is the largest quotient of the magma that is a [[semigroup|0XMEM406JM]].
It is obtained by forcing the associative law and nothing else.
Every homomorphism from the magma to a semigroup factors in exactly one way through this quotient.
So it is the universal way to make a magma associative.

## Explanations

### L1 human

In a magma, $(xy)z$ and $x(yz)$ may be different.
The associative quotient glues these two elements together, for all $x, y, z$, and glues whatever else must follow so that the operation still makes sense.
What remains is a semigroup, where brackets no longer matter.
For example, the free magma on one letter $a$ has the different elements $(aa)a$ and $a(aa)$; in its associative quotient they become the same element $a^3$.
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# Resolvent equation for kernels

## Statement

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]] with $M = \max \abs{K}$, and let $\rho = 1 / (M (b - a))$.
For $\abs{\lambda} < \rho$ let $R(x, s; \lambda)$ be the [[resolvent kernel|ARTJV1XCRX]] of $K$.
Then the following hold.

1. For $\abs{\lambda} < \rho$ and all $x, s \in [a, b]$,
$$R(x, s; \lambda) - K(x, s) = \lambda \int_a^b K(x, t) R(t, s; \lambda) \, dt = \lambda \int_a^b R(x, t; \lambda) K(t, s) \, dt.$$
2. For $\abs{\lambda} < \rho$, $\abs{\mu} < \rho$ and all $x, s \in [a, b]$,
$$R(x, s; \lambda) - R(x, s; \mu) = (\lambda - \mu) \int_a^b R(x, t; \lambda) R(t, s; \mu) \, dt.$$

## Remarks

Proof idea: write $R = \sum_{n \ge 1} \lambda^{n-1} K_n$ with the [[iterated kernels|9QNBFERMHD]] and use $K_{n+1}(x, s) = \int K(x, t) K_n(t, s) \, dt = \int K_n(x, t) K(t, s) \, dt$; uniform convergence allows integration term by term.
For part 2, multiply out the two series and use $K_{n+m}(x, s) = \int K_n(x, t) K_m(t, s) \, dt$; the coefficient identity is $\lambda^{k} - \mu^{k} = (\lambda - \mu) \sum_{i + j = k - 1} \lambda^i \mu^j$.
In operator form, with $R_\lambda = K (I - \lambda K)^{-1}$, the identities read $R_\lambda - K = \lambda K R_\lambda = \lambda R_\lambda K$ and $R_\lambda - R_\mu = (\lambda - \mu) R_\lambda R_\mu$.

Example: for $K(x, s) = x s$ on $[0, 1]$, $R(x, s; \lambda) = 3 x s / (3 - \lambda)$.
Then $\lambda \int_0^1 x t \cdot 3 t s / (3 - \lambda) \, dt = \lambda x s / (3 - \lambda)$, and $3 x s / (3 - \lambda) - x s = \lambda x s / (3 - \lambda)$, as part 1 says.

Both identities stay true, by analytic continuation, for all $\lambda$ and $\mu$ that are not poles of the meromorphic extension of $R$.

## Explanations

### L1 human

The resolvent kernel solves an integral equation of its own, with the kernel $K$ itself as the free term.
A second identity compares the resolvent kernels at two values of the parameter.
These rules let one compute with resolvent kernels as one computes with inverse matrices.

### L2 human

Part 1 is the kernel form of $(I + \lambda R_\lambda)(I - \lambda K) = (I - \lambda K)(I + \lambda R_\lambda) = I$, which says that $I + \lambda R_\lambda$ inverts $I - \lambda K$.
It is often taken as the definition of the resolvent kernel beyond the disc of convergence of the series.
Part 2 is the analogue of the resolvent identity of operator theory, and it shows that $R$ is analytic in $\lambda$ with $\partial_\lambda R(x, s; \lambda) = \int_a^b R(x, t; \lambda) R(t, s; \lambda) \, dt$.
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# Berkovich space

## Summary

A Berkovich space is an analytic space over a complete [[non-Archimedean|1H5F1BP6XS]] field whose points are multiplicative seminorms, see the [[definition|3JFF28P5NJ]].
Vladimir Berkovich introduced these spaces as a version of non-Archimedean analytic geometry with good topological properties.
Unlike the field itself, which is totally disconnected, Berkovich spaces are Hausdorff and locally path-connected, and the spectrum of a Banach algebra is compact.
They contain the points of a [[rigid analytic space|YAH6MZVG7K]] and add many new points.
This makes them a natural setting for [[non-Archimedean dynamics|6RM4RWWSSX]], potential theory and tropical geometry.

## Explanations

### L1 human

Over the $p$-adic numbers, two discs are either disjoint or one contains the other, and the space breaks into dust.
Berkovich's idea is to add a new point for every closed disc, together with some limit points.
A point is described by the way it measures the size of functions.
Over the field $\C_p$, the point of a disc measures a polynomial by its largest size on that disc.
For the closed unit disc, the new point for the whole disc measures $\sum a_n x^n$ by $\max_n \abs{a_n}$.
With these extra points the closed unit disc becomes a connected tree-like space, much closer to the picture of a complex disc.
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# Gauss's law for potentials

## Statement

Let $n \ge 3$ and let $\sigma_n$ be the area of the unit sphere of $\R^n$.
Let $\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\R^n$ with finite mass and compact [[support|FQTXV6BQZG]], and let $U^\mu(x) = \int \abs{x - y}^{2-n} \, d\mu(y)$ be its [[Newtonian potential|DCMFQ01QGS]].
Let $\Omega \subseteq \R^n$ be a bounded open set with $C^1$ boundary $\partial\Omega$ and outward unit normal $\nu$.
Assume that $\partial\Omega$ does not meet the support of $\mu$.
Then
$$\int_{\partial\Omega} \frac{\partial U^\mu}{\partial \nu} \, dS = -(n-2)\,\sigma_n \, \mu(\Omega).$$

## Remarks

Proof idea: first take $\mu = \delta_y$ with $y \notin \partial\Omega$.
If $y \notin \overline{\Omega}$, the function $x \mapsto \abs{x - y}^{2-n}$ is harmonic on a neighbourhood of $\overline{\Omega}$, and the flux is $0$ by the [[divergence theorem|0JX86PSSA9]] (or [[Green's first identity|QDXXWA1QGF]] with $u = 1$).
If $y \in \Omega$, remove a small ball about $y$; the flux through the small sphere is $(2-n) r^{1-n} \cdot \sigma_n r^{n-1} = -(n-2)\sigma_n$.
The general case follows by integrating in $y$ against $\mu$ and exchanging the integrals, which is allowed because $\partial\Omega$ stays at positive distance from the support of $\mu$.
In $\R^3$ the constant is $-4\pi$: with the gravitational field $g = \nabla(G U^\mu)$ pointing towards the masses, the flux of $g$ out of a closed surface is $-4\pi G$ times the enclosed mass.
The hypothesis on $\partial\Omega$ is needed: for a point mass at a point where $\partial\Omega$ is smooth, the flux counts only half of that mass.

## Explanations

### L1 human

Draw any closed surface around some masses.
The total amount of gravitational field going in through the surface only depends on how much mass is inside, not on where it sits or how the surface is shaped.
Masses outside the surface contribute nothing: their field goes in on one side and out on the other.

### L2 human

This is the integral form of the distributional equation $\Delta U^\mu = -(n-2)\sigma_n \mu$, obtained by integrating over $\Omega$ and using the divergence theorem.
It gives a quick proof of the shell theorem by symmetry: for a radial mass distribution, the flux through a sphere fixes the radial derivative of the potential.
The same argument with the logarithmic kernel in the plane gives the constant $-2\pi$.
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<document source="https://tamnd.github.io/math2/node/T2EB8R9HEV.md" tag="T2EB8R9HEV" kind="concept">
# Maximal compact subgroup

## Summary

A maximal compact subgroup of a [[topological group|92700QAB5Y]] is a compact subgroup that is not contained in any larger compact subgroup.
In a connected [[Lie group|7JBRGCVJRB]], maximal compact subgroups exist and are all conjugate.
The group is then, as a space, the product of a maximal compact subgroup with a Euclidean space.
So a maximal compact subgroup carries the topology of the whole group.

## Explanations

### L1 human

Inside a big group of symmetries, look for the largest piece that is bounded.
For the invertible real $n \times n$ matrices, this piece is the group $\mathrm{O}(n)$ of rotations and reflections.
Every invertible matrix is a rotation or reflection times a stretching along perpendicular axes.
The stretchings form a space with no holes, so all the interesting shape of the group comes from $\mathrm{O}(n)$.
</document>

<document source="https://tamnd.github.io/math2/node/T2HPG1780R.md" tag="T2HPG1780R" kind="definition">
# Mean curvature

## Definition

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]], let $p \in S$, and choose a smooth unit normal field $N$ near $p$.
Let $S_p$ be the [[shape operator|DQCD11X1VM]] at $p$ and let $k_1, k_2$ be the [[principal curvatures|FA8QE1SGP4]] at $p$.
The *mean curvature* of $S$ at $p$ is
$$H(p) = \frac{k_1 + k_2}{2} = \frac{1}{2} \tr S_p.$$

## Remarks

In a local parametrisation, with $E, F, G$ the coefficients of the [[first fundamental form|MAJ3DH7MZS]] and $e, f, g$ those of the [[second fundamental form|9KHQR16P1B]],
$$H = \frac{e G - 2 f F + g E}{2 (E G - F^2)}.$$
Example: the cylinder of radius $r$ with the inner normal has $k_1 = 1/r$ and $k_2 = 0$, so $H = 1/(2r)$.
The saddle $z = xy$ has $k_1 = 1$ and $k_2 = -1$ at the origin, so $H = 0$ there.
Non-example: replacing $N$ by $-N$ changes $H$ into $-H$, so on a surface without a global unit normal, such as a Möbius strip, only $\abs{H}$ is defined globally.
Some authors define $H$ as $k_1 + k_2$, without the factor $\frac{1}{2}$.
</document>

<document source="https://tamnd.github.io/math2/node/T2JZ0CKFKW.md" tag="T2JZ0CKFKW" kind="example">
# Fatou–Bieberbach domain

## Example

Let $n \ge 2$.
A *Fatou–Bieberbach domain* is an open set $\Omega \subseteq \C^n$ with $\Omega \ne \C^n$ that is [[biholomorphic|FX92PP3HZR]] to $\C^n$.

A concrete one comes from an [[automorphism of $\C^2$|2J7JB3H3FD]].
Let $F(z, w) = (w/2, \, z/2 + w^2)$.
It is a polynomial map with polynomial inverse $(u, v) \mapsto (2v - 8u^2, \, 2u)$, so it is a holomorphic automorphism of $\C^2$.
The origin is a fixed point, and the Jacobian matrix of $F$ there has eigenvalues $1/2$ and $-1/2$.
So the origin is an attracting fixed point.
Let $\Omega$ be its basin of attraction, the set of points $q$ with $F^m(q) \to 0$ as $m \to \infty$.
It is a known theorem that the basin of attraction of an attracting fixed point of a holomorphic automorphism of $\C^n$ is biholomorphic to $\C^n$.
The point $(3/8, 3/4)$ is a second fixed point of $F$, so it is not in $\Omega$.
Hence $\Omega$ is a proper open subset of $\C^2$ biholomorphic to $\C^2$.

## Remarks

No such domain exists in dimension one.
If $\Omega \subsetneq \C$ were biholomorphic to $\C$, then $\Omega$ would be simply connected, so its complement has at least two points.
A biholomorphism $\C \to \Omega$ would then be an entire function omitting two values, hence constant by Picard's little theorem, which is absurd.

## Explanations

### L1 human

In the plane, the whole complex line cannot be squeezed holomorphically and bijectively onto a smaller region.
In two or more complex dimensions this can happen: all of $\C^n$ fits, holomorphically and without distortion of its complex structure, inside a proper part of itself.
The pieces left out can even contain open sets.
This shows how much more flexible holomorphic maps become in several variables.
</document>

<document source="https://tamnd.github.io/math2/node/T2V7GJJ2EN.md" tag="T2V7GJJ2EN" kind="concept">
# Triangulated category

## Summary

A triangulated category is an [[additive category|B2VP96NJSK]] with a shift functor and a class of distinguished triangles $X \to Y \to Z \to X[1]$ that satisfy a list of axioms.
Distinguished triangles play the role that short exact sequences play in an abelian category.
The [[homotopy category of chain complexes|9Y5Y92R4D5]], the [[derived category|5ST7NVJFSM]] and the [[stable homotopy category|26GTNMAF7B]] are triangulated.

## Explanations

### L1 human

In the derived category, maps need not have kernels and cokernels, so short exact sequences are not available.
Their replacement is the triangle: each map $f \colon X \to Y$ has a cone $Z$, and the sequence $X \to Y \to Z \to X[1]$ behaves like a short exact sequence that continues forever by shifting.
Applying a functor such as $\Hom(W, -)$ to a triangle gives a long exact sequence.
The axioms list the rules that these triangles obey in the examples.
</document>

<document source="https://tamnd.github.io/math2/node/T32W553W95.md" tag="T32W553W95" kind="concept">
# Topological vector space

## Summary

A topological vector space is a [[vector space|000AYPA9AJ]] over $\R$ or $\C$ with a topology in which addition and scalar multiplication are continuous.
It is the common frame of functional analysis: normed spaces, Banach spaces, Hilbert spaces and locally convex spaces are all examples.
Under addition it is a commutative [[topological group|92700QAB5Y]], so its topology is determined by the neighbourhoods of $0$.
Continuity of linear maps, convergence of series and duality are all studied in this setting.

## Explanations

### L1 human

A topological vector space is a space of vectors where you can also say when vectors are close to each other.
Adding vectors and stretching them by numbers must respect this closeness.
The plane $\R^2$ with its usual distance is the simplest example.
Spaces of functions, such as all continuous functions on $[0, 1]$ with the maximum distance, are the examples that matter most in analysis.
</document>

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# Rate of convergence

## Definition

Let $(x_k)_{k \ge 0}$ be a [[real sequence|0CTV783HYT]] that converges to a [[limit|05M5SWSZZD]] $L$, with $x_k \ne L$ for all $k$.
Let $e_k = \abs{x_k - L}$ be the error at step $k$.

1. The sequence *converges linearly* with *rate* $\mu$ if $\lim_{k \to \infty} e_{k+1} / e_k = \mu$ for some $\mu \in (0, 1)$.
2. It *converges superlinearly* if $\lim_{k \to \infty} e_{k+1} / e_k = 0$, and *sublinearly* if $\lim_{k \to \infty} e_{k+1} / e_k = 1$.
3. For a real number $q > 1$, it *converges with order* $q$ if $\lim_{k \to \infty} e_{k+1} / e_k^q = \mu$ for some $\mu > 0$.

Order $2$ is called *quadratic* convergence and order $3$ *cubic* convergence.
The same definitions apply to sequences in a normed space, with $e_k = \norm{x_k - L}$.

## Remarks

Example: $x_k = 2^{-k}$ tends to $0$, and $e_{k+1} / e_k = 1/2$, so it converges linearly with rate $1/2$.
Example: $x_k = 2^{-2^k}$ tends to $0$, and $e_{k+1} / e_k^2 = 2^{-2^{k+1}} / 2^{-2^{k+1}} = 1$, so it converges quadratically.
Non-example of clause 1: $x_k = 1/(k+1)$ tends to $0$, but $e_{k+1} / e_k = (k+1)/(k+2) \to 1$, so it converges only sublinearly.
These notions are often called Q-linear and Q-quadratic convergence, where Q stands for "quotient".
A weaker notion, R-convergence, asks only that $e_k \le \varepsilon_k$ for some sequence $(\varepsilon_k)$ that tends to $0$ with the given order.
</document>

<document source="https://tamnd.github.io/math2/node/T35XFF5AY3.md" tag="T35XFF5AY3" kind="statement">
# Dimension of a polynomial ring

## Statement

Let $R$ be a [[commutative ring|0310REJJY8]] of finite [[Krull dimension|MVGBS3J1P7]] $\dim R$, and let $R[X]$ be the [[polynomial ring|0SDC934R52]] in one variable over $R$.

1. In general, $\dim R + 1 \le \dim R[X] \le 2 \dim R + 1$.
2. If $R$ is a [[Noetherian ring|0670T59HGE]], then $\dim R[X] = \dim R + 1$.

## Remarks

For the lower bound, a chain of [[prime ideals|00MHHZ9SNE]] $\mathfrak p_0 \subsetneq \cdots \subsetneq \mathfrak p_n$ of $R$ gives the chain $\mathfrak p_0[X] \subsetneq \cdots \subsetneq \mathfrak p_n[X] \subsetneq \mathfrak p_n[X] + (X)$ in $R[X]$.
For the upper bound, fix a prime $\mathfrak p$ of $R$ and let $\kappa(\mathfrak p)$ be the fraction field of $R/\mathfrak p$.
The primes of $R[X]$ that contract to $\mathfrak p$ correspond to primes of $\kappa(\mathfrak p)[X]$, so at most two of them lie in one chain.
In the Noetherian case one shows that a prime $\mathfrak P$ of $R[X]$ over $\mathfrak p$ has height at most $\operatorname{ht} \mathfrak p + 1$.
By induction, for a field $k$ the ring $k[X_1, \dots, X_n]$ has dimension $n$, and for a Noetherian ring $R$ of finite dimension, $\dim R[X_1, \dots, X_n] = \dim R + n$.
For example, $\dim \Z[X] = 2$, with the maximal chain $0 \subsetneq (2) \subsetneq (2, X)$.
Without the Noetherian hypothesis the equality in clause 2 can fail: there are non-Noetherian rings with $\dim R[X] > \dim R + 1$.

## Explanations

### L1 human

The Krull dimension counts how long a chain of prime ideals can be.
Adding one polynomial variable adds exactly one to this count when the ring is Noetherian.
This matches geometry: a line has dimension $1$, the plane $2$, and $n$-space $n$.

### L2 human

This result is the algebraic reason why affine $n$-space over a field has dimension $n$.
Together with [[Noether normalization|7Z28JY8QW4]] it shows that a finitely generated domain over a field has dimension equal to the transcendence degree of its fraction field.
The general bounds come from looking at the fibres of $\operatorname{Spec} R[X] \to \operatorname{Spec} R$, which are spectra of polynomial rings in one variable over fields.
</document>

<document source="https://tamnd.github.io/math2/node/T37Q055GDA.md" tag="T37Q055GDA" kind="definition">
# Pseudo-spectral method

## Definition

Let $x_0, \ldots, x_{N-1}$ be collocation points in a domain $\Omega$, and let $V_N$ be a space of global functions with an interpolation property.
For every vector of values $(v_j)$ there is a unique $p \in V_N$ with $p(x_j) = v_j$ for all $j$.
Write $I_N v$ for this interpolant.
A *pseudo-spectral method* for an equation $u_t = \mathcal{N}(u)$, with $\mathcal{N}$ a differential operator, is the following scheme:

1. the unknowns are the values $u_j(t) \approx u(x_j, t)$;
2. each derivative $\partial^m u$ is replaced by $\partial^m (I_N u)$, evaluated at the points $x_j$;
3. products and other nonlinear functions of $u$ and its derivatives are evaluated pointwise at the $x_j$;
4. the equation is required to hold at each point $x_j$.

It is a [[spectral method|MYV657YRDW]] of collocation type.

## Remarks

Periodic case: $\Omega = [0, 2\pi)$, $x_j = 2\pi j / N$, and $V_N$ the [[trigonometric interpolants|SVABG134WE]] of degree about $N/2$.
Derivatives are then computed with the [[discrete Fourier transform|TST0GH66A5]]: transform, multiply the coefficient of $e^{ikx}$ by $ik$, and transform back.
With the [[fast Fourier transform|S966EZX3ZT]] this costs $O(N \log N)$ operations.
On an interval one often uses Chebyshev points $x_j = \cos(\pi j / (N - 1))$ and the [[Chebyshev polynomials|M6EAR1JHPP]].

Example: for $N = 8$ and $u_j = \sin x_j$, the transform has only the coefficients of $e^{\pm ix}$.
Multiplying them by $\pm i$ and transforming back gives $\cos x_j$ exactly.
For Burgers' equation $u_t + u u_x = 0$, one forms $u_x$ this way and then multiplies by $u_j$ at each point.

Non-example: a spectral Galerkin method computes the product $u u_x$ through its exact Fourier coefficients, a convolution of coefficient sequences.
It does not use pointwise evaluation, so clause 3 fails.
</document>

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# Isomorphism of topological groups

## Summary

An isomorphism of topological groups is a group isomorphism that is also a homeomorphism.
Two isomorphic topological groups are the same in every respect that involves both the group law and the topology.
A continuous bijective homomorphism need not be an isomorphism, because its inverse may fail to be continuous.
The open mapping theorem for locally compact groups gives conditions under which this cannot happen.

## Explanations

### L1 human

Two topological groups are isomorphic when one can rename the elements of one to get the other, keeping both the multiplication and the notion of nearness.
For example, the positive real numbers under multiplication and the real numbers under addition are isomorphic.
The logarithm turns products into sums, and both the logarithm and the exponential function are continuous.
So these two groups are copies of each other, even though they look different at first.
</document>

<document source="https://tamnd.github.io/math2/node/T3GGJN70P2.md" tag="T3GGJN70P2" kind="statement">
# Sylvester–Gallai theorem

## Statement

Let $S$ be a finite set of points in the [[Euclidean plane|0XW7V9ZZ05]].
Suppose that the points of $S$ do not all lie on one line.
Then there is a line that contains exactly two points of $S$.

## Remarks

A line that contains exactly two points of $S$ is called an ordinary line.
Kelly's proof considers all pairs of a point $p \in S$ and a line $\ell$ through two points of $S$ with $p \notin \ell$.
It takes a pair with the least distance from $p$ to $\ell$.
Suppose $\ell$ contained three points of $S$.
Then two of them would lie on the same side of the foot of the perpendicular from $p$, or one of them at the foot.
This gives a pair with a smaller positive distance, a contradiction; so $\ell$ is ordinary.
The hypothesis that $S$ is finite is needed: the integer lattice $\Z^2$ has no ordinary line.
The theorem also fails over the complex numbers.
The nine inflection points of a smooth complex cubic curve form a configuration in which every line through two of them contains a third.
For example, the four vertices of a square have ordinary lines: each side contains exactly two of the vertices.
Green and Tao proved that for all sufficiently large $n$, a set of $n$ points not all on a line has at least $n/2$ ordinary lines.

## Explanations

### L1 human

Scatter finitely many points in the plane, not all on one straight line.
Then you can always find a line passing through exactly two of the points.
This is easy to believe but hard to prove without a clever idea.

### L2 human

The Sylvester–Gallai theorem is a starting point of combinatorial incidence geometry, like the [[Szemerédi–Trotter theorem|NQBWFSC1E0]].
A corollary is that $n$ points not all on a line determine at least $n$ distinct lines.
Its failure over $\C$ and over finite fields shows that it depends on the order structure of the real numbers.
</document>

<document source="https://tamnd.github.io/math2/node/T3HAGWYKK7.md" tag="T3HAGWYKK7" kind="statement">
# Reidemeister's theorem

## Statement

Let $L$ be a tame [[link|1A5WRCNTM3]] in $\R^3$.
A *link diagram* of $L$ is a projection of $L$ to a plane that is injective except at finitely many transverse double points, called crossings.
At each crossing the diagram records which strand passes over.
The three *Reidemeister moves* are local changes of a diagram inside a small disk:

1. type I adds or removes a single twist with one crossing;
2. type II adds or removes two crossings where one strand passes entirely over another;
3. type III slides a strand over or under a crossing of two other strands.

Two diagrams represent equivalent links if and only if they are related by a finite sequence of Reidemeister moves and isotopies of the plane.

## Remarks

Each move is realized by a deformation of the link near the small disk, which gives the easy direction.
For the converse, an equivalence of tame links can be taken to be a finite sequence of triangle moves on polygonal links.
Such a move replaces one edge by the two other edges of a triangle, or the reverse, when the triangle meets the link only in these edges.
The effect of each such move on a diagram can be split into Reidemeister moves.
The theorem was proved by Kurt Reidemeister in 1927, and independently by James Alexander and Garland Briggs.
It is the basis of most proofs that a quantity computed from a diagram is a [[knot invariant|GZQ5ZE3X6G]]: one checks invariance under the three moves.
For example, [[tricolorability|KNAGK0W65G]] and the [[Kauffman bracket|JYKVYS4N32]] are proved invariant in this way, the bracket up to a factor under the first move.
All three types are needed: for instance, the number of crossings modulo $2$ is unchanged by moves of type II and III, but changed by a move of type I.

## Explanations

### L1 human

Knots are usually drawn as pictures in the plane with gaps that show which strand goes under.
The same knot has many pictures.
Reidemeister's theorem says that any two pictures of the same knot are related by three simple kinds of local changes.
They are undoing a twist, pulling two strands apart, and sliding a strand past a crossing.

### L2 human

The theorem turns knot theory into a combinatorial subject.
To build a knot invariant, one defines a quantity from a diagram and checks that the three moves leave it unchanged.
This is how the Jones polynomial, the Alexander polynomial via skein relations, and colourings by groups or quandles are shown to be invariants.
The theorem gives no bound on the number of moves needed, so by itself it does not give an algorithm to decide whether two diagrams represent the same knot.
</document>

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# Optimal control

## Summary

Optimal control is the theory of choosing the input of a [[control system|ZPDTVZQN5K]] in the best way.
The aim is to make a cost, which depends on the input and on the resulting trajectory, as small as possible.
An optimal control is an [[admissible control|8M31QKB0GW]] whose cost is not larger than that of any other admissible control.
The theory extends the calculus of variations to problems with constraints on the controls.
Its main tools are [[Pontryagin's maximum principle|7RPV2T29CF]], which gives necessary conditions, and dynamic programming with the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]].
The [[definition|EZ8NBZ8D0E]] states the problem in Bolza form.

## Explanations

### L1 human

A driver wants to reach a destination using as little fuel as possible, and the car obeys the laws of motion.
At each moment the driver chooses the pedal position, and these choices together decide the path and the fuel used.
Optimal control finds the best way to make all these choices.
A simple case: a point on a line moves with speed $u(t)$, where $\abs{u} \le 1$, and should be as far right as possible at time $1$.
The best choice is full speed, $u = 1$, all the time, since the final position is the integral of the speed and the speed is at most $1$.
</document>

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# Alexander's theorem on proper self-maps of the ball

## Statement

Let $n \ge 2$ and let $B \subset \C^n$ be the [[unit ball|PDQ9KW9FAP]].
Every [[proper holomorphic map|TM528Q96ZD]] $F \colon B \to B$ is a [[biholomorphism|FX92PP3HZR]] of $B$ onto itself.

## Remarks

A proper holomorphic self-map of $B$ is a finite branched covering.
One proof shows that it has no branching: near a suitable boundary point $F$ is a local biholomorphism that maps a piece of the sphere $\partial B$ into $\partial B$.
Such a local map is the restriction of an automorphism of $B$, a rigidity property of the sphere that goes back to Poincaré.
By the identity theorem, $F$ itself is that automorphism.

The hypothesis $n \ge 2$ is necessary.
For $n = 1$ the map $z \mapsto z^2$ is a proper holomorphic self-map of the unit disc that is not injective.

The shape of the ball matters too.
The map $(z_1, z_2) \mapsto (z_1, z_2^2)$ is a proper holomorphic self-map of the unit polydisc in $\C^2$ that is not injective.

The target dimension matters as well: the map $(z_1, z_2) \mapsto (z_1^2, \sqrt{2}\, z_1 z_2, z_2^2)$ sends the unit ball of $\C^2$ properly into the unit ball of $\C^3$.
Check: $\abs{z_1}^4 + 2\abs{z_1 z_2}^2 + \abs{z_2}^4 = (\abs{z_1}^2 + \abs{z_2}^2)^2$, which tends to $1$ exactly when $\abs{z_1}^2 + \abs{z_2}^2$ does.

## Explanations

### L1 human

In one variable the unit disc has many maps onto itself that wrap it several times, such as $z \mapsto z^2$.
Alexander's theorem says that the ball in two or more complex variables has none.
Any holomorphic map from the ball to itself that sends the boundary to the boundary is a symmetry of the ball, one to one and onto.

### L2 human

The theorem is a basic example of rigidity in several complex variables.
Combined with the description of the automorphisms of $B$, it lists all proper self-maps of the ball.
They are the maps $z \mapsto U \varphi_a(z)$, with $U$ unitary and $\varphi_a$ the automorphism exchanging $a$ and $0$.
It started the study of proper holomorphic maps between balls of different dimensions, where, as the example above shows, many non-trivial maps exist.
</document>

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# Differential field

## Definition

A *differential field* is a [[field|0QA8CHH8MG]] $K$ with a map $\partial \colon K \to K$ such that:

1. $\partial(a + b) = \partial(a) + \partial(b)$ for all $a, b \in K$.
2. $\partial(ab) = \partial(a) b + a \partial(b)$ for all $a, b \in K$.

So $\partial$ is a [[derivation|R1F0HCCE6F]] of $K$ over $\Z$.
The set $C = \set{a \in K : \partial(a) = 0}$ is the *field of constants* of $(K, \partial)$.

## Remarks

The field of constants is a [[subfield|07QY5RPFBR]] of $K$: clauses 1 and 2 give $\partial(1) = 0$, and for $a \ne 0$ the rule $\partial(a^{-1}) = -\partial(a)/a^2$ holds.
Example: $K = \C(x)$ with $\partial = d/dx$ is a differential field; clauses 1 and 2 are the sum and product rules, and the derivative of a rational function is a rational function.
Its field of constants is $\C$.
Example: every field with $\partial = 0$ is a differential field, with $C = K$.
Non-example: any field with $\partial = \id$ fails clause 2, since $\partial(1 \cdot 1) = 1$ while $\partial(1) \cdot 1 + 1 \cdot \partial(1) = 2 \cdot 1$, and $1 \ne 2 \cdot 1$ in every field.
A field with finitely many commuting derivations is a partial differential field; a differential field in the sense above is also called an ordinary differential field.
</document>

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# Substitution tiling

## Definition

Let $d \ge 1$.
A tile is a compact subset of $\R^d$ that is the closure of its interior.
A tiling of a set $S \subseteq \R^d$ is a family of tiles whose union is $S$ and whose interiors are pairwise disjoint.
Let $T_1, \ldots, T_m$ be tiles, called prototiles, and let $\lambda > 1$.
A *substitution rule* with expansion factor $\lambda$ assigns to each $i$ a finite tiling $\omega(T_i)$ of $\lambda T_i$ by translates of the prototiles.
It extends to translates by $\omega(T_i + a) = \omega(T_i) + \lambda a$, and to tilings by applying it to each tile.
A patch of a tiling is a finite subfamily of its tiles.
A *substitution tiling* for $\omega$ is a tiling $\mathcal{T}$ of $\R^d$ by translates of the prototiles with the following property.
Every patch of $\mathcal{T}$ is a translate of a patch of $\omega^k(T_i)$ for some $k \ge 0$ and some $i$.

## Remarks

Example: in $\R^2$, let $T_1 = [0,1]^2$, $\lambda = 2$, and let $\omega(T_1)$ be the four unit squares $T_1 + a$ with $a \in \set{0,1}^2$.
Then $\omega^k(T_1)$ is the $2^k \times 2^k$ block of unit squares.
The grid of all unit squares $T_1 + a$, $a \in \Z^2$, is a substitution tiling, since each finite patch of it lies in a translate of some block.
Non-example: for the same rule, a tiling of $\R^2$ by rows of unit squares in which the rows are shifted by $1/2$ against each other is not a substitution tiling.
Its patches across two rows are not translates of patches of the grid.
The Penrose tilings, cut into Robinson triangles, are substitution tilings with expansion factor the golden ratio.
Their prototiles are the finitely many rotated and reflected copies of the two triangles.
More generally, $\lambda$ may be replaced by an expanding [[linear map|0RJF668P1T]] of $\R^d$.
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# Primal algebra

## Definition

Let $A$ be an algebra whose universe is a [[finite set|070R982G4B]] with at least two elements.
The algebra $A$ is *primal* if every operation $f \colon A^n \to A$ with $n \ge 1$ is a [[term operation|N9M2N27END]] of $A$.
Equivalently, the set of basic operations of $A$ is [[functionally complete|TDZVCWYJ9F]].

## Remarks

Example: the ring $(\Z_p, +, \cdot, -, 0, 1)$ for a prime $p$ is primal.
Check: the polynomial $\delta(x) = 1 - x^{p - 1}$ equals $1$ at $0$ and $0$ elsewhere, by Fermat's little theorem.
Then $f(a_1, \dots, a_n) = \sum_{b \in \Z_p^n} f(b) \prod_{i} \delta(a_i - b_i)$, and each $f(b)$ is a sum of copies of $1$.
Non-example: the field $\F_4$ with its ring operations is not primal.
Its Frobenius map $x \mapsto x^2$ is an automorphism, and term operations commute with automorphisms.
A constant operation with value outside $\F_2$ does not commute with it, so it is not a term operation.
Every primal algebra is [[quasi-primal|QWSBDFM47K]].
</document>

<document source="https://tamnd.github.io/math2/node/T46ZBWDPHQ.md" tag="T46ZBWDPHQ" kind="concept">
# Nijenhuis tensor

## Summary

Let $J$ be a field of endomorphisms of the tangent bundle.
Its Nijenhuis tensor is a [[tensor field|WX302T93TW]] of type $(1,2)$ built from $J$ and the [[Lie bracket of vector fields|68FGQEBD6J]].
It is, up to a factor $2$, the [[Frölicher–Nijenhuis bracket|D46QECMC10]] of $J$ with itself.
For an [[almost complex structure|9SYWRGVTRS]], it measures the obstruction to integrability.
By the [[Newlander–Nirenberg theorem|VCT848WV1T]], an almost complex structure comes from a complex manifold structure exactly when its Nijenhuis tensor vanishes.

## Explanations

### L1 human

An almost complex structure lets one multiply tangent vectors by $i$ at each point.
One would like coordinates in which this multiplication is the usual one, as on $\C^n$.
The Nijenhuis tensor is a test, computed from derivatives of $J$, that tells whether such coordinates exist.
On $\C^n$ with its usual structure, the test gives zero; on the six-sphere with its standard almost complex structure from the octonions, it does not.
</document>

<document source="https://tamnd.github.io/math2/node/T4BCQHR68Y.md" tag="T4BCQHR68Y" kind="concept">
# Omega-complete partial order

## Summary

An omega-complete partial order, or $\omega$-cpo, is a partially ordered set in which every increasing sequence has a supremum.
It is a weaker form of a [[directed-complete partial order|7TSXP95QN6]] that is enough for the fixed-point semantics of recursive programs.
Let the $\omega$-cpo have a least element $\bot$.
Then a monotone map that preserves suprema of increasing sequences has a least fixed point, the supremum of its iterates on $\bot$.

## Explanations

### L1 human

A computation that runs step by step produces a sequence of better and better approximations $x_0 \le x_1 \le x_2 \le \cdots$.
In an $\omega$-cpo such a sequence always has a limit, its least upper bound.
For example, in $\N \cup \set{\infty}$ every increasing sequence has a supremum, either its largest value or $\infty$.
Only countable increasing sequences are required, which makes the notion easy to use in programming language semantics.
</document>

<document source="https://tamnd.github.io/math2/node/T4EPFGAQ8C.md" tag="T4EPFGAQ8C" kind="definition">
# Topological data analysis

## Definition

*Topological data analysis* is the analysis of data by topological invariants computed through the following steps.

1. The data are a finite set $X = \set{x_1, \dots, x_n}$ in a [[metric space|0VKD1TP7F3]] $(M, d)$.
2. A filtration of [[simplicial complexes|V8GZ0W1ZZ5]] $(K_r)_{r \ge 0}$ is built on $X$, with $K_r \subseteq K_s$ for $r \le s$.
   A standard choice is the Vietoris–Rips complex: its simplices at scale $r$ are the nonempty subsets of $X$ of diameter at most $r$.
3. The [[persistent homology|S8SNBHVPSP]] of the filtration, in each degree $k$ with coefficients in a field, is computed and summarised by a barcode or persistence diagram.
4. Statistical conclusions are drawn from these summaries, for example by comparing them with the bottleneck distance.

## Remarks

The stability theorem gives the summaries a statistical meaning.
For two finite metric spaces, the persistence diagrams of their Vietoris–Rips filtrations are at bottleneck distance at most twice their Gromov–Hausdorff distance.

Example: let $X$ be the $n \ge 4$ vertices of a regular $n$-gon in $\R^2$, with side length $a$ and with length $b > a$ of the diagonals that skip one vertex.
For $a \le r < b$, the Vietoris–Rips complex $K_r$ has an edge between consecutive vertices only, and no triangle, since any three vertices include two that are not consecutive.
So $K_r$ is the $n$-cycle graph, a circle, and its first homology has rank $1$; the bar in degree $1$ that is born at $a$ detects the circle.

Non-example: computing the mean and covariance of $X$ is not topological data analysis, since it uses no filtration and no topological invariant.
</document>

<document source="https://tamnd.github.io/math2/node/T4ESER8G7G.md" tag="T4ESER8G7G" kind="definition">
# 4-manifold

## Definition

A *$4$-manifold* is a [[topological manifold|QQQ9B1DK4P]] of dimension $4$, that is, a topological space $M$ such that:

1. $M$ is Hausdorff,
2. $M$ is second countable, and
3. every point of $M$ has an open neighborhood homeomorphic to an open subset of $\R^4$.

A *smooth $4$-manifold* is a $4$-manifold with a [[smooth structure|R2TQ82PCNB]].

## Remarks

Example: the $4$-sphere $S^4 = \set{x \in \R^5 : \abs{x} = 1}$ is Hausdorff and second countable as a subspace of $\R^5$, and stereographic projection from a point $p$ maps $S^4 \setminus \set{p}$ homeomorphically onto $\R^4$, which gives clause 3.

Non-example: take two copies of $\R^4$ and glue them along the complement of the origin.
Every point has a neighborhood homeomorphic to $\R^4$, but the two origins have no disjoint neighborhoods, so clause 1 fails.

By invariance of domain, no nonempty open subset of $\R^4$ is homeomorphic to an open subset of $\R^n$ with $n \ne 4$, so the dimension is well defined.
The [[$E_8$ manifold|BNDRN0G9BR]] is a closed $4$-manifold with no smooth structure.
</document>

<document source="https://tamnd.github.io/math2/node/T4JEMK1549.md" tag="T4JEMK1549" kind="definition">
# Fourier series on the torus

## Definition

Let $n \ge 1$ and let $\mathbb{T}^n = \R^n / \Z^n$ be the $n$-dimensional torus.
Identify functions on $\mathbb{T}^n$ with functions $f \colon \R^n \to \C$ that are [[periodic|089KWVYD69]] with period $1$ in each variable.
Let $f$ be such a function that is Lebesgue [[integrable|X7GTF3D98F]] on the cube $[0, 1]^n$.
For $k \in \Z^n$, the $k$-th *Fourier coefficient* of $f$ is
$$\hat f(k) = \int_{[0,1]^n} f(x)\, e^{-2\pi i k \cdot x}\, dx,$$
where $k \cdot x = k_1 x_1 + \dots + k_n x_n$.
The *Fourier series* of $f$ is the formal [[series|0NM626FH6Q]]
$$\sum_{k \in \Z^n} \hat f(k)\, e^{2\pi i k \cdot x}.$$

## Remarks

Example: let $n = 2$ and $f(x) = e^{2\pi i (x_1 + 2 x_2)}$.
Then $f$ is periodic in both variables, and $\hat f(k) = \int_0^1 e^{2\pi i (1 - k_1) x_1}\, dx_1 \int_0^1 e^{2\pi i (2 - k_2) x_2}\, dx_2$, which is $1$ for $k = (1, 2)$ and $0$ otherwise.
More generally, if $f(x) = g(x_1) h(x_2)$, then $\hat f(k) = \hat g(k_1) \hat h(k_2)$.

Non-example: $f(x) = x_1$ on $\R^2$ is not periodic in $x_1$, so it is not a function on $\mathbb{T}^2$ and the definition does not apply.

For $n = 1$ this is the classical [[Fourier series|G12KXDHPPS]], written for period $1$ instead of $2\pi$.
The series is formal: for $n \ge 2$ its partial sums can be taken over cubes $\max_j \abs{k_j} \le N$, over balls $\abs{k} \le N$ or over other shapes, and convergence results depend on that choice.
For $f \in L^2(\mathbb{T}^n)$ the series converges to $f$ in $L^2$ for each of these choices, and $\sum_k \abs{\hat f(k)}^2 = \int_{[0,1]^n} \abs{f}^2$.
</document>

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# Nerve theorem

## Statement

Let $X$ be a [[paracompact space|ZH29VRRRJ6]] and let $\mathcal{U} = (U_i)_{i \in I}$ be an [[open cover|049PXYHW1T]] of $X$.
Assume that every nonempty intersection $U_{i_0} \cap \cdots \cap U_{i_k}$ of finitely many members of $\mathcal{U}$ is [[contractible|R5WP8Z69Q8]].
Then $X$ is [[homotopy equivalent|XZG6MTSWCN]] to the geometric realization of the [[nerve|4P3S0DRZXB]] $N(\mathcal{U})$.

## Remarks

The proof uses a partition of unity subordinate to $\mathcal{U}$ to define a map $X \to \abs{N(\mathcal{U})}$, sending $x$ to the point with barycentric coordinates $(\varphi_i(x))_{i \in I}$.
One then shows that this map is a homotopy equivalence.
A standard way compares both spaces with a common space built from the intersections, whose projections to $X$ and to $\abs{N(\mathcal{U})}$ have contractible fibres.

Example: cover $S^1$ by three open arcs, consecutive arcs overlapping in an arc, and no point in all three.
Every nonempty intersection is an arc, which is contractible, and the nerve is the boundary of a triangle, a circle, as the theorem predicts.

The contractibility hypothesis is necessary.
Cover $S^1$ by two open arcs, each a bit larger than a half circle.
Their intersection is two disjoint arcs, which is not contractible, and the nerve is a single edge, which is contractible, while $S^1$ is not.

Versions of the theorem hold for finite covers of a simplicial complex by subcomplexes whose nonempty intersections are contractible.

## Explanations

### L1 human

If a space is covered by simple pieces whose overlaps are also simple, then the overlap pattern alone determines the shape of the space up to homotopy.
The overlap pattern is recorded by the nerve, a combinatorial object.
So one can compute topological invariants of the space from a finite list of which pieces meet.

### L2 human

A cover with contractible finite intersections is called a good cover.
Every smooth manifold has a good cover by geodesically convex balls for a Riemannian metric.
So the nerve theorem shows that a compact smooth manifold has the homotopy type of a finite simplicial complex.
In topological data analysis, the Čech complex of a point cloud is the nerve of a cover by balls.
The nerve theorem shows that it recovers the homotopy type of the union of the balls.
</document>

<document source="https://tamnd.github.io/math2/node/T4PCQ68C5K.md" tag="T4PCQ68C5K" kind="example">
# The alternating group A3 is normal in S3

## Example

In the [[symmetric group|0515A6D2Y1]] $S_3$, let $A_3 = \{e, (1\,2\,3), (1\,3\,2)\}$. This is the [[alternating group|0Z5905Q73D]] of degree $3$.
$A_3$ has $3$ elements and $S_3$ has $6$, so $A_3$ has [[index|04QEBF8S49]] $2$.
The two left cosets are $A_3$ and the set of the three transpositions. The two right cosets are the same two sets.
Thus $gA_3 = A_3g$ for each $g \in S_3$, and $A_3$ is normal in $S_3$.

## Remarks

This shows the result that [[subgroups of index two are normal|0V547YQRMV]].
</document>

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# Sphere packing

## Definition

Let $n \ge 1$ and let $r > 0$.
A *sphere packing* in the [[Euclidean space|0XPR1E85W0]] $\R^n$ with radius $r$ is a set $C \subset \R^n$ of centres such that $\abs{x - y} \ge 2r$ for all distinct $x, y \in C$.
The packing is the family of [[closed balls|0Q0H13G8ZB]] $B(x, r)$ with $x \in C$; their interiors are pairwise disjoint.
The packing is a *lattice packing* if $C$ is a [[lattice|WN9V7WSJKF]], or a translate of one.

## Remarks

Example: in $\R^2$ take $C = \Z^2$ and $r = \tfrac12$.
Two distinct integer points are at distance at least $1 = 2r$, so the discs of radius $\tfrac12$ centred at them form a lattice packing, the square packing.
Non-example: $C = \Z^2$ with $r = 0.6$; the points $(0, 0)$ and $(1, 0)$ are at distance $1 < 1.2$, so their discs overlap.
For a lattice $\Lambda$ with shortest nonzero vector of length $m$, the largest radius is $r = m/2$.
Some authors allow balls of different radii; the term "sphere packing" then refers to any family of balls with disjoint interiors.
</document>

<document source="https://tamnd.github.io/math2/node/T4RFYT8HDN.md" tag="T4RFYT8HDN" kind="statement">
# Gelfand's spectral radius formula

## Statement

Let $A$ be a complex [[Banach algebra|9KYWN94QDN]] with unit $1 \ne 0$, and let $a \in A$.
Let $\sigma(a)$ be the [[spectrum|EH0CZ19RKP]] of $a$, and let $r(a) = \max \set{\abs{\lambda} : \lambda \in \sigma(a)}$ be its [[spectral radius|9V7YQA85PC]].
Then the [[limit|05M5SWSZZD]] of $\norm{a^n}^{1/n}$ exists, and
$$r(a) = \lim_{n \to \infty} \norm{a^n}^{1/n} = \inf_{n \ge 1} \norm{a^n}^{1/n}.$$

## Remarks

The maximum exists because $\sigma(a)$ is compact and, by the [[nonemptiness of the spectrum|QWQC5SV0SD]], nonempty.
Lower bound: by the [[spectral mapping theorem for polynomials|Q71WX8HFYT]], $\lambda^n \in \sigma(a^n)$ for $\lambda \in \sigma(a)$, and every $\mu \in \sigma(b)$ satisfies $\abs{\mu} \le \norm{b}$; so $r(a)^n \le \norm{a^n}$ for every $n$.
Upper bound: for $\abs{\lambda} > r(a)$ the resolvent $(\lambda 1 - a)^{-1}$ is holomorphic, and its Laurent series $\sum_{n \ge 0} a^n \lambda^{-n-1}$ converges there; applying functionals and the uniform boundedness principle gives $\limsup_n \norm{a^n}^{1/n} \le r(a)$.

Example: on $C([0,1])$ with the sup norm, $\norm{f^n} = \norm{f}^n$, so the formula gives $r(f) = \norm{f} = \max \abs{f}$, the largest absolute value of a point of the range.
For a nilpotent element, $a^n = 0$ for large $n$, so $r(a) = 0$ and $\sigma(a) = \set{0}$, even if $\norm{a}$ is large.
For matrices this is [[Gelfand's formula|3KX0KQ7V05]].

## Explanations

### L1 human

The spectral radius measures the largest size of a point of the spectrum.
The norm of $a$ may be bigger than this, but the norms of the powers $a^n$ grow at the exponential rate $r(a)^n$.
Taking $n$-th roots and letting $n$ grow recovers $r(a)$ from norms alone.

### L2 human

The formula shows that $r(a)$ depends only on the norms of the powers of $a$.
So it does not change when the spectrum is computed in a closed subalgebra containing $a$ and $1$.
In a C*-algebra, the C* identity gives $\norm{a^{2^k}} = \norm{a}^{2^k}$ for self-adjoint $a$, so $r(a) = \norm{a}$; this is a key step in the commutative Gelfand–Naimark theorem.
It also shows that the Gelfand transform satisfies $\norm{\hat a}_\infty = r(a)$ in a commutative unital Banach algebra.
</document>

<document source="https://tamnd.github.io/math2/node/T4VP5QTQK6.md" tag="T4VP5QTQK6" kind="statement">
# Recurrence relations for Bessel functions

## Statement

Let $\nu \in \C$ and let $C_\nu$ be any one of $J_\nu$, $Y_\nu$, $H^{(1)}_\nu$, $H^{(2)}_\nu$.
These are a [[Bessel function of the first kind|CZCMRDWDYW]], of the [[second kind|GRSNDJG363]], or a [[Hankel function|N3CC5DRGT1]].
Then for $z \in \C \setminus (-\infty, 0]$:

1. $C_{\nu - 1}(z) + C_{\nu + 1}(z) = \frac{2\nu}{z}\, C_\nu(z)$.
2. $C_{\nu - 1}(z) - C_{\nu + 1}(z) = 2\, C_\nu'(z)$.

## Remarks

For $J_\nu$ both relations follow by comparing coefficients in the power series $J_\nu(z) = \sum_{m \ge 0} \frac{(-1)^m}{m!\, \Gamma(m + \nu + 1)} \left( \frac{z}{2} \right)^{2m + \nu}$.
Equivalent forms are $\left( z^\nu C_\nu \right)' = z^\nu C_{\nu-1}$ and $\left( z^{-\nu} C_\nu \right)' = -z^{-\nu} C_{\nu+1}$.
For $Y_\nu$ they follow from the definition through $J_{\pm\nu}$ and a limit, and for the Hankel functions by linearity.
Example: with $\nu = 0$ the second relation and $J_{-1} = -J_1$ give $J_0' = -J_1$.
Example: from $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$ and $J_{-1/2}(x) = \sqrt{\frac{2}{\pi x}} \cos x$, the first relation with $\nu = \frac{1}{2}$ gives $J_{3/2}(x) = \sqrt{\frac{2}{\pi x}} \left( \frac{\sin x}{x} - \cos x \right)$.

## Explanations

### L1 human

Bessel functions of neighbouring orders are tied together by simple rules.
If one knows two consecutive orders, one gets all the others by the first rule.
The second rule gives the derivative of a Bessel function without any calculus on the function itself.

### L2 human

The first relation is a three-term recurrence in the order, so tables and software compute $J_n(x)$ for many $n$ from a few starting values.
Forward use of it is unstable for $J_n$ when $n > x$, and one runs it backward instead (Miller's algorithm).
The relations also show that Bessel functions of half-integer order are elementary, which gives the spherical Bessel functions.
The modified Bessel functions satisfy the same relations with some signs changed.
</document>

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# Fourier inversion theorem

## Statement

Let $n \ge 1$ and let $f \colon \R^n \to \C$ be [[integrable|X7GTF3D98F]].
Let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi} \, dx$ be its [[Fourier transform|DPVM34ADZV]], and assume that $\hat f$ is also integrable.
Then
$$f(x) = \int_{\R^n} \hat f(\xi) \, e^{2\pi i x \cdot \xi} \, d\xi$$
for [[almost every|RARTMA0ZEC]] $x \in \R^n$, that is, $f$ equals the [[inverse Fourier transform|JB8QMT8DCH]] of $\hat f$ almost everywhere.
If moreover $f$ is continuous, the equality holds for every $x \in \R^n$.

## Remarks

Proof idea: insert the Gaussian factor $e^{-\pi \varepsilon^2 \abs{\xi}^2}$ into the inversion integral.
With this factor the double integral converges absolutely.
Since the Gaussian is its own Fourier transform, Fubini's theorem turns it into the convolution of $f$ with a Gaussian of width $\varepsilon$ and total mass $1$.
As $\varepsilon \to 0$, the left side tends to the inversion integral by dominated convergence, since $\hat f$ is integrable.
The convolutions tend to $f$ in $L^1$, and hence almost everywhere along a subsequence.

Example: $f(x) = e^{-\abs{x}}$ on $\R$ has $\hat f(\xi) = \frac{2}{1 + 4\pi^2 \xi^2}$, which is integrable.
Since $f$ is continuous, $e^{-\abs{x}} = \int_\R \frac{2 e^{2\pi i x \xi}}{1 + 4\pi^2 \xi^2} \, d\xi$ for every $x$.

The integrability of $\hat f$ is needed for the integral to make sense.
For $f = \mathbf{1}_{[-1/2, 1/2]}$, $\hat f(\xi) = \frac{\sin \pi \xi}{\pi \xi}$ is not integrable, so the theorem does not apply as stated.
Then $f$ is recovered only by the symmetric limits $\lim_{R \to \infty} \int_{-R}^{R}$, which give $f(x)$ for $\abs{x} \ne 1/2$ and $1/2$ at the jumps.
A consequence of the theorem is uniqueness: if $f$ is integrable and $\hat f = 0$, then $f = 0$ almost everywhere.

## Explanations

### L1 human

The Fourier transform breaks a function into waves of all frequencies.
The inversion theorem says that adding these waves back together gives the original function.
So no information is lost in passing to the frequency side.

### L2 human

The theorem lets one solve problems on the frequency side, where derivatives become multiplications, and then return.
It is the basis of the Fourier method for linear differential equations with constant coefficients.
On the Schwartz space the transform is a bijection and inversion holds everywhere without extra hypotheses.
The [[Plancherel theorem|HPZHRZ0502]] gives the corresponding statement in $L^2$, where the inverse transform is the adjoint of the transform.
</document>

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# Monotone cubic interpolation

## Statement

Let $x_0 < x_1$, let $y_0, y_1 \in \R$ with $y_0 \ne y_1$, and let $m_0, m_1 \in \R$.
Let $p$ be the [[cubic Hermite interpolant|HCQWPAHYAT]] on $[x_0, x_1]$ with $p(x_i) = y_i$ and $p'(x_i) = m_i$ for $i = 0, 1$.
Let $\Delta = (y_1 - y_0)/(x_1 - x_0)$, $\alpha = m_0/\Delta$ and $\beta = m_1/\Delta$.

1. If $p$ is [[monotone|0A7KV2WB7A]] on $[x_0, x_1]$, then $\alpha \ge 0$ and $\beta \ge 0$.
2. If $0 \le \alpha \le 3$ and $0 \le \beta \le 3$, then $p$ is monotone on $[x_0, x_1]$.

In particular, let $x_0 < \cdots < x_n$ and $y_0 \le \cdots \le y_n$, with slopes $m_0, \dots, m_n$.
On each interval $[x_k, x_{k+1}]$ assume that either $y_k = y_{k+1}$ and $m_k = m_{k+1} = 0$, or the ratios $\alpha_k, \beta_k$ lie in $[0, 3]$.
Then the piecewise cubic Hermite interpolant is non-decreasing on $[x_0, x_n]$.

## Remarks

With $h = x_1 - x_0$ and $t = (x - x_0)/h$, the derivative is
$$\frac{p'(x)}{\Delta} = \alpha (1 - t)(1 - 3t) + \beta \, t (3t - 2) + 6 t (1 - t).$$
At $t = 0$ and $t = 1$ this gives $\alpha$ and $\beta$, which proves part 1.
For fixed $t$, the right side is affine in $(\alpha, \beta)$, so its minimum over the square $[0, 3]^2$ is at a corner.
At the four corners it equals $6t(1 - t)$, $3(1 - t)^2$, $3t^2$ and $3(1 - 2t)^2$, all non-negative, which proves part 2.

The Fritsch–Carlson method (1980) uses this to make a monotone $C^1$ interpolant.
It starts from some slope estimates and sets $m_k = m_{k+1} = 0$ on flat intervals.
Where $\alpha_k^2 + \beta_k^2 > 9$, it multiplies both slopes by $3/\sqrt{\alpha_k^2 + \beta_k^2}$, which moves $(\alpha_k, \beta_k)$ into the disc of radius $3$, a subset of the square.
Without such a limiter, a cubic spline through monotone data can overshoot: for data $0, 0, 1, 1$ at $x = 0, 1, 2, 3$, the natural cubic spline dips below $0$ on $(0, 1)$.
Example: for $y = 0, 1$ on $[0, 1]$ with $m_0 = m_1 = 3$, one has $\alpha = \beta = 3$ and $p(x) = 3x - 6x^2 + 4x^3$.
Then $p'(x) = 3(1 - 2x)^2 \ge 0$, so $p$ is monotone, with a flat point at $x = 1/2$.

## Explanations

### L1 human

Smooth curves through increasing data points can still wiggle down between points, which is bad for data such as cumulative totals or concentrations.
This theorem says how steep the curve may be at the data points without causing such wiggles.
If the slopes at both ends of an interval are between $0$ and $3$ times the slope of the straight line between the points, the curve stays monotone.

### L2 human

The condition is sufficient, and the exact region of monotone $(\alpha, \beta)$ found by Fritsch and Carlson is larger than the square.
Monotone interpolation trades some smoothness for shape preservation: the result is $C^1$ but in general not $C^2$, unlike a cubic spline.
It is the basis of the PCHIP routine in many numerical libraries and of shape-preserving interpolation more generally.
</document>

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# Doléans-Dade exponential

## Definition

Let $X$ be a real [[semimartingale|4EJB6E2AZK]] on a filtered probability space.
The *Doléans-Dade exponential* of $X$ is the unique semimartingale $Z = \mathcal{E}(X)$ such that, for all $t \ge 0$,
$$Z_t = 1 + \int_0^t Z_{s-} \, dX_s.$$
Here the integral is the stochastic integral of [[Itô calculus|JA95NH206E]].
The process $Z_{s-}$ is the left limit of $Z$ at $s$.

## Remarks

If $X$ is continuous, then $\mathcal{E}(X)_t = \exp\left(X_t - X_0 - \tfrac12 [X]_t\right)$, where $[X]$ is the [[quadratic variation|X4DHMZD5JA]] of $X$.
In general, $\mathcal{E}(X)_t = \exp\left(X_t - X_0 - \tfrac12 [X]^c_t\right) \prod_{0 < s \le t} (1 + \Delta X_s) e^{-\Delta X_s}$, where $[X]^c$ is the continuous part of $[X]$ and $\Delta X_s = X_s - X_{s-}$.
Example: let $B$ be a [[Brownian motion|22735FGJV6]] and $X = \sigma B$ with a constant $\sigma$.
Then $Z_t = \exp(\sigma B_t - \tfrac12 \sigma^2 t)$, and [[Itô's lemma|2J1RVQB0CM]] gives $dZ_t = \sigma Z_t \, dB_t = Z_t \, dX_t$.
Non-example: $W_t = \exp(\sigma B_t)$ fails the equation, since Itô's lemma gives $dW_t = W_t \, dX_t + \tfrac12 \sigma^2 W_t \, dt$.
If $X$ is a continuous [[local martingale|XC2YPC2BSW]], then $\mathcal{E}(X)$ is a positive local martingale and hence a supermartingale.
It is a true martingale under Novikov's condition $E\left[\exp\left(\tfrac12 [X]_\infty\right)\right] < \infty$.
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# Direct collocation

## Summary

Direct collocation is a method of [[trajectory optimization|58895A7T4H]] that turns an optimal control problem into a finite nonlinear program.
It represents the state and the control by their values on a time grid.
It imposes the differential equation only at chosen collocation points, through a polynomial model of the state on each interval.
The cost integral becomes a quadrature sum, and the resulting sparse nonlinear program is solved by a general optimiser.
It is "direct" because it discretises first and does not use the [[maximum principle|7RPV2T29CF]].

## Explanations

### L1 human

Instead of solving the equations of motion exactly, direct collocation draws the trajectory as a chain of short polynomial pieces through a few grid points.
It asks only that each piece obey the equations of motion at a few chosen times.
The grid values become the unknowns of an ordinary finite optimisation problem.
With the trapezoidal rule, for example, the condition on one step of length $h$ is $x_{k+1} - x_k = \frac{h}{2} (f_k + f_{k+1})$, where $f_k$ is the velocity at grid point $k$.
Constraints such as speed limits are easy to add, because they are just conditions on the grid values.
</document>

<document source="https://tamnd.github.io/math2/node/T5BEKMG5W8.md" tag="T5BEKMG5W8" kind="statement">
# Verification theorem

## Statement

Let $U \subseteq \R^m$ be a nonempty compact set and $T > 0$.
Let $f \colon [0, T] \times \R^n \times U \to \R^n$ be continuous and Lipschitz in $x$ uniformly in $(t, u)$, and let $L$ and $g$ be continuous.
Let $V$ be the [[value function|K6G0DRQ3CP]] of the problem of minimizing $\int_t^T L(s, y, u) \, ds + g(y(T))$ subject to $y' = f(s, y, u)$, $y(t) = x$.
Let $W \colon [0, T] \times \R^n \to \R$ be continuous, of class $C^1$ on $(0, T) \times \R^n$, with $W(T, x) = g(x)$.
Assume that $W$ solves the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]]
$$W_t(t, x) + \min_{u \in U} \big( L(t, x, u) + \nabla_x W(t, x) \cdot f(t, x, u) \big) = 0 \quad \text{on } (0, T) \times \R^n.$$
Then:

1. $W(t, x) \le V(t, x)$ for all $(t, x) \in [0, T] \times \R^n$.
2. Let $u^*$ be a measurable control on $[t, T]$ with trajectory $y^*$ from $(t, x)$. Suppose that for almost every $s \in [t, T]$ the value $u^*(s)$ attains the minimum in the equation at $(s, y^*(s))$. Then $u^*$ is optimal from $(t, x)$, and $W(t, x) = V(t, x)$.

## Remarks

The proof is a chain rule computation.
For any control $u$ with trajectory $y$, the function $s \mapsto W(s, y(s))$ is absolutely continuous with derivative $W_t + \nabla_x W \cdot f(s, y, u) \ge -L(s, y, u)$ by the equation.
Integrating from $t$ to $T$ gives $g(y(T)) - W(t, x) \ge -\int_t^T L \, ds$, that is, $W(t, x) \le J(t, x; u)$.
For $u^*$ the inequality is an equality at almost every time, so $J(t, x; u^*) = W(t, x) \le V(t, x)$.
Example: for the [[linear-quadratic regulator|HR720TJZZR]], $W(t, x) = x^{T} P(t) x$ with $P$ the solution of the Riccati differential equation solves the equation with $U = \R^m$.
The minimum is attained at $u = -R^{-1} B^{T} P(t) x$, so this feedback is optimal; the proof adapts to the unbounded control set.
The smoothness of $W$ is essential: value functions are often not differentiable, and then one needs the theory of [[viscosity solutions|KREYT31SS1]].

## Explanations

### L1 human

Suppose someone hands you a candidate function for the optimal cost and a candidate control.
The verification theorem gives a test: if the function solves the Hamilton–Jacobi–Bellman equation and the control achieves the minimum in it, then the control is optimal.
It turns a guess into a proof of optimality.

### L2 human

The theorem is the sufficiency half of [[dynamic programming|QPS8VSJGMJ]].
It is the standard way to prove that a feedback law found by solving the HJB equation, for example the linear feedback of the linear-quadratic regulator, is truly optimal.
Pontryagin's maximum principle gives only necessary conditions, so the verification theorem complements it.
When the value function is not smooth, versions of the theorem use viscosity solutions or generalized gradients.
</document>

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# Barbier's theorem

## Statement

Let $w > 0$, and let $C$ be a [[curve of constant width|B4YD1ZXRVH]] $w$ in the plane.
Then the [[length|0TH78S7GBZ]] of $C$ is $\pi w$.

## Remarks

Apply the [[Crofton formula|9D5NT00H6G]] to $C$.
A line that meets the interior of the convex body bounded by $C$ meets $C$ in exactly two points.
The lines with normal direction $\theta$ that meet the body form a band of width $w(\theta)$.
So the length of $C$ is $\frac{1}{2} \int_0^\pi 2 \, w(\theta) \, d\theta = \int_0^\pi w \, d\theta = \pi w$.
Example: a circle of diameter $w$ has length $\pi w$, and the Reuleaux triangle of width $w$, made of three arcs of radius $w$ and angle $\pi/3$, also has length $\pi w$.
The hypothesis of constant width is needed: the square of side $1$ has width $1$ in the direction of its sides, but its perimeter is $4$, not $\pi$.

## Explanations

### L1 human

All shapes that have the same width in every direction also have the same perimeter, namely $\pi$ times the width.
So a Reuleaux triangle of width $1$ and a coin of diameter $1$ need exactly the same length of string to go around them.

### L2 human

The proof shows a more general fact: the perimeter of any plane convex body equals the integral of its width over the directions in $[0, \pi)$, which is Cauchy's formula.
Barbier's theorem is the special case of a constant integrand.
Together with the isoperimetric inequality it shows that the disc has the largest area among bodies of constant width $w$; the smallest area is the subject of the Blaschke–Lebesgue theorem.
</document>

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# Free distributive lattice

## Summary

The free distributive lattice on a set $S$ is the [[distributive lattice|0HM0HQCK6N]] generated by $S$ with no extra relations.
The only equalities in it are those that the lattice laws and distributivity force.
Every map from $S$ into a distributive lattice extends to exactly one lattice homomorphism from it.
Unlike the [[free lattice|WP20S9NZ15]] on three or more generators, the free distributive lattice on a finite set is finite.
On $n$ generators it has $M(n) - 2$ elements, where $M(n)$ is the [[Dedekind number|481ZPQ0GC3]].

## Explanations

### L1 human

Start with letters $x, y, z, \ldots$ and build all expressions with "and" and "or".
Two expressions count as equal when the rules of a distributive lattice prove them equal, as in ordinary logic without negation and without constants.
Every expression can then be written as an "or" of "and"s of letters.
With two letters there are exactly four different elements: $x \wedge y$, $x$, $y$ and $x \vee y$.
</document>

<document source="https://tamnd.github.io/math2/node/T5KZR54SG8.md" tag="T5KZR54SG8" kind="definition">
# Zimmer program

## Definition

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center, all of whose simple factors have real rank at least $2$.
Let $\Gamma$ be a [[lattice|BTYF5Y5PFY]] in $G$.
Let $M$ be a compact [[smooth manifold|F45VQ47GPV]] and $\mathrm{Diff}(M)$ its group of [[diffeomorphisms|0KZZ70ZHXP]].
The *Zimmer program* is the research program that aims to classify the homomorphisms $\rho \colon \Gamma \to \mathrm{Diff}(M)$, with these expected outcomes:

1. If $\dim M$ is small compared with the rank of $G$, then every such $\rho$ has finite image.
2. If $\rho$ preserves a volume form, then, up to finite index, the action is built from isometric actions and from algebraic actions.
   Here an algebraic action is an action on a quotient $\Lambda \backslash H$ of a Lie group $H$ by a lattice $\Lambda$.
   It comes from a homomorphism of $\Gamma$ or $G$ into $H$.

## Remarks

Example of an action in the sense of clause 2: $\mathrm{SL}_n(\Z)$ acts on the torus $\R^n / \Z^n$ by $A \cdot (x + \Z^n) = Ax + \Z^n$.
This is well defined because $A \Z^n = \Z^n$, and it preserves Lebesgue measure because $\det A = 1$.
Clause 1 is sharp for $\mathrm{SL}_n(\Z)$: it acts with infinite image on the real projective space of dimension $n - 1$ through its linear action on lines of $\R^n$.
The program is not a single theorem; clause 2 in particular has no fully agreed precise form.
Precise test cases are [[Zimmer's conjecture|JW71J3AKY4]] for clause 1 and conjectures on actions preserving a rigid geometric structure for clause 2.
</document>

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# Gårding's inequality

## Statement

Let $\Omega \subseteq \R^n$ be a bounded open set and let $k \ge 1$ be an integer.
For multi-indices $\alpha, \beta$ with $\abs{\alpha}, \abs{\beta} \le k$, let $A_{\alpha\beta} \colon \Omega \to \C$ be bounded measurable functions.
Assume that $A_{\alpha\beta}$ is uniformly continuous on $\Omega$ when $\abs{\alpha} = \abs{\beta} = k$.
Assume the uniform ellipticity condition: there is $\theta > 0$ with
$$\operatorname{Re} \sum_{\abs{\alpha} = \abs{\beta} = k} A_{\alpha\beta}(x)\xi^\alpha\xi^\beta \ge \theta\abs{\xi}^{2k} \quad \text{for all } x \in \Omega,\ \xi \in \R^n.$$
For $u, v$ in the [[Sobolev space|M4CEZ5K9HP]] $H^k_0(\Omega)$, let
$$B(u, v) = \sum_{\abs{\alpha}, \abs{\beta} \le k} \int_\Omega A_{\alpha\beta}(x)\, \partial^\beta u(x)\, \overline{\partial^\alpha v(x)} \, dx.$$
Then there are constants $c > 0$ and $G \ge 0$ such that
$$\operatorname{Re} B(u, u) \ge c\norm{u}_{H^k(\Omega)}^2 - G\norm{u}_{L^2(\Omega)}^2 \quad \text{for all } u \in H^k_0(\Omega).$$

## Remarks

The form $B$ is the sesquilinear form of the [[elliptic operator|XY8GEQSWPR]] $Lu = \sum (-1)^{\abs{\alpha}} \partial^\alpha(A_{\alpha\beta}\partial^\beta u)$ in divergence form.
For constant coefficients and only top-order terms, the inequality follows from Plancherel's theorem: the Fourier transform turns $\operatorname{Re} B(u, u)$ into $\int \operatorname{Re}\sigma(\xi)\abs{\hat u(\xi)}^2 \, d\xi \ge \theta \int \abs{\xi}^{2k}\abs{\hat u(\xi)}^2 \, d\xi$.
For variable coefficients, one freezes the coefficients on small pieces of $\Omega$, using uniform continuity, and glues with a partition of unity.
Lower-order terms are absorbed with the interpolation inequality $\norm{u}_{H^j} \le \varepsilon\norm{u}_{H^k} + C_\varepsilon\norm{u}_{L^2}$ for $j < k$.

Example: for $k = 1$ and $B(u, u) = \int_\Omega \abs{\nabla u}^2 \, dx$, the form of $-\Delta$, one has $B(u, u) = \norm{u}_{H^1}^2 - \norm{u}_{L^2}^2$, so $c = 1$ and $G = 1$ work.

Ellipticity cannot be dropped.
In $\R^2$, the form $\int_\Omega \abs{\partial_1 u}^2 \, dx$ is not elliptic, and functions that oscillate fast in the $x_2$ direction make $\norm{u}_{H^1}$ large while the form and $\norm{u}_{L^2}$ stay bounded.

## Explanations

### L1 human

An elliptic operator controls all derivatives of order $k$, not only some of them.
Gårding's inequality makes this precise in terms of energy: the energy $B(u, u)$ bounds the full Sobolev norm of $u$, up to a harmless multiple of $\norm{u}_{L^2}^2$.
It is the basic estimate behind the existence of weak solutions to elliptic boundary problems.

### L2 human

Adding $G$ to the operator, the form $B(u, v) + G\ip{u}{v}$ becomes coercive on $H^k_0(\Omega)$.
The [[Lax–Milgram theorem|85KK79FCN5]] then gives a unique weak solution of $(L + \lambda)u = f$ for every $f$ and every $\lambda \ge G$.
Combined with the compact embedding of $H^k_0(\Omega)$ in $L^2(\Omega)$, this yields the Fredholm alternative and a discrete spectrum for the Dirichlet problem.
The constant $G$ cannot in general be taken to be $0$, because $L$ may have eigenvalues with negative real part.
</document>

<document source="https://tamnd.github.io/math2/node/T5RW1VN8AN.md" tag="T5RW1VN8AN" kind="example">
# Green's function of the ball

## Example

Let $B = \set{x \in \R^3 : \abs{x} < 1}$ be the unit [[open ball|0YE5868HZB]].
For $x \in B$ with $x \ne 0$, let $x^* = x / \abs{x}^2$ be the inverse point of $x$ in the unit sphere; it lies outside $\overline{B}$.
Define, for $y \in \overline{B}$ with $y \ne x$,
$$G(x, y) = \frac{1}{4\pi} \left( \frac{1}{\abs{x - y}} - \frac{1}{\abs{x} \, \abs{y - x^*}} \right) \quad (x \ne 0), \qquad G(0, y) = \frac{1}{4\pi} \left( \frac{1}{\abs{y}} - 1 \right).$$
This is the [[Green's function|M36SD56MWQ]] of $B$, normalized so that $\Delta_y G(x, y) = -\delta_x$.
Check of the three properties, for $x \ne 0$:

1. The first term is the [[fundamental solution|BSJ8X2XRN9]] $\frac{1}{4\pi\abs{x - y}}$, which has the right singularity at $y = x$.
2. The second term is [[harmonic|0DEB0JMN0W]] in $y$ on $B$, because its singularity $x^*$ lies outside $\overline{B}$.
3. For $\abs{y} = 1$ one has $\abs{x}^2 \abs{y - x^*}^2 = \abs{x}^2 - 2 x \cdot y + 1 = \abs{x - y}^2$, so the two terms cancel and $G(x, y) = 0$.

For $x = 0$, the term $1/\abs{y}$ is the fundamental solution times $4\pi$, the constant $1$ is harmonic, and $G(0, y) = 0$ when $\abs{y} = 1$.

## Remarks

In $\R^n$ with $n \ge 3$ the same construction gives $G(x, y) = \Phi(x - y) - \Phi(\abs{x}(y - x^*))$, where $\Phi(z) = \frac{1}{(n-2)\sigma_{n-1}} \abs{z}^{2-n}$ and $\sigma_{n-1}$ is the area of the unit sphere.
The outward normal derivative of $-G(x, \cdot)$ on the sphere is the Poisson kernel of the ball.
The second term is the potential of an image charge of strength $1/\abs{x}$ placed at $x^*$.

## Explanations

### L1 human

To make the potential of a point charge vanish on a sphere, place a second, opposite charge at the mirror point outside the sphere.
The mirror point lies on the same ray, at distance $1/\abs{x}$ from the centre.
On the sphere the two potentials cancel exactly, and inside the ball the sum is the Green's function.
</document>

<document source="https://tamnd.github.io/math2/node/T5TJRYZPFX.md" tag="T5TJRYZPFX" kind="definition">
# Baum–Welch algorithm

## Definition

Consider a [[hidden Markov model|HJVM390354]] with hidden states $\set{1, \dots, N}$ and observation symbols $\set{1, \dots, M}$.
Its parameter $\theta = (\pi, A, B)$ consists of an initial distribution $\pi_i$, a transition matrix $a_{ij}$ and emission probabilities $b_i(k)$.
Let $y_1, \dots, y_T$ with $T \ge 2$ be an observed sequence.

For the current $\theta$, the forward and backward variables are
$$\alpha_t(i) = P_\theta(Y_1 = y_1, \dots, Y_t = y_t, X_t = i), \qquad \beta_t(i) = P_\theta(Y_{t+1} = y_{t+1}, \dots, Y_T = y_T \mid X_t = i),$$
computed by $\alpha_1(i) = \pi_i b_i(y_1)$, $\alpha_{t+1}(j) = \sum_i \alpha_t(i) a_{ij} b_j(y_{t+1})$, $\beta_T(i) = 1$ and $\beta_t(i) = \sum_j a_{ij} b_j(y_{t+1}) \beta_{t+1}(j)$.
With $L = \sum_i \alpha_T(i)$, set
$$\gamma_t(i) = \frac{\alpha_t(i) \beta_t(i)}{L}, \qquad \xi_t(i, j) = \frac{\alpha_t(i) a_{ij} b_j(y_{t+1}) \beta_{t+1}(j)}{L}.$$
The *Baum–Welch algorithm* repeats the following update until the [[likelihood|0VW9E4QXR6]] $L$ stops changing:
$$\pi_i \leftarrow \gamma_1(i), \qquad a_{ij} \leftarrow \frac{\sum_{t=1}^{T-1} \xi_t(i, j)}{\sum_{t=1}^{T-1} \gamma_t(i)}, \qquad b_i(k) \leftarrow \frac{\sum_{t \,:\, y_t = k} \gamma_t(i)}{\sum_{t=1}^{T} \gamma_t(i)}.$$

## Remarks

Here $\gamma_t(i)$ is the [[conditional probability|0990W5PN13]] that $X_t = i$ given the observations, and $\xi_t(i, j)$ is the conditional probability that $X_t = i$ and $X_{t+1} = j$.
The update is one step of the [[expectation–maximization algorithm|11N1AXSXYK]], so the likelihood never decreases from one iteration to the next.
It may converge to a local maximum rather than to the global maximum, so several starting points are often used.

Example: with $N = 1$ every $\gamma_t(1)$ equals $1$, and the update gives $b_1(k) = \#\set{t : y_t = k} / T$.
This is the observed frequency of symbol $k$, which is the maximum likelihood estimate in that case.

Non-example: the Viterbi algorithm uses the same model but returns the single most likely hidden path for fixed parameters.
It does not update $\theta$, so it is not the Baum–Welch algorithm.
</document>

<document source="https://tamnd.github.io/math2/node/T5VW3PE7TZ.md" tag="T5VW3PE7TZ" kind="statement">
# Baouendi–Trèves approximation theorem

## Statement

Let $M$ be a smooth real submanifold of $\C^n$ that is generic, that is, $T_q M + J\, T_q M = T_q \C^n$ for every $q \in M$, where $J$ is multiplication by $i$.
Let $p \in M$.
Then there is a compact neighbourhood $K$ of $p$ in $M$ with the following property.
Let $f$ be a continuous [[CR function|JFKYYK2CEB]] on $M$.
Then there is a sequence $(P_k)$ of complex [[polynomials|07GNX8GCYP]] in $z_1, \dots, z_n$ such that $P_k \to f$ uniformly on $K$.

## Remarks

The theorem is due to M. S. Baouendi and F. Trèves, who proved it for locally integrable structures; the statement above is its form for embedded [[CR manifolds|ZC78JKZX9S]].
The proof convolves $f$ with the Gaussian kernel $e^{-\tau \sum_j (z_j - \zeta_j)^2}$ over a suitable totally real piece of $M$.
The result is an entire function of $z$.
The CR equations and Stokes' theorem show that it tends to $f$ on $K$ as $\tau \to \infty$.
Truncating the power series of these entire functions then gives polynomials.
The neighbourhood $K$ depends on $M$ and $p$ but not on $f$.
For example, let $M = \C^{n-1} \times \R \subset \C^n$.
Its CR functions are the functions holomorphic in $(z_1, \dots, z_{n-1})$ for each fixed real $z_n$.
The theorem says that they are local uniform limits of holomorphic polynomials in all $n$ variables.
The result is local: a CR function on all of $M$ need not be a uniform limit of polynomials on large compact subsets of $M$.

## Explanations

### L1 human

CR functions on a submanifold of complex space are defined by a system of differential equations, and they can be quite rough.
The theorem says that near each point they can nevertheless be approximated as closely as one wants by ordinary polynomials in the complex coordinates.
So every CR function is locally a limit of restrictions of holomorphic functions on the whole space.

### L2 human

This approximation theorem is a basic tool in the local theory of CR functions.
By the maximum principle, polynomials that converge uniformly on $K$ also converge uniformly on the polynomially convex hull $\widehat{K}$ of $K$.
So every continuous CR function on $M$ extends holomorphically to the interior of $\widehat{K}$, which gives a way to prove extension results such as the [[Lewy extension theorem|V2YCGHD719]].
The theorem also shows that CR functions in this sense are exactly the local uniform limits of restrictions of holomorphic functions.
</document>

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# Modular curve

## Summary

A modular curve is the quotient of the [[upper half-plane|X6XGGH5FPW]] by a [[congruence subgroup|1H6R856P3C]] of $\mathrm{SL}_2(\Z)$, compactified by adding finitely many cusps.
The result is a compact [[Riemann surface|5K0N8QMV6Q]] and an algebraic curve defined over a number field.
Modular curves parametrize [[elliptic curves|427N6S6QQW]] with extra level structure, and their holomorphic differentials correspond to cusp [[modular forms|GWKN5VG78A]] of weight $2$.
They are central in the proof of Fermat's Last Theorem through the modularity of elliptic curves.

## Explanations

### L1 human

The group $\mathrm{SL}_2(\Z)$ moves points of the upper half-plane by fractional linear maps $\tau \mapsto (a\tau + b)/(c\tau + d)$.
Gluing together points that are moved into each other produces a surface.
For the full group the surface, with one cusp added, is a sphere, and each point other than the cusp stands for one elliptic curve over $\C$ up to isomorphism.
For smaller groups, such as matrices with $c$ divisible by $N$, one gets curves of higher genus that also remember some points of finite order on the elliptic curve.
</document>

<document source="https://tamnd.github.io/math2/node/T61AW674XD.md" tag="T61AW674XD" kind="statement">
# Ambrose–Singer theorem

## Statement

Let $M$ be a connected smooth manifold, let $E \to M$ be a smooth vector bundle with a [[connection|5ZYRRVE8BW]] $\nabla$, and let $p \in M$.
Let $R$ be the [[curvature|JA29ETCM02]] of $\nabla$, given by $R(X, Y) s = \nabla_X \nabla_Y s - \nabla_Y \nabla_X s - \nabla_{[X, Y]} s$.
Let $P_\gamma$ denote the [[parallel transport|TVAGN3D6W5]] along a piecewise smooth curve $\gamma$.
Then the Lie algebra of the [[holonomy group|WHDZ3MK56B]] $\mathrm{Hol}_p(\nabla) \subset \mathrm{GL}(E_p)$ is the linear span of the endomorphisms
$$P_\gamma^{-1} \circ R_q(u, v) \circ P_\gamma \in \mathrm{End}(E_p).$$
Here $q$ runs over $M$, $\gamma$ runs over the piecewise smooth curves from $p$ to $q$, and $u, v$ run over $T_qM$.

## Remarks

The idea of the proof: transport around a small loop spanned by $u$ and $v$ at $q$ differs from the identity by about $R_q(u, v)$ times the area.
So these conjugated curvatures lie in the holonomy algebra.
Conversely, the set of frames reached from a frame at $p$ by horizontal curves is a reduction of the frame bundle.
Its structure algebra is spanned by these elements, and this is where the [[curvature form|ZAJFW1S078]] of a principal connection enters.
In the language of principal bundles, let $\omega$ be a connection on $P \to M$ with curvature $\Omega$, and let $u \in P$.
Then the Lie algebra of $\mathrm{Hol}_u(\omega)$ is spanned by the values $\Omega_v(X, Y)$.
Here $v$ runs over the points of $P$ joined to $u$ by horizontal curves, and $X, Y$ over the horizontal vectors at $v$.
Example: a connection with $R = 0$ has zero holonomy algebra, so its restricted holonomy group is trivial and parallel transport is locally path independent.
The theorem only sees the identity component: the flat Möbius line bundle over a circle has $R = 0$ but holonomy group $\set{\pm 1}$.

## Explanations

### L1 human

Holonomy is a global notion: it uses loops of any size.
Curvature is local: it is computed at single points.
The Ambrose–Singer theorem says that the infinitesimal part of the holonomy is generated by curvature, once curvature at far away points is carried back along paths.
So if the curvature vanishes everywhere, small loops never turn vectors.

### L2 human

The theorem is the bridge between the holonomy group and the curvature tensor.
It shows that the curvature takes values in the holonomy algebra, which gives strong algebraic restrictions on curvature when the holonomy is small.
This is the starting point of Berger's classification of Riemannian holonomy groups and of the study of manifolds with special holonomy.
It also shows that a connection is flat exactly when its restricted holonomy group is trivial.
</document>

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# n-sphere

## Description

Let $n \ge 0$ be an integer.
The *$n$-sphere* is the subspace
$$S^n = \set{x \in \R^{n+1} : x_1^2 + \cdots + x_{n+1}^2 = 1}$$
of [[Euclidean space|0XPR1E85W0]] $\R^{n+1}$, the set of points at distance $1$ from the origin.
It is a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.
It is [[Hausdorff|0S1FJ8VEX9]] and [[second-countable|03PPNCM9GG]] as a subspace of $\R^{n+1}$.
For the north pole $N = (0, \ldots, 0, 1)$, the stereographic projection
$$\sigma_N(x) = \frac{(x_1, \ldots, x_n)}{1 - x_{n+1}}$$
is a [[homeomorphism|0QT00PB5G2]] from $S^n \setminus \set{N}$ onto $\R^n$, with inverse $y \mapsto \bigl(2y, \abs{y}^2 - 1\bigr) / \bigl(\abs{y}^2 + 1\bigr)$.
The projection from the south pole is defined in the same way, and the two charts cover $S^n$.

## Remarks

$S^0$ is a pair of points, $S^1$ is the [[circle|060W5CBZP6]] and $S^2$ is the usual round sphere.
$S^n$ is [[compact|0TVJZYB6SH]], being closed and bounded in $\R^{n+1}$, and for $n \ge 1$ it is [[path-connected|0WB6DH499T]].
For $n \ge 2$ it is simply connected.
For $n \ge 1$ it is the one-point compactification of $\R^n$, and it is homeomorphic to the closed ball $D^n$ with its boundary sphere collapsed to a point.
Its integer homology is $\Z$ in degrees $0$ and $n$ and zero in the other degrees, for $n \ge 1$.

## Explanations

### L1 human

The $n$-sphere is the set of points at distance one from a centre, in one dimension more than the sphere itself.
The $1$-sphere is a circle in the plane, the $2$-sphere is the surface of a ball in space.
Near each point it looks like flat $n$-dimensional space, but as a whole it closes up on itself.
</document>

<document source="https://tamnd.github.io/math2/node/T62HHSRHP5.md" tag="T62HHSRHP5" kind="statement">
# Maximum principle for harmonic functions

## Statement

Let $U \subseteq \R^n$ be a domain, that is, a connected open set, and let $u \colon U \to \R$ be [[harmonic|0DEB0JMN0W]].

1. (Strong maximum principle.) If there is $a \in U$ with $u(a) = \sup_U u$, then $u$ is constant on $U$.
2. (Weak maximum principle.) If $U$ is bounded and $u$ extends continuously to the closure $\overline{U}$, then $\max_{\overline{U}} u = \max_{\partial U} u$.

## Remarks

Proof idea for part 1: the set $A = \set{x \in U : u(x) = \sup_U u}$ is closed in $U$ by continuity.
It is open: if $x \in A$ and $\overline{B}(x, r) \subseteq U$, the [[mean value property|0HYPR64JM2]] over the ball makes $u(x)$ the average of values $\le u(x)$, which forces $u = u(x)$ on the ball.
As $U$ is connected and $A$ is not empty, $A = U$.
Part 2 follows: the maximum of $u$ on the compact set $\overline{U}$ is attained, and if it were attained only inside, part 1 would make $u$ constant.
Example: $u(x, y) = x^2 - y^2$ on the unit disc takes its maximum $1$ on the boundary, at $(\pm 1, 0)$, and the value $0$ at the centre.
Boundedness is needed in part 2: on the half-plane $\set{x > 0}$, $u = x$ is $0$ on the boundary and positive inside.
Connectedness is needed in part 1.
On two disjoint discs, the function equal to $0$ on one and $1$ on the other is harmonic and attains its maximum without being constant.
This is the harmonic case of the general [[maximum principle|D2JF19HF91]] for elliptic operators.

## Explanations

### L1 human

A harmonic function cannot have a peak inside its domain.
Its largest values on a bounded region are reached on the edge.
Think of the height of a soap film spanned on a wire frame: the highest point lies on the wire.

### L2 human

The weak maximum principle gives uniqueness for the Dirichlet problem.
Two harmonic functions with the same boundary values differ by a harmonic function that is $0$ on the boundary, hence $0$ inside.
It also gives stability: if the boundary values of two solutions differ by at most $\varepsilon$, then the solutions differ by at most $\varepsilon$ everywhere.
Subharmonic functions satisfy the same maximum principle, and Perron's method builds solutions of the Dirichlet problem from it.
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# Right adjoints preserve limits

## Statement

Let $G \colon \mathcal{D} \to \mathcal{C}$ be a functor that has a [[left adjoint|TZHF6ZM92S]] $F \colon \mathcal{C} \to \mathcal{D}$.
Then $G$ [[preserves|H4MY8JN8W4]] every [[limit|CHS5AFJGWE]] that exists in $\mathcal{D}$.
Dually, $F$ preserves every colimit that exists in $\mathcal{C}$.

## Remarks

Let $(L, \lambda)$ be a limit cone of a diagram $K \colon J \to \mathcal{D}$.
For an object $c$, cones from $c$ over $G K$ match, by the adjunction, with cones from $F c$ over $K$.
These match with morphisms $F c \to L$, and so with morphisms $c \to G L$.
So $(G L, G \lambda)$ is a limit cone.

Example: the forgetful functor from groups to sets has a left adjoint, the free group.
So the underlying set of a product or an equalizer of groups is the product or the equalizer of the underlying sets.
The free group functor preserves coproducts: the free group on $X \sqcup Y$ is the free product of the free groups on $X$ and on $Y$.

The theorem gives a test for non-existence.
The forgetful functor from groups to sets does not send the initial trivial group to the initial empty set, so it has no right adjoint.

## Explanations

### L1 human

If a functor has a left adjoint, it behaves well with products, kernels and all other limits.
For example, forgetting the group structure of a product of groups leaves the product of the sets.
Dually, free constructions turn sums into sums.

### L2 human

This is the most used property of adjoint functors.
It explains many familiar facts at once, such as that tensor products distribute over direct sums, since $- \otimes M$ is left adjoint to $\Hom(M, -)$.
It is also the necessary half of the adjoint functor theorems.
Those theorems show that, under size conditions, preserving limits is enough for a functor to have a left adjoint.
</document>

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# Mather's stability theorem

## Statement

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] and let $f \colon M \to N$ be a proper smooth map.
Then $f$ is [[stable|1991WETZF9]] if and only if $f$ is [[infinitesimally stable|C3TFCVE2YK]].

## Remarks

Mather proved the theorem in a series of papers on the stability of smooth mappings.
The proof that infinitesimal stability implies stability joins $f$ to a nearby map $g$ by a smooth path $f_t$ and seeks families of diffeomorphisms $\varphi_t$, $\psi_t$ with $f_t = \psi_t \circ f \circ \varphi_t^{-1}$.
Differentiating in $t$ gives an equation of the same form as in the definition of infinitesimal stability.
The [[Malgrange preparation theorem|2A5HZZRGFT]] is used to solve it for all $t$, not only for $t = 0$.
Example: for a compact $M$, a function $f \colon M \to \R$ is infinitesimally stable exactly when its critical points are nondegenerate with distinct critical values, so these functions are the stable ones.
When $M$ is compact every smooth map $M \to N$ is proper, so the theorem applies to all of them.
Mather also showed that infinitesimal stability can be checked on finite jets at finite sets of points.
He determined the pairs of dimensions for which stable maps are dense in $C^\infty(M, N)$; they are not dense in general.

## Explanations

### L1 human

Stability asks that every nearby map be the same map in new coordinates, which is hard to check directly.
Infinitesimal stability asks only that every tiny push of the map be undone by tiny changes of coordinates, which is a linear equation.
Mather's theorem says that, for proper maps, the easy linear test decides the hard question.

### L2 human

The theorem is the foundation of the modern theory of stable maps.
It reduces stability to the local algebra of germs, where finite determinacy and the preparation theorem make computations finite.
With it, one shows that the fold and cusp of maps of the plane, the cross-cap of maps from the plane to space, and the [[Morin singularities|DF8CJ1A9MF]] are stable.
</document>

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# Discrete Poisson equation

## Definition

Let $h > 0$ and let $\Omega \subset \Z^2$ be a finite set of grid points.
Let $\partial \Omega$ be the set of points of $\Z^2 \setminus \Omega$ at distance $1$ from some point of $\Omega$.
Let $f \colon \Omega \to \R$ and $g \colon \partial \Omega \to \R$ be given [[functions|0EPAT51DXQ]].
The *discrete Poisson equation* with Dirichlet boundary values is the problem of finding $u \colon \Omega \cup \partial \Omega \to \R$ such that
$$\frac{u(i+1, j) + u(i-1, j) + u(i, j+1) + u(i, j-1) - 4u(i, j)}{h^2} = f(i, j) \quad \text{for all } (i, j) \in \Omega,$$
and $u = g$ on $\partial \Omega$.
The left side is $h^{-2}$ times the [[discrete Laplace operator|ZN30Y1ZDB6]] of $u$ at $(i, j)$.

## Remarks

Example: let $\Omega = \set{(0, 0)}$, $h = 1$, $g = 0$ on the four neighbours and $f(0, 0) = -4$.
The equation reads $0 + 0 + 0 + 0 - 4u(0, 0) = -4$, so $u(0, 0) = 1$ is the solution.

Non-example: the function with $u(0,0) = 0$ and $u = 0$ on $\partial\Omega$ satisfies the boundary condition but gives $0 \ne -4$ on the left, so it is not a solution.

For a finite $\Omega$, the problem has exactly one solution for all $f$ and $g$; this follows from the discrete maximum principle.
Ordering the unknowns gives a linear system $A \mathbf{u} = \mathbf{b}$, where $A$ is sparse, symmetric and negative definite.
On a rectangle the matrix has a block tridiagonal form with $-4$ on the diagonal, before scaling by $h^{-2}$.
The same definition works in $\Z^d$ with $2d$ neighbours and $-2d$ in place of $-4$.
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# Descartes' rule of signs

## Statement

Let $f = a_n x^n + \cdots + a_1 x + a_0$ be a nonzero polynomial with real coefficients.
Let $V$ be the number of sign changes in the sequence of nonzero coefficients of $f$, taken in order of degree.
Let $P$ be the number of positive [[roots|00EHX6FWC0]] of $f$, counted with [[multiplicity|03XVP8Q4GY]].
Then $P \le V$, and $V - P$ is even.

## Remarks

The parity part follows by comparing the sign of the leading coefficient with the sign of the lowest nonzero coefficient.
Both $V$ and $P$ are even exactly when these signs agree.
The bound $P \le V$ can be proved by induction on the degree, using [[Rolle's theorem|0194DY35BG]] to compare the positive roots of $f$ with those of $f'$.
Example: $f = x^3 - x^2 - x + 1$ has signs $+, -, -, +$, so $V = 2$; indeed $f = (x - 1)^2 (x + 1)$ has the double positive root $1$, so $P = 2$.
The number of negative roots is bounded in the same way by the sign changes of $f(-x)$.
The bound need not be reached: $x^2 - x + 1$ has $V = 2$ and no real root, so $P = 0$.

## Explanations

### L1 human

Reading the signs of the coefficients of a polynomial gives an upper bound for the number of its positive roots.
Each change from plus to minus, or from minus to plus, allows at most one positive root.
The count can only be too high by an even number.

### L2 human

The rule is a quick first test before one locates roots more precisely.
If all roots of $f$ are real, then the bound is exact: $P = V$.
[[Budan's theorem|7THS519D4A]] extends the rule from the half-line $(0, \infty)$ to any interval, and [[Sturm's theorem|R44CV0KD9T]] gives the exact number of distinct real roots in an interval.
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<document source="https://tamnd.github.io/math2/node/T6HCX959PJ.md" tag="T6HCX959PJ" kind="concept">
# Gorenstein ring

## Summary

A Gorenstein ring is a Noetherian ring whose local rings have finite injective dimension as modules over themselves.
Gorenstein rings sit between [[complete intersection rings|EPM2HKGMZS]] and [[Cohen–Macaulay rings|N1QJQEGBFC]]: every complete intersection is Gorenstein and every Gorenstein ring is Cohen–Macaulay.
They are exactly the Cohen–Macaulay local rings whose [[canonical module|W5ECXPM6FW]] is the ring itself, which makes duality statements particularly clean.

## Explanations

### L1 human

Gorenstein rings have a built-in symmetry.
For a finite-dimensional local algebra over a field, the condition says that the "bottom" of the ring, the part killed by every non-unit, is one-dimensional.
The ring $K[x]/(x^3)$ is Gorenstein: its bottom is spanned by $x^2$.
The ring $K[x,y]/(x^2, xy, y^2)$ is not: its bottom is spanned by both $x$ and $y$.
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# Hodges–Lehmann estimator

## Summary

The Hodges–Lehmann estimator estimates the centre of a symmetric distribution by the [[median|07Y2E7P9NP]] of all pairwise averages of the observations.
It is the estimator that corresponds to the [[Wilcoxon signed-rank test|38SETDT2HN]], and it inherits its good properties.
It resists outliers much better than the mean, and for normal data it is almost as precise, with asymptotic relative efficiency $3/\pi \approx 0.955$.
A two-sample version estimates a shift between two populations by the median of all pairwise differences.

## Explanations

### L1 human

Take every pair of data points, including each point paired with itself, and write down the midpoint of the pair.
The median of all these midpoints is the Hodges–Lehmann estimate.
For the data $1, 2, 9$ the midpoints are $1, 1.5, 2, 5, 5.5, 9$, and their median is $3.5$.
This lies between the median $2$, which ignores how far $9$ is, and the mean $4$, which is pulled by it.
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# Topological modular forms

## Definition

Let $\mathcal{M}_{\mathrm{ell}}$ be the moduli stack of [[elliptic curves|SJ2BVG9F4W]] and $\overline{\mathcal{M}}_{\mathrm{ell}}$ its compactification by generalized elliptic curves.
An $E_\infty$ [[ring spectrum|BAW0T68GWK]] is a ring spectrum with an action of an [[$E_\infty$ operad|T78YG90VVT]].
By the Goerss–Hopkins–Miller theorem there is a sheaf $\mathcal{O}^{\mathrm{top}}$ of $E_\infty$ ring spectra on the étale site of $\overline{\mathcal{M}}_{\mathrm{ell}}$ with the following property.
For each étale map $\operatorname{Spec} R \to \mathcal{M}_{\mathrm{ell}}$, which classifies an elliptic curve $C$ over $R$, the spectrum $\mathcal{O}^{\mathrm{top}}(\operatorname{Spec} R)$ is an [[elliptic cohomology|XH68V2BP4K]] theory for $C$.
From this sheaf one defines three spectra.

1. $\mathrm{Tmf} = \mathcal{O}^{\mathrm{top}}(\overline{\mathcal{M}}_{\mathrm{ell}})$ is the spectrum of global sections.
2. *Topological modular forms* $\mathrm{tmf}$ is the connective cover $\tau_{\ge 0} \mathrm{Tmf}$.
3. Periodic topological modular forms $\mathrm{TMF} = \mathcal{O}^{\mathrm{top}}(\mathcal{M}_{\mathrm{ell}})$ is the spectrum of sections over the open substack of smooth curves.

## Remarks

There is a ring map $\pi_{2k} \mathrm{tmf} \to MF_k$ to the ring of integral [[modular forms|M8NA9X8BTV]] of weight $k$, $MF_* = \Z[c_4, c_6, \Delta]/(c_4^3 - c_6^2 - 1728 \Delta)$.
It becomes an isomorphism after tensoring with $\Q$, so $\pi_* \mathrm{tmf} \otimes \Q \cong \Q[c_4, c_6]$ with $c_4$ in degree $8$ and $c_6$ in degree $12$.
The spectrum $\mathrm{TMF}$ is $576$-periodic: $\Delta^{24}$ lifts to a unit in $\pi_{576} \mathrm{TMF}$.

Example: in degree $8$, $\pi_8 \mathrm{tmf} \otimes \Q \cong \Q$, and the map to modular forms sends a generator to a nonzero multiple of $c_4$, the Eisenstein series of weight $4$; this agrees with $\dim_\Q MF_4 \otimes \Q = 1$.

Non-example: $\pi_* \mathrm{tmf}$ is not the ring $MF_*$ of modular forms.
The group $\pi_3 \mathrm{tmf} \cong \Z/24$ is torsion in odd degree, while $MF_*$ is zero in odd degrees.
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# Turing reduction

## Definition

Let $A$ and $B$ be subsets of $\N$.
A *Turing reduction* from $A$ to $B$ is an [[oracle machine|Z5CCVBME2V]] $M$ such that $M$ with oracle $B$ halts on every input $x \in \N$.
Moreover, the output is $1$ if $x \in A$ and $0$ if $x \notin A$.
The set $A$ is *Turing reducible* to $B$, written $A \le_T B$, if such an $M$ exists.
The sets $A$ and $B$ are *Turing equivalent*, written $A \equiv_T B$, if $A \le_T B$ and $B \le_T A$.

## Remarks

Example: for every $B \subseteq \N$, the complement $\N \setminus B$ satisfies $\N \setminus B \le_T B$.
The machine asks the oracle whether $x \in B$ and outputs the opposite answer.

Non-example: the [[halting problem|Z7TN2B35B0]] $K$ satisfies $K \not\le_T \emptyset$.
An oracle machine with the empty oracle computes only what an ordinary Turing machine computes, and $K$ is not a [[computable set|PPGD5KW8JC]].

The relation $\le_T$ is reflexive and transitive.
A set $A$ satisfies $A \le_T \emptyset$ exactly when $A$ is computable.
Every [[many-one reduction|JTAH37JTHD]] gives a Turing reduction.
The converse fails: $\N \setminus K \le_T K$, but $\N \setminus K \not\le_m K$, since $K$ is computably enumerable and $\N \setminus K$ is not.
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# Steklov eigenvalue problem

## Problem

Let $\Omega \subseteq \R^n$ be a bounded domain with Lipschitz boundary $\partial\Omega$ and outer unit normal $\nu$.
Find all $\sigma \in \R$ for which some nonzero $u \in H^1(\Omega)$ solves
$$\Delta u = 0 \text{ in } \Omega, \qquad \frac{\partial u}{\partial \nu} = \sigma u \text{ on } \partial\Omega,$$
in the weak sense.
Here $\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $H^1(\Omega)$ is a [[Sobolev space|M4CEZ5K9HP]].
Such numbers $\sigma$ are the *Steklov eigenvalues* of $\Omega$, and one asks how they depend on the geometry of $\Omega$.

## Remarks

The Steklov eigenvalues are the eigenvalues of the [[Dirichlet-to-Neumann operator|W783KDGT8E]] of $\Omega$, which maps boundary values $f$ to the normal derivative of the harmonic extension of $f$.
They form a sequence $0 = \sigma_0 < \sigma_1 \le \sigma_2 \le \dots \to \infty$, and $\sigma_0 = 0$ belongs to the constant functions.
Example: for the unit disc, the functions $r^k \cos(k\theta)$ and $r^k \sin(k\theta)$ are harmonic with $\partial_r u = k u$ on the circle, so the eigenvalues are $0, 1, 1, 2, 2, 3, 3, \dots$.
Weinstock proved that among simply connected planar domains of given perimeter, the disc maximises $\sigma_1$.

## Explanations

### L1 human

Here the vibration happens only on the boundary, while the inside is in equilibrium.
The equation says that the flux across the edge is proportional to the value on the edge.
It models, for example, the sloshing of a liquid in a container and heat exchange through a surface.
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# Fractional calculus

## Summary

Fractional calculus studies integrals and [[derivatives|0V1EJCT8G6]] of non-integer order, such as a "half derivative".
Its basic operator is the [[Riemann–Liouville integral|JTKCZ44NN3]], which extends repeated integration to any real order $\alpha > 0$.
Fractional derivatives, such as the Riemann–Liouville, [[Caputo|6VMGQYF8RW]] and [[Grünwald–Letnikov|M9Z8VM4K1Z]] derivatives, are built from it or from limits of differences.
These operators model memory effects and anomalous diffusion, and appear in integral equations such as Abel's equation.

## Explanations

### L1 human

Differentiating once, twice or three times is familiar; fractional calculus asks what it means to differentiate half a time.
The idea is to find a formula for the $n$-th integral that makes sense when $n$ is not a whole number.
[[Cauchy's formula for repeated integration|EYXG11V76E]] gives such a formula, with the factorial replaced by the [[gamma function|0H5SYBSED9]].
For example, the half integral of $1$ from $0$ is $2\sqrt{x/\pi}$, and doing the half integral twice gives the ordinary integral $x$.
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# Periodic point

## Summary

A periodic point of a map is a point that comes back to itself after some positive number of steps.
Its orbit is a finite cycle that repeats forever.
[[Fixed points|R6QVS6P21N]] are the periodic points that return after one step.
Periodic points are the simplest kind of long-time behaviour, and counting them says a lot about a system.
For flows the analogue is a [[periodic orbit|0AA97XA70K]].

## Explanations

### L1 human

Start at a point, apply the map again and again, and watch whether the point ever comes back.
If it comes back after $n$ steps, it will repeat the same loop of $n$ points forever.
For $f(x) = -x$ on the real line, the point $1$ goes to $-1$ and then back to $1$, so it is periodic with period $2$.
For the rotation of a clock face by one hour, every point comes back after $12$ steps.
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<document source="https://tamnd.github.io/math2/node/T761KPKPV6.md" tag="T761KPKPV6" kind="definition">
# Trigonometric integrals

## Definition

Let $\sin$ and $\cos$ be the [[sine and cosine|08WFZP6P89]] functions.
1. The *sine integral* is the function $\operatorname{Si} \colon \R \to \R$ given by
$$\operatorname{Si}(x) = \int_0^x \frac{\sin t}{t}\, dt,$$
where the integrand is given the value $1$ at $t = 0$.
2. The *cosine integral* is the function on $x > 0$ given by the [[improper integral|05KQ3F99BQ]]
$$\operatorname{Ci}(x) = -\int_x^\infty \frac{\cos t}{t}\, dt.$$
3. The *hyperbolic sine integral* and *hyperbolic cosine integral* are $\operatorname{Shi}(x) = \int_0^x \frac{\sinh t}{t}\, dt$ for real $x$ and $\operatorname{Chi}(x) = \gamma + \ln x + \int_0^x \frac{\cosh t - 1}{t}\, dt$ for $x > 0$, where $\gamma$ is the [[Euler–Mascheroni constant|00C5P9JZBK]].

## Remarks

Equivalently, $\operatorname{Ci}(x) = \gamma + \ln x + \int_0^x \frac{\cos t - 1}{t}\, dt$ for $x > 0$.

Example: $\operatorname{Si}$ is odd, $\operatorname{Si}(0) = 0$, and $\lim_{x \to \infty} \operatorname{Si}(x) = \frac{\pi}{2}$ by the [[Dirichlet integral|0GZNSHMG4Q]].
The largest value of $\operatorname{Si}$ on $[0, \infty)$ is $\operatorname{Si}(\pi) \approx 1.8519$.

Non-example: $\int_0^x \frac{\cos t}{t}\, dt$ is not a valid definition of $\operatorname{Ci}$; it diverges at $0$ because $\frac{\cos t}{t} \sim \frac{1}{t}$ there.
This is why $\operatorname{Ci}$ is defined from $\infty$, and it has a logarithmic singularity at $0$.

The shifted function $\operatorname{si}(x) = \operatorname{Si}(x) - \frac{\pi}{2} = -\int_x^\infty \frac{\sin t}{t}\, dt$ is also used.
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# Lattice congruence

## Summary

A lattice congruence is an [[equivalence relation|0G8VEZ3WTK]] on a [[lattice|WH4SFAACAS]] that is compatible with [[join and meet|J2T75ATRV5]].
Its classes can be joined and met by taking representatives, so they form a quotient lattice.
The congruences of a lattice are exactly the kernels of the [[lattice homomorphisms|BNVRR925X8]] defined on it.
This mirrors the role of normal subgroups for groups and of ideals for rings.
The congruences of a lattice form a distributive lattice themselves.

## Explanations

### L1 human

A congruence groups the elements of a lattice into blocks so that the lattice operations do not see the difference inside a block.
If two elements lie in the same block, joining each with a third element gives two results that again lie in a common block.
For example, in the chain $0 < 1 < 2$, merge $1$ and $2$ into one block and keep $0$ alone.
The quotient is the two-element chain, and the map $0 \mapsto 0$, $1 \mapsto 1$, $2 \mapsto 1$ collapses each block to one point.
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<document source="https://tamnd.github.io/math2/node/T78YG90VVT.md" tag="T78YG90VVT" kind="concept">
# E-infinity operad

## Summary

An $E_\infty$ operad is a topological operad whose spaces of $n$-ary operations are contractible and carry a free action of the [[symmetric group|0515A6D2Y1]] $\Sigma_n$.
An algebra over it is a space or spectrum with a multiplication that is commutative and associative up to a coherent system of homotopies.
Such algebras model [[infinite loop spaces|A7VB2NSGJ2]] and $E_\infty$ ring spectra, the homotopy-theoretic version of commutative rings.

## Explanations

### L1 human

Strictly commutative multiplications are too rigid in homotopy theory: many natural products are only commutative up to homotopy.
An $E_\infty$ operad packages all ways of multiplying $n$ elements into a space, and asks that this space be contractible.
Then any two ways are homotopic, any two homotopies are homotopic, and so on.
The free action of $\Sigma_n$ is what keeps the structure homotopically correct instead of strictly commutative.
A standard example has as $n$-ary operations the linear isometric embeddings of $(\R^\infty)^n$ into $\R^\infty$.
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# Projective transformation

## Definition

Let $V$ and $W$ be vector spaces of the same finite dimension over a field $k$.
Let $\mathbb{P}(V)$ and $\mathbb{P}(W)$ be their [[projective spaces|ZT9A6JK8QF]].
For a nonzero vector $v$, write $[v]$ for the point spanned by $v$.
A *projective transformation* (or *homography*) from $\mathbb{P}(V)$ to $\mathbb{P}(W)$ is a map of the form
$$[v] \mapsto [T v],$$
where $T \colon V \to W$ is an invertible linear map.

## Remarks

The map is well defined, since $T(\lambda v) = \lambda T v$, and it is a bijection.
In coordinates, a projective transformation of $\mathbb{P}^n(k)$ is $[x] \mapsto [A x]$ for an [[invertible matrix|0EB9N0S33C]] $A$ of size $n + 1$.
Two matrices give the same map exactly when they differ by a nonzero scalar factor.
So the projective transformations of $\mathbb{P}^n(k)$ form the group $\mathrm{PGL}(n + 1, k) = \mathrm{GL}(n + 1, k) / k^{\times}$, a quotient of the [[general linear group|0FCHE902D7]].

Example: on $\mathbb{P}^1(\C)$, with $z = x_0 / x_1$, the matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $ad - bc \ne 0$ gives $z \mapsto (az + b)/(cz + d)$.
Check: $[x_0 : x_1] \mapsto [a x_0 + b x_1 : c x_0 + d x_1]$, and dividing gives $(az + b)/(cz + d)$.

Non-example: on $\mathbb{P}^2(\C)$ the map $[x_0 : x_1 : x_2] \mapsto [\bar x_0 : \bar x_1 : \bar x_2]$ is a bijection that maps lines to lines, but it is not a projective transformation.
It fixes the four points $[1:0:0]$, $[0:1:0]$, $[0:0:1]$, $[1:1:1]$ without being the identity.
A projective transformation of the plane that fixes four points in general position is the identity, so this map is not one.

Every projective transformation maps projective subspaces to projective subspaces of the same dimension and preserves the [[cross-ratio|09ASMRMD67]].
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# Hammerstein integral equation

## Summary

A Hammerstein integral equation has the form $\varphi(x) = f(x) + \int_\Omega k(x, t)\, g(t, \varphi(t))\, dt$: a linear kernel $k$ acts on a pointwise nonlinear function of the unknown.
In operator form it is $\varphi = f + K N_g \varphi$, where $K$ is a linear integral operator and $N_g$ is the [[Nemytskii operator|W8G3AZ0DNG]] of $g$.
It is a special case of the [[Urysohn equation|WZ4Z26XSVC]].
Its split structure lets one combine linear tools, such as spectral theory of $K$, with monotone operators and fixed point theorems.
Boundary value problems $-u'' = g(x, u)$ with Dirichlet conditions become Hammerstein equations with the Green's function as kernel.

## Explanations

### L1 human

First apply a nonlinear rule to the unknown at each point, for example squaring it.
Then smear the result out with a fixed weight function and integrate.
The Hammerstein equation asks that this give back the unknown, up to a known term.
For example, on $[0, 1]$ the function $\varphi(x) = 4x$ solves $\varphi(x) = \int_0^1 x t\, \varphi(t)^2\, dt$, because $\int_0^1 x t \cdot 16 t^2\, dt = 4x$.
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# Ordered geometry

## Summary

Ordered geometry is the part of geometry that uses only points, lines and the relation of [[betweenness|CE8GX6ADZS]], with no notion of distance, angle or parallel.
Its axioms are the axioms of incidence and order, including [[Pasch's axiom|KJT98G16Y4]].
It is common to [[Euclidean geometry|KRZD96GSVC]], [[absolute geometry|VKFPCSRNDH]], hyperbolic geometry and affine geometry over ordered fields.
Segments, rays, half-planes, convex sets and the inside of a triangle all belong to it.

## Explanations

### L1 human

Ordered geometry asks what one can say about a drawing if one may not measure anything.
One can still say which point lies between which, which side of a line a point is on, and whether a figure is convex.
For example, "the two diagonals of a convex quadrilateral cross each other" is a fact of ordered geometry.
</document>

<document source="https://tamnd.github.io/math2/node/T7JBM8472G.md" tag="T7JBM8472G" kind="definition">
# Generator of a semigroup

## Definition

Let $X$ be a [[Banach space|06GZ956657]] and let $(T(t))_{t \ge 0}$ be a [[strongly continuous semigroup|EVF180QJXY]] of bounded operators on $X$.
The *generator* of $(T(t))_{t \ge 0}$ is the linear [[operator|KRJX5RA0PW]] $A$ with domain
$$D(A) = \set{x \in X : \lim_{t \to 0^+} \frac{T(t)x - x}{t} \text{ exists in } X},$$
given by
$$Ax = \lim_{t \to 0^+} \frac{T(t)x - x}{t}.$$

## Remarks

The generator is closed and densely defined, and it determines the semigroup.
For $x \in D(A)$, the map $u(t) = T(t)x$ is differentiable, stays in $D(A)$, and satisfies $u'(t) = Au(t)$.
One writes $T(t) = e^{tA}$.

Example: let $B$ be a bounded operator on $X$ and $T(t) = e^{tB} = \sum_{k \ge 0} t^k B^k / k!$, the [[exponential of a bounded operator|KX3QRA3CEP]].
Then $\norm{(T(t)x - x)/t - Bx} \le \sum_{k \ge 2} t^{k-1}\norm{B}^k \norm{x}/k! \to 0$, so the generator is $B$ with $D(B) = X$.

Example: on $X = C_0(\R)$, the continuous functions that tend to $0$ at infinity with the sup norm, let $(T(t)f)(s) = f(s + t)$.
Its generator is $Af = f'$ with domain the $C^1$ functions $f$ such that $f$ and $f'$ lie in $C_0(\R)$.

Non-example: the operator $f \mapsto f'$ on smooth functions with compact support is not the generator of this shift semigroup.
It is a proper restriction of $A$: the limit in the definition exists for more functions, and the generator is defined on all of them.

The [[Hille–Yosida theorem|5YR4GJKPCV]] and the [[Lumer–Phillips theorem|Z2N8MPD8DH]] say which operators are generators.
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# Stable theory

## Summary

A complete theory is stable when its models have few [[types|9V71Q3SD1G]].
For some infinite cardinal $\lambda$, every parameter set of size $\lambda$ has at most $\lambda$ complete types over it.
Equivalently, no formula can define an infinite linear order on tuples, a condition called the order property.
Stable theories, introduced and developed by Saharon Shelah, admit a good theory of independence and dimension.
Algebraically closed fields, vector spaces and modules give stable theories, while dense linear orders and real closed fields do not.

## Explanations

### L1 human

Stability measures how complicated the possible behaviours of elements in a theory are.
In a stable theory, fixing a set of parameters leaves relatively few ways for a new element to sit relative to them.
In the rational numbers with their order, every cut of a countable set gives a different behaviour, so there are too many, and the theory is unstable.
In algebraically closed fields, a new element is either algebraic over the parameters, in finitely many ways for each polynomial, or transcendental, and this keeps the count small.
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<document source="https://tamnd.github.io/math2/node/T7KRE7J3B4.md" tag="T7KRE7J3B4" kind="concept">
# Electrical network

## Summary

In discrete potential theory, an electrical network is a graph whose edges carry positive conductances, the reciprocals of resistances.
Voltages on the vertices and currents on the edges obey Ohm's law and Kirchhoff's laws.
A voltage with no sources at a vertex is a [[discrete harmonic function|WRY0M7SW59]] there.
Networks turn questions about [[random walks|04VZE3FZXF]], such as recurrence, into questions about [[effective resistance|XAE5YJ3H68]].

## Explanations

### L1 human

Picture a graph made of wires, where each wire has a resistance.
Connect a battery to two vertices; a current flows and every vertex gets a voltage.
At every other vertex the current in equals the current out, and this makes the voltage there a weighted average of the neighbouring voltages.
For two unit resistors in a row between $a$ and $b$, the voltage at the middle vertex is halfway between the voltages at $a$ and $b$.
The same averaging rule governs a random walker on the graph, which is why circuits help to study random walks.
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<document source="https://tamnd.github.io/math2/node/T7V2VPY61S.md" tag="T7V2VPY61S" kind="concept">
# Fenchel conjugate

## Summary

The Fenchel conjugate, or convex conjugate, of a function $f$ on a vector space is the function $f^*$ on the dual space given by $f^*(y) = \sup_x \big(\ip{y}{x} - f(x)\big)$.
It is always [[convex|07Q7CNM5WF]] and lower semicontinuous, and for proper convex lower semicontinuous $f$ one has $f^{**} = f$.
It extends the classical [[Legendre transformation|ZQYMWVT1TT]] to functions that need not be differentiable.
It is the basic tool of duality in convex optimization.

## Explanations

### L1 human

Think of the graph of $f$ and of all lines of slope $y$ that stay below it.
The highest such line crosses the vertical axis at height $-f^*(y)$.
So $f^*$ records, slope by slope, how the graph of $f$ is supported from below.
For example, the conjugate of $f(x) = \frac12 x^2$ is $f^*(y) = \frac12 y^2$.
A convex function can be rebuilt from its conjugate, which makes this a kind of "Fourier transform" of convex analysis.
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<document source="https://tamnd.github.io/math2/node/T8032P33B7.md" tag="T8032P33B7" kind="definition">
# Inner function

## Definition

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]] and let $f \colon \mathbb{D} \to \C$ be a bounded [[holomorphic function|0DG0Z72CQD]].
By [[Fatou's theorem|Q2N917JQ6A]], the radial limit $f^*(e^{it}) = \lim_{r \to 1^-} f(r e^{it})$ exists for [[almost every|RARTMA0ZEC]] $t \in \R$.
The function $f$ is an *inner function* if $\abs{f^*(e^{it})} = 1$ for almost every $t$.

## Remarks

An inner function satisfies $\abs{f(z)} \le 1$ on $\mathbb{D}$, because $f$ is the Poisson integral of $f^*$.

Example: let $S(z) = \exp\left(-\frac{1 + z}{1 - z}\right)$.
For $z = r e^{it}$ one has $\operatorname{Re} \frac{1 + z}{1 - z} = \frac{1 - r^2}{\abs{1 - z}^2} > 0$, so $\abs{S(z)} < 1$ and $S$ is bounded.
For $t \notin 2\pi\Z$ this real part tends to $0$ as $r \to 1$, so $\abs{S^*(e^{it})} = 1$ and $S$ is inner.
It has no zeros, so it is not a [[Blaschke product|9N6R5ADJXT]].

Non-example: $f(z) = (1 + z)/2$ is bounded and holomorphic, but $\abs{f^*(e^{it})} = \abs{\cos(t/2)} < 1$ for $t \notin \pi\Z$.

Every inner function has the form $c B S_\mu$, where $\abs{c} = 1$ and $B$ is a Blaschke product.
Here $S_\mu(z) = \exp\left(-\int \frac{e^{it} + z}{e^{it} - z} \, d\mu(t)\right)$, and $\mu$ is a finite positive Borel measure on the circle that is singular to Lebesgue measure.
The factor $S_\mu$ is called a singular inner function.
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<document source="https://tamnd.github.io/math2/node/T866256Y1W.md" tag="T866256Y1W" kind="statement">
# Equivariant index theorem

## Statement

Let $G$ be a compact [[Lie group|7JBRGCVJRB]] acting smoothly on a closed smooth manifold $M$.
Let $E$ and $F$ be $G$-equivariant complex vector bundles over $M$, and let $P \colon C^\infty(M; E) \to C^\infty(M; F)$ be a $G$-invariant [[elliptic operator|MT9P2X85V8]].
Then $\ker P$ and $\operatorname{coker} P$ are finite-dimensional [[representations|7JA7WH92K0]] of $G$, and the equivariant analytic index
$$\operatorname{ind}_G P = [\ker P] - [\operatorname{coker} P] \in R(G)$$
lies in the representation ring $R(G)$.
The equivariant analytic index equals the equivariant topological index:
$$\operatorname{ind}_G P = \operatorname{t-ind}_G P.$$
Here $\operatorname{t-ind}_G P$ is defined from the class of the principal symbol in the equivariant K-theory $K_G(T^*M)$.
The definition uses a $G$-equivariant embedding of $M$ into a real representation of $G$.

## Remarks

The proof follows the K-theoretic proof of the non-equivariant theorem.
Both sides are homomorphisms $K_G(T^*M) \to R(G)$ with the same formal properties, and they agree on the basic case of a representation space.
Taking $G$ trivial gives the Atiyah–Singer index theorem, since $R(\{1\}) = \Z$ and the class of a vector space is its dimension.
Evaluating the [[character|RMWMYBY8TC]] of $\operatorname{ind}_G P$ at $g \in G$ gives the Lefschetz number $\operatorname{tr}(g \mid \ker P) - \operatorname{tr}(g \mid \operatorname{coker} P)$.
Atiyah, Segal and Singer computed this number by localization to the fixed point set $M^g$; when $g$ has no fixed points the number is $0$.
Example: the $G$-signature theorem is the case of the signature operator.

## Explanations

### L1 human

When a group acts by symmetries on a manifold and on an operator, the spaces of solutions are not only vector spaces but representations of the group.
The equivariant index records the difference of these representations, not just of their dimensions.
The theorem says that this difference is still computed by topology.

### L2 human

The main use is the fixed point formula: the trace of a symmetry $g$ on the index is a sum of contributions of the fixed points of $g$.
For the de Rham complex this recovers the [[Lefschetz fixed-point theorem|G44NQXCWT9]].
For the Dolbeault complex it gives the holomorphic Lefschetz formula of [[Atiyah and Bott|PKS8JE1PW1]], from which they derived the Weyl character formula for compact connected Lie groups.
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<document source="https://tamnd.github.io/math2/node/T86RSJ8M86.md" tag="T86RSJ8M86" kind="concept">
# Muckenhoupt weights

## Summary

A Muckenhoupt weight is a positive locally integrable function $w$ on $\R^n$ that satisfies the $A_p$ condition.
This condition is a uniform bound, over all cubes, on products of averages of $w$ and of a negative power of $w$.
The class $A_p$, for $1 < p < \infty$, is exactly the class of weights $w$ for which the [[Hardy–Littlewood maximal function|VX1A56P5KQ]] is bounded on $L^p(w \, dx)$.
The same weights make the Hilbert transform and other Calderón–Zygmund operators bounded on $L^p(w \, dx)$.
They are the central objects of weighted norm inequalities.

## Explanations

### L1 human

A weight changes how much each part of space counts in an $L^p$ norm.
The $A_p$ condition says that on every cube the weight is not too unbalanced.
It cannot be very large on average while also being very small on a large part of the cube.
Power weights $\abs{x}^a$ are the main examples; on the line, $\abs{x}^a$ is in $A_2$ exactly when $-1 < a < 1$.
For $a \le -1$ the weight is not even integrable near $0$, and for $a \ge 1$ its reciprocal is not.
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<document source="https://tamnd.github.io/math2/node/T8BVGPPSZP.md" tag="T8BVGPPSZP" kind="concept">
# Cobordism

## Summary

Two closed manifolds of the same dimension are cobordant if together they form the boundary of a compact manifold of one dimension more.
Cobordism is an equivalence relation, much coarser than diffeomorphism, and its classes form groups under disjoint union.
The unoriented groups, and the [[cobordism ring|C02WQY819R]] they form under products, were computed by René Thom through the [[Thom spaces|5MNYT3AFPH]] of universal bundles.
Cobordism links differential topology with stable homotopy theory, and its variants, oriented, complex or framed, are generalized homology theories.

## Explanations

### L1 human

Imagine two closed surfaces as the two ends of a solid piece; the piece is a cobordism between them.
A circle and a pair of circles are cobordant: a pair of pants has one waist and two leg openings, all circles.
A single circle is the boundary of a disc, so it is cobordant to the empty manifold.
Cobordism ignores many details of shape and keeps only what survives such sweeping.
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<document source="https://tamnd.github.io/math2/node/T8EN3CAEQF.md" tag="T8EN3CAEQF" kind="concept">
# Multivariate division algorithm

## Summary

The multivariate division algorithm divides a polynomial in several variables by an ordered list of polynomials, with respect to a fixed [[monomial order|E874NARWPV]].
It writes the polynomial as a combination of the divisors plus a remainder.
No term of the remainder is divisible by the leading term of a divisor.
It extends the [[division algorithm for polynomials|0MES3KS0ZC]] in one variable.
In general the remainder depends on the order of the divisors, but for a [[Gröbner basis|C83KPT47R0]] it is unique, and it then decides ideal membership.

## Explanations

### L1 human

In one variable, long division removes the leading term of the dividend again and again until the degree is too small.
With several variables one does the same.
Whenever the leading term of what is left is a multiple of the leading term of some divisor, subtract a suitable multiple of that divisor.
When no divisor fits, move the leading term to the remainder and go on.
For example, dividing $xy^2 + 1$ by $xy + 1$ and then $y + 1$ gives $xy^2 + 1 = y(xy + 1) - (y + 1) + 2$, with remainder $2$.
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# Residuated mapping

## Definition

Let $(P, \le)$ and $(Q, \le)$ be [[partially ordered sets|011C29Q1S1]].
A map $f \colon P \to Q$ is *residuated* when, for each $y \in Q$, the preimage $f^{-1}(\downarrow y)$ is a principal down-set $\downarrow x$ of $P$, for some $x \in P$.
Here $\downarrow y = \set{z : z \le y}$ is the [[principal down-set|TGW5NTV3AA]] of $y$.

## Remarks

A residuated map is [[monotone|0WFF6AZPBP]].
A map $f$ is residuated if and only if there is a map $g \colon Q \to P$ with $f(x) \le y \iff x \le g(y)$ for all $x \in P$ and $y \in Q$.
Then $g(y)$ is the greatest element of $f^{-1}(\downarrow y)$, and $g$ is called the *residual* of $f$.
For an integer $a > 0$, the map $x \mapsto a x$ on $\Z$ is residuated with residual $y \mapsto \lfloor y / a \rfloor$, since $a x \le y$ if and only if $x \le \lfloor y / a \rfloor$.
The map $f \colon \R \to \R$ with $f(x) = 0$ for $x < 0$ and $f(x) = 1$ for $x \ge 0$ is monotone but not residuated.
Indeed $f^{-1}(\downarrow 0) = (-\infty, 0)$ has no greatest element, so it is not a principal down-set.

## Explanations

### L1 human

A residuated map is one whose inequalities $f(x) \le y$ always have a largest solution $x$.
The largest solution, as a function of $y$, is the residual.
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<document source="https://tamnd.github.io/math2/node/T8NQJ0CBJM.md" tag="T8NQJ0CBJM" kind="definition">
# Neville theta functions

## Definition

Let $0 < k < 1$, $k' = \sqrt{1 - k^2}$, let $K = K(k)$ be the [[complete elliptic integral of the first kind|RM54CPPE12]], and let $q$ be the [[nome|YSAZVCZAZC]] of $k$.
For complex $\zeta$ let
$$\vartheta_1(\zeta) = 2\sum_{n \ge 0} (-1)^n q^{(n + 1/2)^2}\sin((2n + 1)\zeta), \qquad \vartheta_2(\zeta) = 2\sum_{n \ge 0} q^{(n + 1/2)^2}\cos((2n + 1)\zeta),$$
$$\vartheta_3(\zeta) = 1 + 2\sum_{n \ge 1} q^{n^2}\cos(2n\zeta), \qquad \vartheta_4(\zeta) = 1 + 2\sum_{n \ge 1} (-1)^n q^{n^2}\cos(2n\zeta)$$
be the four Jacobi [[theta functions|2F5CYQJVAF]] with nome $q$.
With $\zeta = \pi u/(2K)$, the *Neville theta functions* of $u \in \C$ are
$$\theta_s(u) = \sqrt{\frac{\pi}{2kk'K}}\,\vartheta_1(\zeta), \quad \theta_c(u) = \sqrt{\frac{\pi}{2kK}}\,\vartheta_2(\zeta), \quad \theta_d(u) = \sqrt{\frac{\pi}{2K}}\,\vartheta_3(\zeta), \quad \theta_n(u) = \sqrt{\frac{\pi}{2k'K}}\,\vartheta_4(\zeta).$$

## Remarks

The constants are chosen so that $\theta_c(0) = \theta_d(0) = \theta_n(0) = 1$ and $\theta_s'(0) = 1$.
Check of $\theta_d(0) = 1$: Jacobi's formula $\vartheta_3(0)^2 = 2K/\pi$ gives $\theta_d(0) = \sqrt{\pi/(2K)}\cdot\sqrt{2K/\pi} = 1$.
Similarly $\vartheta_2(0)^2 = 2kK/\pi$ and $\vartheta_4(0)^2 = 2k'K/\pi$ give $\theta_c(0) = \theta_n(0) = 1$.
The [[Jacobi elliptic functions|P071B692JF]] are
$$\operatorname{sn} u = \frac{\theta_s(u)}{\theta_n(u)}, \qquad \operatorname{cn} u = \frac{\theta_c(u)}{\theta_n(u)}, \qquad \operatorname{dn} u = \frac{\theta_d(u)}{\theta_n(u)}.$$
Non-example: $\vartheta_3(\pi u/(2K))$ itself is not $\theta_d(u)$, since its value at $u = 0$ is $\sqrt{2K/\pi} > 1$; the factor $\sqrt{\pi/(2K)}$ is needed.
The function $\theta_d$ is even with period $2K$, and $\theta_s$ is odd with $\theta_s(u + 2K) = -\theta_s(u)$.
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# q-exponential

## Summary

The $q$-exponential $e_q(z) = \sum_{n \ge 0} z^n / [n]_q!$ is the [[q-analog|GFSDMJHHVA]] of the [[exponential function|0J0FCWV32D]].
It is the solution of $D_q f = f$, $f(0) = 1$, for the [[q-derivative|PMJT3FZE0Y]] $D_q$, just as $e^z$ solves $f' = f$.
By the [[q-binomial theorem|VVT27RB46B]] it is an infinite product, and it tends to $e^z$ as $q \to 1$.
A second $q$-exponential $E_q$ satisfies $e_q(z) E_q(-z) = 1$, which replaces the rule $e^z e^{-z} = 1$.

## Explanations

### L1 human

Replace each factorial $n!$ in the series of $e^z$ by its $q$-analogue $[n]_q! = [1]_q [2]_q \cdots [n]_q$, where $[k]_q = 1 + q + \cdots + q^{k-1}$.
The result is the $q$-exponential.
Its first terms are $1 + z + \frac{z^2}{1 + q} + \cdots$, which become $1 + z + \frac{z^2}{2} + \cdots$ at $q = 1$.
Many familiar rules of the exponential survive in a modified form, but $e_q(x) e_q(y) = e_q(x + y)$ fails in general.
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# Monodromy of the hypergeometric equation

## Summary

The monodromy of the [[hypergeometric differential equation|W8VE25R5A7]] records how a basis of solutions changes under analytic continuation along closed loops.
The loops go around the singular points $0$, $1$ and $\infty$.
It is a representation of the [[fundamental group|S1XK3HXHFX]] of the thrice-punctured sphere, a free group on two generators, into $\mathrm{GL}_2(\C)$.
It is the special case of [[monodromy|C8HD62AB6Q]] that Riemann used to study the hypergeometric function.
It decides, for example, when all solutions are algebraic functions.

## Explanations

### L1 human

Solutions of the hypergeometric equation are not single-valued: following a solution once around the point $1$ can bring it back as a different solution.
Since the solutions form a two-dimensional space, each loop acts on it by a $2 \times 2$ matrix.
Collecting these matrices for all loops gives the monodromy group.
For example, when $a = b = 1$ and $c = 2$ the solutions are spanned by $1/z$ and $\log(1 - z)/z$, and going around $z = 1$ adds a multiple of $2\pi i / z$ to the second one.
Riemann showed that this matrix data, together with the local exponents, nearly determines the equation and its solutions.

### L2 human

The local monodromy around each singular point has eigenvalues $e^{2\pi i \rho}$, where $\rho$ runs over the two exponents at that point.
The monodromy is irreducible exactly when none of $a$, $b$, $c - a$, $c - b$ is an integer.
In the irreducible case, the representation is determined up to conjugacy by the three pairs of local eigenvalues, a fact known as rigidity.
Finite monodromy groups give algebraic solutions, and these are classified by [[Schwarz's list|W0MJ51242N]].
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# Rational difference equation

## Summary

A rational difference equation is a [[nonlinear difference equation|7C7FBZGERM]] whose next term is a quotient of two affine expressions in the earlier terms.
It is one of the simplest classes of nonlinear recurrences, yet its solutions show convergence, periodicity and unbounded growth depending on the parameters.
The first-order case is the iteration of a [[Möbius transformation|0SG74WBFA2]] and can be solved in closed form by reduction to a linear recurrence.
Higher-order cases are studied mainly through the stability and periodicity of their solutions.

## Explanations

### L1 human

Each new term is a fraction whose top and bottom are linear in the previous terms.
For example $x_{n+1} = 1/(1 + x_n)$ with $x_0 = 1$ gives $1, 1/2, 2/3, 3/5, 5/8, \dots$.
These are ratios of consecutive Fibonacci numbers, and they approach $(\sqrt{5} - 1)/2 \approx 0.618$.
The division makes the rule nonlinear, but in the first-order case a clever change of variables makes it linear again.
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# Bruhat decomposition

## Statement

Let $G$ be a connected [[reductive|2XS84PWE3S]] [[linear algebraic group|GA81VVQHEP]] over an algebraically closed field $k$.
Let $B$ be a [[Borel subgroup|5JSYA1JNCC]] of $G$, let $T \subseteq B$ be a maximal torus, and let $W = N_G(T)/T$ be the Weyl group.
For $w \in W$, the double coset $B \dot w B$ does not depend on the choice of a representative $\dot w \in N_G(T)$ of $w$, and it is written $BwB$.
Then $G$ is the disjoint union
$$G = \bigsqcup_{w \in W} BwB.$$

## Remarks

For the [[general linear group|0FCHE902D7]] $G = \mathrm{GL}_n(k)$, take $B$ the invertible [[upper triangular matrices|0N4XY7Z3NN]], $T$ the invertible diagonal matrices, and $W$ the [[symmetric group|0515A6D2Y1]] $S_n$ realised by [[permutation matrices|08SP3DF8NB]].
Then the theorem says that each invertible matrix is $b_1 w b_2$ with $b_1, b_2 \in B$ and a unique permutation matrix $w$.
The proof in this case is Gaussian elimination: row and column operations by upper triangular matrices clear an invertible matrix down to a permutation matrix.
The permutation is unique because the ranks of the lower left corner submatrices are unchanged by these operations, and they determine it.

Example: for $n = 2$, $W = \set{1, s}$ with $s = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.
A matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ lies in $B$ when $c = 0$ and in $BsB$ when $c \ne 0$, so $\mathrm{GL}_2(k) = B \sqcup BsB$.

The same decomposition holds for every group with a BN-pair, and this is the basis of the theory of [[buildings|6BQJ50NEJ0]] and of [[groups of Lie type|NPEBABQ49M]].

## Explanations

### L1 human

The Bruhat decomposition cuts a large matrix group into finitely many pieces.
Each piece is labelled by one permutation, or more generally by one element of the Weyl group.
For invertible matrices, the label of a matrix is the permutation that Gaussian elimination with upper triangular operations leaves behind.

### L2 human

For a finite field $\F_q$, the decomposition counts the elements of $\mathrm{GL}_n(\F_q)$ cell by cell, and it gives the order of many finite [[groups of Lie type|NPEBABQ49M]].
In geometry, the images of the double cosets $BwB$ in the flag variety $G/B$ are the Schubert cells.
The closure order of the cells is the Bruhat order on $W$.
Abstracting the two subgroups $B$ and $N_G(T)$ gives the notion of a BN-pair, whose geometric picture is a [[building|6BQJ50NEJ0]].
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# Divided differences

## Definition

Let $K$ be a [[field|04C5ZBG1WT]] and let $f$ be a [[function|0EPAT51DXQ]] from a subset of $K$ to $K$.
Let $x_0, \dots, x_n$ be pairwise distinct points of its domain.
The *divided differences* of $f$ are defined by recursion on the number of points:

1. $f[x_i] = f(x_i)$ for each $i$;
2. for $k \ge 1$ and $0 \le i \le n - k$,
$$f[x_i, \dots, x_{i+k}] = \frac{f[x_{i+1}, \dots, x_{i+k}] - f[x_i, \dots, x_{i+k-1}]}{x_{i+k} - x_i}.$$

## Remarks

The value $f[x_0, \dots, x_n]$ is a symmetric function of the points: it does not depend on their order.
It has the closed form
$$f[x_0, \dots, x_n] = \sum_{j=0}^{n} \frac{f(x_j)}{\prod_{k \ne j} (x_j - x_k)}.$$
It is the coefficient of $x^n$ in the [[interpolation polynomial|SS1YMS2NPC]] of $f$ at $x_0, \dots, x_n$.
For a real function $f$ with enough derivatives, the definition extends to repeated points by continuity; for example $f[x, x] = f'(x)$.

Example: for $f(x) = x^2$ and the points $0, 1, 3$: $f[0, 1] = 1$, $f[1, 3] = \frac{9 - 1}{2} = 4$, and $f[0, 1, 3] = \frac{4 - 1}{3 - 0} = 1$.
This is the leading coefficient of $x^2$, as expected.

Non-example: the recursion cannot be applied to the list $1, 1$ of repeated points, because the denominator $x_1 - x_0$ is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/T94V1HDVNC.md" tag="T94V1HDVNC" kind="problem">
# Lebesgue's universal covering problem

## Problem

A *universal cover* is a [[convex set|02AS45T3H9]] $C$ in the plane with this property.
Every plane set of [[diameter|0KW3PN547K]] $1$ can be moved into $C$ by an [[isometry|02HPP5VCZH]] of the plane.
Find the smallest possible area of a universal cover.

## Remarks

The problem is open: the exact minimal area is not known.
Pál showed that the regular hexagon whose opposite sides are at distance $1$ is a universal cover; its area is $\sqrt{3}/2 \approx 0.866$.
Cutting suitable pieces off this hexagon gives smaller universal covers, and the best known upper bound is about $0.8441$.
A disc of diameter $1$ is not a universal cover: the equilateral triangle of side $1$ has diameter $1$ but circumradius $1/\sqrt{3} > 1/2$.
Lower bounds come from the fact that a universal cover must contain copies of several specific sets of diameter $1$ at once.
Examples are a disc of diameter $1$ and an equilateral triangle of side $1$.
The best known lower bound is about $0.832$.

## Explanations

### L1 human

Lebesgue asked for the smallest convex shape that can cover every shape of diameter one, after moving and turning it.
A regular hexagon of width one works, but it is not the smallest.
More than a century later the best possible area is still only known to lie between about $0.832$ and $0.8441$.
</document>

<document source="https://tamnd.github.io/math2/node/T9640YF4TX.md" tag="T9640YF4TX" kind="definition">
# Wald test

## Definition

Let $X$ be data from a model with parameter $\theta$ in an open set $\Theta \subseteq \R^k$, and let $\theta_0 \in \Theta$.
Let $\hat\theta$ be the [[maximum likelihood estimate|0FREPFKNS7]] of $\theta$, and let $I(\hat\theta)$ be the [[Fisher information|G1Z3WSTE8Q]] of $X$ evaluated at $\hat\theta$, assumed invertible.
The *Wald statistic* for $H_0 \colon \theta = \theta_0$ is
$$W = (\hat\theta - \theta_0)^{T} I(\hat\theta) (\hat\theta - \theta_0).$$
The *Wald test* at level $\alpha$ rejects $H_0$ when $W$ exceeds the $1 - \alpha$ quantile of the [[chi-squared distribution|0RFMMBHFTV]] with $k$ degrees of freedom.

## Remarks

Under $H_0$ and the usual regularity conditions for i.i.d. samples, $W$ converges in distribution to $\chi^2_k$, so the test has asymptotic level $\alpha$.
For $k = 1$ the test rejects when $\abs{\hat\theta - \theta_0}/\widehat{\mathrm{se}} > z_{1 - \alpha/2}$, with $\widehat{\mathrm{se}} = I(\hat\theta)^{-1/2}$ and $z_{1 - \alpha/2}$ a standard normal quantile.
The observed information may replace $I(\hat\theta)$ without changing the limit law.
Example: for $X$ binomial with parameters $n$ and $p$, one has $\hat p = X/n$ and $W = (\hat p - p_0)^2 n/(\hat p(1 - \hat p))$.
Unlike the [[likelihood-ratio test|YM1BNA80RC]], the Wald test is not invariant under a reparametrization of $\theta$.
</document>

<document source="https://tamnd.github.io/math2/node/T97B2GX3JJ.md" tag="T97B2GX3JJ" kind="definition">
# Outer function

## Definition

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].
Let $\varphi$ be a nonnegative measurable function on the unit circle such that $\log \varphi$ is integrable with respect to [[Lebesgue measure|2Z2QVWZ262]].
Let $c \in \C$ with $\abs{c} = 1$.
The *outer function* with modulus $\varphi$ and constant $c$ is the function $F \colon \mathbb{D} \to \C$ given by
$$F(z) = c \exp\left(\frac{1}{2\pi} \int_0^{2\pi} \frac{e^{it} + z}{e^{it} - z} \log \varphi(e^{it}) \, dt\right).$$
A holomorphic function on $\mathbb{D}$ is *outer* if it is of this form for some $\varphi$ and $c$.

## Remarks

$F$ is [[holomorphic|0DG0Z72CQD]] and has no zeros, and $\log \abs{F}$ is the [[Poisson integral|TCYSWCTYYM]] of $\log \varphi$.
Its radial limits satisfy $\abs{F^*} = \varphi$ almost everywhere.
For $0 < p \le \infty$, $F$ lies in the Hardy space $H^p$ if and only if $\varphi \in L^p$ ([[Lp space|SASP14MA1Z]]).

Example: $F(z) = 2 + z$ is outer.
It is holomorphic and has no zeros on a neighbourhood of the closed disc, so $\log \abs{F}$ is harmonic and continuous up to the circle.
Hence $\log \abs{F}$ equals the Poisson integral of $\log \varphi$, where $\varphi = \abs{2 + e^{it}}$.
So $F$ and the outer function with modulus $\varphi$ have the same modulus and no zeros; their quotient is a holomorphic function of constant modulus $1$, hence a constant.

Non-example: $f(z) = z$ is not outer.
Its boundary modulus is $\varphi = 1$, and the outer functions with this modulus are the constants of modulus $1$, which differ from $z$.
</document>

<document source="https://tamnd.github.io/math2/node/T9A1DZ4AQK.md" tag="T9A1DZ4AQK" kind="definition">
# Pontryagin–Thom construction

## Definition

Let $M$ be a smooth manifold and let $N \subset M$ be a compact smooth [[submanifold|VKFW27WXEZ]] without boundary, of codimension $k$, with normal [[vector bundle|P4DZM975BV]] $\nu$.
Choose an inner product on $\nu$ and a closed tubular neighbourhood $U$ of $N$.
Let $e \colon D(\nu) \to U$ be a diffeomorphism from the unit disc bundle that is the identity on the zero section.
Let $\operatorname{Th}(\nu) = D(\nu) / S(\nu)$ be the [[Thom space|5MNYT3AFPH]] of $\nu$ with base point $*$.
The *Pontryagin–Thom collapse map* $c \colon M \to \operatorname{Th}(\nu)$ is the following map:
$$c(x) = \begin{cases} [e^{-1}(x)] & \text{if } x \in U, \\ * & \text{if } x \notin U. \end{cases}$$
Suppose that $\nu$ comes with a framing, that is, a trivialisation $\nu \cong N \times \R^k$.
The map $\operatorname{Th}(N \times \R^k) \to D^k / S^{k-1} \cong S^k$ forgets the point of $N$.
Its composite with $c$ is a map $M \to S^k$.

## Remarks

The map $c$ is continuous because the boundary of $U$ is the image of $S(\nu)$, which goes to the base point; its homotopy class does not depend on the choices.
Example: for $N$ a point of $M = S^n$, the normal bundle is $\R^n$ with its standard framing, and $c \colon S^n \to D^n / S^{n-1} \cong S^n$ has degree $\pm 1$.
Non-example of the framed version: the circle $N = \R P^1$ in $M = \R P^2$ has the Möbius bundle as normal bundle, which has no framing.
The collapse map still exists, with target $\operatorname{Th}(\nu) \cong \R P^2$, but it does not give a map to $S^1$.

For $M = S^{n+k}$, Pontryagin's theorem says that the construction gives a bijection between framed cobordism classes of closed framed $n$-dimensional submanifolds of $S^{n+k}$ and the homotopy group $\pi_{n+k}(S^k)$.
The inverse sends a smooth map, made transverse to a point, to the preimage of that point with the pulled back framing.

## Explanations

### L1 human

Keep a thin tube around the submanifold and squash the rest of the space to one point.
When the tube is a product, the tube with its outer wall squashed is a sphere, so the result is a map to a sphere.
This dictionary between submanifolds and maps lets homotopy theory count manifolds up to cobordism.
</document>

<document source="https://tamnd.github.io/math2/node/T9CJHV6JW3.md" tag="T9CJHV6JW3" kind="statement">
# Szpilrajn extension theorem

## Statement

Let $X$ be a set and let $\le$ be a [[partial order|0QV2JS9RJ3]] on $X$.
Then there is a [[total order|0FRXBVVR03]] $\preceq$ on $X$ such that $x \le y$ implies $x \preceq y$ for all $x, y \in X$.
In other words, every partial order has a [[linear extension|W8DZ5KGGHN]].

## Remarks

For a finite set, take a minimal element of $X$, put it first, and repeat with the remaining elements; this is topological sorting.
For an infinite set, apply [[Zorn's lemma|04RXTQA5ZA]] to the set of partial orders on $X$ that contain $\le$, ordered by inclusion.
A maximal such order is total: if $a$ and $b$ were incomparable in it, adding the pairs $(u, v)$ with $u \preceq a$ and $b \preceq v$ would give a strictly larger partial order.
The same argument shows more: if $a$ and $b$ are incomparable for $\le$, then some linear extension puts $a$ before $b$.
Hence $\le$ is the intersection of all its linear extensions.
Example: the divisibility order on $\set{1, 2, 3, 6}$ has the linear extensions $1, 2, 3, 6$ and $1, 3, 2, 6$, and $2$ and $3$ are incomparable because they appear in both relative orders.

## Explanations

### L1 human

A partial order says that some things come before others, but leaves some pairs undecided.
The theorem says that one can always decide the remaining pairs and line up all the elements in one row without breaking any of the given rules.
For a list of tasks with prerequisites, this is a schedule that does the tasks one at a time.

### L2 human

The theorem is what makes the order dimension of a poset well defined: the set of all linear extensions realizes the order.
For finite posets the proof is an algorithm, and counting the linear extensions is a central problem of order combinatorics.
For infinite sets the proof uses the axiom of choice through Zorn's lemma.
</document>

<document source="https://tamnd.github.io/math2/node/T9DG8DDE75.md" tag="T9DG8DDE75" kind="concept">
# Pointed space

## Summary

A pointed space is a [[topological space|0CQHFPGJ45]] together with a chosen point, its base point.
Maps of pointed spaces are continuous maps that send base point to base point.
Many constructions of homotopy theory need a base point, such as the [[fundamental group|S1XK3HXHFX]], the [[loop space|2JFF5F3EGF]] and the [[wedge sum|5JSWM1HBX2]].

## Explanations

### L1 human

A pointed space is a space with one marked point, like a map of a city with a pin at your home.
Loops in a pointed space start and end at the pin.
For example, the circle with the marked point $1$ is a pointed space, written $(S^1, 1)$.
A map of pointed spaces must send the pin of one space to the pin of the other.
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<document source="https://tamnd.github.io/math2/node/T9H6PT9FHE.md" tag="T9H6PT9FHE" kind="concept">
# Morse theory on Hilbert manifolds

## Summary

Morse theory on Hilbert manifolds extends classical Morse theory from compact manifolds to functions on infinite-dimensional [[Hilbert manifolds|R8DMMW0E9M]].
Richard Palais and Stephen Smale replaced compactness of the manifold by a compactness condition on the function, the [[Palais–Smale condition|4724KRNHAT]].
With it, sublevel sets change only at critical values, and passing a nondegenerate critical point of finite index $\lambda$ attaches a $\lambda$-cell.
The main application is the study of geodesics as critical points of the energy on spaces of paths and loops.

## Explanations

### L1 human

Classical Morse theory reads the shape of a compact manifold off the critical points of a function on it.
Many natural problems live on infinite-dimensional spaces, such as the space of all paths between two points, where the function is the energy of a path.
Such spaces are not compact, so a sequence that looks more and more like a critical point might escape to infinity.
The Palais–Smale condition rules this out, and then the classical theory goes through.
For example, critical points of the energy on paths from $p$ to $q$ are the geodesics from $p$ to $q$, and their indices count conjugate points.
</document>

<document source="https://tamnd.github.io/math2/node/T9PRG9QP8Y.md" tag="T9PRG9QP8Y" kind="definition">
# Bilateral Z-transform

## Definition

Let $x = (x_n)_{n \in \Z}$ be a two-sided [[sequence|0G9EKM0E15]] of complex numbers.
The *bilateral Z-transform* of $x$ is the [[Laurent series|0GXMPDNYX3]]
$$X(z) = \sum_{n = -\infty}^{\infty} x_n z^{-n}.$$
It is a function on its [[region of convergence|1P8WX3EGHF]], the set of nonzero complex $z$ where the series converges absolutely.

## Remarks

Example: let $x_n = 2^{-\abs{n}}$.
The part $\sum_{n \ge 0} (2z)^{-n}$ converges for $\abs{z} > 1/2$ and the part $\sum_{n \ge 1} (z/2)^n$ converges for $\abs{z} < 2$.
So $X(z) = \dfrac{2z}{2z - 1} + \dfrac{z}{2 - z}$ on the annulus $1/2 < \abs{z} < 2$.
Example with the same formula: for $a \ne 0$, $x_n = a^n$ for $n \ge 0$ (and $0$ otherwise) gives $z/(z - a)$ on $\abs{z} > \abs{a}$, while $x_n = -a^n$ for $n \le -1$ (and $0$ otherwise) gives $z/(z - a)$ on $\abs{z} < \abs{a}$.
Non-example: $x_n = 1$ for all $n$ has an empty region of convergence, so the transform is not a function.
The notation $X(z) = \mathcal{Z}\{x\}(z)$ is standard; signal processing texts write $x[n]$ for $x_n$.
</document>

<document source="https://tamnd.github.io/math2/node/TA0NREQMPX.md" tag="TA0NREQMPX" kind="definition">
# Degree of a projective variety

## Definition

Let $k$ be an algebraically closed field and let $X \subseteq \mathbb{P}^n_k$ be a [[projective variety|A4H4CZ2V35]] of [[dimension|36X2WTHYCB]] $d$.
Let $I(X) \subseteq S = k[x_0, \dots, x_n]$ be its homogeneous ideal, and let $P_X$ be the [[Hilbert polynomial|RMH4FBCFDA]] of the graded ring $S/I(X)$.
Then $P_X$ has degree $d$ and leading coefficient $e/d!$ for a positive integer $e$.
The *degree* of $X$ is $\deg X = e$.

## Remarks

Equivalently, $\deg X$ is the number of points of $X \cap L$ for a general linear subspace $L \subseteq \mathbb{P}^n$ of dimension $n - d$.

Example: let $X = V(f) \subseteq \mathbb{P}^2$ with $f$ an irreducible homogeneous polynomial of degree $e$.
The exact sequence $0 \to S(-e) \to S \to S/(f) \to 0$ gives $P_X(t) = \binom{t+2}{2} - \binom{t-e+2}{2} = et - e(e-3)/2$.
Here $d = 1$ and the leading coefficient is $e$, so $\deg X = e$.

Non-example of the geometric form: the conic $X = V(xz - y^2)$ has degree $2$.
The line $y = 0$ meets it in the two points $[1:0:0]$ and $[0:0:1]$.
The tangent line $x = 0$ meets it only in $[0:0:1]$.
The tangent line is not general, so it does not compute the degree; it counts the point with multiplicity $2$.

The degree depends on the embedding: $\mathbb{P}^1$ embedded as a line has degree $1$, and embedded as the conic above it has degree $2$.
For a hypersurface $V(f)$ with $f$ irreducible, the degree of the variety is the degree of $f$.
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<document source="https://tamnd.github.io/math2/node/TA39C6P91W.md" tag="TA39C6P91W" kind="definition">
# Factorization system

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]], and let $E$ and $M$ be classes of morphisms of $\mathcal{C}$.
The pair $(E, M)$ is an *orthogonal factorization system* if these three conditions hold.

1. Both $E$ and $M$ contain all isomorphisms and are closed under composition.
2. Every morphism $f$ of $\mathcal{C}$ factors as $f = m \circ e$ with $e \in E$ and $m \in M$.
3. Every $e \in E$ is orthogonal to every $m \in M$: it has the [[left lifting property|AZK4RD8XV0]] with respect to $m$, with a unique filler.

## Remarks

Conditions 1 to 3 imply that the factorization in clause 2 is unique up to a unique isomorphism.
They also imply that $E$ is exactly the class of morphisms orthogonal to all of $M$, and conversely.
A *weak factorization system* asks instead for factorizations and for liftings without uniqueness.
It also asks that $E$ and $M$ be exactly the classes of maps with the left and the right lifting property against each other.

Example: in the category of sets, let $E$ be the [[surjections|01RVCMSN5N]] and $M$ the [[injections|09NWETVTNY]].
A function $f \colon A \to B$ factors as $A \to f(A) \to B$.
For orthogonality, take a square with $e \colon A \to B$ surjective, $m \colon C \to D$ injective, and $v \circ e = m \circ u$.
Define $d(b) = u(a)$ for any $a$ with $e(a) = b$; this does not depend on $a$, since $m$ is injective and $m(u(a)) = v(b)$.
Then $d$ is the unique filler.

Non-example: in sets, let $E$ and $M$ both be the class of all functions.
Conditions 1 and 2 hold, but condition 3 fails.
For $e \colon \emptyset \to \set{*}$ and $m \colon \set{0, 1} \to \set{*}$, the square with $v = \id$ has two fillers $\set{*} \to \set{0, 1}$.
</document>

<document source="https://tamnd.github.io/math2/node/TA5025GWDQ.md" tag="TA5025GWDQ" kind="definition">
# Integral quadratic form

## Definition

Let $n \ge 1$ be an integer.
An *integral quadratic form* in $n$ variables is a [[quadratic form|0RBQS1JJCR]]
$$Q(x_1, \dots, x_n) = \sum_{1 \le i \le j \le n} a_{ij} x_i x_j$$
whose coefficients $a_{ij}$ all lie in the [[integers|0C4JS9RF5M]].
Two integral quadratic forms $Q$ and $Q'$ in $n$ variables are *equivalent* if $Q'(x) = Q(Mx)$ for some $n \times n$ integer [[matrix|0VD9ZSWWJG]] $M$ with $\det M = \pm 1$.

## Remarks

Every integral quadratic form takes integer values at integer points.
The form can be written $Q(x) = \frac12 x^T A x$ with a symmetric integer matrix $A$ whose diagonal entries $2a_{ii}$ are even.
Some authors call $Q$ integral only when $Q(x) = x^T B x$ for a symmetric integer matrix $B$.
This means that all cross coefficients $a_{ij}$ with $i < j$ are even; such forms are called classically integral.

Example: $x^2 + xy + y^2$ is integral, with $a_{11} = a_{12} = a_{22} = 1$; it is not classically integral, because the coefficient of $xy$ is odd.

Non-example: $x^2 + \frac12 xy + y^2$ is a quadratic form over $\Q$ but not integral, since the coefficient of $xy$ is not an integer; its value at $(1, 1)$ is $5/2$.
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<document source="https://tamnd.github.io/math2/node/TA8NQQCQCF.md" tag="TA8NQQCQCF" kind="definition">
# Hölder summation

## Definition

Let $\sum_{n \ge 0} a_n$ be a series of complex numbers with [[partial sums|0NB643K60Q]] $s_n = a_0 + \cdots + a_n$.
Put $H^0_n = s_n$, and for each integer $k \ge 1$ let $H^k_n$ be the [[Cesàro mean|0P0DJHBGBB]] of the sequence $(H^{k-1}_n)$:
$$H^k_n = \frac{H^{k-1}_0 + H^{k-1}_1 + \cdots + H^{k-1}_n}{n + 1}.$$
The series is *Hölder summable of order $k$*, or $(H, k)$ summable, to $s$ if $H^k_n \to s$ as $n \to \infty$.

## Remarks

The method $(H, 0)$ is ordinary convergence, and $(H, 1)$ is Cesàro summation of order $1$.

Example: for $1 - 2 + 3 - 4 + \cdots$ the partial sums are $1, -1, 2, -2, 3, \dots$.
Then $H^1_{2m} = \frac{m+1}{2m+1}$ and $H^1_{2m+1} = 0$.
So $H^1_n$ is the sequence $\frac12, 0, \frac12, 0, \dots$ up to terms that tend to $0$, and its averages $H^2_n$ tend to $\frac14$.
The series is $(H, 2)$ summable to $1/4$.

Non-example: the same series is not $(H, 1)$ summable, since $H^1_n$ has the two limit points $1/2$ and $0$.

By a theorem of Knopp and Schur, for each $k$ a series is $(H, k)$ summable to $s$ if and only if it is Cesàro $(C, k)$ summable to $s$.
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# Reaction–diffusion system

## Summary

A reaction–diffusion system is a system of [[parabolic|1CCTH8X2FT]] equations $\partial_t u = D \Delta u + R(u)$ for several quantities.
Each quantity diffuses in space and reacts with the others at each point.
It models chemical reactions, population dynamics, the spread of epidemics and the formation of patterns in biology.
Depending on the reaction term, solutions can show travelling fronts, spatial patterns or oscillations.
Turing showed that diffusion can make a stable uniform state unstable and so create patterns.

## Explanations

### L1 human

Picture several chemicals in a dish: each one spreads out by diffusion, and at each point they react and turn into each other.
The equation for each concentration has two parts, a diffusion part that evens it out and a reaction part that depends on the local amounts.
With one quantity and the reaction $u(1 - u)$, it describes a population that grows to a limit and spreads, leading to a wave that invades empty space.
With two quantities that diffuse at different speeds, it can produce spots and stripes like those on animal coats.
</document>

<document source="https://tamnd.github.io/math2/node/TACX13XZST.md" tag="TACX13XZST" kind="definition">
# HNN extension

## Definition

Let $G$ be a [[group|G5S1M9]] with a [[presentation|0JGEQXQ4AC]] $\langle S \mid R \rangle$.
Let $H$ and $K$ be [[subgroups|0X1JXT47GA]] of $G$, and let $\alpha \colon H \to K$ be a group [[isomorphism|08MG7H8JW5]].
Let $t$ be a new symbol not in $S$.
The *HNN extension* of $G$ relative to $\alpha$ is the group
$$G *_{\alpha} = \langle S, t \mid R,\ t h t^{-1} = \alpha(h) \text{ for all } h \in H \rangle,$$
where each $h \in H$ and $\alpha(h) \in K$ is written as a word in $S$.
Equivalently, $G *_{\alpha}$ is the quotient of the [[free product|TM7QS7P7X9]] $G * \langle t \rangle$, with $\langle t \rangle$ infinite cyclic, by the [[normal subgroup|0CB0DX5BVV]] generated by the elements $t h t^{-1} \alpha(h)^{-1}$ for $h \in H$.
The element $t$ is the *stable letter*, and $H$ and $K$ are the *associated subgroups*.

## Remarks

The natural map from $G$ to $G *_{\alpha}$ is injective, so $G$ is a subgroup of $G *_{\alpha}$; this is a theorem of Higman, Neumann and Neumann.
Britton's lemma describes when a word in $S$ and $t$ represents the identity.

Example: let $G = \langle a \mid \ \rangle \cong \Z$, $H = \langle a^m \rangle$ and $K = \langle a^n \rangle$ with $m, n$ nonzero integers, and $\alpha(a^m) = a^n$.
The map $\alpha$ is an isomorphism, since both subgroups are infinite cyclic and $\alpha$ sends a generator to a generator.
The HNN extension is $\langle a, t \mid t a^m t^{-1} = a^n \rangle$, the [[Baumslag–Solitar group|429JJ5ZNBY]] $BS(m, n)$.
For $m = n = 1$ it is $\langle a, t \mid t a t^{-1} = a \rangle \cong \Z^2$.

Non-example: the rule $t h t^{-1} = \varphi(h)$ for a homomorphism $\varphi \colon H \to K$ that is not injective is not an HNN extension.
If $\varphi(h) = 1$ for some $h \ne 1$, then $h = t^{-1} \varphi(h) t = 1$ in the quotient, so $G$ does not embed.
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# Error-correcting code

## Summary

An error-correcting code is a set of words of fixed length, the codewords, chosen far apart from each other in Hamming distance.
A message is sent as a codeword; if a few symbols are corrupted on the way, the receiver still finds the nearest codeword.
A code whose codewords are at distance at least $2t + 1$ from each other corrects any $t$ errors.
Codes link combinatorics with algebra: good codes come from [[block designs|4JCSS2Z1BA]], [[Hadamard matrices|J0N4J606X0]] and finite fields.
The node [[error-correcting code|X122TKFP9G]] gives the precise definition.

## Explanations

### L1 human

Suppose you send one bit, but noise may flip it.
Send it three times instead: $000$ for $0$ and $111$ for $1$.
If the receiver gets $010$, the most likely message is $0$, since only one bit had to flip.
This is the repetition code; it corrects one error at the price of sending three times as much.
Better codes, such as the Hamming code, correct errors at a much lower price.
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<document source="https://tamnd.github.io/math2/node/TAFJV2YWKR.md" tag="TAFJV2YWKR" kind="concept">
# Expansive homeomorphism

## Summary

An expansive homeomorphism is a [[homeomorphism|0QT00PB5G2]] of a metric space that separates any two distinct points by a fixed distance at some time.
This time may lie in the past or in the future.
The fixed distance is the expansivity constant.
Shift spaces and Anosov diffeomorphisms are the basic examples.
On a compact metric space, an expansive homeomorphism has finite [[topological entropy|BDZYBXBFDE]] and at most finitely many periodic points of each period.

## Explanations

### L1 human

Two different starting points can look the same for a while, but an expansive map never lets them stay close forever.
At some moment, forward or backward in time, they are at least a fixed distance apart.
So with a fixed precision one can tell every orbit apart from every other one by watching the whole history.
In a shift space of two-sided sequences, two different sequences differ at some position, and shifting that position to the centre makes them far apart.
A rotation of the circle is not expansive, because it keeps all distances unchanged.
</document>

<document source="https://tamnd.github.io/math2/node/TAHM74ZNWB.md" tag="TAHM74ZNWB" kind="statement">
# Abian–Brown theorem

## Statement

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]] in which every nonempty [[chain|0S8FE7ZMFM]] has a [[supremum|0R3H5PANRQ]].
Let $f \colon P \to P$ be a [[monotone|0A7KV2WB7A]] map, and let $a \in P$ satisfy $a \le f(a)$.
Then $f$ has a fixed point $p$ with $a \le p$.
Moreover, $p$ can be chosen as the least element of the set $\set{x \in P : f(x) = x \text{ and } a \le x}$.

## Remarks

Proof idea: define $x_0 = a$, $x_{\alpha + 1} = f(x_\alpha)$, and $x_\lambda = \sup_{\alpha < \lambda} x_\alpha$ at limit ordinals $\lambda$.
By transfinite induction, $(x_\alpha)$ is increasing and each $x_\alpha$ satisfies $x_\alpha \le f(x_\alpha)$, so the suprema exist.
The sequence cannot be strictly increasing for all ordinals, since $P$ is a set, so $x_{\alpha + 1} = x_\alpha$ for some $\alpha$, and $p = x_\alpha$ is a fixed point.
If $q$ is a fixed point with $a \le q$, induction gives $x_\alpha \le q$ for all $\alpha$, so $p$ is the least such fixed point.
The theorem contains the existence part of the [[Knaster–Tarski theorem|RHXP3HA64M]]: in a complete lattice, take $a$ to be the least element.
The hypothesis $a \le f(a)$ is needed.
On the two-element antichain $\set{a, b}$, every nonempty chain is a single point, and the swap of $a$ and $b$ is monotone and has no fixed point.

## Explanations

### L1 human

If a monotone rule moves some point $a$ upward, keep applying it.
The points climb, and whenever the climb runs through infinitely many steps, jump to the top of what has been reached.
Because every chain has a top, the climb never gets stuck, and since there are only so many points it must stop at a point that the rule fixes.
This fixed point is the lowest one above $a$.

### L2 human

The theorem weakens the completeness hypothesis of the Knaster–Tarski theorem to chain-completeness, at the price of needing a starting point $a$ with $a \le f(a)$.
In a chain-complete poset with a least element, every monotone map has a least fixed point.
Compared with the [[Bourbaki–Witt theorem|9EABA9ZDZS]], it asks for monotonicity instead of $x \le f(x)$ for all $x$, and in return it gives a least fixed point above $a$.
</document>

<document source="https://tamnd.github.io/math2/node/TAHV24ED74.md" tag="TAHV24ED74" kind="concept">
# Bicategory

## Summary

A bicategory is a weak form of a [[2-category|CQACXG10JZ]]: composition of 1-morphisms is associative and unital only up to specified invertible 2-morphisms.
These associators and unitors must satisfy the pentagon and triangle identities, as in a [[monoidal category|M3MSQ3NKAE]].
A bicategory with one object is the same as a monoidal category.
Rings with bimodules, and categories with [[profunctors|EAPHAQVVD5]], form bicategories that are not strict 2-categories.

## Explanations

### L1 human

In many examples, composing arrows in two different orders gives results that are naturally isomorphic but not literally equal.
Tensoring bimodules is the typical case: $(M \otimes N) \otimes P$ and $M \otimes (N \otimes P)$ are isomorphic, not the same.
A bicategory keeps track of these isomorphisms as part of the structure, instead of pretending they are equalities.
Every bicategory is equivalent to a strict 2-category, so not much is lost, but the weak version is the one that occurs in nature.
</document>

<document source="https://tamnd.github.io/math2/node/TAJTC44XC9.md" tag="TAJTC44XC9" kind="definition">
# Fractional integration

## Definition

Let $n \ge 1$ and let $0 < \alpha < n$.
Let $\Gamma$ be the [[gamma function|0H5SYBSED9]] and let
$$\gamma(\alpha) = \frac{\pi^{n/2} \, 2^\alpha \, \Gamma(\alpha/2)}{\Gamma((n - \alpha)/2)}.$$
Let $f$ be a measurable function on $\R^n$.
Its *fractional integral*, or *Riesz potential*, of order $\alpha$ is
$$I_\alpha f(x) = \frac{1}{\gamma(\alpha)} \int_{\R^n} \frac{f(y)}{\abs{x - y}^{n - \alpha}} \, dy,$$
at each point $x \in \R^n$ where the [[Lebesgue integral|DWJF7ABPR1]] converges absolutely.
The integral is with respect to [[Lebesgue measure|2Z2QVWZ262]].

## Remarks

If $f \in L^p(\R^n)$, an element of the [[Lp space|SASP14MA1Z]], with $1 \le p < n/\alpha$, the integral converges absolutely for almost every $x$.
With the [[Fourier transform|DPVM34ADZV]] $\hat f(\xi) = \int f(x) e^{-2\pi i x \cdot \xi} \, dx$, the constant $\gamma(\alpha)$ is chosen so that $\widehat{I_\alpha f}(\xi) = (2\pi\abs{\xi})^{-\alpha} \hat f(\xi)$, for example for Schwartz functions $f$.
In this sense $I_\alpha = (-\Delta)^{-\alpha/2}$, and $I_\alpha I_\beta = I_{\alpha + \beta}$ when $\alpha + \beta < n$.

Example: $n = 3$ and $\alpha = 2$.
Then $\gamma(2) = \pi^{3/2} \cdot 4 \cdot \Gamma(1)/\Gamma(1/2) = 4\pi$, since $\Gamma(1/2) = \sqrt{\pi}$.
So $I_2 f(x) = \frac{1}{4\pi}\int_{\R^3} \frac{f(y)}{\abs{x - y}} \, dy$ is the Newtonian potential, and $-\Delta I_2 f = f$ for smooth $f$ with compact support.

Non-example: the order $\alpha = n$ fails the condition $0 < \alpha < n$.
Then the kernel $\abs{x - y}^{0}$ is the constant $1$, so the formula gives $\int_{\R^n} f(y) \, dy$, which does not depend on $x$ and smooths nothing.
Also $\Gamma((n - \alpha)/2) = \Gamma(0)$ is undefined, so the normalization breaks down.
</document>

<document source="https://tamnd.github.io/math2/node/TAKA1DYBNE.md" tag="TAKA1DYBNE" kind="definition">
# Tautological one-form

## Definition

Let $Q$ be a [[smooth manifold|F45VQ47GPV]] with cotangent bundle $\pi \colon T^*Q \to Q$.
A point of $T^*Q$ is a pair $(q, p)$ with $q \in Q$ and $p \in T_q^* Q$.
The *tautological one-form* is the [[$1$-form|0E3G0GRPZ6]] $\theta$ on $T^*Q$ given by
$$\theta_{(q, p)}(v) = p\big(d\pi_{(q, p)}(v)\big) \quad \text{for } v \in T_{(q,p)}(T^*Q),$$
where $d\pi$ is the [[differential|6P9PXQDSEN]] of $\pi$.

## Remarks

In local coordinates $q^1, \ldots, q^n$ on $Q$ with the induced coordinates $p_1, \ldots, p_n$ on the fibres, $\theta = \sum_i p_i \, dq^i$.
The form $\omega = -d\theta = \sum_i dq^i \wedge dp_i$, with $d$ the [[exterior derivative|004FRCXH6M]], is the canonical [[symplectic form|CAN490EPRT]] of $T^*Q$; some authors use the opposite sign.

Characterization: for every $1$-form $\alpha$ on $Q$, viewed as a section $\alpha \colon Q \to T^*Q$, one has $\alpha^* \theta = \alpha$, and $\theta$ is the only $1$-form on $T^*Q$ with this property.

Example: for $Q = \R$ with coordinate $q$ and fibre coordinate $p$, $\theta = p\, dq$; for $\alpha = c\, dq$ the section is $q \mapsto (q, c)$ and $\alpha^*(p\, dq) = c\, dq = \alpha$.

Non-example: the form $q\, dp$ on $T^*\R$ is not the tautological form.
For the section $\alpha = dq$, that is $q \mapsto (q, 1)$, the pullback of $q\, dp$ is $q \cdot d(1) = 0 \ne dq$.
</document>

<document source="https://tamnd.github.io/math2/node/TAMCTW7SA0.md" tag="TAMCTW7SA0" kind="statement">
# Kawamata–Viehweg vanishing theorem

## Statement

Let $X$ be a [[smooth|EXKGG9X6J0]] [[projective variety|A4H4CZ2V35]] of dimension $n$ over a field of characteristic $0$, with [[canonical bundle|W05VYY4HBR]] $\omega_X$.
Let $L$ be a [[nef line bundle|RZ66X0NAM5]] on $X$ that is big, that is, whose top self-intersection number satisfies $(L^n) > 0$.
Then
$$H^i(X, \omega_X \otimes L) = 0 \quad \text{for all } i > 0,$$
where $H^i$ is [[sheaf cohomology|PQ6Q143ARJ]].

## Remarks

For a nef line bundle, bigness in the sense of maximal Iitaka dimension is equivalent to $(L^n) > 0$.
The idea of the proof: by Kodaira's lemma, some multiple of $L$ is an ample divisor plus an effective divisor.
A cyclic covering trick and a log resolution reduce the statement to the [[Kodaira vanishing theorem|NY77DQ2AX8]] on a covering variety.

There is a more general form for $\Q$-divisors: if $D$ is a nef and big $\Q$-divisor whose fractional part has simple normal crossing support, then $H^i(X, \omega_X \otimes \mathcal{O}_X(\lceil D \rceil)) = 0$ for $i > 0$.

Example: let $n \ge 2$, let $X \to \mathbb{P}^n$ be the blowing up of a point, and let $L$ be the pullback of $\mathcal{O}(1)$.
Then $L$ is nef and big but not ample, since it has degree $0$ on the lines in the exceptional divisor.
The theorem still gives $H^i(X, \omega_X \otimes L) = 0$ for $i > 0$, which Kodaira's theorem does not cover.

Bigness is needed: on $X = C \times \mathbb{P}^1$ with $C$ a curve of genus $g \ge 1$, the pullback $L$ of $\mathcal{O}_{\mathbb{P}^1}(2)$ is nef with $(L^2) = 0$, and $H^1(X, \omega_X \otimes L) \cong H^1(C, \omega_C) \otimes H^0(\mathbb{P}^1, \mathcal{O}) \ne 0$ by the Künneth formula.

## Explanations

### L1 human

The Kodaira vanishing theorem needs a positive (ample) line bundle.
The Kawamata–Viehweg theorem weakens this to a bundle that is never negative on curves and that has positive volume.
Such bundles appear naturally after blowing up, so the weaker hypothesis is much more flexible.

### L2 human

This theorem is the main cohomological tool of the minimal model program.
It is used in the proofs of the basepoint-free theorem and of the cone theorem, where one works with nef and big divisors that are not ample.
Its version for $\Q$-divisors allows rounding of fractional divisors, which is needed when one works with pairs $(X, \Delta)$.
</document>

<document source="https://tamnd.github.io/math2/node/TAN5G5EQ96.md" tag="TAN5G5EQ96" kind="concept">
# Integral geometry

## Summary

Integral geometry studies measures on families of geometric objects, such as the lines or the planes of Euclidean space.
The measures are chosen so that they do not change under rigid motions.
Its main results express a geometric size, such as the length of a curve, as an average over such a family.
For example, the length of a curve is an average of the number of times a random line meets it.
The subject grew out of geometric probability, starting with [[Buffon's needle problem|06X5PF6YFP]], and is closely tied to convex geometry and to stochastic geometry.

## Explanations

### L1 human

Throw a long straight stick at random onto a drawing and count how often it crosses the curves of the drawing.
On average the count is proportional to the total length of the curves, whatever their shape.
Integral geometry makes sense of "a random line" with a measure on the set of all lines.
This measure does not depend on where the origin is or how the axes are turned.
With this measure, lengths, areas and volumes become averages of counts and sections.
</document>

<document source="https://tamnd.github.io/math2/node/TAPCWE72R0.md" tag="TAPCWE72R0" kind="definition">
# Ring of differential operators

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $A$ be a [[commutative|0310REJJY8]] $k$-algebra.
For $a \in A$, write $a$ also for the map $A \to A$, $f \mapsto af$.
For a $k$-linear map $\theta \colon A \to A$, write $[\theta, a] = \theta \circ a - a \circ \theta$.
Define subspaces $D^m(A)$ of the [[endomorphism ring|AWSBZWWMG9]] $\End_k(A)$ by induction on $m \ge 0$:

1. $D^0(A)$ is the set of maps $f \mapsto af$ for $a \in A$.
2. For $m \ge 1$, $D^m(A)$ is the set of $k$-linear maps $\theta \colon A \to A$ with $[\theta, a] \in D^{m-1}(A)$ for every $a \in A$.

The *ring of differential operators* of $A$ is $D(A) = \bigcup_{m \ge 0} D^m(A)$, and the elements of $D^m(A)$ are the differential operators of order at most $m$.

## Remarks

$D(A)$ is a subring of $\End_k(A)$, because $D^m(A) \circ D^l(A) \subseteq D^{m+l}(A)$.
Every $k$-linear [[derivation|3J89REV9WQ]] $\partial$ of $A$ lies in $D^1(A)$, since $[\partial, a]$ is multiplication by $\partial(a)$.
Example: let $k$ have characteristic $0$ and $A = k[x_1, \dots, x_n]$.
Then $D(A)$ is generated by the multiplications by $x_i$ and the [[partial derivatives|09EGC6F8PT]] $\partial/\partial x_i$, and it is isomorphic to the [[Weyl algebra|98QE7KY55W]] $A_n(k)$.
Non-example: let $k = \Q$, $A = \Q[x]$ and let $\tau$ be the shift $f(x) \mapsto f(x + 1)$.
Then $[\tau, x] = \tau$, since $\tau(xf) = (x+1)\tau(f)$.
If $\tau$ were in $D^m(A)$, then $\tau = [\tau, x]$ would be in $D^{m-1}(A)$, and by induction $\tau$ would be in $D^0(A)$, a multiplication map; this is false, so $\tau \notin D(A)$.
In characteristic $p > 0$ the ring $D(k[x])$ is not generated by $x$ and $d/dx$, since $(d/dx)^p = 0$ while the operator $x^p \mapsto 1$, $x^j \mapsto \binom{j}{p} x^{j-p}$ lies in $D^p(k[x])$.
</document>

<document source="https://tamnd.github.io/math2/node/TAQWRWTHX8.md" tag="TAQWRWTHX8" kind="definition">
# Langevin equation

## Definition

Let $d \ge 1$, let $m > 0$ be a mass, let $\gamma > 0$ be a friction coefficient, and let $\beta > 0$ be an inverse temperature.
Let $U \colon \R^d \to \R$ be a continuously differentiable potential, and let $B$ be a $d$-dimensional [[Brownian motion|22735FGJV6]].
The *Langevin equation* is the system of stochastic differential equations for a position $X_t$ and a velocity $V_t$ in $\R^d$:
$$dX_t = V_t \, dt, \qquad m \, dV_t = -\nabla U(X_t) \, dt - \gamma V_t \, dt + \sqrt{2\gamma / \beta} \, dB_t.$$

## Remarks

In physics notation, $\beta = 1 / (k_B T)$ for the temperature $T$ and the Boltzmann constant $k_B$.
The equation is often written $m \ddot{X} = -\nabla U(X) - \gamma \dot{X} + \xi$, where $\xi$ is white noise.
Example: for $U = 0$ and $d = 1$, the velocity solves $dV = -(\gamma/m) V \, dt + (\sqrt{2\gamma/\beta}/m) \, dB$.
This is an [[Ornstein–Uhlenbeck process|HZM2ASCDY4]], and its stationary law is [[normal|05AQE0QR0D]] with mean $0$ and variance $1/(\beta m)$.
Non-example: the equation $m \, dV = -\gamma V \, dt$ lacks the random force, so its solutions decay to $0$ and never reach thermal equilibrium.
The overdamped Langevin equation $dX_t = -\nabla U(X_t) \, dt + \sqrt{2/\beta} \, dB_t$ is the limit of large friction after a change of time scale.
When $e^{-\beta U}$ is integrable, the overdamped equation has a stationary law with density proportional to $e^{-\beta U(x)}$.
Under the same condition, the full equation has a stationary law with density proportional to $e^{-\beta (U(x) + m \abs{v}^2 / 2)}$ in $(x, v)$.
</document>

<document source="https://tamnd.github.io/math2/node/TASMEPWNA6.md" tag="TASMEPWNA6" kind="definition">
# Lévy–Prokhorov metric

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] with its [[Borel σ-algebra|0WQXY32Q8D]] $\mathcal{B}$.
Let $\mathcal{P}(X)$ be the set of Borel [[probability measures|059TJN0NZ1]] on $X$.
For $A \subseteq X$ and $\varepsilon > 0$, let $A^\varepsilon = \set{x \in X : d(x, a) < \varepsilon \text{ for some } a \in A}$.
The *Lévy–Prokhorov metric* is the map $\pi \colon \mathcal{P}(X) \times \mathcal{P}(X) \to [0, 1]$ given by
$$\pi(\mu, \nu) = \inf \set{\varepsilon > 0 : \mu(A) \le \nu(A^\varepsilon) + \varepsilon \text{ and } \nu(A) \le \mu(A^\varepsilon) + \varepsilon \text{ for all } A \in \mathcal{B}}.$$
The value is an [[infimum|0QF1MZ9FRG]].

## Remarks

The function $\pi$ is a metric on $\mathcal{P}(X)$, and $\pi \le 1$ because every $\varepsilon \ge 1$ satisfies both inequalities.
If $X$ is separable, then $\pi(\mu_n, \mu) \to 0$ if and only if $\mu_n$ [[converges weakly|MBW9JCGZC1]] to $\mu$.
If $X$ is separable and complete, then so is $(\mathcal{P}(X), \pi)$.

Example: let $\delta_x$ be the [[Dirac measure|0ACRGYNHSR]] at $x$, and let $x, y \in X$.
If $\varepsilon > d(x, y)$, then every Borel set $A$ containing $x$ has $y \in A^\varepsilon$, so $\delta_x(A) \le \delta_y(A^\varepsilon)$, and symmetrically.
Hence $\pi(\delta_x, \delta_y) = \min(d(x, y), 1)$.

Non-example: for $\varepsilon < \min(d(x, y), 1)$ the value $\varepsilon$ is not admissible.
Take $A = \set{x}$: then $y \notin A^\varepsilon$, and $1 \le 0 + \varepsilon$ fails.

On $\R$, the points $1/n$ and $0$ give $\pi(\delta_{1/n}, \delta_0) = 1/n$, while the total variation distance stays equal to $1$.

## Explanations

### L1 human

The metric allows an error of $\varepsilon$ both in position and in mass.
Its smallest admissible $\varepsilon$ measures how far apart the two measures are.
</document>

<document source="https://tamnd.github.io/math2/node/TAVG968272.md" tag="TAVG968272" kind="definition">
# Direct limit

## Definition

Let $(I, \le)$ be a [[directed set|D9RY4667K6]] and let $\mathcal{C}$ be a [[category|A2DE66937D]].
A *direct system* in $\mathcal{C}$ over $I$ is a family of objects $A_i$ for $i \in I$ with [[morphisms|859F0TFVA3]] $f_{ij} \colon A_i \to A_j$ for all $i \le j$, such that:

1. $f_{ii} = \id_{A_i}$ for all $i$.
2. $f_{jk} \circ f_{ij} = f_{ik}$ for all $i \le j \le k$.

A *direct limit* of this system is an object $A$ with morphisms $\phi_i \colon A_i \to A$ such that:

1. $\phi_j \circ f_{ij} = \phi_i$ for all $i \le j$.
2. For every object $B$ with morphisms $g_i \colon A_i \to B$ such that $g_j \circ f_{ij} = g_i$ for all $i \le j$, there is a unique morphism $u \colon A \to B$ with $u \circ \phi_i = g_i$ for all $i$.

## Remarks

The direct limit is written $\varinjlim A_i$.
It is the [[colimit|BJS1CGTKQ5]] of the diagram given by the system, and it is unique up to a unique isomorphism.
In sets, groups, rings and modules it always exists.
There it is the disjoint union of the $A_i$ divided by a relation: $a_i \in A_i$ and $a_j \in A_j$ are identified when $f_{ik}(a_i) = f_{jk}(a_j)$ for some $k \ge i, j$.

Example: let $A_n = \Z$ for $n \in \N$, and let $f_{n,n+1}$ be multiplication by $2$.
The maps $\phi_n \colon \Z \to \Z[1/2]$, $\phi_n(a) = a/2^n$, make $\Z[1/2]$ a direct limit.
Clause 1 holds because $\phi_{n+1}(2a) = 2a/2^{n+1} = \phi_n(a)$.

Non-example: the maps $\phi_n(a) = a/2^n$ into $\Q$ satisfy clause 1 but not clause 2.
For $B = \Z[1/2]$ with the maps above, there is no group homomorphism $u \colon \Q \to \Z[1/2]$ with $u(a/2^n) = a/2^n$ for all $a$ and $n$.
Indeed $u(1/3)$ would be an element $x$ with $3x = 1$, and $\Z[1/2]$ has no such element.
</document>

<document source="https://tamnd.github.io/math2/node/TAYDF13B23.md" tag="TAYDF13B23" kind="concept">
# Algebraic statistics

## Summary

Algebraic statistics uses algebraic geometry and commutative algebra to study statistical models and procedures.
Many models for discrete data are sets of probability vectors cut out by polynomial equations.
Examples are independence models for [[contingency tables|PANSMHE0SW]], [[graphical models|Q3FQAD1BXG]] and log-linear models.
Tools such as [[polynomial ideals|096Z9MP2ME]], [[Gröbner bases|C83KPT47R0]] and [[algebraic varieties|HV6YBS0YDH]] then give information on identifiability, maximum likelihood estimation and exact tests.
A well-known example is the use of Markov bases, introduced by Persi Diaconis and Bernd Sturmfels, to sample tables with fixed margins.

## Explanations

### L1 human

Toss two coins that may be biased and record the four outcomes HH, HT, TH, TT.
If the coins are independent, the probabilities satisfy $p_{HH} p_{TT} = p_{HT} p_{TH}$.
So the independent models form a curved surface defined by one polynomial equation inside the set of all probability vectors.
Algebraic statistics studies such surfaces with the tools of algebra and uses them to answer statistical questions.
</document>

<document source="https://tamnd.github.io/math2/node/TB0ZRTX7DT.md" tag="TB0ZRTX7DT" kind="example">
# Möbius–Kantor configuration

## Example

The *Möbius–Kantor configuration* is the [[configuration|Q2QFXPP9EE]] $(8_3)$: $8$ points and $8$ lines, with $3$ points on each line and $3$ lines through each point.
Up to isomorphism there is only one $(8_3)$ configuration.
A model has the points $0, 1, \dots, 7$ in $\Z/8\Z$ and the lines $\set{i, i + 1, i + 3}$ for $i \in \Z/8\Z$.

Check: each point $j$ lies on the three lines with $i = j$, $i = j - 1$ and $i = j - 3$.
The differences of elements of $\set{0, 1, 3}$ are $\pm 1, \pm 2, \pm 3$, each obtained once and all distinct modulo $8$.
So two points whose difference is one of these lie on exactly one common line, and two points with difference $4$ lie on none.

Another model: delete one point from the [[affine plane of order $3$|S2NAGK9YHF]], together with the $4$ lines through it.
The remaining $8$ points and $8$ lines have $3$ points on each line, and each point lies on $4 - 1 = 3$ of the remaining lines.

The configuration cannot be realized by points and lines of the real plane, but it can be realized in the complex projective plane.
Its incidence graph is the Möbius–Kantor graph.

## Explanations

### L1 human

Can you draw two quadrilaterals so that each vertex of one lies on a side of the other, and the other way round?
Möbius asked this, and the answer is no with real points and straight lines.
The combinatorial pattern still exists: $8$ points and $8$ lines, each line through $3$ points and each point on $3$ lines.
It can be drawn with complex coordinates, or in the finite plane over the field with $3$ elements.
</document>

<document source="https://tamnd.github.io/math2/node/TB2TR373RC.md" tag="TB2TR373RC" kind="example">
# Square wave

## Example

The *square wave* is the $2\pi$-periodic function $f$ with $f(x) = 1$ for $0 < x < \pi$, $f(x) = -1$ for $-\pi < x < 0$, and $f(0) = f(\pi) = 0$.
It is odd, so its cosine coefficients vanish, and for $n \ge 1$
$$b_n = \frac{2}{\pi} \int_0^\pi \sin nx \, dx = \frac{2 (1 - (-1)^n)}{\pi n},$$
which is $\frac{4}{\pi n}$ for odd $n$ and $0$ for even $n$.
So its Fourier series is
$$\frac{4}{\pi} \left( \sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \cdots \right).$$
The function is of bounded variation, so by the [[Dirichlet conditions|75ZQW12WY0]] the series converges at every $x$ to $\frac{1}{2}(f(x^+) + f(x^-))$.
This is $f(x)$ at every $x$, including the points of [[jump discontinuity|004FH91BN5]] $x \in \pi\Z$, where both sides are $0$.

## Remarks

At $x = \pi/2$ the series gives $1 = \frac{4}{\pi}(1 - \frac{1}{3} + \frac{1}{5} - \cdots)$, which is the Leibniz formula for $\pi/4$.
The coefficients decay only like $1/n$, because of the jumps, so the series does not converge absolutely.
The convergence is uniform on every closed interval that avoids $\pi\Z$, but not on $\R$, since the limit is discontinuous.
Near each jump the partial sums overshoot: their maximum just to the right of $0$ tends to $\frac{2}{\pi} \int_0^\pi \frac{\sin t}{t} \, dt \approx 1.179$, the [[Gibbs phenomenon|VMF169AT67]].

## Explanations

### L1 human

The square wave switches between $-1$ and $1$ every half period, like an ideal on-off signal.
It is built from the odd harmonics only, with amplitudes $1, \frac{1}{3}, \frac{1}{5}, \dots$ times $\frac{4}{\pi}$.
Adding more harmonics makes the flat parts flatter and the jumps steeper, but a small spike of about $9\%$ of the jump remains near each jump.
</document>

<document source="https://tamnd.github.io/math2/node/TB3Q55Q0G4.md" tag="TB3Q55Q0G4" kind="definition">
# Gerstenhaber algebra

## Definition

Let $k$ be a [[field|04C5ZBG1WT]].
Let $A = \bigoplus_{n \in \Z} A^n$ be a [[graded|8KB4SQGC0C]] [[associative algebra|3CTW1P4HBP]] over $k$ that is graded commutative: $ab = (-1)^{|a||b|} ba$ for homogeneous $a, b$ of degrees $\abs{a}, \abs{b}$.
A *Gerstenhaber algebra* is such an $A$ with a bilinear bracket $[\cdot, \cdot] \colon A^p \times A^q \to A^{p+q-1}$ such that, for homogeneous $a, b, c$:

1. $[a, b] = -(-1)^{(|a|-1)(|b|-1)} [b, a]$;
2. $[a, [b, c]\,] = [\,[a, b], c] + (-1)^{(|a|-1)(|b|-1)} [b, [a, c]\,]$;
3. $[a, bc] = [a, b]\, c + (-1)^{(|a|-1)|b|}\, b\, [a, c]$.

## Remarks

Clauses 1 and 2 say that $A$ becomes a graded Lie algebra when each element of $A^n$ is given the degree $n - 1$.
Clause 3 says that $[a, \cdot]$ is a derivation of the product of degree $\abs{a} - 1$.
Example: the polyvector fields $\bigoplus_n \Gamma(\Lambda^n TM)$ on a smooth manifold $M$, with the wedge product and the [[Schouten–Nijenhuis bracket|FJ1MDBZ680]].
Example: Gerstenhaber proved that the Hochschild cohomology $HH^\bullet(B, B)$ of an associative algebra $B$ is a Gerstenhaber algebra, with the cup product and the Gerstenhaber bracket.
Non-example: let $\mathfrak{g}$ have basis $e, f$ with $[e, f] = f$, put it in degree $1$ of $\Lambda(\mathfrak{g})$, and set the bracket to zero on all other degrees.
Then $[e, e \wedge f] = 0$, while clause 3 asks for $[e, e] \wedge f + e \wedge [e, f] = e \wedge f \ne 0$, so clause 3 fails.
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# Faithful module

## Summary

A faithful module over a ring $R$ is a [[module|0NZ8JA25WK]] on which no nonzero element of $R$ acts as zero.
Equivalently, its [[annihilator|HB4XDKJTW8]] is zero, so $R$ embeds in the endomorphism ring of the additive group of the module.
It is the module version of a [[faithful action|0P7G9NGG2G]] of a group.
Faithful simple modules define primitive rings, a central class in the structure theory of rings.

## Explanations

### L1 human

A module is faithful when the ring can be fully seen through its action on the module.
In a faithful module, two different ring elements always act in different ways.
The integers act faithfully on $\Z$, since $n \cdot 1 = n$ is nonzero for $n \ne 0$.
They do not act faithfully on $\Z/5\Z$, since $5$ acts as zero there.
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# Picard–Vessiot extension

## Summary

A Picard–Vessiot extension is the smallest [[differential field|GA5PTQGC6G]] extension that contains a full set of solutions of a linear differential equation.
It adds no new constants.
It plays the role for linear differential equations that the splitting field plays for polynomial equations.
Its group of differential automorphisms is the differential Galois group of the equation, the main object of [[differential Galois theory|EEZS4H8QPH]].

## Explanations

### L1 human

Start with a field of functions, such as the rational functions $\C(x)$, and a [[linear differential equation|0QQRP17Z7S]] with coefficients in it.
Often the solutions are not in the field, so one adjoins them together with all their derivatives.
The result is a bigger field in which the equation can be solved completely.
For $y' = y$ over $\C(x)$, one adjoins $e^x$ and gets the field $\C(x, e^x)$.
The symmetries of this bigger field that fix $\C(x)$ and commute with differentiation tell how the solutions are related to each other.
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# Sheaf

## Summary

A sheaf on a topological space assigns to each open set a set of data, such as functions, with restriction maps to smaller open sets.
The data must be determined by their local pieces, and compatible local pieces must glue together.
Sheaves organise local information on a space and measure how it fails to be global.
They are the language of modern algebraic geometry, where a scheme is a space with a sheaf of rings.
They are also the coefficients of sheaf cohomology in topology and complex analysis.

## Explanations

### L1 human

Think of the continuous functions on the open sets of a space.
A function on a big open set can be restricted to a smaller one.
Suppose you know a function on each set of an open cover, and the pieces agree on the overlaps.
Then there is exactly one function on the union that has these pieces.
A sheaf is any rule of this kind: data on open sets, restriction, and the two local conditions of uniqueness and gluing.
Bounded functions are not a sheaf on $\R$: the function $f(x) = x$ is bounded on each bounded interval, but these pieces glue to a function that is not bounded.
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# Saddle-node bifurcation

## Definition

Let $f \colon \R \times \R \to \R$ be [[smooth|0V33BNKZ9S]], and consider the family of scalar equations $x' = f(x, \mu)$ with parameter $\mu$.
Write $f_x$, $f_\mu$, $f_{xx}$ for the partial derivatives of $f$.
The family has a *saddle-node bifurcation* at $(x, \mu) = (0, 0)$ if:

1. $f(0, 0) = 0$, so $0$ is an [[equilibrium point|0G4X36AE0X]] for $\mu = 0$;
2. $f_x(0, 0) = 0$;
3. $f_{xx}(0, 0) \ne 0$;
4. $f_\mu(0, 0) \ne 0$.

## Remarks

Under these conditions, the equilibria near $(0, 0)$ form a curve $\mu = m(x)$ with $m(0) = m'(0) = 0$ and $m''(0) = -f_{xx}(0, 0) / f_\mu(0, 0) \ne 0$.
So for $\mu$ on one side of $0$ there are two equilibria near $0$, one stable and one unstable, and on the other side there are none.

Example: $f(x, \mu) = \mu - x^2$.
Then $f(0, 0) = 0$, $f_x = -2x$ vanishes at $0$, $f_{xx} = -2 \ne 0$ and $f_\mu = 1 \ne 0$.
The equilibria are $\pm\sqrt{\mu}$ for $\mu > 0$, and there are none for $\mu < 0$.

Non-example: $f(x, \mu) = \mu x - x^2$.
Here $f_\mu(0, 0) = 0$, so clause 4 fails; the two equilibria $0$ and $\mu$ exist for all $\mu$ and only cross, which is a transcritical bifurcation.

For a system in $\R^n$, clause 2 is replaced by a condition on the Jacobian matrix at the equilibrium.
It must have a simple eigenvalue $0$ and no other eigenvalue on the imaginary axis.
The conditions 3 and 4 are then imposed on the equation reduced to a centre manifold.
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# Space form

## Definition

Let $n \ge 2$ and let $K$ be a real number.
A *space form* of curvature $K$ is a Riemannian manifold $(M, g)$ of dimension $n$ such that:

1. $M$ is connected.
2. $M$ is complete as a metric space for the [[Riemannian distance|B4QR306D8G]].
3. The [[sectional curvature|1X42JAA349]] of every 2-plane at every point of $M$ equals $K$.

## Remarks

Example: the flat torus $\R^2 / \Z^2$ with the metric induced from $\R^2$ is connected and compact, hence complete, and its sectional curvature is $0$ everywhere.
So it is a space form of curvature $0$.

Non-example: the open unit disc in $\R^2$ with the Euclidean metric is connected and flat, but the Cauchy sequence $(1 - 1/k, 0)$ has no limit in the disc.
It fails clause 2.

The simply connected space forms are the sphere of radius $1/\sqrt{K}$ for $K > 0$, Euclidean space $\R^n$ for $K = 0$, and hyperbolic space of curvature $K$ for $K < 0$.
Some authors call only these three models space forms.
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# Discriminant of a field extension

## Definition

Let $L/K$ be a [[finite extension|0NRYPM9Y27]] of degree $n$ and let $b_1, \ldots, b_n \in L$.
Let $\mathrm{Tr}_{L/K}$ be the [[field trace|73XV4RP5PT]].
The *discriminant* of the family $(b_1, \ldots, b_n)$ is
$$d_{L/K}(b_1, \ldots, b_n) = \det\left(\mathrm{Tr}_{L/K}(b_i b_j)\right)_{1 \le i, j \le n} \in K,$$
where $\det$ is the [[determinant|0C3CFHDN6R]].
The *discriminant of $L/K$* is the class of $d_{L/K}(b_1, \ldots, b_n)$ modulo $(K^\times)^2$, for any [[basis|0VTQTPQGAV]] $b_1, \ldots, b_n$ of $L$ over $K$.

## Remarks

If $c_1, \ldots, c_n$ is another basis and $c = Pb$ with $P \in GL_n(K)$, then $d(c) = \det(P)^2\, d(b)$, so the class modulo squares is well defined.
If $L/K$ is separable with embeddings $\sigma_1, \ldots, \sigma_n$ into an algebraic closure, then $d(b) = \det\left(\sigma_i(b_j)\right)^2$.
The discriminant of a basis is nonzero if and only if $L/K$ is separable.
If $L = K(\alpha)$ and $f$ is the minimal polynomial of $\alpha$, then $d(1, \alpha, \ldots, \alpha^{n-1})$ is the [[discriminant|0CCPN09EMJ]] of $f$.
For a number field $L$ over $\Q$, the discriminant of a $\Z$-basis of the ring of integers is an integer independent of the basis, the discriminant of the number field.

Example: $L = \Q(i)$, $K = \Q$, basis $1, i$.
Check: $\mathrm{Tr}(1) = 2$, $\mathrm{Tr}(i) = 0$ and $\mathrm{Tr}(i^2) = -2$, so $d(1, i) = \det\begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix} = -4$, the discriminant of $x^2 + 1$.

Non-example: for $K = \F_p(t)$ and $L = K(s)$ with $s^p = t$, every trace is $0$, so the discriminant of every basis is $0$; the extension is not separable.
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# Hamburger moment problem

## Problem

Let $(m_n)_{n \ge 0}$ be a [[sequence|0G9EKM0E15]] of real numbers.
Decide whether there is a positive Borel [[measure|PR11BZQQPG]] $\mu$ on the real line $\R$ such that $x^n$ is $\mu$-integrable and
$$m_n = \int_{\R} x^n \, d\mu(x) \quad \text{for all } n \ge 0.$$
The numbers $m_n$ are then the [[moments|01XAHPADBA]] of $\mu$.

## Remarks

The existence question is solved.
Hamburger proved that such a measure exists if and only if, for every $n \ge 0$, the Hankel matrix $(m_{j+k})_{j,k=0}^{n}$ is [[positive semidefinite|01MMVW8CKC]].
Equivalently, $\sum_{j,k=0}^{n} m_{j+k} c_j c_k \ge 0$ for all $n$ and all real numbers $c_0, \ldots, c_n$, since this sum is $\int \left( \sum_j c_j x^j \right)^2 d\mu$ when $\mu$ exists.
Unlike the Hausdorff moment problem on $[0, 1]$, the measure need not be unique.
The moment problem is called determinate when it is unique.
Carleman's condition $\sum_{n \ge 1} m_{2n}^{-1/(2n)} = \infty$ is sufficient for uniqueness.
The moments of the standard normal distribution satisfy it, so that distribution is determined by its moments.
The log-normal distribution is a classical example whose moments do not determine it.

## Explanations

### L1 human

The problem asks which lists of numbers are the moments of some mass distribution spread over the whole real line.
The answer is a positivity test: certain square tables built from the list must behave like squared lengths, never giving a negative value.
On an unbounded line, two different distributions can share all their moments.
So the moments may fail to pin down the distribution, which matters in probability when one tries to identify a law from its moments.
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# Triebel–Lizorkin space

## Summary

A Triebel–Lizorkin space $F^s_{p,q}(\R^n)$ is a space of [[tempered distributions|AES2T5JPFP]] defined, like a [[Besov space|95QBBPRJ9E]], by splitting the [[Fourier transform|DPVM34ADZV]] into dyadic frequency shells.
The difference is the order of the norms.
One first takes an $\ell^q$ sum over the shells at each point, and then the $L^p$ norm in space.
The scale contains the [[Lp spaces|SASP14MA1Z]] for $1 < p < \infty$, the [[Sobolev spaces|M4CEZ5K9HP]] $W^{k,p}$ for $1 < p < \infty$, and the local Hardy spaces.
It gives one framework for many function spaces of harmonic analysis and partial differential equations.

## Explanations

### L1 human

Split a function into pieces of frequency about $2^j$ and weight the $j$-th piece by $2^{js}$.
A Besov norm measures each weighted piece in $L^p$ and then adds up the results.
A Triebel–Lizorkin norm adds up the weighted pieces at each point first, through a square sum when $q = 2$, and then measures the result in $L^p$.
For $q = 2$ this square function is the classical tool of Littlewood–Paley theory, and it recovers the $L^p$ and Sobolev norms.
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# Krull's height theorem

## Statement

Let $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $I = (x_1, \dots, x_n)$ be an ideal of $R$ [[generated|030TZ0DQWT]] by $n$ elements.
Let $P$ be a [[prime ideal|06HC6HK0Z8]] of $R$ that is [[minimal|S2D8823P8B]] among the prime ideals containing $I$.
Then the [[height|9BKCZPMQT6]] of $P$ is at most $n$.

## Remarks

The proof is by induction on $n$, with [[Krull's principal ideal theorem|SJZW2HN1SC]] as the case $n = 1$.
For the step, take a prime $Q \subsetneq P$ with no prime strictly between them.
Since $P$ is minimal over $I$, some $x_i$ lies outside $Q$, and one shows that $Q$ is minimal over an ideal with $n - 1$ generators.
For example, in $k[x, y, z]$ over a field $k$, the prime $(x, y)$ is minimal over the ideal generated by $x$ and $y$, and its height is $2$.
A converse holds: every prime ideal of height $n$ in a Noetherian ring is minimal over some ideal generated by $n$ elements.
As a consequence, every prime ideal of a Noetherian ring has finite height.
A Noetherian local ring has finite [[Krull dimension|MVGBS3J1P7]], at most the number of generators of its maximal ideal.

## Explanations

### L1 human

Cutting a space by $n$ equations lowers its dimension by at most $n$.
In algebra: a prime that is minimal over an ideal with $n$ generators has height at most $n$.

### L2 human

The theorem is the base of dimension theory of Noetherian local rings.
It gives $\dim R \le$ the minimal number of generators of the maximal ideal, which leads to [[regular local rings|63XXTF6B5M]] and to [[systems of parameters|95QTEPMDA4]].
It also shows that the prime ideals of a Noetherian ring satisfy the descending chain condition.
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# Levi-flat hypersurface

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of complex dimension $n \ge 2$.
Let $M \subset X$ be a real hypersurface of class $C^2$.
Near a point $p \in M$, write $M = \set{\rho = 0}$ in local holomorphic coordinates $z = (z_1, \dots, z_n)$, where $\rho$ is a real $C^2$ function with $d\rho \ne 0$ on $M$.
The [[Levi form|80MTNXWW80]] of $M$ at $p$ is the Hermitian form
$$\mathcal{L}_p(w) = \sum_{j,k=1}^{n} \frac{\partial^2 \rho}{\partial z_j \partial \bar z_k}(p)\, w_j \bar w_k$$
on the complex tangent space $H_p = \set{w \in \C^n : \sum_{j} \frac{\partial \rho}{\partial z_j}(p)\, w_j = 0}$.
The hypersurface $M$ is *Levi-flat* if $\mathcal{L}_p = 0$ on $H_p$ for every $p \in M$.

## Remarks

The condition does not depend on the choice of $\rho$ or of the coordinates.
Example: $M = \set{\operatorname{Im} z_1 = 0} \subset \C^n$ with $\rho = \operatorname{Im} z_1 = (z_1 - \bar z_1)/(2i)$.
All mixed derivatives $\partial^2 \rho / \partial z_j \partial \bar z_k$ vanish, so the Levi form is $0$ and $M$ is Levi-flat.
Non-example: the unit sphere $\set{\abs{z_1}^2 + \cdots + \abs{z_n}^2 = 1}$ with $\rho = \abs{z}^2 - 1$.
Its Levi form is $\abs{w_1}^2 + \cdots + \abs{w_n}^2$, which is not zero on $H_p$, since $H_p \ne 0$ when $n \ge 2$.
For a smooth hypersurface, an equivalent condition is that the complex tangent distribution $p \mapsto H_p$ is integrable.
Then $M$ is locally [[foliated|98HTB7MSPX]] by complex hypersurfaces of $X$; this foliation is called the Levi foliation.
A Levi-flat hypersurface is a [[CR manifold|ZC78JKZX9S]] of hypersurface type.
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# Crown

## Example

Let $n \ge 3$.
The crown $S_n$ is the [[partially ordered set|0QV2JS9RJ3]] on $2n$ elements $a_1, \ldots, a_n, b_1, \ldots, b_n$ whose only strict relations are $a_i < b_j$ for all $i \ne j$.
It is also called the standard example of dimension $n$, because its [[order dimension|JC4J24MF93]] is $n$.
Some authors use the name crown for a different poset of height $2$, whose comparability graph is a cycle of length $2n$.

Lower bound.
For each $i$, the elements $a_i$ and $b_i$ are incomparable, so a realizer needs a [[linear extension|W8DZ5KGGHN]] $L$ with $b_i$ before $a_i$.
Suppose one linear extension $L$ has $b_i$ before $a_i$ and $b_j$ before $a_j$ with $i \ne j$.
Since $a_i < b_j$ and $a_j < b_i$, $L$ would contain the cycle $a_i, b_j, a_j, b_i, a_i$, which is impossible.
So each linear extension reverses at most one pair, and at least $n$ linear extensions are needed.

Upper bound.
For each $i$, let $L_i$ be the following order.
First come the $a_j$ with $j \ne i$, in increasing order of $j$, then $b_i$, then $a_i$.
Last come the $b_j$ with $j \ne i$, in increasing order of $j$.
Each $L_i$ is a linear extension, and every incomparable pair appears in both relative orders among $L_1, \ldots, L_n$, so these $n$ orders form a realizer.

## Remarks

The [[comparability graph|1P4830BH6Z]] of $S_n$ is the crown graph: the complete bipartite graph $K_{n,n}$ with a perfect matching removed.
The crown $S_n$ has height $2$ and width $n$, so it shows that the dimension of a poset of height $2$ can be as large as one likes.

## Explanations

### L1 human

Draw $n$ points in a bottom row and $n$ points in a top row.
Join each bottom point to every top point except the one just above it.
This poset needs $n$ different total orders to describe it exactly, because each total order can account for only one missing link.
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# Cauchy's logarithmic equation

## Definition

*Cauchy's logarithmic equation* is the [[functional equation|SRAN4EKWS0]]
$$f(xy) = f(x) + f(y) \quad \text{for all } x, y > 0,$$
for an unknown [[function|0EPAT51DXQ]] $f \colon (0, \infty) \to \R$.
A solution is a function $f$ that satisfies this identity for all positive $x$ and $y$.

## Remarks

Example: $f(x) = \log_2 x$ is a solution, since $\log_2(xy) = \log_2 x + \log_2 y$.
Non-example: $f(x) = x - 1$ is not a solution, since $f(2 \cdot 2) = 3$ while $f(2) + f(2) = 2$.
The substitution $x = e^s$, $y = e^t$ turns the equation into additivity of $A(s) = f(e^s)$.
So the solutions are $f(x) = A(\ln x)$ with $A$ an [[additive function|KF719SZD5T]], and the continuous ones are $f(x) = c \ln x$.
On the domain $\R \setminus \set{0}$ the solutions are $f(x) = A(\ln \abs{x})$; on all of $\R$ only $f = 0$ is a solution, since $f(0) = f(0) + f(x)$.
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# Universal Turing machine

## Description

Fix an encoding $\langle M, w \rangle$ of a [[Turing machine|758EYCN27B]] $M$ together with an input word $w$ as a single word.
A *universal Turing machine* is a Turing machine $U$ with the following property for every machine $M$ and every input $w$.
The machine $U$ halts on $\langle M, w \rangle$ if and only if $M$ halts on $w$, and then $U$ produces the same output as $M$.
Alan Turing described such a machine in his paper of 1936.

## Remarks

The machine $U$ keeps the description of $M$ on its tape, and the current tape and state of $M$ beside it.
At each step it looks up the matching rule of $M$ in the description and updates the simulated configuration.
In terms of [[computable functions|KZSB5E471K]], it computes the universal function $(e, x) \mapsto \varphi_e(x)$, which is itself partial computable.
Universal machines are the theoretical model of the stored-program computer: one fixed machine runs any program given as data.

## Explanations

### L1 human

A universal Turing machine is a single machine that can imitate every other Turing machine.
You give it, as input, the rule table of another machine and the input for that machine.
It then does whatever that machine would do, step by step.
A modern computer running any program you load is the same idea.
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# Discrete Laplace operator

## Definition

Let $d \ge 1$ and let $e_1, \dots, e_d$ be the standard unit vectors of $\Z^d$.
The *discrete Laplace operator* sends a [[function|0EPAT51DXQ]] $u \colon \Z^d \to \R$ to the function $\Delta u \colon \Z^d \to \R$ given by
$$(\Delta u)(m) = \sum_{i=1}^{d} \big( u(m + e_i) - 2u(m) + u(m - e_i) \big) = \sum_{\abs{m' - m} = 1} \big( u(m') - u(m) \big).$$
More generally, let a graph have vertex set $V$, and let every vertex have finitely many neighbours.
Then the operator sends $u \colon V \to \R$ to
$$(\Delta u)(v) = \sum_{w \sim v} \big( u(w) - u(v) \big), \qquad v \in V,$$
where $w \sim v$ means that $w$ and $v$ are joined by an edge.

## Remarks

Example: on $\Z^2$, the function $u(m, n) = m^2 + n^2$ has $(\Delta u)(m, n) = 2 + 2 = 4$ everywhere, matching $\Delta(x^2 + y^2) = 4$ in the plane.
The function $u(m, n) = mn$ has $\Delta u = 0$, so it is *discrete harmonic*.

Non-example of a discrete harmonic function: $u(m) = 2^m$ on $\Z$ has $(\Delta u)(m) = 2^{m+1} - 2^{m+1} + 2^{m-1} = 2^{m-1} \ne 0$.

On a grid of mesh $h > 0$ one uses $h^{-2} \Delta$, which tends to the Laplacian of $u$ as $h \to 0$ when $u$ is a smooth function sampled on the grid.
On a finite graph with [[Laplacian matrix|TN6M6DVZRP]] $L = D - A$, one has $\Delta u = -L u$ when $u$ is written as a column vector.
Some authors use the opposite sign or divide by the degree.
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# Foster's theorem on primal algebras

## Statement

Let $P$ be a [[primal algebra|XD01Y7B6XW]] and let $V(P)$ be the [[variety|AJNZJD8442]] it generates.
For a [[Boolean algebra|0XADJKM699]] $B$ with [[Stone space|ZV38HA5PAQ]] $X$, let $P[B]^*$ be the set of continuous maps from $X$ to $P$ with the discrete topology.
It is an algebra under the pointwise operations, called a Boolean power of $P$.
Then every algebra in $V(P)$ is isomorphic to $P[B]^*$ for some Boolean algebra $B$.

## Remarks

The result is due to Alfred L. Foster.
The idea is that a primal algebra can express, for each element $a$, a term that tests equality with $a$, and a term that selects according to such tests.
These terms let one split any algebra of $V(P)$ along a Boolean algebra of "decisions" and rebuild it from copies of $P$.
For a finite $B$ with $k$ atoms, $X$ is a discrete set of $k$ points and $P[B]^* \cong P^k$.
So the finite members of $V(P)$ are, up to isomorphism, the finite direct powers $P^k$.
For $P$ the two-element Boolean algebra, the theorem says that every Boolean algebra is isomorphic to the algebra of clopen subsets of its Stone space, which is Stone's representation theorem.
Primality is needed: for the two-element lattice, which is not primal, $V(P)$ contains the three-element chain, which is not a power of $P$.

## Explanations

### L1 human

The theorem says that all algebras built from a primal algebra $P$ are, in a precise sense, families of copies of $P$ indexed by the points of a Stone space.
So once one knows $P$ and Boolean algebras, one knows the whole variety.
This is why Boolean algebras are the simplest model of the general situation.

### L2 human

The map $B \mapsto P[B]^*$ gives an equivalence between the category of Boolean algebras and the variety $V(P)$.
This is the starting point of the theory of Boolean powers and of discriminator varieties.
For quasi-primal algebras, which are more general, a similar description uses Boolean products of subalgebras of $P$.
Foster's theorem makes the structure of Post algebras of each order explicit, since they form the variety generated by a primal algebra.
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<document source="https://tamnd.github.io/math2/node/TCF1PXZ5YN.md" tag="TCF1PXZ5YN" kind="definition">
# Upper nilradical

## Definition

Let $R$ be a [[ring|0894J980RK]].
The *upper nilradical* $\operatorname{Nil}^*(R)$ of $R$ is the sum of all [[nil ideals|NZ1G9WRWFB]] of $R$.
These are the [[two-sided ideals|0DDA6CY7GJ]] $I$ of $R$ such that every element of $I$ is nilpotent.

## Remarks

The sum of two nil ideals $I$ and $I'$ is nil, since $(I + I')/I' \cong I/(I \cap I')$ is nil and $I'$ is nil.
Hence $\operatorname{Nil}^*(R)$ is itself a nil ideal, and it is the largest nil ideal of $R$.
Example: in $\Z/12\Z$ the nil ideals are $0$ and $6\Z/12\Z$, so $\operatorname{Nil}^*(\Z/12\Z) = \set{0, 6}$.
Non-example: in $M_2(\Q)$ the element $e_{12}$ is nilpotent but does not lie in $\operatorname{Nil}^*(M_2(\Q))$.
Indeed the only ideals of $M_2(\Q)$ are $0$ and the whole ring, which contains $1$, so $\operatorname{Nil}^*(M_2(\Q)) = 0$.
In general $\operatorname{Nil}_*(R) \subseteq \operatorname{L}(R) \subseteq \operatorname{Nil}^*(R)$, where $\operatorname{Nil}_*$ is the [[prime radical|WF117EJ2HF]] and $\operatorname{L}$ the [[Levitzki radical|9DD8M9R868]], and both inclusions can be strict.
For a commutative ring the upper nilradical is the [[nilradical|QQDN6EYQNJ]].
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# Systolic geometry

## Definition

Let $(M, g)$ be a [[compact|0TVJZYB6SH]] connected [[Riemannian manifold|7CBT4SM83Y]] that is not simply connected.
A closed curve in $M$ is *noncontractible* if it is not freely homotopic to a constant curve.
The *systole* $\operatorname{sys}(M, g)$ is the infimum of the lengths of the noncontractible closed curves in $M$; it is attained and positive.
If $M$ has dimension $n$, the *systolic ratio* is $\operatorname{sys}(M, g)^n / \operatorname{vol}(M, g)$.
*Systolic geometry* is the study of the systole and of upper bounds for the systolic ratio over all metrics $g$ on a fixed manifold $M$.

## Remarks

Example: the flat torus $\R^2 / \Z^2$ has systole $1$, attained by the images of the coordinate segments, and area $1$, so its systolic ratio is $1$.
[[Loewner's torus inequality|1Y9X1YW6GH]] says that the systolic ratio of any metric on the torus is at most $2/\sqrt{3}$.

Non-example: on the round sphere $S^2$ every closed curve is contractible, so the systole is not defined.

The systolic ratio does not change when $g$ is multiplied by a positive constant.
The same words are used for higher systoles, defined with cycles of higher dimension instead of curves.
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<document source="https://tamnd.github.io/math2/node/TCRZV021XR.md" tag="TCRZV021XR" kind="definition">
# Cotangent space of a local ring

## Definition

Let $(R, \mathfrak m)$ be a [[local ring|MZ02BKPHJA]] with [[residue field|NJ3APQ8BHH]] $k = R/\mathfrak m$.
The quotient $\mathfrak m / \mathfrak m^2$ is an $R$-module killed by $\mathfrak m$, hence a [[vector space|000AYPA9AJ]] over $k$ with $(a + \mathfrak m)(x + \mathfrak m^2) = ax + \mathfrak m^2$.
The *cotangent space* of $R$ is this $k$-vector space $\mathfrak m / \mathfrak m^2$.
Its dual $\Hom_k(\mathfrak m / \mathfrak m^2, k)$ is the *Zariski tangent space* of $R$.

## Remarks

The scalar multiplication is well defined because $\mathfrak m \cdot \mathfrak m \subseteq \mathfrak m^2$.

Example: let $k$ be a field and $R = k[x, y]_{(x, y)}$, the [[localization|X6JD4AG973]] of the polynomial ring at $(x, y)$.
Then $\mathfrak m = (x, y)R$ and $\mathfrak m / \mathfrak m^2$ has the basis given by the classes of $x$ and $y$, so it has dimension $2$.
For the cusp $R = k[x, y]_{(x, y)} / (y^2 - x^3)$, the classes of $x$ and $y$ still form a basis, because $y^2 - x^3 \in \mathfrak m^2$; the dimension is $2$ although the cusp is a curve.

Non-example: the quotient $\mathfrak m / \mathfrak m^3$ is in general not a vector space over $k$.
In $k[\![t]\!]$ with $\mathfrak m = (t)$, the element $t$ acts on $\mathfrak m / \mathfrak m^3$ by sending the class of $t$ to the nonzero class of $t^2$, so $\mathfrak m$ does not kill it.
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<document source="https://tamnd.github.io/math2/node/TCWV185JKY.md" tag="TCWV185JKY" kind="definition">
# F-coalgebra

## Definition

Let $\mathcal{C}$ be a category and let $F \colon \mathcal{C} \to \mathcal{C}$ be a [[functor|28NQ0KRHD3]].
An *$F$-coalgebra* is a pair $(A, \alpha)$ of an object $A$ of $\mathcal{C}$ and a morphism $\alpha \colon A \to F(A)$.
A *homomorphism* of $F$-coalgebras $f \colon (A, \alpha) \to (B, \beta)$ is a morphism $f \colon A \to B$ with $\beta \circ f = F(f) \circ \alpha$.

## Remarks

The $F$-coalgebras and their homomorphisms form a category.
An $F$-coalgebra is the same as an algebra for the functor $F^{\mathrm{op}}$ on $\mathcal{C}^{\mathrm{op}}$.

Example: on the category of sets let $F(X) = \N \times X$, so $F(f) = \id_{\N} \times f$.
The map $\alpha \colon \N \to \N \times \N$ with $\alpha(n) = (n, n + 1)$ makes $(\N, \alpha)$ an $F$-coalgebra.
The set $\N^{\N}$ of streams with $\sigma(s) = (s_0, (s_1, s_2, \dots))$ is the terminal $F$-coalgebra.
The unique homomorphism from $(\N, \alpha)$ sends $n$ to the stream $(n, n + 1, n + 2, \dots)$.

Non-example: the map $f \colon \N \to \N$ with $f(n) = n + 1$ is not a homomorphism from $(\N, \alpha)$ to itself.
Indeed $\alpha(f(n)) = (n + 1, n + 2)$, while $F(f)(\alpha(n)) = (n, n + 2)$.
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<document source="https://tamnd.github.io/math2/node/TCYSWCTYYM.md" tag="TCYSWCTYYM" kind="concept">
# Poisson kernel

## Summary

The Poisson kernel is the integral kernel that produces a [[harmonic function|0DEB0JMN0W]] in a disc from its values on the boundary circle.
On the unit disc it is $P(z, \zeta) = (1 - \abs{z}^2)/\abs{\zeta - z}^2$ for $\abs{z} < 1$ and $\abs{\zeta} = 1$.
Integrating a continuous function on the circle against it solves the [[Dirichlet problem|8D56GQ4AG8]] for the disc.
Similar kernels exist for the upper half-plane and for balls in $\R^n$.
They are central in harmonic analysis and in the theory of [[Hardy spaces|PNNNFJVF6W]].

## Explanations

### L1 human

Suppose you know the temperature at every point of the rim of a round plate and the plate is in equilibrium.
The temperature at an interior point is a weighted average of the rim temperatures.
The Poisson kernel gives the weights: rim points close to the interior point count more.
At the centre all rim points count equally, so the centre temperature is the plain average of the rim.
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<document source="https://tamnd.github.io/math2/node/TD0N49V1SD.md" tag="TD0N49V1SD" kind="definition">
# Projection method for integral equations

## Definition

Let $X$ be a [[Banach space|06GZ956657]], let $K \colon X \to X$ be a bounded linear operator, typically a compact integral operator, and let $f \in X$.
Consider the [[equation of the second kind|8KJCN49V64]] $\varphi - K\varphi = f$.
Let $X_1, X_2, \ldots$ be finite-dimensional subspaces of $X$.
For each $n$ let $P_n \colon X \to X$ be a bounded linear [[projection|033DA1WE8Y]] with image $X_n$, that is $P_n^2 = P_n$ and $P_n(X) = X_n$.
The *projection method* defined by $(X_n, P_n)$ approximates $\varphi$ by the solution $\varphi_n \in X_n$ of
$$\varphi_n - P_n K \varphi_n = P_n f.$$
If $u_1, \ldots, u_N$ is a basis of $X_n$ and $\varphi_n = \sum_j c_j u_j$, this equation is a system of $N$ linear equations for $c_1, \ldots, c_N$.

## Remarks

Two choices of $P_n$ are standard.
In the *collocation method*, $X_n \subseteq C[a, b]$ and $P_n$ is interpolation at points $x_1, \ldots, x_N$, so the equation is required to hold at these points.
In the *Galerkin method*, $X$ is a Hilbert space and $P_n$ is the [[orthogonal projection|0B8KZRT28M]] onto $X_n$.
So the residual is required to be orthogonal to $X_n$; see the [[Galerkin method|Y5B0PEBKE8]].

Example: $X = L^2(0, 1)$, $(K\varphi)(x) = \int_0^1 x t \, \varphi(t) \, dt$, $f(x) = x$, and $X_1$ spanned by $u_1(x) = x$ with $P_1$ orthogonal.
With $\varphi_1 = c x$, one has $K\varphi_1 = \frac{c}{3} x$, so the equation is $c - \frac{c}{3} = 1$ and $c = \frac32$.
Since $K$ maps into $X_1$ and $f \in X_1$, this is the exact solution: $\frac32 x - \frac{x}{3} \cdot \frac32 = x$.

Non-example: the Nyström method replaces the integral in $K$ by a quadrature rule and solves for the values of the unknown at the nodes.
It approximates the operator $K$ instead of projecting the unknown onto a fixed subspace, so it is not a projection method in this sense.
Its analysis uses [[collectively compact operator approximation|AEZE6B2XXY]].

The convergence of projection methods for second-kind equations is described by the [[convergence theorem for second-kind equations|H6KS0E9MF5]].
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# Mikusiński operational calculus

## Summary

The Mikusiński operational calculus, created by Jan Mikusiński, is an algebraic foundation of [[operational calculus|8050R3Z6HA]] that does not use integral transforms.
Functions on $[0, \infty)$ are elements of the [[Mikusiński field of operators|82YAXZMMQ1]], in which differentiation is multiplication by an element $s$ up to initial values.
Linear differential equations with constant coefficients become linear equations in this field, and the solution is read off by algebra.
Unlike the Laplace transform approach, it needs no growth condition on the functions.

## Explanations

### L1 human

Mikusiński replaced the Laplace transform by pure algebra.
He made fractions of functions, with convolution as multiplication, much as one makes fractions of whole numbers.
In this world the derivative is multiplication by a fixed element $s$, corrected by the initial value.
Solving $y' = a y$, $y(0) = 1$ becomes $s y - 1 = a y$, so $y = 1/(s - a)$, and one checks that this fraction is the function $e^{at}$.
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<document source="https://tamnd.github.io/math2/node/TD6VEP3SCE.md" tag="TD6VEP3SCE" kind="concept">
# Ehresmann connection

## Summary

An Ehresmann connection on a [[fibre bundle|CX4M8K61AB]] is a choice of horizontal directions at every point of the total space.
These directions are complementary to the directions along the fibres.
It works for any smooth fibre bundle, not only for vector bundles, and needs no linear structure.
It lets one lift curves in the base to horizontal curves in the total space, which gives parallel transport between fibres.
Connections on vector bundles and on principal bundles are Ehresmann connections with extra symmetry.

## Explanations

### L1 human

Picture a bundle as a family of fibres standing over the points of a base.
At each point of the total space some directions go along the fibre; these are vertical and are given by the bundle.
An Ehresmann connection chooses which of the remaining directions count as horizontal.
For the product of a line and a circle, the obvious choice is to call the line directions horizontal.
Then moving in the base lifts uniquely to moving horizontally upstairs.
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<document source="https://tamnd.github.io/math2/node/TD7S2HF19H.md" tag="TD7S2HF19H" kind="definition">
# Coupling

## Definition

Let $\mu$ be a [[probability measure|059TJN0NZ1]] on a measurable space $(E, \mathcal{E})$ and $\nu$ a probability measure on a measurable space $(F, \mathcal{F})$.
A *coupling* of $\mu$ and $\nu$ is a probability measure $\gamma$ on the product space $(E \times F, \mathcal{E} \otimes \mathcal{F})$ whose [[marginals|0MXT6PRZZ6]] are $\mu$ and $\nu$:
$$\gamma(A \times F) = \mu(A) \quad \text{and} \quad \gamma(E \times B) = \nu(B) \qquad \text{for all } A \in \mathcal{E}, \ B \in \mathcal{F}.$$
Equivalently, a coupling is a pair of [[random variables|0PNRKMRC55]] $(X, Y)$ on one [[probability space|0NFPJVJJHX]] with $X$ of law $\mu$ and $Y$ of law $\nu$.
The joint law of such a pair is a coupling in the first sense.

## Remarks

Example: the [[product measure|77PV0ZAVXT]] $\mu \otimes \nu$ is a coupling, the independent coupling. Indeed $(\mu \otimes \nu)(A \times F) = \mu(A) \nu(F) = \mu(A)$, and in the same way for the second marginal.
Example: for $0 \le p \le q \le 1$, let $U$ be uniform on $[0, 1]$, $X = \mathbf{1}_{\set{U < p}}$ and $Y = \mathbf{1}_{\set{U < q}}$. Then $X$ and $Y$ are [[Bernoulli|0386Y0Z7Y7]] with parameters $p$ and $q$, and $X \le Y$ everywhere.
Non-example: on $\set{0, 1}^2$, the point mass $\delta_{(0, 0)}$ is not a coupling of two Bernoulli laws with parameter $\frac{1}{2}$, because its first marginal is $\delta_0$.
If $E = F$ and the diagonal is measurable, every coupling satisfies the coupling inequality $d_{TV}(\mu, \nu) \le P(X \ne Y)$, where $d_{TV}$ is the [[total variation distance|3SZESKX3N5]].
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<document source="https://tamnd.github.io/math2/node/TD9T02HKPD.md" tag="TD9T02HKPD" kind="definition">
# Magnus expansion

## Definition

Let $A : [0, \infty) \to \C^{n \times n}$ be continuous, and let $Y$ be the solution of the [[linear system|0A6QZ367GX]] $Y' = A(t)\, Y$ with $Y(0) = I$.
For matrices $X, Z$ write $[X, Z] = XZ - ZX$.
The *Magnus expansion* is the series $\Omega(t) = \sum_{k \ge 1} \Omega_k(t)$ with $Y(t) = e^{\Omega(t)}$, the [[matrix exponential|0K665H5V1S]].
Its first terms are
$$\Omega_1(t) = \int_0^t A(t_1)\, dt_1, \qquad \Omega_2(t) = \frac12 \int_0^t \int_0^{t_1} [A(t_1), A(t_2)]\, dt_2\, dt_1,$$
and in general $\Omega_k$ is a sum of $k$-fold iterated integrals of nested commutators of $k$ values of $A$.
The terms are determined recursively by the differential equation $\Omega' = \sum_{j \ge 0} \frac{B_j}{j!} \operatorname{ad}_\Omega^j(A)$, $\Omega(0) = 0$, where $B_j$ are the Bernoulli numbers with $B_1 = -\tfrac12$ and $\operatorname{ad}_\Omega(X) = [\Omega, X]$.

## Remarks

The series converges, and gives the solution, when $\int_0^t \norm{A(s)}_2\, ds < \pi$, where $\norm{\cdot}_2$ is the spectral norm; this is a sufficient condition.
Example: if $A(t_1)$ and $A(t_2)$ commute for all $t_1, t_2$, all terms with $k \ge 2$ vanish and $Y(t) = \exp\left(\int_0^t A\right)$; for $A(t) = a(t) N$ with a fixed matrix $N$ and scalar $a$, this gives $Y = e^{(\int_0^t a) N}$.
Without commutation, $\exp\left(\int_0^t A\right)$ is in general not the solution, and $\Omega_2$ is the first correction.
The reason is that the derivative of $\exp(\Omega_1(t))$ need not be $A(t) \exp(\Omega_1(t))$ when $A(t)$ does not commute with $\Omega_1(t)$.
Let $A(t)$ lie in a matrix Lie algebra for all $t$, such as the skew-symmetric matrices.
Then every truncation of the series also lies in it, so its exponential lies in the corresponding group.
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# Backward differentiation formula

## Summary

A backward differentiation formula, or BDF method, is an implicit [[linear multistep method|BX4D85A79P]].
It differentiates the polynomial through the latest computed values and sets its slope at the new point equal to $f$.
The $k$-step formula has order $k$, and it is zero-stable only for $k \le 6$.
Because of their large regions of stability, BDF methods are the standard methods for [[stiff equations|RR5FCG92PN]].
The one-step formula is the [[backward Euler method|M7XKVNATFA]].

## Explanations

### L1 human

Most methods use the slope at known points to guess the next point.
A BDF method works the other way.
It looks for a new point such that the curve through it and the previous points has, at the new point, the slope that the equation demands.
This has to be solved as an equation at each step, which costs work, but it makes the method very robust when parts of the solution decay very fast.
The simplest case is $y_{k+1} = y_k + h f(t_{k+1}, y_{k+1})$.
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# Multiple integral

## Definition

Let $n \ge 1$ and let $B = [a_1, b_1] \times \cdots \times [a_n, b_n]$ be a closed box in $\R^n$ with $a_i < b_i$.
Its volume is $\abs{B} = (b_1 - a_1) \cdots (b_n - a_n)$.
A partition $P$ of $B$ is given by a partition of each interval $[a_i, b_i]$; it cuts $B$ into finitely many subboxes $B_1, \dots, B_N$.
The mesh of $P$ is the largest diameter of the $B_j$.
A tagged partition also chooses a point $\xi_j \in B_j$ for each $j$.
For a function $f \colon B \to \R$, the [[Riemann sum|0JAEHXTVV5]] of a tagged partition is $\sum_{j=1}^N f(\xi_j) \abs{B_j}$.

The function $f$ is *Riemann integrable* on $B$ if there is a real number $I$ with the following property.
For each $\varepsilon > 0$ there is $\delta > 0$ such that every tagged partition of mesh less than $\delta$ has a Riemann sum within $\varepsilon$ of $I$.
Then $I$ is the *multiple integral* of $f$ over $B$, written
$$\int_B f \, dV = \int \cdots \int_B f(x_1, \dots, x_n) \, dx_1 \cdots dx_n.$$

Let $A \subseteq \R^n$ be bounded and let $f \colon A \to \R$.
Let $f_A$ be the function on a closed box $B \supseteq A$ that equals $f$ on $A$ and $0$ on $B \setminus A$.
Then $f$ is integrable on $A$ if $f_A$ is integrable on $B$, and $\int_A f \, dV = \int_B f_A \, dV$.
This does not depend on the choice of $B$.

## Remarks

For $n = 1$ this is the [[Riemann integral|0W29TBTY12]]; for $n = 2$ and $n = 3$ it is the [[double integral|0WAR4F23B4]] and the [[triple integral|0698M7C7AX]].
A bounded set $A$ is [[Jordan measurable|0KTGJW1ZTH]] exactly when the constant $1$ is integrable on $A$, and then $\int_A 1 \, dV$ is the Jordan measure of $A$.

Example: $f(x, y) = xy$ on $B = [0, 1]^2$.
The function is continuous, so it is integrable, and computing it as an iterated integral gives $\int_0^1 \int_0^1 xy \, dx \, dy = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$.

Non-example: let $f$ on $[0, 1]^2$ be $1$ at points with both coordinates rational and $0$ elsewhere.
Every subbox contains points of both kinds, so some tagged partitions of any mesh have Riemann sum $1$ and others have Riemann sum $0$.
No number $I$ satisfies the condition, so $f$ is not Riemann integrable.
</document>

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# Tietze transformations

## Definition

Let $\langle S \mid R \rangle$ be a [[group presentation|0JGEQXQ4AC]], with $R$ a subset of the [[free group|0KTW26C80J]] $F(S)$.
Let $N(R)$ be the [[normal closure|6XM4CCVTKE]] of $R$ in $F(S)$.
The *Tietze transformations* are the following four operations.

1. Add a relation: replace $R$ by $R \cup \set{r}$, where $r \in N(R)$.
2. Remove a relation: replace $R$ by $R \setminus \set{r}$, where $r \in R$ lies in the normal closure of $R \setminus \set{r}$.
3. Add a generator: for a new symbol $x \notin S$ and a word $w \in F(S)$, replace $\langle S \mid R \rangle$ by $\langle S \cup \set{x} \mid R \cup \set{x w^{-1}} \rangle$.
4. Remove a generator: suppose $x \in S$ and $R$ contains a relation $x w^{-1}$, with $w$ a word in $S \setminus \set{x}$. Delete $x$ and this relation, and replace $x$ by $w$ in all other relations.

## Remarks

Each transformation yields a presentation of an [[isomorphic|08MG7H8JW5]] group.
Moves 1 and 2 keep $N(R)$ unchanged, and in moves 3 and 4 the added or removed letter equals an existing element.
Tietze's theorem: two finite presentations define isomorphic groups if and only if one is obtained from the other by a finite sequence of Tietze transformations.
Example: $\langle a, b \mid a b a^{-1} b^{-2},\ b \rangle$ becomes $\langle a \mid a a^{-1} \rangle$ by removing $b$ with $w = 1$, and then $\langle a \mid \ \rangle$ by removing the trivial relation; so it presents $\Z$.
Non-example: replacing $\langle a \mid a^2 \rangle$ by $\langle a \mid a^4 \rangle$ is not a Tietze transformation, since $a^2$ is not in the normal closure of $a^4$; indeed the groups $\Z/2$ and $\Z/4$ differ.
Tietze transformations do not give an algorithm for the [[group isomorphism problem|MP3F49QT3C]], which is undecidable for finite presentations.
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# Peixoto's theorem

## Statement

Let $M$ be a compact, connected, orientable two-dimensional smooth manifold without boundary.
Let $\mathfrak{X}^1(M)$ be the space of $C^1$ [[vector fields|0299VCW1TH]] on $M$ with the $C^1$ topology.
Let $\varphi_t$ be the flow of $X$.
A point $p$ is *nonwandering* for $X$ if for every neighbourhood $U$ of $p$ and every $T > 0$, some $t > T$ has $\varphi_t(U) \cap U \neq \emptyset$.
A *saddle connection* is an orbit that tends to a saddle point both forward and backward in time.

1. A vector field $X \in \mathfrak{X}^1(M)$ is [[structurally stable|D4MGFAHVMT]] if and only if all four conditions below hold.
   (a) $X$ has finitely many zeros, and each of them is a [[hyperbolic equilibrium point|01TPMXM6NM]].
   (b) $X$ has finitely many [[periodic orbits|0AA97XA70K]], and each of them is hyperbolic.
   This means that the derivative of its [[Poincaré map|60TDCPY4Z4]] at a fixed point has no eigenvalue of modulus $1$.
   (c) $X$ has no saddle connection.
   (d) Every nonwandering point of $X$ is a zero of $X$ or lies on a periodic orbit.
2. The set of structurally stable vector fields is open and [[dense|0BSJSX7ME2]] in $\mathfrak{X}^1(M)$.

## Remarks

Vector fields with properties (a) to (d) are the Morse–Smale vector fields on a surface.
Necessity of each condition comes from small perturbations: a non-hyperbolic zero or orbit can be split or removed, and a saddle connection can be broken.
Sufficiency builds a homeomorphism between phase portraits from the finite combinatorial data of zeros, periodic orbits and separatrices.
Density is the harder part; its proof relies on the Poincaré–Bendixson theory, which is special to dimension two.
On the torus, the linear flow $\dot\theta_1 = 1$, $\dot\theta_2 = \alpha$ with $\alpha$ irrational has every point nonwandering and no periodic orbit, so it fails (d) and is not structurally stable.
In dimension three and higher, structurally stable systems need not be Morse–Smale; Anosov flows are an example.

## Explanations

### L1 human

On a closed surface, a flow is robust under small changes exactly when its picture is simple.
Simple means: finitely many rest points and cycles, all of them strongly attracting or repelling in some direction, no orbit joining two saddles, and no complicated recurrence.
The theorem also says that almost every flow on such a surface is of this kind.

### L2 human

The theorem extends the result of Andronov and Pontryagin for flows on a disc to all compact orientable surfaces.
It is the model for the notion of Morse–Smale systems in higher dimensions.
In a typical course it follows the study of [[hyperbolic equilibria|01TPMXM6NM]], [[limit cycles|0AAQ73WRFB]] and the Poincaré–Bendixson theorem.
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# Direct and inverse theorems of approximation

## Summary

Direct and inverse theorems are the two halves of the link between the smoothness of a function and how fast it can be approximated.
A direct theorem, of Jackson type, says that smooth functions have small errors of best approximation.
An inverse theorem, of Bernstein type, says that small errors of best approximation force smoothness.
Together they often characterise a smoothness class exactly by the rate of approximation.

## Explanations

### L1 human

A smooth function is easy to imitate with polynomials of low degree, and a rough one is hard.
Direct theorems turn "smooth" into "the error goes down fast as the degree grows".
Inverse theorems go back the other way.
For periodic functions and $0 < \alpha < 1$, the error of the best trigonometric polynomial of degree $n$ is of order $n^{-\alpha}$ exactly when the function is Hölder continuous with exponent $\alpha$.
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# Abel inversion formula

## Statement

Let $b > 0$, let $0 < \alpha < 1$, and let $\varphi \colon [0, b] \to \C$ be continuous.
Let
$$f(x) = \int_0^x \frac{\varphi(t)}{(x - t)^{\alpha}} \, dt, \quad x \in [0, b].$$
Then the function $F(x) = \int_0^x \frac{f(t)}{(x - t)^{1 - \alpha}} \, dt$ is continuously differentiable on $[0, b]$, and
$$\varphi(x) = \frac{\sin(\pi \alpha)}{\pi} \, F'(x) = \frac{\sin(\pi \alpha)}{\pi} \, \frac{d}{dx} \int_0^x \frac{f(t)}{(x - t)^{1 - \alpha}} \, dt, \quad x \in [0, b].$$
In particular, the [[Volterra equation of the first kind|H6ZGEDM4VV]] $\int_0^x \varphi(t) (x - t)^{-\alpha} \, dt = f(x)$ has at most one continuous solution $\varphi$.

## Remarks

Proof idea: insert the formula for $f$ into $F$ and exchange the order of integration.
This gives $F(x) = \int_0^x \varphi(s) \left( \int_s^x (x - t)^{\alpha - 1} (t - s)^{-\alpha} \, dt \right) ds$.
The substitution $t = s + u(x - s)$ turns the inner integral into the [[Beta function|QHNWVSC6AQ]] value $B(\alpha, 1 - \alpha) = \Gamma(\alpha) \Gamma(1 - \alpha) = \pi / \sin(\pi \alpha)$, which does not depend on $x$ or $s$.
So $F(x) = \frac{\pi}{\sin(\pi \alpha)} \int_0^x \varphi(s) \, ds$, and the fundamental theorem of calculus gives the formula.

Example: for $\alpha = \tfrac12$ the formula reads $\varphi(x) = \frac{1}{\pi} \frac{d}{dx} \int_0^x \frac{f(t)}{\sqrt{x - t}} \, dt$, the inversion of [[Abel's integral equation|G4GWBYTW7B]].
With $f(x) = 2\sqrt{x}$, the inner integral is $2 x B(\tfrac32, \tfrac12) = \pi x$, so $\varphi = 1$; indeed $\int_0^x (x - t)^{-1/2} \, dt = 2\sqrt{x}$.

The statement starts from a solution $\varphi$; it does not say that every $f$ arises.
A given $f$ is of this form for a continuous $\varphi$ only if $F$ is continuously differentiable; for instance $f$ must vanish at $0$.
If $f$ is continuously differentiable with $f(0) = 0$, then the solution can also be written $\varphi(x) = \frac{\sin(\pi \alpha)}{\pi} \int_0^x \frac{f'(t)}{(x - t)^{1 - \alpha}} \, dt$.

## Explanations

### L1 human

Abel's equation hides the unknown $\varphi$ inside an integral with a kernel that blows up at the endpoint.
The inversion formula recovers $\varphi$ by applying a second integral of the same kind, with the complementary exponent $1 - \alpha$, and then differentiating.
The two integrals together amount to one ordinary integration, which the derivative undoes.

### L2 human

In the language of fractional calculus, the equation says $\Gamma(1 - \alpha) \, I^{1 - \alpha} \varphi = f$, where $I^\beta$ is the [[Riemann–Liouville integral|JTKCZ44NN3]] of order $\beta$.
The formula then reads $\varphi = \frac{1}{\Gamma(1 - \alpha)} \frac{d}{dx} I^{\alpha} f$, using $I^\alpha I^{1 - \alpha} = I^1$; the reflection formula $\Gamma(\alpha) \Gamma(1 - \alpha) = \pi / \sin(\pi \alpha)$ for the [[Gamma function|0H5SYBSED9]] shows that this agrees with the statement.
It solves the [[tautochrone problem|A56P5S5D6Z]], and its variant for the Abel transform is the basis of Abel inversion in tomography of radially symmetric objects.
</document>

<document source="https://tamnd.github.io/math2/node/TDQ4YBMW9C.md" tag="TDQ4YBMW9C" kind="definition">
# Extremal length

## Definition

Let $\Omega \subseteq \C$ be an [[open set|01ZQQDCERP]], and let $\Gamma$ be a family of locally [[rectifiable curves|0C28RT5BG7]] in $\Omega$.
A *metric* on $\Omega$ is a [[Borel measurable|0BVHZXWKCD]] function $\rho \colon \Omega \to [0, \infty]$ with area
$$A(\rho) = \int_\Omega \rho(x + iy)^2 \, dx\, dy$$
satisfying $0 < A(\rho) < \infty$.
The $\rho$-length of $\Gamma$ is
$$L_\rho(\Gamma) = \inf_{\gamma \in \Gamma} \int_\gamma \rho \, \abs{dz},$$
the integral being with respect to arc length.
The *extremal length* of $\Gamma$ is
$$\lambda(\Gamma) = \sup_{\rho} \frac{L_\rho(\Gamma)^2}{A(\rho)} \in [0, \infty],$$
the supremum taken over all metrics $\rho$ on $\Omega$.

## Remarks

The value does not change if $\Omega$ is replaced by a larger open set and $\rho$ is extended by $0$.
If $f \colon \Omega \to \Omega'$ is a conformal map, then $\lambda(f(\Gamma)) = \lambda(\Gamma)$: the metric $\rho$ on $\Omega'$ pulls back to $(\rho \circ f)\abs{f'}$ with the same lengths and area.
Example: let $R = (0, a) \times (0, b)$ and let $\Gamma$ be the curves in $R$ that join the left side to the right side.
For $\rho = 1$, $L_\rho = a$ and $A(\rho) = ab$, so $\lambda(\Gamma) \ge a/b$.
For any metric, each horizontal segment gives $\int_0^a \rho(x + iy)\, dx \ge L_\rho$, so by integrating in $y$ and the Cauchy–Schwarz inequality, $b L_\rho \le \int_R \rho \le \sqrt{ab\, A(\rho)}$, and $\lambda(\Gamma) = a/b$.
Non-example: $\rho = 0$ is not a metric, since its area is $0$; it would give the meaningless ratio $0/0$.
If every curve of $\Gamma_1$ contains a curve of $\Gamma_2$, then $\lambda(\Gamma_1) \ge \lambda(\Gamma_2)$.
</document>

<document source="https://tamnd.github.io/math2/node/TDTMMY7RK1.md" tag="TDTMMY7RK1" kind="statement">
# Kaplansky density theorem

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] with [[algebra of bounded operators|BMDG4JQPEH]] $B(H)$.
Let $A$ be a subalgebra of $B(H)$ that is closed under the [[adjoint|6HSJBJ8TTQ]].
Let $B$ be the closure of $A$ in the [[strong operator topology|KHJ2G2R5G7]].
For a set $X \subseteq B(H)$ write $X_1 = \set{T \in X : \|T\| \le 1}$ and $X_{sa}$ for its self-adjoint elements.
Then:

1. $A_1$ is strongly dense in $B_1$;
2. $(A_{sa})_1$ is strongly dense in $(B_{sa})_1$;
3. if $A$ is a C*-algebra that contains $I$, then the [[unitary|MM0EJ2GDE4]] elements of $A$ are strongly dense in the unitary elements of $B$.

## Remarks

One may replace $A$ by its norm closure, which does not change the strong closure of its unit ball, so $A$ may be taken to be a C*-algebra.
The key step uses the function $f(t) = 2t/(1 + t^2)$.
The map $T \mapsto f(T)$ is strongly continuous on self-adjoint operators.
Also, $f$ maps $\R$ into $[-1, 1]$ and restricts to a homeomorphism of $[-1, 1]$ onto itself.
Write an element of $(B_{sa})_1$ as $f(S)$ with $S$ self-adjoint in $B$.
Approximate $S$ strongly by self-adjoint elements $S_i$ of $A$.
Then the elements $f(S_i)$ of $(A_{sa})_1$ tend strongly to $f(S)$.
This gives part 2; part 1 follows by applying part 2 to the self-adjoint $2 \times 2$ operator matrix with entries $0, T, T^*, 0$.

Without the theorem, an element of $B_1$ is a strong limit of a net in $A$, but the norms along the net might be unbounded.
Such nets are hard to use, since multiplication is jointly strongly continuous only on bounded sets.
Kaplansky's theorem says that elements of $B$ of norm at most $1$ are strong limits of elements of $A$ of norm at most $1$.
When $H$ is separable, the strong topology on bounded sets is metrizable, so one can then even use sequences.

## Explanations

### L1 human

When an algebra of operators is completed by taking strong limits, new elements appear.
This theorem says that each new element of size at most one can be reached by elements of the original algebra that also have size at most one.
So no "blow-up" is needed to approximate.

### L2 human

The theorem is used together with the [[bicommutant theorem|BSA6GFT8SQ]].
If $A$ is a unital C*-algebra on $H$, then the unit ball of $A$ is strongly dense in the unit ball of the von Neumann algebra $A''$.
This is how properties such as approximate finite dimensionality pass from a C*-algebra to the von Neumann algebra it generates.
It also shows that the unit ball of $A''$ is the closure of the unit ball of $A$ in the weak operator topology, which is compact.
</document>

<document source="https://tamnd.github.io/math2/node/TDW023HZ1T.md" tag="TDW023HZ1T" kind="definition">
# Invertible ideal

## Definition

Let $R$ be an [[integral domain|0QVD7WBFGN]] with [[field of fractions|0F2PTGX1A2]] $K$.
A *fractional ideal* of $R$ is a nonzero $R$-[[submodule|05WDSN597C]] $I$ of $K$ such that $dI \subseteq R$ for some nonzero $d \in R$.
The product $IJ$ of two fractional ideals is the set of finite sums $\sum a_k b_k$ with $a_k \in I$ and $b_k \in J$.
A fractional ideal $I$ is *invertible* if there is a fractional ideal $J$ with $IJ = R$.
An [[ideal|096Z9MP2ME]] of $R$ is invertible if it is nonzero and invertible as a fractional ideal.

## Remarks

If $I$ is invertible, its inverse is unique and equals $I^{-1} = (R : I) = \set{x \in K : xI \subseteq R}$.
Equivalently, a fractional ideal is invertible exactly when it is a [[projective|BFFA03GE3V]] $R$-module.
For a Noetherian domain, a nonzero ideal is invertible exactly when its localization at every maximal ideal is principal.

Example: let $R = \Z[\sqrt{-5}]$ and $P = (2, 1 + \sqrt{-5})$.
Then $P^2 = (4, 2 + 2\sqrt{-5}, -4 + 2\sqrt{-5})$ contains $6$ and $4$, hence $2$, and all its generators are multiples of $2$, so $P^2 = (2)$.
Thus $J = \frac{1}{2} P$ is a fractional ideal with $PJ = \frac{1}{2}(2) = R$, and $P$ is invertible.

Non-example: let $R = k[x, y]$ for a field $k$ and $\mathfrak{m} = (x, y)$.
If $f \in K$ satisfies $fx \in R$ and $fy \in R$, write $f = a/x = b/y$ with $a, b \in R$; then $ay = bx$, so $x$ divides $a$ and $f \in R$.
Hence $(R : \mathfrak{m}) = R$ and $\mathfrak{m} (R : \mathfrak{m}) = \mathfrak{m} \ne R$, so $\mathfrak{m}$ is not invertible.
</document>

<document source="https://tamnd.github.io/math2/node/TDY6EY60DV.md" tag="TDY6EY60DV" kind="definition">
# Kodaira–Spencer map

## Definition

Let $\varpi \colon \mathcal{X} \to B$ be a complex analytic family of compact complex manifolds, as in a [[deformation of complex structure|VKQSM3BCE4]], and let $t \in B$ with fibre $X_t$.
For a complex manifold $Y$ let $\Theta_Y$ denote the sheaf of holomorphic sections of its holomorphic [[tangent bundle|WY6K3P4YCG]].
Since $\varpi$ is a submersion, there is a short exact sequence of sheaves on $X_t$
$$0 \to \Theta_{X_t} \to \Theta_{\mathcal{X}}|_{X_t} \xrightarrow{d\varpi} T_t B \otimes \mathcal{O}_{X_t} \to 0.$$
The *Kodaira–Spencer map* of the family at $t$ is the connecting homomorphism of its long exact [[cohomology|PQ6Q143ARJ]] sequence,
$$\rho_t \colon T_t B = H^0(X_t, T_t B \otimes \mathcal{O}_{X_t}) \to H^1(X_t, \Theta_{X_t}).$$

## Remarks

The identification $H^0(X_t, T_t B \otimes \mathcal{O}_{X_t}) = T_t B$ uses that $X_t$ is compact and connected, so its global holomorphic functions are constants.
In Čech terms: cover $\mathcal{X}$ near $X_t$ by charts $(z_i, s)$ with $\varpi(z_i, s) = s$, glued by $z_i = f_{ij}(z_j, s)$.
For $v = \partial / \partial s$, the class $\rho_t(v)$ is represented by the vector fields $\sum_k \frac{\partial f_{ij}^k}{\partial s}(z_j, t)\, \frac{\partial}{\partial z_i^k}$ on the overlaps.

Example: for a trivial family $X \times B \to B$, the sequence splits, and $\rho_t = 0$ for all $t$.

Example: for the family $\C/(\Z + \tau\Z)$ over the upper half-plane, $\Theta_{X_\tau} \cong \mathcal{O}_{X_\tau}$, so $H^1(X_\tau, \Theta_{X_\tau})$ is one-dimensional, and $\rho_\tau$ is an isomorphism at every $\tau$.

A family with $\rho_t$ surjective at $t$ is complete at $t$: every small deformation of $X_t$ is pulled back from it, by a theorem of Kodaira and Spencer.
</document>

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# Bessel function of the first kind

## Definition

Let $\nu$ be a complex number and let $\Gamma$ be the [[gamma function|0H5SYBSED9]], with $\frac{1}{\Gamma(w)}$ read as $0$ when $w \in \set{0, -1, -2, \dots}$.
For $z \in \C \setminus (-\infty, 0]$, with the principal branch of $\left( \frac z2 \right)^\nu$, the *Bessel function of the first kind* of order $\nu$ is
$$J_\nu(z) = \left( \frac z2 \right)^{\nu} \sum_{k = 0}^{\infty} \frac{(-1)^k}{k!\, \Gamma(k + \nu + 1)} \left( \frac z2 \right)^{2k}.$$
If $\nu = n$ is an integer, the same formula defines $J_n$ as an entire function of $z$.

## Remarks

The [[power series|0MZB06ST2B]] in $\left( \frac z2 \right)^2$ converges for all $z$, since the ratio of consecutive terms is $\frac{-(z/2)^2}{(k + 1)(k + \nu + 1)} \to 0$.
The function $J_\nu$ solves [[Bessel's equation|EAV97QH9DF]] $z^2 y'' + z y' + (z^2 - \nu^2) y = 0$.
Example: $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$ for $x > 0$.
Check: with $\Gamma(k + \frac32) = \frac{(2k + 1)!\, \sqrt{\pi}}{4^k\, k!\, 2}$, the term $k$ equals $\left( \frac x2 \right)^{1/2} \frac{2 (-1)^k x^{2k}}{\sqrt{\pi}\, (2k + 1)!}$, and summing gives $\sqrt{\frac{2}{\pi x}} \sum_k \frac{(-1)^k x^{2k + 1}}{(2k + 1)!}$.
For a positive integer $n$ and $\nu = -n$, the terms with $k < n$ vanish, and one gets $J_{-n} = (-1)^n J_n$.
So $J_n$ and $J_{-n}$ are not independent, and a second solution is the Bessel function of the second kind.
For integer $n$, the generating function is $\exp\left( \frac z2 \left( t - \frac1t \right) \right) = \sum_{n \in \Z} J_n(z)\, t^n$ for $t \ne 0$.
Non-example: $\cos z$ is not $J_0(z)$, although both equal $1$ at $0$; their series differ already in the $z^2$ term, $-\frac{z^2}{2}$ for $\cos z$ against $-\frac{z^2}{4}$ for $J_0(z)$.
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# Functional completeness

## Summary

A set of operations on a finite set $A$ is functionally complete if every finitary operation on $A$ can be built from it by composition.
In other words, the [[clone|PTKR703G6A]] it generates is the clone of all operations.
For $A = \set{0, 1}$ this is the familiar notion for sets of [[logical connectives|QS0XVAN9BG]], such as $\set{\wedge, \neg}$.
Criteria for functional completeness are given by [[Post's lattice|MA20X7JCET]] in the two-valued case and by maximal clones in general.

## Explanations

### L1 human

A set of logic gates is functionally complete if every truth table can be realised by a circuit built from those gates.
For example, AND and NOT suffice: OR can be written as $x \vee y = \neg(\neg x \wedge \neg y)$, and every truth table has a disjunctive normal form.
AND and OR together do not suffice, because they can never turn an input of all ones into a zero.
</document>

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# Tarski monster group

## Description

Let $p$ be a prime.
A *Tarski monster group* for $p$ is an infinite [[group|G5S1M9]] $G$ in which every [[subgroup|0X1JXT47GA]] other than $\set{1}$ and $G$ is cyclic of order $p$.
Ol'shanskii proved, with [[small cancellation|3A1D37TVF6]] methods, that a Tarski monster group exists for every prime $p > 10^{75}$.

## Remarks

Let $g \ne 1$ in $G$.
If $\langle g \rangle = G$, then $G \cong \Z$, which has the subgroup $2\Z$ of infinite order; this is impossible.
So $\langle g \rangle$ is a proper subgroup other than $\set{1}$, and $g$ has order $p$.
Two elements $x, y$ that do not lie in one subgroup of order $p$ generate $G$, since $\langle x, y \rangle$ is neither trivial nor cyclic of order $p$.
So $G$ is generated by $2$ elements and has [[exponent|0ZDW0H0VX4]] $p$, and it answers the bounded [[Burnside problem|XWEMJ7T2XV]] in the negative for this $p$.

$G$ is [[simple|0YMYMNQ0QZ]].
Let $N \ne \set{1}$ be a [[normal subgroup|0CB0DX5BVV]] with $N \ne G$; then $N$ has order $p$.
Take $x \notin N$; then $\langle x \rangle N$ is a subgroup of order at most $p^2$, so it is proper since $G$ is infinite.
It contains $N$ and $x$, so it is not cyclic of order $p$, a contradiction.

## Explanations

### L1 human

In a Tarski monster, every subgroup except the trivial one and the whole group has exactly $p$ elements.
So there are no medium-sized subgroups at all: any two elements that are not in the same small subgroup generate the whole infinite group.
Groups of this kind were long expected not to exist, which is why they are called monsters.
</document>

<document source="https://tamnd.github.io/math2/node/TE379HBTY0.md" tag="TE379HBTY0" kind="concept">
# Direct method in the calculus of variations

## Summary

The direct method proves that a functional has a minimiser without solving any equation.
One takes a minimising sequence and uses compactness to extract a subsequence that converges in a weak sense.
Lower semicontinuity then shows that the limit is a minimiser.
It is the standard existence method of the [[calculus of variations|PQ25NHV0KS]].
It explains why minimisers are first found in Sobolev spaces with their weak topologies.
It gave the first rigorous justification of Dirichlet's principle.

## Explanations

### L1 human

To show that an energy has a lowest state, take a sequence of states whose energies go down to the lowest possible value.
If the energies stay bounded, the states cannot run away, and some part of the sequence settles down to a limit.
If the energy cannot jump up in the limit, the limit is a state of lowest energy.
The key points are to choose a notion of convergence weak enough for the sequence to converge, yet strong enough for the energy to behave well.
On the real line the function $x \mapsto x^2$ illustrates the steps: any minimising sequence tends to $0$, and $0$ is the minimiser.
</document>

<document source="https://tamnd.github.io/math2/node/TE3STTNSCK.md" tag="TE3STTNSCK" kind="definition">
# Periodic point

## Definition

Let $X$ be a set, let $f \colon X \to X$ be a map, and let $f^n$ denote the $n$-th [[iterate|AYVTR2ES5B]] of $f$.
A point $x \in X$ is a *periodic point* of $f$ if there is an integer $n \ge 1$ with $f^n(x) = x$.
Such an $n$ is a *period* of $x$.
Equivalently, $x$ is a [[fixed point|R6QVS6P21N]] of $f^n$ for some $n \ge 1$.

## Remarks

If $n$ is a period of $x$, so is every positive multiple of $n$, since $f^{kn}(x) = (f^n)^k(x) = x$.
The orbit $\set{x, f(x), \dots, f^{n-1}(x)}$ of a periodic point is a finite set, called a periodic orbit or cycle.
A point $x$ is *eventually periodic*, or preperiodic, if some $f^m(x)$ with $m \ge 0$ is periodic.

Example: for $f(x) = x^2 - 1$ on $\R$, the point $0$ is periodic with period $2$.
Indeed $f(0) = -1$ and $f(-1) = 0$, so $f^2(0) = 0$.

Non-example: for the same $f$, the point $1$ is not periodic.
One has $f(1) = 0$, after which the orbit alternates between $-1$ and $0$ and never returns to $1$.
So $1$ is eventually periodic but not periodic.
</document>

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# Hirzebruch signature theorem

## Statement

Let $M$ be a closed oriented [[smooth manifold|F45VQ47GPV]] of dimension $4k$ with fundamental class $[M]$.
Let $\sigma(M)$ be the [[signature|07EZA80NEH]] of its [[intersection form|WK8GKRKAVD]] on $H^{2k}(M; \R)$, the symmetric bilinear form $(a, b) \mapsto \langle a \smile b, [M] \rangle$.
Let $p_i = p_i(TM) \in H^{4i}(M; \Q)$ be the [[Pontryagin classes|DN11FKWN63]] of the tangent bundle.
Let $L_k(p_1, \ldots, p_k)$ be the degree $4k$ part of the multiplicative sequence of Hirzebruch with characteristic power series $\sqrt{z} / \tanh \sqrt{z}$.
Then
$$\sigma(M) = \langle L_k(p_1, \ldots, p_k), [M] \rangle.$$

## Remarks

The first two polynomials are $L_1 = \tfrac{1}{3} p_1$ and $L_2 = \tfrac{1}{45}(7 p_2 - p_1^2)$.
Idea of the proof: both sides are invariants of oriented [[cobordism|T8BVGPPSZP]], additive under disjoint union and multiplicative under products.
So both sides are ring homomorphisms from the rational oriented cobordism ring to $\Q$.
Thom showed that this ring is a polynomial ring on the classes of the complex projective spaces $\C P^{2i}$.
The power series is chosen so that the right side equals $1 = \sigma(\C P^{2i})$ on each generator, and so the two homomorphisms agree.

Example: for $M = \C P^2$ with generator $h \in H^2$, the total Pontryagin class is $(1 + h^2)^3$, so $p_1 = 3h^2$ and $\langle L_1, [M] \rangle = 1$.
The intersection form is $(1)$, so $\sigma(\C P^2) = 1$, as the theorem says.
For a closed oriented smooth $4$-manifold the theorem gives $p_1[M] = 3 \sigma(M)$.

## Explanations

### L1 human

The signature is a whole number read off from how middle-dimensional cycles in a manifold intersect.
Pontryagin classes measure how the tangent bundle is twisted.
The theorem says that one fixed combination of the Pontryagin numbers always equals the signature.
So a number defined by pure topology is computed by curvature-type data.

### L2 human

The theorem shows that a certain rational combination of Pontryagin numbers is always an integer, which constrains which manifolds can exist.
Milnor used the case $k = 2$ to show that some $7$-manifolds homeomorphic to $S^7$ are not diffeomorphic to it, the first [[exotic spheres|6WYEMXSHD9]].
The theorem is a special case of the Atiyah–Singer index theorem applied to the [[signature operator|4NCYA7GME1]].
It is also the model for the [[Hirzebruch–Riemann–Roch theorem|G3X02FKQCR]].
In surgery theory, the signature formula is what makes the simply connected surgery obstruction in dimensions $4k$ computable.
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# Hopf invariant

## Summary

The Hopf invariant is an integer $H(f)$ attached to a map $f \colon S^{2n-1} \to S^n$.
It is read off from the [[cup product|JX47MS1R1H]] in the cohomology of the space obtained by attaching a $2n$-cell to $S^n$ along $f$.
It depends only on the homotopy class of $f$ and gives a homomorphism $\pi_{2n-1}(S^n) \to \Z$.
It showed that $\pi_3(S^2)$ is infinite, and the question of which $n$ admit maps of Hopf invariant one is tied to division algebras.

## Explanations

### L1 human

The Hopf invariant measures how much a map from a big sphere to a smaller sphere is twisted.
For the Hopf map $S^3 \to S^2$, the preimages of two different points are two circles that are linked once, and the Hopf invariant is the linking number, $\pm 1$.
For a constant map nothing is linked, and the invariant is $0$.
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# Lagrange polynomial

## Definition

Let $n \ge 0$, let $x_0, x_1, \dots, x_n$ be pairwise distinct real numbers, and let $y_0, y_1, \dots, y_n$ be real numbers.
For $j = 0, \dots, n$, the *Lagrange basis polynomial* is
$$\ell_j(x) = \prod_{\substack{0 \le m \le n \\ m \ne j}} \frac{x - x_m}{x_j - x_m}.$$
The *Lagrange polynomial* of the data $(x_j, y_j)$ is
$$L(x) = \sum_{j=0}^{n} y_j \, \ell_j(x).$$

## Remarks

Each $\ell_j$ has [[degree|0DPF6WRZW6]] $n$, with $\ell_j(x_j) = 1$ and $\ell_j(x_k) = 0$ for $k \ne j$.
So $L$ lies in the [[polynomial ring|0Q9EZS90GY]] $\R[x]$, has degree at most $n$, and satisfies $L(x_k) = y_k$ for all $k$.
It is the unique polynomial with these properties: the difference of two of them has degree at most $n$ and $n + 1$ roots, so it is $0$.
The same formula works over any field; this solves [[polynomial interpolation|SS1YMS2NPC]].

Example: nodes $0, 1, 2$ with values $1, 3, 7$.
Here $\ell_0 = \frac{(x - 1)(x - 2)}{2}$, $\ell_1 = -x(x - 2)$ and $\ell_2 = \frac{x(x - 1)}{2}$.
Then $L = \ell_0 + 3 \ell_1 + 7 \ell_2 = x^2 + x + 1$, and indeed $L(0) = 1$, $L(1) = 3$, $L(2) = 7$.

Non-example: if two nodes coincide, say $x_0 = x_1$, the factor $\frac{x - x_1}{x_0 - x_1}$ in $\ell_0$ has denominator $0$, so the formula is not defined.
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# Resolvent

## Definition

Let $X$ be a complex [[Banach space|06GZ956657]] with identity map $I$, and let $T \colon X \to X$ be a [[continuous linear map|03TM135G0N]] with [[spectrum|BMKATJ802A]] $\sigma(T)$.
The *resolvent set* of $T$ is $\rho(T) = \C \setminus \sigma(T)$.
The *resolvent* of $T$ is the function
$$R(\cdot, T) \colon \rho(T) \to \mathcal{B}(X), \qquad R(\lambda, T) = (T - \lambda I)^{-1},$$
where $\mathcal{B}(X)$ is the space of bounded operators on $X$ and the inverse is the [[bounded inverse|CPAD5W1CGW]].

## Remarks

Some authors use $(\lambda I - T)^{-1}$ instead, which changes the sign.
The resolvent set is open, and $R(\cdot, T)$ is a [[holomorphic function|0NAHXR1FRH]] with values in $\mathcal{B}(X)$.
It satisfies the *resolvent identity*
$$R(\lambda, T) - R(\mu, T) = (\lambda - \mu) R(\lambda, T) R(\mu, T) \qquad (\lambda, \mu \in \rho(T)),$$
and $\norm{R(\lambda, T)} \ge 1 / \operatorname{dist}(\lambda, \sigma(T))$.

Example: for $D(x_n) = (x_n / n)$ on $\ell^2$ and $\lambda \in \rho(D)$, $R(\lambda, D)$ is the diagonal operator with entries $1/(1/n - \lambda)$.

Non-example: $\lambda = 0$ is not in $\rho(D)$, since $D$ is not surjective.
Indeed $(1/n)$ is in $\ell^2$ but $(1)$ is not, so $(1/n)$ is not in the image of $D$.
Hence $R(0, D)$ is not defined.

The same definitions apply to closed operators, with $R(\lambda, T) \colon X \to D(T)$.
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<document source="https://tamnd.github.io/math2/node/TEV7XV14F5.md" tag="TEV7XV14F5" kind="definition">
# Finite ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]].
The ring $R$ is a *finite ring* if its underlying set is finite.
The number of elements of $R$ is the *order* of $R$, written $\abs{R}$.

## Remarks

For each integer $n \ge 1$, the ring $\Z/n\Z$ of [[integers modulo n|03VFZ2WETT]] is a finite ring of order $n$, since there are exactly $n$ residue classes.
The ring $M_2(\F_2)$ of $2 \times 2$ matrices over the [[finite field|0ZXZ82FTX2]] $\F_2$ is a finite ring of order $2^4 = 16$, and it is not commutative.
The ring $\Z$ of integers is not a finite ring, because it has infinitely many elements.
Every finite [[integral domain|0QVD7WBFGN]] is a field: multiplication by a nonzero element is injective, hence bijective on a finite set, so every nonzero element has an inverse.
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<document source="https://tamnd.github.io/math2/node/TEWH75TX8Y.md" tag="TEWH75TX8Y" kind="definition">
# Aleph number

## Definition

Identify each [[cardinal number|0QKPNZKNEA]] of a well-orderable set with the least [[ordinal|0RDJK7EQYY]] of that cardinality.
For an infinite cardinal $\kappa$, let $\kappa^+$ be the least cardinal greater than $\kappa$.
The aleph numbers $\aleph_\alpha$, for ordinals $\alpha$, are defined by [[transfinite recursion|0ZHQMQ2C1N]]:

1. $\aleph_0 = \omega$, the set of natural numbers.
2. $\aleph_{\alpha + 1} = (\aleph_\alpha)^+$.
3. $\aleph_\lambda = \sup_{\beta < \lambda} \aleph_\beta$ for each nonzero limit ordinal $\lambda$, the [[supremum|0BTMQ04556]] of the earlier alephs.

## Remarks

Example: $\aleph_1$ is the set of countable ordinals, which is the least uncountable ordinal $\omega_1$.
Check: each countable ordinal is smaller than $\omega_1$, and $\omega_1$ itself is uncountable, so it is the least cardinal above $\aleph_0$.
Non-example: $\omega + 1$ is an infinite ordinal but not an aleph, since it has the same cardinality as the smaller ordinal $\omega$.
The cardinal $\kappa^+$ exists in ZF by Hartogs' theorem, without the axiom of choice.
The map $\alpha \mapsto \aleph_\alpha$ is a normal function, and every infinite well-orderable cardinal is $\aleph_\alpha$ for exactly one $\alpha$.
The ordinal $\aleph_\alpha$ is also written $\omega_\alpha$ when it is viewed as an ordinal.
</document>

<document source="https://tamnd.github.io/math2/node/TF8HMVXKYC.md" tag="TF8HMVXKYC" kind="concept">
# Descriptive set theory

## Summary

Descriptive set theory studies the definable subsets of [[Polish spaces|584GXXY73M]] such as the real line, the Cantor space and the [[Baire space|P2WS16HKD7]].
It sorts sets by how complicated their definitions are, in the [[Borel hierarchy|4P4Z4ZGXCY]] and the [[projective hierarchy|8DZTNVTGY2]].
Its classical results show that simply defined sets, such as Borel and [[analytic sets|3RWWZYPAFV]], behave well.
For example, they are Lebesgue measurable, and they are either countable or contain a perfect set.
For more complex sets these questions depend on axioms beyond ZFC, such as [[determinacy|QXECDPAYTN]] and [[large cardinals|X1FG7PSCTY]].

## Explanations

### L1 human

Some sets of real numbers are easy to describe, like intervals or the set of rational numbers.
Others are only given by long chains of operations, or not by any explicit rule at all.
Descriptive set theory measures how hard a set is to describe, and shows that easy sets behave well.
For example, an uncountable closed set of reals always contains a copy of the Cantor set, so it has the size of the continuum.
</document>

<document source="https://tamnd.github.io/math2/node/TFADSC9YPZ.md" tag="TFADSC9YPZ" kind="concept">
# Exact category

## Summary

An exact category, in the sense of Quillen, is an [[additive category|B2VP96NJSK]] with a chosen class of kernel–cokernel pairs.
These pairs behave like [[short exact sequences|KJKNJMGKZZ]].
It allows homological algebra and algebraic K-theory in categories that are not abelian, such as the category of finitely generated projective modules.
It is a different notion from an [[exact category in the sense of Barr|G9ZNDNT3YY]].

## Explanations

### L1 human

In an abelian category every monomorphism has a cokernel and fits into a short exact sequence.
Many useful categories, such as vector bundles or projective modules, are additive but not abelian.
An exact category keeps only a chosen family of sequences $A \to B \to C$, declared to be the short exact ones, and asks that this family behave well under composition, pullback and pushout.
For example, in the category of finitely generated projective modules one can choose the split exact sequences.
</document>

<document source="https://tamnd.github.io/math2/node/TFB4BYRZ1W.md" tag="TFB4BYRZ1W" kind="definition">
# Uniform approximation

## Definition

Let $S$ be a set, let $f \colon S \to \R$ be a [[function|0EPAT51DXQ]], and let $A$ be a set of functions $S \to \R$.
For $\varepsilon > 0$, a function $g \in A$ *approximates $f$ uniformly within $\varepsilon$* if
$$\sup_{x \in S} \abs{f(x) - g(x)} \le \varepsilon.$$
The function $f$ *can be uniformly approximated by $A$* if such a $g \in A$ exists for every $\varepsilon > 0$.
Equivalently, there is a sequence $g_n \in A$ that [[converges uniformly|08FPABDQ4C]] to $f$ on $S$.

## Remarks

Example: on $S = [0, 1]$, the function $e^x$ can be uniformly approximated by polynomials.
By Taylor's theorem, $\abs{e^x - \sum_{k=0}^{n} x^k / k!} \le e / (n + 1)!$ for $x \in [0, 1]$, and the bound tends to $0$.

Non-example: on $S = \R$, the function $e^x$ cannot be uniformly approximated by polynomials.
For each polynomial $p$, the difference $e^x - p(x)$ tends to $\infty$ as $x \to \infty$, so the supremum is infinite.

For bounded functions, the quantity $\sup_S \abs{f - g}$ is the distance in the supremum norm, and uniform approximation is [[function approximation|VNWH2KADN7]] in that norm.
Complex-valued functions are treated in the same way.
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<document source="https://tamnd.github.io/math2/node/TFCWC3N41W.md" tag="TFCWC3N41W" kind="definition">
# Pivotal quantity

## Definition

Let $X$ be a random sample whose law $P_\theta$ depends on an unknown parameter $\theta$ in a set $\Theta$.
A *pivotal quantity* is a function $g(X, \theta)$ of the sample and the parameter with the following property.
The [[law|0QB913WM5E]] of the [[random variable|0DKRR22AEQ]] $g(X, \theta)$ under $P_\theta$ is the same for every $\theta \in \Theta$.

## Remarks

Example: let $X_1, \dots, X_n$ be i.i.d. $\mathcal{N}(\mu, \sigma^2)$ with $\theta = (\mu, \sigma^2)$ unknown, let $\bar X$ be the sample mean and $S$ the sample standard deviation.
Then $T = \sqrt{n}(\bar X - \mu)/S$ has Student's t-distribution with $n - 1$ degrees of freedom for every $\theta$, so $T$ is a pivotal quantity.
Non-example: in the same model, $\sqrt{n}(\bar X - \mu)$ is not pivotal, since its law $\mathcal{N}(0, \sigma^2)$ depends on $\sigma^2$.
A pivotal quantity need not be a statistic, since it may involve $\theta$.
If $P_\theta(a \le g(X, \theta) \le b) = 1 - \alpha$ for every $\theta$, then the set $\set{\theta : a \le g(X, \theta) \le b}$ is a [[confidence region|0K4NMMJBG2]] of level $1 - \alpha$.
</document>

<document source="https://tamnd.github.io/math2/node/TFQSF9RA1G.md" tag="TFQSF9RA1G" kind="concept">
# Lie sphere geometry

## Summary

Lie sphere geometry studies oriented spheres, oriented hyperplanes and points of $\R^n$ on an equal footing.
Its transformations are those that preserve oriented contact.
These objects correspond to the points of a [[quadric|Q3Q4FF23VG]] in projective space $\mathbb{RP}^{n+2}$, the Lie quadric, given by a quadratic form of signature $(n+1, 2)$.
Oriented contact becomes orthogonality for that form, and the Lie sphere transformations are the projective maps preserving the quadric.
It contains the Möbius geometry of [[inversive geometry|HFBW3SXQVM]] and Laguerre geometry as subgeometries, and it is named after Sophus Lie.

## Explanations

### L1 human

In ordinary geometry, points, lines and circles are different kinds of objects.
Lie sphere geometry treats them all as circles: a point is a circle of radius zero and a line is a circle of infinite radius.
Give each circle an orientation, a direction of travel, and call two oriented circles in contact when they touch with matching directions at the touching point.
The geometry keeps track only of which circles touch which; its transformations may turn points into circles and circles into lines, as long as touching is preserved.
</document>

<document source="https://tamnd.github.io/math2/node/TFY9CD3019.md" tag="TFY9CD3019" kind="definition">
# Capacity of a set

## Definition

Let $n \ge 3$ and let $K \subseteq \R^n$ be a [[compact set|0R6EWFKRCF]].
Let $C_c^\infty(\R^n)$ be the set of smooth real functions on $\R^n$ with compact support.
The *Newtonian capacity* of $K$ is
$$\operatorname{cap}(K) = \inf \set{\int_{\R^n} \abs{\nabla u}^2 \, dx : u \in C_c^\infty(\R^n), \ u \ge 1 \text{ on } K}.$$

## Remarks

The infimum equals $\int_{\R^n \setminus K} \abs{\nabla u}^2 \, dx$ for the equilibrium potential $u$ of $K$.
This function is harmonic off $K$, equal to $1$ on $K$ when $K$ is regular, and tends to $0$ at infinity.
Some authors multiply by a constant such as $1 / ((n-2) \sigma_{n-1})$, where $\sigma_{n-1}$ is the area of the unit sphere.
Example: for the closed ball $\bar B_r$ of radius $r$, the equilibrium potential is $u(x) = (r / \abs{x})^{n-2}$ for $\abs{x} \ge r$.
Then $\abs{\nabla u} = (n-2) r^{n-2} \abs{x}^{1-n}$, and integration in polar coordinates gives $\operatorname{cap}(\bar B_r) = (n-2) \sigma_{n-1} r^{n-2}$.
For $n = 3$ this is $4 \pi r$.
Non-example of an admissible function: $u = 1/2$ near $K$ does not satisfy $u \ge 1$ on $K$, so it does not enter the infimum.
Capacity is monotone, so a point has capacity at most $(n-2) \sigma_{n-1} r^{n-2}$ for every $r > 0$, hence capacity $0$.
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<document source="https://tamnd.github.io/math2/node/TFYXH8QNR2.md" tag="TFYXH8QNR2" kind="definition">
# Banach algebra

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$.
A *Banach algebra* over $\mathbb{K}$ is a [[normed algebra|7VS7JZ3Y5B]] $(A, \norm{\cdot})$ over $\mathbb{K}$ such that the normed space $(A, \norm{\cdot})$ is complete, that is, a [[Banach space|0BDTTM24DP]].
So $A$ is an associative $\mathbb{K}$-algebra with a complete norm such that $\norm{xy} \le \norm{x}\norm{y}$ for all $x, y \in A$.
A Banach algebra is *unital* if it has a unit $1$, and then one usually also asks $\norm{1} = 1$.

## Remarks

Example: for a compact Hausdorff space $X$, the algebra $C(X)$ of continuous functions $X \to \C$ with $\norm{f} = \sup_{x \in X} \abs{f(x)}$.
Submultiplicativity holds since $\abs{f(x) g(x)} \le \norm{f}\norm{g}$ for every $x$, and completeness holds because a uniform limit of continuous functions is continuous.
Example: the bounded linear operators on a Banach space with the operator norm.
Non-example: the polynomial functions on $[0, 1]$ with the same sup norm form a normed algebra but not a Banach algebra.
They are dense in $C([0, 1])$ by the Weierstrass approximation theorem, and $e^x$ is a uniform limit of polynomials that is not a polynomial, so completeness fails.
In a unital Banach algebra, $1 - x$ is invertible with inverse $\sum_{n \ge 0} x^n$ when $\norm{x} < 1$.
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<document source="https://tamnd.github.io/math2/node/TG6WHHR7CF.md" tag="TG6WHHR7CF" kind="concept">
# Geometric realization

## Summary

The geometric realization $\abs{K}$ of a [[simplicial set|WW81ARB8K9]] $K$ is a topological space.
It has one topological $n$-simplex for each $n$-simplex of $K$, glued as the face and degeneracy maps prescribe.
It is a CW complex with one $n$-cell for each nondegenerate $n$-simplex.
Realization is left adjoint to the [[singular simplicial set|RWBMXQW6S5]] functor, and the two functors identify the homotopy theory of simplicial sets with that of spaces.

## Explanations

### L1 human

A simplicial set is a list of points, edges, triangles and higher simplices with instructions for how they meet.
The realization builds the actual shape: it takes a real triangle for each listed triangle, a real segment for each edge, and glues them together.
Degenerate simplices, which are collapsed copies of lower ones, are squashed and add nothing new.
For example, one vertex and one nondegenerate edge from that vertex to itself realize to a circle.
</document>

<document source="https://tamnd.github.io/math2/node/TGKPC4GV7R.md" tag="TGKPC4GV7R" kind="definition">
# ℓ-adic cohomology

## Definition

Let $k$ be a separably closed field and let $X$ be an [[algebraic variety|HV6YBS0YDH]] over $k$.
Let $\ell$ be a prime number different from the characteristic of $k$.
For $i \ge 0$, the *$\ell$-adic cohomology groups* of $X$ are
$$H^i(X, \Z_\ell) = \varprojlim_n H^i_{\text{ét}}(X, \Z/\ell^n\Z), \qquad H^i(X, \Q_\ell) = H^i(X, \Z_\ell) \otimes_{\Z_\ell} \Q_\ell.$$
Here $H^i_{\text{ét}}$ is [[étale cohomology|7YX76G0838]].
The [[inverse limit|DZB378CNCK]] is taken along the maps induced by the reductions $\Z/\ell^{n+1}\Z \to \Z/\ell^n\Z$.
$\Z_\ell$ is the ring of [[$\ell$-adic integers|GBN9F8FAQ3]], and $\Q_\ell$ is the field of [[$\ell$-adic numbers|CHT5JYM6QW]].

## Remarks

Example: let $k$ be algebraically closed and $X = \mathbb{P}^1_k$.
Then $H^0_{\text{ét}}(X, \Z/\ell^n\Z) = \Z/\ell^n\Z$, $H^1_{\text{ét}} = 0$ and $H^2_{\text{ét}} \cong \Z/\ell^n\Z$, compatibly in $n$.
So $H^i(X, \Q_\ell)$ is $\Q_\ell$ for $i = 0, 2$ and $0$ otherwise, as for the sphere $\mathbb{P}^1(\C)$.
Non-example: the condition $\ell \ne \operatorname{char} k$ is needed.
Let $k$ have characteristic $p > 0$ and let $E$ be an elliptic curve over $k$.
Then $H^1(E, \Q_p)$ has dimension $1$ if $E$ is ordinary and $0$ if $E$ is supersingular, not the expected dimension $2$.
For $X$ proper over $k$, the groups $H^i(X, \Q_\ell)$ are finite-dimensional and vanish for $i > 2 \dim X$.
</document>

<document source="https://tamnd.github.io/math2/node/TGN01264C4.md" tag="TGN01264C4" kind="statement">
# Chevalley's theorem on constructible sets

## Statement

Let $Y$ be a [[Noetherian scheme|ND6XEY68P4]] and let $f \colon X \to Y$ be a [[morphism|JXRKH6P1Z4]] [[of finite type|AHTE5DVX61]].
A subset of a Noetherian scheme is *constructible* if it is a finite union of sets $U \cap Z$.
Here each $U$ is open and each $Z$ is closed in the [[Zariski topology|8SQK7RTB63]].
Then $f(C)$ is a constructible subset of $Y$ for every constructible subset $C$ of $X$.
In particular, the image $f(X)$ is constructible.

## Remarks

The proof reduces to the case where $Y = \operatorname{Spec} A$ is affine and integral and $X$ is affine.
The key step shows that if $f(X)$ is dense in $Y$, then $f(X)$ contains a nonempty open subset of $Y$.
This follows from generic freeness or from Noether normalization over the fraction field of $A$.
Noetherian induction on the closed complement then finishes the proof.
For example, over a field $k$, the morphism $\mathbb{A}^2_k \to \mathbb{A}^2_k$ given by $(x, y) \mapsto (x, xy)$ has image $\set{u \ne 0} \cup \set{(0, 0)}$.
This image is constructible but neither open nor closed, so constructible sets are the right class for images.
Without the finite type hypothesis the image can fail to be constructible.
For instance, $\operatorname{Spec} \Q \to \operatorname{Spec} \Z$ has image the generic point alone, which is not constructible.

## Explanations

### L1 human

The image of an algebraic map need not be open or closed.
Chevalley's theorem says that it is never much worse than that.
It is always built from finitely many pieces, each the part of a closed set that lies in an open set.
In the language of logic, the projection of a set defined by polynomial equations and inequations is again of that form.

### L2 human

Over an algebraically closed field, the theorem is the geometric form of quantifier elimination for algebraically closed fields.
It is used to show that properties of the fibres of a morphism hold on constructible sets of the base.
Together with generic flatness, it gives the usual stratifications of the base of a family into finitely many pieces over which the family is well behaved.
A general version holds for morphisms of finite presentation between arbitrary schemes, with a suitable definition of constructible sets.
</document>

<document source="https://tamnd.github.io/math2/node/TGNMG87Z6T.md" tag="TGNMG87Z6T" kind="definition">
# Join-prime element

## Definition

Let $(L, \vee, \wedge)$ be a [[lattice|WH4SFAACAS]] with order $\le$.
An element $a \in L$ is *join-prime* if:

1. $a$ is not a [[least element|0AANKKNHS4]] of $L$;
2. for all $b, c \in L$, if $a \le b \vee c$, then $a \le b$ or $a \le c$.

## Remarks

Every join-prime element is [[join-irreducible|ZHRQ0AT8PM]]: if $a = b \vee c$, then $a \le b$ or $a \le c$, and since $b, c \le a$ this gives $a = b$ or $a = c$.
In a [[distributive lattice|0HM0HQCK6N]] the converse holds: if $a$ is join-irreducible and $a \le b \vee c$, then $a = a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)$, so $a = a \wedge b$ or $a = a \wedge c$.
Example: in the power set of a set, a singleton $\set{x}$ is join-prime, since $x \in B \cup C$ implies $x \in B$ or $x \in C$.
Non-example: in the diamond lattice $M_3 = \set{0, a, b, c, 1}$, where $a$, $b$, $c$ are pairwise incomparable and the join of any two of them is $1$, the element $a$ is join-irreducible.
It is not join-prime, because $a \le b \vee c = 1$ while $a \not\le b$ and $a \not\le c$.
Meet-prime elements are defined dually.
</document>

<document source="https://tamnd.github.io/math2/node/TGW5NTV3AA.md" tag="TGW5NTV3AA" kind="definition">
# Down-closure

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S \subseteq P$.
The *down-closure* of $S$ is the set
$$\downarrow S = \set{x \in P : x \le s \text{ for some } s \in S}.$$
For $x \in P$, the set $\downarrow x = \downarrow \set{x} = \set{y \in P : y \le x}$ is the *principal down-set* of $x$.

## Remarks

In the subsets of $\set{1, 2, 3}$ ordered by inclusion, $\downarrow \set{\set{1, 2}, \set{3}} = \set{\emptyset, \set{1}, \set{2}, \set{1, 2}, \set{3}}$.
Each listed set is contained in $\set{1, 2}$ or in $\set{3}$, and no other subset is.
The set $\downarrow S$ is a [[lower set|BWYFKCWRW5]] that contains $S$, and it lies inside each lower set that contains $S$.
So $S$ is a lower set if and only if $\downarrow S = S$.
The map $S \mapsto \downarrow S$ is a [[closure operator|17FVAN6HXX]] on the subsets of $P$.
Dually, the up-closure is $\uparrow S = \set{x \in P : s \le x \text{ for some } s \in S}$.

## Explanations

### L1 human

The down-closure fills in everything below the given elements.
It turns any subset into a lower set in the most economical way.
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<document source="https://tamnd.github.io/math2/node/TGWM8G5R0C.md" tag="TGWM8G5R0C" kind="problem">
# Newton's minimal resistance problem

## Problem

A solid body moves with constant speed along its axis through a rarefied medium.
The particles of the medium do not interact with each other and bounce off the body elastically.
Among convex bodies of revolution with base radius $R > 0$ and height $h > 0$, which shape has the least resistance?

In Newton's model, the body is $\set{(x, z) \in \R^2 \times \R : \abs{x} \le R, \ 0 \le z \le u(\abs{x})}$.
Its profile $u \colon [0, R] \to [0, h]$ is [[concave|07Q7CNM5WF]] and nonincreasing.
The resistance is proportional to the [[integral functional|JRQZ803Q5Y]]
$$F(u) = \int_0^R \frac{r}{1 + u'(r)^2} \, dr.$$
The problem is to minimise $F$ over such profiles.

## Remarks

Newton stated the answer in the Principia.
The optimal profile is not pointed: it has a flat disc of positive radius at the front, followed by a curved part that meets the disc at a nonzero angle.
The convexity condition matters.
Without it, profiles with steep zigzags make $F(u)$ as small as one likes.
But then particles would hit the body more than once, and the model no longer describes the resistance.
Later work showed that convex bodies without rotational symmetry can have smaller resistance than Newton's body of the same radius and height.

## Explanations

### L1 human

Newton asked how to shape the nose of a body so that it pushes through a thin gas as easily as possible.
His answer is surprising: the best nose is blunt, with a flat front, and not a sharp point.
It was one of the first problems in which a shape was found by minimising an integral.
</document>

<document source="https://tamnd.github.io/math2/node/TGZATGBHTH.md" tag="TGZATGBHTH" kind="definition">
# Block design

## Definition

Let $t, v, k, \lambda$ be positive integers with $t \le k < v$.
A $t$-$(v, k, \lambda)$ design is a pair $(X, \mathcal{B})$ such that:

1. $X$ is a [[finite set|0BBTH3YSZA]] with $v$ elements, called points.
2. $\mathcal{B}$ is a finite [[family|05Q6G4EH2G]] of subsets of $X$, called blocks, each with exactly $k$ elements.
3. Each subset of $X$ with exactly $t$ elements is contained in exactly $\lambda$ blocks.

A balanced incomplete block design, or $2$-design, is a $2$-$(v, k, \lambda)$ design.

## Remarks

The family $\mathcal{B}$ may contain a block more than once.
The design is simple if it has no repeated block.
In a $2$-$(v, k, \lambda)$ design with $b$ blocks, each point lies in the same number $r$ of blocks, and $bk = vr$ and $r(k - 1) = \lambda (v - 1)$.
Example: let $X = \set{1, \dots, 7}$ and let the blocks be $\set{1,2,3}$, $\set{1,4,5}$, $\set{1,6,7}$, $\set{2,4,6}$, $\set{2,5,7}$, $\set{3,4,7}$, $\set{3,5,6}$.
Each of the $21$ pairs of points lies in exactly one of these blocks, so this is a $2$-$(7, 3, 1)$ design, the [[Fano plane|QQ8GW1ASZM]].
Here each point lies in $r = 3$ blocks, and $b = 7$.
Non-example: on $X = \set{1, 2, 3, 4}$ the blocks $\set{1,2,3}$ and $\set{1,2,4}$ do not form a $2$-design, because the pair $\set{1,2}$ lies in two blocks and the pair $\set{3,4}$ lies in none.
A $t$-$(v, k, 1)$ design is a [[Steiner system|7BPSC4T4M3]] $S(t, k, v)$.
</document>

<document source="https://tamnd.github.io/math2/node/TH2XQGGFWH.md" tag="TH2XQGGFWH" kind="concept">
# Residually small variety

## Summary

A [[variety|AJNZJD8442]] is residually small when the sizes of its [[subdirectly irreducible|DASF939439]] members are bounded by a cardinal.
By Birkhoff's subdirect representation theorem, every algebra of a variety is a subdirect product of subdirectly irreducible members, so these members control the whole variety.
A residually small variety has, up to isomorphism, only a set of subdirectly irreducible members; otherwise it is called residually large.
Residual smallness is related to the existence of enough injective algebras in a variety.

## Explanations

### L1 human

Every algebra of a variety can be broken down into irreducible pieces that cannot be split further.
In a residually small variety these pieces never get too big: there is a fixed size that none of them reaches.
Abelian groups form such a variety: their irreducible pieces are the cyclic groups $\Z/p^n\Z$ and the Prüfer groups $\Z(p^\infty)$, all countable.
The variety of all groups is not residually small, because there are simple groups of every infinite size.
</document>

<document source="https://tamnd.github.io/math2/node/TH4NGC4016.md" tag="TH4NGC4016" kind="concept">
# Integro-partial differential equation

## Summary

An integro-partial differential equation is an equation for an unknown function of several variables.
It involves both partial derivatives of the function and integrals of it.
It combines a [[partial differential equation|GC5BDZ6QBC]] with integral terms, which typically describe nonlocal interactions in space or memory in time.
Main examples are the [[Boltzmann equation|MH6CSWD7F0]] of gas dynamics, the [[radiative transfer equation|EX9YYRMAJA]], [[population balance equations|6Q7H2FQ12Q]] and nonlocal diffusion equations.
Their analysis mixes PDE methods with the theory of integral operators.

## Explanations

### L1 human

In the heat equation, heat at a point flows only to nearby points, which is a statement about derivatives.
In a nonlocal model, heat can jump from a point to any other point, with a rate that depends on the distance.
Then the change in time is given by an integral over all other points, and the equation contains both a time derivative and a space integral.
For example, $\partial_t u(x, t) = \int_{\R} J(x - y) \big(u(y, t) - u(x, t)\big) \, dy$ is such a nonlocal diffusion equation.
</document>

<document source="https://tamnd.github.io/math2/node/TH54SPVN1N.md" tag="TH54SPVN1N" kind="statement">
# Levenberg–Marquardt algorithm

## Statement

Let $D \subseteq \R^n$ be open, let $r \colon D \to \R^m$ be continuously differentiable with [[Jacobian matrix|047NDTEPCV]] $J(x)$, and let $S(x) = \tfrac12 \norm{r(x)}_2^2$.
For $x \in D$ and a damping parameter $\lambda > 0$, the *Levenberg–Marquardt step* $\delta = \delta(\lambda)$ is the solution of
$$\bigl( J(x)^T J(x) + \lambda I \bigr) \delta = -J(x)^T r(x).$$
Write $J = J(x)$, $r = r(x)$ and $g = \nabla S(x) = J^T r$.
Then:

1. $J^T J + \lambda I$ is [[positive definite|0KX66CZMBB]], so $\delta(\lambda)$ exists and is unique;
2. if $g \ne 0$, then $g^T \delta(\lambda) < 0$, so $\delta(\lambda)$ is a descent direction for $S$;
3. $\delta(\lambda)$ minimizes $\norm{r + J p}_2$ over all $p$ with $\norm{p}_2 \le \norm{\delta(\lambda)}_2$;
4. if $J$ has full column rank, then $\delta(\lambda)$ tends to the [[Gauss–Newton|CYV4T43FQJ]] step as $\lambda \to 0$;
5. $\lambda \, \delta(\lambda) \to -g$ as $\lambda \to \infty$.

## Remarks

Part 1 holds because $p^T (J^T J + \lambda I) p = \norm{J p}^2 + \lambda \norm{p}^2 > 0$ for $p \ne 0$.
Part 2 follows: $g^T \delta = -g^T (J^T J + \lambda I)^{-1} g < 0$.
For part 3, $\delta$ satisfies the optimality conditions of the ball-constrained problem with multiplier $\lambda \ge 0$, and the problem is convex.
So the method is a trust region method for the Gauss–Newton model, with $\lambda$ controlling the radius.
Parts 4 and 5 show that it interpolates between Gauss–Newton steps (small $\lambda$) and short steepest descent steps (large $\lambda$).

The algorithm updates $\lambda$ at each step.
If the step reduces $S$ enough, it is accepted and $\lambda$ is decreased; otherwise it is rejected and $\lambda$ is increased.
Marquardt proposed replacing $\lambda I$ by $\lambda \, \mathrm{diag}(J^T J)$, which makes the step invariant under rescaling of the variables.
Example: with $m = n = 1$, $r(x) = x - 1$ and $x = 3$, one has $J = 1$, $g = 2$ and $\delta(\lambda) = -2/(1 + \lambda)$, which is the full Gauss–Newton step $-2$ at $\lambda = 0$ and shrinks as $\lambda$ grows.

## Explanations

### L1 human

Gauss–Newton is fast near a solution but can jump too far when the guess is poor.
Levenberg–Marquardt adds a damping term that shortens the step and turns it toward the downhill direction.
When things go well the damping is reduced and the method becomes Gauss–Newton; when they go badly it is increased.

### L2 human

The method is due to Kenneth Levenberg (1944) and Donald Marquardt (1963).
It is the standard algorithm for small and medium nonlinear least squares problems, such as curve fitting and camera calibration.
Its damped normal equations are the same equations as in Tikhonov regularization of the linearized problem.
</document>

<document source="https://tamnd.github.io/math2/node/TH8EGSCFYY.md" tag="TH8EGSCFYY" kind="definition">
# Proof theory

## Definition

A *formal system* consists of a formal language, a set of axioms, and a finite set of inference rules.
A *formal proof* in such a system is a finite sequence or tree of formulas.
In it, each formula is an axiom or follows from earlier formulas by an inference rule.
*Proof theory* is the branch of mathematical logic whose objects of study are formal proofs.
The proofs are taken in systems of [[predicate logic|086QKPF0A6]] and in theories such as arithmetic.
It studies their structure, their transformations and their length, and what follows for the theories in which they are written.

## Remarks

Example: the question whether every proof in the sequent calculus can be turned into a proof without the cut rule is a question of proof theory.
Its answer is the cut-elimination theorem of Gentzen.
Non-example: the question whether a given group is abelian is not a question of proof theory, because its object is a group and not a formal proof.
Main parts of the field are structural proof theory, ordinal analysis, provability logic, reverse mathematics and proof complexity.
The informal notion that formal proofs model is [[mathematical proof|0B94TKFQJX]].
</document>

<document source="https://tamnd.github.io/math2/node/THCM7DWHNY.md" tag="THCM7DWHNY" kind="concept">
# Resistance form

## Summary

A resistance form is a quadratic energy form on functions on a set which behaves like the energy of an electrical network.
It was introduced by Jun Kigami.
It gives a common frame to energies on finite networks, on intervals and on fractals such as the Sierpiński gasket.
Each resistance form defines a metric, the resistance metric.
It generalizes the [[effective resistance|XAE5YJ3H68]] between two nodes of a network.
Under mild conditions a resistance form is also a [[Dirichlet form|MH20Q2N1AH]].
So it gives a Laplacian, a heat equation and a diffusion process on the set.

## Explanations

### L1 human

Think of a network of wires that joins some points, and put a voltage $u(x)$ at each point $x$.
The energy of $u$ is the heat produced by the currents, a sum of terms $c \, (u(x) - u(y))^2$ over the wires.
A resistance form keeps the main features of this energy but allows infinitely many points, such as all points of a fractal.
The effective resistance between two points $x$ and $y$ is then the largest value of $(u(x) - u(y))^2$ among voltages of energy $1$.
On the interval $[0, 1]$ with energy $\int_0^1 u'(s)^2 \, ds$, this resistance between $x$ and $y$ is just $\abs{x - y}$, like a uniform wire.
</document>

<document source="https://tamnd.github.io/math2/node/THH2YHW8G5.md" tag="THH2YHW8G5" kind="concept">
# Hartogs domain

## Summary

A Hartogs domain is a domain in $\C^n$ that is unchanged when the last coordinate is rotated about the origin.
Such domains are described by the radii of their circles in the last variable over each point of the remaining variables.
They are the standard testing ground for extension phenomena and for convexity notions in several complex variables.
Every [[Reinhardt domain|WA0SE56FVH]] is a Hartogs domain.

## Explanations

### L1 human

Fix the first $n - 1$ coordinates and look at the slice of the domain in the last coordinate.
For a Hartogs domain every such slice is a union of circles centred at $0$, so it is a disc, an annulus, or a union of such rings.
A typical example in $\C^2$ is $\set{(z, w) : \abs{z} < 1, \ \abs{w} < e^{-\varphi(z)}}$ for a continuous function $\varphi$ on the unit disc.
Its slices are discs whose radius $e^{-\varphi(z)}$ varies with $z$.
Such a domain is pseudoconvex exactly when $\varphi$ is [[subharmonic|Z1MHMGFNFW]], which makes these domains very convenient for building examples.
</document>

<document source="https://tamnd.github.io/math2/node/THH4GJFYQN.md" tag="THH4GJFYQN" kind="definition">
# Generalized linear model

## Definition

Let $x_1, \dots, x_n \in \R^p$ be known regressor vectors.
A *generalized linear model* for independent responses $y_1, \dots, y_n$ consists of three parts.

1. Random component: each $y_i$ has the density $f(y; \theta_i, \phi) = \exp\big( (y \theta_i - b(\theta_i))/a(\phi) + c(y, \phi) \big)$ with respect to a fixed measure.
This is an [[exponential family|EP9VPZ0S6G]] with natural parameter $\theta_i$, dispersion parameter $\phi > 0$, and fixed functions $a, b, c$.
2. Linear predictor: $\eta_i = x_i^T \beta$ for an unknown $\beta \in \R^p$.
3. Link: a known strictly monotone differentiable function $g$ such that $g(\mu_i) = \eta_i$, where $\mu_i = E[y_i]$ is the [[expected value|0WQNRCNY3M]] of $y_i$.

## Remarks

In the random component, $\mu_i = b'(\theta_i)$ and $\operatorname{Var}(y_i) = a(\phi) \, b''(\theta_i)$.
The link $g$ with $g(b'(\theta)) = \theta$ is the *canonical link*; with it, $\theta_i = x_i^T \beta$.

Example: the normal distribution $N(\mu, \sigma^2)$ has the form of clause 1 with $\theta = \mu$, $b(\theta) = \theta^2/2$ and $a(\phi) = \phi = \sigma^2$.
With the identity link, $\mu_i = x_i^T \beta$, and the model is normal [[linear regression|41FVC4P5JB]].
Example: for Bernoulli responses with the logit link $g(\mu) = \log(\mu/(1 - \mu))$ one gets logistic regression; for Poisson responses with the log link one gets Poisson regression.

Non-example: the model $y_i = x_i^T \beta + \varepsilon_i$ with independent standard Cauchy errors $\varepsilon_i$ is not a generalized linear model.
The Cauchy location family is not of the form of clause 1, and $y_i$ has no expected value, so clause 3 cannot hold either.

The coefficients are estimated by [[maximum likelihood|0FREPFKNS7]]; with the canonical link the likelihood equations are $\sum_i (y_i - \mu_i) x_i = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/THRHFW3SMP.md" tag="THRHFW3SMP" kind="concept">
# Free category

## Summary

The free category on a [[directed graph|0KFYGSHHF5]] has the vertices of the graph as objects and the finite paths of edges as morphisms.
Paths are composed by joining them end to end.
It is the most general category generated by the graph: no equations between paths hold except those forced by the axioms of a category.
It plays for categories the role that the [[free monoid|CV013KHVJG]] plays for monoids.

## Explanations

### L1 human

Draw some dots and arrows between them.
A morphism of the free category is a route you can travel by following arrows one after another, and composing two routes means doing one and then the other.
Doing nothing at a dot is the identity route.
If the graph has one dot and one loop $a$, the routes are $a^0, a^1, a^2, \dots$, and composing them adds the exponents, so the free category is the monoid $(\N, +)$.
If the graph is $x \to y \to z$, there are exactly six routes: three identities, the two edges, and the route from $x$ to $z$ through $y$.
</document>

<document source="https://tamnd.github.io/math2/node/THSHGNJNPS.md" tag="THSHGNJNPS" kind="concept">
# Differential game

## Summary

A differential game is a game in which several players steer a common state through a [[control system|ZPDTVZQN5K]].
The state obeys a differential equation driven by all their controls.
Each player has a cost or payoff that depends on the resulting trajectory, and the players have different goals.
With one player it is an [[optimal control|T3V6ZNAMDN]] problem.
The two-player zero-sum case leads to the [[Hamilton–Jacobi–Isaacs equation|WHTBZ3K4FB]], and pursuit problems are the classical examples.

## Explanations

### L1 human

Picture a cat chasing a mouse in a field.
Both move continuously, and each changes direction at every instant in reaction to the other.
The cat wants to catch the mouse quickly, the mouse wants to escape or at least delay capture.
A differential game is the mathematical model of such a contest: the positions obey differential equations, and each player chooses its control as time runs.
In a simple version, both move in the plane with bounded speed, the cat faster, and the payoff is the time to capture.
</document>

<document source="https://tamnd.github.io/math2/node/THSVFE7YER.md" tag="THSVFE7YER" kind="statement">
# Siu's semicontinuity theorem

## Statement

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $n$ and let $0 \le p \le n$.
Let $T$ be a closed [[positive current|BYJHP6A906]] on $X$ of bidimension $(p, p)$, that is, of bidegree $(n - p, n - p)$.
For $x \in X$ let $\nu(T, x)$ be the [[Lelong number|WVB60RKC3Z]] of $T$ at $x$.
Then for every real $c > 0$ the set
$$E_c(T) = \set{x \in X : \nu(T, x) \ge c}$$
is a closed [[analytic subset|GB7M582M5Y]] of $X$ of [[dimension|AN0BYRKNRA]] at most $p$.

## Remarks

That $E_c(T)$ is closed is elementary, since $x \mapsto \nu(T, x)$ is upper semicontinuous.
The content is that $E_c(T)$ is analytic.
Siu's original proof was long; a shorter proof by Demailly writes $E_c(T)$ locally as a common zero set of holomorphic functions built with [[Hörmander's L2 estimates|TPSH4ETK67]].

Example: let $A \subseteq X$ be an analytic subset of pure dimension $p$ and $T = [A]$ its current of integration.
By [[Thie's theorem|G5JKJEAKZ9]], $\nu([A], x)$ is the multiplicity of $A$ at $x$, so $E_c([A])$ is the set of points of multiplicity at least $c$, an analytic set.

For a [[plurisubharmonic function|FF2P7ER3QT]] $u$ on an open set of $\C^n$, apply the theorem to $T = dd^c u$ with $p = n - 1$: the sets $\set{x : \nu(u, x) \ge c}$ are analytic of dimension at most $n - 1$.

## Explanations

### L1 human

The Lelong number measures how singular a current is at a point.
The theorem says that the points where this singularity is at least a given size are not scattered at random.
They form an analytic set, cut out locally by holomorphic equations.

### L2 human

The theorem is the starting point of Siu's decomposition of a closed positive current.
It writes the current as a countable sum of currents of integration over analytic sets plus a residual part.
For the residual part, the upper level sets of Lelong numbers have smaller dimension.
It is used with multiplier ideals and with regularization of currents in the study of positivity of line bundles.
</document>

<document source="https://tamnd.github.io/math2/node/THX29SKKPB.md" tag="THX29SKKPB" kind="definition">
# Spectral triple

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.
Let $A$ be a unital $*$-subalgebra of $B(H)$.
Let $D$ be a densely defined [[unbounded|KRJX5RA0PW]] [[self-adjoint operator|0NEGN5CRSC]] on $H$ with domain $\operatorname{dom} D$.
The triple $(A, H, D)$ is a *spectral triple* if:

1. for every $a \in A$, the operator $a$ maps $\operatorname{dom} D$ into itself, and the commutator $[D, a] = Da - aD$ on $\operatorname{dom} D$ extends to a bounded operator on $H$;
2. the resolvent $(D - \lambda I)^{-1}$ is a [[compact operator|M2NEP1K5AE]] for one, and then for every, $\lambda \in \C \setminus \R$.

## Remarks

Example: let $A = C^\infty(S^1)$ act on $H = L^2(S^1)$ by multiplication and let $D = -i \, d/d\theta$ on the Sobolev space of functions with square-integrable derivative.
The functions $e_n(\theta) = e^{in\theta}$, $n \in \Z$, form an orthonormal basis of eigenvectors with $D e_n = n e_n$, so $(D - i)^{-1}$ has eigenvalues $1/(n - i) \to 0$ and is compact.
For $f \in A$ the product rule gives $[D, f] = -i f'$, which is multiplication by a bounded function.
More generally, smooth functions on a compact Riemannian spin manifold, acting on square-integrable spinors, with the [[Dirac operator|3DXGJTEQZH]], form a spectral triple.

Non-example: if $H$ is infinite-dimensional and $D = 0$, every commutator is $0$, so clause 1 holds.
But $(D - i I)^{-1} = i I$ is not compact, so clause 2 fails.

Often $A$ is a dense $*$-subalgebra of a C*-algebra, and $A$ acts on $H$ through a faithful $*$-representation.
A spectral triple is called even if there is a self-adjoint unitary $\gamma$ on $H$ that commutes with $A$ and anticommutes with $D$.
For two states $\varphi$ and $\psi$ of the norm closure of $A$, Connes' distance formula is
$$d(\varphi, \psi) = \sup \set{ \abs{\varphi(a) - \psi(a)} : a \in A, \ \|[D, a]\| \le 1 }.$$
It recovers the geodesic distance when $\varphi$ and $\psi$ are evaluations at two points of a manifold.
</document>

<document source="https://tamnd.github.io/math2/node/THY12GPZ8C.md" tag="THY12GPZ8C" kind="statement">
# Paley–Wiener theorem

## Statement

Let $\sigma > 0$, and let $F$ be an [[entire function|0419794R0T]].
The following two conditions are equivalent.

1. $F$ is of [[exponential type|9P8D8YAHQS]] at most $\sigma$, and its restriction to the real line is square-integrable: $\int_{-\infty}^{\infty} \abs{F(x)}^2 \, dx < \infty$.
2. There is a square-integrable function $\varphi$ on the interval $[-\sigma, \sigma]$ such that
$$F(z) = \int_{-\sigma}^{\sigma} \varphi(t) \, e^{i z t} \, dt \quad \text{for all } z \in \C.$$

## Remarks

The implication from 2 to 1 is direct: $\abs{e^{izt}} = e^{-t \operatorname{Im} z} \le e^{\sigma \abs{z}}$ for $\abs{t} \le \sigma$, and the Plancherel theorem gives the square-integrability on $\R$.
For the converse, $\varphi$ is the Fourier transform of $F$ on $\R$, up to a constant.
It vanishes outside $[-\sigma, \sigma]$; this follows by shifting the line of integration into the upper or lower half-plane.
The growth there is controlled by the [[Phragmén–Lindelöf principle|FYR6V2NH6C]].
For example, with $\varphi = 1/2$ on $[-\sigma, \sigma]$ one gets $F(z) = \sin(\sigma z)/z$, which has exponential type $\sigma$ and is square-integrable on $\R$.
The square-integrability is needed: $\sin(\sigma z)$ has exponential type $\sigma$ but is not of the form 2, since it is not square-integrable on $\R$.

## Explanations

### L1 human

A signal whose frequencies all lie in a bounded band $[-\sigma, \sigma]$ extends from the real line to an entire function of the complex variable.
That entire function grows at most like $e^{\sigma \abs{z}}$.
The theorem says that the converse holds too: such growth, together with finite energy on the real line, forces the frequencies into the band.
So the smallest band $[-\sigma, \sigma]$ that contains the frequencies is given by the exponential type.

### L2 human

The space of functions in the theorem is the Paley–Wiener space, a Hilbert space of entire functions that is the basic example of a [[de Branges space|NYHNTP9AND]].
It is the setting of the sampling theorem of signal processing: such a function is determined by its values at the points $\pi n/\sigma$, $n \in \Z$.
There are related Paley–Wiener theorems for smooth functions and distributions with compact support, where the growth condition also involves $\abs{\operatorname{Im} z}$.
</document>

<document source="https://tamnd.github.io/math2/node/THYBFG527A.md" tag="THYBFG527A" kind="definition">
# Random graph

## Definition

Let $n \ge 1$, let $V = \set{1, \dots, n}$, and let $\mathcal{G}_n$ be the finite set of all simple [[graphs|0JD5C82QMB]] with vertex set $V$.
A random graph on $V$ is a [[probability measure|0VGMEBVW1M]] $\mathbb{P}$ on $\mathcal{G}_n$.
That is, it is a map $\mathbb{P} \colon \mathcal{G}_n \to [0, 1]$ with $\sum_{G \in \mathcal{G}_n} \mathbb{P}(G) = 1$.
Equivalently, it is the law of a random variable on a [[probability space|0ZYBZ34RT2]] with values in $\mathcal{G}_n$.
A model of random graphs is a choice of such a measure $\mathbb{P}_n$ for every $n$.
For $0 \le p \le 1$, the binomial model $G(n, p)$ is the measure with $\mathbb{P}(G) = p^{m} (1 - p)^{\binom{n}{2} - m}$, where $m$ is the number of edges of $G$.

## Remarks

In $G(n, p)$ the $\binom{n}{2}$ possible edges are present independently, each with probability $p$; the probabilities add up to $(p + (1 - p))^{\binom{n}{2}} = 1$ by the binomial theorem.
Example: $G(n, 1/2)$ gives each of the $2^{\binom{n}{2}}$ graphs in $\mathcal{G}_n$ the same probability, so it is the uniform random graph on $V$.
Non-example: giving each of the $8$ graphs on $\set{1, 2, 3}$ the weight $1/2$ is not a random graph, because the weights add up to $4$, not $1$.
A property holds asymptotically almost surely in a model if its probability tends to $1$ as $n \to \infty$.
Another standard model is $G(n, M)$, the uniform measure on the graphs in $\mathcal{G}_n$ with exactly $M$ edges.
</document>

<document source="https://tamnd.github.io/math2/node/TJ04VQP4NB.md" tag="TJ04VQP4NB" kind="concept">
# Group scheme

## Summary

A group scheme over a base scheme $S$ is a scheme over $S$ with multiplication, unit and inverse morphisms that satisfy the group axioms.
Equivalently, its points with values in every $S$-scheme form a group, naturally in the $S$-scheme.
Group schemes include algebraic groups over fields, such as $\mathrm{GL}_n$ and elliptic curves.
They also include infinitesimal objects such as $\mu_p$ in characteristic $p$, which have no analogue among classical groups.

## Explanations

### L1 human

A Lie group is a manifold that is also a group, with smooth operations.
A group scheme is the same idea in algebraic geometry: a scheme whose points can be multiplied by polynomial formulas.
The simplest example is the multiplicative group $\mathbb{G}_m$, whose points are the invertible numbers, multiplied as usual.
Over a general base, a group scheme is a family of such groups that varies with a parameter.
</document>

<document source="https://tamnd.github.io/math2/node/TJ04Y2Z5Q3.md" tag="TJ04Y2Z5Q3" kind="concept">
# Normal form

## Summary

A normal form of a differential equation near an [[equilibrium point|0N8NGKDZDZ]] is a simplest version of the equation.
It is obtained by a change of coordinates close to the identity.
The change removes, degree by degree, every nonlinear term that can be removed, and only the resonant terms remain.
The method goes back to Henri Poincaré.
It is a central tool in bifurcation theory, where normal forms describe the generic local behaviour, as in the Hopf bifurcation.

## Explanations

### L1 human

Near an equilibrium an equation looks like its [[linearisation|09ZWYRQ41J]] plus small nonlinear corrections.
Many of these corrections are only an artefact of the coordinates and disappear in better coordinates.
The normal form is what remains after removing them, and it shows the essential behaviour.
For $x' = x + x^2$ near $0$, the new coordinate $u = x / (1 + x)$ gives $u' = u$, so the quadratic term disappears completely.
The terms that this method cannot remove are called resonant; they come from special relations among the eigenvalues of the linear part.
</document>

<document source="https://tamnd.github.io/math2/node/TJ6EBSS3S7.md" tag="TJ6EBSS3S7" kind="example">
# The alternating group is the kernel of the sign

## Example

Let $n \ge 2$. The [[sign|0SK90238BH]] is a homomorphism $\operatorname{sgn} : S_n \to \{1, -1\}$, where $\{1, -1\}$ is a group under multiplication.
A permutation has sign $1$ when it is a product of an even number of transpositions.
Thus the kernel of $\operatorname{sgn}$ is the set of even permutations. This is the [[alternating group|0Z5905Q73D]] $A_n$.
Thus $A_n$ is a [[normal subgroup|0C9ZKYSPYR]] of $S_n$.

## Remarks

The sign is onto, so $A_n$ has index $2$ in $S_n$.
</document>

<document source="https://tamnd.github.io/math2/node/TJ9XPJ65YX.md" tag="TJ9XPJ65YX" kind="definition">
# Discrete Fourier transform

## Definition

Let $N \ge 1$ be an integer and let $\omega_N = e^{-2\pi i / N}$, a primitive $N$-th [[root of unity|0YV2CKP2AV]], where $e^{z}$ is the [[complex exponential|07GJV36896]].
The *discrete Fourier transform* of a vector $x = (x_0, x_1, \dots, x_{N-1}) \in \C^N$ is the vector $X = (X_0, X_1, \dots, X_{N-1}) \in \C^N$ given by
$$X_k = \sum_{n=0}^{N-1} x_n\, \omega_N^{kn} = \sum_{n=0}^{N-1} x_n\, e^{-2\pi i k n / N}, \qquad k = 0, 1, \dots, N - 1.$$

## Remarks

The map $x \mapsto X$ is linear and invertible, with inverse
$$x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k\, e^{2\pi i k n / N}.$$
This follows from $\sum_{k=0}^{N-1} e^{2\pi i k m / N} = N$ if $N$ divides $m$ and $0$ otherwise.
It also gives the Parseval identity $\sum_k \abs{X_k}^2 = N \sum_n \abs{x_n}^2$, so $x \mapsto N^{-1/2} X$ is unitary.

Example: for $N = 2$, $X_0 = x_0 + x_1$ and $X_1 = x_0 - x_1$; so $(3, 1)$ has transform $(4, 2)$, and the inverse formula gives back $\frac12 (4 + 2, 4 - 2) = (3, 1)$.

Non-example: an infinite sequence $(x_n)_{n \in \Z}$ is not a vector of $\C^N$, so the definition does not apply; its analogue is the discrete-time Fourier transform, a function of a continuous frequency.

Indices are read modulo $N$, so the transform is the Fourier transform on the group $\Z / N\Z$.
It turns the cyclic convolution $(x * y)_n = \sum_{m} x_m y_{n - m}$ into the product $X_k Y_k$, and therefore diagonalizes every circulant matrix.
Some authors put the factor $\frac1N$ or $\frac{1}{\sqrt N}$ in the forward transform, or use $e^{+2\pi i k n/N}$.
</document>

<document source="https://tamnd.github.io/math2/node/TJKANTKASZ.md" tag="TJKANTKASZ" kind="concept">
# Finitely presented group

## Summary

A finitely presented group is a group that has a [[presentation|0JGEQXQ4AC]] with finitely many generators and finitely many relations.
Such a group can be described by a finite amount of data, so it can be handled by algorithms and computers.
Finitely presented groups are the natural objects of combinatorial group theory and arise as [[fundamental groups|S1XK3HXHFX]] of compact manifolds and finite complexes.

## Explanations

### L1 human

A finitely presented group is given by a finite list of letters and a finite list of rules that say which words equal the identity.
For example, $\langle a, b \mid a b a^{-1} b^{-1} \rangle$ has two letters and one rule, $ab = ba$, and it is the group $\Z^2$.
Every finite group is finitely presented, since its whole multiplication table is a finite list of rules.
Some finitely generated groups need infinitely many rules, however one chooses the generators.
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# Ore condition

## Summary

The Ore condition is a condition on a [[multiplicative set|AN9SKEZ3VY]] $S$ of a noncommutative [[ring|0B1261B5ED]].
It allows one to form fractions $a s^{-1}$ with denominators in $S$.
It asks that a "left fraction" $s^{-1} a$ can always be rewritten as a "right fraction" $b t^{-1}$.
Together with a mild cancellation condition, it is exactly what is needed for the [[Ore localization|SG7AQ9T8HD]] to exist.
For commutative rings it always holds, so it explains why ordinary [[localization|4C1AE84N5N]] needs no extra hypothesis.

## Explanations

### L1 human

In a commutative ring, $a / s$ means the same thing whether you divide on the left or on the right.
In a noncommutative ring, $s^{-1} a$ and $a s^{-1}$ are different, and products of fractions must again be fractions of one kind.
To multiply $a s^{-1}$ by $b t^{-1}$ one needs to move $s^{-1}$ past $b$, that is, to write $s^{-1} b$ as $c u^{-1}$.
The Ore condition says that this is always possible.
For example, in the ring of polynomials $k[x]$ every nonzero polynomial can serve as a denominator, but in the free algebra $k\langle x, y \rangle$ the nonzero elements cannot.
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<document source="https://tamnd.github.io/math2/node/TJPNYW7J8E.md" tag="TJPNYW7J8E" kind="concept">
# Catastrophic cancellation

## Summary

Catastrophic cancellation is the large loss of relative accuracy when one subtracts two nearly equal numbers that are only known approximately.
The leading digits cancel, and what is left is dominated by the errors of the inputs.
The subtraction itself is often exact in [[floating-point arithmetic|RTKRSC3KPH]]; the damage comes from earlier [[round-off errors|CPN6SD3BTV]].
It is the most common cause of numerically unstable formulas, and it is avoided by rewriting formulas, for example $\sqrt{x+1} - \sqrt{x} = 1/(\sqrt{x+1} + \sqrt{x})$.

## Explanations

### L1 human

Suppose two lengths are measured as $1.2341$ m and $1.2259$ m, each to the nearest tenth of a millimetre.
Their difference $0.0082$ m has only two significant digits, although each length had five.
If one first rounds both lengths to three digits, both become $1.23$ and the difference becomes $0$, a $100\%$ error.
Subtracting close numbers throws away the digits that agree and keeps the digits that carry the error.
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<document source="https://tamnd.github.io/math2/node/TJQ5HWTXXS.md" tag="TJQ5HWTXXS" kind="definition">
# Hahn series

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $\Gamma$ be an [[abelian group|071DDFVG4J]] with a total order making it a [[linearly ordered group|1P8087YGYQ]].
A *Hahn series* over $K$ with exponents in $\Gamma$ is a map $f \colon \Gamma \to K$, written $f = \sum_{\gamma \in \Gamma} f(\gamma) t^\gamma$, whose support $\set{\gamma \in \Gamma : f(\gamma) \ne 0}$ is [[well-ordered|0F1D6PK9RZ]].
The set of Hahn series is written $K((t^\Gamma))$.
Addition is pointwise, and multiplication is
$$(fg)(\gamma) = \sum_{\alpha + \beta = \gamma} f(\alpha) g(\beta).$$

## Remarks

For each $\gamma$ the sum defining $(fg)(\gamma)$ has finitely many nonzero terms, and the support of $fg$ is again well-ordered.
With these operations $K((t^\Gamma))$ is a field, as Hahn showed.
Example: for $\Gamma = \Z$, the well-ordered subsets are the subsets bounded below, so $K((t^\Z))$ is the [[field of formal Laurent series|2S4K8FA8QY]] $K((t))$.
Example: for $\Gamma = \Q$, the series $\sum_{n \ge 1} t^{1 - 1/n}$ is a Hahn series, since its support $\set{0, 1/2, 2/3, \dots}$ is well-ordered.
Non-example: for $\Gamma = \Z$, the formal sum $\sum_{n \ge 1} t^{-n}$ is not a Hahn series, since its support $\set{-1, -2, \dots}$ has no least element.
The map $v(f) = \min \set{\gamma : f(\gamma) \ne 0}$ for $f \ne 0$ is a [[valuation|SVY882HKSJ]] on $K((t^\Gamma))$ with value group $\Gamma$.
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# Differential game

## Definition

Let $N \ge 2$ and $T > 0$.
For each player $i = 1, \dots, N$, let $U_i \subseteq \R^{m_i}$ be a nonempty compact set.
Let $f \colon [0, T] \times \R^n \times U_1 \times \cdots \times U_N \to \R^n$ be continuous and [[Lipschitz|0QNVQ4VD9T]] in its second variable.
For each $i$, let $L_i \colon [0, T] \times \R^n \times U_1 \times \cdots \times U_N \to \R$ and $g_i \colon \R^n \to \R$ be continuous.
A *differential game* on $[0, T]$ with initial state $x_0 \in \R^n$ consists of these data, used as follows.

1. Each player $i$ chooses a measurable control $u_i \colon [0, T] \to U_i$.
2. The controls drive the state by the [[control system|ZPDTVZQN5K]] $x'(t) = f(t, x(t), u_1(t), \dots, u_N(t))$ with $x(0) = x_0$.
3. Player $i$ wants to minimise its cost $J_i(u_1, \dots, u_N) = \int_0^T L_i(t, x(t), u_1(t), \dots, u_N(t)) \, dt + g_i(x(T))$.

The game is *two-player zero-sum* if $N = 2$ and $J_2 = -J_1$; then player $1$ minimises $J = J_1$ and player $2$ maximises it.

## Remarks

A full definition also says what information each player uses.
In open-loop play each player picks a control as a function of time alone.
In feedback play each player picks $u_i(t)$ as a function of $(t, x(t))$.
In the zero-sum case one often uses nonanticipating strategies, which choose the present control from the opponent's past controls.
A solution concept is a Nash equilibrium, a tuple of strategies from which no player gains by deviating alone.
In the zero-sum case this is a [[saddle point|0JW6MDHAZD]] of $J$, and its value is the value of the game.

Example: $n = 2$, $U_1$ the closed disc of radius $2$, $U_2$ the closed unit disc, $f(t, x, u_1, u_2) = u_2 - u_1$, $L_1 = 0$ and $g_1(x) = \abs{x}$, with $g_2 = -g_1$.
Here $x$ is the position of player $2$ relative to player $1$, and player $1$ wants to end close to player $2$.
All clauses hold: $f$ is continuous and does not depend on $x$, and the sets $U_i$ are compact.

Non-example: in a game where the state changes only at the times $0, 1, 2, \dots$ by a recurrence $x_{k+1} = F(x_k, u_{1,k}, u_{2,k})$, clause 2 fails.
It is a dynamic game in discrete time, not a differential game.

## Explanations

### L1 human

Several players steer the same system at the same time, each with its own goal.
The rules of motion are a differential equation, so the game is played in continuous time.
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<document source="https://tamnd.github.io/math2/node/TK1PT9FDV0.md" tag="TK1PT9FDV0" kind="statement">
# Lichnerowicz–Obata eigenvalue theorem

## Statement

Let $(M, g)$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] without boundary, of dimension $n \ge 2$.
Let $k > 0$, and assume that the [[Ricci curvature|4GTNM0WHDH]] satisfies $\operatorname{Ric} \ge (n - 1) k \, g$.
Let $\lambda_1$ be the smallest nonzero eigenvalue of the [[Laplace–Beltrami operator|F4NWZNTRMD]] $\Delta = -\operatorname{div} \operatorname{grad}$ on $M$.

1. (Lichnerowicz) $\lambda_1 \ge n k$.
2. (Obata) $\lambda_1 = n k$ if and only if $(M, g)$ is isometric to the round [[sphere|P9CM4GQ2T0]] $S^n$ of radius $1/\sqrt{k}$.

## Remarks

The proof of the inequality applies the Bochner formula to an eigenfunction $f$ with $\Delta f = \lambda_1 f$.
With $\nabla^2 f$ the Hessian, the formula reads $-\frac12 \Delta \abs{\nabla f}^2 = \abs{\nabla^2 f}^2 - \ip{\nabla f}{\nabla \Delta f} + \operatorname{Ric}(\nabla f, \nabla f)$.
Integrating over $M$ and using $\abs{\nabla^2 f}^2 \ge (\Delta f)^2 / n$ gives $\lambda_1 \ge nk$.
In the equality case the Hessian of $f$ equals $-k f g$, and Obata showed that a complete manifold carrying a nonconstant solution of this equation is a round sphere.
On the unit sphere $S^n$ one has $\operatorname{Ric} = (n-1) g$ and $\lambda_1 = n$, attained by the restrictions of the coordinate functions, so the bound is sharp.
The hypothesis $k > 0$ is needed: flat tori have $\operatorname{Ric} = 0$, and the flat torus $\R^n / L\Z^n$ has $\lambda_1 = 4\pi^2 / L^2$, which tends to $0$ as $L$ grows.

## Explanations

### L1 human

The first nonzero eigenvalue of the Laplacian measures the lowest pitch at which a closed manifold can vibrate.
A positive lower bound on the Ricci curvature makes the manifold small and round, and this forces the pitch up.
Lichnerowicz's bound says how far up.
Obata's part says that the round sphere is the only shape that sits exactly at the bound.

### L2 human

This is the model comparison theorem of spectral geometry: a curvature bound gives an eigenvalue bound, with rigidity in the extreme case.
It sits beside the Bonnet–Myers theorem, which under the same hypothesis bounds the diameter by $\pi/\sqrt{k}$.
The Bochner technique used in the proof is the same one used for vanishing theorems for harmonic forms.
For a lower bound on $\lambda_1$ without curvature assumptions, see [[Cheeger's inequality|JRGADFNBV2]].
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# Hyperreal number

## Definition

Let $\mathcal{U}$ be a nonprincipal [[ultrafilter|J64Y3JC0RE]] on $\N$.
For [[sequences|0G9EKM0E15]] $a = (a_n)$ and $b = (b_n)$ of [[real numbers|08J2M1JYAG]], put $a \sim b$ if $\set{n \in \N : a_n = b_n} \in \mathcal{U}$.
This is an [[equivalence relation|0G8VEZ3WTK]]; write $[a]$ for the class of $a$.
The set of *hyperreal numbers* is ${}^*\R = \R^{\N} / {\sim}$, with the following structure.

1. $[a] + [b] = [(a_n + b_n)]$ and $[a] \cdot [b] = [(a_n b_n)]$.
2. $[a] < [b]$ if $\set{n \in \N : a_n < b_n} \in \mathcal{U}$.
3. Each real number $r$ is identified with the class of the constant sequence $(r, r, r, \dots)$.

A hyperreal $x$ is *infinitesimal* if $\abs{x} < r$ for every real $r > 0$, and *infinite* if $\abs{x} > r$ for every real $r$.

## Remarks

The operations and the order are well defined, and ${}^*\R$ is an [[ordered field|0P3VCBDEE2]] containing $\R$; it is the [[ultrapower|AX9Q86GB8M]] of $\R$ by $\mathcal{U}$.

Example: $\varepsilon = [(1/(n+1))]$ is a positive infinitesimal.
For a real $r > 0$, the set $\set{n : 1/(n+1) < r}$ is cofinite, and a nonprincipal ultrafilter contains every cofinite set, so $\varepsilon < r$.
Also $\varepsilon > 0$ because every term is positive.
So ${}^*\R$ is not [[Archimedean|0XH79S06J3]].

Non-example: if $\mathcal{U}$ is the principal ultrafilter of the sets containing a fixed $k \in \N$, then $[a] \mapsto a_k$ is an isomorphism onto $\R$, and there are no nonzero infinitesimals.

Under the continuum hypothesis, the ordered field ${}^*\R$ does not depend on the choice of $\mathcal{U}$, up to isomorphism.
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<document source="https://tamnd.github.io/math2/node/TK9WPV0EXS.md" tag="TK9WPV0EXS" kind="definition">
# Analytic continuation in several variables

## Definition

A *function element* in $\C^n$ is a pair $(U, f)$ where $U \subseteq \C^n$ is a nonempty connected [[open set|02BH3MCBQB]] and $f$ is a [[holomorphic function|507C34W0ZT]] on $U$.

1. A function element $(V, g)$ is a *direct analytic continuation* of $(U, f)$ if $U \cap V$ is nonempty and $f = g$ on some [[connected component|0JG9W3KA6B]] of $U \cap V$.
2. A function element $(V, g)$ is an *analytic continuation* of $(U, f)$ along a chain if a finite sequence of function elements runs from $(U_0, f_0) = (U, f)$ to $(U_k, f_k) = (V, g)$. Each $(U_j, f_j)$ must be a direct analytic continuation of $(U_{j-1}, f_{j-1})$.

If $V \supseteq U$ and $g = f$ on $U$, one says that $g$ is an analytic continuation of $f$ to $V$.

## Remarks

By the identity theorem, if $f$ and $g$ agree on a nonempty open subset of a component of $U \cap V$, they agree on that whole component.

Example: let $U = \set{z \in \C^2 : \operatorname{Re} z_1 > 0}$ and $f(z) = \log z_1$ with the principal branch.
Moving through half-spaces in the $z_1$-direction around the plane $z_1 = 0$ gives a chain of direct continuations that returns to $U$ with the function $\log z_1 + 2\pi i$.
So continuation along a closed chain need not give back the original function element.

Non-example: if $U \cap V$ is empty, then $(V, g)$ is never a direct analytic continuation of $(U, f)$, whatever $g$ is.

For $n = 1$ this is the usual [[analytic continuation|0M3YQJ81NB]] of a holomorphic function along a chain of discs or domains.
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<document source="https://tamnd.github.io/math2/node/TKA958Z4KZ.md" tag="TKA958Z4KZ" kind="concept">
# Holomorphic vector bundle

## Summary

A holomorphic vector bundle is a complex [[vector bundle|P4DZM975BV]] over a [[complex manifold|JSGWAN2RJV]].
Its total space is a complex manifold, and its local trivializations are holomorphic.
Equivalently, its transition functions are holomorphic maps into $GL_r(\C)$.
Holomorphic vector bundles carry sheaves of holomorphic sections, and their cohomology is a main tool of complex geometry.
Examples are the holomorphic tangent bundle, the bundles of holomorphic forms, and the line bundles $\mathcal{O}(k)$ on projective space.

## Explanations

### L1 human

Over each point of a complex manifold sits a complex vector space, and these spaces vary holomorphically from point to point.
Locally the bundle looks like a product $U \times \C^r$, and two such local pictures are glued by matrices whose entries are holomorphic functions.
The holomorphic sections are then the "vector-valued holomorphic functions" of the bundle.
On the Riemann sphere, the tautological line bundle, whose fibre over a line in $\C^2$ is that line, is the simplest nontrivial example.
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# Integral transforms of tempered distributions

## Definition

Let $\mathcal{S}(\R^n)$ be the [[Schwartz space|GE020MY43M]] and $\mathcal{S}'(\R^n)$ the space of [[tempered distributions|8TK1JNG87X]], with pairing $\langle u, \varphi \rangle$.
Let $K \colon \R^n \times \R^n \to \C$ be a measurable kernel and let $T$ be the [[integral transform|YCK709QB06]] $(T\varphi)(x) = \int_{\R^n} K(x, y)\, \varphi(y)\, dy$.
Let $T^t$ be the transform with the transposed kernel, $(T^t\varphi)(y) = \int_{\R^n} K(x, y)\, \varphi(x)\, dx$.
Assume that $T$ and $T^t$ map $\mathcal{S}(\R^n)$ continuously into $\mathcal{S}(\R^n)$.
The *transform* of $u \in \mathcal{S}'(\R^n)$ is the tempered distribution $Tu$ given by
$$\langle Tu, \varphi \rangle = \langle u, T^t \varphi \rangle \quad (\varphi \in \mathcal{S}(\R^n)).$$

## Remarks

The definition agrees with the integral on Schwartz functions: for $f, \varphi \in \mathcal{S}(\R^n)$, [[Fubini's theorem|0R0T1VWH5X]] gives $\int (Tf)\varphi = \iint K(x, y) f(y) \varphi(x)\, dy\, dx = \int f\, (T^t\varphi)$, when the double integral converges absolutely.
Example: the [[Fourier transform|6V59T5KJBX]] has the symmetric kernel $K(x, y) = e^{-2\pi i x \cdot y}$, so $T^t = T$ maps $\mathcal{S}$ onto $\mathcal{S}$.
For the Dirac delta $\delta$ at $0$, $\langle T\delta, \varphi \rangle = (T\varphi)(0) = \int \varphi$, so $T\delta = 1$.
Non-example: the kernel $K(x, y) = e^{xy}$ on $\R$ does not qualify, since for the Schwartz function $\varphi(y) = e^{-\sqrt{1 + y^2}}$ the integral $\int_\R e^{2y} \varphi(y)\, dy$ diverges, so $T\varphi$ is not defined at $x = 2$.
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# Filtered category

## Definition

Let $J$ be a [[category|A2DE66937D]].
$J$ is *filtered* if:

1. $J$ has at least one object.
2. For any two objects $a$ and $b$ of $J$, there are an object $c$ and [[morphisms|859F0TFVA3]] $a \to c$ and $b \to c$.
3. For any two morphisms $f, g \colon a \to b$ of $J$, there is a morphism $h \colon b \to c$ with $h \circ f = h \circ g$.

## Remarks

Equivalently, every diagram $F \colon K \to J$ with $K$ a finite category has a cocone.
A colimit over a filtered category is a *filtered colimit*.
The dual notion is a *cofiltered* category.

Example: a [[directed set|D9RY4667K6]] $(I, \le)$, seen as a category with one arrow $i \to j$ when $i \le j$, is filtered.
Clause 1 holds because a directed set is not empty, and clause 2 is the existence of a common upper bound.
Clause 3 holds because there is at most one arrow between two objects, so $f = g$.

Non-example: the category with two objects $a, b$, two arrows $f, g \colon a \to b$ and only identities besides, is not filtered.
Clause 3 fails: the only arrow out of $b$ is $\id_b$, and $\id_b \circ f \neq \id_b \circ g$.
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<document source="https://tamnd.github.io/math2/node/TKK0TV2MYC.md" tag="TKK0TV2MYC" kind="concept">
# Cluster algebra

## Summary

A cluster algebra is a [[commutative ring|0310REJJY8]] inside a field of rational functions, generated by special elements called cluster variables.
The cluster variables are grouped into overlapping sets of the same size, the clusters.
One passes from one cluster to the next by a rule called mutation.
Cluster algebras were introduced by Fomin and Zelevinsky.
They link the [[representation theory of quivers|TXVWFWP6VQ]], total positivity, Teichmüller theory and combinatorics.

## Explanations

### L1 human

Start with a few variables and a table of integers that says how the variables interact.
Mutation replaces one variable by a new rational function of the others, built from the table, and also updates the table.
Repeating mutations in all directions produces a family of variables, possibly infinite, and the cluster algebra is the ring they generate.
A surprising fact is that every new variable is a Laurent polynomial in the starting variables, so the divisions always cancel.
For two variables with the simplest table, mutation gives $x_1$, $x_2$, $(x_2 + 1)/x_1$, $(x_1 + x_2 + 1)/(x_1 x_2)$, $(x_1 + 1)/x_2$, and then the list repeats.
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# Nonharmonic Fourier series

## Summary

A nonharmonic Fourier series is a series $\sum_n c_n e^{i\lambda_n t}$ on an interval, where the frequencies $\lambda_n$ need not be the integers.
Ordinary [[Fourier series|G12KXDHPPS]] on $[-\pi, \pi]$ are the case $\lambda_n = n$.
The main questions ask when the exponentials $e^{i\lambda_n t}$ are complete in $L^2$ of the interval.
They also ask when they form a [[Riesz basis|ZYDYD7FB1Z]] or a frame.
The answers involve the density of the frequencies and entire functions of exponential type.
They matter in sampling theory and control theory.

## Explanations

### L1 human

An ordinary Fourier series builds a function on $[-\pi, \pi]$ from the waves $e^{int}$, whose frequencies are the whole numbers.
What if the frequencies are slightly moved, say $\lambda_n = n + \delta_n$ with small $\delta_n$?
The waves are then no longer orthogonal, but they may still be able to represent every function in a stable way.
A classical answer is Kadec's theorem: if all real $\delta_n$ satisfy $\abs{\delta_n} \le L < 1/4$, the waves still form a Riesz basis.
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<document source="https://tamnd.github.io/math2/node/TKQPE0Q05Y.md" tag="TKQPE0Q05Y" kind="concept">
# Elastic net

## Summary

The elastic net fits a linear model by penalised least squares.
Its penalty combines the $\ell^1$ penalty of the [[lasso|KZ226116HW]] and the squared $\ell^2$ penalty of [[ridge regression|3KZW2930V2]].
It keeps the variable selection of the lasso, since some coefficients are set exactly to zero.
The ridge part makes the problem strictly convex.
It also lets groups of strongly correlated regressors enter the model together, where the lasso tends to pick one of them.
It was proposed by Hui Zou and Trevor Hastie.

## Explanations

### L1 human

The lasso likes to keep one regressor out of a group of near copies and drop the rest.
Ridge regression keeps them all but never drops anything.
The elastic net mixes the two penalties with two tuning parameters, so it can drop useless regressors while sharing the weight among correlated useful ones.
If the ridge parameter is zero, it is the lasso; if the lasso parameter is zero, it is ridge regression.
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# Taft algebra

## Example

Let $n \ge 2$ and let $k$ be a field that contains a [[primitive $n$-th root of unity|0J817RTVG6]] $\omega$.
The *Taft algebra* $T_n(\omega)$ is the $k$-algebra generated by $g$ and $x$ with the relations
$$g^n = 1, \qquad x^n = 0, \qquad x g = \omega\, g x.$$
It has the $k$-basis $g^i x^j$ with $0 \le i, j \le n - 1$, so its dimension is $n^2$.
It is a [[Hopf algebra|7XFYP0N84H]] with
$$\Delta(g) = g \otimes g, \quad \Delta(x) = x \otimes 1 + g \otimes x, \quad \varepsilon(g) = 1, \quad \varepsilon(x) = 0, \quad S(g) = g^{-1}, \quad S(x) = -g^{-1} x.$$
The check that $\Delta$ respects the relations: $\Delta(g)^n = g^n \otimes g^n = 1 \otimes 1$.
For $\Delta(x) \Delta(g) = \omega\, \Delta(g) \Delta(x)$, both sides equal $\omega(gx \otimes g + g^2 \otimes gx)$ after using $xg = \omega gx$.
For $\Delta(x)^n = 0$, put $A = g \otimes x$ and $B = x \otimes 1$; then $BA = \omega AB$, so the $q$-binomial formula with $q = \omega$ gives $(A + B)^n = A^n + B^n$, and $A^n = g^n \otimes x^n = 0$, $B^n = x^n \otimes 1 = 0$.
The antipode check on $x$: $S(x) \cdot 1 + S(g) x = -g^{-1}x + g^{-1}x = 0 = \varepsilon(x)$, and $x S(1) + g S(x) = x - x = 0$.

## Remarks

$T_n(\omega)$ is not commutative, since $xg = \omega gx \ne gx$.
It is not cocommutative, since the flip of $\Delta(x)$ is $1 \otimes x + x \otimes g \ne \Delta(x)$.
The antipode satisfies $S^2(x) = g^{-1} x g = \omega x$, so $S^2 \ne \id$; this shows that the antipode of a finite-dimensional Hopf algebra need not be an involution.
For $n = 2$ and $\omega = -1$ (with $\operatorname{char} k \ne 2$) it is Sweedler's $4$-dimensional Hopf algebra, the smallest Hopf algebra that is neither commutative nor cocommutative.

## Explanations

### L1 human

The Taft algebra is a small, explicit Hopf algebra that is "quantum" in both directions.
It is built from a cyclic group element $g$ of order $n$ and a nilpotent element $x$ that do not commute but $q$-commute.
Moving $x$ past $g$ costs a root of unity.
Because everything is finite and explicit, it is the standard first test case for statements about finite-dimensional Hopf algebras.
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# Function of exponential order

## Summary

A function $f$ on $[0, \infty)$ is of exponential order when it grows no faster than some exponential $M e^{at}$ for large $t$.
This is the usual sufficient condition for the [[Laplace transform|0V49JEZW06]] of $f$ to exist.
For a piecewise continuous $f$ of exponential order $a$, the integral $\int_0^\infty f(t) e^{-st}\, dt$ converges absolutely for $\operatorname{Re} s > a$.
Most functions met in linear differential equations, such as polynomials, exponentials, sines and cosines, are of exponential order.

## Explanations

### L1 human

The factor $e^{-st}$ in the Laplace transform decays exponentially.
It can tame $f$ only if $f$ itself grows at most exponentially.
For example, $t^3$ is of exponential order, because $t^3 \le 6 e^t$ for all $t \ge 0$.
The function $e^{t^2}$ is not: it eventually beats every $e^{at}$, and indeed it has no Laplace transform.
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# Domain decomposition methods

## Definition

Let $A x = b$ be a [[linear system|0GT4WWGWH2]] with an invertible real $n \times n$ matrix $A$, arising from a discretized [[boundary value problem|TM8VQJWGHZ]] on a domain $\Omega$.
Let $\Omega_1, \dots, \Omega_N$ be subdomains with $\Omega = \Omega_1 \cup \cdots \cup \Omega_N$, and let $I_i \subseteq \set{1, \dots, n}$ be the set of unknowns that belong to $\Omega_i$.
Let $R_i$ be the $\abs{I_i} \times n$ matrix that selects the entries in $I_i$, and let $A_i = R_i A R_i^T$ be the local matrix.
A *domain decomposition method* is an [[iterative method|2RPE77ACER]] or [[preconditioner|KE98SW3SBD]] for $A x = b$ in which:

1. each step solves local systems with the matrices $A_i$, or with local matrices built from them;
2. the local solutions are combined through the unknowns shared by, or lying on the interfaces between, the subdomains;
3. optionally, a coarse problem of small size couples all subdomains globally.

The method is *overlapping* if the sets $I_i$ overlap, and *nonoverlapping* if the subdomains meet only along interfaces.

## Remarks

Example: the additive Schwarz preconditioner is $M^{-1} = \sum_{i=1}^{N} R_i^T A_i^{-1} R_i$, for overlapping $I_i$ that cover all unknowns.
Each term solves a local problem, and the terms can be computed in parallel, so clauses 1 and 2 hold.
If $A$ is symmetric [[positive definite|0KX66CZMBB]], each $A_i$ is too, and $M^{-1}$ is symmetric positive definite.
Non-example: the [[finite element method|98GZ9DVHTA]] with a direct solver for the whole of $A$ solves no local systems, so it fails clause 1.
Nonoverlapping methods eliminate the interior unknowns of each subdomain and iterate on the [[Schur complement|6HWFBXYPZP]] for the interface unknowns.
Without a coarse problem, the number of iterations grows with the number of subdomains, because information travels only between neighbours at each step.
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<document source="https://tamnd.github.io/math2/node/TM1RE1G7JK.md" tag="TM1RE1G7JK" kind="statement">
# Schur product theorem

## Statement

Let $n \ge 1$ and let $A$ and $B$ be $n \times n$ complex [[positive semidefinite matrices|01MMVW8CKC]].
Then their [[Hadamard product|SQT6QCEXZ8]] $A \circ B$, the matrix with entries $(A \circ B)_{ij} = a_{ij} b_{ij}$, is positive semidefinite.

## Remarks

By the [[spectral theorem for Hermitian matrices|0DGZBQWZWN]], $B = \sum_k v_k v_k^*$ for some vectors $v_k \in \C^n$.
For one vector $v$, $A \circ (v v^*) = D A D^*$ with $D = \operatorname{diag}(v_1, \ldots, v_n)$, which is positive semidefinite.
So $A \circ B = \sum_k D_k A D_k^*$ is a sum of positive semidefinite matrices.
Example: $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$ are positive definite, and $A \circ B = \begin{pmatrix} 2 & -1 \\ -1 & 4 \end{pmatrix}$ has trace $6$ and determinant $7$, so it is positive definite.
If $A$ and $B$ are both [[positive definite|0KX66CZMBB]], then so is $A \circ B$.
The hypothesis on $B$ is needed: with $A = I_2$ and $B = \operatorname{diag}(1, -1)$, $A \circ B = B$ is not positive semidefinite.

## Explanations

### L1 human

Multiply two matrices entry by entry, not by the usual row-times-column rule.
If both matrices are positive semidefinite, the result is again positive semidefinite.
For the usual product this would fail, since $AB$ need not even be Hermitian.

### L2 human

Positive semidefinite matrices are [[Gram matrices|00SPJSDT3Z]] $a_{ij} = \ip{x_i}{x_j}$, and the entrywise product of two Gram matrices is the Gram matrix of the tensor products $x_i \otimes y_i$; this gives a second proof.
The theorem shows that the entrywise power $(a_{ij}^k)$ and the entrywise exponential $(e^{a_{ij}})$ of a positive semidefinite matrix are positive semidefinite.
This is why sums, products and exponentials of positive definite kernels are again positive definite kernels in machine learning.
</document>

<document source="https://tamnd.github.io/math2/node/TM528Q96ZD.md" tag="TM528Q96ZD" kind="concept">
# Proper holomorphic map

## Summary

A proper holomorphic map is a [[holomorphic map|3MQTWKBBBT]] between domains, or complex manifolds, that is also a [[proper map|15XEBGNGHC]]: preimages of compact sets are compact.
For bounded domains this means that the map sends points near the boundary of the source to points near the boundary of the target.
Proper holomorphic maps between domains of the same dimension are finite branched coverings, so they generalize [[biholomorphisms|FX92PP3HZR]].
In several variables they are rigid: by [[Alexander's theorem|T3VN45KV0D]], every proper holomorphic self-map of the ball in $\C^n$ with $n \ge 2$ is an automorphism.

## Explanations

### L1 human

A proper map cannot hide the boundary: as a point runs to the edge of the source domain, its image runs to the edge of the target.
The map $z \mapsto z^2$ on the unit disc is a typical example.
It wraps the disc twice onto itself, and points near the unit circle go to points near the unit circle.
The inclusion of a smaller disc into the unit disc is not proper, since points near the edge of the small disc land in the middle of the big one.
</document>

<document source="https://tamnd.github.io/math2/node/TM5SYH4XHY.md" tag="TM5SYH4XHY" kind="statement">
# Löwner–John theorem

## Statement

Let $n \ge 1$ and let $K \subseteq \R^n$ be a compact [[convex body|437FSG3419]] with nonempty interior.
An ellipsoid is a set $c + AB$, with $B$ the closed unit ball, $c \in \R^n$ and $A$ an invertible real $n \times n$ [[matrix|0VD9ZSWWJG]]; volume means [[Lebesgue measure|2Z2QVWZ262]].

1. Among the ellipsoids contained in $K$ there is exactly one, $E = c + AB$, of maximal volume.
2. For this $E$, one has $K \subseteq c + nAB$.
3. If moreover $K = -K$, then $c = 0$ and $K \subseteq \sqrt{n}\, E$.
4. Among the ellipsoids containing $K$ there is exactly one of minimal volume.

## Remarks

One may take $A$ symmetric positive definite, by the polar decomposition.
Existence follows from compactness: the pairs $(c, A)$ with $A$ positive semidefinite and $c + AB \subseteq K$ form a compact set, and volume is continuous.
For uniqueness, let $c_1 + A_1 B$ and $c_2 + A_2 B$, with $A_i$ positive definite, both have maximal volume.
Then $\frac{c_1 + c_2}{2} + \frac{A_1 + A_2}{2} B$ also lies in $K$.
Its volume is strictly larger unless $A_1 = A_2$, by the strict log-concavity of the determinant on positive definite matrices.
If $A_1 = A_2$ but $c_1 \neq c_2$, the convex hull of the two ellipsoids contains a larger ellipsoid, stretched in the direction $c_2 - c_1$.
John characterized the case $E = B$ by contact points.
For $K \supseteq B$, the ball $B$ is the maximal ellipsoid if and only if there are $u_1, \dots, u_m \in \partial K$ with $\abs{u_i} = 1$ and weights $c_i > 0$ such that $\sum_i c_i u_i = 0$ and $\sum_i c_i u_i u_i^T = I_n$.
The bounds in 2 and 3 follow from this.
The factor $n$ in 2 is attained by a regular simplex.
The factor $\sqrt{n}$ in 3 is attained by the cube $[-1, 1]^n$, whose maximal ellipsoid is $B$ and which contains the point $(1, \dots, 1)$ of norm $\sqrt{n}$.

## Explanations

### L1 human

Every convex shape with some thickness contains a largest ellipsoid, and it is unique.
The shape is then caught between this ellipsoid and the same ellipsoid enlarged by a factor equal to the dimension.
For shapes that are symmetric about a center, the factor improves to the square root of the dimension.
So, after a linear change of coordinates, every convex body looks like a ball up to a bounded distortion.

### L2 human

Part 3 says that every $n$-dimensional normed space is at [[Banach–Mazur distance|ZJKSYYASPT]] at most $\sqrt{n}$ from Euclidean space, a basic fact of the local theory of Banach spaces.
The contact point condition in John's characterization is a decomposition of the identity.
Ball combined it with the Brascamp–Lieb inequality to prove his reverse isoperimetric inequality.
In optimization, rounding algorithms for convex bodies compute approximate John ellipsoids.
</document>

<document source="https://tamnd.github.io/math2/node/TM7QS7P7X9.md" tag="TM7QS7P7X9" kind="concept">
# Free product

## Summary

The free product $G * H$ of two [[groups|G5S1M9]] is the group generated by $G$ and $H$ with no relations between them.
Its elements are reduced words that alternate between non-identity elements of $G$ and of $H$.
It is the [[coproduct|4FJ0S71P58]] in the category of groups.
The free group of rank $n$ is the free product of $n$ copies of $\Z$.

## Explanations

### L1 human

To form a free product, put two groups side by side and let their elements multiply freely, without imposing any commutation between them.
An element is a word such as $g_1 h_1 g_2 h_2$, with letters taken alternately from the two groups.
For example, $\Z/2 * \Z/2$ is generated by two reflections $s$ and $t$, and the words $st$, $stst$, $ststst$ are all different, so the group is infinite.
In topology, the fundamental group of two well-behaved spaces, such as graphs, glued at a point is the free product of their fundamental groups.
</document>

<document source="https://tamnd.github.io/math2/node/TM8PYTBT6Y.md" tag="TM8PYTBT6Y" kind="statement">
# Borel density theorem

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] without compact factors, and let $\Gamma$ be a [[lattice|BTYF5Y5PFY]] in $G$.
Let $\rho \colon G \to \mathrm{GL}(V)$ be a continuous representation on a finite-dimensional real vector space $V$.
Then:

1. every linear subspace of $V$ that is [[invariant|0ACX5QT8TE]] under $\rho(\Gamma)$ is invariant under $\rho(G)$;
2. the closure of $\rho(\Gamma)$ in the [[Zariski topology|8SQK7RTB63]] of $\mathrm{GL}(V)$ contains $\rho(G)$.

## Remarks

Furstenberg's proof for clause 1 goes as follows.
View a $\Gamma$-invariant subspace $W$ of dimension $k$ as a point of the projective space $P$ of $\Lambda^k V$.
The map $g\Gamma \mapsto \rho(g) W$ pushes the invariant probability measure on $G/\Gamma$ to a $G$-invariant probability measure on $P$.
If $\rho(u)$ is unipotent, the points $\rho(u)^m x$ converge to fixed points of $\rho(u)$, so such a measure lives on these fixed points.
As $G$ is generated by such elements $u$, the measure lives on $G$-fixed points, and this forces $W$ to be $G$-invariant.

Example: $\Gamma = \mathrm{SL}_2(\Z)$ in $G = \mathrm{SL}_2(\R)$, with the standard representation on $\R^2$.
A line invariant under $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ must be $\R e_1$, and a line invariant under $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ must be $\R e_2$.
So no line is invariant under $\Gamma$, as clause 1 predicts, since no line is invariant under $G$.

The hypothesis on compact factors is necessary.
In the compact group $G = \mathrm{SO}(3)$, the trivial subgroup is a lattice, and every subspace of $\R^3$ is invariant under it.
But the axis $\R e_3$ is not invariant under $\mathrm{SO}(3)$.

Consequences: the centraliser of $\Gamma$ in $G$ is the centre of $G$, and $\Gamma$ is not contained in any proper algebraic subgroup.

## Explanations

### L1 human

A lattice is a discrete subgroup with a quotient of finite volume, such as $\mathrm{SL}_2(\Z)$ in $\mathrm{SL}_2(\R)$.
It is a tiny, countable part of the big group.
Still, the theorem says that no polynomial equation can tell the lattice apart from the big group.
Any linear structure preserved by the lattice is preserved by the whole group.

### L2 human

The Borel density theorem is the first step of many rigidity results.
It lets one pass from statements about $\Gamma$ to statements about $G$, for instance in the [[Margulis superrigidity theorem|DSKV65NHZZ]], where Zariski density is part of the setup.
It also shows that the adjoint representation of $G$ restricted to $\Gamma$ is irreducible when $G$ is simple, which is used for local rigidity of lattices.
</document>

<document source="https://tamnd.github.io/math2/node/TM8VQJWGHZ.md" tag="TM8VQJWGHZ" kind="concept">
# Boundary value problem

## Summary

A boundary value problem is a differential equation together with conditions on the solution at the boundary of the domain.
For an ordinary differential equation, these are conditions at both ends of an interval.
It differs from an [[initial value problem|0FXDW992DS]], where all conditions are given at one point.
A boundary value problem may have no solution, exactly one, or infinitely many, and the answer depends on the whole interval.
Such problems describe equilibria, such as a string fixed at both ends, and lead to eigenvalue problems and Sturm–Liouville theory.
See the [[definition|1HGGRS57KT]].

## Explanations

### L1 human

Imagine a rope tied to two posts.
Its shape is decided by conditions at both ends, not by where it starts and in which direction it points.
For $y'' + y = 0$ with $y(0) = 0$ and $y(\frac{\pi}{2}) = 1$, the only solution is $y = \sin x$.
With $y(0) = 0$ and $y(\pi) = 1$ instead there is no solution, since every solution with $y(0) = 0$ is $c \sin x$, which vanishes at $\pi$.
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<document source="https://tamnd.github.io/math2/node/TMAHZSNTA1.md" tag="TMAHZSNTA1" kind="definition">
# Nevanlinna class

## Definition

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].
For $x > 0$ let $\log^+ x = \max(\log x, 0)$, and let $\log^+ 0 = 0$.
The *Nevanlinna class* $N$ is the set of [[holomorphic functions|0DG0Z72CQD]] $f \colon \mathbb{D} \to \C$ such that
$$\sup_{0 < r < 1} \frac{1}{2\pi} \int_0^{2\pi} \log^+ \abs{f(r e^{it})} \, dt < \infty.$$

## Remarks

A holomorphic function on $\mathbb{D}$ lies in $N$ if and only if $f = g/h$ with $g$ and $h$ bounded holomorphic on $\mathbb{D}$ and $h$ without zeros.
Such functions are said to be of bounded type.
Since $\log^+ x \le x^p / p$ for $p > 0$, every Hardy space $H^p$ is contained in $N$.

Example: $f(z) = \exp\left(\frac{1}{1 - z}\right)$.
For $z = r e^{it}$ one has $\log \abs{f(z)} = \operatorname{Re} \frac{1}{1 - z} = \frac{1 - r \cos t}{1 - 2r \cos t + r^2} > 0$.
This equals $\frac{1}{2}\left(1 + \frac{1 - r^2}{1 - 2r \cos t + r^2}\right)$, and the Poisson kernel has mean $1$, so the mean of $\log^+ \abs{f}$ on each circle is exactly $1$.
So $f \in N$, although $f$ lies in no $H^p$.

Non-example: $f(z) = \sin\left(\frac{\pi}{1 - z}\right)$ is holomorphic on $\mathbb{D}$ but not in $N$.
Its zeros are $z_k = 1 - 1/k$ for $k \ge 1$, and $\sum_k (1 - \abs{z_k}) = \sum_k 1/k$ diverges.
By Jensen's formula, the zeros of a function in $N$ that is not identically $0$ satisfy the Blaschke condition $\sum (1 - \abs{z_k}) < \infty$.
</document>

<document source="https://tamnd.github.io/math2/node/TMBCW25DZ4.md" tag="TMBCW25DZ4" kind="definition">
# Scott continuity

## Definition

Let $P$ and $Q$ be [[partially ordered sets|0QV2JS9RJ3]].
A map $f \colon P \to Q$ is *Scott-continuous* if it satisfies the following.
For every [[directed subset|D9RY4667K6]] $D \subseteq P$ with a [[supremum|0R3H5PANRQ]] $\bigvee D$ in $P$, the image $f(D)$ has a supremum in $Q$ and
$$f\left(\bigvee D\right) = \bigvee f(D).$$

## Remarks

A Scott-continuous map is [[monotone|0A7KV2WB7A]]: if $x \le y$, apply the definition to $D = \set{x, y}$, whose supremum is $y$, to get $f(x) \le f(y)$.
A map $f$ is Scott-continuous if and only if it is [[continuous|0X3WGX95FE]] for the [[Scott topologies|ZR8A03FV67]] on $P$ and $Q$.
Example: on the power set of $\N$, the map $A \mapsto A \cup \set{0}$; the image of the union of a directed family $D$ is $\bigcup_{A \in D} (A \cup \set{0})$.
Non-example: on the power set of $\N$, let $f(A) = \N$ if $A$ is infinite and $f(A) = \varnothing$ if $A$ is finite.
This map is monotone but not Scott-continuous.
The finite sets $\set{0, \ldots, n}$ form a directed family with union $\N$, and $f(\N) = \N \ne \varnothing = \bigvee_n f(\set{0, \ldots, n})$.
</document>

<document source="https://tamnd.github.io/math2/node/TMCMTXNZ63.md" tag="TMCMTXNZ63" kind="definition">
# Eichler order

## Definition

Let $F$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\mathcal{O}_F$, and let $B$ be a [[quaternion algebra|ZWXW0T3EAC]] over $F$.
An *Eichler order* in $B$ is an $\mathcal{O}_F$-[[order|1ZBX4TKVZ9]] $\mathcal{O} \subseteq B$ such that $\mathcal{O} = \mathcal{O}_1 \cap \mathcal{O}_2$ for two [[maximal orders|E5DH1CSW1N]] $\mathcal{O}_1, \mathcal{O}_2$ of $B$.

## Remarks

Example: $F = \Q$, $B = M_2(\Q)$, $N \ge 1$ and $g = \operatorname{diag}(1, N)$.
The order $g M_2(\Z) g^{-1}$ consists of the matrices $\begin{pmatrix} a & b/N \\ Nc & d \end{pmatrix}$ with $a, b, c, d \in \Z$.
Its intersection with $M_2(\Z)$ is
$$\mathcal{O}_0(N) = \set{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M_2(\Z) : N \mid c},$$
so $\mathcal{O}_0(N)$ is an Eichler order; $N$ is called its level.
Non-example: the order $\Z + 2 M_2(\Z)$ in $M_2(\Q)$ is not an Eichler order.
Every maximal order of $M_2(\Q)$ is the endomorphism ring of a lattice in $\Q^2$.
By the elementary divisor theorem, two such lattices have a common adapted basis.
So every Eichler order of $M_2(\Q)$ contains an [[idempotent|0N5NJZKKN3]] other than $0$ and $1$.
An idempotent $e$ of $\Z + 2M_2(\Z)$ is congruent to $0$ or $1$ modulo $2M_2(\Z)$; if $e \equiv 0$ then $e = e^k \in 2^k M_2(\Z)$ for all $k$, so $e = 0$, and symmetrically $1 - e \equiv 0$ forces $e = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/TMEKPXMC7V.md" tag="TMEKPXMC7V" kind="statement">
# Separation principle

## Statement

Consider the [[linear-quadratic-Gaussian control problem|DJB8MAV5BH]] on $[0, T]$ with matrices $A, B, C, Q, R, F$, noise covariances $\Xi$ and $\Theta$, and initial mean $m_0$ and covariance $\Sigma_0$.
Let $P$ solve the [[Riccati differential equation|Y4V154JWR7]] $-P' = A^{T} P + P A - P B R^{-1} B^{T} P + Q$ with $P(T) = F$.
Let $\Sigma$ solve the filter Riccati equation $\Sigma' = A \Sigma + \Sigma A^{T} + \Xi - \Sigma C^{T} \Theta^{-1} C \Sigma$ with $\Sigma(0) = \Sigma_0$.
Let $\hat{x}$ be the Kalman–Bucy filter, the solution of
$$d\hat{x} = (A \hat{x} + B u) \, dt + \Sigma C^{T} \Theta^{-1} (dy - C \hat{x} \, dt), \qquad \hat{x}(0) = m_0.$$
Then the control $u^*(t) = -R^{-1} B^{T} P(t) \hat{x}(t)$ is admissible and optimal among all admissible controls.
Here $\hat{x}(t)$ is the [[conditional expectation|0TTDGTEFMN]] of $x(t)$ given the observations $y(s)$, $s \le t$.

## Remarks

The proof has two parts.
First, for every admissible control, the error $x - \hat{x}$ is a Gaussian process whose law does not depend on $u$, with covariance $\Sigma(t)$.
Second, writing $x = \hat{x} + (x - \hat{x})$ splits the cost into a term that does not depend on $u$ and the cost of a fully observed linear-quadratic problem for $\hat{x}$.
The [[linear-quadratic regulator|HR720TJZZR]] then gives the optimal feedback for $\hat{x}$.
The gain $R^{-1} B^{T} P$ is the same as without noise, and the filter gain $\Sigma C^{T} \Theta^{-1}$ does not depend on the cost: estimation and control separate.
A deterministic version holds for observer-based feedback of $x' = A x + B u$, $y = C x$.
With the observer $\hat{x}' = A \hat{x} + B u + L(y - C \hat{x})$ and $u = -K \hat{x}$, the error $e = x - \hat{x}$ solves $e' = (A - L C) e$.
In the coordinates $(x, e)$ the closed-loop matrix is block triangular, so its eigenvalues are those of $A - B K$ together with those of $A - L C$.
The principle depends on linear dynamics, Gaussian noise and quadratic cost; for nonlinear problems the best control may need to act so as to improve future estimates.

## Explanations

### L1 human

When you must control a system that you can only observe through noisy measurements, the separation principle gives a simple recipe.
First build the best estimate of the current state from the measurements.
Then apply the control that would be optimal if that estimate were exactly right.
For linear systems with Gaussian noise and quadratic costs, this two-step recipe is exactly optimal.

### L2 human

The estimator is the Kalman–Bucy filter and the controller is the [[linear-quadratic regulator|HR720TJZZR]] gain, and each comes from its own Riccati equation.
The two Riccati equations are dual: the filter equation is the control equation for $(A^{T}, C^{T})$ run forward in time.
In practice this lets engineers design the estimator and the controller separately.
The deterministic version is the reason why pole placement for the controller and for the observer can be done independently.
</document>

<document source="https://tamnd.github.io/math2/node/TMFG2P912R.md" tag="TMFG2P912R" kind="concept">
# Degree of a continuous map

## Summary

Let $f$ be a continuous map between closed connected oriented manifolds of the same dimension.
Its degree is an integer that counts, with signs, how many times the source wraps around the target.
For maps from the $n$-sphere to itself it is defined by the action on the top homology group $H_n(S^n; \Z) \cong \Z$.
The degree is a homotopy invariant, and for maps to a sphere it is a complete invariant by the [[Hopf degree theorem|902X7EHFSC]].
It underlies the proofs of the Brouwer fixed point theorem, the hairy ball theorem and the fundamental theorem of algebra.

## Explanations

### L1 human

Think of a rubber band wrapped around a circular post.
The degree counts how many times it goes around, counting clockwise turns as negative.
The map $z \mapsto z^3$ on the unit circle in $\C$ wraps the circle three times around itself, so it has degree $3$.
Sliding the band around without cutting it never changes this count.
</document>

<document source="https://tamnd.github.io/math2/node/TMJ0SFW7M7.md" tag="TMJ0SFW7M7" kind="definition">
# Semilocally simply connected space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *semilocally simply connected* if every point $x \in X$ has an open neighbourhood $U$ with the following property.
The homomorphism $\pi_1(U, x) \to \pi_1(X, x)$ of [[fundamental groups|S1XK3HXHFX]] induced by the inclusion $U \hookrightarrow X$ is trivial.
Equivalently, every loop in $U$ based at $x$ is null-homotopic in $X$.

## Remarks

Example: a manifold is semilocally simply connected.
Each point has a neighbourhood $U$ homeomorphic to an open ball, so $\pi_1(U, x)$ is already trivial.

Non-example: the [[Hawaiian earring|73EBKQPP8Q]] $H = \bigcup_{n \ge 1} C_n$, where $C_n \subset \R^2$ is the circle of radius $1/n$ centred at $(1/n, 0)$, is not.
Every neighbourhood $U$ of the origin contains some $C_n$, and a loop around $C_n$ is not null-homotopic in $H$, because $H$ retracts onto $C_n$.

The cone on the Hawaiian earring is contractible, so it is semilocally simply connected, but it is not locally simply connected.
The spelling "semi-locally simply connected" is also common.
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<document source="https://tamnd.github.io/math2/node/TMKXXG8XHR.md" tag="TMKXXG8XHR" kind="definition">
# Semiprimitive ring

## Definition

Let $R$ be a [[ring|0894J980RK]] with identity.
$R$ is *semiprimitive* if its [[Jacobson radical|XA0DCNH7WY]] $J(R)$, the intersection of all maximal [[left ideals|5N2CJZ47D4]] of $R$, is $0$.

## Remarks

Example: $\Z$ is semiprimitive, since its maximal ideals are $p\Z$ for primes $p$ and $\bigcap_p p\Z = 0$.
Non-example: the local ring $\Z_{(p)}$ of fractions with denominator prime to $p$ is not semiprimitive, since $J(\Z_{(p)}) = p\Z_{(p)} \ne 0$.
Equivalently, $R$ is a subdirect product of left [[primitive rings|8F6EHDVJ62]], or $R$ has a faithful semisimple left module.
The definition is left-right symmetric, because $J(R)$ is also the intersection of the maximal right ideals.
Other names are *Jacobson semisimple* and *J-semisimple*.
</document>

<document source="https://tamnd.github.io/math2/node/TMM9SGJMSP.md" tag="TMM9SGJMSP" kind="definition">
# Bifurcation in a family of maps

## Definition

Let $X \subseteq \R^k$, let $I \subseteq \R$ be an open interval, and let $f_\mu \colon X \to X$, $\mu \in I$, be a family of maps such that $(\mu, x) \mapsto f_\mu(x)$ is [[continuous|0X3WGX95FE]].
A parameter $\mu_0 \in I$ is a *bifurcation value* of the family if the following holds.
Every open interval $J \ni \mu_0$ contains some $\mu$ such that $f_\mu$ is not [[topologically conjugate|HNXXAPQKG3]] to $f_{\mu_0}$.
One then says that the family, or the difference equation $x_{n+1} = f_\mu(x_n)$, undergoes a *bifurcation* at $\mu_0$.

## Remarks

Example: for $f_\mu(x) = \mu x$ on $\R$ with $\mu \in (0, \infty)$, the value $\mu_0 = 1$ is a bifurcation value.
The map $f_1$ is the identity, so every point is fixed, while for $\mu \ne 1$ the only fixed point is $0$.
A conjugacy maps fixed points bijectively to fixed points, so $f_\mu$ is not conjugate to $f_1$.

Non-example: in the same family, $\mu_0 = 1/2$ is not a bifurcation value.
For $0 < a, b < 1$ the homeomorphism $h(x) = \sgn(x) \abs{x}^{\ln b / \ln a}$ satisfies $h(a x) = b\, h(x)$, so all $f_\mu$ with $\mu \in (0, 1)$ are conjugate to each other.

Many texts use a local version, in which only the behaviour near a fixed point or a periodic orbit is compared.
A local bifurcation of a fixed point $x^*$ of a smooth family can only occur when an eigenvalue of $D f_{\mu_0}(x^*)$ lies on the unit circle.
For one-dimensional maps this means $f_{\mu_0}'(x^*) = 1$ or $f_{\mu_0}'(x^*) = -1$.
</document>

<document source="https://tamnd.github.io/math2/node/TMNNRRRGAS.md" tag="TMNNRRRGAS" kind="concept">
# Gegenbauer transform

## Summary

The Gegenbauer transform is a finite [[integral transform|YCK709QB06]] on $[-1, 1]$.
It sends a function to its integrals against the [[Gegenbauer polynomials|5TRFA6F79B]] $C_n^{\lambda}$ with the weight $(1 - x^2)^{\lambda - 1/2}$.
It is the symmetric case $\alpha = \beta$ of the [[Jacobi transform|72TSY1Z4VA]], up to constant factors.
It turns the Gegenbauer differential operator into multiplication by $-n(n + 2\lambda)$.
For $\lambda = 1/2$ it is the [[Legendre transform|51PH7Z03TJ]].

## Explanations

### L1 human

Gegenbauer polynomials are orthogonal polynomials on $[-1, 1]$ for a weight that is symmetric about $0$.
They appear naturally for functions on spheres that only depend on one angle.
The Gegenbauer transform lists how much of each polynomial a function contains.
For example, with $\lambda = 1$ the polynomials are the Chebyshev polynomials of the second kind, and the weight is $\sqrt{1 - x^2}$.
</document>

<document source="https://tamnd.github.io/math2/node/TMTPWVETQK.md" tag="TMTPWVETQK" kind="concept">
# Primary ideal

## Summary

A primary ideal of a commutative ring is a proper ideal $Q$ with this property.
Whenever a product $ab$ lies in $Q$ and $a$ does not, some power of $b$ lies in $Q$.
Primary ideals generalise powers of prime numbers: in $\Z$ the nonzero primary ideals are the ideals $(p^n)$ for primes $p$.
The [[radical|437YRCNYRJ]] of a primary ideal is a [[prime ideal|00MHHZ9SNE]].
Primary ideals are the building blocks of [[primary decomposition|ZD34THCXQ0]].

## Explanations

### L1 human

In the integers, every number is a product of prime powers, and the ideals $(p^n)$ are the primary ideals.
A primary ideal is "concentrated at one prime": its radical is a single prime ideal, as $(p^n)$ has radical $(p)$.
For example, $(8) = (2^3)$ is primary in $\Z$, but $(6)$ is not, since $6$ involves two different primes.
</document>

<document source="https://tamnd.github.io/math2/node/TMY3M75EK8.md" tag="TMY3M75EK8" kind="definition">
# Iterated function system

## Definition

Let $(X, d)$ be a [[complete metric space|0BY52JW6NK]].
An *iterated function system* on $X$ is a finite family $f_1, \ldots, f_N \colon X \to X$, with $N \ge 1$, of [[contractions|0QN69YA357]]: there are constants $r_i < 1$ with $d(f_i(x), f_i(y)) \le r_i \, d(x, y)$ for all $x, y \in X$.
Its *attractor* is a nonempty [[compact|00R0X1WGPX]] set $A \subseteq X$ with
$$A = f_1(A) \cup \cdots \cup f_N(A).$$

## Remarks

Hutchinson's theorem says that the attractor exists and is unique.
The proof applies the [[Banach fixed point theorem|0VK6VEH345]] to the map $K \mapsto \bigcup_i f_i(K)$ on the space of nonempty compact subsets of $X$ with the Hausdorff metric.
This space is complete, and the map is a contraction on it with constant $\max_i r_i$.
So for every nonempty compact $K_0$, the iterates $K_{m+1} = \bigcup_i f_i(K_m)$ converge to $A$.

Example: on $\R$, the maps $f_1(x) = x/2$ and $f_2(x) = x/2 + 1/2$ are contractions with constant $\frac{1}{2}$, and $f_1([0, 1]) \cup f_2([0, 1]) = [0, \frac{1}{2}] \cup [\frac{1}{2}, 1] = [0, 1]$.
So the attractor is $[0, 1]$.
With $f_1(x) = x/3$ and $f_2(x) = x/3 + 2/3$ the attractor is the [[Cantor set|0BXNDZFQME]].

Non-example: the family $g_1(x) = x/3$, $g_2(x) = x + 1$ on $\R$ is not an iterated function system, since $g_2$ is not a contraction.
It has no nonempty compact set $A$ with $A = g_1(A) \cup g_2(A)$: the maximum $a$ of such an $A$ would satisfy $a + 1 \in A$, which is impossible.

Suppose that the maps are similarities with ratios $r_i$ and that the pieces $f_i(A)$ do not overlap too much (the open set condition).
Then the [[Hausdorff dimension|SSNFTHZ6YJ]] of $A$ is the number $s$ with $\sum_i r_i^s = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/TN1X6FNT87.md" tag="TN1X6FNT87" kind="definition">
# Neighborly polytope

## Definition

Let $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d$, and let $k \ge 1$ be an integer.
$P$ is *$k$-neighborly* if for every set $S$ of $k$ [[vertices|R7V3S9V891]] of $P$, the convex hull of $S$ is a [[face|KY7SB854VH]] of $P$.
$P$ is *neighborly* if it is $\floor{d/2}$-neighborly.

## Remarks

Every polytope is $1$-neighborly, so in dimensions $2$ and $3$ every polytope is neighborly.
A $2$-neighborly polytope is one whose graph is complete: every two vertices are joined by an edge.
If $P$ is $k$-neighborly for some $k > \floor{d/2}$, then $P$ is a simplex.

Example: the cyclic polytope $C(n, 4)$ for $n \ge 5$ is $2$-neighborly.
For $i < j$, write $p(t) = (t - t_i)^2 (t - t_j)^2 = a_0 + a_1 t + \dots + a_4 t^4$.
The affine function $\ell(y) = a_0 + a_1 y_1 + \dots + a_4 y_4$ satisfies $\ell(x(t)) = p(t) \ge 0$.
It vanishes only at the vertices with parameters $t_i$ and $t_j$, so these two vertices span an edge.

Non-example: the $4$-cube $[0,1]^4$ is not neighborly, since the opposite vertices $0$ and $(1,1,1,1)$ are not joined by an edge.
</document>

<document source="https://tamnd.github.io/math2/node/TN2R1TDE3G.md" tag="TN2R1TDE3G" kind="definition">
# Nome

## Definition

Let $\tau$ be a point of the [[upper half-plane|X6XGGH5FPW]].
The *nome* of $\tau$ is $q = e^{\pi i\tau}$; it satisfies $0 < \abs{q} < 1$.

For a real modulus $0 < k < 1$, let $K = K(k)$ and $K' = K(\sqrt{1 - k^2})$ be [[complete elliptic integrals of the first kind|RM54CPPE12]], and let $\tau = iK'/K$.
The *nome of the modulus* $k$ is
$$q(k) = e^{\pi i\tau} = \exp\!\left(-\frac{\pi K'}{K}\right).$$

## Remarks

For real $k \in (0, 1)$ the nome is real with $0 < q < 1$, and it increases from $0$ to $1$ as $k$ goes from $0$ to $1$.
Example: at $k = 1/\sqrt2$ one has $k = \sqrt{1 - k^2}$, so $K = K'$ and $q = e^{-\pi} \approx 0.0432$.
As $k \to 0$, $K \to \pi/2$ and $K' = \log(4/k) + o(1)$, so $q = k^2/16 + O(k^4)$.
Non-example: $\tau = 0$ or real $\tau$ gives $\abs{q} = 1$, which is not a nome, since the theta series in $q$ do not converge there.
Notation varies: some authors, especially in the theory of [[modular forms|GWKN5VG78A]], use $q = e^{2\pi i\tau}$ instead and call it the nome as well.
With the nome $q = e^{\pi i\tau}$ defined above, the [[Jacobi theta function|2F5CYQJVAF]] is $\vartheta(z, \tau) = \sum_{n \in \Z} q^{n^2} e^{2\pi i n z}$.
</document>

<document source="https://tamnd.github.io/math2/node/TN4GDFWGAR.md" tag="TN4GDFWGAR" kind="statement">
# Friendship theorem

## Statement

Let $G$ be a finite nonempty [[graph|0JD5C82QMB]] in which every two distinct vertices have exactly one common [[neighbour|01T3TMGTTE]].
Then $G$ has a vertex adjacent to all other vertices.
Consequently, for some integer $k \ge 0$, the graph $G$ consists of $k$ triangles that share one common vertex and are otherwise disjoint.

## Remarks

The second sentence follows from the first.
If $w$ is adjacent to all other vertices, each other vertex $u$ has exactly one common neighbour with $w$.
So the vertices other than $w$ fall into pairs of adjacent vertices.
For the first sentence, suppose that no vertex is adjacent to all others.
A counting argument shows that two non-adjacent vertices have the same degree, and from this that $G$ is $k$-regular for some $k$.
Then the [[adjacency matrix|0NZ8MDBN6Q]] $A$ satisfies $A^2 = (k - 1) I + J$, where $J$ is the all-ones matrix, so the eigenvalues of $A$ are $k$ and $\pm \sqrt{k - 1}$.
Since the trace of $A$ is $0$, the number $\sqrt{k - 1}$ must be rational, and the multiplicities force $k = 2$.
Then $G$ is a triangle, which has a vertex adjacent to all others, a contradiction.
The finiteness hypothesis is needed: there are infinite graphs in which every two vertices have exactly one common neighbour but no vertex is adjacent to all others.
Paul Erdős, Alfréd Rényi and Vera Sós proved the theorem in 1966.

## Explanations

### L1 human

Suppose that in a group of people, every two people have exactly one friend in common.
Then someone is friends with everybody else, a "politician".
Moreover, the rest of the group splits into pairs of friends, each pair together with the politician forming a triangle.

### L2 human

The usual proof is a short application of linear algebra to a combinatorial condition: the hypothesis says exactly that $A^2 = (\text{diagonal of degrees}) - I + J$.
It is a typical use of eigenvalues in [[spectral graph theory|5JESM2ZN48]], where integrality of eigenvalue multiplicities rules out most cases.
The graphs it describes, $k$ triangles glued at a vertex, are called friendship graphs or windmill graphs.
</document>

<document source="https://tamnd.github.io/math2/node/TN6M6DVZRP.md" tag="TN6M6DVZRP" kind="concept">
# Laplacian matrix

## Summary

The Laplacian matrix of a finite graph is the degree matrix minus the [[adjacency matrix|08B9CX109G]].
It is symmetric and positive semidefinite, and its eigenvalues carry information about connectivity: the number of connected components is the multiplicity of the eigenvalue $0$.
Through [[Kirchhoff's matrix tree theorem|S1PG1N2VEX]] it counts spanning trees, and its second smallest eigenvalue controls how well the graph is connected.
It is the discrete analogue of the Laplace operator of analysis.

## Explanations

### L1 human

Put a number on each vertex of a graph.
The Laplacian compares each number with the numbers at the neighbouring vertices: it adds up the differences.
If the numbers are all equal on a connected graph, the Laplacian gives zero everywhere, just as a constant function has zero second derivative.
For the path with three vertices, the Laplacian is a $3 \times 3$ matrix with the degrees $1, 2, 1$ on the diagonal and $-1$ for each edge.
</document>

<document source="https://tamnd.github.io/math2/node/TN9G24VABN.md" tag="TN9G24VABN" kind="concept">
# Computably enumerable set

## Summary

A computably enumerable set, also called recursively enumerable or semidecidable, is a set of natural numbers whose elements can be listed by an algorithm.
Equivalently, some algorithm halts exactly on the elements of the set.
Every [[computable set|PPGD5KW8JC]] is computably enumerable, but the halting set is computably enumerable and not computable.
These sets are the $\Sigma^0_1$ sets of the [[arithmetical hierarchy|7CVJCQKY41]] and are studied through their [[Turing degrees|P5N2ND3VCG]].

## Explanations

### L1 human

A set is computably enumerable if a program can print its elements one by one, possibly forever and in any order.
If a number is in the set, you will see it printed after some time.
If it is not in the set, you may wait forever without learning this.
For example, the numbers of programs that halt can be listed by running all programs in parallel and printing each one when it stops.
</document>

<document source="https://tamnd.github.io/math2/node/TNHP7XXYZT.md" tag="TNHP7XXYZT" kind="concept">
# Formal group law in topology

## Summary

A formal group law over a commutative ring $R$ is a power series $F(x, y) = x + y + \cdots$ in two variables.
It behaves like the addition of a group.
In topology, every complex-oriented [[generalized cohomology theory|H28PSZR7XG]] $E$ carries such a law.
The law expresses the first Chern class of a tensor product of line bundles through the Chern classes of the factors.
Ordinary cohomology gives the additive law $x + y$ and complex K-theory gives a multiplicative law.
[[Quillen's theorem|4E518M24MA]] says that [[complex cobordism|5CZHFNY9F3]] carries the universal formal group law, which links stable homotopy theory to the algebraic geometry of formal groups.

## Explanations

### L1 human

For complex line bundles, the first Chern class of $L_1 \otimes L_2$ is $c_1(L_1) + c_1(L_2)$ in ordinary cohomology.
In other cohomology theories the rule is no longer plain addition but a power series, such as $x + y + \beta x y$ in K-theory.
Such a power series still acts like addition: it is commutative and associative and has $0$ as a neutral element.
This "twisted addition" is a formal group law.
Its shape records a lot about the cohomology theory, and the study of these laws by height leads to chromatic homotopy theory.
</document>

<document source="https://tamnd.github.io/math2/node/TNM5CFSZM6.md" tag="TNM5CFSZM6" kind="statement">
# Roe solver

## Statement

Let $\Omega \subseteq \R^m$ be open and let $F \colon \Omega \to \R^m$ be the $C^1$ flux of a [[hyperbolic conservation law|XN081NCPF5]] $u_t + F(u)_x = 0$.
A *Roe matrix* is a map $(u_l, u_r) \mapsto A(u_l, u_r)$ from $\Omega \times \Omega$ to the real $m \times m$ matrices with three properties:

1. $F(u_r) - F(u_l) = A(u_l, u_r)(u_r - u_l)$ for all $u_l, u_r \in \Omega$;
2. $A(u_l, u_r)$ is [[diagonalisable|07SQCNV233]] with real [[eigenvalues|0BQAA2BK3Q]];
3. $A(u, u) = F'(u)$ for all $u \in \Omega$.

Write $A = A(u_l, u_r) = R \Lambda R^{-1}$ with $\Lambda$ diagonal, and $\abs{A} = R \abs{\Lambda} R^{-1}$.
The *Roe flux* is
$$H(u_l, u_r) = \tfrac{1}{2} \big( F(u_l) + F(u_r) \big) - \tfrac{1}{2} \abs{A} (u_r - u_l).$$
Then:

1. $H(u_l, u_r) = F(u_l) + A (w_0 - u_l) = F(u_r) - A (u_r - w_0)$, and $H(u, u) = F(u)$. Here $w_0$ is the value on the line $x = 0$ of the solution of the linear [[Riemann problem|GJWKX22VVE]] $w_t + A w_x = 0$ with data $u_l, u_r$. So $H$ is a consistent [[Riemann solver|ZY4GY93C4B]].
2. Let $u_r \ne u_l$ with $F(u_r) - F(u_l) = s (u_r - u_l)$ for a number $s$. Then $u_r - u_l$ is an eigenvector of $A$ for the eigenvalue $s$. The linear Riemann problem is then solved by the single jump from $u_l$ to $u_r$ with speed $s$.

## Remarks

Write $u_r - u_l = \sum_k \alpha_k r_k$ in eigenvectors $r_k$ of $A$ with eigenvalues $\lambda_k$.
The linear Riemann problem moves each component $\alpha_k r_k$ with speed $\lambda_k$, so $w_0 = u_l + \sum_{\lambda_k < 0} \alpha_k r_k$.
Property 1 of the matrix gives $\tfrac12 (F(u_l) + F(u_r)) = F(u_l) + \tfrac12 \sum_k \lambda_k \alpha_k r_k$, and subtracting $\tfrac12 \sum_k \abs{\lambda_k} \alpha_k r_k$ leaves $F(u_l) + \sum_{\lambda_k < 0} \lambda_k \alpha_k r_k = F(u_l) + A(w_0 - u_l)$.
Claim 2 is property 1 of the matrix applied to the jump condition.

In the scalar case $m = 1$ the Roe matrix is the number $A = \dfrac{f(u_r) - f(u_l)}{u_r - u_l}$ for $u_r \ne u_l$, and $A(u, u) = f'(u)$.
For the Euler equations of gas dynamics, Roe found a Roe matrix in closed form.
It is the flux Jacobian at averages weighted by $\sqrt{\rho_l}$ and $\sqrt{\rho_r}$, such as $\tilde v = \dfrac{\sqrt{\rho_l} v_l + \sqrt{\rho_r} v_r}{\sqrt{\rho_l} + \sqrt{\rho_r}}$ for the velocity.

The Roe solver can keep entropy-violating jumps at transonic rarefactions, where an eigenvalue changes sign.
For $f(u) = u^2/2$ with $u_l = -1$ and $u_r = 1$ one gets $A = 0$ and $H = \tfrac12$, so the scheme keeps the initial jump although the entropy solution is a rarefaction fan.
An entropy fix, such as the one of Harten, changes $\abs{\lambda_k}$ near zero to remove this defect.

## Explanations

### L1 human

Solving the true Riemann problem at every cell wall is costly for systems like gas dynamics.
Roe's idea is to replace the nonlinear equation near each wall by a linear one with a well chosen constant matrix.
The linear problem is solved by splitting the jump into eigenvector components that move left or right.
The matrix is chosen so that a single [[shock|JSDMGGTG7G]] is still captured exactly.

### L2 human

The Roe flux is a building block of [[Godunov-type schemes|43HZJ7DRT0]] and of [[MUSCL schemes|W7QH1BPBS8]], where it is evaluated at the reconstructed interface states.
Its upwind part $-\tfrac12 \abs{A}(u_r - u_l)$ acts as numerical viscosity: each wave family is damped in proportion to the size of its speed.
This keeps stationary shocks and contact discontinuities sharp, but makes the fix near sonic points necessary.
Other approximate solvers, such as HLL, trade sharpness for robustness and need only estimates of the wave speeds.
</document>

<document source="https://tamnd.github.io/math2/node/TNMQDK0RR0.md" tag="TNMQDK0RR0" kind="statement">
# Hadamard three-circle theorem for subharmonic functions

## Statement

Let $0 < r_1 < r_2$ and let $u$ be [[subharmonic|Z1MHMGFNFW]] on an open subset of $\C$ that contains the closed [[annulus|0CW52NV5TC]] $A = \set{z : r_1 \le \abs{z} \le r_2}$.
For $r_1 \le r \le r_2$ let $M(r) = \max_{\abs{z} = r} u(z)$, and assume that $M(r_1)$ and $M(r_2)$ are finite.
Then for every $r$ with $r_1 \le r \le r_2$,
$$M(r) \le \frac{\log(r_2/r)}{\log(r_2/r_1)}\, M(r_1) + \frac{\log(r/r_1)}{\log(r_2/r_1)}\, M(r_2).$$
Equivalently, $M(r)$ is a [[convex function|07Q7CNM5WF]] of $\log r$ on $[r_1, r_2]$.

## Remarks

The maximum $M(r)$ exists because an upper semicontinuous function attains its maximum on a compact circle.
Proof idea: choose real $a, b$ such that $h(z) = a + b \log\abs{z}$ equals $M(r_1)$ on $\abs{z} = r_1$ and $M(r_2)$ on $\abs{z} = r_2$.
The function $h$ is harmonic on $\C \setminus \set{0}$, so $u - h$ is subharmonic on the open annulus, and its upper limits at the two boundary circles are at most $0$.
The [[maximum principle for subharmonic functions|9XJ5FBQD7M]] gives $u \le h$ on $A$, so $M(r) \le a + b \log r$, which is the stated inequality.
Applying the same bound to every sub-annulus gives convexity in $\log r$.

Example: for $u(z) = \log\abs{z}$, $M(r) = \log r$ is linear in $\log r$ and equality holds.
For a holomorphic $f$, the function $\log\abs{f}$ is [[subharmonic|7TXCCCA17A]], and the theorem becomes the classical [[Hadamard three-circle theorem|V00Q05ZW1G]] for $\log \max_{\abs{z}=r} \abs{f(z)}$.

## Explanations

### L1 human

Take the largest value of a subharmonic function on each circle around the origin.
Plotted against the logarithm of the radius, these largest values form a convex curve.
So the maximum on a middle circle is controlled by the maxima on an inner and an outer circle.

### L2 human

The proof is a direct comparison with the harmonic functions $a + b\log\abs{z}$, which are the radial harmonic functions in the plane.
The same argument in $\R^n$ with $n \ge 3$ shows that the maximum over spheres is convex as a function of $r^{2-n}$.
The result is a standard tool for growth estimates of holomorphic and subharmonic functions, together with the Phragmén–Lindelöf principle.
</document>

<document source="https://tamnd.github.io/math2/node/TNPQ9AWVBC.md" tag="TNPQ9AWVBC" kind="object">
# Baby Monster group

## Description

The Baby Monster group $\mathbb{B}$ is the second largest of the $26$ [[sporadic groups|PZ49Z4CBFZ]].
It is a finite [[simple group|0YMYMNQ0QZ]] of [[order|0B85S2V2DR]]
$$\abs{\mathbb{B}} = 2^{41} \cdot 3^{13} \cdot 5^6 \cdot 7^2 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 31 \cdot 47,$$
which is $4154781481226426191177580544000000$, about $4 \cdot 10^{33}$.

## Remarks

The Baby Monster was found by Bernd Fischer.
It is a subquotient of the [[Monster group|C9VY20TAPT]]: the Monster contains an involution whose [[centraliser|02YYV36BFM]] is a double cover $2 \cdot \mathbb{B}$ of the Baby Monster.
The smallest nontrivial irreducible complex representation of $\mathbb{B}$ has dimension $4371$.

## Explanations

### L1 human

The Baby Monster is the second largest exceptional finite simple group.
It sits inside the Monster.
The elements of the Monster that commute with a certain element of order $2$ form, up to a factor of $2$, a copy of the Baby Monster.
</document>

<document source="https://tamnd.github.io/math2/node/TNT3DG0HRJ.md" tag="TNT3DG0HRJ" kind="concept">
# Random graph

## Summary

A random graph is a graph chosen at random according to a probability distribution on a set of graphs.
The most studied model is the [[Erdős–Rényi model|6E83YKANV3]] $G(n, p)$, in which each possible edge on $n$ labelled vertices is present independently with probability $p$.
One studies which properties hold with probability tending to $1$ as $n$ grows.
There are sharp thresholds, for example for connectivity and for the appearance of a giant component.
Random graphs are a main tool of the [[probabilistic method|3B55HNS754]] and a model for large real networks.

## Explanations

### L1 human

Take $n$ dots and, for each pair of dots, flip a coin to decide whether to draw a line between them.
The result is a random graph.
Typical questions are: how likely is the graph to be connected, or to contain a triangle, when $n$ is large?
Often the answer jumps from almost never to almost always as the probability of each edge passes a critical value.
</document>

<document source="https://tamnd.github.io/math2/node/TNYQ5R0NVM.md" tag="TNYQ5R0NVM" kind="statement">
# Fundamental theorem of space curves

## Statement

Let $I \subseteq \R$ be an open interval, and let $\kappa \colon I \to \R$ and $\tau \colon I \to \R$ be smooth functions with $\kappa(s) > 0$ for all $s \in I$.
Then there is a smooth curve $\gamma \colon I \to \R^3$ parametrised by [[arc length|00R1JDDV5X]] whose [[curvature|7JQS6VKFEZ]] is $\kappa$ and whose [[torsion|S4BZ697DSH]] is $\tau$.
If $\tilde\gamma \colon I \to \R^3$ is another such curve, then $\tilde\gamma = \rho \circ \gamma$ for a unique orientation-preserving rigid motion $\rho$ of $\R^3$.
Here a rigid motion is a map $x \mapsto Rx + b$ with $R$ a rotation matrix and $b \in \R^3$.

## Remarks

For existence, solve the [[Frenet–Serret formulas|ZM6W89YNS7]] as a linear system of ordinary differential equations for $(T, N, B)$, with an orthonormal positive initial frame at some $s_0 \in I$.
The solution stays orthonormal because the coefficient matrix is skew-symmetric; then set $\gamma(s) = \int_{s_0}^s T(u) \, du$.
For uniqueness, move $\tilde\gamma$ by a rigid motion so that both curves have the same point and frame at $s_0$.
Then the uniqueness of solutions of the linear system gives equal frames and equal curves.
Example: constant $\kappa = a/(a^2+b^2)$ and $\tau = b/(a^2+b^2)$ with $a > 0$ give exactly the helices $(a \cos t, a \sin t, bt)$, $t = s / \sqrt{a^2 + b^2}$, up to rigid motion.
The hypothesis $\kappa > 0$ matters: if $\kappa$ vanishes on an interval, the torsion is not defined there and the curve is not determined by $\kappa$ alone.
An orientation-reversing motion, such as a reflection, changes the sign of $\tau$, so it does not keep the torsion.

## Explanations

### L1 human

If you know at each moment how sharply a curve bends and how much it twists, you know the curve.
The only freedom left is where you place it in space and which way you point it.
This is like following driving instructions: "turn this much, climb this much" fixes the route once the start is given.

### L2 human

The theorem says that $(\kappa, \tau)$ is a complete system of invariants of unit-speed space curves with positive curvature.
The symmetry group here is the group of orientation-preserving rigid motions, a subgroup of the [[Euclidean group|0H3YY8CPKH]].
For plane curves the analogue is simpler: the signed curvature determines a unit-speed curve up to orientation-preserving rigid motion, through $\theta' = k$ and $\gamma' = (\cos \theta, \sin \theta)$.
The same idea, a frame solving a linear system, reappears in the theory of surfaces, where the Gauss–Codazzi equations are the integrability conditions.
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<document source="https://tamnd.github.io/math2/node/TP1RQPCETV.md" tag="TP1RQPCETV" kind="definition">
# Stable vector bundle

## Definition

Let $k$ be an algebraically closed field and let $C$ be a smooth projective connected [[algebraic curve|XJM2MT4S3Y]] over $k$.
For a nonzero [[vector bundle|P4DZM975BV]] $E$ on $C$ of rank $r$, let $\deg E$ be the degree of a [[divisor|Z9DN23DAE2]] of its determinant line bundle $\det E = \Lambda^r E$.
The *slope* of $E$ is $\mu(E) = \deg E / \rank E$.
The bundle $E$ is *stable* if
$$\mu(F) < \mu(E)$$
for every subbundle $F \subset E$ with $0 < \rank F < \rank E$.
It is *semistable* if $\mu(F) \le \mu(E)$ for every such $F$.

## Remarks

Example: every line bundle is stable, since it has no subbundle of intermediate rank.
Non-example: let $L_1, L_2$ be line bundles with $\deg L_1 > \deg L_2$.
Then $L_1$ is a subbundle of $E = L_1 \oplus L_2$ with $\mu(L_1) = \deg L_1 > (\deg L_1 + \deg L_2)/2 = \mu(E)$, so $E$ is not even semistable.
The bundle $\mathcal{O}_C \oplus \mathcal{O}_C$ is semistable but not stable, since $\mathcal{O}_C$ has the same slope $0$.
One may test with all nonzero proper coherent subsheaves instead of subbundles; the condition is the same.
A stable bundle $E$ is simple: its only endomorphisms are the scalars.
On a smooth projective variety of higher dimension with an ample class $H$, slope stability (Mumford–Takemoto stability) uses torsion-free subsheaves and the degree $c_1(F) \cdot H^{n-1}$.
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<document source="https://tamnd.github.io/math2/node/TP8XXM47CJ.md" tag="TP8XXM47CJ" kind="definition">
# Approximately additive function

## Definition

Let $(G, +)$ be an [[abelian group|082GHNPJ53]], let $(E, \norm{\cdot})$ be a [[normed vector space|03J5RAKB30]], and let $\delta \ge 0$.
A map $f \colon G \to E$ is *$\delta$-additive* if
$$\norm{f(x + y) - f(x) - f(y)} \le \delta \quad \text{for all } x, y \in G.$$
A map is *approximately additive* if it is $\delta$-additive for some $\delta \ge 0$.

## Remarks

The $0$-additive maps are exactly the additive maps.
Example: $f(x) = \floor{x}$ on $\R$ is $1$-additive.
Write $x = \floor{x} + s$ and $y = \floor{y} + t$ with $s, t \in [0, 1)$; then $\floor{x + y} - \floor{x} - \floor{y} = \floor{s + t} \in \set{0, 1}$.
Non-example: $f(x) = x^2$ on $\R$ is not approximately additive, since $f(x + y) - f(x) - f(y) = 2xy$ is unbounded.
Every map $f = A + b$ with $A$ additive and $\norm{b(x)} \le \beta$ for all $x$ is $3\beta$-additive.
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<document source="https://tamnd.github.io/math2/node/TPB4F1VY4Z.md" tag="TPB4F1VY4Z" kind="statement">
# Bürmann–Lagrange series

## Statement

Let $g$ be a [[holomorphic function|0NAHXR1FRH]] on a neighbourhood of a point $a \in \C$ with $g'(a) \ne 0$, and let $b = g(a)$.
Let $H$ be holomorphic on a neighbourhood of $a$.
Put $\varphi(z) = \dfrac{z - a}{g(z) - b}$ for $z \ne a$ and $\varphi(a) = 1/g'(a)$, which is holomorphic near $a$.
Then there is a neighbourhood $U$ of $a$ such that for all $z \in U$
$$H(z) = H(a) + \sum_{n=1}^{\infty} \frac{c_n}{n!} \big( g(z) - b \big)^n, \qquad c_n = \frac{d^{\,n-1}}{dz^{\,n-1}} \Big[ H'(z)\, \varphi(z)^n \Big]_{z = a},$$
and the series converges absolutely on $U$.

## Remarks

Proof idea: since $g'(a) \ne 0$, the function $g$ has a holomorphic inverse $g^{-1}$ near $b$.
The series is the [[Taylor series|0Z49YH6ZWY]] of $w \mapsto H(g^{-1}(w))$ at $w = b$, evaluated at $w = g(z)$.
Its coefficients are found by Cauchy's integral formula for derivatives, followed by the change of variable $w = g(z)$.
An integration by parts then turns the integral into a residue at $z = a$.

With $H(z) = z$ the series is the classical Lagrange inversion formula for $g^{-1}$.
Example: $g(z) = z e^{-z}$ and $a = 0$, so $\varphi(z) = e^{z}$ and $c_n = \frac{d^{\,n-1}}{dz^{\,n-1}} e^{nz} \big|_{z=0} = n^{n-1}$.
So the inverse of $z e^{-z}$ near $0$ is $\sum_{n \ge 1} \frac{n^{n-1}}{n!} w^n$, which is $-W(-w)$ for the [[Lambert W function|FW2JKAN14B]].

The hypothesis $g'(a) \ne 0$ is needed: for $g(z) = z^2$ and $a = 0$ the function $H(z) = z$ is not a power series in $g(z) = z^2$, since $z$ is odd.
The formal version for power series over a field of characteristic $0$ is the [[Lagrange inversion theorem|BANY9YR82A]].

## Explanations

### L1 human

A Taylor series writes a function as a sum of powers of $z - a$.
The Bürmann–Lagrange series instead writes it as a sum of powers of another function $g(z) - g(a)$.
This works whenever $g$ is locally one-to-one at $a$.
The coefficients come from derivatives at the single point $a$, so one never has to invert $g$ explicitly.

### L2 human

The series is the analytic form of Lagrange inversion.
It gives the Taylor coefficients of an inverse function, for example of the solution $w(z)$ of Kepler's equation or of the Lambert $W$ function.
In asymptotic analysis it is used to change variables near a critical point.
For instance, it reduces an integral in [[Laplace's method|9G7K8DBT9Y]] to a Gaussian form with an explicit series of corrections.
The combinatorial version, for formal power series, is the [[Lagrange inversion theorem|BANY9YR82A]], which counts trees.
</document>

<document source="https://tamnd.github.io/math2/node/TPC99RZ316.md" tag="TPC99RZ316" kind="definition">
# Rank-one convexity

## Definition

Let $m, n \ge 1$ and let $f \colon \R^{m \times n} \to \R$.
For $a \in \R^m$ and $b \in \R^n$, write $a \otimes b$ for the $m \times n$ matrix with entries $a_i b_\alpha$; it has [[rank|0ZC9YTXZRY]] at most one.
The function $f$ is *rank-one convex* if for every $\xi \in \R^{m \times n}$, $a \in \R^m$ and $b \in \R^n$ the function
$$t \mapsto f(\xi + t \, a \otimes b), \quad t \in \R,$$
is [[convex|07GR3TNW4H]].

## Remarks

Example: for $m = n = 2$, $f = \det$ is rank-one convex.
By the matrix determinant lemma, $\det(\xi + t \, a \otimes b) = \det \xi + t \, b^T \operatorname{adj}(\xi) \, a$, which is affine in $t$.
But $\det$ is not convex: along $\xi(t) = \operatorname{diag}(t, -t)$ it equals $-t^2$.

Non-example: for $m = n = 1$, $f(\xi) = -\xi^2$ is not rank-one convex, since $t \mapsto -t^2$ is not convex.

For $m = 1$ or $n = 1$ every matrix has rank at most one, so rank-one convexity is ordinary convexity.
If $f$ is $C^2$, it is rank-one convex exactly when $\sum_{i,j,\alpha,\beta} \frac{\partial^2 f}{\partial \xi^i_\alpha \partial \xi^j_\beta}(\xi) \, a_i a_j b_\alpha b_\beta \ge 0$ for all $\xi$, $a$ and $b$, the Legendre–Hadamard condition.
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<document source="https://tamnd.github.io/math2/node/TPG0GQBFHM.md" tag="TPG0GQBFHM" kind="definition">
# Complex powers of elliptic operators

## Definition

Let $M$ be a closed manifold of dimension $n$ with a smooth positive density, and let $E$ be a Hermitian vector bundle over $M$.
Let $A \colon \Gamma(E) \to \Gamma(E)$ be an [[elliptic operator|MT9P2X85V8]] of order $m > 0$ that is formally [[self-adjoint|0NEGN5CRSC]] and positive: $\int_M \ip{Au}{u} > 0$ for every nonzero $u \in \Gamma(E)$.
Then $L^2(E)$ has an orthonormal basis $(\varphi_j)_{j \ge 1}$ of smooth eigensections, $A \varphi_j = \lambda_j \varphi_j$, whose [[eigenvalues|0BQAA2BK3Q]] satisfy $0 < \lambda_1 \le \lambda_2 \le \cdots$ and $\lambda_j \to \infty$.
For $s \in \C$, the *complex power* $A^s$ is the operator
$$A^s u = \sum_{j \ge 1} \lambda_j^{s} \, \ip{u}{\varphi_j} \, \varphi_j, \qquad u \in \Gamma(E),$$
where $\lambda_j^s = e^{s \log \lambda_j}$ with the real logarithm.

## Remarks

The powers satisfy $A^s A^t = A^{s+t}$, $A^0 = \id$ and $A^1 = A$, and $A^{-1}$ is the inverse of $A$.
Seeley proved that $A^s$ is a classical [[pseudo-differential operator|FF27N0BWTP]] of order $ms$.
For $\operatorname{Re} s > n/m$, the operator $A^{-s}$ is [[trace class|Z2CP530WDE]], and the spectral zeta function
$$\zeta_A(s) = \operatorname{tr} A^{-s} = \sum_{j \ge 1} \lambda_j^{-s}$$
is holomorphic there.
It extends to a meromorphic function on $\C$ with at most simple poles, and it is holomorphic at $s = 0$.
This gives the [[zeta-regularized|VJCJYXGTDY]] determinant $\det A = e^{-\zeta_A'(0)}$.
More generally, Seeley defined $A^s$ by a contour integral of $\lambda^s (A - \lambda)^{-1}$ for elliptic $A$ whose principal symbol has no eigenvalues on some ray from $0$, without assuming self-adjointness.

Example: on the circle $S^1 = \R / 2\pi\Z$ with $dx$, let $A = 1 - d^2/dx^2$, which is elliptic of order $2$ and positive.
Its eigensections are $e^{ikx}/\sqrt{2\pi}$, $k \in \Z$, with eigenvalues $1 + k^2$, so $A^{-s} e^{ikx} = (1 + k^2)^{-s} e^{ikx}$.
Then $\zeta_A(s) = \sum_{k \in \Z} (1 + k^2)^{-s}$ converges for $\operatorname{Re} s > \frac{1}{2} = \frac{n}{m}$ with $n = 1$ and $m = 2$.

Non-example: the operator $-d^2/dx^2$ on $S^1$ is elliptic and self-adjoint but not positive, since it kills the constants.
The formula would need $0^s$, which is not defined for all $s$; one first restricts to the orthogonal complement of the kernel.

## Explanations

### L1 human

Complex powers of a positive elliptic operator raise each eigenvalue to the power $s$, keeping the eigenfunctions, just as for a positive matrix.
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<document source="https://tamnd.github.io/math2/node/TPGBASPKE0.md" tag="TPGBASPKE0" kind="example">
# Whitehead link

## Example

Start with a round circle $C_1$ in $\R^3$, and let $D$ be the flat disk that it bounds.
Take a second closed curve $C_2$, disjoint from $C_1$, that goes down through $D$ and then comes back up through $D$, with one twist.
In the standard diagram, $C_2$ has the shape of a figure eight with one self-crossing, and it clasps $C_1$.
The union $C_1 \cup C_2$ is a [[link|1A5WRCNTM3]] with $2$ components, the *Whitehead link*.
Its standard diagram has $5$ crossings: one crossing of $C_2$ with itself, and four crossings between $C_1$ and $C_2$.
The four crossings between the components come in two pairs of opposite signs, so the [[linking number|SFJ329YB9F]] of $C_1$ and $C_2$ is $0$.
Each component on its own is an [[unknot|1TQ06R4Q66]].

## Remarks

Although its linking number is $0$, the Whitehead link is not the unlink of two components: the two curves cannot be separated.
One way to see this uses its complement in $S^3$, which carries a complete hyperbolic metric of finite volume.
The complement of a split link contains a sphere that separates the components.
A hyperbolic complement contains no such sphere.
The complement is built from one regular ideal octahedron, and its volume is about $3.6639$.
Its crossing number is $5$.
It was studied by J. H. C. Whitehead in his work on contractible open $3$-manifolds.
Together with the Hopf link, it shows that the linking number detects some but not all linking.

## Explanations

### L1 human

The Whitehead link has two loops of string, each unknotted on its own.
One loop passes through the other and then hooks back on itself, so that it goes through once in each direction.
Counting with signs, the loops wind around each other zero times, yet they still cannot be pulled apart.
</document>

<document source="https://tamnd.github.io/math2/node/TPH39D9W02.md" tag="TPH39D9W02" kind="definition">
# Proper map

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]].
A map $f \colon X \to Y$ is *proper* if $f$ is [[continuous|0VGFHTM75Y]] and, for every [[compact|0R6EWFKRCF]] subset $K \subseteq Y$, the [[preimage|01AHG4G9RT]] $f^{-1}(K)$ is compact.

## Remarks

Bourbaki uses another definition: $f$ is continuous and, for every topological space $Z$, the map $f \times \id_Z \colon X \times Z \to Y \times Z$ is [[closed|00FSJTXPJM]].
This holds if and only if $f$ is continuous, closed, and every fibre $f^{-1}(y)$ is compact.
A map that is proper in this sense is proper in the sense above.
When $Y$ is locally compact and Hausdorff, the two definitions agree.

Example: $f \colon \R \to \R$, $f(x) = x^2$.
A compact $K \subseteq \R$ lies in some $[-M, M]$ with $M \ge 0$, and $f^{-1}(K)$ is closed and lies in $[-\sqrt{M}, \sqrt{M}]$, so it is compact.

Non-example: $f = \arctan \colon \R \to \R$.
It is continuous, but $f^{-1}([-\pi/2, \pi/2]) = \R$ is not compact.
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<document source="https://tamnd.github.io/math2/node/TPJZCSXSVN.md" tag="TPJZCSXSVN" kind="statement">
# Fejér's theorem

## Statement

Let $f \colon \R \to \C$ be a [[continuous|0P27W9RW0G]] [[periodic function|089KWVYD69]] with period $2\pi$.
For $n \ge 0$ let $S_n f(x) = \sum_{k=-n}^{n} c_k e^{ikx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]], where $c_k = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-ikt} \, dt$.
For $N \ge 1$ let
$$\sigma_N f(x) = \frac{1}{N} \sum_{n=0}^{N-1} S_n f(x)$$
be their [[Cesàro means|0P0DJHBGBB]].
Then $\sigma_N f \to f$ [[uniformly|08FPABDQ4C]] on $\R$ as $N \to \infty$.

## Remarks

The proof writes $\sigma_N f(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} F_N(t) f(x - t) \, dt$ with the [[Fejér kernel|DCCA2T9GKH]] $F_N$.
Since $F_N \ge 0$ and $F_N$ has mean $1$, the error $\sigma_N f(x) - f(x)$ is a weighted average of $f(x - t) - f(x)$.
For small $\abs{t}$ this is small by uniform continuity of $f$, and for $\abs{t} \ge \delta$ the weight $F_N(t)$ is uniformly small for large $N$.

Example: the theorem shows that trigonometric polynomials are uniformly dense in the continuous $2\pi$-periodic functions, since each $\sigma_N f$ is one.
It also shows that two continuous periodic functions with the same Fourier coefficients are equal.

Continuity is needed for uniform convergence: each $\sigma_N f$ is continuous, so a uniform limit of them is continuous, and a function with a jump cannot be such a limit.
The theorem cannot be improved to the partial sums $S_n f$ themselves: there are continuous periodic functions whose Fourier series diverges at a point.

## Explanations

### L1 human

The Fourier series of a continuous periodic function may fail to converge.
Fejér's theorem says that averaging its partial sums always repairs this: the averages converge to the function, evenly over the whole line.
So a continuous periodic function is always recovered from its Fourier coefficients, by Cesàro summation of the series.

### L2 human

Fejér's theorem is the model case of summability methods applied to Fourier series.
Its proof uses only three properties of the Fejér kernel: positivity, unit mean and concentration at $0$.
Any kernel with these properties gives a similar theorem, such as [[Abel–Poisson summation|RZZ78GXP5F]] with the Poisson kernel.
Versions for integrable functions say that $\sigma_N f \to f$ in mean and at every point of continuity.
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<document source="https://tamnd.github.io/math2/node/TPM8JXFWYB.md" tag="TPM8JXFWYB" kind="definition">
# Romberg's method

## Definition

Let $a < b$ be real numbers and let $f$ be a continuous function on $[a, b]$.
For $n \ge 0$ let $h_n = (b - a)/2^n$, and let $R(n, 0)$ be the composite [[trapezoidal rule|FH0538XCZR]] with $2^n$ subintervals:
$$R(n, 0) = h_n \Big( \tfrac12 f(a) + \sum_{k=1}^{2^n - 1} f(a + k h_n) + \tfrac12 f(b) \Big).$$
For $1 \le m \le n$ define
$$R(n, m) = R(n, m - 1) + \frac{R(n, m - 1) - R(n - 1, m - 1)}{4^m - 1}.$$
*Romberg's method* approximates $\int_a^b f(x)\,dx$ by the entries of the triangular table $R(n, m)$, usually by the diagonal entry $R(n, n)$.

## Remarks

Each column is a [[Richardson extrapolation|WMTZG71K2Y]] of the previous one, which removes the term $h^{2m}$ from the error expansion.
If $f$ is smooth enough, the error of $R(n, m)$ is $O(h_n^{2m + 2})$ as $n \to \infty$.
The values $R(n, 0)$ can be computed recursively: $R(n, 0) = \frac12 R(n - 1, 0) + h_n \sum_{k=1}^{2^{n-1}} f(a + (2k - 1) h_n)$, so no function value is computed twice.

Example: $f(x) = x^2$ on $[0, 1]$.
Then $R(0, 0) = \frac12 (0 + 1) = \frac12$ and $R(1, 0) = \frac12 \big( \frac12 \cdot 0 + \frac14 + \frac12 \cdot 1 \big) = \frac38$.
So $R(1, 1) = \frac38 + \frac{3/8 - 1/2}{3} = \frac13$, which is the exact integral, as Simpson's rule is exact for quadratics.

Non-example: replacing $4^m - 1$ by $2^m - 1$ gives a different table that does not cancel the even powers of $h$, so it is not Romberg's method.
</document>

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# Barycentric interpolation formula

## Statement

Let $K$ be a [[field|04C5ZBG1WT]], let $n \ge 0$, let $x_0, \dots, x_n \in K$ be pairwise distinct, and let $y_0, \dots, y_n \in K$.
Let $p$ be the [[interpolation polynomial|SS1YMS2NPC]] of degree at most $n$ with $p(x_j) = y_j$ for all $j$.
Define the *barycentric weights* $w_j = \prod_{k \ne j} (x_j - x_k)^{-1}$ and let $\ell(x) = \prod_{k=0}^{n} (x - x_k)$.
Then for every $x \in K$ that is not a node:

1. (first form) $\displaystyle p(x) = \ell(x) \sum_{j=0}^{n} \frac{w_j}{x - x_j} \, y_j$;
2. (second form) $\displaystyle p(x) = \frac{\sum_{j=0}^{n} \frac{w_j}{x - x_j} \, y_j}{\sum_{j=0}^{n} \frac{w_j}{x - x_j}}$.

## Remarks

The [[Lagrange basis polynomials|DA3CQJ5MJ6]] satisfy $\ell_j(x) = \ell(x) \, w_j / (x - x_j)$, so the first form is the [[Lagrange formula|Y7KSGE97GK]] $p = \sum_j y_j \ell_j$ rewritten.
Applying the first form to the constant data $y_j = 1$, whose interpolant is $1$, gives $1 = \ell(x) \sum_j w_j / (x - x_j)$.
Dividing the first form by this identity gives the second form.

Example: nodes $0, 1$ have weights $w_0 = -1$ and $w_1 = 1$.
The second form gives $p(x) = \dfrac{-y_0/x + y_1/(x - 1)}{-1/x + 1/(x - 1)} = y_0 (1 - x) + y_1 x$, the line through the data.

Once the weights are computed, in $O(n^2)$ operations, each evaluation costs $O(n)$ operations, and adding a new node updates the weights in $O(n)$ operations.
The weights do not depend on the data, so the same weights serve for many data sets.

## Explanations

### L1 human

The Lagrange formula is clear but slow to evaluate, because each basis polynomial is a long product.
The barycentric formula moves the common product out and keeps only simple fractions $w_j / (x - x_j)$.
In the second form the value $p(x)$ is a weighted average of the data, with weights that depend on $x$.

### L2 human

The second form is the standard way to evaluate an interpolation polynomial in floating point arithmetic.
For good node families, such as [[Chebyshev nodes|HYK8J56PSH]], the weights have simple closed forms and the evaluation is numerically stable.
Unlike the [[Newton form|9W77FPTM9E]], it needs no ordering of the nodes and no table of [[divided differences|N5ZHV4HWDC]].
</document>

<document source="https://tamnd.github.io/math2/node/TPSH4ETK67.md" tag="TPSH4ETK67" kind="statement">
# Hörmander's L2 estimates

## Statement

Let $\Omega \subseteq \C^n$ be a [[pseudoconvex|AHYZMSBBNH]] open set.
Let $\varphi \in C^2(\Omega)$ be a real function, and let $c \colon \Omega \to (0, \infty)$ be a continuous function such that
$$\sum_{j,k=1}^n \frac{\partial^2 \varphi}{\partial z_j \partial \bar z_k}(z)\, w_j \bar w_k \ge c(z) \abs{w}^2 \quad \text{for all } z \in \Omega,\ w \in \C^n,$$
so that $\varphi$ is strictly [[plurisubharmonic|FF2P7ER3QT]].
Let $g = \sum_{j=1}^n g_j \, d\bar z_j$ be a [[(0,1)-form|DC81TCR98Y]] with coefficients $g_j$ in $L^2_{\mathrm{loc}}(\Omega)$.
Suppose that $\bar\partial g = 0$ in the sense of distributions and
$$\int_\Omega \frac{\abs{g}^2}{c}\, e^{-\varphi}\, d\lambda < \infty,$$
where $\abs{g}^2 = \sum_j \abs{g_j}^2$ and $\lambda$ is Lebesgue measure on $\C^n = \R^{2n}$.
Then there is a function $u \in L^2_{\mathrm{loc}}(\Omega)$ with $\bar\partial u = g$ in the sense of distributions and
$$\int_\Omega \abs{u}^2 e^{-\varphi}\, d\lambda \le \int_\Omega \frac{\abs{g}^2}{c}\, e^{-\varphi}\, d\lambda.$$

## Remarks

The proof is a duality argument in weighted [[L2 spaces|SASP14MA1Z]].
One views $\bar\partial$ as a closed, densely defined operator between $L^2$ spaces with weight $e^{-\varphi}$.
The key step is the a priori inequality $\norm{\bar\partial^* f}^2 + \norm{\bar\partial f}^2 \ge \int c \abs{f}^2 e^{-\varphi}$ for forms $f$ in the domains of both operators.
This Bochner–Kodaira–Morrey–Kohn type identity is where the plurisubharmonicity of $\varphi$ enters.
The Riesz representation theorem then gives the solution $u$ together with the bound; pseudoconvexity is used to exhaust $\Omega$ by sets on which the argument applies.
There is a version for any plurisubharmonic weight $\varphi$ on a pseudoconvex $\Omega$.
For every $\bar\partial$-closed $(0,1)$-form $g$ with $\int \abs{g}^2 e^{-\varphi} < \infty$ there is $u$ with $\bar\partial u = g$ and $\int \abs{u}^2 e^{-\varphi} (1 + \abs{z}^2)^{-2} \le \int \abs{g}^2 e^{-\varphi}$.
Similar statements hold for $(p,q)$-forms and on Stein and complete Kähler manifolds with Hermitian line bundles.
Example: for $\varphi(z) = \abs{z}^2$ one may take $c = 1$, so every $\bar\partial$-closed $g$ with $\int \abs{g}^2 e^{-\abs{z}^2} < \infty$ is $\bar\partial u$ for some $u$ with $\int \abs{u}^2 e^{-\abs{z}^2} \le \int \abs{g}^2 e^{-\abs{z}^2}$.
Pseudoconvexity cannot be dropped: an open set $\Omega \subseteq \C^n$ on which every smooth $\bar\partial$-closed $(0,q)$-form, $1 \le q \le n - 1$, is $\bar\partial$ of a smooth form is pseudoconvex.

## Explanations

### L1 human

The equation $\bar\partial u = g$ asks for a function whose failure to be holomorphic is prescribed.
Hörmander's theorem says that on a pseudoconvex domain this equation always has a solution, and that the size of the solution is controlled by the size of the data.
Size is measured by integrals with a weight $e^{-\varphi}$, and choosing the weight cleverly is the art of using the theorem.

### L2 human

The estimates turn the main existence problems of several complex variables into a single analytic tool.
To build a holomorphic function with given properties, one first builds a smooth function with these properties.
Then one corrects it by a solution of a $\bar\partial$-equation whose weighted $L^2$ norm is small.
In this way one obtains Oka's solution of the Levi problem, the Cousin problems on pseudoconvex domains, and Cartan's theorem B for the structure sheaf.
The method is the basis of the Ohsawa–Takegoshi extension theorem and of the theory of multiplier ideals.
</document>

<document source="https://tamnd.github.io/math2/node/TPXEFSBQYB.md" tag="TPXEFSBQYB" kind="definition">
# Axiom of dependent choice

## Definition

The axiom of dependent choice is the following statement.
Let $X$ be a nonempty [[set|0SJ3RX9TXA]] and let $R$ be a [[binary relation|08EJC6Q25H]] on $X$.
Assume that for each $x \in X$ there is $y \in X$ with $x \mathrel{R} y$.
Then there is a [[sequence|0QE8STGJH2]] $(x_n)_{n \in \N}$ of elements of $X$ such that $x_n \mathrel{R} x_{n+1}$ for each $n \in \N$.

## Remarks

Example: let $X = \N$ and let $m \mathrel{R} n$ mean $m < n$.
Each $m$ has $m + 1$ with $m \mathrel{R} (m+1)$, and the sequence $x_n = n$ satisfies the conclusion.
Non-example of the hypothesis: on $X = \set{0, 1}$ with $R = \set{(0, 1)}$, the element $1$ has no $y$ with $1 \mathrel{R} y$.
No sequence with $x_n \mathrel{R} x_{n+1}$ for all $n$ exists, so the hypothesis cannot be dropped.
The axiom is abbreviated DC.
Over ZF, the [[axiom of choice|0HTGB13FE2]] implies DC, and DC implies the axiom of countable choice; neither implication reverses.
Over ZF, DC is equivalent to the [[Baire category theorem|0A7KGAYHGV]] for complete metric spaces.
An equivalent form says that a relation $R$ on a nonempty set is [[well-founded|0230ECX1Q5]] if and only if there is no sequence with $x_{n+1} \mathrel{R} x_n$ for all $n$.
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<document source="https://tamnd.github.io/math2/node/TPXYARH912.md" tag="TPXYARH912" kind="statement">
# Federer–Fleming compactness theorem

## Statement

Let $0 \le m \le n$, let $K \subseteq \R^n$ be compact, and let $C > 0$.
Let $(T_j)_{j \ge 1}$ be a sequence of [[integral currents|EG4AJVEFT2]] of dimension $m$ on $\R^n$ with support in $K$.
Assume $\mathbf{M}(T_j) + \mathbf{M}(\partial T_j) \le C$ for all $j$, where $\mathbf{M}$ is the mass and $\partial T_j = 0$ when $m = 0$.
Then there are a subsequence $(T_{j_i})$ and an integral $m$-current $T$ with support in $K$ such that $\mathbf{F}(T_{j_i} - T) \to 0$, where $\mathbf{F}$ is the [[flat norm|Y7P66D41Z8]].
Moreover $\mathbf{M}(T) \le \liminf_i \mathbf{M}(T_{j_i})$ and $\mathbf{M}(\partial T) \le \liminf_i \mathbf{M}(\partial T_{j_i})$.

## Remarks

Weak convergence of a subsequence follows from the mass bound by a diagonal argument on a countable dense set of forms.
The mass is lower semicontinuous under weak convergence, because it is a supremum of the continuous functionals $T \mapsto T(\omega)$.
The hard part is the closure statement: the weak limit is again integral, and the convergence holds in the flat norm.
Federer and Fleming proved it with the deformation theorem, which pushes currents onto a cubical grid; later proofs use slicing.

Example at work: let $B$ be an integral $(m-1)$-current with compact support that is the boundary of some integral $m$-current.
A minimising sequence for the mass among integral currents with boundary $B$ can be projected into a fixed ball without increasing mass, and its boundary masses are all $\mathbf{M}(B)$.
The theorem gives a limit with boundary $B$ and least mass, an [[area-minimizing current|4NKP73KVVW]].

The bound on $\mathbf{M}(\partial T_j)$ is needed.
In $\R$, let $T_j = \sum_{i=0}^{j-1} [\![i/j, \, i/j + 1/(2j)]\!]$, integral $1$-currents in $[0, 1]$ with mass $\frac{1}{2}$ and $\mathbf{M}(\partial T_j) = 2j$.
They converge weakly to $\frac{1}{2} [\![0, 1]\!]$, which has multiplicity $\frac{1}{2}$ and is not an integral current.

## Explanations

### L1 human

Generalised surfaces with integer multiplicities, with bounded area and bounded boundary length, inside a fixed bounded region, always have a convergent subsequence.
The limit is again such a surface, and area can only drop in the limit.
This is what makes it possible to find surfaces of least area by taking limits of better and better ones.

### L2 human

The theorem is the compactness step of the direct method of the calculus of variations for currents.
Combined with lower semicontinuity of mass, it solves the Plateau problem for integral currents in every dimension and codimension.
Regularity theory, such as the results of De Giorgi, Almgren and Allard, then shows that the minimisers are smooth away from a small singular set.
The same compactness, with the flat norm, underlies the theory of flat chains and of minimal surfaces obtained by min-max methods.
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<document source="https://tamnd.github.io/math2/node/TPYXQTYND6.md" tag="TPYXQTYND6" kind="concept">
# Hyers–Ulam stability

## Summary

A [[functional equation|SRAN4EKWS0]] is Hyers–Ulam stable if each approximate solution is near an exact one.
Precisely, a function that satisfies it up to an error $\delta$ lies within $K\delta$ of a solution, for a fixed constant $K$.
The notion comes from a question of Ulam on approximate homomorphisms.
[[Hyers' theorem|VFRJ0P2P15]] shows that the Cauchy equation $f(x+y) = f(x) + f(y)$ is stable, with $K = 1$, for maps from a normed space to a Banach space.
Stability results justify replacing an approximate solution by an exact one.

## Explanations

### L1 human

Measurements and computations are never exact, so a function found in practice satisfies an equation only roughly.
Stability says this is harmless: a rough solution is near a true solution.
For example, $f(x) = x + \sin x$ satisfies $f(x + y) = f(x) + f(y)$ up to an error at most $3$, and it stays within distance $1$ of the true solution $x \mapsto x$.
Some equations are not stable, and then a rough solution can drift arbitrarily far from every true one.
</document>

<document source="https://tamnd.github.io/math2/node/TQ1QDAK2BR.md" tag="TQ1QDAK2BR" kind="conjecture">
# Lang's conjectures

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] and let $X$ be a projective variety of general type over $K$.
This means that the [[Kodaira dimension|ZW00Q0SAGR]] of $X$ equals its dimension.

1. The set $X(K)$ of $K$-[[rational points|SWJYG87QQH]] is not dense in $X$ for the [[Zariski topology|8SQK7RTB63]].
2. There is a nonempty Zariski open subset $U \subseteq X$ such that $U(L)$ is finite for every finite extension $L$ of $K$.

## Remarks

Part 1 is often called the Bombieri–Lang conjecture.
Bombieri posed it for surfaces in a 1980 lecture, and Lang posed it in general.
For curves, general type means genus at least $2$, and both parts follow from [[Faltings's theorem|H0DB07WM20]].
The conjecture is open for surfaces.
It follows from [[Vojta's conjecture|QQ717VSFNN]].
Caporaso, Harris and Mazur showed in 1997 that it implies a uniform bound for curves.
The number of $K$-rational points on a curve of genus $g \ge 2$ over $K$ would be bounded in terms of $g$ and $K$ only.

## Explanations

### L1 human

Faltings proved that a curve of genus at least two has only finitely many rational points.
Lang's conjectures predict the same kind of scarcity in higher dimension, for varieties "of general type", the most curved ones.
Rational points may be infinite, but they should all lie on a smaller subvariety, outside of which there are only finitely many.
</document>

<document source="https://tamnd.github.io/math2/node/TQ2TZDMAPA.md" tag="TQ2TZDMAPA" kind="definition">
# Cyclic polytope

## Definition

Let $d \ge 2$ and $n > d$ be integers.
The *moment curve* in $\R^d$ is the map $x \colon \R \to \R^d$, $x(t) = (t, t^2, \dots, t^d)$.
Let $t_1 < t_2 < \cdots < t_n$ be real numbers.
The *cyclic polytope* $C(n, d)$ is the [[convex hull|0FQSDC2EQ2]] of the points $x(t_1), \dots, x(t_n)$.

## Remarks

$C(n,d)$ is a $d$-dimensional [[convex polytope|300K6CB1KH]] whose vertices are all $n$ points $x(t_i)$.
Its face lattice does not depend on the choice of $t_1 < \cdots < t_n$, so $C(n, d)$ names a well-defined combinatorial type.
Gale's evenness condition describes its [[facets|DPDXN2FJC4]].
A set $S$ of $d$ indices gives a facet if and only if, for any two indices $i < j$ outside $S$, an even number of elements of $S$ lie between them.

Example: for $d = 2$ the moment curve is the parabola $y = x^2$, and $C(n, 2)$ is a convex $n$-gon.
Its facets are the edges $\set{i, i+1}$ and $\set{1, n}$, and these are exactly the pairs passing the evenness condition.

Non-example: the regular octahedron is not of the form $C(6, 3)$.
In $C(6,3)$ the vertex $x(t_1)$ lies in $5$ edges, while every vertex of the octahedron lies in $4$ edges.
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<document source="https://tamnd.github.io/math2/node/TQDZQ964AZ.md" tag="TQDZQ964AZ" kind="concept">
# Lemniscate elliptic functions

## Summary

The lemniscate sine $\operatorname{sl}$ and lemniscate cosine $\operatorname{cl}$ are [[elliptic functions|FF9ZPCZ59K]].
They play for the lemniscate of Bernoulli the role that sine and cosine play for the circle.
The lemniscate sine is the inverse of the integral $\int_0^x dt/\sqrt{1 - t^4}$, which measures arc length on the lemniscate.
Their period lattice is the square lattice of Gaussian integers multiplied by $(1 + i)\varpi$, with $\varpi$ the [[lemniscate constant|V5A2S8QYQQ]].
So they have complex multiplication by $i$.
They were studied by Fagnano, Euler and Gauss, and Abel's work on dividing the lemniscate grew out of them.

## Explanations

### L1 human

The lemniscate of Bernoulli is the figure-eight curve $(x^2 + y^2)^2 = x^2 - y^2$.
If you walk along it from the centre, the arc length you have covered when you are at distance $r$ from the centre is $\int_0^r dt/\sqrt{1 - t^4}$.
The lemniscate sine reverses this: given the arc length, it returns the distance from the centre.
This mirrors the circle, where $\arcsin x = \int_0^x dt/\sqrt{1 - t^2}$ and the sine reverses it.
The whole figure-eight has length $2\varpi \approx 5.244$.
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<document source="https://tamnd.github.io/math2/node/TQFFFSH0BQ.md" tag="TQFFFSH0BQ" kind="concept">
# Local regression

## Summary

Local regression estimates a regression function $m(x) = E[Y \mid X = x]$ by fitting a low-degree polynomial near each point $x$.
The fit at $x$ is a [[weighted least squares|H5CGZHBR58]] fit.
Data points close to $x$ get large weights, and distant points get small or zero weights.
It needs no global formula for $m$, so it is a basic tool of [[nonparametric statistics|VFAJ1EG3X5]].
The methods LOESS and LOWESS are popular versions of it.

## Explanations

### L1 human

Imagine a cloud of points that follows a curved trend.
A single straight line cannot follow the curve, but over a short window the curve looks almost straight.
So at each point $x$ one draws a line through the nearby data only, and keeps the height of that line at $x$.
Sliding the window along the axis traces a smooth curve through the cloud.
For example, with a window of width $1$ around $x = 3$, only data with $X$ between $2$ and $4$ matter for the estimate at $3$.
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# Simplicial approximation theorem

## Statement

Let $K$ and $L$ be [[simplicial complexes|V8GZ0W1ZZ5]] with $K$ finite, and let $\abs{K}$ and $\abs{L}$ be their geometric realizations.
For a point $y \in \abs{L}$, let $\operatorname{supp}(y)$ be the smallest closed simplex of $\abs{L}$ that contains $y$.
A simplicial map $g \colon K \to L$ is a *simplicial approximation* of a continuous map $f \colon \abs{K} \to \abs{L}$ if $\abs{g}(x) \in \operatorname{supp}(f(x))$ for every $x \in \abs{K}$.
Let $f \colon \abs{K} \to \abs{L}$ be continuous.
Let $\operatorname{sd}^m K$ be the $m$-fold [[barycentric subdivision|J5DZEYCEX2]] of $K$, so that $\abs{\operatorname{sd}^m K} = \abs{K}$.
Then there are an integer $m \ge 0$ and a simplicial map $g \colon \operatorname{sd}^m K \to L$ that is a simplicial approximation of $f$.

## Remarks

A vertex map $g$ is a simplicial approximation of $f$ as soon as $f(\operatorname{st} v) \subseteq \operatorname{st} g(v)$ for every vertex $v$, where $\operatorname{st}$ denotes the open star.
The open stars of the vertices of $L$ form an open cover of $\abs{L}$, so their preimages under $f$ form an open cover of the compact metric space $\abs{K}$.
Repeated barycentric subdivision makes the simplices smaller than a Lebesgue number of this cover.
Then each open star of $\operatorname{sd}^m K$ maps into some open star of $L$, and choosing such a vertex for each $v$ gives $g$.

A simplicial approximation $g$ of $f$ is [[homotopic|F2F143KQ2M]] to $f$ by the straight-line homotopy, which stays in $\operatorname{supp}(f(x))$.

Subdivision is necessary.
Let $K = L$ be the boundary of a triangle, so $\abs{K}$ is a circle, and let $f$ wrap this circle twice around itself.
A simplicial map $K \to L$ is either a bijection on vertices, of degree $\pm 1$, or misses a vertex, of degree $0$.
Since a simplicial approximation is homotopic to $f$, which has degree $2$, no simplicial map $K \to L$ approximates $f$.

## Explanations

### L1 human

A continuous map between triangulated spaces can be very irregular.
The theorem says that after cutting the source into small enough triangles, the map can be replaced by one that sends triangles linearly onto triangles.
The new map is close to the old one and homotopic to it.

### L2 human

Simplicial approximation shows that the set of homotopy classes of maps $\abs{K} \to \abs{L}$ is countable when $K$ and $L$ are finite.
It is used to prove that simplicial homology is a homotopy invariant and agrees with singular homology, and to compute $\pi_k(S^n) = 0$ for $k < n$.
The analogue for CW complexes is the cellular approximation theorem.
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<document source="https://tamnd.github.io/math2/node/TQJGPS6VZK.md" tag="TQJGPS6VZK" kind="definition">
# Coplanarity

## Definition

Let $A$ be an [[affine space|09XNAPTJQ7]] over a field $k$, with space of translations $V$.
A set $S$ of points of $A$ is *coplanar* if the vectors $q - p$, for $p, q \in S$, span a [[linear subspace|00YC1D652K]] of [[dimension|03GSZ8RTZJ]] at most $2$.

## Remarks

When $A$ has dimension at least $2$, the set $S$ is coplanar exactly when some affine plane of $A$ contains $S$.

Example: in $\R^3$ the points $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ and $(1,1,0)$ are coplanar.
Check: all differences lie in the subspace $\set{z = 0}$, which has dimension $2$.

Non-example: the points $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ of $\R^3$ are not coplanar.
The differences from the origin are the three standard basis vectors, which span a space of dimension $3$.

In $\R^3$, four points $p_0, p_1, p_2, p_3$ are coplanar exactly when the [[determinant|00KDZRW4C0]] of the $3 \times 3$ matrix with columns $p_1 - p_0$, $p_2 - p_0$, $p_3 - p_0$ is zero.
Every set of at most three points is coplanar.
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<document source="https://tamnd.github.io/math2/node/TQNG7T2RQ6.md" tag="TQNG7T2RQ6" kind="definition">
# Supermanifold

## Definition

Let $p, q \ge 0$ be integers.
Let $\Lambda_q = \Lambda[\theta_1, \dots, \theta_q]$ be the real [[exterior algebra|0MDMZ965VE]] on $q$ generators.
It is graded by parity: a product of an even number of generators is even, and a product of an odd number is odd.
For an open set $U \subseteq \R^p$, let $\mathcal{O}_{p|q}(U) = C^\infty(U) \otimes_\R \Lambda_q$.
The pair $\R^{p|q} = (\R^p, \mathcal{O}_{p|q})$ is the *superdomain* of dimension $p|q$.

A smooth *supermanifold* of dimension $p|q$ is a pair $\mathcal{M} = (\abs{\mathcal{M}}, \mathcal{O}_\mathcal{M})$ with these properties.

1. $\abs{\mathcal{M}}$ is a Hausdorff, second-countable [[topological space|0CQHFPGJ45]].
2. $\mathcal{O}_\mathcal{M}$ is a [[sheaf|TB6TM6R4MD]] of supercommutative [[superalgebras|6JZW4JC68Z]] on $\abs{\mathcal{M}}$; supercommutative means $ab = (-1)^{\abs{a}\abs{b}} ba$ for homogeneous $a, b$ of parities $\abs{a}, \abs{b}$.
3. Every point of $\abs{\mathcal{M}}$ has an open neighbourhood $V$ with the following property. As a space with a sheaf of superalgebras, $(V, \mathcal{O}_\mathcal{M}|_V)$ is isomorphic to $(U, \mathcal{O}_{p|q}|_U)$ for some open $U \subseteq \R^p$.

A morphism of supermanifolds is a morphism of [[locally ringed spaces|B7FPT8JTXR]] whose map on sheaves preserves parity.

## Remarks

Let $\mathcal{J} \subseteq \mathcal{O}_\mathcal{M}$ be the ideal generated by odd elements.
Then $(\abs{\mathcal{M}}, \mathcal{O}_\mathcal{M}/\mathcal{J})$ is an ordinary [[smooth manifold|F45VQ47GPV]] of dimension $p$, the underlying manifold.
Let $E$ be a [[vector bundle|P4DZM975BV]] of rank $q$ over a manifold $M$, and let $\Gamma(\Lambda E^*)$ be the sheaf of sections of its exterior algebra bundle.
By Batchelor's theorem, every smooth supermanifold is isomorphic, not canonically, to some $(M, \Gamma(\Lambda E^*))$.

Example: $\R^{p|q}$ itself, and more generally $(M, \Gamma(\Lambda E^*))$ for a vector bundle $E \to M$ of rank $q$.
Over a trivialising open set $U$ of $E$ with $U$ a chart domain, $\Gamma(\Lambda E^*)|_U \cong C^\infty(U) \otimes \Lambda_q$, which is clause 3.

Non-example: $(\R, \mathcal{O})$ with $\mathcal{O}(U) = C^\infty(U)[\varepsilon]/(\varepsilon^2)$ and $\varepsilon$ declared even.
Every element is even, so the sheaf is not locally isomorphic to $C^\infty \otimes \Lambda_q$ with $q \ge 1$, which has odd elements.
It is not locally isomorphic to $C^\infty$, the case $q = 0$, either, since $C^\infty$ has no nonzero nilpotents.
Clause 3 fails.
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<document source="https://tamnd.github.io/math2/node/TQTDKMBJ0F.md" tag="TQTDKMBJ0F" kind="definition">
# Reverse mathematics

## Definition

Let $\mathsf{RCA}_0$ be the subsystem of [[second-order arithmetic|DC60FYCSPM]] with the basic axioms, induction for $\Sigma^0_1$ formulas, and comprehension for $\Delta^0_1$ formulas.
A sentence $\tau$ of second-order arithmetic is *equivalent over* $\mathsf{RCA}_0$ to a subsystem $S$ if:

1. $S$ proves $\tau$;
2. $\mathsf{RCA}_0 + \tau$ proves every axiom of $S$.

*Reverse mathematics* is the study of which theorems of ordinary mathematics are provable in which subsystems of second-order arithmetic.
In particular, it studies equivalences over $\mathsf{RCA}_0$.

## Remarks

The big five subsystems are $\mathsf{RCA}_0$, $\mathsf{WKL}_0$ (adds weak König's lemma), $\mathsf{ACA}_0$ (arithmetical comprehension), $\mathsf{ATR}_0$ (arithmetical transfinite recursion) and $\Pi^1_1\text{-}\mathsf{CA}_0$ ($\Pi^1_1$ comprehension), each stronger than the one before.

Example: the Bolzano–Weierstrass theorem for sequences in $[0,1]$ is equivalent to $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$.
The Heine–Borel theorem for countable covers of $[0,1]$ by open intervals is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$.

Non-example: weak König's lemma is provable in $\mathsf{ACA}_0$ but not equivalent to $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$.
Clause 2 fails, because $\mathsf{WKL}_0$ does not prove arithmetical comprehension.

The models of $\mathsf{RCA}_0$ with the standard natural numbers include the one whose sets are the computable sets, so $\mathsf{RCA}_0$ corresponds to computable mathematics.
</document>

<document source="https://tamnd.github.io/math2/node/TR4CNS5PMJ.md" tag="TR4CNS5PMJ" kind="concept">
# Root-finding algorithm

## Summary

A root-finding algorithm computes approximations of a zero of a function, that is, a point $x$ with $f(x) = 0$.
Most such algorithms are iterative: they produce a sequence of points, or of shrinking intervals, that tends to a zero.
Bracketing methods such as the [[bisection method|7VMYC7MB3Q]] and [[regula falsi|462AEY7MWF]] keep a sign change and always converge for continuous $f$.
Open methods such as [[Newton's method|07EXVNF9VP]] and the [[secant method|9ZC177PVPC]] converge faster near a simple zero, but only from good starting points.
Methods are compared by their [[rate of convergence|YWZQC5E54B]] and by the number of function evaluations per step.

## Explanations

### L1 human

Solving an equation such as $x^3 = x + 1$ means finding where the function $f(x) = x^3 - x - 1$ is $0$.
There is no simple formula for most such equations, so one guesses and improves the guess step by step.
For this $f$, $f(1) = -1 < 0$ and $f(2) = 5 > 0$, so a zero lies between $1$ and $2$, and halving this interval repeatedly closes in on it, near $1.3247$.
Faster methods use the slope of $f$ to jump close to the zero in a few steps.
</document>

<document source="https://tamnd.github.io/math2/node/TRB69DC8EW.md" tag="TRB69DC8EW" kind="definition">
# Tame Fréchet space

## Definition

A *graded Fréchet space* is a [[Fréchet space|N8TGDDJCFJ]] $F$ with a sequence of [[seminorms|4KX81Q85DT]] $\norm{\cdot}_0 \le \norm{\cdot}_1 \le \norm{\cdot}_2 \le \cdots$ that defines its topology.
Let $L \colon F \to G$ be a [[continuous linear map|0S3SMRH0Z5]] between graded Fréchet spaces.
It is *tame* if there are integers $r, b \ge 0$ and constants $C_n$ with $\norm{L f}_n \le C_n \norm{f}_{n + r}$ for all $f \in F$ and all $n \ge b$.

For a [[Banach space|0BDTTM24DP]] $B$, let $\Sigma(B)$ be the space of sequences $f = (f_k)_{k \ge 0}$ in $B$ with
$$\norm{f}_n = \sum_{k \ge 0} e^{n k} \norm{f_k}_B < \infty \quad \text{for all } n \ge 0,$$
graded by these norms.
A graded Fréchet space $F$ is *tame* if there are a Banach space $B$ and tame linear maps $L \colon F \to \Sigma(B)$ and $M \colon \Sigma(B) \to F$ with $M \circ L = \id_F$.

## Remarks

Example: $\Sigma(B)$ itself is tame, with $L = M = \id$; the identity satisfies the estimate with $r = b = 0$ and $C_n = 1$.
Hamilton proved that $C^\infty(X)$ for a compact manifold $X$, graded by the $C^n$ norms, is tame.

Non-example of a tame map: on $\Sigma(\R)$ let $(L f)_k = f_{\floor{k/2}}$.
Then $\norm{L f}_n \le (1 + e^n) \norm{f}_{2n}$, so $L$ is continuous.
For the unit sequence $\delta_j$ one gets $\norm{L \delta_j}_n \ge e^{2nj}$ while $\norm{\delta_j}_{n+r} = e^{(n+r)j}$, so for $n > r$ no constant $C_n$ works for all $j$, and $L$ is not tame.

The grading is part of the data: equivalent gradings, with $\norm{f}_n$ and $\norm{f}'_n$ bounded by each other up to a fixed shift $r$ and constants, give the same tame maps.
</document>

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# Closure of an operator

## Summary

The closure of a closable operator is its smallest closed extension.
Its graph is the closure of the graph of the original operator.
Taking closures lets one define an operator on a simple domain and then pass to a closed operator, to which the main theorems apply.

## Explanations

### L1 human

An operator given by a formula on nice inputs can be extended by limits: if $x_n \to x$ and $T x_n \to y$, declare $Tx = y$.
For a closable operator this rule is consistent, and the result is the closure.
For example, starting with differentiation on polynomials in $C[0, 1]$, the closure is differentiation on all continuously differentiable functions.
It is the most economical way to make an operator closed without changing what it does on the old domain.
</document>

<document source="https://tamnd.github.io/math2/node/TRD7DSF2KR.md" tag="TRD7DSF2KR" kind="concept">
# Jacobi polynomials

## Summary

The Jacobi polynomials $P_n^{(\alpha, \beta)}$ are the [[classical orthogonal polynomials|47WJ34NC9K]] on $[-1, 1]$ for the weight $(1 - x)^\alpha (1 + x)^\beta$, with $\alpha, \beta > -1$.
They include the [[Gegenbauer polynomials|5TRFA6F79B]], the [[Legendre polynomials|QFYGJVA996]] and both kinds of Chebyshev polynomials as special cases, up to constant factors.
They are terminating [[hypergeometric functions|463X5V31D9]], and they appear in harmonic analysis on spheres and on compact symmetric spaces.

## Explanations

### L1 human

The weight $(1 - x)^\alpha (1 + x)^\beta$ on the interval $[-1, 1]$ can favour one end, the other end, or both, depending on the two exponents.
For each choice there is one family of polynomials, perpendicular to each other for this weight: the Jacobi polynomials.
With $\alpha = \beta = 0$ the weight is constant and the family is the Legendre polynomials.
For example, $P_1^{(0,0)}(x) = x$ is perpendicular to the constant $1$ because $\int_{-1}^1 x \, dx = 0$.
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<document source="https://tamnd.github.io/math2/node/TRFDEJRPAZ.md" tag="TRFDEJRPAZ" kind="definition">
# Pitchfork bifurcation

## Definition

Let $f \colon \R \times \R \to \R$ be [[smooth|0V33BNKZ9S]], and consider the family of scalar equations $x' = f(x, \mu)$ with parameter $\mu$.
Assume that $f$ is [[odd|0FP31SX5EA]] in $x$: $f(-x, \mu) = -f(x, \mu)$ for all $x$ and $\mu$.
Then $f(0, \mu) = 0$, so $0$ is an [[equilibrium point|0G4X36AE0X]] for every $\mu$.
The family has a *pitchfork bifurcation* at $(x, \mu) = (0, 0)$ if:

1. $f_x(0, 0) = 0$;
2. $f_{x\mu}(0, 0) \ne 0$;
3. $f_{xxx}(0, 0) \ne 0$.

## Remarks

Under these conditions, besides $x = 0$, the equilibria near $(0, 0)$ form a curve $\mu = m(x)$ with $m$ even, $m(0) = 0$ and $m''(0) = -f_{xxx}(0, 0) / (3 f_{x\mu}(0, 0))$.
So on one side of $\mu = 0$ there is a symmetric pair of new equilibria, and the stability of $0$ changes at $\mu = 0$.
The bifurcation is supercritical when the new equilibria are stable and subcritical when they are unstable.

Example: $f(x, \mu) = \mu x - x^3$.
It is odd in $x$, $f_x(0, 0) = 0$, $f_{x\mu} = 1$ and $f_{xxx} = -6$.
The new equilibria are $\pm\sqrt{\mu}$ for $\mu > 0$, and they are stable since $f_x(\pm\sqrt{\mu}, \mu) = -2\mu < 0$.
The equation $x' = \mu x + x^3$ gives the subcritical case.

Non-example: $f(x, \mu) = \mu x - x^2$ is not odd in $x$, so the symmetry hypothesis fails; it gives a transcritical bifurcation.
Without the symmetry, a small perturbation such as $\mu x - x^3 + \varepsilon$ breaks the pitchfork into a saddle-node and a separate branch.
</document>

<document source="https://tamnd.github.io/math2/node/TRGBJYPBXH.md" tag="TRGBJYPBXH" kind="statement">
# Sato–Tate conjecture

## Statement

Let $E$ be an [[elliptic curve|427N6S6QQW]] over $\Q$ without [[complex multiplication|D3HEJTVY7G]].
For each prime $p$ of good reduction for $E$, let $\tilde{E}$ be the reduction of $E$ modulo $p$, a curve over the [[finite field|0ZXZ82FTX2]] $\F_p$.
Let $a_p = p + 1 - \#\tilde{E}(\F_p)$.
Let $\theta_p \in [0, \pi]$ be the angle with $a_p = 2\sqrt{p} \cos \theta_p$; it exists by [[Hasse's theorem|7Y26317GJ0]].
Then for all $0 \le \alpha \le \beta \le \pi$,
$$\lim_{x \to \infty} \frac{\#\set{p \le x : p \text{ good}, \ \alpha \le \theta_p \le \beta}}{\#\set{p \le x : p \text{ good}}} = \frac{2}{\pi} \int_\alpha^\beta \sin^2 \theta \, d\theta.$$

## Remarks

In words, the angles $\theta_p$ are equidistributed with respect to the measure $\frac{2}{\pi} \sin^2 \theta \, d\theta$ on $[0, \pi]$.
This is the image of the Haar measure of $\mathrm{SU}(2)$ under the map that sends a matrix with eigenvalues $e^{\pm i\theta}$ to $\theta$.
The conjecture was formulated independently by Mikio Sato and John Tate around 1960.
Tate showed that it follows from the meromorphic continuation and non-vanishing on $\mathrm{Re}(s) \ge 1$ of the symmetric power $L$-functions $L(\mathrm{Sym}^m E, s)$ for all $m \ge 1$.
Clozel, Harris, Shepherd-Barron and Taylor proved it for curves with multiplicative reduction at some prime, in work published in 2008.
Barnet-Lamb, Geraghty, Harris and Taylor removed this condition in 2011, using potential automorphy of the symmetric powers.
The hypothesis of no complex multiplication is necessary.
For a CM curve, $a_p = 0$ for the good primes that are inert in the CM field, which are half of all primes.
So $\theta_p = \pi/2$ for these primes, and the limit distribution is different.

## Explanations

### L1 human

For each prime, the number of points of an elliptic curve modulo $p$ is close to $p + 1$, with an error that is at most $2\sqrt{p}$ in size.
Rescaled to an angle between $0$ and $\pi$, this error looks random as $p$ varies.
The theorem says exactly how it is distributed: angles near $\pi/2$ are most common, and angles near $0$ and $\pi$ are rare, with density proportional to $\sin^2 \theta$.

### L2 human

The proof is a major application of automorphy lifting theorems in the [[Langlands program|J9T2M084W5]].
The statement generalises, through the notion of Sato–Tate group, to other motives and to modular forms.
For example, it is also known for non-CM holomorphic newforms of weight at least $2$.
</document>

<document source="https://tamnd.github.io/math2/node/TRKEGEZGGZ.md" tag="TRKEGEZGGZ" kind="definition">
# Jet

## Definition

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], let $p \in M$ and let $k \ge 0$ be an integer.
Let $f$ and $g$ be [[smooth maps|0V33BNKZ9S]] into $N$, each defined on an open neighbourhood of $p$ in $M$.
Then $f$ and $g$ have the *same $k$-jet at $p$* if both conditions hold.

1. $f(p) = g(p)$.
2. Take a chart $x$ of $M$ around $p$ and a chart $y$ of $N$ around $f(p)$. Then $y \circ f \circ x^{-1}$ and $y \circ g \circ x^{-1}$ have equal [[partial derivatives|09EGC6F8PT]] of every order $1, \dots, k$ at $x(p)$.

This is an equivalence relation.
The *$k$-jet of $f$ at $p$*, written $j^k_p f$, is the equivalence class of $f$.
The set of all $k$-jets at $p$ of maps from $M$ to $N$ is written $J^k_p(M, N)$.

## Remarks

Condition 2 does not depend on the charts, by the chain rule.
Indeed, the derivatives up to order $k$ of a composite depend only on those of its factors.
The jet $j^k_p f$ depends only on the [[germ|7SWSRGDT4A]] of $f$ at $p$.
For $M = \R^n$ and $N = \R$, the $k$-jet of $f$ at $p$ is the same data as its [[Taylor polynomial|0ZZ4RHTR9X]] of degree $k$ at $p$.

Example: on $\R$, $f(x) = \sin x$ and $g(x) = x$ have the same $2$-jet at $0$.
Indeed $f(0) = g(0) = 0$, $f'(0) = g'(0) = 1$ and $f''(0) = g''(0) = 0$.

Non-example: $f(x) = e^x$ and $g(x) = 1 + x$ have the same $1$-jet at $0$ but not the same $2$-jet.
Condition 2 fails for order $2$, since $f''(0) = 1$ and $g''(0) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/TRMYSF91EV.md" tag="TRMYSF91EV" kind="statement">
# Endomorphism ring of a simple module is a division ring

## Statement

Let $R$ be a ring and let $M$ be a [[simple|G00C22AEDH]] left $R$-module.
Then the [[endomorphism ring|AWSBZWWMG9]] $\End_R(M)$ is a [[division ring|0EA5BTA0CX]].

## Remarks

$\End_R(M)$ is nonzero, since $M \ne 0$ gives $\id_M \ne 0$.
By [[Schur's lemma|X4M2FFKQJ6]] every nonzero endomorphism of $M$ is an isomorphism, so its inverse map is again an endomorphism and is its inverse in $\End_R(M)$.
Example: $\End_\Z(\Z/p\Z) \cong \Z/p\Z$, a field, for $p$ prime.
Simplicity is needed: for the semisimple $\Z$-module $M = \Z/p\Z \oplus \Z/p\Z$, the ring $\End_\Z(M) \cong M_2(\Z/p\Z)$ has the nonzero non-invertible element $E_{11}$.
If $R$ is an algebra over an algebraically closed field $k$ and $M$ has finite dimension over $k$, then $\End_R(M) = k$.

## Explanations

### L1 human

The self-maps of a simple module that are not zero can always be undone.
So the self-maps form a ring in which one can divide by every nonzero element, just as in a field, except that the order of multiplication may matter.

### L2 human

This is how division rings enter the structure theory of rings.
In the Artin–Wedderburn theorem, a simple Artinian ring is a matrix ring over the division ring $\End_R(M)^{\mathrm{op}}$ for a simple module $M$.
In representation theory over $\C$, it says that endomorphisms of a finite-dimensional irreducible representation are scalars.
</document>

<document source="https://tamnd.github.io/math2/node/TRRREQX56B.md" tag="TRRREQX56B" kind="definition">
# Hankel matrix

## Definition

Let $R$ be a [[ring|0B1261B5ED]] and let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$.
The matrix $A$ is a *Hankel matrix* when $a_{i, j+1} = a_{i+1, j}$ for all $1 \le i < m$ and $1 \le j < n$.
Equivalently, there are elements $h_k \in R$, for $2 \le k \le m + n$, with $a_{ij} = h_{i+j}$ for all $i, j$.

## Remarks

Example: $\begin{pmatrix} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 5 \end{pmatrix}$ is Hankel, with $h_2, \dots, h_6 = 1, 1, 2, 3, 5$; each anti-diagonal is constant.
Non-example: $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is not Hankel, because $a_{12} = 2 \ne 3 = a_{21}$.
A square Hankel matrix is symmetric, since $a_{ij} = h_{i+j} = a_{ji}$.
The [[Hilbert matrix|2DTWB6F5TA]], with $h_k = 1/(k-1)$, is a Hankel matrix.
If $J$ is the $m \times m$ matrix with ones on the anti-diagonal and zeros elsewhere, then $A$ is Hankel exactly when $JA$ is a [[Toeplitz matrix|GKY2WSHE26]].
</document>

<document source="https://tamnd.github.io/math2/node/TRTN840ZB0.md" tag="TRTN840ZB0" kind="concept">
# Ill-posedness of real Laplace inversion

## Summary

Recovering a function $f$ from the values of its [[Laplace transform|0V49JEZW06]] $F(s)$ at real points $s > 0$ is an ill-posed problem in the sense of Hadamard.
The solution is unique, but it does not depend continuously on the data.
Transforms that are very close can come from functions that are far apart.
So small errors in measured values of $F$ can produce large errors in the computed $f$, and numerical inversion needs regularisation.
The problem fails the stability condition in the definition of a [[well-posed problem|W09B9NZ9B5]].

## Explanations

### L1 human

The Laplace transform smooths a function a great deal: it averages $f$ against the slowly varying weights $e^{-s t}$.
Fast wiggles in $f$ almost cancel in these averages, so they leave nearly no trace in $F$.
For example, $e^{-t} \sin(n t)$ has size about $1/2$ for every $n$, but its transform $n/((s+1)^2 + n^2)$ shrinks to $0$ as $n$ grows.
So from a slightly noisy $F$ one cannot tell whether such wiggles are present, and inversion becomes unstable.
</document>

<document source="https://tamnd.github.io/math2/node/TS1GQGM7ZJ.md" tag="TS1GQGM7ZJ" kind="definition">
# Gauss sum

## Definition

Let $F$ be a [[finite field|0ZXZ82FTX2]] with $q$ elements.
A *multiplicative character* of $F$ is a [[group homomorphism|0JYTG5K9Y4]] $\chi \colon F^\times \to \C^\times$; it is extended to $F$ by $\chi(0) = 0$.
An *additive character* of $F$ is a group homomorphism $\psi \colon (F, +) \to \C^\times$.
The *Gauss sum* of $\chi$ and $\psi$ is
$$g(\chi, \psi) = \sum_{a \in F} \chi(a) \psi(a).$$

## Remarks

The most common case is $F = \Z/p\Z$ for a prime $p$ and $\psi(a) = \zeta^a$ with $\zeta = e^{2\pi i/p}$; one then writes $g(\chi) = \sum_{a=1}^{p-1} \chi(a) \zeta^a$.
If $\chi$ and $\psi$ are both nontrivial, then $\abs{g(\chi, \psi)} = \sqrt{q}$.

Example: let $p = 3$ and let $\chi$ be the [[Legendre symbol|0JTC045NNZ]] modulo $3$, so $\chi(1) = 1$ and $\chi(2) = -1$.
Then $g(\chi) = \zeta - \zeta^2 = i\sqrt{3}$, and $g(\chi)^2 = -3$, in line with the general rule $g(\chi)^2 = \left(\frac{-1}{p}\right) p$ for the quadratic character.

Non-example: the sum $\sum_{a \in \Z/p\Z} \chi(a) e^{2\pi i a^2/p}$ is not a Gauss sum in this sense, because $a \mapsto e^{2\pi i a^2/p}$ is not additive, so it is not an additive character.

For a [[Dirichlet character|7XB5VJKDPG]] $\chi$ modulo $n$ one defines in the same way $g(\chi) = \sum_{a \bmod n} \chi(a) e^{2\pi i a/n}$.
</document>

<document source="https://tamnd.github.io/math2/node/TS442VKWAG.md" tag="TS442VKWAG" kind="problem">
# No-three-in-line problem

## Problem

For an integer $n \ge 1$, consider the $n \times n$ grid $\set{1, \dots, n}^2$ of points in the plane.
Find the largest number of points of the grid that can be chosen so that no three chosen points are [[collinear|0AFNNTRTJE]].
In particular, decide for which $n$ this number equals $2n$.

## Remarks

Henry Dudeney posed the problem in 1900.
Each of the $n$ rows can contain at most $2$ chosen points, so by the [[pigeonhole principle|0HGVCWQGY3]] at most $2n$ points can be chosen.
The bound $2n$ is attained for all $n$ up to at least $46$, by solutions found with computer searches.
For a prime $p$, the $p$ points $(x, x^2 \bmod p)$ with $0 \le x < p$ have no three in a line, an observation of Erdős.
Indeed, three of them on a line would give three roots in the field $\F_p$ of a nonzero polynomial of degree at most $2$.
Hall, Jackson, Sudbery and Wild refined such constructions in 1975.
They showed that for every $\varepsilon > 0$ and all large $n$, at least $(3/2 - \varepsilon)n$ points can be chosen.
It is conjectured that for large $n$ the maximum is strictly less than $2n$.

## Explanations

### L1 human

On an $n$ by $n$ board of dots, choose as many dots as you can so that no three of your dots lie on a straight line, in any direction.
Two dots per row is the most you can hope for, which gives $2n$.
For small boards one can reach $2n$, but whether this works for all boards is unknown.
</document>

<document source="https://tamnd.github.io/math2/node/TS5XH58X86.md" tag="TS5XH58X86" kind="definition">
# Gromov–Hausdorff convergence

## Definition

Let $X_1, X_2, \ldots$ and $X$ be nonempty [[compact|0TVJZYB6SH]] [[metric spaces|0VKD1TP7F3]].
Let $d_{GH}$ be the Gromov–Hausdorff distance, the metric of the [[Gromov–Hausdorff space|ETRF2FHAEE]].
The sequence $(X_n)$ *converges to $X$ in the Gromov–Hausdorff sense* if $d_{GH}(X_n, X) \to 0$ as $n \to \infty$.

## Remarks

Example: let $X_n = \set{k/n : k = 0, 1, \ldots, n}$ and $X = [0, 1]$, both with the distance $\abs{s - t}$.
Both lie in $\R$, and each point of $[0, 1]$ is within $\tfrac{1}{2n}$ of $X_n$.
So $d_{GH}(X_n, X) \le \tfrac{1}{2n} \to 0$, and $X_n$ converges to $[0, 1]$.

Non-example: let $X_n$ be the round sphere of radius $n$ with its great-circle distance.
One has $\abs{\operatorname{diam} Y - \operatorname{diam} Z} \le 2\, d_{GH}(Y, Z)$ for compact $Y$ and $Z$.
The diameters $\pi n$ tend to infinity, so $(X_n)$ converges to no compact space.

The limit is unique up to isometry, because $d_{GH}$ is a metric on isometry classes.
For spaces that are not compact, such as complete Riemannian manifolds, one uses *pointed* Gromov–Hausdorff convergence.
There a base point is fixed in each space, and one asks for convergence of the closed balls of each radius $r$ around the base points.
</document>

<document source="https://tamnd.github.io/math2/node/TS9TBY8ES1.md" tag="TS9TBY8ES1" kind="statement">
# Lindelöf's maximum principle

## Statement

Let $\Omega \subset \C$ be a bounded [[domain|0Q4CBD3HX0]] and let $F \subset \partial\Omega$ be a finite set.
Let $u$ be a [[subharmonic function|Z1MHMGFNFW]] on $\Omega$ that is bounded above.
Assume that $\limsup_{z \to \zeta} u(z) \le 0$ for every $\zeta \in \partial\Omega \setminus F$.
Then $u \le 0$ on $\Omega$.

## Remarks

Proof idea: let $d$ be the diameter of $\Omega$ and let $\varepsilon > 0$.
The function $v = u + \varepsilon \sum_{p \in F} \log(\abs{z - p}/d)$ is subharmonic, and $v \le u$ on $\Omega$ because each logarithm is negative there.
At each $p \in F$, $v \to -\infty$ since $u$ is bounded above; so $\limsup v \le 0$ at every boundary point.
The [[maximum principle for subharmonic functions|9XJ5FBQD7M]] gives $v \le 0$, and letting $\varepsilon \to 0$ gives $u \le 0$.
The bound from above is needed.
On the punctured disc $\Omega = \set{0 < \abs{z} < 1}$ with $F = \set{0}$, the function $u(z) = -\log\abs{z}$ is harmonic, tends to $0$ on the unit circle, and is positive on $\Omega$.
The finite set $F$ may be replaced by any [[polar set|C95QJ7YH5T]] in $\partial\Omega$, with the same conclusion.

## Explanations

### L1 human

The ordinary maximum principle needs control at every boundary point.
This version says that finitely many bad boundary points do not matter, provided the function cannot blow up there.

### L2 human

The lemma is the bounded form of the [[Phragmén–Lindelöf principle|FYR6V2NH6C]]: a growth condition lets one ignore a small part of the boundary.
Applied to $\log\abs{f}$ for a bounded holomorphic $f$, it shows that $\abs{f} \le 1$ on $\Omega$ when $\limsup \abs{f} \le 1$ at all but finitely many boundary points.
A related result for holomorphic functions on a half-strip is [[Lindelöf's theorem|XBM7J5R1VQ]].
In the study of harmonic measure the lemma shows that finite sets of boundary points carry no mass.
</document>

<document source="https://tamnd.github.io/math2/node/TSF8RZDD5B.md" tag="TSF8RZDD5B" kind="concept">
# Topological ring

## Summary

A topological ring is a [[ring|0B1261B5ED]] with a [[topology|0CQHFPGJ45]] in which addition, negation and multiplication are continuous.
It lets one speak of limits, convergent series and completions inside a ring.
The real and complex numbers, the [[$p$-adic numbers|BT6T702C28]] and rings of formal power series with the $(x)$-adic topology are standard examples.
A [[topological field|D7YQW9RG24]] is a topological ring that is a field and in which inversion is also continuous on nonzero elements.

## Explanations

### L1 human

A topological ring is a ring where it makes sense to say that elements are "close" to each other, and the ring operations respect closeness.
If $a$ is close to $a'$ and $b$ is close to $b'$, then $a + b$ is close to $a' + b'$ and $ab$ is close to $a'b'$.
The real numbers with the usual distance are the basic example.
In the integers with the $p$-adic topology, a number is close to $0$ when it is divisible by a high power of $p$.
</document>

<document source="https://tamnd.github.io/math2/node/TSPTN464CM.md" tag="TSPTN464CM" kind="concept">
# Pick matrix

## Summary

The Pick matrix of interpolation data $z_1, \ldots, z_n$ in the [[unit disc|8G5ABXRK85]] and target values $w_1, \ldots, w_n$ is the Hermitian matrix with entries $\frac{1 - w_i \overline{w_j}}{1 - z_i \overline{z_j}}$.
By [[Nevanlinna–Pick interpolation|4VMSSYB5DN]], a holomorphic self-map of the disc with $f(z_i) = w_i$ exists exactly when this matrix is positive semidefinite.
It turns an analytic existence problem into a finite check of linear algebra.

## Explanations

### L1 human

Suppose you want a holomorphic map of the disc into itself that sends some given points to some given values inside the disc.
You write down a square table of numbers built from the points and the values.
If that table is a [[positive semidefinite matrix|0880F9EQCP]], such a map exists; otherwise it does not.
With one point, the table is the single positive number $\frac{1 - \abs{w_1}^2}{1 - \abs{z_1}^2}$, and indeed a [[Möbius transformation|0B346DDAM5]] of the disc sends $z_1$ to $w_1$.
</document>

<document source="https://tamnd.github.io/math2/node/TST0GH66A5.md" tag="TST0GH66A5" kind="concept">
# Discrete Fourier transform

## Summary

The discrete Fourier transform turns a finite list of $N$ complex numbers into another list of $N$ numbers, its frequency components.
It is the Fourier transform of the finite cyclic group $\Z / N\Z$, built from the $N$-th [[roots of unity|0YV2CKP2AV]].
It is invertible, it preserves energy up to the factor $N$, and it turns cyclic convolution into pointwise multiplication.
The fast Fourier transform computes it in $O(N \log N)$ operations, which makes it one of the most used algorithms in numerical computing and signal processing.
See the [[definition|TJ9XPJ65YX]].

## Explanations

### L1 human

Take $N$ samples of a signal at equally spaced times.
The discrete Fourier transform tells how much of each of the $N$ basic waves, which complete $0, 1, \dots, N - 1$ full turns over the sampling window, is present in the samples.
For $N = 4$, the list $(1, 1, 1, 1)$ is a pure constant, and its transform is $(4, 0, 0, 0)$: only the frequency $0$ is present.
The list $(1, 0, 0, 0)$, a single spike, has transform $(1, 1, 1, 1)$: every frequency is present equally.
</document>

<document source="https://tamnd.github.io/math2/node/TST3NVVB58.md" tag="TST3NVVB58" kind="definition">
# Strict hyperbolicity

## Definition

Let $\Omega \subseteq \R^n \times \R$ be open, let $N \ge 1$, and let $A_1, \ldots, A_n \colon \Omega \to \R^{N \times N}$ be continuous.
The first-order system
$$\partial_t u + \sum_{j=1}^{n} A_j(x, t)\, \partial_{x_j} u = 0$$
for $u \colon \Omega \to \R^N$ is *strictly hyperbolic* if for every $(x, t) \in \Omega$ and every $\xi \in \R^n \setminus \set{0}$, the matrix $A(x, t, \xi) = \sum_{j=1}^{n} \xi_j A_j(x, t)$ has $N$ real and distinct [[eigenvalues|0BQAA2BK3Q]].

Let $P u = 0$ be a scalar linear equation of order $m$ in the variables $(x, t)$, with [[principal symbol|F31T650DKY]] $p_m(x, t; \xi, \tau)$.
It is *strictly hyperbolic* with respect to $t$ if, for every $(x, t)$ and every $\xi \ne 0$, the polynomial $\tau \mapsto p_m(x, t; \xi, \tau)$ has $m$ real and distinct roots.

## Remarks

Example: the [[hyperbolic|92C3S03AQZ]] system $u_t + \begin{pmatrix} 0 & -1 \\ -c^2 & 0 \end{pmatrix} u_x = 0$ with $c > 0$, obtained from $w_{tt} = c^2 w_{xx}$ by setting $u = (w_x, w_t)$, has eigenvalues $\pm c$, which are real and distinct.
For the wave operator $\partial_t^2 - c^2 \Delta$, the principal symbol is $-\tau^2 + c^2 \abs{\xi}^2$, with the two distinct real roots $\tau = \pm c\abs{\xi}$ when $\xi \ne 0$.
Non-example: the system $u_t + \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} u_x = 0$ has the double eigenvalue $1$, so it is not strictly hyperbolic; it is not even diagonalizable.
Strict hyperbolicity implies that $A(x, t, \xi)$ is diagonalizable over $\R$, so a strictly hyperbolic system is hyperbolic.
The converse fails: $u_t + I u_x = 0$ with $I$ the $2 \times 2$ identity is hyperbolic but not strictly hyperbolic.
</document>

<document source="https://tamnd.github.io/math2/node/TSX2MAA95V.md" tag="TSX2MAA95V" kind="definition">
# Elimination theory

## Definition

Let $k$ be a [[field|04C5ZBG1WT]], let $S = k[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]], and let $I$ be an [[ideal|096Z9MP2ME]] of $S$.
For $0 \le l < n$, the *$l$-th elimination ideal* of $I$ is
$$I_l = I \cap k[x_{l+1}, \ldots, x_n].$$
Elimination theory is the study of the elimination ideals of ideals and of their zero sets.

## Remarks

$I_l$ is an ideal of $k[x_{l+1}, \ldots, x_n]$; it consists of the consequences of the equations of $I$ in which $x_1, \ldots, x_l$ do not occur.
Elimination theorem: if $G$ is a [[Gröbner basis|C83KPT47R0]] of $I$ for the [[lexicographic order|0B68KNPGCG]] with $x_1 > \cdots > x_n$, then $G \cap k[x_{l+1}, \ldots, x_n]$ is a Gröbner basis of $I_l$.
Closure theorem: let $k$ be [[algebraically closed|0PF6TM8TWT]], and project the zero set of $I$ to the last $n - l$ coordinates.
Then the zero set of $I_l$ in $k^{n-l}$ is the Zariski closure of this projection.

Example: in $k[t, x, y]$ with the lexicographic order $t > x > y$, let $I = (x - t, \; y - t^2)$.
Since $y - t^2 = (y - x^2) - (t - x)(t + x)$, one gets $I = (t - x, \; x^2 - y)$.
These two generators have coprime leading terms $t$ and $x^2$, so they form a Gröbner basis, and the first elimination ideal is $I_1 = (x^2 - y)$: eliminating $t$ gives the parabola $y = x^2$.

Non-example: $x - t$ lies in $I$ but not in $I_1$, because it involves the eliminated variable $t$.
</document>

<document source="https://tamnd.github.io/math2/node/TT0HJQ6QD7.md" tag="TT0HJQ6QD7" kind="definition">
# Initial algebra

## Definition

Let $\mathcal{C}$ be a category and let $F \colon \mathcal{C} \to \mathcal{C}$ be a functor.
An *initial $F$-algebra* is an [[F-algebra|4RB8834SEJ]] $(I, \iota)$ that is an [[initial object|EK85VSSRMR]] of the category of $F$-algebras.
That is, for every $F$-algebra $(B, \beta)$ there is exactly one morphism $h \colon I \to B$ with $h \circ \iota = \beta \circ F h$.

## Remarks

Example: on $\mathbf{Set}$, let $F X = \set{\ast} \sqcup X$, and let $\iota \colon F \N \to \N$ send $\ast$ to $0$ and $n$ to $n + 1$.
Then $(\N, \iota)$ is an initial $F$-algebra.
Given $(B, \beta)$, the condition on $h$ says $h(0) = \beta(\ast)$ and $h(n + 1) = \beta(h(n))$, and these equations define exactly one function $h$ by recursion.

Non-example: $(\Z, \iota')$ with $\iota'(\ast) = 0$ and $\iota'(n) = n + 1$ is an $F$-algebra but not an initial one.
There is no homomorphism $h$ from it to $(\N, \iota)$: it would satisfy $h(0) = 0$ and $h(-1) + 1 = h(0) = 0$, which has no solution in $\N$.

For an algebraic signature $\sigma$ and the functor $F_\sigma X = \bigsqcup_n \sigma_n \times X^n$, the initial $F_\sigma$-algebra is the algebra of closed terms of $\sigma$.
</document>

<document source="https://tamnd.github.io/math2/node/TT2ASTQ0WE.md" tag="TT2ASTQ0WE" kind="definition">
# Logicism

## Definition

*Logicism* is the position in the [[foundations of mathematics|EFFRJFW67Y]] that consists of two theses:

1. every mathematical concept, or every concept of arithmetic in the weaker form, can be defined from purely logical concepts;
2. every mathematical truth, or every truth of arithmetic in the weaker form, can be derived from purely logical laws by purely logical rules of inference.

## Remarks

Frege's definition of number follows the first thesis: the number of a concept $F$ is the class of all concepts that are in one-to-one correspondence with $F$.
The axiom of infinity of Principia Mathematica does not fit the second thesis easily.
It asserts that infinitely many objects exist, and it is not clear that logic alone can assert this.
The underlying logic of logicism is a strong logic, such as higher-order logic or a theory of types, not only [[predicate logic|086QKPF0A6]].
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# Partial correlation

## Summary

The partial correlation of two random variables given some control variables is a [[correlation|0DNN9B6Q2F]].
It is the correlation that remains after the linear effect of the controls is removed from both variables.
It separates a direct linear association from one that only passes through the controls.
For jointly normal variables it is zero exactly when the two variables are [[conditionally independent|0THJ7416DX]] given the controls.
This makes it the basic tool of Gaussian [[graphical models|Q3FQAD1BXG]].

## Explanations

### L1 human

Ice cream sales and sunburn cases rise and fall together, but both follow the temperature.
To see whether they are linked beyond the temperature, predict each one from the temperature by a straight line and keep only the leftover parts.
The partial correlation is the correlation of these two leftovers.
If $X = Z + U$ and $Y = Z + V$ with $Z, U, V$ uncorrelated, then $X$ and $Y$ are correlated, but their partial correlation given $Z$ is $0$.
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# Meromorphic function of several variables

## Definition

Let $D \subseteq \C^n$ be an [[open set|02BH3MCBQB]].
A *meromorphic function* on $D$ is given by an open cover $(U_i)_{i \in I}$ of $D$ by connected open sets.
For each $i$ it gives two [[holomorphic functions|507C34W0ZT]] $g_i, h_i$ on $U_i$ such that:

1. $h_i$ is not identically zero on $U_i$;
2. $g_i h_j = g_j h_i$ on $U_i \cap U_j$ for all $i, j \in I$.

Two such data $(U_i, g_i, h_i)_{i \in I}$ and $(V_k, p_k, q_k)_{k \in K}$ define the same meromorphic function when $g_i q_k = p_k h_i$ on $U_i \cap V_k$ for all $i$ and $k$.

## Remarks

On the dense open set where some $h_i$ is nonzero, a meromorphic function $f$ is the holomorphic function $g_i / h_i$.
Its polar set and its set of indeterminacy points are analytic subsets; for $n \ge 2$ the indeterminacy set has codimension at least $2$.

Example: on $D = \C^2$ take the single set $U_1 = \C^2$ with $g_1(z) = z_1$ and $h_1(z) = z_2$.
Clause 1 holds since $z_2$ is not the zero function, and clause 2 is empty, so $z_1 / z_2$ is meromorphic on $\C^2$.

Non-example: $e^{1/z_1}$, defined where $z_1 \ne 0$, is not the restriction of a meromorphic function on $\C^2$.
Suppose it were $g/h$ near a point $(0, c)$.
Then, for a suitable fixed $z_2$ close to $c$, the one-variable function $z_1 \mapsto e^{1/z_1}$ would be meromorphic at $0$.
But it has an essential singularity there.

For $n = 1$ this definition gives the usual [[meromorphic functions|0ZZ7ZYDR8W]] on an open set of $\C$.
In the language of sheaves, a meromorphic function is a global section of the sheaf of quotient fields of the sheaf of holomorphic functions.
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# Robinson arithmetic

## Definition

Let $L$ be the [[signature|WDVVEYD6AT]] with a constant symbol $0$, a unary function symbol $S$, and binary function symbols $+$ and $\cdot$.
*Robinson arithmetic* $\mathsf{Q}$ is the first-order [[theory|92SWXST1ZM]] in $L$ whose axioms are the universal closures of these seven formulas:

1. $S x \ne 0$;
2. $S x = S y \to x = y$;
3. $x \ne 0 \to \exists y\, (x = S y)$;
4. $x + 0 = x$;
5. $x + S y = S(x + y)$;
6. $x \cdot 0 = 0$;
7. $x \cdot S y = (x \cdot y) + x$.

## Remarks

Example: the natural numbers $\N$ with $0$, $S(n) = n + 1$, and the usual addition and multiplication satisfy all seven axioms, so they form a model of $\mathsf{Q}$.

Non-example: the integers $\Z$ with $S(n) = n + 1$ and the usual operations are not a model of $\mathsf{Q}$.
Axiom 1 fails, because $S(-1) = 0$.

$\mathsf{Q}$ is the [[Peano axioms|0QFT7Z76N5]] without the induction scheme, with axiom 3 added in its place.
$\mathsf{Q}$ does not prove $\forall x\, \forall y\, (x + y = y + x)$; there are models of $\mathsf{Q}$ in which addition is not commutative.
$\mathsf{Q}$ proves every true sentence of the form $\exists x_1 \cdots \exists x_n\, \varphi$ with $\varphi$ quantifier-free, and every computable function is representable in $\mathsf{Q}$.
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# Riesz representation theorem

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $H$ be a [[Hilbert space|HM2HHE3J51]] over $\mathbb{K}$ whose inner product $\ip{\cdot}{\cdot}$ is linear in the first argument.
Let $\varphi \colon H \to \mathbb{K}$ be a continuous [[linear form|07YQ1C78G0]].
Then there is exactly one vector $y \in H$ such that $\varphi(x) = \ip{x}{y}$ for all $x \in H$.
Moreover $\norm{\varphi} = \norm{y}$, where $\norm{\varphi}$ is the operator norm of $\varphi$.
Hence the map that sends $\varphi$ to $y$ is a bijective, conjugate-linear isometry from the [[continuous dual|4A82S5FVRT]] $H'$ onto $H$.

## Remarks

If $\varphi = 0$, take $y = 0$.
Otherwise the kernel $N = \ker \varphi$ is a closed subspace different from $H$.
By the [[Hilbert projection theorem|J5YNTHQMFX]], its [[orthogonal complement|0XJXTBCB2T]] $N^\perp$ contains a vector $z \neq 0$, and $N^\perp$ has dimension $1$.
Then $y = \overline{\varphi(z)}\, z / \norm{z}^2$ works, and uniqueness follows because $\ip{x}{y - y'} = 0$ for all $x$ forces $y = y'$.

Example: on $L^2([0,1])$ the form $\varphi(f) = \int_0^1 f(t)\, dt$ is represented by the constant function $y = 1$, and $\norm{\varphi} = 1$.

Completeness is necessary.
Let $c_{00}$ be the space of finitely supported sequences with the inner product of $\ell^2$.
The form $\varphi(x) = \sum_n x_n / n$ is continuous on $c_{00}$ by the Cauchy–Schwarz inequality.
The only candidate is $y = (1/n)_n$, which is not finitely supported, so no $y \in c_{00}$ represents $\varphi$.

In finite dimension the result reduces to the [[Riesz representation theorem in finite dimensions|04GYPXDXNH]].

## Explanations

### L1 human

On a Hilbert space, every reasonable way to turn vectors into numbers linearly is "take the inner product with a fixed vector".
So the space of such linear measurements looks exactly like the space itself.
For example, taking the average of a function on $[0,1]$ is the inner product with the constant function $1$.

### L2 human

The theorem identifies a Hilbert space with its dual, up to complex conjugation.
It gives the adjoint of a bounded operator: $\ip{Tx}{y} = \ip{x}{T^*y}$ defines $T^*y$ because $x \mapsto \ip{Tx}{y}$ is a continuous linear form.
It also shows that every Hilbert space is reflexive.
The Lax–Milgram theorem, used for weak solutions of elliptic equations, extends it to bounded coercive bilinear forms.
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# Stokes phenomenon

## Summary

The Stokes phenomenon is the fact that one analytic function can have different asymptotic expansions in different sectors of the complex plane.
It occurs for solutions of linear differential equations at an [[irregular singular point|AT1W4MMTYR]].
Across certain rays, the Stokes lines, an exponentially small contribution in the expansion switches on or off.
The constants that measure the jump are the Stokes multipliers.
Together with the formal invariants, the Stokes data classify such equations up to meromorphic equivalence.
See the [[definition|H5V4099GN7]].

## Explanations

### L1 human

Approximate formulas for a function near infinity often hold only in some directions.
For the Airy function, one simple formula holds when $x$ is large and positive, where the function decays like an exponential.
For $x$ large and negative, the function oscillates, and the right formula is a sum of two exponentials, which together make a sine wave.
The function is the same everywhere, but its approximation changes form as one turns around the origin.
Stokes discovered this while computing the Airy integral in optics.
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# Spectral density estimation

## Summary

Spectral density estimation is the problem of estimating the spectral density of a stationary process from a finite stretch of data.
The spectral density describes how the variance of the process is spread over frequencies.
The raw [[periodogram|3ZYM0HXA3E]] is too noisy for this, so practical estimators smooth it over nearby frequencies or fit a parametric model.
Spectral estimates reveal periodic components and the memory structure of a series and are used in signal processing, economics and geophysics.

## Explanations

### L1 human

A stationary signal can be thought of as a mix of waves of many frequencies with random amplitudes.
Its spectral density says how much power each frequency carries, like the bars of a music equalizer.
From a recording of finite length, these bars can only be estimated, and a naive estimate flickers wildly from one frequency to the next.
Averaging over neighbouring frequencies, or over several pieces of the recording, gives a steadier picture at the cost of some blurring.
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# Quantum calculus

## Definition

*Quantum calculus* is the study of [[functions|0EPAT51DXQ]] through the following two difference operators in place of the derivative.

1. For a fixed $q \in \C \setminus \set{0, 1}$, the *$q$-derivative* of $f$ is $D_q f(x) = \dfrac{f(qx) - f(x)}{(q - 1)x}$ for $x \ne 0$; this is the [[q-derivative|PMJT3FZE0Y]].
2. For a fixed $h \in \C \setminus \set{0}$, the *$h$-derivative* of $f$ is $D_h f(x) = \dfrac{f(x + h) - f(x)}{h}$.

The corresponding antiderivatives are the [[Jackson integral|7EMPA6M8S7]] $\int_0^x f(t) \, d_q t = (1 - q) x \sum_{k \ge 0} q^k f(q^k x)$, for $0 < \abs{q} < 1$ when the series converges, and the sum $h \sum_{k=0}^{n-1} f(a + kh)$ over a grid of step $h$.
The part built on $D_q$ is called *$q$-calculus*, and the part built on $D_h$ is called *$h$-calculus*.

## Remarks

Example: for $f(x) = x^n$ with $n \ge 1$, $D_q f(x) = \dfrac{q^n x^n - x^n}{(q - 1)x} = [n]_q x^{n-1}$, where $[n]_q = 1 + q + \cdots + q^{n-1}$.

Non-example: the ordinary derivative $f'(x) = \lim_{h \to 0} (f(x + h) - f(x))/h$ is not an operator of quantum calculus, because it involves a limit rather than a fixed step $q$ or $h$.
It is the common limit of $D_q$ as $q \to 1$ and of $D_h$ as $h \to 0$ for differentiable $f$.

The $q$-derivative satisfies the product rule $D_q(fg)(x) = f(qx) D_q g(x) + g(x) D_q f(x)$.
The Jackson integral inverts $D_q$: if $F(x) = \int_0^x f(t) \, d_q t$ converges at $x$, then $D_q F(x) = f(x)$.
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# Absolutely free algebra

## Summary

An absolutely free algebra is an algebra that is free in the class of all algebras of its signature, with no identities imposed at all.
Up to isomorphism, the absolutely free algebra on a set $X$ is the [[term algebra|3YYKGWJJND]] over $X$, whose elements are the formal terms.
Every free algebra of a variety is a quotient of an absolutely free algebra, so terms are the common raw material of all free algebras.

## Explanations

### L1 human

Take generators $x, y$ and one binary operation $*$.
Build every expression, such as $x * y$, $(x * y) * x$ and $x * (y * x)$, and treat two expressions as equal only when they are written the same way.
This algebra satisfies no law, not even $x * y = y * x$ or associativity.
It is "absolutely" free because no rule restricts it.
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# Kolmogorov's three-series theorem

## Statement

Let $X_1, X_2, \dots$ be [[independent|0GQ938GDE0]] real [[random variables|0PNRKMRC55]].
For $A > 0$, let $Y_n = X_n \mathbf{1}_{\set{\abs{X_n} \le A}}$, the variable $X_n$ truncated at level $A$.
Then the [[series|0NM626FH6Q]] $\sum_{n} X_n$ converges [[almost surely|0ST21PGDAP]] if and only if, for some $A > 0$, the following three series converge:

1. $\sum_{n} P(\abs{X_n} > A)$,
2. $\sum_{n} E[Y_n]$,
3. $\sum_{n} \operatorname{Var}(Y_n)$.

## Remarks

If the three series converge for some $A > 0$, then they converge for every $A > 0$.
By [[Kolmogorov's zero–one law|073Q8Z94KG]], the event that $\sum_n X_n$ converges has probability $0$ or $1$, so when the series do not all converge, $\sum_n X_n$ diverges almost surely.

Idea of the proof of sufficiency.
Series 3 and [[Kolmogorov's inequality|D9B4HE4B4Q]] show that $\sum_n (Y_n - E[Y_n])$ converges almost surely.
Series 2 then gives the convergence of $\sum_n Y_n$.
Series 1 and the [[first Borel–Cantelli lemma|0T6WS7TT77]] show that $X_n = Y_n$ for all but finitely many $n$, almost surely.
The proof of necessity uses a symmetrisation argument and a lower bound for the probability that a sum of bounded centred variables stays small.

Example: let $\varepsilon_n$ be independent signs, equal to $1$ or $-1$ with probability $1/2$ each, and let $X_n = \varepsilon_n / n^{p}$ with $p > 0$.
With $A = 1$, series 1 and 2 are $0$, and series 3 is $\sum_n n^{-2p}$.
So the random series $\sum_n \varepsilon_n / n^p$ converges almost surely if $p > 1/2$ and diverges almost surely if $p \le 1/2$.

## Explanations

### L1 human

When you add up infinitely many independent random numbers, the sum may or may not settle down.
The theorem gives an exact test with three ordinary series.
The large values must be rare, the average sizes of the small values must add up, and their fluctuations must add up.

### L2 human

The theorem completely describes the almost sure convergence of series of independent variables.
It combines [[Kolmogorov's inequality|D9B4HE4B4Q]], the [[Borel–Cantelli lemma|0T6WS7TT77]] and the zero–one law.
For independent variables, almost sure convergence, convergence in probability and convergence in distribution of $\sum_n X_n$ are equivalent, a result of Lévy.
A common use is to show that random series such as $\sum_n \varepsilon_n a_n$ converge almost surely exactly when $\sum_n a_n^2 < \infty$.
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# Atom

## Summary

In a [[partially ordered set|0QV2JS9RJ3]] with a [[least element|0AANKKNHS4]] $0$, an atom is an element just above $0$.
It is above $0$, and nothing lies strictly between $0$ and it.
Atoms are the minimal nonzero elements.
In the lattice of subsets of a set they are the one-element subsets, which is the origin of the name.
Atoms are used to define atomic and atomistic lattices and to describe Boolean algebras.

## Explanations

### L1 human

An atom is a smallest piece that is not nothing.
In the subsets of $\set{1, 2, 3}$, the empty set is nothing, and the atoms are $\set{1}$, $\set{2}$ and $\set{3}$.
In the positive integers ordered by divisibility, the bottom is $1$ and the atoms are the prime numbers, since the only divisors of a prime $p$ are $1$ and $p$.
Some orders have no atoms at all: in the interval $[0, 1]$, between $0$ and any $a > 0$ lies $a / 2$.
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# Bianchi identities

## Statement

1. Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $P \to M$ be a principal $G$-bundle with connection form $\omega$. Let $\Omega = d\omega + \frac12 [\omega \wedge \omega]$ be its [[curvature form|ZAJFW1S078]]. Then
$$d\Omega + [\omega \wedge \Omega] = 0,$$
where $[\omega \wedge \Omega](X, Y, Z)$ is the sum over cyclic permutations $(X, Y, Z)$ of $[\omega(X), \Omega(Y, Z)]$.
2. Let $M$ be a smooth manifold with a torsion-free [[affine connection|JX65KSVHSJ]] $\nabla$ (its [[torsion|JZNQAN00ZJ]] vanishes) and [[curvature tensor|JA29ETCM02]] $R$. Then for all vector fields $X, Y, Z, W$:
   - (first Bianchi identity) $R(X, Y) Z + R(Y, Z) X + R(Z, X) Y = 0$;
   - (second Bianchi identity) $(\nabla_X R)(Y, Z) W + (\nabla_Y R)(Z, X) W + (\nabla_Z R)(X, Y) W = 0$, where $\nabla_X R$ is the [[covariant derivative|5FZDQQHQKM]] of the tensor field $R$.

## Remarks

For part 1, apply $d$ to the structure equation: $d\Omega = \frac12 d[\omega \wedge \omega] = \frac12\big([d\omega \wedge \omega] - [\omega \wedge d\omega]\big)$, substitute $d\omega = \Omega - \frac12[\omega \wedge \omega]$, and use the Jacobi identity to see that $[[\omega \wedge \omega] \wedge \omega] = 0$.
Part 1 is often written $D\Omega = 0$, with $D$ the exterior covariant derivative.
For the first identity in part 2, expand $R$ and use $\nabla_X Y - \nabla_Y X = [X, Y]$; the cyclic sum reduces to the Jacobi identity for the Lie bracket.
Without the torsion-free hypothesis the identities get extra terms involving the torsion and its covariant derivative.
Example: tracing the second identity for the Levi-Civita connection gives $\mathrm{div}\, \mathrm{Ric} = \frac12 \, d\,\mathrm{scal}$, so the Einstein tensor $\mathrm{Ric} - \frac12 \mathrm{scal}\, g$ is divergence-free.

## Explanations

### L1 human

Curvature is not an arbitrary collection of numbers: it satisfies some built-in rules.
The Bianchi identities are these rules.
One is algebraic and says that a certain cyclic sum of curvatures is zero.
The other says that a cyclic sum of derivatives of curvature is zero, much like the rule that the divergence of a curl is zero.

### L2 human

The first identity, together with the symmetries of the Riemann tensor, cuts down its number of independent components to $n^2(n^2 - 1)/12$ in dimension $n$.
The contracted second identity is the reason the left side of Einstein's field equations is divergence-free, which matches conservation of energy and momentum.
In gauge theory, part 1 says that the field strength $F$ of a gauge field satisfies $D F = 0$; for electromagnetism this gives the two homogeneous Maxwell equations.
Part 1 is also the key step in showing that the forms $f(\Omega)$ of Chern–Weil theory are closed.
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# Locally finite poset

## Summary

A locally finite poset is a [[partially ordered set|0QV2JS9RJ3]] in which every closed interval $[x, y] = \set{z : x \le z \le y}$ is finite.
This is the condition under which the [[incidence algebra|1H9SFWHYEB]] and the Möbius function of the poset are defined.
Most posets of enumerative combinatorics are locally finite, even when they are infinite.

## Explanations

### L1 human

A poset may be infinite, but you can ask that between any two comparable elements there are only finitely many others.
The integers satisfy this: between $3$ and $10$ there are eight integers.
The positive integers ordered by divisibility satisfy it too: between $2$ and $12$ lie only the divisors of $12$ that are multiples of $2$.
The rationals fail: between $0$ and $1$ there are infinitely many.
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# Parallel transport

## Summary

Parallel transport moves vectors along a curve in a manifold while keeping them "constant" in the sense of a given [[connection|5ZYRRVE8BW]].
It gives, for each curve, a linear isomorphism between the fibres over its endpoints.
The result usually depends on the curve, not only on its endpoints, and this dependence is measured by curvature and holonomy.
A connection can be recovered from its parallel transports, so the two describe the same data.

## Explanations

### L1 human

Walk along a path on a surface carrying a spear, and never turn the spear relative to the surface as you go.
Where the spear ends up pointing is the parallel transport of its starting direction.
On a flat plane the spear always points the same way, whatever path you take.
On a sphere, walk from the north pole down to the equator, a quarter of the way around the equator, and back to the pole.
The spear comes back turned by a right angle, which shows that the sphere is curved.
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# Orientability

## Definition

Let $n \ge 1$ and let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.
A smooth [[atlas|J7P32JQTVE]] $\set{(U_i, \varphi_i)}$ of $M$ is *oriented* if each transition map $\varphi_j \circ \varphi_i^{-1}$ has a positive [[Jacobian determinant|0ZCSMYHTBP]] at every point of $\varphi_i(U_i \cap U_j)$.
$M$ is *orientable* if it has an oriented smooth atlas.

## Remarks

The circle $S^1$ is orientable.
The stereographic charts $\varphi_N(x, y) = x/(1 - y)$ and $\varphi_S(x, y) = x/(1 + y)$ have transition map $t \mapsto 1/t$, whose derivative $-1/t^2$ is negative; replacing $\varphi_S$ by $-\varphi_S$ gives transition map $t \mapsto -1/t$ with derivative $1/t^2 > 0$.

The open [[Möbius strip|04AJE0M271]] is not orientable.
Following charts around the central circle forces a transition map with negative Jacobian determinant somewhere, so no atlas meets the condition.
The real projective plane and the Klein bottle are not orientable either, since they contain an open Möbius strip.

Equivalently, $M$ is orientable if and only if it has a nowhere vanishing differential form of degree $n$.
An orientation of $M$ is a choice of [[orientation|0QJJAQ6ZYZ]] of each tangent space that is given by the charts of one oriented atlas.
A connected orientable manifold has exactly two orientations.
For a [[topological manifold|QQQ9B1DK4P]], orientability is defined with [[singular homology|M0Q5CFJ7ZH]].
Each local group $H_n(M, M \setminus \set{x}; \Z)$ is infinite cyclic.
An orientation is a choice of generator at each point $x$ that is locally constant in $x$.
For smooth manifolds the two notions agree.
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# Boole's rule

## Statement

Let $h > 0$, let $x_k = x_0 + kh$ for $k = 0, \ldots, 4$, and let $f \in C^6[x_0, x_4]$ with $f_k = f(x_k)$.
Then there is $\xi \in (x_0, x_4)$ with
$$\int_{x_0}^{x_4} f(x) \, dx = \frac{2h}{45} \big( 7 f_0 + 32 f_1 + 12 f_2 + 32 f_3 + 7 f_4 \big) - \frac{8 h^7}{945} f^{(6)}(\xi).$$

## Remarks

The rule is the closed [[Newton–Cotes formula|BQPQAY1FDJ]] with five nodes: its weights are the integrals of the Lagrange basis polynomials of the five equally spaced nodes.
By symmetry it is exact for all polynomials of degree at most $5$, one more than the interpolation degree, so its [[degree of exactness|ZYT1JNT1Y0]] is $5$.
The error constant can be read off from $f(x) = x^6$ on $[0, 4]$ with $h = 1$: the integral is $\frac{4^7}{7}$, the rule gives $\frac{4^7}{7} + \frac{128}{21}$, and $\frac{8}{945} \cdot 720 = \frac{128}{21}$.
Check of the weights: $\frac{2h}{45}(7 + 32 + 12 + 32 + 7) = 4h$, the length of the interval.
The composite rule on $4m$ subintervals has error $O(h^6)$ for $f \in C^6$.

## Explanations

### L1 human

Boole's rule estimates an integral from five equally spaced samples.
It fits a polynomial of degree four through the samples and integrates that polynomial instead.
The weights $7, 32, 12, 32, 7$ come out of this fit, and the rule is exact for all polynomials up to degree five.

### L2 human

Boole's rule is the next member after the trapezoidal rule, Simpson's rule and Simpson's $3/8$ rule in the closed Newton–Cotes family.
It equals one step of [[Richardson extrapolation|WMTZG71K2Y]] applied to Simpson's rule with steps $2h$ and $h$, as in the Romberg method.
Higher Newton–Cotes rules eventually have negative weights and become unstable, so in practice one uses composite rules of low order or Gaussian rules instead.
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# Dual pair

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$.
A *dual pair* over $\mathbb{K}$ is a triple $(X, Y, \langle \cdot, \cdot \rangle)$ such that:

1. $X$ and $Y$ are [[vector spaces|05EGHEQZAW]] over $\mathbb{K}$, and $\langle \cdot, \cdot \rangle \colon X \times Y \to \mathbb{K}$ is a [[bilinear form|0SQZMXB6GP]].
2. For every $x \in X$ with $x \ne 0$ there is $y \in Y$ with $\langle x, y \rangle \ne 0$.
3. For every $y \in Y$ with $y \ne 0$ there is $x \in X$ with $\langle x, y \rangle \ne 0$.

## Remarks

Some authors call any triple satisfying condition 1 a dual pair, and call it *separated* when 2 and 3 hold.
Example: $X = Y = \R^n$ with $\langle x, y \rangle = \sum_i x_i y_i$; if $x \ne 0$ then $\langle x, x \rangle > 0$, which gives 2, and 3 holds in the same way.
Example: for a Hausdorff locally convex space $E$, the triple $(E, E', \langle x, f \rangle = f(x))$ is a dual pair; condition 2 follows from the Hahn–Banach theorem, and condition 3 is trivial.
Non-example: $X = C([0, 1])$, $Y = \R$, $\langle f, t \rangle = t f(0)$ satisfies 1 and 3 but not 2, since the function $f(s) = s$ is nonzero and $\langle f, t \rangle = 0$ for all $t$.
</document>

<document source="https://tamnd.github.io/math2/node/TVYJPBHD0S.md" tag="TVYJPBHD0S" kind="statement">
# Poincaré recurrence theorem

## Statement

Let $(X, \mathcal{A}, \mu)$ be a measure space with $\mu(X) < \infty$, that is, a [[finite measure|63Z1Q4FS23]] space.
Let $T \colon X \to X$ be a [[measure-preserving transformation|556WBDAHNP]], and let $A \in \mathcal{A}$.
Then for [[almost every|RARTMA0ZEC]] $x \in A$, the point $T^k(x)$ lies in $A$ for infinitely many integers $k \ge 1$.

## Remarks

Let $B$ be the set of points of $A$ that never return to $A$, so $B = A \setminus \bigcup_{k \ge 1} T^{-k}(A)$.
The sets $B, T^{-1}(B), T^{-2}(B), \dots$ are pairwise disjoint and all have measure $\mu(B)$.
Since their total measure is at most $\mu(X) < \infty$, one gets $\mu(B) = 0$.
Applying this to $T^m$ for each $m \ge 1$ shows that almost every point of $A$ returns infinitely often.
The hypothesis $\mu(X) < \infty$ is needed: the translation $T(x) = x + 1$ on $\R$ preserves Lebesgue measure, but no point of $A = [0,1)$ ever returns to $A$.
Example: for an irrational rotation of the circle and an arc $A$ of positive length, every point of $A$ comes back to $A$ infinitely often.

## Explanations

### L1 human

A system that keeps its total mass and lives in a bounded space must come back close to where it started.
More precisely, almost every point of a set of states returns to that set again and again.
For a gas in a closed box, this means the gas will, in principle, come back near any earlier state, though the waiting time is enormous.

### L2 human

The theorem uses only finiteness of the measure and invariance, not any mixing property.
It is the starting point of recurrence in ergodic theory, which leads to multiple recurrence and Furstenberg's proof of Szemerédi's theorem.
For infinite measures, the maps for which the conclusion holds are called conservative, and the Mathlib form of the theorem is stated for conservative maps.
</document>

<document source="https://tamnd.github.io/math2/node/TW1KWXYWN0.md" tag="TW1KWXYWN0" kind="definition">
# Congruence-permutable variety

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
The variety $\mathcal{V}$ is *congruence permutable* if every algebra $A$ in $\mathcal{V}$ has [[permuting congruences|2BCES6NPYM]].
That is, for all [[congruences|DCSXXT4SER]] $\alpha$ and $\beta$ of $A$,
$$\alpha \circ \beta = \beta \circ \alpha,$$
where $\circ$ is the [[composition of relations|0PYSR20HGF]].

## Remarks

Example: the variety of groups is congruence permutable.
Congruences of a group $G$ correspond to normal subgroups, and the composition of the congruences of $M$ and $N$ is the congruence relation $a^{-1} b \in MN$.
Since $MN = NM$ for normal subgroups, the two compositions are equal.
Non-example: the variety of lattices is not congruence permutable.
In the chain $0 < 1 < 2$, let $\alpha$ identify $0$ and $1$, and let $\beta$ identify $1$ and $2$.
The pair $(0, 2)$ is in one composition, through $0 \mathrel{\alpha} 1 \mathrel{\beta} 2$.
It is not in the other, because the only element $\beta$-related to $0$ is $0$, and $0$ is not $\alpha$-related to $2$.
For a single algebra, the congruences $\alpha$ and $\beta$ permute if and only if $\alpha \circ \beta$ is the join $\alpha \vee \beta$ in the congruence lattice.

## Explanations

### L1 human

In a congruence-permutable variety, two ways of identifying elements can be applied in either order with the same result.
</document>

<document source="https://tamnd.github.io/math2/node/TW35GWE1XF.md" tag="TW35GWE1XF" kind="definition">
# Torsion tensor

## Definition

Let $M$ be a smooth manifold with an [[affine connection|JX65KSVHSJ]] $\nabla$, and let $\mathfrak{X}(M)$ be the space of smooth vector fields on $M$.
The *torsion* of $\nabla$ is the map $T : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M)$ given by
$$T(X, Y) = \nabla_X Y - \nabla_Y X - [X, Y],$$
where $[X, Y]$ is the [[Lie bracket|68FGQEBD6J]].
The connection $\nabla$ is *torsion-free* if $T = 0$.

## Remarks

$T$ is $C^\infty(M)$-linear in each argument: for a smooth function $f$, $T(fX, Y) = f \nabla_X Y - (Yf) X - f \nabla_Y X - f [X, Y] + (Yf) X = f \, T(X, Y)$.
So $T$ is a [[tensor field|WX302T93TW]] of type $(1, 2)$, and $T(Y, X) = -T(X, Y)$.
In a chart, $T(\partial_i, \partial_j) = \sum_k (\Gamma^k_{ij} - \Gamma^k_{ji}) \partial_k$, since coordinate vector fields commute.
Example: the flat connection of $\R^n$ has $\Gamma^k_{ij} = 0$, so it is torsion-free.
Non-example: on $\R^3$ the connection $\nabla_X Y = D_X Y + X \times Y$, with $D$ the flat connection and $\times$ the cross product, is a connection but not torsion-free, since $T(X, Y) = 2 \, X \times Y \ne 0$ for independent $X, Y$.
</document>

<document source="https://tamnd.github.io/math2/node/TW3ESQB3GQ.md" tag="TW3ESQB3GQ" kind="concept">
# Serial ring

## Summary

A serial ring is a [[ring|0B1261B5ED]] that, as a module over itself on each side, splits into a direct sum of [[uniserial modules|Z1045A0V29]].
Serial rings include [[discrete valuation rings|MGM3CNJW9V]], the rings $\Z/n\Z$, and the rings of [[upper triangular matrices|0N4XY7Z3NN]] over a field.
The finite-dimensional serial algebras over a field are the [[Nakayama algebras|APRFMTZ10K]].
Their module theory is very explicit.
By a theorem of Warfield, every finitely presented module over a serial ring is a direct sum of uniserial modules.

## Explanations

### L1 human

For a serial ring, the ring itself can be cut into a few pieces, and each piece has submodules that line up in a single chain.
So the whole ring is a sum of "ladders" with no branching.
The ring $\Z/12\Z$ is serial: it splits as $\Z/4\Z \times \Z/3\Z$, and each factor has a chain of ideals.
The ring $k[x, y]/(x, y)^2$ over a field $k$ is not serial, because the ideals generated by $x$ and by $y$ cannot be compared.
</document>

<document source="https://tamnd.github.io/math2/node/TWANGP59RB.md" tag="TWANGP59RB" kind="statement">
# Measurable solutions of Cauchy's equation

## Statement

Let $f \colon \R \to \R$ be an [[additive function|KF719SZD5T]], that is, $f(x + y) = f(x) + f(y)$ for all $x, y \in \R$.
Assume that $f$ is [[measurable|YYNCJSR0PJ]] with respect to [[Lebesgue measure|2Z2QVWZ262]].
Then there is a constant $c \in \R$ such that $f(x) = cx$ for all $x \in \R$, and in fact $c = f(1)$.

## Remarks

The proof reduces the claim to boundedness on an interval.
By Lusin's theorem, $f$ is bounded on some compact set $K$ of positive measure.
By the Steinhaus theorem, the difference set $K - K$ contains an open interval around $0$.
Each point $x = a - b$ of that interval with $a, b \in K$ has $f(x) = f(a) - f(b)$, so $f$ is bounded on this interval.
Then [[Darboux's theorem on additive functions|2F30ZHN1RT]] gives $f(x) = cx$.
Measurability cannot be dropped: [[a Hamel basis gives additive functions that are not linear|FFTNS66XG6]], and such functions are not Lebesgue measurable.
This is one of the regularity conditions that settle [[Cauchy's functional equation|ZBMEVWN100]].

## Explanations

### L1 human

An additive function turns sums into sums.
The only additive functions one can draw are the lines $f(x) = cx$ through the origin.
This theorem says that already a very weak regularity, measurability, forces an additive function to be such a line.
So the strange additive functions can never be measured or integrated.

### L2 human

Measurability is weaker than continuity, monotonicity or local boundedness, so this result contains the other classical criteria.
In practice, any additive function that comes from an explicit formula, a limit of continuous functions or a probability model is measurable, so it is linear.
The same argument works for additive maps between $\R^n$ and $\R^m$.
More generally, a Haar measurable homomorphism between second countable locally compact groups is continuous.
The key tools, Lusin's theorem and the Steinhaus theorem, are standard in measure theory.
</document>

<document source="https://tamnd.github.io/math2/node/TWEYQ0XAY4.md" tag="TWEYQ0XAY4" kind="definition">
# Quantum logic

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]].
The *quantum logic* of $H$ is the set $\mathcal{L}(H)$ of closed linear subspaces of $H$, with the following structure:

1. the order is inclusion $\subseteq$;
2. the meet $M \wedge N$ is the intersection $M \cap N$;
3. the join $M \vee N$ is the closure of the sum $M + N$;
4. the negation $M^\perp$ is the [[orthogonal complement|0XJXTBCB2T]] of $M$.

The elements of $\mathcal{L}(H)$ are called *propositions*.
More abstractly, a *quantum logic* is an [[orthomodular lattice|6FPKEFX0X4]], whose elements are the propositions.

## Remarks

With this structure $\mathcal{L}(H)$ is an orthomodular lattice with least element $\set{0}$ and greatest element $H$.
Authors differ on the abstract notion: some use orthomodular partially ordered sets, often with countable joins of orthogonal elements.

Example: in $H = \C^2$, let $A$, $B$ and $C$ be the lines spanned by $(1, 0)$, $(0, 1)$ and $(1, 1)$.
Then $A \wedge (B \vee C) = A \wedge \C^2 = A$, while $(A \wedge B) \vee (A \wedge C) = \set{0}$.
So the distributive law of classical logic fails in $\mathcal{L}(\C^2)$.

Non-example: the hexagon ortholattice $O_6$, with $0 < a < b < 1$, $0 < b^\perp < a^\perp < 1$ and no other relations, is orthocomplemented.
It is not orthomodular, since $a \le b$ but $a \vee (a^\perp \wedge b) = a \vee 0 = a \ne b$; so it is not a quantum logic in the abstract sense.
</document>

<document source="https://tamnd.github.io/math2/node/TWG8QKY8TA.md" tag="TWG8QKY8TA" kind="definition">
# Cyclotomic extension

## Definition

Let $K$ be a field and let $n \ge 1$ be an integer that is not divisible by the characteristic of $K$.
A [[field extension|0WQ1A8SHTZ]] $L/K$ is an *$n$-th cyclotomic extension* if both conditions hold:

1. $L$ contains a [[primitive $n$-th root of unity|0J817RTVG6]] $\zeta_n$;
2. $L = K(\zeta_n)$.

Equivalently, $L$ is a [[splitting field|02NZZ1J0FH]] of $x^n - 1$ over $K$.
More generally, let $S$ be a set of positive integers.
Then $L/K$ is an *$S$-cyclotomic extension* if $L$ contains a primitive $n$-th root of unity for each $n \in S$.
Moreover $L$ must be generated over $K$ by the [[roots of unity|0S61N75H45]] of these orders.

## Remarks

Example: $\Q(i)/\Q$ is a $4$-th cyclotomic extension, since $i$ has order $4$ in $\C^\times$ and $\Q(i)$ is generated by $i$.
Non-example: $\Q(\sqrt[3]{2})/\Q$ is not a $3$-rd cyclotomic extension, since $\Q(\sqrt[3]{2})$ is a subfield of $\R$ and contains no primitive cube root of unity.
The two forms agree because the roots of $x^n - 1$ are the powers of $\zeta_n$.
An $n$-th cyclotomic extension is Galois, and $\sigma \mapsto k$ with $\sigma(\zeta_n) = \zeta_n^k$ is an injective homomorphism $\Gal(L/K) \to (\Z/n\Z)^\times$.
For $K = \Q$ this map is an isomorphism, and $L$ is the cyclotomic field $\Q(\zeta_n)$.
</document>

<document source="https://tamnd.github.io/math2/node/TWH3WVKFF1.md" tag="TWH3WVKFF1" kind="concept">
# Finite type invariant

## Summary

A finite type invariant, or Vassiliev invariant, is a [[knot invariant|GZQ5ZE3X6G]] whose extension to knots with double points vanishes once there are enough double points.
They behave like polynomials on the space of knots, with the number of double points playing the role of the degree.
Many classical invariants, such as the coefficients of the Conway polynomial and of the suitably expanded [[Jones polynomial|4SPC17FPXE]], are of finite type.
The [[Kontsevich integral|727ZEPVSTC]] is universal among rational finite type invariants.

## Explanations

### L1 human

Allow a knot to pass through itself at a point, making a double point.
An invariant can be extended to such singular knots by taking the difference of its values on the two ways to resolve the double point.
An invariant has finite type $n$ when this extension is always zero on knots with more than $n$ double points.
This is like a polynomial of degree $n$, whose $(n+1)$-st differences all vanish.
It is open whether finite type invariants can tell apart every pair of different knots.
</document>

<document source="https://tamnd.github.io/math2/node/TWP1EP7FR1.md" tag="TWP1EP7FR1" kind="definition">
# Étale fundamental group

## Definition

Let $X$ be a connected [[scheme|5A35TYZD02]] and let $\bar{x} \colon \operatorname{Spec} \Omega \to X$ be a geometric point, with $\Omega$ an algebraically closed field.
A *finite étale cover* of $X$ is a morphism $Y \to X$ that is [[finite|N9C4X7XKE4]] and [[étale|H84VHC8PX2]].
Let $\mathrm{FEt}_X$ be the category of finite étale covers of $X$, with morphisms over $X$.
The *fiber functor* $F_{\bar{x}} \colon \mathrm{FEt}_X \to \mathrm{Set}$ sends $Y$ to the finite set of geometric points of $Y$ over $\bar{x}$, that is, of morphisms $\operatorname{Spec} \Omega \to Y$ that lift $\bar{x}$.
The *étale fundamental group* is the automorphism group of this functor:
$$\pi_1^{\text{ét}}(X, \bar{x}) = \Aut(F_{\bar{x}}).$$

## Remarks

The group $\pi_1^{\text{ét}}(X, \bar{x})$ is a [[profinite group|VABPHNZAVV]], as a closed subgroup of $\prod_Y \operatorname{Sym}(F_{\bar{x}}(Y))$.
By [[Grothendieck's Galois theory|218WAD58W3]], $F_{\bar{x}}$ is an equivalence between $\mathrm{FEt}_X$ and the category of finite sets with a continuous action of $\pi_1^{\text{ét}}(X, \bar{x})$.
Example: for $X = \operatorname{Spec} k$ with $k$ a field, the finite étale covers are finite products of finite separable extensions of $k$.
Choosing $\bar{x}$ amounts to choosing a separable closure $k^{s}$, and $\pi_1^{\text{ét}}(X, \bar{x}) \cong \Gal(k^{s}/k)$, the [[absolute Galois group|DPFMXS71D1]].
Non-example: the exponential map $\C \to \C^{*}$ is a covering space with infinitely many sheets, so it is not a finite étale cover and does not enter $\mathrm{FEt}_X$.
Therefore $\pi_1^{\text{ét}}$ of the complex punctured line is $\widehat{\Z}$, the profinite completion of the topological fundamental group $\Z$, and not $\Z$ itself.
For a variety $X$ over $\C$, the Riemann existence theorem gives $\pi_1^{\text{ét}}(X) \cong \widehat{\pi_1(X(\C))}$.
In characteristic $p > 0$ the affine line has nontrivial finite étale covers, such as the Artin–Schreier covers $y^p - y = x$.
</document>

<document source="https://tamnd.github.io/math2/node/TWQFE0FZZS.md" tag="TWQFE0FZZS" kind="concept">
# Continuous-time Markov chain

## Summary

A continuous-time Markov chain is a process with values in a countable set that jumps between states at random times and has the [[Markov property|0B77KZCQ51]].
It stays in each state for an [[exponentially distributed|0E99F51GVB]] time, and then jumps according to fixed probabilities.
Its law is described by a rate matrix $Q$, and on a finite set its transition probabilities form the [[matrix exponential|020M04FN1H]] $e^{tQ}$.
Queues, birth–death processes and the Poisson process are standard examples.

## Explanations

### L1 human

A discrete-time [[Markov chain|0G48PYM96Q]] moves at each tick of a clock.
A continuous-time chain instead waits a random amount of time in each state before it moves.
The waiting time has no memory: having waited a while does not make a jump more or less likely soon.
For example, a machine that breaks down at rate $a$ and is repaired at rate $b$ is a chain on two states, working and broken.
In the long run it is working a fraction $b/(a + b)$ of the time.
</document>

<document source="https://tamnd.github.io/math2/node/TWSZNFKBKA.md" tag="TWSZNFKBKA" kind="definition">
# Deck transformation group

## Definition

Let $p \colon \tilde X \to X$ be a [[covering map|3XZRRE165N]].
A *deck transformation* of $p$ is a [[homeomorphism|0QT00PB5G2]] $f \colon \tilde X \to \tilde X$ such that $p \circ f = p$.
The *deck transformation group* of $p$ is the set of all deck transformations of $p$, with composition of maps as the [[group|G5S1M9]] operation.

## Remarks

Example: for $p \colon \R \to S^1$, $p(t) = (\cos 2\pi t, \sin 2\pi t)$, each translation $t \mapsto t + n$ with $n \in \Z$ is a homeomorphism with $p(t + n) = p(t)$.
These are all the deck transformations, so the group is isomorphic to $\Z$.

Non-example: the translation $t \mapsto t + \tfrac12$ is a homeomorphism of $\R$, but $p(t + \tfrac12) = -p(t) \ne p(t)$, so it is not a deck transformation.

The group is also called the covering transformation group and is written $\operatorname{Deck}(p)$ or $G(\tilde X)$.
If $\tilde X$ is connected, a deck transformation that fixes one point is the identity.
The covering is called normal, or regular, when the group acts transitively on each fibre.
</document>

<document source="https://tamnd.github.io/math2/node/TX0Z2VVKH3.md" tag="TX0Z2VVKH3" kind="statement">
# Frostman's theorem

## Statement

Let $K$ be a [[compact set|0R6EWFKRCF]] in $\C$ with positive logarithmic capacity.
Let $\nu$ be its [[equilibrium measure|464NN2K4FZ]] and let $V(K)$ be its [[Robin constant|W3NVXBEMRQ]].
Let $U^\nu(z) = \int \log \frac{1}{\abs{z - w}} \, d\nu(w)$ be the [[logarithmic potential|46NHWJHW44]] of $\nu$.
Then:

1. $U^\nu(z) \le V(K)$ for every $z \in \C$.
2. $U^\nu(z) = V(K)$ for every $z \in K \setminus E$, where $E$ is an $F_\sigma$ subset of the boundary $\partial K$ of [[capacity zero|W9N1TR1X8P]].

## Remarks

Proof idea: suppose that $U^\nu < V(K)$ on a compact subset of $K$ of positive capacity, or $U^\nu > V(K)$ on a set of positive $\nu$-measure.
Then moving a little mass of $\nu$ lowers the energy, which contradicts minimality.
This gives $U^\nu \ge V(K)$ on $K$ outside a set of capacity zero and $U^\nu \le V(K)$ on the support of $\nu$.
The maximum principle for potentials then extends the upper bound from the support of $\nu$ to all of $\C$.
A separate argument shows that the support of $\nu$ lies in $\partial K$.

Example: for the unit circle, $\nu = d\theta / 2\pi$ and $U^\nu(z) = -\log \max(\abs{z}, 1)$.
This is $0 = V(K)$ on the closed unit disc and negative outside, as the theorem says.
The exceptional set $E$ cannot be dropped in general: it can contain boundary points that are irregular for the Dirichlet problem, where the potential is strictly below $V(K)$.

## Explanations

### L1 human

When a unit charge sits in equilibrium on a conductor $K$, the voltage it creates is the same at almost every point of $K$.
Nowhere in the plane is the voltage higher than this value.
The only exceptions on $K$ form a set so thin that it has capacity zero.

### L2 human

Frostman's theorem is the bridge between the energy definition of capacity and the Green function.
On the unbounded component $\Omega$ of $\C \setminus K$, the function $g(z) = V(K) - U^\nu(z)$ is the Green function of $\Omega$ with pole at infinity.
The exceptional set $E$ is linked to the boundary points that are irregular for the Dirichlet problem.
The Kellogg–Evans theorem says that such points form a set of capacity zero.
The theorem is also the basic tool for computing capacities: a probability measure on $K$ whose potential is constant on all of $K$ must be the equilibrium measure.
</document>

<document source="https://tamnd.github.io/math2/node/TX1W8PRASW.md" tag="TX1W8PRASW" kind="definition">
# Čech cohomology

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]], and let $\mathcal{F}$ be a [[sheaf|TB6TM6R4MD]] of [[abelian groups|082GHNPJ53]] on $X$.
Let $\mathfrak{U} = (U_i)_{i \in I}$ be an [[open cover|049PXYHW1T]] of $X$ with $I$ a well-ordered set.
For $i_0, \dots, i_p \in I$ write $U_{i_0 \dots i_p} = U_{i_0} \cap \dots \cap U_{i_p}$.
The *Čech complex* is the [[cochain complex|WQ82C0M3QB]] with
$$C^p(\mathfrak{U}, \mathcal{F}) = \prod_{i_0 < \dots < i_p} \mathcal{F}(U_{i_0 \dots i_p})$$
and differential $d \colon C^p \to C^{p+1}$ given by
$$(d\alpha)_{i_0 \dots i_{p+1}} = \sum_{j=0}^{p+1} (-1)^j \, \alpha_{i_0 \dots \widehat{i_j} \dots i_{p+1}}|_{U_{i_0 \dots i_{p+1}}},$$
where the hat means that the index is omitted.
The *Čech cohomology* of $\mathcal{F}$ with respect to $\mathfrak{U}$ is $\check{H}^p(\mathfrak{U}, \mathcal{F}) = H^p(C^\bullet(\mathfrak{U}, \mathcal{F}))$.
The Čech cohomology of $X$ with coefficients in $\mathcal{F}$ is the direct limit $\check{H}^p(X, \mathcal{F}) = \varinjlim_{\mathfrak{U}} \check{H}^p(\mathfrak{U}, \mathcal{F})$ over refinements of covers.

## Remarks

The sheaf axioms give $\check{H}^0(\mathfrak{U}, \mathcal{F}) = \mathcal{F}(X)$ for every cover.

Example: let $X = S^1$, covered by two open arcs $U_0, U_1$ whose intersection has two components, and let $\mathcal{F} = \underline{\Z}$ be the constant sheaf.
Then $C^0 = \Z^2$, $C^1 = \underline{\Z}(U_{01}) = \Z^2$, and $d(a, b) = (b - a, b - a)$.
So $\check{H}^0 = \Z$ and $\check{H}^1 = \Z^2 / \set{(c, c)} \cong \Z$.

Non-example: a family of open sets that does not cover $X$ is not allowed; then $\check{H}^0$ would be the sections over the union, not over $X$.

There is a natural map $\check{H}^p(\mathfrak{U}, \mathcal{F}) \to H^p(X, \mathcal{F})$ to [[sheaf cohomology|PQ6Q143ARJ]].
It is an isomorphism when $H^q(U_{i_0 \dots i_p}, \mathcal{F}) = 0$ for all $q > 0$ and all finite intersections (Leray's theorem), for example for a quasi-coherent sheaf on a separated scheme and an affine open cover.
</document>

<document source="https://tamnd.github.io/math2/node/TX32GDQ330.md" tag="TX32GDQ330" kind="concept">
# Beurling–Ahlfors transform

## Summary

The Beurling–Ahlfors transform is a singular integral operator on functions of the complex plane that turns the derivative $\partial f / \partial \bar{z}$ into the derivative $\partial f / \partial z$.
It is the complex analogue of the [[Hilbert transform|NSCEBXM7ZM]], and it is an isometry of $L^2(\C)$ and a bounded operator on $L^p(\C)$ for $1 < p < \infty$.
It is the main analytic tool for solving the [[Beltrami equation|9J7GFJ1GPF]], and hence for the existence theory of [[quasiconformal maps|SB8KG46NHP]].

## Explanations

### L1 human

A function of a complex variable has two Wirtinger derivatives, one with respect to $z$ and one with respect to $\bar{z}$.
For a compactly supported smooth function, the first is determined by the second, and the Beurling–Ahlfors transform is the operator that computes it.
It works like a Fourier multiplier that only rotates the phase, by the factor $\bar{\xi}/\xi$, so it preserves the $L^2$ norm.
For example, it sends $z e^{-\abs{z}^2}$ to $\bar{z} e^{-\abs{z}^2}$.
</document>

<document source="https://tamnd.github.io/math2/node/TX4MSZ0SRP.md" tag="TX4MSZ0SRP" kind="statement">
# JSJ decomposition

## Statement

Let $M$ be a closed connected [[orientable|702SNFQG0W]] irreducible [[3-manifold|R20TB2KEQQ]], where irreducible means that every embedded $2$-sphere bounds a $3$-ball.
A compact $3$-manifold $N$ is *atoroidal* if every [[incompressible|6N9GW98MR6]] torus in $N$ is parallel to a boundary torus of $N$.
Then $M$ contains a finite collection $\mathcal{T}$ of disjoint embedded incompressible tori with the following property.
Each component of $M$ cut along $\mathcal{T}$ is atoroidal or a [[Seifert fiber space|9V1GF5WMKA]].
A minimal collection $\mathcal{T}$ with this property is unique up to isotopy.

## Remarks

The theorem is due to Jaco and Shalen, and independently to Johannson.
Existence rests on Haken's finiteness theorem, which bounds the number of disjoint, pairwise non-parallel incompressible surfaces.
Uniqueness comes from the study of how incompressible tori meet Seifert fibered pieces.
A version of the theorem holds for compact manifolds whose boundary is a union of tori.
For example, take a [[satellite knot|A1E8TDEVJ9]] in $S^3$.
The boundary of a neighborhood of the companion knot is an incompressible torus in the knot complement, so the complement is not atoroidal.
If $M$ is itself Seifert fibered or atoroidal, the minimal collection is empty.

## Explanations

### L1 human

After splitting a $3$-manifold into prime pieces, one can split each piece further along essential tori.
The JSJ theorem says there is a best way to do this, and that it is unique.
Each resulting piece is either built from circles in a regular way or contains no essential torus at all.

### L2 human

The JSJ decomposition is the second step of geometrization, after the prime decomposition.
By the geometrization theorem, the atoroidal pieces that are not Seifert fibered have hyperbolic interiors.
So every closed orientable irreducible $3$-manifold is assembled from Seifert fibered and hyperbolic pieces glued along tori.
</document>

<document source="https://tamnd.github.io/math2/node/TX5AV1JTF5.md" tag="TX5AV1JTF5" kind="statement">
# Contiguous relations

## Statement

Let $a, b, c \in \C$ with $c \notin \set{1, 0, -1, -2, \ldots}$.
Write $F = {}_2F_1(a, b; c; z)$ for the [[hypergeometric function|463X5V31D9]].
Write $F(a \pm 1)$, $F(b \pm 1)$, $F(c \pm 1)$ for the six *contiguous* functions obtained by changing one parameter by $\pm 1$.

1. (Gauss) Let $G$ and $H$ be two of the six contiguous functions.
Then $F$, $G$, $H$ satisfy a linear relation whose coefficients are polynomials in $a, b, c, z$, not all zero.
2. In particular, for all $z \in \C \setminus [1, \infty)$,
$$(c - a) \, F(a - 1) + \big(2a - c + (b - a) z\big) F + a (z - 1) \, F(a + 1) = 0,$$
$$(a - b) \, F = a \, F(a + 1) - b \, F(b + 1).$$

## Remarks

The second relation in part 2 can be checked on the [[hypergeometric series|428M453CR2]] directly.
Write $t_n = \frac{(a)_n (b)_n}{(c)_n \, n!}$ for the coefficients of $F$.
Since $a \, (a+1)_n = (a)_n (a + n)$, the coefficient of $z^n$ in $a F(a + 1)$ is $t_n (a + n)$, and in the same way the coefficient in $b F(b + 1)$ is $t_n (b + n)$.
Their difference is $t_n (a - b)$, the coefficient of $(a - b) F$.
The other relations follow by similar coefficient comparisons, and they extend from $\abs{z} < 1$ to the cut plane by analytic continuation.
There are $\binom{6}{2} = 15$ such relations, and Gauss listed them all.

Example: for $a = 1$, $b = 2$ the second relation gives $-F(1, 2; c; z) = F(2, 2; c; z) - 2 F(1, 3; c; z)$, a relation between three explicit series.

By combining contiguous relations, for generic parameters the function $F(a + m, b + n; c + l; z)$ for integers $m, n, l$ is a combination of $F$ and one contiguous function, with coefficients rational in $a, b, c, z$.

## Explanations

### L1 human

Changing one parameter of a hypergeometric function by $1$ gives a closely related function.
Gauss found that the original function and any two of its six neighbours are tied by a simple linear equation.
This is like the three-term recurrence of orthogonal polynomials, and it lets one step through parameters one unit at a time.

### L2 human

Contiguous relations give recurrences for orthogonal polynomials that are terminating ${}_2F_1$ series, such as Jacobi and Legendre polynomials.
They lead to Gauss's continued fraction for ratios of hypergeometric functions.
They also show that the vector $(F, F')$ at shifted parameters is obtained from the original one by a matrix with rational entries.
This is the starting point of the theory of hypergeometric difference equations.
</document>

<document source="https://tamnd.github.io/math2/node/TXB2F07CC6.md" tag="TXB2F07CC6" kind="definition">
# Riemann zeta function

## Definition

For a complex number $s$ and an integer $n \ge 1$, let $n^{-s} = e^{-s \log n}$.
For $\operatorname{Re} s > 1$, the [[Dirichlet series|8MWTEF6462]]
$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$
converges absolutely, and it defines a [[holomorphic function|0NAHXR1FRH]] on the half-plane $\operatorname{Re} s > 1$.
This function has a unique [[analytic continuation|0M3YQJ81NB]] to a [[meromorphic function|0ZZ7ZYDR8W]] on $\C$.
The *Riemann zeta function* is this meromorphic function $\zeta$.
Its only pole is a simple pole at $s = 1$, with residue $1$.

## Remarks

Example: $\zeta(2) = \pi^2/6$, the value of the series in the [[Basel problem|0DMY75D6FV]].
Non-example of the series form: at $s = 1$ the series is the harmonic series, which diverges, and $\zeta$ has its pole there.
At $s = 0$ or $s = -1$ the series diverges too, but $\zeta$ is defined there by continuation: $\zeta(0) = -1/2$ and $\zeta(-1) = -1/12$.
For $\operatorname{Re} s > 0$ the continuation is given by
$$\zeta(s) = \frac{s}{s - 1} - s \int_1^{\infty} \frac{x - \floor{x}}{x^{s+1}} \, dx,$$
since the integral converges there.
The [[functional equation|NFXKNDJX9C]] gives the continuation to all of $\C$.
The uniqueness of the continuation follows from the [[identity theorem|0YBVKMYX0W]].
</document>

<document source="https://tamnd.github.io/math2/node/TXC6PZT8CF.md" tag="TXC6PZT8CF" kind="example">
# Asymmetric simple exclusion process

## Example

Let $p \in [0, 1]$ and $q = 1 - p$.
The *asymmetric simple exclusion process* (ASEP) on $\Z$ is the [[interacting particle system|GR49AM3PSZ]] on $\set{0, 1}^{\Z}$ in which $\eta(x) = 1$ means that site $x$ holds a particle.
Each particle tries to jump one step to the right at rate $p$ and one step to the left at rate $q$.
A jump is suppressed if the target site is occupied.
Its generator, on functions $f$ that depend on finitely many coordinates, is
$$Lf(\eta) = \sum_{x \in \Z} \Big( p\, \eta(x) \big(1 - \eta(x+1)\big) + q\, \eta(x+1) \big(1 - \eta(x)\big) \Big) \big( f(\eta^{x,x+1}) - f(\eta) \big),$$
where $\eta^{x,x+1}$ exchanges the values at $x$ and $x + 1$.
This is the exclusion process with jump rates $p(x, x+1) = p$ and $p(x+1, x) = q$, so it is an interacting particle system.

## Remarks

Particles are never created or destroyed, and two particles never share a site.
For each $\rho \in [0, 1]$, the product measure under which the $\eta(x)$ are independent Bernoulli variables with mean $\rho$ is invariant.
The case $p = 1$ is the totally asymmetric simple exclusion process (TASEP), and the case $p = q = 1/2$ is the symmetric exclusion process.
In the asymmetric case the current of particles in the Bernoulli state of density $\rho$ is $(p - q)\rho(1 - \rho)$.

## Explanations

### L1 human

Picture cars on a one-lane road divided into cells, at most one car per cell.
Each car tries to move forward at random times, but it cannot move into a cell that is already taken.
This simple rule produces traffic jams and shock waves, and the ASEP is a basic model of such traffic.
</document>

<document source="https://tamnd.github.io/math2/node/TXCA43CMYR.md" tag="TXCA43CMYR" kind="concept">
# K-function

## Summary

The K-function extends the hyperfactorial $1^1 \cdot 2^2 \cdots n^n$ to non-integer arguments.
It satisfies $K(1) = 1$ and $K(z + 1) = z^z K(z)$, so $K(n + 1) = 1^1 2^2 \cdots n^n$ for positive integers $n$.
It is defined through an integral of $\log \Gamma$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]], and is related to the [[Barnes G-function|7E2JJWFF00]].

## Explanations

### L1 human

The factorial multiplies $1, 2, 3, \ldots, n$.
The hyperfactorial multiplies $1^1, 2^2, 3^3, \ldots, n^n$, so it grows even faster.
The K-function is a smooth curve through the values of the hyperfactorial, as the gamma function is a smooth curve through the factorials.
For example, $K(4) = 1^1 \cdot 2^2 \cdot 3^3 = 108$.
</document>

<document source="https://tamnd.github.io/math2/node/TXMV0YDACG.md" tag="TXMV0YDACG" kind="statement">
# Structure theorem for finitely generated modules over a principal ideal domain

## Statement

Let $R$ be a [[principal ideal domain|0CMD0BCH8T]] and let $M$ be a [[finitely generated|05QY1VEBYD]] [[$R$-module|0NZ8JA25WK]].

1. (Invariant factors) There are an integer $r \ge 0$ and nonzero non-unit elements $d_1, \ldots, d_k \in R$ with $d_1 \mid d_2 \mid \cdots \mid d_k$ such that
$$M \cong R^r \oplus R/(d_1) \oplus \cdots \oplus R/(d_k).$$
The integer $r$ and the ideals $(d_1), \ldots, (d_k)$ are uniquely determined by $M$.
2. (Elementary divisors) There are an integer $r \ge 0$, [[prime elements|01VH21EV6F]] $p_1, \ldots, p_s \in R$, not necessarily distinct, and integers $e_1, \ldots, e_s \ge 1$ such that
$$M \cong R^r \oplus R/(p_1^{e_1}) \oplus \cdots \oplus R/(p_s^{e_s}).$$
The integer $r$ and the ideals $(p_j^{e_j})$, counted with multiplicity, are uniquely determined by $M$ up to order.

## Remarks

Since $R$ is Noetherian, $M$ has a finite presentation $R^m \xrightarrow{A} R^n \to M \to 0$.
Row and column operations bring the matrix $A$ to Smith normal form, a diagonal matrix with entries $d_1 \mid d_2 \mid \cdots$; this gives part 1.
Part 2 follows from part 1 by the Chinese remainder theorem, which splits $R/(d)$ into $\bigoplus_p R/(p^{e})$ over the prime powers $p^e$ exactly dividing $d$.
The part $R/(d_1) \oplus \cdots \oplus R/(d_k)$ is the [[torsion submodule|6PBS6KW0NC]] of $M$, and $r$ is the rank of the [[free|0DZ9DKAKZR]] quotient $M$ modulo torsion.

Example: over $\Z$, the module $\Z/2\Z \oplus \Z/6\Z$ has invariant factors $2 \mid 6$ and elementary divisors $2, 2, 3$, since $\Z/6\Z \cong \Z/2\Z \oplus \Z/3\Z$.
For $R = \Z$ the theorem is the [[structure theorem for finitely generated abelian groups|0RY0B72XAS]].
For $R = k[x]$ and $M = k^n$ with $x$ acting by a matrix, it gives the rational canonical form and the [[Jordan normal form|085HQ5G5TE]].

The principal ideal hypothesis is necessary.
Over $R = \Z[x]$, the ideal $I = (2, x)$ is a finitely generated torsion-free module.
Any two elements of $I$ are linearly dependent, so a basis would have one element, and $I$ would be principal.
It is not principal, so $I$ is not free and not of the form above.

## Explanations

### L1 human

Over a principal ideal domain, every finitely generated module breaks into simple pieces.
There is a free part, a copy of $R^r$, and a torsion part made of cyclic modules $R/(d)$.
The list of pieces is unique, so it is a complete invariant: two modules are isomorphic exactly when their lists agree.
For abelian groups, this says each finitely generated abelian group is $\Z^r$ times a product of finite cyclic groups.

### L2 human

The theorem unifies two classical results: the classification of finitely generated abelian groups, for $R = \Z$, and the normal forms of matrices, for $R = k[x]$.
In the second case, a matrix $A$ makes $k^n$ a $k[x]$-module, the invariant factors are the invariant polynomials of $A$, and $d_k$ is the minimal polynomial.
The proof by Smith normal form is algorithmic when $R$ is Euclidean.
Over a Dedekind domain that is not principal, the torsion part still decomposes, but the torsion-free part is a sum of ideals rather than a free module.
</document>

<document source="https://tamnd.github.io/math2/node/TXS0J856RD.md" tag="TXS0J856RD" kind="definition">
# Link

## Definition

Let $m \ge 1$ be an integer, and let $S^1 \sqcup \cdots \sqcup S^1$ be the disjoint union of $m$ copies of the [[circle|060W5CBZP6]].

1. A *link with $m$ components* is a subset $L \subset \R^3$ of [[Euclidean space|0XPR1E85W0]] that is the image of a continuous injective map from $S^1 \sqcup \cdots \sqcup S^1$ to $\R^3$.
2. The *components* of $L$ are the images of the $m$ circles.
3. Two links $L$ and $L'$ are *equivalent* if there is a continuous family of [[homeomorphisms|0QT00PB5G2]] $h_t \colon \R^3 \to \R^3$, $t \in [0, 1]$, with $h_0 = \id$ and $h_1(L) = L'$.

## Remarks

A link with one component is a knot.
As for knots, one usually restricts to tame links, those equivalent to a union of closed polygons.
Example: two disjoint round circles in the plane $z = 0$, far apart, form a link with $2$ components, the *unlink* of two components.
Non-example: two circles that meet in one point do not form a link, because the map from the disjoint union is not injective.
Links are often oriented, by choosing a direction on each component.
</document>

<document source="https://tamnd.github.io/math2/node/TXS2H001R4.md" tag="TXS2H001R4" kind="concept">
# Summation in topological groups

## Summary

Let $(x_i)_{i \in I}$ be a family in a commutative [[topological group|92700QAB5Y]], indexed by any set $I$.
It has sum $s$ when the finite partial sums $\sum_{i \in F} x_i$ converge to $s$ as the finite set $F \subseteq I$ grows.
This defines infinite sums with no order on the index set, and the sum does not depend on any rearrangement.
In a Hausdorff group a family has at most one sum.
This is the notion of summability used in Mathlib (`HasSum`), and it covers series in $\R$, in Banach spaces and in $p$-adic numbers.

## Explanations

### L1 human

To add infinitely many elements of a group without choosing an order, look at all finite sub-collections and their sums.
The family has sum $s$ if these finite sums get as close to $s$ as you like once the sub-collection contains a large enough finite set.
For real numbers this is stronger than ordinary convergence of a series.
For example, $1 - \tfrac12 + \tfrac13 - \cdots$ converges as a series, but as a family it has no sum, because one can pick finite sets of only positive terms with arbitrarily large sums.
</document>

<document source="https://tamnd.github.io/math2/node/TXTR68V4R1.md" tag="TXTR68V4R1" kind="definition">
# Forward difference operator

## Definition

Let $M$ be an additive commutative monoid, let $G$ be an additive abelian group, and let $h \in M$.
The *forward difference operator* with step $h$ sends a function $f \colon M \to G$ to the function $\Delta_h f \colon M \to G$ given by
$$\Delta_h f(x) = f(x + h) - f(x) \qquad (x \in M).$$
When $h = 1$, one writes $\Delta$ for $\Delta_1$.
For a sequence $u \colon \N \to G$, this gives $(\Delta u)_n = u_{n+1} - u_n$.

## Remarks

It is a [[finite difference|7JSQZ126BP]] in the forward direction.
The operator $\Delta_h$ is additive: $\Delta_h(f + g) = \Delta_h f + \Delta_h g$.
Example: for $f(x) = 2^x$ on $\Z$ with $h = 1$, $\Delta f(x) = 2^{x+1} - 2^x = 2^x$, so $f$ is a fixed point of $\Delta$, like $e^x$ for the derivative.
Non-example: $g(x) = f(x+1) - f(x-1)$ is not $\Delta_h f$ for any $h$; it is a central difference with step $2$.
With the [[shift operator|MBK6AE6734]] $E_h f(x) = f(x + h)$ and the identity $I$, one has $\Delta_h = E_h - I$.
</document>

<document source="https://tamnd.github.io/math2/node/TXVWFWP6VQ.md" tag="TXVWFWP6VQ" kind="concept">
# Representation of a quiver

## Summary

A representation of a [[quiver|RE9YFFS3TS]] over a field assigns a [[vector space|000AYPA9AJ]] to each vertex.
It also assigns to each arrow a [[linear map|0NRYHJPQG2]] from the space at its source to the space at its target.
Representations of a finite quiver are the same as modules over its [[path algebra|2PYANBGXFD]].
Classifying the [[indecomposable|E2P29GCD9T]] representations is a central problem; [[Gabriel's theorem|FC78F2E87E]] and [[Kac's theorem|59QJC3ZXA1]] describe their dimension vectors through root systems.
Many problems of linear algebra, such as the classification of pairs of matrices, are problems about quiver representations.

## Explanations

### L1 human

Take a quiver, a diagram of dots and arrows.
Put a vector space at each dot and a matrix along each arrow, going from the space at the start to the space at the end.
That is a representation.
For the quiver with one vertex and one loop, a representation is a single square matrix.
Over the complex numbers its classification up to change of basis is the Jordan normal form.
For one arrow $1 \to 2$, a representation is a linear map, determined up to isomorphism by the two dimensions and its rank.
</document>

<document source="https://tamnd.github.io/math2/node/TY12G0E8KX.md" tag="TY12G0E8KX" kind="problem">
# Measure problem of Lebesgue

## Problem

Is there a function $m$, defined on all bounded subsets of $\R$, with values in $[0, \infty)$, such that:

1. $m$ is [[countably additive|D7KB9HFCGZ]]: $m\big(\bigcup_k A_k\big) = \sum_k m(A_k)$ for pairwise disjoint bounded $A_k$ whose union is bounded.
2. $m$ is invariant under congruence: $m(A) = m(B)$ when $B$ is a [[translation|087VJM5Q77]] or reflection of $A$.
3. $m([0, 1]) = 1$.

## Remarks

[[Lebesgue measure|2Z2QVWZ262]] has all three properties on the Lebesgue measurable sets.
The answer is negative: Vitali showed in 1905, with the Vitali set and the axiom of choice, that no such $m$ exists.
Banach and Ulam studied the question without congruence invariance.
Is there a countably additive measure on all subsets of $\R$ that is $0$ on points and $1$ on $[0, 1]$?
Ulam showed that a positive answer implies that some cardinal at most the continuum is real-valued measurable, a notion close to that of a [[measurable cardinal|AKBWT323RQ]].
Such a cardinal is weakly inaccessible, so ZFC cannot prove a positive answer, if ZFC is consistent.
Dropping countable additivity and keeping finite additivity leads to the [[Banach measure problem|JXK6NRDH4V]].

## Explanations

### L1 human

Lebesgue asked whether every bounded set of real numbers can be given a length.
The length should add up over countably many disjoint pieces and should not change when a set is shifted.
The answer is no, and the obstruction is the existence of very wild sets built with the axiom of choice.
Weaker versions of the question lead deep into set theory, to large cardinals.
</document>

<document source="https://tamnd.github.io/math2/node/TY60P5RE56.md" tag="TY60P5RE56" kind="definition">
# Morita equivalence of Lie groupoids

## Definition

For a [[Lie groupoid|HNVW0E732J]] $G$, write $G_0$ for its manifold of objects and $G_1$ for its manifold of arrows.
Write $s, t \colon G_1 \to G_0$ for the source and target maps.
Let $\phi = (\phi_1, \phi_0) \colon K \to G$ be a morphism of Lie groupoids.
The morphism $\phi$ is a *weak equivalence* if both of the following hold.

1. The map $G_1 \times_{s, G_0, \phi_0} K_0 \to G_0$, $(g, y) \mapsto t(g)$, is a surjective [[submersion|BMY55EW3RF]].
2. The fibre product $(K_0 \times K_0) \times_{G_0 \times G_0} G_1$, formed with $\phi_0 \times \phi_0$ and $(t, s)$, is a [[smooth manifold|F45VQ47GPV]].
The map $k \mapsto ((t(k), s(k)), \phi_1(k))$ from $K_1$ to it is a diffeomorphism.

Two Lie groupoids $G$ and $H$ are *Morita equivalent* if there are a Lie groupoid $K$ and weak equivalences $K \to G$ and $K \to H$.

## Remarks

Clause 1 says that every object of $G$ is isomorphic to an object in the image of $\phi_0$, in a smooth way.
Clause 2 says that the arrows of $K$ between two objects are exactly the arrows of $G$ between their images.
Morita equivalence is an equivalence relation.
An equivalent definition uses a principal bibundle: a manifold $P$ with commuting actions of $G$ and $H$ and surjective submersions as moment maps.
Each action must be free and transitive on the fibres of the moment map of the other action.

Example: let $\pi \colon U \to M$ be a surjective submersion, and let $K$ be the groupoid with $K_0 = U$ and $K_1 = U \times_M U$, where $(u, v)$ is an arrow from $v$ to $u$.
The map $\pi$ gives a weak equivalence from $K$ to the groupoid $M$ that has only identity arrows.
Clause 1 holds because the map in question is $\pi$ itself.
Clause 2 holds because $K_1$ is exactly the set of pairs of points of $U$ with the same image in $M$.
So $K$ and $M$ are Morita equivalent; taking $U$ to be the disjoint union of the sets of an open cover gives the usual Čech groupoid.

Non-example: a nontrivial Lie group $G$, viewed as a groupoid with one object, is not Morita equivalent to the one-point groupoid with only the identity arrow.
Weak equivalences induce isomorphisms of isotropy groups, and the isotropy groups here are $G$ and the trivial group.

## Explanations

### L1 human

Two Lie groupoids are Morita equivalent when one can pass between them by refining or coarsening the space of objects without changing the symmetries between objects.
</document>

<document source="https://tamnd.github.io/math2/node/TY6TPW88MX.md" tag="TY6TPW88MX" kind="definition">
# Strange attractor

## Definition

Let $f \colon X \to X$ be a continuous map of a metric space, or a flow on a metric space, and let $A$ be an [[attractor|7FWX24X5MW]] of it.
The attractor $A$ is *strange* if it has a fractal structure.
In the most common precise form, this means that the [[Hausdorff dimension|SSNFTHZ6YJ]] of $A$, or its box-counting dimension, is not an integer.

## Remarks

There is no single agreed definition.
Many authors use strange attractor and chaotic attractor as synonyms, and then they ask for sensitive dependence on initial conditions on $A$ instead of, or in addition to, fractal structure.
The two conditions are different: there are chaotic attractors that are not fractal, and there are fractal attractors on which nearby orbits do not separate.

Example: the Smale solenoid map doubles the angle $\theta \mapsto 2\theta$ around a solid torus and contracts strongly in the two transverse directions.
Its attractor is locally the product of an interval and a Cantor set.
So its dimension lies strictly between $1$ and $2$.

Non-example: for the flow of $r' = r(1 - r^2)$, $\theta' = 1$ in polar coordinates in the plane, the unit circle is an attractor.
It is a smooth closed curve of Hausdorff dimension $1$, an integer, so it is not strange.
</document>

<document source="https://tamnd.github.io/math2/node/TY7Y32MCH1.md" tag="TY7Y32MCH1" kind="concept">
# Banach analytic space

## Summary

A Banach analytic space is an infinite-dimensional analogue of a [[complex analytic space|HXSZA44J5J]].
It is modelled locally on the zero set of a [[holomorphic map|E8N3EC0YH5]] from an open subset of a complex [[Banach space|06GZ956657]] to another one.
The notion was introduced by Douady to construct the space of compact complex subspaces of a given complex space.
Complex analytic spaces and [[Banach manifolds|9K15AMGBV9]] over $\C$ are Banach analytic spaces.
They are used in deformation theory.
Spaces of maps or of subspaces are first built as Banach analytic spaces, and then shown to be finite-dimensional.

## Explanations

### L1 human

A finite-dimensional complex space is cut out locally by finitely many holomorphic equations in $\C^n$.
In a Banach analytic space the ambient space $\C^n$ is replaced by an infinite-dimensional Banach space, and the equations take values in another Banach space.
For example, any open subset of a complex Banach space is a Banach analytic space with no equations at all.
The equation $\ip{x}{x} = 1$ of the unit sphere in a complex Hilbert space $H$ is not allowed, because $x \mapsto \ip{x}{x}$ is not holomorphic.
But the zero set of any holomorphic map from $H$ to another Banach space is.
</document>

<document source="https://tamnd.github.io/math2/node/TY89DBMJYQ.md" tag="TY89DBMJYQ" kind="definition">
# Von Neumann–Bernays–Gödel set theory

## Definition

*Von Neumann–Bernays–Gödel set theory*, written NBG, is the first-order theory in the language $\set{\in}$ whose variables range over [[classes|W473FH7Z0P]].
A class $X$ is a *set* if $X \in Y$ for some class $Y$, and a *proper class* otherwise.
The axioms of NBG are these.

1. Extensionality: classes with the same elements are equal, as in the [[axiom of extensionality|0TB16T2C97]].
2. Pairing, union, power set and infinity: as in [[ZF|0HE5WDQ9FJ]], stated for sets.
3. Foundation: the [[axiom of regularity|6QXBY7J0M5]], for all nonempty classes.
4. Replacement: if $F$ is a class that is a function and $a$ is a set, then the image $F[a]$ is a set.
5. Class comprehension: for each formula $\varphi(x)$ in which all quantifiers range over sets, possibly with class parameters, there is a class $\set{x : x \text{ is a set and } \varphi(x)}$.
6. Global choice: there is a class function that picks an element from every nonempty set.

## Remarks

Class comprehension can be replaced by finitely many of its instances, so NBG has a finite list of axioms; ZF has no finite axiomatization if it is consistent.
NBG is a conservative extension of ZFC: a statement about sets is provable in NBG if and only if it is provable in ZFC.
Example: the class $V = \set{x : x = x}$ of all sets exists by comprehension, with $\varphi(x)$ the formula $x = x$.
Non-example: $V$ is not a set.
If it were, then comprehension would give the class $R = \set{x : x \notin x}$, which would be a set as a subclass of $V$, and then $R \in R \leftrightarrow R \notin R$.
Morse–Kelley set theory is the stronger theory in which the formula in class comprehension may also quantify over classes.
Some authors drop global choice, or replace it by the [[axiom of choice|0HTGB13FE2]] for sets; NBG without global choice is conservative over ZF.
</document>

<document source="https://tamnd.github.io/math2/node/TYJWZDK485.md" tag="TYJWZDK485" kind="example">
# Basel problem via Fourier series

## Example

Let $f$ be the $2\pi$-periodic function with $f(x) = x$ for $-\pi < x \le \pi$.
Its [[Fourier series|G12KXDHPPS]] is $2 \sum_{n \ge 1} \frac{(-1)^{n+1}}{n} \sin nx$: indeed $f$ is odd, so the cosine coefficients vanish, and integration by parts gives
$$b_n = \frac{2}{\pi} \int_0^\pi x \sin nx \, dx = \frac{2(-1)^{n+1}}{n}.$$
In complex form, $\hat f(0) = 0$ and $\abs{\hat f(n)} = \frac{1}{\abs{n}}$ for $n \ne 0$.
Parseval's identity, from the [[L2 convergence of Fourier series|3DGHC142FT]], gives
$$\frac{1}{2\pi} \int_{-\pi}^{\pi} x^2 \, dx = \sum_{n \ne 0} \frac{1}{n^2}, \qquad \text{that is} \qquad \frac{\pi^2}{3} = 2 \sum_{n=1}^{\infty} \frac{1}{n^2}.$$
So $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$, which is the [[Basel problem|0DMY75D6FV]].

## Remarks

A second route uses the pointwise convergence of a Fourier series instead of Parseval.
The $2\pi$-periodic function equal to $x^2$ on $[-\pi, \pi]$ is continuous and piecewise smooth, and its Fourier series $\frac{\pi^2}{3} + 4 \sum_{n \ge 1} \frac{(-1)^n}{n^2} \cos nx$ converges to it everywhere.
At $x = \pi$ this gives $\pi^2 = \frac{\pi^2}{3} + 4 \sum_{n \ge 1} \frac{1}{n^2}$, the same value.
Parseval applied to $x^2$ gives $\sum_{n \ge 1} \frac{1}{n^4} = \frac{\pi^4}{90}$ in the same way.

## Explanations

### L1 human

The energy of the function $x$ on one period can be computed in two ways: directly, or as the sum of the energies of its waves.
The waves of $x$ have amplitudes $1/n$, so their energies are $1/n^2$.
Comparing the two answers forces the sum $1 + \frac{1}{4} + \frac{1}{9} + \cdots$ to equal $\frac{\pi^2}{6}$.
</document>

<document source="https://tamnd.github.io/math2/node/TYXD1T6FRX.md" tag="TYXD1T6FRX" kind="example">
# Kepler problem

## Example

Let $k > 0$.
The Kepler problem is the system
$$r'' = -\frac{k\, r}{\norm{r}^3}$$
for a position $r(t) \in \R^3 \setminus \set{0}$, with velocity $v = r'$.
It describes a body attracted to a fixed centre by an inverse-square force, such as a planet around the Sun.
It is the [[Hamiltonian system|0EDKGA2FAK]] with $H(r, v) = \frac{1}{2} \norm{v}^2 - \frac{k}{\norm{r}}$.

Three quantities are [[first integrals|0262JP0NV9]]:

1. the energy $E = \frac{1}{2} \norm{v}^2 - k / \norm{r}$;
2. the angular momentum $L = r \times v$;
3. the Laplace–Runge–Lenz vector $A = v \times L - k\, r / \norm{r}$.

Check for $L$: $L' = v \times v + r \times r'' = 0 - \frac{k}{\norm{r}^3}\, r \times r = 0$.

## Remarks

Since $L$ is constant, a motion with $L \ne 0$ stays in the plane through the origin orthogonal to $L$.
In polar coordinates in that plane, with $\theta$ measured from the direction of $A$, the orbit is
$$\norm{r} = \frac{\norm{L}^2 / k}{1 + e \cos \theta}, \qquad e = \norm{A} / k.$$
So the orbit is a [[conic section|00HJCMC2CD]] with a [[focus|0RBKKNH4GV]] at the origin and eccentricity $e$.
It is an [[ellipse|0VKCCEXBJ1]] when $E < 0$, a parabola when $E = 0$, and a branch of a hyperbola when $E > 0$.
For an ellipse with semi-major axis $a$, the period $T$ satisfies $T^2 = 4 \pi^2 a^3 / k$, which is Kepler's third law.

## Explanations

### L1 human

The Kepler problem asks how a body moves when it is pulled toward a fixed point with a force that falls off like the square of the distance.
The answer is that it moves on an ellipse, a parabola or a hyperbola, with the centre of force at a focus.
This explains Kepler's laws of planetary motion from Newton's law of gravitation.
</document>

<document source="https://tamnd.github.io/math2/node/TYYS649FAJ.md" tag="TYYS649FAJ" kind="definition">
# Subdirectly irreducible algebra

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$ having at least two elements, and let $\Delta_A = \set{(a, a) : a \in A}$.
The algebra $\mathbf{A}$ is *subdirectly irreducible* when the set of [[congruences|DCSXXT4SER]] on $\mathbf{A}$ different from $\Delta_A$ has a least element.
This least element is called the *monolith* of $\mathbf{A}$.

## Remarks

Equivalently, $\mathbf{A}$ is subdirectly irreducible when, for every subdirect embedding $\alpha \colon \mathbf{A} \to \prod_{i \in I} \mathbf{A}_i$, some composite $\pi_i \circ \alpha$ is an isomorphism.
This is the form that gives the name, through [[subdirect products|88WN3AD2VQ]].
In terms of the [[congruence lattice|REE07CSXHR]]: the intersection of all congruences other than $\Delta_A$ is not $\Delta_A$.

Example: the group $\Z/4\Z$ is subdirectly irreducible.
Its congruences come from the subgroups $\set{0} \subset \set{0, 2} \subset \Z/4\Z$, which form a chain, so the congruence of $\set{0, 2}$ is the monolith.

Non-example: the group $\Z/6\Z$ is not subdirectly irreducible.
The congruences modulo $2$ and modulo $3$ are both different from $\Delta$, and their intersection is $\Delta$, so there is no least one.

The ring $\Z$ is not subdirectly irreducible either, since the congruences modulo the primes intersect in $\Delta_\Z$.
</document>

<document source="https://tamnd.github.io/math2/node/TZ1N0HAGW7.md" tag="TZ1N0HAGW7" kind="concept">
# Neville theta functions

## Summary

The Neville theta functions $\theta_s$, $\theta_c$, $\theta_d$, $\theta_n$ are rescaled Jacobi [[theta functions|2F5CYQJVAF]].
Their argument is the argument $u$ of the [[Jacobi elliptic functions|P071B692JF]].
With this normalisation the Jacobi functions are simple quotients: $\operatorname{sn} = \theta_s/\theta_n$, $\operatorname{cn} = \theta_c/\theta_n$ and $\operatorname{dn} = \theta_d/\theta_n$.
The letters $s, c, d, n$ match the letters in the names of the twelve Jacobi functions, so any Jacobi function $\operatorname{pq}$ is $\theta_p/\theta_q$.

## Explanations

### L1 human

Theta functions are fast-converging series, and the Jacobi elliptic functions are ratios of them.
In the classical form the theta functions use a rescaled angle $\pi u/(2K)$ and carry awkward constant factors.
Neville chose the scaling so that the constants disappear from the ratios.
Then, for example, $\operatorname{sd} u = \theta_s(u)/\theta_d(u)$, read directly from the letters.
</document>

<document source="https://tamnd.github.io/math2/node/TZ3CT1YA9B.md" tag="TZ3CT1YA9B" kind="concept">
# Complementary error function

## Summary

The complementary error function is $\operatorname{erfc} x = 1 - \operatorname{erf} x$, where $\operatorname{erf}$ is the [[error function|7Y56QCZSZJ]].
It equals $\frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2}\, dt$, the scaled area of the tail of the Gaussian curve.
It is the natural function for small tail probabilities of the [[normal distribution|05AQE0QR0D]], where computing $1 - \operatorname{erf} x$ directly would lose precision.

## Explanations

### L1 human

The error function measures the area under the bell curve from $0$ up to $x$.
The complementary error function measures what is left: the area beyond $x$.
It starts at $1$ for $x = 0$ and drops very fast to $0$ as $x$ grows.
For example, $\operatorname{erfc} 3 \approx 2.2 \times 10^{-5}$, a tiny tail.
</document>

<document source="https://tamnd.github.io/math2/node/TZ6Y9XXQJW.md" tag="TZ6Y9XXQJW" kind="definition">
# Gaussian binomial coefficient

## Definition

Let $q$ be an indeterminate and let $n, k$ be integers with $n \ge 0$.
For $0 \le k \le n$, the *Gaussian binomial coefficient* is
$$\binom{n}{k}_q = \frac{(1 - q^n)(1 - q^{n-1}) \cdots (1 - q^{n-k+1})}{(1 - q)(1 - q^2) \cdots (1 - q^k)} = \frac{[n]_q!}{[k]_q!\, [n - k]_q!},$$
where $[m]_q!$ is the q-factorial of the [[q-analog|GFSDMJHHVA]] node.
For $k < 0$ or $k > n$, $\binom{n}{k}_q = 0$.

## Remarks

The coefficients satisfy the q-Pascal rule $\binom{n}{k}_q = \binom{n-1}{k-1}_q + q^k \binom{n-1}{k}_q$ for $n \ge 1$.
By induction, $\binom{n}{k}_q$ is a polynomial in $q$ with nonnegative integer coefficients, of degree $k(n - k)$.
At $q = 1$ it equals the [[binomial coefficient|0GBZM0MT0F]] $\binom{n}{k}$.
If $q$ is a prime power, it is the number of $k$-dimensional subspaces of $\F_q^n$.

Example: $\binom{4}{2}_q = \frac{(1 - q^4)(1 - q^3)}{(1 - q)(1 - q^2)} = (1 + q^2)(1 + q + q^2) = 1 + q + 2q^2 + q^3 + q^4$.
At $q = 2$ this is $35$.
Counting directly, $\F_2^4$ has $\frac{(2^4 - 1)(2^4 - 2)}{(2^2 - 1)(2^2 - 2)} = \frac{15 \cdot 14}{3 \cdot 2} = 35$ subspaces of dimension $2$.

Non-example: $q^{k(n-k)} \binom{n}{k}$ also equals $\binom{n}{k}$ at $q = 1$, but it is not the Gaussian binomial coefficient.
For $n = 4$, $k = 2$ and $q = 2$ it gives $96$, not $35$.

The coefficient of $q^j$ in $\binom{n}{k}_q$ is the number of integer partitions of $j$ with at most $k$ parts, each at most $n - k$.
Another common notation is $\left[ {n \atop k} \right]_q$.
</document>

<document source="https://tamnd.github.io/math2/node/TZFSE2VNAQ.md" tag="TZFSE2VNAQ" kind="definition">
# Simplicial object

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]], and let $\Delta$ be the [[simplex category|FRFYHJGAN3]] with [[opposite category|DAMK8XE1FR]] $\Delta^{\mathrm{op}}$.
A *simplicial object* in $\mathcal{C}$ is a [[functor|28NQ0KRHD3]] $X \colon \Delta^{\mathrm{op}} \to \mathcal{C}$.
One writes $X_n$ for $X([n])$.
A morphism of simplicial objects $X \to Y$ is a [[natural transformation|KYCPCC57FT]] from $X$ to $Y$.

## Remarks

Equivalently, a simplicial object is a family of objects $X_n$, $n \ge 0$, with face maps $d_i \colon X_n \to X_{n-1}$ and degeneracy maps $s_i \colon X_n \to X_{n+1}$ for $0 \le i \le n$, subject to the simplicial identities:
$$d_i d_j = d_{j-1} d_i \ (i < j), \qquad s_i s_j = s_{j+1} s_i \ (i \le j),$$
$$d_i s_j = s_{j-1} d_i \ (i < j), \qquad d_j s_j = d_{j+1} s_j = \id, \qquad d_i s_j = s_j d_{i-1} \ (i > j+1).$$
Here $d_i = X(\delta_i)$ and $s_i = X(\sigma_i)$ for the face and degeneracy maps $\delta_i$, $\sigma_i$ of $\Delta$.
Example: for an object $A$, the constant functor with $X_n = A$ and every $d_i$, $s_i$ equal to $\id_A$ satisfies all identities, so it is a simplicial object.
Non-example: a covariant functor $\Delta \to \mathcal{C}$ is not a simplicial object, because its maps go from $X_{n-1}$ to $X_n$ for faces; such functors are called cosimplicial objects.
Simplicial objects in $\mathbf{Set}$ are [[simplicial sets|WW81ARB8K9]].
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<document source="https://tamnd.github.io/math2/node/TZHD2WKNSD.md" tag="TZHD2WKNSD" kind="statement">
# Measurable Riemann mapping theorem

## Statement

Let $\mu \colon \C \to \C$ be a measurable function with $\norm{\mu}_\infty < 1$.
Then there is a unique [[quasiconformal|SB8KG46NHP]] homeomorphism $f \colon \C \to \C$ such that:

1. $f$ solves the [[Beltrami equation|9J7GFJ1GPF]] $f_{\bar z} = \mu f_z$ almost everywhere on $\C$.
2. $f(0) = 0$ and $f(1) = 1$.

## Remarks

When $\mu$ has compact support, one looks for $f$ of the form $f(z) = z + \frac{1}{\pi}\int_\C \frac{h(w)}{z - w}\, dA(w)$, where $h = f_{\bar z}$.
The Beltrami equation then becomes $h = \mu + \mu S h$, where $S$ is the [[Beurling–Ahlfors transform|TX32GDQ330]].
The operator $S$ is an isometry of $L^2$, and its norm on $L^p$ is close to $1$ for $p$ close to $2$.
So the equation has a solution given by a Neumann series in $L^p$ for some $p > 2$.
The general case follows by composing solutions.
Uniqueness holds because two normalized solutions differ by a $1$-quasiconformal, hence conformal, self-map of $\C$ fixing $0$ and $1$, which is the identity.
The theorem also gives analytic dependence of $f$ on $\mu$, a fact that is central in Teichmüller theory and holomorphic dynamics.
The condition $\norm{\mu}_\infty < 1$ cannot be weakened to $\abs{\mu} < 1$ almost everywhere: for some such $\mu$ no homeomorphic solution exists.

## Explanations

### L1 human

Imagine that at each point of the plane someone draws a tiny ellipse, with the only rule that the ellipses are never thinner than a fixed ratio.
The theorem says that there is a way to deform the plane so that all these ellipses become circles.
The deformation is unique once two points are pinned down.
So any such field of ellipses is just the round structure of the plane seen through a distorted map.

### L2 human

The theorem is the main existence result for [[quasiconformal mappings|SB8KG46NHP]].
It turns questions about conformal structures into questions about bounded measurable functions, which can be changed freely.
In complex dynamics it is used in quasiconformal surgery: one builds an invariant ellipse field and then straightens it to get a new holomorphic map.
In Teichmüller theory it shows that Beltrami coefficients parametrize deformations of Riemann surfaces, and the holomorphic dependence on $\mu$ gives the complex structure of [[Teichmüller space|ZZGC6BZNJ5]].
For $\mu = 0$ the solution is the identity, so the statement extends the uniqueness of conformal automorphisms of $\C$ fixing two points.
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<document source="https://tamnd.github.io/math2/node/TZHF6ZM92S.md" tag="TZHF6ZM92S" kind="concept">
# Adjoint functors

## Summary

Two [[functors|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathcal{D}$ and $G \colon \mathcal{D} \to \mathcal{C}$ are adjoint when maps $F c \to d$ match naturally with maps $c \to G d$.
Then $F$ is the left adjoint and $G$ the right adjoint.
Adjunctions are everywhere in mathematics: free constructions, products, quotients, tensor and hom, and much more.
Each adjunction gives a [[monad|FF6XXA6D1A]], and right adjoints preserve limits.

## Explanations

### L1 human

Many constructions come with a forgetful partner.
The free group on a set $X$ is built from $X$, and every group has an underlying set.
A group homomorphism from the free group on $X$ to a group $H$ is the same as a plain function from $X$ to the set of $H$.
This kind of matching between two sets of maps is an adjunction.
Here the free group functor is the left adjoint and the forgetful functor is the right adjoint.
</document>

<document source="https://tamnd.github.io/math2/node/TZNR514JQN.md" tag="TZNR514JQN" kind="concept">
# Temperley–Lieb algebra

## Summary

The Temperley–Lieb algebra $TL_n(\delta)$ is an algebra with generators $e_1, \dots, e_{n-1}$ and simple relations depending on a parameter $\delta$.
It first appeared in the work of Temperley and Lieb on lattice models of statistical mechanics.
Jones found it again in the study of subfactors, and it underlies the Jones polynomial of knots.
It has a basis of planar diagrams, and its dimension is the [[Catalan number|0YWZJEABB2]] $\frac{1}{n+1}\binom{2n}{n}$.

## Explanations

### L1 human

Draw $n$ dots on a top line and $n$ dots on a bottom line, and join the $2n$ dots in pairs by curves that do not cross inside the strip.
Such a picture is a basis element of the Temperley–Lieb algebra.
Two pictures are multiplied by stacking one on top of the other and joining the middle dots.
Each closed loop that appears is erased and replaced by a factor $\delta$.
For $n = 2$ there are two pictures, two vertical strands or a cap over a cup, and the cap-cup picture $e_1$ satisfies $e_1^2 = \delta e_1$.
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<document source="https://tamnd.github.io/math2/node/TZT1TTB7CJ.md" tag="TZT1TTB7CJ" kind="statement">
# Cohen's theorem on Noetherian rings

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]].
Then $R$ is [[Noetherian|0MVH4Y1CRZ]] if and only if every [[prime ideal|06HC6HK0Z8]] of $R$ is [[finitely generated|030TZ0DQWT]].

## Remarks

One direction is clear, since in a Noetherian ring every ideal is finitely generated.
For the other, suppose that some ideal is not finitely generated.
By Zorn's lemma there is an ideal $I$ that is maximal among the ideals that are not finitely generated.
Let $a, b \notin I$ with $ab \in I$.
Then $I + (a)$ and the ideal quotient $(I : a) = \set{r \in R : ra \in I}$ are strictly larger than $I$, hence finitely generated.
One checks that $I$ would then be finitely generated, which is impossible; so $I$ is prime.
This prime ideal is not finitely generated, against the hypothesis.
For example, to show that $\Z[x]$ is Noetherian it would be enough to check its prime ideals, although [[Hilbert's basis theorem|0YKA9J3XEE]] gives this more directly.

## Explanations

### L1 human

To check that every ideal of a commutative ring is finitely generated, it is enough to check the prime ideals.
The prime ideals are often easier to list, so this saves work.

### L2 human

The proof is a model of a common method: an ideal maximal for not having a property is often prime.
The same method shows that a commutative ring in which every prime ideal is principal is a principal ideal ring.
The theorem is a test for the [[Noetherian|0670T59HGE]] property, for example for rings of power series and rings of integers.
</document>

<document source="https://tamnd.github.io/math2/node/TZWMEM7GKA.md" tag="TZWMEM7GKA" kind="concept">
# Disjoint union of posets

## Summary

The disjoint union of partially ordered sets places them side by side, with no relation between elements of different parts.
Inside each part the order is unchanged.
It is also called the parallel composition, or cardinal sum, and it is the coproduct of posets.

## Explanations

### L1 human

Put two Hasse diagrams next to each other on the same page, without drawing any new lines.
That picture is the disjoint union.
Two separate two-element chains $a < b$ and $c < d$ give a poset with four elements in which $a$ and $d$ are incomparable.
</document>

<document source="https://tamnd.github.io/math2/node/TZYSW0975T.md" tag="TZYSW0975T" kind="concept">
# Fourier monomial on the circle

## Summary

The Fourier monomials are the functions $e_n(x) = e^{inx}$, $n \in \Z$, viewed as functions on the circle $\mathbb{T} = \R / 2\pi\Z$.
They are the continuous homomorphisms from the [[circle group|04EAFVFTF7]] to the unit circle of $\C$, that is, the characters of the circle.
They form an [[orthonormal basis|0S9GGTR0EX]] of $L^2(\mathbb{T})$, and the coefficients of a function in this basis are its [[Fourier coefficients|5NN2YZZR3V]].

## Explanations

### L1 human

The monomial $e_n$ runs around the unit circle of the complex plane $n$ times while $x$ runs once from $0$ to $2\pi$.
For $n$ negative it runs in the opposite direction, and $e_0$ is the constant $1$.
Its real and imaginary parts are the pure waves $\cos nx$ and $\sin nx$.
Every [[trigonometric polynomial|J3NJQ5FZJ5]] is a finite combination of these monomials, as an ordinary polynomial is a combination of powers.
For example, $e_2 \cdot e_{-3} = e_{-1}$, because exponents add.
</document>

<document source="https://tamnd.github.io/math2/node/V00Q05ZW1G.md" tag="V00Q05ZW1G" kind="statement">
# Hadamard three-circle theorem

## Statement

Let $0 < r_1 < r_2$ and let $f$ be [[holomorphic|0DG0Z72CQD]] on an open set containing the closed annulus $\set{z : r_1 \le \abs{z} \le r_2}$.
For $r_1 \le r \le r_2$ let $M(r) = \max_{\abs{z} = r} \abs{f(z)}$.
Then for each $r$ with $r_1 \le r \le r_2$,
$$\log M(r) \le \frac{\log(r_2 / r)}{\log(r_2 / r_1)} \log M(r_1) + \frac{\log(r / r_1)}{\log(r_2 / r_1)} \log M(r_2).$$
Equivalently, $\log M(r)$ is a [[convex function|07Q7CNM5WF]] of $\log r$.

## Remarks

Proof idea: choose a real $\alpha$ such that $r_1^{\alpha} M(r_1) = r_2^{\alpha} M(r_2)$.
For rational $\alpha = p/q$ with $q > 0$, the [[maximum modulus principle|048F2P90WR]] for $z^p f(z)^q$ on the annulus gives $r^{\alpha} M(r) \le r_1^{\alpha} M(r_1)$.
A limit over rational $\alpha$, or the subharmonicity of $\log \abs{f(z)} + \alpha \log \abs{z}$, handles irrational $\alpha$, and the bound rearranges to the inequality.
Example: for $f(z) = z^n$, $\log M(r) = n \log r$ is linear in $\log r$, so equality holds.
If $M(r) = 0$ for some $r$, then $f$ vanishes on a circle, so $f \equiv 0$ on the annulus and the inequality is trivial.

## Explanations

### L1 human

Take three circles around $0$, one inside another.
The theorem bounds how large $\abs{f}$ can be on the middle circle by its sizes on the inner and outer circles.
Plot the logarithm of the maximum against the logarithm of the radius.
The result is a curve that bends upward, so between two radii it never rises above the straight line joining its values there.

### L2 human

The three-circle theorem is a quantitative form of the maximum modulus principle.
It is the circle version of the [[Hadamard three-lines theorem|WHS58R43N0]]; the map $z \mapsto e^{z}$ carries a vertical strip onto an annulus.
It is used in the theory of entire functions to study the growth of $M(r)$, and it extends to subharmonic functions.
</document>

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# Primary ideal

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
An [[ideal|0MYSA6MYNA]] $Q$ of $R$ is *primary* if these two conditions hold:

1. $Q \ne R$.
2. For all $a, b \in R$ with $ab \in Q$ and $a \notin Q$, there is an integer $n \ge 1$ with $b^n \in Q$.

## Remarks

The [[radical|437YRCNYRJ]] $P = \sqrt{Q}$ of a primary ideal is a [[prime ideal|06HC6HK0Z8]], and one says that $Q$ is $P$-primary.
Example: for a prime $p$ and $n \ge 1$, the ideal $(p^n)$ of $\Z$ is primary; if $p^n \mid ab$ and $p^n \nmid a$, then $p \mid b$, so $p^n \mid b^n$.
Non-example: $(6)$ in $\Z$ fails clause 2, because $2 \cdot 3 \in (6)$ and $2 \notin (6)$, but no power of $3$ is divisible by $6$.
Every prime ideal is primary, and every power of a maximal ideal is primary.
A power of a prime ideal need not be primary: in $R = k[x, y, z]/(xy - z^2)$ over a field $k$, with $P = (x, z)$, one has $xy = z^2 \in P^2$, $x \notin P^2$ and no power of $y$ in $P^2$.

## Explanations

### L1 human

A primary ideal contains a product only for a strong reason.
Either the first factor already lies in it, or some power of the second factor does.
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# Unital

## Summary

A unital of order $n$ is a [[block design|4JCSS2Z1BA]] with $n^3 + 1$ points and blocks of $n + 1$ points.
Any two points lie in exactly one block.
The main examples sit inside a [[projective plane|DANFNN8C4Y]] of order $n^2$, where they meet every line in $1$ or $n + 1$ points.
The classical example is the Hermitian curve, defined by a [[Hermitian form|48P4A1CX85]] over a field with $n^2$ elements.

## Explanations

### L1 human

A unital is a finite geometry of points and blocks in which two points always share exactly one block.
Its numbers are fixed by one parameter $n$: there are $n^3 + 1$ points and each block has $n + 1$ points.
The standard picture is a curve inside a finite plane whose lines cut it either in one point or in $n + 1$ points.
For $n = 2$ one gets $9$ points with blocks of size $3$, sitting inside the plane of order $4$.
</document>

<document source="https://tamnd.github.io/math2/node/V0B6C6TMWD.md" tag="V0B6C6TMWD" kind="concept">
# Triangulation of a point set

## Summary

A triangulation of a finite set of points in the plane divides the [[convex hull|0FQSDC2EQ2]] of the set into triangles.
The corners of the triangles are exactly the given points.
The triangles meet edge to edge, without overlaps.
Triangulations are used to build meshes, to interpolate data and to study the combinatorics of point sets.

## Explanations

### L1 human

Scatter some nails on a board and stretch a rubber band around all of them; the band shows the convex hull.
Now draw straight segments between nails, without crossings, until no more segment can be added.
The region inside the band is then cut into triangles, and this is a triangulation.
Four points at the corners of a square have two triangulations, one for each diagonal.
</document>

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# Radical ideal

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
An [[ideal|0MYSA6MYNA]] $I$ of $R$ is a *radical ideal* if $I = \sqrt{I}$, where $\sqrt{I}$ is the [[radical|437YRCNYRJ]] of $I$.
Equivalently, for all $r \in R$ and all integers $n \ge 1$, if $r^n \in I$ then $r \in I$.

## Remarks

Example: $6\Z$ is a radical ideal of $\Z$; if $6$ divides $r^n$, then $2$ and $3$ divide $r$, so $6$ divides $r$.
Non-example: $12\Z$ is not radical, since $6^2 \in 12\Z$ and $6 \notin 12\Z$.
Every [[prime ideal|06HC6HK0Z8]] is radical: if $r^n \in P$, primality applied $n - 1$ times gives $r \in P$.
The ideal $I$ is radical if and only if the [[quotient ring|0VFW4Y69BT]] $R/I$ is [[reduced|SZHY975FFX]].
The radical $\sqrt{J}$ of any ideal $J$ is a radical ideal.
</document>

<document source="https://tamnd.github.io/math2/node/V0PGBXYJ9R.md" tag="V0PGBXYJ9R" kind="concept">
# Internal set theory

## Summary

Internal set theory, written IST, is an axiomatic form of [[nonstandard analysis|WM62GKCBWN]] introduced by Edward Nelson in 1977.
It adds to [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with choice a new unary predicate "standard" and three axiom schemes: idealization, standardization and transfer.
Instead of building a larger number system, IST finds nonstandard elements, such as infinitesimal reals and unlimited natural numbers, inside the usual sets.
Nelson proved that IST is a conservative extension of ZFC.
So every statement in the language of ZFC that IST proves is already a theorem of ZFC.

## Explanations

### L1 human

In IST the real line is the usual real line, but some of its elements are labelled "standard" and others "nonstandard".
Every number you can name explicitly, such as $0$, $1$ or $\pi$, is standard.
The axioms guarantee that there are also nonstandard numbers, for example a natural number larger than every standard one.
Its reciprocal is then a positive real smaller than every standard positive real, an infinitesimal.
One may use these numbers in proofs, but one may not form "the set of all standard numbers".
</document>

<document source="https://tamnd.github.io/math2/node/V0RBKHYTV7.md" tag="V0RBKHYTV7" kind="concept">
# Valuation on convex bodies

## Summary

A valuation on convex bodies is a function $\varphi$ on the nonempty compact convex sets of $\R^n$ that is additive in the inclusion–exclusion sense.
This means $\varphi(K \cup L) + \varphi(K \cap L) = \varphi(K) + \varphi(L)$ whenever $K \cup L$ is convex.
Volume, surface area, the Euler characteristic and all [[intrinsic volumes|C2CK6KSSTV]] are valuations.
Valuations are the finitely additive "measures" of convex geometry, and their classification, starting with [[Hadwiger's theorem|7V2X68AMEY]], is a central theme of integral geometry.
This notion is unrelated to valuations of fields in algebra.

## Explanations

### L1 human

A valuation is a way to measure convex shapes that does not count overlaps twice.
If two convex shapes together form a convex shape, then the measure of the union plus the measure of the overlap equals the sum of the two measures.
Area and perimeter of plane convex sets behave like this.
The function that gives $1$ to every nonempty convex set also does: $1 + 1 = 1 + 1$.
</document>

<document source="https://tamnd.github.io/math2/node/V0SGQ7E092.md" tag="V0SGQ7E092" kind="definition">
# Clifford–Klein form

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] and let $H$ be a closed subgroup of $G$.
Let $X = G/H$ be the [[homogeneous space|85TWR6E75P]] of left cosets, with $G$ acting by left multiplication.
Let $\Gamma$ be a [[discrete subgroup|5M7D1BSHVZ]] of $G$ whose action on $X$ is [[properly discontinuous|J5X5ED0B2Y]].
The *Clifford–Klein form* of $X$ defined by $\Gamma$ is the quotient space
$$\Gamma \backslash X = \Gamma \backslash G / H,$$
the space of orbits of $\Gamma$ on $X$.
The group $\Gamma$ is then called a discontinuous group for $X$.

## Remarks

In much of the literature $G$ is assumed to be reductive.
If $\Gamma$ is torsion-free, its stabilizers, which are finite by proper discontinuity, are trivial, so $\Gamma \backslash X$ is a manifold locally modelled on $X$.
Example: $G = \mathrm{PSL}_2(\R)$, $H = \mathrm{PSO}(2)$, so that $X$ is the hyperbolic plane, and $\Gamma$ the fundamental group of a closed surface of genus $g \ge 2$ embedded as a cocompact discrete subgroup.
Here $\Gamma$ acts freely and properly discontinuously, and $\Gamma \backslash X$ is the closed hyperbolic surface.
Non-example: $G = \R^2$, $H = \set{(t, \alpha t) : t \in \R}$ with $\alpha$ irrational, and $\Gamma = \Z^2$.
Then $X = G/H \cong \R$ through $(x, y) \mapsto y - \alpha x$, and $(m, n) \in \Gamma$ acts by translation by $n - \alpha m$.
These translations form a dense subgroup of $\R$, so the orbits are not discrete and the action is not properly discontinuous.
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<document source="https://tamnd.github.io/math2/node/V0SH1DY9FJ.md" tag="V0SH1DY9FJ" kind="statement">
# Capacity equals transfinite diameter and Chebyshev constant

## Statement

Let $K$ be a [[compact set|0R6EWFKRCF]] in $\C$.
Let $\operatorname{cap}(K)$ be its [[logarithmic capacity|DBQXRTC0VN]], $\delta(K)$ its [[transfinite diameter|VSGTSQ7RY8]] and $\tau(K)$ its [[Chebyshev constant|DMNN7E2BYA]].
Then
$$\operatorname{cap}(K) = \delta(K) = \tau(K).$$

## Remarks

Proof idea for $\delta(K) \le \tau(K)$: in the Vandermonde determinant of $n$ points of $K$, replace the columns $1, z, \dots, z^{n-1}$ by monic polynomials of small norm on $K$.
The determinant does not change, and by Hadamard's inequality it is at most $n^{n/2}$ times the product of the column maxima.
For $\tau(K) \le \delta(K)$: if $z_1, \dots, z_{n+1}$ are [[Fekete points|F0RVAF11K9]] of $K$, the monic polynomial $\prod_{j \le n} (z - z_j)$ is bounded on $K$ by a ratio of Fekete products.
For $\operatorname{cap}(K) = \delta(K)$: the discrete energy of $n$ Fekete points is close to the least energy of a probability measure.
When $\operatorname{cap}(K) > 0$, the normalized counting measures of Fekete points converge weakly to the equilibrium measure.

Example: for $K = [-1, 1]$, the monic Chebyshev polynomials $2^{1-n} T_n$ have maximum $2^{1-n}$ on $K$ and are extremal.
So $\tau(K) = 1/2$, and the theorem gives $\operatorname{cap}(K) = \delta(K) = 1/2$.

The theorem holds for every compact $K$, including sets of capacity zero, where all three quantities vanish.
Fekete proved the equality of the transfinite diameter and the Chebyshev constant, and Szegő proved their equality with the capacity.

## Explanations

### L1 human

There are three natural ways to measure the size of a plane set $K$ through potential theory.
One uses the least energy of a unit charge on $K$.
One uses how far apart many points of $K$ can be spread.
One uses how small a monic polynomial can stay on $K$.
The theorem says that the three give the same number.

### L2 human

The theorem lets one compute capacity by whichever method is easiest for the set at hand.
Polynomials and Fekete points give upper and lower bounds that can be computed numerically.
It also explains the role of capacity in polynomial approximation: the $n$-th root of the norm of the best monic polynomial on $K$ tends to $\operatorname{cap}(K)$.
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<document source="https://tamnd.github.io/math2/node/V192HFRX5A.md" tag="V192HFRX5A" kind="definition">
# Box–Jenkins method

## Definition

Let $x_1, \dots, x_n$ be an observed time series.
The *Box–Jenkins method* fits an [[ARIMA($p$, $d$, $q$) model|KTDVZP4DF2]] to the series and then uses it to forecast.
It repeats the following three stages until the checks of stage 3 are passed:

1. Identification: choose $d$ so that the differenced series $\nabla^d x_t$ looks stationary. Then choose tentative orders $p$ and $q$ from the sample [[autocorrelation function|YCZHYY8VE5]] and [[partial autocorrelation function|11YRFWD951]] of $\nabla^d x_t$.
2. Estimation: estimate the parameters of the [[ARMA($p$, $q$) model|B1HNX0YT0Z]] for the differenced series by [[maximum likelihood|0FREPFKNS7]] or by nonlinear [[least squares|0BYTQ4X1HC]].
3. Diagnostic checking: test whether the residuals of the fitted model behave like white noise, for instance with a portmanteau test. If they do not, return to stage 1 with a modified model.

## Remarks

The identification stage uses typical patterns.
For an AR($p$) model the partial autocorrelation vanishes beyond lag $p$ while the autocorrelation decays gradually.
For an MA($q$) model the autocorrelation vanishes beyond lag $q$ while the partial autocorrelation decays gradually.

Example: suppose the sample autocorrelation of a series decays very slowly.
Suppose that of its first differences has a single significant value, at lag $1$.
Then stage 1 suggests $d = 1$, $p = 0$ and $q = 1$, that is ARIMA($0$, $1$, $1$).

Non-example: fitting one fixed AR($1$) model by least squares without examining the residuals is not the Box–Jenkins method, because stage 3 and the loop back to stage 1 are missing.

The method prefers parsimonious models, with as few parameters as give an adequate fit.
Seasonal ARIMA models are handled in the same way, with seasonal differences and seasonal orders.
</document>

<document source="https://tamnd.github.io/math2/node/V1C28937NX.md" tag="V1C28937NX" kind="statement">
# Fuglede's theorem

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]], let $N \colon H \to H$ be a bounded normal operator, that is, $N^*N = NN^*$, and let $T \colon H \to H$ be a bounded linear operator.
If $TN = NT$, then $TN^* = N^*T$, where $N^*$ is the [[adjoint|6HSJBJ8TTQ]] of $N$.

## Remarks

Proof idea (Rosenblum): from $TN = NT$ one gets $T = e^{i\bar\lambda N} T e^{-i\bar\lambda N}$ for every $\lambda \in \C$.
Since $N$ and $N^*$ commute, $f(\lambda) = e^{-i\lambda N^*} T e^{i\lambda N^*}$ equals $e^{-iA_\lambda} T e^{iA_\lambda}$ with $A_\lambda = \lambda N^* + \bar\lambda N$ self-adjoint.
So $e^{\pm iA_\lambda}$ are unitary and $f$ is a bounded entire function, hence constant by Liouville's theorem.
Differentiating at $\lambda = 0$ gives $N^*T - TN^* = 0$.

Example: if $N = \mathrm{diag}(1, i)$ on $\C^2$ and $TN = NT$, then $T$ is diagonal because the eigenvalues of $N$ are distinct, and a diagonal $T$ commutes with $N^* = \mathrm{diag}(1, -i)$.

Normality is needed: for the nilpotent $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, the operator $T = N$ commutes with $N$, but $TN^* = NN^* = \mathrm{diag}(1, 0)$ differs from $N^*T = N^*N = \mathrm{diag}(0, 1)$.

Putnam's generalization: if $M$ and $N$ are normal and $TN = MT$, then $TN^* = M^*T$.

## Explanations

### L1 human

For matrices that are not normal, commuting with $N$ need not imply commuting with its adjoint.
Fuglede's theorem says that for normal operators the two conditions go together.
So the operators that commute with a normal operator form an algebra closed under taking adjoints.

### L2 human

In finite dimensions the result follows from the spectral theorem.
An operator $T$ commutes with $N$ if and only if it preserves each eigenspace of $N$.
Then it commutes with $N^*$, which acts on the same eigenspaces by the conjugate eigenvalues.
In infinite dimensions, Fuglede's theorem shows that an operator commuting with $N$ commutes with all spectral projections of $N$, and hence with every bounded Borel function of $N$.
This makes the commutant of a normal operator a von Neumann algebra.
</document>

<document source="https://tamnd.github.io/math2/node/V1FFDDHCH2.md" tag="V1FFDDHCH2" kind="definition">
# Fredholm equation of the first kind

## Definition

Let $a < b$ be real numbers.
Let $K \colon [a, b] \times [a, b] \to \C$ be a [[kernel|GSMBXT7FPK]], for example continuous or square-integrable, and let $f \colon [a, b] \to \C$ be a given function.
Let $\varphi$ denote an unknown function on $[a, b]$.
The *Fredholm equation of the first kind* with kernel $K$ and right-hand side $f$ is
$$\int_a^b K(x, y) \, \varphi(y) \, dy = f(x), \qquad x \in [a, b].$$
A solution is a function $\varphi$ in the chosen function space for which the integral exists and the equality holds.
In $L^2$ the equality is required for almost every $x$, and otherwise for every $x$.

## Remarks

Example: on $[0, 1]$, take $K(x, y) = xy$ and $f(x) = x / 2$.
The constant function $\varphi = 1$ is a solution, because $\int_0^1 x y \, dy = x/2$ for every $x$.
It is not the only one: $\varphi(y) = 1 + (2 - 3y)$ also works, since $\int_0^1 y (2 - 3y) \, dy = 1 - 1 = 0$.
With the same kernel and $f(x) = 1$, there is no solution, since the left side is always a multiple of $x$ and vanishes at $x = 0$.

Non-example: the equation $\varphi(x) - \int_a^b K(x, y) \varphi(y) \, dy = f(x)$ is not of the first kind, because $\varphi$ also appears outside the integral; it is a Fredholm equation of the second kind.

The equation is a special [[Fredholm integral equation|526TCVMSHP]]: the limits of integration are the fixed numbers $a$ and $b$.
In operator notation it reads $T_K \varphi = f$, where $(T_K \varphi)(x) = \int_a^b K(x, y) \varphi(y) \, dy$.
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# Radical of a module

## Definition

Let $R$ be a ring and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.
A *maximal submodule* of $M$ is a [[submodule|05WDSN597C]] $N \ne M$ such that no submodule lies strictly between $N$ and $M$.
The *radical* of $M$, written $\operatorname{rad}(M)$, is the intersection of all maximal submodules of $M$.
If $M$ has no maximal submodule, then $\operatorname{rad}(M) = M$.

## Remarks

Example: let $M = \Z/12\Z$ as a $\Z$-module.
Its maximal submodules are $2\Z/12\Z$ and $3\Z/12\Z$, so $\operatorname{rad}(M) = 6\Z/12\Z$, which has $2$ elements.

Non-example of a proper radical: the $\Z$-module $\Q$ has no maximal submodule, because every quotient $\Q/N$ with $N \ne \Q$ is divisible and so is not simple.
So $\operatorname{rad}(\Q) = \Q$.

For $M = R$ as a left module over itself, $\operatorname{rad}(R)$ is the [[Jacobson radical|XA0DCNH7WY]] $J(R)$.
Equivalently, $\operatorname{rad}(M)$ is the intersection of the kernels of all module homomorphisms from $M$ to simple modules.
For every $M$ one has $J(R) M \subseteq \operatorname{rad}(M)$, with equality when $R$ is left Artinian.
</document>

<document source="https://tamnd.github.io/math2/node/V1HAJNC5RK.md" tag="V1HAJNC5RK" kind="statement">
# Kolmogorov continuity theorem

## Statement

Let $(X_t)_{t \in [0, \infty)}$ be a [[stochastic process|0QH5ENKZM1]] with values in $\R^n$.
Assume that there are constants $\alpha > 0$, $\beta > 0$ and $C > 0$ such that
$$E\left[\abs{X_t - X_s}^\alpha\right] \le C \abs{t - s}^{1 + \beta} \quad \text{for all } s, t \ge 0.$$
Then $X$ has a continuous modification: there is a process $(\tilde X_t)_{t \ge 0}$ such that $P(\tilde X_t = X_t) = 1$ for every $t \ge 0$ and every path $t \mapsto \tilde X_t(\omega)$ is continuous.
Moreover, for every $\gamma \in (0, \beta / \alpha)$, the paths of $\tilde X$ are locally [[Hölder continuous|012PBA0FMD]] of order $\gamma$.

## Remarks

Idea of the proof.
On the dyadic times $k 2^{-m}$, Markov's inequality and the hypothesis give $P(\abs{X_{(k+1)2^{-m}} - X_{k 2^{-m}}} > 2^{-\gamma m}) \le C 2^{-m(1 + \beta - \alpha \gamma)}$.
Summing over the $2^m$ times in $[0, 1]$ and over $m$ gives a convergent series when $\gamma < \beta / \alpha$.
By the [[Borel–Cantelli lemma|0T6WS7TT77]], the increments on dyadic times are eventually small, so $X$ is uniformly Hölder continuous on the dyadic times, and one defines $\tilde X$ as its continuous extension.

Example: for a [[Wiener process|22735FGJV6]] $W$ and an integer $m \ge 2$, $W_t - W_s$ is normal with variance $\abs{t - s}$, so $E[\abs{W_t - W_s}^{2m}] = c_m \abs{t - s}^m$ for a constant $c_m$.
The theorem applies with $\alpha = 2m$ and $\beta = m - 1$, and gives Hölder continuity of every order $\gamma < (m - 1)/(2m)$.
Letting $m \to \infty$, Brownian paths are locally Hölder continuous of every order $\gamma < 1/2$.

The exponent $1 + \beta$ must exceed $1$.
For a Poisson process $N$, $E[\abs{N_t - N_s}] = \abs{t - s}$, so the bound holds with $\alpha = 1$ and exponent $1$, but the paths of $N$ have jumps and $N$ has no continuous modification.

For processes indexed by $t \in [0, 1]^d$, the same conclusion holds with $\abs{t - s}^{d + \beta}$ in place of $\abs{t - s}^{1 + \beta}$.

## Explanations

### L1 human

A random process might be defined only through its values at each fixed time, with no guarantee that its paths are continuous.
The theorem gives a simple test.
If the average size of the jump between two times shrinks fast enough as the times get close, then one can choose a version with continuous paths.
This is how one shows that Brownian motion can be taken to have continuous paths.

### L2 human

The theorem bridges the [[finite-dimensional distributions|Y0CN7V3B7E]] of a process, which say nothing about path properties, and the regularity of its paths.
It is used to construct Brownian motion and the Brownian sheet, to prove continuity of solutions of stochastic differential equations in their initial data, and to obtain Hölder exponents.
Its proof is a chaining argument, a technique that also yields the Kolmogorov–Chentsov tightness criteria in spaces of continuous functions.
</document>

<document source="https://tamnd.github.io/math2/node/V1JG4C7R7B.md" tag="V1JG4C7R7B" kind="statement">
# Schur decomposition

## Statement

Let $A$ be an $n \times n$ complex [[square matrix|0JXZKX526R]].
Then there are a [[unitary matrix|0AR68JTMDT]] $Q$ and an [[upper triangular matrix|0N4XY7Z3NN]] $T$ such that
$$A = Q T Q^*,$$
where $Q^*$ is the [[conjugate transpose|0DXKE10PHF]] of $Q$.
The diagonal entries of $T$ are the [[eigenvalues|0BQAA2BK3Q]] of $A$, each repeated as often as its algebraic multiplicity.

## Remarks

The proof is by induction on $n$.
Over $\C$, the matrix $A$ has an eigenvector $v$ of norm $1$.
Extend $v$ to an orthonormal basis; in this basis the first column of the matrix of $A$ is $(\lambda, 0, \dots, 0)^T$.
Apply the induction hypothesis to the lower right $(n-1) \times (n-1)$ block.

The order of the eigenvalues on the diagonal of $T$ can be chosen freely.
If $A$ is a [[normal matrix|0MR0BYF0K2]], then $T$ is normal and triangular, hence diagonal; this gives the [[spectral theorem for normal matrices|04CW1P44VD]].
The complex field matters.
A real matrix with non-real eigenvalues has no factorisation $A = QTQ^T$ with $Q$ real orthogonal and $T$ real upper triangular.
Indeed, the diagonal of such a $T$ would hold the eigenvalues of $A$.
The real Schur form uses an orthogonal $Q$ and a block upper triangular $T$ with diagonal blocks of size $1$ or $2$ instead.

## Explanations

### L1 human

Every complex square matrix becomes upper triangular after a change to a suitable orthonormal basis.
The eigenvalues then sit on the diagonal.
Unlike diagonalisation, this always works, and the change of basis keeps all lengths and angles.

### L2 human

The Schur decomposition is the unitary form of [[triangularisation|0363JMQ42Y]].
It is stable to compute, and the [[QR algorithm|B6F8YWHYZ5]] for eigenvalues computes it.
It gives quick proofs that the trace is the sum and the determinant is the product of the eigenvalues.
It is also the standard route to the spectral theorem for normal matrices.
</document>

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# Compact-open topology

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], and let $C(X, Y)$ be the set of [[continuous maps|0VGFHTM75Y]] from $X$ to $Y$.
For a [[compact|0R6EWFKRCF]] subset $K \subseteq X$ and an [[open|02BH3MCBQB]] subset $U \subseteq Y$, put
$$V(K, U) = \set{f \in C(X, Y) : f(K) \subseteq U}.$$
The *compact-open topology* on $C(X, Y)$ is the topology generated by the [[subbasis|09S8DV34Y0]] of all sets $V(K, U)$.

## Remarks

If $(Y, d)$ is a metric space, the compact-open topology is the topology of uniform convergence on compact subsets of $X$.
In particular, a sequence $(f_n)$ converges to $f$ if and only if $\sup_{x \in K} d(f_n(x), f(x)) \to 0$ for every compact $K \subseteq X$.
If $X$ is locally compact Hausdorff, the evaluation map $C(X, Y) \times X \to Y$, $(f, x) \mapsto f(x)$, is continuous.

Example: for $X = [0, 1]$ and $Y = \R$, the space $X$ is compact, so convergence in $C([0, 1], \R)$ is uniform convergence on $[0, 1]$.

Non-example: the topology of pointwise convergence on $C([0, 1], \R)$ is not the compact-open topology.
Let $f_n$ be the piecewise linear map with $f_n(0) = 0$, $f_n(1/n) = 1$ and $f_n(x) = 0$ for $x \ge 2/n$.
Then $f_n(x) \to 0$ for each $x$, but $f_n \notin V([0, 1], (-1, 1))$ for every $n$, so $f_n$ does not converge to $0$ in the compact-open topology.
</document>

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# Completion of a topological group

## Summary

The completion of a [[Hausdorff topological group|7GKCGQ7215]] $G$ is a complete Hausdorff topological group that contains $G$ as a dense subgroup.
Completeness refers to the [[uniform structure|P89SJ7JWJE]] of the group, so it makes sense without a metric.
Every Hausdorff topological group has a completion for its two-sided uniformity, unique up to isomorphism; this is the Raikov completion.
For abelian groups this is the familiar completion with respect to Cauchy nets.

## Explanations

### L1 human

The rational numbers have holes: a sequence of rationals can get closer and closer together without having a rational limit.
Filling in all such holes in the additive group of rationals gives the real numbers.
The completion of a topological group does the same thing in general: it adds limits for all the Cauchy nets, and the group law extends to the new points.
For example, the completion of the integers with the $p$-adic topology is the group of $p$-adic integers.
</document>

<document source="https://tamnd.github.io/math2/node/V1QYA9RW73.md" tag="V1QYA9RW73" kind="statement">
# Schwarz–Pick theorem

## Statement

Let $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ be the open [[unit disc|8G5ABXRK85]], and let $f \colon \mathbb{D} \to \mathbb{D}$ be [[holomorphic|0DG0Z72CQD]].
Then for all $z, w \in \mathbb{D}$,
$$\abs{\frac{f(z) - f(w)}{1 - \overline{f(w)} f(z)}} \le \abs{\frac{z - w}{1 - \bar w z}},$$
and for all $z \in \mathbb{D}$,
$$\frac{\abs{f'(z)}}{1 - \abs{f(z)}^2} \le \frac{1}{1 - \abs{z}^2}.$$
Assume equality holds in the first inequality for some $z \neq w$, or in the second for some $z$.
Then $f$ is a bijective [[Möbius transformation|0B346DDAM5]] of $\mathbb{D}$ onto itself.

## Remarks

Proof idea: for $a \in \mathbb{D}$ let $\varphi_a(z) = \frac{z - a}{1 - \bar a z}$, a Möbius transformation of $\mathbb{D}$ onto itself with $\varphi_a(a) = 0$.
The map $g = \varphi_{f(w)} \circ f \circ \varphi_w^{-1}$ sends $\mathbb{D}$ into $\mathbb{D}$ and fixes $0$.
The [[Schwarz lemma|0MB48XDBY3]] gives $\abs{g(\zeta)} \le \abs{\zeta}$, which is the first inequality with $\zeta = \varphi_w(z)$; letting $z \to w$ gives the second.
In terms of the hyperbolic metric $\frac{2 \abs{dz}}{1 - \abs{z}^2}$ of the [[Poincaré disk model|85MM2EA0TK]], the theorem says that $f$ does not increase hyperbolic distances.
The Möbius transformations of $\mathbb{D}$ onto itself are exactly the hyperbolic isometries that preserve orientation.
Example: $f(z) = z^2$ gives $\abs{2z}/(1 - \abs{z}^4) \le 1/(1 - \abs{z}^2)$, that is $2\abs{z} \le 1 + \abs{z}^2$, which is $(1 - \abs{z})^2 \ge 0$.

## Explanations

### L1 human

The Schwarz lemma is about maps of the disc into itself that fix the center.
The Schwarz–Pick theorem drops the condition at the center.
It says that such a map can only bring points closer together, when distance is measured in the right way.
That right way is the hyperbolic distance, for which points near the edge of the disc are very far apart.

### L2 human

The theorem shows that the hyperbolic metric is the natural metric for holomorphic maps of the disc.
Every holomorphic self-map is a contraction for it, and the automorphisms are the isometries.
This point of view leads to the Ahlfors–Schwarz lemma for metrics of negative curvature and to the Kobayashi metric in several complex variables.
It also gives quick proofs of distortion estimates, for example $\abs{f'(0)} \le 1 - \abs{f(0)}^2$.
</document>

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# Voronoi cell

## Summary

Let $S$ be a discrete set of points in $\R^n$.
The Voronoi cell of $p \in S$ is the set of points that are no farther from $p$ than from any other point of $S$.
The Voronoi cells of the points of $S$ cover space and meet only along their boundaries.
For a [[lattice|WN9V7WSJKF]], all cells are translates of the cell of the origin.
This cell is a centrally symmetric convex polytope whose volume is the covolume of the lattice.
Voronoi cells of lattices are central in lattice packing and covering, in quantization and in decoding.

## Explanations

### L1 human

Imagine post offices at the points of $S$, and let everyone go to the nearest post office.
The Voronoi cell of a post office is the region of people who go to it.
For the integer grid $\Z^2$, the cell of the origin is the square of side $1$ centred at the origin.
For the hexagonal grid of a honeycomb, each cell is a regular hexagon.
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<document source="https://tamnd.github.io/math2/node/V1S1223G5E.md" tag="V1S1223G5E" kind="concept">
# Cointegration

## Summary

Several non-stationary time series are cointegrated when some linear combination of them is stationary.
The series then wander, but they wander together, held by a long-run equilibrium relation.
Clive Granger introduced the idea, and Robert Engle and Granger developed its theory and tests in a 1987 paper.
Cointegration is central in the econometrics of prices, interest rates and other economic series.

## Explanations

### L1 human

Picture a person walking a dog on a long leash through a park.
Each path wanders unpredictably, but the distance between them stays bounded because of the leash.
The two paths are cointegrated, and their difference is the stationary combination.
Two independent walkers without a leash are not: their distance can grow without bound.
</document>

<document source="https://tamnd.github.io/math2/node/V1TBNP8R3S.md" tag="V1TBNP8R3S" kind="concept">
# G-delta set

## Summary

A $G_\delta$ set in a [[topological space|0CQHFPGJ45]] is an intersection of countably many [[open sets|02BH3MCBQB]].
The name comes from the German words for open set and intersection.
In a [[metric space|0VKD1TP7F3]] every closed set is $G_\delta$, and $G_\delta$ sets are the next step after open and closed sets in the Borel hierarchy.
The complements of $G_\delta$ sets are the [[F-sigma sets|8MGC7M64GW]].

## Explanations

### L1 human

One open set may be large, but intersecting countably many open sets can carve out a much thinner set.
The irrational numbers form a $G_\delta$ set: for each rational $q$, remove the single point $q$ from $\R$, which leaves an open set, and intersect all these sets.
On the other hand, the rationals are not $G_\delta$ in $\R$, which follows from the [[Baire category theorem|0A7KGAYHGV]].
</document>

<document source="https://tamnd.github.io/math2/node/V1WRSW2AA6.md" tag="V1WRSW2AA6" kind="definition">
# Z-transform

## Definition

Let $x = (x_n)_{n \in \Z}$ be a [[sequence|0G9EKM0E15]] of complex numbers.
The *Z-transform* of $x$ is the [[Laurent series|0GXMPDNYX3]]
$$X(z) = \mathcal{Z}\{x\}(z) = \sum_{n = -\infty}^{\infty} x_n z^{-n}.$$
It is regarded as a function of $z$ on the set of nonzero complex numbers $z$ where the series converges absolutely.
For a sequence $(x_n)_{n \ge 0}$ indexed by the natural numbers, one sets $x_n = 0$ for $n < 0$, and the transform is $X(z) = \sum_{n = 0}^{\infty} x_n z^{-n}$.

## Remarks

Example: for $x_n = a^n$ when $n \ge 0$ and $x_n = 0$ when $n < 0$, with $a \in \C$, the series is geometric with ratio $a z^{-1}$.
It converges absolutely exactly when $\abs{z} > \abs{a}$, and then $X(z) = z/(z - a)$.
Non-example: for $x_n = 1$ for all $n \in \Z$, the series $\sum_{n \in \Z} z^{-n}$ converges for no $z$.
The part with $n \ge 0$ needs $\abs{z} > 1$, and the part with $n < 0$ needs $\abs{z} < 1$, so the sequence has no Z-transform as a function.
The sum over all $n \in \Z$ is the bilateral form, and the sum over $n \ge 0$ is the unilateral form.
A function $X(z)$ alone does not determine $x$: the set of convergence is part of the data.
Substituting $z = e^{i\omega}$ gives the discrete-time Fourier transform, when the unit circle lies in the set of convergence.
</document>

<document source="https://tamnd.github.io/math2/node/V1X38GS4BX.md" tag="V1X38GS4BX" kind="definition">
# Auslander–Reiten quiver

## Definition

Let $k$ be an algebraically closed field and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $k$.
Let $M$ and $N$ be finitely generated left $A$-modules.
Let $\operatorname{rad}(M, N)$ be the space of morphisms $f \colon M \to N$ such that $h f g$ is never an isomorphism, for $X$ [[indecomposable|E2P29GCD9T]], $g \colon X \to M$ and $h \colon N \to X$.
Let $\operatorname{rad}^2(M, N)$ be the space spanned by the composites $f_2 f_1$ with $f_1 \in \operatorname{rad}(M, Y)$ and $f_2 \in \operatorname{rad}(Y, N)$ for some module $Y$.
The *Auslander–Reiten quiver* of $A$ is the [[quiver|RE9YFFS3TS]] $\Gamma_A$ such that:

1. its vertices are the isomorphism classes $[M]$ of indecomposable finitely generated $A$-modules;
2. the number of arrows $[M] \to [N]$ is $\dim_k \operatorname{rad}(M, N) / \operatorname{rad}^2(M, N)$.

It comes with the partial bijection $\tau$ on vertices, the Auslander–Reiten translation, defined on the classes of non-projective modules.

## Remarks

For indecomposable $M$ and $N$, the nonzero elements of $\operatorname{rad}(M, N) \setminus \operatorname{rad}^2(M, N)$ are exactly the irreducible morphisms.
Let $0 \to \tau N \to E \to N \to 0$ be an [[almost split sequence|M5D95FN7G3]], with $E = \bigoplus_r E_r^{m_r}$ and $E_r$ indecomposable.
Then the arrows ending at $[N]$ start exactly at the classes $[E_r]$, and the arrows starting at $[\tau N]$ end exactly there.
Example: for the path algebra of $1 \to 2$, the indecomposable modules are $S_2 = P_2$, $P_1$ and $S_1$.
The quiver is $[S_2] \to [P_1] \to [S_1]$, with $\tau [S_1] = [S_2]$, coming from the almost split sequence $0 \to S_2 \to P_1 \to S_1 \to 0$.
Non-example: there is no arrow $[S_2] \to [S_1]$, since $\Hom(S_2, S_1) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/V20376HXHD.md" tag="V20376HXHD" kind="statement">
# Wiener's Tauberian theorem

## Statement

Let $f \in L^1(\R)$, the Lebesgue space of integrable functions, and let $\widehat{f}(\xi) = \int_\R f(x) e^{-2\pi i x\xi} \, dx$ be its [[Fourier transform|DPVM34ADZV]].
For $a \in \R$ let $f_a(x) = f(x - a)$.
Then the linear span of the translates $\set{f_a : a \in \R}$ is dense in $L^1(\R)$ if and only if $\widehat{f}(\xi) \ne 0$ for every $\xi \in \R$.

## Remarks

The "only if" part is easy: if $\widehat{f}(\xi_0) = 0$, then every $g$ in the closed span has $\widehat{g}(\xi_0) = 0$, because $\widehat{f_a}(\xi) = e^{-2\pi i a\xi}\widehat{f}(\xi)$ and $g \mapsto \widehat{g}(\xi_0)$ is continuous on $L^1$; but some $g \in L^1$ has $\widehat{g}(\xi_0) \ne 0$.
The "if" part uses [[Wiener's lemma|NC768TWZ3D]] type arguments: closed translation-invariant subspaces of $L^1(\R)$ are ideals for convolution, and an ideal whose transforms have no common zero is dense.
Example: the Gaussian $f(x) = e^{-\pi x^2}$ has $\widehat{f}(\xi) = e^{-\pi\xi^2} > 0$, so linear combinations of its translates approximate every integrable function in $L^1$.
The same statement holds for every [[locally compact abelian group|ZR09S9R3AW]] $G$, with $f \in L^1(G)$ and the [[Fourier transform|VHN5VW5Q92]] on the dual group.
There is also an $L^2$ version: the translates of $f \in L^2(\R)$ span a dense subspace of $L^2(\R)$ if and only if the zero set of $\widehat{f}$ has Lebesgue measure $0$.

## Explanations

### L1 human

Take one integrable function and all its shifted copies.
The theorem tells when sums of these copies can approximate every integrable function.
The answer is: exactly when the function contains every frequency, that is, its Fourier transform never vanishes.

### L2 human

Wiener used the theorem to give general Tauberian theorems: if $\phi$ is bounded and $f * \phi(x) \to c \int f$ as $x \to \infty$ for one $f$ with nonvanishing transform, then the same holds for all $g \in L^1$.
This gives a short route to Tauberian results for summation methods and, through the Wiener–Ikehara theorem, to the prime number theorem.
In Banach algebra language, the theorem says that a closed ideal of $L^1(\R)$ whose Fourier transforms have no common zero is all of $L^1(\R)$.
</document>

<document source="https://tamnd.github.io/math2/node/V21XNZ1H0B.md" tag="V21XNZ1H0B" kind="statement">
# Tverberg's theorem

## Statement

Let $d \ge 1$ and $r \ge 2$ be integers, and let $X \subseteq \R^d$ be a finite set with at least $(r - 1)(d + 1) + 1$ points.
Then $X$ can be [[partitioned|08QMKTWF9T]] into $r$ disjoint sets $X_1, \dots, X_r$ whose [[convex hulls|0FQSDC2EQ2]] have a common point:
$$\operatorname{conv}(X_1) \cap \cdots \cap \operatorname{conv}(X_r) \ne \emptyset.$$

## Remarks

For $r = 2$ the bound is $d + 2$, and the theorem is [[Radon's theorem|888R24ZGNY]].

Example: with $d = 2$ and $r = 3$, any $7$ points in the plane can be split into three groups whose convex hulls share a point.
For instance, take the vertices of a large triangle, the vertices of a smaller triangle with the same centroid $c$, and the point $c$ itself.
The three groups formed by the large triangle, the small triangle and $\set{c}$ all have $c$ in their convex hulls.

The bound $(r - 1)(d + 1) + 1$ is sharp: there are sets of $(r - 1)(d + 1)$ points in $\R^d$ with no such partition.

Proof idea: Tverberg's original proof was a direct geometric argument.
A shorter proof by Sarkaria lifts the points to a space of dimension $(r - 1)(d + 1)$ using tensor products with vectors that sum to $0$, and applies the [[colorful Carathéodory theorem|X5GS407VP8]] there.

## Explanations

### L1 human

Radon's theorem splits enough points into two groups whose shapes overlap.
Tverberg's theorem does the same for any number $r$ of groups, all overlapping in one common point.
In the plane, seven points always split into three overlapping groups, and $3(r - 1) + 1$ points split into $r$ such groups.

### L2 human

A common point of a Tverberg partition is a kind of multidimensional median of the set, which links the theorem to the centerpoint theorem.
The topological Tverberg conjecture asked whether continuous maps of a simplex can replace convex hulls.
It holds when $r$ is a prime power and fails for some other values of $r$.
</document>

<document source="https://tamnd.github.io/math2/node/V24RQP5492.md" tag="V24RQP5492" kind="example">
# 1 − 1 + 2 − 6 + 24 − 120 + ⋯

## Example

The alternating series of factorials is
$$\sum_{k=0}^{\infty} (-1)^k k! = 1 - 1 + 2 - 6 + 24 - 120 + \cdots.$$
Its terms do not tend to $0$, since $k! \to \infty$.
So it is a [[divergent series|0JWMJSTBSG]].

## Remarks

The series is [[Borel summable|RQYX6E42D0]].
Its Borel transform is $\sum_{k \ge 0} \frac{(-1)^k k!}{k!} t^k = \sum_{k \ge 0} (-t)^k = \frac{1}{1 + t}$ for $\abs{t} < 1$.
The function $1/(1+t)$ is analytic on a neighbourhood of $[0, \infty)$, and the Borel sum is
$$\int_0^\infty \frac{e^{-t}}{1 + t} \, dt \approx 0.596347.$$
This number is known as the Gompertz constant.
Euler studied this series and gave it the same value.
The formal series $\sum_k (-1)^k k! \, x^k$ is the asymptotic expansion of $\int_0^\infty \frac{e^{-t}}{1 + x t} \, dt$ as $x \to 0^+$, and the example is this series taken formally at $x = 1$.

## Explanations

### L1 human

The terms of this series grow so fast that the partial sums swing wildly between large positive and large negative numbers.
Still, one can give it a meaningful value.
Divide each term by $k!$ to get the geometric series of $1/(1+t)$, then undo the division with the integral $\int_0^\infty e^{-t} t^k \, dt = k!$.
This gives about $0.5963$.
</document>

<document source="https://tamnd.github.io/math2/node/V25YYE26M1.md" tag="V25YYE26M1" kind="definition">
# Mapping cylinder

## Definition

Let $X$ and $Y$ be topological spaces, let $I = [0, 1]$, and let $f \colon X \to Y$ be a [[continuous map|0VGFHTM75Y]].
The *mapping cylinder* of $f$ is the space
$$M_f = \bigl( (X \times I) \sqcup Y \bigr) / \sim,$$
where $X \times I$ has the [[product topology|0GBG8B49AA]] and $\sqcup$ is the [[disjoint union|ZXJKSB30WF]].
The relation $\sim$ is the smallest equivalence relation with $(x, 1) \sim f(x)$ for all $x \in X$.
The space $M_f$ carries the [[quotient topology|0NQ1H2X495]].

## Remarks

The map $j \colon X \to M_f$, $x \mapsto [(x, 0)]$, is an embedding, and $Y$ embeds in $M_f$ as well.
The map $r \colon M_f \to Y$ with $r[(x, t)] = f(x)$ and $r(y) = y$ is a homotopy equivalence, $Y$ is a strong [[deformation retract|CEK4SC20WK]] of $M_f$, and $f = r \circ j$.
Example: for $Y$ a point, $M_f$ is the cone $CX = (X \times I) / (X \times \set{1})$.
Example: for $f = \id_X$, $M_f$ is homeomorphic to $X \times I$.
Non-example: the space $(X \times I) \sqcup Y$ without the identification is not $M_f$; for $X$ and $Y$ nonempty it is disconnected even when $X$ and $Y$ are connected, while $M_f$ is then connected.
</document>

<document source="https://tamnd.github.io/math2/node/V27468ECYW.md" tag="V27468ECYW" kind="definition">
# Genus

## Definition

Let $S$ be a closed [[orientable|0TD904MG06]] [[surface|VGJJVBZ1MQ]], that is, a compact connected orientable surface without boundary.
Call simple closed curves $c_1, \ldots, c_k$ in $S$ *non-separating* when they are pairwise disjoint and $S \setminus (c_1 \cup \cdots \cup c_k)$ is [[connected|0V032VWTHD]].
The *genus* $g(S)$ is the largest integer $k \ge 0$ for which such curves $c_1, \ldots, c_k$ exist.

## Remarks

The genus is finite and satisfies $\chi(S) = 2 - 2g(S)$, where $\chi$ is the Euler characteristic.
By the classification of surfaces, $S$ is homeomorphic to the sphere if $g = 0$, and to the connected sum of $g$ tori if $g \ge 1$.

Example: for the torus $T^2 = S^1 \times S^1$, the curve $c_1 = S^1 \times \set{1}$ has connected complement $S^1 \times (S^1 \setminus \set{1})$, an open annulus.
Any two disjoint simple closed curves in the torus disconnect it, so $g(T^2) = 1$, in accord with $\chi(T^2) = 0$.
Non-example of a non-separating curve: on the sphere $S^2$ every simple closed curve separates the sphere into two pieces, by the Jordan curve theorem, so $g(S^2) = 0$.

For a closed non-orientable surface, the *non-orientable genus* is the number $k$ of copies of $\R P^2$ in its connected sum decomposition; then $\chi = 2 - k$.

## Explanations

### L1 human

The genus counts how many separate cuts along closed loops a surface can survive while staying in one piece.
Each handle allows exactly one such cut, around the handle.
So the genus is the number of handles.
</document>

<document source="https://tamnd.github.io/math2/node/V288YS2F5W.md" tag="V288YS2F5W" kind="definition">
# Term algebra

## Definition

Let $\sigma$ be a [[signature|WDVVEYD6AT]] with only function symbols and let $X$ be a set of variables.
Let $T_\sigma(X)$ be the set of [[terms|6D3WA7N2W4]] over $\sigma$ and $X$.
Assume that $T_\sigma(X)$ is nonempty, that is, $X$ is nonempty or $\sigma$ has a constant symbol.
The *term algebra* $\mathbf{T}_\sigma(X)$ is the algebra of signature $\sigma$ with universe $T_\sigma(X)$.
Its operations are
$$f^{\mathbf{T}}(t_1, \ldots, t_n) = f(t_1, \ldots, t_n)$$
for each function symbol $f$ of arity $n$ and all terms $t_1, \ldots, t_n$.

## Remarks

The term algebra is absolutely free over $X$.
For every algebra $A$ of signature $\sigma$ and every map $\alpha \colon X \to A$, there is a unique [[homomorphism|KBXSZC6BV1]] $\bar\alpha \colon \mathbf{T}_\sigma(X) \to A$ extending $\alpha$, namely $\bar\alpha(t) = t^A[\alpha]$, the [[value|DH30WTA26T]] of $t$.
When $X$ is empty, it is the algebra of closed terms, also called the Herbrand universe.

Example: let $\sigma$ have one binary symbol $\cdot$ and let $X = \set{x}$.
The elements $x$, $x \cdot x$, $(x \cdot x) \cdot x$, $x \cdot (x \cdot x)$, … are pairwise distinct, and the operation sends $(s, t)$ to the term $s \cdot t$.
This is the free magma on one generator.

Non-example: the positive integers with $\cdot$ read as addition are not isomorphic to this term algebra, even though they are generated by $1$.
In the term algebra, $s \cdot t = s' \cdot t'$ implies $s = s'$ and $t = t'$, because terms have a unique reading.
In the positive integers $1 + 2 = 2 + 1$ while $1 \ne 2$, so this property fails.
</document>

<document source="https://tamnd.github.io/math2/node/V2AA2D23JC.md" tag="V2AA2D23JC" kind="definition">
# Banach–Lie group

## Definition

A *Banach–Lie group* is a [[group|G5S1M9]] $G$ with the structure of a smooth [[Banach manifold|VB95VVVDM8]] such that:

1. The multiplication $G \times G \to G$, $(g, h) \mapsto g h$, is smooth.
2. The inversion $G \to G$, $g \mapsto g^{-1}$, is smooth.

## Remarks

Example: let $E$ be a real [[Banach space|0BDTTM24DP]] and $\mathcal{L}(E)$ the Banach space of bounded linear operators with the operator norm.
The group $GL(E)$ of invertible operators is an open subset of $\mathcal{L}(E)$, so it is a Banach manifold with one chart.
Multiplication is the restriction of a continuous bilinear map, hence smooth, and inversion is smooth because near $A$ it is given by the convergent series $(A - X)^{-1} = \sum_{k \ge 0} (A^{-1} X)^k A^{-1}$.

Non-example: on $\R$ with its standard smooth structure, define $x * y = (x^{1/3} + y^{1/3})^3$.
This is a group, isomorphic to $(\R, +)$ through $x \mapsto x^{1/3}$, but the partial derivative of $x * y$ in $x$ at the point $(0, 1)$ does not exist, so clause 1 fails.

The tangent space at the identity carries a Lie bracket and is a Banach–Lie algebra.
Every finite-dimensional Lie group is a Banach–Lie group.
</document>

<document source="https://tamnd.github.io/math2/node/V2H578D4RT.md" tag="V2H578D4RT" kind="definition">
# Singular values of a compact operator

## Definition

Let $H$ and $K$ be complex [[Hilbert spaces|HM2HHE3J51]], and let $T \colon H \to K$ be a [[compact operator|M2NEP1K5AE]] with [[adjoint|6HSJBJ8TTQ]] $T^*$.
Let $\abs{T} = (T^* T)^{1/2}$ be the [[square root|3JY5HNFRMF]] of the [[positive operator|77020CY5ZW]] $T^* T$; it is compact and self-adjoint.
List the nonzero [[eigenvalues|0BQAA2BK3Q]] of $\abs{T}$ in decreasing order, each repeated as many times as the dimension of its eigenspace.
If there are only finitely many, append zeros.
The resulting sequence $s_1(T) \ge s_2(T) \ge \cdots \ge 0$ is the sequence of *singular values* of $T$.

## Remarks

The list is well defined by the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]], and $s_n(T) \to 0$.
One has $s_1(T) = \norm{T}$, and more generally
$$s_n(T) = \min \set{\norm{T - F} : F \text{ of rank less than } n},$$
the distance from $T$ to the [[finite-rank operators|Z8FABSYQTS]] of rank less than $n$.
There are orthonormal families $(u_n)$ in $H$ and $(v_n)$ in $K$ such that $T x = \sum_n s_n(T) \ip{x}{u_n} v_n$ for all $x \in H$, the Schmidt decomposition of $T$.
Example: if $T$ is diagonal in an orthonormal basis $(e_n)$ of $\ell^2$, with $T e_n = d_n e_n$ and $d_n \to 0$, then $\abs{T} e_n = \abs{d_n} e_n$, so the singular values are the numbers $\abs{d_n}$ arranged in decreasing order.
Non-example: the identity of an infinite-dimensional Hilbert space is not compact, so it has no singular values in this sense; the eigenvalue $1$ of $\abs{I} = I$ has infinite multiplicity.
For matrices, this is the usual notion of [[singular value|09VMDSRXWJ]].
</document>

<document source="https://tamnd.github.io/math2/node/V2HG4GN7TN.md" tag="V2HG4GN7TN" kind="concept">
# Bessel potential

## Summary

The Bessel potential of order $\alpha > 0$ is the operator $(I - \Delta)^{-\alpha/2}$ on functions on $\R^n$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
It is a convolution with a positive integrable kernel $G_\alpha$.
Near the origin $G_\alpha$ behaves like the kernel of the [[Riesz potential|1V2N8MFSHC]].
It decays exponentially at infinity, which removes the problems of the Riesz potential at large scales.
The images of $L^p$ under Bessel potentials are the [[Bessel potential spaces|DVJM7NBSF2]], which give Sobolev spaces of fractional order.

## Explanations

### L1 human

The Bessel potential smooths a function by a fractional amount $\alpha$, much like the Riesz potential.
The difference is the extra $I$ in $(I - \Delta)$, which acts like a mass term in physics and makes the smoothing kernel die off quickly far away.
In one dimension with $\alpha = 2$ the kernel is $\frac12 e^{-\abs{x}}$, so the Bessel potential averages $f$ with an exponentially decaying weight.
This operator solves $u - u'' = f$.
</document>

<document source="https://tamnd.github.io/math2/node/V2HTMG4KT1.md" tag="V2HTMG4KT1" kind="statement">
# Weil's local rigidity theorem

## Statement

Let $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center.
Assume that $G$ has no compact factor and no factor locally isomorphic to $\mathrm{SL}_2(\R)$.
Let $\Gamma$ be a uniform [[lattice|BTYF5Y5PFY]] in $G$, so that $G/\Gamma$ is compact, and let $\iota \colon \Gamma \to G$ be the inclusion.
Give the set $\Hom(\Gamma, G)$ of [[group homomorphisms|0JYTG5K9Y4]] the topology of pointwise convergence.
Then $\iota$ is locally rigid: $\iota$ has a neighbourhood $U$ in $\Hom(\Gamma, G)$ with the following property.
For every $\rho \in U$ there is $g \in G$ with $\rho(\gamma) = g \gamma g^{-1}$ for all $\gamma \in \Gamma$.

## Remarks

A uniform lattice is finitely generated, so the topology on $\Hom(\Gamma, G)$ is that of a real algebraic set, given by the images of finitely many generators.
Weil proved a general criterion for a homomorphism $\rho$ from a finitely generated group $\Gamma$ to $G$.
Let $\Gamma$ act on the Lie algebra $\mathfrak{g}$ of $G$ through $\rho$ and the adjoint representation.
If the first cohomology group $H^1(\Gamma, \mathfrak{g})$ vanishes, then $\rho$ is locally rigid.
The tangent space to deformations of $\rho$ is the space of cocycles, and conjugations give the coboundaries, so vanishing of $H^1$ means that every infinitesimal deformation is a conjugation.
He then proved this vanishing for uniform lattices under the hypotheses above.
The hypothesis on $\mathrm{SL}_2(\R)$ is necessary.
The fundamental group of a closed hyperbolic surface sits in $\mathrm{PSL}_2(\R)$ as a uniform lattice, and it has a continuous family of non-conjugate deformations, described by [[Teichmüller space|ZZGC6BZNJ5]].
Garland and Raghunathan extended the result to non-uniform lattices in groups of real rank one other than those locally isomorphic to $\mathrm{SL}_2(\R)$ and $\mathrm{SL}_2(\C)$.

## Explanations

### L1 human

A uniform lattice is a discrete group of symmetries that tiles a big continuous group with compact tiles.
The theorem says that one cannot wiggle such a lattice slightly to get a genuinely new one.
Every small change of the lattice is just the same lattice seen from a different point, that is, conjugated by an element of $G$.
The only exception is the group $\mathrm{SL}_2(\R)$ and its relatives, where surfaces can be deformed.

### L2 human

Local rigidity is the first of a chain of rigidity results for lattices.
[[Mostow rigidity|4BEWKR0EG7]] upgrades it to a global statement: an abstract isomorphism between lattices extends to the ambient groups.
[[Margulis superrigidity|DSKV65NHZZ]] goes further for higher rank, by controlling all homomorphisms of a lattice into other groups.
</document>

<document source="https://tamnd.github.io/math2/node/V2QQKNHVEB.md" tag="V2QQKNHVEB" kind="example">
# Pin group

## Example

Let $n \ge 1$, let $\varepsilon \in \set{1, -1}$, and let $\mathrm{Cl}$ be the real [[Clifford algebra|A38B1QFK20]] of $\R^n$ with the relations $v^2 = \varepsilon \abs{v}^2$ for $v \in \R^n$.
Let $\alpha$ be the automorphism of $\mathrm{Cl}$ with $\alpha(v) = -v$ for $v \in \R^n$.
The *pin group* is the set of all products $u_1 u_2 \cdots u_k$ in $\mathrm{Cl}$ of unit vectors $u_i \in \R^n$, with any number $k \ge 0$ of factors.
It is a group under the Clifford product, since $u_i^{-1} = \varepsilon u_i$.

For $x$ in the pin group and $v \in \R^n$, set $\rho(x)(v) = \alpha(x) v x^{-1}$.
Check for one unit vector $u$: $uv = -vu$ when $v \perp u$, so $\rho(u)(v) = -u v u^{-1}$ equals $v$ for $v \perp u$ and $-u$ for $v = u$.
So $\rho(u)$ is the reflection of $\R^n$ in the hyperplane $u^\perp$.
Since every element of the [[orthogonal group|0TKYAEJ8N7]] $O(n)$ is a product of reflections, $\rho$ is a surjective homomorphism onto $O(n)$, and its kernel is $\set{1, -1}$.

## Remarks

The two signs $\varepsilon = 1$ and $\varepsilon = -1$ give two groups, written $\mathrm{Pin}^+(n)$ and $\mathrm{Pin}^-(n)$, which are not isomorphic in general.
For $n = 1$, with unit vectors $\pm e_1$, one sign gives $\set{\pm 1, \pm e_1}$ with $e_1^2 = 1$, a Klein four-group, and the other gives the same set with $e_1^2 = -1$, a cyclic group of order $4$.
In both cases the products of an even number of unit vectors form the [[spin group|JJSXXJQGEH]] $\mathrm{Spin}(n)$, a subgroup of index $2$.
For $n \ge 2$ the pin groups are compact Lie groups with two connected components, and they are double covers of $O(n)$.

## Explanations

### L1 human

The pin group is to the full orthogonal group what the spin group is to the rotation group.
It doubles the group of all rotations and reflections, so that every such map corresponds to two elements that differ by a sign.
A single unit vector of the Clifford algebra stands for the reflection in the plane perpendicular to it.
Because a reflection can be lifted with either sign convention, there are two different pin groups.
</document>

<document source="https://tamnd.github.io/math2/node/V2S7CHGS26.md" tag="V2S7CHGS26" kind="concept">
# Tight frame

## Summary

A frame of a [[Hilbert space|HM2HHE3J51]] is a family of vectors that recovers every vector stably from its inner products with the family.
Unlike a basis, a frame may be redundant.
A tight frame is a frame whose lower and upper frame bounds are equal.
Then reconstruction is as simple as for an [[orthonormal basis|0S9GGTR0EX]], up to a constant factor.
Tight frames are used in signal processing, coding and quantum information, where redundancy gives robustness.

## Explanations

### L1 human

In the plane, take three unit vectors at angles of $120^\circ$ to each other, like the Mercedes-Benz star.
They are not a basis, since there are three of them in a two-dimensional space.
Still, for every vector $f$ the sum of the squares of its three components along them is exactly $\tfrac32 \norm{f}^2$.
This is the defining property of a tight frame: the energy of $f$ is spread evenly over the frame, whatever direction $f$ has.
</document>

<document source="https://tamnd.github.io/math2/node/V2SBN1FNQV.md" tag="V2SBN1FNQV" kind="concept">
# Epsilon-net of a set system

## Summary

An $\varepsilon$-net of a [[family of sets|0C21Q694AG]] is a subset of the ground set that meets every large member of the family.
Here large means that the member holds at least an $\varepsilon$ fraction of the points.
It is a small sample that is guaranteed to hit every heavy range.
The $\varepsilon$-net theorem of Haussler and Welzl says that families of [[VC dimension|KCFXR9K3GD]] $d$ have $\varepsilon$-nets of size $O\big(\frac{d}{\varepsilon} \log \frac{d}{\varepsilon}\big)$, independent of the number of points.
This makes $\varepsilon$-nets a central tool in computational geometry and learning theory.

## Explanations

### L1 human

Suppose there are many points in the plane and many possible regions, such as all half-planes.
An $\varepsilon$-net is a handful of the points chosen so that every region containing a fair share of all points contains at least one chosen point.
For example, take $100$ points on a line, with intervals as the ranges.
The points in positions $10, 20, \dots, 100$ form a $1/10$-net: any interval with at least $10$ of the points contains one of these.
A random sample of the right size works with high probability when the regions are simple.
This is unrelated to the $\varepsilon$-nets of metric spaces, which cover a space by small balls.
</document>

<document source="https://tamnd.github.io/math2/node/V2YCGHD719.md" tag="V2YCGHD719" kind="statement">
# Lewy extension theorem

## Statement

Let $n \ge 2$ and let $M$ be a smooth real hypersurface in $\C^n$.
Let $p \in M$, and let $\rho$ be a smooth real function on a neighbourhood $V$ of $p$ with $M \cap V = \set{\rho = 0}$ and $d\rho \ne 0$ on $M \cap V$.
Assume that the [[Levi form|80MTNXWW80]] of $\rho$ at $p$, restricted to the complex tangent space of $M$ at $p$, has at least one positive eigenvalue.
Then there is a neighbourhood $U \subset V$ of $p$ with the following property.
Let $f$ be a [[CR function|JFKYYK2CEB]] of class $C^1$ on $M \cap V$.
Then there is a unique continuous function $F$ on $U \cap \set{\rho \le 0}$ with $F = f$ on $M \cap U$ that is [[holomorphic|507C34W0ZT]] on $U \cap \set{\rho < 0}$.

## Remarks

The theorem is due to Hans Lewy.
One proof fills the side $\set{\rho < 0}$ near $p$ with small analytic discs whose boundaries lie in $M$.
The values of $f$ on the boundary circles define $F$ by the Cauchy integral, and the CR equations make $F$ holomorphic.
If the Levi form at $p$ has eigenvalues of both signs, extensions exist on both sides.
Then $f$ is the restriction of a function holomorphic on a full neighbourhood of $p$.
For the unit sphere with $\rho = \abs{z}^2 - 1$, the Levi form is positive, so CR functions extend locally into the ball.
The hypothesis on the Levi form is needed.
On the [[Levi-flat hypersurface|S4842D7MAE]] $\set{\operatorname{Im} z_1 = 0}$, the function $f(z) = g(\operatorname{Re} z_1)$ is CR for every real $C^1$ function $g$.
If $f$ extended holomorphically to one side, the Schwarz reflection principle in $z_1$ would make $g$ real-analytic, so a $g$ that is $C^1$ but not real-analytic gives no extension.

## Explanations

### L1 human

A CR function lives only on a hypersurface, but it satisfies the part of the Cauchy–Riemann equations that makes sense there.
The theorem says that when the hypersurface curves in a complex direction, such a function is automatically the boundary value of a holomorphic function on one side.
So the data on the thin hypersurface secretly carry a holomorphic function on an open set.

### L2 human

The Lewy extension theorem is a local form of the Hartogs phenomenon for CR functions.
It turns questions about CR functions into questions about holomorphic functions on one-sided neighbourhoods.
It is the starting point of the extension theory of CR functions, which treats submanifolds of higher codimension as well.
A global analogue for the boundary of a bounded domain with connected boundary is the [[Hartogs–Bochner extension theorem|8QYVM1JTMF]].
</document>

<document source="https://tamnd.github.io/math2/node/V313FN8HRW.md" tag="V313FN8HRW" kind="statement">
# Homotopy invariance of homology

## Statement

Let $X$ and $Y$ be topological spaces and let $f, g \colon X \to Y$ be continuous maps that are [[homotopic|F2F143KQ2M]].
Then the induced maps on [[singular homology|M0Q5CFJ7ZH]] are equal: $f_* = g_* \colon H_n(X) \to H_n(Y)$ for every $n \ge 0$.
In particular, a [[homotopy equivalence|XZG6MTSWCN]] $f \colon X \to Y$ induces isomorphisms $f_* \colon H_n(X) \to H_n(Y)$ for every $n$.

## Remarks

The proof cuts the prism $\Delta^n \times [0, 1]$ into $(n+1)$-simplices.
A homotopy $H \colon X \times [0,1] \to Y$ from $f$ to $g$ then gives a prism operator $P \colon C_n(X) \to C_{n+1}(Y)$ with $\partial P + P \partial = g_\# - f_\#$.
So $f_\#$ and $g_\#$ are [[chain homotopic|4EBHAWJJ8C]], and chain homotopic maps induce the same map on homology.
The second claim follows by applying the first to $g \circ f \simeq \id_X$ and $f \circ g \simeq \id_Y$.

Example: a [[contractible space|R5WP8Z69Q8]] $X$, such as $\R^n$ or a convex set, is homotopy equivalent to a point.
Hence $H_0(X) \cong \Z$ and $H_n(X) = 0$ for $n \ge 1$.
The same holds with any coefficients and for relative homology of pairs, with homotopies of pairs.

## Explanations

### L1 human

Homology does not change when a space or a map is deformed continuously.
Two spaces that can be squeezed into each other without tearing, such as a thick annulus and a circle, have the same homology.
This is what makes homology a tool for shapes up to deformation rather than for exact geometry.

### L2 human

This is the homotopy axiom of the Eilenberg–Steenrod axioms for singular homology.
It lets one replace a space by a simpler homotopy equivalent one, for example a deformation retract, before computing.
It also shows that singular homology factors through the homotopy category of spaces.
Together with excision, the long exact sequence of a pair and the homology of a point, it is enough to compute the homology of spheres and of CW complexes.
</document>

<document source="https://tamnd.github.io/math2/node/V365PW1HNE.md" tag="V365PW1HNE" kind="definition">
# Caccioppoli set

## Definition

Let $n \ge 1$ and let $\Omega \subseteq \R^n$ be an open set.
A [[Lebesgue measurable|2Z2QVWZ262]] set $E \subseteq \R^n$ is a *Caccioppoli set*, or a *set of finite perimeter*, in $\Omega$ if its [[perimeter|Y68BHXZJBV]] $P(E, \Omega)$ is finite.
The set $E$ has *locally finite perimeter* in $\Omega$ if $P(E, U) < \infty$ for every open set $U$ whose closure is a compact subset of $\Omega$.

## Remarks

Example: the unit ball $B \subseteq \R^n$ is a Caccioppoli set in $\R^n$.
Its boundary is smooth, so $P(B) = \mathcal{H}^{n-1}(\partial B)$, the area of the unit sphere, which is finite.

Non-example: the half-space $H = \set{x \in \R^n : x_n > 0}$ with $n \ge 2$ has $P(H) = \infty$, since its boundary is a whole hyperplane.
It still has locally finite perimeter, since $P(H, U)$ is the area of the part of the hyperplane inside $U$.

Equivalently, $E$ has finite perimeter in $\Omega$ exactly when the indicator function $\chi_E$ is a function of [[bounded variation|9YQ999QHHT]] in $\Omega$.
Then the distributional gradient $D \chi_E$ is a vector-valued [[Radon measure|HZNEQQBCW4]], and $P(E, \Omega) = \abs{D\chi_E}(\Omega)$.
</document>

<document source="https://tamnd.github.io/math2/node/V389A4WGH9.md" tag="V389A4WGH9" kind="concept">
# Heap

## Summary

A heap is a set with a [[ternary operation|WC47MFR6GW]] $[x, y, z]$ that behaves like $x y^{-1} z$ in a [[group|G5S1M9]].
It is a group in which no element has been chosen as the identity.
Every nonempty heap becomes a group as soon as one element is chosen to play the identity, and the groups for different choices are isomorphic.
Heaps describe structures without a natural identity, such as the set of bijections between two sets of the same size.
They are closely related to [[torsors|5CYEQY4SWR]].

## Explanations

### L1 human

Think of the points of a plane, where nothing marks an origin.
You cannot add two points, but from three points $x$, $y$, $z$ you can form $x - y + z$: move $z$ by the arrow that goes from $y$ to $x$.
A heap keeps exactly this three-point operation.
For example, the set of bijections from a set $A$ to a set $B$ is a heap with $[f, g, h] = f \circ g^{-1} \circ h$, though there is no natural "identity bijection" from $A$ to $B$.
</document>

<document source="https://tamnd.github.io/math2/node/V3BFGKBJF7.md" tag="V3BFGKBJF7" kind="concept">
# Full subcategory

## Summary

A full subcategory of a [[category|A2DE66937D]] $\mathcal{C}$ is a [[subcategory|D8C0EYD9ED]] that keeps every morphism of $\mathcal{C}$ between any two of its objects.
So it is fixed by its class of objects alone.
Its inclusion functor is [[fully faithful|GRG8WEBSH5]].
Most categories of "special" objects, such as abelian groups among groups or Hausdorff spaces among topological spaces, are full subcategories.

## Explanations

### L1 human

To form a full subcategory, choose some of the objects and keep all arrows between them.
Nothing about the arrows is decided: once the objects are chosen, the arrows come along.
For example, the finite sets with all functions between them form a full subcategory of the category of sets.
Keeping all sets but only the injective functions gives a subcategory that is not full.
</document>

<document source="https://tamnd.github.io/math2/node/V3G5YQTQ00.md" tag="V3G5YQTQ00" kind="statement">
# Eilenberg–Zilber theorem

## Statement

Let $X$ and $Y$ be topological spaces with singular [[chain complexes|WQ82C0M3QB]] $C_\bullet(X)$ and $C_\bullet(Y)$.
Let $C_\bullet(X) \otimes C_\bullet(Y)$ be the tensor product chain complex, with $(C_\bullet(X) \otimes C_\bullet(Y))_n = \bigoplus_{i + j = n} C_i(X) \otimes_{\Z} C_j(Y)$ and $\partial(a \otimes b) = \partial a \otimes b + (-1)^i a \otimes \partial b$ for $a \in C_i(X)$.
Then there are [[chain maps|HTK9WQZJ1G]]
$$C_\bullet(X \times Y) \rightleftarrows C_\bullet(X) \otimes C_\bullet(Y),$$
natural in $X$ and $Y$, that are inverse to each other up to [[chain homotopy|4EBHAWJJ8C]].
In particular, $H_n(X \times Y) \cong H_n\big(C_\bullet(X) \otimes C_\bullet(Y)\big)$ for every $n$.

## Remarks

A standard choice of maps is the Alexander–Whitney map from $C_\bullet(X \times Y)$ and the shuffle map into it.
The existence and uniqueness up to homotopy of such natural maps follows from the method of acyclic models.
Both functors are free on the models $(\Delta^p, \Delta^q)$ and acyclic on them, because $\Delta^p \times \Delta^q$ is contractible.

Example: with the algebraic [[Künneth theorem|77K0BZQ4MK]], the theorem gives the topological Künneth formula.
Over a field $F$, $H_n(X \times Y; F) \cong \bigoplus_{i + j = n} H_i(X; F) \otimes_F H_j(Y; F)$.
For the torus $S^1 \times S^1$ this gives $H_1 \cong F^2$ and $H_2 \cong F$.

## Explanations

### L1 human

The product of two spaces, such as a square from two intervals, can be cut into pieces in two ways.
One can use simplices of the product, or products of a simplex of each factor.
The theorem says that both ways give the same homology.
So the homology of a product can be computed from the chains of the two factors.

### L2 human

The theorem is the bridge between the homology of a product and the algebra of tensor products of chain complexes.
Combined with the Künneth theorem of homological algebra, it computes the homology of products.
The Alexander–Whitney map, composed with the diagonal $X \to X \times X$, also gives the cup product on singular cohomology.
</document>

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# Mapping cone

## Definition

Let $\mathcal{A}$ be an [[additive category|B2VP96NJSK]] and let $f \colon C_\bullet \to D_\bullet$ be a [[chain map|HTK9WQZJ1G]] of [[chain complexes|WQ82C0M3QB]] in $\mathcal{A}$, with differentials $d^C$ and $d^D$.
The *mapping cone* of $f$ is the chain complex $\mathrm{cone}(f)$ with
$$\mathrm{cone}(f)_n = C_{n-1} \oplus D_n,$$
the [[biproduct|BZEJKTQ76T]], and with differential $\mathrm{cone}(f)_n \to \mathrm{cone}(f)_{n-1}$ given in matrix form by
$$d(c, y) = \left(-d^C_{n-1}(c),\ f_{n-1}(c) + d^D_n(y)\right).$$

## Remarks

The condition $d^2 = 0$ holds: $d^2(c, y) = \left(d^C d^C c,\ -f d^C c + d^D f c + d^D d^D y\right) = (0, 0)$, because $f$ is a chain map.
The inclusion $D_\bullet \to \mathrm{cone}(f)$ and the projection $\mathrm{cone}(f) \to C[-1]_\bullet$, where $C[-1]_n = C_{n-1}$ with differential $-d^C$, form a degreewise split short exact sequence of complexes.
In an abelian category, its long exact homology sequence has connecting maps equal to $H_{n-1}(f)$ up to sign.
So $f$ is a [[quasi-isomorphism|C9JEC2WQKV]] if and only if $\mathrm{cone}(f)$ has zero homology.
Sign conventions differ between texts; they give isomorphic complexes.

Example: let $C_\bullet = D_\bullet = \Z$ in degree $0$ and let $f_0$ be multiplication by $2$.
Then $\mathrm{cone}(f)_1 = \Z$, $\mathrm{cone}(f)_0 = \Z$, and the differential is $c \mapsto 2c$, so $H_0(\mathrm{cone}(f)) = \Z/2\Z \ne 0$ and $f$ is not a quasi-isomorphism.

Non-example: dropping the sign, $d(c, y) = (d^C c, f c + d^D y)$, does not give a complex in general.
For $C_\bullet = D_\bullet$ with $C_1 = C_0 = \Z$, $d_1 = 1_{\Z}$, and $f$ the identity, the element $(c, 0)$ in degree $2$ goes to $(c, c)$ and then to $(0, 2c) \ne 0$.
</document>

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# No small subgroups

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] with identity $e$.
The group $G$ has *no small subgroups* if some [[neighbourhood|0ZCY0Z80B6]] $U$ of $e$ has this property.
The only [[subgroup|0X1JXT47GA]] $H$ of $G$ with $H \subseteq U$ is $H = \set{e}$.

## Remarks

Example: $G = (\R, +)$ with $U = (-1, 1)$.
If $H \subseteq U$ is a subgroup and $x \in H$ with $x \ne 0$, then $nx \in H$ for every integer $n$, and $\abs{nx} \ge 1$ for $n \ge 1/\abs{x}$, so $nx \notin U$.
Hence $H = \set{0}$.
The same argument with the exponential map shows that every [[Lie group|7JBRGCVJRB]] has no small subgroups.

Non-example: the group $\Z_p$ of [[p-adic integers|V6TJAFZ4MD]].
Every neighbourhood of $0$ contains a ball $p^n \Z_p$ for some $n$, and $p^n \Z_p$ is a subgroup different from $\set{0}$.

The property is often abbreviated NSS.
By the Gleason–Yamabe theorem, a locally compact group has no small subgroups if and only if it is a Lie group.
</document>

<document source="https://tamnd.github.io/math2/node/V3QJFN5800.md" tag="V3QJFN5800" kind="concept">
# Causal inference

## Summary

Causal inference is the part of statistics that asks what would happen to an outcome if one intervened on a variable.
This differs from asking which values tend to occur together.
It makes precise the difference between correlation and causation, through frameworks such as the [[Rubin causal model|23ATBZMJHR]] of potential outcomes and causal graphs.
In a randomised experiment causal effects can be estimated by simple comparisons, thanks to [[randomization|734Q6Q51X7]].
With observational data one needs extra assumptions, such as the absence of unmeasured confounders, or special designs such as [[instrumental variables|6ETK64HYK6]].

## Explanations

### L1 human

People who carry umbrellas get wet more often than people who do not, but umbrellas do not cause rain.
Causal inference studies when data can tell us what a change would actually do, for example whether a drug lowers blood pressure.
The key problem is that each person either takes the drug or does not, so we never see both outcomes for the same person.
A coin flip that decides who gets the drug makes the two groups comparable, so the difference of their averages measures the effect.
Without such a coin flip, hidden differences between the groups can create or hide an effect, as in [[Simpson's paradox|T060JHSH05]].
</document>

<document source="https://tamnd.github.io/math2/node/V3RCQ1J2VV.md" tag="V3RCQ1J2VV" kind="definition">
# Polar polytope

## Definition

Let $P \subset \R^d$ be a [[convex polytope|300K6CB1KH]] whose [[interior|0RZH6W98HK]] contains the origin $0$.
Let $x \cdot y$ denote the [[dot product|0S52TNNTH9]] on $\R^d$.
The *polar polytope* of $P$ is
$$P^\circ = \set{y \in \R^d : x \cdot y \le 1 \text{ for all } x \in P}.$$

## Remarks

If $v_1, \dots, v_m$ are the vertices of $P$, then $P^\circ = \set{y : v_i \cdot y \le 1, \ i = 1, \dots, m}$.
$P^\circ$ is again a convex polytope with $0$ in its interior, and $(P^\circ)^\circ = P$.
The map $F \mapsto \set{y \in P^\circ : x \cdot y = 1 \text{ for all } x \in F}$ is an inclusion-reversing bijection from the [[faces|KY7SB854VH]] of $P$ to the faces of $P^\circ$, sending $k$-faces to $(d - 1 - k)$-faces.

Example: let $P = [-1, 1]^2$.
Its vertices are $(\pm 1, \pm 1)$, so $y \in P^\circ$ means $\pm y_1 \pm y_2 \le 1$ for all signs, that is $\abs{y_1} + \abs{y_2} \le 1$.
So $P^\circ$ is the square with vertices $(\pm 1, 0)$ and $(0, \pm 1)$.

Non-example: for $P = [0,1]^2$ the origin is a vertex, not an interior point, so the definition does not apply.
The same formula gives the unbounded set $\set{y : y_1 \le 1, \ y_2 \le 1, \ y_1 + y_2 \le 1}$, which is not a polytope.
</document>

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# Solvable Lie group

## Definition

Let $G$ be a connected real or complex [[Lie group|7JBRGCVJRB]].
Its [[derived series|0Q5G5DJ4P2]] is $G^{(0)} = G$ and $G^{(k+1)} = [G^{(k)}, G^{(k)}]$ for $k \ge 0$, where $[H, H]$ is the subgroup generated by the commutators $h h' h^{-1} h'^{-1}$ with $h, h' \in H$.
The group $G$ is a *solvable Lie group* if $G^{(k)} = \set{e}$ for some $k \ge 0$, that is, if $G$ is a [[solvable group|0FXJ7YFMMX]].

## Remarks

A connected Lie group is solvable if and only if its Lie algebra $\mathfrak{g}$ is a solvable Lie algebra, that is, its derived series $\mathfrak{g} \supseteq [\mathfrak{g}, \mathfrak{g}] \supseteq \cdots$ reaches $0$.

Example: the group $G$ of affine maps $x \mapsto a x + b$ of $\R$ with $a > 0$ and $b \in \R$, written $(a, b)$, with $(a, b)(a', b') = (a a', a b' + b)$.
It is connected, being homeomorphic to $(0, \infty) \times \R$.
The commutator of $(a, 0)$ and $(1, t)$ is $(1, (a - 1)t)$, and every commutator has first entry $1$, so $G^{(1)}$ is the abelian group of translations $(1, b)$.
Then $G^{(2)} = \set{e}$, and $G$ is solvable.

Non-example: $G = SL(2, \R)$.
Its Lie algebra $\mathfrak{sl}(2, \R)$ satisfies $[\mathfrak{sl}(2, \R), \mathfrak{sl}(2, \R)] = \mathfrak{sl}(2, \R)$, since the basis elements $H, E, F$ with $[H, E] = 2E$, $[H, F] = -2F$, $[E, F] = H$ are all brackets.
So the derived series of the Lie algebra never reaches $0$, and $G$ is not solvable.
</document>

<document source="https://tamnd.github.io/math2/node/V3YTF5WJHY.md" tag="V3YTF5WJHY" kind="statement">
# Lerch's uniqueness theorem

## Statement

Let $f, g \colon [0, \infty) \to \C$ be [[Lebesgue integrable|DWJF7ABPR1]] on each interval $[0, T]$.
Assume that the Laplace integrals of $f$ and $g$ converge at some real number $s_0$.
Let $F$ and $G$ be their [[Laplace transforms|05XT5AVJBK]].
If $F(s) = G(s)$ for every real $s > s_0$, then $f = g$ almost everywhere on $[0, \infty)$.
If in addition $f$ and $g$ are continuous, then $f = g$ everywhere on $[0, \infty)$.

## Remarks

By linearity it suffices to show that $h = f - g$ vanishes almost everywhere when its transform vanishes for $s > s_0$.
Let $\varphi(t) = \int_0^t e^{-s_0 \tau} h(\tau) \, d\tau$; it is continuous and bounded, and integration by parts turns $H(s_0 + n) = 0$, $n = 1, 2, \ldots$, into $\int_0^\infty e^{-n t} \varphi(t) \, dt = 0$.
The substitution $x = e^{-t}$ turns these into $\int_0^1 x^{n-1} \psi(x) \, dx = 0$ for a continuous function $\psi$, so all moments of $\psi$ vanish.
By the Weierstrass approximation theorem $\psi = 0$, hence $\varphi = 0$ and $h = 0$ almost everywhere.
The proof shows that equality at the points $s_0 + n$, $n = 1, 2, \ldots$, is already enough.
Without continuity the conclusion "almost everywhere" cannot be improved: changing $f$ at one point does not change $F$.

## Explanations

### L1 human

Two functions with the same Laplace transform are essentially the same function.
So, when one finds a function with a given transform, one has found the only reasonable one.

### L2 human

This theorem is what makes the [[inverse Laplace transform|07WBTX1MHA]] well defined, up to changes on null sets.
It justifies the table method: once $F$ matches a known transform, the original function is determined.
It does not by itself give a formula for $f$; that is the job of the [[Bromwich integral|5KNAXNHRY6]] and of [[Post's inversion formula|D6QKPBS2EN]].
</document>

<document source="https://tamnd.github.io/math2/node/V3ZBMRKC08.md" tag="V3ZBMRKC08" kind="statement">
# Regular local rings are domains

## Statement

Let $(R, \mathfrak m)$ be a [[regular local ring|63XXTF6B5M]].
Then $R$ is an [[integral domain|0QVD7WBFGN]].

## Remarks

Proof idea: let $d = \dim R$ and $k = R/\mathfrak m$.
A regular system of parameters $x_1, \ldots, x_d$ gives a surjection $k[X_1, \ldots, X_d] \to \operatorname{gr}_{\mathfrak m} R = \bigoplus_n \mathfrak m^n / \mathfrak m^{n+1}$ onto the [[associated graded ring|8HFAEGFQT1]].
Comparing dimensions shows that this surjection is an isomorphism, so $\operatorname{gr}_{\mathfrak m} R$ is a [[polynomial ring|0SDC934R52]], hence a domain.
By the [[Krull intersection theorem|1Q1AH2V41S]], $\bigcap_n \mathfrak m^n = 0$, so each nonzero $a \in R$ has an order $n$ with $a \in \mathfrak m^n \setminus \mathfrak m^{n+1}$.
The leading forms of two nonzero elements multiply to a nonzero form, so their product is nonzero.

Example: $k[\![x, y]\!]$ and $\Z_{(p)}[x]_{(p, x)}$ are regular local rings, hence domains.

The hypothesis $\dim R = \operatorname{edim} R$ cannot be dropped.
The Noetherian local ring $R = k[x, y]_{(x, y)}/(xy)$ has dimension $1$ and [[embedding dimension|DP3KSCRKFP]] $2$, and it is not a domain, since $x \cdot y = 0$ with $x, y \ne 0$.

## Explanations

### L1 human

A regular local ring describes a smooth point.
Near a smooth point a space cannot be made of two pieces that cross, so the ring has no zero divisors.
The example $xy = 0$, two lines crossing at the origin, is not smooth at the crossing point, and there $x \cdot y = 0$.

### L2 human

The statement is the first step in the theory of regular local rings.
The stronger theorem that [[regular local rings are UFDs|1HD3EK5392]] uses it together with homological tools such as [[Serre's theorem on regular local rings|93RQRG9R1P]].
It also implies that a regular local ring of dimension $1$ is a [[discrete valuation ring|MGM3CNJW9V]].
</document>

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# Cubic spline

## Summary

A cubic spline is a [[spline|84TJ8KQ926]] of degree $3$.
It is made of cubic polynomial pieces whose values, first derivatives and second derivatives agree at the knots.
It is the most used spline, because it is the lowest degree whose graph looks smooth to the eye, with continuous curvature.
A natural cubic spline also has zero second derivative at both ends.
In [[spline interpolation|D59HBKKRP9]], the natural cubic spline is the interpolant with the [[least total squared second derivative|28C5WWGX9J]], the mathematical form of the bent draftsman's strip.

## Explanations

### L1 human

A cubic spline is a chain of cubic curves joined end to end.
At each joint the two neighbouring pieces have the same height, the same slope and the same bending, so no corner or sudden change of curvature is visible.
For example, the function equal to $0$ for $x \le 0$ and to $x^3$ for $x \ge 0$ is a cubic spline with one knot at $0$.
Cubic splines are used to draw smooth curves through data, to design car bodies and fonts, and to approximate functions.
The [[definition|B7YND5ZVMF]] lists the conditions.
</document>

<document source="https://tamnd.github.io/math2/node/V465WCM25T.md" tag="V465WCM25T" kind="definition">
# Lattice congruence

## Definition

Let $(L, \vee, \wedge)$ be a [[lattice|WH4SFAACAS]].
A *congruence* on $L$ is an [[equivalence relation|0G8VEZ3WTK]] $\theta$ on $L$ such that, for all $x, y, x', y' \in L$ with $x \mathrel{\theta} y$ and $x' \mathrel{\theta} y'$:

1. $(x \vee x') \mathrel{\theta} (y \vee y')$;
2. $(x \wedge x') \mathrel{\theta} (y \wedge y')$.

## Remarks

The quotient set $L / \theta$ is a lattice with $[x] \vee [y] = [x \vee y]$ and $[x] \wedge [y] = [x \wedge y]$; clauses 1 and 2 say that these operations are well defined.
The kernel $\set{(x, y) : f(x) = f(y)}$ of a [[lattice homomorphism|BNVRR925X8]] $f$ is a congruence, and every congruence $\theta$ is the kernel of the quotient map $L \to L / \theta$.
Each class of a congruence is a convex sublattice of $L$.
Example: on the chain $0 < 1 < 2$, the partition $\set{0}, \set{1, 2}$ is a congruence.
It is the kernel of the monotone map $0 \mapsto 0$, $1 \mapsto 1$, $2 \mapsto 1$ onto the chain $0 < 1$.
This map is a lattice homomorphism because join and meet in a chain are $\max$ and $\min$.
Non-example: on the lattice $\set{0, a, b, 1}$ with $0 < a, b < 1$ and $a$, $b$ incomparable, the partition $\set{0, 1}, \set{a}, \set{b}$ is not a congruence.
From $0 \mathrel{\theta} 1$ and $a \mathrel{\theta} a$, clause 1 would give $(0 \vee a) \mathrel{\theta} (1 \vee a)$, that is $a \mathrel{\theta} 1$, which is false.
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<document source="https://tamnd.github.io/math2/node/V4KW8XH2GF.md" tag="V4KW8XH2GF" kind="definition">
# Trace of an integral operator

## Definition

Let $a < b$ and let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]].
Let $T$ be the integral operator $(T x)(s) = \int_a^b K(s, t) \, x(t) \, dt$ of an [[integral equation|R3SG1SN7JQ]] with kernel $K$.
The *trace* of $T$, or of the kernel $K$, is the number
$$\operatorname{tr} K = \int_a^b K(s, s) \, ds.$$

## Remarks

Example: for $K(s, t) = \min(s, t)$ on $[0, 1]$, the kernel is continuous and $\operatorname{tr} K = \int_0^1 s \, ds = 1/2$.
Non-example: the weakly singular kernel $K(s, t) = \abs{s - t}^{-1/2}$ on $[0, 1]$ is not continuous, and it is infinite on the diagonal, so the formula gives no number.
For a degenerate kernel $K(s, t) = \sum_{i=1}^n u_i(s) v_i(t)$, the trace is $\sum_{i=1}^n \int_a^b u_i(s) v_i(s) \, ds$.
This is the [[trace|0XC1RDBY7A]] of the $n \times n$ matrix with entries $\int_a^b v_i(s) u_j(s) \, ds$, which represents $T$ on the span of $u_1, \dots, u_n$.
The traces $\int_a^b K_m(s, s) \, ds$ of the iterated kernels $K_m$, $m \ge 2$, appear in the power series of the Fredholm determinant.
A continuous kernel need not define a [[trace class operator|Z2CP530WDE]] on $L^2(a, b)$; when it does, the number above equals the operator trace.
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<document source="https://tamnd.github.io/math2/node/V4M2HPJQPR.md" tag="V4M2HPJQPR" kind="problem">
# Complex structure on the six-sphere

## Problem

Decide whether the sphere $S^6$ admits a complex structure.
That is, decide whether there is a [[complex manifold|JSGWAN2RJV]] of complex dimension $3$ whose underlying smooth manifold is diffeomorphic to the standard $S^6$.
Equivalently, decide whether $S^6$ carries an integrable [[almost complex structure|9SYWRGVTRS]].

## Remarks

The problem is open; several announced solutions have not been accepted.
It is often called the Hopf problem, after Heinz Hopf, who raised it in a paper published in 1948.

What is known:

1. $S^6$ has almost complex structures; one comes from identifying $S^6$ with the unit imaginary [[octonions|1HRMT3DSX2]] and using octonion multiplication.
2. Borel and Serre showed that $S^2$ and $S^6$ are the only spheres with an almost complex structure; $S^2$ is the Riemann sphere, so $S^6$ is the only open case.
3. The almost complex structure coming from the octonions is not integrable.
4. LeBrun showed that no complex structure on $S^6$ is orthogonal with respect to the round metric.

By the [[Newlander–Nirenberg theorem|VCT848WV1T]], the problem asks for an almost complex structure whose Nijenhuis tensor vanishes.

## Explanations

### L1 human

The two-dimensional sphere is the Riemann sphere, the simplest compact complex manifold.
The question is whether the six-dimensional sphere can also be made into a complex manifold.
It is known to have the weaker structure of an almost complex manifold, which looks complex at each point separately.
Whether this weaker structure can be made to fit together into true complex coordinates is still unknown.
</document>

<document source="https://tamnd.github.io/math2/node/V4PEVFF4F4.md" tag="V4PEVFF4F4" kind="statement">
# Skolem–Noether theorem

## Statement

Let $K$ be a field and let $A$ be a [[central simple algebra|5HRKJBAY2V]] over $K$.
Let $B$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $K$ that is a [[simple ring|62M0NC3E2T]].
Let $f, g \colon B \to A$ be homomorphisms of $K$-algebras.
Then there is an invertible element $u \in A$ such that $g(x) = u f(x) u^{-1}$ for all $x \in B$.
In particular, every automorphism of the $K$-algebra $A$ is an inner automorphism, that is, a conjugation $x \mapsto u x u^{-1}$.

## Remarks

The algebra $B \otimes_K A^{\mathrm{op}}$ is simple and finite-dimensional, so it has only one simple module up to isomorphism, and its modules are determined by their dimension.
The maps $f$ and $g$ make $A$ into two $B \otimes_K A^{\mathrm{op}}$-modules, by $(b \otimes a) \cdot x = f(b) x a$ and $(b \otimes a) \cdot x = g(b) x a$, of the same dimension.
An isomorphism $\varphi$ from the first to the second commutes with right multiplication by $A$, so $\varphi(x) = u x$ with $u = \varphi(1)$ invertible, and compatibility with $B$ gives $u f(b) = g(b) u$.
Example: every automorphism of $M_n(K)$ as a $K$-algebra is conjugation $X \mapsto P X P^{-1}$ by an invertible matrix $P$.
Example: any two elements of $\mathbb{H}$ with the same minimal polynomial $x^2 + 1$ over $\R$, such as $i$ and $j$, are conjugate in $\mathbb{H}$.

## Explanations

### L1 human

Inside a central simple algebra, two copies of the same simple algebra always differ by a change of coordinates.
That change of coordinates is conjugation by an invertible element.
For matrices this says that every symmetry of the matrix algebra that respects scalars comes from a change of basis.

### L2 human

The theorem is used constantly in the theory of central simple algebras.
It shows that $\Aut_K(M_n(K)) \cong \mathrm{PGL}_n(K)$, which identifies central simple algebras split by a Galois extension $L$ with classes in $H^1(\Gal(L/K), \mathrm{PGL}_n(L))$.
Applied to a subfield $L$ of $A$ and its Galois conjugates, it is a key step in building the crossed product description of central simple algebras.
Together with the double centralizer theorem it controls how subfields sit inside central simple algebras.
</document>

<document source="https://tamnd.github.io/math2/node/V4Q6TSY1KX.md" tag="V4Q6TSY1KX" kind="definition">
# Factorial experiment

## Definition

Let $k \ge 2$ and let $L_1, \dots, L_k$ be finite sets with $s_i = \abs{L_i} \ge 2$, the *levels* of factors $1, \dots, k$.
The treatments are the combinations $t = (t_1, \dots, t_k) \in L_1 \times \cdots \times L_k$.
A *full factorial experiment* is an [[experiment|2TE42ZWJ3E]] whose design applies every one of the $s_1 s_2 \cdots s_k$ combinations to at least one unit.
When every $s_i = 2$, it is a *$2^k$ factorial experiment*, and the levels are usually coded $-1$ and $+1$.

## Remarks

In a $2^k$ experiment with one run per combination and responses $y_t$, the *main effect* of factor $i$ is the mean of $y_t$ over $t_i = +1$ minus the mean over $t_i = -1$.
The *interaction* of factors $i$ and $j$ is half of the difference between the effect of factor $i$ at $t_j = +1$ and at $t_j = -1$.
Equivalently, it is the mean of $y_t$ over $t_i t_j = +1$ minus the mean over $t_i t_j = -1$.

Example: a $2^2$ experiment with factors $A$ and $B$ and responses $y_{--} = 10$, $y_{+-} = 14$, $y_{-+} = 12$, $y_{++} = 20$, where the first sign is the level of $A$.
The main effect of $A$ is $(14 + 20)/2 - (10 + 12)/2 = 6$, and that of $B$ is $(12 + 20)/2 - (10 + 14)/2 = 4$.
The interaction is $\big( (20 - 12) - (14 - 10) \big)/2 = 2$, which also equals $(10 + 20)/2 - (14 + 12)/2$.

Non-example: the runs $(-,-)$, $(+,-)$, $(-,+)$ change one factor at a time from a baseline.
The combination $(+,+)$ is missing, so this is not a full factorial experiment, and the interaction of $A$ and $B$ cannot be estimated.

The data of a factorial experiment are usually analysed by analysis of variance, with one sum of squares for each main effect and each interaction.
</document>

<document source="https://tamnd.github.io/math2/node/V50N3JEBDP.md" tag="V50N3JEBDP" kind="definition">
# Cayley–Klein metric

## Definition

Let $n \ge 1$ and let $Q$ be a nondegenerate [[quadric|Q3Q4FF23VG]] in the real [[projective space|ZT9A6JK8QF]] $\mathbb{P}^n$, called the *absolute*.
Let $c$ be a nonzero real or imaginary constant.
Let $x$ and $y$ be two distinct points not on $Q$ such that the line $xy$ is not tangent to $Q$.
Let $a$ and $b$ be the two points where the line $xy$ meets $Q$, taken in the complexification when they are not real.
The *Cayley–Klein distance* of $x$ and $y$ is
$$d(x, y) = c \log (x, y; a, b),$$
where $(x, y; a, b)$ is the [[cross-ratio|09ASMRMD67]] of the four points on the line.
Also $d(x, x) = 0$.

## Remarks

Hyperbolic case: let $Q$ be an ellipsoid and let $x$ and $y$ lie inside it.
Then $a$ and $b$ are real.
In an affine chart with the points in the order $a, x, y, b$ on the line, take $c = \frac{1}{2}$ and the cross-ratio $\frac{\abs{xb} \, \abs{ya}}{\abs{xa} \, \abs{yb}}$.
This ratio is greater than $1$, so $d(x, y) > 0$, and $d(x, y) \to \infty$ as $y$ tends to $b$.
This is the [[Beltrami–Klein model|1WN9B18H86]] of hyperbolic space with curvature $-1$.

Elliptic case: if $Q$ has no real points, such as $x_0^2 + x_1^2 + x_2^2 = 0$, then $a$ and $b$ are complex conjugate, and an imaginary constant $c$ gives the real distance of [[elliptic geometry|H5GB19W2H3]].

Non-example: a degenerate quadric, such as a double plane, does not fit the definition, because a line meets it in one point only.
Euclidean geometry appears only as a limit of the nondegenerate cases.

Conventions for the order in the cross-ratio differ between sources; they change the sign of the logarithm, not the geometry.
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# Lemniscate constant

## Description

The *lemniscate constant* is $\varpi = 2.62205755429\ldots$, defined by
$$\varpi = 2\int_0^1 \frac{dt}{\sqrt{1 - t^4}}.$$
It plays for the lemniscate of Bernoulli $(x^2 + y^2)^2 = x^2 - y^2$ the role that $\pi$ plays for the unit circle.
The lemniscate has total arc length $2\varpi$, as the circle has length $2\pi$.

1. With the [[gamma function|0H5SYBSED9]], $\varpi = \dfrac{\Gamma(1/4)^2}{2\sqrt{2\pi}}$.
2. With the [[complete elliptic integral of the first kind|RM54CPPE12]], $\varpi = \sqrt2\,K(1/\sqrt2)$.
3. With the arithmetic–geometric mean $M$, $\varpi = \pi / M(1, \sqrt2)$; the number $\varpi/\pi = 1/M(1, \sqrt 2) = 0.8346\ldots$ is Gauss's constant.
4. $\varpi$ is a [[transcendental number|0VTQXTCR2G]].

## Remarks

Check of point 1: the substitution $s = t^4$ gives $\int_0^1 (1 - t^4)^{-1/2}\,dt = \frac14 B(1/4, 1/2) = \frac14\,\Gamma(1/4)\Gamma(1/2)/\Gamma(3/4)$.
The reflection formula $\Gamma(1/4)\Gamma(3/4) = \pi/\sin(\pi/4) = \pi\sqrt2$ and $\Gamma(1/2) = \sqrt\pi$ turn this into $\Gamma(1/4)^2/(4\sqrt{2\pi})$, which is $\varpi/2$.
Check of point 2: with $t = \cos\theta$ and $1 - \cos^4\theta = \sin^2\theta\,(1 + \cos^2\theta)$, one gets $\int_0^1 dt/\sqrt{1 - t^4} = \int_0^{\pi/2} d\theta/\sqrt{1 + \cos^2\theta} = \int_0^{\pi/2} d\theta/\sqrt{2 - \sin^2\theta} = K(1/\sqrt2)/\sqrt2$.
The [[lemniscate sine|TQDZQ964AZ]] first reaches $1$ at $\varpi/2$, as the sine first reaches $1$ at $\pi/2$.

## Explanations

### L1 human

Everyone knows $\pi$, the half-length of the unit circle.
The lemniscate constant is the analogous number for the figure-eight lemniscate: half its length is $\varpi \approx 2.622$.
It turns up in elliptic integrals, in the arithmetic–geometric mean of $1$ and $\sqrt 2$, and in values of the gamma function at $1/4$.
</document>

<document source="https://tamnd.github.io/math2/node/V5EPT2PX9Z.md" tag="V5EPT2PX9Z" kind="statement">
# Abel's summation formula

## Statement

Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, and for real $t$ put $A(t) = \sum_{1 \le n \le t} a_n$.
Let $x \ge 1$ be a real number, and let $\varphi$ be a [[continuously differentiable|06SKCHDQT5]] complex-valued function on $[1, x]$.
Then
$$\sum_{1 \le n \le x} a_n \varphi(n) = A(x) \varphi(x) - \int_1^x A(t) \varphi'(t) \, dt.$$

## Remarks

The function $A$ is constant on each interval $[k, k+1)$, so the integral splits as $\sum_k A(k) (\varphi(k+1) - \varphi(k))$ plus a last piece up to $x$.
Rearranging this finite sum gives the left side; this is summation by parts.
The formula is the discrete analogue of [[integration by parts|08Y1BBQTK9]], and it is integration by parts for the Riemann–Stieltjes integral $\int \varphi \, dA$.
Example: with $a_n = 1$ and $\varphi(t) = 1/t$, one has $A(t) = \floor{t}$, and
$$\sum_{1 \le n \le x} \frac{1}{n} = \frac{\floor{x}}{x} + \int_1^x \frac{\floor{t}}{t^2} \, dt.$$
Writing $\floor{t} = t - \{t\}$ shows that the harmonic numbers minus the logarithm converge.

## Explanations

### L1 human

Abel's summation formula turns a weighted sum $\sum a_n \varphi(n)$ into an integral of the partial sums $A(t)$.
It is useful when the partial sums of $a_n$ are well understood but the terms themselves are irregular.
It plays the role of integration by parts for sums.

### L2 human

In analytic number theory one often knows a counting function $A(x)$, such as the number of primes up to $x$, and wants sums like $\sum_{p \le x} 1/p$.
Abel's formula transfers estimates for $A$ into estimates for such weighted sums.
It also extends the [[Riemann zeta function|1XWBY4N0VT]] to a meromorphic function on $\operatorname{Re}(s) > 0$ with a single pole at $s = 1$, by taking $a_n = 1$ and $\varphi(t) = t^{-s}$.
The [[Dirichlet's test|0V9ME8GQE7]] and [[Abel's test|0CN97GADE5]] for series rest on the same summation by parts.
</document>

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# Dynkin system

## Summary

A Dynkin system, or λ-system, on a set $X$ is a family of subsets that contains $X$.
It is closed under complements and under countable unions of disjoint members.
It is weaker than a σ-algebra, because it need not be closed under intersections.
Dynkin systems appear in the π-λ theorem, the standard tool to prove that two measures are equal.

## Explanations

### L1 human

A Dynkin system is the kind of family one gets when one collects all sets on which two probability measures agree.
Such a family is stable under complements and disjoint countable unions, but in general not under intersections.
A small example: on $\set{1, 2, 3, 4}$, the subsets with an even number of elements form a Dynkin system.
It is not a σ-algebra, since $\set{1, 2} \cap \set{2, 3} = \set{2}$ has one element.
</document>

<document source="https://tamnd.github.io/math2/node/V5TBWB2RPQ.md" tag="V5TBWB2RPQ" kind="definition">
# Perron–Wiener–Brelot solution

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]] with [[boundary|05D5EGMFYV]] $\partial\Omega$, and let $f \colon \partial\Omega \to [-\infty, +\infty]$ be a function.

1. The *upper class* $\mathcal{U}_f$ consists of the functions $u$ on $\Omega$, [[superharmonic|X4V2S56YQQ]] or identically $+\infty$ and bounded below, with $\liminf_{x \to \xi} u(x) \ge f(\xi)$ for all $\xi \in \partial\Omega$.
2. The *lower class* $\mathcal{L}_f$ consists of the functions $v$ on $\Omega$, [[subharmonic|Z1MHMGFNFW]] or identically $-\infty$ and bounded above, with $\limsup_{x \to \xi} v(x) \le f(\xi)$ for all $\xi \in \partial\Omega$.
3. The *upper solution* is $\overline{H}_f = \inf_{u \in \mathcal{U}_f} u$ and the *lower solution* is $\underline{H}_f = \sup_{v \in \mathcal{L}_f} v$, using the [[infimum|04KFTA34C5]] and [[supremum|0BTMQ04556]] pointwise.

The function $f$ is *resolutive* if $\overline{H}_f = \underline{H}_f$ and this function is harmonic on $\Omega$.
In that case the *Perron–Wiener–Brelot solution* of $f$ is $H_f = \overline{H}_f = \underline{H}_f$.

## Remarks

Always $\underline{H}_f \le \overline{H}_f$, by the maximum principle applied to $u - v$ for $u \in \mathcal{U}_f$ and $v \in \mathcal{L}_f$.
Each of $\overline{H}_f$ and $\underline{H}_f$ is harmonic, identically $+\infty$, or identically $-\infty$ on each connected component of $\Omega$.
Example: let $f$ be continuous on $\partial\Omega$, and let $u$ be continuous on $\overline{\Omega}$, harmonic on $\Omega$ and equal to $f$ on $\partial\Omega$.
Then $u$ lies in both classes, so $\overline{H}_f \le u \le \underline{H}_f$ and $H_f = u$.
Non-example: let $f \equiv +\infty$ on $\partial\Omega$.
By the minimum principle, a superharmonic function in $\mathcal{U}_f$ would be at least every constant $M$.
So $\mathcal{U}_f$ contains only the constant $+\infty$, and $\overline{H}_f \equiv +\infty$ is not harmonic; hence $f$ is not resolutive.
For $\Omega = \set{0 < \abs{x} < 1}$ in $\R^2$, let $f = 0$ on $\abs{x} = 1$ and $f(0) = 1$.
Then $f$ is resolutive with $H_f = 0$, which does not attain the value $f(0)$.
So the PWB solution need not solve the classical Dirichlet problem.
By Wiener's theorem every continuous $f$ is resolutive.
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# Rigidity of frameworks

## Summary

A framework is a [[graph|00C0BR7D7H]] whose vertices are placed at points of Euclidean space, with the edges read as rigid bars joined at the vertices.
The framework is rigid when it cannot be deformed a little without changing some bar length, except by moving it as a whole.
Rigidity theory decides this from the graph and the positions, and in the plane [[Laman graphs|C81JPZ51NA]] answer the question for generic positions.
It belongs to [[distance geometry|HPW5F0GQ4B]] and has uses in engineering, robotics and molecular biology.

## Explanations

### L1 human

Build a shape from sticks joined by loose hinges at the ends.
A triangle of sticks holds its shape: three side lengths fix the triangle.
A square of four sticks does not: you can push it over into a rhombus while every stick keeps its length.
Adding one diagonal stick to the square makes it rigid again.
Rigidity theory asks which stick figures behave like the triangle and which like the square.
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<document source="https://tamnd.github.io/math2/node/V60F20N6JT.md" tag="V60F20N6JT" kind="definition">
# Interpolation space

## Definition

Let $X_0$ and $X_1$ be [[Banach spaces|06GZ956657]] that are both continuously embedded in a Hausdorff [[topological vector space|T32W553W95]] $Z$.
The pair $(X_0, X_1)$ is a *compatible couple*.
The spaces $X_0 \cap X_1$ and $X_0 + X_1$ are Banach spaces with the norms
$$\norm{x}_\cap = \max(\norm{x}_{X_0}, \norm{x}_{X_1}), \qquad \norm{x}_+ = \inf \set{\norm{x_0}_{X_0} + \norm{x_1}_{X_1} : x = x_0 + x_1}.$$
A Banach space $X$ is an *intermediate space* if $X_0 \cap X_1 \subseteq X \subseteq X_0 + X_1$, with both inclusions continuous.
An intermediate space $X$ is an *interpolation space* if the following holds.
Every linear map $T \colon X_0 + X_1 \to X_0 + X_1$ that restricts to [[bounded linear maps|03TM135G0N]] $X_0 \to X_0$ and $X_1 \to X_1$ maps $X$ into $X$, and the restriction $T \colon X \to X$ is bounded.

## Remarks

Example: $X_0 + X_1$ is an interpolation space.
If $x = x_0 + x_1$, then $T x = T x_0 + T x_1$ with $T x_0 \in X_0$ and $T x_1 \in X_1$.
So $\norm{T x}_+ \le \max(\norm{T}_{X_0}, \norm{T}_{X_1}) \, \norm{x}_+$.

Non-example: take $X_0 = \ell^1$ and $X_1 = \ell^\infty$, and let $X = \ell^1 \oplus \R \mathbf{1}$, where $\mathbf{1} = (1, 1, 1, \dots)$.
With the norm $\norm{y + c \mathbf{1}} = \norm{y}_1 + \abs{c}$, the space $X$ is intermediate.
The map $T(x_1, x_2, x_3, \dots) = (x_1, 0, x_2, 0, x_3, 0, \dots)$ has norm $1$ on $\ell^1$ and on $\ell^\infty$.
But $T \mathbf{1} = (1, 0, 1, 0, \dots)$ is not in $X$, since it differs from every multiple of $\mathbf{1}$ by a sequence that is not summable.

By the Riesz–Thorin theorem, for a measure space and $1 \le p_0 < p < p_1 \le \infty$, the space $L^p$ is an interpolation space for the couple $(L^{p_0}, L^{p_1})$.
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# Exponential stability

## Definition

Let $U \subseteq \R^n$ be open, let $f \colon U \to \R^n$ be [[continuously differentiable|0VW90BBSJ4]], and let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$.
An [[equilibrium point|0G4X36AE0X]] $x^*$ is *exponentially stable* if there are constants $\delta > 0$, $C \ge 1$ and $\alpha > 0$ such that for every $x_0 \in U$ with $\norm{x_0 - x^*} < \delta$:

1. the solution $\varphi_t(x_0)$ exists for all $t \ge 0$;
2. $\norm{\varphi_t(x_0) - x^*} \le C e^{-\alpha t} \norm{x_0 - x^*}$ for all $t \ge 0$.

## Remarks

An exponentially stable equilibrium is asymptotically stable.
If $f$ is of class $C^2$, then $x^*$ is exponentially stable if and only if every eigenvalue of $Df(x^*)$ has negative real part.

Example: $x' = -x$ on $\R$ with $x^* = 0$.
The solution is $x_0 e^{-t}$, so clause 2 holds with $C = 1$ and $\alpha = 1$, for any $\delta$.

Non-example: $x' = -x^3$ on $\R$ with $x^* = 0$.
The solution is $x(t) = x_0 / \sqrt{1 + 2 x_0^2 t}$, since $\frac{d}{dt} x^{-2} = 2$.
It tends to $0$, but $x(t) e^{\alpha t} \to \infty$ for each $\alpha > 0$ and $x_0 \ne 0$, so clause 2 fails.

For a time-dependent system $x' = f(t, x)$ with $f(t, 0) = 0$, the origin is uniformly exponentially stable if $\norm{x(t)} \le C e^{-\alpha (t - t_0)} \norm{x(t_0)}$ for all $t \ge t_0$, all initial times $t_0$ and all $\norm{x(t_0)} < \delta$.
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# Length of a module

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $M$ be a left [[module|09VDS2W1E8]] over $R$.
A chain of [[submodules|0RGZ10DFPG]] $M_0 \subsetneq M_1 \subsetneq \cdots \subsetneq M_n$ of $M$ has length $n$.
The *length* $\ell_R(M)$ of $M$ is the supremum of the lengths of all such chains, an element of $\set{0, 1, 2, \dots} \cup \set{\infty}$.

## Remarks

Example: for a finite-dimensional vector space $V$ over a field $k$, $\ell_k(V) = \dim_k V$.
Indeed, a strict chain of subspaces raises the dimension at each step, and a chain built from a basis has $\dim_k V$ steps.
Non-example of finite length: $\ell_\Z(\Z) = \infty$, because $2^n\Z \subsetneq 2^{n-1}\Z \subsetneq \cdots \subsetneq \Z$ has length $n$ for every $n$.
A composition series of $M$ is a chain $0 = M_0 \subsetneq \cdots \subsetneq M_n = M$ whose factors $M_i/M_{i-1}$ are simple modules.
A module $M$ has finite length if and only if it has a composition series, and then $\ell_R(M) = n$ by the [[Jordan–Hölder theorem for modules|M1ABX18FC4]].
For a short exact sequence $0 \to M' \to M \to M'' \to 0$, one has $\ell_R(M) = \ell_R(M') + \ell_R(M'')$.
The group version of a composition series is the [[composition series of a group|0NB8VPB93V]].

## Explanations

### L1 human

The length of a module counts the steps in the longest strict chain of submodules.
It is the module version of dimension.
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# Convergence of Fourier series

## Summary

Let $S_N f$ be the partial sums of the [[Fourier series|G12KXDHPPS]] of a periodic function $f$.
The convergence of Fourier series is the question whether, and in which sense, $S_N f$ tends to $f$.
The answer depends on the mode of convergence: at a point, uniformly, almost everywhere, or in the norm of $L^p$.
Smoothness or bounded variation of $f$ gives pointwise convergence, while mere continuity does not.
In $L^2$ the partial sums always converge, and by Carleson's theorem they also converge almost everywhere for $f \in L^2$.

## Explanations

### L1 human

Adding more and more waves of the Fourier series should give a better and better picture of the function, but this can fail.
For a smooth function the picture improves everywhere at once.
At a jump the sums settle at the midpoint of the jump and overshoot nearby.
There even exist continuous functions whose Fourier series diverge at some point.
Measured by the average squared error instead of point by point, the picture always improves, for every square integrable function.
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# p-adic integer

## Summary

The $p$-adic integers are the [[p-adic numbers|BT6T702C28]] $x$ with $\abs{x}_p \le 1$.
They form a ring $\Z_p$, which is a [[discrete valuation ring|MGM3CNJW9V]] with maximal ideal $p\Z_p$ and residue field $\F_p$.
The ring $\Z_p$ is compact, and it is the inverse limit of the rings $\Z / p^n \Z$.
It holds, in one object, the information of all congruences modulo powers of $p$.

## Explanations

### L1 human

A $p$-adic integer is a base $p$ expansion $a_0 + a_1 p + a_2 p^2 + \cdots$ that goes on forever to the left.
Cutting it after $n$ digits gives a residue modulo $p^n$, and these residues fit together.
Every ordinary integer is a $p$-adic integer, and so is every fraction whose denominator is not divisible by $p$.
For example, $1/2$ is a $3$-adic integer, but $1/3$ is not.
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# Lasso

## Definition

Let $X$ be a real $n \times p$ matrix, let $y \in \R^n$, and let $\lambda > 0$.
For $b \in \R^p$ write $\norm{b}_1 = \sum_{j=1}^p \abs{b_j}$ and let $\norm{\cdot}_2$ be the Euclidean norm.
A *lasso estimate* with parameter $\lambda$ is a minimiser over $b \in \R^p$ of
$$\frac{1}{2} \norm{y - X b}_2^2 + \lambda \norm{b}_1.$$

## Remarks

The objective is [[convex|07Q7CNM5WF]] and tends to infinity as $\norm{b}_1 \to \infty$, so a minimiser exists.
It need not be unique when the columns of $X$ are linearly dependent.
There is no closed formula in general; the estimate is computed by coordinate descent or by the LARS algorithm.
Authors scale the objective in different ways, for example with $\frac{1}{2n}$ in front of the squared norm, which only rescales $\lambda$.

Example: assume $X^T X = I_p$ and put $z = X^T y$, the [[ordinary least squares|YM708KHV2W]] estimate.
Then $\norm{y - Xb}_2^2 = \norm{z - b}_2^2 + \text{const}$, and the problem splits into the $p$ problems of minimising $\frac{1}{2}(z_j - b_j)^2 + \lambda \abs{b_j}$.
The solution is the soft thresholding $\hat\beta_j = \sgn(z_j) \max(\abs{z_j} - \lambda, 0)$.
So every $j$ with $\abs{z_j} \le \lambda$ gets $\hat\beta_j = 0$.

Non-example: with the squared norm $\norm{b}_2^2$ in place of $\norm{b}_1$, the same orthonormal case gives $\hat\beta_j = z_j / (1 + 2\lambda)$, which is never $0$ when $z_j \ne 0$; that is ridge regression, not the lasso.

Equivalent form: for each $\lambda > 0$ and each lasso estimate $\hat\beta$, the vector $\hat\beta$ minimises $\norm{y - Xb}_2^2$ under the constraint $\norm{b}_1 \le \norm{\hat\beta}_1$.
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# Schur's theorem

## Statement

Let $r \ge 1$ be an integer.
Then there is a positive integer $N$ with the following property.
For every colouring of $\set{1, \ldots, N}$ with $r$ colours, there are $x, y, z \in \set{1, \ldots, N}$ of the same colour with $x + y = z$.
Here $x$ and $y$ need not be distinct.

## Remarks

Let $N + 1$ be at least the Ramsey number for $r$ colours and triangles, given by [[Ramsey's theorem|00TQV3F5S4]].
Colour the edge $\set{i, j}$ of the complete graph on $\set{1, \ldots, N + 1}$ with the colour of $\abs{i - j}$.
A monochromatic triangle $i < j < k$ gives $x = j - i$, $y = k - j$ and $z = k - i$ of the same colour with $x + y = z$.

The least such $N$ is $S(r) + 1$, where the Schur number $S(r)$ is the largest $n$ for which $\set{1, \ldots, n}$ can be split into $r$ [[sum-free sets|7SHV2SPYP6]].
For example $S(2) = 4$: the colouring $\set{1, 4}$, $\set{2, 3}$ has no monochromatic solution, while every $2$-colouring of $\set{1, \ldots, 5}$ has one.
Known values are $S(1) = 1$, $S(2) = 4$, $S(3) = 13$, $S(4) = 44$ and $S(5) = 160$.

## Explanations

### L1 human

However you colour the numbers $1$ to $N$ with a fixed number of colours, if $N$ is large enough, some colour contains two numbers together with their sum.
For two colours, $N = 5$ is enough.
It is one of the first results of [[Ramsey theory|G5TWXV565S]] in the integers.

### L2 human

Schur proved the theorem in 1916 while studying Fermat's equation modulo primes.
It implies that for each $n$ and all large enough primes $p$, the congruence $x^n + y^n \equiv z^n \pmod p$ has a solution with $xyz \not\equiv 0$.
The proof is a direct reduction to [[Ramsey's theorem|00TQV3F5S4]] for triangles.
[[Rado's theorem|BZSGRCTFH4]] extends it to all linear equations whose monochromatic solutions are forced in this way.
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# Josefson's theorem

## Statement

Let $E \subseteq \C^n$ be a [[pluripolar set|N0MAGPW8AP]], that is, locally contained in the $-\infty$ set of a [[plurisubharmonic function|FF2P7ER3QT]].
Then there is a plurisubharmonic function $u$ on $\C^n$, not identically $-\infty$, such that
$$E \subseteq \set{z \in \C^n : u(z) = -\infty}.$$

## Remarks

The theorem was proved by Bo Josefson in 1978.
Proofs build $u$ as a convergent series $\sum_j \varepsilon_j u_j$ of functions of the [[Lelong class|X4ZHQG9995]], with growth estimates that come from the [[Siciak extremal function|QBJ9R06GYE]] of compact pieces of $E$.
A consequence is that a countable union of pluripolar sets is pluripolar, since a suitable series of global functions works for the union.
For example, if $E$ is the union of the lines $\set{z_1 = 1/k}$, $k \ge 1$, in $\C^2$, then each line is pluripolar, and the theorem gives one global function for the whole union.
For $n = 1$ the statement is the classical fact that a locally polar set in $\C$ is polar.

## Explanations

### L1 human

A set is pluripolar if near each of its points it is hidden in the $-\infty$ set of some plurisubharmonic function.
The theorem says that one can then hide the whole set at once in the $-\infty$ set of a single plurisubharmonic function on all of $\C^n$.
So "small locally" and "small globally" mean the same thing.

### L2 human

Without the theorem, one would need two notions of pluripolarity, local and global, and many arguments in pluripotential theory would have to track which one is meant.
The global function $u$ may be chosen of logarithmic growth, $u(z) \le \log(1 + \abs{z}) + C$, which connects pluripolar sets with polynomial approximation through the Siciak extremal function.
Bedford and Taylor proved that pluripolar sets are exactly the sets of zero Monge–Ampère capacity.
Together with Josefson's theorem, this gives a complete description of the negligible sets of the theory.
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# Abelian differential

## Definition

Let $X$ be a [[Riemann surface|5K0N8QMV6Q]].
An *Abelian differential* on $X$ is a meromorphic $1$-form on $X$.
That is, to each chart $z$ of $X$ it assigns a [[meromorphic function|0ZZ7ZYDR8W]] $f$ on the chart domain, written $f(z)\,dz$.
On the overlap of two charts $z$ and $w$ with forms $f(z)\,dz$ and $g(w)\,dw$, these must satisfy
$$f(z) = g(w(z))\, \frac{dw}{dz}.$$
The differential is classified as follows.

1. It is of the *first kind* if every $f$ is holomorphic.
2. It is of the *second kind* if every [[residue|0PHN07KVQT]] of every $f$ is zero.
3. It is of the *third kind* in general, that is, with arbitrary poles and residues.

## Remarks

Example: on the torus $X = \C / \Lambda$ for a lattice $\Lambda$, the form $dz$ is well defined, since the coordinate changes are translations $w = z + \lambda$ with $dw/dz = 1$.
It is of the first kind and has no zeros.

Non-example: on the Riemann sphere, $e^{1/z}\,dz$ is not an Abelian differential, because $e^{1/z}$ has an essential singularity at $0$ and so is not meromorphic there.
Also $dz$ itself on the sphere has a pole of order $2$ at $\infty$: in the chart $w = 1/z$ it becomes $-w^{-2}\,dw$.
So it is an Abelian differential of the second kind but not of the first kind.

Some authors, notably in the study of translation surfaces, use "Abelian differential" for a holomorphic $1$-form only.
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# Radical extension

## Summary

A radical extension is a field extension built by adjoining roots one at a time.
At each step one adjoins an $n$-th root of an element already present.
Each of its elements can be written from elements of the base field with the four field operations and root extraction.
A polynomial is [[solvable by radicals|J7YS0DA3HQ]] when its roots lie in a radical extension, and [[Galois' theorem|VHCWCFA3MX]] turns this into a property of a group.

## Explanations

### L1 human

Start with the rational numbers and allow yourself to take square roots, cube roots and so on, one step at a time.
Every field you reach in this way is a radical extension of $\Q$.
For example, $\Q(\sqrt{2})$ is radical, and so is $\Q(\sqrt{2}, \sqrt[3]{1 + \sqrt{2}})$, built in two steps.
A number like $\sqrt[3]{1 + \sqrt{2}}$ is exactly the kind of expression that a formula for the roots of a polynomial produces.
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# Residuated mapping

## Summary

A residuated mapping is a monotone map $f \colon P \to Q$ between [[partially ordered sets|011C29Q1S1]] with a special property.
For each $y$, the inequality $f(x) \le y$ has a largest solution $x$.
Equivalently, $f$ is the lower adjoint of a [[Galois connection|MBVJ4JQXC5]]; its upper adjoint is called the residual of $f$.
Residuated maps are the basis of residuated lattices, which model division in ordered algebra and implication in logic.

## Explanations

### L1 human

For a positive number $a$, the inequality $a x \le y$ in the reals has the largest solution $x = y / a$.
The map $x \mapsto a x$ is residuated, and its residual is division by $a$.
In general, a residuated map is one for which "how large can $x$ be so that $f(x)$ stays below $y$?" always has an answer.
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<document source="https://tamnd.github.io/math2/node/V7AAT7X2QW.md" tag="V7AAT7X2QW" kind="definition">
# Fekete points

## Definition

Let $K$ be a [[compact set|0R6EWFKRCF]] in $\C$ with at least $n \ge 2$ points.
An $n$-tuple $(z_1, \dots, z_n)$ of points of $K$ is a system of *Fekete points* of $K$ if
$$\prod_{1 \le i < j \le n} \abs{z_i - z_j} = \max \set{ \prod_{1 \le i < j \le n} \abs{w_i - w_j} : w_1, \dots, w_n \in K }.$$

## Remarks

A system of Fekete points exists for each $n$, since $K^n$ is compact and the product is continuous.
It need not be unique: a rotation of Fekete points of a circle centred at $0$ gives Fekete points again.
The points are distinct, because the maximum is positive.
The maximum equals $\delta_n(K)^{n(n-1)/2}$, where $\delta_n(K)$ is the $n$-th diameter in the definition of the [[transfinite diameter|VSGTSQ7RY8]].

Example: on the unit circle, the $n$-th roots of unity $1, \omega, \dots, \omega^{n-1}$ with $\omega = e^{2\pi i/n}$ are Fekete points.
The product of their mutual distances is the absolute value of their Vandermonde determinant, which equals $n^{n/2}$.
Hadamard's inequality bounds the Vandermonde determinant of any $n$ points of the closed unit disc by $n^{n/2}$, so the maximum is attained.

Non-example: for $n = 2$ on the unit circle, the points $1$ and $i$ are not Fekete points, since $\abs{1 - i} = \sqrt 2 < 2 = \abs{1 - (-1)}$.

Maximizing the product is the same as minimizing the discrete logarithmic energy $\sum_{i < j} \log(1/\abs{z_i - z_j})$.
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# Krull topology

## Definition

Let $L/K$ be a [[Galois extension|0DS2F7GK05]], possibly of infinite degree, with [[Galois group|0XVR72M51M]] $G = \Gal(L/K)$.
Let $\mathcal{E}$ be the set of [[intermediate fields|02NBKVQFRQ]] $E$ of $L/K$ with $[E : K]$ finite.
The *Krull topology* on $G$ is the following [[topology|0CNQXVBHCX]].
A subset $U \subseteq G$ is open if and only if for each $\sigma \in U$ there is $E \in \mathcal{E}$ with $\sigma \Gal(L/E) \subseteq U$.
Equivalently, the cosets $\sigma \Gal(L/E)$, for $\sigma \in G$ and $E \in \mathcal{E}$, form a basis of the topology.

## Remarks

With this topology $G$ is a topological group that is [[compact|0QPHW703PV]], [[Hausdorff|0MXB4CKHFW]] and totally disconnected, that is, a [[profinite group|KA0YDB6QXE]].
In fact $G$ is isomorphic, as a topological group, to the inverse limit of the finite groups $\Gal(E/K)$ over the finite Galois subextensions $E/K$.
Example: if $L/K$ is finite, then $L \in \mathcal{E}$ and $\Gal(L/L) = \set{\id}$, so every subset of $G$ is open and the topology is discrete.
Non-example of an open set: for $L$ an algebraic closure of $\F_p$, the one-point set $\set{\id}$ is not open.
Indeed it would contain some $\Gal(L/\F_{p^n})$, which is infinite since $L/\F_{p^n}$ has infinite degree.
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# Parabolic cylinder function

## Summary

Parabolic cylinder functions are solutions of Weber's equation $y'' + \left( \nu + \frac12 - \frac{z^2}{4} \right) y = 0$.
The standard solution $D_\nu(z)$ decays like $z^\nu e^{-z^2/4}$ as $z \to +\infty$.
They arise when the [[Helmholtz equation|MNXKTXJ35G]] or [[Laplace's equation|S8RG6N1PRT]] is solved by [[separation of variables|47NFRVB564]] in parabolic cylindrical coordinates.
For integer $\nu \ge 0$ they reduce to [[Hermite functions|52VS07H6GK]].
They are [[confluent hypergeometric functions|6AZ4ZBN1M8]] in the variable $z^2/2$.

## Explanations

### L1 human

Weber's equation is the Schrödinger equation of the quantum harmonic oscillator, with energy parameter $\nu + \frac12$.
For most values of $\nu$ every solution blows up at $+\infty$ or at $-\infty$.
The function $D_\nu$ is the solution that decays at $+\infty$.
For $\nu = 0$ it is the Gaussian $e^{-z^2/4}$: then $y'' = \left( \frac{z^2}{4} - \frac12 \right) y$, which is exactly the equation with $\nu = 0$.
For $\nu = n$ a nonnegative integer, $D_n$ decays at both ends, and these are the bound states of the oscillator.
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# Morley's categoricity theorem

## Statement

Let $T$ be a [[complete theory|655K8Y7VS8]] in a countable first-order signature, with infinite models.
If $T$ is [[$\kappa$-categorical|1JJCGNX240]] for some [[uncountable|0W7FBYS1F1]] [[cardinal|0EBACQFAZW]] $\kappa$, then $T$ is $\kappa$-categorical for every uncountable cardinal $\kappa$.

## Remarks

Michael Morley proved the theorem in 1965, answering a question of Jerzy Łoś.
A key step shows that a theory categorical in an uncountable cardinal is $\omega$-stable: over each countable set of parameters there are only countably many complete types.
Such a theory has a strongly minimal formula, possibly with parameters.
Each model is determined up to isomorphism by the dimension of the set that this formula defines, as a vector space is determined by its dimension.

Example: the theory of [[algebraically closed fields|MT8TDVPBXB]] of a fixed characteristic is categorical in every uncountable cardinal, with transcendence degree as the dimension.
The theory of infinite vector spaces over $\Q$ is another example.

The countability of the signature matters in this form.
Saharon Shelah extended the theorem: a complete theory $T$ categorical in some cardinal $\kappa > \abs{T}$ is categorical in every cardinal $\kappa > \abs{T}$.
The theorem says nothing about $\aleph_0$.
The theory of algebraically closed fields of characteristic $0$ is uncountably categorical but not $\aleph_0$-categorical.
The theory of dense linear orders without endpoints is $\aleph_0$-categorical but not uncountably categorical.

## Explanations

### L1 human

Some theories have only one model of each uncountable size, up to isomorphism.
Morley's theorem says that, for a countable language, having only one model of a single uncountable size already forces this for every uncountable size.
Algebraically closed fields are the typical case: such a field is fixed by its characteristic and how many independent transcendental elements it has.

### L2 human

The proof introduced tools that became the core of stability theory: Morley rank, $\omega$-stability, and prime models over sets.
A countable complete theory with infinite models has one of four patterns of categoricity.
It is categorical in all infinite cardinals, only in $\aleph_0$, only in the uncountable ones, or in none.
Baldwin and Lachlan later showed that an uncountably categorical theory has either one countable model or countably infinitely many.
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# Maximal ergodic theorem

## Statement

Let $(X, \mathcal{A}, \mu)$ be a [[probability space|0NFPJVJJHX]], let $T \colon X \to X$ be a [[measure-preserving transformation|556WBDAHNP]], and let $f \colon X \to \R$ be [[integrable|X7GTF3D98F]].
For $n \ge 1$ let $S_n f = \sum_{k=0}^{n-1} f \circ T^k$ be the [[Birkhoff sums|8TD2KCVH5N]] of $f$, and let
$$E = \set{x \in X : \sup_{n \ge 1} S_n f(x) > 0}.$$
Then
$$\int_E f \, d\mu \ge 0.$$

## Remarks

Proof idea, after Garsia: let $F_N = \max(0, S_1 f, \dots, S_N f)$.
For $1 \le n \le N$ one has $S_n f \le f + F_N \circ T$, so $F_N \le f + F_N \circ T$ on the set $E_N = \set{F_N > 0}$.
Integrating over $E_N$ and using $F_N \ge 0$ and the invariance of $\mu$ gives $\int_{E_N} f \, d\mu \ge \int_X F_N \, d\mu - \int_X F_N \circ T \, d\mu = 0$.
The sets $E_N$ increase to $E$, and dominated convergence finishes the proof.

Applied to $f - \lambda$ for a real number $\lambda > 0$, it gives the maximal inequality
$$\lambda \, \mu\left(\set{x : \sup_{n \ge 1} \tfrac{1}{n} S_n f(x) > \lambda}\right) \le \int_X \abs{f} \, d\mu.$$
The inequality $\int_E f \, d\mu \ge 0$ holds, with the same proof, for a measure-preserving map of any measure space and any integrable $f$.

Example: if $f \ge 0$ is not almost everywhere $0$, then $E = \set{x : f(T^k(x)) > 0 \text{ for some } k \ge 0}$, and the conclusion $\int_E f \, d\mu \ge 0$ is clear; the content of the theorem lies in functions of both signs.

## Explanations

### L1 human

Consider the points from which the running total of $f$ along the orbit becomes positive at some time.
The theorem says that, on this set, the function $f$ has a nonnegative integral.
In other words, the points where the running sum ever gets ahead cannot, on average, be points where $f$ is negative.

### L2 human

This is the key lemma in the standard proof of Birkhoff's pointwise ergodic theorem.
Through the maximal inequality, it controls the supremum of all Birkhoff averages at once.
This control is what is needed to pass from a dense class of functions to all integrable functions.
It plays the same role as the Hardy–Littlewood maximal inequality in the proof of the Lebesgue differentiation theorem.
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# Collineation

## Summary

A collineation is a bijection between the points of two geometries that maps lines onto lines.
It is the notion of isomorphism for [[projective geometries|N32Z68F0D3]] and other incidence structures.
Every [[projective transformation|P1MKTF7MWQ]] of a projective space of dimension at least $2$ is a collineation.
The [[fundamental theorem of projective geometry|GKV4RVXBRE]] says that all collineations come from semilinear maps.
The collineations of a geometry form its collineation group, its group of symmetries.

## Explanations

### L1 human

A collineation moves the points of a geometry around so that points on a common line stay on a common line.
Any invertible linear change of coordinates does this.
Over the complex numbers, so does replacing every coordinate by its complex conjugate.
Swapping just two points of the plane and fixing all others does not: some line through one of them gets broken.
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# Perfect closure

## Definition

Let $K$ be a field of [[characteristic|0P7CH1CND5]] $p > 0$ and let $\overline{K}$ be an [[algebraic closure|0MCXX29HWY]] of $K$.
The *perfect closure* of $K$ is the set
$$K^{p^{-\infty}} = \set{\alpha \in \overline{K} : \alpha^{p^n} \in K \text{ for some integer } n \ge 0}.$$
For a field $K$ of characteristic $0$, the perfect closure of $K$ is $K$.

## Remarks

The set $K^{p^{-\infty}}$ is a subfield of $\overline{K}$, because $(\alpha + \beta)^{p^n} = \alpha^{p^n} + \beta^{p^n}$ in characteristic $p$.
It is a [[perfect field|MY8BG4EE0F]], it is [[purely inseparable|7FR2PZ141W]] over $K$, and it is contained in every perfect subfield of $\overline{K}$ that contains $K$.
It is unique up to a unique isomorphism over $K$.

Example: for $K = \F_p(t)$, the perfect closure is $\bigcup_{n \ge 0} \F_p(t^{1/p^n})$.
Check: each $\F_p(t^{1/p^n})$ is purely inseparable over $K$, and the union is perfect because every element of $\F_p(t^{1/p^n})$ has a $p$-th root in $\F_p(t^{1/p^{n+1}})$.

Non-example: for $p$ odd, the field $\overline{K}$ is perfect and contains $K$, but it is not the perfect closure.
It contains $\sqrt{t}$, and no power $\sqrt{t}^{\,p^n}$ lies in $K$, since $p^n$ is odd.
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<document source="https://tamnd.github.io/math2/node/V8GZ0W1ZZ5.md" tag="V8GZ0W1ZZ5" kind="concept">
# Simplicial complex

## Summary

An abstract simplicial complex is a family of finite sets closed under taking subsets.
Each set of size $k + 1$ stands for a $k$-dimensional simplex: a point, a segment, a triangle, a tetrahedron, and so on.
Simplicial complexes describe spaces by finite combinatorial data, and they link combinatorics with topology and commutative algebra through [[f-vectors and h-vectors|DCK64GFPD7]], homology and Stanley–Reisner rings.

## Explanations

### L1 human

Build a shape by gluing points, segments, triangles and their higher analogues along common faces.
Record each piece by its set of corner labels.
If a triangle $\set{1, 2, 3}$ is in the list, then its edges $\set{1, 2}$, $\set{1, 3}$, $\set{2, 3}$ and its corners must be in the list too.
The list of the three corners and the three edges of a triangle, without the triangle itself, describes a loop: the boundary of the triangle.
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<document source="https://tamnd.github.io/math2/node/V8HDYN3JMY.md" tag="V8HDYN3JMY" kind="statement">
# Tsen's theorem

## Statement

Let $k$ be an [[algebraically closed field|02SJG1HC32]].
Let $K$ be a [[function field|XKEXQDRTGH]] of one variable over $k$, that is, a finitely generated field extension of $k$ of [[transcendence degree|D7TR0YWR9H]] $1$.
Then $K$ is [[quasi-algebraically closed|D637VGM7GV]].
That is, every [[homogeneous polynomial|SD3M01318E]] over $K$ of degree $d \ge 1$ in $n > d$ variables has a nonzero zero in $K^n$.

## Remarks

For $K = k(t)$ the proof is a count of unknowns and equations.
One may assume the coefficients of the form $f$ are polynomials in $t$ of degree at most $m$.
One looks for a zero whose coordinates are polynomials in $t$ of degree at most $N$ with unknown coefficients.
This gives $n(N + 1)$ unknowns in $k$, and $f = 0$ becomes $dN + m + 1$ homogeneous polynomial equations in them.
Since $n > d$, for large $N$ there are more unknowns than equations, and over the algebraically closed field $k$ such a system has a nonzero solution.
A finite extension of $k(t)$ is handled by a similar count, or by restriction of scalars.

A main consequence is that the [[Brauer group|BEW48TA9WZ]] of $K$ is trivial: every central simple algebra over $K$ is a matrix algebra over $K$.
For example, every quaternion algebra over $\C(t)$ splits.

The hypothesis that $k$ is algebraically closed is necessary.
For $k = \R$, the form $x^2 + y^2 + z^2$ over $\R(t)$ has no nontrivial zero.
Indeed, a nontrivial zero could be taken with polynomial coordinates, not all zero.
Evaluating them at a real $t_0$ where they do not all vanish would give a nontrivial real zero of $x^2 + y^2 + z^2$.

## Explanations

### L1 human

Over $\C$, a polynomial in one variable always has a root.
Tsen's theorem says that the field $\C(t)$ of rational functions is only slightly less closed.
A form over $\C(t)$ with more variables than its degree always has a nonzero solution.
For example, the quadratic form $x^2 + t y^2 + (t + 1) z^2$ has a nonzero zero over $\C(t)$.

### L2 human

Tsen's theorem is the basic example of a $C_1$ field besides finite fields and algebraically closed fields.
In arithmetic geometry it says that the function field of a curve over an algebraically closed field behaves like a field of cohomological dimension at most $1$.
Applied to quadratic forms in $3$ variables, it shows that every conic over such a field has a rational point.
The analogous property for finite fields is the [[Chevalley–Warning theorem|08895Q5W96]].
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<document source="https://tamnd.github.io/math2/node/V8JVWE87TE.md" tag="V8JVWE87TE" kind="statement">
# Brauer's theorem on induced characters

## Statement

Let $G$ be a finite [[group|G5S1M9]].
Let $p$ be a prime.
A subgroup $E$ of $G$ is $p$-elementary if it is a [[direct product|00HF4RJ5QN]] $E = C \times P$ of two subgroups.
Here $C$ is a [[cyclic group|0KVGH0J5B3]] of order prime to $p$ and $P$ is a [[$p$-group|06QYRTX9KQ]].
A subgroup is elementary if it is $p$-elementary for some prime $p$.
Then every complex [[character|RMWMYBY8TC]] of $G$ is an integer linear combination of [[induced|JJP4JFH3C6]] characters $\operatorname{Ind}_E^G \lambda$.
Here $E$ runs over the elementary subgroups of $G$ and $\lambda$ over the degree $1$ characters of $E$.

## Remarks

The proof shows that the trivial character $1_G$ lies in the subgroup $I$ of the ring of virtual characters spanned by characters induced from elementary subgroups.
Since $I$ is an ideal, by the formula $\operatorname{Ind}(\theta) \cdot \psi = \operatorname{Ind}(\theta \cdot \psi|_E)$, it then contains every character.
The passage from all characters of elementary subgroups to degree $1$ characters uses that elementary groups are nilpotent, so their irreducible characters are induced from degree $1$ characters.
The coefficients cannot in general be taken nonnegative.
If $G$ is not elementary, every elementary subgroup $E$ is proper, so each $\operatorname{Ind}_E^G \lambda$ has degree $[G : E] \ge 2$.
Then no combination with nonnegative integer coefficients equals the trivial character, of degree $1$.
Compared with [[Artin's induction theorem|2CABBY4EEB]], which uses cyclic subgroups and rational coefficients, Brauer's theorem uses larger subgroups and gains integer coefficients.

## Explanations

### L1 human

Every character of a finite group can be assembled from very simple pieces: one-dimensional characters of small, almost abelian subgroups, pushed up to the whole group by induction.
The price is that one must allow subtraction as well as addition.

### L2 human

A main use is in number theory: Brauer deduced that Artin L-functions extend to meromorphic functions on the whole complex plane, since this holds for Hecke L-functions of one-dimensional characters.
The theorem also gives Brauer's characterisation of characters: a class function is a virtual character if and only if its restriction to every elementary subgroup is a virtual character.
From this one obtains that every complex representation of $G$ can be realised over the field $\Q(\zeta_m)$, where $m$ is the exponent of $G$.
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# Martin's axiom

## Summary

Martin's axiom, MA, is about partial orders with the countable chain condition.
It says that one filter can meet any fewer than continuum many dense sets of such an order.
It is a statement about [[forcing|F2DTWW9HP7]] that can be used without forcing, and the [[precise definition|Z8XMCCZFA4]] is given separately.
It follows from the [[continuum hypothesis|0MN2MER9WN]], but it is also consistent with its failure.
In that case it decides many questions that ZFC leaves open, for instance [[Suslin's problem|D1T39KHPDN]].

## Explanations

### L1 human

Many constructions in mathematics build an object step by step, meeting countably many requirements one after another.
The Baire category theorem says that such constructions work when there are countably many requirements.
Martin's axiom says they still work with more requirements, as long as there are fewer than the number of real numbers and the steps satisfy a mild compatibility condition.
For example, it implies that the union of fewer than continuum many meagre subsets of the real line is meagre.
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<document source="https://tamnd.github.io/math2/node/V8PEAAE4K1.md" tag="V8PEAAE4K1" kind="definition">
# Stone-type duality for varieties

## Definition

Let $\mathcal{A}$ be a [[variety|AJNZJD8442]] or a [[quasivariety|WRXEKT4NQS]] of algebras, regarded as a category whose morphisms are the homomorphisms.
Let $\mathcal{X}$ be a category whose objects are [[Stone spaces|ZV38HA5PAQ]] with additional relations or operations.
Recall that a Stone space is a compact Hausdorff totally disconnected space.
The morphisms of $\mathcal{X}$ are the continuous maps that preserve the additional structure.
A *Stone-type duality* for $\mathcal{A}$ is an [[equivalence|K7A04ZBD92]] between $\mathcal{A}$ and the [[opposite category|DAMK8XE1FR]] $\mathcal{X}^{\mathrm{op}}$.
Equivalently, it is a pair of contravariant functors $D \colon \mathcal{A} \to \mathcal{X}$ and $E \colon \mathcal{X} \to \mathcal{A}$ with natural isomorphisms $A \cong E(D(A))$ and $X \cong D(E(X))$.

## Remarks

In most examples $D$ and $E$ are represented by one set with two structures.
Then $D(A)$ is the set of homomorphisms from $A$ to a fixed finite algebra.
And $E(X)$ is the set of morphisms from $X$ to the same set with a discrete topology.

Example: Boolean algebras and Stone spaces.
Here $D(A)$ is the set of homomorphisms $A \to \set{0, 1}$, a closed subspace of $\set{0, 1}^A$, and $E(X)$ is the Boolean algebra of clopen subsets of $X$.
Both composites are naturally isomorphic to the identity, by Stone's representation theorem.

Non-example: Pontryagin duality between abelian groups and compact abelian groups is a dual equivalence, but not of this type.
The circle group $\R / \Z$ is the dual of $\Z$, and it is connected, so it is not a Stone space.

Bounded distributive lattices are dual to Priestley spaces, which are Stone spaces with a partial order satisfying a separation condition.
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# Householder transformation

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $v \in \mathbb{K}^n$ be a vector with $v^* v = 1$, where $v^*$ is the [[conjugate transpose|0EX045E2BP]] of $v$.
The *Householder transformation* (or *Householder reflector*) defined by $v$ is the $n \times n$ matrix
$$P = I - 2 v v^*.$$

## Remarks

$P$ is Hermitian and [[unitary|0D6RMJ10M9]], and $P^2 = I$; for $\mathbb{K} = \R$ it is a symmetric [[orthogonal matrix|0QRBVFJ50X]] with determinant $-1$.
It fixes every vector orthogonal to $v$ and sends $v$ to $-v$, so it is the [[reflection|025158VEYZ]] in the hyperplane $v^\perp$.
If $x, y \in \R^n$ are distinct with $\norm{x} = \norm{y}$, then $v = (x - y)/\norm{x - y}$ gives $P x = y$.
Example: in $\R^2$, take $x = (3, 4)$ and $y = (5, 0)$; then $v = (-2, 4)/\sqrt{20}$, and
$$P = I - \frac{2}{20} \begin{pmatrix} 4 & -8 \\ -8 & 16 \end{pmatrix} = \frac{1}{5} \begin{pmatrix} 3 & 4 \\ 4 & -3 \end{pmatrix},$$
and indeed $P (3, 4)^T = (5, 0)^T$.
Non-example: $v = (2, 0)$ is not a unit vector, and $I - 2 v v^T = \begin{pmatrix} -7 & 0 \\ 0 & 1 \end{pmatrix}$ is not orthogonal.
For a nonzero $u$ that is not normalized, one writes $P = I - 2 u u^* / (u^* u)$.
In computations, $P$ is never formed: $P A = A - 2 v (v^* A)$ costs $O(n m)$ operations for an $n \times m$ matrix $A$.

## Explanations

### L1 human

A Householder transformation is a mirror reflection written as a matrix.
The unit vector $v$ points straight out of the mirror.
Reflecting twice gives back the original vector, and lengths never change.
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<document source="https://tamnd.github.io/math2/node/V8ZZ7A2FMF.md" tag="V8ZZ7A2FMF" kind="concept">
# von Mises–Fisher distribution

## Summary

The von Mises–Fisher distribution is a probability distribution on the unit [[sphere|T62369RFH6]] that is concentrated around a mean direction.
Its density is proportional to $\exp(\kappa \mu^T x)$, where $\mu$ is the mean direction and $\kappa \ge 0$ controls the concentration.
It plays the role of the normal distribution in [[directional statistics|8GZJQBPBY0]].
On the circle it is the von Mises distribution, and on the ordinary sphere it is the Fisher distribution.

## Explanations

### L1 human

Imagine arrows that mostly point north but wobble a little around it.
The von Mises–Fisher distribution is the standard model for such arrows in any dimension.
The number $\kappa$ says how tightly the arrows cluster: $\kappa = 0$ means no preferred direction at all, and large $\kappa$ means a tight bundle around $\mu$.
For large $\kappa$, the arrows near $\mu$ look like a normal distribution on the tangent plane.
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<document source="https://tamnd.github.io/math2/node/V9079C40AZ.md" tag="V9079C40AZ" kind="definition">
# Sullivan minimal model

## Definition

Work over $\Q$ with cochain [[differential graded algebras|GQVAXW5X09]] (dgas) that are graded-commutative: $ab = (-1)^{\abs{a}\abs{b}} ba$ for homogeneous $a$ and $b$.
For a graded vector space $V = \bigoplus_{n \ge 1} V^n$, let $\Lambda V$ be the free graded-commutative algebra on $V$.
It is the tensor product of the [[symmetric algebra|YZCBSRCZC7]] on the even part and the [[exterior algebra|08HK38GCSM]] on the odd part.
Let $\Lambda^{\ge 2} V$ be its ideal of decomposable elements, the span of products of at least two elements of $V$.

1. A *Sullivan algebra* is a dga $(\Lambda V, d)$ with $V = V^{\ge 1}$ and an increasing chain of graded subspaces $V(0) \subseteq V(1) \subseteq \cdots$ with union $V$.
   The chain must satisfy $d(V(0)) = 0$ and $d(V(k)) \subseteq \Lambda V(k-1)$ for $k \ge 1$.
2. A Sullivan algebra is *minimal* if $d(V) \subseteq \Lambda^{\ge 2} V$.
3. Let $(A, d_A)$ be a graded-commutative dga with $H^0(A) = \Q$.
   A *Sullivan minimal model* of it is a minimal Sullivan algebra $(\Lambda V, d)$ with a dga map $m \colon (\Lambda V, d) \to (A, d_A)$ that is an isomorphism in cohomology.

The minimal model of a path-connected space $X$ is the minimal model of Sullivan's algebra $A_{PL}(X)$ of rational polynomial differential forms on $X$.
The cohomology of $A_{PL}(X)$ is the rational [[singular cohomology|6N7BC70CX0]] $H^*(X; \Q)$.

## Remarks

If $X$ is simply connected and each $H^n(X; \Q)$ is finite-dimensional, then $V^n \cong \Hom(\pi_n(X), \Q)$ for all $n \ge 2$, and $V^1 = 0$.

Example: for $n$ odd, $(\Lambda(x), 0)$ with $\abs{x} = n$ is the minimal model of $S^n$.
Here $\Lambda(x) = \Q \oplus \Q x$ since $x^2 = 0$ by graded commutativity, which is $H^*(S^n; \Q)$.

Example: for $n$ even, the minimal model of $S^n$ is $(\Lambda(x, y), d)$ with $\abs{x} = n$, $\abs{y} = 2n - 1$, $dx = 0$ and $dy = x^2$.
Its cohomology has basis $1, x$: each $x^k$ with $k \ge 2$ equals $d(x^{k-2} y)$, and $d(x^k y) = x^{k+2} \ne 0$.

Non-example: $(\Lambda(x, y), d)$ with $\abs{y} = 1$, $\abs{x} = 2$ and $dy = x$ is a Sullivan algebra but not minimal, since $dy = x$ is not decomposable.
It is acyclic in positive degrees, and its minimal model is $\Q$.
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<document source="https://tamnd.github.io/math2/node/V90RGBV52W.md" tag="V90RGBV52W" kind="concept">
# Newtonian space

## Summary

The Newtonian space $N^{1,p}(X)$ is a version of the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}$ for functions on a metric measure space $X$.
It replaces the gradient by an upper gradient, a function that bounds the change of $u$ along every rectifiable curve.
It was introduced by Nageswari Shanmugalingam, and it is a [[Banach space|06GZ956657]] for $1 \le p < \infty$.
It is the basic function space of nonlinear potential theory on metric spaces, where it carries $p$-capacities and [[p-harmonic functions|7BKQT71FYZ]].

## Explanations

### L1 human

In $\R^n$ a Sobolev function is one whose gradient has finite size in an average sense.
A general metric space has no gradient, but one can still measure how fast a function changes along curves.
An upper gradient of $u$ is a function $g$ with this property: along any curve, the change of $u$ between the ends is at most the integral of $g$.
For example, on the real line the function $u(x) = \abs{x}$ has the upper gradient $g = 1$, because $\abs{\abs{b} - \abs{a}} \le \abs{b - a}$.
The Newtonian space collects the functions that, together with some upper gradient, have finite $L^p$ norm.
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<document source="https://tamnd.github.io/math2/node/V93Y0XYPZD.md" tag="V93Y0XYPZD" kind="concept">
# Serre's twisting sheaf

## Summary

Let $S$ be a graded ring.
Serre's twisting sheaf $\mathcal{O}(1)$ on $\operatorname{Proj} S$ is the sheaf attached to the graded module $S$ shifted by one degree.
On projective space $\mathbb{P}^n_k$ it is the line bundle whose global sections are the linear forms in the homogeneous coordinates.
Its powers $\mathcal{O}(m)$ turn graded modules into sheaves and sheaves back into graded modules, and they control embeddings into projective space.

## Explanations

### L1 human

A homogeneous polynomial of degree $m$ is not a function on projective space, since its value changes when one rescales the coordinates.
It is a section of a line bundle instead, and this line bundle is $\mathcal{O}(m)$.
For example, on the projective line, the linear forms $x_0$ and $x_1$ are sections of $\mathcal{O}(1)$, and $x_0^2 + x_1^2$ is a section of $\mathcal{O}(2)$.
The zeros of such a section are well defined, and they form a hypersurface of projective space.
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<document source="https://tamnd.github.io/math2/node/V94VYJ2RR1.md" tag="V94VYJ2RR1" kind="definition">
# Two-sided Laplace transform

## Definition

Let $f \colon \R \to \C$ be [[integrable|DWJF7ABPR1]] on each bounded interval.
The *two-sided Laplace transform* of $f$ is the function
$$\mathcal{B}\{f\}(s) = \int_{-\infty}^{\infty} f(t) \, e^{-s t} \, dt = \lim_{T_1, T_2 \to \infty} \int_{-T_1}^{T_2} f(t) \, e^{-s t} \, dt.$$
Here $T_1$ and $T_2$ tend to $\infty$ independently.
The transform is defined at each complex $s$ for which this [[improper integral|05KQ3F99BQ]] converges.

## Remarks

Example: for $f(t) = e^{-\abs{t}}$, the part over $[0, \infty)$ is $1/(1 + s)$ for $\operatorname{Re} s > -1$, and the part over $(-\infty, 0]$ is $1/(1 - s)$ for $\operatorname{Re} s < 1$.
So $\mathcal{B}\{f\}(s) = \frac{2}{1 - s^2}$ for $-1 < \operatorname{Re} s < 1$.
Non-example: for the constant $f = 1$, the part over $[0, \infty)$ needs $\operatorname{Re} s > 0$ and the part over $(-\infty, 0]$ needs $\operatorname{Re} s < 0$, so the integral converges for no $s$.
If $f(t) = 0$ for $t < 0$, the two-sided transform is the one-sided [[Laplace transform|05XT5AVJBK]] of $f$ restricted to $[0, \infty)$.
For a random variable $X$ with density $p$, $\mathcal{B}\{p\}(-s) = \mathbb{E}[e^{s X}]$ is its moment generating function wherever it converges.
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# van der Waerden's theorem

## Statement

Let $r \ge 1$ and $k \ge 1$ be integers.
Then there is a positive integer $N$ with the following property.
Let $\set{1, \ldots, N}$ be coloured with $r$ colours.
Then there are integers $a \ge 1$ and $d \ge 1$ such that the $k$ numbers $a, a + d, \ldots, a + (k-1)d$ lie in $\set{1, \ldots, N}$ and all have the same colour.

## Remarks

The least such $N$ is the van der Waerden number $W(r, k)$.
For example $W(2, 3) = 9$: the colouring $1, 2, 5, 6$ red and $3, 4, 7, 8$ blue of $\set{1, \ldots, 8}$ has no monochromatic three-term progression, while every $2$-colouring of $\set{1, \ldots, 9}$ has one.
The classical proof is a double induction with "colour focusing" and gives bounds that grow like the Ackermann function.
The theorem also follows from the [[Hales–Jewett theorem|14SGZ5HA4Y]]: a combinatorial line in $\set{0, \ldots, k-1}^n$ is sent to an arithmetic progression by the map $w \mapsto w_1 + \cdots + w_n$.
Shelah gave primitive recursive bounds, and Gowers gave the bound $W(r, k) \le 2^{2^{r^{2^{2^{k + 9}}}}}$ through his proof of [[Szemerédi's theorem|B23B0QEMN5]].

## Explanations

### L1 human

However you colour the positive integers with finitely many colours, one colour contains arithmetic progressions of every length.
The theorem says more: for a given length and number of colours, a fixed finite stretch $1, \ldots, N$ already suffices.

### L2 human

Van der Waerden's theorem is a cornerstone of [[Ramsey theory|G5TWXV565S]] in the integers.
Its density form is [[Szemerédi's theorem|B23B0QEMN5]]: a set of integers of positive upper density contains arbitrarily long progressions.
The colour classes of a colouring cannot all have density zero, so the density form implies the colouring form.
The [[Green–Tao theorem|H8BAHWSEMC]] gives progressions in the primes, which have density zero.
Exact values of $W(r, k)$ are known only for a few small cases.
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<document source="https://tamnd.github.io/math2/node/V9BRV06TVY.md" tag="V9BRV06TVY" kind="definition">
# Unital

## Definition

Let $n \ge 2$ be an integer.
A *unital of order $n$* is a [[block design|4JCSS2Z1BA]] $(X, \mathcal{B})$ such that:

1. the point set $X$ has $n^3 + 1$ elements;
2. every block $B \in \mathcal{B}$ has $n + 1$ elements;
3. any two distinct points of $X$ lie in exactly one common block.

## Remarks

Equivalently, a unital of order $n$ is a [[Steiner system|7BPSC4T4M3]] $S(2, n + 1, n^3 + 1)$.

Let $\Pi$ be a [[projective plane|DANFNN8C4Y]] of order $n^2$.
A unital is *embedded* in $\Pi$ if $X$ is a set of points of $\Pi$ and the blocks are the intersections of $X$ with lines that have size $n + 1$.
In that case every line of $\Pi$ meets $X$ in $1$ or $n + 1$ points.

Example: let $q$ be a prime power and let $\F_{q^2}$ be the [[finite field|0ZXZ82FTX2]] with $q^2$ elements.
In $\mathrm{PG}(2, q^2)$, the Hermitian curve $x^{q+1} + y^{q+1} + z^{q+1} = 0$, the set of isotropic points of a non-degenerate [[Hermitian form|48P4A1CX85]], has $q^3 + 1$ points.
Each line meets it in $1$ or $q + 1$ points, so with the secant lines as blocks it is a unital of order $q$.

Non-example: in the Fano plane, blocks have size $3$ and any two points lie in exactly one block.
But it has $7$ points rather than $2^3 + 1 = 9$, so clause 1 fails.
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# X-ray transform

## Summary

The X-ray transform sends a function $f$ on $\R^n$ to the function that gives the integral of $f$ along each line.
In the plane it is the [[Radon transform|VJNZJ352PC]], up to the parametrization of lines.
In higher dimensions lines are smaller than hyperplanes, and the two transforms differ.
It models what X-ray tomography measures in three dimensions, where the data are line integrals.
It is injective on integrable functions, and in dimension $n \ge 3$ the data are overdetermined, since the set of lines has dimension $2n - 2 > n$.

## Explanations

### L1 human

An X-ray passes through a body along a straight line, and the detector measures the total density along that line.
The X-ray transform collects these totals for all lines.
For example, the integral of a ball of uniform density $1$ and radius $a$ along a line at distance $d < a$ from the center is the chord length $2\sqrt{a^2 - d^2}$.
From all these totals one can recover the density, which is what three-dimensional scanners do.
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# Malcev completion

## Summary

The Malcev completion of a finitely generated torsion-free nilpotent group $\Gamma$ is a simply connected [[nilpotent Lie group|DK87YN458D]] $N$ that contains $\Gamma$ as a lattice.
It exists and is unique up to isomorphism, by a theorem of Malcev.
It turns questions about the discrete group $\Gamma$ into questions about the Lie algebra of $N$.
In this way compact nilmanifolds correspond to finitely generated torsion-free nilpotent groups.

## Explanations

### L1 human

The integer lattice $\Z^n$ sits inside the space $\R^n$, and $\R^n$ is the smallest connected group in which $\Z^n$ is spread out evenly.
The Malcev completion does the same for groups that are nearly commutative, the nilpotent groups.
For example, the $3 \times 3$ upper triangular integer matrices with ones on the diagonal sit inside the same matrices with real entries.
The real group is the Malcev completion of the integer group.
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# Connection on a vector bundle

## Definition

Let $M$ be a smooth manifold, and let $\pi : E \to M$ be a smooth real vector bundle.
Let $\Gamma(E)$ be the space of smooth [[sections|WWX7H5W1RP]] of $E$, and let $\mathfrak{X}(M)$ be the space of smooth vector fields on $M$.
A *connection* on $E$ is a map
$$\nabla : \mathfrak{X}(M) \times \Gamma(E) \to \Gamma(E), \quad (X, s) \mapsto \nabla_X s,$$
such that for all $X, Y \in \mathfrak{X}(M)$, $s, t \in \Gamma(E)$, $f, h \in C^\infty(M)$ and $a, b \in \R$:

1. $\nabla_{fX + hY} s = f \nabla_X s + h \nabla_Y s$.
2. $\nabla_X (a s + b t) = a \nabla_X s + b \nabla_X t$.
3. $\nabla_X (f s) = (X f) \, s + f \nabla_X s$.

## Remarks

Example: on the trivial bundle $E = M \times \R^r$, a section is a smooth map $s = (s^1, \dots, s^r)$, and $\nabla_X s = (X s^1, \dots, X s^r)$ is a connection; clause 3 is the product rule $X(f s^i) = (Xf) s^i + f X s^i$.
Non-example: on the [[tangent bundle|WY6K3P4YCG]], the map $(X, Y) \mapsto [X, Y]$ satisfies clause 2 but fails clause 1, since $[fX, Y] = f[X, Y] - (Yf) X$.
By clause 1, $(\nabla_X s)(p)$ depends only on $X_p$, so $\nabla$ can be seen as a map $\Gamma(E) \to \Gamma(T^*M \otimes E)$.
In a local frame $e_1, \dots, e_r$ of $E$, $\nabla e_j = \sum_i \omega^i_j \otimes e_i$ for a matrix $(\omega^i_j)$ of $1$-forms, and locally $\nabla = d + \omega$.
The difference of two connections is $C^\infty(M)$-linear in both arguments, so it is a section of $T^*M \otimes \mathrm{End}(E)$; thus connections form an affine space over this space of sections.
Complex vector bundles are treated the same way with complex scalars.
</document>

<document source="https://tamnd.github.io/math2/node/V9K9TG0K3J.md" tag="V9K9TG0K3J" kind="statement">
# Plancherel theorem for abelian groups

## Statement

Let $G$ be a [[locally compact abelian group|ZR09S9R3AW]] with a [[Haar measure|VY0HXD3GC4]] $\mu$, and let $\widehat{G}$ be its [[Pontryagin dual|FMW57WNPDB]].
For $f \in L^1(G)$ let $\widehat{f}$ be its [[Fourier transform|VHN5VW5Q92]].
Then there is a unique Haar measure $\nu$ on $\widehat{G}$ such that:

1. for every $f \in L^1(G) \cap L^2(G)$, the function $\widehat{f}$ lies in $L^2(\widehat{G}, \nu)$ and $\norm{\widehat{f}}_{L^2(\nu)} = \norm{f}_{L^2(\mu)}$;
2. the map $f \mapsto \widehat{f}$ on $L^1(G) \cap L^2(G)$ extends uniquely to a unitary isomorphism of [[Hilbert spaces|HM2HHE3J51]] $L^2(G, \mu) \to L^2(\widehat{G}, \nu)$.

## Remarks

The measure $\nu$ is called the dual measure of $\mu$; scaling $\mu$ by $c > 0$ scales $\nu$ by $1/c$.
One proof applies [[Bochner's theorem|ZRHAG4NXHC]] to functions $f * \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$, whose transform is $\abs{\widehat{f}}^2$.
This yields a Haar measure on $\widehat{G}$ for which the inversion formula holds for such functions.
Evaluating $f * \tilde{f}$ at $0$ then gives the isometry, and density of $L^1 \cap L^2$ in $L^2$ gives the extension.
Example: for $G = \mathbb{T}$ with normalised arc length, $\widehat{G} = \Z$, $\nu$ is counting measure, and the theorem is Parseval's identity $\frac{1}{2\pi}\int_0^{2\pi} \abs{f(\theta)}^2 \, d\theta = \sum_{n \in \Z} \abs{\widehat{f}(n)}^2$.
For $G = \R$ with Lebesgue measure and characters $x \mapsto e^{2\pi i \xi x}$, the dual measure is again Lebesgue measure.
The general [[Plancherel theorem for groups|DH3PYQJXH0]] extends the result to some non-abelian groups, where the dual is the unitary dual and $\nu$ becomes the Plancherel measure.

## Explanations

### L1 human

The energy of a function, the integral of its squared size, is the same as the total energy of its Fourier transform.
No energy is lost or created when a function is split into waves.
On the circle this is the familiar fact that the mean square of a periodic function is the sum of the squares of its Fourier coefficients.

### L2 human

The theorem makes the Fourier transform a unitary operator on $L^2$, so one can transform any square-integrable function, not only integrable ones.
It fixes a normalisation of the Haar measure on the dual group, which is the one used in the Fourier inversion formula.
Together with [[Pontryagin duality|QTVC6F2P9D]] it shows that $L^2(G)$ is diagonalised by the characters: translation operators become multiplication operators on $L^2(\widehat{G})$.
</document>

<document source="https://tamnd.github.io/math2/node/V9R8K4JCVM.md" tag="V9R8K4JCVM" kind="definition">
# Numerical methods for linear least squares

## Definition

Let $A \in \R^{m \times n}$ and $b \in \R^m$.
The *linear least squares problem* is to find $x \in \R^n$ that minimizes $\norm{Ax - b}_2$.
When $A$ has rank less than $n$, one asks for the minimizer of smallest norm.
This is $x = A^\dagger b$, with $A^\dagger$ the [[Moore–Penrose inverse|0YY8VX28JR]].
A *numerical method for linear least squares* is an algorithm that computes this minimizer from $A$ and $b$.
It is exact in exact arithmetic, and accurate up to rounding errors in floating point arithmetic.
The standard methods are:

1. the normal equations $A^T A x = A^T b$, solved by a [[Cholesky decomposition|0WKG8CYY6A]] of $A^T A$ when $A$ has full column rank;
2. a [[QR decomposition|0ERWG4NXQ9]] $A = QR$ with $Q \in \R^{m \times n}$ having orthonormal columns and $R$ upper triangular, when $A$ has full column rank, followed by the solution of $R x = Q^T b$;
3. the singular value decomposition $A = U \Sigma V^T$, followed by $x = V \Sigma^\dagger U^T b$.

## Remarks

Example: for fitting a line $y = c_0 + c_1 t$ to the points $(0, 0)$, $(1, 1)$, $(2, 1)$, take $A$ with rows $(1, 0)$, $(1, 1)$, $(1, 2)$ and $b = (0, 1, 1)$.
Then $A^T A = \begin{pmatrix} 3 & 3 \\ 3 & 5 \end{pmatrix}$ and $A^T b = (2, 3)$, so $c_0 = 1/6$ and $c_1 = 1/2$; the residual $(-1/6, 1/3, -1/6)$ is orthogonal to both columns of $A$, as it must be.
Non-example: solving the square system formed by the first $n$ rows of $Ax = b$ fits those rows exactly, but it does not give the minimizer in general.
Here it gives $c_0 = 0$, $c_1 = 1$, with residual norm $1$ instead of $1/\sqrt{6}$.
The normal equations are the cheapest method but square the condition number, $\kappa(A^T A) = \kappa(A)^2$.
Householder QR is backward stable and is the standard choice for dense full-rank problems; the SVD is the most robust for rank-deficient problems, at a higher cost.
For large sparse problems one uses iterative methods such as LSQR or the conjugate gradient method applied to the normal equations.

## Explanations

### L1 human

All these methods compute the same answer in exact arithmetic: the best fit in the sense of squared errors.
They differ in speed and in how many digits they lose to rounding.
</document>

<document source="https://tamnd.github.io/math2/node/VA2JEWWB61.md" tag="VA2JEWWB61" kind="statement">
# Remmert's proper mapping theorem

## Statement

Let $X$ and $Y$ be [[complex analytic spaces|HXSZA44J5J]] and let $f \colon X \to Y$ be a holomorphic map that is [[proper|15XEBGNGHC]].
Let $A \subseteq X$ be an [[analytic subset|GB7M582M5Y]].
Then $f(A)$ is an analytic subset of $Y$, and its [[dimension|AN0BYRKNRA]] is at most the dimension of $A$.

## Remarks

Properness is necessary.
The projection $f \colon \C^2 \to \C$, $(z_1, z_2) \mapsto z_1$, maps the analytic set $A = \set{z_1 z_2 = 1}$ onto $\C \setminus \set{0}$, which is not closed and so not analytic; here $f$ restricted to $A$ is not proper.

Example: the projection $\mathbb{P}^m \times Y \to Y$ is proper because $\mathbb{P}^m$ is compact, so the image of any analytic subset of $\mathbb{P}^m \times Y$ in $Y$ is analytic.
This is the analytic form of the main theorem of elimination theory.

One proof derives it from the coherence of direct images.
Give $A$ its reduced structure.
Then $f_* \mathcal{O}_A$ is a coherent sheaf on $Y$ by Grauert's direct image theorem, and its support is $f(A)$.
The support of a coherent sheaf is analytic.

## Explanations

### L1 human

If a holomorphic map does not let points escape to infinity, then it sends sets defined by holomorphic equations to sets of the same kind.
The image cannot be larger in dimension than the set itself.

### L2 human

The theorem is a basic tool for building analytic sets.
For example, the graph of a holomorphic map is analytic, and its image under a proper projection is analytic.
Remmert proved it before the direct image theorem was known, and it was an early milestone of the global theory of complex spaces.
</document>

<document source="https://tamnd.github.io/math2/node/VA3DY2YX83.md" tag="VA3DY2YX83" kind="statement">
# Grothendieck's local duality

## Statement

Let $(R, \mathfrak{m}, k)$ be a [[Cohen–Macaulay|N1QJQEGBFC]] [[local ring|MZ02BKPHJA]] of dimension $d$ with [[residue field|NJ3APQ8BHH]] $k$, and assume that $R$ has a [[canonical module|W5ECXPM6FW]] $\omega_R$.
Let $E = E_R(k)$ be the [[injective hull|QCJN05GN64]] of $k$.
Let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module and let $i$ be an integer.
Then there is an isomorphism, natural in $M$,
$$H^i_{\mathfrak{m}}(M) \cong \Hom_R\left(\operatorname{Ext}^{d-i}_R(M, \omega_R), E\right),$$
where $H^i_{\mathfrak{m}}$ is [[local cohomology|6JB30BXDA2]] and $\operatorname{Ext}$ is the [[Ext functor|4P7GBWJ4RR]].

## Remarks

If $R$ is Gorenstein, then $\omega_R = R$ and the formula reads $H^i_{\mathfrak{m}}(M) \cong \Hom_R(\operatorname{Ext}^{d-i}_R(M, R), E)$.
For $M = R$ and $i = d$ it gives $H^d_{\mathfrak{m}}(R) \cong \Hom_R(\omega_R, E)$.
Example: for $R = K[\![x]\!]$, one has $d = 1$, $\omega_R = R$ and $E = K[\![x]\!]_x / K[\![x]\!]$, and indeed $H^1_{\mathfrak{m}}(R) = R_x / R \cong E = \Hom_R(R, E)$.
When $R$ is complete, [[Matlis duality|2V9BR5XX6A]] turns the formula around: $\operatorname{Ext}^{d-i}_R(M, \omega_R) \cong \Hom_R(H^i_{\mathfrak{m}}(M), E)$.

## Explanations

### L1 human

Local cohomology modules are large and hard to see directly.
Local duality says that they are the duals of Ext modules, which are finitely generated and computable.
It is an algebraic analogue of Serre duality for projective varieties.

### L2 human

Combined with the vanishing pattern of local cohomology, local duality shows that $M$ is Cohen–Macaulay of dimension $t$ exactly when $\operatorname{Ext}^{d-i}_R(M, \omega_R) = 0$ for all $i \ne t$.
It also identifies the canonical module as the Matlis dual of the top local cohomology $H^d_{\mathfrak{m}}(R)$ when $R$ is complete.
</document>

<document source="https://tamnd.github.io/math2/node/VA41ZY2HTF.md" tag="VA41ZY2HTF" kind="definition">
# Term evaluation

## Definition

Let $\sigma$ be a [[signature|WDVVEYD6AT]] with only function symbols and let $X$ be a set of variables.
Let $A$ be a [[structure|983GZBQYT4]] of signature $\sigma$, that is, an algebra, with operations $f^A$.
Let $\alpha \colon X \to A$ be a map, called an *assignment*.
The *value* $t^A[\alpha]$ of a [[term|6D3WA7N2W4]] $t$ over $\sigma$ and $X$ is defined by recursion on $t$:

1. if $t$ is a variable $x \in X$, then $t^A[\alpha] = \alpha(x)$;
2. if $t$ is a constant symbol $c$, then $t^A[\alpha] = c^A$;
3. if $t = f(t_1, \ldots, t_n)$ for a function symbol $f$ of arity $n \ge 1$, then $t^A[\alpha] = f^A(t_1^A[\alpha], \ldots, t_n^A[\alpha])$.

## Remarks

The recursion is well defined because every term has a unique reading.
The value depends only on the values of $\alpha$ at the variables that occur in $t$.
If these variables are among $x_1, \ldots, x_n$ and $\alpha(x_i) = a_i$, one writes $t^A(a_1, \ldots, a_n)$.

Example: in the ring $(\Z, +, \cdot, 0, 1)$, let $t = x \cdot (y + 1)$ and $\alpha(x) = 3$, $\alpha(y) = 4$.
Clause 1 gives $x \mapsto 3$ and $y \mapsto 4$, clause 2 gives $1 \mapsto 1$, and clause 3 twice gives $y + 1 \mapsto 5$ and $t \mapsto 15$.

Non-example: computing $x \cdot (y + 1)$ at the same assignment as $3 \cdot 4 + 1 = 13$ is not term evaluation.
The outermost symbol of $t$ is $\cdot$ applied to $x$ and $y + 1$, so clause 3 requires the product of the values $3$ and $5$.

Homomorphisms commute with evaluation: if $h \colon A \to B$ is a homomorphism, then $h(t^A[\alpha]) = t^B[h \circ \alpha]$.
In Mathlib term evaluation is `FirstOrder.Language.Term.realize`.
</document>

<document source="https://tamnd.github.io/math2/node/VA6BNNY2YC.md" tag="VA6BNNY2YC" kind="statement">
# Church's theorem on undecidability of first-order logic

## Statement

There is a finite signature $L$ with the following property.
The set of valid first-order [[sentences|DKFGQTRYZE]] of $L$, that is, of the $L$-sentences true in every $L$-structure, is not a [[computable set|PPGD5KW8JC]].
Equivalently, no algorithm decides, for an arbitrary $L$-sentence, whether it is valid in [[first-order logic|086QKPF0A6]].

## Remarks

Church proved the result in 1936 with the lambda calculus, and Turing proved it independently with [[Turing machines|758EYCN27B]].
In Turing's approach, one builds from each Turing machine $M$ a sentence $\varphi_M$ that is valid exactly when $M$ halts on the empty input.
A decision procedure for validity would then solve the [[halting problem|Z7TN2B35B0]], which is impossible.

The theorem gives a negative answer to Hilbert's Entscheidungsproblem.
It holds for every signature that contains a binary relation symbol.
It fails for signatures with only unary relation symbols: monadic first-order logic is decidable.

By [[Gödel's completeness theorem|WQ1ZE255SF]], the valid sentences are exactly the provable ones, so they form a computably enumerable set.
So validity is semi-decidable but not decidable.

## Explanations

### L1 human

The theorem says that there is no general mechanical method to decide whether a first-order sentence is logically true.
A computer can list all proofs and so confirm every valid sentence eventually, but it cannot always confirm that a sentence is not valid.

### L2 human

Church's theorem is the first-order counterpart of the undecidability of the halting problem.
It is the reason why automated theorem provers can only search for proofs, and may run forever.
Decidable fragments, such as monadic logic, and decidable theories, such as [[Presburger arithmetic|B66JYY3K58]], are therefore valuable in computer science.
</document>

<document source="https://tamnd.github.io/math2/node/VA7AK1R5AN.md" tag="VA7AK1R5AN" kind="definition">
# Type

## Definition

Let $L$ be a [[signature|WDVVEYD6AT]], let $T$ be an $L$-[[theory|92SWXST1ZM]], and let $n \ge 1$.
An *$n$-type* of $T$ is a set $p(x_1, \dots, x_n)$ of $L$-[[formulas|HYJ10AF6RT]] whose free variables are among $x_1, \dots, x_n$, such that $T \cup p$ is satisfiable.
This means that some [[model|MVCH6N9SNW]] $M$ of $T$ has a tuple $(a_1, \dots, a_n)$ that satisfies every formula of $p$.
The type $p$ is *complete* if for every $L$-formula $\varphi(x_1, \dots, x_n)$, either $\varphi \in p$ or $\neg \varphi \in p$.
A model $M$ of $T$ *realizes* $p$ if some tuple of $M$ satisfies every formula of $p$, and *omits* $p$ otherwise.
The set of complete $n$-types of $T$ is written $S_n(T)$.

Let $M$ be a model and $A$ a set of elements of $M$.
Let $L_A$ be $L$ extended by a new constant for each element of $A$.
A type over $A$ is a type of the $L_A$-theory of $M$.
The set of complete $n$-types over $A$ is written $S_n^M(A)$.

## Remarks

For a tuple $\bar{a}$ of a model $M$, the set $\operatorname{tp}(\bar{a})$ of all formulas satisfied by $\bar{a}$ in $M$ is a complete type of the theory of $M$.

Example: let $T$ be the theory of $(\N, 0, 1, +, \cdot, <)$ and let $p(x) = \set{x > 1 + \cdots + 1 \ (k \text{ terms}) : k \ge 1}$.
Each finite part of $p$ is satisfied in $\N$ by a large number, so by compactness $T \cup p$ has a model, and $p$ is a $1$-type.
The standard model $\N$ omits $p$, while every nonstandard model of $T$ realizes it.

Non-example: in the theory of fields, $\set{x = 0, \; x \cdot x = 1}$ is not a type, since $x = 0$ gives $x \cdot x = 0 \ne 1$ in every field.

The set $S_n(T)$ carries a natural topology, with basic open sets $\set{p : \varphi \in p}$; it is a compact, totally disconnected Hausdorff space by the compactness theorem.
</document>

<document source="https://tamnd.github.io/math2/node/VA7MH05Z3B.md" tag="VA7MH05Z3B" kind="concept">
# Zeilberger's algorithm

## Summary

Zeilberger's algorithm, also called the method of creative telescoping, takes a definite sum $f(n) = \sum_k F(n, k)$ of a hypergeometric term.
It finds a linear [[recurrence|0FYX44GP92]] in $n$ with polynomial coefficients that $f$ satisfies.
It works by running a parametrised version of [[Gosper's algorithm|NHMJQBS04M]] on a combination of shifts $F(n + j, k)$.
With the recurrence and a few initial values, identities for definite sums can be proved or discovered automatically.
It is a main tool of the "A = B" approach to hypergeometric identities.

## Explanations

### L1 human

Many identities say that a sum over $k$ of binomial-like terms has a simple value depending on $n$.
Zeilberger's algorithm does not guess the value; it finds a recurrence in $n$ that the sum must satisfy.
For $\sum_k \binom{n}{k}$ it finds $f(n + 1) = 2f(n)$, and with $f(0) = 1$ this gives $2^n$.
If the claimed value satisfies the same recurrence and the same initial values, the identity is proved.
</document>

<document source="https://tamnd.github.io/math2/node/VA8R0TZCKB.md" tag="VA8R0TZCKB" kind="statement">
# Masur–Veech theorem

## Statement

Let $g \ge 1$ and let $k_1, \dots, k_n \ge 0$ be integers with $k_1 + \cdots + k_n = 2g - 2$.
Let $\mathcal{H}_1(k_1, \dots, k_n)$ be the stratum of [[translation surfaces|CEHHBBD6NT]] $(X, \omega)$ of genus $g$ and area $\int_X \frac{i}{2} \omega \wedge \bar\omega = 1$ whose form $\omega$ has $n$ zeros of orders $k_1, \dots, k_n$.
Let $\nu$ be the Masur–Veech measure on it, and let $g_t$, $t \in \R$, be the Teichmüller geodesic flow, the action of the diagonal matrices $\begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}$.
Let $\mathcal{C}$ be a connected component of $\mathcal{H}_1(k_1, \dots, k_n)$.
Then:

1. The total measure $\nu(\mathcal{C})$ is finite.
2. The flow $g_t$ on $\mathcal{C}$ is ergodic with respect to $\nu$.

## Remarks

The stratum $\mathcal{H}(k_1, \dots, k_n)$ of all such surfaces, of any area, has local period coordinates.
Take a basis $\gamma_1, \dots, \gamma_{2g + n - 1}$ of the relative homology $H_1(X, \Sigma; \Z)$, where $\Sigma$ is the set of zeros.
Then the map $(X, \omega) \mapsto (\int_{\gamma_j} \omega)_j \in \C^{2g + n - 1}$ is a local chart.
The Masur–Veech measure on $\mathcal{H}$ is Lebesgue measure in these charts, normalized so that the lattice of periods in $\Z + i\Z$ has covolume $1$.
The measure $\nu$ on $\mathcal{H}_1$ is the cone measure: $\nu(E)$ is the Lebesgue measure of the set of $r \cdot (X, \omega)$ with $(X, \omega) \in E$ and $0 < r \le 1$.
The group $SL(2, \R)$ acts on translation surfaces by acting linearly on the polygons, it preserves area, and the measure $\nu$ is invariant under it.

Masur and Veech proved the finiteness of the volume and the main consequence below independently in 1982, and the strata are not compact, so finiteness is not automatic.
Consequence: for $\nu$-almost every translation surface, the vertical straight-line flow is [[uniquely ergodic|NZZE9KCAD8]].
Through the first return map to a transversal, this gives Keane's conjecture: almost every [[interval exchange transformation|19JFF3GY9F]] with an irreducible permutation is uniquely ergodic.

## Explanations

### L1 human

Translation surfaces of a given shape type form a space, and this space has a natural notion of volume.
The theorem says that this volume is finite, and that a natural flow on the space, which stretches each surface horizontally and squeezes it vertically, mixes up the space thoroughly.
From this one learns that for a randomly chosen surface, a straight line in a fixed direction spreads out evenly over the whole surface.

### L2 human

The strata $\mathcal{H}_1(k_1, \dots, k_n)$ are moduli spaces of abelian differentials, fibred over the [[moduli space|FJZRW9P0K3]] of Riemann surfaces of genus $g$.
The flow $g_t$ projects to geodesics for the Teichmüller metric on [[Teichmüller space|ZZGC6BZNJ5]].
Ergodicity of $g_t$ is a starting point for counting problems, such as the asymptotic number of cylinders of closed geodesics on a typical surface.
The passage from ergodicity of $g_t$ to unique ergodicity of almost every vertical flow uses Masur's criterion.
If the orbit $g_t(X, \omega)$ returns infinitely often to a compact set as $t \to \infty$, then the vertical flow is uniquely ergodic.
</document>

<document source="https://tamnd.github.io/math2/node/VABPHNZAVV.md" tag="VABPHNZAVV" kind="definition">
# Profinite group

## Definition

Let $(G_i)_{i \in I}$ be an inverse system of finite groups indexed by a directed set $I$, each with the discrete topology.
Give its [[inverse limit|DZB378CNCK]] $\varprojlim_{i \in I} G_i$ the subspace topology of the product $\prod_{i \in I} G_i$.
A *profinite group* is a [[topological group|92700QAB5Y]] $G$ that is isomorphic, as a topological group, to such an inverse limit.

## Remarks

A topological group is profinite if and only if it is [[compact|0TVJZYB6SH]], [[Hausdorff|0S1FJ8VEX9]] and [[totally disconnected|0RJSBR9GW1]].
Then $G$ is the inverse limit of its finite quotients $G/N$, where $N$ runs over the open normal subgroups.

Example: the [[p-adic integers|V6TJAFZ4MD]] $\Z_p = \varprojlim_n \Z/p^n\Z$, with the reduction maps $\Z/p^{n+1}\Z \to \Z/p^n\Z$.
Each $\Z/p^n\Z$ is finite and discrete, so $\Z_p$ is profinite.

Non-example: $\Z$ with the discrete topology is Hausdorff and totally disconnected but not compact, so it is not profinite.
Its profinite completion $\hat\Z = \varprojlim_n \Z/n\Z$ is profinite and contains $\Z$ as a dense subgroup.

Every finite group with the discrete topology is profinite.
Closed subgroups, quotients by closed normal subgroups, and arbitrary products of profinite groups are profinite.
</document>

<document source="https://tamnd.github.io/math2/node/VADXDK3TKM.md" tag="VADXDK3TKM" kind="statement">
# Krein–Rutman theorem

## Statement

Let $X$ be a real [[Banach space|06GZ956657]] and let $K \subseteq X$ be a closed [[convex cone|6VE4KA12VD]] with $K \cap (-K) = \set{0}$.
Assume that $K$ is total, that is, the set $K - K = \set{u - v : u, v \in K}$ is dense in $X$.
Let $T \colon X \to X$ be a [[compact operator|M2NEP1K5AE]] with $T(K) \subseteq K$ and with [[spectral radius|9V7YQA85PC]] $r(T) > 0$.
Then $r(T)$ is an eigenvalue of $T$ with an [[eigenvector|0N6960W0BV]] in $K$: there is $u \in K$, $u \ne 0$, with $Tu = r(T) u$.

## Remarks

Proof idea: for $\lambda > r(T)$ the resolvent $(\lambda I - T)^{-1} = \sum_{n \ge 0} T^n / \lambda^{n+1}$ maps $K$ into $K$.
Using totality of the cone one shows that $r(T)$ lies in the spectrum and that the resolvent blows up along vectors of $K$ as $\lambda$ decreases to $r(T)$.
Normalizing $(\lambda I - T)^{-1} x$ for suitable $x \in K$ and using compactness of $T$ to pass to a limit gives a nonzero eigenvector in $K$.

Example: let $X = \R^n$, let $K$ be the vectors with nonnegative entries, and let $T$ be a matrix with nonnegative entries and $r(T) > 0$.
The theorem gives the weak form of the [[Perron–Frobenius theorem|62NS7PV1M9]]: $r(T)$ is an eigenvalue with a nonnegative eigenvector.

The hypothesis $r(T) > 0$ is needed: the Volterra operator $(Vf)(x) = \int_0^x f(t)\,dt$ on $C[0, 1]$ is compact and maps nonnegative functions to nonnegative functions, but $r(V) = 0$ and $V$ has no eigenvalue at all.

A stronger form holds when $K$ has nonempty interior and $T$ maps $K \setminus \set{0}$ into the interior of $K$.
Then $r(T) > 0$, it is a simple eigenvalue with an eigenvector in the interior of $K$, and every other eigenvalue $\lambda$ satisfies $\abs{\lambda} < r(T)$.

## Explanations

### L1 human

A positive operator here is one that sends nonnegative things to nonnegative things, such as a matrix with nonnegative entries acting on vectors with nonnegative entries.
The theorem says that, under a compactness condition, the largest growth rate of such an operator is achieved by a nonnegative eigenvector.
It extends the Perron–Frobenius theorem from matrices to infinite-dimensional spaces.

### L2 human

The theorem is a main tool for principal eigenvalues of elliptic operators.
The inverse of such an operator, with suitable boundary conditions, is compact and positive on a space of continuous functions, so it has a positive eigenfunction.
This gives the existence of a positive first eigenfunction of the Dirichlet Laplacian and similar results in population dynamics and bifurcation theory.
</document>

<document source="https://tamnd.github.io/math2/node/VAF6B01Q8W.md" tag="VAF6B01Q8W" kind="statement">
# Extension of Lagrange's theorem

## Statement

If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then

$$[G:K] = [G:H]\,[H:K].$$
</document>

<document source="https://tamnd.github.io/math2/node/VAJPHCQBHR.md" tag="VAJPHCQBHR" kind="definition">
# Direct method in the calculus of variations

## Definition

Let $X$ be a set with a notion of convergence of sequences, and let $J \colon X \to \R \cup \set{+\infty}$ be a functional with $\inf_X J < +\infty$.
The *direct method* is the following way to prove that $J$ has a minimiser.

1. Take a minimising sequence $(u_k)$, that is, $J(u_k) \to \inf_X J$.
2. Use *coercivity* to show that $(u_k)$ lies in a set that is sequentially compact, and extract a subsequence $u_{k_j} \to u$ in $X$.
3. Use sequential *lower semicontinuity*, $J(u) \le \liminf_j J(u_{k_j})$, to conclude that $J(u) = \inf_X J$.

The standard abstract form is this.
Let $X$ be a [[reflexive|NN8RWPC8WG]] Banach space and let $J$ be coercive, meaning $J(u) \to +\infty$ as $\norm{u} \to \infty$.
If $J$ is sequentially [[lower semicontinuous|95BSKT0S6M]] for the [[weak topology|P6KHVSTPZV]], then $J$ attains its minimum on $X$.

## Remarks

In the abstract form, coercivity makes a minimising sequence bounded, and bounded sequences in a reflexive Banach space have weakly convergent subsequences.
For integral functionals $J(u) = \int_\Omega L(x, u, \nabla u) \, dx$ of the [[calculus of variations|PQ25NHV0KS]] on $W^{1,q}(\Omega)$ with $1 < q < \infty$, weak lower semicontinuity holds when $L$ is continuous, bounded below and convex in $\nabla u$.

Example: let $\Omega \subseteq \R^n$ be bounded and open, $f \in L^2(\Omega)$, and $J(u) = \int_\Omega \big(\frac{1}{2}\abs{\nabla u}^2 - f u\big) \, dx$ on $H^1_0(\Omega)$.
By the Poincaré inequality $J(u) \ge c\norm{u}_{H^1}^2 - C\norm{u}_{H^1}$, so $J$ is coercive; it is convex and continuous, hence weakly lower semicontinuous.
So $J$ has a minimiser, which is the weak solution of $-\Delta u = f$ with $u = 0$ on $\partial\Omega$.

Non-example: $J(u) = \int_0^1 \big((u'^2 - 1)^2 + u^2\big) \, dx$ on $W^{1,4}(0, 1)$ is coercive, but its integrand is not convex in $u'$, so it is not weakly lower semicontinuous.
Indeed $\inf J = 0$, approached by sawtooth functions with slopes $\pm 1$ and small height, but no function has $J(u) = 0$, since this would need $u = 0$ and $\abs{u'} = 1$ almost everywhere.

## Explanations

### L1 human

The direct method finds a minimiser by taking a minimising sequence, extracting a convergent subsequence, and passing to the limit.
Coercivity gives the convergent subsequence and lower semicontinuity makes the limit a minimiser.
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# Yau's conjecture on infinitely many minimal surfaces

## Statement

Let $(M, g)$ be a closed [[Riemannian manifold|7CBT4SM83Y]] of dimension $n + 1$ with $3 \le n + 1 \le 7$.
Then $M$ contains infinitely many distinct smooth, closed, embedded [[minimal hypersurfaces|41X0YW2R9N]].

## Remarks

Shing-Tung Yau asked in his 1982 list of problems in differential geometry whether every closed $3$-manifold contains infinitely many immersed minimal surfaces.
The statement above, which gives embedded minimal hypersurfaces in all dimensions from $3$ to $7$, was proved by Antoine Song, building on work of Marques and Neves.

The proofs use [[Almgren–Pitts min-max theory|6KVJHG1KRZ]] for $k$-parameter sweepouts.
Marques and Neves showed that the min-max values $\omega_k(M)$ for $k = 1, 2, \dots$ are areas of minimal hypersurfaces with multiplicities; by results of Gromov and Guth they grow like $k^{1/(n+1)}$.
Under positive Ricci curvature this growth forces infinitely many distinct hypersurfaces, which gave the first cases of the conjecture.
Liokumovich, Marques and Neves proved a Weyl law for the $\omega_k$.
With it, Irie, Marques and Neves showed that for generic metrics the union of the closed embedded minimal hypersurfaces is dense in $M$.
Song treated arbitrary metrics with a min-max theory on non-compact manifolds obtained from pieces of $M$ bounded by stable minimal hypersurfaces.

Example: in the round sphere $S^3$ the great $2$-spheres are infinitely many embedded minimal surfaces; the theorem asserts the same for every metric on every closed $3$-manifold.

## Explanations

### L1 human

A minimal surface is a soap-film-like surface that is in balance: it cannot lower its area by small changes.
The theorem says that any closed curved space of dimension three to seven contains infinitely many different such closed surfaces, however irregular the shape of the space is.
In a round sphere one sees infinitely many of them at once, the great spheres; in a bumpy space they are hidden, and min-max methods find them.

### L2 human

The theorem is the analogue for minimal hypersurfaces of the existence of infinitely many closed geodesics on surfaces.
Its proof is a showcase of Almgren–Pitts theory: the sequence of min-max widths behaves like the spectrum of the Laplacian, with a Weyl-type growth law.
The dimension restriction comes from regularity: in dimension $8$ and higher, min-max hypersurfaces may have singular points.
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# Trigonometric definition of Chebyshev polynomials

## Statement

Let $T_0 = 1$, $T_1 = x$ and $T_{n+1} = 2x \, T_n - T_{n-1}$ for $n \ge 1$ be the [[Chebyshev polynomials|M6EAR1JHPP]] of the first kind.
Then for every integer $n \ge 0$ and every real $\theta$,
$$T_n(\cos \theta) = \cos(n \theta).$$
Equivalently, $T_n(x) = \cos(n \arccos x)$ for all $x \in [-1, 1]$.
Moreover, $T_n$ is the only polynomial with this property.

## Remarks

Proof idea: the cases $n = 0$ and $n = 1$ are clear.
The sum formula for the [[cosine|CFWMB0S5NK]] gives $\cos((n + 1)\theta) + \cos((n - 1)\theta) = 2 \cos \theta \cos(n \theta)$.
This is the recursion of the $T_n$ evaluated at $x = \cos \theta$, so induction on $n$ gives the identity.
Uniqueness holds because two polynomials that agree on the infinite set $[-1, 1]$ are equal.

Example: $\cos(3\theta) = 4 \cos^3 \theta - 3 \cos \theta$, which matches $T_3 = 4x^3 - 3x$.

Many texts take the identity as the definition of $T_n$ and derive the recursion from it.
Outside $[-1, 1]$ the same polynomial satisfies $T_n(\cosh t) = \cosh(n t)$ for real $t$, by the same induction.

## Explanations

### L1 human

The cosine of $n$ times an angle can always be written as a polynomial in the cosine of the angle.
The Chebyshev polynomial $T_n$ is exactly that polynomial.
So on $[-1, 1]$, $T_n$ is a cosine wave seen through the change of variable $x = \cos \theta$.

### L2 human

The identity explains most properties of $T_n$ at once.
It gives $\abs{T_n} \le 1$ on $[-1, 1]$, the zeros $\cos \frac{(2k - 1)\pi}{2n}$ and the extreme points $\cos \frac{k\pi}{n}$.
It also turns expansions in $T_n$ on $[-1, 1]$ into cosine series in $\theta$, so Fourier methods apply to Chebyshev series.
The composition rule $T_m(T_n(x)) = T_{mn}(x)$ follows from $\cos(m \cdot n\theta) = \cos(mn\theta)$.
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# Operator space

## Summary

An operator space is a closed linear subspace of the [[algebra of bounded operators on a Hilbert space|BMDG4JQPEH]].
It comes with the norms it inherits on all matrices with entries in it.
These matrix norms carry more information than the Banach space norm alone.
The natural maps between operator spaces are the completely bounded maps, and the theory is often described as a "quantized" Banach space theory.
Ruan's theorem characterizes operator spaces abstractly by two axioms on the matrix norms.

## Explanations

### L1 human

A Banach space only knows the length of each vector.
An operator space also knows the length of every matrix whose entries are its vectors, because such a matrix acts as one big operator.
Two spaces can look the same as Banach spaces but differ as operator spaces.
For example, the row vectors and the column vectors in $M_n(\C)$, for $n \ge 2$, give the same Banach space $\C^n$ with the Euclidean norm, but they are different operator spaces.
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# Minkowski plane

## Summary

A Minkowski plane is an incidence structure of points and circles with two parallelisms, modelled on the plane sections of a hyperboloid of one sheet.
Each hyperboloid of one sheet carries two families of lines, and these give the two parallelisms.
It is one of the three kinds of [[Benz plane|BG5A4CB6NB]], next to Möbius and Laguerre planes.

## Explanations

### L1 human

A hyperboloid of one sheet, the shape of a cooling tower, is covered by two families of straight lines.
Points on the same line of the first family are called $(+)$-parallel, and points on the same line of the second family $(-)$-parallel.
The circles are the plane sections that are not made of lines.
Through three points, no two of them parallel, there is exactly one circle.
An equivalent picture: the points are pairs $(x, y)$ of points of the projective line, and the circles are the graphs of the projective maps $x \mapsto (ax + b)/(cx + d)$.
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# Trigonometric integrals

## Summary

The trigonometric integrals are the sine integral $\operatorname{Si}(x) = \int_0^x \frac{\sin t}{t}\, dt$ and the cosine integral $\operatorname{Ci}(x) = -\int_x^\infty \frac{\cos t}{t}\, dt$, with their hyperbolic versions $\operatorname{Shi}$ and $\operatorname{Chi}$.
They are antiderivatives of $\frac{\sin x}{x}$ and $\frac{\cos x}{x}$, which have no elementary antiderivatives.
The sine integral tends to $\frac{\pi}{2}$, the value of the [[Dirichlet integral|0GZNSHMG4Q]], and its overshoot describes the Gibbs phenomenon of Fourier series.
They are closely related to the [[exponential integral|EV6ZDQ0NDY]].

## Explanations

### L1 human

The [[sinc function|P1T98BECWR]] $\frac{\sin t}{t}$ has a tall hump at $0$ and smaller and smaller ripples.
The sine integral adds up its area from $0$ to $x$.
It first rises above $\frac{\pi}{2}$, then swings below, and settles at $\frac{\pi}{2}$ with smaller and smaller swings.
Its first peak, at $x = \pi$, is its largest value, about $1.852$, roughly $18\%$ above the limit.
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# Morita duality

## Definition

Let $R$ and $S$ be [[rings|0B1261B5ED]] with identity and let $U$ be an $R$-$S$-[[bimodule|EM7FHAFD3N]].
A left $R$-module $C$ is a *cogenerator* if every left $R$-module embeds in a direct product of copies of $C$; likewise on the right.
The bimodule $U$ defines a *Morita duality* between $R$ and $S$ if these hold:

1. $U$ is an [[injective|85H7PRHDPZ]] cogenerator as a left $R$-module and as a right $S$-module.
2. The map $R \to \End(U_S)$ sending $r$ to left multiplication by $r$ is a ring isomorphism.
3. The map $S \to \End({}_R U)^{\mathrm{op}}$ sending $s$ to right multiplication by $s$ is a ring isomorphism.

Here $\End(U_S)$ and $\End({}_R U)$ are the [[endomorphism rings|AWSBZWWMG9]] of $U$ as a right $S$-module and as a left $R$-module.
A ring $R$ *has a Morita duality* if such $S$ and $U$ exist.

## Remarks

A left $R$-module $M$ is *reflexive* if the evaluation map $M \to \Hom_S(\Hom_R(M, U), U)$ is an isomorphism, and likewise for right $S$-modules.
Under a Morita duality, the functors $\Hom_R(-, U)$ and $\Hom_S(-, U)$ are mutually inverse contravariant equivalences between the reflexive modules on the two sides.
The reflexive modules include all [[finitely generated|05QY1VEBYD]] modules and all finitely cogenerated modules.
Example: for a field $k$, take $R = S = U = k$.
The space $k$ is injective and is a cogenerator of $k$-vector spaces, the multiplication maps $k \to \End(k)$ are isomorphisms, and the duality is $V \mapsto V^*$ on finite-dimensional spaces.
Non-example: $R = S = U = \Z$ fails clause 1, since $\Z$ is not an injective $\Z$-module; the map $2\Z \to \Z$, $2 \mapsto 1$, does not extend to $\Z$.
Over a quasi-Frobenius ring $R$, the bimodule $U = R$ gives a Morita duality between $R$ and itself.
Over a complete Noetherian local commutative ring with residue field $k$, the [[injective hull|QCJN05GN64]] of $k$ gives a Morita duality, known as Matlis duality.
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# Banach manifold

## Definition

Let $M$ be a [[Hausdorff space|0MXB4CKHFW]] and let $k \ge 1$ be an integer or $\infty$.
A *chart* on $M$ is a pair $(U, \varphi)$ with $U \subseteq M$ open.
Here $\varphi$ is a [[homeomorphism|0QT00PB5G2]] from $U$ onto an [[open subset|02BH3MCBQB]] of a real [[Banach space|06GZ956657]] $E$.
A *$C^k$ atlas* on $M$ is a family of charts $(U_i, \varphi_i)_{i \in I}$ such that:

1. The sets $U_i$ cover $M$.
2. For all $i, j \in I$, the transition map $\varphi_j \circ \varphi_i^{-1} \colon \varphi_i(U_i \cap U_j) \to \varphi_j(U_i \cap U_j)$ is of class $C^k$ in the sense of the [[Fréchet derivative|06EH97V2A7]].

Two $C^k$ atlases are equivalent if their union is a $C^k$ atlas.
A *Banach manifold of class $C^k$* is a space $M$ with an equivalence class of $C^k$ atlases.
If all charts take values in the same Banach space $E$, the manifold is *modelled on* $E$.

## Remarks

When every model space is $\R^n$, this is the definition of a [[smooth manifold|F45VQ47GPV]] of class $C^k$.
When every model space is a Hilbert space, $M$ is a Hilbert manifold.
Some authors start from a set instead of a Hausdorff space and give it the topology defined by the charts.

Example: an open subset $U$ of a Banach space $E$, with the single chart $(U, \id)$, is a Banach manifold.
So the invertible elements of the Banach algebra of bounded operators on $E$ form a Banach manifold, since they form an open subset.
The unit sphere of a Hilbert space is a Banach manifold, with charts given by projections onto tangent hyperplanes.

Non-example: on $\R$, the charts $(\R, \id)$ and $(\R, x \mapsto x^3)$ do not form a $C^1$ atlas together.
The transition map $x \mapsto x^{1/3}$ is not differentiable at $0$, so clause 2 fails.
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# q-convex manifold

## Summary

A $q$-convex manifold is a complex manifold with a smooth exhaustion function whose [[Levi form|80MTNXWW80]] has many positive eigenvalues outside a compact set.
When this holds everywhere, the manifold is called $q$-complete.
For $q = 1$ one gets the strongly pseudoconvex manifolds and the Stein manifolds.
These classes measure how far a manifold is from being Stein, and the [[Andreotti–Grauert theorem|67RJ0S8F74]] controls their cohomology.

## Explanations

### L1 human

A Stein manifold has an exhaustion function that is convex in every complex direction.
A $q$-convex manifold has one that is convex at least in $n - q + 1$ independent complex directions at each point far out, where $n$ is the dimension.
So the larger $q$ is, the weaker the condition.
For example, $\C^n$ with the origin removed is not Stein when $n \ge 2$.
The function $\abs{z}^2 + 1/\abs{z}^2$ grows to infinity near the origin and near infinity, and its Levi form is positive in the radial direction $w = z$.
So $\C^n \setminus \set{0}$ is $n$-complete in the convention used here.
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# n-skeleton

## Summary

The $n$-skeleton of a CW complex or of a simplicial complex is the part made of the cells or simplices of dimension at most $n$.
The skeleta form an increasing sequence of subcomplexes whose union is the whole complex.
Each one is built from the previous one by attaching $n$-dimensional cells.
Induction over the skeleta is the main technique for proving facts about CW complexes.
The $n$-skeleton already determines the homotopy groups of the complex below degree $n$.

## Explanations

### L1 human

Think of a cube built in stages.
First come its $8$ corners, the $0$-skeleton.
Then the $12$ edges are added, giving a wire frame, the $1$-skeleton.
Then the $6$ square faces are glued in, giving the hollow box, the $2$-skeleton.
Finally, filling in the solid gives the whole cube.
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# Cole–Hopf transformation

## Summary

The Cole–Hopf transformation turns the viscous [[Burgers' equation|NEGHXN8RE1]] into the linear [[heat equation|VP2HN86VAG]].
It sets $u = -2\nu\, \partial_x \log \varphi$, where $\varphi$ is a positive solution of the heat equation.
It gives an explicit formula for the solution of the initial value problem for Burgers' equation.
Letting the viscosity tend to zero in that formula recovers the entropy solution of the inviscid equation.
See the [[definition|PPM7V7WWHQ]].

## Explanations

### L1 human

Burgers' equation is nonlinear, so solutions cannot simply be added together.
The Cole–Hopf trick is a change of unknown that removes the nonlinearity.
After the change, the equation becomes the heat equation, which is solved by a known integral formula.
Changing back gives the solution of Burgers' equation with positive viscosity for given initial values.
For example, $\varphi(x, t) = x$ solves the heat equation, and it gives the solution $u = -2\nu / x$ of Burgers' equation for $x > 0$.
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# Universal property

## Definition

Let $F \colon \mathcal{C} \to \mathcal{D}$ be a [[functor|28NQ0KRHD3]] between [[categories|A2DE66937D]] and let $X$ be an object of $\mathcal{D}$.
A *universal morphism from $X$ to $F$* is a pair $(A, u)$, with $A$ in $\mathcal{C}$ and $u \colon X \to F(A)$ in $\mathcal{D}$, with this property.
For every object $A'$ of $\mathcal{C}$ and every morphism $f \colon X \to F(A')$, there is exactly one $g \colon A \to A'$ with $F(g) \circ u = f$.
Dually, a *universal morphism from $F$ to $X$* is a pair $(A, v)$ with $v \colon F(A) \to X$, with the following property.
For every $A'$ and every $f \colon F(A') \to X$, there is exactly one $g \colon A' \to A$ with $v \circ F(g) = f$.
An object or morphism is said to have a *universal property* when it is part of such a pair.

## Remarks

Equivalently, $(A, u)$ is an initial object of the comma category $(X \downarrow F)$, whose objects are pairs $(A', f \colon X \to F(A'))$; in the dual case it is a terminal object of $(F \downarrow X)$.
Since [[initial objects|EK85VSSRMR]] are unique up to unique isomorphism, so are universal morphisms.

Example: let $U \colon \mathbf{Grp} \to \mathbf{Set}$ send a group to its underlying set, and let $S$ be a set.
The [[free group|0KTW26C80J]] $F(S)$ with the inclusion $u \colon S \to U(F(S))$ of generators is a universal morphism from $S$ to $U$.
Indeed, every function $f \colon S \to U(H)$ to a group $H$ extends to exactly one [[group homomorphism|0JYTG5K9Y4]] $g \colon F(S) \to H$ with $g \circ u = f$.

Non-example: take $S = \{s\}$ and the pair $(\Z \times \Z, \ s \mapsto (1, 0))$.
The function $s \mapsto 1 \in \Z$ extends to the homomorphisms $(m, n) \mapsto m$ and $(m, n) \mapsto m + n$, so uniqueness fails and this pair is not universal.
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# Closed geodesic

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]].
A *closed geodesic* is a nonconstant [[geodesic|NSYMQMHPDN]] $\gamma \colon \R \to M$ that is periodic: there is $T > 0$ with $\gamma(t + T) = \gamma(t)$ for all $t \in \R$.
Two closed geodesics are *geometrically distinct* if their images are different sets.

## Remarks

Equivalently, a closed geodesic is a smooth map $\gamma \colon \R / T\Z \to M$ that satisfies the geodesic equation and is not constant.
A geodesic $\gamma$ is closed if and only if $\gamma(T) = \gamma(0)$ and $\dot\gamma(T) = \dot\gamma(0)$ for some $T > 0$, by uniqueness of geodesics with given initial point and velocity.
The closed geodesics are the critical points with positive energy of the energy functional $E(\gamma) = \frac{1}{2} \int_0^1 \abs{\dot\gamma}^2 \, dt$ on the [[free loop space|SGGXZ3TBSA]] of $M$.
Each closed geodesic of period $T$ gives infinitely many critical points, its iterates $t \mapsto \gamma(kt)$ for $k \ge 1$, which is why counting geometrically distinct ones is hard.

Example: on the unit sphere $S^2 \subset \R^3$, the great circle $\gamma(t) = (\cos t, \sin t, 0)$ is a geodesic with period $2\pi$, so it is a closed geodesic.

Non-example: on the flat torus $\R^2 / \Z^2$, the geodesic $\gamma(t) = (t, \sqrt{2}\, t) \bmod \Z^2$ is not periodic.
Indeed $\gamma(t + T) = \gamma(t)$ would need $T \in \Z$ and $\sqrt{2}\, T \in \Z$, which is impossible for $T > 0$.
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# Ancillary statistic

## Summary

An ancillary statistic is a [[statistic|0Q5CG2NEQ8]] whose distribution does not depend on the parameter.
On its own it carries no information about the parameter, yet it can tell how precise the data are.
Fisher proposed to condition inference on ancillary statistics, and [[Basu's theorem|6GATRFMJYY]] shows that they are independent of complete sufficient statistics.

## Explanations

### L1 human

Some features of the data do not depend on the quantity you want to learn.
For measurements with an unknown true value and known noise, the difference between two measurements does not depend on the true value.
Such a feature is ancillary: its distribution is known in advance.
It cannot tell you where the true value is, but it can tell you how spread out your data happened to be.
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# Lyndon–Hochschild–Serre spectral sequence

## Statement

Let $G$ be a group, let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$, and let $M$ be a $G$-module.
For each $q \ge 0$, the [[cohomology group|YXJXA0Q52R]] $H^q(N, M)$ is a module over the [[quotient group|0KYX2N69J0]] $G/N$, through the action of $G$ by conjugation.
Then there is a first-quadrant cohomological [[spectral sequence|GG8AYV847V]]
$$E_2^{p,q} = H^p\big(G/N, H^q(N, M)\big) \Longrightarrow H^{p+q}(G, M).$$

## Remarks

The spectral sequence is the Grothendieck spectral sequence of the composite of the functors $M \mapsto M^N$ and $A \mapsto A^{G/N}$, whose composite is $M \mapsto M^G$.
It applies because $M \mapsto M^N$ sends injective $G$-modules to $G/N$-modules that are acyclic for $A \mapsto A^{G/N}$.
There is a homological version $E^2_{p,q} = H_p(G/N, H_q(N, M)) \Rightarrow H_{p+q}(G, M)$ for [[group homology|CFAG3VRXEA]].

The terms of low degree give the inflation-restriction exact sequence
$$0 \to H^1(G/N, M^N) \to H^1(G, M) \to H^1(N, M)^{G/N} \to H^2(G/N, M^N) \to H^2(G, M).$$

Example: suppose that $H^q(N, M) = 0$ for all $q \ge 1$, as happens when $N$ is finite and multiplication by $\abs{N}$ is bijective on $M$.
Then the $E_2$ page is concentrated in the row $q = 0$, the sequence collapses, and $H^p(G, M) \cong H^p(G/N, M^N)$ for all $p$.

## Explanations

### L1 human

When a group is built from a normal subgroup and a quotient, its cohomology is assembled from the cohomology of the two pieces.
The spectral sequence is the bookkeeping device for this assembly.
It does not always give the answer at once, but it bounds it and often determines it.

### L2 human

The five-term exact sequence of the homological version already gives [[Hopf's formula|S31M1ZYJRK]] for $H_2$.
For a [[polycyclic group|9RBA56NXX3]], one can apply the spectral sequence step by step along a normal series with cyclic factors.
Its topological counterpart is the Serre spectral sequence of the fibration $BN \to BG \to B(G/N)$ of [[classifying spaces|WA796ZCNRX]].
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# Kohn–Rossi cohomology

## Summary

Kohn–Rossi cohomology is the cohomology of the tangential Cauchy–Riemann complex $\bar\partial_b$ on a [[CR manifold|ZC78JKZX9S]].
It plays for CR manifolds the role that [[Dolbeault cohomology|Y3900T2EY5]] plays for complex manifolds.
Its degree $0$ group is the space of smooth [[CR functions|JFKYYK2CEB]], and the higher groups measure the obstruction to solving the equation $\bar\partial_b u = f$.

## Explanations

### L1 human

On a complex manifold one studies the equation $\bar\partial u = f$, and Dolbeault cohomology records when it cannot be solved.
A CR manifold, such as a real hypersurface in $\C^n$, has only part of the complex directions, so only the tangential part $\bar\partial_b$ of $\bar\partial$ makes sense on it.
Kohn–Rossi cohomology records when $\bar\partial_b u = f$ cannot be solved, given the necessary condition $\bar\partial_b f = 0$.
Take the unit sphere $S^{2n-1}$ in $\C^n$ with $n \ge 2$.
Its degree $0$ group consists of the boundary values of holomorphic functions on the ball that are smooth up to the boundary.
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# Fractional Brownian motion

## Summary

Fractional Brownian motion is a continuous centred [[Gaussian process|22WEM6RTW0]] with stationary increments that is [[self-similar|RASDTDEKP7]] with a Hurst index $H \in (0, 1)$.
For $H = 1/2$ it is the [[Wiener process|22735FGJV6]].
For $H \ne 1/2$ its increments are correlated.
For $H > 1/2$ this gives long memory, which is used to model data in hydrology, finance and network traffic.

## Explanations

### L1 human

Ordinary Brownian motion forgets its past: each new step is independent of the earlier ones.
Fractional Brownian motion has a memory controlled by one number $H$ between $0$ and $1$.
For $H > 1/2$, a rise tends to be followed by a rise, so paths look smoother and trend more.
For $H < 1/2$, a rise tends to be followed by a fall, so paths look rougher and more jagged.
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<document source="https://tamnd.github.io/math2/node/VC7EH3FZX3.md" tag="VC7EH3FZX3" kind="definition">
# BK-space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $X$ be a [[sequence space|XYTN19EPF2]] over $\mathbb{K}$.
A *BK-space* is such an $X$ together with a norm $\norm{\cdot}$ on $X$ such that:

1. $(X, \norm{\cdot})$ is a [[Banach space|0BDTTM24DP]].
2. For each $n$, the coordinate map $X \to \mathbb{K}$, $x \mapsto x_n$, is [[continuous|0X3WGX95FE]].

## Remarks

Equivalently, a BK-space is an [[FK-space|71DMQHW6JZ]] whose topology is given by a norm.

Example: $\ell^\infty$ with $\norm{x}_\infty = \sup_n \abs{x_n}$ is a BK-space.
It is a Banach space, and $\abs{x_n} \le \norm{x}_\infty$ shows that each coordinate map is continuous.
In the same way, the [[spaces ℓ^p|08P6T2WH6R]] for $1 \le p < \infty$, the space $c$ of convergent sequences and the space $c_0$ of null sequences are BK-spaces.

Non-example: the space $\omega$ of all sequences is an FK-space but not a BK-space.
Its FK topology, the topology of coordinatewise convergence, is not given by any norm, because no neighbourhood of $0$ in it is bounded.
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<document source="https://tamnd.github.io/math2/node/VC8286D7TZ.md" tag="VC8286D7TZ" kind="definition">
# Blocking

## Definition

Let $T$ be a finite set of treatments and $U$ a finite set of experimental units.
*Blocking* is a partition $U = B_1 \cup \cdots \cup B_b$ of the units into disjoint *blocks* of units that are expected to respond alike.
The partition is fixed before the treatments are assigned, and treatments are then assigned and compared within blocks.
A [[design|2TE42ZWJ3E]] $d \colon U \to T$ is a *randomized complete block design* if:

1. each block has exactly $\abs{T}$ units;
2. each treatment occurs exactly once in each block, that is, $d$ restricted to $B_j$ is a bijection onto $T$ for every $j$;
3. the bijections on the different blocks are chosen independently and uniformly at random, by [[randomization|734Q6Q51X7]] within blocks.

## Remarks

The usual model for a randomized complete block design is $y_{ij} = \mu + \tau_i + \beta_j + \varepsilon_{ij}$, with treatment effects $\tau_i$, block effects $\beta_j$ and independent errors.
Differences $y_{ij} - y_{i'j}$ within block $j$ do not contain $\beta_j$, so block effects drop out of treatment comparisons.

Example: three varieties $T = \set{A, B, C}$ on four fields, each cut into three plots.
The fields are the blocks, each of size $3$; in each field the three varieties are put on the three plots in an order chosen at random.

Non-example: if each field receives a single variety on all its plots, clause 2 fails.
The variety effect is then confounded with the field effect, and the two cannot be separated.

When blocks are smaller than the number of treatments, clause 1 fails and one uses incomplete block designs, such as balanced incomplete block designs.
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# Banach–Mazur distance

## Definition

Let $n \ge 1$, and let $X$ and $Y$ be real [[normed vector spaces|03J5RAKB30]] of the same dimension $n$.
The *Banach–Mazur distance* between $X$ and $Y$ is
$$d(X, Y) = \inf_T \norm{T} \, \norm{T^{-1}}.$$
Here $T$ runs over the [[linear isomorphisms|0B8T10CE7N]] from $X$ to $Y$, and $\norm{\cdot}$ is the [[operator norm|06EKW1AKXF]].

## Remarks

Always $d(X, Y) \ge 1$, $d(X, Y) = d(Y, X)$ and $d(X, Z) \le d(X, Y) \, d(Y, Z)$; so $\log d$ is a pseudometric.
The infimum is attained by compactness, and $d(X, Y) = 1$ if and only if $X$ and $Y$ are isometric.
In terms of the unit balls $B_X$ and $B_Y$, $d(X, Y)$ is the least $\lambda \ge 1$ with $B_X \subseteq T B_Y \subseteq \lambda B_X$ for some linear isomorphism $T \colon \R^n \to \R^n$, after identifying both spaces with $\R^n$.
Example: let $X = \ell_1^2$ and $Y = \ell_\infty^2$, that is, $\R^2$ with the norms $\abs{x_1} + \abs{x_2}$ and $\max(\abs{x_1}, \abs{x_2})$.
The map $T(x_1, x_2) = (x_1 + x_2, x_1 - x_2)$ satisfies $\max(\abs{x_1 + x_2}, \abs{x_1 - x_2}) = \abs{x_1} + \abs{x_2}$, so it is an isometry and $d(X, Y) = 1$.
Non-example: if $\dim X \ne \dim Y$, there is no linear isomorphism, and the distance is not defined.
John's theorem gives $d(X, \ell_2^n) \le \sqrt{n}$ for every $n$-dimensional normed space $X$.
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<document source="https://tamnd.github.io/math2/node/VCGF9YR36X.md" tag="VCGF9YR36X" kind="conjecture">
# Riemann hypothesis

## Statement

Let $\zeta$ be the [[Riemann zeta function|1XWBY4N0VT]].
Every zero $s$ of $\zeta$ with $0 < \operatorname{Re} s < 1$ satisfies $\operatorname{Re} s = \frac{1}{2}$.

## Remarks

The zeros of $\zeta$ are the trivial zeros $-2, -4, -6, \dots$ and the nontrivial zeros, which lie in the strip $0 < \operatorname{Re} s < 1$.
This follows from the [[functional equation|NFXKNDJX9C]] and the [[non-vanishing on the line Re s = 1|CXES7915R7]].
So the conjecture says that every nontrivial zero lies on the critical line $\operatorname{Re} s = 1/2$.
The first nontrivial zeros are at about $\frac{1}{2} \pm 14.1347 i$.
The conjecture is open.
Riemann stated it in his 1859 paper on the number of primes below a given size.
It is one of Hilbert's problems of 1900 and one of the Clay Millennium Prize Problems.
Known partial results: Hardy proved that infinitely many zeros lie on the critical line, and Selberg proved that a positive proportion of them do.
An equivalent form: $\pi(x) = \operatorname{li}(x) + O(\sqrt{x} \log x)$, where $\pi$ is the [[prime-counting function|0RQSNVTFVZ]] and $\operatorname{li}(x) = \int_2^x dt / \log t$.
The [[generalized Riemann hypothesis|DNKA4WBKDS]] states the same for [[Dirichlet L-functions|9D2Z543EJQ]] and other L-functions.

## Explanations

### L1 human

The zeta function has infinitely many zeros in a vertical strip of the complex plane.
Riemann guessed that all of them lie exactly on the middle line of this strip.
If true, the primes are distributed as regularly as possible: the number of primes up to $x$ differs from its smooth approximation by roughly $\sqrt{x}$ only.
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<document source="https://tamnd.github.io/math2/node/VCHJVR9NGH.md" tag="VCHJVR9NGH" kind="concept">
# Word metric

## Summary

The word metric on a [[group|G5S1M9]] with a generating set $S$ is a distance on the group.
The distance from $g$ to $h$ is the least number of generators and their inverses whose product is $g^{-1} h$.
It turns the group into a [[metric space|0VKD1TP7F3]], and it equals the path distance in the [[Cayley graph|0GYQK9NDJF]].
For a [[finitely generated group|082DWWXMRJ]], changing the finite generating set changes the metric only up to bounded factors.
So the metric is well defined up to [[quasi-isometry|6FKYSH729T]].
It is the starting point of [[geometric group theory|2T3218RXTD]].

## Explanations

### L1 human

Think of the generators as the allowed steps.
The length of an element is the fewest steps needed to reach it from the identity.
The distance between two elements is the fewest steps from one to the other.
In $\Z$ with the steps $\pm 1$, the distance between $m$ and $n$ is $\abs{m - n}$.
With the steps $\pm 2$ and $\pm 3$ instead, the number $1$ is at distance $2$ from $0$, since $1 = 3 - 2$ and $1$ itself is not a step.
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<document source="https://tamnd.github.io/math2/node/VCHS7PH0NF.md" tag="VCHS7PH0NF" kind="concept">
# Dual operator

## Summary

The dual operator of a bounded operator $T \colon X \to Y$ between normed spaces is the operator $T' \colon Y' \to X'$ between the continuous dual spaces given by $T'(f) = f \circ T$.
It is the analytic form of the [[dual map|0AZ5ARP287]] of linear algebra, also called the transpose or Banach space adjoint.
Many properties of $T$, such as having closed or dense range, can be read off from $T'$.

## Explanations

### L1 human

A continuous linear functional measures vectors, giving one number for each vector.
Given an operator $T$ from $X$ to $Y$, any measurement $f$ on $Y$ gives a measurement on $X$: first apply $T$, then measure with $f$.
That is the dual operator, and it goes in the opposite direction.
For a matrix acting on column vectors, the dual operator is given by the transposed matrix.
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<document source="https://tamnd.github.io/math2/node/VCMS1CP91N.md" tag="VCMS1CP91N" kind="statement">
# Hook length formula

## Statement

Let $\lambda$ be a [[partition|0JV30XN73Z]] of $n$ with [[Young diagram|096AHMB3P4]] $D$.
For a cell $c$ of $D$, let $a(c)$ be the number of cells of $D$ to the right of $c$ in its row.
Let $l(c)$ be the number of cells of $D$ below $c$ in its column.
The hook length of $c$ is $h(c) = a(c) + l(c) + 1$.
Let $f^\lambda$ be the number of standard [[Young tableaux|Q6TH9PH2R7]] of shape $\lambda$.
Then
$$f^\lambda = \frac{n!}{\prod_{c \in D} h(c)}.$$

## Remarks

The formula is due to Frame, Robinson and Thrall.
Let $F(\lambda)$ denote the right-hand side of the formula.
A short proof by Greene, Nijenhuis and Wilf uses a random "hook walk" inside the diagram.
They show that the walk ends at each corner $c$ with probability $F(\lambda \setminus c) / F(\lambda)$, so these ratios add up to $1$.
The largest entry $n$ of a standard tableau sits in a corner.
So $f^\lambda$ is the sum of $f^{\lambda \setminus c}$ over the corners $c$, and $F$ satisfies the same recursion; induction on $n$ finishes the proof.

Example: for $\lambda = (2, 2)$ the hook lengths are $3, 2$ in the first row and $2, 1$ in the second row.
The formula gives $4! / (3 \cdot 2 \cdot 2 \cdot 1) = 2$, and indeed the standard tableaux are $\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}$ and $\begin{smallmatrix} 1 & 3 \\ 2 & 4 \end{smallmatrix}$.
For $\lambda = (n)$ the hook lengths are $n, n-1, \dots, 1$ and $f^\lambda = 1$.

## Explanations

### L1 human

The formula counts the ways to fill a Young diagram with $1, \dots, n$ so that rows and columns increase.
Each box has a hook: the box itself, the boxes to its right and the boxes below it.
Divide $n!$ by the product of all hook sizes and the answer is the count.
It is surprising that such a simple product gives an exact count.

### L2 human

The number $f^\lambda$ is the dimension of the irreducible representation of the symmetric group $S_n$ indexed by $\lambda$.
So the hook length formula computes these dimensions directly.
Together with the [[Robinson–Schensted correspondence|1SCF8X47G1]], which gives $\sum_{\lambda} (f^\lambda)^2 = n!$, it is a basic tool of the representation theory of $S_n$.
There are versions for shifted shapes and for trees, and a $q$-analogue that counts tableaux by a statistic.
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<document source="https://tamnd.github.io/math2/node/VCT848WV1T.md" tag="VCT848WV1T" kind="statement">
# Newlander–Nirenberg theorem

## Statement

Let $(M, J)$ be an [[almost complex manifold|9SYWRGVTRS]] of dimension $2n$, with $J$ smooth.
Let $N_J$ be the Nijenhuis tensor of $J$.
Then the following are equivalent.

1. $N_J = 0$ on $M$.
2. $M$ has an atlas of charts $\varphi \colon U \to \C^n$ whose transition maps are holomorphic and such that $d\varphi \circ J = i \cdot d\varphi$ on each $U$.

In that case $J$ is the almost complex structure of a unique complex manifold structure on $M$.

## Remarks

The implication from 2 to 1 is a direct computation: in holomorphic charts $J$ has constant coefficients, so all the brackets in $N_J$ cancel.
The hard direction is from 1 to 2.
When $J$ is real analytic, consider the bundle of $(0,1)$-vectors.
The condition $N_J = 0$ says that this bundle is closed under the bracket, and a complex form of the [[Frobenius theorem|5Q4BHBFXJY]] gives the charts.
For smooth $J$ one must solve a nonlinear system of partial differential equations of $\bar\partial$ type, which is the analytic heart of the proof.

In real dimension $2$ the tensor $N_J$ always vanishes, because $N_J(X, JX) = 0$ and $N_J$ is antisymmetric.
So every almost complex structure on a surface is integrable.
In particular, an oriented surface with a [[Riemannian metric|Q61VFHCH2J]], with $J$ the rotation by a right angle, is a Riemann surface.

## Explanations

### L1 human

An almost complex structure lets one multiply tangent vectors by $i$, but it may not come from complex coordinates.
The theorem gives a test: a certain tensor built from $J$ and brackets of vector fields must vanish.
When it vanishes, complex coordinates exist near every point.

### L2 human

The theorem turns the question "is $M$ a complex manifold" into the vanishing of a tensor, a condition that can be checked locally.
It is used to show that Kähler and many other geometric structures defined by tensors are really complex manifolds.
On surfaces it contains the existence of isothermal coordinates.
In higher dimension many almost complex manifolds have non-integrable structures; on $S^6$ the octonionic structure is not integrable, and whether $S^6$ admits any complex structure is an open problem.
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<document source="https://tamnd.github.io/math2/node/VD2REA47BN.md" tag="VD2REA47BN" kind="definition">
# Gabor transform

## Definition

Let $f \in L^2(\R)$ (see [[Lp space|SASP14MA1Z]]).
The *Gabor transform* of $f$ is the function $G f \colon \R^2 \to \C$,
$$G f(\tau, \omega) = \int_{\R} f(t) \, e^{-\pi (t - \tau)^2} \, e^{-2\pi i \omega t} \, dt,$$
for $\tau \in \R$ (time) and $\omega \in \R$ (frequency).

## Remarks

It is the [[short-time Fourier transform|3DHNP5NEWH]] $V_g f$ with the Gaussian window $g(t) = e^{-\pi t^2}$.
Some authors allow a Gaussian $e^{-\alpha \pi (t - \tau)^2}$ of any width $\alpha > 0$, or normalize the window to have $L^2$ norm $1$.

Example: let $f(t) = e^{-\pi t^2}$.
Then $f(t) e^{-\pi (t - \tau)^2} = e^{-\pi \tau^2/2} e^{-2\pi (t - \tau/2)^2}$, and the Fourier transform of $e^{-2\pi s^2}$ at $\omega$ is $2^{-1/2} e^{-\pi \omega^2/2}$.
So $G f(\tau, \omega) = 2^{-1/2} e^{-\pi \tau^2/2} e^{-\pi \omega^2/2} e^{-\pi i \tau \omega}$, and $\abs{G f}$ is a Gaussian bump centred at the origin of the time–frequency plane.

Non-example: the short-time Fourier transform with the window $1_{[-1/2, 1/2]}$ is not a Gabor transform, since its window is not Gaussian.
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# Cyclotomic extension

## Summary

A cyclotomic extension of a field $K$ is obtained by adjoining [[roots of unity|0S61N75H45]] to $K$.
The extension $K(\zeta_n)/K$ generated by a [[primitive $n$-th root of unity|0J817RTVG6]] is Galois, and its Galois group embeds in $(\Z/n\Z)^\times$, so it is [[abelian|EQDEXBWGJH]].
Over $\Q$ the embedding is an isomorphism, because the [[cyclotomic polynomials|0W0KVBM54S]] are [[irreducible over the rationals|0SK1VHZ5R2]].
The fields $\Q(\zeta_n)$ are the cyclotomic fields of number theory.

## Explanations

### L1 human

The $n$-th roots of unity are the $n$ complex numbers whose $n$-th power is $1$; they sit at the corners of a regular $n$-gon on the unit circle.
Adjoining one primitive root $\zeta_n$ gives all the others, because they are powers of it.
Each symmetry of $\Q(\zeta_n)$ sends $\zeta_n$ to a power $\zeta_n^k$ with $k$ coprime to $n$.
For $n = 4$, $\zeta_4 = i$ and the extension is $\Q(i)$, with the two symmetries $i \mapsto i$ and $i \mapsto -i$.
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<document source="https://tamnd.github.io/math2/node/VDBAW9PN88.md" tag="VDBAW9PN88" kind="concept">
# Generalized hypergeometric function

## Summary

The generalized hypergeometric function ${}_pF_q$ is a power series whose coefficient ratio is a rational function of $n$.
That rational function has $p$ numerator parameters and $q$ denominator parameters.
The [[hypergeometric function|463X5V31D9]] ${}_2F_1$ is the case $p = 2$, $q = 1$, and the exponential, the binomial series and the Bessel functions come from smaller $p$ and $q$.
Many classical summation formulas, such as those of Pfaff–Saalschütz, Dixon and Dougall, are evaluations of ${}_3F_2$ and higher series at $z = 1$.

## Explanations

### L1 human

A series $\sum t_n$ is called hypergeometric when the ratio $t_{n+1}/t_n$ of consecutive terms is a rational function of $n$.
After factoring numerator and denominator into linear factors, such a series can be written in a standard form ${}_pF_q$ with $p$ upper and $q$ lower parameters.
For example, $e^z = \sum z^n / n!$ has ratio $\frac{z}{n+1}$, so it is ${}_0F_0(;;z)$.
The numbers $p$ and $q$ control growth: the series is entire when $p \le q$, converges in the unit disc when $p = q + 1$, and diverges when $p > q + 1$, except when it is a polynomial.
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<document source="https://tamnd.github.io/math2/node/VDBDJZWPEA.md" tag="VDBDJZWPEA" kind="concept">
# Order polynomial

## Summary

The order polynomial of a finite poset $P$ counts the [[order-preserving maps|0A7KV2WB7A]] from $P$ to a chain with $m$ elements, as a function of $m$.
This count is a polynomial in $m$ of degree $\abs{P}$, and its leading coefficient is the number of [[linear extensions|W8DZ5KGGHN]] of $P$ divided by $\abs{P}!$.
It was introduced by Richard Stanley, who proved a reciprocity theorem linking it to the count of strictly order-preserving maps.

## Explanations

### L1 human

Label the elements of a poset with numbers from $1$ to $m$ so that larger elements never get smaller labels.
The number of ways to do this depends on $m$, and it is always given by a polynomial in $m$.
For a set of $p$ elements with no relations, every labelling is allowed, so the count is $m^p$.
For a chain of $2$ elements, a labelling is a pair $a \le b$, and there are $m(m+1)/2$ of them.
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<document source="https://tamnd.github.io/math2/node/VDEFF7A5Q4.md" tag="VDEFF7A5Q4" kind="definition">
# Errors-in-variables model

## Definition

The *simple linear errors-in-variables model* for observed pairs $(x_i, y_i)$, $i = 1, \dots, n$, states that
$$y_i = \alpha + \beta \xi_i + \varepsilon_i, \qquad x_i = \xi_i + \eta_i,$$
where:

1. $\alpha, \beta \in \R$ are unknown parameters;
2. $\xi_i$ is the true value of the regressor, which is not observed;
3. $\varepsilon_i$ and $\eta_i$ are random errors with mean $0$, and $\eta_i$ is the *measurement error* of the regressor.

The multivariate version replaces $\beta \xi_i$ by $\beta^T \xi_i$ with $\xi_i, \eta_i \in \R^p$.

## Remarks

Assume that the $\xi_i$ are independent and identically distributed with [[variance|0477R9PYAE]] $\sigma_\xi^2 > 0$, and that the $\eta_i$ have variance $\sigma_\eta^2$.
Assume also that all the $\xi_i$, $\eta_i$, $\varepsilon_i$ are independent, with finite variances, that the $\eta_i$ are identically distributed, and that the $\varepsilon_i$ are identically distributed.
Then $\operatorname{Cov}(x_i, y_i) = \beta \sigma_\xi^2$ and $\operatorname{Var}(x_i) = \sigma_\xi^2 + \sigma_\eta^2$.
So the [[ordinary least squares|YM708KHV2W]] slope of $y$ on $x$ converges in probability to
$$\beta \, \frac{\sigma_\xi^2}{\sigma_\xi^2 + \sigma_\eta^2},$$
which differs from $\beta$ when $\beta \ne 0$ and $\sigma_\eta^2 > 0$; the slope is not [[consistent|0AS800VAYX]].

Example: $\beta = 2$ and $\sigma_\xi^2 = \sigma_\eta^2 = 1$ give the limit $1$.

Special case: if $\eta_i = 0$ for all $i$, the regressor is observed exactly and the model is ordinary [[linear regression|41FVC4P5JB]].
Error in the response alone is absorbed into $\varepsilon_i$ and does not bias the slope.

With normal $\xi_i$, $\eta_i$, $\varepsilon_i$, the parameters are not identified from the joint distribution of $(x_i, y_i)$ unless extra information is given, such as the ratio $\sigma_\varepsilon^2 / \sigma_\eta^2$.
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<document source="https://tamnd.github.io/math2/node/VDF7CGPC0P.md" tag="VDF7CGPC0P" kind="concept">
# Geometric lattice

## Summary

A geometric lattice is a finite lattice that is [[atomistic|99DVVV7BH2]] and [[semimodular|C1NET23GG0]].
These lattices are exactly the lattices of flats of finite [[matroids|PX31ZNYS3Z]].
So they give an order-theoretic description of matroids, in which the atoms are the points and the rank function measures dimension.
Lattices of subspaces of a finite vector space, Boolean lattices of subsets of a finite set and lattices of partitions are geometric.

## Explanations

### L1 human

Take finitely many points in space and consider the "flats" they span: the points, the lines through two of them, the planes through three, and so on.
Ordered by inclusion, these flats form a geometric lattice.
Every flat is the join of the points it contains, and joining one more point raises the dimension by at most one.
For three points in general position in the plane, the lattice has the empty flat, three points, three lines and the whole plane.
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<document source="https://tamnd.github.io/math2/node/VDGDD8Q61N.md" tag="VDGDD8Q61N" kind="definition">
# Markov kernel

## Definition

Let $(X, \mathcal{A})$ and $(Y, \mathcal{B})$ be [[measurable spaces|N4KDT83V3S]].
A *Markov kernel* from $X$ to $Y$ is a map $\kappa \colon X \times \mathcal{B} \to [0, 1]$ such that:

1. for each $B \in \mathcal{B}$, the map $x \mapsto \kappa(x, B)$ is [[measurable|YYNCJSR0PJ]] from $(X, \mathcal{A})$ to $[0, 1]$;
2. for each $x \in X$, the map $B \mapsto \kappa(x, B)$ is a [[probability measure|059TJN0NZ1]] on $(Y, \mathcal{B})$.

## Remarks

Example: let $X = Y = \set{1, \ldots, n}$ with all subsets measurable, and let $P = (p_{ij})$ be a matrix with nonnegative entries and row sums $1$.
Then $\kappa(i, B) = \sum_{j \in B} p_{ij}$ is a Markov kernel: clause 1 holds because every function on a finite discrete space is measurable, and clause 2 because row $i$ is a probability vector.

Non-example: on $\R$ with its Borel sets, $\kappa(x, B) = 2 \, \mathbf{1}_B(x)$ fails clause 2, since $\kappa(x, \R) = 2$; it also takes values outside $[0, 1]$.

Another standard example: for a measurable map $f \colon X \to Y$, the kernel $\kappa(x, B) = \mathbf{1}_B(f(x))$ is a Markov kernel, the deterministic kernel of $f$.
A kernel acts on bounded measurable functions by $(\kappa g)(x) = \int_Y g(y) \, \kappa(x, dy)$ and on probability measures by $(\mu \kappa)(B) = \int_X \kappa(x, B) \, \mu(dx)$.
Markov kernels are also called stochastic kernels, transition kernels or probability kernels.
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<document source="https://tamnd.github.io/math2/node/VDHW982H70.md" tag="VDHW982H70" kind="definition">
# Residually small variety

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
The variety $\mathcal{V}$ is *residually small* if there is a cardinal $\kappa$ such that every [[subdirectly irreducible algebra|DASF939439]] in $\mathcal{V}$ has fewer than $\kappa$ elements.
A variety that is not residually small is *residually large*.

## Remarks

Equivalently, the subdirectly irreducible members of $\mathcal{V}$ form a set up to isomorphism.
The variety is residually finite when $\kappa$ can be chosen with every subdirectly irreducible member finite, and it has finite residual bound when $\kappa$ can be chosen finite.

Example: the variety of abelian groups is residually small.
Its subdirectly irreducible members are the groups $\Z/p^n\Z$ and $\Z(p^\infty)$ for primes $p$ and $n \ge 1$, which are all countable, so $\kappa = \aleph_1$ works.

Non-example: the variety of all groups is residually large.
For an infinite set $X$, the group of even permutations of $X$ that move only finitely many points is [[simple|9CJCXNYNK8]] and has $\abs{X}$ elements.
A nontrivial simple algebra is subdirectly irreducible, so no cardinal $\kappa$ bounds the sizes.
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# Curse of dimensionality

## Definition

For each dimension $d \ge 1$, let $F_d$ be a class of real functions on $[0, 1]^d$ and let $S_d \colon F_d \to \R$ be a functional, such as the integral $S_d(f) = \int_{[0,1]^d} f$.
An algorithm with $n$ function values computes $f(x_1), \dots, f(x_n)$, where each point may depend on the earlier values, and outputs a number computed from these values.
Its worst-case error is the supremum of $\abs{S_d(f) - A(f)}$ over $f \in F_d$.
Let $n(\varepsilon, d)$ be the smallest $n$ such that some algorithm with $n$ function values has worst-case error at most $\varepsilon$.
The family of problems $(S_d, F_d)$ *suffers from the curse of dimensionality* if there are constants $C, \gamma, \varepsilon_0 > 0$ as follows.
For all $\varepsilon \le \varepsilon_0$ and infinitely many $d$,
$$n(\varepsilon, d) \ge C (1 + \gamma)^d.$$

## Remarks

Example: let $F_d$ be the functions $f$ on $[0, 1]^d$ with $\abs{f} \le 1$ and $\abs{f(x) - f(y)} \le \norm{x - y}_\infty$, and let $S_d$ be the integral.
Take an algorithm with $n \le 2^{d-1}$ values, and let $P$ be the points it uses for the input $0$.
The function $g(x) = \min\big(1, \operatorname{dist}_\infty(x, P)\big)$ lies in $F_d$ and vanishes on $P$, so the algorithm uses the same points and outputs the same number for $g$, $-g$ and $0$.
The set where $g < \frac14$ is covered by $n$ cubes of side $\frac12$, of total volume at most $n 2^{-d} \le \frac12$, so $\int g \ge \frac14 \cdot \frac12 = \frac18$.
Hence the error is at least $\frac18$ for $g$ or for $-g$, and $n(\varepsilon, d) > 2^{d-1}$ for $\varepsilon < \frac18$: the problem suffers from the curse.

Non-example: let $F_d$ be the functions $f(x) = h(x_1)$ that depend only on the first coordinate, where $h \colon [0, 1] \to \R$ satisfies $\abs{h} \le 1$ and $\abs{h(s) - h(t)} \le \abs{s - t}$.
A [[quadrature rule|ADSVMCY77A]] in the first variable alone reaches error $\varepsilon$ with a number of points independent of $d$, so no exponential lower bound holds.

In the information-based complexity literature the error is often normalized by the error of the zero algorithm; this changes constants but not the idea.
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<document source="https://tamnd.github.io/math2/node/VDMCEPTTXB.md" tag="VDMCEPTTXB" kind="definition">
# Convolution on the half-line

## Definition

Let $f, g \colon [0, \infty) \to \C$ be locally integrable, that is, Lebesgue integrable on $[0, T]$ for every $T > 0$.
The *convolution* of $f$ and $g$ is the function $f * g$ on $[0, \infty)$ given by
$$(f * g)(t) = \int_0^t f(\tau) \, g(t - \tau) \, d\tau$$
at each $t \ge 0$ at which the [[Lebesgue integral|DWJF7ABPR1]] converges absolutely.

## Remarks

The integral converges for almost every $t$, and $f * g$ is again locally integrable.
If $f$ and $g$ are continuous, then $f * g$ is defined everywhere and continuous, with $(f * g)(0) = 0$.
Extending $f$ and $g$ by $0$ on $(-\infty, 0)$ turns this into the [[convolution on the line|ERP5737VZ1]].

Example: for integers $m, n \ge 0$, the functions $t^m$ and $t^n$ give
$$(t^m * t^n)(t) = \int_0^t \tau^m (t - \tau)^n \, d\tau = \frac{m! \, n!}{(m + n + 1)!} \, t^{m + n + 1},$$
by the Beta integral; for $m = n = 0$ this is $1 * 1 = t$.

Non-example: $f(t) = 1/t$ for $t > 0$ is not integrable on $[0, 1]$, so it is not locally integrable on $[0, \infty)$.
Indeed $(f * 1)(t) = \int_0^t d\tau / \tau$ diverges for every $t > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/VDPTQ35JWE.md" tag="VDPTQ35JWE" kind="definition">
# Brunn–Minkowski theory

## Definition

Let $n \ge 1$, let $\mathcal{K}^n$ be the set of [[convex bodies|437FSG3419]] in $\R^n$, and let $V$ be the volume, that is, [[Lebesgue measure|2Z2QVWZ262]].
*Brunn–Minkowski theory* is the study of the functionals on $\mathcal{K}^n$ obtained from $V$ and [[Minkowski addition|SXS6D87PHE]] $K + L = \set{x + y : x \in K, \ y \in L}$ together with scaling by nonnegative reals.
Its basis is Minkowski's theorem: for $K_1, \ldots, K_m \in \mathcal{K}^n$ and $\lambda_1, \ldots, \lambda_m \ge 0$,
$$V(\lambda_1 K_1 + \cdots + \lambda_m K_m) = \sum_{i_1, \ldots, i_n = 1}^{m} \lambda_{i_1} \cdots \lambda_{i_n} V(K_{i_1}, \ldots, K_{i_n}),$$
where the coefficients $V(K_{i_1}, \ldots, K_{i_n})$, symmetric in their arguments, are the [[mixed volumes|QB3Q5FBJ08]].
The theory studies the mixed volumes and their special cases, and inequalities between them such as the [[Brunn–Minkowski theorem|9B02RPN7XD]].

## Remarks

Example: take $K_1 = K$ and $K_2 = B$, the unit ball.
Then $V(K + t B)$ is a polynomial of degree $n$ in $t \ge 0$; this is Steiner's formula.
Its coefficients are, up to binomial factors, the quermassintegrals of $K$, among them the volume and a multiple of the surface area.
Non-example: Lutwak's dual Brunn–Minkowski theory replaces Minkowski addition by radial addition of star bodies; it studies sections rather than projections and is a different theory.
The $L_p$ Brunn–Minkowski theory is an extension that replaces Minkowski addition by $L_p$ addition of support functions.
</document>

<document source="https://tamnd.github.io/math2/node/VDQ32JT7ZS.md" tag="VDQ32JT7ZS" kind="concept">
# Random integral equation

## Summary

A random integral equation is an [[integral equation|R3SG1SN7JQ]] whose kernel, forcing term or both depend on a random parameter.
For each outcome of the randomness one gets an ordinary integral equation, and the solution is a [[stochastic process|0QH5ENKZM1]].
The basic questions are existence, uniqueness and measurability of such random solutions, and the laws of the solutions.
Such equations model systems with uncertain coefficients, such as materials with random properties.
They differ from stochastic integral equations with Itô integrals, where the integral against a [[Wiener process|22735FGJV6]] is not defined path by path.

## Explanations

### L1 human

Suppose that the coefficients of an integral equation are measured with uncertainty and are modelled as random variables.
For each possible value of the coefficients the equation has a solution, so the solution itself is random.
For example, with a random number $Y$ between $0$ and $1$, the equation $x(t) = 1 + \frac{Y}{2} \int_0^1 x(s) \, ds$ has the solution $x(t) = 2/(2 - Y)$.
The solution is a random variable, and one can ask for its mean or its distribution.
</document>

<document source="https://tamnd.github.io/math2/node/VDQAZDHFF7.md" tag="VDQAZDHFF7" kind="statement">
# Small object argument

## Statement

Let $\mathcal{C}$ be a category with all small [[colimits|BJS1CGTKQ5]], and let $I$ be a set of morphisms of $\mathcal{C}$.
Let $I\text{-cell}$ be the class of transfinite composites of [[pushouts|Q699KKMVJD]] of [[coproducts|4FJ0S71P58]] of morphisms of $I$.
Let $I\text{-inj}$ be the class of morphisms with the right [[lifting property|AZK4RD8XV0]] with respect to every morphism of $I$.
Assume that the domain of each morphism of $I$ is small relative to $I\text{-cell}$.
Then every morphism $f \colon X \to Y$ of $\mathcal{C}$ factors as $f = p \circ i$ with $i \in I\text{-cell}$ and $p \in I\text{-inj}$, and this factorization can be chosen functorial in $f$.

## Remarks

The proof builds $X = X_0 \to X_1 \to X_2 \to \cdots$ over $Y$ by transfinite induction.
At each step one takes all commutative squares from a morphism of $I$ to $X_\beta \to Y$.
One pushout of a coproduct then glues in a solution for every square at once.
At limit stages one takes the colimit.
One stops at an ordinal $\kappa$ so large that every map from a domain of $I$ into $X_\kappa$ factors through some earlier $X_\beta$.
The smallness hypothesis guarantees such a $\kappa$; then any lifting problem against $X_\kappa \to Y$ is solved at stage $\beta + 1$.
Without smallness the transfinite construction need not stop at any stage, and the argument does not apply.

A consequence is that every morphism with the left lifting property with respect to $I\text{-inj}$ is a retract of a morphism of $I\text{-cell}$.
In a locally presentable category every object is small, so the hypothesis holds for every set $I$.

Example: in topological spaces let $I$ be the set of boundary inclusions $S^{n-1} \to D^n$ for $n \ge 0$.
The theorem factors every map into a relative cell complex followed by a map with the right lifting property against these inclusions.
The second map is then a weak homotopy equivalence and a Serre fibration.
This is how the cofibration and trivial fibration factorization of the Quillen [[model structure|SDTZVZ6B4Q]] is obtained.

## Explanations

### L1 human

Model categories need a way to break any map into a nice inclusion followed by a nice projection.
The small object argument does this by brute force: whenever some lifting problem has no solution, glue in a piece that solves it.
Gluing creates new problems, so one repeats, possibly transfinitely many times.
Smallness of the pieces guarantees that the process stops, and at the end every problem has a solution.

### L2 human

The theorem is the main tool to build cofibrantly generated model structures.
One gives a set $I$ of generating cofibrations and a set $J$ of generating trivial cofibrations, and applies the argument to both.
The [[cofibrations|HSV1J7BJXX]] are then the retracts of $I$-cell maps, and the fibrations are the $J$-injective maps.
This is how the standard model structures on topological spaces, simplicial sets and chain complexes are constructed.
The functoriality of the factorization comes from the fact that the gluing step uses all lifting problems, not a chosen one.
</document>

<document source="https://tamnd.github.io/math2/node/VDQX55E7D4.md" tag="VDQX55E7D4" kind="definition">
# Betti number

## Definition

Let $X$ be a topological space, let $k \ge 0$ be an integer, and let $H_k(X)$ be the $k$-th [[singular homology|M0Q5CFJ7ZH]] group of $X$ with integer coefficients.
The *$k$-th Betti number* of $X$ is the [[rank|EMCTDD3HZ1]] of the abelian group $H_k(X)$:
$$b_k(X) = \rank H_k(X) = \dim_{\Q} \left( H_k(X) \otimes_{\Z} \Q \right),$$
a nonnegative integer or $\infty$.

## Remarks

When $H_k(X)$ is finitely generated, it is isomorphic to $\Z^{b_k} \oplus T$ with $T$ finite, so $b_k$ is the number of copies of $\Z$.
By the [[universal coefficient theorem|GTA8ZFGZ3N]], $b_k(X) = \dim_{\Q} H_k(X; \Q)$.
Torsion is invisible to Betti numbers.

Example: for the torus $T^2 = S^1 \times S^1$, $H_0 = \Z$, $H_1 = \Z^2$ and $H_2 = \Z$, so $b_0 = 1$, $b_1 = 2$, $b_2 = 1$.

Non-example of a finite Betti number: an infinite discrete space $X$ has $H_0(X)$ free abelian of infinite rank, so $b_0(X) = \infty$.
For the real projective plane, $H_1 = \Z/2$ is nonzero but has rank $0$, so $b_1 = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/VDRCDZ048J.md" tag="VDRCDZ048J" kind="statement">
# von Neumann coordinatization theorem

## Statement

Let $L$ be a [[complemented|ZT47BK9Y6K]] [[modular lattice|QX7WZ92QAK]] with least element $0$ and greatest element $1$.
Elements $a, b \in L$ are *perspective* if some $x \in L$ satisfies $a \wedge x = b \wedge x = 0$ and $a \vee x = b \vee x$.
Elements $a_1, \dots, a_n \in L$ are *independent* if $(a_1 \vee \cdots \vee a_k) \wedge a_{k+1} = 0$ for each $k$ with $1 \le k < n$.
A *homogeneous basis* of $L$ is a family $a_1, \dots, a_n$ of independent elements with $a_1 \vee \cdots \vee a_n = 1$.
In addition, $a_i$ and $a_j$ must be perspective for all $i, j$.

Suppose that $L$ has a homogeneous basis with $n \ge 4$ elements.
Then there is a [[von Neumann regular ring|DXX7GNY3HE]] $R$ such that $L$ is isomorphic to the lattice of principal right ideals of $R$.
Here the principal right ideals are ordered by inclusion.

## Remarks

For a regular ring $R$, every principal right ideal is generated by an idempotent, and the principal right ideals form a complemented modular lattice; the theorem gives the converse.
The ring is built from $L$ much as a division ring is built from a projective space of dimension at least $3$.
The bound $n \ge 4$ plays the role of that dimension bound.

Example: for a division ring $D$ and $n \ge 4$, the lattice of subspaces of $D^n$ has the homogeneous basis of coordinate lines.
It is isomorphic to the lattice of principal right ideals of the matrix ring $M_n(D)$, which is regular.

Von Neumann applied the theorem to [[continuous geometries|5CET1KWWWH]], where it yields continuous regular rings.

## Explanations

### L1 human

Classical projective geometry shows that a projective space of dimension at least $3$ always comes from a division ring, so its points and lines can be given coordinates.
Von Neumann's theorem does the same for complemented modular lattices with a large enough homogeneous basis.
The coordinates now live in a regular ring, a ring in which every element $a$ has some $x$ with $axa = a$.

### L2 human

The theorem identifies complemented modular lattices that have a homogeneous basis with at least $4$ elements with lattices of principal right ideals of regular rings.
It explains why regular rings are the natural algebraic counterpart of complemented modular lattices.
Its main application is to [[continuous geometries|5CET1KWWWH]], whose coordinate rings are the continuous regular rings, including rings built from type II$_1$ factors.
The independence and perspectivity conditions are lattice versions of a direct sum decomposition into isomorphic summands, like $D^n = D \oplus \cdots \oplus D$.
</document>

<document source="https://tamnd.github.io/math2/node/VDTEARFVMB.md" tag="VDTEARFVMB" kind="concept">
# Hausdorff content

## Summary

The $s$-dimensional Hausdorff content of a set is the least total of the $s$-th powers of the diameters of countable covers of the set.
Unlike the [[Hausdorff measure|Q1TGZK8RN6]], it allows covers by sets of any size.
It is an [[outer measure|BJ314H17HC]] that is usually not additive, but it has the same null sets as the Hausdorff measure.
So it can be used to compute [[Hausdorff dimension|SSNFTHZ6YJ]], and it is easier to estimate.

## Explanations

### L1 human

To estimate the $s$-dimensional size of a set, cover it with countably many pieces and add up each diameter raised to the power $s$.
The Hausdorff content is the smallest total that one can reach.
For a segment of length $1$ and $s = 1$ it is $1$.
For a filled unit square and $s = 1$ it is at most $\sqrt 2$, the diameter of the square, although the square has infinite length.
This shows that the content does not see fine structure, but it still sees whether a set is negligible.
</document>

<document source="https://tamnd.github.io/math2/node/VDVQD2HFSD.md" tag="VDVQD2HFSD" kind="problem">
# Smale's fourteenth problem

## Problem

Consider the [[Lorenz system|WYRC41CQN9]] in $\R^3$,
$$\dot x = \sigma (y - x), \qquad \dot y = x(\rho - z) - y, \qquad \dot z = xy - \beta z,$$
with the classical parameters $\sigma = 10$, $\rho = 28$ and $\beta = 8/3$.
Does the flow of this system have a [[strange attractor|9WKCZGG906]], with the dynamics of the geometric Lorenz attractor?

## Remarks

The geometric Lorenz attractor is a model flow, built in the 1970s from the features seen in numerical simulations.
For the model, the existence of a robust [[attractor|7FWX24X5MW]] with sensitive dependence on initial conditions was proved.
The question was whether the actual Lorenz equations behave like the model.
Smale included it as the fourteenth problem of his 1998 list of problems for the next century.
Warwick Tucker gave a positive answer with a computer-assisted proof, published in 2002.
His proof combines rigorous numerics with interval arithmetic, far from the origin, with normal form theory near the equilibrium at the origin.

## Explanations

### L1 human

The Lorenz system is a simple model of convection whose solutions trace the famous butterfly-shaped figure on a computer.
Pictures are not proofs: rounding errors might hide a stable cycle, and the figure might then be an illusion.
Smale asked for a proof that the butterfly is a true chaotic attractor.
The answer is yes, and its proof was a major use of computers to settle a question in dynamics.
</document>

<document source="https://tamnd.github.io/math2/node/VDWKE1KK4H.md" tag="VDWKE1KK4H" kind="definition">
# Method of characteristics

## Definition

Let $U \subseteq \R^n$ be open, and let $a = (a_1, \dots, a_n) \colon U \times \R \to \R^n$ and $c \colon U \times \R \to \R$ be $C^1$ functions.
Consider the first-order [[quasilinear equation|7X9N9H3RK6]]
$$\sum_{i=1}^n a_i(x, u)\, \partial_i u = c(x, u)$$
for an unknown $C^1$ function $u$ on $U$, where $\partial_i u$ is the [[partial derivative|09EGC6F8PT]] in $x_i$.
Its *characteristic equations* are the [[ordinary differential equations|0NT64JV96A]]
$$\frac{dx}{ds} = a(x, z), \qquad \frac{dz}{ds} = c(x, z)$$
for a curve $s \mapsto (x(s), z(s))$ in $U \times \R$; the curves $s \mapsto x(s)$ are the *characteristics*.
The *method of characteristics* solves the equation with data $u = g$ on a hypersurface $\Gamma \subseteq U$ as follows.

1. For each $y \in \Gamma$, solve the characteristic equations with [[initial values|0FXDW992DS]] $x(0) = y$ and $z(0) = g(y)$.
2. Invert the map $(y, s) \mapsto x(y, s)$ near $\Gamma$, which is possible when $\Gamma$ is non-characteristic, that is, $a(y, g(y))$ is not tangent to $\Gamma$.
3. Set $u(x(y, s)) = z(y, s)$.

## Remarks

The key fact is that if $u$ is a $C^1$ solution and $x(s)$ solves $x' = a(x, u(x))$, then $z(s) = u(x(s))$ satisfies $z' = \sum_i a_i\, \partial_i u = c(x, z)$.
So the graph of a solution is a union of solutions of the characteristic system.
For a fully nonlinear equation $F(Du, u, x) = 0$ the method uses the larger system $\dot x = D_p F$, $\dot z = D_p F \cdot p$, $\dot p = -D_x F - D_z F\, p$, where $p$ stands for $Du$.

Example: for $u_t + u\, u_x = 0$ in the variables $(x, t)$ with $u(x, 0) = g(x)$, the characteristic equations are $\dot t = 1$, $\dot x = z$, $\dot z = 0$.
So $z$ is constant and the characteristics are the lines $x = y + g(y) t$, which gives the implicit solution $u = g(x - ut)$.
Check: differentiating $u = g(x - ut)$ gives $u_t = -g'\,(u + t u_t)$ and $u_x = g'\,(1 - t u_x)$, so $u_t = -u g'/(1 + t g')$ and $u_x = g'/(1 + t g')$, and $u_t + u u_x = 0$.

Non-example: data given on a characteristic curve.
Take $\partial_1 u = 0$ on $\R^2$ with data $u(x_1, 0) = g(x_1)$ on $\Gamma = \set{x_2 = 0}$.
Here $a = (1, 0)$ is tangent to $\Gamma$, so step 2 fails.
Indeed no solution exists unless $g$ is constant, and if $g$ is constant there are infinitely many, such as $u = g + h(x_2)$ with $h(0) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/VDX0PQ2GJ4.md" tag="VDX0PQ2GJ4" kind="definition">
# Asymptotic analysis

## Definition

Let $D \subseteq \R$ and let $x_0$ be a limit point of $D$, possibly $\pm\infty$.
A sequence of functions $\varphi_0, \varphi_1, \varphi_2, \ldots$ on $D$ is an *asymptotic sequence* as $x \to x_0$ if $\varphi_{n+1}(x) = o(\varphi_n(x))$ as $x \to x_0$ for every $n$.
Let $f$ be a function on $D$ and let $a_0, a_1, \ldots$ be numbers.
The formal series $\sum_{n \ge 0} a_n \varphi_n$ is an *asymptotic expansion* of $f$ as $x \to x_0$, written
$$f(x) \sim \sum_{n=0}^{\infty} a_n \varphi_n(x) \quad (x \to x_0),$$
if for every $N \ge 0$,
$$f(x) - \sum_{n=0}^{N} a_n \varphi_n(x) = o(\varphi_N(x)) \quad (x \to x_0).$$
*Asymptotic analysis* is the study of such expansions and of [[asymptotic equivalence|0QR92XBV6H]] $f \sim g$, meaning $f / g \to 1$.
Its estimates are written with [[big O|0TGV80E6A4]] and [[little o|00T5EPFQB9]] notation.

## Remarks

Example: $f(x) = 1 / (x - 1)$ on $(1, \infty)$ as $x \to \infty$, with $\varphi_n(x) = x^{-n-1}$.
Here $f(x) - \sum_{n=0}^{N} x^{-n-1} = x^{-N-1} / (x - 1) = o(x^{-N-1})$, so $f(x) \sim \sum_{n \ge 0} x^{-n-1}$.

A divergent example: as $x \to 0^+$, $\int_0^\infty \frac{e^{-t}}{1 + x t} \, dt \sim \sum_{n \ge 0} (-1)^n n! \, x^n$, although the series converges for no $x \neq 0$.

Non-example: $\varphi_n(x) = x^n$ is not an asymptotic sequence as $x \to \infty$, since $x^{n+1}$ is not $o(x^n)$ there; it is one as $x \to 0$.

Given the asymptotic sequence, the coefficients $a_n$ are unique.
Different functions can have the same expansion: $e^{-x}$ has all coefficients $0$ with respect to $x^{-n}$ as $x \to \infty$.

## Explanations

### L1 human

An asymptotic expansion is a list of terms of decreasing size.
Stopping after any term leaves an error smaller than the last term kept, when $x$ is close enough to the limit point.
</document>

<document source="https://tamnd.github.io/math2/node/VDZ5FKYC2T.md" tag="VDZ5FKYC2T" kind="concept">
# Logarithmic kernel

## Summary

The logarithmic kernel is the function $k(z, w) = \log(1/\abs{z - w})$ of two points of the plane.
It is the planar counterpart of the [[Newtonian kernel|6TJTWQ5HCR]].
It equals $2\pi$ times the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] of the plane, translated to the pole $w$.
Integrating it against a measure gives the logarithmic potential, which links potential theory with complex analysis, polynomial approximation and the distribution of zeros of polynomials.
Unlike the Newtonian kernel, it takes negative values when the two points are more than one unit apart.

## Explanations

### L1 human

In the plane, the potential of a point charge does not decay like one over the distance.
It changes like the logarithm of the distance, which grows without bound but very slowly.
The logarithmic kernel is $\log(1/\abs{z - w})$: it is large and positive near $w$, zero on the unit circle around $w$, and negative farther out.
For example, at distance $e$ from the pole its value is $-1$.
</document>

<document source="https://tamnd.github.io/math2/node/VE4SJTXFBV.md" tag="VE4SJTXFBV" kind="definition">
# Day terms

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras and let $n \ge 1$.
*Day terms* for $\mathcal{V}$ are quaternary terms $m_0, m_1, \dots, m_n$ such that $\mathcal{V}$ satisfies the following [[identities|PKD440G2XW]]:

1. $m_0(x, y, z, u) = x$ and $m_n(x, y, z, u) = u$.
2. $m_i(x, y, y, x) = x$ for every $i \le n$.
3. $m_i(x, x, u, u) = m_{i+1}(x, x, u, u)$ for every even $i < n$.
4. $m_i(x, y, y, u) = m_{i+1}(x, y, y, u)$ for every odd $i < n$.

## Remarks

Day's theorem says that $\mathcal{V}$ is a [[congruence-modular variety|QAQAS3JWXW]] if and only if $\mathcal{V}$ has Day terms for some $n$.
Example: if $p$ is a [[Mal'cev term|Y203PPTS61]] for $\mathcal{V}$, then $m_0 = x$, $m_1 = p(u, z, y)$, $m_2 = u$ are Day terms.
The checks are $m_1(x, y, y, x) = p(x, y, y) = x$, $m_1(x, x, u, u) = p(u, u, x) = x = m_0(x, x, u, u)$, and $m_1(x, y, y, u) = p(u, y, y) = u = m_2(x, y, y, u)$.
For groups this gives $m_1(x, y, z, u) = u z^{-1} y$.
Non-example: with $n = 1$, the terms $m_0 = x$ and $m_1 = u$ are not Day terms in any nontrivial variety.
Clause 3 for $i = 0$ reads $x = u$, which holds only in one-element algebras.

## Explanations

### L1 human

Day terms are a chain of four-input formulas that witness, by equations alone, that a variety has modular congruence lattices.
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# Clenshaw–Curtis quadrature

## Summary

Clenshaw–Curtis quadrature integrates a function on $[-1, 1]$ by interpolating it at the points $\cos(k\pi/N)$ and integrating the interpolating polynomial exactly.
The substitution $x = \cos\theta$ turns the problem into a cosine series, so the weights can be computed fast with the discrete cosine transform.
Its nodes for $N$ are contained in its nodes for $2N$, so doubling $N$ reuses all earlier function values.
For many smooth functions it is about as accurate as [[Gauss–Legendre quadrature|RGGXDRCJVE]] with the same number of points.

## Explanations

### L1 human

Sample the function at points that are the shadows on the $x$-axis of equally spaced points on a half circle.
These points crowd towards the ends of the interval, which avoids the wild oscillations of interpolation at equally spaced points.
Fit the polynomial through the samples and integrate that polynomial exactly.
For example, with $N = 2$ the points are $-1, 0, 1$, and the result is [[Simpson's rule|MNXC26G3BQ]].
</document>

<document source="https://tamnd.github.io/math2/node/VEBDNCJJTN.md" tag="VEBDNCJJTN" kind="concept">
# Bayes risk

## Summary

The Bayes risk of a decision rule is its [[risk function|N26TSVRH2S]] averaged over a [[prior distribution|00NBBSXRQS]] on the parameter.
It reduces the comparison of rules to the comparison of single numbers.
The smallest Bayes risk over all rules is the Bayes risk of the prior, and a rule attaining it is a [[Bayes estimator|PP1A0JB47Q]].

## Explanations

### L1 human

The risk of a rule depends on the unknown truth, so it is a whole curve rather than one number.
If one has an opinion about which values of the truth are likely, one can average the curve with those weights.
That weighted average is the Bayes risk.
For example, estimating a coin's heads probability $\theta$ by the proportion of heads in $n$ tosses has risk $\theta(1-\theta)/n$, and its average over a uniform prior is $1/(6n)$.
</document>

<document source="https://tamnd.github.io/math2/node/VEFE5WZMS3.md" tag="VEFE5WZMS3" kind="concept">
# Self-distributive operation

## Summary

A self-distributive operation is a [[binary operation|0M418FE7BB]] that distributes over itself, on the left or on the right.
Left self-distributivity reads $x * (y * z) = (x * y) * (x * z)$.
Conjugation in a group, the join and meet of a lattice, and the operation $2x - y$ on an abelian group are self-distributive.
Left self-distributive magmas are called [[shelves|PDFC0MRYR8]]; together with racks and quandles they are used in knot theory and in set theory through the [[Laver tables|9H65YDXMGJ]].

## Explanations

### L1 human

The usual [[distributive law|0KBEMAJAYA]] says that multiplication spreads over addition: $a(b + c) = ab + ac$.
A self-distributive operation spreads over itself.
Acting with $x$ on a combination of $y$ and $z$ is the same as combining the results of acting with $x$ on each of them.
Think of $x * y$ as "move $y$ by the symmetry attached to $x$".
Then the rule says that moving a combined object is the same as moving its parts and combining them.
For example, $x * y = 2x - y$ reflects $y$ through $x$, and reflections behave in this way.
</document>

<document source="https://tamnd.github.io/math2/node/VEGBD6Y61A.md" tag="VEGBD6Y61A" kind="definition">
# Rational homotopy theory

## Definition

Consider simply connected [[CW complexes|5W95QFHXPR]].
A map $f \colon X \to Y$ between them is a *rational homotopy equivalence* if the maps $\pi_n(f) \otimes \Q \colon \pi_n(X) \otimes \Q \to \pi_n(Y) \otimes \Q$ on [[homotopy groups|P976WYEFMB]] are isomorphisms for all $n \ge 2$.
Two such spaces have the same *rational homotopy type* if they are joined by a finite chain of rational homotopy equivalences, in either direction.
Equivalently, their [[rationalizations|D0ZHVD797D]] $X_\Q$ and $Y_\Q$ are [[homotopy equivalent|XZG6MTSWCN]].
*Rational homotopy theory* is the study of simply connected spaces up to rational homotopy type.
In other words, it studies the homotopy category of simply connected spaces with the rational homotopy equivalences inverted.

## Remarks

The condition on $f$ is equivalent to: $H_n(f; \Q)$ is an isomorphism in [[singular homology|M0Q5CFJ7ZH]] for all $n$.

Example: the rationalization $S^3 \to K(\Q, 3)$ is a rational homotopy equivalence, since both sides have rational homotopy $\Q$ in degree $3$ and $0$ in other degrees.
So $S^3$ and $K(\Q, 3)$ have the same rational homotopy type.

Non-example: the map $S^2 \to K(\Q, 2)$ that represents a generator of $H^2(S^2; \Q)$ is an isomorphism on $\pi_2 \otimes \Q$.
It is not a rational homotopy equivalence: $\pi_3(S^2) \otimes \Q \cong \Q$, generated by the Hopf map, while $\pi_3 K(\Q, 2) = 0$.

Sullivan showed that this homotopy category, with finiteness conditions, is equivalent to a homotopy category of commutative differential graded algebras over $\Q$.
Quillen showed that it is also equivalent to a homotopy category of differential graded Lie algebras over $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/VEKV1EM77B.md" tag="VEKV1EM77B" kind="object">
# Modular group

## Description

The modular group is the [[special linear group|0V9PMN0RBH]]
$$\mathrm{SL}_2(\Z) = \set{\begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \Z, \ ad - bc = 1}$$
under matrix multiplication.
It acts on the [[upper half-plane|X6XGGH5FPW]] $\mathbb{H}$ by [[Möbius transformations|0SG74WBFA2]] $\tau \mapsto (a\tau + b)/(c\tau + d)$.
The matrices $I$ and $-I$ both act as the identity map of $\mathbb{H}$, so many authors call the [[quotient group|0KYX2N69J0]] $\mathrm{PSL}_2(\Z) = \mathrm{SL}_2(\Z) / \set{\pm I}$ the modular group.
The group is generated by
$$S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix},$$
which act by $S\tau = -1/\tau$ and $T\tau = \tau + 1$.

## Remarks

The generators satisfy $S^2 = -I$ and $(ST)^3 = -I$.
Check: $ST = \begin{pmatrix} 0 & -1 \\ 1 & 1 \end{pmatrix}$, $(ST)^2 = \begin{pmatrix} -1 & -1 \\ 1 & 0 \end{pmatrix}$, and $(ST)^3 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$.
The group $\mathrm{PSL}_2(\Z)$ is the free product of a cyclic group of order $2$ and a cyclic group of order $3$.
The set $\set{\tau \in \mathbb{H} : \abs{\tau} \ge 1, \ \abs{\operatorname{Re} \tau} \le 1/2}$ is a fundamental domain for its action on $\mathbb{H}$.
The [[modular forms|GWKN5VG78A]] are the functions on $\mathbb{H}$ with a prescribed behaviour under this group.

## Explanations

### L1 human

The modular group is the group of $2 \times 2$ integer matrices with determinant $1$.
Each such matrix moves the points of the upper half-plane by a fraction $(a\tau + b)/(c\tau + d)$.
Every such move is built from two simple ones: the shift $\tau \mapsto \tau + 1$ and the inversion $\tau \mapsto -1/\tau$.
The group also describes the ways to change an oriented basis of a lattice in the plane.
</document>

<document source="https://tamnd.github.io/math2/node/VEMZC8Y458.md" tag="VEMZC8Y458" kind="definition">
# Independent component analysis

## Definition

Let $p \ge 2$.
The *independent component analysis model* for a random vector $X$ in $\R^p$ is
$$X = A S,$$
where:

1. $A$ is an unknown [[invertible|0EB9N0S33C]] $p \times p$ matrix, the *mixing matrix*.
2. $S = (S_1, \ldots, S_p)$ is an unobserved random vector whose components are [[mutually independent|0GQ938GDE0]] and non-constant.
3. At most one of the $S_i$ has a [[normal distribution|05AQE0QR0D]].

*Independent component analysis* estimates, from observations of $X$, an unmixing matrix $W$ such that $W X$ has independent components.
Then $W X$ is the estimate of the sources $S$.

## Remarks

Under clauses 1 to 3, $A$ is determined up to a permutation of its columns and a nonzero scaling of each column.
So the sources are recovered up to order and scale.
Example: let $S_1, S_2$ be independent and uniform on $[-1, 1]$, and $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.
Then $X = (S_1 + S_2, S_2)$ has dependent components, and $W = A^{-1}$ gives back $W X = S$.
Non-example: if $S_1$ and $S_2$ are independent standard normal, clause 3 fails.
For every rotation matrix $Q$, the vector $Q S$ is again standard normal with independent components, so $A$ and $A Q^T$ give the same law of $X$ and cannot be told apart.
Algorithms usually whiten $X$ first and then maximise a measure of non-Gaussianity, such as kurtosis or negentropy.
</document>

<document source="https://tamnd.github.io/math2/node/VEPJN1NK3C.md" tag="VEPJN1NK3C" kind="statement">
# Differentiation in the transform variable

## Statement

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$, and let $F$ be its [[Laplace transform|05XT5AVJBK]].
Let $a$ be a real number such that $\int_0^\infty \abs{f(t)} e^{-\sigma t} \, dt < \infty$ for every $\sigma > a$.
Then $F$ is [[holomorphic|0DG0Z72CQD]] on the half-plane $\operatorname{Re} s > a$, and for every integer $n \ge 1$ and every $s$ with $\operatorname{Re} s > a$,
$$F^{(n)}(s) = \int_0^\infty (-t)^n f(t) \, e^{-s t} \, dt = \mathcal{L}\{(-t)^n f(t)\}(s).$$

## Remarks

The hypothesis says that the Laplace integral converges absolutely to the right of $a$; the least such $a$ is the [[abscissa of absolute convergence|YCSH2WK69M]].
For $\operatorname{Re} s \ge \sigma > \sigma' > a$, one has $t^n e^{-\sigma t} \le C e^{-\sigma' t}$, so $t^n \abs{f(t)} e^{-\sigma t}$ is integrable.
This domination justifies differentiation under the integral sign.
Example: from $\mathcal{L}\{e^{b t}\}(s) = 1/(s - b)$ one gets $\mathcal{L}\{t e^{b t}\}(s) = 1/(s - b)^2$.
From $\mathcal{L}\{\sin \omega t\}(s) = \omega/(s^2 + \omega^2)$ one gets $\mathcal{L}\{t \sin \omega t\}(s) = 2 \omega s / (s^2 + \omega^2)^2$.

## Explanations

### L1 human

Multiplying a function by $-t$ corresponds to differentiating its transform.
So the transform of $t \cdot f(t)$ is minus the slope of the transform of $f$.

### L2 human

This rule produces the transforms of $t^n e^{b t}$, $t \cos \omega t$ and similar functions, which appear as resonant solutions of linear equations.
It also shows that a Laplace transform is infinitely differentiable, and in fact holomorphic, in its half-plane of convergence.
The reverse operation is [[integration in the transform variable|GXX2WPVRJ2]].
</document>

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# Brody's lemma

## Statement

Let $X$ be a compact [[complex manifold|JSGWAN2RJV]] with a [[Hermitian metric|CK1DCPC3D3]].
Let $\norm{\cdot}$ denote the norm of tangent vectors of $X$ for this metric.

1. If $X$ is not [[Kobayashi hyperbolic|HRPAX8V7HA]], then there is a nonconstant [[holomorphic map|3MQTWKBBBT]] $g \colon \C \to X$ such that $\norm{g'(t)} \le \norm{g'(0)} = 1$ for all $t \in \C$.
2. Consequently, $X$ is Kobayashi hyperbolic if and only if it is [[Brody hyperbolic|Y7W36GJDJD]].

## Remarks

Robert Brody proved the theorem in 1978.
Proof idea: if a compact $X$ is not hyperbolic, then there are holomorphic maps $f_k \colon \mathbb{D} \to X$ with $\norm{f_k'(0)} \to \infty$.
The reparametrisation lemma recentres and rescales each $f_k$ at a point where $(1 - \abs{t}^2) \norm{f_k'(t)}$ is nearly maximal, which gives maps $g_k$ from discs of radii $R_k \to \infty$ with $\norm{g_k'(0)} = 1$ and $\norm{g_k'(t)} \le R_k^2 / (R_k^2 - \abs{t}^2)$.
These maps are equicontinuous, and $X$ is compact, so by the Arzelà–Ascoli theorem a subsequence converges locally uniformly to the required $g$.
Part 2 follows: a hyperbolic manifold has no entire curve because $k_\C \equiv 0$, and part 1 gives the converse.
Example: a smooth plane curve of degree $d \ge 4$ has genus at least $3$, so it has no entire curve, and by the theorem it is hyperbolic.
Compactness is necessary: there are noncompact complex manifolds that contain no entire curve but are not Kobayashi hyperbolic.

## Explanations

### L1 human

To decide whether a compact complex manifold is hyperbolic, it is enough to look for copies of the complex plane inside it.
If there are none, the manifold is hyperbolic.
If the manifold is not hyperbolic, the lemma even produces a copy of the plane whose speed is bounded everywhere by its speed at the origin.

### L2 human

The lemma turns hyperbolicity, a metric property, into a statement about entire curves, which is much easier to test and to deform.
It implies that hyperbolicity is an open condition in families of compact complex manifolds: a limit of entire curves in nearby fibres gives an entire curve in the limit fibre.
It is the starting point of the strategy for the [[Kobayashi conjecture|1X3EV2TZNN]].
There one shows that entire curves in a generic hypersurface must satisfy algebraic differential equations, and then that no such curves exist.
</document>

<document source="https://tamnd.github.io/math2/node/VEVPDJ8S7B.md" tag="VEVPDJ8S7B" kind="concept">
# Curvelet

## Summary

Curvelets are a multiscale system of functions on the plane that are indexed by scale, orientation and position.
At fine scales they are thin needles whose width is roughly the square of their length, a rule called parabolic scaling.
This shape matches smooth curves well, so images that are smooth apart from smooth edges have sparse curvelet expansions.
For such images, the best approximation with $N$ curvelets has squared error of order $N^{-2}(\log N)^3$, while $N$ wavelets give only order $N^{-1}$.

## Explanations

### L1 human

Two-dimensional wavelets are small square blobs, so following a curved edge needs many of them.
Curvelets are long, thin and oriented, like short strokes of a pen.
A smooth edge, seen up close, is almost a straight segment, and a single curvelet aligned with it covers it.
As the scale gets finer, the strokes get thinner faster than they get shorter, and there are more possible directions.
This is why curvelets describe images with clean contours using far fewer terms than wavelets.
</document>

<document source="https://tamnd.github.io/math2/node/VEY29CS26X.md" tag="VEY29CS26X" kind="concept">
# Wasserstein metric

## Summary

The Wasserstein metric is a distance between [[probability measures|059TJN0NZ1]] on a [[metric space|0VKD1TP7F3]].
It is the least cost of moving one mass distribution onto the other.
The cost of moving a unit of mass is a power of the distance it travels.
It is the central distance of optimal transport, and it metrizes weak convergence together with convergence of moments.

## Explanations

### L1 human

Think of one measure as a pile of earth and the other as a hole of the same volume.
A transport plan says how much earth goes from each place to each other place.
The Wasserstein distance is the cheapest total cost of such a plan.
This is why the $1$-Wasserstein distance is also called the earth mover's distance.
For two point masses at $x$ and $y$, all the mass must travel from $x$ to $y$, so the distance is $d(x, y)$.
</document>

<document source="https://tamnd.github.io/math2/node/VF17XWCK8Z.md" tag="VF17XWCK8Z" kind="definition">
# Isogeometric analysis

## Definition

Let $\hat\Omega = (0, 1)^d$ be the parameter domain.
Let $R_1, \ldots, R_n \colon \hat\Omega \to \R$ be [[NURBS|Y8B2N9ERPE]] basis functions, built from [[B-splines|62AFX32JGT]] on given knot vectors with positive weights.
Let $P_1, \ldots, P_n \in \R^d$ be control points.
Assume the geometry map $\mathbf{F}(\xi) = \sum_{i=1}^{n} R_i(\xi) P_i$ is a bijection from $\hat\Omega$ onto the physical domain $\Omega$, with a smooth inverse.
*Isogeometric analysis* is the [[Galerkin method|Y5B0PEBKE8]] for a boundary value problem on $\Omega$ with the discrete space
$$V_h = \operatorname{span} \set{ R_i \circ \mathbf{F}^{-1} : 1 \le i \le n }.$$
Refinement is done by knot insertion and degree elevation, which leave $\mathbf{F}$ unchanged.

## Remarks

The same functions $R_i$ define the geometry and the solution, as in isoparametric [[finite elements|98GZ9DVHTA]], hence the name.
With $p$ the spline degree and knots of multiplicity one, the basis functions are $C^{p-1}$ across knots.

Example: the quarter circle $\set{(x, y) : x^2 + y^2 = 1, \ x, y \ge 0}$ is the quadratic NURBS curve with control points $(1, 0)$, $(1, 1)$, $(0, 1)$ and weights $1$, $\tfrac{\sqrt{2}}{2}$, $1$.
A quarter annulus is then given exactly by a geometry map that is quadratic in the angular parameter and linear in the radial one.

Non-example: a standard finite element method on a mesh of straight-sided triangles approximates the annulus by a polygon.
Its geometry is not given by the solution basis of a NURBS map from $\hat\Omega$, so it is not isogeometric.
</document>

<document source="https://tamnd.github.io/math2/node/VF3PV13BA1.md" tag="VF3PV13BA1" kind="concept">
# Ore extension

## Summary

An Ore extension $R[x; \sigma, \delta]$ of a [[ring|0B1261B5ED]] $R$ is a ring of polynomials $\sum a_i x^i$ in which the variable does not commute with the coefficients.
Instead, $x a = \sigma(a) x + \delta(a)$ for an endomorphism $\sigma$ of $R$ and a twisted [[derivation|3J89REV9WQ]] $\delta$.
Ore extensions give many of the basic examples of [[noncommutative rings|D1SA8CHJV0]], such as the [[Weyl algebras|98QE7KY55W]] and quantum planes.
They keep good properties of $R$, such as being a domain or being Noetherian, under mild conditions.

## Explanations

### L1 human

In an ordinary polynomial ring the variable $x$ commutes with every coefficient.
In an Ore extension, moving $x$ past a coefficient $a$ changes $a$ to $\sigma(a)$ and adds a correction term $\delta(a)$.
Every element can still be written uniquely as $a_0 + a_1 x + \cdots + a_n x^n$, so the ring looks like a polynomial ring, only with a new rule of multiplication.
For example, with $R = k[y]$, $\sigma$ the identity and $\delta = d/dy$, the rule $xy = yx + 1$ gives the first Weyl algebra.
</document>

<document source="https://tamnd.github.io/math2/node/VF43FTDA1H.md" tag="VF43FTDA1H" kind="example">
# Geodesic flow

## Example

Let $M$ be a complete [[Riemannian manifold|5HDG31YEPW]] with [[tangent bundle|DQB1BVQJWD]] $TM$.
Let $SM$ be its unit tangent bundle, the set of tangent vectors of length $1$.
For $v \in SM$ let $\gamma_v$ be the [[geodesic|NSYMQMHPDN]] with $\gamma_v(0)$ the base point of $v$ and $\gamma_v'(0) = v$.
The geodesic flow is $\varphi_t(v) = \gamma_v'(t)$ for $t \in \R$.
It is defined for all $t$ because $M$ is complete, and $\varphi_{s+t} = \varphi_s \circ \varphi_t$ because $t \mapsto \gamma_v(s + t)$ is the geodesic with initial vector $\gamma_v'(s)$.
So it is a [[smooth dynamical system|116TPXR5FS]] in continuous time on $SM$.

Check on the flat torus $M = \R^2/\Z^2$: geodesics are straight lines, so $SM = M \times S^1$ and $\varphi_t(x, v) = (x + t v, v)$.
For a direction $v$ of irrational slope, the orbit of $(x, v)$ is dense in the torus $M \times \set{v}$.
For a direction of rational slope, every orbit is periodic.

## Remarks

The geodesic flow preserves the Liouville measure on $SM$, the product of the Riemannian volume and the uniform measure on each unit sphere.
If $M$ is compact and its [[sectional curvature|1X42JAA349]] is negative everywhere, the geodesic flow is an [[Anosov flow|8FQJXSA4DT]], as Anosov proved, and it is ergodic for the Liouville measure.
For a surface of constant curvature $-1$ it can be written as a homogeneous flow, see [[geodesic flow on a locally symmetric space|5TKBSHY6VC]].
Identify $TM$ with the cotangent bundle $T^*M$ by the metric.
Then the geodesic flow is the Hamiltonian flow of the kinetic energy $H(v) = \frac12\norm{v}^2$, restricted to an energy level.

## Explanations

### L1 human

Put a frictionless marble on a curved surface and push it in some direction.
It rolls along the straightest possible path, a geodesic, at constant speed.
The geodesic flow records where the marble is and which way it moves after time $t$.
On a flat torus the motion is simple, but on a surface curved like a saddle everywhere, nearby paths diverge exponentially and the motion is chaotic.
</document>

<document source="https://tamnd.github.io/math2/node/VF6CA0VE41.md" tag="VF6CA0VE41" kind="concept">
# Nonexpansive map

## Summary

A nonexpansive map between metric spaces is a map that never increases distances: $d(f(x), f(y)) \le d(x, y)$.
These are exactly the maps that are [[Lipschitz|0AW1E6YTJY]] with constant $1$; they are also called short maps or metric maps.
They include isometries and [[contraction mappings|0QN69YA357]].
In fixed point theory they are the borderline case where the Banach fixed point theorem no longer applies and the geometry of the space matters.

## Explanations

### L1 human

A nonexpansive map may move points around, but it never pulls two points further apart.
Rotations of the plane, reflections, and the map sending each point to the nearest point of a closed disc are all nonexpansive.
The map $x \mapsto 2x$ on the real line is not, since it doubles distances.
Unlike contractions, nonexpansive maps need not have fixed points: the shift $x \mapsto x + 1$ on the line moves every point.
</document>

<document source="https://tamnd.github.io/math2/node/VFAJ1EG3X5.md" tag="VFAJ1EG3X5" kind="concept">
# Nonparametric statistics

## Summary

Nonparametric statistics is the part of statistics that does not assume the law of the data belongs to a family described by finitely many parameters.
It uses models of infinite dimension, such as all continuous distributions on $\R$ or all smooth densities.
It also includes distribution-free methods, whose behaviour under the null hypothesis does not depend on the unknown law.
Typical tools are the empirical distribution function, rank tests, [[kernel density estimation|ZV65XMJ9A4]] and the bootstrap.

## Explanations

### L1 human

A parametric method assumes, for example, that the data are normal and then only estimates a mean and a variance.
A nonparametric method lets the data speak for the shape of the law.
For instance, a histogram or a kernel density estimate shows the shape of the data without assuming a bell curve.
The price is that more data are needed for the same precision.
</document>

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# Perfect graph

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]].
For a graph $H$, let $\chi(H)$ be its [[chromatic number|0C5M7AM6FH]] and $\omega(H)$ its [[clique number|0WCKRBHFFD]].
The graph $G$ is perfect if $\chi(H) = \omega(H)$ for every [[induced subgraph|0V05FC4GKQ]] $H$ of $G$.

## Remarks

Example: every bipartite graph $G$ is perfect.
An induced subgraph $H$ of $G$ is again bipartite, so $\chi(H) = \omega(H) = 2$ if $H$ has an edge, and $\chi(H) = \omega(H) = 1$ if $H$ has vertices but no edge.
Non-example: the cycle $C_5$ is not perfect, because $\omega(C_5) = 2$ while $\chi(C_5) = 3$, so the condition fails for $H = C_5$ itself.
The condition must hold for all induced subgraphs, not just for $G$: the disjoint union of $C_5$ and a triangle has $\chi = \omega = 3$ but is not perfect.
By the weak perfect graph theorem of Lovász, the complement of a perfect graph is perfect.
</document>

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# Cyclic extension

## Summary

A cyclic extension is a [[Galois extension|0DS2F7GK05]] whose [[Galois group|0XVR72M51M]] is [[cyclic|0B85Q41RP6]].
Cyclic extensions are the building blocks of radical extensions.
If a field contains $n$ distinct $n$-th roots of unity, adjoining an $n$-th root of one of its elements gives a cyclic extension.
[[Kummer theory|XZQENYV15Y]] and [[Artin–Schreier theory|7YNMNRGH3T]] describe cyclic extensions of a given degree.

## Explanations

### L1 human

A cyclic extension has one basic symmetry, and every other symmetry is a repeat of it.
Every finite extension of finite fields is cyclic, generated by the map $x \mapsto x^q$, where $q$ is the size of the smaller field.
For example, $\Q(\sqrt{5})$ over $\Q$ is cyclic of degree $2$, generated by $\sqrt{5} \mapsto -\sqrt{5}$.
Every cyclic extension is abelian, but $\Q(\sqrt{2}, \sqrt{3})$ over $\Q$ is abelian and not cyclic.
</document>

<document source="https://tamnd.github.io/math2/node/VFKNNPVMTX.md" tag="VFKNNPVMTX" kind="definition">
# Complete Boolean algebra

## Definition

Let $B$ be a [[Boolean algebra|0XADJKM699]] with order $\le$.
The algebra $B$ is *complete* if every subset $S \subseteq B$ has a supremum $\bigvee S$ in $B$, that is, a least upper bound for $\le$.

## Remarks

Then every subset also has an infimum, $\bigwedge S = \neg \bigvee \set{\neg s : s \in S}$, so $B$ is a [[complete lattice|D7A0ZGZT5S]].
The infinite distributive laws $a \wedge \bigvee S = \bigvee \set{a \wedge s : s \in S}$ hold in every complete Boolean algebra.

Example: the power set of a set $X$, ordered by inclusion.
The supremum of a family of subsets is its union, which is again a subset of $X$.

Non-example: the algebra $B$ of finite and cofinite subsets of $\N$ is not complete.
Let $S = \set{\set{2n} : n \in \N}$.
An upper bound of $S$ in $B$ contains all even numbers, so it is infinite and hence cofinite, and it contains infinitely many odd numbers.
Removing one odd number gives a smaller upper bound in $B$, so there is no least one.

In Mathlib the notion is `CompleteBooleanAlgebra`.
</document>

<document source="https://tamnd.github.io/math2/node/VFPGADHJYA.md" tag="VFPGADHJYA" kind="definition">
# Ultraproduct

## Definition

Let $L$ be a [[signature|WDVVEYD6AT]], let $I$ be a nonempty set, and let $\mathcal{U}$ be an [[ultrafilter|J64Y3JC0RE]] on $I$.
For each $i \in I$ let $M_i$ be an [[$L$-structure|983GZBQYT4]] with nonempty domain.
Let $P = \prod_{i \in I} M_i$ be the [[Cartesian product|09CEDJYBKP]] of the domains.
For $a, b \in P$ write $a \sim_{\mathcal{U}} b$ when $\set{i \in I : a(i) = b(i)} \in \mathcal{U}$; this is an [[equivalence relation|0G8VEZ3WTK]] on $P$, and $[a]$ denotes the class of $a$.
The *ultraproduct* $\prod_{i \in I} M_i / \mathcal{U}$ is the $L$-structure with domain $P / {\sim_{\mathcal{U}}}$ and the following interpretations:

1. for an $n$-ary function symbol $f$, $f([a_1], \dots, [a_n]) = [b]$, where $b(i) = f^{M_i}(a_1(i), \dots, a_n(i))$;
2. for an $n$-ary relation symbol $R$, $R([a_1], \dots, [a_n])$ holds if and only if $\set{i \in I : (a_1(i), \dots, a_n(i)) \in R^{M_i}} \in \mathcal{U}$;
3. for a constant symbol $c$, its value is $[b]$, where $b(i) = c^{M_i}$.

If all $M_i$ equal one structure $M$, the ultraproduct is called the *ultrapower* $M^I / \mathcal{U}$.

## Remarks

The interpretations do not depend on the chosen representatives, because $\mathcal{U}$ is closed under finite intersections and supersets.

Example: let $\mathcal{U}$ be a non-principal ultrafilter on $\N$ and let $M_i = \R$ for all $i$.
The ultrapower $\R^{\N} / \mathcal{U}$ is the ordered field of [[hyperreal numbers|421QMTFG9T]].
The class of $a = (0, 1, 2, 3, \dots)$ is larger than the class of each constant sequence $(r, r, r, \dots)$, since $\set{i : i > r}$ is cofinite and so lies in $\mathcal{U}$.

Non-example: replace $\mathcal{U}$ by the Fréchet filter $\mathcal{F}$ of cofinite subsets of $\N$, which is a filter but not an ultrafilter.
The same construction gives a ring $\R^{\N} / \mathcal{F}$, but it is not a field.
The classes of $(1, 0, 1, 0, \dots)$ and $(0, 1, 0, 1, \dots)$ are both nonzero, because neither sequence is zero on a cofinite set, but their product is the class of $0$.

If $\mathcal{U}$ is the principal ultrafilter at $i_0 \in I$, the ultraproduct is isomorphic to $M_{i_0}$.
</document>

<document source="https://tamnd.github.io/math2/node/VFPP3RZGK3.md" tag="VFPP3RZGK3" kind="definition">
# Bockstein homomorphism

## Definition

Let $X$ be a topological space and let
$$0 \to A \xrightarrow{i} B \xrightarrow{j} C \to 0$$
be a short [[exact sequence|CBDM280TDF]] of [[abelian groups|071DDFVG4J]].
Let $C^*(X; G) = \Hom(C_*(X), G)$ be the singular cochain complex with coefficients in $G$.
Since each group $C_n(X)$ of singular chains is free abelian, the induced sequence of cochain complexes
$$0 \to C^*(X; A) \to C^*(X; B) \to C^*(X; C) \to 0$$
is exact.
The *Bockstein homomorphism* is its connecting homomorphism
$$\beta \colon H^n(X; C) \to H^{n+1}(X; A)$$
in [[singular cohomology|6N7BC70CX0]].
Explicitly, let $c$ be a cocycle representing a class in $H^n(X; C)$, and choose a cochain $b$ with values in $B$ with $j \circ b = c$.
Then $\delta b = i \circ a$ for a unique cocycle $a$, and $\beta[c] = [a]$.

## Remarks

The same construction on chains gives a homology Bockstein $H_n(X; C) \to H_{n-1}(X; A)$ in [[homology with coefficients|XT7VWNTKGB]].
The most used case is $0 \to \Z/p \to \Z/p^2 \to \Z/p \to 0$, which gives a natural map $\beta \colon H^n(X; \F_p) \to H^{n+1}(X; \F_p)$ with $\beta \circ \beta = 0$.

Example: let $X = \R P^2$ and $p = 2$, so $H^1(X; \Z/2) \cong H^2(X; \Z/2) \cong \Z/2$, while $H^1(X; \Z/4) \cong \Z/2$ and $H^2(X; \Z/4) \cong \Z/2$.
The map $H^1(X; \Z/4) \to H^1(X; \Z/2)$ is zero, since the class of order two in $H^1(X; \Z/4)$ reduces to zero mod $2$.
So by exactness the generator of $H^1(X; \Z/2)$ has nonzero Bockstein, which is the generator of $H^2(X; \Z/2)$.

Non-example of a nonzero Bockstein: for the split sequence $0 \to \Z/2 \to \Z/2 \oplus \Z/2 \to \Z/2 \to 0$, every class lifts, and the Bockstein is the zero map.
In general $\beta = 0$ whenever the coefficient sequence splits.

For $p = 2$ the Bockstein on mod $2$ cohomology equals the Steenrod square $Sq^1$.
For the sequence $0 \to \Z \xrightarrow{p} \Z \to \Z/p \to 0$ one gets the integral Bockstein $H^n(X; \Z/p) \to H^{n+1}(X; \Z)$, whose image consists of $p$-torsion classes.
</document>

<document source="https://tamnd.github.io/math2/node/VFRJ0P2P15.md" tag="VFRJ0P2P15" kind="statement">
# Hyers' theorem

## Statement

Let $E_1$ be a real [[normed vector space|03J5RAKB30]], let $E_2$ be a real [[Banach space|06GZ956657]], and let $\delta \ge 0$.
Let $f \colon E_1 \to E_2$ be [[$\delta$-additive|J6GPWAX3X2]], that is, $\norm{f(x + y) - f(x) - f(y)} \le \delta$ for all $x, y \in E_1$.
Then:

1. For each $x \in E_1$ the limit $A(x) = \lim_{n \to \infty} 2^{-n} f(2^n x)$ exists in $E_2$.
2. The map $A \colon E_1 \to E_2$ is [[additive|KF719SZD5T]] and satisfies $\norm{f(x) - A(x)} \le \delta$ for all $x \in E_1$.
3. $A$ is the only additive map $E_1 \to E_2$ such that $f - A$ is bounded.
4. If moreover, for each $x \in E_1$, the map $t \mapsto f(tx)$ from $\R$ to $E_2$ is continuous, then $A$ is [[linear|0NRYHJPQG2]].

## Remarks

Setting $x = y$ gives $\norm{f(2x) - 2f(x)} \le \delta$, so consecutive terms of the sequence $2^{-n} f(2^n x)$ differ by at most $2^{-n-1}\delta$.
The sequence is therefore Cauchy, it converges by completeness of $E_2$, and summing the geometric bounds gives $\norm{f(x) - A(x)} \le \delta$.
Additivity follows from $\norm{2^{-n}(f(2^n(x + y)) - f(2^n x) - f(2^n y))} \le 2^{-n}\delta$.
For uniqueness, the difference $D$ of two such additive maps is additive and bounded, and $D(x) = D(nx)/n \to 0$.
Example: $f(x) = x + \sin x$ on $\R$ is $3$-additive, and the theorem gives $A(x) = x$ with $\abs{f(x) - x} \le 1$.
Completeness of $E_2$ is used to make the limit exist.
D. H. Hyers proved the theorem in 1941, answering [[Ulam's stability problem|CBYB39FTFG]] for the Cauchy equation.

## Explanations

### L1 human

If a map between normed spaces adds vectors correctly up to a fixed error, then it is close to a map that adds them exactly.
The exact map is found by scaling up, applying $f$, and scaling back down, so the fixed error becomes negligible.
The distance between the two maps is at most the error itself.

### L2 human

The theorem shows that the Cauchy equation $f(x + y) = f(x) + f(y)$ is [[Hyers–Ulam stable|TPYXQTYND6]] with constant $1$.
The method, now called the direct method, works for many other equations, such as the Jensen and quadratic equations, after a suitable rescaling.
Replacing the constant $\delta$ by a bound that grows with $\norm{x}$ and $\norm{y}$ leads to the [[Aoki–Rassias theorem|93WX6PZC8R]].
</document>

<document source="https://tamnd.github.io/math2/node/VFYVR2H4QS.md" tag="VFYVR2H4QS" kind="example">
# The odd integers are not a subgroup of the integers

## Example

Let $G = (\mathbb{Z}, +)$ be the [[integers|0C4JS9RF5M]] with addition, and let $O$ be the set of odd integers.
The identity $0$ is not odd, so $0 \notin O$.
Also, $1 \in O$ and $1 + 1 = 2 \notin O$. Thus $O$ is not closed under addition.
Thus $O$ is not a [[subgroup|0N0PXCTYA7]] of $G$.

## Remarks

$O$ is a coset of the even integers. A coset $g + H$ is a subgroup only when $g \in H$.
</document>

<document source="https://tamnd.github.io/math2/node/VFZRVZ9SD8.md" tag="VFZRVZ9SD8" kind="concept">
# Minimal period

## Summary

The minimal period, or prime period, of a [[periodic point|T75CZ5H47N]] is the least number of steps after which it returns to itself.
It equals the number of distinct points in its orbit.
Every period of the point is a multiple of the minimal period.
Results such as Sharkovskii's theorem are stated in terms of minimal periods.

## Explanations

### L1 human

A periodic point comes back to its start after some steps, and then again and again.
The minimal period is the length of one trip around the loop.
For the rotation of a clock face by $3$ hours, the point at $12$ visits $3$, $6$, $9$ and returns to $12$.
So its minimal period is $4$, even though it is also back after $8$ or $12$ steps.
</document>

<document source="https://tamnd.github.io/math2/node/VG2P7R91Z7.md" tag="VG2P7R91Z7" kind="concept">
# Signed measure

## Summary

A signed measure is a countably additive set function on a σ-algebra that may take negative values.
It models quantities such as electric charge, which can be positive in some regions and negative in others.
Every signed measure is the difference of two ordinary [[measures|PR11BZQQPG]], one of them finite.
Signed measures appear as differences of measures, as integrals $E \mapsto \int_E f \, d\mu$ of integrable functions, and as elements of dual spaces of function spaces.

## Explanations

### L1 human

An ordinary measure gives each set a size that is zero or positive.
A signed measure also allows negative sizes, like a bank account in which some regions bring money and others cost money.
It must still add up correctly over countably many disjoint pieces.
For example, on $[-1, 1]$ the rule $\nu(E) = \int_E x \, dx$ gives $\nu([0, 1]) = \tfrac12$ and $\nu([-1, 0]) = -\tfrac12$.
</document>

<document source="https://tamnd.github.io/math2/node/VG4ZVTGJ9W.md" tag="VG4ZVTGJ9W" kind="definition">
# Smooth structure

## Definition

Let $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.
A chart $(U, \varphi)$ on $M$ is an open set $U \subseteq M$ with a [[homeomorphism|0QT00PB5G2]] $\varphi$ from $U$ onto an open subset of $\R^n$.
Two charts $(U, \varphi)$ and $(V, \psi)$ are *smoothly compatible* if $U \cap V = \emptyset$, or if the transition map $\psi \circ \varphi^{-1} \colon \varphi(U \cap V) \to \psi(U \cap V)$ is [[smooth|0V33BNKZ9S]] with smooth inverse.
A *smooth [[atlas|J7P32JQTVE]]* on $M$ is a family of pairwise smoothly compatible charts whose domains cover $M$.
A *smooth structure* on $M$ is a smooth atlas $\mathcal{A}$ that is maximal.
This means that every chart that is smoothly compatible with all charts of $\mathcal{A}$ already belongs to $\mathcal{A}$.

## Remarks

Every smooth atlas is contained in a unique maximal smooth atlas, so a smooth structure is usually given by any one atlas.
Example: on $\R^n$ the single chart $(\R^n, \id)$ is a smooth atlas, and the maximal atlas containing it is the standard smooth structure.
Non-example: the atlas $\set{(\R, \id), (\R, t \mapsto t^3)}$ on $\R$ is not a smooth atlas.
The transition map $t \mapsto t^{1/3}$ is not differentiable at $0$, so the two charts are not smoothly compatible.
Each of the two charts alone defines a smooth structure; the two structures are different but diffeomorphic.
Two smooth structures on $M$ are usually counted as the same when they are diffeomorphic, not only when they are equal.
</document>

<document source="https://tamnd.github.io/math2/node/VGB11C8TTN.md" tag="VGB11C8TTN" kind="statement">
# Bourgain's dimension bound for harmonic measure

## Statement

Let $n \ge 2$.
There is a number $\varepsilon_n > 0$, depending only on $n$, with the following property.
Let $\Omega \subseteq \R^n$ be a bounded domain, let $x \in \Omega$, and let $\omega^x$ be the [[harmonic measure|CFD7BPM51W]] of $\Omega$ at $x$.
Then there is a Borel set $F \subseteq \partial\Omega$ with $\omega^x(\partial\Omega \setminus F) = 0$ and [[Hausdorff dimension|SSNFTHZ6YJ]] at most $n - \varepsilon_n$.

## Remarks

In short, $\dim \omega^x \le n - \varepsilon_n$: harmonic measure never fills a set of full dimension $n$, even when the boundary itself has dimension $n$.
By mutual absolute continuity of harmonic measures, the conclusion does not depend on $x$.
Idea of the proof: a multiscale argument looks at cubes that meet the boundary.
In each such cube, harmonic measure is spread unevenly over the subcubes by a uniform amount.
This loss at each scale accumulates into the dimension drop $\varepsilon_n$.
Example: in the plane one can take $\varepsilon_2 = 1$.
This is the theorem of Jones and Wolff that planar harmonic measure is carried by a set of dimension at most $1$.
For Jordan domains it also follows from [[Makarov's theorem|CM9BDNEA10]].
The value of the best constant $\varepsilon_n$ is unknown for $n \ge 3$.
It cannot be taken equal to $1$ in $\R^3$: Wolff constructed domains in $\R^3$ whose harmonic measure has dimension larger than $2$.
The theorem was proved by J. Bourgain in 1987.

## Explanations

### L1 human

Start a random path in a domain in space and record where it first hits the boundary.
Even if the boundary is so rough that it has full dimension, the hitting points concentrate on a set of strictly smaller dimension.
The drop in dimension is at least a fixed amount that depends only on the dimension of the space.

### L2 human

This is the main general result on the dimension of harmonic measure in higher dimensions.
In the plane the sharp answer is known: dimension at most $1$ for all domains, and exactly $1$ for simply connected ones.
In space, Wolff's examples show that the planar answer $n - 1$ is wrong, and Bourgain's bound is the general upper estimate.
The question of the optimal $\varepsilon_n$ connects potential theory with geometric measure theory and the study of rough boundaries.
</document>

<document source="https://tamnd.github.io/math2/node/VGCTDCGS6V.md" tag="VGCTDCGS6V" kind="example">
# Pseudo-arc

## Example

A [[continuum|NJQ196M1WR]] $X$ is *chainable* if for every $\varepsilon > 0$ there is a finite cover of $X$ by open sets $U_1, \dots, U_n$ of the following kind.
Each $U_i$ has diameter less than $\varepsilon$, and $U_i$ meets $U_j$ only when $\abs{i - j} \le 1$.
The *pseudo-arc* is a nondegenerate chainable continuum in which every subcontinuum is [[indecomposable|56J7DSVAZH]].
It can be built in the plane as the intersection of a decreasing sequence of chains of discs.
Each chain is folded back and forth inside the previous one in a specific crooked way.
Knaster gave the first such continuum, and Bing proved that any two such continua are homeomorphic.
The pseudo-arc is indecomposable, since by its definition every subcontinuum, including the whole space, is indecomposable.

## Remarks

The pseudo-arc is homogeneous: for any two points there is a homeomorphism of the space onto itself that sends one to the other.
It is homeomorphic to each of its nondegenerate subcontinua, as Moise proved.
The arc $[0, 1]$ is also chainable and also homeomorphic to its nondegenerate subcontinua, but it is decomposable; this is why the name "pseudo-arc" fits.

## Explanations

### L1 human

The pseudo-arc looks like an arc from far away, since it can be covered by a thin chain of small links.
Up close it is folded back on itself at every scale, so that no piece of it is a simple arc.
Every piece of it is as tangled as the whole, and every piece looks the same as the whole.
</document>

<document source="https://tamnd.github.io/math2/node/VGJJVBZ1MQ.md" tag="VGJJVBZ1MQ" kind="concept">
# Surface

## Summary

A surface is a [[topological manifold|QQQ9B1DK4P]] of dimension $2$: a space that looks like a piece of the plane near each of its points.
Spheres, tori, the Möbius strip and the Klein bottle are surfaces.
Closed surfaces are completely classified by orientability and Euler characteristic, see the [[classification of surfaces|2SFM73YET0]].
Surfaces are the first nontrivial case of manifold topology and appear throughout geometry, complex analysis and dynamics.

## Explanations

### L1 human

A surface is a space in which a tiny creature, looking around itself, would always believe it lives on a flat sheet.
The skin of a ball, the skin of a doughnut and an infinite plane are surfaces.
Two cones touching at their tips do not form a surface, because at the tips the creature would see two sheets meeting at a point.
Globally a surface can close up on itself, have holes, or even have no consistent inside and outside, like the Möbius strip.
</document>

<document source="https://tamnd.github.io/math2/node/VGPTRNE3FF.md" tag="VGPTRNE3FF" kind="definition">
# Bifurcation set

## Definition

Let $n, k \ge 1$ and let $V \colon \R^n \times \R^k \to \R$ be a [[smooth function|0V33BNKZ9S]], written $V(x, c)$ with state $x \in \R^n$ and control $c \in \R^k$.
A point $x$ is a *degenerate critical point* of $V(\cdot, c)$ if:

1. $\nabla_x V(x, c) = 0$, so $x$ is a [[critical point|0YTCMFB4HC]] of $V(\cdot, c)$, and
2. $\det H_x V(x, c) = 0$, where $H_x V$ is the [[Hessian matrix|0693Y2METT]] of second derivatives in the variables $x$.

The *bifurcation set* of the family $V$ is the set
$$B_V = \set{c \in \R^k : V(\cdot, c) \text{ has a degenerate critical point}}.$$

## Remarks

Example: the cusp family $V(x, a, b) = x^4/4 + a x^2/2 + b x$ with $n = 1$, $k = 2$.
Clause 1 reads $x^3 + a x + b = 0$ and clause 2 reads $3x^2 + a = 0$.
Then $a = -3x^2$ and $b = 2x^3$, and eliminating $x$ gives $B_V = \set{(a, b) : 4a^3 + 27b^2 = 0}$, a curve with a cusp at the origin.

Non-example: for $V(x, c) = x^2 + c x$ the only critical point is $x = -c/2$, and $V'' = 2 \ne 0$, so clause 2 always fails and $B_V$ is empty.

Let $M = \set{(x, c) : \nabla_x V(x, c) = 0}$ and let $\chi \colon M \to \R^k$ be the projection $(x, c) \mapsto c$.
When $M$ is a smooth submanifold of dimension $k$, which holds for generic families, $B_V$ is the set of critical values of $\chi$.
Some authors also add the Maxwell set, where two critical values of $V(\cdot, c)$ are equal; this node uses the narrower meaning.
</document>

<document source="https://tamnd.github.io/math2/node/VGV9FAPCHN.md" tag="VGV9FAPCHN" kind="statement">
# Jury stability criterion

## Statement

Let $P(z) = a_n z^n + \cdots + a_1 z + a_0$ be a polynomial of degree $n \ge 1$ with real coefficients and $a_n > 0$.
Build rows of numbers as follows.
The first row is $(a_0, a_1, \dots, a_n)$.
If a row $(x_0, x_1, \dots, x_m)$ has $m \ge 3$, the next row is $(y_0, \dots, y_{m-1})$ with
$$y_k = x_0 x_k - x_m x_{m-k}, \qquad k = 0, 1, \dots, m-1.$$
Stop at the first row that has exactly $3$ entries; for $n \le 2$ only the first row is built.
Then $P$ is [[Schur stable|X11Q0G8PNA]] if and only if all of the following hold.

1. $P(1) > 0$.
2. $(-1)^n P(-1) > 0$.
3. $\abs{a_0} < a_n$.
4. Every row $(y_0, \dots, y_{m-1})$ built after the first satisfies $\abs{y_0} > \abs{y_{m-1}}$.

## Remarks

The proof rests on the Schur–Cohn reduction.
Write $P^*(z) = z^n P(1/z)$, so that $\abs{P^*(z)} = \abs{P(z)}$ on the unit circle because the coefficients are real.
When $\abs{a_0} < a_n$, [[Rouché's theorem|0D430QR02W]] shows that $P$ is Schur stable if and only if the polynomial $(a_n P(z) - a_0 P^*(z))/z$ of degree $n - 1$ is Schur stable.
The next row of the table lists the coefficients of this reduced polynomial in reverse order and with the opposite sign, so the table repeats the reduction until degree $2$.
The conditions 1 and 2 are necessary, since a real polynomial with $a_n > 0$ and no roots in $[1, \infty)$ or $(-\infty, -1]$ has the stated signs at $\pm 1$.
For $n = 2$ the criterion reads $\abs{a_0} < a_2$ and $\abs{a_1} < a_0 + a_2$.
For instance $z^2 - z + \tfrac{1}{2}$ passes: $\tfrac{1}{2} < 1$ and $1 < \tfrac{3}{2}$.
The polynomial $z^2 - \tfrac{5}{2} z + 1 = (z - 2)(z - \tfrac{1}{2})$ fails condition 1, since $P(1) = -\tfrac{1}{2}$.

## Explanations

### L1 human

The criterion decides whether all roots of a real polynomial lie inside the unit circle without finding the roots.
It checks a few signs and inequalities that come from simple products of the coefficients.
Engineers use it to see whether a digital filter or a sampled control loop is stable.

### L2 human

The criterion is the discrete-time analogue of the [[Routh–Hurwitz stability criterion|KJF8Y19BNV]], which tests whether roots lie in the left half-plane.
Combined with the stability theorem for linear recurrences, it decides from the coefficients whether every solution of a linear recurrence tends to zero.
A degree $n$ polynomial needs $n + 1$ conditions in total: three simple ones and $n - 2$ from the table.
When a parameter enters the coefficients, the conditions give the region of parameters where the system is stable.
</document>

<document source="https://tamnd.github.io/math2/node/VGYXWXV5M4.md" tag="VGYXWXV5M4" kind="statement">
# Laplace transform of an integral

## Statement

Let $f \colon [0, \infty) \to \C$ be piecewise continuous and of [[exponential order|TKZAV3KR86]] $a$, that is, $\abs{f(t)} \le M e^{a t}$ for all $t \ge 0$ and some constants $M$ and $a$.
Let $F$ be the [[Laplace transform|05XT5AVJBK]] of $f$, and let $g(t) = \int_0^t f(\tau) \, d\tau$.
Then for every complex $s$ with $\operatorname{Re} s > \max(a, 0)$,
$$\mathcal{L}\{g\}(s) = \frac{F(s)}{s}.$$

## Remarks

The function $g$ is continuous, $g(0) = 0$, and $g' = f$ at each point where $f$ is continuous.
The bound on $f$ gives a bound $\abs{g(t)} \le M' e^{b t}$ for every $b > \max(a, 0)$, so $\mathcal{L}\{g\}$ exists on $\operatorname{Re} s > \max(a, 0)$.
The [[rule for derivatives|0K090GVHVN]] then gives $F(s) = s \, \mathcal{L}\{g\}(s) - g(0) = s \, \mathcal{L}\{g\}(s)$.
Example: with $f = 1$, one gets $\mathcal{L}\{t\}(s) = 1 / s^2$; repeating gives $\mathcal{L}\{t^n / n!\}(s) = 1 / s^{n + 1}$.

## Explanations

### L1 human

Integrating in time corresponds to dividing by $s$ in the transform.
It is the opposite of differentiation, which corresponds to multiplying by $s$.

### L2 human

Read backwards, the rule inverts transforms of the form $F(s) / s$: the answer is the running integral of $\mathcal{L}^{-1}\{F\}$.
For example $1 / (s (s^2 + 1))$ inverts to $\int_0^t \sin \tau \, d\tau = 1 - \cos t$.
It is a special case of the [[convolution theorem|0JNMJEDD93]], since $g = f * 1$ and $\mathcal{L}\{1\} = 1/s$.
</document>

<document source="https://tamnd.github.io/math2/node/VGZRSDNEBK.md" tag="VGZRSDNEBK" kind="definition">
# Measure with density

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $f \colon X \to [0, \infty]$ be a [[measurable function|YYNCJSR0PJ]].
The *measure with density* $f$ is the map $f\mu \colon \Sigma \to [0, \infty]$ given by
$$(f\mu)(A) = \int_A f \, d\mu = \int_X f \, \mathbf{1}_A \, d\mu,$$
where the integral is the [[integral of a non-negative function|DG76G5Z6XW]].

## Remarks

The map $f\mu$ is a [[measure|PR11BZQQPG]]: $(f\mu)(\emptyset) = 0$, and countable additivity follows from the monotone convergence theorem applied to $f \mathbf{1}_{A_1 \cup \dots \cup A_n}$.
It is absolutely continuous with respect to $\mu$, since $\mu(A) = 0$ implies $\int_A f \, d\mu = 0$.

Example: on $\R$ with Lebesgue measure and $f = \mathbf{1}_{[0,1]}$, the measure $f\mu$ is Lebesgue measure restricted to $[0, 1]$.
Check: $(f\mu)(A) = \int \mathbf{1}_{A \cap [0,1]} \, dx$, the length of $A \cap [0, 1]$.

Non-example: $f(x) = x$ on $\R$ takes negative values, so it is not an admissible density.
The formula $A \mapsto \int_A x \, dx$ is not even defined for $A = \R$.

For measurable $g \ge 0$, $\int g \, d(f\mu) = \int g f \, d\mu$.
Notation: $f\mu$, $f \cdot \mu$, or $d\nu = f \, d\mu$.
</document>

<document source="https://tamnd.github.io/math2/node/VH12PN9V13.md" tag="VH12PN9V13" kind="concept">
# Stone-type duality for varieties

## Summary

A Stone-type duality for a variety or quasivariety of algebras is a dual equivalence of categories.
It matches the algebras with [[Stone spaces|ZV38HA5PAQ]] that carry additional structure.
Under it, homomorphisms of algebras correspond to continuous structure-preserving maps in the opposite direction.
The prototype is [[Stone duality|B9T33D5MJK]] between Boolean algebras and Stone spaces; [[Priestley duality|SA22SEC75K]] and Esakia duality for Heyting algebras are further examples.
Such dualities turn algebraic questions into topological ones, and the theory of [[natural dualities|W5VG6HS8KW]] gives a uniform method to build many of them.

## Explanations

### L1 human

A Boolean algebra can be pictured as the clopen subsets of a compact, totally disconnected space, and this picture loses nothing.
A Stone-type duality is a dictionary of this kind for another class of algebras.
Algebras become spaces, subalgebras become quotient spaces, quotients become subspaces, and products of finitely many algebras become disjoint unions of spaces.
For example, a finite distributive lattice corresponds to the finite ordered set of its join-irreducible elements.
</document>

<document source="https://tamnd.github.io/math2/node/VH4FD4ZHEQ.md" tag="VH4FD4ZHEQ" kind="definition">
# Period map

## Definition

Let $\varpi \colon \mathcal{X} \to B$ be a complex analytic family of compact complex manifolds, as in a [[deformation|VKQSM3BCE4]].
Assume that $B$ is simply connected and that every fibre $X_t$ is a [[Kähler manifold|X60JG9AEQJ]].
Fix $k \ge 0$, a base point $0 \in B$, and put $H = H^k(X_0, \C)$.
By Ehresmann's theorem the family is smoothly locally trivial, and since $B$ is simply connected this gives canonical isomorphisms $H^k(X_t, \C) \cong H$ for all $t$.
By the [[Hodge decomposition|HABCJ4Y0SR]], $H^k(X_t, \C)$ has the Hodge filtration $F^p H^k(X_t) = \bigoplus_{r \ge p} H^{r, k-r}(X_t)$, whose dimensions $f^p$ do not depend on $t$.
The *period map* of the family is
$$\mathcal{P} \colon B \to \mathcal{F}, \qquad t \mapsto \left(F^k H^k(X_t) \subseteq F^{k-1} H^k(X_t) \subseteq \cdots \subseteq F^1 H^k(X_t)\right).$$
Here each $F^p H^k(X_t)$ is viewed as a subspace of $H$, and $\mathcal{F}$ is the [[flag variety|NHRWCQQKTH]] of flags in $H$ with dimensions $f^k \le \cdots \le f^1$.
Its components $\mathcal{P}^p \colon t \mapsto F^p H^k(X_t)$ map to the [[Grassmannian|CYPA882RB9]] of $f^p$-dimensional subspaces of $H$.

## Remarks

For a base that is not simply connected, one defines the period map on the universal cover.
Alternatively, it is a map to the quotient of a period domain by the monodromy group.
When the fibres are projective with a fixed polarization, the period map on primitive cohomology takes values in a period domain of polarized [[Hodge structures|ZSM083M97C]].

Example: the family $\C/(\Z + \tau\Z)$ over the upper half-plane, with $k = 1$.
Identify $H = H^1(X_\tau, \C)$ with $\C^2$ by integrating over the loops $[0, 1]$ and $[0, \tau]$.
Then $F^1 = H^{1,0}$ is spanned by the class of $dz$, with periods $(1, \tau)$, so $\mathcal{P}(\tau) = [1 : \tau] \in \mathbb{P}^1$, which is injective.

Example: a trivial family $X \times B \to B$ has a constant period map.
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# Huntington's axioms

## Definition

Let $A$ be a set with a binary operation $+$ and a unary operation $n$.
The structure $(A, +, n)$ satisfies *Huntington's axioms* if for all $x, y, z \in A$:

1. $x + y = y + x$.
2. $(x + y) + z = x + (y + z)$.
3. $n(n(x) + y) + n(n(x) + n(y)) = x$.

The third axiom is called *Huntington's equation*.

## Remarks

A structure satisfies Huntington's axioms if and only if it comes from a [[Boolean algebra|0XADJKM699]] with $x + y = x \vee y$ and $n(x) = \neg x$.
The other operations are recovered as $x \wedge y = n(n(x) + n(y))$, $1 = x + n(x)$ and $0 = n(1)$, where the last two do not depend on $x$.

Example: $\set{0, 1}$ with $x + y = x \vee y$ and $n(x) = \neg x$.
By De Morgan's laws, the left side of axiom 3 is $(x \wedge \neg y) \vee (x \wedge y) = x \wedge (\neg y \vee y) = x$.

Non-example: $\set{0, 1}$ with $x + y = x \vee y$ and $n(x) = x$.
Axioms 1 and 2 hold, but axiom 3 becomes $(x \vee y) \vee (x \vee y) = x$, which fails for $x = 0$, $y = 1$.
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# Weaire–Phelan structure

## Example

The *Weaire–Phelan structure* is a periodic partition of $\R^3$ into cells of equal volume, of two kinds.

1. Irregular dodecahedra, with $12$ pentagonal faces.
2. Tetrakaidecahedra, with $14$ faces: $2$ hexagons and $12$ pentagons.

A cubic repeating unit contains $8$ cells: $2$ dodecahedra and $6$ tetrakaidecahedra.
The faces are slightly curved, so that the cell walls meet in threes at $120^\circ$ and the edges meet in fours, as [[Plateau's laws|381RB634WS]] require.

The check for the [[Kelvin problem|CVTKSCP55A]] is a computation that Weaire and Phelan made in 1993.
For the same cell volume, the total area of the walls per cell is about $0.3\%$ less than for Kelvin's partition into truncated octahedra.
So this structure shows that Kelvin's proposed answer is not optimal.

## Remarks

Without curving the faces, the cells form a [[honeycomb|HTJ03RFKFH]] of flat-faced [[polyhedra|0325SGSJAR]] related to the crystal structure of some clathrate hydrates.
The structure was the model for the walls of the Beijing National Aquatics Centre built for the 2008 Olympic Games.
It is not known whether the Weaire–Phelan structure solves the Kelvin problem.

## Explanations

### L1 human

Imagine a foam where every bubble has the same volume.
Kelvin thought the best such foam used one shape of bubble with fourteen faces.
Weaire and Phelan found, with computer help, a foam mixing two shapes of bubble that uses slightly less soap film.
It is still the best known answer.
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<document source="https://tamnd.github.io/math2/node/VHAMHH5SPC.md" tag="VHAMHH5SPC" kind="concept">
# List coloring

## Summary

List coloring is a form of [[graph colouring|0NCPXH194Y]] in which each vertex must take its colour from its own list of allowed colours.
A graph is $k$-choosable if it can be coloured from any lists of size $k$, and the least such $k$ is its choice number.
The choice number is at least the [[chromatic number|0VHJRXNAAS]], and it can be much larger.

## Explanations

### L1 human

Imagine scheduling exams, where two exams that share a student need different time slots.
In ordinary colouring every exam may use any slot.
In list coloring each exam has its own list of possible slots, and you must still find a clash-free choice.
For example, the complete bipartite graph $K_{3,3}$ can be coloured with $2$ colours, but there are lists of size $2$ from which it cannot be coloured.
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# Galois' theorem on solvability

## Statement

Let $K$ be a field of [[characteristic|1ZB13N78XW]] $0$, and let $f \in K[x]$ be a nonconstant polynomial.
Then $f$ is [[solvable by radicals|J7YS0DA3HQ]] over $K$ if and only if the [[Galois group of $f$|0JZY3R908H]] over $K$ is a [[solvable group|0P3KZK0FTV]].

## Remarks

For one direction, put a radical extension containing the roots of $f$ inside a Galois radical extension that first adjoins enough roots of unity.
Each later step then adjoins an $n$-th root in the presence of the $n$-th roots of unity, which gives an abelian extension.
So the big Galois group is solvable, and so is its quotient, the Galois group of $f$.
For the other direction, a solvable group has a chain of subgroups with cyclic quotients of prime order.
After adjoining roots of unity, each cyclic step of prime degree $q$ is generated by a $q$-th root, found with a [[Lagrange resolvent|2GDFFH7Y7T]].
Example: $x^5 - 4x + 2$ over $\Q$ has Galois group $S_5$, which is not solvable, so its roots cannot be written with radicals.
The characteristic hypothesis matters: in characteristic $p$, the [[Artin–Schreier|7YNMNRGH3T]] extensions are cyclic of degree $p$ but in general not radical.

## Explanations

### L1 human

The theorem answers an old question: when can the roots of a polynomial be written with $+$, $-$, $\times$, $\div$ and $n$-th roots?
Galois' answer is to look at the group of permutations of the roots that respect all algebraic relations.
The roots have such a formula exactly when this group can be broken down into commutative pieces.

### L2 human

The theorem reduces a question about formulas to a question about finite groups.
Since $S_n$ is solvable for $n \le 4$, every polynomial of degree at most $4$ is solvable by radicals, which matches the [[cubic formula|FB6J5WAT5V]] and the [[quartic formula|4CDFMKKZ01]].
Since [[$S_n$ is not solvable for $n \ge 5$|0J53NE674D]], the [[Abel–Ruffini theorem|163A2WQQJX]] follows once a polynomial with Galois group $S_n$ is found.
The proof uses the [[fundamental theorem of Galois theory|0T1Q8D0FW8]] together with [[Kummer theory|XZQENYV15Y]] for the cyclic steps.
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# Algebraic set

## Summary

An algebraic set is the set of common zeros, in the affine space $k^n$ over a field $k$, of a family of polynomials.
The polynomials are in $n$ variables.
Algebraic sets are the basic geometric objects of classical algebraic geometry.
They are the closed sets of the Zariski topology on $k^n$.
Over an algebraically closed field they correspond to radical ideals of the polynomial ring.

## Explanations

### L1 human

Write down some polynomial equations in $x_1, \dots, x_n$ and look at all points that solve every one of them at once.
That set of solutions is an algebraic set.
In the plane, the equation $x^2 + y^2 - 1 = 0$ gives the unit circle, and the two equations $y = 0$ and $x = 0$ together give just the origin.
Lines, conics, finite sets of points and the whole plane are algebraic sets, but a disc or a half-line is not.
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<document source="https://tamnd.github.io/math2/node/VHHMZPNVEA.md" tag="VHHMZPNVEA" kind="concept">
# f-ring

## Summary

An f-ring is a lattice-ordered ring in which multiplying by a nonnegative element keeps disjoint elements disjoint.
The notion was introduced by Birkhoff and Pierce.
Rings of continuous real functions with the pointwise order, and products of totally ordered rings, are f-rings.
Every f-ring is a subdirect product of totally ordered rings, so f-rings behave in many ways like rings of real functions.

## Explanations

### L1 human

Two nonnegative functions are disjoint when at each point at least one of them is zero.
If you multiply one of them by any nonnegative function, it stays zero where it was zero, so the two stay disjoint.
An f-ring is a ring with a lattice order in which this property holds.
The letter f stands for function, since these rings behave like rings of functions.
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<document source="https://tamnd.github.io/math2/node/VHKMX3DR53.md" tag="VHKMX3DR53" kind="definition">
# Shrinkage estimator

## Definition

Let $X$ be an observation from a statistical model with parameter $\theta \in \R^p$.
Let $\delta_0(X)$ be an [[estimator|0SSS6ERQ0S]] of $\theta$ and let $\nu \in \R^p$ be a fixed target.
Let $w(X)$ be a [[statistic|0Q5CG2NEQ8]] with values in $[0, 1]$.
The estimator
$$\delta(X) = \nu + w(X) \big( \delta_0(X) - \nu \big)$$
is a *shrinkage estimator* of $\theta$, obtained by shrinking $\delta_0$ toward $\nu$.

## Remarks

The weight $w$ may be a constant or may depend on the data.
The term is also used more broadly for any estimator that pulls a basic estimate toward a target, for example by a penalty in a regression.
The plain James–Stein estimator has a weight that can be negative; its positive-part version fits the definition above.

Example: let $X$ be [[normal|05AQE0QR0D]] with mean $\theta$ and known variance $\sigma^2$, and give $\theta$ the normal prior with mean $0$ and variance $\tau^2$.
The [[posterior|0KJ7ZS8AJA]] mean is $\frac{\tau^2}{\tau^2 + \sigma^2} X$.
This is a shrinkage estimator with $\delta_0 = X$, $\nu = 0$ and constant weight $w = \tau^2/(\tau^2 + \sigma^2) \in (0, 1)$.

Non-example: $\delta(X) = 2X$ in the same model.
It has the form above with $\nu = 0$ and $w = 2$, which is not in $[0, 1]$, so it pushes the estimate away from the target rather than toward it.
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# Gegenbauer transform

## Definition

Fix a real number $\lambda > -1/2$ with $\lambda \ne 0$, and let $w(x) = (1 - x^2)^{\lambda - 1/2}$ for $-1 < x < 1$.
Let $C_n^{\lambda}$ be the [[Gegenbauer polynomial|5TRFA6F79B]] of degree $n$.
Let $F \colon (-1, 1) \to \C$ be a function such that $w F$ is integrable on $(-1, 1)$.
The *Gegenbauer transform* of $F$ is the sequence
$$f(n) = \int_{-1}^{1} w(x)\, C_n^{\lambda}(x)\, F(x)\, dx, \qquad n = 0, 1, 2, \dots$$

## Remarks

The polynomials $C_n^\lambda$ are orthogonal for $w$, and $F \in L^2(w)$ is recovered as $F = \sum_{n \ge 0} \frac{f(n)}{h_n} C_n^{\lambda}$ with $h_n = \int_{-1}^1 w\, (C_n^\lambda)^2\, dx$.
If $F$ is smooth enough and the boundary terms vanish, the transform of $w^{-1} \frac{d}{dx}\left( (1 - x^2)^{\lambda + 1/2} F' \right)$ is $-n(n + 2\lambda) f(n)$.

Example: $\lambda = 1/2$ gives $w = 1$ and $C_n^{1/2} = P_n$, the Legendre polynomials.
For $F(x) = x$ the transform is $f(1) = \int_{-1}^1 x^2\, dx = 2/3$ and $f(n) = 0$ for $n \ne 1$, since $x = P_1(x)$.

Non-example: for $\lambda = -1/2$ the weight $(1 - x^2)^{-1}$ is not integrable near $\pm 1$, so $F = 1$ has no transform.

The value $\lambda = 0$ is excluded because $C_n^0 = 0$ for $n \ge 1$ in the usual normalization; the Chebyshev polynomials of the first kind take its place.
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# Fourier transform on locally compact abelian groups

## Summary

The Fourier transform on a locally compact abelian group $G$ sends an integrable function $f$ on $G$ to a function $\widehat{f}$ on the dual group.
The value of $\widehat{f}$ at a character is the integral of $f$ against the conjugate of that character.
It unifies Fourier series on the circle, the Fourier transform on $\R^n$ and the discrete Fourier transform on $\Z/n\Z$.
It turns convolution into pointwise multiplication, and with the right normalisation it is unitary on $L^2$.

## Explanations

### L1 human

The Fourier transform measures how much of each wave is present in a function.
On a group, the waves are the characters, and the amount of the wave $\chi$ in $f$ is the average of $f$ times the conjugate of $\chi$.
On the circle this gives the Fourier coefficients $\frac{1}{2\pi}\int_0^{2\pi} f(\theta) e^{-in\theta} \, d\theta$.
On the group $\Z/n\Z$ it gives the discrete Fourier transform used in signal processing.
One definition covers all these cases.
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# Artinian module

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $M$ be a left [[module|09VDS2W1E8]] over $R$.
The module $M$ is *Artinian* if the set of its [[submodules|0RGZ10DFPG]], ordered by inclusion, satisfies the descending chain condition.
That is, for every chain $N_1 \supseteq N_2 \supseteq \cdots$ of submodules of $M$, there is an index $k$ with $N_n = N_k$ for all $n \ge k$.

## Remarks

Equivalently, every nonempty set of submodules of $M$ has a minimal element for inclusion.
Example: a finite-dimensional vector space $V$ over a field is Artinian, because a strictly decreasing chain of subspaces has at most $\dim V + 1$ terms.
Non-example: $\Z$ as a $\Z$-module is not Artinian, because $2\Z \supsetneq 4\Z \supsetneq 8\Z \supsetneq \cdots$ never stops.
Example: for a prime $p$, the Prüfer group $\Z[1/p]/\Z$ is an Artinian $\Z$-module, since its proper subgroups form the chain $0 \subset \frac{1}{p}\Z/\Z \subset \frac{1}{p^2}\Z/\Z \subset \cdots$.
That chain also shows that it is not a [[Noetherian module|AK1ZRCXR32]].
Submodules, quotients and extensions of Artinian modules are Artinian.

## Explanations

### L1 human

An Artinian module is a module with no endless decreasing chain of submodules.
It is the mirror image of a Noetherian module.
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# Riesz basis

## Definition

Let $H$ be a separable [[Hilbert space|HM2HHE3J51]].
A sequence $(x_n)_{n \ge 1}$ in $H$ is a *Riesz sequence* if there are constants $0 < A \le B < \infty$ such that for every finite sequence of scalars $c_1, \dots, c_N$,
$$A \sum_{n=1}^{N} \abs{c_n}^2 \le \Big\| \sum_{n=1}^{N} c_n x_n \Big\|^2 \le B \sum_{n=1}^{N} \abs{c_n}^2.$$
A *Riesz basis* of $H$ is a Riesz sequence whose closed linear span is $H$.

## Remarks

Equivalently, $(x_n)$ is a Riesz basis if and only if $x_n = T e_n$ for an [[orthonormal basis|0S9GGTR0EX]] $(e_n)$ of $H$ and a bounded linear operator $T \colon H \to H$ with a bounded inverse.
Then every $x \in H$ has a unique expansion $x = \sum_n c_n x_n$, the series converges unconditionally, and $A \sum \abs{c_n}^2 \le \norm{x}^2 \le B \sum \abs{c_n}^2$.
The best constants are $A = \norm{T^{-1}}^{-2}$ and $B = \norm{T}^2$.

Example: let $(e_n)$ be an orthonormal basis of $H$ and $x_n = e_n + \frac12 e_{n+1}$.
Then $x_n = T e_n$ with $T = I + \frac12 S$, where $S$ is the shift $e_n \mapsto e_{n+1}$, of norm $1$.
Since $\norm{\frac12 S} < 1$, the Neumann series shows that $T$ is invertible with bounded inverse, so $(x_n)$ is a Riesz basis.

Non-example: $x_n = e_n / n$.
Its closed span is $H$ and the upper bound holds with $B = 1$.
The lower bound fails: for $c = (0, \dots, 0, 1)$ with the $1$ in place $n$, the middle term is $1/n^2$, which is not bounded below by a positive $A$.
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# Completely monotone function

## Summary

A completely monotone function is an infinitely differentiable function on $(0, \infty)$ whose derivatives alternate in sign: $f \ge 0$, $f' \le 0$, $f'' \ge 0$, and so on.
It is the mirror image of an [[absolutely monotonic function|QQSPWH4NG9]] under $x \mapsto -x$.
By [[Bernstein's theorem|ZBS1JA62QP]], the completely monotone functions are exactly the [[Laplace transforms|0V49JEZW06]] of positive measures on $[0, \infty)$.
This makes them central in probability and potential theory.

## Explanations

### L1 human

A completely monotone function is positive, decreasing, convex, and every further derivative keeps this alternating pattern.
The function $e^{-x}$ is the basic example: its derivatives are $\pm e^{-x}$ with alternating signs.
Any sum of decaying exponentials $c_1 e^{-t_1 x} + c_2 e^{-t_2 x}$ with $c_i \ge 0$, $t_i \ge 0$ is again completely monotone.
So is $1/x$, which is a continuous mixture of such exponentials.
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<document source="https://tamnd.github.io/math2/node/VJ3BVREDWN.md" tag="VJ3BVREDWN" kind="concept">
# Lindeberg's condition

## Summary

Lindeberg's condition is a condition on a sequence of independent random variables with finite variances.
It says that no single variable contributes a non-negligible part of the total variance through its large values.
Under this condition, the normalised sums satisfy the [[central limit theorem|0EPZC4Z1FC]], even when the variables do not have the same distribution.
It is close to necessary: by a theorem of Feller, it is necessary when each variance is a negligible part of the total.

## Explanations

### L1 human

The central limit theorem says that a sum of many small independent effects looks normal.
For this, no single effect may dominate the sum.
Lindeberg's condition makes "no effect dominates" precise: the extreme values of each variable, taken together, carry almost none of the total variance.
For example, a sum of independent coin flips with different biases satisfies it as long as the total variance grows without bound.
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# Risch algorithm

## Summary

The Risch algorithm is a procedure of symbolic integration.
Given an elementary function, it decides whether the function has an elementary [[antiderivative|0BYWRWFT4F]], and it finds one when it exists.
It rests on [[Liouville's theorem|8VRTPECBPZ]] of differential algebra, which fixes the shape that such an antiderivative must have.
It is named after Robert Henry Risch, and parts of it are implemented in computer algebra systems.

## Explanations

### L1 human

Integration in calculus courses looks like a collection of tricks: substitution, integration by parts, partial fractions.
The Risch algorithm replaces the tricks by a procedure that always ends with an answer.
The answer is either an explicit antiderivative or a proof that none can be written with the usual functions.
For example, it returns $(x - 1) e^x$ for $x e^x$, and it reports that $e^{x^2}$ has no elementary antiderivative.
It views a function as an element of a tower of fields, built from the rational functions by adding logarithms, exponentials and algebraic functions one at a time.
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# Askey–Gasper inequality

## Statement

Let $P_k^{(\alpha, \beta)}$ denote the [[Jacobi polynomials|TRD7DSF2KR]].
Let $\alpha, \beta$ be real numbers with $\beta \ge 0$ and $\alpha + \beta \ge -2$, and let $n \ge 0$ be an integer.
Then for every $x \in [-1, 1]$,
$$\sum_{k=0}^{n} \frac{P_k^{(\alpha, \beta)}(x)}{P_k^{(\beta, \alpha)}(1)} \ge 0.$$

## Remarks

The denominators are positive: $P_k^{(\beta, \alpha)}(1) = \frac{(\beta + 1)_k}{k!} > 0$ for $\beta \ge 0$.
For $\beta = 0$ all denominators equal $1$, and the inequality says $\sum_{k=0}^{n} P_k^{(\alpha, 0)}(x) \ge 0$ on $[-1, 1]$.
Check for $n = 1$ and $\alpha = \beta = 0$: the sum is $1 + x$, which is nonnegative on $[-1, 1]$ and vanishes at $x = -1$.
Askey and Gasper proved it in 1976.
A short proof by Ekhad (1993) treats the case $\beta = 0$ in its ${}_3F_2$ form.
An identity found with the Wilf–Zeilberger method writes this ${}_3F_2$ as a positive combination of other ${}_3F_2$ series.
These are nonnegative by Clausen's formula, which expresses them as squares of ${}_2F_1$ series.

## Explanations

### L1 human

Jacobi polynomials take both positive and negative values on $[-1, 1]$.
The inequality says that a certain weighted running sum of them never goes below zero.
It is surprising because the single terms change sign many times.
Its fame comes from its use by Louis de Branges in his 1985 proof of the Bieberbach conjecture.

### L2 human

Sums of the form $\sum_k c_k P_k^{(\alpha, \beta)}(x)$ are partial sums of Jacobi expansions, and their positivity is a question of positive summability kernels.
The case $\beta = 0$ can be rewritten as the positivity of a terminating ${}_3F_2$ hypergeometric series on $0 \le t < 1$.
De Branges needed this ${}_3F_2$ form for integer values of $\alpha$ to finish his proof.
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# Zeta function regularization

## Summary

Zeta function regularization assigns finite values to divergent sums $\sum \lambda_n$ and divergent products $\prod \lambda_n$ of positive numbers.
One forms the zeta function $\zeta_\lambda(s) = \sum \lambda_n^{-s}$, which converges for large $\operatorname{Re} s$, and continues it by [[analytic continuation|0M3YQJ81NB]].
The regularized sum is $\zeta_\lambda(-1)$, and the regularized product is $e^{-\zeta_\lambda'(0)}$.
For $\lambda_n = n$ the [[Riemann zeta function|1XWBY4N0VT]] gives the regularized value $-\frac{1}{12}$ of $1 + 2 + 3 + \cdots$.
The method is used for determinants of Laplacians and in quantum field theory.

## Explanations

### L1 human

The sum $1 + 2 + 3 + \cdots$ diverges, but the sum $1^{-s} + 2^{-s} + 3^{-s} + \cdots$ converges when $s > 1$ and defines a function of $s$.
That function has a unique analytic continuation to all complex $s \ne 1$.
Plugging $s = -1$ into the extension formally gives back $1 + 2 + 3 + \cdots$, and the value obtained is $-\frac{1}{12}$.
Zeta regularization does the same for any sequence of positive numbers that grows fast enough.
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# Opposite ring

## Definition

Let $(R, +, \cdot)$ be a [[ring|0894J980RK]].
The *opposite ring* $R^{\mathrm{op}}$ is the set $R$ with the same addition and the product $*$ given by
$$a * b = b \cdot a$$
for all $a, b \in R$.

## Remarks

$R^{\mathrm{op}}$ is a ring with the same zero and the same unit element as $R$.
Associativity holds because $(a * b) * c = c \cdot (b \cdot a) = (c \cdot b) \cdot a = a * (b * c)$.
Example: for a commutative ring $K$, the transpose map $A \mapsto A^T$ is a ring isomorphism $M_n(K) \to M_n(K)^{\mathrm{op}}$, since $(AB)^T = B^T A^T = A^T * B^T$.
Non-example: for a nonzero commutative ring $K$, the identity map of $M_2(K)$ is not a ring isomorphism $M_2(K) \to M_2(K)^{\mathrm{op}}$, because it would need $AB = BA$ for all $A, B$.
A right $R$-module is the same thing as a left $R^{\mathrm{op}}$-module, with $r \cdot m = m r$.
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# Littlewood's L1 norm conjecture

## Statement

There is an absolute constant $c > 0$ with the following property.
Let $N \ge 1$ and let $n_1 < n_2 < \cdots < n_N$ be distinct integers.
Then the [[trigonometric polynomial|J3NJQ5FZJ5]] $f(x) = \sum_{k=1}^{N} e^{i n_k x}$ satisfies
$$\frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{f(x)} \, dx \ge c \log N.$$

## Remarks

Littlewood conjectured this bound, and it was proved in 1981, independently by McGehee, Pigno and Smith and by Konyagin.
McGehee, Pigno and Smith proved more: for all complex $a_1, \ldots, a_N$,
$$\frac{1}{2\pi} \int_{-\pi}^{\pi} \Big\lvert \sum_{k=1}^{N} a_k e^{i n_k x} \Big\rvert \, dx \ge c \sum_{k=1}^{N} \frac{\abs{a_k}}{k}.$$
Taking all $a_k = 1$ gives the conjecture, because $\sum_{k=1}^{N} 1/k \ge \log N$.
Their proof constructs, by duality, a bounded function whose pairing with $f$ is large.

The order $\log N$ is sharp.
For $n_k = k$ one has $\abs{f(x)} = \abs{\sin(Nx/2) / \sin(x/2)}$, the absolute value of a Dirichlet-type kernel, and its $L^1$ norm grows like $\frac{4}{\pi^2} \log N$, as for the [[Lebesgue constants|RWKQ5EP4YF]].

## Explanations

### L1 human

Add $N$ pure waves of size one with different whole-number frequencies.
At $x = 0$ they all line up and the sum has size $N$, but at most points they partly cancel.
The theorem says that the cancellation can never be too strong on average: the average size of the sum is at least a constant times $\log N$.

### L2 human

In the $L^2$ norm the answer is exact: the norm is $\sqrt N$ by Parseval's identity, whatever the frequencies.
The $L^1$ norm is much harder because no orthogonality is available.
The McGehee–Pigno–Smith inequality is a version of Hardy's inequality for functions whose Fourier coefficients are supported on an arbitrary set of integers.
Before the full solution, P. J. Cohen proved a lower bound tending to infinity with $N$, as part of his work on idempotent measures.
</document>

<document source="https://tamnd.github.io/math2/node/VJNZJ352PC.md" tag="VJNZJ352PC" kind="concept">
# Radon transform

## Summary

The Radon transform sends a function $f$ on $\R^n$ to the function that gives the integral of $f$ over each hyperplane.
In the plane the hyperplanes are lines.
There the Radon transform records what an X-ray scanner measures: the total absorption along each line through a body.
The [[projection-slice theorem|J7VA040MC3]] links it to the [[Fourier transform|DPVM34ADZV]], and the [[Radon inversion formula|HN93ST5CBT]] recovers $f$ from its Radon transform.
It is the mathematical basis of computed tomography and a central object of integral geometry.

## Explanations

### L1 human

Shine parallel X-rays through a body and measure how much each ray is weakened.
Each measurement is the integral of the density of the body along one line.
Doing this for every direction and every offset gives the Radon transform of the density.
For example, the indicator of the unit disk has the integral $2\sqrt{1 - s^2}$ over each line at distance $s < 1$ from the center, the length of the chord.
The surprising fact is that these line integrals determine the density completely.
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<document source="https://tamnd.github.io/math2/node/VJVDDR8BCE.md" tag="VJVDDR8BCE" kind="concept">
# Köthe–Toeplitz dual

## Summary

The Köthe–Toeplitz duals of a [[sequence space|XYTN19EPF2]] $X$ are the sequence spaces of multipliers $a$ for which $\sum a_n x_n$ behaves well for every $x$ in $X$.
The $\alpha$-dual asks for absolute convergence of $\sum a_n x_n$, and the $\beta$-dual asks for convergence.
They play the role of a dual space that is again a space of sequences, without any topology.
For example, the $\beta$-dual of $\ell^1$ is $\ell^\infty$, which matches the dual space of $\ell^1$ in functional analysis.

## Explanations

### L1 human

Given a family of sequences $X$, ask which sequences $a$ can be multiplied term by term with every $x$ in $X$ so that the series $\sum a_n x_n$ still converges.
These $a$ form a new space of sequences, the dual of $X$.
If $X$ consists of the sequences with $\sum \abs{x_n} < \infty$, then any bounded $a$ works, and no unbounded $a$ does.
The bigger $X$ is, the smaller its dual.
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<document source="https://tamnd.github.io/math2/node/VJXS8AK67E.md" tag="VJXS8AK67E" kind="concept">
# Hom functor

## Summary

In a locally small [[category|A2DE66937D]] $\mathcal{C}$, each object $A$ gives a [[functor|28NQ0KRHD3]] $\Hom(A, -)$ from $\mathcal{C}$ to the category of sets.
It sends an object $X$ to the set of morphisms $A \to X$.
Dually, $\Hom(-, A)$ is a contravariant functor.
Hom functors are the basic examples of functors into sets, and the Yoneda lemma says that they control all other such functors.

## Explanations

### L1 human

An object $A$ can be studied by the arrows that leave it: for each object $X$, collect all arrows from $A$ to $X$ into a set.
An arrow $X \to Y$ then turns each arrow $A \to X$ into an arrow $A \to Y$, by following one and then the other.
This turns $\Hom(A, -)$ into a functor.
For example, in the category of sets with $A = \{*\}$ a one-point set, a map $\{*\} \to X$ is the same as a point of $X$, so $\Hom(\{*\}, -)$ is essentially the identity functor.
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<document source="https://tamnd.github.io/math2/node/VK1XXMR2WW.md" tag="VK1XXMR2WW" kind="definition">
# Graph embedding

## Definition

Let $G = (V, E)$ be a finite [[graph|0JD5C82QMB]] and let $\Sigma$ be a [[surface|VGJJVBZ1MQ]].
An embedding of $G$ in $\Sigma$ consists of a point $p_v \in \Sigma$ for each vertex $v$ and a subset $\gamma_e \subseteq \Sigma$ for each edge $e$ such that:

1. The points $p_v$ are distinct.
2. For each edge $e = uv$, the set $\gamma_e$ is the image of a [[topological embedding|08SVRVHJNY]] of $[0, 1]$ into $\Sigma$ with ends $p_u$ and $p_v$.
3. An arc $\gamma_e$ contains no point $p_w$ other than its two ends.
4. Two distinct arcs meet at most in common ends.

## Remarks

Example: drawing $K_4$ in the plane as a triangle with a central vertex joined to its three corners satisfies all four clauses.
Non-example: drawing $K_4$ as a square with both diagonals is not an embedding, because the diagonals cross at an inner point, which violates clause 4.
A graph is planar exactly when it has an embedding in the plane, or equivalently in the sphere.
The genus of $G$ is the least $g$ such that $G$ embeds in the [[orientable surface|0EKZW08FQ5]] of genus $g$; for example $K_5$ and $K_{3,3}$ have genus $1$.
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<document source="https://tamnd.github.io/math2/node/VK39X07EYK.md" tag="VK39X07EYK" kind="conjecture">
# Bing–Borsuk conjecture

## Statement

Let $n \ge 1$ and let $X$ be an [[absolute neighborhood retract|DK3MT3HYX5]] of [[covering dimension|A3W7RGXXK9]] $n$.
Suppose that $X$ is homogeneous: for all points $x, y \in X$ there is a [[homeomorphism|0QT00PB5G2]] $h \colon X \to X$ with $h(x) = y$.
Then $X$ is a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.

## Remarks

Every connected topological manifold is a homogeneous absolute neighborhood retract, so the conjecture asks for a converse.
The conjecture is known to hold for $n = 1$ and $n = 2$.
In dimension $3$, it is known to imply the Poincaré conjecture.
In general it is open.
The finite dimension is needed: the Hilbert cube is a homogeneous compact absolute retract of infinite dimension that is not a manifold of any finite dimension.

## Explanations

### L1 human

A manifold looks the same at every point, in the sense that any point can be moved to any other by a self-homeomorphism.
It is also a well-behaved space in the sense of homotopy theory.
The conjecture says that these two properties, in finite dimension, are enough to force a space to be a manifold.
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<document source="https://tamnd.github.io/math2/node/VK3ZMD1BCP.md" tag="VK3ZMD1BCP" kind="concept">
# Perfectly normal space

## Summary

A perfectly normal space is a [[normal space|0NSR9V8TED]] in which every closed set is a countable intersection of open sets.
Equivalently, every closed set is the zero set of a continuous real function.
Every perfectly normal space is [[completely normal|9HBDQ2W9CF]].
Every metric space is perfectly normal.

## Explanations

### L1 human

In a perfectly normal space, each closed set can be described exactly by one continuous function: the set where the function is zero.
In a metric space, the distance to the closed set is such a function.
For example, the closed set $[0, 1]$ in $\R$ is the zero set of the distance $x \mapsto d(x, [0, 1])$.
It is also the intersection of the open sets $(-1/n, 1 + 1/n)$.
</document>

<document source="https://tamnd.github.io/math2/node/VK50GVVFTN.md" tag="VK50GVVFTN" kind="object">
# Thompson groups

## Description

Thompson's group $F$ is the [[group|G5S1M9]], under composition, of all [[homeomorphisms|0QT00PB5G2]] $f \colon [0, 1] \to [0, 1]$ such that:

1. $f$ is piecewise linear, with finitely many break points;
2. every break point is a dyadic rational, a number $k / 2^j$ with integers $k$ and $j \ge 0$;
3. every slope of $f$ is an integer power of $2$.

Thompson's group $T$ is the group of such piecewise linear maps of the circle $[0, 1] / \set{0, 1}$, which also map the dyadic rationals onto themselves.
Thompson's group $V$ is the group of right-continuous bijections of $[0, 1)$ that are piecewise linear in the same sense.
They may have finitely many jumps, all at dyadic rationals.
The inclusions $F \subset T \subset V$ hold.

For example, the map $A$ with $A(x) = x / 2$ on $[0, 1/2]$, $A(x) = x - 1/4$ on $[1/2, 3/4]$ and $A(x) = 2x - 1$ on $[3/4, 1]$ lies in $F$.
The pieces agree at the break points: $1/2 \mapsto 1/4$ and $3/4 \mapsto 1/2$, and the slopes are $1/2$, $1$ and $2$.

## Remarks

All three groups are [[finitely presented|TJKANTKASZ]].
$F$ has the infinite presentation $\langle x_0, x_1, x_2, \ldots \mid x_i^{-1} x_n x_i = x_{n+1} \text{ for } i < n \rangle$, and it is generated by $x_0$ and $x_1$ with two relations.
The commutator subgroup $[F, F]$ is simple, and $F / [F, F] \cong \Z^2$.
$F$ is torsion-free and contains no [[free subgroup|0KTW26C80J]] of rank $2$.
Whether $F$ is [[amenable|CEAT9BESKK]] is a well-known open problem.
$T$ and $V$ are infinite [[simple groups|0YMYMNQ0QZ]] that are finitely presented; they were the first known examples of such groups.

## Explanations

### L1 human

Cut the interval $[0, 1]$ into pieces by repeatedly halving, and do the same with another copy of $[0, 1]$, with the same number of pieces.
Stretch each piece of the first copy linearly onto the matching piece of the second.
The maps obtained this way form Thompson's group $F$.
The groups $T$ and $V$ allow the pieces to be rotated cyclically or shuffled in any order.
These simple-looking groups answer many questions in group theory and serve as counterexamples to many guesses.
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# Torsion submodule

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].
An element $m \in M$ is a *torsion element* if $r m = 0$ for some $r \in R$ that is not a [[zero divisor|0NNRDW10RZ]] of $R$.
The *torsion submodule* of $M$ is the set
$$T(M) = \set{m \in M : r m = 0 \text{ for some non-zero-divisor } r \in R}.$$
The module $M$ is *torsion-free* if $T(M) = 0$, and it is a *torsion module* if $T(M) = M$.

## Remarks

The set $T(M)$ is a [[submodule|05WDSN597C]]: if $r m = 0$ and $s n = 0$ with $r, s$ non-zero-divisors, then $rs$ is a non-zero-divisor and $rs(m + n) = 0$, and $r(am) = a(rm) = 0$ for $a \in R$.
Example: for $R = \Z$ and $M = \Z \oplus \Z/6\Z$, one has $T(M) = 0 \oplus \Z/6\Z$, since $6 (k, c) = (6k, 0)$ is zero exactly when $k = 0$.
Non-example: the element $(1, 0)$ of this $M$ is not torsion, because $n (1, 0) = (n, 0) \ne 0$ for every nonzero integer $n$.
When $R$ is an [[integral domain|0QVD7WBFGN]], the non-zero-divisors are the nonzero elements, and $M / T(M)$ is torsion-free.
For $R = \Z$ the torsion submodule is the [[torsion subgroup|0DBWXQRXQJ]].
</document>

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# Reductive group

## Definition

Let $k$ be an algebraically closed field, and let $G$ be a smooth connected [[linear algebraic group|GA81VVQHEP]] over $k$.
An element $g \in \mathrm{GL}_n(k)$ is *unipotent* if $g - I$ is a [[nilpotent matrix|07BSPZPSMF]].
A linear algebraic group is *unipotent* if all its points over $k$ are unipotent in some, equivalently every, faithful representation.
The *unipotent radical* $R_u(G)$ is the largest smooth connected unipotent [[normal subgroup|0CB0DX5BVV]] of $G$.
The group $G$ is *reductive* if $R_u(G)$ is trivial.

## Remarks

Over an arbitrary field $k$, a smooth connected linear algebraic group $G$ is called reductive if $G \times_k \bar{k}$ is reductive.

Example: $\mathrm{GL}_n$ is reductive.
Let $U$ be a connected normal unipotent subgroup.
By Kolchin's theorem, the vectors fixed by $U$ form a nonzero subspace of $k^n$.
Since $U$ is normal, $\mathrm{GL}_n$ preserves this subspace, so it is all of $k^n$ and $U$ is trivial.

Non-example: the group $B$ of invertible upper triangular $2 \times 2$ matrices is not reductive.
The subgroup of matrices $\begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix}$ is normal in $B$, connected and unipotent.

In characteristic $0$, a smooth connected linear algebraic group is reductive if and only if all its finite-dimensional representations are semisimple.
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<document source="https://tamnd.github.io/math2/node/VKFPCSRNDH.md" tag="VKFPCSRNDH" kind="concept">
# Absolute geometry

## Summary

Absolute geometry, also called neutral geometry, is the geometry that [[Hilbert's axioms|KM1FB8XZ2R]] give without the axiom of parallels.
Its theorems hold both in [[Euclidean geometry|KRZD96GSVC]] and in [[hyperbolic geometry|GN4N94GQ0Z]].
It contains the congruence criteria for triangles, the [[exterior angle theorem|0NTSE7EREA]] and the [[Saccheri–Legendre theorem|1Y2XK0QJDX]], but not the angle sum $180^\circ$.

## Explanations

### L1 human

Absolute geometry is what remains of school geometry when one forgets the rule about parallel lines.
Everything proved there is true both on a flat plane and in the curved world of hyperbolic geometry.
For example, in any triangle an exterior angle is larger than each of the two opposite interior angles.
On the other hand, "the angles of a triangle add up to $180^\circ$" is not a theorem of absolute geometry.
</document>

<document source="https://tamnd.github.io/math2/node/VKFW27WXEZ.md" tag="VKFW27WXEZ" kind="concept">
# Submanifold

## Summary

A submanifold of a [[smooth manifold|F45VQ47GPV]] $M$ is a subset that is itself a smooth manifold in a way compatible with $M$.
The most common kind, an embedded submanifold, looks near each of its points like a flat coordinate subspace $\R^k \subseteq \R^n$.
Level sets of submersions, images of embeddings and graphs of smooth maps are submanifolds.
A wider notion, the immersed submanifold, is the image of an injective [[immersion|6RW68JGBHC]] with its own topology.

## Explanations

### L1 human

A submanifold is a smooth curve, surface, or higher-dimensional piece sitting inside a bigger smooth space.
Seen through a good magnifying glass, it looks like a flat line or plane inside flat space.
The unit circle in the plane and the unit sphere in $\R^3$ are submanifolds.
The union of the two coordinate axes in $\R^2$ is not, because at the origin it looks like a cross, not like a line.
</document>

<document source="https://tamnd.github.io/math2/node/VKJ8GQ07H7.md" tag="VKJ8GQ07H7" kind="definition">
# Volterra integral equation

## Definition

Let $a < b$, let $\Delta = \set{(t, s) : a \le s \le t \le b}$, let $K \colon \Delta \to \R$ be continuous, and let $f \colon [a, b] \to \R$ be continuous.
A *linear Volterra integral equation of the first kind* is
$$f(t) = \int_a^t K(t, s) \, x(s) \, ds, \quad t \in [a, b].$$
A *linear Volterra integral equation of the second kind* is
$$x(t) = f(t) + \lambda \int_a^t K(t, s) \, x(s) \, ds, \quad t \in [a, b],$$
with a constant $\lambda$.
In both, $x \colon [a, b] \to \R$ is the unknown continuous function and $K$ is the *kernel*.
The upper limit of the [[integral|0W29TBTY12]] is the variable $t$.

## Remarks

Example: $x(t) = 1 + \int_0^t x(s) \, ds$ on $[0, b]$, second kind with $K = 1$ and $\lambda = 1$.
The function $x = e^t$ solves it, since $1 + \int_0^t e^s \, ds = e^t$.

Non-example: $x(t) = f(t) + \int_a^b K(t, s) x(s) \, ds$ has a fixed upper limit $b$; it is a Fredholm integral equation, not a Volterra one.

For the second kind, the solution exists and is unique for every $\lambda$.
It is the sum of the Neumann series $x = \sum_{j \ge 0} \lambda^j V^j f$, where $(V x)(t) = \int_a^t K(t, s) x(s) \, ds$; the bound $\abs{(V^j f)(t)} \le \norm{K}_\infty^j (t - a)^j \norm{f}_\infty / j!$ makes it converge.
The same scheme gives existence for nonlinear equations $x(t) = f(t) + \int_a^t K(t, s, x(s)) \, ds$ with $K$ Lipschitz in $x$, which include the integral form of initial value problems.

## Explanations

### L1 human

The unknown appears under an integral whose upper limit is the current time.
So the equation can be solved step by step forward in time.
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# Deformation of complex structure

## Summary

A deformation of the complex structure of a compact [[complex manifold|JSGWAN2RJV]] $X$ is a holomorphic family of compact complex manifolds.
The family is parametrized by a base, and its fibre over a chosen base point is $X$.
By Ehresmann's theorem all fibres are diffeomorphic, so the family varies only the complex structure on a fixed smooth manifold.
Kodaira and Spencer showed that first-order deformations are measured by $H^1(X, \Theta_X)$, where $\Theta_X$ is the sheaf of holomorphic vector fields, and obstructions lie in $H^2(X, \Theta_X)$.
The theory is the analytic part of [[deformation theory|SWFR1GD6QT]] and leads to local [[moduli spaces|FJZRW9P0K3]].

## Explanations

### L1 human

A complex manifold is a smooth manifold together with a complex structure, and one can try to change that structure a little.
A deformation is a family of complex manifolds $X_t$ depending holomorphically on a parameter $t$, with $X_0 = X$.
Elliptic curves give the basic picture: the torus $\C / (\Z + \tau \Z)$ changes its complex structure as $\tau$ moves in the upper half-plane, while staying the same smooth torus.
Some manifolds, such as projective space, cannot be deformed at all; they are called rigid.
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# Azumaya algebra

## Summary

An Azumaya algebra over a [[commutative ring|0310REJJY8]] $R$ is an [[associative algebra|3CTW1P4HBP]] that behaves like a [[central simple algebra|5HRKJBAY2V]] over every point of $R$.
It is a faithful, finitely generated projective $R$-module $A$ for which $A \otimes_R A^{\mathrm{op}} \to \End_R(A)$ is an isomorphism.
Over a field the Azumaya algebras are exactly the central simple algebras.
Azumaya algebras define the [[Brauer group|BEW48TA9WZ]] of a commutative ring and, in geometry, of a scheme.

## Explanations

### L1 human

A central simple algebra over a field looks like a matrix algebra after a change of field.
An Azumaya algebra is the same idea with the field replaced by a commutative ring.
Its fibre at each maximal ideal $\mathfrak{m}$ of $R$ is a central simple algebra over the field $R/\mathfrak{m}$, and these fibres vary nicely.
The simplest example is the ring $M_n(R)$ of $n \times n$ matrices over $R$.
</document>

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# Ridders' method

## Statement

Let $f \colon [x_0, x_2] \to \R$ be continuous with $f(x_0) f(x_2) < 0$, and let $x_1 = (x_0 + x_2)/2$.
*Ridders' method* computes
$$x_3 = x_1 + (x_1 - x_0) \, \frac{\sgn\big(f(x_0) - f(x_2)\big) \, f(x_1)}{\sqrt{f(x_1)^2 - f(x_0) f(x_2)}}.$$
Then the following hold.

1. The square root is real and positive, and $x_3$ lies in the open interval $(x_0, x_2)$.
2. If $f(x_1) = 0$, then $x_3 = x_1$.
3. If $f(x_3) \ne 0$, then $f$ changes sign between $x_3$ and some point of $\set{x_0, x_1, x_2}$. The method keeps the shortest such interval as the next bracket.

## Remarks

Claim 1: since $f(x_0) f(x_2) < 0$, the radicand is larger than $f(x_1)^2$, so the fraction has absolute value less than $1$ and $\abs{x_3 - x_1} < x_1 - x_0$.
Claim 3: $f(x_3)$ has the sign opposite to $f(x_0)$ or to $f(x_2)$, unless it is $0$.
So the bracket shrinks at each step and a zero stays inside it, by the [[intermediate value theorem|00KAFBR32E]].
The formula comes from a trick: choose $Q$ so that the three points of $h(x) = f(x) e^{Qx}$ at $x_0, x_1, x_2$ lie on a line, then apply [[regula falsi|462AEY7MWF]] to $h$.
For $f(x) = x - c$ on $[0, 2]$ with $0 < c < 2$, the radicand is $1$ and the formula gives $x_3 = c$ at once.
For a smooth $f$ with a simple zero, the [[convergence|YWZQC5E54B]] is quadratic per step, and each step costs two evaluations of $f$.

## Explanations

### L1 human

Ridders' method starts like bisection by evaluating $f$ at the midpoint.
Then it bends the function by an exponential factor so that the three values fall on a straight line, and it takes the zero of that line.
The new point always stays inside the current interval, so the root is never lost.

### L2 human

The method is as safe as bisection but usually much faster.
It needs no derivatives, and it is a simple alternative to Brent's method when a bracket is known.
Its cost is two function values per step, so its order per function evaluation is $\sqrt{2}$.
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# Condenser capacity

## Definition

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be an open set, and let $K \subseteq \Omega$ be a [[compact set|0R6EWFKRCF]].
The pair $(K, \Omega)$ is called a *condenser*.
Let $C_c^\infty(\Omega)$ be the set of smooth real functions with compact support in $\Omega$.
The *capacity* of the condenser $(K, \Omega)$ is
$$\operatorname{cap}(K, \Omega) = \inf \set{ \int_\Omega \abs{\nabla u}^2 \, dx : u \in C_c^\infty(\Omega), \ u \ge 1 \text{ on } K }.$$

## Remarks

When $\Omega$ is bounded and its boundary and that of $K$ are regular, the infimum is attained in a larger class by the potential $u$ of the condenser.
This is the [[harmonic function|0DEB0JMN0W]] on $\Omega \setminus K$ with $u = 1$ on $\partial K$ and $u = 0$ on $\partial \Omega$.

Example: in $\R^3$ let $0 < r < R$, $K = \bar B_r$ and $\Omega = B_R$.
The potential is $u(x) = (1/\abs{x} - 1/R) / (1/r - 1/R)$ for $r \le \abs{x} \le R$, and $\abs{\nabla u} = \abs{x}^{-2} / (1/r - 1/R)$.
Integration over spheres gives $4 \pi \int_r^R s^{-2} \, ds / (1/r - 1/R)^2 = 4 \pi / (1/r - 1/R)$.
So $\operatorname{cap}(\bar B_r, B_R) = 4 \pi r R / (R - r)$.
In the plane the same computation with $u(x) = \log(R/\abs{x}) / \log(R/r)$ gives $\operatorname{cap}(\bar B_r, B_R) = 2 \pi / \log(R/r)$.

Non-example: if $K$ is compact but not contained in $\Omega$, the pair is not a condenser.
No $u \in C_c^\infty(\Omega)$ can be at least $1$ at a point of $K$ outside $\Omega$.

Some authors normalize by a constant, such as $1/(4\pi)$ in $\R^3$ or $1/(2\pi)$ in the plane, or define the capacity with $\abs{\nabla u}^n$, which makes it invariant under conformal maps.
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# Closed operator

## Summary

A closed operator is an unbounded operator whose graph is a closed subset of the product space.
Closedness is the weak substitute for continuity that most differential operators have.
On Banach spaces, the closed graph theorem says that a closed operator defined on the whole space is bounded.
So interesting closed operators are usually defined on a proper subspace.

## Explanations

### L1 human

A continuous map sends convergent sequences to convergent sequences.
A closed operator only promises something weaker: if $x_n \to x$ and the outputs $T x_n$ also converge, then $x$ is in the domain and the limit of the outputs is $Tx$.
Differentiation is a typical example: if functions converge and their derivatives converge uniformly, then the limit is differentiable with the limit derivative.
So the operator never gives a wrong answer in the limit, even though it may blow up.
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# Pesin theory

## Definition

Let $M$ be a compact Riemannian [[manifold|F45VQ47GPV]] and let $f \colon M \to M$ be a $C^{1+\alpha}$ [[diffeomorphism|0KZZ70ZHXP]] for some $\alpha > 0$.
Let $\mu$ be an $f$-[[invariant|B232EZ1FMB]] Borel probability measure on $M$.
Let $x \in M$ and let $v \in T_x M$ be nonzero.
The [[Lyapunov exponent|NJ1FJSNK6A]] of $v$ is
$$\lambda(x, v) = \lim_{n \to \infty} \frac{1}{n} \log \norm{D f^n_x \, v}$$
when the limit exists.
By the [[Oseledets theorem|NJ1PDJ3S4R]], for $\mu$-almost every $x$ the limit exists for every $v$ and takes finitely many values.
The measure $\mu$ is *hyperbolic* if, for $\mu$-almost every $x$, the number $0$ is not among the values $\lambda(x, v)$.
*Pesin theory* is the theory of the pairs $(f, \mu)$ of this kind with $\mu$ hyperbolic, also called *nonuniformly hyperbolic* systems.

## Remarks

Example: let $f$ be the linear map of the torus $\R^2 / \Z^2$ given by the matrix $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$, and let $\mu$ be Lebesgue measure.
The exponents are $\log \frac{3 + \sqrt 5}{2} > 0$ and $\log \frac{3 - \sqrt 5}{2} < 0$ at every point, so $\mu$ is hyperbolic.

Non-example: an irrational rotation of the torus $\R^2/\Z^2$ with Lebesgue measure is an isometry, so both exponents are $0$ everywhere and the measure is not hyperbolic.

The main results of the theory, for hyperbolic $\mu$ and $C^{1+\alpha}$ maps, include the following.
Local stable and unstable manifolds exist at $\mu$-almost every point.
If $\mu$ is absolutely continuous with respect to volume, the entropy of $\mu$ equals the integral of the sum of the positive exponents, counted with multiplicity (the Pesin entropy formula).
The hypothesis $\alpha > 0$ is needed for the stable manifold theorem: the theory does not extend in general to $C^1$ maps.
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# Complex analytic variety

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]], for example an open subset of $\C^n$.
A subset $A \subseteq X$ is a *complex analytic variety* in $X$, or an analytic subset of $X$, if:

1. $A$ is [[closed|0ZCK8DQYN1]] in $X$, and
2. every point $a \in X$ has an open neighbourhood $U$ and finitely many [[holomorphic functions|507C34W0ZT]] $f_1, \dots, f_k$ on $U$ with
$$A \cap U = \set{z \in U : f_1(z) = \cdots = f_k(z) = 0}.$$

## Remarks

For points $a \notin A$ the second condition holds with $U = X \setminus A$ and $f_1 = 1$, so it only matters at points of $A$.
Example: in $\C^2$, $A = \set{z_1 z_2 = 0}$ is an analytic variety, with $U = \C^2$ and $f_1 = z_1 z_2$; it is the union of the two coordinate axes.
Non-example: the closed unit disc $\set{\abs{z} \le 1}$ in $\C$ is closed but not analytic.
Let $a$ be a point with $\abs{a} = 1$ and let $U$ be a connected neighbourhood of $a$.
A holomorphic function on $U$ that vanishes on the part of the disc in $U$ vanishes on an open set, hence on all of $U$.
So the common zero set would be $U$, not the part of the disc in $U$.
A set that satisfies condition 2 only at its own points is called locally analytic.
It need not be closed, for example the punctured line $\set{z_1 = 0} \setminus \set{0}$ in $\C^2$.
Some authors reserve "variety" for irreducible analytic sets; here no irreducibility is required.
The empty set and $X$ itself are analytic varieties, and finite unions and arbitrary intersections of analytic varieties are analytic varieties.
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# Gibbs phenomenon

## Summary

The Gibbs phenomenon is the overshoot of the partial sums of a [[Fourier series|G12KXDHPPS]] near a [[jump discontinuity|004FH91BN5]] of the function.
As more terms are added, the overshoot moves closer to the jump but does not shrink.
It tends to about $9\%$ of the size of the jump on each side.
It shows that the [[convergence of a Fourier series|V6Q24CHJG8]] at a jump cannot be uniform.
It matters in signal processing, where truncating frequencies causes ringing near sharp edges.

## Explanations

### L1 human

Try to draw a step with a finite number of smooth waves.
The flat parts come out nearly right, but just before and after the step the curve shoots past the target and wiggles.
With more waves the wiggles get narrower and crowd towards the step, yet the first spike stays about $9\%$ of the step height too high.
For the square wave that jumps from $-1$ to $1$, the partial sums peak near $1.179$ instead of $1$.
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# Thom–Mather stratified space

## Definition

Let $V$ be a [[Hausdorff|0MXB4CKHFW]], locally compact topological space with a countable basis of open sets.
A *stratification* of $V$ is a locally finite partition $\mathcal{S}$ of $V$ into locally closed subsets, called strata.
Each stratum must be a [[smooth manifold|F45VQ47GPV]] in the subspace topology.
The frontier condition must hold: if $X, Y \in \mathcal{S}$ and $X \cap \overline{Y} \neq \emptyset$, then $X \subseteq \overline{Y}$.
One writes $X < Y$ when $X \ne Y$ and $X \subseteq \overline{Y}$.
*Control data* for $\mathcal{S}$ consist of three things for each stratum $X$.
They are an [[open|02BH3MCBQB]] neighbourhood $T_X$ of $X$ in $V$, a continuous retraction $\pi_X \colon T_X \to X$, and a continuous function $\rho_X \colon T_X \to [0, \infty)$ with $\rho_X^{-1}(0) = X$.
A *Thom–Mather stratified space* is a triple $(V, \mathcal{S}, \mathcal{J})$, where $\mathcal{J}$ is control data for $\mathcal{S}$, such that:

1. For all strata $X < Y$, the map $(\pi_X, \rho_X) \colon T_X \cap Y \to X \times (0, \infty)$ is a smooth [[submersion|GTMT5ZFV00]].
2. For all strata $X < Y$ and every stratum $Z$, the control conditions $\pi_X(\pi_Y(v)) = \pi_X(v)$ and $\rho_X(\pi_Y(v)) = \rho_X(v)$ hold for all $v \in T_X \cap T_Y \cap Z$ for which both sides are defined.

## Remarks

Example: let $V \subseteq \R^2$ be the union of the two coordinate axes, with strata $X = \set{0}$ and the four open half-axes.
Take $T_X = V$, $\pi_X(v) = 0$, and $\rho_X(v) = \abs{v}^2$.
On each half-axis, $\rho_X$ has non-zero derivative, so $(\pi_X, \rho_X)$ is a submersion onto $X \times (0, \infty)$.
For each half-axis $Y$ take $T_Y = Y$, $\pi_Y = \id_Y$ and $\rho_Y = 0$.
Then the only control conditions to check are $\pi_X(\pi_Y(v)) = 0 = \pi_X(v)$ and $\rho_X(\pi_Y(v)) = \rho_X(v)$ for $v \in Y$, which hold since $\pi_Y(v) = v$.


Non-example: give $V = \R$ the strata $X = \set{0}$ and $Y = \R \setminus \set{0}$, with $T_X = \R$ and $\pi_X = 0$.
Let $\rho_X(v) = \abs{v \sin(1/v)}$ for $v \ne 0$ and $\rho_X(0) = 0$; this function is continuous.
It vanishes at the points $v = 1/(k\pi)$ of $Y$ for every integer $k \ne 0$, so $\rho_X^{-1}(0) \ne X$ and these are not control data.

By a theorem of Mather, every closed subset of a smooth manifold with a Whitney stratification admits control data that make it a Thom–Mather stratified space.
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# Laman graph

## Definition

Let $G = (V, E)$ be a finite simple [[graph|0JD5C82QMB]] with $n = \abs{V} \ge 2$ vertices.
The graph $G$ is a *Laman graph* if both conditions hold.

1. $\abs{E} = 2n - 3$.
2. For every set $S \subseteq V$ with $k = \abs{S} \ge 2$, the [[induced subgraph|0V05FC4GKQ]] on $S$ has at most $2k - 3$ edges.

## Remarks

Example: the complete graph $K_4$ minus one edge has $4$ vertices and $5 = 2 \cdot 4 - 3$ edges.
Each $3$-vertex subset spans at most $3$ edges, and each pair spans at most $1$ edge, so condition 2 holds.

Non-example: two triangles that share one vertex have $5$ vertices and $6$ edges, but $2 \cdot 5 - 3 = 7$, so condition 1 fails.
Such a framework in the plane can rotate one triangle about the shared vertex.

Laman's theorem concerns a generic placement of the vertices in $\R^2$.
It states that $G$ is a Laman graph if and only if this framework is rigid and becomes flexible when any edge is removed.
Hilda Pollaczek-Geiringer found this result earlier, so it is also called the Geiringer–Laman theorem.
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# Exposed point

## Summary

An exposed point of a set $C$ is a point at which some linear function attains its maximum over $C$ and attains it nowhere else.
Geometrically, some [[supporting hyperplane|K7N565NKCV]] touches $C$ at that point only.
Every exposed point is an [[extreme point|3MSGFJM2NX]], but not conversely.
Exposed points are the solutions found by linear optimisation over a convex set when the optimum is unique.

## Explanations

### L1 human

A point of a shape is exposed if you can lay a flat wall against the shape so that it touches the shape at that point and nowhere else.
The corners of a square are exposed: a wall placed diagonally at a corner touches only that corner.
Every point of a circle is an exposed point of the disc.
Now take a stadium, a rectangle with two half-discs glued on its short sides.
The points where a straight side meets a round end are extreme, but every supporting line there contains the whole straight side, so they are not exposed.
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# Heat equation

## Definition

Let $n \ge 1$, let $\Omega \subseteq \R^n$ be an open set, and let $T > 0$.
Let $u \colon \Omega \times (0, T) \to \R$ have one continuous [[partial derivative|09EGC6F8PT]] in $t$ and two continuous partial derivatives in $x$.
The *heat equation* on $\Omega \times (0, T)$ is the [[parabolic|1CCTH8X2FT]] equation
$$\frac{\partial u}{\partial t}(x, t) = \Delta u(x, t), \qquad x \in \Omega, \ 0 < t < T.$$
Here $\Delta = \partial_{x_1}^2 + \cdots + \partial_{x_n}^2$ is the [[Laplacian|0E0B6RA8K0]] in $x$.

## Remarks

Many authors write $\partial_t u = \alpha \Delta u$ with a constant $\alpha > 0$, the diffusivity; the change of variable $t \mapsto \alpha t$ reduces it to $\alpha = 1$.
Example: for $k > 0$, $u(x, t) = e^{-k^2 t} \sin(k x)$ solves the heat equation on $\R \times (0, \infty)$, since $\partial_t u = -k^2 u$ and $\partial_x^2 u = -k^2 u$.
Non-example: $u(x, t) = e^{k^2 t} \sin(k x)$ is not a solution, since $\partial_t u = k^2 u$ while $\partial_x^2 u = -k^2 u$; it solves the backward heat equation $\partial_t u = -\Delta u$.
The equation is usually completed by an initial condition $u(x, 0) = g(x)$ and, when $\Omega \ne \R^n$, by boundary conditions on $\partial \Omega$.
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# Irreducible representation

## Definition

Let $G$ be a group, let $F$ be a field, and let $\rho \colon G \to GL(V)$ be a [[representation|WYGRXCG2RN]] of $G$ on a [[vector space|000AYPA9AJ]] $V$ over $F$.
A subspace $W \subseteq V$ is $G$-invariant if $\rho(g)(W) \subseteq W$ for all $g \in G$.
The representation $\rho$ is *irreducible* if $V \ne 0$ and the only $G$-invariant subspaces of $V$ are $0$ and $V$.

## Remarks

Example: let $G = \Z/4\Z$ act on $\R^2$, with the class of $1$ acting as the rotation $R$ by $90$ degrees.
A $G$-invariant line would be spanned by an eigenvector of $R$, but $R$ has characteristic polynomial $x^2 + 1$ with no real root.
So the only invariant subspaces are $0$ and $\R^2$, and the representation is irreducible.
Non-example: let $G = \Z$ act on $\C^2$, with $n$ acting by the matrix $\begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}$.
The line spanned by $(1, 0)$ is invariant, so the representation is not irreducible.
It is also not a direct sum of irreducible representations, because this line has no invariant complement.
Over $\C$, the same rotation representation of $\Z/4\Z$ on $\C^2$ is not irreducible: it splits into the eigenlines for $i$ and $-i$.
So irreducibility depends on the field.
An irreducible representation is also called simple, since it is the same as a simple module over the [[group algebra|MNP2HBWB5Q]].
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# Fuchsian group

## Summary

A Fuchsian group is a discrete group of orientation-preserving isometries of the hyperbolic plane, that is, a discrete subgroup of $\mathrm{PSL}_2(\R)$.
It acts properly discontinuously on the [[upper half-plane|X6XGGH5FPW]], and the quotient is a Riemann surface, possibly with cone points and cusps.
By the uniformization theorem, every compact Riemann surface of genus at least $2$ arises as such a quotient.
The [[modular group|VEKV1EM77B]] and its congruence subgroups are the Fuchsian groups behind modular forms.

## Explanations

### L1 human

The hyperbolic plane can be drawn as the upper half of the complex plane, where lines are half-circles and the scale shrinks near the bottom edge.
A Fuchsian group is a set of rigid motions of this plane that is closed under composition and has no motions arbitrarily close to doing nothing.
Its images of a single tile then cover the plane without crowding, like the famous tilings in Escher's "Circle Limit" pictures.
Gluing the edges of one tile gives a surface, and in this way every closed surface with two or more handles gets a hyperbolic shape.
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# Horseshoe lemma

## Statement

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]].
Let $0 \to A' \xrightarrow{i} A \xrightarrow{\pi} A'' \to 0$ be a [[short exact sequence|KJKNJMGKZZ]] in $\mathcal{A}$.
Let $P'_\bullet \to A'$ and $P''_\bullet \to A''$ be [[projective resolutions|HK2AA1YSQ2]].
Then there is a projective resolution $P_\bullet \to A$ with $P_n = P'_n \oplus P''_n$ for every $n \ge 0$ with the following property.
The inclusions $P'_n \to P_n$ and the projections $P_n \to P''_n$ form a short exact sequence of [[chain complexes|WQ82C0M3QB]]
$$0 \to P'_\bullet \to P_\bullet \to P''_\bullet \to 0$$
that lifts the sequence $0 \to A' \to A \to A'' \to 0$.

## Remarks

The proof builds $P_\bullet$ one degree at a time.
Since $P''_0$ is [[projective|YXMXAGVC1Q]], the map $P''_0 \to A''$ lifts through $\pi$ to a map $P''_0 \to A$.
Together with $P'_0 \to A' \to A$ this gives $P'_0 \oplus P''_0 \to A$, which is an epimorphism by the snake lemma.
The kernels again form a short exact sequence, and the same step is repeated with them.
The differential of $P_\bullet$ is in general not the direct sum of the differentials: it has an extra component $P''_n \to P'_{n-1}$.
For example, over $\Z$ take $0 \to \Z \xrightarrow{\cdot 2} \Z \to \Z/2\Z \to 0$ with $P'_\bullet = \Z$ and $P''_\bullet = (\Z \xrightarrow{\cdot 2} \Z)$; the lemma gives a resolution of $\Z$ by $\Z^2$ in degree $0$ and $\Z$ in degree $1$.
The dual statement holds for injective resolutions.

## Explanations

### L1 human

If the two outer terms of a short exact sequence have projective resolutions, the middle term gets one too.
The new resolution is built from the two old ones, placed side by side.
The picture of the two resolutions on the left and right, with the new one in the middle, looks like a horseshoe.

### L2 human

The lemma is the key step to show that a short exact sequence gives a long exact sequence of left [[derived functors|NPMC3K6J2F]].
Applying an additive functor to the split sequences $0 \to P'_n \to P_n \to P''_n \to 0$ keeps them exact, and the long exact sequence in homology does the rest.
The same construction, applied degree by degree, is used to build [[Cartan–Eilenberg resolutions|SX8V8GR5ZG]].
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# Linear programming

## Summary

Linear programming is the optimization of a linear function over the set of solutions of finitely many linear inequalities.
The feasible set is a convex polyhedron.
When an optimum exists, it is attained on a face, and at a vertex when the polyhedron has vertices.
Each linear program has a dual program.
If one of the two has an optimal solution, so does the other, with the same optimal value; this is tied to [[Farkas' lemma|FYZC634H2D]].
Linear programs are solved in practice by the [[simplex algorithm|MH272QHWPM]] and by interior point methods.
The ellipsoid method and interior point methods solve them in polynomial time.

## Explanations

### L1 human

A factory makes two products, each using some amounts of limited resources, and earns some profit on each.
How much of each should it make to earn the most?
The limits are straight-line conditions, so the allowed plans form a polygon, and the best plan is found at one of its corners.
Linear programming is the study of such problems, with any number of variables and conditions.
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# Nyström method

## Statement

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \R$ and $f \colon [a, b] \to \R$ be [[continuous|0X3WGX95FE]], and consider the [[Fredholm integral equation of the second kind|526TCVMSHP]]
$$u(x) - \int_a^b K(x, y)\, u(y)\, dy = f(x), \qquad x \in [a, b].$$
Assume that its homogeneous version, with $f = 0$, has only the solution $u = 0$ in $C[a, b]$.
For each $n \ge 1$ let $Q_n(g) = \sum_{j=1}^{n} w_{j}^{(n)} g(x_{j}^{(n)})$ be a [[quadrature rule|ADSVMCY77A]] with nodes $x_j^{(n)} \in [a, b]$, such that $Q_n(g) \to \int_a^b g(y)\, dy$ for every $g \in C[a, b]$.
The *Nyström method* solves the linear system
$$u_i^{(n)} - \sum_{j=1}^{n} w_j^{(n)} K(x_i^{(n)}, x_j^{(n)})\, u_j^{(n)} = f(x_i^{(n)}), \qquad i = 1, \ldots, n,$$
and sets $u_n(x) = f(x) + \sum_{j=1}^{n} w_j^{(n)} K(x, x_j^{(n)})\, u_j^{(n)}$.
Then:

1. the integral equation has a unique solution $u \in C[a, b]$;
2. there is $n_0$ such that for $n \ge n_0$ the linear system has a unique solution;
3. there is a constant $C$ such that for $n \ge n_0$
$$\max_{x \in [a,b]} \abs{u_n(x) - u(x)} \le C \max_{x \in [a, b]} \Bigl| \int_a^b K(x, y) u(y)\, dy - \sum_{j=1}^{n} w_j^{(n)} K(x, x_j^{(n)}) u(x_j^{(n)}) \Bigr|,$$
and the right side tends to $0$ as $n \to \infty$.

## Remarks

Clause 1 is the [[Fredholm alternative|4HH8C9V8JQ]] for the compact integral operator $(\mathcal K u)(x) = \int_a^b K(x, y) u(y)\, dy$ on $C[a, b]$.
Write $(\mathcal K_n u)(x) = \sum_j w_j^{(n)} K(x, x_j^{(n)}) u(x_j^{(n)})$; the values $u_n(x_i^{(n)})$ equal $u_i^{(n)}$, and $u_n$ solves $u_n - \mathcal K_n u_n = f$.
By the [[uniform boundedness principle|YG0042V8CJ]] the sums $\sum_j \abs{w_j^{(n)}}$ are bounded, so $\mathcal K_n \to \mathcal K$ pointwise and the family $(\mathcal K_n)$ is collectively compact.
Anselone's theory of collectively compact operators then shows that $I - \mathcal K_n$ is invertible for large $n$ with uniformly bounded inverses, and $u_n - u = (I - \mathcal K_n)^{-1}(\mathcal K_n - \mathcal K) u$ gives clause 3.
So the error is of the order of the quadrature error on the smooth functions $y \mapsto K(x, y) u(y)$: with Gauss–Legendre nodes and smooth $K$ and $f$ it decays very fast.
The formula for $u_n$ is the Nyström interpolation; it extends the nodal values to all of $[a, b]$ with the accuracy of the nodal values.
Without unique solvability of the homogeneous equation the method has no limit to converge to: for $K = 1$ on $[0, 1]$ and $f = 0$ every constant solves the equation.

## Explanations

### L1 human

An integral equation asks for an unknown function that appears inside an integral.
The Nyström method replaces the integral by a quadrature rule, a weighted sum of values at a few nodes.
The equation, written at those same nodes, becomes an ordinary linear system for the values of the unknown there.
Suppose the original equation is uniquely solvable and the quadrature rule converges.
Then the theorem says that the discrete solutions converge to the true one, as fast as the quadrature rule does.

### L2 human

The Nyström method is the simplest and often the most accurate way to solve second-kind Fredholm equations with smooth kernels.
Its error analysis goes through the operator $\mathcal K_n$, which is not norm-convergent to $\mathcal K$ but is collectively compact, which is enough.
It contrasts with projection methods such as collocation and Galerkin methods, which approximate $u$ in a finite-dimensional subspace.
For weakly singular kernels one uses product integration rules, where the singular factor is integrated exactly.
Equations of the first kind are ill-posed and need regularization, so this theorem does not cover them.
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# Densely defined operator

## Definition

Let $X$ and $Y$ be normed spaces, and let $T$ be an [[unbounded operator|KRJX5RA0PW]] from $X$ to $Y$ with domain $D(T)$.
The operator $T$ is *densely defined* if $D(T)$ is a [[dense subset|0BSJSX7ME2]] of $X$, that is, the closure of $D(T)$ in $X$ is $X$.

## Remarks

Example: on $X = C[0, 1]$ with the supremum norm, let $D(T)$ be the space of polynomials and $Tp = p'$.
By the Weierstrass approximation theorem, every continuous function on $[0, 1]$ is a uniform limit of polynomials, so $T$ is densely defined.

Non-example: on $X = L^2(0, 1)$, let $D(T)$ be the closed subspace of functions that vanish almost everywhere on $(0, \tfrac12)$, and $Tf = f$.
The closure of $D(T)$ is $D(T)$ itself, which is not $X$, so $T$ is not densely defined.

A bounded operator that is densely defined extends uniquely to a bounded operator on all of $X$ when $Y$ is a Banach space.
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# Extreme point

## Definition

Let $X$ be a [[vector space|000AYPA9AJ]] over $\R$ or $\C$, and let $K \subseteq X$ be a [[convex set|02AS45T3H9]].
A point $x \in K$ is an *extreme point* of $K$ if, whenever $x = t y + (1 - t) z$ with $y, z \in K$ and $0 < t < 1$, one has $y = z = x$.
The set of extreme points of $K$ is written $\operatorname{ext} K$.

## Remarks

In $\R^2$, let $K = [0, 1]^2$.
The corner $(0, 0)$ is extreme: if $(0, 0) = t y + (1 - t) z$ with $y, z \in K$ and $0 < t < 1$, then each coordinate is a combination of nonnegative numbers with positive weights equal to $0$, so $y = z = (0, 0)$.
The point $(1/2, 0)$ is not extreme, because it is the midpoint of $(0, 0)$ and $(1, 0)$, which lie in $K$ and differ from it.
Equivalently, $x$ is extreme when $x$ is not the midpoint of two distinct points of $K$.
An open ball in a nonzero normed space has no extreme points.
The closed unit ball of the sequence space $c_0$ also has no extreme points.
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# Wandering set

## Summary

A wandering set of a map is a set that the map moves completely off itself: none of its later images meets it again.
A point is a wandering point when it has a neighbourhood that is a wandering set.
Wandering points carry the transient part of the motion, the part that happens once and is never repeated.
Their complement, the [[non-wandering set|EY2W34JZA4]], is where all the recurrent behaviour of the system lives.

## Explanations

### L1 human

Think of a ball rolling down a slope into a valley.
A small region on the slope is carried down and never comes back to overlap with itself.
That region is a wandering set, and the points in it are wandering points.
For the map $x \mapsto x + 1$ on the real line, the interval $(0, 1)$ is a wandering set, since its images $(n, n+1)$ never meet it.
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# Function approximation

## Summary

Function approximation is the task of choosing, from a fixed class of simple functions, one that is close to a given target function.
Closeness is measured by a [[norm|0V2WP9B1MF]], for instance the largest pointwise difference or a mean square difference.
The task appears in numerical analysis, in signal processing and in machine learning, where the target is often known only through samples.

## Explanations

### L1 human

Imagine a complicated curve that you want to replace by something easy to store and evaluate.
You choose a family of easy curves, such as polynomials of degree $2$, and pick one whose graph stays near the target.
For instance, on $[0, 1]$ the polynomial $1 + x + x^2/2$ stays within $0.22$ of $e^x$.
How good such a replacement can be depends on the family you allow and on how you measure the gap.
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<document source="https://tamnd.github.io/math2/node/VNX79CRH90.md" tag="VNX79CRH90" kind="concept">
# Algebraic stack

## Summary

An algebraic stack is a generalization of an algebraic space in which points may have automorphism groups.
Formally it is a stack in groupoids over the category of schemes that admits a smooth surjection from a scheme.
Algebraic stacks are the natural home of moduli problems, such as the moduli of curves, where the objects classified have nontrivial symmetries.

## Explanations

### L1 human

A moduli space should have one point for each object of a given kind, for example each curve of genus $g$.
When objects have symmetries, no ordinary space can do this job well: families get glued in ways that lose information.
A stack keeps track of the symmetries by remembering, for each point, a group of automorphisms.
The simplest example is the classifying stack $BG$ of a finite group $G$: it has one point, with automorphism group $G$.
</document>

<document source="https://tamnd.github.io/math2/node/VNXJRNE33W.md" tag="VNXJRNE33W" kind="definition">
# Conjugate points

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]], let $\gamma \colon [a, b] \to M$ be a [[geodesic|NSYMQMHPDN]], and let $t_1 < t_2$ in $[a, b]$.
The points $\gamma(t_1)$ and $\gamma(t_2)$ are *conjugate along* $\gamma$ if there is a [[Jacobi field|9AHMM41PJ3]] $J$ along $\gamma$, not identically zero, with $J(t_1) = 0$ and $J(t_2) = 0$.
The *multiplicity* of the pair is the dimension of the space of such Jacobi fields.

## Remarks

Example: on the unit sphere $S^n$, let $\gamma(t)$ be a unit-speed great circle and $E$ a parallel unit field normal to $\gamma'$.
The Jacobi field $J = \sin(t) E$ vanishes at $t = 0$ and $t = \pi$, so $\gamma(0)$ and $\gamma(\pi)$, which are antipodal, are conjugate; there are $n - 1$ independent choices of $E$, so the multiplicity is $n - 1$.

Non-example: in $\R^n$ with the Euclidean metric, the Jacobi fields along $\gamma(t) = p + tv$ are $J(t) = A + tB$ with constant vectors $A, B$.
Such a field vanishing at two times is zero, so no two points are conjugate.

Equivalently, with $p = \gamma(0)$ and $\gamma(t) = \exp_p(tv)$, the point $\gamma(t_2)$ is conjugate to $p$ along $\gamma$ if and only if the differential of the [[exponential map|Y7Z0AGY1A7]] $\exp_p$ is singular at $t_2 v$.
The multiplicity is at most $n - 1$, where $n = \dim M$.
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<document source="https://tamnd.github.io/math2/node/VNYXJXX44H.md" tag="VNYXJXX44H" kind="definition">
# Reinhardt domain

## Definition

A *domain* in $\C^n$ is a nonempty [[open|02BH3MCBQB]] and [[connected|0V032VWTHD]] subset of $\C^n$.
A domain $D \subseteq \C^n$ is a *Reinhardt domain* (with centre $0$) if for every $z \in D$ and every $\theta = (\theta_1, \dots, \theta_n) \in \R^n$ the point
$$(e^{i\theta_1} z_1, \dots, e^{i\theta_n} z_n)$$
lies in $D$.
A Reinhardt domain $D$ is *complete* if for every $z \in D$ and every $w \in \C^n$ with $\abs{w_j} \le \abs{z_j}$ for $j = 1, \dots, n$, the point $w$ lies in $D$.

## Remarks

Example: the unit polydisc $\set{\abs{z_1} < 1, \abs{z_2} < 1}$ is a complete Reinhardt domain, since both conditions only involve $\abs{z_1}$ and $\abs{z_2}$ and stay true when these decrease.
The annulus product $\set{1 < \abs{z_1} < 2, \ 1 < \abs{z_2} < 2}$ is a Reinhardt domain that is not complete: it contains $(3/2, 3/2)$ but not $(0, 3/2)$.
Non-example: the ball $\set{z \in \C^2 : \abs{z_1 - 1/2}^2 + \abs{z_2}^2 < 1}$ contains $(7/5, 0)$, but its rotation $(-7/5, 0)$ is at distance $19/10 > 1$ from the centre, so it is not Reinhardt.
A Reinhardt domain with centre $a$ is a set $a + D$ with $D$ a Reinhardt domain with centre $0$.
A Reinhardt domain is determined by its image under $z \mapsto (\abs{z_1}, \dots, \abs{z_n})$.
</document>

<document source="https://tamnd.github.io/math2/node/VP1N9S4J4F.md" tag="VP1N9S4J4F" kind="definition">
# Algebraic stack

## Definition

Let $S$ be a [[scheme|5A35TYZD02]], and let $\mathcal{C}$ be the category of schemes over $S$ with the fppf topology.
An *algebraic stack* over $S$ is a category $\mathcal{X}$ with a functor $p \colon \mathcal{X} \to \mathcal{C}$ such that:

1. $\mathcal{X}$ is a stack in groupoids over $\mathcal{C}$: every fibre category is a groupoid, and objects and morphisms glue along fppf coverings.
2. The diagonal $\mathcal{X} \to \mathcal{X} \times_{\mathcal{C}} \mathcal{X}$ is representable by [[algebraic spaces|MTV9TFKF3X]].
3. There are an $S$-scheme $U$ and a 1-morphism from $U$ to $\mathcal{X}$ that is surjective and [[smooth|EXKGG9X6J0]].

In clause 3, a scheme $U$ is seen as the stack of $S$-schemes over $U$.
Clause 2 makes this 1-morphism representable by algebraic spaces, so surjective and smooth make sense.

## Remarks

Let $k$ be a field and $G$ a finite group, seen as a constant group scheme over $k$.
The classifying stack $BG$, whose objects over $T$ are $G$-torsors over $T$, is an algebraic stack.
The map $\operatorname{Spec} k \to BG$ given by the trivial torsor is surjective and étale, hence smooth.
Every algebraic space is an algebraic stack whose fibre categories have only identity morphisms.
A functor that fails the sheaf condition, such as the constant presheaf with value $\set{0, 1}$, fails clause 1.
If in clause 3 the map can be chosen étale, the stack is called a Deligne–Mumford stack.
This is the definition of the Stacks Project; some authors also require the diagonal to be quasi-compact and separated.
</document>

<document source="https://tamnd.github.io/math2/node/VP2HN86VAG.md" tag="VP2HN86VAG" kind="concept">
# Heat equation

## Summary

The heat equation $\partial_t u = \Delta u$ describes how temperature, or the concentration of a diffusing substance, evolves in time.
Here $\Delta$ is the [[Laplacian|0E0B6RA8K0]] in the space variables.
It is the model [[parabolic partial differential equation|1CCTH8X2FT]].
Its solutions become smooth at once, spread information at infinite speed, and decay toward equilibrium.
It is linked to probability through the [[Wiener process|22735FGJV6]], and to geometry through the spectrum of the Laplacian.

## Explanations

### L1 human

Heat flows from warm places to cold places, at a rate that depends on how uneven the temperature is.
The Laplacian measures how much the temperature at a point differs from the average around it.
The heat equation says that a point warms up when it is colder than its surroundings and cools down when it is warmer.
On a rod, $u(x, t) = e^{-t} \sin x$ is a solution: a sine-shaped temperature profile that keeps its shape and fades away.
</document>

<document source="https://tamnd.github.io/math2/node/VP32769PVR.md" tag="VP32769PVR" kind="statement">
# Bernstein–Kushnirenko theorem

## Statement

Let $n \ge 1$ and let $f_1, \dots, f_n$ be Laurent polynomials in $\C[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$.
For each $i$, let $P_i \subset \R^n$ be the [[Newton polytope|CYTTX8KHC8]] of $f_i$.
It is the convex hull of the exponents of the monomials of $f_i$ with nonzero coefficient.
Let $\operatorname{MV}(P_1, \dots, P_n)$ be the mixed volume, the coefficient of $\lambda_1 \lambda_2 \cdots \lambda_n$ in the polynomial $\operatorname{vol}_n(\lambda_1 P_1 + \cdots + \lambda_n P_n)$ in $\lambda_1, \dots, \lambda_n \ge 0$.
Then:

1. The number of isolated common zeros of $f_1, \dots, f_n$ in the [[algebraic torus|66FQFAJNS4]] $(\C^*)^n$, counted with multiplicity, is at most $\operatorname{MV}(P_1, \dots, P_n)$.
2. Fix the supports and take generic coefficients. Then all common zeros in $(\C^*)^n$ are simple, and their number equals $\operatorname{MV}(P_1, \dots, P_n)$.
3. (Kushnirenko) In particular, if $P_1 = \cdots = P_n = P$, the generic number of common zeros in $(\C^*)^n$ is $n! \operatorname{vol}_n(P)$.

## Remarks

Bernstein's proof deforms the system along a curve and studies the solutions as Puiseux series; the leading exponents of these series are governed by the faces of the Newton polytopes.
Bernstein also gave an explicit sufficient condition on the coefficients, stated with initial forms.
Suppose that, for every nonzero $w \in \R^n$, the initial forms of $f_1, \dots, f_n$ in direction $w$ have no common zero in $(\C^*)^n$.
Then all common zeros in $(\C^*)^n$ are isolated, and their number with multiplicity equals $\operatorname{MV}(P_1, \dots, P_n)$.
Take $n = 1$ and $f = a_0 + a_1 x + \cdots + a_d x^d$ with $a_0 a_d \ne 0$.
The Newton polytope is $[0, d]$, of length $d$, and the theorem says $f$ has $d$ roots in $\C^*$ counted with multiplicity.
For two bilinear equations $a + bx + cy + dxy = 0$ and $a' + b'x + c'y + d'xy = 0$, both polytopes are the unit square, so the generic number of solutions in $(\C^*)^2$ is $2! \cdot 1 = 2$.
Let $f_i$ be a generic dense polynomial of degree $d_i$, so that $P_i$ is $d_i$ times the standard simplex.
Then the mixed volume is $d_1 d_2 \cdots d_n$, and the bound agrees with [[Bézout's theorem|KWNVF194D8]].
For sparse systems the mixed volume can be much smaller than the Bézout number, which is why the theorem is useful.
The count concerns only zeros in the torus: zeros with some coordinate equal to $0$ are not counted.

## Explanations

### L1 human

How many solutions does a system of $n$ polynomial equations in $n$ unknowns have?
The theorem answers this using only the shapes formed by the exponents of the monomials that appear.
Each equation gives a polygon or polytope, and a volume computed from these shapes is the number of solutions with all coordinates nonzero, for typical coefficients.
So a question of algebra becomes a question of geometry of polytopes.

### L2 human

The theorem is the sparse refinement of Bézout's theorem: it sees which monomials actually occur.
It is central in toric geometry, where the mixed volume becomes an intersection number of divisors on a toric variety attached to the polytopes.
In tropical geometry, the same count appears as the number of intersection points of tropical hypersurfaces, counted with lattice multiplicities.
In numerical algebraic geometry, polyhedral homotopy methods start from the mixed volume to track exactly the generic number of solutions.
Computing the mixed volume uses mixed subdivisions of the Minkowski sum $P_1 + \cdots + P_n$.
</document>

<document source="https://tamnd.github.io/math2/node/VP40T7MHGM.md" tag="VP40T7MHGM" kind="concept">
# Romberg's method

## Summary

Romberg's method computes a definite integral by applying [[Richardson extrapolation|WMTZG71K2Y]] again and again to the [[trapezoidal rule|FH0538XCZR]] with step sizes $h, h/2, h/4, \dots$.
It works because the error of the trapezoidal rule for a smooth function has an expansion in even powers of $h$.
This expansion comes from the [[Euler–Maclaurin formula|9P839VB8S7]].
Each extrapolation step removes one more term of this expansion.
The results are arranged in a triangular table whose diagonal converges fast for smooth integrands.

## Explanations

### L1 human

The trapezoidal rule is crude, but its error shrinks in a very regular way when the step is halved.
Since the leading error is about four times smaller after halving, a suitable combination of two results cancels it.
Repeating this trick removes the next error term, then the next one.
For example, combining the trapezoidal values for $1$ and $2$ subintervals in this way gives exactly Simpson's rule.
</document>

<document source="https://tamnd.github.io/math2/node/VP5PW8VSHZ.md" tag="VP5PW8VSHZ" kind="object">
# Moduli space of instantons

## Description

Let $X$ be a compact oriented Riemannian [[4-manifold|8H58Q6YJCH]] and $P \to X$ a [[principal|8E37E74XVP]] $SU(2)$-bundle with second [[Chern class|PAD37GKDPD]] $c_2(P) = k$.
The *moduli space of instantons* on $P$ is
$$\mathcal{M}_k = \set{A : A \text{ a connection on } P, \ F_A^+ = 0} / \mathcal{G}(P),$$
the set of anti-self-dual connections, the [[instantons|YWBTGC7MFA]], modulo the [[gauge group|707DEWCCFX]].
It is a subset of the [[space of connections modulo gauge|888SJ2508Y]].

## Remarks

The linearisation of the equation $F_A^+ = 0$, together with the Coulomb gauge condition, is an elliptic operator.
Its index, computed by the Atiyah–Singer index theorem, gives the expected dimension
$$\dim \mathcal{M}_k = 8k - 3(1 - b_1(X) + b^+(X)).$$
Here $b_1$ is the first Betti number and $b^+$ the dimension of a maximal positive subspace for the intersection form.
Near an irreducible instanton $A$ at which this linearised operator is surjective, $\mathcal{M}_k$ is a smooth manifold of this dimension.
For $X = S^4$ with its round metric and $k = 1$, the formula gives $8 - 3 = 5$.
Indeed $\mathcal{M}_1$ is a $5$-dimensional manifold, diffeomorphic to an open $5$-ball, and each of its instantons is determined by a centre in $S^4$ and a scale.


The moduli space is not compact in general: instantons can concentrate at points, and Uhlenbeck's compactness theorem describes the limits.
Donaldson used these moduli spaces to prove his theorem on definite intersection forms of smooth $4$-manifolds and to define new invariants.

## Explanations

### L1 human

Instantons are the most efficient twisted connections on a bundle over a four-dimensional space.
Counting them only up to gauge transformations, which do not change the geometry, gives the moduli space.
It is a finite-dimensional space even though connections form an infinite-dimensional space, because the instanton equation is elliptic.
Its shape carries information about the smooth structure of the underlying four-manifold.
</document>

<document source="https://tamnd.github.io/math2/node/VP98WSTZWT.md" tag="VP98WSTZWT" kind="concept">
# Logarithmic integral

## Summary

The logarithmic integral is $\operatorname{li}(x) = \int_0^x \frac{dt}{\ln t}$, taken as a principal value at $t = 1$.
By the [[prime number theorem|0EHBY5Y84N]], it is asymptotic to, and an excellent approximation of, the [[prime-counting function|0RQSNVTFVZ]] $\pi(x)$.
The offset version $\operatorname{Li}(x) = \int_2^x \frac{dt}{\ln t}$ avoids the singularity at $1$.
It is the [[exponential integral|EV6ZDQ0NDY]] composed with the logarithm: $\operatorname{li}(x) = \operatorname{Ei}(\ln x)$.

## Explanations

### L1 human

Near a large number $t$, about one number in $\ln t$ is prime.
Adding up these densities from $2$ to $x$ gives the integral $\int_2^x \frac{dt}{\ln t}$.
So the logarithmic integral predicts how many primes there are up to $x$.
For $x = 10^6$ the prediction is about $78\,627$, while the true count is $78\,498$.
</document>

<document source="https://tamnd.github.io/math2/node/VP9S3PNJQ0.md" tag="VP9S3PNJQ0" kind="concept">
# Kriging

## Summary

Kriging is a method to predict the value of a random field at an unobserved location from its values at observed locations.
The prediction is the best linear unbiased predictor: the weighted average of the data with the smallest mean squared prediction error.
The weights come from the spatial dependence of the field, described by its [[variogram|ZF8AA2JVKV]] or its covariance function.
It is named after the mining engineer Danie Krige, and in the language of [[Gaussian processes|22WEM6RTW0]] it is essentially Gaussian process regression.

## Explanations

### L1 human

Suppose you measured the gold content of rock at a few drill holes and want a guess for a new spot.
Nearby holes should count more than far ones, and holes that cluster together should not be counted twice.
Kriging chooses the weights to make the expected squared error of the guess as small as possible.
At a drill hole itself, kriging returns the measured value, so it interpolates the data.
It also gives the size of the expected error, which is larger far from the data.
</document>

<document source="https://tamnd.github.io/math2/node/VPA41PWRR1.md" tag="VPA41PWRR1" kind="concept">
# Infinitesimal generator of a Markov process

## Summary

The infinitesimal generator of a Markov process is the operator that gives the rate of change of expected values at time $0$.
It is the derivative at $t = 0$ of the transition [[semigroup|0XMEM406JM]] of the process.
For a [[Feller process|4HS3C12472]] the generator determines the law of the process.
It links Markov processes with differential operators: Brownian motion has generator $\tfrac12 \Delta$.

## Explanations

### L1 human

Take a function $f$ of the state and ask how fast its average value changes when the process has just started at $x$.
The answer, as a function of $x$, is $Af(x)$, and $A$ is the generator.
For a chain with finitely many states, the generator is the matrix of jump rates.
For a particle that jumps from $x$ to $x + 1$ at rate $\lambda$, one gets $Af(x) = \lambda (f(x+1) - f(x))$.
For Brownian motion it is half the second derivative, which is why the heat equation appears.
</document>

<document source="https://tamnd.github.io/math2/node/VPAQZ0XW74.md" tag="VPAQZ0XW74" kind="concept">
# Algebraic combinatorics

## Summary

Algebraic combinatorics uses methods of abstract algebra to study combinatorial structures, and combinatorial methods to study algebraic objects, see the [[definition|5VDCJTZ8SW]].
Central topics are [[symmetric functions|PAJEHWXW6M]], [[Young tableaux|Q6TH9PH2R7]] and the [[representation theory of the symmetric group|1VQC05EEYW]].
Other topics are [[association schemes|C338GKB4VJ]], [[incidence algebras|1H9SFWHYEB]] of partially ordered sets, and the algebra of simplicial complexes.
Many of its results are identities that have both an algebraic proof and a bijective proof.

## Explanations

### L1 human

Algebraic combinatorics is a two-way street between counting and algebra.
In one direction, a counting problem is solved by building a group, a ring or a matrix and computing with it.
In the other direction, an algebraic quantity, such as the dimension of a representation, is computed by counting pictures.
For example, the dimensions of the irreducible representations of the symmetric group are counted by standard Young tableaux.
</document>

<document source="https://tamnd.github.io/math2/node/VPBSH2KDFD.md" tag="VPBSH2KDFD" kind="definition">
# Pexider equation

## Definition

The *Pexider equation* is the [[functional equation|SRAN4EKWS0]]
$$f(x + y) = g(x) + h(y) \quad \text{for all } x, y \in \R,$$
for three unknown [[functions|0EPAT51DXQ]] $f, g, h \colon \R \to \R$.
A solution is a triple $(f, g, h)$ that satisfies this identity for all real $x$ and $y$.

## Remarks

Example: $f(x) = 5x + 3$, $g(x) = 5x + 1$, $h(x) = 5x + 2$ is a solution, since both sides equal $5x + 5y + 3$.
Non-example: $f(x) = x^2$, $g(x) = x^2$, $h(x) = x^2$ is not a solution, since $f(1 + 1) = 4$ while $g(1) + h(1) = 2$.
Setting $y = 0$ and then $x = 0$ gives $g(x) = f(x) - h(0)$ and $h(y) = f(y) - g(0)$.
Then $A(x) = f(x) - g(0) - h(0)$ is an [[additive function|KF719SZD5T]], which gives the general solution $f = A + a + b$, $g = A + a$, $h = A + b$ with $a = g(0)$ and $b = h(0)$.
The same method solves the equation for maps from a group to an abelian group.
</document>

<document source="https://tamnd.github.io/math2/node/VPJCSVRY6S.md" tag="VPJCSVRY6S" kind="concept">
# Associated Legendre polynomials

## Summary

The associated Legendre polynomials $P_\ell^m$ are the [[Legendre functions|69XMERM6KJ]] of integer degree $\ell$ and integer order $m$ on the interval $[-1, 1]$.
They are obtained from the [[Legendre polynomials|QFYGJVA996]] by differentiating $m$ times and multiplying by $(1 - x^2)^{m/2}$.
They give the dependence on the polar angle of the [[spherical harmonics|3F32Y21HP4]] on the sphere in $\R^3$.

## Explanations

### L1 human

Spherical harmonics describe patterns on a sphere, such as the shapes of atomic orbitals.
Each pattern splits into a wave around the equator and a profile from the north pole to the south pole.
The associated Legendre polynomials are these north-to-south profiles.
For example, $P_1^1(\cos \theta) = -\sin \theta$ for $0 \le \theta \le \pi$, a profile that is zero at both poles and largest in size at the equator.
Despite the name, they are not polynomials when $m$ is odd, because of the square root $(1 - x^2)^{1/2}$.
</document>

<document source="https://tamnd.github.io/math2/node/VPMHPRRKZH.md" tag="VPMHPRRKZH" kind="definition">
# Positive-definite function

## Definition

Let $n \ge 1$.
A function $f \colon \R^n \to \C$ is *positive-definite* if for every $N \ge 1$, all points $x_1, \dots, x_N \in \R^n$ and all $c_1, \dots, c_N \in \C$,
$$\sum_{i=1}^{N} \sum_{j=1}^{N} c_i \, \overline{c_j} \, f(x_i - x_j) \ge 0.$$

## Remarks

Equivalently, each matrix $\big( f(x_i - x_j) \big)_{i,j=1}^{N}$ is [[Hermitian|0EKZCX4K3P]] and [[positive semidefinite|0880F9EQCP]].
Taking $N = 1$ and $N = 2$ gives $f(0) \ge 0$, $f(-x) = \overline{f(x)}$ and $\abs{f(x)} \le f(0)$.
This is the case $G = \R^n$ of a [[positive-definite function on a group|3PRZ02APFA]], except that continuity is not part of the definition here.
Some authors call these functions positive semidefinite and reserve "positive-definite" for strict inequality when the points are distinct and $c \ne 0$.

Example: $f(x) = \cos x$ on $\R$.
Since $\cos(x_i - x_j) = \frac12 \big( e^{ix_i} e^{-ix_j} + e^{-ix_i} e^{ix_j} \big)$,
$$\sum_{i,j} c_i \overline{c_j} \cos(x_i - x_j) = \frac12 \Big| \sum_i c_i e^{ix_i} \Big|^2 + \frac12 \Big| \sum_i c_i e^{-ix_i} \Big|^2 \ge 0.$$

Non-example: $f = 1_{[-1,1]}$ on $\R$.
For the points $0, 1, 2$ the matrix $\big(f(x_i - x_j)\big)$ is
$$\begin{pmatrix} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{pmatrix},$$
whose determinant is $-1$.
A positive semidefinite matrix has nonnegative determinant, so $f$ is not positive-definite.
</document>

<document source="https://tamnd.github.io/math2/node/VPMRHXP6NM.md" tag="VPMRHXP6NM" kind="statement">
# Control variates

## Statement

Let $Y$ and $C$ be random variables with finite variance, where $\mu = \mathbb E C$ is known and $\operatorname{Var} C > 0$.
For a real number $\beta$ let $Z_\beta = Y - \beta (C - \mu)$.
Then:

1. $\mathbb E Z_\beta = \mathbb E Y$ for every $\beta$;
2. $\operatorname{Var} Z_\beta = \operatorname{Var} Y - 2\beta \operatorname{Cov}(Y, C) + \beta^2 \operatorname{Var} C$;
3. this variance is smallest for $\beta^* = \operatorname{Cov}(Y, C) / \operatorname{Var} C$, and $\operatorname{Var} Z_{\beta^*} = (1 - \rho^2) \operatorname{Var} Y$, where $\rho$ is the [[correlation coefficient|0DNN9B6Q2F]] of $Y$ and $C$.

## Remarks

Clause 1 holds because $\mathbb E(C - \mu) = 0$, and clause 2 is the expansion of the [[variance|0477R9PYAE]] of a difference.
Clause 3 minimises the quadratic in $\beta$; substituting $\beta^*$ gives $\operatorname{Var} Y - \operatorname{Cov}(Y, C)^2 / \operatorname{Var} C = (1 - \rho^2)\operatorname{Var} Y$.
Example: to estimate $\mathbb E e^U = e - 1$ for $U$ uniform on $(0, 1)$, use $C = U$ with $\mu = \frac12$.
Here $\operatorname{Cov}(e^U, U) = 1 - \frac{e - 1}{2} \approx 0.1409$ and $\operatorname{Var} U = \frac{1}{12}$, so $\beta^* \approx 1.690$ and $\operatorname{Var} Z_{\beta^*} \approx 0.0039$, against $\operatorname{Var} e^U \approx 0.2420$.
In practice $\beta^*$ is estimated from the same samples, by least squares regression of $Y$ on $C$; this introduces a bias of order $1/n$, which is usually negligible.
If $C$ is uncorrelated with $Y$, then $\rho = 0$ and no gain is possible.

## Explanations

### L1 human

Suppose a quantity $Y$ is hard to average, but a related quantity $C$ has an exactly known average.
On each sample, see how far $C$ is from its known average, and correct $Y$ by a multiple of that error.
If $Y$ and $C$ move together, the correction removes much of the noise in $Y$.
The better $C$ tracks $Y$, the larger the gain: a correlation of $0.99$ cuts the variance by a factor of about $50$.

### L2 human

Control variates is a [[variance reduction|A01GEGWKVB]] technique that turns side knowledge, the exact mean of $C$, into accuracy.
In option pricing, the price of a similar option with a closed-form value is a classic control.
With several controls $C_1, \ldots, C_k$ the optimal coefficients solve a linear least squares problem, and the variance becomes $(1 - R^2)\operatorname{Var} Y$, with $R^2$ the squared multiple correlation.
The multilevel Monte Carlo method can be seen as a recursive use of control variates, with coarse approximations as controls for fine ones.
</document>

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# Proj construction

## Definition

Let $S = \bigoplus_{d \ge 0} S_d$ be a commutative [[graded ring|8KB4SQGC0C]], and let $S_+ = \bigoplus_{d > 0} S_d$.
As a set, $\operatorname{Proj} S$ is the set of [[homogeneous|6ZBNGS0P98]] [[prime ideals|00MHHZ9SNE]] $\mathfrak{p}$ of $S$ with $S_+ \not\subseteq \mathfrak{p}$.
Its closed sets are the sets $V_+(\mathfrak{a}) = \set{\mathfrak{p} \in \operatorname{Proj} S : \mathfrak{a} \subseteq \mathfrak{p}}$ for homogeneous ideals $\mathfrak{a}$.
For a homogeneous $f \in S_+$, let $D_+(f) = \set{\mathfrak{p} \in \operatorname{Proj} S : f \notin \mathfrak{p}}$.
Let $S_{(f)}$ be the ring of elements of degree $0$ in the [[localization|4C1AE84N5N]] $S_f$, that is, the fractions $a / f^n$ with $a$ homogeneous of degree $n \deg f$.

The structure sheaf $\mathcal{O}$ on $\operatorname{Proj} S$ is the unique sheaf of rings such that, for every homogeneous $f \in S_+$:

1. $\mathcal{O}(D_+(f)) = S_{(f)}$;
2. $(D_+(f), \mathcal{O}|_{D_+(f)})$ is isomorphic to the [[affine scheme|B3V3XJ2W8G]] $\operatorname{Spec} S_{(f)}$, by $\mathfrak{p} \mapsto \mathfrak{p} S_f \cap S_{(f)}$.

The sets $D_+(f)$ cover $\operatorname{Proj} S$, so $(\operatorname{Proj} S, \mathcal{O})$ is a [[scheme|5A35TYZD02]].

## Remarks

Example: for a ring $A$, projective $n$-space over $A$ is $\mathbb{P}^n_A = \operatorname{Proj} A[x_0, \ldots, x_n]$, with $\deg x_i = 1$.
It is covered by the $n+1$ affine pieces $D_+(x_i) \cong \operatorname{Spec} A[x_0/x_i, \ldots, x_n/x_i]$.
Non-example: for $S = k[x_0, \ldots, x_n]$ over a field $k$, the ideal $S_+ = (x_0, \ldots, x_n)$ is a homogeneous prime ideal.
It is not a point of $\operatorname{Proj} S$, since it contains $S_+$; it would correspond to the excluded origin of $k^{n+1}$.
For a homogeneous polynomial $F$ in $x_0, \ldots, x_n$, the scheme $\operatorname{Proj} k[x_0, \ldots, x_n]/(F)$ is the projective hypersurface $F = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/VQ2BTYFST4.md" tag="VQ2BTYFST4" kind="concept">
# Geometric integrator

## Summary

A geometric integrator is a numerical method for differential equations that keeps exactly a geometric property of the exact [[flow|06738GGWWK]].
Examples of such properties are symplecticity, a conserved quantity, volume, time-reversibility, and staying on a manifold.
General-purpose methods only approximate such properties, and the small violations add up over long times.
Keeping the structure exactly gives much better qualitative behaviour in long simulations, for example in celestial mechanics and molecular dynamics.
[[Symplectic integrators|B4C25SXFRE]] are the best known class.

## Explanations

### L1 human

The exact motion of a frictionless pendulum keeps its energy, and its phase-space picture has a precise geometric structure.
A typical numerical method makes tiny errors in this structure at each step, and after many steps the computed pendulum may spiral outwards or inwards.
A geometric integrator is designed so that the structure is kept exactly, step by step.
Its trajectories may still be slightly off, but they look like trajectories of a nearby system of the same kind.
For example, the symplectic Euler method on the harmonic oscillator maps each step by a matrix of determinant $1$, so areas in the phase plane are kept.
The explicit Euler method inflates them by the factor $1 + h^2$ at every step.
</document>

<document source="https://tamnd.github.io/math2/node/VQ31D68MDD.md" tag="VQ31D68MDD" kind="definition">
# Annihilator

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $M$ be an $R$-[[module|09VDS2W1E8]].
The *annihilator* of $M$ is the set
$$\operatorname{Ann}_R(M) = \set{r \in R : rm = 0 \text{ for all } m \in M}.$$
For an element $m \in M$, the annihilator of $m$ is $\operatorname{Ann}_R(m) = \set{r \in R : rm = 0}$.

## Remarks

$\operatorname{Ann}_R(M)$ and $\operatorname{Ann}_R(m)$ are [[ideals|0MYSA6MYNA]] of $R$.
Example: $\operatorname{Ann}_\Z(\Z/6\Z) = 6\Z$; multiplication by $r$ kills the class of $1$ exactly when $6 \mid r$, and then it kills every class.
Non-example: $3 \notin \operatorname{Ann}_\Z(\Z/6\Z)$, since $3 \cdot 1 = 3 \ne 0$ in $\Z/6\Z$.
$M$ is a *faithful* module when $\operatorname{Ann}_R(M) = 0$.
For an ideal $J$ of $R$, the annihilator of the module $J$ is the [[ideal quotient|16J9DM8V57]] $(0 : J)$.
$M$ is naturally a module over $R/\operatorname{Ann}_R(M)$.
</document>

<document source="https://tamnd.github.io/math2/node/VQ4Q2V45RP.md" tag="VQ4Q2V45RP" kind="concept">
# Line segment

## Summary

The line segment between two points is the set of all points on the straight path from one to the other, endpoints included.
In a real vector space it is the set of [[convex combinations|21SXYJYC59]] of the two points.
A set is [[convex|02AS45T3H9]] exactly when it contains the segment between any two of its points, so segments are the building blocks of convexity.

## Explanations

### L1 human

Take two points and stretch a string tightly between them.
The string traces the line segment.
In coordinates, the segment from $x$ to $y$ is swept out by $(1 - t) x + t y$ as $t$ runs from $0$ to $1$.
For example, the segment from $(0, 0)$ to $(2, 2)$ in the plane is the set of points $(s, s)$ with $0 \le s \le 2$.
</document>

<document source="https://tamnd.github.io/math2/node/VQ4SSKS53D.md" tag="VQ4SSKS53D" kind="definition">
# Torsion-free group

## Definition

Let $G$ be a [[group|G5S1M9]] with identity $e$.
$G$ is *torsion-free* if every element $g \ne e$ of $G$ has infinite [[order|0WE2NHZJ7B]], that is, $g^n \ne e$ for all integers $n \ge 1$.

## Remarks

Example: the [[integers|0C4JS9RF5M]] $\Z$ under addition.
If $m \ne 0$ and $n \ge 1$, then $n m \ne 0$.
The groups $\Z^k$, the [[rational numbers|0NCJ5WXN6Z]] under addition, and [[free groups|0KTW26C80J]] are also torsion-free.

Non-example: $\Z \times \Z/2\Z$.
The element $(0, 1)$ is not the identity, but $2 \cdot (0, 1) = (0, 0)$.

For an abelian group $A$ with [[torsion subgroup|0DBWXQRXQJ]] $T$, the quotient $A/T$ is torsion-free.
By the [[structure theorem for finitely generated abelian groups|0RY0B72XAS]], a finitely generated torsion-free abelian group is isomorphic to $\Z^n$ for some $n \ge 0$.
In a nonabelian group the elements of finite order need not form a subgroup.
In the infinite dihedral group, two reflections have order $2$, but their product is a translation of infinite order.
</document>

<document source="https://tamnd.github.io/math2/node/VQ7D5N3MV7.md" tag="VQ7D5N3MV7" kind="example">
# Cusp catastrophe

## Example

The *cusp catastrophe* is the two-parameter family of functions
$$V_{a,b}(x) = x^4 + a x^2 + b x, \qquad x, a, b \in \R.$$
Its [[critical points|0YTCMFB4HC]] solve $4x^3 + 2a x + b = 0$, and such a point is degenerate when also $12 x^2 + 2a = 0$.

Bifurcation set: a degenerate critical point $x$ has $a = -6x^2$ and then $b = -4x^3 - 2ax = 8x^3$.
Eliminating $x$ gives the curve
$$8 a^3 + 27 b^2 = 0,$$
a semicubical parabola with a cusp at the origin, lying in the half-plane $a \le 0$.
Check: with $a = -6x^2$ and $b = 8x^3$, one has $8a^3 = -1728 x^6$ and $27 b^2 = 1728 x^6$.

Number of critical points: the cubic $4x^3 + 2ax + b$ has discriminant $-16(8a^3 + 27b^2)$.
So for $8a^3 + 27b^2 < 0$, inside the cusp, there are three critical points, two local minima and one local maximum between them.
For $8a^3 + 27b^2 > 0$, outside the cusp, there is one critical point, a minimum.
The family is the [[universal unfolding|KSAFWZVB8K]] of the germ $x^4$, whose codimension is $2$.

## Explanations

### L1 human

Now the curve can have two valleys, and the ball sits in one of them.
Inside the cusp-shaped region of the two knobs $a$ and $b$, both valleys exist; outside it, only one does.
If you turn the knob $b$ slowly, the ball stays in its valley until that valley disappears at the edge of the cusp.
Then it jumps to the other valley.
Turning $b$ back, the ball does not jump back at the same place but only at the opposite edge, which is called hysteresis.
The cusp is the simplest catastrophe with this memory effect, and its [[bifurcation diagram|2XDRZV3GSA]] is the cusp curve.
</document>

<document source="https://tamnd.github.io/math2/node/VQB2N60ARE.md" tag="VQB2N60ARE" kind="statement">
# Slice theorem

## Statement

Let $G$ be a compact Lie group with a smooth [[action|0YSE802YCQ]] on a [[smooth manifold|R2TQ82PCNB]] $M$.
Let $x \in M$, let $G x$ be its [[orbit|00F2JFA41J]], and let $H = G_x$ be its [[stabiliser|0RGGF8ZS6D]], a closed subgroup of $G$.
Let $V = T_x M / T_x(G x)$ be the normal space of the orbit at $x$.
The group $H$ acts linearly on $V$ by the derivatives at $x$ of the maps $y \mapsto h y$ for $h \in H$; this is a [[representation|RK8RV2C998]] of $H$.
Let $G \times_H V$ be the quotient of $G \times V$ by the action $h \cdot (g, v) = (g h^{-1}, h v)$ of $H$, with $G$ acting by $g' [g, v] = [g' g, v]$.
Then there are a $G$-invariant open neighbourhood $U$ of $G x$ in $M$ and a $G$-equivariant [[diffeomorphism|0Z1Y7JHGBR]] $\Phi \colon G \times_H V \to U$.
Moreover $\Phi([e, 0]) = x$, where $e$ is the identity of $G$.

## Remarks

The idea of the proof is to average a Riemannian metric on $M$ over $G$, which is possible because $G$ is compact.
The new metric is $G$-invariant, so the exponential map on the normal bundle of the orbit $G x$ is $G$-equivariant.
The orbit is a compact submanifold diffeomorphic to $G / H$, and its normal bundle is $G \times_H V$.
The [[tubular neighbourhood theorem|GCBY80CSC6]] gives $\Phi$ from this exponential map on vectors of small length.
The image $S = \Phi(\set{[e, v] : v \in V})$ is called a *slice* at $x$: it is $H$-invariant, and $g S \cap S \neq \emptyset$ implies $g \in H$.
Example: let $G = SO(2)$ act on $\R^2$ by rotations and let $x = (1, 0)$.
Then $H$ is trivial, $V$ is the radial line $\R$, and $\Phi([g, t]) = g (e^t, 0)$ is an equivariant diffeomorphism from $SO(2) \times \R$ onto $U = \R^2 \setminus \set{0}$.
Compactness can be replaced by properness of the action, a result of Palais.
Without such a hypothesis the theorem fails.
Let $\R$ act on the torus $\R^2 / \Z^2$ by translations along a line of irrational slope.
Every orbit is dense, so no orbit has a neighbourhood of the form $G \times_H V$ in which it is closed.

## Explanations

### L1 human

When a compact group of symmetries moves a manifold around, the theorem describes how the manifold looks near one orbit.
Near the orbit, the manifold is a thickened copy of the orbit.
The thickening is a small disk at each point of the orbit, and the symmetries that fix the point turn this disk by linear maps.

### L2 human

The theorem reduces local questions about smooth actions of compact groups to linear algebra.
It shows that every orbit has a neighbourhood that retracts onto it equivariantly, and that the stabilisers of nearby points are conjugate to subgroups of $H$.
From it one derives the principal orbit type theorem.
It also shows that the quotient of a manifold by a free action of a compact Lie group is again a manifold.
When $G$ is finite and acts faithfully, it shows that $M / G$ is an [[orbifold|RPAVY0W0XS]].
</document>

<document source="https://tamnd.github.io/math2/node/VQK5HQFYEM.md" tag="VQK5HQFYEM" kind="statement">
# Twenty-seven lines on a cubic surface

## Statement

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]].
Let $X \subset \mathbb{P}^3$ be a smooth cubic surface over $k$.
This means that $X$ is the zero set in [[projective space|ZT9A6JK8QF]] of a homogeneous polynomial $F$ of degree $3$.
Moreover, the partial derivatives of $F$ have no common zero on $X$.
Then $X$ contains exactly $27$ lines of $\mathbb{P}^3$.

## Remarks

One proof identifies $X$ with the [[blow-up|C4KGYAFPN7]] of $\mathbb{P}^2$ at six points $p_1, \ldots, p_6$ in general position, so that $X$ is a [[del Pezzo surface|AV2ZAAZXN9]] of degree $3$.
The lines on $X$ are exactly the curves $E$ with $E^2 = -1$ and $-K_X \cdot E = 1$.
They are the $6$ exceptional curves over the $p_i$ and the strict transforms of the $15$ lines through two of the points.
The other $6$ are the strict transforms of the conics through five of the points, and $6 + 15 + 6 = 27$.
Each of the $27$ lines meets exactly $10$ of the others, and the symmetry group of the incidence configuration is the Weyl group of type $E_6$, of order $51840$.
Smoothness is needed: the cone over a smooth plane cubic is a singular cubic surface.
It contains infinitely many lines, one through the vertex and each point of the base curve.
The result goes back to Cayley and Salmon in 1849.

## Explanations

### L1 human

One might expect a curved surface given by a cubic equation to contain no straight lines at all.
But every smooth cubic surface contains exactly $27$ straight lines, no more and no fewer.
For the Fermat cubic $x^3 + y^3 + z^3 + w^3 = 0$ they can be listed: $x = -\zeta y$, $z = -\zeta' w$ with cube roots of unity $\zeta, \zeta'$, and the analogous lines for the other pairings of the variables.

### L2 human

The theorem is one of the first results of enumerative geometry and a model for counting curves on varieties.
The lines are the $(-1)$-curves of the del Pezzo surface of degree $3$.
Similar counts hold for the other degrees, for instance $16$ such curves on a del Pezzo surface of degree $4$ and $240$ on one of degree $1$.
Over the real numbers a smooth cubic surface has $3$, $7$, $15$ or $27$ real lines.
</document>

<document source="https://tamnd.github.io/math2/node/VQPN24VDH1.md" tag="VQPN24VDH1" kind="statement">
# Chow–Rashevskii theorem

## Statement

Let $M$ be a connected [[smooth manifold|F45VQ47GPV]] and let $X_1, \ldots, X_k$ be smooth [[vector fields|0299VCW1TH]] on $M$.
Assume the bracket-generating condition.
For every $p \in M$, the values at $p$ of the $X_i$ and of their iterated [[Lie brackets|68FGQEBD6J]] $[X_i, X_j]$, $[X_i, [X_j, X_l]]$, and so on, span $T_p M$.
Consider the driftless control system
$$x' = \sum_{i=1}^{k} u_i X_i(x), \qquad u = (u_1, \ldots, u_k) \in \R^k.$$
This system is [[controllable|A9WFM77Z4P]]: any two points of $M$ are joined by a trajectory of the system for a piecewise constant control.
Equivalently, any two points of $M$ are joined by a piecewise smooth curve whose velocity lies at each point in the span of $X_1, \ldots, X_k$.

## Remarks

The set of points reachable from $p$ is the set of endpoints of finite compositions of [[flows|06738GGWWK]] of the fields $\pm X_i$.
Since the system is symmetric under $u \mapsto -u$, reachability is an equivalence relation, and the equivalence classes partition $M$.
Flowing back and forth along two fields moves a point, to second order, in the direction of their bracket.
With the bracket condition, this shows that for suitable $Y_j$ among the $\pm X_i$, the map $(t_1, \ldots, t_N) \mapsto e^{t_N Y_N} \circ \cdots \circ e^{t_1 Y_1}(p)$ has an image with nonempty interior.
So each class is open, and connectedness of $M$ leaves only one class.
Example: on $\R^3$, the fields $X_1 = \partial_x - \frac{y}{2} \partial_z$ and $X_2 = \partial_y + \frac{x}{2} \partial_z$ satisfy $[X_1, X_2] = \partial_z$, so $X_1, X_2, [X_1, X_2]$ span $\R^3$ at each point and every point is reachable, although only two directions are controlled.
The bracket condition cannot be dropped: for $X_1 = \partial_x$ and $X_2 = \partial_y$ on $\R^3$ all brackets vanish, and the trajectories from $p$ stay in the plane through $p$ where $z$ is constant.

## Explanations

### L1 human

A car can move forward, backward and turn, but it cannot slide sideways.
Still, by a sequence of small forward and turning motions, it can park in a gap beside it.
The theorem explains why: alternating two allowed motions produces a motion in a new direction, given by their Lie bracket.
If the allowed motions and their brackets point in all directions, then every position can be reached.

### L2 human

The theorem is the basic controllability result of geometric control theory for systems without drift.
In [[sub-Riemannian geometry|97NYHNBZ2D]] it shows that the distance defined by curves tangent to a bracket-generating distribution is finite, and it is the starting point of that theory.
For systems with drift $x' = X_0(x) + \sum u_i X_i(x)$ the bracket condition only gives that the reachable sets have nonempty interior, not controllability.
</document>

<document source="https://tamnd.github.io/math2/node/VQQM4Y550T.md" tag="VQQM4Y550T" kind="concept">
# Null set

## Summary

A null set in a measure space is a measurable set of measure zero.
Null sets are the sets that measure theory can ignore: changing a function on a null set does not change its integral.
For Lebesgue measure, every countable set is null, and so is the Cantor set.

## Explanations

### L1 human

A null set is a set that is too thin to have any size.
A single point on the line has length zero, and so does any countable list of points.
Even some uncountable sets, such as the Cantor set, have length zero.
In measure theory, what happens on a null set does not matter.
</document>

<document source="https://tamnd.github.io/math2/node/VQXDB9T838.md" tag="VQXDB9T838" kind="statement">
# Kolmogorov–Arnold representation theorem

## Statement

Let $n \ge 2$.
There are [[continuous functions|0X3WGX95FE]] $\varphi_{q,p} \colon [0, 1] \to \R$, for $0 \le q \le 2n$ and $1 \le p \le n$, with the following property.
For every continuous function $f \colon [0, 1]^n \to \R$ there are continuous functions $\Phi_q \colon \R \to \R$, for $0 \le q \le 2n$, such that
$$f(x_1, \dots, x_n) = \sum_{q=0}^{2n} \Phi_q\left( \sum_{p=1}^{n} \varphi_{q,p}(x_p) \right) \quad \text{for all } (x_1, \dots, x_n) \in [0, 1]^n.$$

## Remarks

The inner functions $\varphi_{q,p}$ do not depend on $f$; only the outer functions $\Phi_q$ do.
The inner functions can be chosen increasing.
So the only genuine function of several variables needed to build every continuous function of $n$ variables is addition.
Example: for $f(x, y) = xy$ on $[0, 1]^2$ there is an explicit formula of the same kind, $xy = \frac{1}{4}\left((x + y)^2 - (x - y)^2\right)$, with fewer terms than the theorem uses.
In Hilbert's thirteenth problem, Hilbert conjectured that some continuous functions of three variables are not superpositions of continuous functions of two variables.
The theorem refutes this conjecture: every continuous function of several variables is a superposition of continuous functions of one variable and addition.

## Explanations

### L1 human

Any continuous function of many variables can be built from functions of one variable and the operation of adding numbers.
First apply fixed one-variable functions to each coordinate and add the results; then apply one more one-variable function and add again.
The fixed inner functions work for every target function at once.

### L2 human

The theorem belongs to the study of superpositions of functions.
It contrasts with results of Vitushkin for smooth functions: superpositions of $C^k$ functions of fewer variables do not give every $C^k$ function of $n$ variables.
The inner functions are continuous but quite irregular, so the representation is not a practical way to compute $f$, though it has inspired work on approximation by networks.
</document>

<document source="https://tamnd.github.io/math2/node/VR7RHP41M0.md" tag="VR7RHP41M0" kind="concept">
# Steenrod square

## Summary

The Steenrod squares $\operatorname{Sq}^i$, $i \ge 0$, are [[cohomology operations|K8T15QHN3W]] on mod $2$ cohomology that raise degree by $i$.
They are additive, stable under suspension, and extend the cup square: $\operatorname{Sq}^n x = x \smile x$ when $x$ has degree $n$.
They generate the [[Steenrod algebra|JWTTXSGD0E]] and give the mod $2$ cohomology of a space the structure of a module over it.
They detect maps that cup products miss, such as the suspensions of the Hopf maps, and they compute [[Stiefel–Whitney classes|JQCK7A210W]] through [[Wu's formula|7KEDZ9TFEE]].

## Explanations

### L1 human

In mod $2$ cohomology, squaring a class with the cup product doubles its degree.
Steenrod squares fill in the steps between: $\operatorname{Sq}^i$ adds $i$ to the degree.
They start with the identity at $i = 0$ and end with the cup square when $i$ equals the degree.
For the complex projective plane, $\operatorname{Sq}^2$ of the degree $2$ generator is the nonzero top class, and this shows that the Hopf map $S^3 \to S^2$ is not null-homotopic.
</document>

<document source="https://tamnd.github.io/math2/node/VR89YHC0TX.md" tag="VR89YHC0TX" kind="concept">
# Eigenvalue perturbation

## Summary

Eigenvalue perturbation studies how the [[eigenvalues|0BQAA2BK3Q]] and [[eigenvectors|0N6960W0BV]] of a matrix change when the matrix is changed by a small amount.
For a simple eigenvalue, the change is smooth, and its first-order term is given by left and right eigenvectors.
Bounds such as [[Weyl's inequality|RACSNQXDQV]] and the [[Bauer–Fike theorem|AK9J4RNZ51]] control the change without expansions.
The subject matters for numerical computation, where matrices are known only up to rounding errors.
It also matters in physics, where it is the base of perturbation theory.

## Explanations

### L1 human

Change the entries of a matrix a little and ask how its eigenvalues move.
For a symmetric matrix they move by at most the size of the change.
For example, the eigenvalues of $\begin{pmatrix} 1 & \varepsilon \\ \varepsilon & 2 \end{pmatrix}$ are $1 - \varepsilon^2 + \dots$ and $2 + \varepsilon^2 + \dots$, so they barely move.
For some non-symmetric matrices a tiny change can move the eigenvalues a lot.
</document>

<document source="https://tamnd.github.io/math2/node/VRBP76MM84.md" tag="VRBP76MM84" kind="statement">
# Operator Cauchy equation and one-parameter semigroups

## Statement

Let $X$ be a [[Banach space|06GZ956657]] and let $\mathcal{L}(X)$ be the space of bounded linear operators on $X$ with the [[operator norm|06EKW1AKXF]].
Let $T \colon [0, \infty) \to \mathcal{L}(X)$ satisfy:

1. $T(0) = I$, the identity operator;
2. $T(s + t) = T(s) T(t)$ for all $s, t \ge 0$;
3. $\lim_{t \to 0^+} \norm{T(t) - I} = 0$.

Then there is a unique $A \in \mathcal{L}(X)$ such that $T(t) = e^{tA}$ for all $t \ge 0$, where $e^{tA}$ is the [[exponential|KX3QRA3CEP]] of $tA$.
Conversely, for every $A \in \mathcal{L}(X)$ the map $t \mapsto e^{tA}$ satisfies conditions 1 to 3.

## Remarks

Conditions 2 and 3 make $T$ continuous in operator norm on $[0, \infty)$.
Let $V(t) = \int_0^t T(s) \, ds$; since $V(t)/t \to I$ in norm, $V(t_0)$ is invertible for a small $t_0 > 0$.
The functional equation gives $(T(h) - I) V(t_0) = V(t_0 + h) - V(h) - V(t_0)$, and dividing by $h$ shows that $T$ is differentiable at $0$ with derivative $A = (T(t_0) - I) V(t_0)^{-1}$.
Then $T'(t) = A T(t)$, and uniqueness for this linear equation gives $T(t) = e^{tA}$.
The operator $A$ is the [[generator|60HSWZBTK9]] of the semigroup.
For $X = \R^n$ this says that the continuous solutions $F \colon [0, \infty) \to \R^{n \times n}$ of $F(s + t) = F(s) F(t)$ with $F(0) = I$ are exactly $F(t) = e^{tA}$.
Condition 3 cannot be weakened to strong continuity.
The left shifts $(T(t)f)(s) = f(s + t)$ on $L^2(\R)$ form a [[$C_0$-semigroup|EVF180QJXY]] with the unbounded generator $f \mapsto f'$, so they are not of the form $e^{tA}$ with $A$ bounded.
Condition 1 is needed: for a projection $P \ne I$, the constant map $T(t) = P$ satisfies condition 2 and is norm continuous, but $T(0) = P \ne e^{0A}$.

## Explanations

### L1 human

The only continuous real functions with $f(s + t) = f(s) f(t)$ and $f(0) = 1$ are $f(t) = e^{at}$.
This theorem is the same fact for operators: a family of operators that composes like exponentials, and moves continuously in norm, is an exponential $e^{tA}$.
The operator $A$ is the rate of change of the family at time $0$.

### L2 human

The theorem identifies the norm-continuous one-parameter semigroups with the bounded operators $A$.
Many evolution equations, such as the heat equation, give semigroups that are only strongly continuous, and then the generator is an unbounded operator.
For those semigroups the Hille–Yosida theorem takes the place of this result.
</document>

<document source="https://tamnd.github.io/math2/node/VRP0HESPYD.md" tag="VRP0HESPYD" kind="definition">
# Lie pseudogroup

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] and let $k \ge 0$ be an integer.
Let $J^k_\times(M)$ be the set of $k$-[[jets|1QE0Z0756D]] $j^k_x f$ of local [[diffeomorphisms|0KZZ70ZHXP]] $f$ of $M$ at points $x$ of their domains.
It is a [[Lie groupoid|HNVW0E732J]] over $M$, with composition of jets as multiplication.
Let $\Gamma$ be a [[pseudogroup|3017J7P6JD]] of local diffeomorphisms of $M$.
It is a *Lie pseudogroup of order $k$* if there is a subgroupoid $R \subseteq J^k_\times(M)$ with these properties.

1. $R$ is a closed embedded submanifold of $J^k_\times(M)$.
2. A local diffeomorphism $f \colon U \to V$ of $M$ lies in $\Gamma$ if and only if $j^k_x f \in R$ for every $x \in U$.

## Remarks

The condition $j^k_x f \in R$ for all $x$ is a system of partial differential equations of order $k$ for $f$, and $\Gamma$ is its set of local invertible solutions.
Definitions in the literature differ in details, for example in requiring the equations to be analytic or involutive.

Example: let $g$ be a Riemannian metric on $M$, and let $R \subseteq J^1_\times(M)$ be the set of $1$-jets whose derivative $T_x M \to T_y M$ is a linear isometry from $g_x$ to $g_y$.
It is a closed subgroupoid and a submanifold, and a local diffeomorphism has all its $1$-jets in $R$ exactly when it is a local isometry.
So local isometries form a Lie pseudogroup of order $1$.
In the same way, area-preserving maps of $\R^2$ are the local diffeomorphisms whose $1$-jets satisfy $\det Df = 1$.

Non-example: the pseudogroup of all local homeomorphisms of $\R^n$ is not a Lie pseudogroup.
It contains maps that are not smooth, so it is not even a pseudogroup of local diffeomorphisms.

## Explanations

### L1 human

A Lie pseudogroup is the set of all local invertible solutions of a system of differential equations.
Examples are the maps that preserve a metric or an area form.
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# Askey scheme

## Definition

The *Askey scheme* is a directed graph whose vertices are families of hypergeometric orthogonal polynomials.
The vertices are grouped in levels by the type of hypergeometric series that defines them:

1. ${}_4F_3$: the [[Wilson|1CPF7SWQ3S]] and [[Racah polynomials|P6JYNDX5DR]].
2. ${}_3F_2$: the continuous dual Hahn, continuous Hahn, [[Hahn|Y1M82VZYS2]] and dual Hahn polynomials.
3. ${}_2F_1$: the Meixner–Pollaczek, [[Jacobi|TRD7DSF2KR]], [[Meixner|CEC3DESCFT]] and [[Krawtchouk polynomials|CJGKW80KBR]].
4. ${}_1F_1$ and ${}_2F_0$: the [[Laguerre|WAMRHWW82R]], [[Bessel|M5NXATQ48S]] and [[Charlier polynomials|6KH8KVYS3R]].
5. ${}_2F_0$: the [[Hermite polynomials|MGACY98E9H]].

There is an arrow from a family $A$ to a family $B$ when the polynomials of $B$ are limits of rescaled polynomials of $A$.
In such a limit, parameters of $A$ tend to special values or to infinity.

## Remarks

Example of an arrow, Jacobi to Laguerre: $\lim_{\beta \to \infty} P_n^{(\alpha, \beta)}\left( 1 - \frac{2x}{\beta} \right) = L_n^{(\alpha)}(x)$.
Check for $n = 1$: $P_1^{(\alpha, \beta)}(1 - 2x/\beta) = (\alpha + 1) - (\alpha + \beta + 2) \frac{x}{\beta} \to \alpha + 1 - x = L_1^{(\alpha)}(x)$.
Example of an arrow, Krawtchouk to Charlier: $\lim_{N \to \infty} K_n\left( x; \frac{a}{N}, N \right) = C_n(x; a)$; for $n = 1$ both sides equal $1 - \frac{x}{a}$.
Special cases inside a vertex, such as the Gegenbauer, Legendre and Chebyshev polynomials inside the Jacobi family, are usually not drawn as separate vertices.
Non-example: the Bernoulli polynomials are not in the scheme, because they are not orthogonal for any positive measure on $\R$.
Among Appell sequences, only the Hermite polynomials, up to rescaling and shifting the variable, are orthogonal.
The $q$-analogue of the scheme, with the [[Askey–Wilson polynomials|RSNZNG0WA1]] at the top, is called the $q$-Askey scheme.
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# Periodic solution

## Definition

Let $U$ be an open subset of $\R^n$ and let $f \colon \R \times U \to \R^n$ be continuous.
A [[solution|07VX34QXAV]] $x \colon \R \to U$ of $x' = f(t, x)$ is periodic if there is a real number $T > 0$ with
$$x(t + T) = x(t) \quad \text{for all } t \in \R.$$
Such a $T$ is a period of $x$.
If $f$ is $T$-periodic in $t$, that is, $f(t + T, x) = f(t, x)$ for all $t$ and $x$, a $T$-periodic solution is a periodic solution with period $T$.

## Remarks

A periodic solution is a [[periodic function|0ATF6Q0633]] of $t$ that solves the equation.
For an autonomous equation $x' = f(x)$, each constant solution is periodic with every period, and the trajectory of a nonconstant periodic solution is a [[periodic orbit|0AA97XA70K]].
Example: for $x' = -x + \cos t$, the function $x(t) = \frac{1}{2}(\cos t + \sin t)$ has $x'(t) = \frac{1}{2}(\cos t - \sin t) = -x(t) + \cos t$.
It is a $2\pi$-periodic solution of an equation with $2\pi$-periodic right-hand side.
Non-example: the other solutions $x(t) = \frac{1}{2}(\cos t + \sin t) + c e^{-t}$ with $c \ne 0$ are not periodic, because $c e^{-t}$ is not.
A maximal solution of a $T$-periodic equation with $x(t_0 + T) = x(t_0)$ for one $t_0$ is $T$-periodic when solutions are unique.

## Explanations

### L1 human

A periodic solution returns to the same state after a fixed time and repeats forever.
For a periodically forced system, the most interesting periodic solutions have the same period as the forcing.
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<document source="https://tamnd.github.io/math2/node/VRVVKMXDGV.md" tag="VRVVKMXDGV" kind="concept">
# Uniform integrability

## Summary

A family of integrable functions is uniformly integrable when the part of each function above a large level carries uniformly little integral.
It prevents mass from escaping into tall spikes, in the same way for every member of the family.
On a finite measure space it is exactly the condition that upgrades convergence almost everywhere or in measure to convergence in $L^1$.
It plays a central role in probability, for instance in martingale convergence.

## Explanations

### L1 human

A single integrable function puts only a small part of its integral on the set where it is very large.
Uniform integrability asks this for a whole family at once, with the same threshold for all members.
For example, all functions bounded by the constant $1$ on $[0, 1]$ form a uniformly integrable family.
The spikes $n \mathbf{1}_{(0, 1/n)}$ do not, since each one puts all of its integral $1$ above the level $n - 1$.
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<document source="https://tamnd.github.io/math2/node/VRZDAWNTCR.md" tag="VRZDAWNTCR" kind="definition">
# Sheaf cohomology in topology

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $\mathrm{Sh}(X)$ be the category of [[sheaves|TB6TM6R4MD]] of [[abelian groups|082GHNPJ53]] on $X$.
The global sections functor $\Gamma(X, -) \colon \mathrm{Sh}(X) \to \mathrm{Ab}$, $F \mapsto F(X)$, is left exact, and $\mathrm{Sh}(X)$ has enough injective objects.
For a sheaf $F$, choose an [[injective resolution|ZW3W6FX4PW]] $0 \to F \to I^0 \to I^1 \to I^2 \to \cdots$ in $\mathrm{Sh}(X)$.
The *$q$-th cohomology group of $X$ with coefficients in $F$* is
$$H^q(X, F) = H^q\big(\Gamma(X, I^0) \to \Gamma(X, I^1) \to \Gamma(X, I^2) \to \cdots\big).$$
So $H^q(X, -)$ is the $q$-th right [[derived functor|NPMC3K6J2F]] of $\Gamma(X, -)$; it does not depend on the choice of resolution up to canonical isomorphism.

## Remarks

Always $H^0(X, F) = F(X)$, by left exactness of $\Gamma$.
A short exact sequence of sheaves gives a long exact sequence of cohomology groups.

Example: if $X$ is a single point, a sheaf is just an abelian group $A$, and $\Gamma$ is the identity functor, which is exact.
So $H^0(X, A) = A$ and $H^q(X, A) = 0$ for $q > 0$.

Non-example: the rule $U \mapsto$ (constant functions $U \to \Z$) is a presheaf but in general not a sheaf.
Two different constants on two disjoint open sets do not glue to a constant function on their union.
The definition needs a sheaf, here its sheafification, the sheaf of locally constant functions.

In practice one uses acyclic resolutions instead of injective ones: flasque, soft or fine resolutions compute the same groups.
If $X$ is paracompact, Hausdorff and locally contractible, then $H^q(X, \underline{A})$ for the constant sheaf $\underline{A}$ is isomorphic to singular cohomology $H^q(X; A)$.
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# Complete elliptic integral of the second kind

## Summary

The complete elliptic integral of the second kind is the function $E(k) = \int_0^{\pi/2} \sqrt{1 - k^2\sin^2\theta}\,d\theta$ of the modulus $k$.
It gives the perimeter of an ellipse: an ellipse with semi-major axis $a$ and eccentricity $e$ has perimeter $4aE(e)$.
Together with the [[complete integral of the first kind|RM54CPPE12]] it enters [[Legendre's relation|8GSKDNZ4TK]].

## Explanations

### L1 human

A circle of radius $a$ has perimeter $2\pi a$, but there is no such simple formula for an ellipse.
The perimeter of an ellipse is four times the length of a quarter arc, and the quarter arc is $a E(e)$.
For a circle $e = 0$, and $E(0) = \pi/2$ gives back $2\pi a$.
For a flat ellipse squeezed to a segment of length $2a$, $e = 1$ and $E(1) = 1$, so the perimeter is $4a$, the segment traced there and back.
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<document source="https://tamnd.github.io/math2/node/VSC3F53DX4.md" tag="VSC3F53DX4" kind="concept">
# Elliptic surface

## Summary

An elliptic surface is an [[algebraic surface|KFTECAATKG]] fibred over a curve so that almost every fibre is a smooth curve of genus one.
So it is a family of [[elliptic curves|427N6S6QQW]], or of genus-one curves, parametrized by a curve, degenerating over finitely many points.
Kodaira classified the possible singular fibres.
Elliptic surfaces include all minimal surfaces of Kodaira dimension $1$ and many K3 and rational surfaces.
They connect geometry with the arithmetic of elliptic curves over function fields.

## Explanations

### L1 human

Picture a curve, and above each of its points place a doughnut-shaped elliptic curve that varies with the point.
At a few special points the doughnut is pinched or broken into pieces.
The whole family forms a surface.
A simple example comes from the equation $y^2 = x^3 + t$, which for each nonzero value of the parameter $t$ is an elliptic curve and degenerates to a cusp at $t = 0$.
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# Modified Bessel functions

## Definition

Let $\nu \in \C$, let $\Gamma$ be the [[gamma function|0H5SYBSED9]], and let $z \in \C \setminus (-\infty, 0]$ with the principal branch of $z^\nu$.

1. The *modified Bessel function of the first kind* of order $\nu$ is
$$I_\nu(z) = \sum_{m=0}^{\infty} \frac{1}{m!\, \Gamma(m + \nu + 1)} \left( \frac{z}{2} \right)^{2m + \nu},$$
where $\frac{1}{\Gamma(m + \nu + 1)}$ is read as $0$ when $m + \nu + 1$ is a non-positive integer.
2. For $\nu \notin \Z$, the *modified Bessel function of the second kind* of order $\nu$ is
$$K_\nu(z) = \frac{\pi}{2}\, \frac{I_{-\nu}(z) - I_\nu(z)}{\sin(\nu \pi)},$$
and for an integer $n$ it is $K_n(z) = \lim_{\nu \to n} K_\nu(z)$.

## Remarks

Both functions solve the modified Bessel equation $z^2 y'' + z y' - (z^2 + \nu^2) y = 0$.
In terms of the [[Bessel function of the first kind|CZCMRDWDYW]], $I_\nu(z) = i^{-\nu} J_\nu(iz)$ for $-\pi < \arg z \le \frac{\pi}{2}$.
Example: for $x > 0$, $I_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sinh x$ and $K_{1/2}(x) = \sqrt{\frac{\pi}{2x}}\, e^{-x}$.
For $x > 0$ there is the integral form $K_\nu(x) = \int_0^\infty e^{-x \cosh t} \cosh(\nu t)\, dt$.
A non-example: $I_{-n}$ for an integer $n$ is not a second solution, since $I_{-n} = I_n$; this is why $K_n$ is defined as a limit.
$K_\nu$ is also called the Macdonald function.
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# Totally unimodular matrix

## Summary

A totally unimodular matrix is an integer matrix in which every square submatrix has determinant $0$, $1$ or $-1$.
In particular every entry is $0$, $1$ or $-1$.
A linear program whose constraint matrix is totally unimodular and whose right-hand side is integral has integral vertices.
This explains why many problems on graphs, such as bipartite matching and network flows, can be solved exactly by linear programming.

## Explanations

### L1 human

Linear programs are easy to solve, but problems that ask for whole-number answers are usually hard.
For some special constraint matrices, the two coincide: whenever the right-hand side is whole-number and the linear program has a best solution, it has a best solution in whole numbers.
Totally unimodular matrices are exactly such special matrices: every square piece of them has determinant $0$, $1$ or $-1$.
The incidence matrix of a bipartite graph, with a row for each vertex and a column for each edge, is a standard example.
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<document source="https://tamnd.github.io/math2/node/VSGTSQ7RY8.md" tag="VSGTSQ7RY8" kind="concept">
# Transfinite diameter

## Summary

The transfinite diameter of a compact set $K$ in the plane measures how far apart many points of $K$ can be placed at once.
For each $n$ one maximizes the geometric mean of the mutual distances of $n$ points of $K$, and then lets $n$ tend to infinity.
For $n = 2$ this is the ordinary diameter, which explains the name.
A theorem of Fekete and Szegő says that the transfinite diameter equals the [[logarithmic capacity|DBQXRTC0VN]] of $K$.

## Explanations

### L1 human

Place $n$ points in $K$ so that they are as spread out as possible, in the sense that the product of all their mutual distances is largest.
Take the geometric mean of these distances.
For two points this is just the largest distance in $K$, the diameter.
As $n$ grows, the points must crowd together, and the mean distance decreases to a limit, the transfinite diameter.
For the unit circle the best points are the $n$-th roots of unity, and the limit is $1$.
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<document source="https://tamnd.github.io/math2/node/VSNQV29KFJ.md" tag="VSNQV29KFJ" kind="concept">
# Interval order

## Summary

An interval order is a partial order whose elements can be drawn as [[intervals|0Q1D85B8G6]] of the real line.
One element is below another exactly when its interval lies entirely to the left.
Interval orders model events in time and preferences with a threshold of indifference.
A finite poset is an interval order exactly when it contains no copy of two incomparable two-element chains, a result of Fishburn.

## Explanations

### L1 human

Give each element a time slot, such as a meeting from $9$ to $10$ o'clock.
Say that one meeting comes before another when it ends before the other starts.
Overlapping meetings are then incomparable.
For the slots $a = [0, 1]$, $b = [2, 3]$ and $c = [0.5, 2.5]$, the only relation is $a < b$, while $c$ overlaps both.
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# Numerical differentiation

## Definition

Let $k \ge 1$ be an integer, let $f$ be a real function defined near a point $x \in \R$, and let $s_1, \ldots, s_N$ be distinct real numbers.
A *numerical differentiation formula* for the [[$k$-th derivative|04TKTF8KDK]] $f^{(k)}(x)$ is an expression
$$D_h f(x) = \frac{1}{h^k} \sum_{j=1}^{N} a_j \, f(x + s_j h), \qquad h > 0,$$
with real coefficients $a_1, \ldots, a_N$ that do not depend on $h$ or $f$.
It has *order* $p \ge 1$ if $D_h f(x) - f^{(k)}(x) = O(h^p)$ as $h \to 0$ for every $f$ of class $C^{k+p}$ near $x$, in [[big O notation|0TGV80E6A4]].
*Numerical differentiation* is the approximation of $f^{(k)}(x)$ by such formulas, or more generally by derivatives of functions fitted to finitely many values of $f$.

## Remarks

Example: the central difference $D_h f(x) = \frac{f(x + h) - f(x - h)}{2h}$ for $k = 1$, with stencil $-1, 1$ and $a = (-\tfrac12, \tfrac12)$, has order $2$.
Check with Taylor's theorem for $f \in C^3$: $f(x \pm h) = f(x) \pm h f'(x) + \frac{h^2}{2} f''(x) \pm \frac{h^3}{6} f'''(\xi_\pm)$, so $D_h f(x) - f'(x) = O(h^2)$.
The forward difference $\frac{f(x + h) - f(x)}{h}$ has only order $1$, with error $\frac{h}{2} f''(\xi)$.
Non-example: $\frac{f(x+h) + f(x)}{h}$ approximates no derivative; for $f = 1$ it equals $\frac{2}{h}$, which does not tend to $f'(x) = 0$.
In floating-point arithmetic with unit round-off $u$, the forward difference has total error about $\frac{h}{2}\abs{f''} + \frac{2u\abs{f}}{h}$, which is smallest for $h$ of the order of $\sqrt{u}$.

## Explanations

### L1 human

A numerical differentiation formula combines a few nearby values of a function to estimate one of its derivatives.
The order says how fast the error shrinks when the spacing $h$ shrinks: order $2$ means that halving $h$ divides the error by about $4$.
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# Sine and cosine addition equations

## Definition

The *sine and cosine addition equations* are the system of [[functional equations|SRAN4EKWS0]]
$$S(x + y) = S(x) C(y) + C(x) S(y), \qquad C(x + y) = C(x) C(y) - S(x) S(y) \quad \text{for all } x, y \in \R,$$
for two unknown [[functions|0EPAT51DXQ]] $S, C \colon \R \to \R$.
The first equation is the *sine addition equation* and the second is the *cosine addition equation*.
A solution is a pair $(S, C)$ that satisfies both identities for all real $x$ and $y$.

## Remarks

Example: $S = \sin$ and $C = \cos$ form a solution, by the angle addition formulas of [[sine and cosine|08WFZP6P89]].
Non-example: $S(x) = x$, $C(x) = 1$ is not a solution, since the cosine equation gives $1 = 1 - xy$, which fails for $x = y = 1$.
Set $E = C + iS \colon \R \to \C$.
The two equations together say exactly that $E(x + y) = E(x) E(y)$.
So the continuous solutions are $S = C = 0$ and $S(x) = e^{bx} \sin(ax)$, $C(x) = e^{bx} \cos(ax)$ with $a, b \in \R$.
Each of the two equations is also studied alone, with one or both functions unknown.
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<document source="https://tamnd.github.io/math2/node/VT2DVEEVRH.md" tag="VT2DVEEVRH" kind="definition">
# Polish space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *Polish* if:

1. $X$ is [[separable|0Z2P41SVDP]]: it has a countable dense subset.
2. $X$ is completely metrizable: there is a [[metric|0VKD1TP7F3]] $d$ on $X$ that induces the topology of $X$ and makes $(X, d)$ a [[complete metric space|0BY52JW6NK]].

## Remarks

The real line $\R$ is Polish: the rationals are dense, and the usual metric is complete.
The open interval $(0, 1)$ is Polish, because the metric $d(x, y) = \abs{\tan(\pi x - \pi/2) - \tan(\pi y - \pi/2)}$ induces its topology and is complete.
The rationals $\Q$ are a non-example: they are separable and metrizable, but no complete metric induces their topology.
Indeed, by the Baire category theorem a nonempty complete metric space is not a countable union of nowhere dense sets, while $\Q$ is the countable union of its points.
An uncountable discrete space is another non-example: it is completely metrizable but not separable.
A subset of a Polish space is Polish in the subspace topology if and only if it is a $G_\delta$ set.
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# Entourage

## Definition

Let $(X, \mathcal{U})$ be a [[uniform space|A54T86KS63]].
An *entourage* of $X$ is a member of the uniformity $\mathcal{U}$, that is, a subset $U \subseteq X \times X$ with $U \in \mathcal{U}$.
For an entourage $U$ and a point $x \in X$, the set $U[x] = \set{y \in X : (x, y) \in U}$ is the $U$-neighbourhood of $x$.
An entourage $U$ is *symmetric* if $(x, y) \in U$ implies $(y, x) \in U$.

## Remarks

On $\R$ with the uniformity of the metric $\abs{x - y}$, the set $\set{(x, y) : \abs{x - y} < 1}$ is an entourage, and $U[x] = (x - 1, x + 1)$.
The set $\set{(x, y) : \abs{x - y} < 1 / (1 + x^2)}$ contains the diagonal, but it is not an entourage of that uniformity.
It contains no set $\set{(x, y) : \abs{x - y} < \varepsilon}$ with $\varepsilon > 0$, because its width tends to $0$ as $\abs{x}$ grows.
Every entourage contains a symmetric entourage, namely $U \cap U^{-1}$.
The sets $U[x]$, for $U$ an entourage, form a neighbourhood base at $x$ of the uniform topology.

## Explanations

### L1 human

An entourage is a set of pairs of points that a uniform space counts as close.
Picture the plane $X \times X$; an entourage is a region that contains the whole diagonal $y = x$.
For $\R$ with its usual distance, the region between the lines $y = x - 1$ and $y = x + 1$ is an entourage.
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# Jónsson's lemma

## Statement

Let $\mathcal{K}$ be a class of algebras of one signature.
Assume that the variety $\mathcal{V} = \mathbf{HSP}(\mathcal{K})$ is a [[congruence-distributive variety|9GZ7F6ETC6]].
Then every [[subdirectly irreducible algebra|DASF939439]] in $\mathcal{V}$ belongs to $\mathbf{HSP}_U(\mathcal{K})$.
Here $\mathbf{H}$, $\mathbf{S}$ and $\mathbf{P}_U$ are the [[class operators|S7MJVSP79N]] of homomorphic images, subalgebras and [[ultraproducts|AX9Q86GB8M]].
In particular, if $\mathcal{K}$ is a finite set of finite algebras, then every subdirectly irreducible algebra in $\mathcal{V}$ belongs to $\mathbf{HS}(\mathcal{K})$.

## Remarks

A subdirectly irreducible $B \in \mathcal{V}$ has the form $C / \psi$, where $C$ is a subalgebra of a product $\prod_{i \in I} A_i$ with $A_i \in \mathcal{K}$.
For each subset $J \subseteq I$, the kernel of the projection of $C$ onto $\prod_{i \in J} A_i$ is a congruence $\eta_J$ of $C$.
Distributivity of the congruence lattice of $C$ and the irreducibility of $\psi$ show that the sets $J$ with $\eta_J \le \psi$ contain an ultrafilter $U$ on $I$.
Then $\psi$ contains the congruence induced by $U$, so $B$ is a homomorphic image of a subalgebra of the ultraproduct $\prod_{i \in I} A_i / U$.
When $\mathcal{K}$ is a finite set of finite algebras, every ultraproduct of members of $\mathcal{K}$ is isomorphic to a member of $\mathcal{K}$, which gives the second claim.

Example: the variety of distributive lattices is generated by the two-element lattice $\mathbf{2}$.
Lattices are congruence distributive, so every subdirectly irreducible distributive lattice lies in $\mathbf{HS}(\mathbf{2})$.
The only such lattice with more than one element is $\mathbf{2}$ itself.

## Explanations

### L1 human

Jónsson's lemma controls the building blocks of a variety whose congruences are distributive.
The irreducible algebras of the variety cannot be new: they all come from the generating algebras by ultraproducts, subalgebras and quotients.
For a variety generated by one finite algebra, they are quotients of subalgebras of that algebra, so up to isomorphism there are only finitely many of them.

### L2 human

By Birkhoff's subdirect representation theorem, a variety is determined by its subdirectly irreducible members, so Jónsson's lemma describes the whole variety.
It gives a description of the [[lattice of subvarieties|GJAYCVYZYF]] of a congruence-distributive variety: each subvariety is generated by its subdirectly irreducible members.
For example, for a finite lattice $L$, the subvarieties of $\mathbf{HSP}(L)$ correspond to the down-sets of the finitely many subdirectly irreducible lattices in $\mathbf{HS}(L)$, up to isomorphism.
Here $M$ lies below $N$ when $M \in \mathbf{HS}(N)$.
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# Braided monoidal category

## Summary

A braided monoidal category is a [[monoidal category|M3MSQ3NKAE]] with a natural isomorphism $A \otimes B \to B \otimes A$, the braiding, compatible with the associator.
The braiding need not square to the identity, and its behaviour is modelled by the [[braid groups|BNR4H5GCTP]] rather than the symmetric groups.
Braided monoidal categories arise from representations of quantum groups and give invariants of knots and links.
When the braiding squares to the identity the category is symmetric.

## Explanations

### L1 human

In a braided monoidal category two objects placed side by side can swap places.
Picture each object as a string: swapping them makes one string cross over the other.
Crossing over and crossing under are different moves, so swapping twice need not bring things back to the start.
For vector spaces the flip $v \otimes w \mapsto w \otimes v$ is a braiding, and there swapping twice does give the identity.
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<document source="https://tamnd.github.io/math2/node/VT98YJJJ47.md" tag="VT98YJJJ47" kind="definition">
# Grunsky matrix

## Definition

Let $\Delta = \set{z \in \C : \abs{z} > 1}$.
Let $g$ be an injective meromorphic function on $\Delta \cup \set{\infty}$ with a simple pole at $\infty$ and [[Laurent expansion|0X3XD2ATB0]]
$$g(z) = z + b_0 + \sum_{n=1}^{\infty} b_n z^{-n}, \qquad \abs{z} > 1.$$
The function $Q(z, \zeta) = \dfrac{g(z) - g(\zeta)}{z - \zeta}$ for $z \ne \zeta$, with $Q(z, z) = g'(z)$, is holomorphic and nowhere zero on $\Delta \times \Delta$ and tends to $1$ as $z, \zeta \to \infty$.
So it has a holomorphic [[logarithm|0X2FY491PF]] with value $0$ at $(\infty, \infty)$, with an expansion
$$\log \frac{g(z) - g(\zeta)}{z - \zeta} = -\sum_{m, n \ge 1} c_{mn}\, z^{-m} \zeta^{-n}, \qquad \abs{z}, \abs{\zeta} > 1.$$
The numbers $c_{mn}$ are the *Grunsky coefficients* of $g$, and the infinite matrix $(c_{mn})_{m, n \ge 1}$ is the *Grunsky matrix* of $g$.

## Remarks

The Grunsky matrix is symmetric: $c_{mn} = c_{nm}$, because $Q(z, \zeta) = Q(\zeta, z)$.
The coefficients are polynomials in $b_1, b_2, \ldots$; for example $c_{11} = b_1$.
The Grunsky inequalities say that for every finite sequence $\lambda_1, \ldots, \lambda_N \in \C$,
$$\Bigl| \sum_{m, n = 1}^{N} \sqrt{mn}\, c_{mn} \lambda_m \lambda_n \Bigr| \le \sum_{n=1}^{N} \abs{\lambda_n}^2.$$
Example: $g(z) = z + 1/z$ is injective on $\Delta$, since $g(z) = g(w)$ with $z \ne w$ forces $zw = 1$.
Here $Q(z, \zeta) = 1 - 1/(z\zeta)$, so $\log Q = -\sum_{n \ge 1} \frac{1}{n} (z\zeta)^{-n}$, and $c_{nn} = 1/n$ while $c_{mn} = 0$ for $m \ne n$.
The weighted matrix $(\sqrt{mn}\, c_{mn})$ is the identity, so the Grunsky inequalities reduce to $\abs{\sum_n \lambda_n^2} \le \sum_n \abs{\lambda_n}^2$, with equality for real $\lambda_n$.
Non-example: $g(z) = z + 2/z$ is not injective on $\Delta$, since $g(i\sqrt{2}) = g(-i\sqrt{2}) = 0$.
Then $Q(z, \zeta) = 1 - 2/(z\zeta)$ vanishes at points of $\Delta \times \Delta$, and no holomorphic logarithm exists there.
Sign and normalisation conventions differ between authors; some use $\log\frac{g(z) - g(\zeta)}{z - \zeta}$ with a plus sign or with the factors $\sqrt{mn}$ built in.
</document>

<document source="https://tamnd.github.io/math2/node/VT9QWDTKA4.md" tag="VT9QWDTKA4" kind="definition">
# Normalization

## Definition

Let $X$ be an [[integral scheme|17NVR4XBHK]] with function field $K$.
For each open affine subscheme $U = \operatorname{Spec} A$ of $X$, let $\tilde{A}$ be the [[integral closure|G0V86SJBQW]] of $A$ in its [[field of fractions|0F2PTGX1A2]] $K$.
The *normalization* of $X$ is the integral scheme $\tilde{X}$ with a [[morphism|JXRKH6P1Z4]] $\nu \colon \tilde{X} \to X$ such that, for every such $U$:

1. $\nu^{-1}(U)$ is isomorphic to $\operatorname{Spec} \tilde{A}$;
2. under this isomorphism, $\nu$ restricted to $\nu^{-1}(U)$ is the morphism $\operatorname{Spec} \tilde{A} \to \operatorname{Spec} A$ induced by the inclusion $A \subseteq \tilde{A}$.

It is obtained by [[gluing|ABVBHCA9VW]] the schemes $\operatorname{Spec} \tilde{A}$, and it is unique up to unique isomorphism over $X$.

## Remarks

The scheme $\tilde{X}$ is [[normal|EK261G1XP9]].
For every normal integral scheme $Z$ and every dominant morphism $f \colon Z \to X$, there is a unique morphism $g \colon Z \to \tilde{X}$ with $f = \nu \circ g$.
Example: let $A = k[x, y]/(y^2 - x^3) \cong k[t^2, t^3]$ over a field $k$, with $x = t^2$ and $y = t^3$.
Its fraction field is $k(t)$, and $t$ is integral over $A$ since $t^2 = x$; also $k[t]$ is integrally closed, so $\tilde{A} = k[t]$.
The normalization of the cusp is $\mathbb{A}^1_k \to \operatorname{Spec} A$, $t \mapsto (t^2, t^3)$.
Non-example: the identity morphism of the cusp is not its normalization, since the cusp is not normal and $A \ne \tilde{A}$.
If $X$ is normal, then $\nu$ is an isomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/VTE2H1GT4A.md" tag="VTE2H1GT4A" kind="example">
# Cantor algebra

## Example

Let $C = \set{0, 1}^{\N}$ be the [[Cantor space|W9VYJRBF9N]] of infinite binary sequences, with the product topology.
The *countable Cantor algebra* is the [[Boolean algebra|0XADJKM699]] of all clopen subsets of $C$, with union, intersection and complement in $C$.

For a finite binary word $w$, let $[w]$ be the set of sequences that begin with $w$.
Each set $[w]$ is clopen, and every clopen subset of $C$ is a finite union of such sets, by compactness of $C$.

Check the main properties.

1. It is [[countable|01TQB88H4C]]. There are countably many finite words, hence countably many finite unions of sets $[w]$.
2. It has no atoms. A nonempty clopen set contains some $[w]$. The set $[w]$ is the disjoint union of the nonempty sets $[w0]$ and $[w1]$, so it is not minimal.
3. It is free on the generators $G_n = \set{s \in C : s_n = 1}$ for $n \in \N$. Each $[w]$ is an intersection of finitely many sets $G_n$ and complements of sets $G_n$, so these sets generate the algebra. Any such intersection over distinct indices $n$ is nonempty. So no nontrivial Boolean relation holds among the $G_n$.

## Remarks

Up to isomorphism, the countable Cantor algebra is the only countable atomless Boolean algebra with more than one element.
The name Cantor algebra is also used for its completion, the *complete Cantor algebra*.
The completion is isomorphic to the algebra of Borel subsets of $\R$ modulo meagre sets.
A finite Boolean algebra with more than one element always has atoms, so it is a non-example of an atomless algebra.

## Explanations

### L1 human

Think of an infinite sequence of coin tosses.
A clopen set is an event that can always be decided after a fixed finite number of tosses, such as "the first two tosses are heads".
These events form a Boolean algebra under "or", "and" and "not".
Every nonempty event can be split into two smaller nonempty events by looking at one more toss, so there is no smallest event.
The events "toss $n$ is heads" are independent generators, and every event is built from finitely many of them.
</document>

<document source="https://tamnd.github.io/math2/node/VTFQYCP1WA.md" tag="VTFQYCP1WA" kind="definition">
# Tensor algebra

## Definition

Let $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
For $k \ge 1$, let $T^k(V) = V \otimes \cdots \otimes V$ be the [[tensor product|0BF2DE2SYK]] of $k$ copies of $V$, and let $T^0(V) = F$.
The *tensor algebra* of $V$ is the vector space
$$T(V) = \bigoplus_{k \ge 0} T^k(V),$$
with the bilinear multiplication determined by
$$(v_1 \otimes \cdots \otimes v_k) \cdot (w_1 \otimes \cdots \otimes w_l) = v_1 \otimes \cdots \otimes v_k \otimes w_1 \otimes \cdots \otimes w_l$$
for $v_i, w_j \in V$, and with unit $1 \in T^0(V)$.

## Remarks

$T(V)$ is an associative unital algebra over $F$, graded by the degree $k$, and $V = T^1(V)$ sits inside it.
Universal property: for every associative unital $F$-algebra $A$ and every linear map $f \colon V \to A$, there is exactly one unital algebra homomorphism $\bar f \colon T(V) \to A$ with $\bar f(v) = f(v)$ for $v \in V$.
Example: if $V = F x$ has dimension $1$, then $T^k(V) = F x^{\otimes k}$, and $x^{\otimes k} \mapsto x^k$ is an isomorphism $T(V) \cong F[x]$; the check is that both products add degrees.
Non-example: for $V = F^2$ with basis $x, y$, the polynomial ring $F[x, y]$ is not $T(V)$, because $x \otimes y \ne y \otimes x$ in $T(V)$ while $xy = yx$ in $F[x, y]$.
If $V$ has a basis $(e_i)_{i \in I}$, then $T(V)$ is the free associative algebra on the set $I$.
The same construction works for a module over a commutative ring.
</document>

<document source="https://tamnd.github.io/math2/node/VTGYEES3VS.md" tag="VTGYEES3VS" kind="concept">
# Isolated hypersurface singularity

## Summary

An isolated hypersurface singularity is the germ at a point of the zero set of a holomorphic function $f$ of $n + 1$ variables, see the [[definition|4KAWK0FD4V]].
The gradient of $f$ vanishes at the point but nowhere else nearby.
It is the basic object of the local theory of [[singularities|42V5XGZS4F]] of complex spaces.
Its topology is described by the [[Milnor fibration|KTFG1S2SMG]], the [[link|C831AN3RFM]] and the [[Milnor number|1ZE51P7GRQ]].
It is a special kind of [[singular point|ZW5A88PCPK]] of a [[complex analytic variety|GB7M582M5Y]].

## Explanations

### L1 human

A hypersurface in $\C^{n+1}$ is the set where one holomorphic function is zero.
At most points it is smooth, like a curved sheet.
At an isolated singularity it has a single bad point, such as the tip of a cone, while every nearby point is smooth.
The cone $z_0^2 + z_1^2 + z_2^2 = 0$ in $\C^3$ is the simplest example: its only singular point is the origin.
</document>

<document source="https://tamnd.github.io/math2/node/VTJV7JVJEA.md" tag="VTJV7JVJEA" kind="concept">
# Schröder's equation

## Summary

Schröder's equation asks for a function $\Psi$ with $\Psi(f(x)) = \lambda \Psi(x)$, where the map $f$ and the constant $\lambda$ are given.
A solution is a change of coordinates in which $f$ becomes multiplication by $\lambda$.
It is one of the basic [[iterative functional equations|BCGBQ3HK26]]: once it is solved, every [[iterate|AYVTR2ES5B]] of $f$ has the closed form $f^n = \Psi^{-1} \circ (\lambda^n \Psi)$.
Near a [[fixed point|R6QVS6P21N]] of a holomorphic map, the natural choice of $\lambda$ is the derivative of $f$ at that point.

## Explanations

### L1 human

Iterating a map is often hard, but iterating multiplication by a number is easy: $n$ steps multiply by $\lambda^n$.
Schröder's equation looks for a new coordinate in which the given map is just such a multiplication.
For example, take $f(x) = x^2 + 2x$ on $x > -1$ and $\Psi(x) = \log(1 + x)$.
Then $\Psi(f(x)) = \log\big((1 + x)^2\big) = 2 \Psi(x)$, so in the coordinate $\Psi$ the map $f$ doubles.
This gives at once $f^n(x) = (1 + x)^{2^n} - 1$.
</document>

<document source="https://tamnd.github.io/math2/node/VTN8JHGDS5.md" tag="VTN8JHGDS5" kind="statement">
# Kodaira–Nirenberg–Spencer existence theorem

## Statement

Let $X$ be a compact [[complex manifold|JSGWAN2RJV]], and let $\Theta_X$ be the sheaf of holomorphic vector fields on $X$.
Assume that $H^2(X, \Theta_X) = 0$, and let $m = \dim_\C H^1(X, \Theta_X)$.
Then there is a [[deformation|VKQSM3BCE4]] $\varpi \colon \mathcal{X} \to B$ of $X$ with $X_0 \cong X$, over an open neighbourhood $B$ of $0$ in $\C^m$.
Its [[Kodaira–Spencer map|EWZ1G10MAY]] at $0$,
$$\rho_0 \colon T_0 B \to H^1(X, \Theta_X)$$
is an isomorphism.

## Remarks

The proof writes a deformed complex structure as a vector-valued $(0,1)$-form $\varphi(t)$ with $\varphi(0) = 0$.
Integrability of the new structure is the equation $\bar\partial \varphi + \tfrac12 [\varphi, \varphi] = 0$.
One solves it by a power series in $t$ whose linear term runs over harmonic representatives of $H^1(X, \Theta_X)$.
At each order the obstruction to the next step lies in $H^2(X, \Theta_X)$, which is zero, and estimates for elliptic operators give convergence.
The Newlander–Nirenberg theorem then turns the integrable almost complex structures into complex manifolds.

By the completeness theorem of Kodaira and Spencer, a family with surjective Kodaira–Spencer map at $0$ is complete there.
So the family of the theorem contains all small deformations of $X$.

Example: for a compact Riemann surface of genus $g \ge 2$, $H^2(X, \Theta_X) = 0$ for dimension reasons, and $\dim H^1(X, \Theta_X) = 3g - 3$ by Riemann–Roch and Serre duality.
So $X$ has a complete family with a smooth $(3g-3)$-dimensional base.

Without the hypothesis $H^2(X, \Theta_X) = 0$, the conclusion can fail: the base of a complete family with bijective Kodaira–Spencer map may have to be singular.

## Explanations

### L1 human

The space $H^1(X, \Theta_X)$ lists all the directions in which the complex structure of $X$ could move to first order.
The theorem says that, if a certain obstruction space vanishes, every such direction is realized by an actual family.
So the small deformations of $X$ are parametrized by a ball in $H^1(X, \Theta_X)$.

### L2 human

This is the first general existence result in the deformation theory of complex manifolds, from 1958.
Kuranishi later removed the hypothesis by allowing singular bases: his family lives over the zero set of an obstruction map from $H^1$ to $H^2$.
The theorem applies to curves and to many surfaces, and it is the local model behind moduli spaces of these objects.
It does not apply to complex tori of dimension at least $2$, where $H^2(X, \Theta_X) \ne 0$, although their deformations are still unobstructed.
</document>

<document source="https://tamnd.github.io/math2/node/VTWSNXB5S7.md" tag="VTWSNXB5S7" kind="definition">
# Limit

## Definition

Let $J$ and $\mathcal{C}$ be [[categories|A2DE66937D]] and let $D \colon J \to \mathcal{C}$ be a [[diagram|HJK75HDWFE]].
A *limit* of $D$ is a [[cone|7TPBQPPN97]] $(L, \varphi)$ over $D$, with $\varphi_j \colon L \to D(j)$, that satisfies the following condition.
For every cone $(N, \psi)$ over $D$ there is exactly one morphism $h \colon N \to L$ with $\varphi_j \circ h = \psi_j$ for every object $j$ of $J$.
The object $L$ is written $\lim D$ or $\varprojlim D$, and the $\varphi_j$ are its *projections*.

## Remarks

Equivalently, a limit of $D$ is a [[terminal object|EK85VSSRMR]] of the category of cones over $D$.
So a limit is unique up to a unique isomorphism of cones.
When $J$ is empty, a cone is just an object, and a limit of the empty diagram is a terminal object of $\mathcal{C}$.

Example: in the category of sets, every diagram $D$ with $J$ small has a limit.
It is the subset
$$L = \set{(x_j)_j \in \textstyle\prod_j D(j) : D(u)(x_j) = x_k \text{ for all } u \colon j \to k}$$
of the [[Cartesian product|09CEDJYBKP]] of the sets $D(j)$, with $\varphi_j$ the coordinate projections.
A cone $(N, \psi)$ factors through $L$ by $h(n) = (\psi_j(n))_j$, which lies in $L$ by the cone condition, and $h$ is forced by $\varphi_j \circ h = \psi_j$.

Non-example: take the diagram of two non-empty sets $X$ and $Y$ with no arrows.
The cone $X \times Y \times \{0, 1\}$ with the projections to $X$ and $Y$ is not a limit.
The cone $X \times Y$ factors through it in two ways, by $(x, y) \mapsto (x, y, 0)$ and $(x, y) \mapsto (x, y, 1)$, so uniqueness fails.
</document>

<document source="https://tamnd.github.io/math2/node/VTYJ9419C7.md" tag="VTYJ9419C7" kind="concept">
# Character sum

## Summary

A character sum is a sum of the values of a [[Dirichlet character|7XB5VJKDPG]] over a range of integers, such as $\sum_{M < n \le M + N} \chi(n)$.
Over a full period the sum is $0$ for a nonprincipal character.
So the interest lies in incomplete sums and in sums of $\chi(f(n))$ for polynomials $f$.
Bounds for character sums control the distribution of quadratic residues and of primitive roots, and they enter the study of [[Dirichlet L-functions|9D2Z543EJQ]].

## Explanations

### L1 human

A Dirichlet character assigns to each integer a root of unity or $0$, following a periodic and multiplicative pattern.
For the Legendre symbol modulo a prime $p$, the value is $+1$ on quadratic residues and $-1$ on non-residues.
A character sum adds these values over an interval, and so counts residues minus non-residues there.
A small sum means that residues and non-residues are well mixed in the interval.
</document>

<document source="https://tamnd.github.io/math2/node/VTZPSW07PQ.md" tag="VTZPSW07PQ" kind="definition">
# Empirical process

## Definition

Let $X_1, X_2, \dots$ be [[independent and identically distributed|00MRN0FARV]] real random variables with [[cumulative distribution function|0H8H7ZTHW2]] $F$.
For $n \ge 1$, the *empirical distribution function* is
$$F_n(x) = \frac{1}{n} \sum_{i=1}^{n} \mathbf{1}\{X_i \le x\}, \qquad x \in \R.$$
The *empirical process* is the [[stochastic process|0QH5ENKZM1]] indexed by $x \in \R$ given by
$$G_n(x) = \sqrt{n}\,\bigl(F_n(x) - F(x)\bigr).$$

## Remarks

For each fixed $x$, $n F_n(x)$ is binomial with parameters $n$ and $F(x)$.
So $E[G_n(x)] = 0$ and $\operatorname{Var}(G_n(x)) = F(x)(1 - F(x))$, and by the central limit theorem $G_n(x)$ converges in distribution to a normal law with these parameters.
For the uniform law on $[0, 1]$, [[Donsker's theorem|818832PMSV]] says that $G_n$ converges in distribution, as a random function, to a [[Brownian bridge|3MCVMBAGD1]].
A non-example: if $X_i = X_1$ for all $i$, the variables are not independent, $F_n(x) = \mathbf{1}\{X_1 \le x\}$ for all $n$, and $\sqrt{n}(F_n - F)$ does not have the behaviour above.
More generally, for a probability measure $P$ and a class $\mathcal{F}$ of functions, the empirical process indexed by $\mathcal{F}$ is $f \mapsto \sqrt{n}\bigl(\frac{1}{n}\sum_{i=1}^n f(X_i) - \int f \, dP\bigr)$; the case above takes the indicator functions $\mathbf{1}_{(-\infty, x]}$.
</document>

<document source="https://tamnd.github.io/math2/node/VVT27RB46B.md" tag="VVT27RB46B" kind="statement">
# q-binomial theorem

## Statement

Let $a, q, z \in \C$ with $\abs{q} < 1$ and $\abs{z} < 1$.
Then
$$\sum_{n=0}^{\infty} \frac{(a; q)_n}{(q; q)_n} z^n = \frac{(az; q)_\infty}{(z; q)_\infty},$$
where $(\cdot\,; q)_n$ and $(\cdot\,; q)_\infty$ are [[q-Pochhammer symbols|Y1X8ZZQ7VM]].
In the notation of [[basic hypergeometric series|EB4N9RQR51]], ${}_1\phi_0(a; -; q, z) = \frac{(az; q)_\infty}{(z; q)_\infty}$.

## Remarks

Proof idea: let $f(z) = \sum_n c_n z^n$ be the left side, with $c_n = \frac{(a; q)_n}{(q; q)_n}$.
Comparing the coefficients of $z^n$ on both sides, with $c_n (1 - q^n) = c_{n-1} (1 - a q^{n-1})$, gives $(1 - z) f(z) = (1 - az) f(qz)$.
Iterating this functional equation gives $f(z) = \frac{(az; q)_N}{(z; q)_N} f(q^N z)$, and $f(q^N z) \to f(0) = 1$ as $N \to \infty$.
Special cases: $a = 0$ gives $\sum_n \frac{z^n}{(q; q)_n} = \frac{1}{(z; q)_\infty}$, and replacing $a$ by $a/b$ and $z$ by $bz$ and letting $b \to 0$ gives $\sum_n \frac{(-1)^n q^{\binom{n}{2}} a^n z^n}{(q; q)_n} = (az; q)_\infty$.
With $a = q^{-N}$ and $z$ replaced by $-q^N z$, one obtains the finite form $\prod_{k=0}^{N-1} (1 + q^k z) = \sum_{k=0}^{N} q^{\binom{k}{2}} \binom{N}{k}_q z^k$ with the [[Gaussian binomial coefficients|YS09Z1YNEK]].
The hypothesis $\abs{z} < 1$ is needed for $a \notin \set{1, q^{-1}, q^{-2}, \dots}$, since the series then has radius of convergence $1$.

## Explanations

### L1 human

The binomial series says $\sum_n \binom{-a}{n} (-z)^n = (1 - z)^{-a}$.
The $q$-binomial theorem is its $q$-analogue: the power $(1 - z)^{-a}$ becomes a ratio of two infinite products.
Letting $q \to 1$ with $a$ replaced by $q^{a}$ recovers the [[binomial theorem|0EHWPWGHB2]] for real exponents.

### L2 human

This is the simplest summation formula for basic hypergeometric series, and many others follow from it.
Together with the case $a = 0$ it gives Euler's two expansions of $1/(z; q)_\infty$ and $(z; q)_\infty$, which are generating functions for partitions into parts of a given kind.
Its bilateral extension is [[Ramanujan's 1psi1 summation|RK6DCJVJCR]].
</document>

<document source="https://tamnd.github.io/math2/node/VVVJ655R0J.md" tag="VVVJ655R0J" kind="definition">
# Tricomi confluent hypergeometric function

## Definition

Let $a$ and $b$ be complex numbers, let $z \in \C \setminus (-\infty, 0]$, and let $z^{1 - b}$ be the principal branch.
Let $M$ be [[Kummer's function|6AZ4ZBN1M8]] and $\Gamma$ the [[gamma function|0H5SYBSED9]], with $1/\Gamma$ read as $0$ at the poles of $\Gamma$.
If $b \notin \Z$, the *Tricomi confluent hypergeometric function* is
$$U(a, b, z) = \frac{\Gamma(1 - b)}{\Gamma(a - b + 1)} M(a, b, z) + \frac{\Gamma(b - 1)}{\Gamma(a)} z^{1 - b} M(a - b + 1, 2 - b, z).$$
If $b \in \Z$, it is defined by continuity: $U(a, b, z) = \lim_{\beta \to b} U(a, \beta, z)$.

## Remarks

The function $U(a, b, \cdot)$ solves [[Kummer's equation|NT917KFR57]], since both terms of the definition do.
It is characterised among the solutions by $U(a, b, z) \sim z^{-a}$ as $z \to \infty$ with $\abs{\arg z} < \frac{3\pi}{2} - \delta$, for each $\delta > 0$.
For $\operatorname{Re} a > 0$ and $\operatorname{Re} z > 0$ it has the integral form
$$U(a, b, z) = \frac{1}{\Gamma(a)} \int_0^\infty e^{-zt}\, t^{a - 1} (1 + t)^{b - a - 1}\, dt.$$
Example: $U(a, a + 1, z) = z^{-a}$; for $\operatorname{Re} a > 0$ and $\operatorname{Re} z > 0$ the integral is $\frac{1}{\Gamma(a)} \int_0^\infty e^{-zt} t^{a - 1}\, dt = z^{-a}$.
Non-example: $M(a, b, z)$ itself is in general not a multiple of $U(a, b, z)$; for $a = b$ it is $e^z$, which grows faster than any power as $z \to +\infty$.
The two functions satisfy Kummer's second transformation $U(a, b, z) = z^{1 - b} U(a - b + 1, 2 - b, z)$.
</document>

<document source="https://tamnd.github.io/math2/node/VW2H7873YM.md" tag="VW2H7873YM" kind="statement">
# Amitsur's theorem on the radical of polynomial rings

## Statement

Let $R$ be a [[ring|0894J980RK]] and let $R[x]$ be the [[polynomial ring|0Q9EZS90GY]] in a central variable $x$ over $R$.
Let $J(R[x])$ be the [[Jacobson radical|XA0DCNH7WY]] of $R[x]$ and let $N = J(R[x]) \cap R$.
Then $N$ is a [[nil ideal|NZ1G9WRWFB]] of $R$ and $J(R[x]) = N[x]$, the set of polynomials with all coefficients in $N$.

## Remarks

The harder part is to show that $J(R[x])$ is generated by its constant polynomials.
The nil part is then direct: for $a \in N$, the element $ax$ lies in $J(R[x])$, so $1 - ax$ is a [[unit|019NFGFPR5]] of $R[x]$.
Its inverse must be the power series $\sum_{n \ge 0} a^n x^n$, which is a polynomial only if $a^n = 0$ for large $n$.
Example: for a commutative ring $R$, $J(R[x])$ is $\mathfrak{N}[x]$, where $\mathfrak{N}$ is the [[nilradical|QQDN6EYQNJ]] of $R$; for instance $J(\Z[x]) = 0$, although $\Z$ has many maximal ideals.
In particular, if $R$ has no nonzero nil ideals, then $R[x]$ is [[semiprimitive|1B2DBKV1C9]].

## Explanations

### L1 human

The Jacobson radical of a ring of polynomials is always made of polynomials whose coefficients are nilpotent.
So adjoining a variable kills most of the radical: only the nilpotent part survives.

### L2 human

The theorem is a standard tool for producing semiprimitive rings, since $R[x]$ is semiprimitive whenever $R$ has no nonzero nil ideals.
It also connects the Jacobson radical with nil ideals and with the [[Köthe conjecture|RT2D9JEVRR]], which in Krempa's form asks whether $J(R[x]) = R[x]$ for every nil ring $R$.
</document>

<document source="https://tamnd.github.io/math2/node/VWBVKT794W.md" tag="VWBVKT794W" kind="concept">
# Perfect field

## Summary

A perfect field is a field over which every [[irreducible polynomial|03FKX7Z5DC]] has distinct roots.
Equivalently, it has characteristic $0$, or it has characteristic $p > 0$ and every element is a $p$-th power.
Over a perfect field every algebraic extension is [[separable|0WCX7NB4N7]], so Galois theory applies without restrictions.
The fields $\Q$, $\R$, $\C$, all [[finite fields|0ZXZ82FTX2]] and all algebraically closed fields are perfect.

## Explanations

### L1 human

In characteristic $p$ the map $x \mapsto x^p$ is the [[Frobenius endomorphism|0T3WVTD675]], and it is always one-to-one.
A field of characteristic $p$ is perfect when this map is also onto, so each element has a $p$-th root.
In a finite field a one-to-one map from the field to itself is automatically onto, so finite fields are perfect.
The field $\F_p(t)$ of rational functions is not perfect, since $t$ has no $p$-th root in it.
Imperfect fields are where inseparable polynomials such as $x^p - t$ appear.
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# Exponential type

## Definition

Let $f$ be an [[entire function|0419794R0T]].
The function $f$ is of *exponential type* if there are constants $C > 0$ and $\tau \ge 0$ such that
$$\abs{f(z)} \le C e^{\tau \abs{z}} \quad \text{for all } z \in \C.$$
In that case the *type* of $f$ is
$$\sigma = \limsup_{r \to \infty} \frac{\log M(r)}{r}, \qquad M(r) = \max_{\abs{z} = r} \abs{f(z)},$$
with $\log 0 = -\infty$, and $f$ is said to be of exponential type $\sigma$.

## Remarks

Example: $\abs{\sin z} \le e^{\abs{z}}$ for all $z$, and $M(r) = \sinh r$ for $\sin z$, so $\sin z$ is of exponential type with type $1$.
Non-example: $f(z) = e^{z^2}$ has $\abs{f(x)} = e^{x^2}$ on the real line, and $e^{x^2} \le C e^{\tau \abs{x}}$ fails for large $x$ for every $C$ and $\tau$.
In terms of the [[order|SHF8Z7GMFX]], $f$ is of exponential type exactly when its order is less than $1$, or equal to $1$ with finite type.
For $f(z) = \sum_n a_n z^n$, the type is $\sigma = \frac{1}{e} \limsup_{n \to \infty} n \abs{a_n}^{1/n}$; for $e^z$ this gives $1$.
The notion is also used for functions holomorphic in a sector or a half-plane, with the bound required only there.
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# Convenient vector space

## Definition

Let $E$ be a real Hausdorff [[locally convex space|BW9BBF244W]].
A sequence $(x_n)$ in $E$ is *Mackey–Cauchy* if there are an absolutely convex [[bounded set|3BAJ8ZTVPX]] $B \subseteq E$ and real numbers $\mu_{n,m}$ with this property.
One has $\mu_{n,m} \to 0$ as $n, m \to \infty$, and $x_n - x_m \in \mu_{n,m} B$ for all $n, m$.
The space $E$ is *convenient*, or *$c^\infty$-complete*, if every Mackey–Cauchy sequence in $E$ converges in $E$.

## Remarks

Kriegl and Michor give equivalent conditions.
One is that a curve $c \colon \R \to E$ is smooth as soon as $\ell \circ c$ is smooth for every continuous linear functional $\ell$.
Another is that the Riemann integral $\int_0^1 c(t) \, dt$ exists in $E$ for every smooth curve $c$.


Example: every Fréchet space is convenient.
A Mackey–Cauchy sequence is Cauchy, because $\mu_{n,m} B$ is eventually inside any given neighbourhood of $0$ when $B$ is bounded, and a Fréchet space is complete.
More generally every sequentially complete locally convex space is convenient.

Non-example: the space $P$ of real polynomials on $[0, 1]$ with the supremum norm is not convenient.
In a normed space, a sequence is Mackey–Cauchy if and only if it is Cauchy, since the unit ball is bounded.
The Taylor polynomials of $e^x$ form a Cauchy sequence in $P$ whose uniform limit $e^x$ is not a polynomial, so the sequence does not converge in $P$.

A normed space is therefore convenient exactly when it is a [[Banach space|0BDTTM24DP]].
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# Binomial proportion confidence interval

## Definition

Let $n \ge 1$, let $X$ have the [[binomial distribution|0VRWZDWW6B]] with parameters $n$ and unknown $p \in [0, 1]$, and let $\alpha \in (0, 1)$.
A *binomial proportion confidence interval* of level $1 - \alpha$ is an interval $[L(X), U(X)]$ with endpoints computed from $X$ alone.
Its coverage function is $C(p) = P_p\big(L(X) \le p \le U(X)\big)$.
It is a [[confidence interval|0K4NMMJBG2]] of level at least $1 - \alpha$ when $C(p) \ge 1 - \alpha$ for every $p \in [0, 1]$.
It is an approximate interval when $C(p)$ tends to $1 - \alpha$ as $n \to \infty$, for each fixed $p \in (0, 1)$.

## Remarks

Let $\hat p = X/n$ and let $z$ be the $1 - \alpha/2$ quantile of the standard normal law.
The Wald interval $\hat p \pm z \sqrt{\hat p(1 - \hat p)/n}$ is an approximate interval, by the normal approximation to the binomial law.
The Wilson interval is the set of $p$ with $\abs{\hat p - p} \le z \sqrt{p(1 - p)/n}$; it is also approximate, with better coverage for small $n$.
The Clopper–Pearson interval has level at least $1 - \alpha$ for every $p$; its endpoints are quantiles of beta laws.
Non-example: the Wald interval is not of level at least $1 - \alpha$ for all $p$.
If $X = 0$, it is the single point $\set{0}$, so for small $p > 0$ the coverage $C(p)$ is close to $P_p(X \ge 1) = 1 - (1 - p)^n$, which tends to $0$ as $p \to 0$.
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# Gelfand transform

## Summary

The Gelfand transform turns each element $a$ of a commutative complex [[Banach algebra|9KYWN94QDN]] into a continuous function $\hat a$ on the [[character space|M7JVFJ3MMX]], by $\hat a(\varphi) = \varphi(a)$.
It is an algebra homomorphism into the functions vanishing at infinity, and it does not increase norms.
For a unital algebra, the range of $\hat a$ is the spectrum of $a$.
For commutative C*-algebras it is an isometric isomorphism onto all such functions, by the [[commutative Gelfand–Naimark theorem|NT564Y315M]].

## Explanations

### L1 human

The idea is to view the elements of an abstract commutative algebra as functions.
The points are the characters, and the value of $a$ at a character $\varphi$ is the number $\varphi(a)$.
For sequences $(a_n)_{n \in \Z}$ with $\sum \abs{a_n} < \infty$ and convolution as product, the characters are the points $z$ of the unit circle.
There the Gelfand transform is the Fourier series $\sum_n a_n z^n$, which turns convolution into ordinary multiplication.
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# Routh's theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane with area $S$.
Let $D$, $E$, $F$ be points in the interiors of the sides $BC$, $CA$, $AB$, and let $x = BD/DC$, $y = CE/EA$ and $z = AF/FB$.
Then the three segments $AD$, $BE$, $CF$ bound a triangle whose [[area|065JWMZW86]] is
$$S \cdot \frac{(xyz - 1)^2}{(xy + x + 1)(yz + y + 1)(zx + z + 1)}.$$
When $xyz = 1$, this triangle degenerates to a point.

## Remarks

A proof places the triangle in barycentric coordinates, computes the three intersection points of the cevians, and takes the ratio of determinants.
Example: with $x = y = z = 2$, the area is $S \cdot 49/343 = S/7$, the well-known one-seventh triangle.
The case $xyz = 1$ is exactly the condition of [[Ceva's theorem|0KN93VDSXA]] for the three cevians to be concurrent, and Routh's formula gives area $0$ there.

## Explanations

### L1 human

Draw three lines from the corners of a triangle to points on the opposite sides.
In general the three lines enclose a small triangle in the middle.
Routh's theorem tells exactly how big that small triangle is, using only the ratios in which the sides are cut.

### L2 human

Routh's theorem refines Ceva's theorem: Ceva says when the inner triangle is a point, and Routh measures it in all other cases.
Because affine maps preserve ratios of areas and ratios along lines, it is enough to prove it for one triangle, for example $A = (0, 0)$, $B = (1, 0)$, $C = (0, 1)$.
The theorem is a good exercise in barycentric coordinates and in the use of the [[determinant|00KDZRW4C0]] to compute areas.
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# Holonomy

## Definition

Let $M$ be a connected smooth manifold, let $E \to M$ be a smooth vector bundle with a [[connection|5ZYRRVE8BW]] $\nabla$, and let $p \in M$.
For a piecewise smooth loop $\gamma$ based at $p$, let $P_\gamma : E_p \to E_p$ be the [[parallel transport|TVAGN3D6W5]] along $\gamma$.
The *holonomy group* of $\nabla$ at $p$ is
$$\mathrm{Hol}_p(\nabla) = \set{P_\gamma : \gamma \text{ a piecewise smooth loop based at } p} \subset \mathrm{GL}(E_p).$$
The *restricted holonomy group* $\mathrm{Hol}^0_p(\nabla)$ is the subset given by the loops that are null-homotopic.

## Remarks

$\mathrm{Hol}_p(\nabla)$ is a group, because $P_{\gamma \cdot \delta} = P_\delta \circ P_\gamma$ and $P_{\gamma^{-1}} = P_\gamma^{-1}$.
It is a [[Lie subgroup|NM8XCPS1Q3]] of $\mathrm{GL}(E_p)$, and $\mathrm{Hol}^0_p(\nabla)$ is its identity component; the quotient is a quotient of $\pi_1(M, p)$.
Changing the base point along a curve $\sigma$ from $p$ to $q$ conjugates the group: $\mathrm{Hol}_q = P_\sigma \mathrm{Hol}_p P_\sigma^{-1}$.
Example: the flat connection on $T\R^n$ has $P_\gamma = \id$ for every loop, so $\mathrm{Hol}_p$ is trivial.
For the Levi-Civita connection of the round sphere $S^2$, $\mathrm{Hol}_p = \mathrm{SO}(2)$: the transport around a loop is the rotation by the enclosed area.
For the Levi-Civita connection of any Riemannian $n$-manifold, $\mathrm{Hol}_p \subset \mathrm{O}(T_pM)$, and $\mathrm{Hol}_p \subset \mathrm{SO}(T_pM)$ if $M$ is orientable.
Non-example: the flat connection on the Möbius line bundle over the circle has trivial restricted holonomy, but $\mathrm{Hol}_p = \set{\pm 1}$ is not trivial; so flatness alone does not force trivial holonomy.
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# Tensor decomposition

## Summary

A tensor decomposition writes a multi-way array as a combination of simpler pieces.
It plays the role that a matrix factorization plays for matrices.
The canonical polyadic (CP) decomposition writes the tensor as a sum of simple tensors.
The Tucker decomposition combines a small core tensor with one factor matrix per mode.
The shortest CP decomposition has length equal to the [[tensor rank|W84NTF77M0]].
Tensor decompositions are used to compress data and to find hidden structure in signal processing, statistics and chemistry.

## Explanations

### L1 human

Think of a cube of numbers, such as ratings indexed by user, product and month.
A CP decomposition writes the cube as a sum of a few "outer products" of three vectors, one for each direction.
Each term is a simple pattern: a group of users who like a group of products during certain months.
For example, the cube with entries $a_{ijk} = x_i y_j z_k$ is a single term, so it has a decomposition with one piece.
This extends the idea of the [[singular value decomposition|000NPTMX6J]] from tables to cubes.
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# Comma category

## Definition

Let $S \colon \mathcal{A} \to \mathcal{C}$ and $T \colon \mathcal{B} \to \mathcal{C}$ be [[functors|28NQ0KRHD3]].
The *comma category* $(S \downarrow T)$ is defined as follows:

1. its objects are the triples $(A, B, h)$ with $A$ an object of $\mathcal{A}$, $B$ an object of $\mathcal{B}$ and $h \colon S(A) \to T(B)$ a [[morphism|859F0TFVA3]] of $\mathcal{C}$;
2. a morphism $(A, B, h) \to (A', B', h')$ is a pair $(f, g)$ with $f \colon A \to A'$ in $\mathcal{A}$ and $g \colon B \to B'$ in $\mathcal{B}$ such that $T(g) \circ h = h' \circ S(f)$;
3. composition is $(f', g') \circ (f, g) = (f' \circ f, g' \circ g)$, and the identity of $(A, B, h)$ is $(1_A, 1_B)$.

## Remarks

The condition in clause 2 says that the square with sides $S(f)$, $h'$, $h$ and $T(g)$ is a [[commutative diagram|WV898P4X16]]; pasting two such squares shows that composites are again morphisms.
Example: with $S = T = 1_{\mathcal{C}}$, the comma category is the arrow category of $\mathcal{C}$, whose objects are the morphisms $h \colon X \to Y$.
In the arrow category of sets, the pair $(f, g)$ with $f = g = (x \mapsto -x)$ is a morphism from $h = (x \mapsto x^3)$ on $\R$ to itself, since $(-x)^3 = -x^3$.
Non-example: for $h = (x \mapsto x^2)$ on $\R$, the same pair $(f, g)$ is not a morphism from $h$ to $h$, since $-x^2 \ne (-x)^2$ when $x \ne 0$.
When $\mathcal{A}$ is the category with one object and one morphism, $S$ picks an object $X$ of $\mathcal{C}$ and one writes $(X \downarrow T)$.
There are projection functors from $(S \downarrow T)$ to $\mathcal{A}$ and to $\mathcal{B}$.
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# Hardy–Littlewood maximal function

## Summary

Let $f$ be a locally integrable function on $\R^n$.
Its Hardy–Littlewood maximal function $Mf$ gives at each point the largest average of $\abs{f}$ over balls centred there.
It dominates $\abs{f}$ almost everywhere and controls all local averages of $f$ at once.
The maximal operator $f \mapsto Mf$ is bounded on $L^p$ for $1 < p \le \infty$ and of weak type $(1,1)$.
These bounds give the Lebesgue differentiation theorem and are a basic tool in harmonic analysis.

## Explanations

### L1 human

At each point, look at all balls centred there and average $\abs{f}$ over each ball.
The maximal function records the biggest of these averages.
It is a worst-case summary of how large $f$ is near the point at every scale.
For the indicator of $[0, 1]$ on the line, the maximal function at a point $x > 1$ is $1/(2x)$, so it decays slowly far away.
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# Interpolation space

## Summary

Let $X_0$ and $X_1$ be two Banach spaces that sit in a common space.
An interpolation space for them is a Banach space between them with this property.
Every linear map bounded on both $X_0$ and $X_1$ is bounded on it too.
The model case is the Riesz–Thorin theorem.
It says that the [[Lp spaces|SASP14MA1Z]] with $p$ between $p_0$ and $p_1$ are interpolation spaces for $L^{p_0}$ and $L^{p_1}$.
The real method of Lions and Peetre and the complex method of Calderón build interpolation spaces in general.
Interpolation gives bounds for operators such as the Fourier transform and singular integrals, and it defines fractional Sobolev spaces.

## Explanations

### L1 human

Suppose you know that an operation is well behaved for two extreme ways of measuring size.
Interpolation lets you conclude that it is also well behaved for the measures of size in between.
For example, the Fourier transform is bounded from $L^1$ to $L^\infty$ and from $L^2$ to $L^2$.
Interpolation then gives the Hausdorff–Young inequality: it is bounded from $L^p$ to $L^q$ for $1 \le p \le 2$ and $1/p + 1/q = 1$.
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# Strong operator topology

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.
The *strong operator topology* on $B(H)$ is the coarsest topology such that each map $B(H) \to H$, $T \mapsto T x$, with $x \in H$, is continuous.
Here $H$ carries its norm topology.

## Remarks

Equivalently, it is the [[topology generated by the seminorms|ZVFXQ7S6FY]] $T \mapsto \|T x\|$ for $x \in H$.
A net $(T_i)$ converges to $T$ in this topology if and only if $\|T_i x - T x\| \to 0$ for every $x \in H$.

Example: on $H = \ell^2(\N)$ with standard basis $(e_k)$, let $P_n$ be the orthogonal projection onto the span of $e_0, \dots, e_{n-1}$.
For every $x$, $\|x - P_n x\|^2 = \sum_{k \ge n} \abs{x_k}^2 \to 0$, so $P_n \to I$ strongly.
But $\|I - P_n\| = 1$ for all $n$, so there is no convergence in norm.

Non-example: let $S$ be the right shift on $\ell^2(\N)$.
The powers $S^n$ tend to $0$ in the [[weak operator topology|4XM1VN1C1N]].
If $S^n$ converged strongly to some $T$, then $T$ would also be the weak limit, so $T = 0$; but $\|S^n x\| = \|x\|$ does not tend to $0$ for $x \ne 0$.
So $(S^n)$ has no strong limit, while the adjoint powers $(S^*)^n$, which delete the first $n$ entries, do tend to $0$ strongly.
In particular $T \mapsto T^*$ is not continuous in the strong operator topology.

The strong operator topology is finer than the weak operator topology and coarser than the norm topology.
A convex subset of $B(H)$ has the same closure in both the weak and the strong operator topology.
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# Kigami's construction of Laplacians on post-critically finite fractals

## Statement

Let $K \subset \R^d$ be the attractor of an [[iterated function system|75B8VHDGSA]] of contracting similitudes $F_1, \dots, F_N$, and assume that $K$ is connected and post-critically finite.
Let $V_0 \subset K$ be its finite boundary set, let $V_{m+1} = \bigcup_{i=1}^N F_i(V_m)$, and for a word $w = w_1 \cdots w_m$ write $F_w = F_{w_1} \circ \cdots \circ F_{w_m}$ and $r_w = r_{w_1} \cdots r_{w_m}$.
Let $\mathcal{E}_0(u) = \sum_{\set{p, q} \subset V_0} c_{pq} (u(p) - u(q))^2$ be the energy of an electrical network on $V_0$ with conductances $c_{pq} = c_{qp} \ge 0$ that connect $V_0$, and let $r_1, \dots, r_N \in (0, 1)$.
For $u \colon V_m \to \R$ set
$$\mathcal{E}_m(u) = \sum_{\abs{w} = m} r_w^{-1} \, \mathcal{E}_0(u \circ F_w|_{V_0}).$$
Assume the renormalization condition: for every $u \colon V_0 \to \R$, the minimum of $\mathcal{E}_1(v)$ over all $v \colon V_1 \to \R$ with $v|_{V_0} = u$ equals $\mathcal{E}_0(u)$.
Then:

1. For every function $u \colon K \to \R$, the sequence $\mathcal{E}_m(u|_{V_m})$ is nondecreasing in $m$; let $\mathcal{E}(u)$ be its limit in $[0, \infty]$.
2. Let $\mathcal{F} = \set{u \colon K \to \R : u \text{ continuous}, \ \mathcal{E}(u) < \infty}$, and define $\mathcal{E}(u, v)$ from $\mathcal{E}$ by polarization. Then $(\mathcal{E}, \mathcal{F})$ is a [[resistance form|THCM7DWHNY]] on $K$, and its resistance metric induces the usual topology of $K$.
3. For every finite Borel measure $\mu$ on $K$ whose support is $K$, $(\mathcal{E}, \mathcal{F})$ is a local regular [[Dirichlet form|MH20Q2N1AH]] on $L^2(K, \mu)$.
4. The Laplacian $\Delta_\mu$ is defined by: $\Delta_\mu u = f$, with $u \in \mathcal{F}$ and $f$ continuous, if $\mathcal{E}(u, v) = -\int_K f v \, d\mu$ for every $v \in \mathcal{F}$ that vanishes on $V_0$.

## Remarks

The renormalization condition says that $\mathcal{E}_0$ is the trace of $\mathcal{E}_1$ on $V_0$, and by self-similarity $\mathcal{E}_m$ is then the trace of $\mathcal{E}_{m+1}$ on $V_m$ for every $m$.
This gives the monotonicity in clause 1 and shows that each $u$ on $V_m$ has a unique extension to $V_{m+1}$ with the same energy, the harmonic extension.
The bound $(u(x) - u(y))^2 \le R(x, y) \, \mathcal{E}(u)$ with a resistance $R$ that is small on small cells gives the continuity of functions of finite energy; this uses $r_i < 1$.
For the Sierpiński gasket, with $\mathcal{E}_0$ the unit-conductance triangle and all $r_i = 3/5$, the renormalization condition is the computation of the "$1/5$–$2/5$ rule", as in [[Laplacian on the Sierpiński gasket|W9V9JSAWYK]].
Finding $\mathcal{E}_0$ and $r_i$ that satisfy the renormalization condition is a nonlinear fixed-point problem, and the theorem takes a solution as given.
A pair $(\mathcal{E}_0, (r_i))$ that satisfies the renormalization condition is called a harmonic structure, and it is called regular when all $r_i < 1$.
The construction is due to Kigami.

## Explanations

### L1 human

Many fractals can be cut into finitely many small copies of themselves which touch each other at only finitely many points.
On such a fractal, one can put a finite electrical network on the touching points at each level of detail.
The theorem says that if the network at level $0$ is chosen so that it agrees with the network at level $1$, then the energies of all levels fit together.
Their limit is an energy on the whole fractal, and from the energy and a measure one gets a Laplacian, much as the usual Laplacian comes from the energy $\int \abs{\nabla u}^2$.

### L2 human

The result turns the analysis of a fractal into linear algebra on a sequence of finite graphs, each step governed by the same [[Laplacian matrix|TN6M6DVZRP]].
Harmonic functions are determined by their values on $V_0$ and are computed level by level through fixed harmonic extension matrices.
Once the Laplacian exists, one studies its spectrum, the heat equation $\partial_t u = \Delta_\mu u$ and its heat kernel, and the diffusion process attached to the Dirichlet form by the Fukushima correspondence.
The resistance metric is the natural metric for these estimates.
The approach complements the probabilistic one, in which Brownian motion on the gasket is built as a limit of random walks on the graphs $\Gamma_m$.
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# Exact couple

## Summary

An exact couple is a pair of abelian groups with three maps between them that form an exact triangle.
From it one builds a derived exact couple, and repeating this gives a [[spectral sequence|GG8AYV847V]].
Many spectral sequences of algebraic topology arise in this way.
Examples are the spectral sequence of a filtered space, the Bockstein spectral sequence and the Adams spectral sequence.

## Explanations

### L1 human

Long exact sequences in topology often come in families, one for each step of a filtration.
An exact couple packages such a family into a triangle of two groups and three maps, exact at each corner.
The composite of two of the maps squares to zero, so one can take its homology.
Doing this again and again produces the pages of a spectral sequence.
For example, multiplication by a prime $p$ on integral homology and the reduction to mod $p$ homology form an exact couple that gives the Bockstein spectral sequence.
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# Baskakov operator

## Summary

The Baskakov operators are [[positive linear operators|AQDXWH4812]] that approximate functions on $[0, \infty)$ by sampling them at the points $k/n$.
Their weights come from the negative binomial distribution.
They belong to a family that also contains the [[Bernstein polynomials|0JT92VK307]] and the [[Szász–Mirakyan operators|GRQJ49XQNR]].

## Explanations

### L1 human

Like the Bernstein and Szász–Mirakyan operators, the Baskakov operator replaces $f(x)$ by a weighted average of the samples $f(k/n)$.
The weights are probabilities, they add up to $1$, and they concentrate near $k/n \approx x$ for large $n$.
Here they come from counting failures before the $n$-th success in repeated trials.
Because the weights have mean $x$, linear functions are reproduced exactly.
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# Unknotting problem

## Problem

The input is a finite diagram of a tame [[knot|QYEAZTYGPT]] in $\R^3$, for example a polygon with finitely many vertices given by rational coordinates.
The question is whether the knot is the [[unknot|1TQ06R4Q66]], that is, whether it is ambient isotopic to a round circle.
The problem asks for an algorithm that answers this question for every input, and for the cost of such an algorithm.

## Remarks

Dehn's work of 1910 began the search for such an algorithm, through the fundamental group of the knot complement.
Haken proved with his theory of normal surfaces, published in 1961, that the problem is decidable.
His method searches for a disk bounded by the knot among finitely many normal surfaces in the knot complement.
Hass, Lagarias and Pippenger proved in 1999 that the problem lies in the complexity class NP.
Lackenby proved without extra hypotheses that it also lies in co-NP.
It is not known whether the problem can be solved in polynomial time.
[[Reidemeister moves|T3HAGWYKK7]] give another approach: a diagram of the unknot can be reduced to a circle by such moves, and there are explicit bounds on the number of moves needed.

## Explanations

### L1 human

Take a tangled loop of string whose ends are glued together.
The question is whether you can untangle it into a plain circle without cutting it.
A messy picture of the plain circle can look very knotted, so the answer is not obvious.
Mathematicians know that a computer can always decide this, but they do not know whether it can be done quickly for large diagrams.
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# Stable distribution

## Definition

Let $X$ be a real [[random variable|0PNRKMRC55]] that is not almost surely constant.
Let $X_1, X_2$ be independent random variables with the same law as $X$.
The law of $X$ is *stable* if for all $a, b > 0$ there are $c > 0$ and $d \in \R$ such that $a X_1 + b X_2$ has the same law as $c X + d$.
It is *strictly stable* if moreover $d = 0$ can be chosen for all $a, b > 0$.

## Remarks

For a stable law there is a unique $\alpha \in (0, 2]$, the index, such that $c^\alpha = a^\alpha + b^\alpha$ always.
The [[characteristic function|0V5B6D6BQH]] of a stable law has the form $\varphi(t) = \exp\left(i \mu t - \abs{\gamma t}^\alpha (1 - i \beta \sgn(t) \Phi(t))\right)$, with $\Phi(t) = \tan(\pi\alpha/2)$ for $\alpha \ne 1$ and $\Phi(t) = -\frac{2}{\pi} \log \abs{t}$ for $\alpha = 1$, where $\beta \in [-1, 1]$, $\gamma > 0$ and $\mu \in \R$.
Example: let $X$ be [[normal|05AQE0QR0D]] with mean $0$ and variance $1$.
Then $a X_1 + b X_2$ is normal with mean $0$ and variance $a^2 + b^2$.
So it has the law of $c X$ with $c = \sqrt{a^2 + b^2}$, and the index is $2$.
Example: for the standard [[Cauchy|0GPG24Q3GR]] law, $\varphi(t) = e^{-\abs{t}}$, so $a X_1 + b X_2$ has characteristic function $e^{-(a + b)\abs{t}}$, the law of $(a + b) X$; this is index $1$.
Non-example: let $X$ be Poisson with mean $\lambda > 0$. Then $X_1 + X_2$ is Poisson with mean $2\lambda$, and takes all values in $\N$. If it had the law of $cX + d$, comparing the two smallest values of the supports would force $c = 1$ and $d = 0$, but $X$ has mean $\lambda \ne 2\lambda$.
For $\alpha < 2$, $E[\abs{X}^p] < \infty$ if and only if $p < \alpha$; so the variance is infinite, and the mean is finite only for $\alpha > 1$.
Some authors also call the point masses stable.
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# Haar measure

## Summary

A Haar measure on a [[locally compact group|AHV05ZBCYW]] is a nonzero [[Radon measure|HZNEQQBCW4]] that does not change under left translation by elements of the group.
Every locally compact group has one, and it is unique up to a positive constant.
It extends Lebesgue measure on $\R^n$ and counting measure on discrete groups, and it is the basic tool for integration and harmonic analysis on groups.

## Explanations

### L1 human

On the real line, the length of an interval does not change when the interval is shifted.
A Haar measure is a way of assigning sizes to subsets of a group with the same property: moving a set by a group element keeps its size.
On the circle it is arc length, on a finite group it counts points, and on the positive reals under multiplication it is $dx/x$.
Indeed, the interval $[a, b]$ has measure $\log(b/a)$ for $dx/x$, and multiplying it by $c > 0$ gives $[ca, cb]$ with the same measure.
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<document source="https://tamnd.github.io/math2/node/VY180V8QY2.md" tag="VY180V8QY2" kind="concept">
# Exponential stability

## Summary

An equilibrium is exponentially stable when nearby solutions approach it at least as fast as a decaying exponential.
It is a strong form of [[asymptotic stability|0NB689782W]] that comes with an explicit rate.
For a linear system $x' = Ax$, asymptotic and exponential stability coincide, and both mean that every eigenvalue of $A$ has negative real part.
Exponential stability survives small perturbations of the system, which makes it the preferred notion in control theory.
See the [[definition|V6E2R7JNN8]].

## Explanations

### L1 human

A solution that is disturbed from rest returns to rest, and the size of the disturbance shrinks by a fixed factor in each unit of time.
For $x' = -x$, the solution $x_0 e^{-t}$ loses about two thirds of its size every unit of time.
For $x' = -x^3$, solutions also tend to $0$, but only like $1/\sqrt{2t}$, which is slower than any exponential.
So the origin of $x' = -x^3$ is asymptotically stable but not exponentially stable.
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<document source="https://tamnd.github.io/math2/node/VY2TTB854G.md" tag="VY2TTB854G" kind="definition">
# Spherical code

## Definition

Let $n \ge 1$ and let $S^{n-1} = \set{x \in \R^n : x \cdot x = 1}$ be the unit sphere, where $x \cdot y$ is the [[dot product|0S52TNNTH9]].
A *spherical code* in $\R^n$ is a finite nonempty subset $C \subset S^{n-1}$.
Let $0 < \theta \le \pi$.
The code $C$ has *minimal angle* at least $\theta$ if $x \cdot y \le \cos \theta$ for all distinct $x, y \in C$.
Equivalently, every two distinct points of $C$ make an [[angle|0ZEHZY85E6]] at least $\theta$ at the origin.
The largest size of a spherical code in $\R^n$ with minimal angle at least $\theta$ is denoted $A(n, \theta)$.

## Remarks

Example: the $2n$ points $\pm e_1, \dots, \pm e_n$, where $e_i$ are the standard basis vectors, form a spherical code with minimal angle $\pi/2$.
Two distinct points among them have dot product $0$ or $-1$, both at most $\cos(\pi/2) = 0$.
Non-example: in $\R^2$, the points $(1, 0)$, $(0, 1)$ and $(1/\sqrt 2, 1/\sqrt 2)$ do not have minimal angle at least $\pi/2$, since the dot product of the first and the third is $1/\sqrt 2 > 0$.
The [[kissing number|B408FZYXF9]] is $\tau(n) = A(n, \pi/3)$.
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# Lax–Friedrichs method

## Summary

The Lax–Friedrichs method is an explicit [[finite difference method|X7HYEBA5SA]] for hyperbolic equations such as [[conservation laws|8RA7AMWSAF]].
It replaces the value at the current point by the average of its two neighbours and then applies a centred difference of the flux.
The averaging adds numerical viscosity, which makes the scheme stable under the [[Courant–Friedrichs–Lewy condition|FQVVNYV6N8]].
It is first-order accurate and smears shocks, but it is simple and robust.
See the [[definition|53H8EVRC1J]].

## Explanations

### L1 human

To update a value on a grid, a naive scheme uses the old value plus a centred estimate of the change, and this blows up.
The Lax–Friedrichs method uses the average of the two neighbouring old values instead of the old value itself.
This small change acts like a little diffusion, and it damps the growing oscillations.
For the equation $u_t + a u_x = 0$, the new value is a weighted average of the two neighbours when $\abs{a} \Delta t \le \Delta x$.
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# Boolean algebras canonically defined

## Summary

The canonical definition of a [[Boolean algebra|0XADJKM699]] avoids choosing a list of axioms.
It uses the operations of the [[two-element Boolean algebra|0WG43AC0EV]].
A Boolean algebra is then any algebra with these operations that satisfies every equation true in the two-element algebra.
This gives the same class of structures as the usual axioms.
It also explains why many different lists of axioms work.
A list of equations true in $\set{0, 1}$ is a complete axiom system exactly when it proves every equation true in $\set{0, 1}$.
It also connects Boolean algebra with truth tables and with the equational logic of universal algebra.

## Explanations

### L1 human

An equation such as $x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$ can be tested on true and false by a truth table.
The canonical view takes all equations that pass this test as the laws of Boolean algebra.
A Boolean algebra is then any set with "and", "or", "not", $0$ and $1$ that obeys all these laws.
The power set of a set is an example, because each law can be checked one point at a time.
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<document source="https://tamnd.github.io/math2/node/VY6ZCZY8DA.md" tag="VY6ZCZY8DA" kind="concept">
# Synthetic geometry

## Summary

Synthetic geometry is geometry done from axioms about figures, without coordinates or formulas.
Its proofs reason directly with points, lines, circles and the relations between them, as in [[Euclid's Elements|83WHDAV68J]].
It is opposite to analytic geometry, which translates figures into equations.
Axiom systems such as [[Hilbert's axioms|KM1FB8XZ2R]] and the theory of [[projective planes|DANFNN8C4Y]] are modern forms of synthetic geometry.

## Explanations

### L1 human

There are two ways to prove that the medians of a triangle meet in one point.
One can give the vertices coordinates and solve equations; that is analytic geometry.
Or one can argue with the figure itself, using similar triangles; that is synthetic geometry.
Synthetic proofs often show why a result is true, while analytic proofs are often more mechanical.
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<document source="https://tamnd.github.io/math2/node/VYB03ZZ8H6.md" tag="VYB03ZZ8H6" kind="example">
# Wilson–Cowan neural field equation

## Example

A neural field describes the activity of a sheet of neurons, with a position variable $x \in \R^d$ and time $t$.
Let $u_e(x, t)$ and $u_i(x, t)$ be the mean activities of the excitatory and the inhibitory populations, with time constants $\tau_e, \tau_i > 0$.
Let $w_{ab} \colon \R^d \to [0, \infty)$ be integrable connectivity kernels, let $S_e, S_i$ be increasing sigmoid firing-rate functions, and let $h_e, h_i$ be external inputs.
A commonly used form of the Wilson–Cowan neural field equations, with refractory terms neglected, is
$$\tau_e \, \partial_t u_e = -u_e + S_e\big(w_{ee} * u_e - w_{ei} * u_i + h_e\big), \qquad \tau_i \, \partial_t u_i = -u_i + S_i\big(w_{ie} * u_e - w_{ii} * u_i + h_i\big),$$
where $(w * u)(x, t) = \int_{\R^d} w(x - y) \, u(y, t) \, dy$.
Each equation has a time derivative of the unknowns and a space integral of them, so the system is a nonlinear [[integro-differential equation|YRKW2KX2CW]].

Check: for constant inputs, a spatially constant state $(\bar u_e, \bar u_i)$ is a steady state exactly when
$$\bar u_e = S_e\big(W_{ee} \bar u_e - W_{ei} \bar u_i + h_e\big), \qquad \bar u_i = S_i\big(W_{ie} \bar u_e - W_{ii} \bar u_i + h_i\big),$$
where $W_{ab} = \int_{\R^d} w_{ab}(z) \, dz$, because $w * \bar u = W \bar u$ for a constant $\bar u$.
These are the equations of the space-clamped Wilson–Cowan model.

## Remarks

The convolution structure makes the linearization at a homogeneous steady state diagonal in Fourier modes $e^{i k \cdot x}$.
This gives conditions for pattern-forming (Turing-type) instabilities, used to model visual hallucination patterns and travelling waves of cortical activity.
A single-population version $\tau \partial_t u = -u + w * S(u) + h$ is also widely studied, under the name Amari equation.

## Explanations

### L1 human

Neurons in the cortex excite or inhibit other neurons, more strongly when they are close.
The model follows the average activity at each point of the cortex.
This activity decays on its own and is driven by a weighted sum of the activity at all other points.
The weighted sum is an integral over space, so the model is an integro-differential equation, whose solutions can form bumps, stripes and waves.
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# CUSUM

## Definition

Let $X_1, X_2, \dots$ be observed one at a time, in the setting of the [[change detection|Z5H8ZZHVMC]] problem with pre-change density $f_0$ and post-change density $f_1$.
Let $\ell(x) = \log \big( f_1(x) / f_0(x) \big)$ be the log [[likelihood ratio|0VW9E4QXR6]] of one observation.
The *CUSUM statistic* is defined by
$$W_0 = 0, \qquad W_n = \max\left(0, \; W_{n-1} + \ell(X_n)\right) \quad (n \ge 1).$$
For a threshold $h > 0$, the *CUSUM procedure* is the [[stopping time|WND87FBAKC]]
$$T_h = \min \set{n \ge 1 : W_n \ge h},$$
at which it raises an alarm.

## Remarks

Equivalently, $W_n = \max_{1 \le k \le n + 1} \sum_{i=k}^{n} \ell(X_i)$, with the empty sum for $k = n + 1$ equal to $0$.
So $W_n$ is the largest log-likelihood ratio of "a change happened at time $k$" against "no change yet", over all candidate times $k$.

Example: for a shift of a normal mean from $\mu_0$ to $\mu_1 > \mu_0$ with variance $\sigma^2$, one finds $\ell(x) = \frac{\mu_1 - \mu_0}{\sigma^2} \left( x - \frac{\mu_0 + \mu_1}{2} \right)$.
After rescaling, the statistic becomes $S_n = \max(0, S_{n-1} + X_n - \mu_0 - k)$ with the reference value $k = (\mu_1 - \mu_0)/2$, the form used in quality control.

Non-example: the plain cumulative sum $\sum_{i=1}^n \ell(X_i)$ without the reset at $0$ is not the CUSUM statistic.
Before the change it drifts down to $-\infty$, so after a late change it needs a long time to climb back to the threshold.

The mean time to a false alarm satisfies $E_\infty[T_h] \ge e^h$, which guides the choice of $h$.
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# G-structure

## Summary

A $G$-structure on an $n$-manifold is a choice, at every point, of a preferred family of frames of the tangent space.
Any two preferred frames at a point are related by an element of a fixed subgroup $G$ of $\mathrm{GL}(n, \R)$.
Formally it is a reduction of the [[frame bundle|F4HWTRNXAP]] to the group $G$.
Many geometric structures are $G$-structures: Riemannian metrics for $G = \mathrm{O}(n)$, orientations for $\mathrm{GL}^+(n, \R)$, almost complex structures for $\mathrm{GL}(m, \C)$ with $n = 2m$.
The notion gives one language for comparing and classifying these structures.

## Explanations

### L1 human

At each point of a manifold one can choose many bases of the tangent space.
A geometric structure picks out some of them as the good ones.
A Riemannian metric picks out the orthonormal bases, and any two of these differ by a rotation or reflection, an element of $\mathrm{O}(n)$.
An orientation picks out the positively oriented bases, which differ by matrices of positive determinant.
A $G$-structure is this idea for an arbitrary group $G$.
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<document source="https://tamnd.github.io/math2/node/VYP60TA2EC.md" tag="VYP60TA2EC" kind="definition">
# Hausdorff dimension

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]], and for $s \ge 0$ let $\mathcal{H}^s$ be the $s$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] on $X$.
The *Hausdorff dimension* of a set $A \subseteq X$ is
$$\dim_H A = \inf \set{ s \ge 0 : \mathcal{H}^s(A) = 0 },$$
with the convention $\inf \emptyset = \infty$.

## Remarks

If $\mathcal{H}^s(A) < \infty$ and $t > s$, then $\mathcal{H}^t(A) = 0$.
So $\mathcal{H}^s(A) = \infty$ for $s < \dim_H A$ and $\mathcal{H}^s(A) = 0$ for $s > \dim_H A$.
At $s = \dim_H A$ the value can be $0$, positive and finite, or $\infty$.
Equivalently, $\dim_H A = \sup \set{s \ge 0 : \mathcal{H}^s(A) = \infty}$, with $\sup \emptyset = 0$.

Example: the middle-thirds [[Cantor set|0BXNDZFQME]] $C$ has $\dim_H C = \log 2 / \log 3$.
At the $k$-th step, $C$ is covered by $2^k$ intervals of length $3^{-k}$, so $\mathcal{H}^s_{3^{-k}}(C) \le 2^k 3^{-ks}$, which equals $1$ for $s = \log 2 / \log 3$.
This gives $\dim_H C \le \log 2 / \log 3$; the matching lower bound needs a separate argument.

A dense set can have dimension $0$: $\Q$ is dense in $\R$, yet $\dim_H \Q = 0$.
Each point has $\mathcal{H}^s$ measure $0$ for $s > 0$, and $\mathcal{H}^s$ is countably subadditive.

For subsets of $\R^n$, one has $\dim_H A \le n$, and every nonempty open set has dimension $n$.

## Explanations

### L1 human

The Hausdorff dimension is the exponent $s$ at which the $s$-dimensional size of a set drops from infinite to zero.
It measures how a set fills space at small scales.
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# Macdonald polynomial

## Summary

Macdonald polynomials $P_\lambda(x; q, t)$ are [[symmetric functions|PAJEHWXW6M]] with coefficients in the field $\Q(q, t)$ of rational functions in two parameters, indexed by [[partitions|0JV30XN73Z]].
They are characterized by a triangularity condition and orthogonality for a two-parameter inner product.
Special values and limits of $q$ and $t$ give the [[Schur polynomials|43VFMJCDYJ]], the Hall–Littlewood polynomials and the Jack polynomials.
Macdonald introduced them in the 1980s, and they connect combinatorics with representation theory and special functions.

## Explanations

### L1 human

Many classical families of symmetric polynomials are special cases of one family with two extra parameters $q$ and $t$.
Setting $q = t$ gives back the Schur polynomials, and setting $q = 0$ gives the Hall–Littlewood polynomials.
For example, for the partition $(2)$ the Macdonald polynomial is $m_{(2)} + \frac{(1+q)(1-t)}{1-qt} m_{(1,1)}$, where $m_{(2)} = \sum_i x_i^2$ and $m_{(1,1)} = \sum_{i<j} x_i x_j$.
At $q = t$ the fraction equals $1$, and the result is the Schur polynomial $s_{(2)}$.
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# Graph minor

## Summary

A graph $H$ is a minor of a graph $G$ if $H$ can be obtained from $G$ by deleting vertices, deleting edges and contracting edges.
The minor relation is a partial order on finite graphs up to isomorphism.
Many natural classes, such as planar graphs and forests, are closed under taking minors.
[[Wagner's theorem|1B2ZJPVV03]] describes planar graphs by two forbidden minors, and the [[Robertson–Seymour theorem|WP3GDNWJJW]] shows that every minor-closed class has a finite list of forbidden minors.

## Explanations

### L1 human

Take a graph and allow three moves: erase a vertex, erase an edge, or shrink an edge so that its two ends merge into one vertex.
Any graph you can reach this way is a minor of the original.
For example, shrinking the five spokes of the Petersen graph turns it into the complete graph $K_5$.
Minors capture the idea of one graph being hidden inside another in a coarse, flexible way.
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# Function field of a variety

## Definition

Let $k$ be an algebraically closed field and let $X$ be an [[algebraic variety|HV6YBS0YDH]] over $k$.
Consider pairs $(U, f)$ where $U \subseteq X$ is a nonempty [[open subset|02BH3MCBQB]] and $f$ is a [[regular function|4M94FZH9Y8]] on $U$.
Two pairs $(U, f)$ and $(V, g)$ are equivalent if $f = g$ on $U \cap V$.
The *function field* $k(X)$ of $X$ is the set of equivalence classes, called *rational functions* on $X$, with the operations
$$[(U, f)] + [(V, g)] = [(U \cap V, f + g)], \qquad [(U, f)] \cdot [(V, g)] = [(U \cap V, fg)].$$

## Remarks

Since $X$ is irreducible, $U \cap V$ is nonempty and dense, and two regular functions that agree on a nonempty open subset of $U \cap V$ agree on all of $U \cap V$.
This makes the relation transitive and the operations well defined.
$k(X)$ is a field: if $f \ne 0$ on $U$, then $1/f$ is regular on the nonempty open set $U \setminus V(f)$.

If $X$ is an [[affine variety|JERVQT4C12]], then $k(X)$ is the [[field of fractions|0F2PTGX1A2]] of the [[coordinate ring|G9WB27VWP7]] $k[X]$.
If $U$ is a nonempty open subset of $X$, then $k(U) = k(X)$.
$k(X)$ is a [[function field|XKEXQDRTGH]] over $k$ in the algebraic sense: a finitely generated field extension of $k$.

Example: $k(\mathbb{A}^n) = k(x_1, \dots, x_n)$, the [[field of rational functions|4V81611P9N]], since $k[\mathbb{A}^n]$ is the polynomial ring.
Example: for the parabola $X = V(y - x^2)$, $k[X] \cong k[x]$, so $k(X) \cong k(x)$.

Non-example: the construction needs irreducibility.
In $X = V(xy) \subseteq k^2$, the open sets $\set{x \ne 0}$ and $\set{y \ne 0}$ of $X$ are nonempty and disjoint.
So the relation above is not transitive there, and the analogous ring built from $k[X] = k[x, y]/(xy)$, its total ring of fractions, is $k(x) \times k(y)$, which is not a field.
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# Trefoil knot

## Example

The *trefoil knot* is the image of the map $f \colon \R / 2\pi\Z \to \R^3$ given by
$$f(t) = \bigl((2 + \cos 3t) \cos 2t, \ (2 + \cos 3t) \sin 2t, \ \sin 3t\bigr).$$
The map is continuous, and it is injective.
Indeed, if $f(t) = f(s)$, the last coordinate and the distance $2 + \cos 3t$ from the $z$-axis give $3t \equiv 3s \pmod{2\pi}$.
The angle around the $z$-axis gives $2t \equiv 2s \pmod{2\pi}$.
Then $t - s$ is a multiple of $2\pi/3$ and of $\pi$, so of $2\pi$.
Hence the image is a [[knot|QYEAZTYGPT]].
It lies on a torus, and it winds $2$ times around the axis of the torus and $3$ times around its core.
So it is the [[torus knot|NS60VV786W]] of type $(2, 3)$.

## Remarks

Its [[crossing number|SWJZ4QRF9W]] is $3$, and it is the only knot with crossing number $3$ up to mirror image.
It is [[tricolorable|KNAGK0W65G]] while the unknot is not, so it is not the unknot.
Its [[knot group|AQVHM3FP0D]] has the presentation $\langle x, y \mid xyx = yxy \rangle$, which is isomorphic to $\langle a, b \mid a^2 = b^3 \rangle$.
Its Alexander polynomial is $t^{-1} - 1 + t$, and its genus is $1$.
It is not equivalent to its mirror image, a fact that the Jones polynomial detects.

## Explanations

### L1 human

The trefoil is the simplest real knot: tie an overhand knot in a string and glue the ends together.
Any picture of it has at least three crossings, and the standard picture has three lobes like a clover leaf.
It comes in two mirror forms, left-handed and right-handed, and these cannot be moved into each other.
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# Transfer homomorphism

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of finite [[index|0DTXBQPA2S]] $n$.
Let $H'$ be the [[commutator subgroup|0ASGSTXSBW]] of $H$, so that $H/H'$ is the [[abelianisation|0AQZV0905N]] of $H$.
Choose elements $t_1, \dots, t_n$ of $G$ such that $G$ is the disjoint union of the [[left cosets|03DEH43F4H]] $t_1 H, \dots, t_n H$.
For $g \in G$ and each $i$, there are a unique index $\sigma_g(i)$ and a unique $h_i(g) \in H$ with $g t_i = t_{\sigma_g(i)} h_i(g)$.
The *transfer* is the map $V \colon G \to H/H'$ given by
$$V(g) = h_1(g) h_2(g) \cdots h_n(g) \, H'.$$

## Remarks

The product is taken in the abelian group $H/H'$, so its order does not matter.
One proves that $V$ is a [[group homomorphism|0JYTG5K9Y4]] and that it does not depend on the choice of $t_1, \dots, t_n$.

Example: if $G$ is abelian, then $H' = \set{e}$ and $h_i(g) = t_{\sigma_g(i)}^{-1} g t_i$.
Since $\sigma_g$ permutes $\set{1, \dots, n}$, the factors $t_i$ cancel, and $V(g) = g^n$.
For $G = \Z$ and $H = n\Z$, the transfer is $V(k) = nk$.

Non-example: take the product $h_1(g) \cdots h_n(g)$ in $H$ itself, without passing to $H/H'$.
In general it depends on the order of the factors and on the choice of the $t_i$, so it does not give a well-defined homomorphism to $H$.

A main application is Burnside's normal $p$-complement theorem.
Let $P$ be a [[Sylow $p$-subgroup|0053CQE9Y2]] of a finite group $G$ that lies in the [[centre|0DZ93493VN]] of its [[normaliser|0YGN3VAA0D]].
Then $G$ has a normal subgroup $K$ with $G = KP$ and $K \cap P = \set{e}$.
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# Akra–Bazzi theorem

## Statement

Let $k \ge 1$, and let $a_1, \dots, a_k > 0$ and $0 < b_1, \dots, b_k < 1$ be real constants.
Let $g \colon [1, \infty) \to [0, \infty)$ satisfy the polynomial growth condition: there are constants $c_1, c_2 > 0$ such that $c_1 g(x) \le g(u) \le c_2 g(x)$ for all $x \ge 1$, all $i$ and all $u \in [b_i x, x]$.
Let $h_1, \dots, h_k$ be real functions with $\abs{h_i(x)} = O(x / (\log x)^2)$, in [[big O notation|0TGV80E6A4]].
Let $x_0$ be large enough, and let $T \colon [1, \infty) \to \R$ be positive and bounded above and below by positive constants on $[1, x_0]$, and satisfy the recurrence
$$T(x) = g(x) + \sum_{i=1}^{k} a_i\, T\big(b_i x + h_i(x)\big) \qquad (x > x_0).$$
Let $p$ be the unique real number with $\sum_{i=1}^{k} a_i b_i^p = 1$.
Then
$$T(x) = \Theta\left( x^p \left( 1 + \int_1^x \frac{g(u)}{u^{p+1}}\, du \right) \right),$$
where $f(x) = \Theta(F(x))$ means that $c F(x) \le f(x) \le C F(x)$ for some constants $0 < c \le C$ and all large $x$.

## Remarks

The number $p$ exists and is unique because $s \mapsto \sum a_i b_i^s$ is continuous and strictly decreasing, with limits $\infty$ and $0$.
The proof is by induction on $x$: one guesses the bound with suitable constants and checks that the recurrence preserves it.
The growth condition on $g$ is used to compare $g$ at $b_i x$ and at $x$.
The perturbations $h_i$ cover floors and ceilings, such as $T(\lceil x/2 \rceil)$, which appear in real algorithms.
Example: for $T(x) = T(x/3) + T(2x/3) + x$, one has $p = 1$ since $1/3 + 2/3 = 1$, and $\int_1^x u/u^2\, du = \log x$, so $T(x) = \Theta(x \log x)$.
The growth condition on $g$ is needed.
For $T(x) = T(x/2) + 2^x$, one has $p = 0$, and the formula would give the order $2^x / x$, since $\int_1^x 2^u u^{-1}\, du$ grows like $2^x/(x \log 2)$; but $T(x) \ge 2^x$.
Here $g(x) = 2^x$ fails the growth condition, as $g(x)/g(x/2) = 2^{x/2}$ is unbounded.

## Explanations

### L1 human

Divide-and-conquer algorithms split a problem of size $x$ into smaller pieces of sizes $b_i x$, solve them, and pay $g(x)$ to split and combine.
Their running time satisfies a recurrence like the one above.
The theorem gives the order of growth of the running time directly, without solving the recurrence.
For merge sort, $T(x) = 2 T(x/2) + x$ gives $p = 1$ and $T(x) = \Theta(x \log x)$.

### L2 human

The theorem generalizes the master theorem of algorithm analysis, which handles only equal subproblem sizes $T(x) = a T(x/b) + g(x)$.
Here the pieces may have different sizes $b_i x$, and small perturbations $h_i$ are allowed.
To use it, first solve $\sum a_i b_i^p = 1$ for $p$, then compute the integral of $g(u)/u^{p+1}$.
If $g(x) = x^q$ with $q < p$, the integral stays bounded and $T = \Theta(x^p)$; if $q = p$, one gets $\Theta(x^p \log x)$; if $q > p$, one gets $\Theta(x^q)$.
In Mathlib a version is `AkraBazziRecurrence.isTheta_asympBound`.
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# Herbrand's theorem

## Statement

Consider first-order logic without equality over a [[signature|WDVVEYD6AT]] $\sigma$ with at least one constant symbol.
Let $\varphi = \forall x_1 \cdots \forall x_n \, \psi(x_1, \dots, x_n)$ be a sentence in [[Skolem normal form|GCPQGMJYEV]], with $\psi$ quantifier-free.
Then $\varphi$ is unsatisfiable if and only if there are finitely many tuples $\bar t_1, \dots, \bar t_k$ of closed [[terms|6D3WA7N2W4]] with the following property.
The conjunction $\psi(\bar t_1) \land \cdots \land \psi(\bar t_k)$ is unsatisfiable as a propositional formula, with its atomic sentences read as propositional variables.

## Remarks

The instances $\psi(\bar t)$ with closed terms $\bar t$ are the ground instances of $\varphi$.
For the proof, suppose every finite set of ground instances is propositionally [[satisfiable|QN7M1JCGT9]].
By propositional compactness, all ground instances are satisfiable at once by some assignment.
This assignment defines a Herbrand structure, whose elements are the closed terms, and this structure is a model of $\varphi$.
The other direction holds because $\varphi$ implies each of its ground instances.
Example: let $\varphi = \forall x \, (P(x) \land \neg P(f(x)))$ with a constant $c$.
The instances for $x = c$ and $x = f(c)$ contain $\neg P(f(c))$ and $P(f(c))$, so their conjunction is unsatisfiable, and so is $\varphi$.
If $\sigma$ has no constant symbol, one adds one; without closed terms there are no ground instances.

## Explanations

### L1 human

The theorem reduces a question about all possible structures to a question about finitely many concrete cases.
To show that a universal sentence is contradictory, it is enough to plug in finitely many concrete terms and find a contradiction by truth tables.
This is the basis of many automatic theorem provers.

### L2 human

Combined with Skolemization, the theorem gives a semi-decision procedure for validity in first-order logic.
To test whether $\chi$ is valid, put $\neg \chi$ in Skolem normal form and search through larger and larger sets of ground instances.
If $\chi$ is valid, the search ends; by [[Church's theorem|VA6BNNY2YC]] it cannot always end when $\chi$ is not valid.
The theorem is closely related to the [[compactness theorem|1NR3HAWG9D]] and to [[Gödel's completeness theorem|WQ1ZE255SF]].
With equality, the same holds after adding the equality axioms for $\sigma$ to $\varphi$.
</document>

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# Whittaker transform

## Summary

A Whittaker transform is an [[integral transform|YCK709QB06]] on $(0, \infty)$ whose kernel is built from a [[Whittaker function|7EFDMJSVDP]] $W_{\kappa, \mu}$ of the product $xy$.
Several normalizations are in use in the literature.
The form recorded here contains the [[Laplace transform|0V49JEZW06]] as the special case $\kappa = \mu + 1/2$.
Whittaker functions are confluent hypergeometric functions, so these transforms belong to the study of transforms with special function kernels.

## Explanations

### L1 human

Whittaker functions are solutions of a second order differential equation that also produces the exponential function, the error function and Laguerre polynomials as special cases.
A Whittaker transform integrates a function against such a solution instead of against $e^{-xy}$.
With two parameters $\kappa$ and $\mu$ it gives a family of transforms that bends the Laplace transform in two directions.
For one choice of the parameters the kernel becomes exactly $e^{-xy}$, and nothing new happens.
</document>

<document source="https://tamnd.github.io/math2/node/VZ5QQMDV2D.md" tag="VZ5QQMDV2D" kind="definition">
# Pseudoanalytic function

## Definition

Let $\Omega \subset \C$ be a [[domain|0EWMDD6C14]].
A *generating pair* on $\Omega$ is a pair $(F, G)$ of [[Hölder continuous|0WZTM0EFR6]] functions $\Omega \to \C$ with $\operatorname{Im}\left( \overline{F(z)}\, G(z) \right) > 0$ for all $z \in \Omega$.
Then every function $w \colon \Omega \to \C$ can be written uniquely as $w = \varphi F + \psi G$ with $\varphi, \psi \colon \Omega \to \R$.
The function $w$ is *$(F, G)$-pseudoanalytic* if $\varphi$ and $\psi$ have continuous first partial derivatives and
$$\varphi_{\bar z}\, F + \psi_{\bar z}\, G = 0 \quad \text{on } \Omega,$$
where $\partial_{\bar z} = \frac{1}{2}\left( \partial_x + i \partial_y \right)$ is the [[Wirtinger derivative|WMXDVF36ZT]].

## Remarks

The condition $\operatorname{Im}(\bar F G) > 0$ says that $F(z)$ and $G(z)$ are linearly independent over $\R$, which gives the uniqueness of $\varphi$ and $\psi$.

Example: for $(F, G) = (1, i)$, the condition reads $(\varphi + i \psi)_{\bar z} = 0$.
So the $(1, i)$-pseudoanalytic functions are exactly the [[holomorphic functions|0NAHXR1FRH]] with continuous derivative.

Non-example: $w(z) = \bar z$ is not $(1, i)$-pseudoanalytic.
Here $\varphi = x$ and $\psi = -y$, so $\varphi_{\bar z} + i \psi_{\bar z} = \frac{1}{2} + i \cdot \left( -\frac{i}{2} \right) = 1 \ne 0$.

If $F$ and $G$ are continuously differentiable, then $w_{\bar z} = \varphi F_{\bar z} + \psi G_{\bar z}$ for a pseudoanalytic $w$.
Since $\varphi$ and $\psi$ are real-linear in $w$, this is an equation $w_{\bar z} = a w + b \bar w$ with coefficients $a, b$ determined by $F$ and $G$.
Some authors ask for less regularity of $\varphi$ and $\psi$ and use a weak form of the condition.
</document>

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# Ehresmann connection

## Definition

Let $\pi : E \to M$ be a smooth [[fibre bundle|CX4M8K61AB]], and let $TE$ be the [[tangent bundle|WY6K3P4YCG]] of $E$.
The *vertical bundle* is $V = \ker d\pi \subset TE$, where $d\pi$ is the [[differential|6P9PXQDSEN]] of $\pi$.
Its fibre $V_e$ is the tangent space at $e$ of the fibre through $e$.
An *Ehresmann connection* on $E$ is a smooth subbundle $H \subset TE$, the *horizontal bundle*, such that $T_eE = H_e \oplus V_e$ for every $e \in E$.

## Remarks

Since $d\pi_e$ is onto with kernel $V_e$, it restricts to an isomorphism $H_e \to T_{\pi(e)}M$.
So each tangent vector of $M$ has a unique horizontal lift at each point of the fibre above it.
Example: for the product bundle $E = M \times F$ with $\pi$ the first projection, $H_{(p, y)} = T_pM \times \set{0}$ is an Ehresmann connection; $V_{(p,y)} = \set{0} \times T_yF$, and the two are complementary.
Non-example: $H = TE$ fails the condition when the fibres have positive dimension, because $H_e \cap V_e = V_e \ne \set{0}$.
Equivalently, an Ehresmann connection is a smooth vector-valued $1$-form $\Phi$ on $E$ with $\Phi \circ \Phi = \Phi$ and image $V$; then $H = \ker \Phi$.
Horizontal lifts of curves exist locally; if the fibres are not compact they may fail to exist for all times.
On a vector bundle, the Ehresmann connections invariant under the scalings of the fibres and compatible with addition correspond to connections in the sense of covariant derivatives.
</document>

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# Medial magma

## Summary

A medial magma is a [[magma|9689SFPVE9]] that satisfies the medial law $(x y)(u v) = (x u)(y v)$: the two middle elements of a product of four may be swapped.
Every commutative semigroup is medial, and so is every operation of the form $x \cdot y = \varphi(x) + \psi(y) + c$ on an abelian group with commuting endomorphisms $\varphi, \psi$.
For quasigroups, the [[Toyoda–Bruck theorem|W0KBNZBCRR]] says that these affine operations are the only medial ones.
Mediality is the binary case of the [[entropic property|9T2XVJNBBY]] of general algebras.

## Explanations

### L1 human

Write four elements in a square: $x$ and $y$ in the top row, $u$ and $v$ in the bottom row.
You can combine each row and then combine the two results, or combine each column and then combine the results.
A magma is medial when both ways give the same answer.
For example, the average $x \circ y = (x + y)/2$ on the rationals is medial: both ways give the average of all four numbers.
</document>

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# Boolean product

## Definition

Let $X$ be a [[Stone space|ZV38HA5PAQ]], that is, a compact Hausdorff totally disconnected space.
Let $(\mathbf{A}_x)_{x \in X}$ be a family of [[algebras|QJGPT2W0FT]] of the same signature.
A *Boolean product* of this family is a [[subdirect product|88WN3AD2VQ]] $\mathbf{A}$ of the [[direct product|WJ73XYBNE4]] $\prod_{x \in X} \mathbf{A}_x$ with these two properties:

1. for all $a, b \in A$, the set $\set{x \in X : a(x) = b(x)}$ is [[clopen|DQ4E55T3BW]] in $X$;
2. for all $a, b \in A$ and every clopen set $N \subseteq X$, the function $c$ with $c(x) = a(x)$ for $x \in N$ and $c(x) = b(x)$ for $x \notin N$ lies in $A$.

## Remarks

Clause 2 is called the patchwork property.
An algebra is said to be representable as a Boolean product when it is isomorphic to a Boolean product of some family.

Example: let $\mathbf{B}$ be a Boolean algebra with Stone space $X$, and let $\mathbf{2}$ be the two-element Boolean algebra.
The continuous maps $X \to \mathbf{2}$, with $\mathbf{2}$ discrete, form a Boolean product of copies of $\mathbf{2}$ isomorphic to $\mathbf{B}$.
Two continuous maps to a discrete space agree on a clopen set, and pasting two continuous maps along a clopen set gives a continuous map.

Non-example: let $X = \set{p, q}$ be discrete, and let $A = \set{(0, 0), (1, 1)}$ inside $\mathbf{2} \times \mathbf{2}$.
This is a subdirect product and satisfies clause 1, but pasting $(0, 0)$ on $\set{p}$ with $(1, 1)$ on $\set{q}$ gives $(0, 1) \notin A$, so clause 2 fails.
</document>

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# Cascade algorithm

## Summary

The cascade algorithm computes the [[scaling function|YH4BERB952]] of a wavelet from its filter coefficients by iterating the refinement equation.
It starts from a simple function, such as the indicator of $[0,1)$.
Each step builds a new function from shrunken and shifted copies of the old one.
For good filters, such as the filters of the Daubechies wavelets, the iterates converge to the scaling function.
Most compactly supported wavelets have no closed formula, so this is the usual way to draw them.

## Explanations

### L1 human

A scaling function is defined by a self-similarity rule: it is a weighted sum of half-size copies of itself.
The cascade algorithm uses this rule as a recipe.
Start with a box, apply the rule to get a new shape, then apply it again to the new shape, and so on.
For the Haar filter the box is already a fixed point: two half-width boxes side by side make the box again.
For Daubechies filters the shapes become finer and more jagged at each step and settle down to a fractal-looking curve.
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# Germ

## Summary

The germ of a function at a point records the behaviour of the function on arbitrarily small [[neighbourhoods|0ZCY0Z80B6]] of the point and forgets the rest.
Two functions have the same germ at $x$ when they agree on some neighbourhood of $x$.
Germs make local properties, such as derivatives and local power series, into properties of single objects.
They are the stalks of the sheaf of functions and appear throughout analysis, geometry and singularity theory.

## Explanations

### L1 human

Look at a function through a microscope at one point, and keep zooming in.
Two functions have the same germ at the point if, after enough zoom, they look identical.
For example, $t \mapsto t$ and $t \mapsto \abs{t}$ on $\R$ have the same germ at $1$, since they agree on $(0, 2)$.
They have different germs at $0$, because every interval around $0$ contains negative numbers, where they differ.
</document>

<document source="https://tamnd.github.io/math2/node/W007WWX7AN.md" tag="W007WWX7AN" kind="definition">
# Quasi-compact morphism

## Definition

A topological space is *quasi-compact* if every open cover of it has a finite subcover, as for a [[compact space|0TVJZYB6SH]] without any separation axiom.
A [[morphism of schemes|JXRKH6P1Z4]] $f \colon X \to Y$ is *quasi-compact* if for every open [[affine|B3V3XJ2W8G]] subscheme $V \subseteq Y$ the open set $f^{-1}(V)$ is quasi-compact.

## Remarks

It is enough to check this for the sets $V_i$ of one cover of $Y$ by open affine subschemes.
Every affine scheme is quasi-compact, by [[quasi-compactness of the spectrum|J4YW23MD6N]].
Example: every closed immersion $f \colon Z \to Y$ is quasi-compact.
For $V$ open affine, $f^{-1}(V)$ is homeomorphic to a closed subset of the quasi-compact space $V$, and closed subsets of quasi-compact spaces are quasi-compact.
Example: if $X$ is a Noetherian scheme, every morphism $X \to Y$ is quasi-compact, since every subset of a Noetherian space is quasi-compact.
Non-example: let $X$ be the disjoint union of infinitely many copies of $\operatorname{Spec} k$ for a field $k$, and $f \colon X \to \operatorname{Spec} k$ the obvious map.
Then $f^{-1}(\operatorname{Spec} k) = X$ is covered by the copies, with no finite subcover, so $f$ is not quasi-compact.
</document>

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# Lévy's upward theorem

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_n)_{n \ge 0}$, and let $\mathcal{F}_\infty$ be the [[σ-algebra|0SV8K0TDGY]] generated by $\bigcup_{n} \mathcal{F}_n$.
Let $X$ be a real random variable with $E[\abs{X}] < \infty$.
Then
$$E[X \mid \mathcal{F}_n] \to E[X \mid \mathcal{F}_\infty] \quad \text{as } n \to \infty,$$
[[almost surely|0AAXWGT5TD]] and in $L^1$, where $E[\cdot \mid \cdot]$ is the [[conditional expectation|0YAMJRK04E]].

## Remarks

The sequence $M_n = E[X \mid \mathcal{F}_n]$ is the [[Doob martingale|Z57CG6W9S6]] of $X$, and it is uniformly integrable.
By [[Doob's martingale convergence theorems|MRXBC4ZNRR]] it converges almost surely and in $L^1$ to some $M_\infty$, which is $\mathcal{F}_\infty$-measurable.
For $A \in \mathcal{F}_m$ and $n \ge m$, $E[M_n \mathbf{1}_A] = E[X \mathbf{1}_A]$, and the $L^1$ limit gives $E[M_\infty \mathbf{1}_A] = E[X \mathbf{1}_A]$.
A π–λ argument extends this to all $A \in \mathcal{F}_\infty$, so $M_\infty = E[X \mid \mathcal{F}_\infty]$.
Taking $X = \mathbf{1}_A$ with $A \in \mathcal{F}_\infty$ gives Lévy's zero–one law: $P(A \mid \mathcal{F}_n) \to \mathbf{1}_A$ almost surely.

## Explanations

### L1 human

As you learn more and more, your best guess of a random quantity approaches the best guess based on all the information you will ever have.
If all the information together determines the quantity, the guesses converge to the true value.
For example, learn the binary digits of a uniform random number one by one.
Your best guess of any integrable function of the number then converges to its true value almost surely.

### L2 human

The theorem is a probabilistic counterpart of the Lebesgue differentiation theorem.
On $[0, 1)$ with the dyadic filtration, $E[X \mid \mathcal{F}_n]$ is the average of $X$ over the dyadic interval of length $2^{-n}$ containing the point.
Lévy's zero–one law follows at once, and it gives Kolmogorov's zero–one law for tail events of independent sequences.
A companion result, Lévy's downward theorem, treats decreasing families of σ-algebras.
</document>

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# Canonical bundle

## Summary

The canonical bundle $\omega_X$ of a smooth variety $X$ of dimension $n$ is the line bundle of differential forms of top degree $n$.
It is the $n$-th exterior power of the [[sheaf of differentials|6VC6HQTFSA]].
A divisor $K_X$ with $\mathcal{O}_X(K_X) \cong \omega_X$ is a canonical divisor.
The canonical bundle appears in [[Serre duality|N3QPQATSRN]], in the [[adjunction formula|RYB6F7JFGV]] and in the [[Kodaira dimension|ZW00Q0SAGR]].
Its sign governs the broad classification of varieties: Fano varieties have anti-ample $\omega_X$, Calabi–Yau varieties trivial $\omega_X$, and varieties of general type big $\omega_X$.

## Explanations

### L1 human

On a surface, a differential form of top degree looks locally like $g \, dx \wedge dy$, a way to measure small areas.
The canonical bundle is the line bundle whose sections are such top-degree forms.
On a curve these are the forms $g \, dz$.
On the projective line, the form $dz$ has a double pole at infinity, since $w = 1/z$ gives $dz = -dw / w^2$.
So the canonical bundle of $\mathbb{P}^1$ has degree $-2$, and for a curve of genus $g$ its degree is $2g - 2$.
</document>

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# Well-posed problem

## Summary

A problem is well posed in the sense of Hadamard if it has a unique solution that depends continuously on the data.
These conditions say that a model of a physical process gives one answer, stable under small errors in the data.
A problem that fails one of them is ill posed; such problems, like the backward heat equation or many inverse problems, need extra care.
See the [[definition|FZ64956KX2]].

## Explanations

### L1 human

Data from the real world always carry small errors.
If a tiny change of the data could change the answer completely, the model would be useless for prediction.
A well-posed problem has exactly one answer, and nearby data give nearby answers.
For example, fixing the temperature on the edge of a disc determines the steady temperature inside.
Changing the edge values by at most $0.1$ degrees changes the inside by at most $0.1$ degrees.
</document>

<document source="https://tamnd.github.io/math2/node/W0B0BBAH35.md" tag="W0B0BBAH35" kind="concept">
# Universally closed morphism

## Summary

A morphism of schemes is universally closed if it stays a closed map of topological spaces after every base change.
Being closed alone is too weak a condition, because the product topology is not the Zariski topology of a fibre product.
Universal closedness is the topological half of the definition of a proper morphism.

## Explanations

### L1 human

A closed map sends closed sets to closed sets.
For schemes one asks more: the map must stay closed even after one adds extra parameters to the base.
The map from the affine line to a point is closed, since the image of anything is the point or empty.
After one adds a parameter, it becomes the projection of the plane to a line.
This projection sends the hyperbola $xy = 1$ onto the line with the origin removed, which is not closed.
So the affine line is not universally closed over a point, which reflects the missing point at infinity.
</document>

<document source="https://tamnd.github.io/math2/node/W0B5CQVV8V.md" tag="W0B5CQVV8V" kind="concept">
# Whittle likelihood

## Summary

The Whittle likelihood approximates the Gaussian likelihood of a stationary time series using the [[periodogram|3ZYM0HXA3E]] and the model spectral density.
It replaces a large covariance matrix by a sum over frequencies, which makes estimation fast.
Peter Whittle introduced it in the 1950s, and maximizing it gives estimators with the same first-order asymptotic behaviour as Gaussian maximum likelihood.

## Explanations

### L1 human

Neighbouring values of a time series are dependent, which makes the exact likelihood awkward to compute.
After a Fourier transform, the values at different frequencies are nearly independent.
The Whittle likelihood treats them as independent, each with a spread given by the model's spectrum at that frequency.
Fitting a model then means matching the model spectrum to the observed periodogram, frequency by frequency.
</document>

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# Helmholtz equation

## Definition

Let $n \ge 1$, let $\Omega \subseteq \R^n$ be an open set, and let $k > 0$ be a real constant.
Let $u \colon \Omega \to \C$ be twice differentiable.
The *Helmholtz equation* on $\Omega$ with wave number $k$ is the [[partial differential equation|GC5BDZ6QBC]]
$$\Delta u + k^2 u = 0,$$
where $\Delta = \partial_1^2 + \cdots + \partial_n^2$ is the [[Laplacian|0E0B6RA8K0]].
The *inhomogeneous Helmholtz equation* is $\Delta u + k^2 u = -f$ for a given function $f$ on $\Omega$.

## Remarks

Example: in $\R^n$, for a unit vector $\omega$, the plane wave $u(x) = e^{i k \, \omega \cdot x}$ is a solution, since $\Delta u = (ik)^2 \abs{\omega}^2 u = -k^2 u$.
Non-example: $u(x) = e^{k x_1}$ is not a solution, since $\Delta u + k^2 u = 2k^2 u \ne 0$; it solves instead $\Delta u - k^2 u = 0$, the modified Helmholtz equation.
If $v(x, t) = u(x) e^{-i \omega t}$ solves the wave equation $\partial_t^2 v = c^2 \Delta v$, then $u$ solves the Helmholtz equation with $k = \omega / c$.
On a bounded domain with zero boundary values, the equation has a nonzero solution exactly when $k^2$ is an eigenvalue of $-\Delta$ with that boundary condition.
Some authors allow complex $k$.
</document>

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# Projective subspace

## Definition

Let $V$ be a vector space over a field $k$, and let $\mathbb{P}(V)$ be the [[projective space|ZT9A6JK8QF]] of one-dimensional subspaces of $V$.
For a nonzero vector $v$, write $[v]$ for the point spanned by $v$.
A *projective subspace* of $\mathbb{P}(V)$ is a set of the form
$$\mathbb{P}(W) = \set{[w] : w \in W, \ w \ne 0},$$
where $W$ is a [[linear subspace|00YC1D652K]] of $V$.
Its *projective dimension* is $\dim W - 1$.

## Remarks

Equivalently, a set $S$ of points is a projective subspace when, for all $[v], [w] \in S$ and all scalars $a, b$ with $a v + b w \ne 0$, the point $[a v + b w]$ lies in $S$.
In words, $S$ contains the whole line through any two of its distinct points.

Example: in $\mathbb{P}^2(\R)$, the set $\set{[x_0 : x_1 : x_2] : x_0 = 0}$ is a projective subspace of dimension $1$, a line.
Check: it is $\mathbb{P}(W)$ for the plane $W = \set{x_0 = 0}$ of $\R^3$, which has [[dimension|03GSZ8RTZJ]] $2$.

Non-example: the set $\set{[1:0:0], [0:1:0]}$ is not a projective subspace.
The line through its two points also contains $[1:1:0]$, which is not in the set.

The empty set is $\mathbb{P}(0)$, of dimension $-1$, and $\mathbb{P}(V)$ itself is a projective subspace.
For finite-dimensional $V$, a projective subspace of dimension $\dim V - 2$ is a hyperplane.
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# Toyoda–Bruck theorem

## Statement

Let $(Q, \cdot)$ be a quasigroup.
Then $(Q, \cdot)$ is [[medial|VZHN6GA3TA]], that is, $(x \cdot y) \cdot (u \cdot v) = (x \cdot u) \cdot (y \cdot v)$ for all $x, y, u, v \in Q$, if and only if there exist:

1. an [[abelian group|082GHNPJ53]] structure $(Q, +)$ on the same set;
2. two [[automorphisms|DSQ17TNW1C]] $\varphi$ and $\psi$ of $(Q, +)$ with $\varphi \circ \psi = \psi \circ \varphi$;
3. an element $c \in Q$;

such that $x \cdot y = \varphi(x) + \psi(y) + c$ for all $x, y \in Q$.

## Remarks

The converse direction is a direct computation.
With $x \cdot y = \varphi(x) + \psi(y) + c$, the product $(x \cdot y) \cdot (u \cdot v)$ equals $\varphi^2(x) + \varphi\psi(y) + \psi\varphi(u) + \psi^2(v) + \varphi(c) + \psi(c) + c$.
Swapping $y$ and $u$ gives the same value because $\varphi\psi = \psi\varphi$, and the operation is a quasigroup because $\varphi$ and $\psi$ are bijective.
For the main direction, one fixes an element of $Q$ and defines a new addition on $Q$ from the quasigroup operation by an isotopy.
Mediality forces this addition to be associative and commutative, and the quasigroup operation then becomes affine in each argument.

Example: on $\Z/5\Z$ the operation $x \cdot y = 2x + 3y$ is a medial quasigroup, with $\varphi(x) = 2x$, $\psi(y) = 3y$ and $c = 0$.
Example: on $\Z/3\Z$ the operation $x \cdot y = -x - y$ is a medial quasigroup, with $\varphi = \psi = -\id$ and $c = 0$.

The quasigroup hypothesis is needed.
The commutative semigroup $(\N, \cdot)$ is medial, but it is not of this form.
In an operation $\varphi(x) + \psi(y) + c$ each map $y \mapsto x \cdot y$ is a bijection, while in $(\N, \cdot)$ the map $y \mapsto 0 \cdot y$ is constant.

## Explanations

### L1 human

A quasigroup is a set with an operation whose "multiplication table" is a Latin square: every element appears exactly once in each row and each column.
The theorem says that such an operation obeys the medial law only when it comes from an abelian group in a simple way.
One twists each argument by a symmetry of the group, adds, and shifts by a constant.
So a single identity forces the whole structure to be essentially linear.

### L2 human

The theorem is the main structure result for medial quasigroups.
It reduces questions about medial quasigroups to questions about abelian groups with two commuting automorphisms.
It is also called the Bruck–Murdoch–Toyoda theorem.
If the medial quasigroup is moreover idempotent, so that $x \cdot x = x$, then putting $x = 0$ gives $c = 0$, and then $\varphi + \psi = \id$.
Such quasigroups are the affine combinations $x \cdot y = \varphi(x) + (\id - \varphi)(y)$, as in the midpoint operation $(x + y)/2$ on $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/W0M20JWC9K.md" tag="W0M20JWC9K" kind="statement">
# Quartic reciprocity

## Statement

Let $\Z[i]$ be the ring of [[Gaussian integers|074P7C85XH]], with norm $N(a + bi) = a^2 + b^2$.
Let $\pi$ be a prime element of $\Z[i]$ that does not divide $1 + i$.
For $\alpha \in \Z[i]$ not divisible by $\pi$, the quartic residue character $\left(\frac{\alpha}{\pi}\right)_4$ is the unique element of $\set{1, i, -1, -i}$ with
$$\alpha^{(N(\pi) - 1)/4} \equiv \left(\frac{\alpha}{\pi}\right)_4 \pmod{\pi}.$$
A prime $\pi$ is called primary if $\pi \equiv 1 \pmod{(1 + i)^3}$.
Let $\lambda$ and $\pi$ be primary primes of $\Z[i]$ that are not associates.
Then
$$\left(\frac{\lambda}{\pi}\right)_4 = \left(\frac{\pi}{\lambda}\right)_4 \cdot (-1)^{\frac{N(\lambda) - 1}{4} \cdot \frac{N(\pi) - 1}{4}}.$$

## Remarks

The proof follows the same pattern as for cubic reciprocity, with quartic [[Gauss sums|EQ6A3ZN48K]] and [[Jacobi sums|2767J23XHN]] over the field with $p$ elements for $p \equiv 1 \pmod 4$.
Each prime of $\Z[i]$ not dividing $1 + i$ has exactly one primary associate.
The sign is $-1$ only when both $\lambda$ and $\pi$ are congruent to $3 + 2i$ modulo $4$.
A classical consequence: for a prime $p \equiv 1 \pmod 8$, the integer $2$ is a fourth power modulo $p$ if and only if $p = x^2 + 64y^2$ for some integers $x, y$.
For example, $73 = 3^2 + 64 \cdot 1^2$, and $2^{18} \equiv 1 \pmod{73}$, so $2$ is a fourth power modulo $73$.
The prime $17$ is not of this form, and the nonzero fourth powers modulo $17$ are $1$, $4$, $13$ and $16$.

## Explanations

### L1 human

Quartic reciprocity is the analogue of [[quadratic reciprocity|085MJS6907]] for fourth powers.
It compares "is $\lambda$ a fourth power modulo $\pi$?" with "is $\pi$ a fourth power modulo $\lambda$?".
The natural place to state it is the Gaussian integers, which contain the fourth roots of unity $\pm 1, \pm i$.
The two answers agree up to an explicit sign.

### L2 human

With supplementary laws for $i$ and $1 + i$, the law lets one compute any quartic residue character by swapping and reducing.
It answers questions such as when $2$ is a fourth power modulo a prime, which quadratic reciprocity cannot decide.
Like cubic reciprocity, it is a special case of the [[Artin reciprocity law|7938EKMMVR]].
</document>

<document source="https://tamnd.github.io/math2/node/W0MJ51242N.md" tag="W0MJ51242N" kind="statement">
# Schwarz's list

## Statement

Let $a, b, c$ be real, and assume that the [[monodromy of the hypergeometric equation|T8SBE28DM1]] with these parameters is irreducible.
Let $\lambda = \abs{1 - c}$, $\mu = \abs{c - a - b}$ and $\nu = \abs{a - b}$ be the exponent differences at $0$, $1$ and $\infty$.
Then all solutions of the [[hypergeometric differential equation|W8VE25R5A7]] are algebraic functions of $z$ if and only if the triple $(\lambda, \mu, \nu)$ belongs to Schwarz's list.
Here the triple is taken up to permutation, after a standard normalization by sign changes and integer shifts.
This list consists of one infinite family, $(\frac12, \frac12, \nu)$ with $\nu$ rational, and $14$ further triples.
Of these, $2$ have tetrahedral, $2$ octahedral and $10$ icosahedral monodromy.
Among them are $(\frac12, \frac13, \frac13)$, $(\frac12, \frac13, \frac14)$ and $(\frac12, \frac13, \frac15)$.

## Remarks

All solutions are algebraic exactly when the monodromy group is finite.
The ratio of two solutions, the [[Schwarz triangle function|XMD3Q49Z2C]], maps the upper half-plane onto a circular-arc triangle with angles $\pi\lambda$, $\pi\mu$, $\pi\nu$.
The monodromy group modulo scalars is generated by products of reflections in its sides.
Finite monodromy forces the triangle to be spherical and the projective monodromy group to be a finite subgroup of $\mathrm{PSL}_2(\C)$: cyclic, dihedral, or the rotation group of a [[Platonic solid|017E5PRS1Q]].
The cyclic case is excluded by irreducibility, the dihedral case gives the infinite family, and the tetrahedral, octahedral and icosahedral groups give the remaining entries.

Example: $(\frac12, \frac13, \frac14)$ gives a spherical triangle with angles $\frac{\pi}{2}, \frac{\pi}{3}, \frac{\pi}{4}$, whose reflections tile the sphere in $48$ triangles, matching the symmetries of the octahedron.

A necessary condition, for the normalized triple, is $\lambda + \mu + \nu > 1$, which says that the triangle is spherical.
Irreducibility fails exactly when one of $a$, $b$, $c - a$, $c - b$ is an integer.
That case is treated separately, since then one solution is elementary and the other is an integral of it.

## Explanations

### L1 human

Usually the hypergeometric function is transcendental: it cannot be defined by a polynomial equation with its variable.
Schwarz asked when it is algebraic, like $\sqrt{1 - z}$.
He found that this happens only for a short explicit list of parameter values, and the list matches the symmetry groups of the regular polyhedra.

### L2 human

The proof combines the monodromy of the hypergeometric equation with the geometry of reflection groups on the sphere.
It is the first instance of a general question: when does a linear differential equation have a full set of algebraic solutions?
Beukers and Heckman answered it for the higher generalized hypergeometric equations.
The same triangles, when $\lambda + \mu + \nu < 1$, lead instead to hyperbolic triangle groups and automorphic functions.
</document>

<document source="https://tamnd.github.io/math2/node/W0PAJ23MQ0.md" tag="W0PAJ23MQ0" kind="example">
# Pappus configuration

## Example

The *Pappus configuration* is a [[configuration|Q2QFXPP9EE]] $(9_3)$ in the projective plane.
Take distinct points $A, B, C$ on a line $g$ and distinct points $a, b, c$ on another line $h$, none of them on both lines.
Let $X = Ab \cap aB$, $Y = Ac \cap aC$ and $Z = Bc \cap bC$.
By [[Pappus's hexagon theorem|B6C3J2GJWP]], $X$, $Y$, $Z$ lie on a line, the Pappus line.

Its $9$ points are $A, B, C, a, b, c, X, Y, Z$.
Its $9$ lines are $g$, $h$, the six lines $Ab$, $aB$, $Ac$, $aC$, $Bc$, $bC$ and the Pappus line.
Check: each line has $3$ of the points, for instance $Ab$ contains $A$, $b$, $X$ and $Bc$ contains $B$, $c$, $Z$.
Each point lies on $3$ lines.
For instance, $A$ lies on $g$, $Ab$, $Ac$, the point $b$ on $h$, $Ab$, $bC$, and $X$ on $Ab$, $aB$ and the Pappus line.
Its incidence graph is the Pappus graph.

## Explanations

### L1 human

Put three points on one line and three on another, and join them crosswise in pairs.
The three crossing points always land on a common line.
The finished picture has $9$ points and $9$ lines, with $3$ points on each line and $3$ lines through each point.
</document>

<document source="https://tamnd.github.io/math2/node/W0RYRVJY5Z.md" tag="W0RYRVJY5Z" kind="concept">
# Free spectrum of a variety

## Summary

The free spectrum of a [[variety|AJNZJD8442]] $\mathcal{V}$ is a function on the natural numbers.
It sends $n$ to the number of elements of the free algebra of $\mathcal{V}$ on $n$ generators.
It is finite for all $n$ exactly when $\mathcal{V}$ is locally finite.
Its growth rate measures how much freedom the identities of $\mathcal{V}$ leave, and it is used to separate and classify locally finite varieties.

## Explanations

### L1 human

Count how many different expressions in $n$ letters a variety can tell apart.
For Boolean algebras on $n$ letters, every expression is a truth function of $n$ inputs, so there are $2^{2^n}$ of them.
For semilattices, an expression is determined by the set of letters it uses, so there are $2^n - 1$ of them.
For groups, already one letter gives infinitely many elements $x, x^2, x^3, \ldots$, so the free spectrum is infinite.
</document>

<document source="https://tamnd.github.io/math2/node/W0VD6EAHDA.md" tag="W0VD6EAHDA" kind="definition">
# Nonassociative ring

## Definition

A *nonassociative ring* is an [[abelian group|071DDFVG4J]] $(A, +)$ with a binary operation $A \times A \to A$, $(a, b) \mapsto ab$, such that
$$a(b + c) = ab + ac \quad \text{and} \quad (a + b)c = ac + bc$$
for all $a, b, c \in A$.
No associativity and no unit element are required.

## Remarks

The two distributive laws say that the multiplication is biadditive.
An associative ring is a nonassociative ring whose product is associative; the word "nonassociative" means "not necessarily associative".
Some authors, and the Lean library Mathlib (`NonAssocRing`), also require a two-sided unit element.
Example: $\R^3$ with the cross product $\times$.
The product is bilinear, hence biadditive, but $(e_1 \times e_1) \times e_2 = 0 \ne -e_2 = e_1 \times (e_1 \times e_2)$, so it is not associative.
Non-example: $\Z$ with the product $a * b = a - b$.
Distributivity fails: $1 * (1 + 1) = -1$, while $1 * 1 + 1 * 1 = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/W10ERK1ADE.md" tag="W10ERK1ADE" kind="definition">
# Cup product

## Definition

Let $X$ be a topological space and let $R$ be a [[commutative ring|0310REJJY8]] with $1$.
For a singular simplex $\sigma \colon \Delta^{n} \to X$ with vertices $v_0, \ldots, v_n$, write $\sigma|_{[v_i, \ldots, v_j]}$ for its restriction to the face spanned by $v_i, \ldots, v_j$.
For cochains $\varphi \in C^k(X; R)$ and $\psi \in C^l(X; R)$, the *cup product* $\varphi \smile \psi \in C^{k+l}(X; R)$ is
$$(\varphi \smile \psi)(\sigma) = \varphi\big(\sigma|_{[v_0, \ldots, v_k]}\big) \cdot \psi\big(\sigma|_{[v_k, \ldots, v_{k+l}]}\big)$$
for every singular simplex $\sigma \colon \Delta^{k+l} \to X$.
It satisfies $\delta(\varphi \smile \psi) = \delta\varphi \smile \psi + (-1)^k \varphi \smile \delta\psi$.
Hence it induces a bilinear map $\smile \colon H^k(X; R) \times H^l(X; R) \to H^{k+l}(X; R)$ on [[singular cohomology|6N7BC70CX0]], also called the cup product.

## Remarks

The cup product on cohomology is associative, the class of the constant $0$-cochain $1$ is a unit, and $a \smile b = (-1)^{kl}\, b \smile a$ for $a \in H^k$ and $b \in H^l$.
Graded commutativity holds on cohomology but not on cochains.
It is natural: $f^*(a \smile b) = f^*(a) \smile f^*(b)$ for a continuous map $f$.
The same formula gives relative products such as $H^k(X; R) \times H^l(X, A; R) \to H^{k+l}(X, A; R)$.

Example: for the torus $T^2 = S^1 \times S^1$, $H^1(T^2; \Z) \cong \Z^2$ with generators $\alpha$ and $\beta$ dual to the two circles.
Then $\alpha \smile \beta$ generates $H^2(T^2; \Z) \cong \Z$, and $\alpha \smile \alpha = 0$.

Non-example: the pointwise product $(\varphi\psi)(\sigma) = \varphi(\sigma)\psi(\sigma)$ only makes sense when $\varphi$, $\psi$ and $\sigma$ all have the same dimension.
So it does not define a product $C^k \times C^l \to C^{k+l}$.
The cup product avoids this by splitting each simplex into a front face and a back face.
</document>

<document source="https://tamnd.github.io/math2/node/W11JCMFK4D.md" tag="W11JCMFK4D" kind="statement">
# Superstability of the exponential equation

## Statement

Let $(S, +)$ be a [[semigroup|0XMEM406JM]], let $\delta \ge 0$, and let $f \colon S \to \C$ satisfy
$$\abs{f(x + y) - f(x) f(y)} \le \delta \quad \text{for all } x, y \in S.$$
Then one of the following holds:

1. $\abs{f(x)} \le \frac{1 + \sqrt{1 + 4\delta}}{2}$ for all $x \in S$;
2. $f(x + y) = f(x) f(y)$ for all $x, y \in S$.

## Remarks

Write $K = \frac{1 + \sqrt{1 + 4\delta}}{2}$, so that $K^2 - \delta = K$ and $K \ge 1$.
If $\abs{f(a)} = K + \eta$ with $\eta > 0$, then $\abs{f(2a)} \ge (K + \eta)^2 - \delta \ge K + 2\eta$, and iterating shows that $f$ is unbounded.
If $f$ is unbounded, compare $f(x + y + z)$ with $f(x + y) f(z)$ and with $f(x) f(y + z)$ to get
$$\abs{f(x + y) - f(x) f(y)} \abs{f(z)} \le 2\delta + \abs{f(x)} \delta,$$
and let $\abs{f(z)} \to \infty$.
So an approximate solution of [[Cauchy's exponential equation|CEGF8W64VQ]] is either bounded or an exact solution.
This is called superstability: it is stronger than [[Hyers–Ulam stability|TPYXQTYND6]], since an unbounded approximate solution must be exact.
Example: on $S = (\R, +)$, the constant $f = 1/2$ satisfies the hypothesis with $\delta = 1/4$, and it is bounded by $K = (1 + \sqrt{2})/2$ but not exponential.
The result was first proved by J. Baker, J. Lawrence and F. Zorzitto in 1979 for real functions on a vector space.

## Explanations

### L1 human

Suppose a function satisfies $f(x + y) = f(x) f(y)$ up to a fixed error.
Either the function stays small everywhere, or the error is in fact zero.
A large value of $f$ would magnify any error without bound, which the fixed bound forbids.

### L2 human

For the additive Cauchy equation, approximate solutions are near exact ones but need not be exact.
The exponential equation behaves differently because it is multiplicative: an error is amplified by large values of $f$.
Similar superstability results hold for d'Alembert's equation and other multiplicative equations.
</document>

<document source="https://tamnd.github.io/math2/node/W168DS966V.md" tag="W168DS966V" kind="concept">
# Completion of a valued field

## Summary

The completion of a field with an absolute value is a larger such field, see the [[definition|KFYW0WN9AD]].
In it every Cauchy sequence converges, and the original field is dense.
It is the [[completion of the metric space|06RW4RKWH7]] of the field, with the field operations extended by continuity.
The real numbers are the completion of $\Q$ for the usual absolute value, and the [[p-adic numbers|BT6T702C28]] are its completion for the [[p-adic absolute value|WGN82TCEF2]].
Many problems over a valued field are solved first in its completion, where limits exist and [[Hensel's lemma|010QVB71T2]] applies.

## Explanations

### L1 human

A field such as $\Q$ has holes: the decimal approximations $1, 1.4, 1.41, 1.414, \ldots$ of $\sqrt 2$ get closer and closer to each other but have no limit in $\Q$.
Completing means adding a limit for every such sequence, and doing arithmetic with these limits.
Which sequences count as "getting closer" depends on the absolute value.
For the $5$-adic absolute value, the integers $1, 1 + 5, 1 + 5 + 25, \ldots$ get closer, and their limit in the completion is $-1/4$, because $1 + 5 + 25 + \cdots = 1/(1 - 5)$ in this sense.
</document>

<document source="https://tamnd.github.io/math2/node/W18EY4E5TH.md" tag="W18EY4E5TH" kind="concept">
# Mackey topology

## Summary

The Mackey topology $\tau(X, Y)$ of a [[dual pair|R68HNB5SVR]] $(X, Y)$ is a topology on $X$.
It is the topology of uniform convergence on the convex, balanced, [[weakly|P6KHVSTPZV]] compact subsets of $Y$.
By the [[Mackey–Arens theorem|RKKWS7CDNF]], it is the finest locally convex topology on $X$ whose continuous dual is $Y$.
So, among all locally convex topologies with a given dual, the [[weak topology|P6KHVSTPZV]] is the coarsest and the Mackey topology is the finest.

## Explanations

### L1 human

Many topologies on a space can have the same continuous linear forms.
They form a range, from the weakest one to the strongest one.
The Mackey topology is the strongest one in this range.
For a normed space the norm topology is already the strongest, so there the Mackey topology is the norm topology.
</document>

<document source="https://tamnd.github.io/math2/node/W19129PVKQ.md" tag="W19129PVKQ" kind="statement">
# De Finetti's theorem

## Statement

Let $S$ be a [[Polish space|584GXXY73M]] with its [[Borel σ-algebra|0WQXY32Q8D]], and let $\mathcal{P}(S)$ be the set of [[probability measures|059TJN0NZ1]] on $S$.
Give $\mathcal{P}(S)$ the smallest σ-algebra that makes the maps $m \mapsto m(A)$ measurable for all Borel sets $A \subseteq S$.
Let $X_1, X_2, \dots$ be an infinite [[exchangeable|4MR4YJS1ZD]] sequence of random variables with values in $S$.
Then there is a unique probability measure $\nu$ on $\mathcal{P}(S)$ such that for every $n \ge 1$ and all Borel sets $A_1, \dots, A_n \subseteq S$,
$$P(X_1 \in A_1, \dots, X_n \in A_n) = \int_{\mathcal{P}(S)} m(A_1) \cdots m(A_n) \, \nu(dm).$$

## Remarks

In words, the sequence is a mixture of [[i.i.d.|00MRN0FARV]] sequences: first draw a law $m$ from $\nu$, then draw $X_1, X_2, \dots$ i.i.d. with law $m$.
For $S = \set{0, 1}$ the theorem says that there is a unique probability measure $\mu$ on $[0, 1]$ with
$$P(X_1 = x_1, \dots, X_n = x_n) = \int_0^1 p^k (1 - p)^{n - k} \, \mu(dp), \qquad k = x_1 + \dots + x_n,$$
so the sequence is a mixture of [[Bernoulli|0386Y0Z7Y7]] sequences.
In this case the random parameter is the almost sure limit of the frequencies $(X_1 + \dots + X_n)/n$, and its law is $\mu$.
One proof shows that, given the σ-algebra of events invariant under finite permutations, the variables are conditionally i.i.d.; a reverse martingale argument gives the limit of the empirical measures.
The theorem fails for finite sequences: drawing both balls without replacement from an urn with one red and one blue ball gives an exchangeable pair $(X_1, X_2)$ with $P(X_1 = X_2 = 1) = 0$ and $P(X_1 = X_2 = 0) = 0$.
A mixture would need $\int p^2 \, \mu(dp) = 0$ and $\int (1 - p)^2 \, \mu(dp) = 0$, which is impossible.
In the $\set{0,1}$ case, $\operatorname{Cov}(X_1, X_2) = \int p^2 \, d\mu - \left(\int p \, d\mu\right)^2 \ge 0$, so infinite exchangeable sequences of this kind are never negatively correlated.

## Explanations

### L1 human

Suppose you toss a coin of unknown bias many times, and you believe that the order of the tosses does not matter.
De Finetti's theorem says that this belief alone forces a simple picture.
There is a hidden bias $p$, drawn once from some law, and given $p$ the tosses are independent with success probability $p$.
So symmetry under reordering is the same as independence given an unknown parameter.

### L2 human

The theorem is the main justification for the Bayesian model "parameter with a prior, data i.i.d. given the parameter": the measure $\nu$ plays the role of the prior.
The [[Pólya urn model|DHPXDFEEMF]] gives an example: its colour sequence is exchangeable, and the mixing measure is a [[beta distribution|0EW824Z7G2]].
For general exchangeable sequences, the [[Dirichlet process|XT4SWG6J7N]] is a standard choice of $\nu$ in nonparametric Bayesian statistics.
The [[Hewitt–Savage zero–one law|JTXNFQNQ4J]] is a related result: for an i.i.d. sequence, the exchangeable σ-algebra is trivial, which matches $\nu$ being a point mass.
</document>

<document source="https://tamnd.github.io/math2/node/W19567AVYG.md" tag="W19567AVYG" kind="definition">
# Poisson kernel

## Definition

Let $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ be the [[unit disc|8G5ABXRK85]] and $\mathbb{T} = \set{\zeta \in \C : \abs{\zeta} = 1}$ its boundary circle.
The *Poisson kernel* of $\mathbb{D}$ is the function $P \colon \mathbb{D} \times \mathbb{T} \to \R$ given by
$$P(z, \zeta) = \frac{1 - \abs{z}^2}{\abs{\zeta - z}^2}.$$
In polar form, for $0 \le r < 1$ and real $\theta$, one writes $P_r(\theta) = P(r e^{i\theta}, 1) = \dfrac{1 - r^2}{1 - 2r\cos\theta + r^2}$.

## Remarks

The kernel is positive, and $P(z, \zeta) = \operatorname{Re} \dfrac{\zeta + z}{\zeta - z}$, so $z \mapsto P(z, \zeta)$ is [[harmonic|0DEB0JMN0W]] on $\mathbb{D}$.
It has the expansion $P_r(\theta) = \sum_{n \in \Z} r^{\abs{n}} e^{i n\theta}$, from which $\frac{1}{2\pi}\int_0^{2\pi} P_r(\theta)\, d\theta = 1$.
Example: at $z = 0$ one gets $P(0, \zeta) = 1$ for every $\zeta$, so the Poisson integral at the centre is the plain mean over the circle.
For a continuous function $\varphi$ on $\mathbb{T}$, the Poisson integral $u(z) = \frac{1}{2\pi}\int_0^{2\pi} P(z, e^{it}) \varphi(e^{it})\, dt$ is harmonic on $\mathbb{D}$ and extends continuously to $\varphi$ on $\mathbb{T}$.
Non-example: the function $(z, \zeta) \mapsto 1/\abs{\zeta - z}^2$ is not the Poisson kernel; it lacks the factor $1 - \abs{z}^2$, it is not harmonic in $z$, and its integral over the circle is $1/(1 - \abs{z}^2)$, not $1$.
For the upper half-plane, the Poisson kernel is $\dfrac{1}{\pi}\dfrac{y}{(x - t)^2 + y^2}$ for $x + iy$ with $y > 0$ and real $t$.
</document>

<document source="https://tamnd.github.io/math2/node/W1C0W5MCX0.md" tag="W1C0W5MCX0" kind="statement">
# Briançon–Skoda theorem

## Statement

Let $\mathcal{O}_n = \C\{z_1, \dots, z_n\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]] in $n$ variables.
Let $I$ be an [[ideal|096Z9MP2ME]] of $\mathcal{O}_n$.
Its *integral closure* $\overline{I}$ is the set of $h \in \mathcal{O}_n$ that satisfy an equation
$$h^m + a_1 h^{m-1} + \cdots + a_m = 0 \quad \text{with } a_j \in I^j \text{ for } j = 1, \dots, m,$$
for some $m \ge 1$; it is an ideal containing $I$.
Then, for every ideal $I$ of $\mathcal{O}_n$ and every integer $k \ge 1$,
$$\overline{I^{\,n + k - 1}} \subseteq I^k.$$
In particular $\overline{I^n} \subseteq I$.

## Remarks

Joël Briançon and Henri Skoda proved the theorem in 1974, using Skoda's $L^2$ estimates for the division problem.
A modern proof uses [[multiplier ideals|BKM0TQSCD9]]: one has $\overline{I^{\,l}} \subseteq \mathcal{J}(I^{\,l})$ for all $l$, and Skoda's theorem gives $\mathcal{J}(I^{\,l}) \subseteq I \cdot \mathcal{J}(I^{\,l-1})$ for $l \ge n$, so induction gives the inclusion.
Lipman and Teissier, and Lipman and Sathaye, extended the result to regular local rings.

A corollary answers a question of Mather.
If $f \in \mathcal{O}_n$ and $f(0) = 0$, then $f$ lies in the integral closure of its Jacobian ideal $J(f) = (\partial f / \partial z_1, \dots, \partial f / \partial z_n)$, and hence $f^n \in J(f)$.

The exponent cannot be lowered to $1$ in general.
For $n = 2$ and $I = (x^2, y^2)$, the element $xy$ satisfies $(xy)^2 - x^2 y^2 = 0$ with $x^2 y^2 \in I^2$, so $xy \in \overline{I}$, but $xy \notin I$.
The theorem gives instead $\overline{I^2} \subseteq I$.

## Explanations

### L1 human

The integral closure of an ideal adds the elements that are "as small as" elements of the ideal.
This means that they satisfy a polynomial equation with coefficients from powers of the ideal.
This can be strictly bigger than the ideal.
The theorem says that taking enough powers makes up for the difference, and the number of powers needed depends only on the number of variables.

### L2 human

In analytic terms, $h \in \overline{I}$ when $\abs{h} \le C (\abs{g_1} + \cdots + \abs{g_r})$ near $0$, where $g_1, \dots, g_r$ generate $I$.
The theorem turns such a size bound into an algebraic membership in a power of $I$.
This is how it is used in the study of the Jacobian ideal and of the [[Milnor number|1ZE51P7GRQ]].
It is a model for many effective results in commutative algebra and for applications of $L^2$ methods in algebraic geometry.
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<document source="https://tamnd.github.io/math2/node/W1F5C7RZ0M.md" tag="W1F5C7RZ0M" kind="concept">
# Inertial manifold

## Summary

An inertial manifold of a dissipative evolution equation is a finite-dimensional Lipschitz manifold in the infinite-dimensional phase space.
It is invariant and attracts all solutions exponentially fast.
It contains the [[global attractor|4SH2ZB984E]].
Restricted to it, the equation becomes a system of finitely many ordinary differential equations, the inertial form, which carries the full long-time dynamics.
Inertial manifolds were introduced by Foias, Sell and Temam in the 1980s.
They exist for many dissipative PDEs whose linear part has large gaps in its spectrum.

## Explanations

### L1 human

Many physical systems, like a fluid or a chemical reaction spread over space, need infinitely many numbers to describe their state.
Yet after a short time their behaviour often looks as if only a few numbers mattered.
An inertial manifold makes this precise: it is a finite-dimensional surface in the space of states that all solutions approach very fast.
Once a solution is near it, the motion is governed by finitely many equations.
</document>

<document source="https://tamnd.github.io/math2/node/W1JXH6NCKX.md" tag="W1JXH6NCKX" kind="concept">
# Doubling measure

## Summary

A doubling measure on a metric space is a Borel measure with the following property, for a fixed constant $C$.
Doubling the radius of any ball multiplies its measure by at most $C$.
Lebesgue measure on $\R^n$ is the model example, with constant $2^n$.
Many tools of real analysis on $\R^n$ extend to metric spaces with a doubling measure.
Examples are the Vitali covering lemma, maximal function bounds and the Lebesgue differentiation theorem.
A metric space that carries a doubling measure is a doubling metric space.

## Explanations

### L1 human

Take any ball and look at the ball with the same centre and twice the radius.
For a doubling measure, the big ball never has more than $C$ times the mass of the small ball, with the same $C$ for all balls.
So mass is spread out without sudden thin spots next to heavy ones.
On the line with ordinary length, doubling an interval doubles its length, so $C = 2$ works.
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<document source="https://tamnd.github.io/math2/node/W1VV1MP5EG.md" tag="W1VV1MP5EG" kind="definition">
# Coproduct

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $(X_i)_{i \in I}$ be a family of objects of $\mathcal{C}$ indexed by a set $I$.
A *coproduct* of the family is an object $C$ with [[morphisms|859F0TFVA3]] $\iota_i \colon X_i \to C$ for $i \in I$, called *injections*, with the following property.
For every object $Y$ and every family of morphisms $f_i \colon X_i \to Y$, there is exactly one morphism $f \colon C \to Y$ with $f \circ \iota_i = f_i$ for all $i$.
The object $C$ is written $\coprod_{i \in I} X_i$, or $X_1 \sqcup X_2$ for two objects, and $f$ is written $[f_1, f_2]$.

## Remarks

A coproduct is the [[colimit|BJS1CGTKQ5]] of the diagram whose shape is the set $I$ with only identity morphisms, and it is a [[product|S5744QZV4N]] in the opposite category.
The coproduct of the empty family is an initial object.

Example: in the category of [[abelian groups|082GHNPJ53]], the [[direct sum|0S7W6EG32V]] $A \oplus B$ with $\iota_A(a) = (a, 0)$ and $\iota_B(b) = (0, b)$ is a coproduct.
Given homomorphisms $f_A \colon A \to Y$ and $f_B \colon B \to Y$, the map $f(a, b) = f_A(a) + f_B(b)$ is a homomorphism because $Y$ is abelian, it satisfies $f \circ \iota_A = f_A$ and $f \circ \iota_B = f_B$, and it is forced since $(a, b) = (a, 0) + (0, b)$.
In the category of sets the coproduct is the disjoint union, and in the category of groups it is the [[free product|TM7QS7P7X9]].

Non-example: in the category of fields, $\mathbb{F}_2$ and $\mathbb{F}_3$ have no coproduct, since no field receives homomorphisms from both: a field homomorphism preserves the characteristic.
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<document source="https://tamnd.github.io/math2/node/W1VVPNBPMX.md" tag="W1VVPNBPMX" kind="definition">
# Properly discontinuous action

## Definition

Let $\Gamma$ be a group and let $X$ be a topological space.
Let $\Gamma$ [[act|0WCY42EV9V]] on $X$ by [[homeomorphisms|0QT00PB5G2]].
The action is *properly discontinuous* if, for all [[compact|0TVJZYB6SH]] subsets $K, L \subseteq X$, the set
$$\set{\gamma \in \Gamma : (\gamma \cdot K) \cap L \ne \emptyset}$$
is finite.

## Remarks

When $X$ is Hausdorff and locally compact, the action is properly discontinuous if and only if the action of $\Gamma$ with the [[discrete topology|0S2TEHX4E1]] is [[proper|MDH7VXDCVF]].
In covering space theory, some authors use the term for the weaker condition that each point has a neighbourhood $U$ with $\gamma U \cap U = \emptyset$ for all $\gamma \ne 1$.

Example: let $\Z^n$ act on $\R^n$ by translations.
If $K$ and $L$ lie in the ball of radius $R$ about $0$, then $(v + K) \cap L \ne \emptyset$ forces $\norm{v} \le 2R$.
Only finitely many $v \in \Z^n$ satisfy this, so the action is properly discontinuous.

Non-example: let $\Z$ act on $\R$ by $n \cdot x = 2^n x$.
For $K = L = \set{0}$, every $n \in \Z$ satisfies $(n \cdot K) \cap L \ne \emptyset$, so the set is infinite.
So the action is not properly discontinuous.
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<document source="https://tamnd.github.io/math2/node/W1X984PKJ9.md" tag="W1X984PKJ9" kind="concept">
# Wreath product

## Summary

The wreath product $A \wr_\Omega G$ builds a new group from a group $A$ and a group $G$ that acts on a set $\Omega$.
It is the [[semidirect product|0H3WZXCZNF]] of the group of all functions $\Omega \to A$ with $G$, where $G$ moves the functions by moving the points of $\Omega$.
Wreath products describe the symmetries of objects made of several copies of a smaller object.
Each [[group extension|MF4N3VGGFV]] $E$ of a group $B$ by a group $A$ embeds in the unrestricted regular wreath product $A \wr B$.

## Explanations

### L1 human

Think of $m$ identical boxes, each holding a copy of some object with symmetry group $A$.
A symmetry of the whole picture may turn each box by its own element of $A$, and then permute the boxes by an element of $G$.
The wreath product is the group of all such combined moves.
For example, take two coins on a table, each of which can be flipped, and allow the two coins to swap places.
Each coin gives a group of order $2$, the swap gives another group of order $2$, and the group of all moves is $C_2 \wr C_2$, of order $2^2 \cdot 2 = 8$.
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<document source="https://tamnd.github.io/math2/node/W1YH3YD3S4.md" tag="W1YH3YD3S4" kind="statement">
# Harris inequality

## Statement

Let $n \ge 1$ and let $p_1, \ldots, p_n \in [0, 1]$.
Let $\Omega = \set{0, 1}^n$ with the product [[probability measure|059TJN0NZ1]] $\mu$ under which the coordinates are independent and the $i$-th coordinate equals $1$ with probability $p_i$.
Order $\Omega$ coordinatewise: $x \le y$ when $x_i \le y_i$ for all $i$.
Let $f, g \colon \Omega \to \R$ be [[increasing|0A7KV2WB7A]] functions, that is, $x \le y$ implies $f(x) \le f(y)$ and $g(x) \le g(y)$.
Then
$$\mathbb{E}[f g] \ge \mathbb{E}[f] \, \mathbb{E}[g],$$
where $\mathbb{E}$ is the [[expected value|0WQNRCNY3M]] with respect to $\mu$.

## Remarks

The proof is by induction on $n$.
For $n = 1$, the inequality is equivalent to $p_1 (1 - p_1) (f(1) - f(0)) (g(1) - g(0)) \ge 0$, which holds because both differences are nonnegative.
The induction step conditions on the last coordinate and applies the case $n = 1$ to the conditional expectations, which are again increasing.

For indicator functions of increasing events $A$ and $B$, the inequality reads $\mu(A \cap B) \ge \mu(A) \, \mu(B)$: increasing events are positively correlated.
For example, in the [[Erdős–Rényi model|6E83YKANV3]] $G(n, p)$ the events "the graph is connected" and "the graph has a triangle" are increasing, so they are positively correlated.
If $f$ is increasing and $g$ is decreasing, the inequality is reversed.
The product structure of $\mu$ matters.
For the measure giving probability $1/2$ to each of $(1, 0)$ and $(0, 1)$, the increasing events $\set{x_1 = 1}$ and $\set{x_2 = 1}$ have probability $1/2$ each but are disjoint.

## Explanations

### L1 human

Under independent coin flips, two "more is better" quantities tend to go up together.
If knowing that some coins show heads makes $f$ larger, it also makes $g$ larger, so the average of the product is at least the product of the averages.

### L2 human

The Harris inequality is the case of a product measure in the [[FKG inequality|DCXJGS6E6D]], which allows any measure on a distributive lattice satisfying the log-supermodularity condition.
Both follow from the [[Ahlswede–Daykin inequality|4085B8A528]].
For the uniform measure on subsets of a finite set, it is Kleitman's lemma: two up-sets $\mathcal{A}, \mathcal{B}$ of subsets of $\set{1, \ldots, n}$ satisfy $2^n |\mathcal{A} \cap \mathcal{B}| \ge |\mathcal{A}| \, |\mathcal{B}|$.
It is a standard tool in percolation theory and random graphs, often together with the [[Janson inequality|J3CT5D7FSZ]], which gives a bound in the other direction.
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<document source="https://tamnd.github.io/math2/node/W251JWP6PS.md" tag="W251JWP6PS" kind="definition">
# Finitely generated algebra

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.
For $X \subseteq A$, let $\operatorname{Sg}(X)$ denote the [[subuniverse generated by|2TQD8NMG9P]] $X$.
It is the smallest subset of $A$ that contains $X$ and is closed under all fundamental operations of $\mathbf{A}$.
The algebra $\mathbf{A}$ is *finitely generated* when there is a [[finite set|070R982G4B]] $X \subseteq A$ with $\operatorname{Sg}(X) = A$.

## Remarks

Example: the group $(\Z, +, -, 0)$ is finitely generated.
Indeed $\operatorname{Sg}(\set{1})$ contains $0$, all sums $1 + \cdots + 1$, and their negatives, so it is all of $\Z$.

Non-example: the group $(\Q, +, -, 0)$ is not finitely generated.
If $X = \set{p_1/q, \dots, p_k/q}$ with a common denominator $q \ge 1$, then every element of $\operatorname{Sg}(X)$ is an integer multiple of $1/q$, so $1/(2q) \notin \operatorname{Sg}(X)$.

Every finite algebra is finitely generated, with $X = A$.
In model theory the same notion is used for structures in a signature with relation symbols, where the relations play no role in generation.
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<document source="https://tamnd.github.io/math2/node/W26MGR4DA3.md" tag="W26MGR4DA3" kind="statement">
# Feynman–Kac formula

## Statement

Let $T > 0$ and let $W$ be a one-dimensional [[Wiener process|22735FGJV6]].
Let $\mu, \sigma \colon \R \times [0, T] \to \R$ be continuous and Lipschitz in $x$ with linear growth, uniformly in $t$.
For $(x, t) \in \R \times [0, T)$, let $X^{x,t}$ be the solution of the [[stochastic differential equation|B219G9SHJA]]
$$dX_s = \mu(X_s, s)\, ds + \sigma(X_s, s)\, dW_s, \quad s \ge t, \qquad X_t = x.$$
Let $V \colon \R \times [0, T] \to [0, \infty)$ be continuous, and let $f \colon \R \times [0, T] \to \R$ and $\psi \colon \R \to \R$ be continuous with polynomial growth in $x$.
Let $u \colon \R \times [0, T] \to \R$ be continuous, of class $C^{2,1}$ on $\R \times [0, T)$, of polynomial growth in $x$ uniformly in $t$, and a solution of the [[parabolic problem|1CCTH8X2FT]]
$$u_t + \mu\, u_x + \frac{1}{2}\sigma^2 u_{xx} - V u + f = 0 \ \text{ on } \R \times [0, T), \qquad u(x, T) = \psi(x).$$
Then for all $(x, t) \in \R \times [0, T]$,
$$u(x, t) = \mathbb{E}\Big[ \int_t^T e^{-\int_t^r V(X_\tau, \tau)\, d\tau} f(X_r, r)\, dr + e^{-\int_t^T V(X_\tau, \tau)\, d\tau}\, \psi(X_T) \Big],$$
where $X = X^{x,t}$.

## Remarks

The proof applies [[Itô's formula|2J1RVQB0CM]] to $Y_s = e^{-\int_t^s V(X_\tau, \tau) d\tau} u(X_s, s)$.
By the equation for $u$, the drift of $Y_s$ equals $-e^{-\int_t^s V d\tau} f(X_s, s)$, so $Y_s$ plus the integral of this term is a local martingale.
The growth conditions and the moment bounds for $X$ make it a true martingale, after a localization by stopping times.
Taking expectations between $s = t$ and $s = T$ gives the formula.
Example: with $\mu = 0$, $\sigma = 1$, $V = f = 0$, the formula says that the solution of the backward [[heat equation|VP2HN86VAG]] $u_t + \frac{1}{2}u_{xx} = 0$ with $u(\cdot, T) = \psi$ is $u(x, t) = \mathbb{E}[\psi(x + W_{T} - W_t)]$, a Gaussian average of $\psi$.
Without a growth condition on $u$ the conclusion can fail, because the heat equation has nonzero solutions with zero data that grow very fast in $x$.
The same formula holds in $\R^n$ with $\frac{1}{2}\sum_{i,j} (\sigma\sigma^T)_{ij}\, \partial_{x_i}\partial_{x_j}$ in place of $\frac{1}{2}\sigma^2 \partial_x^2$.

## Explanations

### L1 human

The Feynman–Kac formula says that the solution of a certain parabolic equation is an average over random paths.
Start a random walker at the point $x$ at time $t$, let it wander until time $T$, and record a reward collected along the way and at the end.
The average reward is the value $u(x, t)$.
This lets one compute solutions of equations by simulating random paths, and it lets one study random paths by solving equations.

### L2 human

The formula is the bridge between [[stochastic differential equations|B219G9SHJA]] and second-order parabolic equations.
The operator $\mu \partial_x + \frac{1}{2}\sigma^2 \partial_x^2$ is the [[infinitesimal generator|VPA41PWRR1]] of the diffusion $X$, and the potential $V$ acts as a killing rate.
In finance it gives the risk-neutral pricing formula: the Black–Scholes equation is a special case with $V = r$ constant.
The dual statement, for the density of $X$ rather than for expectations, is the Fokker–Planck equation.
Its proof is the standard use of Itô's formula together with the optional stopping theorem.
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<document source="https://tamnd.github.io/math2/node/W27Z60Q1PF.md" tag="W27Z60Q1PF" kind="statement">
# Serre's criterion for normality

## Statement

Let $A$ be a [[Noetherian ring|0670T59HGE]].
For a [[prime ideal|00MHHZ9SNE]] $\mathfrak{p}$, write $A_{\mathfrak{p}}$ for the [[localization|X6JD4AG973]] at $\mathfrak{p}$, $\operatorname{ht} \mathfrak{p}$ for its [[height|87WH65FJB8]], and $\operatorname{depth} A_{\mathfrak{p}}$ for the [[depth|M2RADBFW99]] of $A_{\mathfrak{p}}$.
Consider the conditions:

- $(R_1)$: $A_{\mathfrak{p}}$ is a [[regular local ring|63XXTF6B5M]] for every prime $\mathfrak{p}$ with $\operatorname{ht} \mathfrak{p} \le 1$.
- $(S_2)$: $\operatorname{depth} A_{\mathfrak{p}} \ge \min(2, \dim A_{\mathfrak{p}})$ for every prime $\mathfrak{p}$.

Call $A$ a [[normal ring|KR7SWBYHRQ]] if $A_{\mathfrak{p}}$ is an [[integrally closed domain|H20P600HQS]] for every prime $\mathfrak{p}$.
Then $A$ is normal if and only if $A$ satisfies $(R_1)$ and $(S_2)$.

## Remarks

The proof works locally.
A normal local ring of dimension $1$ is a discrete valuation ring, hence regular, which gives $(R_1)$.
For $(S_2)$, let $\dim A_{\mathfrak{p}} \ge 2$ and let $x \in \mathfrak{p}A_{\mathfrak{p}}$ be a nonzerodivisor.
Suppose the maximal ideal were associated to $A_{\mathfrak{p}}/(x)$.
Then some quotient $y/x$ would be integral but not in the ring, or it would make the maximal ideal principal and the dimension $1$.
Conversely, $(S_2)$ says that $A$ is the intersection of its localizations at height one primes, and $(R_1)$ says those localizations are discrete valuation rings, hence integrally closed.
The companion statement says that $A$ is reduced if and only if it satisfies $(R_0)$ and $(S_1)$.

Example: $A = k[x, y, z]/(z^2 - xy)$ over a field $k$ of characteristic not $2$ is normal.
It is a hypersurface, hence [[Cohen–Macaulay|N1QJQEGBFC]], so $(S_2)$ holds; its only singular point is the origin, whose prime has height $2$, so $(R_1)$ holds.

Both conditions are needed.
The cusp $k[x, y]/(y^2 - x^3)$ is Cohen–Macaulay but fails $(R_1)$ at the origin, a prime of height $1$; indeed $t = y/x$ is integral, since $t^2 = x$, but not in the ring.
Two planes meeting in one point, $k[x, y, u, v]/((x, y) \cap (u, v))$, satisfy $(R_1)$.
But they have depth $1$ at the origin, where the dimension is $2$, so $(S_2)$ fails.
The ring is not normal, because its local ring at the origin is not a domain.

## Explanations

### L1 human

A normal ring is one with no "missing fractions", which in geometry means mild singularities.
Serre's criterion splits normality into two simpler checks.
The first check says the space has no singularities in codimension one, so it is smooth away from a small set.
The second says that the ring has enough depth, so functions defined away from that small set extend over it.

### L2 human

In practice, $(S_2)$ is often automatic: it holds for every [[Cohen–Macaulay ring|N1QJQEGBFC]], in particular for complete intersections.
Then normality reduces to $(R_1)$, which one checks with the Jacobian criterion by computing the codimension of the singular locus.
This is the standard way to prove that a hypersurface with isolated singularities in dimension at least $2$ is normal.
The criterion also explains the algebraic Hartogs property of normal rings: a normal Noetherian domain is the intersection of its localizations at height one primes.
</document>

<document source="https://tamnd.github.io/math2/node/W29AA0266Z.md" tag="W29AA0266Z" kind="statement">
# Hopkins–Levitzki theorem

## Statement

Let $R$ be a [[semiprimary ring|XVP517QRH5]]: its [[Jacobson radical|XA0DCNH7WY]] $J = J(R)$ is a [[nilpotent ideal|C2H26BYWHE]] and $R/J$ is a [[semisimple ring|NY25HXA478]].
Let $M$ be a right $R$-module.
Then the following conditions are equivalent.

1. $M$ is [[Noetherian|AK1ZRCXR32]].
2. $M$ is [[Artinian|YQNZTQGP1X]].
3. $M$ has a [[composition series|0XV2A49X6Q]].

## Remarks

Over any ring, condition 3 holds exactly when $M$ is both Noetherian and Artinian, so the point is that each chain condition alone gives the other.
Choose $n$ with $J^n = 0$ and look at the filtration $M \supseteq MJ \supseteq MJ^2 \supseteq \cdots \supseteq MJ^n = 0$.
Each factor $MJ^{i}/MJ^{i+1}$ is killed by $J$, so it is a module over the semisimple ring $R/J$, hence a [[semisimple module|N3MRHDKDGC]].
A semisimple module satisfies one chain condition only when it is a finite direct sum of simple modules, and then it has finite length.
So each factor has finite length, and so does $M$.

Main corollary: a right [[Artinian ring|20QEAM8AX2]] is semiprimary, because [[its Jacobson radical is nilpotent|H7RAXJV9CP]] and $R/J$ is Artinian and semiprimitive, hence semisimple.
So every right Artinian ring is right [[Noetherian|0670T59HGE]], and the same holds on the left.

The hypothesis on $R$ cannot be dropped.
The ring $\Z$ is not semiprimary, and the Prüfer group $\Z[1/p]/\Z$ is an Artinian $\Z$-module that is not Noetherian.
Also $\Z$ itself is a Noetherian $\Z$-module that is not Artinian.

For commutative rings, the [[Akizuki–Hopkins–Levitzki theorem|C7X657TF19]] gives the sharper description of Artinian rings as the Noetherian rings in which every prime ideal is maximal.

## Explanations

### L1 human

For modules in general, "no infinite descending chains" and "no infinite ascending chains" are different conditions.
Over a semiprimary ring the two conditions agree, and both mean that the module has finite length.
The best-known case is the ring itself: a ring with the descending chain condition on right ideals also has the ascending one.

### L2 human

The proof peels the module into finitely many layers using the powers of the radical.
Each layer is a module over a semisimple ring, where both chain conditions just say "finitely many simple summands".
This argument is the reason why right Artinian rings are right Noetherian, a fact used all through the structure theory of Artinian rings, together with the [[Artin–Wedderburn theorem|R1F271180N]].
The converse fails: $\Z$ is Noetherian but not Artinian, so Artinian rings are a strictly smaller class.
</document>

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# Intermittency

## Summary

Intermittency is a behaviour of dynamical systems in which long phases of nearly regular motion are interrupted at irregular times by short chaotic bursts.
It typically appears just past a bifurcation at which a stable [[periodic orbit|0AA97XA70K]] disappears or loses stability.
Pomeau and Manneville classified the common cases into types I, II and III, after the kind of bifurcation involved.
Intermittency is one of the standard routes from regular motion to chaos, next to period doubling.

## Explanations

### L1 human

Picture a signal that looks almost periodic for a long time, then goes wild for a moment, then calms down again.
The calm phases happen because the system passes through a narrow channel near a periodic orbit that has just vanished, and it moves slowly there.
The bursts happen when it leaves the channel and wanders chaotically before being reinjected.
The closer the parameter is to the bifurcation, the longer the calm phases last.
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<document source="https://tamnd.github.io/math2/node/W2ZD4XP74R.md" tag="W2ZD4XP74R" kind="concept">
# Fine topology

## Summary

The fine topology on $\R^n$ is the coarsest topology that makes every [[superharmonic function|X4V2S56YQQ]] continuous.
It is strictly finer than the Euclidean topology.
Henri Cartan introduced it to study [[thin sets|7DGF6MMEGX]]: a set is a fine neighbourhood of a point exactly when its complement is thin there.
Many results of potential theory, such as the behaviour of solutions at irregular boundary points, are stated most cleanly with fine limits.

## Explanations

### L1 human

Superharmonic functions can be discontinuous in the usual sense, for instance a Newtonian potential that is infinite on a sequence of points.
The fine topology adds just enough open sets to make all of them continuous.
In this topology, removing a set of capacity zero, such as a sequence of points, from an open set leaves a set that is still open.
So the fine topology ignores sets that harmonic functions cannot see.
</document>

<document source="https://tamnd.github.io/math2/node/W35ZP39S4R.md" tag="W35ZP39S4R" kind="definition">
# Dunford–Pettis property

## Definition

Let $X$ be a real or complex [[Banach space|06GZ956657]] with [[continuous dual|4A82S5FVRT]] $X^*$.
The space $X$ has the *Dunford–Pettis property* if the following holds.
Let $(x_n)$ be a sequence in $X$ that converges to $0$ in the [[weak topology|P6KHVSTPZV]] $\sigma(X, X^*)$.
Let $(f_n)$ be a sequence in $X^*$ that converges to $0$ in the weak topology $\sigma(X^*, X^{**})$.
Then $f_n(x_n) \to 0$.

## Remarks

Example: $\ell^1$ has the Dunford–Pettis property.
By the Schur property of $\ell^1$, a weakly null sequence $(x_n)$ in $\ell^1$ tends to $0$ in norm.
A weakly convergent sequence $(f_n)$ is bounded, by the uniform boundedness principle.
So $\abs{f_n(x_n)} \le \norm{f_n} \, \norm{x_n} \to 0$.

Non-example: $\ell^2$ does not have it.
The unit vectors $e_n$ tend weakly to $0$, and so do the functionals $f_n = \ip{\cdot}{e_n}$ in the dual of $\ell^2$.
But $f_n(e_n) = 1$ for all $n$.
More generally, no infinite-dimensional reflexive space has the property.

An equivalent form: every weakly compact operator from $X$ to a Banach space $Y$ maps weakly convergent sequences to norm convergent sequences.
The spaces $L^1(\mu)$, $C(K)$ for compact Hausdorff $K$, and $c_0$ have the property.
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# Gelfand pair

## Summary

A Gelfand pair is a [[locally compact group|AHV05ZBCYW]] $G$ with a compact subgroup $K$ such that convolution of $K$-bi-invariant functions on $G$ is commutative.
Equivalently, each irreducible unitary representation of $G$ has at most a one-dimensional space of $K$-fixed vectors.
For a Gelfand pair, harmonic analysis on the homogeneous space $G/K$ is governed by [[spherical functions|BKZNC76RC8]].
They play the role of the exponentials in Fourier analysis.
Examples are $(\mathrm{SO}(3), \mathrm{SO}(2))$, related to the sphere, and $(\mathrm{SL}_2(\R), \mathrm{SO}(2))$, related to the hyperbolic plane.

## Explanations

### L1 human

Think of functions on the sphere that only depend on the latitude.
They can be seen as functions on the rotation group that do not change under rotations around the north pole, on either side.
For such functions, convolution does not depend on the order of the factors.
This commutativity makes their analysis almost as simple as Fourier series.
A group with a subgroup that behaves like this is called a Gelfand pair.
The pair of all rotations and the rotations around one axis is the main example.
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# Exceptional collection

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $\mathcal{T}$ be a $k$-linear [[triangulated category|T2V7GJJ2EN]] with shift $X \mapsto X[1]$.
An object $E$ of $\mathcal{T}$ is *exceptional* if $\Hom(E, E) = k$ and $\Hom(E, E[l]) = 0$ for every integer $l \ne 0$.
An *exceptional collection* is a sequence $(E_1, \dots, E_n)$ of objects of $\mathcal{T}$ such that:

1. each $E_i$ is exceptional;
2. $\Hom(E_j, E_i[l]) = 0$ for all $j > i$ and all integers $l$.

The collection is *full* if the smallest triangulated subcategory of $\mathcal{T}$ that contains $E_1, \dots, E_n$ and is closed under isomorphisms is $\mathcal{T}$ itself.
It is *strong* if moreover $\Hom(E_i, E_j[l]) = 0$ for all $i, j$ and all $l \ne 0$.

## Remarks

Example: on the projective space $\mathbb{P}^n$ over $k$, the line bundles $(\mathcal{O}, \mathcal{O}(1), \dots, \mathcal{O}(n))$ form a full strong exceptional collection.
It lives in the [[bounded derived category of coherent sheaves|2VP2D51KB8]], and this is a theorem of Beilinson.
Clause 2 holds since $\Hom(\mathcal{O}(j), \mathcal{O}(i)[l]) = H^l(\mathbb{P}^n, \mathcal{O}(i - j))$, and for $-n \le i - j < 0$ all cohomology of $\mathcal{O}(i - j)$ vanishes.
Non-example: on an elliptic curve $C$ over $k$, no line bundle $L$ is exceptional, since $\Hom(L, L[1]) = H^1(C, \mathcal{O}_C) \cong k \ne 0$.
A full exceptional collection gives a semiorthogonal decomposition $\mathcal{T} = \langle \mathcal{A}_1, \dots, \mathcal{A}_n \rangle$, where $\mathcal{A}_i$ is the subcategory generated by $E_i$, equivalent to the bounded derived category of finite-dimensional vector spaces over $k$.
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# Consistency of summability methods

## Summary

Two [[summability methods|8X25Q4XYMN]] are consistent when they never give different values to the same series.
If a series is summable by both methods, the two values must agree.
Consistency is what makes it reasonable to speak of "the" generalized sum of a divergent series.
The [[Mazur–Orlicz theorem|YE5439NH1T]] gives a condition under which two regular matrix methods are consistent on bounded sequences.

## Explanations

### L1 human

There are many ways to give a value to a divergent series, such as averaging the partial sums or plugging $x \to 1$ into a power series.
One wants these ways to agree whenever they both work.
For the series $1 - 1 + 1 - 1 + \cdots$, both [[Cesàro summation|N8ZGG7EW6J]] and [[Abel summation|EZXPY4A19C]] give $1/2$.
In fact, every series that Cesàro summation sums is summed by Abel summation to the same value, so the two methods are consistent.
</document>

<document source="https://tamnd.github.io/math2/node/W3BX66Y47G.md" tag="W3BX66Y47G" kind="concept">
# Hilbert–Schmidt operator

## Summary

A Hilbert–Schmidt operator is a bounded operator between [[Hilbert spaces|HM2HHE3J51]] whose images of an orthonormal basis have a finite sum of squared norms.
These operators are [[compact|M2NEP1K5AE]], and they form a Hilbert space themselves, with the Hilbert–Schmidt norm.
On $L^2$ spaces they are exactly the [[integral operators with square-integrable kernels|CB5G212Q03]].
This makes them the natural class for the Fredholm theory of integral equations.

## Explanations

### L1 human

Think of an operator as an infinite matrix.
It is Hilbert–Schmidt when the sum of the squares of the absolute values of all the entries is finite.
This is the infinite version of the Frobenius norm of a matrix.
For example, the diagonal operator on sequences that multiplies the $n$-th entry by $1/n$ is Hilbert–Schmidt, since $\sum 1/n^2$ is finite.
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<document source="https://tamnd.github.io/math2/node/W3DCQ22JA3.md" tag="W3DCQ22JA3" kind="concept">
# Dandelin spheres

## Summary

Cut a right circular cone by a plane.
The Dandelin spheres of this plane section are the spheres that touch the cone along a circle and also touch the cutting plane.
For an ellipse there are two, and their points of contact with the plane are the two foci of the ellipse.
They give a short proof that the plane sections of a cone are the curves defined by foci and distances.
So the two classical definitions of the [[conic sections|00HJCMC2CD]] agree.
They are named after Germinal Pierre Dandelin.

## Explanations

### L1 human

Slice an ice-cream cone with a tilted plane: the cut is an ellipse.
Now drop one ball into the cone above the plane and one below it.
Each ball sits snugly against the cone along a circle and just touches the plane.
The two balls touch the plane at two points, and these points are exactly the foci of the ellipse.
The reason is that the two tangent segments from any point to a ball have the same length.
So the sum of distances to the foci equals the distance between the two circles along the cone.
This distance is the same for every point of the ellipse.
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<document source="https://tamnd.github.io/math2/node/W3NVXBEMRQ.md" tag="W3NVXBEMRQ" kind="concept">
# Robin constant

## Summary

The Robin constant of a compact set $K$ in the plane is the least logarithmic energy of a unit charge placed on $K$.
It is a number in $(-\infty, \infty]$, and it equals $\infty$ exactly when every unit charge on $K$ has infinite energy.
The [[logarithmic capacity|DBQXRTC0VN]] of $K$ is $e^{-V}$, where $V$ is the Robin constant.
The Robin constant also appears in the expansion at infinity of the Green function of the outside of $K$.

## Explanations

### L1 human

Spread a unit charge on a plane conductor $K$ in the cheapest way, with the logarithmic law of repulsion.
The energy of this best arrangement is the Robin constant.
A big set gives the charge room to spread, so its Robin constant is small, possibly negative.
A single point gives no room at all, and its Robin constant is $+\infty$.
For a disc of radius $r$ the Robin constant is $-\log r$.
</document>

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# Riesz transform

## Definition

Let $n \ge 1$ and $j \in \set{1, \ldots, n}$.
Let $c_n = \Gamma\big(\frac{n+1}{2}\big) / \pi^{(n+1)/2}$, where $\Gamma$ is the gamma function.
For a [[Schwartz function|CG9JEATW3Q]] $f$ on $\R^n$, the *$j$-th Riesz transform* of $f$ is the [[principal value|0SYR10HD4R]]
$$R_j f(x) = c_n \lim_{\varepsilon \to 0^+} \int_{\abs{y} > \varepsilon} \frac{y_j}{\abs{y}^{n+1}} f(x - y) \, dy.$$

## Remarks

With $\hat f(\xi) = \int f(x) e^{-2\pi i x \cdot \xi} \, dx$, the [[Fourier transform|DPVM34ADZV]] of $R_j f$ is
$$\widehat{R_j f}(\xi) = -i \frac{\xi_j}{\abs{\xi}} \hat f(\xi).$$
So $\sum_{j=1}^n R_j^2 = -I$ on $L^2(\R^n)$, and each $R_j$ is bounded on $L^2$ with norm $1$.
Each $R_j$ extends to a bounded operator on $L^p(\R^n)$ for $1 < p < \infty$.

Example: for $n = 1$, $c_1 = \Gamma(1)/\pi = 1/\pi$ and the kernel is $\frac{1}{\pi}\frac{y}{\abs{y}^2} = \frac{1}{\pi y}$.
So $R_1$ is the [[Hilbert transform|NSCEBXM7ZM]], with multiplier $-i\sgn\xi$.

Non-example: the kernel $\abs{y}^{-n}$, with the same homogeneity but without the odd factor $y_j/\abs{y}$, does not define a principal value operator.
It has no cancellation, and $\int_{\varepsilon < \abs{y} < 1} \abs{y}^{-n} \, dy$ grows like $\log(1/\varepsilon)$.

The kernel $c_n y_j/\abs{y}^{n+1}$ is odd, homogeneous of degree $-n$, and smooth away from $0$, which makes $R_j$ a [[singular integral operator|HTMCEARGDD]].
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# Matrix norm

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, let $m, n \ge 1$, and let $\mathbb{K}^{m \times n}$ be the vector space of $m \times n$ [[matrices|0VD9ZSWWJG]] over $\mathbb{K}$.
A *matrix norm* on $\mathbb{K}^{m \times n}$ is a map $\norm{\cdot} \colon \mathbb{K}^{m \times n} \to \R$ such that, for all $A, B \in \mathbb{K}^{m \times n}$ and $c \in \mathbb{K}$:

1. $\norm{A} \ge 0$, and $\norm{A} = 0$ only if $A = 0$.
2. $\norm{cA} = \abs{c} \norm{A}$.
3. $\norm{A + B} \le \norm{A} + \norm{B}$.

A matrix norm on $\mathbb{K}^{n \times n}$ is *submultiplicative* if moreover $\norm{AB} \le \norm{A} \norm{B}$ for all $A, B \in \mathbb{K}^{n \times n}$.

## Remarks

Clauses 1 to 3 say that $\norm{\cdot}$ is a [[norm|0V2WP9B1MF]] on $\mathbb{K}^{m \times n}$; many authors include submultiplicativity in the word "matrix norm".
Example: for a vector norm $\abs{\cdot}$ on $\mathbb{K}^n$, the [[operator norm|06EKW1AKXF]] $\norm{A} = \max_{\abs{x} = 1} \abs{Ax}$ is a submultiplicative matrix norm, since $\abs{ABx} \le \norm{A} \abs{Bx} \le \norm{A}\norm{B}\abs{x}$.
Non-example: $\norm{A}_{\max} = \max_{i,j} \abs{a_{ij}}$ is a norm but not submultiplicative.
For $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ one has $A^2 = 2A$, so $\norm{A^2}_{\max} = 2 > 1 = \norm{A}_{\max}^2$.
All norms on $\mathbb{K}^{m \times n}$ are equivalent, because the space has finite dimension.
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# Fundamental theorem of affine geometry

## Statement

Let $k$ be a [[field|04C5ZBG1WT]] with at least $3$ elements, and let $n \ge 2$.
Let $A$ and $B$ be [[affine spaces|09XNAPTJQ7]] of dimension $n$ over $k$, with spaces of translations $V$ and $W$.
Let $f \colon A \to B$ be a bijection that maps every line of $A$ onto a line of $B$.
Then there are a field automorphism $\sigma$ of $k$ and a bijection $\varphi \colon V \to W$ such that:

1. $\varphi(u + v) = \varphi(u) + \varphi(v)$ and $\varphi(c v) = \sigma(c) \varphi(v)$ for all $u, v \in V$ and $c \in k$;
2. $f(p + v) = f(p) + \varphi(v)$ for all $p \in A$ and $v \in V$.

In particular, if $k = \R$, then $f$ is an [[affine map|0JF0TVYGWT]].

## Remarks

A map with properties 1 and 2 is called semi-affine.
The proof first shows that $f$ maps planes to planes and hence parallel lines to parallel lines.
Parallelograms then go to parallelograms, which makes the induced map $\varphi$ on vectors well defined and additive.
Comparing $f$ on the points of one line with its action on scalars defines $\sigma$ and shows that it is a field automorphism.
For $k = \R$ the only field automorphism is the identity, which gives the last sentence.

The hypothesis on $k$ is needed: over $\F_2$ each line has exactly two points, so every bijection maps lines to lines.
For $n \ge 3$ there are bijections of $\F_2^n$ that are not affine, for example the map that swaps two basis vectors $e_1$ and $e_2$ and fixes every other point.
It is not affine, since it fixes $0$, $e_3$ and $e_1 + e_3$ but moves $e_1$.
The hypothesis $n \ge 2$ is needed too: an affine line has only one line, so every bijection of it qualifies.
Over $\C$, the map $(z_1, \dots, z_n) \mapsto (\bar z_1, \dots, \bar z_n)$ maps lines to lines but is not affine, so $\sigma$ cannot be dropped.

## Explanations

### L1 human

Suppose a bijection of the plane or of space sends every straight line to a straight line.
Over the real numbers, the theorem says that it must then be an affine map: a linear map followed by a translation.
So keeping lines straight forces the map to keep ratios along lines and parallelism.
Over other fields the map can also twist the scalars by a field automorphism, such as complex conjugation.

### L2 human

The theorem shows that the incidence structure of points and lines determines the affine structure of $k^n$, up to the automorphisms of $k$.
It is the affine version of the [[fundamental theorem of projective geometry|GKV4RVXBRE]], and one can pass between the two by adding or removing a hyperplane at infinity.
It shows that the [[collineations|V7PRPN24CV]] of a real affine space are exactly the invertible affine maps.
It also explains why the automorphism group of a field enters the symmetry groups of geometries over that field.
</document>

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# Jacobi elliptic functions

## Definition

Let $0 < k < 1$, and let $\operatorname{am}(u) = \operatorname{am}(u, k)$ be the [[Jacobi amplitude|WDHHJHVP7N]].
For real $u$, the *Jacobi elliptic functions* with modulus $k$ are
$$\operatorname{sn} u = \sin\operatorname{am} u, \qquad \operatorname{cn} u = \cos\operatorname{am} u, \qquad \operatorname{dn} u = \sqrt{1 - k^2\sin^2\operatorname{am} u}.$$
Each of them extends by [[analytic continuation|0M3YQJ81NB]] to a meromorphic function on $\C$, still written $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$.

## Remarks

Let $K = K(k)$ and $K' = K(\sqrt{1 - k^2})$ be [[complete elliptic integrals of the first kind|RM54CPPE12]].
The extended functions are [[elliptic functions|FF9ZPCZ59K]] with these periods:
$\operatorname{sn}$ has periods $4K$ and $2iK'$; $\operatorname{cn}$ has periods $4K$ and $2K + 2iK'$; $\operatorname{dn}$ has periods $2K$ and $4iK'$.
They satisfy
$$\operatorname{sn}^2 u + \operatorname{cn}^2 u = 1, \qquad \operatorname{dn}^2 u + k^2\operatorname{sn}^2 u = 1,$$
$$\frac{d}{du}\operatorname{sn} u = \operatorname{cn} u \operatorname{dn} u, \quad \frac{d}{du}\operatorname{cn} u = -\operatorname{sn} u \operatorname{dn} u, \quad \frac{d}{du}\operatorname{dn} u = -k^2\operatorname{sn} u \operatorname{cn} u.$$
Example: at $u = K$ one has $\operatorname{am} K = \pi/2$, so $\operatorname{sn} K = 1$, $\operatorname{cn} K = 0$ and $\operatorname{dn} K = k'$, where $k' = \sqrt{1 - k^2}$.
Check of the first derivative rule: $\frac{d}{du}\sin\operatorname{am} u = \cos(\operatorname{am} u)\cdot\frac{d\operatorname{am}}{du} = \operatorname{cn} u \operatorname{dn} u$.
Non-example: $\sin$ is not one of these functions for $0 < k < 1$, since $\sin$ is entire while $\operatorname{sn}$ has poles, at $u = iK'$ for example.
Limits: for $k \to 0$ they become $\sin$, $\cos$, $1$, and the formulas with $k = 1$ give $\tanh$, $\operatorname{sech}$, $\operatorname{sech}$.
</document>

<document source="https://tamnd.github.io/math2/node/W448SK1RTY.md" tag="W448SK1RTY" kind="concept">
# Elementary theory of the category of sets

## Summary

The Elementary Theory of the Category of Sets (ETCS) is an axiomatic set theory proposed by Lawvere in 1964.
It describes sets and functions through the way functions compose, instead of through a membership relation.
Its models are the nondegenerate well-pointed [[toposes|6837RS7N82]] with a natural numbers object in which every epimorphism splits.
It is weaker than [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with choice, but strong enough for most ordinary mathematics.

## Explanations

### L1 human

In ETCS a set is not a bag of members; it is an object of a [[category|A2DE66937D]], and functions are its arrows.
An element of a set $X$ is a function from a one-point set into $X$.
The axioms say what one can build: products, function sets, subsets classified by truth values, the natural numbers, and sections of surjections.
For example, the subsets of $X$ correspond to functions from $X$ to a two-element set of truth values.
</document>

<document source="https://tamnd.github.io/math2/node/W44XETYC4Q.md" tag="W44XETYC4Q" kind="proof">
# Proof of Lagrange's theorem

## Proof

1. Let $G$ be a finite [[group|G5S1M9]] and let $H$ be a subgroup of $G$.
2. For $a, b \in G$, let $a \sim b$ when $a^{-1}b \in H$. Since $H$ contains the identity, inverses and products, this relation is reflexive, symmetric and transitive.
3. The class of $a$ under $\sim$ is the [[left coset|03DEH43F4H]] $aH$, because $a^{-1}b \in H$ exactly when $b \in aH$.
4. The [[equivalence classes form a partition|01HT2CH8BH]] of $G$. Thus $G$ is the disjoint union of $k$ distinct left cosets for some integer $k \ge 1$.
5. [[Each left coset has $|H|$ elements|0DKMBWTD7P]].
6. Thus $|G| = k\,|H|$, and $|H|$ divides $|G|$.

## Idea

The left cosets of the subgroup cut the group into blocks of equal size. The order of the group is the number of blocks times the size of one block.
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<document source="https://tamnd.github.io/math2/node/W453BM8T6P.md" tag="W453BM8T6P" kind="concept">
# Coherent ring

## Summary

A coherent ring is a ring in which every finitely generated ideal is [[finitely presented|3EH4C3M5GV]] as a module.
Every [[Noetherian ring|0670T59HGE]] is coherent, but coherence is weaker, and it is kept in many constructions that lose the Noetherian property.
Over a coherent ring, the finitely presented modules form an abelian category, which makes it a good setting for homological algebra without the Noetherian hypothesis.

## Explanations

### L1 human

In a Noetherian ring every ideal has finitely many generators, and the relations among the generators are also finitely generated.
A coherent ring keeps only the second half for finitely generated ideals: once an ideal has finitely many generators, it has finitely many relations among them too.
For example, the polynomial ring $k[x_1, x_2, x_3, \dots]$ in infinitely many variables over a field $k$ is coherent but not Noetherian.
</document>

<document source="https://tamnd.github.io/math2/node/W46BP9W06H.md" tag="W46BP9W06H" kind="definition">
# Fixed point

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a [[map|0EPAT51DXQ]].
A point $x \in X$ is a *fixed point* of $f$ if $f(x) = x$.
The set of fixed points of $f$ is $\operatorname{Fix}(f) = \set{x \in X : f(x) = x}$.

## Remarks

If $x$ is a fixed point of $f$, then $f^n(x) = x$ for every $n \ge 0$, so the orbit of $x$ is the single point $\set{x}$.

Example: $f(x) = \cos x$ on $\R$ has a fixed point $x_0 \in (0, 1)$.
Indeed $g(x) = \cos x - x$ is continuous with $g(0) = 1 > 0$ and $g(1) = \cos 1 - 1 < 0$, so $g$ vanishes somewhere in $(0, 1)$.

Non-example: the translation $f(x) = x + 1$ on $\R$ has no fixed point, since $x + 1 \ne x$ for every $x$.

For a function of one real variable, $x$ is a fixed point exactly when the graph of $f$ meets the line $y = x$ at $(x, x)$.
In Lean's Mathlib the predicate is `Function.IsFixedPt f x`.
</document>

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# Class

## Definition

Work in [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].
Let $\varphi(x, p_1, \dots, p_n)$ be a formula of [[first-order logic|086QKPF0A6]] in the language $\set{\in}$, and let $p_1, \dots, p_n$ be sets, the parameters.
The *class* defined by $\varphi$ with these parameters is the collection
$$C = \set{x : \varphi(x, p_1, \dots, p_n)},$$
and $x \in C$ is an abbreviation for $\varphi(x, p_1, \dots, p_n)$.
A class $C$ is a *set* if there is a [[set|0SJ3RX9TXA]] $a$ with $\forall x\, (x \in a \leftrightarrow x \in C)$.
Otherwise $C$ is a *proper class*.

In class theories such as von Neumann–Bernays–Gödel set theory, classes are instead objects of the theory.
There, a set is a class that is an element of some class.

## Remarks

Example: the class $\set{x : x \subseteq \omega}$ is a set, namely the power set of $\omega$.
Non-example: the class $V = \set{x : x = x}$ of all sets is not a set, so it is a proper class.
If $V$ were a set $a$, then separation would give the set $\set{x \in a : x \notin x}$, and [[Russell's paradox|0HESMMSYEG]] would follow.
The class of all [[ordinals|0RDJK7EQYY]] is also a proper class, by the Burali-Forti argument.
Every set $a$ is a class, defined by the formula $x \in p$ with parameter $p = a$.
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# Jakobson's theorem

## Statement

For $r \in [0, 4]$ let $f_r(x) = r x (1 - x)$ be the [[logistic map|QN9X9BNKTX]] on $[0, 1]$.
Let $A$ be the set of parameters $r \in [0, 4]$ for which $f_r$ has an [[invariant|B232EZ1FMB]] probability measure that is [[absolutely continuous|P0P21KDV6E]] with respect to [[Lebesgue measure|2Z2QVWZ262]].
Then $A$ has positive Lebesgue measure.
Moreover, $4$ is a one-sided Lebesgue density point of $A$:
$$\lim_{\varepsilon \to 0^+} \frac{\lambda(A \cap [4 - \varepsilon, 4])}{\varepsilon} = 1,$$
where $\lambda$ is Lebesgue measure on $\R$.

## Remarks

Michael Jakobson proved the theorem in 1981.
The proof is a parameter exclusion argument.
Near $r = 4$, one follows the orbit of the critical point $1/2$ and removes the parameters for which this orbit comes too close to $1/2$ too early.
For the remaining parameters the derivative along the critical orbit grows exponentially, and this growth gives the absolutely continuous invariant measure; the removed set is shown to be small.
Benedicks and Carleson later gave another proof of this kind, which they extended to the Hénon maps.

Example: $r = 4$ itself is in $A$: the measure with density $\frac{1}{\pi \sqrt{x (1 - x)}}$ on $(0, 1)$ is invariant for $f_4$.

For parameters in $A$ the measure is ergodic and has a positive [[Lyapunov exponent|NJ1FJSNK6A]], so the dynamics is chaotic in a measurable sense.
This is in contrast with parameters where $f_r$ has an attracting periodic orbit; by [[Singer's theorem|Y2DKZH3CCT]] there is at most one, and then almost every orbit converges to it.
Such parameters form an open and dense subset of $[0, 4]$, but by Jakobson's theorem they do not have full measure.
Lyubich showed in 2002 that almost every parameter $r$ is of one of these two kinds.

## Explanations

### L1 human

In the logistic family, some parameters give a stable cycle and others give chaos.
Stable cycles are robust: a small change of the parameter keeps the cycle, so these parameters form open intervals that are everywhere in the parameter line.
One might expect chaos to be rare then, like a set of measure zero.
Jakobson's theorem says no: chaotic parameters, with a smooth invariant density, fill a set of positive length, and near $r = 4$ almost all parameters are of this kind.

### L2 human

The theorem shows that the chaotic behaviour seen at $r = 4$ persists, in a measure-theoretic sense, for many nearby parameters.
This holds even though the topological picture changes at a dense set of parameters.
It became a model for the study of nonuniformly hyperbolic systems with critical points.
The tool of following the critical orbit and excluding bad parameters was later used for the Hénon family and other strange attractors.
</document>

<document source="https://tamnd.github.io/math2/node/W4B84CGEG1.md" tag="W4B84CGEG1" kind="definition">
# Elliptic operator on a compact manifold

## Definition

Let $M$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary, and let $E$ and $F$ be smooth [[vector bundles|P4DZM975BV]] over $M$.
Let $P \colon \Gamma(E) \to \Gamma(F)$ be a [[differential operator|CGBX1N9V8H]] of order $m$ with [[principal symbol|S8GC34D5ZG]] $\sigma_m(P)$.
The operator $P$ is *elliptic* if the linear map
$$\sigma_m(P)(x, \xi) \colon E_x \to F_x$$
is invertible for every $x \in M$ and every nonzero $\xi \in T^*_x M$.

## Remarks

When $\dim M \ge 1$, the condition forces $E$ and $F$ to have the same rank, since an invertible linear map exists only between spaces of equal dimension.
On an open subset of $\R^n$ with $E$ and $F$ trivial of rank $1$, it is the usual condition for an [[elliptic operator|XY8GEQSWPR]].

Example: let $(M, g)$ be a closed Riemannian manifold and let $\Delta = -\operatorname{div} \operatorname{grad}$ be the [[Laplace–Beltrami operator|F4NWZNTRMD]] on functions.
Its principal symbol is $\sigma_2(\Delta)(x, \xi) = -\abs{\xi}_g^2$, which is a nonzero number for $\xi \ne 0$, so $\Delta$ is elliptic.

Non-example: on a closed manifold $M$ of dimension $n \ge 2$, the [[exterior derivative|0ZHSTAG8ZY]] $d$ from functions to $1$-forms is not elliptic.
Its symbol at $\xi$ is the map $\R \to T^*_x M$, $c \mapsto c \, \xi$, which is injective but not surjective, since $T^*_x M$ has dimension $n > 1$.

## Explanations

### L1 human

An operator on bundles over a closed manifold is elliptic when its top-order part is an invertible matrix in every nonzero frequency direction.
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# Cellular homology

## Summary

Cellular homology is a homology theory for [[CW complexes|5W95QFHXPR]] whose chain groups are free abelian groups on the cells.
Its boundary maps are built from the degrees of the attaching maps of the cells.
It agrees with [[singular homology|M0Q5CFJ7ZH]], so it is often the fastest way to compute the singular homology of a space with a cell structure.

## Explanations

### L1 human

A CW complex is built by gluing points, then intervals, then discs, and so on.
Cellular homology counts these building blocks in each dimension and records how the boundary of each cell wraps around the cells one dimension lower.
The sphere $S^2$ can be built from one point and one disc whose whole boundary is collapsed to the point.
There is one cell in dimension $0$, one in dimension $2$, and no boundary map can be nonzero.
So the homology is $\Z$ in dimensions $0$ and $2$ and zero elsewhere.
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<document source="https://tamnd.github.io/math2/node/W4E25FVXV1.md" tag="W4E25FVXV1" kind="statement">
# Schmidt's singular value decomposition of a kernel

## Statement

Let $a < b$ and let $K \in L^2([a, b] \times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]], with [[Hilbert–Schmidt integral operator|CB5G212Q03]] $T_K$ on $L^2[a, b]$ and [[adjoint|6HSJBJ8TTQ]] $T_K^*$.
Then there are a finite or countable index set $N$ and orthonormal families $(u_n)_{n \in N}$ and $(v_n)_{n \in N}$ in $L^2[a, b]$.
There are also numbers $s_1 \ge s_2 \ge \cdots > 0$, with $s_n \to 0$ if $N$ is infinite, such that:

1. $T_K v_n = s_n u_n$ and $T_K^* u_n = s_n v_n$ for every $n \in N$;
2. $T_K \varphi = \sum_{n \in N} s_n \ip{\varphi}{v_n} u_n$ for every $\varphi \in L^2[a, b]$;
3. $K(x, y) = \sum_{n \in N} s_n \, u_n(x) \overline{v_n(y)}$, with convergence in $L^2([a, b] \times [a, b])$;
4. $\sum_{n \in N} s_n^2 = \int_a^b \int_a^b \abs{K(x, y)}^2 \, dx \, dy$.

The numbers $s_n$ are the [[singular values|QSJKPKPM7H]] of $T_K$: their squares are the nonzero eigenvalues of $T_K^* T_K$, counted with multiplicity.

## Remarks

The operator $T_K^* T_K$ is compact, self-adjoint and has nonnegative eigenvalues; it is the integral operator with the kernel $\int_a^b \overline{K(t, x)} K(t, y) \, dt$.
The [[spectral theorem for compact self-adjoint operators|A1G25R99PY]] gives orthonormal eigenfunctions $v_n$ with eigenvalues $s_n^2 > 0$.
Then $u_n = T_K v_n / s_n$ is an orthonormal family, since $\ip{T_K v_n}{T_K v_m} = \ip{T_K^* T_K v_n}{v_m} = s_n^2 \delta_{nm}$.
Parts 3 and 4 follow as for the bilinear series of a symmetric kernel, using the orthonormal family $u_n(x) \overline{v_m(y)}$.

Example: the Volterra kernel on $[0, 1]$, equal to $1$ for $y \le x$ and $0$ otherwise, has no eigenvalues, but it has singular values $s_n = \frac{1}{(n - \frac12)\pi}$, with $v_n(y) = \sqrt{2} \cos\left((n - \tfrac12)\pi y\right)$ and $u_n(x) = \sqrt{2} \sin\left((n - \tfrac12)\pi x\right)$.
Part 4 gives $\sum_n \frac{1}{(n - \frac12)^2 \pi^2} = \frac12$, the area of the triangle where the kernel is $1$.

For a Hermitian kernel, the $s_n$ are the absolute values of the nonzero eigenvalues, and one may take $u_n = \pm v_n$.

## Explanations

### L1 human

Every square-integrable kernel, symmetric or not, can be written as a sum of simple products $u_n(x) \overline{v_n(y)}$ with decreasing positive weights.
The two families of functions are each orthonormal, but they are different families.
This is the integral version of the [[singular value decomposition|000NPTMX6J]] of a matrix.

### L2 human

The decomposition is the right tool for non-symmetric kernels, where eigenfunctions may fail to exist or to be orthogonal.
Keeping the first $m$ terms gives the best approximation of $K$ in the $L^2$ norm by a degenerate kernel of rank at most $m$, with error $\left(\sum_{n > m} s_n^2\right)^{1/2}$.
For a Fredholm equation of the first kind $T_K \varphi = f$, it gives the formal solution $\varphi = \sum_n s_n^{-1} \ip{f}{u_n} v_n$, and the growth of $1/s_n$ shows why such equations are ill-posed.
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# Monte Carlo method

## Definition

Let $q$ be a real number to be computed.
A *(plain) Monte Carlo method* for $q$ consists of the following data:

1. a [[random variable|0PNRKMRC55]] $Y$ with $\mathbb E\abs{Y} < \infty$ and [[expected value|0WQNRCNY3M]] $\mathbb E Y = q$;
2. a procedure that produces [[independent and identically distributed|00MRN0FARV]] random variables $Y_1, Y_2, \ldots$ with the distribution of $Y$;
3. for each sample size $n \ge 1$, the estimate
$$\hat q_n = \frac{1}{n} \sum_{i=1}^{n} Y_i.$$

More generally, a *Monte Carlo method* is any algorithm whose output is a random variable used to approximate a deterministic quantity.

## Remarks

Example: $q = \pi/4$ is the probability that a point $(U, V)$ uniform on $[0, 1]^2$ satisfies $U^2 + V^2 \le 1$.
So $Y = 1$ if $U^2 + V^2 \le 1$ and $Y = 0$ otherwise satisfies clause 1, since $Y$ is bounded and $\mathbb E Y = \pi/4$, and independent uniform points give clause 2.
Non-example: $Y = 1/U$ with $U$ uniform on $(0, 1)$ has $\mathbb E\abs{Y} = \int_0^1 u^{-1}\, du = \infty$, so clause 1 fails and the averages $\hat q_n$ tend to $+\infty$ almost surely.
The estimate $\hat q_n$ is unbiased, $\mathbb E \hat q_n = q$, and if $\operatorname{Var} Y = \sigma^2 < \infty$ its standard deviation is $\sigma/\sqrt{n}$.
In practice clause 2 is realised with a pseudorandom number generator, so the samples are only imitations of independent random variables.
Methods based on Markov chains, such as Markov chain Monte Carlo, fit the general sense but not the plain one, since their samples are dependent.
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<document source="https://tamnd.github.io/math2/node/W4TT4W95GR.md" tag="W4TT4W95GR" kind="definition">
# Normal complex space

## Definition

Let $(X, \mathcal{O}_X)$ be a [[complex analytic space|HXSZA44J5J]].
A point $x \in X$ is *normal* if the [[stalk|JTAKAPTT8H]] $\mathcal{O}_{X,x}$ is an [[integrally closed domain|H20P600HQS]].
The space $X$ is *normal* if every point of $X$ is normal.

## Remarks

A normal space is [[reduced|831N2Q4GCB]], since an integral domain has no nonzero nilpotents.

Example: the cusp $A = \set{z_1^2 = z_2^3} \subseteq \C^2$ is not normal at $0$.
Its local ring is isomorphic to $\C\{t^2, t^3\}$ by $z_1 \mapsto t^3$, $z_2 \mapsto t^2$.
The element $t = z_1 / z_2$ of the field of fractions satisfies $t^2 - z_2 = 0$, so it is integral, but it does not lie in $\C\{t^2, t^3\}$.

Non-example of another kind: the cross $\set{z_1 z_2 = 0}$ is not normal at $0$, because its local ring has the zero divisors $z_1$ and $z_2$, so it is not a domain.

Example: $\C^n$ is normal, since the ring $\C\{z_1, \dots, z_n\}$ of convergent power series is factorial, and factorial domains are integrally closed.

For a reduced space $X$, there is an equivalent form.
The space $X$ is normal if and only if the following holds for every open $U \subseteq X$.
Every function on $U$ that is holomorphic on the regular points and locally bounded is holomorphic on $U$.
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<document source="https://tamnd.github.io/math2/node/W57F9ENPMV.md" tag="W57F9ENPMV" kind="definition">
# Split exact sequence

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ be a [[short exact sequence|KJKNJMGKZZ]] in $\mathcal{A}$.
Let $A \oplus C$ be the [[biproduct|BZEJKTQ76T]] of $A$ and $C$, with inclusion $i_A \colon A \to A \oplus C$ and projection $p_C \colon A \oplus C \to C$.
The sequence is *split* if there is an isomorphism $h \colon B \to A \oplus C$ with $h \circ f = i_A$ and $p_C \circ h = g$.

## Remarks

Equivalently, there is a morphism $t \colon B \to A$ with $t \circ f = 1_A$, or there is a morphism $u \colon C \to B$ with $g \circ u = 1_C$.
Every additive functor sends a split exact sequence to a split exact sequence.

Example: $0 \to \Z \xrightarrow{i} \Z \oplus \Z/2\Z \xrightarrow{p} \Z/2\Z \to 0$, with $i(n) = (n, 0)$ and $p(n, a) = a$, is split.
Take $h$ to be the identity of $\Z \oplus \Z/2\Z$; then $h \circ i = i_A$ and $p_C \circ h = p$.

Non-example: $0 \to \Z \xrightarrow{2} \Z \to \Z/2\Z \to 0$ is short exact but not split.
A split sequence would give an isomorphism $\Z \cong \Z \oplus \Z/2\Z$, but $\Z$ has no nonzero element of finite order.
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<document source="https://tamnd.github.io/math2/node/W58NYR1DGR.md" tag="W58NYR1DGR" kind="example">
# Reeb foliation

## Example

Let $D = \set{u \in \R^2 : \abs{u} < 1}$ be the open unit disc.
On $D \times \R$ consider, for each $c \in \R$, the surface
$$L_c = \set{(u, t) \in D \times \R : t = c + e^{1/(1 - \abs{u}^2)}}.$$
These surfaces are disjoint graphs over $D$, and they fill $D \times \R$.
Together with the boundary cylinder $\partial D \times \R$, they form a codimension one [[foliation|98HTB7MSPX]] of the closed solid cylinder $\bar D \times \R$.
Each $L_c$ is a plane that bends upwards and approaches the boundary cylinder as $\abs{u} \to 1$, since $e^{1/(1 - \abs{u}^2)} \to \infty$.
The translation $t \mapsto t + 1$ maps $L_c$ to $L_{c+1}$, so the foliation passes to the quotient solid torus $\bar D \times (\R / \Z)$.
This foliated solid torus is a *Reeb component*.
Its boundary torus is one leaf, and every interior leaf is a plane that spirals towards the boundary.

The $3$-sphere is the union of two solid tori glued along their boundary tori.
The gluing sends the meridian of each solid torus to the longitude of the other.
Putting a Reeb component on each solid torus gives a smooth codimension one foliation of $S^3$, the *Reeb foliation*.
The leaves are one torus and infinitely many planes.

## Remarks

The construction shows that $S^3$, which has Euler characteristic $0$, carries a codimension one foliation.
Novikov proved that every codimension one foliation of class $C^2$ on $S^3$ has a compact leaf, a torus bounding a Reeb component.
The torus leaf has infinite fundamental group, so the [[Reeb stability theorem|Z93KMZ9R37]] does not force the other leaves to be compact, and indeed they are planes.

## Explanations

### L1 human

Imagine a stack of bowls inside a solid cylinder, each bowl a little higher than the last, with rims climbing ever higher along the wall.
Roll the cylinder into a doughnut, and the bowls become sheets spiralling towards the doughnut's surface.
Two such doughnuts glued together make up the $3$-sphere, foliated by one torus and many planes.
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# Packing measure

## Summary

Packing measure gives the $s$-dimensional size of a set in a [[metric space|0VKD1TP7F3]].
It packs disjoint small balls centred in the set, instead of covering the set.
It is a dual of [[Hausdorff measure|Q1TGZK8RN6]], which uses covers.
It leads to the packing dimension, which is at least the Hausdorff dimension, and the two can differ for irregular fractals.

## Explanations

### L1 human

Hausdorff measure asks for the most economical way to cover a set by small pieces.
Packing measure asks the opposite question: how much room can one fill with disjoint small balls whose centres lie in the set.
Each ball counts by its diameter to the power $s$.
For a line segment of length $1$ and $s = 1$, the best packings fill the segment almost completely, and the packing measure is $1$.
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<document source="https://tamnd.github.io/math2/node/W5BHHXCYXA.md" tag="W5BHHXCYXA" kind="concept">
# Lusternik–Schnirelmann category

## Summary

The Lusternik–Schnirelmann category of a space $X$ is the least number of open sets, minus one, that cover $X$ and are each contractible inside $X$.
It is a homotopy invariant that measures how far a space is from being contractible.
Lusternik and Schnirelmann introduced it to bound from below the number of critical points of a smooth function on a closed manifold.
It is itself bounded below by the cup-length of the cohomology ring.

## Explanations

### L1 human

Try to cover a space by as few open patches as possible, where each patch can be shrunk to a point inside the whole space.
A contractible space needs one patch, so its category is $0$.
A sphere cannot be shrunk in one piece.
But the sphere minus the north pole and the sphere minus the south pole each shrink to a point, so the sphere has category $1$.
The more patches a space needs, the more complicated its shape.
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<document source="https://tamnd.github.io/math2/node/W5BZZ59CVM.md" tag="W5BZZ59CVM" kind="statement">
# Balog–Szemerédi–Gowers theorem

## Statement

There are absolute constants $c > 0$ and $C > 0$ with the following property.
Let $G$ be an [[abelian group|082GHNPJ53]], let $K \ge 1$, and let $A, B \subseteq G$ be finite sets with $\abs{A} = \abs{B} = n$.
Suppose that the [[additive energy|MAQKH72P6V]] satisfies $E(A, B) \ge n^3 / K$.
Then there are subsets $A' \subseteq A$ and $B' \subseteq B$ such that
$$\abs{A'} \ge c K^{-C} n, \qquad \abs{B'} \ge c K^{-C} n, \qquad \abs{A' + B'} \le C K^{C} n.$$

## Remarks

The converse direction is easy: if $\abs{A + B} \le Kn$, then Cauchy–Schwarz gives $E(A, B) \ge n^4 / (Kn) = n^3/K$.
Large energy alone does not give a small sumset for the whole sets.
For example, let $A$ be the union of an arithmetic progression of length $n/2$ and $n/2$ random-looking integers.
Then $E(A, A)$ is at least of order $n^3$, but $\abs{A + A}$ is of order $n^2$; the theorem finds the progression part.
The proof is graph-theoretic.
One forms the bipartite graph on $A \cup B$ with an edge $ab$ when $a + b$ is a popular sum.
A dependent-random-choice argument then finds $A'$ and $B'$ joined by many paths of length three, which bounds $\abs{A' + B'}$.
Balog and Szemerédi proved the theorem first, and Gowers obtained polynomial dependence on $K$.

## Explanations

### L1 human

Additive energy counts how often sums of elements of two sets coincide.
If there are many coincidences, one would like to say that the sets are structured.
The theorem says that large pieces of the two sets are indeed structured, with few distinct sums.

### L2 human

The theorem lets one pass from a statistical statement, large energy, to a combinatorial one, a small sumset for large subsets.
After that, [[Freiman's theorem|KEPFYE6WYZ]] gives structure.
Gowers used this chain in his quantitative proof of [[Szemerédi's theorem|B23B0QEMN5]].
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<document source="https://tamnd.github.io/math2/node/W5ECXPM6FW.md" tag="W5ECXPM6FW" kind="concept">
# Canonical module

## Summary

The canonical module $\omega_R$ of a [[Cohen–Macaulay|N1QJQEGBFC]] local ring $R$ is a finitely generated module that serves as a dualizing object for the Cohen–Macaulay $R$-modules.
It is the algebraic counterpart of the sheaf of top differential forms on a smooth variety.
A Cohen–Macaulay local ring is [[Gorenstein|T6HCX959PJ]] exactly when $\omega_R \cong R$, and the canonical module appears in [[Grothendieck's local duality|VA3DY2YX83]].

## Explanations

### L1 human

For a Gorenstein ring the ring itself has perfect duality properties.
For a general Cohen–Macaulay ring this fails, but a substitute module $\omega_R$ often exists that has those properties.
It is unique up to isomorphism when it exists.
For a finite-dimensional local algebra over a field, the canonical module is the dual vector space of the algebra, with the induced module structure.
</document>

<document source="https://tamnd.github.io/math2/node/W5FHT07SJJ.md" tag="W5FHT07SJJ" kind="definition">
# Symplectic vector space

## Definition

Let $F$ be a field and let $V$ be a [[finite-dimensional vector space|0ZHWKQMVX5]] over $F$.
A *symplectic form* on $V$ is a [[bilinear form|0SQZMXB6GP]] $\omega \colon V \times V \to F$ such that:

1. $\omega$ is [[alternating|0DGDS5XY84]]: $\omega(v, v) = 0$ for all $v \in V$.
2. $\omega$ is [[nondegenerate|01NHFZF2SK]]: if $\omega(v, w) = 0$ for all $w \in V$, then $v = 0$.

A *symplectic vector space* is a pair $(V, \omega)$ of such a space $V$ and a symplectic form $\omega$ on $V$.

## Remarks

Example: on $F^{2n}$ with coordinates $(x_1, \dots, x_n, y_1, \dots, y_n)$, the standard form is $\omega(u, u') = \sum_{i=1}^n (x_i y'_i - x'_i y_i)$.
It is alternating, since each term vanishes when $u = u'$.
It is nondegenerate: if $u$ has $x_i \ne 0$, then $\omega(u, f_i) = x_i \ne 0$, and if $u$ has $y_i \ne 0$, then $\omega(u, e_i) = -y_i \ne 0$.
Here $e_i$ and $f_i$ are the standard basis vectors for the coordinates $x_i$ and $y_i$.
Non-example: the form $\omega(u, u') = x_1 y'_1 - x'_1 y_1$ on $F^3$ with coordinates $(x_1, y_1, z)$ is alternating but degenerate, because $(0, 0, 1)$ pairs to zero with every vector.
Every symplectic vector space has even dimension $2n$ and a basis $e_1, \dots, e_n, f_1, \dots, f_n$ with $\omega(e_i, f_j) = \delta_{ij}$ and $\omega(e_i, e_j) = \omega(f_i, f_j) = 0$.
In such a basis, the matrix of $\omega$ is $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$.
In characteristic other than $2$, alternating is the same as skew-symmetric: $\omega(v, w) = -\omega(w, v)$.
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# Isospectral manifolds

## Definition

Let $(M_1, g_1)$ and $(M_2, g_2)$ be closed [[Riemannian manifolds|7CBT4SM83Y]].
They are *isospectral* if they have the same [[spectrum of the Laplace–Beltrami operator|B8TYKV7AA1]].
This means that each real number $\lambda$ has the same multiplicity as an eigenvalue of the Laplacian of $M_1$ and of $M_2$.

## Remarks

An [[isometry|R9YWQS15AA]] $M_1 \to M_2$ carries eigenfunctions to eigenfunctions, so isometric manifolds are isospectral.
By Weyl's law, isospectral manifolds have the same dimension and the same volume.
Variants compare the spectra of the Laplacian on $p$-forms, or the Dirichlet or Neumann spectra of domains with boundary.

Example: [[Milnor's two flat tori|JNR9YJTT54]] $\R^{16}/\Gamma_1$ and $\R^{16}/\Gamma_2$, built from two non-isometric even unimodular lattices, are isospectral and not isometric.

Non-example: the round spheres $S^2$ of radius $1$ and of radius $2$ are not isospectral.
Their eigenvalues are $k(k+1)$ and $k(k+1)/4$, so the first nonzero eigenvalues $2$ and $\tfrac12$ differ.
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# Zero-dimensional space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *zero-dimensional* if it has a base consisting of [[clopen sets|DQ4E55T3BW]], that is, of sets that are both open and closed.

## Remarks

Example: $\Q$ with the subspace topology from $\R$ is zero-dimensional.
For irrational numbers $a < b$, the set $\Q \cap (a, b) = \Q \cap [a, b]$ is clopen in $\Q$.
These sets form a base, since irrational numbers are dense in $\R$.

Non-example: $\R$ is not zero-dimensional.
Since $\R$ is [[connected|0V032VWTHD]], its only clopen sets are $\emptyset$ and $\R$, and they do not form a base.

Some authors define zero-dimensional by [[Lebesgue covering dimension|A3W7RGXXK9]] zero instead.
The two definitions agree for separable metric spaces, but not for all spaces.
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# Natural duality

## Summary

A natural duality represents each algebra in the [[quasivariety|WRXEKT4NQS]] generated by a finite algebra $M$.
The algebra becomes an algebra of continuous structure-preserving maps on a topological structure.
The dual of an algebra $A$ is the set of homomorphisms from $A$ to $M$.
It is viewed inside a power of a topological structure $\underset{\sim}{M}$ on the same set as $M$.
[[Stone duality|B9T33D5MJK]] for Boolean algebras and [[Priestley duality|SA22SEC75K]] for bounded distributive lattices are the model cases.
The theory gives general conditions under which such a duality exists.
It also tells when the duality is a dual equivalence of categories, a [[Stone-type duality|VH12PN9V13]].

## Explanations

### L1 human

Stone showed that a Boolean algebra can be rebuilt from the set of its homomorphisms into the two-element Boolean algebra, once that set carries the right topology.
Natural duality asks the same question for other finite "building block" algebras $M$.
Each algebra $A$ built from copies of $M$ is turned into a space of homomorphisms $A \to M$, with extra relations or operations remembered from $M$.
The goal is that $A$ can be recovered as the continuous structure-preserving maps from that space back to $M$.
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# Cofibration

## Summary

A cofibration is a continuous map $i \colon A \to X$ along which homotopies extend: it has the [[homotopy extension property|D42X0F1EG1]] for every target space.
Inclusions of subcomplexes of CW complexes are the main examples.
Cofibrations are well behaved for gluing, and for a cofibration the quotient $X / A$ has the homotopy type of the mapping cone.
They are dual, in the sense of [[Eckmann–Hilton duality|1EK3355WMC]], to fibrations.

## Explanations

### L1 human

A cofibration is a nicely attached subspace.
Whenever a map on the whole space is given and you deform it on the subspace, the deformation can be carried along on the whole space.
The boundary circle of a disk is attached in this nice way.
An open interval inside a closed interval is not, because its closure adds new points.
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# Submultiplicative function

## Definition

Let $(S, \cdot)$ be a [[semigroup|0XMEM406JM]].
A function $f \colon S \to [0, \infty)$ is *submultiplicative* if
$$f(x \cdot y) \le f(x) \, f(y) \quad \text{for all } x, y \in S.$$

## Remarks

For $S = (\N_{\ge 1}, +)$ the condition reads $a_{m+n} \le a_m a_n$ for a sequence $(a_n)_{n \ge 1}$, and one speaks of a submultiplicative sequence.
Example: on the multiplicative semigroup $(0, \infty)$, let $f(x) = x + 1$.
Then $f(xy) = xy + 1 \le xy + x + y + 1 = f(x) f(y)$, since $x, y > 0$.
Example: for a square matrix $A$ and a [[matrix norm|54T6Y6DSY4]] with $\norm{BC} \le \norm{B} \norm{C}$, the sequence $a_n = \norm{A^n}$ is submultiplicative.
Non-example: the constant function $f = 1/2$ on any semigroup fails the condition, since $1/2 > 1/4 = f(x) f(y)$.
If $f$ takes only positive values, then $f$ is submultiplicative exactly when $\log f$ satisfies $\log f(xy) \le \log f(x) + \log f(y)$.
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# Existence of transcendence bases

## Statement

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
Let $S \subseteq T \subseteq L$ be subsets such that $S$ is [[algebraically independent|KV4PT6KW10]] over $K$ and $L$ is [[algebraic|09EYMKMSB9]] over $K(T)$.
Then there is a [[transcendence basis|E45PA267S4]] $B$ of $L/K$ with $S \subseteq B \subseteq T$.
In particular, taking $S = \emptyset$ and $T = L$, every field extension has a transcendence basis.

## Remarks

Apply [[Zorn's lemma|04RXTQA5ZA]] to the algebraically independent sets $A$ with $S \subseteq A \subseteq T$, ordered by inclusion; a union of a chain is independent, since a polynomial relation involves only finitely many elements.
Let $B$ be a maximal one.
Each $t \in T$ is algebraic over $K(B)$, since otherwise $B \cup \set{t}$ would be a larger independent set.
So $K(T)$ is algebraic over $K(B)$, and $L$ is algebraic over $K(B)$ by transitivity of algebraic extensions.
For a finitely generated extension $L = K(t_1, \dots, t_n)$, Zorn's lemma is not needed, and one gets a transcendence basis among $t_1, \dots, t_n$.

## Explanations

### L1 human

Every field extension can be split into a free part and an algebraic part.
One can choose the free variables among any given generators, and one can start from any free variables one already has.

### L2 human

This is the analogue of the theorem that [[every vector space has a basis|00MS9VMQ9Z]], and the proof is the same Zorn's lemma argument.
Together with the exchange property, it gives that all transcendence bases have the same cardinality, which defines the [[transcendence degree|D7TR0YWR9H]].
It is used to write every extension as a [[purely transcendental extension|G5PVJMM37Z]] followed by an algebraic one.
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<document source="https://tamnd.github.io/math2/node/W6AVEAW2WR.md" tag="W6AVEAW2WR" kind="statement">
# Five lemma

## Statement

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], such as the category of modules over a ring.
Consider a [[commutative diagram|WV898P4X16]]
$$\begin{array}{ccccccccc} A & \to & B & \to & C & \to & D & \to & E \\ \downarrow{\scriptstyle l} & & \downarrow{\scriptstyle m} & & \downarrow{\scriptstyle n} & & \downarrow{\scriptstyle p} & & \downarrow{\scriptstyle q} \\ A' & \to & B' & \to & C' & \to & D' & \to & E' \end{array}$$
in $\mathcal{A}$ whose two rows are [[exact|JB7ZBFBA9C]].
Suppose that $m$ and $p$ are isomorphisms, that $l$ is an [[epimorphism|RVXDSY04KG]] and that $q$ is a [[monomorphism|Y2NP1JFR86]].
Then $n$ is an isomorphism.

## Remarks

The lemma is the conjunction of the two halves of the [[four lemma|JMCKACGCQ0]].
The left four columns, with $m, p$ monomorphisms and $l$ an epimorphism, show that $n$ is a monomorphism.
The right four columns, with $m, p$ epimorphisms and $q$ a monomorphism, show that $n$ is an epimorphism.
Each half is a diagram chase.
A typical use: a map between two short exact sequences that is an isomorphism on the outer terms is an isomorphism on the middle term, by taking $A = A' = E = E' = 0$.
The existence of the map $n$ making the diagram commute is essential: $0 \to \Z/2\Z \to \Z/4\Z \to \Z/2\Z \to 0$ and $0 \to \Z/2\Z \to \Z/2\Z \oplus \Z/2\Z \to \Z/2\Z \to 0$ have isomorphic outer terms, but $\Z/4\Z \not\cong \Z/2\Z \oplus \Z/2\Z$.

## Explanations

### L1 human

Picture two exact rows of five objects, with vertical maps between them.
If the four outer vertical maps are isomorphisms, then so is the middle one.
In fact less is needed at the two ends: the far left map only needs to be onto, and the far right map only needs to be one-to-one.

### L2 human

The five lemma is used constantly to show that a natural map is an isomorphism.
One fits the map into a ladder of long exact sequences, for example in homology, where the other maps are already known to be isomorphisms.
This is how one proves that a map of spaces that induces isomorphisms on the homology of pieces induces an isomorphism on the whole, as in Mayer–Vietoris arguments.
It also proves the splitting lemma and the invariance of derived functors under the choice of resolution.
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# Euler summation

## Definition

Let $\sum_{k \ge 0} a_k$ be a [[series|0JH4MN4K1C]] of complex numbers, and let $q \ge 0$ be real.
The series is *Euler summable* with parameter $q$, or $(E, q)$ summable, to $s$ if the series
$$\sum_{p \ge 0} \frac{1}{(1 + q)^{p + 1}} \sum_{k=0}^{p} \binom{p}{k} q^{p - k} a_k$$
converges and its sum is $s$.
Here $\binom{p}{k}$ is the [[binomial coefficient|092GEF38YD]].
The case $q = 1$ is called Euler summation without qualification:
$$\sum_{p \ge 0} \frac{1}{2^{p + 1}} \sum_{k=0}^{p} \binom{p}{k} a_k = s.$$

## Remarks

For $q = 0$ the inner sum is $a_p$ alone, and $(E, 0)$ is ordinary convergence.

Example: take $a_k = (-1)^k (k + 1)$, that is $1 - 2 + 3 - 4 + \cdots$, and $q = 1$.
The identities $\sum_{k=0}^{p} \binom{p}{k} (-1)^k = 0$ for $p \ge 1$ and $\sum_{k=0}^{p} \binom{p}{k} (-1)^k k = 0$ for $p \ge 2$ show that only $p = 0$ and $p = 1$ contribute.
These give $\frac12 \cdot 1 + \frac14 \cdot (1 - 2) = \frac14$, so the series is $(E, 1)$ summable to $1/4$.

Non-example: for $1 + 1 + 1 + \cdots$ and $q = 1$, the inner sum is $2^p$, and every term of the new series is $\frac12$.
That series diverges, so $1 + 1 + 1 + \cdots$ is not $(E, 1)$ summable.
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# Homogeneous integral equation

## Summary

A homogeneous integral equation is a [[linear integral equation|AYZCKG34QK]] whose free term $f$ is zero, such as $\varphi(x) = \lambda \int_a^b K(x, s) \varphi(s) \, ds$.
It always has the zero solution, and its solutions form a vector space.
The values of $\lambda$ for which it has a nonzero solution are the characteristic values of the kernel.
By the Fredholm alternative, these values decide whether the inhomogeneous equation is uniquely solvable.

## Explanations

### L1 human

Homogeneous means that the equation has no term without the unknown.
It is the integral version of a system $A x = 0$ of linear equations.
Zero is always a solution; the interesting question is whether there are others.
For example, $\varphi(x) = \lambda \int_0^1 \varphi(s) \, ds$ has the nonzero solution $\varphi = 1$ exactly when $\lambda = 1$.
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# Tennenbaum's theorem

## Statement

Let $M = (\N, 0^M, S^M, \oplus, \otimes)$ be a [[model|MVCH6N9SNW]] of the [[Peano axioms|0QFT7Z76N5]] $\mathsf{PA}$ whose domain is $\N$, and suppose $M$ is not isomorphic to the standard model.
Then neither $\oplus \colon \N^2 \to \N$ nor $\otimes \colon \N^2 \to \N$ is a [[computable function|KZSB5E471K]].
In particular, no [[countable|01TQB88H4C]] non-standard model of $\mathsf{PA}$ is computable.

## Remarks

Proof idea: take two computably enumerable sets $A$ and $B$ that are disjoint but cannot be separated by any computable set.
Let $p_n$ be the $n$-th prime.
Using coding of finite sets inside $M$ and overspill, one finds an element $c$ of $M$ such that $X = \set{n \in \N : M \models p_n \mid c}$ satisfies $A \subseteq X$ and $X \cap B = \emptyset$.
Suppose $\oplus$ is computable.
For a standard number $p_n$, the product of $p_n$ and $q$ in $M$ is the sum of $p_n$ copies of $q$, so it is computable from $q$.
In $M$ there are unique $q$ and $r < p_n$ with $c = p_n q + r$, and $r$ is standard, so a search for $q$ and $r$ finds them and decides whether $p_n$ divides $c$.
Then $X$ would be a computable set separating $A$ and $B$, a contradiction.
The case of $\otimes$ needs a further argument.

The theorem uses that the model is non-standard: the standard model $\N$ has computable addition and multiplication.

## Explanations

### L1 human

The natural numbers can be built on a computer: addition and multiplication are computable.
Other models of the axioms of arithmetic exist, but none of them can be built on a computer in this way.
So among the countable models of arithmetic, the standard one is the only one that is computable.

### L2 human

The theorem gives a computational characterization of the standard model among countable [[non-standard models of arithmetic|1QWN5CV4CG]].
It contrasts with weaker theories: for Presburger arithmetic, or for Robinson arithmetic $\mathsf{Q}$, there are computable non-standard models.
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# G-signature theorem

## Statement

Let $M$ be a closed oriented smooth manifold of dimension $4k$.
Let a compact Lie group $G$ act smoothly on $M$, and let $g \in G$ act by an orientation-preserving diffeomorphism.
The [[intersection form|WK8GKRKAVD]] on $H^{2k}(M; \R)$ is $g$-invariant, so $H^{2k}(M; \R) = H^+ \oplus H^-$ with $g$-invariant subspaces on which the form is positive and negative definite.
The $g$-signature is $\operatorname{sign}(g, M) = \operatorname{tr}(g \mid H^+) - \operatorname{tr}(g \mid H^-)$.
Let $M^g$ be the fixed point set of $g$, a disjoint union of closed submanifolds $F$.
Then
$$\operatorname{sign}(g, M) = \sum_{F} \langle L(g, F), [F] \rangle.$$
Here each $L(g, F) \in H^*(F; \C)$ is a characteristic class.
It is determined by the tangent bundle of $F$, the normal bundle of $F$ in $M$ and the action of $g$ on that normal bundle.
In particular:

1. If $g$ has no fixed points, then $\operatorname{sign}(g, M) = 0$.
2. If $g$ is the identity, the formula is the [[Hirzebruch signature theorem|TEG1ZX1AGJ]].

## Remarks

The theorem is the [[equivariant index theorem|T866256Y1W]] for the [[signature operator|4NCYA7GME1]], whose equivariant index evaluated at $g$ is $\operatorname{sign}(g, M)$; the topological side localizes to the fixed point set.
For an isolated fixed point the contribution is an explicit product of cotangents of half the rotation angles of $g$ on the tangent space, up to a power of $i$.
Example: let a finite group $\Gamma$ act freely and preserving orientation on $M$.
Then every $g \ne 1$ has $\operatorname{sign}(g, M) = 0$, and averaging the characters gives $\sigma(M) = \abs{\Gamma} \, \sigma(M / \Gamma)$, where $\sigma$ is the [[signature|07EZA80NEH]] of the intersection form.

## Explanations

### L1 human

A symmetry of a manifold acts on its middle cohomology and preserves the intersection form.
The $g$-signature measures this action by a trace.
The theorem says that this trace can be computed only by looking near the points that the symmetry fixes.

### L2 human

The $G$-signature theorem of Atiyah and Singer is a basic tool in the study of group actions on manifolds.
It gives obstructions to actions with prescribed fixed point data, through the cotangent expressions that appear at isolated fixed points.
It is a model for later fixed point formulas from the equivariant index theorem.
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# Kummer's equation

## Definition

Let $a$ and $b$ be complex numbers.
*Kummer's equation* with parameters $a$ and $b$ is the [[linear differential equation|0QQRP17Z7S]] of second order
$$z \frac{d^2 w}{dz^2} + (b - z) \frac{dw}{dz} - a w = 0$$
for a function $w$ of a complex variable $z$.

## Remarks

The point $z = 0$ is a [[regular singular point|H96HXG6CA0]] with exponents $0$ and $1 - b$.
The point $z = \infty$ is an [[irregular singularity|AT1W4MMTYR]], and there are no other singular points.
Example: for $a = b$, the function $w = e^z$ is a solution, since $z e^z + (b - z) e^z - b e^z = 0$.
Non-example: for $a \ne b$, the same function is not a solution, since the left side equals $(b - a) e^z \ne 0$.
For $b$ not in $\set{0, -1, -2, \dots}$, the solution analytic at $0$ with value $1$ is Kummer's function $M(a, b, z) = {}_1F_1(a; b; z)$.
A second standard solution is the Tricomi function $U(a, b, z)$, singled out by its behaviour $U(a, b, z) \sim z^{-a}$ as $z \to \infty$ in the right half-plane.
The substitution $w = z^{-b/2} e^{z/2} v$ turns the equation into Whittaker's equation, which has no first derivative term.
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# Prüfer domain

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]] with [[field of fractions|0VF11ZTTMD]] $K$.
$R$ is a *Prüfer domain* if every nonzero finitely generated [[ideal|0MYSA6MYNA]] $I$ of $R$ is [[invertible|E190TAV2S6]].
That is, there is an $R$-submodule $J$ of $K$ with $IJ = R$.

## Remarks

Equivalent form: for every prime ideal $P$ of $R$, the localization $R_P$ is a valuation ring, that is, for every nonzero $x \in K$, $x \in R_P$ or $x^{-1} \in R_P$.
Example: every [[principal ideal domain|0RZYDVZ475]] is Prüfer; a nonzero ideal $aR$ has inverse $a^{-1}R$.
Non-example: in $\Z[x]$, the ideal $I = (2, x)$ is not invertible; the only $y \in \Q(x)$ with $yI \subseteq \Z[x]$ are the elements of $\Z[x]$, so any $J$ with $IJ \subseteq \Z[x]$ satisfies $IJ \subseteq I \ne \Z[x]$.
A Noetherian Prüfer domain is a Dedekind domain; a Prüfer domain that is a [[GCD domain|ACDEK1GD9G]] is a [[Bézout domain|HGEX40P0Z0]].
Prüfer domains are integrally closed.
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# Turing machine

## Definition

A *Turing machine* is a tuple $M = (Q, \Gamma, \sqcup, \Sigma, \delta, q_0, F)$ with the following parts.

1. $Q$ is a finite nonempty [[set|0BSJ7RQ9BW]] of states.
2. $\Gamma$ is a finite set of tape symbols, and $\sqcup \in \Gamma$ is the blank symbol.
3. $\Sigma \subseteq \Gamma \setminus \set{\sqcup}$ is the set of input symbols.
4. $q_0 \in Q$ is the initial state, and $F \subseteq Q$ is the set of halting states.
5. $\delta$ is a partial [[function|0EPAT51DXQ]] from $(Q \setminus F) \times \Gamma$ to $Q \times \Gamma \times \set{L, R}$, the transition function.

A configuration consists of a state $q$, a tape $t \colon \Z \to \Gamma$ with only finitely many non-blank cells, and a head position $p \in \Z$.
If $\delta(q, t(p)) = (q', s, D)$, the next configuration has state $q'$, the symbol $s$ written at cell $p$, and head position $p - 1$ if $D = L$ and $p + 1$ if $D = R$.
The machine *halts* when no transition applies, in particular when it reaches a state in $F$.
On an input word $w \in \Sigma^*$, the computation starts in state $q_0$ with the head at cell $0$.
At the start, $w$ is written from cell $0$ onward and all other cells are blank.

## Remarks

Example: let $Q = \set{q_0, h}$, $\Gamma = \set{0, 1, \sqcup}$, $\Sigma = \set{0, 1}$ and $F = \set{h}$.
Let $\delta(q_0, 0) = (q_0, 1, R)$, $\delta(q_0, 1) = (q_0, 0, R)$ and $\delta(q_0, \sqcup) = (h, \sqcup, R)$.
Each clause holds, and on input $0110$ the machine moves right, writes $1001$, and halts at the first blank.

Non-example: a "machine" with one state for each natural number is not a Turing machine, because clause 1 requires finitely many states.

Many variants are in use: a tape infinite in one direction only, several tapes, or a single halting state.
They all compute the same partial functions, though their running times differ.
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# Meromorphic function of several variables

## Summary

A meromorphic function of several variables is locally a quotient of two [[holomorphic functions|507C34W0ZT]] on an open set of $\C^n$.
The denominator must not be identically zero.
It extends the one-variable notion of a [[meromorphic function|0ZZ7ZYDR8W]].
A new feature for $n \ge 2$ is the presence of points of indeterminacy, where the function has no value at all, not even $\infty$.
Meromorphic functions are the natural analogue of rational functions on complex manifolds and analytic spaces.

## Explanations

### L1 human

Rational functions such as $z_1 / z_2$ are the simplest meromorphic functions on $\C^2$.
This function has poles along the line $z_2 = 0$, except at the origin.
Near the origin it takes every complex value: along the line $z_1 = c z_2$ it is constantly equal to $c$.
So at the origin it cannot be given any value, finite or infinite; the origin is a point of indeterminacy.
In one variable this never happens, because poles are isolated points where the function tends to $\infty$.
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# d'Alembert's equation

## Example

Let $f$ and $g$ be continuously differentiable functions on an open interval $J$.
*d'Alembert's equation*, also called Lagrange's equation, is the implicit first-order equation
$$y = x\, f(y') + g(y'),$$
for an unknown $y$ of $x$ with $y' \in J$.
When $f(p) = p$ for all $p$, it is [[Clairaut's equation|0T6D4PE51C]].
Method: write $p = y'$ and differentiate, which gives $p = f(p) + \left(x f'(p) + g'(p)\right) \frac{dp}{dx}$.
Where $p \ne f(p)$, view $x$ as a function of $p$; then $(p - f(p))\, \frac{dx}{dp} = f'(p)\, x + g'(p)$ is a linear equation for $x(p)$.
Each root $p_0$ of $p = f(p)$ gives the line $y = p_0 x + g(p_0)$, which may be a [[singular solution|0CZA0NCFWP]].

Instance: $y = x\, (y')^2$, so $f(p) = p^2$ and $g = 0$.
For $p \ne 0, 1$, the linear equation is $(p - p^2)\, dx/dp = 2 p x$, so $dx/dp = 2x/(1 - p)$ and $x = C/(1 - p)^2$.
This leads to the solutions $y = (\sqrt{x} - k)^2$ for $x > 0$ and a constant $k$.
Check: $y' = 1 - k/\sqrt{x}$, so $x (y')^2 = (\sqrt{x} - k)^2 = y$.
The roots $p = 0$ and $p = 1$ of $p = p^2$ give the lines $y = 0$ and $y = x$, which also solve the equation: $x \cdot 0^2 = 0$ and $x \cdot 1^2 = x$.

## Explanations

### L1 human

d'Alembert's equation gives $y$ as a combination of $x$ and the slope $y'$, rather than giving the slope directly.
The trick is to treat the slope as the new variable: then $x$ satisfies a linear equation, which can be solved.
Besides the [[general solution|0SKYFQAFYD]] there can be special straight-line solutions, as in Clairaut's equation.
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# Composition operator

## Definition

Let $\Omega$ be a set and let $X$ be a vector space of functions $\Omega \to \C$.
Let $\varphi \colon \Omega \to \Omega$ be a map such that $f \circ \varphi \in X$ for every $f \in X$.
The *composition operator* with symbol $\varphi$ is the [[linear map|0NRYHJPQG2]]
$$C_\varphi \colon X \to X, \qquad C_\varphi f = f \circ \varphi,$$
where $\circ$ is [[function composition|07HF9YCWPP]].

## Remarks

Linearity is automatic: $(af + bg) \circ \varphi = a (f \circ \varphi) + b (g \circ \varphi)$.

Example: let $X$ be the [[Hardy space|PNNNFJVF6W]] $H^2$ of the unit disc $\mathbb{D}$ and $\varphi(z) = z/2$.
For $f(z) = \sum_{n \ge 0} a_n z^n$ in $H^2$, $f \circ \varphi = \sum_{n \ge 0} 2^{-n} a_n z^n$, whose coefficients are again square-summable; so $C_\varphi$ is defined, and it is the diagonal operator with entries $2^{-n}$ on the basis $z^n$.

Non-example: $\varphi(z) = 2z$ does not map $\mathbb{D}$ into $\mathbb{D}$.
For $f(z) = \sum_{n \ge 0} z^n / (n + 1)$, which lies in $H^2$, the function $f(2z)$ is not defined on all of $\mathbb{D}$, since the series diverges for $\abs{z} > 1/2$.

When $X$ is a normed space one usually asks in addition that $C_\varphi$ be bounded.
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# Quaternion-Kähler manifold

## Definition

Let $n \ge 2$ and let $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $4n$.
Let $Sp(n) \cdot Sp(1) = (Sp(n) \times Sp(1)) / \set{\pm 1}$, acting on $\mathbb{H}^n = \R^{4n}$ by $v \mapsto a v \bar{b}$ for $a \in Sp(n)$ and unit quaternions $b \in Sp(1)$.
The manifold $M$ is *quaternion-Kähler* if its [[holonomy|WHDZ3MK56B]] group is contained in $Sp(n) \cdot Sp(1) \subseteq SO(4n)$.

Equivalently, there is a rank $3$ subbundle $Q \subseteq \End(TM)$ such that:

1. $Q$ is preserved by the [[Levi-Civita connection|CAZYGE1T4E]];
2. near each point, $Q$ has a local frame $I, J, K$ of [[almost complex structures|9SYWRGVTRS]] that are orthogonal for $g$ and satisfy $IJ = K$.

## Remarks

Example: the quaternionic projective space $\mathbb{H}P^n$ with its standard symmetric metric, whose holonomy is all of $Sp(n) \cdot Sp(1)$.

Non-example: no manifold of dimension $6$ is quaternion-Kähler, since the dimension must be a multiple of $4$.

For $n \ge 2$, a quaternion-Kähler manifold is [[Einstein|41S2EZ0KR3]].
[[Hyperkähler manifolds|87AWY7NQS0]] satisfy the definition, with holonomy in $Sp(n) \subseteq Sp(n) \cdot Sp(1)$; many authors exclude them by asking for nonzero scalar curvature.
In dimension $4$ the holonomy condition is empty, since $Sp(1) \cdot Sp(1) = SO(4)$, so one defines quaternion-Kähler $4$-manifolds instead as self-dual Einstein manifolds.
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# Schrödinger operator

## Definition

Let $I \subseteq \R$ be an open interval and let $V \colon I \to \R$ be locally integrable.
The one-dimensional Schrödinger operator with potential $V$ is the differential expression
$$H y = -y'' + V y,$$
acting on functions $y \colon I \to \C$ such that $y$ and $y'$ are locally absolutely continuous.
More generally, for an open set $\Omega \subseteq \R^d$ and a real function $V$ on $\Omega$, the Schrödinger operator is $H = -\Delta + V$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]].
The equation $H y = \lambda y$ with $\lambda \in \C$ is the time-independent Schrödinger equation.

## Remarks

Example: for $I = \R$ and $V(x) = x^2$, the function $y(x) = e^{-x^2/2}$ has $y'' = (x^2 - 1) e^{-x^2/2}$.
So $H y = -y'' + x^2 y = y$, and $y$ solves the Schrödinger equation with $\lambda = 1$.
Non-example: $-y'' + i x y$ is not a Schrödinger operator in this sense, because the potential $i x$ is not real.
In units where Planck's constant and the mass are normalised, $H$ is the energy operator of a quantum particle.
To get a self-adjoint operator on $L^2(I)$ one must also fix boundary conditions at the ends of $I$ when they are needed.

## Explanations

### L1 human

A Schrödinger operator is minus the second derivative plus multiplication by a real function.
It is the energy of a quantum particle that moves in a landscape of height $V$.
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# Caccioppoli set

## Summary

A Caccioppoli set, or set of finite perimeter, is a measurable subset of $\R^n$ whose [[perimeter|Y68BHXZJBV]] is finite.
Equivalently, its indicator function has [[bounded variation|9YQ999QHHT]].
These sets are the measure-theoretic replacement for regions with smooth boundary.
They are closed under the limits that appear when one minimizes area.
So they give a setting where problems such as the isoperimetric problem have solutions.

## Explanations

### L1 human

Many regions that come up as limits of nice regions have wild boundaries, and the classical notion of surface area breaks down for them.
A Caccioppoli set is a region whose boundary has finite area in a weak sense that still works for such limits.
Any bounded region in the plane with a polygonal boundary is a Caccioppoli set.
The union of the disjoint discs of radius $2^{-k}$ centred at $(3k, 0)$, for $k = 1, 2, \dots$, is also one, since its perimeter is $\sum_k 2\pi 2^{-k} = 2\pi$.
The union of the discs of radius $1$ centred at the same points is not, since its perimeter is infinite.
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# Cerf theory

## Summary

Cerf theory studies one-parameter families of smooth real-valued functions on a manifold and the ways in which they fail to be [[Morse functions|KPQ96640ZJ]].
In a generic family, the function is a Morse function with distinct critical values at all but finitely many times.
At those times either a pair of critical points is born or dies, or two critical values cross.
Jean Cerf developed it to study diffeomorphisms of manifolds, in particular to prove pseudo-isotopy results.
Through the correspondence between Morse functions and [[handle decompositions|9V11N8VVTP]], it describes how any two handle decompositions of a manifold are related.

## Explanations

### L1 human

Morse theory looks at one function on a manifold; Cerf theory looks at a film in which the function changes with time.
Most frames of the film show a Morse function, with clean peaks, pits and saddles.
At a few moments something happens: a peak and a saddle melt together and disappear, or appear out of nothing, or two of them swap heights.
For example, $f_t(x) = x^3 - t x$ on the line has no critical points for $t < 0$ and two for $t > 0$, which are born at $t = 0$.
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# Kellogg–Evans theorem

## Statement

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a bounded [[open set|02BH3MCBQB]].
Call a point of the [[boundary|05D5EGMFYV]] $\partial\Omega$ irregular if it is not a [[regular boundary point|B7948M365K]].
Then the set of irregular points of $\partial\Omega$ is a [[polar set|C95QJ7YH5T]].

## Remarks

Polar sets have capacity zero, so they are very small: they have Lebesgue measure zero, and in $\R^n$ with $n \ge 3$ they have Hausdorff dimension at most $n - 2$.
Consequence: for each continuous $f$ on $\partial\Omega$, the Perron–Wiener–Brelot solution $H_f$ tends to $f(\xi)$ at every boundary point $\xi$ outside a polar set, that is, quasi-everywhere.
So the generalized solution solves the Dirichlet problem up to an exceptional set of capacity zero.
Example: for the punctured ball, the irregular set is $\set{0}$, a single point, which is polar.
Example: for the [[Lebesgue spine|EFFC7688KR]], the only irregular point is the tip of the spike.

## Explanations

### L1 human

Some boundary points can be bad for the Dirichlet problem, but there are always very few of them.
They form a set so thin that electric charge cannot sit on it.
So the generalized solution takes the right boundary values almost everywhere, in the sense of capacity.

### L2 human

The theorem turns the Perron–Wiener–Brelot solution into a genuine solution off a negligible set.
It is the basis for treating boundary values quasi-everywhere in modern potential theory and in the theory of Sobolev spaces.
Its proof uses the fine topology or the equilibrium potential of compact parts of the boundary, together with the barrier characterization of regular points.
</document>

<document source="https://tamnd.github.io/math2/node/W6V340HYNK.md" tag="W6V340HYNK" kind="definition">
# Chow group

## Definition

Let $X$ be an [[algebraic variety|HV6YBS0YDH]] over a field and let $r \ge 0$ be an integer.
Let $Z_r(X)$ be the group of $r$-dimensional [[algebraic cycles|9VFWPZEVGN]] on $X$ and $\operatorname{Rat}_r(X)$ its subgroup of cycles [[rationally equivalent|K76N026JFK]] to $0$.
The *Chow group* of $r$-cycles on $X$ is the quotient
$$CH_r(X) = Z_r(X) / \operatorname{Rat}_r(X).$$
If $X$ is equidimensional of dimension $n$, the Chow group of codimension $j$ cycles is $CH^j(X) = CH_{n-j}(X)$.

## Remarks

Example: for $X = \mathbb{P}^n$ over a field and $0 \le r \le n$, $CH_r(\mathbb{P}^n) \cong \Z$, generated by the class of a linear subspace of dimension $r$.
A subvariety of dimension $r$ and degree $d$ has class $d$ times the generator.
For $r = 0$ and $X$ proper, the degree map $\sum n_i [P_i] \mapsto \sum n_i [k(P_i) : k]$ is well defined on $CH_0(X)$.
Non-example: replacing rational equivalence by algebraic equivalence gives a different group.
For an elliptic curve $E$ over an algebraically closed field $k$, $CH_0(E) \cong \Z \oplus E(k)$, while modulo algebraic equivalence the $0$-cycles form only $\Z$, detected by the degree.
For a smooth variety, $CH^1(X)$ is isomorphic to the Picard group of $X$.
</document>

<document source="https://tamnd.github.io/math2/node/W6VE30V10Y.md" tag="W6VE30V10Y" kind="example">
# Legendre's equation

## Example

Let $\lambda \in \C$. *Legendre's equation* is
$$(1 - z^2)\, y'' - 2 z\, y' + \lambda\, y = 0,$$
usually written with $\lambda = n (n + 1)$.
In normalised form, $p(z) = \frac{-2z}{1 - z^2}$ and $q(z) = \frac{\lambda}{1 - z^2}$, which are rational.
They have simple poles at $z = \pm 1$, so $\pm 1$ are regular singular points, and every other finite point is ordinary.
As $z \to \infty$, $z p(z) \to 2$ and $z^2 q(z) \to -\lambda$, so $\infty$ is ordinary or regular singular.
Hence Legendre's equation is a [[Fuchsian equation|ZBVH2YE50B]], with singular points among $-1$, $1$ and $\infty$.

## Remarks

At the [[ordinary point|QK2J038VB4]] $0$, a [[power series solution|08RT9GDDPQ]] $\sum a_k z^k$ satisfies $(k + 2)(k + 1)\, a_{k+2} = \left( k (k + 1) - \lambda \right) a_k$.
The series converge for $\abs{z} < 1$, the distance from $0$ to the nearest singular point.
If $\lambda = n (n + 1)$ with $n \ge 0$ an integer, the recurrence stops, and one solution is the Legendre polynomial $P_n$ of degree $n$.
For $n = 2$, $P_2(z) = \frac{3 z^2 - 1}{2}$: then $y'' = 3$ and $y' = 3z$, and $(1 - z^2) \cdot 3 - 2z \cdot 3z + 6 \cdot \frac{3 z^2 - 1}{2} = 0$.
The equation arises from the Laplace equation in spherical coordinates.

## Explanations

### L1 human

Legendre's equation appears when one studies gravity or electric potential around a sphere.
Its only bad points are the poles $z = \pm 1$ of the sphere and the point at infinity, and all of them are tame.
For special values of the parameter it has polynomial solutions, the Legendre polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/W6Z6CWRE25.md" tag="W6Z6CWRE25" kind="example">
# Airfoil equation

## Example

The *airfoil equation* asks for a function $\varphi$ on $(-1, 1)$ with
$$\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\varphi(t)}{t - x}\, dt = f(x), \quad -1 < x < 1,$$
for a given function $f$, where $\mathrm{PV}$ is the [[Cauchy principal value|0SYR10HD4R]].
So it inverts the [[finite Hilbert transform|5JYBXH9BP6]], and it is a [[singular integral equation|E30EA6ETZ4]] of the first kind.
In thin airfoil theory the chord of the wing is scaled to $[-1, 1]$, and $\varphi$ is the density of vorticity along the chord.
The right side $f$ is fixed by the free stream speed, the angle of attack and the slope of the camber line.

For $f$ smooth enough, the solutions are
$$\varphi(x) = -\frac{1}{\pi \sqrt{1 - x^2}}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\sqrt{1 - t^2}\, f(t)}{t - x}\, dt + \frac{C}{\sqrt{1 - x^2}},$$
with an arbitrary constant $C$.

Check on Chebyshev polynomials.
Let $T_n$ and $U_{n-1}$ be the Chebyshev polynomials of the first and second kind, $n \ge 1$.
For $-1 < x < 1$ one has
$$\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{T_n(t)}{\sqrt{1 - t^2}\,(t - x)}\, dt = U_{n-1}(x), \qquad \frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\sqrt{1 - t^2}\, U_{n-1}(t)}{t - x}\, dt = -T_n(x).$$
For $f = U_{n-1}$ the second identity turns the formula into $\varphi = T_n / \sqrt{1 - x^2}$ plus the free term, and the first identity confirms that this solves the equation.
The free term solves the homogeneous equation, since $\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{dt}{\sqrt{1 - t^2}\,(t - x)} = 0$ for $-1 < x < 1$.

## Remarks

The constant $C$ is not fixed by the equation.
In aerodynamics it is fixed by the Kutta condition, which asks that $\varphi$ stay bounded at the trailing edge $x = 1$.
The lift of the airfoil is proportional to $\int_{-1}^{1} \varphi(t)\, dt$.

## Explanations

### L1 human

A thin wing in a steady flow is replaced by a line of small vortices along its chord.
The flow must slide along the wing, and this condition gives one equation for the strength of the vortices.
That equation is the airfoil equation, and it has an explicit solution.
The solution is unique only after one adds a physical condition at the rear edge of the wing.
</document>

<document source="https://tamnd.github.io/math2/node/W72CA3R9MP.md" tag="W72CA3R9MP" kind="concept">
# Sequent calculus

## Summary

A sequent calculus is a proof system whose lines are sequents $\Gamma \vdash \Delta$.
Such a sequent is read as "if all formulas of $\Gamma$ hold, then some formula of $\Delta$ holds".
Gentzen introduced the systems LK for classical logic and LJ for intuitionistic logic in 1934.
Each connective has rules that introduce it on the left and on the right.
This symmetry makes the calculus the main tool of structural [[proof theory|N4EDJNMTEF]].

## Explanations

### L1 human

A sequent is a pair of lists of formulas, written $\Gamma \vdash \Delta$.
It says that the assumptions on the left together force at least one of the claims on the right.
Proofs are trees of sequents that grow downward, from axioms $A \vdash A$ to the sequent one wants.
For example, from $A \vdash A$ the rule for "and" on the left gives $A \wedge B \vdash A$.
Every rule only adds structure, except one rule, the cut, which removes a formula.
</document>

<document source="https://tamnd.github.io/math2/node/W783KDGT8E.md" tag="W783KDGT8E" kind="concept">
# Dirichlet-to-Neumann operator

## Summary

The Dirichlet-to-Neumann operator of a domain sends boundary values to normal derivatives.
Given a function $f$ on the boundary, it solves the [[Dirichlet problem|8D56GQ4AG8]] for [[Laplace's equation|S8RG6N1PRT]] with data $f$.
Then it returns the normal derivative of the solution on the boundary; conductivity equations work the same way.
It is a basic example of a [[pseudo-differential operator|FF27N0BWTP]] on the boundary.
It carries the boundary measurements of electrical impedance tomography, in the [[Calderón problem|R8Y8DDTWBM]], and its eigenvalues are the Steklov eigenvalues.

## Explanations

### L1 human

Fix the temperature, or the voltage, on the boundary of a body and let it settle to equilibrium inside.
Then measure how much heat, or current, flows out through each part of the boundary.
The Dirichlet-to-Neumann operator is the rule that turns the first list of numbers into the second.
On the unit disk, the boundary data $\cos(k\theta)$ extend to the harmonic function $r^k \cos(k\theta)$, whose radial derivative at $r = 1$ is $k \cos(k\theta)$.
So on the disk the operator multiplies the $k$-th Fourier mode by $\abs{k}$.
</document>

<document source="https://tamnd.github.io/math2/node/W7CK84MZFV.md" tag="W7CK84MZFV" kind="concept">
# Neural network approximation

## Summary

Neural network approximation studies how well functions can be approximated by the functions that neural networks compute.
A shallow network computes a sum $\sum_{k=1}^n c_k \sigma(w_k \cdot x + b_k)$ of ridge functions built from one activation function $\sigma$; a deep network composes several such layers.
The universal approximation theorem says that shallow networks with a continuous non-polynomial activation are dense in the [[continuous functions|0X3WGX95FE]] on every [[compact set|0TVJZYB6SH]] of $\R^d$.
Rates of approximation in terms of the number of neurons or parameters are a main topic of [[nonlinear approximation|YS1GRQRVY7]].

## Explanations

### L1 human

A neural network adds up many copies of one simple function, each one shifted, stretched and pointed in its own direction.
With enough copies, such sums can follow any continuous shape on a bounded region as closely as one likes.
With the ReLU activation $\sigma(t) = \max(t, 0)$, each copy is a hinge, and sums of hinges in one variable are exactly the continuous piecewise linear functions with finitely many pieces.
For example, $\abs{x} = \sigma(x) + \sigma(-x)$ is a network with two neurons.
Approximation theory asks how many neurons or layers are needed to reach a given accuracy, and for which functions.
</document>

<document source="https://tamnd.github.io/math2/node/W7MPTRPXYD.md" tag="W7MPTRPXYD" kind="concept">
# Matrix function

## Summary

A matrix function extends a scalar function $f$ to square matrices, giving a matrix $f(A)$ for a matrix $A$.
For a function given by a [[power series|0MZB06ST2B]], $f(A)$ is obtained by putting $A$ into the series.
The [[matrix exponential|020M04FN1H]], the [[logarithm of a matrix|FS339MCFEQ]] and the [[square root of a matrix|63FAKMYZ2E]] are the main examples.
They are used to solve linear differential equations and in numerical analysis, control theory and quantum mechanics.

## Explanations

### L1 human

Many functions are sums of powers, like $e^x = 1 + x + x^2/2 + \cdots$.
Powers of a square matrix make sense, so one can put a matrix in place of $x$.
For a diagonal matrix, this simply applies $f$ to each diagonal entry.
For a general matrix, $f(A)$ acts on each eigenvector of $A$ by multiplying it by $f$ of the eigenvalue.
</document>

<document source="https://tamnd.github.io/math2/node/W7NFRDY0GA.md" tag="W7NFRDY0GA" kind="definition">
# Unary operation

## Definition

Let $A$ be a set.
A *unary operation* on $A$ is a [[finitary operation|EJ8YJVM9BW]] of arity $1$, that is, a [[function|0EPAT51DXQ]] $f \colon A \to A$.

## Remarks

Example: the absolute value $x \mapsto \abs{x}$ is a unary operation on $\R$, because $\abs{x} \in \R$ for every $x \in \R$.

Non-example: the map $x \mapsto 1/x$ is not a unary operation on $\R$, because it is not defined at $0$.
It is a unary operation on $\R \setminus \set{0}$.

A unary operation is often written in prefix notation $f(x)$ or $-x$, in postfix notation $x^{-1}$ or $x!$, or with a bar $\bar{x}$.
Here $A^1$ is identified with $A$.
</document>

<document source="https://tamnd.github.io/math2/node/W7P6CCSMX7.md" tag="W7P6CCSMX7" kind="concept">
# Baum–Welch algorithm

## Summary

The Baum–Welch algorithm estimates the parameters of a [[hidden Markov model|HJVM390354]] from a sequence of observations alone.
It is the [[expectation–maximization algorithm|11N1AXSXYK]] for this model.
It alternates between computing the expected hidden path under the current parameters and re-estimating the parameters from those expectations.
The expectations are computed efficiently by the forward–backward recursions.
It is a standard tool in speech recognition and in biological sequence analysis.

## Explanations

### L1 human

Imagine a machine that switches between hidden moods and emits visible symbols, with each mood favouring different symbols.
You see only the symbols and want to learn how often the moods switch and what each mood emits.
Baum–Welch starts from a guess, works out for each time how likely each mood was, and then updates the guess using these soft counts.
Repeating this never lowers the likelihood of the observed data.
With only one mood, a single step gives the observed frequencies of the symbols.
</document>

<document source="https://tamnd.github.io/math2/node/W7PYDJZ8BC.md" tag="W7PYDJZ8BC" kind="concept">
# Eichler order

## Summary

An Eichler order is an [[order|1ZBX4TKVZ9]] in a [[quaternion algebra|ZWXW0T3EAC]] that is the intersection of two [[maximal orders|E5DH1CSW1N]].
Every maximal order is an Eichler order, as the intersection of itself with itself.
In $M_2(\Q)$ the standard Eichler order of level $N$ is the ring of integer matrices whose lower left entry is divisible by $N$.
Its elements of determinant $1$ form the congruence subgroup $\Gamma_0(N)$.
Eichler orders in indefinite and definite quaternion algebras give Shimura curves and the Brandt matrices used to study modular forms.

## Explanations

### L1 human

In the rational $2 \times 2$ matrices, the integer matrices form a maximal order, and so does each of its conjugates.
Intersecting two of these gives a slightly smaller ring with a congruence condition.
For example, the integer matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $c$ divisible by $N$ form such an intersection.
This ring is behind the group $\Gamma_0(N)$, which is basic in the theory of modular forms.
Eichler orders carry the same idea to other quaternion algebras.
</document>

<document source="https://tamnd.github.io/math2/node/W7QH1BPBS8.md" tag="W7QH1BPBS8" kind="concept">
# MUSCL scheme

## Summary

A MUSCL scheme is a [[finite volume method|HGKN8E09RS]] for [[conservation laws|8RA7AMWSAF]] that replaces the constant cell values of [[Godunov's scheme|43HZJ7DRT0]] by limited linear reconstructions.
The name stands for Monotonic Upstream-centred Scheme for Conservation Laws, and the approach goes back to Bram van Leer.
The reconstruction gives second-order accuracy where the solution is smooth.
A [[flux limiter|NZ8XFNCKQW]] switches the slopes off near jumps, so shocks are captured without spurious oscillations.
See the [[definition|SB1TM11S5Q]].

## Explanations

### L1 human

Godunov's scheme pretends that the solution is flat inside each cell, which smears out every feature.
A MUSCL scheme draws a short straight line in each cell instead, with a slope taken from the neighbouring cells.
Near a jump such a line could overshoot, so a limiter flattens the slope there.
For data on a straight line, such as $u_j = j$, the reconstruction hits the exact value at every cell edge.
For a step from $0$ to $1$, the limiter sets the slopes to zero next to the step, and no new maximum or minimum appears.
</document>

<document source="https://tamnd.github.io/math2/node/W7RCXS3CHT.md" tag="W7RCXS3CHT" kind="example">
# Stiefel manifold

## Example

Let $1 \le k \le n$.
The real Stiefel manifold $V_k(\R^n)$ is the set of orthonormal $k$-frames in $\R^n$:
$$V_k(\R^n) = \set{(v_1, \ldots, v_k) \in (\R^n)^k : \ip{v_i}{v_j} = \delta_{ij} \text{ for all } i, j},$$
with the [[subspace topology|0WS9P12FQR]] of $(\R^n)^k$.
It is a compact [[smooth manifold|F45VQ47GPV]] of dimension $nk - k(k+1)/2$.

Check: $V_k(\R^n)$ is closed and bounded in $(\R^n)^k$, hence compact.
It is the level set of the map $(v_1, \ldots, v_k) \mapsto (\ip{v_i}{v_j})_{i \le j}$ to the space of symmetric $k \times k$ matrices at the identity matrix, and this value is regular.
The space of symmetric $k \times k$ matrices has dimension $k(k+1)/2$, which gives the dimension.

## Remarks

Special cases: $V_1(\R^n) = S^{n-1}$, and $V_n(\R^n)$ is the orthogonal group $O(n)$.
For $V_2(\R^3)$ the dimension is $6 - 3 = 3$, and indeed $(v_1, v_2) \mapsto (v_1, v_2, v_1 \times v_2)$ identifies it with the rotation group $SO(3)$.

Stiefel manifolds give standard [[fiber bundles|CX4M8K61AB]].
The map $V_k(\R^n) \to S^{n-1}$, $(v_1, \ldots, v_k) \mapsto v_1$, is a fiber bundle with fiber $V_{k-1}(\R^{n-1})$.
The map $V_k(\R^n) \to G_k(\R^n)$ that sends a frame to the subspace it spans is a [[principal $O(k)$-bundle|8E37E74XVP]] over the [[Grassmannian|CYPA882RB9]].
Using these bundles one shows that $V_k(\R^n)$ is $(n - k - 1)$-connected; as $n \to \infty$, the bundle $V_k(\R^\infty) \to G_k(\R^\infty)$ becomes the [[universal bundle|1HDH716Z24]] for $O(k)$.

## Explanations

### L1 human

A point of the Stiefel manifold $V_k(\R^n)$ is a list of $k$ unit vectors in $\R^n$ that are all perpendicular to each other.
For one vector, this is just a point of the unit sphere.
For two perpendicular unit vectors in space, the third vector of a right-handed frame is fixed by the first two.
So these pairs correspond to the rotations of space.
The question whether the sphere $S^{n-1}$ has $k$ independent tangent vector fields is the question whether the bundle $V_{k+1}(\R^n) \to S^{n-1}$ has a section.
</document>

<document source="https://tamnd.github.io/math2/node/W7VXNYZQN2.md" tag="W7VXNYZQN2" kind="statement">
# Sturm comparison theorem

## Statement

Let $a < b$ and let $q_1, q_2 \colon [a, b] \to \R$ be continuous with $q_2(x) \ge q_1(x)$ for all $x \in [a, b]$.
Let $y$ be a solution of $y'' + q_1(x) y = 0$ that is not identically zero and satisfies $y(a) = y(b) = 0$.
Let $z$ be a solution of $z'' + q_2(x) z = 0$.
Then $z$ has a zero in the open interval $(a, b)$, unless $q_1 = q_2$ on $[a, b]$ and $z$ is a constant multiple of $y$.

## Remarks

The proof uses the function $W = y' z - y z'$, whose derivative is $(q_2 - q_1) y z$.
If $z$ had no zero in $(a, b)$, one may take $y > 0$ and $z > 0$ on $(a, b)$.
Integrating $W'$ over $[a, b]$ gives $y'(b) z(b) - y'(a) z(a) \ge 0$, while the signs $y'(a) > 0$, $y'(b) < 0$ and $z \ge 0$ force the left side to be $\le 0$.
Analysing the case of equality gives $q_1 = q_2$ and $z(a) = z(b) = 0$, so $z$ is a multiple of $y$.
Example: $y = \sin x$ solves $y'' + y = 0$ with zeros $0$ and $\pi$; every solution of $z'' + 4 z = 0$, such as $\cos 2x$, has a zero in $(0, \pi)$.
The inequality $q_2 \ge q_1$ is needed: with $q_2 = 0$, the solution $z = 1$ of $z'' = 0$ has no zero at all.
Mauro Picone extended the theorem to equations $(p_i y')' + q_i y = 0$ with $p_1 \ge p_2 > 0$ and $q_2 \ge q_1$; this is the Sturm–Picone comparison theorem.

## Explanations

### L1 human

A larger coefficient $q$ in $y'' + q y = 0$ pulls the solution back to the axis harder, so it oscillates faster.
The theorem makes this precise: between two zeros of a solution of the slower equation, every solution of the faster equation has a zero.

### L2 human

The theorem lets one bound the zeros of solutions by comparison with equations with constant coefficients.
If $m^2 \le q(x) \le M^2$ with $m, M > 0$, then the distance between consecutive zeros of a nonzero solution of $y'' + q y = 0$ lies between $\pi / M$ and $\pi / m$.
It is the main tool of [[oscillation theory|ZH2S50NSZ6]] and it underlies the count of zeros of eigenfunctions in Sturm–Liouville theory.
For equations of the form $y'' + q y = 0$, the case $q_1 = q_2$ gives the [[Sturm separation theorem|YEBPTXZ1FS]].
</document>

<document source="https://tamnd.github.io/math2/node/W7WDXEQH65.md" tag="W7WDXEQH65" kind="concept">
# Limiting subdifferential

## Summary

The limiting subdifferential, also called the Mordukhovich or basic subdifferential, is a set of generalized gradients of a function at a point.
It is built in two steps.
One first takes the regular (Fréchet) subgradients at nearby points, then all limits of these as the points approach the given point.
For a convex function it equals the [[subdifferential|9C5WXYYQEK]] of convex analysis, and for a locally Lipschitz function its convex hull is the [[Clarke generalized gradient|20967BXZKA]].
It is usually smaller than the Clarke generalized gradient and has a good calculus, which makes it a main tool of variational analysis.
The [[definition|YXJDQM9VVD]] gives the precise construction.

## Explanations

### L1 human

At a smooth point, the gradient is the only reasonable slope.
At a kink, one looks at the slopes that work just beside the point and keeps all their limits.
For $-\abs{x}$ at $0$, no slope works at the peak itself, but the nearby slopes are $1$ and $-1$.
So the limiting subdifferential is the two-point set $\set{-1, 1}$, while the Clarke generalized gradient fills in the whole interval $[-1, 1]$.
</document>

<document source="https://tamnd.github.io/math2/node/W82R6N0GM9.md" tag="W82R6N0GM9" kind="concept">
# Universal algebra

## Summary

Universal algebra is the part of algebra that studies [[algebraic structures|QJGPT2W0FT]] in general, rather than groups, rings or lattices one at a time.
Its objects are sets with a family of [[finitary operations|EJ8YJVM9BW]], described by a [[signature|WDVVEYD6AT]].
It treats [[homomorphisms|KBXSZC6BV1]], [[subalgebras|676H7EGTKJ]], [[congruences|DCSXXT4SER]] and products for all such structures at once.
Its central results, such as [[Birkhoff's HSP theorem|5083DHSK82]], describe the classes of algebras defined by identities.

## Explanations

### L1 human

Groups, rings, vector spaces and lattices are all sets with some operations that obey some rules.
Many facts about them have the same proof: for example, the image of a homomorphism is always closed under the operations.
Universal algebra proves such facts once, for every kind of structure.
For example, a group is a set with a binary operation $\cdot$, a unary operation ${}^{-1}$ and a constant $e$.
A ring is a set with two binary operations, one unary operation and two constants.
The general theory then applies to both.
</document>

<document source="https://tamnd.github.io/math2/node/W83D78WN7T.md" tag="W83D78WN7T" kind="problem">
# Finite lattice representation problem

## Problem

Let $L$ be a finite [[lattice|WH4SFAACAS]].
Is there always a finite [[algebra|QJGPT2W0FT]] $A$ whose [[congruence lattice|REE07CSXHR]] is isomorphic to $L$?

## Remarks

Pálfy and Pudlák proved in 1980 that the problem has an equivalent form in group theory.
The problem has a positive answer if and only if the following holds.
Every finite lattice is isomorphic to an interval $[H, G] = \set{K : H \le K \le G}$ in the [[lattice of subgroups|0B8BW4YWTF]] of some finite group $G$.
Here $H$ is a subgroup of $G$, and the interval is ordered by inclusion.
The problem is open.

Without the finiteness condition on $A$, the answer is yes.
By a theorem of Grätzer and Schmidt, every algebraic lattice, and so every finite lattice, is the congruence lattice of some algebra.
The difficulty is to find a finite algebra.
For a finite distributive lattice the answer is also yes, since it is the congruence lattice of a finite lattice.

## Explanations

### L1 human

Each finite algebraic structure comes with a lattice of [[congruences|DCSXXT4SER]]: the ways of gluing its elements together that respect the operations.
The question is whether every finite lattice can be obtained in this way from some finite structure.
If one allows infinite structures, the answer is yes.
For finite structures nobody knows, and the question turns out to be the same as a question about subgroups of finite groups.
</document>

<document source="https://tamnd.github.io/math2/node/W84NTF77M0.md" tag="W84NTF77M0" kind="concept">
# Tensor rank

## Summary

The rank of a tensor is the smallest number of simple tensors $v_1 \otimes \cdots \otimes v_d$ whose sum is the tensor.
For tensors of order $2$, that is matrices, it is the usual [[rank of a matrix|0ZC9YTXZRY]].
For order $3$ and higher it is much harder to understand: it can depend on the field, and computing it is NP-hard.
The rank of the matrix multiplication tensor governs the [[matrix multiplication exponent|X0G02CGQ5J]].

## Explanations

### L1 human

A simple tensor is the most basic kind of tensor, an outer product of vectors, like a rank-one matrix $u v^T$.
The rank counts how many such building blocks are needed to write a given tensor.
A matrix of rank $r$ is a sum of $r$ rank-one matrices and no fewer.
For a three-way array, the tensor $e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_2 \otimes e_2$ needs two blocks, so its rank is $2$.
</document>

<document source="https://tamnd.github.io/math2/node/W890NRHGS4.md" tag="W890NRHGS4" kind="concept">
# Dehn twist

## Summary

A Dehn twist is a self-homeomorphism of a [[surface|VGJJVBZ1MQ]].
It cuts the surface along a simple closed curve, twists one side by a full turn, and glues back.
It is the identity outside a thin annulus around the curve.
Dehn twists are the basic elements of the [[mapping class group|JV5NYG913V]] of a surface: by the [[Dehn–Lickorish theorem|BG3EH18KRE]] finitely many of them generate it.

## Explanations

### L1 human

Take a garden hose, hold one end fixed, and turn the other end once around.
The hose itself is twisted in a narrow band, and away from that band nothing moves.
On a surface, do the same in a thin band around a loop.
A straight line crossing the band gets wrapped once around the loop.
On a torus, a twist around the meridian turns the longitude into a curve that goes once around each way.
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<document source="https://tamnd.github.io/math2/node/W8DZ5KGGHN.md" tag="W8DZ5KGGHN" kind="concept">
# Linear extension

## Summary

A linear extension of a [[partial order|0QV2JS9RJ3]] is a [[total order|0FRXBVVR03]] on the same set that keeps every relation of the partial order.
Every partial order has one, and the order is recovered as the intersection of all of them.
Linear extensions are the basis of order dimension, of sorting problems, and of many counting questions about finite posets.

## Explanations

### L1 human

A partial order is a set of rules of the form "this comes before that".
A linear extension is a way to put all the elements in a single row so that every rule holds.
For the tasks "dress" before "leave" and "eat" before "leave", both rows dress, eat, leave and eat, dress, leave are linear extensions.
The row leave, dress, eat is not, since it breaks a rule.
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<document source="https://tamnd.github.io/math2/node/W8DZQZQTJA.md" tag="W8DZQZQTJA" kind="definition">
# False discovery rate and the Benjamini–Hochberg procedure

## Definition

Let $H_1, \dots, H_m$ be null hypotheses tested by a procedure that rejects $R$ of them.
Let $V$ be the number of rejected hypotheses that are true.
The *false discovery proportion* is $Q = V/R$ if $R \ge 1$ and $Q = 0$ if $R = 0$.
The *false discovery rate* is $\mathrm{FDR} = E[Q]$.

Let $p_1, \dots, p_m$ be [[p-values|0EHYZFZ4PK]] for the $H_i$, ordered as $p_{(1)} \le \dots \le p_{(m)}$ with hypotheses $H_{(1)}, \dots, H_{(m)}$, and let $q \in (0, 1)$.
The *Benjamini–Hochberg procedure* at level $q$ lets $k$ be the largest $i$ with $p_{(i)} \le i q/m$.
It rejects $H_{(1)}, \dots, H_{(k)}$, and it rejects nothing if no such $i$ exists.

## Remarks

Benjamini and Hochberg proved the following in 1995.
If the p-values of the $m_0$ true nulls are independent, uniform on $[0, 1]$, and independent of the other p-values, then the procedure has $\mathrm{FDR} \le m_0 q/m \le q$.
Benjamini and Yekutieli later extended the bound to a form of positive dependence.
If all null hypotheses are true, then $Q = 1$ whenever $R \ge 1$, so $\mathrm{FDR} = P(R \ge 1)$ is the [[family-wise error rate|GQCY1N8JZT]].
In general $\mathrm{FDR} \le \mathrm{FWER}$, since $Q \le 1$ and $Q = 0$ when $V = 0$.
Example: with $m = 4$, $q = 0.05$ and p-values $0.001, 0.02, 0.03, 0.5$, the thresholds are $0.0125, 0.025, 0.0375, 0.05$.
The first three p-values are below their thresholds, so $k = 3$ and $H_{(1)}, H_{(2)}, H_{(3)}$ are rejected.
Non-example: the rule "reject $H_{(i)}$ when $p_{(i)} \le q/m$" is the Bonferroni rule, not the Benjamini–Hochberg procedure; here it rejects only $H_{(1)}$.
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# Tension field

## Summary

The tension field of a smooth map between [[Riemannian manifolds|7CBT4SM83Y]] is a vector field along the map.
It is the trace of the covariant derivative of the differential of the map.
It generalizes the Laplacian of a function and the acceleration of a curve.
Minus the tension field is the gradient of the [[energy|E6HDE1DFXD]], so a map is [[harmonic|KZJYG9DS3M]] exactly when its tension field vanishes.
The [[harmonic map heat flow|MD99PYBCE6]] moves a map in the direction of its tension field.

## Explanations

### L1 human

The tension field tells, at each point, in which direction the map is pulled if one wants to lower its energy.
For a curve, it is the acceleration as seen from inside the target, the part of the bending that is not forced by the curvature of the target.
For a real function on a domain, it is the Laplacian, which compares the value at a point with the average of nearby values.
For example, the map $t \mapsto (\cos t, \sin t)$ into the plane has tension $(-\cos t, -\sin t)$, pointing inward, but the same circle seen as a map into the unit circle has tension $0$.
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# Nemytskii operator

## Summary

A Nemytskii operator, also called a superposition operator, turns a function $g(x, u)$ of two variables into a map on functions.
It sends $u$ to the function $x \mapsto g(x, u(x))$.
It is the simplest kind of nonlinear operator on function spaces, since it acts point by point.
Nonlinear integral equations such as the Hammerstein equation are compositions of a linear integral operator with a Nemytskii operator.
Much of the theory is about when such an operator maps one [[$L^p$ space|SASP14MA1Z]] into another and is continuous.

## Explanations

### L1 human

Take a function $u$ and apply a fixed rule, such as squaring or taking the sine, to its value at every point.
The result is a new function, and the map from $u$ to this new function is a Nemytskii operator.
For example, with $g(x, u) = u^2$, the function $u(x) = x$ is sent to $x^2$.
The rule may also depend on the point, as in $g(x, u) = x \sin u$.
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# Lambert summation

## Summary

Lambert summation is a [[summability method|8X25Q4XYMN]] built on [[Lambert series|HATVWNZNMR]] $\sum n a_n \frac{x^n}{1 - x^n}$, multiplied by $1 - x$, with $x \to 1$ from below.
It is regular, so it gives the usual sum to every convergent series.
It is closely tied to the [[prime number theorem|0EHBY5Y84N]].
The series $\sum_{n \ge 1} \mu(n)/n$ is Lambert summable to $0$ by a short computation, and a Tauberian theorem upgrades this to convergence.

## Explanations

### L1 human

A Lambert series groups the terms of a series along divisors: $\frac{x^n}{1 - x^n} = x^n + x^{2n} + x^{3n} + \cdots$.
Lambert summation uses this grouping as a damping device.
It suits series from number theory, whose terms are built from divisors and primes.
For the Möbius function, the grouping makes almost everything cancel, which gives a quick Lambert sum.
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<document source="https://tamnd.github.io/math2/node/W8J42M1G28.md" tag="W8J42M1G28" kind="statement">
# Impossibility of the classical construction problems

## Statement

Start from two points at distance $1$ in the plane, and use only straightedge and compass.

1. It is impossible to construct a segment of length $\sqrt[3]{2}$, that is, to double the volume of a cube with edge $1$.
2. It is impossible to trisect every angle; in particular the angle $\pi/3$ cannot be trisected.
3. It is impossible to construct a square with the same area as a circle of radius $1$, that is, a segment of length $\sqrt{\pi}$.

## Remarks

A length is constructible exactly when it is a [[constructible number|0FMN2MN7TM]], and [[the degree of a constructible number over $\Q$ is a power of $2$|0SSSM1D475]].
For clause 1, $\sqrt[3]{2}$ has minimal polynomial $x^3 - 2$, of degree $3$; see [[impossibility of doubling the cube|0CQ357M0QS]].
For clause 2, $2\cos(\pi/9)$ has minimal polynomial $x^3 - 3x - 1$, of degree $3$; see [[impossibility of trisecting the angle|0EWXPZCSZ0]].
For clause 3, constructible numbers are algebraic, while $\pi$, and hence $\sqrt{\pi}$, is transcendental by the [[Lindemann–Weierstrass theorem|0JNCBFMPXR]]; see [[impossibility of squaring the circle|0X7CVDE3BG]].
Some angles can be trisected, such as $\pi/2$, since $\pi/6$ is constructible; clause 2 says that no method works for all angles.

## Explanations

### L1 human

The ancient Greeks posed three famous drawing problems: double a cube, divide any angle into three equal parts, and turn a circle into a square of the same area.
They allowed only an unmarked ruler and a compass.
Two thousand years later, algebra showed that all three are impossible.
The reason is that these tools can only produce numbers reached by repeated square roots.

### L2 human

Each straightedge-and-compass step solves linear or quadratic equations, so each new point lies in an extension of degree $1$ or $2$ of the field of the previous points.
A constructible number therefore lies at the top of a tower of quadratic extensions, and by the tower law its degree is a power of $2$.
Clauses 1 and 2 then follow from degree $3$ minimal polynomials, while clause 3 needs the much deeper transcendence of $\pi$.
The same method, with [[cyclotomic extensions|VD6VC3ZMMY]], decides which regular polygons are constructible.
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# H-representation and V-representation

## Summary

A [[convex polytope|300K6CB1KH]] can be described in two ways.
A V-representation lists finitely many points whose [[convex hull|0FQSDC2EQ2]] is the polytope.
An H-representation lists finitely many linear inequalities, that is closed half-spaces, whose common solutions form the polytope.
Every convex polytope has both, and computations with polytopes, as in linear programming and computational geometry, depend on which one is at hand.

## Explanations

### L1 human

To describe a box you can give its eight corners, or say that each coordinate lies between two bounds.
The list of corners is a V-representation, and the list of bounds is an H-representation.
Some questions are easy with one description and hard with the other.
Checking that a point lies in the box is easy with the inequalities; finding the corner farthest in a given direction is easy with the corner list.
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# Prime ideals of a localization

## Statement

Let $R$ be a [[commutative ring|0310REJJY8]], let $S \subseteq R$ be a [[multiplicative set|AN9SKEZ3VY]], and let $\lambda \colon R \to S^{-1}R$, $a \mapsto a/1$, be the canonical map to the [[localization|4C1AE84N5N]].
Then the maps
$$\mathfrak q \mapsto \lambda^{-1}(\mathfrak q), \qquad \mathfrak p \mapsto S^{-1}\mathfrak p = \set{a/s : a \in \mathfrak p, \ s \in S}$$
are mutually inverse bijections between the [[prime ideals|00MHHZ9SNE]] $\mathfrak q$ of $S^{-1}R$ and the prime ideals $\mathfrak p$ of $R$ with $\mathfrak p \cap S = \emptyset$.
Both maps preserve inclusion.

## Remarks

The preimage of a prime ideal under a ring homomorphism is prime, and $\lambda^{-1}(\mathfrak q)$ misses $S$ because elements of $S$ become units.
Conversely, if $\mathfrak p \cap S = \emptyset$, then $a/s \in S^{-1}\mathfrak p$ forces $ua \in \mathfrak p$ for some $u \in S$, hence $a \in \mathfrak p$ since $u \notin \mathfrak p$.
This shows $\lambda^{-1}(S^{-1}\mathfrak p) = \mathfrak p$ and that $S^{-1}\mathfrak p$ is a proper prime ideal.
Finally every ideal $J$ of $S^{-1}R$ is the [[extension|5AFH94P4S3]] of its contraction, which gives $S^{-1}\lambda^{-1}(\mathfrak q) = \mathfrak q$.

Example: for $R = \Z$ and $S = \set{1, 2, 4, \ldots}$, the prime ideals of $\Z[1/2]$ are $(0)$ and $p\Z[1/2]$ for the odd primes $p$; the prime $2\Z$ meets $S$ and disappears.

For $S = R \setminus \mathfrak p$ with $\mathfrak p$ prime, the primes of $R_{\mathfrak p}$ correspond to the primes of $R$ contained in $\mathfrak p$.
So $R_{\mathfrak p}$ has the unique maximal ideal $\mathfrak p R_{\mathfrak p}$, that is, it is a [[local ring|MZ02BKPHJA]].

The hypothesis that $\mathfrak p$ be prime matters: for a non-prime ideal $I$ with $I \cap S = \emptyset$ one can have $\lambda^{-1}(S^{-1}I) \ne I$.
For instance, with $R = \Z$, $I = 6\Z$ and $S = \set{1, 2, 4, \ldots}$, one gets $\lambda^{-1}(S^{-1}I) = 3\Z$.

## Explanations

### L1 human

When you localize, some prime ideals vanish and the others survive unchanged.
The ones that vanish are exactly those that contain an allowed denominator, since such an ideal would contain a unit.
In the language of geometry, localizing keeps an open part of the space of prime ideals, or the primes near a chosen point.

### L2 human

This result is the reason why localization is a "local" operation on the [[spectrum|FB45BHHA2J]].
For $S = \set{1, f, f^2, \ldots}$ it identifies $\operatorname{Spec} R_f$ with the [[basic open set|N6JX9BCKQP]] $D(f)$.
For $S = R \setminus \mathfrak p$ it identifies $\operatorname{Spec} R_{\mathfrak p}$ with the primes inside $\mathfrak p$, so the [[height|87WH65FJB8]] of $\mathfrak p$ equals the [[Krull dimension|MVGBS3J1P7]] of $R_{\mathfrak p}$.
Combined with the description of the primes of a quotient $R/I$, it describes the primes of rings such as $R_{\mathfrak p}/I R_{\mathfrak p}$ for an ideal $I$ of $R$.
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# Particle filter

## Definition

Consider a state-space model: a [[Markov chain|0G48PYM96Q]] $(X_t)_{t \ge 0}$ with initial density $\mu$ and transition density $f(x_t \mid x_{t-1})$.
The observations $Y_t$ are conditionally independent given the states, with density $g(y_t \mid x_t)$.
Fix a number $N \ge 1$ of particles and observed values $y_1, y_2, \dots$.
The *bootstrap particle filter* (sequential importance resampling) runs as follows.

1. Draw $x_0^{(1)}, \dots, x_0^{(N)}$ independently from $\mu$.
2. At time $t \ge 1$, draw $\tilde x_t^{(i)}$ from $f(\cdot \mid x_{t-1}^{(i)})$ for each $i$.
3. Compute the weights $w_t^{(i)} = g(y_t \mid \tilde x_t^{(i)}) / \sum_{k=1}^{N} g(y_t \mid \tilde x_t^{(k)})$.
4. Draw $x_t^{(1)}, \dots, x_t^{(N)}$ from the weighted points, choosing $\tilde x_t^{(i)}$ with probability $w_t^{(i)}$.

The filtering distribution of $X_t$ given $y_1, \dots, y_t$ is approximated by the weighted empirical measure $\sum_{i=1}^{N} w_t^{(i)} \delta_{\tilde x_t^{(i)}}$.

## Remarks

Steps 2 and 3 are a Monte Carlo version of the prediction step and of the update by [[Bayes' theorem|0NM6E9VW04]] in the exact filtering recursion.
General particle filters draw $\tilde x_t^{(i)}$ from another proposal density $q$ and multiply the weight by $f / q$.
As $N \to \infty$, integrals against the particle approximation converge to the exact filtering integrals under mild conditions.

Example: let $X_t = X_{t-1} + U_t$ and $Y_t = X_t + V_t$ with independent standard normal $U_t, V_t$.
Then $g(y \mid x)$ is proportional to $e^{-(y - x)^2 / 2}$, so particles close to $y_t$ get the largest weights in step 3.
For this linear Gaussian model the exact filter is the Kalman filter, and the particle filter approximates it.

Non-example: if step 4 is left out, the method is sequential importance sampling.
Then after a few steps almost all the weight sits on one particle, which is why the resampling step is part of the definition.
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# Sinai–Ruelle–Bowen measure

## Definition

Let $M$ be a compact Riemannian [[manifold|F45VQ47GPV]] and let $f \colon M \to M$ be a $C^2$ [[diffeomorphism|0KZZ70ZHXP]].
Let $\mu$ be an $f$-[[invariant|B232EZ1FMB]] Borel probability measure on $M$.
The measure $\mu$ is a *Sinai–Ruelle–Bowen measure* (an *SRB measure*) if both conditions hold:

1. For $\mu$-almost every $x$, at least one [[Lyapunov exponent|NJ1FJSNK6A]] of $f$ at $x$ is positive.
2. The conditional measures of $\mu$ on the local unstable manifolds given by [[Pesin theory|ZHKS81A7QN]] are [[absolutely continuous|P0P21KDV6E]] with respect to their Riemannian volume.

## Remarks

Example: let $f$ be the map of $\R^2 / \Z^2$ given by the matrix $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ and $\mu$ Lebesgue measure.
The exponent $\log \frac{3 + \sqrt 5}{2}$ is positive at every point.
The unstable manifolds are the lines of the expanding eigendirection, and the conditional measures of $\mu$ on them are multiples of length.
So $\mu$ is an SRB measure.

Non-example: for the same map, the Dirac measure at the fixed point $0$ is invariant and has a positive exponent.
Its conditional measure on the unstable line through $0$ is a point mass, which is not absolutely continuous with respect to length, so clause 2 fails.

Some authors use a different definition: $\mu$ is SRB, or *physical*, if the set of points $x$ with $\frac{1}{n} \sum_{k=0}^{n-1} \varphi(f^k x) \to \int \varphi \, d\mu$ for every continuous $\varphi$ has positive volume.
For an Axiom A attractor of a $C^2$ diffeomorphism, there is a unique SRB measure on each basic piece of the attractor, and it is physical in this sense.
Ledrappier and Young proved a characterization for $C^2$ diffeomorphisms.
An invariant measure $\mu$ that satisfies clause 1 is SRB in the sense above if and only if its entropy equals the integral of the sum of its positive exponents.
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# Hypergeometric differential equation

## Summary

The hypergeometric differential equation is the linear equation $z(1-z)w'' + [c - (a+b+1)z]w' - ab\,w = 0$.
It is a [[Fuchsian equation|ZBVH2YE50B]] with exactly three [[regular singular points|H96HXG6CA0]], at $0$, $1$ and $\infty$.
The [[hypergeometric function|463X5V31D9]] is its solution that is holomorphic at $0$ with value $1$ there.
Every second-order Fuchsian equation with three singular points reduces to this form.
One uses a Möbius change of variable and a change of the unknown function.

## Explanations

### L1 human

Many equations of physics, after separating variables, become second-order linear equations whose coefficients blow up at a few points.
The hypergeometric equation is the simplest interesting case: it has three such points, $0$, $1$ and $\infty$, and near each of them solutions behave like powers.
Solving it by a power series around $0$ gives the hypergeometric series.
For example, with $a = 1$, $b = c$ the equation is solved by $(1 - z)^{-1}$, which is the geometric series.

### L2 human

The local exponents are $0, 1 - c$ at $z = 0$; $0, c - a - b$ at $z = 1$; and $a, b$ at $z = \infty$.
Their sum is $1$, as Fuchs's relation requires for three singular points.
Studying how the solutions change around the singular points leads to the [[monodromy of the hypergeometric equation|T8SBE28DM1]].
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# Weak derivative

## Definition

Let $U \subseteq \R^n$ be open.
A function $u \colon U \to \R$ is *locally integrable* if it is measurable and $\int_K \abs{u} \, dx < \infty$ for every compact set $K \subset U$.
Let $C_c^\infty(U)$ be the space of smooth functions on $U$ with compact support in $U$, the test functions.
For a multi-index $\alpha = (\alpha_1, \ldots, \alpha_n)$ write $\abs{\alpha} = \alpha_1 + \cdots + \alpha_n$ and $\partial^\alpha = \partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n}$.
Let $u$ and $v$ be locally integrable on $U$.
The function $v$ is the *$\alpha$-th weak derivative* of $u$, written $D^\alpha u = v$, if
$$\int_U u \, \partial^\alpha \varphi \, dx = (-1)^{\abs{\alpha}} \int_U v \, \varphi \, dx \quad \text{for all } \varphi \in C_c^\infty(U).$$

## Remarks

The weak derivative, when it exists, is unique up to a set of measure zero.
If $u$ is of class $C^{\abs{\alpha}}$ on $U$, then its classical [[partial derivative|09EGC6F8PT]] $\partial^\alpha u$ is also its weak derivative, by integration by parts.

Example: on $U = (-1, 1)$, the function $u(x) = \abs{x}$ has weak derivative $v = \sgn x$.
For $\varphi \in C_c^\infty(-1, 1)$, integration by parts on $(-1, 0)$ and on $(0, 1)$ gives $\int_{-1}^{1} \abs{x} \varphi'(x) \, dx = \int_{-1}^{0} \varphi \, dx - \int_{0}^{1} \varphi \, dx = -\int_{-1}^{1} \sgn(x) \varphi(x) \, dx$, since the boundary terms at $0$ vanish.

Non-example: on $(-1, 1)$, the function $\sgn x$ has no weak derivative.
For any $\varphi \in C_c^\infty(-1, 1)$ one finds $\int \sgn(x) \varphi'(x) \, dx = -2 \varphi(0)$.
A locally integrable $v$ with $\int v \varphi \, dx = 2 \varphi(0)$ for all $\varphi$ would vanish almost everywhere on $(-1, 0)$ and on $(0, 1)$, and then the integral would be $0$, not $2 \varphi(0)$.
In the language of distributions, the derivative of $\sgn x$ is $2\delta$, which is not a function.
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# Tube lemma

## Statement

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], with $Y$ [[compact|0TVJZYB6SH]], and give $X \times Y$ the [[product topology|0GBG8B49AA]].
Let $x_0 \in X$, and let $N$ be an [[open set|02BH3MCBQB]] of $X \times Y$ with $\set{x_0} \times Y \subseteq N$.
Then there is an open set $W \subseteq X$ with $x_0 \in W$ and $W \times Y \subseteq N$.

## Remarks

For each $y \in Y$, there is a basic open box $U_y \times V_y \subseteq N$ that contains $(x_0, y)$.
The sets $V_y$ cover the compact space $Y$, so finitely many $V_{y_1}, \ldots, V_{y_n}$ cover it.
Then $W = U_{y_1} \cap \cdots \cap U_{y_n}$ is open, contains $x_0$, and satisfies $W \times Y \subseteq N$.

The lemma gives the proof that the [[product of two compact spaces is compact|09ZKE5XQ74]].
It also shows that the projection $X \times Y \to X$ is a [[closed map|00FSJTXPJM]] when $Y$ is compact.

Compactness of $Y$ is necessary.
Let $X = Y = \R$, $x_0 = 0$, and $N = \set{(x, y) : \abs{xy} < 1}$.
Then $\set{0} \times \R \subseteq N$, but for any $\varepsilon > 0$ the point $(\varepsilon / 2, 4 / \varepsilon)$ lies in $(-\varepsilon, \varepsilon) \times \R$ and not in $N$.

## Explanations

### L1 human

Picture $X \times Y$ with $X$ horizontal and $Y$ vertical, and a vertical slice $\set{x_0} \times Y$ inside an open set $N$.
The lemma says that a whole vertical tube around the slice also fits inside $N$, as long as $Y$ is compact.
Without compactness, $N$ can get thinner and thinner as you go up, and no tube fits.

### L2 human

The tube lemma is the key step in the proof that a finite product of compact spaces is compact.
A more general form says that if $A \subseteq X$ and $B \subseteq Y$ are compact and $A \times B \subseteq N$ with $N$ open, then there are open $U \supseteq A$ and $V \supseteq B$ with $U \times V \subseteq N$.
This general form is often used to show that compact sets in Hausdorff spaces can be separated by open sets.
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# Term operation

## Definition

Let $A$ be an algebra of a signature $\sigma$ with only function symbols, and let $n \ge 1$.
Let $t$ be a [[term|6D3WA7N2W4]] over $\sigma$ whose variables are among $x_1, \ldots, x_n$.
The $n$-ary *term operation* of $t$ on $A$ is the map $t^A \colon A^n \to A$ given by
$$t^A(a_1, \ldots, a_n) = t^A[\alpha].$$
Here $\alpha(x_i) = a_i$, and $t^A[\alpha]$ is the [[value|DH30WTA26T]] of $t$ at the assignment $\alpha$.
An operation on $A$ is a *term operation* of $A$ if it equals $t^A$ for some $n$ and some term $t$.

## Remarks

The term operation of the variable $x_i$ is the projection $(a_1, \ldots, a_n) \mapsto a_i$.
The term operation of $f(t_1, \ldots, t_k)$ is the composition $f^A(t_1^A, \ldots, t_k^A)$.
So the set $\operatorname{Clo}(A)$ of all term operations is the smallest set of operations on $A$ that contains the projections and the basic operations and is closed under composition.
Every homomorphism $h \colon A \to B$ satisfies $h(t^A(a_1, \ldots, a_n)) = t^B(h(a_1), \ldots, h(a_n))$.

Example: in the group $(\Z, +, -, 0)$ the map $(a, b) \mapsto 2a - b$ is the term operation of $(x_1 + x_1) + (-x_2)$.

Non-example: the map $a \mapsto a + 1$ on $(\Z, +, -, 0)$ is not a term operation.
Every term operation $t^A$ of this group satisfies $t^A(0, \ldots, 0) = 0$, since $0 + 0 = -0 = 0$, while $0 + 1 \ne 0$.
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# Rees algebra

## Summary

The Rees algebra of an [[ideal|096Z9MP2ME]] $I$ of a [[commutative ring|0310REJJY8]] $R$ is the [[graded ring|8KB4SQGC0C]] $R[It] = \bigoplus_{n \ge 0} I^n t^n$, a subring of the polynomial ring $R[t]$.
It packs all the powers of $I$ into one ring, so that statements about all $I^n$ become statements about a single ring.
Its quotient by $I R[It]$ is the [[associated graded ring|8HFAEGFQT1]], and in geometry it gives the [[blowing up|C4KGYAFPN7]] along $I$.

## Explanations

### L1 human

The powers $I, I^2, I^3, \dots$ of an ideal form an infinite list.
The Rees algebra stores the whole list in one ring, using a new variable $t$ to label which power each element comes from.
For $R = k[x]$ and $I = (x)$, the Rees algebra is $k[x, xt]$, a polynomial ring in the two elements $x$ and $xt$.
The [[definition|6YCKECNR3T]] makes this precise.
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# Frobenius method

## Summary

The Frobenius method finds solutions of a second-order linear differential equation near a [[regular singular point|H96HXG6CA0]] in the form $(z - z_0)^r \sum_{k \ge 0} a_k (z - z_0)^k$.
The exponent $r$ comes from a quadratic equation, the indicial equation, and the coefficients $a_k$ from a recurrence.
It extends the [[power series method|08RT9GDDPQ]], which works only at ordinary points.
It produces Bessel functions, Legendre functions and hypergeometric functions.
See the [[definition|7D47PADY9N]].

## Explanations

### L1 human

Near a regular singular point the solutions look like a power $z^r$ times an ordinary power series.
The power $r$ need not be a whole number, and it is found first by looking at the lowest terms of the equation.
Then the series coefficients follow one after another.
For $z^2 y'' + z y' + (z^2 - \frac{1}{4}) y = 0$, the exponents are $r = \pm \frac{1}{2}$, and the method gives $z^{-1/2} \sin z$ and $z^{-1/2} \cos z$.
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# Stein equation

## Definition

Let $n \ge 1$, let $A$ be an $n \times n$ complex [[matrix|0JXZKX526R]], and let $Q$ be an $n \times n$ [[Hermitian matrix|0P17NGQQMQ]].
Write $A^*$ for the [[conjugate transpose|0DXKE10PHF]] of $A$.
The *Stein equation* for $A$ and $Q$ is
$$X - A X A^* = Q$$
for an unknown $n \times n$ complex matrix $X$.

## Remarks

It is the discrete [[Lyapunov equation|92GVPZH9FT]] $AXA^* - X + Q = 0$.
More generally, the name is used for $X - AXB = C$ with given $A$, $B$, $C$.
This equation has a unique solution exactly when $\lambda \mu \ne 1$ for every eigenvalue $\lambda$ of $A$ and every eigenvalue $\mu$ of $B$.
If the [[spectral radius|9V7YQA85PC]] of $A$ is less than $1$, the unique solution is $X = \sum_{k=0}^{\infty} A^k Q (A^*)^k$, and $X$ is [[positive definite|0KX66CZMBB]] when $Q$ is.
Example: for $n = 1$, $A = 1/2$ and $Q = 3$, the equation reads $X - X/4 = 3$, so $X = 4$; the series gives $3 \sum_{k \ge 0} 4^{-k} = 4$ as well.
Non-example of solvability: for $n = 1$, $A = 1$ and $Q = 1$, the equation reads $0 = 1$ and has no solution, since $\lambda \bar{\lambda} = 1$ for the eigenvalue $\lambda = 1$.
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# Completely distributive lattice

## Definition

Let $L$ be a [[complete lattice|D7A0ZGZT5S]].
Let $J$ be a set, let $(K_j)_{j \in J}$ be a family of sets, and let $x_{j,k} \in L$ for $j \in J$ and $k \in K_j$.
Let $F$ be the set of choice functions $f$ on $J$ with $f(j) \in K_j$ for every $j \in J$.
The lattice $L$ is *completely distributive* if for all such data
$$\bigwedge_{j \in J} \bigvee_{k \in K_j} x_{j,k} = \bigvee_{f \in F} \bigwedge_{j \in J} x_{j, f(j)}.$$

## Remarks

Example: the [[power set|00E0YZCERH]] of a set $X$.
A point $x$ lies in $\bigcap_j \bigcup_k x_{j,k}$ if and only if for each $j$ there is $k = f(j)$ with $x \in x_{j, f(j)}$.
This is membership in the right side, where the axiom of choice picks $f$.
Non-example: the lattice of open subsets of $\R$ is a frame but not completely distributive.
Take $J = \N_{\ge 1}$, $K_j = \set{1, 2}$, $x_{j,1} = (-1/j, 1/j)$ and $x_{j,2} = \R \setminus \set{0}$.
The left side is $\R$.
Each meet on the right is the interior of an intersection and never contains $0$, so the right side is $\R \setminus \set{0}$.
Complete distributivity is self-dual: the condition is equivalent to the same identity with $\bigwedge$ and $\bigvee$ exchanged.
Every completely distributive lattice is a [[distributive lattice|0HM0HQCK6N]].
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# Hamiltonian vector field

## Definition

Let $(M, \omega)$ be a [[symplectic manifold|CAN490EPRT]] and let $H \colon M \to \R$ be a smooth function.
The *Hamiltonian vector field* of $H$ is the unique [[vector field|0299VCW1TH]] $X_H$ on $M$ with
$$\iota_{X_H} \omega = dH.$$
That is, $\omega(X_H, Y) = dH(Y)$ for every tangent vector $Y$ at every point.
A vector field is called *Hamiltonian* if it equals $X_H$ for some smooth $H$.

## Remarks

The vector field exists and is unique because $\omega$ is nondegenerate, so $v \mapsto \iota_v \omega$ is an isomorphism from each tangent space onto the cotangent space.

Example: on $\R^{2n}$ with coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ and $\omega = \sum_i dq_i \wedge dp_i$, set
$$X_H = \sum_i \left( \frac{\partial H}{\partial p_i} \frac{\partial}{\partial q_i} - \frac{\partial H}{\partial q_i} \frac{\partial}{\partial p_i} \right).$$
Then $\iota_{X_H} \omega = \sum_i \left( dq_i(X_H)\, dp_i - dp_i(X_H)\, dq_i \right) = \sum_i \left( \frac{\partial H}{\partial p_i} dp_i + \frac{\partial H}{\partial q_i} dq_i \right) = dH$.
So the integral curves of $X_H$ solve Hamilton's equations $q_i' = \partial H / \partial p_i$, $p_i' = -\partial H / \partial q_i$.

Non-example: on $\R^4$ the form $\omega = dq_1 \wedge dp_1$ is closed but degenerate.
For $H = q_2$ there is no vector field $X$ with $\iota_X \omega = dq_2$, because $\iota_X \omega$ is always a combination of $dq_1$ and $dp_1$.

Some authors use the opposite sign, $\iota_{X_H} \omega = -dH$, together with $\omega = \sum_i dp_i \wedge dq_i$; this gives the same vector field on $\R^{2n}$.
The function $H$ is constant along the flow of $X_H$, since $dH(X_H) = \omega(X_H, X_H) = 0$.
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# Enumerative geometry

## Definition

*Enumerative geometry* is the branch of algebraic geometry that determines the number of geometric objects of a given kind that satisfy given conditions.
The objects are typically subvarieties of [[projective space|ZT9A6JK8QF]], such as lines, conics or rational curves.
The conditions are chosen in general position, and an answer is a number that is the same for all general choices.
A standard method treats the objects as points of a [[moduli space|FJZRW9P0K3]] and each condition as a subvariety.
The count is then an [[intersection number|4HV4WTAX9P]].

## Remarks

Example: in $\mathbb{P}^3$, exactly $2$ lines meet four lines in general position.
In the Grassmannian of lines, each condition is a hypersurface, and the product of four such classes is $2$ times the class of a point.

Example: there are $3264$ smooth conics in $\mathbb{P}^2$ tangent to five general conics.
Steiner gave the answer $7776$, which counts degenerate double lines as well, and the correct number $3264$ is due to Chasles.

Non-example: the number of lines meeting four lines that all lie in one plane is infinite.
Every line in that plane meets all four, so the conditions are not in general position.

Other classical numbers are $27$ lines on a smooth cubic surface and $2875$ lines on a general quintic threefold.
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# Besicovitch set of measure zero

## Example

There is a [[Kakeya set|6PEN7AKC7M]] in $\R^2$ of [[Lebesgue measure|2Z2QVWZ262]] zero.
Besicovitch first showed this; the construction below, with Cantor sets, is due to Jean-Pierre Kahane.

Let $C \subseteq [0, 1]$ be the set of numbers $\sum_{i \ge 1} a_i 4^{-i}$ with every digit $a_i \in \set{0, 3}$.
It is the Cantor set obtained by keeping the first and last quarter of each interval.
Let $E$ be the union of all line segments from $(x, 0)$ to $(y/2, 1)$ with $x, y \in C$.

1. $E$ is compact, as the image of the compact set $C \times C \times [0, 1]$ under a continuous map.
2. $E$ contains segments of every slope in a range of directions. The horizontal shift $y/2 - x$ of such a segment has base-$4$ digits $b/2 - a$ with $a, b \in \set{0, 3}$, that is $\frac{3}{2} d$ with $d \in \set{-2, -1, 0, 1}$. Since $\set{-2, -1, 0, 1}$ is a full set of residues modulo $4$, the shifts fill the interval $[-1, 1/2]$. So $E$ contains a segment from height $0$ to height $1$, of length at least $1$, in each direction $(s, 1)$ with $-1 \le s \le 1/2$.
3. $E$ has measure zero. Its horizontal slice at height $t \in (0, 1)$ is $\set{(1 - t)x + t y/2 : x, y \in C}$. Up to scaling, this is the projection of the four-corner Cantor set $C \times C$ onto a line whose direction depends smoothly and injectively on $t$. The set $C \times C$ has finite and positive one-dimensional Hausdorff measure and is purely unrectifiable. By the [[Besicovitch–Federer projection theorem|8J8AW0TG9M]], almost all of its projections have length zero. So almost every slice has length zero, and [[Fubini's theorem|0R0T1VWH5X]] gives that $E$ has area zero.

The directions $(s, 1)$ with $-1 \le s \le 1/2$ fill an arc of angle $\pi/4 + \arctan(1/2)$, which is more than $\pi/3$.
So three rotated copies of $E$ contain unit segments in all directions.
Their union is compact, has measure zero, and is a Kakeya set.

## Explanations

### L1 human

One can find a set in the plane that contains a needle of length one pointing in every direction and yet has area zero.
Here the set is made of all segments joining two thin Cantor dusts on two parallel lines.
The dust is so thin that each horizontal line meets the set in a set of length zero, yet the segments point in a whole range of directions.
Three rotated copies then cover every direction.
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# Kaplan–Meier estimator

## Definition

Let $(\tilde t_1, \delta_1), \dots, (\tilde t_n, \delta_n)$ be right-[[censored|MY8ZZP84NP]] observations of a lifetime.
Here $\tilde t_i \ge 0$ is the observed time, and $\delta_i = 1$ if it is an event and $\delta_i = 0$ if it is censored.
Let $t_{(1)} < \dots < t_{(m)}$ be the distinct times $\tilde t_i$ with $\delta_i = 1$.
For each $j$, let $d_j = \#\set{i : \tilde t_i = t_{(j)}, \delta_i = 1}$ be the number of events at $t_{(j)}$ and $n_j = \#\set{i : \tilde t_i \ge t_{(j)}}$ the number at risk just before $t_{(j)}$.
The *Kaplan–Meier estimator* of the [[survival function|QTM1HZM53Q]] is
$$\hat S(t) = \prod_{j \,:\, t_{(j)} \le t} \left(1 - \frac{d_j}{n_j}\right), \qquad t \ge 0,$$
with the empty product equal to $1$.

## Remarks

It is also called the product-limit [[estimator|0SSS6ERQ0S]].
Its variance is usually estimated by Greenwood's formula $\hat S(t)^2 \sum_{j : t_{(j)} \le t} d_j / (n_j (n_j - d_j))$.
If no observation is censored, $\hat S(t) = 1 - \hat F_n(t)$, where $\hat F_n$ is the [[empirical distribution function|A05WR6TEK5]].

Example: take the observations $2, 3^+, 5, 7$, where $3^+$ is censored.
Then $(n_j, d_j)$ is $(4, 1)$ at $2$, $(2, 1)$ at $5$ and $(1, 1)$ at $7$.
So $\hat S = 1$ on $[0, 2)$, $3/4$ on $[2, 5)$, $3/8$ on $[5, 7)$ and $0$ from $7$ on.

Non-example: dropping the censored observation and using $1 - \hat F_3$ on $2, 5, 7$ gives $1/3$ on $[5, 7)$ instead of $3/8$.
This ignores the information that the censored subject survived past $3$, and it is biased downwards in general.
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# Stopping time

## Definition

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in T}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]], where $T \subseteq [0, \infty)$, for example $T = \N$ or $T = [0, \infty)$.
A *stopping time* with respect to $(\mathcal{F}_t)$ is a map $\tau \colon \Omega \to T \cup \set{\infty}$ such that
$$\set{\omega \in \Omega : \tau(\omega) \le t} \in \mathcal{F}_t \quad \text{for every } t \in T.$$

## Remarks

In discrete time, $T = \N$, the condition is equivalent to $\set{\tau = n} \in \mathcal{F}_n$ for every $n \in \N$, since $\set{\tau = n} = \set{\tau \le n} \setminus \set{\tau \le n - 1}$ and $\set{\tau \le n} = \bigcup_{k \le n} \set{\tau = k}$.
The value $\infty$ means that the event never happens.

Example: let $(S_n)_{n \in \N}$ be a [[random walk|04VZE3FZXF]] with natural filtration $\mathcal{F}_n = \sigma(S_0, \dots, S_n)$, let $a \in \R$, and let $\tau = \min\set{n : S_n \ge a}$, with $\min \emptyset = \infty$.
Then $\set{\tau \le n} = \bigcup_{k \le n} \set{S_k \ge a} \in \mathcal{F}_n$, so $\tau$ is a stopping time.

Non-example: with the same walk started at $S_0 = 0$ and with steps $\pm 1$, let $N \ge 2$ and $\sigma = \max\set{n \le N : S_n = 0}$, the last visit to $0$ before time $N$.
The event $\set{\sigma \le 0}$ is the event that $S_n \ne 0$ for $1 \le n \le N$, which depends on steps after time $0$ and is not in $\mathcal{F}_0 = \set{\emptyset, \Omega}$.
So $\sigma$ is not a stopping time.

Constant times are stopping times, and the minimum and the maximum of two stopping times are stopping times.
The *$\sigma$-algebra of the past* at $\tau$ is $\mathcal{F}_\tau = \set{A \in \mathcal{F} : A \cap \set{\tau \le t} \in \mathcal{F}_t \text{ for all } t \in T}$.
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# Nielsen–Schreier theorem

## Statement

Let $F$ be a [[free group|0KTW26C80J]].
Then every [[subgroup|0X1JXT47GA]] $H$ of $F$ is a free group.
Moreover, if $F$ has finite rank $n$ and $H$ has finite [[index|0DTXBQPA2S]] $e = [F : H]$, then $H$ is free of rank $1 + e(n - 1)$.

## Remarks

A topological proof: $F$ is the [[fundamental group|S1XK3HXHFX]] of a bouquet of circles, a graph.
Then $H$ is the fundamental group of a [[covering space|3XZRRE165N]], which is again a graph.
The fundamental group of a connected graph is free.
For the rank formula, the covering graph of degree $e$ of a bouquet of $n$ circles has $e$ vertices and $en$ edges.
So its Euler characteristic is $e(1 - n)$.
A connected graph with Euler characteristic $\chi$ has free fundamental group of rank $1 - \chi$.
An algebraic proof uses a Schreier transversal and gives explicit free generators, the Schreier generators.
Example: in the free group $F(a, b)$, let $H$ be the kernel of the map to $\Z/2$ sending $a$ and $b$ to $1$.
It has index $2$, so it is free of rank $1 + 2 \cdot 1 = 3$; a free basis is $a^2$, $ab$ and $ab^{-1}$.
So a subgroup of a free group can have larger rank than the group.
The free group of rank $2$ contains free subgroups of every finite rank, and of countably infinite rank, such as its [[commutator subgroup|0ASGSTXSBW]].
The finite index hypothesis is needed in the rank formula: the commutator subgroup of $F(a, b)$ has infinite index and is not finitely generated.

## Explanations

### L1 human

A free group is a group with no relations among its generators other than the obvious cancellations.
The theorem says that every subgroup of such a group is again free.
This is surprising, since subgroups of a free group can be complicated, but they never acquire genuine relations.

### L2 human

The theorem is one of the starting points of combinatorial and geometric group theory.
The covering space proof is the model for [[Bass–Serre theory|6M731X66RD]], where groups acting on trees are studied, and the [[Kurosh subgroup theorem|RY8K5RQJSR]] is the analogue for subgroups of [[free products|TM7QS7P7X9]].
The rank formula is a form of the multiplicativity of the Euler characteristic under finite coverings.
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# Newton–Cotes formulas

## Definition

Let $a < b$ and let $n \ge 1$ be an integer.
The *closed Newton–Cotes formula* with $n + 1$ nodes on $[a, b]$ is the [[quadrature rule|ADSVMCY77A]]
$$Q_n(f) = \sum_{i=0}^{n} w_i\, f(x_i), \qquad x_i = a + i h, \quad h = \frac{b - a}{n}, \qquad w_i = \int_a^b \ell_i(x)\, dx,$$
where $\ell_0, \dots, \ell_n$ are the [[Lagrange basis polynomials|DA3CQJ5MJ6]] for the nodes $x_0, \dots, x_n$.
For $n \ge 0$, the *open Newton–Cotes formula* with $n + 1$ nodes uses instead the nodes $x_i = a + i h$, $i = 1, \dots, n + 1$, with $h = (b - a)/(n + 2)$, and the weights $\int_a^b \ell_i(x)\, dx$ for these nodes.

## Remarks

So $Q_n(f)$ is the integral of the interpolating polynomial of $f$ at the nodes, and it is exact for polynomials of degree at most $n$.
The closed formula has [[degree of exactness|ZYT1JNT1Y0]] $n$ for odd $n$ and $n + 1$ for even $n$, by symmetry of the nodes.
Example: $n = 1$ gives the trapezoidal rule $\frac{h}{2}(f(a) + f(b))$, $n = 2$ gives Simpson's rule $\frac{h}{3}(f_0 + 4 f_1 + f_2)$, and $n = 3$ gives Simpson's $3/8$ rule $\frac{3h}{8}(f_0 + 3 f_1 + 3 f_2 + f_3)$; the open formula with $n = 0$ is the midpoint rule.
Non-example: the two-point Gauss–Legendre rule on $[-1, 1]$ has nodes $\pm 1/\sqrt{3}$ and weights $1$.
It is interpolatory but not a Newton–Cotes formula.
Its nodes are neither the endpoints $\pm 1$, as a closed formula needs, nor $\pm 1/3$, as the open formula with two nodes needs.
For closed formulas with $n \ge 8$ some weights are negative, and as $n \to \infty$ the formulas need not converge even for analytic integrands, much as in Runge's phenomenon.
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# Néron model

## Definition

Let $R$ be a [[discrete valuation ring|MGM3CNJW9V]] with fraction field $K$.
Let $A_K$ be an [[abelian variety|R7W03TXRS9]] over $K$.
A *Néron model* of $A_K$ is an $R$-scheme $N$ with an isomorphism $N \times_R K \cong A_K$ such that:

1. $N$ is [[smooth|EXKGG9X6J0]], [[separated|X30G4V7VAJ]] and [[of finite type|AHTE5DVX61]] over $R$;
2. for every smooth $R$-scheme $Y$ and every $K$-morphism $u_K \colon Y \times_R K \to A_K$, there is a unique $R$-morphism $u \colon Y \to N$ that extends $u_K$.

Clause 2 is called the Néron mapping property.

## Remarks

A Néron model exists for every abelian variety $A_K$, and it is unique up to unique isomorphism by clause 2.
Clause 2 extends the group law of $A_K$, so $N$ is a group scheme over $R$.
Taking $Y = \operatorname{Spec} R$ in clause 2 shows that every point in $A_K(K)$ extends to a point in $N(R)$, so $N(R) = A_K(K)$.

Example: if $A_K$ extends to an [[abelian scheme|JDDXZ89N33]] $A$ over $R$, then $A$ is the Néron model of $A_K$.
This is the case of good reduction.

Non-example: let $E$ be an elliptic curve over $K$ with bad reduction, and let $W$ be the closure in $\mathbb{P}^2_R$ of a Weierstrass equation for $E$ with coefficients in $R$.
The special fibre of $W$ is a singular cubic, so $W$ is not smooth over $R$ and fails clause 1.
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# Sets of capacity zero

## Summary

A set of capacity zero is a set so thin that no unit charge of finite energy can be placed on it.
In the plane the relevant capacity is the [[logarithmic capacity|DBQXRTC0VN]], and in $\R^n$ with $n \ge 3$ it is the Newtonian [[capacity|4YZVFGJVSZ]].
For Borel sets, having capacity zero is the same as being a [[polar set|C95QJ7YH5T]].
These sets are the exceptional sets of potential theory: theorems often hold everywhere except on a set of capacity zero.

## Explanations

### L1 human

Volume and area cannot tell apart a point, a curve and a surface inside space: all have volume zero.
Capacity is finer.
In space a point and a straight segment have capacity zero, but a flat disc does not.
In the plane a point and a countable set have capacity zero, but a segment does not.
Sets of capacity zero are invisible to harmonic functions that stay bounded.
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# End and coend

## Summary

An end is a universal way of extracting from a [[functor|28NQ0KRHD3]] $S \colon \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathcal{D}$ an object that is compatible with all the diagonal values $S(c, c)$.
A coend is the dual notion.
Ends are special [[limits|CHS5AFJGWE]] and coends are special [[colimits|BJS1CGTKQ5]].
The set of [[natural transformations|KYCPCC57FT]] between two functors is an end, and many constructions such as tensor products and Kan extensions are coends or ends.

## Explanations

### L1 human

Some functors take two inputs from the same category, one in a backward and one in a forward way, like $\Hom(c, c')$.
An end picks out the families of diagonal elements, one for each object $c$, that are consistent along all arrows.
For example, a natural transformation between two functors is a family of maps, one per object, consistent with all arrows.
So the natural transformations form the end of the functor $(c, c') \mapsto \Hom(F c, G c')$.
A coend glues the diagonal values together instead, and plays the role of a tensor product.
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# Laplacian on the Sierpiński gasket

## Example

Let $p_1, p_2, p_3$ be the vertices of an equilateral triangle in $\R^2$ and let $F_i(x) = (x + p_i)/2$ for $i = 1, 2, 3$.
The Sierpiński gasket $K$ is the unique nonempty compact set with $K = F_1(K) \cup F_2(K) \cup F_3(K)$, the attractor of this [[iterated function system|75B8VHDGSA]].
Let $V_0 = \set{p_1, p_2, p_3}$ and $V_{m+1} = F_1(V_m) \cup F_2(V_m) \cup F_3(V_m)$.
Let $\Gamma_m$ be the graph on $V_m$ in which $x \sim_m y$ when $x$ and $y$ are two vertices of one triangle $F_{w_1} \circ \cdots \circ F_{w_m}(\text{triangle } p_1 p_2 p_3)$.
For $u \colon V_m \to \R$, set
$$\mathcal{E}_m(u) = \left(\frac{5}{3}\right)^m \sum_{x \sim_m y} (u(x) - u(y))^2,$$
the sum over edges of $\Gamma_m$.

The factor $5/3$ is the one that makes these energies compatible.
Take $u(p_1) = 1$ and $u(p_2) = u(p_3) = 0$, so $\mathcal{E}_0(u) = 2$.
Among all extensions of $u$ to $V_1$, the one of least energy $\mathcal{E}_1$ follows the "$1/5$–$2/5$ rule".
At the midpoint of $p_j p_k$ its value is $(2 u(p_j) + 2 u(p_k) + u(p_i))/5$, where $i$ is the third index.
Here this gives $2/5$ at the midpoints of $p_1 p_2$ and $p_1 p_3$, and $1/5$ at the midpoint of $p_2 p_3$.
The three small triangles contribute $18/25$, $6/25$ and $6/25$ to the sum, in total $6/5$, and $(5/3) \cdot (6/5) = 2 = \mathcal{E}_0(u)$.

Because of this compatibility, $\mathcal{E}_m(u|_{V_m})$ is nondecreasing in $m$ for every $u$, and its limit $\mathcal{E}(u)$ is a [[resistance form|THCM7DWHNY]] on $K$.
Let $\mu$ be the self-similar probability measure that gives mass $3^{-m}$ to each level-$m$ cell.
For a suitable function $u$ and a point $x \in V_m \setminus V_0$, the Laplacian of $u$ with respect to $\mu$ is
$$\Delta_\mu u(x) = \frac{3}{2} \lim_{m \to \infty} 5^m \sum_{y \sim_m x} (u(y) - u(x)).$$
The factor $5^m = (5/3)^m \cdot 3^m$ combines the energy scaling with the measure scaling.
The function on $V_1$ found above satisfies $\sum_{y \sim_1 x} (u(y) - u(x)) = 0$ at each of the three midpoints.
For example, at the midpoint of $p_2 p_3$ the neighbours have values $0, 0, 2/5, 2/5$, and $4/5 - 4 \cdot (1/5) = 0$.
Its continuation by the same rule at all levels is a harmonic function on $K$, that is, $\Delta_\mu u = 0$ off $V_0$.

## Remarks

The eigenvalue counting function of $-\Delta_\mu$ with Dirichlet conditions on $V_0$ grows like $\lambda^{d_s/2}$ up to bounded factors, with spectral dimension $d_s = 2 \log 3 / \log 5 \approx 1.365$.
This differs from the [[Hausdorff dimension|SSNFTHZ6YJ]] $\log 3 / \log 2 \approx 1.585$ of the gasket.

## Explanations

### L1 human

The Sierpiński gasket is a triangle with holes at every scale, so ordinary derivatives make no sense on it.
Instead, one looks at finer and finer triangle networks inside the gasket and computes how the value at each vertex differs from the values at its neighbours.
A small calculation on a single triangle shows that energies must be multiplied by $5/3$ at each step for the networks to agree.
With this scaling the network Laplacians converge, and their limit is the Laplacian of the gasket.
Harmonic functions are then easy to compute: the value at each new midpoint is a fixed mix of $2/5$, $2/5$ and $1/5$ of the corner values.
</document>

<document source="https://tamnd.github.io/math2/node/W9VAD8QYZF.md" tag="W9VAD8QYZF" kind="definition">
# Geometric graph

## Definition

A *geometric graph* is a pair $(V, E)$ such that:

1. $V$ is a finite set of points in $\R^2$ in general position, that is, no three of them lie on a common line;
2. $E$ is a set of closed [[line segments|VQ4Q2V45RP]] each of whose two endpoints lie in $V$.

The points of $V$ are its vertices and the segments of $E$ are its edges.
Its underlying abstract [[graph|00C0BR7D7H]] has vertex set $V$, with $u$ and $v$ adjacent when the segment from $u$ to $v$ is in $E$.
Two edges cross if they meet at a point that is not a common endpoint.

## Remarks

Example: let $V$ be the four points $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$ and let $E$ be all six segments between them.
No three of these points are collinear and every segment has its endpoints in $V$, so this is a geometric graph.
Its underlying graph is $K_4$, and the two diagonals cross at $(1/2, 1/2)$.

Non-example: the points $(0,0)$, $(1,0)$, $(2,0)$ with the segment from $(0,0)$ to $(2,0)$ fail clause 1, since the three points are collinear, and the segment passes through a third vertex.

A geometric graph is called plane when no two of its edges cross.
Some authors drop the general position condition and only require that no edge passes through a vertex other than its endpoints.
</document>

<document source="https://tamnd.github.io/math2/node/W9VYJRBF9N.md" tag="W9VYJRBF9N" kind="object">
# Cantor space

## Description

The *Cantor space* is the set $2^{\N} = \set{0, 1}^{\N}$ of all infinite sequences of $0$s and $1$s, with the [[product topology|0GBG8B49AA]] of the discrete space $\set{0, 1}$.
Its basic open sets are the sets of all sequences that begin with a given finite word.
The topology comes from the metric $d(x, y) = 2^{-n}$, where $n$ is the first index with $x_n \neq y_n$, and $d(x, x) = 0$.
The space is [[compact|0TVJZYB6SH]] by Tychonoff's theorem, [[totally disconnected|0RJSBR9GW1]], and has no isolated points.
More generally, a topological space is called a Cantor space when it is homeomorphic to $2^{\N}$.

## Remarks

The [[Cantor set|0BXNDZFQME]] is a Cantor space: it [[is homeomorphic to the space of binary sequences|0RQ8DHHR8V]].
By [[Brouwer's theorem|3NKBCNH5JS]], a space is a Cantor space if and only if it is nonempty, compact, [[metrizable|091GJ3FYD7]], totally disconnected and without isolated points.
For this reason $2^{\N} \times 2^{\N}$ and $(2^{\N})^{\N}$ are again Cantor spaces.
Every nonempty compact metric space is a continuous image of the Cantor space.

## Explanations

### L1 human

A point of the Cantor space is an endless list of yes or no answers.
Two points are close when their lists agree for a long time.
The space looks the same everywhere and is made only of dust, with no two points joined by a connected piece, yet it is compact like a closed interval.
</document>

<document source="https://tamnd.github.io/math2/node/W9YR470VY2.md" tag="W9YR470VY2" kind="concept">
# Kelvin transform

## Summary

The Kelvin transform turns a function $u$ on a region of $\R^n \setminus \set{0}$ into the function $x \mapsto \abs{x}^{2-n} u(x / \abs{x}^2)$ on the inverted region.
It sends [[harmonic functions|0DEB0JMN0W]] to harmonic functions, and it exchanges the neighbourhoods of $0$ and of infinity.
It is the main tool to study harmonic functions near infinity, exterior Dirichlet problems and the Green's function of a ball.
In the plane it is just composition with the inversion $z \mapsto 1/\bar z$.

## Explanations

### L1 human

Inversion in the unit sphere maps a point at distance $r$ from the origin to the point in the same direction at distance $1/r$.
Composing a harmonic function with this map does not keep it harmonic in dimension three or more; one has to multiply by the weight $\abs{x}^{2-n}$.
With that weight, harmonicity survives.
For instance, in $\R^3$ the Kelvin transform of the constant $1$ is $1/\abs{x}$, the potential of a point charge.
</document>

<document source="https://tamnd.github.io/math2/node/WA0NAANTAH.md" tag="WA0NAANTAH" kind="definition">
# Sieve

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $c$ be an object of $\mathcal{C}$.
A *sieve* on $c$ is a set $S$ of morphisms with codomain $c$ with the following property.
For every $f \colon d \to c$ in $S$ and every morphism $g \colon e \to d$, the composite $f \circ g$ is in $S$.

## Remarks

Equivalently, a sieve on $c$ is a subfunctor $S \subseteq \Hom_{\mathcal{C}}(-, c)$ of the [[representable functor|KCCHH87H6E]], with $S(d)$ the set of members of $S$ with domain $d$.
The *maximal sieve* on $c$ is the set of all morphisms with codomain $c$.
For a sieve $S$ on $c$ and a morphism $h \colon d \to c$, the *pullback sieve* $h^*S = \set{g : h \circ g \in S}$ is a sieve on $d$.
Example: view the poset of open subsets of a space $X$ as a category, and let $\set{U_i}$ be an open cover of $U$.
The open sets $V \subseteq U$ that lie in some $U_i$ form a sieve on $U$, since $W \subseteq V \subseteq U_i$ gives $W \subseteq U_i$.
Non-example: in the same category, the set $\set{U_1}$ alone is not a sieve on $U$ when $U_1$ has a smaller open subset $W \ne U_1$, since the composite $W \subseteq U_1 \subseteq U$ is missing.
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<document source="https://tamnd.github.io/math2/node/WA0SE56FVH.md" tag="WA0SE56FVH" kind="concept">
# Reinhardt domain

## Summary

A Reinhardt domain is a domain in $\C^n$ that is unchanged when each coordinate is rotated by its own angle.
Such a domain is determined by the set of absolute values $(\abs{z_1}, \dots, \abs{z_n})$ of its points.
[[Polydiscs|DJBWP7D3M8]] and the [[unit ball|PDQ9KW9FAP]] are Reinhardt domains.
They are the natural domains of convergence of [[power series in several variables|R3QTCPTJK7]].
They also give the first examples of the extension phenomena of [[several complex variables|BH57D2FBNR]].

## Explanations

### L1 human

Multiply the first coordinate of a point by $e^{i\theta_1}$, the second by $e^{i\theta_2}$, and so on.
This turns each coordinate around the origin of its own complex plane without changing its size.
A Reinhardt domain contains, with each point, all the points obtained in this way.
So one can draw it as a region in the "absolute value space" of points $(\abs{z_1}, \dots, \abs{z_n})$, which is a part of $[0, \infty)^n$.
For example, the unit polydisc of $\C^2$ is drawn as the square $[0, 1) \times [0, 1)$, and the unit ball as a quarter of a disc.
</document>

<document source="https://tamnd.github.io/math2/node/WA3974X06N.md" tag="WA3974X06N" kind="definition">
# Differential dynamic programming

## Definition

Let $F \colon \R^n \times \R^m \to \R^n$, $\ell \colon \R^n \times \R^m \to \R$ and $\ell_f \colon \R^n \to \R$ be twice continuously differentiable, and let $N \ge 1$.
Consider the problem of minimising $\sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_f(x_N)$ over $u_0, \dots, u_{N-1} \in \R^m$, where $x_{k+1} = F(x_k, u_k)$ and $x_0$ is given.
Subscripts denote partial derivatives, $\nabla^2$ a [[Hessian|0693Y2METT]] and $\nabla$ a [[gradient|0ABK9V9Y64]].
One iteration of *differential dynamic programming* starts from controls $\bar u_k$ with states $\bar x_k$ and does the following.

1. Backward pass: set $V_x = \nabla \ell_f(\bar x_N)$ and $V_{xx} = \nabla^2 \ell_f(\bar x_N)$. For $k = N - 1, \dots, 0$, take all derivatives at $(\bar x_k, \bar u_k)$ and $V_x, V_{xx}$ from step $k + 1$. Compute
$$Q_x = \ell_x + F_x^T V_x, \quad Q_u = \ell_u + F_u^T V_x,$$
$$Q_{xx} = \ell_{xx} + F_x^T V_{xx} F_x + V_x \cdot F_{xx}, \quad Q_{uu} = \ell_{uu} + F_u^T V_{xx} F_u + V_x \cdot F_{uu}, \quad Q_{ux} = \ell_{ux} + F_u^T V_{xx} F_x + V_x \cdot F_{ux},$$
where $V_x \cdot F_{xx}$ is the matrix $\sum_i (V_x)_i \nabla_x^2 F_i$ and similarly for the others. If $Q_{uu}$ is positive definite, set $d_k = -Q_{uu}^{-1} Q_u$, $K_k = -Q_{uu}^{-1} Q_{ux}$, and update $V_x \leftarrow Q_x - Q_{ux}^T Q_{uu}^{-1} Q_u$ and $V_{xx} \leftarrow Q_{xx} - Q_{ux}^T Q_{uu}^{-1} Q_{ux}$.
2. Forward pass: set $x_0^{\text{new}} = x_0$ and, for $k = 0, \dots, N - 1$, set $u_k^{\text{new}} = \bar u_k + d_k + K_k (x_k^{\text{new}} - \bar x_k)$ and $x_{k+1}^{\text{new}} = F(x_k^{\text{new}}, u_k^{\text{new}})$.

The new controls replace $\bar u_k$, and the iteration is repeated until the corrections $d_k$ are small.

## Remarks

The function $Q$ is the second-order expansion of $\delta x, \delta u \mapsto \ell(\bar x_k + \delta x, \bar u_k + \delta u) + V_{k+1}(F(\bar x_k + \delta x, \bar u_k + \delta u))$, where $V_{k+1}$ is the quadratic model of the cost-to-go.
The correction $\delta u = d_k + K_k \delta x$ minimises this quadratic model in $\delta u$, and the update of $V_x, V_{xx}$ gives the model of the minimum.
Dropping the terms with second derivatives of $F$ gives the iterative linear-quadratic regulator.
In practice one scales $d_k$ by a step size and adds a multiple of the identity to $Q_{uu}$ for robustness.

Example: if $F(x, u) = A x + B u$ and $\ell$, $\ell_f$ are quadratic forms with $\ell_{uu}$ positive definite, the expansions are exact.
Then the backward pass is the discrete Riccati recursion of the linear-quadratic regulator, and one iteration gives the optimal controls from any starting guess.

Non-example: take $\ell = -u^2$, $F(x, u) = x$ and $\ell_f = 0$, so that $Q_{uu} = -2$ is not positive definite.
The quadratic model has no minimiser in $\delta u$, and the step of clause 1 is not defined.

## Explanations

### L1 human

Each iteration fits quadratic models of the costs around the current trajectory, solves them backward in time, and then simulates the improved controls forward.
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<document source="https://tamnd.github.io/math2/node/WA3M3RZDQA.md" tag="WA3M3RZDQA" kind="definition">
# de Rham cohomology

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]], and for $k \ge 0$ let $\Omega^k(M)$ be the real vector space of smooth [[differential forms|0E3G0GRPZ6]] of degree $k$ on $M$.
Let $d \colon \Omega^k(M) \to \Omega^{k+1}(M)$ be the [[exterior derivative|0ZHSTAG8ZY]]; it satisfies $d \circ d = 0$.
A form $\omega$ is *closed* if $d\omega = 0$, and *exact* if $\omega = d\eta$ for some $\eta \in \Omega^{k-1}(M)$; for $k = 0$ only the zero form counts as exact.
The $k$-th *de Rham cohomology group* of $M$ is the quotient vector space
$$H^k_{\mathrm{dR}}(M) = \frac{\ker\big(d \colon \Omega^k(M) \to \Omega^{k+1}(M)\big)}{\im\big(d \colon \Omega^{k-1}(M) \to \Omega^k(M)\big)}.$$

## Remarks

Since $d \circ d = 0$, every exact form is closed, so the quotient makes sense.
$H^0_{\mathrm{dR}}(M)$ is the space of locally constant functions, so when $M$ has finitely many connected components its dimension is their number.
Example: on $\R^n$ every closed $k$-form with $k \ge 1$ is exact, by the Poincaré lemma, so $H^k_{\mathrm{dR}}(\R^n) = 0$ for $k \ge 1$ and $H^0_{\mathrm{dR}}(\R^n) = \R$.
Non-example of exactness: on $M = \R^2 \setminus \set{0}$ the form $\omega = (x \, dy - y \, dx)/(x^2 + y^2)$ is closed but not exact.
Its integral over the unit circle is $2\pi$, while the integral of an exact form $df$ over a closed loop is $0$; so $H^1_{\mathrm{dR}}(\R^2 \setminus \set{0}) \neq 0$, and in fact it is $\R$.
A smooth map $f \colon M \to N$ induces linear maps $f^* \colon H^k_{\mathrm{dR}}(N) \to H^k_{\mathrm{dR}}(M)$ by pullback of forms, and the wedge product of forms makes $\bigoplus_k H^k_{\mathrm{dR}}(M)$ a graded ring.
</document>

<document source="https://tamnd.github.io/math2/node/WA60RW9E04.md" tag="WA60RW9E04" kind="statement">
# Fully faithful essentially surjective functor is an equivalence

## Statement

Assume the axiom of choice, applied to the classes of objects involved.
Let $F \colon \mathcal{C} \to \mathcal{D}$ be a [[functor|28NQ0KRHD3]].
Then $F$ is an [[equivalence of categories|K7A04ZBD92]] if and only if $F$ is [[fully faithful|GRG8WEBSH5]] and [[essentially surjective|6ERBSBVF14]].

## Remarks

Suppose first that $F$ has a quasi-inverse $G$ with [[natural isomorphisms|KYCPCC57FT]] $\eta \colon 1 \Rightarrow GF$ and $\varepsilon \colon FG \Rightarrow 1$.
Then $\varepsilon_D \colon F(G(D)) \cong D$ gives essential surjectivity, and naturality $\eta_B \circ f = G(F(f)) \circ \eta_A$ shows that $F$ is faithful, and likewise $G$.
For $h \colon F(A) \to F(B)$, the morphism $f = \eta_B^{-1} \circ G(h) \circ \eta_A$ satisfies $G(F(f)) = G(h)$, so $F(f) = h$ because $G$ is faithful; this direction needs no choice.
Conversely, choose for each object $D$ of $\mathcal{D}$ an object $G(D)$ of $\mathcal{C}$ and an isomorphism $\varepsilon_D \colon F(G(D)) \to D$.
For $g \colon D \to D'$, full faithfulness gives a unique $G(g)$ with $F(G(g)) = \varepsilon_{D'}^{-1} \circ g \circ \varepsilon_D$, which makes $G$ a functor and $\varepsilon$ natural.
The unit $\eta_A$ is the unique morphism with $F(\eta_A) = \varepsilon_{F(A)}^{-1}$, and it is an isomorphism because fully faithful functors reflect isomorphisms.
Example at work: let $K$ be a field, and include the full subcategory of spaces $K^n$ into all finite-dimensional vector spaces over $K$.
This inclusion is fully faithful and essentially surjective, so it is an equivalence.
Without choice, the converse can fail, since a quasi-inverse requires picking one preimage up to isomorphism for every object.
In fact the converse for all functors between small categories is equivalent to the axiom of choice.

## Explanations

### L1 human

To check that two categories are "the same up to isomorphism", one does not need to build a functor going back.
It is enough that the functor matches the arrows exactly and reaches every object up to isomorphism.
The return functor can then be built by picking, for each target object, one source object that lands near it.

### L2 human

This criterion is how most equivalences are proved in practice, for instance that every category is equivalent to its skeleton.
The quasi-inverse constructed in the proof can be chosen so that $\eta$ and $\varepsilon$ satisfy the triangle identities, so every equivalence is part of an adjoint equivalence.
Properties defined by universal properties, such as having products or being abelian, are invariant under equivalence, so the theorem lets one transfer them along any fully faithful essentially surjective functor.
</document>

<document source="https://tamnd.github.io/math2/node/WA74AY36GC.md" tag="WA74AY36GC" kind="definition">
# Sidon set

## Definition

Let $G$ be an [[abelian group|082GHNPJ53]], for example $\Z$, and let $A$ be a subset of $G$.
The set $A$ is a *Sidon set* if for all $a, b, c, d \in A$ with $a + b = c + d$, one has $\set{a, b} = \set{c, d}$.

## Remarks

If $G$ has no elements of order $2$, as in $\Z$, the definition is equivalent to this one: the differences $a - b$ with $a, b \in A$ and $a \ne b$ are pairwise different.
The set $\set{1, 2, 4, 8}$ is a Sidon set in $\Z$.
Its sums $a + b$ with $a \le b$ are $2, 3, 4, 5, 6, 8, 9, 10, 12, 16$, and these ten numbers are distinct.
The set $\set{1, 2, 3, 4}$ is not a Sidon set, since $1 + 4 = 2 + 3$ but $\set{1, 4} \ne \set{2, 3}$.
An infinite Sidon set of positive integers, listed in increasing order, is called a Sidon sequence.
Sidon sets in $\Z$ are also called Golomb rulers when they are finite.
</document>

<document source="https://tamnd.github.io/math2/node/WA796ZCNRX.md" tag="WA796ZCNRX" kind="concept">
# Classifying space of a group

## Summary

A classifying space of a discrete [[group|G5S1M9]] $G$ is a path-connected space $BG$ whose [[fundamental group|S1XK3HXHFX]] is $G$ and whose higher [[homotopy groups|P976WYEFMB]] all vanish.
It is the [[Eilenberg–MacLane space|AJHEYJE5XY]] $K(G, 1)$, and among CW complexes it is unique up to [[homotopy equivalence|XZG6MTSWCN]].
Through $BG$, group theory and topology meet: the [[homology of the group|CFAG3VRXEA]] is the homology of the space $BG$.
For topological groups, the same symbol denotes a space that classifies principal $G$-bundles, see [[principal bundle|8E37E74XVP]].

## Explanations

### L1 human

A classifying space turns a group into a shape.
The loops of the shape behave like the elements of the group, and the shape has no other holes of any kind.
For the group $\Z$, the circle works: going around $k$ times is the element $k$.
For $\Z^2$, the torus works, with one kind of loop for each generator.
</document>

<document source="https://tamnd.github.io/math2/node/WAE4ZGJXJ7.md" tag="WAE4ZGJXJ7" kind="statement">
# Cameron–Martin theorem

## Statement

Let $T > 0$ and let $C_0 = C_0([0, T])$ be the [[classical Wiener space|5Z00K7K1HY]] of continuous functions $\omega \colon [0, T] \to \R$ with $\omega(0) = 0$.
Let $W$ be Wiener measure on $C_0$, the law of a [[Wiener process|22735FGJV6]] on $[0, T]$.
Let $H$ be the Cameron–Martin space: the set of absolutely continuous $h \in C_0$ with $h' \in L^2([0, T])$.
For $h \in C_0$, let $W_h$ be the [[pushforward|3N66TM3TSQ]] of $W$ under the translation $\omega \mapsto \omega + h$.

1. If $h \in H$, then $W_h$ and $W$ are [[mutually absolutely continuous|P0P21KDV6E]], and for $W$-almost every $\omega$
$$\frac{dW_h}{dW}(\omega) = \exp\left( \int_0^T h'(t) \, d\omega(t) - \frac12 \int_0^T h'(t)^2 \, dt \right).$$
2. If $h \notin H$, then $W_h$ and $W$ are [[mutually singular|EBGECZNCV5]].

## Remarks

Here $dW_h / dW$ is the [[Radon–Nikodym derivative|1NTKG64F7N]], and $\int_0^T h'(t) \, d\omega(t)$ is the Wiener integral of the deterministic function $h'$, defined as an $L^2(W)$ limit.
In one dimension the formula is a direct computation: the density of $N(a, 1)$ with respect to $N(0, 1)$ is $e^{a x - a^2/2}$.
The general case reduces to finite-dimensional projections of the path, since the Wiener integral is a normal variable with variance $\int_0^T h'(t)^2 \, dt$.

Example: for $h(t) = c t$, one has $h' = c$, and the density is $\exp(c\, \omega(T) - c^2 T / 2)$.
So Brownian motion with drift $c$ has a law equivalent to that of Brownian motion on $[0, T]$.

The analogous result holds for every centered [[Gaussian measure|P0DV2TCH1N]] on a separable Banach space, with $H$ its Cameron–Martin space.
In infinite dimensions $H$ has measure zero, so most translations give singular measures.

## Explanations

### L1 human

Shifting a Brownian path by a fixed function changes its law.
If the shift is smooth enough, with square-integrable derivative, the new law has a density with respect to the old one, given by an explicit formula.
If the shift is rougher, the two laws live on disjoint sets of paths.

### L2 human

The theorem is the starting point of Girsanov's theorem, where the deterministic shift is replaced by an adapted random drift.
It shows that Wiener measure is quasi-invariant only along the directions of $H$, which is a dense subspace of measure zero.
Integration by parts on Wiener space, the base of Malliavin calculus, comes from differentiating the density in $h$.
</document>

<document source="https://tamnd.github.io/math2/node/WAGJVK2NWE.md" tag="WAGJVK2NWE" kind="statement">
# Jordan decomposition of a signed measure

## Statement

Let $(X, \Sigma)$ be a measurable space and let $\nu$ be a [[signed measure|VG2P7R91Z7]] on it.
Then there are unique [[measures|PR11BZQQPG]] $\nu^+$ and $\nu^-$ on $(X, \Sigma)$ such that $\nu^+$ and $\nu^-$ are [[mutually singular|EBGECZNCV5]] and
$$\nu = \nu^+ - \nu^-.$$
At least one of $\nu^+$ and $\nu^-$ is finite.

## Remarks

Existence comes from a [[Hahn decomposition|2PKHANZ8JS]] $X = P \cup N$: put $\nu^+(E) = \nu(E \cap P)$ and $\nu^-(E) = -\nu(E \cap N)$.
These are measures, they are concentrated on the disjoint sets $P$ and $N$, and their difference is $\nu$.
Since $\nu$ takes at most one infinite value, one of them is finite.
Uniqueness: any such pair gives a Hahn decomposition, and two Hahn decompositions differ by a $\nu$-null set.

Example: for $\nu(E) = \int_E f \, d\mu$ with $f \in L^1(\mu)$ real, $\nu^+(E) = \int_E f^+ \, d\mu$ and $\nu^-(E) = \int_E f^- \, d\mu$, where $f^+ = \max(f, 0)$ and $f^- = \max(-f, 0)$.

Mutual singularity is what makes the decomposition unique.
Without it, $\nu = (\nu^+ + \mu) - (\nu^- + \mu)$ for any finite measure $\mu$.

## Explanations

### L1 human

Every signed measure is the difference between a positive part and a negative part.
The two parts live on separate regions of the space, so they do not cancel each other anywhere.
It is the measure version of writing a number as its positive part minus its negative part.

### L2 human

The Jordan decomposition reduces signed measures to ordinary measures: one integrates against $\nu$ by integrating against $\nu^+$ and $\nu^-$.
The sum $\nu^+ + \nu^-$ is the [[total variation|EPBND51CR5]] $\abs{\nu}$, the natural size of $\nu$.
The name comes from the [[Jordan decomposition of a function of bounded variation|KDT3E0VE0K]] as a difference of increasing functions.
For the measure $dF$ of such a function $F$ the two decompositions correspond.
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<document source="https://tamnd.github.io/math2/node/WAMRHWW82R.md" tag="WAMRHWW82R" kind="concept">
# Laguerre polynomials

## Summary

The Laguerre polynomials $L_n^{(\alpha)}$ are the [[classical orthogonal polynomials|47WJ34NC9K]] on the half-line $[0, \infty)$ for the weight $x^\alpha e^{-x}$.
The case $\alpha = 0$ gives the ordinary Laguerre polynomials $L_n$.
They solve the Laguerre differential equation.
They appear in the radial wave functions of the hydrogen atom and in Gaussian quadrature on $[0, \infty)$.

## Explanations

### L1 human

On the half-line from $0$ to infinity, the weight $e^{-x}$ gives more importance to small numbers than to large ones.
The Laguerre polynomials are the polynomials that are perpendicular to each other for this weight.
The first ones are $L_0 = 1$, $L_1 = 1 - x$ and $L_2 = \frac{x^2 - 4x + 2}{2}$.
For instance, $L_1$ is perpendicular to $L_0$ because $\int_0^\infty (1 - x) e^{-x} \, dx = 1 - 1 = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/WAQJJ5GX68.md" tag="WAQJJ5GX68" kind="concept">
# Many-sorted algebra

## Summary

A many-sorted algebra has several carrier sets, one for each sort, and operations whose arguments and values may lie in different carriers.
It is also called a heterogeneous algebra.
Many-sorted algebras describe structures with several kinds of objects.
Examples are a module with its ring of scalars, and an automaton with its states and its input letters.
They are the standard setting for the algebraic specification of [[abstract data types|86144NHH59]] in computer science.

## Explanations

### L1 human

An ordinary algebra has one set of elements, and every operation takes elements of this set to an element of the same set.
Many real structures mix several kinds of things.
A vector space has vectors and scalars, and scalar multiplication takes one scalar and one vector to a vector.
A many-sorted algebra keeps one set for each kind, or sort, and records for each operation which sorts it takes and which sort it returns.
This avoids nonsense such as multiplying two vectors when only scalar times vector is meant.
</document>

<document source="https://tamnd.github.io/math2/node/WAS3TN7VKD.md" tag="WAS3TN7VKD" kind="definition">
# Kohn Laplacian

## Definition

Let $M$ be a [[CR manifold|ZC78JKZX9S]] with CR structure $T^{0,1}M \subseteq \C TM$.
Fix a Hermitian metric on $T^{0,1}M$ and a smooth volume form on $M$.
For each integer $q \ge 0$, let $\mathcal{E}^{0,q}_b(M)$ be the space of smooth sections of $\Lambda^q (T^{0,1}M)^*$, and let
$$\bar\partial_b \colon \mathcal{E}^{0,q}_b(M) \to \mathcal{E}^{0,q+1}_b(M)$$
be the [[tangential Cauchy–Riemann operator|X7GP9BECD8]].
The metric and the volume form give an $L^2$ inner product on compactly supported forms in $\mathcal{E}^{0,q}_b(M)$.
Let $\vartheta_b$ be the formal adjoint of $\bar\partial_b$ for these inner products, so that $\ip{\bar\partial_b u}{v} = \ip{u}{\vartheta_b v}$ for compactly supported $u$ and $v$.
The *Kohn Laplacian* on $(0,q)$-forms is
$$\square_b = \bar\partial_b \vartheta_b + \vartheta_b \bar\partial_b \colon \mathcal{E}^{0,q}_b(M) \to \mathcal{E}^{0,q}_b(M).$$

## Remarks

The adjoint $\vartheta_b$ is often written $\bar\partial_b^*$, and $\square_b$ is also called the Kohn–Rossi Laplacian or the boundary Laplacian.
On functions ($q = 0$) the first term is absent, and $\square_b u = \vartheta_b \bar\partial_b u$.
Example: on the Heisenberg group $\mathbb{H} = \C \times \R$ with coordinates $(z, t)$, take $T^{0,1}\mathbb{H}$ spanned by $\bar Z = \partial/\partial \bar z - i z\, \partial/\partial t$, declare $\bar Z$ to have length $1$, and use Lebesgue measure.
The vector field $\bar Z$ has divergence $0$, so integration by parts shows that the formal adjoint of $\bar Z$ is $-Z$, where $Z = \partial/\partial z + i \bar z\, \partial/\partial t$ is its complex conjugate.
Hence $\square_b u = -Z \bar Z u$ for functions $u$.
Non-example: the operator $-\bar Z Z$ is not the Kohn Laplacian on functions of $\mathbb{H}$.
It differs from $-Z \bar Z$ by the commutator $[Z, \bar Z] = -2i\, \partial/\partial t$, which is not zero.
The Kohn Laplacian is not elliptic, because its principal symbol vanishes on the nonzero real covectors that annihilate $T^{1,0}M \oplus T^{0,1}M$.

## Explanations

### L1 human

The Kohn Laplacian is built from the derivatives along the complex directions tangent to a real hypersurface.
Like the ordinary Laplacian, it is a sum of a first order operator followed by its adjoint, done in both orders.
It is the operator whose solutions describe the boundary values of holomorphic functions and forms.
</document>

<document source="https://tamnd.github.io/math2/node/WAS9EANSJJ.md" tag="WAS9EANSJJ" kind="concept">
# Random walk on a group

## Summary

A random walk on a group $G$ starts at the identity and multiplies by independent random elements of $G$.
All these elements are drawn from one [[probability measure|059TJN0NZ1]] $\mu$.
It is a [[Markov chain|0G48PYM96Q]] on $G$ whose transition probabilities are invariant under the group.
When $\mu$ is uniform on a symmetric generating set, it is the simple [[random walk|04VZE3FZXF]] on the [[Cayley graph|0GYQK9NDJF]].
The long-run behaviour of the walk reflects the algebra and geometry of the group.
For example, let $G$ be finitely generated and let $\mu$ be symmetric with support generating $G$.
By a theorem of Kesten, the return probabilities then decay exponentially exactly when $G$ is not [[amenable|CEAT9BESKK]].

## Explanations

### L1 human

Picture a walker on the grid of integer points in the plane who, at each step, moves one unit north, south, east or west at random.
This is a random walk on the group $\Z^2$, and it comes back to its start with probability $1$.
Shuffling a deck of cards by repeated random swaps is a random walk on the group of permutations of the cards.
The question of how many shuffles mix the deck well is a question about this random walk.
</document>

<document source="https://tamnd.github.io/math2/node/WAS9NHKMYD.md" tag="WAS9NHKMYD" kind="definition">
# Eilenberg–MacLane space

## Definition

Let $n \ge 1$ and let $G$ be a [[group|G5S1M9]], [[abelian|082GHNPJ53]] if $n \ge 2$.
An *Eilenberg–MacLane space of type $K(G, n)$* is a [[path-connected space|0WB6DH499T]] $X$ with base point $x_0$ such that
$$\pi_n(X, x_0) \cong G \quad \text{and} \quad \pi_k(X, x_0) = 0 \text{ for all } k \ne n, \ k \ge 1,$$
where $\pi_k$ is the $k$-th [[homotopy group|P976WYEFMB]].

## Remarks

The condition that $G$ be abelian for $n \ge 2$ is needed, since $\pi_n$ is abelian for $n \ge 2$.
For every such $G$ and $n$ there is a CW complex of type $K(G, n)$, and any two CW complexes of type $K(G, n)$ are homotopy equivalent.
One often writes $K(G, n)$ for any such CW complex.

Example: the circle $S^1$ is a $K(\Z, 1)$.
It is path-connected, $\pi_1(S^1) \cong \Z$, and $\pi_k(S^1) = 0$ for $k \ge 2$ because its universal cover $\R$ is contractible.

Non-example: the sphere $S^2$ is path-connected with $\pi_2(S^2) \cong \Z$ and $\pi_1(S^2) = 0$, but $\pi_3(S^2) \cong \Z$, so it is not a $K(\Z, 2)$.
</document>

<document source="https://tamnd.github.io/math2/node/WASXFTTKK6.md" tag="WASXFTTKK6" kind="statement">
# Range conditions for the Radon transform

## Statement

Let $n \ge 2$, and let $R$ be the [[Radon transform|VJNZJ352PC]] on the [[Schwartz space|GE020MY43M]] $\mathcal{S}(\R^n)$.
Let $\mathcal{S}(S^{n-1} \times \R)$ be the space of smooth functions $g(\theta, s)$ on $S^{n-1} \times \R$ that decay rapidly in $s$ with all derivatives.
That is, each derivative of $g$ in $\theta$ along the [[sphere|P9CM4GQ2T0]] and in $s$, times any power of $\abs{s}$, is bounded.
A function $g \in \mathcal{S}(S^{n-1} \times \R)$ is of the form $g = R f$ for some $f \in \mathcal{S}(\R^n)$ if and only if both conditions hold:

1. $g(-\theta, -s) = g(\theta, s)$ for all $(\theta, s)$.
2. For every integer $k \ge 0$, the function $\theta \mapsto \int_\R g(\theta, s)\, s^k\, ds$ is the restriction to $S^{n-1}$ of a homogeneous polynomial of degree $k$ in $\theta = (\theta_1, \dots, \theta_n)$.

## Remarks

Necessity is a direct computation.
Condition 1 holds because $(\theta, s)$ and $(-\theta, -s)$ give the same hyperplane.
For condition 2, Fubini's theorem gives $\int_\R R f(\theta, s)\, s^k\, ds = \int_{\R^n} f(x)\, (x \cdot \theta)^k\, dx$, and expanding $(x \cdot \theta)^k$ shows a homogeneous polynomial of degree $k$ in $\theta$.
Sufficiency is the deep part.
Following the projection-slice theorem, one defines $\hat f(\sigma\theta)$ as the Fourier transform of $g(\theta, \cdot)$.
The moment conditions are what make this function smooth at the origin $\xi = 0$.

Example: for $n = 2$, $g(\theta, s) = \sqrt{\pi}\, e^{-s^2}$ is the Radon transform of $e^{-\abs{x}^2}$.
Its moments vanish for odd $k$ and are constants $c_k$ for even $k$.
On the circle, $c_k = c_k (\theta_1^2 + \theta_2^2)^{k/2}$, a homogeneous polynomial of degree $k$.

Non-example: for $n = 2$, $g(\theta, s) = \theta_1^2 e^{-s^2}$ satisfies condition 1.
But for $k = 0$ its integral in $s$ is $\sqrt{\pi}\, \theta_1^2$, which is not constant on the circle.
So condition 2 fails, and $g$ is not a Radon transform.

These are often called the Helgason–Ludwig consistency conditions.

## Explanations

### L1 human

Not every function of lines is the Radon transform of something.
The theorem lists the conditions.
The data must not depend on how a line is labeled, and its moments in the offset must be polynomials of the right degree in the direction.
For example, the total integral of the data over each parallel family of lines must be the same number for every direction, namely the total mass of the function.

### L2 human

In tomography, measured data are noisy and usually violate these conditions.
Projecting the data onto the range, or checking the conditions, helps to detect errors, to estimate unknown calibration parameters and to fill in missing data.
The case $k = 0$ says that the mass is the same in every direction, and $k = 1$ says that the center of mass is a single point of $\R^n$.
The theorem describes the range of $R$ on $\mathcal{S}(\R^n)$; for other function spaces the range is described by similar conditions.
</document>

<document source="https://tamnd.github.io/math2/node/WB25VJQEXJ.md" tag="WB25VJQEXJ" kind="definition">
# Neyman construction

## Definition

Let $X$ be data with values in a set $\mathcal{X}$, whose law $P_\theta$ depends on a parameter $\theta \in \Theta$, and let $\alpha \in (0, 1)$.
For each $\theta \in \Theta$, choose an acceptance region $A(\theta) \subseteq \mathcal{X}$ with $P_\theta(X \in A(\theta)) \ge 1 - \alpha$.
The *Neyman construction* is the set-valued statistic
$$C(X) = \set{\theta \in \Theta : X \in A(\theta)}.$$

## Remarks

For each $\theta$, the events $\set{\theta \in C(X)}$ and $\set{X \in A(\theta)}$ are equal, so $P_\theta(\theta \in C(X)) \ge 1 - \alpha$.
So $C(X)$ is a [[confidence set|0K4NMMJBG2]] of level $1 - \alpha$; its coverage is exactly $1 - \alpha$ when each $A(\theta)$ has probability exactly $1 - \alpha$.
Equivalently, $A(\theta)$ is the acceptance region of a [[test|087P03NEND]] at level $\alpha$ of the hypothesis that $\theta$ is the true value.
Then $C(X)$ is the set of values that these tests do not reject.
Example: for $X \sim \mathcal{N}(\theta, 1)$ and $A(\theta) = [\theta - 1.96, \theta + 1.96]$, the construction gives $C(X) = [X - 1.96, X + 1.96]$, of level about $0.95$.
Non-example: for $\alpha = 0.05$, the choice $A(\theta) = [\theta - 1, \theta + 1]$ fails the condition, since $P_\theta(X \in A(\theta)) \approx 0.68$; the resulting set has level only about $0.68$.
</document>

<document source="https://tamnd.github.io/math2/node/WBCN9A847H.md" tag="WBCN9A847H" kind="concept">
# Orlicz space

## Summary

Fix a convex growth function $\Phi$.
The Orlicz space $L^\Phi(\mu)$ consists of the measurable functions $f$ for which $\Phi(\abs{f})$ is integrable after scaling $f$.
The choice $\Phi(t) = t^p$ gives the [[Lp space|SASP14MA1Z]] $L^p$, so Orlicz spaces generalise the $L^p$ scale.
They are Banach spaces with the Luxemburg norm.
They describe integrability conditions that lie between the $L^p$ spaces, such as $L \log L$, which appears in the theory of maximal functions and singular integrals.

## Explanations

### L1 human

The space $L^p$ asks that $\abs{f}^p$ have finite integral.
An Orlicz space replaces the power $t^p$ by another convex function $\Phi(t)$.
With $\Phi(t) = e^t - 1$ one asks for exponential integrability, which is stronger than every $L^p$ condition on a finite measure space.
With $\Phi(t) = t \log(1 + t)$ one asks, on a finite measure space, for a little more than integrability, but less than any $L^p$ with $p > 1$.
</document>

<document source="https://tamnd.github.io/math2/node/WBKCGFMT0P.md" tag="WBKCGFMT0P" kind="statement">
# Delzant's theorem on toric manifolds

## Statement

A *symplectic toric manifold* is a compact connected [[symplectic manifold|CAN490EPRT]] $(M, \omega)$ of dimension $2n$ with two data.
These are an effective action of the torus $T^n = (S^1)^n$ and a [[moment map|ZARM34FZ6Z]] $\mu \colon M \to \R^n$ for it.
A *Delzant polytope* in $\R^n$ is a [[convex polytope|300K6CB1KH]] $\Delta$ such that at each vertex $v$:

1. exactly $n$ edges meet at $v$;
2. the edges at $v$ have the form $v + t u_i$, $t \ge 0$, with $u_1, \ldots, u_n \in \Z^n$;
3. these vectors $u_1, \ldots, u_n$ can be chosen to form a basis of the lattice $\Z^n$.

Then the image $\mu(M)$ of a symplectic toric manifold is a Delzant polytope.
The map $(M, \omega, \mu) \mapsto \mu(M)$ is a bijection from symplectic toric manifolds onto Delzant polytopes in $\R^n$.
Here toric manifolds are taken up to $T^n$-equivariant symplectomorphisms that intertwine the moment maps.

## Remarks

That $\mu(M)$ is a convex polytope is the [[Atiyah–Guillemin–Sternberg convexity theorem|5CTHC6612N]]; the local normal form at fixed points gives the three conditions.
For surjectivity, Delzant builds $M$ from $\Delta$ as a [[symplectic reduction|SBJVATVH44]] of $\C^d$ by a subtorus of $T^d$, where $d$ is the number of facets of $\Delta$.
Injectivity uses the local structure near each orbit and a gluing argument.

Example: the simplex with vertices $0, e_1, \ldots, e_n$ is a Delzant polytope, and it corresponds to $\mathbb{CP}^n$; the square $[0,1]^2$ corresponds to $S^2 \times S^2$.
The triangle with vertices $(0,0)$, $(1,0)$, $(0,2)$ is not Delzant.
At $(1,0)$ the primitive edge vectors are $(-1,0)$ and $(-1,2)$, with determinant $-2$, so they do not form a basis of $\Z^2$.

## Explanations

### L1 human

A toric manifold is a symplectic manifold with as much torus symmetry as its dimension allows.
Delzant's theorem says that such a manifold is completely described by a polytope with integer edge directions, its moment polytope.
So questions about these manifolds can be turned into questions about polytopes.

### L2 human

The theorem gives a dictionary: faces of the polytope correspond to orbit closures, and vertices to fixed points of the torus.
Every symplectic toric manifold has a compatible complex structure that makes it a smooth [[toric variety|YWB981MEF2]].
Rational simple polytopes with a positive integer label on each facet correspond in the same way to toric orbifolds, by work of Lerman and Tolman.
</document>

<document source="https://tamnd.github.io/math2/node/WBS4NM1TFC.md" tag="WBS4NM1TFC" kind="statement">
# Gamma function has no zeros

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\C$.
For every $z \in \C \setminus \set{0, -1, -2, \ldots}$, $\Gamma(z) \ne 0$.

## Remarks

For $z \notin \Z$, the [[reflection formula|WF7KQAQTS2]] gives $\Gamma(z)\Gamma(1 - z) = \pi/\sin(\pi z)$, and the right side is finite and nonzero, so $\Gamma(z) \ne 0$.
For a positive integer $n$, $\Gamma(n) = (n-1)! \ne 0$.
The points $0, -1, -2, \ldots$ are excluded because $\Gamma$ has poles there, not values.
A consequence is that the [[reciprocal gamma function|PY6K9AWR33]] $1/\Gamma$ is entire, with zeros exactly at $0, -1, -2, \ldots$.
In Mathlib the statement is `Complex.Gamma_ne_zero`.

## Explanations

### L1 human

The gamma function never takes the value zero.
It can become very large near its poles and very small for large negative arguments, but it never vanishes.

### L2 human

Because $\Gamma$ has no zeros, its logarithmic derivative, the [[digamma function|HWRRXB0YXS]] $\psi = \Gamma'/\Gamma$, is meromorphic with poles only at $0, -1, -2, \ldots$.
It also allows a holomorphic logarithm of $\Gamma$ on the slit plane, the [[log-gamma function|JT95B44G4G]].
</document>

<document source="https://tamnd.github.io/math2/node/WBYJ7A0HY7.md" tag="WBYJ7A0HY7" kind="definition">
# Hyperbolic conservation law

## Definition

Let $\partial_t u + \sum_{j=1}^n \partial_{x_j} F_j(u) = 0$ be a system of [[conservation laws|8RA7AMWSAF]], with $C^1$ fluxes $F_j \colon \R^m \to \R^m$ and unknown $u$ with values in an open set $\mathcal{U} \subseteq \R^m$.
For $u \in \mathcal{U}$ and $\xi \in \R^n$ let $B(u; \xi) = \sum_{j=1}^n \xi_j\, DF_j(u)$, an $m \times m$ real matrix, where $DF_j$ is the Jacobian matrix of $F_j$.
The system is *hyperbolic* on $\mathcal{U}$ if, for every $u \in \mathcal{U}$ and every unit vector $\xi \in \R^n$, the matrix $B(u; \xi)$ is diagonalizable over $\R$.
That is, $B(u; \xi)$ has $m$ real eigenvalues $\lambda_1(u; \xi) \le \cdots \le \lambda_m(u; \xi)$ and a basis of $\R^m$ of corresponding eigenvectors.

## Remarks

It is *strictly hyperbolic* if moreover the eigenvalues are distinct, $\lambda_1 < \cdots < \lambda_m$.
In one space dimension $B(u; \xi) = \xi\, DF(u)$, and the condition is that $DF(u)$ is diagonalizable with real eigenvalues; these are the characteristic speeds.
The definition makes the system a [[hyperbolic|92C3S03AQZ]] system of first-order equations when written in the non-conservative form $u_t + \sum_j DF_j(u)\, u_{x_j} = 0$.

Example: every scalar conservation law ($m = 1$) is hyperbolic, since $B(u; \xi) = F'(u) \cdot \xi$ is a real $1 \times 1$ matrix.
Example: the $p$-system $v_t - w_x = 0$, $w_t + p(v)_x = 0$ with $p' < 0$ has $DF = \begin{pmatrix} 0 & -1 \\ p'(v) & 0 \end{pmatrix}$.
Its eigenvalues $\pm \sqrt{-p'(v)}$ are real and distinct, so the system is strictly hyperbolic.

Non-example: the linear system $u_t + A u_x = 0$ with $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, so that $F(u) = Au$.
The eigenvalues of $A$ are $\pm i$, which are not real, so the system is not hyperbolic; its components satisfy the Cauchy–Riemann equations, which are elliptic.
</document>

<document source="https://tamnd.github.io/math2/node/WBYTAFMWJR.md" tag="WBYTAFMWJR" kind="definition">
# Commutator theory

## Definition

Let $A$ be an algebra and let $\alpha$, $\beta$, $\delta$ be [[congruences|DCSXXT4SER]] of $A$.
The congruence $\alpha$ *centralizes* $\beta$ *modulo* $\delta$, written $C(\alpha, \beta; \delta)$, if the following holds.
For every $n \ge 1$, every $(n+1)$-ary [[term operation|N9M2N27END]] $t$ of $A$, every pair $(a, b) \in \alpha$ and all tuples $\bar{c} = (c_1, \dots, c_n)$ and $\bar{d} = (d_1, \dots, d_n)$ with $(c_i, d_i) \in \beta$ for each $i$:
$$t(a, \bar{c}) \mathrel{\delta} t(a, \bar{d}) \implies t(b, \bar{c}) \mathrel{\delta} t(b, \bar{d}).$$
The *commutator* $[\alpha, \beta]$ is the least congruence $\delta$ of $A$ such that $C(\alpha, \beta; \delta)$.

## Remarks

The least such $\delta$ exists: $C(\alpha, \beta; 1_A)$ holds for the full congruence $1_A$, and if $C(\alpha, \beta; \delta_i)$ holds for every $i$, then $C(\alpha, \beta; \bigcap_i \delta_i)$ holds.
The definition is often called the term condition, and this commutator the term-condition commutator.
Example: in a group $G$, let $\alpha$ and $\beta$ be the congruences of normal subgroups $M$ and $N$.
Then $[\alpha, \beta]$ is the congruence of the commutator subgroup $[M, N]$.
In particular $[1_G, 1_G]$ is the congruence of the [[commutator subgroup|0ASGSTXSBW]] of $G$.
Non-example of $C(1_G, 1_G; 0_G)$: let $G$ be a nonabelian group with $b c \ne c b$, and let $t(x, y) = x y x^{-1} y^{-1}$.
With $a = e$, $\bar{c} = c$ and $\bar{d} = e$, both $t(e, c)$ and $t(e, e)$ equal $e$, but $t(b, c) = b c b^{-1} c^{-1} \ne e = t(b, e)$.
In a congruence-modular variety the commutator is symmetric, $[\alpha, \beta] = [\beta, \alpha]$, and $[\alpha, \beta] \le \alpha \wedge \beta$.

## Explanations

### L1 human

The commutator of two congruences is the smallest congruence that makes them commute with each other inside all term operations.
</document>

<document source="https://tamnd.github.io/math2/node/WBZHXGKNSJ.md" tag="WBZHXGKNSJ" kind="definition">
# Smooth manifold

## Definition

Let $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $n$, so $M$ is Hausdorff, second countable and locally homeomorphic to $\R^n$.
A *chart* on $M$ is a pair $(U, \varphi)$ where $U \subset M$ is open and $\varphi \colon U \to \varphi(U)$ is a [[homeomorphism|0QT00PB5G2]] onto an open subset of $\R^n$.
Two charts $(U, \varphi)$ and $(V, \psi)$ are *smoothly compatible* if $U \cap V = \emptyset$ or the transition map
$$\psi \circ \varphi^{-1} \colon \varphi(U \cap V) \to \psi(U \cap V)$$
is a [[smooth|0V33BNKZ9S]] ($C^\infty$) map with smooth inverse.
A *smooth atlas* on $M$ is a family of pairwise smoothly compatible charts whose domains cover $M$.
A *smooth structure* is a smooth atlas that is maximal: it contains every chart that is smoothly compatible with all of its charts.
A *smooth manifold* of dimension $n$ is a topological manifold of dimension $n$ together with a smooth structure.

## Remarks

The unit circle $S^1 \subset \R^2$ is a smooth manifold.
Let $N = (0, 1)$ and $S = (0, -1)$, and let $\sigma_N \colon S^1 \setminus \set{N} \to \R$ and $\sigma_S \colon S^1 \setminus \set{S} \to \R$ be the stereographic projections, $\sigma_N(x, y) = x/(1 - y)$ and $\sigma_S(x, y) = x/(1 + y)$.
On the overlap, $\sigma_N \sigma_S = x^2/(1 - y^2) = 1$, so the transition map is $t \mapsto 1/t$ on $\R \setminus \set{0}$, which is smooth with smooth inverse.

The charts $(\R, \id)$ and $(\R, t \mapsto t^3)$ on $\R$ are not smoothly compatible.
The transition map $t \mapsto t^{1/3}$ is not differentiable at $0$.
Each of them alone defines a smooth structure on $\R$, and the two structures are different, although the resulting manifolds are diffeomorphic.

Every smooth atlas is contained in a unique maximal one, so in practice a smooth manifold is given by any smooth atlas.
Replacing $C^\infty$ by $C^k$ gives $C^k$ manifolds.

## Explanations

### L1 human

A smooth manifold is a space covered by coordinate charts in which every change of coordinates is a smooth map.
</document>

<document source="https://tamnd.github.io/math2/node/WC0QNABPZ4.md" tag="WC0QNABPZ4" kind="concept">
# Wilf–Zeilberger pair

## Summary

A Wilf–Zeilberger pair, or WZ pair, is a pair of hypergeometric terms $F(n, k)$ and $G(n, k)$ with $F(n + 1, k) - F(n, k) = G(n, k + 1) - G(n, k)$.
Summing this relation over $k$ makes the right side telescope, which proves that $\sum_k F(n, k)$ does not depend on $n$.
Many binomial and hypergeometric identities can be proved this way, and the pair can be found by computer, for example with [[Gosper's algorithm|NHMJQBS04M]].
The rational function $G/F$ is a short certificate that anyone can check by algebra.

## Explanations

### L1 human

To prove that a sum such as $\sum_k \binom{n}{k}$ equals $2^n$, divide by the right side so that the claim becomes "this sum is always $1$".
It is then enough to show that the sum does not change when $n$ goes to $n + 1$.
A WZ pair shows this by writing each change in a term as a difference of neighbouring terms of a second sequence, so the changes cancel in the sum.
The second sequence is the original one times a rational function, the certificate, and checking the proof reduces to simplifying a fraction.
</document>

<document source="https://tamnd.github.io/math2/node/WC16TBY3FN.md" tag="WC16TBY3FN" kind="object">
# Tits group

## Description

The Tits group ${}^2F_4(2)'$ is the [[commutator subgroup|0ASGSTXSBW]] of the [[Ree group|YB62XEQJAC]] ${}^2F_4(2)$.
It has index $2$ in ${}^2F_4(2)$ and its order is
$$17971200 = 2^{11} \cdot 3^3 \cdot 5^2 \cdot 13.$$
It is a finite [[simple group|0YMYMNQ0QZ]], found by Jacques Tits.

## Remarks

The groups ${}^2F_4(2^{2n+1})$ with $n \ge 1$ are simple, but ${}^2F_4(2)$ is not, and its simple part is the Tits group.
So the Tits group is not strictly a group of Lie type, and it is not counted among the $26$ [[sporadic groups|PZ49Z4CBFZ]] either.
Some authors list it as a group of Lie type, and others count it as a $27$th sporadic group.

## Explanations

### L1 human

The Tits group is a single finite simple group that sits at the edge of the classification.
It is half of a group of Lie type, so it does not quite fit the families of Lie type and is not usually counted as sporadic.
</document>

<document source="https://tamnd.github.io/math2/node/WC47MFR6GW.md" tag="WC47MFR6GW" kind="concept">
# Ternary operation

## Summary

A ternary operation on a set $A$ is a [[finitary operation|EJ8YJVM9BW]] of [[arity|MSXX2P86CY]] $3$, a function from $A^3$ to $A$.
Ternary operations appear naturally in universal algebra.
Examples are the operation $x y^{-1} z$ of a [[group|G5S1M9]], which turns a group into a [[heap|V389A4WGH9]], and the [[ternary discriminator|72PG6XCQ7D]].
Many conditions on [[varieties|AJNZJD8442]] are stated with ternary terms, such as a [[Mal'cev term|Y203PPTS61]] or a [[majority term|9879ZMW5AA]].

## Explanations

### L1 human

A ternary operation takes three elements and gives one.
On the integers, $(x, y, z) \mapsto x - y + z$ is an example.
On a totally ordered set, the median of three elements, the one in the middle, is another example.
The rule "if $x$ then $y$ else $z$" on truth values is a third example.
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<document source="https://tamnd.github.io/math2/node/WC4QAGSMWR.md" tag="WC4QAGSMWR" kind="definition">
# Penny graph

## Definition

A finite [[graph|00C0BR7D7H]] $G = (V, E)$ is a *penny graph* if there is a map $p \colon V \to \R^2$ such that:

1. $\norm{p(u) - p(v)} \ge 1$ for all distinct $u, v \in V$;
2. $uv \in E$ if and only if $\norm{p(u) - p(v)} = 1$.

Equivalently, $G$ is the contact graph of a family of closed discs of diameter $1$ with pairwise disjoint interiors, centred at the points $p(v)$.

## Remarks

Example: the triangle $K_3$ is a penny graph.
Put $p$ at the vertices of an equilateral triangle of side $1$; all distances are equal to $1$, so both clauses hold.

Non-example: $K_4$ is not a penny graph.
Three of its vertices would have to sit at the corners of an equilateral triangle of side $1$.
No point of the plane is at distance $1$ from all three corners, so clause 2 fails for the fourth vertex.

Every penny graph is planar and has maximum degree at most $6$.
Take a vertex whose point is a corner of the convex hull of $p(V)$.
Its neighbours are at angles at least $60^\circ$ apart inside an angle less than $180^\circ$, so it has degree at most $3$.
Hence every penny graph can be properly coloured with $4$ colours, by colouring greedily after removing such vertices one at a time.
</document>

<document source="https://tamnd.github.io/math2/node/WC5JZVG82R.md" tag="WC5JZVG82R" kind="definition">
# Axiom of determinacy

## Definition

The *axiom of determinacy* (AD) is the following statement in the language of set theory.
For every set $X \subseteq \N^{\N}$ of sequences of natural numbers, the game $G(\N, X)$ is [[determined|QXECDPAYTN]].
Here, in $G(\N, X)$, players I and II alternately choose natural numbers $a_0, a_1, a_2, \dots$, and player I wins if $(a_0, a_1, a_2, \dots) \in X$.

## Remarks

Example: for each Borel set $X$ of the [[Baire space|P2WS16HKD7]] $\N^{\N}$, the instance of AD for $X$ holds.
It is even a theorem of ZFC, by the Borel determinacy theorem.

Non-example: AD is false in [[ZFC|0HE5WDQ9FJ]].
The [[axiom of choice|0HTGB13FE2]] gives a well-ordering of all strategies.
A diagonal argument along it builds a set $X$ that defeats every strategy of each player.

AD is therefore studied over ZF without choice.
It still implies the axiom of countable choice for sets of reals.
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<document source="https://tamnd.github.io/math2/node/WC8M1YAYCD.md" tag="WC8M1YAYCD" kind="definition">
# Hyperelliptic curve

## Definition

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]], and let $\mathbb{P}^1$ be the [[projective line|ZT9A6JK8QF]] over $k$.
A smooth projective [[algebraic curve|XJM2MT4S3Y]] $X$ over $k$ is *hyperelliptic* if:

1. its [[genus|6W8TDV9VVN]] $g$ satisfies $g \ge 2$, and
2. there is a finite [[morphism|CDS1VWB7A3]] $X \to \mathbb{P}^1$ of degree $2$.

## Remarks

If the characteristic of $k$ is not $2$, such an $X$ is the smooth projective model of an affine curve $y^2 = f(x)$.
Here $f \in k[x]$ is squarefree of degree $2g + 1$ or $2g + 2$, and the degree-$2$ map is $(x, y) \mapsto x$.
The degree-$2$ map is unique up to automorphisms of $\mathbb{P}^1$, and it defines the hyperelliptic involution $y \mapsto -y$.
Example: $y^2 = x^5 - x$ over $\C$ has $f$ squarefree of degree $5 = 2 \cdot 2 + 1$, so it defines a hyperelliptic curve of genus $2$.
In fact every curve of genus $2$ is hyperelliptic, because its canonical divisor gives a map of degree $2$ to $\mathbb{P}^1$.
Non-example: a smooth plane quartic has genus $3$ but is not hyperelliptic, because its canonical map is the embedding into the plane and so has degree $1$, not $2$.
Curves of genus $1$ with a point are also double covers of $\mathbb{P}^1$, but they fail clause 1 and are called elliptic curves.
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# Sparse approximation

## Summary

Let $A$ be a matrix with more columns than rows and let $b$ be a vector.
Sparse approximation looks for a vector $x$ with few nonzero entries such that $Ax$ is close to $b$.
The columns of $A$ form a redundant dictionary, so $b$ has many representations, and one wants the sparsest.
The exact problem is NP-hard in general.
In practice one uses greedy methods such as [[matching pursuit|8KJ9P9KQ5H]], or convex relaxations such as basis pursuit and the [[Lasso|KZ226116HW]].
It is the core of signal compression, denoising and [[compressed sensing|6V8E0AQNHA]].

## Explanations

### L1 human

Think of the columns of a [[matrix|0VD9ZSWWJG]] $A$ as building blocks, and of $b$ as a signal to build.
When there are more blocks than dimensions, the [[system of linear equations|0GT4WWGWH2]] $Ax = b$ has many solutions.
Sparse approximation asks for a solution, or a near solution, that uses as few blocks as possible.
For example, with $A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$ and $b = (1, 1)$, both $x = (1, 1, 0)$ and $x = (0, 0, 1)$ solve $Ax = b$.
The second one uses a single column, so it is the sparsest solution.
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<document source="https://tamnd.github.io/math2/node/WC92BCCFHV.md" tag="WC92BCCFHV" kind="definition">
# Nonlinear difference equation

## Definition

Let $K$ be a field and let $k \ge 1$.
Let $f \colon \N \times K^k \to K$ be a function, and consider the [[recurrence relation|07TZV87Y0T]]
$$u_{n+k} = f(n, u_n, u_{n+1}, \dots, u_{n+k-1}) \qquad (n \in \N).$$
It is *linear* if there are functions $a_0, \dots, a_{k-1}, g \colon \N \to K$ with
$$f(n, y_0, \dots, y_{k-1}) = a_0(n) y_0 + \cdots + a_{k-1}(n) y_{k-1} + g(n)$$
for all $n \in \N$ and all $y_0, \dots, y_{k-1} \in K$.
It is a *nonlinear difference equation* if it is not linear.

## Remarks

Example: the [[logistic map|QN9X9BNKTX]] recurrence $u_{n+1} = r u_n (1 - u_n)$ over $\R$ with $r \ne 0$ is nonlinear.
Indeed, for fixed $n$ an affine function $y \mapsto a y + c$ has constant second difference zero, while $f(y) = r y - r y^2$ gives $f(2) - 2 f(1) + f(0) = -2r \ne 0$.
Non-example: the Fibonacci recurrence $u_{n+2} = u_{n+1} + u_n$ is linear, with $a_0 = a_1 = 1$ and $g = 0$.
The same definitions apply with $K$ replaced by $\R^m$ and $f$ by a map to $\R^m$, which gives nonlinear systems.
An autonomous first-order equation $u_{n+1} = f(u_n)$ is the iteration of the map $f$, so its study belongs to discrete dynamical systems.
</document>

<document source="https://tamnd.github.io/math2/node/WC9WZ53ZNS.md" tag="WC9WZ53ZNS" kind="definition">
# Invariant basis number

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with $1$.
For an integer $n \ge 0$, let $R^n$ be the [[free left module|0DZ9DKAKZR]] of rank $n$ over $R$.
The ring $R$ has *invariant basis number* (IBN) if, for all integers $m, n \ge 0$, an isomorphism $R^m \cong R^n$ of left $R$-[[modules|0NZ8JA25WK]] implies $m = n$.

## Remarks

Example: every nonzero commutative ring $R$ has IBN.
To check it, take a maximal ideal $\mathfrak{m}$ of $R$; an isomorphism $R^m \cong R^n$ gives an isomorphism $(R/\mathfrak{m})^m \cong (R/\mathfrak{m})^n$ of vector spaces over the field $R/\mathfrak{m}$, so $m = n$ by dimension.

Non-example: let $V$ be a vector space of countably infinite dimension over a field $k$ and let $R = \End_k(V)$.
Since $V \cong V \oplus V$, one gets $R \cong R^2$ as left $R$-modules, so $R$ does not have IBN.

The zero ring does not have IBN, since $0 = R^m$ for all $m$.
The condition can be stated with matrices: $R$ has IBN if and only if there are no matrices $A \in R^{m \times n}$ and $B \in R^{n \times m}$ with $m \ne n$, $AB = I_m$ and $BA = I_n$.
This form is symmetric in left and right, so the definition does not depend on the side.
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<document source="https://tamnd.github.io/math2/node/WCM87QEEZW.md" tag="WCM87QEEZW" kind="statement">
# Wielandt's theorem

## Statement

Let $H = \set{z \in \C : \operatorname{Re} z > 0}$ and let $F \colon H \to \C$ be a [[holomorphic function|0NAHXR1FRH]] such that:

1. $F(z + 1) = z F(z)$ for every $z \in H$;
2. $F$ is bounded on the strip $\set{z \in \C : 1 \le \operatorname{Re} z \le 2}$.

Then $F(z) = F(1)\,\Gamma(z)$ for every $z \in H$, where $\Gamma$ is the [[gamma function|0V3FZMVB4K]].

## Remarks

The gamma function satisfies both conditions: the functional equation, and $\abs{\Gamma(z)} \le \Gamma(\operatorname{Re} z) \le 1$ on the strip, by Euler's integral.
For the proof, put $v = F - F(1)\Gamma$; it satisfies condition 1 and $v(1) = 0$.
The equation $v(z) = v(z + 1)/z$ extends $v$ to a meromorphic function on $\C$, and the possible poles at $0, -1, -2, \ldots$ are removable because $v(1) = 0$, so $v$ is entire and bounded on $0 \le \operatorname{Re} z \le 1$.
Then $u(z) = v(z) v(1 - z)$ is entire, satisfies $u(z + 1) = -u(z)$, and is bounded on that strip, so it is bounded on $\C$ and constant by [[Liouville's theorem|0EHKVRF4J6]]; since $u(1) = 0$, $u = 0$ and hence $v = 0$.
Condition 2 cannot be dropped: $F(z) = \Gamma(z)\big(1 + \sin(2\pi z)\big)$ satisfies condition 1 but is not a multiple of $\Gamma$, and it is unbounded on the strip.

## Explanations

### L1 human

Wielandt's theorem says that the gamma function is the only holomorphic solution of $F(z + 1) = z F(z)$ on the right half-plane that stays bounded on a vertical strip, up to a constant factor.
It is a complex analogue of the Bohr–Mollerup theorem, with boundedness in place of log-convexity.

### L2 human

The theorem is a quick tool for proving identities: to show that a function $F$ equals $\Gamma$, check the functional equation, boundedness on the strip $1 \le \operatorname{Re} z \le 2$, and $F(1) = 1$.
For example, the [[Euler–Gauss limit formula|SKRP9NDXR4]] and the [[Legendre duplication formula|CND4E1PY9V]] can be proved this way.
Compare with the [[Bohr–Mollerup theorem|QFF87AXWN2]], which works on the real line.
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<document source="https://tamnd.github.io/math2/node/WCMAN2AJYR.md" tag="WCMAN2AJYR" kind="definition">
# Complex multiplication

## Definition

Let $K$ be a field of characteristic $0$ with [[algebraic closure|0X6AB5P3G1]] $\overline{K}$, and let $E$ be an [[elliptic curve|427N6S6QQW]] over $K$.
Let $\End(E)$ be the ring of [[endomorphisms|0J7H0YBFDJ]] of $E$ defined over $\overline{K}$, that is, morphisms $E \to E$ over $\overline{K}$ that send the origin $O$ to $O$.
For $n \in \Z$ let $[n] \colon E \to E$ be multiplication by $n$; the map $n \mapsto [n]$ embeds $\Z$ in $\End(E)$.
The curve $E$ has *complex multiplication*, or is a *CM elliptic curve*, if $\End(E) \ne \Z$.
That is, $E$ has an endomorphism that is not of the form $[n]$.

## Remarks

In characteristic $0$, $\End(E)$ is either $\Z$ or an order $\mathcal{O}$ in an imaginary [[quadratic field|7SV4WA92SM]] $F$, and one then says that $E$ has CM by $\mathcal{O}$.
Over $\C$, write $E \cong \C/\Lambda$ for a lattice $\Lambda$; then $\End(E) \cong \set{\alpha \in \C : \alpha \Lambda \subseteq \Lambda}$, which explains the name.

Example: $E \colon y^2 = x^3 - x$ over $\Q$.
The map $[i](x, y) = (-x, iy)$ sends $E$ to itself, since $(iy)^2 = -(x^3 - x) = (-x)^3 - (-x)$.
It fixes $O$, and $[i] \circ [i] = [-1]$, so $[i] \ne [n]$ for every integer $n$; hence $E$ has CM, by $\Z[i]$.

Non-example: $y^2 = x^3 + x + 1$ over $\Q$ has $j$-invariant $1728 \cdot \frac{4}{31} = \frac{6912}{31}$, which is not an algebraic integer.
The $j$-invariant of a CM curve is always an algebraic integer, so this curve has endomorphism ring $\Z$.

Over a finite field the endomorphism ring of every elliptic curve is larger than $\Z$, so the notion is mainly used in characteristic $0$.
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<document source="https://tamnd.github.io/math2/node/WCSZYPBZY4.md" tag="WCSZYPBZY4" kind="example">
# Laplace transform method for the heat equation

## Example

Consider the [[heat equation|VP2HN86VAG]] on the half-line, for an unknown function $u(x, t)$ with $x > 0$ and $t > 0$:
$$u_t = u_{xx}, \qquad u(x, 0) = 0, \qquad u(0, t) = 1, \qquad u \text{ bounded}.$$
This models a cold rod, initially at temperature $0$, whose end $x = 0$ is held at temperature $1$ from time $0$ on.
Let $U(x, s) = \int_0^\infty u(x, t) e^{-s t} \, dt$ be the [[Laplace transform|05XT5AVJBK]] in the time variable, for real $s > 0$.
Since $u(x, 0) = 0$, the transform of $u_t$ is $s U$, so the equation becomes the ordinary differential equation $U_{xx} = s U$ in $x$.
Its solutions are $U = A(s) e^{-x \sqrt{s}} + B(s) e^{x \sqrt{s}}$.
Boundedness in $x$ forces $B(s) = 0$, and the boundary condition gives $U(0, s) = \mathcal{L}\{1\}(s) = 1/s$, so $A(s) = 1/s$.
Hence $U(x, s) = e^{-x \sqrt{s}} / s$.
The table pair $\mathcal{L}\{\operatorname{erfc}(x / (2 \sqrt{t}))\}(s) = e^{-x \sqrt{s}} / s$, for fixed $x > 0$, gives the [[inverse transform|0JRW4X1MDC]]:
$$u(x, t) = \operatorname{erfc}\left(\frac{x}{2 \sqrt{t}}\right),$$
where $\operatorname{erfc}$ is the [[complementary error function|TZ3CT1YA9B]].
Check: as $t \to 0^+$ with $x > 0$ fixed, $\operatorname{erfc}(x/(2\sqrt{t})) \to \operatorname{erfc}(+\infty) = 0$; at $x = 0$, $\operatorname{erfc}(0) = 1$; and $0 \le u \le 1$ for $x \ge 0$, so $u$ is bounded.

## Explanations

### L1 human

The Laplace transform in time removes the time derivative from the heat equation.
What is left is an ordinary differential equation in space, which one can solve by hand.
Transforming back gives the temperature profile $\operatorname{erfc}(x / (2 \sqrt{t}))$: the heat front spreads like $\sqrt{t}$.
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# Azumaya algebra

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $R$ with identity.
Let $A^{\mathrm{op}}$ be the [[opposite ring|M8Q24XB0AF]] of $A$.
The algebra $A$ is an *Azumaya algebra* over $R$ if the following hold.

1. $A$ is a [[finitely generated|0XHM30WBVB]] [[projective|CRE6VEKT32]] $R$-module.
2. $A$ is a [[faithful|TB40Y3SNSX]] $R$-module.
3. The $R$-linear map $A \otimes_R A^{\mathrm{op}} \to \End_R(A)$ that sends $a \otimes b$ to the map $x \mapsto axb$ is an isomorphism.

Here $\otimes_R$ is the [[tensor product of modules|SMFZN9E2ZB]] and $\End_R(A)$ is the [[endomorphism ring|AWSBZWWMG9]] of the $R$-module $A$.

## Remarks

Example: $A = M_n(R)$ with $n \ge 1$.
It is free of rank $n^2$, hence projective and faithful, and the map $M_n(R) \otimes_R M_n(R)^{\mathrm{op}} \to \End_R(M_n(R))$ sends the basis elements $e_{ij} \otimes e_{kl}$ to the maps $x \mapsto e_{ij} x e_{kl}$, which form a basis of $\End_R(M_n(R)) \cong M_{n^2}(R)$.
Non-example: let $L/K$ be a field extension of degree $2$ and $A = L$ over $R = K$.
Clauses 1 and 2 hold, but $L \otimes_K L$ is commutative while $\End_K(L) \cong M_2(K)$ is not, so the map in clause 3 is not an isomorphism.
Over a field $K$, the Azumaya algebras are exactly the [[central simple algebras|5HRKJBAY2V]] over $K$.
Equivalently, $A$ is an Azumaya algebra when it is a central [[separable algebra|X8H7A98Q55]] over $R$.
Another equivalent form: clauses 1 and 2 hold and $A / \mathfrak{m}A$ is central simple over $R/\mathfrak{m}$ for every maximal ideal $\mathfrak{m}$ of $R$.
</document>

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# Weyl tensor

## Definition

Let $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] of dimension $n \ge 3$.
Let $\operatorname{Rm}(X, Y, Z, V) = g(R(X, Y)Z, V)$ be its [[Riemann curvature tensor|JA29ETCM02]], with $R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$.
Let $\operatorname{Ric}$ be its [[Ricci curvature|4GTNM0WHDH]] and $S$ its [[scalar curvature|ZHX2VYQDZ3]].
The *Schouten tensor* is
$$P = \frac{1}{n - 2} \left( \operatorname{Ric} - \frac{S}{2(n - 1)} \, g \right).$$
For symmetric $(0,2)$-tensors $h, k$, their Kulkarni–Nomizu product, written here $h \odot k$, is the $(0,4)$-tensor
$$(h \odot k)(X, Y, Z, V) = h(X, V) k(Y, Z) + h(Y, Z) k(X, V) - h(X, Z) k(Y, V) - h(Y, V) k(X, Z).$$
The *Weyl tensor* of $(M, g)$ is the $(0,4)$-tensor
$$W = \operatorname{Rm} - P \odot g.$$

## Remarks

Example: the unit sphere $S^n$, $n \ge 3$, has $R(X, Y)Z = g(Y, Z)X - g(X, Z)Y$, so $\operatorname{Rm} = \tfrac{1}{2} \, g \odot g$, $\operatorname{Ric} = (n - 1)g$ and $S = n(n - 1)$.
Then $P = \tfrac{1}{2} g$ and $P \odot g = \tfrac{1}{2} \, g \odot g = \operatorname{Rm}$, so $W = 0$.

Non-example: for $n = 2$ the Schouten tensor has the factor $1/(n - 2)$ and is not defined, so surfaces have no Weyl tensor in this sense.

The Weyl tensor has the symmetries of the curvature tensor and all its traces vanish.
For $n = 3$ it is always zero.
For $n \ge 4$, $W = 0$ if and only if each point has a neighbourhood on which $g$ is a positive function times a flat metric.
Under a change $\tilde g = e^{2f} g$, the Weyl tensor changes by $\tilde W = e^{2f} W$.
Sign and index conventions for $\operatorname{Rm}$ differ between texts, and the formula for $W$ changes accordingly.
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# Outer measure

## Definition

Let $X$ be a set and let $\mathcal{P}(X)$ be its [[power set|00E0YZCERH]].
An *outer measure* on $X$ is a map $\mu^* \colon \mathcal{P}(X) \to [0, \infty]$, with values in the [[extended real line|0GSZ7B0CGR]], such that:

1. $\mu^*(\varnothing) = 0$.
2. If $A \subseteq B \subseteq X$, then $\mu^*(A) \le \mu^*(B)$.
3. For every sequence $A_1, A_2, \ldots$ of subsets of $X$,
$$\mu^*\Big(\bigcup_{n=1}^{\infty} A_n\Big) \le \sum_{n=1}^{\infty} \mu^*(A_n).$$

## Remarks

Example: Lebesgue outer measure on $\R$ is
$$\lambda^*(A) = \inf \Big\{ \sum_{n} (b_n - a_n) : A \subseteq \bigcup_{n} (a_n, b_n) \Big\},$$
the infimum over all countable covers of $A$ by open intervals.
Clause 1 holds since $\varnothing$ is covered by intervals of arbitrarily small total length.
Clause 2 holds since a cover of $B$ also covers $A$.
Clause 3 follows by covering each $A_n$ within $\varepsilon / 2^n$ of $\lambda^*(A_n)$.
Non-example: on $X = \set{1, 2}$, the map with $\mu(\varnothing) = 0$, $\mu(\set{1}) = \mu(\set{2}) = 1$ and $\mu(X) = 3$ is monotone, but it fails clause 3 because $3 > 1 + 1$.
Some authors, such as Federer, call outer measures simply measures.
The outer measures defined here are the ones to which [[Carathéodory's criterion|JDGQ135YK2]] applies.
</document>

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# Indecomposable module

## Definition

Let $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].
The module $M$ is *indecomposable* if:

1. $M \ne 0$;
2. whenever $M = N_1 \oplus N_2$ for [[submodules|0RGZ10DFPG]] $N_1, N_2$ of $M$, one has $N_1 = 0$ or $N_2 = 0$.

## Remarks

Equivalently, $M \ne 0$ and the only [[idempotents|0X4JKZXJ86]] of the [[endomorphism ring|AWSBZWWMG9]] $\End_R(M)$ are $0$ and $1$.
Example: the $\Z$-module $\Z$ is indecomposable.
The check: two nonzero submodules $a\Z$ and $b\Z$ both contain $ab \ne 0$, so their sum is never direct.
Example: $\Z/p^n\Z$ is indecomposable for $p$ prime and $n \ge 1$, since its subgroups form a chain.
Non-example: $\Z/6\Z = \set{0, 3} \oplus \set{0, 2, 4}$ fails clause 2.
Every simple module is indecomposable, but $\Z/4\Z$ is indecomposable and not simple.
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# Extreme set

## Summary

An extreme set of a set $A$ is a subset $B$ of $A$ with the following property.
Whenever a point of $B$ lies strictly between two points of $A$, both of those points lie in $B$.
It generalises the notion of [[extreme point|3MSGFJM2NX]]: a point is extreme exactly when the one-point set it forms is an extreme set.
The convex extreme subsets of a convex set are its [[faces|KY7SB854VH]].

## Explanations

### L1 human

Think of a filled square.
An edge of the square is an extreme set.
If a point of the edge is the middle of a segment inside the square, the whole segment must run along that edge.
A corner is an extreme set with one point.
The diagonal is not an extreme set, because its centre is also the middle of the other diagonal, which leaves it.
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# Complex measure

## Definition

Let $(X, \Sigma)$ be a [[measurable space|N4KDT83V3S]].
A *complex measure* on $(X, \Sigma)$ is a map $\nu \colon \Sigma \to \C$ with the following property.
For every sequence $(E_n)$ of pairwise disjoint sets in $\Sigma$,
$$\nu\Big( \bigcup_{n} E_n \Big) = \sum_{n} \nu(E_n),$$
where the series of [[complex numbers|09FTAET5Z4]] converges.

## Remarks

Taking all $E_n = \emptyset$ gives $\nu(\emptyset) = 0$.
The left side does not depend on the order of the $E_n$, so every rearrangement of the series converges to the same value, and the series converges absolutely.

Example: let $\mu$ be a measure and $f \in L^1(\mu)$ complex-valued.
Then $\nu(E) = \int_E f \, d\mu$ is a complex measure; countable additivity follows from the dominated convergence theorem with dominating function $\abs{f}$.

Non-example: Lebesgue measure on $\R$ is not a complex measure, because $\lambda(\R) = \infty$ is not a complex number.

Writing $\nu = \nu_r + i \nu_i$, the real and imaginary parts $\nu_r$ and $\nu_i$ are finite [[signed measures|VG2P7R91Z7]].
Integration against $\nu$ is defined by $\int f \, d\nu = \int f \, d\nu_r + i \int f \, d\nu_i$.
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<document source="https://tamnd.github.io/math2/node/WDH6QSH1Z6.md" tag="WDH6QSH1Z6" kind="example">
# Space of null sequences

## Example

Let $c_0$ be the set of sequences $x = (x_n)_{n \ge 1}$ of real (or complex) numbers with $x_n \to 0$, with the norm $\norm{x}_\infty = \sup_n \abs{x_n}$.
Then $c_0$ is a [[sequence space|XYTN19EPF2]] and a [[Banach space|0BDTTM24DP]].

Check: $c_0$ is the kernel of the linear functional $x \mapsto \lim_n x_n$ on the [[space of convergent sequences|GWCBGYG5RA]] $c$.
This functional satisfies $\abs{\lim_n x_n} \le \norm{x}_\infty$, so it is continuous, and its kernel is a closed linear subspace of $c$.
A closed subspace of a Banach space is complete, and $c$ is a Banach space, so $c_0$ is a Banach space.

## Remarks

The unit vectors $e_k$, with $1$ in place $k$ and $0$ elsewhere, form a [[Schauder basis|528G11SDWR]] of $c_0$: for $x \in c_0$, $\norm{x - \sum_{k \le N} x_k e_k}_\infty = \sup_{k > N} \abs{x_k} \to 0$.
In particular $c_0$ is separable, unlike the space $\ell^\infty$ of [[bounded sequences|G3F2CWQSQH]].
The dual space of $c_0$ is isometrically isomorphic to $\ell^1$, through the pairing $(a, x) \mapsto \sum_n a_n x_n$.
The space $c$ is the direct sum of $c_0$ and the line of constant sequences.

## Explanations

### L1 human

The null sequences are the sequences that tend to $0$, such as $1, 1/2, 1/3, \dots$.
They form a vector space, and the largest absolute value of the terms is a good way to measure their size.
This space is one of the basic examples of an infinite-dimensional Banach space.
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<document source="https://tamnd.github.io/math2/node/WDHHJHVP7N.md" tag="WDHHJHVP7N" kind="concept">
# Jacobi amplitude

## Summary

The Jacobi amplitude $\operatorname{am}(u, k)$ is the inverse function of the incomplete [[elliptic integral|NXMCYGCB59]] of the first kind $u = F(\varphi, k)$.
It is the angle $\varphi$ whose sine and cosine are the [[Jacobi elliptic functions|P071B692JF]] $\operatorname{sn} u$ and $\operatorname{cn} u$.
For modulus $k = 0$ it is the identity, and for $k = 1$ it is the Gudermannian function.

## Explanations

### L1 human

The integral $F(\varphi, k)$ measures something like a distorted angle: it adds up $d\theta / \sqrt{1 - k^2\sin^2\theta}$ from $0$ to $\varphi$.
The amplitude goes the other way, from the value $u$ back to the angle $\varphi$.
For a pendulum, $u$ is proportional to time and the amplitude tracks the angle of an auxiliary variable that turns unevenly.
When $k = 0$ the integrand is $1$, so $F(\varphi, 0) = \varphi$ and $\operatorname{am}(u, 0) = u$.
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<document source="https://tamnd.github.io/math2/node/WDHWCQ6Y2Y.md" tag="WDHWCQ6Y2Y" kind="statement">
# Birkhoff's completeness theorem

## Statement

Let $\sigma$ be a signature with only function symbols and let $X$ be an infinite set of variables.
Let $\Sigma$ be a set of [[identities|PKD440G2XW]] over $\sigma$ and $X$.
For every identity $s \approx t$ over $\sigma$ and $X$, the following are equivalent:

1. $\Sigma \vdash s \approx t$, that is, $s \approx t$ is derivable from $\Sigma$ in [[equational logic|72S3YSQFZ7]];
2. $\Sigma \models s \approx t$, that is, every algebra of signature $\sigma$ that satisfies all identities of $\Sigma$ satisfies $s \approx t$.

## Remarks

Soundness, from 1 to 2, holds because each rule of equational logic preserves truth in every algebra.
For completeness, let $\theta$ be the set of pairs $(u, v)$ of terms with $\Sigma \vdash u \approx v$.
The rules say that $\theta$ is a congruence of the term algebra that is invariant under substitutions, and the quotient of the term algebra by $\theta$ satisfies $\Sigma$.
Evaluating $s$ and $t$ in this quotient at the assignment $x \mapsto x/\theta$ gives $s/\theta$ and $t/\theta$, so if $\Sigma \models s \approx t$ then $(s, t) \in \theta$.
The quotient is the free algebra on $X$ of the variety defined by $\Sigma$.

Consequence: a set of identities is an [[equational theory|F1M51B4MDG]] exactly when it contains all identities $t \approx t$ and is closed under the rules of symmetry, transitivity, replacement and substitution.

## Explanations

### L1 human

The theorem says that equational reasoning is enough.
If a law follows from some given laws in every possible algebra, then it can be proved from them by plugging in, replacing equals by equals, and chaining equalities.
Nothing more powerful, such as case analysis or quantifier tricks, is ever needed for this kind of conclusion.

### L2 human

This is the equational analogue of Gödel's completeness theorem, with a much simpler set of rules.
The proof builds a canonical model: the term algebra divided by provable equality, which is the free algebra of the variety defined by $\Sigma$.
Together with Birkhoff's HSP theorem, it sets up the correspondence between varieties of algebras and equational theories.
It also justifies term rewriting: a rewriting proof of $s \approx t$ is a derivation in equational logic, and every true consequence has such a derivation, though finding it may be undecidable.
</document>

<document source="https://tamnd.github.io/math2/node/WDJSNMG1EZ.md" tag="WDJSNMG1EZ" kind="statement">
# Harmonic measure as exit distribution

## Statement

Let $d \ge 2$ and let $D \subset \R^d$ be a bounded [[domain|0Q4CBD3HX0]] whose boundary points are all regular for the Dirichlet problem.
For $x \in D$, let $\omega_x$ be the [[harmonic measure|CFD7BPM51W]] of $D$ at $x$.
It is the [[Borel probability measure|DYR1QFNGNJ]] on $\partial D$ such that $\int \varphi \, d\omega_x = u_\varphi(x)$ for every continuous $\varphi$ on $\partial D$.
Here $u_\varphi$ is the solution of the Dirichlet problem with boundary data $\varphi$.
Let $(B_t)$ be a [[Brownian motion|22735FGJV6]] in $\R^d$ started at $x$, with law $P^x$, and let $\tau = \inf\set{t > 0 : B_t \notin D}$.
Then for every Borel set $A \subseteq \partial D$,
$$P^x(B_\tau \in A) = \omega_x(A).$$

## Remarks

Proof idea: by [[Kakutani's theorem|CCEKM4241P]], $E^x[\varphi(B_\tau)]$ is the solution of the Dirichlet problem with data $\varphi$, for each continuous $\varphi$.
So the law of $B_\tau$ and $\omega_x$ give the same integral to each continuous function on the compact set $\partial D$, and the two Borel measures are equal.
Example: for the ball $\abs{y} < 1$ in $\R^d$, the exit point from $x$ has density, with respect to normalized surface measure, equal to the [[Poisson kernel|TCYSWCTYYM]] $(1 - \abs{x}^2)/\abs{x - \zeta}^d$.
From the centre $x = 0$ this density is $1$, so the exit point is uniform on the sphere.
Without regularity, the identity still holds when $\omega_x$ is the harmonic measure defined through Perron solutions.
The set of irregular boundary points is then polar, and it gets no mass.

## Explanations

### L1 human

Release a particle at a point inside a region and let it move at random until it reaches the boundary.
The harmonic measure of a piece of the boundary is the chance that the particle first arrives there.
Pieces close to the start point, or wide open to it, get more probability; pieces hidden in narrow fjords get very little.

### L2 human

The probabilistic picture explains many properties of harmonic measure at once.
It is a probability measure, it depends harmonically on the starting point, and it is monotone in the domain.
In the plane, Brownian motion is conformally invariant up to a time change, which explains the conformal invariance of harmonic measure.
Estimates such as Beurling's projection theorem have natural proofs and interpretations in this language.
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<document source="https://tamnd.github.io/math2/node/WDQTBSK6ES.md" tag="WDQTBSK6ES" kind="definition">
# Matrix function

## Definition

Let $f(z) = \sum_{k=0}^{\infty} c_k z^k$ be a complex [[power series|0MZB06ST2B]] with radius of convergence $R > 0$, where $R = \infty$ is allowed.
Let $n \ge 1$ and let $A$ be an $n \times n$ complex [[matrix|0JXZKX526R]] whose [[spectral radius|9V7YQA85PC]] satisfies $\rho(A) < R$.
Then the series $\sum_{k=0}^{\infty} c_k A^k$ converges, with $A^0 = I$, and its sum is the matrix $f(A)$.

## Remarks

If $A = P D P^{-1}$ with $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$, then $f(A) = P \operatorname{diag}(f(\lambda_1), \ldots, f(\lambda_n)) P^{-1}$.
The matrix $f(A)$ is always a polynomial in $A$, so it commutes with $A$.
Example: $f(z) = 1/(1 - z) = \sum_k z^k$ has $R = 1$, and $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ has $\rho(N) = 0$.
Since $N^2 = 0$, $f(N) = I + N$, and indeed $(I - N)(I + N) = I - N^2 = I$.
Non-example: for the same $f$ and $A = 2I$, $\rho(A) = 2 > 1$, and the series $\sum_k 2^k I$ diverges.
For $f$ holomorphic near the spectrum of $A$, the Cauchy integral or the Jordan form gives a more general definition; see the [[holomorphic functional calculus|YFKA3KVTGR]].
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<document source="https://tamnd.github.io/math2/node/WDVVEYD6AT.md" tag="WDVVEYD6AT" kind="concept">
# Signature

## Summary

A signature lists the non-logical symbols of a first-order language: function symbols, relation symbols and constant symbols, each with its number of arguments.
It fixes what one can talk about, before any axioms are chosen.
Terms, formulas and [[structures|983GZBQYT4]] are always taken over a given signature.
Another name is vocabulary, and in Mathlib it is called a first-order language.

## Explanations

### L1 human

Before writing axioms for groups, one has to say which symbols are allowed.
For groups, one may take a binary symbol $\cdot$, a unary symbol ${}^{-1}$ and a constant $e$.
This list of symbols with their numbers of arguments is the signature.
The axioms, such as associativity, come later and are not part of the signature.
</document>

<document source="https://tamnd.github.io/math2/node/WDZFV2ACXR.md" tag="WDZFV2ACXR" kind="concept">
# Quadratic polynomial dynamics

## Summary

Quadratic polynomial dynamics is the study of the iteration of the maps $f_c(z) = z^2 + c$ on the complex plane.
Up to an affine change of variable, every quadratic polynomial is one of these maps.
It is the simplest nontrivial case of [[complex dynamics|BQQZ0ZCGQF]] and the testing ground for most of its ideas.
Its parameter space is organised by the [[Mandelbrot set|QSWSVYRCS1]].

## Explanations

### L1 human

Take the rule $z \mapsto z^2 + c$ for a fixed complex number $c$ and follow the orbit of each starting point.
Some orbits run off to infinity and others stay bounded.
The bounded ones form the filled Julia set, and its border is the [[Julia set|5M4S7P04GR]].
For $c = 0$ the filled Julia set is the closed unit disc.
For $c = -1$ it is a shape made of infinitely many touching blobs, called the basilica.
The single orbit of $0$ already decides whether the Julia set is one piece or a cloud of dust.
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<document source="https://tamnd.github.io/math2/node/WE0RQTR68H.md" tag="WE0RQTR68H" kind="statement">
# Chen's theorem

## Statement

Call a positive integer a *$P_2$ number* if it is a [[prime|01826KMG2H]] or a product of two primes.
Equivalently, its [[number of prime factors counted with multiplicity|0KXQKES075]] is $1$ or $2$.

1. There is an integer $N_0$ such that every even integer $N > N_0$ can be written as $N = p + m$ with $p$ prime and $m$ a $P_2$ number.
2. There are infinitely many primes $p$ such that $p + 2$ is a $P_2$ number.

## Remarks

Both parts are due to Chen Jingrun.
The proof is a weighted sieve argument.
A lower bound sieve counts the $m = N - p$ with no small prime factor, and an upper bound sieve removes those with three or more prime factors.
The [[Bombieri–Vinogradov theorem|B3JMY54GJF]] controls the remainder terms on average over the moduli.
A switching argument, which counts the same configurations with the roles of the factors exchanged, is the key new step.
Example: $100 = 3 + 97 = 7 + 93$, where $97$ is prime and $93 = 3 \cdot 31$ is a $P_2$ number.
Part 1 is a weak form of [[Goldbach's conjecture|063JNBSMJ0]], and part 2 a weak form of the [[twin prime conjecture|0FER36N8SB]].

## Explanations

### L1 human

Goldbach's conjecture says that every even number larger than $2$ is a sum of two primes.
Chen proved the next best thing: every large even number is a prime plus a number that is either prime or the product of two primes.
In the same way, he proved that there are infinitely many primes $p$ for which $p + 2$ is "almost prime".

### L2 human

Sieve methods cannot by themselves separate numbers with an odd number of prime factors from those with an even number; this is the parity problem.
So a sieve can at best reach products of two primes, and Chen's theorem reaches exactly that limit.
Earlier sieve results of the same kind, starting with the [[Brun sieve|M4G4RXASTY]], allowed more prime factors on both sides.
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<document source="https://tamnd.github.io/math2/node/WE89NFZ84Q.md" tag="WE89NFZ84Q" kind="example">
# Van der Pol oscillator

## Example

Let $\mu > 0$.
The van der Pol oscillator is the equation
$$x'' - \mu (1 - x^2)\, x' + x = 0.$$
It is a [[Liénard equation|D2KC8NEAPS]] with $f(x) = -\mu (1 - x^2)$ and $g(x) = x$, so $F(x) = \mu (x^3/3 - x)$.
The function $F$ has the single positive zero $\sqrt{3}$, is negative on $(0, \sqrt{3})$, and is increasing on $(1, \infty)$.
By [[Liénard's theorem|178GG6TXKQ]], the equation has exactly one [[limit cycle|0AAQ73WRFB]], and it is stable.

## Remarks

The only equilibrium is the origin.
The [[linearisation|09ZWYRQ41J]] there is $x'' - \mu x' + x = 0$, whose characteristic roots $\frac{1}{2}\left(\mu \pm \sqrt{\mu^2 - 4}\right)$ have positive real part.
So the origin is unstable, and every other solution tends to the limit cycle.
For $\mu = 0$ the equation is the [[harmonic oscillator|0YG0ZJPTWG]].
For small $\mu > 0$, the limit cycle is close to the circle of radius $2$ in the $(x, x')$-plane.
For large $\mu$, the oscillation is a [[relaxation oscillation|2NHWCM0DK8]]: slow drifts alternate with fast jumps.

## Explanations

### L1 human

The van der Pol oscillator is an oscillator with negative friction for small swings and positive friction for large swings.
Small motions grow and large motions shrink, so all motions settle on one oscillation of a definite size.
It was first studied as a model of electrical circuits with vacuum tubes.
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<document source="https://tamnd.github.io/math2/node/WEBBT18D4X.md" tag="WEBBT18D4X" kind="concept">
# Sigma-finite measure

## Summary

A σ-finite measure is a measure for which the whole space is a countable union of sets of finite measure.
Lebesgue measure on $\R^n$ is the main example.
Many basic theorems, such as the Fubini–Tonelli theorem and the Radon–Nikodym theorem, assume σ-finiteness.

## Explanations

### L1 human

A σ-finite measure may give the whole space infinite size, but the space can be cut into countably many pieces of finite size.
The real line has infinite length, yet it is the union of the intervals $[-n, n]$, each of finite length.
Counting measure on the real line is not σ-finite: a set of finite counting measure is finite, and countably many finite sets cannot cover the line.
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<document source="https://tamnd.github.io/math2/node/WEDP73Y7GK.md" tag="WEDP73Y7GK" kind="concept">
# Generic filter

## Summary

A generic filter is a filter in a [[partial order|0QV2JS9RJ3]] that meets every dense subset belonging to a given model.
It is the new object that the method of forcing adds to the model.
Because it meets every dense set of the model, it has every property that the model can see as typical.
In nontrivial cases it cannot be an element of the model.
When the model is countable, generic filters exist by the Rasiowa–Sikorski lemma.

## Explanations

### L1 human

Think of the elements of the partial order as finite pieces of information about an unknown object.
A filter is a consistent collection of such pieces.
A dense set is a requirement that can always be met by adding more information.
A generic filter meets all the requirements that the old model can write down.
For example, in a generic sequence of zeros and ones, every finite pattern appears, because "the pattern appears" is a requirement that can always be met.
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<document source="https://tamnd.github.io/math2/node/WEGEXJBX26.md" tag="WEGEXJBX26" kind="definition">
# Fundamental domain

## Definition

Let $(X, \mathcal{A}, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $G$ be a [[countable|01TQB88H4C]] group that [[acts|0WCY42EV9V]] on $X$ by [[measure-preserving|556WBDAHNP]] measurable maps $x \mapsto gx$.
A measurable set $F \subseteq X$ is a *fundamental domain* for this action if:

1. the translates cover almost all of $X$: $\mu\big(X \setminus \bigcup_{g \in G} gF\big) = 0$;
2. distinct translates are almost disjoint: $\mu(gF \cap hF) = 0$ for all $g, h \in G$ with $g \ne h$.

## Remarks

Example: $G = \Z$ acting on $X = \R$ with Lebesgue measure by $n \cdot x = x + n$, and $F = [0, 1)$.
The translates $F + n = [n, n+1)$ cover $\R$ and are pairwise disjoint, so both clauses hold.

Non-example: for the same action, $F = [0, 2)$ covers $\R$ but fails clause 2, since $F \cap (F + 1) = [1, 2)$ has measure $1$.

Without a measure, a *strict fundamental domain* is a set that meets each [[orbit|0YQPZB5Q1G]] in exactly one point.
In geometry one often asks instead for a closed region $D$ whose translates cover $X$ and whose interiors are pairwise disjoint.
Any two measurable fundamental domains of the same action have the same measure.
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<document source="https://tamnd.github.io/math2/node/WEGT4EZCX2.md" tag="WEGT4EZCX2" kind="definition">
# Cobordism ring

## Definition

Let $n \ge 0$.
Let $M$ and $N$ be closed [[smooth manifolds|F45VQ47GPV]] of dimension $n$, that is, compact and without boundary.
They are *cobordant* if there is a compact smooth [[manifold with boundary|7DS6B2BFK8]] $W$ of dimension $n + 1$ whose boundary is diffeomorphic to $M \sqcup N$.
This is an equivalence relation; let $\mathfrak{N}_n$ be the set of its classes.
The *unoriented cobordism ring* is the [[graded ring|CAPQBQDRA7]]
$$\mathfrak{N}_* = \bigoplus_{n \ge 0} \mathfrak{N}_n,$$
with addition $[M] + [N] = [M \sqcup N]$, zero the class of the empty manifold, and multiplication $[M] \cdot [N] = [M \times N]$.

## Remarks

Every element has order $2$, since $M \sqcup M$ is the boundary of $M \times [0, 1]$; so $\mathfrak{N}_*$ is an algebra over $\F_2$.
Thom proved that $\mathfrak{N}_* \cong \F_2[x_i : i \ge 2,\ i \neq 2^j - 1]$ with $\deg x_i = i$, and that $\mathfrak{N}_n$ is isomorphic to the homotopy group $\pi_n(MO)$ of the Thom spectrum $MO$.

The *oriented cobordism ring* $\Omega^{SO}_*$ uses closed oriented manifolds and requires $\partial W \cong M \sqcup (-N)$ as oriented manifolds, where $-N$ is $N$ with the opposite orientation.
It is graded commutative, and $\Omega^{SO}_* \otimes \Q$ is a polynomial ring over $\Q$ on the classes of the complex projective spaces $\C P^{2k}$, $k \ge 1$.

Example: $\mathfrak{N}_1 = 0$, since every closed $1$-manifold is a finite union of circles and each circle bounds a disc.
Also $\mathfrak{N}_2 \cong \Z/2$, generated by $\R P^2$, which does not bound because its Euler characteristic is odd.

Non-example: $\R P^2$ is a closed surface that is not cobordant to the empty manifold.
Indeed, if $\R P^2 = \partial W$ for a compact $3$-manifold $W$, then the Euler characteristic of $\R P^2$ would be even, but it is $1$.

Further variants use manifolds with a stably complex structure, a spin structure or a framing, and each is the coefficient ring of a Thom spectrum.
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# Hypersingular integral equation

## Definition

Let $a, b \colon (-1, 1) \to \R$ and $f \colon (-1, 1) \to \R$ be given functions, and let $K$ be a [[kernel|GSMBXT7FPK]] on $(-1, 1)^2$ that is integrable in $t$ for each $x$.
A *hypersingular integral equation* on $(-1, 1)$ for an unknown $\varphi$ is
$$a(x)\, \varphi(x) + \frac{b(x)}{\pi}\, \mathrm{f.p.}\!\int_{-1}^{1} \frac{\varphi(t)}{(t - x)^2}\, dt + \int_{-1}^{1} K(x, t)\, \varphi(t)\, dt = f(x), \quad -1 < x < 1,$$
where $\mathrm{f.p.}$ denotes the [[Hadamard finite part|G7VE41JHHH]] and $b$ does not vanish identically.
The unknown $\varphi$ is sought among functions for which the finite part exists, for instance functions whose derivative is Hölder continuous on each compact subinterval.
If $a = 0$, the equation is of the *first kind*.

## Remarks

More generally, an integral equation on a domain or surface of dimension $m$ is called hypersingular when its kernel grows like $\abs{x - t}^{-m-1}$ or faster near the diagonal.

Example: the first kind equation $\frac{1}{\pi}\, \mathrm{f.p.}\!\int_{-1}^{1} \frac{\varphi(t)}{(t - x)^2}\, dt = -1$ has the solution $\varphi(t) = \sqrt{1 - t^2}$.
Check: the finite part is the $x$-derivative of $\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\sqrt{1 - t^2}}{t - x}\, dt = -x$, which is $-1$.
Here $\mathrm{PV}$ is the [[Cauchy principal value|0SYR10HD4R]].
Equations of this form, with the condition $\varphi(\pm 1) = 0$, describe the opening of a straight crack under uniform pressure.

Non-example: $\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\varphi(t)}{t - x}\, dt = f(x)$ has a Cauchy kernel of order $1$, so it is a singular but not a hypersingular equation.

The finite part is the derivative of a principal value.
So, when $\varphi(\pm 1) = 0$, an integration by parts turns a hypersingular equation of the first kind for $\varphi$ into a Cauchy singular equation for $\varphi'$.
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# Hartley transform

## Summary

The Hartley transform is an integral transform like the [[Fourier transform|DPVM34ADZV]], but with the real kernel $\cos(\omega t) + \sin(\omega t)$ in place of $e^{-i\omega t}$.
It sends real functions to real functions, and with the symmetric normalization it is its own inverse.
For a real function it carries the same information as the Fourier transform.
For a real function $f$, the even part of the Hartley transform of $f$ is the real part of the Fourier transform of $f$.
Its odd part is minus the imaginary part of the Fourier transform of $f$.
Its discrete version is used as a real-arithmetic alternative to the discrete Fourier transform.

## Explanations

### L1 human

The Fourier transform describes a real signal by complex numbers.
The Hartley transform packs the same information into real numbers, by adding the cosine and the sine parts instead of keeping them apart.
Applying it twice gives back the signal.
For example, the Gaussian $e^{-t^2/2}$ is even, so its sine part vanishes and its Hartley transform is again $e^{-\omega^2/2}$.
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<document source="https://tamnd.github.io/math2/node/WEMJBN339R.md" tag="WEMJBN339R" kind="concept">
# Axiom of dependent choice

## Summary

The axiom of dependent choice, DC, allows a sequence of countably many choices.
Each choice may depend on the choices made before.
It is weaker than the full [[axiom of choice|0HTGB13FE2]] and stronger than the axiom of countable choice.
It is the form of choice that most of real analysis needs, and the [[precise statement|TPXEFSBQYB]] uses a binary relation with no dead ends.

## Explanations

### L1 human

Imagine a game where from every position there is at least one legal move.
Dependent choice says you can then play forever: there is an infinite sequence of positions, each one a legal move from the one before.
For a finite number of moves this needs no axiom, but an infinite run needs infinitely many choices made one after another.
For example, let $R$ be a relation on the real numbers such that each $x$ is related to at least one $y > x$.
Then DC gives a sequence $x_0 < x_1 < x_2 < \cdots$ in which each term is related to the next.
When there is no explicit rule that picks $y$ from $x$, the axiom is what makes the sequence exist.
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<document source="https://tamnd.github.io/math2/node/WER1M071EK.md" tag="WER1M071EK" kind="definition">
# Natural boundary

## Definition

Let $U \subseteq \C$ be a [[domain|097KNJFJ7V]] and let $f$ be [[holomorphic|0DG0Z72CQD]] on $U$.
A point $\zeta$ of the [[boundary|0JJ9J1XZ8G]] $\partial U$ is a *regular point* of $f$ if the following holds.
There are an open disc $B$ centered at $\zeta$ and a holomorphic function $g$ on $B$ such that $g = f$ on some connected component of $B \cap U$.
That component must have $\zeta$ in its closure.
The boundary $\partial U$ is a *natural boundary* of $f$ if no point of $\partial U$ is a regular point of $f$.

## Remarks

The most common case is a power series with radius of convergence $R$, with $U$ its open disc of convergence.
On the circle of convergence there is always at least one point that is not regular.
Example: $f(z) = \sum_{n \ge 0} z^{2^n}$ on the unit disc has the unit circle as natural boundary.
If $\zeta^{2^k} = 1$, then $f(r\zeta) = \sum_{n < k} (r\zeta)^{2^n} + \sum_{n \ge k} r^{2^n} \to \infty$ as $r \to 1^-$.
Such $\zeta$ are dense in the circle, while $f$ would be bounded near a regular point.
Non-example: $\frac{1}{1 - z} = \sum_{n \ge 0} z^n$ does not have the unit circle as natural boundary.
Each point $\zeta \neq 1$ with $\abs{\zeta} = 1$ is regular, since $\frac{1}{1 - z}$ is holomorphic on a disc around $\zeta$.
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<document source="https://tamnd.github.io/math2/node/WF08BMN6Q6.md" tag="WF08BMN6Q6" kind="concept">
# Aitken's delta-squared process

## Summary

Aitken's delta-squared process is a [[series acceleration|B1S2XJPGYD]] method that replaces three consecutive terms of a sequence by one improved estimate of its limit.
It is exact for sequences whose error is exactly geometric.
It speeds up sequences whose error is close to geometric, such as those from [[fixed-point iteration|K6ZF91051B]].
It is the first case of the [[Shanks transformation|C0TXDZZMH7]].

## Explanations

### L1 human

If the distance to the limit shrinks by the same factor at each step, three consecutive terms are enough to find the factor and hence the limit.
Aitken's process does this calculation and returns the limit it predicts.
For $s_n = 1 + 2^{-n}$ it returns exactly $1$ from any three consecutive terms.
For sequences whose errors are only nearly geometric, the predictions are not exact but approach the limit much faster than the original terms.
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<document source="https://tamnd.github.io/math2/node/WF117EJ2HF.md" tag="WF117EJ2HF" kind="concept">
# Prime radical

## Summary

The prime radical of a ring is the intersection of all its [[prime ideals|E8K2MN5V1E]].
It is also called the lower nilradical or the Baer radical.
For a [[commutative ring|0310REJJY8]] it is the [[nilradical|QQDN6EYQNJ]].
It is the smallest of the classical radicals of a noncommutative ring, and it vanishes exactly for [[semiprime rings|5F0YMCV50R]].

## Explanations

### L1 human

In a commutative ring the nilpotent elements form an ideal, but in a noncommutative ring they need not.
The prime radical is a good replacement for this ideal.
It is a two-sided ideal made of nilpotent elements, cut out by the prime ideals.
For example, in the ring of upper triangular $2 \times 2$ matrices over a field, the prime radical is the set of strictly upper triangular matrices.
</document>

<document source="https://tamnd.github.io/math2/node/WF22KK7C3R.md" tag="WF22KK7C3R" kind="statement">
# Stable manifold theorem

## Statement

Let $U$ be an open subset of $\R^n$, let $k \ge 1$, and let $f \colon U \to \R^n$ be of class $C^k$.
Let $x_0 \in U$ be a [[hyperbolic equilibrium point|01TPMXM6NM]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$, with [[flow|02903SSA48]] $\varphi$.
Let $A = Df(x_0)$ be the [[Jacobian matrix|047NDTEPCV]].
Let $E^s$ be the sum of the generalised eigenspaces of $A$ for eigenvalues with negative real part.
Let $E^u$ be the sum of those for eigenvalues with positive real part.
Then there is an open neighbourhood $V$ of $x_0$ in $U$ such that:

1. The local stable manifold $W^s_{\mathrm{loc}} = \set{x \in V : \varphi_t(x) \in V \text{ for all } t \ge 0}$ is an embedded $C^k$ submanifold of $\R^n$ of dimension $\dim E^s$ whose tangent space at $x_0$ is $E^s$.
2. For each $x \in W^s_{\mathrm{loc}}$, $\varphi_t(x) \to x_0$ as $t \to +\infty$.
3. The local unstable manifold $W^u_{\mathrm{loc}} = \set{x \in V : \varphi_t(x) \in V \text{ for all } t \le 0}$ is an embedded $C^k$ submanifold of dimension $\dim E^u$ with tangent space $E^u$ at $x_0$.
4. For each $x \in W^u_{\mathrm{loc}}$, $\varphi_t(x) \to x_0$ as $t \to -\infty$.

## Remarks

One proof writes the solutions that stay near $x_0$ as fixed points of an integral equation built from the variation of constants formula.
Then it applies the contraction mapping principle; another proof uses the graph transform.
The global stable manifold $W^s(x_0) = \set{x \in U : \varphi_t(x) \to x_0 \text{ as } t \to \infty}$ is the union of the sets $\varphi_{-t}(W^s_{\mathrm{loc}})$ for $t \ge 0$, and it is an [[invariant set|WWZTW11664]].
Example: for $x' = -x$, $y' = y + x^2$, the origin is hyperbolic with $E^s$ the $x$-axis and $E^u$ the $y$-axis.
On the curve $y = -x^2/3$ one has $y' = (-2x/3)(-x) = 2x^2/3 = y + x^2$, so the curve is invariant; it is the stable manifold, tangent to $E^s$, while the unstable manifold is the $y$-axis.
Hyperbolicity is needed: for $x' = x^2$, $y' = -y$, the set of points that tend to the origin is the closed half-plane $x \le 0$, which is not a curve.

## Explanations

### L1 human

Near a saddle point, the points that flow into the saddle form a smooth surface, and the points that come out of it form another.
These surfaces are curved copies of the flat stable and unstable directions of the linearised system.

### L2 human

The theorem is the nonlinear version of the splitting of $\R^n$ into $E^s \oplus E^u$ for a linear system.
Together with the [[Hartman–Grobman theorem|0S8WX8A4V9]], it gives the local picture near a hyperbolic equilibrium.
Stable and unstable manifolds organise the global phase portrait: their intersections give homoclinic and heteroclinic orbits.
The same result holds for hyperbolic fixed points of diffeomorphisms and for hyperbolic periodic orbits.
</document>

<document source="https://tamnd.github.io/math2/node/WF7KQAQTS2.md" tag="WF7KQAQTS2" kind="statement">
# Reflection formula for the gamma function

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\C$.
For every $z \in \C \setminus \Z$,
$$\Gamma(z)\,\Gamma(1 - z) = \frac{\pi}{\sin(\pi z)},$$
where $\sin$ is the [[complex sine|0A7YDKRNR6]].

## Remarks

For $0 < \operatorname{Re} z < 1$, the product $\Gamma(z)\Gamma(1-z)$ equals the beta integral $\int_0^\infty \frac{t^{z-1}}{1+t}\,dt$.
This integral is computed by the residue theorem, or for real $z$ by a substitution, and equals $\pi / \sin(\pi z)$.
Both sides are meromorphic on $\C$, so the identity extends from the strip to all of $\C \setminus \Z$ by the identity theorem.
Another proof multiplies the Weierstrass products of $1/\Gamma(z)$ and $1/\Gamma(1-z)$ and compares with [[Euler's sine product|DQEHD56649]].
At $z = 1/2$ the formula gives $\Gamma(1/2)^2 = \pi$, so $\Gamma(1/2) = \sqrt{\pi}$, because $\Gamma(1/2) > 0$.
The integers are excluded: at each $n \in \Z$ one factor on the left has a [[pole|0WPQYW8MTE]] and $\sin(\pi n) = 0$.
In Mathlib it is `Complex.Gamma_mul_Gamma_one_sub`.

## Explanations

### L1 human

The gamma function at $z$ times the gamma function at $1 - z$ is a simple expression in the sine function.
The two points $z$ and $1 - z$ are symmetric about the point $1/2$, so the formula relates the values of $\Gamma$ on the two sides of $1/2$, hence the name.
The formula explains why $\Gamma(1/2) = \sqrt{\pi}$.

### L2 human

The right side $\pi/\sin(\pi z)$ is never $0$, so the formula shows that $\Gamma$ has no zeros off the integers.
Since also $\Gamma(n) = (n-1)! \ne 0$ for $n \ge 1$, the function $1/\Gamma$ is entire with zeros exactly at $0, -1, -2, \ldots$.
Together with the functional equation $\Gamma(z+1) = z\Gamma(z)$ and the duplication formula, it is one of the basic identities of the gamma function.
It is used, for example, in the functional equation of the Riemann zeta function, where values of $\Gamma$ at $s/2$ and $(1-s)/2$ appear.
</document>

<document source="https://tamnd.github.io/math2/node/WF95J6ZXV3.md" tag="WF95J6ZXV3" kind="definition">
# Rate function

## Definition

Let $\mathcal{X}$ be a [[topological space|0CQHFPGJ45]].
A *rate function* on $\mathcal{X}$ is a function $I \colon \mathcal{X} \to [0, \infty]$ that is [[lower semicontinuous|RN4G8SGP6P]].
Equivalently, for every $a \ge 0$, the sublevel set $\set{x \in \mathcal{X} : I(x) \le a}$ is [[closed|0ZCK8DQYN1]].
A rate function $I$ is *good* if every sublevel set $\set{x : I(x) \le a}$, for $a \ge 0$, is [[compact|0TVJZYB6SH]].

## Remarks

Example: $I(x) = x^2/2$ on $\R$ is a good rate function.
It is continuous, so lower semicontinuous, and $\set{I \le a} = [-\sqrt{2a}, \sqrt{2a}]$ is compact.

Example: on $\R$, let $I(x) = x \log x + (1 - x) \log(1 - x) + \log 2$ for $x \in [0, 1]$, with $0 \log 0 = 0$, and $I(x) = \infty$ otherwise.
It is a good rate function, the one for fair coin tossing in [[Cramér's theorem|RZZK51CXMN]].

Non-example: on $\R$, let $I(0) = 1$ and $I(x) = 0$ for $x \neq 0$.
Then $\set{I \le 0} = \R \setminus \set{0}$ is not closed, so $I$ is not lower semicontinuous.

Non-example of goodness: $I = 0$ on $\R$ is a rate function, but $\set{I \le 0} = \R$ is not compact.

In a large deviation principle, the rate function is unique when $\mathcal{X}$ is a regular topological space, for example a metric space.
A good rate function attains its infimum on every non-empty closed set.
</document>

<document source="https://tamnd.github.io/math2/node/WFFGB4RK60.md" tag="WFFGB4RK60" kind="concept">
# Jacobson topology

## Summary

The Jacobson topology, or hull-kernel topology, is a topology on the set $\operatorname{Prim}(R)$ of [[primitive ideals|M0JDQR0V5S]] of a ring $R$.
A set of primitive ideals is closed when it consists of all primitive ideals that contain some fixed ideal.
For a commutative ring it is the [[Zariski topology|8SQK7RTB63]] restricted to maximal ideals.
So the space $\operatorname{Prim}(R)$ is a noncommutative analogue of the [[spectrum of a ring|FB45BHHA2J]].
For a commutative C*-algebra $C_0(X)$ it recovers the space $X$.
For this reason it is the standard topology on the primitive spectrum of a C*-algebra.

## Explanations

### L1 human

For a commutative ring the maximal ideals behave like points, and the Zariski topology turns them into a geometric space.
In a noncommutative ring the right replacement for points is the primitive ideals, the annihilators of simple modules.
The Jacobson topology declares a set of these points closed when it is "cut out by an ideal": it is all the points that contain that ideal.
For the integers the points are the ideals $p\Z$, and the closed sets are the finite sets of primes together with the whole space.
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<document source="https://tamnd.github.io/math2/node/WFG4YMBYVQ.md" tag="WFG4YMBYVQ" kind="concept">
# Hilbert symbol

## Summary

The Hilbert symbol $(a, b)_K$ of two nonzero elements of a [[local field|J85FZEQG38]] $K$ of characteristic other than $2$ is $1$ or $-1$.
It is $1$ exactly when the equation $z^2 = a x^2 + b y^2$ has a solution other than $(0, 0, 0)$.
It is symmetric and bimultiplicative, and it is a basic tool for [[quadratic forms|0RBQS1JJCR]] over local fields.
Over $\Q$, the product of all local Hilbert symbols is $1$, a statement equivalent to [[quadratic reciprocity|085MJS6907]].

## Explanations

### L1 human

The Hilbert symbol answers a yes-or-no question: does a certain quadratic equation have a nonzero solution in the field?
Over the real numbers, $z^2 = a x^2 + b y^2$ has a nonzero solution unless both $a$ and $b$ are negative.
So $(a, b)_\R = -1$ exactly when $a < 0$ and $b < 0$.
Over $\Q_p$ the answer depends on [[Legendre symbols|0JTC045NNZ]] modulo $p$, and it can be computed by a short formula.
</document>

<document source="https://tamnd.github.io/math2/node/WFJ4TVF15P.md" tag="WFJ4TVF15P" kind="definition">
# Bateman transform

## Definition

Let $U \subseteq \R^4$ be open and let $C$ be a piecewise smooth closed curve in $\C$.
For $x = (x_1, x_2, x_3, x_4) \in U$ and $\zeta \in \C$ put
$$A(x, \zeta) = (x_1 + i x_2) + (x_4 + i x_3)\, \zeta, \qquad B(x, \zeta) = (x_4 - i x_3) - (x_1 - i x_2)\, \zeta.$$
Let $\Omega \subseteq \C^3$ be an open set that contains the points $(A(x, \zeta), B(x, \zeta), \zeta)$ for all $x \in U$ and $\zeta \in C$.
Let $f(w_1, w_2, \zeta)$ be a [[holomorphic function|0NAHXR1FRH]] on $\Omega$.
The *Bateman transform* of $f$ is the function $u \colon U \to \C$ given by
$$u(x) = \oint_C f\big(A(x, \zeta), B(x, \zeta), \zeta\big)\, d\zeta.$$

## Remarks

The function $u$ is [[harmonic|0DEB0JMN0W]] on $U$, that is, it solves [[Laplace's equation|S8RG6N1PRT]] $\sum_{k=1}^4 \partial_k^2 u = 0$.
Indeed, by the chain rule, $\sum_k \partial_k^2$ applied to $f(A, B, \zeta)$ is a combination of second derivatives of $f$ with coefficients $\sum_k (\partial_k A)^2$, $\sum_k \partial_k A\, \partial_k B$ and $\sum_k (\partial_k B)^2$.
The gradients are $\nabla A = (1, i, i\zeta, \zeta)$ and $\nabla B = (-\zeta, i\zeta, -i, 1)$, and all three sums vanish: $1 - 1 - \zeta^2 + \zeta^2 = 0$, $-\zeta - \zeta + \zeta + \zeta = 0$ and $\zeta^2 - \zeta^2 - 1 + 1 = 0$.
Differentiation under the integral sign then gives $\Delta u = 0$.
When $\Omega$ also contains the points $(A(x, \zeta), B(x, \zeta), \zeta)$ with $x_4$ replaced by $i t$, the same integral defines $u(x_1, x_2, x_3, i t)$.
Replacing $x_4$ by $i t$ turns $\partial_{x_4}^2$ into $-\partial_t^2$, so $u(x_1, x_2, x_3, it)$ solves the [[wave equation|5EHD2XBQDG]] $\partial_t^2 u = \partial_1^2 u + \partial_2^2 u + \partial_3^2 u$.

Example: $f(w_1, w_2, \zeta) = \frac{w_1}{\zeta^2}$ and $C$ the unit circle.
Then $u(x) = \oint_C \Big( \frac{x_1 + i x_2}{\zeta^2} + \frac{x_4 + i x_3}{\zeta} \Big) d\zeta = 2\pi i\, (x_4 + i x_3)$, a linear, hence harmonic, function.

Non-example: if $f(w_1, w_2, \zeta) = \overline{w_1}$, which is not holomorphic, the function $\oint_C \overline{A(x, \zeta)}\, d\zeta$ is not covered by the definition, and the chain rule computation above fails.
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<document source="https://tamnd.github.io/math2/node/WFPYZ0XDRN.md" tag="WFPYZ0XDRN" kind="definition">
# Anderson localization

## Definition

Let $d \ge 1$ and let $H$ be a bounded [[self-adjoint operator|0NEGN5CRSC]] on $\ell^2(\Z^d)$.
Let $I \subseteq \R$ be an interval.
The operator $H$ exhibits *Anderson localization* in $I$ if:

1. the [[spectrum|BMKATJ802A]] of $H$ in $I$ is pure point: the spectral subspace of $H$ for $I$ has an orthonormal basis of eigenvectors;
2. every eigenvector $\psi$ of $H$ with [[eigenvalue|0BQAA2BK3Q]] in $I$ decays exponentially: there are $C_\psi > 0$ and $\mu_\psi > 0$ with $\abs{\psi(n)} \le C_\psi e^{-\mu_\psi\abs{n}}$ for all $n \in \Z^d$.

The main case is the *Anderson model* $H_\omega = -\Delta + \lambda V_\omega$, where $(\Delta u)(n) = \sum_{\abs{m - n} = 1}(u(m) - u(n))$, $\lambda > 0$, and $V_\omega(n)$, $n \in \Z^d$, are [[independent identically distributed random variables|00MRN0FARV]].
One says the model exhibits Anderson localization in $I$ if $H_\omega$ does so for almost every $\omega$.

## Remarks

Example: a diagonal operator $(Hu)(n) = v(n)u(n)$ with a bounded real sequence $v$ exhibits localization in every interval.
Its eigenvectors $\delta_n$ have one nonzero entry, so they decay faster than any exponential.

Non-example: the free operator $-\Delta$ on $\ell^2(\Z)$ has spectrum $[0, 4]$ and no eigenvalues at all, since its Fourier transform is multiplication by $2 - 2\cos k$.
So clause 1 fails in every interval that meets $[0, 4]$.

For the Anderson model with a bounded density of the $V_\omega(n)$, localization holds in $d = 1$ for the whole spectrum, and in every dimension for large $\lambda$.
The strongest form, dynamical localization, also bounds the spreading of $e^{-itH}\psi$ uniformly in time.
</document>

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# Mehler–Fock transform

## Summary

The Mehler–Fock transform is an [[integral transform|YCK709QB06]] on the half-line $[1, \infty)$ whose kernel is a conical [[Legendre function|69XMERM6KJ]] $P_{-1/2 + i\tau}(x)$.
It is an index transform: the transform variable $\tau$ sits in the degree of the Legendre function, not in its argument.
It plays for the Legendre operator the role that the Fourier transform plays for $d^2/dx^2$.
It is used to solve boundary value problems for cones and in harmonic analysis on the hyperbolic plane.

## Explanations

### L1 human

The functions $P_{-1/2 + i\tau}$, one for each $\tau \ge 0$, are the basic waves of the Legendre differential operator on $[1, \infty)$.
The transform measures how much of each wave a function contains, as a Fourier transform measures frequencies.
The inverse transform adds the waves back with the right weights.
For example, the transform of the function equal to $1$ on $[1, 2]$ and $0$ elsewhere is $\int_1^2 P_{-1/2 + i\tau}(x)\, dx$, a smooth function of $\tau$.
</document>

<document source="https://tamnd.github.io/math2/node/WFREP1QRPQ.md" tag="WFREP1QRPQ" kind="definition">
# Riemann surface

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]].
A *complex chart* on $X$ is a [[homeomorphism|05GZ11PWRY]] $\varphi \colon U \to V$ from an open set $U \subseteq X$ onto an open set $V \subseteq \C$.
Two charts $\varphi_1 \colon U_1 \to V_1$ and $\varphi_2 \colon U_2 \to V_2$ are *compatible* if the transition map
$$\varphi_2 \circ \varphi_1^{-1} \colon \varphi_1(U_1 \cap U_2) \to \varphi_2(U_1 \cap U_2)$$
is [[holomorphic|0DG0Z72CQD]], and likewise $\varphi_1 \circ \varphi_2^{-1}$.
A *complex atlas* on $X$ is a family of pairwise compatible charts whose domains form an [[open cover|0NVPNC12JY]] of $X$.
Two atlases are equivalent if their union is again an atlas.
A *complex structure* on $X$ is an equivalence class of complex atlases.
A *Riemann surface* is a topological space $X$ with a complex structure such that:

1. $X$ is [[Hausdorff|0MXB4CKHFW]], and
2. $X$ is [[connected|0D939CMME7]].

## Remarks

Some authors also require $X$ to be second countable; by Radó's theorem this holds automatically for every Riemann surface.
So a Riemann surface is the same as a connected [[complex manifold|JSGWAN2RJV]] of dimension one.

Example: the [[Riemann sphere|0CDMXGPSES]] $\hat\C = \C \cup \set{\infty}$ with the two charts $\varphi_1(z) = z$ on $\C$ and $\varphi_2(z) = 1/z$ on $\hat\C \setminus \set{0}$, where $\varphi_2(\infty) = 0$.
On $\varphi_1(\C \setminus \set{0}) = \C \setminus \set{0}$ the transition map is $z \mapsto 1/z$, which is holomorphic there; its inverse is the same map.

Non-example: the closed unit disc $\set{z : \abs{z} \le 1}$ is not a Riemann surface for any atlas.
A boundary point has no open neighbourhood homeomorphic to an open subset of $\C$, so the domains of charts cannot cover $X$.
</document>

<document source="https://tamnd.github.io/math2/node/WFRGSBFZGV.md" tag="WFRGSBFZGV" kind="concept">
# Periodic tiling

## Summary

A periodic tiling is a [[tessellation|GKJK293DXS]] of $\R^n$ that is carried onto itself by [[translations|087VJM5Q77]] in $n$ independent directions.
Such a tiling is determined by a finite patch repeated along a lattice.
The symmetries of periodic patterns are described by [[crystallographic groups|7KTJKFQT2N]].
Tilings that admit no period lead to the theory of [[aperiodic sets of prototiles|Z3FCJZF6WK]].

## Explanations

### L1 human

A periodic tiling looks like wallpaper: one block of the pattern is copied over and over in two directions.
If you slide the whole tiling by the right amount to the right, or by the right amount upward, every tile lands exactly on another tile.
The square grid is periodic, since a shift by one unit to the right or one unit up maps it onto itself.
A tiling with only a single direction of shifts, such as a strip pattern, is not periodic in this sense.
</document>

<document source="https://tamnd.github.io/math2/node/WFS3PHFHA1.md" tag="WFS3PHFHA1" kind="statement">
# Necklace splitting theorem

## Statement

Let $k \ge 2$ and $t \ge 1$ be integers.
Let a necklace be a row of beads of $t$ colours.
Suppose colour $i$ occurs exactly $k a_i$ times, for some integer $a_i \ge 0$.
Then the necklace can be cut in at most $(k - 1) t$ places so that the pieces can be divided among $k$ people fairly.
Fairly means that each person receives exactly $a_i$ beads of colour $i$, for every $i$.

## Remarks

For $k = 2$ the result follows from the [[Borsuk–Ulam theorem|QHJ0GBQE2E]].
Model the necklace as the interval $[0, 1]$, with each bead a short subinterval.
A point $x$ of the sphere $S^t \subset \R^{t+1}$ encodes a cut into $t+1$ consecutive pieces of lengths $x_1^2, \dots, x_{t+1}^2$.
The sign of $x_j$ tells which of the two people gets piece $j$.
For each colour, take the amount the first person gets minus half the total; this defines an odd map $S^t \to \R^t$.
By Borsuk–Ulam it has a zero, which is a fair split with at most $t$ cuts.
A final step moves the cuts to bead boundaries.
The general case is due to Noga Alon, by induction on the prime factors of $k$ using a generalisation of Borsuk–Ulam.

The bound is sharp: if the beads of each colour lie in one contiguous block, each block needs $k - 1$ cuts.

## Explanations

### L1 human

Thieves steal a necklace with several kinds of jewels and want to share each kind equally, using as few cuts as possible.
With two thieves and $t$ kinds of jewels, $t$ cuts always suffice, wherever the jewels sit.
The surprising proof uses a theorem from topology about maps on spheres.

### L2 human

This is a standard showcase of topological methods in combinatorics, next to Lovász's proof of the Kneser conjecture.
The continuous version, where colours are replaced by measures on an interval, is the form to which Borsuk–Ulam applies directly, much as in the [[ham sandwich theorem|GDXX60SNG5]].
The bound $(k - 1) t$ cannot be lowered in general, as the example of contiguous colour blocks shows.
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<document source="https://tamnd.github.io/math2/node/WFVC137TYB.md" tag="WFVC137TYB" kind="concept">
# Infinitary operation

## Summary

An infinitary operation on a set $A$ takes an infinite family of elements of $A$ and returns an element of $A$.
Its [[arity|MSXX2P86CY]] is an infinite cardinal or ordinal, in contrast to a [[finitary operation|EJ8YJVM9BW]].
The supremum of an arbitrary subset of a complete lattice and countable unions in a $\sigma$-algebra of sets are typical infinitary operations.
Algebras with infinitary operations need care, because some results of finitary [[universal algebra|W82R6N0GM9]] fail for them.

## Explanations

### L1 human

Most operations take two or three inputs, but some take infinitely many.
In the interval $[0, 1]$, every sequence $x_0, x_1, x_2, \dots$ has a least upper bound, which is again in $[0, 1]$.
So "take the supremum of a sequence" is an operation with countably many inputs.
In contrast, "add up a sequence" is not an operation on the real numbers, because many series do not converge.
</document>

<document source="https://tamnd.github.io/math2/node/WG4GVC9HR8.md" tag="WG4GVC9HR8" kind="statement">
# Unique solvability of Volterra equations of the second kind

## Statement

Let $a < b$, let $\Delta = \set{(t, s) : a \le s \le t \le b}$, let $K \colon \Delta \to \C$ be continuous, let $f \colon [a, b] \to \C$ be continuous, and let $\lambda \in \C$.
Then the [[Volterra equation of the second kind|ATPSAHYG7C]]
$$x(t) = f(t) + \lambda \int_a^t K(t, s) \, x(s) \, ds, \quad t \in [a, b],$$
has exactly one continuous solution $x \colon [a, b] \to \C$.
It is $x(t) = f(t) + \lambda \int_a^t R(t, s; \lambda) \, f(s) \, ds$, where $R$ is the [[resolvent kernel|ARTJV1XCRX]] of $K$.

## Remarks

Existence is given by the [[convergence of the Neumann series for Volterra kernels|3QDDSSH4FB]].
For uniqueness, let $y$ be the difference of two continuous solutions; then $y(t) = \lambda \int_a^t K(t, s) \, y(s) \, ds$.
With $M = \max \abs{K}$ and $C = \max \abs{y}$, induction gives $\abs{y(t)} \le C \frac{(\abs{\lambda} M (t - a))^n}{n!}$ for every $n$, so $y = 0$.
In particular, the homogeneous equation has only the zero solution, so a continuous Volterra kernel has no [[characteristic values|DC7HMHVREP]].
Example: $x(t) = \cos t - \int_0^t x(s) \, ds$ has the unique solution $x(t) = \tfrac12 (\cos t - \sin t + e^{-t})$; it solves $x' = -\sin t - x$ with $x(0) = 1$.
The continuity of $K$ can be weakened to boundedness with measurability, or to a weak singularity such as $\abs{K(t, s)} \le C (t - s)^{-\alpha}$ with $0 \le \alpha < 1$, with the same conclusion.
For the equation of the first kind the statement fails: $\int_0^t x(s) \, ds = 1$ has no continuous solution, because the left side vanishes at $t = 0$.

## Explanations

### L1 human

A linear Volterra equation of the second kind with a continuous kernel can always be solved, and in only one way.
This holds for every value of the parameter $\lambda$, with no exceptions.
The reason is that the solution is built forward in time, each value being determined by the earlier ones.

### L2 human

For a Fredholm equation of the second kind, the [[Fredholm alternative|4HH8C9V8JQ]] says that solvability fails exactly at the characteristic values of the kernel.
A Volterra kernel has none, because the Volterra integral operator is quasinilpotent: its spectrum is $\set{0}$.
The theorem covers the integral form of every linear initial value problem $y^{(n)} + a_{n-1}(t) y^{(n-1)} + \cdots + a_0(t) y = g(t)$ with continuous coefficients, and thus gives global existence and uniqueness for those.
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<document source="https://tamnd.github.io/math2/node/WG6BY4SWTV.md" tag="WG6BY4SWTV" kind="definition">
# Positive cone

## Definition

Let $V$ be a real [[vector space|05EGHEQZAW]].
A *positive cone* in $V$ is a subset $P \subseteq V$ such that:

1. $x + y \in P$ for all $x, y \in P$;
2. $\lambda x \in P$ for all $x \in P$ and all real $\lambda \ge 0$;
3. $P \cap (-P) = \set{0}$.

## Remarks

A positive cone $P$ defines a [[partial order|011C29Q1S1]] on $V$ by $x \le y \iff y - x \in P$.
This order satisfies $x \le y \implies x + z \le y + z$ and $x \le y \implies \lambda x \le \lambda y$ for $\lambda \ge 0$, and $P = \set{x : 0 \le x}$.
Conversely, for each partial order with these two properties, the set $\set{x : 0 \le x}$ is a positive cone.
Example: $P = \set{x \in \R^n : x_i \ge 0 \text{ for all } i}$ is a positive cone, and it gives the componentwise order.
Example: the [[positive semidefinite matrices|0880F9EQCP]] form a positive cone in the space of real symmetric $n \times n$ matrices.
Non-example: the half-plane $\set{(x, y) \in \R^2 : y \ge 0}$ satisfies clauses 1 and 2, but contains $(1, 0)$ and $(-1, 0)$, so it fails clause 3.
Some authors drop clause 3 and call $P$ a *wedge* or a convex cone; then the relation $\le$ is only a preorder.
</document>

<document source="https://tamnd.github.io/math2/node/WGBR7JHBV4.md" tag="WGBR7JHBV4" kind="statement">
# Post's functional completeness theorem

## Statement

Let $F$ be a set of [[Boolean functions|0X05GS525E]], that is, of operations on $\set{0, 1}$ of arity at least $1$.
Then $F$ is [[functionally complete|TDZVCWYJ9F]] if and only if $F$ has the following five properties.

1. Some $f \in F$ does not preserve $0$, that is, $f(0, \dots, 0) = 1$.
2. Some $f \in F$ does not preserve $1$, that is, $f(1, \dots, 1) = 0$.
3. Some $f \in F$ is not [[monotone|0A7KV2WB7A]] for the order $0 \le 1$ applied coordinatewise.
4. Some $f \in F$ is not self-dual, where $f$ is self-dual if $f(\neg x_1, \dots, \neg x_n) = \neg f(x_1, \dots, x_n)$ for all inputs.
5. Some $f \in F$ is not affine, where $f$ is affine if $f(x_1, \dots, x_n) = c_0 + c_1 x_1 + \cdots + c_n x_n$ modulo $2$ for some $c_i \in \set{0, 1}$.

## Remarks

Each of the five classes is a clone and does not contain all Boolean functions, so a complete $F$ cannot lie inside one of them.
The converse is the real content: these five clones are the maximal clones of [[Post's lattice|MA20X7JCET]].
One can also prove it directly by building $\neg$ and the constants from the functions that escape the first four classes, and then $\wedge$ from a non-affine function.
Example: the NAND operation $x \uparrow y = \neg(x \wedge y)$ maps $(0, 0)$ to $1$ and $(1, 1)$ to $0$.
It is not monotone, since $0 \uparrow 0 = 1 > 0 = 1 \uparrow 1$, it is not self-dual, since $0 \uparrow 1 = 1 = 1 \uparrow 0$, and it is not affine.
So $\set{\uparrow}$ is functionally complete.

## Explanations

### L1 human

The theorem gives five simple tests for a set of logic gates.
The gates can build every truth table exactly when, for each test, at least one gate fails it.
The tests ask whether gates keep all-zeros at zero, keep all-ones at one, respect the order $0 < 1$, commute with negation, or are sums modulo $2$.

### L2 human

The result comes from Emil Post's study of the lattice of clones on $\set{0, 1}$.
It is the two-element case of a general picture: a set of operations on a finite set is complete exactly when it is contained in no maximal clone.
For larger finite sets the maximal clones are listed by [[Rosenberg's classification|GF1RK91VED]], and the analogue of the theorem has more cases.
It explains at once why $\set{\wedge, \vee}$ and $\set{\wedge, \vee, \to}$ are not complete, while $\set{\to, 0}$ is, with $0$ read as a constant unary function.
</document>

<document source="https://tamnd.github.io/math2/node/WGDAZ1YY3J.md" tag="WGDAZ1YY3J" kind="definition">
# Dual Radon transform

## Definition

Let $n \ge 2$, let $S^{n-1}$ be the unit [[sphere|P9CM4GQ2T0]] in $\R^n$, and let $d\theta$ be its surface measure.
Let $g \colon S^{n-1} \times \R \to \C$ be continuous.
The *dual Radon transform* of $g$ is the function on $\R^n$ given by
$$(R^* g)(x) = \int_{S^{n-1}} g(\theta, x \cdot \theta)\, d\theta.$$

## Remarks

The pair $(\theta, x \cdot \theta)$ describes the hyperplane with normal $\theta$ through $x$, so $R^* g(x)$ integrates $g$ over all hyperplanes through $x$.
For $f$ continuous with compact support and $g$ continuous, Fubini's theorem gives the duality
$$\int_{S^{n-1}} \int_\R (R f)(\theta, s)\, g(\theta, s)\, ds\, d\theta = \int_{\R^n} f(x)\, (R^* g)(x)\, dx,$$
where $R$ is the [[Radon transform|VJNZJ352PC]].

Example: $g(\theta, s) = s^2$.
Then $R^* g(x) = \int_{S^{n-1}} (x \cdot \theta)^2\, d\theta = \frac{\omega_{n-1}}{n} \abs{x}^2$, where $\omega_{n-1}$ is the area of $S^{n-1}$, because the $n$ integrals $\int \theta_j^2\, d\theta$ are equal and add up to $\omega_{n-1}$.

Non-example: $R^*$ is not an inverse of $R$.
For $n = 2$ and $f$ continuous with compact support, polar coordinates give $R^* R f(x) = 2 \int_{\R^2} \frac{f(x + z)}{\abs{z}}\, dz$, a smoothed version of $f$, not $f$ itself.

Other notations are $R^\#$ and $R^t$; some authors divide by the area of the sphere.
</document>

<document source="https://tamnd.github.io/math2/node/WGGZEKF224.md" tag="WGGZEKF224" kind="problem">
# Lüroth problem

## Problem

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $X$ be an integral [[algebraic variety|HV6YBS0YDH]] over $k$.
Suppose that $X$ is [[unirational|YF1SE8WJK3]].
Must $X$ be [[rational|XV4K9GREJ9]]?

## Remarks

In dimension $1$ the answer is yes, by [[Lüroth's theorem|AVCZ6HVBA7]].
In dimension $2$ over an algebraically closed field of characteristic $0$, the answer is yes, by [[Castelnuovo's rationality criterion|QXHMMQX1D7]].
In dimension $2$ over an algebraically closed field of characteristic $p > 0$, the answer is no: Zariski found unirational surfaces that are not rational.
In dimension $3$ over $\C$, the answer is no.
Clemens and Griffiths showed that a smooth cubic threefold in $\mathbb{P}^4$ is unirational but not rational.
Iskovskikh and Manin showed that a smooth quartic threefold in $\mathbb{P}^4$ is not rational, and some of these are unirational.
Artin and Mumford gave another unirational threefold that is not rational, detected by torsion in its third cohomology group.
So the problem is settled in the negative in general, and the open questions now concern specific families of varieties.

## Explanations

### L1 human

A variety is rational when it can be parametrised one-to-one, up to small exceptions, by free parameters.
It is unirational when it can be covered by such a parametrisation, perhaps many-to-one.
The problem asks whether a many-to-one parametrisation can always be improved to a one-to-one parametrisation.
For curves and for complex surfaces it can.
For complex threefolds it cannot, and the cubic threefold is the standard example.
</document>

<document source="https://tamnd.github.io/math2/node/WGN82TCEF2.md" tag="WGN82TCEF2" kind="concept">
# p-adic absolute value

## Summary

For a prime $p$, the $p$-adic absolute value of a nonzero rational number $x$ is $\abs{x}_p = p^{-v_p(x)}$, where $v_p$ is the [[p-adic valuation|03HDF50VN4]].
It is a [[non-Archimedean absolute value|1H5F1BP6XS]] on $\Q$.
Completing $\Q$ for it gives the field of [[p-adic numbers|BT6T702C28]].
Together with the usual [[absolute value|0FFXKTKKQC]], these are all the nontrivial absolute values on $\Q$ up to equivalence.

## Explanations

### L1 human

The $p$-adic absolute value measures how divisible a number is by $p$.
The more factors of $p$ a number has, the smaller its $p$-adic size.
For example, $\abs{12}_2 = 1/4$ because $12 = 2^2 \cdot 3$, and $\abs{3/8}_2 = 8$ because $8 = 2^3$ is in the denominator.
Numbers not divisible by $p$, in numerator or denominator, have size $1$.
</document>

<document source="https://tamnd.github.io/math2/node/WGY75H4VWW.md" tag="WGY75H4VWW" kind="statement">
# Titchmarsh convolution theorem

## Statement

Let $T > 0$, and let $\varphi, \psi \colon [0, T] \to \C$ be Lebesgue integrable.
Suppose that the [[convolution|FCKX8KFVGN]]
$$(\varphi * \psi)(x) = \int_0^x \varphi(t) \, \psi(x - t) \, dt$$
is $0$ for almost every $x \in (0, T)$.
Then there are $\lambda \ge 0$ and $\mu \ge 0$ with $\lambda + \mu \ge T$ such that $\varphi = 0$ almost everywhere on $(0, \lambda)$ and $\psi = 0$ almost everywhere on $(0, \mu)$.

## Remarks

Equivalently, write $\lambda(\varphi)$ for the largest $\lambda$ such that $\varphi$ vanishes almost everywhere on $(0, \lambda)$.
Then for integrable $\varphi, \psi$ on $[0, T]$ one has $\lambda(\varphi * \psi) = \lambda(\varphi) + \lambda(\psi)$ whenever the right side is less than $T$.
A standard form concerns functions or distributions with compact support on $\R$.
It says that the convex hull of the support of $\varphi * \psi$ is the sum of the convex hulls of the supports of $\varphi$ and $\psi$.
The classical proof uses complex analysis: the Laplace or Fourier transforms of such functions are entire functions of exponential type, and the theorem reduces to growth estimates for them.
Real-variable proofs are also known.

Example: with $T = 2$ and $\varphi = \psi = \mathbf{1}_{(1, 2)}$, the convolution vanishes on $(0, 2)$, and the conclusion holds with $\lambda = \mu = 1$, so $\lambda + \mu = T$ exactly.

The easy direction is the converse: if $\varphi = 0$ on $(0, \lambda)$ and $\psi = 0$ on $(0, \mu)$, then each product $\varphi(t) \psi(x - t)$ with $x < \lambda + \mu$ vanishes, so $\varphi * \psi = 0$ on $(0, \lambda + \mu)$.

## Explanations

### L1 human

Convolving two signals that start at time $0$ can only produce silence for as long as the first signal waits plus the second signal waits.
There is no way for the two signals to cancel each other out early.
So if the convolution is zero for a long time, at least one of the signals must have been silent for a correspondingly long time.

### L2 human

Letting $T \to \infty$ gives the corollary that the [[convolution ring of continuous functions on the half-line has no zero divisors|E9QNNPQD2P]].
This is the algebraic foundation of the [[Mikusiński operational calculus|TD5XK5AHNV]], which builds the field of fractions of that ring.
The support form of the theorem is used in the theory of distributions and in the uniqueness theory of partial differential equations.
</document>

<document source="https://tamnd.github.io/math2/node/WGYXR97NPK.md" tag="WGYXR97NPK" kind="concept">
# Projective cover

## Summary

A projective cover of a [[module|0NZ8JA25WK]] $M$ is a surjection from a [[projective module|BFFA03GE3V]] onto $M$ that is as small as possible.
This means that its kernel is a superfluous submodule.
When it exists it is unique up to isomorphism.
It is the dual notion of the [[injective hull|QCJN05GN64]], which always exists, whereas projective covers exist only over special rings.
The rings over which every finitely generated module has a projective cover are the [[semiperfect rings|ZRC2S3ECFK]].
Those over which every left module has one are the left [[perfect rings|49Y9Q949PK]].

## Explanations

### L1 human

Every module is a quotient of a projective module, for instance of a free module, but usually in many ways and with a lot of waste.
A projective cover is a most economical way to do it: no proper piece of the projective module already maps onto $M$.
For example, over the ring $\Z/4$, the map $\Z/4 \to \Z/2$ is a projective cover, since the kernel $2\Z/4\Z$ is too small to help any proper submodule fill everything.
Over the integers, the module $\Z/2$ has no projective cover.
</document>

<document source="https://tamnd.github.io/math2/node/WH10W2GWZX.md" tag="WH10W2GWZX" kind="definition">
# Order polynomial

## Definition

Let $(P, \le)$ be a finite [[partially ordered set|0QV2JS9RJ3]] with $p$ elements.
For an integer $m \ge 1$, let $[m] = \set{1, \ldots, m}$ with the usual order.
Let $\Omega(P, m)$ be the number of [[order-preserving|0A7KV2WB7A]] maps $f \colon P \to [m]$, that is, maps with $f(x) \le f(y)$ whenever $x \le y$.
The *order polynomial* of $P$ is the unique polynomial $\Omega(P, t)$ in $t$ that takes the value $\Omega(P, m)$ at every integer $m \ge 1$.

## Remarks

Such a polynomial exists, and it has degree $p$ with leading coefficient $e(P)/p!$, where $e(P)$ is the number of [[linear extensions|W8DZ5KGGHN]] of $P$.

Example: for the chain $x_1 < x_2 < \cdots < x_p$, $\Omega(P, m) = \binom{m + p - 1}{p}$.
Check: an order-preserving map is a weakly increasing sequence $f(x_1) \le \cdots \le f(x_p)$ in $[m]$, that is, a multiset of size $p$ from $m$ values; the leading coefficient is $1/p!$, and $e(P) = 1$.

Non-example: the count $m^p$ of all maps $P \to [m]$ is not the order polynomial of the chain when $p \ge 2$.
It also counts maps that break the order, such as $f(x_1) = 2$, $f(x_2) = 1$.

The strict order polynomial $\Omega^\circ(P, m)$ counts the maps with $x < y \Rightarrow f(x) < f(y)$.
Stanley's reciprocity theorem says $\Omega^\circ(P, t) = (-1)^p \, \Omega(P, -t)$.
</document>

<document source="https://tamnd.github.io/math2/node/WH1MR8VHE3.md" tag="WH1MR8VHE3" kind="definition">
# Inner model

## Definition

Work in [[ZF|0HE5WDQ9FJ]], and let $M$ be a [[class|Y48FA69563]].
The class $M$ is an *inner model* if:

1. $M$ is transitive: if $x \in M$ and $y \in x$, then $y \in M$.
2. $M$ contains every [[ordinal|0YSCSX5YZC]].
3. $(M, \in)$ satisfies each axiom of ZF, that is, the relativization $\varphi^M$ holds for each axiom $\varphi$ of ZF.

## Remarks

The universe $V$ and the constructible universe $L$ are inner models.
For $L$, clause 1 holds because each level $L_\alpha$ is transitive, and clause 2 holds because $\alpha \in L_{\alpha + 1}$.
The set $L_{\omega_1}$ is a non-example: it is transitive, but it fails clause 2, since $\omega_1 \notin L_{\omega_1}$.
Clause 3 is a schema, one statement for each axiom, because satisfaction in a proper class is not expressed by a single formula.
A transitive proper class that satisfies ZF contains all ordinals, so some authors drop clause 2 and ask instead that $M$ be a proper class.
Some authors ask for the axioms of ZFC; the name "inner model of ZFC" makes this choice explicit.
</document>

<document source="https://tamnd.github.io/math2/node/WH3Z5M5D2R.md" tag="WH3Z5M5D2R" kind="definition">
# Completeness in order theory

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]] and let $\mathcal{C}$ be a class of subsets of $P$.
The poset $P$ is *complete for* $\mathcal{C}$, or has all $\mathcal{C}$-joins, if every $S \in \mathcal{C}$ has a [[supremum|0R3H5PANRQ]] in $P$.
Dually, $P$ has all $\mathcal{C}$-meets if every $S \in \mathcal{C}$ has an [[infimum|0QF1MZ9FRG]] in $P$.
The standard choices of $\mathcal{C}$ are:

1. all subsets, giving a [[complete lattice|D7A0ZGZT5S]];
2. all finite nonempty subsets, giving a join-semilattice, and with meets as well, a lattice;
3. all nonempty subsets with an upper bound, giving (with the dual condition on lattices) a [[conditionally complete lattice|CZPBQY67ZJ]];
4. all [[directed subsets|D9RY4667K6]], giving a [[directed-complete partial order|7TSXP95QN6]];
5. all chains, giving a chain-complete poset.

## Remarks

Example: the power set of a set is complete for every class $\mathcal{C}$, since unions always exist.
Non-example: $\N$ with its usual order has all finite nonempty joins but is not directed-complete, because $\N$ itself is directed and has no supremum.
The empty set has a supremum if and only if $P$ has a least element.
If $P$ has all joins of arbitrary subsets, it also has all meets, so in case 1 one condition suffices.
</document>

<document source="https://tamnd.github.io/math2/node/WH48X83YPJ.md" tag="WH48X83YPJ" kind="concept">
# Internal language

## Summary

The internal language of a category with enough structure is a formal language read inside the category.
Its types are the objects of the category, and its terms are interpreted as morphisms.
For a [[topos|6837RS7N82]] it is the Mitchell–Bénabou language, a higher-order [[intuitionistic logic|Q11N8X99FK]]; for a [[Cartesian closed category|BFBNQ36QS8]] it is the simply typed [[lambda calculus|BD756KR8FM]].
It lets one reason about objects of the category as if they were sets, with elements, functions and subsets.
The reasoning then stays valid in the category.
It is one of the main tools of [[categorical logic|NES4B9CWW4]].

## Explanations

### L1 human

Working with diagrams of arrows can be heavy.
The internal language lets one write ordinary-looking statements such as "for every $x$ in $A$ there is $y$ in $B$ with $f(y) = x$" and read them inside a category.
Each such statement is translated into a statement about arrows, such as "$f$ is an epimorphism".
For example, in the topos of sheaves on a space, a statement holds when it holds locally, and the law of excluded middle may fail.
</document>

<document source="https://tamnd.github.io/math2/node/WH4SFAACAS.md" tag="WH4SFAACAS" kind="concept">
# Lattice

## Summary

A lattice is a [[partially ordered set|0QV2JS9RJ3]] in which every two elements have a [[join and a meet|J2T75ATRV5]].
These are a least upper bound and a greatest lower bound.
Equivalently, it is a set with two binary operations $\vee$ and $\wedge$ that are commutative and associative and satisfy the absorption laws.
Lattices occur all over mathematics.
Subsets of a set, subgroups of a group, subspaces of a vector space and open sets of a space all form lattices.
Special classes such as [[distributive lattices|0HM0HQCK6N]] and [[Boolean algebras|0XADJKM699]] are central in algebra and logic.

## Explanations

### L1 human

A lattice is an ordering in which any two elements have a best common upper bound and a best common lower bound.
For the subsets of a set ordered by inclusion, these are the union and the intersection.
For positive integers ordered by divisibility, they are the least common multiple and the greatest common divisor.
In a picture of the order, from any two points one can go up to a lowest common point and down to a highest common point.
Not every ordering is a lattice: two incomparable elements with two incomparable common upper bounds have no best one.
</document>

<document source="https://tamnd.github.io/math2/node/WHDZ3MK56B.md" tag="WHDZ3MK56B" kind="concept">
# Holonomy

## Summary

The holonomy of a [[connection|5ZYRRVE8BW]] records how [[parallel transport|TVAGN3D6W5]] around closed loops transforms the fibre at a base point.
These transformations form a group, the holonomy group, which is a Lie subgroup of the general linear group of the fibre.
The holonomy group measures curvature globally: it is trivial for flat connections on simply connected manifolds.
For Riemannian manifolds the possible holonomy groups are classified, and special holonomy groups correspond to special geometries such as Kähler, Calabi–Yau and $G_2$ manifolds.

## Explanations

### L1 human

Carry an arrow around a closed loop, keeping it parallel all the way.
When you come back, the arrow may have turned.
The holonomy is the set of all turns you can get from all loops at that point.
On a flat plane no loop turns the arrow, so the holonomy is trivial.
On a round sphere every rotation of the tangent plane can occur, so the holonomy is the rotation group $\mathrm{SO}(2)$.
</document>

<document source="https://tamnd.github.io/math2/node/WHHMJERDFG.md" tag="WHHMJERDFG" kind="definition">
# Integral current

## Definition

Let $0 \le m \le n$, and let $T$ be an $m$-dimensional [[current|360H7MD9RD]] on $\R^n$, with mass $\mathbf{M}(T)$ and boundary $\partial T$.
The current $T$ is *integer multiplicity rectifiable* if there are:

1. an $\mathcal{H}^m$-measurable countably $m$-[[rectifiable set|7N0GFMW05R]] $S \subseteq \R^n$, where $\mathcal{H}^m$ is the [[Hausdorff measure|Q1TGZK8RN6]];
2. a locally $\mathcal{H}^m$-integrable function $\theta \colon S \to \set{1, 2, 3, \dots}$;
3. an $\mathcal{H}^m$-measurable map $\xi$ that assigns to $\mathcal{H}^m$-almost every $x \in S$ a unit simple $m$-vector $\xi(x)$ orienting the approximate tangent space of $S$ at $x$;

such that $T(\omega) = \int_S \ip{\omega(x)}{\xi(x)} \, \theta(x) \, d\mathcal{H}^m(x)$ for every smooth compactly supported $m$-form $\omega$.
The current $T$ is an *integral current* if it is integer multiplicity rectifiable, has compact support, satisfies $\mathbf{M}(T) < \infty$, and, when $m \ge 1$, satisfies $\mathbf{M}(\partial T) < \infty$.

## Remarks

By the boundary rectifiability theorem of Federer and Fleming, the boundary of an integral current is again an integral current.
Federer's original definition asks instead that $T$ and $\partial T$ be rectifiable currents, and the two definitions agree by that theorem.
The integral $0$-currents are the finite sums $\sum_i a_i \delta_{p_i}$ with $a_i \in \Z$.

Example: let $D$ be the closed unit disc in $\R^2 \subseteq \R^3$, oriented by $e_1 \wedge e_2$.
Then $[\![D]\!](\omega) = \int_D \omega$ is an integral $2$-current: take $S = D$, $\theta = 1$, $\xi = e_1 \wedge e_2$.
Its mass is $\pi$, and its boundary is the circle, of mass $2\pi$.

Non-example: in $\R$, let $T = \sum_{k \ge 1} [\![I_k]\!]$, where $I_k = [1/(2k + 1), 1/(2k)]$ is oriented by $e_1$.
It is integer multiplicity rectifiable with compact support in $[0, 1]$ and $\mathbf{M}(T) = \sum_k \frac{1}{2k(2k + 1)} < \infty$.
But $\partial T$ is a sum of infinitely many unit point masses, so $\mathbf{M}(\partial T) = \infty$ and $T$ is not an integral current.

## Explanations

### L1 human

An integral current is integration over an oriented rectifiable set with positive integer multiplicities, with finite area and finite boundary length.
</document>

<document source="https://tamnd.github.io/math2/node/WHK5B8X2CQ.md" tag="WHK5B8X2CQ" kind="definition">
# Quadratic functional equation

## Definition

Let $G$ be an [[abelian group|082GHNPJ53]], written additively.
The *quadratic functional equation* on $G$ is the [[functional equation|SRAN4EKWS0]]
$$f(x + y) + f(x - y) = 2 f(x) + 2 f(y) \quad \text{for all } x, y \in G,$$
for an unknown [[function|0EPAT51DXQ]] $f \colon G \to \R$.
A solution is called a *quadratic function* on $G$.

## Remarks

Example: on $G = \R^2$, $f(x_1, x_2) = x_1^2 + 3 x_1 x_2$ is a solution, since it is a [[quadratic form|0RBQS1JJCR]] and the cross terms in $f(x + y) + f(x - y)$ cancel.
Non-example: $f(x) = x$ on $\R$ is not a solution, since the left side is $2x$ and the right side is $2x + 2y$.
If $f$ is a solution, then $B(x, y) = \frac{1}{4}\big(f(x + y) - f(x - y)\big)$ is symmetric and additive in each variable, and $f(x) = B(x, x)$.
Conversely, $f(x) = B(x, x)$ is a solution for every symmetric biadditive $B \colon G \times G \to \R$.
On $G = \R$ the continuous solutions are $f(x) = cx^2$.
</document>

<document source="https://tamnd.github.io/math2/node/WHK8D383YT.md" tag="WHK8D383YT" kind="statement">
# Midpoint method

## Statement

Let $f \colon [t_0, T] \times \R^d \to \R^d$ be of class $C^2$, and consider the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$ with exact solution $y$.
The *explicit midpoint method* with step $h > 0$ is
$$y_{n+1} = y_n + h \, f\Big(t_n + \frac{h}{2}, \; y_n + \frac{h}{2} f(t_n, y_n)\Big).$$
Its [[local truncation error|T220HF0562]] is $O(h^3)$, so it is a Runge–Kutta method of order $2$.

## Remarks

Proof idea: by [[Taylor's theorem|03QHBVW4TT]], $y(t + h) = y + h f + \frac{h^2}{2}(f_t + f_y f) + O(h^3)$, with $f$ and its partial derivatives taken at $(t, y(t))$.
Expanding the increment gives $h f\big(t + \frac h2, y + \frac h2 f\big) = h f + \frac{h^2}{2}(f_t + f_y f) + O(h^3)$, and the two expansions agree up to $h^2$.
In Butcher form it has $c = (0, \tfrac12)$, $a_{21} = \tfrac12$ and $b = (0, 1)$.
Example: for $y' = \lambda y$ it gives $y_{n+1} = \big(1 + z + \frac{z^2}{2}\big) y_n$ with $z = h\lambda$, the Taylor polynomial of $e^z$ of degree $2$.
The *implicit midpoint method* $y_{n+1} = y_n + h f\big(t_n + \frac h2, \frac{y_n + y_{n+1}}{2}\big)$ also has order $2$; it is the one-stage Gauss–Legendre method.

## Explanations

### L1 human

The Euler method follows the slope at the start of the step all the way to the end.
The midpoint method first uses that slope to walk half a step, measures the slope there, and then makes the whole step with this better slope.
This costs two slope evaluations per step but makes the error shrink like $h^2$ instead of $h$.

### L2 human

The explicit midpoint method is one of the simplest second-order [[Runge–Kutta methods|7DBPJKYC6C]], together with [[Heun's method|RJEK60G28F]].
Both satisfy the order conditions $\sum b_i = 1$ and $\sum b_i c_i = \frac12$, and they differ in the choice of the second stage.
The implicit midpoint method is symplectic and preserves quadratic invariants, which makes it useful for Hamiltonian systems.
</document>

<document source="https://tamnd.github.io/math2/node/WHME2NGANG.md" tag="WHME2NGANG" kind="concept">
# Filtration of a module

## Summary

A filtration of a [[module|0NZ8JA25WK]] $M$ is a decreasing sequence of [[submodules|05WDSN597C]] $M = M_0 \supseteq M_1 \supseteq M_2 \supseteq \cdots$.
For an ideal $I$, an $I$-filtration is one with $I M_n \subseteq M_{n+1}$, and it is stable when equality holds from some point on.
The basic example is the $I$-adic filtration $M_n = I^n M$; filtrations lead to the [[associated graded ring|8HFAEGFQT1]], the [[$I$-adic topology|2VF2TSJ3PY]] and the [[Artin–Rees lemma|30H6B0G2FT]].

## Explanations

### L1 human

A filtration sorts the elements of a module by how small they are.
The deeper the submodule $M_n$, the smaller its elements are thought to be.
For the integers with the prime $p$, the submodule $p^n \Z$ contains the integers divisible by $p^n$, and these count as small when $n$ is large.
The [[definition|GEJYZEY7Y0]] makes this precise.
</document>

<document source="https://tamnd.github.io/math2/node/WHS58R43N0.md" tag="WHS58R43N0" kind="statement">
# Hadamard three-lines theorem

## Statement

Let $S = \set{z \in \C : 0 \le \operatorname{Re} z \le 1}$ be the closed vertical strip.
Let $f \colon S \to \C$ be continuous and bounded on $S$ and [[holomorphic|0DG0Z72CQD]] on the interior of $S$.
For $0 \le x \le 1$ let $M(x) = \sup_{y \in \R} \abs{f(x + iy)}$.
Then for each $x$ with $0 \le x \le 1$,
$$M(x) \le M(0)^{1 - x} M(1)^{x}.$$

## Remarks

Proof idea: assume $M(0), M(1) > 0$ and put $g(z) = f(z) M(0)^{z - 1} M(1)^{-z}$, so that $\abs{g} \le 1$ on both boundary lines.
For $\varepsilon > 0$ the function $g(z) e^{\varepsilon z^2}$ tends to $0$ as $\abs{\operatorname{Im} z} \to \infty$ in $S$, since $g$ is bounded.
The [[maximum modulus principle|048F2P90WR]] on long rectangles gives $\abs{g(z)} e^{\varepsilon (x^2 - y^2)} \le e^{\varepsilon}$, and $\varepsilon \to 0$ gives $\abs{g} \le 1$.
The boundedness hypothesis is needed.
The function $f(z) = \exp(i e^{-i\pi z})$ satisfies $\abs{f(x + iy)} = \exp(\sin(\pi x) e^{\pi y})$.
It has modulus $1$ on both lines but is unbounded on the line $\operatorname{Re} z = \frac{1}{2}$.
This theorem is the key step in the Riesz–Thorin interpolation theorem.

## Explanations

### L1 human

Look at a holomorphic function on a vertical strip, and record its largest size on each vertical line.
The theorem says that the logarithm of this size is a convex function of the position across the strip.
So the size in the middle is controlled by the sizes on the two edges, in a geometric-mean way.

### L2 human

The theorem is the strip form of the [[Hadamard three-circle theorem|V00Q05ZW1G]] and a special case of the [[Phragmén–Lindelöf principle|FYR6V2NH6C]].
Its main use is in interpolation: the Riesz–Thorin theorem bounds an operator on $L^{p}$ for intermediate $p$ by applying it to an analytic family of functions.
The same idea appears in complex interpolation of Banach spaces and in convexity bounds for the Riemann zeta function in the critical strip.
</document>

<document source="https://tamnd.github.io/math2/node/WHTBZ3K4FB.md" tag="WHTBZ3K4FB" kind="concept">
# Hamilton–Jacobi–Isaacs equation

## Summary

The Hamilton–Jacobi–Isaacs equation is the partial differential equation for the value of a two-player zero-sum [[differential game|THSHGNJNPS]].
It is the analogue for games of the [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] of optimal control, with a min-max in place of a minimum.
Under standard hypotheses the upper and lower values of the game are its [[viscosity solutions|H1ERJ3T75T]].
When the min and the max commute, the game has a value and the equation characterises it.

## Explanations

### L1 human

In a game between two players with opposite goals, each one reacts to the other at every instant.
The value of the game, as a function of time and position, satisfies one equation.
In it, the minimiser and the maximiser each make their best choice against the other at the present instant.
For example, if each player controls its own part of the motion and pays its own part of the cost, the order of the two choices does not matter.
Then the equation has a single form, and solving it gives both the value and the best feedback of each player.
</document>

<document source="https://tamnd.github.io/math2/node/WJ0D8QAMW0.md" tag="WJ0D8QAMW0" kind="problem">
# Congruence lattice problem

## Problem

Is every [[distributive|0HM0HQCK6N]] [[algebraic lattice|3J0BTXSCVN]] isomorphic to the [[congruence lattice|REE07CSXHR]] of some [[lattice|WH4SFAACAS]]?

## Remarks

The question is natural because the congruence lattice of every lattice is distributive and algebraic.
An equivalent form asks whether every distributive join-semilattice with a least element is isomorphic to the semilattice of compact [[lattice congruences|T782PJYVEZ]] of some lattice.

The answer is positive for finite lattices: every finite distributive lattice is isomorphic to the congruence lattice of a finite lattice.
It is also positive when the semilattice of compact elements has at most $\aleph_1$ elements.

The problem was solved in the negative by Friedrich Wehrung, in a paper published in 2007.
He built a counterexample whose semilattice of compact elements has $\aleph_{\omega+1}$ elements.
Pavel Růžička later lowered the size of a counterexample to $\aleph_2$, which is optimal by the positive result above.

## Explanations

### L1 human

Every lattice has a lattice of congruences, and this lattice always has two special properties: it is distributive and algebraic.
Dilworth asked whether these two properties are the whole story, that is, whether every lattice with them arises from some lattice.
For finite and for not too large lattices the answer is yes.
For very large ones it is no, which was proved after about sixty years.
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<document source="https://tamnd.github.io/math2/node/WJ5WRGG5DE.md" tag="WJ5WRGG5DE" kind="statement">
# Castelnuovo's contraction theorem

## Statement

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $X$ be a smooth projective [[surface|KFTECAATKG]] over $k$.
Let $E \subset X$ be a curve with $E \cong \mathbb{P}^1$ and self-[[intersection number|4HV4WTAX9P]] $E \cdot E = -1$.
Then there are a smooth projective surface $Y$, a point $p \in Y$ and a morphism $f \colon X \to Y$ with the following property.
The morphism $f$ is isomorphic to the [[blow-up|C4KGYAFPN7]] of $Y$ at $p$, with exceptional curve $E$.

## Remarks

The idea of the proof is to choose a very ample divisor $H$ on $X$ with $H^1(X, \mathcal{O}_X(H)) = 0$, put $m = H \cdot E$, and use the linear system $\abs{H + mE}$.
It has degree $0$ on $E$, so it contracts $E$ to a point, and a computation of sections shows that the image surface is smooth at that point.
Example: a smooth cubic surface over $k$ contains $27$ lines, each a $(-1)$-curve, and contracting six disjoint lines among them gives $\mathbb{P}^2$.
The condition $E \cdot E = -1$ cannot be weakened to $E \cdot E < 0$.
A smooth rational curve with $E \cdot E = -2$ can sometimes be contracted, but the image point is then singular, an $A_1$ singularity.

## Explanations

### L1 human

Blowing up a point of a smooth surface creates a new curve, a copy of the projective line with self-intersection $-1$.
The theorem says the converse: any such curve comes from a blow-up and can be blown down.
So the curves of this kind are exactly the traces left by blowing up.

### L2 human

The theorem is the basic step of the minimal model program for surfaces.
Contracting $(-1)$-curves one at a time lowers the Picard number, so the process stops, at a surface with no $(-1)$-curves.
So every smooth projective surface is obtained from a surface without $(-1)$-curves by finitely many blow-ups of points.
The surfaces without $(-1)$-curves are then sorted by the Enriques–Kodaira classification.
</document>

<document source="https://tamnd.github.io/math2/node/WJ73XYBNE4.md" tag="WJ73XYBNE4" kind="concept">
# Direct product of algebras

## Summary

The direct product of a family of [[algebras|QJGPT2W0FT]] of the same signature is the [[Cartesian product|09CEDJYBKP]] of their universes with the operations computed coordinatewise.
It generalizes the [[direct product of groups|00HF4RJ5QN]], of rings and of vector spaces.
Products are one of the three basic constructions of universal algebra, with subalgebras and homomorphic images, and varieties are the classes closed under all three.

## Explanations

### L1 human

To multiply two algebras, take pairs, one element from each, and do every operation separately in each coordinate.
For example, in $\Z/2\Z \times \Z/3\Z$ one adds $(1, 2) + (1, 2) = (0, 1)$, computing modulo $2$ in the first place and modulo $3$ in the second.
Any identity that holds in every factor, such as commutativity, still holds in the product.
Properties that are not identities can fail: a product of two fields is not a field, since $(1, 0) \cdot (0, 1) = (0, 0)$.
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<document source="https://tamnd.github.io/math2/node/WJ7PDMGQQW.md" tag="WJ7PDMGQQW" kind="concept">
# Hilbert sum

## Summary

The Hilbert sum of a family of Hilbert spaces consists of the families of vectors, one from each space, with finite sum of squared norms.
It is the right notion of direct sum for Hilbert spaces, because the algebraic direct sum of infinitely many of them is not complete.
Every Hilbert space with an orthonormal basis indexed by $I$ is the Hilbert sum of $I$ copies of the scalar field.

## Explanations

### L1 human

Think of putting several Hilbert spaces side by side at right angles to each other.
A vector of the sum has one component in each space, and its squared length is the sum of the squared lengths of the components.
With infinitely many spaces, one only keeps families for which this sum is finite.
For example, the Hilbert sum of countably many copies of $\R$ is the space $\ell^2$ of square-summable real sequences.
</document>

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# Nakano positivity

## Summary

Nakano positivity is a notion of positive curvature for a holomorphic Hermitian vector bundle, stronger than [[Griffiths positivity|FVCDGMNSAN]].
It views the curvature of the [[Chern connection|7RC9CN2Q49]] as a Hermitian form on the tensor product of the tangent space and the fibre.
Then it asks that this form be positive definite.
It is the hypothesis under which the Bochner–Kodaira–Nakano identity gives vanishing of cohomology, as in Nakano's vanishing theorem $H^q(X, K_X \otimes E) = 0$ for $q \ge 1$ on compact Kähler $X$.
For line bundles it agrees with Griffiths positivity.

## Explanations

### L1 human

The curvature of a vector bundle eats a tangent direction and a fibre vector.
Griffiths positivity only tests "pure" pairs, one tangent vector together with one fibre vector.
Nakano positivity tests all combinations, that is, all elements of the tensor product of the two spaces, including sums of such pairs.
So it is a stronger requirement, and in return it gives stronger vanishing theorems.
</document>

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# Parabolic partial differential equation

## Definition

Let $U \subseteq \R^n$ be open, let $T > 0$, and write the variables in $U \times (0, T)$ as $(x, t)$.
Consider a [[linear PDE|6DZETZ5T9G]] in $U \times (0, T)$ of the following form.
$$\partial_t u - \sum_{i,j=1}^{n} a_{ij}(x, t) \, \partial_i \partial_j u + \sum_{i=1}^{n} b_i(x, t) \, \partial_i u + c(x, t) \, u = f(x, t).$$
Here all coefficients are real, $a_{ij} = a_{ji}$, and $\partial_i$ is the derivative in $x_i$.
The equation is a *second-order linear parabolic equation* if its spatial part is [[uniformly elliptic|B4TTNHSTRX]].
That is, there is $\theta > 0$ with
$$\sum_{i,j=1}^{n} a_{ij}(x, t) \, \xi_i \xi_j \ge \theta \abs{\xi}^2 \quad \text{for all } (x, t) \in U \times (0, T), \ \xi \in \R^n.$$

## Remarks

Example: the [[heat equation|VP2HN86VAG]] $\partial_t u - \Delta u = 0$ is parabolic with $a_{ij} = \delta_{ij}$ and $\theta = 1$.
Non-example: the backward heat equation $\partial_t u + \Delta u = 0$ has $a_{ij} = -\delta_{ij}$, which fails the inequality for every $\theta > 0$, so it is not parabolic in this sense.
View a parabolic equation as a second-order equation in the $n + 1$ variables $(x, t)$.
Its coefficient matrix has one zero eigenvalue, in the $t$ direction, and $n$ eigenvalues of the same sign.
This matches the [[classification of second-order PDEs|5FC6BMM1NA]].
Higher-order parabolic equations, such as $\partial_t u + \Delta^2 u = 0$, are defined with an elliptic spatial operator of higher order.

## Explanations

### L1 human

A parabolic equation is a first-order time derivative plus a uniformly elliptic operator in space, with the signs arranged as in the heat equation.
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<document source="https://tamnd.github.io/math2/node/WJK0YA4GP2.md" tag="WJK0YA4GP2" kind="concept">
# Provability logic

## Summary

Provability logic is the [[modal logic|C3RZY20ETK]] in which $\square\varphi$ is read as "$\varphi$ is provable" in a formal theory such as Peano arithmetic.
Its main system $\mathrm{GL}$ is the normal modal logic given by Löb's axiom $\square(\square p \to p) \to \square p$.
By Solovay's theorem, $\mathrm{GL}$ proves exactly the modal laws of provability in Peano arithmetic, so questions about provability become finite combinatorics of Kripke frames.

## Explanations

### L1 human

Gödel showed that a formal theory can talk about its own proofs.
Provability logic collects the general laws of "it is provable that" into a small modal system.
One law is "if $p$ is provable, then it is provable that $p$ is provable", the formula $\square p \to \square\square p$.
"If $p$ is provable, then $p$" is not a law of the system.
With $p$ the false sentence $0 = 1$, it says that the theory is consistent, and by Gödel's second incompleteness theorem the theory cannot prove this.
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# Calabi conjecture

## Statement

Let $M$ be a compact complex manifold of complex dimension $n$ with a Kähler metric $g$, whose fundamental form is $\omega$; see [[Hermitian manifold|CK1DCPC3D3]].
Let $c_1(M)$ be the first Chern class of $M$ in real de Rham cohomology.
Let $\rho'$ be a real closed $(1,1)$-form on $M$ whose [[de Rham class|0ZV14XRNF0]] is $2\pi c_1(M)$.
Then there is a unique Kähler form $\omega'$ on $M$ in the same cohomology class as $\omega$ whose Ricci form is $\rho'$.

## Remarks

Calabi proved uniqueness, and Yau proved existence; the result is often called the Calabi–Yau theorem.
Write $\omega' = \omega + i\partial\bar\partial\varphi$ for a real function $\varphi$, by the $\partial\bar\partial$-lemma.
Since the Ricci form is $-i\partial\bar\partial \log$ of the volume density, the problem becomes the complex Monge–Ampère equation
$$(\omega + i\partial\bar\partial\varphi)^n = e^{F} \omega^n,$$
for a given function $F$ normalized so that $\int_M e^F \omega^n = \int_M \omega^n$.
Yau solved it by the continuity method, and the key step is an a priori bound on $\sup \abs{\varphi}$.

The main application is the case $c_1(M) = 0$ in real cohomology.
Take $\rho' = 0$: each Kähler class then contains a unique Ricci-flat Kähler metric.
Such manifolds are the Calabi–Yau manifolds, for example K3 surfaces and quintic threefolds in $\C P^4$.
The same method gives [[Kähler–Einstein metrics|6ETEBA8827]] with negative constant when $c_1(M) < 0$, a result of Aubin and Yau.

## Explanations

### L1 human

The curvature of a Kähler metric is tied to the topology of the manifold through the first Chern class.
Calabi asked whether every curvature form allowed by this topology is actually the Ricci curvature of some Kähler metric.
Yau proved that the answer is yes, and that the metric is unique once its Kähler class is fixed.

### L2 human

The theorem is a statement about a fully nonlinear elliptic equation, and its proof was a landmark of geometric analysis.
Its best-known consequence is the existence of Ricci-flat metrics on compact Kähler manifolds with vanishing first Chern class.
These metrics have restricted holonomy in $SU(n)$ and are used in string theory and in mirror symmetry.
The positive case $c_1(M) > 0$ is not covered and is the subject of the [[Yau–Tian–Donaldson conjecture|DHYSPSYSCB]].
</document>

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# Lebesgue's lemma on near-best approximation

## Statement

Let $X$ be a [[normed vector space|03J5RAKB30]] and let $V \subseteq X$ be a linear subspace.
Let $P \colon X \to X$ be a bounded linear map with image in $V$ and $P v = v$ for all $v \in V$, with [[operator norm|06EKW1AKXF]] $\norm{P}$.
Then for every $f \in X$,
$$\norm{f - P f} \le (1 + \norm{P}) \inf_{v \in V} \norm{f - v}.$$

## Remarks

Proof: for any $v \in V$, $P v = v$ gives $f - P f = (f - v) - P(f - v)$.
So $\norm{f - P f} \le (1 + \norm{P}) \norm{f - v}$, and taking the infimum over $v$ gives the claim.

Example: let $P$ be polynomial interpolation at $n + 1$ nodes in $[a, b]$, acting on $C[a, b]$ with the uniform norm.
Then $\norm{P}$ is the [[Lebesgue constant|7V7KWNK9HN]] $\Lambda_n$, and the lemma gives $\norm{f - P f}_\infty \le (1 + \Lambda_n) E_n(f)$, where $E_n(f)$ is the error of best uniform approximation by polynomials of degree at most $n$.
For Chebyshev nodes, $\Lambda_n \le \frac{2}{\pi} \log(n + 1) + 1 < 4$ for $n < 100$, so interpolation loses less than a factor $5$ compared with the best polynomial.

The same lemma applies to Fourier partial sums, with $\Lambda_n$ the Lebesgue constants of the Dirichlet kernel.

## Explanations

### L1 human

A linear [[projection|033DA1WE8Y]] onto a space of approximants is never much worse than the best approximation from that space.
The loss factor is at most one plus the norm of the projection.
So a cheap method like interpolation is nearly optimal whenever its Lebesgue constant is small.

### L2 human

This is how Lebesgue constants enter convergence proofs.
Combined with Jackson's inequality, it shows that interpolation at Chebyshev nodes converges uniformly when $\omega(f, 1/n) \log n \to 0$, the Dini–Lipschitz condition.
Conversely, unbounded norms of projections lead through the uniform boundedness principle to divergence results such as Faber's theorem.
</document>

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# Morphism of varieties

## Definition

Let $k$ be an algebraically closed field and let $X$ and $Y$ be [[algebraic varieties|HV6YBS0YDH]] over $k$.
A *morphism* $\varphi \colon X \to Y$ is a [[continuous map|0VGFHTM75Y]] for the Zariski topologies with the following property.
For every [[open set|02BH3MCBQB]] $V \subseteq Y$ and every [[regular function|4M94FZH9Y8]] $f \colon V \to k$, the function $f \circ \varphi \colon \varphi^{-1}(V) \to k$ is regular.
An *isomorphism* is a morphism that has an inverse morphism.

## Remarks

If $X \subseteq k^m$ and $Y \subseteq k^n$ are [[affine varieties|JERVQT4C12]], a map $\varphi \colon X \to Y$ is a morphism if and only if $\varphi = (\varphi_1, \dots, \varphi_n)$ with each $\varphi_i$ a polynomial function on $X$.
A morphism $\varphi$ gives the [[ring homomorphism|0VTH8KPCG2]] $\varphi^* \colon k[Y] \to k[X]$, $f \mapsto f \circ \varphi$, on [[coordinate rings|G9WB27VWP7]].
This gives a bijection between morphisms $X \to Y$ and $k$-algebra homomorphisms $k[Y] \to k[X]$.

Example: $\varphi \colon k \to V(y - x^2)$, $t \mapsto (t, t^2)$, is an isomorphism with inverse $(x, y) \mapsto x$.

Non-example: $\psi \colon k \to C = V(y^2 - x^3)$, $t \mapsto (t^2, t^3)$, is a bijective morphism but not an isomorphism.
Its comorphism $k[C] \to k[t]$ has image $k[t^2, t^3]$, which does not contain $t$, so $\psi^*$ is not surjective and $\psi$ has no inverse morphism.
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<document source="https://tamnd.github.io/math2/node/WJS9TPSGVM.md" tag="WJS9TPSGVM" kind="definition">
# Local ring

## Definition

A *local ring* is a [[commutative ring|0310REJJY8]] $R$ that has exactly one [[maximal ideal|0PV74TR5FP]] $\mathfrak m$.
One writes $(R, \mathfrak m)$, or $(R, \mathfrak m, k)$ with the [[residue field|NJ3APQ8BHH]] $k = R/\mathfrak m$.

## Remarks

For a commutative ring $R$, the following are equivalent:

1. $R$ is local;
2. $R \ne 0$ and the set of non-[[units|0XVEZDE9J8]] of $R$ is an ideal;
3. $R \ne 0$ and, for every $a \in R$, $a$ or $1 - a$ is a unit.

In a local ring, the maximal ideal is exactly the set of non-units, because every non-unit lies in some maximal ideal.

Example: let $p$ be a prime and $\Z_{(p)}$ the ring of rationals $a/b$ with $p \nmid b$.
The element $a/b$ in lowest terms is a unit exactly when $p \nmid a$.
So the non-units are the elements of $p\Z_{(p)}$, which is an ideal, and $\Z_{(p)}$ is local.
Other examples are every field, the ring $\Z/p^n\Z$, the formal power series ring $k[\![x]\!]$ over a field $k$, and every [[localization at a prime ideal|X6JD4AG973]].

Non-example: $\Z$ is not local, because $2\Z$ and $3\Z$ are two different maximal ideals.
Equivalently, $3$ and $1 - 3 = -2$ are both non-units of $\Z$.
The zero ring is not local either, since it has no maximal ideal.

Some authors call a Noetherian local ring a local ring, and call the general case a quasi-local ring.
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# Matrix congruence

## Summary

Two square matrices $A$ and $B$ are congruent when $B = P^{T} A P$ for some invertible matrix $P$.
Congruent matrices are the matrices of one [[bilinear form|0SQZMXB6GP]] in two different bases.
Congruence is an equivalence relation, and its classification for real symmetric matrices is [[Sylvester's law of inertia|0WFED3K3DQ]].
It differs from [[similarity|0E89HQATM8]], which uses $P^{-1}$ in place of $P^{T}$.

## Explanations

### L1 human

A bilinear form, such as a dot product, gives a matrix once you choose a basis.
If you change the basis with a matrix $P$, the new matrix of the same form is $P^{T} A P$.
So congruent matrices describe the same form seen in two coordinate systems.
For example, $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 4 & 0 \\ 0 & 9 \end{pmatrix}$ are congruent, with $P = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$.
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# Hopf algebra structure on cohomology

## Summary

The cohomology with field coefficients of a connected [[H-space|EASHTX8EG5]] of finite type is a [[Hopf algebra|7XFYP0N84H]].
The cup product is its multiplication, and the map induced by the H-space multiplication is its comultiplication.
Hopf discovered this structure in 1941.
He used it to show that the rational cohomology of a compact connected Lie group is an exterior algebra on generators of odd degree.
The structure theory of Hopf algebras, due to Borel, Milnor and Moore, then restricts which spaces can be H-spaces.

## Explanations

### L1 human

On an H-space one can multiply points, and this multiplication induces a map in cohomology that goes the other way, from the cohomology of $X$ to that of $X \times X$.
So the cohomology gets both a product, the cup product, and a "coproduct".
The two operations are compatible, which is the definition of a Hopf algebra.
For the $3$-sphere, seen as the unit quaternions, the cohomology is spanned by $1$ and a class $x$ of degree $3$, and the coproduct sends $x$ to $x \otimes 1 + 1 \otimes x$.
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# Simplicial object

## Summary

A simplicial object in a [[category|A2DE66937D]] $\mathcal{C}$ is a contravariant [[functor|28NQ0KRHD3]] from the [[simplex category|FRFYHJGAN3]] to $\mathcal{C}$.
It is a sequence of objects $X_0, X_1, X_2, \dots$ of $\mathcal{C}$ with face and degeneracy maps between them that obey the simplicial identities.
In $\mathcal{C} = \mathbf{Set}$ these are the [[simplicial sets|WW81ARB8K9]], and in abelian groups they are the simplicial abelian groups of the [[Dold–Kan correspondence|PY4EWJ8BQA]].
They let one do homotopy theory with objects of $\mathcal{C}$.

## Explanations

### L1 human

Think of $X_n$ as the "$n$-dimensional pieces" of a shape built from points, edges, triangles and higher simplices.
The face maps send a piece to its faces, and the degeneracy maps view a piece as a flattened piece one dimension up.
In a simplicial object, each $X_n$ is not a mere set but an object of some category, such as a group or a space.
For example, the constant simplicial object on an object $A$ has $X_n = A$ for all $n$ and all face and degeneracy maps equal to the identity.
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# Commuting elements

## Definition

Let $\ast$ be a [[binary operation|0M418FE7BB]] on a set $S$ and let $a, b \in S$.
The elements $a$ and $b$ *commute* if $a \ast b = b \ast a$.

## Remarks

Example: in the set of $2 \times 2$ real matrices with matrix multiplication, every matrix $A$ commutes with $2A$, because $A(2A) = 2A^2 = (2A)A$.

Non-example: the matrices
$$A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$$
do not commute, because $AB = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ while $BA = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$.

The relation "commutes with" is reflexive and symmetric but not transitive in general.
In the example, $A$ commutes with the identity matrix $I$ and $I$ commutes with $B$, but $A$ and $B$ do not commute.
The operation $\ast$ is commutative if and only if every two elements of $S$ commute.
</document>

<document source="https://tamnd.github.io/math2/node/WK5HPW5Y24.md" tag="WK5HPW5Y24" kind="concept">
# Smoothness classes

## Summary

The smoothness classes $C^0, C^1, C^2, \ldots, C^\infty$ sort functions by how many continuous derivatives they have.
A function of class $C^k$ has $k$ derivatives, and the $k$-th one is continuous.
The classes are nested, each strictly inside the one before, and their intersection is the class $C^\infty$ of [[smooth functions|0V33BNKZ9S]].
Theorems in analysis and geometry state their hypotheses in terms of these classes.

## Explanations

### L1 human

Some functions are continuous but have corners, like $\abs{x}$.
Some have a derivative, but the derivative has a corner, like $x \abs{x}$, whose derivative is $2\abs{x}$.
Counting how many times you can differentiate before a corner shows up gives the smoothness class.
So $\abs{x}$ is of class $C^0$, $x\abs{x}$ is of class $C^1$, and a polynomial is of class $C^\infty$.
</document>

<document source="https://tamnd.github.io/math2/node/WK5V372TG3.md" tag="WK5V372TG3" kind="definition">
# Double sequence

## Definition

Let $X$ be a set.
A double sequence in $X$ is a [[function|0EPAT51DXQ]] $x \colon \N \times \N \to X$.
Its value at $(m, n)$ is written $x_{m,n}$, and the double sequence is written $(x_{m,n})_{m, n \in \N}$.

## Remarks

A double sequence is a [[sequence|0QE8STGJH2]] indexed by $\N \times \N$ instead of $\N$.
Example: $x_{m,n} = m/(m + n + 1)$ is a double sequence of real numbers.
For fixed $m$ it tends to $0$ as $n \to \infty$, and for fixed $n$ it tends to $1$ as $m \to \infty$.
So $\lim_m \lim_n x_{m,n} = 0$ and $\lim_n \lim_m x_{m,n} = 1$: the two iterated limits differ.
Non-example: a triangular array $(x_{m,n})_{0 \le n \le m}$ is not a double sequence, since it is defined only for $n \le m$; it becomes one after extension by $0$.
The joint limit in the sense of [[Pringsheim convergence|QYTGCW1GTK]] is the usual notion of limit for double sequences.

## Explanations

### L1 human

A double sequence assigns an element to each pair $(m, n)$ of natural numbers, like an infinite matrix.
</document>

<document source="https://tamnd.github.io/math2/node/WK7MKDWRFF.md" tag="WK7MKDWRFF" kind="statement">
# Envelope theorem

## Statement

Let $X$ be a set, let $T \subseteq \R^k$ be open, and let $f \colon X \times T \to \R$ be a function.
Let $V(t) = \sup_{x \in X} f(x, t)$ be the value function, and assume that $V$ is finite on $T$.
Let $t_0 \in T$ and let $x_0 \in X$ be a maximizer at $t_0$, that is, $f(x_0, t_0) = V(t_0)$.
If $V$ and $t \mapsto f(x_0, t)$ are both [[differentiable|0135XZZQ8M]] at $t_0$, then
$$\nabla V(t_0) = \nabla_t f(x_0, t_0).$$

## Remarks

For every $t$ one has $V(t) \ge f(x_0, t)$, with equality at $t = t_0$.
So $t \mapsto V(t) - f(x_0, t)$ has a [[local minimum|09WRNGPTMV]] at the interior point $t_0$, and its [[gradient|0ABK9V9Y64]] vanishes there.
In words: to first order, the change of the optimal value equals the direct effect of the parameter, with the maximizer held fixed.

Example: $f(x, t) = tx - x^2$ on $\R \times \R$.
The maximizer is $x = t/2$ and $V(t) = t^2/4$, so $V'(t) = t/2$.
The theorem predicts $V'(t) = \partial_t f(x, t) = x = t/2$, which agrees.

The classical version assumes that $X$ is an open subset of $\R^n$, that $f$ is continuously differentiable, and that a maximizer $x^*(t)$ depends differentiably on $t$.
Then the chain rule gives $\nabla V = \nabla_t f + (\nabla_x f) \, Dx^*$, and the second term vanishes because $\nabla_x f(x^*(t), t) = 0$.
For a constrained problem, the same idea shows that the derivative of the optimal value with respect to a constraint level is the corresponding Lagrange multiplier.
Differentiability of $V$ is a real hypothesis: for $X = \set{-1, 1}$ and $f(x, t) = xt$, the value $V(t) = \abs{t}$ is not differentiable at $0$.

## Explanations

### L1 human

Suppose you choose the best option for every value of a parameter, and you watch how the best value changes when the parameter moves.
The envelope theorem says that for a small change you may ignore that the best option also changes.
Only the direct effect of the parameter on the payoff counts.
The name comes from a picture.
The value function is the upper envelope of the graphs of $t \mapsto f(x, t)$ for all $x$, and it touches the graph of the optimal $x$ there.

### L2 human

The theorem is used throughout economics, for example to derive Hotelling's lemma and Roy's identity, and in optimization to compute sensitivities of optimal values.
In optimal control and the calculus of variations, the same reasoning shows that the costate is the gradient of the value function, when the latter is differentiable.
The version stated here, which needs no differentiability of the maximizer, is due to Paul Milgrom and Ilya Segal.
When $V$ is not differentiable, one-sided derivatives of $V$ are given by Danskin's theorem under compactness assumptions.
</document>

<document source="https://tamnd.github.io/math2/node/WK8GKRKAVD.md" tag="WK8GKRKAVD" kind="concept">
# Intersection form

## Summary

The intersection form of a closed oriented $4$-manifold $M$ is a symmetric bilinear form on $H^2(M; \Z)$.
It is the cup product followed by evaluation on the [[fundamental class|ZEA0PEM3B2]].
Geometrically, it counts with signs the intersection points of two surfaces in $M$ in general position.
By Poincaré duality it is unimodular on $H^2(M; \Z)$ modulo torsion.
It is the central invariant of $4$-manifold topology: Freedman's classification and Donaldson's theorem are statements about it.

## Explanations

### L1 human

In a $4$-dimensional space, two surfaces in general position meet in isolated points, just as two curves do in the plane.
Each point counts as $+1$ or $-1$ according to orientations, and the total is the intersection number.
The intersection form records this number for every pair of surface classes.
For the complex projective plane, a line meets another line in exactly one point, so the form is the $1 \times 1$ matrix $(1)$.
</document>

<document source="https://tamnd.github.io/math2/node/WKD2S2XA9C.md" tag="WKD2S2XA9C" kind="definition">
# Coherent analytic sheaf

## Definition

Let $(X, \mathcal{O}_X)$ be a [[complex analytic space|HXSZA44J5J]] and let $\mathcal{F}$ be a [[sheaf of $\mathcal{O}_X$-modules|S33B2H3MYD]].
The sheaf $\mathcal{F}$ is of *finite type* if each point of $X$ has an open neighbourhood $U$ with a surjective morphism $\mathcal{O}_U^p \to \mathcal{F}|_U$ for some $p \ge 0$.
The sheaf $\mathcal{F}$ is a *coherent analytic sheaf* if:

1. $\mathcal{F}$ is of finite type;
2. for every open $U \subseteq X$, every integer $q \ge 0$ and every morphism $\varphi \colon \mathcal{O}_U^q \to \mathcal{F}|_U$ of sheaves of modules, the kernel of $\varphi$ is of finite type.

## Remarks

Example: the sheaf $\mathcal{O}_X^p$ is coherent for every complex space $X$; for $X = \C^n$ this is the Oka coherence theorem, and the general case follows from it.
Given it, a sheaf is coherent if and only if it has, locally, a presentation $\mathcal{O}_U^q \to \mathcal{O}_U^p \to \mathcal{F}|_U \to 0$.

Non-example: let $X = \C$, $U = \C \setminus \set{0}$, and let $\mathcal{F}$ be the sheaf of holomorphic functions on $U$ extended by zero to $\C$.
Its stalk at $0$ is $0$ and its stalks at other points are $\mathcal{O}_{\C,x}$.
A section of $\mathcal{F}$ on a connected open set $V \ni 0$ is a holomorphic function on $V \setminus \set{0}$ that vanishes near $0$, so it is zero by the identity theorem.
So no finite set of sections over a neighbourhood of $0$ generates the nonzero stalks near $0$, and clause 1 fails.
</document>

<document source="https://tamnd.github.io/math2/node/WKG2DZDEBT.md" tag="WKG2DZDEBT" kind="definition">
# Nil ideal

## Definition

Let $R$ be a [[ring|0894J980RK]].
Let $I$ be a [[left ideal|5N2CJZ47D4]], right ideal or [[two-sided ideal|0DDA6CY7GJ]] of $R$.
The ideal $I$ is *nil* if every element of $I$ is [[nilpotent|0N9R5NN6TW]].
This means that for each $a \in I$ there is an integer $n \ge 1$ with $a^n = 0$.

## Remarks

Example: in $\Z/8\Z$, the ideal $I = 2\Z/8\Z$ is nil.
The check: every element of $I$ is $2b$ for some residue $b$, and $(2b)^3 = 8b^3 = 0$.
Non-example: in $\Z$, the ideal $2\Z$ is not nil, since $2^n \ne 0$ for all $n \ge 1$.
In a commutative ring the nil ideals are the ideals contained in the [[nilradical|2K5GKVZR78]].
The exponent $n$ may depend on the element, and there need not be a common bound.
</document>

<document source="https://tamnd.github.io/math2/node/WKJ4EP6E1X.md" tag="WKJ4EP6E1X" kind="definition">
# Alexander polynomial

## Definition

Let $K \subset S^3$ be an oriented tame [[knot|QYEAZTYGPT]], and let $F$ be an oriented [[Seifert surface|K866ND3WF7]] of $K$ of genus $g$.
Let $a_1, \ldots, a_{2g}$ be oriented simple closed curves on $F$ whose classes form a basis of $H_1(F; \Z)$.
For a curve $a$ on $F$, let $a^+$ be the curve $a$ pushed off $F$ slightly in the positive normal direction.
The *Seifert matrix* is the $2g \times 2g$ integer matrix $V$ with entries $V_{ij} = \operatorname{lk}(a_i, a_j^+)$, where $\operatorname{lk}$ is the linking number.
The *Alexander polynomial* of $K$ is
$$\Delta_K(t) = \det\big(V - t V^{T}\big) \in \Z[t, t^{-1}],$$
taken up to multiplication by a unit $\pm t^{k}$ with $k \in \Z$.

## Remarks

Different Seifert surfaces and bases change $\Delta_K$ only by such a unit.
The usual normalization, due to Conway, is the representative with $\Delta_K(t^{-1}) = \Delta_K(t)$ and $\Delta_K(1) = 1$.
Example: the [[trefoil knot|VYTYSVHN4A]] has a genus-one Seifert surface with $V = \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix}$.
Then $\det(V - tV^T) = (t - 1)^2 + t = t^2 - t + 1$, normalized to $t - 1 + t^{-1}$.
The [[unknot|1TQ06R4Q66]] bounds a disk with $g = 0$, so its polynomial is the empty determinant $1$.
Non-example of detection: the [[Conway knot|ZSAMVFPWN6]] is knotted but also has $\Delta = 1$, so $\Delta_K = 1$ does not imply that $K$ is the unknot.
The [[figure-eight knot|C01XBGAHC6]] has $\Delta = -t + 3 - t^{-1}$.
The degree span of $\Delta_K$ is at most $2g(K)$.
</document>

<document source="https://tamnd.github.io/math2/node/WKK2DC7JE9.md" tag="WKK2DC7JE9" kind="definition">
# Chattering control

## Definition

Let $U \subseteq \R^m$ be a nonempty set, let $a < b$, and let $u \colon [a, b] \to U$ be a measurable control.
A *switching time* of $u$ is a point $s \in [a, b]$ with this property.
On no neighbourhood of $s$ in $[a, b]$ is $u$ almost everywhere equal to a constant.
The control $u$ is *chattering* if its set of switching times is infinite but has only finitely many accumulation points.

## Remarks

The set of switching times is closed, and on each interval between consecutive switching times the control is almost everywhere constant.
Since $[a, b]$ is compact, the switching times of a chattering control have an accumulation point.
The condition on accumulation points excludes, for example, continuous nonconstant controls such as $u(t) = t$, for which every point is a switching time.
Example: on $[0, 1]$ let $u(t) = (-1)^k$ for $1 - 2^{-k} \le t < 1 - 2^{-k-1}$, $k = 0, 1, 2, \ldots$, and $u(1) = 1$.
Each point $1 - 2^{-k}$ with $k \ge 1$ is a switching time, since $u$ takes the value $1$ on one side and $-1$ on the other.
So $u$ is a chattering [[bang–bang control|GK1Z3Q1WRZ]], and its switching times accumulate at $1$.
Non-example: $u(t) = -1$ for $t < 1/2$ and $u(t) = 1$ for $t \ge 1/2$ has only one switching time, $1/2$.
Some authors also use the word for a sequence of fast-switching ordinary controls that approximates a relaxed control.
</document>

<document source="https://tamnd.github.io/math2/node/WKM1B317YB.md" tag="WKM1B317YB" kind="definition">
# Fredholm minors

## Definition

Let $a < b$ and let $K \colon [a, b] \times [a, b] \to \C$ be a [[continuous kernel|ZM5G144PZG]].
For points $z_1, \dots, z_m$ and $w_1, \dots, w_m$ of $[a, b]$, write
$$K\begin{pmatrix} z_1, \dots, z_m \\ w_1, \dots, w_m \end{pmatrix} = \det[K(z_i, w_j)]_{i,j=1}^{m},$$
the [[determinant|00KDZRW4C0]] of the $m \times m$ matrix with entries $K(z_i, w_j)$.
Let $n \ge 1$, and let $x_1, \dots, x_n, y_1, \dots, y_n \in [a, b]$.
The *Fredholm minor of order $n$* of $K$ is the function
$$D_n\begin{pmatrix} x_1, \dots, x_n \\ y_1, \dots, y_n \end{pmatrix}(\lambda) = \sum_{k=0}^{\infty} \frac{(-\lambda)^k}{k!} \int_a^b \cdots \int_a^b K\begin{pmatrix} x_1, \dots, x_n, t_1, \dots, t_k \\ y_1, \dots, y_n, t_1, \dots, t_k \end{pmatrix} dt_1 \cdots dt_k$$
of $\lambda \in \C$, where the term with $k = 0$ is $K\begin{pmatrix} x_1, \dots, x_n \\ y_1, \dots, y_n \end{pmatrix}$.
The minor of order $1$ is the *first minor*, written $D(x, y; \lambda)$.

## Remarks

The series converges for every $\lambda$, uniformly in the points, by Hadamard's bound on determinants.
So each minor is an entire function of $\lambda$ and a continuous function of the points.
With $n = 0$ the same formula gives the [[Fredholm determinant|PPX0AK81QV]] $D(\lambda)$.

Example: for $K(x, y) = xy$ on $[0, 1]$, every matrix $[z_i w_j]$ of size at least $2$ has rank $1$, so its determinant is $0$.
Hence $D(x, y; \lambda) = xy$, and all minors of order $n \ge 2$ vanish identically.

Non-example: the series $\sum_k \frac{(-\lambda)^k}{k!} \int K\begin{pmatrix} t_1, \dots, t_k \\ t_1, \dots, t_k \end{pmatrix} dt$, with no fixed points, is not a minor of order $n \ge 1$; it is the determinant $D(\lambda)$ itself.

Some authors put an extra factor $\lambda^n$ in front of the minor of order $n$; the formulas that involve the minors change accordingly.
</document>

<document source="https://tamnd.github.io/math2/node/WKYG28Z8ER.md" tag="WKYG28Z8ER" kind="definition">
# Unirational variety

## Definition

Let $k$ be a field and let $X$ be an integral [[algebraic variety|HV6YBS0YDH]] over $k$.
The variety $X$ is *unirational* over $k$ if there are an integer $N \ge 1$ and a dominant [[rational map|4D073CCYGV]] $\mathbb{P}^N_k \dashrightarrow X$ from [[projective space|ZT9A6JK8QF]].

## Remarks

Equivalently, the [[function field|2D5SS6WM14]] $k(X)$ embeds over $k$ into a purely transcendental extension $k(t_1, \ldots, t_N)$.
Example: every rational variety is unirational, since a birational map is dominant.
Example: the curve $y^2 = x^3$ in $\mathbb{A}^2$ is unirational.
The map $t \mapsto (t^2, t^3)$ is a dominant morphism from $\mathbb{A}^1$, an open subset of $\mathbb{P}^1$, since $(t^3)^2 = (t^2)^3$ and its image is the whole curve.
Non-example: a smooth projective curve $E$ of genus $1$ over $\C$ is not unirational.
A dominant rational map $\mathbb{P}^N \dashrightarrow E$ restricts to a nonconstant rational map on a general line, and this extends to a nonconstant morphism $f \colon \mathbb{P}^1 \to E$.
The Riemann–Hurwitz formula would then give $-2 = 2g(\mathbb{P}^1) - 2 \ge \deg f \cdot (2g(E) - 2) = 0$, which is false.
</document>

<document source="https://tamnd.github.io/math2/node/WM4QWH5F65.md" tag="WM4QWH5F65" kind="definition">
# Pseudocompact space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *pseudocompact* if every [[continuous map|0VGFHTM75Y]] $f \colon X \to \R$ is bounded, that is, there is a real number $M$ with $\abs{f(x)} \le M$ for all $x \in X$.

## Remarks

Example: a [[compact space|0TVJZYB6SH]] $X$ is pseudocompact.
The image $f(X)$ of a continuous $f \colon X \to \R$ is compact in $\R$, so it is bounded.

Non-example: the real line $\R$ is not pseudocompact.
The identity map $f(x) = x$ is continuous and takes arbitrarily large values.

Some authors require in addition that $X$ be a [[Tychonoff space|C5N2J2K6Z3]].
A normal Hausdorff pseudocompact space is [[countably compact|S3N3CHZV89]].
</document>

<document source="https://tamnd.github.io/math2/node/WM5FNBB7K0.md" tag="WM5FNBB7K0" kind="statement">
# Gamma of one half

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]].
Then
$$\Gamma\!\left(\tfrac12\right) = \sqrt{\pi}.$$

## Remarks

By definition $\Gamma(\tfrac12) = \int_0^\infty t^{-1/2} e^{-t}\, dt$.
The substitution $t = x^2$, $dt = 2x\,dx$, turns this into $2\int_0^\infty e^{-x^2}\,dx$, and the [[Gaussian integral|0DFT1GP1JP]] gives $2 \cdot \tfrac{\sqrt{\pi}}{2} = \sqrt{\pi}$.
Another proof uses the [[reflection formula|WF7KQAQTS2]] at $z = \tfrac12$: $\Gamma(\tfrac12)^2 = \pi/\sin(\tfrac{\pi}{2}) = \pi$, and $\Gamma(\tfrac12) > 0$.
With the functional equation it gives all half-integer values, for example $\Gamma(\tfrac32) = \tfrac{\sqrt\pi}{2}$ and $\Gamma(-\tfrac12) = -2\sqrt{\pi}$.
In Mathlib the statement is `Complex.Gamma_one_half_eq`.

## Explanations

### L1 human

The gamma function at one half equals the square root of $\pi$, about $1.7725$.
In factorial language, this says that "$(-\tfrac12)!$" is $\sqrt{\pi}$.

### L2 human

The value is the gamma-function form of the Gaussian integral $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$.
It is the reason $\pi$ appears in the volume of balls in odd dimension and in the normalisation of the normal and chi-squared distributions.
</document>

<document source="https://tamnd.github.io/math2/node/WM62GKCBWN.md" tag="WM62GKCBWN" kind="concept">
# Nonstandard analysis

## Summary

Nonstandard analysis is a way of doing analysis with genuine infinitesimal and infinite numbers.
Abraham Robinson founded it in the 1960s, using model theory to build an ordered field of [[hyperreal numbers|421QMTFG9T]] that extends the [[real numbers|08J2M1JYAG]].
The [[transfer principle|3RT74FQYG1]] lets first-order statements pass between the reals and the hyperreals.
The [[standard part function|1ZBY4W4E81]] rounds a finite hyperreal to the nearest real.
With these tools, limits, derivatives and integrals get short definitions close to the intuition of Leibniz.
The theory proves the same theorems about real numbers as ordinary analysis.
An axiomatic version is Nelson's [[internal set theory|V0PGBXYJ9R]].

## Explanations

### L1 human

The founders of calculus spoke of infinitely small quantities, but for a long time no one could make this precise.
Nonstandard analysis makes it precise: there is a larger number system with positive numbers smaller than every positive real.
For example, to differentiate $f(x) = x^2$, take an infinitesimal $\varepsilon \ne 0$.
Then $(f(x + \varepsilon) - f(x)) / \varepsilon = 2x + \varepsilon$, and rounding away the infinitesimal gives $f'(x) = 2x$.
</document>

<document source="https://tamnd.github.io/math2/node/WMB2MAEDX4.md" tag="WMB2MAEDX4" kind="concept">
# Sequential space

## Summary

A sequential space is a topological space whose topology is determined by its convergent sequences.
A set is closed in it exactly when it contains the limits of all its convergent sequences.
Every [[first-countable space|0CP5NDD21C]], in particular every metric space, is sequential.
In a sequential space, sequences are enough to test closedness and continuity, so nets or filters are not needed.

## Explanations

### L1 human

In a metric space, you can decide whether a set is closed by checking sequences: a closed set keeps the limits of its sequences.
In a general space this test can fail, because sequences may be too short to reach some points.
Sequential spaces are the spaces where the sequence test still works.
For example, in $\R$ the set $(0, 1]$ is not closed, and the sequence $1/n$ in it shows this, because its limit $0$ is outside.
</document>

<document source="https://tamnd.github.io/math2/node/WMBJSNG57F.md" tag="WMBJSNG57F" kind="statement">
# Baker's theorem

## Statement

Let $\overline{\Q}$ be the field of complex [[algebraic numbers|0MNF0BYFQC]].
Let $\alpha_1, \dots, \alpha_n$ be nonzero algebraic numbers, and let $\lambda_i$ be a value of the [[complex logarithm|02QGGD5G87]] of $\alpha_i$ for each $i$.
Suppose that $\lambda_1, \dots, \lambda_n$ are [[linearly independent|0CRMGPAH83]] over $\Q$.
Then $1, \lambda_1, \dots, \lambda_n$ are linearly independent over $\overline{\Q}$.

## Remarks

The proof extends the method of Gelfond.
It builds an auxiliary function in several variables that vanishes to high order at many points, then uses an extrapolation step and an estimate to reach a contradiction.
The method also gives an effective lower bound.
Let $\Lambda = \beta_0 + \beta_1 \lambda_1 + \dots + \beta_n \lambda_n$ with algebraic $\beta_i$, and suppose $\Lambda \ne 0$.
Then $\abs{\Lambda}$ is bounded below by an explicit function of the degrees and heights of the $\alpha_i$ and $\beta_i$.
The [[Gelfond–Schneider theorem|RN58Z2NWAC]] is a special case.
Another consequence: $e^{\beta_0} \alpha_1^{\beta_1} \cdots \alpha_n^{\beta_n}$ is [[transcendental|0VTQXTCR2G]] for nonzero algebraic $\alpha_i$ and $\beta_i$.
For example, $\pi + \log 2$ is transcendental: indeed $\pi = -i \log(-1)$, and $\log(-1) = i\pi$ and $\log 2$ are linearly independent over $\Q$.
The hypothesis of independence over $\Q$ is necessary: $\log 4 - 2 \log 2 = 0$ is a relation with algebraic coefficients.

## Explanations

### L1 human

Logarithms of algebraic numbers behave as independently as they possibly can.
If no relation with rational coefficients links them, then no relation with algebraic coefficients does either.
The quantitative form says that such a combination, when it is not zero, cannot be extremely close to zero.

### L2 human

The effective lower bounds for linear forms in logarithms are the key input for many effective results in Diophantine equations.
They give explicit bounds for the solutions of [[Thue equations|3Q6CFPQZDF]] and of $S$-unit equations.
They were also used in the solution of the [[class number problem|KTGAS9DGRX]] for imaginary quadratic fields of class number one.
The theorem is a step toward [[Schanuel's conjecture|65PV4VFP9Y]], which would give algebraic independence instead of linear independence.
</document>

<document source="https://tamnd.github.io/math2/node/WMGA92P2G4.md" tag="WMGA92P2G4" kind="statement">
# Corona theorem

## Statement

Let $\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]] and let $H^\infty$ be the algebra of bounded [[holomorphic functions|0DG0Z72CQD]] on $\mathbb{D}$.
Let $f_1, \ldots, f_n \in H^\infty$, and assume there is $\delta > 0$ such that
$$\abs{f_1(z)} + \cdots + \abs{f_n(z)} \ge \delta \quad \text{for all } z \in \mathbb{D}.$$
Then there are $g_1, \ldots, g_n \in H^\infty$ such that
$$f_1 g_1 + \cdots + f_n g_n = 1 \quad \text{on } \mathbb{D}.$$

## Remarks

The hypothesis is necessary: if $\sum f_j g_j = 1$ with $\abs{g_j} \le C$, then $\sum \abs{f_j} \ge 1/C$ on $\mathbb{D}$.
The theorem is equivalent to the density of $\mathbb{D}$ in the space of [[maximal ideals|09J6KGT65G]] of the [[Banach algebra|9KYWN94QDN]] $H^\infty$.
Here $z \in \mathbb{D}$ corresponds to the ideal of functions vanishing at $z$.
The name comes from this picture: without the theorem, there could be a "corona" of maximal ideals away from the closure of the disc.

Proof idea: Carleson proved the theorem in 1962 with a construction that led to [[Carleson measures|SQHGJ9C0QE]].
A shorter proof due to Wolff solves the problem first with smooth functions, then corrects them by solving a $\bar\partial$-equation with bounded solutions, using a Carleson measure estimate.

Example: for $n = 1$ the theorem says that if $\abs{f_1} \ge \delta$ on $\mathbb{D}$, then $1/f_1$ is bounded and holomorphic, which is clear.

## Explanations

### L1 human

If you have finitely many bounded holomorphic functions that never get small at the same time, the theorem says you can combine them to get the constant $1$.
The combination uses bounded holomorphic coefficients.
For polynomials the analogous fact is the statement that polynomials with no common zero generate the unit ideal.
In the disc, "no common zero" must be replaced by "not all close to zero", because zeros can escape to the boundary.

### L2 human

In Banach algebra language, the theorem identifies where the disc sits inside the maximal ideal space of $H^\infty$: it is dense.
Its proof introduced [[Carleson measures|SQHGJ9C0QE]] and interpolating sequences, which became central tools in function theory.
The analogue for the unit ball in $\C^n$ with $n \ge 2$ is a well known open problem.
</document>

<document source="https://tamnd.github.io/math2/node/WMHMGJD261.md" tag="WMHMGJD261" kind="concept">
# Luzin N property

## Summary

A function has the Luzin N property when it maps every set of Lebesgue measure zero to a set of Lebesgue measure zero.
The property singles out the [[absolutely continuous|ZT6F4C2YHK]] functions among the continuous functions of [[bounded variation|A8251BHCMP]], by the [[Banach–Zarecki theorem|ZMEYSAHNX3]].
It also appears in the change of variables formula for Lebesgue integrals and in the study of Sobolev maps.

## Explanations

### L1 human

A [[null set|VQQM4Y550T]] is a set so thin that it has length zero, such as a countable set or the [[Cantor set|0BXNDZFQME]].
A function with the N property never blows up such a thin set into a set of positive length.
Every Lipschitz function has the property, because it stretches lengths by at most a fixed factor.
The [[Cantor function|NPBYYK63FE]] does not: it maps the Cantor set, of length $0$, onto the whole interval $[0, 1]$.
</document>

<document source="https://tamnd.github.io/math2/node/WMNEQ6A8FB.md" tag="WMNEQ6A8FB" kind="definition">
# Regular category

## Definition

A morphism is a *regular epimorphism* if it is the [[coequalizer|DVSTWGVCSZ]] of some pair of morphisms.
A category $\mathcal{C}$ is *regular* if these three conditions hold.

1. $\mathcal{C}$ is [[finitely complete|PYSBNWCHCF]].
2. For every morphism $f$ of $\mathcal{C}$, the [[kernel pair|E70DQGM8X5]] of $f$ has a coequalizer.
3. Regular epimorphisms are stable under [[pullback|Q05K04QJ71]]: the pullback of a regular epimorphism $e \colon A \to B$ along any $g \colon C \to B$ is a regular epimorphism.

## Remarks

A regular epimorphism is an [[epimorphism|RVXDSY04KG]].
In a regular category every morphism $f$ factors as $f = m \circ e$ with $e$ a regular epimorphism and $m$ a monomorphism.
Here $e$ is the coequalizer of the kernel pair of $f$, and $m$ is called the image of $f$.

Example: the category of sets is regular.
It has all finite limits, and the kernel pair of $f \colon A \to B$ is the relation $\set{(a, a') : f(a) = f(a')}$, whose coequalizer is the quotient of $A$ by it.
Regular epimorphisms are the surjections, and a pullback of a surjection is a surjection.

Non-example: the category of topological spaces satisfies clauses 1 and 2 but not clause 3.
Its regular epimorphisms are the quotient maps, and a pullback of a quotient map need not be a quotient map.
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<document source="https://tamnd.github.io/math2/node/WMPCR723MJ.md" tag="WMPCR723MJ" kind="concept">
# Weak and strong extremum

## Summary

In the calculus of variations a local minimum depends on what "nearby" means for functions.
A strong local minimum beats every admissible function whose values are close.
A weak local minimum only beats those whose values and slopes are both close.
Every strong local extremum is a weak one, but not conversely.
The [[Euler–Lagrange equation|PS0VV149ET]] and the [[Legendre condition|4HZX2EN319]] are necessary for weak extrema, while the [[Weierstrass excess function|R8TWRPSPNV]] gives a further necessary condition for strong ones.

## Explanations

### L1 human

Two curves can be close in height but very different in slope, like a straight line and a fine zigzag near it.
A weak minimum only has to win against curves that are close in height and in slope.
A strong minimum must also win against wiggly curves that stay close only in height.
Take the functional $\int_0^1 y'(x)^3 \, dx$ with ends $y(0) = 0$ and $y(1) = 1$.
The line $y = x$ wins against all gentle perturbations.
But a curve that drops steeply for a very short time and then climbs gives a much lower value.
</document>

<document source="https://tamnd.github.io/math2/node/WMR73J0S26.md" tag="WMR73J0S26" kind="concept">
# Pseudocomplement

## Summary

Let $L$ be a [[lattice|WH4SFAACAS]] with a least element $0$, and let $a \in L$.
The pseudocomplement of $a$ is the largest element whose meet with $a$ is $0$, when such an element exists.
It is written $a^*$.
In a [[Boolean algebra|0XADJKM699]] it is the usual complement, and in a [[Heyting algebra|NG943XNQWH]] it is $a \to 0$.
Pseudocomplements give a weak, intuitionistic form of negation.

## Explanations

### L1 human

The pseudocomplement of $a$ is the biggest thing that does not overlap $a$ at all.
For the open sets of the real line, the pseudocomplement of an open set $U$ is the interior of its complement.
If $U = (0, 1)$, this is $(-\infty, 0) \cup (1, \infty)$.
The points $0$ and $1$ belong neither to $U$ nor to $U^*$, so $U \cup U^*$ is not the whole line, unlike a true complement.
</document>

<document source="https://tamnd.github.io/math2/node/WMTZG71K2Y.md" tag="WMTZG71K2Y" kind="concept">
# Richardson extrapolation

## Summary

Richardson extrapolation combines two approximations computed with step sizes $h$ and $h/t$ so that the leading term of their error cancels.
It applies whenever the error of a method has a known expansion in powers of the step size.
Repeated, it turns the trapezoidal rule into Romberg integration, and it is used throughout numerical differentiation, integration and the solution of differential equations.

## Explanations

### L1 human

Suppose a method has error roughly proportional to $h^2$, where $h$ is a step size.
Halving $h$ divides the error by about $4$.
Knowing this, one can combine the two results so that the $h^2$ parts cancel exactly: $\frac{4 A(h/2) - A(h)}{3}$.
What remains is a much smaller error, of higher order in $h$.
</document>

<document source="https://tamnd.github.io/math2/node/WMVAT9MBZ1.md" tag="WMVAT9MBZ1" kind="statement">
# Riemann–Roch theorem

## Statement

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]], and let $X$ be a smooth projective [[algebraic curve|XJM2MT4S3Y]] over $k$ of [[genus|6W8TDV9VVN]] $g$.
Let $k(X)$ be the field of rational functions on $X$.
For a [[divisor|Z9DN23DAE2]] $D$ on $X$, let
$$L(D) = \set{f \in k(X)^\times : \operatorname{div}(f) + D \ge 0} \cup \set{0},$$
which is a finite-dimensional vector space over $k$, and let $\ell(D) = \dim_k L(D)$.
Let $K$ be a [[canonical divisor|W05VYY4HBR]] of $X$.
Then for every divisor $D$ on $X$,
$$\ell(D) - \ell(K - D) = \deg D - g + 1.$$

## Remarks

In sheaf language, $\ell(D) = h^0(X, \mathcal{O}_X(D))$ and $\ell(K - D) = h^1(X, \mathcal{O}_X(D))$ by Serre duality, so the formula reads $\chi(\mathcal{O}_X(D)) = \deg D + 1 - g$.
One proof checks the formula for $D = 0$, where it says $h^1(\mathcal{O}_X) = g$.
Then it shows that both sides change by $1$ when a point $P$ is added to $D$.
This uses the exact sequence $0 \to \mathcal{O}_X(D) \to \mathcal{O}_X(D + P) \to k_P \to 0$.
Taking $D = 0$ and $D = K$ gives $\ell(K) = g$ and $\deg K = 2g - 2$.
If $\deg D > 2g - 2$, then $\deg(K - D) < 0$, so $\ell(K - D) = 0$ and $\ell(D) = \deg D - g + 1$ exactly.
Example: on the projective line, $g = 0$ and $K = -2P$ for any point $P$.
For $D = nP$ with $n \ge 0$ the formula gives $\ell(D) = n + 1$.
This is the dimension of the space of polynomials of degree at most $n$ in an affine coordinate.

## Explanations

### L1 human

A divisor prescribes where a function may have poles and how bad they may be, and where it must vanish.
The theorem counts how many independent functions obey such a prescription.
The answer is the degree of the prescription, corrected by the genus of the curve and by a second count that involves the canonical divisor.
For prescriptions of large degree the correction term disappears, and the count is simply $\deg D - g + 1$.

### L2 human

Riemann–Roch is the main computational tool of the theory of curves.
It shows that a divisor of degree at least $2g + 1$ gives an embedding into projective space.
It also shows that every curve of genus $0$ is the projective line, and that every curve of genus $1$ with a point is a plane cubic.
It feeds directly into [[Clifford's theorem|RSW29AVKGM]], the [[Riemann–Hurwitz formula|42A5ZT6AWN]] and Brill–Noether theory.
Its higher-dimensional generalizations are the Hirzebruch–Riemann–Roch and Grothendieck–Riemann–Roch theorems.
</document>

<document source="https://tamnd.github.io/math2/node/WMXDVF36ZT.md" tag="WMXDVF36ZT" kind="concept">
# Wirtinger derivatives

## Summary

The Wirtinger derivatives $\partial / \partial z$ and $\partial / \partial \bar z$ are two first-order differential operators on functions of $z = x + iy$, see the [[definition|698R12MY4G]].
They let one treat $z$ and $\bar z$ as if they were independent variables.
A continuously differentiable function is [[holomorphic|0NAHXR1FRH]] exactly when its $\partial / \partial \bar z$ derivative is zero, which is a compact form of the [[Cauchy–Riemann equations|0T71SSYZ39]].

## Explanations

### L1 human

A function of a complex variable is also a function of two real variables $x$ and $y$.
The Wirtinger derivatives repackage the two partial derivatives in $x$ and $y$ into two complex ones.
On a polynomial in $z$ and $\bar z$, the derivative in $z$ differentiates in $z$ while holding $\bar z$ fixed.
The derivative in $\bar z$ does the opposite.
For example, $\abs{z}^2 = z \bar z$ has derivative $\bar z$ in $z$ and derivative $z$ in $\bar z$, just as if $z$ and $\bar z$ were separate letters.
</document>

<document source="https://tamnd.github.io/math2/node/WN43DRPNBX.md" tag="WN43DRPNBX" kind="concept">
# Morley rank

## Summary

Morley rank is an ordinal-valued dimension for definable sets in a first-order structure.
A definable set has rank at least $\alpha + 1$ when it contains infinitely many pairwise disjoint definable subsets of rank at least $\alpha$.
In an algebraically closed field, the Morley rank of a constructible set equals its dimension in the sense of algebraic geometry.
A complete theory in a countable language is $\omega$-stable exactly when every formula has ordinal Morley rank.
The notion is central in [[stable theories|T7JW5QY6KR]] and in Morley's proof of his categoricity theorem.

## Explanations

### L1 human

Morley rank measures the size of a definable set by how many times one can split it into infinitely many smaller pieces.
A finite nonempty set has rank $0$, since it cannot be split into infinitely many nonempty pieces.
The complex line has rank $1$: it splits into infinitely many points, but its definable subsets are finite or cofinite, so it does not split into infinitely many infinite pieces.
The complex plane has rank $2$, since it splits into infinitely many disjoint lines.
</document>

<document source="https://tamnd.github.io/math2/node/WN6M8B3ZKB.md" tag="WN6M8B3ZKB" kind="concept">
# Series in Banach spaces

## Summary

A series $\sum x_n$ of vectors in a [[Banach space|06GZ956657]] converges when its partial sums converge in norm.
It converges absolutely when the real series $\sum \norm{x_n}$ converges.
[[Absolute convergence implies convergence|082MGNWNA8]], because the space is complete.
In infinite dimensions the converse fails more sharply than in $\R$: there are series that converge after every rearrangement but not absolutely.
Series of this kind are used everywhere in analysis, for example in the Neumann series $\sum T^n$ for an operator $T$ of norm less than $1$.

## Explanations

### L1 human

A series of vectors is added term by term, like a series of numbers, and one asks whether the partial sums approach a vector.
The size of a vector is its norm, so convergence means that the norm of the difference between the partial sums and the limit tends to $0$.
If the sizes $\norm{x_n}$ add up to a finite number, the series converges, just as for numbers.
For example, in the space $\ell^2$ of square-summable sequences, with $e_n$ the $n$-th unit vector, the series $e_1 + e_2/2 + e_3/3 + \cdots$ can converge even though $1 + 1/2 + 1/3 + \cdots$ diverges.
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<document source="https://tamnd.github.io/math2/node/WN8PK6RQZR.md" tag="WN8PK6RQZR" kind="concept">
# Divisor class group

## Summary

Let $A$ be a [[Krull domain|CB9WJCN9BM]].
Its divisor class group is the group of formal integer combinations of its height-one primes, modulo the divisors of nonzero fractions.
It is trivial exactly when the ring is a [[unique factorisation domain|06YMXJMXYS]], so it measures the failure of unique factorization.
For a Dedekind domain it is the ideal class group, and it contains the [[Picard group|DZW58R6T2Q]] of the ring.

## Explanations

### L1 human

In a ring with unique factorization, every height-one prime ideal is generated by one prime element.
In other rings some height-one primes are not principal, and the class group records which combinations of them are principal.
In $\Z[\sqrt{-5}]$, the prime $P = (2, 1 + \sqrt{-5})$ is not principal, but $P^2 = (2)$ is.
So the class of $P$ is a nonzero element of order $2$; in fact the class group of $\Z[\sqrt{-5}]$ has exactly two elements.
</document>

<document source="https://tamnd.github.io/math2/node/WN9V7WSJKF.md" tag="WN9V7WSJKF" kind="concept">
# Lattice (group)

## Summary

A lattice in $\R^n$ is the set of all integer combinations of $n$ linearly independent vectors.
Equivalently, it is a discrete subgroup of $\R^n$ that spans $\R^n$.
As a group it is [[free abelian|0WJX8MGG37]] of rank $n$, and its covolume, the volume of a fundamental parallelepiped, measures how dense it is.
Lattices are the basic objects of the geometry of numbers, through [[Minkowski's theorem|0K387S2PKQ]], and they appear in number fields, in crystallography and in cryptography.

## Explanations

### L1 human

Picture the points with integer coordinates in the plane: a regular grid.
Any grid obtained by taking two independent arrows and all their integer combinations is a lattice.
Squares, slanted parallelograms and the hexagonal pattern of a honeycomb all come from lattices.
For example, the vectors $(1, 0)$ and $(1/2, \sqrt{3}/2)$ generate the hexagonal lattice.
Different pairs of arrows can generate the same lattice, but the area of the parallelogram they span is always the same.
</document>

<document source="https://tamnd.github.io/math2/node/WND87FBAKC.md" tag="WND87FBAKC" kind="concept">
# Stopping time

## Summary

A stopping time is a random time $\tau$ with the following property.
At each time $t$, one can decide whether $\tau$ has already happened using only the information available at time $t$.
First hitting times of sets by adapted processes are the main examples.
Stopping times are central in martingale theory, through the [[optional stopping theorem|DC9HZT2HVJ]], and in the strong Markov property and optimal stopping.

## Explanations

### L1 human

A stopping time is a rule for when to stop that never needs to look into the future.
"Stop the first time the coin shows heads" is such a rule: when it happens, you know it.
"Stop at the last head before the hundredth toss" is not, because you cannot know that a head is the last one until later.
For a random walk started at $0$, the first time it reaches $5$ is a stopping time.
</document>

<document source="https://tamnd.github.io/math2/node/WNFEPEBART.md" tag="WNFEPEBART" kind="statement">
# Antithetic variates

## Statement

Let $f \colon (0, 1) \to \R$ be monotone (non-decreasing or non-increasing) with $\int_0^1 f(u)^2\, du < \infty$, and let $U$ have the [[uniform distribution|0AREFQ9Z65]] on $(0, 1)$.
Let $Z = \frac12 \bigl( f(U) + f(1 - U) \bigr)$.
Then:

1. $\mathbb E Z = \mathbb E f(U) = \int_0^1 f(u)\, du$;
2. $\operatorname{Cov}\bigl(f(U), f(1 - U)\bigr) \le 0$;
3. $\operatorname{Var} Z \le \frac12 \operatorname{Var} f(U)$.

Both $n$ copies of $Z$ and $2n$ copies of $f(U)$ need $2n$ evaluations of $f$.
The average of $n$ independent copies of $Z$ is at least as accurate as that of $2n$ independent copies of $f(U)$.

## Remarks

Clause 1 holds because $1 - U$ is uniform on $(0, 1)$ too.
For clause 2, if $f$ is non-decreasing then $u \mapsto f(1 - u)$ is non-increasing, and for a non-decreasing $a$ and a non-increasing $b$ one has $\mathbb E[a(U) b(U)] \le \mathbb E a(U)\, \mathbb E b(U)$, the integral form of [[Chebyshev's sum inequality|S8X2N1DW8P]].
This in turn follows from $\mathbb E\bigl[(a(U) - a(V))(b(U) - b(V))\bigr] \le 0$ for an independent copy $V$ of $U$, since the two factors have opposite signs.
Clause 3: $\operatorname{Var} Z = \frac14 \bigl( 2 \operatorname{Var} f(U) + 2 \operatorname{Cov}(f(U), f(1-U)) \bigr) \le \frac12 \operatorname{Var} f(U)$, using $\operatorname{Var} f(1 - U) = \operatorname{Var} f(U)$.
Example: for $f(u) = e^u$, $\operatorname{Var} f(U) \approx 0.2420$ while $\operatorname{Var} Z \approx 0.0039$, a gain of a factor of about $30$ over $2$ independent samples.
Monotonicity is needed: for $f(u) = (u - \frac12)^2$ one has $f(1 - u) = f(u)$, so $Z = f(U)$ and $\operatorname{Var} Z = \operatorname{Var} f(U)$, twice the variance of the average of two independent samples.

## Explanations

### L1 human

The idea is to sample in pairs that are opposite to each other: $U$ and $1 - U$.
If one value of the function is too high, its partner tends to be too low, and the errors partly cancel in the average.
For a function that always increases or always decreases, the method is guaranteed not to hurt and often helps a lot.

### L2 human

Antithetic variates is one of the simplest [[variance reduction|A01GEGWKVB]] techniques, with no extra modelling and almost no extra cost.
The same idea applies to normal inputs with the pair $Z$ and $-Z$, and to simulations driven by many uniforms, replacing each $U_j$ by $1 - U_j$.
In the multivariate case the guarantee holds when the output is a monotone function of each input separately, but for general outputs the method can increase the variance.
Combined with inverse transform sampling, which is monotone, it yields negatively correlated pairs from any distribution.
</document>

<document source="https://tamnd.github.io/math2/node/WNHXE7FCG4.md" tag="WNHXE7FCG4" kind="concept">
# John ellipsoid

## Summary

The John ellipsoid of a convex body $K$ in $\R^n$ with nonempty interior is the ellipsoid of largest volume contained in $K$.
It exists and is unique by the [[Löwner–John theorem|TM5SYH4XHY]], and $K$ is contained in its dilate by the factor $n$ about its center.
It gives a canonical affine position of $K$.
It is the starting point of many estimates in convex geometry and the geometry of normed spaces.
Its dual partner is the Löwner ellipsoid, the ellipsoid of least volume containing $K$.

## Explanations

### L1 human

Inside any convex shape one can fit ellipses (or ellipsoids, in space) of many sizes and directions.
Among them there is exactly one of largest area: the John ellipsoid.
For a square it is the inscribed circle, and for a long rectangle it is the inscribed ellipse that touches all four sides at their midpoints.
Every convex shape is squeezed between its John ellipsoid and a copy of it blown up by a factor equal to the dimension.
</document>

<document source="https://tamnd.github.io/math2/node/WNHZC07XVP.md" tag="WNHZC07XVP" kind="conjecture">
# Feder–Vardi dichotomy conjecture

## Statement

Let $\mathbb{B}$ be a finite [[relational structure|MJF9M19E3E]] with finitely many relations.
The problem $\mathrm{CSP}(\mathbb{B})$ takes as input a finite structure $\mathbb{A}$ of the same signature.
It asks whether there is a [[homomorphism|KBXSZC6BV1]] from $\mathbb{A}$ to $\mathbb{B}$.
The conjecture states that every such constraint satisfaction problem $\mathrm{CSP}(\mathbb{B})$ is either solvable in polynomial time or NP-complete.

## Remarks

The conjecture is now a theorem: it follows from the [[CSP dichotomy theorem|ZA4YXJ9T1P]], proved independently by Andrei Bulatov and by Dmitriy Zhuk.
It is not a consequence of general complexity theory.
By Ladner's theorem, if $\mathrm{P} \ne \mathrm{NP}$ then NP contains problems that are neither in P nor NP-complete.
Feder and Vardi studied a logically defined fragment of NP called monotone monadic SNP, defined by three syntactic restrictions.
They showed that its problems are equivalent to fixed-template CSPs.
They also showed that dropping any one of the three restrictions gives a class that contains, up to polynomial-time equivalence, every problem in NP.
So, assuming $\mathrm{P} \ne \mathrm{NP}$, a dichotomy can only hold for the restricted class, and CSPs are its natural representatives.
Before the proof, the conjecture was known for structures with a two-element domain (Schaefer), for undirected graphs (Hell and Nešetřil), and for three-element domains (Bulatov).

## Explanations

### L1 human

Fix a finite set of labels and a finite list of rules saying which combinations of labels are allowed.
The task is then: given a finite input, can its points be labelled so that every rule is obeyed?
Feder and Vardi guessed that each such task is either easy, with a fast algorithm, or as hard as the hardest problems in NP.
There would be no task of intermediate difficulty in this family, even though such tasks exist elsewhere in NP if P differs from NP.
</document>

<document source="https://tamnd.github.io/math2/node/WNJPRHZ680.md" tag="WNJPRHZ680" kind="statement">
# Szegő limit theorems

## Statement

Let $f \colon \R \to [0, \infty)$ be a $2\pi$-periodic function, [[integrable|X7GTF3D98F]] on $[-\pi, \pi]$, with [[Fourier coefficients|5NN2YZZR3V]] $\hat f(k) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(\theta) e^{-ik\theta} \, d\theta$.
For $n \ge 1$ let $T_n(f) = (\hat f(j - k))_{j,k=0}^{n-1}$ be the $n \times n$ [[Toeplitz matrix|GKY2WSHE26]] of $f$, and let $D_n(f) = \det T_n(f)$ be its [[determinant|00KDZRW4C0]].

1. (First Szegő theorem.) If $\log f$ is integrable, then
$$\lim_{n \to \infty} D_n(f)^{1/n} = G(f), \qquad G(f) = \exp\left( \frac{1}{2\pi} \int_{-\pi}^{\pi} \log f(\theta) \, d\theta \right).$$
2. (Strong Szegő theorem.) Assume $f = e^{g}$ with $g$ real-valued and continuous, with Fourier coefficients satisfying $\sum_{k \in \Z} \abs{\hat g(k)} < \infty$ and $\sum_{k \in \Z} \abs{k} \abs{\hat g(k)}^2 < \infty$.
Then
$$\lim_{n \to \infty} \frac{D_n(f)}{G(f)^n} = \exp\left( \sum_{k=1}^{\infty} k \, \hat g(k) \hat g(-k) \right) = \exp\left( \sum_{k=1}^{\infty} k \abs{\hat g(k)}^2 \right).$$

## Remarks

Proof idea for the first theorem: the ratio $D_{n+1}(f) / D_n(f)$ is the minimum of $\frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{p(e^{i\theta})}^2 f(\theta) \, d\theta$ over polynomials $p(z) = 1 + a_1 z + \cdots + a_n z^n$.
These minima decrease to $G(f)$ by [[Szegő's theorem|SZ8ARMDEMY]] on prediction, and then the geometric means $D_n^{1/n}$ have the same limit.
The strong theorem gives the next term of the asymptotics, $D_n(f) \sim E(f) \, G(f)^n$ with the constant $E(f)$ above.

Example: let $f(\theta) = \abs{1 - a e^{i\theta}}^2$ with $\abs{a} < 1$.
Then $T_n(f)$ is tridiagonal with diagonal $1 + \abs{a}^2$, and $D_n(f) = \frac{1 - \abs{a}^{2(n+1)}}{1 - \abs{a}^2}$.
Here $g = \log f$ has $\hat g(0) = 0$ and $\hat g(k) = -a^k / k$ for $k \ge 1$, so $G(f) = 1$ and the strong theorem predicts the limit $\exp\left(\sum_{k \ge 1} \abs{a}^{2k}/k\right) = \frac{1}{1 - \abs{a}^2}$, as the formula confirms.

Szegő proved the first theorem in 1915 and the strong theorem in 1952, motivated by the computation of the spontaneous magnetization of the two-dimensional Ising model.
The strong theorem was later proved under weaker hypotheses, essentially only the condition $\sum \abs{k} \abs{\hat g(k)}^2 < \infty$.

## Explanations

### L1 human

A Toeplitz matrix has constant entries along each diagonal, and these entries are the Fourier coefficients of a function on the circle.
The Szegő theorems say how the determinants of larger and larger such matrices grow.
To first order, they grow like $G^n$, where $G$ is the geometric mean of the function.
The strong theorem gives the precise constant factor in front.

### L2 human

The theorems are the analogue for Toeplitz matrices of the fact that the determinant of a diagonal matrix is the product of its entries.
The first theorem is closely tied to [[orthogonal polynomials on the unit circle|3F9G3DRQ62]] and to prediction theory of stationary sequences.
The strong theorem has applications in statistical mechanics and in random matrix theory, where such determinants compute averages over the unitary group.
</document>

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# Galois theory

## Summary

Galois theory studies a field extension through its group of automorphisms.
Its central result, the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], matches the intermediate fields of a finite [[Galois extension|0DS2F7GK05]] with the subgroups of its [[Galois group|0XVR72M51M]].
It explains which polynomial equations are [[solvable by radicals|J7YS0DA3HQ]] and which constructions with straightedge and compass are possible.

## Explanations

### L1 human

The roots of a polynomial can often be permuted without breaking any algebraic relation among them.
These permutations form a group, and Galois theory reads properties of the equation off this group.
For $x^2 - 2$, the two roots $\sqrt{2}$ and $-\sqrt{2}$ can be exchanged, and the group has two elements.
Over a field of characteristic $0$, a polynomial can be solved with roots and the four basic operations exactly when its group is [[solvable|0P3KZK0FTV]].
This is why there is a formula for degree $4$ and none for the general equation of degree $5$.
</document>

<document source="https://tamnd.github.io/math2/node/WNQWERBWBB.md" tag="WNQWERBWBB" kind="problem">
# Hilbert's thirteenth problem

## Problem

Consider the equation of degree seven
$$x^7 + a x^3 + b x^2 + c x + 1 = 0,$$
and its solution $x$ as a function of the three coefficients $a, b, c$.
Can this function be written as a finite [[composition|0THBK9V825]] of [[continuous functions|0X3WGX95FE]] of two variables?

## Remarks

Hilbert expected the answer to be negative.
Every equation of degree seven can be brought to the form above by Tschirnhaus transformations, so the question asks whether the general septic is solvable by functions of two variables.

In 1957 Andrey Kolmogorov and Vladimir Arnold showed that the answer is positive for continuous functions, in a strong form.
Arnold proved that every continuous function of three variables is a composition of continuous functions of two variables.
Kolmogorov's superposition theorem goes further: every continuous $f \colon [0, 1]^n \to \R$ can be written as
$$f(x_1, \dots, x_n) = \sum_{q=0}^{2n} \Phi_q\Big( \sum_{p=1}^{n} \varphi_{q, p}(x_p) \Big)$$
with continuous functions $\Phi_q$ and $\varphi_{q, p}$ of one variable.
So addition is the only function of two variables that is needed.

Many authors read Hilbert's question as being about algebraic functions instead of continuous ones; in that form the problem is still open.

## Explanations

### L1 human

Some functions of several variables secretly need only one or two variables at a time, such as $\sin(x + y z)$, built from multiplication, addition and the sine.
Hilbert asked whether the solution of a particular seventh-degree equation, which depends on three numbers, can be built in this way from continuous functions of two variables.
Kolmogorov and Arnold showed that, surprisingly, every continuous function of any number of variables can be built from continuous functions of one variable and addition.
</document>

<document source="https://tamnd.github.io/math2/node/WNSAZD8EZC.md" tag="WNSAZD8EZC" kind="definition">
# Contact manifold

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of odd dimension $2n + 1$.
A *contact structure* on $M$ is a smooth field $\xi$ of hyperplanes $\xi_x \subset T_x M$ with the following property.
Each point has an open neighborhood $U$ and a [[differential $1$-form|0E3G0GRPZ6]] $\alpha$ on $U$ with $\xi = \ker \alpha$ on $U$ and
$$\alpha \wedge (d\alpha)^n \ne 0 \quad \text{at every point of } U.$$
Here $d$ is the [[exterior derivative|0ZHSTAG8ZY]].
A *contact manifold* is a pair $(M, \xi)$ with $\xi$ a contact structure.
A $1$-form $\alpha$ defined on all of $M$ with $\ker \alpha = \xi$ is a *contact form* for $\xi$.

## Remarks

The condition does not depend on the choice of $\alpha$: if $\alpha' = h \alpha$ with $h$ nowhere zero, then $\alpha' \wedge (d\alpha')^n = h^{n+1} \alpha \wedge (d\alpha)^n$.
Equivalently, $d\alpha$ restricted to each hyperplane $\xi_x$ is nondegenerate.
A contact form exists globally exactly when $\xi$ is coorientable.

Example: on $\R^3$ with coordinates $(x, y, z)$, let $\alpha = dz - y\,dx$.
Then $d\alpha = dx \wedge dy$ and $\alpha \wedge d\alpha = dz \wedge dx \wedge dy = dx \wedge dy \wedge dz \ne 0$, so $\ker \alpha$ is a contact structure, the standard one.
The unit sphere $S^{2n+1} \subset \C^{n+1}$ with $\alpha = \sum_j (x_j\, dy_j - y_j\, dx_j)$ restricted to it is another standard example.

Non-example: on $\R^3$ the form $\alpha = dz$ has $d\alpha = 0$, so $\alpha \wedge d\alpha = 0$; its kernel is tangent to the planes $z = \text{const}$.
By the [[Frobenius theorem|5Q4BHBFXJY]], $\alpha \wedge d\alpha = 0$ is exactly the condition for $\ker \alpha$ to be tangent to a family of surfaces.
</document>

<document source="https://tamnd.github.io/math2/node/WNVACS2S2Z.md" tag="WNVACS2S2Z" kind="concept">
# Incomplete beta function

## Summary

The incomplete beta function $B(x; a, b)$ is the integral of $t^{a-1}(1-t)^{b-1}$ from $0$ to $x$, for $0 \le x \le 1$.
At $x = 1$ it is the [[beta function|QHNWVSC6AQ]] $B(a, b)$.
Divided by $B(a, b)$ it becomes the regularised incomplete beta function $I_x(a, b)$, the distribution function of the [[beta distribution|0EW824Z7G2]].
It also gives the distribution functions of the binomial distribution and of Student's $t$-distribution.

## Explanations

### L1 human

The beta function is the area under the curve $t^{a-1}(1-t)^{b-1}$ between $0$ and $1$.
The incomplete beta function keeps only the part of this area to the left of a point $x$.
As $x$ moves from $0$ to $1$, it grows from $0$ to the full beta function.
For $a = b = 1$ the curve is the constant $1$, and the area up to $x$ is just $x$.
</document>

<document source="https://tamnd.github.io/math2/node/WP0YB7VC59.md" tag="WP0YB7VC59" kind="example">
# Category of sets

## Example

The *category of sets*, written $\mathbf{Set}$, has these parts.
Its objects are all [[sets|0BSJ7RQ9BW]].
A morphism from $A$ to $B$ is a [[function|0EPAT51DXQ]] $f \colon A \to B$.
Composition is the composition of functions, $(g \circ f)(a) = g(f(a))$, and the identity of $A$ is the identity function $\id_A$.

This is a [[category|A2DE66937D]].
Composition of functions is associative, since both $(h \circ g) \circ f$ and $h \circ (g \circ f)$ send $a$ to $h(g(f(a)))$.
The identity laws hold, since $f \circ \id_A$ and $\id_B \circ f$ both send $a$ to $f(a)$.
The collection of all sets is not a set, so $\mathbf{Set}$ is a large category.
Each hom-set $\mathbf{Set}(A, B)$ is a set, so $\mathbf{Set}$ is locally small.

## Remarks

The empty set is an initial object and every one-point set is a terminal object.
The monomorphisms are the [[injective functions|09NWETVTNY]] and the epimorphisms are the [[surjective functions|01RVCMSN5N]].
Products are cartesian products and coproducts are disjoint unions; in fact $\mathbf{Set}$ has all small limits and colimits.
It is cartesian closed, with the set of functions $B^A$ as exponential object, and it is an elementary topos whose subobject classifier is a two-element set.

## Explanations

### L1 human

The category of sets is the most basic category: its objects are sets and its arrows are functions.
Most other familiar categories, such as groups or topological spaces, are sets with extra structure and functions that respect it.
Many notions of category theory are modelled on what happens in sets, such as products, unions, images and quotients.
</document>

<document source="https://tamnd.github.io/math2/node/WP12H9FG91.md" tag="WP12H9FG91" kind="definition">
# Whitehead product

## Definition

Let $(X, x_0)$ be a [[pointed space|T9DG8DDE75]], and let $p, q \ge 1$.
Give $S^p \times S^q$ the [[CW structure|5W95QFHXPR]] with cells of dimensions $0$, $p$, $q$ and $p + q$, so that its $(p+q-1)$-skeleton is the [[wedge sum|5JSWM1HBX2]] $S^p \vee S^q$.
Let $w \colon S^{p+q-1} \to S^p \vee S^q$ be the attaching map of the top cell.
For $\alpha = [f] \in \pi_p(X, x_0)$ and $\beta = [g] \in \pi_q(X, x_0)$, the *Whitehead product* is
$$[\alpha, \beta] = [(f \vee g) \circ w] \in \pi_{p+q-1}(X, x_0),$$
where $f \vee g \colon S^p \vee S^q \to X$ is $f$ on $S^p$ and $g$ on $S^q$, and $\pi_k$ are the [[homotopy groups|P976WYEFMB]].

## Remarks

The class $[\alpha, \beta]$ depends only on $\alpha$ and $\beta$.
For $p = q = 1$, the map $w$ traces the boundary of the square in the torus, and $[\alpha, \beta]$ is the commutator $\alpha \beta \alpha^{-1} \beta^{-1}$, up to the choice of conventions.

Example: let $\iota_2 \in \pi_2(S^2)$ be the class of the identity.
Then $[\iota_2, \iota_2] \in \pi_3(S^2) \cong \Z$ has Hopf invariant $\pm 2$, so it is twice a generator, up to sign.

Non-example of a nonzero product: if $X$ is an H-space, for instance a topological group, then all Whitehead products in $X$ vanish.
In particular, every Whitehead product in $S^1$ or $S^3$ is zero.

For $p, q \ge 2$ the product is bilinear and satisfies $[\alpha, \beta] = (-1)^{pq} [\beta, \alpha]$, together with a graded Jacobi identity.
</document>

<document source="https://tamnd.github.io/math2/node/WP20S9NZ15.md" tag="WP20S9NZ15" kind="concept">
# Free lattice

## Summary

The free lattice on a set $X$ is the [[lattice|WH4SFAACAS]] generated by $X$ in which no relations hold except those forced by the lattice laws.
It has a universal property: every map from $X$ into a lattice extends uniquely to a [[lattice homomorphism|BNVRR925X8]] from the free lattice.
Its elements can be described by lattice terms in the letters of $X$.
Deciding when two terms are equal is the [[word problem for free lattices|Z86VGP5RMG]].
The free lattice on two generators has four elements, while the free lattice on three generators is already infinite.

## Explanations

### L1 human

Take some letters $x$, $y$, $z$ and form all expressions with $\vee$ and $\wedge$, such as $x \wedge (y \vee z)$.
Two expressions count as equal only when the rules of every lattice force them to be equal.
The result is the free lattice on these letters: the most general lattice that these letters can generate.
With two letters $x$ and $y$ only four elements appear: $x$, $y$, $x \wedge y$ and $x \vee y$.
With three letters, infinitely many different elements appear.
</document>

<document source="https://tamnd.github.io/math2/node/WP3GDNWJJW.md" tag="WP3GDNWJJW" kind="statement">
# Robertson–Seymour theorem

## Statement

The finite graphs are [[well-quasi-ordered|NH25G6GC07]] by the [[minor|VYQ32ZBPEE]] relation.
That is, for every infinite sequence $G_1, G_2, G_3, \dots$ of finite graphs, there are indices $i < j$ such that $G_i$ is a minor of $G_j$.

## Remarks

An equivalent form: let $\mathcal{C}$ be a class of finite graphs closed under isomorphism and under taking minors.
Then there is a finite set $\mathcal{F}$ of graphs such that a graph lies in $\mathcal{C}$ if and only if no member of $\mathcal{F}$ is a minor of it.
For planar graphs $\mathcal{F} = \set{K_5, K_{3,3}}$, by [[Wagner's theorem|1B2ZJPVV03]], and for forests $\mathcal{F} = \set{K_3}$.
Neil Robertson and Paul Seymour proved the theorem in a long series of papers called Graph Minors, published between 1983 and 2004.
The proof rests on a structure theorem for graphs that exclude a fixed minor, built on tree decompositions and embeddings in surfaces.
The same project gives a polynomial-time algorithm that tests whether a fixed graph $H$ is a minor of an input graph.
So every minor-closed class can be recognized in polynomial time.
This proof of polynomial-time recognition is not constructive, since the theorem does not say how to find the set $\mathcal{F}$.

## Explanations

### L1 human

A minor of a graph is what remains after deleting and shrinking parts of it.
The theorem says that you cannot have an infinite list of graphs in which none is a minor of a later one.
As a result, every property that survives deleting and shrinking is described by a finite list of forbidden patterns.

### L2 human

For trees, the analogous statement is [[Kruskal's tree theorem|XVHKCX4F5Y]].
The proof of the general case extends ideas from it through [[treewidth|CD2XB4Y1SF]].
Graphs of bounded treewidth are handled like trees, and graphs of large treewidth contain large grid minors.
The theorem is one of the deepest results of combinatorics and gives finite obstruction sets for embeddability in any fixed surface.
</document>

<document source="https://tamnd.github.io/math2/node/WP4TGXJQJG.md" tag="WP4TGXJQJG" kind="concept">
# Sufficient statistic

## Summary

A [[statistic|0Q5CG2NEQ8]] is sufficient for a parameter when it keeps all the information that the sample contains about that parameter.
Once its value is known, the rest of the data has a distribution free of the parameter, so it says nothing more about it.
Sufficiency allows data reduction without loss, and it underlies the [[Rao–Blackwell theorem|Y3TX4J41TP]] and the theory of optimal estimators.
The [[Fisher–Neyman factorization theorem|Z6MF9ZGCMW]] gives a practical test for it.

## Explanations

### L1 human

Toss a coin $10$ times to learn its chance of heads.
Only the number of heads matters, not the order in which heads and tails came.
Given that there were $4$ heads, every order of $4$ heads and $6$ tails is equally likely, whatever the coin.
So the number of heads is a sufficient statistic: it compresses the data without losing anything about the coin.
</document>

<document source="https://tamnd.github.io/math2/node/WP4YAHJ6V4.md" tag="WP4YAHJ6V4" kind="statement">
# Penrose–Hawking singularity theorems

## Statement

A spacetime here is a connected, time-oriented, four-dimensional [[Lorentzian manifold|BY4QYYS788]] $(M, g)$ of signature $(-, +, +, +)$.
It is timelike (resp. null) geodesically complete if every inextendible timelike (resp. null) [[geodesic|NSYMQMHPDN]] is defined for all values of its affine parameter.
A closed trapped surface is a compact spacelike surface without boundary in $M$.
On it, both families of future-directed null geodesics orthogonal to it must have negative expansion.
Let $\operatorname{Ric}$ be the [[Ricci curvature|4GTNM0WHDH]] of $g$.

1. (Hawking) Assume that $(M, g)$ is globally hyperbolic and that $\operatorname{Ric}(V, V) \ge 0$ for all timelike vectors $V$.
Let $\Sigma$ be a smooth spacelike Cauchy surface in $M$.
Let $\theta$ be the expansion on $\Sigma$ of the past-directed normal geodesics, that is, the mean curvature for the past-directed unit normal.
Assume that $\theta \le C$ on $\Sigma$ for a constant $C < 0$.
Then every past-directed timelike curve starting on $\Sigma$ has length at most $3/\abs{C}$.
In particular, $(M, g)$ is not timelike geodesically complete.
2. (Penrose) Assume that $(M, g)$ is globally hyperbolic with a noncompact Cauchy surface.
Assume that $\operatorname{Ric}(K, K) \ge 0$ for all null vectors $K$, and that $M$ contains a closed trapped surface.
Then $(M, g)$ is not null geodesically complete.
3. (Hawking and Penrose) Assume that $\operatorname{Ric}(V, V) \ge 0$ for all timelike and null vectors $V$.
Assume that $M$ has no closed timelike curves, and that every timelike or null geodesic satisfies the generic condition.
Assume also that at least one of the conditions (a), (b), (c) below holds.
(a) $M$ contains a compact achronal set without edge.
(b) $M$ contains a closed trapped surface.
(c) There is a point $p$ such that the null geodesics from $p$ all start to reconverge.
This means that the expansion of the future-directed null geodesics from $p$ becomes negative along each of them.
The same with past-directed geodesics also suffices.
Then $(M, g)$ contains an incomplete timelike or null geodesic.

## Remarks

The common core of the proofs is the Raychaudhuri equation.
By it, the curvature conditions force the expansion of a congruence of geodesics to become $-\infty$ in finite affine parameter, which creates conjugate or focal points.
Such points prevent geodesics from maximizing length beyond them.
On the other hand, global causal assumptions, such as global hyperbolicity, guarantee the existence of maximizing geodesics.
The two facts together show that some geodesic cannot be extended.
The Schwarzschild black hole contains closed trapped surfaces inside its horizon, and the theorem of Penrose shows that the singularity there persists under perturbations that keep a trapped surface.
The theorems prove incompleteness of geodesics; they do not say that the curvature becomes infinite.

## Explanations

### L1 human

These theorems say that, under reasonable physical conditions, general relativity predicts singularities.
Gravity attracts, so light rays and free-falling particles that start to converge keep converging.
Once matter collapses far enough to form a trapped surface, or the whole universe is expanding uniformly, some paths of particles or light must end after a finite time.
This is how the theory predicts black holes and a Big Bang.

### L2 human

The curvature conditions are energy conditions on matter, translated through the Einstein equations into inequalities on the Ricci curvature.
The proofs use the comparison theory of geodesics, much like the proof of Myers's theorem in Riemannian geometry, where positive Ricci curvature bounds the length of minimizing geodesics.
Penrose's theorem dates from 1965, and the joint theorem of Hawking and Penrose from 1970.
The third theorem trades global hyperbolicity for the weaker chronology condition, at the cost of the generic condition.
</document>

<document source="https://tamnd.github.io/math2/node/WPC2XRMB9Y.md" tag="WPC2XRMB9Y" kind="concept">
# Essential supremum

## Summary

The essential supremum of a measurable function is the smallest number that the function exceeds only on a set of measure zero.
It is the supremum that ignores what happens on null sets.
It gives the norm of the space $L^\infty$, and it is the natural bound when functions are only known up to equality almost everywhere.

## Explanations

### L1 human

The ordinary supremum is the least upper bound of all the values of a function.
A single wild value at one point can make it huge, even though one point does not matter for integrals.
The essential supremum throws away such values when they live on a set of measure zero.
For example, let $f(x) = 0$ for $x \ne 0$ and $f(0) = 100$ on the real line with Lebesgue measure.
Its supremum is $100$, but its essential supremum is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/WPE5Z53QYJ.md" tag="WPE5Z53QYJ" kind="concept">
# Field trace

## Summary

Let $L/K$ be a [[finite extension|0C1Y04SZ1W]] and let $\alpha \in L$.
Multiplication by $\alpha$ is a $K$-linear map of $L$, and the field trace $\mathrm{Tr}_{L/K}(\alpha)$ is its [[trace|0KZDNGA561]].
It is a $K$-linear map $L \to K$.
For a separable extension it is the sum of the images of $\alpha$ under all embeddings of $L$ into an algebraic closure.
The trace gives the [[trace form|YWX2R53ED6]] and the [[discriminant|3BCYVS9E17]] of an extension.
The trace map is not identically zero exactly when the extension is separable.

## Explanations

### L1 human

Multiplying by a fixed element $\alpha$ is a linear map of the field $L$, seen as a vector space over $K$.
The trace of $\alpha$ is the trace of this map, the sum of the diagonal entries of its matrix.
For Gaussian numbers $a + bi$ over $\Q$, the trace is $(a + bi) + (a - bi) = 2a$.
So the trace adds up all the conjugates of $\alpha$, while the [[norm|KRNFEZWPY0]] multiplies them.
</document>

<document source="https://tamnd.github.io/math2/node/WPESA9XQ8X.md" tag="WPESA9XQ8X" kind="example">
# Laplace transform of the Heaviside step function

## Example

Let $H$ be the [[Heaviside step function|09C4JP7Y0M]], with $H(t) = 0$ for $t < 0$ and $H(t) = 1$ for $t > 0$, and let $a \ge 0$.
The delayed step $f(t) = H(t - a)$ is $0$ on $[0, a)$ and $1$ for $t > a$.
Its [[Laplace transform|05XT5AVJBK]] is
$$\mathcal{L}\{H(t - a)\}(s) = \frac{e^{-as}}{s} \quad (\operatorname{Re} s > 0).$$
Check: for $\operatorname{Re} s > 0$,
$$\int_0^\infty H(t - a) e^{-st}\, dt = \int_a^\infty e^{-st}\, dt = \lim_{T \to \infty} \frac{e^{-as} - e^{-sT}}{s} = \frac{e^{-as}}{s}.$$
The value of $H$ at the single point $0$ does not change the integral.
For $a = 0$ this is $\mathcal{L}\{1\} = 1/s$.
A rectangular pulse $H(t - a) - H(t - b)$ with $0 \le a < b$ has transform $(e^{-as} - e^{-bs})/s$, by linearity.

## Explanations

### L1 human

The step function models a switch turned on at time $a$.
Delaying a signal by $a$ multiplies its transform by $e^{-as}$, and here the undelayed signal is the constant $1$ with transform $1/s$.
Sums of steps describe piecewise constant inputs, and their transforms are sums of terms $e^{-as}/s$.
</document>

<document source="https://tamnd.github.io/math2/node/WPQPBXNN5F.md" tag="WPQPBXNN5F" kind="problem">
# Hilbert's fifteenth problem

## Problem

Give a rigorous foundation for the enumerative calculus of Hermann Schubert.
Determine exactly the range of validity of his methods, and check the numbers of [[enumerative geometry|ZPZA1FKZZC]] that he and his contemporaries found.

## Remarks

Hilbert posed this problem in his list of 23 problems presented in 1900.
Schubert had computed many enumerative numbers with symbolic rules, but these rules lacked proofs, and some classical answers were later found to need correction.
The modern foundations are [[intersection theory|4HV4WTAX9P]] and the cohomology of Grassmannians and flag varieties, in which [[Schubert calculus|FWGHMADJR4]] becomes a set of theorems.
Much of the problem is regarded as settled in this way, but checking all of Schubert's numbers is a separate task, and the problem is often listed as partially solved.

## Explanations

### L1 human

In the nineteenth century, geometers counted figures, such as lines meeting other lines or conics touching other conics, with clever but unproven rules.
The rules usually gave right answers, but sometimes wrong ones, and nobody knew why.
Hilbert asked for a solid foundation that explains when the rules work.
The answer came through the theory of intersections and the topology of Grassmannians, developed during the twentieth century.
</document>

<document source="https://tamnd.github.io/math2/node/WPR0NVC9PC.md" tag="WPR0NVC9PC" kind="concept">
# Stieltjes function

## Summary

A Stieltjes function is a non-decreasing, right-continuous function from the real line to itself.
Each Stieltjes function $F$ gives a [[Lebesgue–Stieltjes measure|QV75XJBFCS]] that assigns the mass $F(b) - F(a)$ to each interval $(a, b]$.
Conversely, every Borel measure on $\R$ that is finite on bounded sets comes from a Stieltjes function, unique up to an additive constant.
Cumulative distribution functions of random variables are the main examples.

## Explanations

### L1 human

Picture $F(x)$ as the total amount of sand piled up to the left of the point $x$.
As $x$ moves right, the amount can only grow, so $F$ is non-decreasing.
Where a single grain of positive mass sits at a point, $F$ jumps there, and right-continuity fixes the value at the jump to the upper one.
For example, $F(x) = x$ is a Stieltjes function and gives Lebesgue measure.
The step function equal to $0$ for $x < 0$ and $1$ for $x \ge 0$ is another, and it gives a unit mass at $0$.
</document>

<document source="https://tamnd.github.io/math2/node/WPYG78BSD5.md" tag="WPYG78BSD5" kind="concept">
# Stochastic difference equation

## Summary

A stochastic difference equation is a difference equation driven by random inputs, such as $X_{n+1} = f(X_n, \xi_{n+1})$ with a sequence of random noise terms $\xi_n$.
Its solution is a [[stochastic process|0QH5ENKZM1]] in discrete time rather than a single sequence.
The [[autoregressive model|2JC96FMGA1]] of time series analysis and the [[random walk|04VZE3FZXF]] are the basic linear examples.
When the noise terms are independent of each other and of the start, the solution is a [[Markov chain|0G48PYM96Q]].

## Explanations

### L1 human

A deterministic recursion computes the next state from the present one by a fixed rule.
A stochastic recursion also adds a fresh random shock at each step.
For example, $X_{n+1} = \tfrac{1}{2} X_n + \xi_{n+1}$ with independent coin-flip shocks $\xi_n = \pm 1$ keeps pulling the state toward $0$ while the shocks push it around.
One then studies averages, spreads and long-run distributions instead of exact values.
</document>

<document source="https://tamnd.github.io/math2/node/WQ1ZE255SF.md" tag="WQ1ZE255SF" kind="statement">
# Gödel's completeness theorem

## Statement

Let $\sigma$ be a signature, let $\Gamma$ be a set of first-order [[sentences|DKFGQTRYZE]] over $\sigma$, and let $\varphi$ be a sentence over $\sigma$.
Let $\vdash$ be derivability in a standard [[Hilbert system|1MGMQX43B8]] for first-order logic with equality.
If $\Gamma \models \varphi$, as a [[logical consequence|1HK0VYWTV7]], then $\Gamma \vdash \varphi$.

## Remarks

Together with [[soundness|4A84VHDTHV]], the theorem gives that $\Gamma \vdash \varphi$ if and only if $\Gamma \models \varphi$.
An equivalent form: every consistent set of sentences has a [[model|MVCH6N9SNW]].
The usual proof is Henkin's.
Extend a consistent $\Gamma$ to a maximal consistent set with witnesses: for each sentence $\exists x \, \chi$ in it, a constant $c$ with $\chi(c)$.
The closed terms, modulo provable equality, then form a model.
Since derivations are finite, the theorem implies the [[compactness theorem|1NR3HAWG9D]].
For example, every sentence true in all groups is derivable from the group axioms.
The theorem is about first-order logic; full second-order logic has no sound and complete proof system of this kind.
Kurt Gödel proved the theorem in his doctoral thesis of 1929.

## Explanations

### L1 human

The theorem says that the rules of first-order proof are strong enough.
Whatever follows from some axioms, in the sense that it is true in every situation where the axioms are true, can actually be proved from them.
So "true in all models" and "provable" mean the same thing.

### L2 human

The completeness theorem links syntax with semantics.
To show that something is unprovable, it is enough to build a model where the axioms hold and the statement fails.
This is how independence results, such as those in set theory, are proved.
It should not be confused with the [[incompleteness theorems|X9WYNYE29C]]: these say that some theories are incomplete, meaning that they neither prove nor refute some sentence.
Both results hold at once, since such a sentence is true in some models of the theory and false in others.
</document>

<document source="https://tamnd.github.io/math2/node/WQ46BD3ARY.md" tag="WQ46BD3ARY" kind="concept">
# Ideal class group

## Summary

The ideal class group of a [[Dedekind domain|7NPY2CN5KB]] is the group of nonzero fractional ideals modulo the principal ones.
The main case is the ring of integers of a [[number field|MWBDFS2KJ5]].
It is trivial exactly when the ring is a [[principal ideal domain|0CMD0BCH8T]].
For a Dedekind domain this is the same as being a [[unique factorisation domain|0W4X9G6263]].
So it measures how far unique factorisation of elements fails.
For a number field it is a finite abelian group, and its order is the [[class number|KTKK9B1X41]].

## Explanations

### L1 human

In a number ring, numbers may factor in several ways, but ideals factor uniquely.
Some ideals are generated by a single number, and those behave just like numbers; others are not.
The class group lumps together ideals that differ by a principal ideal and records how they multiply.
For $\Z[\sqrt{-5}]$ there are two classes: the principal ideals, and the class of the ideal $(2, 1 + \sqrt{-5})$, whose square is the principal ideal $(2)$.
</document>

<document source="https://tamnd.github.io/math2/node/WQ82C0M3QB.md" tag="WQ82C0M3QB" kind="concept">
# Chain complex

## Summary

A chain complex is a sequence of objects, such as abelian groups or modules, joined by morphisms $d_n \colon C_n \to C_{n-1}$ with $d_{n-1} \circ d_n = 0$.
The condition says that the image of each map lies in the kernel of the next, so the [[homology|2J4JJJYP22]] $\ker d_n / \im d_{n+1}$ is defined.
Chain complexes are the central objects of [[homological algebra|SAHAEZNCVD]].
They arise from topological spaces as in [[singular homology|M0Q5CFJ7ZH]], and from modules through resolutions.

## Explanations

### L1 human

A chain complex is a row of objects with maps going down one step at a time, such that two steps in a row always give zero.
In topology, $C_n$ is built from $n$-dimensional pieces of a space and $d_n$ takes the boundary of a piece.
The boundary of a boundary is empty, which is the rule $d \circ d = 0$.
A small algebraic example is $\cdots \to \Z/4\Z \xrightarrow{2} \Z/4\Z \xrightarrow{2} \Z/4\Z \to \cdots$, where doing "multiply by $2$" twice multiplies by $4$, which is $0$ in $\Z/4\Z$.
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# Jackknife

## Summary

The jackknife is a resampling method that recomputes a [[statistic|0Q5CG2NEQ8]] on each of the $n$ samples obtained by leaving out one observation.
From these leave-one-out values it estimates the [[bias|0BWGTN92FG]] and the variance of the statistic and gives a bias-corrected estimate.
Quenouille introduced it for bias reduction, and Tukey proposed its use for variance estimation.
It works well for smooth statistics, and it is a forerunner of the [[bootstrap|40ATJEVR32]].

## Explanations

### L1 human

Take your sample, remove the first observation, and compute your estimate again.
Put it back, remove the second one instead, and repeat until each observation has been left out once.
If these $n$ estimates move a lot, your estimate is sensitive to individual data points and thus uncertain.
Their systematic shift from the full-sample estimate also tells you roughly how biased the estimate is.
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# Easton's theorem

## Statement

Let $M$ be a countable transitive model of ZFC in which the generalized continuum hypothesis holds.
Let $F$ be a class function, definable in $M$, from the [[regular cardinals|ZCE333FAWC]] of $M$ to the [[cardinals|0QKPNZKNEA]] of $M$.
Assume that in $M$ the function $F$ satisfies both conditions:

1. If $\kappa \le \lambda$, then $F(\kappa) \le F(\lambda)$.
2. $\operatorname{cf}(F(\kappa)) > \kappa$ for each $\kappa$, where $\operatorname{cf}$ is the [[cofinality|NTE5C260S1]].

Then there is a [[forcing|F2DTWW9HP7]] extension $N$ of $M$ with the same cardinals and the same cofinalities as $M$.
In $N$, $2^\kappa = F(\kappa)$ for every regular cardinal $\kappa$.

## Remarks

The two conditions are exactly the constraints that ZFC proves for $\kappa \mapsto 2^\kappa$ on regular cardinals.
Monotonicity is clear, and $\operatorname{cf}(2^\kappa) > \kappa$ is König's theorem.
The proof uses a class product of Cohen-style forcings, one for each regular $\kappa$, with supports chosen so that the whole product preserves cardinals.
Example: one can have $2^{\aleph_0} = \aleph_2$ and $2^{\aleph_1} = \aleph_2$, or $2^{\aleph_0} = \aleph_{\omega + 1}$.
But $2^{\aleph_0} = \aleph_\omega$ is impossible, since $\operatorname{cf}(\aleph_\omega) = \omega$.
The theorem says nothing about singular cardinals, where the situation is far more constrained, as in Silver's theorem and the singular cardinals hypothesis.

## Explanations

### L1 human

For an infinite cardinal $\kappa$, the number $2^\kappa$ counts the subsets of a set of size $\kappa$.
The usual axioms of set theory say very little about it.
Easton's theorem makes this precise for regular cardinals: apart from two simple rules, any pattern of values is possible.
For example, the axioms do not decide whether there are $\aleph_1$ or $\aleph_2$ real numbers.

### L2 human

Forcing with Cohen reals shows that $2^{\aleph_0}$ can be any cardinal of uncountable cofinality.
Easton's theorem extends this to all regular cardinals at once.
It shows that monotonicity and König's theorem are the only ZFC constraints on the function $\kappa \mapsto 2^\kappa$ on regular cardinals.
At singular cardinals the story changes: the values of $2^\kappa$ below a singular cardinal can restrict its value at that cardinal.
This is the subject of Silver's theorem, of the singular cardinals hypothesis and of Shelah's pcf theory.
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# Arnold tongue

## Summary

An Arnold tongue is a region in the parameter plane of the [[circle map|N2SVK69S53]] where the [[rotation number|DP4WWQBJJF]] takes one fixed rational value $p/q$.
Each tongue starts at the single point $\Omega = p/q$ when there is no nonlinearity, and widens as the nonlinearity $K$ grows.
Inside a tongue the system is mode locked: it has periodic orbits of period $q$.
Arnold tongues describe synchronization of oscillators with a periodic forcing.

## Explanations

### L1 human

Drive an oscillator, such as a swing or a pendulum clock, with a periodic push.
For some combinations of push strength and frequency, the oscillator locks into a rhythm, for example two swings for every three pushes.
Drawing these locked regions in the plane of frequency and strength gives thin wedge shapes that grow wider as the push gets stronger.
They look like tongues hanging down to the axis of zero strength, hence the name.
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# Metacompact space

## Summary

A metacompact space is a [[topological space|0CQHFPGJ45]] in which every [[open cover|049PXYHW1T]] has a point-finite open refinement.
Point-finite means that each point lies in only finitely many members of the refinement.
This is weaker than [[paracompactness|8R29G6NG9Q]], where each point needs a neighbourhood meeting only finitely many members.
So every paracompact space, and in particular every compact space, is metacompact.

## Explanations

### L1 human

In a metacompact space, you can always shrink an open cover so that no point is covered too many times.
Each single point sits in only finitely many of the new sets.
Points close to it may still be covered by many different sets.
That is the difference from paracompact spaces, where a whole neighbourhood must see only finitely many sets.
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# Wilks' theorem

## Statement

Let $X_1, \dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] with density $f_\theta$, where $\theta$ lies in an open set $\Theta \subseteq \R^k$.
Assume the standard regularity conditions for the asymptotic normality of the maximum likelihood estimator, with a positive definite Fisher information.
Let $\Theta_0 \subseteq \Theta$ be the set of $\theta$ whose first $q$ coordinates equal fixed values, with $1 \le q \le k$.
Let $\lambda_n$ be the [[likelihood-ratio statistic|YM1BNA80RC]] of $H_0 \colon \theta \in \Theta_0$ based on $X_1, \dots, X_n$.
If the true $\theta$ lies in $\Theta_0$, then $-2 \log \lambda_n$ [[converges in distribution|0JDAQHWA9N]] to the [[chi-squared distribution|0RFMMBHFTV]] with $q$ degrees of freedom as $n \to \infty$.

## Remarks

The number $q$ is the difference between the dimension of $\Theta$ and the dimension of $\Theta_0$.
Proof idea: expand the log-likelihood to second order around the unrestricted and the restricted maximum likelihood estimates.
The difference becomes a quadratic form in an asymptotically normal vector, which has a limiting chi-squared law with $q$ degrees of freedom.
Example: for an i.i.d. $\mathcal{N}(\mu, \sigma^2)$ sample and $H_0 \colon \mu = 0$, here $k = 2$ and $q = 1$, so one rejects when $-2 \log \lambda_n$ exceeds the $1 - \alpha$ quantile of $\chi^2_1$.
The hypothesis that the true parameter is an interior point matters.
For a variance component tested at the boundary value $0$, the limit law is a mixture of $\chi^2_0$ and $\chi^2_1$ rather than $\chi^2_1$.

## Explanations

### L1 human

The likelihood-ratio test needs a threshold, and for that one needs the law of its statistic when the null hypothesis holds.
Wilks' theorem says that for large samples this law is almost always the same: a chi-squared law.
Its only parameter is the number of constraints that the null hypothesis imposes.

### L2 human

This is why likelihood-ratio tests are used everywhere: one computes $-2 \log \lambda_n$ and compares it with a chi-squared quantile.
The score test and the Wald test have the same limiting chi-squared law under the null hypothesis, and the three tests are asymptotically equivalent.
The theorem needs regularity: smooth densities, identifiability, and a true parameter in the interior of the parameter set.
It is an asymptotic statement, so for small samples the chi-squared approximation can be poor.
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# Funk transform

## Summary

The Funk transform sends a function on the sphere $S^2$ to the function that gives its integral over each great circle.
It is the spherical analogue of the [[Radon transform|VJNZJ352PC]], with great circles in place of lines.
It kills odd functions, and it is injective on even continuous functions.
On [[spherical harmonics|3F32Y21HP4]] it acts by multiplication by explicit constants, which gives its inversion.

## Explanations

### L1 human

Think of a function on the surface of the Earth, such as temperature.
For every great circle, such as the equator or a meridian circle through both poles, add up the temperature along it.
The Funk transform records these totals, indexed by the pole of each great circle.
A function with $f(-x) = -f(x)$ gives total $0$ on every great circle, so it is invisible.
For functions with $f(-x) = f(x)$, the totals determine the function.
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# McKay conjecture

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a prime, and let $P$ be a [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$ with [[normaliser|0YGN3VAA0D]] $N_G(P)$.
For a finite group $H$, let $\operatorname{Irr}_{p'}(H)$ be the set of irreducible complex [[characters|RMWMYBY8TC]] of $H$ whose degree is not divisible by $p$.
Then
$$\abs{\operatorname{Irr}_{p'}(G)} = \abs{\operatorname{Irr}_{p'}(N_G(P))}.$$

## Remarks

John McKay stated the conjecture in the early 1970s.
Isaacs, Malle and Navarro reduced it in 2007 to a stronger condition on finite simple groups, the inductive McKay condition.
Malle and Späth completed the case $p = 2$ in 2016.
Cabanes and Späth completed the proof for all primes in 2024, so the statement is now a theorem, though it keeps its name.
Example: let $G = S_3$ and $p = 3$, with $P = A_3$ normal.
Then $N_G(P) = G$, so both sides count the same set.
For $p = 2$ and $G = S_3$, take $P = \set{1, (1\,2)}$; then $N_G(P) = P$ has $2$ characters of odd degree, and $S_3$ has the characters of degrees $1, 1, 2$, so $2$ of odd degree.

## Explanations

### L1 human

The conjecture compares a finite group with a much smaller subgroup, the normaliser of a Sylow subgroup.
It says that both have the same number of irreducible characters whose degree is not a multiple of $p$.
So a global count is determined by local data near the prime $p$.

### L2 human

It is the prototype of the local-global conjectures of modular representation theory, with Alperin's weight conjecture and Dade's conjecture.
For $p$-solvable groups it was proved earlier, by Okuyama and Wajima.
The proof for all groups goes through the [[classification of finite simple groups|06J6P03MG7]] and the character theory of groups of Lie type, using [[Deligne–Lusztig theory|ZH21D5B62C]].
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# Tight and overtwisted contact structures

## Summary

Contact structures on $3$-manifolds split into two classes.
An overtwisted contact structure contains an overtwisted disc, a disc whose boundary circle is tangent to the contact planes in a special twisted way.
A tight one contains no such disc.
Eliashberg showed that overtwisted structures are classified by homotopy theory alone, so they are flexible.
Tight structures are rigid and carry the deep information of three-dimensional [[contact topology|HBQ8BZ7XB5]].

## Explanations

### L1 human

A contact structure is a field of planes on a three-dimensional space that twists everywhere.
In an overtwisted one, the planes twist so much that one can find a special disc.
Its rim runs along the planes, and all along the rim the planes lie flat on the disc.
A tight contact structure never twists that much.
The standard structure on $\R^3$ is tight, by a theorem of Bennequin, while for example the planes $\ker(\cos r\, dz + r \sin r\, d\theta)$ in cylindrical coordinates are overtwisted.
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# Jackson's inequality

## Statement

Let $r \ge 0$ be an integer.
For a continuous $2\pi$-[[periodic function|0ATF6Q0633]] $g$, let $\omega(g, \delta) = \sup_{\abs{x - y} \le \delta} \abs{g(x) - g(y)}$ be its [[modulus of continuity|AXCJBSQZWF]].
For $n \ge 1$, let $E_n(f) = \inf_T \max_x \abs{f(x) - T(x)}$, the infimum over real [[trigonometric polynomials|J3NJQ5FZJ5]] $T$ of degree at most $n$.
Then there is a constant $C_r$, depending only on $r$, such that every $2\pi$-periodic $f$ with $r$ continuous derivatives satisfies
$$E_n(f) \le \frac{C_r}{n^r} \, \omega\left(f^{(r)}, \frac{1}{n}\right) \qquad \text{for all } n \ge 1.$$

## Remarks

Proof idea for $r = 0$: approximate $f$ by its convolution with the [[Jackson kernel|9MZFTX77CD]] $J_m$ of degree at most $n$.
The kernel is nonnegative with mean $1$, so $\abs{f - f * J_m} \le \frac{1}{2\pi} \int \omega(f, \abs{t}) J_m(t) \, dt$.
With $\omega(f, \abs{t}) \le (1 + n\abs{t}) \, \omega(f, 1/n)$ and $\frac{1}{2\pi} \int \abs{t} J_m(t) \, dt = O(1/n)$, the bound follows.
The case $r \ge 1$ follows by induction on $r$, using $E_n(f) \le \frac{C}{n} E_n(f')$ for continuously differentiable $2\pi$-periodic $f$.

Example: if $f$ is Lipschitz with constant $L$, then $\omega(f, 1/n) \le L/n$, so $E_n(f) \le C_0 L / n$.

The same bounds hold for algebraic polynomials of degree at most $n$ on $[-1, 1]$ when $r = 0$, through the substitution $x = \cos \theta$.
For $\abs{f^{(r)}} \le 1$ with $r \ge 1$, the sharp bound is $E_n(f) \le K_r / (n + 1)^r$, where $K_r$ is the [[Favard constant|9K1W9Q49NR]], by [[Favard's theorem|JYX6N8A2P6]].

## Explanations

### L1 human

The smoother a periodic function is, the better trigonometric polynomials of a given degree can approximate it.
Jackson's inequality makes this precise: each derivative gains a factor $1/n$, and the modulus of continuity of the last derivative gives the rest.

### L2 human

This is the basic direct theorem of approximation theory.
Its converse, Bernstein's inverse theorem, shows that the rates it gives are the right ones when $0 < \alpha < 1$ in the Hölder scale.
Together with [[Bernstein's inequality|F0YGC89S9R]], it underlies the characterisation of smoothness classes by approximation rates.
For interpolation and Fourier partial sums, it combines with Lebesgue constants to give convergence results such as the Dini–Lipschitz test.
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# Null-homotopic map

## Definition

Let $X$ and $Y$ be topological spaces.
A continuous map $f \colon X \to Y$ is *null-homotopic* if it is [[homotopic|F2F143KQ2M]] to a constant map $X \to Y$.

## Remarks

Example: every continuous map $f \colon X \to \R^n$ is null-homotopic, via $H(x, t) = (1 - t) f(x)$.
Non-example: the identity of the circle $S^1$ is not null-homotopic, since a constant map induces the trivial map on the [[fundamental group|S1XK3HXHFX]] while the identity induces the identity of $\Z$.
A map $f \colon S^n \to Y$ is null-homotopic if and only if it extends to a continuous map on the closed ball $D^{n+1}$.
A space $X$ is [[contractible|R5WP8Z69Q8]] if and only if it is nonempty and $\id_X$ is null-homotopic.
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# Screw theory

## Definition

Work in $\R^3$ with the [[dot product|0S52TNNTH9]] $\cdot$ and the [[cross product|0N5N15HWZ7]] $\times$.

1. A *screw* is a pair $S = (\omega, v)$ of vectors of $\R^3$, an element of $\R^6$.
2. A *twist* is a screw read as the velocity field $u(x) = v + \omega \times x$ of a moving rigid body. Here $\omega$ is the angular velocity and $v$ is the velocity of the body point at the origin.
3. A *wrench* is a screw $(f, m)$ read as a system of forces with resultant force $f$ and moment $m$ about the origin.
4. If $\omega \ne 0$, the *pitch* of $(\omega, v)$ is $h = \omega \cdot v / \abs{\omega}^2$, and its *axis* is the line $\set{\omega \times v / \abs{\omega}^2 + t \omega : t \in \R}$.

The *reciprocal product* of a twist $(\omega, v)$ and a wrench $(f, m)$ is $\omega \cdot m + v \cdot f$, the power of the wrench on the motion.
*Screw theory* is the study of screws, twists and wrenches with these operations.

## Remarks

On the axis, the velocity is parallel to $\omega$: for $p = \omega \times v / \abs{\omega}^2$ one finds $\omega \times p = \omega (\omega \cdot v)/\abs{\omega}^2 - v$, so $u(p) = h \omega$.
So the body turns about the axis and slides along it with speed $h \abs{\omega}$, which is the infinitesimal form of Chasles' theorem.
Example: for $\omega = (0, 0, 1)$ and $v = (0, -1, 2)$, the pitch is $h = 2$ and $p = \omega \times v = (1, 0, 0)$.
Check: $u(p) = (0, -1, 2) + (0, 0, 1) \times (1, 0, 0) = (0, -1, 2) + (0, 1, 0) = (0, 0, 2) = 2\omega$; the axis is the vertical line through $(1, 0, 0)$.
If $\omega = 0$, the twist is a pure translation with velocity $v$, a screw of infinite pitch.
Non-example: the field $u(x) = A x$ with $A = \operatorname{diag}(1, 0, 0)$ is not a twist, since $A$ is not skew-symmetric; it stretches the body along the first axis instead of moving it rigidly.
The twists form the Lie algebra of the [[Euclidean group|0H3YY8CPKH]], with bracket $[(\omega_1, v_1), (\omega_2, v_2)] = (\omega_1 \times \omega_2, \omega_1 \times v_2 - \omega_2 \times v_1)$.
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# Finitely generated algebra

## Summary

An [[algebra|QJGPT2W0FT]] is finitely generated when some finite subset of it generates the whole algebra.
Every element can then be built from finitely many fixed elements by repeated use of the operations.
Finitely generated groups, rings and lattices are all special cases.
In the [[subalgebra lattice|T0TNG7AGXT]], the finitely generated subalgebras are exactly the compact elements.

## Explanations

### L1 human

Pick a few elements of an algebra and apply the operations to them again and again.
If every element of the algebra shows up sooner or later, the algebra is finitely generated.
For example, the integers with addition, negation and $0$ are generated by the single element $1$.
Every integer is $1 + 1 + \cdots + 1$, its negative, or $0$.
The rational numbers with the same operations are not finitely generated, because finitely many fractions only reach the multiples of one fixed fraction.
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# Kolmogorov extension theorem

## Statement

Let $T$ be a nonempty index set, and for each $t \in T$ let $E_t$ be a [[Polish space|584GXXY73M]] with its Borel σ-algebra.
For $J \subseteq T$ let $E_J = \prod_{t \in J} E_t$ with the [[product σ-algebra|6HVCTJGB61]], and for $K \subseteq J$ let $\pi^J_K \colon E_J \to E_K$ be the restriction of coordinates.
For each finite nonempty $J \subseteq T$ let $\mu_J$ be a [[probability measure|059TJN0NZ1]] on $E_J$.
Assume the family is consistent: for all finite nonempty $K \subseteq J \subseteq T$, the [[pushforward|3N66TM3TSQ]] of $\mu_J$ under $\pi^J_K$ equals $\mu_K$.
Then there is a unique probability measure $\mu$ on $E_T$ such that, for every finite nonempty $J \subseteq T$, the pushforward of $\mu$ under $\pi^T_J$ equals $\mu_J$.

## Remarks

The measure $\mu$ is first defined on cylinder sets, where consistency makes it well defined and finitely additive.
The key step is countable additivity on the algebra of cylinder sets.
It uses inner regularity: each $\mu_J$ is approximated from inside by compact sets, and a decreasing sequence of nonempty compact cylinders has nonempty intersection.
[[Carathéodory's extension theorem|Q8EXG7EKYW]] then extends $\mu$ to the product σ-algebra, and uniqueness holds because cylinder sets form a π-system.

Example: let $T = [0, \infty)$, $E_t = \R$, and let $\mu_J$ be the centred Gaussian measure on $\R^J$ with covariance $\min(s, t)$ for $s, t \in J$.
These measures are consistent, so there is a [[stochastic process|0QH5ENKZM1]] $(X_t)_{t \ge 0}$ with these finite-dimensional laws.
This is the first step in the construction of Brownian motion; the continuity of paths needs a further argument.

A topological assumption on the factors is needed.
For arbitrary measurable spaces in place of Polish spaces, consistent families exist that have no extension.
No such assumption is needed when the $\mu_J$ are products, as for the [[infinite product measure|2HWVFHM13R]].
It is not needed either when they are built from kernels, as in the [[Ionescu-Tulcea theorem|94ASD187AQ]].

## Explanations

### L1 human

To describe a random process, such as the position of a particle at every time, it is enough to say how it behaves at finitely many times at once.
If these finite descriptions do not contradict each other, the theorem gives a single probability measure on the whole space of paths that matches all of them.
So a process can be built from its finite-dimensional distributions.

### L2 human

The theorem is the standard existence result for stochastic processes with prescribed finite-dimensional distributions: Gaussian processes, Markov processes from transition functions, and sequences of independent variables.
The resulting σ-algebra only contains events that depend on countably many coordinates.
So for uncountable $T$, sets such as "the path is continuous" are not measurable, and one has to pass to a modification, for instance with the [[Kolmogorov continuity theorem|V1HAJNC5RK]].
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# Abstract polytope

## Summary

An abstract polytope is a partially ordered set that behaves like the set of faces of a polytope ordered by inclusion.
It keeps only the incidence pattern of vertices, edges and higher faces and forgets all geometry.
The axioms ask for a grading by dimension, a connectedness condition, and the diamond condition.
The diamond condition says that between two faces whose dimensions differ by two there are exactly two faces.
Abstract polytopes include the face lattices of convex polytopes, but also tilings and objects such as the hemicube that no convex polytope realises.

## Explanations

### L1 human

Write down the vertices, edges and faces of a shape, and record only which ones touch which.
This list of incidences, ordered by "is part of", is the skeleton that an abstract polytope keeps.
In a polygon, each vertex is on exactly two edges, and each edge has exactly two vertices; the diamond rule generalises this to all dimensions.
Shapes like the hemicube, made by gluing opposite faces of a cube together, fit the rules even though they cannot be built as convex solids.
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# Uniqueness of solutions given initial values

## Statement

Let $R$ be a commutative ring and let $d \ge 1$ be an integer.
Let $E$ be a [[linear recurrence with constant coefficients|081THG13JZ]] of order $d$ over $R$, with coefficients $c_0, \dots, c_{d-1} \in R$.
Let $u$ and $v$ be [[solutions|0ZV2YKKZ58]] of $E$, that is, sequences $\N \to R$ with $u_{n+d} = \sum_{i=0}^{d-1} c_i u_{n+i}$ and $v_{n+d} = \sum_{i=0}^{d-1} c_i v_{n+i}$ for all $n \in \N$.
If $u_i = v_i$ for $i = 0, 1, \dots, d-1$, then $u_n = v_n$ for all $n \in \N$.

## Remarks

The proof is a strong induction on $n$.
For $n < d$ the claim is the hypothesis.
For $n \ge d$, the recurrence writes $u_n$ and $v_n$ by the same formula in the $d$ previous terms, which agree by induction.
Existence also holds: for each choice of $u_0, \dots, u_{d-1}$ in $R$, the recurrence defines the later terms one by one.
So the map $u \mapsto (u_0, \dots, u_{d-1})$ is a bijection from the solutions onto $R^d$; see the [[space of solutions|099GXMHV40]].
Example: let $\varphi$ and $\psi$ be the roots of $X^2 - X - 1$, and let $v_n = (\varphi^n - \psi^n)/\sqrt{5}$.
Then $v$ solves $u_{n+2} = u_{n+1} + u_n$ over $\R$, and $v_0 = 0$, $v_1 = 1$.
The [[Fibonacci sequence|0GA9K57A39]] solves the same recurrence with the same two initial values, so it equals $v$; this proves Binet's formula.
Fewer initial values are not enough: $(0, 1, 1, 2, 3, \dots)$ and $(0, 2, 2, 4, 6, \dots)$ both solve $u_{n+2} = u_{n+1} + u_n$ and agree in the first term only.

## Explanations

### L1 human

A linear recurrence of order $d$ tells you how to compute each term from the $d$ terms before it.
So once the first $d$ terms are known, there is no freedom left: every later term is forced.
Two sequences that obey the same rule and start the same way are the same sequence.

### L2 human

This fact is the basic tool for proving closed formulas for recurrent sequences.
To check a formula, one checks that it satisfies the recurrence and that it has the right first $d$ terms.
Together with existence, it shows that the solutions form a free module of rank $d$ with the initial values as coordinates.
In Mathlib it is `LinearRecurrence.eq_mk_of_is_sol_of_eq_init`.
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# Triangle center

## Summary

A triangle center is a point attached to every [[triangle|0CTNFGYMM2]] by a fixed rule.
The rule must not depend on the labelling of the vertices, and it must commute with similarities.
Classical examples are the [[centroid|0X7EEWES5G]], the [[incentre|0M9MX8YW0A]], the [[circumcentre|0F98FZK14A]] and the [[orthocentre|0WB619CA0V]].
The notion is made precise with trilinear coordinates, and Clark Kimberling's Encyclopedia of Triangle Centers catalogues many thousands of them.

## Explanations

### L1 human

Every triangle has several natural "middle points": the balance point, the centre of the inscribed circle, the centre of the circle through the corners, and so on.
A triangle center is any such point given by a fixed recipe.
The recipe must treat the three corners in the same way, so renaming the corners gives the same point.
It must also behave well under scaling: enlarge the triangle and the point moves with it.
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# Maximal order

## Definition

Let $R$ be a Noetherian integral domain with field of fractions $K$, and let $A$ be a finite-dimensional algebra over $K$.
An $R$-[[order|1ZBX4TKVZ9]] $\Lambda$ in $A$ is *maximal* if there is no $R$-order $\Lambda'$ in $A$ with $\Lambda \subsetneq \Lambda'$.

## Remarks

Example: $M_n(\Z)$ is a maximal $\Z$-order in $M_n(\Q)$.
Example: for an [[algebraic number field|MWBDFS2KJ5]] $F$, the ring of integers $\mathcal{O}_F$ is the unique maximal $\Z$-order in $F$.
The reason is that every element of an order is integral over $\Z$ and so lies in $\mathcal{O}_F$.
Non-example: $\Z[\sqrt{-3}]$ is an order in $\Q(\sqrt{-3})$, but it is strictly contained in the order $\Z[\omega]$ with $\omega = (1 + \sqrt{-3})/2$, so it is not maximal.
Maximal orders need not be unique: with $g = \operatorname{diag}(1, 2)$, the order $g M_2(\Z) g^{-1}$ is maximal in $M_2(\Q)$ and contains the matrix $\begin{pmatrix} 0 & 1/2 \\ 0 & 0 \end{pmatrix}$, which is not in $M_2(\Z)$.
If $R$ is a [[Dedekind domain|7NPY2CN5KB]] and $A$ is a [[separable algebra|X8H7A98Q55]] over $K$, every $R$-order in $A$ is contained in a maximal $R$-order.
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# Quasivariety

## Summary

A quasivariety is a class of algebras of one signature that is axiomatized by [[quasi-identities|94GVY3APPQ]].
Every [[variety|AJNZJD8442]] is a quasivariety, but quasivarieties need not be closed under homomorphic images.
By [[Mal'cev's characterization|DFNM3BGMAY]], they are the classes closed under isomorphic copies, subalgebras, direct products and ultraproducts.
They are the algebraic counterpart of universal Horn logic and appear in algebraic logic, where many logics are described by quasivarieties rather than varieties.

## Explanations

### L1 human

A variety is described by equations that must always hold.
A quasivariety is described by conditional equations: "if these equations hold, then that one holds too".
Torsion-free groups form a quasivariety: for each $n \ge 1$, the rule "if $x^n = 1$ then $x = 1$" is a quasi-identity.
They do not form a variety, because the torsion-free group $\Z$ has the quotient $\Z / 2\Z$, which has torsion.
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<document source="https://tamnd.github.io/math2/node/WRY0M7SW59.md" tag="WRY0M7SW59" kind="concept">
# Discrete harmonic function

## Summary

A discrete harmonic function is a function on the vertices of a graph equal at each vertex to a weighted average of its neighbouring values.
It is the discrete analogue of a [[harmonic function|0DEB0JMN0W]], and it is exactly a function killed by the [[graph Laplacian|TN6M6DVZRP]].
Discrete harmonic functions describe voltages in an [[electrical network|T7KRE7J3B4]] and hitting probabilities of a [[random walk|04VZE3FZXF]].
They satisfy discrete versions of the mean value property and the maximum principle.

## Explanations

### L1 human

Put a number on every vertex of a graph.
The function is harmonic at a vertex if the number there is the average of the numbers at the neighbouring vertices.
On the integers, with neighbours $n - 1$ and $n + 1$, the harmonic functions are the straight lines $f(n) = an + b$.
A harmonic function cannot have a strict maximum at an interior vertex, since an average cannot exceed all its terms.
If a random walker moves to a uniformly chosen neighbour at each step, then a harmonic function keeps its expected value along the walk.
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<document source="https://tamnd.github.io/math2/node/WRYP2XCKJB.md" tag="WRYP2XCKJB" kind="definition">
# Bridge number

## Definition

Let $K$ be a tame [[knot|QYEAZTYGPT]] in $\R^3$, and let $h \colon \R^3 \to \R$ be the height function $h(x, y, z) = z$.
Call a knot $K'$ in $\R^3$ *generic* if the restriction of $h$ to $K'$ is a Morse function.
The *bridge number* $b(K)$ is the least number of local maxima of $h$ on $K'$, over all generic knots $K'$ ambient isotopic to $K$.

## Remarks

A function on a circle has at least one maximum, so $b(K) \ge 1$.
Example: a round circle in a vertical plane has exactly one maximum, so the [[unknot|1TQ06R4Q66]] has bridge number $1$.
Conversely $b(K) = 1$ only for the unknot, since a knot with one maximum and one minimum bounds an embedded disk.
The [[trefoil knot|VYTYSVHN4A]] has a position with two maxima and is knotted, so its bridge number is $2$.
Non-example: a position of the trefoil with three maxima is generic but does not realize the minimum, so the count $3$ is only an upper bound.
Equivalently, $b(K)$ is the least number of overpasses (bridges) in a diagram of $K$ whose over-strands are disjoint arcs carrying all overcrossings.
Schubert proved $b(K_1 \# K_2) = b(K_1) + b(K_2) - 1$ for the [[knot sum|AGZSZW7GYE]].
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# Gale diagram

## Definition

Let $d \ge 1$ and $n \ge d + 1$, and let $v_1, \dots, v_n$ be points of $\R^d$ whose [[affine span|0YMB1676R1]] is $\R^d$.
Let $A$ be the $(d+1) \times n$ matrix whose $i$-th column is $(v_i, 1) \in \R^{d+1}$.
Its [[null space|0PHDYNR24D]] $N = \set{\lambda \in \R^n : A \lambda = 0}$, the space of affine dependences of the points, has dimension $n - d - 1$.
Choose a basis of $N$ and let $B$ be the $n \times (n - d - 1)$ matrix with these basis vectors as columns.
The *Gale transform* of $v_1, \dots, v_n$ is the sequence $g_1, \dots, g_n \in \R^{n-d-1}$ of rows of $B$.

## Remarks

The Gale transform is determined up to an invertible linear map of $\R^{n-d-1}$, the change of basis of $N$.
Many authors call the configuration $g_1, \dots, g_n$ a *Gale diagram*, and some first rescale each nonzero $g_i$ by a positive factor; this does not affect the property below.

Key property: let $T \subseteq \set{1, \dots, n}$ be nonempty and $S$ its complement.
Consider an affine function $\ell$ on $\R^d$ with $\ell(v_i) = 0$ for $i \in S$ and $\ell(v_i) > 0$ for $i \in T$.
Such an $\ell$ exists if and only if $0$ lies in the relative interior of the [[convex hull|0FQSDC2EQ2]] of $\set{g_i : i \in T}$.
Indeed, the vectors $(\ell(v_1), \dots, \ell(v_n))$ are exactly the vectors $y$ with $\sum_i y_i g_i = 0$.
So when the $v_i$ are the vertices of a [[convex polytope|300K6CB1KH]] $P$, the sets $\set{v_i : i \in S}$ that are vertex sets of faces of $P$ are read off from the $g_i$.

Example: for the $4$ vertices $0, e_1, e_2, e_1 + e_2$ of a square in $\R^2$, $n - d - 1 = 1$, and $N$ is spanned by $(1, -1, -1, 1)$.
So the Gale transform is $1, -1, -1, 1$ in $\R$.
For $T = \set{1, 4}$, the hull of $g_1 = g_4 = 1$ is $\set{1}$, which misses $0$, so $\set{e_1, e_2}$ is not a face: it is a diagonal.
For $T = \set{1, 2}$, the hull of $1$ and $-1$ contains $0$ in its interior, so $\set{e_2, e_1 + e_2}$ is an edge.

Non-example: the points $0, e_1, 2e_1$ in $\R^2$ do not span $\R^2$ affinely, so the construction does not apply.
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# Solution of recurrences by generating functions

## Summary

This method turns a recurrence for a sequence $(u_n)$ into an equation for its [[generating function|0PHAC833JF]] $U(x) = \sum_{n \ge 0} u_n x^n$.
For a linear recurrence with constant coefficients, the equation is linear and its solution $U$ is a rational function.
Expanding $U$ again, for example by partial fractions, gives a closed formula for $u_n$.
The method is a standard tool of enumerative combinatorics and of the analysis of algorithms.

## Explanations

### L1 human

A sequence is an infinite list of numbers, which is hard to handle at once.
Packing the whole list into one power series turns the step-by-step rule into an algebraic equation.
For example, if $u_0 = 1$ and $u_{n+1} = 2 u_n$, then $U(x) - 1 = 2 x U(x)$.
So $U(x) = 1/(1 - 2x) = \sum 2^n x^n$, and $u_n = 2^n$.
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# Linear and stable summation methods

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $M$ be a [[summability method|X80DZEXQDQ]].
Write $M\left(\sum a_n\right)$ for the $M$-sum of an $M$-summable series $\sum_{n \ge 0} a_n$ with terms in $\mathbb{K}$.

1. $M$ is *linear* if, whenever $\sum a_n$ and $\sum b_n$ are $M$-summable and $\lambda, \mu \in \mathbb{K}$, the series $\sum (\lambda a_n + \mu b_n)$ is $M$-summable and $M\left(\sum (\lambda a_n + \mu b_n)\right) = \lambda M\left(\sum a_n\right) + \mu M\left(\sum b_n\right)$.
2. $M$ is *stable* if, for every sequence $(a_n)_{n \ge 0}$, the series $a_0 + a_1 + a_2 + \cdots$ is $M$-summable if and only if $a_1 + a_2 + \cdots$ is $M$-summable, and then $M(a_0 + a_1 + a_2 + \cdots) = a_0 + M(a_1 + a_2 + \cdots)$.

## Remarks

Example: ordinary summation of convergent series is linear and stable, by the limit rules for sums and because shifting the partial sums by $a_0$ does not affect convergence.
Cesàro summation is also linear and stable.
Non-example: zeta function regularization assigns $1 + 1 + 1 + \cdots$ the value $\zeta(0) = -\frac12$; stability would give $x = 1 + x$ for this value, so this method is not stable.
In terms of the partial sums $s_n$, linearity says that $D_M$ is a vector space and $M$ is a linear map on it.

## Explanations

### L1 human

Linear means sums and multiples are respected; stable means the first term can be split off.
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# Propagation of uncertainty

## Summary

Propagation of uncertainty computes how the uncertainties of input quantities carry over to a quantity computed from them.
Inputs are modelled as random variables with known means, [[variances|0477R9PYAE]] and [[covariances|0KBFAWYRTE]].
For a smooth function one usually linearizes it at the means.
The output covariance is then $J \Sigma J^{T}$, where $J$ is the [[Jacobian matrix|047NDTEPCV]] and $\Sigma$ the input covariance matrix.
It is the standard tool for error bars in experimental science; for guaranteed bounds one uses [[interval arithmetic|QPKKTE1ZY0]] instead.

## Explanations

### L1 human

Every measurement comes with an uncertainty, such as $2.00 \pm 0.01$ m.
If one computes an area or a speed from measured values, the result is uncertain too, and one wants to know by how much.
For a sum of two independent measurements, the standard deviations combine like the sides of a right triangle: $\sigma^2 = \sigma_1^2 + \sigma_2^2$.
For a product, the relative uncertainties combine in the same way.
For example, a rectangle measured as $2.00 \pm 0.02$ by $3.00 \pm 0.03$ has area $6.00$ with relative uncertainty about $\sqrt{0.01^2 + 0.01^2} \approx 1.4\%$, so about $\pm 0.08$.
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# Mean-field game

## Summary

A mean-field game is a differential game with a very large number of small, identical players.
Each player reacts only to the distribution of all the others.
In the limit of infinitely many players, a Nash equilibrium is described by a coupled system of two partial differential equations.
A [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] backward in time gives the optimal strategy of one player.
A [[Fokker–Planck equation|FVQWR7NS00]] forward in time gives the evolution of the distribution of players.
The theory was introduced independently by Jean-Michel Lasry and Pierre-Louis Lions and by Minyi Huang, Roland Malhamé and Peter Caines.

## Explanations

### L1 human

Think of a crowd leaving a stadium: each person chooses a path to the exit, and the best path depends on where everyone else goes.
No single person changes the crowd, but the crowd shapes every choice.
An equilibrium is a crowd movement in which nobody can do better by changing their own path, given the movement of the crowd.
Mean-field games compute such equilibria by treating the crowd as a continuous density.
The same idea models traders in a market, drivers in traffic and firms competing for a resource.
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<document source="https://tamnd.github.io/math2/node/WS9MRAZMFW.md" tag="WS9MRAZMFW" kind="concept">
# Costate equation

## Summary

The costate equation is the differential equation satisfied by the multipliers, or costates, of an optimal control problem along an optimal trajectory.
It reads $p' = -\partial H / \partial x$, where $H$ is the [[Hamiltonian of the problem|RACVWG5QTK]].
Together with the state equation $x' = \partial H / \partial p$ it forms a Hamiltonian system in [[Pontryagin's maximum principle|7RPV2T29CF]].
It is affine in $p$, and its linear part is the adjoint of the linearized state equation.

## Explanations

### L1 human

The costate $p(t)$ tells how much the optimal remaining cost would change if the state at time $t$ were nudged.
It is like a price that changes with time.
The costate equation says how this price evolves, and it is usually solved backward from the final time.
For example, if nothing in the cost or dynamics depends on the state, the price stays constant.
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# Menger sponge

## Example

Let $M_0 = [0, 1]^3$.
Divide each cube of side $3^{-k}$ in $M_k$ into $27$ equal cubes of side $3^{-k-1}$.
Remove the interiors of the central cube and of the $6$ cubes that share a face with it.
The union of the other $20$ cubes is $M_{k+1}$.
The *Menger sponge* is $M = \bigcap_{k \ge 0} M_k$.

The set $M$ is a [[continuum|NJQ196M1WR]].
Each $M_k$ is a finite union of closed cubes, so it is [[compact|0TVJZYB6SH]].
It is [[connected|0V032VWTHD]]: the $20$ kept cubes in each divided cube are the $12$ edge cubes and $8$ corner cubes, which form a connected frame.
A decreasing intersection of nonempty compact connected sets is nonempty, compact and connected.

## Remarks

The set $M_k$ has volume $(20/27)^k$, so $M$ has volume $0$.
Its Hausdorff dimension is $\log 20 / \log 3 \approx 2.727$.
Its [[Lebesgue covering dimension|A3W7RGXXK9]] is $1$, so topologically it is a curve.
Each face of the cube meets $M$ in a [[Sierpiński carpet|PCM3H6J3EE]].
Menger showed that every compact metric space of dimension at most $1$ embeds in $M$, so $M$ is a universal curve.

## Explanations

### L1 human

Take a cube, cut it into a three by three by three grid, and remove the middle cube and the six cubes in the middle of the faces.
Repeat with each of the twenty small cubes left, and keep going forever.
The result is a sponge with tunnels at every scale.
It has no volume and it is made of curves only, yet every curve one can imagine fits inside it.
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# Henselian ring

## Definition

Let $(R, \mathfrak m)$ be a [[local ring|MZ02BKPHJA]] with [[residue field|NJ3APQ8BHH]] $k = R/\mathfrak m$.
For $f \in R[x]$, write $\bar f \in k[x]$ for the polynomial obtained by reducing the coefficients modulo $\mathfrak m$.
The ring $R$ is *Henselian* if the following holds for every [[monic polynomial|0C3QBCR46M]] $f \in R[x]$.
If $\bar f = g_0 h_0$ with $g_0, h_0 \in k[x]$ monic and coprime, then there are monic $g, h \in R[x]$ with $f = gh$, $\bar g = g_0$ and $\bar h = h_0$.

## Remarks

An equivalent condition: for every monic $f \in R[x]$ and every $a_0 \in k$ with $\bar f(a_0) = 0$ and $\bar f'(a_0) \ne 0$, there is $a \in R$ with $f(a) = 0$ and $a \equiv a_0 \pmod{\mathfrak m}$.
That is, simple roots modulo $\mathfrak m$ lift to roots.

Example: every complete Noetherian local ring is Henselian, by the general form of [[Hensel's lemma|010QVB71T2]]; this includes the $p$-adic integers $\Z_p$ and $k[\![t]\!]$ for a field $k$.
Every field is Henselian, since $\mathfrak m = 0$ and $\bar f = f$.

Non-example: $R = \Z_{(5)}$ is not Henselian.
For $f = x^2 + 1$, one has $\bar f = (x - 2)(x - 3)$ in $\Z/5\Z[x]$, and the factors $x - 2$ and $x - 3$ are coprime because $2 \ne 3$.
A lift would give a root $a \in \Z_{(5)} \subseteq \Q$ of $x^2 + 1$, but $-1$ has no rational square root.
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# Lebesgue's density theorem

## Statement

Let $n \ge 1$, let $\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$, and let $A \subseteq \R^n$ be Lebesgue measurable.
For $x \in \R^n$ and $r > 0$, let $B(x, r)$ be the [[open ball|0YE5868HZB]] of centre $x$ and radius $r$.
Then for [[almost every|RARTMA0ZEC]] $x \in A$,
$$\lim_{r \to 0^+} \frac{\lambda(A \cap B(x, r))}{\lambda(B(x, r))} = 1,$$
and for almost every $x \in \R^n \setminus A$ the same limit equals $0$.

## Remarks

Apply the Lebesgue differentiation theorem to the indicator function $\mathbf{1}_{A \cap B(0, R)}$ for each integer $R$.
Its ball averages are exactly the ratios in the statement, and its values are $1$ on $A \cap B(0,R)$ and $0$ off it.

Example: for $A = [0, \infty) \subseteq \R$, the density is $1$ at every $x > 0$, $0$ at every $x < 0$, and $1/2$ at $x = 0$.
The exceptional set $\{0\}$ has measure zero, as the theorem allows.

The theorem says that a measurable set cannot be spread out evenly at small scales: near almost every one of its points it fills almost all of a small ball.
For instance, no measurable $A \subseteq [0,1]$ can satisfy $\lambda(A \cap I) = \lambda(I)/2$ for every interval $I \subseteq [0,1]$.

## Explanations

### L1 human

Pick a measurable set and zoom in on one of its points.
For almost every point of the set, the set fills almost the whole view as one zooms in.
For almost every point outside the set, the set almost disappears from view.

### L2 human

The points where the density is $1$ are called density points.
The density theorem is used to show that measurable sets are, up to null sets, sets of density points, which leads to the density topology on $\R$.
It can also be used to prove the Steinhaus theorem that $A - A$ contains an interval around $0$ when $\lambda(A) > 0$.
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# Bounded symmetric domain

## Definition

Let $D \subset \C^n$ be a bounded domain, that is, a bounded connected open set.
The domain $D$ is a *bounded symmetric domain* if for every $p \in D$ there is a [[biholomorphism|FX92PP3HZR]] $s_p \colon D \to D$ such that:

1. $s_p \circ s_p = \id_D$;
2. $p$ is an isolated fixed point of $s_p$.

## Remarks

The map $s_p$ is called the symmetry at $p$, and it is unique.
Its derivative at $p$ is $-\id$.
A bounded symmetric domain is homogeneous: its automorphism group acts transitively.
A bounded domain that is homogeneous and has a symmetry at one point is symmetric.

Example: the unit disc $\Delta = \set{z \in \C : \abs{z} < 1}$.
At $0$ take $s_0(z) = -z$: it is an involution with only the fixed point $0$.
At $a \in \Delta$ take $s_a = \varphi_a \circ s_0 \circ \varphi_a$, where $\varphi_a(z) = (a - z)/(1 - \bar a z)$ is an automorphism of $\Delta$ with $\varphi_a \circ \varphi_a = \id$ and $\varphi_a(a) = 0$.
Then $s_a$ is an involution whose only fixed point is $a$.

Non-example: the annulus $A = \set{z : r < \abs{z} < 1}$ with $0 < r < 1$.
Its automorphisms are the rotations $z \mapsto e^{i\theta} z$ and the maps $z \mapsto r e^{i\theta}/z$.
So each orbit of $\Aut(A)$ is a union of at most two circles, $A$ is not homogeneous, and it cannot be symmetric.

The unit ball of $\C^n$ and the polydisc $\Delta^n$ are bounded symmetric domains, and so is any product of bounded symmetric domains.
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# Classifying space

## Summary

The classifying space $BG$ of a [[topological group|92700QAB5Y]] $G$ is the base of a [[universal bundle|1HDH716Z24]] $EG \to BG$.
Principal $G$-bundles over a CW complex $X$ correspond to homotopy classes of maps $X \to BG$.
Cohomology classes of $BG$ give the [[characteristic classes|7HACJC0JE2]] of such bundles.
For a discrete group $G$, the space $BG$ is the [[classifying space of the group|WA796ZCNRX]] in the sense of an Eilenberg–MacLane space $K(G, 1)$.

## Explanations

### L1 human

A classifying space is a catalogue of all bundles with a given structure group.
Each map from $X$ into the catalogue picks out a bundle over $X$, and deforming the map does not change the bundle.
For real line bundles the catalogue is the infinite projective space $\R P^\infty$, the space of lines through the origin in $\R^\infty$.
Mapping a point of $X$ to a line tells which line sits over that point.
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# Euler transform

## Summary

The Euler transform rewrites an [[alternating series|0QED0GPYKY]] $\sum (-1)^n a_n$ as a new series built from the iterated differences of the $a_n$, with weights $2^{-n-1}$.
When the $a_n$ vary smoothly, the new series converges much faster, often geometrically.
It is a classical tool of [[series acceleration|B1S2XJPGYD]], and the same formula gives the [[Euler summation|81Q4ERP594]] method for some divergent series.

## Explanations

### L1 human

The terms of an alternating series cancel each other in pairs.
The Euler transform groups this cancellation in a systematic way using differences of consecutive terms.
For the [[alternating harmonic series|0JQ5Y3WWVE]] $1 - \frac12 + \frac13 - \cdots = \ln 2$, the transformed series is $\sum_{n \ge 1} \frac{1}{n 2^n}$.
Its terms shrink by about half each time, so a few terms already give several correct digits.
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# G-structure

## Definition

Let $M$ be a smooth manifold of dimension $n$, and let $F(M) \to M$ be its [[frame bundle|F4HWTRNXAP]].
It is the [[principal bundle|8E37E74XVP]] with group $\mathrm{GL}(n, \R)$ whose fibre over $p$ is the set of ordered bases of $T_pM$.
Let $G$ be a [[Lie subgroup|NM8XCPS1Q3]] of $\mathrm{GL}(n, \R)$.
A *$G$-structure* on $M$ is a principal $G$-subbundle $P \subset F(M)$.
That is, $P$ is a smooth submanifold of $F(M)$ that meets every fibre and is preserved by the right action of $G$.
Moreover $G$ acts freely and transitively on each fibre of $P$.

## Remarks

Example: a Riemannian metric $g$ gives an $\mathrm{O}(n)$-structure, the bundle of $g$-orthonormal frames.
Two orthonormal frames at $p$ differ by an orthogonal matrix, so $\mathrm{O}(n)$ acts simply transitively on each fibre.
Conversely every $\mathrm{O}(n)$-structure is the orthonormal frame bundle of exactly one Riemannian metric.
A $\set{e}$-structure is a global frame, that is, a parallelization of $M$.
Non-example: the sphere $S^2$ has no $\set{e}$-structure, since a global frame would contain a nowhere vanishing vector field, which $S^2$ does not have.
Other standard cases: $\mathrm{GL}^+(n, \R)$ gives orientations, $\mathrm{GL}(m, \C) \subset \mathrm{GL}(2m, \R)$ gives almost complex structures, and $\mathrm{Sp}(2m, \R)$ gives nondegenerate $2$-forms.
A $G$-structure is called integrable, or flat, when near each point there is a chart whose coordinate frames lie in $P$.
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<document source="https://tamnd.github.io/math2/node/WT65A6ARGN.md" tag="WT65A6ARGN" kind="definition">
# Inverse scattering transform

## Definition

Consider the [[Korteweg–De Vries equation|KTT75QC72Z]] $u_t - 6 u u_x + u_{xxx} = 0$ for $u(x, t)$ real, smooth and decaying fast as $\abs{x} \to \infty$.
For each $t$, let $L_t = -\partial_x^2 + u(\cdot, t)$ be the one-dimensional [[Schrödinger operator|89AA6XJY5K]] with potential $u(\cdot, t)$.
Its *scattering data* $S(t)$ consist of:

1. the finitely many eigenvalues $-\kappa_1^2, \dots, -\kappa_N^2$ of $L_t$, with $\kappa_n > 0$;
2. the norming constants $c_n(t) > 0$, where the normalised eigenfunction $\psi_n$ satisfies $\psi_n(x) \sim c_n e^{-\kappa_n x}$ as $x \to +\infty$;
3. the reflection coefficient $b(k, t)$ for $k \in \R$, where a solution of $L_t \psi = k^2 \psi$ satisfies $\psi(x) \sim e^{-ikx} + b(k, t) e^{ikx}$ as $x \to +\infty$ and $\psi(x) \sim a(k, t) e^{-ikx}$ as $x \to -\infty$.

The *inverse scattering transform* solves the initial value problem with data $u(x, 0)$ in three steps.

1. Direct scattering: compute $S(0)$ from $u(\cdot, 0)$.
2. Time evolution: $\kappa_n$ stays constant, $c_n(t) = c_n(0) e^{4 \kappa_n^3 t}$ and $b(k, t) = b(k, 0) e^{8 i k^3 t}$.
3. Inverse scattering: set $F(x) = \sum_{n} c_n(t)^2 e^{-\kappa_n x} + \frac{1}{2\pi} \int_{\R} b(k, t) e^{ikx} \, dk$, solve the Gelfand–Levitan–Marchenko equation
$$K(x, y) + F(x + y) + \int_x^\infty K(x, z) F(z + y) \, dz = 0 \quad (y > x)$$
for $K$, and put $u(x, t) = -2 \frac{d}{dx} K(x, x)$.

## Remarks

The time evolution in step 2 comes from the [[Lax pair|G3A4S7CSB3]] of the equation: $L_t$ evolves by conjugation, so its eigenvalues do not move.

Example: take $N = 1$ and $b = 0$, so that $F(x) = c^2 e^{-\kappa x}$ with $c = c_1(t)$.
The kernel $K(x, y) = -\dfrac{c^2 e^{-\kappa (x + y)}}{1 + (c^2/2\kappa) e^{-2\kappa x}}$ solves the equation.
With $w = (c^2/2\kappa) e^{-2\kappa x}$ this gives $u = -8\kappa^2 w/(1 + w)^2 = -2\kappa^2 \operatorname{sech}^2(\kappa(x - x_0))$, where $w = e^{-2\kappa(x - x_0)}$.
Since $c^2$ grows like $e^{8\kappa^3 t}$, the centre is $x_0 = 4\kappa^2 t + \text{const}$: this is the one-[[soliton|63D4AXYCT4]] solution moving with speed $4\kappa^2$.

Non-example: the method needs a Lax pair, and most nonlinear evolution equations have none.
For instance, the generalised KdV equation $u_t + u^p u_x + u_{xxx} = 0$ with an integer $p \ge 3$ is not integrable, and the steps above do not apply to it.

Other equations with a Lax pair, such as the [[nonlinear Schrödinger equation|GEJHRXC176]] and the [[sine-Gordon equation|802QSF9NM3]], have inverse scattering transforms built on other spectral problems.

## Explanations

### L1 human

The inverse scattering transform turns a hard nonlinear evolution into three linear steps.
Map the initial wave to spectral data, move the data forward by explicit exponentials, and map back by solving a linear integral equation.
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<document source="https://tamnd.github.io/math2/node/WT7K28D1CD.md" tag="WT7K28D1CD" kind="definition">
# Wigner distribution function

## Definition

Let $f \in L^2(\R)$ (see [[Lp space|SASP14MA1Z]]).
The *Wigner distribution* of $f$ is the function $Wf \colon \R^2 \to \C$,
$$Wf(x, \omega) = \int_{\R} f\!\left(x + \tfrac{t}{2}\right) \overline{f\!\left(x - \tfrac{t}{2}\right)} \, e^{-2\pi i t \omega} \, dt,$$
for $x \in \R$ (time) and $\omega \in \R$ (frequency).

## Remarks

The integral converges absolutely for each $x$, by the Cauchy–Schwarz inequality.
$Wf$ is real-valued, since replacing $t$ by $-t$ turns the integral into its complex conjugate.
For $f$ in the Schwartz space, $\int_\R Wf(x, \omega) \, d\omega = \abs{f(x)}^2$ and $\int_\R Wf(x, \omega) \, dx = \abs{\hat f(\omega)}^2$, where $\hat f$ is the [[Fourier transform|DPVM34ADZV]].
In quantum mechanics the Wigner function of a wave function is usually written with Planck's constant and a normalizing factor.

Example: let $f(t) = 2^{1/4} e^{-\pi t^2}$, so that $\norm{f}_2 = 1$.
Then $f(x + t/2) \overline{f(x - t/2)} = \sqrt2 \, e^{-2\pi x^2} e^{-\pi t^2/2}$, and the Fourier transform of $e^{-\pi t^2/2}$ is $\sqrt2 \, e^{-2\pi \omega^2}$.
So $Wf(x, \omega) = 2 e^{-2\pi (x^2 + \omega^2)}$, and integrating in $\omega$ gives $\sqrt2 e^{-2\pi x^2} = \abs{f(x)}^2$.

Non-example: the Wigner distribution is quadratic in $f$, not linear.
For instance $W(2f) = 4 Wf \ne 2 Wf$ when $f \ne 0$.
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<document source="https://tamnd.github.io/math2/node/WTCNKST134.md" tag="WTCNKST134" kind="example">
# Diffeomorphism group

## Example

Let $M$ be a [[smooth manifold|F45VQ47GPV]].
The *diffeomorphism group* $\operatorname{Diff}(M)$ is the set of all [[diffeomorphisms|0KZZ70ZHXP]] $f \colon M \to M$, with composition as the operation.
It is a [[group|G5S1M9]].
The composition of two diffeomorphisms is a diffeomorphism, and composition is associative.
The identity map $\id_M$ is the identity element, and the inverse of $f$ is the inverse map $f^{-1}$, a diffeomorphism too.

When $M$ is compact, $\operatorname{Diff}(M)$ is an infinite-dimensional Lie group.
It is a [[Fréchet manifold|EBMCFQT0JN]] modelled on the space of smooth [[vector fields|0299VCW1TH]] on $M$, with smooth composition and inversion.
Its Lie algebra is the space of vector fields; with the usual conventions, the bracket is the negative of the Lie bracket of vector fields.

## Remarks

Unlike a finite-dimensional [[Lie group|7JBRGCVJRB]], $\operatorname{Diff}(M)$ has an [[exponential map|H3QX1ZFC8K]], given by the time-one flow of a vector field, that need not be onto any neighbourhood of the identity.
This happens already for $M = S^1$.
So some results of finite-dimensional Lie theory fail for diffeomorphism groups.

## Explanations

### L1 human

The diffeomorphism group of a space collects all ways to deform the space smoothly and reversibly onto itself.
For the circle, these are the smooth bijections of the circle with smooth inverse, such as rotations and smooth stretchings.
Doing one deformation after another is again a deformation, and each can be undone, so they form a group.
This group is too big to be a Lie group in the usual sense, since it has infinitely many independent directions, one for each vector field.
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<document source="https://tamnd.github.io/math2/node/WTCP8A87P1.md" tag="WTCP8A87P1" kind="counterexample">
# Borsuk's conjecture

## Counterexample

Borsuk's conjecture concerns bounded sets $S \subseteq \R^n$ with more than one point, for $n \ge 1$.
It says that every such $S$ can be split into $n + 1$ sets, each of smaller [[diameter|0KW3PN547K]] than $S$.
The conjecture is false.
Kahn and Kalai proved this in 1993.
For all large $n$, they found finite sets in $\R^n$ that cannot be split into fewer than $1.2^{\sqrt{n}}$ parts of smaller diameter.
This number exceeds $n + 1$ for large $n$.

Their sets consist of the matrices $x x^T$, for $x$ in a suitable family of vectors in $\set{-1, 1}^m$.
These matrices are viewed as points of $\R^{m \times m}$ with the Euclidean norm.
For two such vectors, $\norm{x x^T - y y^T}^2 = 2 m^2 - 2 (x \cdot y)^2$.
So the diameter is attained exactly by the pairs with $x \cdot y = 0$, and a part of smaller diameter contains no orthogonal pair.
The Frankl–Wilson theorem shows that such parts are exponentially small in $m$.
So exponentially many parts are needed, while the dimension is only of order $m^2$.

## Remarks

The conjecture is true for $n = 2$ and $n = 3$, and for smooth convex bodies in every dimension.
The number $n + 1$ cannot be lowered: by the [[Borsuk–Ulam theorem|QHJ0GBQE2E]], the unit ball of $\R^n$ cannot be split into $n$ closed sets of smaller diameter.
Later work lowered the dimension of counterexamples; the smallest dimension known to admit a counterexample is $64$, found by Jenrich and Brouwer.
The vector $x x^T$ is the [[tensor product|0BF2DE2SYK]] $x \otimes x$, and the formula above uses $\norm{x \otimes x}^2 = \abs{x}^4 = m^2$ and $\langle x \otimes x, y \otimes y \rangle = (x \cdot y)^2$.

## Explanations

### L1 human

Borsuk asked whether any shape in $n$-dimensional space can be cut into $n + 1$ pieces that are all strictly smaller across.
In the plane a disc can be cut into three such pieces, and three is needed.
For a long time people believed this in every dimension.
In 1993 Kahn and Kalai found strange high-dimensional point sets where far more pieces are needed, so the conjecture fails.
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<document source="https://tamnd.github.io/math2/node/WTRPSJKAZH.md" tag="WTRPSJKAZH" kind="definition">
# Cobordism

## Definition

Let $n \ge 0$ and let $M$ and $N$ be closed (compact, without boundary) [[smooth manifolds|F45VQ47GPV]] of dimension $n$.
A *cobordism* between $M$ and $N$ is a compact smooth [[manifold with boundary|7DS6B2BFK8]] $W$ of dimension $n + 1$ with $\partial W$ diffeomorphic to $M \sqcup N$.
The manifolds $M$ and $N$ are *cobordant* if a cobordism between them exists.

## Remarks

Being cobordant is an [[equivalence relation|0G8VEZ3WTK]]: $M \times [0, 1]$ gives reflexivity, and two cobordisms can be glued along a common boundary component for transitivity.
The set of classes $\Omega_n^O$ is an abelian group under disjoint union, and every element has order $2$, since $M \sqcup M$ bounds $M \times [0, 1]$.
In the oriented version, $W$ is oriented and $\partial W \cong M \sqcup (-N)$, where $-N$ is $N$ with the opposite orientation; this gives the groups $\Omega_n^{SO}$.

Example: the circle $S^1$ bounds the disc $D^2$, so $S^1$ is cobordant to the empty manifold.
Every sphere $S^n = \partial D^{n+1}$ is likewise null-cobordant.

Non-example: the real projective plane $\R P^2$ is not the boundary of any compact $3$-manifold.
A closed manifold $M$ that bounds a compact manifold $W$ has even Euler characteristic, but $\chi(\R P^2) = 1$.

A manifold cobordant to the empty manifold is called null-cobordant.
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# Estimating equations

## Summary

The method of estimating equations defines an estimator as a solution of $\sum_i g(X_i, \theta) = 0$.
Here the function $g$ has mean zero at the true parameter.
The likelihood equations, the moment equations of the method of moments and the normal equations of least squares are all estimating equations.
The approach needs only a model for some moments, not a full probability model.
Its asymptotic theory is the same as that of M-estimators, with a sandwich formula for the variance.

## Explanations

### L1 human

Write down an equation that the true parameter satisfies on average, such as "the mean of $X - \theta$ is zero".
Replace the average over the population by the average over the sample and solve for $\theta$.
In the example this gives $\theta = \bar{X}$, the sample mean.
Because the sample average is close to the population average, the solution is close to the true parameter.
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<document source="https://tamnd.github.io/math2/node/WTXMRG6WR4.md" tag="WTXMRG6WR4" kind="definition">
# Gosper's algorithm

## Definition

Let $t_k$ be a hypergeometric term in $k$ over a field of characteristic $0$.
This means that $t_k \ne 0$ and that $r(k) = t_{k+1}/t_k$ is a [[rational function|0XDK9CGYKJ]] of $k$.
*Gosper's algorithm* takes $t_k$ and either returns a hypergeometric term $T_k$ with $T_{k+1} - T_k = t_k$, or reports that no such term exists.
It runs as follows.

1. Write $r(k) = \dfrac{a(k)}{b(k)}\cdot\dfrac{c(k+1)}{c(k)}$ with polynomials $a, b, c$ such that $\gcd\big(a(k), b(k + h)\big) = 1$ for every integer $h \ge 0$ (a [[polynomial gcd|Q75YRG25JK]] condition).
2. Search for a polynomial $x(k)$ with
$$a(k)\,x(k + 1) - b(k - 1)\,x(k) = c(k),$$
using a bound on its degree computed from $a$, $b$, $c$ and linear algebra on its coefficients.
3. If such an $x$ exists, return $T_k = \dfrac{b(k - 1)\,x(k)}{c(k)}\,t_k$. Otherwise report that $t_k$ has no hypergeometric indefinite sum.

## Remarks

Check of step 3: with $y(k) = b(k-1)x(k)/c(k)$, one has $T_{k+1} - T_k = \big(y(k + 1)r(k) - y(k)\big)t_k$, and $y(k+1)r(k) - y(k) = \big(a(k)x(k+1) - b(k-1)x(k)\big)/c(k) = 1$ by step 2.
Gosper's theorem says that the method is complete: if some hypergeometric $T_k$ works, then $T_k/t_k$ is rational and the equation in step 2 has a polynomial solution.
Example: $t_k = k\cdot k!$ has $r(k) = (k + 1)^2/k$; take $a(k) = k + 1$, $b(k) = 1$, $c(k) = k$, which satisfy the gcd condition.
The equation $(k + 1)x(k + 1) - x(k) = k$ has the solution $x = 1$, so $T_k = k!$ for $k \ge 1$, and indeed $(k + 1)! - k! = k\cdot k!$.
Non-example: for $t_k = 1/(k + 1)$ the algorithm reports failure, since the partial sums are harmonic numbers, which are not a hypergeometric term.
The equation of step 2 is a linear [[recurrence|0FYX44GP92]] with polynomial coefficients for the unknown polynomial $x$.
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# Regularization of ill-posed problems

## Summary

A problem is ill-posed in the sense of Hadamard when it is not a [[well-posed problem|W09B9NZ9B5]].
This means that a solution does not exist, is not unique, or does not depend continuously on the data.
Regularization replaces an ill-posed problem by a family of nearby well-posed problems, indexed by a parameter.
The parameter is chosen from the noise level so that the regularized solutions converge to the true solution as the noise tends to zero.
It is the basic tool for [[inverse problems|PCMW7KN5P4]] such as image deblurring, tomography and parameter identification.

## Explanations

### L1 human

Some problems amplify small errors in the data enormously, so that the naive answer is useless.
Undoing a blur in a photograph is an example: tiny noise gets magnified into wild patterns.
Regularization adds a mild preference, such as "the answer should not be too large or too rough", which tames this amplification.
The strength of the preference is tuned to the amount of noise.
Too little regularization leaves the noise in, and too much smooths away the real details.
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<document source="https://tamnd.github.io/math2/node/WV00RFGTVT.md" tag="WV00RFGTVT" kind="definition">
# Rogers–Ramanujan continued fraction

## Definition

Let $\tau$ be in the [[upper half-plane|X6XGGH5FPW]], let $q = e^{2\pi i \tau}$, and let $q^{1/5} = e^{2\pi i \tau / 5}$.
The *Rogers–Ramanujan continued fraction* is
$$R(\tau) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^2}{1 + \cfrac{q^3}{1 + \cdots}}}},$$
the limit of the finite [[continued fractions|04P2RJ4WPS]] obtained by stopping after the term $q^n$, as $n \to \infty$.

## Remarks

Since $\abs{q} < 1$ the limit exists, and Rogers and Ramanujan showed
$$R(\tau) = q^{1/5} \prod_{n=1}^{\infty} \frac{(1 - q^{5n-1})(1 - q^{5n-4})}{(1 - q^{5n-2})(1 - q^{5n-3})} = q^{1/5}\, \frac{H(q)}{G(q)},$$
with $G(q) = \sum_{n \ge 0} \frac{q^{n^2}}{(q; q)_n}$ and $H(q) = \sum_{n \ge 0} \frac{q^{n^2 + n}}{(q; q)_n}$, the two sides of the [[Rogers–Ramanujan identities|M24H4GJPPW]].

Example: $\tau = i$, so $q = e^{-2\pi} \approx 0.001867$.
Then $R(i) = \sqrt{\frac{5 + \sqrt5}{2}} - \frac{1 + \sqrt5}{2} \approx 0.284079$.
A numerical check: $q^{1/5} = e^{-2\pi/5} \approx 0.284609$, and the continued fraction $1 + q/(1 + q^2/(1 + \cdots))$ is about $1.001867$, whose quotient is about $0.284079$.

Non-example: with numerators $1, 1, 1, \dots$ instead of $q, q^2, q^3, \dots$, the fraction $\cfrac{1}{1 + \cfrac{1}{1 + \cdots}}$ is $\frac{\sqrt5 - 1}{2}$, the inverse of the golden ratio, and it does not depend on $q$; it is not $R$.

Another common notation is $R(q)$ in place of $R(\tau)$.
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# Fujiki class C

## Definition

Let $X$ be a compact, connected [[complex manifold|JSGWAN2RJV]].
Then $X$ is *in Fujiki class $\mathcal{C}$* if there are a compact [[Kähler manifold|X60JG9AEQJ]] $Y$ and a compact complex manifold $Z$ with the following property.
There are [[modifications|SB3MY7XQM9]] $p \colon Z \to X$ and $q \colon Z \to Y$.
Here a surjective holomorphic map $f \colon Z \to V$ is a *modification* if it satisfies the following condition.
There is a nowhere dense analytic subset $A \subset V$ such that $f^{-1}(A)$ is nowhere dense in $Z$ and $f$ maps $Z \setminus f^{-1}(A)$ biholomorphically onto $V \setminus A$.
In other words, $X$ is bimeromorphic to a compact Kähler manifold.

## Remarks

Fujiki's original definition, for reduced compact complex spaces, asks that $X$ be the image of a compact Kähler space under a meromorphic map.
By a theorem of Varouchas, for compact complex manifolds this is equivalent to the definition above.

Example: every compact Kähler manifold $X$ is in class $\mathcal{C}$, with $Y = Z = X$ and $p = q = \id$.
Every [[Moishezon manifold|8ZW9P9JRQD]] is in class $\mathcal{C}$, since it is bimeromorphic to a projective manifold, which is Kähler.

Non-example: a [[Hopf surface|PNFNGBXRM3]] is not in class $\mathcal{C}$.
Manifolds in class $\mathcal{C}$ have even odd Betti numbers, as compact Kähler manifolds do, while a Hopf surface has $b_1 = 1$.
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# Forgetful functor

## Summary

A forgetful functor sends a mathematical object to a less structured object by dropping some of its structure.
It sends each morphism to the same map, regarded in the poorer setting.
The basic example sends an algebra to its underlying set and a homomorphism to its underlying function.
Forgetful functors often have left adjoints, the [[free functors|N353JF8RBK]], and the pair produces a monad that encodes the algebraic theory.
They also make a category of algebras into a [[concrete category|ZD0GTPSJS1]].

## Explanations

### L1 human

A group is a set together with a multiplication.
Forgetting the multiplication leaves just the set, and a group homomorphism is still a function between the sets.
The forgetful functor from groups to sets is this act of forgetting, done for all groups and homomorphisms at once.
One can forget less: a ring can be sent to its additive group, forgetting only the multiplication.
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# Pseudovariety

## Definition

Let $\sigma$ be an algebraic signature.
A class $V$ of finite $\sigma$-algebras is a *pseudovariety* if it satisfies these three conditions.

1. If $A \in V$ and $B$ is a homomorphic image of $A$, that is, isomorphic to a [[quotient algebra|N5KE7FH6CX]] of $A$, then $B \in V$.
2. If $A \in V$ and $B$ is a [[subalgebra|7CHG6SD365]] of $A$, then $B \in V$.
3. If $A_1, \ldots, A_n \in V$ with $n \ge 0$, then the [[direct product|WJ73XYBNE4]] $A_1 \times \cdots \times A_n$ is in $V$; for $n = 0$ it is the one-element algebra.

## Remarks

Example: the class of finite abelian groups is a pseudovariety.
Quotients and subgroups of finite abelian groups are finite and abelian, and so are finite products.

Non-example: the class of finite cyclic groups fails condition 3.
The product $\Z / 2\Z \times \Z / 2\Z$ is finite but not cyclic, since each of its elements has order at most $2$.

For each variety $W$, the finite members of $W$ form a pseudovariety, but not every pseudovariety arises this way.
A standard pseudovariety of semigroups is the class of finite aperiodic semigroups, those with only trivial subgroups.
By Eilenberg's variety theorem, pseudovarieties of finite monoids correspond to varieties of regular languages; aperiodic monoids correspond to star-free languages by Schützenberger's theorem.
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<document source="https://tamnd.github.io/math2/node/WV4S47X6QC.md" tag="WV4S47X6QC" kind="statement">
# Cartan's test for involutivity

## Statement

Let $\mathcal{I}$ be an [[exterior differential system|901CCKQHVY]] on a [[smooth manifold|F45VQ47GPV]] $M$, and assume $\mathcal{I}$ contains no nonzero functions.
For $p \in M$, a linear subspace $E \subseteq T_pM$ is an *integral element* if $\varphi|_E = 0$ for every $\varphi \in \mathcal{I}$.
Let $V_k(\mathcal{I})$ be the set of $k$-dimensional integral elements, a subset of the bundle $G_k(TM)$ of $k$-planes in tangent spaces, whose fibers are [[Grassmannians|CYPA882RB9]].
For an integral element $E$ with basis $e_1, \dots, e_k$, its *polar space* is
$$H(E) = \set{ v \in T_pM : \varphi(v, e_1, \dots, e_k) = 0 \text{ for all } \varphi \in \mathcal{I} \text{ of degree } k + 1 },$$
and $c(E) = \dim M - \dim H(E)$ is its codimension.
An integral element $E$ is *regular* if, near $E$, the set $V_k(\mathcal{I})$ is a smooth [[submanifold|VKFW27WXEZ]] of $G_k(TM)$ and $\dim H(E')$ is constant for $E' \in V_k(\mathcal{I})$.

Let $E \in V_n(\mathcal{I})$ and let $\set{0} = E_0 \subset E_1 \subset \dots \subset E_n = E$ be a flag with $\dim E_k = k$.
Put $c_k = c(E_k)$.
Then:

1. Near $E$, the set $V_n(\mathcal{I})$ lies in a submanifold of $G_n(TM)$ of codimension $c_0 + c_1 + \dots + c_{n-1}$.
2. The following are equivalent. Near $E$, $V_n(\mathcal{I})$ is a smooth submanifold of $G_n(TM)$ of codimension exactly $c_0 + \dots + c_{n-1}$. Each $E_k$ with $k < n$ is a regular integral element.

An integral element $E$ that has a flag as in 2 is called *ordinary*.

## Remarks

The idea is that each extension $E_k \subset E_{k+1}$ is a choice of a vector in the polar space $H(E_k)$, and each such choice imposes at least $c_k$ independent conditions.
Counting these conditions gives the lower bound for the codimension of $V_n(\mathcal{I})$, and equality means that no hidden conditions appear.

When the system comes with an independence condition, an integral element that passes this test is called involutive.
Ordinary integral elements are the starting points of the Cartan–Kähler theorem, which produces integral manifolds through them in the real-analytic case.

Example: let $\mathcal{I}$ be generated by $\theta = dz - y\, dx$ on $\R^3$, and let $n = 1$.
For $E_0 = \set{0}$, the polar space is $\ker \theta$, of dimension $2$, so $c_0 = 1$.
The $1$-dimensional integral elements at a point are the lines in the plane $\ker \theta$, a circle's worth inside the projective plane of lines, so $V_1(\mathcal{I})$ has codimension $1 = c_0$ in $G_1(T\R^3)$.
The test is passed, and every integral line is ordinary.

## Explanations

### L1 human

To decide whether a system of differential equations has many solutions, Cartan counts how many conditions each step of building a solution imposes.
The test compares this count with the actual number of conditions defining the possible tangent planes of solutions.
If the two agree, the system is in good shape: no hidden conditions will appear later.

### L2 human

Cartan's test is a purely algebraic, pointwise computation with polar spaces.
It is the practical way to check the hypothesis of the Cartan–Kähler theorem.
When a system fails the test, one passes to its prolongation, adding the derivatives as new variables, and tests again.
The Cartan–Kuranishi prolongation theorem says that, under regularity assumptions, this process ends with an involutive system after finitely many steps.
The Cartan characters, read off from the numbers $c_k$, then describe how many functions of how many variables the general solution depends on.
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# Commutative diagram

## Summary

A commutative diagram is a drawing of objects and [[morphisms|859F0TFVA3]] of a [[category|A2DE66937D]].
In it, any two directed paths with the same start and the same end compose to the same morphism.
Diagrams are the main notation of category theory and homological algebra: one picture records many equations between composites.
The commutative square $g \circ f = k \circ h$ and the commutative triangle $g \circ f = h$ are the basic shapes.
The definition of a [[natural transformation|KYCPCC57FT]] is the commutativity of one square for each morphism.

## Explanations

### L1 human

Draw objects as points and morphisms as arrows between them.
Following a path of arrows means composing them.
The diagram commutes if any two routes from one point to another give the same result.
For example, with $f(x) = x + 1$ and $g(x) = 2x$ on the real numbers, the square "first $f$ then $g$" against "first $g$ then $f$" does not commute.
The first route gives $2x + 2$ and the second gives $2x + 1$.
</document>

<document source="https://tamnd.github.io/math2/node/WVB60RKC3Z.md" tag="WVB60RKC3Z" kind="concept">
# Lelong number

## Summary

The Lelong number of a [[plurisubharmonic function|FF2P7ER3QT]] $u$ at a point $x$ measures how fast $u$ tends to $-\infty$ near $x$, compared with $\log \abs{z - x}$.
The same name is used for the analogous density of a closed [[positive current|BYJHP6A906]] at a point.
For $u = \log \abs{f}$ with $f$ holomorphic, the Lelong number is the order of vanishing of $f$ at $x$.
Lelong numbers are the basic local invariant of singularities in pluripotential theory.

## Explanations

### L1 human

A plurisubharmonic function may have poles where it equals $-\infty$.
The Lelong number tells how strong such a pole is, on a logarithmic scale.
The function $\log \abs{z}$ on $\C$ has Lelong number $1$ at $0$, and $\log \abs{z^3} = 3 \log \abs{z}$ has Lelong number $3$.
At a point where the function is finite, the Lelong number is $0$.
</document>

<document source="https://tamnd.github.io/math2/node/WVD069C7QF.md" tag="WVD069C7QF" kind="statement">
# Cox–de Boor recursion

## Statement

Let $(t_j)$ be a nondecreasing sequence of real knots.
Let $i$ be an index and $k \ge 1$ an order.
The normalized [[B-spline|62AFX32JGT]] is
$$B_{i,k}(x) = (t_{i+k} - t_i)\, [t_i, \dots, t_{i+k}]\, (\cdot - x)_+^{k-1}.$$
Here $[t_i, \dots, t_{i+k}]$ is the [[divided difference|N5ZHV4HWDC]] in the first variable, $(u)_+^{k-1} = u^{k-1}$ for $u > 0$ and $0$ otherwise, also for $k = 1$.
Set $B_{i,k} = 0$ when $t_i = t_{i+k}$.
Then
$$B_{i,1}(x) = \begin{cases} 1 & \text{if } t_i \le x < t_{i+1}, \\ 0 & \text{otherwise,} \end{cases}$$
and for $k \ge 2$ and all real $x$,
$$B_{i,k}(x) = \frac{x - t_i}{t_{i+k-1} - t_i} B_{i,k-1}(x) + \frac{t_{i+k} - x}{t_{i+k} - t_{i+1}} B_{i+1,k-1}(x),$$
where a term whose denominator is $0$ is read as $0$.

## Remarks

The proof applies the Leibniz rule for divided differences to the product $(t - x)_+^{k-1} = (t - x) \cdot (t - x)_+^{k-2}$ and uses the recursive definition of divided differences.
Many texts take the recursion itself as the definition of B-splines, and then the divided difference formula becomes a theorem.
Example: with knots $0, 1, 2$, the recursion gives $B_{0,2}(x) = x$ on $[0, 1)$, $2 - x$ on $[1, 2)$ and $0$ elsewhere, the hat function.
The recursion shows at once that $B_{i,k}$ is nonnegative, vanishes outside $[t_i, t_{i+k})$, and is a piecewise polynomial of degree at most $k - 1$.
It is numerically stable, since for $x$ in the support both coefficients lie in $[0, 1]$.

## Explanations

### L1 human

B-splines of higher degree are built from those of lower degree.
Each one is a blend of two neighbours of one lower degree, with weights that grow or shrink linearly across the support.
Starting from step functions that are $1$ on one knot interval, one gets hat functions, then smooth quadratic bumps, then cubic bumps.

### L2 human

The recursion, found independently by Cox and by de Boor in 1972, is the standard way to evaluate B-splines and their derivatives.
Applying it to a whole spline gives [[de Boor's algorithm|5JHFB384BF]], which evaluates $\sum_j c_j B_{j,k}(x)$ by repeated convex combinations of the coefficients.
From it one also derives the derivative formula for B-splines and the partition of unity $\sum_j B_{j,k}(x) = 1$ on the basic interval.
</document>

<document source="https://tamnd.github.io/math2/node/WVJQXA3K2E.md" tag="WVJQXA3K2E" kind="example">
# Morlet wavelet

## Example

Fix a parameter $\sigma > 0$ and put $\kappa_\sigma = e^{-\sigma^2/2}$.
The *Morlet wavelet* is the complex-valued function
$$\psi_\sigma(t) = c_\sigma \, \pi^{-1/4} e^{-t^2/2} \left(e^{i\sigma t} - \kappa_\sigma\right), \qquad t \in \R,$$
where $c_\sigma > 0$ is the constant that makes $\norm{\psi_\sigma}_2 = 1$.
It is a plane wave $e^{i\sigma t}$ under a Gaussian envelope, with a small correction term.

Check of the zero mean.
Since $\int_{\R} e^{-t^2/2} e^{i\sigma t} \, dt = \sqrt{2\pi} \, e^{-\sigma^2/2}$ and $\int_{\R} e^{-t^2/2} \, dt = \sqrt{2\pi}$, the integral of $\psi_\sigma$ is $c_\sigma \pi^{-1/4} \sqrt{2\pi} \left(e^{-\sigma^2/2} - \kappa_\sigma\right) = 0$.
Its Fourier transform is a combination of two Gaussians and vanishes at frequency $0$.
Together with the Gaussian decay, this gives the [[admissibility condition|2BBB8NY5ZA]], so $\psi_\sigma$ can be used in the [[continuous wavelet transform|57FR21NDGV]].

## Remarks

The correction $\kappa_\sigma$ is tiny when $\sigma$ is large; for $\sigma \ge 5$ it is below $10^{-5}$, and in practice it is often dropped.
Without the correction, the function $e^{-t^2/2} e^{i\sigma t}$ has nonzero integral, so it is not admissible in the strict sense.
The Morlet wavelet does not generate an orthonormal basis; it is used for the continuous transform and for time–frequency analysis of signals.
Its real part is even and its imaginary part is odd.
Since the wavelet is complex-valued, the modulus of the transform gives an amplitude and the argument gives a phase.

## Explanations

### L1 human

The Morlet wavelet is a short burst of oscillation: a wave of frequency $\sigma$ that fades out like a bell curve on both sides.
Comparing a signal with stretched and shifted copies of it tells how much of each frequency is present near each time.
The small constant subtracted from the wave makes the average of the function exactly zero, which a wavelet needs.
</document>

<document source="https://tamnd.github.io/math2/node/WVKDC011KP.md" tag="WVKDC011KP" kind="statement">
# Riemann–Lebesgue lemma

## Statement

Let $n \ge 1$ and let $f \colon \R^n \to \C$ be [[integrable|X7GTF3D98F]] for [[Lebesgue measure|2Z2QVWZ262]].
For $\xi \in \R^n$ let
$$\hat{f}(\xi) = \int_{\R^n} f(x) \, e^{-2 \pi i \, x \cdot \xi} \, dx,$$
a [[Lebesgue integral|DWJF7ABPR1]], where $x \cdot \xi$ is the dot product.
Then $\hat{f}(\xi) \to 0$ as $\abs{\xi} \to \infty$.

## Remarks

Proof idea: for the indicator function of a box, $\hat{f}$ is a product of terms like $(e^{-2\pi i b \xi_k} - e^{-2\pi i a \xi_k}) / (-2\pi i \xi_k)$, which tends to $0$.
Linear combinations of such indicators, the step functions, are dense in $L^1(\R^n)$.
Since $\abs{\hat{f}(\xi) - \hat{g}(\xi)} \le \int \abs{f - g}$ for every $\xi$, the property passes to all integrable $f$.

Example: for $f = \mathbf{1}_{[-1/2, 1/2]}$ on $\R$, $\hat{f}(\xi) = \sin(\pi \xi) / (\pi \xi)$ for $\xi \ne 0$, which tends to $0$.

The same proof gives the version for Fourier coefficients.
If $f$ is integrable on $[0, 2\pi]$, then $\int_0^{2\pi} f(x) e^{-ikx} \, dx \to 0$ as $\abs{k} \to \infty$ over the integers.
The lemma gives no rate of decay in general.

## Explanations

### L1 human

Multiplying a function by a fast oscillating wave and integrating gives almost complete cancellation.
The positive and negative parts of the wave nearly balance on each small piece where the function is almost constant.
So the Fourier transform of an integrable function becomes small at high frequencies.

### L2 human

The lemma says that the Fourier transform maps $L^1(\R^n)$ into the space $C_0(\R^n)$ of continuous functions vanishing at infinity.
In Fourier series it is a first step in the study of pointwise convergence, through the Dini and Jordan criteria.
The image of $L^1$ is a proper subset of $C_0$, so the decay at infinity alone does not characterise Fourier transforms of integrable functions.
</document>

<document source="https://tamnd.github.io/math2/node/WVKE1QS423.md" tag="WVKE1QS423" kind="definition">
# Bézout domain

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]].
$R$ is a *Bézout domain* if every finitely generated [[ideal|0MYSA6MYNA]] of $R$ is [[principal|00A2EDFT9Q]].

## Remarks

Equivalent form: for all $a, b \in R$ the ideal $(a, b)$ is principal; the general case follows by induction on the number of generators.
Then $(a, b) = (d)$ for a gcd $d$ of $a$ and $b$, and $d = ax + by$ for some $x, y \in R$.
Example: every [[principal ideal domain|0RZYDVZ475]], such as $\Z$ or $\Q[x]$, is a Bézout domain.
Non-example: in $\Z[x]$ the ideal $(2, x)$ is not principal.
A generator would divide $2$, so it would be $\pm 1$ or $\pm 2$.
But $(2, x) \ne \Z[x]$, because its elements have even constant term, and $2$ does not divide $x$.
A Bézout domain is a [[GCD domain|ACDEK1GD9G]]; a Noetherian Bézout domain is a principal ideal domain.
</document>

<document source="https://tamnd.github.io/math2/node/WVV4MFB2SE.md" tag="WVV4MFB2SE" kind="statement">
# Riesz–Fischer theorem

## Statement

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $1 \le p \le \infty$.
Then the space [[$L^p(\mu)$|SASP14MA1Z]] with the norm $\norm{\cdot}_p$ is complete.
So $L^p(\mu)$ is a [[Banach space|06GZ956657]].

## Remarks

The proof for $p < \infty$ starts from a Cauchy sequence $(f_n)$ and picks a subsequence with $\norm{f_{n_{k+1}} - f_{n_k}}_p < 2^{-k}$.
The function $g = \abs{f_{n_1}} + \sum_k \abs{f_{n_{k+1}} - f_{n_k}}$ lies in $L^p$ by Minkowski's inequality and the [[monotone convergence theorem|G2S6H789RG]].
So the subsequence converges [[almost everywhere|RARTMA0ZEC]] to a function $f$, and [[Fatou's lemma|S95TR54TCG]] shows that $f \in L^p$ and $f_n \to f$ in $L^p$.
A by-product: for $p < \infty$, every sequence that converges in $L^p$ has a subsequence that converges almost everywhere.
The case $p = \infty$ is a direct argument with uniform convergence outside a null set.

The original form concerns $p = 2$.
If $(\varphi_n)$ is an orthonormal sequence in $L^2(\mu)$ and $(c_n)$ is a sequence of scalars with $\sum_n \abs{c_n}^2 < \infty$, then there is $f \in L^2(\mu)$ with $\ip{f}{\varphi_n} = c_n$ for all $n$.
For the trigonometric system on $[0, 2\pi]$, every square-summable sequence is the sequence of Fourier coefficients of some square-integrable function.

The Lebesgue integral is essential here.
The continuous functions on $[0, 1]$ with the norm $\int_0^1 \abs{f}\, dx$ do not form a complete space.

## Explanations

### L1 human

The theorem says that the spaces $L^p$ have no holes.
If the functions of a sequence get closer and closer to each other in the $L^p$ distance, then they get close to an actual function of the space.
This is the same property that the real numbers have and the rational numbers lack.
It is the main reason why analysts use the Lebesgue integral and not the Riemann integral.

### L2 human

Completeness makes the tools of Banach space theory available on $L^p$: the Baire category theorem, the open mapping theorem and the uniform boundedness principle.
For $p = 2$ it makes $L^2(\mu)$ a Hilbert space, which is the setting of Fourier series and of quantum mechanics.
The proof also gives the useful fact that $L^p$ convergence implies almost everywhere convergence along a subsequence, but not for the whole sequence.
The "typewriter" sequence of indicator functions of shrinking intervals that sweep across $[0, 1]$ converges to $0$ in $L^1$ but at no point.
</document>

<document source="https://tamnd.github.io/math2/node/WVYYQCFS4Q.md" tag="WVYYQCFS4Q" kind="definition">
# Schottky group

## Definition

Let $g \ge 1$ be an integer.
Let $D_1, D_1', \ldots, D_g, D_g'$ be $2g$ open Jordan discs in the [[Riemann sphere|0J78KBPFZB]] $\hat{\C}$ whose closures are pairwise disjoint.
Let $C_i$ and $C_i'$ be the boundary curves of $D_i$ and $D_i'$.
For each $i$, let $A_i$ be a [[Möbius transformation|0SG74WBFA2]] that maps $C_i$ onto $C_i'$ and maps $\hat{\C} \setminus \overline{D_i}$ onto $D_i'$.
The group generated by $A_1, \ldots, A_g$ is a *Schottky group* of genus $g$.
It is a *classical* Schottky group if the curves $C_i$ and $C_i'$ can be chosen to be circles.

## Remarks

Example: let $\lambda \in \C$ with $\abs{\lambda} > 1$, $A_1(z) = \lambda z$, $D_1 = \set{\abs{z} < 1}$ and $D_1' = \set{\abs{z} > \abs{\lambda}} \cup \set{\infty}$.
Their closures are disjoint, and $A_1$ maps the circle $\abs{z} = 1$ onto $\abs{z} = \abs{\lambda}$ and the region $\abs{z} > 1$ onto $D_1'$.
So $\langle A_1 \rangle$ is a Schottky group of genus $1$; the quotient of $\C \setminus \set{0}$ by it is a torus.

Non-example: the group generated by $z \mapsto z + 1$ is not a Schottky group.
Its generator is parabolic, while every element other than the identity of a Schottky group is loxodromic.

A Schottky group of genus $g$ is a [[free group|0KTW26C80J]] of rank $g$ on $A_1, \ldots, A_g$ and is a [[Kleinian group|27QRARKWD5]].
The common exterior of the $2g$ discs is a fundamental domain for its action on the domain of discontinuity $\Omega$.
The quotient $\Omega / G$ is a compact Riemann surface of genus $g$.
</document>

<document source="https://tamnd.github.io/math2/node/WVZ5EZN9V2.md" tag="WVZ5EZN9V2" kind="statement">
# GAGA

## Statement

Let $X$ be a projective scheme over $\C$, for example a [[projective variety|A4H4CZ2V35]].
Let $X^{an}$ be the associated complex analytic space, with its sheaf $\mathcal{O}_{X^{an}}$ of holomorphic functions.
For a [[coherent sheaf|56RXEJ8Z6S]] $\mathcal{F}$ on $X$, let $\mathcal{F}^{an} = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_{X^{an}}$, a coherent analytic sheaf on $X^{an}$.

1. For every coherent sheaf $\mathcal{F}$ on $X$ and every $i \ge 0$, the natural map $H^i(X, \mathcal{F}) \to H^i(X^{an}, \mathcal{F}^{an})$ of [[sheaf cohomology|PQ6Q143ARJ]] groups is an isomorphism.
2. The functor $\mathcal{F} \mapsto \mathcal{F}^{an}$ is an equivalence between the category of coherent sheaves on $X$ and the category of coherent analytic sheaves on $X^{an}$.

## Remarks

The idea of the proof: both statements are first checked on $\mathbb{P}^n$, by comparing the cohomology of the sheaves $\mathcal{O}(d)$, which is the same in both settings.
For a general coherent sheaf one uses that it has a resolution by sums of sheaves $\mathcal{O}(d)$, together with Serre's theorems on twisting and induction on dimension.

Example: by part 2 applied to ideal sheaves, every closed analytic subset of $\mathbb{P}^n_\C$ is algebraic; this is Chow's theorem.
In particular, every compact [[complex manifold|JSGWAN2RJV]] that embeds holomorphically in $\mathbb{P}^n_\C$ is a projective algebraic variety, and every holomorphic map between projective varieties is algebraic.

The properness of $X$ is essential.
On $X = \mathbb{A}^1_\C$, the holomorphic function $e^z$ is a global section of $\mathcal{O}_{X^{an}}$ that is not a polynomial.
So part 1 fails already for $i = 0$ and $\mathcal{F} = \mathcal{O}_X$.
The theorem extends to all proper schemes over $\C$, by a result of Grothendieck.

## Explanations

### L1 human

GAGA, named after Serre's paper "Géométrie algébrique et géométrie analytique", says that for projective varieties over the complex numbers, algebraic and holomorphic objects are the same.
Holomorphic vector bundles, holomorphic sections and their cohomology all come from algebra.
So one can use analysis to prove theorems about algebraic varieties and conversely.

### L2 human

GAGA is what allows results proved with analytic tools, such as Hodge theory and the Kodaira vanishing theorem, to be stated and used for algebraic varieties.
It also shows that the Picard group of a projective variety equals the group of holomorphic line bundles on the associated complex manifold.
For non-proper varieties the two theories differ, as the example of $e^z$ on the affine line shows.
</document>

<document source="https://tamnd.github.io/math2/node/WVZ7F41S9Q.md" tag="WVZ7F41S9Q" kind="definition">
# Medial magma

## Definition

Let $(M, \cdot)$ be a [[magma|9689SFPVE9]].
It is *medial* if
$$(x \cdot y) \cdot (u \cdot v) = (x \cdot u) \cdot (y \cdot v)$$
for all $x, y, u, v \in M$.

## Remarks

Example: $\Q$ with $x \circ y = (x + y)/2$.
Both $(x \circ y) \circ (u \circ v)$ and $(x \circ u) \circ (y \circ v)$ equal $(x + y + u + v)/4$.
This magma is medial but not associative.

Example: every commutative semigroup is medial, since $(xy)(uv) = x(yu)v = x(uy)v = (xu)(yv)$.

Non-example: $\Z$ with $x \cdot y = xy + 1$.
Take $x = y = 1$ and $u = v = 0$.
Then $1 \cdot 1 = 2$ and $0 \cdot 0 = 1$, so the left side is $2 \cdot 1 = 3$.
Also $1 \cdot 0 = 1$, so the right side is $1 \cdot 1 = 2 \ne 3$.

A magma is medial if and only if the map $M \times M \to M$, $(x, y) \mapsto x \cdot y$, is a homomorphism from the product magma $M \times M$ to $M$.
</document>

<document source="https://tamnd.github.io/math2/node/WW53KG71KH.md" tag="WW53KG71KH" kind="statement">
# Fubini's theorem for series

## Statement

Let $(a_{m,n})_{m, n \in \N}$ be a [[double sequence|WK5V372TG3]].

1. If $a_{m,n} \ge 0$ for all $m, n$, then $\sum_{(m,n) \in \N \times \N} a_{m,n} = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} a_{m,n}$ as elements of $[0, \infty]$, where the left side is the supremum of the finite sums.
2. If $a_{m,n} \in \C$ and the family $(a_{m,n})_{(m,n) \in \N \times \N}$ is [[summable|Z4H7272S9S]], then for each $m$ the series $\sum_n a_{m,n}$ converges [[absolutely|0W3K3QPEWT]], the series $\sum_m \sum_n a_{m,n}$ converges absolutely, and
$$\sum_{(m,n) \in \N \times \N} a_{m,n} = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} a_{m,n}.$$

## Remarks

For part 1, every finite subset of $\N \times \N$ lies in a rectangle, and every iterated partial sum $\sum_{m \le M} \sum_{n \le N}$ is a finite sum; comparing suprema gives the equality.
For part 2, split the real and imaginary parts into positive and negative parts, each bounded by $\abs{a_{m,n}}$, and apply part 1 to each.
Summability of a complex family over $\N \times \N$ is equivalent to $\sum_{m} \sum_{n} \abs{a_{m,n}} < \infty$, by part 1.
The statement is the case of the counting measure in [[Tonelli's theorem|067SYD2J4B]] and [[Fubini's theorem|0R0T1VWH5X]].
Example: for $\abs{x} < 1$ and $\abs{y} < 1$, the family $x^m y^n$ is summable, and its sum is $\frac{1}{1 - x} \cdot \frac{1}{1 - y}$.
Without summability the conclusion fails: [[interchange of summation order|MZM1AFXY82]] gives a double sequence whose two iterated sums differ.

## Explanations

### L1 human

To add all the entries of an infinite table, you may add each row first and then add the row totals.
This is safe when all entries are nonnegative, or when the sum of the absolute values is finite.
Without such a condition, adding row by row can give a different answer than adding column by column.

### L2 human

The theorem is the discrete form of Fubini's theorem, and it is used constantly with power series and Dirichlet series.
Typical uses are the [[Cauchy product of absolutely convergent series|0AN28XQ45Z]], the rearrangement of a double power series, and the exchange of two sums in number theory.
The nonnegative case needs no hypothesis at all, which makes it the usual first step: one checks $\sum_m \sum_n \abs{a_{m,n}} < \infty$ and then applies part 2.
</document>

<document source="https://tamnd.github.io/math2/node/WW7H1KNXK8.md" tag="WW7H1KNXK8" kind="concept">
# Jack function

## Summary

Jack functions $J_\lambda^{(\alpha)}$ are [[symmetric functions|PAJEHWXW6M]] indexed by an [[integer partition|0JV30XN73Z]] $\lambda$ and depending on a parameter $\alpha$.
They are fixed by being triangular with respect to the monomial basis and orthogonal for a deformation of the Hall inner product.
For $\alpha = 1$ they are multiples of the [[Schur polynomials|43VFMJCDYJ]], and for $\alpha = 2$ they are the zonal polynomials of multivariate statistics.
They are the limit $q = t^\alpha$, $t \to 1$ of the [[Macdonald polynomials|VYP9PB000X]], and they are eigenfunctions of the Calogero–Sutherland model of quantum particles on a circle.

## Explanations

### L1 human

Schur polynomials are a very good basis of the symmetric polynomials, because they are orthogonal and have a meaning in representation theory.
Jack functions keep the orthogonality but add a dial $\alpha$ that changes the inner product.
As $\alpha$ moves, the basis moves from the elementary symmetric functions ($\alpha = 0$) through the Schur functions ($\alpha = 1$) to the monomial symmetric functions ($\alpha \to \infty$).
For example, in degree $2$ the monic Jack function is $m_{(2)} + \frac{2}{1 + \alpha} m_{(1,1)}$, which equals $s_{(2)} = m_{(2)} + m_{(1,1)}$ at $\alpha = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/WW7HPC3WKA.md" tag="WW7HPC3WKA" kind="definition">
# Kullback–Leibler divergence

## Definition

Let $(\Omega, \mathcal{F})$ be a measurable space, and let $P$ and $Q$ be [[probability measures|059TJN0NZ1]] on it.
If $P$ is [[absolutely continuous|P0P21KDV6E]] with respect to $Q$, let $dP/dQ$ be the [[Radon–Nikodym derivative|1NTKG64F7N]], and define
$$D(P \,\|\, Q) = \int_{\Omega} \log \frac{dP}{dQ} \, dP \in [0, \infty].$$
If $P$ is not absolutely continuous with respect to $Q$, define $D(P \,\|\, Q) = \infty$.
The number $D(P \,\|\, Q)$ is the *Kullback–Leibler divergence* of $P$ from $Q$.

## Remarks

The integral is well defined: with $f = dP/dQ$, it equals $\int f \log f \, dQ$, and $f \log f \ge -1/e$, so its negative part is integrable.
By Jensen's inequality it is non-negative, with equality exactly when $P = Q$; this is Gibbs' inequality.

For distributions on a countable set with [[probability mass functions|0D6J4B671D]] $p$ and $q$,
$$D(P \,\|\, Q) = \sum_{x} p(x) \log \frac{p(x)}{q(x)},$$
with the conventions $0 \log \frac{0}{q} = 0$ and $p \log \frac{p}{0} = \infty$ for $p > 0$.
With densities $p$ and $q$ with respect to a common measure, the sum becomes an integral.
The base of the [[logarithm|04DXAFN2ES]] fixes the unit: base $e$ gives nats, base $2$ gives bits.

Example: let $P$ be the Bernoulli law with parameter $1/2$ and $Q$ the Bernoulli law with parameter $1/4$.
Then $D(P \,\|\, Q) = \frac{1}{2} \log \frac{1/2}{1/4} + \frac{1}{2} \log \frac{1/2}{3/4} = \frac{1}{2} \log \frac{4}{3}$.
In the other order, $D(Q \,\|\, P) = \frac{1}{4} \log \frac{1}{2} + \frac{3}{4} \log \frac{3}{2} = \frac{1}{4} \log \frac{27}{16}$.
These differ, so $D$ is not symmetric and is not a metric.

Example of the infinite case: let $Q$ be the Dirac measure at $0$ and $P$ the Bernoulli law with parameter $1/2$.
Then $P(\set{1}) > 0 = Q(\set{1})$, so $P$ is not absolutely continuous with respect to $Q$, and $D(P \,\|\, Q) = \infty$.

Pinsker's inequality bounds the total variation distance by $\sqrt{D(P \,\|\, Q)/2}$.
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<document source="https://tamnd.github.io/math2/node/WW81ARB8K9.md" tag="WW81ARB8K9" kind="concept">
# Simplicial set

## Summary

A simplicial set is a [[presheaf|AND2WHZC8P]] on the [[simplex category|FRFYHJGAN3]], that is, a [[simplicial object|WK0A32TZHB]] in the category of sets.
It records a set $X_n$ of $n$-simplices for each $n$, together with face and degeneracy maps.
Simplicial sets are a combinatorial model for topological spaces up to homotopy, and they extend [[simplicial complexes|V8GZ0W1ZZ5]] by allowing degenerate and repeated simplices.
Through the [[nerve|RV0MKJ3CQ6]] they also contain all small categories, which makes them the basis of [[quasi-categories|9328WEVJEK]].

## Explanations

### L1 human

Imagine building a shape from points, edges, triangles and higher-dimensional triangles, glued along their faces.
A simplicial set lists the pieces of each dimension and says which pieces are faces of which.
It also keeps "flattened" copies of lower pieces, such as an edge whose two ends are the same point, which makes the bookkeeping uniform.
The [[singular simplicial set|RWBMXQW6S5]] of a space $Y$ is an example: its $n$-simplices are the continuous maps from the standard $n$-simplex into $Y$.
The simplest example is the standard $n$-simplex $\Delta^n$, whose $m$-simplices are the order-preserving maps $[m] \to [n]$.
</document>

<document source="https://tamnd.github.io/math2/node/WWCJ7CR2R3.md" tag="WWCJ7CR2R3" kind="concept">
# Hilbert class field

## Summary

The Hilbert class field of a number field $K$ is the largest abelian extension of $K$ that is unramified at every place.
By [[class field theory|BEKCEYHCHX]], its Galois group over $K$ is isomorphic to the [[ideal class group|WQ46BD3ARY]] of $K$, so its degree is the [[class number|KTKK9B1X41]].
A prime ideal of $K$ splits completely in it exactly when it is principal.
So the class group, an arithmetic object, is realised as a Galois group.

## Explanations

### L1 human

The class group measures how far the ring of integers of a field is from having unique factorisation.
The Hilbert class field is a bigger field that makes this measure visible as a group of symmetries.
For $K = \Q(\sqrt{-5})$, with class number $2$, the Hilbert class field is $K(i) = \Q(\sqrt{-5}, i)$, an extension of degree $2$.
Every ideal of $K$ becomes principal in the Hilbert class field: for example $(2, 1 + \sqrt{-5})$ is generated there by $1 + i$.
</document>

<document source="https://tamnd.github.io/math2/node/WWE9H2948V.md" tag="WWE9H2948V" kind="definition">
# Method of steepest descent

## Definition

Let $f$ and $g$ be [[holomorphic functions|0NAHXR1FRH]] on a domain $\Omega \subseteq \C$, let $C$ be a [[contour|06ZGPYTWK8]] in $\Omega$, and let
$$I(\lambda) = \int_C g(z)\, e^{\lambda f(z)}\,dz, \qquad \lambda > 0.$$
A simple [[saddle point|0JW6MDHAZD]] of $f$ is a point $z_0$ with $f'(z_0) = 0$ and $f''(z_0) \ne 0$.
A *steepest descent path* through $z_0$ is a path through $z_0$ on which $\operatorname{Im} f$ is constant.
Along it, $\operatorname{Re} f$ must decrease strictly as one moves away from $z_0$.
The *method of steepest descent* finds the behaviour of $I(\lambda)$ as $\lambda \to \infty$ in three steps:

1. by Cauchy's theorem, replace $C$ by a contour with the same ends, or ending in the same valleys of $\operatorname{Re} f$ at infinity.
   The new contour consists of steepest descent paths through saddle points of $f$ and of pieces where $\operatorname{Re} f$ is smaller;
2. on each steepest descent path, where $e^{\lambda f}$ has constant phase, apply [[Laplace's method|9G7K8DBT9Y]];
3. keep the contributions of the saddle points with the largest value of $\operatorname{Re} f$.

## Remarks

Let $z_0$ be a simple saddle with $g(z_0) \ne 0$, and let $\theta = \frac12 \big(\pi - \arg f''(z_0)\big)$.
The angle $\theta$, defined modulo $\pi$, is the direction in which the path crosses $z_0$, taken with the orientation of the contour.
The contribution of $z_0$ is
$$g(z_0)\, e^{\lambda f(z_0)}\, e^{i\theta} \sqrt{\frac{2\pi}{\lambda \abs{f''(z_0)}}}\,\big(1 + o(1)\big).$$
Indeed $f(z_0 + r e^{i\theta}) \approx f(z_0) + \frac12 f''(z_0) r^2 e^{2i\theta}$, and $f''(z_0) e^{2i\theta}$ is real and negative.

Example: $I(\lambda) = \int_{\R} e^{i\lambda x^2}\,dx$, with $f(z) = i z^2$ and $g = 1$.
The only saddle is $z_0 = 0$ with $f''(0) = 2i$, so $\theta = \pi/4$.
On the line $z = r e^{i\pi/4}$ one has $f(z) = -r^2$, which is real and decreasing in $\abs{r}$, so this line is the steepest descent path.
Rotating the contour to it gives exactly $I(\lambda) = e^{i\pi/4} \sqrt{\pi/\lambda}$.

Non-example: for $f(z) = z^2$ the real axis passes through the saddle $0$, but $\operatorname{Re} f = x^2$ increases away from $0$ along it.
So it is a path of steepest ascent; the steepest descent path is the imaginary axis.
</document>

<document source="https://tamnd.github.io/math2/node/WWMWX7AD4V.md" tag="WWMWX7AD4V" kind="concept">
# Square-integrable kernel

## Summary

A square-integrable kernel is a [[kernel|GSMBXT7FPK]] $K(x, s)$ whose squared absolute value has a finite double integral.
Such kernels are also called Hilbert–Schmidt kernels or $L^2$ kernels.
The integral operator of a square-integrable kernel is a bounded and [[compact operator|M2NEP1K5AE]] on the [[Lp space|SASP14MA1Z]] $L^2$, namely a [[Hilbert–Schmidt integral operator|CB5G212Q03]].
This class contains all continuous kernels on a square and also many unbounded ones.

## Explanations

### L1 human

The kernel may be infinite at some points, but not too badly: its square must still have a finite total integral.
This is the natural condition when one measures functions by their energy $\int \abs{\varphi}^2$.
For example, $K(x, s) = \abs{x - s}^{-1/4}$ on $[0, 1]^2$ is unbounded but square-integrable.
The kernel $\abs{x - s}^{-1/2}$ is not, because the double integral of $1/\abs{x - s}$ diverges.
</document>

<document source="https://tamnd.github.io/math2/node/WWQJQXJECK.md" tag="WWQJQXJECK" kind="problem">
# Shape optimization

## Problem

Let $D \subseteq \R^d$ be a bounded open set, called the design region.
Let $\mathcal{O}$ be a class of open subsets $\Omega \subseteq D$, the admissible shapes, for example those with a given volume.
For each $\Omega \in \mathcal{O}$ let $y_\Omega$ be the solution of a boundary value problem for a [[partial differential equation|GC5BDZ6QBC]] on $\Omega$.
Let $J(\Omega) = j(\Omega, y_\Omega)$ be a real-valued shape [[functional|1EQM0DJ4NC]].
The *shape optimization problem* is to find $\Omega^* \in \mathcal{O}$ with $J(\Omega^*) \le J(\Omega)$ for every $\Omega \in \mathcal{O}$.

## Remarks

It is a form of [[optimal control of partial differential equations|S2BBZPAFDR]] in which the control is the domain itself.
Example: minimize the first Dirichlet eigenvalue $\lambda_1(\Omega)$ of the Laplacian among open sets $\Omega \subseteq \R^d$ of a given volume.
The Rayleigh–Faber–Krahn inequality solves it: the minimizers are the balls.
In general minimizers need not exist in a given class of shapes, since minimizing sequences can develop holes or oscillating boundaries.
One then restricts the class, for example by a uniform cone condition or a perimeter bound, or relaxes the problem; relaxation leads to topology optimization.
Gradient methods use the [[shape derivative|9XFEG6ANGM]] of $J$, often computed with an adjoint equation.

## Explanations

### L1 human

Shape optimization asks for the best shape of an object: a wing with least drag, a beam with greatest stiffness, a drum with lowest tone.
The quality of a shape is measured through the solution of a physical equation on that shape.
For example, among all drums of the same area, the round drum has the lowest fundamental tone.
</document>

<document source="https://tamnd.github.io/math2/node/WWRWYSJDET.md" tag="WWRWYSJDET" kind="concept">
# Closed immersion

## Summary

A closed immersion is a [[morphism of schemes|JXRKH6P1Z4]] that identifies its source with a closed subscheme of its target.
For affine schemes, closed immersions into $\operatorname{Spec} A$ are exactly the maps $\operatorname{Spec} A/I \to \operatorname{Spec} A$ for ideals $I$.
They make precise the idea of a subvariety cut out by equations, and they include non-reduced subschemes such as a double point.

## Explanations

### L1 human

A closed immersion is the scheme version of "the set of solutions of some equations inside a bigger space".
The equations $I$ in $A$ cut out $\operatorname{Spec} A/I$, and every function on it comes from a function on the big space.
The equation $x^2 = 0$ on the line cuts out the origin with a double structure.
This is a different closed subscheme from the origin cut out by $x = 0$, though both have the same single point.
</document>

<document source="https://tamnd.github.io/math2/node/WWSYQJF7TZ.md" tag="WWSYQJF7TZ" kind="conjecture">
# Eremenko's conjecture

## Statement

Let $f \colon \C \to \C$ be a transcendental [[entire function|0G4F0XB9X2]].
Let $I(f) = \set{z \in \C : f^n(z) \to \infty \text{ as } n \to \infty}$ be its escaping set.
Then every [[connected component|0JG9W3KA6B]] of $I(f)$ is unbounded.

## Remarks

Eremenko proved that $I(f)$ is nonempty and that the [[Julia set|5M4S7P04GR]] of $f$ is the boundary of $I(f)$.
He also proved that every component of the closure of $I(f)$ is unbounded.
Rippon and Stallard proved in 2005 that $I(f)$ always has at least one unbounded component.
A stronger form, that every escaping point can be joined to $\infty$ by a curve in $I(f)$, is false.
Rottenfußer, Rückert, Rempe and Schleicher gave entire functions whose escaping sets contain no curves at all.
In 2021 Martí-Pete, Rempe and Waterman constructed a counterexample to the conjecture itself.
For a polynomial of degree at least $2$ the analogous statement holds.
There the escaping set is the complement of the filled Julia set $K$, which is compact and has connected complement.
So the escaping set is connected and unbounded.

## Explanations

### L1 human

For a transcendental entire function such as $e^z$, some points run off to infinity under iteration; together they form the escaping set.
Eremenko guessed that this set has no bounded pieces: every piece should reach out to infinity.
For many families, such as the exponential maps, the escaping set is indeed made of curves that go to infinity.
In general the guess turned out to be false, but the question shaped much of the study of transcendental dynamics.
</document>

<document source="https://tamnd.github.io/math2/node/WWVK5WF6WY.md" tag="WWVK5WF6WY" kind="definition">
# Parabolic and hyperbolic Riemann surfaces

## Definition

Let $X$ be a connected [[Riemann surface|5K0N8QMV6Q]].
A function on $X$ is subharmonic if it is a [[subharmonic function|Z1MHMGFNFW]] in every holomorphic chart, as for [[harmonic functions on Riemann surfaces|P214QJ0S3B]].

1. $X$ is *hyperbolic* if there is a nonconstant subharmonic function $u$ on $X$ with $u < 0$ on $X$.
2. $X$ is *parabolic* if $X$ is not compact and not hyperbolic.

## Remarks

Equivalently, $X$ is hyperbolic if and only if it has a Green's function.
This is the smallest positive harmonic function on $X \setminus \set{a}$ with a logarithmic pole at $a$.
A compact Riemann surface is never hyperbolic, because a subharmonic function on it attains its maximum and is therefore constant; such surfaces are sometimes called elliptic.
Example: the unit disc $\mathbb{D}$ is hyperbolic, since $u(z) = \abs{z}^2 - 1$ is subharmonic, nonconstant and negative on $\mathbb{D}$.
Non-example: the plane $\C$ is not hyperbolic.
By the subharmonic form of [[Liouville's theorem|T12FSF2J2V]], a subharmonic function on $\C$ that is bounded above is constant.
As $\C$ is not compact, it is parabolic.
A domain $\Omega \subseteq \C$ is hyperbolic if and only if its boundary is not a [[polar set|C95QJ7YH5T]].
Its Green's function is then the classical [[Green's function|M36SD56MWQ]] of $\Omega$.
This notion differs from the one in the uniformization theorem, which classifies $X$ by its [[universal cover|Y1YPE6PM52]].
For example, $\C \setminus \set{0, 1}$ has the disc as universal cover, but it is parabolic, because a subharmonic function bounded above extends across the two punctures to all of $\C$.
</document>

<document source="https://tamnd.github.io/math2/node/WWW053WXP2.md" tag="WWW053WXP2" kind="concept">
# Riemann P-symbol

## Summary

The Riemann P-symbol is a notation for the solutions of [[Riemann's differential equation|RS8Z48E784]].
It is a table listing the three singular points, the two exponents at each, and the variable.
Since the equation is determined by these data, the symbol names its whole two-dimensional space of solutions.
Simple rules tell how the symbol changes under a [[Möbius transformation|0SG74WBFA2]] of the variable and under multiplication by powers.
These rules turn many transformation formulas of the [[hypergeometric function|463X5V31D9]] into bookkeeping.

## Explanations

### L1 human

Instead of writing a long differential equation, one writes a small table with three columns.
Each column gives a singular point and the two powers that the solutions show near it.
The table stands for all solutions of the equation with that data.
For example, the hypergeometric function belongs to the table with columns $(0; 0, 1-c)$, $(\infty; a, b)$ and $(1; 0, c-a-b)$.
Moving the points by a Möbius map or multiplying by powers just changes entries of the table in a predictable way.
</document>

<document source="https://tamnd.github.io/math2/node/WWX7H5W1RP.md" tag="WWX7H5W1RP" kind="concept">
# Section of a vector bundle

## Summary

A section of a vector bundle chooses one vector in the fiber over each point of the base.
The choice varies smoothly with the point.
Vector fields are sections of the tangent bundle, and differential forms are sections of bundles of alternating forms.
The sections of a vector bundle form a vector space, and in fact a module over the ring of smooth functions on the base.
Many objects of geometry and physics, such as fields in gauge theory, are sections of suitable bundles.

## Explanations

### L1 human

A vector bundle attaches a vector space to each point of a space.
A section picks one vector from each of these spaces, without jumps.
Wind directions on the surface of the Earth form a section of the tangent bundle of the sphere: at each place you get an arrow tangent to the surface.
On a trivial bundle $M \times \R^k$, a section is just a smooth function from $M$ to $\R^k$.
</document>

<document source="https://tamnd.github.io/math2/node/WWX8HCN260.md" tag="WWX8HCN260" kind="statement">
# Wang sequence

## Statement

Let $n \ge 2$ and let $p \colon E \to S^n$ be a [[Serre fibration|PB9JXQ2KC3]] over the [[sphere|T62369RFH6]] $S^n$, with fibre $F = p^{-1}(x_0)$ over a point $x_0 \in S^n$.
Then there is a long [[exact sequence|JB7ZBFBA9C]] in integral [[singular homology|M0Q5CFJ7ZH]]
$$\cdots \to H_k(F) \xrightarrow{\ i_*\ } H_k(E) \to H_{k-n}(F) \to H_{k-1}(F) \xrightarrow{\ i_*\ } H_{k-1}(E) \to \cdots,$$
where $i \colon F \to E$ is the inclusion of the fibre.

## Remarks

The base $S^n$ is simply connected because $n \ge 2$, so the [[Serre spectral sequence|AHW4B5480K]] has $E^2_{s,t} = H_s(S^n; H_t(F))$.
This is nonzero only for $s = 0$ and $s = n$, so the only possible differential is $d^n \colon E^n_{n,t} \to E^n_{0,t+n-1}$.
Unfolding this two-column spectral sequence gives the long exact sequence, and the map $H_{k-n}(F) \to H_{k-1}(F)$ is this differential.
A dual sequence holds in cohomology.

Example: the [[path space fibration|D3BN7MBKTR]] $\Omega S^n \to P S^n \to S^n$ has contractible total space, where $\Omega S^n$ is the [[loop space|2JFF5F3EGF]].
Exactness then gives isomorphisms $H_{k-n}(\Omega S^n) \cong H_{k-1}(\Omega S^n)$ for $k \ge 2$.
Since $H_0(\Omega S^n) = \Z$, one gets $H_j(\Omega S^n) \cong \Z$ when $j$ is a multiple of $n - 1$ and $H_j(\Omega S^n) = 0$ otherwise.

## Explanations

### L1 human

A fibration over a sphere is a space that looks locally like a product of a piece of sphere with a fixed fibre, but may be twisted globally.
The Wang sequence links the homology of the total space with the homology of the fibre.
All the twisting is concentrated in a single map from the homology of the fibre to itself, which lowers the degree by $n - 1$.

### L2 human

The Wang sequence is the fibration analogue of the Mayer–Vietoris sequence, obtained by cutting the sphere into two discs.
It is the base-sphere counterpart of the Gysin sequence, which treats fibrations whose fibre is a sphere.
Its classical use is the computation of the homology of loop spaces of spheres, as in the example above.
</document>

<document source="https://tamnd.github.io/math2/node/WWZTW11664.md" tag="WWZTW11664" kind="concept">
# Invariant set

## Summary

An invariant set of a differential equation is a set of states that no solution can leave or enter.
A solution that starts in the set stays in it for all times.
Equilibria, [[periodic orbits|0AA97XA70K]], limit sets and stable manifolds are invariant sets.
A positively invariant set only needs to be kept for positive times.
Such sets are used to trap solutions and to prove stability and existence of periodic orbits.

## Explanations

### L1 human

Picture the state of a system moving along its [[trajectory|07TZ2Q4C1M]] in the phase plane.
A set is invariant if every trajectory that touches it lies entirely inside it.
For the rotation $x' = -y$, $y' = x$, every disc $x^2 + y^2 \le r^2$ is invariant, since each trajectory is a circle around the origin.
The half-plane $x \ge 0$ is not invariant, because the circles leave it.
A union of whole trajectories is always invariant.
</document>

<document source="https://tamnd.github.io/math2/node/WX0HMW4GT7.md" tag="WX0HMW4GT7" kind="statement">
# Koebe quarter theorem

## Statement

Let $\mathbb{D}$ be the [[unit disc|8G5ABXRK85]], and let $f \colon \mathbb{D} \to \C$ be a [[univalent function|R91E83SDBD]].
Then the image $f(\mathbb{D})$ contains the open disc with centre $f(0)$ and radius $\abs{f'(0)}/4$.

## Remarks

After replacing $f$ by $(f - f(0))/f'(0)$, one may assume $f(z) = z + a_2 z^2 + \cdots$ is in the class $S$.
If $w \notin f(\mathbb{D})$, then $g = wf/(w - f)$ is again in $S$, and its second coefficient is $a_2 + 1/w$.
Bieberbach's inequality $\abs{a_2} \le 2$ for functions in $S$, which follows from the [[Grönwall area theorem|BYZDAZNHP4]], gives $\abs{a_2 + 1/w} \le 2$ and $\abs{a_2} \le 2$, hence $\abs{1/w} \le 4$, that is, $\abs{w} \ge 1/4$.
The constant $1/4$ is sharp: the [[Koebe function|EZQFWKF03Q]] $k(z) = z/(1-z)^2$ is in $S$ and omits the value $-1/4$.
Injectivity is necessary: $f(z) = (e^{Nz} - 1)/N$ has $f(0) = 0$ and $f'(0) = 1$, but omits the value $-1/N$, which is close to $0$ for large $N$.

## Explanations

### L1 human

A univalent function on the unit disc cannot squeeze its image too much near the image of the centre.
After normalising so that $f(0) = 0$ and $f'(0) = 1$, the image always contains the disc of radius $1/4$ around $0$.
The Koebe function shows that a quarter is the best one can say.

### L2 human

The theorem is the basic covering theorem of univalent function theory.
It gives the distance estimate $\frac{1}{4}\abs{f'(z)}(1 - \abs{z}^2) \le \operatorname{dist}(f(z), \partial f(\mathbb{D})) \le \abs{f'(z)}(1 - \abs{z}^2)$ for univalent $f$, where the upper bound comes from the Schwarz lemma.
This compares the hyperbolic metric of a simply connected domain with the reciprocal of the distance to its boundary, up to a factor $4$.
Together with the Koebe distortion theorem, it gives the compactness of the class $S$ in the topology of locally uniform convergence.
For maps that need not be injective, [[Bloch's theorem|KW31KDX7TB]] gives a much weaker covering result.
</document>

<document source="https://tamnd.github.io/math2/node/WX1Z2F3DCB.md" tag="WX1Z2F3DCB" kind="example">
# Borromean rings

## Example

Take three [[ellipses|0VKCCEXBJ1]] in the three coordinate planes of $\R^3$:
$$C_1 = \set{(x, y, 0) : \tfrac{x^2}{4} + y^2 = 1}, \quad C_2 = \set{(0, y, z) : \tfrac{y^2}{4} + z^2 = 1}, \quad C_3 = \set{(x, 0, z) : x^2 + \tfrac{z^2}{4} = 1}.$$
They are pairwise disjoint.
For instance, a common point of $C_1$ and $C_2$ has $x = 0$ and $z = 0$.
Then $y = \pm 1$ from $C_1$ and $y = \pm 2$ from $C_2$, which is impossible.
The other pairs are similar.
So $C_1 \cup C_2 \cup C_3$ is a [[link|1A5WRCNTM3]] with $3$ components, the *Borromean rings*.
Any two of the components form the unlink of two components.
For example, $C_2$ meets the plane $z = 0$ only at $(0, \pm 2, 0)$, outside the region bounded by $C_1$.
So $C_1$ bounds a flat disk disjoint from $C_2$, and it can be shrunk in that disk to a tiny circle far from $C_2$.
But the three components together cannot be pulled apart: the link is not equivalent to the unlink of three components.

## Remarks

All three pairwise [[linking numbers|SFJ329YB9F]] are $0$, so linking numbers do not detect the Borromean rings.
Their nontriviality is detected by Milnor's triple linking number, or by the fact that their link group is not free.
A nontrivial link in which removing any one component leaves an unlink is called Brunnian, and the Borromean rings are the simplest example.
They have a diagram with $6$ crossings that is alternating.
Their complement in $S^3$ carries a complete hyperbolic metric of finite volume, and it can be built from two regular ideal octahedra.
Michael Freedman and Richard Skora showed that the Borromean rings cannot be built from three round circles.

## Explanations

### L1 human

The Borromean rings are three rings that hold together as a group, although no two of them are linked.
If any one ring is cut and removed, the other two fall apart.
They appear in old coats of arms, among them that of the Borromeo family of Italy, which gives the name.
</document>

<document source="https://tamnd.github.io/math2/node/WX302T93TW.md" tag="WX302T93TW" kind="concept">
# Tensor field

## Summary

A tensor field on a [[smooth manifold|F45VQ47GPV]] assigns to each point a tensor built from the tangent space and its dual.
The assignment varies smoothly with the point.
Vector fields, differential forms, Riemannian metrics and curvature tensors are all tensor fields.
Tensor fields are the basic objects of differential geometry, because their meaning does not depend on the choice of coordinates.

## Explanations

### L1 human

At each point of a curved space there is a flat space of directions, the tangent space.
A tensor at a point is a multilinear gadget that eats some directions and some covectors and returns a number.
A tensor field places one such gadget at every point, smoothly.
For example, the dot product $dx \otimes dx + dy \otimes dy$ at every point of the plane is a tensor field that measures lengths of arrows.
A wind map that puts an arrow at each point is a simpler tensor field, of type $(1,0)$.
</document>

<document source="https://tamnd.github.io/math2/node/WXFTF05MHK.md" tag="WXFTF05MHK" kind="statement">
# Poisson integral for the half-space

## Statement

Let $n \ge 2$, write points of $\R^n$ as $x = (x', x_n)$ with $x' \in \R^{n-1}$, and let $H = \set{x \in \R^n : x_n > 0}$ be the upper half-space.
Let $\sigma_{n-1}$ be the [[surface area|0WQE3G7B88]] of the unit sphere in $\R^n$, and for $x \in H$ and $y \in \R^{n-1}$ put
$$P(x, y) = \frac{2 x_n}{\sigma_{n-1} \left( \abs{x' - y}^2 + x_n^2 \right)^{n/2}}.$$
Let $g \colon \R^{n-1} \to \R$ be bounded and [[continuous|0X3WGX95FE]], and define
$$u(x) = \int_{\R^{n-1}} P(x, y) \, g(y) \, dy, \qquad x \in H.$$
Then $u$ is bounded and [[harmonic|0DEB0JMN0W]] on $H$, and $u(x) \to g(y_0)$ as $x \to (y_0, 0)$ with $x \in H$, for every $y_0 \in \R^{n-1}$.

## Remarks

Proof idea: for fixed $y$, $P(\cdot, y)$ is harmonic in $H$.
It is, up to a constant factor, the normal derivative at the boundary of the Green's function of $H$ built by reflection.
The kernel is positive and $\int_{\R^{n-1}} P(x, y) \, dy = 1$, so $\abs{u} \le \sup \abs{g}$.
As $x_n \to 0$ the kernel concentrates at $x'$, which gives the boundary values.
For $n = 2$ the kernel is $\frac{1}{\pi} \frac{x_2}{(x_1 - y)^2 + x_2^2}$, the Poisson kernel of the upper half-plane.
Boundedness matters for uniqueness: $u(x) = x_n$ is harmonic on $H$ with boundary values $0$, but it is not the Poisson integral of $0$.

## Explanations

### L1 human

This formula fills in a harmonic function on a half-space from its values on the flat boundary.
The value at a point is a weighted average of the boundary values, with weights concentrated below the point.
The closer the point is to the boundary, the sharper the concentration.

### L2 human

The half-space formula is the model for boundary behaviour of harmonic functions, nontangential limits and Hardy spaces.
In $\R^2$ it is the link between harmonic functions on the upper half-plane and the Hilbert transform.
The family $P(\cdot, y)$ for $x_n = t$ fixed is the Poisson semigroup, an approximate identity used throughout harmonic analysis.
</document>

<document source="https://tamnd.github.io/math2/node/WXGM3VWEK6.md" tag="WXGM3VWEK6" kind="definition">
# Algebraic number field

## Definition

Let $K$ be a [[field|04C5ZBG1WT]] that contains the field $\Q$ of [[rational numbers|0NCJ5WXN6Z]].
Then $K$ is an *algebraic number field* (or *number field*) if it has finite dimension as a vector space over $\Q$.
This dimension is the [[degree|0EAJ2XXADC]] $[K : \Q]$ of the [[field extension|0EQWVHF3W5]] $K / \Q$, also called the degree of $K$.

## Remarks

Example: $K = \Q(i) = \set{a + bi : a, b \in \Q}$, a subfield of $\C$.
It contains $\Q$, and $1, i$ is a basis over $\Q$, so $[K : \Q] = 2$ and $K$ is a number field.
Non-example: $\R$ contains $\Q$, but it is not of finite dimension over $\Q$, since a finite-dimensional space over $\Q$ is countable and $\R$ is not.
Another non-example: the field of all algebraic numbers is an algebraic extension of $\Q$ of infinite degree.
Every element of a number field is an [[algebraic number|0MNF0BYFQC]], and by the primitive element theorem every number field is $\Q(\alpha)$ for one algebraic number $\alpha$.
The elements of $K$ that are roots of a [[monic polynomial|0C3QBCR46M]] with integer coefficients form a subring $\mathcal{O}_K$, the ring of integers of $K$.
</document>

<document source="https://tamnd.github.io/math2/node/WXJAVQKFR6.md" tag="WXJAVQKFR6" kind="problem">
# Euler's thirty-six officers problem

## Problem

Thirty-six officers come from six regiments, with one officer of each of six ranks in each regiment.
Arrange them in a $6 \times 6$ square with two conditions.
Each row and each column contains exactly one officer of each regiment and exactly one officer of each rank.

## Remarks

Writing the regiment and the rank of the officer in each cell gives two Latin squares of order $6$.
The arrangement asked for exists if and only if these two squares are [[orthogonal|KHD2QMNR7R]].
So the problem asks for a pair of orthogonal Latin squares of order $6$.
Euler posed the problem in 1782 and believed it had no solution.
Gaston Tarry proved in 1901, by a case analysis, that there is no solution.
Euler also conjectured that no orthogonal pair exists for any order $n \equiv 2 \pmod 4$.
Bose, Shrikhande and Parker disproved this around 1959 to 1960: an orthogonal pair of order $n$ exists for every $n$ except $2$ and $6$.

## Explanations

### L1 human

The puzzle is a two-colour Sudoku without boxes: every row and column must show each regiment once and each rank once.
For $25$ officers in a $5 \times 5$ square the analogous puzzle has a solution, but for $36$ officers it has none.
It is surprising that $6$ fails while $10$, $14$ and every larger order work.
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# Pseudocomplement

## Definition

Let $L$ be a [[lattice|WH4SFAACAS]] with a least element $0$, and let $a \in L$.
An element $a^* \in L$ is a *pseudocomplement* of $a$ if for every $x \in L$,
$$a \wedge x = 0 \iff x \le a^*.$$
The lattice $L$ is *pseudocomplemented* if every element has a pseudocomplement.

## Remarks

Equivalently, $a^*$ is the greatest element of the set $\set{x \in L : a \wedge x = 0}$, so it is unique when it exists.
In a pseudocomplemented lattice, $a \le a^{**}$ and $a^{***} = a^*$ for all $a$.

Example: the open sets of a topological space $X$, ordered by inclusion.
For an open set $U$, put $U^* = \operatorname{int}(X \setminus U)$.
An open set $V$ satisfies $U \cap V = \emptyset$ exactly when $V \subseteq X \setminus U$, that is, when $V \subseteq \operatorname{int}(X \setminus U)$, since $V$ is open.

Non-example: in the diamond lattice $M_3$ with atoms $p, q, r$, the element $p$ has no pseudocomplement.
The elements $x$ with $p \wedge x = 0$ are $0$, $q$ and $r$, and this set has no greatest element, since $q \vee r = 1$ and $p \wedge 1 = p \ne 0$.
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# Yoneda Ext and extensions

## Statement

Let $R$ be a [[ring|0B1261B5ED]] and let $M$ and $N$ be [[$R$-modules|0NZ8JA25WK]].
An *extension* of $M$ by $N$ is a [[short exact sequence|KJKNJMGKZZ]] $0 \to N \to E \to M \to 0$ of $R$-modules.
Two extensions with middle terms $E$ and $E'$ are *equivalent* if some homomorphism $E \to E'$ commutes with the maps from $N$ and to $M$.

1. There is a natural bijection between $\operatorname{Ext}^1_R(M, N)$ and the set of equivalence classes of extensions of $M$ by $N$. Under it, $0$ corresponds to the class of the split extension $0 \to N \to N \oplus M \to M \to 0$.
2. Let $n \ge 1$, and consider exact sequences $0 \to N \to E_n \to \cdots \to E_1 \to M \to 0$. Use the equivalence relation generated by maps of such sequences that are the identity on $N$ and on $M$. Then there is a natural bijection between $\operatorname{Ext}^n_R(M, N)$ and the set of classes.

## Remarks

Given an extension, apply $\Hom_R(-, N)$ and take the image of $\id_N$ under the connecting map $\Hom_R(N, N) \to \operatorname{Ext}^1_R(M, N)$; this gives the class of the extension.
Conversely, a class in $\operatorname{Ext}^1_R(M, N)$ is represented by a map $P_1 \to N$ from the first term of a projective resolution of $M$, and a pushout along this map gives the extension.
The group law on $\operatorname{Ext}^1$ becomes the Baer sum of extensions.
Example: for a prime $p$, $\operatorname{Ext}^1_\Z(\Z/p\Z, \Z/p\Z) \cong \Z/p\Z$.
Its $p$ elements are the split extension and the $p - 1$ classes of $0 \to \Z/p\Z \xrightarrow{\cdot p} \Z/p^2\Z \to \Z/p\Z \to 0$, whose quotient maps differ by the automorphisms of $\Z/p\Z$.
The description in part 2 makes sense in any [[abelian category|Y9CZ3MTNF1]], even without enough projectives or injectives; this is the *Yoneda Ext*.

## Explanations

### L1 human

The group $\operatorname{Ext}^1(M, N)$ has a concrete meaning: it lists the ways to glue $N$ and $M$ into a bigger module $E$ with $N$ inside and $M$ as quotient.
The trivial way, the direct sum, is the zero element.
For example, $\Z/p\Z$ and $\Z/p\Z$ can be glued into $\Z/p\Z \oplus \Z/p\Z$ or into $\Z/p^2\Z$.

### L2 human

This is how Ext got its name: it classifies extensions.
The higher groups $\operatorname{Ext}^n$ classify longer exact sequences, and the Yoneda product, which splices sequences end to end, corresponds to the composition of morphisms $M \to N[n]$ in the [[derived category|5ST7NVJFSM]].
In group theory, the same idea shows that $H^2$ of a group with coefficients in a module classifies extensions with abelian kernel, see [[group cohomology|YXJXA0Q52R]].
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# Asymptotic stability of linear recurrences

## Statement

Let $d \ge 1$ and let $c_0, \dots, c_{d-1}$ be complex numbers with $c_0 \ne 0$.
Consider the [[linear recurrence|AA7XRZ4VAG]]
$$x_{n+d} = c_{d-1} x_{n+d-1} + \cdots + c_1 x_{n+1} + c_0 x_n, \qquad n \ge 0,$$
and its [[characteristic polynomial|0YA7X86K05]] $\chi(z) = z^d - c_{d-1} z^{d-1} - \cdots - c_0$.
The following are equivalent.

1. Every solution $(x_n)$ satisfies $x_n \to 0$ as $n \to \infty$.
2. The zero solution is asymptotically [[stable|CHEQ235HW9]].
3. The polynomial $\chi$ is [[Schur stable|X11Q0G8PNA]], that is, every root $\lambda$ of $\chi$ satisfies $\abs{\lambda} < 1$.

## Remarks

Every solution is a finite sum of terms $p(n) \lambda^n$, where $\lambda$ is a root of $\chi$ and $p$ is a polynomial of degree less than the multiplicity of $\lambda$.
If all roots satisfy $\abs{\lambda} < 1$, each such term tends to $0$, because $\abs{\lambda}^n$ decays faster than any polynomial grows.
If some root has $\abs{\lambda} \ge 1$, then $x_n = \lambda^n$ is a solution that does not tend to $0$.
For example, $x_{n+2} = x_{n+1} - \tfrac{1}{2} x_n$ has characteristic polynomial $z^2 - z + \tfrac{1}{2}$ with roots of modulus $1/\sqrt{2}$, so all its solutions tend to $0$.
In contrast, $x_{n+2} = x_n$ has roots $\pm 1$ on the unit circle, and its solution $x_n = 1$ stays bounded but does not tend to $0$.
Suppose all roots satisfy $\abs{\lambda} \le 1$, some root lies on the unit circle, and the roots on the circle are simple.
Then the zero solution is stable but not asymptotically stable.

## Explanations

### L1 human

A linear recurrence builds each term from a fixed combination of the previous ones.
This theorem says when every such sequence dies out, whatever the starting values.
The answer is read off from one polynomial: all its roots must lie strictly inside the unit circle.
It turns a question about infinitely many sequences into a question about finitely many numbers.

### L2 human

The result is the discrete counterpart of the fact that $x' = Ax$ is asymptotically stable exactly when all eigenvalues of $A$ have negative real part.
Writing the recurrence as $X_{n+1} = C X_n$ with the companion matrix $C$, the condition says that the spectral radius of $C$ is less than $1$.
In practice one does not compute the roots; tests such as the [[Jury stability criterion|VGV9FAPCHN]] decide Schur stability from the coefficients alone.
The same criterion governs the local stability of a fixed point of a nonlinear map through its linearization.
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# Tangent bundle

## Summary

The tangent bundle of a smooth manifold collects all tangent vectors at all points into one space.
A point of the tangent bundle is a pair: a point of the manifold and a velocity vector at that point.
The tangent bundle is itself a smooth manifold of twice the dimension, and it is the basic example of a vector bundle.
Vector fields are its sections, and the derivative of a smooth map is a map between tangent bundles.

## Explanations

### L1 human

At each point of a sphere there is a tangent plane, the flat plane that just touches the sphere there.
The tangent bundle puts all these planes together, one for each point.
To specify a point of it, you give a place on the sphere and an arrow lying in the tangent plane at that place.
For the plane $\R^2$ itself, the tangent bundle is just $\R^2 \times \R^2$: a position and an arrow.
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# Proximity space

## Summary

A proximity space is a set with a relation that says when two subsets are near each other.
Every metric space gives a proximity, in which two sets are near when their distance is $0$.
A proximity determines a topology, and it lies between a topology and a [[uniform space|A54T86KS63]] in the amount of structure it carries.

## Explanations

### L1 human

A topology tells when a point is near a set: when it lies in the closure of the set.
A proximity tells when two sets are near each other.
In the plane, the open disc of radius $1$ and the point $(1, 0)$ are near, since their distance is $0$, though they do not meet.
The open disc and the point $(2, 0)$ are not near.
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# Brill–Noether theory

## Definition

Let $X$ be a smooth projective [[algebraic curve|XJM2MT4S3Y]] of [[genus|6W8TDV9VVN]] $g$ over an algebraically closed field.
For an integer $d$, let $\operatorname{Pic}^d(X)$ be the set of isomorphism classes of [[line bundles|7JCF2TE3V5]] of degree $d$ on $X$.
It is a subset of the [[Picard group|3Z9DCWYXQZ]] of $X$.
For integers $r \ge 0$ and $d$:

1. The *Brill–Noether locus* is $W^r_d(X) = \set{L \in \operatorname{Pic}^d(X) : h^0(X, L) \ge r + 1}$, the set of line bundles of degree $d$ whose complete [[linear system|GHRPE8CXJ9]] has dimension at least $r$.
2. The *Brill–Noether number* is $\rho(g, r, d) = g - (r + 1)(g - d + r)$.

*Brill–Noether theory* is the study of the loci $W^r_d(X)$, in particular of when they are nonempty and of their dimension.

## Remarks

The locus $W^r_d(X)$ is a closed subvariety of $\operatorname{Pic}^d(X)$, which is a copy of the Jacobian, of dimension $g$.
The number $\rho$ is the expected dimension of $W^r_d(X)$: it is $g$ minus the number of conditions $(r + 1)(g - d + r)$ that the existence of $r + 1$ sections imposes, as predicted by Riemann–Roch.
Existence theorem (Kempf, Kleiman–Laksov): if $\rho \ge 0$, then $W^r_d(X)$ is nonempty for every curve $X$.
Griffiths–Harris theorem: for a general curve $X$ of genus $g$, $W^r_d(X)$ is empty if $\rho < 0$, and it has dimension $\rho$ if $0 \le \rho \le g$.
Example: for $g = 3$, $r = 1$, $d = 2$, one gets $\rho = 3 - 2 \cdot 2 = -1$, so a general curve of genus $3$ has no pencil of degree $2$ and is not hyperelliptic.
Example: for $r = 0$ and $0 \le d \le g$, one gets $\rho = d$, and $W^0_d(X)$ is the image of the $d$-fold symmetric product of $X$, of dimension $d$.
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# Group representation

## Summary

A representation of a [[group|G5S1M9]] $G$ on a [[vector space|000AYPA9AJ]] $V$ is a [[group homomorphism|0JYTG5K9Y4]] $G \to GL(V)$.
Here $GL(V)$ is the group of invertible [[linear maps|0NRYHJPQG2]] $V \to V$.
It lets each element of $G$ act on $V$ as a linear transformation, compatibly with the group operation.
Representations turn questions about groups into questions of linear algebra, where matrices, traces and eigenvalues are available.
They are central in group theory, number theory, physics and chemistry.

## Explanations

### L1 human

A representation lets a group act by matrices.
Each group element becomes an invertible matrix, and multiplying group elements corresponds to multiplying the matrices.
For example, the cyclic group of order $4$ can be represented on the plane $\R^2$ by sending a generator to the rotation by $90$ degrees.
Then the four group elements become the four rotations by $0$, $90$, $180$ and $270$ degrees.
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# Vitali set

## Counterexample

On $[0, 1]$, let $x \sim y$ mean that $x - y$ is a rational number.
This is an [[equivalence relation|0G8VEZ3WTK]].
By the [[axiom of choice|0HTGB13FE2]], there is a set $V \subseteq [0, 1]$ that contains exactly one element of each equivalence class.
Such a set $V$ is called a Vitali set.
No Vitali set is measurable for [[Lebesgue measure|2Z2QVWZ262]] $\lambda$.

## Remarks

Let $q_1, q_2, \ldots$ list the [[countable set|01TQB88H4C]] $\Q \cap [-1, 1]$, and put $V_k = V + q_k = \set{v + q_k : v \in V}$.
The sets $V_k$ are pairwise disjoint: if $v + q_k = w + q_j$ with $v, w \in V$, then $v \sim w$, so $v = w$ and $q_k = q_j$.
Every $x \in [0, 1]$ is equivalent to some $v \in V$, and then $x - v \in \Q \cap [-1, 1]$; so $[0, 1] \subseteq \bigcup_k V_k \subseteq [-1, 2]$.
If $V$ were measurable, each $V_k$ would be measurable with $\lambda(V_k) = \lambda(V)$, by translation invariance.
Countable additivity would give $1 \le \sum_{k} \lambda(V) \le 3$.
This is impossible: the sum is $0$ if $\lambda(V) = 0$ and $\infty$ otherwise.
The same argument shows that there is no translation-invariant, countably additive measure on all subsets of $\R$ that gives $[0, 1]$ a finite positive value.
Some form of choice is needed.
Solovay showed that, if an inaccessible cardinal is consistent with ZFC, then so is ZF with dependent choice and all sets of reals Lebesgue measurable.

## Explanations

### L1 human

Group the numbers of $[0, 1]$ into classes, putting two numbers in the same class when they differ by a fraction.
Pick one number from each class to form the set $V$.
Countably many shifted copies of $V$ fill $[0, 1]$ without overlap and stay inside $[-1, 2]$.
If $V$ had a length, these equal lengths would have to add up to a number between $1$ and $3$.
But countably many equal numbers add up to $0$ or to $\infty$.
So $V$ cannot be given a length at all.
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# Majorization

## Definition

Let $n \ge 1$ and let $x, y \in \R^n$.
Write $x^{\downarrow}$ for the vector with the entries of $x$ in decreasing order, so $x^{\downarrow}_1 \ge x^{\downarrow}_2 \ge \cdots \ge x^{\downarrow}_n$.
The vector $x$ *majorizes* $y$, written $y \prec x$, if:

1. $\sum_{i=1}^{k} y^{\downarrow}_i \le \sum_{i=1}^{k} x^{\downarrow}_i$ for each $k = 1, \ldots, n - 1$.
2. $\sum_{i=1}^{n} y_i = \sum_{i=1}^{n} x_i$.

## Remarks

Example: $y = (\frac{1}{2}, \frac{1}{2}, 0) \prec x = (1, 0, 0)$.
Check: $\frac{1}{2} \le 1$ for $k = 1$, $1 \le 1$ for $k = 2$, and both sums equal $1$.
Non-example: $(1, 0) \not\prec (2, 0)$, because clause 1 holds but the sums $1$ and $2$ differ, so clause 2 fails.
The relation is a preorder: $y \prec x$ and $x \prec y$ hold together exactly when $y$ is a permutation of $x$.
By a theorem of Hardy, Littlewood and Pólya, $y \prec x$ if and only if $y = Dx$ for some $n \times n$ [[doubly stochastic matrix|JJA90EB2PS]] $D$.
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# Fox–Wright function

## Definition

Let $p, q \ge 0$ be integers.
Let $a_1, \ldots, a_p, b_1, \ldots, b_q \in \C$ and let $A_1, \ldots, A_p, B_1, \ldots, B_q > 0$ be real.
Assume that $a_i + A_i n$ is never a pole of the [[gamma function|0H5SYBSED9]] $\Gamma$, for $n = 0, 1, 2, \ldots$, and use the convention $1/\Gamma(-m) = 0$ for $m = 0, 1, 2, \ldots$.
The *Fox–Wright function* is the [[power series|0MZB06ST2B]]
$${}_p\Psi_q\left[\begin{matrix} (a_1, A_1), \ldots, (a_p, A_p) \\ (b_1, B_1), \ldots, (b_q, B_q) \end{matrix}; z\right] = \sum_{n=0}^{\infty} \frac{\Gamma(a_1 + A_1 n) \cdots \Gamma(a_p + A_p n)}{\Gamma(b_1 + B_1 n) \cdots \Gamma(b_q + B_q n)} \, \frac{z^n}{n!}.$$

## Remarks

Let $\Delta = 1 + \sum_j B_j - \sum_i A_i$.
If $\Delta > 0$ the series converges for every $z \in \C$.
If $\Delta = 0$ it converges for $\abs{z} < \prod_i A_i^{-A_i} \prod_j B_j^{B_j}$.

Example: with $p = q = 1$, $(a_1, A_1) = (1, 1)$ and $(b_1, B_1) = (\beta, \alpha)$, the coefficient is $\frac{\Gamma(1 + n)}{\Gamma(\beta + \alpha n)} \frac{1}{n!} = \frac{1}{\Gamma(\alpha n + \beta)}$, so the series is the [[Mittag-Leffler function|12G9M694YG]] $E_{\alpha, \beta}(z)$; here $\Delta = \alpha > 0$.

When all $A_i$ and $B_j$ equal $1$,
$${}_p\Psi_q = \frac{\Gamma(a_1) \cdots \Gamma(a_p)}{\Gamma(b_1) \cdots \Gamma(b_q)} \, {}_pF_q(a_1, \ldots, a_p; b_1, \ldots, b_q; z),$$
a multiple of the [[generalized hypergeometric function|VDBAW9PN88]].

Non-example: with $p = 1$, $q = 0$ and $A_1 = 2$, one has $\Delta = -1 < 0$, and the series $\sum \Gamma(1 + 2n) z^n / n!$ diverges for every $z \neq 0$.
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# Lorenz system

## Example

Let $\sigma, \rho, \beta > 0$.
The Lorenz system is the [[autonomous system|0673D85QKZ]] in $\R^3$
$$x' = \sigma (y - x), \qquad y' = x (\rho - z) - y, \qquad z' = x y - \beta z.$$
The classical parameter values are $\sigma = 10$, $\beta = 8/3$ and $\rho = 28$.

The origin is an [[equilibrium point|0G4X36AE0X]] for all parameters.
For $\rho > 1$ there are two more equilibria $C_\pm = \left(\pm\sqrt{\beta(\rho - 1)}, \pm\sqrt{\beta(\rho - 1)}, \rho - 1\right)$.
Check: $x = y$ makes the first equation vanish, then $z = \rho - 1$ makes the second vanish, and $x^2 = \beta(\rho - 1)$ makes the third vanish.

## Remarks

The divergence of the vector field is $-(\sigma + 1 + \beta) < 0$, so the flow shrinks volumes at a constant exponential rate.
At $\rho = 1$, the two equilibria $C_\pm$ branch off the origin in a [[pitchfork bifurcation|P88X34MSVV]].
For $\sigma > \beta + 1$, the linearisation at $C_\pm$ has a pair of purely imaginary eigenvalues at $\rho = \sigma(\sigma + \beta + 3)/(\sigma - \beta - 1)$.
For $\sigma = 10$ and $\beta = 8/3$ this value is $470/19 \approx 24.74$, and there $C_\pm$ lose stability in a subcritical [[Hopf bifurcation|F5H8GW1XYS]].
At the classical values, numerical solutions approach a butterfly-shaped set, the Lorenz attractor, and nearby solutions separate quickly.
Warwick Tucker gave a computer-assisted proof in 2002 that this set is a strange attractor.

## Explanations

### L1 human

Edward Lorenz found these three equations in 1963 as a very simplified model of convection in the atmosphere.
Solutions wind around two lobes and switch between them in an unpredictable way.
Two solutions that start very close soon look completely different.
This sensitivity is the "butterfly effect" and the Lorenz system is its standard example.
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# Minkowski–Bouligand dimension

## Summary

The Minkowski–Bouligand dimension is also called the box-counting dimension.
For a bounded set, it measures how fast the number of small boxes needed to cover the set grows as the boxes shrink.
If about $\varepsilon^{-d}$ boxes of side $\varepsilon$ are needed, the dimension is $d$.
It is easy to estimate numerically, and it is at least the [[Hausdorff dimension|SSNFTHZ6YJ]].

## Explanations

### L1 human

Lay a grid of squares of side $\varepsilon$ over a picture of the set and count the squares that meet it.
For a segment, halving $\varepsilon$ doubles the count; for a filled square, it multiplies the count by four.
The exponent in this growth is the box-counting dimension: $1$ for the segment and $2$ for the square.
For the middle-thirds Cantor set, dividing $\varepsilon$ by $3$ doubles the count, which gives dimension $\log 2 / \log 3$.
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# Kernel

## Definition

Let $\mathcal{C}$ be a category with [[zero morphisms|YF9F9EG6M3]], and let $f \colon X \to Y$ be a morphism of $\mathcal{C}$.
A *kernel* of $f$ is an object $K$ with a morphism $k \colon K \to X$ such that:

1. $f \circ k = 0$;
2. for every morphism $u \colon T \to X$ with $f \circ u = 0$ there is a unique morphism $v \colon T \to K$ with $u = k \circ v$.

## Remarks

Equivalently, $k$ is an [[equalizer|CTJ5AD8ST5]] of $f$ and $0 \colon X \to Y$.
A kernel is unique up to a unique isomorphism, and is written $\ker f \to X$.
Every kernel is a [[monomorphism|Y2NP1JFR86]], by the uniqueness in clause 2.

Example: in the category of [[modules|0NZ8JA25WK]] over a ring, the inclusion $k \colon \set{x \in X : f(x) = 0} \to X$ of the [[kernel of the homomorphism|0W88CEGQP1]] is a kernel.
If $f \circ u = 0$, then $u$ takes values in this submodule, so $u$ factors through $k$, and uniquely since $k$ is injective.

Non-example: for the reduction $f \colon \Z \to \Z/2$ of abelian groups, the inclusion $k \colon 4\Z \to \Z$ satisfies clause 1 but not clause 2.
The inclusion $u \colon 2\Z \to \Z$ satisfies $f \circ u = 0$ but does not factor through $4\Z$, since $2 \notin 4\Z$.
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# Locally finite variety

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
The variety $\mathcal{V}$ is *locally finite* if every [[finitely generated algebra|WR4WF1K2KW]] in $\mathcal{V}$ is finite.

## Remarks

An equivalent condition is that the [[free algebra|GTS4DRRMZR]] $F_{\mathcal{V}}(n)$ of $\mathcal{V}$ on $n$ generators is finite for every natural number $n$.
This holds because every algebra of $\mathcal{V}$ generated by $n$ elements is a homomorphic image of $F_{\mathcal{V}}(n)$.
Example: the variety of Boolean algebras is locally finite.
A Boolean algebra generated by $n$ elements is an image of the free Boolean algebra on $n$ generators, which has $2^{2^n}$ elements.
Non-example: the variety of groups is not locally finite.
The group $(\Z, +)$ is generated by the single element $1$ and is infinite.

## Explanations

### L1 human

In a locally finite variety, a finite set of elements can never generate an infinite algebra.
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# Kakeya needle problem

## Problem

A needle is a line segment of length $1$ in the plane $\R^2$.
Find the smallest area, in the sense of [[Lebesgue measure|2Z2QVWZ262]], of a region $R \subset \R^2$ with the following property.
The needle can be moved continuously inside $R$ so that it ends in its starting position with its two ends exchanged.

## Remarks

The problem is solved, and the answer is that there is no positive minimum.
Abram Besicovitch showed that for every $\varepsilon > 0$ there is a region of area less than $\varepsilon$ in which the needle can be turned around.
His construction builds on a set of measure zero that contains a unit segment in every direction, an example of a [[Kakeya set|6PEN7AKC7M]].
If $R$ is required to be convex, the minimum exists: it is the equilateral triangle of height $1$, of area $1/\sqrt{3}$, a result of Julius Pál.

## Explanations

### L1 human

Imagine a stick of length one lying on a table, and you must turn it to face the other way while keeping it inside a painted region.
A disc of diameter one works, with area about $0.785$, and a triangle of height one does better.
Surprisingly, with clever back-and-forth moves the painted region can be made as small in area as you like.
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# Girth

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
If $G$ contains a [[cycle|07CT9RDY7Q]], the girth of $G$ is the smallest length of a cycle in $G$.
If $G$ contains no cycle, that is, if $G$ is [[acyclic|0XNN9Z8E5A]], the girth of $G$ is $\infty$.

## Remarks

In a simple graph each cycle has length at least $3$, so the girth is at least $3$.
Example: the [[cycle graph|0QQ0F12962]] $C_n$ with $n \ge 3$ has girth $n$, since its only cycle has length $n$.
The [[complete graph|03WKB2H474]] $K_n$ with $n \ge 3$ has girth $3$, and the [[complete bipartite graph|0V0GC7BQAN]] $K_{m,n}$ with $m, n \ge 2$ has girth $4$, since a [[bipartite graph|07DVHJ8N0A]] has no odd cycle.
The [[Petersen graph|0NR95CTFCZ]] has girth $5$.
Non-example: in the path graph $u - v - w$, the closed walk $u, v, u$ has length $2$ but is not a cycle, because it uses the edge $uv$ twice.
The path graph has no cycle, so its girth is $\infty$.
Every [[tree|0MJ0JS72DZ]] has girth $\infty$.
The girth is sometimes written $g(G)$.
</document>

<document source="https://tamnd.github.io/math2/node/WZ4Z26XSVC.md" tag="WZ4Z26XSVC" kind="concept">
# Urysohn integral equation

## Summary

A Urysohn integral equation is a nonlinear integral equation in which the unknown function enters the integrand through a general function of three variables: $\varphi(x) = f(x) + \int_\Omega K(x, t, \varphi(t))\, dt$.
It is the most general form of a [[nonlinear integral equation|BSCYBP6AS6]] of Fredholm type in one unknown.
It contains the Hammerstein equation as a special case.
Existence results for it usually come from fixed point theorems, such as the [[Schauder fixed-point theorem|DCPFVVZZ54]], applied to the operator on the right side.
The name is also written Uryson.

## Explanations

### L1 human

In a linear integral equation the unknown is multiplied by a kernel and then integrated.
In a Urysohn equation the kernel may depend on the unknown in any way, for example through $\sin(x\, \varphi(t))$.
The equation still says that the unknown at a point is a weighted total of its values elsewhere.
For instance, $\varphi(x) = x$ solves $\varphi(x) = x + \int_0^1 \sin\bigl(x(\varphi(t) - t)\bigr)\, dt$, since the sine is then $0$.
</document>

<document source="https://tamnd.github.io/math2/node/WZC2MN3CX3.md" tag="WZC2MN3CX3" kind="definition">
# Fiber of a morphism

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]] and let $y \in Y$.
Let $\kappa(y)$ be the [[residue field|G51E8MW7J1]] of $Y$ at $y$.
Let $\iota_y \colon \operatorname{Spec} \kappa(y) \to Y$ be the canonical morphism with image $y$.
The *fiber* of $f$ over $y$ is the [[fiber product|1DZMT4S9FB]]
$$X_y = X \times_Y \operatorname{Spec} \kappa(y),$$
taken with respect to $f$ and $\iota_y$.
It is a scheme over $\kappa(y)$.

## Remarks

The projection $X_y \to X$ is a homeomorphism of the underlying space of $X_y$ onto $f^{-1}(y)$.
Example: let $f \colon \operatorname{Spec} \Z[i] \to \operatorname{Spec} \Z$ and $y = (p)$ for a prime $p$.
Then $X_y = \operatorname{Spec}(\Z[i] \otimes_\Z \F_p) = \operatorname{Spec} \F_p[x]/(x^2 + 1)$.
For $p = 5$ this is $\operatorname{Spec} \F_5[x]/((x-2)(x+2))$, two points.
For $y = (0)$ the fiber is $\operatorname{Spec} \Q(i)$, the generic fiber.
Non-example: the fiber is in general not the reduced point structure on $f^{-1}(y)$.
Let $k$ be a field with $2 \ne 0$, let $f \colon \operatorname{Spec} k[t] \to \operatorname{Spec} k[x]$ be given by $x \mapsto t^2$, and let $y = (x)$.
Then $X_y = \operatorname{Spec} k[t]/(t^2)$, which is not reduced, while the single point of $f^{-1}(y)$ with its reduced structure is $\operatorname{Spec} k$.
</document>

<document source="https://tamnd.github.io/math2/node/WZC4J62B1G.md" tag="WZC4J62B1G" kind="concept">
# Cauchy filter

## Summary

A Cauchy filter on a [[uniform space|A54T86KS63]] is a [[filter|15PF73BKY6]] that contains arbitrarily small sets, where "small" is measured by the [[entourages|FVFFKNKEDF]].
It is the uniform version of a [[Cauchy sequence|0PWX18FM4G]].
A uniform space is complete when every Cauchy filter converges, and this gives completeness without a metric.

## Explanations

### L1 human

A Cauchy sequence is a sequence whose terms eventually all lie very close together.
A filter is a family of "large" sets.
A Cauchy filter has, for each degree of closeness, a member whose points are all that close to each other.
In $\R$, the filter of all sets that contain an interval $(-\varepsilon, \varepsilon)$ for some $\varepsilon > 0$ is a Cauchy filter.
In the rational numbers, the filter of neighbourhoods of $\sqrt{2}$ traced on $\Q$ is a Cauchy filter that does not converge in $\Q$.
</document>

<document source="https://tamnd.github.io/math2/node/WZJ99RSXNN.md" tag="WZJ99RSXNN" kind="definition">
# Multidimensional Laplace transform

## Definition

Let $n \ge 1$ and let $f \colon [0, \infty)^n \to \C$ be locally integrable.
For $s = (s_1, \dots, s_n) \in \C^n$ and $x \in [0, \infty)^n$ write $s \cdot x = s_1 x_1 + \cdots + s_n x_n$.
The *$n$-dimensional Laplace transform* of $f$ is the function
$$F(s) = \int_{[0, \infty)^n} e^{-s \cdot x} f(x)\, dx,$$
defined at those $s \in \C^n$ where the integral converges absolutely.

## Remarks

For $n = 1$ this is the [[Laplace transform|0V49JEZW06]].
If the integral converges absolutely at $s$, it converges absolutely at every $s'$ with $\operatorname{Re} s'_j \ge \operatorname{Re} s_j$ for all $j$.

Example: $f(x_1, x_2) = e^{a x_1 + b x_2}$ with real $a, b$.
For $\operatorname{Re} s_1 > a$ and $\operatorname{Re} s_2 > b$ the integral splits into a product, and $F(s_1, s_2) = \frac{1}{(s_1 - a)(s_2 - b)}$.

Non-example: $f(x_1, x_2) = e^{x_1 x_2}$ has no Laplace transform.
For any $s$, let $c = \abs{s_1} + \abs{s_2}$.
If $x_1, x_2 \ge c$, then $x_1 x_2 - \operatorname{Re}(s_1) x_1 - \operatorname{Re}(s_2) x_2 \ge x_1 x_2 - c x_1 - c x_2 = (x_1 - c)(x_2 - c) - c^2$.
So $\abs{e^{-s \cdot x} f(x)} \ge 1$ on the set where $(x_1 - c)(x_2 - c) \ge c^2$, which has infinite area, and the integral does not converge absolutely.

The transform is often written $\mathcal{L}_n f$, and the variables are kept real in many applications.
</document>

<document source="https://tamnd.github.io/math2/node/WZKPDWBW2Z.md" tag="WZKPDWBW2Z" kind="definition">
# Maximal chain

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
A [[chain|0GP9ASVQ1X]] $C \subseteq P$ is *maximal* when there is no chain $D \subseteq P$ with $C \subsetneq D$.

## Remarks

In the subsets of $\set{1, 2, 3}$ ordered by inclusion, $\set{\emptyset, \set{1}, \set{1, 2}, \set{1, 2, 3}}$ is a maximal chain.
Every other subset is incomparable with one of its members, so no subset can be added.
The chain $\set{\emptyset, \set{1}, \set{1, 2, 3}}$ is not maximal, since adding $\set{1, 2}$ gives a larger chain.
The subsets of $\set{1, \dots, n}$ have exactly $n!$ maximal chains, one for each order in which the elements are added.
A maximal chain is a maximal element of the set of chains ordered by inclusion.
The statement that every chain of a poset lies in a maximal chain is the [[Hausdorff maximal principle|0KM4KVR9MV]], which is equivalent to [[Zorn's lemma|04RXTQA5ZA]].

## Explanations

### L1 human

A maximal chain is a chain that cannot be made longer by adding any element of the poset.
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<document source="https://tamnd.github.io/math2/node/WZKRG7H3FX.md" tag="WZKRG7H3FX" kind="definition">
# Lie symmetry method

## Definition

Let $U \subseteq \R^2$ be open and let $\omega : U \to \R$ be smooth; consider the [[ordinary differential equation|0P97V20RW7]] $y' = \omega(x, y)$.
Let $\Gamma_\varepsilon : (x, y) \mapsto (\hat x, \hat y)$ be a smooth local one-parameter group of maps, with $\Gamma_0 = \id$ and $\Gamma_\varepsilon \circ \Gamma_\delta = \Gamma_{\varepsilon + \delta}$.
It is a *Lie point symmetry* of the equation if it maps graphs of solutions to graphs of solutions.
Its infinitesimal generator is the vector field $X = \xi \partial_x + \eta \partial_y$ with $\xi = \partial \hat x / \partial \varepsilon$ and $\eta = \partial \hat y / \partial \varepsilon$ at $\varepsilon = 0$.
The group is a symmetry if and only if the linearized symmetry condition holds:
$$\eta_x + (\eta_y - \xi_x)\, \omega - \xi_y\, \omega^2 = \xi\, \omega_x + \eta\, \omega_y,$$
where subscripts denote [[partial derivatives|05AXG3QRKR]].
The *Lie symmetry method* solves the equation from a known symmetry, near a point where $X \ne 0$, as follows.

1. Find canonical coordinates $(r, s)$, functions of $(x, y)$ with $X r = 0$ and $X s = 1$.
2. In these coordinates the equation takes the form $ds/dr = \Omega(r)$, with no dependence on $s$.
3. Integrate: $s = \int \Omega(r)\, dr + C$, then return to $(x, y)$.

## Remarks

Example: for $\omega(x, y) = F(y/x)$, the scalings $(x, y) \mapsto (e^\varepsilon x, e^\varepsilon y)$ have generator $X = x \partial_x + y \partial_y$.
Check of the condition: the left side is $0 + (1 - 1)\omega - 0 = 0$, and the right side is $x \cdot (-y/x^2) F' + y \cdot (1/x) F' = 0$.
Canonical coordinates are $r = y/x$ and $s = \ln\abs{x}$, since $X r = -y/x + y/x = 0$ and $X s = 1$.
Then $ds/dr = 1/(F(r) - r)$, and the equation is solved by an [[antiderivative|0BW9N5CDSB]].
Non-example: the translations $(x, y) \mapsto (x, y + \varepsilon)$ are not symmetries of $y' = x y$, since $\xi = 0$, $\eta = 1$ give left side $0$ and right side $\omega_y = x \ne 0$.
For equations of order $n$, a symmetry reduces the order by one.
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<document source="https://tamnd.github.io/math2/node/WZNF6Q5JBS.md" tag="WZNF6Q5JBS" kind="definition">
# Pseudo-Anosov map

## Definition

Let $S$ be a closed orientable [[surface|VGJJVBZ1MQ]].
A *singular measured foliation* on $S$ is a [[foliation|98HTB7MSPX]] $\mathcal{F}$ of $S \setminus \Sigma$, for a finite set $\Sigma$, with a transverse measure $\mu$.
Here $\mu$ is a measure on arcs transverse to the leaves, invariant under sliding along the leaves.
Near each point of $\Sigma$ the foliation must be a $p$-prong singularity for some $p \ge 3$.
A [[homeomorphism|0QT00PB5G2]] $f \colon S \to S$ is *pseudo-Anosov* if there are $\lambda > 1$ and singular measured foliations $(\mathcal{F}^s, \mu^s)$, $(\mathcal{F}^u, \mu^u)$ on $S$ such that:

1. $\mathcal{F}^s$ and $\mathcal{F}^u$ have the same singular set $\Sigma$ and are transverse away from $\Sigma$.
2. $f$ maps the leaves of $\mathcal{F}^s$ to leaves of $\mathcal{F}^s$, and $f_* \mu^s = \lambda^{-1} \mu^s$.
3. $f$ maps the leaves of $\mathcal{F}^u$ to leaves of $\mathcal{F}^u$, and $f_* \mu^u = \lambda \mu^u$.

The number $\lambda$ is the *stretch factor* (or dilatation) of $f$.

## Remarks

Example: on the torus $\R^2 / \Z^2$, the map induced by $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$, [[Arnold's cat map|BSCZN8Y1AN]], is pseudo-Anosov with no singular points.
The eigenvalues of $A$ are $\lambda = \frac{3 + \sqrt 5}{2} > 1$ and $\lambda^{-1}$.
The foliations by lines parallel to the two eigenvectors, with transverse measures given by distance across the lines, are contracted by $\lambda^{-1}$ and stretched by $\lambda$.

Non-example: the [[Dehn twist|W890NRHGS4]] induced by $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ on the torus has only the eigenvalue $1$.
It preserves the horizontal foliation without stretching it, so no $\lambda > 1$ fits clauses 2 and 3.

On a closed surface of genus $g \ge 2$, the Euler–Poincaré formula forces singular points, since the foliations cannot be nonsingular.
For surfaces with punctures, $1$-prong singularities are also allowed at the punctures.
The stretch factor $\lambda$ is an algebraic integer, and $\log \lambda$ is the topological entropy of $f$, the least entropy among all homeomorphisms isotopic to $f$.
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<document source="https://tamnd.github.io/math2/node/WZQ12PPVA2.md" tag="WZQ12PPVA2" kind="definition">
# Cut locus

## Definition

Let $(M, g)$ be a complete connected [[Riemannian manifold|7CBT4SM83Y]] with [[Riemannian distance|B4QR306D8G]] $d$, and let $p \in M$.
For a unit vector $v \in T_pM$, let $\gamma_v(t) = \exp_p(tv)$ be the unit-speed [[geodesic|NSYMQMHPDN]] from $p$ in direction $v$, using the [[exponential map|Y7Z0AGY1A7]], and let
$$t_c(v) = \sup \set{ t > 0 : d(p, \gamma_v(t)) = t } \in (0, \infty].$$
If $t_c(v) < \infty$, the point $\gamma_v(t_c(v))$ is the *cut point* of $p$ along $\gamma_v$.
The *cut locus* of $p$ is the set of all cut points of $p$, over all unit vectors $v \in T_pM$.

## Remarks

Example: on the unit sphere $S^n$, every unit-speed geodesic from $p$ is a great circle with $d(p, \gamma_v(t)) = t$ for $t \le \pi$ and $d(p, \gamma_v(t)) < t$ just after $t = \pi$.
So $t_c(v) = \pi$ for every $v$, and the cut locus of $p$ is the single point $-p$.

Non-example: in $\R^n$ with the Euclidean metric, $d(p, p + tv) = t$ for all $t > 0$, so $t_c(v) = \infty$ for all $v$, there are no cut points, and the cut locus is empty.

Let $q = \gamma_v(t_c(v))$ be the cut point of $p$ along $\gamma_v$.
Then $q$ is the first point [[conjugate|ZPXE5VSZ35]] to $p$ along $\gamma_v$, or there are two distinct minimizing geodesics from $p$ to $q$, or both.
The cut locus is a closed set of measure zero, and $\exp_p$ maps the open star-shaped set $\set{tv : \abs{v} = 1, \ 0 \le t < t_c(v)}$ diffeomorphically onto $M$ minus the cut locus of $p$.
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<document source="https://tamnd.github.io/math2/node/WZRXSVBVQC.md" tag="WZRXSVBVQC" kind="definition">
# Continuous group homomorphism

## Definition

Let $G$ and $H$ be [[topological groups|92700QAB5Y]].
A *continuous group homomorphism* from $G$ to $H$ is a map $f \colon G \to H$ such that:

1. $f$ is a [[group homomorphism|0JYTG5K9Y4]], that is, $f(xy) = f(x) f(y)$ for all $x, y \in G$;
2. $f$ is [[continuous|0VGFHTM75Y]].

## Remarks

Example: $f \colon (\R, +) \to (\C \setminus \set{0}, \cdot)$, $f(t) = e^{2\pi i t}$.
It satisfies $f(s + t) = f(s) f(t)$, and it is continuous as a composition of continuous functions.
Non-example: the identity map from $(\R, +)$ with the usual topology to $(\R, +)$ with the discrete topology.
It is a homomorphism, but the preimage of the open set $\set{0}$ is $\set{0}$, which is not open in the usual topology.
A homomorphism $f$ is continuous if and only if it is continuous at the identity $e$ of $G$.
Indeed, near a point $a$ one writes $f(x) = f(a) f(a^{-1} x)$ and uses that translations are homeomorphisms.
</document>

<document source="https://tamnd.github.io/math2/node/X001F42KH8.md" tag="X001F42KH8" kind="concept">
# Adaptive mesh refinement

## Summary

Adaptive mesh refinement makes a computational mesh finer only where the solution needs it, guided by the computed solution itself.
In [[finite element methods|98GZ9DVHTA]] it runs a loop: solve, estimate the error on each element, mark elements with large error, and refine them.
For time-dependent hyperbolic problems, block-structured refinement in space and time was developed by Marsha Berger and Joseph Oliger.
Adaptivity recovers optimal convergence rates for solutions with singularities, where uniform refinement is slow.
See the [[definition|C5JM6D0841]].

## Explanations

### L1 human

A uniform mesh spends as much effort on calm regions as on regions where the solution changes sharply.
Adaptive refinement looks at the current approximation and adds detail only where the error seems large.
Near a sharp corner of a domain, or near a shock wave, the mesh becomes very fine, while elsewhere it stays coarse.
For the Poisson equation on an L-shaped domain, the elements cluster around the inner corner, where the solution is singular.
</document>

<document source="https://tamnd.github.io/math2/node/X0243CZD9D.md" tag="X0243CZD9D" kind="concept">
# Resolvent set

## Summary

The resolvent set of an element $a$ of a unital algebra is the set of scalars $\lambda$ for which $a - \lambda 1$ is invertible.
The same notion applies to linear operators.
It is the complement of the spectrum.
On it the resolvent $\lambda \mapsto (a - \lambda 1)^{-1}$ is defined.
In a complex Banach algebra the resolvent set is open, and the resolvent is holomorphic on it.
These properties of the resolvent are the main tools to study the spectrum.

## Explanations

### L1 human

The resolvent set collects the numbers $\lambda$ for which the equation $(a - \lambda) x = y$ can always be solved in a unique and stable way.
For a matrix, these are all numbers that are not eigenvalues.
For multiplication by the function $t$ on $[0,1]$, these are the complex numbers outside the interval $[0,1]$.
</document>

<document source="https://tamnd.github.io/math2/node/X057MN9YY9.md" tag="X057MN9YY9" kind="conjecture">
# Borel conjecture

## Statement

A path-connected space $X$ is *aspherical* when its [[homotopy groups|P976WYEFMB]] $\pi_k(X)$ vanish for all $k \ge 2$; equivalently, its [[universal cover|Y1YPE6PM52]] is contractible.
Let $M$ and $N$ be closed connected aspherical [[topological manifolds|QQQ9B1DK4P]].
Then every [[homotopy equivalence|XZG6MTSWCN]] $f \colon M \to N$ is homotopic to a [[homeomorphism|0QT00PB5G2]].
In particular, two closed aspherical manifolds with isomorphic fundamental groups are homeomorphic.

## Remarks

The conjecture is open in general.
It holds in dimensions at most $2$ by the classification of surfaces, and in dimension $3$ as a consequence of the geometrization theorem.
Farrell and Jones proved it for closed Riemannian manifolds of nonpositive sectional curvature of dimension at least $5$.
Bartels and Lück proved it in dimensions at least $5$ when the fundamental group is hyperbolic or CAT(0).
In terms of the [[surgery exact sequence|Q7C480HQ9J]], it says that the structure set of a closed aspherical manifold of dimension at least $5$ has one element.
The analogue for diffeomorphisms is false: for $n \ge 5$ there are smooth manifolds homeomorphic to the $n$-torus but not diffeomorphic to it.

## Explanations

### L1 human

An aspherical manifold is one whose shape is completely recorded by its loops: all higher-dimensional holes are absent.
The torus and closed hyperbolic manifolds are examples.
The conjecture says that for such manifolds the loops determine everything: if two of them have the same homotopy type, they are the same manifold up to homeomorphism.
</document>

<document source="https://tamnd.github.io/math2/node/X0G02CGQ5J.md" tag="X0G02CGQ5J" kind="problem">
# Matrix multiplication exponent

## Problem

Fix a field $F$.
Consider the real numbers $\tau$ such that an algebraic algorithm multiplies two $n \times n$ matrices over $F$ with $O(n^\tau)$ operations in $F$ (see [[big O notation|0TGV80E6A4]]).
The *matrix multiplication exponent* $\omega$ is the infimum of these $\tau$.
Determine the value of $\omega$; in particular, decide whether $\omega = 2$.

## Remarks

The bound $\omega \ge 2$ holds because the output has $n^2$ entries, and the schoolbook [[matrix multiplication|032Q9FB8MT]] gives $\omega \le 3$.
The [[Strassen algorithm|3CVWXDS4JQ]] gave $\omega \le \log_2 7 \approx 2.807$ in 1969.
Coppersmith and Winograd obtained $\omega < 2.376$ in 1990.
Refinements of their method by several groups since 2010 have brought the upper bound below $2.372$.
The problem is open: no lower bound better than $2$ is known.
Strassen showed that $\omega$ can be read off from the [[tensor rank|W84NTF77M0]] of the matrix multiplication tensor $\langle n, n, n \rangle$.
It is the infimum of the $\tau$ such that this rank is $O(n^\tau)$.
Many other problems of linear algebra, such as inverting a matrix or computing a determinant, can be solved with $O(n^{\omega + \varepsilon})$ operations for every $\varepsilon > 0$.

## Explanations

### L1 human

Multiplying two $n \times n$ matrices the usual way needs about $n^3$ multiplications.
The exponent $\omega$ measures how much better one can do in the best case, as a power of $n$.
Strassen showed that the power $3$ is not optimal, and later work has pushed it down to about $2.37$.
Nobody knows whether one can get as low as $2$, which would mean that multiplying matrices costs hardly more than reading them.
</document>

<document source="https://tamnd.github.io/math2/node/X0Q69ZS09T.md" tag="X0Q69ZS09T" kind="concept">
# Convex integrand

## Summary

A convex integrand is a Lagrangian $L(x, u, \xi)$ of an integral functional $\int_\Omega L(x, u, \nabla u) \, dx$ that is a [[convex function|07GR3TNW4H]] of the gradient variable $\xi$.
Convexity in $\xi$ is the hypothesis that makes such a functional weakly lower semicontinuous on Sobolev spaces, for scalar functions $u$.
It is therefore the basic structural assumption of existence theorems such as Tonelli's.
For vector-valued $u$ it can be weakened to quasiconvexity.

## Explanations

### L1 human

The integrand describes the cost of each local slope of a function.
If this cost is convex in the slope, mixing two slopes in fine oscillations never costs less than using their average slope.
So oscillating functions cannot undercut smooth ones, and a minimiser can be found as a limit.
The Dirichlet energy, with $L = \frac{1}{2} \abs{\xi}^2$, has a convex integrand.
The double-well cost $(\xi^2 - 1)^2$ is not convex, and indeed fine zigzags with slopes $\pm 1$ beat the flat function.
</document>

<document source="https://tamnd.github.io/math2/node/X0X1Q5JZFK.md" tag="X0X1Q5JZFK" kind="definition">
# Maurer–Cartan form

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] with identity $e$ and Lie algebra $\mathfrak{g} = T_eG$, the [[tangent space|E7A5DHFT01]] at $e$.
For $g \in G$, let $L_g \colon G \to G$, $L_g(h) = gh$, be left translation.
The *(left) Maurer–Cartan form* is the $\mathfrak{g}$-valued [[one-form|02Z5B2HA3F]] $\omega$ on $G$ with
$$\omega_g = (dL_{g^{-1}})_g \colon T_gG \to \mathfrak{g},$$
for each $g \in G$, where $(dL_{g^{-1}})_g$ is the [[differential|6P9PXQDSEN]] of $L_{g^{-1}}$ at $g$.

## Remarks

Example: for $G = GL_n(\R)$, left translation by $g^{-1}$ is the linear map $V \mapsto g^{-1}V$ of $M_n(\R)$.
So $\omega_g(V) = g^{-1} V$, written $\omega = g^{-1}\, dg$.
Then $d\omega = -g^{-1}\,dg\, g^{-1} \wedge dg = -\omega \wedge \omega$, which is the Maurer–Cartan equation for matrix groups.

Non-example: the form $\omega'_g = (dR_{g^{-1}})_g$, with $R_g(h) = hg$, also takes values in $\mathfrak{g}$, but it is not the left Maurer–Cartan form.
For $GL_n(\R)$ it is $dg\, g^{-1}$, and at $g = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$, $V = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ one gets $V g^{-1} = \tfrac12 V \ne V = g^{-1}V$.
It is the right Maurer–Cartan form.

The form $\omega$ is left-invariant: $L_h^*\omega = \omega$ for all $h \in G$.
For each [[left-invariant vector field|SNA3DVPK0C]] $X$, the function $\omega(X)$ is constant, equal to $X_e$.
In general it satisfies $d\omega + \tfrac12[\omega \wedge \omega] = 0$.
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<document source="https://tamnd.github.io/math2/node/X117TDBR51.md" tag="X117TDBR51" kind="definition">
# Convergence along an ultrafilter

## Definition

Let $S$ be a set and let $\mathcal{U}$ be an [[ultrafilter|J64Y3JC0RE]] on $S$.
Let $X$ be a [[topological space|0CQHFPGJ45]], let $f \colon S \to X$ be a map, and let $x \in X$.
The map $f$ *converges to $x$ along $\mathcal{U}$* if
$$\set{s \in S : f(s) \in V} \in \mathcal{U}$$
for every [[neighbourhood|0ZCY0Z80B6]] $V$ of $x$.
When this limit is unique it is written $\lim_{\mathcal{U}} f$ or $\mathcal{U}\text{-}\lim_{s} f(s)$.

## Remarks

This is the [[filter limit|F6QDHKE6J3]] of $f$ along the filter $\mathcal{U}$.

Example: for $a \in S$ and the principal ultrafilter $\mathcal{U}_a = \set{A \subseteq S : a \in A}$, every $f$ converges to $f(a)$ along $\mathcal{U}_a$.
Indeed, for each neighbourhood $V$ of $f(a)$ the set $\set{s : f(s) \in V}$ contains $a$.

Non-example: let $\mathcal{U}$ be a non-principal ultrafilter on $\N$ and $f(n) = n$ in $\R$.
For any $x \in \R$, the set $\set{n : \abs{n - x} < 1}$ is finite, and a non-principal ultrafilter contains no finite set.
So $f$ converges to no point along $\mathcal{U}$.

If $X$ is [[compact|0TVJZYB6SH]], then every $f \colon S \to X$ converges along $\mathcal{U}$ to some point; this is closely related to [[compactness via ultrafilters|DT5PH9QVQ7]].
If $X$ is [[Hausdorff|0S1FJ8VEX9]], the limit is unique.
For a non-principal ultrafilter $\mathcal{U}$ on $\N$, the map $x \mapsto \lim_{\mathcal{U}} x_n$ on bounded real sequences is linear and multiplicative, and its value lies between $\liminf_n x_n$ and $\limsup_n x_n$.
It is not a [[Banach limit|SYF2ND6J97]]: it gives $(1, 0, 1, 0, \dots)$ the value $0$ or $1$, while every Banach limit gives $1/2$.
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# Schur stable polynomial

## Summary

A Schur stable polynomial is a polynomial whose [[roots|019MA1VTR2]] all lie strictly inside the unit circle of the complex plane.
It is the discrete-time partner of a Hurwitz stable polynomial, whose roots lie in the open left half-plane.
A linear recurrence with constant coefficients has all its solutions tending to zero exactly when its [[characteristic polynomial|0YA7X86K05]] is Schur stable.
So the notion is the basic test for stability of discrete-time linear systems and of numerical schemes.

## Explanations

### L1 human

Look at all the complex roots of a polynomial and draw them in the plane.
The polynomial is Schur stable when every root lies inside the circle of radius $1$ around the origin, and none lies on the circle.
For example, $z^2 - \tfrac{1}{4}$ has roots $\pm \tfrac{1}{2}$, so it is Schur stable.
The polynomial $z - 2$ is not, because its root $2$ lies outside the circle.
Powers $\lambda^n$ of a root with $\abs{\lambda} < 1$ shrink to zero, which is why these polynomials describe systems that settle down.
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<document source="https://tamnd.github.io/math2/node/X122TKFP9G.md" tag="X122TKFP9G" kind="definition">
# Error-correcting code

## Definition

Let $A$ be a [[finite set|0BBTH3YSZA]] with at least two elements, the alphabet, and let $n$ be a positive integer.
For words $x, y \in A^n$, the Hamming distance $d(x, y)$ is the number of positions $i$ with $x_i \ne y_i$.
A code of length $n$ over $A$ is a [[subset|0BJR9QR08E]] $C \subseteq A^n$ with at least two elements; its elements are the codewords.
The minimum distance of $C$ is $d(C) = \min \set{d(x, y) : x, y \in C,\ x \ne y}$.
For an integer $t \ge 0$, the code $C$ is $t$-error-correcting if for each word $y \in A^n$ there is at most one codeword $c \in C$ with $d(c, y) \le t$.

## Remarks

The code $C$ is $t$-error-correcting if and only if $d(C) \ge 2t + 1$.
One direction uses the triangle inequality of the Hamming distance: if $d(c, y) \le t$ and $d(c', y) \le t$, then $d(c, c') \le 2t$.
Example: the binary repetition code $C = \set{000, 111}$ has $d(C) = 3$, so it is $1$-error-correcting.
A word with at most one $1$ is within distance $1$ of $000$ only, and a word with at least two $1$'s is within distance $1$ of $111$ only.
Non-example: $C = \set{000, 011}$ is not $1$-error-correcting, because $001$ is at distance $1$ from both codewords.
A code with $M$ codewords, length $n$ and minimum distance $d$ is called an $(n, M, d)$ code.
When $A$ is a finite field and $C$ is a subspace of $A^n$, the code is a linear code.
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<document source="https://tamnd.github.io/math2/node/X126J76RH7.md" tag="X126J76RH7" kind="definition">
# Knot Floer homology

## Definition

Let $K \subset S^3$ be an oriented tame [[knot|QYEAZTYGPT]].
Let $(\Sigma, \boldsymbol{\alpha}, \boldsymbol{\beta}, w, z)$ be a doubly pointed Heegaard diagram for $(S^3, K)$, built from a [[Heegaard splitting|EDE9919B5R]] of genus $g$.
Here $\boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_g)$ and $\boldsymbol{\beta} = (\beta_1, \ldots, \beta_g)$ are attaching curves, and $w, z$ are basepoints off them.
Let $\mathbb{T}_\alpha = \alpha_1 \times \cdots \times \alpha_g$ and $\mathbb{T}_\beta = \beta_1 \times \cdots \times \beta_g$, viewed as tori in the symmetric product $\operatorname{Sym}^g(\Sigma)$.
Let $\widehat{CFK}(K)$ be the $\F_2$-vector space with basis $\mathbb{T}_\alpha \cap \mathbb{T}_\beta$.
Its differential is
$$\partial x = \sum_{y} \# \big( \mathcal{M}(\phi) / \R \big) \, y,$$
summed over $y$ and over Whitney disk classes $\phi$ from $x$ to $y$ with Maslov index $1$ and $n_w(\phi) = n_z(\phi) = 0$.
Here $\mathcal{M}(\phi)$ is the moduli space of holomorphic representatives of $\phi$, counted modulo $2$.
The *knot Floer homology* $\widehat{HFK}(K)$ is the homology of $(\widehat{CFK}(K), \partial)$.

## Remarks

It carries a Maslov grading $m$ and an Alexander grading $a$, and up to isomorphism it does not depend on the diagram.
The graded Euler characteristic is the Conway-normalized [[Alexander polynomial|JKA25KDPQP]]: $\sum_{m,a} (-1)^m \dim \widehat{HFK}_m(K, a) \, t^a = \Delta_K(t)$.
Example: the [[unknot|1TQ06R4Q66]] has a genus-$0$ diagram on $S^2$ with one generator, so $\widehat{HFK}$ is $\F_2$ in bidegree $(0, 0)$.
Non-example: if the condition $n_z(\phi) = 0$ is dropped, the complex computes $\widehat{HF}(S^3) \cong \F_2$, which forgets the knot.
Ozsváth and Szabó proved that the Seifert genus of $K$ is the largest $a$ with $\widehat{HFK}(K, a) \ne 0$.
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# Symplectic group

## Description

Let $\mathbb{K}$ be a [[field|04C5ZBG1WT]], let $n \ge 1$, and let
$$\Omega = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix},$$
a $2n \times 2n$ [[block matrix|0ASDFEQS6P]], where $I_n$ is the $n \times n$ identity matrix.
The *symplectic group* is
$$\operatorname{Sp}(2n, \mathbb{K}) = \set{M \in \operatorname{GL}(2n, \mathbb{K}) : M^T \Omega M = \Omega}.$$
It is a [[group|G5S1M9]] under matrix multiplication, a subgroup of the [[general linear group|0FCHE902D7]].
Check: $I^T \Omega I = \Omega$; if $M^T \Omega M = \Omega$ and $N^T \Omega N = \Omega$, then $(MN)^T \Omega (MN) = N^T (M^T \Omega M) N = \Omega$; and multiplying $M^T \Omega M = \Omega$ by $(M^T)^{-1}$ on the left and $M^{-1}$ on the right gives $(M^{-1})^T \Omega M^{-1} = \Omega$.

## Remarks

The group $\operatorname{Sp}(2n, \mathbb{K})$ is the group of linear maps of $\mathbb{K}^{2n}$ that preserve the standard symplectic form $\omega(x, y) = x^T \Omega y$; see [[symplectic vector space|PQ7JJY0VPH]].
Every symplectic matrix has determinant $1$, so $\operatorname{Sp}(2n, \mathbb{K})$ lies in the [[special linear group|0V9PMN0RBH]].
For $n = 1$, every $2 \times 2$ matrix $M$ satisfies $M^T \Omega M = (\det M)\, \Omega$, so $\operatorname{Sp}(2, \mathbb{K}) = \operatorname{SL}(2, \mathbb{K})$.
For $\mathbb{K} = \R$ it is a connected noncompact [[Lie group|7JBRGCVJRB]] of dimension $n(2n + 1)$.
The notation $\operatorname{Sp}(n)$ is also used for the compact symplectic group, the quaternionic unitary group, which is a different group.

## Explanations

### L1 human

Symplectic matrices are the linear changes of coordinates that preserve a certain signed area between pairs of position and momentum coordinates.
They are the linear maps allowed in Hamiltonian mechanics, because they keep the form of Hamilton's equations.
In the plane, preserving signed area is the same as having determinant $1$.
In higher dimensions the condition is stronger than determinant $1$.
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# Computability theory

## Summary

Computability theory, also called recursion theory, studies which functions and sets of natural numbers can be computed by an algorithm.
Its basic objects are [[computable functions|KZSB5E471K]], defined through [[Turing machines|758EYCN27B]] or through [[general recursive functions|HBK5QHC1GC]].
It proves that some problems, such as the [[halting problem|Z7TN2B35B0]], have no algorithmic solution.
It then compares unsolvable problems by reductions, which leads to [[Turing degrees|P5N2ND3VCG]] and to hierarchies of definable sets.

## Explanations

### L1 human

An algorithm is a finite list of exact instructions that a machine can follow.
Computability theory asks which questions about numbers can be answered by such a list, with no limit on time or memory.
Checking whether a number is prime is computable: try all smaller divisors.
Deciding whether an arbitrary program stops on a given input is not computable, and this can be proved.
The theory also measures how far an unsolvable problem is from being solvable.
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# Dispersive partial differential equation

## Definition

Let $n \ge 1$.
Consider a linear evolution [[partial differential equation|GC5BDZ6QBC]] with constant coefficients for $u \colon \R^n \times \R \to \C$.
Suppose that its plane-wave solutions $u(x, t) = e^{i(\xi \cdot x - \omega t)}$ are exactly those with $\omega = \omega_j(\xi)$ for finitely many functions $\omega_j \colon \R^n \to \C$.
These functions are the branches of the dispersion relation.
The equation is *dispersive* if:

1. each $\omega_j(\xi)$ is real for all real $\xi$, so plane waves neither grow nor decay;
2. for some $j$, the [[Hessian matrix|0693Y2METT]] of $\omega_j$ is not identically zero, so the group velocity $\nabla \omega_j(\xi)$ depends on $\xi$.

A nonlinear equation is called dispersive when its linear part is.

## Remarks

Example: the Schrödinger equation $i u_t + \Delta u = 0$ has $\omega(\xi) = \abs{\xi}^2$, which is real, with Hessian $2I \ne 0$, so it is dispersive.
The Airy equation $u_t + u_{xxx} = 0$, the linear part of the Korteweg–De Vries equation, has $\omega(\xi) = -\xi^3$, with $\omega''(\xi) = -6\xi$, so it is dispersive.
Non-example: the transport equation $u_t + c u_x = 0$ has $\omega(\xi) = c\xi$, whose second derivative vanishes, so clause 2 fails; solutions move without changing shape.
The heat equation $u_t = \Delta u$ has $\omega(\xi) = -i\abs{\xi}^2$, which is not real, so clause 1 fails; it is dissipative rather than dispersive.
The one-dimensional wave equation, with $\omega = \pm c\xi$, is not dispersive.
In dimension $n \ge 2$ the branches $\pm c\abs{\xi}$ have nonzero Hessian, so the wave equation meets clause 2, although it is often not called dispersive.
By the [[Fourier transform|DPVM34ADZV]], the solution is $\hat u(\xi, t) = e^{-i\omega(\xi) t}\hat u(\xi, 0)$ for a single branch, and a nondegenerate Hessian, as for the Schrödinger equation, is what yields decay estimates in $L^\infty$.
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# Kolmogorov–Sinai entropy

## Summary

The Kolmogorov–Sinai entropy, or metric entropy, of a [[measure-preserving transformation|556WBDAHNP]] measures how much new information each step of the map produces.
It is defined from the Shannon entropy of finite partitions, refined along the orbit and taken per step.
It is an isomorphism invariant.
It tells apart systems with the same spectral data, such as two Bernoulli shifts whose probability vectors have different entropies.
By the [[Ornstein isomorphism theorem|8H4GF58E9M]] it is a complete invariant for Bernoulli shifts.

## Explanations

### L1 human

Watch a system through a coarse lens that only tells you in which of a few boxes the point is.
Record the sequence of boxes visited over many steps.
The entropy is the number of bits per step that you need, in the long run, to write down this record, for the best choice of boxes.
Tossing a fair coin produces $\log 2$ of entropy per step, one bit.
A rotation of a circle produces none, because after a while the past record predicts the future.
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<document source="https://tamnd.github.io/math2/node/X1DB08QJHK.md" tag="X1DB08QJHK" kind="definition">
# Fractional iteration

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a [[function|0EPAT51DXQ]].
A *fractional iteration* of $f$ is a family $(f^t)_{t \ge 0}$ of functions $f^t \colon X \to X$ such that:

1. $f^0 = \id_X$, the [[identity function|0PSKT5W817]] of $X$.
2. $f^1 = f$.
3. $f^s \circ f^t = f^{s+t}$ for all real $s, t \ge 0$.

## Remarks

Example: on $X = (0, \infty)$, let $f(x) = \frac{x}{1 + x}$ and $f^t(x) = \frac{x}{1 + tx}$.
Clauses 1 and 2 hold by substitution, and clause 3 holds since $f^s(f^t(x)) = \frac{x/(1+tx)}{1 + sx/(1+tx)} = \frac{x}{1 + (s+t)x}$.

Non-example: on $X = \R$, the family $f^t(x) = (1 + t) x$ satisfies clauses 1 and 2 for $f(x) = 2x$, but fails clause 3, since $f^1(f^1(x)) = 4x \ne 3x = f^2(x)$.

Clause 3 with integer $s, t$ forces $f^n$ to be the usual [[iterate|AYVTR2ES5B]], and $f^{1/n}$ is an [[iterative root|27C6J3YNZY]] of order $n$.
In terms of $F(x, t) = f^t(x)$, clause 3 is the [[translation equation|CPE1ANDDJ0]] $F(F(x, t), s) = F(x, s + t)$.
If $\Psi \colon X \to \R$ is a bijective solution of [[Schröder's equation|VTJV7JVJEA]] $\Psi \circ f = \lambda \Psi$ with $\lambda > 0$, then $f^t = \Psi^{-1} \circ (\lambda^t \Psi)$ is a fractional iteration.
If $A \colon X \to \R$ is a bijective solution of the [[Abel equation|68SHB7Z721]] $A \circ f = A + 1$, then $f^t = A^{-1} \circ (A + t)$ is one.
Some authors also call a family indexed by all real $t$, or only by rational $t$, a fractional iteration.
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# Large cardinal

## Summary

A large cardinal is an infinite [[cardinal number|0EBACQFAZW]] with a property that makes it very large.
If ZFC is consistent, ZFC cannot prove that such cardinals exist.
Examples are [[inaccessible|NW4DREQ5DQ]], [[Mahlo|CJFEPPP9KA]], [[weakly compact|1RHTVMH34G]], [[measurable|AKBWT323RQ]], [[Woodin|CMC2C1D53W]] and [[supercompact|91GE4C1BC5]] cardinals.
There is no precise definition of the term; it names a family of axioms that is, as far as is known, ordered by consistency strength.
Large cardinal axioms measure the strength of other statements, and they decide many questions about sets of reals, such as [[projective determinacy|2GY3131ZPS]].

## Explanations

### L1 human

The usual axioms of set theory give infinitely many sizes of infinity, one after another.
A large cardinal is a size of infinity so big that the usual axioms cannot reach it from below.
For example, an inaccessible cardinal cannot be reached by taking power sets or unions of fewer sets.
Adding a large cardinal as a new axiom gives a stronger theory, and set theorists compare statements by asking which large cardinal they need.
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<document source="https://tamnd.github.io/math2/node/X1GZ9WKQEW.md" tag="X1GZ9WKQEW" kind="definition">
# Wolff potential

## Definition

Let $n \ge 1$, let $1 < p < \infty$ and let $0 < \alpha < n/p$.
Let $\mu$ be a positive [[Radon measure|HZNEQQBCW4]] on $\R^n$, and let $B(x, t)$ be the open ball of center $x$ and radius $t$.
The *Wolff potential* of $\mu$ is the function $W^{\mu}_{\alpha, p} \colon \R^n \to [0, \infty]$ given by
$$W^{\mu}_{\alpha, p}(x) = \int_0^\infty \left( \frac{\mu\big(B(x, t)\big)}{t^{\,n - \alpha p}} \right)^{1/(p - 1)} \frac{dt}{t}.$$
For $R > 0$, the *truncated Wolff potential* $W^{\mu}_{\alpha, p}(x, R)$ is the same integral over $0 < t < R$.

## Remarks

Example: let $\mu = \delta_0$ be the unit point mass at $0$.
For $x \ne 0$, $\mu(B(x, t)) = 0$ when $t \le \abs{x}$ and $= 1$ when $t > \abs{x}$, so
$$W^{\delta_0}_{\alpha, p}(x) = \int_{\abs{x}}^\infty t^{-\frac{n - \alpha p}{p - 1} - 1} \, dt = \frac{p - 1}{n - \alpha p}\, \abs{x}^{-\frac{n - \alpha p}{p - 1}},$$
while $W^{\delta_0}_{\alpha, p}(0) = \infty$.
For $p = 2$, exchanging the order of integration gives $W^{\mu}_{\alpha, 2}(x) = \frac{1}{n - 2\alpha} \int \abs{x - y}^{2\alpha - n} \, d\mu(y)$, a multiple of the [[Riesz potential|1V2N8MFSHC]] of order $2\alpha$ of $\mu$.
The condition $\alpha < n/p$ is needed: if $\alpha p \ge n$ and $\mu \ne 0$, then $\mu(B(x, t))$ is bounded below for large $t$, the integrand does not decay fast enough, and $W^{\mu}_{\alpha, p} = \infty$ everywhere.
The truncated potential has no such problem and is used for local estimates.
The case $\alpha = 1$ is the one tied to the $p$-Laplace equation.
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# Anosov diffeomorphism

## Definition

Let $M$ be a [[compact|0QPHW703PV]] Riemannian manifold and let $f \colon M \to M$ be a $C^1$ [[diffeomorphism|0Z1Y7JHGBR]].
Then $f$ is an *Anosov diffeomorphism* if $M$ itself is a [[hyperbolic set|NY9MCZXY6N]] of $f$.
That is, there is a $Df$-invariant splitting $T_x M = E^s_x \oplus E^u_x$ for all $x \in M$, and there are constants $C > 0$ and $0 < \lambda < 1$.
For all $n \ge 0$, they satisfy $\norm{Df^n v} \le C\lambda^n \norm{v}$ for $v \in E^s$ and $\norm{Df^{-n} v} \le C\lambda^n\norm{v}$ for $v \in E^u$.

## Remarks

The property does not depend on the choice of Riemannian metric, since $M$ is compact.
Example: let $A$ be a $2 \times 2$ integer matrix with $\det A = \pm 1$ and no eigenvalue of modulus $1$, and let $f_A$ be the induced map of the torus $\R^2/\Z^2$.
Then $Df_A = A$ at every point, and the eigenlines of $A$ give $E^s$ and $E^u$ with $C = 1$ after a linear change of metric.
For $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ the eigenvalues are $(3 \pm \sqrt5)/2$, which gives [[Arnold's cat map|BSCZN8Y1AN]].
Non-example: for $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ the map $f_A$ is a diffeomorphism of the torus, but both eigenvalues equal $1$, so no direction is contracted and the condition fails.
All known Anosov diffeomorphisms are, up to topological conjugacy, hyperbolic automorphisms of tori, nilmanifolds or infranilmanifolds; whether this is always the case is an open question.
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# Schreier coset graph

## Definition

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $S$ be a subset of $G$.
The *Schreier coset graph* of $(G, H, S)$ is the [[directed graph|0KFYGSHHF5]] with labelled edges such that:

1. its vertices are the [[right cosets|0NE1DK7H4T]] $Hg$ of $H$ in $G$;
2. for each vertex $Hg$ and each $s \in S$, there is one edge with label $s$ from $Hg$ to $Hgs$.

## Remarks

The rule is well defined: if $Hg = Hg'$, then $g' = hg$ for some $h \in H$, so $Hg's = Hhgs = Hgs$.
The number of vertices is the index of $H$ in $G$, and every vertex has exactly one outgoing edge with each label.
If $S$ generates $G$, the graph is connected.
For $H = \set{1}$ it is the [[Cayley graph|0GYQK9NDJF]] of $G$ with respect to $S$.

Example: let $G = \Z$, $H = 3\Z$ and $S = \set{1}$.
The vertices are $3\Z$, $1 + 3\Z$ and $2 + 3\Z$, and the edges go $3\Z \to 1 + 3\Z \to 2 + 3\Z \to 3\Z$, since adding $1$ moves each coset to the next one.
The graph is a directed cycle of length $3$.

Non-example: the rule with left cosets $gH$ and edges $gH \to gsH$ is not well defined in general.
In $S_3$, with composition from right to left, take $H = \set{1, (1\,2)}$ and $s = (2\,3)$.
Then $1 \cdot H = (1\,2) H$, but $(2\,3) H = \set{(2\,3), (1\,3\,2)}$ and $(1\,2)(2\,3) H = (1\,2\,3) H = \set{(1\,2\,3), (1\,3)}$ are different cosets.
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<document source="https://tamnd.github.io/math2/node/X24AGT6TK9.md" tag="X24AGT6TK9" kind="concept">
# Horn hypergeometric series

## Summary

A Horn hypergeometric series is a double power series $\sum_{m, n} A_{m, n} x^m y^n$ with a special kind of coefficients.
Moving one step in either index multiplies the coefficient by a rational function of $(m, n)$.
This is the two-variable form of the defining property of the Gauss [[hypergeometric series|428M453CR2]], whose consecutive coefficients have a rational ratio.
Horn classified the convergent series of this type of order two.
His list has 34 essentially distinct series, including the four Appell series and their confluent forms.

## Explanations

### L1 human

In one variable, a hypergeometric series is a power series where each coefficient is the previous one times a fixed rational function of the index.
In two variables the coefficients form a grid, and one can step right (in $m$) or up (in $n$).
Horn's condition asks that each of the two steps multiplies the coefficient by a rational function of the position.
For instance $A_{m, n} = \frac{(m + n)!}{m!\, n!}$, the coefficients of $\sum (x + y)^N$, give the ratios $\frac{m + n + 1}{m + 1}$ and $\frac{m + n + 1}{n + 1}$.
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# Analytic capacity

## Definition

Let $K \subset \C$ be a [[compact|0QPHW703PV]] set.
Let $\mathcal{A}(K)$ be the set of [[holomorphic|0DG0Z72CQD]] functions $f \colon \C \setminus K \to \C$ with $\abs{f(z)} \le 1$ for all $z$ and $f(z) \to 0$ as $z \to \infty$.
For $f \in \mathcal{A}(K)$ put $f'(\infty) = \lim_{z \to \infty} z f(z)$; this limit exists.
The *analytic capacity* of $K$ is
$$\gamma(K) = \sup \set{\abs{f'(\infty)} : f \in \mathcal{A}(K)}.$$

## Remarks

The limit $f'(\infty)$ exists because $w \mapsto f(1/w)$ has a removable singularity at $w = 0$ with value $0$.
Example: for the closed disc $K = \set{\abs{z} \le r}$, the function $r/z$ lies in $\mathcal{A}(K)$ with $f'(\infty) = r$.
For any $f \in \mathcal{A}(K)$, the function $g(w) = f(r/w)$ extends to the unit disc with $g(0) = 0$ and $g'(0) = f'(\infty)/r$.
The [[Schwarz lemma|0MB48XDBY3]] gives $\abs{g'(0)} \le 1$, so $\gamma(K) = r$.
Non-example of positive capacity: for a single point $K = \set{a}$, each $f \in \mathcal{A}(K)$ extends to a bounded entire function by [[Riemann's theorem on removable singularities|0BSQDZWF34]].
By Liouville's theorem it is constant, hence $0$, so $\gamma(\set{a}) = 0$.
A segment of length $\ell$ has analytic capacity $\ell/4$.
Ahlfors proved that $\gamma(K) = 0$ if and only if every bounded holomorphic function on $U \setminus K$ extends holomorphically to $U$, for each open $U \supseteq K$.
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# Synthetic differential geometry

## Definition

Let $\mathcal{E}$ be a [[topos|6837RS7N82]] and let $R$ be a [[commutative ring|0310REJJY8]] object in $\mathcal{E}$.
Let $D = \set{ d \in R : d^2 = 0 }$ be the subobject of square-zero elements, and let $R^D$ be the object of maps from $D$ to $R$.
The pair $(\mathcal{E}, R)$ satisfies the *Kock–Lawvere axiom* if the map
$$\alpha \colon R \times R \to R^D, \qquad (a, b) \mapsto (d \mapsto a + b d)$$
is an isomorphism.
*Synthetic differential geometry* is the development of differential calculus and differential geometry inside such a pair $(\mathcal{E}, R)$.
Its reasoning uses the internal logic of $\mathcal{E}$, which is intuitionistic.

## Remarks

The axiom says that every map $g \colon D \to R$ is of the form $g(d) = a + bd$ for unique $a, b \in R$.
For $f \colon R \to R$ and $x \in R$, applying it to $d \mapsto f(x + d)$ defines the derivative $f'(x)$ by $f(x + d) = f(x) + d f'(x)$ for all $d \in D$.
Tangent vectors at a point $x$ of an object $M$ are maps $t \colon D \to M$ with $t(0) = x$.

Example: there are toposes containing the category of smooth manifolds in which the axiom holds, the well-adapted models.
They are built from [[C-infinity rings|4EJED0YZBB]]; for instance, the ring $C^\infty(\R)[\varepsilon]/(\varepsilon^2)$ plays the role of functions on $R \times D$.

Non-example: in the topos of sets, with $R = \R$, the axiom fails.
Since $\R$ has no [[nilpotent elements|0ZB2JG1M9C]] except $0$, $D = \set{0}$, so $R^D \cong R$, and $\alpha$ is not injective: $(0, 1)$ and $(0, 0)$ have the same image.
The axiom forces $D \ne \set{0}$, which is incompatible with classical logic in a field-like $R$; this is why the internal logic must be intuitionistic.
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# Stable ∞-category

## Summary

A stable ∞-category is an [[(∞,1)-category|YFY1FWA1MY]] with a [[zero object|AQET9YHH6F]] and with fibers and cofibers of all morphisms.
In it, fiber sequences and cofiber sequences coincide.
Its homotopy category is a [[triangulated category|T2V7GJJ2EN]] in a natural way.
Stable ∞-categories give a well-behaved replacement for triangulated categories: cones are functorial, and all finite limits and finite colimits exist.
Main examples are the ∞-category of [[spectra|1GHF4EKBXS]] and the derived ∞-category of a ring.

## Explanations

### L1 human

In a stable ∞-category, every map has a kernel-like object, the fiber, and a cokernel-like object, the cofiber.
Stability says that these two constructions fit together perfectly: a sequence is exact from the left exactly when it is exact from the right.
So the loop and suspension operations are inverse to each other, as in the world of chain complexes.
For chain complexes, the cofiber of a map is its mapping cone, and the fiber is the cone shifted down by one.
Pointed topological spaces are not stable: the loop space of the suspension of a space is in general not the space itself.
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# Regularization of ill-posed problems

## Definition

Let $X, Y$ be [[Hilbert spaces|HM2HHE3J51]], let $T \colon X \to Y$ be a bounded linear operator, and let $T^\dagger$ be its [[Moore–Penrose generalized inverse|0YY8VX28JR]], defined on $\mathcal{D}(T^\dagger) = \mathcal{R}(T) + \mathcal{R}(T)^\perp$.
A *regularization* of $T^\dagger$ is a family $(R_\alpha)_{\alpha > 0}$ of maps $Y \to X$ together with a *parameter choice rule* $\alpha \colon (0, \infty) \times Y \to (0, \infty)$ such that:

1. each $R_\alpha$ is continuous;
2. for every $y \in \mathcal{D}(T^\dagger)$, $\sup \set{ \alpha(\delta, y^\delta) : y^\delta \in Y, \ \norm{y^\delta - y} \le \delta } \to 0$ as $\delta \to 0$;
3. for every $y \in \mathcal{D}(T^\dagger)$, $\sup \set{ \norm{R_{\alpha(\delta, y^\delta)} y^\delta - T^\dagger y} : y^\delta \in Y, \ \norm{y^\delta - y} \le \delta } \to 0$ as $\delta \to 0$.

The pair is then called a *convergent regularization method*.

## Remarks

Example: [[Tikhonov regularization|M68V0M5A3R]] $R_\alpha = (T^* T + \alpha I)^{-1} T^*$ with an a priori rule $\alpha = \alpha(\delta)$ such that $\alpha(\delta) \to 0$ and $\delta^2 / \alpha(\delta) \to 0$.
Each $R_\alpha$ is bounded with $\norm{R_\alpha} \le 1/(2\sqrt{\alpha})$, which gives clause 1.
The error splits as $\norm{R_\alpha y^\delta - T^\dagger y} \le \norm{R_\alpha y - T^\dagger y} + \delta/(2\sqrt{\alpha})$.
The first term tends to $0$ as $\alpha \to 0$ by spectral calculus, and the second by the choice of $\alpha$, which gives clause 3.
Non-example: $R_\alpha = T^\dagger$ for all $\alpha$ fails clause 1 whenever $\mathcal{R}(T)$ is not closed, because $T^\dagger$ is then unbounded.
This happens for every injective [[compact operator|M2NEP1K5AE]] with infinite-dimensional range, the typical model of an ill-posed problem.
Other standard methods are truncated singular value decomposition, [[Landweber iteration|87DV5ZTJ9J]] with the iteration index as parameter, and the discrepancy principle as a parameter choice rule.

## Explanations

### L1 human

A regularization method replaces the unstable inverse by stable approximations, one for each value of a parameter.
The parameter is chosen from the noise level.
The method is convergent when the answers approach the true solution as the noise goes to zero.
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# Information-based complexity

## Definition

Let $F$ be a set, let $G$ be a [[normed vector space|03J5RAKB30]], and let $S \colon F \to G$ be a map, the *solution operator*.
Let $\Lambda$ be a set of maps $F \to \R$, the *admissible information functionals*, for example the evaluations $f \mapsto f(t)$.
For $n \ge 0$, an *algorithm using $n$ pieces of information* is a map $A = \phi \circ N$ where:

1. $N(f) = \big(L_1(f), \ldots, L_n(f)\big)$, with $L_1 \in \Lambda$ and each $L_k \in \Lambda$ chosen as a function of the values $L_1(f), \ldots, L_{k-1}(f)$;
2. $\phi$ is any map from the set of possible values of $N$ to $G$.

The *worst-case error* of $A$ is $e(A) = \sup_{f \in F} \norm{S(f) - A(f)}$.
The *$n$-th minimal error* is $e(n) = \inf_A e(A)$ over all algorithms $A$ using $n$ pieces of information.
For $\varepsilon > 0$, the *information complexity* of the problem is
$$n(\varepsilon) = \min \set{n \ge 0 : e(n) \le \varepsilon},$$
with $n(\varepsilon) = \infty$ if no such $n$ exists.
*Information-based complexity* is the study of $e(n)$ and $n(\varepsilon)$, and of the total cost $\mathrm{comp}(\varepsilon)$ that also counts the arithmetic operations of $\phi$.

## Remarks

Example: let $F$ be the set of functions $f \colon [0,1] \to \R$ with $\abs{f(s) - f(t)} \le \abs{s - t}$, let $S(f) = \int_0^1 f(t) \, dt \in \R$, and let $\Lambda$ be the function evaluations.
The midpoint rule $A(f) = \frac{1}{n} \sum_{k=1}^{n} f\big(\frac{2k-1}{2n}\big)$ has $e(A) = \frac{1}{4n}$, since on each subinterval of length $1/n$ the error is at most $\int \abs{t - c} \, dt = \frac{1}{4n^2}$, where $c$ is its midpoint.
No algorithm does better, so $e(n) = \frac{1}{4n}$ and $n(\varepsilon) = \ceil{1/(4\varepsilon)}$.
Non-example: the map $A(f) = \int_0^1 f$ itself has error $0$, but it uses infinitely many pieces of information.
Indeed two functions in $F$ that agree at finitely many points can have different integrals.
Besides the worst-case setting, there are average-case, probabilistic and randomized settings, where the supremum over $F$ is replaced by an expectation or the algorithm may use random choices.
Often $\Lambda$ is the class of all continuous [[linear functionals|07YQ1C78G0]] on a normed space containing $F$.
A problem is called *polynomially tractable* when $n(\varepsilon)$ for the $d$-variate version grows at most polynomially in $d$ and $1/\varepsilon$.
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# Complex torus

## Definition

Let $g \ge 1$ be an integer.
A *lattice* in $\C^g$ is a subgroup $\Lambda \subseteq \C^g$ generated by $2g$ vectors that are linearly independent over $\R$.
A *complex torus* of dimension $g$ is the [[quotient group|0KYX2N69J0]] $X = \C^g / \Lambda$ for a [[lattice|WN9V7WSJKF]] $\Lambda$ in $\C^g$.
It carries the complex manifold structure for which the projection $\C^g \to X$ is a local biholomorphism.

## Remarks

Example: $\Lambda = \Z + \Z \tau$ with $\tau \in \C$ and $\operatorname{Im} \tau > 0$.
The numbers $1$ and $\tau$ are linearly independent over $\R$, so $\Lambda$ is a lattice in $\C$, and $\C / \Lambda$ is a complex torus of dimension $1$.
Every complex torus of dimension $1$ is the set of complex points of an elliptic curve.

Non-example: $\Z \subseteq \C$ is a discrete subgroup, but it is generated by only one vector, not two.
The quotient $\C / \Z$ is not compact; the map $z \mapsto e^{2 \pi i z}$ identifies it with $\C^{*}$.

Let $X = \C^g / \Lambda$ be a complex torus.
Then $X$ comes from an [[abelian variety|R7W03TXRS9]] if and only if $\C^g$ has a positive definite Hermitian form $H$ with $\operatorname{Im} H(\Lambda \times \Lambda) \subseteq \Z$.
Such a form is called a Riemann form or a polarization.
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# Mellin's inverse formula

## Statement

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$, and let $\gamma$ be a real number with $\int_0^\infty \abs{f(t)} e^{-\gamma t} \, dt < \infty$.
Let $F$ be the [[Laplace transform|05XT5AVJBK]] of $f$.
Let $t > 0$, and assume that $f$ has [[bounded variation|F34S6P61SW]] on some interval around $t$.
Then the [[Bromwich integral|9WRE8KEJJX]] of $F$ along the line $\operatorname{Re} s = \gamma$, that is, the symmetric limit below, exists at $t$, and
$$\frac{1}{2 \pi i} \lim_{T \to \infty} \int_{\gamma - i T}^{\gamma + i T} e^{s t} F(s) \, ds = \frac{f(t^+) + f(t^-)}{2}.$$
In particular the integral equals $f(t)$ at each $t > 0$ where $f$ is continuous.
For every $t < 0$ the same integral exists and equals $0$.

## Remarks

Put $g(\tau) = f(\tau) e^{-\gamma \tau}$ for $\tau \ge 0$ and $g(\tau) = 0$ for $\tau < 0$; then $g$ is integrable on $\R$.
On the line $s = \gamma + i y$, one has $F(\gamma + i y) = \hat{g}(y) = \int_\R g(\tau) e^{-i y \tau} \, d\tau$, and the Bromwich integral is $e^{\gamma t} \cdot \frac{1}{2\pi} \lim_{T \to \infty} \int_{-T}^{T} \hat{g}(y) e^{i y t} \, dy$.
So the statement is the [[Fourier inversion theorem|T4Z8YQZKZV]] in its Dirichlet–Jordan form, applied to $g$.
Example: for $f(t) = e^{b t}$ with $b$ real and $\gamma > b$, $F(s) = 1/(s - b)$, and the residue of $e^{s t}/(s - b)$ at $b$ gives back $e^{b t}$ for $t > 0$.
At a jump of $f$ the integral gives the midpoint of the jump, not either one-sided value.

## Explanations

### L1 human

If you know the Laplace transform $F$ of a function, this formula gives the function back.
You integrate $e^{s t} F(s)$ along a vertical line far enough to the right in the complex plane.
At a jump, the formula returns the average of the values on the two sides.

### L2 human

The formula shows that the Laplace transform is a Fourier transform in disguise: along a vertical line it is the Fourier transform of $f(t) e^{-\gamma t}$.
In practice one rarely integrates along the line directly; one deforms the contour and uses residues, as in [[residue calculus for Laplace inversion|E14W2V09HM]].
The same formula, after the change of variables $x = e^{-t}$, gives the inversion theorem for the Mellin transform.
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# Separated morphism

## Summary

A separated morphism $X \to Y$ is a [[morphism of schemes|JXRKH6P1Z4]] whose diagonal $X \to X \times_Y X$ is a [[closed immersion|WWRWYSJDET]].
It is the scheme version of the Hausdorff property, which fails for the Zariski topology itself.
A scheme is separated if its morphism to $\operatorname{Spec} \Z$ is separated; all affine and all projective schemes are separated.

## Explanations

### L1 human

A topological space is Hausdorff exactly when the diagonal $\set{(x, x)}$ is closed in $X \times X$ with the product topology.
Zariski topologies are almost never Hausdorff, but the same diagonal test works when one uses the fiber product of schemes.
The standard failure is the line with two origins: two copies of the line glued everywhere except at $0$.
There the two copies of the line agree away from $0$ but not at $0$, much like a sequence with two limits, and the diagonal is not closed.
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# Formally real field

## Summary

A formally real field is a [[field|04C5ZBG1WT]] in which $-1$ is not a sum of squares.
By a theorem of Artin and Schreier, these are exactly the fields that admit an [[ordering|0P3VCBDEE2]].
They are the setting of real algebra, which studies the real numbers through algebraic properties alone.

## Explanations

### L1 human

In the real numbers, a sum of squares is never negative, so it is never $-1$.
A formally real field is a field with this one property of the real numbers.
The rational numbers and the field $\R(t)$ of real rational functions are formally real.
The complex numbers are not, because $-1 = i^2$.
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# Ordered vector space

## Definition

Let $V$ be a real [[vector space|05EGHEQZAW]] and let $\le$ be a [[partial order|011C29Q1S1]] on $V$.
The pair $(V, \le)$ is an *ordered vector space* if for all $x, y, z \in V$ and all real $\lambda \ge 0$:

1. if $x \le y$, then $x + z \le y + z$;
2. if $x \le y$, then $\lambda x \le \lambda y$.

## Remarks

Example: $\R^n$ with the componentwise order is an ordered vector space.
If $x_i \le y_i$ for all $i$, then $x_i + z_i \le y_i + z_i$ and $\lambda x_i \le \lambda y_i$ for $\lambda \ge 0$.
Non-example: on $\R$, let $x \preceq y$ when $x = y$ or $(x, y) = (0, 1)$.
This is a partial order, but $0 \preceq 1$ while $1 \not\preceq 2$, so clause 1 fails.
The set $\set{x \in V : 0 \le x}$ is a [[positive cone|WG6BY4SWTV]], and the order is recovered from it by $x \le y \iff y - x \ge 0$.
So giving an ordered vector space is the same as giving a real vector space with a positive cone.
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# Gelfand–Levitan–Marchenko equation

## Summary

The Gelfand–Levitan–Marchenko equation is a linear [[integral equation|R3SG1SN7JQ]] that recovers a potential $u$ in the one-dimensional Schrödinger operator $-\frac{d^2}{dx^2} + u(x)$ from its scattering data.
The data are the reflection coefficient, the bound state energies and their norming constants; they enter the equation through a single function $F$.
The solution $K(x, y)$ gives the potential by $u(x) = -2 \frac{d}{dx} K(x, x)$.
It is the central step of the [[inverse scattering transform|6GY3X4Y8X5]], which solves the [[Korteweg–De Vries equation|KTT75QC72Z]] and other integrable equations.

## Explanations

### L1 human

Send waves at an unknown bump in a potential and record what comes back: how much of each wave is reflected and which waves stay trapped.
The inverse problem asks for the bump from these echoes.
The Gelfand–Levitan–Marchenko equation turns this question into a linear integral equation, which is much easier than the nonlinear problem it replaces.
When there is no reflection and one trapped state, the equation can be solved by hand and gives the potential $-2\kappa^2 \operatorname{sech}^2(\kappa(x - x_0))$, the shape of a [[soliton|63D4AXYCT4]].
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# Semilinear PDE

## Definition

Let $\Omega$ be an open set in $\R^n$ and let $m \ge 1$.
A [[partial differential equation|GC5BDZ6QBC]] of [[order|E7JDFDPM1X]] $m$ in $\Omega$ is *semilinear* if it has the following form.
$$\sum_{\abs{\alpha} = m} a_\alpha(x) \, \partial^\alpha u + a_0\left(D^{m-1} u, \dots, D u, u, x\right) = 0.$$
Here the coefficients $a_\alpha$ of the highest-order derivatives depend on $x$ only, and $a_0$ is a given function, which may be nonlinear.
Here $\alpha$ runs over [[multi-indices|0PJXBR6QA3]], and $D^j u$ denotes the collection of all partial derivatives of order $j$.

## Remarks

Example: $\Delta u = u^3$ in $\R^3$, with $\Delta = \partial_1^2 + \partial_2^2 + \partial_3^2$, is semilinear of order $2$; its highest-order part $\Delta u$ has constant coefficients, and $a_0(u) = -u^3$.
Non-example: $\partial_t u + u \, \partial_x u = 0$ is not semilinear, because the coefficient $u$ of the highest-order derivative $\partial_x u$ depends on the unknown; it is [[quasilinear|7X9N9H3RK6]].
Every [[linear PDE|6DZETZ5T9G]] is semilinear, with $a_0$ affine in the lower-order derivatives.

## Explanations

### L1 human

Semilinear means linear in the highest derivatives, with coefficients that depend only on the position, and arbitrary in the rest.
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# Lie symmetry method

## Summary

The Lie symmetry method solves differential equations by using their continuous symmetries.
A Lie point symmetry is a one-parameter group of changes of variables that maps solutions to solutions.
In adapted coordinates, called canonical coordinates, the symmetry becomes a translation and the equation can be integrated by quadrature.
The method, due to Sophus Lie, explains most of the classical tricks for first-order equations, such as the substitution $y = x v$ for a [[homogeneous first-order equation|0RN88KZBJ3]].
See the [[definition|WZKRG7H3FX]].

## Explanations

### L1 human

Many solving tricks for differential equations look unrelated, but most of them use a hidden symmetry.
For example, the equation $y' = F(y/x)$ does not change if one scales $x$ and $y$ by the same factor.
So scaling a solution curve gives another solution curve.
In coordinates in which the scaling is a shift, such as $r = y/x$ and $s = \ln\abs{x}$, the equation no longer depends on $s$ and can be integrated directly.
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# Hearing the shape of a drum

## Problem

Let $\Omega_1$ and $\Omega_2$ be bounded domains in $\R^2$.
Let $\Delta$ be the [[Laplacian|0E0B6RA8K0]].
Suppose that $-\Delta$ with the [[Dirichlet boundary condition|EETVA6QYX4]] has the same [[eigenvalues|0BQAA2BK3Q]] on $\Omega_1$ and on $\Omega_2$, with the same multiplicities.
Must $\Omega_1$ and $\Omega_2$ be congruent?

## Remarks

Mark Kac popularised the question in his 1966 article "Can one hear the shape of a drum?".
The answer is no.
In 1964 John Milnor had already found two flat tori of dimension $16$ that have the same spectrum but are not isometric.
In 1992 Carolyn Gordon, David Webb and Scott Wolpert constructed two plane domains, both polygons, that have the same Dirichlet spectrum but are not congruent.
Some features can be heard: by [[Weyl's law|2JK9YMZV2B]] the spectrum determines the area of the domain, and for domains with smooth boundary it also determines the perimeter.
Among plane domains with smooth boundary, the disc is therefore determined by its spectrum, by the equality case of the isoperimetric inequality.

## Explanations

### L1 human

A drumhead of a given shape can play certain pitches, which are the eigenvalues of the Laplacian.
Kac asked whether a listener who knows all these pitches can tell the shape of the drum.
The answer is no: there are two differently shaped drums that sound exactly alike, though some features, such as the area, can be heard.
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# Exotic R4

## Description

An *exotic $\R^4$* is a [[smooth manifold|F45VQ47GPV]] that is [[homeomorphic|0QT00PB5G2]] to the [[Euclidean space|0XPR1E85W0]] $\R^4$ but not [[diffeomorphic|0KZZ70ZHXP]] to $\R^4$ with its standard smooth structure.
Exotic $\R^4$'s exist.
They were found in the early 1980s as a consequence of [[Freedman's classification|HSD6FKCYRC]] of topological $4$-manifolds and [[Donaldson's theorem|34B7K77P31]] on smooth ones.

## Remarks

For $n \ne 4$, every smooth manifold homeomorphic to $\R^n$ is diffeomorphic to $\R^n$, so dimension $4$ is the only exception.
Clifford Taubes showed that there are uncountably many pairwise non-diffeomorphic exotic $\R^4$'s.
Some of them, the small exotic $\R^4$'s, embed smoothly as open subsets of the standard $\R^4$; others, the large ones, do not.
The constructions use handle decompositions and Casson handles rather than explicit coordinates.

## Explanations

### L1 human

Four-dimensional space can be given smooth structures, ways of doing calculus, that are different from the usual one.
Each of them is the same as ordinary $\R^4$ for continuous deformation, but no smooth change of coordinates turns it into ordinary $\R^4$.
This happens in no other dimension, and it is one of the strangest facts of geometry.
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# Center manifold

## Summary

A center manifold of an [[equilibrium point|0N8NGKDZDZ]] is an [[invariant|WWZTW11664]] manifold through the equilibrium.
It is tangent to the directions where the linearisation neither grows nor decays.
The center manifold theorem says that such a manifold exists, though it need not be unique.
For an equation of class $C^k$ with finite $k$, it can be taken of class $C^k$.
The interesting dynamics near a non-hyperbolic equilibrium happens on the center manifold.
So the study of stability and bifurcations reduces to a system of lower dimension.

## Explanations

### L1 human

At an equilibrium, some directions pull solutions in fast, some push them out fast, and some do neither at the linear level.
The neutral directions are where the nonlinear terms decide what happens.
The center manifold is a curved surface along these neutral directions that solutions near the equilibrium follow.
For $x' = x^2$, $y' = -y$, the $y$-direction contracts and the $x$-axis is a center manifold, on which the motion is $x' = x^2$.
So all the delicate behaviour can be read from a one-dimensional equation.
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# Quadratic variation

## Summary

The quadratic variation of a process is the limit of the sums of squared increments over finer and finer partitions of a time interval.
For smooth paths it is zero, but for Brownian motion it equals the elapsed time.
It is the correction term in [[Itô's lemma|2J1RVQB0CM]] and a basic tool for [[semimartingales|4EJB6E2AZK]].

## Explanations

### L1 human

Cut the time interval $[0, t]$ into many small pieces, square the change of the path on each piece, and add.
For a smooth curve each change is about the length of the piece, so the squares are tiny and the sum goes to zero.
A Brownian path is so rough that its change over a piece of length $h$ is about $\sqrt{h}$.
Then each square is about $h$, and the sum tends to $t$.
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# Calabi–Weil rigidity

## Statement

Let $n \ge 3$ and let $G = \mathrm{SO}_0(n, 1)$ be the group of orientation-preserving isometries of hyperbolic space $\mathbb{H}^n$.
Let $\Gamma \subset G$ be a discrete torsion-free subgroup such that the hyperbolic manifold $M = \Gamma \backslash \mathbb{H}^n$ is compact.
Then the inclusion $\Gamma \to G$ is locally rigid.
That is, let $\rho \colon \Gamma \to G$ be a [[homomorphism|0JYTG5K9Y4]] close enough to the inclusion on a finite generating set of $\Gamma$.
Then $\rho$ is conjugate to the inclusion by an element of $G$.
Equivalently, the hyperbolic structure of $M$ admits no nontrivial small deformations.

## Remarks

Calabi proved this for compact hyperbolic manifolds, and Weil extended it to all uniform lattices in semisimple groups.
It follows from [[Weil's local rigidity theorem|V2HTMG4KT1]], since $\Gamma$ is a uniform [[lattice|BTYF5Y5PFY]] in $G$, which for $n \ge 3$ is simple, has trivial center and is not locally isomorphic to $\mathrm{SL}_2(\R)$.
The hypothesis $n \ge 3$ is necessary: for $n = 2$ the hyperbolic structures on a closed surface of genus $g \ge 2$ form a [[Teichmüller space|ZZGC6BZNJ5]] of real dimension $6g - 6$.
For $n = 3$ compactness is also needed: hyperbolic structures on cusped $3$-manifolds deform to incomplete structures, which give hyperbolic Dehn filling.

## Explanations

### L1 human

A closed surface of genus two can carry many different hyperbolic shapes, which can be changed continuously.
In dimension three and higher this freedom disappears.
A closed hyperbolic manifold cannot be bent slightly into a different hyperbolic manifold.
Its shape is locally fixed by its topology.

### L2 human

This theorem is the local version of [[Mostow rigidity|4BEWKR0EG7]], which states that the hyperbolic structure is determined by the fundamental group globally.
In the language of representations, the point of $\Hom(\Gamma, G)/G$ defined by $M$ is isolated.
The proof goes through the vanishing of the cohomology group $H^1(\Gamma, \mathfrak{g})$ with coefficients in the Lie algebra of $G$.
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# Tanh-sinh quadrature

## Statement

Let $\varphi \colon \R \to (-1, 1)$ be $\varphi(t) = \tanh\big(\tfrac{\pi}{2} \sinh t\big)$, so that
$$\varphi'(t) = \frac{\frac{\pi}{2} \cosh t}{\cosh^2\big(\frac{\pi}{2} \sinh t\big)}.$$
Then:

1. $\varphi$ is a smooth increasing bijection of $\R$ onto $(-1, 1)$, and for every Lebesgue integrable $f$ on $(-1, 1)$,
$$\int_{-1}^{1} f(x) \, dx = \int_{-\infty}^{\infty} f(\varphi(t)) \, \varphi'(t) \, dt;$$
2. if moreover $\abs{f(x)} \le M$ for all $x \in (-1, 1)$, then for all real $t$,
$$\abs{f(\varphi(t)) \, \varphi'(t)} \le 2 \pi M \cosh t \; e^{-\pi \sinh \abs{t}}.$$

The *tanh-sinh rule* with step $h > 0$ and $2N + 1$ points is $T_{h,N}(f) = h \sum_{k=-N}^{N} \varphi'(kh) \, f(\varphi(kh))$, the [[trapezoidal rule|FH0538XCZR]] for the transformed integral, cut off at $\abs{t} \le Nh$.

## Remarks

Clause 1 is the change of variables formula, since $\varphi' > 0$ and $\varphi(t) \to \pm 1$ as $t \to \pm\infty$.
Clause 2 follows from $\cosh u \ge \frac12 e^{\abs{u}}$, which gives $\cosh^{-2}\big(\frac{\pi}{2}\sinh t\big) \le 4 e^{-\pi \sinh\abs{t}}$.
So the transformed integrand decays double exponentially, and the cut-off at $\abs{t} \le Nh$ loses almost nothing; this is why the method is also called double exponential quadrature.
The trapezoidal rule on $\R$ is extremely accurate for functions analytic in a strip around the real axis.
For integrands analytic in a suitable region around $(-1, 1)$, the error of $T_{h,N}$ with a good choice of $h$ decreases roughly like $e^{-cN/\log N}$ for some $c > 0$.
Example: for $f(x) = (1 - x^2)^{-1/2}$, which is unbounded, the transformed integrand is $\frac{\pi}{2} \cosh t / \cosh\big(\frac{\pi}{2} \sinh t\big)$, still double exponentially decaying, and its integral is $\pi$.
The method was introduced by Hidetosi Takahashi and Masatake Mori in 1974.

## Explanations

### L1 human

Tanh-sinh quadrature stretches the interval $(-1, 1)$ to the whole real line with a clever change of variable.
After the change, the function to be integrated dies off extremely fast at both ends, even if the original one blew up at an endpoint.
Then the simplest rule, adding equally spaced samples, gives an excellent answer.

### L2 human

The method is robust against endpoint singularities, which defeat Gaussian rules, and it reuses all old samples when the step $h$ is halved.
For these reasons it is a standard choice in arbitrary-precision computation, where hundreds of correct digits are needed.
It needs care near the endpoints in floating-point arithmetic, since $\varphi(kh)$ is extremely close to $\pm 1$ and $1 \mp \varphi(kh)$ should be computed directly.
Related double exponential substitutions treat integrals over $[0, \infty)$ and $\R$.
</document>

<document source="https://tamnd.github.io/math2/node/X4M2FFKQJ6.md" tag="X4M2FFKQJ6" kind="statement">
# Schur's lemma

## Statement

Let $R$ be a ring and let $M$ and $N$ be [[simple|G00C22AEDH]] left [[$R$-modules|09VDS2W1E8]].
Let $f \colon M \to N$ be a homomorphism of $R$-modules.
Then either $f = 0$ or $f$ is an isomorphism.

## Remarks

The kernel of $f$ is a submodule of $M$, so it is $0$ or $M$.
The image of $f$ is a submodule of $N$, so it is $0$ or $N$.
If $f \ne 0$, the kernel is not $M$ and the image is not $0$, so $f$ is injective and surjective.
Example: a nonzero group homomorphism $\Z/p\Z \to \Z/p\Z$, for $p$ prime, is a bijection.
Simplicity is needed: the map $\Z/4\Z \to \Z/4\Z$, $x \mapsto 2x$, is nonzero and not an isomorphism.
A common form concerns an algebraically closed field $k$, a $k$-algebra $A$, and a simple $A$-module $M$ of finite dimension over $k$.
Then every endomorphism of $M$ is multiplication by a scalar.

## Explanations

### L1 human

A simple module has no proper nonzero pieces.
A map between two simple modules therefore has no room to be "partly" zero: it either kills everything or matches the two modules exactly.

### L2 human

Schur's lemma is the first tool of representation theory.
It shows that the endomorphism ring of a simple module is a division ring.
For representations of a group over $\C$ it gives that the endomorphisms of an irreducible representation are scalars.
This is the key step in the orthogonality relations for characters and in the decomposition of semisimple modules into isotypic parts.
</document>

<document source="https://tamnd.github.io/math2/node/X4R2GY1F6J.md" tag="X4R2GY1F6J" kind="statement">
# Mittag-Leffler's theorem

## Statement

Let $U \subseteq \C$ be an [[open set|01ZQQDCERP]], and let $A \subseteq U$ be a set with no accumulation point in $U$.
For each $a \in A$, let
$$p_a(z) = \sum_{k=1}^{N_a} \frac{c_{a,k}}{(z - a)^k}$$
be a nonzero finite sum, with $N_a \ge 1$ and complex coefficients $c_{a,k}$.
Then there is a [[meromorphic function|0NSD1TPHR0]] $f$ on $U$ such that:

1. $f$ is [[holomorphic|0DG0Z72CQD]] on $U \setminus A$;
2. for each $a \in A$, the function $f - p_a$ is holomorphic near $a$.

In particular $f$ has a [[pole|0WPQYW8MTE]] at each $a \in A$ with principal part $p_a$.

## Remarks

For $U = \C$ the proof is explicit.
List $A$ as $a_1, a_2, \dots$ with $\abs{a_n}$ nondecreasing, and subtract from $p_{a_n}$ a Taylor polynomial $q_n$ at $0$ (empty when $a_n = 0$) so that $\abs{p_{a_n} - q_n} \le 2^{-n}$ on the disc $\abs{z} \le \abs{a_n}/2$.
Then $\sum_n (p_{a_n} - q_n)$ converges uniformly on compact sets away from $A$ and gives $f$.
For a general open set, Runge's theorem on rational approximation replaces the Taylor polynomials.
A classical example is
$$\frac{\pi^2}{\sin^2(\pi z)} = \sum_{n \in \Z} \frac{1}{(z - n)^2},$$
where here the plain sum already converges.

## Explanations

### L1 human

The theorem lets one place poles at any points that do not pile up inside the region, with any chosen principal parts.
Summing the principal parts directly may diverge, so each term gets a correction that is holomorphic and does not change the pole.
It is the counterpart for poles of the [[Weierstrass factorization theorem|11VC51YHQJ]], which places zeros.

### L2 human

The function $f$ is not unique: one can add any holomorphic function on $U$.
The theorem gives partial fraction expansions of meromorphic functions, such as $\pi \cot(\pi z) = 1/z + \sum_{n \ne 0} \left(1/(z-n) + 1/n\right)$.
Combined with Weierstrass products, it shows that one can prescribe both the principal parts at $A$ and finitely many Taylor coefficients at other points.
In the language of sheaf cohomology, it says that the first Cousin problem is solvable on every open subset of $\C$; see also [[Runge's theorem|6DG4CTGXJD]].
</document>

<document source="https://tamnd.github.io/math2/node/X4V2S56YQQ.md" tag="X4V2S56YQQ" kind="concept">
# Superharmonic function

## Summary

A superharmonic function is a lower semicontinuous function whose value at each point is at least its average over spheres around that point.
A function $u$ is superharmonic exactly when $-u$ is [[subharmonic|Z1MHMGFNFW]], so the two theories are mirror images.
The potentials of positive charges, such as the Newtonian kernel $\abs{x}^{2-n}$ in $\R^n$ for $n \ge 3$, are the basic examples.
Superharmonic functions form the upper class of the Perron method and are the main objects of classical [[potential theory|9PM8Q2E30Q]].

## Explanations

### L1 human

A [[harmonic function|0DEB0JMN0W]] equals its average over every small sphere around a point.
A superharmonic function sits at or above that average, like a tent pole pushing a membrane up from below.
It cannot have a strict local minimum inside its domain.
The gravitational potential $1/\abs{x}$ of a point mass in space is superharmonic, with the value $+\infty$ at the mass.
At the origin the average of $1/\abs{x}$ over the sphere of radius $r$ is $1/r$, which is below the value $+\infty$.
</document>

<document source="https://tamnd.github.io/math2/node/X4ZHQG9995.md" tag="X4ZHQG9995" kind="concept">
# Lelong class

## Summary

The Lelong class $\mathcal{L}$ is the set of [[plurisubharmonic functions|FF2P7ER3QT]] on $\C^n$ that grow at most like $\log\abs{z}$ at infinity.
It is the analogue, for functions, of the set of polynomials of bounded degree.
For every polynomial $p$ of degree $d \ge 1$, the function $\frac{1}{d} \log\abs{p}$ lies in $\mathcal{L}$.
The [[Siciak extremal function|QBJ9R06GYE]] of a set is defined as an upper envelope of functions of the Lelong class.
The subclass $\mathcal{L}^+$ of functions that also grow at least like $\log\abs{z}$ contains the regularised extremal functions of compact sets that are not pluripolar.

## Explanations

### L1 human

A polynomial of degree $d$ grows like $\abs{z}^d$, so its logarithm divided by $d$ grows like $\log\abs{z}$.
The Lelong class collects all plurisubharmonic functions with this kind of logarithmic growth.
For example, $\log(1 + \abs{z})$ and $\max(\log\abs{z_1}, \log\abs{z_2})$ are in the Lelong class of $\C^2$.
The function $\abs{z_1}$ is plurisubharmonic but grows too fast, so it is not in the class.
</document>

<document source="https://tamnd.github.io/math2/node/X4ZZKC319Q.md" tag="X4ZZKC319Q" kind="definition">
# Compositum of fields

## Definition

Let $\Omega$ be a field and let $E$ and $F$ be [[subfields|07QY5RPFBR]] of $\Omega$.
The *compositum* $EF$ of $E$ and $F$ is the intersection of all subfields of $\Omega$ that contain $E \cup F$.
More generally, the compositum of a family of subfields of $\Omega$ is the intersection of all subfields of $\Omega$ containing every member.

## Remarks

The compositum $EF$ is the smallest subfield of $\Omega$ containing $E$ and $F$, so $EF = E(F) = F(E)$ in the notation of [[fields generated by adjoining elements|0MA5B744K3]].
Its elements are the quotients $a/b$ with $b \ne 0$, where $a$ and $b$ are finite sums of products $xy$ with $x \in E$ and $y \in F$.
If $K \subseteq E \cap F$ and $E/K$ is finite, then $[EF:F] \le [E:K]$.

Example: in $\Omega = \C$, the compositum of $E = \Q(\sqrt{2})$ and $F = \Q(\sqrt{3})$ is $\Q(\sqrt{2}, \sqrt{3})$.
Check: this field contains $E$ and $F$, and any subfield of $\C$ containing $\sqrt{2}$ and $\sqrt{3}$ contains $\Q(\sqrt{2}, \sqrt{3})$.

Non-example: the union $E \cup F$ is not the compositum, because it is not a field.
Indeed $\sqrt{2} + \sqrt{3}$ lies in neither $E$ nor $F$.
</document>

<document source="https://tamnd.github.io/math2/node/X54RV6SENQ.md" tag="X54RV6SENQ" kind="concept">
# Rees matrix semigroup

## Summary

A Rees matrix semigroup is a [[semigroup|0XMEM406JM]] built from a [[group|G5S1M9]] $G$ and two index sets $I$ and $\Lambda$.
It also uses a matrix $P$ with entries in $G$.
Its elements are triples $(i, g, \lambda)$, and the matrix $P$ is inserted in the middle of each product.
Rees's theorem says that the completely simple semigroups are exactly the Rees matrix semigroups over groups.
A variant with a zero element describes the completely $0$-simple semigroups, which are the building blocks of finite semigroups.

## Explanations

### L1 human

A Rees matrix semigroup is like a grid of copies of a group.
The rows are labelled by $I$ and the columns by $\Lambda$.
When you multiply two elements, you keep the row of the first and the column of the second.
You multiply the group parts with a twist taken from the matrix $P$.
If the group is trivial, a product just keeps the row of the left factor and the column of the right factor.
</document>

<document source="https://tamnd.github.io/math2/node/X5GS407VP8.md" tag="X5GS407VP8" kind="statement">
# Colorful Carathéodory theorem

## Statement

Let $d \ge 1$ and let $C_1, \dots, C_{d+1}$ be subsets of $\R^d$.
Let $x \in \R^d$ be a point that lies in the [[convex hull|0FQSDC2EQ2]] of each $C_i$, for $i = 1, \dots, d + 1$.
Then there are points $c_1 \in C_1, \dots, c_{d+1} \in C_{d+1}$ such that
$$x \in \operatorname{conv}\set{c_1, \dots, c_{d+1}}.$$

## Remarks

Proof idea: by [[Carathéodory's theorem|0201Q4P2B5]] one may assume each $C_i$ is finite.
Among all choices of $c_1 \in C_1, \dots, c_{d+1} \in C_{d+1}$, pick one for which the distance from $x$ to $\operatorname{conv}\set{c_1, \dots, c_{d+1}}$ is smallest.
If this distance were positive, the nearest point $y$ would lie in the hull of at most $d$ of the $c_j$, so some $c_i$ could be dropped.
The hyperplane through $y$ perpendicular to $x - y$ has all the $c_j$ on one side.
Since $x \in \operatorname{conv}(C_i)$, some point of $C_i$ lies strictly on the side of $x$.
Replacing $c_i$ by that point brings the hull closer to $x$, a contradiction.

When all the $C_i$ are the same set $C$, the theorem is Carathéodory's theorem for $C$.

Example: in the plane, let $C_1$, $C_2$, $C_3$ be red, green and blue point sets, each with the origin in its convex hull.
Then some triangle with one red, one green and one blue vertex contains the origin.

## Explanations

### L1 human

Colour points with $d + 1$ colours, and suppose that the points of each colour surround a given point $x$.
Then you can pick one point of each colour so that these $d + 1$ points already surround $x$.
In the plane: three colour classes each surrounding $x$ give a rainbow triangle around $x$.

### L2 human

The theorem is due to Imre Bárány.
It is the key tool in Sarkaria's proof of Tverberg's theorem, and Bárány used it to prove the first selection lemma on simplices that contain a common point.
It also has a colourful counterpart for Helly's theorem.
</document>

<document source="https://tamnd.github.io/math2/node/X5HMTTGB62.md" tag="X5HMTTGB62" kind="statement">
# Milnor fibre is a bouquet of spheres

## Statement

Let $n \ge 1$ and let $f$ be a holomorphic function near $0$ in $\C^{n+1}$ with an [[isolated singularity|VTGYEES3VS]] at $0$.
Let $F$ be its Milnor fibre, the fibre of the [[Milnor fibration|KTFG1S2SMG]] of $f$ at $0$.
Let $\mu$ be the [[Milnor number|1ZE51P7GRQ]] of $f$ at $0$.
Then $F$ is [[homotopy equivalent|XZG6MTSWCN]] to a [[wedge sum|5JSWM1HBX2]] of $\mu$ copies of the [[sphere|P9CM4GQ2T0]] $S^n$.

## Remarks

The proof has three steps.
Morse theory shows that $F$ has the homotopy type of a finite CW complex of dimension at most $n$.
Milnor shows that, for an isolated singularity, $F$ is $(n-1)$-connected.
A finite CW complex of dimension $n$ that is $(n-1)$-connected is homotopy equivalent to a wedge of $n$-spheres.
For $n \ge 2$ this follows from the Hurewicz and Whitehead theorems, and for $n = 1$ a connected graph is homotopy equivalent to a wedge of circles.
The number of spheres is the rank of $H_n(F; \Z)$, which Milnor identifies with $\mu$ through the degree of the gradient map.

Example: for $f = z_0^2 + \cdots + z_n^2$, $\mu = 1$, and the fibre $\set{z_0^2 + \cdots + z_n^2 = \delta}$ near $0$ is homotopy equivalent to the single sphere $\set{x \in \R^{n+1} : x_0^2 + \cdots + x_n^2 = \delta}$ for $\delta > 0$.

The hypothesis that the singularity is isolated is needed.
For $f(z_0, z_1, z_2) = z_0 z_1$ on $\C^3$, the Milnor fibre is homotopy equivalent to a circle, not to a wedge of $2$-spheres.

## Explanations

### L1 human

The Milnor fibre is a smooth model of a singular hypersurface near its singular point.
The theorem says that, up to continuous deformation, this model is just a bunch of spheres of the middle dimension glued at one point.
The number of spheres is the Milnor number, which can be computed by algebra.

### L2 human

The theorem shows that the reduced homology of the Milnor fibre is concentrated in degree $n$ and is free of rank $\mu$.
So the monodromy of the [[Milnor fibration|KTFG1S2SMG]] is described by a single $\mu \times \mu$ integer matrix, and the intersection form on $H_n(F; \Z)$ is a $\mu \times \mu$ integer form.
A basis of $H_n(F; \Z)$ can be chosen of [[vanishing cycles|M5EK9AKXNR]], one for each nondegenerate critical point of a generic perturbation of $f$.
This is the setting of [[Picard–Lefschetz theory|FFYQAVT7MB]].
</document>

<document source="https://tamnd.github.io/math2/node/X5JYH26J3M.md" tag="X5JYH26J3M" kind="example">
# Laplace transform of the exponential function

## Example

Let $a \in \C$ and $f(t) = e^{at}$ for $t \ge 0$, an [[exponential function|0RMVS7RXZX]].
Its [[Laplace transform|05XT5AVJBK]] is
$$\mathcal{L}\{e^{at}\}(s) = \frac{1}{s - a} \quad (\operatorname{Re} s > \operatorname{Re} a).$$
Check: for $s \ne a$ and $T > 0$,
$$\int_0^T e^{at} e^{-st}\, dt = \frac{1 - e^{-(s - a)T}}{s - a}.$$
Since $\abs{e^{-(s - a)T}} = e^{-\operatorname{Re}(s - a) T}$, this tends to $0$ as $T \to \infty$ exactly when $\operatorname{Re} s > \operatorname{Re} a$, which gives the formula.
When $\operatorname{Re} s < \operatorname{Re} a$ the term $e^{-(s-a)T}$ grows without bound.
When $\operatorname{Re} s = \operatorname{Re} a$ it stays on the unit circle without a limit (or the integrand is $1$ if $s = a$), so the integral diverges.
So the abscissa of convergence is $\operatorname{Re} a$.
The case $a = 0$ gives $\mathcal{L}\{1\}(s) = 1/s$.

## Explanations

### L1 human

The exponential $e^{at}$ is the basic building block of Laplace transform tables.
Its transform $1/(s - a)$ has a single pole at $s = a$, and the location of the pole records the growth rate of $e^{at}$.
Many other entries, such as those for sine, cosine and hyperbolic functions, follow from this one by linearity.
</document>

<document source="https://tamnd.github.io/math2/node/X5KD3X3VJ3.md" tag="X5KD3X3VJ3" kind="definition">
# Algorithmic randomness

## Definition

Let $2^\omega$ be [[Cantor space|W9VYJRBF9N]], the set of infinite binary sequences.
For a finite binary string $\sigma$ of length $\abs{\sigma}$, let $[\sigma] \subseteq 2^\omega$ be the set of sequences that begin with $\sigma$.
Let $\mu$ be the uniform measure on $2^\omega$, with $\mu([\sigma]) = 2^{-\abs{\sigma}}$.
A *Martin-Löf test* is a sequence $(U_n)_{n \in \N}$ of subsets of $2^\omega$ such that:

1. there is a [[computably enumerable|TN9G24VABN]] set $W$ of pairs $(n, \sigma)$ with $U_n = \bigcup \set{[\sigma] : (n, \sigma) \in W}$ for every $n$;
2. $\mu(U_n) \le 2^{-n}$ for every $n$.

A sequence $X \in 2^\omega$ is *Martin-Löf random*, or *algorithmically random*, if $X \notin \bigcap_n U_n$ for every Martin-Löf test $(U_n)$.

## Remarks

Example: the binary expansion of Chaitin's halting probability $\Omega$ is Martin-Löf random.
Also, the set of non-random sequences has measure $0$, because there is a universal Martin-Löf test whose intersection contains every non-random sequence.

Non-example: a computable sequence $X$ is not random.
Let $X \upharpoonright n$ be the string of the first $n$ bits of $X$.
The sets $U_n = [X \upharpoonright n]$ form a Martin-Löf test: they are uniformly computable and $\mu(U_n) = 2^{-n}$.
And $X$ lies in all of them.

The Levin–Schnorr theorem gives an equivalent form with [[Kolmogorov complexity|8PY3K6NCKF]].
Let $K$ be prefix-free complexity.
Then $X$ is Martin-Löf random if and only if there is a constant $c$ with $K(X \upharpoonright n) \ge n - c$ for all $n$.
</document>

<document source="https://tamnd.github.io/math2/node/X5R7FV9KWH.md" tag="X5R7FV9KWH" kind="concept">
# Nörlund mean

## Summary

A Nörlund mean is a weighted average of the first terms of a sequence.
Its weights are read backwards from a fixed sequence $p_0, p_1, p_2, \dots$.
It defines a [[summability method|8X25Q4XYMN]] for sequences and, through partial sums, for series.
With all weights equal to $1$ it is the [[Cesàro mean|08R2V9GZF7]].
The method gives the usual limit to every convergent sequence exactly when $p_n$ is small compared with $p_0 + \cdots + p_n$.

## Explanations

### L1 human

To average $s_0, \dots, s_n$, the Nörlund mean gives the weight $p_0$ to the newest term $s_n$, the weight $p_1$ to $s_{n-1}$, and so on back to $p_n$ for $s_0$.
Then it divides by the total weight.
With all weights equal it is the plain average.
Weights that grow, such as $p_k = k + 1$, give more weight to the early terms of the sequence than the plain average does.
</document>

<document source="https://tamnd.github.io/math2/node/X5WKGC5Q6T.md" tag="X5WKGC5Q6T" kind="concept">
# Cut locus

## Summary

The cut locus of a point $p$ in a complete [[Riemannian manifold|7CBT4SM83Y]] is the set of points where [[geodesics|NSYMQMHPDN]] from $p$ stop being shortest paths.
Its complement is the diffeomorphic image, under the [[exponential map|Y7Z0AGY1A7]], of an open star-shaped region of the tangent space.
So the complement of the cut locus is contractible, and the topology of the manifold is concentrated in the cut locus.
The distance from $p$ to its cut locus is the [[injectivity radius|Y6ZSKVAE3V]] at $p$.

## Explanations

### L1 human

Walk from $p$ along a geodesic.
At first you are on a shortest route from $p$.
At some moment, a different route becomes just as short, and from then on your path is no longer shortest.
The place where this happens is a cut point.
On the unit sphere, the only cut point of the north pole is the south pole.
Make a flat torus from a square centred at a point $p$ by gluing opposite sides.
Then the cut locus of $p$ is the image of the boundary of the square.
</document>

<document source="https://tamnd.github.io/math2/node/X600GRHDQ7.md" tag="X600GRHDQ7" kind="definition">
# Givens rotation

## Definition

Let $n \ge 2$, let $1 \le i < j \le n$, and let $c, s$ be real numbers with $c^2 + s^2 = 1$.
The *Givens rotation* $G(i, j; c, s)$ is the $n \times n$ real matrix that agrees with the [[identity matrix|06DW9BAWB1]] except in the four entries
$$g_{ii} = c, \qquad g_{ij} = -s, \qquad g_{ji} = s, \qquad g_{jj} = c.$$
Writing $c = \cos\theta$ and $s = \sin\theta$, it is the [[rotation|0XNZWJ96HK]] by the angle $\theta$ in the plane of the coordinates $i$ and $j$.

## Remarks

$G = G(i, j; c, s)$ is an [[orthogonal matrix|0QRBVFJ50X]] with determinant $c^2 + s^2 = 1$.
For $x \in \R^n$, the product $G x$ changes only the entries $i$ and $j$: $(Gx)_i = c x_i - s x_j$ and $(Gx)_j = s x_i + c x_j$.
If $(x_i, x_j) \ne (0, 0)$, put $r = \sqrt{x_i^2 + x_j^2}$, $c = x_i / r$ and $s = -x_j / r$; then $(Gx)_i = r$ and $(Gx)_j = 0$.
Example: for $x = (3, 4)$ in $\R^2$, $r = 5$, $c = 3/5$, $s = -4/5$, and $G x = (9/5 + 16/5, \ -12/5 + 12/5) = (5, 0)$.
Non-example: the matrix with $c = s = 1$ in these four places is not a Givens rotation, since $c^2 + s^2 = 2$, and it is not orthogonal.
Other books put $-s$ in position $(j, i)$ instead; the two conventions differ by the sign of $\theta$.

## Explanations

### L1 human

A Givens rotation turns a vector inside the plane of two coordinate axes and does nothing to the other coordinates.
Its matrix is the identity with a $2 \times 2$ rotation block written into rows and columns $i$ and $j$.
The angle can be chosen so that one of the two entries becomes zero.
</document>

<document source="https://tamnd.github.io/math2/node/X60JG9AEQJ.md" tag="X60JG9AEQJ" kind="concept">
# Kähler manifold

## Summary

A Kähler manifold is a [[complex manifold|JSGWAN2RJV]] with a Hermitian metric whose associated $2$-form is closed.
Its Riemannian, complex and symplectic structures are then compatible in the strongest way.
Complex projective space and all its complex submanifolds are Kähler, so Kähler geometry contains smooth projective algebraic geometry.
Compact Kähler manifolds satisfy strong topological restrictions, such as the Hodge decomposition of their cohomology and the evenness of odd Betti numbers.

## Explanations

### L1 human

On a complex manifold one can measure lengths with a Hermitian metric, and every such metric has a $2$-form attached to it.
A Kähler metric is one where this $2$-form has zero derivative.
This condition means that, to second order at each point, the metric looks like the flat metric of $\C^n$ in suitable holomorphic coordinates.
The flat space $\C^n$, every Riemann surface and complex projective space with its Fubini–Study metric are Kähler.
Many compact complex manifolds, such as Hopf manifolds, are not.
</document>

<document source="https://tamnd.github.io/math2/node/X629NF2CQS.md" tag="X629NF2CQS" kind="statement">
# Abel–Dini theorem

## Statement

Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that $\sum a_n$ [[diverges|0NBRP6093V]].
Let $s_n = a_1 + \cdots + a_n$ be the [[partial sums|0NB643K60Q]], and let $\alpha$ be a real number.
Then $\sum_{n=1}^{\infty} a_n / s_n^{\alpha}$ [[converges|0CVVKY7GY2]] if $\alpha > 1$, and diverges if $\alpha \le 1$.

## Remarks

For $\alpha = 1$ and $m > n$, the terms with index from $n+1$ to $m$ satisfy $\sum_{k=n+1}^{m} a_k / s_k \ge (s_m - s_n)/s_m$.
Since $s_m \to \infty$, this tends to $1$ as $m \to \infty$, so the [[Cauchy criterion|06PVXMY0TH]] fails.
For $\alpha < 1$, one has $s_n \ge 1$ for large $n$, so $a_n / s_n^{\alpha} \ge a_n / s_n$.
For $\alpha > 1$, one has $a_n / s_n^{\alpha} \le \int_{s_{n-1}}^{s_n} t^{-\alpha} \, dt$ for $n \ge 2$, and these integrals add up to at most $\int_{s_1}^{\infty} t^{-\alpha} \, dt < \infty$.
Example: with $a_n = 1$, one has $s_n = n$, and the theorem says that $\sum 1/n^{\alpha}$ converges exactly when $\alpha > 1$.

## Explanations

### L1 human

Start from any divergent series of positive terms.
Dividing each term by the partial sum so far gives a new series that still diverges, but more slowly.
Dividing by a slightly higher power of the partial sum makes it converge.

### L2 human

The theorem shows that there is no slowest divergent series: from any divergent series of positive terms, $\sum a_n / s_n$ diverges, and its terms are much smaller than $a_n$.
It also gives a supply of convergent series near the boundary.
So no single comparison series can decide all series of positive terms, which is one reason why there is a whole scale of [[convergence tests|FJKV01J9F1]].
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# Differentiable curve

## Definition

Let $n \ge 1$ be an integer, let $k \ge 1$ be an integer or $k = \infty$, and let $I \subseteq \R$ be an [[interval|0TK8Y3J5RW]] with more than one point.
Let $\gamma \colon I \to \R^n$, $\gamma(t) = (\gamma_1(t), \dots, \gamma_n(t))$, be a [[parametrised curve|0JF1XXGWB4]].
The curve $\gamma$ is a *differentiable curve of class $C^k$* if each coordinate function $\gamma_i$ is of class $C^k$ (see [[smoothness classes|WK5HPW5Y24]]) on $I$.
This means that $\gamma_i$ is $k$ times differentiable and its derivatives up to order $k$ are continuous.
At an end point of $I$ that belongs to $I$, the derivatives are one-sided.
A *smooth curve* is a differentiable curve of class $C^\infty$.

## Remarks

Example: $\gamma(t) = (t, t^2)$, $t \in \R$, is a smooth curve, since both coordinates are polynomials.
Non-example: $\gamma(t) = (t, \abs{t})$, $t \in \R$, is a continuous curve but not of class $C^1$, because the second coordinate has no derivative at $t = 0$.
A differentiable curve can still have a cusp in its image: $t \mapsto (t^2, t^3)$ is smooth, but its velocity vanishes at $t = 0$.
Curves whose velocity never vanishes are the [[regular curves|06JMR2XWXW]].
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# Finite ring

## Summary

A finite ring is a [[ring|0B1261B5ED]] with only finitely many elements.
The rings $\Z/n\Z$ of [[integers modulo n|03VFZ2WETT]], the [[finite fields|0ZXZ82FTX2]] and the matrix rings over them are the basic examples.
Finiteness forces strong structure: by [[Wedderburn's little theorem|0RRNVX1VKA]], every finite division ring is a field.
Finite rings are used in number theory, combinatorics and coding theory.

## Explanations

### L1 human

A finite ring is a number system with only finitely many numbers, where you can add, subtract and multiply.
Clock arithmetic is the main picture: on a clock with $12$ hours, $9 + 5 = 2$ and $3 \cdot 4 = 0$.
This gives the ring $\Z/12\Z$ with $12$ elements.
In a finite ring, repeated addition of $1$ must come back to $0$, so the ring always has a positive characteristic.
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# Additive Schwarz method

## Statement

Let $V$ be a finite-dimensional real vector space with an inner product $a(\cdot, \cdot)$.
Let $V_0, V_1, \dots, V_N$ be subspaces of $V$ with $V = V_0 + V_1 + \cdots + V_N$.
Let $P_i \colon V \to V_i$ be the $a$-[[orthogonal projection|0B8KZRT28M]] onto $V_i$, and let $P = P_0 + P_1 + \cdots + P_N$ be the *additive Schwarz operator*.
Then $P$ is self-adjoint and positive definite with respect to $a$, and for every $v \in V$
$$a(P^{-1} v, v) = \min \Bigl\{ \sum_{i=0}^{N} a(v_i, v_i) : v_i \in V_i, \ \sum_{i=0}^{N} v_i = v \Bigr\}.$$
Suppose that every $v \in V$ has a decomposition $v = \sum_i v_i$ with $v_i \in V_i$ and $\sum_i a(v_i, v_i) \le C_0^2 \, a(v, v)$.
Then the eigenvalues of $P$ lie in $[C_0^{-2}, N + 1]$.
So the [[condition number|AB3S6MHJ8X]] of $P$ is at most $(N + 1) C_0^2$.

## Remarks

Each $P_i$ is self-adjoint and satisfies $0 \le a(P_i w, w) \le a(w, w)$, so $P$ is self-adjoint with eigenvalues at most $N + 1$.
If $a(Pw, w) = 0$, then $P_i w = 0$ for all $i$, so $w$ is orthogonal to every $V_i$ and hence to $V$, and $w = 0$.
For the identity, the choice $v_i = P_i P^{-1} v$ gives $\sum_i a(v_i, v_i) = a(v, P^{-1} v)$.
For any other decomposition, $a(v, P^{-1} v) = \sum_i a(v_i, P_i P^{-1} v)$, and the Cauchy–Schwarz inequality gives $a(v, P^{-1} v) \le \sum_i a(v_i, v_i)$.

In matrix form, let $A$ be the symmetric positive definite stiffness matrix and let $R_i$ be the restriction to the unknowns of subdomain $i$.
Then the method uses the [[preconditioner|KE98SW3SBD]] $M^{-1} = \sum_i R_i^T (R_i A R_i^T)^{-1} R_i$, and $M^{-1} A$ is the matrix of $P$.
One applies it inside the [[conjugate gradient method|AFXZGQB4EN]], and the local solves can run in parallel.
Take finite elements on overlapping subdomains of diameter $H$ with overlap $\delta$, and a coarse space $V_0$ on the coarse grid.
Then the standard bound is $\kappa(P) \le C (1 + H/\delta)$, with $C$ independent of the mesh size and the number of subdomains.
Without the coarse space, the bound degrades as the number of subdomains grows, because a single subdomain solve cannot carry information across the whole domain.

## Explanations

### L1 human

The additive Schwarz method splits a big problem into many overlapping small ones, solves all of them at the same time, and adds the corrections.
Used alone this would overshoot, so it is used as a preconditioner: it turns a hard linear system into an easier one for an iterative solver.
The theorem says the result is well conditioned as long as every function can be split into local pieces whose total energy is not much more than its own.

### L2 human

The identity for $a(P^{-1} v, v)$ is often called Lions' lemma, and it reduces the analysis to finding a stable splitting.
It is the parallel counterpart of the [[Schwarz alternating method|97VNBS15CD]], just as the Jacobi method is the parallel counterpart of Gauss–Seidel.
Coloring the subdomains so that overlapping ones get different colors improves the bound $N + 1$ to one plus the number of colors.
The same abstract framework covers [[multigrid methods|AC9PY6FBYF]] viewed as subspace corrections.
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# Exponential functions on semigroups

## Definition

Let $S$ be a [[semigroup|0XMEM406JM]], with its operation written as $(x, y) \mapsto xy$.
An *exponential function* on $S$, or *exponential*, is a [[function|0EPAT51DXQ]] $m \colon S \to \C$ such that
$$m(xy) = m(x)\, m(y) \quad \text{for all } x, y \in S.$$

## Remarks

Example: on $S = (\R, +)$, $m(x) = e^{(1 + 2i) x}$ is an exponential, since $e^{(1 + 2i)(x + y)} = e^{(1 + 2i) x} e^{(1 + 2i) y}$.
Non-example: on $S = (\R, +)$, $m(x) = 2 e^x$ is not an exponential, since $m(0 + 0) = 2$ while $m(0) m(0) = 4$.
The zero function is an exponential; some authors exclude it.
If $S$ is a group and $m$ is not identically zero, then $m(e) = 1$, $m$ never vanishes, and $m$ is a homomorphism into the multiplicative group $\C^{\times}$.
An exponential on an abelian group whose values all have absolute value $1$ is a [[character|H1WNVN5V7F]] of the group.
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# Fourier–Bessel series

## Definition

Let $\nu > -1$ and let $J_\nu$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $\nu$, a solution of [[Bessel's equation|EAV97QH9DF]].
Let $0 < j_{\nu,1} < j_{\nu,2} < \cdots$ be the positive zeros of $J_\nu$.
Let $f$ be a measurable function on $(0, 1)$ with $\int_0^1 x \abs{f(x)}^2 \, dx < \infty$.
The *Fourier–Bessel series* of $f$ of order $\nu$ is
$$\sum_{k=1}^{\infty} c_k \, J_\nu(j_{\nu,k} x), \qquad c_k = \frac{2}{J_{\nu+1}(j_{\nu,k})^2} \int_0^1 x \, f(x) \, J_\nu(j_{\nu,k} x) \, dx.$$

## Remarks

The functions $J_\nu(j_{\nu,k} x)$ are orthogonal in the weighted [[Lebesgue space|SASP14MA1Z]] $L^2((0, 1), x \, dx)$, with
$$\int_0^1 x \, J_\nu(j_{\nu,k} x) \, J_\nu(j_{\nu,m} x) \, dx = \frac{\delta_{km}}{2} J_{\nu+1}(j_{\nu,k})^2.$$
So the series is the [[generalized Fourier series|SNJC87TBHD]] of $f$ with respect to this orthogonal family, and the formula for $c_k$ is the usual quotient of inner products.
The family is complete in $L^2((0, 1), x \, dx)$, so the series converges to $f$ in this space.
On an interval $[0, b]$ one uses $J_\nu(j_{\nu,k} x / b)$ instead.

Example: $\nu = 0$ and $f = 1$.
Since $\frac{d}{dt}\big(t J_1(t)\big) = t J_0(t)$, the substitution $t = j x$ gives $\int_0^1 x J_0(jx) \, dx = J_1(j)/j$.
Hence $c_k = \frac{2}{j_{0,k} J_1(j_{0,k})}$ and
$$1 = \sum_{k=1}^{\infty} \frac{2}{j_{0,k} \, J_1(j_{0,k})} \, J_0(j_{0,k} x) \quad \text{in } L^2((0,1), x\,dx).$$

Non-example: the functions $J_0(k\pi x)$, $k = 1, 2, \dots$, do not give a Fourier–Bessel series.
The numbers $k\pi$ are not zeros of $J_0$; for instance $J_0(\pi) \approx -0.304 \ne 0$.
</document>

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# Localization at a prime ideal

## Summary

The localization of a [[commutative ring|0310REJJY8]] $R$ at a [[prime ideal|00MHHZ9SNE]] $\mathfrak p$ is the ring $R_{\mathfrak p}$ of fractions whose denominators lie outside $\mathfrak p$.
It is a [[local ring|MZ02BKPHJA]] with maximal ideal $\mathfrak p R_{\mathfrak p}$.
It keeps the information about $R$ near the point $\mathfrak p$ of the [[spectrum|FB45BHHA2J]] and forgets the rest.
Many questions in commutative algebra reduce to questions about these local rings.

## Explanations

### L1 human

Pick a prime number $p$ and allow as denominators all integers not divisible by $p$.
The result $\Z_{(p)}$ contains $1/3$ and $5/7$ when $p = 2$, but not $1/2$.
In this ring every prime other than $p$ has become invertible, so only divisibility by $p$ still matters.
This is what "local at $p$" means: the ring sees only the prime $p$.
The [[definition|N37SPDWYGS]] does the same for any prime ideal of any commutative ring.
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# Rough path

## Summary

A rough path is a path together with extra data that play the role of its iterated integrals.
For paths with [[Hölder regularity|012PBA0FMD]] of exponent at most $1/2$, such as Brownian paths, these iterated integrals are not determined by the path itself.
Once they are given, differential equations driven by the path can be solved path by path, and the solution depends continuously on the rough path.
Rough path theory, introduced by Terry Lyons, thus separates the probabilistic and the analytic parts of stochastic calculus.

## Explanations

### L1 human

To solve an equation driven by a very irregular signal, it is not enough to know the signal; one also needs its "areas", the integrals of the signal against itself.
For a smooth signal these areas can be computed, but for a Brownian path different natural choices give different answers.
The Itô and the Stratonovich integrals are two such choices.
A rough path fixes the choice once and for all, and then the equation has a unique, stable solution.
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<document source="https://tamnd.github.io/math2/node/X6QC23AAB9.md" tag="X6QC23AAB9" kind="definition">
# Harmonic map heat flow

## Definition

Let $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]] without boundary, let $(N, h)$ be a Riemannian manifold, and let $u_0 \colon M \to N$ be a smooth map.
Let $0 < T \le \infty$.
A solution of the *harmonic map heat flow* with initial map $u_0$ on $[0, T)$ is a map $u \colon M \times [0, T) \to N$ with these properties.
It is smooth on $M \times (0, T)$, continuous up to $t = 0$, and
$$\frac{\partial u}{\partial t}(x, t) = \tau\big(u(\cdot, t)\big)(x) \quad \text{on } M \times (0, T), \qquad u(x, 0) = u_0(x),$$
where $\tau$ is the [[tension field|W8F8S8DX5X]].

## Remarks

The [[energy|E6HDE1DFXD]] decreases along a solution:
$$\frac{d}{dt} E\big(u(\cdot, t)\big) = -\int_M \abs{\tau(u(\cdot, t))}^2 \, dV_g \le 0.$$
If $N$ is compact, a solution exists on some interval $[0, T)$ with $T > 0$, since the equation is a semilinear parabolic system.
A map $u_0$ is a fixed point of the flow exactly when it is a harmonic map.

Example: for $N = \R$ the Christoffel symbols vanish, and the flow is the [[heat equation|VP2HN86VAG]] $\partial_t u = \Delta_g u$ on $M$.
Its solutions exist for all time and converge to the constant function equal to the mean of $u_0$.

Non-example: a family with $\partial_t u = -\tau(u)$ is not a solution of the harmonic map heat flow but of the backward flow, along which the energy increases.
For $N = \R$ this is the backward heat equation, which is ill-posed.
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# Upper half-plane

## Description

The upper half-plane is the set
$$\mathbb{H} = \set{\tau \in \C : \operatorname{Im} \tau > 0}$$
of [[complex numbers|09FTAET5Z4]] with positive imaginary part.
It is an open subset of $\C$, so it is a [[Riemann surface|5K0N8QMV6Q]].
The group $\mathrm{SL}_2(\R)$ acts on $\mathbb{H}$ by [[Möbius transformations|0SG74WBFA2]] $\tau \mapsto (a\tau + b)/(c\tau + d)$.

## Remarks

The action is well defined because
$$\operatorname{Im}\left(\frac{a\tau + b}{c\tau + d}\right) = \frac{(ad - bc) \operatorname{Im} \tau}{\abs{c\tau + d}^2},$$
which is positive when $ad - bc = 1$ and $\operatorname{Im} \tau > 0$.
The holomorphic bijections of $\mathbb{H}$ onto itself are exactly these Möbius transformations, and they form the group $\mathrm{PSL}_2(\R)$.
The map $\tau \mapsto (\tau - i)/(\tau + i)$ is a biholomorphic map of $\mathbb{H}$ onto the open unit disc.
With the metric $ds^2 = (dx^2 + dy^2)/y^2$, where $\tau = x + iy$, the upper half-plane is a model of the hyperbolic plane, and $\mathrm{PSL}_2(\R)$ acts by isometries.
[[Modular forms|GWKN5VG78A]] are functions on $\mathbb{H}$.

## Explanations

### L1 human

The upper half-plane is the part of the complex plane above the real axis.
It is the stage on which modular forms live.
The [[modular group|VEKV1EM77B]] moves its points around, for example by $\tau \mapsto \tau + 1$ and $\tau \mapsto -1/\tau$, and never pushes them below the axis.
With a suitable way of measuring length, it is a picture of non-Euclidean, hyperbolic geometry.
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# Inner-outer factorization

## Statement

Let $0 < p \le \infty$, and let $f$ be a function in the [[Hardy space|PNNNFJVF6W]] $H^p$ of the [[unit disc|8G5ABXRK85]] $\mathbb{D}$, with $f$ not identically $0$.
Then $f = I F$, where $I$ is an [[inner function|NN8XH5RJYB]] and $F$ is an [[outer function|KGJD4A4ATY]] that lies in $H^p$.
The factors are unique up to constants of modulus $1$, and $\abs{F^*} = \abs{f^*}$ almost everywhere on the circle.
Moreover $I = c B S_\mu$, where $\abs{c} = 1$ and $B$ is the [[Blaschke product|9N6R5ADJXT]] of the zeros of $f$, with multiplicity.
The last factor is
$$S_\mu(z) = \exp\left(-\int_0^{2\pi} \frac{e^{it} + z}{e^{it} - z} \, d\mu(t)\right)$$
for a finite positive Borel measure $\mu$ on the circle that is singular with respect to Lebesgue measure.

## Remarks

Proof idea: the zeros of $f$ satisfy the Blaschke condition, by Jensen's formula.
Dividing by their Blaschke product $B$ gives a function $g = f/B$ in $H^p$ with no zeros and the same norm.
The outer function $F$ with modulus $\abs{g^*}$ dominates $g$ in the sense that $\abs{g} \le \abs{F}$, because $\log \abs{g(z)}$ is at most the Poisson integral of $\log \abs{g^*}$ for $g \in H^p$.
Then $g/F$ is bounded, has no zeros and has boundary modulus $1$, and such a function is a singular inner function $c S_\mu$.

Example: $f(z) = z (2 + z)$ is bounded, its inner factor is $z$ and its outer factor is $2 + z$.

The hypothesis $f \not\equiv 0$ is needed, since the zero function has no inner factor.

## Explanations

### L1 human

Every function in a Hardy space splits into two parts.
The outer part is fixed by the size of the function on the boundary circle.
The inner part has size $1$ on the circle and holds the zeros of the function and a hidden "singular" part.
This is the disc analogue of writing a number as a sign times a size.

### L2 human

The factorization is the main structural tool for Hardy spaces.
It reduces many questions to zero-free functions, for which one can take logarithms and roots: for example $f = B g$ with $g$ zero-free gives $g^{p/2} \in H^2$, which transfers results from $H^2$ to $H^p$.
It underlies [[Beurling's theorem|2V1DXXM9FA]] on shift-invariant subspaces, where outer functions are exactly the cyclic vectors of the shift on $H^2$.
The same factorization holds in the [[Nevanlinna class|M3WQTN3FH2]] if one allows quotients of singular inner functions.
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# Envelope of holomorphy

## Summary

The envelope of holomorphy of a domain $D \subseteq \C^n$ is the largest domain to which every [[holomorphic function|507C34W0ZT]] on $D$ continues.
This domain may be spread over $\C^n$ in several sheets.
It exists for every domain and is unique up to isomorphism.
In general it is a [[Riemann domain over $\C^n$|A15BF6QHDR]] and not a subset of $\C^n$.
A domain is a [[domain of holomorphy|4487CSC0MB]] exactly when it is its own envelope of holomorphy.

## Explanations

### L1 human

Given a region, ask how far all of its holomorphic functions can be pushed at once.
The answer is the envelope of holomorphy: the region together with all its hidden extensions.
For the [[Hartogs figure|GHJC53RFBT]], a polydisc with a piece cut out, the envelope is the full polydisc.
For a ball with a smaller closed ball removed, the envelope is the whole ball.
In one variable nothing new appears: every domain is its own envelope.
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# Spectral theorem for bounded normal operators

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]], and let $N$ be a bounded operator on $H$.
Assume that $N$ is normal: $N N^* = N^* N$, where $N^*$ is the [[adjoint|6HSJBJ8TTQ]] of $N$.
Let $\sigma(N) \subseteq \C$ be the [[spectrum|BMKATJ802A]] of $N$.
Then there is a unique [[projection-valued measure|ATGT25E5FF]] $E$ on the [[Borel sets|0WQXY32Q8D]] of $\sigma(N)$ such that
$$\ip{N x}{y} = \int_{\sigma(N)} \lambda \, d E_{x,y}(\lambda) \quad \text{for all } x, y \in H,$$
where $E_{x,y}(B) = \ip{E(B) x}{y}$.
In short, $N = \int_{\sigma(N)} \lambda \, dE(\lambda)$.

## Remarks

One proof starts from the [[continuous functional calculus|JB7S6WPZ77]] $f \mapsto f(N)$ on $C(\sigma(N))$.
For fixed $x, y$, the map $f \mapsto \ip{f(N) x}{y}$ is a bounded linear functional on $C(\sigma(N))$, so by the Riesz–Markov theorem it is integration against a complex measure $E_{x,y}$.
One then defines $E(B)$ by $\ip{E(B) x}{y} = E_{x,y}(B)$ and checks that it is a projection-valued measure.
The projections $E(B)$ commute with every bounded operator that commutes with $N$.
An equivalent form holds when $H$ is separable.
Then there are a $\sigma$-finite measure space $(\Omega, \mu)$, a bounded measurable function $\varphi$ on $\Omega$ and a unitary map $U \colon H \to L^2(\Omega, \mu)$ with $U N U^{-1} f = \varphi f$ for all $f \in L^2(\Omega, \mu)$.
Example: on $L^2([0,1])$, the operator $(Nf)(t) = t f(t)$ is self-adjoint with spectrum $[0,1]$, and $E(B)$ is multiplication by the indicator function of $B$.
In finite dimension, $E(B)$ is the orthogonal projection onto the sum of the eigenspaces for the eigenvalues in $B$, and the theorem is the [[spectral theorem for normal matrices|04CW1P44VD]].
Normality is necessary: the right shift on $\ell^2$ is not normal, and it is not of this form.

## Explanations

### L1 human

A normal matrix is diagonal in a suitable orthonormal basis.
An operator on an infinite-dimensional space may have no eigenvectors, so a basis of eigenvectors may not exist.
The spectral theorem replaces the list of eigenvalues by a measure on the spectrum, whose values are projections.
In this sense, every normal operator is a continuous version of a diagonal matrix: it is multiplication by a function.

### L2 human

The theorem makes it possible to define $f(N) = \int f(\lambda) \, dE(\lambda)$ for every bounded Borel function $f$ on $\sigma(N)$, the Borel functional calculus.
With $f$ an indicator function, this gives the spectral projections of $N$, which split $H$ into pieces where $N$ acts with spectrum in a given set.
The self-adjoint case covers the observables of quantum mechanics with bounded values; the unitary case gives the spectral analysis of unitary operators, used in ergodic theory.
For unbounded self-adjoint operators, a similar theorem holds with a projection-valued measure on $\R$.
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# Robbins conjecture

## Statement

Let $A$ be a set with a binary operation $+$ and a unary operation $n$ such that for all $x, y, z \in A$:

1. $x + y = y + x$;
2. $(x + y) + z = x + (y + z)$;
3. $n(n(x + y) + n(x + n(y))) = x$.

Such a structure is called a *Robbins algebra*, and axiom 3 is *Robbins' equation*.
Then $A$ satisfies [[Huntington's equation|M3GFCVX549]] $n(n(x) + y) + n(n(x) + n(y)) = x$, so $(A, +, n)$ is a [[Boolean algebra|0XADJKM699]] with $+$ as join and $n$ as complement.

## Remarks

Every Boolean algebra satisfies Robbins' equation, so the theorem says that the Robbins algebras are exactly the Boolean algebras.
The question was raised by Herbert Robbins in the 1930s and stayed open for about sixty years.
William McCune settled it in 1996 with the automated theorem prover EQP, which found a derivation of a condition that earlier work had shown to be sufficient.
The proof is purely equational; human-readable versions of it were written later.

## Explanations

### L1 human

Boolean algebra can be built from "or" and "not" with a few rules.
Robbins proposed a rule that looks very much like Huntington's rule, and asked whether it is just as strong.
For decades nobody could prove or refute it, until a computer program found a proof in 1996.

### L2 human

The result is a standard example of a mathematical question solved by an automated theorem prover.
It shows that the equational theory of Boolean algebras has several very short bases, and the search for the shortest single axioms for Boolean algebra builds on such results.
The proof works inside equational logic: it uses only substitution and replacement of equals by equals, starting from the three axioms.
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# Rack and quandle

## Summary

A rack is a set with a [[binary operation|0M418FE7BB]] that distributes over itself and acts on the set by bijections.
A quandle is a rack in which every element $a$ satisfies $a \triangleleft a = a$.
The model example is conjugation in a group, $a \triangleleft b = a b a^{-1}$, which keeps the properties of conjugation and forgets the group multiplication.
The three quandle axioms match the three Reidemeister moves of [[knot|QYEAZTYGPT]] diagrams, so each knot has a fundamental quandle, which is a strong [[knot invariant|GZQ5ZE3X6G]].

## Explanations

### L1 human

In a group, conjugating $b$ by $a$ moves $b$ to $a b a^{-1}$.
A quandle keeps only this operation and its basic rules.
Knot theory uses quandles by giving a label to each arc of a knot diagram.
At each crossing, the label of the arc that passes under changes by the operation with the label of the arc that passes over.
The rules of a quandle are exactly the rules that make this labelling unchanged by the usual moves of knot diagrams.
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# Multiplier ideal

## Definition

Let $X$ be a complex manifold.
Let $\varphi$ be a [[plurisubharmonic function|FF2P7ER3QT]] on $X$, or more generally a function that is locally a plurisubharmonic function plus a smooth function.
The *multiplier ideal sheaf* $\mathcal{I}(\varphi) \subseteq \mathcal{O}_X$ is the subsheaf of $\mathcal{O}_X$ defined as follows.
Its stalk at $x \in X$ is the set of [[germs|7PJ8KVZZCQ]] $f \in \mathcal{O}_{X,x}$ that satisfy the following condition.
There is a coordinate neighbourhood $U$ of $x$ with $\int_U \abs{f}^2 e^{-2\varphi} \, dV < \infty$, where $dV$ is Lebesgue measure in the coordinates of $U$.

## Remarks

The condition does not depend on the coordinates, since a change of coordinates multiplies $dV$ by a smooth positive function.
The stalks are ideals, so $\mathcal{I}(\varphi)$ is an [[ideal sheaf|N3PGYJJQ7W]]; Nadel's theorem says it is coherent.
Example: $X = \C$, $\varphi = c \log\abs{z}$ with $c > 0$.
In polar coordinates, $\int_{\abs{z} < r} \abs{z}^{2k} \abs{z}^{-2c} \, dV = 2\pi \int_0^r t^{2k - 2c + 1} \, dt$, which is finite exactly when $k > c - 1$.
So the stalk at $0$ is $(z^{\floor{c}})$; for example $\mathcal{I}(\log\abs{z})_0 = (z)$.
Non-example: for $\varphi = \log\abs{z}$, the germ $f = 1$ is not in the stalk at $0$, since $\int \abs{z}^{-2} \, dV$ diverges near $0$.
If $\varphi$ is locally bounded, then $\mathcal{I}(\varphi) = \mathcal{O}_X$.
Algebraic form: for an ideal $\mathfrak{a} = (g_1, \dots, g_k)$ and a real $c > 0$, $\mathcal{J}(c \cdot \mathfrak{a}) = \mathcal{I}\left(\tfrac{c}{2} \log \sum_i \abs{g_i}^2\right)$, the germs with $\abs{f}^2 / (\sum_i \abs{g_i}^2)^c$ locally integrable.
Some authors use the weight $e^{-\varphi}$ instead of $e^{-2\varphi}$.
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<document source="https://tamnd.github.io/math2/node/X7GP9BECD8.md" tag="X7GP9BECD8" kind="concept">
# Tangential Cauchy–Riemann equations

## Summary

The tangential Cauchy–Riemann equations on a real submanifold of $\C^n$ are a restricted form of the [[Cauchy–Riemann equations|9PT772R94A]], see the [[definition|KQWJ8SGJEZ]].
They are taken only in the anti-holomorphic directions that are tangent to the submanifold.
Their solutions are the [[CR functions|JFKYYK2CEB]], and every restriction of a holomorphic function is one.
They form an overdetermined system of first-order linear partial differential equations.
Hans Lewy's study of them led to his example of a linear equation with no solutions.

## Explanations

### L1 human

A holomorphic function satisfies $\partial f / \partial \bar z_j = 0$ in every direction.
On a hypersurface, a function is only defined on the hypersurface, so only the derivatives along it make sense.
The tangential equations keep those combinations of the $\partial / \partial \bar z_j$ that point along the hypersurface.
On a real curve in $\C$ no such combination is left, so there are no equations at all; on a hypersurface in $\C^2$ there is one equation.
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<document source="https://tamnd.github.io/math2/node/X7GTF3D98F.md" tag="X7GTF3D98F" kind="concept">
# Integrable function

## Summary

A measurable function on a measure space is integrable when the integral of its absolute value is finite.
These are the functions for which the Lebesgue integral is a well-defined finite number.
Integrable functions form the space $L^1$, and the convergence theorems of integration are stated for them.

## Explanations

### L1 human

A function is integrable when the total area between its graph and the axis is finite.
The area above the axis and the area below the axis must each be finite, so that their difference makes sense.
For example, $e^{-\abs{x}}$ is integrable on the real line, with total area $2$.
The function $1/x$ on $[1, \infty)$ is not integrable, because its area grows like $\ln x$ without bound.
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<document source="https://tamnd.github.io/math2/node/X7HYEBA5SA.md" tag="X7HYEBA5SA" kind="concept">
# Finite difference method

## Summary

The finite difference method solves differential equations approximately on a grid of points.
It replaces each derivative by a difference quotient of the values at nearby grid points, which turns the equation into a system of algebraic equations.
It is the oldest and simplest numerical method for [[partial differential equations|GC5BDZ6QBC]].
Its convergence rests on consistency and stability, as in the [[Lax equivalence theorem|D5M7TRD3MY]].
For time-dependent problems, stability often requires a step restriction such as the [[Courant–Friedrichs–Lewy condition|FQVVNYV6N8]].

## Explanations

### L1 human

A derivative is a limit of slopes $(u(x + h) - u(x))/h$ as $h \to 0$.
The finite difference method simply stops before the limit and uses a small fixed $h$.
Put points $0, h, 2h, \ldots$ on the line and keep track only of the values of the unknown function there.
For example, $u''(x)$ is replaced by $(u(x + h) - 2u(x) + u(x - h))/h^2$, which is exact for quadratic polynomials.
The equation then becomes one linear equation per grid point, which a computer can solve.
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# Archimedean solid

## Description

An *Archimedean solid* is a convex [[polyhedron|0325SGSJAR]] with two properties.
Its faces are [[regular polygons|01KJJ9A400]] of at least two different kinds.
Its [[symmetry group|0ZG3Z5JZ85]] acts transitively on its vertices.
The prisms and antiprisms also have these properties and are excluded.
Up to similarity there are thirteen Archimedean solids.
Seven have the symmetry of the tetrahedron or the cube: the truncated tetrahedron, cuboctahedron, truncated cube, truncated octahedron, rhombicuboctahedron, truncated cuboctahedron and snub cube.
Six have the symmetry of the icosahedron: the icosidodecahedron, truncated dodecahedron, truncated icosahedron, rhombicosidodecahedron, truncated icosidodecahedron and snub dodecahedron.

## Remarks

The truncated tetrahedron has $4$ triangles and $4$ hexagons as faces, $12$ vertices and $18$ edges.
[[Euler's polyhedron formula|09HT3P35YP]] checks out: $12 - 18 + 8 = 2$.
The truncated icosahedron, the pattern of a classic football, has $12$ pentagons, $20$ hexagons, $60$ vertices and $90$ edges, and $60 - 90 + 32 = 2$.
The snub cube and the snub dodecahedron come in two mirror image forms, and some authors count these separately.
Asking only that all vertices look alike locally is weaker: the elongated square gyrobicupola satisfies that condition but is not vertex-transitive.
The [[Platonic solids|017E5PRS1Q]] are the vertex-transitive convex polyhedra with a single kind of regular face.

## Explanations

### L1 human

The Archimedean solids are the next most regular solids after the five Platonic solids.
Their faces are regular polygons of more than one shape, such as triangles and squares.
Every corner looks exactly like every other corner, and a rotation or reflection of the solid can move any corner to any other.
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<document source="https://tamnd.github.io/math2/node/X7R15C00ET.md" tag="X7R15C00ET" kind="statement">
# Artin's theorem on fixed fields

## Statement

Let $L$ be a field and let $G$ be a finite subgroup of the [[automorphism group|0K1SCPBNKJ]] of $L$.
Let $K = L^G$ be the [[fixed field|0KMJ6919WS]] of $G$.
Then:

1. the [[degree|0GS74FGFEK]] $[L:K]$ equals the order $\abs{G}$;
2. $L/K$ is a [[Galois extension|0DS2F7GK05]];
3. the [[Galois group|0XVR72M51M]] $\Gal(L/K)$ equals $G$.

## Remarks

For $\alpha \in L$, let $\alpha_1, \ldots, \alpha_r$ be the distinct elements of the orbit $G\alpha$.
The polynomial $\prod_{i=1}^r (x - \alpha_i)$ is fixed by $G$, so it has coefficients in $K$; it is separable and splits in $L$.
Hence every element of $L$ is separable over $K$ of degree at most $\abs{G}$, and $L/K$ is normal and separable.
By the [[primitive element theorem|0SVX98HYV1]], every finite subextension of $L/K$ is then generated by one element, so it has degree at most $\abs{G}$.
Hence $[L:K] \le \abs{G}$.
Conversely, $G \subseteq \Gal(L/K)$ and $\abs{\Gal(L/K)} \le [L:K]$ give the reverse inequality.
Another proof uses [[Dedekind's lemma on independence of characters|N43G50YHG2]].

Example: let $L = \C(t)$ and let $G = \set{\id, \tau}$ with $\tau(f)(t) = f(-t)$.
The fixed field is $\C(t^2)$, and indeed $[\C(t) : \C(t^2)] = 2$.

The hypothesis that $G$ is finite is necessary.
The automorphism of $\C(t)$ with $t \mapsto t + 1$ generates an infinite group whose fixed field is $\C$, and $\C(t)/\C$ is not even algebraic.

## Explanations

### L1 human

Start with a field and a finite group of its symmetries.
The elements that no symmetry moves form a smaller field.
Artin's theorem says that the big field is a Galois extension of the small one.
Its degree is the number of symmetries, and its Galois group consists exactly of these symmetries.

### L2 human

This theorem is one half of the [[fundamental theorem of Galois theory|0T1Q8D0FW8]].
It shows that every subgroup $H$ of a finite Galois group $\Gal(L/K)$ is the Galois group of $L$ over the fixed field $L^H$, so the map $H \mapsto L^H$ is injective.
It also produces Galois extensions directly: the symmetric group $S_n$ acting on $k(x_1, \ldots, x_n)$ shows that $k(x_1, \ldots, x_n)$ is Galois over the field of symmetric rational functions with group $S_n$.
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# Bondy's theorem

## Statement

Let $n \ge 1$ and let $X$ be a set with $n$ elements.
Let $S_1, \ldots, S_n$ be $n$ distinct subsets of $X$.
Then there is an element $x \in X$ such that the sets $S_1 \setminus \set{x}, \ldots, S_n \setminus \set{x}$ are still distinct.

## Remarks

Suppose that no such $x$ exists.
Then for each $x \in X$ there are $i \ne j$ with [[symmetric difference|0ME21QN125]] $S_i \mathbin{\triangle} S_j = \set{x}$; join $S_i$ and $S_j$ by an edge labelled $x$.
This gives a graph on the $n$ sets with $n$ edges, all with distinct labels, so it contains a cycle.
Going around the cycle, the symmetric differences of the edges combine to $\emptyset$, but a symmetric difference of distinct singletons is not empty, a contradiction.

Example: $X = \set{1, 2}$ with $S_1 = \emptyset$ and $S_2 = \set{1}$.
Removing $x = 2$ leaves $\emptyset$ and $\set{1}$, which are still distinct.
The bound is sharp: for $X = \set{1, \ldots, n}$ the $n + 1$ sets $\emptyset, \set{1}, \ldots, \set{n}$ become non-distinct after removing any element $x$, since $\set{x}$ becomes $\emptyset$.

## Explanations

### L1 human

Describe $n$ different objects by $n$ yes/no features.
Bondy's theorem says that one of the features is redundant: you can drop it and still tell all the objects apart.
With $n + 1$ objects this can fail.

### L2 human

The theorem is a small result of extremal set theory with a short graph proof.
It says that $n$ distinct rows of a $0$–$1$ matrix with $n$ columns stay distinct after deleting a suitable column.
Its proof is a typical use of a graph to encode a family of sets.
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# Lie algebra of a compact group is reductive

## Statement

Let $G$ be a [[compact Lie group|9TFC6ADC1C]], and let $\mathfrak{g}$ be its Lie algebra, with center $\mathfrak{z}$ and Killing form $B$.
Then:

1. $\mathfrak{g}$ is a reductive Lie algebra: $\mathfrak{g} = \mathfrak{z} \oplus [\mathfrak{g}, \mathfrak{g}]$ as Lie algebras, and $[\mathfrak{g}, \mathfrak{g}]$ is semisimple;
2. $B(X, X) \le 0$ for all $X \in \mathfrak{g}$, with equality exactly when $X \in \mathfrak{z}$.

## Remarks

The proof averages an inner product on $\mathfrak{g}$ over $G$ with the normalized [[Haar measure|VY0HXD3GC4]].
This gives an inner product invariant under $\mathrm{Ad}(G)$, so each $\operatorname{ad} X$ is skew-symmetric for it.
The orthogonal complement of an ideal is then an ideal, so $\mathfrak{g}$ is a direct sum of simple ideals and of the abelian ideal $\mathfrak{z}$.
For 2, the eigenvalues of the skew-symmetric map $\operatorname{ad} X$ are purely imaginary, so $B(X, X) = \tr\big((\operatorname{ad} X)^2\big) \le 0$, with equality only when $\operatorname{ad} X = 0$.

Example: for $G = \mathrm{U}(n)$, the Lie algebra $\mathfrak{u}(n)$ of skew-Hermitian matrices is $i\R \cdot I \oplus \mathfrak{su}(n)$, the center plus a simple ideal for $n \ge 2$.

Compactness is needed.
The group of affine maps $x \mapsto ax + b$ of $\R$ with $a > 0$ is a noncompact Lie group whose Lie algebra is solvable but not abelian, so it is not reductive.

## Explanations

### L1 human

The infinitesimal version of a compact group is very well behaved.
It splits into a commutative part and a part made of simple pieces, with no mixing between them.
The reason is that a compact group can be averaged over, which produces a symmetric inner product that the group respects.

### L2 human

The theorem is the first step of the structure theory of compact Lie groups.
A real Lie algebra with an inner product for which each $\operatorname{ad} X$ is skew-symmetric is called a compact Lie algebra.
The compact Lie algebras are exactly the reductive ones with negative semidefinite Killing form.
Conversely, each semisimple Lie algebra with negative definite Killing form is the Lie algebra of a compact Lie group, by Weyl's theorem that its simply connected group is compact.
Together with Weyl's unitary trick, this lets one move between compact groups and complex semisimple Lie algebras.
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# Marsden–Weinstein reduction

## Definition

Let $(M, \omega)$ be a [[symplectic manifold|CAN490EPRT]] with a Hamiltonian action of a [[Lie group|7JBRGCVJRB]] $G$ and a [[moment map|ZARM34FZ6Z]] $\mu \colon M \to \mathfrak{g}^*$.
Assume that $G$ acts freely and properly on the level set $Z = \mu^{-1}(0)$.
Then $Z$ is a submanifold of $M$, the [[quotient|0BR3FY5Q8M]] $M /\!\!/ G = Z / G$ is a smooth manifold, and the quotient map $\pi \colon Z \to Z / G$ is smooth.
Let $i \colon Z \to M$ be the inclusion.
The *Marsden–Weinstein reduction* (or *symplectic quotient*) of $M$ by $G$ is the manifold $M /\!\!/ G$ with the unique $2$-form $\omega_{\mathrm{red}}$ such that
$$\pi^* \omega_{\mathrm{red}} = i^* \omega.$$

## Remarks

The Marsden–Weinstein–Meyer theorem says that $\omega_{\mathrm{red}}$ exists, is unique, and is symplectic, and that $\dim M /\!\!/ G = \dim M - 2 \dim G$.
Freeness of the action on $Z$ implies that $0$ is a regular value of $\mu$, which is why $Z$ is a submanifold.
Properness holds automatically when $G$ is compact.
Reduction at another value $\lambda \in \mathfrak{g}^*$ uses $\mu^{-1}(\lambda)$ divided by the stabilizer of $\lambda$ under the coadjoint action.

Example: let $S^1$ act on $\C^{n+1}$ by $e^{i\theta} \cdot z = e^{i\theta} z$, with the standard symplectic form.
A moment map is $\mu(z) = -\frac{1}{2}\abs{z}^2 + \frac{1}{2}$.
Then $\mu^{-1}(0)$ is the unit sphere $S^{2n+1}$, the circle acts freely on it, and the reduction is $\mathbb{CP}^n$ with a multiple of the Fubini–Study form.

Non-example: take the same circle action on $\C$ with $\mu(z) = -\frac{1}{2}\abs{z}^2$.
The level set $\mu^{-1}(0) = \set{0}$ is a fixed point, so the action on it is not free and the definition does not apply.
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# Summability method

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $\mathbb{K}^{\N}$ be the set of all [[sequences|0QE8STGJH2]] in $\mathbb{K}$.
A *summability method* $M$ consists of a set $D_M \subseteq \mathbb{K}^{\N}$ and a map $M \colon D_M \to \mathbb{K}$.
A sequence $s$ is *$M$-summable* (or *$M$-limitable*) to $M(s)$ if $s \in D_M$.
A [[series|0JH4MN4K1C]] $\sum a_n$ is $M$-summable with $M$-sum $L$ if its sequence of [[partial sums|0NB643K60Q]] $s_n = a_0 + \cdots + a_n$ is $M$-summable to $L$.

## Remarks

Example: the ordinary limit, with $D_M$ the set of convergent sequences and $M(s) = \lim s_n$, is a summability method; it sums exactly the convergent series.
Example: Cesàro summation takes $D_M$ to be the sequences whose [[Cesàro means|0P0DJHBGBB]] converge, and $M(s)$ the limit of these means; it sums $1 - 1 + 1 - \cdots$ to $\frac12$.
Non-example: "take the limit of some convergent subsequence" is not a summability method.
The sequence $0, 1, 0, 1, \dots$ would get both $0$ and $1$, so the rule does not define a map.
Some authors define methods directly on series $(a_n)$ instead of on partial sums; the two points of view are equivalent.

## Explanations

### L1 human

A summability method is any well-defined rule that turns certain sequences into numbers.
Further conditions, such as agreement with the ordinary limit, single out the useful methods.
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<document source="https://tamnd.github.io/math2/node/X83ECKQ6G8.md" tag="X83ECKQ6G8" kind="concept">
# Chern–Simons form

## Summary

A Chern–Simons form is an odd-degree differential form built from a connection whose exterior derivative is a characteristic form of Chern–Weil theory.
The best-known one is the $3$-form $\mathrm{tr}(A \wedge dA + \frac23 A \wedge A \wedge A)$, whose derivative is $\mathrm{tr}(F \wedge F)$.
Chern–Simons forms give secondary invariants, defined when the primary characteristic forms vanish.
Their integrals over $3$-manifolds are the action of Chern–Simons gauge theory, which is central in low-dimensional topology and physics.

## Explanations

### L1 human

The characteristic forms of [[Chern–Weil theory|82ZPRD39J7]] are closed, and on a trivial bundle they are even exact.
A Chern–Simons form is an explicit form whose derivative is such a characteristic form.
It is like an antiderivative of the curvature expression.
On a closed $3$-manifold the integral of the Chern–Simons $3$-form is a number that changes in a controlled way under gauge transformations.
This number is used to define invariants of knots and of $3$-manifolds.
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# Integral element

## Definition

Let $B$ be a [[commutative ring|0310REJJY8]] and let $A$ be a [[subring|0GNCRPYXYV]] of $B$.
An element $b \in B$ is *integral over $A$* if there are an integer $n \ge 1$ and elements $a_0, \ldots, a_{n-1} \in A$ such that
$$b^n + a_{n-1} b^{n-1} + \cdots + a_1 b + a_0 = 0.$$
Equivalently, $b$ is a root of a [[monic polynomial|0C3QBCR46M]] in $A[x]$.
More generally, for a ring homomorphism $f \colon A \to B$, an element $b$ is integral over $A$ if it is integral over the subring $f(A)$.

## Remarks

Example: $i \in \C$ is integral over $\Z$, because $i^2 + 1 = 0$ and $x^2 + 1$ is monic with integer coefficients.

Non-example: $1/2 \in \Q$ is not integral over $\Z$.
By the [[rational root theorem|0819CQ1V8H]], a rational root of a monic polynomial with integer coefficients is an integer.

The following are equivalent for $b \in B$:

1. $b$ is integral over $A$.
2. The subring $A[b]$ is a [[finitely generated|05QY1VEBYD]] $A$-module.
3. $A[b]$ is contained in a subring $C$ of $B$ that is a finitely generated $A$-module.

The proof of 3 implies 1 uses the [[determinant trick|A8HRT9H3XA]].
When $A$ and $B$ are fields, an element is integral exactly when it is [[algebraic|0H628RSGWX]], since a nonzero polynomial over a field can be divided by its leading coefficient.
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# Jensen inequality as a functional inequality

## Statement

Let $I \subseteq \R$ be an [[interval|0Q1D85B8G6]], and let $f \colon I \to \R$ be [[midpoint convex|1TQ1JQXEWP]], that is,
$$f\left(\frac{x + y}{2}\right) \le \frac{f(x) + f(y)}{2} \quad \text{for all } x, y \in I.$$
Let $n \ge 1$, let $x_1, \ldots, x_n \in I$, and let $q_1, \ldots, q_n$ be nonnegative rational numbers with $q_1 + \cdots + q_n = 1$.
Then
$$f\left(\sum_{i=1}^n q_i x_i\right) \le \sum_{i=1}^n q_i f(x_i).$$

## Remarks

The idea of the proof is Cauchy's forward and backward induction.
By induction on $k$, the inequality holds for $m = 2^k$ points with equal weights $1/m$.
For the step, split the points into two halves of $2^{k-1}$ points and apply midpoint convexity to the two averages.
For $n < 2^k$ points with mean $\bar x = \frac{1}{n} \sum x_i$, add $2^k - n$ copies of $\bar x$; the case $m = 2^k$ gives $f(\bar x) \le \frac{1}{2^k}\big(\sum f(x_i) + (2^k - n) f(\bar x)\big)$, which rearranges to $f(\bar x) \le \frac1n \sum f(x_i)$.
For rational weights $q_i = k_i / N$, repeat each point $x_i$ exactly $k_i$ times and use the equal-weight case with $N$ points.
The step that rearranges uses that $f(\bar x)$ is a real number.

Example: with $f(x) = x^2$, $q_1 = 1/3$ and $q_2 = 2/3$, the statement gives $\big(\frac{x_1 + 2x_2}{3}\big)^2 \le \frac{x_1^2 + 2x_2^2}{3}$.
The statement is the form of [[Jensen's inequality|0RWMRTPC87]] that holds without any regularity, as a consequence of the [[functional inequality|YQB2B0EXY8]] of midpoint convexity alone.
For real weights it can fail: a discontinuous additive function is midpoint convex, but it is not convex, so the inequality fails for some real weights.

## Explanations

### L1 human

Midpoint convexity is an inequality about averages of two points.
This result upgrades it to weighted averages of any finite number of points, as long as the weights are rational.
So the midpoint rule alone already controls all rational [[convex combinations|21SXYJYC59]].

### L2 human

The result is the first step in the regularity theory of midpoint convex functions.
Passing from rational to real weights needs some continuity, and results such as the [[Bernstein–Doetsch theorem|MQA1VVCJ63]] and [[Sierpiński's theorem|YBVSDT89T6]] supply it under weak hypotheses.
The same forward and backward induction proves the inequality of arithmetic and geometric means.
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<document source="https://tamnd.github.io/math2/node/X898DPBSY7.md" tag="X898DPBSY7" kind="statement">
# Baker–Campbell–Hausdorff formula

## Statement

Let $G$ be a real [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{g}$ and [[exponential map|H3QX1ZFC8K]] $\exp \colon \mathfrak{g} \to G$.
There is a neighbourhood $U$ of $0$ in $\mathfrak{g}$ such that for all $X, Y \in U$,
$$\exp(X)\exp(Y) = \exp(Z(X, Y)),$$
where $Z(X, Y)$ is the sum of a convergent series
$$Z(X, Y) = X + Y + \tfrac12 [X, Y] + \tfrac1{12}[X, [X, Y]] - \tfrac1{12}[Y, [X, Y]] - \tfrac1{24}[Y, [X, [X, Y]]] + \cdots.$$
Each term of the series is a rational multiple of an iterated Lie bracket of $X$ and $Y$.
The coefficients do not depend on $G$.

## Remarks

The formula is first proved for formal power series in two non-commuting variables, where one shows that $\log(e^X e^Y)$ is a series of Lie polynomials.
Dynkin gave an explicit formula for all the coefficients.
Convergence for $X$ and $Y$ near $0$ then follows from estimates on the norms of iterated brackets.
For matrices, $e^X e^Y = e^{Z(X, Y)}$ with the [[matrix exponential|020M04FN1H]] when $X$ and $Y$ have small enough norm.
If $X$ and $Y$ commute, all brackets vanish and the formula reduces to $e^X e^Y = e^{X + Y}$.
If $[X, Y]$ commutes with both $X$ and $Y$, as in the Heisenberg group, the series stops after the first bracket: $e^X e^Y = e^{X + Y + \frac12[X, Y]}$.
Without the smallness condition the formula can fail.
In $SL_2(\R)$, take $X = \begin{pmatrix} 0 & -\pi \\ \pi & 0 \end{pmatrix}$ and $Y = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$; then $e^X = -I$ and $e^X e^Y = \begin{pmatrix} -1 & -1 \\ 0 & -1 \end{pmatrix}$, which is not the exponential of any matrix of trace $0$.

## Explanations

### L1 human

For numbers, $e^x e^y = e^{x + y}$.
For matrices, and in Lie groups generally, this rule fails because the order of multiplication matters.
The formula says how to repair it: the exponent of the product is $X + Y$ plus correction terms built only from commutators $[X, Y] = XY - YX$ and commutators of commutators.

### L2 human

The formula shows that the group law near the identity is determined by the Lie bracket alone.
This is the key step in proving that a Lie algebra homomorphism integrates to a local homomorphism of groups, and in building a local Lie group from a Lie algebra.
For nilpotent Lie algebras the series is finite, so it defines a group law on the whole Lie algebra; this gives the simply connected nilpotent Lie groups.
Related formulas, such as the Lie product formula $e^{X + Y} = \lim_{n \to \infty} (e^{X/n} e^{Y/n})^n$, also express the interplay between the group and the bracket.
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# Mayer–Vietoris sequence

## Statement

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $A, B \subseteq X$ be subspaces whose [[interiors|0RZH6W98HK]] cover $X$.
Let $H_n$ denote [[singular homology|M0Q5CFJ7ZH]] with coefficients in an abelian group.
Then there are homomorphisms $\partial \colon H_n(X) \to H_{n-1}(A \cap B)$ such that the sequence
$$\cdots \to H_n(A \cap B) \xrightarrow{\Phi} H_n(A) \oplus H_n(B) \xrightarrow{\Psi} H_n(X) \xrightarrow{\partial} H_{n-1}(A \cap B) \to \cdots \to H_0(X) \to 0$$
is [[exact|JB7ZBFBA9C]], where $\Phi(x) = (i_* x, -j_* x)$ and $\Psi(y, z) = k_* y + l_* z$ for the inclusions $i \colon A \cap B \to A$, $j \colon A \cap B \to B$, $k \colon A \to X$ and $l \colon B \to X$.

## Remarks

Let $C_\bullet^{A+B}(X)$ be the complex of singular chains that are sums of simplices lying in $A$ or in $B$.
There is a short exact sequence of chain complexes $0 \to C_\bullet(A \cap B) \to C_\bullet(A) \oplus C_\bullet(B) \to C_\bullet^{A+B}(X) \to 0$.
Its long exact sequence in homology is the Mayer–Vietoris sequence, once one knows that the inclusion $C_\bullet^{A+B}(X) \to C_\bullet(X)$ induces isomorphisms in homology.
This last fact uses barycentric subdivision and needs the interiors of $A$ and $B$ to cover $X$.
Without this hypothesis the sequence can fail.
Take $X = [0, 1]$, $A = [0, 1/2]$ and $B = (1/2, 1]$, so that $A \cap B = \emptyset$.
Exactness at $H_0(A) \oplus H_0(B)$ would make the map $\Z^2 \to \Z = H_0(X)$ injective, which is false.
Example: write the sphere $S^n$, $n \ge 1$, as the union of two open caps, each contractible, whose intersection deformation retracts to $S^{n-1}$.
The sequence in reduced homology gives $\tilde H_k(S^n) \cong \tilde H_{k-1}(S^{n-1})$, hence $H_n(S^n; \Z) \cong \Z$.
There is a reduced version when $A \cap B \neq \emptyset$, and a dual version for singular cohomology with the arrows reversed.

## Explanations

### L1 human

If a space is glued from two pieces, its homology can be computed from the homology of the pieces and of their overlap.
The Mayer–Vietoris sequence is the exact rule for this computation.
Holes in the whole space come either from holes in the pieces or from the overlap.
If the overlap has several parts, or holes that get filled in the pieces, these create a new hole one dimension higher in the whole space.
For example, a circle is two arcs overlapping in two small arcs, and the two components of the overlap produce the loop of the circle.

### L2 human

It is the main computational tool of basic algebraic topology.
It is the analogue for open covers of the long exact sequence of a pair, and it can also be derived from that sequence by excision.
It gives the homology of spheres, of wedge sums and of connected sums, and it is the inductive step in proofs about manifolds, such as Poincaré duality.
The same pattern appears for de Rham cohomology, for sheaf cohomology and in any cohomology theory satisfying the Eilenberg–Steenrod axioms.
</document>

<document source="https://tamnd.github.io/math2/node/X8CF1K8QB4.md" tag="X8CF1K8QB4" kind="definition">
# Eikonal equation

## Definition

Let $\Omega \subseteq \R^n$ be an open set and let $f \colon \Omega \to (0, \infty)$ be a continuous function.
The *eikonal equation* with speed $1/f$ is
$$\abs{\nabla u(x)} = f(x), \qquad x \in \Omega,$$
for an unknown function $u \colon \Omega \to \R$.
Here $\nabla u$ is the [[gradient|0ABK9V9Y64]] of $u$ and $\abs{\cdot}$ is the Euclidean norm.
It is usually completed by a boundary condition such as $u = 0$ on $\partial \Omega$.

## Remarks

It is the stationary [[Hamilton–Jacobi equation|5TRXQKP7DY]] with Hamiltonian $H(p, x) = \abs{p} - f(x)$.
In optics $f$ is the refractive index, $u$ is the optical path length, and the level sets of $u$ are the wave fronts.

Example: on $\Omega = \R^n \setminus \set{0}$ with $f = 1$, the function $u(x) = \abs{x}$ is a classical solution.
Check: $\nabla u(x) = x / \abs{x}$, which has length $1$.

Non-example: on the open unit disc $\Omega$ in $\R^2$ with $f = 1$ and $u = 0$ on the circle, there is no $C^1$ solution.
Indeed a function that is continuous on the closed disc, $C^1$ in the open disc and zero on the circle is identically $0$ or has an interior maximum or minimum.
In both cases $\nabla u = 0$ somewhere in $\Omega$, against $\abs{\nabla u} = 1$.
The function $u(x) = 1 - \abs{x}$ solves the equation except at $0$, where it is not differentiable.
Written as $\abs{\nabla u} - 1 = 0$, the problem has $1 - \abs{x}$, the distance to the boundary, as its unique [[viscosity solution|H1ERJ3T75T]].
</document>

<document source="https://tamnd.github.io/math2/node/X8H7A98Q55.md" tag="X8H7A98Q55" kind="concept">
# Separable algebra

## Summary

A separable algebra over a commutative ring $K$ is an [[associative algebra|3CTW1P4HBP]] $A$ whose multiplication map $A \otimes_K A^{\mathrm{op}} \to A$ splits as a map of bimodules.
Over a field, for finite-dimensional $A$, it means that $A$ stays [[semisimple|NY25HXA478]] after every extension of the base field.
It generalizes the notion of a [[separable field extension|0WCX7NB4N7]] to noncommutative algebras, and it is the starting point of the theory of Azumaya algebras.

## Explanations

### L1 human

A semisimple algebra can lose its semisimplicity when the base field is enlarged, in the same way as an irreducible polynomial can acquire a repeated root.
Separable algebras are the ones for which this never happens.
For example, $\C$ is separable over $\R$: after tensoring with $\C$ it becomes $\C \times \C$, which is still semisimple.
Over fields of characteristic $0$, and over finite fields, every finite-dimensional semisimple algebra is separable.
</document>

<document source="https://tamnd.github.io/math2/node/X8K0Z7484R.md" tag="X8K0Z7484R" kind="statement">
# Stinespring dilation theorem

## Statement

Let $A$ be a unital C*-algebra, let $H$ be a [[Hilbert space|HM2HHE3J51]], and let $B(H)$ be the C\*-algebra of bounded operators on $H$.
Let $\varphi \colon A \to B(H)$ be a [[completely positive map|CGY299QMC3]].
Then there exist a Hilbert space $K$, a unital $*$-homomorphism $\pi \colon A \to B(K)$, and a bounded operator $V \colon H \to K$ such that
$$\varphi(a) = V^* \pi(a) V \quad \text{for all } a \in A.$$
Moreover $\norm{\varphi} = \norm{V}^2 = \norm{\varphi(1)}$.

## Remarks

Proof idea: on the algebraic tensor product $A \odot H$ define
$$\ip{a \otimes \xi}{b \otimes \eta} = \ip{\varphi(b^* a) \xi}{\eta}.$$
Complete positivity of $\varphi$ is exactly what makes this sesquilinear form positive semidefinite.
Let $K$ be the completion of the quotient by the null vectors, let $\pi(a)$ act by left multiplication, and let $V \xi$ be the class of $1 \otimes \xi$.

The converse holds: every map of the form $a \mapsto V^* \pi(a) V$ is completely positive.
If $\varphi(1) = 1$, then $V$ is an isometry and $H$ can be viewed as a subspace of $K$, so $\varphi$ is the compression of $\pi$ to $H$.
One can choose $K$ minimal, meaning that the vectors $\pi(a) V \xi$ span a dense subspace; the minimal dilation is unique up to unitary equivalence.

Example: for $H = \C$, a completely positive map $A \to \C$ with $\varphi(1) = 1$ is a state, and the theorem gives $\varphi(a) = \ip{\pi(a) \xi}{\xi}$ for a unit vector $\xi = V 1$; this is the Gelfand–Naimark–Segal construction.

## Explanations

### L1 human

A $*$-homomorphism into operators is the most rigid kind of map between operator algebras.
Stinespring's theorem says that every completely positive map is obtained from such a homomorphism, by enlarging the Hilbert space and then cutting back down to the original one.
So completely positive maps are "shadows" of representations.

### L2 human

In quantum information, the theorem says that every quantum channel comes from a unitary interaction with an environment followed by discarding the environment.
For finite-dimensional algebras it is equivalent to the Kraus form of [[Choi's theorem|EQG2K8ZFM4]].
It is a basic tool in the theory of operator systems and operator spaces.
</document>

<document source="https://tamnd.github.io/math2/node/X8MCVW56QP.md" tag="X8MCVW56QP" kind="statement">
# Balancing domain decomposition

## Statement

Let $\Omega \subset \R^d$, $d \in \set{2, 3}$, be a polygonal or polyhedral domain.
Divide it into $N$ non-overlapping subdomains $\Omega_i$ of diameter about $H$, triangulated by a quasi-uniform mesh of size $h$.
Let $\rho > 0$ be constant on each $\Omega_i$.
Consider the [[finite element|98GZ9DVHTA]] discretization of $-\nabla \cdot (\rho \nabla u) = f$ by continuous piecewise linear functions, with homogeneous Dirichlet conditions on part of $\partial \Omega$.
Let $S = \sum_i R_i^T S_i R_i$ be the interface [[Schur complement|6HWFBXYPZP]], where $R_i$ restricts interface values to $\partial \Omega_i$ and $S_i$ is the local Schur complement of $\Omega_i$.
Let $D_i$ be diagonal weight matrices with $\sum_i R_i^T D_i R_i = I$.
Let $Z_i$ have columns spanning the null space of $S_i$ (the constants for a floating subdomain), and let $R_0^T$ have the columns $R_i^T D_i Z_i$.
With $Q_0 = R_0^T (R_0 S R_0^T)^{-1} R_0$, the *balancing domain decomposition* preconditioner is
$$M^{-1} = Q_0 + (I - Q_0 S) \Bigl( \sum_{i=1}^{N} R_i^T D_i S_i^{\dagger} D_i R_i \Bigr) (I - S Q_0),$$
where $S_i^\dagger$ is a generalized inverse of $S_i$.
Choose the weights $D_i$ from the values of $\rho$ as in the standard theory.
Then the [[condition number|AB3S6MHJ8X]] of $M^{-1} S$ satisfies
$$\kappa(M^{-1} S) \le C \Bigl( 1 + \log \frac{H}{h} \Bigr)^2,$$
with $C$ independent of $h$, $H$, $N$ and the values of $\rho$.

## Remarks

The local operator $\sum_i R_i^T D_i S_i^\dagger D_i R_i$ is the Neumann–Neumann preconditioner: each $S_i^\dagger$ solves a Neumann problem on $\Omega_i$.
A Neumann problem on a subdomain that does not touch the Dirichlet boundary is solvable only if its right-hand side is orthogonal to the constants.
The coarse correction $Q_0$ "balances" the residual so that these compatibility conditions hold, and at the same time it passes global information between all subdomains.
The proof bounds the energy of the weighted average of local functions by trace and discrete harmonic extension estimates, which bring in the factor $(1 + \log(H/h))^2$.

The bound depends on $H/h$ only, so refining the mesh and adding subdomains in proportion keeps the number of [[conjugate gradient|AFXZGQB4EN]] iterations nearly constant.
The method was introduced by Jan Mandel in 1993.

## Explanations

### L1 human

Balancing domain decomposition is a way to precondition the interface problem of the [[Schur complement method|R997ZAD141]].
Each piece of the domain solves a local problem with free (Neumann) edges, and the results are averaged on the interfaces.
A small global problem fixes the parts that the local solves cannot see.
The theorem says the number of iterations grows only like a squared logarithm of the number of mesh cells per subdomain.

### L2 human

The plain Neumann–Neumann method has no coarse space and scales poorly with the number of subdomains.
BDD adds a coarse space, and the later BDDC method uses a different coarse space built from primal constraints.
Its condition number bound is of the same polylogarithmic type as that of FETI methods, and in fact the two families are closely related.
It is one of the [[domain decomposition methods|AN054RHT1K]] that are robust with respect to large jumps of the coefficient across subdomains.
</document>

<document source="https://tamnd.github.io/math2/node/X8P0H71EET.md" tag="X8P0H71EET" kind="definition">
# Frölicher–Nijenhuis bracket

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]].
Let $\Omega(M) = \bigoplus_k \Omega^k(M)$ be the graded algebra of [[differential forms|0E3G0GRPZ6]] with the [[wedge product|0GRAATMGP1]].
Let $\Omega^k(M, TM)$ be the space of $k$-forms with values in $TM$, that is, smooth sections of $\Lambda^k T^*M \otimes TM$.
A graded [[derivation|3J89REV9WQ]] of degree $j$ of $\Omega(M)$ is a linear map $D$ with $D(\Omega^k) \subseteq \Omega^{k+j}$ and $D(\alpha \wedge \beta) = D\alpha \wedge \beta + (-1)^{jk} \alpha \wedge D\beta$ for $\alpha \in \Omega^k(M)$.
The graded commutator of derivations $D_1, D_2$ of degrees $j_1, j_2$ is $[D_1, D_2] = D_1 D_2 - (-1)^{j_1 j_2} D_2 D_1$.

For $K \in \Omega^k(M, TM)$, let $\iota_K$ be the derivation of degree $k - 1$ given on decomposable $K = \alpha \otimes X$ by $\iota_K \omega = \alpha \wedge \iota_X \omega$.
Let $\mathcal{L}_K = [\iota_K, d] = \iota_K d - (-1)^{k-1} d\, \iota_K$, a derivation of degree $k$, where $d$ is the [[exterior derivative|0ZHSTAG8ZY]].

For $K \in \Omega^k(M, TM)$ and $L \in \Omega^l(M, TM)$, the *Frölicher–Nijenhuis bracket* $[K, L] \in \Omega^{k+l}(M, TM)$ is the unique element with
$$\mathcal{L}_{[K, L]} = [\mathcal{L}_K, \mathcal{L}_L].$$

## Remarks

The bracket exists and is unique because every graded derivation of $\Omega(M)$ that commutes with $d$ in the graded sense is $\mathcal{L}_K$ for exactly one vector-valued form $K$.
It is graded antisymmetric, $[K, L] = -(-1)^{kl} [L, K]$, and satisfies the graded Jacobi identity.

For $\alpha \in \Omega^k(M)$, $\beta \in \Omega^l(M)$ and vector fields $X, Y$, an explicit formula is
$$[\alpha \otimes X, \beta \otimes Y] = \alpha \wedge \beta \otimes [X, Y] + \alpha \wedge \mathcal{L}_X \beta \otimes Y - \mathcal{L}_Y \alpha \wedge \beta \otimes X + (-1)^k \big( d\alpha \wedge \iota_X \beta \otimes Y + \iota_Y \alpha \wedge d\beta \otimes X \big).$$

Example: for $k = l = 0$, take $\alpha = \beta = 1$.
The formula gives $[X, Y]$, the [[Lie bracket of vector fields|68FGQEBD6J]], and $\mathcal{L}_X$ is the usual [[Lie derivative|PJWK0PKN6Q]].
Another example: the identity $\mathrm{Id} \in \Omega^1(M, TM)$ has $\iota_{\mathrm{Id}} \omega = j \omega$ for $\omega \in \Omega^j(M)$.
Hence $\mathcal{L}_{\mathrm{Id}} \omega = (j + 1) d\omega - j\, d\omega = d\omega$, so $\mathcal{L}_{\mathrm{Id}} = d$ and $[\mathrm{Id}, L] = 0$ for every $L$.

Non-example: the naive rule $[\alpha \otimes X, \beta \otimes Y] = \alpha \wedge \beta \otimes [X, Y]$ does not define a bracket on $\Omega(M, TM)$.
For a function $f$, the elements $f\alpha \otimes X$ and $\alpha \otimes fX$ are equal, but the rule gives different results for them, since $[fX, Y] \ne f[X, Y]$ in general.
</document>

<document source="https://tamnd.github.io/math2/node/X8TVJSVTD4.md" tag="X8TVJSVTD4" kind="statement">
# Hilbert's tenth problem

## Statement

Consider polynomials $p(x_1, \ldots, x_n)$ with integer coefficients, in any number $n$ of variables.
There is no algorithm that decides, for every such $p$, whether the [[Diophantine equation|08B1RTW7VZ]] $p(x_1, \ldots, x_n) = 0$ has a solution in integers.

In other words, the set of codes of [[polynomials in several variables|07GNX8GCYP]] over $\Z$ that have an integer zero is not a [[computable set|PPGD5KW8JC]].

## Remarks

The result follows from the Matiyasevich–Robinson–Davis–Putnam (MRDP) theorem.
A set $S \subseteq \N$ is *Diophantine* if there is a polynomial $p(a, y_1, \ldots, y_m)$ with integer coefficients such that $a \in S$ exactly when $p(a, y_1, \ldots, y_m) = 0$ for some $y_1, \ldots, y_m \in \N$.
The MRDP theorem says that the Diophantine sets are exactly the [[computably enumerable sets|TN9G24VABN]].
Taking a Diophantine definition of a c.e. set that is not computable, such as the [[halting problem|Z7TN2B35B0]], shows that solvability cannot be decided.
Matiyasevich completed the proof in 1970 by showing that exponentiation is Diophantine, the last step after work of Davis, Putnam and Julia Robinson.

The question in natural numbers and the question in integers are equivalent.
One direction uses Lagrange's four-square theorem: every natural number is a sum of four squares.
For equations in one variable the problem is decidable, since the integer roots of a nonzero polynomial lie in a finite range bounded in terms of its coefficients.

## Explanations

### L1 human

Hilbert asked in 1900 for a method that tells, for any polynomial equation with whole-number coefficients, whether it has a whole-number solution.
The answer turned out to be no: no computer program can do this for all equations.
The reason is that polynomial equations are expressive enough to describe the run of any computer program.

### L2 human

This is a central example of an undecidable problem from ordinary mathematics, outside of logic.
The key tool is the MRDP theorem, which identifies Diophantine sets with computably enumerable sets.
A consequence is that there is a single polynomial whose positive values, at natural number arguments, are exactly the prime numbers.
Whether the analogous problem over $\Q$ is decidable is still open.
</document>

<document source="https://tamnd.github.io/math2/node/X8VKW5Y1CZ.md" tag="X8VKW5Y1CZ" kind="statement">
# Chebotarev's density theorem

## Statement

Let $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]] of [[algebraic number fields|MWBDFS2KJ5]] with [[Galois group|0XTY86BZKY]] $G$.
Let $C \subseteq G$ be a [[conjugacy class|0DEBXAJ5SA]].
For a nonzero prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ [[unramified|H5RC67SD24]] in $L$, let $\mathrm{Frob}_{\mathfrak{p}}$ be its [[Frobenius|H5145REFMW]] conjugacy class, and let $N(\mathfrak{p})$ be its [[norm|M54ADM0CVF]].
Then
$$\lim_{x \to \infty} \frac{\#\set{\mathfrak{p} : N(\mathfrak{p}) \le x, \ \mathfrak{p} \text{ unramified in } L, \ \mathrm{Frob}_{\mathfrak{p}} = C}}{\#\set{\mathfrak{p} : N(\mathfrak{p}) \le x}} = \frac{\abs{C}}{\abs{G}}.$$

## Remarks

The proof first treats abelian extensions, where the statement reduces, through class field theory, to the non-vanishing at $s = 1$ of Hecke $L$-functions, as in [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]].
The general case is reduced to the cyclic case by passing to the fixed field of the cyclic subgroup generated by an element of $C$.
Chebotarev's original proof also adjoins roots of unity to reduce to cyclotomic extensions.

Example: for $K = \Q$ and $L = \Q(\zeta_n)$, the group $G$ is $(\Z/n\Z)^\times$ and $\mathrm{Frob}_p$ is the class of $p$ modulo $n$.
The theorem gives that the primes are equidistributed among the $\varphi(n)$ classes modulo $n$ coprime to $n$, which is Dirichlet's theorem in its density form.

Example: taking $C = \set{1}$, the primes of $K$ that split completely in $L$ have density $1/[L : K]$.
For $L = \Q(i)$, half the primes split, namely those $\equiv 1 \pmod 4$.

## Explanations

### L1 human

In a Galois extension, each unramified prime has a Frobenius symmetry that records how it splits.
The theorem says that these symmetries are spread out evenly over the Galois group, in proportion to the size of each conjugacy class.
So in the long run, every possible splitting pattern happens with a predictable frequency.

### L2 human

The theorem generalises [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]] from cyclotomic extensions of $\Q$ to all Galois extensions.
It is used to show that a Galois extension is determined by the set of primes that [[split completely|B7KNZCBFYT]] in it.
It also gives the density of primes for which a polynomial has a given factorisation pattern modulo $p$, by looking at the cycle types in its Galois group.
Effective versions with explicit error terms depend on zero-free regions of [[Dedekind zeta functions|KXS5DQDFYD]].
</document>

<document source="https://tamnd.github.io/math2/node/X8WD0ABG07.md" tag="X8WD0ABG07" kind="definition">
# Lattice

## Definition

A *lattice* is a [[partially ordered set|0QV2JS9RJ3]] $(L, \le)$ in which every two elements $x, y \in L$ have a [[join|J2T75ATRV5]] $x \vee y$ and a meet $x \wedge y$.

Algebraic form: a *lattice* is a set $L$ with two binary operations $\vee$ and $\wedge$ such that, for all $x, y, z \in L$:

1. $x \vee y = y \vee x$ and $x \wedge y = y \wedge x$;
2. $(x \vee y) \vee z = x \vee (y \vee z)$ and $(x \wedge y) \wedge z = x \wedge (y \wedge z)$;
3. $x \vee (x \wedge y) = x$ and $x \wedge (x \vee y) = x$.

## Remarks

The two forms are equivalent.
From a lattice order, the join and meet satisfy clauses 1 to 3.
From the operations, the order is recovered by $x \le y \iff x \vee y = y$, which holds exactly when $x \wedge y = x$.
The idempotent laws $x \vee x = x$ and $x \wedge x = x$ follow from the absorption laws in clause 3.
Example: $\N_{>0}$ ordered by divisibility is a lattice with $a \vee b = \lcm(a, b)$ and $a \wedge b = \gcd(a, b)$.
Indeed $\lcm(a, b)$ is a common multiple of $a$ and $b$ and divides every common multiple.
Non-example: the poset $\set{a, b, c, d}$ with $a, b < c$ and $a, b < d$ and no other strict relations is not a lattice.
The elements $a$ and $b$ have the two upper bounds $c$ and $d$, which are incomparable, so $a \vee b$ does not exist.
</document>

<document source="https://tamnd.github.io/math2/node/X9072A41NK.md" tag="X9072A41NK" kind="concept">
# Residually finite algebra

## Summary

An algebra is residually finite when any two distinct elements can be told apart in some finite homomorphic image.
Equivalently, it is a [[subdirect product|88WN3AD2VQ]] of finite algebras.
For groups this is the notion of a [[residually finite group|HEE922EHQ8]].
Residual finiteness lets one study an infinite algebra through its finite quotients, and it is linked to the solvability of word problems.

## Explanations

### L1 human

Imagine you can only look at finite pictures of an algebra.
The algebra is residually finite if, whenever two elements are different, some finite picture already shows the difference.
The integers are an example: two different integers $a$ and $b$ have different remainders modulo any $n > \abs{a - b}$.
The rationals under addition are not: they have no nontrivial finite quotient, so no finite picture separates $0$ from $1$.
</document>

<document source="https://tamnd.github.io/math2/node/X93AEE88PX.md" tag="X93AEE88PX" kind="statement">
# Bangert–Franks theorem on closed geodesics on the sphere

## Statement

Let $g$ be a smooth [[Riemannian metric|Q61VFHCH2J]] on the two-dimensional [[sphere|P9CM4GQ2T0]] $S^2$.
Then $(S^2, g)$ has infinitely many geometrically distinct [[closed geodesics|4MCCP8GG4J]].

## Remarks

The proof combines two results.
Franks treated, with a theorem on area-preserving homeomorphisms of the open annulus, the metrics for which a simple closed geodesic gives an annulus of section for the geodesic flow.
Bangert treated the remaining metrics by variational methods on the loop space.
The [[theorem of the three geodesics|AB9FNZD18B]] gives a simple closed geodesic to start from.

The [[Gromoll–Meyer theorem|XC8BRYJN93]] does not apply here, because the Betti numbers of the free loop space of $S^2$ are bounded.
The hypothesis that the metric is Riemannian matters: Katok constructed non-reversible Finsler metrics on $S^2$ with only finitely many closed geodesics.

## Explanations

### L1 human

However one bends a sphere, it carries infinitely many different closed loops that go straight ahead, though they may cross themselves.
On a round sphere this is clear, since every great circle is one.
The theorem says that no smooth deformation, however irregular, can destroy all but finitely many of them.

### L2 human

The result answers a long-standing question for the simplest closed surface.
It mixes two methods: dynamics of the geodesic flow, through a surface of section and a return map of an annulus, and Morse theory of the energy on loop spaces.
For higher-dimensional spheres the corresponding question for Riemannian metrics is open.
</document>

<document source="https://tamnd.github.io/math2/node/X9B1XETBPF.md" tag="X9B1XETBPF" kind="concept">
# Permanent

## Summary

The permanent of a square matrix is the sum, over all permutations, of the products of one entry from each row and each column.
It is the [[determinant|00KDZRW4C0]] formula without the signs.
For the $0/1$ biadjacency matrix of a [[bipartite graph|0ZVQKTRKVQ]], it counts the [[perfect matchings|0NHK9MFFW9]].
Unlike the determinant, it is believed to be hard to compute: computing the permanent of a $0/1$ matrix is #P-complete.

## Explanations

### L1 human

Put $n$ rooks on an $n \times n$ board so that no two share a row or a column, and multiply the entries under the rooks.
The permanent is the sum of these products over all such placements.
The determinant uses the same products but adds some of them with a minus sign.
For $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ the permanent is $ad + bc$, while the determinant is $ad - bc$.
</document>

<document source="https://tamnd.github.io/math2/node/X9F8SQFQGP.md" tag="X9F8SQFQGP" kind="definition">
# Adjoint state method

## Definition

Let $F \colon \R^n \times \R^p \to \R^n$ and $j \colon \R^n \times \R^p \to \R$ be continuously differentiable, written $F(u, \theta)$ and $j(u, \theta)$.
Let $\theta_0 \in \R^p$ and let $u(\theta)$ be a continuously differentiable map defined near $\theta_0$ with $F(u(\theta), \theta) = 0$.
Assume that the [[Jacobian matrix|047NDTEPCV]] $F_u = \partial F / \partial u$ is invertible at $(u(\theta), \theta)$.
Let $J(\theta) = j(u(\theta), \theta)$ be the reduced objective.
The *adjoint state* at $\theta$ is the vector $\lambda \in \R^n$ that solves the *adjoint equation*
$$F_u(u(\theta), \theta)^T \lambda = \nabla_u j(u(\theta), \theta).$$
The *adjoint state method* computes the [[gradient|0ABK9V9Y64]] of $J$ by the formula
$$\nabla J(\theta) = \nabla_\theta j(u(\theta), \theta) - F_\theta(u(\theta), \theta)^T \lambda.$$

## Remarks

The formula follows from the [[chain rule|0RXX2PT1MQ]].
Differentiating $F(u(\theta), \theta) = 0$ gives $F_u \, u'(\theta) = -F_\theta$, so $\nabla J^T = j_u u'(\theta) + j_\theta = -j_u F_u^{-1} F_\theta + j_\theta = -\lambda^T F_\theta + j_\theta$.
By the [[implicit function theorem|0BXBADR94C]], invertibility of $F_u$ at a solution gives the smooth map $u(\theta)$ locally.
The forward method needs $p$ linear solves with $F_u$, one for each column of $u'(\theta)$; the adjoint method needs one solve with $F_u^T$.
For a time-dependent state equation the adjoint equation runs backward in time; in optimal control it is the [[costate equation|WS9MRAZMFW]].

Example: $F(u, \theta) = A u - \theta$ with $A$ an invertible $n \times n$ matrix and $p = n$, and $j(u, \theta) = \frac{1}{2} \abs{u - d}^2$ for a fixed $d \in \R^n$.
Then $F_u = A$, $F_\theta = -I$ and $\nabla_\theta j = 0$, so $A^T \lambda = u - d$ and $\nabla J(\theta) = \lambda$.
Directly, $J(\theta) = \frac{1}{2} \abs{A^{-1} \theta - d}^2$ has gradient $A^{-T} (A^{-1} \theta - d) = \lambda$, as it should.

Non-example: let $n = p = 1$ and $F(u, \theta) = u^2 - \theta$, at the point $u = \theta = 0$, where $F_u = 0$ is singular.
The adjoint equation $0 \cdot \lambda = j_u$ need not have a solution, and the state is not a differentiable function of $\theta$ there.

## Explanations

### L1 human

The gradient of an objective constrained by an equation is found from one extra linear solve with the transposed Jacobian, whatever the number of parameters.
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<document source="https://tamnd.github.io/math2/node/X9HEQ29Q49.md" tag="X9HEQ29Q49" kind="concept">
# Direct limit

## Summary

A direct limit, or inductive limit, is the [[colimit|BJS1CGTKQ5]] of a system of objects indexed by a [[directed set|D9RY4667K6]].
Its maps go from earlier objects to later ones.
It builds a large object as the union of a growing family of smaller ones.
Many objects of algebra are built this way, such as the [[Prüfer group|F0XAMM2PPS]] and stalks of sheaves.
The dual notion is the [[inverse limit|DZB378CNCK]].

## Explanations

### L1 human

Picture a chain of boxes, each one placed inside the next.
The direct limit is the big box that holds all of them at once.
Two elements of the boxes count as the same once they become equal in some later box.
For example, the groups $\frac{1}{2^n}\Z$ for $n = 0, 1, 2, \dots$ sit inside each other.
Their direct limit is the group of all fractions whose denominator is a power of $2$.
</document>

<document source="https://tamnd.github.io/math2/node/X9HVQPWT6T.md" tag="X9HVQPWT6T" kind="example">
# Padua points

## Example

Let $n \ge 1$ and consider the curve $\gamma \colon [0, \pi] \to [-1, 1]^2$,
$$\gamma(t) = \bigl(-\cos((n + 1) t), \ -\cos(n t)\bigr).$$
The *Padua points* of degree $n$ are the points $\gamma\bigl(k\pi / (n(n + 1))\bigr)$ for $k = 0, 1, \dots, n(n + 1)$.
These are the self-intersections of $\gamma$ together with its points on the boundary of the square.
For $n \ge 2$ several of these parameter values give the same point, and there are exactly $(n + 1)(n + 2)/2$ distinct Padua points.

This number is the dimension of the space of [[polynomials in two variables|07GNX8GCYP]] of total degree at most $n$.
The Padua points are a [[unisolvent point set|5276MDSDQV]] for this space.
So each choice of values at the Padua points is taken by exactly one polynomial of total degree at most $n$.

Check for $n = 1$: the parameters $0$, $\pi/2$, $\pi$ give the points $(-1, -1)$, $(1, 0)$, $(-1, 1)$.
They are not collinear, so they are unisolvent for the polynomials $a + bx + cy$.

The [[Lebesgue constant|7V7KWNK9HN]] of interpolation at the Padua points is $O(\log^2 n)$ as $n \to \infty$.

## Explanations

### L1 human

In one variable, Chebyshev nodes are a near-ideal choice of points for polynomial interpolation on an interval.
In the square, finding a good set of points that also gives a unique interpolating polynomial is much harder.
The Padua points solve this: they are the crossing points of a Lissajous-type curve that sweeps through the square, plus its touching points on the edges.
Interpolation at these points is unique and numerically well behaved.
</document>

<document source="https://tamnd.github.io/math2/node/X9PBTW2J8D.md" tag="X9PBTW2J8D" kind="definition">
# Hecke operator

## Definition

Let $k$ be an integer, let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]], and let $M_k$ be the space of [[modular forms|GWKN5VG78A]] of weight $k$ for $\mathrm{SL}_2(\Z)$.
For an integer $n \ge 1$, the *Hecke operator* $T_n \colon M_k \to M_k$ is given by
$$(T_n f)(\tau) = n^{k-1} \sum_{\substack{ad = n \\ a, d \ge 1}} \ \sum_{b = 0}^{d - 1} d^{-k} f\left(\frac{a\tau + b}{d}\right).$$

## Remarks

If $f = \sum_{m \ge 0} c(m) q^m$ with $q = e^{2 \pi i \tau}$, then $T_n f = \sum_{m \ge 0} \gamma(m) q^m$ with
$$\gamma(m) = \sum_{d \mid \gcd(m, n)} d^{k-1} c(mn/d^2).$$
The operators commute, $T_m T_n = T_{mn}$ when $\gcd(m, n) = 1$, and $T_p T_{p^r} = T_{p^{r+1}} + p^{k-1} T_{p^{r-1}}$ for primes $p$ and $r \ge 1$.
They map the [[cusp forms|A50Y012RBP]] $S_k$ into $S_k$.
Example: $S_{12} = \C \Delta$ is one-dimensional, so $T_n \Delta$ is a multiple of $\Delta$; comparing the coefficients of $q$ gives $T_n \Delta = \tau(n) \Delta$.
Non-example: the map $f(\tau) \mapsto f(n\tau)$ alone does not preserve $M_k$.
For instance, $E_4(2\tau) = 1 + 240 q^2 + \cdots$ is not a multiple of $E_4 = 1 + 240 q + \cdots$, while $M_4 = \C E_4$.
</document>

<document source="https://tamnd.github.io/math2/node/X9R1PCKJ0R.md" tag="X9R1PCKJ0R" kind="concept">
# Sub-Gaussian distribution

## Summary

A real random variable is sub-Gaussian when its tails decay at least as fast as those of a [[normal distribution|05AQE0QR0D]], that is, like $e^{-c t^2}$.
Normal variables and bounded variables are sub-Gaussian.
Sums of independent sub-Gaussian variables are again sub-Gaussian, with explicit constants, and this gives concentration bounds of [[Hoeffding|0WPDNXGSSN]] type.
The notion is central in high-dimensional probability, random matrix theory and statistical learning.

## Explanations

### L1 human

A sub-Gaussian variable is one that is very unlikely to be far from $0$: the chance of exceeding $t$ drops like $e^{-c t^2}$, as for a bell curve.
A fair $\pm 1$ coin is sub-Gaussian, since it never exceeds $1$ at all.
An [[exponential|0E99F51GVB]] variable is not, since $P(X > t) = e^{-t}$ decays only like $e^{-t}$.
The average of many independent sub-Gaussian variables is sharply concentrated around its mean.
</document>

<document source="https://tamnd.github.io/math2/node/X9SRCEENV3.md" tag="X9SRCEENV3" kind="concept">
# Laguerre–Pólya class

## Summary

The Laguerre–Pólya class is the class of entire functions that are limits, uniformly on compact sets, of real polynomials with only real zeros.
Its members are described by an explicit Hadamard-type product with real zeros and a Gaussian factor.
The class is closed under differentiation, so it is the natural home of real-rootedness for entire functions.
It appears in the study of the zeros of the Riemann zeta function and of special functions.

## Explanations

### L1 human

A real polynomial whose roots are all real has a graph that meets the real axis at each of its roots.
The functions in the Laguerre–Pólya class are the entire functions that are limits of such polynomials.
Examples are $\sin z$, $\cos z$ and $e^{-z^2}$, the last being a limit of $(1 - z^2/n)^n$.
The function $e^{z^2}$ is not in the class, even though it has no zeros at all.
</document>

<document source="https://tamnd.github.io/math2/node/X9V1E3V145.md" tag="X9V1E3V145" kind="statement">
# F. and M. Riesz theorem

## Statement

Let $\mu$ be a [[complex measure|NY6TYHRGMS]] on the Borel sets of $[0, 2\pi)$, viewed as the unit circle.
For $n \in \Z$ let $\hat\mu(n) = \int e^{-int} \, d\mu(t)$.
Assume that $\hat\mu(n) = 0$ for every integer $n < 0$.
Then $\mu$ is [[absolutely continuous|P0P21KDV6E]] with respect to [[Lebesgue measure|2Z2QVWZ262]].
Moreover $d\mu = h \, dt$ for an integrable function $h$ whose [[Poisson integral|TCYSWCTYYM]] is holomorphic on the unit disc and belongs to the [[Hardy space|PNNNFJVF6W]] $H^1$.

## Remarks

Proof idea: the Poisson integral of $\mu$ is $F(z) = \frac{1}{2\pi} \sum_{n \ge 0} \hat\mu(n) z^n$, a holomorphic function.
Its integrals $\int \abs{F(r e^{it})} \, dt$ are bounded by the total variation of $\mu$, so $F \in H^1$.
A function in $H^1$ has boundary values $F^*$ with $F(r e^{it}) \to F^*(e^{it})$ in $L^1$, a fact that uses the [[inner-outer factorization|X6ZQPAQF76]].
Hence the measures $F(r e^{it}) \, dt$ converge both to $\mu$ and to $F^* \, dt$, and $\mu = F^* \, dt$.

Example: if $\mu$ is a point mass at $t = 0$, then $\hat\mu(n) = 1$ for all $n$, so the hypothesis fails, as it must since a point mass is not absolutely continuous.

A consequence: if $\Omega$ is a domain bounded by a rectifiable Jordan curve, then harmonic measure on $\partial\Omega$ and arc length have the same null sets.

## Explanations

### L1 human

A measure on the circle has Fourier coefficients, one for each integer.
If all the coefficients with negative index vanish, the measure is "analytic".
The theorem says that an analytic measure cannot be concentrated on a small set: it must have a density.
So analytic measures are much more regular than general measures.

### L2 human

The theorem identifies the analytic measures on the circle with the boundary values of $H^1$ functions.
It is a key step in the theory of [[Hardy spaces|PNNNFJVF6W]] and in the study of boundary behaviour of conformal maps.
Its consequence for rectifiable Jordan domains links the [[harmonic measure|CFD7BPM51W]] with arc length.
</document>

<document source="https://tamnd.github.io/math2/node/X9WYNYE29C.md" tag="X9WYNYE29C" kind="statement">
# Gödel's incompleteness theorems

## Statement

Let $T$ be a consistent first-order [[theory|92SWXST1ZM]] in the language of arithmetic whose set of axioms is [[computably enumerable|TN9G24VABN]].

1. (First incompleteness theorem.) If $T$ contains [[Robinson arithmetic|3K1AKGXF8A]] $\mathsf{Q}$, then there is a [[sentence|DKFGQTRYZE]] $G$ such that $T \nvdash G$ and $T \nvdash \neg G$. So $T$ is not [[complete|655K8Y7VS8]].
2. (Second incompleteness theorem.) If $T$ contains the [[Peano axioms|0QFT7Z76N5]] $\mathsf{PA}$, then $T \nvdash \mathrm{Con}(T)$. Here $\mathrm{Prov}_T$ is the standard provability predicate of $T$ for a [[Gödel numbering|MFJXPAF0R0]], and $\mathrm{Con}(T)$ is $\neg \mathrm{Prov}_T(\ulcorner 0 = S0 \urcorner)$.

## Remarks

Proof idea for 1: by the [[diagonal lemma|3PRW76471J]] there is a sentence $G$ with $T \vdash G \leftrightarrow \neg \mathrm{Prov}_T(\ulcorner G \urcorner)$.
If $T \vdash G$, then $T \vdash \mathrm{Prov}_T(\ulcorner G \urcorner)$, and $T$ is inconsistent.
Gödel's argument that $T \nvdash \neg G$ used $\omega$-consistency; Rosser's variant of the sentence needs only consistency, which gives the form stated here.
Proof idea for 2: the argument for "if $T$ is consistent then $T \nvdash G$" can be carried out inside $T$, so $T \vdash \mathrm{Con}(T) \to G$; thus $T \vdash \mathrm{Con}(T)$ would give $T \vdash G$.

The hypotheses matter.
The theory of the true sentences of $\N$ is complete and consistent, but its axioms are not computably enumerable.
[[Presburger arithmetic|B66JYY3K58]], with addition but no multiplication, is consistent, computably axiomatized and complete.

## Explanations

### L1 human

The first theorem says that any reasonable list of axioms for arithmetic leaves some statements undecided: neither they nor their negations can be proved.
The second says that such a system cannot prove its own consistency.
So no single computer-checkable set of axioms captures all truths about the natural numbers.

### L2 human

The proofs turn statements about proofs into statements about numbers by a [[Gödel numbering|MFJXPAF0R0]], and then use the [[diagonal lemma|3PRW76471J]] to build a self-referential sentence.
The first theorem implies that every consistent computably axiomatized extension of $\mathsf{Q}$ is incomplete, and with more work that such theories are undecidable.
The second theorem ended the hope of Hilbert's program to prove the consistency of mathematics by finitary means within mathematics itself.
Concrete true statements unprovable in $\mathsf{PA}$ include [[Goodstein's theorem|XQMNJF241J]] and the [[Paris–Harrington theorem|509ZWA0R26]].
</document>

<document source="https://tamnd.github.io/math2/node/X9XRGMDCQZ.md" tag="X9XRGMDCQZ" kind="statement">
# Weyl's theorem on compact perturbations

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]].
Let $A$ be a bounded self-adjoint operator on $H$, that is, $A = A^*$ with $A^*$ the [[adjoint|6HSJBJ8TTQ]] of $A$.
Let $K$ be a [[compact operator|M2NEP1K5AE]] on $H$ with $K = K^*$.
Then
$$\sigma_{\mathrm{ess}}(A + K) = \sigma_{\mathrm{ess}}(A),$$
where $\sigma_{\mathrm{ess}}$ denotes the [[essential spectrum|FRZG0E15SK]].

## Remarks

For a self-adjoint operator, $\lambda$ lies in the essential spectrum if and only if there is a Weyl sequence: an orthonormal sequence $(x_n)$ in $H$ with $\norm{(A - \lambda) x_n} \to 0$.
An orthonormal sequence tends to $0$ weakly, and a compact operator maps weakly null sequences to norm null sequences, so $\norm{K x_n} \to 0$.
Hence a Weyl sequence for $A$ at $\lambda$ is also one for $A + K$, and the symmetric argument gives the reverse inclusion.
The theorem also holds for unbounded self-adjoint $A$, with $A + K$ defined on the domain of $A$.
In other words, a compact perturbation of a self-adjoint operator can only change its [[discrete spectrum|PJWY3JJQQ7]], the isolated eigenvalues of finite multiplicity.
With the definition of the essential spectrum through Fredholm operators, the equality holds for every bounded operator $A$ on a Banach space and every compact $K$.
Indeed, $A + K - \lambda I$ is Fredholm exactly when $A - \lambda I$ is.
Example: on $\ell^2$, the zero operator has essential spectrum $\set{0}$, and so does every compact self-adjoint operator, such as the diagonal operator with entries $1/n$.
Without compactness the conclusion fails: a self-adjoint projection $P$ of infinite rank and infinite corank has $\sigma_{\mathrm{ess}}(0 + P) = \set{0, 1}$, while $\sigma_{\mathrm{ess}}(0) = \set{0}$.

## Explanations

### L1 human

The essential spectrum is the robust part of the spectrum of an operator, the part that does not come from finitely many isolated eigenvalues.
Weyl's theorem says that adding a compact operator, a small perturbation in a precise sense, never changes this robust part.
Only the isolated eigenvalues of finite multiplicity can move, appear or disappear.

### L2 human

The theorem is a basic tool of mathematical physics.
For a Schrödinger operator $-\Delta + V$ with a potential $V$ that decays at infinity, one shows that $V$ is a relatively compact perturbation of $-\Delta$.
A version of Weyl's theorem for relatively compact perturbations then gives $\sigma_{\mathrm{ess}}(-\Delta + V) = [0, \infty)$, so the negative spectrum consists of isolated eigenvalues, the bound states.
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<document source="https://tamnd.github.io/math2/node/XA0DCNH7WY.md" tag="XA0DCNH7WY" kind="concept">
# Jacobson radical

## Summary

The Jacobson radical of a commutative ring is the intersection of all its [[maximal ideals|09J6KGT65G]].
An element lies in it exactly when $1 - xy$ is a [[unit|019NFGFPR5]] for every $y$ in the ring.
It contains the [[nilradical|QQDN6EYQNJ]], and it is the ideal that appears in [[Nakayama's lemma|15JHBYR6KH]].
For a [[local ring|MZ02BKPHJA]] it is the unique maximal ideal.

## Explanations

### L1 human

The Jacobson radical collects the elements that lie in every maximal ideal.
Such an element is "small" in a precise sense: adding a multiple of it to $1$ never destroys invertibility.
In $\Z$ the maximal ideals are $p\Z$ for the primes $p$, and no nonzero integer is divisible by every prime, so the Jacobson radical of $\Z$ is $\set{0}$.
In $\Z/12\Z$ the maximal ideals are generated by $2$ and $3$, and the Jacobson radical is $\set{0, 6}$.
For noncommutative rings the same name is used for the intersection of the maximal left ideals.
</document>

<document source="https://tamnd.github.io/math2/node/XA2H0YDBG2.md" tag="XA2H0YDBG2" kind="definition">
# Logarithmic potential

## Definition

Identify $\R^2$ with $\C$.
Let $\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\C$ with finite total mass and compact [[support|FQTXV6BQZG]].
The *logarithmic potential* of $\mu$ is the function $p_\mu \colon \C \to (-\infty, +\infty]$,
$$p_\mu(z) = \int \log \frac{1}{\abs{z - w}} \, d\mu(w).$$
It is the [[potential|JHKSF3XFY2]] of $\mu$ with respect to the [[logarithmic kernel|VDZ5FKYC2T]].

## Remarks

Example: let $\mu$ be the uniform probability measure on the unit circle.
Then $p_\mu(z) = 0$ for $\abs{z} \le 1$ and $p_\mu(z) = -\log\abs{z}$ for $\abs{z} \ge 1$, by the mean value property.
For $\abs{z} > 1$ apply it to the harmonic function $w \mapsto \log\abs{z - w}$ on the disc $\abs{w} < \abs{z}$.
For $\abs{z} < 1$ write $\abs{z - e^{it}} = \abs{1 - z e^{-it}}$ and apply it to the harmonic function $\zeta \mapsto \log\abs{1 - z\zeta}$ on the disc $\abs{\zeta} < 1/\abs{z}$; the case $\abs{z} = 1$ follows from $\int_0^{2\pi} \log\abs{1 - e^{it}} \, dt = 0$.
Example: for $\mu = \sum_{j=1}^d \delta_{z_j}$ one gets $p_\mu(z) = -\log\abs{(z - z_1)\cdots(z - z_d)}$.
Non-example: for Lebesgue measure on all of $\C$ the integral diverges at every point, which is why the definition asks for a finite measure with compact support.
The function $p_\mu$ is [[harmonic|0DEB0JMN0W]] on $\C \setminus \operatorname{supp}\mu$, and $p_\mu(z) = -\mu(\C) \log\abs{z} + o(1)$ as $\abs{z} \to \infty$.
In the sense of distributions, $\Delta p_\mu = -2\pi\mu$.
Some authors use the opposite sign, $\int \log\abs{z - w} \, d\mu(w)$, which is subharmonic.
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<document source="https://tamnd.github.io/math2/node/XA35DT40XS.md" tag="XA35DT40XS" kind="problem">
# Hilbert's fourth problem

## Problem

Let $U$ be an open convex subset of $\R^n$.
Determine all [[metrics|0VKD1TP7F3]] $d$ on $U$ for which every straight [[line|0MP9BCF6DP]] segment in $U$ is a shortest path.
That is, determine all $d$ such that $d(x, z) = d(x, y) + d(y, z)$ whenever $y$ lies on the segment from $x$ to $z$.

## Remarks

Such metrics are called projective metrics, and their [[geodesics|NSYMQMHPDN]] include the straight lines.
Examples are the Euclidean metric and the metrics given by any norm on $\R^n$.
Another example is the Hilbert metric on a bounded convex domain; when the domain is a disc, it gives the Beltrami–Klein model of the hyperbolic plane.
Hilbert posed the question in his list of problems of 1900.
Herbert Busemann gave a construction of such metrics from measures on the space of hyperplanes.
Aleksei Pogorelov proved that in the plane every continuous projective metric arises in this way.

## Explanations

### L1 human

In ordinary geometry the shortest way between two points is the straight line.
Hilbert asked which other ways of measuring distance keep this property.
There are many: for example, the taxicab distance $\abs{x_1 - y_1} + \abs{x_2 - y_2}$ in the plane also makes every straight segment a shortest path, although it is not the only shortest path.
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<document source="https://tamnd.github.io/math2/node/XA8CY1X2BE.md" tag="XA8CY1X2BE" kind="definition">
# Matrix Lie group

## Definition

Let $n \ge 1$ and let $GL_n(\C)$ be the [[general linear group|0FCHE902D7]] of invertible complex $n \times n$ matrices, with the topology of $\C^{n^2}$.
A *matrix Lie group* is a [[subgroup|0X1JXT47GA]] $G$ of $GL_n(\C)$ that is a [[closed subset|0ZCK8DQYN1]] of $GL_n(\C)$.
That is, if $A_k \in G$ and $A_k \to A$ entrywise with $A \in GL_n(\C)$, then $A \in G$.

## Remarks

Example: $SL_n(\R) = \set{A \in GL_n(\C) : A \text{ real}, \det A = 1}$ is a subgroup, and it is closed because the entries being real and $\det$ being $1$ are conditions preserved under limits, as $\det$ is continuous.

Non-example: let $\alpha$ be irrational and $G = \set{\operatorname{diag}(e^{it}, e^{i\alpha t}) : t \in \R}$.
It is a subgroup of $GL_2(\C)$, but it is not closed: its closure is the set of all $\operatorname{diag}(e^{i\theta}, e^{i\varphi})$, which contains $\operatorname{diag}(1, -1) \notin G$.

The condition is closedness in $GL_n(\C)$, not in the space of all matrices: $GL_n(\C)$ itself is a matrix Lie group.
Closed subgroups of $GL_n(\R)$ are matrix Lie groups too.
By the [[closed subgroup theorem|3NJGY8QSTE]], a matrix Lie group is an embedded Lie subgroup of $GL_n(\C)$.
Its Lie algebra is $\mathfrak{g} = \set{X \in M_n(\C) : e^{tX} \in G \text{ for all } t \in \R}$, with the [[matrix exponential|020M04FN1H]] and the bracket $[X, Y] = XY - YX$.
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# Sub-Riemannian geodesic problem

## Problem

Let $M$ be a connected smooth manifold, and let $\Delta \subseteq TM$ be a smooth subbundle of the tangent bundle, the horizontal distribution.
Let $g$ be a smooth inner product on the fibres of $\Delta$, so that $(M, \Delta, g)$ is a [[sub-Riemannian manifold|97NYHNBZ2D]].
A curve $\gamma \colon [0, 1] \to M$ is *horizontal* if it is [[absolutely continuous|ZT6F4C2YHK]] and $\gamma'(t) \in \Delta_{\gamma(t)}$ for almost every $t$.
Its length is $\ell(\gamma) = \int_0^1 \sqrt{g(\gamma'(t), \gamma'(t))} \, dt$.
Let $a, b \in M$.
The *sub-Riemannian geodesic problem* is to find a horizontal curve from $a$ to $b$ of least length among all horizontal curves from $a$ to $b$.

## Remarks

If $\Delta$ is bracket generating, the [[Chow–Rashevskii theorem|VQPN24VDH1]] shows that any two points are joined by a horizontal curve.
So the infimum of lengths, the sub-Riemannian distance, is finite.
If moreover the distance is complete, any two points are joined by a length minimizer, by a compactness argument.
Locally, with an orthonormal frame $X_1, \dots, X_k$ of $\Delta$, the problem is the [[optimal control|T3V6ZNAMDN]] problem $\gamma' = \sum_{i=1}^k u_i X_i(\gamma)$ with cost $\int_0^1 \sum_i u_i^2 \, dt$.
[[Pontryagin's maximum principle|7RPV2T29CF]] gives normal geodesics, which are projections of solutions of a Hamiltonian system on $T^*M$, and abnormal extremals.
Unlike in Riemannian geometry, some length minimizers can be abnormal and fail to be normal geodesics.
Example: in the Heisenberg group, $\R^3$ with $\Delta$ spanned by $\partial_x - \frac{y}{2} \partial_z$ and $\partial_y + \frac{x}{2} \partial_z$, the geodesics project to lines and circles in the $(x, y)$-plane.
The length of a horizontal curve is the length of its projection.
Take a horizontal curve that starts at the origin.
Its change in $z$ equals the signed area enclosed by its projection and the segment from the endpoint back to the origin.
So the problem becomes a version of Dido's isoperimetric problem.

## Explanations

### L1 human

In a sub-Riemannian space one may only move in certain allowed directions at each point, like a car that cannot slide sideways.
Still, by combining allowed moves, one can reach every point, just as a car can parallel park.
The geodesic problem asks for the shortest allowed path between two points.
These shortest paths can look very different from straight lines, and they connect geometry with optimal control.
</document>

<document source="https://tamnd.github.io/math2/node/XAANCQH8A4.md" tag="XAANCQH8A4" kind="definition">
# Proportional hazards model

## Definition

Let $T$ be an event time with values in $[0, \infty)$ and let $x \in \R^p$ be a vector of covariates.
The *Cox proportional hazards model* is the following [[statistical model|K3AMMSY8DD]].
The conditional [[hazard function|Y959NKQ3XW]] of $T$ given $x$ is
$$h(t \mid x) = h_0(t) \, e^{\beta^T x}, \qquad t \ge 0,$$
where $\beta \in \R^p$ is an unknown parameter and the baseline hazard $h_0 \colon [0, \infty) \to [0, \infty)$ is an unknown function.

## Remarks

For two covariate vectors $x$ and $x'$, the hazard ratio $h(t \mid x) / h(t \mid x') = e^{\beta^T (x - x')}$ does not depend on $t$; this is the proportional hazards property.
Equivalently, the survival functions satisfy $S(t \mid x) = S_0(t)^{e^{\beta^T x}}$, where $S_0$ is the baseline survival function.

Given right-[[censored|MY8ZZP84NP]] data $(\tilde t_i, \delta_i, x_i)$ without tied times, $\beta$ is estimated by maximising the partial [[likelihood|0VW9E4QXR6]]
$$L(\beta) = \prod_{i \,:\, \delta_i = 1} \frac{e^{\beta^T x_i}}{\sum_{j \,:\, \tilde t_j \ge \tilde t_i} e^{\beta^T x_j}}.$$

Example: the Weibull model with shape $k > 0$ and hazard $h(t \mid x) = k t^{k-1} \lambda^{-k} e^{\beta^T x}$ is a proportional hazards model with $h_0(t) = k t^{k-1} \lambda^{-k}$.

Non-example: if a treatment has hazard $h_1(t) = 2t$ and the control has $h_0(t) = 1$, the ratio $2t$ changes with time and the curves cross at $t = 1/2$.
No constant $\beta$ satisfies $h_1 = h_0 e^{\beta}$, so these two groups do not follow a proportional hazards model.
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<document source="https://tamnd.github.io/math2/node/XAARVNKXF1.md" tag="XAARVNKXF1" kind="concept">
# Uniform approximation

## Summary

Uniform approximation is [[function approximation|VNWH2KADN7]] in which the error is the largest pointwise difference, the supremum norm $\sup_x \abs{f(x) - g(x)}$.
A function is uniformly approximable by a class when it is a [[uniform limit|08FPABDQ4C]] of functions from that class.
It is the most demanding common way to measure an approximation, because it controls the error at every point at once.

## Explanations

### L1 human

Draw a thin band of width $\varepsilon$ around the graph of $f$.
A uniform approximation within $\varepsilon$ is a function whose whole graph stays inside the band.
For example, on $[0, 1]$ the polynomial $x - 1/8$ approximates $x^2$ uniformly within $1/8$.
Being close on average is not enough: a function can be close to $f$ in mean but have a tall narrow spike, and then it is far from $f$ uniformly.
</document>

<document source="https://tamnd.github.io/math2/node/XAC78GN2YJ.md" tag="XAC78GN2YJ" kind="concept">
# Commutative magma

## Summary

A commutative magma is a [[magma|9689SFPVE9]] whose operation does not depend on the order of the two arguments.
Commutativity does not imply associativity, and commutative magmas are the natural setting to see the two conditions apart.
Examples include the average of two numbers and the operation of "rock, paper, scissors".

## Explanations

### L1 human

In a commutative magma, combining $x$ with $y$ gives the same as combining $y$ with $x$.
But brackets may still matter.
Take the average of two numbers: the average of $0$ and $4$ is the average of $4$ and $0$.
Yet averaging $0$ with $4$ and then with $8$ gives $5$, while averaging $0$ with the average of $4$ and $8$ gives $3$.
</document>

<document source="https://tamnd.github.io/math2/node/XAE5YJ3H68.md" tag="XAE5YJ3H68" kind="concept">
# Effective resistance

## Summary

The effective resistance between two vertices of an [[electrical network|T7KRE7J3B4]] is the resistance of the whole network seen from those two vertices.
It is the voltage difference needed to push a unit current from one vertex to the other.
It is a metric on the vertices, called the resistance distance, and it can be computed from the pseudoinverse of the [[graph Laplacian|TN6M6DVZRP]].
For random walks it gives escape probabilities, commute times and criteria for recurrence.

## Explanations

### L1 human

Replace the whole network between $a$ and $b$ by a single resistor that behaves the same way from the outside: its resistance is the effective resistance.
Resistors in series add: two unit resistors in a row have effective resistance $2$.
Resistors in parallel combine by adding conductances: two unit resistors side by side have effective resistance $\tfrac12$.
Many paths between $a$ and $b$ make the effective resistance small, and a single long path makes it large.
</document>

<document source="https://tamnd.github.io/math2/node/XAEEDWV3PM.md" tag="XAEEDWV3PM" kind="conjecture">
# Hodge conjecture

## Statement

Let $X$ be a [[smooth|EXKGG9X6J0]] [[projective variety|A4H4CZ2V35]] over $\C$, and let $p \ge 0$.
Let $H^{p,p}(X)$ be the $(p, p)$ part of the [[Hodge decomposition|HABCJ4Y0SR]] of $H^{2p}(X, \C)$.
Here $H^{2p}(X, \Q)$ is [[singular cohomology|6N7BC70CX0]], viewed inside $H^{2p}(X, \C)$.
Let $\mathrm{Hdg}^p(X) = H^{2p}(X, \Q) \cap H^{p,p}(X)$ be the group of rational Hodge classes.
Then every class in $\mathrm{Hdg}^p(X)$ is a $\Q$-linear combination of classes of closed subvarieties of $X$ of codimension $p$.
In other words, it is the class of an [[algebraic cycle|9VFWPZEVGN]] with rational coefficients.

## Remarks

The conjecture is open.
It is one of the Millennium Prize Problems of the Clay Mathematics Institute.
The class of a subvariety of codimension $p$ always lies in $\mathrm{Hdg}^p(X)$, so the conjecture asks for the converse.

The case $p = 1$ is true: it is the Lefschetz theorem on $(1,1)$-classes, which follows from the exponential sequence.
It shows that every class in $H^{2}(X, \Z)$ whose image lies in $H^{1,1}$ is the first Chern class of a line bundle.
By the hard Lefschetz theorem, the case $p = 1$ implies the case $p = n - 1$, where $n = \dim X$.
So the conjecture holds for varieties of dimension at most $3$.

The version with integer coefficients is false: Atiyah and Hirzebruch found torsion classes that are not algebraic, and Kollár found non-torsion counterexamples.
The analogous statement for compact Kähler manifolds, with analytic cycles or Chern classes of coherent sheaves, is false by an example of Voisin.

## Explanations

### L1 human

The Hodge conjecture asks which cohomology classes of a complex projective variety come from geometry, namely from subvarieties.
Hodge theory singles out a natural candidate set, the rational classes of type $(p, p)$, and every subvariety gives such a class.
The conjecture says that there are no others: every class that looks algebraic from the point of view of Hodge theory is algebraic.
</document>

<document source="https://tamnd.github.io/math2/node/XAQSNBJSX4.md" tag="XAQSNBJSX4" kind="definition">
# Inverse limit

## Definition

Let $(I, \le)$ be a [[directed set|D9RY4667K6]] and let $\mathcal{C}$ be a [[category|A2DE66937D]].
An *inverse system* in $\mathcal{C}$ over $I$ is a family of objects $A_i$ for $i \in I$ with [[morphisms|859F0TFVA3]] $f_{ij} \colon A_j \to A_i$ for all $i \le j$, such that:

1. $f_{ii} = \id_{A_i}$ for all $i$.
2. $f_{ij} \circ f_{jk} = f_{ik}$ for all $i \le j \le k$.

An *inverse limit* of this system is an object $A$ with morphisms $\pi_i \colon A \to A_i$ such that:

1. $f_{ij} \circ \pi_j = \pi_i$ for all $i \le j$.
2. For every object $B$ with morphisms $g_i \colon B \to A_i$ such that $f_{ij} \circ g_j = g_i$ for all $i \le j$, there is a unique morphism $u \colon B \to A$ with $\pi_i \circ u = g_i$ for all $i$.

## Remarks

The inverse limit is written $\varprojlim A_i$.
It is the [[limit|CHS5AFJGWE]] of the diagram given by the system, and it is unique up to a unique isomorphism.
In sets, groups, rings and modules it always exists.
There it is the set of families $(a_i) \in \prod_i A_i$ with $f_{ij}(a_j) = a_i$ for all $i \le j$, with the projections as $\pi_i$.

Example: the rings $\Z/p^n\Z$ for $n \ge 1$, with the reduction maps, form an inverse system over $\N$.
Its inverse limit is the ring $\Z_p$ of $p$-adic integers.
Clause 1 holds because reducing a compatible sequence at level $n+1$ gives its entry at level $n$.

Non-example: $\Z$ with the reduction maps $\Z \to \Z/p^n\Z$ satisfies clause 1 but not clause 2.
Taking $B = \Z_p$, a morphism $u \colon \Z_p \to \Z$ with $\pi_n \circ u$ equal to reduction would have to be injective, which is impossible since $\Z_p$ is uncountable.
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<document source="https://tamnd.github.io/math2/node/XAW51D9E62.md" tag="XAW51D9E62" kind="definition">
# Quasi-Monte Carlo method

## Definition

Let $s \ge 1$ be an integer and let $f \colon [0, 1]^s \to \R$ be a [[Riemann integrable|0W29TBTY12]] function.
A *quasi-Monte Carlo method* for $\int_{[0,1]^s} f(x)\, dx$ is an equal-weight [[quadrature rule|ADSVMCY77A]]
$$Q_N(f) = \frac{1}{N} \sum_{i=1}^{N} f(x_i),$$
with $N \ge 1$.
Here $x_1, x_2, \ldots$ are the points of a deterministic [[low-discrepancy sequence|XSPX1KW9AQ]] in $[0, 1)^s$, or the $N$ points of a deterministic point set of low discrepancy.

## Remarks

Example: $s = 1$, $f(x) = x$ and the van der Corput points $\frac12, \frac14, \frac34, \frac18$; for $N = 4$ one gets $Q_4(f) = \frac{1}{4}\left(\frac12 + \frac14 + \frac34 + \frac18\right) = \frac{13}{32}$, against the exact value $\frac12$, and for $N = 3$ one gets exactly $\frac12$.
Non-example: the same average with independent uniform random points $x_i$ is plain Monte Carlo integration.
It is not a quasi-Monte Carlo method, because the points are random rather than a fixed low-discrepancy sequence.
Riemann integrability makes $Q_N(f) \to \int f$ for every uniformly distributed sequence; error bounds of order $(\log N)^s/N$ need more, for instance bounded variation in the sense of Hardy and Krause.
*Randomized* quasi-Monte Carlo methods apply a random shift or scrambling to a low-discrepancy point set.
Each point becomes uniform while the set keeps its low discrepancy, and this allows statistical error estimates.
Rank-1 lattice rules, with $x_i = \{ i z / N \}$ for an integer vector $z$, are point sets of this kind used for periodic integrands.
</document>

<document source="https://tamnd.github.io/math2/node/XB24W24B20.md" tag="XB24W24B20" kind="concept">
# Khovanov homology

## Summary

Khovanov homology is an invariant of oriented [[links|1A5WRCNTM3]] that takes values in bigraded abelian groups.
Mikhail Khovanov built it from a cube of smoothings of a diagram, as a categorification of the [[Jones polynomial|4SPC17FPXE]].
Its graded Euler characteristic is the Jones polynomial, but it carries strictly more information.
Kronheimer and Mrowka proved that it detects the [[unknot|1TQ06R4Q66]].

## Explanations

### L1 human

The [[Kauffman bracket|JYKVYS4N32]] computes the Jones polynomial by summing over all ways to smooth the crossings of a diagram.
Khovanov homology replaces each term of that sum by a group and each sign by a map between groups.
The result is a chain complex, and its homology does not depend on the diagram.
Counting dimensions with signs gives back the Jones polynomial.
The extra structure lets it tell apart knots that the Jones polynomial cannot.
</document>

<document source="https://tamnd.github.io/math2/node/XB91QS2AR0.md" tag="XB91QS2AR0" kind="concept">
# Pseudo-spectral method

## Summary

A pseudo-spectral method is a [[spectral method|MYV657YRDW]] that imposes the equation at grid points, by collocation.
It does not use integrals against the basis functions.
Derivatives are computed in coefficient space, for example with the [[fast Fourier transform|S966EZX3ZT]], while products and other nonlinear terms are computed pointwise on the grid.
This keeps the accuracy of spectral methods at a much lower cost for nonlinear problems.
The price is aliasing errors from the pointwise products, which are often removed by padding or truncation.
See the [[definition|T37Q055GDA]].

## Explanations

### L1 human

Store a function by its values at evenly spaced points around a circle.
To differentiate, turn the values into wave amplitudes, multiply each amplitude by its frequency times $i$, and turn the result back into values.
To multiply two functions, just multiply their values point by point.
Switching back and forth between values and amplitudes is fast, which makes the method efficient.
</document>

<document source="https://tamnd.github.io/math2/node/XB9QTY77ZA.md" tag="XB9QTY77ZA" kind="concept">
# CAT(0) group

## Summary

A CAT(0) group is a group that acts properly and cocompactly by isometries on a [[CAT(0) space|1FAF2VNA4G]].
Such a space is a geodesic space whose triangles are at least as thin as Euclidean triangles.
It is the group-theoretic form of nonpositive curvature.
Examples include $\Z^n$, free groups, fundamental groups of closed nonpositively curved Riemannian manifolds, [[right-angled Artin groups|6D1DV09D6P]] and [[Coxeter groups|G6FND7X69J]].
CAT(0) groups are finitely presented, have a solvable word problem, and have a [[Dehn function|Q4HGMRA9JQ]] at most quadratic.

## Explanations

### L1 human

Draw a triangle on a saddle surface: its sides bend inwards compared with a flat triangle of the same side lengths.
A CAT(0) space is a space where every triangle is at least this thin, never fatter than a flat one.
A CAT(0) group is a group of symmetries of such a space.
It moves points around in a discrete way, and the translates of one compact piece cover the whole space.
For example, $\Z^2$ acts on the flat plane by translations, and the unit square tiles the plane.
</document>

<document source="https://tamnd.github.io/math2/node/XBDB3YNV9S.md" tag="XBDB3YNV9S" kind="definition">
# Schnirelmann density

## Definition

Let $A$ be a subset of the [[natural numbers|0MWKJ44V5A]].
For each integer $n \ge 1$, let $A(n)$ be the number of elements $a \in A$ with $1 \le a \le n$.
The *Schnirelmann density* of $A$ is
$$\sigma(A) = \inf_{n \ge 1} \frac{A(n)}{n}.$$

## Remarks

Always $0 \le \sigma(A) \le 1$, and $A(n) \ge \sigma(A)\, n$ for every $n \ge 1$.
The element $0$, if present, does not count.
One has $\sigma(A) = 1$ if and only if $A$ contains every positive integer.

Example: for the set $A$ of odd positive integers, $A(n)/n = 1/2$ for even $n$ and $A(n)/n = (n+1)/(2n) > 1/2$ for odd $n$; so $\sigma(A) = 1/2$.
Contrast: for the even positive integers, $A(1)/1 = 0$, so $\sigma(A) = 0$, although their [[natural density|RVG9Z8NYA6]] is $1/2$.

Schnirelmann's inequality: if $0 \in A$ and $0 \in B$, then $\sigma(A + B) \ge \sigma(A) + \sigma(B) - \sigma(A)\sigma(B)$, where $A + B = \set{a + b : a \in A, b \in B}$.
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<document source="https://tamnd.github.io/math2/node/XBJEDC18B1.md" tag="XBJEDC18B1" kind="statement">
# Cotlar–Stein lemma

## Statement

Let $H_1$ and $H_2$ be [[Hilbert spaces|HM2HHE3J51]], let $N \ge 1$, and let $T_1, \dots, T_N \colon H_1 \to H_2$ be bounded linear operators.
Write $T_i^*$ for the [[adjoint|6HSJBJ8TTQ]] of $T_i$ and $\norm{\cdot}$ for the [[operator norm|06EKW1AKXF]].
Let $A \ge 0$ and assume that
$$\sup_{i} \sum_{j=1}^{N} \norm{T_i^* T_j}^{1/2} \le A \quad \text{and} \quad \sup_{i} \sum_{j=1}^{N} \norm{T_i T_j^*}^{1/2} \le A.$$
Then
$$\Big\Vert \sum_{i=1}^{N} T_i \Big\Vert \le A.$$

## Remarks

The bound does not depend on $N$.
So let $(T_i)$ be an infinite sequence with the same two hypotheses.
Then the partial sums are bounded by $A$, and the series $\sum_i T_i$ converges in the strong operator topology to an operator of norm at most $A$.

Idea of the proof.
Let $T = \sum_i T_i$ and let $m \ge 1$.
Since $\norm{T}^{2m} = \norm{(T^* T)^m}$, expand $(T^*T)^m$ as a sum of $N^{2m}$ products $T_{i_1}^* T_{i_2} T_{i_3}^* \cdots T_{i_{2m}}$.
Bound each product in two ways, by grouping the factors in pairs $T_{i_1}^* T_{i_2}, T_{i_3}^* T_{i_4}, \dots$ or in pairs $T_{i_2} T_{i_3}^*, \dots$, and take the geometric mean of the two bounds.
Summing over the indices gives $\norm{T}^{2m} \le N A^{2m}$, and letting $m \to \infty$ gives $\norm{T} \le A$.

Example.
If $T_i^* T_j = 0$ and $T_i T_j^* = 0$ for all $i \ne j$, the operators act on mutually orthogonal pieces.
The two hypotheses then reduce to $\norm{T_i} \le A$ for each $i$, and the lemma gives $\norm{\sum_i T_i} \le \max_i \norm{T_i}$.

One hypothesis alone is not enough.
Let $(e_k)$ be an orthonormal basis of a Hilbert space $H$ of dimension at least $N$, and let $T_i x = \ip{x, e_1} e_i$ for $i = 1, \dots, N$.
Then $T_i^* T_j = 0$ for $i \ne j$ and $\norm{T_i^* T_i} = 1$, so the first sum equals $1$.
But $\norm{T_i T_j^*} = 1$ for all $i, j$, so the second sum is $N$.
Indeed $\sum_i T_i$ sends $e_1$ to $e_1 + \cdots + e_N$, so its norm is $\sqrt{N}$.

## Explanations

### L1 human

Adding many operators can make a large operator, even if each piece is small.
The lemma says that this does not happen when the pieces are almost orthogonal to each other, both on the input side and on the output side.
The amount of interaction between two pieces is measured by $\norm{T_i^* T_j}$ and $\norm{T_i T_j^*}$, and these must decay fast enough.

### L2 human

The lemma is the basic tool of almost orthogonality in harmonic analysis.
A singular integral or pseudodifferential operator is cut into pieces that live at one scale or one frequency band, and the lemma gives $L^2$ boundedness of the sum.
A classical use is the proof of the Calderón–Vaillancourt theorem on pseudodifferential operators.
It only works on Hilbert spaces, since the proof uses the identity $\norm{T}^2 = \norm{T^*T}$.
</document>

<document source="https://tamnd.github.io/math2/node/XBM7J5R1VQ.md" tag="XBM7J5R1VQ" kind="statement">
# Lindelöf's theorem

## Statement

Let $x_1 < x_2$ and $y_0$ be real numbers, and let $\Omega = \set{z \in \C : x_1 \le \operatorname{Re} z \le x_2, \ \operatorname{Im} z \ge y_0}$ be a closed half-strip.
Let $f \colon \Omega \to \C$ be continuous on $\Omega$ and [[holomorphic|0DG0Z72CQD]] on its interior.
Assume that $\abs{f(z)} \le M$ for each $z$ on the boundary of $\Omega$.
Assume also that there are constants $A, B > 0$ with $\abs{f(x + iy)} \le B (1 + \abs{y})^{A}$ for all $x + iy \in \Omega$.
Then $\abs{f(z)} \le M$ for all $z \in \Omega$.

## Remarks

Proof idea: for $\varepsilon > 0$ let $g_\varepsilon(z) = f(z) e^{i \varepsilon (z - i y_0)}$, so that $\abs{g_\varepsilon(x + iy)} = \abs{f(x + iy)} e^{-\varepsilon (y - y_0)}$.
Then $\abs{g_\varepsilon} \le \abs{f}$ on $\Omega$, and $g_\varepsilon \to 0$ as $\operatorname{Im} z \to \infty$ because $f$ grows at most polynomially.
The [[maximum modulus principle|048F2P90WR]] on the rectangles $\Omega \cap \set{\operatorname{Im} z \le T}$ with $T$ large gives $\abs{g_\varepsilon} \le M$, and $\varepsilon \to 0$ gives $\abs{f} \le M$.
A growth hypothesis is needed.
On the half-strip $\abs{\operatorname{Re} z} \le \frac{\pi}{2}$, $\operatorname{Im} z \ge 0$, the function $\exp(e^{-iz})$ has modulus $\exp(e^{y} \cos x)$.
It is bounded on the boundary, where $\cos x = 0$ or $y = 0$, but unbounded on the line $x = 0$.
The theorem is a case of the [[Phragmén–Lindelöf principle|FYR6V2NH6C]].

## Explanations

### L1 human

On a half-strip, which goes off to infinity in one direction, the maximum principle can fail for fast-growing functions.
Lindelöf's theorem says that polynomial growth is slow enough.
Then a bound on the edges of the half-strip is a bound everywhere inside.

### L2 human

The theorem is used in analytic number theory for functions such as the Riemann zeta function on vertical half-strips.
Together with the [[Hadamard three-lines theorem|WHS58R43N0]] it gives the convexity bounds for the growth of $\zeta(\sigma + it)$ as $t \to \infty$.
In the proof, the factor $e^{i\varepsilon z}$ is the standard device to force decay at infinity before applying the maximum modulus principle.
</document>

<document source="https://tamnd.github.io/math2/node/XBNAJ1RRET.md" tag="XBNAJ1RRET" kind="concept">
# Euler–Poisson equation for higher derivatives

## Summary

The Euler–Poisson equation is the [[Euler–Lagrange equation|PS0VV149ET]] of an integral functional whose Lagrangian depends on derivatives of order higher than one.
If the Lagrangian involves derivatives up to order $k$, the equation is an ordinary differential equation of order at most $2k$.
The order is exactly $2k$ when the Lagrangian really depends on the $k$-th derivative.
It governs, for example, the bending of beams and the shape of cubic splines.

## Explanations

### L1 human

Some energies depend on how much a curve bends, not only on its slope.
Bending is measured by the second derivative, so the energy contains $y''$.
Asking that small changes of the curve do not lower the energy gives an equation of order four instead of two.
For the energy $\frac{1}{2} \int y''(x)^2 \, dx$ the equation is $y'''' = 0$, whose solutions are the cubic polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/XBPK7Y559D.md" tag="XBPK7Y559D" kind="concept">
# Incomplete LU factorization

## Summary

An incomplete LU factorization of a sparse square matrix $A$ is a pair of sparse triangular matrices $L$ and $U$ with $L U \approx A$.
It is computed like the [[LU decomposition|0RKG5YW66Y]], but entries outside a prescribed sparsity pattern are dropped.
It is one of the most common [[preconditioners|KE98SW3SBD]] for Krylov methods on nonsymmetric systems.

## Explanations

### L1 human

Exact elimination on a sparse matrix creates many new nonzeros, called fill.
The incomplete factorization throws away the fill that falls outside allowed positions.
The factors stay as sparse as $A$, or nearly so, and $L U$ is close to $A$.
Solving with $L$ and then $U$ is cheap, so $L U$ acts as an easy approximate inverse inside an iterative solver.
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<document source="https://tamnd.github.io/math2/node/XBPKCBX4EW.md" tag="XBPKCBX4EW" kind="definition">
# J-homomorphism

## Definition

Let $r, n \ge 1$, and let $SO(n)$ be the [[special orthogonal group|0CGYNDCWRR]].
Write $S^{r+n} = \partial(D^{r+1} \times D^n) = (S^r \times D^n) \cup (D^{r+1} \times S^{n-1})$, and $S^n = D^n / \partial D^n$ with base point the image of $\partial D^n$.
For a map $f \colon S^r \to SO(n)$, define $J(f) \colon S^{r+n} \to S^n$ as follows:

1. on $S^r \times D^n$, set $J(f)(x, v) = [f(x) v]$, the class of $f(x) v \in D^n$;
2. on $D^{r+1} \times S^{n-1}$, let $J(f)$ be constant at the base point.

The *J-homomorphism* is the map $J \colon \pi_r(SO(n)) \to \pi_{r+n}(S^n)$, $[f] \mapsto [J(f)]$, on [[homotopy groups|P976WYEFMB]].

## Remarks

The two clauses agree on $S^r \times S^{n-1}$, since $f(x)$ preserves the unit sphere, so $f(x) v \in \partial D^n$.
The map $J$ is a well-defined group homomorphism.

Example: for $n = 2$ and $r = 1$, let $f \colon S^1 \to SO(2)$ send $z$ to rotation by $z$, viewing $\R^2$ as $\C$.
Then $J(f) \colon S^3 \to S^2$ is the Hopf map, up to sign and homotopy, and it generates $\pi_3(S^2) \cong \Z$.

Non-example: suppose $f(x)$ is allowed to be any invertible linear map.
Then $f(x) v$ need not lie on $\partial D^n$ for $v \in S^{n-1}$, and may even leave $D^n$, so the two clauses do not define a map.

The maps $J$ commute with the stabilisation maps $SO(n) \to SO(n+1)$ and with suspension, so they give the stable J-homomorphism $J \colon \pi_r(SO) \to \pi_r^s$.
</document>

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# Harmonic space

## Definition

Let $X$ be a [[locally compact|0FE2SB8WVJ]] Hausdorff space with a countable base.
To each open set $U \subseteq X$, let a linear subspace $\mathcal{H}(U)$ of the real [[continuous functions|0X3WGX95FE]] on $U$ be given.
Its elements are called *harmonic* on $U$.
An open set $V$ with compact closure $\overline{V} \subseteq X$ is *regular* if the following holds for every continuous $f \colon \partial V \to \R$.
There is a unique continuous $h \colon \overline{V} \to \R$ with $h|_V \in \mathcal{H}(V)$ and $h = f$ on $\partial V$, and $h \ge 0$ whenever $f \ge 0$.
The pair $(X, \mathcal{H})$ is a *harmonic space* if:

1. $\mathcal{H}$ is a [[sheaf|TB6TM6R4MD]]. Restrictions of harmonic functions to smaller open sets are harmonic. A continuous function on $U = \bigcup_i U_i$ that is harmonic on each $U_i$ is harmonic on $U$.
2. The regular sets form a base of the topology of $X$.
3. For every open $U$ and every increasing sequence $(h_n)$ in $\mathcal{H}(U)$ whose limit $h$ is locally bounded on $U$, the function $h$ lies in $\mathcal{H}(U)$.

## Remarks

These are the axioms in the form given by Bauer; several authors add a positivity or separation axiom, and some impose a stronger convergence axiom.
Example: $X = \R^n$ with $\mathcal{H}(U)$ the classical [[harmonic functions|0DEB0JMN0W]] on $U$.
Clause 1 is clear, since harmonicity is a local condition.
Every ball is regular, by the Poisson integral and the maximum principle, so clause 2 holds.
Clause 3 follows from [[Harnack's principle|P15K8RKHDV]].
A second example is $X = \R^n \times \R$ with $\mathcal{H}(U)$ the solutions of the [[heat equation|VP2HN86VAG]] $\partial_t u = \Delta u$ on $U$.
Non-example: on $X = \R^n$, let $\mathcal{H}(U)$ be all continuous functions on $U$.
No open set $V$ with nonempty boundary is regular, since a continuous extension of boundary values is never unique, so clause 2 fails.
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# Associated graded ring

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $I$ be an [[ideal|096Z9MP2ME]] of $R$, with $I^0 = R$.
The *associated graded ring* of $R$ with respect to $I$ is the abelian group
$$\operatorname{gr}_I(R) = \bigoplus_{n \ge 0} I^n / I^{n+1}$$
with the following multiplication.
For $a \in I^m$ and $b \in I^n$ it is
$$(a + I^{m+1})(b + I^{n+1}) = ab + I^{m+n+1},$$
and it is extended to all elements by distributivity.

## Remarks

The product is well defined: if $a' - a \in I^{m+1}$ and $b' - b \in I^{n+1}$, then $a'b' - ab = (a' - a) b' + a (b' - b) \in I^{m+n+1}$.
With this product $\operatorname{gr}_I(R)$ is a commutative [[graded ring|8KB4SQGC0C]] whose degree $0$ part is the [[quotient ring|0076HWHGHQ]] $R/I$.
Example: for $R = \Z$ and $I = p\Z$ with $p$ prime, each $p^n\Z/p^{n+1}\Z$ is $\F_p$ spanned by the class of $p^n$, and $p^m p^n = p^{m+n}$, so $\operatorname{gr}_I(\Z) \cong \F_p[t]$.
Non-example: the direct sum $\bigoplus_{n \ge 0} I^n$ with the same rule is not $\operatorname{gr}_I(R)$; it is the [[Rees algebra|W8ZAH8TTHS]] $R[It]$, and $\operatorname{gr}_I(R)$ is its quotient by $I R[It]$.
Let $(R, \mathfrak m)$ be a Noetherian local ring with residue field $k$.
Then the ring $\operatorname{gr}_{\mathfrak m}(R)$ is a polynomial ring over $k$ in $\dim R$ variables if and only if $R$ is a [[regular local ring|63XXTF6B5M]].
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# Local martingale

## Summary

A local martingale is a process that becomes a [[martingale|0D0GBNDKDH]] when it is stopped at suitable stopping times that increase to infinity.
Every martingale is a local martingale, but not conversely.
Stochastic integrals with respect to Brownian motion are local martingales, which is why the notion is central in [[Itô calculus|JA95NH206E]].

## Explanations

### L1 human

A local martingale is a fair game as long as you stop it before it runs out of control.
You choose a sequence of stopping rules, such as "stop when the size reaches $n$", and each stopped game is fair.
The full game may still fail to be fair, because rare huge swings can push the expected value away from its start.
For example, a Brownian motion started at $0$ is a martingale and hence a local martingale.
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# Kovacic's algorithm

## Summary

Kovacic's algorithm decides whether a second-order linear differential equation with rational function coefficients has a [[Liouvillian|N5PM2NCPWF]] solution, and finds one when it exists.
It works on the reduced form $y'' = r(x)\, y$ with $r \in \C(x)$.
It is based on the classification of the algebraic subgroups of $\mathrm{SL}_2(\C)$, the possible [[differential Galois groups|EEZS4H8QPH]] of such equations.
It is implemented in computer algebra systems and is the standard tool for closed-form solution of these equations.
See the [[definition|MMEX2JZWGR]].

## Explanations

### L1 human

Given an equation like $y'' = (x^2 + 1)\, y$, one wants to know whether it has a solution built from exponentials, integrals and algebraic functions.
Here the answer is yes: $y = e^{x^2/2}$ works, since $y'' = (x^2 + 1)\, e^{x^2/2}$.
Kovacic's algorithm answers this question in finitely many steps.
It tries a short list of possible shapes for a solution, one for each kind of symmetry group the equation could have.
If no shape fits, the equation has no closed-form solution of this kind, as for the Airy equation $y'' = x y$.
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# Nuclear space

## Summary

A nuclear space is a [[locally convex space|B6GY2KJMWW]] whose seminorms can be compared through [[nuclear operators|XPANM7C1D3]].
These are maps given by absolutely summable series of rank-one maps.
Nuclear spaces were introduced by Alexander Grothendieck in his work on topological tensor products.
They behave in many ways like finite-dimensional spaces: tensor products are well behaved, and the [[Schwartz kernel theorem|2ZKWZ0A4MT]] holds for them.
The [[Schwartz space|CG9JEATW3Q]] and spaces of smooth functions on open sets of $\R^n$ are nuclear, while no infinite-dimensional normed space is.

## Explanations

### L1 human

In a nuclear space, passing from one seminorm to a stronger one makes things very much smaller, in a summable way.
For smooth functions, a bound on many derivatives forces the function to be very close to a finite combination of simple functions.
This is why nuclear spaces feel almost finite-dimensional.
For example, in the Schwartz space of rapidly decreasing smooth functions, bounded sets are relatively compact, as in $\R^n$.
Infinite-dimensional Banach spaces are never nuclear.
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# Gromoll–Meyer theorem on closed geodesics

## Statement

Let $M$ be a compact, simply connected smooth manifold without boundary, and let $\Lambda M$ be its [[free loop space|SGGXZ3TBSA]].
Assume that for some field $F$ the sequence of [[Betti numbers|ST71ZV3FES]] $b_k(\Lambda M; F) = \dim_F H_k(\Lambda M; F)$, $k \ge 0$, is unbounded.
Then for every Riemannian metric $g$ on $M$, the [[Riemannian manifold|7CBT4SM83Y]] $(M, g)$ has infinitely many geometrically distinct [[closed geodesics|4MCCP8GG4J]].

## Remarks

The idea of the proof is this.
Suppose there were only finitely many geometrically distinct closed geodesics.
Each critical orbit of the energy on $\Lambda M$, with all its iterates, contributes to the homology of $\Lambda M$ only in a controlled way.
The indices of iterates grow linearly, and the local homology of iterates takes only finitely many forms.
This forces the Betti numbers of $\Lambda M$ to be bounded, a contradiction.

By a theorem of Vigué-Poirrier and Sullivan, for $F = \Q$ the Betti numbers of $\Lambda M$ are unbounded exactly when the rational cohomology ring $H^*(M; \Q)$ needs at least two generators.
So the hypothesis holds, for example, for $M = S^2 \times S^2$.
The hypothesis fails for spheres and for compact rank one symmetric spaces, whose rational cohomology is generated by one element.
For $S^2$ the conclusion still holds by the theorem of [[Bangert and Franks|X93AEE88PX]].

## Explanations

### L1 human

Suppose the space of all loops on a manifold has very complicated topology.
Then every Riemannian metric on the manifold has infinitely many essentially different closed geodesics.
The complexity is measured by the Betti numbers of the loop space, which count holes of each dimension.
The point is that finitely many closed geodesics, even with all their repetitions, cannot account for unbounded numbers of holes.

### L2 human

The theorem is a basic result of Morse theory on the free loop space.
There closed geodesics are critical orbits of the energy, for the circle action that rotates the parameter.
Its main technical inputs are the growth of the Morse index under iteration and a finiteness result for local homology of degenerate critical orbits.
It is a well-known open question whether every compact simply connected Riemannian manifold of dimension at least $2$ has infinitely many closed geodesics.
The Gromoll–Meyer theorem settles it when the loop space homology grows.
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# Prime ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
$R$ is a *prime ring* if $R \ne 0$ and, for all $a, b \in R$, the condition $aRb = 0$ implies $a = 0$ or $b = 0$.
Here $aRb = \set{arb : r \in R}$.

## Remarks

Example: $M_n(k)$ over a field $k$ is prime.
If $a \ne 0$ and $b \ne 0$, choose entries $a_{ij} \ne 0$ and $b_{kl} \ne 0$; then $e_{jk} \in R$ and the $(i, l)$ entry of $a e_{jk} b$ is $a_{ij} b_{kl} \ne 0$.
Non-example: $k \times k$ is not prime, since $(1, 0)\, r\, (0, 1) = 0$ for every $r \in k \times k$.
Equivalent forms: the zero ideal is a [[prime ideal|E8K2MN5V1E]] of $R$; or, for two-sided ideals $A, B$ of $R$, $AB = 0$ implies $A = 0$ or $B = 0$.
A commutative ring is prime if and only if it is an integral domain.
Every [[primitive ring|8F6EHDVJ62]] is prime.
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# Probit model

## Summary

The probit model is a regression model for a binary response.
The probability of the outcome $1$ is the standard normal [[cumulative distribution function|0H8H7ZTHW2]] evaluated at a linear combination of regressors.
It can be read as a latent variable model: an unobserved normal quantity crosses a threshold exactly when the outcome is $1$.
It is a [[generalized linear model|K6C48KMRKC]] for Bernoulli responses with the probit link, and it is fitted by [[maximum likelihood|0FREPFKNS7]].
It is close to [[logistic regression|JN1PJ5AJJS]] in practice, and the choice between them is often a matter of convention in the field.

## Explanations

### L1 human

Think of each person as having a hidden score, equal to a linear formula in their characteristics plus a normal random term.
The person says yes when the hidden score is above zero.
The probability of a yes is then the normal distribution function at the linear formula.
For example, if the linear formula equals $0$, the probability is $1/2$, and if it equals $1.96$, the probability is about $0.975$.
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# Blocking

## Summary

Blocking is the grouping of experimental units into blocks of similar units, with treatments compared within each block.
Variation between blocks, such as differences between fields, days or batches of material, is then removed from the comparison of treatments.
This makes the comparison more precise.
It is the local control principle of [[design of experiments|2TE42ZWJ3E]], and is usually combined with [[randomization|734Q6Q51X7]] inside each block.
The randomized complete block design, Latin squares and balanced incomplete block designs are the standard block designs.

## Explanations

### L1 human

To compare three wheat varieties, a farmer has four fields of different quality.
If each variety got its own fields, a good field could make a poor variety look good.
Instead, each field is split into three plots, and every variety is grown once in every field, in plots chosen at random.
Differences between fields then affect all varieties equally and cancel out of the comparison.
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# Primal algebra

## Summary

A primal algebra is a finite algebra in which every finitary operation on its universe is a term operation.
Equivalently, its basic operations form a [[functionally complete|TDZVCWYJ9F]] set.
The two-element Boolean algebra and the rings $\Z_p$ for a prime $p$ are primal.
By [[Foster's theorem|TCD8SZW95Z]], the variety generated by a primal algebra looks just like the variety of Boolean algebras.

## Explanations

### L1 human

In a primal algebra, the given operations can express every function on the set.
The two-element Boolean algebra is the model: every truth table is a Boolean formula.
Another example is arithmetic modulo a prime $p$: every function on $\set{0, 1, \dots, p - 1}$ is a polynomial with integer coefficients, reduced modulo $p$.
For instance, modulo $3$ the function equal to $1$ at $0$ and $0$ elsewhere is $1 - x^2$.
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# A subgroup of order two in S3 is not normal

## Example

In the [[symmetric group|0515A6D2Y1]] $S_3$, let $H = \{e, (1\,2)\}$. Compose permutations from right to left.
$H$ is a subgroup, because $(1\,2)(1\,2) = e$.
The [[left coset|05FD317V2W]] of $(1\,2\,3)$ is $(1\,2\,3)H = \{(1\,2\,3), (1\,3)\}$.
The right coset of $(1\,2\,3)$ is $H(1\,2\,3) = \{(1\,2\,3), (2\,3)\}$.
These cosets are different. Thus $H$ is not [[normal|0W77CVBCHQ]] in $S_3$.

## Remarks

$H$ has index $3$. Thus the result for index two does not apply to $H$.
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# Unary algebra

## Summary

A unary algebra is a set with a family of [[unary operations|Q2W26ZY10B]], maps from the set to itself, and no operations of other arities.
It is the simplest kind of [[algebraic structure|QJGPT2W0FT]], and yet it already contains the theory of functional graphs and of automata.
A unary algebra with operations indexed by a set $I$ is the same as an action of the free monoid on $I$.
A unary algebra with one operation is called a mono-unary algebra.

## Explanations

### L1 human

Take a set and some ways of moving each element to another element.
That is a unary algebra.
With one operation $f$, draw an arrow from each $x$ to $f(x)$, so that every point has exactly one outgoing arrow.
For a finite set, the picture is made of cycles with trees hanging off them.
For example, the integers with the successor map $n \mapsto n + 1$ form a unary algebra whose picture is one infinite line.
A [[semiautomaton|RTMYXRDGWR]] is a unary algebra with one operation for each input letter.
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# Affine plane

## Definition

An *affine plane* is an [[incidence structure|FGKNAFNHA3]] of points and lines such that:

1. Any two distinct points lie on exactly one common line.
2. For every line $\ell$ and every point $p$ not on $\ell$, exactly one line through $p$ has no point in common with $\ell$.
3. There are three points that are not [[collinear|0AFNNTRTJE]].

Two lines are *parallel* if they are equal or have no common point.

## Remarks

Example: for a field $k$, the points of $k^2$ and the lines $\set{(x, y) : ax + by = c}$ with $(a, b) \ne (0, 0)$ form an affine plane.
Check: two distinct points determine one such equation up to a factor.
The parallels to $ax + by = c$ are the lines $ax + by = c'$, and exactly one of them passes through a given point.
Finally, $(0,0)$, $(1,0)$, $(0,1)$ are not collinear.
For $k = \F_2$ this is the smallest affine plane, with $4$ points and $6$ lines.

Non-example: the real [[projective plane|DANFNN8C4Y]] satisfies clauses 1 and 3 but not clause 2, because any two of its lines meet.

Clause 2 is the parallel axiom in the form of [[Playfair's axiom|PR7Q3QMEHZ]].
Parallelism is an equivalence relation on the lines of an affine plane.
In a finite affine plane all lines have the same number $n$ of points, called the order, and there are $n^2$ points.
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# Fubini's theorem on differentiation

## Statement

Let $a < b$ be real numbers and let $f_n \colon [a, b] \to \R$, for $n \ge 1$, be nondecreasing functions.
Assume that the series $F(x) = \sum_{n=1}^{\infty} f_n(x)$ converges for every $x \in [a, b]$.
Then for [[almost every|RARTMA0ZEC]] $x \in [a, b]$, with respect to [[Lebesgue measure|2Z2QVWZ262]], the [[derivatives|0F1WR9C83K]] $F'(x)$ and $f_n'(x)$ exist for all $n$, and
$$F'(x) = \sum_{n=1}^{\infty} f_n'(x).$$

## Remarks

Replacing $f_n$ by $f_n - f_n(a)$ changes nothing, so one may assume $f_n \ge 0$.
By [[Lebesgue's theorem|1Y4N1DDQNR]], $F$ and all $f_n$ are differentiable almost everywhere.
Since $F$ minus a partial sum is nondecreasing, the partial sums of $\sum f_n'$ are bounded by $F'$ almost everywhere, so the series of derivatives converges.
Choosing partial sums $S_{n_k}$ with $F(b) - S_{n_k}(b) < 2^{-k}$ and applying the same argument to $\sum_k (F - S_{n_k})$ shows that the terms $F' - S_{n_k}'$ tend to $0$ almost everywhere.
Monotonicity is essential: take $f_1(x) = \sin x$ and $f_n(x) = \sin(n^2 x)/n^2 - \sin((n-1)^2 x)/(n-1)^2$ for $n \ge 2$; the partial sums $\sin(n^2 x)/n^2$ converge uniformly to $0$, but their derivatives $\cos(n^2 x)$ do not converge to $0$.

## Explanations

### L1 human

A convergent series of increasing functions can be differentiated term by term, except on a set of length zero.
No uniform convergence and no smoothness is needed, only that each term is increasing.

### L2 human

The theorem is a tool in the theory of monotone functions and functions of [[bounded variation|A8251BHCMP]].
It shows, for example, that a jump function, a convergent sum of step functions, has derivative $0$ almost everywhere.
It contrasts with the classical termwise differentiation theorem, which asks for uniform convergence of the derivatives.
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# Metric outer measure

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]].
For nonempty $A, B \subseteq X$ let $d(A, B) = \inf \set{ d(a, b) : a \in A, \ b \in B }$, an [[infimum|0QF1MZ9FRG]].
A *metric outer measure* on $X$ is an [[outer measure|BJ314H17HC]] $\mu^*$ on $X$ such that
$$\mu^*(A \cup B) = \mu^*(A) + \mu^*(B) \quad \text{whenever } A, B \subseteq X \text{ are nonempty with } d(A, B) > 0.$$

## Remarks

Example: Lebesgue outer measure $\lambda^*$ on $\R^n$ is a metric outer measure.
Let $d(A, B) = \delta > 0$.
A cover of $A \cup B$ by boxes can be refined into boxes of diameter less than $\delta$, and then no box meets both $A$ and $B$.
So the cover splits into a cover of $A$ and a cover of $B$, which gives $\ge$; subadditivity gives $\le$.
Non-example: on $\R$, put $\mu^*(\varnothing) = 0$ and $\mu^*(A) = 1$ for every nonempty $A$.
This is an outer measure, but $\mu^*(\set{0, 1}) = 1 \ne \mu^*(\set{0}) + \mu^*(\set{1})$ although $d(\set{0}, \set{1}) = 1 > 0$.
Carathéodory's theorem on metric outer measures: an outer measure on a metric space is a metric outer measure if and only if every Borel set is Carathéodory measurable for it.
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# Harmonic polynomial

## Definition

Let $n \ge 1$ and let $p \in \R[x_1, \dots, x_n]$ be a [[polynomial|1G0B73WPX8]] with real coefficients.
The polynomial $p$ is *harmonic* if its [[Laplacian|0E0B6RA8K0]] is the zero polynomial:
$$\Delta p = \frac{\partial^2 p}{\partial x_1^2} + \dots + \frac{\partial^2 p}{\partial x_n^2} = 0.$$

## Remarks

Example: $p(x, y, z) = x^2 + y^2 - 2z^2$ is harmonic, since $\Delta p = 2 + 2 - 4 = 0$.
Non-example: $p(x, y) = x^2 + y^2$ is not harmonic, since $\Delta p = 4$.
The Laplacian maps a [[homogeneous polynomial|2C3PAQ7H50]] of degree $k$ to one of degree $k - 2$, so a polynomial is harmonic if and only if each of its homogeneous parts is harmonic.
The homogeneous harmonic polynomials of degree $k$ in $n$ variables form a vector space $\mathcal{H}_k(\R^n)$ of dimension $\binom{n+k-1}{k} - \binom{n+k-3}{k-2}$, where the second term is $0$ for $k < 2$.
For $n = 2$ and $k \ge 1$ this dimension is $2$, with basis $\operatorname{Re}(x + iy)^k$ and $\operatorname{Im}(x + iy)^k$; for $n = 3$ it is $2k + 1$.
The restrictions of the elements of $\mathcal{H}_k(\R^n)$ to the unit sphere are the [[spherical harmonics|3F32Y21HP4]] of degree $k$.
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# Branch point

## Definition

Let $a \in \C$ and $r > 0$, and let $D^* = \set{z : 0 < \abs{z - a} < r}$ be a punctured disc.
Let $F$ be a [[multivalued function|03CPD1HGF3]] on $D^*$ given by a [[holomorphic|0DG0Z72CQD]] function element that admits [[analytic continuation|0M3YQJ81NB]] along every path in $D^*$.
The point $a$ is a *branch point* of $F$ if some function element of $F$ changes after continuation once around a circle $\abs{z - a} = \rho$ with $0 < \rho < r$.
The point $\infty$ of the [[Riemann sphere|0CDMXGPSES]] is a branch point of $F$ if $0$ is a branch point of $w \mapsto F(1/w)$.

## Remarks

If $n$ turns around $a$ bring each element back to itself for some $n \ge 2$, the branch point is *algebraic*, and the least such $n$ is its order.
Otherwise $a$ is a *transcendental* branch point; a *logarithmic* branch point, where no number of turns brings any element back, is the main example.
Some authors call $n - 1$ the order instead of $n$.
Example: for $F(z) = \sqrt{z}$, continuing the element $e^{(\log z)/2}$ once around $\abs{z} = 1$ multiplies it by $e^{i\pi} = -1$.
So $0$ is an algebraic branch point of order $2$, and so is $\infty$, because $\sqrt{1/w}$ also changes sign around $w = 0$.
The [[complex logarithm|02QGGD5G87]] has logarithmic branch points at $0$ and $\infty$, since each turn adds $2\pi i$.
Non-example: $0$ is not a branch point of $1/z$, which is single-valued on $\C \setminus \set{0}$; it is a pole.
For a holomorphic map between [[Riemann surfaces|5K0N8QMV6Q]], a point where the map is not locally injective is also called a branch point or ramification point.
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# Square root of a positive operator

## Definition

Let $H$ be a [[Hilbert space|HM2HHE3J51]] and let $A \colon H \to H$ be a [[positive operator|77020CY5ZW]].
A *positive square root* of $A$ is a positive operator $B \colon H \to H$ with $B^2 = A$.
Every positive $A$ has exactly one positive square root; it is called *the square root* of $A$ and written $A^{1/2}$ or $\sqrt{A}$.

## Remarks

Example: on $L^2[0, 1]$ let $A$ be multiplication by $x$, so $(Af)(x) = x f(x)$.
Multiplication by $\sqrt{x}$ is a positive operator $B$, since $\ip{Bf}{f} = \int_0^1 \sqrt{x} \abs{f(x)}^2 \, dx \ge 0$, and $B^2 f = x f$; so $A^{1/2} = B$.

Non-example: $-I$ satisfies $(-I)^2 = I$, but $\ip{-x}{x} = -\norm{x}^2 < 0$ for $x \ne 0$, so $-I$ is a square root of $I$ but not the positive square root.

The square root is given by the continuous functional calculus, $A^{1/2} = f(A)$ with $f(t) = \sqrt{t}$ on the spectrum of $A$.
It is also the norm limit of polynomials in $A$, so it commutes with every bounded operator that commutes with $A$.
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# Nagata ring

## Definition

Let $A$ be an [[integral domain|0QVD7WBFGN]] with [[field of fractions|0F2PTGX1A2]] $K$.
The domain $A$ is *N-2* if, for every [[finite extension|0C1Y04SZ1W]] $L$ of $K$, the [[integral closure|G0V86SJBQW]] of $A$ in $L$ is a [[finitely generated|05QY1VEBYD]] $A$-module.
An N-2 domain is also called *Japanese*.
A commutative ring $R$ is a *Nagata ring* if:

1. $R$ is a [[Noetherian ring|0670T59HGE]].
2. For every [[prime ideal|00MHHZ9SNE]] $\mathfrak{p}$ of $R$, the domain $R/\mathfrak{p}$ is N-2.

## Remarks

Example: $\Z$ is a Nagata ring.
It is Noetherian, and its quotients by primes are $\Z$ and the fields $\F_p$.
A field $F$ is N-2, because the integral closure of $F$ in a finite extension $L$ is $L$, an $F$-module of finite dimension.
The integral closure of $\Z$ in a number field $L$ is the ring of integers of $L$, a free $\Z$-module of rank $[L : \Q]$.

Non-example: there are one-dimensional Noetherian local domains whose integral closure in their own field of fractions is not a finitely generated module.
Such a domain fails clause 2 at the prime $\mathfrak{p} = 0$.

A theorem of Nagata says that a Noetherian ring $R$ is a Nagata ring if and only if every finitely generated $R$-algebra that is a domain is N-2.
Such rings are also called universally Japanese.
Older texts call Nagata rings pseudo-geometric rings.
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# Brauer–Manin obstruction

## Summary

A variety over a number field can have points over every completion of the field and still have no rational point.
The Brauer–Manin obstruction explains many such failures of the [[Hasse principle|EMAR4X3W8D]].
It uses the [[Brauer group|BEW48TA9WZ]] of the variety and the reciprocity law of [[class field theory|BEKCEYHCHX]].
These cut out a subset of the adelic points that must contain all rational points.
Manin introduced it in 1970, and it accounts for most known counterexamples to the Hasse principle.

## Explanations

### L1 human

To show an equation has no rational solution, one can check it has no real solution or no solution modulo powers of some prime.
Sometimes these local tests all pass and still there is no rational solution.
The Brauer–Manin obstruction is a finer test.
Each element of the Brauer group attaches to each local solution a number, and a genuine rational solution must make these numbers add up to zero.
If no family of local solutions passes this test, there is no rational point.
</document>

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# Fox–Wright function

## Summary

The Fox–Wright function ${}_p\Psi_q$ is a power series whose coefficients are ratios of products of values $\Gamma(a + A n)$.
Here $\Gamma$ is the [[gamma function|0H5SYBSED9]] and the scale factors $A$ are positive.
When all scale factors equal $1$ it reduces to a [[generalized hypergeometric function|VDBAW9PN88]], up to a constant factor.
It contains the [[Mittag-Leffler function|12G9M694YG]] and the Wright function as special cases, and it is a special case of the [[Fox H-function|EAZMKENA4N]].

## Explanations

### L1 human

In a hypergeometric series, the $n$-th coefficient is built from rising factorials such as $a(a+1)\cdots(a+n-1) = \Gamma(a+n)/\Gamma(a)$.
The Fox–Wright function allows $\Gamma(a + An)$ instead, so the argument of the gamma function may grow at any positive speed $A$.
This one extra parameter per factor covers many functions of fractional calculus.
For example, one numerator factor $\Gamma(1 + n)$ and one denominator factor $\Gamma(\beta + \alpha n)$ give the Mittag-Leffler function $\sum z^n / \Gamma(\alpha n + \beta)$.
</document>

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# Section of a vector bundle

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] and let $k \ge 0$.
A *smooth vector bundle* of rank $k$ over $M$ is a smooth [[fiber bundle|CX4M8K61AB]] $\pi \colon E \to M$ with fiber $\R^k$ and the following structure.
Each fiber $E_p = \pi^{-1}(p)$ is a real [[vector space|000AYPA9AJ]], and the local trivializations $\pi^{-1}(U) \to U \times \R^k$ can be chosen linear on each fiber.
A *smooth section* of $E$ is a [[smooth|0V33BNKZ9S]] map $s \colon M \to E$ such that
$$\pi \circ s = \id_M,$$
that is, $s(p) \in E_p$ for every $p \in M$.
The set of smooth sections is written $\Gamma(E)$.

## Remarks

The zero section $p \mapsto 0_p$ is a section of every vector bundle.
For the trivial bundle $E = M \times \R^k$, the sections are the maps $p \mapsto (p, u(p))$ with $u \colon M \to \R^k$ smooth.
A smooth vector field on $M$ is a section of the [[tangent bundle|WY6K3P4YCG]] $TM$.

Non-example: on $E = S^1 \times \R$, the map $s(p) = (-p, 0)$ is smooth, but $\pi(s(p)) = -p \neq p$, so it is not a section.

Sections are added and multiplied by smooth functions pointwise, so $\Gamma(E)$ is a module over $C^\infty(M)$.
The same definition, without linearity, gives sections of any fiber bundle; continuous sections are defined with continuous maps.

## Explanations

### L1 human

A section of a vector bundle is a smooth choice of one vector in the fiber over each point of the base.
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# Face lattice

## Example

Let $P \subset \R^d$ be a convex polytope, that is, the [[convex hull|0FQSDC2EQ2]] of finitely many points.
A face of $P$ is $P$ itself, the empty set, or a set $P \cap H$ for a supporting hyperplane $H$.
Here $H$ is supporting when $P$ lies in one closed half-space bounded by $H$.
The *face lattice* of $P$ is the set of all faces of $P$, ordered by inclusion.
It is a [[lattice|WH4SFAACAS]]: the meet of two faces is their intersection, and the join is the smallest face that contains both.
It is [[graded|1C7TKGTB2F]] by $\operatorname{rank}(F) = \dim F + 1$, with the empty face of rank $0$ and $P$ of rank $\dim P + 1$.
It is an [[Eulerian poset|N1Q0BJAEFX]].

Check for a square $Q$.
The faces are the empty face, $4$ vertices, $4$ edges and $Q$ itself, so the lattice has $10$ elements.
Two vertices have the empty face as meet; two adjacent vertices have their common edge as join, and two opposite vertices have $Q$ as join.
In the interval from the empty face to $Q$, the elements of even rank are the empty face and the $4$ edges.
The elements of odd rank are the $4$ vertices and $Q$.
There are five of each, as the Eulerian condition asks.

## Remarks

The face lattice records the [[combinatorial type|879SBTVE7V]] of the polytope: two polytopes are combinatorially equivalent exactly when their face lattices are isomorphic.
The face lattice of a $d$-simplex is the lattice of all subsets of its $d + 1$ vertices.
Abstract polytopes are defined by taking some properties of face lattices as axioms.

## Explanations

### L1 human

List all the pieces of a polytope: its vertices, edges, two-dimensional faces and so on, together with the empty set and the whole polytope.
Order them by "is contained in".
For a square you get the empty set at the bottom, four corners, four sides, and the square at the top.
This diagram forgets the exact shape and keeps only how the pieces fit together.
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# Lebesgue–Stieltjes measure

## Definition

Let $F \colon \R \to \R$ be a [[non-decreasing|0A7KV2WB7A]] function that is right-continuous at every point, that is, a [[Stieltjes function|WPR0NVC9PC]].
The *Lebesgue–Stieltjes measure* of $F$ is the unique [[measure|PR11BZQQPG]] $\mu_F$ on the [[Borel sigma-algebra|0WQXY32Q8D]] of $\R$ such that
$$\mu_F((a, b]) = F(b) - F(a) \quad \text{for all } a < b.$$

## Remarks

Existence comes from the Carathéodory extension theorem applied to the premeasure $(a, b] \mapsto F(b) - F(a)$ on half-open intervals; right-continuity is what makes it countably additive.
Uniqueness holds because $\R$ is a countable union of intervals $(-n, n]$ of finite measure.
Many authors also call the completion of $\mu_F$ the Lebesgue–Stieltjes measure.
Example: $F(x) = x$ gives Lebesgue measure on the Borel sets.
Example: for $F = \mathbf{1}_{[0, \infty)}$, which is right-continuous, $\mu_F$ is the Dirac measure at $0$, and $\mu_F(\set{a}) = F(a) - \lim_{x \to a^-} F(x)$ in general.
Non-example: $F(x) = -x$ is not non-decreasing, and the formula would give $(0, 1]$ the negative value $-1$.
Two functions $F$ and $G$ give the same measure if and only if $F - G$ is constant.
The notation $dF$ is common, as in $\int g \, dF$.
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# Faddeeva function

## Definition

Let $\operatorname{erfc}$ be the [[complementary error function|TZ3CT1YA9B]], extended to complex arguments as an [[entire function|0G4F0XB9X2]].
The *Faddeeva function* is the entire function $w \colon \C \to \C$ given by
$$w(z) = e^{-z^2} \operatorname{erfc}(-iz).$$

## Remarks

For $\operatorname{Im} z > 0$ it has the integral form
$$w(z) = \frac{i}{\pi} \int_{-\infty}^{\infty} \frac{e^{-t^2}}{z - t}\, dt.$$

Example: $w(0) = \operatorname{erfc} 0 = 1$.
For real $x$, $\operatorname{erfc}(-ix) = 1 + i \operatorname{erfi} x$ with $\operatorname{erfi} x = \frac{2}{\sqrt{\pi}}\int_0^x e^{t^2}\, dt$, so
$$w(x) = e^{-x^2} + \frac{2i}{\sqrt{\pi}} F(x),$$
where $F$ is the [[Dawson function|EQ4QB1QW60]].

Non-example: $z \mapsto e^{-z^2} \operatorname{erfc}(z)$ is a different function, because the argument of $\operatorname{erfc}$ is not rotated.
At $z = i$ it equals $e \operatorname{erfc}(i) = e\,(1 - i \operatorname{erfi} 1)$, which is not real, while $w(i) = e \operatorname{erfc} 1$ is real.

The real part of $w(x + iy)$ for $y > 0$ is, up to a constant factor, the Voigt profile, the convolution of a Gaussian with a Lorentzian.
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# Modulation space

## Definition

Let $d \ge 1$ and let $g$ be a nonzero function in the [[Schwartz space|CG9JEATW3Q]] $\mathcal{S}(\R^d)$, called the window.
For a [[tempered distribution|AES2T5JPFP]] $f \in \mathcal{S}'(\R^d)$, let $V_g f$ be its [[short-time Fourier transform|3DHNP5NEWH]]:
$$V_g f(x, \omega) = \int_{\R^d} f(t) \, \overline{g(t - x)} \, e^{-2\pi i t \cdot \omega} \, dt, \quad x, \omega \in \R^d,$$
understood as the action of $f$ on the Schwartz function $t \mapsto \overline{g(t - x)} e^{-2\pi i t \cdot \omega}$.
Let $1 \le p, q < \infty$.
The *modulation space* $M^{p,q}(\R^d)$ is the set of $f \in \mathcal{S}'(\R^d)$ with
$$\norm{f}_{M^{p,q}} = \left( \int_{\R^d} \left( \int_{\R^d} \abs{V_g f(x, \omega)}^p \, dx \right)^{q/p} d\omega \right)^{1/q} < \infty.$$
If $p = \infty$ or $q = \infty$, the corresponding integral is replaced by an essential supremum.

## Remarks

The space does not depend on the choice of the window $g$, and different windows give equivalent norms.
With this norm $M^{p,q}(\R^d)$ is a Banach space.
One writes $M^p = M^{p,p}$.
Special cases are $M^{2} = L^2(\R^d)$ with an equivalent norm, and $M^{1}$, the Feichtinger algebra.
Weighted versions $M^{p,q}_m$ insert a weight $m(x, \omega)$ into the integrand.

Example: let $d = 1$ and $\varphi(t) = e^{-\pi t^2}$, and take $g = \varphi$.
Completing the square in $t$ gives
$$\abs{V_\varphi \varphi(x, \omega)} = 2^{-1/2} e^{-\pi (x^2 + \omega^2)/2}.$$
This is integrable to every power in $x$ and in $\omega$, so $\varphi \in M^{p,q}(\R)$ for all $p, q$.

Non-example: the constant function $f = 1$ is not in $M^{p,q}(\R)$ for $p < \infty$.
Here $\abs{V_g 1(x, \omega)} = \abs{\hat{\overline{g}}(\omega)}$ does not depend on $x$, so the inner integral is infinite whenever $\hat{\overline{g}}(\omega) \ne 0$.
It does lie in $M^{\infty, 1}(\R)$.
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# Gauss–Laguerre quadrature

## Statement

Let $n \ge 1$ and let $L_n$ be the [[Laguerre polynomial|WAMRHWW82R]] of degree $n$, normalised by $L_n(0) = 1$.
Let $x_1, \ldots, x_n$ be the zeros of $L_n$; they are simple and lie in $(0, \infty)$.
Set
$$w_i = \frac{x_i}{(n+1)^2 \, L_{n+1}(x_i)^2} = \frac{1}{x_i \, L_n'(x_i)^2}, \qquad i = 1, \ldots, n.$$
Then $w_i > 0$ for each $i$, and for every polynomial $p$ of degree at most $2n - 1$,
$$\int_{0}^{\infty} e^{-x} p(x) \, dx = \sum_{i=1}^{n} w_i \, p(x_i).$$

## Remarks

This is the [[Gaussian quadrature|FP37BTP93K]] rule for the weight $e^{-x}$ on $[0, \infty)$, whose orthogonal polynomials are the $L_n$; the proof is the general one, by division by $L_n$ and orthogonality.
Example: for $n = 2$, $L_2(x) = \frac12 (x^2 - 4x + 2)$ has zeros $2 \pm \sqrt{2}$ and $L_2'(x) = x - 2 = \pm\sqrt{2}$ there, so $w = \frac{1}{2(2 \pm \sqrt{2})} = \frac{2 \mp \sqrt{2}}{4}$.
Check: the weights add up to $1 = \int_0^\infty e^{-x} dx$, and $\sum_i w_i x_i^2 = \frac{2 + \sqrt{2}}{2} + \frac{2 - \sqrt{2}}{2} = 2 = \int_0^\infty x^2 e^{-x} dx$.
The degree $2n - 1$ cannot be raised: for $p = L_n^2$ the rule gives $0$ and the integral is positive.
With the generalized Laguerre polynomials $L_n^{(\alpha)}$, $\alpha > -1$, the same construction gives rules for the weight $x^{\alpha} e^{-x}$.

## Explanations

### L1 human

Integrals from $0$ to infinity with a decaying factor $e^{-x}$ appear in physics, probability and the Laplace transform.
Gauss–Laguerre quadrature replaces such an integral by a weighted sum of a few values of the remaining factor at special points.
With $n$ points it integrates every polynomial of degree up to $2n - 1$ exactly.

### L2 human

To integrate $\int_0^\infty g(x)\,dx$, one writes $g(x) = e^{-x} \big(e^x g(x)\big)$ and applies the rule to $f(x) = e^x g(x)$.
This works well when $f$ is smooth and grows at most like a polynomial.
The rule is the half-line companion of [[Gauss–Hermite quadrature|89S5KGQEMJ]] on the whole line and of Gauss–Legendre quadrature on a bounded interval.
</document>

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# Square root of a matrix

## Definition

Let $\mathbb{K}$ be a field, let $n \ge 1$, and let $A$ be an $n \times n$ [[matrix|0JXZKX526R]] over $\mathbb{K}$.
A *square root* of $A$ is an $n \times n$ matrix $B$ over $\mathbb{K}$ such that $B^2 = A$, where $B^2 = BB$ is the [[matrix product|032Q9FB8MT]].

## Remarks

Example: for every real $\theta$, $B = \begin{pmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{pmatrix}$ is a square root of $I_2$.
Check: the entries of $B^2$ are $\cos^2\theta + \sin^2\theta = 1$ on the diagonal and $\cos\theta\sin\theta - \sin\theta\cos\theta = 0$ off it.
Non-example: $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ has no square root.
If $B^2 = N$, then $B^4 = 0$, so $B$ is [[nilpotent|07BSPZPSMF]]; a nilpotent $2 \times 2$ matrix satisfies $B^2 = 0 \ne N$.
A [[positive semidefinite matrix|01MMVW8CKC]] has a unique positive semidefinite square root, written $A^{1/2}$; see [[square root of a positive semidefinite matrix|0BJVDCF38Y]].
A complex matrix with no eigenvalue in $(-\infty, 0]$ has a unique square root whose eigenvalues have positive real part, the *principal square root*.
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# Best approximation

## Definition

Let $X$ be a [[normed vector space|03J5RAKB30]], let $A \subseteq X$ be nonempty, and let $x \in X$.
A point $a^* \in A$ is a *best approximation* of $x$ from $A$ if
$$\norm{x - a^*} \le \norm{x - a} \quad \text{for all } a \in A.$$
Equivalently, $\norm{x - a^*} = \inf_{a \in A} \norm{x - a}$, an [[infimum|04KFTA34C5]] that is attained at $a^*$.

## Remarks

Example: in $\R^2$ with the Euclidean norm, let $A$ be the horizontal axis and $x = (1, 2)$.
For $a = (t, 0)$ one has $\norm{x - a}^2 = (1 - t)^2 + 4 \ge 4$, with equality only at $t = 1$, so $(1, 0)$ is the best approximation.

Non-example: in $\R$, let $A = (0, 1]$ and $x = 0$.
The infimum of $\abs{0 - a}$ over $A$ is $0$, but no $a \in A$ attains it, so $x$ has no best approximation from $A$.

Other names are nearest point and, in the uniform norm, minimax approximation.
The set of best approximations of $x$ from $A$ can be empty, a single point, or larger.
</document>

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# Asymptotic scale

## Summary

An asymptotic scale, also called an asymptotic sequence, is a sequence of functions $\varphi_0, \varphi_1, \dots$ near a point $x_0$.
Each function in it is negligible compared with the one before it.
It is the system of reference functions in which [[asymptotic expansions|BJQTEPEZ9G]] are written.
The most common scale is the powers $x^{-n}$ as $x \to \infty$, but scales with logarithms, fractional powers or exponentials are also needed.
Fixing a scale makes the coefficients of an expansion unique.

## Explanations

### L1 human

An asymptotic scale is a ladder of functions, each one much smaller than the previous one when $x$ is close to the limit point.
It plays the role that the powers $1, x, x^2, \dots$ play in a Taylor series.
An approximation is then described by saying how many steps of the ladder it gets right.
For example, $1, \frac1x, \frac{1}{x^2}, \dots$ is a scale as $x \to \infty$, because $\frac{1/x^{n+1}}{1/x^n} = \frac1x \to 0$.
</document>

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# Siegel–Walfisz theorem

## Statement

Let $q \ge 1$ and $a$ be integers, and let $x \ge 2$ be real.
Let
$$\psi(x; q, a) = \sum_{\substack{n \le x \\ n \equiv a \pmod q}} \Lambda(n),$$
where $\Lambda$ is the [[von Mangoldt function|0NK8JTE1NN]], and let $\varphi$ be [[Euler's totient function|0DAMHCT17R]].
For every real number $N > 0$ there is a constant $C_N > 0$ such that
$$\psi(x; q, a) = \frac{x}{\varphi(q)} + O\!\left(x\, e^{-C_N \sqrt{\log x}}\right)$$
for all $x \ge 2$, all $q \le (\log x)^N$ and all $a$ coprime to $q$.
The constant $C_N$ and the implied constant in the [[big O|0TGV80E6A4]] depend only on $N$.

## Remarks

The proof follows the proof of the [[prime number theorem|0EHBY5Y84N]] for each [[Dirichlet L-function|9D2Z543EJQ]] $L(s, \chi)$ with $\chi$ modulo $q$.
The classical zero-free region handles all zeros except a possible [[Siegel zero|5VBZBXSFDQ]].
Siegel's theorem says that for every $\varepsilon > 0$ such a real zero $\beta$ satisfies $\beta < 1 - c(\varepsilon) q^{-\varepsilon}$, and for $q \le (\log x)^N$ this is enough to make its contribution small.
Siegel's theorem is ineffective, so $C_N$ cannot be computed from the proof.
For example, with $N = 2$ and $q = 4$, the theorem gives $\psi(x; 4, 1) = x/2 + O(x e^{-c\sqrt{\log x}})$, a strong form of [[Dirichlet's theorem on primes in arithmetic progressions|0GNVN3CA2P]] for primes $p \equiv 1 \pmod 4$.
The range $q \le (\log x)^N$ is very short; for larger $q$ one has only average results such as the [[Bombieri–Vinogradov theorem|B3JMY54GJF]].

## Explanations

### L1 human

Primes are spread evenly over the residue classes modulo $q$ that they can reach.
This theorem gives a precise error bound for that fact, valid as long as the modulus $q$ is small compared to $x$, at most a power of $\log x$.
It is the prime number theorem for arithmetic progressions, uniform in small moduli.

### L2 human

The Siegel–Walfisz theorem is the standard input for the "small moduli" part of many arguments in analytic number theory.
In proofs of the [[Bombieri–Vinogradov theorem|B3JMY54GJF]] and in the circle method for [[Vinogradov's theorem|C01Z9QRGZY]], the major arcs with small denominators are handled by it.
Its constants are ineffective because of the possible [[Siegel zero|5VBZBXSFDQ]]; this is why results that rely on it, such as Vinogradov's theorem in its first form, give no explicit bound.
Under the [[generalized Riemann hypothesis|DNKA4WBKDS]], the error improves to $O(x^{1/2} (\log x)^2)$ for all $q \le x$.
</document>

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# F-sigma set

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A subset $A \subseteq X$ is an *$F_\sigma$ set* if there are [[closed sets|0ZCK8DQYN1]] $F_1, F_2, F_3, \dots$ of $X$ with
$$A = \bigcup_{n=1}^{\infty} F_n.$$

## Remarks

A set is $F_\sigma$ if and only if its complement is a [[G-delta set|V1TBNP8R3S]], by De Morgan's laws.

Example: $\Q = \bigcup_{q \in \Q} \set{q}$ in $\R$.
Each $\set{q}$ is closed and $\Q$ is [[countable|01TQB88H4C]], so $\Q$ is $F_\sigma$.

Example: $(0, 1) = \bigcup_{n \ge 3} [1/n, 1 - 1/n]$, so this open interval is $F_\sigma$.
In a [[metric space|0VKD1TP7F3]], every open set is $F_\sigma$.

Non-example: the set $\R \setminus \Q$ of irrational numbers is not $F_\sigma$ in $\R$.
Otherwise its complement $\Q$ would be a $G_\delta$ set, and it is not.
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# Leech lattice

## Description

The Leech lattice $\Lambda_{24}$ is a [[lattice|WN9V7WSJKF]] in the [[Euclidean space|0XPR1E85W0]] $\R^{24}$, discovered by John Leech.
Up to isometry, it is the unique lattice $L \subset \R^{24}$ with these properties:

1. $L$ is unimodular: it has a basis whose Gram matrix has determinant $1$;
2. $L$ is even: $x \cdot x$ is an even integer for every $x \in L$;
3. $L$ has no vector $x$ with $x \cdot x = 2$.

So its shortest nonzero vectors have norm $x \cdot x = 4$.
There are $196560$ of them.

## Remarks

The Leech lattice can be constructed from the extended binary [[Golay code|SM1TJSAR75]].
Its automorphism group $\mathrm{Co}_0$ has order $8315553613086720000$, and its quotient by $\set{\pm 1}$ is the [[Conway group|366D4ETGQH]] $\mathrm{Co}_1$.
The $196560$ shortest vectors show that the [[kissing number|B408FZYXF9]] in dimension $24$ is at least $196560$, and in fact it equals $196560$.
The [[sphere packing|HF32CBN5MA]] obtained from the Leech lattice is the densest sphere packing in $\R^{24}$; this was proved by Cohn, Kumar, Miller, Radchenko and Viazovska.
Even unimodular lattices exist only in dimensions divisible by $8$.
In dimension $24$ there are $24$ of them up to isometry, and the Leech lattice is the only one without vectors of norm $2$.

## Explanations

### L1 human

A lattice is a regular grid of points, like the integer points in the plane, but it can be tilted and stretched.
The Leech lattice is a very special grid in $24$ dimensions.
If you put equal balls at its points, each ball touches $196560$ others, the most possible in $24$ dimensions, and the packing is the densest possible there.
Its symmetry group is huge and gave rise to several new finite simple groups.
</document>

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# Filippov's existence theorem

## Statement

Let $U \subseteq \R^m$ be a nonempty [[compact set|0R6EWFKRCF]], let $T > 0$, and let $f \colon [0, T] \times \R^n \times U \to \R^n$ be continuous.
Let $x_0 \in \R^n$, let $S \subseteq \R^n$ be a closed set, and let $\varphi \colon \R^n \to \R$ be continuous.
Call a measurable map $u \colon [0, T] \to U$ an [[admissible control|8M31QKB0GW]] if the [[control system|ZPDTVZQN5K]] $x'(t) = f(t, x(t), u(t))$ with $x(0) = x_0$ has an absolutely continuous solution $x$ on $[0, T]$ with $x(T) \in S$.
Assume that:

1. at least one admissible control exists;
2. there is a constant $R$ such that $\abs{x(t)} \le R$ for every $t \in [0, T]$ and every trajectory $x$ of an admissible control;
3. for every $(t, x) \in [0, T] \times \R^n$ the set of velocities $f(t, x, U) = \set{f(t, x, v) : v \in U}$ is [[convex|02AS45T3H9]].

Then the [[Mayer problem|1J8TYT7C49]] of minimizing $\varphi(x(T))$ over all admissible controls has a solution.
That is, there is an admissible control $u^*$ with trajectory $x^*$ such that $\varphi(x^*(T)) \le \varphi(x(T))$ for every admissible trajectory $x$.

## Remarks

The proof is a compactness argument.
Take a minimizing sequence of trajectories; they are uniformly bounded with uniformly bounded derivatives, so the Arzelà–Ascoli theorem gives a subsequence that converges uniformly to an absolutely continuous limit $x^*$.
The convexity of the velocity sets shows that $x^*$ solves the [[differential inclusion|9ZSMF6NYKM]] $x'(t) \in f(t, x(t), U)$ for almost every $t$.
A measurable selection result, known as Filippov's lemma, then gives a measurable control $u^*$ with $x^{*\prime}(t) = f(t, x^*(t), u^*(t))$, and continuity of $\varphi$ shows that $u^*$ is optimal.

Convexity cannot be dropped.
Take $n = 2$, $U = \set{-1, 1}$, $f(t, x, u) = (u, x_1^2)$, $x_0 = (0, 0)$, $T = 1$, $S = \set{0} \times \R$ and $\varphi(x) = x_2$, so that the cost is $\int_0^1 x_1(t)^2 \, dt$ with $x_1' = \pm 1$ and $x_1(1) = 0$.
Controls that switch faster and faster keep $x_1$ close to $0$, so the infimum of the cost is $0$.
A cost $0$ would need $x_1 \equiv 0$, hence $x_1' = 0 \notin \set{-1, 1}$, so no optimal control exists.
Here the velocity sets $\set{(-1, x_1^2), (1, x_1^2)}$ are not convex.

A Lagrange cost $\int_0^T L(t, x, u) \, dt$ is handled by adding the state $y' = L(t, x, u)$; the convexity hypothesis then concerns the sets $\set{(f(t, x, v), L(t, x, v) + \gamma) : v \in U, \gamma \ge 0}$.
Hypothesis 2 holds, for example, when $\abs{f(t, x, v)} \le C(1 + \abs{x})$ for a constant $C$, by Grönwall's inequality.

## Explanations

### L1 human

An optimal control problem can fail to have a best solution, because cheaper and cheaper controls may wiggle faster and faster without reaching a limit control.
Filippov's theorem says that this cannot happen when the control values form a compact set and, at each state, the possible velocities form a convex set.
Then a minimizing sequence settles down to a genuine optimal control.

### L2 human

The theorem is the optimal control version of the direct method of the calculus of variations.
Compactness comes from bounded velocities, and convexity of the velocity sets plays the role of lower semicontinuity.
When convexity fails, one passes to relaxed controls, which convexify the velocity sets.
The relaxed problem then has a minimizer, and under mild extra hypotheses its value is the infimum of the original problem.
Existence is the first step before using necessary conditions such as [[Pontryagin's maximum principle|7RPV2T29CF]], which only describe minimizers that exist.
For linear systems $x' = A x + B u$ with $U$ compact and convex, the velocity sets are automatically convex.
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# Adjoint integral equation

## Summary

Let an [[integral equation of the second kind|8KJCN49V64]] have kernel $K(x, s)$ and parameter $\lambda$.
Its adjoint is the equation of the same kind with the kernel $\overline{K(s, x)}$ and the parameter $\bar\lambda$.
Its integral operator is the [[adjoint|0R9NMSM1ZP]] of the original integral operator for the $L^2$ inner product.
The Fredholm alternative uses it: the original equation is solvable exactly when its right side is orthogonal to the solutions of the homogeneous adjoint equation.
For a real kernel the adjoint kernel is the transposed kernel $K(s, x)$.

## Explanations

### L1 human

The adjoint equation swaps the roles of the two variables of the kernel, like transposing a matrix.
For complex kernels one also takes complex conjugates, as in the conjugate transpose of a matrix.
The original equation and its adjoint have solvability properties that mirror each other.
For example, the adjoint of $\varphi(x) = f(x) + \lambda \int_0^1 x \, \varphi(s) \, ds$ is $\psi(x) = g(x) + \bar\lambda \int_0^1 s \, \psi(s) \, ds$.
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# Kontorovich–Lebedev transform

## Summary

The Kontorovich–Lebedev transform integrates a function $f$ on $(0, \infty)$ against the [[modified Bessel function|2ET068BZZH]] $K_{iy}(x)$ of imaginary order $iy$.
Unlike most transforms, the transform variable $y$ enters through the order of the Bessel function, not through its argument.
It has an explicit inversion formula with the weight $y \sinh(\pi y)$.
It is used to solve boundary value problems in wedge-shaped and conical regions, where the separation of variables leads to Bessel functions of imaginary order.

## Explanations

### L1 human

Many transforms take a family of functions, such as waves of every frequency, and measure how much of each one a given function contains.
Here the family is $x \mapsto K_{iy}(x)$, indexed by a real number $y$.
These functions oscillate faster and faster near $x = 0$ as $y$ grows and decay exponentially for large $x$.
The transform records the overlap of $f$ with each of them.
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<document source="https://tamnd.github.io/math2/node/XFJVAMSAQ2.md" tag="XFJVAMSAQ2" kind="definition">
# Superalgebra

## Definition

Let $K$ be a [[commutative ring|0GYGHKX834]].
Let $A$ be a [[$K$-module|09VDS2W1E8]] with a direct sum decomposition $A = A_0 \oplus A_1$.
Then $A$ with a $K$-bilinear product $A \times A \to A$ is a *superalgebra* over $K$ if
$$A_i A_j \subseteq A_{i + j}$$
for all $i, j \in \Z/2\Z$, with indices added modulo $2$.
Elements of $A_0$ are *even*, elements of $A_1$ are *odd*, and a nonzero element of $A_0 \cup A_1$ is *homogeneous* of parity $\abs{a} \in \{0, 1\}$.
Unless stated otherwise the product is associative with a unit.

## Remarks

A superalgebra is *supercommutative* if $ab = (-1)^{\abs{a}\abs{b}} ba$ for all homogeneous $a, b$.
Example: the exterior algebra $\Lambda(V)$ of a vector space $V$, with $A_0$ the sum of the even powers $\Lambda^{2m} V$ and $A_1$ the sum of the odd powers.
The product maps $\Lambda^p V \times \Lambda^r V$ into $\Lambda^{p+r} V$, and $p + r$ has the parity of the sum, so the condition holds; moreover $\Lambda(V)$ is supercommutative.
Example: the matrix ring $M_2(K)$, with $A_0$ the diagonal matrices and $A_1$ the off-diagonal ones, is a superalgebra that is not supercommutative.
Non-example: $K[x]$ with $A_0 = K$ and $A_1 = xK[x]$ fails the condition, since $x \in A_1$ but $x \cdot x = x^2 \notin A_0$.
Equivalently, a superalgebra is a monoid object in the category of $\Z/2\Z$-graded modules, with the tensor product of graded modules.
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# Continuous q-Hermite polynomials

## Summary

The continuous $q$-Hermite polynomials $H_n(x \mid q)$ are a family of orthogonal polynomials on $[-1, 1]$ depending on a base $q$.
They are $q$-analogues of the [[Hermite polynomials|MGACY98E9H]], to which they tend after rescaling as $q \to 1$.
At $q = 0$ they are the Chebyshev polynomials of the second kind.
They are one of the simplest families of the $q$-Askey scheme, a special case of the [[Rogers polynomials|7H7SACRYEF]] and of the [[Askey–Wilson polynomials|RSNZNG0WA1]].
They appear in $q$-deformed probability and in the theory of $q$-Gaussian processes.

## Explanations

### L1 human

Write $x = \cos\theta$ and form the sum of $e^{i(n - 2k)\theta}$ over $k = 0, \dots, n$, each weighted by a [[Gaussian binomial coefficient|YS09Z1YNEK]] $\binom{n}{k}_q$.
The result is a polynomial in $x$ of degree $n$, the continuous $q$-Hermite polynomial.
At $q = 0$ all weights are $1$, and the sum is $\frac{\sin((n+1)\theta)}{\sin\theta}$.
For $n = 2$ one gets $H_2(x \mid q) = 4x^2 - (1 - q)$.
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# Sturmian word

## Example

A Sturmian word is an infinite word $s = s_0 s_1 s_2 \cdots$ over $\set{0, 1}$ that has exactly $n + 1$ distinct factors of length $n$ for every $n \ge 0$.
A factor of length $n$ is a block $s_i s_{i+1} \cdots s_{i+n-1}$.
Equivalently, there are an [[irrational number|080T2TXVQV]] $\alpha \in (0, 1)$ and a real $\rho$ with the following property.
For all $n$, $s_n = \floor{(n+1)\alpha + \rho} - \floor{n \alpha + \rho}$, or the same holds with $\ceil{\cdot}$ in place of $\floor{\cdot}$.
Then $s_n = 1$ exactly when the point $n\alpha + \rho$ of the circle $\R/\Z$ lies in a fixed arc of length $\alpha$.
So $s$ codes the [[irrational rotation|2Q1J7RZ4TK]] by $\alpha$.

The Fibonacci word is the main instance.
Define $S_0 = 0$, $S_1 = 01$ and $S_{k} = S_{k-1} S_{k-2}$ for $k \ge 2$.
Each $S_{k-1}$ is a prefix of $S_k$, and the limit is
$$f = 0100101001001010010100100101 \cdots .$$
Check for small $n$: the factors of length $1$ are $0, 1$, and those of length $2$ are $00, 01, 10$, since $11$ never occurs.
So there are $2$ and $3$ of them, as required.
For $k \ge 1$ the word $S_k$ has $F_k$ zeros and $F_{k-1}$ ones, with $F_k$ the Fibonacci numbers ($F_0 = F_1 = 1$).
So the frequency of the letter $1$ is $\lim F_{k-1}/F_{k+1} = 1/\varphi^2$, where $\varphi$ is the [[golden ratio|0SC8T7548D]].

## Remarks

A word with at most $n$ factors of length $n$ for some $n$ is eventually periodic, by a theorem of Morse and Hedlund.
So Sturmian words are the aperiodic words of the lowest possible complexity.
The closure of the shift orbit of a Sturmian word is a minimal shift space on which the shift has zero entropy.

## Explanations

### L1 human

Draw a straight line of irrational slope across a grid of squares.
Write $0$ each time the line crosses a vertical grid line and $1$ each time it crosses a horizontal one.
The resulting word is Sturmian: it never repeats periodically, yet it is as simple as a non-repeating word can be.
The Fibonacci word, built by gluing the two previous words together, is the best known example.
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# Naive set theory

## Summary

Naive set theory is [[set theory|RTHHCX84HN]] done in ordinary language, without a formal list of axioms.
In its original form, every property defines a [[set|0BSJ7RQ9BW]], the set of all objects with that property.
This principle leads to contradictions, the best known being [[Russell's paradox|0HESMMSYEG]].
The axiomatic theories, such as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], restrict how sets may be formed to avoid them.
The name is also used for the informal set theory taught at the start of most courses, which is consistent with those axioms.

## Explanations

### L1 human

Naive set theory treats a set as any collection you can describe.
This works well for everyday sets, such as the even numbers or the points of a circle.
But consider the set of all sets that are not members of themselves.
If it is a member of itself, then it is not, and if it is not, then it is.
So "any collection you can describe" is too generous, and the axioms of modern set theory say more carefully which sets exist.
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# Hadamard's two-constants theorem

## Statement

Let $\Omega \subset \C$ be a bounded [[domain|0Q4CBD3HX0]] on which the [[Dirichlet problem|8D56GQ4AG8]] is solvable for continuous boundary data.
Let $E \subseteq \partial\Omega$ be a closed set, and let $\omega(z) = \omega(z, E, \Omega)$ be its [[harmonic measure|CFD7BPM51W]] at $z \in \Omega$.
Let $0 < m \le M$, and let $f$ be a [[holomorphic function|0NAHXR1FRH]] on $\Omega$ such that:

1. $\abs{f(z)} \le M$ for all $z \in \Omega$.
2. $\limsup_{z \to \zeta} \abs{f(z)} \le m$ for every $\zeta \in E$.

Then for every $z \in \Omega$,
$$\abs{f(z)} \le m^{\omega(z)} M^{1 - \omega(z)}.$$

## Remarks

Proof idea: the function $\log\abs{f}$ is [[subharmonic|Z1MHMGFNFW]], and $h = \omega \log m + (1 - \omega) \log M$ is harmonic.
Near points of $E$ the bound for $\log\abs{f}$ is $\log m$, elsewhere it is $\log M$, and $h$ takes these boundary values in the generalized (Perron) sense.
A maximum principle for $\log\abs{f} - h$, in the form valid for Perron solutions, gives $\log\abs{f} \le h$.
Example: on the annulus $r_1 < \abs{z} < r_2$ with $E$ the inner circle, $\omega(z) = \log(r_2/\abs{z}) / \log(r_2/r_1)$.
With $m$ and $M$ the maxima of $\abs{f}$ on the two circles, this gives the [[Hadamard three-circle theorem|V00Q05ZW1G]].
On a strip, the same argument gives the [[Hadamard three-lines theorem|WHS58R43N0]].
If $E$ is the whole boundary, then $\omega \equiv 1$ and the theorem reduces to the maximum modulus principle.

## Explanations

### L1 human

Suppose a holomorphic function is at most $M$ everywhere and at most a smaller number $m$ on part of the boundary.
Then inside the region the bound improves, and it improves more at points that "see" more of that part of the boundary.
Harmonic measure measures exactly how much of the boundary a point sees.

### L2 human

The theorem is the main way harmonic measure enters estimates of holomorphic functions.
It turns lower bounds for harmonic measure into upper bounds for holomorphic functions.
Taking logarithms, it says that $\log\abs{f}$ lies below the harmonic interpolation between $\log m$ and $\log M$.
It is a special case of the principle that a subharmonic function lies below the harmonic function with the same boundary bounds.
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# Colimit

## Definition

Let $J$ and $\mathcal{C}$ be [[categories|A2DE66937D]] and let $D \colon J \to \mathcal{C}$ be a [[diagram|HJK75HDWFE]].
A *colimit* of $D$ is a [[cocone|7TPBQPPN97]] $(C, \iota)$ under $D$, with $\iota_j \colon D(j) \to C$, that satisfies the following condition.
For every cocone $(N, \psi)$ under $D$ there is exactly one morphism $h \colon C \to N$ with $h \circ \iota_j = \psi_j$ for every object $j$ of $J$.
The object $C$ is written $\operatorname{colim} D$ or $\varinjlim D$, and the $\iota_j$ are its *injections*.

## Remarks

Equivalently, a colimit is an [[initial object|EK85VSSRMR]] of the category of cocones under $D$, so it is unique up to unique isomorphism.
A colimit of $D$ in $\mathcal{C}$ is the same as a [[limit|CHS5AFJGWE]] of the corresponding diagram $D^{\mathrm{op}} \colon J^{\mathrm{op}} \to \mathcal{C}^{\mathrm{op}}$ in the [[opposite category|DAMK8XE1FR]].
The colimit of the empty diagram is an initial object.

Example: in the category of sets, every diagram $D$ with $J$ small has a colimit.
Let $\sim$ be the smallest [[equivalence relation|0G8VEZ3WTK]] on the disjoint union $\bigsqcup_j D(j)$ that identifies $x \in D(j)$ with $D(u)(x) \in D(k)$ for every $u \colon j \to k$.
The colimit is the quotient $\left(\bigsqcup_j D(j)\right) / {\sim}$.
The injections send $x$ to its class, and a cocone $(N, \psi)$ induces $h([x]) = \psi_j(x)$, which is well defined by the cocone condition.

Non-example: for the diagram of two parallel functions $f, g \colon X \rightrightarrows Y$ with $f \neq g$, the set $Y$ with $\iota_Y = \id_Y$ and $\iota_X = f$ is not a cocone, since $\iota_Y \circ g = g \neq \iota_X$; so it cannot be a colimit.
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# Cartan–Kähler theorem

## Statement

Let $M$ be a real-analytic manifold and let $\mathcal{I}$ be an [[exterior differential system|901CCKQHVY]] on $M$ generated by real-[[analytic|081WKEDK6C]] differential forms.
Let $p \in M$ and let $E \subseteq T_pM$ be an $n$-dimensional ordinary integral element of $\mathcal{I}$, in the sense of [[Cartan's test|WV4S47X6QC]].
So $E$ has a flag $\set{0} = E_0 \subset E_1 \subset \dots \subset E_n = E$ of integral elements with $\dim E_k = k$ and $E_k$ regular for $k < n$.
Then there is a real-analytic $n$-dimensional integral manifold $N \subseteq M$ of $\mathcal{I}$, a [[submanifold|VKFW27WXEZ]] with $p \in N$ and $T_pN = E$.

## Remarks

The proof builds $N$ one dimension at a time.
Start from a point.
An integral manifold of dimension $k$, tangent to $E_k$, is extended to one of dimension $k + 1$, tangent to $E_{k+1}$.
This step solves an initial value problem with the [[Cauchy–Kovalevskaya theorem|97HMV9BT2W]].
Regularity of $E_k$ ensures that the equations to solve are in Cauchy–Kovalevskaya form, and closure of $\mathcal{I}$ under $d$ ensures that the new manifold is again integral.
At each step there is freedom in the choice of initial data, and counting it gives the size of the space of solutions.

Example: let $\mathcal{I}$ be generated by $1$-forms $\theta^1, \dots, \theta^s$, independent at each point of an $m$-dimensional manifold.
If $d\theta^a \equiv 0$ modulo $\theta^1, \dots, \theta^s$, then every $(m - s)$-plane in the kernel of the $\theta^a$ is ordinary.
The theorem then gives integral manifolds of dimension $m - s$, as does the [[Frobenius theorem|5Q4BHBFXJY]], which holds without analyticity.

Analyticity is needed in general.
The method rests on the Cauchy–Kovalevskaya theorem, which has no smooth version.
Lewy's example gives a linear partial differential equation with polynomial coefficients and a smooth right-hand side with no solution near any point.
Writing such an equation as an exterior differential system shows that the smooth analogue of the theorem fails.

## Explanations

### L1 human

The theorem gives solutions to systems of partial differential equations written in the language of differential forms.
It says that if the algebraic test of Cartan is passed at a point, then a genuine solution passes through that point with the predicted tangent plane.
It needs all data to be real-analytic, that is, given locally by convergent power series.

### L2 human

The theorem is the main existence result of the theory of exterior differential systems.
It contains the Cauchy–Kovalevskaya theorem and, in the analytic case, the Frobenius theorem.
It was used, for instance, to show that every real-analytic Riemannian manifold of dimension $n$ embeds locally isometrically into Euclidean space of dimension $n(n+1)/2$, the Cartan–Janet theorem.
In practice one first checks Cartan's test, prolonging the system if the test fails, and then applies the theorem.
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# Regular semigroup

## Summary

A regular semigroup is a [[semigroup|0XMEM406JM]] in which each element $a$ has a partner $x$ with $axa = a$.
Groups, the full [[transformation semigroups|7ZB7MRKPQ9]], and the semigroups of all $n \times n$ matrices over a field under multiplication are regular.
Regular semigroups are the class in which [[Green's relations|S7WJW2YES2]] give the most detailed structure theory.
The [[inverse semigroups|3M724K3B50]] are the regular semigroups in which each $a$ has exactly one $x$ with $axa = a$ and $xax = x$.

## Explanations

### L1 human

In a group, every element can be undone by its inverse.
In a regular semigroup, every element can at least be undone partly: some $x$ brings $a$ back to itself in the product $axa$.
For a function $f$ that is not a bijection, one can take $x$ to map each value of $f$ back to one of its preimages.
Then applying $f$, then $x$, then $f$ again gives the same result as $f$.
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# Stability conjecture

## Statement

Let $M$ be a compact smooth manifold without boundary and let $f \colon M \to M$ be a $C^1$ diffeomorphism, with [[non-wandering set|EY2W34JZA4]] $\Omega(f)$.
Say that $f$ satisfies the strong transversality condition if, for all $x, y \in \Omega(f)$, the stable manifold $W^s(x)$ and the unstable manifold $W^u(y)$ intersect [[transversally|EWVATW6EYZ]].
Then $f$ is [[structurally stable|KQX43WB57R]] in the $C^1$ topology if and only if $f$ satisfies [[Axiom A|S31RTTJPDC]] and the strong transversality condition.

## Remarks

The statement was conjectured by Palis and Smale and is now a theorem.
The direction from Axiom A and strong transversality to structural stability was proved by Robbin for $C^2$ diffeomorphisms and by Robinson in the $C^1$ case.
The converse direction, that $C^1$ structural stability forces Axiom A and strong transversality, was proved by Mañé.
Its proof uses closing-lemma techniques, in particular Mañé's ergodic version of the [[closing lemma|E1Z444F6TV]], and a delicate perturbation argument to obtain hyperbolicity from robustness.
Example: hyperbolic toral automorphisms such as [[Arnold's cat map|BSCZN8Y1AN]] satisfy both conditions, so they are structurally stable.
An irrational rotation of the circle is not structurally stable, as nearby rational rotations have periodic orbits.
Accordingly it does not satisfy Axiom A.
The analogous statement for flows was proved by Hayashi.

## Explanations

### L1 human

A system is structurally stable if small changes of the rule do not change the pattern of its orbits.
The theorem says exactly which systems have this property.
They are the systems that stretch and squeeze uniformly on their recurrent part, and whose stable and unstable directions cross cleanly.
So robustness and hyperbolicity turn out to be the same thing.

### L2 human

The theorem closes a programme started by Smale in the 1960s: to describe the structurally stable systems.
The easy half shows that hyperbolic systems are robust, using the persistence of hyperbolic sets under perturbation.
The hard half shows that robustness forces hyperbolicity; it relies on perturbation tools such as the [[closing lemma|E1Z444F6TV]].
A related result is the $\Omega$-stability theorem, which characterises the stability of the dynamics on $\Omega(f)$ by Axiom A and the absence of cycles.
In the $C^r$ topology with $r \ge 2$ the necessity direction is still open.
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# Colombeau algebra

## Summary

A Colombeau algebra is an algebra of generalized functions that contains the distributions and in which any two elements can be multiplied.
Its elements are classes of families of smooth functions indexed by a small parameter, with bounds that are polynomial in the inverse of the parameter.
It keeps the product of smooth functions and has a derivative obeying the Leibniz rule.
It is used for nonlinear equations with singular data.

## Explanations

### L1 human

The delta function cannot be squared as a distribution.
A Colombeau algebra replaces the delta by the whole family of narrower and narrower spikes that approximate it.
Families can be multiplied term by term, so products such as the square of the delta make sense.
Families that become small faster than any power of the parameter are treated as zero.
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# Analytic combinatorics

## Definition

Let $(a_n)_{n \ge 0}$ be a sequence of counts with [[generating function|0PHAC833JF]] $A(z) = \sum_{n} a_n z^n$ or $\sum_n a_n z^n / n!$.
*Analytic combinatorics* is the branch of [[enumerative combinatorics|EX382NQTGP]] that derives the asymptotic behaviour of $a_n$ as $n \to \infty$ from $A$.
Here $A$ is viewed as an [[analytic function|081WKEDK6C]] of a complex variable $z$.
Its main methods are:

1. location of the dominant singularities, which fixes the exponential growth rate;
2. singularity analysis, which transfers the local behaviour of $A$ near a dominant singularity to the coefficients;
3. saddle-point estimates of the Cauchy coefficient integral, mainly for entire functions $A$.

## Remarks

The basic principle is the Cauchy–Hadamard formula.
If $A$ has [[radius of convergence|0N288P2TB5]] $\rho$ with $0 < \rho < \infty$, then $\limsup_{n \to \infty} \abs{a_n}^{1/n} = 1/\rho$.
When $a_n \ge 0$, Pringsheim's theorem says that $z = \rho$ is itself a singularity of $A$.

Example: the Catalan generating function is $C(z) = \frac{1 - \sqrt{1 - 4z}}{2z}$.
Its dominant singularity is $z = 1/4$, where $C$ behaves like $2 - 2\sqrt{1 - 4z}$.
Singularity analysis then gives $C_n \sim \frac{4^n}{\sqrt{\pi}\, n^{3/2}}$.

Non-example: a proof that $\binom{2n}{n} \le 4^n$ by counting subsets is not analytic combinatorics.
It uses no property of a generating function as a function of a complex variable.
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# Kähler manifold

## Definition

Let $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $n$.
Let $h$ be a Hermitian metric on $X$: a smooth family of positive definite [[Hermitian forms|48P4A1CX85]] $h_p$ on the holomorphic tangent spaces $T^{1,0}_p X$.
In local holomorphic coordinates write $h = \sum_{j,k} h_{j\bar k}\, dz_j \otimes d\bar z_k$, where $(h_{j\bar k})$ is a positive definite Hermitian matrix of smooth functions.
The *fundamental form* of $h$ is the real $(1,1)$-[[form|0E3G0GRPZ6]]
$$\omega = \frac{i}{2} \sum_{j,k} h_{j\bar k}\, dz_j \wedge d\bar z_k.$$
The metric $h$ is *Kähler* if $\omega$ is [[closed|0ZV14XRNF0]], that is, $d\omega = 0$.
A *Kähler manifold* is a complex manifold together with a Kähler metric; $\omega$ is then called its Kähler form.

## Remarks

In coordinates, $d\omega = 0$ holds if and only if $\dfrac{\partial h_{j\bar k}}{\partial z_l} = \dfrac{\partial h_{l\bar k}}{\partial z_j}$ for all $j, k, l$.
Equivalent forms: locally $\omega = i\partial\bar\partial\varphi$ for a real smooth strictly plurisubharmonic function $\varphi$; or the complex structure is parallel for the Levi-Civita connection of the underlying Riemannian metric $\operatorname{Re} h$.
A compact complex manifold is called Kähler if it admits some Kähler metric.

Example: on $\C^n$, $h_{j\bar k} = \delta_{jk}$ gives $\omega = \frac{i}{2}\sum_j dz_j \wedge d\bar z_j$, which has constant coefficients and so is closed.
On a Riemann surface every Hermitian metric is Kähler, because every $3$-form on a manifold of real dimension $2$ is zero.
The Fubini–Study form on $\mathbb{CP}^n$, given on $\set{z_0 \ne 0}$ by $\omega = \frac{i}{2}\partial\bar\partial \log(1 + \abs{w}^2)$ in the coordinates $w_j = z_j/z_0$, is closed because $d = \partial + \bar\partial$ and $\partial^2 = \bar\partial^2 = 0$, $\partial\bar\partial = -\bar\partial\partial$.

Non-example: on $W = \C^2 \setminus \set{0}$, the form $\omega = \frac{i}{2}\abs{z}^{-2}\sum_j dz_j \wedge d\bar z_j$ is the fundamental form of a Hermitian metric invariant under $z \mapsto \lambda z$, so it descends to a [[Hopf surface|PNFNGBXRM3]].
It is not closed: $d\omega = d(\abs{z}^{-2}) \wedge \omega_0$ with $\omega_0 = \frac{i}{2}\sum_j dz_j \wedge d\bar z_j$, and the wedge product of a nonzero $1$-form with $\omega_0$ is nonzero in complex dimension $2$.
So this metric is a [[Hermitian metric|CK1DCPC3D3]] that is not Kähler; in fact the Hopf surface has no Kähler metric at all.
</document>

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# Preconditioner

## Definition

Let $A$ be an [[invertible|0EB9N0S33C]] real $n \times n$ matrix, and consider the [[linear system|0GT4WWGWH2]] $A x = b$.
A *preconditioner* for $A$ is an invertible $n \times n$ matrix $P$, used through the map $r \mapsto P^{-1} r$.
The *left preconditioned* system is $P^{-1} A x = P^{-1} b$.
The *right preconditioned* system is $A P^{-1} y = b$ with $x = P^{-1} y$.
If $P = P_1 P_2$, the *split preconditioned* system is $P_1^{-1} A P_2^{-1} y = P_1^{-1} b$ with $x = P_2^{-1} y$.

## Remarks

All three systems have the same solution $x$ as $A x = b$.
The definition is formal; a preconditioner is useful when two informal requirements hold.
First, $P^{-1} r$ is cheap to compute for any $r$.
Second, $P^{-1} A$ is close to the identity, for example it has a small [[condition number|AB3S6MHJ8X]] or few clusters of [[eigenvalues|0BQAA2BK3Q]].
The two extremes are $P = I$, which is cheap but changes nothing, and $P = A$, which is ideal but as costly as the original problem.
Example: the Jacobi preconditioner $P = \operatorname{diag}(a_{11}, \dots, a_{nn})$ is invertible when all diagonal entries are nonzero.
For $A$ with rows $(100, 1)$ and $(1, 1)$, the ratio of the extreme eigenvalues of $A$ is about $101$.
The matrix $P^{-1} A$ has rows $(1, 0.01)$ and $(1, 1)$ and eigenvalues $0.9$ and $1.1$, with ratio about $1.2$.
Non-example: the matrix $P$ with rows $(1, 1)$ and $(1, 1)$ is singular, so it is not a preconditioner.
For a symmetric [[positive definite|0KX66CZMBB]] $A$ one takes $P$ symmetric positive definite, so that methods for symmetric systems still apply.
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# Elliptic cohomology

## Summary

An elliptic cohomology theory is a [[generalized cohomology theory|H28PSZR7XG]] whose [[formal group law|TNHP7XXYZT]] comes from an [[elliptic curve|427N6S6QQW]].
Ordinary cohomology corresponds to the additive group and complex K-theory to the multiplicative group; elliptic curves are the next family of one-dimensional group varieties.
Elliptic cohomology theories detect height $2$ phenomena in chromatic homotopy theory.
They are tied to elliptic genera of manifolds and to [[topological modular forms|JN73KE96YB]].

## Explanations

### L1 human

Many cohomology theories come with a rule for adding Chern classes of line bundles, given by a power series.
For ordinary cohomology the rule is plain addition, and for K-theory it is like multiplication.
Elliptic curves are curves that also carry an addition law, and near the identity this law is a power series too.
An elliptic cohomology theory is one whose rule is the addition law of an elliptic curve.
Such theories see more of the stable homotopy groups of spheres than K-theory does.
</document>

<document source="https://tamnd.github.io/math2/node/XH6FFFR8AZ.md" tag="XH6FFFR8AZ" kind="concept">
# Train track

## Summary

A train track on a [[surface|VGJJVBZ1MQ]] is an embedded graph whose edges meet smoothly at the vertices, like railway lines merging at switches.
William Thurston introduced train tracks to describe simple closed curves and measured laminations by finitely many numbers, the weights on the edges.
They are a combinatorial tool in the study of the [[mapping class group|JV5NYG913V]], of pseudo-Anosov maps and of Teichmüller space.

## Explanations

### L1 human

Picture a railway map drawn on a surface.
At each switch, tracks merge tangentially, so a train passing through never makes a sharp turn.
A loop on the surface that runs close to the tracks, following them like a train, is said to be carried by the track.
Recording how many times the loop runs along each piece of track gives a list of numbers that describes the loop.
</document>

<document source="https://tamnd.github.io/math2/node/XHHTS6TDYZ.md" tag="XHHTS6TDYZ" kind="definition">
# Geometric realization

## Definition

Let $K$ be a [[simplicial set|WW81ARB8K9]], with $n$-simplices $K_n$ and structure maps $\theta^* \colon K_n \to K_m$ for order-preserving $\theta \colon [m] \to [n]$.
Let $\abs{\Delta^n}$ be the standard topological $n$-simplex, and let $\theta_* \colon \abs{\Delta^m} \to \abs{\Delta^n}$ be the affine map sending the vertex $e_i$ to $e_{\theta(i)}$.
Give each set $K_n$ the discrete topology.
The *geometric realization* of $K$ is the [[quotient space|0V8046R0P9]]
$$\abs{K} = \Bigl( \bigsqcup_{n \ge 0} K_n \times \abs{\Delta^n} \Bigr) \Big/ \sim$$
of the [[disjoint union|A486VR787Z]], where $\sim$ is the equivalence relation generated by
$$(\theta^* x, t) \sim (x, \theta_* t) \quad \text{for all } \theta \colon [m] \to [n], \ x \in K_n, \ t \in \abs{\Delta^m}.$$

## Remarks

A simplicial map $f \colon K \to L$ induces a continuous map $\abs{f} \colon \abs{K} \to \abs{L}$, so realization is a functor.
It is left adjoint to the [[singular simplicial set|RWBMXQW6S5]] functor: continuous maps $\abs{K} \to X$ correspond to simplicial maps $K \to \operatorname{Sing} X$.
Each point of $\abs{K}$ has a unique representative $(x, t)$ with $x$ nondegenerate and $t$ in the interior of $\abs{\Delta^n}$.
It follows that $\abs{K}$ is a [[CW complex|5W95QFHXPR]] with one $n$-cell for each nondegenerate $n$-simplex.

Example: for the standard simplex $\Delta^n$, the realization $\abs{\Delta^n}$ is the topological $n$-simplex, since every simplex of $\Delta^n$ is $\theta^*$ of the top simplex $\id_{[n]}$.

Example: the quotient $\Delta^1 / \partial \Delta^1$ has one nondegenerate $0$-simplex and one nondegenerate $1$-simplex whose two faces are equal, so its realization is a circle.

Non-example: the disjoint union $\bigsqcup_n K_n \times \abs{\Delta^n}$ without the relation $\sim$ is not the realization; for $K = \Delta^1$ it has infinitely many components, while $\abs{\Delta^1}$ is an interval.

Some authors use the category of compactly generated spaces, where the realization functor preserves finite products.
</document>

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# Noether's theorem

## Statement

Let $L \colon \R^n \times \R^n \to \R$, written $L(q, v)$, be of class $C^2$.
Let $(h^s)_{s \in \R}$ be the [[flow|06738GGWWK]] of a complete $C^1$ vector field $X$ on $\R^n$, so that each $h^s$ is a [[diffeomorphism|0KZZ70ZHXP]] and $X(q) = \frac{d}{ds} h^s(q) \big|_{s = 0}$.
Assume that $L$ is invariant under the flow: $L\big(h^s(q), Dh^s(q) \, v\big) = L(q, v)$ for all $s \in \R$, $q \in \R^n$ and $v \in \R^n$.
Then the function
$$I(q, v) = \sum_{i=1}^n \frac{\partial L}{\partial v_i}(q, v) \, X_i(q)$$
is a [[first integral|0262JP0NV9]] of the [[Euler–Lagrange equations|PS0VV149ET]] $\frac{d}{dt} \frac{\partial L}{\partial v}(q, \dot q) = \frac{\partial L}{\partial q}(q, \dot q)$.
That is, $t \mapsto I(q(t), \dot q(t))$ is constant for every $C^2$ solution $q(t)$.

## Remarks

Along a solution, the Euler–Lagrange equations give $\frac{d}{dt} I = \frac{\partial L}{\partial q} \cdot X(q) + \frac{\partial L}{\partial v} \cdot DX(q) \, \dot q$.
The right side is the derivative in $s$ at $s = 0$ of $L(h^s(q), Dh^s(q) \, \dot q)$, which is $0$ by invariance.

Example: in the plane, let $L(q, v) = \frac{1}{2} \abs{v}^2 - U(\abs{q}^2)$ for a $C^2$ function $U \colon \R \to \R$.
It is invariant under the rotations $h^s$, with $X(q) = (-q_2, q_1)$.
The first integral is the angular momentum $I = q_1 v_2 - q_2 v_1$, which is conserved in every central force motion.

A related fact covers time: if $L$ does not depend on $t$, the energy $\dot q \cdot \frac{\partial L}{\partial v} - L$ is conserved.
The field-theory version says that each continuous symmetry of an action $\int L(x, u, \nabla u) \, dx$ gives a [[conservation law|8RA7AMWSAF]] in divergence form.

## Explanations

### L1 human

Noether's theorem links symmetries with conserved quantities.
If a mechanical system looks the same after you rotate it, then its angular momentum stays constant.
If it looks the same after you shift it in space, its momentum stays constant.
If its laws do not change in time, its energy stays constant.

### L2 human

The theorem turns a geometric fact, invariance of the Lagrangian under a one-parameter group, into a first integral that can be written down explicitly.
First integrals lower the order of the equations of motion, and in Hamiltonian terms the conserved quantity generates the symmetry through Poisson brackets.
Applied to a geodesic problem with a Killing field, it gives Clairaut's relation on surfaces of revolution.
In field theory, the same argument produces conserved currents, such as the energy–momentum tensor from translations.
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# Skorokhod's embedding theorem

## Statement

Let $(B_t)_{t \ge 0}$ be a standard one-dimensional [[Brownian motion|22735FGJV6]] with $B_0 = 0$, and let $(\mathcal{F}_t)$ be the filtration generated by $B$.
Let $\mu$ be a [[probability measure|059TJN0NZ1]] on $\R$ with $\int x^2 \, d\mu(x) < \infty$ and $\int x \, d\mu(x) = 0$.
Then there is a [[stopping time|WND87FBAKC]] $T$ for $(\mathcal{F}_t)$ such that $B_T$ has law $\mu$ and
$$E[T] = \int_{\R} x^2 \, d\mu(x).$$

## Remarks

For a law on two points $a < 0 < b$, the exit time of $B$ from $(a, b)$ works.
The [[optional stopping theorem|DC9HZT2HVJ]] for the martingales $B_t$ and $B_t^2 - t$ gives the right weights and $E[T] = -ab$.
The general case writes $\mu$ as a mixture of such two-point laws, or builds $T$ by a sequence of exit times from nested intervals.
Applied repeatedly, the theorem gives stopping times $0 = T_0 \le T_1 \le T_2 \le \cdots$ with i.i.d. increments such that $(B_{T_n})_{n \ge 0}$ has the law of a [[random walk|04VZE3FZXF]] with step law $\mu$.
This embedding is one route to [[Donsker's theorem|818832PMSV]] and to limit laws for random walks.
The mean zero condition is necessary: if $E[T] < \infty$, then Wald's identities give $E[B_T] = 0$ and $E[B_T^2] = E[T]$.

## Explanations

### L1 human

Any centred distribution with finite variance can be produced by watching a Brownian path and stopping it at a cleverly chosen moment.
The decision to stop uses only the path seen so far.
For a fair coin with values $\pm 1$, one stops the first time the path reaches $1$ or $-1$.

### L2 human

The theorem lets one transfer questions about sums of independent variables to questions about a single Brownian path.
A random walk becomes Brownian motion sampled at random times $T_n$, and the strong law of large numbers gives $T_n / n \to \int x^2 \, d\mu$.
So the walk and the path stay close on the scale $\sqrt{n}$, which yields functional limit theorems.
Many explicit constructions of $T$ exist, with different extra properties.
</document>

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# Beckman–Quarles theorem

## Statement

Let $n \ge 2$ and let $f \colon \R^n \to \R^n$ be any map.
Assume that for all $x, y \in \R^n$ with $\norm{x - y} = 1$ one has $\norm{f(x) - f(y)} = 1$.
Then $f$ is an isometry: $\norm{f(x) - f(y)} = \norm{x - y}$ for all $x, y \in \R^n$.

## Remarks

No continuity, injectivity or surjectivity of $f$ is assumed; preserving the single distance $1$ is enough.
The theorem was published by F. S. Beckman and D. A. Quarles in 1953.
The proof uses rigid configurations made of unit segments, such as unit equilateral triangles and simplices.
They show that $f$ also preserves other distances, and then that it preserves all distances.
Since an isometry of [[Euclidean space|0XPR1E85W0]] is affine, $f$ is then of the form $x \mapsto Ax + b$ with $A$ orthogonal.

The hypothesis $n \ge 2$ is necessary.
On $\R$, let $h \colon [0, 1) \to [0, 1)$ be any bijection, for instance $h(t) = t^2$, and set $f(x) = \floor{x} + h(x - \floor{x})$.
If $\abs{x - y} = 1$, then $x$ and $y$ have the same fractional part and their integer parts differ by $1$, so $\abs{f(x) - f(y)} = 1$.
But $f(0) = 0$ and $f(1/2) = 1/4$, so $f$ is not an [[isometry|02HPP5VCZH]].

## Explanations

### L1 human

Suppose a map of the plane keeps every pair of points at distance $1$ at distance $1$.
Nothing is said about other distances, and the map need not even be continuous.
The theorem says that the map must then keep all distances, so it is just a rigid motion.
In the plane, unit triangles and other rigid unit frameworks lock everything in place.

### L2 human

The theorem is a rigidity result in [[Euclidean distance|0NPPGHQNXQ]] geometry: one preserved distance forces the whole metric to be preserved.
It contrasts with the classical characterization of isometries, which assumes that all distances are preserved.
It is connected with unit distance graphs: a unit-distance-preserving map is a graph homomorphism of the unit distance graph of $\R^n$ to itself.
The one-dimensional counterexample shows that the plane's richer supply of unit triangles is what makes the result work.
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# Decay of Fourier coefficients and smoothness

## Statement

Let $f \colon \R \to \C$ be a [[periodic function|0ATF6Q0633]] of period $2\pi$ that is integrable on $[-\pi, \pi]$, with [[Fourier coefficients|5NN2YZZR3V]] $\hat f(n)$, and let $k \ge 0$ be an integer.

1. If $f$ is $k$ times continuously differentiable, then $\widehat{f^{(k)}}(n) = (in)^k \hat f(n)$ for all $n$, and $\abs{n}^k \abs{\hat f(n)} \to 0$ as $\abs{n} \to \infty$.
2. If $\sum_{n \in \Z} \abs{n}^k \abs{\hat f(n)} < \infty$, then $f$ is equal almost everywhere to a $k$ times continuously differentiable function $g$, and $g^{(j)}(x) = \sum_{n \in \Z} (in)^j \hat f(n) e^{inx}$ for $0 \le j \le k$ and all $x$.
3. $f$ is equal almost everywhere to a [[smooth function|0V33BNKZ9S]] if and only if, for every $m \ge 0$, some constant $C_m$ satisfies $\abs{\hat f(n)} \le C_m (1 + \abs{n})^{-m}$ for all $n$.

## Remarks

For part 1, integration by parts over a period gives $\widehat{f'}(n) = in \hat f(n)$, since the boundary terms cancel by periodicity; induction gives the formula for $f^{(k)}$.
The [[Riemann–Lebesgue lemma|WVKDC011KP]] applied to $f^{(k)}$ then gives $\widehat{f^{(k)}}(n) \to 0$.
For part 2, the series $\sum (in)^j \hat f(n) e^{inx}$ converge uniformly by the Weierstrass M-test, so their sums are continuous.
This justifies term-by-term differentiation.
The sum $g$ has the same Fourier coefficients as $f$, so $g = f$ almost everywhere, as in the [[uniform convergence of absolutely summable Fourier series|CSYZGA5YKS]].
Part 3 follows from parts 1 and 2, since a bound $C_{m+2} (1 + \abs{n})^{-m-2}$ makes $\sum \abs{n}^m \abs{\hat f(n)}$ finite.
Example: the sawtooth $f(x) = x$ on $(-\pi, \pi)$ has a jump, and $\abs{\hat f(n)} = 1/\abs{n}$ decays only like $1/\abs{n}$.
The triangle wave $\abs{x}$ is continuous with a piecewise continuous derivative, and its coefficients decay like $1/n^2$.
Part 1 gives only $o(\abs{n}^{-k})$ and part 2 needs slightly more than $O(\abs{n}^{-k-1})$, so there is a gap between the two directions for a fixed $k$.

## Explanations

### L1 human

A smooth function has small high-frequency content, and a rough function has a lot of it.
Each derivative that $f$ possesses makes its Fourier coefficients decay faster by one power of $\abs{n}$.
A jump or a corner in the graph shows up as slow decay of the coefficients.

### L2 human

This dictionary between regularity and decay is used constantly.
It explains why Fourier series of smooth functions converge very fast, and why the Fourier series of a function with a jump converges slowly and shows the Gibbs phenomenon.
It is the starting point of Sobolev spaces on the circle, where smoothness of order $s$ is measured by $\sum (1 + n^2)^s \abs{\hat f(n)}^2 < \infty$.
For real analytic functions the coefficients decay exponentially, and conversely.
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# Interacting particle system

## Definition

Let $S$ be a countable set of sites, such as $\Z^d$, and let $W$ be a finite set of local states.
A configuration is an element $\eta \in X = W^S$, and $X$ carries the product topology, in which it is compact.
An *interacting particle system* on $X$ is a [[Feller process|4HS3C12472]] $(\eta_t)_{t \ge 0}$ on $X$ whose transitions change finitely many coordinates at a time.
The rates of these transitions depend on the configuration.
The two main classes, with $W = \set{0, 1}$, are given by their [[generators|VPA41PWRR1]] on functions $f$ that depend on finitely many coordinates:

1. a *spin system* with flip rates $c(x, \eta) \ge 0$ has
$$Lf(\eta) = \sum_{x \in S} c(x, \eta) \big( f(\eta^x) - f(\eta) \big),$$
where $\eta^x$ is $\eta$ with the coordinate at $x$ changed;
2. an *exclusion process* with jump rates $p(x, y) \ge 0$ has
$$Lf(\eta) = \sum_{x, y \in S} p(x, y)\, \eta(x) \big(1 - \eta(y)\big) \big( f(\eta^{x,y}) - f(\eta) \big),$$
where $\eta^{x,y}$ is $\eta$ with the coordinates at $x$ and $y$ exchanged.

## Remarks

The rates must be such that this formula defines the generator of a Feller process.
For a spin system it suffices that $\sup_x c(x, \cdot)$ is bounded, each $c(x, \cdot)$ is continuous, and $\sup_x \sum_{u \in S} \sup_\eta \abs{c(x, \eta^u) - c(x, \eta)} < \infty$.
Example: on $S = \Z^d$, the rates $c(x, \eta) = 1$ if $\eta(x) = 1$ and $c(x, \eta) = \lambda \#\set{y : \abs{y - x} = 1, \eta(y) = 1}$ if $\eta(x) = 0$ define the [[contact process|9RVKQJQ5JJ]]; they are bounded by $\max(1, 2d\lambda)$ and depend only on the $2d + 1$ nearest sites.
Non-example: a probabilistic cellular automaton, in which all sites update together at each discrete time step, is not of this form, since each of its transitions changes infinitely many coordinates.
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# Polyharmonic spline

## Summary

A polyharmonic spline is a sum of translates of one [[radial basis function|ZBSXSTQM2J]] $\varphi(\norm{x})$ on $\R^d$ plus a low degree polynomial.
The radial function is $r^{2m-d}$ or $r^{2m-d} \log r$, a multiple of the [[fundamental solution|K2NK9A8M5F]] of the iterated [[Laplacian|0E0B6RA8K0]] $\Delta^m$ in $\R^d$.
Polyharmonic splines solve scattered data interpolation with minimal "energy", and the [[thin plate spline|P9ZPH7EC8K]] is the best known case.

## Explanations

### L1 human

Data are often measured at scattered points in the plane or in space, not on a grid.
A polyharmonic spline puts a radially symmetric bump at each data point, adds a simple polynomial, and tunes the sizes so that the sum passes through the data.
In one dimension with $m = 2$ the bump is $\abs{x}^3$, and the result is a natural cubic spline.
In the plane with $m = 2$ the bump is $r^2 \log r$, which gives the thin plate spline.
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# Separated morphism

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]], and let $X \times_Y X$ be the [[fiber product|1DZMT4S9FB]] of $f$ with itself, with projections $p_1, p_2$.
The *diagonal* of $f$ is the unique morphism $\Delta \colon X \to X \times_Y X$ with $p_1 \circ \Delta = p_2 \circ \Delta = \id_X$.
The morphism $f$ is *separated* if $\Delta$ is a [[closed immersion|WWRWYSJDET]].
A scheme $X$ is *separated* if the unique morphism $X \to \operatorname{Spec} \Z$ is separated.

## Remarks

Example: every morphism $\operatorname{Spec} A \to \operatorname{Spec} B$ is separated.
The diagonal is $\operatorname{Spec}$ of the multiplication map $A \otimes_B A \to A$, $a \otimes a' \mapsto a a'$, which is surjective, so the diagonal is a closed immersion.
Non-example: let $X$ be the affine line over a field $k$ with a doubled origin, the [[gluing|ABVBHCA9VW]] of two copies of $\mathbb{A}^1_k$ along the identity of $\mathbb{A}^1_k \setminus \set{0}$.
Then $X \to \operatorname{Spec} k$ is not separated.
Let $j_1, j_2 \colon \mathbb{A}^1_k \to X$ be the two inclusions, and let $(j_1, j_2) \colon \mathbb{A}^1_k \to X \times_k X$.
The preimage of the diagonal under $(j_1, j_2)$ is $\mathbb{A}^1_k \setminus \set{0}$, the set where $j_1$ and $j_2$ agree.
This set is not closed in $\mathbb{A}^1_k$, so the image of the diagonal is not closed.
A morphism is separated if and only if its diagonal has closed image, since the diagonal is always an immersion.
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# Normal fan

## Summary

The normal fan of a [[convex polytope|300K6CB1KH]] $P$ is the collection of its normal cones, one cone for each face.
The normal cone of a face collects the linear functionals that reach their maximum on $P$ at that face.
The normal fan records the combinatorics of $P$ in a dual form.
When $P$ is full-dimensional, its vertices give the full-dimensional cones and its facets give the rays.
For a full-dimensional lattice polytope it is a rational fan, and it defines the projective [[toric variety|YWB981MEF2]] of the polytope.

## Explanations

### L1 human

Stand at a corner of a polytope and ask in which directions that corner is the farthest point of the polytope.
These directions form a cone, the normal cone of the corner.
Doing the same for edges and the other faces cuts the space of directions into cones that fit together with no gaps or overlaps.
For a square centred at the origin, the four corners give the four quadrants and the four sides give the four half-axes.
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# Gleason–Yamabe theorem

## Statement

Let $G$ be a [[locally compact group|AHV05ZBCYW]].
Then there is an [[open subgroup|QYCFH68RD1]] $G'$ of $G$ with the following property.
Let $U$ be any open neighbourhood of the identity in $G'$.
Then there is a compact [[normal subgroup|0CB0DX5BVV]] $K$ of $G'$ with $K \subseteq U$ such that $G'/K$ is isomorphic, as a topological group, to a [[Lie group|7JBRGCVJRB]].

## Remarks

The proof has two parts.
Gleason showed that a locally compact group with [[no small subgroups|HPMAJV2RDY]] is a Lie group, by building one-parameter subgroups and a Lie algebra from the group itself.
Yamabe then showed that every locally compact group can be reduced to this case by dividing out small compact normal subgroups, after passing to an open subgroup.
Example: for $G = \Z_p$, the $p$-adic integers, take $G' = G$ and $K = p^n \Z_p$ inside a given neighbourhood of $0$; then $G/K = \Z/p^n\Z$ is a finite group, hence a Lie group of dimension $0$.
The open subgroup is needed.
The group $G = PSL_2(\Q_p)$ is simple as an abstract group, so its only compact normal subgroup is trivial.
But $G$ is totally disconnected and not discrete, so it is not a Lie group.
Here one takes for $G'$ a compact open subgroup such as the image of $SL_2(\Z_p)$, which is profinite.
A corollary is that a locally compact group with no small subgroups is a Lie group.

## Explanations

### L1 human

The theorem says that every locally compact group is close to a Lie group.
After passing to an open piece of the group and ignoring an arbitrarily small compact part, what is left is a smooth group.
So the smooth groups of calculus and geometry, together with compact groups such as profinite groups, are the building blocks of all locally compact groups.

### L2 human

The theorem gives the solution of Hilbert's fifth problem: a locally Euclidean group has no small subgroups, so it is a Lie group.
It is also the starting point of the structure theory of locally compact groups.
For a connected locally compact group one can take $G' = G$, so $G$ is a projective limit of Lie groups.
The theorem has found use outside group theory, for example in Breuillard, Green and Tao's classification of approximate groups and in Gromov's theorem on groups of polynomial growth.
</document>

<document source="https://tamnd.github.io/math2/node/XJM2MT4S3Y.md" tag="XJM2MT4S3Y" kind="concept">
# Algebraic curve

## Summary

An algebraic curve is an [[algebraic variety|HV6YBS0YDH]] of dimension one, such as the set of solutions of one polynomial equation in two unknowns.
Curves are the first case of algebraic geometry beyond points, and their theory is the most complete.
Their main invariant is the [[genus|6W8TDV9VVN]].
Over $\C$, smooth projective curves are the same as compact [[Riemann surfaces|5K0N8QMV6Q]].

## Explanations

### L1 human

A curve is a geometric object with one degree of freedom, cut out by polynomial equations.
The circle $x^2 + y^2 = 1$ and the cubic $y^2 = x^3 - x$ are curves in the plane.
The twisted cubic $\set{(t, t^2, t^3)}$ is a curve in space that does not lie in any plane.
Over the complex numbers each curve is a real surface, and a smooth projective connected curve looks like a sphere with some number of handles; that number is its genus.
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<document source="https://tamnd.github.io/math2/node/XJMGD9V64J.md" tag="XJMGD9V64J" kind="definition">
# Curvelet

## Definition

This is the second-generation curvelet system on $\R^2$.
Let $W \colon (0,\infty) \to [0,\infty)$ and $V \colon \R \to [0,\infty)$ be smooth windows with $W$ supported in $(1/2, 2)$ and $V$ supported in $[-1, 1]$, such that
$$\sum_{j \in \Z} W(2^j r)^2 = 1 \text{ for } r > 0, \qquad \sum_{l \in \Z} V(t - l)^2 = 1 \text{ for } t \in \R.$$
For an integer $j \ge 0$, let $\varphi_j$ be the function whose [[Fourier transform|DPVM34ADZV]] is, in polar coordinates $(r, \theta)$ of the frequency plane,
$$\hat\varphi_j(r, \theta) = 2^{-3j/4} \, W(2^{-j} r) \, V\left(\frac{2^{\floor{j/2}}\theta}{2\pi}\right).$$
Let $R_\theta$ be the rotation of $\R^2$ by the angle $\theta$.
For $l$ with $0 \le l < 2^{\floor{j/2}}$, put $\theta_l = 2\pi \cdot 2^{-\floor{j/2}} \, l$.
For $k = (k_1, k_2) \in \Z^2$, put $x_k^{(j,l)} = R_{\theta_l}^{-1}(k_1 2^{-j}, k_2 2^{-j/2})$.
The *curvelet* at scale $2^{-j}$, orientation $\theta_l$ and position $x_k^{(j,l)}$ is
$$\varphi_{j,l,k}(x) = \varphi_j\left(R_{\theta_l}\left(x - x_k^{(j,l)}\right)\right), \qquad x \in \R^2.$$
The curvelet coefficients of $f \in L^2(\R^2)$ are $c(j,l,k) = \ip{f}{\varphi_{j,l,k}}$.

## Remarks

Each $\hat\varphi_j$ lives on a polar wedge with radial size about $2^j$ and angular width about $2^{-j/2}$.
In space, $\varphi_j$ is then a needle of length about $2^{-j/2}$ and width about $2^{-j}$.
This is parabolic scaling: width $\approx$ length$^2$.
The number of orientations doubles at every second scale.
Together with a coarse-scale family at low frequencies, the curvelets form a tight frame of $L^2(\R^2)$ in the sense that $\sum \abs{c(j,l,k)}^2 = \norm{f}_2^2$.

Example: any pair of smooth windows with these supports and partition conditions, such as the Meyer-type windows used for the Meyer wavelet, gives a curvelet system.

Non-example: if the angular window is not narrowed with scale, that is, $V(2^{\floor{j/2}}\theta/(2\pi))$ is replaced by a fixed angular window, the elements keep the same aspect ratio at all scales.
They then behave like directional wavelets and lose the parabolic scaling, which is the defining feature of curvelets.

## Explanations

### L1 human

Curvelets are built in frequency space from wedges of a polar grid.
Finer scales use wedges farther out and narrower in angle, and the inverse Fourier transform turns each wedge into a thin oriented needle in the plane.
</document>

<document source="https://tamnd.github.io/math2/node/XJMYGY0ZZF.md" tag="XJMYGY0ZZF" kind="concept">
# p-Laplacian

## Summary

The $p$-Laplacian is the nonlinear operator $\Delta_p u = \operatorname{div}(\abs{\nabla u}^{p-2} \nabla u)$, for a real number $1 < p < \infty$.
For $p = 2$ it is the ordinary [[Laplacian|0E0B6RA8K0]].
It is the Euler–Lagrange operator of the energy $\int \abs{\nabla u}^p \, dx$, just as the Laplacian is for $\int \abs{\nabla u}^2 \, dx$.
It is the model of a quasilinear elliptic operator that degenerates where $\nabla u = 0$ for $p > 2$ and becomes singular there for $p < 2$.

## Explanations

### L1 human

The Laplacian measures how a function differs from its averages, with all slopes treated alike.
The $p$-Laplacian gives more weight to steep slopes when $p > 2$ and less weight when $p < 2$.
Functions with $\Delta_p u = 0$, called $p$-harmonic functions, minimize the total amount of $\abs{\nabla u}^p$ for given boundary values.
Every affine function $u(x) = a \cdot x + b$ is $p$-harmonic, because its gradient is constant and so the divergence vanishes.
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# Eilenberg–MacLane spectrum

## Description

Let $G$ be an [[abelian group|071DDFVG4J]].
The Eilenberg–MacLane spectrum $HG$ is the [[spectrum|1GHF4EKBXS]] with $(HG)_n = K(G, n)$, an [[Eilenberg–MacLane space|AJHEYJE5XY]], for $n \ge 0$.
Its structure maps $\Sigma K(G, n) \to K(G, n+1)$ are adjoint to weak homotopy equivalences $K(G, n) \to \Omega K(G, n+1)$, so $HG$ is an $\Omega$-spectrum.
Its homotopy groups are $\pi_0(HG) = G$ and $\pi_k(HG) = 0$ for $k \neq 0$, and these properties determine $HG$ up to isomorphism in the [[stable homotopy category|26GTNMAF7B]].

## Remarks

The spectrum $HG$ represents ordinary [[singular cohomology|6N7BC70CX0]] with coefficients in $G$: for a CW complex $X$ and $n \ge 0$, $\tilde H^n(X; G) \cong [\Sigma^\infty X, \Sigma^n HG]$.
It also represents homology: $\tilde H_n(X; G) \cong \pi_n(HG \wedge X)$.
For $p$ prime, the graded ring $[H\F_p, \Sigma^* H\F_p]$ of stable self-maps is the mod $p$ Steenrod algebra.
If $R$ is a commutative ring, the product of $R$ makes $HR$ a commutative ring spectrum.

## Explanations

### L1 human

The Eilenberg–MacLane spectrum of a group $G$ is the simplest spectrum with homotopy $G$: it has $G$ in degree zero and nothing else.
It is the stable object behind ordinary cohomology with coefficients in $G$.
Comparing other spectra with it is how one connects generalized cohomology theories with ordinary cohomology.
</document>

<document source="https://tamnd.github.io/math2/node/XJS07AJ50Y.md" tag="XJS07AJ50Y" kind="concept">
# Semisimple Lie group

## Summary

A semisimple Lie group is a connected [[Lie group|7JBRGCVJRB]] whose Lie algebra is a semisimple Lie algebra, that is, has no nonzero solvable ideals.
Equivalently, its Lie algebra is a direct sum of simple Lie algebras.
So, up to a covering and a central quotient, the group is a product of simple Lie groups.
Semisimple Lie groups have a rich structure theory, with Cartan, Iwasawa and Bruhat decompositions, and a deep representation theory.
By the Levi decomposition, a finite-dimensional real or complex Lie algebra is a semisimple subalgebra plus its largest solvable ideal.
So semisimple and solvable groups are the two basic kinds.

## Explanations

### L1 human

A semisimple Lie group is a Lie group built from simple Lie groups, with no commutative or "almost commutative" continuous part.
The group $SO(4)$ is a typical example.
Its Lie algebra splits into two copies of the Lie algebra of $SO(3)$, one for each of two independent kinds of rotation of four-dimensional space.
The unitary group $U(n)$ is not semisimple, because the scalar matrices $e^{i\theta} I$ form a commutative continuous part.
Removing that part by fixing the determinant gives the semisimple group $SU(n)$.
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# Mellin inversion theorem

## Statement

Let $f \colon (0, \infty) \to \C$ be measurable and let $c \in \R$.
Assume that $f$ is [[Mellin convergent|SGZZ4Y0D36]] at $c$, so that the [[Mellin transform|N5PVGRJE1T]] $\mathcal{M}f(s) = \int_0^\infty x^{s-1} f(x) \, dx$ is defined on the line $\operatorname{Re} s = c$.
Assume also that $y \mapsto \mathcal{M}f(c + iy)$ is [[integrable|X7GTF3D98F]] on $\R$.
Then, for every $x > 0$ at which $f$ is continuous,
$$f(x) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} x^{-s} \, \mathcal{M}f(s) \, ds.$$
The right side is the [[inverse Mellin transform|CHESR1W1CY]] of $\mathcal{M}f$ along $\operatorname{Re} s = c$.

## Remarks

Put $x = e^{u}$ and $g(u) = e^{cu} f(e^u)$.
Mellin convergence at $c$ says that $g$ is integrable on $\R$, and $\mathcal{M}f(c + iy) = \int_{\R} g(u) e^{iyu} \, du$ is a Fourier transform of $g$.
The [[Fourier inversion theorem|T4Z8YQZKZV]] then recovers $g$ almost everywhere, hence at every point where $g$ is continuous, and this is the formula above.
Example: $f(x) = e^{-x}$ with $c > 0$ gives $e^{-x} = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \Gamma(s) x^{-s} \, ds$, the Cahen–Mellin integral.
Continuity at $x$ is needed: if $f(x) = e^{-x}$ is changed at the single point $x = 1/2$, its Mellin transform is still $\Gamma$.
So the formula gives $e^{-1/2}$ there and not the changed value.
For $f = \mathbf{1}_{(0, 1]}$, the transform $\mathcal{M}f(s) = 1/s$ is not integrable on vertical lines, so the integrability hypothesis fails.
The formula still holds at $x \ne 1$ as a limit of integrals over $[c - iT, c + iT]$, but that needs a separate argument.

## Explanations

### L1 human

The Mellin transform loses no information: a reasonable function can be rebuilt from its transform on any single vertical line where the transform is defined.
The rebuilding is an integral of the powers $x^{-s}$ against the transform.

### L2 human

The theorem is the Fourier inversion theorem written on the multiplicative group $(0, \infty)$.
It is how one passes from a Mellin transform back to a function, as in Perron's formula and in [[Barnes integrals|1PDGBGEHM2]].
There the transform is often a product of gamma functions or a Dirichlet series.
In practice one then shifts the line of integration and collects residues, which turns the inversion integral into series or asymptotic expansions.
Without integrability of $\mathcal{M}f$ on the line, as for jump functions, the integral must be taken as a symmetric limit.
For functions of bounded variation near a jump, this limit is the average of the one-sided limits.
</document>

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# Abelian algebra

## Definition

Let $A$ be an algebra.
The algebra $A$ is *abelian* if the following holds.
For every $n \ge 1$, every $(n+1)$-ary [[term operation|N9M2N27END]] $t$ of $A$, all elements $a, b \in A$ and all tuples $\bar{c}, \bar{d} \in A^n$:
$$t(a, \bar{c}) = t(a, \bar{d}) \implies t(b, \bar{c}) = t(b, \bar{d}).$$

## Remarks

In the language of [[commutator theory|6DSWPHSWQW]], $A$ is abelian if and only if $[1_A, 1_A] = 0_A$, where $1_A$ and $0_A$ are the largest and the least congruences of $A$.
Example: every module $M$ over a ring $R$ is abelian.
Each term operation has the form $t(x, \bar{y}) = r x + s_1 y_1 + \cdots + s_n y_n$ with $r, s_i \in R$.
So $t(a, \bar{c}) = t(a, \bar{d})$ means $\sum_i s_i c_i = \sum_i s_i d_i$, and this gives $t(b, \bar{c}) = t(b, \bar{d})$ for every $b$.
Non-example: a lattice $L$ with elements $0 < 1$ is not abelian.
Take $t(x, y) = x \wedge y$, $a = 0$, $c = 0$ and $d = 1$: then $t(0, 0) = 0 = t(0, 1)$, but $t(1, 0) = 0 \ne 1 = t(1, 1)$.
A group is abelian in this sense exactly when it is commutative.

## Explanations

### L1 human

An algebra is abelian when the effect of changing some inputs of a term operation never depends on the value of another input.
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# Frobenius's theorem on Frobenius kernels

## Statement

Let $G$ be a [[Frobenius group|JXS91FGFHY]] acting on a finite set $\Omega$, and let $H = G_\omega$ be the [[stabiliser|0AYMP41THN]] of a point $\omega \in \Omega$.
Let $K$ be the set of elements of $G$ that fix no point of $\Omega$, together with the identity $e$.
Then $K$ is a [[normal subgroup|0CB0DX5BVV]] of $G$, $K$ acts regularly on $\Omega$, and $G$ is the [[semidirect product|0H3WZXCZNF]] $K \rtimes H$.

## Remarks

Counting is easy.
Let $n = \abs{\Omega}$; the $n$ point stabilisers are the [[conjugates|0H11ZPX4ND]] of $H$, and any two of them meet only in $e$.
So their union has $n(\abs{H} - 1) + 1$ elements, and $K$ has $\abs{G} - n\abs{H} + n = n$ elements.
The hard part is that $K$ is closed under multiplication.
The standard proof uses characters.
Each irreducible character of $H$ is extended to a character of $G$, and $K$ is the intersection of the kernels of these characters.

Example: for the maps $x \mapsto ax + b$ with $a \ne 0$ acting on $\F_q$, the kernel $K$ is the group of translations $x \mapsto x + b$, and the complement $H$ is the group of maps $x \mapsto ax$.
For $S_3$ acting on $3$ points, $K = A_3$.

John Thompson later proved that the Frobenius kernel is always [[nilpotent|0Q7MMQHC1P]].

## Explanations

### L1 human

In a Frobenius group, the elements that move every point look like a loose collection.
Frobenius's theorem says they form a subgroup, and in fact a normal one.
So every Frobenius group splits into two pieces: the kernel, which moves points around freely, and a point stabiliser.

### L2 human

The theorem is a famous example of the power of character theory: its statement is purely about permutations, but the standard proof needs characters.
It gives the structure $G = K \rtimes H$ with $\abs{K} = \abs{\Omega}$ and $\abs{H}$ dividing $\abs{K} - 1$, since $H$ acts on $K \setminus \set{e}$ by conjugation with all orbits of size $\abs{H}$.
The Frobenius kernel and complement are central in the study of finite groups, for example in the proof of the Feit–Thompson theorem.
</document>

<document source="https://tamnd.github.io/math2/node/XK7W504YEW.md" tag="XK7W504YEW" kind="definition">
# Generalized least squares

## Definition

Let $X$ be a fixed real $n \times p$ matrix of rank $p$, and let $y = X\beta + \varepsilon$ with $\beta \in \R^p$ unknown.
Assume $E[\varepsilon] = 0$ and $\operatorname{Cov}(\varepsilon) = \sigma^2 \Omega$, where $\sigma^2 > 0$ and $\Omega$ is a known symmetric [[positive definite|0KX66CZMBB]] $n \times n$ matrix.
The *generalized least squares estimator* of $\beta$ is the minimiser of
$$(y - X b)^T \Omega^{-1} (y - X b)$$
over $b \in \R^p$, which is
$$\hat\beta_{\mathrm{GLS}} = (X^T \Omega^{-1} X)^{-1} X^T \Omega^{-1} y.$$

## Remarks

Equivalent form: let $P$ be an invertible matrix with $P^T P = \Omega^{-1}$, for example $P = \Omega^{-1/2}$.
Then $P y = P X \beta + P \varepsilon$ with $\operatorname{Cov}(P\varepsilon) = \sigma^2 I_n$.
The estimator $\hat\beta_{\mathrm{GLS}}$ is the [[ordinary least squares|YM708KHV2W]] estimator of this transformed model.
By the [[Gauss–Markov theorem|N3YMNMWANM]] applied to the transformed model, it is the best linear unbiased estimator; this is Aitken's theorem.
Its covariance matrix is $\sigma^2 (X^T \Omega^{-1} X)^{-1}$.

Example: if $\Omega = I_n$, the estimator reduces to ordinary least squares.
If $\Omega$ is diagonal, it is the weighted least squares estimator with weights $1/\Omega_{ii}$.

Non-example: if $\Omega$ is only positive semidefinite and singular, then $\Omega^{-1}$ does not exist and the formula is not defined.

When $\Omega$ is unknown and is replaced by an estimate, the method is called *feasible* generalized least squares.
</document>

<document source="https://tamnd.github.io/math2/node/XK866CAZHE.md" tag="XK866CAZHE" kind="object">
# Hilbert scheme

## Description

Let $k$ be a field and let $X \subset \mathbb{P}^n_k$ be a closed subscheme of [[projective space|ZT9A6JK8QF]], for example a [[projective variety|A4H4CZ2V35]].
The *Hilbert functor* of $X$ sends a $k$-scheme $T$ to the set of closed subschemes $Z \subset X \times_k T$ such that the projection $Z \to T$ is [[flat|5M6KS8F47R]] and [[proper|9537RKBZPP]].
Grothendieck proved that this functor is [[representable|KCCHH87H6E]].
So there are a $k$-scheme $\operatorname{Hilb}(X)$ and a universal flat family $\mathcal{Z} \subset X \times \operatorname{Hilb}(X)$.
Every family $Z \subset X \times T$ as above is the pullback of $\mathcal{Z}$ along a unique morphism $T \to \operatorname{Hilb}(X)$.
This is the *Hilbert scheme* of $X$.
It is a disjoint union $\operatorname{Hilb}(X) = \coprod_P \operatorname{Hilb}^P(X)$ over the [[Hilbert polynomials|5FR6JX04RS]] $P$ of the fibers, and each $\operatorname{Hilb}^P(X)$ is a projective scheme.
So the Hilbert scheme is a fine [[moduli space|FJZRW9P0K3]] for closed subschemes of $X$, in the sense of the [[functor of points|SWWW9X4A3N]].

## Remarks

The hypersurfaces of degree $d$ in $\mathbb{P}^n$ are the zero sets of nonzero homogeneous polynomials of degree $d$ up to scalars.
Their Hilbert scheme is the projective space $\mathbb{P}^N$ with $N = \binom{n+d}{d} - 1$, whose points are these polynomials up to scaling.
The Hilbert scheme also exists for quasi-projective $X$, with quasi-projective components.
The Hilbert scheme of $m$ points in the plane, $\operatorname{Hilb}^m(\mathbb{A}^2)$, is smooth and irreducible of dimension $2m$ by a theorem of Fogarty.
It resolves the singularities of the symmetric product of $m$ copies of $\mathbb{A}^2$.

## Explanations

### L1 human

The Hilbert scheme is a space whose points are the subvarieties of a fixed projective variety, together with their degenerate limits.
Families of subvarieties become curves or surfaces inside this space.
For example, the lines in $\mathbb{P}^3$ form a four-dimensional Hilbert scheme, the Grassmannian of lines.
It is one of the first and most important examples of a moduli space.
</document>

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# Galois insertion

## Definition

Let $(P, \le)$ and $(Q, \le)$ be [[partially ordered sets|011C29Q1S1]].
A *Galois insertion* of $Q$ into $P$ is a [[Galois connection|MBVJ4JQXC5]] with lower adjoint $f \colon P \to Q$ and upper adjoint $g \colon Q \to P$ such that $f(g(y)) = y$ for all $y \in Q$.

## Remarks

Let $G$ be a group, let $P$ be the set of subsets of $G$, and let $Q$ be the set of subgroups of $G$, both ordered by inclusion.
Let $f(S)$ be the [[subgroup generated by|0RG9SPQSWM]] $S$, and let $g(H) = H$ as a subset.
Then $f(S) \subseteq H$ if and only if $S \subseteq H$, and $f(g(H)) = H$, so this is a Galois insertion.
The pair of the inclusion $\Z \to \R$ and the floor function $\R \to \Z$ is a Galois connection but not a Galois insertion.
There the lower adjoint is the inclusion, and the inclusion of $\lfloor 1/2 \rfloor = 0$ is $0 \ne 1/2$.
For a Galois connection, these conditions are equivalent: $f \circ g$ is the identity of $Q$; $f$ is surjective; $g$ is injective.
The dual notion, with $g(f(x)) = x$ for all $x \in P$, is a Galois coinsertion.

## Explanations

### L1 human

A Galois insertion is a Galois connection in which the smaller side sits faithfully inside the larger side.
Going in and coming back out changes nothing.
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# Szász–Mirakyan operator

## Definition

Let $f \colon [0, \infty) \to \R$ satisfy $\abs{f(t)} \le C e^{A t}$ for some constants $C, A \ge 0$ and all $t \ge 0$.
For an integer $n \ge 1$, the *Szász–Mirakyan operator* $S_n$ maps $f$ to the function
$$S_n(f)(x) = e^{-nx} \sum_{k=0}^{\infty} \frac{(nx)^k}{k!} \, f\left(\frac{k}{n}\right), \qquad x \ge 0.$$

## Remarks

The growth bound makes the series converge for all $x \ge 0$.
Each $S_n$ is a [[positive linear operator|AQDXWH4812]].
The weights $e^{-nx} (nx)^k / k!$ are the probabilities of the [[Poisson distribution|0XB9GQTN9G]] with mean $nx$.

Example: if $N$ is Poisson with mean $nx$, then $E[N] = nx$ and $E[N^2] = nx + n^2 x^2$.
So $S_n(1) = 1$, $S_n(t)(x) = x$ and $S_n(t^2)(x) = x^2 + x/n$.

Non-example: $f(t) = e^{t^2}$ is not in the domain, since $e^{k^2/n^2} (nx)^k / k!$ does not tend to $0$ as $k \to \infty$ when $x > 0$.

From the three moments and Korovkin's theorem, $S_n f \to f$ uniformly on each interval $[0, b]$ for every bounded continuous $f$.
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# Exchangeable random variables

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $(S, \mathcal{S})$ be a measurable space.
Let $X_1, \dots, X_n$ be [[random variables|0PNRKMRC55]] on $\Omega$ with values in $S$.
The finite sequence $(X_1, \dots, X_n)$ is *exchangeable* if for every [[permutation|05RWFKVTJ7]] $\sigma$ of $\set{1, \dots, n}$ the random vector $(X_{\sigma(1)}, \dots, X_{\sigma(n)})$ has the same [[joint distribution|0DR81R1F2B]] as $(X_1, \dots, X_n)$.
An infinite sequence $X_1, X_2, \dots$ is *exchangeable* if $(X_1, \dots, X_n)$ is exchangeable for every $n \ge 1$.

## Remarks

Equivalently, an infinite sequence is exchangeable if its joint law on $S^{\N}$ is invariant under every permutation of $\N$ that moves only finitely many indices.
Example: if $X_1, X_2, \dots$ are [[i.i.d.|00MRN0FARV]], the joint law of $(X_1, \dots, X_n)$ is the product $\mu^{\otimes n}$ of the common law $\mu$, and a product of equal factors does not change when the factors are permuted.
Another example: in the [[Pólya urn model|DHPXDFEEMF]], the sequence of colours drawn is exchangeable but not independent.
Non-example: let $Y$ be a [[Bernoulli|0386Y0Z7Y7]] variable with parameter $\frac{1}{2}$, and let $X_1 = Y$ and $X_2 = 2Y$.
Then $P((X_1, X_2) = (1, 2)) = \frac{1}{2}$ but $P((X_2, X_1) = (1, 2)) = 0$, so the transposition of $1$ and $2$ changes the law.
Each $X_i$ of an exchangeable sequence has the same law, and each pair $(X_i, X_j)$ with $i \ne j$ has the same joint law.
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# Function field

## Summary

A function field over a field $k$ is a finitely generated field extension of $k$.
One views it as the field of rational functions on some algebraic variety.
Its [[transcendence degree|D7TR0YWR9H]] over $k$ is the number of variables, which is the dimension of the variety.
Function fields in one variable correspond to algebraic curves, and over finite fields they behave much like number fields.

## Explanations

### L1 human

On the plane, the rational functions are fractions of polynomials in $x$ and $y$.
On a curve such as $y^2 = x^3 + 1$, one uses the same fractions, but $y^2$ and $x^3 + 1$ count as equal.
The field of all such functions is the function field of the curve.
Many questions about a curve, such as whether two curves are birational, become questions about its function field.
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# Empirical process

## Summary

The empirical process of a sample is the gap between its empirical distribution function and the true [[cumulative distribution function|0H8H7ZTHW2]], times $\sqrt{n}$.
It is the main object behind uniform laws of large numbers and functional central limit theorems in statistics.
The [[Glivenko–Cantelli theorem|E7B78X780X]] and [[Donsker's theorem|818832PMSV]] describe its behaviour as the sample size grows.

## Explanations

### L1 human

Draw $n$ numbers at random from a distribution and plot the fraction of them below each level $x$.
This staircase is the empirical distribution function, and it should be close to the true distribution function $F$.
The empirical process is the error between the two curves, multiplied by $\sqrt{n}$ so that it stays of order one.
For one fixed $x$ it is just the rescaled error of a proportion, which the central limit theorem says is roughly normal.
Seen as a random curve in $x$, it looks like a [[Brownian bridge|3MCVMBAGD1]] for large $n$.
</document>

<document source="https://tamnd.github.io/math2/node/XKFKTHW8GS.md" tag="XKFKTHW8GS" kind="definition">
# Quillen adjunction

## Definition

Let $\mathcal{M}$ and $\mathcal{N}$ be [[model categories|SDTZVZ6B4Q]], and let $F \colon \mathcal{M} \rightleftarrows \mathcal{N} \colon G$ be an [[adjunction|TZHF6ZM92S]] with $F$ left adjoint to $G$.
The adjunction is a *Quillen adjunction* if the following equivalent conditions hold.

1. $F$ preserves [[cofibrations|HSV1J7BJXX]] and trivial cofibrations.
2. $G$ preserves fibrations and trivial fibrations.
3. $F$ preserves cofibrations and $G$ preserves fibrations.

Then $F$ is called a *left Quillen functor* and $G$ a *right Quillen functor*.

## Remarks

For an adjunction, $F(i)$ has the left lifting property against $p$ if and only if $i$ has the left lifting property against $G(p)$.
Since the classes of a model structure are characterized by lifting properties, the three conditions are equivalent.

A Quillen adjunction induces an adjunction $\mathbf{L}F \dashv \mathbf{R}G$ between the [[homotopy categories|48QQ7T98Z9]].
Here $\mathbf{L}F$ is computed by applying $F$ to a cofibrant replacement, and $\mathbf{R}G$ by applying $G$ to a fibrant replacement.
This works because $F$ sends [[weak equivalences|E2870TEDG7]] between cofibrant objects to weak equivalences, by Ken Brown's lemma.

Example: [[geometric realization|TG6WHHR7CF]] $\abs{-}$ from simplicial sets to topological spaces is left adjoint to the [[singular simplicial set|RWBMXQW6S5]] functor $\mathrm{Sing}$.
With the Kan–Quillen and Quillen model structures, $\mathrm{Sing}$ sends Serre fibrations to Kan fibrations and trivial ones to trivial ones, so the pair is a Quillen adjunction.

Non-example: let $\mathcal{M}$ and $\mathcal{N}$ both be the category of sets.
Give $\mathcal{M}$ the trivial model structure: $W$ the bijections, and all maps fibrations and cofibrations.
Give $\mathcal{N}$ the model structure with $W$ all maps, cofibrations the bijections, and fibrations all maps.
The identity adjunction fails clause 1, since a non-bijective map is a cofibration of $\mathcal{M}$ but not of $\mathcal{N}$.
</document>

<document source="https://tamnd.github.io/math2/node/XKGEV5RP3T.md" tag="XKGEV5RP3T" kind="definition">
# Coefficient of determination

## Definition

Let $y_1, \dots, y_n$ be observed values of a response, not all equal, and let $\hat y_1, \dots, \hat y_n$ be the fitted values of a regression model.
Let $\bar y = \frac{1}{n} \sum_{i=1}^n y_i$ be the [[sample mean|03Y6AQ2S0J]].
The *total sum of squares* is $SS_{\mathrm{tot}} = \sum_{i=1}^n (y_i - \bar y)^2$.
The *residual sum of squares* is $SS_{\mathrm{res}} = \sum_{i=1}^n (y_i - \hat y_i)^2$.
The *coefficient of determination* is
$$R^2 = 1 - \frac{SS_{\mathrm{res}}}{SS_{\mathrm{tot}}}.$$

## Remarks

Example: take the points $(x_i, y_i) = (0,0), (1,1), (2,3)$ and fit $y = a + bx$ by [[ordinary least squares|YM708KHV2W]].
The fit is $b = 3/2$, $a = -1/6$, with fitted values $-1/6$, $4/3$, $17/6$.
Then $SS_{\mathrm{res}} = 1/36 + 4/36 + 1/36 = 1/6$ and $SS_{\mathrm{tot}} = 16/9 + 1/9 + 25/9 = 14/3$, so $R^2 = 1 - 1/28 = 27/28$.

When the fit is by ordinary least squares with an intercept, $SS_{\mathrm{tot}} = SS_{\mathrm{reg}} + SS_{\mathrm{res}}$ with $SS_{\mathrm{reg}} = \sum_i (\hat y_i - \bar y)^2$.
So $R^2 = SS_{\mathrm{reg}} / SS_{\mathrm{tot}}$ lies in $[0, 1]$.
In that case $R^2$ is the square of the sample correlation between $y$ and $\hat y$.
For a straight line fit it is the square of the sample correlation between $x$ and $y$.

Non-example: if the fitted values do not come from least squares with an intercept, the formula can give a negative number.
With the same data and the constant prediction $\hat y_i = 3$, one gets $SS_{\mathrm{res}} = 13 > SS_{\mathrm{tot}}$, so $R^2 < 0$.

The *adjusted* $R^2$ for a model with $p$ coefficients, including the intercept, is $1 - (1 - R^2)(n - 1)/(n - p)$.
It penalises extra regressors.
</document>

<document source="https://tamnd.github.io/math2/node/XKMBKAN1MR.md" tag="XKMBKAN1MR" kind="definition">
# Cohomological dimension of a field

## Definition

Let $K$ be a field with [[absolute Galois group|PGKWBJDTKX]] $G_K$, and let $p$ be a [[prime number|0SN0MAEKYX]].
For a discrete torsion $G_K$-module $M$, let $H^q(K, M)(p)$ be the $p$-primary part of the [[Galois cohomology|EESNTXFWHZ]] group $H^q(K, M)$.
The *$p$-cohomological dimension* $\operatorname{cd}_p(K)$ is the least integer $n \ge 0$ with the following property.
For all $q > n$ and all discrete torsion $G_K$-modules $M$, $H^q(K, M)(p) = 0$.
If there is no such $n$, then $\operatorname{cd}_p(K) = \infty$.
The *cohomological dimension* of $K$ is $\operatorname{cd}(K) = \sup_p \operatorname{cd}_p(K)$.

## Remarks

These are the [[cohomological dimensions|EF4ZWACHD8]] $\operatorname{cd}_p(G_K)$ and $\operatorname{cd}(G_K)$ of the profinite group $G_K$.
It suffices to test one degree and simple modules: $\operatorname{cd}_p(K) \le n$ if and only if $H^{n+1}(K, M) = 0$ for every simple discrete $G_K$-module $M$ with $pM = 0$.
Example: a finite field $\F_q$ has $\operatorname{cd}(\F_q) = 1$, since $G_{\F_q}$ is the profinite completion $\widehat{\Z}$ of $\Z$, whose cohomology with torsion coefficients vanishes above degree $1$, while $H^1(\F_q, \Z/p\Z) = \Hom(\widehat{\Z}, \Z/p\Z) \ne 0$.
Example: $\operatorname{cd}(K) = 0$ if and only if $K$ is separably closed.
Example: $\R$ has $\operatorname{cd}_2(\R) = \infty$, since $G_\R$ has order $2$ and $H^q(\Z/2\Z, \Z/2\Z) \cong \Z/2\Z$ for all $q \ge 0$.
More generally, a field with a nontrivial finite absolute Galois group has infinite cohomological dimension.
If $K$ has characteristic $p > 0$, then $\operatorname{cd}_p(K) \le 1$.
A finite extension of $\Q_p$ has cohomological dimension $2$.
</document>

<document source="https://tamnd.github.io/math2/node/XKMWYSGBN0.md" tag="XKMWYSGBN0" kind="definition">
# Prime end

## Definition

Let $\Omega \subset \C$ be a bounded [[simply connected domain|075C629XTH]] with [[boundary|0JJ9J1XZ8G]] $\partial\Omega$.
A *crosscut* of $\Omega$ is a set $C \subset \Omega$ that is homeomorphic to the open interval $(0, 1)$ and whose closure is $C \cup \set{a, b}$ with $a, b \in \partial\Omega$.
Then $\Omega \setminus C$ has exactly two [[connected components|0R37DFJKFW]].

A *null chain* is a sequence $(C_n)_{n \ge 0}$ of crosscuts of $\Omega$ such that:

1. the closures $\overline{C_n}$ and $\overline{C_{n+1}}$ are disjoint for all $n$;
2. for each $n \ge 1$, the crosscut $C_n$ separates $C_0$ from $C_{n+1}$ in $\Omega$;
3. the diameter of $C_n$ tends to $0$.

For a null chain, let $V_n$ be the component of $\Omega \setminus C_n$ that does not contain $C_0$, so $V_1 \supset V_2 \supset \cdots$.
Two null chains $(C_n)$ and $(C'_n)$, with domains $V_n$ and $V'_n$, are *equivalent* if for each $m$ there is an $n$ with $V_n \subseteq V'_m$ and $V'_n \subseteq V_m$.
A *prime end* of $\Omega$ is an equivalence class of null chains.

## Remarks

Example: for the unit disc $\mathbb{D}$ and a point $\zeta$ with $\abs{\zeta} = 1$, the arcs $C_n = \set{z \in \mathbb{D} : \abs{z - \zeta} = 2^{-n}}$ for $n \ge 1$, relabelled from $0$, form a null chain.
Their closures are disjoint, each arc separates the previous one from the next, and their diameters tend to $0$.
Every null chain of $\mathbb{D}$ shrinks to a single boundary point, and the prime ends of $\mathbb{D}$ correspond to the points of the unit circle.
Non-example: in $\mathbb{D}$, the vertical chords $C_n = \set{z \in \mathbb{D} : \operatorname{Re} z = 1/(n+2)}$ satisfy clauses 1 and 2, but their diameters $2\sqrt{1 - (n+2)^{-2}}$ tend to $2$, so clause 3 fails.
The *impression* of a prime end is the intersection of the closures $\overline{V_n}$; it is a nonempty compact connected subset of $\partial\Omega$.
Carathéodory's prime end theorem says that a conformal map of $\mathbb{D}$ onto $\Omega$ induces a bijection from the unit circle onto the set of prime ends of $\Omega$.
</document>

<document source="https://tamnd.github.io/math2/node/XKRR37N916.md" tag="XKRR37N916" kind="concept">
# Grunsky matrix

## Summary

Let $g$ be a univalent function on the outside of the unit disc.
Its Grunsky matrix lists the coefficients of the double series of $\log\frac{g(z) - g(\zeta)}{z - \zeta}$.
Univalence is reflected in the Grunsky inequalities, which say that a weighted form of this matrix is a contraction on the sequence space $\ell^2$.
These inequalities refine the [[Grönwall area theorem|BYZDAZNHP4]] and were a main tool in the work on coefficient problems before the proof of the [[Bieberbach conjecture|ZMRFSYVWQT]].

## Explanations

### L1 human

A univalent function $g$ never takes the same value twice, so the difference quotient $(g(z) - g(\zeta))/(z - \zeta)$ never vanishes.
Its logarithm is then a nice function of two variables, which can be expanded in powers of $1/z$ and $1/\zeta$.
The Grunsky matrix lists the coefficients of this expansion.
For $g(z) = z + 1/z$ the expansion is $\log(1 - 1/(z\zeta))$, and the matrix is diagonal with entries $1, 1/2, 1/3, \ldots$.
</document>

<document source="https://tamnd.github.io/math2/node/XM4NTS5BGG.md" tag="XM4NTS5BGG" kind="statement">
# Pentagon identity

## Statement

Let $(\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)$ be a [[monoidal category|M3MSQ3NKAE]] and let $A, B, C, D$ be objects of $\mathcal{C}$.
Then the following [[diagram commutes|WV898P4X16]], as an equality of morphisms $((A \otimes B) \otimes C) \otimes D \to A \otimes (B \otimes (C \otimes D))$:
$$(\id_A \otimes \alpha_{B,C,D}) \circ \alpha_{A, B \otimes C, D} \circ (\alpha_{A,B,C} \otimes \id_D) = \alpha_{A,B,C \otimes D} \circ \alpha_{A \otimes B, C, D}.$$

## Remarks

The identity is one of the two axioms in the definition of a monoidal category, the other being the triangle identity.
The five objects obtained by bracketing $A \otimes B \otimes C \otimes D$ in all ways are the vertices of a pentagon, and the five instances of $\alpha$ are its edges.
The left side goes around three edges and the right side around the other two.
In $(\mathbf{Set}, \times)$ both sides send $(((a,b),c),d)$ to $(a,(b,(c,d)))$.
By Mac Lane's coherence theorem, the pentagon and triangle identities imply that all formal diagrams built from $\alpha$, $\lambda$ and $\rho$ commute.
Without the pentagon this fails: an associator with an extra sign on real vector spaces satisfies naturality but makes the two routes around the pentagon differ by $-1$.

## Explanations

### L1 human

There are five ways to put brackets in a product of four things, and the associator moves brackets one step at a time.
Going from the leftmost bracketing to the rightmost one can be done in two different ways around a pentagon.
The pentagon identity says both ways give the same result.
This is what makes "rebracketing" a well-defined operation that does not depend on the order of the moves.

### L2 human

The pentagon identity is the first nontrivial coherence condition for an associativity isomorphism.
Together with the triangle identity it implies, by Mac Lane's coherence theorem, that all other diagrams of associators and unitors commute, so one never has to check longer ones.
The same pentagon reappears in the definition of bicategories, in the 3-cocycle condition for associators on $\mathrm{Vec}_G$, and as the Stasheff associahedron $K_4$ in the theory of $A_\infty$-structures.
</document>

<document source="https://tamnd.github.io/math2/node/XM695ZZ4FJ.md" tag="XM695ZZ4FJ" kind="statement">
# Theta transformation formula

## Statement

Let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]], and let $\vartheta(z, \tau) = \sum_{n \in \Z} \exp(\pi i n^2\tau + 2\pi i n z)$ be the Jacobi [[theta function|2F5CYQJVAF]] for $z \in \C$ and $\tau \in \mathbb{H}$.
Then for all $z \in \C$ and $\tau \in \mathbb{H}$:

1. $\vartheta(z, \tau + 1) = \vartheta(z + \tfrac12, \tau)$;
2. $$\vartheta\!\left(\frac{z}{\tau}, -\frac{1}{\tau}\right) = \sqrt{-i\tau}\;\exp\!\left(\frac{\pi i z^2}{\tau}\right)\vartheta(z, \tau),$$
where $\sqrt{-i\tau}$ is the principal branch, which is positive when $\tau$ is on the positive imaginary axis.

## Remarks

Part 1 holds term by term, because $e^{\pi i n^2} = (-1)^n = e^{2\pi i n/2}$.
Proof idea for part 2: for fixed $\tau$, apply the [[Poisson summation formula|GARGJYSGFG]] to the Gaussian $x \mapsto \exp(\pi i x^2\tau + 2\pi i x z)$, whose Fourier transform is again a Gaussian, with $-1/\tau$ in place of $\tau$.
Both sides are holomorphic in $\tau$, so it is enough to check $\tau = it$ with $t > 0$ and continue analytically.
Example: at $z = 0$ and $\tau = it$ the formula reads $\sum_{n} e^{-\pi n^2/t} = \sqrt{t}\sum_n e^{-\pi n^2 t}$, the classical identity behind the functional equation of the Riemann zeta function.
The hypothesis $\tau \in \mathbb{H}$ is needed: for real $\tau$ the series does not converge.

## Explanations

### L1 human

The theta function is a sum of Gaussian terms.
Replacing $\tau$ by $-1/\tau$ turns narrow Gaussians into wide ones, and the formula says the sum changes only by a simple explicit factor.
Together with the shift $\tau \mapsto \tau + 1$ this makes theta a modular object.

### L2 human

The maps $\tau \mapsto \tau + 1$ and $\tau \mapsto -1/\tau$ generate the [[modular group|VEKV1EM77B]], so the formula describes how $\vartheta$ behaves under it.
The theta constant $\vartheta(0, \tau)$ is invariant under $\tau \mapsto \tau + 2$ and picks up the factor $\sqrt{-i\tau}$ under $\tau \mapsto -1/\tau$.
So it is a [[modular form|GWKN5VG78A]] of weight $1/2$ for the subgroup generated by these two maps.
Raising to the fourth power and using part 2 gives the transformation law used in counting sums of four squares.
In the integral $\int_0^\infty (\vartheta(0, it) - 1)\,t^{s/2 - 1}\,dt$, the formula of the example yields Riemann's proof of the functional equation of $\zeta(s)$.
</document>

<document source="https://tamnd.github.io/math2/node/XM6DH57BWY.md" tag="XM6DH57BWY" kind="concept">
# Equidistribution modulo one

## Summary

A sequence of real numbers is equidistributed modulo one when its fractional parts fill the interval $[0, 1)$ evenly in the long run.
Each subinterval then receives a proportion of the terms equal to its length.
The notion connects Diophantine approximation, Fourier analysis and probability, and Weyl's criterion reduces it to estimates of [[exponential sums|97PBMQCPPM]].

## Explanations

### L1 human

Wrap the real line around a circle of length $1$, so that only the fractional part of a number matters.
Place the terms of a sequence on this circle one by one.
The sequence is equidistributed if, after many steps, every arc holds a share of the points equal to its length.
For example the multiples $n \sqrt{2}$ are equidistributed modulo one, while the multiples $n/2$ only visit the two points $0$ and $1/2$.
</document>

<document source="https://tamnd.github.io/math2/node/XM772FD0BS.md" tag="XM772FD0BS" kind="statement">
# Hilbert's theorem on the hyperbolic plane

## Statement

Let $S$ be a complete two-dimensional [[Riemannian manifold|7CBT4SM83Y]] whose [[Gaussian curvature|SQVQ5DQT6W]] is a negative constant.
Then there is no smooth isometric [[immersion|6RW68JGBHC]] of $S$ into $\R^3$.
In particular, the hyperbolic plane cannot be realized as a smooth surface in $\R^3$ with its induced metric.

## Remarks

After scaling, one may assume $K = -1$; one may also pass to the universal cover, which is the hyperbolic plane.
For an immersed surface with $K < 0$, the two asymptotic directions at each point give coordinates in which the angle $\omega$ between the coordinate curves satisfies the sine-Gordon equation $\omega_{uv} = \sin\omega$.
Completeness lets these coordinates cover the whole plane, and the equation then bounds the area of every coordinate quadrilateral by $2\pi$.
This contradicts the fact that the hyperbolic plane has infinite area.

The [[pseudosphere|44WEBQSKEY]] has constant curvature $-1$ and lies in $\R^3$, but it is not complete: it ends in a sharp edge.
The smoothness hypothesis matters: by the Nash–Kuiper theorem there are $C^1$ isometric embeddings of the hyperbolic plane into $\R^3$.
For constant positive curvature the situation is different, since round spheres are complete and lie in $\R^3$.

## Explanations

### L1 human

The hyperbolic plane is a surface that curves like a saddle at every point, by the same amount everywhere, and goes on forever.
Hilbert's theorem says that you cannot build such a surface in ordinary space without creases or edges.
Every smooth attempt, like the pseudosphere, must stop at some edge.
This is why hyperbolic geometry is usually drawn with models such as the Poincaré disc, which distort distances.

### L2 human

The theorem explains why the hyperbolic plane is studied as an abstract Riemannian manifold rather than as a surface in space.
Efimov later showed that a complete surface with curvature bounded above by a negative constant cannot be smoothly immersed in $\R^3$ either.
In contrast, the Nash embedding theorems show that every Riemannian manifold embeds isometrically in some higher-dimensional Euclidean space.
The proof is a good example of how the Gauss equation turns a geometric problem into a partial differential equation.
</document>

<document source="https://tamnd.github.io/math2/node/XMCAHWK1YB.md" tag="XMCAHWK1YB" kind="problem">
# Equilibrium problem with an external field

## Problem

Let $E \subseteq \C$ be a closed set.
Let $w \colon E \to [0, \infty)$ be an upper semicontinuous weight such that the set $\set{z \in E : w(z) > 0}$ has positive [[logarithmic capacity|DBQXRTC0VN]].
If $E$ is unbounded, assume also that $\abs{z}\, w(z) \to 0$ as $\abs{z} \to \infty$ in $E$.
Let $Q = -\log w$ be the *external field*.
For a Borel [[probability measure|059TJN0NZ1]] $\mu$ with support in $E$, set
$$I_w(\mu) = \int \int \log \frac{1}{\abs{z - t}\, w(z)\, w(t)}\, d\mu(z)\, d\mu(t).$$
The problem is to find $V_w = \inf_\mu I_w(\mu)$, a measure $\mu_w$ that attains it, and its support $S_w$.

## Remarks

The weighted energy equals $I(\mu) + 2\int Q\, d\mu$, where $I(\mu)$ is the logarithmic energy.
So the field $Q$ pushes the charge away from where $Q$ is large.
For $w = 1$ on a compact set $E$ this is the problem of the [[equilibrium measure|464NN2K4FZ]] of $E$.
Gauss considered the analogous problem for the Newtonian kernel in space.
The weighted logarithmic version in the plane is the subject of the book of Saff and Totik (1997).
Existence and uniqueness are known under the hypotheses above: $V_w$ is finite, there is a unique minimiser $\mu_w$, and $S_w$ is compact with positive capacity.
The minimiser is characterised by Frostman-type conditions on its [[logarithmic potential|46NHWJHW44]] $U^{\mu_w}$.
For a constant $F_w$, one has $U^{\mu_w} + Q \ge F_w$ quasi-everywhere on $E$ and $U^{\mu_w} + Q \le F_w$ on $S_w$.
Finding $S_w$ for a given weight is the hard part, and it is done explicitly only for special weights.
Example: on $E = \R$ with $w(x) = e^{-x^2}$, the support is $[-1, 1]$ and $\mu_w$ has the semicircle density $\frac{2}{\pi}\sqrt{1 - x^2}$.

## Explanations

### L1 human

Put a unit of charge on a conductor in the plane, but add an outside force that pulls the charge towards some places and pushes it from others.
The charge settles in the arrangement of least total energy.
The question is what this arrangement is, and in particular which part of the conductor it covers.
Such arrangements describe where zeros of orthogonal polynomials and eigenvalues of large random matrices accumulate.
</document>

<document source="https://tamnd.github.io/math2/node/XMD3Q49Z2C.md" tag="XMD3Q49Z2C" kind="concept">
# Schwarz triangle function

## Summary

The Schwarz triangle function is the ratio of two independent solutions of the [[hypergeometric differential equation|W8VE25R5A7]].
For suitable real parameters it is a [[conformal map|0YYT36R4YD]] from the upper half-plane onto a triangle bounded by circular arcs or line segments.
The angles are $\pi\lambda$, $\pi\mu$, $\pi\nu$, where $\lambda$, $\mu$, $\nu$ are the exponent differences.
Its inverse functions, when the angles are $\pi/p$, $\pi/q$, $\pi/r$, are automorphic functions for triangle groups.
When all three angles are zero, the inverse function is the modular lambda function.

## Explanations

### L1 human

A ratio of two solutions of a second-order linear equation forgets the scaling of the solutions and keeps only their shape.
For the hypergeometric equation, this ratio maps the upper half-plane onto a curved triangle, whose sides are arcs of circles.
The three vertices are the images of $0$, $1$ and $\infty$, and the angles there are fixed by the parameters $a$, $b$, $c$.
When the angles are of the form $\pi/n$, reflecting the triangle across its sides again and again tiles a disc, the plane or the sphere.
The case of the sphere is how Schwarz found the algebraic hypergeometric functions.
</document>

<document source="https://tamnd.github.io/math2/node/XMHQCDWZW1.md" tag="XMHQCDWZW1" kind="statement">
# AKS primality test

## Statement

Let $n \ge 2$ be an integer and let $r \ge 2$ be an integer.
Assume that $n$ has no [[prime|01826KMG2H]] factor $p \le r$.
Then $\gcd(n, r) = 1$, so the [[multiplicative order|02RRHJ7R98]] $\ord_r(n)$ of $n$ modulo $r$ is defined.
Assume also that $\ord_r(n) > (\log_2 n)^2$.
Let $\varphi$ be [[Euler's totient function|0DAMHCT17R]] and let $\ell = \floor{\sqrt{\varphi(r)} \log_2 n}$.
Then the following hold.

1. The integer $n$ is prime if and only if $n$ is not a perfect power and
$$(X + a)^n \equiv X^n + a \pmod{X^r - 1,\ n}$$
for every integer $a$ with $1 \le a \le \ell$.
Here the congruence is an equality in the [[quotient ring|0076HWHGHQ]] $\Z[X]/(X^r - 1, n)$ of the [[polynomial ring|0SDC934R52]] $\Z[X]$.
A perfect power is an integer $m^k$ with integers $m, k \ge 2$.
2. For every $n \ge 2$ there is an integer $r \le \max\set{3, \ceil{(\log_2 n)^5}}$ with $\ord_r(n) > (\log_2 n)^2$ or $\gcd(n, r) > 1$.
3. Hence a deterministic algorithm decides whether $n$ is prime with a number of bit operations polynomial in $\log n$.

## Remarks

The algorithm is due to Manindra Agrawal, Neeraj Kayal and Nitin Saxena, in the paper "PRIMES is in P".
It starts from a simple fact: for $\gcd(a, n) = 1$, the integer $n$ is prime if and only if $(X + a)^n \equiv X^n + a \pmod n$ in $\Z[X]$.
The direct check of this identity costs about $n$ operations, so the test reduces it modulo $X^r - 1$ for a small $r$.
If $n$ is composite with a prime factor $p$, the congruences for all $a \le \ell$ give a group of polynomials modulo $p$.
This group is forced to be both small and large at once, unless $n$ is a power of $p$.
The algorithm finds the smallest suitable $r$, checks $\gcd(a, n)$ for $a \le r$, tests for perfect powers, and then checks the $\ell$ congruences.

For $n$ prime, part 1 holds by [[Fermat's little theorem|0M25PRN3WQ]] and the [[freshman's dream|0396VGFA78]] $(X + a)^p = X^p + a^p$ in characteristic $p$.
The test needs no unproved hypothesis and is never wrong, unlike the [[Miller–Rabin primality test|0SZCSYHW60]], which is randomised.

## Explanations

### L1 human

The AKS test decides whether a whole number is prime, and it is always right.
Its running time grows only like a power of the number of digits of $n$, not like $n$ itself.
Before it, no known test was at the same time deterministic, free of unproved conjectures and polynomial in the number of digits.
It answered the old question whether primality can be decided in polynomial time.

### L2 human

The test turns the binomial identity $(X + a)^p = X^p + a$ modulo $p$ into a criterion that is cheap to check.
The working ring $\Z[X]/(X^r - 1, n)$ has elements of size about $r \log n$ bits, and $r$ is polynomial in $\log n$ by part 2.
The proof uses only elementary algebra of finite fields and cyclotomic polynomials, together with a bound on the least common multiple of $1, \ldots, m$.
In practice the test is slower than the [[Miller–Rabin primality test|0SZCSYHW60]] or elliptic curve methods, so its importance is mainly theoretical.
It is a [[primality test|0EGJZHZ02C]], and it does not find factors; the [[integer factorization|YPE65X5HJ4]] problem is believed to be much harder.
</document>

<document source="https://tamnd.github.io/math2/node/XMJ0V0FCPY.md" tag="XMJ0V0FCPY" kind="definition">
# Associated bundle

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] and let $\pi \colon P \to B$ be a [[principal $G$-bundle|8E37E74XVP]], with $G$ acting on $P$ from the right.
Let $F$ be a topological space with a continuous left [[action|0WCY42EV9V]] of $G$.
On $P \times F$, with the [[product topology|0GBG8B49AA]], let $G$ act by $g \cdot (p, f) = (p g^{-1}, g f)$.
The *associated bundle* with fiber $F$ is the [[quotient space|0BR3FY5Q8M]]
$$P \times_G F = (P \times F) / G,$$
together with the map $\pi_F \colon P \times_G F \to B$, $\pi_F([p, f]) = \pi(p)$.

## Remarks

The map $\pi_F$ is well defined because $\pi(p g^{-1}) = \pi(p)$.
It is a fiber bundle with fiber $F$: a local trivialization $\pi^{-1}(U) \cong U \times G$ of $P$ induces $\pi_F^{-1}(U) \cong U \times F$.
The class $[p, f]$ is also written $p \times_G f$; in it, $[p g, f] = [p, g f]$.

Example: let $G = \Z/2\Z$ and $P = S^1 \to S^1$, $z \mapsto z^2$, a principal $G$-bundle where the generator acts by $z \mapsto -z$.
Let the generator act on $F = \R$ by $t \mapsto -t$.
Then $P \times_G \R$ is the quotient of $S^1 \times \R$ by $(z, t) \sim (-z, -t)$, which is the open Möbius band, a nontrivial line bundle over the circle.

Non-example: the product $P \times F$ itself, with the map $(p, f) \mapsto \pi(p)$, is not the associated bundle when $G$ is nontrivial.
Its fibers are copies of $G \times F$, not of $F$, because the quotient by $G$ was not taken.

If $F = \R^n$ with $G = GL_n(\R)$ acting linearly, then $P \times_G \R^n$ is a real vector bundle of rank $n$.
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# Triangle removal lemma

## Statement

For every real number $\varepsilon > 0$ there is a real number $\delta > 0$ with the following property.
Let $G$ be a finite simple [[graph|00C0BR7D7H]] with $n$ vertices that contains at most $\delta n^3$ triangles.
Then one can delete at most $\varepsilon n^2$ edges of $G$ so that the resulting graph has no triangle.

## Remarks

The standard proof applies the [[Szemerédi regularity lemma|ETMAPNQY7T]] with a small accuracy.
Delete the edges inside parts, between irregular pairs, and between pairs of low density; these are at most $\varepsilon n^2$ edges.
If a triangle survives, its three parts are pairwise regular and dense, and the counting lemma then gives more than $\delta n^3$ triangles.
The resulting $\delta$ is very small: the proof gives $1/\delta$ of tower type in $1/\varepsilon$.

A consequence: if every edge of a graph on $n$ vertices lies in exactly one triangle, then the graph has $o(n^2)$ edges.
Indeed, if it has $e$ edges, it has $e/3 \le n^2/6$ triangles, which is less than $\delta n^3$ for large $n$.
The triangles are edge-disjoint, so destroying them needs $e/3$ deletions, and the lemma gives $e/3 \le \varepsilon n^2$.
Applied to a graph built from a subset of $\Z/N\Z$, this gives [[Roth's theorem|E7H8NWXESF]] on three-term arithmetic progressions.

## Explanations

### L1 human

A graph with few triangles, compared with the $n^3$ triangles that are possible, is close to a graph with no triangles at all.
"Close" means that a small fraction of the $n^2$ edges is enough to destroy every triangle.
This sounds obvious but is not: the few triangles could share edges in a clever way, and the lemma says they cannot.

### L2 human

The lemma is the simplest case of the graph removal lemma.
There, any fixed graph $H$ takes the place of the triangle, and $\delta n^{|V(H)|}$ copies take the place of $\delta n^3$ triangles.
It is a central tool linking [[extremal graph theory|SC5QB5F4TE]] and [[additive combinatorics|3NM362A2JG]].
Its use for Roth's theorem shows how a statement about graphs can prove a statement about integers.
Better bounds for $\delta$ than tower type in this generality are a well-known open question.
</document>

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# Bounded gaps between primes

## Statement

Let $p_n$ denote the $n$-th [[prime number|01826KMG2H]].
There is a constant $H$ such that $p_{n+1} - p_n \le H$ for infinitely many $n$.
Equivalently,
$$\liminf_{n \to \infty} \, (p_{n+1} - p_n) < \infty,$$
where $\liminf$ is the [[limit inferior|033RZ32T4Q]].

## Remarks

Yitang Zhang proved the statement in 2013 with $H = 7 \cdot 10^7$.
James Maynard, and independently Terence Tao, gave a simpler proof with $H = 600$, and the Polymath project lowered the bound to $H = 246$.
Maynard's method also shows that for every $m \ge 1$, $\liminf_{n \to \infty} (p_{n+m} - p_n) < \infty$.
The proof builds on the method of Goldston, Pintz and Yıldırım: a [[Selberg sieve|5252P3NHT7]] with multidimensional weights detects a tuple $n + h_1, \ldots, n + h_k$ that contains at least two primes.
The input on primes in arithmetic progressions is the [[Bombieri–Vinogradov theorem|B3JMY54GJF]] in Maynard's version and a strengthening of it for smooth moduli in Zhang's version.
The [[twin prime conjecture|0FER36N8SB]] is the claim that $H = 2$ works; it remains open.

## Explanations

### L1 human

The primes become rarer as numbers grow: on average the gap after $p$ is about $\log p$.
Still, this theorem says that the primes keep coming in close pairs forever: some fixed distance $H$ is reached again and again.
The twin prime conjecture predicts the smallest possible distance $H = 2$, as in $11, 13$ or $41, 43$; the best proven value is $246$.

### L2 human

The statement is the first unconditional step towards the twin prime conjecture.
Goldston, Pintz and Yıldırım had shown that bounded gaps would follow from a level of distribution of the primes greater than $1/2$, which is beyond the [[Bombieri–Vinogradov theorem|B3JMY54GJF]].
Zhang proved such a level for a restricted class of moduli; Maynard changed the sieve weights so that the level $1/2$ is already enough.
Under the Elliott–Halberstam conjecture and its generalisations, the method gives smaller values of $H$, down to $6$.
</document>

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# Néron–Severi group

## Summary

The Néron–Severi group of a smooth projective variety is the group of divisors modulo algebraic equivalence, equivalently the [[Picard group|3Z9DCWYXQZ]] modulo its identity component.
The theorem of the base, due to Severi over $\C$ and Néron in general, says that it is a finitely generated abelian group.
Its rank is the Picard number, and over $\C$ it is the group of integral cohomology classes of type $(1, 1)$.

## Explanations

### L1 human

The Picard group of a variety can be huge, because line bundles come in continuous families.
The Néron–Severi group forgets the continuous part and keeps only the discrete data.
On an elliptic curve, line bundles are classified by their degree and a point of the curve; the Néron–Severi group keeps the degree only and is $\Z$.
On a product of two curves, it also records how a divisor sits with respect to both factors.
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<document source="https://tamnd.github.io/math2/node/XN081NCPF5.md" tag="XN081NCPF5" kind="concept">
# Hyperbolic conservation law

## Summary

A hyperbolic conservation law is a system of [[conservation laws|8RA7AMWSAF]] whose flux Jacobian, in every direction, has real eigenvalues and a full set of eigenvectors.
The eigenvalues are the speeds at which small disturbances travel, so information moves at finite speed along several families of waves.
The equations of gas dynamics, shallow water and traffic flow are of this type.
Their solutions typically form [[shock waves|JSDMGGTG7G]], and the main results concern weak entropy solutions.
See the [[definition|WBYJ7A0HY7]].

## Explanations

### L1 human

In a gas, a small push creates sound waves that travel left and right at finite speeds.
A hyperbolic conservation law is a conservation law with this behaviour: every small disturbance splits into waves with real speeds.
The condition is that a certain matrix built from the flux has only real eigenvalues and enough eigenvectors.
For a single equation $u_t + F(u)_x = 0$ the matrix is the number $F'(u)$, which is always real, so every scalar conservation law is hyperbolic.
</document>

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# Substitution

## Definition

Let $\sigma$ be a [[signature|WDVVEYD6AT]] with only function symbols and let $X$ be a set of variables.
Let $T(X)$ be the set of [[terms|6D3WA7N2W4]] over $\sigma$ and $X$.
A *substitution* is a map $\theta \colon X \to T(X)$.
Its *application* to a term $t$, written $t\theta$, is defined by recursion on $t$:

1. $x\theta = \theta(x)$ for each variable $x \in X$;
2. $c\theta = c$ for each constant symbol $c$;
3. $f(t_1, \ldots, t_n)\theta = f(t_1\theta, \ldots, t_n\theta)$ for each function symbol $f$ of arity $n \ge 1$.

## Remarks

Usually $\theta(x) = x$ for all but finitely many $x$, and $\theta$ is written $\set{x_1 \mapsto s_1, \ldots, x_k \mapsto s_k}$.
A term $s$ is an *instance* of $t$ if $s = t\theta$ for some substitution $\theta$.
The substitutions are exactly the endomorphisms of the term algebra $T(X)$, restricted to the variables.
For an identity $s \approx t$, the identity $s\theta \approx t\theta$ holds in every algebra in which $s \approx t$ holds.

Example: with $\theta = \set{x \mapsto z \cdot z, \ y \mapsto 1}$, clause 3 and clause 1 give $(x + y)\theta = (z \cdot z) + 1$.

Non-example: the replacement of the first occurrence of $x$ in $x + x$ by $y$ is not a substitution.
By clauses 1 and 3, every substitution sends $x + x$ to $\theta(x) + \theta(x)$, which never equals $y + x$.
</document>

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# Admissible decision rule

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a statistical model with a loss function, and let $R(\theta, \delta)$ be the [[risk function|N26TSVRH2S]] of a decision rule $\delta$.
A rule $\delta'$ *dominates* a rule $\delta$ if both conditions hold:

1. $R(\theta, \delta') \le R(\theta, \delta)$ for every $\theta \in \Theta$;
2. $R(\theta_0, \delta') < R(\theta_0, \delta)$ for some $\theta_0 \in \Theta$.

A rule $\delta$ is *admissible* if no rule dominates it, and *inadmissible* otherwise.

## Remarks

Admissibility depends on the [[decision problem|CPFSBBHXT5]]: on the model, on the loss and on the class of rules allowed.

Example: let $X$ be [[normal|05AQE0QR0D]] with mean $\theta \in \R$ and variance $1$, with squared error loss, and let $\delta \equiv \theta_0$ be a constant.
Its risk is $(\theta - \theta_0)^2$, which is $0$ at $\theta_0$.
A rule $\delta'$ with $R(\theta_0, \delta') \le 0$ must equal $\theta_0$ almost surely under $P_{\theta_0}$.
All $P_\theta$ have the same null sets, so $\delta' = \theta_0$ almost surely for every $\theta$, and $\delta'$ has the same risk as $\delta$.
So nothing dominates $\delta$, and it is admissible.

Non-example: in the same problem, $\delta(X) = 2X$ has risk $E_\theta[(2X - \theta)^2] = 4 + \theta^2$.
The rule $X$ has risk $1 < 4 + \theta^2$ for every $\theta$, so it dominates $2X$, which is inadmissible.
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<document source="https://tamnd.github.io/math2/node/XNAWD33HN1.md" tag="XNAWD33HN1" kind="example">
# Torus as a group

## Example

Let $n \ge 1$ and let $\mathbb{T} = \set{z \in \C : \abs{z} = 1}$ be the [[circle group|04EAFVFTF7]].
The *$n$-torus* is the product group $\mathbb{T}^n$ with coordinatewise multiplication and the product topology.
It is a [[locally compact abelian group|ZR09S9R3AW]].
It is abelian because $\mathbb{T}$ is, Hausdorff as a subspace of $\C^n$, and compact as a closed bounded subset of $\C^n$.
The map $(t_1, \dots, t_n) \mapsto (e^{2\pi i t_1}, \dots, e^{2\pi i t_n})$ is a continuous surjective homomorphism $\R^n \to \mathbb{T}^n$ with kernel $\Z^n$, and it induces an isomorphism of topological groups $\R^n / \Z^n \cong \mathbb{T}^n$.

## Remarks

As a topological space, $\mathbb{T}^2$ is the [[torus|03NM7DZXXC]] surface; the group structure is the extra data.
The torus $\mathbb{T}^n$ is a compact connected abelian Lie group of dimension $n$, and every compact connected abelian Lie group is isomorphic to some $\mathbb{T}^n$.
Its [[Pontryagin dual|FMW57WNPDB]] is $\Z^n$: the continuous characters are $(z_1, \dots, z_n) \mapsto z_1^{k_1} \cdots z_n^{k_n}$ with $(k_1, \dots, k_n) \in \Z^n$.
Fourier analysis on $\mathbb{T}^n$ is the theory of multiple Fourier series.

## Explanations

### L1 human

The $n$-torus is a list of $n$ angles, where one adds angles in each slot and forgets full turns.
For $n = 2$ the two angles locate a point on the surface of a doughnut: one goes around the hole, the other around the tube.
Adding angles slides points along the surface, so the doughnut surface becomes a group.
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<document source="https://tamnd.github.io/math2/node/XNBN31HQWB.md" tag="XNBN31HQWB" kind="concept">
# Sumset

## Summary

The sumset $A + B$ of two subsets of an abelian group is the set of all sums $a + b$ with $a \in A$ and $b \in B$.
It is the basic object of [[additive combinatorics|3NM362A2JG]].
Comparing $\abs{A + B}$ with $\abs{A}$ and $\abs{B}$ measures how much additive structure the sets have.
For subsets of a vector space the same operation is called Minkowski addition.

## Explanations

### L1 human

Take two sets of numbers and write down every sum of one number from the first set and one from the second.
The collection of these sums is the sumset.
For example, $\set{0, 1} + \set{0, 10} = \set{0, 1, 10, 11}$.
Sets that are evenly spaced have few different sums, while scattered sets have many.
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<document source="https://tamnd.github.io/math2/node/XNE7F0BAGE.md" tag="XNE7F0BAGE" kind="concept">
# Factorization of polynomials over finite fields

## Summary

Over a [[finite field|0ZXZ82FTX2]] every polynomial can be [[factored into irreducibles|JWB6B13Q8Z]] by efficient algorithms.
The usual methods remove repeated factors, then group the irreducible factors by degree, then split each group.
They rest on the fact that $x^{q^d} - x$ is the product of all monic irreducible polynomials over $\F_q$ whose degree divides $d$.
These algorithms are also the first step of factoring over $\Z$ and $\Q$, and they are used in coding theory and cryptography.

## Explanations

### L1 human

Over a finite field there are only finitely many polynomials of each degree, so a factorization can always be found by trying.
Good algorithms are much faster than trying.
They use $\gcd$ computations with special polynomials like $x^{q^d} - x$ to cut the polynomial into pieces.
For example, over $\F_2$, $x^3 + 1 = (x + 1)(x^2 + x + 1)$, and $x^2 + x + 1$ has no root in $\F_2$.
The [[definition|XXQ378NQE2]] describes the three stages.
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# Hemicontinuity

## Summary

Hemicontinuity is the notion of continuity for [[multivalued functions|03CPD1HGF3]], which assign to each point a set of values.
Upper hemicontinuity says that the value set cannot suddenly grow, and lower hemicontinuity says that it cannot suddenly shrink.
These notions appear in fixed-point theorems and in selection theorems such as the [[Michael selection theorem|3RJ1K36A9A]].
They also appear in economics and game theory, where best responses are often set-valued.

## Explanations

### L1 human

A multivalued function sends each point to a whole set, for example a set of best choices.
As the point moves a little, this set should change only a little.
Upper hemicontinuity forbids the set from exploding outward in a jump: near a point, the sets stay inside any open region around the set at that point.
Lower hemicontinuity forbids the set from losing parts in a jump: every part of the set at a point is still nearly reached by the sets at nearby points.
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# De la Vallée Poussin lower bound

## Statement

Let $a < b$, let $f \colon [a, b] \to \R$ be continuous, let $n \ge 0$, and let $p$ be a real [[polynomial function|1G0B73WPX8]] of degree at most $n$.
Let $a \le x_0 < x_1 < \cdots < x_{n+1} \le b$ be points at which the values $f(x_i) - p(x_i)$ are nonzero and alternate in sign.
Then the [[error of best approximation|YDN0T4MEX0]] of $f$ by polynomials of degree at most $n$ in the supremum norm satisfies
$$E_n(f) \ge \min_{0 \le i \le n+1} \abs{f(x_i) - p(x_i)}.$$

## Remarks

The idea of the proof: let $\mu$ be the minimum on the right and suppose some polynomial $q$ of degree at most $n$ has $\norm{f - q}_\infty < \mu$.
At each $x_i$, the difference $q - p = (f - p) - (f - q)$ has the sign of $f(x_i) - p(x_i)$, because $\abs{f(x_i) - q(x_i)} < \mu \le \abs{f(x_i) - p(x_i)}$.
So $q - p$ alternates in sign at $n + 2$ points and has at least $n + 1$ zeros, hence $q = p$.
But then $\norm{f - q}_\infty \ge \mu$, a contradiction.

Example: for $f(x) = x^2$ on $[0, 1]$, $n = 1$ and $p(x) = x - 1/10$, the error $x^2 - x + 1/10$ takes the values $1/10, -3/20, 1/10$ at $0, 1/2, 1$.
So $E_1(f) \ge 1/10$, while $\norm{f - p}_\infty = 3/20$ gives $E_1(f) \le 3/20$; the true value is $1/8$.

The number of points matters: with only $n + 1$ alternation points no lower bound follows.
For $f(x) = x$, $n = 1$ and $p = 1/2$, the error $x - 1/2$ takes the values $-1/2, 1/2$ at $0, 1$, but $E_1(f) = 0$.

## Explanations

### L1 human

If an approximation's error changes sign enough times, then no approximation of the same degree can be much better.
The theorem turns this into a number: the best possible error is at least the smallest of the alternating error values.
Together with the actual error of $p$, it gives a two-sided bound on the best achievable error.

### L2 human

The bound is the easy half of the equioscillation theorem: when all the alternating values equal $\norm{f - p}_\infty$, it shows $p$ is best.
In the Remez algorithm it certifies progress, since each iteration produces a polynomial and a set of $n + 2$ alternation points, and the lower and upper bounds squeeze $E_n(f)$.
The same argument works for any Haar subspace of dimension $n + 1$.
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# Convex polytope

## Definition

Let $n \ge 1$.
A *convex polytope* in $\R^n$ is a set of the form
$$P = \mathrm{conv}\set{v_1, \ldots, v_m} = \set{\lambda_1 v_1 + \cdots + \lambda_m v_m : \lambda_i \ge 0, \ \lambda_1 + \cdots + \lambda_m = 1},$$
the [[convex hull|0FQSDC2EQ2]] of finitely many points $v_1, \ldots, v_m$ of the [[Euclidean space|0XPR1E85W0]] $\R^n$.
The *dimension* of $P$ is the dimension of its affine hull.

## Remarks

Example: the square $[0, 1]^2$ is $\mathrm{conv}\set{(0, 0), (1, 0), (0, 1), (1, 1)}$, since every point $(x, y)$ of the square is $(1 - x)(1 - y) (0, 0) + x (1 - y) (1, 0) + (1 - x) y (0, 1) + x y (1, 1)$, with nonnegative weights of sum $1$.
Non-example: the closed unit disc is compact and convex but not a convex polytope.
The convex hull of finitely many points has only finitely many extreme points, while every point of the unit circle is an extreme point of the disc.
Equivalently, a convex polytope is a bounded set of the form $\set{x \in \R^n : A x \le b}$ for a real $m \times n$ matrix $A$ and $b \in \R^m$; this equivalence is the Minkowski–Weyl theorem.
Many authors allow the empty set as a convex polytope, the hull of no points.
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# Dirac equation

## Summary

The Dirac equation is the relativistic wave equation for particles of spin one half, such as the electron.
It is a first-order system for a function with four complex components, and its coefficients are the Dirac matrices, which generate a [[Clifford algebra|A38B1QFK20]].
Each component of a solution solves the [[Klein–Gordon equation|AYCCDQWCJF]], so the Dirac equation is a kind of square root of it.
It predicted antimatter and explains the spin and magnetic moment of the electron.

## Explanations

### L1 human

The Schrödinger equation treats space and time differently, so it does not fit special relativity.
Dirac looked for an equation that is first order in time and in space, like a square root of the relativistic energy relation $E^2 = p^2 + m^2$.
Ordinary numbers cannot do this, so he used $4 \times 4$ matrices whose squares are $\pm 1$ and which anticommute.
The four components of the solution describe the two spin states of the particle and of its antiparticle.
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# Dehn twist

## Definition

Let $A = S^1 \times [0, 1]$ be the annulus, with $S^1 = \R / 2\pi\Z$ and the orientation of the $(\theta, t)$-plane.
The *twist map* $T \colon A \to A$ is $T(\theta, t) = (\theta + 2\pi t, t)$; it is the identity on both boundary circles.

Let $S$ be an oriented [[surface|VGJJVBZ1MQ]] and let $c$ be a simple closed curve in the interior of $S$.
Choose an orientation-preserving embedding $\varphi \colon A \to S$ with $\varphi(S^1 \times \set{1/2}) = c$.
The *Dehn twist* about $c$ is the [[homeomorphism|0QT00PB5G2]] $T_c \colon S \to S$ given by
$$T_c(x) = \begin{cases} \varphi \circ T \circ \varphi^{-1}(x) & \text{if } x \in \varphi(A), \\ x & \text{otherwise.} \end{cases}$$

## Remarks

The isotopy class of $T_c$ does not depend on the choice of $\varphi$.
It depends only on the isotopy class of $c$, and it is an element of the [[mapping class group|JV5NYG913V]] of $S$.
Its definition does not depend on an orientation of $c$, but it does depend on the orientation of $S$: reversing it replaces $T_c$ by its inverse.

Example: on the torus $T^2 = \R^2 / \Z^2$, let $c$ be the image of the horizontal line.
The Dehn twist $T_c$ acts on $H_1(T^2; \Z) = \Z^2$ by the matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, up to the sign convention.
Since this matrix has infinite order, $T_c$ has infinite order in the mapping class group.

Non-example: if $c$ bounds a disc in $S$, then $T_c$ is isotopic to the identity, so it is trivial in the mapping class group.
And on a non-orientable surface, a curve whose neighbourhood is a Möbius strip has no annulus neighbourhood, so no Dehn twist about it is defined.

## Explanations

### L1 human

Cut the surface along the curve, rotate one side through a full turn, and glue it back.
To make this continuous, the rotation is spread over a thin band, from no rotation on one edge to a full turn on the other.
Every curve that crosses the band picks up one extra lap around $c$.
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# Branch point

## Summary

A branch point of a [[multivalued function|03CPD1HGF3]] is a point with this property: going once around it along a small loop changes the value.
See the [[definition|XDNRXKH3KT]].
The square root has branch points at $0$ and $\infty$, and so does the [[complex logarithm|02QGGD5G87]].
Branch points are the reason for [[branch cuts|0W8QHVSN95]].
When the function is made single-valued on a [[Riemann surface|5K0N8QMV6Q]], they become the ramification points of a [[branched covering|YARENZRXTD]].

## Explanations

### L1 human

Follow $\sqrt{z}$ continuously while $z$ walks once around the circle $\abs{z} = 1$, starting at $z = 1$ with $\sqrt{1} = 1$.
When $z = e^{it}$ goes from $t = 0$ to $t = 2\pi$, the chosen root $e^{it/2}$ goes from $1$ to $-1$.
So you come back to the same $z$ with a different value of the root.
The point $0$ that you walked around is a branch point.
For $1/z$ nothing like this happens, because $1/z$ has only one value at each point.
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# Polynomial operation

## Summary

A polynomial operation of an algebra is a [[term operation|N9M2N27END]] in which some arguments are fixed to constant elements of the algebra.
For a commutative ring, the polynomial operations are the [[polynomial functions|1G0B73WPX8]] in several variables, which explains the name.
Polynomial operations describe the congruences of an algebra: an equivalence relation is a congruence exactly when it is preserved by all unary polynomial operations.

## Explanations

### L1 human

Term operations may use only the operations that the algebra comes with.
Polynomial operations may also use any element of the algebra as a fixed constant.
In the group $(\Z, +)$, the map $x \mapsto 2x$ is a term operation, while $x \mapsto 2x + 5$ is a polynomial operation that is not a term operation.
In a ring, $x \mapsto 3x^2 + 7$ is a polynomial operation, matching the usual polynomial.
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# Nuclear operator

## Summary

A nuclear operator is a bounded linear map between Banach spaces that can be written as an absolutely summable series of rank-one maps.
The notion, due to Grothendieck, extends the trace class operators of Hilbert spaces to Banach spaces.
Nuclear operators are compact, and they are the basis of Grothendieck's theory of nuclear spaces and of trace and determinant theory on Banach spaces.

## Explanations

### L1 human

The simplest operators are those of rank one: measure a vector by a linear functional, then multiply a fixed vector by the result.
A nuclear operator is an infinite sum of such pieces whose sizes add up to a finite number.
So it is a limit of matrices in a strong, summable sense.
For example, on square-summable sequences, multiplying the $n$-th coordinate by $2^{-n}$ is nuclear, since it is the sum of the rank-one pieces $x \mapsto 2^{-n} x_n e_n$ of sizes $2^{-n}$.
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# Nesbitt's inequality

## Statement

Let $a, b, c$ be positive real numbers.
Then
$$\frac{a}{b + c} + \frac{b}{c + a} + \frac{c}{a + b} \ge \frac{3}{2},$$
with equality if and only if $a = b = c$.

## Remarks

Proof idea: let $s = a + b + c$.
Adding $1$ to each fraction gives $\frac{a}{b+c} + 1 = \frac{s}{b+c}$, so the left side equals $s \big( \frac{1}{b+c} + \frac{1}{c+a} + \frac{1}{a+b} \big) - 3$.
By the [[Cauchy–Schwarz inequality|0RFWW56Y22]], or the inequality between arithmetic and harmonic means, $\big((b+c) + (c+a) + (a+b)\big)\big(\frac{1}{b+c} + \frac{1}{c+a} + \frac{1}{a+b}\big) \ge 9$.
Since the first factor is $2s$, the left side is at least $\frac{9}{2} - 3 = \frac{3}{2}$.

Example: $a = b = 1$, $c = 2$ gives $\frac{1}{3} + \frac{1}{3} + 1 = \frac{5}{3} \ge \frac{3}{2}$.

There is no upper bound: for $a = b = 1$ and $c \to \infty$ the left side tends to $\infty$.

## Explanations

### L1 human

Divide each of three positive numbers by the sum of the other two and add the results.
You always get at least $\frac{3}{2}$, and you get exactly $\frac{3}{2}$ only when the three numbers are equal.

### L2 human

Nesbitt's inequality is a standard first example of a symmetric inequality in three variables.
It has many proofs: by the means inequalities, by the [[rearrangement inequality|6E9TAF8HCY]], or by clearing denominators and using [[Muirhead's inequality|PBPFVY20WT]].
Comparing the proofs is a good way to learn the toolbox of elementary inequalities.
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# Unimodular group

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] with [[modular function|63TX60RTRQ]] $\Delta \colon G \to (0, \infty)$.
The group $G$ is *unimodular* if $\Delta(g) = 1$ for every $g \in G$.
Equivalently, every left [[Haar measure|VY0HXD3GC4]] $\mu$ on $G$ is also right-invariant: $\mu(Ag) = \mu(A)$ for every $g \in G$ and every Borel set $A$.

## Remarks

Example: every compact group $K$ is unimodular.
The modular function is a continuous homomorphism, so $\Delta(K)$ is a compact subgroup of $(0, \infty)$.
The only compact subgroup of $(0, \infty)$ is $\set{1}$, so $\Delta \equiv 1$.

Non-example: the group of maps $x \mapsto ax + b$ with $a > 0$ and $b \in \R$ has modular function $\Delta(a, b) = 1/a$, which is not identically $1$.

Every abelian group is unimodular, since $Ag = gA$ for all $g$.
Every discrete group is unimodular, since counting measure is invariant on both sides.
On a unimodular group, $\int_G f(x^{-1}) \, d\mu(x) = \int_G f(x) \, d\mu(x)$ for every integrable $f$.
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# Contraction semigroup

## Summary

A contraction semigroup is a [[strongly continuous semigroup|EVF180QJXY]] $(T(t))_{t \ge 0}$ of operators on a Banach space with $\norm{T(t)} \le 1$ for every $t \ge 0$.
It describes a linear evolution that never increases the norm of the state.
The [[Lumer–Phillips theorem|Z2N8MPD8DH]] says that its generators are exactly the densely defined m-dissipative operators.
Heat flows, diffusions and unitary groups restricted to $t \ge 0$ give contraction semigroups.

## Explanations

### L1 human

A semigroup moves a state forward in time: $T(t)x$ is the state at time $t$ when the start is $x$.
For a contraction semigroup, the size of the state can only shrink or stay the same.
The heat semigroup is an example: the total squared temperature of a rod with ends kept at zero goes down with time.
On the real line, $T(t)x = e^{-t}x$ is a contraction semigroup, while $T(t)x = e^{t}x$ is not.
Every strongly continuous semigroup with $\norm{T(t)} \le M e^{\omega t}$ becomes a contraction semigroup after rescaling and a change of norm, which is why contractions are a central case.
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# Fredholm's theorem

## Statement

Let $a < b$ and let $K \in L^2([a, b] \times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]].
Let $K^*(x, y) = \overline{K(y, x)}$ be the adjoint kernel, and let $\mu \in \C$.
Consider, in $L^2[a, b]$, the [[Fredholm equation of the second kind|DEYYNVCV9J]] and its [[adjoint equation|XEZXGBT42J]]
$$\varphi(x) - \mu \int_a^b K(x, y) \varphi(y) \, dy = f(x), \qquad \psi(x) - \overline{\mu} \int_a^b K^*(x, y) \psi(y) \, dy = g(x).$$
Then:

1. Either the first equation has exactly one solution $\varphi \in L^2[a, b]$ for every $f \in L^2[a, b]$, or the [[homogeneous equation|W6CYS54QXS]] with $f = 0$ has a nonzero solution.
2. The homogeneous equation ($f = 0$) and the homogeneous adjoint equation ($g = 0$) have the same finite number of linearly independent solutions.
3. Assume the homogeneous equation has a nonzero solution. Then the first equation is solvable if and only if $\int_a^b f \overline{\psi} = 0$ for every solution $\psi$ of the homogeneous adjoint equation.
4. The set of [[characteristic values|DC7HMHVREP]] of $K$ has no accumulation point in $\C$. These are the $\mu$ for which the homogeneous equation has a nonzero solution.

## Remarks

Let $T$ be the integral operator with kernel $K$; it is a [[compact operator|M2NEP1K5AE]] on $L^2[a, b]$, and its adjoint is the integral operator with kernel $K^*$.
For $\mu = 0$ the statements are trivial, and for $\mu \ne 0$ they are statements about $I - \mu T$ with $\mu T$ compact.
One proof approximates $K$ in $L^2$ by a degenerate kernel $K_0$ with $\abs{\mu} \norm{K - K_0} < 1$, with operator $T_0$.
Then $I - \mu (T - T_0)$ is invertible by a Neumann series, and the equation reduces to a finite linear system.
The [[Fredholm alternative|4HH8C9V8JQ]] for matrices applies to that system.

Example: on $[0, 1]$ with $K(x, y) = xy$ and $\mu = 3$, the homogeneous equation has the solution $\varphi(x) = x$, and $K^* = K$.
The equation $\varphi(x) - 3 \int_0^1 x y \varphi(y) \, dy = f(x)$ is solvable exactly when $\int_0^1 f(x) \, x \, dx = 0$; for instance $f = 1$ fails this, and indeed integrating $\varphi - 3x \int_0^1 y \varphi = 1$ against $x$ gives the contradiction $0 = 1/2$.

The square-integrability of $K$ matters because it makes $T$ compact.
For the identity operator on $L^2[a, b]$, which is bounded but not compact and not given by a square-integrable kernel, the equation $\varphi - \varphi = 0$ has infinitely many independent solutions, against part 2.

## Explanations

### L1 human

An integral equation of the second kind behaves like a square system of linear equations.
Either it always has exactly one solution, or its homogeneous version has nonzero solutions.
In the second case, a right-hand side is allowed exactly when it is perpendicular to all solutions of the adjoint homogeneous equation, just as for matrices.

### L2 human

These are Fredholm's results, obtained first with his determinants for continuous kernels and later extended to square-integrable kernels.
In the language of operators, they say that $I - \mu T$ is a [[Fredholm operator|8HWGKWD6ZM]] of index $0$ when $T$ is compact.
They also say that its image is the orthogonal complement of the kernel of its adjoint.
Part 4 is the statement that the nonzero spectrum of a compact operator is discrete, as in the [[spectral theory of compact operators|JEPDM5B5FC]].
They are the reason why boundary value problems that reduce to integral equations of the second kind have a solvability theory as clean as linear algebra.
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# Hyperbolic partial differential equation

## Definition

Let $U \subseteq \R^n$ be open, let $T > 0$, and write the variables in $U \times (0, T)$ as $(x, t)$.
Consider a [[linear PDE|6DZETZ5T9G]] in $U \times (0, T)$ of the following form.
$$\partial_t^2 u - \sum_{i,j=1}^{n} a_{ij}(x, t) \, \partial_i \partial_j u + \sum_{i=1}^{n} b_i(x, t) \, \partial_i u + b_0(x, t) \, \partial_t u + c(x, t) \, u = f(x, t).$$
Here all coefficients are real and $a_{ij} = a_{ji}$.
The equation is a *second-order linear hyperbolic equation* if its spatial part is [[uniformly elliptic|B4TTNHSTRX]].
That is, there is $\theta > 0$ with $\sum_{i,j} a_{ij}(x, t) \xi_i \xi_j \ge \theta \abs{\xi}^2$ for all $(x, t)$ and all $\xi \in \R^n$.

More generally, let $P$ be a linear operator of order $m$ in the variables $(x, t)$, with [[principal symbol|F31T650DKY]] $\sigma_m$.
It is *hyperbolic with respect to the time direction* at a point $y$ if both conditions hold:

1. $\sigma_m(y, (0, \dots, 0, 1)) \ne 0$;
2. for every $\xi \in \R^n$ the polynomial $\tau \mapsto \sigma_m(y, (\xi, \tau))$ has only real roots.

It is *strictly hyperbolic* if, in addition, these $m$ roots are distinct for every $\xi \ne 0$.

## Remarks

Example: the [[wave equation|5EHD2XBQDG]] $\partial_t^2 u - c^2 \Delta u = 0$ with $c > 0$ is hyperbolic with $a_{ij} = c^2 \delta_{ij}$; its symbol $\tau^2 - c^2 \abs{\xi}^2$ has the distinct real roots $\tau = \pm c \abs{\xi}$ for $\xi \ne 0$.
Non-example: the equation $\partial_t^2 u + \Delta u = 0$ is not hyperbolic; its symbol $\tau^2 + \abs{\xi}^2$ has the non-real roots $\tau = \pm i \abs{\xi}$, and the equation is elliptic.
A first-order system $\partial_t u + \sum_j A_j(x, t) \partial_j u = 0$ is called hyperbolic when every matrix $\sum_j \xi_j A_j$ with $\xi \in \R^n$ has real eigenvalues and a basis of eigenvectors; see also [[strict hyperbolicity|P3VXR8FW17]].

## Explanations

### L1 human

A hyperbolic equation is built like the wave equation: a second time derivative minus a uniformly elliptic operator in space.
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# Intuitionism

## Summary

Intuitionism is the view, founded by L. E. J. Brouwer, that mathematics is a construction of the human mind.
A mathematical object exists only when it can be constructed, and a statement is true only when it has a proof.
As a result, intuitionists do not accept the law of excluded middle in general.
Its logic, [[intuitionistic logic|Q11N8X99FK]], is a central example of [[constructivism|G958EQ8CWW]].

## Explanations

### L1 human

For an intuitionist, saying "there is a number with this property" means "I can show you such a number".
Proving that "no such number" leads to a contradiction is not enough.
Take a real number $x$ given by an infinite list of digits.
The statement "either $x = 0$ or $x \ne 0$" cannot always be settled by a finite check, so an intuitionist does not assert it.
This makes intuitionistic mathematics more careful, and every intuitionistic proof contains a method of construction.
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# Hochschild homology

## Definition

Let $k$ be a commutative ring and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$ with identity.
Let $M$ be an $A$-[[bimodule|EM7FHAFD3N]] on which $k$ acts the same way on both sides.
All [[tensor products|F0HQXQ0HCD]] are over $k$.
The *Hochschild complex* is the [[chain complex|WQ82C0M3QB]] with $C_n(A, M) = M \otimes A^{\otimes n}$ for $n \ge 0$ and boundary $b \colon C_n \to C_{n-1}$ given by
$$b(m \otimes a_1 \otimes \cdots \otimes a_n) = m a_1 \otimes a_2 \otimes \cdots \otimes a_n + \sum_{i=1}^{n-1} (-1)^i m \otimes a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n + (-1)^n a_n m \otimes a_1 \otimes \cdots \otimes a_{n-1}.$$
The *Hochschild homology* of $A$ with coefficients in $M$ is $HH_n(A, M) = H_n(C_\bullet(A, M), b)$.
One writes $HH_n(A) = HH_n(A, A)$.

## Remarks

One checks that $b \circ b = 0$, so the homology is defined.
In degree $0$, $b(m \otimes a) = ma - am$, so $HH_0(A, M)$ is the quotient of $M$ by the $k$-submodule spanned by all $ma - am$, and $HH_0(A) = A/[A, A]$.
Example: for $A = M_n(k)$ with $k$ a field, $[A, A]$ is the space of trace-zero matrices, so the [[trace|0XC1RDBY7A]] gives $HH_0(A) \cong k$.
If $A$ is projective as a $k$-module, then $HH_n(A, M) \cong \operatorname{Tor}_n^{A^e}(A, M)$, where $A^e = A \otimes A^{\mathrm{op}}$ uses the [[opposite ring|M8Q24XB0AF]]; see the [[Tor functor|DDQRERAQCV]].
Without such a flatness hypothesis the complex still makes sense, but this description by Tor can fail.
Hochschild cohomology $HH^n(A, M)$ is defined dually from $\Hom_k(A^{\otimes n}, M)$, and it equals $\operatorname{Ext}^n_{A^e}(A, M)$ when $A$ is projective over $k$.
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# Goodstein's theorem

## Statement

Let $b \ge 2$ be an integer.
The *hereditary base-$b$ notation* of $n \ge 1$ writes $n$ as a sum $b^{e_1} c_1 + \cdots + b^{e_r} c_r$ with $e_1 > \cdots > e_r$ and $0 < c_i < b$.
Then it writes each exponent $e_i$ in hereditary base-$b$ notation again.
Let $B_b(n)$ be the number obtained by replacing every $b$ in this notation by $b + 1$, with $B_b(0) = 0$.
The *Goodstein sequence* of $m \in \N$ is $G_1 = m$ and $G_{k+1} = B_{k+1}(G_k) - 1$ while $G_k > 0$.

1. For every $m \in \N$ the Goodstein sequence of $m$ reaches $0$.
2. Statement 1, written as a sentence of first-order arithmetic, is not provable in [[Peano arithmetic|0QFT7Z76N5]] $\mathsf{PA}$.

## Remarks

Proof idea for 1: replace each base in the hereditary notation of $G_k$ by the [[ordinal|0YSCSX5YZC]] $\omega$.
This gives an ordinal $o_k < \varepsilon_0$, and the change of base does not change it while subtracting $1$ makes it strictly smaller.
A strictly decreasing sequence of ordinals is finite, so the sequence reaches $0$.
Goodstein proved part 1 in 1944; Kirby and Paris proved part 2 in 1982, by showing that $\mathsf{PA}$ plus statement 1 proves the consistency of $\mathsf{PA}$.

Example: for $m = 3$, the sequence is $3 = 2 + 1$, then $3 + 1 - 1 = 3$, then in base $3$, $3 = 3^1$ becomes $4 - 1 = 3$, then $2$, $1$, $0$.
For $m = 4$ the sequence also reaches $0$, but only after more than $10^{100{,}000{,}000}$ steps.

## Explanations

### L1 human

Start with a number and write it in base $2$, using only $2$s even in the exponents.
Change all $2$s to $3$s and subtract $1$; then repeat with bases $3$ and $4$, and so on.
The numbers first grow enormously, yet the theorem says they always come back to $0$.
The proof uses infinite ordinals, and in fact the theorem cannot be proved with the usual axioms of arithmetic alone.

### L2 human

The theorem shows how ordinals below $\varepsilon_0$ measure the strength of $\mathsf{PA}$: transfinite induction up to [[epsilon nought|KT50SMH3EC]] proves it, while $\mathsf{PA}$ does not.
This ties it to [[Gentzen's consistency proof|QPV0HNP22S]] and to [[ordinal analysis|C5E5TG8PGP]].
Like the [[Paris–Harrington theorem|509ZWA0R26]], it is a natural true statement independent of $\mathsf{PA}$.
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# Pseudo-differential operator

## Definition

Let $n \ge 1$ and $m \in \R$, and let $a \in S^m(\R^n \times \R^n)$ be a symbol of order $m$ in the [[symbol class|AM4RNWVV4Q]] $S^m = S^m_{1,0}$.
For $u$ in the [[Schwartz space|CG9JEATW3Q]] $\mathcal{S}(\R^n)$, let $\hat{u}(\xi) = \int_{\R^n} e^{-i x \cdot \xi} u(x) \, dx$ be its [[Fourier transform|DPVM34ADZV]].
The *pseudo-differential operator* with symbol $a$ is the operator $\mathrm{Op}(a) \colon \mathcal{S}(\R^n) \to \mathcal{S}(\R^n)$ given by
$$\mathrm{Op}(a) u(x) = \frac{1}{(2\pi)^n} \int_{\R^n} e^{i x \cdot \xi} a(x, \xi) \, \hat{u}(\xi) \, d\xi.$$
An operator of this form is a pseudo-differential operator *of order $m$*, and the set of them is denoted $\Psi^m(\R^n)$.

## Remarks

Example: let $P = \sum_{\abs{\alpha} \le k} a_\alpha(x) D^\alpha$ with $D^\alpha = (-i)^{\abs{\alpha}} \partial^\alpha$ and coefficients $a_\alpha$ smooth and bounded with all their derivatives.
By Fourier inversion, $D^\alpha u(x) = (2\pi)^{-n} \int e^{i x \cdot \xi} \xi^\alpha \hat{u}(\xi) \, d\xi$, so $P = \mathrm{Op}(p)$ with $p(x, \xi) = \sum_\alpha a_\alpha(x) \xi^\alpha \in S^k$.

Non-example: the operator $u \mapsto \mathcal{F}^{-1}\big( e^{i\abs{\xi}^2} \hat{u} \big)$, a Schrödinger evolution operator, is not a pseudo-differential operator of any order.
The symbol of an operator of this form is determined by the operator, here $e^{i \abs{\xi}^2}$, and it is in no class $S^m$.

The composition of $\mathrm{Op}(a)$ with $a \in S^m$ and $\mathrm{Op}(b)$ with $b \in S^{m'}$ is $\mathrm{Op}(c)$ with $c \in S^{m + m'}$, and $c - ab \in S^{m + m' - 1}$.
Operators of order $0$ are bounded on $L^2(\R^n)$, and operators of order $-\infty$ are smoothing.
The same formula with a symbol in $S^m_{\rho, \delta}$ defines the larger classes $\Psi^m_{\rho, \delta}$.
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# Montessus de Ballore's theorem

## Statement

Let $R > 0$ and let $n \ge 1$ be an integer.
Let $f$ be a [[meromorphic function|0ZZ7ZYDR8W]] on the disc $D = \set{z \in \C : \abs{z} < R}$ that is analytic at $0$ and has exactly $n$ [[poles|066E7PJCYE]] in $D$, counted with multiplicity.
Let $P$ be the set of these poles.
Let $[m/n]_f$ be the [[Padé approximant|1F16G4V64X]] of $f$ at $0$ with numerator degree at most $m$ and denominator degree at most $n$.
Then:

1. as $m \to \infty$, $[m/n]_f$ converges to $f$ [[uniformly|08VZ57PT26]] on every compact subset of $D \setminus P$;
2. for $m$ large, $[m/n]_f$ has exactly $n$ poles counted with multiplicity, and as $m \to \infty$ they converge to the poles of $f$, each with its multiplicity.

## Remarks

The idea of the proof: let $Q(z) = \prod_{p \in P} (z - p)$, so that $Qf$ is analytic on $D$.
The denominators $q_m$ of $[m/n]_f$ are determined by $n$ linear conditions on the Taylor coefficients of $f$ of high order.
The asymptotics of these coefficients are dominated by the poles in $P$, and one shows that $q_m \to Q$ after normalization, at a geometric rate.
Then $q_m f - p_m$ is small by construction, and dividing by $q_m$ gives the convergence away from $P$.
Example: $f(z) = e^z / (1 - z)$ has exactly one pole in every disc of radius $R > 1$.
So the column $[m/1]_f$ converges to $f$ on compact subsets of $\C \setminus \set{1}$, and the pole of $[m/1]_f$ tends to $1$.
The hypothesis that the number of poles equals $n$ matters.
With fewer poles the approximants may have spurious poles, and with more poles a denominator of degree $n$ cannot capture them all.
The theorem was proved by Robert de Montessus de Ballore in 1902.

## Explanations

### L1 human

A Taylor polynomial cannot see beyond the first pole of a function: it converges only in a disc that stops there.
Suppose the function has exactly $n$ poles in a larger disc, and the approximations may have $n$ poles.
Then the Padé approximants find those poles and converge in the whole larger disc.
So rational approximation extends the reach of power series past their singularities.

### L2 human

This is the basic convergence theorem for columns of the [[Padé table|MDM8F48X18]], where the denominator degree is fixed and the numerator degree grows.
It gives a practical way to locate the poles of a function from its Taylor coefficients, as the limits of the poles of the approximants.
For diagonal sequences $[n/n]_f$, where both degrees grow, convergence is much harder; uniform convergence can fail, which is the subject of the [[Baker–Gammel–Wills conjecture|YPTNBRWRY7]].
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# Laplace transform method for convolution equations

## Definition

Let $k$ and $f$ be locally integrable functions on $[0, \infty)$ whose [[Laplace transforms|0V49JEZW06]] $\hat k(p) = \int_0^\infty e^{-pt} k(t) \, dt$ and $\hat f(p)$ converge absolutely for $\operatorname{Re} p > c$, for some real $c$.
Consider the [[Volterra equation of convolution type|SNFYTC1TTA]]
$$x(t) = f(t) + \int_0^t k(t - s) \, x(s) \, ds, \quad t \ge 0.$$
The *Laplace transform method* for this equation consists of the following steps.

1. Assume that the solution $x$ has a Laplace transform $\hat x$, and apply the [[convolution theorem|0JNMJEDD93]] to get $\hat x(p) = \hat f(p) + \hat k(p) \, \hat x(p)$.
2. Solve: $\hat x(p) = \dfrac{\hat f(p)}{1 - \hat k(p)}$ for $\operatorname{Re} p$ large enough that $\hat k(p) \ne 1$.
3. Recover $x$ as the [[inverse Laplace transform|07WBTX1MHA]] of $\hat x$, and check that it solves the equation.

For an equation of the first kind $\int_0^t k(t - s) \, x(s) \, ds = f(t)$, step 2 becomes $\hat x = \hat f / \hat k$.

## Remarks

Example: $x(t) = t - \int_0^t (t - s) \, x(s) \, ds$, with $k(u) = -u$ and $f(t) = t$.
Here $\hat k(p) = -1/p^2$ and $\hat f(p) = 1/p^2$, so $\hat x(p) = \frac{1/p^2}{1 + 1/p^2} = \frac{1}{p^2 + 1}$ and $x = \sin$.
Check: $\int_0^t (t - s) \sin s \, ds = t - \sin t$, so the right side is $\sin t$.
Non-example: the method does not apply directly to $x(t) = 1 + \int_0^t ts \, x(s) \, ds$, since the kernel $ts$ is not a function of $t - s$, and the transform of the integral term is not a product.
The step 1 assumption is harmless when $k$ and $f$ are as above: the unique locally integrable solution then has a Laplace transform for $\operatorname{Re} p$ large.
The function $r$ with $\hat r = \hat k / (1 - \hat k)$ is the resolvent kernel, and $x = f + r * f$.
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# Ideal completion

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
An *order ideal* of $P$ is a subset $I \subseteq P$ such that:

1. $I$ is nonempty;
2. $I$ is a down-set: if $x \in I$ and $y \le x$, then $y \in I$;
3. $I$ is [[directed|D9RY4667K6]]: for all $x, y \in I$ there is $z \in I$ with $x \le z$ and $y \le z$.

The *ideal completion* of $P$ is the set $\operatorname{Idl}(P)$ of all order ideals of $P$, ordered by inclusion.
It comes with the map $x \mapsto {\downarrow} x = \set{y \in P : y \le x}$ from $P$ to $\operatorname{Idl}(P)$.

## Remarks

$\operatorname{Idl}(P)$ is a [[directed-complete partial order|7TSXP95QN6]]: the supremum of a directed family of ideals is its union.
The map $x \mapsto {\downarrow} x$ is an order embedding, and the principal ideals ${\downarrow} x$ are exactly the [[compact elements|9F7TYES4KE]] of $\operatorname{Idl}(P)$, so $\operatorname{Idl}(P)$ is an algebraic dcpo.
When $P$ is a lattice, the order ideals are its [[lattice ideals|5GW2T2AM1A]] and $\operatorname{Idl}(P)$ is an algebraic lattice if $P$ has a least element.
Example: for $P = \N$, the ideals are the sets $\set{0, \ldots, n}$ and $\N$ itself, so $\operatorname{Idl}(\N) \cong \N \cup \set{\infty}$.
Non-example: in $\N$, the set $\set{0, 2}$ is not an ideal, because $1 \le 2$ but $1 \notin \set{0, 2}$.
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# Hodge index theorem

## Statement

Let $S$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field.
For [[divisors|Z9DN23DAE2]] $D$ and $E$ on $S$, let $D \cdot E$ be their [[intersection number|4HV4WTAX9P]].
Let $H$ be an [[ample divisor|RRK8P6TXG2]] on $S$, and let $D$ be a divisor with $D \cdot H = 0$.
Then $D^2 \le 0$, and $D^2 = 0$ holds if and only if $D \cdot E = 0$ for every divisor $E$ on $S$.

## Remarks

Equivalently, the intersection form on the real vector space $\operatorname{NS}(S) \otimes \R$, where $\operatorname{NS}(S)$ is the [[Néron–Severi group|XMZ09SBM0X]], has [[signature|07EZA80NEH]] $(1, \rho - 1)$, where $\rho$ is its dimension.
The proof uses Riemann–Roch on the surface, which shows that if $D^2 > 0$, then $nD$ or $-nD$ is linearly equivalent to an effective divisor for some $n \ge 1$.
A nonzero effective divisor meets the ample $H$ positively, which contradicts $D \cdot H = 0$.
A useful consequence is the inequality $(D^2)(H^2) \le (D \cdot H)^2$ for $H$ ample and $D$ arbitrary, obtained by applying the theorem to $(H^2) D - (D \cdot H) H$.
Example: on $S = \mathbb{P}^1 \times \mathbb{P}^1$ with fibre classes $F_1, F_2$, one has $F_1^2 = F_2^2 = 0$ and $F_1 \cdot F_2 = 1$; the ample $H = F_1 + F_2$ is orthogonal to $D = F_1 - F_2$, and indeed $D^2 = -2 < 0$.
Over $\C$, Hodge theory gives the more general fact that the cup product on $H^2(S, \R)$ has signature $(2 p_g + 1, h^{1,1} - 1)$, where $p_g = h^{2,0}$.

## Explanations

### L1 human

On a surface, curves can be intersected and their intersection numbers form a quadratic form.
The theorem says that this form has only one positive direction, given by an ample curve, and is negative on everything orthogonal to it.
So its shape is like the light cone in special relativity, with one time direction and many space directions.

### L2 human

The Hodge index theorem is the main structural fact about intersection theory on surfaces.
It is used in a proof of the Riemann hypothesis for curves over finite fields, applied to the graph of Frobenius on a product of curves.
It is also used in the study of the Néron–Severi lattice of K3 surfaces and in many inequalities between Chern numbers.
Its higher-dimensional analogues are the Hodge–Riemann bilinear relations and the hard Lefschetz theorem.
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# Finiteness of associated primes

## Statement

Let $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $M$ be a [[finitely generated|0XHM30WBVB]] $R$-module.
Then the set $\operatorname{Ass}_R(M)$ of [[associated primes|1RJJ33KKSS]] of $M$ is finite.

## Remarks

The proof builds a chain of submodules $0 = M_0 \subsetneq M_1 \subsetneq \cdots \subsetneq M_n = M$ with $M_i / M_{i-1} \cong R/P_i$ for prime ideals $P_i$.
Each step picks an associated prime of $M/M_{i-1}$, which exists when this module is nonzero, and the chain stops because $M$ is a [[Noetherian module|AK1ZRCXR32]].
Then $\operatorname{Ass}_R(M) \subseteq \set{P_1, \dots, P_n}$, because associated primes of an extension lie among those of the two pieces.
For example, $\operatorname{Ass}_\Z(\Z/n\Z)$ is the finite set of ideals $(p)$ for the primes $p$ dividing $n \ge 2$.
Finite generation is needed: the $\Z$-module $\bigoplus_p \Z/p\Z$, over all primes $p$, has every $(p)$ as an associated prime.

## Explanations

### L1 human

A finitely generated module over a Noetherian ring has only finitely many associated primes.
So it is "supported" by finitely many basic primes, like an integer with finitely many prime factors.

### L2 human

Associated primes of a nonzero module also exist, so $\operatorname{Ass}_R(M)$ is finite and nonempty when $M \ne 0$.
With the result that [[zero divisors are the union of associated primes|S37DQGW3TZ]], this lets one use [[prime avoidance|FMP5AWG6PR]] to find non-zero-divisors outside finitely many primes.
This is a key step in building regular sequences and in the theory of depth.
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# Pseudovariety

## Summary

A pseudovariety is a class of finite algebras of one signature that is closed under homomorphic images, subalgebras and finite direct products.
It is the analogue of a [[variety|AJNZJD8442]] for finite algebras.
Pseudovarieties are not always defined by identities, but [[Reiterman's theorem|NT8W7R2KDR]] shows that they are defined by pseudoidentities.
Pseudovarieties of finite semigroups and monoids classify regular languages in algebraic automata theory.

## Explanations

### L1 human

Start from some finite algebras and allow three moves: take a quotient, take a subalgebra, or take the product of finitely many.
A class that is stable under these moves is a pseudovariety.
Finite groups, finite abelian groups and finite nilpotent groups are examples.
Finite groups regarded as semigroups also form a pseudovariety, yet no set of semigroup identities captures exactly the finite groups among finite semigroups.
The reason is that the infinite semigroup $(\set{1, 2, 3, \ldots}, +)$ lies in every variety of semigroups containing all finite groups, and it has finite quotients that are not groups.
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<document source="https://tamnd.github.io/math2/node/XRBD5BT2T9.md" tag="XRBD5BT2T9" kind="statement">
# Schwarz reflection principle for harmonic functions

## Statement

Let $n \ge 2$ and write points of $\R^n$ as $x = (x', x_n)$ with $x' \in \R^{n-1}$ and $x_n \in \R$.
Let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]] that is symmetric in the hyperplane $x_n = 0$: if $(x', x_n) \in \Omega$, then $(x', -x_n) \in \Omega$.
Let $\Omega^+ = \set{x \in \Omega : x_n > 0}$ and $\sigma = \set{x \in \Omega : x_n = 0}$.
Let $u$ be a [[continuous function|0X3WGX95FE]] on $\Omega^+ \cup \sigma$ that is [[harmonic|0DEB0JMN0W]] on $\Omega^+$ and equal to $0$ on $\sigma$.
Define $U(x', x_n) = u(x', x_n)$ for $x_n \ge 0$ and $U(x', x_n) = -u(x', -x_n)$ for $x_n < 0$.
Then $U$ is harmonic on $\Omega$.

## Remarks

Proof idea: $U$ is continuous on $\Omega$ and harmonic off $\sigma$, so only points of $\sigma$ need care.
Take a closed ball centred at a point of $\sigma$ inside $\Omega$, and let $v$ be the Poisson integral of $U$ over its boundary sphere.
The boundary data are odd in $x_n$, so $v$ is odd in $x_n$ and vanishes on the hyperplane.
On each half ball, $U - v$ is harmonic and has boundary values $0$, so $U = v$ by the maximum principle.
Example: $u(x) = x_n$ on the upper half-space vanishes on $x_n = 0$, and its reflection is $U(x) = x_n$ on all of $\R^n$, which is harmonic.
The hypothesis $u = 0$ on $\sigma$ is needed: for $u \equiv 1$ the odd reflection jumps from $1$ to $-1$ across the hyperplane and is not continuous.
For the plane, this principle and the [[Schwarz reflection principle|P0R6YEGNB5]] for holomorphic functions imply each other through harmonic conjugates.

## Explanations

### L1 human

A harmonic function on one side of a flat wall that is zero on the wall can be continued through the wall.
On the other side it takes the mirror value with the opposite sign.
The two pieces fit together into one harmonic function.

### L2 human

The reflection principle is a tool to extend harmonic functions across flat pieces of the boundary.
It shows, for example, that a harmonic function on a half-ball that vanishes on the flat part is real analytic up to that flat part.
Combined with the Kelvin transform, the same idea works for reflection in a sphere.
The method of images for half-spaces rests on the same odd reflection.
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<document source="https://tamnd.github.io/math2/node/XRBPS2QYE3.md" tag="XRBPS2QYE3" kind="conjecture">
# Graph reconstruction conjecture

## Statement

For a finite [[graph|0JD5C82QMB]] $G$ and a vertex $v$ of $G$, let $G - v$ be the [[induced subgraph|0V05FC4GKQ]] on the vertices other than $v$.
The deck of $G$ is the multiset of the [[isomorphism|00CT0PTH74]] classes of the graphs $G - v$, one for each vertex $v$.
Conjecture: if $G$ and $H$ are finite graphs with at least $3$ vertices and the same deck, then $G$ and $H$ are isomorphic.

## Remarks

The bound on the number of vertices is needed.
The graph with $2$ vertices and one edge and the graph with $2$ vertices and no edge have the same deck, made of two single vertices.
The conjecture is due to Kelly and Ulam, and it is open.
Many invariants can be read off the deck, for example the number of edges and the degree sequence.
Several classes of graphs are known to be reconstructible from their decks, among them regular graphs, disconnected graphs and trees.
Computer searches have confirmed the conjecture for all graphs with few vertices.

## Explanations

### L1 human

Take a graph and make one copy of it for each vertex, with that vertex and its edges erased.
Shuffle the copies and forget which vertex was erased in each.
The conjecture says that, as soon as there are at least three vertices, the shuffled pile of copies always determines the original graph.
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<document source="https://tamnd.github.io/math2/node/XRGSKJVF0M.md" tag="XRGSKJVF0M" kind="definition">
# Descending chain condition

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
The poset $P$ satisfies the *descending chain condition* when every [[sequence|0G9EKM0E15]] $x_1 \ge x_2 \ge x_3 \ge \cdots$ in $P$ is eventually constant.
That is, there is an index $n$ such that $x_m = x_n$ for all $m \ge n$.

## Remarks

Example: the ideals of the ring $\Z / 12\Z$, ordered by inclusion, satisfy the descending chain condition.
Check: there are only six ideals, so a decreasing sequence of ideals takes finitely many values and must be eventually constant.
Rings whose ideals satisfy the condition are the [[Artinian rings|20QEAM8AX2]].

Non-example: the ideals of $\Z$ do not satisfy it, since $2\Z \supsetneq 4\Z \supsetneq 8\Z \supsetneq \cdots$ never stops.

The condition is the [[ascending chain condition|0GZ6Y1H4RN]] for the opposite order.
A poset satisfies it if and only if every nonempty subset has a minimal element, that is, the order is [[well-founded|55WD0H03CZ]].
The direction from the descending chain condition to minimal elements uses the axiom of dependent choice.
The abbreviation DCC is common, and the minimal-element form is called the minimal condition.
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<document source="https://tamnd.github.io/math2/node/XS0Q09SV2T.md" tag="XS0Q09SV2T" kind="definition">
# Method of fundamental solutions

## Definition

Let $\Omega \subset \R^d$ be a bounded domain and $L$ a linear elliptic differential operator with constant coefficients.
Let $G$ be a [[fundamental solution|K2NK9A8M5F]] of $L$, so that $L G = \delta$; it is smooth on $\R^d \setminus \set{0}$.
Consider $L u = 0$ in $\Omega$ with the condition $u = g$ on $\partial \Omega$.
The *method of fundamental solutions* has three ingredients:

1. source points $y_1, \ldots, y_n$ in $\R^d \setminus \overline{\Omega}$;
2. the trial function $u_n(x) = \sum_{j=1}^{n} c_j \, G(x - y_j)$, which satisfies $L u_n = 0$ in $\Omega$;
3. boundary points $x_1, \ldots, x_m \in \partial \Omega$, with $m \ge n$.

The coefficients $c_j$ solve $u_n(x_i) = g(x_i)$ for all $i$, exactly if $m = n$ and in the least squares sense if $m > n$.

## Remarks

The sources lie outside $\overline{\Omega}$, so each $G(\cdot - y_j)$ is smooth on $\overline{\Omega}$ and solves the equation there.
The error comes only from the boundary condition.
The sources are often placed on a curve or surface surrounding $\Omega$ at some distance.
The linear system becomes very ill-conditioned as $n$ grows, which limits the accuracy reached in floating point arithmetic.

Example: the [[Dirichlet problem|8D56GQ4AG8]] for the Laplace equation $\Delta u = 0$, with $\Delta$ the [[Laplacian|0E0B6RA8K0]], in the unit disk of $\R^2$.
Take $L = -\Delta$, with fundamental solution $G(x) = -\frac{1}{2\pi} \ln \abs{x}$.
Take sources $y_j = R (\cos \theta_j, \sin \theta_j)$ with $\theta_j = 2\pi j / n$ and $R > 1$.
Take as many boundary points $x_k = (\cos \alpha_k, \sin \alpha_k)$, with $\alpha_k = 2\pi (k - \tfrac12) / n$.
Every $u_n$ is harmonic in the disk, and the $c_j$ are fixed by matching $g$ at the $x_k$.

Non-example: with a source point inside $\Omega$, clause 1 fails.
Then $u_n$ has a logarithmic singularity in $\Omega$ and does not solve the equation there.
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<document source="https://tamnd.github.io/math2/node/XS3VFNSY9N.md" tag="XS3VFNSY9N" kind="definition">
# Separable degree

## Definition

Let $L/K$ be an [[algebraic extension|09EYMKMSB9]].
Let $\Omega$ be an [[algebraically closed field|02SJG1HC32]] and let $\sigma \colon K \to \Omega$ be a [[field homomorphism|EZVN87C88J]].
The *separable degree* $[L:K]_s$ of $L/K$ is the cardinality of the set of field homomorphisms $\tau \colon L \to \Omega$ with $\tau|_K = \sigma$.
This cardinality does not depend on the choice of $\Omega$ and $\sigma$.

## Remarks

Let $L/K$ be finite.
Then $[L:K]_s$ divides the [[degree|0GS74FGFEK]] $[L:K]$.
If $K$ has characteristic $p > 0$, the quotient $[L:K]_i = [L:K] / [L:K]_s$ is a power of $p$, called the inseparable degree; in characteristic $0$ it is $1$.
The extension $L/K$ is separable if and only if $[L:K]_s = [L:K]$.
Separable degree is multiplicative in towers: $[M:K]_s = [M:L]_s [L:K]_s$.
It equals the degree over $K$ of the [[separable closure|JJ91A1V8NQ]] of $K$ in $L$.

Example: $L = \Q(\sqrt{2})$ over $\Q$ with $\Omega = \C$.
Check: an embedding is determined by the image of $\sqrt{2}$, which must be a root of $x^2 - 2$, so there are exactly $2$ embeddings and $[L:\Q]_s = 2 = [L:\Q]$.

Non-example of the equality $[L:K]_s = [L:K]$: let $K = \F_p(t)$ and $L = K(s)$ with $s^p = t$.
Over an algebraic closure, $x^p - t = (x - s)^p$ has one root, so $[L:K]_s = 1$ while $[L:K] = p$.
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# Cyclic term theorem

## Statement

Let $A$ be a finite [[idempotent|9FS1JFZSX4]] algebra, meaning that every basic operation $f$ of $A$ satisfies $f(x, \dots, x) = x$.
A *cyclic term* of $A$ of arity $n \ge 2$ is a [[term|N9M2N27END]] $c(x_1, \dots, x_n)$ such that $A$ satisfies the [[identity|PKD440G2XW]] $c(x_1, x_2, \dots, x_n) \approx c(x_2, \dots, x_n, x_1)$.
Then the following are equivalent:

1. $A$ has a [[Taylor term|QEAMRNRJJD]].
2. $A$ has a cyclic term.
3. $A$ has a cyclic term of arity $p$ for every [[prime number|01826KMG2H]] $p > \abs{A}$.

## Remarks

The theorem is due to Libor Barto and Marcin Kozik.
A cyclic term is idempotent here because $A$ is idempotent.
An idempotent cyclic term $c$ of arity $n$ is a Taylor term.
For each place $i$, put $y$ in place $i$ and $x$ elsewhere.
The cyclic identity equates this with $c$ having $y$ in the previous place and $x$ elsewhere, and the two sides differ in place $i$.
So the real content is that a Taylor term forces cyclic terms; the proof uses the theory of absorbing subalgebras.
Example: a finite semilattice has the cyclic term $x_1 \wedge \cdots \wedge x_n$ for every $n$.
Example: the idempotent algebra $(\Z_3; x - y + z)$ has as term operations the affine combinations $\sum a_i x_i$ with $\sum a_i \equiv 1 \pmod 3$.
For the prime $p = 5 > 3$, the operation $2(x_1 + \cdots + x_5)$ is such a combination, since $2 \cdot 5 = 10 \equiv 1$, and it is cyclic.

## Explanations

### L1 human

A cyclic term is an operation that does not change when its inputs are rotated, like a symmetric way of combining $n$ items arranged in a circle.
The theorem says that, for finite idempotent algebras, such an operation exists as soon as the algebra has any Taylor term at all.
Moreover, one can choose the number of inputs to be any prime larger than the size of the algebra.

### L2 human

Cyclic terms give a clean test for tractability in the algebraic approach to constraint satisfaction.
A finite core structure has a polynomial-time constraint satisfaction problem exactly when its idempotent polymorphisms include a cyclic operation, by this theorem together with the [[CSP dichotomy theorem|ZA4YXJ9T1P]].
The theorem fits with other characterizations of Taylor algebras by a single identity, such as the [[Siggers term|JYW1H3J8SX]].
Unlike the Siggers identity, the cyclic identity has an arity that may be chosen freely among large primes, which is useful in arguments that combine several terms.
</document>

<document source="https://tamnd.github.io/math2/node/XS8KB53QKA.md" tag="XS8KB53QKA" kind="definition">
# Shilov boundary

## Definition

Let $A$ be a commutative complex [[Banach algebra|9KYWN94QDN]] with unit, and let $\Delta(A)$ be its [[character space|M7JVFJ3MMX]], a compact Hausdorff space.
For $a \in A$, let $\hat a$ be its [[Gelfand transform|VWNSHKJT92]].
A *boundary* for $A$ is a subset $F \subseteq \Delta(A)$ such that, for every $a \in A$,
$$\max_{\varphi \in \Delta(A)} \abs{\hat a(\varphi)} = \sup_{\varphi \in F} \abs{\hat a(\varphi)}.$$
The *Shilov boundary* $\partial A$ is the smallest closed boundary for $A$: a closed boundary contained in every closed boundary.

## Remarks

The existence of a smallest closed boundary is a theorem of Shilov: the intersection of all closed boundaries is again a boundary.

Example: the disc algebra $A(\mathbb{D})$ of continuous functions on the closed unit disc $\overline{\mathbb{D}}$ that are holomorphic in the open disc, with the sup norm.
Its character space is $\overline{\mathbb{D}}$, through the evaluations at points.
By the maximum modulus principle, the unit circle $\mathbb{T}$ is a closed boundary.
No proper closed subset $F \subsetneq \mathbb{T}$ is a boundary.
Indeed, let $\zeta \in \mathbb{T} \setminus F$; the function $f(z) = (1 + \overline{\zeta} z) / 2$ satisfies $\abs{f} < 1$ on $\overline{\mathbb{D}} \setminus \set{\zeta}$ and $f(\zeta) = 1$.
So its supremum on the compact set $F$ is less than $1$.
Hence $\partial A(\mathbb{D}) = \mathbb{T}$.

Non-example: for the disc algebra, the circle $\set{\abs{z} = 1/2}$ is closed but not a boundary, since $f(z) = z$ has maximum modulus $1/2$ there and $1$ on $\overline{\mathbb{D}}$.

For $A = C(X)$ with $X$ compact Hausdorff, the Shilov boundary is all of $X$.
Indeed, if $F \subsetneq X$ is closed and $x \notin F$, Urysohn's lemma gives $f \in C(X)$ with $f(x) = 1$ and $f = 0$ on $F$, so $F$ is not a boundary.
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<document source="https://tamnd.github.io/math2/node/XS9GETCZX2.md" tag="XS9GETCZX2" kind="concept">
# Hotelling's T-squared distribution

## Summary

Hotelling's T-squared distribution is the multivariate analogue of the square of Student's $t$-distribution.
It is the distribution of a squared Mahalanobis-type length $m\, x^T M^{-1} x$ of a normal vector $x$, measured with an independent Wishart matrix $M$.
A rescaling of it is an F-distribution.
It is the null distribution of Hotelling's $T^2$ test for the mean vector of a multivariate normal sample.

## Explanations

### L1 human

To test whether the mean of one variable is zero, one divides the sample mean by its estimated standard error and gets a $t$ statistic.
With several variables at once, the variables have different scales and are correlated.
Hotelling's statistic measures the distance of the sample mean vector from the hypothesized mean in units given by the sample covariance matrix.
Its distribution, when the hypothesis holds, is Hotelling's T-squared distribution.
With a single variable it reduces to the square of the $t$ statistic.
</document>

<document source="https://tamnd.github.io/math2/node/XSCSX39GEZ.md" tag="XSCSX39GEZ" kind="concept">
# Amalgamation property

## Summary

A class of structures has the amalgamation property when any two members that share a common part fit together in a third member.
In the third member, the two copies of the common part must coincide.
It is one of the three conditions in Fraïssé's theorem, which builds a countable homogeneous structure, the [[Fraïssé limit|HJVRBDQSGG]], from a class of finite structures.
It also appears in algebra, for example in the study of amalgamated free products of groups.

## Explanations

### L1 human

Think of two drawings that both contain the same small picture.
Amalgamation means one can draw a bigger picture that contains both drawings and in which the two copies of the small picture are the same.
For finite graphs this is easy: take two graphs $B$ and $C$ that both contain a graph $A$, glue them along $A$, and add no other edges.
The glued graph contains $B$ and $C$, and the two copies of $A$ coincide.
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<document source="https://tamnd.github.io/math2/node/XSPX1KW9AQ.md" tag="XSPX1KW9AQ" kind="concept">
# Low-discrepancy sequence

## Summary

A low-discrepancy sequence is a sequence of points in the unit cube $[0, 1)^s$ whose initial segments fill the cube as evenly as possible.
Evenness is measured by the star discrepancy.
It compares the share of the first $N$ points in each box anchored at the origin with the volume of the box.
For a low-discrepancy sequence the largest difference is of order $(\log N)^s / N$.
For large $N$ this is much smaller than the typical value of order $N^{-1/2}$ for random points.
Such sequences are the point sets of quasi-Monte Carlo integration.
Every low-discrepancy sequence is uniformly distributed in the cube; for $s = 1$ this is [[equidistribution modulo one|XM6DH57BWY]].

## Explanations

### L1 human

Random points tend to clump in some places and leave gaps in others.
A low-discrepancy sequence places each new point in a region that is still empty, so that at every stage the points are spread out evenly.
The simplest example on $[0, 1)$ is the van der Corput sequence $\frac12, \frac14, \frac34, \frac18, \frac58, \frac38, \frac78, \ldots$, obtained by writing $1, 2, 3, \ldots$ in binary and mirroring the digits at the binary point.
In higher dimension, the Halton and Sobol sequences play the same role.
Averages of a function over such points approximate its integral with a smaller error than random points.
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<document source="https://tamnd.github.io/math2/node/XSQX8X2YQ2.md" tag="XSQX8X2YQ2" kind="concept">
# Green's function of a Markov process

## Summary

The Green's function of a Markov process is the density of its potential kernel with respect to a reference measure.
Its value $u(x, y)$ measures how much time, on average, the process started at $x$ spends near $y$.
For Brownian motion it is a constant multiple of the classical Green's function of the Laplacian.
For a Markov chain it is the expected number of visits.

## Explanations

### L1 human

Start a random particle at $x$ and count, on average, how long it stays near another point $y$, per unit of volume.
This density is the Green's function.
For a random walk on a graph, $u(x, y)$ is simply the expected number of visits to $y$ by a walk started at $x$.
For Brownian motion in three-dimensional space, it is $1/(2\pi \abs{x - y})$, which decreases with distance as one would expect.
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<document source="https://tamnd.github.io/math2/node/XSZ49NF2AE.md" tag="XSZ49NF2AE" kind="conjecture">
# Manin conjecture

## Statement

Let $K$ be an [[algebraic number field|MWBDFS2KJ5]] and let $V$ be a smooth [[Fano variety|FMHA649HJ9]] over $K$.
Assume that $V(K)$ is dense in $V$ for the [[Zariski topology|8SQK7RTB63]].
Let $H$ be a [[height|F1FF505PWH]] on $V(K)$ relative to the anticanonical divisor $-K_V$, and let $\rho$ be the rank of the [[Picard group|3Z9DCWYXQZ]] of $V$.
Then there are a nonempty Zariski open subset $U \subseteq V$ and a constant $c > 0$ such that the number
$$N_U(B) = \#\set{P \in U(K) : H(P) \le B}$$
satisfies
$$N_U(B) \sim c \, B (\log B)^{\rho - 1} \quad \text{as } B \to \infty.$$

## Remarks

Franke, Manin and Tschinkel stated it in 1989, and Batyrev and Manin extended it to other heights; Peyre later gave a conjectural formula for the constant $c$.
The open set $U$ is needed, since a subvariety such as a line on a cubic surface can contain more points of bounded height than the rest of the variety.
For $V = \mathbb{P}^n$ over $\Q$ one has $\rho = 1$, and the anticanonical height is $H_0^{n+1}$ with $H_0$ the usual height.
So the prediction $N(B) \sim cB$ follows from Schanuel's count of points of bounded height.
The conjecture is proved for flag varieties, toric varieties and several other families, and is open in general.
Batyrev and Tschinkel found examples in which removing a Zariski closed set is not enough, and current formulations remove a thin set instead.

## Explanations

### L1 human

A Fano variety, such as projective space or a smooth cubic surface, usually has infinitely many rational points.
The Manin conjecture predicts how many of them have height at most $B$: roughly a constant times $B$ times a power of $\log B$.
The power of the logarithm is fixed by the geometry, namely by the number of independent divisor classes.
It turns a counting problem in number theory into a statement about the geometry of the variety.
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<document source="https://tamnd.github.io/math2/node/XSZ9C68SPG.md" tag="XSZ9C68SPG" kind="definition">
# System of parameters

## Definition

Let $(R, \mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] of [[Krull dimension|MVGBS3J1P7]] $d$.
Elements $x_1, \dots, x_d$ of $\mathfrak m$ form a *system of parameters* of $R$ if the ideal $(x_1, \dots, x_d)$ is [[$\mathfrak m$-primary|TMTPWVETQK]].

## Remarks

The ideal $(x_1, \dots, x_d)$ is $\mathfrak m$-primary if and only if $\mathfrak m^n \subseteq (x_1, \dots, x_d)$ for some $n \ge 1$, and if and only if $R/(x_1, \dots, x_d)$ has finite [[length|CZSCNZH8VY]].
Example: in $R = k[x, y]_{(x, y)}$, with $k$ a field, the elements $x^2, y^3$ form a system of parameters, because $\dim R = 2$ and $\mathfrak m^4 \subseteq (x^2, y^3)$: each monomial $x^a y^b$ with $a + b = 4$ has $a \ge 2$ or $b \ge 3$.
Non-example: in the same ring, $x, x + x^2$ is not a system of parameters, because it generates the ideal $(x)$, and no power of $y$ lies in $(x)$.
If $x_1, \dots, x_d$ generate $\mathfrak m$ itself, then $R$ is a [[regular local ring|63XXTF6B5M]] and the list is called a regular system of parameters.
</document>

<document source="https://tamnd.github.io/math2/node/XT034AHK9E.md" tag="XT034AHK9E" kind="statement">
# Hadamard's lemma

## Statement

Let $U \subseteq \R^n$ be an open set that is star-shaped with respect to a point $a \in U$.
This means that the segment from $a$ to $x$ lies in $U$ for every $x \in U$.
Let $f \colon U \to \R$ be a [[smooth function|0V33BNKZ9S]].
Then there are smooth functions $g_1, \dots, g_n \colon U \to \R$ such that
$$f(x) = f(a) + \sum_{i=1}^{n} (x_i - a_i)\, g_i(x) \quad \text{for all } x \in U,$$
and $g_i(a) = \dfrac{\partial f}{\partial x_i}(a)$ for $i = 1, \dots, n$, where $\partial f / \partial x_i$ is the [[partial derivative|09EGC6F8PT]].

## Remarks

The proof is one line of calculus.
For $x \in U$, the fundamental theorem of calculus and the chain rule give
$$f(x) - f(a) = \int_0^1 \frac{d}{dt} f(a + t(x - a))\, dt = \sum_{i=1}^n (x_i - a_i) \int_0^1 \frac{\partial f}{\partial x_i}(a + t(x - a))\, dt.$$
Set $g_i(x) = \int_0^1 \frac{\partial f}{\partial x_i}(a + t(x - a))\, dt$.
Differentiation under the integral sign shows that $g_i$ is smooth, and $g_i(a) = \frac{\partial f}{\partial x_i}(a)$.

Example: for $f(x) = x^3$ on $\R$ and $a = 1$, one has $x^3 = 1 + (x - 1)(x^2 + x + 1)$, so $g_1(x) = x^2 + x + 1$ and $g_1(1) = 3 = f'(1)$.

If $f$ is only of class $C^k$ with $k \ge 1$, the same formula gives functions $g_i$ of class $C^{k-1}$ only.
For example, $f(x) = x \abs{x}$ is $C^1$ on $\R$, and with $a = 0$ the formula gives $g_1(x) = \abs{x}$, which is not differentiable at $0$.

## Explanations

### L1 human

A smooth function that vanishes at a point can be divided by the coordinates of that point, and the quotients are still smooth.
In one variable it says: if $f(a) = 0$, then $f(x) = (x - a) g(x)$ with $g$ smooth.
This is the smooth analogue of the factor theorem for polynomials.

### L2 human

In terms of [[germs|7SWSRGDT4A]], the lemma says that the maximal ideal $\mathfrak{m}_a$ of the ring of smooth germs at $a$ is generated by the germs of $x_1 - a_1, \dots, x_n - a_n$.
Applying the lemma twice gives $f(x) = f(a) + \sum_i \frac{\partial f}{\partial x_i}(a)(x_i - a_i) + \sum_{i,j} (x_i - a_i)(x_j - a_j) h_{ij}(x)$ with $h_{ij}$ smooth.
This is the key step in the proof that derivations of the ring of germs are exactly the tangent vectors, and in the proof of the Morse lemma.
</document>

<document source="https://tamnd.github.io/math2/node/XT32T13KBY.md" tag="XT32T13KBY" kind="problem">
# Heilbronn triangle problem

## Problem

Let $n \ge 3$ and let $P$ be a set of $n$ points in the unit square $[0, 1]^2$.
Let $\delta(P)$ be the smallest area of a [[triangle|0CTNFGYMM2]] whose vertices are three points of $P$.
Three collinear points count as a triangle of area $0$.
Let $\Delta(n)$ be the supremum of $\delta(P)$ over all such $P$.
Determine the order of growth of $\Delta(n)$ as $n \to \infty$.

## Remarks

Hans Heilbronn conjectured that $\Delta(n) = O(1/n^2)$, with $O$ the [[big O notation|0RYTGBJAGN]]; the problem first appeared in print in a 1951 paper of Klaus Roth.
Erdős showed $\Delta(n) \ge c/n^2$ for a constant $c > 0$, so $1/n^2$ would be the right order.
Komlós, Pintz and Szemerédi disproved the conjecture in 1982 by showing $\Delta(n) \ge c \log n / n^2$.
On the other side, Roth proved $\Delta(n) = o(1/n)$, and Komlós, Pintz and Szemerédi proved $\Delta(n) \le n^{-8/7 + o(1)}$ in 1981.
In 2023 Cohen, Pohoata and Zakharov improved the exponent $-8/7$ for the first time.
The exact order of growth remains open.

## Explanations

### L1 human

Scatter $n$ points in a square so that every triangle formed by three of them has a large area.
Since there are many points, some triangle must be small; the question is how small.
The answer lies somewhere between about $(\log n)/n^2$ and a little below $n^{-8/7}$, and nobody knows exactly where.
</document>

<document source="https://tamnd.github.io/math2/node/XT4SWG6J7N.md" tag="XT4SWG6J7N" kind="concept">
# Dirichlet process

## Summary

A Dirichlet process is a random [[probability measure|059TJN0NZ1]] whose values on every finite partition of the space follow a [[Dirichlet distribution|RX4CVSSRRJ]].
It depends on a concentration parameter $\alpha > 0$ and a base probability measure $H$.
It is the basic prior distribution on probability measures in nonparametric [[Bayesian inference|0JWRJX2RMP]].
Its samples are discrete measures almost surely, and the posterior after observing data is again a Dirichlet process.

## Explanations

### L1 human

A Dirichlet process draws a whole probability distribution at random.
On average the drawn distribution equals the base measure $H$, and a large $\alpha$ keeps it close to $H$.
The drawn distribution puts all its mass on countably many points, like a random histogram with infinitely many thin bars.
For example, with $H$ uniform on $[0, 1]$, the mass that a draw gives to $[0, 1/2]$ is a random number with mean $1/2$.
Drawing values from such a random distribution again and again produces ties, and the pattern of ties is the [[Chinese restaurant process|2SEF8QTYQF]].
</document>

<document source="https://tamnd.github.io/math2/node/XT6EEXPD4P.md" tag="XT6EEXPD4P" kind="statement">
# Jordan–Schur theorem

## Statement

For each positive integer $n$ there is an integer $J(n)$ with the following property.
Every finite subgroup $G$ of the [[general linear group|0FCHE902D7]] $\mathrm{GL}_n(\C)$ has an [[abelian|082GHNPJ53]] [[normal subgroup|0CB0DX5BVV]] $A$ of [[index|0DTXBQPA2S]] $[G : A] \le J(n)$.

## Remarks

Jordan proved the theorem for finite groups.
Schur extended it to [[torsion groups|JZNQTTMG58]] of matrices: every torsion subgroup of $\mathrm{GL}_n(\C)$ has an abelian normal subgroup of finite index.

Idea of the proof: a finite group $G$ preserves a Hermitian inner product, so after a change of basis $G$ lies in the [[unitary group|0R9R0Y1HDK]] $U(n)$.
If two unitary matrices are close enough to the identity, their commutator is even closer to the identity.
In a finite group this forces the elements in a small ball around the identity to generate an abelian subgroup $A$.
This $A$ is normal, since conjugation by unitary matrices preserves distances to the identity.
Its index is at most the number of small balls needed to cover $U(n)$, which depends only on $n$.

Example: for $n = 2$, the dihedral group of order $2m$ acts on $\C^2$ by rotations and reflections of the plane.
It is not abelian for $m \ge 3$, but its rotations form a cyclic normal subgroup of index $2$.

The finiteness or torsion hypothesis is necessary.
The group $\mathrm{SL}_2(\Z) \subseteq \mathrm{GL}_2(\C)$ contains a nonabelian [[free group|0KTW26C80J]] of finite index, and a nonabelian free group has no abelian subgroup of finite index.

## Explanations

### L1 human

Finite groups of invertible matrices of a fixed size are almost abelian.
Each one contains an abelian piece whose number of cosets is bounded by a number that depends only on the size of the matrices.
So a finite group of $3 \times 3$ matrices can be huge, but most of its complexity is commutative.

### L2 human

The bound $J(n)$ does not depend on the order of $G$, and this is the whole point.
The same commutator argument in the Euclidean group is behind the first of the [[Bieberbach theorems|YE87R5CDJY]] on [[crystallographic groups|7KTJKFQT2N]].
The theorem also shows that a finite group with no abelian normal subgroup of small index, such as a large alternating group, has no faithful complex representation of small degree.
</document>

<document source="https://tamnd.github.io/math2/node/XT7VWNTKGB.md" tag="XT7VWNTKGB" kind="concept">
# Homology with coefficients

## Summary

Homology with coefficients in an abelian group $G$ is [[singular homology|M0Q5CFJ7ZH]] computed with chains whose coefficients lie in $G$ instead of $\Z$.
Coefficients in a field such as $\Q$ or $\Z/p$ give vector spaces, which are easier to handle, and coefficients in $\Z/2$ remove all sign questions.
The [[universal coefficient theorem|GTA8ZFGZ3N]] recovers the homology with any coefficients from the integral homology.

## Explanations

### L1 human

In ordinary homology a chain is a sum of simplices with integer multiplicities.
With coefficients in $G$ the multiplicities are taken from $G$ instead.
With $\Z/2$ coefficients, one only records whether a simplex is present or not, so orientations no longer matter.
For the real projective plane this makes a difference: its top homology is $0$ with integer coefficients but $\Z/2$ with $\Z/2$ coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/XT8PA31VAX.md" tag="XT8PA31VAX" kind="concept">
# Weyl–Titchmarsh theory

## Summary

Weyl–Titchmarsh theory is the spectral theory of singular [[Sturm–Liouville|7WV6B2502K]] operators built on the Weyl–Titchmarsh function $m(z)$.
The function $m$ records the boundary values of the square integrable solution of $-y'' + q y = z y$ at the regular end.
It is a Herglotz function, and the measure in its integral representation is the spectral measure of the operator.
So the spectrum, the eigenfunction expansion and, by the Borg–Marchenko theorem, even the potential $q$ can be read off from $m$.
The theory goes back to Hermann Weyl and was developed by Edward Charles Titchmarsh.

## Explanations

### L1 human

For an equation on a half-line, most solutions blow up at infinity and only one direction of solutions stays square integrable.
The Weyl–Titchmarsh function is the ratio of slope to value of that good solution at the starting point.
It is one complex function of the spectral parameter, yet it carries all the spectral information of the operator.
For $-y'' = z y$ on $[0, \infty)$ the good solution is $e^{i\sqrt{z}\, x}$ and the function is $m(z) = i \sqrt{z}$.
</document>

<document source="https://tamnd.github.io/math2/node/XV4K9GREJ9.md" tag="XV4K9GREJ9" kind="concept">
# Rational variety

## Summary

A rational variety is a variety that is birationally equivalent to projective space.
Its points can be parametrised, on a dense open set, by rational functions of independent parameters.
Deciding which varieties are rational is one of the oldest problems of [[birational geometry|51YVV85GWB]].
Every rational variety is [[unirational|YF1SE8WJK3]], and the converse fails in dimension $3$ over $\C$.

## Explanations

### L1 human

A variety is rational when you can describe almost all of its points by free coordinates, with no hidden constraints.
The circle $x^2 + y^2 = 1$ is rational: each point other than $(-1, 0)$ is
$$\left( \frac{1 - t^2}{1 + t^2}, \frac{2t}{1 + t^2} \right)$$
for exactly one $t$, which is the slope of the line through $(-1, 0)$ and the point.
A smooth plane cubic, such as $y^2 = x^3 - x$, is not rational.
Its genus is $1$, while a rational curve has genus $0$.
</document>

<document source="https://tamnd.github.io/math2/node/XVBZJVQ3WA.md" tag="XVBZJVQ3WA" kind="definition">
# Conformal geometric algebra

## Definition

Let $n \ge 1$, and let $\R^{n+1,1} = \R^n \oplus \R e_+ \oplus \R e_-$, where $\R^n$ has the dot product, $e_+ \cdot e_+ = 1$, $e_- \cdot e_- = -1$, and the three summands are pairwise orthogonal.
Put $n_0 = \tfrac{1}{2}(e_- - e_+)$ and $n_\infty = e_- + e_+$.
The *conformal geometric algebra* of $\R^n$ is the [[geometric algebra|N8H460W3E2]] $\mathcal{G}(\R^{n+1,1})$, together with the embedding
$$x \mapsto X = x + \tfrac{1}{2} \abs{x}^2 n_\infty + n_0, \qquad x \in \R^n.$$

## Remarks

From $e_\pm \cdot e_\pm = \pm 1$ one gets $n_0 \cdot n_0 = n_\infty \cdot n_\infty = 0$ and $n_0 \cdot n_\infty = -1$.
Hence $X \cdot X = \abs{x}^2 - \abs{x}^2 = 0$, so points become null vectors, and $X \cdot Y = x \cdot y - \tfrac{1}{2}(\abs{x}^2 + \abs{y}^2) = -\tfrac{1}{2} \abs{x - y}^2$.
Example: for $c \in \R^n$ and $r > 0$, put $S = C - \tfrac{1}{2} r^2 n_\infty$ with $C$ the image of $c$.
Since $X \cdot n_\infty = -1$, one gets $X \cdot S = \tfrac{1}{2}(r^2 - \abs{x - c}^2)$, which vanishes exactly when $x$ lies on the sphere with centre $c$ and radius $r$.
Likewise, for a unit vector $m \in \R^n$ and $\delta \in \R$, the vector $m + \delta n_\infty$ satisfies $X \cdot (m + \delta n_\infty) = x \cdot m - \delta$, and it represents the plane $x \cdot m = \delta$.
Non-example: the vector $e_- \in \R^{n+1,1}$ is not the image of a point, since $e_- \cdot e_- = -1 \ne 0$.
The versors of the algebra, the products of invertible vectors, act on $\R^{n+1,1}$ by conjugation, up to sign.
They induce exactly the Möbius transformations of $\R^n \cup \set{\infty}$, the maps generated by translations, rotations, dilations and inversions in spheres.
</document>

<document source="https://tamnd.github.io/math2/node/XVEWR93EMV.md" tag="XVEWR93EMV" kind="definition">
# Lebesgue integral

## Definition

Let $(X, \Sigma, \mu)$ be a measure space and let $f \colon X \to \R$ be an [[integrable function|X7GTF3D98F]].
Let $f^+ = \max(f, 0)$ and $f^- = \max(-f, 0)$, so that $f = f^+ - f^-$.
The *Lebesgue integral* of $f$ is
$$\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu,$$
where both terms are [[integrals of non-negative functions|DG76G5Z6XW]] and are finite.
For an integrable $f \colon X \to \C$, the integral is $\int_X f \, d\mu = \int_X \operatorname{Re} f \, d\mu + i \int_X \operatorname{Im} f \, d\mu$.

## Remarks

Example: on $[-1, 1]$ with Lebesgue measure, let $f(x) = x$.
Then $f^+ = x \mathbf{1}_{[0, 1]}$ and $f^- = -x \mathbf{1}_{[-1, 0]}$ both have integral $1/2$, so $\int f = 0$.

Non-example: $g(x) = x$ on all of $\R$ is not integrable.
Both $\int g^+$ and $\int g^-$ are infinite, so the difference $\infty - \infty$ has no meaning.

The same formula is used when only one of $\int f^+$ and $\int f^-$ is finite, with value $\pm \infty$.
The integral is linear and monotone on integrable functions, and it satisfies $\abs{\int_X f \, d\mu} \le \int_X \abs{f} \, d\mu$.
On a bounded interval, every Riemann integrable function is Lebesgue integrable and the two integrals agree.
Common notations are $\int f \, d\mu$, $\int f(x) \, d\mu(x)$ and $\int f(x) \, \mu(dx)$.
</document>

<document source="https://tamnd.github.io/math2/node/XVG7FPG227.md" tag="XVG7FPG227" kind="concept">
# Well-powered category

## Summary

A [[category|A2DE66937D]] is well-powered when every object has only a set of [[subobjects|FS1FKAE89H]], not a proper class of them.
This size condition holds in most categories of everyday algebra and topology.
It is a hypothesis of the special adjoint functor theorem and of several existence results for factorisations and limits.

## Explanations

### L1 human

In a large category, there can be a proper class of monomorphisms into an object, because there is a proper class of sets that could serve as domains.
After identifying monomorphisms that differ by an isomorphism, one usually gets back a set.
For example, the subobjects of a set $X$ correspond to the subsets of $X$, and these form the power set of $X$, which is a set.
A well-powered category is one where this collapse to a set happens for every object.
</document>

<document source="https://tamnd.github.io/math2/node/XVHKCX4F5Y.md" tag="XVHKCX4F5Y" kind="statement">
# Kruskal's tree theorem

## Statement

For finite [[rooted trees|0C9J6W2A8A]] $T$ and $T'$, write $u \wedge v$ for the nearest common ancestor of vertices $u, v$ of $T$.
Say that $T$ embeds in $T'$ if some injective map $f$ from the vertices of $T$ to those of $T'$ satisfies $f(u \wedge v) = f(u) \wedge f(v)$ for all $u, v$.
Then the finite rooted trees are [[well-quasi-ordered|NH25G6GC07]] by embedding.
That is, for every infinite sequence $T_1, T_2, T_3, \dots$ of finite rooted trees, there are indices $i < j$ such that $T_i$ embeds in $T_j$.

## Remarks

The labelled form also holds.
Let the vertices carry labels from a well-quasi-ordered set $(X, \le)$, and ask also that each label of $T$ is at most the label of its image.
Then the trees are still well-quasi-ordered.
Joseph Kruskal proved the theorem in 1960, and Crispin Nash-Williams gave a short proof by the minimal bad sequence argument.
That argument takes a bad sequence whose trees are as small as possible at each step.
It then applies Higman's lemma to the sets of subtrees at the roots to reach a contradiction.
Harvey Friedman showed that a finite form of the theorem, although true, is not provable in the subsystem $\mathrm{ATR}_0$ of second-order arithmetic.
The [[Robertson–Seymour theorem|WP3GDNWJJW]] on graph minors builds on its methods.

## Explanations

### L1 human

Write down an endless list of finite trees.
The theorem says that, no matter how you choose them, some tree in the list will be contained, in a branch-preserving way, in a later tree.
You cannot keep producing trees that avoid all earlier ones.

### L2 human

The theorem is a basic example of a well-quasi-order on structured objects, after Higman's lemma for finite words.
It is used in term rewriting to prove that certain orderings, such as recursive path orderings, are well-founded.
Its finite form grows so fast that it escapes strong formal systems, which makes it a standard example in proof theory.
</document>

<document source="https://tamnd.github.io/math2/node/XVJA0JD79P.md" tag="XVJA0JD79P" kind="concept">
# Gromov–Witten invariant

## Summary

A Gromov–Witten invariant is a rational number that counts curves in a smooth projective variety or a symplectic manifold.
The curves have a given genus and homology class, and they pass through given cycles.
It is defined by integrating cohomology classes over the virtual fundamental class of a [[moduli space|FJZRW9P0K3]] of stable maps.
These invariants solve classical questions of [[enumerative geometry|ZPZA1FKZZC]], they define quantum cohomology, and they are one side of [[mirror symmetry|YK3DMN2NSK]].

## Explanations

### L1 human

Ask how many curves of a fixed kind pass through a fixed set of points.
For lines in the plane through two points the answer is $1$, for conics through five points it is also $1$.
For rational plane cubics through eight general points the answer is $12$.
Gromov–Witten invariants give a uniform way to define and compute such numbers, by integrating over a space that parametrizes all the curves at once.
They stay the same when the variety is deformed, which makes them invariants.
</document>

<document source="https://tamnd.github.io/math2/node/XVNB6HV2Y1.md" tag="XVNB6HV2Y1" kind="statement">
# Blaschke–Lebesgue theorem

## Statement

Let $w > 0$, and let $K$ be a plane convex body of constant width $w$.
So $K$ is bounded by a [[curve of constant width|B4YD1ZXRVH]] $w$.
Then the area of $K$ is at least $\frac{\pi - \sqrt{3}}{2} w^2$.
Equality holds if and only if $K$ is a [[Reuleaux triangle|KMNV11B91V]] of width $w$.

## Remarks

One classical route approximates $K$ by Reuleaux polygons of width $w$, which are bodies of constant width bounded by finitely many circular arcs of radius $w$.
For Reuleaux polygons one shows that the area can be lowered by deforming the polygon towards a Reuleaux triangle, and a limit argument finishes the proof.
Example: a disc of diameter $w$ has area $\frac{\pi}{4} w^2 \approx 0.785 \, w^2$, while the Reuleaux triangle has area $\frac{\pi - \sqrt{3}}{2} w^2 \approx 0.705 \, w^2$.
The constant width hypothesis is needed: a segment of length $w$ has diameter $w$ and area $0$.

## Explanations

### L1 human

Among all shapes that have the same width $w$ in every direction, the Reuleaux triangle uses the least area.
It is the "thinnest" such shape, while the round disc is the "fattest".

### L2 human

The maximum among bodies of constant width $w$ is the disc, which follows from [[Barbier's theorem|T5D9RSE0X2]] and the isoperimetric inequality, since all such bodies have perimeter $\pi w$.
The minimum is much harder, and the Blaschke–Lebesgue theorem identifies it.
In three dimensions the question of the body of constant width with least volume is open.
</document>

<document source="https://tamnd.github.io/math2/node/XVNJPJ53W8.md" tag="XVNJPJ53W8" kind="concept">
# Cardinal function

## Summary

A cardinal function in topology assigns a [[cardinal number|0EBACQFAZW]] to each topological space so that homeomorphic spaces get the same cardinal.
Standard examples are the weight, the density, the character and the cellularity.
Cardinal functions turn countability properties into numbers, so that, for example, [[second-countable|03PPNCM9GG]] means weight at most $\aleph_0$.
Many theorems of general topology are inequalities between cardinal functions.

## Explanations

### L1 human

A cardinal function measures the size of some feature of a space.
The weight counts how many open sets you need in a base, and the density counts how many points you need in a dense subset.
For the real line, both numbers are countable: open intervals with rational endpoints form a base, and the rationals are dense.
For an uncountable set with the discrete topology, both numbers equal the number of points.
</document>

<document source="https://tamnd.github.io/math2/node/XVP517QRH5.md" tag="XVP517QRH5" kind="concept">
# Semiprimary ring

## Summary

A semiprimary ring is a ring $R$ such that $R/J(R)$ is semisimple and the [[Jacobson radical|XA0DCNH7WY]] $J(R)$ is nilpotent.
Every left Artinian ring is semiprimary, because its [[Jacobson radical is nilpotent|H7RAXJV9CP]].
Semiprimary rings are left and right [[perfect|49Y9Q949PK]].
They are the setting of the [[Hopkins–Levitzki theorem|W29AA0266Z]]: a semiprimary ring is left Artinian exactly when it is left Noetherian.

## Explanations

### L1 human

A semiprimary ring is semisimple up to a radical that dies after finitely many multiplications.
This is the structure of a left Artinian ring, but without any chain condition.
For example, let $R$ be the ring $k[x_1, x_2, \ldots]$ with all products $x_i x_j$ set equal to $0$.
It is semiprimary: its radical is spanned by the $x_i$, it squares to zero, and the quotient is the field $k$.
This ring is not Artinian, because the ideals $(x_n, x_{n+1}, \ldots)$ form an infinite descending chain.
</document>

<document source="https://tamnd.github.io/math2/node/XVPJ44E0KW.md" tag="XVPJ44E0KW" kind="definition">
# Supporting hyperplane

## Definition

Let $n \ge 1$ and let $S \subseteq \R^n$.
A hyperplane $H = \set{x \in \R^n : a \cdot x = b}$, with $a \in \R^n \setminus \set{0}$ and $b \in \R$, is a *supporting hyperplane* of $S$ if:

1. $S$ lies in one of the closed half-spaces of $H$, say $a \cdot x \le b$ for all $x \in S$;
2. $H$ contains at least one point of the [[boundary|05D5EGMFYV]] of $S$.

Here $a \cdot x$ is the [[dot product|0S52TNNTH9]].
If $x_0 \in H$ is such a boundary point, $H$ supports $S$ at $x_0$.

## Remarks

Example: for the closed unit disc $D \subseteq \R^2$ and $a = (0, 1)$, $b = 1$, the line $y = 1$ supports $D$ at $(0, 1)$.
Each $(x, y) \in D$ has $y \le 1$, and $(0, 1)$ is a boundary point of $D$ on the line.

Non-example: the line $y = 2$ satisfies clause 1 for $D$, but it misses $D$ and its boundary, so clause 2 fails.
The line $y = 0$ meets the boundary of $D$ but has points of $D$ on both sides, so clause 1 fails.

In terms of the [[support function|1H92CYA4ZC]] $h_S$, when $S$ is closed and bounded and nonempty, the hyperplane $a \cdot x = h_S(a)$ is the supporting hyperplane with outer normal $a$.
</document>

<document source="https://tamnd.github.io/math2/node/XVQN0GT988.md" tag="XVQN0GT988" kind="definition">
# Disk algebra

## Definition

Let $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ be the open [[unit disc|8G5ABXRK85]] and $\overline{\mathbb{D}}$ its closure.
The *disk algebra* $A(\mathbb{D})$ is the set of functions $f \colon \overline{\mathbb{D}} \to \C$ such that

1. $f$ is [[continuous|0P27W9RW0G]] on $\overline{\mathbb{D}}$, and
2. the restriction of $f$ to $\mathbb{D}$ is [[holomorphic|0DG0Z72CQD]],

with pointwise addition and multiplication and the norm $\norm{f} = \sup_{z \in \overline{\mathbb{D}}} \abs{f(z)}$.

## Remarks

$A(\mathbb{D})$ is a closed subalgebra of $C(\overline{\mathbb{D}})$, hence a commutative [[Banach algebra|9KYWN94QDN]] with unit.
By the [[maximum modulus principle|048F2P90WR]], the norm equals $\sup_{\abs{z} = 1} \abs{f(z)}$, so $A(\mathbb{D})$ may also be viewed as an algebra of functions on the circle.
The polynomials are dense: $f(rz) \to f(z)$ uniformly as $r \to 1$, and each $f(rz)$ is the uniform limit of its Taylor polynomials on $\overline{\mathbb{D}}$.
The maximal ideals of $A(\mathbb{D})$ are the sets $\set{f : f(a) = 0}$ for $a \in \overline{\mathbb{D}}$.

Example: $f(z) = \sum_{n \ge 1} z^n / n^2$.
The terms are bounded by $1/n^2$ on $\overline{\mathbb{D}}$, so the series converges uniformly there; hence $f$ is continuous on $\overline{\mathbb{D}}$ and holomorphic in $\mathbb{D}$.

Non-example: $S(z) = \exp\left(-\frac{1 + z}{1 - z}\right)$ is bounded and holomorphic in $\mathbb{D}$, but it has no continuous extension to $\overline{\mathbb{D}}$.
Its radial limit at $1$ is $0$, while its radial limits at the points $e^{it} \ne 1$ have modulus $1$, so no continuous extension to $\overline{\mathbb{D}}$ exists.
</document>

<document source="https://tamnd.github.io/math2/node/XVSVNP3K9G.md" tag="XVSVNP3K9G" kind="statement">
# Bombieri–De Giorgi–Giusti theorem

## Statement

Write points of $\R^8$ as $(x, y)$ with $x, y \in \R^4$, and let $E = \set{(x, y) : \abs{x} < \abs{y}}$.

1. The set $E$ minimizes [[perimeter|Y68BHXZJBV]] in $\R^8$: $P(E, B) \le P(F, B)$ for every open ball $B$ and every measurable $F$ with $F \setminus B = E \setminus B$.
2. The [[Simons cone|ED3DAZVQZ4]] $\partial E = \set{\abs{x} = \abs{y}}$ is therefore an area-minimizing hypersurface with a singular point at the origin.
3. There is a smooth function $u \colon \R^8 \to \R$ that is not affine and solves the [[minimal surface equation|3GV2E2YTZX]]
$$\operatorname{div} \left( \frac{\nabla u}{\sqrt{1 + \abs{\nabla u}^2}} \right) = 0$$
on all of $\R^8$.

## Remarks

Enrico Bombieri, Ennio De Giorgi and Enrico Giusti proved these results in 1969.
For parts 1 and 2 they compared $E$ with a family of smooth minimal hypersurfaces that fill each side of the cone.
Such a family forces any competitor to have at least as much perimeter.
For part 3 they built a subsolution and a supersolution of the minimal surface equation, and found an entire solution between them.
At infinity both behave like a homogeneous function that vanishes on the Simons cone.
The function $u$ of part 3 gives, for every $m \ge 8$, a non-affine solution on $\R^m$ that does not depend on the last $m - 8$ variables.
So the answer to [[Bernstein's problem|NP948D6FR0]] is negative in all dimensions $m \ge 8$, while Simons had shown it is positive for $m \le 7$.
Part 2 shows that the bound $n - 8$ on the dimension of the singular set of area-minimizing hypersurfaces in $\R^n$ is sharp.

## Explanations

### L1 human

A minimal graph is a soap-film-like surface that lies over the whole of a flat space as the graph of a function.
In low dimensions such a graph must be flat.
This theorem shows that in eight dimensions there are curved minimal graphs.
It also shows that least-area surfaces in eight dimensions can have a sharp point, the tip of the Simons cone.

### L2 human

The minimizing cone and the minimal graph are linked through a blow-down argument.
If a non-affine entire minimal graph exists over $\R^m$, a blow-down argument of Fleming and De Giorgi produces a non-flat area-minimizing cone in $\R^m$.
Simons showed that there is no such cone in $\R^m$ for $m \le 7$, which gives Bernstein's theorem for $m \le 7$.
The theorem shows that this obstruction disappears in $\R^8$, and constructs the corresponding graph.
It is a standard example of how the regularity theory of minimal surfaces and the Liouville-type theorems for the minimal surface equation depend on the dimension.
</document>

<document source="https://tamnd.github.io/math2/node/XVVHKCYF2Z.md" tag="XVVHKCYF2Z" kind="statement">
# Balian–Low theorem

## Statement

Let $g \in L^2(\R)$, and let $\hat g(\xi) = \int_\R g(t) e^{-2\pi i t \xi} \, dt$ be its [[Fourier transform|DPVM34ADZV]].
Suppose that the [[Gabor system|M9WHY8E2PM]] $\left(e^{2\pi i n t} g(t - k)\right)_{k, n \in \Z}$ is a [[Riesz basis|ZYDYD7FB1Z]] of $L^2(\R)$.
Then
$$\left( \int_\R \abs{t \, g(t)}^2 \, dt \right) \left( \int_\R \abs{\xi \, \hat g(\xi)}^2 \, d\xi \right) = \infty.$$

## Remarks

So a window of a Gabor Riesz basis at the critical density $ab = 1$ cannot be well localized both in time and in frequency.
Roger Balian and, independently, Francis Low stated the result for orthonormal bases; later work completed their proofs and extended it to Riesz bases.
A standard route uses the [[Zak transform|S026HQSGY9]]: the system is a Riesz basis exactly when $Zg$ is bounded above and below by positive constants almost everywhere.
On the other hand $Zg$ is quasi-periodic, and a continuous quasi-periodic function must vanish somewhere; when $g$ is well localized this contradicts the lower bound.
The hypothesis $ab = 1$ matters: for $ab < 1$ the Gaussian $e^{-\pi t^2}$, which is as well localized as possible, generates a Gabor frame.

Example: $g = 1_{[0, 1)}$ generates an orthonormal basis.
Here $\int \abs{t g(t)}^2 \, dt = \tfrac13$ is finite, but $\abs{\hat g(\xi)} = \abs{\sin(\pi \xi)}/(\pi \abs{\xi})$, so $\abs{\xi \hat g(\xi)}^2 = \sin^2(\pi \xi)/\pi^2$ is not integrable.

## Explanations

### L1 human

A Gabor basis builds every signal from shifted and modulated copies of one window, placed on a grid as sparse as possible.
The theorem says that this window cannot be nice: it must be badly spread out either in time or in frequency.
The box function is sharp in time but its spectrum decays too slowly.
It is a strong form of the uncertainty principle for bases.

### L2 human

The theorem explains why practical Gabor expansions are redundant frames with $ab < 1$ rather than bases.
It is proved with the [[Zak transform|S026HQSGY9]], which turns the Gabor system at $a = b = 1$ into multiplication by a quasi-periodic function, and the question becomes one about zeros of that function.
It contrasts with wavelets: there exist orthonormal wavelet bases, such as the Meyer wavelets, whose generator is smooth and decays fast together with its Fourier transform.
Compare with the [[uncertainty principle in Fourier analysis|S1FB2K14NV]], which bounds the product of the two integrals from below for one function.
</document>

<document source="https://tamnd.github.io/math2/node/XW2BS4VG3P.md" tag="XW2BS4VG3P" kind="definition">
# Schramm–Loewner evolution

## Definition

Let $\kappa > 0$, and let $(B_t)_{t \ge 0}$ be a standard one-dimensional [[Brownian motion|22735FGJV6]] with $B_0 = 0$.
Let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]].
*Chordal $\mathrm{SLE}_\kappa$* in $\mathbb{H}$ from $0$ to $\infty$ is a random family of conformal maps $(g_t)_{t \ge 0}$ and hulls $(K_t)_{t \ge 0}$.
It is given by the chordal [[Loewner differential equation|51G4QJZ345]]
$$\frac{\partial}{\partial t} g_t(z) = \frac{2}{g_t(z) - \sqrt{\kappa}\, B_t}, \qquad g_0(z) = z,$$
with the driving function $U(t) = \sqrt{\kappa}\, B_t$.
Let $D \ne \C$ be a simply connected domain with two distinct boundary points $a$ and $b$.
Let $\phi$ be a [[conformal map|0VHNKP1N3K]] of $\mathbb{H}$ onto $D$ that sends $0$ to $a$ and $\infty$ to $b$.
Chordal $\mathrm{SLE}_\kappa$ in $D$ from $a$ to $b$ is the image under $\phi$ of chordal $\mathrm{SLE}_\kappa$ in $\mathbb{H}$.

## Remarks

The definition in $D$ does not depend on the choice of $\phi$, because two choices differ by a scaling $z \mapsto cz$ of $\mathbb{H}$ with $c > 0$.
Brownian scaling shows that such a scaling only changes the time parameter of the hulls.
Almost surely the hulls are generated by a continuous curve $\gamma$: $\mathbb{H} \setminus K_t$ is the unbounded component of $\mathbb{H} \setminus \gamma[0, t]$.
This was proved by Rohde and Schramm for $\kappa \ne 8$, and by Lawler, Schramm and Werner for $\kappa = 8$.
The curve is simple for $\kappa \le 4$, touches itself for $4 < \kappa < 8$, and fills space for $\kappa \ge 8$.
Example: for $\kappa = 2$, $\mathrm{SLE}_2$ is the scaling limit of loop-erased random walk, a theorem of Lawler, Schramm and Werner.
For $\kappa = 0$ the driving function is $0$ and the hulls are deterministic vertical segments; this degenerate case is sometimes included as $\mathrm{SLE}_0$.
Radial $\mathrm{SLE}_\kappa$ is defined in the same way from the radial Loewner equation.
</document>

<document source="https://tamnd.github.io/math2/node/XW80KQJ62C.md" tag="XW80KQJ62C" kind="definition">
# Conformal geometry

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n \ge 2$.
Two [[Riemannian metrics|Q61VFHCH2J]] $g$ and $g'$ on $M$ are *conformally equivalent* when $g' = e^{2f} g$ for some smooth function $f \colon M \to \R$.
A *conformal structure* on $M$ is an equivalence class $[g]$ for this relation, and $(M, [g])$ is a *conformal manifold*.
A [[diffeomorphism|0KZZ70ZHXP]] $\varphi \colon (M, [g]) \to (N, [h])$ is *conformal* when $\varphi^* h \in [g]$.
*Conformal geometry* is the study of the properties of conformal manifolds that are invariant under conformal diffeomorphisms.

## Remarks

The angle between two tangent vectors at a point is the same for all metrics in $[g]$, so angles are defined in a conformal manifold, while lengths are not.
Accordingly, conformal diffeomorphisms are exactly the diffeomorphisms that are [[conformal maps|0YYT36R4YD]], that is, that preserve angles.
Example: stereographic projection from the north pole identifies $S^n$ minus a point with $\R^n$, and the round metric becomes $\dfrac{4}{(1 + \abs{x}^2)^2} \abs{dx}^2$.
This is $e^{2f}$ times the flat metric with $f(x) = \ln 2 - \ln(1 + \abs{x}^2)$, so the round sphere minus a point is conformally equivalent to Euclidean space.
Non-example: the map $(x, y) \mapsto (2x, y)$ of $\R^2$ pulls back $dx^2 + dy^2$ to $4\, dx^2 + dy^2$, which is not a multiple of $dx^2 + dy^2$; it does not preserve the angle between the vectors $(1, 1)$ and $(1, -1)$.
On an oriented surface, a conformal structure is the same thing as a structure of Riemann surface.
</document>

<document source="https://tamnd.github.io/math2/node/XWCAS1ESAA.md" tag="XWCAS1ESAA" kind="definition">
# Étale morphism

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].
The morphism $f$ is *étale* if it satisfies the following three conditions.

1. $f$ is locally of finite presentation: over affine opens $\operatorname{Spec} B \to \operatorname{Spec} A$, the $A$-algebra $B$ is [[finitely presented|764GZHTGV8]].
2. $f$ is [[flat|5M6KS8F47R]].
3. $f$ is [[unramified|1JQX6HD8E5]].

## Remarks

Let $k$ be a field of characteristic not $2$, and let $A = k[t, t^{-1}]$.
The morphism $\operatorname{Spec} A \to \operatorname{Spec} A$ given by $t \mapsto t^2$ is étale.
The ring $A$ is free of rank $2$ over the subring $k[t^2, t^{-2}]$ with basis $1, t$, so the map is flat and finitely presented.
Its module of relative differentials is generated by $dt$ with the relation $2t \, dt = 0$, and $2t$ is a unit, so it is zero and the map is unramified.
The same map on $\operatorname{Spec} k[t]$ is not étale: at $t = 0$ the relation $2t \, dt = 0$ does not kill $dt$, so clause 3 fails.
Equivalently, an étale morphism is a [[smooth morphism|EXKGG9X6J0]] of relative dimension $0$.
The ring version of this notion is an [[étale morphism of rings|FTRW0679E2]].
</document>

<document source="https://tamnd.github.io/math2/node/XWEMJ7T2XV.md" tag="XWEMJ7T2XV" kind="problem">
# Burnside problem

## Problem

1. General Burnside problem: let $G$ be a [[finitely generated group|082DWWXMRJ]] in which every element has finite [[order|0WE2NHZJ7B]]. Must $G$ be finite?
2. Bounded Burnside problem: let $m \ge 1$ and $n \ge 1$ be integers. Let $B(m, n)$ be the quotient of the [[free group|0KTW26C80J]] $F_m$ of rank $m$ by the normal subgroup generated by all $w^n$ with $w \in F_m$. For which $m$ and $n$ is $B(m, n)$ finite?

## Remarks

The group $B(m, n)$ is the free Burnside group: every group with $m$ generators and [[exponent|0ZDW0H0VX4]] dividing $n$ is a quotient of it.
So the bounded problem asks whether all groups with $m$ generators and exponent $n$ are finite of bounded order.
Burnside posed the problem in 1902.

The general problem has a negative answer.
Golod and Shafarevich built infinite finitely generated $p$-groups in 1964, and the [[Grigorchuk group|M0WAT1VQZH]] is a later explicit example.

For the bounded problem, $B(m, n)$ is finite for $n = 2, 3, 4, 6$ and all $m$.
For instance $B(m, 2) \cong (\Z/2)^m$, since $x^2 = 1$ for all $x$ forces the group to be abelian.
Novikov and Adian proved that $B(m, n)$ is infinite for $m \ge 2$ and all odd $n$ large enough, and later work extended this to large even $n$.
[[Tarski monster groups|TE19BGFFF1]] are infinite $2$-generated groups of large prime exponent.
Whether $B(2, 5)$ is finite is open.

## Explanations

### L1 human

In a finite group, every element returns to the identity after finitely many repetitions.
Burnside asked about the converse: if a group is generated by finitely many elements and every element returns to the identity, must the group be finite?
The answer is no in general, and even if all elements satisfy $x^n = 1$ for one fixed $n$, the answer is no for many large $n$.
The question for $2$ generators and $n = 5$ is still open.
</document>

<document source="https://tamnd.github.io/math2/node/XWJ786Z701.md" tag="XWJ786Z701" kind="definition">
# Three-term recurrence relation

## Definition

Let $K$ be a field, and let $a_n, b_n, c_n \in K$ for $n \ge 1$ be given, with $a_n \ne 0$ and $c_n \ne 0$ for all $n$.
A *three-term recurrence relation* is the homogeneous [[linear recurrence|8Y28JQB0Q5]]
$$a_n\, u_{n+1} + b_n\, u_n + c_n\, u_{n-1} = 0 \qquad (n \ge 1)$$
for a [[sequence|0G9EKM0E15]] $(u_n)_{n \ge 0}$ in $K$.

## Remarks

Example: the [[Chebyshev polynomials|M6EAR1JHPP]] satisfy $T_{n+1}(x) - 2x\, T_n(x) + T_{n-1}(x) = 0$ in $\R[x]$, here read in the field $\R(x)$ of rational functions with $a_n = 1$, $b_n = -2x$, $c_n = 1$.
Check for $n = 1$: $T_2(x) = 2x^2 - 1 = 2x \cdot x - 1$.
Non-example: $u_{n+1} = u_n + u_{n-1} + u_{n-2}$ links four terms, so it is not a three-term recurrence.
The condition $a_n \ne 0$ lets one solve for $u_{n+1}$, and $c_n \ne 0$ makes the order exactly two; so the solutions form a $K$-vector space of dimension $2$, fixed by $u_0$ and $u_1$.
A common normalized form is $u_{n+1} = (x - \alpha_n) u_n - \beta_n u_{n-1}$, which is the [[three-term recurrence for orthogonal polynomials|YRYWCP9TYV]].
</document>

<document source="https://tamnd.github.io/math2/node/XWMKYE038F.md" tag="XWMKYE038F" kind="concept">
# Dynamical billiards

## Summary

A dynamical billiard is the motion of a point particle in a region, called the table.
The particle moves in straight lines at constant speed.
At the boundary it bounces, with angle of reflection equal to angle of incidence.
The shape of the table alone decides the behaviour: some tables give regular, integrable motion, others give chaotic, ergodic motion.
Billiards are model systems for the ergodic hypothesis of statistical mechanics and for quantum chaos.
The [[Sinai billiard|GX9Q2PYNZE]] and the [[Bunimovich stadium|YYB1JP7TWJ]] are the classical chaotic examples.

## Explanations

### L1 human

Think of a billiard ball with no friction and no spin on a table of some shape.
It rolls straight, hits the cushion, and leaves at the same angle it came in, like light on a mirror.
On a round table every shot keeps the same angle to the cushion forever, so the motion is very regular.
On a table with a round obstacle in the middle, tiny changes of the first shot lead to completely different paths, and the motion is chaotic.
</document>

<document source="https://tamnd.github.io/math2/node/XWV6VS7BQM.md" tag="XWV6VS7BQM" kind="statement">
# Landau's necessary density conditions for sampling

## Statement

Let $d \ge 1$ and let $S \subset \R^d$ be a compact set of positive measure whose boundary has Lebesgue measure zero.
Let $PW_S$ be the space of [[band-limited functions|P3T104G564]] with spectrum in $S$: the $f \in L^2(\R^d)$ whose [[Fourier transform|DPVM34ADZV]] $\hat f(\xi) = \int f(x) e^{-i x \cdot \xi} \, dx$ vanishes outside $S$.
Let $\Lambda \subset \R^d$ be uniformly discrete, that is, $\inf \set{\abs{\lambda - \lambda'} : \lambda \ne \lambda' \in \Lambda} > 0$.
Define the lower and upper Beurling densities
$$D^-(\Lambda) = \liminf_{r \to \infty} \inf_{x \in \R^d} \frac{\#(\Lambda \cap (x + [0, r]^d))}{r^d}, \qquad D^+(\Lambda) = \limsup_{r \to \infty} \sup_{x \in \R^d} \frac{\#(\Lambda \cap (x + [0, r]^d))}{r^d}.$$

1. If $\Lambda$ is a sampling set for $PW_S$, that is, there are $0 < A \le B$ with $A \norm{f}_2^2 \le \sum_{\lambda \in \Lambda} \abs{f(\lambda)}^2 \le B \norm{f}_2^2$ for all $f \in PW_S$, then $D^-(\Lambda) \ge \abs{S} / (2\pi)^d$.
2. If $\Lambda$ is an interpolation set for $PW_S$, that is, for every square-summable $(a_\lambda)_{\lambda \in \Lambda}$ there is $f \in PW_S$ with $f(\lambda) = a_\lambda$ for all $\lambda$, then $D^+(\Lambda) \le \abs{S} / (2\pi)^d$.

Here $\abs{S}$ is the Lebesgue measure of $S$.

## Remarks

Henry Landau proved these conditions in 1967.
The proof compares the eigenvalues of two concentration operators: the operator that restricts $f \in PW_S$ to a large cube, and its discrete analogue on the samples in that cube.
The number of eigenvalues of the first operator that are not close to $0$ or $1$ is small compared with the volume of the cube.
The main count is $\abs{S} r^d / (2\pi)^d$.

Example: for $d = 1$ and $S = [-\pi, \pi]$, the integers form a sampling and interpolation set, and $D^-(\Z) = D^+(\Z) = 1 = \abs{S}/2\pi$.
This is the critical case of the [[Nyquist–Shannon sampling theorem|BMXDMHRXBG]].
So uniform sampling at the Nyquist rate is optimal, and no irregular scheme can sample stably at a lower average rate.

Since $f(\lambda) = (2\pi)^{-d} \int_S \hat f(\xi) e^{i\lambda \cdot \xi} \, d\xi$, the set $\Lambda$ is a sampling set if and only if the exponentials $e^{i\lambda \cdot \xi}$, $\lambda \in \Lambda$, form a frame of $L^2(S)$.
It is a sampling and interpolation set if and only if they form a [[Riesz basis|ZYDYD7FB1Z]] of $L^2(S)$, and then both parts give $D^-(\Lambda) = D^+(\Lambda) = \abs{S}/(2\pi)^d$.

## Explanations

### L1 human

To record a signal whose frequencies lie in a set $S$, one measures its values at a set of points.
Landau's theorem says that stable recording needs, on average, at least $\abs{S}/(2\pi)^d$ points per unit volume, however cleverly the points are placed.
And one cannot prescribe arbitrary values at more than that many points per unit volume.

### L2 human

The Shannon sampling theorem gives a sufficient condition for one regular lattice.
Landau's theorem gives a necessary condition for every uniformly discrete set, measured by the Beurling densities.
It shows that the Nyquist rate is sharp also for irregular sampling.
It also implies that a Riesz basis of exponentials for $L^2(S)$ must have uniform density exactly $\abs{S}/(2\pi)^d$, which links it with the theory of [[nonharmonic Fourier series|TKNA7PR99K]].
The theorem has analogues for Gabor systems and for other spaces of functions with localized transforms.
</document>

<document source="https://tamnd.github.io/math2/node/XWXZ0DGW3Z.md" tag="XWXZ0DGW3Z" kind="concept">
# Numerical range

## Summary

The numerical range of an operator $T$ on a complex Hilbert space is the set of values $\ip{Tx}{x}$ for unit vectors $x$.
It is a convex subset of the complex plane whose closure contains the spectrum.
It gives easy bounds for the spectrum and for the norm, and it characterises self-adjoint operators as those with real numerical range.

## Explanations

### L1 human

For a unit vector $x$, the number $\ip{Tx}{x}$ measures how much $Tx$ points back in the direction of $x$.
In quantum mechanics it is the expected value of the observable $T$ in the state $x$.
Collecting these numbers over all unit vectors gives the numerical range.
For a real symmetric $2 \times 2$ matrix with eigenvalues $1$ and $3$, the numerical range is the interval $[1, 3]$.
</document>

<document source="https://tamnd.github.io/math2/node/XWYXSRM5YS.md" tag="XWYXSRM5YS" kind="statement">
# Poncelet's closure theorem

## Statement

Let $C$ and $D$ be [[ellipses|0VKCCEXBJ1]] in the Euclidean plane with $D$ in the interior of $C$.
For $P_0 \in C$, build a sequence $P_0, P_1, P_2, \ldots$ on $C$ as follows.
From $P_k$ there are two [[tangent lines|01R1HW3Y42]] to $D$.
For $k \ge 1$, take the one other than the line $P_{k-1} P_k$; for $k = 0$, take either one.
Let $P_{k+1}$ be the second point where this tangent line meets $C$.
Suppose that for one starting point $P_0$ the sequence satisfies $P_n = P_0$ for some $n \ge 3$.
Then $P_n = P_0$ holds for every starting point $P_0 \in C$.

## Remarks

The polygon $P_0 P_1 \cdots P_{n-1}$ is inscribed in $C$ and circumscribed about $D$.
A modern proof considers the curve of pairs $(P, \ell)$ with $P \in C$ and $\ell$ a tangent line of $D$ through $P$.
Over the complex numbers this curve is an [[elliptic curve|427N6S6QQW]], and one step of the construction is a translation on it.
A translation that has a point of period $n$ has every point of period $n$.
Example: for two concentric circles with radii $R$ and $R/2$, every inscribed equilateral triangle is circumscribed about the inner circle, so every start closes after $3$ steps.
For a triangle and two circles, closure happens exactly when [[Euler's theorem|RFCQT6TKZF]] $d^2 = R(R - 2r)$ holds between the radii and the distance $d$ of the centres.

## Explanations

### L1 human

Draw an oval with a smaller oval inside it.
Start at a point of the outer oval and walk along straight lines that just graze the inner oval, bouncing from the outer one each time.
If, from one starting point, the path closes after some number of steps, then it closes after the same number of steps from every starting point.

### L2 human

Poncelet's theorem is a porism: the closing property is all or nothing.
It holds more generally for two smooth conics in general position in the complex projective plane.
Its proof through elliptic curves is a standard illustration of how the group law of a cubic curve solves a problem of elementary geometry.
</document>

<document source="https://tamnd.github.io/math2/node/XX5MY5CCKX.md" tag="XX5MY5CCKX" kind="definition">
# Enriched category

## Definition

Let $(\mathcal{V}, \otimes, I, \alpha, \lambda, \rho)$ be a [[monoidal category|M3MSQ3NKAE]].
A *$\mathcal{V}$-enriched category* (or $\mathcal{V}$-category) $\mathcal{C}$ consists of:

1. a class $\mathrm{Ob}(\mathcal{C})$ of objects;
2. for each pair of objects $a, b$, an object $\mathcal{C}(a, b)$ of $\mathcal{V}$, the hom-object;
3. for each triple $a, b, c$, a morphism $M_{a,b,c} \colon \mathcal{C}(b, c) \otimes \mathcal{C}(a, b) \to \mathcal{C}(a, c)$ of $\mathcal{V}$, the composition;
4. for each object $a$, a morphism $j_a \colon I \to \mathcal{C}(a, a)$ of $\mathcal{V}$, the identity.

These data satisfy, for all objects $a, b, c, d$:

1. associativity: $M_{a,b,d} \circ (M_{b,c,d} \otimes \id) = M_{a,c,d} \circ (\id \otimes M_{a,b,c}) \circ \alpha$ as morphisms $(\mathcal{C}(c, d) \otimes \mathcal{C}(b, c)) \otimes \mathcal{C}(a, b) \to \mathcal{C}(a, d)$;
2. units: $M_{a,b,b} \circ (j_b \otimes \id) = \lambda$ and $M_{a,a,b} \circ (\id \otimes j_a) = \rho$ as morphisms $I \otimes \mathcal{C}(a, b) \to \mathcal{C}(a, b)$ and $\mathcal{C}(a, b) \otimes I \to \mathcal{C}(a, b)$.

## Remarks

Example: for $\mathcal{V} = (\mathbf{Set}, \times, \set{*})$, a $\mathcal{V}$-category is a locally small [[category|A2DE66937D]]; for $\mathcal{V} = (\mathbf{Ab}, \otimes, \Z)$, it is a [[preadditive category|SC6FD5F4J0]].
Example: let $\mathcal{V} = [0, \infty]$, with a morphism $x \to y$ exactly when $x \ge y$, tensor product $+$ and unit $0$.
A $\mathcal{V}$-category is a set $X$ with numbers $d(x, y)$ such that $d(y, z) + d(x, y) \ge d(x, z)$ (composition) and $0 \ge d(x, x)$ (identities), so every [[metric space|0VKD1TP7F3]] is one.
Non-example: on $\R$, the function $d(x, y) = (x - y)^2$ does not give a $[0, \infty]$-category: composition fails, since $d(0, 1) + d(1, 2) = 2$ is less than $d(0, 2) = 4$.
The associativity and unit axioms are automatic in this example, because $[0, \infty]$ is a preorder.
</document>

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# Kontsevich formality theorem

## Statement

Let $M = \R^d$.
Let $T_{\mathrm{poly}}$ be the differential graded Lie algebra of polyvector fields on $M$, with zero differential and the [[Schouten–Nijenhuis bracket|FJ1MDBZ680]].
Let $D_{\mathrm{poly}}$ be the differential graded Lie algebra of polydifferential operators on $M$, with the Hochschild differential and the Gerstenhaber bracket.
Both carry the grading in which $k$-vector fields, and polydifferential operators with $k$ arguments, have degree $k - 1$.

1. There is an $L_\infty$ quasi-isomorphism $T_{\mathrm{poly}} \to D_{\mathrm{poly}}$ whose first component is the Hochschild–Kostant–Rosenberg map.
2. As a consequence, every [[Poisson manifold|A2KQ49Y4NF]] structure on $M$ admits a [[star product|3MKSJ1J04K]].
Moreover, equivalence classes of star products on $M$ correspond bijectively to classes of formal Poisson structures $\hbar \pi_1 + \hbar^2 \pi_2 + \cdots$ under formal diffeomorphisms.

## Remarks

The Hochschild–Kostant–Rosenberg map identifies the cohomology of $D_{\mathrm{poly}}$ with $T_{\mathrm{poly}}$, but it does not respect the brackets; the theorem corrects it by higher terms.
Kontsevich built these terms from integrals over configuration spaces of points in the upper half-plane, indexed by graphs.
Star products correspond to Maurer–Cartan elements of $\hbar D_{\mathrm{poly}}[\![\hbar]\!]$, formal Poisson structures to those of $\hbar T_{\mathrm{poly}}[\![\hbar]\!]$, and an $L_\infty$ quasi-isomorphism matches their equivalence classes.
For a constant Poisson bivector on $\R^{2n}$, [[Kontsevich's quantization formula|JZEK70CH3G]] gives the Moyal product.
The result was later extended from $\R^d$ to all smooth manifolds.

## Explanations

### L1 human

The theorem says that every Poisson manifold, a space with a classical bracket of observables, can be quantized in the formal sense.
One can always find a noncommutative product of functions whose first correction is the given bracket.
It also says that the two kinds of data, Poisson structures and star products, are classified by the same thing.
Maxim Kontsevich proved it with an explicit formula built from graphs.

### L2 human

Deformation theory says that deformations of an algebraic structure are governed by a differential graded Lie algebra through its Maurer–Cartan equation.
For the algebra of functions on $M$, this algebra is $D_{\mathrm{poly}}$, whose cohomology is $T_{\mathrm{poly}}$ by the Hochschild–Kostant–Rosenberg theorem.
Formality means that $D_{\mathrm{poly}}$ is equivalent, as an $L_\infty$ algebra, to its cohomology with the induced bracket.
Equivalent $L_\infty$ algebras have the same deformation theory, which gives the classification of star products.
The same circle of ideas links to the Gerstenhaber structure on Hochschild cohomology and to the Deligne conjecture.
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# Degree of exactness of a quadrature rule

## Definition

Let $Q(f) = \sum_{i=1}^{n} w_i f(x_i)$ be a [[quadrature rule|ADSVMCY77A]] on $[a, b]$, and let $I(f) = \int_a^b f(x)\, dx$.
The rule has *degree of exactness* $d \ge 0$ if:

1. $Q(p) = I(p)$ for every real [[polynomial|MAM9H3SY3Y]] $p$ of [[degree|0DPF6WRZW6]] at most $d$;
2. $Q(p) \ne I(p)$ for some polynomial $p$ of degree $d + 1$.

## Remarks

By linearity, clause 1 holds if and only if $Q(x^k) = I(x^k)$ for $k = 0, 1, \dots, d$.
The degree of exactness of an $n$-node rule is at most $2n - 1$: for $p(x) = \prod_i (x - x_i)^2$, of degree $2n$, one has $Q(p) = 0 < I(p)$.
A rule has degree of exactness at least $n - 1$ if and only if it is interpolatory.
Example: the midpoint rule $Q(f) = (b - a) f(m)$ with $m = (a + b)/2$ has degree $1$: it is exact for $1$ and $x - m$, while for $(x - m)^2$ it gives $0$ instead of $(b - a)^3/12$.
Non-example: the left endpoint rule $Q(f) = (b - a) f(a)$ does not have degree $1$, since for $p(x) = x - a$ it gives $0$ instead of $(b - a)^2/2$; its degree is $0$.
Some authors call it the degree of precision or the algebraic degree of accuracy.
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# Factorization of polynomials over finite fields

## Definition

Let $\F_q$ be the [[finite field|08RTDHX1SK]] with $q$ elements and let $f \in \F_q[x]$ be a nonzero polynomial.
Factoring $f$ over $\F_q$ means computing its [[factorization into irreducibles|EFDBS6ZME8]] $f = c \, p_1^{e_1} \cdots p_k^{e_k}$ in $\F_q[x]$.
The standard algorithms do this in three stages:

1. square-free factorization: write the monic part of $f$ as $g_1 g_2^2 \cdots g_s^s$, with the $g_i$ square-free and pairwise coprime, using $\gcd(f, f')$ for the [[formal derivative|0AD4VS02H3]] $f'$;
2. distinct-degree factorization: for a square-free monic $g$, compute for each $d \ge 1$ the product $h_d$ of the monic irreducible factors of $g$ of degree $d$;
3. equal-degree factorization: split each $h_d$ into its irreducible factors of degree $d$.

## Remarks

Stage 2 uses that $x^{q^d} - x$ is the product of all monic irreducible polynomials in $\F_q[x]$ whose degree divides $d$.
So $h_1 = \gcd(g, x^q - x)$, and $h_d$ is the [[gcd|99J1WXG3F7]] of $x^{q^d} - x$ with $g$ divided by $h_1 \cdots h_{d-1}$.
In stage 1, if $f' = 0$ then $f$ is a $p$-th power, where $p$ is the characteristic, since every element of $\F_q$ is a $p$-th power.
Stage 3 is usually done with a randomised method, such as the Cantor–Zassenhaus algorithm; [[Berlekamp's algorithm|EM8MFGFAGN]] is a deterministic alternative to stages 2 and 3.
Example: over $\F_2$, $g = x^3 + 1$ is square-free, $\gcd(g, x^2 - x) = x + 1$, so $h_1 = x + 1$ and $h_2 = x^2 + x + 1$.
Non-example: $(x + 1)(x^3 + 1) = (x + 1)^2 (x^2 + x + 1)$ is not square-free, so stage 2 does not apply before stage 1.

## Explanations

### L1 human

First take out repeated factors, then sort the factors by degree, then separate factors of the same degree.
Each stage uses only greatest common divisors and powers modulo $f$.
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<document source="https://tamnd.github.io/math2/node/XXQ92H7T83.md" tag="XXQ92H7T83" kind="concept">
# Delta derivative on a time scale

## Summary

The delta derivative is the derivative of [[time-scale calculus|30G1M35TK1]], defined for functions on a time scale, that is, on a nonempty closed subset of $\R$.
At a point that is a limit from the right it is the usual derivative.
At a point isolated on the right it is a difference quotient across the gap.
On $\R$ it is the ordinary [[derivative|0V1EJCT8G6]], and on $\Z$ it is the forward difference, so it unifies differential and difference equations.

## Explanations

### L1 human

Picture a time line in which some stretches are continuous and some moments are separated by gaps.
At a moment followed by a gap, the delta derivative compares the value now with the value at the next moment, and divides by the length of the gap.
At a moment with no gap to its right, it is the usual slope.
For example, on the time scale $2\Z = \set{\dots, -2, 0, 2, 4, \dots}$, the delta derivative of $f(t) = t^2$ is $\frac{(t+2)^2 - t^2}{2} = 2t + 2$.
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<document source="https://tamnd.github.io/math2/node/XXXSST2DR5.md" tag="XXXSST2DR5" kind="definition">
# Spectrum of an operator

## Definition

Let $X$ be a complex [[Banach space|06GZ956657]], let $I$ be its identity map, and let $T \colon X \to X$ be a [[continuous linear map|03TM135G0N]].
The *spectrum* of $T$ is the set
$$\sigma(T) = \set{\lambda \in \C : T - \lambda I \text{ is not invertible}},$$
where invertible means [[invertible as a bounded operator|CPAD5W1CGW]].

## Remarks

By the bounded inverse theorem, $\lambda \in \sigma(T)$ if and only if $T - \lambda I$ is not bijective.
If $X \ne \set{0}$, then $\sigma(T)$ is a nonempty compact subset of the closed disc $\set{\lambda : \abs{\lambda} \le \norm{T}}$.
For $X = \C^n$, $\sigma(T)$ is the [[spectrum of the matrix|0XPX5NC67E]] of $T$, its set of eigenvalues.

Example: let $D(x_n) = (x_n / n)$ on $\ell^2$.
For $\lambda \notin \set{1/n : n \ge 1} \cup \set{0}$, the numbers $1/(1/n - \lambda)$ are bounded, so the diagonal operator with these entries is a bounded inverse of $D - \lambda I$.
Each $1/n$ is an eigenvalue, and $D$ is not surjective, so $\sigma(D) = \set{1/n : n \ge 1} \cup \set{0}$.

Non-example: the number $2$ is not in $\sigma(D)$, since $\abs{1/(1/n - 2)} \le 1$ for all $n$, so $D - 2I$ has a bounded inverse.

For a [[closed operator|VM7B38PA87]] $T$ with domain $D(T)$, $\lambda$ is outside $\sigma(T)$ when $T - \lambda I \colon D(T) \to X$ is bijective with bounded inverse.
The spectrum is then closed, but it can be empty or unbounded.
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# Tilting theory

## Summary

Tilting theory compares the module categories of two algebras $A$ and $B = \End_A(T)$ by means of a tilting module $T$.
It generalizes [[Morita equivalence|RD5WKA1VK7]], which is the case where $T$ is a progenerator.
The two module categories are not equivalent in general, but they share large pieces, and by Happel's theorem their bounded [[derived categories|5ST7NVJFSM]] are equivalent.
It grew out of the [[reflection functors|MG4RNMA8AR]] of Bernstein, Gelfand and Ponomarev.

## Explanations

### L1 human

Two algebras can have different modules and still be closely related.
Tilting theory gives a way to pass from one to the other through a special module $T$, the tilting module.
Part of the modules of the first algebra correspond to part of the modules of the second, and the rest is "tilted" across.
At the level of derived categories, which keep track of complexes of modules, the two algebras look exactly the same.
This lets one transfer hard questions from a complicated algebra to a simpler one.
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<document source="https://tamnd.github.io/math2/node/XXZTMNBYFZ.md" tag="XXZTMNBYFZ" kind="definition">
# Knot concordance

## Definition

Let $K_0$ and $K_1$ be oriented tame [[knots|QYEAZTYGPT]] in $S^3$.
They are *concordant* if there is a smoothly and properly embedded annulus $A \cong S^1 \times [0, 1]$ in $S^3 \times [0, 1]$ such that $\partial A = (K_0 \times \set{0}) \cup (K_1 \times \set{1})$.
The orientation of $A$ must induce the given orientation on $K_0$ and the opposite one on $K_1$.
A *concordance class* is an equivalence class under this relation.

## Remarks

Isotopic knots are concordant, since the trace of an isotopy is a concordance.
A knot is concordant to the [[unknot|1TQ06R4Q66]] if and only if it is [[slice|P68G28E4AD]].
The concordance classes form an abelian group $\mathcal{C}$ under [[knot sum|AGZSZW7GYE]], with the class of the unknot as zero.
The inverse of $K$ is $K^*$, the mirror image of $K$ with reversed orientation.
Example: $K \# K^*$ is slice, so it is concordant to the unknot.
Non-example: the [[trefoil knot|VYTYSVHN4A]] is not concordant to the unknot, since its signature is nonzero.
Signature is additive under knot sum, so the trefoil has infinite order in $\mathcal{C}$.
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<document source="https://tamnd.github.io/math2/node/XY2MRKDA23.md" tag="XY2MRKDA23" kind="definition">
# Volterra equation of the second kind

## Definition

Let $a < b$, let $\Delta = \set{(t, s) : a \le s \le t \le b}$, let $K \colon \Delta \to \C$ be a [[kernel|GSMBXT7FPK]], let $f \colon [a, b] \to \C$ be a given function, and let $\lambda \in \C$.
A *linear Volterra equation of the second kind* is the equation
$$x(t) = f(t) + \lambda \int_a^t K(t, s) \, x(s) \, ds, \quad t \in [a, b],$$
for an unknown function $x$ on $[a, b]$.
The unknown appears outside the [[integral|0W29TBTY12]] with coefficient $1$, and the upper limit of integration is the variable $t$.
The equation is *homogeneous* when $f = 0$.

## Remarks

Example: $x(t) = t + \int_0^t (s - t) \, x(s) \, ds$ on $[0, 1]$, with $K(t, s) = s - t$ and $\lambda = 1$.
The function $x = \sin$ solves it: $\int_0^t (s - t) \sin s \, ds = \sin t - t$, so the right side is $\sin t$.
This is the integral form of $x'' = -x$, $x(0) = 0$, $x'(0) = 1$.
Non-example: $x(t) = f(t) + \int_a^b K(t, s) \, x(s) \, ds$ has the fixed upper limit $b$, so it is a Fredholm equation, not a Volterra one.
The usual standing hypotheses are that $K$ is continuous on $\Delta$ and $f$ is continuous on $[a, b]$; then the [[unique solvability theorem|WG4GVC9HR8]] applies.
Setting $K(t, s) = 0$ for $s > t$ shows that a Volterra equation is a Fredholm equation with a kernel that vanishes above the diagonal.
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# Automorphism

## Definition

Let $\mathbf{A}$ be a [[structure|983GZBQYT4]] for a signature $\sigma$.
An *automorphism* of $\mathbf{A}$ is an [[isomorphism|PZMGQMDN25]] from $\mathbf{A}$ to $\mathbf{A}$.
The set of automorphisms of $\mathbf{A}$ is written $\Aut(\mathbf{A})$.

## Remarks

Example: the map $z \mapsto \bar{z}$ is an automorphism of the field $(\C, +, \cdot, 0, 1)$, because it is a bijection of $\C$ to itself with $\overline{z + w} = \bar{z} + \bar{w}$, $\overline{zw} = \bar{z}\,\bar{w}$, $\bar{0} = 0$ and $\bar{1} = 1$.

Non-example: the map $x \mapsto 2x$ on $(\Z, +, 0)$ is an injective homomorphism but not an automorphism, because it is not surjective: $1$ is not in its image.

The set $\Aut(\mathbf{A})$ is a group under composition, with the identity map as identity element.
The only automorphism of the field $\R$ is the identity.
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<document source="https://tamnd.github.io/math2/node/XY6XAJN7Z7.md" tag="XY6XAJN7Z7" kind="definition">
# Cospherical points

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $S \subseteq X$ be a set of points.
The set $S$ is *cospherical* if there are a point $c \in X$ and a real number $r$ such that $d(x, c) = r$ for every $x \in S$.
That is, $S$ is contained in the [[sphere|P9CM4GQ2T0]] with centre $c$ and radius $r$.

## Remarks

This is the form of Mathlib's `EuclideanGeometry.Cospherical`.
In the Euclidean plane, cospherical points are the same as [[concyclic points|049DCJZ448]].
In $\R^n$, any $n + 1$ affinely independent points are cospherical, and their sphere is the [[circumsphere|MRC4WMVTAH]] of the simplex they span.

Example: the eight vertices $(\pm 1, \pm 1, \pm 1)$ of a cube in $\R^3$ are cospherical.
Each lies at distance $\sqrt 3$ from $c = (0, 0, 0)$.

Non-example: the points $(0, 0)$, $(1, 0)$, $(2, 0)$ of $\R^2$ are not cospherical.
A point equidistant from the first two has $x$-coordinate $1/2$, and a point equidistant from the last two has $x$-coordinate $3/2$, so no centre exists.
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<document source="https://tamnd.github.io/math2/node/XY7H05ANPD.md" tag="XY7H05ANPD" kind="definition">
# Symmetric kernel

## Definition

Let $a < b$.
A *symmetric kernel* on $[a, b]$ is a real-valued [[kernel|GSMBXT7FPK]] $K \colon [a, b] \times [a, b] \to \R$ such that
$$K(x, s) = K(s, x) \quad \text{for all } x, s \in [a, b].$$

## Remarks

Example: $K(x, s) = \min(x, s)$ on $[0, 1]$.
The check: the minimum of two numbers does not depend on their order.

Non-example: $K(x, s) = x - s$.
Here $K(1, 0) = 1$ and $K(0, 1) = -1$.

A symmetric kernel is a Hermitian kernel that takes real values.
For square-integrable kernels the condition is often required only for almost all $(x, s)$.
Some authors call a complex kernel with $K(x, s) = K(s, x)$ symmetric as well; that condition is different from the Hermitian one.
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# Bôcher's theorem

## Statement

Let $n \ge 2$, let $r > 0$, and let $u$ be a positive [[harmonic function|0DEB0JMN0W]] on the punctured ball $B(0, r) \setminus \set{0}$ in $\R^n$.
Then there are a constant $b \ge 0$ and a harmonic function $v$ on the whole ball $B(0, r)$ such that, for $0 < \abs{x} < r$,
$$u(x) = v(x) + b \log \frac{1}{\abs{x}} \quad (n = 2), \qquad u(x) = v(x) + \frac{b}{\abs{x}^{n-2}} \quad (n \ge 3).$$

## Remarks

So an isolated singularity of a positive harmonic function is a nonnegative multiple of the [[fundamental solution of the Laplacian|BSJ8X2XRN9]], up to a harmonic term.
If $b = 0$, the singularity is removable, in agreement with the [[removable singularity theorem|3H6SQA6JWM]].
Proof idea: averaging $u$ over the spheres about $0$ gives a radial harmonic function on the punctured ball.
It has the form $a + b\abs{x}^{2-n}$, or $a + b \log(1/\abs{x})$ in the plane.
Positivity of $u$ is then used to show that $b \ge 0$ and that the difference of $u$ and the singular term has a removable singularity at $0$.
Positivity is needed: $u(x) = x_1 / \abs{x}^n$ is harmonic on $\R^n \setminus \set{0}$, takes both signs, and is not of the stated form, since it grows like $\abs{x}^{1-n}$ near $0$.
Example: $u(x) = 1 + \abs{x}^{2-n}$ is positive on the punctured unit ball for $n \ge 3$, with $v = 1$ and $b = 1$.

## Explanations

### L1 human

A positive harmonic function can only blow up at an isolated point in one way: like the potential of a point charge.
Every other part of it is a perfectly regular harmonic function.
So positivity forbids wild singularities such as dipoles, which take both large positive and large negative values.

### L2 human

The theorem classifies isolated singularities of positive harmonic functions, and it is the local form of the Riesz decomposition of positive superharmonic functions.
A global corollary: a positive harmonic function on $\R^n \setminus \set{0}$, $n \ge 3$, has the form $a + b\abs{x}^{2-n}$ with $a, b \ge 0$, by combining Bôcher's theorem with the Kelvin transform and Liouville's theorem.
It is also the model for the description of the Martin boundary at an isolated boundary point.
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# Elliptic operator

## Summary

An elliptic operator is a linear differential operator whose [[principal symbol|F31T650DKY]] does not vanish at any nonzero covector.
The [[Laplacian|0E0B6RA8K0]] is the model example.
Elliptic operators have strong regularity properties: solutions of $Lu = f$ are as smooth as the data allow.
On compact manifolds they are Fredholm, which is the setting of index theory.

## Explanations

### L1 human

A linear differential operator is a sum of derivatives of a function, each multiplied by a coefficient.
The terms of highest order decide most of its behaviour, and the principal symbol records them by replacing each derivative $\partial_j$ by a variable $\xi_j$.
The operator is elliptic when this polynomial in $\xi$ vanishes only at $\xi = 0$.
For the Laplacian $\partial_1^2 + \partial_2^2$ the symbol is $\xi_1^2 + \xi_2^2$, which is zero only at the origin.
For the wave operator $\partial_t^2 - \partial_x^2$ the symbol $\tau^2 - \xi^2$ vanishes on the lines $\tau = \pm \xi$, so it is not elliptic.
Indeed, waves can carry sharp edges, while solutions of Laplace's equation are smooth.
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<document source="https://tamnd.github.io/math2/node/XYCD53PB5K.md" tag="XYCD53PB5K" kind="statement">
# Classification of regular polytopes

## Statement

Up to similarity, the convex [[regular polytopes|F60PD912D5]] of dimension $d$ are exactly the following, listed by their [[Schläfli symbols|MAE6BF4DFK]].

1. For $d = 2$: the [[regular polygon|01KJJ9A400]] $\set{p}$ for each integer $p \ge 3$.
2. For $d = 3$: the five [[Platonic solids|017E5PRS1Q]] $\set{3,3}$, $\set{4,3}$, $\set{3,4}$, $\set{5,3}$ and $\set{3,5}$.
3. For $d = 4$: six polytopes, the $5$-cell $\set{3,3,3}$, the tesseract $\set{4,3,3}$, the $16$-cell $\set{3,3,4}$, the [[24-cell|SH9NBM8JRR]] $\set{3,4,3}$, the [[120-cell|3RQEK50CP0]] $\set{5,3,3}$ and the [[600-cell|8MNQDK5X5W]] $\set{3,3,5}$.
4. For $d \ge 5$: three polytopes, the regular simplex $\set{3, \dots, 3}$, the [[hypercube|P9PYM71GMM]] $\set{4, 3, \dots, 3}$ and the [[cross-polytope|5DYYRZWER6]] $\set{3, \dots, 3, 4}$, each symbol having $d-1$ entries.

## Remarks

The facets and the vertex figures of a regular polytope are regular, so the possible symbols are built from shorter ones that are already known.
The new condition at each step is an angle condition: the facets around a face of dimension $d-3$ must have dihedral angles that add up to less than $2\pi$.
In dimension $3$ this gives $(p-2)(q-2) < 4$, which leaves the five Platonic symbols, as in the [[classification of the Platonic solids|0M4TCS0RRG]].
Each surviving symbol is then realised by an explicit construction, and the symbol determines the polytope up to similarity.

Example: the regular dodecahedron $\set{5,3}$ has dihedral angle $\arccos(-1/\sqrt5) \approx 116.57^\circ$.
Three dodecahedra around an edge use about $349.7^\circ < 360^\circ$, so $\set{5,3,3}$ is allowed, and it is the $120$-cell.
Four would use more than $360^\circ$, so there is no $\set{5,3,4}$.
For the cube, four around an edge use exactly $360^\circ$, so $\set{4,3,4}$ gives the cubic [[honeycomb|HTJ03RFKFH]] of space and not a polytope.

Convexity is needed for the list: allowing star polytopes adds the four [[Kepler–Poinsot polyhedra|CKQRZVBGTS]] in dimension $3$ and ten regular star polytopes in dimension $4$.

## Explanations

### L1 human

There are infinitely many regular polygons, one for each number of sides.
In three dimensions there are only five regular solids, and in four dimensions there are six.
From dimension five on, only three remain: the higher versions of the tetrahedron, the cube and the octahedron.
So the most symmetric shapes become rarer, not more common, as the dimension grows.

### L2 human

The theorem is proved by combining the [[Schläfli symbol|MAE6BF4DFK]] with an angle count, and dimension four is special because the dodecahedron and icosahedron still fit in.
The six four-dimensional polytopes include three without analogues elsewhere: the $24$-cell, which is self-dual, and the dual pair of the $120$-cell and the $600$-cell.
The symmetry groups of the regular polytopes are finite reflection groups: the dihedral groups and the types $A_d$, $B_d$, $F_4$, $H_3$ and $H_4$.
So the classification is tied to the classification of finite [[Coxeter groups|G6FND7X69J]].
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<document source="https://tamnd.github.io/math2/node/XYH30V13SE.md" tag="XYH30V13SE" kind="definition">
# Day convolution

## Definition

Let $(\mathcal{C}, \otimes, I)$ be a small [[monoidal category|M3MSQ3NKAE]], and let $[\mathcal{C}, \mathbf{Set}]$ be the [[functor category|1X0WKFQX8K]] of functors $\mathcal{C} \to \mathbf{Set}$.
The *Day convolution* of $F, G \in [\mathcal{C}, \mathbf{Set}]$ is the functor $F \ast G \colon \mathcal{C} \to \mathbf{Set}$ given by the [[coend|W9QTFV0Z8D]]
$$(F \ast G)(c) = \int^{a, b \in \mathcal{C}} \Hom_{\mathcal{C}}(a \otimes b, c) \times F(a) \times G(b).$$
With unit object the [[hom functor|VJXS8AK67E]] $\Hom_{\mathcal{C}}(I, -)$, Day convolution makes $[\mathcal{C}, \mathbf{Set}]$ a monoidal category.

## Remarks

Equivalently, $F \ast G$ is the left [[Kan extension|RN9D96ESWN]] of the functor $(a, b) \mapsto F(a) \times G(b)$ on $\mathcal{C} \times \mathcal{C}$ along $\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$.
Natural transformations $F \ast G \to H$ correspond to families of maps $F(a) \times G(b) \to H(a \otimes b)$ natural in $a$ and $b$.
Example: let $\mathcal{C}$ be the discrete category on $\N$ with $\otimes = +$ and $I = 0$, so a functor $F$ is a sequence of sets $F(0), F(1), \ldots$.
Since $\Hom(a + b, n)$ is a point when $a + b = n$ and empty otherwise, $(F \ast G)(n) = \bigsqcup_{a + b = n} F(a) \times G(b)$, the Cauchy product.
Non-example: the pointwise product $(F \times G)(c) = F(c) \times G(c)$ is also a monoidal structure, but it is not the Day convolution.
In the example above it pairs elements of equal degree, and its unit is the constant functor with value a point, not $\Hom(0, -)$.
The monoidal category $[\mathcal{C}, \mathbf{Set}]$ with Day convolution is [[closed|EKJBDD07TY]], and $a \mapsto \Hom_{\mathcal{C}}(a, -)$ is a strong monoidal functor $\mathcal{C}^{\mathrm{op}} \to [\mathcal{C}, \mathbf{Set}]$.
The same construction works for functors into a cocomplete closed symmetric monoidal category $\mathcal{V}$ in place of $\mathbf{Set}$.
</document>

<document source="https://tamnd.github.io/math2/node/XYJ6T5CKR6.md" tag="XYJ6T5CKR6" kind="concept">
# Umbilical point

## Summary

An umbilical point, or umbilic, of a surface is a point where the two [[principal curvatures|FA8QE1SGP4]] are equal.
There the surface bends by the same amount in every direction, like a sphere or a plane.
Every point of a sphere or a plane is umbilical.
Conversely, a connected surface made only of umbilical points is a piece of a sphere or of a plane.

## Explanations

### L1 human

At most points of a surface there is a direction of strongest bending and a direction of weakest bending.
At an umbilical point there is no preferred direction: the surface looks the same in every direction, up to second order.
On a ball every point is like this.
An egg-shaped ellipsoid with three different axes has exactly four such special points.
</document>

<document source="https://tamnd.github.io/math2/node/XYPJ07PT5G.md" tag="XYPJ07PT5G" kind="definition">
# Almost periodic function

## Definition

Let $G$ be an abelian [[topological group|92700QAB5Y]], written additively.
Let $C_b(G)$ be the space of bounded [[continuous functions|0X3WGX95FE]] $G \to \C$ with the norm $\norm{f}_\infty = \sup_{x \in G} \abs{f(x)}$.
For $f \in C_b(G)$ and $a \in G$, let $f_a(x) = f(x + a)$ be the translate of $f$ by $a$.
A function $f \in C_b(G)$ is *almost periodic* if the set of its translates $\set{f_a : a \in G}$ is relatively [[compact|0R6EWFKRCF]] in $C_b(G)$.
This means that its closure in the norm $\norm{\cdot}_\infty$ is compact.

## Remarks

Equivalently, every sequence of translates $f_{a_1}, f_{a_2}, \dots$ has a subsequence that converges uniformly on $G$.
This is Bochner's form of the definition.
For $G = \R$ it is equivalent to Bohr's definition through relatively dense sets of almost periods, see [[Bohr almost periodic function|ED69H4C3PK]].
For a locally compact abelian group $G$, the almost periodic functions are exactly the functions $F \circ b$, where $b \colon G \to bG$ is the [[Bohr compactification|G852FHZ3W1]] and $F$ is continuous on $bG$.

Example: on $\R$, let $f(x) = \cos x + \cos(\sqrt{2} x)$.
The translates of $\cos$ are $\cos(x + a) = \cos a \cos x - \sin a \sin x$, the image of the circle $\set{(\cos a, \sin a)}$ under a continuous map into $C_b(\R)$, so they form a compact set.
The same holds for $\cos(\sqrt{2} x)$, and the translates of $f$ lie in the sum of these two compact sets, which is compact.
So $f$ is almost periodic.

Non-example: $f(x) = \arctan x$ on $\R$ is bounded and continuous, but not almost periodic.
The translates $f_n$, $n = 1, 2, \dots$, converge pointwise to the constant $\pi/2$.
A uniformly convergent subsequence would have the same limit, but $\abs{f_n(-n) - \pi/2} = \pi/2$ for every $n$.
</document>

<document source="https://tamnd.github.io/math2/node/XYTN19EPF2.md" tag="XYTN19EPF2" kind="concept">
# Sequence space

## Summary

A sequence space is a vector space whose elements are infinite sequences of real or complex numbers, with termwise addition and scalar multiplication.
Standard examples are the [[spaces ℓ^p|08P6T2WH6R]], the [[space of convergent sequences|GWCBGYG5RA]] and the [[space of null sequences|WDH6QSH1Z6]].
Sequence spaces give the simplest examples of infinite-dimensional normed and topological vector spaces.
In summability theory, the sequences summable by a matrix method form a sequence space, and functional analysis is used to study it.

## Explanations

### L1 human

Think of a sequence as a vector with infinitely many coordinates.
Adding two sequences term by term and multiplying a sequence by a number work just as for vectors in $\R^n$.
A sequence space is any collection of sequences that is closed under these two operations.
For example, the sequences that tend to $0$ form a sequence space, because the sum of two such sequences also tends to $0$.
</document>

<document source="https://tamnd.github.io/math2/node/XYVPYNVYBX.md" tag="XYVPYNVYBX" kind="example">
# List of small groups

## Example

Up to isomorphism, the [[groups|G5S1M9]] of order $n \le 15$ are the following.
Here $C_n$ is the [[cyclic group|0KVGH0J5B3]] of order $n$ and $\times$ is the [[direct product|00HF4RJ5QN]].
The group $D_{2m}$ is the [[dihedral group|0JZ27DMJ5D]] of order $2m$, and $Q_8$ is the [[quaternion group|0NRJGW864D]].
The group $A_4$ is the [[alternating group|0Z5905Q73D]] on $4$ letters, and $\mathrm{Dic}_{12} = C_3 \rtimes C_4$ is the dicyclic group of order $12$.

1. Orders $1$, $2$, $3$, $5$, $7$, $11$, $13$: only $C_n$.
2. Order $4$: $C_4$ and the [[Klein four-group|0PMRQJVTME]] $C_2 \times C_2$.
3. Order $6$: $C_6$ and $D_6 \cong S_3$, the [[symmetric group|0515A6D2Y1]].
4. Order $8$: $C_8$, $C_4 \times C_2$, $C_2 \times C_2 \times C_2$, $D_8$ and $Q_8$.
5. Order $9$: $C_9$ and $C_3 \times C_3$.
6. Order $10$: $C_{10}$ and $D_{10}$.
7. Order $12$: $C_{12}$, $C_6 \times C_2$, $A_4$, $D_{12}$ and $\mathrm{Dic}_{12}$.
8. Order $14$: $C_{14}$ and $D_{14}$.
9. Order $15$: only $C_{15}$.

So the numbers of groups of orders $1, 2, \dots, 15$ are $1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1$.
For order $16$ there are already $14$ groups.

## Remarks

Groups of prime order $p$ are cyclic, which explains the first line.
For primes $p < q$, a group of order $pq$ is cyclic unless $p$ divides $q - 1$.
This explains why $15 = 3 \cdot 5$ gives only $C_{15}$, while $6$, $10$ and $14$ also give a dihedral group.
A group of order $p^2$ is abelian, so it is $C_{p^2}$ or $C_p \times C_p$, as for orders $4$ and $9$.
In each order the abelian groups are given by the [[classification of finite abelian groups|07CQR9BW3R]].

## Explanations

### L1 human

This is the start of a catalogue of all finite groups, sorted by size.
Small sizes allow few groups: a prime number of elements forces a cyclic group.
The first interesting sizes are $8$ and $12$, which already show non-abelian groups such as the quaternion group and the alternating group $A_4$.
</document>

<document source="https://tamnd.github.io/math2/node/XYWMB0G011.md" tag="XYWMB0G011" kind="definition">
# Conservation law

## Definition

Let $n, m \ge 1$ and let $F = (F_1, \dots, F_n)$ with each $F_j \colon \R^m \to \R^m$ a $C^1$ function, called the *flux*.
The *system of conservation laws* with flux $F$ is the [[partial differential equation|GC5BDZ6QBC]]
$$\partial_t u + \sum_{j=1}^n \partial_{x_j} \big( F_j(u) \big) = 0$$
for an unknown $u \colon \R^n \times (0, \infty) \to \R^m$, where $\partial_t$ and $\partial_{x_j}$ are [[partial derivatives|09EGC6F8PT]].
For $m = 1$ it is called a *scalar conservation law*.
The components of $u$ are the *conserved quantities*.

## Remarks

The name comes from the integral form.
Let $u$ be a $C^1$ solution, and let $V \subseteq \R^n$ be a bounded region with smooth boundary and outer unit normal $\nu$.
Then
$$\frac{d}{dt} \int_V u \, dx = - \int_{\partial V} \sum_{j=1}^n F_j(u)\, \nu_j \, dS,$$
by the divergence theorem, so the total amount of $u$ in $V$ changes only through the flux across $\partial V$.
The integral form, or the equivalent weak form with test functions, still makes sense for discontinuous $u$, and it is the basis of weak solutions.

Example: the inviscid [[Burgers' equation|NEGHXN8RE1]] $u_t + (u^2/2)_x = 0$ is a scalar conservation law with $n = m = 1$ and $F(u) = u^2/2$.
The Euler equations of gas dynamics are a system with $m = n + 2$ conserved quantities: mass, momentum and energy.

Non-example: the equation $u_t + x\, u_x = 0$ is not of conservation form with a flux depending on $u$ only.
Indeed it equals $u_t + (x u)_x = u$, so the amount of $u$ is not conserved: $\int_\R u(x, t)\, dx$ grows like $e^{t}$ for integrable solutions, as $u(x, t) = g(x e^{-t})$ shows.
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<document source="https://tamnd.github.io/math2/node/XYZWTQD6NN.md" tag="XYZWTQD6NN" kind="definition">
# Skeleton of a category

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
A *skeleton* of $\mathcal{C}$ is a [[full subcategory|V3BFGKBJF7]] $\mathcal{S}$ of $\mathcal{C}$ such that every object of $\mathcal{C}$ is [[isomorphic|859F0TFVA3]] to exactly one object of $\mathcal{S}$.

## Remarks

A category is *skeletal* if isomorphic objects in it are equal; a skeleton is a skeletal full subcategory that meets every isomorphism class.
Under the axiom of choice every category has a skeleton, and its inclusion is fully faithful and essentially surjective.
So the inclusion is an [[equivalence of categories|K7A04ZBD92]] by [[the criterion for equivalences|WA60RW9E04]].
Any two skeletons of $\mathcal{C}$ are isomorphic categories.
Example: in the category of finite sets, the full subcategory on the sets $[n] = \set{1, \dots, n}$, $n \ge 0$, with $[0] = \emptyset$, is a skeleton.
A finite set with $n$ elements is in bijection with $[n]$, and $[m]$ and $[n]$ are not in bijection when $m \ne n$.
Non-example: the full subcategory on the sets $[n]$ together with $\set{a, b}$ is not a skeleton, since $\set{a, b}$ and $[2]$ are two of its objects in the same isomorphism class.
</document>

<document source="https://tamnd.github.io/math2/node/XZ1EXQPJ6S.md" tag="XZ1EXQPJ6S" kind="statement">
# Generalized mean inequality

## Statement

Let $x_1, \dots, x_n > 0$ be real numbers and let $w_1, \dots, w_n > 0$ be weights with $w_1 + \dots + w_n = 1$.
For a real number $p \ne 0$, let
$$M_p = \Big( \sum_{i=1}^{n} w_i x_i^{p} \Big)^{1/p}, \qquad M_0 = \prod_{i=1}^{n} x_i^{w_i}.$$
If $p < q$, then $M_p \le M_q$, with equality if and only if $x_1 = x_2 = \dots = x_n$.

## Remarks

Proof idea: for $0 < p < q$, the function $t \mapsto t^{q/p}$ is strictly convex on $(0, \infty)$, and [[Jensen's inequality|0RWMRTPC87]] applied to the numbers $x_i^p$ gives $M_p^q \le M_q^q$.
The case $p < q < 0$ follows by applying this to the numbers $1 / x_i$, since $M_{-r}(x) = 1 / M_r(1/x)$.
The cases with $p = 0$ or $q = 0$ follow from the weighted [[AM–GM inequality|0DRF0F3XC9]], or from the limit $M_p \to M_0$ as $p \to 0$.

Example: the cases $p = -1, 0, 1, 2$ give harmonic mean $\le$ geometric mean $\le$ arithmetic mean $\le$ quadratic mean.
For $x = (1, 4)$ with equal weights these are $8/5$, $2$, $5/2$ and $\sqrt{17/2} \approx 2.92$.

Mathlib states a related form as `Real.rpow_arith_mean_le_arith_mean_rpow`.

## Explanations

### L1 human

There are many ways to average positive numbers: the usual average, the geometric mean, the root mean square, and more.
They form one family indexed by an exponent $p$.
The inequality says that a larger exponent always gives a larger average, unless all the numbers are equal.

### L2 human

The power mean inequality packages the AM–GM inequality, the AM–HM inequality and the AM–QM inequality into one statement.
It is a direct consequence of Jensen's inequality for the power functions.
As $p \to \infty$ the mean $M_p$ tends to $\max_i x_i$, and as $p \to -\infty$ it tends to $\min_i x_i$, so all the means lie between the smallest and the largest number.
</document>

<document source="https://tamnd.github.io/math2/node/XZ33P3HBB9.md" tag="XZ33P3HBB9" kind="concept">
# Radial basis function interpolation

## Summary

Radial basis function interpolation fits scattered data in $\R^d$ with a combination of translates of one [[radial basis function|ZBSXSTQM2J]].
There is one translate centred at each data point.
The coefficients solve a square linear system whose matrix collects the values of the radial function at all pairwise distances.
For many radial functions, such as the Gaussian and the inverse multiquadric, this matrix is invertible for every set of distinct points in every dimension.
This makes it one of the main methods of [[multivariate interpolation|BM703GV34G]], free of the restrictions on point position that polynomial interpolation has.

## Explanations

### L1 human

Place a bump at each data point, all of the same shape, and choose their heights so that the sum of the bumps passes through all the data values.
Finding the heights means solving a linear system with one equation per data point.
In one dimension with the function $\varphi(r) = r$, the sum of shifted copies of $\abs{x - x_j}$ is a piecewise linear function, which gives the broken line through the data.
The [[definition|NH59FNJQG1]] writes down the system.
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<document source="https://tamnd.github.io/math2/node/XZ3BMJARBR.md" tag="XZ3BMJARBR" kind="definition">
# Fermat point

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
The *Fermat point* of $ABC$ is the unique point $F$ of the plane that minimises
$$f(P) = d(P, A) + d(P, B) + d(P, C).$$

## Remarks

A minimiser exists because $f$ is continuous and tends to infinity far away; it is unique because $f$ is strictly convex when $A$, $B$, $C$ are not collinear.
If each angle of $ABC$ is less than $120^\circ$, then $F$ is the interior point with $\angle AFB = \angle BFC = \angle CFA = 120^\circ$.
In that case $F$ is also the common point of the lines that join each vertex to the apex of the outward equilateral triangle on the opposite side.
If the angle at a vertex is at least $120^\circ$, then $F$ is that vertex.
The point is also called the Torricelli point.

Example: in an equilateral triangle the centre $O$ sees each side under $120^\circ$ by symmetry, so $F = O$.

Non-example: take $A = (0, 0)$, $B = (1, 0)$, $C = (0, 1)$ and the [[centroid|0X7EEWES5G]] $G = (1/3, 1/3)$.
Then $f(G) = (\sqrt 2 + 2\sqrt 5)/3 \approx 1.962$.
All angles are below $120^\circ$, and the minimum of $f$ is $\sqrt{2 + \sqrt 3} \approx 1.932$, so $G$ is not the Fermat point.
</document>

<document source="https://tamnd.github.io/math2/node/XZ6NNH7JED.md" tag="XZ6NNH7JED" kind="definition">
# Morphism

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $A$, $B$ be objects of $\mathcal{C}$.
A *morphism* from $A$ to $B$ is a member $f$ of $\Hom_{\mathcal{C}}(A, B)$, written $f \colon A \to B$.
The object $A$ is its domain and the object $B$ is its codomain.
Special kinds of morphisms are defined as follows:

1. $f \colon A \to B$ is an *isomorphism* if there is $g \colon B \to A$ with $g \circ f = 1_A$ and $f \circ g = 1_B$;
2. $f$ is an *endomorphism* if $A = B$;
3. $f$ is an *automorphism* if it is both an endomorphism and an isomorphism.

## Remarks

The morphism $g$ in clause 1 is unique: if $g'$ also works, then $g' = g' \circ (f \circ g) = (g' \circ f) \circ g = g$.
It is called the inverse $f^{-1}$, and $A$ and $B$ are then *isomorphic*, written $A \cong B$.
Example: in the category of groups, the morphisms are the [[group homomorphisms|0JYTG5K9Y4]].
The isomorphisms are the bijective homomorphisms, since the inverse map of a bijective homomorphism is again a homomorphism.
Non-example: in the category of topological spaces with [[continuous maps|0VGFHTM75Y]], let $X$ be $[0, 1]$ with the discrete topology and $Y$ be $[0, 1]$ with the usual topology.
The identity map $X \to Y$ is a continuous bijection, but its inverse $Y \to X$ is not continuous, so it is not an isomorphism.
The endomorphisms of $A$ form a monoid $\End(A)$ under composition, and the automorphisms form a group $\Aut(A)$.
</document>

<document source="https://tamnd.github.io/math2/node/XZBPKQWD0P.md" tag="XZBPKQWD0P" kind="concept">
# Toric ideal

## Summary

A toric ideal is the ideal of all polynomial relations among a finite set of monomials in Laurent variables.
It is a prime ideal generated by binomials, that is, by differences of two monomials.
Its zero set is an affine toric variety, so toric ideals connect commutative algebra with polyhedral geometry and with integer programming and algebraic statistics.

## Explanations

### L1 human

Take some monomials, such as $s$, $st$ and $st^2$.
They satisfy relations: here $s \cdot st^2 = (st)^2$.
Naming the monomials $x_1, x_2, x_3$, the relation becomes $x_1 x_3 - x_2^2 = 0$.
The toric ideal collects all such relations, and each relation says that two products of the monomials have the same exponent vector.
So finding the toric ideal is a question about linear relations among integer vectors.
</document>

<document source="https://tamnd.github.io/math2/node/XZF9BFJFXB.md" tag="XZF9BFJFXB" kind="statement">
# Erdős–Stone theorem

## Statement

For integers $r \ge 1$ and $t \ge 1$, let $K_{r+1}(t)$ be the complete $(r+1)$-partite [[graph|00C0BR7D7H]] with $r + 1$ parts of $t$ vertices each.
Let $r \ge 1$ and $t \ge 1$ be integers and let $\varepsilon > 0$ be a real number.
Then there is an integer $n_0$ such that every graph with $n \ge n_0$ vertices and at least $\left(1 - \frac{1}{r} + \varepsilon\right) \frac{n^2}{2}$ edges contains a [[subgraph|0WBFFC8XZ9]] isomorphic to $K_{r+1}(t)$.

## Remarks

The main consequence is a formula for the [[extremal number|DRGS5TT784]].
Let $H$ be a graph with [[chromatic number|0VHJRXNAAS]] $\chi(H) = r + 1 \ge 2$.
Then $H$ is a subgraph of $K_{r+1}(t)$ for $t = |V(H)|$, and the [[Turán graph|0VTCABADBE]] $T(n, r)$ is $H$-free, so
$$\mathrm{ex}(n, H) = \left(1 - \frac{1}{r}\right) \frac{n^2}{2} + o(n^2) \quad (n \to \infty).$$
For example, the Petersen graph has chromatic number $3$, so a graph on $n$ vertices with no Petersen subgraph has at most $n^2/4 + o(n^2)$ edges.
For bipartite $H$ ($r = 1$) the formula only says $\mathrm{ex}(n, H) = o(n^2)$; the true order is the subject of the [[Zarankiewicz problem|KT9WJFRQ3F]].
A modern proof applies the [[Szemerédi regularity lemma|ETMAPNQY7T]] to find a dense $K_{r+1}$ in the reduced graph, and then a counting lemma to blow it up to $K_{r+1}(t)$.

## Explanations

### L1 human

[[Turán's theorem|0JG1KZSPFZ]] says how many edges force a complete graph $K_{r+1}$.
The Erdős–Stone theorem says that only a tiny bit more forces a much larger graph: a "blown-up" $K_{r+1}$ with many vertices in each part.
As a result, the largest number of edges avoiding any fixed graph $H$ is governed, up to a small error, by its chromatic number alone.

### L2 human

The theorem is often called the fundamental theorem of [[extremal graph theory|SC5QB5F4TE]].
It gives $\mathrm{ex}(n, H) / \binom{n}{2} \to 1 - \frac{1}{\chi(H) - 1}$ for every graph $H$ with at least one edge.
It is sharp for non-bipartite $H$ and degenerate for bipartite $H$, where the extremal number is $o(n^2)$.
Stability results refine it: an $H$-free graph with nearly $\mathrm{ex}(n, H)$ edges is close in structure to the Turán graph.
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<document source="https://tamnd.github.io/math2/node/XZFYB2PBPD.md" tag="XZFYB2PBPD" kind="statement">
# Fenchel–Moreau theorem

## Statement

Let $X$ be a real [[normed space|03J5RAKB30]] with [[continuous dual space|4A82S5FVRT]] $X^*$.
Let $f \colon X \to [-\infty, +\infty]$ be a function with values in the [[extended real line|0GSZ7B0CGR]], with [[Fenchel conjugate|T7V2VPY61S]] $f^* \colon X^* \to [-\infty, +\infty]$.
Define the biconjugate $f^{**} \colon X \to [-\infty, +\infty]$ by $f^{**}(x) = \sup_{y \in X^*} \big(y(x) - f^*(y)\big)$.
Then $f^{**} = f$ if and only if one of the following holds:

1. $f$ is proper, [[convex|07Q7CNM5WF]] and lower [[semicontinuous|RN4G8SGP6P]], where proper means that $f$ never takes the value $-\infty$ and is not identically $+\infty$;
2. $f$ is identically $+\infty$;
3. $f$ is identically $-\infty$.

## Remarks

For every $f$, the function $f^{**}$ is a supremum of continuous affine functions, so it is convex and lower semicontinuous, and the Fenchel–Young inequality gives $f^{**} \le f$.
If $f$ takes the value $-\infty$ somewhere, then $f^*$ is identically $+\infty$ and $f^{**}$ is identically $-\infty$, so $f^{**} = f$ forces case 3.
Otherwise $f^{**} = f$ forces $f$ to be convex and lower semicontinuous, which is case 1 or case 2.
For the "if" direction, take $x_0$ and a real $r < f(x_0)$.
The point $(x_0, r)$ lies outside the closed convex [[epigraph|8BJ5M76FSX]] of $f$, and the Hahn–Banach separation theorem gives a continuous affine minorant of $f$ with value above $r$ at $x_0$.
Every such minorant is below $f^{**}$, so $f^{**}(x_0) \ge f(x_0)$.

Example: $f(x) = \abs{x}$ on $\R$ has $f^* = 0$ on $[-1, 1]$ and $+\infty$ elsewhere, and $f^{**}(x) = \sup_{\abs{y} \le 1} xy = \abs{x}$.
Lower semicontinuity is necessary: for $f = 0$ on $(0, 1)$ and $f = +\infty$ elsewhere on $\R$, the function is convex and proper, but $f^{**}$ is $0$ on the closed interval $[0, 1]$.
Convexity is necessary: for $f(x) = \min\big((x - 1)^2, (x + 1)^2\big)$, the biconjugate is $0$ on $[-1, 1]$ and agrees with $f$ outside.

## Explanations

### L1 human

Taking the conjugate twice returns the function you started with, as long as the function is convex and has no gaps in its graph from below.
In general, the double conjugate is the largest convex, lower semicontinuous function that stays below $f$.
So a convex closed function is completely described by the family of lines that lie below it.

### L2 human

The theorem is the basis of duality in convex optimization.
A primal problem and its dual are related through conjugates, and the theorem says that dualizing twice gives back the primal problem.
It also shows that $f \mapsto f^*$ is a bijection between proper convex lower semicontinuous functions on $X$ and on the dual side, when $X$ is reflexive.
It extends the involution property of the classical Legendre transformation to nonsmooth functions.
The same statement holds in every Hausdorff locally convex space.
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# Homotopy equivalence

## Summary

A homotopy equivalence is a continuous map $f \colon X \to Y$ that has an inverse up to homotopy.
This is a map $g \colon Y \to X$ with $g \circ f$ [[homotopic|F2F143KQ2M]] to $\id_X$ and $f \circ g$ homotopic to $\id_Y$.
It is a weaker notion than a [[homeomorphism|0QT00PB5G2]].
Homotopy invariants, such as the fundamental group and homology, cannot distinguish spaces that are related by a homotopy equivalence.

## Explanations

### L1 human

A homotopy equivalence says that two spaces have the same shape once you allow stretching, shrinking and squashing, but no tearing.
A thick object can be squashed to a thin one: a solid cylinder squashes to a disk, and further to a point.
For example, the plane without the origin is homotopy equivalent to the circle: push each point radially onto the unit circle.
The two spaces are not homeomorphic, but they have the same holes.
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# Kummer theory

## Summary

Kummer theory describes the abelian extensions of exponent dividing $n$ of a field $K$ that contains the $n$-th [[roots of unity|0YV2CKP2AV]].
Every such extension is obtained by adjoining $n$-th roots of elements of $K$, and these extensions correspond to subgroups of $K^\times$ containing $K^{\times n}$.
The Galois group is dual to the corresponding group of radicals through the Kummer pairing.
Kummer theory is a basic tool in [[class field theory|BEKCEYHCHX]] and in [[Iwasawa theory|Y2SPPSQ5D4]].

## Explanations

### L1 human

Let $K$ be a field of characteristic other than $2$, so that $\pm 1$ are two distinct square roots of unity.
Then every extension of $K$ of degree $2$ is obtained by adjoining a square root, as with $\Q(\sqrt{2})$ over $\Q$.
Kummer theory is the same idea for $n$-th roots.
Suppose that $K$ contains $n$ distinct $n$-th roots of unity.
Then adjoining an $n$-th root $\sqrt[n]{a}$ gives a cyclic extension.
All abelian extensions whose Galois group has exponent dividing $n$ arise by adjoining several such roots.
For example, $\Q(i)$ contains the fourth roots of unity, and $\Q(i, \sqrt[4]{2})$ is a cyclic extension of $\Q(i)$ of degree $4$.
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# Tight closure

## Definition

Let $p$ be a prime and let $R$ be a [[Noetherian ring|0670T59HGE]] of [[characteristic|0F6ES3BRYP]] $p$.
Let $R^\circ$ be the set of elements of $R$ that lie in no [[minimal prime ideal|S2D8823P8B]] of $R$.
For an [[ideal|096Z9MP2ME]] $I$ of $R$ and a power $q = p^e$ with $e \ge 0$, let $I^{[q]}$ be the ideal generated by the elements $a^q$ with $a \in I$.
An element $x \in R$ is in the *tight closure* $I^*$ of $I$ if there is $c \in R^\circ$ such that $c x^q \in I^{[q]}$ for all $q = p^e$ with $e$ large enough.

## Remarks

The ideal $I^{[q]}$ is the ideal generated by the image of $I$ under the $e$-th power of the [[Frobenius endomorphism|0T3WVTD675]].
If $I = (a_1, \ldots, a_m)$, then $I^{[q]} = (a_1^q, \ldots, a_m^q)$.
$I^*$ is an ideal, $I \subseteq I^*$, and $(I^*)^* = I^*$.
The ideal $I$ is *tightly closed* if $I^* = I$.

Example: let $k$ be a field of characteristic $p$, let $R = k[x, y]/(x^2)$, and let $I = (0)$.
The only minimal prime is $(x)$, so $1 \in R^\circ$, and $1 \cdot x^q = 0 \in I^{[q]}$ for $q \ge 2$.
So $x \in (0)^*$; more generally every [[nilpotent element|0ZB2JG1M9C]] lies in $(0)^*$.

Non-example: let $R = k[x]$ and $I = (x^2)$; then $x \notin I^*$.
Indeed $c x^q \in (x^{2q})$ means $x^q$ divides $c$, and if this holds for infinitely many $q$ then $c = 0$, which is not in $R^\circ$.
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# Mathematical statistics

## Summary

Mathematical statistics is the part of mathematics that studies how to learn about an unknown probability distribution from data.
It models data as the values of random variables whose law belongs to a [[statistical model|JZ6R7J36K5]].
It then studies procedures such as estimators, tests and confidence sets through their behaviour under each law of the model.
It gives the rigorous basis for [[statistical inference|FXCQWFY5RV]].

## Explanations

### L1 human

Data are noisy, so a conclusion drawn from data can be wrong.
Mathematical statistics uses probability theory to measure how often a method of drawing conclusions goes wrong.
Picture a coin with an unknown chance $p$ of heads, tossed $100$ times.
The proportion of heads is a natural guess for $p$, and mathematical statistics tells how far this guess typically lies from $p$: about $\sqrt{p(1-p)/100}$, at most $0.05$.
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# Primitive recursive function

## Definition

The class of *primitive recursive functions* is the smallest class of [[functions|0EPAT51DXQ]] $f \colon \N^k \to \N$, for $k \ge 0$, with the following properties.

1. It contains the constant function $0$ of arity $0$, and the successor function $S(x) = x + 1$.
2. It contains the projections $P^k_i(x_1, \ldots, x_k) = x_i$ for $1 \le i \le k$.
3. It is closed under composition: if $g \colon \N^m \to \N$ and $h_1, \ldots, h_m \colon \N^k \to \N$ are in the class, so is $\bar x \mapsto g(h_1(\bar x), \ldots, h_m(\bar x))$.
4. It is closed under primitive recursion: if $g \colon \N^k \to \N$ and $h \colon \N^{k+2} \to \N$ are in the class, so is the function $f \colon \N^{k+1} \to \N$ given below.

$$f(0, \bar x) = g(\bar x), \qquad f(y + 1, \bar x) = h(y, f(y, \bar x), \bar x).$$

Here $\N = \set{0, 1, 2, \ldots}$ is the set of [[natural numbers|0MWKJ44V5A]].

## Remarks

Example: addition is primitive recursive.
Take $g = P^1_1$ and $h(y, z, x) = S(P^3_2(y, z, x))$, which are in the class by clauses 1, 2 and 3.
Clause 4 gives $f(0, x) = x$ and $f(y + 1, x) = f(y, x) + 1$, so $f(y, x) = x + y$.
In the same way, multiplication, exponentiation, the factorial and truncated subtraction are primitive recursive.

Non-example: the [[Ackermann function|N470JF7ZJX]] is total and computable but not primitive recursive.
Its diagonal $n \mapsto A(n, n)$ eventually exceeds every unary function in the class.

A set or relation is primitive recursive when its characteristic function is.
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# Functional equations on groups

## Definition

Let $G$ be a [[group|G5S1M9]] with identity $e$, and let $Y$ be a set.
A *functional equation on $G$* is a [[functional equation|SRAN4EKWS0]] for one or more unknown [[functions|0EPAT51DXQ]] $f_1, \dots, f_m \colon G \to Y$ in which:

1. the variables $x_1, \dots, x_k$ range over all of $G$;
2. each unknown function is evaluated at words in $x_1, \dots, x_k$, their inverses and fixed elements of $G$;
3. the two sides of the equation are combined from these values by fixed operations on $Y$.

A solution is a tuple $(f_1, \dots, f_m)$ for which the equation holds for all $x_1, \dots, x_k \in G$.

## Remarks

Example: for $G$ a group and $Y = \C$, the equation $f(xy) + f(xy^{-1}) = 2 f(x) f(y)$ is d'Alembert's equation on $G$.
It has the solution $f(x) = (\chi(x) + \chi(x)^{-1})/2$ for every homomorphism $\chi \colon G \to \C^{\times}$, as a direct expansion shows.
Non-example: the equation $f(f(x)) = x$ for $f \colon G \to G$ does not fit the definition.
There the unknown function is evaluated at a value of itself, not at a word in the variables, so it is an iterative equation.
For $G = (\R, +)$, this setting includes the classical equations of Cauchy, d'Alembert and Pexider, and Jensen's equation in the equivalent form $f(x + y) + f(x - y) = 2 f(x)$.
Solutions on $G$ are often required to be continuous or measurable when $G$ is a topological group.
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# Hardy space

## Definition

Let $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ be the [[unit disc|8G5ABXRK85]] and let $0 < p < \infty$.
The *Hardy space* $H^p$ is the set of [[holomorphic functions|0NAHXR1FRH]] $f \colon \mathbb{D} \to \C$ such that
$$\norm{f}_{H^p} = \sup_{0 < r < 1} \left( \frac{1}{2\pi}\int_0^{2\pi} \abs{f(r e^{i\theta})}^p \, d\theta \right)^{1/p} < \infty.$$
The space $H^\infty$ is the set of bounded holomorphic functions on $\mathbb{D}$, with $\norm{f}_{H^\infty} = \sup_{z \in \mathbb{D}} \abs{f(z)}$.

## Remarks

The integral means increase with $r$, because $\abs{f}^p$ is [[subharmonic|Z1MHMGFNFW]], so the supremum is a limit as $r \to 1$.
For $f(z) = \sum_{n \ge 0} a_n z^n$, a [[power series|0D6KXBHKC0]] on $\mathbb{D}$, Parseval's identity gives $\frac{1}{2\pi}\int_0^{2\pi} \abs{f(re^{i\theta})}^2 d\theta = \sum \abs{a_n}^2 r^{2n}$, so $\norm{f}_{H^2}^2 = \sum_{n \ge 0} \abs{a_n}^2$.
Example: $f(z) = \log(1 - z) = -\sum_{n \ge 1} z^n/n$ has $\sum 1/n^2 < \infty$, so $f \in H^2$.
Non-example: $g(z) = 1/(1 - z) = \sum_{n \ge 0} z^n$ has all coefficients $1$, so $g \notin H^2$; in fact $g \in H^p$ exactly for $p < 1$.
For $1 \le p \le \infty$, $H^p$ is a [[Banach space|06GZ956657]]; for $0 < p < 1$ it is a complete metric space but not a normed space.
There are also Hardy spaces of the upper half-plane and of domains in $\C^n$ and $\R^n$.
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# Confluent hypergeometric function

## Definition

Let $a$ and $b$ be complex numbers with $b \notin \set{0, -1, -2, \dots}$.
Write $(x)_n = x (x + 1) \cdots (x + n - 1)$ for the [[Pochhammer symbol|YZHM3HSM0F]], with $(x)_0 = 1$.
*Kummer's confluent hypergeometric function* is the [[power series|0MZB06ST2B]]
$$M(a, b, z) = {}_1F_1(a; b; z) = \sum_{n = 0}^{\infty} \frac{(a)_n}{(b)_n} \frac{z^n}{n!}, \qquad z \in \C.$$
More generally, a *confluent hypergeometric function* is any solution of [[Kummer's equation|NT917KFR57]] $z w'' + (b - z) w' - a w = 0$.

## Remarks

The ratio of consecutive terms is $\frac{(a + n) z}{(b + n)(n + 1)}$, which tends to $0$, so the series converges for every $z$ and $M(a, b, \cdot)$ is entire.
Inserting the series into Kummer's equation shows that $M(a, b, z)$ is its solution that is analytic at $0$ with $M(a, b, 0) = 1$.
Example: $M(a, a, z) = e^z$, since $(a)_n / (a)_n = 1$.
Example: $M(1, 2, z) = \frac{e^z - 1}{z}$, since $\frac{(1)_n}{(2)_n\, n!} = \frac{1}{(n + 1)!}$.
Non-example: for $b = -1$ the series is not defined, since $(b)_2 = 0$ appears in a denominator.
If $a = -m$ with $m$ a nonnegative integer, the series stops and $M(-m, b, z)$ is a polynomial of degree $m$; for $b = \alpha + 1$ it is a multiple of the generalized Laguerre polynomial $L_m^{(\alpha)}(z)$.
Another notation is $\Phi(a; b; z)$.
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# Finite-dimensional distributions

## Summary

The finite-dimensional distributions of a [[stochastic process|0QH5ENKZM1]] $(X_t)_{t \in T}$ are the joint laws of the random vectors $(X_{t_1}, \dots, X_{t_k})$ for finitely many times $t_1, \dots, t_k$.
They determine the law of the process on the product $\sigma$-algebra.
By the [[Kolmogorov extension theorem|WR4X7MMTH2]], a consistent family of such laws comes from some process, which is the usual way to construct processes.
They do not determine path properties such as continuity.

## Explanations

### L1 human

A random process is a random quantity that evolves over time.
To describe it, one can list, for each finite set of times, the joint distribution of its values at those times.
These are the finite-dimensional distributions.
For example, take a sequence of independent fair coin tosses.
The values at any $k$ times are independent and each is fair.
So each such joint law gives probability $2^{-k}$ to each of the $2^k$ outcomes.
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# Involution

## Definition

Let $X$ be a set.
An *involution* on $X$ is a [[function|0EPAT51DXQ]] $f \colon X \to X$ such that $f(f(x)) = x$ for all $x \in X$.
Equivalently, $f \circ f = \id_X$, where $\circ$ is [[composition|0THBK9V825]] and $\id_X$ is the [[identity function|0PSKT5W817]] of $X$.

## Remarks

Example: complex conjugation $z \mapsto \bar{z}$ on $\C$ is an involution, because $\overline{\overline{a + bi}} = \overline{a - bi} = a + bi$.

Non-example: the map $x \mapsto x + 1$ on $\Z$ is not an involution, because applying it twice gives $x + 2 \ne x$.

Every involution is a bijection, and it is its own inverse.
The identity map is an involution.
On a finite set, the number of points not fixed by an involution is even, since they come in pairs $\set{x, f(x)}$.
In a group, the map $x \mapsto x^{-1}$ is an involution, because $(x^{-1})^{-1} = x$.
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# Tannery's theorem

## Statement

For each natural number $n$, let $(a_k(n))_{k \ge 0}$ be a sequence of complex numbers.
Assume that:

1. for each $k$, the limit $a_k = \lim_{n \to \infty} a_k(n)$ exists;
2. there are nonnegative real numbers $M_k$ with $\abs{a_k(n)} \le M_k$ for all $k$ and $n$, and $\sum_k M_k < \infty$.

Then the series $\sum_k a_k(n)$ and $\sum_k a_k$ [[converge absolutely|0W3K3QPEWT]], and
$$\lim_{n \to \infty} \sum_{k=0}^{\infty} a_k(n) = \sum_{k=0}^{\infty} a_k.$$

## Remarks

Given $\varepsilon > 0$, choose $K$ with $\sum_{k > K} M_k < \varepsilon$.
The tails beyond $K$ of both series are then smaller than $\varepsilon$, since $\abs{a_k} \le M_k$ too.
The finitely many terms with $k \le K$ converge, so the difference of the two sums is less than $3\varepsilon$ for large $n$.
Example: for real $x$ put $a_k(n) = \binom{n}{k} x^k / n^k$ for $k \le n$ and $a_k(n) = 0$ for $k > n$.
Then $\sum_k a_k(n) = (1 + x/n)^n$, $a_k(n) \to x^k/k!$, and $\abs{a_k(n)} \le \abs{x}^k / k!$.
So $(1 + x/n)^n \to \sum_k x^k / k! = e^x$.
Without the bound of clause 2 the conclusion fails: $a_k(n) = 1$ for $k = n$ and $0$ otherwise gives sums equal to $1$ but limits $a_k = 0$.

## Explanations

### L1 human

Tannery's theorem allows you to take a limit inside an infinite sum.
It needs one fixed summable bound for all the terms, uniformly in $n$.
Then no mass can escape to infinity, so the limit of the sums is the sum of the limits.

### L2 human

Tannery's theorem is the [[dominated convergence theorem|BYNV84PF86]] for the counting measure on $\N$, but it predates measure theory and has a short direct proof.
A typical use is to pass from finite products or binomial expansions to their limits, as in the proof that $(1 + x/n)^n \to e^x$.
Compare the [[Weierstrass M-test|0NX5NWAFC9]], which uses the same kind of bound to get uniform convergence of a series of functions.
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# Connected sum

## Summary

The connected sum of two connected $n$-manifolds is formed by cutting a small open ball out of each.
The two resulting boundary spheres are then glued together.
The result is again a connected $n$-manifold, written $M \# N$.
The operation is the basic way to build new manifolds from old ones.
The classification of surfaces and the prime decomposition of $3$-manifolds are stated in terms of it.

## Explanations

### L1 human

Take two surfaces, for example two doughnut surfaces.
Punch a small round hole in each.
Join the two holes with a short tube, or simply sew the edges of the holes together.
The result is one surface with two handles, the surface of a pretzel with two holes.
Taking the connected sum with a sphere changes nothing, since the sphere with a hole is just a disk that plugs the hole again.
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# Poisson–Mellin–Newton cycle

## Summary

The Poisson–Mellin–Newton cycle is a chain of three transforms that links a sequence $(f_n)$, its Poisson generating function, and the [[Mellin transform|N5PVGRJE1T]] of that function.
Going round the cycle recovers the sequence from the Mellin transform by a Nörlund–Rice integral, which is a contour integral over a Newton-type kernel.
It explains why alternating binomial sums and the Mellin transform have the same kind of asymptotic analysis.
It is used in the average-case analysis of algorithms such as digital search trees and tries.

## Explanations

### L1 human

Start with a sequence $f_0, f_1, f_2, \dots$ of numbers.
Mix it with a Poisson weight to get a smooth function $g(t) = e^{-t} \sum_n f_n t^n / n!$, which is the mean of $f_N$ when $N$ is a Poisson random number with mean $t$.
Take the Mellin transform of $g$; its poles describe how $g(t)$ behaves for small and large $t$.
A contour integral then turns the Mellin transform back into the numbers $f_n$, and this closes the cycle.
For example, for the constant sequence $f_n = 1$ one gets $g(t) = 1$ for all $t$.
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# Simple random sample

## Summary

A simple random sample of size $n$ from a finite population is a random subset of $n$ units.
It is chosen so that every subset of size $n$ is equally likely.
It is the basic design of survey sampling, against which other designs are compared.
Under it, the sample mean is an unbiased estimator of the population mean with an explicit variance.

## Explanations

### L1 human

Write the name of every member of the population on a ticket, mix the tickets well, and draw $n$ of them without putting any back.
The people drawn form a simple random sample.
Every person has the same chance to be chosen, and moreover every group of $n$ people has the same chance to be the sample.
For instance, from $4$ people, a sample of $2$ is one of $6$ possible pairs, each with probability $1/6$.
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# Toric ideal

## Definition

Let $k$ be a [[field|04C5ZBG1WT]], let $d, n \ge 1$, and let $a_1, \ldots, a_n \in \Z^d$.
For $a = (c_1, \ldots, c_d) \in \Z^d$, write $t^a = t_1^{c_1} \cdots t_d^{c_d}$ in the Laurent polynomial ring $k[t_1^{\pm 1}, \ldots, t_d^{\pm 1}]$.
Let $\varphi \colon k[x_1, \ldots, x_n] \to k[t_1^{\pm 1}, \ldots, t_d^{\pm 1}]$ be the $k$-algebra [[homomorphism|0VTH8KPCG2]] with $\varphi(x_i) = t^{a_i}$.
The *toric ideal* of $A = (a_1, \ldots, a_n)$ is the [[kernel|0JN0SQG2JH]] $I_A = \ker \varphi$.

## Remarks

The ideal $I_A$ is [[prime|00MHHZ9SNE]], since $k[x_1, \ldots, x_n]/I_A$ is isomorphic to a subring of a domain.
Let $A$ also denote the $d \times n$ integer matrix with columns $a_i$.
Then $I_A$ is generated, as a $k$-vector space and hence as an ideal, by the binomials $x^u - x^v$ with $u, v \in \N^n$ and $Au = Av$.

Example: $d = 2$, $n = 3$ and $a_1 = (1, 0)$, $a_2 = (1, 1)$, $a_3 = (1, 2)$, so $\varphi(x_1) = t_1$, $\varphi(x_2) = t_1 t_2$, $\varphi(x_3) = t_1 t_2^2$.
Since $\varphi(x_1 x_3) = t_1^2 t_2^2 = \varphi(x_2^2)$, the binomial $x_1 x_3 - x_2^2$ lies in $I_A$, and in fact $I_A = (x_1 x_3 - x_2^2)$.

Non-example: if the characteristic of $k$ is not $2$, the binomial ideal $(x^2 - y^2)$ of $k[x, y]$ is not a toric ideal.
It is not prime, because $(x - y)(x + y) \in (x^2 - y^2)$ while neither factor is.
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# Invertible operator

## Definition

Let $X$ and $Y$ be [[normed spaces|03J5RAKB30]] over $\R$ or $\C$, and let $T \colon X \to Y$ be a [[continuous linear map|03TM135G0N]].
The operator $T$ is *invertible* if there is a continuous linear map $S \colon Y \to X$ such that
$$S T = \id_X \quad \text{and} \quad T S = \id_Y.$$
The map $S$ is then unique; it is the *inverse* of $T$, written $T^{-1}$.

## Remarks

So $T$ is an [[inverse element|0ZNZH0EXFX]] for composition, with the inverse required to be bounded.

Example: on $\ell^2$, the diagonal operator $D(x_n) = ((1 + 1/n) x_n)$ is invertible.
Its inverse is $D^{-1}(x_n) = (\tfrac{n}{n+1} x_n)$, which is bounded with norm at most $1$, and both compositions are the identity.

Non-example: the diagonal operator $E(x_n) = (x_n / n)$ on $\ell^2$ is injective, but it is not invertible.
It is not surjective, since $(1/n)$ lies in $\ell^2$ but $(1)$ does not, so $(1/n)$ is not in the image of $E$.

If $X$ and $Y$ are [[Banach spaces|06GZ956657]], every bijective bounded operator is invertible, by the [[bounded inverse theorem|MR5FJXR348]].
For an operator $T$ on one space $X$, invertibility means invertibility in the algebra of bounded operators on $X$.
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# Percolation threshold

## Definition

Let $G = (V, E)$ be an infinite connected [[graph|00C0BR7D7H]] with countably many vertices, each of finite degree, and let $o \in V$.
For $p \in [0, 1]$, let $\theta(p)$ be the probability that the open cluster of $o$ is infinite in Bernoulli bond [[percolation|BH7YPXCYDK]] on $G$ with parameter $p$.
The *bond percolation threshold* of $G$ is
$$p_c(G) = \sup \set{p \in [0, 1] : \theta(p) = 0}.$$
The site percolation threshold is defined in the same way with site percolation.

## Remarks

Since $G$ is connected, $p_c$ does not depend on the choice of $o$.
By Kolmogorov's zero–one law, an infinite open cluster exists with probability $0$ when $p < p_c$ and with probability $1$ when $p > p_c$.
Example: for the line graph $\Z$, $\theta(p) = 0$ for every $p < 1$, so $p_c(\Z) = 1$.
Example: take the infinite tree in which every vertex has $d \ge 3$ neighbours.
After the first step, the open cluster of $o$ grows as a branching process with mean offspring $(d-1)p$, so $p_c = 1/(d-1)$.
Non-example: a finite graph does not satisfy the hypothesis; there every cluster is finite, so $\theta \equiv 0$ and the supremum carries no information.
For the lattice $\Z^d$ with $d \ge 2$, one has $0 < p_c < 1$.
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# Primitive permutation group

## Summary

A permutation group is primitive when it is [[transitive|02GK6HZYBT]] and preserves no nontrivial [[block system|SF738NKWJ0]], that is, no nontrivial splitting of the set into parts.
Primitive groups are the building blocks of all finite permutation groups.
A transitive group of degree $n$ that is not primitive embeds in a [[wreath product|W1X984PKJ9]] $S_k \wr S_m$ with $km = n$ and $1 < k, m < n$.
A transitive group is primitive exactly when its point [[stabilisers|0AYMP41THN]] are maximal subgroups.
The [[O'Nan–Scott theorem|Z4WVANZPK4]] sorts the finite primitive groups into a few types.

## Explanations

### L1 human

Imagine a group that shuffles a set of points.
If the group always keeps some grouping of the points together, for example pairs of opposite corners of a square, then it is not primitive.
A primitive group has no such hidden grouping: the only groupings it respects are the trivial ones, all points in one part or each point alone.
The full group of all permutations of a set is primitive.
The symmetries of a square, acting on its corners, are not primitive, because they keep the two diagonals as pairs.
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# Rayleigh quotient iteration

## Statement

Let $A \in \R^{n \times n}$ be [[symmetric|0VR671PHE5]], and let $v$ be a unit [[eigenvector|0N6960W0BV]] of $A$ for a simple [[eigenvalue|0BQAA2BK3Q]] $\lambda$.
*Rayleigh quotient iteration* starts from a unit vector $x_0$.
For $k \ge 0$ it sets
$$\mu_k = x_k^T A x_k, \qquad x_{k+1} = \frac{(A - \mu_k I)^{-1} x_k}{\norm{(A - \mu_k I)^{-1} x_k}}.$$
It stops if $A - \mu_k I$ is singular, in which case $\mu_k$ is an eigenvalue.
Let $\theta_k \in [0, \pi/2]$ be the angle between $x_k$ and $v$.
Then there are constants $\delta > 0$ and $C > 0$ such that $\theta_k \le \delta$ implies $\theta_{k+1} \le C \theta_k^3$.
In particular, if $x_0$ is close enough to $\pm v$, then $x_k \to \pm v$ in direction cubically, and $\mu_k \to \lambda$.

## Remarks

Write $x_k = \cos\theta_k \, v + \sin\theta_k \, w$ with $w$ a unit vector orthogonal to $v$.
For symmetric $A$ the [[Rayleigh quotient|06WHP8SE98]] is stationary at eigenvectors, so $\abs{\mu_k - \lambda} \le \norm{A - \lambda I} \sin^2\theta_k = O(\theta_k^2)$.
Let the gap be the distance from $\mu_k$ to the other eigenvalues.
One step of [[inverse iteration|70QYCEWHVY]] with shift $\mu_k$ multiplies the ratio of the $w$-component to the $v$-component by at most $\abs{\mu_k - \lambda}/\mathrm{gap}$.
So $\tan\theta_{k+1} \le O(\theta_k^2) \tan\theta_k$, which is cubic.

Example: for $A = \mathrm{diag}(1, 3)$ and $x_k = (\cos\theta, \sin\theta)$, one finds $\mu_k = 1 + 2\sin^2\theta$ and $\tan\theta_{k+1} = \tan^3\theta_k$ exactly.
Symmetry is needed for cubic convergence: for a general matrix with a simple eigenvalue the Rayleigh quotient error is only $O(\theta_k)$, and the local convergence is quadratic.
Which eigenvector the iteration finds depends on $x_0$ in a complicated way, so the method is used to refine a good approximation rather than to locate eigenvalues from scratch.

## Explanations

### L1 human

Inverse iteration needs a guess $\mu$ of the eigenvalue and converges faster the better the guess.
Rayleigh quotient iteration updates the guess at every step with the best estimate available from the current vector.
For symmetric matrices the number of correct digits roughly triples at each step once the vector is close.

### L2 human

Each step costs a fresh factorization of $A - \mu_k I$, unlike inverse iteration with a fixed shift, but very few steps are needed.
The cubic rate comes from two facts.
The Rayleigh quotient is a second-order accurate eigenvalue estimate for symmetric matrices, and inverse iteration converges at a rate proportional to the shift error.
The shifted [[QR algorithm|B6F8YWHYZ5]] with Rayleigh quotient shifts is closely related, and this explains its fast convergence on symmetric matrices.
</document>

<document source="https://tamnd.github.io/math2/node/Y19WM1EMS7.md" tag="Y19WM1EMS7" kind="definition">
# Robin boundary condition

## Definition

Let $\Omega \subseteq \R^d$ be a bounded domain, that is, a connected open set, with boundary $\partial \Omega$ of class $C^1$ and outward unit [[normal vector|0WQJJCCTC8]] $\nu$.
Let $c_1, c_2$ be nonzero real constants and let $g \colon \partial \Omega \to \R$ be a given function.
For an equation for $u$ on $\Omega$, the *Robin boundary condition* is
$$c_1\, u(x) + c_2\, \frac{\partial u}{\partial \nu}(x) = g(x) \quad \text{for all } x \in \partial \Omega.$$
For a second-order ordinary differential equation on $[a, b]$, where the outward normal derivatives are $-y'(a)$ and $y'(b)$, it reads
$$c_1\, y(a) - c_2\, y'(a) = g_a, \qquad c_1\, y(b) + c_2\, y'(b) = g_b.$$

## Remarks

If $c_2 = 0$, the condition becomes a [[Dirichlet condition|EETVA6QYX4]]; if $c_1 = 0$, it becomes a [[Neumann condition|GS3KT2H645]].
Some authors allow these cases and also let $c_1, c_2$ be functions on the boundary.

Example: $y'' = 0$ on $[0, 1]$ with $y(0) - y'(0) = 0$ and $y(1) + y'(1) = 3$, so $c_1 = c_2 = 1$.
The function $y = 1 + x$ gives $y(0) - y'(0) = 1 - 1 = 0$ and $y(1) + y'(1) = 2 + 1 = 3$, so it solves this [[boundary value problem|TM8VQJWGHZ]].

Non-example: the conditions $y(0) = 0$ and $y(1) = 0$ are not of Robin type, because no derivative appears; they are Dirichlet conditions.

A Sturm–Liouville problem usually has separated conditions $\alpha_1 y(a) + \alpha_2 y'(a) = 0$ and $\beta_1 y(b) + \beta_2 y'(b) = 0$, which include the Robin case.
</document>

<document source="https://tamnd.github.io/math2/node/Y1BVY7J6AE.md" tag="Y1BVY7J6AE" kind="concept">
# Falling and rising factorials

## Summary

The falling factorial $x^{\underline{n}}$ multiplies $n$ consecutive factors going down from $x$, and the rising factorial $x^{\overline{n}}$ multiplies $n$ consecutive factors going up from $x$.
Both are polynomials of degree $n$ in $x$, and both reduce to the [[factorial|0XWG0JGDFW]] $n!$ for a suitable choice of $x$.
The falling factorial counts ordered selections without repetition, and the rising factorial (the Pochhammer symbol of special functions) appears in hypergeometric series.
The node [[falling factorial|0SJNCJCCQG]] treats the falling case alone.

## Explanations

### L1 human

Start at a number and multiply it by the next few numbers below it: that is a falling factorial.
Multiply it by the next few numbers above it instead: that is a rising factorial.
For example, $5^{\underline{3}} = 5 \cdot 4 \cdot 3 = 60$ and $5^{\overline{3}} = 5 \cdot 6 \cdot 7 = 210$.
The number $60$ counts the ways to give gold, silver and bronze medals to $3$ of $5$ runners.
Both products start at $x$ and use $n$ factors, so with $x = n$ falling or $x = 1$ rising they give $n!$.
</document>

<document source="https://tamnd.github.io/math2/node/Y1E9SJC3Y0.md" tag="Y1E9SJC3Y0" kind="statement">
# Carleson–Sjölin theorem

## Statement

Let $\delta > 0$ and $1 \le p \le \infty$.
Let $S^\delta$ be the [[Bochner–Riesz operator|AFS2DCWCD3]] of order $\delta$ on $\R^2$, the operator with multiplier $(1 - \abs{\xi}^2)_+^\delta$.
Then $S^\delta$ is bounded on $L^p(\R^2)$ if and only if
$$\Big| \frac{1}{p} - \frac{1}{2} \Big| < \frac{1 + 2\delta}{4}.$$
In particular, $S^\delta$ is bounded on $L^p(\R^2)$ for every $\delta > 0$ and every $4/3 \le p \le 4$.

## Remarks

Lennart Carleson and Per Sjölin proved the sufficiency of the condition in 1972.
The necessity holds in every dimension $n$.
The kernel of $S^\delta$ decays like $\abs{x}^{-(n + 1)/2 - \delta}$ with oscillation, and it lies in $L^p$ only when $1/p - 1/2 < (1 + 2\delta)/(2n)$.
Boundedness of $S^\delta$ on $L^p$ forces its kernel into $L^p$.
Since $S^\delta$ is self-adjoint, boundedness on $L^p$ is equivalent to boundedness on $L^{p'}$, and this gives the other half of the condition.
The proof of sufficiency decomposes the multiplier into pieces near the circle $\abs{\xi} = 1$ and uses an $L^4$ estimate for oscillatory integrals whose phase has curved level sets.

Example: for $\delta = 1/4$ the condition reads $\abs{1/p - 1/2} < 3/8$, that is $8/7 < p < 8$.
Without $\delta > 0$ the result fails: for $\delta = 0$, [[Fefferman's ball multiplier theorem|4A5MJY4XPV]] shows that $S^0$ is bounded only on $L^2(\R^2)$.

The same condition in dimension $n$, namely $\abs{1/p - 1/2} < (1 + 2\delta)/(2n)$, is the Bochner–Riesz conjecture; the theorem settles it for $n = 2$, while it is open for $n \ge 3$.

## Explanations

### L1 human

In the plane, rebuilding a function from its frequencies by a sharp round cut fails in every $L^p$ norm except $L^2$.
This theorem says that fading the frequencies out with any tiny amount of smoothness, any $\delta > 0$, already works for all $p$ between $4/3$ and $4$.
It also gives the exact range of $p$ for each $\delta$.

### L2 human

The theorem is the two-dimensional case of the Bochner–Riesz problem, and its proof has the same structure as the two-dimensional case of the [[restriction conjecture|9CE11G48JY]].
For $\delta > 0$ and $4/3 \le p \le 4$, it gives $L^p$ convergence of the Bochner–Riesz means $S^\delta_R f \to f$ for $f \in L^p(\R^2)$.
In higher dimensions, the [[Stein–Tomas restriction theorem|ZZVWHNFY6F]] gives a partial range, and the full range is open.
</document>

<document source="https://tamnd.github.io/math2/node/Y1EGR8DF84.md" tag="Y1EGR8DF84" kind="concept">
# Liouville property of graphs

## Summary

A graph has the Liouville property if every bounded [[discrete harmonic function|WRY0M7SW59]] on it is constant.
The name comes from Liouville's theorem, whose harmonic form says that bounded harmonic functions on $\R^n$ are constant.
The lattices $\Z^d$ have the property, while regular trees of degree at least $3$ do not.
For Cayley graphs of groups it says that the [[Poisson boundary|6EEF2E8XPF]] of the simple random walk is trivial.

## Explanations

### L1 human

A harmonic function on a graph gives each vertex the average of the values at its neighbours.
On a graph with the Liouville property, the only way to do this with values that stay bounded is to use the same value everywhere.
On the integers, the harmonic functions are the lines $an + b$, and only the constant ones are bounded.
On a tree that branches into two or more new directions at every vertex, a random walk drifts off along one branch.
The probability of ending up in a fixed branch is then a bounded harmonic function that is not constant.
</document>

<document source="https://tamnd.github.io/math2/node/Y1G3NNJSQH.md" tag="Y1G3NNJSQH" kind="statement">
# Symmetric successive over-relaxation

## Statement

Let $A$ be a real symmetric $n \times n$ matrix with positive diagonal entries, and let $b \in \R^n$.
Write $A = D + L + L^T$, with $D$ diagonal and $L$ strictly lower triangular, and let $0 < \omega < 2$.
One step of *symmetric successive over-relaxation* (SSOR) is a forward [[SOR|6KQJR85ZC6]] sweep in the order $1, \dots, n$.
It is followed by a backward SOR sweep in the order $n, \dots, 1$.
In matrix form it is $x^{(k+1)} = x^{(k)} + M_\omega^{-1} \left( b - A x^{(k)} \right)$, where
$$M_\omega = \frac{1}{\omega (2 - \omega)} \, (D + \omega L) \, D^{-1} \, (D + \omega L^T).$$
Then $M_\omega$ is symmetric and positive definite.
If moreover $A$ is [[positive definite|0KX66CZMBB]], then the iteration matrix $I - M_\omega^{-1} A$ has [[spectral radius|9V7YQA85PC]] less than $1$.
So for every $b$ and every $x^{(0)}$ the iterates converge to $A^{-1} b$.

## Remarks

The matrix $M_\omega$ is symmetric by its form, and it is positive definite because $D$ is and $D + \omega L$ is invertible.
When $A$ is positive definite, each of the two SOR sweeps strictly decreases the energy norm $\norm{e}_A$ of the error, so their product contracts too.
Example: for $n = 1$ and $A = (a)$ with $a > 0$, one has $M_\omega = a / (\omega (2 - \omega))$.
The iteration matrix is $1 - \omega (2 - \omega) = (1 - \omega)^2$, which lies in $[0, 1)$ exactly when $0 < \omega < 2$.
As a stand-alone method, SSOR is usually no faster than SOR.
Its value is that $M_\omega$ is a symmetric positive definite [[preconditioner|KE98SW3SBD]], so it can be combined with the conjugate gradient method, unlike plain SOR.

## Explanations

### L1 human

SOR sweeps through the unknowns in one direction, which treats the system unevenly.
SSOR sweeps forward and then backward, so the combined step is symmetric.
This symmetry is what makes it useful together with methods that need symmetric operators.

### L2 human

The formula for $M_\omega$ shows that applying SSOR costs one forward and one backward triangular solve with the sparsity of $A$.
For $\omega = 1$ the method is symmetric Gauss–Seidel, with $M_1 = (D + L) D^{-1} (D + L^T)$.
SSOR belongs to the family of stationary [[iterative methods|2RPE77ACER]] given by a splitting $A = M_\omega - (M_\omega - A)$.
</document>

<document source="https://tamnd.github.io/math2/node/Y1M03JQKKR.md" tag="Y1M03JQKKR" kind="concept">
# Monotone operator

## Summary

A monotone operator is a map $T$ from a Banach space $X$ to its dual $X^*$ with $\ip{T(u) - T(v)}{u - v} \ge 0$ for all $u, v$.
It generalises the increasing functions of one real variable, the [[monotone functions|0A7KV2WB7A]], to infinite dimensions.
Gradients of convex functionals and many nonlinear elliptic operators are monotone.
Monotone operator theory, with results such as the [[Browder–Minty theorem|NWK9Z80BT2]], solves nonlinear equations $T(u) = f$ without compactness.

## Explanations

### L1 human

A function $f$ on the real line is increasing when $(f(u) - f(v))(u - v) \ge 0$: larger inputs never give smaller outputs.
A monotone operator asks the same thing in many dimensions, with the product replaced by a pairing.
For example, the gradient of a convex function on $\R^n$ is monotone, because a convex function bends upwards in every direction.
The map $T(u) = -u$ is not monotone, since it reverses order.
</document>

<document source="https://tamnd.github.io/math2/node/Y1M6T8DEXK.md" tag="Y1M6T8DEXK" kind="definition">
# Vlasov equation

## Definition

Let $d \ge 1$, let $T > 0$, and let $F$ assign to each function $f \colon [0, T) \times \R^d \times \R^d \to [0, \infty)$ a force field $F[f] \colon [0, T) \times \R^d \to \R^d$.
The *Vlasov equation* with force $F$ is the [[partial differential equation|GC5BDZ6QBC]]
$$\partial_t f + v \cdot \nabla_x f + F[f](t, x) \cdot \nabla_v f = 0.$$
The unknown is a density $f(t, x, v) \ge 0$, and $\nabla_x$ and $\nabla_v$ are the [[gradients|0ABK9V9Y64]] in $x$ and in $v$.
The main case is the *Vlasov–Poisson system* in $d = 3$, where $\gamma \in \set{1, -1}$ and
$$F[f](t, x) = \frac{\gamma}{4\pi} \int_{\R^3} \frac{x - y}{\abs{x - y}^3} \rho(t, y) \, dy, \qquad \rho(t, y) = \int_{\R^3} f(t, y, v) \, dv.$$
The sign $\gamma = 1$ gives a repulsive force, as for a plasma of one species, and $\gamma = -1$ gives an attractive force, as for a self-gravitating system.

## Remarks

The force of the Vlasov–Poisson system is $F = -\gamma \nabla_x \phi$, where $-\Delta \phi = \rho$.
With a force that does not depend on $f$, the equation is a linear [[transport equation|PVD7VZJZ9B]] along the trajectories of $\dot x = v$, $\dot v = F$.

Example: for $d = 1$ and the external force $F(t, x) = -x$, every function $f(t, x, v) = h(x^2 + v^2)$ with $h$ differentiable is a stationary solution.
Indeed $v \, \partial_x f - x \, \partial_v f = 2xv \, h' - 2xv \, h' = 0$.

Non-example: with the same force, $f = h(x^2 + 2v^2)$ is not a solution when $h' \ne 0$, because $v \, \partial_x f - x \, \partial_v f = -2xv \, h'(x^2 + 2v^2)$.
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<document source="https://tamnd.github.io/math2/node/Y1M82VZYS2.md" tag="Y1M82VZYS2" kind="concept">
# Hahn polynomials

## Summary

The Hahn polynomials $Q_n(x; \alpha, \beta, N)$ are a finite family of orthogonal polynomials on the points $0, 1, \dots, N$.
Their weight is the beta-binomial distribution, a discrete version of the Jacobi weight.
They are terminating ${}_3F_2$ hypergeometric series and sit in the [[Askey scheme|RR2S05C09B]] just below the Wilson and Racah polynomials.
As $N \to \infty$ after rescaling they tend to the [[Jacobi polynomials|TRD7DSF2KR]], and they also appear as Clebsch–Gordan coefficients of $SU(2)$.

## Explanations

### L1 human

The Jacobi polynomials are perpendicular for a weight on a continuous interval.
Replace the interval by the $N + 1$ points $0, 1, \dots, N$ and the weight by a matching discrete weight.
The polynomials that are perpendicular for this discrete weight are the Hahn polynomials.
For each degree $n$ from $0$ to $N$ there is exactly one of them, normalised to take the value $1$ at $x = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/Y1MH7WPRDZ.md" tag="Y1MH7WPRDZ" kind="concept">
# Pseudoanalytic function

## Summary

A pseudoanalytic function is a complex function that is "holomorphic relative to a pair of basis functions" $(F, G)$ instead of relative to the pair $(1, i)$.
The notion was introduced by Lipman Bers to extend the theory of [[holomorphic functions|0NAHXR1FRH]] to solutions of first-order elliptic systems in the plane.
For smooth generating pairs, pseudoanalytic functions satisfy equations of the form $w_{\bar z} = a w + b \bar w$, and many theorems of complex analysis carry over to them.

## Explanations

### L1 human

Every complex number $w$ can be written as $\varphi \cdot 1 + \psi \cdot i$ with $\varphi, \psi$ real, and $w$ is holomorphic when these real coefficients vary in a special way.
A pseudoanalytic function replaces the fixed pair $(1, i)$ by a pair of functions $(F, G)$ that are independent over the reals at each point.
One writes $w = \varphi F + \psi G$ and asks the real coefficients to satisfy the analogue of the Cauchy–Riemann condition.
With $F = 1$ and $G = i$ this is exactly ordinary holomorphy.
</document>

<document source="https://tamnd.github.io/math2/node/Y1N80NR1CG.md" tag="Y1N80NR1CG" kind="definition">
# Priority method

## Definition

A *finite-injury priority construction* builds sets $A \subseteq \N$ by a computable process in stages $s = 0, 1, 2, \ldots$, as follows.

1. The goal is a list of *requirements* $R_0, R_1, R_2, \ldots$, and $R_i$ has *higher priority* than $R_j$ when $i < j$.
2. At each stage $s$, finitely many numbers are added to the finite approximation $A_s$, so $A_0 \subseteq A_1 \subseteq \cdots$ and $A = \bigcup_s A_s$ is [[computably enumerable|TN9G24VABN]].
3. At a stage, an action taken for $R_i$ may undo the work done for some $R_j$ with $j > i$; then $R_j$ is *injured* and starts again.
4. An action for $R_j$ never injures any $R_i$ with $i < j$.

The construction is *finite-injury* if each requirement is injured only finitely often and acts only finitely often.
The *priority method* is the use of such constructions to prove the existence of computably enumerable sets with given properties.
It also covers extensions in which requirements may be injured infinitely often.

## Remarks

Example: the proof of the [[Friedberg–Muchnik theorem|RA358M9QCN]] is a finite-injury construction with requirements $A \ne \Phi_e^B$ and $B \ne \Phi_e^A$.
By induction on $i$, after some stage no requirement before $R_i$ acts, so $R_i$ is injured only finitely often and is met.

Non-example: a construction of $A$ that at each stage asks whether a given program halts is not a priority construction in this sense.
It is not a computable process, so it does not make $A$ computably enumerable.

The proof that a finite-injury construction succeeds is usually an induction on the priority order.
Infinite-injury methods, such as those used for the Sacks density theorem, allow some requirements to be injured infinitely often.
</document>

<document source="https://tamnd.github.io/math2/node/Y1P96JQN34.md" tag="Y1P96JQN34" kind="definition">
# Group ring

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity and let $G$ be a [[group|G5S1M9]].
The *group ring* $R[G]$ is the set of functions $a \colon G \to R$ such that $a(g) \ne 0$ for only finitely many $g \in G$.
Write such a function as a finite formal sum $\sum_{g \in G} a_g \, g$ with $a_g = a(g)$.
Addition is termwise, and multiplication is
$$\Big(\sum_{g} a_g \, g\Big)\Big(\sum_{h} b_h \, h\Big) = \sum_{x \in G} \Big(\sum_{gh = x} a_g b_h\Big) x.$$

## Remarks

With these operations $R[G]$ is a ring with identity $1_R \, e$, where $e$ is the identity of $G$.
The maps $r \mapsto r e$ and $g \mapsto 1_R \, g$ embed $R$ as a subring and, when $R \ne 0$, $G$ into the group of units of $R[G]$.
If $R$ is commutative, $R[G]$ is an algebra over $R$, and if $R$ is a field, it is the [[group algebra|MNP2HBWB5Q]].

Example: let $G = \langle t \rangle$ be infinite cyclic.
A finite sum $\sum_n a_n t^n$ is a Laurent polynomial, and the product formula is the usual product of Laurent polynomials, so $R[G] \cong R[t, t^{-1}]$.

Non-example: for infinite $G$, the set of all functions $G \to R$ with the same product formula is not a ring in general.
For $G = \Z$ and $a = b = \sum_{n \in \Z} t^n$, the coefficient of $t^0$ in $ab$ would be the infinite sum $\sum_{n} 1$, which is not defined in $R = \Z$.

If $G$ has an element $g \ne e$ of finite order $n$, then $(1 - g)(1 + g + \cdots + g^{n-1}) = 1 - g^n = 0$, so $R[G]$ has [[zero divisors|0NNRDW10RZ]] when $R \ne 0$.
The group ring is the [[monoid ring|1HM7ZYMPD7]] of $G$ over $R$.
</document>

<document source="https://tamnd.github.io/math2/node/Y1RAX7VVBX.md" tag="Y1RAX7VVBX" kind="definition">
# Full subcategory

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
A *full subcategory* of $\mathcal{C}$ is a [[subcategory|D8C0EYD9ED]] $\mathcal{D}$ such that $\Hom_{\mathcal{D}}(A, B) = \Hom_{\mathcal{C}}(A, B)$ for all objects $A$, $B$ of $\mathcal{D}$.

## Remarks

For every subclass $S$ of $\mathrm{Ob}(\mathcal{C})$ there is exactly one full subcategory with object class $S$, the full subcategory *spanned* by $S$.
Example: the [[abelian groups|082GHNPJ53]] form a full subcategory of the category of [[groups|G5S1M9]].
A morphism between two abelian groups in the category of groups is a group homomorphism, and every such homomorphism is a morphism of the subcategory.
Non-example: the subcategory of sets with all sets as objects and only the injective functions as morphisms is not full.
The constant map $\set{0, 1} \to \set{0}$ is a morphism of the category of sets between two of its objects, but it is not injective.
</document>

<document source="https://tamnd.github.io/math2/node/Y1X8ZZQ7VM.md" tag="Y1X8ZZQ7VM" kind="concept">
# q-Pochhammer symbol

## Summary

The $q$-Pochhammer symbol $(a; q)_n = (1 - a)(1 - aq) \cdots (1 - aq^{n-1})$ is a $q$-analogue of the rising factorial.
Its infinite version $(a; q)_\infty$ is a convergent [[infinite product|JYV1Y8XX16]] for $\abs{q} < 1$.
It is the basic building block of $q$-series, basic hypergeometric series and generating functions for integer partitions.
The exact definition is in [[q-Pochhammer symbol|K8PHD2NAW9]].

## Explanations

### L1 human

The $q$-Pochhammer symbol is a product of factors $1 - a$, $1 - aq$, $1 - aq^2$, and so on, each with one more power of $q$.
For example, $(q; q)_3 = (1 - q)(1 - q^2)(1 - q^3)$.
Such products appear when one counts partitions of integers: the coefficients of $1/(q; q)_\infty$, expanded in powers of $q$, are the partition numbers.
</document>

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# Szegő kernel

## Summary

Let $\Omega \subset \C^n$ be a bounded domain with smooth boundary.
Its Szegő kernel is the reproducing kernel of the [[Hardy space|PNNNFJVF6W]] of holomorphic functions with square-integrable boundary values on $\partial\Omega$.
It plays for boundary integrals the role that the [[Bergman kernel|8RWDHW0CXC]] plays for volume integrals.
It gives a canonical integral formula that recovers a holomorphic function inside $\Omega$ from its values on the boundary.
On the unit disc it is the Cauchy kernel, and on the unit ball it has an explicit closed form.

## Explanations

### L1 human

A holomorphic function on a nice domain is fixed by its values on the boundary.
The Szegő kernel is the weight that tells how much each boundary point contributes to the value at an inside point $z$.
To get $f(z)$, average the boundary values of $f$ against this weight.
On the unit disc the weight is $\frac{1}{2\pi(1 - z\bar\zeta)}$, and this average is exactly Cauchy's integral formula.
</document>

<document source="https://tamnd.github.io/math2/node/Y1YPE6PM52.md" tag="Y1YPE6PM52" kind="concept">
# Universal cover

## Summary

A universal cover of a connected space $X$ is a [[covering space|3XZRRE165N]] of $X$ whose total space is simply connected.
It exists for every connected, locally path-connected and [[semilocally simply connected|R3SHZX0NZR]] space, and it is unique up to isomorphism.
It covers every other connected covering space of $X$, and its group of deck transformations is isomorphic to the [[fundamental group|S1XK3HXHFX]] of $X$.

## Explanations

### L1 human

The universal cover unwraps every loop of a space that cannot be shrunk.
The real line wound around the circle as a helix is the universal cover of the circle.
Going once around the circle moves one step up the helix, so no loop survives upstairs.
The plane is the universal cover of the torus, through the map that reduces both coordinates modulo $1$.
The universal cover is the largest connected covering, and every other connected covering sits between it and the space.
</document>

<document source="https://tamnd.github.io/math2/node/Y203PPTS61.md" tag="Y203PPTS61" kind="concept">
# Mal'cev term

## Summary

A Mal'cev term of a [[variety|AJNZJD8442]] is a ternary term $p$ such that $p(x, y, y) = x$ and $p(x, x, y) = y$ hold in the variety.
By Mal'cev's theorem, a variety has a Mal'cev term exactly when it is a [[congruence-permutable variety|1B3N9T68YV]].
Groups, rings, modules and quasigroups all have Mal'cev terms.

## Explanations

### L1 human

In a group, the formula $x y^{-1} z$ cancels the middle input against either neighbour.
If the last two inputs are equal, it returns the first, and if the first two are equal, it returns the last.
A Mal'cev term is any formula with this cancelling behaviour.
Its existence makes congruences, the ways of identifying elements, combine in either order with the same result.
</document>

<document source="https://tamnd.github.io/math2/node/Y20PEX049W.md" tag="Y20PEX049W" kind="example">
# Orders of the elements of S3

## Example

The [[symmetric group|0515A6D2Y1]] $S_3$ has order $6$.
The identity has [[order|0JPYSD5E0B]] $1$.
Each transposition $(1\,2)$, $(1\,3)$ and $(2\,3)$ has order $2$.
Each $3$-cycle $(1\,2\,3)$ and $(1\,3\,2)$ has order $3$.
Each of the orders $1$, $2$ and $3$ divides $6$, as the [[result on element orders|08V7JHS3QZ]] states.

## Remarks

No element has order $6$, so $S_3$ is not cyclic.
</document>

<document source="https://tamnd.github.io/math2/node/Y2467453Q5.md" tag="Y2467453Q5" kind="concept">
# Algebraic K-theory of schemes

## Summary

The algebraic K-theory of a scheme $X$ is a sequence of abelian groups $K_0(X), K_1(X), \ldots$ built from the category of [[vector bundles|P4DZM975BV]] on $X$.
The group $K_0(X)$ is the Grothendieck group of vector bundles, and the higher groups are defined by Quillen's Q-construction.
K-theory is the setting of the [[Grothendieck–Riemann–Roch theorem|3HKREN3Y3F]].
It is linked to [[Chow groups|EBEF6PEK4N]] by the Chern character and to motivic cohomology by spectral sequences.

## Explanations

### L1 human

The group $K_0(X)$ treats vector bundles like numbers: one can add them by direct sum, and a bundle built from two others by an extension counts as their sum.
For a point, vector bundles are vector spaces, and $K_0$ is $\Z$, measured by the dimension.
On the projective line, the bundles $\mathcal{O}(n)$ have both a rank and a degree, and $K_0(\mathbb{P}^1) \cong \Z^2$ records exactly these two numbers.
The higher groups $K_n$ capture finer information, such as units of rings for $K_1$.
</document>

<document source="https://tamnd.github.io/math2/node/Y277TCXZ9Z.md" tag="Y277TCXZ9Z" kind="statement">
# Laguerre's method

## Statement

Let $p$ be a polynomial of degree $n \ge 2$ with complex coefficients.
For a point $x$ with $p(x) \ne 0$, put
$$G = \frac{p'(x)}{p(x)}, \qquad H = G^2 - \frac{p''(x)}{p(x)}, \qquad a = \frac{n}{G \pm \sqrt{(n - 1)(nH - G^2)}},$$
where the sign is chosen so that the denominator has the larger absolute value.
*Laguerre's method* is the iteration $x_{k+1} = x_k - a$, with $G, H, a$ computed at $x = x_k$.
Then:

1. If $r$ is a [[root|019MA1VTR2]] of $p$ of [[multiplicity|047SX3TD69]] $1$, then for $x_0$ close enough to $r$ the iterates [[converge|YWZQC5E54B]] to $r$ with order $3$.
2. If all roots of $p$ are real, then for every real $x_0$ the iterates stay real and converge to a root of $p$.

## Remarks

The formula comes from a model of $p$ near $x$.
Assume that one root $r$ is at distance $a = x - r$, and that the other $n - 1$ roots all sit at a common distance $b$.
The identities $G = \sum_i \frac{1}{x - r_i}$ and $H = \sum_i \frac{1}{(x - r_i)^2}$ over the roots $r_i$ then give two equations for $a$ and $b$, whose solution is the formula above.
For polynomials with only real roots, a real starting point between two consecutive roots leads to one of these two roots.
A real starting point outside all roots leads to the nearest root on that side.
At a multiple root the convergence is only linear.
The square root can be non-real, so the method also finds the [[complex|09FTAET5Z4]] roots of real polynomials.

## Explanations

### L1 human

Laguerre's method is a root finder made for polynomials.
It pretends that one root is near and all the others are bunched together far away, and it solves this simplified picture exactly.
The answer is a very good next guess, and near a simple root the number of correct digits roughly triples at each step.

### L2 human

The method uses $p$, $p'$ and $p''$, which Horner's scheme evaluates cheaply.
Its convergence from almost any start makes it a robust choice for finding all roots one at a time, with deflation after each root.
Its behaviour on polynomials with only real roots is a rare case of guaranteed global convergence.
</document>

<document source="https://tamnd.github.io/math2/node/Y28DDJ9F79.md" tag="Y28DDJ9F79" kind="definition">
# Covering space

## Definition

Let $E$ and $X$ be topological spaces and let $p \colon E \to X$ be a [[continuous map|0VGFHTM75Y]].
An open set $U \subseteq X$ is *evenly covered* by $p$ if $p^{-1}(U)$ is a union of pairwise disjoint open sets $V_i \subseteq E$, $i \in J$, with one more property.
For each $i$, the restriction $p|_{V_i} \colon V_i \to U$ is a [[homeomorphism|0QT00PB5G2]], where $V_i$ and $U$ carry the [[subspace topology|0WS9P12FQR]].
The map $p$ is a *covering map* if every point of $X$ has an evenly covered open neighbourhood.
Then $E$ is a *covering space* of $X$.

## Remarks

The sets $V_i$ are the *sheets* over $U$.
Many authors also require $p$ to be surjective, or $E$ to be nonempty and connected; this varies between textbooks.
Every covering map is a [[local homeomorphism|MDH71ZZ8XX]].
Example: $p \colon \R \to S^1$, $p(t) = e^{2\pi i t}$, is a covering map.
For $z = e^{2\pi i a}$, the arc $U = p((a - 1/2, a + 1/2))$ is evenly covered, with sheets $(a - 1/2 + n, a + 1/2 + n)$ for $n \in \Z$.
Non-example: the restriction of this $p$ to $(0, 2)$ is a surjective local homeomorphism but not a covering map.
For every open arc $U$ around $1$, the part of $p^{-1}(U)$ near $0$ is a half-open piece $(0, \varepsilon)$ that is not mapped onto $U$.
</document>

<document source="https://tamnd.github.io/math2/node/Y2DKZH3CCT.md" tag="Y2DKZH3CCT" kind="statement">
# Singer's theorem

## Statement

Let $I \subset \R$ be a compact interval and let $f \colon I \to I$ be a map of class $C^3$ with finitely many [[critical points|0YTCMFB4HC]].
Assume that $f$ has negative [[Schwarzian derivative|6ZPKWE0VAZ]]:
$$Sf(x) = \frac{f'''(x)}{f'(x)} - \frac{3}{2} \left( \frac{f''(x)}{f'(x)} \right)^2 < 0 \quad \text{for every } x \in I \text{ with } f'(x) \neq 0.$$
Let $\mathcal{O}$ be an attracting [[periodic orbit|0AA97XA70K]] of $f$, that is, the orbit of a point $p$ with $f^n(p) = p$ and $\abs{(f^n)'(p)} < 1$.
The immediate basin of $\mathcal{O}$ is the union of the connected components of its basin of attraction that contain a point of $\mathcal{O}$.
Then the immediate basin of $\mathcal{O}$ contains a critical point of $f$ or a boundary point of $I$.
In particular, the number of attracting periodic orbits of $f$ is at most the number of critical points of $f$ plus $2$.

## Remarks

The key fact is that the Schwarzian derivative of a composition satisfies $S(g \circ f) = (Sg \circ f) \, (f')^2 + Sf$, so negative Schwarzian passes to all iterates $f^n$.
Then $\abs{(f^n)'}$ has no positive local minimum on an interval where $f^n$ is monotone (the minimum principle).
Suppose the component $J$ of the immediate basin that contains $p$ had no critical point and no boundary point.
Then $f^n$ would map $J$ diffeomorphically onto itself, and the minimum principle applied there gives a contradiction.

Example: for the [[logistic map|QN9X9BNKTX]] $f_r(x) = r x (1 - x)$ with $1 < r \le 4$, one computes $S f_r = -\frac{3}{2} \left( \frac{-2}{1 - 2x} \right)^2 < 0$.
Its only critical point is $1/2$, and the boundary points are not in the basin of an interior attracting orbit, since $0$ is a repelling fixed point and $f_r(1) = 0$.
So $f_r$ has at most one attracting periodic orbit in $(0, 1)$, and if it has one, the orbit of $1/2$ converges to it.
This is why, in numerical experiments, one iterates the critical point to find the attracting orbit.

The hypothesis on the Schwarzian derivative is necessary.
For $N \ge 3$ and $0 < a < 1$, the map $f(x) = x + \frac{a}{2\pi N} \sin(2 \pi N x)$ is an increasing diffeomorphism of $[0, 1]$ with no critical points, and it has the $N$ attracting fixed points $\frac{2j + 1}{2N}$, $0 \le j < N$, where $f' = 1 - a$.
This is more than $0 + 2$, so the Schwarzian derivative of this map cannot be negative everywhere.

## Explanations

### L1 human

For many smooth maps of an interval, every stable cycle must attract one of the turning points of the map.
So a map with only one turning point, like the logistic map, can have at most one stable cycle inside the interval, and following the turning point reveals it.
The condition that makes this work is a sign condition on a particular combination of derivatives, the Schwarzian derivative.

### L2 human

David Singer proved the theorem in 1978.
Negative Schwarzian derivative is the standard hypothesis in one-dimensional dynamics: it holds for all real quadratic maps and is preserved under composition.
Combined with kneading theory, it gives a fine picture of the logistic family, where at each parameter there is at most one attracting cycle.
It is also used in the study of neutral cycles, which with negative Schwarzian are attracting from at least one side.
</document>

<document source="https://tamnd.github.io/math2/node/Y2GYM93C1W.md" tag="Y2GYM93C1W" kind="statement">
# Borg–Marchenko uniqueness theorem

## Statement

Let $q_1, q_2 \colon [0, \infty) \to \R$ be integrable on each interval $[0, R]$ with $R > 0$.
Assume that both [[Schrödinger operators|89AA6XJY5K]] $-y'' + q_j y$ are in the [[limit-point case|6J9FXT6HAV]] at $\infty$.
Let $m_1$ and $m_2$ be their [[Weyl–Titchmarsh functions|XT8PA31VAX]].
If $m_1(z) = m_2(z)$ for all $z \in \C \setminus \R$, then $q_1 = q_2$ almost everywhere on $[0, \infty)$.

## Remarks

The function $m$ determines the spectral measure through its Herglotz representation, and the spectral measure determines $q$ by Gelfand–Levitan theory; this gives one proof when $q$ is regular enough.
Example: if $m(z) = i\sqrt{z}$ for all non-real $z$, then $q = 0$ almost everywhere, since this is the $m$-function of $q = 0$.
The hypothesis cannot be weakened to equality of the spectra: in general the spectrum alone does not determine $q$.
There is also a local version: if $m_1(z) - m_2(z)$ decays fast enough as $z \to \infty$ along suitable rays, then $q_1 = q_2$ on an initial interval $[0, a]$.

## Explanations

### L1 human

The theorem says that one complex function, the $m$-function, pins down the potential completely.
Two different potentials on a half-line cannot have the same $m$-function.

### L2 human

This is the basic uniqueness result of inverse spectral theory for one-dimensional Schrödinger operators.
It sits beside the [[Gelfand–Levitan|R4MRNP9EQT]] reconstruction, which turns the same data into an explicit formula for $q$.
The local version shows that the asymptotic behaviour of $m(z)$ for large $\abs{z}$ already determines $q$ on an initial interval.
</document>

<document source="https://tamnd.github.io/math2/node/Y2HXSEA2EE.md" tag="Y2HXSEA2EE" kind="concept">
# Newton series

## Summary

The Newton series of a function $f$ at a point $a$ is the series $\sum_{k \ge 0} \binom{x-a}{k} \Delta^k f(a)$, built from the forward differences of $f$ at $a$.
It is the difference analogue of the Taylor series, with generalized binomial coefficients in place of powers.
For a polynomial it is a finite sum equal to the polynomial; for other functions it may or may not converge to $f$.

## Explanations

### L1 human

A Taylor series rebuilds a function from its derivatives at one point.
A Newton series tries to rebuild it from its differences at one point, that is, from its values at $a, a+1, a+2, \dots$.
At the integers $x = a + n$ the series stops after $n + 1$ terms and gives exactly $f(a + n)$.
Between the integers it is an interpolation, which can work well or fail: the function $\sin(\pi x)$ vanishes at every integer, so its Newton series at $0$ is the zero series.
</document>

<document source="https://tamnd.github.io/math2/node/Y2NP1JFR86.md" tag="Y2NP1JFR86" kind="concept">
# Monomorphism

## Summary

A monomorphism is a [[morphism|859F0TFVA3]] $f$ that can be cancelled on the left: $f \circ g_1 = f \circ g_2$ implies $g_1 = g_2$.
It is the categorical version of an [[injective function|09NWETVTNY]].
In the categories of sets, groups, rings and vector spaces, the monomorphisms are exactly the injective homomorphisms.
Its dual notion is the [[epimorphism|RVXDSY04KG]].

## Explanations

### L1 human

An injective function never sends two different points to the same point.
A category has no points, so a monomorphism is defined by looking at maps into the domain instead.
The arrow $f$ is a monomorphism if two maps $g_1$, $g_2$ that look the same after $f$ were already the same.
For sets, test with maps from a one-point set: they pick out points $a_1$, $a_2$, and $f(a_1) = f(a_2)$ must force $a_1 = a_2$.
So in the category of sets, monomorphisms are the injective functions.
</document>

<document source="https://tamnd.github.io/math2/node/Y2S9GWG1J1.md" tag="Y2S9GWG1J1" kind="concept">
# Homotopy principle

## Summary

The homotopy principle, or h-principle, is a property of a partial differential relation.
It says that every formal solution can be deformed through formal solutions into a genuine solution.
A formal solution assigns to each point a value and candidate derivatives that satisfy the relation pointwise.
It does not require the candidates to be the true derivatives of the values.
When the h-principle holds, the existence and classification of solutions reduce to a problem in homotopy theory.
It was formulated by Mikhail Gromov, building on results of Smale, Hirsch and Nash.
It covers immersions, many symplectic and contact problems, and isometric $C^1$ embeddings.

## Explanations

### L1 human

Many geometric problems ask for a map whose derivative obeys some condition, for example a map whose derivative is never zero.
The easy part is to choose, at each point, a value and a matrix that obeys the condition, with no link between the two.
The hard part is to find a map whose real derivative obeys it.
The h-principle says that whenever the easy problem can be solved, so can the hard one, and up to deformation in the same number of ways.
A famous consequence is that a sphere can be turned inside out through immersions.
</document>

<document source="https://tamnd.github.io/math2/node/Y2SPPSQ5D4.md" tag="Y2SPPSQ5D4" kind="concept">
# Iwasawa theory

## Summary

Iwasawa theory studies arithmetic objects, such as [[ideal class groups|WQ46BD3ARY]], along an infinite tower of number fields $K = K_0 \subset K_1 \subset K_2 \subset \cdots$.
The union of the tower has Galois group isomorphic to the additive group of [[p-adic integers|V6TJAFZ4MD]] $\Z_p$.
Such a tower is a $\Z_p$-extension, and the main example is obtained from the fields $\Q(\zeta_{p^n})$ of $p$-power roots of unity.
The objects attached to the whole tower are modules over the Iwasawa algebra $\Z_p[\![T]\!]$, whose structure theory gives regular growth formulas.
The [[Iwasawa main conjecture|67A4D4FAYJ]] links these modules to $p$-adic $L$-functions.

## Explanations

### L1 human

Instead of studying one number field at a time, Iwasawa theory looks at an infinite tower of fields, each of degree $p$ over the one below.
The class numbers along the tower grow in a very regular way.
Iwasawa proved that the power of $p$ dividing the $n$-th class number is $p^{e_n}$ with $e_n = \mu p^n + \lambda n + \nu$ for all large $n$, for some fixed integers $\lambda, \mu, \nu$.
For example, if $p$ divides no class number in the tower $\Q(\zeta_{p^{n+1}})$, all three of these numbers are $0$.
Studying the whole tower at once reveals structure that is hidden in each single field.
</document>

<document source="https://tamnd.github.io/math2/node/Y2XHZKR0TX.md" tag="Y2XHZKR0TX" kind="definition">
# Finitely presented algebra

## Definition

Let $A$ be a commutative ring and let $B$ be a commutative [[$A$-algebra|DPBJ04JMH8]].
The algebra $B$ is *finitely presented* over $A$ if it has a presentation with finitely many generators and relations.
That means there are an integer $n \ge 0$ and a finitely generated ideal $I = (g_1, \ldots, g_m)$ of the [[polynomial ring|0SDC934R52]] $A[x_1, \ldots, x_n]$ with $B \cong A[x_1, \ldots, x_n]/I$ as $A$-algebras.
Here $A[x_1, \ldots, x_n]/I$ is the [[quotient ring|0076HWHGHQ]].

## Remarks

Example: $\Z[i]$ is finitely presented over $\Z$, since $\Z[i] \cong \Z[x]/(x^2 + 1)$ and the ideal $(x^2 + 1)$ is generated by one element.

Non-example: let $A = k[y_1, y_2, \ldots]$ be a polynomial ring over a field $k$ in countably many variables, and let $B = A/(y_1, y_2, \ldots) \cong k$.
Then $B$ is of [[finite type|AC5TFFQT9M]] over $A$, generated by no elements at all, but it is not finitely presented.
If an algebra is finitely presented, then the kernel of every surjection $A[x_1, \ldots, x_n] \to B$ is finitely generated, and the kernel of $A \to B$ is $(y_1, y_2, \ldots)$, which is not.

If $A$ is Noetherian, every finite type $A$-algebra is finitely presented, because every ideal of $A[x_1, \ldots, x_n]$ is finitely generated by [[Hilbert's basis theorem|0YKA9J3XEE]].
</document>

<document source="https://tamnd.github.io/math2/node/Y30JE2DC6Y.md" tag="Y30JE2DC6Y" kind="concept">
# Enriques surface

## Summary

An Enriques surface is a smooth projective [[algebraic surface|KFTECAATKG]] with no holomorphic $1$-forms and no holomorphic $2$-forms.
Its canonical divisor is not trivial, but twice it is trivial.
Over $\C$, every Enriques surface is the quotient of a [[K3 surface|Q3V566R6K2]] by an involution without fixed points.
Enriques surfaces form one of the four classes of minimal surfaces of Kodaira dimension $0$.
They are non-rational surfaces with $q = p_g = 0$, which shows that Castelnuovo's rationality criterion needs the second plurigenus.

## Explanations

### L1 human

Take a K3 surface with a symmetry of order two that moves every point.
Identifying each point with its image gives a new surface, half as big, which is an Enriques surface.
Its canonical bundle is not trivial, but it becomes trivial after squaring it.
So an Enriques surface is a close relative of a K3 surface that has lost its holomorphic volume form.
</document>

<document source="https://tamnd.github.io/math2/node/Y33AJ7GZ6R.md" tag="Y33AJ7GZ6R" kind="definition">
# Curvature form

## Definition

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, let $P \to M$ be a principal $G$-bundle, and let $\omega$ be a [[connection form|NJ21PN5ZGJ]] on $P$.
For $\mathfrak{g}$-valued $1$-forms $\alpha, \beta$ on $P$, let $[\alpha \wedge \beta]$ be the $\mathfrak{g}$-valued $2$-form $[\alpha \wedge \beta](X, Y) = [\alpha(X), \beta(Y)] - [\alpha(Y), \beta(X)]$.
The *curvature form* of $\omega$ is the $\mathfrak{g}$-valued $2$-form
$$\Omega = d\omega + \tfrac12 [\omega \wedge \omega],$$
where $d$ is the [[exterior derivative|0ZHSTAG8ZY]].
The connection is *flat* if $\Omega = 0$.

## Remarks

Unwinding the bracket, $\Omega(X, Y) = d\omega(X, Y) + [\omega(X), \omega(Y)]$.
This is the structure equation; an equivalent form is $\Omega(X, Y) = d\omega(hX, hY)$, where $h$ is the projection of $TP$ onto the horizontal bundle $\ker \omega$.
The form $\Omega$ vanishes on vertical vectors and satisfies $R_g^* \Omega = \mathrm{Ad}(g^{-1}) \Omega$.
For a matrix group, $\frac12 [\omega \wedge \omega] = \omega \wedge \omega$ with the matrix [[wedge product|0GRAATMGP1]], so $\Omega = d\omega + \omega \wedge \omega$.
Example: on $M \times G$ with $\omega$ the pullback of the [[Maurer–Cartan form|P1JVA39H5E]] $\theta$, the Maurer–Cartan equation $d\theta + \frac12 [\theta \wedge \theta] = 0$ gives $\Omega = 0$, so this connection is flat.
Non-example of flatness: for $G = \mathrm{U}(1)$ the bracket vanishes and $\Omega = d\omega$; for a connection on the Hopf bundle $S^3 \to S^2$, $\Omega$ descends to a $2$-form on $S^2$ with nonzero integral, so $\Omega \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/Y3900T2EY5.md" tag="Y3900T2EY5" kind="concept">
# Dolbeault cohomology

## Summary

Dolbeault cohomology is the cohomology of the complex of smooth [[(p,q)-forms|DC81TCR98Y]] on a [[complex manifold|JSGWAN2RJV]] with the [[dbar operator|M1VKCW3RFW]] as differential.
It measures to what extent the equation $\bar\partial u = f$ fails to be solvable for $\bar\partial$-closed data $f$.
By the [[Dolbeault isomorphism|G2HC1SFQ4P]] it computes the sheaf cohomology of the sheaves of holomorphic $p$-forms.
On compact Kähler manifolds it gives the Hodge decomposition of de Rham cohomology.

## Explanations

### L1 human

On a complex manifold, a differential form splits into pieces of type $(p,q)$: $p$ holomorphic differentials $dz_j$ and $q$ antiholomorphic differentials $d\bar z_k$.
The operator $\bar\partial$ differentiates in the antiholomorphic directions and squares to zero.
Dolbeault cohomology is the space of $\bar\partial$-closed forms modulo $\bar\partial$-exact forms.
For example, $H^{0,0}$ is the space of holomorphic functions, since a function with $\bar\partial f = 0$ is holomorphic.
On a polydisc in $\C^n$ all groups $H^{p,q}$ with $q \ge 1$ vanish, while on a compact Riemann surface of genus $g$ the group $H^{0,1}$ has dimension $g$.
</document>

<document source="https://tamnd.github.io/math2/node/Y3KYKTAMCT.md" tag="Y3KYKTAMCT" kind="definition">
# Henstock–Kurzweil integral

## Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function.
A *gauge* on $[a, b]$ is a function $\delta \colon [a, b] \to (0, \infty)$.
A tagged [[partition|069P20KYQT]] $a = x_0 < x_1 < \cdots < x_n = b$ with tags $t_i \in [x_{i-1}, x_i]$ is *$\delta$-fine* if $[x_{i-1}, x_i] \subseteq (t_i - \delta(t_i), t_i + \delta(t_i))$ for every $i$.
The function $f$ is *Henstock–Kurzweil integrable* on $[a, b]$, with integral $A \in \R$, if the following holds.
For every $\varepsilon > 0$ there is a gauge $\delta$ such that
$$\Bigl\lvert \sum_{i=1}^{n} f(t_i)(x_i - x_{i-1}) - A \Bigr\rvert < \varepsilon$$
for every $\delta$-fine tagged partition of $[a, b]$.

## Remarks

Cousin's lemma says that for every gauge a $\delta$-fine tagged partition exists, so the integral, when it exists, is unique.
If the gauges can always be taken constant, the definition is that of the [[Riemann integral|0QM4TXT3DR]].

Example: let $F(x) = x^2 \sin(1/x^2)$ for $x \ne 0$ and $F(0) = 0$, and $f = F'$ on $[0, 1]$.
The function $F$ is differentiable everywhere, so $f$ is Henstock–Kurzweil integrable with integral $F(1) - F(0) = \sin 1$.
Here $f(x) = 2x \sin(1/x^2) - (2/x)\cos(1/x^2)$ for $x \ne 0$, and $\abs{f}$ is not Lebesgue integrable.

Non-example: $f(x) = 1/x$ for $0 < x \le 1$ and $f(0) = 0$.
A nonnegative Henstock–Kurzweil integrable function is Lebesgue integrable, and $\int_0^1 dx/x = \infty$, so $f$ is not integrable.
</document>

<document source="https://tamnd.github.io/math2/node/Y3TX4J41TP.md" tag="Y3TX4J41TP" kind="statement">
# Rao–Blackwell theorem

## Statement

Let $\set{P_\theta : \theta \in \Theta}$ be a [[parametric model|MSC0MR8VMH]] for an observation $X$, and let $g \colon \Theta \to \R$.
Let $T = T(X)$ be a [[sufficient statistic|WP4TGXJQJG]] for $\theta$.
Let $\delta(X)$ be an [[estimator|0SSS6ERQ0S]] of $g(\theta)$ with $E_\theta[\delta(X)^2] < \infty$ for every $\theta \in \Theta$.
Put $\delta_1 = E[\delta(X) \mid T]$, the [[conditional expectation|0TTDGTEFMN]] given $T$.
Then:

1. $\delta_1$ does not depend on $\theta$, so it is an estimator, and $E_\theta[\delta_1] = E_\theta[\delta(X)]$ for every $\theta$.
2. For every $\theta \in \Theta$, the [[mean squared errors|05NZ1HHQMV]] satisfy
$$E_\theta\big[(\delta_1 - g(\theta))^2\big] \le E_\theta\big[(\delta(X) - g(\theta))^2\big].$$
3. Equality holds in 2 for a given $\theta$ if and only if $\delta(X) = \delta_1$ with $P_\theta$-probability $1$.

## Remarks

Sufficiency means that the conditional law of $X$ given $T$ is the same for all $\theta$, so $\delta_1$ is computed without knowing $\theta$.
The tower property gives the same mean, and the law of total variance gives $\operatorname{Var}_\theta(\delta) = \operatorname{Var}_\theta(\delta_1) + E_\theta[\operatorname{Var}(\delta \mid T)]$.
The last term is nonnegative, and it is zero exactly when $\delta$ is already a function of $T$ almost surely.
The same argument with the conditional Jensen inequality shows that $\delta_1$ is at least as good as $\delta$ for every loss that is convex in the estimate.

Example: let $X_1, \dots, X_n$ be independent Bernoulli variables with parameter $p$.
The estimator $\delta = X_1$ is unbiased for $p$, and $T = X_1 + \cdots + X_n$ is sufficient.
By symmetry $E[X_1 \mid T] = T/n$, so the Rao–Blackwell step turns $X_1$, of variance $p(1-p)$, into the sample mean, of variance $p(1-p)/n$.

Sufficiency is needed: conditioning on a statistic that is not sufficient gives a quantity that depends on the unknown $\theta$ in general, and so it is not an estimator.

## Explanations

### L1 human

Start with any rough estimate, then average it over all the data sets that carry the same summary information.
The averaged estimate is never worse, in mean squared error, than the rough one.
So a good estimator should only depend on the data through a sufficient statistic.

### L2 human

The theorem is the main tool for improving unbiased estimators: conditioning on a sufficient statistic keeps the bias and lowers the variance.
Combined with completeness of the sufficient statistic, it gives the Lehmann–Scheffé theorem, which identifies the unique minimum-variance unbiased estimator.
In practice one often guesses a crude unbiased estimator, such as a single observation or an indicator, and computes its conditional expectation given the sufficient statistic.
</document>

<document source="https://tamnd.github.io/math2/node/Y3WG213N0Q.md" tag="Y3WG213N0Q" kind="definition">
# Box spline

## Definition

Let $d \le n$ and let $\Xi = (\xi_1, \dots, \xi_n)$ be a real $d \times n$ matrix whose columns span $\R^d$.
The *box spline* $M_\Xi \colon \R^d \to \R$ is the function such that
$$\int_{\R^d} f(x)\, M_\Xi(x)\, dx = \int_{[0,1]^n} f(\Xi u)\, du$$
for every continuous function $f$ on $\R^d$ with compact support.
Equivalently, $M_\Xi$ is the density of the image of the uniform probability measure on $[0, 1]^n$ under the linear map $u \mapsto \Xi u$.

## Remarks

If $n = d$, then $M_\Xi = \abs{\det \Xi}^{-1}\, \mathbf{1}_{\Xi [0,1)^d}$, a multiple of the [[indicator function|0RCQZ36MME]] of a parallelepiped, by the change of variables formula with the [[determinant|0C3CFHDN6R]].
Adding a column gives the recursion $M_{(\Xi, \xi)}(x) = \int_0^1 M_\Xi(x - t\xi)\, dt$, a directional [[convolution|2Z29JN3VES]].
Example: $d = 1$ and $\Xi = (1, 1, \dots, 1)$ with $n$ columns gives the cardinal [[B-spline|62AFX32JGT]] of order $n$ with knots $0, 1, \dots, n$; for $n = 2$ this is the hat function $\min(x, 2 - x)$ on $[0, 2]$, zero elsewhere.
Non-example: $d = 2$ and $\Xi$ with both columns equal to $(1, 0)^T$ fails the spanning condition; the image measure lives on a line and has no density on $\R^2$.
$M_\Xi$ is nonnegative, supported on the zonotope $\Xi [0, 1]^n$, of integral $1$, and piecewise polynomial of degree $n - d$.
It has continuous derivatives of order $m - 2$, where $m$ is the least number of columns whose removal leaves a set that does not span $\R^d$.
</document>

<document source="https://tamnd.github.io/math2/node/Y3Z99461HJ.md" tag="Y3Z99461HJ" kind="statement">
# Fisher–Tippett–Gnedenko theorem

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|00MRN0FARV]] real random variables, and let $M_n = \max(X_1, \dots, X_n)$.
Assume that there are constants $a_n > 0$ and $b_n \in \R$ such that $(M_n - b_n) / a_n$ [[converges in distribution|0SRYJGXKR4]] to a random variable $Y$.
Assume that the law of $Y$ is not concentrated at one point.
Let $G$ be the [[cumulative distribution function|0H8H7ZTHW2]] of $Y$.
Then there are $a > 0$ and $b \in \R$ such that $x \mapsto G(ax + b)$ is one of the following three functions:

1. the Gumbel distribution function $\Lambda(x) = \exp(-e^{-x})$ for $x \in \R$;
2. a Fréchet distribution function $\Phi_\alpha(x) = \exp(-x^{-\alpha})$ for $x > 0$ and $\Phi_\alpha(x) = 0$ for $x \le 0$, for some $\alpha > 0$;
3. a reversed Weibull distribution function $\Psi_\alpha(x) = \exp(-(-x)^{\alpha})$ for $x \le 0$ and $\Psi_\alpha(x) = 1$ for $x > 0$, for some $\alpha > 0$.

## Remarks

The three families combine into the generalised extreme value distributions $G_\xi(x) = \exp\left(-(1 + \xi x)^{-1/\xi}\right)$ for $1 + \xi x > 0$, with $G_0 = \Lambda$ as the limit case $\xi \to 0$.
The sign of $\xi$ gives the type: $\xi > 0$ is Fréchet, $\xi = 0$ is Gumbel, and $\xi < 0$ is reversed Weibull.

Idea of the proof.
If $F$ is the distribution function of $X_1$, then $P(M_n \le a_n x + b_n) = F(a_n x + b_n)^n$.
Comparing the limits along $n$ and $nk$ shows that $G$ is max-stable: for each $k \ge 1$ there are $\alpha_k > 0$ and $\beta_k$ with $G(x)^k = G(\alpha_k x + \beta_k)$.
Solving this functional equation gives the three families.

Example: let the $X_k$ be [[exponential|0E99F51GVB]] with rate $1$, so $F(x) = 1 - e^{-x}$ for $x \ge 0$.
Then $P(M_n - \log n \le x) = (1 - e^{-x}/n)^n \to \exp(-e^{-x})$ for every $x$, so the Gumbel law appears with $a_n = 1$ and $b_n = \log n$.

The hypothesis that a non-degenerate limit exists is needed.
For some laws, such as the geometric and the Poisson distributions, no choice of $a_n$ and $b_n$ gives a non-degenerate limit.

## Explanations

### L1 human

The largest of many independent measurements has, after a shift and a rescaling, only three possible limit shapes.
Which shape appears depends on the tail of the distribution.
Heavy polynomial tails give the Fréchet law, and light tails such as the exponential or normal give the Gumbel law.
A finite upper end point can give the Weibull law.
This is the basis for modelling floods, record temperatures and other extremes.

### L2 human

The theorem plays for maxima the role that the [[central limit theorem|0EPZC4Z1FC]] plays for sums.
The set of laws whose maxima converge to a given type is its domain of attraction, and Gnedenko characterised these domains through conditions on the tail $1 - F$.
In statistics one fits the generalised extreme value family to block maxima of data.
</document>

<document source="https://tamnd.github.io/math2/node/Y3ZNV24V2C.md" tag="Y3ZNV24V2C" kind="concept">
# Evolute

## Summary

The evolute of a plane curve is the curve traced by its centres of curvature.
Equivalently, it is the envelope of the normal lines of the curve.
The evolute stops moving, and typically has a cusp, at the vertices of the curve, the points where the derivative of the curvature vanishes.

## Explanations

### L1 human

At each point of a curve, draw the [[osculating circle|NWSFMCRWHJ]], the circle that hugs the curve best.
As the point moves, the centre of that circle moves too, and its path is the evolute.
For a circle, the centre never moves, so the evolute is a single point.
For the parabola $y = x^2$, the evolute is a curve with a sharp cusp, called a semicubical parabola.
</document>

<document source="https://tamnd.github.io/math2/node/Y403TS95KG.md" tag="Y403TS95KG" kind="definition">
# Fox H-function

## Definition

Let $m, n, p, q$ be integers with $0 \le m \le q$ and $0 \le n \le p$.
Let $a_1, \dots, a_p$ and $b_1, \dots, b_q$ be complex numbers and let $A_1, \dots, A_p$ and $B_1, \dots, B_q$ be positive real numbers.
Let $\Gamma$ be the [[gamma function|0H5SYBSED9]].
Fix $z \ne 0$ and a branch of $\log z$, and put $z^{-s} = e^{-s \log z}$.
Assume that no pole $s = -(b_j + \ell)/B_j$ of a factor $\Gamma(b_j + B_j s)$ with $j \le m$ coincides with a pole $s = (1 - a_k + \ell)/A_k$ of a factor $\Gamma(1 - a_k - A_k s)$ with $k \le n$, where $\ell \in \N$.
Let $L$ be a path that separates the poles of the factors $\Gamma(b_j + B_j s)$ from the poles of the factors $\Gamma(1 - a_k - A_k s)$.
Choose $L$ so that the integral below converges.
The *Fox H-function* is the [[Barnes integral|1PDGBGEHM2]]
$$H^{m, n}_{p, q}\left[ z \,\middle|\, \begin{matrix} (a_1, A_1), \dots, (a_p, A_p) \\ (b_1, B_1), \dots, (b_q, B_q) \end{matrix} \right] = \frac{1}{2\pi i} \int_L \frac{\prod_{j = 1}^{m} \Gamma(b_j + B_j s) \prod_{k = 1}^{n} \Gamma(1 - a_k - A_k s)}{\prod_{j = m + 1}^{q} \Gamma(1 - b_j - B_j s) \prod_{k = n + 1}^{p} \Gamma(a_k + A_k s)}\, z^{-s}\, ds.$$

## Remarks

When all $A_k$ and $B_j$ equal $1$, the substitution $s \mapsto -s$ turns the integral into that of the [[Meijer G-function|19JNNN5KV7]], so $H^{m, n}_{p, q}$ equals $G^{m, n}_{p, q}$ with the same parameters $a_k, b_j$.
Example: $H^{1, 0}_{0, 1}\left[ z \,\middle|\, (0, 1) \right] = e^{-z}$ for $z > 0$.
Check: the integrand is $\Gamma(s) z^{-s}$ on a line $\operatorname{Re} s = c > 0$; closing to the left, the residues $\frac{(-1)^\ell}{\ell!} z^\ell$ at $s = -\ell$ sum to $e^{-z}$.
Non-example: a scale $B_1 = 0$ is not allowed; then $\Gamma(b_1 + B_1 s)$ is a constant and the integrand has no poles to collect, so the scales must be positive.
The Fox–Wright function and the Mittag-Leffler function $E_\alpha(z) = \sum_{k \ge 0} \frac{z^k}{\Gamma(\alpha k + 1)}$ are special H-functions.
</document>

<document source="https://tamnd.github.io/math2/node/Y40ZRJJNH0.md" tag="Y40ZRJJNH0" kind="statement">
# Serre's finiteness theorem

## Statement

Let $n \ge 1$ and $k \ge 1$, and let $\pi_k(S^n)$ be the [[homotopy group|P976WYEFMB]] of the $n$-sphere.

1. If $k \ne n$, and it is not the case that $n$ is even and $k = 2n - 1$, then $\pi_k(S^n)$ is a finite group.
2. If $n$ is even, then $\pi_{2n-1}(S^n) \cong \Z \oplus F$ for a finite abelian group $F$.

## Remarks

Since $\pi_n(S^n) \cong \Z$, the theorem says exactly which [[homotopy groups of spheres|YD6VAXXSBM]] are infinite: $\pi_n(S^n)$, and $\pi_{2n-1}(S^n)$ for even $n$.
The proof uses the Serre spectral sequence and Serre's theory of classes of abelian groups.
For odd $n$, it shows that a map $S^n \to K(\Z, n)$ representing a generator induces isomorphisms on $\pi_k \otimes \Q$.
For even $n$, the Whitehead product $[\iota_n, \iota_n] \in \pi_{2n-1}(S^n)$ has Hopf invariant $\pm 2$, so it has infinite order.
Example: $\pi_3(S^2) \cong \Z$ is the case $n = 2$, $k = 3$, with $F = 0$, and $\pi_7(S^4) \cong \Z \oplus \Z/12$.
In particular, every stable homotopy group $\pi_k^s$ with $k \ge 1$ is finite.

## Explanations

### L1 human

Most homotopy groups of spheres are finite: a map from one sphere to another, repeated enough times, can be undone.
There are only two kinds of exceptions, where an infinite family of different maps exists.
One is the wrapping of a sphere around itself, counted by the degree.
The other occurs for spheres of even dimension $n$, in dimension $2n - 1$, like the Hopf map $S^3 \to S^2$.

### L2 human

The theorem says that the rational homotopy groups of spheres are very simple, even though the integral groups are not.
It is a starting point of rational homotopy theory.
It reduces the study of homotopy groups of spheres to their $p$-primary parts, one prime at a time.
</document>

<document source="https://tamnd.github.io/math2/node/Y41SH8BJVT.md" tag="Y41SH8BJVT" kind="definition">
# Kolmogorov–Sinai entropy

## Definition

Let $(X, \mathcal{B}, \mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \colon X \to X$ be a [[measure-preserving transformation|556WBDAHNP]].
A finite measurable partition is a finite [[partition|08QMKTWF9T]] $\mathcal{P}$ of $X$ into sets of $\mathcal{B}$.
Its entropy is
$$H(\mathcal{P}) = -\sum_{P \in \mathcal{P}} \mu(P) \log \mu(P),$$
with the convention $0 \log 0 = 0$.
For partitions $\mathcal{P}$ and $\mathcal{Q}$, the join $\mathcal{P} \vee \mathcal{Q}$ is the partition into the nonempty sets $P \cap Q$ with $P \in \mathcal{P}$ and $Q \in \mathcal{Q}$, their common [[refinement|0SPZ1H3N9V]].
Let $T^{-i}\mathcal{P}$ be the partition into the sets $T^{-i}P$ with $P \in \mathcal{P}$.

1. The entropy of $T$ with respect to $\mathcal{P}$ is
$$h(T, \mathcal{P}) = \lim_{n \to \infty} \frac{1}{n} H\Big(\bigvee_{i=0}^{n-1} T^{-i} \mathcal{P}\Big).$$
2. The *Kolmogorov–Sinai entropy* of $T$ is $h_\mu(T) = \sup_{\mathcal{P}} h(T, \mathcal{P})$, over all finite measurable partitions $\mathcal{P}$; it lies in $[0, \infty]$.

## Remarks

The limit in clause 1 exists because the sequence $a_n = H(\bigvee_{i=0}^{n-1} T^{-i} \mathcal{P})$ is subadditive, $a_{m+n} \le a_m + a_n$.

Example: let $T(x) = 2x \bmod 1$ on $[0,1)$ with Lebesgue measure, and $\mathcal{P} = \set{[0, 1/2), [1/2, 1)}$.
The join $\bigvee_{i=0}^{n-1} T^{-i} \mathcal{P}$ is the partition into the $2^n$ dyadic intervals of length $2^{-n}$, so its entropy is $n \log 2$, and $h(T, \mathcal{P}) = \log 2$.
This partition generates the Borel sets under $T$, and the Kolmogorov–Sinai generator theorem gives $h_\mu(T) = \log 2$.

Non-example of positive entropy: the identity map has $\bigvee_{i=0}^{n-1} T^{-i} \mathcal{P} = \mathcal{P}$, so $h(T, \mathcal{P}) = \lim H(\mathcal{P})/n = 0$ for every $\mathcal{P}$, and $h_\mu(\mathrm{id}) = 0$.
Rotations of the circle also have entropy $0$.

Isomorphic systems have the same entropy.
With the natural logarithm the unit is the nat; with $\log_2$ it is the bit.
</document>

<document source="https://tamnd.github.io/math2/node/Y4759V8CY6.md" tag="Y4759V8CY6" kind="statement">
# Gauss multiplication formula

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\C$, and let $m$ be a positive integer.
For every $z \in \C$ with $mz \notin \set{0, -1, -2, \ldots}$,
$$\prod_{k=0}^{m-1} \Gamma\!\left(z + \frac{k}{m}\right) = (2\pi)^{(m-1)/2} \, m^{1/2 - mz} \, \Gamma(mz),$$
where $m^{1/2 - mz} = e^{(1/2 - mz)\ln m}$.

## Remarks

For $x > 0$, the function $f(x) = \frac{m^{x - 1/2}}{(2\pi)^{(m-1)/2}} \prod_{k=0}^{m-1} \Gamma\!\left(\frac{x + k}{m}\right)$ satisfies $f(x+1) = x f(x)$ and $\ln f$ is convex.
So $f = c\,\Gamma$ for a constant $c$, by the argument of the [[Bohr–Mollerup theorem|QFF87AXWN2]]; the constant is found from $f(1) = 1$, which uses $\prod_{k=1}^{m-1}\Gamma(k/m) = (2\pi)^{(m-1)/2}/\sqrt{m}$, a consequence of the [[reflection formula|WF7KQAQTS2]].
Alternatively, the constant follows from [[Stirling's formula|53750WSJ6K]] as $z \to \infty$.
For $m = 1$ the formula is trivial, and for $m = 2$ it is the [[Legendre duplication formula|CND4E1PY9V]], since $(2\pi)^{1/2} 2^{1/2 - 2z} = 2^{1-2z}\sqrt{\pi}$.

## Explanations

### L1 human

The multiplication formula splits $\Gamma$ at $m$ times a number into a product of $m$ values of $\Gamma$ at evenly spaced points.
It generalizes the duplication formula from $2$ to any whole number $m$.

### L2 human

Taking logarithms and differentiating gives the matching multiplication formula for the [[digamma function|HWRRXB0YXS]].
Similar multiplication theorems hold for other special functions, such as the Hurwitz zeta function and the Bernoulli polynomials.
Setting $z = 1/m$ gives the product $\prod_{k=1}^{m-1}\Gamma(k/m) = (2\pi)^{(m-1)/2}/\sqrt{m}$.
</document>

<document source="https://tamnd.github.io/math2/node/Y48FA69563.md" tag="Y48FA69563" kind="concept">
# Class

## Summary

In set theory, a class is a collection of sets defined by a property that all its members share.
Every [[set|0BSJ7RQ9BW]] is a class, but some classes are too large to be sets; these are the proper classes.
Examples of proper classes are the class of all sets and the class of all [[ordinals|0YSCSX5YZC]].
Classes let one speak about such large collections without the contradictions of [[Russell's paradox|0HESMMSYEG]].

## Explanations

### L1 human

A class is the collection of all sets with some property, such as "being a set" or "being a group".
Small collections, such as the even numbers, are sets.
Some collections are too big to be a set: if the collection of all sets were a set, one could form Russell's paradoxical set from it.
So one calls them proper classes, and does not allow them as elements of other collections.
In [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], a class is just a way of talking about a formula; other theories treat classes as objects.
</document>

<document source="https://tamnd.github.io/math2/node/Y48R8EJQT8.md" tag="Y48R8EJQT8" kind="statement">
# Kirchberger's theorem

## Statement

Let $d \ge 1$ and let $P, Q \subseteq \R^d$ be finite sets.
Call $P' \subseteq P$ and $Q' \subseteq Q$ strictly separated if the following holds.
There are $a \in \R^d$ and $b \in \R$ with $a \cdot p < b$ for all $p \in P'$ and $a \cdot q > b$ for all $q \in Q'$, where $a \cdot x$ is the [[dot product|0S52TNNTH9]].
Suppose that for every set $T \subseteq P \cup Q$ with at most $d + 2$ points, the sets $T \cap P$ and $T \cap Q$ are strictly separated.
Then $P$ and $Q$ are strictly separated.

## Remarks

Proof idea: for each $p \in P$, the set of pairs $(a, b) \in \R^{d+1}$ with $a \cdot p < b$ is an open half-space, hence [[convex|02AS45T3H9]], and likewise for each $q \in Q$ with $a \cdot q > b$.
The hypothesis says that every $d + 2$ of these half-spaces in $\R^{d+1}$ have a common point.
By [[Helly's theorem|F1W3QRBNGJ]] in $\R^{d+1}$, all of them have a common point $(a, b)$, which separates $P$ from $Q$.

Example: take finite sets of red and blue points in the plane.
If every four of the points can be split by a line according to their colours, then so can all of them.

The number $d + 2$ cannot be lowered.
In the plane, take the red points $(0, 0)$, $(1, 1)$ and the blue points $(1, 0)$, $(0, 1)$.
No line separates them, since the two diagonals cross, while every three of the four points can be separated.

## Explanations

### L1 human

You have red and blue points and want a straight wall with all red points on one side and all blue points on the other.
Kirchberger's theorem says you only need to test small groups: in the plane, groups of four points.
If every small group can be split correctly, the whole set can.

### L2 human

The theorem is a Helly-type result for linear separability.
In machine learning it bounds the size of a certificate that two finite classes are not linearly separable: some $d + 2$ points already show it.
The proof shows a general pattern: a separation problem becomes a common-point problem for convex sets in a space of parameters, where Helly's theorem applies.
</document>

<document source="https://tamnd.github.io/math2/node/Y4AQFH6N3Y.md" tag="Y4AQFH6N3Y" kind="definition">
# Multialgebra

## Definition

Let $\Sigma$ be a signature, a set of function symbols each with an [[arity|MSXX2P86CY]] $n \ge 0$.
For a [[set|0SJ3RX9TXA]] $A$, let $\mathcal{P}^*(A)$ be the set of nonempty subsets of $A$, a part of the [[power set|00E0YZCERH]].
A *multialgebra* of signature $\Sigma$ is a set $A$ together with a map $f^A \colon A^n \to \mathcal{P}^*(A)$ for each symbol $f \in \Sigma$ of arity $n$.
Here $A^n$ is the [[Cartesian power|09CEDJYBKP]] of $A$.
Each map $f^A$ is a *multioperation* on $A$.

## Remarks

An ordinary algebra is the same as a multialgebra in which every value $f^A(a_1, \dots, a_n)$ has exactly one element.
Some authors allow the empty set as a value; then partial algebras are also multialgebras.

Example: $A = \R$ with the binary multioperation $x \circ y = [\min(x, y), \max(x, y)]$.
Each value is a closed interval that contains $x$, so it is nonempty.

Non-example: $A = \set{0, 1, 2}$ with $x \circ y = \set{z \in A : z > x + y}$.
For $x = y = 1$ the value is $\set{z \in A : z > 2} = \emptyset$, so $\circ$ is not a multioperation in the sense above.
</document>

<document source="https://tamnd.github.io/math2/node/Y4FQT5WPQJ.md" tag="Y4FQT5WPQJ" kind="statement">
# Sauer–Shelah lemma

## Statement

Let $X$ be a finite set with $n$ elements, let $d \ge 0$ be an integer, and let $\mathcal{F}$ be a [[family|0C21Q694AG]] of subsets of $X$.
Suppose that $\mathcal{F}$ has [[VC dimension|14HX3QF45H]] at most $d$, that is, no subset of $X$ with $d + 1$ elements is shattered by $\mathcal{F}$.
Then
$$|\mathcal{F}| \le \sum_{i=0}^{d} \binom{n}{i}.$$

## Remarks

A stronger statement holds: every family $\mathcal{F}$ shatters at least $|\mathcal{F}|$ subsets of $X$, where the empty set counts and is shattered by every nonempty family.
It is proved by induction on $n$, splitting $\mathcal{F}$ according to a fixed element $x$.
If the VC dimension is at most $d$, all shattered sets have at most $d$ elements, which gives the lemma.

The bound is sharp: for $d \le n$, the family of all subsets of $X$ with at most $d$ elements has VC dimension $d$ and exactly $\sum_{i \le d} \binom{n}{i}$ members.
For fixed $d$ the bound is a polynomial of degree $d$ in $n$, while $X$ has $2^n$ subsets.

## Explanations

### L1 human

If a family of sets cannot realise every pattern on any $d + 1$ points, then it is small: its size grows only like $n^d$ instead of $2^n$.
So a single local obstruction forces a strong global limit.

### L2 human

In learning theory, the lemma bounds the number of ways a class of classifiers with finite [[VC dimension|KCFXR9K3GD]] can label $n$ points.
This polynomial growth is the key step in the proof that such classes are learnable from a number of samples that depends on the dimension.
In combinatorial geometry it bounds the complexity of set systems such as half-planes, which leads to the existence of small $\varepsilon$-nets.
</document>

<document source="https://tamnd.github.io/math2/node/Y4JBCX161G.md" tag="Y4JBCX161G" kind="definition">
# Non-atomic measure

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
An *atom* of $\mu$ is a set $A \in \Sigma$ with $\mu(A) > 0$ such that for every $B \in \Sigma$ with $B \subseteq A$, either $\mu(B) = 0$ or $\mu(A \setminus B) = 0$.
The [[measure|PR11BZQQPG]] $\mu$ is *non-atomic*, or *atomless*, if it has no atoms.

## Remarks

Lebesgue measure $\lambda$ on $\R$ is non-atomic.
If $\lambda(A) > 0$, the function $t \mapsto \lambda(A \cap (-\infty, t])$ is continuous and increases from $0$ to $\lambda(A)$.
So there is $t$ with $B = A \cap (-\infty, t]$ such that $\lambda(B) > 0$ and $\lambda(A \setminus B) > 0$.
The [[Dirac measure|0ACRGYNHSR]] $\delta_x$ is not non-atomic, since $\set{x}$ is an atom: its only measurable subsets are $\varnothing$ and $\set{x}$.
Some authors define an atom by the condition that every measurable $B \subseteq A$ has $\mu(B) = 0$ or $\mu(B) = \mu(A)$.
For sets $A$ of finite measure the two conditions agree.
</document>

<document source="https://tamnd.github.io/math2/node/Y4NSSJT4RC.md" tag="Y4NSSJT4RC" kind="concept">
# Class VII surface

## Summary

A class VII surface is a compact complex surface with [[Kodaira dimension|ZW00Q0SAGR]] $-\infty$ and first [[Betti number|ST71ZV3FES]] $1$.
The name comes from Kodaira's list of classes in the [[Enriques–Kodaira classification|1GVF19KFAK]].
These surfaces are never Kähler, since $b_1$ is odd.
They form the part of the classification of compact complex surfaces that is still not complete.
[[Hopf surfaces|PNFNGBXRM3]] and [[Inoue surfaces|GPSVJK3R4X]] are the minimal examples with second Betti number $0$.

## Explanations

### L1 human

Compact complex surfaces are sorted into classes by a few numbers, such as how many independent holomorphic forms of top degree their powers carry and the number of independent loops.
Class VII collects the surfaces with no such forms and with a single independent loop.
The simplest one is the product of a circle and a three-dimensional sphere, given a complex structure as a Hopf surface.
Such surfaces cannot be described by polynomial equations, and many questions about them are still open.
</document>

<document source="https://tamnd.github.io/math2/node/Y4T3GVK8GV.md" tag="Y4T3GVK8GV" kind="counterexample">
# Sorgenfrey line

## Counterexample

The Sorgenfrey line $\R_\ell$ is the set $\R$ with the topology generated by the base of half-open intervals $[a, b)$ with $a < b$.
It is [[separable|0Z2P41SVDP]] and [[first-countable|0CP5NDD21C]], but it is not [[second-countable|03PPNCM9GG]] and not [[metrizable|091GJ3FYD7]].

## Remarks

The rationals are dense, since each $[a, b)$ contains a rational number.
The intervals $[x, x + 1/n)$ for $n \ge 1$ form a countable neighbourhood base at $x$.
Let $\mathcal{B}$ be any base.
For each $x \in \R$ choose $B_x \in \mathcal{B}$ with $x \in B_x \subseteq [x, x + 1)$.
Then $x$ is the least element of $B_x$, so $x \mapsto B_x$ is injective and $\mathcal{B}$ is uncountable.
A separable metrizable space is second-countable, so $\R_\ell$ is not metrizable.
The Sorgenfrey line is finer than the usual topology, it is [[Lindelöf|SW0SN1NEJE]] and normal, and it is [[totally disconnected|0RJSBR9GW1]] because each $[a, b)$ is both open and closed.

## Explanations

### L1 human

Here an open set around a point may stretch to the right of the point but must start exactly at it.
This small change to the real line breaks the link between "countable dense set" and "countable base".
It is the standard example that separable spaces need not be metrizable even when they are very regular.
</document>

<document source="https://tamnd.github.io/math2/node/Y4V154JWR7.md" tag="Y4V154JWR7" kind="concept">
# Riccati differential equation of optimal control

## Summary

The Riccati differential equation of optimal control is a quadratic matrix differential equation for a symmetric matrix function $P(t)$.
It is solved backward in time from a terminal condition.
Its solution gives the optimal feedback law and the optimal cost of the [[linear-quadratic regulator|HR720TJZZR]].
It is the matrix analogue of the scalar [[Riccati equation|0VT2QK6X6Q]], and its steady states solve the [[algebraic Riccati equation|GGZV75ZZMV]].

## Explanations

### L1 human

For a linear system with a quadratic cost, the optimal remaining cost from state $x$ at time $t$ is a quadratic form $x^T P(t) x$.
The matrix $P(t)$ answers the question "how expensive is it to be at $x$ at time $t$?".
The Riccati equation tells how this matrix changes as one moves backward from the final time.
Once $P(t)$ is known, the best control is a simple linear function of the current state.
</document>

<document source="https://tamnd.github.io/math2/node/Y4WACJKC3S.md" tag="Y4WACJKC3S" kind="definition">
# Attainable null space

## Definition

Let $\mathcal{S}$ be the [[Schwartz space|CG9JEATW3Q]], and let $\mathcal{S}'$ be the space of its [[tempered distributions|AES2T5JPFP]], with the topology of convergence against each element of $\mathcal{S}$.

Let $q$ be a continuous [[seminorm|FP5GVCN8GV]] on $\mathcal{S}$.

The **attainable null space** of $q$ is the set

$$\mathcal{N}_q = \{\, p \in \mathcal{S}' : p = \lim_{k} \chi_k \text{ in } \mathcal{S}' \text{ for some sequence } (\chi_k) \subset \mathcal{S} \text{ with } q(\chi_k) \to 0 \,\}.$$

Let $\mathrm{B} \colon \mathcal{S} \to \mathcal{S}'$ be a positive-semi-definite operator. This means that $\langle \mathrm{B}\varphi, \varphi \rangle$ is real and nonnegative for every $\varphi \in \mathcal{S}$.

The seminorm induced by $\mathrm{B}$ is $|\varphi|_{\mathrm{B}} = \langle \mathrm{B}\varphi, \varphi \rangle^{1/2}$.

For this case, $\mathcal{N}_{\mathrm{B}}$ denotes the attainable null space of $|\cdot|_{\mathrm{B}}$.
</document>

<document source="https://tamnd.github.io/math2/node/Y4ZE8FVZ8D.md" tag="Y4ZE8FVZ8D" kind="concept">
# Intersection body

## Summary

Let $K$ be a [[convex body|437FSG3419]] with the origin in its interior.
Its intersection body is the star-shaped body whose radius in each direction $u$ is the volume of the central section of $K$ orthogonal to $u$.
It records all the hyperplane sections of $K$ through the origin in one geometric object.
Intersection bodies were introduced by Lutwak, and they are the key to the answer of the [[Busemann–Petty problem|MV58A1F52E]].

## Explanations

### L1 human

Cut a solid through its centre with a flat knife held perpendicular to a direction $u$, and measure the area of the cut.
Draw a point in direction $u$ at that distance from the centre.
Doing this for every direction draws the boundary of the intersection body.
For a round ball every central cut is the same disc, so the intersection body of a ball is again a ball.
</document>

<document source="https://tamnd.github.io/math2/node/Y50G59S75Q.md" tag="Y50G59S75Q" kind="definition">
# Eigenvalue perturbation

## Definition

Let $A$ and $B$ be $n \times n$ complex matrices, and let $A(\varepsilon) = A + \varepsilon B$ for $\varepsilon \in \C$.
Let $\lambda_0$ be a simple [[eigenvalue|0BQAA2BK3Q]] of $A$, that is, a root of [[multiplicity|03XVP8Q4GY]] $1$ of the characteristic polynomial.
Let $x$ be a right [[eigenvector|0N6960W0BV]] and $y$ a left eigenvector of $A$ for $\lambda_0$: $Ax = \lambda_0 x$ and $y^* A = \lambda_0 y^*$, with $x, y \ne 0$.
Then $y^* x \ne 0$.
There are a neighbourhood $D$ of $0$ in $\C$ and a unique analytic function $\lambda \colon D \to \C$ with $\lambda(0) = \lambda_0$ and the following property.
For each $\varepsilon \in D$, the number $\lambda(\varepsilon)$ is a simple eigenvalue of $A(\varepsilon)$.
The *eigenvalue perturbation* of $\lambda_0$ in the direction $B$ is this function $\lambda$.
Its *first-order coefficient* is
$$\lambda'(0) = \frac{y^* B x}{y^* x}.$$

## Remarks

So $\lambda(\varepsilon) = \lambda_0 + \varepsilon \, \dfrac{y^* B x}{y^* x} + O(\varepsilon^2)$.
To obtain the coefficient, differentiate $A(\varepsilon) x(\varepsilon) = \lambda(\varepsilon) x(\varepsilon)$ at $0$ and multiply on the left by $y^*$; the terms with $x'(0)$ cancel.

Example: for a Hermitian $A$ one may take $y = x$, so $\lambda'(0) = x^* B x / x^* x$.
For $A = \operatorname{diag}(1, 2)$ and $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, with $\lambda_0 = 1$ and $x = y = e_1$, this gives $\lambda'(0) = 0$.
Indeed the eigenvalues of $A + \varepsilon B$ are $\frac{3}{2} \pm \sqrt{\frac{1}{4} + \varepsilon^2}$, and the smaller one is $1 - \varepsilon^2 + O(\varepsilon^4)$.

Non-example: for $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, the eigenvalue $0$ is not simple.
With $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, the eigenvalues of $A + \varepsilon B$ are $\pm \sqrt{\varepsilon}$, which are not analytic in $\varepsilon$ at $0$.
Here the right eigenvector $x = e_1$ and the left eigenvector $y = e_2$ satisfy $y^* x = 0$, so the formula for $\lambda'(0)$ has no meaning.

## Explanations

### L1 human

When a matrix depends smoothly on a small parameter, a simple eigenvalue moves smoothly too.
Its speed at the start is given by a short formula with the left and right eigenvectors.
A small value of $y^* x$ means that the eigenvalue is sensitive to changes.
</document>

<document source="https://tamnd.github.io/math2/node/Y50YJMKRAM.md" tag="Y50YJMKRAM" kind="definition">
# Associative quotient of a magma

## Definition

Let $(M, \cdot)$ be a [[magma|9689SFPVE9]].
Let $\sim$ be the smallest [[congruence|DCSXXT4SER]] on $M$ such that $(x \cdot y) \cdot z \sim x \cdot (y \cdot z)$ for all $x, y, z \in M$.
The *associative quotient* of $M$ is the [[quotient magma|N5KE7FH6CX]] $M/{\sim}$, with the quotient map $\pi \colon M \to M/{\sim}$, $\pi(x) = [x]$.

## Remarks

The quotient $M/{\sim}$ is a [[semigroup|0XMEM406JM]], since $([x][y])[z] = [(xy)z] = [x(yz)] = [x]([y][z])$.
Universal property: for every semigroup $S$ and every magma homomorphism $f \colon M \to S$, there is exactly one homomorphism $\bar f \colon M/{\sim} \to S$ with $\bar f \circ \pi = f$.
This holds because the kernel of $f$ is a congruence that contains all the pairs $((xy)z, x(yz))$, so it contains $\sim$.

Example: if $M$ is already a semigroup, then $\sim$ is equality and the associative quotient is $M$ itself.
Example: for the free magma on one letter $a$, the associative quotient is the free semigroup on $a$.
It is isomorphic to the positive integers under addition, with a bracketed word sent to its length.

Non-example: the one-element semigroup is a quotient of every nonempty magma, but it is not the associative quotient of the free magma on $a$.
The length map to $(\Z_{>0}, +)$ is a homomorphism to a semigroup that does not factor through the one-element quotient, so the universal property fails.

In Mathlib this construction is `Magma.AssocQuotient`.
</document>

<document source="https://tamnd.github.io/math2/node/Y553MSYDAH.md" tag="Y553MSYDAH" kind="statement">
# Schur–Horn theorem

## Statement

Let $n \ge 1$ and let $\lambda = (\lambda_1, \ldots, \lambda_n)$ and $d = (d_1, \ldots, d_n)$ be vectors in $\R^n$.
Then the following are equivalent.

1. There is an $n \times n$ [[Hermitian matrix|0P17NGQQMQ]] with [[eigenvalues|0BQAA2BK3Q]] $\lambda_1, \ldots, \lambda_n$, counted with multiplicity, and diagonal entries $d_1, \ldots, d_n$.
2. The vector $d$ is [[majorized|5K7VT2VKZT]] by $\lambda$, that is, $d \prec \lambda$.

## Remarks

Schur proved that the condition is necessary, and Horn proved that it is sufficient.
For necessity, write $A = U \Lambda U^*$ with $U$ [[unitary|0AR68JTMDT]] and $\Lambda$ diagonal; then $d_i = \sum_j \abs{u_{ij}}^2 \lambda_j$.
The matrix $(\abs{u_{ij}}^2)$ is [[doubly stochastic|JJA90EB2PS]], so $d \prec \lambda$ by the Hardy–Littlewood–Pólya characterisation of majorization.
The converse is proved by constructing the matrix, for example by induction on $n$.
Example: $\lambda = (2, 0)$ and $d = (1, 1)$ satisfy $d \prec \lambda$, and $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ has eigenvalues $2, 0$ and diagonal $1, 1$.
The diagonal $(3, -1)$ is impossible for the eigenvalues $(2, 0)$, since $3 > 2$.
The same statement holds with real symmetric matrices in place of Hermitian matrices.

## Explanations

### L1 human

The diagonal entries of a Hermitian matrix are averages of its eigenvalues.
So they are less spread out than the eigenvalues, in the precise sense of majorization.
The theorem says that this is the only restriction: every diagonal that is less spread out than the eigenvalues actually occurs.

### L2 human

The case $k = 1$ of the majorization gives $\max_i d_i \le \max_j \lambda_j$, a special case of the Rayleigh quotient bound.
The theorem identifies the set of diagonals of the matrices $U \Lambda U^*$, $U$ unitary, with the permutohedron, the convex hull of the permutations of $\lambda$.
In this form it is a special case of Kostant's convexity theorem and of the convexity theorem of Atiyah and of Guillemin and Sternberg for moment maps.
In frame theory it is used to build frames with prescribed norms and frame operator.
</document>

<document source="https://tamnd.github.io/math2/node/Y5B0PEBKE8.md" tag="Y5B0PEBKE8" kind="concept">
# Galerkin method

## Summary

The Galerkin method approximates the solution of a problem in weak form, $B(u, v) = f(v)$ for all $v$ in a [[Hilbert space|HM2HHE3J51]].
It solves the same problem on a finite-dimensional subspace.
This turns a partial differential equation into a finite system of linear equations.
For a bounded and [[coercive|86KRHS6X22]] form, the [[Lax–Milgram theorem|85KK79FCN5]] gives existence and uniqueness of the approximation.
Its error is within a fixed factor of the best possible approximation from the subspace.
The finite element method and spectral methods are Galerkin methods, and Galerkin approximations are also used to prove existence of [[weak solutions|SJZX81A783]].

## Explanations

### L1 human

The true solution lives in an infinite-dimensional space of functions.
The Galerkin method picks a few simple functions, such as polynomials or piecewise linear "hat" functions, and looks for the best combination of them.
"Best" means that the equation holds when tested against each of the chosen functions.
This gives as many equations as unknown coefficients, which a computer can solve.
</document>

<document source="https://tamnd.github.io/math2/node/Y5BF0G0T9X.md" tag="Y5BF0G0T9X" kind="definition">
# Mal'cev term

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
A *Mal'cev term* for $\mathcal{V}$ is a ternary term $p(x, y, z)$ such that $\mathcal{V}$ satisfies the [[identities|PKD440G2XW]]
$$p(x, y, y) = x \quad \text{and} \quad p(x, x, y) = y.$$
An algebra $A$ has a *Mal'cev operation* if some ternary [[term operation|N9M2N27END]] of $A$ satisfies these two equations for all elements of $A$.

## Remarks

Example: for groups, $p(x, y, z) = x y^{-1} z$ is a Mal'cev term, since $x y^{-1} y = x$ and $x x^{-1} z = z$.
For modules and rings, $p(x, y, z) = x - y + z$ is a Mal'cev term.
Non-example: the variety of semilattices has no Mal'cev term.
Every semilattice term in the variables $x, y, z$ equals the join of the set $S$ of variables that occur in it.
If $p(x, y, y) = x$, then neither $y$ nor $z$ is in $S$, so $p = x$, and then $p(x, x, y) = x \ne y$ in a semilattice with two elements.
By Mal'cev's theorem, a variety has a Mal'cev term if and only if it is a [[congruence-permutable variety|1B3N9T68YV]].

## Explanations

### L1 human

A Mal'cev term is a three-input formula that returns the outer input when the other two inputs agree.
</document>

<document source="https://tamnd.github.io/math2/node/Y5CC9MSGYM.md" tag="Y5CC9MSGYM" kind="definition">
# Function of exponential order

## Definition

Let $f \colon [0, \infty) \to \C$ be a [[function|0XHDTE5YMZ]] and let $a \in \R$.
The function $f$ is *of exponential order $a$* if there are constants $M > 0$ and $T \ge 0$ such that
$$\abs{f(t)} \le M e^{at} \quad \text{for all } t \ge T,$$
where $e^{at}$ is the [[exponential function|0RMVS7RXZX]].
The function $f$ is *of exponential order* if it is of exponential order $a$ for some real $a$.

## Remarks

Example: $f(t) = t^n$ with $n \in \N$ is of exponential order $1$, since $e^t \ge t^n / n!$ for $t \ge 0$ by the power series of $e^t$; so $\abs{f(t)} \le n!\, e^t$ with $T = 0$.
In fact $t^n$ is of exponential order $a$ for every $a > 0$.
Non-example: $f(t) = e^{t^2}$ is not of exponential order, since $e^{t^2}/e^{at} = e^{t(t - a)} \to \infty$ for every real $a$.
A bounded function, such as $\sin t$, is of exponential order $0$.
Some books require the bound for all $t \ge 0$ ($T = 0$); for functions that are bounded on bounded intervals the two versions agree, after enlarging $M$.
</document>

<document source="https://tamnd.github.io/math2/node/Y5HMW0BDTP.md" tag="Y5HMW0BDTP" kind="concept">
# Score test

## Summary

The score test is a [[hypothesis test|087P03NEND]] that uses the slope of the log-likelihood at the null value of the parameter.
If the null hypothesis is true, the [[score|CP97GRVTQ0]] there should be close to zero, measured against the [[Fisher information|G1Z3WSTE8Q]].
It needs only the fit under the null hypothesis, and it is also called Rao's score test or the Lagrange multiplier test.

## Explanations

### L1 human

At the true parameter, the log-likelihood is flat on average: it has no tendency to go up or down.
The score test checks whether the log-likelihood is steep at the value claimed by the null hypothesis.
A steep slope means that moving away from that value would raise the likelihood of the data by a large amount.
For a coin with $60$ heads in $100$ tosses and $H_0 \colon p = 0.5$, the score statistic is $(60 - 50)^2/25 = 4$, which is beyond the $95\%$ chi-squared quantile $3.84$.
</document>

<document source="https://tamnd.github.io/math2/node/Y5V5CN8FGN.md" tag="Y5V5CN8FGN" kind="definition">
# Semi-continuity

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]], let $f \colon X \to [-\infty, \infty]$ be a function to the [[extended real line|0GSZ7B0CGR]], and let $x_0 \in X$.

1. The function $f$ is *upper semicontinuous at* $x_0$ if for each real $y > f(x_0)$ there is a [[neighbourhood|01T3TMGTTE]] $U$ of $x_0$ with $f(x) < y$ for all $x \in U$.
2. The function $f$ is *lower semicontinuous at* $x_0$ if for each real $y < f(x_0)$ there is a neighbourhood $U$ of $x_0$ with $f(x) > y$ for all $x \in U$.
3. The function $f$ is *upper* (or *lower*) *semicontinuous* if it is upper (or lower) semicontinuous at each point of $X$.

## Remarks

For $X \subseteq \R$, upper semicontinuity at $x_0$ means $\limsup_{x \to x_0} f(x) \le f(x_0)$, and lower semicontinuity means $\liminf_{x \to x_0} f(x) \ge f(x_0)$.
The function $f$ is upper semicontinuous if and only if the set $\set{x \in X : f(x) < y}$ is open for each real $y$.
It is lower semicontinuous if and only if $\set{x \in X : f(x) > y}$ is open for each real $y$.
The function $f$ is upper semicontinuous if and only if $-f$ is lower semicontinuous.
A real function is [[continuous|0P27W9RW0G]] at $x_0$ if and only if it is both upper and lower semicontinuous at $x_0$.

Example: the [[indicator function|0VSB0XF4G7]] $f = \mathbf 1_{[0, 1]}$ on $\R$ is upper semicontinuous.
The set $\set{f < y}$ is empty for $y \le 0$, is $\R \setminus [0, 1]$ for $0 < y \le 1$, and is $\R$ for $y > 1$, and each of these is open.

Non-example: the same function is not lower semicontinuous at $0$.
Take $y = 1/2 < 1 = f(0)$; each neighbourhood of $0$ contains a point $x < 0$ with $f(x) = 0 < y$.
</document>

<document source="https://tamnd.github.io/math2/node/Y60D7BN1BN.md" tag="Y60D7BN1BN" kind="statement">
# Generic flatness

## Statement

Let $Y$ be an [[integral scheme|17NVR4XBHK]] that is [[Noetherian|ND6XEY68P4]].
Let $f \colon X \to Y$ be a [[morphism|JXRKH6P1Z4]] [[of finite type|AHTE5DVX61]].
Then there is a nonempty open subset $U \subseteq Y$ such that the restriction $f^{-1}(U) \to U$ is [[flat|5M6KS8F47R]].

## Remarks

The statement reduces to the affine case and to the generic freeness lemma.
That lemma says the following: let $A$ be a Noetherian domain, $B$ a finitely generated $A$-algebra, and $M$ a finitely generated $B$-module.
Then there is a nonzero $a \in A$ such that $M_a$ is a free $A_a$-module.
Its proof uses Noether normalization over the fraction field of $A$ and induction on dimension.
For example, the blow-up of $\mathbb{A}^2_k$ at the origin is flat over $U = \mathbb{A}^2_k \setminus \set{0}$, where it is an isomorphism.
The integrality of $Y$ cannot be dropped.
For $Y = \operatorname{Spec} k[\varepsilon]/(\varepsilon^2)$ and the closed immersion $\operatorname{Spec} k \to Y$, the only nonempty open subset of $Y$ is $Y$, and $k$ is not flat over $k[\varepsilon]/(\varepsilon^2)$.
The same proof gives a sheaf version: a coherent sheaf on $X$ is flat over $Y$ above some nonempty open subset of $Y$.

## Explanations

### L1 human

A family of spaces can behave badly over some special points of the base.
Generic flatness says that this bad behaviour is confined to a proper closed subset.
Over a dense open part of the base, the family is flat, so its fibres vary in a controlled way.

### L2 human

Generic flatness is the starting point of many stratification arguments.
One applies it on $Y$, then again on the closed complement with its reduced structure, and so on; Noetherian induction ends the process.
The result is a finite partition of $Y$ into locally closed pieces over each of which $f$ is flat.
This is how one builds flattening stratifications and proves that Hilbert polynomials of fibres take finitely many values.
The theorem also gives a quick proof of Chevalley's theorem on constructible sets.
</document>

<document source="https://tamnd.github.io/math2/node/Y63847Z06N.md" tag="Y63847Z06N" kind="definition">
# Completion of a topological group

## Definition

Let $G$ be a [[Hausdorff topological group|7GKCGQ7215]].
The *two-sided uniformity* of $G$ is the coarsest uniform structure finer than both the left and the right [[uniform structures|P89SJ7JWJE]] of $G$.
$G$ is *complete* if every [[Cauchy filter|WZC4J62B1G]] for the two-sided uniformity converges.
A *completion* of $G$ is a pair $(\hat G, \iota)$ such that:

1. $\hat G$ is a complete Hausdorff topological group;
2. $\iota \colon G \to \hat G$ is an [[isomorphism of topological groups|T3A3JTW6Q4]] from $G$ onto the subgroup $\iota(G)$ with the subspace topology;
3. $\iota(G)$ is dense in $\hat G$.

## Remarks

Every Hausdorff topological group has a completion, and any two completions are isomorphic by a unique isomorphism compatible with the maps $\iota$.
Example: $(\R, +)$ with the inclusion of $(\Q, +)$.
The group $\R$ is complete, $\Q$ carries the subspace topology, and $\Q$ is dense in $\R$.
Non-example: $(\R, +)$ with the inclusion of $(\Z, +)$ fails clause 3, since $\Z$ is closed in $\R$ and not dense.
For an abelian group the left, right and two-sided uniformities agree.
In general the completion for the left uniformity alone need not be a group.
</document>

<document source="https://tamnd.github.io/math2/node/Y68BHXZJBV.md" tag="Y68BHXZJBV" kind="concept">
# Perimeter of a set

## Summary

The perimeter of a measurable set $E \subseteq \R^n$ is a measure-theoretic version of the area of its boundary.
It is the [[total variation|8QA5DEWNYD]] of the distributional gradient of the indicator function of $E$.
For a set with smooth boundary it equals the $(n-1)$-dimensional area of the boundary, but it is defined for every Lebesgue measurable set.
It is lower semicontinuous under convergence in $L^1_{\mathrm{loc}}$, which makes it the natural quantity to minimize in problems such as the isoperimetric problem.

## Explanations

### L1 human

To measure the boundary of a rough set, test the set against all small vector fields and record the largest flux that can leave through it.
For a smooth region this flux is largest when the field points straight out of the boundary, and it then equals the area of the boundary.
For example, the unit disc in the plane has perimeter $2\pi$, its circumference.
Removing a single point or a segment from the disc does not change its perimeter, because the perimeter only sees the set up to sets of measure zero.
</document>

<document source="https://tamnd.github.io/math2/node/Y6C40MHZZC.md" tag="Y6C40MHZZC" kind="definition">
# Structure

## Definition

Let $L$ be a [[signature|WDVVEYD6AT]]: a set of function symbols, relation symbols and constant symbols.
Each function symbol and each relation symbol has an arity $n \ge 1$.
An $L$-structure $\mathcal{M}$ consists of the following data.

1. A nonempty [[set|0BSJ7RQ9BW]] $M$, the domain or universe of $\mathcal{M}$.
2. For each $n$-ary function symbol $f$ of $L$, a [[function|0EPAT51DXQ]] $f^{\mathcal{M}} \colon M^n \to M$, where $M^n$ is the [[Cartesian product|09CEDJYBKP]] of $n$ copies of $M$.
3. For each $n$-ary relation symbol $R$ of $L$, a subset $R^{\mathcal{M}} \subseteq M^n$.
4. For each constant symbol $c$ of $L$, an element $c^{\mathcal{M}} \in M$.

The objects $f^{\mathcal{M}}$, $R^{\mathcal{M}}$ and $c^{\mathcal{M}}$ are the interpretations of the symbols in $\mathcal{M}$.

## Remarks

Example: let $L = \set{+, -, 0}$ with $+$ binary, $-$ unary and $0$ a constant.
The integers $\Z$ with addition, negation and the number $0$ form an $L$-structure, because each symbol receives a function of the right arity on $\Z$ or an element of $\Z$.

Non-example: for the same signature, the natural numbers $\N$ with $x \mapsto -x$ do not form an $L$-structure, because $x \mapsto -x$ is not a function from $\N$ to $\N$.

A structure is often written as a tuple, such as $(\Z, +, -, 0)$.
Some authors, and the Mathlib library, allow the empty set as domain.

## Explanations

### L1 human

A structure gives a meaning to each symbol of a formal language.
The symbol $<$ alone says nothing; in the structure $(\Q, <)$ it means the usual order of the rational numbers.
Once each symbol has a meaning, each sentence of the language becomes true or false in the structure.
</document>

<document source="https://tamnd.github.io/math2/node/Y6D894R6HB.md" tag="Y6D894R6HB" kind="definition">
# Divisor class group

## Definition

Let $A$ be a [[Krull domain|CB9WJCN9BM]] with [[field of fractions|0VF11ZTTMD]] $K$, and let $X$ be the set of prime ideals of $A$ of [[height|87WH65FJB8]] one.
For $P \in X$, the localization $A_P$ is a [[discrete valuation ring|MGM3CNJW9V]]; let $v_P \colon K^\times \to \Z$ be its valuation.

1. The *divisor group* $\operatorname{Div}(A)$ is the [[free abelian group|0YMSEZE1VH]] on the set $X$.
2. For $x \in K^\times$, the *principal divisor* of $x$ is $\operatorname{div}(x) = \sum_{P \in X} v_P(x) P$; the sum is finite because $A$ is a Krull domain.
3. The map $x \mapsto \operatorname{div}(x)$ is a group homomorphism $K^\times \to \operatorname{Div}(A)$; its image is the subgroup $\operatorname{Prin}(A)$ of principal divisors.
4. The *divisor class group* of $A$ is the [[quotient group|0YYWCNTPWR]] $\operatorname{Cl}(A) = \operatorname{Div}(A)/\operatorname{Prin}(A)$.

## Remarks

$\operatorname{Cl}(A) = 0$ if and only if $A$ is a [[unique factorisation domain|06YMXJMXYS]].
Example: for $A = \Z$, the height-one primes are $p\Z$, and the divisor $\sum n_p \, p\Z$ is $\operatorname{div}\left(\prod p^{n_p}\right)$, so every divisor is principal and $\operatorname{Cl}(\Z) = 0$.
Non-example of a principal divisor: in $A = \Z[\sqrt{-5}]$, the divisor $P$ with $P = (2, 1 + \sqrt{-5})$ is not principal, since no element of $A$ has norm $2$; but $2P = \operatorname{div}(2)$, because $P^2 = (2)$.
For a Dedekind domain, $\operatorname{Cl}(A)$ is the ideal class group, the group of nonzero fractional ideals modulo principal ones.
The notation $\operatorname{Cl}(A)$ is standard.
</document>

<document source="https://tamnd.github.io/math2/node/Y6EPBG6S8G.md" tag="Y6EPBG6S8G" kind="concept">
# Sine and cosine series

## Summary

The Fourier sine series and the Fourier cosine series expand a function given on the interval $[0, \pi]$ in the functions $\sin nx$ alone or $\cos nx$ alone.
They are the [[Fourier series|G12KXDHPPS]] of the odd and of the even $2\pi$-periodic extension of the function.
They appear when separation of variables is applied to the heat or wave equation on an interval.
Dirichlet boundary conditions lead to sine series, and Neumann conditions to cosine series.

## Explanations

### L1 human

A function given only on $[0, \pi]$ can be extended to $[-\pi, \pi]$ in two natural ways: as an odd function or as an even function.
The odd extension has a Fourier series with only sines, and the even extension one with only cosines.
For example, the constant $1$ on $(0, \pi)$ has the cosine series $1$ and the sine series $\frac{4}{\pi}\left(\sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \cdots\right)$.
The sine series vanishes at $0$ and $\pi$, which suits a vibrating string fixed at both ends.
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# Injectivity radius

## Summary

Let $p$ be a point of a [[Riemannian manifold|7CBT4SM83Y]].
Its injectivity radius at $p$ is the largest $r$ such that the [[exponential map|Y7Z0AGY1A7]] at $p$ is a diffeomorphism on the $r$-ball in $T_pM$.
Within this radius, [[normal coordinates|RRHKMGNKG9]] at $p$ exist and each point is joined to $p$ by a unique shortest geodesic.
The injectivity radius of the whole manifold is the infimum over all points.
Lower bounds for it are a key ingredient in compactness and finiteness theorems.

## Explanations

### L1 human

From a point $p$, send out geodesics in all directions at the same speed.
For a while they spread out without meeting, and every nearby point is reached in exactly one way.
The injectivity radius is how far one can go before this breaks down.
On the unit sphere all geodesics from the north pole meet again at the south pole, at distance $\pi$, so the injectivity radius is $\pi$.
On a thin cylinder of circumference $c$, geodesics going around in opposite directions meet after distance $c/2$, so the injectivity radius is $c/2$.
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# Glimm scheme

## Summary

The Glimm scheme, or random choice method, builds approximate solutions of a hyperbolic system of [[conservation laws|8RA7AMWSAF]] in one space dimension.
It pieces together exact solutions of [[Riemann problems|GJWKX22VVE]] on a grid and samples them at random points.
Glimm used it in 1965 to prove global existence of weak solutions for initial data of small total variation.
See the [[definition|5JQG4S5JVH]].

## Explanations

### L1 human

Cut the initial data into small steps, so it is constant on each cell.
At each cell boundary the data has a jump, and the solution of a jump is known: it is a Riemann problem.
Run these local solutions for a short time, before they can meet.
Then flatten the result again into steps, taking on each cell the value at one randomly chosen point.
Repeat this many times, and let the grid become finer.
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# Pseudoconvexity

## Definition

Let $D \subseteq \C^n$ be an [[open set|02BH3MCBQB]].
Then $D$ is *pseudoconvex* if there is a continuous [[plurisubharmonic function|FF2P7ER3QT]] $\varphi \colon D \to \R$ with the following property.
For every $c \in \R$, the set $\set{z \in D : \varphi(z) < c}$ has [[compact|0R6EWFKRCF]] closure contained in $D$.

A function $\varphi$ with this last property is called an exhaustion function of $D$.

## Remarks

Example: the unit ball $B = \set{z \in \C^n : \abs{z} < 1}$ with $\varphi(z) = -\log(1 - \abs{z}^2)$.
The function $\abs{z}^2$ is plurisubharmonic, and $t \mapsto -\log(1 - t)$ is convex and increasing on $[0, 1)$, so $\varphi$ is plurisubharmonic and continuous.
The set $\set{\varphi < c}$ is the ball of radius $\sqrt{1 - e^{-c}}$ (empty for $c \le 0$), whose closure is compact and contained in $B$.

Non-example: $D = \set{z \in \C^2 : 0 < \abs{z} < 1}$.
Let $\varphi$ be any continuous plurisubharmonic function on $D$, and let $M$ be its maximum on the compact set $\set{\abs{z_1} \le 1/4, \ \abs{z_2} = 1/2} \subseteq D$.
For $0 < \abs{t} \le 1/4$, the disc $\zeta \mapsto (t, \zeta/2)$, $\abs{\zeta} \le 1$, lies in $D$, so by subharmonicity $\varphi(t, 0) \le M$.
The points $(t, 0)$ accumulate at the origin, which is not in $D$, so $\set{\varphi < M + 1}$ does not have compact closure in $D$, and no exhaustion of the required kind exists.

Equivalent forms: for $D \ne \C^n$, $D$ is pseudoconvex exactly when $-\log d(z)$ is plurisubharmonic on $D$, where $d(z)$ is the distance from $z$ to the boundary.
If $D$ has $C^2$ boundary, pseudoconvexity is equivalent to Levi pseudoconvexity: the Levi form of a defining function is positive semidefinite on the complex tangent space at each boundary point.
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# Numerical range

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] with inner product $\ip{\cdot}{\cdot}$, and let $T \colon H \to H$ be a [[continuous linear map|03TM135G0N]].
The *numerical range* of $T$ is the set
$$W(T) = \set{\ip{Tx}{x} : x \in H, \ \norm{x} = 1} \subseteq \C.$$
The *numerical radius* of $T$ is $w(T) = \sup \set{\abs{z} : z \in W(T)}$.

## Remarks

The numerical range is convex; this is the [[Toeplitz–Hausdorff theorem|8TV0DJBBAY]].
The closure of $W(T)$ contains the [[spectrum|BMKATJ802A]] of $T$.
The numerical radius satisfies $\tfrac12 \norm{T} \le w(T) \le \norm{T}$, with the [[operator norm|06EKW1AKXF]].
The operator $T$ is self-adjoint if and only if $W(T) \subseteq \R$.

Example: let $T = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ on $\C^2$.
For $x = (x_1, x_2)$ with $\abs{x_1}^2 + \abs{x_2}^2 = 1$, $\ip{Tx}{x} = x_2 \overline{x_1}$, whose absolute value is at most $\tfrac12$ and takes every value with $\abs{z} \le \tfrac12$.
So $W(T)$ is the closed disc of radius $\tfrac12$, and $w(T) = \tfrac12 \norm{T}$.

Non-example: for the same $T$ and $x = (1, 1)$, $\ip{Tx}{x} = 1$, which is not in $W(T)$; the vector $x$ is not a unit vector.
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# Castelnuovo–Mumford regularity

## Summary

The Castelnuovo–Mumford regularity is an integer that bounds the degrees in the minimal free resolution of a finitely generated graded module over a polynomial ring.
It measures how complicated the module is from the point of view of syzygies.
It bounds the degree from which the Hilbert function agrees with the [[Hilbert polynomial|RMH4FBCFDA]].
It also bounds the complexity of [[Gröbner basis|C83KPT47R0]] computations.

## Explanations

### L1 human

A module over a polynomial ring is described by generators, relations among the generators, relations among the relations, and so on.
Each of these comes in certain degrees.
The regularity records how far the degrees drift upward as one goes down this chain, after a shift of one degree for each step.
For the quotient $k[x]/(x^d)$, the generator has degree $0$ and the single relation has degree $d$, so the regularity is $d - 1$.
A small regularity means the module is simple in this sense.
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# Hurewicz fibration

## Summary

A Hurewicz fibration is a continuous map $p \colon E \to B$ that has the [[homotopy lifting property|G4QDM01P6J]] with respect to all topological spaces.
Fibrations behave in homotopy theory like fibre bundles.
The fibres over points of a path-connected base are homotopy equivalent, and there is a long exact sequence of homotopy groups.
A [[Serre fibration|PB9JXQ2KC3]] is the weaker notion that asks for the lifting property only for cubes, or equivalently for CW complexes.

## Explanations

### L1 human

Think of $E$ as a space lying over $B$, with a fibre over each point.
Suppose a family of points moves in $B$ and you know where it starts in $E$.
The map is a fibration if you can always lift the whole motion to $E$ continuously.
Product projections, covering maps and fibre bundles over nice bases are fibrations.
The path space fibration, sending a path in $B$ to its end point, is a central example in homotopy theory.
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# Universal coalgebra

## Definition

Let $F$ be a [[functor|28NQ0KRHD3]] from the category of [[sets|0BSJ7RQ9BW]] to itself.
An *$F$-coalgebra* is a pair $(X, \alpha)$ where $X$ is a set and $\alpha \colon X \to F(X)$ is a [[function|0EPAT51DXQ]].
The set $X$ is the carrier, or set of states, and $\alpha$ is the structure map.
A *homomorphism* of $F$-coalgebras from $(X, \alpha)$ to $(Y, \beta)$ is a function $h \colon X \to Y$ such that $\beta \circ h = F(h) \circ \alpha$.
*Universal coalgebra* is the study of $F$-coalgebras and their homomorphisms, for an arbitrary functor $F$ of this kind.

## Remarks

Example: let $A$ be a set and $F(X) = A \times X$, with $F(h)(a, x) = (a, h(x))$.
An $F$-coalgebra is a set $X$ with a map $\alpha(x) = (o(x), n(x))$, an output $o(x) \in A$ and a next state $n(x) \in X$.
The set $A^{\N}$ of streams with $\beta(s) = (s_0, (s_1, s_2, \dots))$ is such a coalgebra.
For every coalgebra $(X, \alpha)$, the map $h(x) = (o(x), o(n(x)), o(n(n(x))), \dots)$ satisfies $\beta \circ h = F(h) \circ \alpha$, and it is the only such map, so the streams form a final $F$-coalgebra.

Non-example of a homomorphism: let $A = \set{a, b}$, let $X = \set{x}$ with $\alpha(x) = (a, x)$, and let $Y = \set{y_1, y_2}$ with $\beta(y_1) = (a, y_2)$ and $\beta(y_2) = (b, y_2)$.
The map $h(x) = y_1$ keeps the output of $x$, but $\beta(h(x)) = (a, y_2)$ while $F(h)(\alpha(x)) = (a, y_1)$, so the square does not commute.

The same definition, with any category in place of sets, gives the [[F-coalgebra|6BS0300CMV]].
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# Circle average

## Summary

The circle average of a function is its mean value over a circle, taken with respect to normalized arc length.
It is the basic quantity in the mean value property of harmonic functions and in the sub-mean value property of subharmonic functions.
Many results of planar potential theory, such as Jensen's formula, are identities between circle averages.

## Explanations

### L1 human

Walk once around a circle at constant speed and record the value of a function along the way.
The circle average is the average of all the recorded values.
For $f(z) = \abs{z}^2$ and the circle of radius $r$ around $0$, every recorded value is $r^2$, so the average is $r^2$.
For $f(z) = z$ and the same circle, the values point in all directions equally and cancel, so the average is $0$, the value at the centre.
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# Homogeneity equation

## Summary

The homogeneity equation is the [[functional equation|SRAN4EKWS0]] $f(tx) = t^k f(x)$, required for all positive scalars $t$ and all points $x$.
Its solutions are the positively homogeneous functions of degree $k$.
They are the functions that scale in a fixed way when all inputs are scaled by the same factor.
[[Homogeneous polynomials|2C3PAQ7H50]] of degree $k$ are the basic examples.

## Explanations

### L1 human

Doubling every input of a homogeneous function of degree $k$ multiplies the output by $2^k$.
The area of a rectangle, $f(a, b) = ab$, is homogeneous of degree $2$: doubling both sides multiplies the area by $4$.
On a line, the solution is fixed by its values at $1$ and $-1$, since $f(x) = x^k f(1)$ for $x > 0$.
In several variables the solution can be any function on the unit sphere, extended along rays.
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# de Branges space

## Definition

Let $\mathbb{H} = \set{z \in \C : \operatorname{Im} z > 0}$ be the [[upper half-plane|X6XGGH5FPW]].
For an [[entire function|0419794R0T]] $F$ let $F^\#(z) = \overline{F(\bar z)}$, which is again entire.
Let $E$ be an entire function such that $\abs{E(\bar z)} < \abs{E(z)}$ for all $z \in \mathbb{H}$.
The *de Branges space* $\mathcal{H}(E)$ is the set of entire functions $F$ such that

1. $F/E$ belongs to the [[Hardy space|PNNNFJVF6W]] $H^2(\mathbb{H})$, and
2. $F^\#/E$ belongs to $H^2(\mathbb{H})$,

with the inner product
$$\ip{F}{G} = \int_{-\infty}^{\infty} \frac{F(x) \overline{G(x)}}{\abs{E(x)}^2} \, dx.$$

## Remarks

$\mathcal{H}(E)$ is a [[Hilbert space|HM2HHE3J51]] and a [[reproducing kernel Hilbert space|4CK2GSRAGD]].
For $w \in \C$ the reproducing kernel is
$$K_w(z) = \frac{\overline{E(w)} E(z) - \overline{E^\#(w)} E^\#(z)}{2\pi i (\bar w - z)},$$
so that $F(w) = \ip{F}{K_w}$ for all $F \in \mathcal{H}(E)$.
The condition on $E$ implies that $E$ has no zeros in $\mathbb{H}$, so condition 1 makes sense.

Example: $E(z) = e^{-iaz}$ with $a > 0$.
For $z = x + iy$ with $y > 0$ one has $\abs{E(z)} = e^{ay} > e^{-ay} = \abs{E(\bar z)}$.
Here $E^\#(z) = e^{iaz}$, and the kernel becomes $K_w(z) = \frac{\sin(a(\bar w - z))}{\pi (\bar w - z)}$.
$\mathcal{H}(E)$ is the Paley–Wiener space of entire functions of exponential type at most $a$ that are square integrable on $\R$.

Non-example: $E(z) = 1$ gives $\abs{E(\bar z)} = \abs{E(z)}$, so the strict inequality fails and $E$ does not define a de Branges space.
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# Lagrange polynomial

## Summary

Take $n + 1$ points in the plane with distinct first coordinates.
Their Lagrange polynomial is the unique [[polynomial function|1G0B73WPX8]] of degree at most $n$ whose graph passes through all of them.
It is written as a sum of basis polynomials, each equal to $1$ at one node and $0$ at the others.
It gives an explicit solution of [[polynomial interpolation|SS1YMS2NPC]] and is the starting point of numerical differentiation and quadrature rules.
The [[definition|TEQNAME853]] gives the formula.

## Explanations

### L1 human

Through two points there is one line, and through three points with different $x$-values there is one parabola or line.
The Lagrange formula writes this curve down directly.
For each point it builds a small polynomial that is $1$ at that point and $0$ at all the other points.
Then it adds these pieces, each multiplied by the height it must reach.
For the points $(0, 1)$, $(1, 3)$, $(2, 7)$ the result is $x^2 + x + 1$.
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# Complex torus

## Summary

A complex torus of dimension $g$ is the quotient of $\C^g$ by a [[lattice|WN9V7WSJKF]] of rank $2g$.
It is a compact connected complex manifold and a commutative complex Lie group.
Every complex [[abelian variety|R7W03TXRS9]] is a complex torus, but for $g \ge 2$ most complex tori are not algebraic.

## Explanations

### L1 human

Take the plane $\C$ and the grid of points $m + n i$ with integers $m$ and $n$.
Glue together any two points that differ by a grid point.
Each square of the grid is folded up into the surface of a doughnut, the quotient $\C / (\Z + \Z i)$.
A complex torus does the same in $\C^g$, with a grid that has $2g$ independent directions.
Addition in $\C^g$ passes to the quotient, so the points of a complex torus form a group.
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# Flat norm

## Summary

The flat norm measures the distance between two [[currents|360H7MD9RD]] by the least total size of a decomposition.
The difference is split into a small piece plus the boundary of a small filling.
Two surfaces are close in the flat norm when they differ by a small area or bound a thin region.
This holds even if their masses differ.
Convergence in the flat norm is the natural convergence of generalised surfaces, and the [[Federer–Fleming compactness theorem|TPXYARH912]] is stated with it.
It also gives the distance used in the comparison and approximation of [[integral currents|EG4AJVEFT2]].

## Explanations

### L1 human

Two nearby points, one counted positively and one negatively, look very different to the mass, which counts each point once and gives a total of $2$.
But they are the two ends of a short segment, so they are almost the same once one allows filling in.
The flat norm captures this: the difference is the boundary of a segment of length $\varepsilon$, so its flat norm is at most $\varepsilon$.
In the same way, a wiggly curve close to a straight one is near it in the flat norm, because the two curves bound a thin region.
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# Bounded symmetric domain

## Summary

A bounded symmetric domain is a bounded domain $D$ in $\C^n$ with many symmetries.
At each point $p$ of $D$ there is a holomorphic involution of $D$ with $p$ as an isolated fixed point.
Examples are the unit ball, the polydisc and the bounded model of the [[Siegel upper half-space|5PTYE97PF6]].
With its Bergman metric, a bounded symmetric domain is a [[Hermitian symmetric space|NT4797Z9H8]] of noncompact type, and every such space arises in this way.
Cartan classified them, and they are the domains on which automorphic forms in several variables live.

## Explanations

### L1 human

The unit disc has a holomorphic "point reflection" $z \mapsto -z$ through its centre, and by moving points around with Möbius maps it has one through every point.
A bounded symmetric domain is a bounded region in several complex variables with the same property.
These reflections are so many that they move any point to any other, so the domain looks the same everywhere.
Products of discs and the unit ball in $\C^n$ are basic examples.
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# Canonical height

## Definition

Let $\overline{\Q}$ be an algebraic closure of $\Q$, let $N \ge 1$ and let $h \colon \mathbb{P}^N(\overline{\Q}) \to \R$ be the absolute logarithmic [[height|F1FF505PWH]] on [[projective space|ZT9A6JK8QF]].
Let $\phi \colon \mathbb{P}^N \to \mathbb{P}^N$ be a morphism of degree $d \ge 2$ defined over $\overline{\Q}$, and write $\phi^n$ for its $n$-fold composition.
The *canonical height* of $\phi$ is the function $\hat{h}_\phi \colon \mathbb{P}^N(\overline{\Q}) \to \R$ given by
$$\hat{h}_\phi(P) = \lim_{n \to \infty} \frac{h(\phi^n(P))}{d^n}.$$

## Remarks

The limit exists because $h(\phi(Q)) = d \, h(Q) + O(1)$ uniformly in $Q$, so the sequence $d^{-n} h(\phi^n(P))$ is Cauchy, by a telescoping argument.
It has these properties, due to Call and Silverman:

1. $\hat{h}_\phi(\phi(P)) = d \, \hat{h}_\phi(P)$;
2. $\hat{h}_\phi - h$ is bounded on $\mathbb{P}^N(\overline{\Q})$;
3. $\hat{h}_\phi(P) \ge 0$, with equality exactly when $P$ is preperiodic for $\phi$.

Property 3 uses Northcott's theorem: a point of canonical height $0$ has an orbit of bounded height inside a fixed [[number field|MWBDFS2KJ5]], and such an orbit is finite.

Example: for $\phi(z) = z^2$ on $\mathbb{P}^1$, $h(\phi(P)) = 2 h(P)$ exactly, so $\hat{h}_\phi = h$.
Its zeros are $0$, $\infty$ and the roots of unity, which are exactly the preperiodic points of $z \mapsto z^2$.

Non-example: for $\psi(z) = z^2 + 1$ the naive height $h$ is not the canonical height of $\psi$.
It fails property 1: $h(1) = 0$, but $h(\psi(1)) = h(2) = \log 2 \ne 2 h(1)$.

For an elliptic curve $E$ over a number field, the [[Lattès map|9PK608W8MS]] that describes doubling on $x$-coordinates has degree $4$.
Its canonical height, up to a normalizing factor, is the Néron–Tate height of $E$.
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# Cheeger constant

## Summary

The Cheeger constant of a finite graph measures its worst bottleneck.
It is the least ratio of the number of edges leaving a set of vertices to the size of the set.
The minimum runs over the sets with at most half of the vertices.
It is zero exactly for disconnected graphs, and it is large when every set of vertices has many edges to its complement.
It is also called the edge expansion or isoperimetric number, and bounded-degree graphs with Cheeger constant bounded below are [[expander graphs|DXJXVFV79H]].
The Cheeger inequalities compare it with the second smallest eigenvalue of the [[Laplacian matrix|TN6M6DVZRP]].

## Explanations

### L1 human

Try to cut a graph into two parts so that few edges cross the cut, while the smaller part is still big.
The Cheeger constant is the best ratio you can get: crossing edges divided by the size of the smaller part.
A long path has a tiny Cheeger constant, since one edge cuts it in half.
A complete graph has a large one, since every cut crosses many edges.
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# Brody hyperbolicity

## Summary

A complex manifold is Brody hyperbolic if every holomorphic map from $\C$ into it is constant.
Nonconstant holomorphic maps $\C \to X$ are called entire curves, so Brody hyperbolicity means that $X$ contains no entire curve.
Every [[Kobayashi hyperbolic manifold|HRPAX8V7HA]] is Brody hyperbolic, and for compact manifolds the converse holds by [[Brody's lemma|VEV5E2077C]].
The notion is named after Robert Brody, who proved the converse in 1978.

## Explanations

### L1 human

Liouville's theorem says that a bounded holomorphic function on $\C$ is constant.
Brody hyperbolicity asks the same thing of a complex manifold: the only holomorphic maps from the whole complex plane into it are the constant maps.
The unit disc is Brody hyperbolic because of Liouville's theorem.
The plane minus two points is Brody hyperbolic by the little Picard theorem.
A complex torus $\C / \Lambda$ is not, because the quotient map $\C \to \C / \Lambda$ is a nonconstant holomorphic map.
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# Hypergeometric differential equation

## Definition

Let $a, b, c \in \C$.
The *hypergeometric differential equation* is the second-order linear [[ordinary differential equation|0NT64JV96A]] for a function $w$ of a complex variable $z$:
$$z(1 - z) \, \frac{d^2 w}{dz^2} + \big[c - (a + b + 1) z\big] \frac{dw}{dz} - ab \, w = 0.$$

## Remarks

Its singular points on the Riemann sphere are $0$, $1$ and $\infty$, and all three are [[regular singular points|H96HXG6CA0]].
The exponents are $\set{0, 1 - c}$ at $0$, $\set{0, c - a - b}$ at $1$ and $\set{a, b}$ at $\infty$.

Example: if $c \notin \set{0, -1, -2, \ldots}$, the [[hypergeometric function|463X5V31D9]] $w = {}_2F_1(a, b; c; z)$ is a solution.
To check it, put $w = \sum_n t_n z^n$; the equation gives $(n+1)(n+c)\, t_{n+1} = (n+a)(n+b)\, t_n$, which is exactly the recursion of the coefficients $t_n = \frac{(a)_n (b)_n}{(c)_n n!}$.
If moreover $c \notin \Z$, a second, independent solution near $0$ is $z^{1-c} \, {}_2F_1(a - c + 1, b - c + 1; 2 - c; z)$.

Non-example: [[Kummer's equation|NT917KFR57]] $z w'' + (c - z) w' - a w = 0$ is not of this form; it has a regular singular point at $0$, but its singular point at $\infty$ is irregular.

In the notation of the Riemann P-symbol, the solutions form
$$P\begin{Bmatrix} 0 & 1 & \infty & \\ 0 & 0 & a & z \\ 1 - c & c - a - b & b & \end{Bmatrix}.$$
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# Exponential map

## Summary

Let $p$ be a point of a [[Riemannian manifold|7CBT4SM83Y]].
The exponential map at $p$ sends $v \in T_pM$ to the point reached at time $1$ by the [[geodesic|NSYMQMHPDN]] from $p$ with velocity $v$.
It maps a small ball around $0$ in the tangent space diffeomorphically onto a neighbourhood of $p$.
So it gives coordinates adapted to the geometry, the [[normal coordinates|RRHKMGNKG9]].
It is central to the study of completeness, conjugate points and the cut locus.
For a bi-invariant metric on a Lie group, it agrees at the identity with the [[exponential map of the Lie group|H3QX1ZFC8K]].

## Explanations

### L1 human

Stand at a point $p$, choose a direction and a distance, and walk straight ahead, along a geodesic, for that distance.
The point where you stop is the exponential map of the vector with that direction and length.
On the unit sphere, starting at the north pole, the vectors of length $\pi/2$ go to the equator and all vectors of length $\pi$ go to the south pole.
This is like wrapping the flat tangent plane around the sphere so that straight lines through the origin become great circles through $p$.
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# Non-uniform rational B-spline

## Summary

A non-uniform rational B-spline, or NURBS, is a curve or surface given as a weighted average of control points.
The weights are [[B-spline|62AFX32JGT]] basis functions times positive numbers, divided by their sum.
The weights make the representation rational, which lets it reproduce conic sections such as circles exactly.
NURBS are the standard representation of curves and surfaces in computer-aided design.

## Explanations

### L1 human

A designer moves a few control points, and the curve follows them smoothly.
In a NURBS curve each control point also has a weight, and a larger weight pulls the curve closer to that point.
With all weights equal, one gets an ordinary B-spline curve.
With suitable weights, three control points at the corners $(1, 0)$, $(1, 1)$, $(0, 1)$ give an exact quarter of the unit circle, which no polynomial curve can do.
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<document source="https://tamnd.github.io/math2/node/Y8H0JWZ2ZQ.md" tag="Y8H0JWZ2ZQ" kind="definition">
# Brauer–Manin obstruction

## Definition

Let $k$ be an [[algebraic number field|MWBDFS2KJ5]] with [[adele ring|SEH68BA1QQ]] $\mathbb{A}_k$, and for each place $v$ let $k_v$ be the [[completion|W168DS966V]] of $k$ at $v$.
Let $X$ be a smooth projective geometrically integral variety over $k$, and let $\operatorname{Br} X = H^2_{\text{ét}}(X, \mathbb{G}_m)$ be its [[Brauer group|BEW48TA9WZ]].
For each place $v$ let $\operatorname{inv}_v \colon \operatorname{Br} k_v \to \Q/\Z$ be the local invariant map of [[class field theory|BEKCEYHCHX]].
For $\alpha \in \operatorname{Br} X$ and $P \in X(k_v)$ let $\alpha(P) \in \operatorname{Br} k_v$ be the pullback of $\alpha$ along $P$.
The *Brauer–Manin set* is
$$X(\mathbb{A}_k)^{\operatorname{Br}} = \set{(P_v)_v \in X(\mathbb{A}_k) : \textstyle\sum_v \operatorname{inv}_v \alpha(P_v) = 0 \text{ for all } \alpha \in \operatorname{Br} X}.$$
There is a *Brauer–Manin obstruction to the Hasse principle* for $X$ if $X(\mathbb{A}_k) \neq \emptyset$ and $X(\mathbb{A}_k)^{\operatorname{Br}} = \emptyset$.

## Remarks

Since $X$ is projective, $X(\mathbb{A}_k) = \prod_v X(k_v)$.
For fixed $\alpha$, the term $\operatorname{inv}_v \alpha(P_v)$ is $0$ for all but finitely many $v$, so the sum is finite.
The reciprocity law says that $\sum_v \operatorname{inv}_v \beta = 0$ for every $\beta \in \operatorname{Br} k$.
Applied to $\beta = \alpha(P)$ for a [[rational point|SWJYG87QQH]] $P \in X(k)$, it gives the inclusion
$$X(k) \subseteq X(\mathbb{A}_k)^{\operatorname{Br}} \subseteq X(\mathbb{A}_k).$$
So a Brauer–Manin obstruction implies $X(k) = \emptyset$ although $X$ has points over every $k_v$.

Example: suppose that $\operatorname{Br} X$ is the image of $\operatorname{Br} k$, as when $X$ is a projective space.
Then the sum vanishes for every adelic point by reciprocity, so $X(\mathbb{A}_k)^{\operatorname{Br}} = X(\mathbb{A}_k)$ and there is no obstruction.
Non-example of a sufficient test: Skorobogatov constructed a surface with $X(\mathbb{A}_k)^{\operatorname{Br}} \ne \emptyset$ and $X(k) = \emptyset$, so the Brauer–Manin obstruction does not explain all failures of the Hasse principle.
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# Prime knot

## Definition

Let $K$ be a tame [[knot|QYEAZTYGPT]] in $S^3$.
The knot $K$ is *prime* if:

1. $K$ is not the [[unknot|1TQ06R4Q66]];
2. whenever $K$ is isotopic to a [[knot sum|AGZSZW7GYE]] $K_1 \# K_2$, one of $K_1$ and $K_2$ is the unknot.

## Remarks

Equivalently, $K$ is nontrivial and every $2$-sphere in $S^3$ that meets $K$ transversely in two points bounds a ball meeting $K$ in an unknotted arc.
Example: the [[trefoil knot|VYTYSVHN4A]] has Seifert genus $1$.
Genus is additive under knot sum and only the unknot has genus $0$, so a decomposition $K_1 \# K_2$ forces one summand to be trivial.
Non-example: the granny knot, the sum of two trefoils, fails clause 2.
The unknot fails clause 1.
By Schubert's theorem, every nontrivial knot is a knot sum of finitely many prime knots, unique up to order and isotopy.
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<document source="https://tamnd.github.io/math2/node/Y8RZ73N0JA.md" tag="Y8RZ73N0JA" kind="definition">
# Infinite compositions of functions

## Definition

Let $D$ be a subset of a [[metric space|0VKD1TP7F3]], for example of $\C$, and let $f_1, f_2, \dots$ be functions from $D$ to $D$.
The *forward* (or *inner*) compositions are $F_n = f_1 \circ f_2 \circ \cdots \circ f_n$.
The *backward* (or *outer*) compositions are $G_n = f_n \circ f_{n-1} \circ \cdots \circ f_1$.
The infinite forward composition converges at a point $z \in D$ if the sequence $(F_n(z))$ has a [[limit|0XKWSSY8P2]], and similarly for the backward composition.
When the limit exists for all $z$ in $D$, it defines the function $F(z) = \lim_{n \to \infty} F_n(z)$ (or $G(z) = \lim_{n \to \infty} G_n(z)$).

## Remarks

Example: on $D = \R$ let $f_n(x) = \frac{x}{2} + 1$ for all $n$.
Then $G_n(0) = 2 - 2^{1-n}$, so the backward composition converges at $0$ to the fixed point $2$ of $f_1$.
Non-example: with $f_n(x) = -x$ for all $n$, the sequence $F_n(1) = (-1)^n$ has no limit, so the infinite composition does not converge at $1$.
For a [[continued fraction|0BK6JBK58W]] with $f_n(w) = \frac{a_n}{b_n + w}$, the values $F_n(0)$ are the convergents; backward compositions do not have this interpretation.
When all $f_n$ equal a single map $f$, both $F_n$ and $G_n$ are the iterate $f^{\circ n}$.

## Explanations

### L1 human

Forward compositions add each new function on the inside, next to the starting value.
Backward compositions add it on the outside, so they describe a point moved step by step by a changing rule.
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# Second-order arithmetic

## Definition

The language of second-order arithmetic has number variables $n, m, x, y, \ldots$ and set variables $X, Y, \ldots$.
It has the constants $0$ and $1$, the operations $+$ and $\cdot$, and the relation $<$ on numbers.
It also has the membership relation $n \in X$ between a number and a set.
*Second-order arithmetic* $\mathsf{Z}_2$ is the [[theory|92SWXST1ZM]] in this two-sorted first-order language with these axioms:

1. the basic axioms: the axioms of a discretely ordered commutative semiring with least element $0$ and successor $n + 1$, in the style of [[Robinson arithmetic|3K1AKGXF8A]];
2. the induction axiom: $\forall X\, \big( (0 \in X \wedge \forall n\, (n \in X \to n + 1 \in X)) \to \forall n\, (n \in X) \big)$;
3. the comprehension scheme: $\exists X\, \forall n\, (n \in X \leftrightarrow \varphi(n))$ for every [[formula|HYJ10AF6RT]] $\varphi(n)$ in which $X$ does not occur free.

## Remarks

Example: the full structure, with number domain $\N$ and set domain the power set of $\N$, is a model of $\mathsf{Z}_2$.
Each clause holds because every subset of $\N$ is available, and ordinary induction holds in $\N$.

Non-example: the structure with number domain $\N$ and set domain the computable subsets of $\N$ satisfies clauses 1 and 2 but not clause 3.
The formula $\varphi(e)$ that says "program $e$ halts on input $e$" uses only number quantifiers, and the set it defines is the halting problem, which is not computable.

Subsystems are obtained by restricting comprehension and induction to classes of formulas, for example arithmetical comprehension in $\mathsf{ACA}_0$.
Despite the name, $\mathsf{Z}_2$ is a first-order theory: its models may have a set domain smaller than the full power set.
Together with clause 2, the comprehension scheme proves induction for every formula, so the first-order [[Peano axioms|0QFT7Z76N5]] are provable in $\mathsf{Z}_2$.
</document>

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# Hodge structure

## Definition

Let $n$ be an integer.
A *pure Hodge structure of weight $n$* consists of:

1. a finitely generated [[abelian group|082GHNPJ53]] $H_\Z$, with complex [[vector space|000AYPA9AJ]] $H_\C = H_\Z \otimes_\Z \C$;
2. a [[direct sum|0XYYT5KNTX]] decomposition $H_\C = \bigoplus_{p + q = n} H^{p,q}$ into complex subspaces;
3. the symmetry $\overline{H^{p,q}} = H^{q,p}$ for all $p, q$, where the bar is complex conjugation on $H_\C$ with respect to the real form $H_\Z \otimes \R$.

A *rational Hodge structure* is defined in the same way with a finite-dimensional $\Q$-vector space $H_\Q$ in place of $H_\Z$.

## Remarks

Equivalently, the data can be given by the Hodge [[filtration|HNK1Y62G7V]] $F^p = \bigoplus_{p' \ge p} H^{p', n - p'}$, a decreasing filtration of $H_\C$ with $H_\C = F^p \oplus \overline{F^{n-p+1}}$ for all $p$; then $H^{p,q} = F^p \cap \overline{F^q}$.

Example: let $X$ be a compact Kähler manifold, for instance a smooth projective complex variety.
Then $H^k(X, \Z)$ modulo torsion, with the Hodge decomposition $H^k(X, \C) = \bigoplus H^{p,q}(X)$, is a Hodge structure of weight $k$: clauses 2 and 3 are parts 1 and 2 of the Hodge decomposition theorem.
The Tate structure $\Z(1)$ is $H_\Z = 2\pi i \Z \subseteq \C$ with $H_\C = H^{-1,-1}$, a Hodge structure of weight $-2$.

Non-example: let $H_\Z = \Z^2$ and $n = 1$, with $H^{1,0} = H_\C$ and $H^{0,1} = 0$.
Clause 2 holds, but clause 3 fails, since $\overline{H^{1,0}} = H_\C \ne 0 = H^{0,1}$.
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# Hazard function

## Summary

The hazard function of a lifetime $T$ gives the instantaneous rate of the event at time $t$.
The rate is taken among the subjects that have not had the event yet.
It equals the density of $T$ divided by its [[survival function|QTM1HZM53Q]].
Its integral, the cumulative hazard, determines the survival function through $S(t) = e^{-H(t)}$.
Hazards are the natural scale of many models in [[survival analysis|443ZWDX88P]], such as the [[proportional hazards model|N029V3JX60]].
In reliability theory the hazard is called the failure rate.

## Explanations

### L1 human

The hazard at age $70$ answers: among people alive at $70$, how fast are they dying right now, per year?
It is a rate, not a probability, so it can be larger than $1$.
A machine part that wears out has a hazard that grows with age.
A part that fails only by pure accident, like a fuse hit by random surges, has a constant hazard; its lifetime is exponential.
Newborn mortality followed by a long flat period and then ageing gives the well-known bathtub-shaped hazard.
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# Continuous and residual spectrum

## Definition

Let $X$ be a complex Banach space with identity map $I$, and let $T \colon X \to X$ be a bounded operator with [[spectrum|BMKATJ802A]] $\sigma(T)$ and [[point spectrum|7FNVA0Q6YQ]] $\sigma_p(T)$.
For $\lambda \in \sigma(T) \setminus \sigma_p(T)$, the operator $T - \lambda I$ is injective.

1. The *continuous spectrum* $\sigma_c(T)$ is the set of $\lambda \in \sigma(T) \setminus \sigma_p(T)$ such that the [[image|0WSFKKT6FF]] of $T - \lambda I$ is [[dense|0BSJSX7ME2]] in $X$.
2. The *residual spectrum* $\sigma_r(T)$ is the set of $\lambda \in \sigma(T) \setminus \sigma_p(T)$ such that the image of $T - \lambda I$ is not dense in $X$.

## Remarks

By construction, $\sigma(T)$ is the disjoint union $\sigma_p(T) \cup \sigma_c(T) \cup \sigma_r(T)$.
For $\lambda \in \sigma_c(T)$, the image of $T - \lambda I$ is dense but not all of $X$, since a bounded bijection between Banach spaces has a bounded inverse.

Example: for $D(x_n) = (x_n / n)$ on $\ell^2$, $0 \in \sigma_c(D)$.
Indeed $D$ is injective, it is not surjective, and its image contains every finitely supported sequence, a dense set.

Non-example: for the right shift $S(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$ on $\ell^2$, $0$ is not in $\sigma_c(S)$.
The image of $S$ is the closed subspace $\set{x : x_1 = 0}$, which is not dense, so $0 \in \sigma_r(S)$.

For the right shift, $\sigma_p(S) = \emptyset$, $\sigma_r(S)$ is the open unit disc and $\sigma_c(S)$ is the unit circle.
A self-adjoint operator on a Hilbert space has empty residual spectrum.
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# Measure with density

## Summary

Given a [[measure space|P15VQS5DJ9]] $(X, \Sigma, \mu)$ and a measurable function $f \ge 0$, the measure with density $f$ gives each set $A$ the mass $\int_A f \, d\mu$.
It is written $f \mu$ or $f \, d\mu$.
It is the basic way to build new measures from old ones.
The Radon–Nikodym theorem says that, for σ-finite measures, every measure absolutely continuous with respect to $\mu$ is of this form.

## Explanations

### L1 human

Start with length on the real line and a weight function $f$.
The new measure of an interval is the area under the graph of $f$ over that interval.
Where $f$ is large, the new measure packs more mass.
With $f(x) = e^{-x^2/2} / \sqrt{2\pi}$ and Lebesgue measure, one gets the standard normal distribution.
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# Germ

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]] and let $x \in X$.
Consider pairs $(U, f)$, where $U$ is an open [[neighbourhood|0ZCY0Z80B6]] of $x$ and $f \colon U \to Y$ is a map.
Two such pairs $(U, f)$ and $(V, g)$ are *equivalent at $x$* if there is an open neighbourhood $W \subseteq U \cap V$ of $x$ with $f(w) = g(w)$ for all $w \in W$.
This is an [[equivalence relation|0G8VEZ3WTK]].
The *germ* of $(U, f)$ at $x$ is its [[equivalence class|0W95W33CMP]].

## Remarks

One often restricts to continuous, smooth or holomorphic maps, and writes $[f]_x$ for the germ.
The value $f(x)$ depends only on the germ, and so do derivatives of all orders at $x$ when $f$ is smooth.
More generally, let $\mathcal{F}$ be a filter on a set $S$.
Two maps on $S$ have the same germ along $\mathcal{F}$ if they agree on some member of $\mathcal{F}$.
The case above uses the neighbourhood filter of $x$.

Example: on $\R$, the maps $f(t) = t$ and $g(t) = \abs{t}$ have the same germ at $1$, since they agree on $W = (0, 2)$.

Non-example: on $\R$, let $f = 0$ and let $g(t) = e^{-1/t^2}$ for $t \ne 0$, with $g(0) = 0$.
Both are smooth, and all their derivatives agree at $0$.
Yet $g(t) \ne 0$ for every $t \ne 0$, so they agree on no neighbourhood of $0$, and their germs at $0$ differ.
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<document source="https://tamnd.github.io/math2/node/Y99MCTGZHZ.md" tag="Y99MCTGZHZ" kind="definition">
# Stochastic differential equation

## Definition

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, P)$ be a [[probability space|0NFPJVJJHX]] with a [[filtration|HNK1Y62G7V]] that satisfies the usual conditions.
Let $W = (W_t)_{t \ge 0}$ be an $m$-dimensional [[Wiener process|22735FGJV6]] with respect to this filtration.
Let $T > 0$, and let $b \colon [0, T] \times \R^n \to \R^n$ and $\sigma \colon [0, T] \times \R^n \to \R^{n \times m}$ be Borel measurable.
Let $\xi$ be an $\R^n$-valued $\mathcal{F}_0$-measurable random variable.
The *stochastic differential equation* with drift $b$, diffusion coefficient $\sigma$ and initial value $\xi$ is
$$dX_t = b(t, X_t) \, dt + \sigma(t, X_t) \, dW_t, \qquad X_0 = \xi.$$
A *strong solution* is a continuous [[adapted process|YRNFE016FF]] $X$ with values in $\R^n$ with two properties.
Almost surely, $\int_0^T \left( \abs{b(s, X_s)} + \abs{\sigma(s, X_s)}^2 \right) ds < \infty$.
Almost surely,
$$X_t = \xi + \int_0^t b(s, X_s) \, ds + \int_0^t \sigma(s, X_s) \, dW_s \quad \text{for all } t \in [0, T],$$
where the last integral is the Itô integral of [[Itô calculus|JA95NH206E]].

## Remarks

Assume that $b$ and $\sigma$ are [[Lipschitz|0AW1E6YTJY]] in $x$ uniformly in $t$ and grow at most linearly in $x$, and that $\xi$ has a finite second moment.
Then a strong solution exists and is unique up to indistinguishability.
Example: for $n = m = 1$, $b(t, x) = \mu x$ and $\sigma(t, x) = s x$ with constants $\mu, s$, the process $X_t = X_0 \exp\left( (\mu - s^2/2) t + s W_t \right)$ is a solution; [[Itô's lemma|2J1RVQB0CM]] gives $dX_t = \mu X_t \, dt + s X_t \, dW_t$.
This is [[geometric Brownian motion|D48DR46B97]].
Non-example: $Y_t = e^{W_t}$ does not solve $dY_t = Y_t \, dW_t$, $Y_0 = 1$, because Itô's lemma gives $dY_t = Y_t \, dW_t + \frac{1}{2} Y_t \, dt$, with an extra drift term.
A *weak solution* allows the probability space and the Wiener process to be constructed together with $X$.
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# Abelian category

## Summary

An abelian category is an [[additive category|B2VP96NJSK]] in which every morphism has a [[kernel|S417YNEJ6M]] and a [[cokernel|H8W38933NC]].
In it, monomorphisms and epimorphisms behave as in abelian groups.
In it, exact sequences, homology and the standard diagram lemmas make sense.
Modules over a ring and sheaves of abelian groups on a space are abelian categories.
Abelian categories were introduced by Buchsbaum and by Grothendieck to give a common framework for homological algebra.

## Explanations

### L1 human

Abelian groups have a rich toolkit: subgroups, quotients, images, and exact sequences that track how they fit together.
An abelian category is any category where the same toolkit works, described only through maps.
In such a category every injective-like map is the kernel of something, and every surjective-like map is the cokernel of something.
So a theorem of homological algebra proved once for abelian categories applies at once to modules, sheaves and representations.
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# Girard's theorem

## Statement

Let $R > 0$, and let $T$ be a spherical triangle on a sphere of radius $R$, with angles $\alpha$, $\beta$, $\gamma$.
Then the area of $T$ is
$$\operatorname{Area}(T) = R^2 (\alpha + \beta + \gamma - \pi).$$

## Remarks

The quantity $E = \alpha + \beta + \gamma - \pi$ is the *spherical excess*; the theorem shows that it is positive.
Proof idea: a lune, the region between two half great circles meeting at angle $\theta$, has area $2\theta R^2$, since it is the fraction $\theta / (2\pi)$ of the sphere of area $4\pi R^2$.
The three great circles through the sides of $T$ cut the sphere into eight triangles.
Let $T'$ be the antipodal triangle; the six lunes with angles $\alpha$, $\beta$, $\gamma$ at the vertices of $T$ and of $T'$ cover the sphere.
They cover $T$ and $T'$ three times each and every other point once, so $2(2\alpha + 2\beta + 2\gamma) R^2 = 4\pi R^2 + 4 \operatorname{Area}(T)$, which gives the formula.
Example: the octant triangle with vertices $(R, 0, 0)$, $(0, R, 0)$, $(0, 0, R)$ has three right angles, so its area is $R^2 (3\pi/2 - \pi) = \pi R^2 / 2$, one eighth of $4\pi R^2$.

## Explanations

### L1 human

On a sphere, the angles of a triangle add up to more than $180^\circ$.
Girard's theorem says that the extra amount, the excess, is exactly the area of the triangle divided by $R^2$.
So on a globe one can find the area of a triangle just by measuring its three angles.

### L2 human

The theorem is the spherical case of the [[Gauss–Bonnet theorem|ESGEHKPD60]]: the integral of the Gaussian curvature $1/R^2$ over the triangle equals the angle excess.
The hyperbolic counterpart says that a [[hyperbolic triangle|GCZ8EEDCHS]] in curvature $-1$ has area $\pi - (\alpha + \beta + \gamma)$, the defect.
Since the excess is positive and tends to $0$ for small triangles, it explains why the angle sum of a small spherical triangle is close to $\pi$.
It also shows that two spherical triangles with the same angles have the same area.
In fact such triangles are congruent, by the law of cosines for angles.
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# Stieltjes moment problem

## Problem

Let $(m_n)_{n \ge 0}$ be a sequence of real numbers.
Decide whether there is a positive Borel [[measure|PR11BZQQPG]] $\mu$ on $[0, \infty)$ such that every $x^n$ is $\mu$-integrable and
$$m_n = \int_{[0, \infty)} x^n \, d\mu(x) \quad \text{for all } n \ge 0.$$
This is the [[moment problem|CS7TWHY7JM]] on the half-line $K = [0, \infty)$, and the $m_n$ are the [[moments|01XAHPADBA]] of $\mu$.

## Remarks

The existence question is solved.
Such a measure exists if and only if, for every $n \ge 0$, both [[Hankel matrices|PQ1D8FQB3E]] $(m_{j+k})_{j,k=0}^{n}$ and $(m_{j+k+1})_{j,k=0}^{n}$ are [[positive semidefinite|01MMVW8CKC]].
A measure with infinite support exists if and only if all the determinants of these matrices are strictly positive.
Necessity is direct: for real $c_0, \ldots, c_n$, the quadratic forms equal $\int p(x)^2 \, d\mu$ and $\int x \, p(x)^2 \, d\mu$ with $p(x) = \sum_j c_j x^j$, and $x \ge 0$ on the support.
The solution need not be unique; the log-normal distribution gives a sequence with many solutions.
A Stieltjes moment sequence can be determinate as a Stieltjes problem but indeterminate as a Hamburger problem.

## Explanations

### L1 human

The problem asks which lists of numbers are the moments of a mass distribution placed on the nonnegative half-line.
The answer is a positivity test on two square tables built from the list.
The second table is the first one with the list shifted by one place, and its positivity reflects that the mass sits where $x \ge 0$.
Stieltjes met this problem in his work on continued fractions.
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# Affine variety

## Definition

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $n \ge 0$.
Give $k^n$ the Zariski topology, whose closed sets are the [[algebraic sets|VHFMQQYSTN]].
An *affine variety* in $k^n$ is an algebraic set $X \subseteq k^n$ that is [[irreducible|SY2MWC434Z]].
This means that $X$ is nonempty and that $X$ is not the union of two algebraic sets strictly smaller than $X$.

## Remarks

An algebraic set $X$ is irreducible if and only if its [[ideal|HH6VEVM6Z0]] $I(X)$ is a [[prime ideal|00MHHZ9SNE]].
For instance, if $X$ is irreducible and $fg \in I(X)$, then $X = (X \cap V(f)) \cup (X \cap V(g))$, and irreducibility forces $f \in I(X)$ or $g \in I(X)$.
With [[Hilbert's Nullstellensatz|D85VZWS9QK]], affine varieties in $k^n$ correspond to prime ideals of $k[x_1, \dots, x_n]$.

Example: the parabola $X = V(y - x^2) \subseteq k^2$ is an affine variety.
Its ideal is $(y - x^2)$, and $k[x, y]/(y - x^2) \cong k[x]$ is an integral domain, so the ideal is prime.

Non-example: $V(xy) \subseteq k^2$ is an algebraic set but not an affine variety.
It is the union of the two axes, each strictly smaller than $V(xy)$.

Many texts, especially those based on schemes, call any algebraic set an affine variety, or use the term for a variety isomorphic to a closed subvariety of some $k^n$.
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# Permutation test

## Definition

Let $Z = (Z_1, \dots, Z_N)$ be a random vector, and let $S_N$ be the [[symmetric group|0RJMNEN3BJ]], acting on $\R^N$ by $(\pi z)_i = z_{\pi^{-1}(i)}$.
Let $H_0$ be a null hypothesis under which $Z$ is exchangeable, that is, $\pi Z$ has the same law as $Z$ for every $\pi \in S_N$.
Let $T \colon \R^N \to \R$ be a [[statistic|036756NFD6]].
The *permutation p-value* is
$$p(Z) = \frac{1}{N!} \, \#\set{\pi \in S_N : T(\pi Z) \ge T(Z)}.$$
The *permutation test* at level $\alpha$ rejects $H_0$ when $p(Z) \le \alpha$.

## Remarks

Under $H_0$, $P(p(Z) \le \alpha) \le \alpha$ for every $\alpha \in [0, 1]$, so the test has level $\alpha$, whatever the law of $Z$.
In the two-sample problem, $Z$ is the pooled sample $X_1, \dots, X_m, Y_1, \dots, Y_n$ with $N = m + n$, and $H_0$ says that all $N$ values are i.i.d.
Since $T$ usually depends only on which values fall in the first group, one may count the $\binom{N}{m}$ splits instead of the $N!$ permutations.
When $N!$ is too large, one uses a random sample of permutations, which gives a Monte Carlo approximation of $p(Z)$.

Example: $X = (3, 5)$, $Y = (1, 4)$ and $T = \bar X - \bar Y$, so $T = 4 - 2.5 = 1.5$.
The $6$ splits of $\set{1, 3, 4, 5}$ give the $X$-pairs $\set{1,3}, \set{1,4}, \set{1,5}, \set{3,4}, \set{3,5}, \set{4,5}$ and the values $-2.5, -1.5, -0.5, 0.5, 1.5, 2.5$.
Two of them are at least $1.5$, so $p = 2/6 = 1/3$.

Non-example: if under $H_0$ the two groups have the same mean but different variances, the pooled vector is not exchangeable.
The clause on exchangeability fails, and the test of equal means need not have level $\alpha$.

The [[Mann–Whitney U test|CKF9KDXBP7]] is a permutation test whose statistic depends only on the ranks.
</document>

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# Rigid analytic space

## Summary

A rigid analytic space is the analogue of a complex analytic space over a complete non-archimedean field such as the [[$p$-adic numbers|BT6T702C28]].
It is glued from affinoid spaces, the maximal spectra of quotients of algebras of convergent power series, along a Grothendieck topology of admissible opens.
John Tate introduced the theory to describe elliptic curves with split multiplicative reduction as quotients of the multiplicative group.
It underlies $p$-adic uniformization and $p$-adic Hodge theory.

## Explanations

### L1 human

Over the $p$-adic numbers, the usual topology is totally disconnected: every disc splits into smaller disjoint open discs.
With this topology, functions defined piece by piece behave badly, and there is no good notion of analytic continuation.
Rigid analytic geometry fixes this by allowing only certain coverings, the admissible ones.
The closed unit disc, for instance, is not covered admissibly by the open unit disc and the circle $\abs{x} = 1$.
With this restriction, analytic functions on $p$-adic spaces behave much like holomorphic functions on complex spaces.
</document>

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# Minkowski–Hlawka theorem

## Statement

Let $n \ge 2$ and let $\zeta$ be the [[Riemann zeta function|1XWBY4N0VT]].
Then there is a [[lattice|WN9V7WSJKF]] [[sphere packing|HF32CBN5MA]] of equal balls in $\R^n$ whose [[density|4P5PSWADJD]] is at least
$$\frac{\zeta(n)}{2^{n-1}}.$$

## Remarks

Minkowski stated the result, and Edmund Hlawka gave the first proof in 1943.
A standard proof averages over lattices.
Siegel's mean value theorem says that for a random lattice $\Lambda$ of covolume $1$ and an integrable $f$, the mean of $\sum_{v \in \Lambda \setminus \set{0}} f(v)$ is $\int_{\R^n} f$.
Applied to the indicator of a ball of volume $V < 1$, this gives a lattice of covolume $1$ with no nonzero point in the ball.
Balls of half the radius centred at its points then form a packing of density $V/2^n$.
Counting only primitive vectors and using the symmetry $v \mapsto -v$ improves the condition to $V < 2\zeta(n)$, which gives the bound $\zeta(n)/2^{n-1}$.
For $n = 2$ the bound is $\zeta(2)/2 = \pi^2/12 \approx 0.82$, below the true optimum $\pi/\sqrt{12} \approx 0.91$.
The argument is not constructive: it shows that good lattices exist without exhibiting one.

## Explanations

### L1 human

In high dimensions it is hard to pack balls well, and good explicit packings are rare.
The theorem guarantees that some lattice arrangement fills at least about $2^{-n+1}$ of space.
It does so by showing that a randomly chosen lattice does this well on average, so a good one must exist.

### L2 human

The lower bound $\zeta(n) 2^{1-n}$ is roughly $2^{-n}$, and upper bounds such as [[Rogers' bound|QHPXB39SN7]] and the Kabatiansky–Levenshtein bound are also exponentially small, but with a different exponent.
Closing this exponential gap is a central open problem.
Later work improved the lower bound by factors linear in $n$, for example Ball's bound $2(n - 1) \zeta(n) 2^{-n}$, and Venkatesh obtained a further $\log \log n$ factor in infinitely many dimensions.
The same averaging idea underlies existence proofs for good error-correcting codes.
</document>

<document source="https://tamnd.github.io/math2/node/YARENZRXTD.md" tag="YARENZRXTD" kind="concept">
# Branched covering

## Summary

A branched covering is a map that is a [[covering map|3XZRRE165N]] except over a discrete set of branch values, where several sheets come together.
For [[Riemann surfaces|5K0N8QMV6Q]], every non-constant proper [[holomorphic map|T19P7J53A1]] is a branched covering of finite degree.
Branched coverings of the [[Riemann sphere|0CDMXGPSES]] describe compact Riemann surfaces as the surfaces of multivalued algebraic functions, and the [[Riemann–Hurwitz formula|42A5ZT6AWN]] computes their genus.

## Explanations

### L1 human

The map $z \mapsto z^2$ on the complex plane is two-to-one away from $0$: each nonzero $w$ has two square roots.
Over $w = 0$ the two sheets meet in the single point $z = 0$, a [[branch point|XP5JC1VXKZ]].
Away from such special points the map is an honest covering, like a stack of identical sheets.
A branched covering is a map that behaves like this: a covering, except at isolated points where sheets are glued together like $z \mapsto z^k$.
</document>

<document source="https://tamnd.github.io/math2/node/YATEFAKZYB.md" tag="YATEFAKZYB" kind="definition">
# Dirichlet eigenvalue

## Definition

Let $\Omega \subseteq \R^n$ be a nonempty bounded open set and let $H^1_0(\Omega)$ be the [[Sobolev space|M4CEZ5K9HP]] of functions with zero boundary values.
Let $\Delta$ be the [[Laplacian|0E0B6RA8K0]].
A real number $\lambda$ is a *Dirichlet eigenvalue* of $\Omega$ if some $u \in H^1_0(\Omega)$ with $u \ne 0$ is a [[weak solution|SJZX81A783]] of
$$-\Delta u = \lambda u \quad \text{in } \Omega.$$
That is, $\int_\Omega \nabla u \cdot \nabla v \, dx = \lambda \int_\Omega u v \, dx$ for every $v \in H^1_0(\Omega)$.
Such a $u$ is a *Dirichlet eigenfunction* for $\lambda$.

## Remarks

The Dirichlet eigenvalues form a sequence $0 < \lambda_1(\Omega) < \lambda_2(\Omega) \le \lambda_3(\Omega) \le \dots \to \infty$, listed with multiplicity, when $\Omega$ is connected.
They are the [[eigenvalues|0BQAA2BK3Q]] of the Dirichlet Laplacian, a self-adjoint operator with compact inverse.
The first one is the minimum of the Rayleigh quotient: $\lambda_1(\Omega) = \min \set{ \int_\Omega \abs{\nabla u}^2 \, dx / \int_\Omega u^2 \, dx : u \in H^1_0(\Omega), u \ne 0 }$.

Example: for $\Omega = (0, \pi)$, the function $u(x) = \sin(kx)$ with $k \ge 1$ an integer vanishes at $0$ and $\pi$ and satisfies $-u'' = k^2 u$.
So $k^2$ is a Dirichlet eigenvalue, and these are all of them.

Non-example: on $(0, \pi)$, the number $0$ is not a Dirichlet eigenvalue.
The solutions of $-u'' = 0$ are $u = a + bx$, and the boundary conditions force $a = b = 0$.
With the Neumann condition $u' = 0$ instead, the constant function shows that $0$ is a Neumann eigenvalue.
</document>

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# Lattice of subvarieties

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.
The *lattice of subvarieties* of $\mathcal{V}$ is the set $L(\mathcal{V})$ of all [[subvarieties|4YSCK1DF26]] of $\mathcal{V}$, ordered by inclusion.
In it, the meet of a family $(\mathcal{W}_i)_{i \in I}$ is the intersection $\bigcap_i \mathcal{W}_i$.
The join is the smallest subvariety that contains $\bigcup_i \mathcal{W}_i$, which is $HSP\left(\bigcup_i \mathcal{W}_i\right)$.

## Remarks

An intersection of varieties is a variety, since it is defined by the union of their identities.
So $L(\mathcal{V})$ is a [[complete lattice|D7A0ZGZT5S]], with least element the trivial variety and greatest element $\mathcal{V}$.
The maps $\mathcal{W} \mapsto \operatorname{Id}(\mathcal{W})$ and $T \mapsto \operatorname{Mod}(T)$ are inverse dual isomorphisms between $L(\mathcal{V})$ and the [[lattice of equational theories|2R8QQTP1CE]] that contain $\operatorname{Id}(\mathcal{V})$.
Example: for the variety of Boolean algebras, $L(\mathcal{V})$ has exactly two elements, the trivial variety and $\mathcal{V}$ itself.
Example: for abelian groups, the subvarieties are $\mathcal{A}_n = \operatorname{Mod}(nx \approx 0)$ for $n \ge 1$ and the whole variety, and $\mathcal{A}_n \subseteq \mathcal{A}_m$ if and only if $n \mid m$.
Non-example: the join is not the union in general, since the union of $\mathcal{A}_2$ and $\mathcal{A}_3$ does not contain $\Z / 2\Z \times \Z / 3\Z \in \mathcal{A}_6$.
</document>

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# Euler's continued fraction formula

## Statement

Let $n \ge 1$ and let $a_0, a_1, \dots, a_n$ be complex numbers such that no division by zero occurs on the right-hand side below.
Then
$$a_0 + a_0 a_1 + a_0 a_1 a_2 + \cdots + a_0 a_1 \cdots a_n = \cfrac{a_0}{1 - \cfrac{a_1}{1 + a_1 - \cfrac{a_2}{1 + a_2 - \cfrac{\cdots}{\cdots + a_{n-1} - \cfrac{a_n}{1 + a_n}}}}}.$$
Consequently, let $(a_k)$ be an infinite sequence, and consider the infinite [[continued fraction|0BK6JBK58W]] of the same shape.
Its $n$-th [[convergent|0SRWNR42BQ]] equals the $n$-th [[partial sum|0NB643K60Q]] of the [[series|0JH4MN4K1C]] $\sum_{k \ge 0} a_0 a_1 \cdots a_k$.
So the continued fraction converges exactly when the series does, and to the same value.

## Remarks

Proof idea: by induction on $n$.
Replacing $a_n$ by $a_n(1 + a_{n+1})$ turns the left side for $n$ into the left side for $n + 1$.
On the right side the same replacement turns the last fraction $\frac{a_n}{1 + a_n}$ into $\frac{a_n}{1 + a_n - \frac{a_{n+1}}{1 + a_{n+1}}}$, because both equal $\frac{a_n(1 + a_{n+1})}{1 + a_n + a_n a_{n+1}}$.
Check for $n = 1$: $\cfrac{a_0}{1 - \frac{a_1}{1 + a_1}} = a_0(1 + a_1)$.
Example: with $a_0 = 1$ and $a_k = x$ for $k \ge 1$, the series is the [[geometric series|01X5CVV1Q5]], and for $\abs{x} < 1$
$$\frac{1}{1 - x} = \cfrac{1}{1 - \cfrac{x}{1 + x - \cfrac{x}{1 + x - \cdots}}}.$$

## Explanations

### L1 human

Any series whose terms are running products $a_0$, $a_0 a_1$, $a_0 a_1 a_2$, and so on, can be rewritten as a continued fraction with the same partial values.
So results about series transfer directly to continued fractions of this shape, and back.

### L2 human

Power series have this form with $a_k = c_k x / c_{k-1}$ when the coefficients $c_k$ are nonzero.
In this way the formula turns the series of $e^x$, $\log(1 + x)$ or $\arctan x$ into continued fractions.
These continued fractions converge at the same rate as the series; other transformations are needed to obtain faster convergence.
</document>

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# Fractional ideal

## Summary

A fractional ideal of an integral domain $R$ is a subset of its field of fractions.
After clearing denominators, it becomes an [[ideal|096Z9MP2ME]] of $R$.
Fractional ideals can be multiplied, and in a [[Dedekind domain|7NPY2CN5KB]] every nonzero fractional ideal has an inverse.
So the nonzero fractional ideals of a Dedekind domain form a group, and its quotient by the principal ones is the [[ideal class group|WQ46BD3ARY]].

## Explanations

### L1 human

Ideals of the integers are the sets $n\Z$ of multiples of a number $n$.
They can be multiplied, $(2\Z)(3\Z) = 6\Z$, but not divided: there is no ideal $I$ of $\Z$ with $(2\Z) I = \Z$.
Fractional ideals allow fractions: $\tfrac12 \Z = \set{\ldots, -\tfrac12, 0, \tfrac12, 1, \ldots}$ is one, and $(2\Z)(\tfrac12\Z) = \Z$.
So fractional ideals are to ideals what fractions are to integers.
</document>

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# Ree groups

## Description

The Ree groups are two infinite families of finite [[groups of Lie type|NPEBABQ49M]], found by Rimhak Ree.

1. For $q = 3^{2n+1}$ with $n \ge 0$, the group ${}^2G_2(q)$ is the subgroup of $G_2(\F_q)$ fixed by a twisted automorphism of characteristic $3$.
2. For $q = 2^{2n+1}$ with $n \ge 0$, the group ${}^2F_4(q)$ is the subgroup of $F_4(\F_q)$ fixed by a twisted automorphism of characteristic $2$.

Each twisted automorphism combines the exceptional graph automorphism of the Dynkin diagram with a field automorphism.
Their orders are
$$\abs{{}^2G_2(q)} = q^3 (q^3 + 1)(q - 1), \qquad \abs{{}^2F_4(q)} = q^{12} (q^6 + 1)(q^4 - 1)(q^3 + 1)(q - 1).$$
For $n \ge 1$ these groups are non-abelian [[simple groups|0YMYMNQ0QZ]].

## Remarks

The construction is possible because the algebraic group $G_2$ in characteristic $3$, and the algebraic group $F_4$ in characteristic $2$, has an exceptional endomorphism that exchanges long and short roots.
The smallest cases are not simple.
${}^2G_2(3)$ has order $1512$ and its [[commutator subgroup|0ASGSTXSBW]] is $\mathrm{SL}_2(\F_8)$, of index $3$.
${}^2F_4(2)$ has order $35942400$ and its commutator subgroup, of index $2$, is the [[Tits group|WC16TBY3FN]], which is simple.
The [[Suzuki groups|36X3GET13B]] arise from $B_2$ in characteristic $2$ in the same way.

## Explanations

### L1 human

The Ree groups are twisted versions of two exceptional matrix groups, $G_2$ and $F_4$, over special finite fields.
The twist is a symmetry that exists only in characteristic $3$ for $G_2$ and characteristic $2$ for $F_4$.
They give two more infinite families of finite simple groups.
</document>

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# Hellinger distance

## Definition

Let $P$ and $Q$ be [[probability measures|059TJN0NZ1]] on a measurable space $(\mathcal{X}, \mathcal{F})$.
Let $\lambda$ be a [[σ-finite measure|WEBBT18D4X]] such that $P$ and $Q$ are absolutely continuous with respect to $\lambda$, for example $\lambda = P + Q$.
Let $p = dP/d\lambda$ and $q = dQ/d\lambda$ be the [[Radon–Nikodym derivatives|1NTKG64F7N]].
The *Hellinger distance* between $P$ and $Q$ is the number $H(P, Q) \ge 0$ given by
$$H^2(P, Q) = \frac{1}{2} \int_{\mathcal{X}} \big( \sqrt{p} - \sqrt{q} \big)^2 \, d\lambda.$$

## Remarks

The value does not depend on the choice of $\lambda$.
Expanding the square gives $H^2(P, Q) = 1 - \int \sqrt{p q} \, d\lambda$, so $0 \le H \le 1$.
The map $H$ is a [[metric|0VKD1TP7F3]] on the set of probability measures on $(\mathcal{X}, \mathcal{F})$.
Some authors omit the factor $\frac{1}{2}$, which multiplies $H$ by $\sqrt{2}$.
With the convention above, the [[total variation distance|3SZESKX3N5]] $\delta$ satisfies $H^2(P, Q) \le \delta(P, Q) \le \sqrt{2} \, H(P, Q)$.

Example: on $\set{0, 1}$ with counting measure, let $P$ give mass $\frac{1}{2}$ to each point and let $Q$ be the point mass at $1$.
Then $\int \sqrt{pq} = \sqrt{0 \cdot \tfrac{1}{2}} + \sqrt{1 \cdot \tfrac{1}{2}} = 1/\sqrt{2}$, so $H^2(P, Q) = 1 - 1/\sqrt{2}$.

Non-example: the quantity $\frac{1}{2} \int (p - q)^2 \, d\lambda$, without square roots, is not the Hellinger distance.
It changes with $\lambda$: replacing $\lambda$ by $2\lambda$ halves $p$ and $q$ and divides the quantity by $2$.
</document>

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# Sz.-Nagy's dilation theorem

## Statement

Let $H$ be a [[Hilbert space|HM2HHE3J51]] and let $T \colon H \to H$ be a contraction, that is, a bounded linear operator with $\norm{T} \le 1$.
Then there is a Hilbert space $K$ that contains $H$ as a closed subspace.
There is also a [[unitary operator|MM0EJ2GDE4]] $U \colon K \to K$ such that
$$T^n = P_H U^n|_H \quad \text{for all integers } n \ge 0,$$
where $P_H$ is the [[orthogonal projection|0B8KZRT28M]] of $K$ onto $H$.
Moreover, $K$ can be chosen as the closed span of $\bigcup_{n \in \Z} U^n H$.
With this choice, the pair $(K, U)$ is unique up to a unitary isomorphism that fixes $H$.

## Remarks

Proof idea: the defect operator $D = (I - T^*T)^{1/2}$ satisfies $\norm{Tx}^2 + \norm{Dx}^2 = \norm{x}^2$.
On $\bigoplus_{n \ge 0} H$, the operator $(x_0, x_1, \ldots) \mapsto (T x_0, D x_0, x_1, x_2, \ldots)$ is an isometry, and it compresses to the powers of $T$ on the first copy of $H$.
Every isometry extends to a unitary operator on a larger space, which gives $U$.

Example: the unilateral shift on $\ell^2(\N)$ is a contraction, and the bilateral shift on $\ell^2(\Z)$ is a unitary dilation of it.

The compression property holds only for powers $n \ge 0$; for negative $n$ one has $T^{*\abs{n}} = P_H U^n|_H$ instead.

## Explanations

### L1 human

A contraction is an operator that never increases lengths.
The theorem says that every contraction is the shadow of a unitary operator, which preserves lengths, acting on a bigger space.
To see $T^n x$, apply $U$ to $x$ $n$ times in the big space and then project back onto $H$.

### L2 human

The dilation turns questions about a contraction into questions about a unitary operator, where the spectral theorem is available.
An immediate consequence is [[von Neumann's inequality|7KS1W5K54S]].
The minimal unitary dilation is the starting point of the Sz.-Nagy–Foias model theory of contractions and of the [[commutant lifting theorem|3TVXB7X266]].
</document>

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# Incomplete gamma function

## Summary

The incomplete gamma functions split the integral of the [[gamma function|0H5SYBSED9]] at a point $x$.
The lower one $\gamma(s, x)$ integrates $t^{s-1} e^{-t}$ from $0$ to $x$, and the upper one $\Gamma(s, x)$ integrates it from $x$ to $\infty$.
Their sum is $\Gamma(s)$.
They give the distribution function of the [[gamma distribution|0B80PAQD3C]] and contain the [[exponential integral|EV6ZDQ0NDY]] and the [[error function|7Y56QCZSZJ]] as special cases.

## Explanations

### L1 human

The gamma function is the area under the curve $t^{s-1} e^{-t}$ on the whole positive half line.
Cut this area at a point $x$.
The part on the left is the lower incomplete gamma function, and the part on the right is the upper one.
For $s = 1$ the curve is $e^{-t}$, and the two parts are $1 - e^{-x}$ and $e^{-x}$.
</document>

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# Grushko theorem

## Statement

Let $A$ and $B$ be groups, and let $G = A * B$ be their [[free product|TM7QS7P7X9]].
For a [[finitely generated group|082DWWXMRJ]] $X$, let $\rank X$ be the least number of elements that generate $X$.

1. Let $F$ be a [[free group|0KTW26C80J]] of finite rank, and let $\varphi \colon F \to G$ be a [[surjective|01RVCMSN5N]] [[homomorphism|0JYTG5K9Y4]]. Then $F = F_1 * F_2$ for some subgroups $F_1$ and $F_2$ of $F$ with $\varphi(F_1) = A$ and $\varphi(F_2) = B$.
2. If $A$ and $B$ are finitely generated, then $\rank(A * B) = \rank A + \rank B$.

## Remarks

Part 2 follows from part 1.
If $G$ is generated by $n$ elements, take $F$ free of rank $n$ and $\varphi$ onto $G$.
Then $F_1$ and $F_2$ are free factors of $F$, so their ranks add up to $n$, and they map onto $A$ and $B$, so $\rank A + \rank B \le n$.
The other inequality is clear, since generators of $A$ and of $B$ together generate $A * B$.
Proofs of part 1 use Stallings foldings or [[Bass–Serre theory|6M731X66RD]].

Example: $\Z/2 * \Z/3$ has rank $2$, and $\Z^2 * \Z$ has rank $3$.

The free product cannot be replaced by the direct product.
The groups $\Z/2$ and $\Z/3$ have rank $1$ each, but $\Z/2 \times \Z/3 \cong \Z/6$ is cyclic, so its rank is $1$, not $2$.

## Explanations

### L1 human

The rank of a group is the smallest number of elements that generate it.
The theorem says that for a free product, the ranks just add.
No clever choice of generators can save anything when two groups are glued with no relations between them.

### L2 human

The first part says that every way of mapping a free group onto $A * B$ splits along a free product decomposition of the free group.
The rank formula is the usual form of the theorem.
It implies that every nontrivial finitely generated group is a free product of finitely many freely indecomposable groups.
Here a group is freely indecomposable if it is nontrivial and not a free product of two nontrivial groups.
The decomposition stops because each split lowers the ranks of the factors.
It fits with the [[Kurosh subgroup theorem|RY8K5RQJSR]] as one of the basic facts about free products.
</document>

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# Fisher's exact test

## Definition

Consider a $2 \times 2$ [[contingency table|PANSMHE0SW]] of counts $a, b$ in the first row and $c, d$ in the second row, with total $n = a + b + c + d$.
Fix the row totals $r_1 = a + b$, $r_2 = c + d$ and the column totals $k_1 = a + c$, $k_2 = b + d$.
Assume the null hypothesis that the row factor and the column factor are independent.
Then, conditionally on these totals, $a$ has the [[hypergeometric distribution|0SPBPW3K61]]
$$P(a) = \binom{r_1}{a} \binom{r_2}{k_1 - a} \Big/ \binom{n}{k_1}.$$
*Fisher's exact test* computes the [[p-value|0EHYZFZ4PK]] as a sum of $P(a')$ over tables with the same totals.
The sum runs over the tables at least as extreme as the observed one.
The one-sided version sums over $a' \ge a$; the usual two-sided version sums over all $a'$ with $P(a') \le P(a)$.

## Remarks

Example: in the lady tasting tea experiment, $r_1 = r_2 = k_1 = k_2 = 4$, $n = 8$, and she classifies all cups correctly, so $a = 4$.
Then $P(4) = \binom{4}{4}\binom{4}{0}/\binom{8}{4} = 1/70$, and the one-sided p-value is $1/70 \approx 0.014$.
Non-example: replacing the hypergeometric law by its chi-squared approximation gives Pearson's [[chi-squared test|0EN0V3NY20]], whose p-value is only approximate.
The test is conservative: since the conditional law is discrete, the probability of a type I error is at most $\alpha$ and often smaller.
</document>

<document source="https://tamnd.github.io/math2/node/YBAE5QCH1A.md" tag="YBAE5QCH1A" kind="example">
# Voter model

## Example

Let $d \ge 1$.
The *voter model* on $\Z^d$ is the spin system on $\set{0, 1}^{\Z^d}$ with flip rates
$$c(x, \eta) = \frac{1}{2d} \, \#\set{y \in \Z^d : \abs{y - x} = 1, \ \eta(y) \ne \eta(x)}.$$
Here $\eta(x)$ is the opinion of the voter at site $x$.
At rate $1$, each voter picks one of its $2d$ neighbours uniformly at random and adopts that neighbour's opinion.
The rates lie in $[0, 1]$ and depend on $2d + 1$ sites, so it is an [[interacting particle system|GR49AM3PSZ]].

## Remarks

The two constant configurations, all $0$ and all $1$, are absorbing.
The model is dual to a system of coalescing [[random walks|04VZE3FZXF]].
Tracing back whose opinion a voter holds gives a random walk path, and two such paths merge when they meet.
Because of this duality, the long-time behaviour depends on whether simple random walk on $\Z^d$ is recurrent, which by [[Pólya's recurrence theorem|BEQH6AZPYM]] holds exactly for $d \le 2$.
Started from independent opinions with density $\rho \in (0, 1)$, for $d \le 2$ any two sites tend to agree, $P(\eta_t(x) \ne \eta_t(y)) \to 0$ as $t \to \infty$.
For $d \ge 3$ there is a one-parameter family of invariant measures in which both opinions keep a positive density.

## Explanations

### L1 human

Each person on a grid holds one of two opinions.
From time to time a person asks a random neighbour and copies their opinion.
Nobody is persuaded by arguments; opinions simply spread by imitation.
On a line or in the plane, large regions of agreement form and grow, while in three or more dimensions disagreement can persist forever.
</document>

<document source="https://tamnd.github.io/math2/node/YBAPZPEDWQ.md" tag="YBAPZPEDWQ" kind="definition">
# Thurston norm

## Definition

Let $M$ be a compact [[orientable|702SNFQG0W]] [[3-manifold|R20TB2KEQQ]].
For a compact surface $S$ with connected components $S_1, \ldots, S_k$, set
$$\chi_-(S) = \sum_{i=1}^{k} \max(0, -\chi(S_i)),$$
where $\chi$ is the [[Euler characteristic|RNDDWDAFJ5]].
For a class $a \in H_2(M, \partial M; \Z)$ in [[homology|M0Q5CFJ7ZH]], the *Thurston norm* of $a$ is
$$x(a) = \min \set{\chi_-(S) : S \text{ a properly embedded oriented surface in } M \text{ with } [S] = a}.$$
The function $x$ satisfies $x(n a) = n\, x(a)$ for integers $n \ge 0$ and $x(a + b) \le x(a) + x(b)$, and it extends uniquely to a continuous [[seminorm|FP5GVCN8GV]] on $H_2(M, \partial M; \R)$, also called the Thurston norm.

## Remarks

Example: let $F$ be a closed orientable surface of genus $g \ge 2$ and $M = F \times S^1$.
The fiber $F \times \set{1}$ represents a class $a$ with $\chi_-(F) = 2g - 2$, and Thurston showed that fibers of a fibration over the circle minimise $\chi_-$ in their class, so $x(a) = 2g - 2$.

Non-example of definiteness: in $M = S^2 \times S^1$ the class $a$ of $S^2 \times \set{1}$ is nonzero, but $\chi_-(S^2) = 0$, so $x(a) = 0$.
So $x$ is in general only a seminorm, not a norm.
It is a norm when no nonzero class is represented by spheres, disks, tori and annuli, for example when $M$ is hyperbolic.
</document>

<document source="https://tamnd.github.io/math2/node/YBBDYHBADD.md" tag="YBBDYHBADD" kind="statement">
# Böttcher's equation

## Statement

Let $U \subseteq \C$ be an open set containing $0$, and let $f \colon U \to \C$ be [[holomorphic|0NAHXR1FRH]].
Assume $f(z) = a_n z^n + a_{n+1} z^{n+1} + \cdots$ near $0$, with $n \ge 2$ and $a_n \ne 0$.
Then there is a holomorphic map $\varphi$ on a neighbourhood $V$ of $0$ with $\varphi(0) = 0$ and $\varphi'(0) \ne 0$ that solves *Böttcher's equation*
$$\varphi(f(z)) = \varphi(z)^n \quad \text{for all } z \in V \text{ with } f(z) \in V.$$
Such a $\varphi$ is unique up to multiplication by an $(n-1)$-st root of unity.

## Remarks

So near a superattracting [[fixed point|R6QVS6P21N]] of local degree $n$, the map $f$ is [[conjugate|HNXXAPQKG3]] to $w \mapsto w^n$.
The idea of the proof: after a linear change of variable one may take $a_n = 1$.
Then one sets $\varphi(z) = \lim_{k \to \infty} \big(f^k(z)\big)^{1/n^k}$, choosing the branches of the roots that are tangent to the identity.
The limit exists because $f^k(z)$ tends to $0$ very fast, and the correction factors form a convergent infinite product.
Example: $f(z) = z^2 / (1 - 2z + 2z^2)$ satisfies the equation with $n = 2$ and $\varphi(z) = z/(1 - z)$.
Indeed $\varphi(f(z)) = f(z)/(1 - f(z)) = z^2 / (1 - 2z + z^2) = \varphi(z)^2$.
For a monic [[polynomial|1G0B73WPX8]] $P$ of degree $d \ge 2$, the same result at the superattracting point $\infty$ gives a map $\varphi$ near $\infty$ with $\varphi(z) \sim z$ and $\varphi(P(z)) = \varphi(z)^d$.
This is the Böttcher coordinate used to define external rays.

## Explanations

### L1 human

The simplest map with a very strongly attracting fixed point is $w \mapsto w^n$, which squares or cubes distances to $0$ at each step.
The theorem says that every holomorphic map with such a fixed point looks exactly like $w \mapsto w^n$ near it, after a change of coordinates.

### L2 human

The equation was studied by Lucjan Böttcher in the early twentieth century.
It is the superattracting counterpart of Koenigs linearization, which handles multipliers $\lambda$ with $0 < \abs{\lambda} < 1$.
For polynomials the Böttcher coordinate at infinity is the main tool for describing the basin of infinity.
When the filled Julia set is connected, it extends to a conformal map from the basin of infinity onto the outside of the closed unit disc.
</document>

<document source="https://tamnd.github.io/math2/node/YBBS082ZY1.md" tag="YBBS082ZY1" kind="definition">
# Kadomtsev–Petviashvili equation

## Definition

Let $\sigma \in \set{1, -1}$.
The *Kadomtsev–Petviashvili equation* is the [[partial differential equation|GC5BDZ6QBC]]
$$\partial_x \big( u_t + 6 u u_x + u_{xxx} \big) + 3 \sigma u_{yy} = 0$$
for an unknown real function $u(x, y, t)$ of $(x, y) \in \R^2$ and $t \in \R$, where subscripts denote [[partial derivatives|09EGC6F8PT]].
The case $\sigma = 1$ is called *KP-II* and the case $\sigma = -1$ is called *KP-I*.

## Remarks

Example: let $v(x, t)$ be a solution of the [[Korteweg–De Vries equation|KTT75QC72Z]] $v_t + 6 v v_x + v_{xxx} = 0$, and put $u(x, y, t) = v(x, t)$.
Then the bracket vanishes and $u_{yy} = 0$, so $u$ solves both KP-I and KP-II.

Non-example: $u(x, y, t) = y^2$ is not a solution, because the first term is $0$ while $3 \sigma u_{yy} = 6 \sigma \ne 0$.

Other normalisations are used, for example $\partial_x(u_t + u u_x + \varepsilon^2 u_{xxx}) + \lambda u_{yy} = 0$ with $\lambda = \pm 1$.
They reduce to the form above by scaling the variables, and the sign of the $u_{yy}$ term relative to the dispersive term decides between KP-I and KP-II.
</document>

<document source="https://tamnd.github.io/math2/node/YBCZ9VY7SX.md" tag="YBCZ9VY7SX" kind="statement">
# Sharkovskii's theorem

## Statement

Order the positive integers by the *Sharkovskii ordering* $\triangleright$:
$$3 \triangleright 5 \triangleright 7 \triangleright \cdots \triangleright 2 \cdot 3 \triangleright 2 \cdot 5 \triangleright \cdots \triangleright 2^2 \cdot 3 \triangleright 2^2 \cdot 5 \triangleright \cdots \triangleright 2^3 \triangleright 2^2 \triangleright 2 \triangleright 1.$$
First come the odd numbers greater than $1$ in increasing order, then $2$ times them, then $2^2$ times them, and so on.
At the end come the powers of $2$ in decreasing order.
Let $I \subseteq \R$ be an [[interval|0Q1D85B8G6]] and let $f \colon I \to I$ be a [[continuous map|0VGFHTM75Y]].

1. If $f$ has a [[periodic point|T75CZ5H47N]] of [[minimal period|VFZRVZ9SD8]] $m$ and $m \triangleright n$, then $f$ has a periodic point of minimal period $n$.
2. Conversely, for each integer $m \ge 1$ some continuous $f \colon \R \to \R$ has a point of minimal period $m$ but none of minimal period $k$ when $k \triangleright m$.

## Remarks

The proof of part 1 uses the intermediate value theorem in the form: if $f(J) \supseteq K$ for closed intervals $J, K$, then $J$ contains a closed interval mapped onto $K$.
The points of a periodic orbit cut the interval into pieces, and a directed graph records which pieces cover which.
Cycles in this graph give periodic points of the required periods.
Example: since $3$ is the first number in the ordering, a continuous map of an interval with a point of period $3$ has points of every period.
The hypothesis that the space is an interval is necessary: the rotation of the circle by $1/3$ has every point of period $3$ and no other periods.
For a continuous map of a compact interval, the set of minimal periods is therefore of one of two kinds.
It is either a tail $\set{k : m \trianglerighteq k}$ of the ordering for some $m$, or the set of all powers of $2$.

## Explanations

### L1 human

The theorem says that the periods of a continuous map of a line segment come in a fixed order.
If a period appears, then every period further down the list appears as well.
Period $3$ is at the top of the list, so period $3$ forces all periods.
Period $1$, a fixed point, is at the bottom: it is forced by any other period.

### L2 human

The theorem is purely one-dimensional: its proof uses the order of the real line through the intermediate value theorem, and it fails on the circle and in the plane.
It refines the Li–Yorke result that period three implies chaos.
In families like the logistic map, the passage from maps with only power-of-two periods to the others is where period doubling turns into chaos.
</document>

<document source="https://tamnd.github.io/math2/node/YBERRJMXT6.md" tag="YBERRJMXT6" kind="concept">
# Schottky group

## Summary

A Schottky group is a group of [[Möbius transformations|0SG74WBFA2]] generated by maps that pair off disjoint discs in the [[Riemann sphere|0J78KBPFZB]].
It is a [[Kleinian group|27QRARKWD5]] that is a [[free group|0KTW26C80J]] and has only loxodromic elements besides the identity.
The quotient of its domain of discontinuity is a compact [[Riemann surface|5K0N8QMV6Q]].
Every compact Riemann surface arises in this way, which gives a concrete way to build them.

## Explanations

### L1 human

Draw $2g$ disjoint circles on the Riemann sphere and pair them up.
For each pair, choose a Möbius map that sends the outside of the first circle onto the inside of the second.
The group generated by these $g$ maps moves the region outside all the circles around without overlap.
Gluing each circle to its partner turns that region into a surface with $g$ handles.
For $g = 1$ and the map $z \mapsto 2z$, with circles $\abs{z} = 1$ and $\abs{z} = 2$, the glued annulus is a torus.
</document>

<document source="https://tamnd.github.io/math2/node/YBG22HS28V.md" tag="YBG22HS28V" kind="definition">
# Order filter

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
A *filter* of $P$ is a subset $F \subseteq P$ such that:

1. $F$ is nonempty;
2. if $x \in F$, $y \in P$ and $x \le y$, then $y \in F$;
3. if $x, y \in F$, then there is $z \in F$ with $z \le x$ and $z \le y$.

The filter $F$ is *proper* if $F \ne P$.

## Remarks

Clause 2 says that $F$ is an [[upper set|ETCE287E2R]].
If $P$ is a lattice, clause 3 can be replaced by: $x, y \in F$ implies $x \wedge y \in F$, using the [[meet|J2T75ATRV5]].
For $a \in P$, the set $\uparrow a = \set{x \in P : a \le x}$ is a filter, the *principal filter* of $a$.
Example: in the power set of $\N$, the cofinite sets $\set{A \subseteq \N : \N \setminus A \text{ is finite}}$ form a proper filter.
It contains $\N$, a superset of a cofinite set is cofinite, and $A \cap B$ is cofinite when $A$ and $B$ are, since $\N \setminus (A \cap B) = (\N \setminus A) \cup (\N \setminus B)$.
Non-example: in the power set of $\set{1, 2}$, the set $\set{\set{1}, \set{2}, \set{1, 2}}$ satisfies clauses 1 and 2 but not clause 3.
The only common lower bound of $\set{1}$ and $\set{2}$ is $\emptyset$, which is not in the set.
A filter on a set $X$ in the sense of [[set filters|15PF73BKY6]] is a proper filter of the power set of $X$.
</document>

<document source="https://tamnd.github.io/math2/node/YBJ2VQHFDQ.md" tag="YBJ2VQHFDQ" kind="statement">
# Zariski's lemma

## Statement

Let $k$ be a [[field|04C5ZBG1WT]] and let $K$ be a field extension of $k$ that is [[finitely generated|AC5TFFQT9M]] as a $k$-algebra.
Then $K$ is a [[finite extension|0C1Y04SZ1W]] of $k$.

## Remarks

By [[Noether normalization|7Z28JY8QW4]], $K$ is finite over a polynomial ring $k[y_1, \ldots, y_d]$.
By [[integral extensions of fields|5GCANV7337]], $k[y_1, \ldots, y_d]$ is then a field, which forces $d = 0$, since $y_1$ has no inverse in a polynomial ring when $d \ge 1$.
So $K$ is finite over $k$.
Another proof uses the [[Artin–Tate lemma|4MBR0W0S6N]].

Example: $\Q(\sqrt{2}) = \Q[\sqrt{2}]$ is a field generated by one element as a $\Q$-algebra, and it is finite over $\Q$ of degree $2$.

The lemma is about generation as an algebra.
The field $k(t)$ of rational functions is generated by $t$ as a field, but it is not a finitely generated $k$-algebra, and it is not finite over $k$.

## Explanations

### L1 human

A field that can be built from a smaller field with finitely many elements, using only addition and multiplication, is a finite extension of it.
Inverses come for free in such a field, so no transcendental element can occur.

### L2 human

Zariski's lemma is the algebraic core of the [[weak Nullstellensatz|ZWS0YWGF17]].
If $\mathfrak{m}$ is a maximal ideal of $k[x_1, \ldots, x_n]$, then $k[x_1, \ldots, x_n]/\mathfrak{m}$ is a field finitely generated as a $k$-algebra, so it is finite over $k$.
When $k$ is algebraically closed, this residue field is $k$ itself, and $\mathfrak{m}$ is the ideal of a point.
</document>

<document source="https://tamnd.github.io/math2/node/YBNZWCG8AC.md" tag="YBNZWCG8AC" kind="statement">
# Balinski's theorem

## Statement

Let $d \ge 1$ and let $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d$.
Let $G(P)$ be the [[graph|00C0BR7D7H]] whose vertices are the vertices of $P$ and whose edges are the edges of $P$.
Then $G(P)$ is [[d-vertex-connected|7FZB7YYBXA]]: it has more than $d$ vertices, and it stays connected after the removal of any $d - 1$ vertices.

## Remarks

The theorem was proved by Michel Balinski in 1961.
Proof idea: remove a set $S$ of $d - 1$ vertices, and pass a hyperplane through $S$ and one further vertex.
From every vertex that does not maximise a linear function on $P$ there is an edge along which the function increases, as in the simplex method.
Use this for the linear function defining the hyperplane and for its negative.
It connects every remaining vertex, avoiding $S$, to the top face or the bottom face of $P$.
These pieces are then joined through the extra vertex.

Example: the graph of the $3$-cube is $3$-connected; removing the $3$ neighbours of a vertex isolates it, so the bound $d$ is sharp.

## Explanations

### L1 human

Take a convex polytope in $d$ dimensions and look at its network of corners and edges.
Balinski's theorem says that this network is very robust: you must delete at least $d$ corners to cut it into pieces.
For a solid in space you need to remove at least three corners.

### L2 human

For $d = 3$ this is the easy half of Steinitz's theorem on graphs of $3$-polytopes.
The bound is sharp for every simple polytope, since a vertex has only $d$ neighbours.
It is one of the few general facts known about graphs of polytopes in high dimension, along with results and open questions about their diameter, such as the Hirsch conjecture.
</document>

<document source="https://tamnd.github.io/math2/node/YBSCJY6BY2.md" tag="YBSCJY6BY2" kind="definition">
# q-gamma function

## Definition

Let $0 < q < 1$, and let $(a; q)_\infty = \prod_{k=0}^\infty (1 - a q^k)$ be the infinite [[q-Pochhammer symbol|Y1X8ZZQ7VM]].
For a real number $x$ that is not in $\set{0, -1, -2, \ldots}$, the *$q$-gamma function* is
$$\Gamma_q(x) = (1 - q)^{1 - x} \frac{(q; q)_\infty}{(q^x; q)_\infty}.$$

## Remarks

The products converge because $\sum q^k < \infty$, and $(q^x; q)_\infty \ne 0$ because $q^{x+k} \ne 1$ for all $k \ge 0$.

Example: $\Gamma_q(1) = 1$, since the two products are equal.
Also $\Gamma_q(2) = (1-q)^{-1} \frac{(q;q)_\infty}{(q^2;q)_\infty} = \frac{1 - q}{1 - q} = 1$ and $\Gamma_q(3) = \frac{(1-q)(1-q^2)}{(1-q)^2} = 1 + q$.
In general $\Gamma_q(x + 1) = [x]_q\, \Gamma_q(x)$ with $[x]_q = \frac{1 - q^x}{1 - q}$, so $\Gamma_q(n + 1) = [1]_q [2]_q \cdots [n]_q$.

Non-example of a valid argument: for $x = -1$ the factor $1 - q^{x+1}$ of $(q^x; q)_\infty$ is $0$, so $\Gamma_q$ has a pole there.

For each fixed $x$, $\Gamma_q(x) \to \Gamma(x)$ as $q \to 1^-$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
A related definition exists for $q > 1$.
</document>

<document source="https://tamnd.github.io/math2/node/YBVSDT89T6.md" tag="YBVSDT89T6" kind="statement">
# Sierpiński's theorem on measurable midpoint convex functions

## Statement

Let $I \subseteq \R$ be an open [[interval|0Q1D85B8G6]], and let $f \colon I \to \R$ be [[midpoint convex|1TQ1JQXEWP]].
Assume that $f$ is Lebesgue [[measurable|YYNCJSR0PJ]].
Then $f$ is [[continuous|0X3WGX95FE]] on $I$, and $f$ is [[convex|07Q7CNM5WF]].

## Remarks

The idea of the proof: the sets $\set{x \in I : f(x) \le n}$, for $n = 1, 2, \ldots$, are measurable and cover $I$, so one of them, say $A$, has positive [[Lebesgue measure|2Z2QVWZ262]].
By a theorem of Steinhaus type, the set of midpoints $\set{(a + b)/2 : a, b \in A}$ contains a nonempty open interval $J$.
For $a, b \in A$, midpoint convexity gives $f\big(\frac{a + b}{2}\big) \le \frac{f(a) + f(b)}{2} \le n$, so $f$ is bounded above on $J$.
The [[Bernstein–Doetsch theorem|MQA1VVCJ63]] then gives continuity and convexity.

Measurability cannot be dropped: a discontinuous additive function $a \colon \R \to \R$ is midpoint convex, but it is not convex, and by this theorem it is not measurable.
Example of use: if $f$ is midpoint convex and is a pointwise limit of continuous functions, then $f$ is measurable, hence convex.

## Explanations

### L1 human

Midpoint convexity alone allows very irregular functions, but all of them are so wild that they cannot even be measured.
Sierpiński's theorem says that any midpoint convex function that is measurable is a genuine convex function.
In practice this means that every midpoint convex function one meets in analysis is convex.

### L2 human

The theorem follows the pattern of the regularity results for Cauchy's equation, where a measurable additive function is linear.
The key step turns measurability into an upper bound on an interval, and then the [[Bernstein–Doetsch theorem|MQA1VVCJ63]] takes over.
The same argument works for functions that are bounded above on a set of positive measure, and an analogue holds with Baire category instead of measure.
</document>

<document source="https://tamnd.github.io/math2/node/YBYE4TBNJV.md" tag="YBYE4TBNJV" kind="statement">
# Approximation by convolution with mollifiers

## Statement

Let $\varphi$ be a [[mollifier|CQMK4YPE1J]] on $\R^n$, let $\varphi_\varepsilon(x) = \varepsilon^{-n} \varphi(x/\varepsilon)$ for $\varepsilon > 0$, and let $f$ be a locally integrable function on $\R^n$.
Put $f_\varepsilon = \varphi_\varepsilon * f$, the [[convolution|2Z29JN3VES]] $f_\varepsilon(x) = \int_{\R^n} \varphi_\varepsilon(x - y) f(y) \, dy$.
Then:

1. $f_\varepsilon$ is a [[smooth function|0V33BNKZ9S]], and $\partial^\alpha f_\varepsilon = (\partial^\alpha \varphi_\varepsilon) * f$ for every multi-index $\alpha$.
2. If $f$ vanishes almost everywhere outside a closed set $K$, then $f_\varepsilon$ vanishes outside $\set{x : \operatorname{dist}(x, K) \le \varepsilon}$.
3. If $f$ is [[continuous|0X3WGX95FE]], then $f_\varepsilon \to f$ uniformly on every compact set as $\varepsilon \to 0$.
4. If $f \in L^p(\R^n)$ with $1 \le p < \infty$, then $\norm{f_\varepsilon}_{L^p} \le \norm{f}_{L^p}$ and $\norm{f_\varepsilon - f}_{L^p} \to 0$ as $\varepsilon \to 0$.

## Remarks

Part 1 follows by differentiating under the integral sign, which is allowed because $\varphi_\varepsilon$ has bounded derivatives and compact support.
Part 2 holds because $\varphi_\varepsilon(x - y) = 0$ when $\abs{x - y} > \varepsilon$.
For parts 3 and 4, write $f_\varepsilon(x) - f(x) = \int \varphi_\varepsilon(y) \big( f(x - y) - f(x) \big) dy$, an average of differences over $\abs{y} \le \varepsilon$.
For continuous $f$, uniform continuity on compact sets makes it small.
For $f \in L^p$, Minkowski's integral inequality bounds its norm by $\sup_{\abs{y} \le \varepsilon} \norm{f(\cdot - y) - f}_{L^p}$.
This tends to $0$ by the continuity of translation in $L^p$.
The bound $\norm{f_\varepsilon}_{L^p} \le \norm{f}_{L^p}$ is [[Young's convolution inequality|JXDKJCBXJ8]] with $\norm{\varphi_\varepsilon}_{L^1} = 1$.

Consequence: for $1 \le p < \infty$ the [[bump functions|HS65DW0HXH]] are dense in $L^p(\R^n)$.
Indeed, first cut $f$ off to a large ball, then mollify, and use parts 1, 2 and 4.

Part 4 fails for $p = \infty$.
If $f$ is the indicator function of $[0, \infty)$ on $\R$, then each $f_\varepsilon$ is continuous, so $\norm{f_\varepsilon - f}_{L^\infty} \ge 1/2$ for all $\varepsilon$.

## Explanations

### L1 human

Blurring a function with a narrow smooth blob gives a smooth function.
If the blob is narrow enough, the blurred function is as close as one wants to the original.
The closeness is uniform for continuous functions and in the $L^p$ sense for $L^p$ functions.
So rough functions can be replaced by smooth ones, at a small cost.

### L2 human

This theorem is the workhorse behind density arguments in analysis.
To prove an identity or an inequality for all $f \in L^p$ or all $f$ in a [[Sobolev space|M4CEZ5K9HP]], one proves it for smooth functions and passes to the limit.
Mollification commutes with weak derivatives, $\partial^\alpha (\varphi_\varepsilon * f) = \varphi_\varepsilon * \partial^\alpha f$, so the same argument shows that smooth functions are dense in $W^{k,p}(\R^n)$.
At a Lebesgue point of $f$, $f_\varepsilon(x) \to f(x)$, so the convergence also holds almost everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/YC90FXM4SA.md" tag="YC90FXM4SA" kind="definition">
# Buchberger's algorithm

## Definition

Let $k$ be a [[field|04C5ZBG1WT]], let $S = k[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]], and fix a [[monomial order|E874NARWPV]].
For nonzero $f \in S$, let $\operatorname{LT}(f)$ be its leading term and $\operatorname{LM}(f)$ its leading monomial.
For nonzero $f, g \in S$, let $x^\gamma$ be the [[least common multiple|007NPQX34K]] of $\operatorname{LM}(f)$ and $\operatorname{LM}(g)$; the *S-polynomial* of $f$ and $g$ is
$$S(f, g) = \frac{x^\gamma}{\operatorname{LT}(f)} \, f - \frac{x^\gamma}{\operatorname{LT}(g)} \, g.$$
*Buchberger's algorithm* takes a finite set $F$ of nonzero elements of $S$ and proceeds as follows.

1. Set $G = F$.
2. For each pair $f \ne g$ in $G$, compute the remainder $r$ of $S(f, g)$ under the [[multivariate division algorithm|T8EN3CAEQF]] by $G$, listed in a fixed order.
3. If some such $r$ is nonzero, add all the nonzero remainders to $G$ and go back to step 2.
4. If all such remainders are zero, return $G$.

## Remarks

The algorithm rests on Buchberger's criterion.
Let $G$ be a finite set of generators of an ideal $I$.
Then $G$ is a [[Gröbner basis|C83KPT47R0]] of $I$ if and only if each $S(f, g)$ with $f, g \in G$ has remainder zero on division by $G$.
The algorithm stops for the following reason.
Each pass with a nonzero remainder makes the [[monomial ideal|4G25S7PJ22]] generated by the leading terms of $G$ strictly larger.
An ascending chain of ideals of $S$ is stationary by [[Hilbert's basis theorem|0YKA9J3XEE]].
The output is a Gröbner basis of the ideal generated by $F$.

Example: in $k[x, y]$ with the lexicographic order $x > y$, let $F = \set{x + y, \; x}$.
Then $S(x + y, x) = (x + y) - x = y$, which has remainder $y \ne 0$ on division by $F$, so $y$ is added.
For $G = \set{x + y, x, y}$, one finds $S(x + y, y) = y(x + y) - x y = y^2$, with remainder $0$, and $S(x, y) = 0$, and $S(x + y, x) = y$ now has remainder $0$.
So $G$ is a Gröbner basis of $(x, y)$.

Non-example: the set $F$ above fails the test in step 4, since $S(x + y, x)$ has nonzero remainder; this is why $F$ is not a Gröbner basis.
</document>

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# Evolute

## Definition

Let $\gamma \colon I \to \R^2$ be a [[regular|06JMR2XWXW]] curve of class $C^2$ on an interval $I$.
Let $N(t)$ be its unit tangent rotated by a right angle counterclockwise, and let $k(t)$ be its signed [[curvature|7JQS6VKFEZ]].
Assume that $k(t) \ne 0$ for all $t \in I$.
The *evolute* of $\gamma$ is the curve
$$E(t) = \gamma(t) + \frac{1}{k(t)} N(t), \qquad t \in I.$$
So $E(t)$ is the centre of the [[osculating circle|NWSFMCRWHJ]] of $\gamma$ at $t$.

## Remarks

Example: for $\gamma(t) = (t, t^2)$, one has $k(t) = 2 (1 + 4t^2)^{-3/2}$ and $N(t) = (-2t, 1) / \sqrt{1 + 4t^2}$.
Then $E(t) = (-4t^3, 1/2 + 3t^2)$, so the evolute is the semicubical parabola $27 x^2 = 16 (y - 1/2)^3$.
Non-example: for $\gamma(t) = (t, t^3)$ on $\R$, the curvature vanishes at $t = 0$, so the evolute is not defined on the whole interval; near $t = 0$ the centres of curvature go off to infinity.
The evolute of a circle of radius $r$ is the constant curve at its centre.
If $\gamma$ is of class $C^3$, then $E'(t)$ is parallel to $N(t)$, and $E'(t) = 0$ exactly where $k'(t) = 0$, that is, at the vertices of $\gamma$.
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# Hessenberg matrix

## Summary

An upper Hessenberg matrix is a square matrix whose entries below the first subdiagonal are zero; a lower Hessenberg matrix is its transpose.
It is almost [[triangular|GTG8MJGDKC]], with one extra diagonal.
Every complex square matrix is unitarily similar to an upper Hessenberg matrix, and this reduction takes finitely many steps.
For this reason the [[QR algorithm|B6F8YWHYZ5]] for eigenvalues first brings a matrix to Hessenberg form.

## Explanations

### L1 human

Start from an upper triangular matrix and allow one more diagonal of nonzero entries just below the main diagonal.
The result is an upper Hessenberg matrix, such as $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 0 & 7 & 8 \end{pmatrix}$.
Exact triangular form usually needs the eigenvalues, which are hard to find.
Hessenberg form is the closest one can get with a fixed, finite amount of work, and it makes each later step of eigenvalue algorithms much cheaper.
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# Boundary layer

## Summary

In a [[singular perturbation|G6J21TGRMM]] problem, a boundary layer is a thin region next to the boundary where the solution changes rapidly.
Outside the layer, the solution of the reduced problem is a good approximation; inside it, this approximation fails.
The layer thickness tends to $0$ with the small parameter.
The notion comes from fluid dynamics, where Prandtl introduced it for the thin layer of fluid near a wall in which viscosity matters.

## Explanations

### L1 human

Air flowing past a wing moves almost freely, except in a very thin sheet close to the surface where it is slowed down to rest.
That sheet is a boundary layer.
Take the equation $\varepsilon y'' + y' = 0$ with $y(0) = 0$ and $y(1) = 1$.
Its solution is close to $1$ on most of the interval, but it falls to $0$ in a region of width about $\varepsilon$ next to $x = 0$.
As $\varepsilon$ gets smaller, the layer gets thinner and steeper.
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# Integral transform

## Summary

An integral transform turns a function $f$ into a new function $Tf$ by integrating $f$ against a fixed function of two variables, the [[kernel|9M0NNNFJGM]].
In symbols, $(Tf)(u) = \int f(t) K(t, u)\, dt$.
The [[Laplace transform|0V49JEZW06]], the [[Fourier transform|DPVM34ADZV]] and the [[Mellin transform|N5PVGRJE1T]] are the main examples.
A good transform turns a hard operation, such as differentiation or convolution, into an easy one, such as multiplication.
It can be undone by an [[inverse transform|GC96MQC0PP]].

## Explanations

### L1 human

Think of the kernel as a family of test patterns, one for each value of $u$.
The number $(Tf)(u)$ measures how much $f$ looks like the pattern with label $u$.
For the Laplace transform the patterns are the decaying exponentials $e^{-st}$.
For example, the constant function $1$ on $[0, \infty)$ has Laplace transform $\int_0^\infty e^{-st}\, dt = 1/s$ for $s > 0$.
A problem about $f$ often becomes simpler when it is rewritten as a problem about $Tf$.
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# Hilbert's twenty-third problem

## Problem

Hilbert's twenty-third problem asks for the further development of the methods of the [[calculus of variations|PQ25NHV0KS]].
Unlike most of Hilbert's problems, it does not ask a single question with a definite answer.
It is a call to extend and deepen the theory of necessary and sufficient conditions for extrema of functionals.

## Remarks

In the text of the problem Hilbert presented his own contribution, the [[invariant integral|D1EBAAFEWV]], which gives a short route to sufficient conditions for a minimum in the Weierstrass theory.
Because it is a programme rather than a statement, the problem is not regarded as solved or unsolved.
The closely related [[twentieth problem|187NP5Z002]] asks for the existence of solutions of variational boundary value problems.

## Explanations

### L1 human

The last problem on Hilbert's famous list of 1900 is not a precise question.
It asks mathematicians to keep developing the calculus of variations, the theory of finding curves and surfaces that make a quantity smallest or largest.
That theory has grown a great deal since then, in directions that Hilbert could not foresee.
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# Homology with coefficients

## Definition

Let $X$ be a topological space and let $G$ be an [[abelian group|082GHNPJ53]].
The group $C_n(X; G)$ of singular $n$-chains with coefficients in $G$ is the set of finite formal sums $\sum_i g_i \sigma_i$.
Here $g_i \in G$ and $\sigma_i \colon \Delta^n \to X$ are singular $n$-simplices; equivalently, $C_n(X; G) = C_n(X) \otimes_{\Z} G$.
The boundary map is $\partial\left(\sum_i g_i \sigma_i\right) = \sum_i \sum_{k=0}^{n} (-1)^k g_i \, \sigma_i|_{[v_0, \ldots, \hat v_k, \ldots, v_n]}$, where the hat means that the vertex is omitted.
This gives a [[chain complex|WQ82C0M3QB]], and the *[[homology|2J4JJJYP22]] of $X$ with coefficients in $G$*, written $H_n(X; G)$, is its homology in degree $n$.
For a subspace $A \subseteq X$, the relative groups $H_n(X, A; G)$ come from $C_n(X; G) / C_n(A; G)$ in the same way.

## Remarks

For $G = \Z$ this is ordinary [[singular homology|M0Q5CFJ7ZH]].
If $R$ is a commutative ring, then $H_n(X; R)$ is an $R$-module; for a field it is a vector space.
The [[universal coefficient theorem|GTA8ZFGZ3N]] gives a split exact sequence $0 \to H_n(X) \otimes G \to H_n(X; G) \to \operatorname{Tor}(H_{n-1}(X), G) \to 0$, with [[Tor|DDQRERAQCV]] the Tor functor.

Example: for the [[real projective plane|1DDYB6VVSR]], $H_1(\R P^2) = \Z/2$ and $H_2(\R P^2) = 0$.
The universal coefficient theorem gives $H_2(\R P^2; \Z/2) \cong 0 \oplus \operatorname{Tor}(\Z/2, \Z/2) \cong \Z/2$.

Non-example: the construction needs $G$ to be abelian, since the boundary formula adds coefficients in an arbitrary order.
The proof that $\partial \circ \partial = 0$ rearranges the double sum and cancels terms in pairs, which uses that addition in $G$ is commutative.
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# Abscissa of absolute convergence

## Summary

The abscissa of absolute convergence $\sigma_a$ of a [[Laplace transform|0V49JEZW06]] marks the left edge of the half-plane where $\int_0^\infty \abs{f(t)} e^{-\operatorname{Re}(s) t}\, dt$ is finite.
It is never smaller than the [[abscissa of convergence|FTHFSJ30WJ]] $\sigma_c$.
In the half-plane of absolute convergence the transform can be handled with the standard theorems of Lebesgue integration, such as dominated convergence and Fubini's theorem.
The strip $\sigma_c < \operatorname{Re} s < \sigma_a$, where the integral converges only conditionally, can be nonempty.

## Explanations

### L1 human

Convergence of an integral can come from two sources: the size of the integrand gets small, or positive and negative parts cancel.
Absolute convergence uses only the size.
The abscissa of absolute convergence tells where size alone is enough.
For $f(t) = e^{t} \sin(e^{t})$ the integral converges for $\operatorname{Re} s > 0$ thanks to cancellation, but converges absolutely only for $\operatorname{Re} s > 1$.
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# Autocorrelation function

## Summary

The autocorrelation function of a weakly [[stationary process|DT0QXJE705]] gives, for each lag $h$, the correlation between $X_t$ and $X_{t+h}$.
It summarizes the linear dependence in a [[time series|EA3MF4VTEH]] in one function of the lag.
Its sample version is the first tool for identifying a time series model.

## Explanations

### L1 human

Take a time series and lay a copy of it next to the original, shifted by $h$ steps.
The autocorrelation at lag $h$ measures how well the shifted copy lines up with the original, on a scale from $-1$ to $1$.
For daily temperatures, the autocorrelation at lag $1$ is high, because today is usually like yesterday.
For independent coin tosses, the autocorrelation is $0$ at every lag $h \ne 0$.
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# Homotopy groups of spheres

## Summary

The homotopy groups of spheres are the groups $\pi_k(S^n)$ of homotopy classes of based maps from the $k$-sphere to the $n$-sphere.
Some are simple: $\pi_k(S^n) = 0$ for $k < n$, and $\pi_n(S^n) \cong \Z$, detected by the degree.
For $k > n$ they are mostly finite but highly irregular, and no complete formula is known.
Their computation has driven much of algebraic topology, through fibrations, spectral sequences and stable homotopy theory.

## Explanations

### L1 human

These groups ask in how many essentially different ways one sphere can be wrapped around another.
Wrapping a circle around a circle is counted by the winding number, an integer.
Surprisingly, a $3$-sphere can be wrapped around a $2$-sphere in infinitely many different ways, by the Hopf map and its multiples, even though the $3$-sphere has higher dimension.
Small changes in the dimensions give very different answers, which is why these groups are famous.
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# Dini derivative

## Definition

Let $I \subseteq \R$ be an open interval, let $f \colon I \to \R$, and let $x \in I$.
The four *Dini derivatives* of $f$ at $x$ are the following elements of the [[extended real line|0GSZ7B0CGR]].

1. The upper right Dini derivative $D^+ f(x) = \limsup_{h \to 0^+} \frac{f(x + h) - f(x)}{h}$.
2. The lower right Dini derivative $D_+ f(x) = \liminf_{h \to 0^+} \frac{f(x + h) - f(x)}{h}$.
3. The upper left Dini derivative $D^- f(x) = \limsup_{h \to 0^-} \frac{f(x + h) - f(x)}{h}$.
4. The lower left Dini derivative $D_- f(x) = \liminf_{h \to 0^-} \frac{f(x + h) - f(x)}{h}$.

Here the upper and lower limits are the [[limit superior|0P8BA7T7S5]] and [[limit inferior|0H4R3RESGV]] of the quotient.
They are taken over $h$ in the given one-sided range with $x + h \in I$.

## Remarks

Always $D_+ f(x) \le D^+ f(x)$ and $D_- f(x) \le D^- f(x)$.
The [[right derivative|0HXEX0DPRA]] of $f$ at $x$ exists and is finite exactly when $D_+ f(x) = D^+ f(x)$ is finite, and similarly on the left.
So $f$ is differentiable at $x$ exactly when all four Dini derivatives are equal and finite.

Example: let $f(x) = x \sin(1/x)$ for $x \ne 0$ and $f(0) = 0$.
For $h \ne 0$, the [[difference quotient|AZBDB5F4M4]] at $0$ is $\frac{h \sin(1/h)}{h} = \sin(1/h)$.
For $h > 0$ and for $h < 0$ alike, it takes every value in $[-1, 1]$ for arbitrarily small $\abs{h}$.
So $D^+ f(0) = D^- f(0) = 1$ and $D_+ f(0) = D_- f(0) = -1$, and $f$ is not differentiable at $0$.

Non-example of finiteness: for $f(x) = \sqrt{\abs{x}}$ at $0$, the quotient for $h > 0$ is $h^{-1/2}$, so $D^+ f(0) = D_+ f(0) = +\infty$.
The Dini derivatives exist but are not real numbers.
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# Adjoint state method

## Summary

The adjoint state method computes the [[gradient|0ABK9V9Y64]] of an objective that depends on parameters through the solution of an equation.
A typical example of such an equation is a discretised differential equation.
Instead of differentiating the state with respect to each parameter, it solves one extra linear equation, the adjoint equation, whose solution is the adjoint state.
The cost of the gradient is then about that of one or two state solves, whatever the number of parameters.
It is the basis of PDE-constrained optimization, of seismic and medical inversion, and of reverse-mode differentiation.

## Explanations

### L1 human

Suppose a simulation turns a million parameters into a state, and you measure how far the state is from data.
Changing each parameter one at a time and re-running the simulation would take a million runs.
The adjoint method runs the simulation once, then runs one extra backward problem that tells how sensitive the misfit is to the state.
From that single answer it reads off the sensitivity to all parameters at once.
In the simplest case, the state is $u = A^{-1} \theta$ and the misfit is $\frac{1}{2} \abs{u - d}^2$; then the gradient is the solution $\lambda$ of $A^T \lambda = u - d$.
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# Barnes G-function

## Definition

Let $\gamma$ be the [[Euler–Mascheroni constant|00C5P9JZBK]].
The *Barnes G-function* is the [[entire function|0G4F0XB9X2]] $G$ given for $z \in \C$ by
$$G(1 + z) = (2\pi)^{z/2} \exp\left(-\frac{z + (1+\gamma) z^2}{2}\right) \prod_{k=1}^\infty \left(1 + \frac{z}{k}\right)^k \exp\left(\frac{z^2}{2k} - z\right).$$

## Remarks

The product converges uniformly on compact sets.
The function $G$ satisfies $G(1) = 1$ and
$$G(z + 1) = \Gamma(z)\, G(z)$$
for all $z$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
At a pole of $\Gamma$ the right side is read as a limit, since $G$ has a zero there.

Example: the recurrence gives $G(2) = \Gamma(1) G(1) = 1$, $G(3) = \Gamma(2) G(2) = 1$, $G(4) = \Gamma(3) G(3) = 2$ and $G(5) = \Gamma(4) G(4) = 12$.
In general $G(n) = \prod_{k=0}^{n-2} k!$ for integers $n \ge 2$.

Non-example: the function $F(z) = G(z) \cdot (1 + \sin^2 \pi z)$ also satisfies $F(1) = 1$ and $F(z+1) = \Gamma(z) F(z)$, but it is not $G$.
So the recurrence alone does not determine $G$; one needs an extra condition, for example that $\frac{d^3}{dx^3} \log G(x) \ge 0$ for $x > 0$.

From the product, $G$ has zeros exactly at $z = -n$ for integers $n \ge 0$, of order $n + 1$.
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# Artin–Schreier theorem on formally real fields

## Statement

Let $F$ be a [[field|0QA8CHH8MG]].
Then $F$ has a total order that makes it an [[ordered field|010JCV9J4E]] if and only if $F$ is [[formally real|2362AFNG8X]].
That is, such an order exists if and only if $-1$ is not a sum of squares in $F$.

## Remarks

One direction is short: in an ordered field every square is nonnegative, so every sum of squares is nonnegative and cannot equal $-1$.
For the converse, let $\Sigma$ be the set of sums of squares of $F$.
If $F$ is formally real, then $\Sigma$ is closed under addition and multiplication and does not contain $-1$.
By Zorn's lemma, there is a maximal subset $P \supseteq \Sigma$ with these properties, and one shows that $P$ is the set of nonnegative elements of an ordering of $F$.
Example: $\Q(\sqrt{2})$ is formally real, and it has two orderings, one with $\sqrt{2} > 0$ and one with $\sqrt{2} < 0$.
Example: $\C$ is not formally real, since $-1 = i^2$, and so $\C$ has no field ordering.

## Explanations

### L1 human

Some fields can be ordered like the real numbers, and some cannot, like the complex numbers.
The theorem gives a simple algebraic test: a field can be ordered exactly when $-1$ is not a sum of squares.
So a purely algebraic condition decides whether an order exists.

### L2 human

The theorem is the starting point of real algebra and of the theory of real closed fields developed by Artin and Schreier.
Its proof shows more: the orderings of a formally real field correspond to the maximal subsets $P$ described above.
The same theory shows that every formally real field has an algebraic extension that is real closed.
Artin used this theory in his solution of Hilbert's seventeenth problem, which asks whether a nonnegative real polynomial is a sum of squares of rational functions.
The [[Artin–Schreier theorem|58P0EGAFVV]] on fields with algebraic closure of finite degree belongs to the same theory.
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# Amitsur–Levitzki theorem

## Statement

Let $K$ be a [[commutative ring|0310REJJY8]] with identity and let $n \ge 1$.
For $m \ge 1$, the *standard polynomial* of degree $m$ is
$$s_m(x_1, \ldots, x_m) = \sum_{\sigma \in S_m} \sgn(\sigma) \, x_{\sigma(1)} x_{\sigma(2)} \cdots x_{\sigma(m)},$$
where $S_m$ is the [[symmetric group|0515A6D2Y1]] and $\sgn$ is the [[sign of a permutation|02DKV69ZH0]].

1. For all [[matrices|0JXZKX526R]] $A_1, \ldots, A_{2n} \in M_n(K)$, one has $s_{2n}(A_1, \ldots, A_{2n}) = 0$.
2. If $K \ne 0$, then $M_n(K)$ satisfies no monic [[polynomial identity|C0EDT81EGH]] of degree less than $2n$.

## Remarks

Part 2 is the easy part.
A monic identity of degree $d < 2n$ can be replaced by a multilinear monic one of the same degree.
One evaluates it on the first $d$ matrix units of the staircase $e_{11}, e_{12}, e_{22}, e_{23}, \ldots, e_{n-1,n}, e_{nn}$.
Only the product in the staircase order is nonzero, so the identity would have to have coefficient $0$ on that monomial, and after renaming variables this contradicts monicity.
For part 1 there are several proofs: the original combinatorial one, a short proof by Rosset that works in the [[exterior algebra|0MDMZ965VE]], and proofs based on the [[Cayley–Hamilton theorem|0F765KFC2J]].

Example: for $n = 1$, part 1 says $ab - ba = 0$ for scalars $a, b$, which is commutativity.
For $n = 2$, it says that the alternating sum of the $24$ products of four $2 \times 2$ matrices in all orders is $0$.

The degree $2n$ is the best possible by part 2: $M_n(K)$ does not satisfy $s_{2n-1}$.

## Explanations

### L1 human

Matrix multiplication is not commutative, but it is not completely free either.
Take any $2n$ matrices of size $n \times n$ and multiply them in all possible orders.
Add the products, each with the sign given by the parity of its order: the result is always the zero matrix.
For $1 \times 1$ matrices this is just $ab = ba$.

### L2 human

The theorem says that $M_n(K)$ is a [[polynomial identity ring|C0EDT81EGH]] with the standard identity $s_{2n}$, and that $2n$ is the minimal degree of an identity.
This fact is the base of the theory of PI rings, since by Kaplansky's and Posner's theorems many PI rings embed in matrix rings over commutative rings or fields.
The minimal degree $2n$ of an identity is then a measure of the size of the matrices needed.
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# Brjuno number

## Definition

Let $\alpha$ be an [[irrational number|0DZXAXEJQ9]].
Let $p_n / q_n$, for $n \ge 0$, be the [[convergents|0D474C7ES1]] of the [[continued fraction|04P2RJ4WPS]] expansion of $\alpha$, with $q_n \ge 1$.
The number $\alpha$ is a *Brjuno number* if
$$\sum_{n = 0}^{\infty} \frac{\log q_{n+1}}{q_n} < \infty.$$

## Remarks

Example: for the golden mean $\alpha = (\sqrt{5} - 1)/2$, the $q_n$ are Fibonacci numbers.
They satisfy $q_{n+1} \le 2 q_n$ and $q_n \ge (3/2)^{n-1}$ for $n \ge 1$.
So the terms are at most $\log(2 q_n) / q_n$, which decay geometrically, and the series converges.
Non-example: choose partial quotients $a_n \ge 1$ inductively with $a_{n+1} \ge e^{q_n}$.
Then $q_{n+1} \ge a_{n+1} q_n \ge e^{q_n}$, so each term is at least $1$ and the series diverges.
Every Diophantine number, and hence almost every real number, is a Brjuno number.
Brjuno proved that if $\alpha$ is a Brjuno number, every germ $f(z) = e^{2\pi i \alpha} z + O(z^2)$ is linearizable at $0$.
Yoccoz proved that for the quadratic germ $e^{2\pi i \alpha} z + z^2$ the condition is also necessary.
So $e^{2\pi i \alpha} z + z^2$ has a [[Siegel disc|KNDYA9T8ZK]] around $0$ exactly when $\alpha$ is a Brjuno number.
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# Successive minima

## Summary

The successive minima of a [[lattice|WN9V7WSJKF]] with respect to a symmetric convex body are $n$ numbers $\lambda_1 \le \dots \le \lambda_n$.
The $k$-th one is the smallest scaling factor at which the scaled body contains $k$ linearly independent lattice vectors.
They measure how long the short vectors of the lattice are, in all independent directions at once.
[[Minkowski's second theorem|5H4XZVF9B9]] bounds their product in terms of the volume of the body and the covolume of the lattice.

## Explanations

### L1 human

Take a grid of points and a symmetric convex shape centred at the origin, such as a disc or a box.
Inflate the shape slowly.
At some moment it first catches a grid point other than the origin; the scaling factor at that moment is $\lambda_1$.
Keep inflating until it holds two grid points that do not lie on one line through the origin; that factor is $\lambda_2$, and so on.
For the integer grid $\Z^2$ and the box $[-1, 1] \times [-2, 2]$, the point $(0, 1)$ enters at factor $1/2$ and the point $(1, 0)$ at factor $1$, so $\lambda_1 = 1/2$ and $\lambda_2 = 1$.
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# Error of best approximation

## Summary

The error of best approximation of $x$ from a set $A$ is the distance from $x$ to $A$, the infimum of $\norm{x - a}$ over $a \in A$.
For a function $f$ and the polynomials of degree at most $n$ it is written $E_n(f)$.
It measures how well $f$ can be approximated at all, whether or not a [[best approximation|7N4988HN3P]] exists.
Its decay as $n \to \infty$ reflects the smoothness of $f$.

## Explanations

### L1 human

It is the smallest gap you can achieve with the tools you are allowed to use.
For $f(x) = x^2$ on $[0, 1]$ and uniform distance, the best constant is $1/2$ with gap $1/2$, the best line has gap $1/8$, and a quadratic gives gap $0$.
Allowing more complicated tools never makes the gap larger, so $E_0(f) \ge E_1(f) \ge E_2(f) \ge \cdots$.
For a continuous function on a closed interval these numbers tend to $0$.
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# Frequentist inference

## Summary

Frequentist inference is the school of [[statistical inference|JZ2S01BHJJ]] in which the unknown parameter is a fixed quantity, not a random one.
A procedure is judged by how it behaves over hypothetical repetitions of the experiment.
That is, it is judged by its [[sampling distribution|656Y8FA3RY]] under each value of the parameter.
Confidence intervals, p-values, significance tests and unbiased estimators are its main tools.
It contrasts with [[Bayesian inference|0JWRJX2RMP]], which puts a probability distribution on the parameter.

## Explanations

### L1 human

A frequentist asks: if I used this method again and again on fresh data, how often would it give the right answer?
A $95\%$ confidence interval is a method that captures the true value in $95\%$ of repetitions, whatever that true value is.
It does not say that the true value lies in today's interval with probability $95\%$; the true value is fixed, and today's interval either contains it or not.
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# Going-down theorem

## Statement

Let $A \subseteq B$ be [[integral domains|0QVD7WBFGN]] such that $A$ is [[integrally closed|H20P600HQS]] and $B$ is [[integral over|B7X0KFY1S1]] $A$.
Let $\mathfrak{p}_1 \supseteq \mathfrak{p}_2$ be [[prime ideals|00MHHZ9SNE]] of $A$, and let $\mathfrak{q}_1$ be a prime ideal of $B$ with $\mathfrak{q}_1 \cap A = \mathfrak{p}_1$.
Then there is a prime ideal $\mathfrak{q}_2$ of $B$ with $\mathfrak{q}_2 \subseteq \mathfrak{q}_1$ and $\mathfrak{q}_2 \cap A = \mathfrak{p}_2$.

## Remarks

The proof shows that $\mathfrak{p}_2 B_{\mathfrak{q}_1} \cap A = \mathfrak{p}_2$; a prime of $B_{\mathfrak{q}_1}$ over $\mathfrak{p}_2$ then exists, and it contracts to the wanted $\mathfrak{q}_2$ in $B$.
The key step uses that $A$ is integrally closed.
If an element of $B$ is integral over the ideal $\mathfrak{p}_2$, then the coefficients of its minimal polynomial over the fraction field of $A$ lie in $\mathfrak{p}_2$.
Here one also uses that $B$ is a domain, so that nonzero elements of $A$ are not zero divisors on $B$.

Example: $\Z \subseteq \Z[i]$ satisfies the hypotheses, and for $\mathfrak{p}_1 = 5\Z \supseteq \mathfrak{p}_2 = 0$ and $\mathfrak{q}_1 = (2 + i)$, the prime $\mathfrak{q}_2 = 0$ works.

The hypothesis that $A$ is integrally closed cannot be dropped: there are integral extensions of domains in which going-down fails.
A different sufficient condition is flatness: going-down holds for every flat ring extension.

## Explanations

### L1 human

Going-down is the reverse of [[going-up|GHKYR5PBQC]]: given a prime upstairs, one wants to descend along a chain of smaller primes downstairs.
Unlike going-up, this needs the small ring to be integrally closed.

### L2 human

Going-down is used to prove that heights behave well in integral extensions over a normal base: $\operatorname{ht} \mathfrak{q} = \operatorname{ht}(\mathfrak{q} \cap A)$ for every prime $\mathfrak{q}$ of $B$.
It is a key ingredient in dimension theory of finitely generated algebras over a field, for example to show that such domains are catenary.
In geometry it gives that a finite surjective map from an irreducible variety onto a normal variety is an open map.
</document>

<document source="https://tamnd.github.io/math2/node/YE5439NH1T.md" tag="YE5439NH1T" kind="statement">
# Mazur–Orlicz theorem

## Statement

Let $A = (a_{nk})$ and $B = (b_{nk})$ be [[regular|29C7C41A52]] [[matrix summability methods|M6311DWJ1E]].
A sequence $x = (x_k)$ is $A$-summable if each series $(Ax)_n = \sum_{k} a_{nk} x_k$ converges and the limit $\lim_A x = \lim_{n \to \infty} (Ax)_n$ exists; the same holds for $B$.
Assume that every [[bounded sequence|05FQ3VGAR9]] that is $A$-summable is also $B$-summable.
Then $\lim_A x = \lim_B x$ for every bounded $A$-summable sequence $x$.

## Remarks

The result is also called the bounded consistency theorem: inclusion of the bounded summability fields forces [[consistency|W3BVN1HDXW]] on them.
Its proof is functional-analytic.
It treats the bounded $A$-summable sequences as a space of sequences with the supremum norm.
On this space it studies the linear functional $\lim_B - \lim_A$, which vanishes on convergent sequences because both methods are regular.

Example: every bounded sequence that is Cesàro summable of order $1$ is Cesàro summable of order $2$.
The theorem then says at once that the two Cesàro limits agree on such sequences.

The conclusion is only about bounded sequences.
The theorem says nothing about unbounded sequences in the summability field of $A$.

## Explanations

### L1 human

A matrix method turns a sequence into a new sequence of weighted averages and takes its limit.
Suppose one such method can handle every bounded sequence that a second method can handle.
The theorem says that then the two methods also give the same answer on those sequences.
So, for bounded sequences, "summing more" never means "summing differently".

### L2 human

The theorem belongs to the functional-analytic side of summability theory, where summability fields are studied as sequence spaces.
It complements the [[Silverman–Toeplitz theorem|S4J7SRPXMZ]], which characterizes regular matrices.
Inclusion theorems between classical methods, such as Cesàro and Hölder methods, become consistency theorems on bounded sequences through it.
</document>

<document source="https://tamnd.github.io/math2/node/YE5F548WMS.md" tag="YE5F548WMS" kind="statement">
# Arnold's classification of simple singularities

## Statement

Let $n \ge 1$ and let $\mathcal{O}_{n+1} = \C\{x, y, z_2, \dots, z_n\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]] in $n + 1$ variables.
Two germs $f, g \in \mathcal{O}_{n+1}$ are *right equivalent* if $g = f \circ \phi$ for a biholomorphic germ $\phi$ of $(\C^{n+1}, 0)$.
Let $f$ be a germ with $f(0) = 0$ and an [[isolated singularity|VTGYEES3VS]] at $0$.
It is *simple* if, for $k$ large, a neighbourhood of its $k$-jet in the space of $k$-jets meets only finitely many right equivalence classes.
Write $Q = z_2^2 + \cdots + z_n^2$.
Then $f$ is simple if and only if it is right equivalent to exactly one germ of the following list:

1. $A_k$: $x^{k+1} + y^2 + Q$, for $k \ge 1$;
2. $D_k$: $x^2 y + y^{k-1} + Q$, for $k \ge 4$;
3. $E_6$: $x^3 + y^4 + Q$;
4. $E_7$: $x^3 + x y^3 + Q$;
5. $E_8$: $x^3 + y^5 + Q$.

## Remarks

Vladimir Arnold proved the classification in 1972.
In each case the index is the [[Milnor number|1ZE51P7GRQ]]: $\mu(A_k) = \mu(D_k) = k$ and $\mu(E_k) = k$.
The proof uses finite determinacy to reduce to polynomials and the splitting lemma to remove the nondegenerate quadratic part $Q$.
Then a case analysis of the low-degree terms of the remaining function of one or two variables finishes it.
A function of two variables whose $3$-jet is zero is never simple: it lies near the family $x^4 + t x^2 y^2 + y^4$, whose cross-ratio of the four lines is a continuous modulus.

Example: $f = x^3 + y^3$ has $\mu = 4$ and is right equivalent to $D_4$, since its cubic part has three distinct linear factors, as does $x^2 y + y^3 = y(x^2 + y^2)$.

## Explanations

### L1 human

Most critical points of functions are simple, like the bottom of a bowl.
Arnold asked which singular functions cannot be continuously deformed into infinitely many inequivalent nearby types.
The answer is a short list, labelled by letters $A$, $D$, $E$ and an index.
The same letters label the simply-laced Dynkin diagrams of Lie theory, and this is no accident.

### L2 human

The simple singularities appear in many guises.
For $n = 2$, they are the [[ADE singularities|HQNXQBZ6C7]], the surface singularities $\C^2 / G$ for finite subgroups $G$ of $\mathrm{SL}_2(\C)$, also called Kleinian or Du Val singularities.
For $n = 2$, the intersection form on $H_2$ of the Milnor fibre is, up to sign, the Cartan matrix of the Dynkin diagram of the same name.
Arnold continued with the unimodal and bimodal singularities, which include the $14$ exceptional unimodal singularities.
The elementary catastrophes of [[catastrophe theory|6FKJ6SXP3G]] are real forms of simple singularities of small Milnor number.
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<document source="https://tamnd.github.io/math2/node/YE6WZR46RQ.md" tag="YE6WZR46RQ" kind="concept">
# Stationary set

## Summary

A stationary set in a regular uncountable cardinal $\kappa$ is a subset of $\kappa$ that meets every [[club set|YW8YSCFPG5]].
If club sets are the "almost all" sets, stationary sets are the sets that are "not negligible".
They are central in combinatorial set theory, through [[Fodor's lemma|55W1M20Q3F]], the [[diamond principle|HFQSSB1Z23]] and forcing axioms; the [[precise definition|6FDYJVG1CK]] is given separately.

## Explanations

### L1 human

Among the countable ordinals, the club sets are those that are "almost everything".
A stationary set is one that you cannot avoid with a club set: whichever club set you pick, the two share a point.
Every club set is stationary.
One level up, the set of all ordinals of cofinality $\omega$ below $\omega_2$ is stationary in $\omega_2$.
A bounded set, for instance all ordinals below $\omega \cdot 2$, is not stationary in $\omega_1$, because the ordinals above it form a club set that misses it.
</document>

<document source="https://tamnd.github.io/math2/node/YE71ZHHGKW.md" tag="YE71ZHHGKW" kind="definition">
# Chebyshev constant

## Definition

Let $K$ be a [[compact set|0R6EWFKRCF]] in $\C$ with at least one point.
For a polynomial $p$, let $\norm{p}_K = \max_{z \in K} \abs{p(z)}$.
For an integer $n \ge 1$, let
$$t_n(K) = \inf \set{ \norm{p}_K : p \text{ a monic polynomial of degree } n }.$$
The *Chebyshev constant* of $K$ is $\tau(K) = \lim_{n \to \infty} t_n(K)^{1/n}$.

## Remarks

The infimum is attained, and a [[monic polynomial|0C3QBCR46M]] $p$ of degree $n$ with $\norm{p}_K = t_n(K)$ is a Chebyshev polynomial of $K$.
Since a product of monic polynomials is monic, $t_{m+n}(K) \le t_m(K) \, t_n(K)$.
So the sequence $\log t_n(K)$ is subadditive, and [[Fekete's lemma|8A4GGX1MEY]] shows that the limit exists and equals $\inf_n t_n(K)^{1/n}$.

Example: let $K$ be the unit circle.
For a monic $p(z) = z^n + a_{n-1} z^{n-1} + \cdots + a_0$, Parseval's identity gives $\frac{1}{2\pi} \int_0^{2\pi} \abs{p(e^{i\theta})}^2 \, d\theta = 1 + \sum_k \abs{a_k}^2 \ge 1$.
So $\norm{p}_K \ge 1$, with equality for $p = z^n$, and $t_n(K) = 1$ and $\tau(K) = 1$.

Non-example: the polynomial $2 z^n$ has maximum $2$ on the unit circle, but it is not monic, so it does not enter the infimum.
Without the monic condition the infimum would be $0$.

If $K$ is a finite set with $m$ points, then $p(z) = \prod_{a \in K} (z - a)$ vanishes on $K$, so $t_n(K) = 0$ for $n \ge m$ and $\tau(K) = 0$.
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<document source="https://tamnd.github.io/math2/node/YE7H8DGDPY.md" tag="YE7H8DGDPY" kind="example">
# Hill differential equation

## Example

Let $T > 0$ and let $f : \R \to \R$ be continuous and $T$-periodic.
*Hill's equation* is the second-order [[linear differential equation|0PB7K8M7FG]]
$$y'' + f(t)\, y = 0.$$
The Mathieu equation $y'' + (a - 2q \cos 2t)\, y = 0$, with real constants $a$ and $q$, is the special case $f(t) = a - 2q\cos 2t$, of period $\pi$.

Floquet analysis: let $y_1, y_2$ be the solutions with $y_1(0) = 1$, $y_1'(0) = 0$ and $y_2(0) = 0$, $y_2'(0) = 1$.
The monodromy matrix of the equivalent system for $(y, y')$ is $M$ with columns $(y_1(T), y_1'(T))$ and $(y_2(T), y_2'(T))$.
The system matrix has trace $0$, so by [[Liouville's formula|09PS3S0EF6]] $\det M = 1$.
Hence the multipliers are the roots of $\mu^2 - \Delta \mu + 1 = 0$, where $\Delta = y_1(T) + y_2'(T)$ is the discriminant.

1. If $\abs{\Delta} < 2$, the multipliers are two distinct conjugate numbers of modulus $1$, and all solutions are bounded.
2. If $\abs{\Delta} > 2$, there is a real multiplier of modulus greater than $1$, and some solution is unbounded.

Check with constant $f = \omega^2$, $\omega > 0$, viewed as $T$-periodic: $y_1 = \cos \omega t$ and $y_2 = \sin(\omega t)/\omega$, so $\Delta = 2\cos \omega T$ and $\abs{\Delta} \le 2$, matching the bounded oscillations of the [[harmonic oscillator|0YG0ZJPTWG]].

## Explanations

### L1 human

Hill's equation describes an oscillator whose stiffness changes periodically in time.
Depending on the parameters, the oscillation stays bounded or grows without limit; the growth is called parametric resonance.
A single number $\Delta$, computed from two solutions over one period, decides which case occurs.
George William Hill studied the equation for the motion of the Moon, and it also describes particles in periodic potentials.
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<document source="https://tamnd.github.io/math2/node/YE87R5CDJY.md" tag="YE87R5CDJY" kind="statement">
# Bieberbach theorems

## Statement

Let $n \ge 1$, and let crystallographic groups be as in [[crystallographic group|7KTJKFQT2N]].

1. Let $\Gamma$ be a crystallographic group of dimension $n$ and let $T$ be the subgroup of [[translations|087VJM5Q77]] in $\Gamma$. Then $T$ is a [[free abelian group|0WJX8MGG37]] of rank $n$ whose elements span $\R^n$, and $T$ is a [[normal subgroup|0CB0DX5BVV]] of finite [[index|0DTXBQPA2S]] in $\Gamma$.
2. Let $\Gamma_1$ and $\Gamma_2$ be crystallographic groups of dimension $n$. If $\Gamma_1$ and $\Gamma_2$ are [[isomorphic|08MG7H8JW5]] as abstract groups, then there is a bijective [[affine map|0JF0TVYGWT]] $\alpha \colon \R^n \to \R^n$ with $\Gamma_2 = \alpha \Gamma_1 \alpha^{-1}$.
3. For each $n$, there are only finitely many crystallographic groups of dimension $n$ up to isomorphism.

## Remarks

The proof of part 1 uses a commutator estimate.
If two isometries have linear parts close to the identity, their commutator has a linear part closer still.
Discreteness then forces enough elements to be pure translations.
Part 3 follows from part 1, because the finite quotient $\Gamma / T$ acts faithfully on $T \cong \Z^n$.
There are finitely many finite subgroups of $\mathrm{GL}_n(\Z)$ up to conjugacy, and each gives finitely many extensions.

Example: in dimension $1$ there are two crystallographic groups up to isomorphism.
They are the translations $x \mapsto x + k$ with $k \in \Z$, and the infinite dihedral group of all maps $x \mapsto \pm x + k$ with $k \in \Z$.
In both, the translation subgroup is $\Z$, of index $1$ and $2$.

The compactness of the quotient is necessary.
The translations $(x, y) \mapsto (x + k, y)$ with $k \in \Z$ form a discrete group of isometries of $\R^2$ whose translations span only a line.

## Explanations

### L1 human

Every crystal pattern in $n$-dimensional space repeats along $n$ independent directions.
Its symmetry group is mostly made of translations, with only finitely many kinds of extra rotations and reflections on top.
Two such symmetry groups that look the same as abstract groups really are the same, after a change of coordinates.
And in each dimension the list of possible crystal symmetry groups is finite.

### L2 human

Part 3 is the reason why there are exactly $17$ [[wallpaper groups|MY3CBHARYQ]] in the plane and $219$ space groups in dimension $3$ up to affine conjugacy.
A torsion-free crystallographic group acts freely on $\R^n$, and its quotient is a compact flat Riemannian manifold.
So the theorems say that each compact flat manifold is finitely covered by a torus, and that there are finitely many of them in each dimension up to affine equivalence.
The argument behind part 1 is close to the proof of the [[Jordan–Schur theorem|XT6EEXPD4P]] on finite groups of matrices.
</document>

<document source="https://tamnd.github.io/math2/node/YEBPTXZ1FS.md" tag="YEBPTXZ1FS" kind="statement">
# Sturm separation theorem

## Statement

Let $I \subseteq \R$ be an interval and let $p, q \colon I \to \R$ be continuous.
Let $y_1$ and $y_2$ be linearly independent real solutions of the [[linear differential equation|0QQRP17Z7S]]
$$y'' + p(x) y' + q(x) y = 0 \quad \text{on } I.$$
Then the zeros of $y_1$ and $y_2$ in $I$ strictly alternate.
That is, $y_1$ and $y_2$ have no common zero.
Moreover, if $x_1 < x_2$ are two consecutive zeros of $y_1$ in $I$, then $y_2$ has exactly one zero in the open interval $(x_1, x_2)$.

## Remarks

The [[Wronskian|0KC1NMG90B]] $W = y_1 y_2' - y_1' y_2$ of two independent solutions never vanishes, so it has a constant sign on $I$.
A common zero would make $W$ vanish there.
If $y_2$ had no zero in $(x_1, x_2)$, the function $y_1 / y_2$ would be differentiable on $[x_1, x_2]$ with derivative $-W / y_2^2$ of constant sign.
Yet it vanishes at $x_1$ and $x_2$, which contradicts Rolle's theorem.
Applying the same argument with the roles exchanged shows that there is at most one zero.
Example: for $y'' + y = 0$, the solutions $\sin x$ and $\cos x$ have zeros at $k\pi$ and at $\pi/2 + k\pi$, which alternate.
Linear independence is needed: $y_1 = \sin x$ and $y_2 = 2 \sin x$ have the same zeros.

## Explanations

### L1 human

Take two genuinely different solutions of a second-order linear equation.
Between two crossings of the axis by one of them, the other crosses the axis exactly once.
So the zeros of the two solutions interlace like the zeros of sine and cosine.

### L2 human

The theorem shows that oscillation is a property of the equation and not of the solution.
If one nonzero solution has infinitely many zeros on an interval, then every nonzero solution does.
It also shows that the numbers of zeros of two solutions on an interval differ by at most one.
For equations $y'' + q y = 0$ it also follows from the [[Sturm comparison theorem|W7VXNYZQN2]] with $q_1 = q_2$, and it is a basic tool in Sturm–Liouville theory.
</document>

<document source="https://tamnd.github.io/math2/node/YEFR2GGGJN.md" tag="YEFR2GGGJN" kind="definition">
# Random dynamical system

## Definition

Let $\mathbb{T}$ be $\Z$, $\N$, $\R$ or $[0, \infty)$, and let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]].
Let $(\theta_t)_{t \in \mathbb{T}}$ be a family of [[measure-preserving maps|556WBDAHNP]] $\theta_t \colon \Omega \to \Omega$ such that $(t, \omega) \mapsto \theta_t \omega$ is measurable, $\theta_0 = \id$ and $\theta_{s + t} = \theta_s \circ \theta_t$ for all $s, t \in \mathbb{T}$.
Let $X$ be a measurable space.
A *random dynamical system* on $X$ over $\theta$ is a measurable map $\varphi \colon \mathbb{T} \times \Omega \times X \to X$, written $\varphi(t, \omega) = \varphi(t, \omega, \cdot)$, such that for all $\omega \in \Omega$:

1. $\varphi(0, \omega) = \id_X$;
2. $\varphi(s + t, \omega) = \varphi(t, \theta_s \omega) \circ \varphi(s, \omega)$ for all $s, t \in \mathbb{T}$.

## Remarks

Clause 2 is the cocycle property.
When $\Omega$ has one point, $\varphi$ is an ordinary [[dynamical system|BBZP1R872N]].
Often $X$ is a topological space and each $\varphi(t, \omega)$ is required to be continuous.

Example: let $f_0, f_1 \colon X \to X$ be measurable maps.
Let $\Omega = \set{0, 1}^{\Z}$ with the product of the measures giving weight $\frac{1}{2}$ to $0$ and to $1$, and let $\theta_t$ be the $t$-th power of the left shift, for $t \in \N$.
Put $\varphi(0, \omega) = \id_X$ and $\varphi(n, \omega) = f_{\omega_{n-1}} \circ \cdots \circ f_{\omega_1} \circ f_{\omega_0}$ for $n \ge 1$.
Then $\varphi(m + n, \omega)$ applies $f_{\omega_0}, \ldots, f_{\omega_{m-1}}$ and then $f_{\omega_m}, \ldots, f_{\omega_{m+n-1}}$, and the second block is $\varphi(n, \theta_m \omega)$, so clause 2 holds.

Non-example: with the same data, $\psi(n, \omega) = f_{\omega_0}^{\,n}$ fails clause 2 when $f_0^{\,2} \ne f_1 \circ f_0$.
Indeed, for $\omega_0 = 0$ and $\omega_1 = 1$, $\psi(2, \omega) = f_0 \circ f_0$, while $\psi(1, \theta_1 \omega) \circ \psi(1, \omega) = f_1 \circ f_0$.

Stochastic differential equations with suitable coefficients generate random dynamical systems with $\Omega$ the space of Brownian paths and $\theta_t$ the shift of paths.
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<document source="https://tamnd.github.io/math2/node/YEJVKCY78E.md" tag="YEJVKCY78E" kind="counterexample">
# Nagata's example of a Noetherian ring of infinite dimension

## Counterexample

Let $k$ be a field and let $A = k[x_1, x_2, x_3, \ldots]$ be the [[polynomial ring|0SDC934R52]] in countably many variables.
Split the variables into consecutive blocks $B_1, B_2, B_3, \ldots$ where the block $B_i$ has exactly $i$ variables.
Let $\mathfrak{p}_i$ be the [[prime ideal|00MHHZ9SNE]] of $A$ generated by the variables of $B_i$.
Let $S = A \setminus \bigcup_{i \ge 1} \mathfrak{p}_i$, a [[multiplicative set|AN9SKEZ3VY]], and let $R = S^{-1} A$ be the [[localization|4C1AE84N5N]].
Then $R$ is a [[Noetherian ring|0670T59HGE]], but its [[Krull dimension|MVGBS3J1P7]] is infinite.

## Remarks

The maximal ideals of $R$ are exactly the ideals $\mathfrak{p}_i R$.
The localization of $R$ at $\mathfrak{p}_i R$ is a localization of a polynomial ring in $i$ variables over the field generated by the other variables.
So it is Noetherian of dimension $i$.
Each nonzero element of $R$ involves finitely many variables, so it lies in only finitely many maximal ideals.
A ring whose localizations at maximal ideals are all Noetherian and whose nonzero elements lie in finitely many maximal ideals is Noetherian.
The [[height|87WH65FJB8]] of $\mathfrak{p}_i R$ is $i$, so the dimension of $R$ is at least $i$ for every $i$.
So the statement "a Noetherian ring has finite Krull dimension" is false, although every Noetherian local ring has finite dimension.

## Explanations

### L1 human

In a Noetherian ring every ascending chain of ideals stops, so one might expect the length of chains of prime ideals to be bounded.
Each single chain of primes is finite here, but the chains can be as long as one likes.
The ring is glued from local pieces of dimension $1, 2, 3, \ldots$, each of them well behaved, and there is no common bound.
</document>

<document source="https://tamnd.github.io/math2/node/YENN31BFZA.md" tag="YENN31BFZA" kind="definition">
# von Neumann bounded set

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a [[topological vector space|T32W553W95]] over $\mathbb{K}$.
A subset $B \subseteq E$ is *von Neumann bounded*, or *bounded*, if the following holds.
For every [[neighbourhood|0ZCY0Z80B6]] $U$ of $0$ in $E$ there is $s > 0$ such that $B \subseteq t U$ for every $t \in \mathbb{K}$ with $\abs{t} > s$.

## Remarks

Since the balanced neighbourhoods of $0$ form a neighbourhood base, an equivalent condition is: for every neighbourhood $U$ of $0$ there is $t > 0$ with $B \subseteq t U$.
If the topology of $E$ is generated by a family $\mathcal{P}$ of seminorms, then $B$ is bounded if and only if $\sup_{x \in B} p(x) < \infty$ for every $p \in \mathcal{P}$.
Example: in a normed space, the closed unit ball $B$ is bounded; a neighbourhood $U$ of $0$ contains a ball of some radius $\varepsilon > 0$, and then $B \subseteq tU$ for all $\abs{t} > 1 / \varepsilon$.
Non-example: in $\R^2$ with the Euclidean norm, the line $L = \R \times \set{0}$ is not bounded; for $U$ the open unit disc, $tU$ is the disc of radius $\abs{t}$, which never contains $L$.
Subsets of bounded sets, finite unions of bounded sets, closures of bounded sets and compact sets are bounded.
</document>

<document source="https://tamnd.github.io/math2/node/YES06WTNRZ.md" tag="YES06WTNRZ" kind="definition">
# Schläfli symbol

## Definition

The *Schläfli symbol* of a convex [[regular polytope|F60PD912D5]] $P$ of dimension $d \ge 2$ is defined by induction on $d$.

1. If $d = 2$, then $P$ is a [[regular polygon|01KJJ9A400]] with $p$ sides for some $p \ge 3$, and its symbol is $\set{p}$.
2. If $d \ge 3$, then all [[facets|DPDXN2FJC4]] of $P$ are congruent regular polytopes of dimension $d-1$, with a common symbol $\set{p_1, \dots, p_{d-2}}$.
Let $p_{d-1}$ be the number of facets of $P$ that contain a given [[face|KY7SB854VH]] of dimension $d-3$.
By regularity it does not depend on the face.
The symbol of $P$ is $\set{p_1, \dots, p_{d-2}, p_{d-1}}$.

## Remarks

Example: the cube has square facets, with symbol $\set{4}$, and exactly $3$ facets contain each vertex, so its symbol is $\set{4, 3}$.
The tesseract has cubes as facets and $3$ cubes around each edge, so its symbol is $\set{4, 3, 3}$.
The regular $d$-simplex has symbol $\set{3, 3, \dots, 3}$ with $d-1$ entries.

Non-example: the string $\set{6, 3}$ is not the symbol of a convex polyhedron.
Three regular hexagons at a vertex have angles summing to $3 \cdot 120^\circ = 360^\circ$, so they lie flat, and the pattern gives the hexagonal tiling of the plane instead of a polytope.

If $P$ has symbol $\set{p_1, \dots, p_{d-1}}$, its vertex figure has symbol $\set{p_2, \dots, p_{d-1}}$, and its dual polytope has symbol $\set{p_{d-1}, \dots, p_1}$.
The same notation is used for regular tilings, such as $\set{4, 4}$ for the square tiling, and for star polytopes with fractional entries, such as $\set{5/2}$ for the pentagram.
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<document source="https://tamnd.github.io/math2/node/YEYERRY37K.md" tag="YEYERRY37K" kind="concept">
# HNN extension

## Summary

An HNN extension of a [[group|G5S1M9]] $G$ adds one new element $t$.
This element conjugates a [[subgroup|0X1JXT47GA]] $H$ of $G$ onto another subgroup $K$ by a given [[isomorphism|08MG7H8JW5]].
The group $G$ embeds in the new group, so two isomorphic subgroups of $G$ become conjugate in a larger group.
The construction is named after Graham Higman, Bernhard Neumann and Hanna Neumann.
Together with the [[free product|TM7QS7P7X9]] with amalgamation, it is a basic way to build groups from smaller ones.
It is used in the proof of the [[Higman embedding theorem|P7TZ694FCM]].

## Explanations

### L1 human

Start with a group $G$ in which two subgroups $H$ and $K$ look alike, through an isomorphism $\alpha$ from $H$ to $K$.
Inside $G$ there may be no element that turns $H$ into $K$ by conjugation.
The HNN extension adds a new letter $t$ and the rules $t h t^{-1} = \alpha(h)$, and nothing else.
For example, take $G = \Z$ written as the powers of $a$, take $H = \langle a \rangle$, $K = \langle a^2 \rangle$ and $\alpha(a) = a^2$.
The HNN extension is $\langle a, t \mid t a t^{-1} = a^2 \rangle$, the [[Baumslag–Solitar group|429JJ5ZNBY]] $BS(1, 2)$.
</document>

<document source="https://tamnd.github.io/math2/node/YF1KAWB5MB.md" tag="YF1KAWB5MB" kind="concept">
# Quasi-analytic function

## Summary

A quasi-analytic class is a class of [[smooth functions|0V33BNKZ9S]] in which a function is determined by its derivatives at a single point.
Real [[analytic functions|081WKEDK6C]] form such a class, but there are larger ones.
Quasi-analytic classes are defined by bounds on the growth of derivatives.
The [[Denjoy–Carleman theorem|KV0GW9K7JS]] says exactly which bounds give a quasi-analytic class.

## Explanations

### L1 human

For an analytic function, the derivatives at one point fix the function everywhere.
For general smooth functions this fails: a smooth function can vanish to all orders at a point and still be nonzero.
Quasi-analytic functions keep the good property of analytic functions without being analytic.
They are smooth functions whose derivatives do not grow too fast.
</document>

<document source="https://tamnd.github.io/math2/node/YF1SE8WJK3.md" tag="YF1SE8WJK3" kind="concept">
# Unirational variety

## Summary

A unirational variety is a variety that is dominated by projective space through a rational map.
Its points can be parametrised by rational functions, but the parametrisation may cover each point several times.
Every [[rational variety|XV4K9GREJ9]] is unirational.
Whether the converse holds is the Lüroth problem.
It holds for curves, and for surfaces over an algebraically closed field of characteristic $0$, but it fails for some threefolds over $\C$.

## Explanations

### L1 human

A rational variety has a parametrisation that is one-to-one on a dense open set.
A unirational variety only needs a parametrisation that covers a dense open set, possibly many times over.
For curves the two notions agree, by [[Lüroth's theorem|AVCZ6HVBA7]].
Over $\C$, a smooth cubic threefold in $\mathbb{P}^4$ is unirational but not rational, by a theorem of Clemens and Griffiths.
</document>

<document source="https://tamnd.github.io/math2/node/YF1XA47RYT.md" tag="YF1XA47RYT" kind="definition">
# Robin constant

## Definition

Let $K$ be a [[compact set|0R6EWFKRCF]] in $\C$.
For a Borel [[probability measure|059TJN0NZ1]] $\mu$ on $K$, let
$$I(\mu) = \int_K \int_K \log \frac{1}{\abs{z - w}} \, d\mu(z) \, d\mu(w)$$
be its logarithmic [[energy|SFSRQCM3EG]], with values in $(-\infty, \infty]$.
The *Robin constant* of $K$ is
$$V(K) = \inf \set{ I(\mu) : \mu \text{ a Borel probability measure on } K },$$
with $V(K) = \infty$ when $I(\mu) = \infty$ for every such $\mu$.

## Remarks

The logarithmic capacity of $K$ is $\operatorname{cap}(K) = e^{-V(K)}$.
If $V(K) < \infty$ and $\Omega$ is the unbounded component of $\hat\C \setminus K$, the Green function of $\Omega$ with pole at infinity satisfies $g_\Omega(z, \infty) = \log \abs{z} + V(K) + o(1)$ as $z \to \infty$.

Example: let $K$ be the circle of radius $r$ centred at $0$ and let $\mu$ be normalized arc length on it.
For $\abs{z} = r$ the mean value of $\log(1/\abs{z - w})$ over the circle is $-\log r$, so $I(\mu) = -\log r$.
This $\mu$ is the minimizer, so $V(K) = -\log r$.
The Green function $\log(\abs{z}/r)$ of the outside of the circle agrees with the expansion above.

Non-example of a finite value: for $K = \set{0}$ the only probability measure is $\delta_0$, and $I(\delta_0) = \log(1/0) = \infty$, so $V(K) = \infty$.

Some authors use the opposite sign or the energy kernel $\log \abs{z - w}$, so that the Robin constant becomes $\log \operatorname{cap}(K)$.
</document>

<document source="https://tamnd.github.io/math2/node/YF4XH2BEG5.md" tag="YF4XH2BEG5" kind="concept">
# Limit set

## Summary

The $\omega$-limit set of a point is the set of states that its solution approaches again and again as time goes to $+\infty$.
The $\alpha$-limit set is the same notion for time going to $-\infty$.
Limit sets describe the long-time behaviour of a dynamical system: they can be equilibria, [[periodic orbits|0AA97XA70K]] or more complicated sets.
They are closed [[invariant sets|WWZTW11664]], and for a bounded forward orbit the $\omega$-limit set is nonempty, compact and connected.

## Explanations

### L1 human

Run the system for a long time and watch where the state keeps returning.
Those places form the $\omega$-limit set.
For the equation $x' = -x$, every solution tends to $0$, so the $\omega$-limit set of every point is $\set{0}$.
For a system whose solutions spiral onto the unit circle, the $\omega$-limit set of every point except the origin is the whole circle.
It is the "final destination" of the motion, even when the motion never stops.
</document>

<document source="https://tamnd.github.io/math2/node/YF7YFQ58QJ.md" tag="YF7YFQ58QJ" kind="concept">
# Excellent ring

## Summary

An excellent ring is a [[Noetherian ring|0670T59HGE]] that satisfies three extra conditions on chains of primes, on completions and on regular loci.
The class was introduced by Grothendieck to collect the Noetherian rings that behave as well as the rings met in geometry and number theory.
Fields, the integers, complete Noetherian local rings and all finitely generated algebras over them are excellent.
For excellent rings, properties such as being reduced or normal pass from a local ring to its completion.
Also, the integral closure of a reduced excellent ring is a finite module over it.

## Explanations

### L1 human

Noetherian rings can be strange: some have maximal chains of primes of different lengths between the same two primes, and some local domains have a completion with nilpotent elements.
Excellent rings exclude these pathologies.
Almost every ring that appears in practice, such as a polynomial ring over a field or a quotient of it, is excellent.
So the word "excellent" marks the rings for which the usual geometric intuition works.
</document>

<document source="https://tamnd.github.io/math2/node/YF9F9EG6M3.md" tag="YF9F9EG6M3" kind="concept">
# Zero morphism

## Summary

A zero morphism is a distinguished [[morphism|859F0TFVA3]] between two objects that behaves like multiplication by zero.
Composing it with any morphism on either side gives again a zero morphism.
In a category with a [[zero object|AQET9YHH6F]], the zero morphism $X \to Y$ is the composite $X \to 0 \to Y$.
Zero morphisms are needed to define kernels and cokernels, and they make sense of the statement that a composite is zero.

## Explanations

### L1 human

In the category of groups, the zero morphism from $G$ to $H$ is the homomorphism that sends every element of $G$ to the identity of $H$.
It is the "do nothing useful" map, and it exists between any two groups.
If you follow it with any homomorphism, or precede it with any homomorphism, you still send everything to the identity.
For linear maps between vector spaces, the zero morphism is the zero map $v \mapsto 0$.
</document>

<document source="https://tamnd.github.io/math2/node/YFAKVC0WEJ.md" tag="YFAKVC0WEJ" kind="definition">
# Generalized cohomology theory

## Definition

Consider pairs of spaces $(X, A)$ and maps of pairs, as in the [[Eilenberg–Steenrod axioms|YHK51GHK2T]], and write $X$ for $(X, \emptyset)$.
A *generalized cohomology theory* consists of contravariant [[functors|28NQ0KRHD3]] $h^n$, for $n \in \Z$, from pairs of spaces to [[abelian groups|082GHNPJ53]], and [[natural transformations|KYCPCC57FT]] $\delta \colon h^n(A) \to h^{n+1}(X, A)$, such that:

1. (Homotopy) [[Homotopic|F2F143KQ2M]] maps of pairs induce the same homomorphisms.
2. (Exactness) For each pair, the sequence $\cdots \to h^n(X, A) \to h^n(X) \to h^n(A) \xrightarrow{\delta} h^{n+1}(X, A) \to \cdots$, with the unnamed maps induced by inclusions, is [[exact|JB7ZBFBA9C]].
3. (Excision) If $Z \subseteq A$ and the closure of $Z$ lies in the interior of $A$, then the inclusion $(X \setminus Z, A \setminus Z) \to (X, A)$ induces isomorphisms $h^n(X, A) \to h^n(X \setminus Z, A \setminus Z)$.
4. (Additivity) If $X$ is the disjoint union of spaces $X_\alpha$, the inclusions induce an isomorphism $h^n(X) \to \prod_\alpha h^n(X_\alpha)$.

No condition is put on the groups $h^n(P)$ of a one-point space $P$.

## Remarks

Often the axioms are only asked for CW pairs, or for pairs of spaces with the homotopy type of CW pairs.
A *generalized homology theory* is defined in the same way with covariant functors.

Example: [[singular cohomology|6N7BC70CX0]] $H^n(-; G)$ with coefficients in an abelian group $G$ is a generalized cohomology theory, and it also satisfies the dimension axiom $H^n(P; G) = 0$ for $n \ne 0$.

Example: complex topological K-theory is a generalized cohomology theory with $K^n(P) \cong \Z$ for $n$ even and $K^n(P) = 0$ for $n$ odd, by Bott periodicity; it is not an ordinary theory.

Non-example: the homotopy groups $\pi_n(X, A, x_0)$ of pointed pairs satisfy a homotopy axiom and have long exact sequences, but they fail excision, so they do not form a generalized homology theory.
</document>

<document source="https://tamnd.github.io/math2/node/YFAQGE2ATQ.md" tag="YFAQGE2ATQ" kind="statement">
# Isomorphism theorem for measure algebras

## Statement

Let $\mathfrak{A}$ be a [[Boolean algebra|0XADJKM699]] in which every countable subset has a supremum, and let $m \colon \mathfrak{A} \to [0, 1]$.
The pair $(\mathfrak{A}, m)$ is a *probability measure algebra* if three conditions hold.

1. $m(1) = 1$, and $m(a) = 0$ only for $a = 0$.
2. $m\left(\bigvee_k a_k\right) = \sum_k m(a_k)$ for every sequence of pairwise disjoint elements $a_k$.
3. The set $\mathfrak{A}$ is separable for the metric $d(a, b) = m(a \mathbin{\triangle} b)$.

Assume also that $(\mathfrak{A}, m)$ is atomless: every $a \ne 0$ has some $b \le a$ with $0 < m(b) < m(a)$.
Let $\mathfrak{L}$ be the algebra of [[Lebesgue measurable|2Z2QVWZ262]] subsets of $[0,1]$ modulo [[null sets|VQQM4Y550T]], with $\lambda$ induced by Lebesgue measure.
Then there is a Boolean algebra isomorphism $\Phi \colon \mathfrak{A} \to \mathfrak{L}$ with $\lambda(\Phi(a)) = m(a)$ for all $a \in \mathfrak{A}$.

## Remarks

Idea of proof: take a dense sequence in $\mathfrak{A}$ and the finite partitions of $1$ that it generates.
Since the algebra is atomless, $m$ takes all values between $0$ and $m(a)$ on elements below $a$.
This lets one send each partition to a partition of $[0,1]$ into finite unions of intervals with the same measures, compatibly with refinement.
The map then extends by continuity in the metric $d$.
Without atomlessness the result fails: the algebra with two atoms of measure $1/2$ has four elements, while $\mathfrak{L}$ is infinite.
Without separability it fails as well: the measure algebra of the product measure on $\set{0,1}^I$ with $I$ uncountable is atomless but not separable.
As a consequence, the measure algebra of every atomless [[standard probability space|QDDRDW9PXS]] is isomorphic to that of $[0,1]$.

## Explanations

### L1 human

Ignore sets of measure zero.
Then there is essentially only one way to spread probability evenly with no lumps, as long as the space is not too large.
Every such system of events looks like the system of measurable subsets of the unit interval with length as probability.
This is why the unit interval can serve as a model for many random experiments.

### L2 human

The theorem lets ergodic theory move between point maps and automorphisms of measure algebras.
Together with the theory of standard probability spaces, it shows that many probability spaces of analysis are isomorphic to $[0,1]$ up to null sets.
An example is $[0,1]^\N$ with the product of Lebesgue measures.
Maharam's theorem classifies the non-separable case.
</document>

<document source="https://tamnd.github.io/math2/node/YFF8B5A0QG.md" tag="YFF8B5A0QG" kind="statement">
# Hilbert's Theorem 90

## Statement

1. Let $L/K$ be a finite [[cyclic extension|VFE5B71E7Y]] and let $\sigma$ be a generator of its [[Galois group|0XVR72M51M]].
   An element $\alpha \in L$ has [[norm|8HHY1K3XDC]] $N_{L/K}(\alpha) = 1$ if and only if $\alpha = \beta / \sigma(\beta)$ for some $\beta \in L^\times$.
2. Let $L/K$ be a finite [[Galois extension|0DS2F7GK05]] with Galois group $G$.
   Let $f \colon G \to L^\times$ be a map with $f(\sigma\tau) = f(\sigma)\, \sigma(f(\tau))$ for all $\sigma, \tau \in G$.
   Then there is $\beta \in L^\times$ with $f(\sigma) = \beta / \sigma(\beta)$ for all $\sigma \in G$.
   In the language of group cohomology, $H^1(G, L^\times) = 1$.

## Remarks

For part 2, [[Dedekind's lemma|N43G50YHG2]] gives $\gamma \in L$ such that $\beta = \sum_{\tau \in G} f(\tau)\, \tau(\gamma)$ is nonzero.
Then $\sigma(\beta) = \sum_\tau \sigma(f(\tau))\, \sigma\tau(\gamma) = f(\sigma)^{-1} \sum_\tau f(\sigma\tau)\, \sigma\tau(\gamma) = f(\sigma)^{-1} \beta$.
For part 1, if $N(\alpha) = 1$ then $f(\sigma^i) = \alpha\, \sigma(\alpha) \cdots \sigma^{i-1}(\alpha)$ is well defined on the cyclic group and satisfies the condition of part 2; conversely $N(\beta / \sigma(\beta)) = N(\beta)/N(\sigma(\beta)) = 1$.

Example: take $L = \Q(i)$ and $K = \Q$, with $\sigma$ complex conjugation.
The elements of norm $1$ are the $a + bi$ with $a^2 + b^2 = 1$, and the theorem writes each as $(m + ni)/(m - ni) = (m^2 - n^2 + 2mni)/(m^2 + n^2)$ with integers $m, n$ not both zero.
This recovers the parametrization of rational points on the unit circle and of Pythagorean triples.

Part 1 needs a generator $\sigma$ of the Galois group, so it applies only to cyclic extensions; for a general finite Galois extension one uses part 2.

## Explanations

### L1 human

The norm of an element multiplies it with all its conjugates.
Hilbert's Theorem 90 says that, in a cyclic extension, the elements of norm $1$ are exactly the quotients of an element by its conjugate.
For the Gaussian numbers this explains why every rational point on the circle $a^2 + b^2 = 1$ has the form $\left(\frac{m^2 - n^2}{m^2 + n^2}, \frac{2mn}{m^2 + n^2}\right)$.

### L2 human

The cohomological form $H^1(\Gal(L/K), L^\times) = 1$ is the starting point of [[Galois cohomology|9MPCZ3EAJM]].
It is the main step in Kummer theory.
When $K$ contains a primitive $n$-th root of unity, it shows that every cyclic extension of degree $n$ is obtained by adjoining an $n$-th root.
There is also an additive version: in a cyclic extension, the elements of [[trace|WPE5Z53QYJ]] $0$ are exactly the $\beta - \sigma(\beta)$, which gives Artin–Schreier theory in characteristic $p$.
</document>

<document source="https://tamnd.github.io/math2/node/YFFEKD37RP.md" tag="YFFEKD37RP" kind="definition">
# Order of a PDE

## Definition

Let $F\left(D^k u, \dots, D u, u, x\right) = 0$ be a [[partial differential equation|GC5BDZ6QBC]] in an open set $\Omega \subseteq \R^n$.
Here $D^j u$ is the collection of the partial derivatives $\partial^\alpha u$ with $\abs{\alpha} = j$, in [[multi-index notation|0PJXBR6QA3]].
Its *order* is the largest integer $m$ such that $F$ actually depends on some derivative $\partial^\alpha u$ with $\abs{\alpha} = m$.
For a system of equations, the order is the largest order of a derivative that occurs in some equation of the system.

## Remarks

Example: the [[minimal surface equation|3GV2E2YTZX]] in two variables involves $\partial_x^2 u$, $\partial_x \partial_y u$ and $\partial_y^2 u$ but no third derivatives, so it has order $2$.
Non-example of a misreading: $(\partial_x u)^2 + (\partial_y u)^2 = 1$ has order $1$, not $2$, because the square is a power of a first derivative and not a second derivative.
This is the same notion as the [[order of a differential equation|0XK0G19SK8]], applied to equations in several variables.
A mixed derivative $\partial^\alpha u$ counts with its total order $\abs{\alpha} = \alpha_1 + \dots + \alpha_n$.

## Explanations

### L1 human

The order of a PDE is the largest total number of differentiations applied to the unknown in any term.
</document>

<document source="https://tamnd.github.io/math2/node/YFKA3KVTGR.md" tag="YFKA3KVTGR" kind="concept">
# Holomorphic functional calculus

## Summary

Let $a$ be an element of a unital [[Banach algebra|9KYWN94QDN]], and let $f$ be a [[holomorphic function|0NAHXR1FRH]] on a neighbourhood of the [[spectrum|EH0CZ19RKP]] of $a$.
The holomorphic functional calculus gives a meaning to $f(a)$.
It is defined by a Cauchy integral of $f$ against the [[resolvent|J5YNNKSB9P]] of $a$.
The map $f \mapsto f(a)$ is an algebra homomorphism that extends the evaluation of polynomials, and it satisfies the spectral mapping theorem $\sigma(f(a)) = f(\sigma(a))$.
It works for all bounded operators on a Banach space, not only normal ones, and it gives spectral projections, logarithms and square roots of operators.

## Explanations

### L1 human

For a polynomial $p$ and a square matrix $A$, the matrix $p(A)$ is clear: replace $z$ by $A$.
The holomorphic functional calculus does the same for any function given locally by power series, using Cauchy's integral formula instead of a formula for $f$.
The value $f(\lambda)$ is replaced by the integral of $f(\lambda)$ times $(\lambda - a)^{-1}$ around the spectrum.
For example, if $A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ and $f = \exp$, the calculus gives the diagonal matrix with entries $e^2$ and $e^3$.
If the spectrum splits into two parts, the function equal to $1$ near one part and $0$ near the other gives a projection onto the matching invariant subspace.
</document>

<document source="https://tamnd.github.io/math2/node/YFSKBA5BDZ.md" tag="YFSKBA5BDZ" kind="definition">
# Malcev completion

## Definition

Let $\Gamma$ be a finitely generated, torsion-free [[nilpotent group|0Q7MMQHC1P]].
A *Malcev completion* of $\Gamma$ is a pair $(N, \iota)$ with three properties.

1. $N$ is a connected and simply connected [[nilpotent Lie group|DK87YN458D]].
2. $\iota \colon \Gamma \to N$ is an injective group homomorphism.
3. $\iota(\Gamma)$ is a [[lattice|BTYF5Y5PFY]] in $N$: it is discrete and $N/\iota(\Gamma)$ is compact.

## Remarks

Malcev proved that a Malcev completion exists and is unique up to a unique isomorphism compatible with $\iota$.
More generally, each homomorphism between two such groups $\Gamma$ extends uniquely to a continuous homomorphism of their completions.

Example: $\Gamma = \Z^n$ with $N = \R^n$ and $\iota$ the inclusion.
The group $\R^n$ is abelian, hence nilpotent, and simply connected; $\Z^n$ is discrete, and $\R^n/\Z^n$ is the compact torus.
In the same way, the integer [[Heisenberg group|ZZYCV5EX9J]] has the real Heisenberg group as Malcev completion.

Non-example: $\Gamma = \Z$ with $N = \R^2$ and $\iota(k) = (k, 0)$.
Clauses 1 and 2 hold, but $\R^2/\iota(\Z)$ is a cylinder, which is not compact, so clause 3 fails.

The name is also used for the rational Malcev completion of $\Gamma$.
It is a torsion-free nilpotent group containing $\Gamma$ in which each element has a unique $k$-th root for each $k \ge 1$.
Moreover, some positive power of each of its elements lies in $\Gamma$.
</document>

<document source="https://tamnd.github.io/math2/node/YFT50501A0.md" tag="YFT50501A0" kind="statement">
# Birkhoff–Frink representation of algebraic closure systems

## Statement

Let $A$ be a set and let $C$ be a [[closure operator|2HJGRABSPZ]] on $A$.
Assume that $C$ is algebraic: $C(X) = \bigcup \set{C(Y) : Y \subseteq X,\ Y \text{ finite}}$ for every $X \subseteq A$.
Then there is an [[algebra|QJGPT2W0FT]] $\mathbf{A}$ with universe $A$ and finitary operations with the following property.
A set $X \subseteq A$ is a subuniverse of $\mathbf{A}$ if and only if $C(X) = X$.

## Remarks

Proof idea.
For each finite set $B = \set{b_1, \dots, b_n} \subseteq A$ with $n$ distinct elements and each $c \in C(B)$, add an $n$-ary operation.
It sends $(b_1, \dots, b_n)$ to $c$ and sends every other tuple to its first entry; for $n = 0$ it is the constant $c$.
A closed set is closed under these operations because $C(B) \subseteq C(X)$ when $B \subseteq X$.
A set closed under them contains $C(B)$ for each finite $B$ inside it, so it is closed because $C$ is algebraic.

Consequence: every [[algebraic lattice|3J0BTXSCVN]] is isomorphic to the [[subalgebra lattice|T0TNG7AGXT]] of some algebra.
Indeed, an algebraic lattice is isomorphic to the lattice of closed sets of some algebraic closure operator.

The hypothesis that $C$ is algebraic is necessary.
For any algebra with finitary operations, the union of a chain of subuniverses is a subuniverse.
On $A = \N$, let $C(X) = X$ for finite $X$ and $C(X) = \N$ for infinite $X$.
This is a closure operator, and it is not algebraic.
The closed sets $\set{0, 2, \dots, 2k}$ form a chain whose union, the set of even numbers, is not closed.
So no algebra on $\N$ has exactly the closed sets of $C$ as its subuniverses.

## Explanations

### L1 human

A closure operator takes a set of elements and adds everything that the set "forces".
It is algebraic when whatever a set forces is already forced by some finite part of it.
The theorem says that every such operator comes from an algebra: the closed sets are exactly the subsets closed under some operations.
So the abstract idea of "generating" is no more general than generating with operations.

### L2 human

The theorem is the converse of a basic fact: in any algebra, the subuniverse operator $\operatorname{Sg}$ is an algebraic closure operator.
Together they show that algebraic closure systems, subuniverse lattices and algebraic lattices describe the same objects.
A similar but harder result for congruences is the theorem of Grätzer and Schmidt, which says that every algebraic lattice is a [[congruence lattice|REE07CSXHR]].
</document>

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# (∞,1)-category

## Summary

An $(\infty,1)$-category is a higher category with $k$-morphisms for every $k \ge 1$, in which all $k$-morphisms with $k \ge 2$ are invertible.
It behaves like a [[category|A2DE66937D]] whose hom-sets are replaced by spaces, taken up to homotopy.
Several precise models exist, such as [[quasi-categories|9328WEVJEK]], categories enriched in Kan complexes, and complete Segal spaces, and these models are known to be equivalent.
$(\infty,1)$-categories are the natural setting for homotopy theory, derived algebraic geometry and stable homotopy theory.

## Explanations

### L1 human

In topology, two maps can be homotopic, two homotopies can be homotopic, and so on.
An $(\infty,1)$-category keeps all of this information instead of collapsing homotopic maps to a single class.
The "$1$" means that only the ordinary morphisms may fail to be invertible; everything above them is a kind of homotopy, which can be reversed.
For example, topological spaces with continuous maps, homotopies between maps and homotopies between homotopies form an $(\infty,1)$-category.
</document>

<document source="https://tamnd.github.io/math2/node/YG0042V8CJ.md" tag="YG0042V8CJ" kind="statement">
# Uniform boundedness principle

## Statement

Let $X$ be a [[Banach space|06GZ956657]] and let $Y$ be a [[normed vector space|03J5RAKB30]], both over $\R$ or $\C$.
Let $\mathcal{F}$ be a family of [[continuous linear maps|03TM135G0N]] from $X$ to $Y$.
If $\sup_{T \in \mathcal{F}} \norm{Tx} < \infty$ for every $x \in X$, then $\sup_{T \in \mathcal{F}} \norm{T} < \infty$, where $\norm{T}$ is the [[operator norm|06EKW1AKXF]].

## Remarks

For $k \ge 1$ let $F_k = \set{x \in X : \norm{Tx} \le k \text{ for all } T \in \mathcal{F}}$.
Each $F_k$ is closed, and the hypothesis says that the $F_k$ cover $X$.
Since $X$ is complete, the [[Baire category theorem|0A7KGAYHGV]] gives some $F_k$ that contains a ball $B(x_0, r)$.
Then for $\norm{x} < r$ and $T \in \mathcal{F}$, $\norm{Tx} \le \norm{T(x_0 + x)} + \norm{T x_0} \le 2k$, so $\norm{T} \le 2k/r$.

A main use is the Banach–Steinhaus theorem: if continuous linear maps $T_n \colon X \to Y$ converge at every point to $Tx$, then $T$ is linear and continuous, since $\norm{Tx} \le \sup_n \norm{T_n}\,\norm{x}$.
Completeness of $X$ is necessary.
Let $X$ be the space of finitely supported real sequences with the norm $\norm{x}_\infty = \sup_k \abs{x_k}$, and let $T_n(x) = n x_n$.
For each $x$ the values $T_n x$ are eventually $0$, so they are bounded, but $\norm{T_n} = n$.

## Explanations

### L1 human

Suppose a family of linear operators on a complete space is bounded at each single point.
The theorem says that the family is then bounded uniformly, by one constant times the norm.
Pointwise control turns into global control, for free.

### L2 human

It is one of three consequences of the Baire category theorem at the core of a first course on Banach spaces.
The other two are the [[bounded inverse theorem|MR5FJXR348]] and the [[closed graph theorem|GXM8BX8YM7]].
A classical application shows that there is a continuous periodic function whose Fourier series diverges at a point: the partial-sum functionals at that point have unbounded norms.
It also shows that weakly convergent sequences in a normed space are bounded, by applying it to the sequence viewed in the bidual.
</document>

<document source="https://tamnd.github.io/math2/node/YG3HWE6GEX.md" tag="YG3HWE6GEX" kind="concept">
# Finite field arithmetic

## Summary

Finite field arithmetic is the way to add, multiply and divide elements of a [[finite field|0ZXZ82FTX2]] in practice.
The usual model of $\F_{p^n}$ is the set of polynomials of degree less than $n$ over $\F_p$.
One computes with them modulo a fixed [[irreducible polynomial|03FKX7Z5DC]] of degree $n$.
It is the basis of error-correcting codes and of much of modern cryptography.

## Explanations

### L1 human

In $\F_p$ one computes with integers modulo a prime $p$.
For a field with $p^n$ elements, one computes with polynomials instead, and one reduces modulo a fixed polynomial, as one reduces integers modulo $p$.
Over $\F_2$, an element of $\F_{2^8}$ is a string of $8$ bits, and addition is the bitwise exclusive or.
For example, in $\F_8$ built with $x^3 + x + 1$, one has $x^3 = x + 1$, so $(x^2 + 1)(x + 1) = x^3 + x^2 + x + 1 = x^2$.
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# Dirichlet energy

## Summary

The Dirichlet energy of a function $u$ on a domain is $\frac12 \int \abs{\nabla u}^2 \, dx$, half the integral of the squared length of its gradient.
It measures how much the function varies.
Its critical points with fixed boundary values are the [[harmonic functions|0DEB0JMN0W]], which is the content of Dirichlet's principle.
The energy is the quadratic form behind the [[Laplacian|0E0B6RA8K0]], the Sobolev space $H^1$ and the theory of Dirichlet forms.

## Explanations

### L1 human

Think of a rubber membrane whose height over the point $x$ is $u(x)$.
For small slopes, the stretching energy of the membrane is roughly proportional to the Dirichlet energy.
A flat membrane has energy $0$, and a wrinkled one has large energy.
For example, $u(x, y) = x$ on the unit disc has gradient of length $1$ everywhere, so its energy is $\frac12$ times the area $\pi$, that is, $\pi/2$.
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# Dissipative operator

## Definition

Let $X$ be a real or complex [[Banach space|06GZ956657]] and let $A$ be a linear [[operator|KRJX5RA0PW]] in $X$ with domain $D(A)$.
The operator $A$ is *dissipative* if
$$\norm{(\lambda I - A)x} \ge \lambda \norm{x} \quad \text{for all } \lambda > 0 \text{ and all } x \in D(A).$$
A dissipative operator $A$ is *maximal dissipative*, or *m-dissipative*, if moreover $\lambda I - A$ maps $D(A)$ onto $X$ for some $\lambda > 0$.

## Remarks

If $X$ is a Hilbert space with inner product $\ip{\cdot}{\cdot}$, then $A$ is dissipative if and only if $\operatorname{Re}\ip{Ax}{x} \le 0$ for all $x \in D(A)$.
In a general Banach space, $A$ is dissipative if and only if for each $x \in D(A)$ there is a functional $j$ in the [[dual space|07JBSZE0V4]] with $j(x) = \norm{x}^2 = \norm{j}^2$ and $\operatorname{Re} j(Ax) \le 0$.

Example: on $L^2(0, 1)$, let $Af = f''$ on smooth functions with compact support in $(0, 1)$.
Integration by parts gives $\ip{Af}{f} = -\int_0^1 \abs{f'}^2 \, dx \le 0$, so $A$ is dissipative.

Non-example: the identity $I$ on a nonzero Hilbert space is not dissipative, since $\ip{x}{x} > 0$ for $x \ne 0$.
Directly, for $1/2 < \lambda < 1$ and $x \ne 0$, $\norm{(\lambda I - I)x} = (1 - \lambda)\norm{x} < \lambda\norm{x}$.
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# Choquet capacity

## Definition

Let $X$ be a [[Hausdorff space|0S1FJ8VEX9]], and let $\mathcal{P}(X)$ be the set of all subsets of $X$.
A *Choquet capacity* on $X$ is a map $I \colon \mathcal{P}(X) \to [0, \infty]$ with the following properties.

1. If $A \subseteq B \subseteq X$, then $I(A) \le I(B)$.
2. If $A_1 \subseteq A_2 \subseteq \cdots$ are subsets of $X$ with union $A$, then $I(A_k) \to I(A)$.
3. If $K_1 \supseteq K_2 \supseteq \cdots$ are [[compact subsets|0TVJZYB6SH]] of $X$ with intersection $K$, then $I(K_k) \to I(K)$.

A set $A \subseteq X$ is *capacitable* for $I$ if $I(A) = \sup \set{I(K) : K \subseteq A, \ K \text{ compact}}$.

## Remarks

Example: let $\mu$ be a finite Borel measure on $\R^n$ and $\mu^*$ its outer measure, $\mu^*(A) = \inf \set{\mu(B) : B \supseteq A \text{ Borel}}$.
Clause 1 is clear.
Clause 2 holds because each $A_k$ has a Borel hull $B_k$ with $\mu(B_k) = \mu^*(A_k)$, and the sets $\bigcap_{j \ge k} B_j$ increase to a Borel hull of $A$.
Clause 3 is continuity from above of the finite measure $\mu$.
Non-example: on $\R$ let $I(A) = 1$ if $A$ is infinite and $I(A) = 0$ otherwise.
The finite sets $\set{1, \dots, k}$ increase to the infinite set $\N \setminus \set{0}$, but $I = 0$ along the sequence, so clause 2 fails.
Choquet's capacitability theorem: if $X$ is a Polish space, every analytic subset of $X$ is capacitable for every Choquet capacity.
Some authors allow values in $[-\infty, \infty]$ or define capacities relative to a paving other than the compact sets.
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<document source="https://tamnd.github.io/math2/node/YG9NR68ZH3.md" tag="YG9NR68ZH3" kind="concept">
# Operator theory

## Summary

Operator theory is the part of functional analysis that studies linear operators between normed spaces, above all bounded operators on Banach and Hilbert spaces.
It asks how far the theory of matrices, with eigenvalues, adjoints and diagonalisation, extends to infinite dimensions.
Its tools are used for differential and integral equations, quantum mechanics and signal processing.

## Explanations

### L1 human

A matrix turns a vector of numbers into another vector of numbers.
An operator does the same thing for objects with infinitely many coordinates, such as sequences or functions.
A good picture is the derivative, which turns a function into another function and respects sums and scalar multiples.
A simple example is the right shift on sequences, $(x_1, x_2, \ldots) \mapsto (0, x_1, x_2, \ldots)$.
It never loses information, yet it is not onto, something no square matrix can do.
Operator theory explains such new effects of infinite dimension.
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# Projective object

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]].
An object $P$ of $\mathcal{C}$ is *projective* if, for every [[epimorphism|RVXDSY04KG]] $e \colon E \to X$ and every [[morphism|859F0TFVA3]] $f \colon P \to X$, there is a morphism $g \colon P \to E$ with $e \circ g = f$.

## Remarks

Equivalently, the [[hom functor|VJXS8AK67E]] $\Hom(P, -)$ sends epimorphisms to surjections of sets.
In an [[abelian category|Y9CZ3MTNF1]], $P$ is projective if and only if $\Hom(P, -)$ is an exact functor to abelian groups, and if and only if every short exact sequence $0 \to A \to B \to P \to 0$ splits.
In the category of modules over a ring, the projective objects are the [[projective modules|BFFA03GE3V]], that is, the direct summands of free modules.
In general categories some authors use only regular or strong epimorphisms in the definition; in abelian categories all these versions agree.

Example: $\Z$ is projective in the category of abelian groups.
Given an epimorphism $e \colon E \to X$ and $f \colon \Z \to X$, choose $u \in E$ with $e(u) = f(1)$, and let $g(n) = nu$; then $e(g(n)) = n f(1) = f(n)$.

Non-example: $\Z/2\Z$ is not projective in the category of abelian groups.
For the reduction $e \colon \Z \to \Z/2\Z$ and $f = 1_{\Z/2\Z}$, a lift $g \colon \Z/2\Z \to \Z$ would have to be $0$, since $\Z$ has no nonzero element of finite order, but then $e \circ g = 0 \ne f$.
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# Chebyshev–Markov–Stieltjes inequalities

## Statement

Let $m \ge 1$ and let $\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\R$ with finite [[moments|01XAHPADBA]] $m_k = \int_\R x^k \, d\mu(x)$ for $0 \le k \le 2m - 1$.
Assume that the [[Hankel matrix|PQ1D8FQB3E]] $(m_{i+j})_{i,j=0}^{m-1}$ is [[positive definite|0KX66CZMBB]].
Let $P_m$ be the monic polynomial of degree $m$ that is [[orthogonal|8TXASARQN1]] to all polynomials of degree less than $m$ in $L^2(\mu)$.
Then $P_m$ has $m$ simple real zeros $\xi_1 < \xi_2 < \cdots < \xi_m$.
Let $\rho_1, \ldots, \rho_m > 0$ be the Christoffel numbers, that is, the unique numbers with $\int_\R p \, d\mu = \sum_{j=1}^{m} \rho_j \, p(\xi_j)$ for every real polynomial $p$ of degree at most $2m - 1$.
Let $\nu$ be any positive Borel measure on $\R$ with $\int_\R x^k \, d\nu(x) = m_k$ for $0 \le k \le 2m - 1$.
Then for each $k$ with $1 \le k \le m$,
$$\sum_{j=1}^{k-1} \rho_j \;\le\; \nu\bigl((-\infty, \xi_k)\bigr) \;\le\; \nu\bigl((-\infty, \xi_k]\bigr) \;\le\; \sum_{j=1}^{k} \rho_j.$$

## Remarks

For the upper bound one builds, by Hermite interpolation, a polynomial $q$ of degree $2m - 2$ with $q(\xi_j) = 1$ for $j \le k$, $q(\xi_j) = 0$ for $j > k$, and $q'(\xi_j) = 0$ for $j \ne k$.
One shows that $q \ge 1$ on $(-\infty, \xi_k]$ and $q \ge 0$ on $\R$.
Then $\nu((-\infty, \xi_k]) \le \int q \, d\nu = \int q \, d\mu = \sum_j \rho_j q(\xi_j) = \sum_{j \le k} \rho_j$, since the integrals of $q$ depend only on the moments; the lower bound follows by the reflection $x \mapsto -x$.
The measure $\sum_j \rho_j \delta_{\xi_j}$ of Gaussian quadrature has the given moments, and it attains every bound.
In particular, the mass that any such $\nu$ puts at the point $\xi_k$ is at most $\rho_k$.
For example, for $m = 1$ with $m_0 = 1$ and $m_1 = 0$, one gets $\xi_1 = 0$ and $\rho_1 = 1$, and the inequalities say only $0 \le \nu((-\infty, 0)) \le \nu((-\infty, 0]) \le 1$.
The positive definiteness of the Hankel matrix holds exactly when $\mu$ is supported on at least $m$ points; it is needed for $P_m$ and the numbers $\rho_j$ to exist.

## Explanations

### L1 human

Suppose only the first few moments of a mass distribution on the line are known.
Then the distribution is not known, but the amount of mass to the left of certain special points is trapped between two explicit numbers.
The special points are the zeros of an orthogonal polynomial, and the numbers are sums of the weights of Gaussian quadrature.

### L2 human

The inequalities say that the distribution functions of all solutions of a truncated moment problem pass through a band around the staircase of the Gaussian quadrature measure.
The bound on point masses, $\nu(\set{\xi_k}) \le \rho_k$, also follows from the extremal property of the Christoffel function.
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# Reciprocal gamma function

## Definition

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a meromorphic function on $\C$ with poles at $0, -1, -2, \ldots$ and no zeros.
The *reciprocal gamma function* is the function $\C \to \C$ given by
$$\frac{1}{\Gamma}(z) = \begin{cases} 1/\Gamma(z) & \text{if } z \notin \set{0, -1, -2, \ldots}, \\ 0 & \text{if } z \in \set{0, -1, -2, \ldots}. \end{cases}$$

## Remarks

It is an [[entire function|0419794R0T]]: near a pole $-n$ of $\Gamma$, the quotient $1/\Gamma$ has a removable singularity with value $0$, because the pole is simple.
Its zeros are exactly $0, -1, -2, \ldots$, and they are simple, with derivative $(-1)^n n!$ at $-n$, the reciprocal of the residue of $\Gamma$.
Example: at $z = 3$ the value is $1/\Gamma(3) = 1/2! = \tfrac12$.
Non-example: $\Gamma$ itself is not entire, since it is not defined at $0$ and $\abs{\Gamma(z)} \to \infty$ as $z \to 0$.
It is an entire function of [[order|SHF8Z7GMFX]] $1$ and of maximal type.

## Explanations

### L1 human

The reciprocal gamma function is $1/\Gamma(z)$, with the value $0$ at the points where $\Gamma$ blows up.
These values make it smooth everywhere in the complex plane.
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# Heat kernel

## Definition

Let $n \ge 1$.
The *heat kernel* of $\R^n$ is the function $\Phi \colon \R^n \times (0, \infty) \to (0, \infty)$ given by
$$\Phi(x, t) = \frac{1}{(4 \pi t)^{n/2}} \exp\left( -\frac{\abs{x}^2}{4t} \right).$$
It solves the [[heat equation|VP2HN86VAG]] $\partial_t \Phi = \Delta \Phi$ on $\R^n \times (0, \infty)$, satisfies $\int_{\R^n} \Phi(x, t) \, dx = 1$ for every $t > 0$, and $\Phi(\cdot, t)$ tends to the Dirac measure at $0$ as $t \to 0^+$.

## Remarks

Check for $n = 1$: $\partial_t \Phi = \Phi \cdot \big( -\tfrac{1}{2t} + \tfrac{x^2}{4t^2} \big)$ and $\partial_x \Phi = -\tfrac{x}{2t} \Phi$, so $\partial_x^2 \Phi = \Phi \cdot \big( \tfrac{x^2}{4t^2} - \tfrac{1}{2t} \big)$, which is the same.
For bounded continuous $g$, the function $u(x, t) = \int_{\R^n} \Phi(x - y, t) g(y) \, dy$ solves the heat equation for $t > 0$ and tends to $g(x)$ as $t \to 0^+$.
For each $t$, $\Phi(\cdot, t)$ is the density of a [[normal distribution|05AQE0QR0D]] with mean $0$ and covariance $2t$ times the identity.
Non-example: $p(x, t) = (2 \pi t)^{-n/2} e^{-\abs{x}^2/(2t)}$ is not the heat kernel of $\partial_t = \Delta$; it solves $\partial_t p = \tfrac12 \Delta p$ and is the transition density of Brownian motion.
More generally, a domain with a boundary condition, or a Riemannian manifold, has a heat kernel $K(x, y, t)$.
It is the function such that $u(x, t) = \int K(x, y, t) g(y) \, dy$ solves the heat equation with that boundary condition and initial value $g$.
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# Period map

## Summary

The period map of a family of compact [[Kähler manifolds|X60JG9AEQJ]] records how the [[Hodge structure|ZSM083M97C]] on the cohomology of the fibres moves.
All fibres have the same cohomology group, and only the position of the Hodge filtration inside it changes.
So the period map goes from the base of the family to a space of flags, a period domain.
Griffiths showed that it is holomorphic and satisfies a transversality condition.
Classical periods of elliptic curves and abelian integrals are the simplest case.

## Explanations

### L1 human

When the complex structure of a manifold changes, its topology and so its cohomology stay the same.
What changes is how the cohomology splits into pieces of different types $(p, q)$.
The period map follows these pieces as subspaces of one fixed vector space.
For the elliptic curve $\C/(\Z + \tau\Z)$, the holomorphic form $dz$ has periods $1$ and $\tau$ along the two basic loops, and the period map is $\tau \mapsto [1 : \tau]$.
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<document source="https://tamnd.github.io/math2/node/YGP3Q2EF31.md" tag="YGP3Q2EF31" kind="definition">
# Bonferroni correction

## Definition

Let $H_1, \dots, H_m$ be null hypotheses with [[p-values|0EHYZFZ4PK]] $p_1, \dots, p_m$, and let $\alpha \in (0, 1)$.
The *Bonferroni correction* at level $\alpha$ is the procedure that rejects $H_i$ exactly when $p_i \le \alpha/m$.

## Remarks

Assume each p-value is valid: if $H_i$ is true, then $P(p_i \le u) \le u$ for all $u \in [0, 1]$.
Let $I_0$ be the set of true null hypotheses and $m_0$ its size.
By [[Boole's inequality|0EGSEE9J9R]], $P(\text{some } H_i \text{ with } i \in I_0 \text{ is rejected}) \le \sum_{i \in I_0} P(p_i \le \alpha/m) \le m_0 \alpha/m \le \alpha$.
So the correction controls the [[family-wise error rate|GQCY1N8JZT]] strongly at level $\alpha$, with no assumption on the joint law of the p-values.
Example: with $m = 10$ and $\alpha = 0.05$, a hypothesis is rejected when its p-value is at most $0.005$.
Non-example: rejecting when $p_i \le \alpha$ is not the correction, and with $m$ independent true nulls its FWER is $1 - (1 - \alpha)^m > \alpha$.
Equivalently, one rejects $H_i$ when the adjusted p-value $\min(1, m p_i)$ is at most $\alpha$.
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<document source="https://tamnd.github.io/math2/node/YGPR4VR3GP.md" tag="YGPR4VR3GP" kind="concept">
# Quantum invariant of 3-manifolds

## Summary

Quantum invariants of $3$-manifolds are invariants of closed oriented [[3-manifolds|R20TB2KEQQ]].
They are built from the representation theory of [[quantum groups|AP2ZDXANKC]] at roots of unity, or more generally from modular tensor categories.
The main examples are the Witten–Reshetikhin–Turaev invariants, which Witten predicted from quantum [[Chern–Simons theory|F28G886PWD]] and Reshetikhin and Turaev constructed rigorously.
They are computed from a surgery presentation of the manifold, using colored versions of the [[Jones polynomial|4SPC17FPXE]].
They are part of [[topological quantum field theories|DAP92B80J2]] in dimension $3$.

## Explanations

### L1 human

Every closed oriented $3$-manifold can be obtained by surgery on a link in the $3$-sphere, but in many different ways.
A quantum invariant assigns a number to the link, built from knot polynomials, and corrects it so that the number does not depend on which link is chosen.
The result is a number attached to the manifold itself.
These numbers can be computed from a picture of the link, yet their geometric meaning is still only partly understood.
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# Reflexive operator algebra

## Definition

Let $H$ be a Hilbert space and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.
For a set $\mathcal{S} \subseteq B(H)$, let $\operatorname{Lat} \mathcal{S}$ be the set of closed subspaces of $H$ that are [[invariant|0ACX5QT8TE]] under every operator in $\mathcal{S}$.
For a set $\mathcal{L}$ of closed subspaces of $H$, let
$$\operatorname{Alg} \mathcal{L} = \set{T \in B(H) : T(L) \subseteq L \text{ for all } L \in \mathcal{L}}.$$
A subalgebra $\mathcal{A} \subseteq B(H)$ is *reflexive* if $\mathcal{A} = \operatorname{Alg} \operatorname{Lat} \mathcal{A}$.

## Remarks

Always $\mathcal{A} \subseteq \operatorname{Alg} \operatorname{Lat} \mathcal{A}$, so reflexivity is the reverse inclusion.
A reflexive algebra contains the identity and is closed in the weak operator topology, because each $\operatorname{Alg} \mathcal{L}$ has these properties.
Example: $\mathcal{A} = B(H)$ is reflexive, since $\operatorname{Lat} B(H) = \set{\set{0}, H}$ and $\operatorname{Alg} \set{\set{0}, H} = B(H)$.
Non-example: in $M_2(\C)$, let $\mathcal{A}$ be the matrices $\begin{pmatrix} a & b \\ 0 & a \end{pmatrix}$.
Its invariant subspaces are those of $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, namely $\set{0}$, $\C e_1$ and $\C^2$, so $\operatorname{Alg} \operatorname{Lat} \mathcal{A}$ is the algebra of all upper triangular matrices, which is strictly larger.
Every von Neumann algebra is reflexive: its invariant subspaces are the ranges of the projections in its commutant, and the bicommutant theorem gives the equality.
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# Gromov's compactness theorem

## Statement

Let $n \ge 2$ be an integer, let $k$ be a real number and let $D > 0$.
Let $\mathcal{M}$ be the class of closed [[Riemannian manifolds|7CBT4SM83Y]] $M$ of dimension $n$ whose [[Ricci curvature|4GTNM0WHDH]] satisfies $\operatorname{Ric} \ge (n - 1) k$ and whose diameter is at most $D$.
Then every sequence in $\mathcal{M}$ has a subsequence that converges in the [[Gromov–Hausdorff sense|GT56RTG9X5]] to a compact metric space.

## Remarks

The proof rests on a criterion of Gromov.
A class of compact metric spaces is precompact for $d_{GH}$ when the diameters are bounded and, for each $\varepsilon > 0$, the number of $\varepsilon$-balls needed to cover each space is bounded.
This is a uniform version of being [[totally bounded|020Q06TYYH]].
For the manifolds in $\mathcal{M}$, the [[Bishop–Gromov inequality|82F5Q4F1S6]] bounds the number of disjoint balls of radius $\varepsilon/2$, and hence the number of covering balls of radius $\varepsilon$.

The limit space need not be a manifold, and its dimension may drop.
For example, the flat tori $\R^2 / (\Z \times \tfrac{1}{m}\Z)$ have $\operatorname{Ric} = 0$ and bounded diameter, and they converge to a circle of length $1$ as $m \to \infty$.
Without the lower curvature bound the statement fails.
Take spheres with $m$ long thin spikes of length $1$ whose tips are pairwise at distance at least $2$.
The diameters stay bounded, but at least $m$ balls of radius $\tfrac{1}{2}$ are needed to cover, so no subsequence converges.

## Explanations

### L1 human

Manifolds whose curvature is not too negative and whose size is bounded cannot be too complicated at any scale.
The theorem says that from any sequence of them one can pick a subsequence that settles down to a limit shape.
The limit may be rougher than the manifolds, for example lower-dimensional.

### L2 human

This is the starting point of the study of limit spaces of manifolds with Ricci curvature bounded below.
Under a lower bound on sectional curvature the limits are [[Alexandrov spaces|71AWVBG6KH]].
The theorem is used to prove finiteness and pinching results by contradiction: take a bad sequence, pass to a limit, and study the limit.
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# A Dedekind UFD is a PID

## Statement

Let $R$ be a [[Dedekind domain|7NPY2CN5KB]] that is also a [[unique factorization domain|0W4X9G6263]].
Then $R$ is a [[principal ideal domain|0CMD0BCH8T]].

## Remarks

Let $P$ be a nonzero [[prime ideal|00MHHZ9SNE]] and let $a \in P$ be nonzero.
The element $a$ is not a unit, and one of its [[irreducible|0YMT0D2442]] factors $\pi$ lies in $P$.
In a UFD, $\pi$ is a [[prime element|01VH21EV6F]], so $(\pi)$ is a nonzero prime ideal inside $P$; as nonzero primes of a Dedekind domain are maximal, $(\pi) = P$.
So every nonzero prime is principal, and by the [[unique factorization of ideals|S6PHT75MEK]] every nonzero ideal is a product of principal ideals, hence principal.

Example: $\Z[\sqrt{-5}]$ is a Dedekind domain, but $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ gives two different factorizations into irreducibles, so it is not a UFD; consistent with this, $(2, 1 + \sqrt{-5})$ is not principal.
The Dedekind hypothesis cannot be dropped: $k[x, y]$ for a field $k$ is a UFD, but the ideal $(x, y)$ is not principal.

## Explanations

### L1 human

For rings of algebraic integers, unique factorization of numbers and "every ideal has one generator" are the same property.
One direction holds in every ring: a principal ideal domain always has unique factorization.
This result gives the other direction for Dedekind domains.
So a ring of integers of a number field has unique factorization exactly when its ideals are all principal.

### L2 human

In terms of the [[ideal class group|WQ46BD3ARY]], a Dedekind domain is a UFD exactly when its class group is trivial, and this is the same as being a PID.
The class group thus measures the failure of unique factorization of elements.
For example, the class group of $\Z[\sqrt{-5}]$ has order $2$.
The proof uses only that nonzero primes are maximal and that ideals factor into primes, so it is a good first exercise with the ideal factorization theorem.
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# Finitely based algebra

## Summary

An algebra is finitely based when all of its [[identities|PKD440G2XW]] follow from finitely many of them, that is, when it has a finite [[equational basis|9Q008GMZSW]].
Every finite group, every finite ring and every finite lattice is finitely based, but some finite algebras are not.
Deciding which finite algebras are finitely based is the subject of Tarski's finite basis problem.

## Explanations

### L1 human

Any algebra satisfies infinitely many laws.
It is finitely based if a finite list of these laws is enough to derive all the others.
The integers under addition are finitely based: the laws of abelian groups give every law they satisfy.
Surprisingly, there are algebras with only a few elements whose laws cannot be captured by any finite list.
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# Triangulation

## Summary

A triangulation of a topological space is a homeomorphism from the geometric realization of a simplicial complex onto the space.
It cuts the space into vertices, edges, triangles and their higher-dimensional analogues, glued along faces.
Triangulations turn topological questions into combinatorial ones, which is how simplicial homology and the Euler characteristic were first computed.
Whether every manifold can be triangulated, and whether triangulations are essentially unique, are classical questions of geometric topology.

## Explanations

### L1 human

A geodesic dome or a soccer ball made of panels is a sphere cut into flat pieces.
A triangulation does the same with triangles, and in higher dimensions with tetrahedra and their analogues.
The surface of a tetrahedron is a triangulation of the sphere with $4$ vertices, $6$ edges and $4$ triangles.
The count $4 - 6 + 4 = 2$ is the Euler characteristic of the sphere, and every triangulation of the sphere gives the same number.
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# Scaling function and refinement equation

## Summary

The scaling function $\varphi$ of a [[multiresolution analysis|1GHQMBR7DE]] is a function whose integer translates form an orthonormal basis of the base space $V_0$.
Since $V_0 \subseteq V_1$, it satisfies a refinement equation $\varphi(x) = \sqrt2 \sum_k h_k \varphi(2x - k)$: it is a combination of half-size copies of itself.
The coefficients $h_k$ form the low-pass filter of the [[discrete wavelet transform|P3H4MMC4ZH]], and they also determine the wavelet.
Conversely, one constructs wavelets such as the [[Daubechies wavelets|90V8S5CXE6]] by choosing the coefficients first and solving the refinement equation.

## Explanations

### L1 human

The box function $1_{[0,1)}$ is made of two boxes of half the width: the one on $[0, \tfrac12)$ and the one on $[\tfrac12, 1)$.
So $\varphi(x) = \varphi(2x) + \varphi(2x - 1)$.
An equation of this kind, where a function equals a combination of squeezed and shifted copies of itself, is a refinement equation.
Many of its solutions are not simple pictures: they are fractal-like curves fixed by the equation, and the box is only the easiest one.
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# Synthetic geometry

## Definition

*Synthetic geometry* is the study of geometry from axioms whose primitive notions are geometric objects and relations, such as points, lines, incidence, betweenness and congruence.
Its proofs derive theorems from the axioms directly, without coordinates in a field or a ring.

## Remarks

Example: [[Hilbert's axioms|KM1FB8XZ2R]] give a synthetic geometry, and a proof of the congruence criterion side-side-side from them is a synthetic proof.
Non-example: write the vertices of a parallelogram as vectors $0$, $u$, $v$, $u + v$, and check that both diagonals have midpoint $(u + v)/2$.
This proof that the diagonals bisect each other is analytic, not synthetic.
Coordinates may come back as a result: for example, Desargues's theorem in a projective plane allows one to build a division ring of coordinates from the axioms.
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# Semilocal ring

## Definition

Let $R$ be a [[ring|0894J980RK]] with identity and $J(R)$ its [[Jacobson radical|XA0DCNH7WY]].
$R$ is *semilocal* if the quotient ring $R/J(R)$ is [[semisimple|NY25HXA478]], that is, semisimple as a left module over itself.

## Remarks

By the [[Artin–Wedderburn theorem|R1F271180N]], $R$ is semilocal exactly when $R/J(R) \cong M_{n_1}(D_1) \times \cdots \times M_{n_k}(D_k)$ for division rings $D_i$ and integers $n_i \ge 1$.
Example: $M_n(\Z_{(p)})$ is semilocal, since its Jacobson radical is $M_n(p\Z_{(p)})$ and the quotient is $M_n(\mathbb{F}_p)$.
Non-example: $\Z$ is not semilocal: $J(\Z) = 0$ and $\Z$ is not semisimple, since the ideal $2\Z$ has no complement.
A commutative ring is semilocal if and only if it has finitely many maximal ideals; this is the [[commutative notion|6DYAGW68AV]].
The definition is left-right symmetric, and the spelling *semi-local* is also used.
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# One-parameter subgroup

## Summary

A one-parameter subgroup of a topological group $G$ is a continuous [[group homomorphism|0JYTG5K9Y4]] from the additive group $\R$ into $G$.
In a [[Lie group|7JBRGCVJRB]], it is a smooth curve through the identity along which the group law becomes addition of the parameter.
One-parameter subgroups of a Lie group correspond to the tangent vectors at the identity, which is how the exponential map is built.

## Explanations

### L1 human

A one-parameter subgroup is a way to move steadily through a group.
Moving for time $s$ and then for time $t$ is the same as moving for time $s + t$.
Turning a wheel at constant speed is a picture: the rotation after time $t$ is a rotation by the angle $ct$.
In the group of invertible matrices, $t \mapsto e^{tA}$ is a one-parameter subgroup for each square matrix $A$.
For $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, it gives the rotations by angle $t$ of the plane.
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# Hilbert's axioms

## Definition

The primitive objects are points, lines and planes.
The primitive relations are incidence, betweenness of three points, and congruence $\cong$ of segments and of angles.
*Hilbert's axioms* are the following five groups of axioms on these notions.

1. Incidence: two distinct points lie on exactly one [[line|0MP9BCF6DP]]. Three non-collinear points lie on exactly one [[plane|0V3XAMF3C1]]. A line with two points in a plane lies in the plane. Two planes with a common point have a second common point. Each line has at least two points, each plane has three non-collinear points, and there are four points not in one plane.
2. Order: the [[betweenness|CE8GX6ADZS]] relation is symmetric in $A$ and $C$ and holds only for three distinct collinear points. For distinct $A$ and $C$ some point $B$ has $C$ between $A$ and $B$. Of three collinear points, at most one lies between the other two. [[Pasch's axiom|KJT98G16Y4]] holds.
3. Congruence: a segment can be laid off on a given ray in a unique way, and congruence of segments is transitive. Congruent adjacent segments add up to congruent segments. An angle can be laid off on a given side of a given ray in a unique way. If two triangles have two sides and the included angle congruent, then the other angles are congruent too.
4. Parallels: [[Playfair's axiom|PR7Q3QMEHZ]] holds.
5. Continuity: the axiom of Archimedes holds for segments. The points of a line admit no extension that keeps all the other axioms (line completeness).

## Remarks

Example: $\R^3$ with its usual points, lines, planes, betweenness and congruence by isometries is a model.
For instance, Pasch's axiom holds there because a line in a plane is the zero set of an affine function.
Up to isomorphism, it is the only model, so the system is categorical.
Non-example: the space $\Q^3$ with the same notions fails the first congruence axiom.
The segment from $(0,0,0)$ to $(1,1,0)$ has length $\sqrt 2$, so it cannot be laid off on the positive $x$-axis from the origin, since $(\sqrt 2, 0, 0) \notin \Q^3$.
The plane version keeps only the axioms of groups 1 to 5 that speak of one plane.
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# Langlands classification

## Statement

Let $G$ be a linear connected [[real reductive Lie group|42ZCE14AD2]], and fix a minimal parabolic subgroup $P_0$ of $G$.
Consider the triples $(P, \sigma, \nu)$ such that:

1. $P \supseteq P_0$ is a [[parabolic subgroup|Z2FEBW5MQM]] with [[Langlands decomposition|BM3TWJ0D4P]] $P = MAN$;
2. $\sigma$ is an equivalence class of irreducible tempered representations of $M$;
3. $\nu \in \mathfrak a^*_\C$, where $\mathfrak a$ is the Lie algebra of $A$, and $\operatorname{Re} \nu$ is strictly positive on every root of $\mathfrak a$ in the Lie algebra of $N$.

For such a triple, the normalised [[induced representation|6Q7AHAZ6Q4]] $\mathrm{Ind}_P^G(\sigma \otimes e^\nu \otimes 1)$ has a unique irreducible quotient $J(P, \sigma, \nu)$, the Langlands quotient.
Every irreducible [[admissible representation|SZV8Y806Y7]] of $G$ is infinitesimally equivalent to exactly one $J(P, \sigma, \nu)$.

## Remarks

Infinitesimal equivalence means that the [[Harish-Chandra modules|2MKHH636C3]] are isomorphic.
The proof compares the leading exponents of the matrix coefficients of a representation with the parameter $\nu$.
The quotient $J(P, \sigma, \nu)$ is the image of a standard intertwining operator, which converges because $\operatorname{Re} \nu$ lies in the positive chamber.
If $G$ is semisimple and $P = G$, then $A$ is trivial, so $\nu = 0$, and $J(G, \sigma, 0) = \sigma$ runs over the tempered representations themselves.
Example: let $G = \mathrm{SL}_2(\R)$ and let $P$ be the upper triangular matrices.
The constant functions form a subrepresentation of the [[principal series|CWDF4KGHS3]] $I(\mathrm{triv}, -\rho)$, and dually the trivial representation is a quotient of $I(\mathrm{triv}, \rho)$.
So $J(P, \mathrm{triv}, \rho)$ is the trivial representation.
The positivity of $\operatorname{Re} \nu$ is needed: on the unitary axis $\operatorname{Re} \nu = 0$ the induced representation can be reducible with several irreducible quotients, so uniqueness fails.

## Explanations

### L1 human

The theorem reduces the classification of all irreducible representations to that of the tempered ones, which are the representations that appear in $L^2(G)$.
Each irreducible representation is built from a tempered representation of a smaller group and a parameter that measures its growth.

### L2 human

Langlands proved the classification for real groups; the analogous result for $p$-adic groups is due to Borel–Wallach and Silberger.
Tempered representations are in turn built from discrete series of Levi subgroups, so the whole picture rests on Harish-Chandra's discrete series.
The classification is the starting point of the local Langlands correspondence for real groups, which attaches $L$-parameters to the triples $(P, \sigma, \nu)$.
</document>

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# Automorphism group of a bounded domain is a Lie group

## Statement

Let $D \subset \C^n$ be a bounded domain, that is, a bounded connected open set.
Let $\Aut(D)$ be the group of [[biholomorphisms|FX92PP3HZR]] $D \to D$, with the topology of uniform convergence on compact subsets.
Then $\Aut(D)$ is a real [[Lie group|7JBRGCVJRB]], and it acts properly on $D$.
In particular, for each $p \in D$ the isotropy group $\set{f \in \Aut(D) : f(p) = p}$ is compact.

## Remarks

The theorem is due to Henri Cartan.
Since $D$ is bounded, [[Montel's theorem|ZVFVBVJMCB]] makes families of automorphisms normal, and a limit argument shows that the action is proper.
By [[Cartan's uniqueness theorem|EQHPSKC5SY]], an automorphism $f$ with $f(p) = p$ is determined by its derivative $f'(p)$.
So the isotropy group embeds as a compact subgroup of $GL_n(\C)$, and the whole group is locally compact and finite-dimensional.

Example: for the unit disc $\Delta$, $\Aut(\Delta)$ consists of the maps $z \mapsto e^{i\theta}(z - a)/(1 - \bar a z)$ with $\theta \in \R$ and $\abs{a} < 1$.
It is a $3$-dimensional Lie group, isomorphic to $PSU(1, 1)$, and the isotropy group of $0$ is the circle of rotations.

Boundedness cannot be dropped.
For $n \ge 2$, $\Aut(\C^n)$ contains the maps $(z_1, z_2, \dots) \mapsto (z_1, z_2 + g(z_1), \dots)$ for every entire function $g$, so it is not a finite-dimensional Lie group.

## Explanations

### L1 human

The holomorphic symmetries of a bounded region in $\C^n$ form a group.
The theorem says this group is as nice as a group of matrices: it is a smooth finite-dimensional manifold, and its operations are smooth.
Moreover, the symmetries fixing a point form a compact group.

### L2 human

This is the starting point for the study of homogeneous and symmetric bounded domains.
If $\Aut(D)$ acts transitively, then $D \cong \Aut(D)/K$ with $K$ compact, and $D$ becomes a homogeneous space of a Lie group.
Combined with the invariant Bergman metric, this leads to Cartan's classification of bounded symmetric domains.
For unbounded domains the group can be infinite-dimensional, as for $\C^n$.
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# Poincaré–Lindstedt method

## Summary

The Poincaré–Lindstedt method finds approximate [[periodic solutions|FHNDX22WMK]] of weakly nonlinear oscillators.
It expands both the solution and its frequency in powers of the small parameter.
The frequency corrections are chosen to remove secular terms, which grow without bound in time and spoil a naive expansion.
The method shows how nonlinearity shifts the frequency of an oscillation with its amplitude.

## Explanations

### L1 human

A pendulum with a large swing has a slightly different period from one with a tiny swing.
If one expands the motion in a small parameter but keeps the old period, the error grows with time.
The method lets the period change too, and adjusts it so that the error stays bounded.
For $x'' + x + \varepsilon x^3 = 0$ with amplitude $a$, it gives the frequency $1 + \frac{3}{8} \varepsilon a^2$ up to $O(\varepsilon^2)$.
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<document source="https://tamnd.github.io/math2/node/YHK51GHK2T.md" tag="YHK51GHK2T" kind="concept">
# Eilenberg–Steenrod axioms

## Summary

The Eilenberg–Steenrod axioms list the properties that a homology theory on pairs of spaces should have.
They are functoriality, homotopy invariance, a long exact sequence for each pair, excision, and a dimension axiom fixing the homology of a point.
[[Singular homology|M0Q5CFJ7ZH]] satisfies them, and on finite CW pairs any two theories that satisfy them with the same coefficient group agree.
Dropping the dimension axiom gives generalized homology and cohomology theories such as K-theory and cobordism.

## Explanations

### L1 human

There are many ways to build homology: from simplices, from cells, from singular chains.
The axioms say which properties really matter, so that one can prove things about all these constructions at once.
They act like a contract: any construction that keeps the contract gives the same answer on reasonable spaces.
For example, from the axioms alone one can compute that the $n$-sphere has homology only in degrees $0$ and $n$.
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<document source="https://tamnd.github.io/math2/node/YHKSDF59HK.md" tag="YHKSDF59HK" kind="concept">
# Perspective projection

## Summary

A perspective projection maps the points of space onto a picture plane along the lines through a fixed centre, the eye.
It is the geometry behind realistic drawing, photography and 3D computer graphics.
Straight lines stay straight, but parallel lines that are not parallel to the picture plane converge to a common vanishing point.
In projective terms it is a central projection, and its restriction to a plane is a [[perspectivity|QTZWZR1KWQ]] between planes.

## Explanations

### L1 human

Put a glass window between your eye and a street, and trace on the glass what you see.
Each point of the street is drawn where the ray from it to your eye crosses the glass.
The two edges of a straight road are parallel, but on the glass they run together toward a single point on the horizon.
Far objects come out small and near objects large, which is what makes the picture look real.
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<document source="https://tamnd.github.io/math2/node/YHSSQ67GDS.md" tag="YHSSQ67GDS" kind="statement">
# Doob decomposition theorem

## Statement

Let $(X_n)_{n \ge 0}$ be a sequence of real random variables, [[adapted|YRNFE016FF]] to a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_n)_{n \ge 0}$, with $E[\abs{X_n}] < \infty$ for all $n$.
Then there are a [[martingale|0CDVK9RQBH]] $(M_n)_{n \ge 0}$ and a sequence $(A_n)_{n \ge 0}$ of integrable random variables with $A_0 = 0$ and $A_n$ $\mathcal{F}_{n-1}$-measurable for $n \ge 1$, such that
$$X_n = M_n + A_n \quad \text{for all } n \ge 0.$$
This decomposition is unique up to equality almost surely.
Explicitly,
$$A_n = \sum_{k=1}^{n} \bigl(E[X_k \mid \mathcal{F}_{k-1}] - X_{k-1}\bigr).$$
Moreover, $(A_n)$ is almost surely non-decreasing if and only if $(X_n)$ is a [[submartingale|EQ055QMACW]].

## Remarks

Existence: with $A_n$ as above, $M_n = X_n - A_n$ satisfies $E[M_n - M_{n-1} \mid \mathcal{F}_{n-1}] = E[X_n \mid \mathcal{F}_{n-1}] - X_{n-1} - (A_n - A_{n-1}) = 0$.
Uniqueness: if $M_n + A_n = M'_n + A'_n$, then $M - M'$ is a martingale that is predictable, so $M_n - M'_n = E[M_n - M'_n \mid \mathcal{F}_{n-1}] = M_{n-1} - M'_{n-1}$, and it is constant equal to $0$.
Example: for the simple symmetric random walk $S_n$, $S_n^2 = (S_n^2 - n) + n$, so $A_n = n$.
A sequence $(A_n)$ with $A_n$ $\mathcal{F}_{n-1}$-measurable is called predictable.
Without the predictability of $A$ the decomposition is not unique: $X_n = 0 + X_n$ is another split into a martingale and an adapted process.

## Explanations

### L1 human

Any random process with finite means splits into a fair-game part and a trend part.
The trend is the part you can predict one step ahead, and the fair game is the surprise.
For a submartingale the trend only goes up.

### L2 human

The predictable part $A$ of $M^2$ for a square-integrable martingale $M$ is the predictable quadratic variation $\langle M \rangle$, a basic tool for martingale central limit theorems and stochastic integration.
The decomposition is the discrete-time model for the [[Doob–Meyer decomposition theorem|SX7ZPP72EM]], whose continuous-time proof is much harder.
It is also the discrete form of the split of a [[semimartingale|MZNFPJJMY9]] into a local martingale and a finite variation part.
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# Lax–Wendroff method

## Summary

The Lax–Wendroff method is an explicit [[finite difference|X7HYEBA5SA]] scheme of second order in space and time for hyperbolic equations and [[conservation laws|8RA7AMWSAF]].
It comes from a Taylor expansion in time in which the time derivatives are replaced by space derivatives using the equation.
It is stable under the CFL condition and accurate for smooth solutions, but it produces oscillations next to discontinuities.
This is why it is often combined with a first-order scheme through a flux limiter.

## Explanations

### L1 human

For the [[transport equation|PVD7VZJZ9B]] $u_t + a u_x = 0$, the value at the next time is $u + \Delta t\, u_t + \frac{\Delta t^2}{2} u_{tt}$ up to higher terms.
The equation says $u_t = -a u_x$ and $u_{tt} = a^2 u_{xx}$, so one only needs space derivatives, and these are replaced by centred differences.
The result uses three neighbouring grid values and is accurate to second order.
Near a jump, it overshoots and undershoots, creating small wiggles.
</document>

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# Fredholm integral equation

## Definition

Let $a < b$, let $K \colon [a, b] \times [a, b] \to \C$ be a [[kernel|GSMBXT7FPK]], let $f \colon [a, b] \to \C$ be given, and let $\lambda \in \C$.
A *Fredholm integral equation of the first kind* is the equation
$$\int_a^b K(x, s) \varphi(s) \, ds = f(x) \quad \text{for all } x \in [a, b].$$
A *Fredholm integral equation of the second kind* is the equation
$$\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds \quad \text{for all } x \in [a, b].$$
In both, $\varphi \colon [a, b] \to \C$ is the unknown function, and both limits of integration are the fixed numbers $a$ and $b$.
A *Fredholm integral equation* is a [[linear integral equation|AYZCKG34QK]] of one of these two forms.

## Remarks

Example: $\varphi(x) = x + \int_0^1 x s \, \varphi(s) \, ds$ on $[0, 1]$ is of the second kind.
A solution has the form $\varphi(x) = c x$ with $c = 1 + c/3$, so $\varphi(x) = 3x/2$.

Non-example: $\varphi(x) = x + \int_0^x x s \, \varphi(s) \, ds$ is not a Fredholm equation, since the upper limit is the variable $x$; it is a Volterra equation.

A Volterra equation can be written as a Fredholm equation with the kernel set to $0$ for $s > x$, but this kernel is in general discontinuous on the diagonal.
The same definitions are used with $[a, b]$ replaced by a region of $\R^m$ or a measure space.
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# Lommel function

## Summary

The Lommel functions $s_{\mu,\nu}$ and $S_{\mu,\nu}$ are particular solutions of Bessel's equation with right-hand side $z^{\mu + 1}$.
They contain the [[Struve function|K290RBT8GB]] and the [[Anger function|9P87VQZ0P7]] as special cases, up to adding Bessel functions.
They give a common frame for the inhomogeneous Bessel equations met in physics.

## Explanations

### L1 human

Bessel's equation with zero right-hand side has the Bessel functions as solutions.
If the right-hand side is a power $z^{\mu+1}$, one needs one extra solution to describe all the others.
The Lommel function $s_{\mu,\nu}$ is such a solution, written as a power series that starts with $z^{\mu+1}$.
For example, with $\mu = 1$ and $\nu = 0$ the constant $1$ already solves the equation, since $z^2 \cdot 0 + z \cdot 0 + z^2 \cdot 1 = z^2$.
The Lommel function is the solution that starts with $\frac{z^2}{4}$, namely $s_{1,0}(z) = 1 - J_0(z)$.
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# Chandrasekhar's X- and Y-functions

## Example

Let $\tau_1 > 0$ be the optical thickness of a plane-parallel slab.
Let $\Psi$ be an even polynomial, the characteristic function, with $\Psi \ge 0$ on $[0, 1]$ and $\int_0^1 \Psi\, d\mu \le \frac{1}{2}$.
*Chandrasekhar's X- and Y-functions* are the functions $X$ and $Y$ on $[0, 1]$ that solve the system
$$X(\mu) = 1 + \mu \int_0^1 \frac{\Psi(\mu')}{\mu + \mu'} \bigl[X(\mu) X(\mu') - Y(\mu) Y(\mu')\bigr]\, d\mu',$$
$$Y(\mu) = e^{-\tau_1/\mu} + \mu \int_0^1 \frac{\Psi(\mu')}{\mu - \mu'} \bigl[Y(\mu) X(\mu') - X(\mu) Y(\mu')\bigr]\, d\mu'.$$
In the second equation the bracket vanishes at $\mu' = \mu$, so the integrand has no singularity there.
This is a system of two coupled [[nonlinear integral equations|BSCYBP6AS6]] for the two unknowns.

Check of two limits.
For $\tau_1 = 0$, the constant functions $X = 1$ and $Y = 1$ solve the system: both brackets vanish and $e^{0} = 1$.
This is a slab of thickness $0$, which lets all light through.
As $\tau_1 \to \infty$, the term $e^{-\tau_1/\mu}$ tends to $0$; with $Y = 0$ the first equation becomes the equation of [[Chandrasekhar's H-function|99879Z5Y1K]], $X = H$.

## Remarks

The functions $X$ and $Y$ give the light reflected and transmitted by a slab of finite thickness in the [[radiative transfer equation|EX9YYRMAJA]], while $H$ handles a half-space.
In the conservative case $\int_0^1 \Psi\, d\mu = \frac{1}{2}$ the system does not determine $X$ and $Y$ uniquely, and an extra condition selects the physical solution.

## Explanations

### L1 human

When light falls on a slab of fog of finite thickness, part of it is reflected and part of it goes through.
Chandrasekhar described both parts with two functions, $X$ for the reflected light and $Y$ for the transmitted light.
They solve two equations that are tied together.
For a slab of zero thickness everything passes, and for an infinitely thick slab nothing passes and $X$ becomes the H-function.
</document>

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# Symmetric decreasing rearrangement

## Definition

Let $\abs{\cdot}$ denote [[Lebesgue measure|2Z2QVWZ262]] on $\R^n$.
For a measurable set $A \subseteq \R^n$ of finite measure, its *symmetric rearrangement* $A^*$ is the open ball centred at $0$ with $\abs{A^*} = \abs{A}$, and $A^* = \varnothing$ if $\abs{A} = 0$.
Let $f \colon \R^n \to \C$ be a [[measurable function|YYNCJSR0PJ]] that vanishes at infinity, that is, $\abs{\set{x : \abs{f(x)} > t}} < \infty$ for every $t > 0$.
The *symmetric decreasing rearrangement* of $f$ is the function $f^* \colon \R^n \to [0, \infty]$ given by
$$f^*(x) = \int_0^\infty \mathbf{1}_{\set{\abs{f} > t}^*}(x) \, dt,$$
where $\mathbf{1}_B$ is the indicator function of $B$.

## Remarks

The function $f^*$ is nonnegative, radial, non-increasing in $\abs{x}$ and lower semicontinuous.
For every $t > 0$, $\set{f^* > t} = \set{\abs{f} > t}^*$, so $f^*$ and $\abs{f}$ have the same distribution function, and $\norm{f^*}_p = \norm{f}_p$ for $1 \le p \le \infty$.
The Pólya–Szegő inequality states that $\norm{\nabla f^*}_2 \le \norm{\nabla f}_2$ for $f \in H^1(\R^n)$.
The Riesz rearrangement inequality states that $\iint f(x) g(x - y) h(y) \, dx \, dy \le \iint f^*(x) g^*(x - y) h^*(y) \, dx \, dy$ for nonnegative $f, g, h$.

Example: in $\R$, let $f = \mathbf{1}_{[1, 3]}$.
For $0 < t < 1$ the set $\set{f > t}$ is $[1, 3]$, of length $2$, so its rearrangement is $(-1, 1)$; for $t \ge 1$ it is empty.
Hence $f^* = \mathbf{1}_{(-1, 1)}$.

Non-example: the constant function $f = 1$ on $\R^n$ does not vanish at infinity, since $\set{f > 1/2}$ has infinite measure, so $f^*$ is not defined.

The one-variable [[decreasing rearrangement|BWS042NC2P]] $f^\sharp$ on $[0, \infty)$ is related by $f^*(x) = f^\sharp(\omega_n \abs{x}^n)$, where $\omega_n$ is the volume of the unit ball, up to values on a null set.
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# Sphere packing in dimensions 8 and 24

## Statement

1. Every [[sphere packing|HF32CBN5MA]] of equal balls in $\R^8$ has upper [[packing density|4P5PSWADJD]] at most $\pi^4/384 \approx 0.2537$. This is the density of the lattice packing given by the [[E8 lattice|2CXH65EKY7]].
2. Every sphere packing of equal balls in $\R^{24}$ has upper packing density at most $\pi^{12}/12! \approx 0.00193$. This is the density of the lattice packing given by the [[Leech lattice|XETV3657DG]].

## Remarks

Maryna Viazovska proved the case of dimension $8$ in 2016.
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko and Viazovska proved the case of dimension $24$ soon after.
Both proofs use the linear programming bound of Cohn and Elkies.
If $f \colon \R^n \to \R$ is a suitable integrable function with $f(0) = \hat f(0) > 0$, $f(x) \le 0$ for $\abs{x} \ge r$, and $\hat f \ge 0$, then every packing of balls of radius $r/2$ has density at most $\operatorname{vol}(B(0, r/2))$.
Viazovska built such a function for $r = \sqrt 2$ in dimension $8$ from modular forms and quasimodular forms, and the second paper did the same in dimension $24$.
The density values follow from the lattices.
$E_8$ has covolume $1$ and minimal length $\sqrt 2$, so the balls have radius $\sqrt 2/2$.
The Leech lattice has covolume $1$ and minimal length $2$, so the balls have radius $1$.

## Explanations

### L1 human

In most dimensions nobody knows the best way to pack equal balls.
Dimensions $8$ and $24$ are special: there are two extremely symmetric arrangements there, and these are now proved to be the best.
The proof is short compared with the proof in dimension $3$, but it uses deep tools from number theory.

### L2 human

The theorem completes the list of dimensions $1$, $2$, $3$, $8$, $24$ in which the sphere packing problem is solved.
The Cohn–Elkies bound is a Fourier analytic version of the [[Delsarte bound|7RPEDHAT7R]] for codes.
Numerical computations by Cohn and Elkies suggested that it is sharp in dimensions $2$, $8$ and $24$.
The work was to find the exact auxiliary function in dimensions $8$ and $24$.
Viazovska received the Fields Medal in 2022, partly for this work.
</document>

<document source="https://tamnd.github.io/math2/node/YJ961M051W.md" tag="YJ961M051W" kind="concept">
# Combinatorics on words

## Summary

Combinatorics on words studies finite and infinite sequences of letters from an alphabet, see the [[definition|N815PEPPFA]].
Typical topics are repetitions such as squares and overlaps, periodicity, unavoidable patterns, and the number of distinct factors of a word.
Classical objects of the field are the [[Thue–Morse sequence|7JZVT385PC]] and the [[de Bruijn sequences|GY8F3M39R8]].
It has links with algebra through the free [[monoid|0N6R902FG1]] of words, with number theory, and with the theory of formal languages.

## Explanations

### L1 human

A word is just a string of letters, such as $abaab$.
Combinatorics on words asks which patterns must appear in long words and which can be avoided.
For example, every word of length $4$ on two letters contains a block repeated twice in a row, such as $aa$ or $abab$.
On three letters, by contrast, there are infinite words with no such repeated block.
</document>

<document source="https://tamnd.github.io/math2/node/YJEBN8N3G8.md" tag="YJEBN8N3G8" kind="concept">
# WKB approximation

## Summary

The WKB approximation gives approximate solutions of linear differential equations whose highest derivative is multiplied by a small parameter, such as $\varepsilon^2 y'' = Q(x) y$.
It writes the solution as the exponential of a series in $\varepsilon$ and keeps the first terms.
It is also called the Liouville–Green method, and the name WKB refers to Wentzel, Kramers and Brillouin.
It is the basis of semiclassical approximations in quantum mechanics and of the asymptotics of special functions.

## Explanations

### L1 human

When a wave travels through a medium that changes slowly, it looks locally like a pure wave.
The WKB approximation lets the local wavelength and amplitude change slowly along the way.
For $\varepsilon^2 y'' = -k(x)^2 y$ with $k > 0$, the solutions are close to $k(x)^{-1/2} \cos\bigl(\frac{1}{\varepsilon} \int k(x) \, dx + c\bigr)$.
If $k$ is constant, this is exactly a cosine of frequency $k / \varepsilon$.
The approximation breaks down where $k$ vanishes, at turning points.
</document>

<document source="https://tamnd.github.io/math2/node/YJFP002B62.md" tag="YJFP002B62" kind="problem">
# Trigonometric moment problem

## Problem

Let $(c_k)_{k \in \Z}$ be complex numbers with $c_{-k} = \overline{c_k}$ for all $k$.
Decide whether there is a positive finite [[Borel measure|DYR1QFNGNJ]] $\mu$ on the [[circle|04EAFVFTF7]] $\mathbb{T} = \R / 2\pi\Z$ such that
$$c_k = \frac{1}{2\pi} \int_{\mathbb{T}} e^{-ik\theta} \, d\mu(\theta) \quad \text{for all } k \in \Z.$$
The numbers $c_k$ are then the trigonometric moments, or Fourier–Stieltjes coefficients, of $\mu$.

## Remarks

The problem is solved.
Such a measure exists if and only if, for every $n \ge 0$, the Hermitian [[Toeplitz matrix|GKY2WSHE26]] $T_n = (c_{j-k})_{j,k=0}^{n}$ is [[positive semidefinite|01MMVW8CKC]].
This is the Carathéodory–Toeplitz theorem, also known in the form of Herglotz's theorem on positive-definite sequences.
Necessity is a short computation: for complex numbers $\xi_0, \ldots, \xi_n$,
$$\sum_{j,k=0}^{n} c_{j-k} \xi_j \overline{\xi_k} = \frac{1}{2\pi} \int_{\mathbb{T}} \Big\lvert \sum_{j=0}^{n} \xi_j e^{-ij\theta} \Big\rvert^2 d\mu(\theta) \ge 0.$$
When the measure exists it is unique, because trigonometric polynomials are dense in the continuous functions on $\mathbb{T}$; this contrasts with the [[Hamburger moment problem|TBNCFXTMMW]].
The truncated problem, with only $c_0, \ldots, c_n$ given, is also solvable exactly when $T_n$ is positive semidefinite.

Example: $c_0 = 1$ and $c_k = 0$ for $k \ne 0$ give $T_n = I$, and $\mu$ is Lebesgue measure $d\theta$.
The sequence $c_0 = 1$, $c_{\pm 1} = 1$, $c_k = 0$ otherwise is not a moment sequence: the matrix $T_1 = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is positive semidefinite, but $T_2$ has determinant $-1$.

## Explanations

### L1 human

A measure on the circle can be summarized by its Fourier coefficients, one complex number for each integer frequency.
The trigonometric moment problem asks which sequences of numbers arise in this way from a positive measure.
The answer is a positivity test on matrices built from the sequence, the Toeplitz matrices, which have constant diagonals.
</document>

<document source="https://tamnd.github.io/math2/node/YJK4WF6YWV.md" tag="YJK4WF6YWV" kind="definition">
# Ramification group

## Definition

Let $L / K$ be a finite [[Galois extension|0QMZAQ6QQ4]] of non-Archimedean [[local fields|J85FZEQG38]], with [[Galois group|0XTY86BZKY]] $G$.
Let $\mathcal{O}_L$ be the valuation ring of $L$ and let $v_L$ be the normalized discrete valuation of $L$, with $v_L(L^\times) = \Z$.
For an integer $i \ge -1$, the $i$-th *ramification group* (in the lower numbering) is
$$G_i = \set{\sigma \in G : v_L(\sigma(a) - a) \ge i + 1 \text{ for all } a \in \mathcal{O}_L}.$$

## Remarks

Each $G_i$ is a normal [[subgroup|0X1JXT47GA]] of $G$, and $G_{-1} = G \supseteq G_0 \supseteq G_1 \supseteq \cdots$.
The group $G_0$ is the inertia group, and $G_i = 1$ for $i$ large.
If $p$ is the characteristic of the [[residue field|NJ3APQ8BHH]], then $G_0 / G_1$ is cyclic of order prime to $p$, and $G_i / G_{i+1}$ is an elementary abelian $p$-group for $i \ge 1$.
If $\mathcal{O}_L = \mathcal{O}_K[\alpha]$, then $\sigma \in G_i$ exactly when $v_L(\sigma(\alpha) - \alpha) \ge i + 1$.
Example: for $L = \Q_2(i)$ and $K = \Q_2$, one has $\mathcal{O}_L = \Z_2[i]$ and $v_L(2) = 2$.
For complex conjugation $\sigma$, $v_L(\sigma(i) - i) = v_L(-2i) = 2$, so $\sigma \in G_1$ and $\sigma \notin G_2$.
Non-example: if $L / K$ is unramified and $\sigma \neq 1$, then $\sigma$ acts nontrivially on the residue field, so $v_L(\sigma(a) - a) = 0$ for some $a$ and $\sigma \notin G_0$.
There is also an upper numbering $G^u$, defined through the Herbrand function, which behaves well under quotients.
</document>

<document source="https://tamnd.github.io/math2/node/YJK71RKVEB.md" tag="YJK71RKVEB" kind="concept">
# Analytic combinatorics

## Summary

Analytic combinatorics finds the asymptotic size of combinatorial counts from their [[generating functions|0PHAC833JF]], see the [[definition|XGZ2YYCNWK]].
It views a generating function as a function of a complex variable.
The singularities closest to the origin decide the exponential growth of the coefficients, and the type of these singularities decides the correction factor.
Together with the [[symbolic method|JP11D90EJC]], it is the subject of the book "Analytic Combinatorics" by Philippe Flajolet and Robert Sedgewick.

## Explanations

### L1 human

A generating function packs a whole sequence of counts into one function.
If one plugs complex numbers into that function, it blows up or misbehaves at some point.
The position of the nearest bad point tells how fast the counts grow.
For example, the Catalan generating function first breaks down at $z = 1/4$, so the Catalan numbers grow roughly like $4^n$.
The shape of the breakdown, here a square root, gives the extra factor $n^{-3/2}$.
</document>

<document source="https://tamnd.github.io/math2/node/YJM07BT0KK.md" tag="YJM07BT0KK" kind="statement">
# Wedderburn's theorem on simple Artinian rings

## Statement

Let $R$ be a [[simple ring|62M0NC3E2T]] that is left [[Artinian|20QEAM8AX2]].
Then there are an integer $n \ge 1$ and a [[division ring|0EA5BTA0CX]] $D$ such that $R \cong M_n(D)$, the [[matrix ring|0FHVPRAE60]] of $n \times n$ matrices over $D$.
The integer $n$ is unique, and $D$ is unique up to isomorphism.

## Remarks

Let $L$ be a minimal left ideal of $R$, which exists because $R$ is Artinian.
The sum of the right translates $La$ is a nonzero two-sided ideal, so it equals $R$.
Hence $R$ is a finite direct sum of copies of $L$, say $n$ of them.
Then $R^{\mathrm{op}} \cong \End_R(R) \cong M_n(\End_R(L))$, and $\End_R(L)$ is a division ring by Schur's lemma, so $R \cong M_n(D)$ with $D = \End_R(L)^{\mathrm{op}}$.
For example, a finite-dimensional simple algebra over $\R$ is $M_n(\R)$, $M_n(\C)$ or $M_n(\mathbb{H})$.
The Artinian hypothesis is needed: the Weyl algebra over $\C$ is simple and is not a matrix ring over a division ring.

## Explanations

### L1 human

Every simple ring that satisfies the finiteness condition "Artinian" is a ring of square matrices over a division ring.
For finite-dimensional algebras over a field this condition holds automatically.
So all such simple rings are known as soon as the division rings are known.

### L2 human

The theorem is the case of one factor in the Artin–Wedderburn theorem.
It is the starting point of the theory of central simple algebras: a central simple algebra over a field $K$ is $M_n(D)$ for a central division algebra $D$ over $K$.
This is why the Brauer group, which collects these $D$, classifies central simple algebras up to matrix size.
Uniqueness of $D$ follows because $D^{\mathrm{op}}$ is the endomorphism ring of the unique simple left module of $R$.
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<document source="https://tamnd.github.io/math2/node/YJMGEKHYEC.md" tag="YJMGEKHYEC" kind="definition">
# Ruled surface

## Definition

Let $k$ be an algebraically closed field, and let $\mathbb{P}^1$ be the [[projective line|ZT9A6JK8QF]] over $k$.
A *ruled surface* over $k$ is a smooth projective [[algebraic surface|KFTECAATKG]] $X$ together with a surjective [[morphism|CDS1VWB7A3]] $\pi \colon X \to C$, with the following properties.
The target $C$ is a smooth projective [[curve|XJM2MT4S3Y]], and every fibre $\pi^{-1}(c)$, $c \in C$, is isomorphic to $\mathbb{P}^1$.

## Remarks

Some books also ask for a section of $\pi$; over an algebraically closed field a section always exists, by Tsen's theorem.
Every ruled surface is isomorphic over $C$ to the projectivization $\mathbb{P}(\mathcal{E})$ of a vector bundle $\mathcal{E}$ of rank $2$ on $C$.
Another usage calls a surface *ruled* when it is only birational to $C \times \mathbb{P}^1$; the surfaces of the definition are then called geometrically ruled.
Example: for any smooth projective curve $C$, the projection $C \times \mathbb{P}^1 \to C$ has every fibre $\set{c} \times \mathbb{P}^1 \cong \mathbb{P}^1$, so $C \times \mathbb{P}^1$ is ruled.
Non-example: the projective plane $\mathbb{P}^2$ is not ruled, because every morphism from $\mathbb{P}^2$ to a curve is constant.
Indeed, two fibres of a nonconstant one would be disjoint curves in $\mathbb{P}^2$, but any two curves in $\mathbb{P}^2$ meet.
</document>

<document source="https://tamnd.github.io/math2/node/YJRBCWQV6B.md" tag="YJRBCWQV6B" kind="concept">
# Commuting elements

## Summary

Two elements $a$ and $b$ commute for a [[binary operation|0M418FE7BB]] $\ast$ when $a \ast b = b \ast a$.
An operation is [[commutative|099E72KTF8]] exactly when every pair of elements commutes, but in a noncommutative structure some pairs commute and some do not.
In a [[group|G5S1M9]], $a$ and $b$ commute if and only if their [[commutator|0PXW0BN7DW]] is the identity.
[[Commuting matrices|8G3X8MPG7E]] are an important special case.

## Explanations

### L1 human

Two things commute when doing them in either order gives the same result.
Turning a square by $90$ degrees and turning it by $180$ degrees commute: both orders give a turn by $270$ degrees.
A turn by $90$ degrees and a flip across a diagonal do not commute: the two orders give different pictures.
Every element commutes with itself and with the identity element.
</document>

<document source="https://tamnd.github.io/math2/node/YJRJVZ998D.md" tag="YJRJVZ998D" kind="concept">
# 3-j symbol

## Summary

The Wigner 3-j symbol $\begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix}$ is a number that describes how two irreducible representations of [[SU(2)|E5P8AG6CK7]] combine into a third.
It is the Clebsch–Gordan coefficient of the tensor product, rescaled and with a sign, so that it has simple symmetries under permutation of its columns.
It vanishes unless $m_1 + m_2 + m_3 = 0$ and $j_1, j_2, j_3$ satisfy the triangle inequality.
3-j symbols appear in the coupling of angular momenta in quantum mechanics and in products of [[spherical harmonics|3F32Y21HP4]].

## Explanations

### L1 human

Two spinning particles with spins $j_1$ and $j_2$ together can have total spin $j_3$ only for a few values of $j_3$.
The 3-j symbols are the coefficients that express a state of total spin in terms of the states of the two particles.
They are arranged in a table with three columns, one per spin, and the bottom row lists the projections on an axis.
For two spins $\frac12$ that combine to spin $0$, the symbols are $\pm 1/\sqrt{2}$, which gives the singlet state $(\uparrow\downarrow - \downarrow\uparrow)/\sqrt{2}$.
</document>

<document source="https://tamnd.github.io/math2/node/YJS6X8JN7D.md" tag="YJS6X8JN7D" kind="problem">
# Inverse Galois problem

## Problem

Is every finite group $G$ isomorphic to the [[Galois group|0XVR72M51M]] $\Gal(L/\Q)$ of some [[Galois extension|0DS2F7GK05]] $L$ of the rational numbers $\Q$?

## Remarks

The problem is open.
Every finite abelian group is a Galois group over $\Q$, realised inside a [[cyclotomic extension|VD6VC3ZMMY]] $\Q(\zeta_n)$.
The symmetric groups $S_n$ are Galois groups over $\Q$; one route uses [[Hilbert's irreducibility theorem|N1V3YEYHNP]].
Every finite solvable group is a Galois group over $\Q$, by [[Shafarevich's theorem|SDV692Z915]].
The same question can be asked over other fields; over $\C(t)$ every finite group is a Galois group.
Over a finite field $\F_q$ only cyclic groups occur, since every finite extension of $\F_q$ is cyclic.

## Explanations

### L1 human

Galois theory attaches a finite group to each polynomial with rational coefficients.
The inverse problem asks whether every finite group appears in this way.
Many families of groups are known to appear, but no proof covers all finite groups.
</document>

<document source="https://tamnd.github.io/math2/node/YK3DMN2NSK.md" tag="YK3DMN2NSK" kind="concept">
# Mirror symmetry

## Summary

Mirror symmetry is a correspondence, first predicted by string theory, between pairs of [[Calabi–Yau manifolds|STYAB4PFDW]] $X$ and $\check{X}$.
The symplectic geometry of $X$, encoded by its [[Gromov–Witten invariants|XVJA0JD79P]], matches the complex geometry of $\check{X}$, encoded by the variation of its [[Hodge structures|ZSM083M97C]].
For Calabi–Yau threefolds the Hodge numbers are exchanged: $h^{1,1}(X) = h^{2,1}(\check{X})$ and $h^{2,1}(X) = h^{1,1}(\check{X})$.
It is not a single theorem but a program with several precise forms, such as [[homological mirror symmetry|B5N9WJH5F0]].

## Explanations

### L1 human

Two spaces that look unrelated can carry the same hidden information, read in two different languages.
On one side one counts curves inside a space, which is a hard enumerative problem.
On the mirror side one computes certain integrals of differential forms, called periods, which is much easier.
In 1991 Candelas, de la Ossa, Green and Parkes used this to predict the number of rational curves of each degree on a quintic threefold in $\mathbb{P}^4$.
The predicted numbers include $2875$ lines and $609250$ conics.
These predictions were later proved by mathematicians, and mirror symmetry became a major field of geometry.
</document>

<document source="https://tamnd.github.io/math2/node/YK8121V1PN.md" tag="YK8121V1PN" kind="definition">
# Ambiguity function

## Definition

Let $f \in L^2(\R)$ (see [[Lp space|SASP14MA1Z]]).
The *ambiguity function* of $f$ is the function $Af \colon \R^2 \to \C$,
$$Af(x, \omega) = \int_{\R} f\!\left(t + \tfrac{x}{2}\right) \overline{f\!\left(t - \tfrac{x}{2}\right)} \, e^{-2\pi i \omega t} \, dt,$$
for a time shift $x \in \R$ and a frequency shift $\omega \in \R$.

## Remarks

The substitution $t = s - x/2$ gives $Af(x, \omega) = e^{\pi i x \omega} V_f f(x, \omega)$, where $V_f f$ is the [[short-time Fourier transform|3DHNP5NEWH]] of $f$ with window $f$.
Hence $\abs{Af(x, \omega)} \le Af(0, 0) = \norm{f}_2^2$, by the Cauchy–Schwarz inequality.
In radar texts the ambiguity function is often written as $\chi(\tau, \nu) = \int s(t) \overline{s(t - \tau)} e^{2\pi i \nu t} \, dt$.
It differs from $Af$ by a sign convention and a phase factor, and $\abs{\chi(\tau, \nu)} = \abs{Af(\tau, -\nu)}$ for $s = f$.

Example: let $f(t) = 2^{1/4} e^{-\pi t^2}$.
Then $f(t + x/2) \overline{f(t - x/2)} = \sqrt2 \, e^{-\pi x^2/2} e^{-2\pi t^2}$, and the Fourier transform of $e^{-2\pi t^2}$ is $2^{-1/2} e^{-\pi \omega^2/2}$.
So $Af(x, \omega) = e^{-\pi (x^2 + \omega^2)/2}$, a single peak of height $1 = \norm{f}_2^2$ at the origin.

Non-example: the autocorrelation $R(x) = \int_\R f(t) \overline{f(t - x)} \, dt$ is not the ambiguity function.
The substitution $t = s + x/2$ shows $R(x) = Af(x, 0)$, so $R$ is only the slice $\omega = 0$ of $Af$ and does not see frequency shifts.
</document>

<document source="https://tamnd.github.io/math2/node/YKETR75ZYW.md" tag="YKETR75ZYW" kind="problem">
# Discrete Dirichlet problem

## Problem

Let $G = (V, E)$ be a finite connected [[graph|00C0BR7D7H]] with symmetric positive edge weights $c(x, y)$.
Let $B \subseteq V$ be a nonempty set of boundary vertices, and let $I = V \setminus B$ be the set of interior vertices.
Let $g \colon B \to \R$ be a function.
Find a function $f \colon V \to \R$ that is [[harmonic|WRY0M7SW59]] at every vertex of $I$ and satisfies $f = g$ on $B$.

## Remarks

The problem always has exactly one solution.
Uniqueness follows from the maximum principle: the difference $h$ of two solutions is harmonic on $I$ and zero on $B$.
At an interior vertex where $h$ is largest, harmonicity forces the same value at all neighbours.
Connectedness spreads this value to a vertex of $B$, so $\max h = 0$, and in the same way $\min h = 0$.
The conditions form a square system of $\abs{I}$ linear equations in the $\abs{I}$ unknowns $f(x)$, $x \in I$, so uniqueness gives existence.
The solution has the probabilistic form $f(x) = \mathbb{E}_x\big[g(X_\tau)\big]$, where $(X_n)$ is the random walk with $p(x, y) = c(x, y)/c(x)$ started at $x$ and $\tau$ is its first hitting time of $B$.
Example: on the path with vertices $0, 1, \dots, n$, unit weights and $B = \set{0, n}$, the solution is the linear interpolation $f(k) = g(0) + \tfrac{k}{n}\big(g(n) - g(0)\big)$.
Connectedness matters: if a component of the graph does not meet $B$, any constant can be added to $f$ on that component, and the solution is not unique.

## Explanations

### L1 human

Fix numbers on some vertices of a graph, the boundary.
The task is to fill in the other vertices so that each of them carries the average of its neighbours.
Think of the boundary numbers as fixed voltages, or fixed temperatures, and of the filled-in numbers as the equilibrium state.
On a connected finite graph there is always exactly one way to do it.
It can be found by solving a linear system or by running random walks to the boundary.
</document>

<document source="https://tamnd.github.io/math2/node/YKN9BKTCYN.md" tag="YKN9BKTCYN" kind="definition">
# Kauffman bracket

## Definition

Let $D$ be an unoriented diagram of a tame [[link|1A5WRCNTM3]] in $\R^3$, with crossing set $X$.
At a crossing, the *A-regions* are the two regions swept out when the over-strand is turned counterclockwise.
The *A-smoothing* joins the two A-regions, and the *B-smoothing* joins the other two regions.
A *state* $s$ of $D$ chooses a smoothing at each crossing; let $a(s)$ and $b(s)$ count its A- and B-smoothings.
Let $\abs{s}$ be the number of circles in the resulting diagram without crossings.
The *Kauffman bracket* of $D$ is
$$\langle D \rangle = \sum_{s} A^{a(s) - b(s)} \, (-A^2 - A^{-2})^{\abs{s} - 1} \in \Z[A, A^{-1}],$$
where the sum runs over all $2^{\abs{X}}$ states.

## Remarks

Equivalently, $\langle O \rangle = 1$ for a single circle, adding a disjoint circle multiplies by $-A^2 - A^{-2}$, and each crossing expands as $A$ times its A-smoothing plus $A^{-1}$ times its B-smoothing.
Example: a one-crossing diagram of the [[unknot|1TQ06R4Q66]] has one state with one circle and one with two circles.
Depending on the sign of the kink, the bracket is $A + A^{-1}(-A^2 - A^{-2}) = -A^{-3}$ or its mirror value $-A^{3}$.
Non-example of invariance: this kinked diagram and the round circle depict the same knot but have brackets $-A^{\mp 3}$ and $1$, so $\langle \cdot \rangle$ is not a link invariant.
It is invariant under the second and third [[Reidemeister moves|T3HAGWYKK7]], and the first move multiplies it by $-A^{\pm 3}$.
For an oriented diagram with writhe $w(D)$, the product $(-A^3)^{-w(D)} \langle D \rangle$ is an invariant, and $A = t^{-1/4}$ turns it into the [[Jones polynomial|4SPC17FPXE]].
</document>

<document source="https://tamnd.github.io/math2/node/YKSA8SJ81V.md" tag="YKSA8SJ81V" kind="statement">
# Theorema Egregium

## Statement

Let $S$ and $\bar S$ be [[regular surfaces|6080X3FAD4]] in $\R^3$, and let $\varphi \colon S \to \bar S$ be a smooth map.
Assume that $\varphi$ is a local isometry: for each $p \in S$ and $v, w \in T_p S$, $\ip{d\varphi_p(v)}{d\varphi_p(w)} = \ip{v}{w}$.
Let $K$ and $\bar K$ be the [[Gaussian curvatures|SQVQ5DQT6W]] of $S$ and $\bar S$.
Then $\bar K(\varphi(p)) = K(p)$ for each $p \in S$.

## Remarks

The key step is that $K$ can be written, in any local parametrisation, through $E, F, G$ of the [[first fundamental form|MAJ3DH7MZS]] and their partial derivatives of order at most $2$.
This follows from the Gauss equation among the [[Gauss–Codazzi equations|68TVVHCQ3Z]], which expresses $eg - f^2$ through the Christoffel symbols, and these depend only on $E, F, G$.
A local isometry has, in suitable parametrisations, the same functions $E, F, G$, hence the same $K$.
Example: the map $(u, v) \mapsto (\cos u, \sin u, v)$ wraps the plane onto the unit cylinder as a local isometry, and indeed both have $K = 0$.
Consequence: no open piece of a sphere, where $K > 0$, is isometric to a piece of the plane, so every flat map of a region of the Earth distorts some distances.
The converse fails: a map that preserves $K$ need not be a local isometry.
For example, the map $(x, y, 0) \mapsto (2x, y, 0)$ of the plane keeps $K = 0$ but doubles some lengths.

## Explanations

### L1 human

Gaussian curvature can be measured by someone who lives on the surface and only measures lengths on it.
Bending a surface without stretching it, like rolling a sheet of paper, never changes its Gaussian curvature.
This is why you cannot flatten an orange peel without tearing it, and why every flat map of the Earth is distorted.

### L2 human

Gauss called this result remarkable because $K$ was defined through the normal and the second fundamental form, which depend on how the surface sits in space.
The theorem shows that $K$ is intrinsic, and it is the starting point of Riemannian geometry, where curvature is defined from the metric alone.
In orthogonal coordinates, where $F = 0$, one has
$$K = -\frac{1}{2\sqrt{EG}} \left( \frac{\partial}{\partial u} \frac{G_u}{\sqrt{EG}} + \frac{\partial}{\partial v} \frac{E_v}{\sqrt{EG}} \right).$$
The mean curvature is not intrinsic: the plane and the cylinder are locally isometric but have $H = 0$ and $H \ne 0$.
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# Developable surface

## Summary

A developable surface is a surface in space whose Gaussian curvature is zero at every point.
Such a surface can be flattened onto a plane, at least locally, without stretching or tearing.
Cylinders, cones and the surfaces swept by the tangent lines of a space curve are the basic examples.
They matter in practice because they are the shapes one can bend out of a flat sheet of paper or metal.

## Explanations

### L1 human

Roll a sheet of paper into a tube or a cone: you get a developable surface.
No length on the paper changes when you do this, so you can unroll it flat again.
A sphere is not developable, which is why every flat map of the Earth distorts some distances.
The number that decides this is the Gaussian curvature: it is zero on a cylinder and positive on a sphere.
</document>

<document source="https://tamnd.github.io/math2/node/YKZ1FP1PAG.md" tag="YKZ1FP1PAG" kind="definition">
# Circle average

## Definition

Let $E$ be $\R$ or $\C$, let $c \in \C$ and let $r \ge 0$.
Let $f$ be a function with values in $E$, defined at least on the [[circle|060W5CBZP6]] $\set{z \in \C : \abs{z - c} = r}$.
Assume that $\theta \mapsto f(c + r e^{i\theta})$ is [[Lebesgue integrable|DWJF7ABPR1]] on $[0, 2\pi]$.
The *circle average* of $f$ on the circle of centre $c$ and radius $r$ is
$$M(f; c, r) = \frac{1}{2\pi} \int_0^{2\pi} f(c + r e^{i\theta}) \, d\theta.$$

## Remarks

Example: let $f(z) = z$, with $c \in \C$ and $r > 0$.
Then $M(f; c, r) = c + \frac{r}{2\pi} \int_0^{2\pi} e^{i\theta} \, d\theta = c$, since the integral of $e^{i\theta}$ over a full period is $0$.
Non-example: let $f(z) = 1/\abs{z - 1}$ on $\C \setminus \set{1}$, with $c = 0$ and $r = 1$.
Then $f(e^{i\theta}) = 1/(2\abs{\sin(\theta/2)})$ for $0 < \theta < 2\pi$, which is not integrable near $\theta = 0$, so the circle average is not defined.
For $r = 0$ the circle average is $f(c)$.
The [[mean value property|0HYPR64JM2]] says that $M(u; c, r) = u(c)$ when $u$ is harmonic on a neighbourhood of the closed disc of centre $c$ and radius $r$.
For $r > 0$, $M(f; c, r)$ is the integral of $f$ against arc length on the circle, divided by the length $2\pi r$.
</document>

<document source="https://tamnd.github.io/math2/node/YM1BNA80RC.md" tag="YM1BNA80RC" kind="concept">
# Likelihood-ratio test

## Summary

The likelihood-ratio test compares the best fit of a model under a null hypothesis with its best fit without that restriction.
It rejects the null hypothesis when the maximized [[likelihood|0VW9E4QXR6]] under the null is much smaller than the overall maximum.
It is the most widely used general method to build a [[hypothesis test|087P03NEND]], and [[Wilks' theorem|WQPB6GXD0M]] gives its approximate null law.

## Explanations

### L1 human

Fit the model twice: once with the constraint of the null hypothesis, and once freely.
The free fit is always at least as good, so the question is whether it is much better.
If the constrained fit explains the data far worse, the constraint is probably false.
For a coin with $60$ heads in $100$ tosses, the fair-coin likelihood is compared with the likelihood at $p = 0.6$, the best value.
</document>

<document source="https://tamnd.github.io/math2/node/YM3JZ34PME.md" tag="YM3JZ34PME" kind="statement">
# Rellich–Kondrachov theorem

## Statement

Let $n \ge 2$ and let $U \subset \R^n$ be a bounded open set with $C^1$ boundary.

1. Let $1 \le p < n$ and $p^* = \frac{np}{n - p}$. For every $q$ with $1 \le q < p^*$, the inclusion of the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(U)$ into the [[Lp space|SASP14MA1Z]] $L^q(U)$ is compact.
2. Let $n \le p \le \infty$. Then the inclusion of $W^{1,p}(U)$ into $L^p(U)$ is compact.

Here compact means that every bounded sequence in $W^{1,p}(U)$ has a subsequence that converges in $L^q(U)$, respectively in $L^p(U)$.

## Remarks

The inclusion is bounded by the [[Sobolev inequality|B03CA692K3]].
For compactness, one mollifies: the mollified functions $u^\varepsilon$ form an equicontinuous and bounded family for each fixed $\varepsilon$, so the [[Arzelà–Ascoli theorem|0381S8Z5P0]] applies to them.
The error $\norm{u^\varepsilon - u}_{L^1}$ is at most $\varepsilon \norm{\nabla u}_{L^1}$, uniformly on bounded sets, and interpolation with the bound in $L^{p^*}$ gives the same uniform control in $L^q$ for $q < p^*$.
So the inclusion is a [[compact operator|M2NEP1K5AE]].

The hypotheses are needed.
The inclusion into $L^{p^*}(U)$ itself is not compact: the rescaled functions $\lambda^{(n-p)/p} \varphi(\lambda x)$ for a fixed bump $\varphi$ are bounded in $W^{1,p}$ but concentrate at a point as $\lambda \to \infty$.
On an unbounded set, such as $U = \R^n$, translates $\varphi(x - k e_1)$ of a bump are bounded in $W^{1,p}$ but have no convergent subsequence in $L^p$.

## Explanations

### L1 human

A bounded family of functions with bounded slopes cannot oscillate wildly.
The theorem turns this into a compactness statement: from any such sequence one can extract a subsequence that converges in the mean.
It is the infinite-dimensional analogue of the fact that a bounded sequence of numbers has a convergent subsequence.

### L2 human

Compactness is how one passes to the limit in approximations of nonlinear equations: a weak limit in $W^{1,p}$ becomes a strong limit in $L^q$, so nonlinear terms in $u$ converge.
It also implies that the inverse of an elliptic operator such as $-\Delta$ with Dirichlet conditions is compact on $L^2(U)$, which gives a discrete spectrum of eigenvalues.
The Poincaré inequality for functions with zero mean is proved from it by contradiction.
</document>

<document source="https://tamnd.github.io/math2/node/YM68BCKBXC.md" tag="YM68BCKBXC" kind="statement">
# Multiplication theorem

## Statement

Let $m \ge 1$ be an integer.

1. For the [[gamma function|0H5SYBSED9]] and every $z \in \C$ with $mz \notin \set{0, -1, -2, \dots}$,
$$\prod_{k=0}^{m-1} \Gamma\!\left(z + \frac{k}{m}\right) = (2\pi)^{(m-1)/2}\, m^{1/2 - mz}\, \Gamma(mz).$$
2. For the [[digamma function|HWRRXB0YXS]] $\psi = \Gamma'/\Gamma$ and the same $z$,
$$\psi(mz) = \ln m + \frac{1}{m} \sum_{k=0}^{m-1} \psi\!\left(z + \frac{k}{m}\right).$$
3. For the [[Bernoulli polynomials|BXQKBJJGFG]] $B_n$, $n \ge 0$, and every $x \in \C$,
$$B_n(mx) = m^{n-1} \sum_{k=0}^{m-1} B_n\!\left(x + \frac{k}{m}\right).$$
4. For the [[Hurwitz zeta function|G4SW7HJG88]] $\zeta(s, a) = \sum_{j \ge 0} (j + a)^{-s}$, every real $x > 0$ and every $s$ with $\operatorname{Re} s > 1$,
$$\sum_{k=0}^{m-1} \zeta\!\left(s, x + \frac{k}{m}\right) = m^{s} \zeta(s, mx).$$

## Remarks

Part 1 is the [[Gauss multiplication formula|Y4759V8CY6]], and part 2 follows from it by taking the logarithmic derivative.
Part 4 comes from splitting the sum defining $\zeta(s, mx) = m^{-s} \sum_{j \ge 0} (j/m + x)^{-s}$ according to the residue $k$ of $j$ modulo $m$.
Part 3 follows from the generating function $t e^{xt}/(e^t - 1) = \sum_n B_n(x) t^n/n!$: summing over $k$ gives a geometric sum that collapses to the generating function at $mx$ with $t$ replaced by $t/m$.
For $m = 2$, part 1 is the Legendre duplication formula, and part 3 gives $B_n(2x) = 2^{n-1}\big(B_n(x) + B_n(x + \tfrac{1}{2})\big)$.
At $x = 0$ and $n = 1$ this reads $B_1(0) = B_1(0) + B_1(\tfrac{1}{2})$, consistent with $B_1(\tfrac{1}{2}) = 0$.

## Explanations

### L1 human

Several classical functions have a rule for what happens when the argument is multiplied by a whole number $m$.
The value at $mx$ can be recovered from the values at the $m$ evenly spaced points $x, x + 1/m, \dots, x + (m-1)/m$.
These rules hold for the gamma function, for the Bernoulli polynomials and for the Hurwitz zeta function, among others.

### L2 human

All these functions satisfy a simple difference equation in $x$, such as $\Gamma(x + 1) = x\Gamma(x)$ or $B_n(x + 1) - B_n(x) = n x^{n-1}$.
The sum or product over the $m$ shifted points satisfies the same kind of equation in the variable $mx$.
A uniqueness argument for that difference equation identifies it with the function at $mx$.
For the gamma function this uniqueness is the Bohr–Mollerup theorem.
The formulas are used to compute special values, such as $\Gamma(1/2) = \sqrt{\pi}$, and they connect to distribution relations in number theory.
</document>

<document source="https://tamnd.github.io/math2/node/YM708KHV2W.md" tag="YM708KHV2W" kind="concept">
# Ordinary least squares

## Summary

Ordinary least squares estimates the coefficients of a [[linear regression|41FVC4P5JB]] by minimising the sum of squared residuals.
The estimate has the closed form $\hat\beta = (X^T X)^{-1} X^T y$ when the design matrix has full column rank.
Geometrically, the fitted values are the [[orthogonal projection|0B8KZRT28M]] of the data onto the column space of the design matrix.
By the [[Gauss–Markov theorem|N3YMNMWANM]], it is the best linear unbiased estimator under the classical assumptions.

## Explanations

### L1 human

Draw a line through a cloud of points.
For each point, measure the vertical gap between the point and the line, and square it.
Ordinary least squares picks the line that makes the sum of these squared gaps as small as possible.
For the points $(0, 0)$, $(1, 1)$ and $(2, 3)$, the best line is $y = -1/6 + 3x/2$.
</document>

<document source="https://tamnd.github.io/math2/node/YMDADSX0E1.md" tag="YMDADSX0E1" kind="definition">
# Local asymptotic normality

## Definition

Let $\Theta \subseteq \R^k$ be open, and for each $n$ let $\set{P_{n,\theta} : \theta \in \Theta}$ be a family of probability measures on a measurable space $(\Omega_n, \mathcal{F}_n)$.
Let $\theta \in \Theta$.
The sequence of models is *locally asymptotically normal* (LAN) at $\theta$ if the following holds.
There are random vectors $\Delta_{n,\theta}$ on $\Omega_n$ and a $k \times k$ [[positive definite matrix|0QQCG3JWJ4]] $I_\theta$ such that:

1. $\Delta_{n,\theta} \xrightarrow{d} \mathcal{N}_k(0, I_\theta)$ under $P_{n,\theta}$, in the sense of [[convergence in distribution|0JDAQHWA9N]] to a [[multivariate normal|0G558GJQMW]] law;
2. for every sequence $h_n \to h$ in $\R^k$,
$$\log \frac{dP_{n, \theta + h_n/\sqrt{n}}}{dP_{n,\theta}} = h^T \Delta_{n,\theta} - \frac{1}{2} h^T I_\theta h + o_{P_{n,\theta}}(1).$$
Here the log-likelihood ratio uses the [[Radon–Nikodym derivative|1NTKG64F7N]] of the absolutely continuous part.
The term $o_{P_{n,\theta}}(1)$ is a sequence of random variables that tends to $0$ in [[probability|0EFDDXTVD6]] under $P_{n,\theta}$.

## Remarks

Example: for i.i.d. $\mathcal{N}(\theta, 1)$ observations, $P_{n,\theta} = \mathcal{N}(\theta, 1)^{\otimes n}$, the log-likelihood ratio is exactly $h\, \Delta_n - h^2/2$ with $\Delta_n = n^{-1/2}\sum_i (X_i - \theta) \sim \mathcal{N}(0, 1)$, so the model is LAN with $I_\theta = 1$ and no remainder.

More generally, if $X_1, \dots, X_n$ are i.i.d. from a model that is differentiable in quadratic mean at $\theta$ with nonsingular [[Fisher information|G1Z3WSTE8Q]] $I_\theta$, the model is LAN at $\theta$ with $\Delta_{n,\theta} = n^{-1/2}\sum_i s(\theta; X_i)$, where $s$ is the [[score|CP97GRVTQ0]].

Non-example: for i.i.d. Uniform$(0, \theta)$ observations and $h \ne 0$, $P_{n, \theta + h/\sqrt{n}}$ and $P_{n,\theta}$ become asymptotically singular.
The reason is that the scale at which they differ is $1/n$; clause 2 fails, and the model is not LAN.

Under LAN, $P_{n,\theta + h/\sqrt{n}}$ and $P_{n,\theta}$ are mutually [[contiguous|DQGJMP9WHC]], since the limit law $\mathcal{N}(-\frac{1}{2} h^T I_\theta h, \, h^T I_\theta h)$ of the log-likelihood ratio has $E e^{L} = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/YMGHNBDZCP.md" tag="YMGHNBDZCP" kind="definition">
# Oscillation of a function

## Definition

Let $D \subseteq \R$, let $f \colon D \to \R$ be a [[function|0XHDTE5YMZ]], and let $A \subseteq D$ be nonempty.
The *oscillation of $f$ on $A$* is
$$\operatorname{osc}_f(A) = \sup_{x \in A} f(x) - \inf_{x \in A} f(x) = \sup \set{\abs{f(x) - f(y)} : x, y \in A},$$
a value in $[0, \infty]$.
For $x_0 \in D$, the *oscillation of $f$ at $x_0$* is
$$\operatorname{osc}_f(x_0) = \inf_{\varepsilon > 0} \operatorname{osc}_f\big(D \cap (x_0 - \varepsilon, x_0 + \varepsilon)\big).$$

## Remarks

The quantity $\operatorname{osc}_f(D \cap (x_0 - \varepsilon, x_0 + \varepsilon))$ decreases as $\varepsilon$ decreases, so the infimum is also the limit as $\varepsilon \to 0^+$.
The function $f$ is [[continuous at|0EEE2CZB2A]] $x_0$ if and only if $\operatorname{osc}_f(x_0) = 0$.
For each $c > 0$, the set $\set{x \in D : \operatorname{osc}_f(x) \ge c}$ is closed in $D$.
So the set of discontinuities of $f$ is a countable union of closed subsets of $D$.

Example: let $f(x) = x^2$ on $\R$ and $x_0 = 1$.
On $(1 - \varepsilon, 1 + \varepsilon)$ with $\varepsilon < 1$, the values fill $((1 - \varepsilon)^2, (1 + \varepsilon)^2)$, a spread of $4\varepsilon$.
This tends to $0$, so $\operatorname{osc}_f(1) = 0$, as continuity predicts.

Non-example of continuity: the [[Dirichlet function|0T4EJ8ZJM9]], equal to $1$ on $\Q$ and $0$ elsewhere, takes both values on every interval.
So its oscillation is $1$ at every point, and it is continuous nowhere.
</document>

<document source="https://tamnd.github.io/math2/node/YMHYAEJQ80.md" tag="YMHYAEJQ80" kind="statement">
# Bochner–Martinelli formula

## Statement

Let $n \ge 1$ and let $D \subseteq \C^n$ be a bounded domain with $C^1$ boundary $\partial D$, oriented as the boundary of $D$.
For $\zeta \ne z$ in $\C^n$, let $\omega(\zeta, z)$ be the Bochner–Martinelli kernel, the $(n, n-1)$-form in $\zeta$
$$\omega(\zeta, z) = \frac{(n-1)!}{(2\pi i)^n} \frac{1}{\abs{\zeta - z}^{2n}} \sum_{j=1}^n (\bar\zeta_j - \bar z_j) \, d\zeta_j \wedge \bigwedge_{k \ne j} \left( d\bar\zeta_k \wedge d\zeta_k \right).$$
Let $f$ be a function of class $C^1$ on a neighbourhood of the closure $\overline{D}$.
Then for every $z \in D$,
$$f(z) = \int_{\partial D} f(\zeta) \, \omega(\zeta, z) - \int_{D} \bar\partial f(\zeta) \wedge \omega(\zeta, z),$$
where $\bar\partial$ is the [[$\bar\partial$ operator|M1VKCW3RFW]] in $\zeta$.

## Remarks

The forms $d\bar\zeta_k \wedge d\zeta_k$ have degree $2$, so the order of the factors in the product does not matter.
For fixed $z$, the kernel satisfies $d_\zeta \, \omega(\zeta, z) = 0$ on $\zeta \ne z$.
Apply Stokes' theorem on $D$ minus a small ball around $z$ and let the radius tend to $0$; the small sphere contributes $f(z)$ in the limit.
If $f$ is [[holomorphic|507C34W0ZT]] near $\overline{D}$, then $\bar\partial f = 0$ and $f(z) = \int_{\partial D} f(\zeta) \, \omega(\zeta, z)$.
For $n = 1$, $\omega(\zeta, z) = \frac{1}{2\pi i} \frac{d\zeta}{\zeta - z}$ and the formula is the Cauchy–Pompeiu formula, which contains the Cauchy integral formula.
For $n \ge 2$ the kernel is not holomorphic in $z$, unlike the Cauchy kernel of the polydisc, but it works on every bounded domain with $C^1$ boundary.

## Explanations

### L1 human

The formula recovers the values of a function inside a domain from its values on the boundary.
A correction term measures how far the function is from being holomorphic.
For holomorphic functions the correction is zero.
It is a version of the Cauchy integral formula that works on any reasonable bounded domain of $\C^n$, not only on products of discs.

### L2 human

The kernel is built from the fundamental solution of the Laplacian on $\R^{2n}$, which is why it is not holomorphic in $z$ when $n \ge 2$.
The formula is a main tool for solving the equation $\bar\partial u = g$ and for proving extension results such as the Hartogs–Bochner theorem.
It is the special case $\eta_j = (\bar\zeta_j - \bar z_j)/\abs{\zeta - z}^2$ of the [[Cauchy–Fantappiè formula|R9JJ44TMQB]] for holomorphic functions.
</document>

<document source="https://tamnd.github.io/math2/node/YMK0D06F4B.md" tag="YMK0D06F4B" kind="definition">
# Site

## Definition

A *site* is a pair $(\mathcal{C}, J)$, where $\mathcal{C}$ is a [[category|A2DE66937D]] and $J$ is a [[Grothendieck topology|GDRFSJ8E2B]] on $\mathcal{C}$.
The site is *small* when $\mathcal{C}$ is a small category.

## Remarks

Example: for a topological space $X$, let $\mathcal{O}(X)$ be the category of open sets with inclusions.
Let a sieve on $U$ cover when the union of its members is $U$.
This rule satisfies the three axioms of a Grothendieck topology, so $(\mathcal{O}(X), J)$ is a small site.
Non-example: the category $\mathcal{O}(X)$ alone, with no choice of covering sieves, is not a site, since it does not say which families cover.
Some authors define a site with a Grothendieck pretopology, that is, with covering families of morphisms instead of covering sieves; each pretopology determines a topology with the same sheaves.
</document>

<document source="https://tamnd.github.io/math2/node/YMP12DXCSP.md" tag="YMP12DXCSP" kind="statement">
# Bochner's tube theorem

## Statement

Let $B \subseteq \R^n$ be a nonempty [[connected|0SPE8WNWFV]] [[open set|02BH3MCBQB]], and let $\operatorname{ch}(B)$ be its [[convex hull|0FQSDC2EQ2]].
For a set $S \subseteq \R^n$ let $T_S = \set{x + iy \in \C^n : x \in S, \ y \in \R^n}$ be the tube over $S$.
Then every [[holomorphic function|507C34W0ZT]] on $T_B$ is the restriction of a unique holomorphic function on $T_{\operatorname{ch}(B)}$.

## Remarks

Uniqueness holds because $T_{\operatorname{ch}(B)}$ is connected and contains the nonempty open set $T_B$.
Classical proofs push the function across the convex hull with Cauchy integrals over families of analytic discs, in the spirit of the continuity principle.
Another route uses that a tube which is a domain of holomorphy has a convex base.
On such a tube, $-\log$ of the distance to the boundary is plurisubharmonic.
It does not depend on $y$, so it is a convex function of $x$.

Example: let $B = \set{x \in \R^2 : 1 < \abs{x} < 2}$, an annulus.
Its convex hull is the disc $\abs{x} < 2$, so every holomorphic function on $T_B$ extends to the tube over the disc.

Consequently, a tube $T_B$ with connected open base is a domain of holomorphy exactly when $B$ is convex.
For $n = 1$ the theorem is empty, since connected open subsets of $\R$ are intervals and already convex.

## Explanations

### L1 human

A tube domain is determined by its real part: the imaginary part may be anything.
The theorem says that holomorphic functions on a tube automatically extend to the tube over the convex hull of the base.
So, for tubes, being a natural domain of holomorphic functions is the same as ordinary convexity of the base.

### L2 human

Tube domains appear in Fourier and Laplace analysis: the Laplace transform $z \mapsto \int e^{z_1 t_1 + \cdots + z_n t_n} \, d\mu(t)$ of a measure $\mu$ on $\R^n$ is holomorphic on a tube whose base is convex.
The theorem explains why convex bases are the natural ones.
It is a basic example of the computation of an envelope of holomorphy: the envelope of $T_B$ is $T_{\operatorname{ch}(B)}$.
</document>

<document source="https://tamnd.github.io/math2/node/YN280Q5G49.md" tag="YN280Q5G49" kind="concept">
# Bredon cohomology

## Summary

Bredon cohomology is a cohomology theory for spaces with an action of a group $G$.
Its coefficients are not one group but a system of groups, one for each orbit type $G/H$.
It is computed from the cell structure of a $G$-CW complex and keeps track of the fixed point sets $X^H$ of all subgroups $H$.
It is the basic ordinary cohomology theory of equivariant homotopy theory, and the obstructions of equivariant obstruction theory lie in it.

## Explanations

### L1 human

When a group acts on a space, some points are moved freely and others are fixed by part of the group.
Borel's [[equivariant cohomology|C7MZ3SFGYF]] uses one coefficient group for the whole space.
Bredon cohomology looks separately at the points fixed by each subgroup and lets the coefficients depend on that subgroup.
For example, for a group acting on a single point, it returns the coefficient group attached to the whole group $G$.
</document>

<document source="https://tamnd.github.io/math2/node/YNBTV2S7SR.md" tag="YNBTV2S7SR" kind="definition">
# Frequentist inference

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a [[statistical model|JZ6R7J36K5]] and let $X$ be the data, with law $P_\theta$ for a fixed unknown $\theta$.
*Frequentist inference* is the approach to [[statistical inference|FXCQWFY5RV]] in which:

1. the parameter $\theta$ is treated as fixed, and no probability distribution is placed on $\Theta$;
2. a procedure $\delta(X)$ is assessed through its [[sampling distribution|FC9NRV5GSP]] under $P_\theta$;
3. a guarantee about $\delta$ must hold for every $\theta \in \Theta$, or be stated as a function of $\theta$.

## Remarks

Example: a [[confidence set|0K4NMMJBG2]] of level $1 - \alpha$ is a set-valued statistic $C(X)$ with $P_\theta(\theta \in C(X)) \ge 1 - \alpha$ for every $\theta \in \Theta$.
For $n$ i.i.d. $\mathcal{N}(\theta, 1)$ observations, $C(X) = [\bar{X}_n - z/\sqrt{n}, \bar{X}_n + z/\sqrt{n}]$, with $z$ the $1 - \alpha/2$ quantile of $\mathcal{N}(0, 1)$, has coverage exactly $1 - \alpha$ for every $\theta$, since $\sqrt{n}(\bar{X}_n - \theta) \sim \mathcal{N}(0, 1)$.
Non-example: a Bayesian credible interval is computed from a posterior distribution on $\Theta$; it violates clause 1, and in general it has no coverage guarantee that holds for every $\theta$.
A test of level $\alpha$ is likewise a procedure whose probability of rejecting a true hypothesis is at most $\alpha$ under every $\theta$ of the hypothesis.
</document>

<document source="https://tamnd.github.io/math2/node/YNFT400EV9.md" tag="YNFT400EV9" kind="definition">
# Incidence matrix

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]] with vertices $v_1, \dots, v_n$ and edges $e_1, \dots, e_m$.
The incidence matrix of $G$ is the $n \times m$ matrix $B$ with $B_{ij} = 1$ if $v_i$ is an end of $e_j$, and $B_{ij} = 0$ otherwise.
Suppose that each edge $e_j$ is given an orientation, from a tail to a head.
The oriented incidence matrix is the $n \times m$ matrix $N$ with $N_{ij} = 1$ if $v_i$ is the head of $e_j$.
Also $N_{ij} = -1$ if $v_i$ is the tail of $e_j$, and $N_{ij} = 0$ otherwise.

## Remarks

Example: for the triangle with edges $e_1 = v_1 v_2$, $e_2 = v_2 v_3$, $e_3 = v_1 v_3$,
$$B = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix},$$
and each column has exactly two entries $1$, at the ends of its edge.
Non-example: a $0$–$1$ matrix with a column containing three entries $1$ is not the incidence matrix of a graph, since an edge has exactly two ends; such matrices describe hypergraphs.
Let $D$ be the diagonal matrix of degrees and $A$ the [[adjacency matrix|0NZ8MDBN6Q]] of a simple graph.
Then $B B^{T} = D + A$, and $N N^{T} = D - A$ is the [[Laplacian matrix|TN6M6DVZRP]] for every orientation.
Some authors use the transpose, with rows indexed by edges, or the opposite sign convention for $N$.
Mathlib defines the unoriented incidence matrix as `SimpleGraph.incMatrix`.
</document>

<document source="https://tamnd.github.io/math2/node/YNH832D50W.md" tag="YNH832D50W" kind="statement">
# Harnack's inequality

## Statement

Let $a \in \C$, let $R > 0$, and let $u$ be a [[harmonic function|0DEB0JMN0W]] on the disc $\set{z \in \C : \abs{z - a} < R}$ with $u \ge 0$.
Then for every $z$ with $r = \abs{z - a} < R$,
$$\frac{R - r}{R + r}\, u(a) \le u(z) \le \frac{R + r}{R - r}\, u(a).$$

## Remarks

For $\rho$ with $r < \rho < R$, the Poisson formula on the disc of radius $\rho$ gives $u(z)$ as an average of $u$ on the circle $\abs{w - a} = \rho$ against the [[Poisson kernel|W19567AVYG]] $(\rho^2 - r^2)/\abs{w - z}^2$.
Since $\rho - r \le \abs{w - z} \le \rho + r$, this kernel lies between $(\rho - r)/(\rho + r)$ and $(\rho + r)/(\rho - r)$, and the average of $u$ over the circle is $u(a)$.
Letting $\rho \to R$ gives the inequality.
The hypothesis $u \ge 0$ is necessary: $u(z) = \operatorname{Re} z$ on the unit disc has $u(0) = 0$ but takes nonzero values.
In $\R^n$ with $n \ge 2$, the same argument gives $\dfrac{R^{n-2}(R - r)}{(R + r)^{n-1}}\, u(a) \le u(x) \le \dfrac{R^{n-2}(R + r)}{(R - r)^{n-1}}\, u(a)$ for $\abs{x - a} = r < R$.

## Explanations

### L1 human

A nonnegative harmonic function cannot change too fast.
If you know its value at the centre of a disc, you know its values at other points up to a factor.
This factor depends only on how close the points are to the edge.
For example, at half the radius the value is between one third and three times the value at the centre.

### L2 human

A chain of discs gives the general form on a connected open set $\Omega$ with a compact subset $K$.
There is a constant $C$ such that $\sup_K u \le C \inf_K u$ for every nonnegative harmonic $u$ on $\Omega$.
This is the key estimate behind [[Harnack's principle|P15K8RKHDV]] on increasing sequences of harmonic functions.
It is used in Perron's method for the [[Dirichlet problem|8D56GQ4AG8]] and in the study of positive harmonic functions, such as the [[Herglotz representation theorem|9ZBAX4ZPX0]].
Moser extended Harnack's inequality to solutions of elliptic equations in divergence form with measurable coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/YNM6V0R269.md" tag="YNM6V0R269" kind="definition">
# Gumm terms

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras and let $n \ge 0$.
*Gumm terms* for $\mathcal{V}$ are ternary terms $d_0, d_1, \dots, d_n$ and $p$ such that $\mathcal{V}$ satisfies the following [[identities|PKD440G2XW]]:

1. $d_0(x, y, z) = x$.
2. $d_i(x, y, x) = x$ for every $i \le n$.
3. $d_i(x, x, z) = d_{i+1}(x, x, z)$ for every even $i < n$.
4. $d_i(x, z, z) = d_{i+1}(x, z, z)$ for every odd $i < n$.
5. $d_n(x, z, z) = p(x, z, z)$.
6. $p(x, x, z) = z$.

## Remarks

A variety has Gumm terms if and only if it is a [[congruence-modular variety|QAQAS3JWXW]].
Example: if $p$ is a [[Mal'cev term|Y203PPTS61]], then $n = 0$, $d_0 = x$ and $p$ are Gumm terms, since $p(x, z, z) = x = d_0(x, z, z)$ and $p(x, x, z) = z$.
Example: lattices have Gumm terms with $n = 1$, $d_0 = x$, $d_1 = (x \wedge y) \vee (y \wedge z) \vee (x \wedge z)$ and $p(x, y, z) = z$.
Here $d_0(x, x, z) = x = d_1(x, x, z)$, $d_1(x, z, z) = z = p(x, z, z)$, and $p(x, x, z) = z$.
These examples show the two extreme cases: a pure Mal'cev step, and a chain of [[Jónsson terms|3M8Y76E4AG]] with $p$ the third projection.
Non-example: with $n = 0$, the terms $d_0 = x$ and $p = x$ are not Gumm terms in a nontrivial variety, since clause 6 reads $x = z$.

## Explanations

### L1 human

Gumm terms are a chain of three-input formulas, ending in one Mal'cev-like formula, that witness modularity of all congruence lattices by equations alone.
</document>

<document source="https://tamnd.github.io/math2/node/YP1PMW0HE3.md" tag="YP1PMW0HE3" kind="concept">
# Synchronization of chaos

## Summary

Synchronization of chaos is the phenomenon in which two or more coupled chaotic systems come to follow the same motion, or related motions.
This happens although each of them alone is sensitive to initial conditions.
The simplest kind is complete synchronization of identical systems, where the difference of the states tends to zero.
Other kinds are phase synchronization and generalized synchronization.
The phenomenon was studied by Fujisaka and Yamada in 1983 and made widely known by Pecora and Carroll in 1990.
It has applications to the modelling of coupled lasers, neurons and electronic circuits.

## Explanations

### L1 human

Two [[chaotic|DBKXCGDKTD]] systems started at slightly different points quickly move apart.
That seems to rule out any agreement between them.
But if they are coupled, each one being pulled a little towards the other, the coupling can win and the two can move together forever, still in a chaotic way.
For example, two identical electronic circuits that each produce chaotic signals can be wired so that their signals become identical.
</document>

<document source="https://tamnd.github.io/math2/node/YP5TQ6WFVN.md" tag="YP5TQ6WFVN" kind="concept">
# Nonassociative ring

## Summary

A nonassociative ring is an abelian group with a multiplication that distributes over addition on both sides, without the requirement that it be associative.
"Nonassociative" means "not necessarily associative", so associative rings are a special case.
Lie algebras, [[Jordan algebras|6C2MCVB8C7]] and the [[octonions|1HRMT3DSX2]] are the main examples where associativity fails.
A nonassociative algebra over a field is the same notion with a compatible vector space structure.

## Explanations

### L1 human

In most familiar number systems, $(ab)c = a(bc)$, so brackets in products do not matter.
A nonassociative ring drops this rule and keeps the others: addition works as usual, and multiplication distributes over addition.
The cross product of vectors in space is an example: $(e_1 \times e_1) \times e_2 = 0$, but $e_1 \times (e_1 \times e_2) = e_1 \times e_3 = -e_2$.
Such products are typical in geometry and physics, where they measure rotations or symmetries rather than repeated scaling.
</document>

<document source="https://tamnd.github.io/math2/node/YPCH4Q0V34.md" tag="YPCH4Q0V34" kind="concept">
# Chevalley group

## Summary

A Chevalley group is a group of automorphisms of a simple Lie algebra rebuilt over an arbitrary field.
The rebuilding uses the integral structure given by a Chevalley basis.
Claude Chevalley introduced this construction in 1955.
It gives, in a uniform way, a simple group for each simple complex Lie algebra and each field, apart from a few small exceptions.
Over finite fields these groups give most of the families of finite simple groups of Lie type, including new families of exceptional type.

## Explanations

### L1 human

A complex simple Lie algebra, such as the algebra of $2 \times 2$ matrices of trace $0$, has a basis in which all the structure constants are integers.
Because of this, one can reduce it modulo a prime, or more generally build a copy of it over any field.
The Chevalley group is generated by exponentials of certain nilpotent operators on this copy.
For the algebra of type $A_1$ and a field $K$, the result is the group $\mathrm{PSL}_2(K)$.
</document>

<document source="https://tamnd.github.io/math2/node/YPE65X5HJ4.md" tag="YPE65X5HJ4" kind="problem">
# Integer factorization

## Problem

Given a [[composite number|008BH74D7R]] $n$, written in binary, find integers $a$ and $b$ with $1 < a, b < n$ and $n = ab$.
Repeating this step gives the [[prime factorization|02EKEPPJ3R]] of $n$.
The open question is whether there is a classical algorithm that does this in a number of bit operations bounded by a polynomial in $\log n$.

## Remarks

The prime factorization exists and is unique by the [[existence of prime factorization|020K0X7HVR]] and the [[fundamental theorem of arithmetic|084HH1NDCB]], so the problem is about the cost of finding it.
Trial division by all integers up to $\sqrt n$ always works, but it needs about $\sqrt n$ steps, which is exponential in the number of digits.
No polynomial-time classical algorithm is known.
The fastest known general method for large numbers is the general number field sieve, whose heuristic running time is
$$\exp\left( \left( \sqrt[3]{64/9} + o(1) \right) (\ln n)^{1/3} (\ln \ln n)^{2/3} \right).$$
The decision version, which asks whether $n$ has a factor $d$ with $1 < d < k$ for a given $k$, lies in both NP and co-NP.
For this reason it is not expected to be NP-complete, but this is not proved.
On a quantum computer, Shor's algorithm factors $n$ in polynomial time.
Deciding whether $n$ is prime is easier: the [[AKS primality test|XMHQCDWZW1]] does it in deterministic polynomial time, without finding any factor.

## Explanations

### L1 human

Multiplying two large primes is quick, but recovering the two primes from their product seems to be very slow.
For example, $91 = 7 \cdot 13$ is easy to factor by hand, but a product of two primes with three hundred digits each is beyond every known method.
The security of the [[RSA cryptosystem|0HM584C465]] rests on this gap.
Nobody has proved that factoring is really hard, so this remains an open problem.
</document>

<document source="https://tamnd.github.io/math2/node/YPE8E9CVXP.md" tag="YPE8E9CVXP" kind="concept">
# Affine arithmetic

## Summary

Affine arithmetic is a method of range analysis that represents each uncertain quantity as an affine function of shared noise symbols, each ranging over $[-1, 1]$.
Because quantities that depend on the same source of error share a noise symbol, their correlations are kept.
This avoids much of the overestimation of [[interval arithmetic|QPKKTE1ZY0]], where $x - x$ for $x \in [0, 1]$ gives $[-1, 1]$ instead of $0$.
Nonlinear operations are approximated by affine ones, and the approximation error becomes a new noise symbol, so the bounds remain guaranteed.

## Explanations

### L1 human

Suppose one only knows that a quantity $x$ lies between $1$ and $3$.
Interval arithmetic forgets where a range came from, so it treats $x$ and a second copy of $x$ as unrelated, and $x - x$ seems to lie anywhere between $-2$ and $2$.
Affine arithmetic writes $x = 2 + \varepsilon$ with an unknown $\varepsilon \in [-1, 1]$, and keeps the same $\varepsilon$ wherever $x$ appears.
Then $x - x = 0$ exactly, and sums such as $x + 2x = 6 + 3\varepsilon$ also come out sharp.
</document>

<document source="https://tamnd.github.io/math2/node/YPJSH79SHQ.md" tag="YPJSH79SHQ" kind="concept">
# Approximation theory

## Summary

Approximation theory studies how well complicated functions can be replaced by simpler ones, such as polynomials, splines or rational functions.
It asks whether a [[best approximation|7N4988HN3P]] exists, whether it is unique, and how to recognize and compute it.
It also asks how fast the [[error of best approximation|YDN0T4MEX0]] decreases.
Its results give the error bounds behind numerical methods for computing functions, integrals and solutions of equations.

## Explanations

### L1 human

A computer can only add and multiply, so it evaluates $\sin x$ or $e^x$ through a simpler formula, often a polynomial, that stays close to the true function.
Approximation theory makes "close" precise, usually with a norm such as the largest gap between the two graphs.
For example, the [[Weierstrass approximation theorem|0JQE130DHW]] says that every continuous function on $[a, b]$ can be imitated by polynomials as closely as one likes.
The theory then asks which polynomial of a given degree is the best one and how small its error is.
</document>

<document source="https://tamnd.github.io/math2/node/YPPPPHE6TN.md" tag="YPPPPHE6TN" kind="definition">
# Subfactor

## Definition

A *factor* is a [[von Neumann algebra|PSFWQVZ0FS]] $M$ whose center $M \cap M'$ is $\C I$, where $M'$ is the [[commutant|1VC7EJ1ZBZ]].
A *subfactor* of a factor $M$ is a von Neumann subalgebra $N \subseteq M$ that is itself a factor and contains the identity of $M$.
One often writes the subfactor as the inclusion $N \subseteq M$.

If $M$ is of type $\mathrm{II}_1$ with faithful normal tracial state $\tau$, let $L^2(M)$ be the Hilbert space completion of $M$ for $\langle x, y \rangle = \tau(y^* x)$.
The *Jones index* of $N \subseteq M$ is
$$[M : N] = \dim_N L^2(M) \in [1, \infty],$$
the Murray–von Neumann dimension of $L^2(M)$ as a module over $N$ acting by left multiplication.

## Remarks

Example: let $N$ be a $\mathrm{II}_1$ factor and $k \ge 1$.
Then $N \otimes I_k$, the matrices $\operatorname{diag}(x, \dots, x)$ with $x \in N$, is a subfactor of $M = N \otimes M_k(\C)$, the $k \times k$ matrices over $N$.
Both are factors with the same unit, and $L^2(M)$ is the direct sum of $k^2$ copies of $L^2(N)$ as an $N$-module, so $[M:N] = k^2$.

Non-example: the diagonal matrices $D_2 \subseteq M_2(\C)$ form a unital von Neumann subalgebra of the factor $M_2(\C)$.
But $D_2$ is commutative and two-dimensional, so its center is $D_2 \ne \C I$, and $D_2$ is not a factor.

Another example: if a finite group $G$ acts on a $\mathrm{II}_1$ factor $M$ by outer automorphisms, the fixed-point algebra $M^G$ is a subfactor with $[M : M^G] = \abs{G}$.
The index is $1$ exactly when $N = M$.
</document>

<document source="https://tamnd.github.io/math2/node/YPRZTC01QC.md" tag="YPRZTC01QC" kind="example">
# 1 + 1 + 1 + 1 + ⋯

## Example

The series
$$\sum_{n=1}^{\infty} 1 = 1 + 1 + 1 + 1 + \cdots$$
has $N$-th partial sum $s_N = N$, which tends to $+\infty$.
So it is a [[divergent series|0JWMJSTBSG]].

## Remarks

The [[Cesàro means|N8ZGG7EW6J]] $(s_1 + \cdots + s_N)/N = (N+1)/2$ also tend to $+\infty$.
For $0 < x < 1$, $\sum_{n \ge 1} x^n = x/(1-x)$, which tends to $+\infty$ as $x \to 1^-$, so the series is not [[Abel summable|EZXPY4A19C]].
[[Zeta function regularization|VJCJYXGTDY]] assigns it the value $-1/2$.
Indeed $\sum_n n^{-s} = \zeta(s)$ for $\operatorname{Re} s > 1$, the series of the example is formally the case $s = 0$, and the analytic continuation of the [[Riemann zeta function|1XWBY4N0VT]] has $\zeta(0) = -1/2$.
No linear and stable method can give it a finite value $S$: stability gives $S = 1 + S$, which is impossible.

## Explanations

### L1 human

Adding $1$ forever gives totals $1, 2, 3, \ldots$ that grow without bound.
This is the simplest divergent series.
The value $-1/2$ that sometimes appears for it comes from the Riemann zeta function at $0$, not from ordinary addition.
</document>

<document source="https://tamnd.github.io/math2/node/YPTNBRWRY7.md" tag="YPTNBRWRY7" kind="conjecture">
# Baker–Gammel–Wills conjecture

## Statement

Let $D = \set{z \in \C : \abs{z} < 1}$ be the unit disc.
Let $f$ be a [[meromorphic function|0ZZ7ZYDR8W]] on $D$ that is analytic at $0$, and let $P$ be its set of [[poles|066E7PJCYE]] in $D$.
Let $[n/n]_f$ be the diagonal [[Padé approximants|1F16G4V64X]] of $f$ at $0$.
Then there is an infinite sequence $n_1 < n_2 < \cdots$ such that $[n_j/n_j]_f$ converges to $f$ [[uniformly|08VZ57PT26]] on every compact subset of $D \setminus P$.

## Remarks

The conjecture is false in this generality.
In 2003 D. S. Lubinsky published a counterexample built from the Rogers–Ramanujan continued fraction.
Convergence of the full diagonal fails in general, because the approximants can have spurious poles that wander through the disc; this is why the conjecture only asks for a subsequence.
A weaker form holds: for functions meromorphic in the whole plane, the diagonal sequence converges in logarithmic capacity (the Nuttall–Pommerenke theorem).
For columns of the [[Padé table|MDM8F48X18]] with a fixed number of poles, [[Montessus de Ballore's theorem|XQQWP9ZPEW]] gives uniform convergence.

## Explanations

### L1 human

Diagonal Padé approximants, with equal degrees of numerator and denominator, are the most used rational approximations to a power series.
They often converge very well in practice, but sometimes an approximant has an unwanted pole in a bad place.
The conjecture predicted that one can always skip the bad approximants and keep infinitely many good ones that converge uniformly.
This turned out to be false, although convergence holds in a weaker averaged sense.
</document>

<document source="https://tamnd.github.io/math2/node/YQ21C2N2HX.md" tag="YQ21C2N2HX" kind="definition">
# Hurwitz zeta function

## Definition

Let $a$ be a real number with $a > 0$, and let $s$ be a complex number with $\operatorname{Re} s > 1$.
The *Hurwitz zeta function* is
$$\zeta(s, a) = \sum_{n=0}^{\infty} \frac{1}{(n + a)^{s}},$$
where $(n + a)^{s} = e^{s \log(n + a)}$ with the real logarithm.
The series converges absolutely for $\operatorname{Re} s > 1$.
For fixed $a$, the function $s \mapsto \zeta(s, a)$ has an [[analytic continuation|0M3YQJ81NB]] to a [[meromorphic function|0ZZ7ZYDR8W]] on $\C$, and this continuation is also written $\zeta(s, a)$.

## Remarks

For $a = 1$, the series is $\sum_{n \ge 1} n^{-s}$, so $\zeta(s, 1)$ is the [[Riemann zeta function|1XWBY4N0VT]] $\zeta(s)$.
For $a = 1/2$, the terms are $2^s (2n + 1)^{-s}$, so $\zeta(s, 1/2) = 2^s \sum_{m \text{ odd}} m^{-s} = (2^s - 1)\zeta(s)$.
The value $s = 1$ is not allowed in the series: there the terms behave like $1/n$ and the series diverges.
The continuation has a single pole, at $s = 1$; it is simple with residue $1$.
For a [[Dirichlet character|7XB5VJKDPG]] $\chi$ modulo $q$, grouping $n$ by its residue gives
$$L(s, \chi) = q^{-s} \sum_{r=1}^{q} \chi(r)\, \zeta\!\left(s, \frac{r}{q}\right).$$
Some authors also allow complex $a$ with $\operatorname{Re} a > 0$.
</document>

<document source="https://tamnd.github.io/math2/node/YQ5C1SRE0W.md" tag="YQ5C1SRE0W" kind="concept">
# Locally finite measure

## Summary

A Borel measure on a topological space is locally finite if every point has an open neighbourhood of finite measure.
It may give infinite measure to the whole space, but never to a small region.
On locally compact Hausdorff spaces this is the same as being finite on compact sets.
Local finiteness is part of the definition of a Radon measure.

## Explanations

### L1 human

Locally finite means that, around each point, a small enough window holds only a finite amount of mass.
Length on the real line is locally finite: the whole line has infinite length, but each bounded interval has finite length.
Counting measure on the real line is not: every open interval holds infinitely many points.
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<document source="https://tamnd.github.io/math2/node/YQ5HQP6HGE.md" tag="YQ5HQP6HGE" kind="definition">
# Finitely presented module

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].
The module $M$ is *finitely presented* if there are integers $m, n \ge 0$ and an [[exact sequence|JB7ZBFBA9C]] of $R$-modules
$$R^m \xrightarrow{\ \varphi\ } R^n \xrightarrow{\ \pi\ } M \to 0.$$
Such a sequence is a *finite presentation* of $M$.

## Remarks

Equivalently, $M$ is [[finitely generated|05QY1VEBYD]] and the kernel of some surjection $R^n \to M$ is finitely generated.
By Schanuel's lemma, the kernel of every surjection $R^n \to M$ is then finitely generated.
The map $\varphi$ is given by an $n \times m$ matrix over $R$, so $M$ is the quotient of $R^n$ by the column span of a matrix.
Over a [[Noetherian ring|0670T59HGE]], every finitely generated module is finitely presented, since submodules of $R^n$ are finitely generated.

Example: over $R = \Z$, the module $\Z/6\Z$ is finitely presented.
The sequence $\Z \xrightarrow{6} \Z \to \Z/6\Z \to 0$, with the second map the reduction modulo $6$, is exact, because the image of multiplication by $6$ is $6\Z$, which is the kernel of the reduction.

Non-example: let $k$ be a field, $R = k[x_1, x_2, x_3, \ldots]$ the polynomial ring in infinitely many variables, and $M = R/(x_1, x_2, \ldots) \cong k$.
Then $M$ is finitely generated, by the class of $1$.
The kernel of $R \to M$ is the ideal $(x_1, x_2, \ldots)$, which is not finitely generated, so $M$ is not finitely presented.
</document>

<document source="https://tamnd.github.io/math2/node/YQ626FK82G.md" tag="YQ626FK82G" kind="definition">
# Commutant

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.
The *commutant* of a subset $S \subseteq B(H)$ is
$$S' = \set{T \in B(H) : TA = AT \text{ for all } A \in S}.$$
The *bicommutant*, or double commutant, of $S$ is $S'' = (S')'$.

## Remarks

Example: let $D \subseteq M_n(\C) = B(\C^n)$ be the algebra of diagonal matrices.
If $T$ commutes with $E = \operatorname{diag}(e_1, \dots, e_n)$ for distinct $e_k$, then $(e_i - e_j) T_{ij} = 0$, so $T$ is diagonal; hence $D' = D$.
Likewise, $M_n(\C)' = \C I$, since commuting with all matrix units forces $T$ to be scalar.

Non-example: the matrix unit $E_{12}$ does not lie in $D'$, because $E_{12} \operatorname{diag}(1, 2) = 2E_{12}$ while $\operatorname{diag}(1, 2) E_{12} = E_{12}$.

For all subsets, $S \subseteq S''$, $S_1 \subseteq S_2$ implies $S_2' \subseteq S_1'$, and $S''' = S'$.
The commutant $S'$ is always a unital subalgebra of $B(H)$ that is closed in the weak operator topology.
If $S$ is closed under adjoints, then $S'$ is also closed under adjoints, so $S'$ is a von Neumann algebra.
The same definition in any ring or algebra gives the centralizer of a subset.
</document>

<document source="https://tamnd.github.io/math2/node/YQB2B0EXY8.md" tag="YQB2B0EXY8" kind="concept">
# Functional inequality

## Summary

A functional inequality is an inequality between expressions in an unknown function.
It is required to hold for all values of the variables.
It is the inequality analogue of a [[functional equation|SRAN4EKWS0]].
Standard examples are subadditivity $f(x + y) \le f(x) + f(y)$ and midpoint convexity $f\big(\tfrac{x + y}{2}\big) \le \tfrac{f(x) + f(y)}{2}$.
Typical questions are which functions satisfy a given inequality, and when mild regularity forces its solutions to be nice.

## Explanations

### L1 human

A functional equation asks for functions that make two expressions equal everywhere; a functional inequality only asks that one expression is never larger than the other.
For example, the inequality $f(x + y) \le f(x) + f(y)$ says that the value at a sum is at most the sum of the values.
The square root on $[0, \infty)$ satisfies it, while the squaring map does not.
Convex functions, norms and many quantities in probability and analysis are described by such inequalities.
</document>

<document source="https://tamnd.github.io/math2/node/YQNZTQGP1X.md" tag="YQNZTQGP1X" kind="concept">
# Artinian module

## Summary

An Artinian module is a module in which every decreasing chain of submodules stops after finitely many steps.
It is the condition dual to that of a [[Noetherian module|AK1ZRCXR32]].
A module is both Artinian and Noetherian exactly when it has finite [[length|CZSCNZH8VY]].

## Explanations

### L1 human

In an Artinian module you cannot keep shrinking submodules forever.
A finite abelian group, seen as a $\Z$-module, is Artinian, since it has only finitely many subgroups.
The group $\Z$ is not Artinian, because $2\Z \supset 4\Z \supset 8\Z \supset \cdots$ never stops.
The two chain conditions are different: the group of all complex roots of unity of order a power of a prime $p$ is Artinian but not Noetherian as a $\Z$-module.
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<document source="https://tamnd.github.io/math2/node/YQP8V1KKMW.md" tag="YQP8V1KKMW" kind="concept">
# Hahn series

## Summary

A Hahn series is a formal sum $\sum_\gamma a_\gamma t^\gamma$ with coefficients in a field and exponents in a [[linearly ordered group|P2E8BNYC7B]].
Its set of exponents must be [[well-ordered|01CGSJ0GNQ]].
Hahn series generalise [[formal power series|033WGN00BF]] and [[formal Laurent series|YZWYF7DXTW]] to exponents that may be fractions, reals, or elements of any ordered abelian group.
They form a field with a natural [[valuation|B85T5VKR3M]], and these fields are a main source of examples of valued and ordered fields.

## Explanations

### L1 human

A Laurent series in $t$ uses integer powers, starting from some lowest power and going up.
A Hahn series allows any exponents from an ordered group, like $t^{1/2}$ or $t^{\sqrt{2}}$, as long as each nonempty set of exponents used has a smallest element.
For example $t^{1/2} + t^{2/3} + t^{3/4} + \cdots$ is a Hahn series with rational exponents.
The rule about smallest elements is exactly what makes products of two such series well defined.
</document>

<document source="https://tamnd.github.io/math2/node/YQQW8X2ZZJ.md" tag="YQQW8X2ZZJ" kind="definition">
# Bessel polynomials

## Definition

Let $n \ge 0$ be an integer.
The *Bessel polynomial* of degree $n$ is the [[polynomial|1G0B73WPX8]]
$$y_n(x) = \sum_{k=0}^{n} \frac{(n + k)!}{(n - k)! \, k!} \left( \frac{x}{2} \right)^k.$$
The *reverse Bessel polynomial* is $\theta_n(x) = x^n y_n(1/x)$.

## Remarks

Examples: $y_0 = 1$, $y_1 = 1 + x$ and $y_2 = 1 + 3x + 3x^2$.
Check for the top coefficient of $y_2$: $\frac{4!}{0! \, 2!} \cdot \frac14 = 3$.
Each $y_n$ solves $x^2 y'' + (2x + 2) y' - n(n + 1) y = 0$; for $y_1$ this reads $0 + (2x + 2) - 2(1 + x) = 0$.
Non-example: $1 + 2x$ has degree $1$ but is not a solution for $n = 1$, since $(2x + 2) \cdot 2 - 2(1 + 2x) = 2 \ne 0$.
For $x > 0$ they are linked to the modified Bessel function of the second kind: $y_n(x) = \sqrt{\frac{2}{\pi x}} \, e^{1/x} K_{n + 1/2}(1/x)$.
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# Observability

## Definition

Let $x' = f(x, u)$, $y = h(x)$ be a [[control system|ZPDTVZQN5K]] on $\R^n$ with output map $h \colon \R^n \to \R^p$.
Let $x_0, \tilde{x}_0 \in \R^n$.
For an admissible control $u$, let $x$ and $\tilde{x}$ be the trajectories with $x(0) = x_0$ and $\tilde{x}(0) = \tilde{x}_0$.
The states $x_0$ and $\tilde{x}_0$ are *indistinguishable* if, for every $u$, $h(x(t)) = h(\tilde{x}(t))$ for all $t \ge 0$ at which both trajectories are defined.
The system is *observable* if any two indistinguishable initial states are equal.

## Remarks

Consider a linear system $x' = A x + B u$, $y = C x$.
The output difference of two initial states does not depend on $u$.
The system is observable if and only if, for any $T > 0$, the output $y$ on $[0, T]$ determines $x(0)$.
Observability of the linear system is equivalent to $\rank \begin{pmatrix} C \\ CA \\ \vdots \\ CA^{n-1} \end{pmatrix} = n$.
Example: $x_1' = x_2$, $x_2' = 0$, $y = x_1$ is observable, since $y(t) = x_1(0) + t \, x_2(0)$ gives $x_1(0) = y(0)$ and $x_2(0) = y'(0)$.
Non-example: the same system with output $y = x_2$ is not observable, since $(0, 0)$ and $(1, 0)$ give the same output $y \equiv 0$.
</document>

<document source="https://tamnd.github.io/math2/node/YR2MKGA2CE.md" tag="YR2MKGA2CE" kind="definition">
# Diagonally dominant matrix

## Definition

Let $A = (a_{ij})$ be an $n \times n$ complex [[square matrix|0JXZKX526R]].
The matrix $A$ is *diagonally dominant* (by rows) when
$$\abs{a_{ii}} \ge \sum_{j \ne i} \abs{a_{ij}} \qquad \text{for all } i = 1, \dots, n.$$
It is *strictly diagonally dominant* when this inequality is strict for all $i$.

## Remarks

Example: $\begin{pmatrix} 3 & -1 \\ 1 & 2 \end{pmatrix}$ is strictly diagonally dominant, because $3 > 1$ and $2 > 1$.
Non-example: $\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ is not diagonally dominant, because in the first row $\abs{1} < \abs{2}$.
Diagonal dominance by columns is the same condition for the transpose $A^T$.
A strictly diagonally dominant matrix is invertible, by the [[Levy–Desplanques theorem|PWKN6ANA0G]].
A diagonally dominant matrix that is not strictly so can be singular, for example $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$.
</document>

<document source="https://tamnd.github.io/math2/node/YR42E2W19Q.md" tag="YR42E2W19Q" kind="problem">
# Free boundary problem

## Problem

A free boundary problem is a problem for a [[partial differential equation|GC5BDZ6QBC]] in which part of the domain is unknown.
A typical stationary form is the following.
Let $D \subseteq \R^n$ be a bounded open set and let $g$ be given boundary data on $\partial D$.
Find an open set $\Omega \subseteq D$ and a function $u$ on $\overline{\Omega}$ such that:

1. $u$ solves a given partial differential equation in $\Omega$;
2. $u = g$ on the known part $\partial \Omega \cap \partial D$ of the boundary;
3. $u$ satisfies two conditions on the unknown part $\Gamma = \partial \Omega \cap D$, the free boundary.

One condition on $\Gamma$ alone would fix $u$ for each choice of $\Omega$.
The second condition is what determines $\Gamma$.
A standard example is the one-phase Bernoulli problem: find $u \ge 0$ with $\Delta u = 0$ in $\Omega = \set{u > 0}$, $u = 0$ on $\Gamma$ and $\abs{\nabla u} = 1$ on $\Gamma$.
When the free boundary moves in time, one also speaks of a moving boundary problem.

## Remarks

Free boundary problems are nonlinear even when the equation in clause 1 is linear, since the domain depends on the solution.
The [[Stefan problem|3B0HB3DHW1]] for melting ice is the classical evolution example.
The heat equation holds in the liquid, and the melting front moves with a speed fixed by the heat flux.
The obstacle problem is the classical stationary example.
There the free boundary separates the region where a membrane touches an obstacle from the region where it is [[harmonic|0DEB0JMN0W]].
A common method is to recast the problem as a minimization or a variational inequality on a fixed domain, solve it there in a weak sense, and recover $\Gamma$ afterwards.
The main questions are then existence and uniqueness of a solution, the regularity of $u$, and the regularity of the free boundary $\Gamma$.
Lamé and Clapeyron studied the solidification of a liquid sphere in 1831, a problem of this type.

## Explanations

### L1 human

In most problems for differential equations the region is given, and only the function is unknown.
In a free boundary problem the edge of the region is also unknown and must be found together with the function.
Think of ice melting in water: the temperature in the water is unknown, and so is the position of the ice front.
Because the region has an extra unknown, the problem needs one extra condition on its edge.
</document>

<document source="https://tamnd.github.io/math2/node/YR43719RWE.md" tag="YR43719RWE" kind="definition">
# Controllability

## Definition

Let $x' = f(x, u)$ be a [[control system|ZPDTVZQN5K]] on $\R^n$ with control set $U$.
Let $x_0, x_1 \in \R^n$ and $T > 0$.
The state $x_1$ is *reachable* from $x_0$ in time $T$ if some admissible control $u$ on $[0, T]$ has a trajectory with $x(0) = x_0$ and $x(T) = x_1$.
The system is *controllable* if for all $x_0, x_1 \in \R^n$ there is $T > 0$ such that $x_1$ is reachable from $x_0$ in time $T$.

## Remarks

For a linear system $x' = A x + B u$ with $U = \R^m$, controllability in some time implies controllability in every time $T > 0$.
Example: the double integrator $x_1' = x_2$, $x_2' = u$ with $U = \R$ is controllable.
Given $(a_0, b_0)$, $(a_1, b_1)$ and $T > 0$, take the cubic polynomial $p$ with $p(0) = a_0$, $p'(0) = b_0$, $p(T) = a_1$, $p'(T) = b_1$; then $u = p''$ gives the trajectory $(x_1, x_2) = (p, p')$, which joins the two states.
Non-example: the system $x_1' = u$, $x_2' = 0$ is not controllable, because $x_2$ stays constant along every trajectory, so $(0, 1)$ is not reachable from $(0, 0)$.
Weaker notions are local controllability near an equilibrium and small-time local controllability.
</document>

<document source="https://tamnd.github.io/math2/node/YR5RAT9HWZ.md" tag="YR5RAT9HWZ" kind="statement">
# Likelihood principle

## Statement

Let $E_1$ and $E_2$ be two experiments about the same unknown parameter $\theta \in \Theta$, each given by a [[statistical model|JZ6R7J36K5]] with densities or mass functions.
Let $x_1$ be an outcome of $E_1$ and $x_2$ an outcome of $E_2$, with [[likelihood functions|05FSTXM92W]] $L_1(\theta \mid x_1)$ and $L_2(\theta \mid x_2)$.
Suppose there is a constant $c > 0$, not depending on $\theta$, such that $L_1(\theta \mid x_1) = c \, L_2(\theta \mid x_2)$ for all $\theta \in \Theta$.
The likelihood principle asserts that the outcomes $(E_1, x_1)$ and $(E_2, x_2)$ carry the same evidence about $\theta$.
So every inference about $\theta$ should be the same in both cases.

## Remarks

This is a principle of inference, not a theorem: it is a requirement that a method of inference may satisfy or violate.
Birnbaum showed that it is implied by the sufficiency principle together with the conditionality principle; this argument and its reading are still debated.
Example: one experiment tosses a coin $12$ times and sees $3$ heads; another tosses until the $3$rd head, which comes at toss $12$.
The likelihoods are $\binom{12}{3} p^3 (1 - p)^9$ and $\binom{11}{2} p^3 (1 - p)^9$, which are proportional, so the principle demands the same conclusion about $p$.
Bayesian inference with a fixed prior obeys the principle, since the posterior depends on the data only through the likelihood.
Many frequentist procedures violate it: in the example, the two p-values for the hypothesis $p = 1/2$ differ, because they depend on outcomes that could have occurred but did not.

## Explanations

### L1 human

The likelihood function tells how well each parameter value explains the data that were actually seen.
The likelihood principle says that this function is all that matters for learning about the parameter.
In particular, how the experimenter planned to stop, or which other results could have occurred, should not change the conclusion.

### L2 human

The principle separates the two main schools of statistics.
A Bayesian posterior is proportional to prior times likelihood, so it respects the principle automatically, and so does a comparison of likelihood values.
P-values, confidence levels and unbiasedness are defined by averaging over all possible data sets of the experiment, so they depend on the sampling design and usually violate the principle.
The binomial versus negative binomial example above, with a stopping rule, is the standard illustration of this conflict.
</document>

<document source="https://tamnd.github.io/math2/node/YR6HCMABTE.md" tag="YR6HCMABTE" kind="concept">
# Model complete theory

## Summary

A theory is model complete when every embedding between two of its models is elementary.
Then a smaller model agrees with a larger one on all first-order statements about its own elements.
Abraham Robinson introduced the notion as an algebraic counterpart of [[quantifier elimination|7FDDXV3JK7]], which implies it.
Model completeness is equivalent to the property that every formula is equivalent to an existential formula.
The theory of real closed fields in the language of rings is a standard model complete theory without quantifier elimination.

## Explanations

### L1 human

Suppose one model of a theory sits inside another.
The theory is model complete when the inner model cannot tell the difference.
Any first-order statement about its elements is true in the inner model exactly when it is true in the outer one.
For example, the real algebraic numbers sit inside the real numbers, and both are real closed fields.
A polynomial equation with real algebraic coefficients has a solution in one field exactly when it has one in the other.
</document>

<document source="https://tamnd.github.io/math2/node/YRBPQS8GHM.md" tag="YRBPQS8GHM" kind="definition">
# Functional differential equation

## Definition

Let $r \ge 0$ and $n \ge 1$, and let $C = C([-r, 0], \R^n)$ be the space of continuous functions with the supremum norm.
For a continuous function $x$ defined on an interval containing $[t - r, t]$, let $x_t \in C$ be the segment $x_t(\theta) = x(t + \theta)$ for $\theta \in [-r, 0]$.
Let $\Omega \subseteq \R \times C$ be open and let $F \colon \Omega \to \R^n$.
A *(retarded) functional differential equation* is the equation
$$x'(t) = F(t, x_t).$$
A *solution* through $(\sigma, \phi) \in \Omega$ is a continuous $x \colon [\sigma - r, \sigma + T) \to \R^n$, for some $T > 0$, such that $x_\sigma = \phi$, $(t, x_t) \in \Omega$ and $x'(t) = F(t, x_t)$ for $t \in (\sigma, \sigma + T)$.

## Remarks

Example: $x'(t) = -\int_{-1}^{0} x(t + \theta) \, d\theta$ is of this form with $r = 1$ and $F(t, \psi) = -\int_{-1}^{0} \psi(\theta) \, d\theta$.

Non-example: $x'(t) = x(t + 1)$ is not of this form, since $x'(t)$ depends on the value at the later time $t + 1$, which is not part of $x_t$.

For $r = 0$, $C$ is $\R^n$ and the equation is an [[ordinary differential equation|0NT64JV96A]].
If $F$ is continuous and locally Lipschitz in its second argument, the [[initial value problem|0FXDW992DS]] has a unique local solution through each $(\sigma, \phi) \in \Omega$.
Equations whose derivative term also involves the past, such as $\frac{d}{dt}\bigl(x(t) - c\, x(t - r)\bigr) = F(t, x_t)$, are called neutral.

## Explanations

### L1 human

The derivative at time $t$ is any rule that reads the solution on the window $[t - r, t]$.
The window slides forward in time and plays the role of the state.
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<document source="https://tamnd.github.io/math2/node/YRBT3ZHSV3.md" tag="YRBT3ZHSV3" kind="statement">
# Nevanlinna parametrization of solutions

## Statement

Let $(m_n)_{n \ge 0}$ be a sequence of real numbers for which the [[Hamburger moment problem|TBNCFXTMMW]] is [[indeterminate|GCTN4BZ1ET]].
Let $V$ be the set of positive [[Borel measures|DYR1QFNGNJ]] $\mu$ on $\R$ with $\int_\R x^n \, d\mu(x) = m_n$ for all $n \ge 0$.
A Pick function is a [[holomorphic function|0NAHXR1FRH]] $\varphi$ on the [[upper half-plane|X6XGGH5FPW]] $\mathbb{H} = \set{z \in \C : \operatorname{Im} z > 0}$ with $\operatorname{Im} \varphi(z) \ge 0$ for all $z \in \mathbb{H}$.
Let $\mathcal{P}$ be the set of Pick functions together with the constant $\infty$.
Then there are four [[entire functions|0G4F0XB9X2]] $A, B, C, D$, real on $\R$ and determined by $(m_n)$, with $A(z) D(z) - B(z) C(z) = 1$ for all $z \in \C$, such that the following holds.
For each $\varphi \in \mathcal{P}$ there is exactly one $\mu_\varphi \in V$ with
$$\int_\R \frac{d\mu_\varphi(x)}{x - z} = -\frac{A(z) \varphi(z) - C(z)}{B(z) \varphi(z) - D(z)} \quad \text{for all } z \in \mathbb{H},$$
where for $\varphi = \infty$ the right side means $-A(z)/B(z)$.
The map $\varphi \mapsto \mu_\varphi$ is a bijection from $\mathcal{P}$ onto $V$.

## Remarks

The functions $A, B, C, D$ are built from the orthonormal polynomials $p_n$ of the moment sequence and the polynomials $q_n$ of the second kind.
For example, $B(z) = -1 + z \sum_{k \ge 0} q_k(0) p_k(z)$ and $D(z) = z \sum_{k \ge 0} p_k(0) p_k(z)$, and these series converge locally uniformly exactly because the problem is indeterminate.
The left side is the Stieltjes transform of $\mu_\varphi$, which determines $\mu_\varphi$ by the [[Stieltjes inversion formula|C0X2A02MRE]].
The parameters $\varphi = t$ with $t \in \R \cup \set{\infty}$ give the N-extremal solutions, for which the polynomials are dense in $L^2(\mu_t)$.
Each $\mu_t$ is a discrete measure supported on the zeros of $B t - D$, or on the zeros of $B$ when $t = \infty$.
The hypothesis of indeterminacy is needed: in the determinate case $V$ has only one element, so there is nothing to parametrize.
Sign conventions for $A, B, C, D$ differ between books.

## Explanations

### L1 human

When a list of moments does not single out one distribution, there are infinitely many distributions with those moments.
The theorem describes all of them at once.
Four fixed functions turn each choice of a free parameter, a Pick function, into exactly one solution, and every solution arises in this way.

### L2 human

The parametrization reduces the study of an indeterminate moment problem to the study of four entire functions, the Nevanlinna matrix.
These entire functions have order at most one and minimal exponential type.
The parameters $t \in \R \cup \set{\infty}$ give solutions that are spectral measures of self-adjoint extensions of the Jacobi operator of the moment sequence.
The Pick functions are the functions of the [[Herglotz representation theorem|9ZBAX4ZPX0]] transplanted from the disc to the upper half-plane, so the parameter set has a concrete integral form.
</document>

<document source="https://tamnd.github.io/math2/node/YREPR562ES.md" tag="YREPR562ES" kind="definition">
# Koszul algebra

## Definition

Let $k$ be a [[field|04C5ZBG1WT]].
Let $A = \bigoplus_{n \ge 0} A_n$ be a [[graded|8KB4SQGC0C]] [[associative algebra|3CTW1P4HBP]] over $k$ with $A_0 = k$ and each $A_n$ finite-dimensional.
Let $k$ also denote the graded $A$-module $A/A_{>0}$, where $A_{>0} = \bigoplus_{n > 0} A_n$.
Consider a graded [[projective resolution|HK2AA1YSQ2]] of $k$ by graded free $A$-modules:
$$\cdots \to P_2 \to P_1 \to P_0 \to k \to 0.$$
The algebra $A$ is *Koszul* if $k$ has such a resolution in which each $P_i$, $i \ge 0$, is generated by homogeneous elements of degree $i$.

## Remarks

Example: $A = k[x]$ with $\deg x = 1$.
The resolution $0 \to A(-1) \xrightarrow{x} A \to k \to 0$ has $P_0 = A$ generated in degree $0$ and $P_1 = A(-1)$ generated in degree $1$.
More generally, the [[Koszul complex|YZY11W08SD]] is a linear resolution of $k$ over $k[x_1, \dots, x_n]$.
Non-example: $A = k[x]/(x^3)$ with $\deg x = 1$.
The minimal resolution starts with $P_0 = A$ and $P_1 = A(-1)$, mapping the generator to $x$.
The kernel of multiplication by $x$ on $A$ is spanned by $x^2$, so $P_2$ is generated in degree $1 + 2 = 3$, not $2$.
Equivalent form: $A$ is Koszul if and only if $\operatorname{Ext}^i_A(k, k)$ is concentrated in internal degree $i$ for every $i$.
Every Koszul algebra is quadratic: it is generated by $A_1$ with relations in degree $2$.
</document>

<document source="https://tamnd.github.io/math2/node/YRHJA7C7HS.md" tag="YRHJA7C7HS" kind="concept">
# Kolmogorov automorphism

## Summary

A Kolmogorov automorphism, or K-automorphism, is an invertible [[measure-preserving transformation|556WBDAHNP]] with a sub-σ-algebra that plays the role of "the past".
This σ-algebra grows under the map, it generates everything, and its remote part is trivial.
K-automorphisms are exactly the systems of completely positive [[entropy|X1D1C7VH6P]]: every nontrivial factor has positive entropy.
They are [[mixing|S3RJQ9CX2P]], indeed mixing of all orders.
Every Bernoulli shift is a K-automorphism, but there are K-automorphisms that are not isomorphic to Bernoulli shifts.

## Explanations

### L1 human

Think of a sequence of independent coin tosses running from the infinite past to the infinite future.
The information about the past grows as time moves on, and together all the pasts describe the whole sequence.
Yet what lies in the very remote past tells you nothing about the present.
A K-automorphism is any measure-preserving system with this structure.
It has the strongest kind of randomness in its long-range behaviour, with no predictable part at all.
</document>

<document source="https://tamnd.github.io/math2/node/YRKW2KX2CW.md" tag="YRKW2KX2CW" kind="concept">
# Integro-differential equation

## Summary

An integro-differential equation involves both derivatives and integrals of the unknown function.
A typical form is $x'(t) = f(t, x(t)) + \int_{t_0}^{t} K(t, s, x(s)) \, ds$, in which the integral describes memory of the past.
Such equations appear in viscoelasticity, population dynamics with age structure, circuit theory and epidemiology.
They are studied with the Laplace transform, by turning them into integral equations, or as functional differential equations.

## Explanations

### L1 human

In an ordinary differential equation, the rate of change depends on the present state.
In an integro-differential equation, it can also depend on an accumulated total over the past.
In an electric circuit with a resistor, an inductor and a capacitor, the voltage across the capacitor is the integral of the current.
So the circuit law mixes the current, its derivative and its integral.
The equation $x'(t) = -\int_0^t x(s) \, ds$ with $x(0) = 1$ has the solution $x(t) = \cos t$.
</document>

<document source="https://tamnd.github.io/math2/node/YRNFE016FF.md" tag="YRNFE016FF" kind="concept">
# Adapted process

## Summary

A [[stochastic process|0QH5ENKZM1]] is adapted to a [[filtration|HNK1Y62G7V]] when its value at each time is determined by the information available at that time.
Adapted processes cannot see the future.
Adaptedness is part of the definition of a [[martingale|0D0GBNDKDH]] and of the stochastic integral, where the integrand must not anticipate the noise.

## Explanations

### L1 human

Think of a gambler who chooses each bet using only the results seen so far.
The gambler's fortune over time is an adapted process: at every moment it depends only on the past and present.
A strategy that peeks at the next coin toss is not adapted.
Every process is adapted to its own history, called its natural filtration.
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<document source="https://tamnd.github.io/math2/node/YRTCH4Q9GA.md" tag="YRTCH4Q9GA" kind="definition">
# Order of an entire function

## Definition

Let $f$ be an [[entire function|0419794R0T]] that is not constant.
For $r > 0$, let $M(r) = \max_{\abs{z} = r} \abs{f(z)}$.
The *order* of $f$ is
$$\rho = \limsup_{r \to \infty} \frac{\log \log M(r)}{\log r} \in [0, \infty],$$
where $\limsup$ is the [[limit superior|0P8BA7T7S5]].
By convention, constant functions have order $0$.

## Remarks

Equivalently, $\rho$ is the infimum of the numbers $\lambda \ge 0$ such that $M(r) < \exp(r^\lambda)$ for all sufficiently large $r$.
Example: for $f(z) = e^{z^2}$, $M(r) = e^{r^2}$, so $\log \log M(r)/\log r = 2$ and the order is $2$.
Non-example for a bound: $e^{z^2}$ does not satisfy $M(r) < \exp(r^{\lambda})$ for large $r$ when $\lambda < 2$, so its order is not at most $1$.
For a nonconstant polynomial, $\log M(r) \sim d \log r$ with $d$ the degree, so the order is $0$.
In terms of the Taylor coefficients of $f(z) = \sum_n a_n z^n$, the order is $\limsup_{n \to \infty} \dfrac{n \log n}{\log(1/\abs{a_n})}$, where terms with $a_n = 0$ are omitted.
If $0 < \rho < \infty$, the *type* of $f$ is $\sigma = \limsup_{r \to \infty} r^{-\rho} \log M(r)$; functions of order $1$ and finite type, or of order less than $1$, are those of [[exponential type|9P8D8YAHQS]].
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<document source="https://tamnd.github.io/math2/node/YRTFS7NHM8.md" tag="YRTFS7NHM8" kind="definition">
# Convex body

## Definition

Let $n \ge 1$ and consider [[Euclidean space|0XPR1E85W0]] $\R^n$.
A *convex body* in $\R^n$ is a subset $K \subseteq \R^n$ such that:

1. $K$ is [[convex|02AS45T3H9]];
2. $K$ is [[compact|0R6EWFKRCF]];
3. $K$ has nonempty [[interior|0RZH6W98HK]].

## Remarks

Example: the closed unit ball $B = \set{x \in \R^n : \abs{x} \le 1}$.
It is convex by the triangle inequality, compact because it is closed and bounded, and its interior contains the open unit ball, so it is not empty.

Non-example: the closed segment $[0, 1] \times \set{0}$ in $\R^2$ is convex and compact, but its interior in $\R^2$ is empty, so clause 3 fails.

Some authors, for example in the theory of mixed volumes, call every nonempty compact convex set a convex body and drop clause 3.
With that convention, the bodies with nonempty interior are called full-dimensional.
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<document source="https://tamnd.github.io/math2/node/YRV6SYK2BY.md" tag="YRV6SYK2BY" kind="statement">
# Tarski's undefinability theorem

## Statement

Let $\mathcal{N} = (\N, 0, S, +, \cdot)$ be the standard model of arithmetic on the [[natural numbers|0MWKJ44V5A]], and fix a [[Gödel numbering|MFJXPAF0R0]] of the [[formulas|HYJ10AF6RT]] of the language of arithmetic.
There is no formula $\mathrm{True}(x)$ with one free variable such that, for every [[sentence|DKFGQTRYZE]] $\varphi$,
$$\mathcal{N} \models \mathrm{True}(\ulcorner \varphi \urcorner) \iff \mathcal{N} \models \varphi.$$
In other words, the set of codes of sentences true in $\mathcal{N}$ is not definable in $\mathcal{N}$.

## Remarks

Proof idea: suppose $\mathrm{True}(x)$ exists.
The [[diagonal lemma|3PRW76471J]], applied in the theory of all sentences true in $\mathcal{N}$, gives a sentence $\lambda$ with $\mathcal{N} \models \lambda \leftrightarrow \neg \mathrm{True}(\ulcorner \lambda \urcorner)$.
Then $\lambda$ is true exactly when it is not true, a contradiction.
This is a precise form of the liar paradox.

By contrast, for each $n$ the set of true $\Sigma_n$ sentences is definable in $\mathcal{N}$; only full truth is not.
A syntactic form also holds: if $T$ is a consistent theory that contains Robinson arithmetic, there is no formula $\mathrm{True}(x)$ with $T \vdash \mathrm{True}(\ulcorner \varphi \urcorner) \leftrightarrow \varphi$ for every sentence $\varphi$.

## Explanations

### L1 human

Arithmetic can talk about its own formulas through codes, but it cannot define which of them are true.
Any attempt runs into the sentence "this sentence is false".
So the notion of arithmetical truth lies outside arithmetic itself.

### L2 human

Compare with [[Gödel's incompleteness theorems|X9WYNYE29C]]: provability in a computably axiomatized theory is definable in arithmetic, but truth is not.
So truth and provability differ, which gives another proof of the first incompleteness theorem.
In terms of the [[arithmetical hierarchy|7CVJCQKY41]], the set of true sentences is not $\Sigma^0_n$ for any $n$.
</document>

<document source="https://tamnd.github.io/math2/node/YRYWCP9TYV.md" tag="YRYWCP9TYV" kind="statement">
# Three-term recurrence for orthogonal polynomials

## Statement

Let $\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\R$ with infinite support and with $\int_\R \abs{x}^k\, d\mu(x) < \infty$ for every $k \ge 0$.
Let $\ip{f}{g} = \int_\R f g\, d\mu$, and let $(p_n)_{n \ge 0}$ be the monic orthogonal polynomials for $\mu$, so that $\deg p_n = n$, $p_n$ has leading coefficient $1$, and $\ip{p_m}{p_n} = 0$ for $m \ne n$.
Set $p_{-1} = 0$.
Then there are real numbers $a_n$ for $n \ge 0$ and $b_n > 0$ for $n \ge 1$ such that, for every $n \ge 0$,
$$p_{n+1}(x) = (x - a_n)\, p_n(x) - b_n\, p_{n-1}(x),$$
namely $a_n = \frac{\ip{x p_n}{p_n}}{\ip{p_n}{p_n}}$ and $b_n = \frac{\ip{p_n}{p_n}}{\ip{p_{n-1}}{p_{n-1}}}$.

## Remarks

Infinite support makes $\ip{\cdot}{\cdot}$ an [[inner product|0NB0AWGJBW]] on all polynomials, so the [[Gram–Schmidt process|0VT6P0GNQR]] applied to $1, x, x^2, \dots$ gives the $p_n$.
The polynomial $x p_n - p_{n+1}$ has degree at most $n$, so it is a combination of $p_0, \dots, p_n$.
Its coefficient on $p_k$ is $\frac{\ip{x p_n}{p_k}}{\ip{p_k}{p_k}} = \frac{\ip{p_n}{x p_k}}{\ip{p_k}{p_k}}$, which is $0$ for $k \le n - 2$ because $x p_k$ has degree less than $n$.
This key step uses that multiplication by $x$ is symmetric for $\ip{\cdot}{\cdot}$.
The same symmetry gives $\ip{x p_n}{p_{n-1}} = \ip{p_n}{x p_{n-1}} = \ip{p_n}{p_n}$, which yields the formula for $b_n$.
Example: for the [[Hermite polynomials|MGACY98E9H]] made monic, $a_n = 0$ and $b_n = \frac{n}{2}$, so $p_2 = x^2 - \frac{1}{2}$.

## Explanations

### L1 human

Orthogonal polynomials are not just any list of polynomials: each one is built from the previous two by a short rule.
To get the next one, multiply the current one by $x$, shift by a multiple of it, and subtract a multiple of the one before.
This makes them cheap to compute and gives them many special properties.

### L2 human

The recurrence coefficients encode the measure: they form a symmetric tridiagonal Jacobi matrix whose spectral measure is $\mu$, normalised to mass $1$, when $\mu$ is determined by its moments.
The converse, that any such recurrence with $b_n > 0$ comes from a measure, is Favard's theorem.
The zeros of $p_n$ are the eigenvalues of the $n \times n$ truncated Jacobi matrix, which is the basis of the Golub–Welsch method for Gaussian quadrature.
The Christoffel–Darboux formula is a direct consequence of the recurrence.
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# Gaussian binomial coefficient

## Summary

The Gaussian binomial coefficient $\binom{n}{k}_q$ is the [[q-analog|GFSDMJHHVA]] of the [[binomial coefficient|0GBZM0MT0F]] $\binom{n}{k}$, see the [[definition|TZ6Y9XXQJW]].
It is a polynomial in $q$ with nonnegative integer coefficients, and it equals $\binom{n}{k}$ at $q = 1$.
When $q$ is a prime power, it counts the $k$-dimensional subspaces of the [[vector space|000AYPA9AJ]] $\F_q^n$.
Its coefficients count the [[integer partitions|0JV30XN73Z]] whose [[Young diagram|096AHMB3P4]] fits in a $k \times (n - k)$ box.

## Explanations

### L1 human

The ordinary binomial coefficient counts subsets of a set.
The Gaussian binomial coefficient counts subspaces of a vector space over a finite field with $q$ elements.
For example, $\binom{4}{2}_q = 1 + q + 2q^2 + q^3 + q^4$.
At $q = 1$ it gives $6$, the number of $2$-element subsets of a $4$-element set.
At $q = 2$ it gives $35$, the number of planes through the origin in $\F_2^4$.
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# Nonlinear approximation

## Summary

Nonlinear approximation approximates a function by elements of a set that is not a linear space.
A typical such set is the sums of $n$ terms chosen freely from a dictionary.
The approximant may adapt to the function, for example by placing spline knots or wavelets where the function is rough.
Typical cases are [[n-term approximation|GEK0NP5WQM]], [[rational approximation|EHFQK1WXKS]], splines with free knots and neural networks.
For functions with isolated singularities it can be much more efficient than linear approximation with the same number of parameters.

## Explanations

### L1 human

In linear approximation the building blocks are fixed in advance, for example the polynomials of degree at most $n$.
In nonlinear approximation one first looks at the function and then picks the building blocks that suit it best.
The set of all such choices is not closed under addition: the sum of two $n$-term sums can have $2n$ terms.
For example, to approximate a function that is smooth except for one jump, one can put many small wavelets near the jump and few elsewhere.
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# Curse of dimensionality

## Summary

The curse of dimensionality is the exponential growth in the dimension $d$ of the cost of solving a numerical problem to a fixed accuracy.
The phrase is due to Richard Bellman.
In integration and approximation it means that the number of function values needed for error $\varepsilon$ grows like $c^d$ with $c > 1$.
Methods such as [[sparse grids|84BR7X6HET]], Monte Carlo and quasi-Monte Carlo methods, and restrictions to special classes of functions are ways to avoid it.

## Explanations

### L1 human

To sample a line segment with spacing $0.1$ one needs $10$ points, a square needs $100$, and a cube needs $1000$.
In $d$ dimensions the same spacing needs $10^d$ points, which is hopeless for $d = 50$.
High-dimensional space is so large that any reasonable set of sample points leaves most of it far from every point.
So an unknown function can hide almost anything between the samples.
Take the functions on $[0, 1]^d$ bounded by $1$ and with Lipschitz constant $1$.
For example, no method that uses at most $2^{d-1}$ of their values can always compute the integral with error below $1/8$.
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# Axiomatic potential theory

## Summary

Axiomatic potential theory studies [[potential theory|9PM8Q2E30Q]] from a short list of axioms on a sheaf of "harmonic" functions on a locally compact space.
The axioms keep the solvability of the Dirichlet problem on small sets and a convergence property of increasing sequences.
From them one recovers superharmonic functions, potentials, balayage, the fine topology and the Dirichlet problem.
The theory covers the Laplace equation, more general elliptic equations, the heat equation and Markov processes in one framework.

## Explanations

### L1 human

Classical potential theory is built on the Laplace equation, but many of its theorems use only a few of its features.
Axiomatic potential theory picks out those features and takes them as axioms.
Any equation or random process that satisfies the axioms then inherits the whole theory at once.
The main frameworks are the [[harmonic spaces|DWN0XEQHHK]] of Brelot and of Bauer, and later more abstract ones such as [[balayage spaces|G9M1H5NNF6]] and [[H-cones|Q5C1E10CSQ]].
For example, the solutions of the heat equation fit Bauer's axioms, so results like the minimum principle for supersolutions follow from the general theory.
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# Nome

## Summary

The nome is the variable $q = e^{\pi i\tau}$ in which [[theta functions|2F5CYQJVAF]] and elliptic functions are expanded as power series.
For a real modulus $0 < k < 1$ it is $q = \exp(-\pi K'/K)$, where $K$ and $K'$ are [[complete elliptic integrals of the first kind|RM54CPPE12]].
It ties the modulus of the Jacobi elliptic functions to the period ratio of their lattice.

## Explanations

### L1 human

Elliptic functions depend on a shape parameter, which can be given as the modulus $k$ or as the ratio $\tau$ of two periods.
The nome $q = e^{\pi i\tau}$ is a third way to give the same parameter, chosen so that series expansions are power series in $q$.
For moderate $k$, $q$ is very small, about $k^2/16$ when $k$ is small, so these series converge very fast.
For example, at $k = 1/\sqrt2$ the nome is $e^{-\pi} \approx 0.0432$.
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# Lambert summation

## Definition

Let $\sum_{n \ge 1} a_n$ be a [[series|0JH4MN4K1C]] of complex numbers.
The series is *Lambert summable* to $s$ if the following two conditions hold.

1. The series $\sum_{n \ge 1} n a_n \frac{x^n}{1 - x^n}$ converges for every real $x$ with $0 < x < 1$.
2. The [[one-sided limit|0909PWZR62]]
$$\lim_{x \to 1^-} (1 - x) \sum_{n \ge 1} n a_n \frac{x^n}{1 - x^n}$$
exists and equals $s$.

## Remarks

Example: let $\mu$ be the [[Möbius function|0C8QENHYZW]] and $a_n = \mu(n)/n$.
Expanding $\frac{x^n}{1 - x^n} = \sum_{m \ge 1} x^{nm}$ and grouping by $N = nm$ gives
$$\sum_{n \ge 1} \mu(n) \frac{x^n}{1 - x^n} = \sum_{N \ge 1} x^N \sum_{d \mid N} \mu(d) = x,$$
since $\sum_{d \mid N} \mu(d)$ is $1$ for $N = 1$ and $0$ for $N > 1$.
The rearrangement is valid because the double series converges absolutely for $0 < x < 1$.
So $(1 - x) \cdot x \to 0$, and $\sum \mu(n)/n$ is Lambert summable to $0$.

Non-example: for $1 + 1 + 1 + \cdots$, the inequality $1 - x^n \le n (1 - x)$ gives $n \frac{x^n}{1 - x^n} \ge \frac{x^n}{1 - x}$.
Hence $(1 - x) \sum_{n \ge 1} n \frac{x^n}{1 - x^n} \ge \sum_{n \ge 1} x^n = \frac{x}{1 - x} \to +\infty$, and condition 2 fails.
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# Fuchsian equation

## Definition

Let $p_1, \ldots, p_n$ be [[rational functions|0XDK9CGYKJ]] of $z$ with complex coefficients.
The linear equation
$$y^{(n)} + p_1(z)\, y^{(n-1)} + \cdots + p_n(z)\, y = 0$$
is *Fuchsian* (or of Fuchsian class) if it satisfies the following condition.
Every point of the [[Riemann sphere|0J78KBPFZB]] $\C \cup \set{\infty}$ is an [[ordinary point|QK2J038VB4]] or a [[regular singular point|H96HXG6CA0]] of the equation.

## Remarks

At a finite point $a$, the condition says that each $p_k$ has at most a pole of order $k$ at $a$.
At $\infty$, it says that $z^k p_k(z)$ stays bounded as $z \to \infty$, for each $k$.
Since the $p_k$ are rational, there are only finitely many singular points.

Example: the hypergeometric equation $z (1 - z) y'' + \left( c - (a + b + 1) z \right) y' - a b\, y = 0$ is Fuchsian.
Its normalised coefficients are $p = \frac{c - (a + b + 1) z}{z (1 - z)}$ and $q = \frac{-a b}{z (1 - z)}$.
They have simple poles at $0$ and $1$, and $z p(z) \to a + b + 1$ and $z^2 q(z) \to a b$ as $z \to \infty$.
So the singular points $0$, $1$ and $\infty$ are all regular.

Non-example: the Bessel equation $z^2 y'' + z y' + (z^2 - \nu^2) y = 0$ has $q(z) = 1 - \frac{\nu^2}{z^2}$.
Then $z^2 q(z) = z^2 - \nu^2$ is unbounded as $z \to \infty$, so $\infty$ is an irregular singular point and the equation is not Fuchsian.
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# Compact quantum group

## Definition

Let $A$ be a unital C*-algebra and let $A \otimes A$ denote its minimal C*-tensor product with itself.
A *compact quantum group* is a pair $(A, \Delta)$ where $\Delta \colon A \to A \otimes A$ is a unital $*$-homomorphism, called the comultiplication, such that:

1. $(\Delta \otimes \id) \circ \Delta = (\id \otimes \Delta) \circ \Delta$ (coassociativity);
2. the linear spans of $\set{\Delta(a)(1 \otimes b) : a, b \in A}$ and of $\set{\Delta(a)(b \otimes 1) : a, b \in A}$ are both norm-dense in $A \otimes A$ (cancellation).

## Remarks

Example: let $G$ be a [[compact|0TVJZYB6SH]] Hausdorff group and $A = C(G)$, so that $A \otimes A = C(G \times G)$.
Set $\Delta(f)(s, t) = f(st)$.
Coassociativity is the associativity $(rs)t = r(st)$.
The products $\Delta(f)(1 \otimes g)$ are the functions $(s, t) \mapsto f(st) g(t)$.
Since $(s, t) \mapsto (st, t)$ is a homeomorphism of $G \times G$, their span is a self-adjoint subalgebra that separates points and contains $1$.
So it is dense by the Stone–Weierstrass theorem, and the other span is handled in the same way.

Non-example: let $S = [0, 1]$ with ordinary multiplication, a compact semigroup that is not a group, and define $\Delta$ on $C(S)$ by the same formula.
Clause 1 holds, but every function $(s, t) \mapsto f(st) g(t)$ is constant in $s$ on the line $t = 0$.
This property passes to linear combinations and uniform limits, so the span is not dense, and clause 2 fails.

For commutative $A$, every compact quantum group is of the form $C(G)$ for a compact group $G$.
Every compact quantum group has a unique state $h$, the Haar state, with $(h \otimes \id)\Delta(a) = (\id \otimes h)\Delta(a) = h(a) 1$ for all $a \in A$; it plays the role of the [[Haar measure|VY0HXD3GC4]].
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# Zeta-regularized determinant

## Summary

The zeta-regularized determinant of a positive elliptic operator is the regularized product of its eigenvalues, $\exp(-\zeta'(0))$, where $\zeta(s) = \sum \lambda^{-s}$ is its spectral zeta function.
Ray and Singer introduced it for Laplacians in order to define [[analytic torsion|MZT9D34B0A]].
It is an instance of [[zeta function regularization|VJCJYXGTDY]] and appears in string theory, quantum field theory and the geometry of [[determinant line bundles|5DST3VKEMX]].

## Explanations

### L1 human

The determinant of a matrix is the product of its eigenvalues.
A differential operator has infinitely many eigenvalues that grow to infinity, so the product diverges.
The zeta function trick replaces the logarithm of the product, $\sum \log \lambda$, by $-\zeta'(0)$, which is finite.
For the Laplacian on the circle of length $2\pi$, with the zero eigenvalue removed, this gives the value $4\pi^2$.
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# Product order

## Definition

Let $I$ be a set and let $(P_i, \le_i)$ be a [[partially ordered set|0QV2JS9RJ3]] for each $i \in I$.
The *product order* on the [[Cartesian product|09CEDJYBKP]] $\prod_{i \in I} P_i$ is the relation $\le$ given by
$$x \le y \iff x_i \le_i y_i \text{ for all } i \in I.$$

## Remarks

It is a partial order: reflexivity, antisymmetry and transitivity hold in each coordinate, hence for tuples.
For two factors, $(a, b) \le (a', b')$ means $a \le a'$ and $b \le b'$.
Example: in $\N \times \N$, $(1, 2) \le (3, 2)$, since $1 \le 3$ and $2 \le 2$.
Non-example: $(1, 2)$ and $(2, 1)$ are incomparable, since the first coordinate rises and the second falls.
For finitely many factors, the product order is contained in the lexicographic order on the same set, which is total when all factors are total.
The power set of a set $I$, ordered by inclusion, is isomorphic to the product order on $\set{0, 1}^I$.
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# Nevanlinna characteristic

## Summary

The Nevanlinna characteristic $T(r, f)$ of a [[meromorphic function|0NSD1TPHR0]] $f$ on $\C$ measures its growth on the disc of radius $r$.
It adds a term for the size of $f$ on the circle $\abs{z} = r$ and a term that counts the poles inside.
For entire functions it plays the role of $\log M(r)$, and for meromorphic functions it replaces the degree of a rational function.
It is the central quantity of [[Nevanlinna theory|1EHH63VNAD]].

## Explanations

### L1 human

For an entire function, growth is measured by the maximum of $\abs{f}$ on circles.
A meromorphic function has poles, where it is infinite, so the maximum is useless.
Nevanlinna's idea is to average $\log \abs{f}$ where it is large on the circle and to add a logarithmic count of the poles inside.
For a rational function of degree $d$, $T(r, f)$ is about $d \log r$.
For $e^z$, $T(r, f) = r/\pi$.
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# Expansion in spherical harmonics

## Statement

Let $n \ge 2$, let $R > 0$, and let $B = \set{x \in \R^n : \abs{x} < R}$ be the [[open ball|0YE5868HZB]] of radius $R$ centred at $0$.
Let $u$ be a [[harmonic function|0DEB0JMN0W]] on $B$.
Then there is a unique sequence $(p_k)_{k \ge 0}$ with the following property.
Each $p_k$ is a [[harmonic|MVYFSYMECG]] [[homogeneous polynomial|2C3PAQ7H50]] of degree $k$ on $\R^n$, and
$$u(x) = \sum_{k=0}^{\infty} p_k(x), \qquad x \in B.$$
The series converges absolutely and [[uniformly|08VZ57PT26]] on each compact subset of $B$.

## Remarks

Proof idea: on a smaller closed ball, $u$ is the Poisson integral of its boundary values.
The [[Poisson kernel|TCYSWCTYYM]] of the ball expands as a series of zonal harmonics, each a homogeneous harmonic polynomial in $x$; integrating term by term gives the $p_k$.
Uniqueness follows by comparing homogeneous parts of the Taylor series of $u$ at $0$, since $p_k$ is the degree $k$ part.
In polar form $x = r\zeta$ with $\abs{\zeta} = 1$, the expansion reads $u(r\zeta) = \sum_k r^k p_k(\zeta)$, and the restriction of $p_k$ to the unit sphere is a [[spherical harmonic|3F32Y21HP4]] of degree $k$.
The restrictions to the sphere of harmonic polynomials of different degrees are orthogonal in $L^2$ of the sphere.
In fact, $L^2$ of the sphere is the orthogonal direct sum of the spaces of spherical harmonics of each degree.
Example: on the unit disc in $\R^2 = \C$, $u = \operatorname{Re} \frac{1}{1 - z}$ is harmonic, and $u(r e^{i\theta}) = \sum_{k \ge 0} r^k \cos k\theta$, where $r^k \cos k\theta = \operatorname{Re} z^k$ is a homogeneous harmonic polynomial of degree $k$.

## Explanations

### L1 human

A harmonic function on a ball can be written as an infinite sum of harmonic polynomials, one for each degree.
This is like a Taylor series, but each piece is itself harmonic.
On the sphere, the pieces become the spherical harmonics, the analogues of $\cos k\theta$ and $\sin k\theta$ on a circle.

### L2 human

This is the higher-dimensional version of the Fourier series of a harmonic function on the disc.
It solves the Dirichlet problem on a ball by separation of variables: expand the boundary data in spherical harmonics and multiply the degree $k$ part by $(\abs{x}/R)^k$.
In $\R^3$ the space of spherical harmonics of degree $k$ has dimension $2k + 1$, and the expansion is the one used for gravitational and electrostatic potentials.
The [[Fischer decomposition|JWNRB34ZGV]] describes how all polynomials, not only harmonic ones, split into harmonic pieces.
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# Time-optimal control

## Problem

Let $U \subseteq \R^m$ be a nonempty compact set, and let $f \colon \R^n \times U \to \R^n$ be continuous.
Let $x_0, x_1 \in \R^n$.
For a measurable control $u \colon [0, T] \to U$, let $x$ be the trajectory of the [[control system|ZPDTVZQN5K]] $x' = f(x, u)$ with $x(0) = x_0$.
The *time-optimal control problem* is to find a final time $T^* \ge 0$ and an [[admissible control|8M31QKB0GW]] $u^*$ on $[0, T^*]$ whose trajectory satisfies $x(T^*) = x_1$.
It is required that no admissible control reaches $x_1$ at any time $T < T^*$.

## Remarks

For linear systems $x' = A x + B u$ with $U = [-1, 1]^m$ the theory is complete.
If $x_1$ can be reached from $x_0$ at some time, then a time-optimal control exists, because the reachable sets are compact and convex and depend continuously on time.
By [[Pontryagin's maximum principle|7RPV2T29CF]] a time-optimal control satisfies $u_j(t) = \sgn\big(b_j^{T} p(t)\big)$ wherever $b_j^{T} p(t) \neq 0$, where $b_j$ is the $j$-th column of $B$ and $p' = -A^{T} p$.
If each pair $(A, b_j)$ satisfies the [[Kalman rank condition|9NSCZ2KBEB]], then the time-optimal control is unique and [[bang–bang|GK1Z3Q1WRZ]] with finitely many switches.
If moreover all eigenvalues of $A$ are real, each component $u_j$ switches at most $n - 1$ times.
Example: for the double integrator $x_1' = x_2$, $x_2' = u$, $\abs{u} \le 1$, steered to the origin, the optimal control is $u = -1$ above the switching curve $x_1 = -\frac12 x_2 \abs{x_2}$ and $u = 1$ below it.
On the curve itself it is the constant that drives the state along the curve into the origin, and every optimal control switches at most once.

## Explanations

### L1 human

A time-optimal control problem asks how to bring a system from one state to another as fast as possible with limited control power.
A car that must stop at a wall in least time is an example.
For linear systems the answer is to use full power all the time, switching direction at well-chosen moments.
For the car, one accelerates fully and then brakes fully, with exactly one switch.
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# Classifying topos

## Definition

Let $\mathbb{T}$ be a geometric theory.
This is a first-order theory whose axioms are sequents between formulas built from finite conjunctions, arbitrary disjunctions and existential quantifiers.
For a [[Grothendieck topos|S3YVE24AQK]] $\mathcal{E}$, let $\mathbb{T}\text{-}\mathrm{Mod}(\mathcal{E})$ be the category of models of $\mathbb{T}$ in $\mathcal{E}$.
For Grothendieck toposes $\mathcal{E}$ and $\mathcal{F}$, let $\mathbf{Geom}(\mathcal{E}, \mathcal{F})$ be the category of [[geometric morphisms|N5R1WAEPFB]] $\mathcal{E} \to \mathcal{F}$, and write $f^*$ for the inverse image part of $f$.
A *classifying topos* of $\mathbb{T}$ is a Grothendieck topos $\mathbf{Set}[\mathbb{T}]$ together with a model $U_{\mathbb{T}}$ of $\mathbb{T}$ in it with the following property.
For every Grothendieck topos $\mathcal{E}$ the functor
$$\mathbf{Geom}(\mathcal{E}, \mathbf{Set}[\mathbb{T}]) \to \mathbb{T}\text{-}\mathrm{Mod}(\mathcal{E}), \qquad f \mapsto f^*(U_{\mathbb{T}})$$
is an [[equivalence of categories|K7A04ZBD92]].
The model $U_{\mathbb{T}}$ is called the generic model of $\mathbb{T}$.

## Remarks

Example: let $\mathbb{O}$ be the theory with one sort, no symbols and no axioms, so that a model of $\mathbb{O}$ in $\mathcal{E}$ is just an object of $\mathcal{E}$.
The functor category $[\mathbf{FinSet}, \mathbf{Set}]$, with the inclusion functor as generic model, is a classifying topos for $\mathbb{O}$; it is called the object classifier.
Non-example: $\mathbf{Set}$ is not a classifying topos for $\mathbb{O}$.
For every Grothendieck topos $\mathcal{E}$ there is, up to isomorphism, only one geometric morphism $\mathcal{E} \to \mathbf{Set}$, while $\mathcal{E}$ has in general many non-isomorphic objects.
Every geometric theory has a classifying topos, and it is unique up to equivalence.
For a discrete group $G$, the topos of $G$-sets classifies $G$-torsors.
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# Mellin transform of a dilation

## Statement

Let $f \colon (0, \infty) \to \C$ be measurable, let $a > 0$ and let $g(x) = f(ax)$ for $x > 0$.
Let $s \in \C$.
Then $g$ is [[Mellin convergent|SGZZ4Y0D36]] at $s$ if and only if $f$ is.
In that case
$$\mathcal{M}g(s) = a^{-s} \, \mathcal{M}f(s),$$
where $\mathcal{M}$ is the [[Mellin transform|N5PVGRJE1T]] and $a^{-s} = e^{-s \log a}$ is a [[complex power|0RNGDMHT7M]].

## Remarks

The proof is the substitution $u = ax$:
$$\int_0^\infty x^{s-1} f(ax) \, dx = \int_0^\infty \left(\frac{u}{a}\right)^{s-1} f(u) \, \frac{du}{a} = a^{-s} \int_0^\infty u^{s-1} f(u) \, du.$$
The same computation with $\abs{f}$ and $\operatorname{Re} s$ gives the equivalence of convergence.
Example: $f(x) = e^{-x}$ gives $\int_0^\infty x^{s-1} e^{-ax} \, dx = a^{-s} \Gamma(s)$ for $\operatorname{Re} s > 0$.
Applied to $a = n$ and summed over $n$, this is the step that links the gamma function with Dirichlet series.
The hypothesis $a > 0$ is needed so that $x \mapsto ax$ maps $(0, \infty)$ onto itself.

## Explanations

### L1 human

Stretching or squeezing the input of $f$ by a factor $a$ multiplies its Mellin transform by $a^{-s}$.
So the Mellin transform turns the scaling of the variable into a simple factor, in the same way the Fourier transform turns a shift into a phase factor.

### L2 human

This is the reason the Mellin transform suits problems with scale invariance, such as homogeneous kernels and harmonic sums $\sum_k \lambda_k f(\mu_k x)$.
For a harmonic sum the rule gives $\mathcal{M}\left[\sum_k \lambda_k f(\mu_k x)\right](s) = \left(\sum_k \lambda_k \mu_k^{-s}\right) \mathcal{M}f(s)$, when the sum and integral can be exchanged.
The factor in brackets is a Dirichlet series, and its poles drive the asymptotic behaviour of the sum.
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# Quasi-everywhere

## Definition

Let $n \ge 2$, let $A \subseteq \R^n$, and let $P$ be a property that each point of $A$ may or may not have.
The property $P$ holds *quasi-everywhere* on $A$, abbreviated q.e., if the set of points of $A$ where $P$ fails has [[capacity zero|W9N1TR1X8P]].

## Remarks

Example: let $K$ be the closed unit disc in $\C$ and let $u(z) = \log \abs{z}$.
Then $u > -\infty$ quasi-everywhere on $K$, since the exceptional set $\set{0}$ is finite and so has capacity zero.

Non-example: let $u$ be the function on $\C$ with $u = 1$ on the segment $[0, 1]$ and $u = 0$ elsewhere.
Then $u = 0$ almost everywhere with respect to area, but not quasi-everywhere, because the segment has positive logarithmic capacity.

In $\R^n$ with $n \ge 3$ the capacity used is the Newtonian capacity, and in the plane it is the logarithmic capacity.
In nonlinear potential theory one speaks of $p$-quasi-everywhere, with the $p$-capacity in place of the Newtonian capacity.
</document>

<document source="https://tamnd.github.io/math2/node/YVBHE2SEE6.md" tag="YVBHE2SEE6" kind="conjecture">
# Vaught conjecture

## Statement

Let $T$ be a [[complete theory|655K8Y7VS8]] in a countable first-order signature.
Let $I(T, \aleph_0)$ be the number of [[countable|01TQB88H4C]] [[models|MVCH6N9SNW]] of $T$ up to isomorphism.
Then $I(T, \aleph_0)$ is finite, or equal to [[$\aleph_0$|0V2H5PGJ9B]], or equal to $2^{\aleph_0}$.

## Remarks

The conjecture is open.
Morley proved in 1970 that $I(T, \aleph_0)$ is at most $\aleph_1$ or equal to $2^{\aleph_0}$.
So the conjecture holds under the continuum hypothesis, and the open question is whether a theory can have exactly $\aleph_1$ countable models when $\aleph_1 < 2^{\aleph_0}$.
Shelah, Harrington and Makkai proved the conjecture for $\omega$-stable theories.
Vaught himself proved that $I(T, \aleph_0)$ is never equal to $2$.

All the allowed values occur.
The theory of dense linear orders without endpoints has exactly one countable model.
The theory of algebraically closed fields of characteristic $0$ has $\aleph_0$ countable models, one for each transcendence degree $0, 1, 2, \dots, \aleph_0$.
The complete theory of $(\Z, <)$ has $2^{\aleph_0}$ countable models.

## Explanations

### L1 human

A complete theory can have many non-isomorphic countable models.
The conjecture says that their number is never strictly between "countably many" and "as many as there are real numbers".
Without the continuum hypothesis, there could be a size in between, and the question is whether some theory has exactly that many countable models.
</document>

<document source="https://tamnd.github.io/math2/node/YVDWN0KVG4.md" tag="YVDWN0KVG4" kind="definition">
# Clenshaw–Curtis quadrature

## Definition

Let $N \ge 1$ and let $x_k = \cos(k\pi/N)$ for $k = 0, 1, \dots, N$.
For a continuous function $f$ on $[-1, 1]$, let $p_N$ be the interpolating polynomial of degree at most $N$ with $p_N(x_k) = f(x_k)$ for all $k$.
The $(N + 1)$-point *Clenshaw–Curtis quadrature* rule is the [[quadrature rule|ADSVMCY77A]]
$$\int_{-1}^{1} f(x)\,dx \approx \int_{-1}^{1} p_N(x)\,dx = \sum_{k=0}^{N} w_k f(x_k),$$
where $w_k = \int_{-1}^{1} \ell_k(x)\,dx$ and $\ell_k$ is the Lagrange basis polynomial of the node $x_k$.

## Remarks

The nodes are the extreme points of the [[Chebyshev polynomial|M6EAR1JHPP]] $T_N$ on $[-1, 1]$.
Writing $p_N(\cos\theta) = \sum_{j=0}^{N} c_j \cos(j\theta)$ gives $\int_{-1}^{1} p_N = \sum_j c_j \int_0^\pi \cos(j\theta)\sin\theta\,d\theta$, and $\int_0^\pi \cos(j\theta)\sin\theta\,d\theta$ is $2/(1 - j^2)$ for even $j$ and $0$ for odd $j$.
The rule is exact for polynomials of degree at most $N$, and of degree at most $N + 1$ when $N$ is even, by symmetry.

Example: $N = 2$.
The nodes are $1, 0, -1$, and the interpolatory weights are $\frac13, \frac43, \frac13$.
This is Simpson's rule; for $f = x^2$ it gives $\frac13 + 0 + \frac13 = \frac23 = \int_{-1}^{1} x^2\,dx$.

Non-example: the interpolatory rule at the equally spaced points $-1, -\frac13, \frac13, 1$ is a Newton–Cotes rule, not a Clenshaw–Curtis rule, because its nodes are not of the form $\cos(k\pi/3)$, which are $1, \frac12, -\frac12, -1$.
</document>

<document source="https://tamnd.github.io/math2/node/YVEA283NZW.md" tag="YVEA283NZW" kind="object">
# Mersenne Twister

## Description

The Mersenne Twister is a [[pseudorandom number generator|C5FG6ARWVV]] designed by Makoto Matsumoto and Takuji Nishimura.
Its standard version, MT19937, works with $32$-bit words.
Its state is an array of $624$ words, that is $19\,968$ bits, of which $19\,937$ bits are effective.
The transition function is linear over the [[field with two elements|0ZXZ82FTX2]] $\F_2$.
It combines three words of the array by shifts, masks and exclusive or, in a twisted generalised feedback shift register recurrence.
The output function applies a fixed invertible linear map, the tempering, to the newest word.
It fits the definition of a generator: the state space is finite, and the transition and output are fixed maps.
The seed determines the initial array.
Its period is $2^{19937} - 1$, which is a [[Mersenne prime|0F3Z22M2GT]]; this is where the name comes from.
Its outputs are $623$-dimensionally equidistributed to $32$-bit accuracy.
That is, over a full period every $623$-tuple of consecutive $32$-bit outputs occurs equally often, except that the all-zero tuple occurs once less.

## Remarks

Because the generator is linear over $\F_2$, $624$ consecutive outputs determine the whole state, so all later outputs can be predicted.
It is therefore not suitable for cryptography.
Linearity also makes it fail some statistical tests based on linear complexity, although these rarely matter in simulation.
The large state means it can take many steps to recover from a seed with mostly zero bits.

## Explanations

### L1 human

The Mersenne Twister is the default random number generator of many programming languages and libraries, for example in Python.
It keeps a large hidden table of bits and stirs it with cheap bit operations at each step.
The sequence repeats only after $2^{19937} - 1$ outputs, a number with about $6000$ decimal digits, so no computation will ever see it repeat.
It is fast and statistically good for simulations, but a clever observer who sees enough outputs can predict the rest.
</document>

<document source="https://tamnd.github.io/math2/node/YVH149HCAG.md" tag="YVH149HCAG" kind="concept">
# Continuous and residual spectrum

## Summary

The continuous and residual spectra are the parts of the spectrum of an operator that consist of numbers which are not eigenvalues.
At such a $\lambda$, the operator $T - \lambda I$ is injective but not invertible.
The spectrum is continuous at $\lambda$ when the range is dense and residual when it is not.
Together with the [[point spectrum|7FNVA0Q6YQ]], they split the spectrum into three disjoint pieces.

## Explanations

### L1 human

Outside the eigenvalues, $T - \lambda I$ loses no information, yet it still cannot be inverted with a bounded inverse.
Either its range comes close to every vector but misses some, which is the continuous case, or its range stays away from a whole direction, which is the residual case.
For multiplication by $x$ on square-integrable functions on $[0, 1]$, every number in $[0, 1]$ is in the continuous spectrum.
For the right shift on sequences, every number of absolute value less than $1$ is in the residual spectrum.
</document>

<document source="https://tamnd.github.io/math2/node/YVJ15PGKDM.md" tag="YVJ15PGKDM" kind="concept">
# Simplicial polytope

## Summary

A simplicial polytope is a [[convex polytope|300K6CB1KH]] all of whose [[facets|DPDXN2FJC4]] are simplices.
Its boundary is then a [[simplicial complex|V8GZ0W1ZZ5]] homeomorphic to a sphere.
Simplicial polytopes are the natural setting of the face-counting theory of polytopes, including the [[Dehn–Sommerville equations|4WKHF5BCTA]] and the [[g-theorem|4CZMWQVC30]].
Points in general position always have a simplicial convex hull, so "most" polytopes are simplicial.

## Explanations

### L1 human

A simplex is the simplest solid of its dimension: a segment, a triangle, a tetrahedron, and so on.
A polytope is simplicial when every face on its surface is a simplex.
In three dimensions this means every face is a triangle.
The octahedron and the icosahedron are simplicial; the cube is not, because its faces are squares.
</document>

<document source="https://tamnd.github.io/math2/node/YVPTBY45PK.md" tag="YVPTBY45PK" kind="object">
# Surreal number

## Description

The *surreal numbers* are built in stages, called days.
A surreal number is written $x = \{ L \mid R \}$, where $L$ and $R$ are sets of surreal numbers built on earlier days.
The condition is that no element of $L$ is greater than or equal to any element of $R$.
The order is defined at the same time.
The rule is: $x \le y$ holds when no left option of $x$ is $\ge y$ and no right option of $y$ is $\le x$.
Two forms $x$ and $y$ name the same number when $x \le y$ and $y \le x$.

On day $0$, only $0 = \{ \mid \}$ exists, with both sets empty.
On day $1$, one gets $1 = \{ 0 \mid \}$ and $-1 = \{ \mid 0 \}$.
On day $2$, one gets $2 = \{ 1 \mid \}$, $\tfrac12 = \{ 0 \mid 1 \}$, and their negatives.
After infinitely many days one gets $\omega = \{ 0, 1, 2, \dots \mid \}$, which is larger than every natural number.
One also gets $\varepsilon = \{ 0 \mid 1, \tfrac12, \tfrac14, \dots \}$, which is positive but smaller than every positive real number.

Addition and multiplication are defined by recursion on the options.
With them the surreal numbers satisfy all the axioms of an [[ordered field|0P3VCBDEE2]].
The one caveat is size: they form a proper class, not a set, so strictly they are an ordered field only in this class sense.
This field contains a copy of the real numbers and a copy of every [[ordinal number|0YSCSX5YZC]].
It is not [[Archimedean|0XH79S06J3]], since $\omega$ exceeds every natural number, and $\omega \cdot \varepsilon = 1$.
It is a [[real closed field|DDY3Q3PJ6W]].

## Remarks

Each real number $r$ is the surreal number $\{ L \mid R \}$ with $L$ the rationals below $r$ with power-of-two denominators and $R$ those above $r$; this mirrors a [[Dedekind cut|0G3RRM41X9]].
Like the [[hyperreal numbers|421QMTFG9T]], the surreals contain infinite and infinitesimal elements.
The surreals were introduced by John Conway, and the name comes from Donald Knuth's book *Surreal Numbers* (1974).

## Explanations

### L1 human

Surreal numbers start from nothing and grow by one rule: a new number sits between a set of numbers on its left and a set on its right.
From the empty pair one gets $0$, then $1$ and $-1$, then $2$, $\tfrac12$ and so on.
Every real number appears, and so do infinite numbers like $\omega$ and tiny numbers like $1/\omega$.
You can still add, subtract, multiply and divide them, and compare any two of them.
So they are one huge number line that holds both the reals and the ordinals.
</document>

<document source="https://tamnd.github.io/math2/node/YVQ9CKGA0T.md" tag="YVQ9CKGA0T" kind="definition">
# Valuation ring

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]] with [[field of fractions|0VF11ZTTMD]] $K$.
The ring $R$ is a *valuation ring* if for every nonzero $x \in K$, at least one of $x$ and $x^{-1}$ lies in $R$.

## Remarks

Example: for a prime $p$, the ring $\Z_{(p)} = \set{a/b : a, b \in \Z,\ p \nmid b}$ is a valuation ring.
Each nonzero rational number is $x = p^n a/b$ with $n \in \Z$ and $a, b$ prime to $p$; if $n \ge 0$ then $x \in \Z_{(p)}$, and if $n < 0$ then $x^{-1} = p^{-n} b/a \in \Z_{(p)}$.
Non-example: $\Z$ is not a valuation ring, since neither $2/3$ nor $3/2$ is an integer.
Every field is a valuation ring of itself.
If $v$ is a [[valuation|B85T5VKR3M]] on a field $K$, then $\set{x \in K : v(x) \ge 0}$ is a valuation ring with fraction field $K$.
Conversely, let $R$ be a valuation ring with fraction field $K$.
Then $K^\times / R^\times$, ordered by $xR^\times \le yR^\times$ when $y/x \in R$, is a totally ordered group.
The quotient map $K^\times \to K^\times/R^\times$ extends to a valuation whose ring is $R$.
A valuation ring is a local ring, with maximal ideal the set of non-units, and any two of its ideals are comparable by inclusion.
</document>

<document source="https://tamnd.github.io/math2/node/YVQRAPSF77.md" tag="YVQRAPSF77" kind="statement">
# d'Alembert's formula

## Statement

Let $c > 0$, let $g \in C^2(\R)$ and let $h \in C^1(\R)$.
Define $u \colon \R \times [0, \infty) \to \R$ by
$$u(x, t) = \frac{g(x + ct) + g(x - ct)}{2} + \frac{1}{2c} \int_{x - ct}^{x + ct} h(s)\, ds.$$
Then $u \in C^2(\R \times [0, \infty))$, $u$ solves the [[wave equation|5EHD2XBQDG]] $u_{tt} = c^2 u_{xx}$ on $\R \times (0, \infty)$, and $u(x, 0) = g(x)$, $u_t(x, 0) = h(x)$ for all $x \in \R$.
Moreover, $u$ is the only $C^2$ solution of this initial value problem.

## Remarks

The change of variables $\xi = x + ct$, $\eta = x - ct$ turns the equation into $u_{\xi\eta} = 0$.
Hence every $C^2$ solution has the form $u = F(x + ct) + G(x - ct)$, and the two initial conditions determine $F$ and $G$ up to a constant that cancels.
This argument also proves uniqueness.
Example: with $g = 0$ and $h = \cos$, the formula gives $u(x, t) = \frac{1}{2c}\big(\sin(x + ct) - \sin(x - ct)\big) = \frac{1}{c}\cos x \sin(ct)$, a standing wave.
The value $u(x, t)$ depends only on the data in the interval $[x - ct, x + ct]$, which shows finite speed of propagation in one dimension.
If $g$ is only continuous, the formula still defines a function, but it is a weak solution rather than a classical one.

## Explanations

### L1 human

The formula solves the vibrating string problem on an infinitely long string in closed form.
The initial shape $g$ splits into two halves, one moving left and one moving right at speed $c$.
The initial velocity $h$ adds the average of $h$ over the interval that a signal can reach in time $t$.
So what happens at a point depends only on what happened nearby, within distance $ct$.

### L2 human

d'Alembert's formula is the first explicit solution formula in a PDE course and the basis for the other dimensions.
By the method of spherical means, it gives Kirchhoff's formula in dimension $3$, and then by descent Poisson's formula in dimension $2$.
On the half line, the formula combined with an odd or even reflection of the data solves the problems with Dirichlet or Neumann boundary condition.
For the inhomogeneous equation $u_{tt} - c^2 u_{xx} = f$, Duhamel's principle adds the term $\frac{1}{2c}\int_0^t \int_{x - c(t - s)}^{x + c(t - s)} f(y, s)\, dy\, ds$.
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<document source="https://tamnd.github.io/math2/node/YW5P7NEKR4.md" tag="YW5P7NEKR4" kind="concept">
# Topology of uniform convergence

## Summary

The topology of uniform convergence is the topology on a set of functions into a metric space in which convergence means [[uniform convergence|08VZ57PT26]].
It comes from the uniform metric, which measures the largest distance between the values of two functions.
It is finer than the [[topology of pointwise convergence|FVGB8ZP67B]].
On continuous maps, uniform limits stay continuous, which is why this topology is so useful.

## Explanations

### L1 human

Two functions are close in this topology when their graphs stay close everywhere at once.
A band of fixed width around the graph of $f$ must contain the whole graph of $g$.
For example, $f_n(x) = x / n$ on $[0, 1]$ converges uniformly to $0$, since $\abs{f_n(x)} \le 1/n$ for all $x$.
But $f_n(x) = x^n$ on $[0, 1)$ does not, because near $x = 1$ the values stay close to $1$.
</document>

<document source="https://tamnd.github.io/math2/node/YW6W2JXMQB.md" tag="YW6W2JXMQB" kind="definition">
# Jacobi field

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] with Levi-Civita connection $\nabla$ and curvature $R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$, see [[Riemann curvature tensor|JA29ETCM02]].
Let $\gamma \colon I \to M$ be a [[geodesic|NSYMQMHPDN]], and let $D_t$ be the [[covariant derivative|5FZDQQHQKM]] along $\gamma$.
A smooth vector field $J$ along $\gamma$ is a *Jacobi field* if
$$D_t^2 J + R(J, \gamma')\gamma' = 0 \quad \text{on } I.$$

## Remarks

Example: on the unit sphere $S^2$, let $\gamma$ be a unit-speed geodesic and $E$ a parallel unit field along $\gamma$ orthogonal to $\gamma'$.
For $J = \sin(t) E$, one has $D_t^2 J = -\sin(t) E$ and, since the curvature is $1$, $R(J, \gamma')\gamma' = \sin(t) E$; the sum is $0$, so $J$ is a Jacobi field.

Non-example: on the same sphere, $J = tE$ has $D_t^2 J = 0$ but $R(J, \gamma')\gamma' = tE \ne 0$ for $t \ne 0$, so $J$ is not a Jacobi field, although it is one in the flat plane.

If $\Gamma(s, t)$ is a smooth family of geodesics $t \mapsto \Gamma(s, t)$, then $J(t) = \partial_s \Gamma(0, t)$ is a Jacobi field, and every Jacobi field along a geodesic on a compact interval arises in this way.
A Jacobi field is determined by $J(t_0)$ and $D_t J(t_0)$ at one time, so the Jacobi fields along $\gamma$ form a vector space of dimension $2n$, where $n = \dim M$.
The fields $\gamma'$ and $t \gamma'$ are always Jacobi fields; the interesting ones are normal to $\gamma$.
With the opposite sign convention for $R$, the equation reads $D_t^2 J + R(\gamma', J)\gamma' = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/YW8HBW1GJX.md" tag="YW8HBW1GJX" kind="definition">
# Deformation retract

## Definition

Let $X$ be a topological space and let $A \subseteq X$ carry the [[subspace topology|0WS9P12FQR]].
The subspace $A$ is a *deformation retract* of $X$ if there is a [[homotopy|9YA6S6HD9V]] $H \colon X \times [0, 1] \to X$ such that:

1. $H(x, 0) = x$ for all $x \in X$;
2. $H(x, 1) \in A$ for all $x \in X$;
3. $H(a, 1) = a$ for all $a \in A$.

The subspace $A$ is a *strong deformation retract* of $X$ if moreover $H(a, t) = a$ for all $a \in A$ and $t \in [0, 1]$.

## Remarks

The map $r = H(\cdot, 1) \colon X \to A$ is a retraction, and the inclusion $i \colon A \to X$ is a [[homotopy equivalence|XZG6MTSWCN]] with homotopy inverse $r$.
Some textbooks, such as Hatcher's, use the name deformation retraction for the strong version.
Example: $S^{n-1}$ is a strong deformation retract of $\R^n \setminus \set{0}$, via $H(x, t) = (1 - t) x + t x / \norm{x}$; clause 3 holds since $\norm{a} = 1$ for $a \in S^{n-1}$.
Non-example: a point $p$ of the circle $S^1$ is not a deformation retract of $S^1$, since otherwise $S^1$ would be [[contractible|R5WP8Z69Q8]].
</document>

<document source="https://tamnd.github.io/math2/node/YW8YSCFPG5.md" tag="YW8YSCFPG5" kind="concept">
# Club set

## Summary

A club set is a subset of a limit ordinal that is closed under limits and unbounded, and the name abbreviates "closed unbounded".
Below a [[regular cardinal|13ZV2F8JAR]] $\kappa > \omega$, the club sets play the role of "large" sets: the intersection of fewer than $\kappa$ of them is again club.
They define the [[stationary sets|YE6WZR46RQ]], and the [[precise definition|GWS5QXH7WW]] is given separately.

## Explanations

### L1 human

Think of the countable ordinals as a very long line.
A club set is a subset that keeps appearing as far as you go, and that contains every point it approaches from below.
For example, the limit ordinals below $\omega_1$, such as $\omega$, $\omega \cdot 2$ and $\omega^2$, form a club set.
The successor ordinals do not: they keep appearing, but the limits they approach, like $\omega$, are not successors.
Two club sets of countable ordinals always meet, and in fact their intersection is again club.
</document>

<document source="https://tamnd.github.io/math2/node/YWB981MEF2.md" tag="YWB981MEF2" kind="concept">
# Toric variety

## Summary

A toric variety is a variety that contains an [[algebraic torus|66FQFAJNS4]] as a dense open subset.
The action of the torus on itself must extend to the whole variety.
Normal toric varieties correspond to combinatorial objects called fans, built from rational polyhedral cones.
So geometric questions about them turn into questions about cones and polytopes, such as [[Newton polytopes|CYTTX8KHC8]].
They are a standard testing ground for conjectures and constructions in algebraic geometry.

## Explanations

### L1 human

Start with the points $(x, y)$ where both coordinates are nonzero; this is a two-dimensional torus.
Adding back the coordinate axes gives the plane, and adding a line at infinity gives the projective plane.
In each case the torus sits inside as a dense open part, and scaling the coordinates still makes sense on the whole space.
Such spaces are toric varieties, and the way they are glued from the torus is recorded by a picture of cones in a lattice.
</document>

<document source="https://tamnd.github.io/math2/node/YWBTGC7MFA.md" tag="YWBTGC7MFA" kind="concept">
# Instanton

## Summary

In differential geometry, an instanton is a connection on a principal bundle over an oriented Riemannian 4-manifold whose curvature is anti-self-dual for the [[Hodge star|1Y4WX4XQWJ]].
Over a compact 4-manifold, instantons are absolute minima of the Yang–Mills functional among connections on a given bundle, so they solve the Yang–Mills equations.
In physics they are finite-action solutions of the Euclidean field equations that describe tunnelling between vacua.
Their moduli spaces were used by Donaldson to discover new invariants of smooth 4-manifolds.

## Explanations

### L1 human

A connection on a bundle tells how to move internal states, such as the phase of a particle, from point to point, and its curvature measures the twist.
In four dimensions the Hodge star sends 2-forms to 2-forms, so a curvature can equal minus its own star.
Instantons are the connections with this property.
They are the most economical way to twist a bundle with a given topology.
The basic example is the BPST instanton, a connection for the group $SU(2)$ on $\R^4$ whose curvature is concentrated near one point.
</document>

<document source="https://tamnd.github.io/math2/node/YWTERZZME5.md" tag="YWTERZZME5" kind="definition">
# Isomorphism

## Definition

Let $\sigma$ be a signature and let $\mathbf{A}$ and $\mathbf{B}$ be $\sigma$-[[structures|983GZBQYT4]] with domains $A$ and $B$.
An *isomorphism* from $\mathbf{A}$ to $\mathbf{B}$ is a [[bijection|0GG9TVW90M]] $h \colon A \to B$ such that both $h$ and its inverse $h^{-1} \colon B \to A$ are [[homomorphisms|KBXSZC6BV1]].
The structures $\mathbf{A}$ and $\mathbf{B}$ are *isomorphic*, written $\mathbf{A} \cong \mathbf{B}$, if there is an isomorphism from $\mathbf{A}$ to $\mathbf{B}$.

## Remarks

The condition on $h^{-1}$ means that $h$ also reflects relations: $(a_1, \dots, a_n) \in R^{\mathbf{A}}$ if and only if $(h(a_1), \dots, h(a_n)) \in R^{\mathbf{B}}$.
For a signature without relation symbols, the inverse of a bijective homomorphism is automatically a homomorphism, so an isomorphism is the same as a bijective homomorphism.

Example: the map $x \mapsto e^x$ is an isomorphism from $(\R, +, 0)$ to $(\R_{>0}, \cdot, 1)$, because it is a bijection with $e^{x + y} = e^x e^y$ and $e^0 = 1$; its inverse is $\ln$.

Non-example: let $A = \set{1, 2}$ with the empty binary relation and $B = \set{1, 2}$ with the relation $\le$.
The identity map is a bijective homomorphism from $A$ to $B$, but its inverse does not preserve the relation, since $1 \le 2$ in $B$ while $(1, 2)$ is not related in $A$.
</document>

<document source="https://tamnd.github.io/math2/node/YWTT6FD50Y.md" tag="YWTT6FD50Y" kind="definition">
# Oriented matroid

## Definition

Let $E$ be a finite set.
A *signed subset* of $E$ is a pair $X = (X^+, X^-)$ of disjoint subsets of $E$; its support is $\underline{X} = X^+ \cup X^-$, and $-X = (X^-, X^+)$.
An *oriented matroid* on $E$, given by its signed circuits, is a set $\mathcal{C}$ of signed subsets of $E$ such that:

1. $(\emptyset, \emptyset) \notin \mathcal{C}$.
2. If $X \in \mathcal{C}$, then $-X \in \mathcal{C}$.
3. If $X, Y \in \mathcal{C}$ and $\underline{X} \subseteq \underline{Y}$, then $X = Y$ or $X = -Y$.
4. If $X, Y \in \mathcal{C}$, $X \ne -Y$, and $e \in X^+ \cap Y^-$, then there is $Z \in \mathcal{C}$ with $Z^+ \subseteq (X^+ \cup Y^+) \setminus \set{e}$ and $Z^- \subseteq (X^- \cup Y^-) \setminus \set{e}$.

## Remarks

The supports of the signed circuits are the circuits of a [[matroid|PX31ZNYS3Z]] on $E$, the underlying matroid.
Axiom 4 is the signed form of circuit elimination.
There are equivalent axiom systems in terms of covectors, of vectors and of chirotopes, which are signed versions of the bases.

Example: let $v_1, \dots, v_n$ be vectors in $\R^r$ and $E = \set{1, \dots, n}$.
For each linear dependence $\sum_i \lambda_i v_i = 0$ whose support $\set{i : \lambda_i \ne 0}$ is minimal and nonempty, put $X^+ = \set{i : \lambda_i > 0}$ and $X^- = \set{i : \lambda_i < 0}$.
These signed sets satisfy the axioms; such oriented matroids are called realizable.
For $v_1 = e_1$, $v_2 = e_2$, $v_3 = e_1 + e_2$ in $\R^2$, the only minimal dependence up to scaling is $v_1 + v_2 - v_3 = 0$, so $\mathcal{C} = \set{(\set{1,2}, \set{3}), (\set{3}, \set{1,2})}$.

Non-example: on $E = \set{1, 2}$, the set $\mathcal{C} = \set{(\set{1}, \set{2}), (\set{1,2}, \emptyset), (\set{2}, \set{1}), (\emptyset, \set{1,2})}$ fails axiom 3, since all four sets have the same support but $(\set{1}, \set{2}) \ne \pm (\set{1,2}, \emptyset)$.
</document>

<document source="https://tamnd.github.io/math2/node/YWTXGDC427.md" tag="YWTXGDC427" kind="definition">
# Chebyshev nodes

## Definition

Let $n \ge 1$ be an integer.
The *Chebyshev nodes* of order $n$ in $[-1, 1]$ are the $n$ numbers
$$x_k = \cos\left(\frac{(2k - 1)\pi}{2n}\right), \qquad k = 1, \dots, n.$$
They are the zeros of the [[Chebyshev polynomial|M6EAR1JHPP]] $T_n$ of the first kind.
For an [[interval|0TK8Y3J5RW]] $[a, b]$ with $a < b$, the Chebyshev nodes of order $n$ in $[a, b]$ are the numbers $\frac{a + b}{2} + \frac{b - a}{2} x_k$, $k = 1, \dots, n$.

## Remarks

Example: for $n = 3$ the nodes are $\cos(\pi/6), \cos(\pi/2), \cos(5\pi/6)$, that is $\sqrt{3}/2$, $0$ and $-\sqrt{3}/2$.
Check: $T_3(x) = 4x^3 - 3x = x(4x^2 - 3)$ vanishes exactly there.

Non-example: the equally spaced points $-1, 0, 1$ are not the Chebyshev nodes of order $3$, since $T_3(1) = 1 \ne 0$.

The node polynomial is $\prod_{k=1}^{n} (x - x_k) = 2^{1-n} T_n(x)$, whose maximum on $[-1, 1]$ is $2^{1-n}$, the least possible for a monic polynomial of degree $n$.
Hence, if $f$ has $n$ continuous derivatives and $p$ interpolates $f$ at these nodes, then $\max_{[-1, 1]} \abs{f - p} \le \frac{\max \abs{f^{(n)}}}{2^{n-1} n!}$.
The points $\cos(k\pi/n)$, $k = 0, \dots, n$, where $T_n = \pm 1$, are the Chebyshev extreme points, also called Chebyshev–Lobatto points.
</document>

<document source="https://tamnd.github.io/math2/node/YWV6J23AP6.md" tag="YWV6J23AP6" kind="statement">
# Weierstrass–Enneper parameterization

## Statement

Let $U \subset \C$ be a [[simply connected domain|0QW64ARSHB]] and let $z_0 \in U$.
Let $f$ be a [[holomorphic function|0NAHXR1FRH]] on $U$ and $g$ a [[meromorphic function|0ZZ7ZYDR8W]] on $U$.
Assume that the zeros of $f$ are exactly the poles of $g$.
Assume also that at a pole of $g$ of order $m$, the function $f$ has a zero of order exactly $2m$.
For $z \in U$, let
$$x_1(z) = \operatorname{Re} \int_{z_0}^{z} \frac{f(w)\,(1 - g(w)^2)}{2} \, dw, \quad x_2(z) = \operatorname{Re} \int_{z_0}^{z} \frac{i f(w)\,(1 + g(w)^2)}{2} \, dw, \quad x_3(z) = \operatorname{Re} \int_{z_0}^{z} f(w) g(w) \, dw.$$
Then $x = (x_1, x_2, x_3) \colon U \to \R^3$ is a [[conformal|0YYT36R4YD]] immersion whose image is a [[minimal surface|41X0YW2R9N]].
Conversely, after a rotation of $\R^3$ if needed, every conformal minimal immersion of $U$ into $\R^3$ arises in this way.

## Remarks

The idea of the proof: for a conformal immersion $x$, write $\varphi_k = 2\,\partial x_k / \partial z$.
The surface is minimal exactly when each $x_k$ is harmonic, that is, when each $\varphi_k$ is holomorphic.
Conformality is the equation $\varphi_1^2 + \varphi_2^2 + \varphi_3^2 = 0$.
The substitution $f = \varphi_1 - i\varphi_2$ and $g = \varphi_3 / (\varphi_1 - i\varphi_2)$ solves this equation, and it gives back the integrands above.
The condition on zeros and poles makes the integrands holomorphic and keeps $(\varphi_1, \varphi_2, \varphi_3)$ nonzero, so $x$ is an immersion.

The data $f = 2$ and $g(z) = z$ on $\C$ give the [[Enneper surface|73XN6AS0WX]].
The function $g$ is the Gauss map of the surface followed by stereographic projection.
Simple connectivity of $U$ makes the integrals independent of the path; on other domains one must check that the real parts of the periods vanish.

## Explanations

### L1 human

This formula produces minimal surfaces from two functions of a complex variable.
Choose the functions, compute three integrals, and take real parts: the result is always a minimal surface.
So the study of minimal surfaces in space turns into a study of complex functions.

### L2 human

The representation is the main tool for constructing complete minimal surfaces, such as [[Costa's minimal surface|EF0QMQB2J1]].
On a Riemann surface that is not simply connected, the same formulas work once the real periods of the three forms vanish; this period problem is usually the hardest step.
Because $g$ is the Gauss map, curvature questions become questions about meromorphic functions.
For example, the total curvature of a complete minimal surface of finite total curvature is $-4\pi$ times the degree of $g$.
</document>

<document source="https://tamnd.github.io/math2/node/YWW42G978D.md" tag="YWW42G978D" kind="definition">
# Uniform space

## Definition

Let $X$ be a set and let $\Delta = \set{(x, x) : x \in X}$ be the diagonal of $X \times X$.
For relations $U, V \subseteq X \times X$, write $U^{-1} = \set{(y, x) : (x, y) \in U}$ and $V \circ U = \set{(x, z) : \text{there is } y \text{ with } (x, y) \in U \text{ and } (y, z) \in V}$.
A *uniformity* on $X$ is a nonempty family $\mathcal{U}$ of subsets of $X \times X$ such that:

1. Every $U \in \mathcal{U}$ contains $\Delta$.
2. If $U \in \mathcal{U}$ and $U \subseteq V \subseteq X \times X$, then $V \in \mathcal{U}$.
3. If $U, V \in \mathcal{U}$, then $U \cap V \in \mathcal{U}$.
4. If $U \in \mathcal{U}$, then there is $V \in \mathcal{U}$ with $V \circ V \subseteq U$.
5. If $U \in \mathcal{U}$, then $U^{-1} \in \mathcal{U}$.

A *uniform space* is a pair $(X, \mathcal{U})$ of a set $X$ and a uniformity $\mathcal{U}$ on $X$.

## Remarks

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $U_\varepsilon = \set{(x, y) : d(x, y) < \varepsilon}$.
The sets that contain some $U_\varepsilon$ with $\varepsilon > 0$ form a uniformity.
Clauses 1, 2, 3 and 5 follow from $d(x, x) = 0$, from $U_\varepsilon \cap U_\delta = U_{\min(\varepsilon, \delta)}$ and from symmetry of $d$.
Clause 4 holds with $V = U_{\varepsilon/2}$, by the [[triangle inequality|01M6CW9VM6]].
On $\R$, the family of sets that contain $\set{(x, y) : \abs{x - y} < 1}$ satisfies clauses 1, 2, 3 and 5 but not 4.
For $V$ in this family, $V \circ V$ contains $\set{(x, y) : \abs{x - y} < 2}$, which does not lie in $\set{(x, y) : \abs{x - y} < 1}$.
The members of $\mathcal{U}$ are its [[entourages|FVFFKNKEDF]].
The uniform topology on $X$ declares $O \subseteq X$ open when for each $x \in O$ there is $U \in \mathcal{U}$ with $\set{y : (x, y) \in U} \subseteq O$.

## Explanations

### L1 human

A uniformity is a list of "closeness relations" on a set.
Each relation is a set of pairs of points that count as close.
The clauses say that every point is close to itself and that closeness relations can be enlarged and combined.
Each relation can be turned around, and inside each relation lies a smaller one that, applied twice, stays inside the first, like $\varepsilon/2$ and $\varepsilon$ in a metric space.
</document>

<document source="https://tamnd.github.io/math2/node/YWX2R53ED6.md" tag="YWX2R53ED6" kind="concept">
# Trace form

## Summary

The trace form of a finite extension $L/K$ is the [[symmetric bilinear form|0AXZNE5XNH]] $(x, y) \mapsto \mathrm{Tr}_{L/K}(xy)$ on $L$, built from the [[field trace|WPE5Z53QYJ]].
It is [[nondegenerate|01NHFZF2SK]] exactly when $L/K$ is separable.
Its Gram determinant in a basis is the [[discriminant|3BCYVS9E17]] of that basis.
Over $\Q$ the trace form is a quadratic form whose signature counts the real embeddings of a number field.

## Explanations

### L1 human

The trace turns each element of $L$ into an element of $K$.
Applying it to products $xy$ gives a way to pair two elements of $L$ and get a number in $K$, like a dot product.
For $\Q(i)$ over $\Q$, the pairing of $a + bi$ with $c + di$ is $2(ac - bd)$.
In characteristic $p$ this pairing can vanish completely, and that happens exactly for inseparable extensions.
</document>

<document source="https://tamnd.github.io/math2/node/YWX4FCFPAP.md" tag="YWX4FCFPAP" kind="statement">
# Smn theorem

## Statement

Fix a standard [[Gödel numbering|MFJXPAF0R0]] of programs, and let $\varphi^{(k)}_e$ be the partial [[computable function|KZSB5E471K]] of $k$ arguments computed by program $e$.
Let $m, n \ge 1$.
Then there is a [[primitive recursive|5NVA2JRHPV]] function $s^m_n \colon \N^{m+1} \to \N$ such that for all $e \in \N$, $\bar x \in \N^m$ and $\bar y \in \N^n$:
$$\varphi^{(n)}_{s^m_n(e, \bar x)}(\bar y) \simeq \varphi^{(m+n)}_e(\bar x, \bar y).$$
Here $\simeq$ means that both sides are undefined, or both are defined and equal.

## Remarks

The program $s^m_n(e, \bar x)$ first writes the fixed numbers $\bar x$ in front of its input $\bar y$, then runs program $e$.
Building this program from $e$ and $\bar x$ is a simple manipulation of codes, so $s^m_n$ is primitive recursive.

Example: let $e$ be a program for $(x, y) \mapsto x + y$.
Then $s^1_1(e, 5)$ is the number of a program for $y \mapsto 5 + y$, and $x \mapsto s^1_1(e, x)$ computes an index for "add $x$" from $x$.

The theorem is also called the parameter theorem, and it is due to Kleene.

## Explanations

### L1 human

If a program takes two inputs and you fix the first one, you get a program with one input.
The smn theorem says that the new program can be produced from the old one and the fixed input by a simple mechanical rule.
In programming terms, it is partial application, or currying, done on source code.

### L2 human

The smn theorem is used to turn a construction "uniform in a parameter" into a computable function on indices.
It is the main tool in the proof of [[Kleene's recursion theorem|F3Z30M606H]].
It is also used to build many-one reductions, for example in the proof of [[Rice's theorem|AKDR5YBBE2]].
Numberings that satisfy the smn theorem and have a universal function are called acceptable, and any two of them are computably isomorphic by a theorem of Rogers.
</document>

<document source="https://tamnd.github.io/math2/node/YWZQC5E54B.md" tag="YWZQC5E54B" kind="concept">
# Rate of convergence

## Summary

The rate of convergence of a convergent sequence measures how fast its distance to the limit shrinks.
One compares the error at step $k+1$ with a power of the error at step $k$.
The exponent is the order of convergence, such as linear or quadratic.
It is the standard way to compare iterative methods in numerical analysis.
Two examples are [[fixed-point iteration|K6ZF91051B]] and [[Newton's method|07EXVNF9VP]].

## Explanations

### L1 human

Two sequences can have the same limit and still approach it at very different speeds.
With linear convergence, each step multiplies the error by about the same factor, so one gains correct digits at a steady pace.
With quadratic convergence, each error is about the square of the previous error, so the number of correct digits roughly doubles at each step.
For example, $2^{-k}$ tends to $0$ linearly, while $2^{-2^k}$ tends to $0$ quadratically.
</document>

<document source="https://tamnd.github.io/math2/node/YX0J6TVGZ8.md" tag="YX0J6TVGZ8" kind="definition">
# Koszul duality

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $V$ be a finite-dimensional vector space over $k$, with [[dual space|07JBSZE0V4]] $V^*$.
Let $R \subseteq V \otimes V$ be a subspace.
Let $A = T(V)/(R)$ be the quotient of the [[tensor algebra|P4J5JZDW0B]] by the two-sided ideal generated by $R$; such an $A$ is a *quadratic algebra*.
Identify $V^* \otimes V^*$ with $(V \otimes V)^*$, and let $R^\perp \subseteq V^* \otimes V^*$ be the set of forms that vanish on $R$.
The *quadratic dual* of $A$ is $A^! = T(V^*)/(R^\perp)$.
*Koszul duality* is the correspondence $A \leftrightarrow A^!$ between [[Koszul algebras|FTKFCCZ8YA]], together with the induced relations between their module categories.

## Remarks

The construction is an involution: $(A^!)^! \cong A$, because $(R^\perp)^\perp = R$ in finite dimension.
If $A$ is Koszul, then $A^!$ is Koszul, and the algebra $\operatorname{Ext}^\bullet_A(k, k)$ is isomorphic to $A^!$ (some conventions give the opposite algebra).
For Koszul $A$ the Hilbert series satisfy $h_A(t) \, h_{A^!}(-t) = 1$.
Example: let $A = S(V)$ be the symmetric algebra, so $R$ is spanned by the tensors $x \otimes y - y \otimes x$.
A form on $V \otimes V$ vanishes on $R$ exactly when it is symmetric, so $R^\perp$ is the space of symmetric tensors, spanned by the tensors $\xi \otimes \xi$ and $\xi \otimes \eta + \eta \otimes \xi$.
Hence $A^! = \Lambda(V^*)$ is the exterior algebra, and $h_A(t) = (1 - t)^{-n}$, $h_{A^!}(-t) = (1 - t)^n$ for $n = \dim V$.
</document>

<document source="https://tamnd.github.io/math2/node/YX1DJK2QAG.md" tag="YX1DJK2QAG" kind="concept">
# Simplex

## Summary

An $n$-simplex is the [[convex hull|0FQSDC2EQ2]] of $n + 1$ points in general position, the simplest convex shape of dimension $n$.
A point, a segment, a [[triangle|0CTNFGYMM2]] and a [[tetrahedron|JFX8ASQS3E]] are the simplices of dimensions $0$ to $3$.
Simplices are the building blocks of [[simplicial complexes|V8GZ0W1ZZ5]] in topology and the cells of triangulations in geometry.
Many notions of triangle geometry, such as the circumsphere and the centroid, extend to simplices.

## Explanations

### L1 human

Start with a point.
Add a second point and fill in between: a segment.
Add a third point off the line and fill in: a triangle.
Add a fourth point off the plane and fill in: a tetrahedron.
Each step adds one point in a new direction, and the $n$-simplex is what you get after $n$ steps.
</document>

<document source="https://tamnd.github.io/math2/node/YX386FVZQR.md" tag="YX386FVZQR" kind="definition">
# Loewner differential equation

## Definition

Let $\mathbb{H} = \set{z \in \C : \operatorname{Im} z > 0}$ be the [[upper half-plane|X6XGGH5FPW]], and let $U \colon [0, \infty) \to \R$ be a [[continuous function|0X3WGX95FE]], the *driving function*.
The *chordal Loewner differential equation* with driving function $U$ is the initial value problem
$$\frac{\partial}{\partial t} g_t(z) = \frac{2}{g_t(z) - U(t)}, \qquad g_0(z) = z,$$
for each $z \in \overline{\mathbb{H}}$.
For each $z$, let $T_z \in (0, \infty]$ be the supremum of the times up to which the solution exists with $g_t(z) - U(t) \ne 0$; for $z = U(0)$ set $T_z = 0$.
For $t \ge 0$, the *hull* at time $t$ is $K_t = \set{z \in \mathbb{H} : T_z \le t}$.

## Remarks

For each $t$, the map $g_t$ is a [[conformal map|0VHNKP1N3K]] of $\mathbb{H} \setminus K_t$ onto $\mathbb{H}$ with the normalisation $g_t(z) = z + \frac{2t}{z} + O(\abs{z}^{-2})$ as $z \to \infty$.
The hulls $K_t$ increase with $t$, and the number $2t$ is their half-plane capacity.
Example: for $U = 0$, the equation gives $\partial_t (g_t(z)^2) = 4$, so $g_t(z) = \sqrt{z^2 + 4t}$ with the branch that maps $\mathbb{H}$ into $\mathbb{H}$.
A point $iy$ with $y > 0$ has $g_t(iy)^2 = 4t - y^2$, which reaches $0 = U(t)$ at $t = y^2/4$; so $K_t$ is the vertical segment from $0$ to $2i\sqrt{t}$.
Non-example: the translations $f_t(z) = z + t$ are conformal self-maps of $\mathbb{H}$, but they do not come from the equation, since $f_t(z) - z = t$ does not tend to $0$ at infinity.
The *radial* Loewner equation, which Loewner introduced first, is the analogue for the unit disc, with a driving function with values on the unit circle and maps normalised at $0$.

## Explanations

### L1 human

Each point of the half-plane flows according to a simple rule: it is pushed away from the moving point $U(t)$ on the real line, faster when it is closer.
A point that is pushed into $U(t)$ is swallowed and becomes part of the hull.
The equation turns a real function of time into a growing set in the plane and a family of maps that straighten it out.
</document>

<document source="https://tamnd.github.io/math2/node/YX5FFSCR8P.md" tag="YX5FFSCR8P" kind="example">
# Exceptional Lie group G2

## Example

Let $\mathbb{O}$ be the [[octonions|1HRMT3DSX2]], the $8$-dimensional real normed division algebra.
The compact Lie group $G_2$ is the automorphism group of $\mathbb{O}$:
$$G_2 = \set{g \in GL(\mathbb{O}) : g(xy) = g(x)\, g(y) \text{ for all } x, y \in \mathbb{O}}.$$
It is a closed subgroup of $GL_8(\R)$, since it is cut out by polynomial equations, so it is a Lie group.
Each $g \in G_2$ fixes $1$ and preserves the octonion norm, so it maps the $7$-dimensional space $\operatorname{Im} \mathbb{O}$ of imaginary octonions to itself by a rotation.
This makes $G_2$ a closed subgroup of [[SO(7)|0CGYNDCWRR]], hence compact.

$G_2$ is a connected and simply connected [[simple Lie group|AM7YJP8AR6]] of dimension $14$ and rank $2$, with trivial centre.

## Remarks

$G_2$ is the smallest of the five exceptional compact simple Lie groups $G_2, F_4, E_6, E_7, E_8$.
Its Dynkin diagram has two nodes joined by a triple edge, and its root system has $12$ roots, of two lengths whose squares have ratio $3$.
It acts transitively on the unit sphere $S^6$ of $\operatorname{Im} \mathbb{O}$, with stabilizer isomorphic to [[SU(3)|CZCWFPNS4N]], so $G_2 / SU(3) \cong S^6$.
The name $G_2$ also denotes the complex simple Lie group of the same type and its noncompact split real form.
Riemannian $7$-manifolds whose holonomy group lies in $G_2$ are called $G_2$-manifolds.

## Explanations

### L1 human

The classical Lie groups come in infinite families of matrix groups, while $G_2$ is one of five exceptional groups outside those families.
It is the symmetry group of the octonions, the largest of the four normed division algebras.
It has dimension $14$, so a symmetry of the octonions is fixed by $14$ real parameters.
Every such symmetry is a rotation of the $7$ imaginary directions, but most rotations of those directions do not respect octonion multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/YX985SQG26.md" tag="YX985SQG26" kind="problem">
# Hadwiger–Nelson problem

## Problem

Let $G$ be the [[graph|00C0BR7D7H]] whose vertices are the points of the plane $\R^2$.
Two points $p$ and $q$ are adjacent exactly when their [[Euclidean distance|0NPPGHQNXQ]] is $\norm{p - q} = 1$.
Determine the [[chromatic number|0VHJRXNAAS]] $\chi(G)$ of this graph.
Equivalently, colour every point of the plane so that any two points at distance $1$ get different colours.
Find the least number of colours that makes this possible.
This number is called the chromatic number of the plane.

## Remarks

The problem is open.
The known bounds are $5 \le \chi(G) \le 7$.
The upper bound $7$ comes from a tiling of the plane by regular hexagons of diameter slightly less than $1$, coloured with $7$ colours in a periodic pattern.
The lower bound $4$ follows from a finite unit-distance graph with $7$ vertices, the Moser spindle, which has no proper [[colouring|0NCPXH194Y]] with $3$ colours.
In 2018 Aubrey de Grey found a finite unit-distance graph that needs $5$ colours, which raised the lower bound to $5$.
By the de Bruijn–Erdős theorem (assuming the axiom of choice), $\chi(G)$ equals the largest chromatic number of a finite subgraph of $G$.

## Explanations

### L1 human

Paint every point of the plane with some colour.
The rule is that two points exactly one unit apart must get different colours.
What is the smallest number of colours that makes this possible?
Three colours are not enough, and seven colours are enough.
Today we know that at least five are needed, but whether the answer is $5$, $6$ or $7$ is unknown.
</document>

<document source="https://tamnd.github.io/math2/node/YXEVE39Q31.md" tag="YXEVE39Q31" kind="concept">
# Uniform module

## Summary

A uniform module is a nonzero [[module|0NZ8JA25WK]] in which any two nonzero submodules intersect in a nonzero submodule.
Equivalently, every nonzero submodule is an [[essential submodule|8MS51FASMX]].
Uniform modules are the units for measuring the uniform (Goldie) dimension of a module, which counts how many independent uniform submodules fit inside it.
They are a main tool in the proof of Goldie's theorem on rings of quotients of semiprime Noetherian rings.

## Explanations

### L1 human

In a uniform module, nonzero pieces can never be disjoint: any two of them share a nonzero element.
So the module cannot contain two independent directions, as $\Z \oplus \Z$ does.
The integers form a uniform $\Z$-module, since for nonzero $a$ and $b$ the subgroups $a\Z$ and $b\Z$ share $ab \ne 0$.
Every uniform module is [[indecomposable|E2P29GCD9T]], since two summands of a direct sum meet only in zero.
</document>

<document source="https://tamnd.github.io/math2/node/YXJ10HDDBJ.md" tag="YXJ10HDDBJ" kind="definition">
# Regularity of minimizers

## Definition

Let $\Omega \subseteq \R^n$ be a bounded open set, let $m \ge 1$ and $1 < p < \infty$.
Let $J(u) = \int_\Omega L(x, u, \nabla u) \, dx$ be an [[integral functional|JRQZ803Q5Y]] for maps $u \colon \Omega \to \R^m$.
A map $u$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\Omega; \R^m)$ is a *minimiser* of $J$ if $J(u) \le J(u + \varphi)$ for all $\varphi \in W^{1,p}_0(\Omega; \R^m)$.
Let $k \ge 1$ and $0 < \alpha < 1$.

1. The minimiser $u$ is *regular* (of class $C^{k,\alpha}$) if it agrees almost everywhere with a map in $C^{k,\alpha}(\Omega; \R^m)$.
2. It is *partially regular* if there is a relatively closed set $\Sigma \subseteq \Omega$ of Lebesgue measure zero such that $u$ is regular on $\Omega \setminus \Sigma$.

The *regularity problem* for $J$ asks under which conditions on $L$ every minimiser is regular or partially regular.

## Remarks

Here $C^{k,\alpha}$ is the space of $C^k$ maps whose $k$-th derivatives are locally [[Hölder continuous|012PBA0FMD]] with exponent $\alpha$.

Example: for $m = 1$ and $L(\xi) = \frac{1}{2} \abs{\xi}^2$, minimisers are weak solutions of $\Delta u = 0$.
By Weyl's lemma they are harmonic, hence smooth, so they are regular for every $k$ and $\alpha$.

Non-example: for $n = 1$, $m = 1$ and $L(x, u, \xi) = (\xi^2 - 1)^2$ on $\Omega = (0, 2)$, the function $u(x) = \min(x, 2 - x)$ is a minimiser among maps with its boundary values, since $J(u) = 0$.
It is Lipschitz but not $C^1$, so it is not regular in the sense of clause 1.
Here $L$ is not convex in $\xi$.

For $m = 1$ and smooth, uniformly convex $L(\xi)$ with bounded second derivatives, every minimiser in $W^{1,2}$ is smooth, by the De Giorgi–Nash–Moser theorem and Schauder estimates.
For $m \ge 2$ this fails in general, and partial regularity is the typical result for smooth, uniformly convex or strongly quasiconvex integrands with suitable growth.
</document>

<document source="https://tamnd.github.io/math2/node/YXJDQM9VVD.md" tag="YXJDQM9VVD" kind="definition">
# Limiting subdifferential

## Definition

Let $f \colon \R^n \to (-\infty, +\infty]$ be a function with values in the [[extended real line|0GSZ7B0CGR]], and let $\bar x \in \R^n$ with $f(\bar x)$ finite.

1. A vector $v \in \R^n$ is a *regular subgradient* of $f$ at a point $x$ with $f(x)$ finite if
$$\liminf_{y \to x, \ y \ne x} \frac{f(y) - f(x) - \ip{v}{y - x}}{\abs{y - x}} \ge 0,$$
where $\liminf$ is the [[limit inferior|033RZ32T4Q]].
The set of these vectors is the *regular subdifferential* $\hat\partial f(x)$.
2. The *limiting subdifferential* $\partial f(\bar x)$ is the set of vectors $v \in \R^n$ for which there are [[sequences|0G9EKM0E15]] $x_k \to \bar x$ and $v_k \to v$ with $f(x_k) \to f(\bar x)$ and $v_k \in \hat\partial f(x_k)$ for all $k$.

## Remarks

Taking $x_k = \bar x$ and $v_k = v$ shows that $\hat\partial f(\bar x) \subseteq \partial f(\bar x)$.
If $f$ is differentiable near $\bar x$ with continuous gradient, then $\partial f(\bar x) = \set{\nabla f(\bar x)}$.
If $f$ is convex, both sets equal the [[subdifferential|9C5WXYYQEK]] of convex analysis.
If $\bar x$ is a local minimum of $f$, then $0 \in \hat\partial f(\bar x) \subseteq \partial f(\bar x)$.
Other names are the Mordukhovich subdifferential and the basic subdifferential.

Example: $f(x) = -\abs{x}$ on $\R$ and $\bar x = 0$.
For $x > 0$ the function is differentiable with derivative $-1$, so $\hat\partial f(x) = \set{-1}$; for $x < 0$, $\hat\partial f(x) = \set{1}$.
Hence $\set{-1, 1} \subseteq \partial f(0)$, and in fact $\partial f(0) = \set{-1, 1}$.

Non-example: the vector $v = 0$ for the same function.
It is not a regular subgradient at $0$, since $(f(y) - f(0) - 0)/\abs{y} = -1$ for all $y \ne 0$.
It is not a limit of regular subgradients at nearby points either, since these are all $\pm 1$.
So $0 \notin \partial f(0)$, although $0$ lies in the Clarke generalized gradient $[-1, 1]$.
</document>

<document source="https://tamnd.github.io/math2/node/YXJPPC9CXS.md" tag="YXJPPC9CXS" kind="definition">
# Riesz potential operator

## Definition

Let $n \ge 1$ and let $\alpha$ be a real number with $0 < \alpha < n$.
Let
$$c_\alpha = \frac{\pi^{n/2} \, 2^\alpha \, \Gamma(\alpha/2)}{\Gamma((n - \alpha)/2)},$$
where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
Let $f$ be a measurable function on $\R^n$.
At each point $x \in \R^n$ where the integral converges absolutely, the *Riesz potential* of order $\alpha$ of $f$ is
$$I_\alpha f(x) = \frac{1}{c_\alpha} \int_{\R^n} \frac{f(y)}{\abs{x - y}^{n - \alpha}} \, dy.$$
The integral is taken with respect to [[Lebesgue measure|2Z2QVWZ262]].
The map $f \mapsto I_\alpha f$ is the *Riesz potential operator* $I_\alpha$.

## Remarks

If $f \in L^p(\R^n)$ with $1 \le p < n/\alpha$, then $I_\alpha f(x)$ converges absolutely for almost every $x$.
For $1 < p < n/\alpha$ and $1/q = 1/p - \alpha/n$, the [[Hardy–Littlewood–Sobolev inequality|HG65C9R1SP]] gives $\norm{I_\alpha f}_q \le A \norm{f}_p$ with $A$ depending only on $n$, $\alpha$ and $p$, see [[Lp space|SASP14MA1Z]].

With the Fourier transform $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi} \, dx$, the constant $c_\alpha$ is chosen so that $\widehat{I_\alpha f}(\xi) = (2\pi \abs{\xi})^{-\alpha} \hat f(\xi)$ for nice $f$.
So $I_\alpha = (-\Delta)^{-\alpha/2}$, the inverse of the [[fractional Laplacian|F8RSNQBMBE]], and $I_\alpha I_\beta = I_{\alpha + \beta}$ when $\alpha + \beta < n$.

Example: $n = 3$ and $\alpha = 2$.
Then $c_2 = \pi^{3/2} \cdot 4 \cdot \Gamma(1) / \Gamma(1/2) = 4\pi$, since $\Gamma(1/2) = \sqrt{\pi}$.
So $I_2 f(x) = \frac{1}{4\pi} \int_{\R^3} \frac{f(y)}{\abs{x - y}} \, dy$, the Newtonian potential, which solves $-\Delta u = f$ for smooth $f$ with compact support.

Non-example: for $f = 1$ the integral $\int_{\R^n} \abs{x - y}^{\alpha - n} \, dy$ diverges at every $x$, because $\alpha - n > -n$, so $I_\alpha 1$ is not defined.

## Explanations

### L1 human

The Riesz potential averages $f$ with a weight that is large near the point $x$ and decays like a power of the distance.
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<document source="https://tamnd.github.io/math2/node/YXJXA0Q52R.md" tag="YXJXA0Q52R" kind="concept">
# Group cohomology

## Summary

Group cohomology attaches to a [[group|G5S1M9]] $G$ and a $G$-module $M$ a sequence of abelian groups $H^n(G, M)$, for $n \ge 0$.
They are the right [[derived functors|NPMC3K6J2F]] of the functor of invariants $M \mapsto M^G$, or equivalently the groups $\operatorname{Ext}^n_{\Z[G]}(\Z, M)$.
In low degrees they have classical meanings.
$H^0$ is the invariants, $H^1$ classifies crossed homomorphisms up to principal ones, and $H^2$ classifies extensions of $G$ by $M$.
It is dual to [[group homology|CFAG3VRXEA]], and [[Galois cohomology|9MPCZ3EAJM]] is the case of Galois groups.

## Explanations

### L1 human

A group acting on an abelian group $M$ leaves some elements fixed; these form $H^0(G, M)$.
Taking fixed points does not respect quotients well, and the higher cohomology groups record this failure.
They can be computed with cochains, which are functions of several group elements with values in $M$.
When every element of $G$ acts on $M$ as the identity, $H^1(G, M)$ is just the group of homomorphisms from $G$ to $M$.
For a cyclic group of order $m$ acting on $\Z$ as the identity, the cohomology is $\Z/m\Z$ in every positive even degree and $0$ in every odd degree.
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<document source="https://tamnd.github.io/math2/node/YXMXAGVC1Q.md" tag="YXMXAGVC1Q" kind="concept">
# Projective object

## Summary

A projective object of a [[category|A2DE66937D]] is an object $P$ such that maps from $P$ lift through every [[epimorphism|RVXDSY04KG]].
In a category of modules, the projective objects are exactly the [[projective modules|BFFA03GE3V]], for example the free modules.
Projective objects are used to build projective resolutions, from which left [[derived functors|NPMC3K6J2F]] are computed.
The dual notion is the [[injective object|5QRFTBF4PG]].

## Explanations

### L1 human

Think of an epimorphism $E \to X$ as a way of covering $X$ by a bigger object $E$.
An object $P$ is projective if any map from $P$ into $X$ can be pulled back up to a map into $E$.
Free objects have this property because one only has to choose where the generators go, and each generator can be sent to any preimage.
For example, $\Z$ is projective among abelian groups, but $\Z/2\Z$ is not: the identity of $\Z/2\Z$ does not lift along $\Z \to \Z/2\Z$, since $\Z$ has no element of order $2$.
</document>

<document source="https://tamnd.github.io/math2/node/YXST7NGMTC.md" tag="YXST7NGMTC" kind="concept">
# Inversion in a circle

## Summary

Inversion in a circle with center $O$ and radius $r$ sends a point $P \ne O$ to a point $P'$ on the ray from $O$ through $P$.
The point $P'$ is chosen so that $\abs{OP} \cdot \abs{OP'} = r^2$.
It swaps the inside and the outside of the circle and fixes the circle pointwise.
It maps circles and lines to circles and lines, and it preserves angles.
This makes it a basic tool of [[inversive geometry|HFBW3SXQVM]] and of geometric constructions.

## Explanations

### L1 human

Inversion is a mirror for circles instead of lines.
A point close to the center is sent far away, a point far away is sent close to the center, and points on the circle stay put.
For the unit circle centered at the origin, the point $(2, 0)$ goes to $(1/2, 0)$, since $2 \cdot 1/2 = 1$.
Doing the inversion twice brings every point back, just as reflecting twice in a mirror does.
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<document source="https://tamnd.github.io/math2/node/YXWGV80Z2V.md" tag="YXWGV80Z2V" kind="concept">
# Arakelov theory

## Summary

Arakelov theory is an intersection theory for arithmetic varieties.
These are schemes that are flat and of finite type over the ring of integers of a [[number field|MWBDFS2KJ5]].
Such a scheme behaves like a family over a curve that is missing its points at infinity, which correspond to the archimedean places.
Arakelov added these fibers at infinity by equipping the associated complex manifolds with Hermitian metrics and Green functions, so that intersection numbers become well defined.
The theory gives a geometric meaning to [[heights|F1FF505PWH]] and is a main tool of Diophantine geometry.

## Explanations

### L1 human

The ring of integers $\Z$ is in many ways like a curve, with one point for each prime.
But this curve is not complete: there is no point for the usual absolute value, the "infinite prime".
Arakelov theory fills in this missing point with analysis on complex manifolds.
After that, one can intersect divisors on arithmetic surfaces as on ordinary projective surfaces, and prove analogues of the Riemann–Roch theorem and the Hodge index theorem.
Heights of rational points appear as arithmetic degrees in this theory.
</document>

<document source="https://tamnd.github.io/math2/node/YY277RBF8Z.md" tag="YY277RBF8Z" kind="statement">
# Classification of covering spaces

## Statement

Let $X$ be a [[path-connected|0WB6DH499T]], [[locally path-connected|1ZVR8BRK65]] and [[semilocally simply connected|R3SHZX0NZR]] space, and let $x_0 \in X$.
Two pointed covering maps $p_1 \colon (\tilde X_1, \tilde x_1) \to (X, x_0)$ and $p_2 \colon (\tilde X_2, \tilde x_2) \to (X, x_0)$ are *isomorphic* if there is a homeomorphism $h \colon \tilde X_1 \to \tilde X_2$ with $p_2 \circ h = p_1$ and $h(\tilde x_1) = \tilde x_2$.
Then:

1. Consider pointed [[covering maps|3XZRRE165N]] $p \colon (\tilde X, \tilde x_0) \to (X, x_0)$ with $\tilde X$ path-connected. The rule $p \mapsto p_*\bigl(\pi_1(\tilde X, \tilde x_0)\bigr)$ is a bijection from their isomorphism classes onto the subgroups of the [[fundamental group|S1XK3HXHFX]] $\pi_1(X, x_0)$.
2. Consider isomorphisms that need not preserve base points. Then the same rule gives a bijection from isomorphism classes of covering maps with path-connected total space onto the [[conjugacy classes|0H11ZPX4ND]] of subgroups of $\pi_1(X, x_0)$.

## Remarks

The covering that belongs to a subgroup $H$ is built like the universal cover from homotopy classes of paths that start at $x_0$.
Two such classes are identified when the paths end at the same point and differ by a loop whose class lies in $H$.
Injectivity uses the lifting criterion: a map from a path-connected, locally path-connected space lifts through a covering exactly when its image on $\pi_1$ lies in the image of the covering.
Changing the base point inside the fibre replaces the subgroup by a conjugate, which explains the second clause.
For $X = S^1$, the subgroups of $\pi_1(S^1) = \Z$ are $n\Z$ with $n \ge 0$; they belong to the coverings $z \mapsto z^n$ for $n \ge 1$ and to the universal cover $\R \to S^1$ for $n = 0$.
Without the semilocal condition the trivial subgroup may have no covering: the [[Hawaiian earring|73EBKQPP8Q]] has no simply connected covering space.

## Explanations

### L1 human

For a reasonable space, the connected coverings are described completely by the subgroups of its fundamental group.
The whole group belongs to the space itself, the trivial subgroup to the universal cover, and the other subgroups to the coverings in between.
So a topological question, which coverings exist, becomes an algebraic one, which subgroups exist.

### L2 human

The theorem is the topological analogue of the fundamental theorem of Galois theory, with the universal cover in the role of an algebraic closure.
It is used to build coverings with given properties.
For example, each subgroup of a free group gives a covering of a graph, and this yields a quick proof that subgroups of free groups are free.
Further properties of the correspondence, about deck transformations and normal subgroups, are collected in the [[Galois correspondence for coverings|MTWNAJMH1D]].
</document>

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# Analysis on fractals

## Summary

Analysis on fractals builds Laplacians, energies, harmonic functions, heat equations and diffusion processes on fractal sets such as the Sierpiński gasket.
On such sets there are no smooth curves or tangent vectors, so the [[Laplacian|0E0B6RA8K0]] cannot be defined by derivatives.
Instead one takes limits of renormalized energies and [[Laplacian matrices|TN6M6DVZRP]] of finite graphs that approximate the fractal, as in [[Kigami's construction|VXB4G47Z3T]].
Another route builds Brownian motion on the fractal as a limit of random walks on these graphs.
The key tools are [[Dirichlet forms|MH20Q2N1AH]] and [[resistance forms|THCM7DWHNY]].
The results show new features, such as a spectral dimension that differs from the [[Hausdorff dimension|SSNFTHZ6YJ]].

## Explanations

### L1 human

Imagine a triangle from which the middle triangle is removed, and then the middle of each smaller triangle, forever: this is the Sierpiński gasket.
Heat cannot flow along straight lines in it, because almost every line segment hits a hole.
But one can model the gasket by a finite network of resistors, one network for each level of detail, and compute how heat or current flows in each network.
When the networks are rescaled in the right way, these computations converge, and the limit is a Laplacian on the gasket itself.
For the [[Laplacian on the Sierpiński gasket|W9V9JSAWYK]], the right scaling multiplies energies by $5/3$ at each level, a number that comes from a small computation on a triangle.
</document>

<document source="https://tamnd.github.io/math2/node/YY4XSJX81P.md" tag="YY4XSJX81P" kind="concept">
# Fractional differential equation

## Summary

A fractional differential equation is a [[differential equation|003R0V5M7V]] in which derivatives of non-integer order of the unknown function appear.
These derivatives come from [[fractional calculus|T73XK15SA2]], most often the Riemann–Liouville or the [[Caputo derivative|6VMGQYF8RW]].
Because a fractional derivative is an integral over the whole past, such equations model processes with memory, such as viscoelastic materials and anomalous diffusion.

## Explanations

### L1 human

An ordinary derivative only looks at the function near one point.
A fractional derivative, such as a derivative of order $1/2$, is a weighted average over all earlier times.
So the rate of change in a fractional equation depends on the whole history of the solution.
For example, the equation $D^{1/2} y = 1$ with $y(0) = 0$, with the Caputo derivative, has the solution $y(t) = 2\sqrt{t/\pi}$, the half integral of the constant $1$.
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# Local truncation error

## Definition

Let $y$ be the exact solution of an [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$, and let $t_n = t_0 + nh$.
Consider a [[one-step method|558XADG64M]] $y_{n+1} = y_n + h \, \Phi(t_n, y_n, h)$ with increment function $\Phi$.
The *local truncation error* at step $n + 1$ is
$$\tau_{n+1} = y(t_{n+1}) - y(t_n) - h \, \Phi\big(t_n, y(t_n), h\big),$$
the error of one step started from the exact value $y(t_n)$.
The method is *consistent of order* $p$ if $\tau_{n+1} = O(h^{p+1})$ as $h \to 0$, uniformly in $n$, for every sufficiently smooth $f$.

## Remarks

For an implicit method or a linear multistep method $\sum_{j=0}^{k} \alpha_j y_{n+j} = h \sum_{j=0}^{k} \beta_j f(t_{n+j}, y_{n+j})$, one inserts the exact solution in the same way: $\tau_{n+k} = \sum_j \alpha_j y(t_{n+j}) - h \sum_j \beta_j y'(t_{n+j})$.
Some books divide by $h$ and call $\tau_{n+1}/h$ the truncation error; then order $p$ means $O(h^p)$.
Example: for the Euler method, $\Phi(t, y, h) = f(t, y)$, and Taylor's theorem gives $\tau_{n+1} = y(t_n + h) - y(t_n) - h y'(t_n) = \frac{h^2}{2} y''(\xi_n)$, so the method is consistent of order $1$.
Non-example: $\Phi(t, y, h) = 2 f(t, y)$ is not consistent, since $\tau_{n+1} = -h y'(t_n) + O(h^2)$ is not even $O(h^2)$.
If $\Phi$ is [[Lipschitz|0AW1E6YTJY]] in $y$ with constant $L > 0$ and $\abs{\tau_n} \le C h^{p+1}$ for all $n$, and if the method starts from $y_0 = y(t_0)$, then the global error satisfies $\abs{y(t_n) - y_n} \le \frac{C h^p}{L}\big(e^{L(t_n - t_0)} - 1\big)$.

## Explanations

### L1 human

The local truncation error asks: if the method starts a step at the exact answer, how wrong is it one step later?
A method of order $p$ makes an error of size about $h^{p+1}$ per step.
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<document source="https://tamnd.github.io/math2/node/YY8CG0NZ9C.md" tag="YY8CG0NZ9C" kind="definition">
# Gelfand–Kirillov dimension

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$ with identity.
For a finite-dimensional [[subspace|000AYPA9AJ]] $V \subseteq A$ and $n \ge 1$, let $V^n$ be the span of all products of $n$ elements of $V$.
The *Gelfand–Kirillov dimension* of $A$ is
$$\operatorname{GKdim} A = \sup_{V} \, \limsup_{n \to \infty} \frac{\log \dim_k \big( k + V + V^2 + \cdots + V^n \big)}{\log n},$$
where the supremum runs over all finite-dimensional subspaces $V$ of $A$ and $\limsup$ is the [[limit superior|0T3RZQXDYD]].
It is a number in $[0, \infty]$.

## Remarks

If $A$ is finitely generated and $V$ is a finite-dimensional generating subspace with $1 \in V$, then the supremum is attained at $V$: $\operatorname{GKdim} A = \limsup_n \log d_V(n) / \log n$, where $d_V$ is the [[growth function|EHY4PYSMGZ]] of $A$.
So $\operatorname{GKdim} A$ depends only on the growth of $A$.

Example: let $A = k[x_1, \ldots, x_m]$ be a [[polynomial ring|0SDC934R52]] and $V = k + k x_1 + \cdots + k x_m$.
The space $V^n$ is spanned by the monomials of degree at most $n$, so $\dim V^n = \binom{n + m}{m}$, which is a polynomial of degree $m$ in $n$.
Hence $\operatorname{GKdim} A = m$.

Example: if $A$ is finite-dimensional, then $\dim V^n \le \dim A$ for all $n$, so $\operatorname{GKdim} A = 0$.

Non-example of finiteness: for the free algebra $A = k\langle x, y \rangle$ and $V = k + kx + ky$, one has $\dim V^n = 2^{n+1} - 1$.
Then $\log \dim V^n / \log n \to \infty$, so $\operatorname{GKdim} A = \infty$.

The base of the logarithm does not matter, because only the ratio of two logarithms appears.
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<document source="https://tamnd.github.io/math2/node/YYB1JP7TWJ.md" tag="YYB1JP7TWJ" kind="example">
# Bunimovich stadium

## Example

Let $a > 0$ and $r > 0$.
The *Bunimovich stadium* is the table $Q \subset \R^2$ made of the rectangle $[-a, a] \times [-r, r]$ and two half-discs of radius $r$.
The half-discs are centred at $(-a, 0)$ and $(a, 0)$ and sit on the short sides of the rectangle.
Its boundary consists of two parallel segments of length $2a$ and two semicircles of radius $r$.
These pieces have a common tangent at the four junctions, so the boundary is a $C^1$ curve.
The table is compact with piecewise smooth boundary, so it carries a [[billiard|XWMKYE038F]] flow.

## Remarks

Bunimovich proved that the billiard in the stadium is ergodic for every $a > 0$.
This was a surprise, because the boundary has no dispersing part: the semicircles are focusing and the segments are flat, unlike the [[Sinai billiard|GX9Q2PYNZE]].
The mechanism is defocusing: a beam focused by a semicircle passes through a focal point and spreads out again before the next reflection, if the free path is long enough.
For $a = 0$ the table is a disc and the billiard is integrable, so the length $a > 0$ of the straight parts is essential.
The stadium also has a family of periodic orbits bouncing between the two flat sides; they fill a set of zero measure, so they do not contradict ergodicity.

## Explanations

### L1 human

Take a round table, cut it in half, and pull the halves apart, joining them with two straight sides; the result looks like a running track.
A ball on a round table keeps the same angle forever, but on the stadium the motion becomes chaotic.
Almost every shot eventually explores the whole table, spending time in each region in proportion to its area.
The stadium is a standard example in the study of quantum chaos.
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# Kirby–Siebenmann class

## Definition

Let $M$ be a [[topological manifold|QQQ9B1DK4P]].
Let $\mathrm{TOP}$ and $\mathrm{PL}$ be the stable groups of homeomorphisms and of piecewise linear homeomorphisms of Euclidean spaces, with classifying spaces $B\mathrm{TOP}$ and $B\mathrm{PL}$.
The stable tangent microbundle of $M$ has a classifying map $\tau_M \colon M \to B\mathrm{TOP}$.
The forgetful map $B\mathrm{PL} \to B\mathrm{TOP}$ is a fibration up to homotopy, with fibre $\mathrm{TOP}/\mathrm{PL}$.
By a theorem of Kirby and Siebenmann, $\mathrm{TOP}/\mathrm{PL}$ is an Eilenberg–MacLane space $K(\Z/2, 3)$: its only nonzero [[homotopy group|P976WYEFMB]] is $\pi_3 \cong \Z/2$.
The *Kirby–Siebenmann class* $\operatorname{ks}(M) \in H^4(M; \Z/2)$ is the obstruction class, in [[singular cohomology|6N7BC70CX0]] with coefficients $\pi_3(\mathrm{TOP}/\mathrm{PL}) \cong \Z/2$, to lifting $\tau_M$ to a map $M \to B\mathrm{PL}$.

## Remarks

The $4$-dimensional $E_8$ manifold $M_{E_8}$ has $\operatorname{ks}(M_{E_8}) \ne 0$ in $H^4(M_{E_8}; \Z/2) \cong \Z/2$.
For a closed simply connected topological $4$-manifold with even intersection form, $\operatorname{ks}$ is the signature divided by $8$, taken modulo $2$, and the $E_8$ form has signature $8$.

A space that is not a manifold, such as a finite graph with a vertex of degree three, has no tangent microbundle, so the definition does not apply to it.

If $M$ has a [[piecewise linear structure|15MKGRNM9K]], or a smooth structure, then $\operatorname{ks}(M) = 0$.
Conversely, if $\dim M \ge 5$ and $\operatorname{ks}(M) = 0$, then $M$ has a piecewise linear structure; for any $M$, $\operatorname{ks}(M) = 0$ if and only if $M \times \R$ has one.
The class is natural: for an open subset $U \subseteq M$, $\operatorname{ks}(U)$ is the restriction of $\operatorname{ks}(M)$.
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# J-homomorphism

## Summary

The J-homomorphism $J \colon \pi_r(SO(n)) \to \pi_{r+n}(S^n)$ turns a family of rotations of $\R^n$ parametrised by an $r$-sphere into a map of spheres.
It connects the homotopy of the rotation groups, known through Bott periodicity, with the homotopy groups of spheres.
In the stable range it gives a homomorphism $\pi_r(SO) \to \pi_r^s$ into the [[stable homotopy groups of spheres|HGZAYPQBXW]], whose image is the [[image of J|GX8K2SCWVQ]].

## Explanations

### L1 human

A loop of rotations of the plane, such as one full turn, can be used to twist a sphere around another sphere.
The J-homomorphism makes this precise in every dimension.
Its value on one full turn of the plane is the Hopf map $S^3 \to S^2$, up to sign.
It explains a regular part of the otherwise chaotic homotopy groups of spheres.
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<document source="https://tamnd.github.io/math2/node/YYNCJSR0PJ.md" tag="YYNCJSR0PJ" kind="concept">
# Measurable function

## Summary

A measurable function is a map between [[measurable spaces|N4KDT83V3S]] under which the preimage of every measurable set is measurable.
These are the functions that one can integrate in Lebesgue's theory, and in probability they are the random variables.
Sums, products, limits and suprema of sequences of real measurable functions are again measurable.

## Explanations

### L1 human

To integrate a function, one needs to know how big the set is where the function lies in a given range.
A measurable function is one for which all these sets can be measured.
For example, on the real line, every continuous function is measurable, and so is the function that is $1$ on the rational numbers and $0$ elsewhere.
Measurable functions are the analogue for measure theory of continuous functions in topology.
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# Connes embedding problem

## Problem

Let $R$ be the [[hyperfinite type II1 factor|NZ4NY8SK1C]] with its faithful normal tracial state $\tau$, and let $\omega$ be a free [[ultrafilter|J64Y3JC0RE]] on $\N$.
Let $\ell^\infty(R)$ be the algebra of norm-bounded sequences in $R$, and let $I_\omega$ be the ideal of sequences $(x_n)$ with $\lim_{n \to \omega} \tau(x_n^* x_n) = 0$.
The ultrapower $R^\omega = \ell^\infty(R) / I_\omega$ is a type $\mathrm{II}_1$ factor with trace $\tau_\omega((x_n)) = \lim_{n \to \omega} \tau(x_n)$.
The problem asks: does every type $\mathrm{II}_1$ factor $M$ with separable predual, with trace $\tau_M$, admit an injective unital $*$-homomorphism $\pi \colon M \to R^\omega$ with $\tau_\omega \circ \pi = \tau_M$?

## Remarks

Connes raised the question in his 1976 paper on the classification of injective factors.
Over the following decades it was shown to be equivalent to several problems in other areas.
Among them are Kirchberg's QWEP conjecture on C*-algebras and Tsirelson's problem in quantum information theory.
In 2020, Ji, Natarajan, Vidick, Wright and Yuen announced a proof of the complexity result $\mathrm{MIP}^* = \mathrm{RE}$.
It implies a negative answer to Tsirelson's problem and hence to the Connes embedding problem.
So there exist $\mathrm{II}_1$ factors with separable predual that do not embed into $R^\omega$, but no explicit example is known.
Many classes of factors are known to embed, for example the group von Neumann algebras of amenable and of free groups.

## Explanations

### L1 human

The hyperfinite factor $R$ is built from finite matrices, and its ultrapower collects all limits of sequences of such approximations.
Connes asked whether every reasonable $\mathrm{II}_1$ factor can be approximated by finite matrices in this way, at the level of traces of products.
For many years most experts expected the answer to be yes.
The problem was then linked to questions about quantum correlations, and a result in computational complexity theory showed that the answer is no.
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<document source="https://tamnd.github.io/math2/node/YYQG5P32GX.md" tag="YYQG5P32GX" kind="statement">
# Laplace's integral for Legendre polynomials

## Statement

Let $n \ge 0$ be an integer and let $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$.
For every $x \in \C$,
$$P_n(x) = \frac{1}{\pi} \int_0^\pi \left( x + \sqrt{x^2 - 1} \, \cos \varphi \right)^n \, d\varphi,$$
where $\sqrt{x^2 - 1}$ is either square root of $x^2 - 1$.

## Remarks

The choice of square root does not matter: the substitution $\varphi \mapsto \pi - \varphi$ changes the sign of $\cos \varphi$.
Proof idea: multiply by $t^n$ and sum over $n$ for small $t$.
The integral $\int_0^\pi \frac{d\varphi}{a - b \cos \varphi} = \frac{\pi}{\sqrt{a^2 - b^2}}$ with $a = 1 - tx$ and $b = t \sqrt{x^2 - 1}$ gives $(1 - 2xt + t^2)^{-1/2}$.
This is the generating function of the Legendre polynomials, so the coefficients agree.
Check for $n = 2$: the integral equals $x^2 + \frac{x^2 - 1}{2} = \frac{3x^2 - 1}{2} = P_2(x)$, since the mean of $\cos \varphi$ is $0$ and the mean of $\cos^2 \varphi$ is $\frac12$.
For $-1 \le x \le 1$ the integrand is $\left( x + i \sqrt{1 - x^2} \cos \varphi \right)^n$, whose modulus is at most $\left( x^2 + (1 - x^2) \right)^{n/2} = 1$.
Hence $\abs{P_n(x)} \le 1$ on $[-1, 1]$.

## Explanations

### L1 human

The formula writes each Legendre polynomial as an average of $n$-th powers.
The quantity being raised to the power moves along a segment as the angle $\varphi$ goes from $0$ to $\pi$.
It turns a polynomial into an integral that is easy to bound, which is useful for estimates.

### L2 human

Laplace's integral is one of the standard integral representations of $P_n$, next to Rodrigues' formula and the generating function.
Its main use is in estimates: it gives the bound $\abs{P_n(x)} \le 1$ on $[-1, 1]$ in one line.
For $x > 1$ and the positive root, the integrand is largest at $\varphi = 0$, where it equals $\left( x + \sqrt{x^2 - 1} \right)^n$.
So $P_n(x) \le \left( x + \sqrt{x^2 - 1} \right)^n$, and [[Laplace's method|9G7K8DBT9Y]] gives the precise growth as $n \to \infty$.
The same idea, with Gegenbauer or Jacobi weights, gives integral representations for those families.
</document>

<document source="https://tamnd.github.io/math2/node/YZ4268SK8G.md" tag="YZ4268SK8G" kind="statement">
# Morse index theorem for geodesics

## Statement

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $\gamma \colon [0, 1] \to M$ be a [[geodesic|NSYMQMHPDN]] from $p = \gamma(0)$ to $q = \gamma(1)$.
Let $V$ be the vector space of continuous, piecewise smooth vector fields $W$ along $\gamma$ with $W(0) = 0$ and $W(1) = 0$.
Let $I$ be the Hessian at $\gamma$ of the [[energy functional on paths|Q12AB1MJ64]] from $p$ to $q$.
It is the symmetric bilinear form on $V$ given by
$$I(W_1, W_2) = \int_0^1 \Big( g\big(\tfrac{D W_1}{dt}, \tfrac{D W_2}{dt}\big) - g\big(R(W_1, \dot\gamma) \dot\gamma, W_2\big) \Big) \, dt.$$
Here $D/dt$ is the covariant derivative along $\gamma$ and $R$ is the Riemann curvature tensor.
The sign is such that, for $W \perp \dot\gamma$, $g(R(W, \dot\gamma)\dot\gamma, W)$ is $\abs{W}^2 \abs{\dot\gamma}^2$ times the sectional curvature of the plane of $W$ and $\dot\gamma$.
Then:

1. The [[Morse index|MAX2ZYG6H9]] of $I$, the largest dimension of a subspace of $V$ on which $I$ is negative definite, is finite.
2. It equals the number of parameters $t \in (0, 1)$ such that $\gamma(t)$ is [[conjugate|ZPXE5VSZ35]] to $\gamma(0)$ along $\gamma$, each counted with its multiplicity.
3. The nullity of $I$ equals the multiplicity of $q$ as a conjugate point of $p$ along $\gamma$.
It is $0$ if $q$ is not conjugate to $p$.

## Remarks

Integration by parts gives $I(W_1, W_2) = -\int_0^1 g\big(\tfrac{D^2 W_1}{dt^2} + R(W_1, \dot\gamma)\dot\gamma, W_2\big)\,dt$ plus terms at the breaks of $W_1$, so the null space of $I$ consists of the [[Jacobi fields|9AHMM41PJ3]] vanishing at both ends.
For the index, one splits $[0, 1]$ into short intervals on which $\gamma$ is minimizing.
Then $V$ decomposes into the finite-dimensional space of broken Jacobi fields and a complement on which $I$ is positive definite.
Then one tracks how the index of $I$ restricted to $[0, \tau]$ changes as $\tau$ grows: it jumps exactly at conjugate points, by their multiplicity.

Example: on the unit sphere $S^n$, a unit-speed great circle arc of length $L$ with $\pi < L < 2\pi$ passes one conjugate point, the antipode of its start, with multiplicity $n - 1$.
So its index is $n - 1$, and for $n \ge 2$ it is not a local minimum of the energy.
In particular, a geodesic that passes a conjugate point in its interior is not locally length-minimizing.

## Explanations

### L1 human

A geodesic is a critical point of the energy among paths with the same ends, but it need not be a minimum.
The index counts the independent ways of shortening it by a small deformation.
The theorem says that this number is the number of conjugate points along the way, where nearby geodesics from the start refocus.
On a sphere, a great circle arc longer than half the circle passes the antipode of its start, and it can be shortened by sliding it over the sphere.

### L2 human

The theorem turns a variational question into a question about an ordinary differential equation, the Jacobi equation.
It is the key input of Morse theory on path spaces: together with the Palais–Smale condition, it computes the cells attached at each geodesic.
For example, on a manifold of nonpositive sectional curvature there are no conjugate points, so every geodesic has index $0$.
Bott used it to compute indices of geodesics in compact Lie groups in his proof of periodicity.
</document>

<document source="https://tamnd.github.io/math2/node/YZ8AN4J2KD.md" tag="YZ8AN4J2KD" kind="definition">
# Manifold with boundary

## Definition

Let $n \ge 1$ be an integer and let $\mathbb{H}^n = \set{x \in \R^n : x_n \ge 0}$ be the closed upper half-space of [[Euclidean space|0XPR1E85W0]], with the subspace topology.
A [[topological space|0CQHFPGJ45]] $M$ is a *topological manifold with boundary of dimension $n$* if:

1. $M$ is [[Hausdorff|0S1FJ8VEX9]].
2. $M$ is [[second-countable|03PPNCM9GG]].
3. Each point $x \in M$ has an [[open|02BH3MCBQB]] neighbourhood that is [[homeomorphic|0QT00PB5G2]] to an open subset of $\mathbb{H}^n$.

A point $x \in M$ is a *boundary point* if some homeomorphism $\varphi$ from an open neighbourhood of $x$ onto an open subset of $\mathbb{H}^n$ has $\varphi(x) \in \set{x_n = 0}$.
The set of boundary points is the *boundary* $\partial M$, and $M \setminus \partial M$ is the *interior* of $M$.

## Remarks

The closed disk $D^2 = \set{x \in \R^2 : \abs{x} \le 1}$ is a manifold with boundary of dimension $2$, with $\partial D^2 = S^1$.
Near a point of the open disk it is an open subset of $\R^2$, which is homeomorphic to an open subset of the interior of $\mathbb{H}^2$.
Near a point of $S^1$, polar coordinates $(r, \theta) \mapsto (\theta, 1 - r)$ map a neighbourhood onto an open subset of $\mathbb{H}^2$, sending $S^1$ into the line $\set{x_2 = 0}$.

The cross $\set{(x, y) \in \R^2 : xy = 0}$ satisfies clauses 1 and 2 but fails clause 3 at the origin, where four arcs meet.

By [[invariance of domain|6X2D0JDFQW]], no point is a boundary point for one chart and an interior point for another, so the boundary is well defined.
The interior is a [[topological manifold|QQQ9B1DK4P]] of dimension $n$, and $\partial M$ is a topological manifold of dimension $n - 1$ with empty boundary.
A topological manifold is the same as a manifold with boundary whose boundary is empty.
Some authors use $\set{x_1 \ge 0}$ or $\set{x_1 \le 0}$ as the model half-space; the notion does not change.
</document>

<document source="https://tamnd.github.io/math2/node/YZ97GKYY3P.md" tag="YZ97GKYY3P" kind="statement">
# Lying over theorem

## Statement

Let $A \subseteq B$ be commutative rings with $B$ [[integral over|B7X0KFY1S1]] $A$.
Let $\mathfrak{p}$ be a [[prime ideal|00MHHZ9SNE]] of $A$.
Then there is a prime ideal $\mathfrak{q}$ of $B$ with $\mathfrak{q} \cap A = \mathfrak{p}$.

## Remarks

One [[localizes|4C1AE84N5N]] at $\mathfrak{p}$: the ring $B_{\mathfrak{p}}$ is integral over the [[local ring|MZ02BKPHJA]] $A_{\mathfrak{p}}$ and is not zero, since $A_{\mathfrak{p}} \subseteq B_{\mathfrak{p}}$.
Take a maximal ideal $\mathfrak{n}$ of $B_{\mathfrak{p}}$.
Then $B_{\mathfrak{p}}/\mathfrak{n}$ is a field integral over $A_{\mathfrak{p}}/(\mathfrak{n} \cap A_{\mathfrak{p}})$, so the latter is a field by [[integral extensions of fields|5GCANV7337]], and $\mathfrak{n} \cap A_{\mathfrak{p}}$ is the maximal ideal $\mathfrak{p} A_{\mathfrak{p}}$.
The [[contraction|5AFH94P4S3]] of $\mathfrak{n}$ to $B$ is the wanted $\mathfrak{q}$.

Example: for $\Z \subseteq \Z[i]$ and $\mathfrak{p} = 5\Z$, the prime $\mathfrak{q} = (2 + i)$ satisfies $\mathfrak{q} \cap \Z = 5\Z$, since $5 = (2 + i)(2 - i)$.

Integrality is needed: for $\Z \subseteq \Q$, the only prime ideal of $\Q$ is $0$, so no prime of $\Q$ lies over $2\Z$.
Injectivity is needed too: for the integral map $\Z \to \Z/2\Z$, no prime lies over $3\Z$.

## Explanations

### L1 human

The theorem says that every prime ideal of the small ring is the trace of a prime ideal of the big ring.
In geometry: the map of spaces given by an integral extension is surjective, every point downstairs has a point above it.

### L2 human

Lying over is the first of the Cohen–Seidenberg theorems; [[going-up|GHKYR5PBQC]], [[incomparability|Q2XBHJ96GV]] and [[going-down|YE52MQ13MV]] build on it.
In number theory it says that each prime number $p$ is divisible by some prime ideal of the ring of integers of a number field.
Together with going-up and incomparability it gives $\dim A = \dim B$ for an integral extension.
</document>

<document source="https://tamnd.github.io/math2/node/YZA1V04R96.md" tag="YZA1V04R96" kind="concept">
# Measure algebra

## Summary

A measure algebra is a [[Boolean algebra|0XADJKM699]] in which countable suprema exist, together with a countably additive [[measure|PR11BZQQPG]] that vanishes only at $0$.
The main example is the algebra of measurable sets of a [[measure space|P15VQS5DJ9]] modulo [[null sets|VQQM4Y550T]].
Passing to the measure algebra forgets the points and keeps only what the measure can see.
By the [[isomorphism theorem for measure algebras|YFAQGE2ATQ]], all separable atomless probability measure algebras are isomorphic to that of $[0, 1]$.

## Explanations

### L1 human

In measure theory, two sets that differ only by a set of measure zero usually behave the same.
A measure algebra glues such sets together, so that each element is a measurable set "up to measure zero".
For instance, the intervals $[0, 1]$ and $(0, 1)$ become the same element of the measure algebra of Lebesgue measure.
What is left is an algebraic object on which the measure is still defined and is positive on every nonzero element.
</document>

<document source="https://tamnd.github.io/math2/node/YZCBSRCZC7.md" tag="YZCBSRCZC7" kind="definition">
# Symmetric algebra

## Definition

Let $F$ be a field, let $V$ be a [[vector space|000AYPA9AJ]] over $F$, and let $T(V)$ be its [[tensor algebra|P4J5JZDW0B]].
Let $I$ be the [[two-sided ideal|0CD87SFM2N]] of $T(V)$ generated by the elements $v \otimes w - w \otimes v$ with $v, w \in V$.
The *symmetric algebra* of $V$ is the [[quotient ring|0076HWHGHQ]]
$$S(V) = T(V) / I.$$

## Remarks

$S(V)$ is a commutative associative unital $F$-algebra, graded by degree; its degree $k$ part $S^k(V)$ is the $k$-th symmetric power of $V$.
Universal property: every linear map from $V$ to a commutative associative unital $F$-algebra $A$ extends uniquely to a unital algebra homomorphism $S(V) \to A$.
Example: if $V$ has basis $x_1, \ldots, x_n$, then $S(V) \cong F[x_1, \ldots, x_n]$, and $S^k(V)$ has dimension $\binom{n + k - 1}{k}$; for $n = 2$, $k = 2$ the basis is $x_1^2, x_1 x_2, x_2^2$.
Non-example: for $\dim V \ge 2$, the tensor algebra $T(V)$ itself is not commutative, so it is not $S(V)$.
The [[exterior algebra|0MDMZ965VE]] uses instead the ideal generated by the elements $v \otimes v$.
</document>

<document source="https://tamnd.github.io/math2/node/YZHM3HSM0F.md" tag="YZHM3HSM0F" kind="concept">
# Pochhammer symbol

## Summary

The Pochhammer symbol $(a)_n$ is the rising factorial $a (a + 1) \cdots (a + n - 1)$, a product of $n$ consecutive terms that go up by $1$ starting at $a$.
It is the basic building block of the [[hypergeometric series|428M453CR2]], whose coefficients are quotients of Pochhammer symbols.
For $a$ not a non-positive integer it equals $\Gamma(a + n)/\Gamma(a)$, which ties it to the [[gamma function|0H5SYBSED9]].
In combinatorics the same notation $(x)_n$ often stands for the [[falling factorial|0SJNCJCCQG]] instead, so one has to check the convention of each text.

## Explanations

### L1 human

The [[factorial|0XWG0JGDFW]] $n!$ multiplies $1, 2, \ldots, n$.
The Pochhammer symbol does the same but may start at any number $a$: it multiplies $a, a + 1, \ldots$ for $n$ steps.
For example $(3)_2 = 3 \cdot 4 = 12$ and $(\tfrac12)_2 = \tfrac12 \cdot \tfrac32 = \tfrac34$.
Starting at $a = 1$ gives back the factorial: $(1)_n = n!$.
</document>

<document source="https://tamnd.github.io/math2/node/YZMPRMQ6GA.md" tag="YZMPRMQ6GA" kind="definition">
# Calabi–Yau manifold

## Definition

Let $X$ be a compact connected [[Kähler manifold|X60JG9AEQJ]] of complex dimension $n$, and let $K_X$ be its [[canonical bundle|W05VYY4HBR]], the bundle of holomorphic $n$-forms.
The manifold $X$ is a *Calabi–Yau manifold* if $K_X$ is holomorphically trivial, that is, if $X$ has a holomorphic $n$-form that vanishes nowhere.

## Remarks

Several stricter definitions are in use.
Many authors also ask that $H^0(X, \Omega^i_X) = 0$ for $0 < i < n$, or that $X$ be simply connected, or that the holonomy group of a Ricci-flat Kähler metric be exactly $SU(n)$.
Example: a smooth quintic hypersurface $X \subset \mathbb{P}^4$ has $K_X = \mathcal{O}_X(5 - 5) = \mathcal{O}_X$ by the adjunction formula, so it is Calabi–Yau.
Example: an elliptic curve $\C/\Lambda$ has the nowhere vanishing form $dz$.
Non-example: $\mathbb{P}^1$ has $K = \mathcal{O}(-2)$, which has no nonzero sections, so it is not Calabi–Yau.
A complex torus of dimension $2$ has trivial canonical bundle, but $H^0(X, \Omega^1) \cong \C^2$, so it fails the stricter definition.
</document>

<document source="https://tamnd.github.io/math2/node/YZNFWJAKA4.md" tag="YZNFWJAKA4" kind="concept">
# Periodicity equation

## Summary

The periodicity equation is the [[functional equation|SRAN4EKWS0]] $f(x + p) = f(x)$ for a fixed nonzero number $p$ and an unknown function $f$ on $\R$.
Its solutions are exactly the [[periodic functions|0ATF6Q0633]] with period $p$.
It is the simplest linear difference equation with constant coefficients, and many other equations reduce to it.
For example, the general solution of $g(x + 1) = g(x) + 1$ is $g(x) = x + f(x)$, with $f$ a solution of the periodicity equation for $p = 1$.

## Explanations

### L1 human

The equation says that the function repeats itself after a shift by $p$.
A solution is fixed by its values on one interval of length $p$, and these values can be chosen freely.
For example, $\sin x$ solves it for $p = 2\pi$.
The fractional part $x - \floor{x}$ solves it for $p = 1$.
</document>

<document source="https://tamnd.github.io/math2/node/YZWYF7DXTW.md" tag="YZWYF7DXTW" kind="concept">
# Field of formal Laurent series

## Summary

The field of formal Laurent series $k((X))$ over a field $k$ consists of formal sums $\sum_{n \ge N} a_n X^n$, see the [[definition|2S4K8FA8QY]].
Only finitely many negative powers of $X$ occur.
It is the [[field of fractions|0F2PTGX1A2]] of the ring $k[\![X]\!]$ of [[formal power series|033WGN00BF]].
With the order of the lowest term it is a complete field for a [[discrete valuation|A2KKZAV29W]], the equal-characteristic analogue of the [[p-adic numbers|BT6T702C28]].
It is the formal counterpart of the [[Laurent series|0GXMPDNYX3]] of complex analysis, without any condition of convergence.

## Explanations

### L1 human

A formal Laurent series is like a power series that may start with a few negative powers, such as $X^{-2} + 3X^{-1} + 1 + X + X^2 + \cdots$.
One adds them term by term and multiplies them like polynomials; each coefficient of a product is a finite sum because there are only finitely many negative powers.
Every nonzero series has an inverse: $1/(X - X^2) = X^{-1}(1 + X + X^2 + \cdots)$.
So these series form a field.
No question of convergence arises: the series are only lists of coefficients.
</document>

<document source="https://tamnd.github.io/math2/node/YZXPWC2MT1.md" tag="YZXPWC2MT1" kind="definition">
# Arity

## Definition

Let $A$ and $B$ be sets and let $n \ge 0$ be an integer.
Write $A^n$ for the [[Cartesian product|09CEDJYBKP]] of $n$ copies of $A$, the set of $n$-tuples of elements of $A$.
Here $A^0$ is the set whose only element is the empty tuple.

1. A [[function|0EPAT51DXQ]] $f \colon A^n \to B$ has *arity* $n$.
2. A relation $R \subseteq A^n$ has *arity* $n$.

A function or relation of arity $n$ is called $n$-ary.

## Remarks

Example: the map $(x, y, z) \mapsto x y + z$ from $\R^3$ to $\R$ has arity $3$, because its domain is $\R^3$.

Non-example: the map $\R \to \R^2$, $x \mapsto (x, x)$, has arity $1$ and not $2$, because the arity counts the factors of the domain, not of the codomain.

The arity of a function depends on how its domain is written as a power.
A map on $\R^2$ can be seen as a binary function on $\R$ or as a unary function on the set $\R^2$.
The words nullary, unary, binary and ternary are used for arities $0$, $1$, $2$ and $3$.
</document>

<document source="https://tamnd.github.io/math2/node/YZXV7CHF5R.md" tag="YZXV7CHF5R" kind="definition">
# Slice category

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]] and let $X$ be an object of $\mathcal{C}$.
The *slice category* $\mathcal{C}/X$ is defined as follows:

1. its objects are the pairs $(A, h)$ with $h \colon A \to X$ a [[morphism|859F0TFVA3]] of $\mathcal{C}$;
2. a morphism $(A, h) \to (A', h')$ is a morphism $f \colon A \to A'$ of $\mathcal{C}$ with $h' \circ f = h$;
3. composition and identities are those of $\mathcal{C}$.

## Remarks

It is the [[comma category|PC6HG1JEGM]] $(1_{\mathcal{C}} \downarrow X)$, where $X$ also denotes the functor from the one-morphism category that picks $X$.
Dually, the *coslice category* $X/\mathcal{C}$ has as objects the morphisms $X \to A$, and morphisms $f$ with $f \circ h = h'$.
Example: in the category of sets, an object of $\mathbf{Set}/\set{0, 1}$ is a set $A$ with a map $h \colon A \to \set{0, 1}$, that is, a set split into two labelled parts.
Write $e$ for the identity map of $\set{0, 1}$; then $e$ is a morphism from $(\set{0, 1}, e)$ to itself, since $e \circ e = e$.
Non-example: let $s \colon \set{0, 1} \to \set{0, 1}$ swap $0$ and $1$.
The identity map $e$ is not a morphism from $(\set{0, 1}, e)$ to $(\set{0, 1}, s)$, since $s \circ e = s \ne e$.
The object $(X, 1_X)$ is a terminal object of $\mathcal{C}/X$.
</document>

<document source="https://tamnd.github.io/math2/node/YZY11W08SD.md" tag="YZY11W08SD" kind="concept">
# Koszul complex

## Summary

The Koszul complex of elements $x_1, \ldots, x_n$ of a commutative ring $R$ is a complex of free modules built from the [[exterior algebra|0MDMZ965VE]] of $R^n$.
Its zeroth homology is $R/(x_1, \ldots, x_n)$, and when the $x_i$ form a [[regular sequence|69R5R2FCS5]] it is a [[free resolution|HK2AA1YSQ2]] of this quotient.
It computes [[depth|M2RADBFW99]], gives the proof of [[Hilbert's syzygy theorem|R0DSZPRG6T]], and is a basic tool throughout commutative algebra.

## Explanations

### L1 human

The Koszul complex is the simplest free complex one can write down from a list of elements.
For one element $x$ it is $0 \to R \xrightarrow{x} R \to 0$.
For two elements $x, y$ it is $0 \to R \to R^2 \to R \to 0$, where the last map sends $(a, b)$ to $ax + by$ and the first map sends $c$ to $(-cy, cx)$.
The composite is zero because $-cyx + cxy = 0$, and this simple cancellation is the heart of the construction.
</document>

<document source="https://tamnd.github.io/math2/node/YZYHWC6C5F.md" tag="YZYHWC6C5F" kind="concept">
# Kolmogorov–Smirnov test

## Summary

The Kolmogorov–Smirnov test is a [[hypothesis test|087P03NEND]] of whether a sample comes from a given continuous distribution.
A two-sample version tests whether two samples come from the same distribution.
Its statistic is the largest vertical distance between an [[empirical distribution function|A05WR6TEK5]] and the hypothesised [[cumulative distribution function|0H8H7ZTHW2]], or between two empirical distribution functions.
Under the null hypothesis its law does not depend on the continuous distribution being tested, so one table of critical values serves all cases.
It is one of the basic tests of [[nonparametric statistics|VFAJ1EG3X5]].

## Explanations

### L1 human

Draw the staircase made by the data: it rises by $1/n$ at each data point.
Draw on the same picture the smooth curve of the distribution that you expect.
The test looks for the place where the staircase and the curve are farthest apart.
If that gap is too large for the sample size, the data probably do not come from the expected distribution.
For example, one observation $0.3$ tested against the uniform law on $[0, 1]$ gives a largest gap of $0.7$, at $0.3$ itself.
</document>

<document source="https://tamnd.github.io/math2/node/Z014Q8KPDD.md" tag="Z014Q8KPDD" kind="concept">
# Schubert variety

## Summary

A Schubert variety is a subvariety of a [[Grassmannian|CYPA882RB9]] or a [[flag variety|NHRWCQQKTH]] defined by conditions on how a subspace meets a fixed flag.
In a Grassmannian they are indexed by partitions that fit in a rectangle.
Their classes form a basis of the cohomology ring, and computing products of these classes is [[Schubert calculus|FWGHMADJR4]].
Schubert varieties are usually singular, and their singularities are studied through combinatorics.

## Explanations

### L1 human

Fix a line $\ell$ in three-dimensional projective space.
The lines that meet $\ell$ form a three-dimensional family inside the four-dimensional space of all lines.
This is a Schubert variety: it is defined by one incidence condition with a fixed subspace.
Imposing stronger conditions, such as lying in a fixed plane or passing through a fixed point, gives smaller Schubert varieties.
</document>

<document source="https://tamnd.github.io/math2/node/Z0K8AXW8Z1.md" tag="Z0K8AXW8Z1" kind="definition">
# Semicontinuity

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]], let $f \colon X \to [-\infty, +\infty]$ be a function with values in the [[extended real line|0GSZ7B0CGR]], and let $x_0 \in X$.

1. $f$ is *upper semicontinuous* at $x_0$ if for every real $y > f(x_0)$ there is a [[neighbourhood|0ZCY0Z80B6]] $U$ of $x_0$ with $f(x) < y$ for all $x \in U$.
2. $f$ is *lower semicontinuous* at $x_0$ if for every real $y < f(x_0)$ there is a neighbourhood $U$ of $x_0$ with $f(x) > y$ for all $x \in U$.

The function $f$ is upper (lower) semicontinuous if it is upper (lower) semicontinuous at every point of $X$.

## Remarks

The function $f$ is upper semicontinuous if and only if $\set{x : f(x) < y}$ is [[open|02BH3MCBQB]] for every real $y$.
It is lower semicontinuous if and only if $\set{x : f(x) > y}$ is open for every real $y$.
In terms of limits, $f$ is upper semicontinuous at $x_0$ if and only if $\limsup_{x \to x_0} f(x) \le f(x_0)$, and lower semicontinuous at $x_0$ if and only if $\liminf_{x \to x_0} f(x) \ge f(x_0)$.
A real-valued $f$ is [[continuous|0VGFHTM75Y]] at $x_0$ if and only if it is both upper and lower semicontinuous at $x_0$.

Example: the [[floor function|08GR54CKEA]] $\floor{\cdot} \colon \R \to \R$ is upper semicontinuous.
At an integer $n$ and for $y > n$, the neighbourhood $(n - 1, n + 1)$ works, since $\floor{x} \le n < y$ there.
At a non-integer, the floor function is locally constant.

Non-example: the indicator function $f$ of $[0, 1]$ in $\R$ is not lower semicontinuous at $0$.
Here $f(0) = 1$; for $y = 1/2$, every neighbourhood of $0$ contains some $x < 0$ with $f(x) = 0 < y$.
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# Sard–Smale theorem

## Statement

Let $M$ and $N$ be [[Banach manifolds|9K15AMGBV9]], with $M$ connected and second countable.
Let $f \colon M \to N$ be a [[Fredholm map|GFXQ6HEM65]] of class $C^k$ with index $\operatorname{ind} f$, where $k > \max(\operatorname{ind} f, 0)$.
A point $y \in N$ is a *regular value* of $f$ if $d f_x$ is surjective for every $x \in f^{-1}(y)$.
Then the set of regular values of $f$ is a [[residual set|GRH8MCVN3N]] in $N$, that is, it contains a countable intersection of open dense sets.

## Remarks

The theorem is due to Smale and extends [[Sard's theorem|Z7R8J71SJN]] to infinite dimension.
Measure zero has no meaning in an infinite-dimensional Banach space, so it is replaced by the Baire notion of a residual set.
The proof is local.
Near a point $x$, the Fredholm property lets one write $f$ in charts as $(u, v) \mapsto (u, g(u, v))$, with $u$ in a Banach space and $g$ with values in a finite-dimensional space.
The critical values near $x$ are then those of finite-dimensional maps, which have measure zero by Sard's theorem.
Fredholm maps are locally proper, so the critical values coming from a small closed neighbourhood form a closed set with empty interior.
Second countability gives countably many such pieces.

At a regular value $y$, the implicit function theorem shows that $f^{-1}(y)$ is a $C^k$ submanifold of $M$ of dimension $\operatorname{ind} f$, empty if the index is negative.
Example: if $f \colon M \to N$ is a smooth map between finite-dimensional manifolds, it is Fredholm of index $\dim M - \dim N$ and the theorem reduces to Sard's theorem, with residual in place of measure zero.

## Explanations

### L1 human

Sard's theorem says that almost every value of a smooth map is regular, so that its preimage is a nice manifold.
In infinite dimension there is no good notion of volume.
For Fredholm maps one can still say that regular values are generic: they form a set that is as large as possible in the sense of Baire.

### L2 human

The theorem is the basic transversality tool of global analysis.
Take a nonlinear elliptic equation, written as $f(x) = y$ with $f$ Fredholm.
The theorem shows that for generic data $y$ the solution set is a smooth finite-dimensional manifold.
One can also apply it to the projection onto the parameter space of the zero set of a universal family $F(x, \lambda)$, where $\lambda$ is a metric or a perturbation.
This gives the parametric transversality used for moduli spaces of instantons, of pseudoholomorphic curves and of Morse flow lines.
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<document source="https://tamnd.github.io/math2/node/Z0RWBST5BN.md" tag="Z0RWBST5BN" kind="definition">
# Chern–Simons form

## Definition

Let $M$ be a smooth manifold, and let $\mathfrak{gl}_n(\C)$ be the space of complex $n \times n$ matrices.
Let $A$ be a smooth $\mathfrak{gl}_n(\C)$-valued [[$1$-form|0E3G0GRPZ6]] on $M$; it is a connection form on the trivial bundle $M \times \C^n$.
Let $F = dA + A \wedge A$ be its [[curvature|ZAJFW1S078]], where $d$ is the [[exterior derivative|0ZHSTAG8ZY]] and $\wedge$ is the matrix [[wedge product|0GRAATMGP1]].
The *Chern–Simons $3$-form* of $A$ is
$$\mathrm{CS}(A) = \tr\Big(A \wedge dA + \tfrac23 \, A \wedge A \wedge A\Big).$$

## Remarks

The defining property is $d\, \mathrm{CS}(A) = \tr(F \wedge F)$.
Check: $d \tr(A \wedge dA) = \tr(dA \wedge dA)$ and $d \tr(A \wedge A \wedge A) = 3 \tr(dA \wedge A \wedge A)$ by cyclicity of the graded trace; while $\tr(F \wedge F) = \tr(dA \wedge dA) + 2 \tr(dA \wedge A \wedge A) + \tr(A^4)$ and $\tr(A^4) = 0$.
Example: if $n = 1$, then $A \wedge A = 0$, $F = dA$ and $\mathrm{CS}(A) = A \wedge dA$; on $M = \R^4$ with coordinates $x_1, \dots, x_4$ and $A = x_1 \, dx_2 + x_3 \, dx_4$, one gets $dA = dx_1 \wedge dx_2 + dx_3 \wedge dx_4$, and both $d(A \wedge dA)$ and $F \wedge F$ equal $2 \, dx_1 \wedge dx_2 \wedge dx_3 \wedge dx_4$.
Non-example: for non-abelian $A$, the form $\tr(A \wedge dA)$ alone does not have derivative $\tr(F \wedge F)$ in general, since the term $2 \tr(dA \wedge A \wedge A)$ is missing; the cubic term is needed.
More generally, let $f$ be an invariant polynomial of degree $k$ as in the [[Chern–Weil homomorphism|82ZPRD39J7]], and let $\omega$ be a connection on a principal bundle $P$.
Put $\Omega_t = t\, d\omega + \frac{t^2}{2} [\omega \wedge \omega]$ and $Tf(\omega) = k \int_0^1 f(\omega, \Omega_t, \dots, \Omega_t) \, dt$.
Then $Tf(\omega)$ is a $(2k-1)$-form on $P$ with $d \, Tf(\omega) = f(\Omega, \dots, \Omega)$.
The $3$-form above is the case $f(X, Y) = \tr(XY)$ in a trivialization.
Under a gauge transformation $A \mapsto g^{-1} A g + g^{-1} dg$, $\mathrm{CS}(A)$ changes by an exact form plus $-\frac13 \tr\big((g^{-1}dg)^3\big)$, whose integral over a closed $3$-manifold is an integer multiple of $8\pi^2$ for $G = \mathrm{SU}(n)$.
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# Supermanifold

## Summary

A supermanifold is a space whose functions depend on ordinary commuting coordinates and also on anticommuting coordinates.
Formally, it is a topological space with a [[sheaf|TB6TM6R4MD]] of supercommutative [[superalgebras|6JZW4JC68Z]] that looks locally like smooth functions with values in an [[exterior algebra|0MDMZ965VE]].
Supermanifolds are the geometric setting of supersymmetry, where bosons are described by even and fermions by odd coordinates.
Every smooth supermanifold is built, up to isomorphism, from an ordinary [[vector bundle|P4DZM975BV]] over an ordinary manifold.

## Explanations

### L1 human

On an ordinary manifold, functions are built from coordinates $x_1, \dots, x_p$ that commute.
A supermanifold adds coordinates $\theta_1, \dots, \theta_q$ that anticommute: $\theta_i \theta_j = -\theta_j \theta_i$, so in particular $\theta_i^2 = 0$.
A function in one even and one odd variable is then always of the form $f(x) + g(x)\, \theta$, since higher powers of $\theta$ vanish.
These odd coordinates have no points of their own; they live only in the algebra of functions.
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# Line segment

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]] and let $x, y \in V$.
The *line segment* (or closed segment) between $x$ and $y$ is the set
$$[x, y] = \set{(1 - t) x + t y : t \in \R,\ 0 \le t \le 1}.$$
The points $x$ and $y$ are its endpoints.

## Remarks

Example: in $\R^2$, the point $(1, 2)$ lies in $[(0, 0), (2, 4)]$, since $(1, 2) = \tfrac12 (0, 0) + \tfrac12 (2, 4)$ with $t = 1/2 \in [0, 1]$.

Non-example: the point $(3, 6)$ lies on the line through $(0, 0)$ and $(2, 4)$, but $(3, 6) = (1 - t)(0,0) + t (2, 4)$ forces $t = 3/2$, which is not in $[0, 1]$.
So $(3, 6) \notin [(0,0), (2,4)]$.

The segment $[x, y]$ is the set of [[convex combinations|21SXYJYC59]] of $x$ and $y$, and $[x, y] = [y, x]$.
If $x = y$, then $[x, y] = \set{x}$.
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<document source="https://tamnd.github.io/math2/node/Z1045A0V29.md" tag="Z1045A0V29" kind="concept">
# Uniserial module

## Summary

A uniserial module is a [[module|0NZ8JA25WK]] whose [[submodules|05WDSN597C]] form a chain: of any two submodules, one contains the other.
Uniserial modules are the building blocks of serial modules and [[serial rings|TW3ESQB3GQ]].
Over a [[discrete valuation ring|MGM3CNJW9V]] and over many finite-dimensional algebras, the indecomposable modules of finite length are uniserial, which makes these modules easy to classify.

## Explanations

### L1 human

Picture the submodules of a uniserial module as the rungs of a single ladder.
There are no side branches: you can only go up or down the ladder.
The abelian group $\Z/8\Z$ is uniserial, because its only subgroups are $0 \subset 4\Z/8\Z \subset 2\Z/8\Z \subset \Z/8\Z$.
The group $\Z/6\Z$ is not uniserial, because its subgroups of order $2$ and $3$ do not contain each other.
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<document source="https://tamnd.github.io/math2/node/Z11GGXV5KD.md" tag="Z11GGXV5KD" kind="definition">
# Thin plate spline

## Definition

Let $\varphi(r) = r^2 \log r$ for $r > 0$ and $\varphi(0) = 0$.
Let $x_1, \dots, x_N \in \R^2$ be distinct points, not all on one line, and let $y_1, \dots, y_N$ be real numbers.
The *thin plate spline* interpolant of these data is the function
$$s(x) = a_0 + a^T x + \sum_{i=1}^{N} w_i\, \varphi(\norm{x - x_i}), \qquad x \in \R^2,$$
with $a_0 \in \R$, $a \in \R^2$ and $w \in \R^N$ determined by:

1. $s(x_i) = y_i$ for $i = 1, \dots, N$;
2. $\sum_{i=1}^{N} w_i = 0$;
3. $\sum_{i=1}^{N} w_i x_i = 0$ in $\R^2$.

## Remarks

The conditions form a square linear system, and it has exactly one solution when the points are not collinear.
This is the [[polyharmonic spline|XJ0JGPBC8R]] with $d = 2$ and $m = 2$, built from the [[radial basis function|ZBSXSTQM2J]] $\varphi$.
Among all functions $f$ with square integrable second derivatives and $f(x_i) = y_i$, the spline $s$ minimizes the bending energy
$$\iint_{\R^2} \big( f_{x_1 x_1}^2 + 2 f_{x_1 x_2}^2 + f_{x_2 x_2}^2 \big)\, dx.$$
Away from the points, $s$ satisfies the [[biharmonic equation|AS8A9F5SKE]] $\Delta^2 s = 0$.
Example: if $y_i = c_0 + c^T x_i$ for an affine function, then $w = 0$, $a_0 = c_0$, $a = c$ satisfy all three clauses, so $s$ is that affine function.
Non-example: $f(x) = \varphi(\norm{x})$ with the single centre $0$ fails clause 2, since its only weight is $1$.
Thin plate splines are also used with one map per coordinate to define smooth deformations of the plane from matched landmark points.
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<document source="https://tamnd.github.io/math2/node/Z15FXM9WM1.md" tag="Z15FXM9WM1" kind="concept">
# Order filter

## Summary

A filter of a [[partially ordered set|0QV2JS9RJ3]] is a nonempty subset that is closed upward.
Moreover, any two of its elements have a common lower bound inside it.
In a [[lattice|WH4SFAACAS]] this means: closed upward and closed under meets.
Filters are the order duals of [[lattice ideals|5GW2T2AM1A]].
A [[filter on a set|15PF73BKY6]] $X$ is a proper order filter in the lattice of subsets of $X$.

## Explanations

### L1 human

An order filter is a collection of "large" elements.
Anything above a large element is large, and two large elements always have a large element below both of them.
For a fixed element $a$, the set of all elements above $a$ is a filter.
In the subsets of the natural numbers, the sets with finite complement form a filter, since the intersection of two such sets again has finite complement.
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# Duality in category theory

## Definition

Let $\Sigma$ be a statement in the first-order language of [[categories|A2DE66937D]], built from objects, morphisms, domains, codomains, identities and composition.
The *dual statement* $\Sigma^{\mathrm{op}}$ is obtained from $\Sigma$ by these changes:

1. each domain is replaced by the codomain, and each codomain by the domain;
2. each composite $g \circ f$ is replaced by $f \circ g$.

The *duality principle* states that $\Sigma$ holds in the [[opposite category|DAMK8XE1FR]] $\mathcal{C}^{\mathrm{op}}$ if and only if $\Sigma^{\mathrm{op}}$ holds in $\mathcal{C}$.
In particular, if $\Sigma$ holds in every category, then so does $\Sigma^{\mathrm{op}}$.

## Remarks

Two notions are *dual* when the defining statement of one is the dual of the defining statement of the other.
Example: "$f$ is a [[monomorphism|Y2NP1JFR86]]" says that $f \circ g_1 = f \circ g_2$ implies $g_1 = g_2$.
Its dual says that $g_1 \circ f = g_2 \circ f$ implies $g_1 = g_2$, which is "$f$ is an [[epimorphism|RVXDSY04KG]]".
Hence the fact "a composite of monomorphisms is a monomorphism" gives at once "a composite of epimorphisms is an epimorphism".
Likewise a section and a retraction are dual, as in [[section and retraction|DXJH3DGXBC]].
Non-example: the dual of a statement true in one particular category need not be true in that category.
In the category of sets, every morphism $X \to \emptyset$ is an isomorphism, since $X$ must then be empty.
The dual statement says that every morphism $\set{0} \to X$ out of the one-point set is an isomorphism, and it fails for $X = \set{0, 1}$.
Duality in this sense is different from the duality of vector spaces, although the [[dual space|07JBSZE0V4]] functor turns linear maps around.
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# Racah polynomials

## Definition

Let $N \ge 0$ be an integer, let $\alpha, \beta, \gamma, \delta$ be real numbers, and assume that one of $\alpha + 1$, $\beta + \delta + 1$, $\gamma + 1$ equals $-N$.
Write $(a)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]] and set $\lambda(x) = x(x + \gamma + \delta + 1)$.
For $0 \le n \le N$, the *Racah polynomial* of degree $n$ is
$$R_n(\lambda(x); \alpha, \beta, \gamma, \delta) = \sum_{k=0}^{n} \frac{(-n)_k \, (n + \alpha + \beta + 1)_k \, (-x)_k \, (x + \gamma + \delta + 1)_k}{(\alpha + 1)_k \, (\beta + \delta + 1)_k \, (\gamma + 1)_k \, k!},$$
that is, the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_4F_3\left( -n, n + \alpha + \beta + 1, -x, x + \gamma + \delta + 1; \alpha + 1, \beta + \delta + 1, \gamma + 1; 1 \right)$, assuming the denominators do not vanish.

## Remarks

Since $(-x)_k (x + \gamma + \delta + 1)_k = \prod_{j=0}^{k-1} \left( j(j + \gamma + \delta + 1) - \lambda(x) \right)$, each $R_n$ is a polynomial of degree $n$ in $\lambda(x)$.
Example: $R_1 = 1 + \frac{(\alpha + \beta + 2) \, \lambda(x)}{(\alpha + 1)(\beta + \delta + 1)(\gamma + 1)}$.
They are orthogonal on $x = 0, 1, \dots, N$ for the weight
$$w(x) = \frac{(\alpha + 1)_x (\beta + \delta + 1)_x (\gamma + 1)_x (\gamma + \delta + 1)_x \left( \frac{\gamma + \delta + 3}{2} \right)_x}{(-\alpha + \gamma + \delta + 1)_x (-\beta + \gamma + 1)_x \left( \frac{\gamma + \delta + 1}{2} \right)_x (\delta + 1)_x \, x!},$$
that is, $\sum_{x=0}^{N} w(x) R_m(\lambda(x)) R_n(\lambda(x)) = 0$ for $m \ne n$; the weight is positive for suitable ranges of the parameters.
The balanced condition holds: the sum of the lower parameters exceeds the sum of the upper ones by $1$.
Non-example: with $\alpha + 1 = -N$, the degree $n = N + 1$ is not allowed, since the term $k = N + 1$ has the factor $(\alpha + 1)_{N+1} = (-N)_{N+1} = 0$ in its denominator.
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# Cotangent space of a local ring

## Summary

The cotangent space of a [[local ring|MZ02BKPHJA]] $(R, \mathfrak m, k)$ is the [[vector space|000AYPA9AJ]] $\mathfrak m / \mathfrak m^2$ over the [[residue field|NJ3APQ8BHH]] $k$.
Its dual is the Zariski tangent space at the point that $R$ describes.
Its [[dimension|03GSZ8RTZJ]] is the [[embedding dimension|DP3KSCRKFP]] of $R$, which is used to define [[regular local rings|63XXTF6B5M]], that is, smooth points.

## Explanations

### L1 human

Near a point of a curve or a surface, the elements of $\mathfrak m$ are the functions that vanish at the point.
Two such functions have the same image in $\mathfrak m / \mathfrak m^2$ when they agree to first order, so the quotient records only their linear parts.
For the plane at the origin, the linear parts are spanned by $x$ and $y$, so the cotangent space has dimension $2$.
The [[definition|TCRZV021XR]] gives the construction.
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# Difference algebra

## Summary

Difference algebra studies [[rings|0B1261B5ED]] and [[fields|04C5ZBG1WT]] equipped with a distinguished endomorphism, called difference rings and difference fields.
It is the algebraic theory of difference equations, such as $f(x+1) - f(x) = g(x)$, in the same way that differential algebra is the algebraic theory of differential equations.
The endomorphism plays the role that the [[derivation|3J89REV9WQ]] plays in a [[differential field|GA5PTQGC6G]].

## Explanations

### L1 human

A difference equation relates the values of an unknown function at shifted points, such as $f(x+1)$ and $f(x)$.
The shift $f(x) \mapsto f(x+1)$ respects sums and products of functions, so it is a ring homomorphism.
Difference algebra keeps only this structure: a ring together with one such map.
For example, on the rational functions $\C(x)$ the shift sends $x^2$ to $x^2 + 2x + 1$.
Questions such as "can this sum be written in closed form?" then become questions about difference fields.
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# Abel summation

## Definition

Let $\sum_{n \ge 0} a_n$ be a [[series|0JH4MN4K1C]] of complex numbers.
The series is *Abel summable* to $s$ if the following two conditions hold.

1. The [[power series|0VP56B4EMX]] $f(x) = \sum_{n \ge 0} a_n x^n$ converges for every real $x$ with $0 \le x < 1$.
2. The [[one-sided limit|0909PWZR62]] $\lim_{x \to 1^-} f(x)$ exists and equals $s$.

## Remarks

Example: for $a_n = (-1)^n (n + 1)$, that is $1 - 2 + 3 - 4 + \cdots$, one has $f(x) = \sum_{n \ge 0} (-1)^n (n+1) x^n = \frac{1}{(1 + x)^2}$ for $0 \le x < 1$.
This tends to $\frac14$ as $x \to 1^-$, so the series is Abel summable to $1/4$.

Non-example: for $1 + 1 + 1 + \cdots$, condition 1 holds with $f(x) = \frac{1}{1 - x}$.
But $f(x) \to +\infty$ as $x \to 1^-$, so condition 2 fails and the series is not Abel summable.

If $\sum a_n$ converges to $s$, then it is Abel summable to $s$; this is Abel's theorem on power series.
In the notation of summability, one writes $A\text{-}\sum a_n = s$.
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# Subharmonic function

## Summary

A subharmonic function is an upper semicontinuous function whose value at each point is at most its average over small circles around that point.
Subharmonic functions play the role in two real dimensions that convex functions play in one.
For a holomorphic function $f$, the functions $\log\abs{f}$ and $\abs{f}^p$ with $p > 0$ are subharmonic, which links complex analysis with [[potential theory|9PM8Q2E30Q]].
They are the main tool in Perron's method for the [[Dirichlet problem|8D56GQ4AG8]] and in the study of growth of holomorphic functions.

## Explanations

### L1 human

A [[harmonic function|0DEB0JMN0W]] equals its average over every circle around a point.
A subharmonic function is at most that average, so it sags below its surroundings and cannot have a strict local maximum inside its domain.
A good picture is a rubber sheet pulled down at some points by weights hanging from it.
For example, $\abs{z}^2$ is subharmonic: its value $0$ at the origin is below its average $r^2$ over the circle of radius $r$.
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# Frobenius group

## Definition

Let a finite group $G$ act [[transitively|02GK6HZYBT]] on a finite set $\Omega$ with $\abs{\Omega} \ge 2$.
$G$ is a *Frobenius group* (for this action) if:

1. no element $g \ne e$ of $G$ fixes two distinct points of $\Omega$, and
2. some element $g \ne e$ of $G$ fixes a point of $\Omega$.

## Remarks

Equivalently, a finite group $G$ is a Frobenius group if it has a [[subgroup|0X1JXT47GA]] $H$ with $\set{e} \ne H \ne G$ and $H \cap gHg^{-1} = \set{e}$ for all $g \in G \setminus H$.
Then $H$ is a point stabiliser for the action on the left cosets of $H$, and it is called a *Frobenius complement*.
The set $K$ of elements that fix no point, together with $e$, is the *Frobenius kernel*; by [[Frobenius's theorem|XK7H3YXBQT]] it is a normal subgroup.

Example: the group of maps $x \mapsto ax + b$ with $a \ne 0$, acting on a [[finite field|0ZXZ82FTX2]] $\F_q$ with $q \ge 3$.
If $a \ne 1$, the equation $ax + b = x$ has exactly one solution, and if $a = 1$ and $b \ne 0$ it has none, so clause 1 holds.
The map $x \mapsto ax$ with $a \ne 0, 1$ fixes $0$, so clause 2 holds.
For $q = 3$ this is $S_3$ acting on $3$ points.
For odd $n \ge 3$, the [[dihedral group|0JZ27DMJ5D]] of order $2n$ acting on the corners of a regular $n$-gon is also a Frobenius group.

Non-example: $S_4$ acting on $\set{1, 2, 3, 4}$.
The transposition $(1\,2)$ fixes the two points $3$ and $4$, so clause 1 fails.
The cyclic group generated by $(1\,2\,3)$ acting on $\set{1, 2, 3}$ satisfies clause 1 but fails clause 2.
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<document source="https://tamnd.github.io/math2/node/Z219Y8PJHD.md" tag="Z219Y8PJHD" kind="example">
# Lieb–Liniger integral equation

## Example

Consider the Lieb–Liniger model of bosons on a line with delta interaction of strength $c > 0$, in units with $\hbar = 2m = 1$.
In the thermodynamic limit its ground state is described by a density $\rho$ of quasi-momenta on an interval $[-K, K]$, with $K > 0$.
This density solves
$$\rho(k) = \frac{1}{2\pi} + \frac{1}{\pi} \int_{-K}^{K} \frac{c}{c^2 + (k - q)^2} \, \rho(q) \, dq, \quad -K \le k \le K.$$
The particle density and the ground state energy per length are $n = \int_{-K}^{K} \rho(k) \, dk$ and $e = \int_{-K}^{K} k^2 \rho(k) \, dk$.

The equation is a [[Fredholm equation of the second kind|DEYYNVCV9J]] on $[-K, K]$ with the continuous [[difference kernel|B5AB451TB7]] $\frac{1}{\pi} \frac{c}{c^2 + (k - q)^2}$.
This kernel is positive and $\frac{1}{\pi} \int_{-\infty}^{\infty} \frac{c}{c^2 + u^2} \, du = 1$.
So for each $k$ the integral of the kernel over the bounded interval $[-K, K]$ is strictly less than $1$.
By continuity and compactness, its maximum over $k$ is some $\theta < 1$.
Hence the integral operator has norm $\theta < 1$ on $C[-K, K]$ with the supremum norm.
Then the [[Liouville–Neumann series|H1HWSWME6B]] converges to the unique continuous solution, which is positive.

## Remarks

In the limit $c \to \infty$ the kernel tends to $0$ uniformly on $[-K, K]^2$, so $\rho \to \frac{1}{2\pi}$.
Then $n = \frac{K}{\pi}$ and $e = \frac{K^3}{3\pi} = \frac{\pi^2 n^3}{3}$, the energy of free fermions with the same density.

## Explanations

### L1 human

Bosons that repel each other on a line can be solved exactly with the Bethe ansatz.
For many particles the allowed momenta fill an interval with some density, and this density solves a linear integral equation.
The kernel is a Lorentzian bump of width $c$, so for strong repulsion the integral term is small and the density is almost constant.
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# Operator exponential and translation

## Statement

Let $\mathcal{M}$ be the [[Mikusiński field of operators|82YAXZMMQ1]], the field of fractions of the convolution ring of locally integrable functions on $[0, \infty)$.
Let $s$ be the inverse of the constant function $1$ in $\mathcal{M}$.
For $\lambda \ge 0$, let $h_\lambda(t) = H(t - \lambda)$ for $t \ge 0$, where $H$ is the [[Heaviside step function|09C4JP7Y0M]], and define the translation operator $e^{-\lambda s} = s \, h_\lambda$.
Then:

1. For every locally integrable $f$ on $[0, \infty)$, the product $e^{-\lambda s} f$ is the function $t \mapsto H(t - \lambda) f(t - \lambda)$.
2. $e^{-0 s} = 1$, and $e^{-\lambda s} e^{-\mu s} = e^{-(\lambda + \mu) s}$ for all $\lambda, \mu \ge 0$.

## Remarks

For $t \ge \lambda$, $(h_\lambda * f)(t) = \int_0^{t - \lambda} f(\tau) \, d\tau$, and for $t < \lambda$ it is $0$.
So $h_\lambda * f$ is the [[integral from zero|BDBDJSP6DF]] of the delayed function $f_\lambda(t) = H(t - \lambda) f(t - \lambda)$, that is, $h_\lambda f = h f_\lambda$.
Multiplying by $s = 1/h$ gives clause 1.
Clause 2 follows from clause 1, since a delay by $\mu$ followed by a delay by $\lambda$ is a delay by $\lambda + \mu$, and $\mathcal{M}$ has no zero divisors.
For example, $e^{-\lambda s} h = h_\lambda$: delaying the unit step $h$ by $\lambda$ gives the jump of size one at time $\lambda$.
The notation matches the Laplace transform, where a delay by $\lambda$ multiplies the transform by $e^{-\lambda s}$.

## Explanations

### L1 human

Shifting a signal later in time by $\lambda$ is, in the field of operators, multiplication by one fixed operator.
That operator behaves like an exponential: two shifts compose by adding their lengths.
This is why it is written $e^{-\lambda s}$.

### L2 human

The result explains Heaviside's formal rule that $e^{-\lambda p}$ acts as a delay, where $p$ is differentiation.
In Mikusiński's setting no power series in $s$ is needed: the operator is defined directly from the delayed step.
It lets one solve equations with delays and treat piecewise defined inputs, such as a switch that turns on at time $\lambda$.
The same rule for the Laplace transform is the time shift theorem.
</document>

<document source="https://tamnd.github.io/math2/node/Z23FCT4WTZ.md" tag="Z23FCT4WTZ" kind="definition">
# CR function

## Definition

Let $(M, L)$ be a [[CR manifold|ZC78JKZX9S]], with $L \subseteq TM \otimes \C$ its CR structure and $\overline{L}$ the conjugate subbundle.
A function $f \colon M \to \C$ of class $C^1$ is a *CR function* if $X f = 0$ for every smooth section $X$ of $\overline{L}$.

## Remarks

The bundle $L$ plays the role of the holomorphic tangent vectors $\partial / \partial z_j$, so $\overline{L}$ plays the role of the $\partial / \partial \bar z_j$, and the condition says $\bar\partial_b f = 0$.
One also defines CR distributions by the same equations in the sense of distributions.

Example: let $M \subset \C^n$ be a real hypersurface with its induced CR structure, and let $F$ be holomorphic on a neighbourhood of $M$.
A section $X$ of $\overline{L}$ is a combination of the $\partial / \partial \bar z_j$ that is tangent to $M$.
So $X (F|_M) = (X F)|_M = 0$, and $F|_M$ is a CR function.

Non-example: on the unit sphere $S^3 \subset \C^2$, the bundle $\overline{L}$ is spanned by $\overline{Z} = z_2 \, \partial / \partial \bar z_1 - z_1 \, \partial / \partial \bar z_2$.
It is tangent to $S^3$, since $\overline{Z}(\abs{z_1}^2 + \abs{z_2}^2) = z_2 z_1 - z_1 z_2 = 0$.
For $f = \bar z_1$ one finds $\overline{Z} f = z_2$, which is not identically zero on $S^3$, so $\bar z_1$ is not a CR function.
</document>

<document source="https://tamnd.github.io/math2/node/Z26Z4XZFVA.md" tag="Z26Z4XZFVA" kind="concept">
# Axiom A

## Summary

A diffeomorphism satisfies Axiom A if its [[non-wandering set|EY2W34JZA4]] is a [[hyperbolic set|MPFD24TYW2]] in which periodic points are dense.
Smale introduced the condition to single out a large class of systems whose dynamics can be described completely.
For such systems the non-wandering set splits into finitely many basic pieces by the [[spectral decomposition theorem|6MTVY6NYY7]].
Each piece is coded by a [[subshift of finite type|624XDXX8FN]].
[[Anosov diffeomorphisms|RCQH8P6DPQ]], [[Morse–Smale systems|7AMQ8GS78Y]] and the [[Smale horseshoe|N63Z6EA6YS]] satisfy Axiom A.

## Explanations

### L1 human

Most points of a dynamical system wander off and never come back near where they started.
The interesting dynamics happens on the set of points that keep returning, the non-wandering set.
Axiom A asks that on this set the system stretches and squeezes uniformly, and that periodic orbits are everywhere in it.
Then the long-term behaviour can be cut into finitely many pieces, each understood by symbolic codes.
See the [[definition|S31RTTJPDC]].
</document>

<document source="https://tamnd.github.io/math2/node/Z2AJFZQTWT.md" tag="Z2AJFZQTWT" kind="definition">
# Maximal torus

## Definition

Let $G$ be a [[compact Lie group|9TFC6ADC1C]].
A *torus* in $G$ is a compact connected abelian Lie subgroup of $G$.
Each such subgroup is isomorphic to the [[torus group|XNAWD33HN1]] $\mathbb{T}^k$ for some $k \ge 0$.
A *maximal torus* of $G$ is a torus $T \subseteq G$ such that every torus $T'$ with $T \subseteq T' \subseteq G$ satisfies $T' = T$.

## Remarks

Example: in $G = U(n)$, the subgroup $T$ of diagonal matrices $\operatorname{diag}(z_1, \dots, z_n)$ with $\abs{z_j} = 1$ is a maximal torus.
It is a torus, isomorphic to $\mathbb{T}^n$.
Let $T' \supseteq T$ be a torus.
Each element of $T'$ commutes with a diagonal matrix with distinct entries, so it is diagonal; thus $T' = T$.
Non-example: in $U(2)$, the subgroup $\set{\operatorname{diag}(z, 1) : \abs{z} = 1}$ is a torus, but it is not maximal, since it lies in the diagonal torus $\mathbb{T}^2$.
The dimension of a maximal torus is the rank of $G$; for $U(n)$ it is $n$, and for $SU(n)$ it is $n - 1$.
Every torus of $G$ lies in some maximal torus, by a dimension argument.
</document>

<document source="https://tamnd.github.io/math2/node/Z2CP530WDE.md" tag="Z2CP530WDE" kind="concept">
# Trace class operator

## Summary

A trace class operator is a bounded operator on a Hilbert space whose singular values have a finite sum.
For such operators the trace $\sum_i \ip{T e_i}{e_i}$ converges absolutely and does not depend on the orthonormal basis, so the trace of matrices extends to them.
Trace class operators are central in the theory of Fredholm determinants.
In quantum mechanics, density matrices are positive trace class operators of trace $1$.

## Explanations

### L1 human

For a matrix, the trace is the sum of the diagonal entries.
For an operator on an infinite-dimensional space, this sum may diverge or depend on the chosen basis.
Trace class operators are those small enough that the sum always converges and gives the same answer in every basis.
For example, the operator on square-summable sequences that multiplies the $n$-th coordinate by $1/n^2$ is trace class, with trace $\sum 1/n^2 = \pi^2 / 6$.
</document>

<document source="https://tamnd.github.io/math2/node/Z2EWA10YMB.md" tag="Z2EWA10YMB" kind="definition">
# Horner's method

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]], let $f = a_n x^n + \dots + a_1 x + a_0 \in R[x]$ with $n \ge 0$, and let $c \in R$.
Horner's method computes the elements $b_n, b_{n-1}, \dots, b_0$ of $R$ by
$$b_n = a_n, \qquad b_k = a_k + c \, b_{k+1} \quad (k = n - 1, \dots, 0),$$
and returns $b_0$.

## Remarks

The output is the value: $b_0 = f(c)$, see [[polynomial evaluation|0GJ7T0NK49]].
Moreover $f = (x - c)(b_n x^{n-1} + \dots + b_2 x + b_1) + b_0$, so the method also gives the quotient of $f$ by $x - c$.
Example: for $f = 2x^3 - 6x^2 + 2x - 1$ and $c = 3$, one gets $b_3 = 2$, $b_2 = 0$, $b_1 = 2$, $b_0 = 5$.
Check: $f(3) = 54 - 54 + 6 - 1 = 5$, and $f = (x - 3)(2x^2 + 2) + 5$.
Non-example: computing each power $c^k$ separately and then the sum gives the same value, but it is not Horner's method and uses more multiplications.
The method uses $n$ multiplications and $n$ additions.

## Explanations

### L1 human

Horner's method reads the coefficients from the top and, at each step, multiplies by $c$ and adds the next coefficient.
</document>

<document source="https://tamnd.github.io/math2/node/Z2FEBW5MQM.md" tag="Z2FEBW5MQM" kind="concept">
# Parabolic subgroup

## Summary

A parabolic subgroup of a real [[semisimple Lie group|XJS07AJ50Y]] is a closed subgroup that contains a conjugate of the minimal parabolic subgroup $MAN$.
In $\mathrm{SL}_n(\R)$ the parabolic subgroups are the conjugates of the groups of block upper triangular matrices.
They are the stabilizers of partial flags and the building blocks of the Langlands decomposition.
Inducing representations from parabolic subgroups is the main way to build representations of $G$.

## Explanations

### L1 human

In the group of invertible matrices, look at the matrices that are upper triangular in blocks.
Such a matrix maps the span of the first few basis vectors to itself.
So it keeps a chain of subspaces in place, a partial flag.
The groups of this shape, and their conjugates, are the parabolic subgroups.
For $2 \times 2$ matrices of determinant $1$, the only proper ones are the conjugates of the upper triangular matrices.
</document>

<document source="https://tamnd.github.io/math2/node/Z2N8MPD8DH.md" tag="Z2N8MPD8DH" kind="statement">
# Lumer–Phillips theorem

## Statement

Let $X$ be a real or complex [[Banach space|06GZ956657]] and let $A$ be a [[densely defined|BCJHNHKQFG]] linear operator in $X$ with domain $D(A)$.
Then the following are equivalent.

1. $A$ is the [[generator|60HSWZBTK9]] of a [[contraction semigroup|XPWE94BV6N]] on $X$.
2. $A$ is [[dissipative|3S7HJ5GQRF]], and there is $\lambda_0 > 0$ such that $\lambda_0 I - A$ maps $D(A)$ onto $X$.

## Remarks

If $A$ generates a contraction semigroup $T(t)$, then for $\lambda > 0$ the resolvent is the Laplace transform $(\lambda I - A)^{-1}x = \int_0^\infty e^{-\lambda t} T(t)x \, dt$.
This gives surjectivity of $\lambda I - A$ and the bound $\norm{(\lambda I - A)^{-1}} \le 1/\lambda$, which is dissipativity.
Conversely, dissipativity and one surjective $\lambda_0 I - A$ give $\norm{(\lambda I - A)^{-1}} \le 1/\lambda$ for all $\lambda > 0$, and the [[Hille–Yosida theorem|5YR4GJKPCV]] then produces the semigroup.

Example: on $L^2(0, 1)$, let $Af = f''$ with domain $H^2(0, 1) \cap H^1_0(0, 1)$.
It is dissipative since $\ip{Af}{f} = -\int_0^1 \abs{f'}^2 \, dx$, and $I - A$ is onto because the boundary problem $f - f'' = g$, $f(0) = f(1) = 0$ has a solution for every $g \in L^2(0, 1)$.
So $A$ generates the heat semigroup with Dirichlet boundary conditions, and it is a contraction semigroup.

The range condition cannot be dropped.
The operator $f \mapsto f''$ on smooth functions with compact support in $(0, 1)$ is densely defined and dissipative, but it is not closed, so it is not a generator.

## Explanations

### L1 human

The theorem tells when the equation $u' = Au$ has solutions that never grow in norm.
Two things are needed.
The operator must lose energy, which is dissipativity.
And the equation $\lambda u - Au = g$ must be solvable for every right side $g$, which rules out an operator with too small a domain.

### L2 human

The Hille–Yosida theorem asks for the resolvent bound $\norm{(\lambda - A)^{-1}} \le 1/\lambda$ for every $\lambda > 0$.
The Lumer–Phillips theorem replaces this bound by dissipativity, which is often a one-line integration by parts, plus solvability for one $\lambda_0$.
For a second-order elliptic operator with Dirichlet conditions, the solvability usually comes from the Lax–Milgram theorem.
For a self-adjoint operator $A \le 0$ on a Hilbert space, both conditions hold, and the semigroup is $e^{tA}$ given by the spectral theorem.
</document>

<document source="https://tamnd.github.io/math2/node/Z2R1JY1CHS.md" tag="Z2R1JY1CHS" kind="statement">
# Morrey's inequality

## Statement

Let $n \ge 1$, let $n < p \le \infty$, and let $\gamma = 1 - n/p$.
There is a constant $C = C(n, p)$ such that every continuously differentiable function $u \colon \R^n \to \R$ satisfies
$$\sup_{x \in \R^n} \abs{u(x)} + \sup_{x \ne y} \frac{\abs{u(x) - u(y)}}{\abs{x - y}^{\gamma}} \le C \norm{u}_{W^{1,p}(\R^n)}.$$
Consequently, each $u$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\R^n)$ agrees almost everywhere with a function in the [[Hölder space|P101AX1ZSC]] $C^{0,\gamma}(\R^n)$.
The norm of this function in $C^{0,\gamma}(\R^n)$ is at most $C \norm{u}_{W^{1,p}(\R^n)}$.

## Remarks

The key step is the bound $\frac{1}{\abs{B}} \int_B \abs{u(y) - u(x)} \, dy \le C \int_B \frac{\abs{\nabla u(y)}}{\abs{x - y}^{n - 1}} \, dy$ for a ball $B$ centred at $x$.
By [[Hölder's inequality|JQ9WXC729D]], the kernel $\abs{x - y}^{1 - n}$ lies in $L^{p'}(B)$ exactly when $p > n$, and this gives a bound by $r^{1 - n/p} \norm{\nabla u}_{L^p}$ for the ball of radius $r$.
Comparing $u(x)$ and $u(y)$ with the average of $u$ on a ball containing both gives the [[Hölder continuity|012PBA0FMD]] bound.

The hypothesis $p > n$ is needed.
For $n = 2$ and $p = 2$, the function $\log \log (1/\abs{x})$, cut off away from the origin, lies in $W^{1,2}$ but is unbounded near $0$.

For a bounded open set $U$ with $C^1$ boundary, the same holds with $\R^n$ replaced by $U$ and $C$ depending also on $U$.
Together with the [[Sobolev inequality|B03CA692K3]] it forms the [[Sobolev embedding theorem|GZTT6FT18G]].

## Explanations

### L1 human

When the slope of a function is integrable to a high enough power, larger than the dimension, the function cannot jump or blow up.
It is then continuous, and even Hölder continuous with an exponent that measures how much $p$ exceeds $n$.
In one dimension any $p > 1$ is enough, which matches the fact that $\abs{u(x) - u(y)} \le \abs{x - y}^{1 - 1/p} \norm{u'}_{L^p}$ by Hölder's inequality.

### L2 human

Morrey's inequality is how one shows that weak solutions are continuous functions.
In dimension $n$, a function in $W^{k,p}$ with $kp > n$ is continuous, by combining Morrey's inequality with the Sobolev inequality for the lower derivatives.
This is the step that turns $L^2$ regularity results, such as those of [[elliptic regularity|2PVQY9PCT4]], into classical smoothness.
</document>

<document source="https://tamnd.github.io/math2/node/Z2T5R88QDB.md" tag="Z2T5R88QDB" kind="concept">
# Trace operator

## Summary

The trace operator gives boundary values to functions in a [[Sobolev space|M4CEZ5K9HP]], which in general are only defined up to sets of measure zero.
Let $U$ be a bounded open set with $C^1$ boundary, and let $1 \le p < \infty$.
The trace operator extends the restriction $u \mapsto u|_{\partial U}$ to all of $W^{1,p}(U)$, as a bounded linear map into $L^p(\partial U)$.
It gives a precise meaning to boundary conditions such as $u = g$ on $\partial U$ for weak solutions of [[boundary value problems|TM8VQJWGHZ]].

## Explanations

### L1 human

The boundary of a region has volume zero, so the value of an $L^p$ function there is meaningless: one can change it freely.
A Sobolev function has a controlled slope, and this is enough to make its boundary values well defined in an averaged sense.
For a continuous function up to the boundary, the trace is just the restriction to the boundary.
The functions with zero trace are exactly the limits of smooth functions that vanish near the boundary.
</document>

<document source="https://tamnd.github.io/math2/node/Z2XXJ5Z4NF.md" tag="Z2XXJ5Z4NF" kind="definition">
# Hadamard product

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $m, n \ge 1$.
Let $A = (a_{ij})$ and $B = (b_{ij})$ be $m \times n$ [[matrices|0VD9ZSWWJG]] over $R$.
The *Hadamard product* of $A$ and $B$ is the $m \times n$ matrix $A \circ B$ with entries
$$(A \circ B)_{ij} = a_{ij} b_{ij}$$
for $1 \le i \le m$ and $1 \le j \le n$.

## Remarks

Example: for $A = \begin{pmatrix} 1 & 0 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 5 & -1 \end{pmatrix}$, the entries of $A \circ B$ are $1 \cdot 3$, $0 \cdot 5$ and $2 \cdot (-1)$, so $A \circ B = \begin{pmatrix} 3 & 0 & -2 \end{pmatrix}$.
Non-example: a $2 \times 2$ matrix and a $2 \times 3$ matrix have no Hadamard product, because the two matrices must have the same size.
The Hadamard product is commutative, associative and distributive over addition.
Its identity element is the $m \times n$ matrix $J$ with all entries $1$, not the identity matrix.
Other notations are $A \odot B$ and $A * B$.
</document>

<document source="https://tamnd.github.io/math2/node/Z2Z9BE946H.md" tag="Z2Z9BE946H" kind="definition">
# Osculating circle

## Definition

Let $\gamma \colon I \to \R^2$ be a [[regular|06JMR2XWXW]] curve of class $C^2$ on an interval $I$, and let $t_0 \in I$.
Let $T$ be its unit [[tangent vector|0D98771SM0]], let $N$ be $T$ rotated by a right angle counterclockwise, and let $k$ be its signed [[curvature|7JQS6VKFEZ]].
Assume that $k(t_0) \ne 0$.
The *centre of curvature* at $t_0$ is $c = \gamma(t_0) + N(t_0) / k(t_0)$, and the *radius of curvature* is $1 / \abs{k(t_0)}$.
The *osculating circle* of $\gamma$ at $t_0$ is the [[circle|045DN06F3S]] with centre $c$ and radius $1 / \abs{k(t_0)}$.

## Remarks

Example: for $\gamma(t) = (t, t^2)$ at $t_0 = 0$, one has $T = (1, 0)$, $N = (0, 1)$ and $k(0) = 2$.
So the osculating circle has centre $(0, 1/2)$ and radius $1/2$.
Non-example: for $\gamma(t) = (t, t^3)$ at $t_0 = 0$, the curvature is $0$, so there is no osculating circle there; the tangent line plays its role.
The osculating circle passes through $\gamma(t_0)$, has the same tangent there, and is the only circle with contact of order at least $2$ with the curve.
For a curve in $\R^3$ with $\kappa(t_0) > 0$, the osculating circle is defined in the same way, with $N$ the principal normal.
It lies in the osculating plane, spanned by $T$ and $N$.
</document>

<document source="https://tamnd.github.io/math2/node/Z344RY5XX1.md" tag="Z344RY5XX1" kind="concept">
# Positive cone

## Summary

A positive cone in a real [[vector space|000AYPA9AJ]] is a subset closed under addition and under multiplication by nonnegative scalars.
It must also contain no line through the origin.
Positive cones are the same thing as compatible partial orders: the cone is the set of elements that are at least $0$.
So one can describe an order on a vector space just by saying which vectors are positive.
The same name is used for the set of nonnegative elements of an ordered group, ring or field.

## Explanations

### L1 human

In the plane, take the quarter plane of points with both coordinates nonnegative.
Adding two such points, or stretching one by a nonnegative factor, gives another such point.
The quarter plane contains no full line through the origin.
Saying that $u \le v$ when $v - u$ lies in the quarter plane gives the componentwise order.
</document>

<document source="https://tamnd.github.io/math2/node/Z35TD4ZZRD.md" tag="Z35TD4ZZRD" kind="statement">
# Littlewood's Tauberian theorem

## Statement

Let $(a_n)_{n \ge 0}$ be complex numbers such that $\sum_{n \ge 0} a_n x^n$ converges for $0 \le x < 1$.
Assume that the series $\sum a_n$ is [[Abel summable|EZXPY4A19C]] to $s$, that is, $\sum_{n \ge 0} a_n x^n \to s$ as $x \to 1^-$.
Assume also that $a_n = O(1/n)$ in [[big O notation|0RYTGBJAGN]].
Then $\sum_{n \ge 0} a_n$ [[converges|0CVVKY7GY2]] and its sum is $s$.

## Remarks

Littlewood's original proof was long.
Karamata found a short proof.
It first controls the sums $\sum a_n g(x^n)$ for polynomials $g$ and then approximates the indicator function of an interval from above and below by polynomials, which isolates the partial sum $s_N$.
The same method proves the [[Hardy–Littlewood Tauberian theorem|N42JDBT0VA]].

The Tauberian condition cannot be dropped.
For $a_n = (-1)^n$, $\sum a_n x^n = \frac{1}{1+x} \to \frac12$, but $\sum (-1)^n$ diverges; here $n a_n$ is unbounded.

## Explanations

### L1 human

Abel summation damps the $n$-th term by $x^n$ and lets $x \to 1$.
It can give a value to a divergent series.
Littlewood's theorem says that this cannot happen when the terms shrink at least like $1/n$: then the Abel value is the true sum.

### L2 human

The theorem strengthens [[Tauber's theorem|8SK3GQ4S0M]], where the condition is $n a_n \to 0$, and it is the Abel analogue of [[Hardy's Tauberian theorem for Cesàro means|BWWZB1F0HY]].
Because Abel summation is stronger than Cesàro summation, it implies Hardy's theorem.
Littlewood's theorem was a starting point of the systematic study of Tauberian theorems by Hardy and Littlewood.
</document>

<document source="https://tamnd.github.io/math2/node/Z367AGJRWH.md" tag="Z367AGJRWH" kind="definition">
# Lattice in a Lie group

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] and let $\Gamma$ be a [[discrete subgroup|5M7D1BSHVZ]] of $G$.
The group $G$ acts on the coset space $G/\Gamma$ by left multiplication.
The subgroup $\Gamma$ is a *lattice* in $G$ if there is a nonzero finite Borel measure on $G/\Gamma$ that is invariant under this action.
A lattice $\Gamma$ is *uniform*, or *cocompact*, if $G/\Gamma$ is compact, and *non-uniform* otherwise.

## Remarks

When such a measure exists, it is unique up to a positive factor and is induced by a [[Haar measure|VY0HXD3GC4]] $\mu$ of $G$.
The total mass of $G/\Gamma$ for the measure induced by $\mu$ is the *covolume* of $\Gamma$.
A Lie group that contains a lattice is unimodular.
Example: $\Z^n$ is a uniform lattice in $\R^n$, because $\R^n/\Z^n$ is a compact torus whose Lebesgue measure is finite and invariant under translations.
Example: $\mathrm{SL}_2(\Z)$ is a non-uniform lattice in $\mathrm{SL}_2(\R)$; the modular surface has finite hyperbolic area $\pi/3$ but is not compact.
Non-example: $\Z \times \set{0}$ is discrete in $\R^2$, but $\R^2/(\Z \times \set{0})$ is an infinite cylinder, and any invariant measure is a multiple of its infinite area measure.
Non-example: the subgroup $\Z + \sqrt{2}\,\Z$ of $\R$ is dense, not discrete, so it is not a lattice.
</document>

<document source="https://tamnd.github.io/math2/node/Z38TW6347X.md" tag="Z38TW6347X" kind="definition">
# Spectrum

## Definition

A *spectrum* $E$ is a sequence $(E_n)_{n \ge 0}$ of [[pointed spaces|T9DG8DDE75]] together with pointed maps
$$\sigma_n \colon \Sigma E_n \to E_{n+1}, \qquad n \ge 0,$$
called structure maps, where $\Sigma$ is the reduced [[suspension|2VD7GR64JJ]].
A *map of spectra* $f \colon E \to F$ is a sequence of pointed maps $f_n \colon E_n \to F_n$ with $f_{n+1} \circ \sigma_n = \sigma_n \circ \Sigma f_n$ for all $n$.
For $k \in \Z$, the $k$-th *homotopy group* of $E$ is
$$\pi_k(E) = \operatorname{colim}_n \, \pi_{k+n}(E_n),$$
where the colimit runs over $n \ge \max(0, 2 - k)$ along the maps $\pi_{k+n}(E_n) \to \pi_{k+n+1}(\Sigma E_n) \to \pi_{k+n+1}(E_{n+1})$.

## Remarks

A spectrum is an *$\Omega$-spectrum* if each adjoint map $E_n \to \Omega E_{n+1}$ into the [[loop space|2JFF5F3EGF]] is a [[weak homotopy equivalence|4Q3WY1RX91]].
Then $\pi_k(E) = \pi_{k+n}(E_n)$ for all $n \ge 0$ with $k + n \ge 0$.
A map of spectra is a *stable weak equivalence* if it induces isomorphisms on all $\pi_k$.

Example: for a pointed space $X$, the *suspension spectrum* $\Sigma^\infty X$ has $(\Sigma^\infty X)_n = \Sigma^n X$ and identity structure maps.
Its homotopy groups are the stable homotopy groups $\pi_k^s(X)$ for $k \ge 0$, and they vanish for $k < 0$.

Example: for an abelian group $G$, choose [[Eilenberg–MacLane spaces|AJHEYJE5XY]] $K(G, n)$ and weak equivalences $K(G, n) \to \Omega K(G, n+1)$; their adjoints make $(K(G, n))_n$ an $\Omega$-spectrum $HG$ with $\pi_0(HG) = G$ and $\pi_k(HG) = 0$ for $k \neq 0$.

Non-example: a sequence of pointed spaces alone is not a spectrum, since the structure maps are part of the data.
For instance, $E_n = S^n$ with every $\sigma_n$ constant is a spectrum, but it is not the sphere spectrum.
All its homotopy groups are zero, because the maps in the colimit are zero.

An $\Omega$-spectrum $E$ defines a generalized cohomology theory by $E^n(X) = [X, E_n]_*$ for $n \ge 0$, extended to all $n \in \Z$ by suspension.
Other models, such as symmetric spectra and orthogonal spectra, add structure to get a well-behaved smash product.
</document>

<document source="https://tamnd.github.io/math2/node/Z3AY2NEVW3.md" tag="Z3AY2NEVW3" kind="definition">
# Plane curve

## Definition

Let $k$ be a field.
A *projective plane curve* over $k$ is the closed subscheme $C = V(F)$ of the [[projective plane|ZT9A6JK8QF]] $\mathbb{P}^2_k$ defined by a nonconstant [[homogeneous polynomial|2C3PAQ7H50]] $F \in k[x, y, z]$.
Its *degree* is $\deg F$.
An *affine plane curve* is the closed subscheme $V(f) \subset \mathbb{A}^2_k$ defined by a nonconstant polynomial $f$ in the [[polynomial ring|0SDC934R52]] $k[x, y]$.

## Remarks

The polynomial is determined by $C$ up to a nonzero scalar factor.
A plane curve is an [[algebraic curve|XJM2MT4S3Y]]: it has dimension $1$.
It is reduced exactly when $F$ has no repeated irreducible factor, and integral exactly when $F$ is irreducible.
Example: $F = x^2 + y^2 - z^2$ defines a conic of degree $2$ in $\mathbb{P}^2$, the projective closure of the circle $x^2 + y^2 = 1$.
Non-example: the twisted cubic $\set{[s^3 : s^2 t : s t^2 : t^3]} \subset \mathbb{P}^3$ is a curve in $\mathbb{P}^3$ that lies in no plane of $\mathbb{P}^3$, so it is not a plane curve inside $\mathbb{P}^3$.
As an abstract curve it is isomorphic to $\mathbb{P}^1$, which is a line in $\mathbb{P}^2$; being a plane curve is a property of the embedding.
A nonzero linear form vanishing on it would give a linear relation among $s^3, s^2 t, s t^2, t^3$, and there is none.
</document>

<document source="https://tamnd.github.io/math2/node/Z3AYQ5M633.md" tag="Z3AYQ5M633" kind="definition">
# Inverse tangent integral

## Definition

The *inverse tangent integral* is the function $\operatorname{Ti}_2 \colon \R \to \R$ given by
$$\operatorname{Ti}_2(x) = \int_0^x \frac{\arctan t}{t} \, dt.$$
Here $\arctan$ is the [[arctangent|0CE768XNVP]], and the integrand has the value $1$ at $t = 0$.

## Remarks

For $\abs{x} \le 1$, integrating the series of $\arctan t / t$ term by term gives
$$\operatorname{Ti}_2(x) = \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k+1}}{(2k+1)^2}.$$
The function is odd, and $\operatorname{Ti}_2(x) = \operatorname{Im} \operatorname{Li}_2(ix)$ for real $x$, where $\operatorname{Li}_2$ is the [[dilogarithm|G0X0NWSDB7]].

Example: $\operatorname{Ti}_2(1) = 1 - \frac19 + \frac1{25} - \cdots = G$, Catalan's constant.

Inversion check: for $x > 0$, the derivative of $\operatorname{Ti}_2(x) - \operatorname{Ti}_2(1/x)$ is $\frac{\arctan x + \arctan(1/x)}{x} = \frac{\pi}{2x}$, and the difference is $0$ at $x = 1$, so
$$\operatorname{Ti}_2(x) - \operatorname{Ti}_2\left(\tfrac1x\right) = \frac{\pi}{2} \log x.$$

Non-example: $\int_0^x \frac{\arctan t}{t^2} \, dt$ is not a variant of this function; it diverges at $t = 0$, because $\arctan t / t^2 \sim 1/t$ there.
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<document source="https://tamnd.github.io/math2/node/Z3C8B6QTS6.md" tag="Z3C8B6QTS6" kind="concept">
# Weierstrass equation

## Summary

A Weierstrass equation is a cubic equation $y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$ with coefficients in a field.
When its discriminant is nonzero, it defines an [[elliptic curve|427N6S6QQW]], with the point at infinity as the origin.
Every elliptic curve can be written in this form, so Weierstrass equations are the standard way to compute with elliptic curves.
The name comes from the differential equation of the [[Weierstrass elliptic function|361NAK68QK]], which yields such an equation over $\C$.

## Explanations

### L1 human

A Weierstrass equation says that $y^2$, up to some correction terms, equals a cubic polynomial in $x$.
The simplest case is $y^2 = x^3 + ax + b$.
Over the real numbers its graph is symmetric about the $x$-axis, and when it is smooth it has one or two pieces.
For example, $y^2 = x^3 - x$ is a smooth curve, while $y^2 = x^3$ has a sharp cusp at the origin and does not count as an elliptic curve.
</document>

<document source="https://tamnd.github.io/math2/node/Z3DE4R4RET.md" tag="Z3DE4R4RET" kind="concept">
# Algebraic logic

## Summary

Algebraic logic studies logical systems by means of algebraic structures.
Formulas are identified when they are provably equivalent, and the logical connectives become operations on the resulting classes, as in the [[Lindenbaum–Tarski algebra|QC9M3Q8B8Q]].
[[Classical propositional logic|09K9MWD2QQ]] corresponds to [[Boolean algebras|0XADJKM699]], [[intuitionistic logic|Q11N8X99FK]] to [[Heyting algebras|NG943XNQWH]], and Łukasiewicz infinite-valued logic to [[MV-algebras|8T4M7RN9AC]].
For [[predicate logic|086QKPF0A6]] there are [[cylindric algebras|1MDT9BF81H]] and polyadic algebras, and [[relation algebras|GQK4DSA11K]] treat the calculus of binary relations.

## Explanations

### L1 human

In school algebra one calculates with numbers by rules such as $a + b = b + a$.
Algebraic logic calculates with statements in the same way: "$p$ and $q$" equals "$q$ and $p$", and "not not $p$" equals "$p$" in classical logic.
The statements, up to logical equivalence, form an algebra, and questions about the logic become questions about that algebra.
For example, a formula is a theorem of classical propositional logic exactly when it takes the value $1$ in every Boolean algebra under every assignment of its variables.
</document>

<document source="https://tamnd.github.io/math2/node/Z3FCJZF6WK.md" tag="Z3FCJZF6WK" kind="concept">
# Aperiodic set of prototiles

## Summary

An aperiodic set of prototiles is a finite set of shapes that can [[tile|GKJK293DXS]] the plane, but only in ways that never repeat.
No tiling by these shapes is [[periodic|WFRGSBFZGV]].
Such sets show that local rules can force global order without periodicity, a phenomenon related to quasicrystals.

## Explanations

### L1 human

Most shapes that tile the floor, such as squares, can be laid out in a repeating pattern.
An aperiodic set is a set of tiles that do cover the whole plane, but any covering by them is forced to be non-repeating.
The [[Penrose tiles|MBE83FTRBG]], two shapes with marked edges, are the best known example.
In 2023 a single tile with this property was found, which answered the [[einstein problem|F456KQGPFV]].
</document>

<document source="https://tamnd.github.io/math2/node/Z3KPTM2YWX.md" tag="Z3KPTM2YWX" kind="definition">
# Henselian field

## Definition

Let $K$ be a field with a [[valuation|B85T5VKR3M]] $v$.
Let $\mathcal{O} = \set{x \in K : v(x) \ge 0}$ be its [[valuation ring|QSX12D8RQM]] and $\mathfrak{m} = \set{x \in K : v(x) > 0}$ its [[maximal ideal|09J6KGT65G]].
The valued field $(K, v)$ is *Henselian* if the following holds.
For every monic $f \in \mathcal{O}[X]$ and every $a \in \mathcal{O}$ with $f(a) \in \mathfrak{m}$ and $f'(a) \notin \mathfrak{m}$, there is $b \in \mathcal{O}$ with $f(b) = 0$ and $b - a \in \mathfrak{m}$.

Here $f'$ is the formal [[derivative|0AD4VS02H3]] of $f$.

## Remarks

Example: for a prime $p$, the field $\Q_p$ with the $p$-adic valuation is Henselian; this is [[Hensel's lemma|010QVB71T2]] in its $p$-adic form.
More generally, a field that is complete for a valuation with values in $\R$ is Henselian.
Non-example: $\Q$ with the $7$-adic valuation $v_7$ is not Henselian.
Take $f = X^2 - 2$ and $a = 3$: $f(3) = 7 \in \mathfrak{m}$ and $f'(3) = 6 \notin \mathfrak{m}$, but $f$ has no root in $\Q$.
Equivalent forms: $(K, v)$ is Henselian if and only if $\mathcal{O}$ is a [[Henselian ring|RNXPV3KC5J]].
It is Henselian if and only if $v$ extends to a unique valuation on each algebraic extension of $K$.
The element $b$ is then unique.
Every field with the trivial valuation is Henselian.
</document>

<document source="https://tamnd.github.io/math2/node/Z3MQ1D5Q38.md" tag="Z3MQ1D5Q38" kind="definition">
# Pushout

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]], and let $f \colon Z \to X$ and $g \colon Z \to Y$ be [[morphisms|859F0TFVA3]] of $\mathcal{C}$.
A *pushout* of $f$ and $g$ is an object $P$ with morphisms $i_1 \colon X \to P$ and $i_2 \colon Y \to P$ such that:

1. $i_1 \circ f = i_2 \circ g$.
2. For every object $Q$ and morphisms $j_1 \colon X \to Q$ and $j_2 \colon Y \to Q$ with $j_1 \circ f = j_2 \circ g$, there is a unique morphism $u \colon P \to Q$ with $u \circ i_1 = j_1$ and $u \circ i_2 = j_2$.

## Remarks

Equivalently, $(P, i_1, i_2)$ is a [[colimit|BJS1CGTKQ5]] of the diagram $X \xleftarrow{f} Z \xrightarrow{g} Y$.
A pushout is unique up to a unique isomorphism, and one writes $P = X \sqcup_Z Y$ or $X +_Z Y$.

Example: in the category of sets, let $\sim$ be the smallest equivalence relation on $X \sqcup Y$ with $f(z) \sim g(z)$ for all $z \in Z$.
Let $P = (X \sqcup Y)/{\sim}$, and let $i_1, i_2$ send an element to its class.
Clause 1 holds by the choice of the relation.
For clause 2, the map $u$ must send the class of $x$ to $j_1(x)$ and the class of $y$ to $j_2(y)$; this is well defined because $j_1 \circ f = j_2 \circ g$.

Non-example: take $P = X \sqcup Y$ with the two inclusions and no identification.
If $Z$ is not empty, then $i_1(f(z))$ and $i_2(g(z))$ lie in different parts, so clause 1 fails.

In the category of groups the pushout is the amalgamated free product $X *_Z Y$.
</document>

<document source="https://tamnd.github.io/math2/node/Z3VRMW8ETQ.md" tag="Z3VRMW8ETQ" kind="statement">
# Myers–Steenrod theorem

## Statement

Let $(M, g)$ and $(N, h)$ be connected [[Riemannian manifolds|7CBT4SM83Y]] with [[Riemannian distances|B4QR306D8G]] $d_g$ and $d_h$.

1. If $\varphi \colon M \to N$ is a surjective map with $d_h(\varphi(p), \varphi(q)) = d_g(p, q)$ for all $p, q \in M$, then $\varphi$ is smooth and is an [[isometry of Riemannian manifolds|R9YWQS15AA]].
2. The group $\operatorname{Isom}(M, g)$ of all isometries of $(M, g)$ onto itself, with the compact-open topology, is a [[Lie group|7JBRGCVJRB]], and its action on $M$ is smooth.
3. If $M$ is [[compact|0TVJZYB6SH]], then $\operatorname{Isom}(M, g)$ is compact.

## Remarks

The idea of part 1 is that a distance-preserving map sends minimizing geodesics to minimizing geodesics, so in normal coordinates it acts as a linear isometry, and this forces smoothness.
For part 2, note that an isometry of a connected Riemannian manifold is determined by its value and its differential at one point.
So $\operatorname{Isom}(M, g)$ embeds in the orthonormal frame bundle of $M$, and one shows that the image is a closed submanifold.
For the round sphere $S^n$, the theorem gives $\operatorname{Isom}(S^n) = O(n + 1)$, a compact Lie group of dimension $n(n + 1)/2$.
The theorem is about Riemannian (positive definite) metrics; a pseudo-Riemannian metric defines no distance function.

## Explanations

### L1 human

If a map between two curved spaces keeps all distances, it automatically keeps the whole smooth geometry, with no further hypothesis.
Also, the symmetries of a curved space form a group that behaves like a group of matrices, with a finite number of independent directions of motion.

### L2 human

Part 1 lets one pass freely between the metric-space view of a Riemannian manifold and its smooth view.
Part 2 is the starting point for the study of homogeneous and symmetric spaces.
A Riemannian manifold whose isometry group acts transitively is $G/H$, with $G$ a Lie group and $H$ a compact subgroup.
The Lie algebra of $\operatorname{Isom}(M, g)$ is identified with the complete [[Killing vector fields|5AKNNW3W4Z]] of $M$.
</document>

<document source="https://tamnd.github.io/math2/node/Z45HQ5VNF4.md" tag="Z45HQ5VNF4" kind="definition">
# Cubic Hermite spline

## Definition

Let $x_0 < x_1 < \cdots < x_n$ be real nodes, and let $p_0, \dots, p_n$ and $m_0, \dots, m_n$ be real numbers.
Let $h_i = x_{i+1} - x_i$.
The *cubic Hermite spline* for these data is the function $s \colon [x_0, x_n] \to \R$ given on each $[x_i, x_{i+1}]$ by
$$s(x) = h_{00}(u)\, p_i + h_{10}(u)\, h_i m_i + h_{01}(u)\, p_{i+1} + h_{11}(u)\, h_i m_{i+1},$$
where $u = (x - x_i)/h_i$ and
$$h_{00}(u) = 2u^3 - 3u^2 + 1, \quad h_{10}(u) = u^3 - 2u^2 + u, \quad h_{01}(u) = -2u^3 + 3u^2, \quad h_{11}(u) = u^3 - u^2.$$

## Remarks

The piece on $[x_i, x_{i+1}]$ is the unique cubic with $s(x_i) = p_i$, $s'(x_i) = m_i$, $s(x_{i+1}) = p_{i+1}$ and $s'(x_{i+1}) = m_{i+1}$, that is, a [[Hermite interpolant|9Z50M2WPTP]].
Check: $h_{00}(0) = 1$, $h_{00}(1) = 0$, $h_{01}(1) = 1$, $h_{10}'(0) = 1$, $h_{11}'(1) = 1$, and the other values and slopes at $0$ and $1$ vanish.
So $s$ is a [[spline|84TJ8KQ926]] of class [[$C^1$|0VW90BBSJ4]], but in general not of class $C^2$.
Example: nodes $0, 1$ with $p = (0, 1)$ and $m = (0, 0)$ give $s(x) = 3x^2 - 2x^3$.
Non-example: the piecewise linear interpolant of the same values is not a cubic Hermite spline for these slopes, since its slope on $[0, 1]$ is $1$, not $0$.
When the slopes are not given, they are chosen from the values, for instance by finite differences (Catmull–Rom spline) or by monotonicity-preserving rules.
</document>

<document source="https://tamnd.github.io/math2/node/Z45TBD32BR.md" tag="Z45TBD32BR" kind="definition">
# Exact category in the sense of Barr

## Definition

Let $\mathcal{C}$ be a finitely complete category and $X$ an object of $\mathcal{C}$.
An *internal equivalence relation* on $X$ is a pair $r_1, r_2 \colon R \to X$ with this property.
For every object $T$, the set $\set{(r_1 \circ t, r_2 \circ t) : t \colon T \to R}$ is an [[equivalence relation|0G8VEZ3WTK]] on the set $\mathcal{C}(T, X)$, and $t$ is determined by this pair.
The relation is *effective* if it is the [[kernel pair|E70DQGM8X5]] of some morphism $f \colon X \to Y$.
The category $\mathcal{C}$ is *exact in the sense of Barr* if both conditions hold.

1. $\mathcal{C}$ is a [[regular category|2SM3JRJXA4]].
2. Every internal equivalence relation in $\mathcal{C}$ is effective.

## Remarks

The second part of the first clause says that $(r_1, r_2) \colon R \to X \times X$ is a [[monomorphism|Y2NP1JFR86]].
The kernel pair of any morphism is always an internal equivalence relation, so clause 2 asks for the converse.

Example: the category of sets is Barr-exact.
An equivalence relation $R \subseteq X \times X$ is the kernel pair of the quotient map $X \to X / R$.

Non-example: the category of torsion-free abelian groups.
In $\Z$, the subgroup $R = \set{(a, b) : a \equiv b \pmod 2}$ of $\Z \times \Z$ is torsion-free and is an internal equivalence relation.
Let $f \colon \Z \to Y$ be a homomorphism into a torsion-free group with $f(2) = 0$.
Then $2 f(1) = 0$, so $f(1) = 0$ and the kernel pair of $f$ is all of $\Z \times \Z$.
So $R$ is not effective and clause 2 fails.
</document>

<document source="https://tamnd.github.io/math2/node/Z49RAKVCWG.md" tag="Z49RAKVCWG" kind="statement">
# Hairy ball theorem

## Statement

Let $n \ge 1$ and let $S^{2n} = \set{x \in \R^{2n+1} : \norm{x} = 1}$ be the [[sphere|T62369RFH6]] of even dimension $2n$.
Let $v \colon S^{2n} \to \R^{2n+1}$ be a continuous tangent [[vector field|0299VCW1TH]], that is, a continuous map with $v(x) \cdot x = 0$ for all $x \in S^{2n}$.
Then $v(x) = 0$ for some $x \in S^{2n}$.

## Remarks

Suppose $v$ has no zero, and put $w = v / \norm{v}$.
Then $H(x, t) = \cos(\pi t) \, x + \sin(\pi t) \, w(x)$ lies on the sphere and is a homotopy from the identity to the antipodal map.
The identity has [[degree|TMFG2P912R]] $1$ and the antipodal map of $S^{2n}$ has degree $(-1)^{2n+1} = -1$, which contradicts the homotopy invariance of degree.

Example: model the wind on the Earth as a continuous tangent vector field on the sphere $S^2$.
Then at any moment there is a point where the horizontal wind speed is zero.

The dimension must be even: on $S^{2n-1} \subset \R^{2n}$, the field $v(x_1, x_2, \ldots, x_{2n-1}, x_{2n}) = (-x_2, x_1, \ldots, -x_{2n}, x_{2n-1})$ is tangent and has no zero.

## Explanations

### L1 human

You cannot comb the hair on a ball flat without leaving a cowlick or a bald spot.
In mathematical terms, every continuous field of tangent arrows on an ordinary sphere has a point where the arrow is zero.

### L2 human

The theorem follows from the degree of the antipodal map.
It is also the special case of the [[Poincaré–Hopf theorem|4XTPY82PP4]] for $S^{2n}$, since the Euler characteristic of $S^{2n}$ is $2 \neq 0$.
It is the first case of the question how many linearly independent vector fields exist on a sphere, which Adams answered in the theorem on [[vector fields on spheres|3ES79NWRS7]].
It also implies that $S^{2n}$ carries no structure of a Lie group, since left translates of a nonzero tangent vector at the identity would give a nowhere vanishing vector field.
</document>

<document source="https://tamnd.github.io/math2/node/Z4AMJS5KA6.md" tag="Z4AMJS5KA6" kind="concept">
# Stein space

## Summary

A Stein space is a [[complex analytic space|HXSZA44J5J]] with enough global holomorphic functions: they separate points and give compact holomorphically convex hulls of compact sets.
It extends the notion of Stein manifold to spaces with singularities, as affine schemes extend affine varieties.
Stein spaces are exactly the complex spaces on which every coherent analytic sheaf has vanishing higher cohomology.
Closed complex subspaces of $\C^N$ are the main examples.

## Explanations

### L1 human

A Stein space is a space defined locally by holomorphic equations, possibly with singular points, that carries many global holomorphic functions.
The cone $\set{(x, y, z) \in \C^3 : x^2 + y^2 = z^2}$ is an example: it has a singular point at the origin, and the coordinate functions already separate its points.
Stein spaces are the setting where the theorems of Cartan and Oka about holomorphic functions work in full generality.
</document>

<document source="https://tamnd.github.io/math2/node/Z4FSES3CZ0.md" tag="Z4FSES3CZ0" kind="definition">
# Lattice (group)

## Definition

Let $n \ge 1$ and consider the [[Euclidean space|0XPR1E85W0]] $\R^n$ as a group under addition.
A *lattice* in $\R^n$ is a [[subgroup|0X1JXT47GA]] of the form
$$\Lambda = \Z v_1 + \Z v_2 + \dots + \Z v_n = \set{k_1 v_1 + \dots + k_n v_n : k_1, \dots, k_n \in \Z},$$
where $v_1, \dots, v_n$ are [[linearly independent|0CRMGPAH83]] vectors of $\R^n$.
The family $(v_1, \dots, v_n)$ is a *basis* of $\Lambda$.
The *covolume* (or *determinant*) of $\Lambda$ is $\det \Lambda = \abs{\det(v_1, \dots, v_n)}$.

## Remarks

A subgroup $\Lambda$ of $\R^n$ is a lattice if and only if it is discrete and spans $\R^n$ as a real vector space.
Two bases of the same lattice differ by an integer matrix of [[determinant|00KDZRW4C0]] $\pm 1$, so the covolume does not depend on the basis.
It equals the volume of the fundamental parallelepiped $\set{t_1 v_1 + \dots + t_n v_n : 0 \le t_i < 1}$.

Example: $\Z^n$ is a lattice with basis the standard vectors $e_1, \dots, e_n$ and covolume $1$.

Non-example: $\Z + \Z\sqrt{2} \subset \R$ is a subgroup generated by two vectors, but two vectors in $\R^1$ are never linearly independent.
In fact this subgroup is dense in $\R$, so it is not discrete.
Another non-example: $\Z \times \set{0}$ in $\R^2$ is discrete but does not span $\R^2$.
</document>

<document source="https://tamnd.github.io/math2/node/Z4H7272S9S.md" tag="Z4H7272S9S" kind="definition">
# Summation of families over arbitrary index sets

## Definition

Let $X$ be a [[normed vector space|00A0RPAQ8A]] over $\R$ or $\C$.
Let $I$ be a set, and let $(x_i)_{i \in I}$ be a family of elements of $X$.
For each [[finite subset|0BBTH3YSZA]] $J \subseteq I$ put $s_J = \sum_{i \in J} x_i$.
The family is summable with sum $s \in X$ if the following holds.
For every $\varepsilon > 0$ there is a finite set $J_0 \subseteq I$ with $\norm{s_J - s} < \varepsilon$ for every finite $J$ with $J_0 \subseteq J \subseteq I$.
Then one writes $s = \sum_{i \in I} x_i$.

## Remarks

Equivalently, the [[net|4DVJQJV0DG]] $(s_J)$, indexed by the finite subsets of $I$ ordered by inclusion, converges to $s$.
The sum is unique when it exists.
The same definition works in any Hausdorff abelian topological group, with neighbourhoods of $s$ in place of balls; see [[summation in topological groups|TXS2H001R4]].
If all $x_i$ are nonnegative reals, the family is summable if and only if the finite sums $s_J$ are bounded, and then $s = \sup_J s_J$.
Example: the family $(2^{-m} 3^{-n})_{(m, n) \in \N \times \N}$ is summable with sum $3$.
Its finite sums are bounded by $\sum_{m \le M} 2^{-m} \sum_{n \le M} 3^{-n} \le 2 \cdot \tfrac32$, and the squares $\set{0, \ldots, M}^2$ give sums tending to $3$.
Non-example: the family $((-1)^n / (n+1))_{n \in \N}$ is not summable, although the series $\sum (-1)^n/(n+1)$ converges.
Any finite set $J_0$ can be enlarged by many even indices $n$, which adds more than $1$ to $s_J$, because $\sum 1/(2k+1)$ diverges.
In a Banach space, the family is summable if and only if for every $\varepsilon > 0$ there is a finite $J_0$ with $\norm{s_K} < \varepsilon$ for every finite $K$ disjoint from $J_0$.
A summable family in a normed space has at most countably many nonzero terms.

## Explanations

### L1 human

The sum of a family is the limit of its finite partial sums, where the finite batches of terms grow in every direction at once.
This removes any dependence on the order of the terms.
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<document source="https://tamnd.github.io/math2/node/Z4JZPEBM68.md" tag="Z4JZPEBM68" kind="definition">
# Condensed set

## Definition

A *profinite set* is a [[Stone space|ZV38HA5PAQ]], that is, a compact Hausdorff totally disconnected space.
Fix an uncountable strong limit cardinal $\kappa$, and let $\mathbf{ProFin}_\kappa$ be the category of profinite sets of cardinality less than $\kappa$, with continuous maps.
A *$\kappa$-condensed set* is a [[presheaf|AND2WHZC8P]] $T \colon \mathbf{ProFin}_\kappa^{\mathrm{op}} \to \mathbf{Set}$ such that:

1. $T(\emptyset)$ is a one-point set.
2. For all $S_1, S_2$, the map $T(S_1 \sqcup S_2) \to T(S_1) \times T(S_2)$ is a bijection.
3. For each continuous surjection $p \colon S' \to S$, the map $T(p)$ is a bijection from $T(S)$ onto the [[equalizer|CTJ5AD8ST5]] of the two maps $T(S') \rightrightarrows T(S' \times_S S')$ induced by the projections.

A *condensed set* is a $\kappa$-condensed set for some such $\kappa$.
Formally, the category of condensed sets is the union (colimit) over $\kappa$ of the categories of $\kappa$-condensed sets.

## Remarks

The three conditions say exactly that $T$ is a [[sheaf|RT8A8FJ566]] for the Grothendieck topology whose covering families are the finite jointly surjective families.
The cardinal $\kappa$ only avoids set-theoretic size problems; other authors use sheaves on all compact Hausdorff spaces with a fixed universe.
Example: for a topological space $X$, the functor $\underline{X}(S) = C(S, X)$ of continuous maps is a condensed set.
A continuous map on $S_1 \sqcup S_2$ is a pair of continuous maps.
A continuous map on $S'$ that agrees on the two pullbacks descends to $S$, because a surjection of compact Hausdorff spaces is a quotient map.
Non-example: for a set $A$ with at least two points, the constant presheaf $T(S) = A$ fails clause 1, since $T(\emptyset) = A$.
The value $T(*)$ at a point is the underlying set of the condensed set.
</document>

<document source="https://tamnd.github.io/math2/node/Z4NV7VK9T9.md" tag="Z4NV7VK9T9" kind="concept">
# Little q-Jacobi polynomials

## Summary

The little $q$-Jacobi polynomials $p_n(x; a, b \mid q)$ are a family of orthogonal polynomials defined by a terminating [[basic hypergeometric series|EB4N9RQR51]].
They are orthogonal for a discrete weight carried by the points $q^k$ with $k = 0, 1, 2, \ldots$.
As $q \to 1$ with $a = q^\alpha$ and $b = q^\beta$, they tend to the [[Jacobi polynomials|TRD7DSF2KR]] in a shifted variable.
They belong to the $q$-analogue of the [[Askey scheme|RR2S05C09B]].

## Explanations

### L1 human

Ordinary Jacobi polynomials are orthogonal for an integral over an interval.
In the little $q$-Jacobi family the integral is replaced by a sum over the points $1, q, q^2, \ldots$, which pile up at $0$.
The parameter $q$ between $0$ and $1$ controls how dense these points are.
The first nontrivial member is the linear polynomial $p_1(x) = 1 - \frac{1 - abq^2}{1 - aq} x$.
Letting $q$ approach $1$ fills the gaps between the points and gives back the classical Jacobi polynomials.
</document>

<document source="https://tamnd.github.io/math2/node/Z4PGTJFGYE.md" tag="Z4PGTJFGYE" kind="definition">
# Character of a representation

## Definition

Let $G$ be a group and let $F$ be a field.
Let $\rho \colon G \to GL(V)$ be a [[representation|WYGRXCG2RN]] of $G$ on a vector space $V$ over $F$ of finite [[dimension|03GSZ8RTZJ]].
The *character* of $\rho$ is the function $\chi_\rho \colon G \to F$ given by
$$\chi_\rho(g) = \tr \rho(g),$$
where $\tr$ is the [[trace|0XC1RDBY7A]] of a linear map.
The degree of $\chi_\rho$ is $\dim V$.
When $F$ has characteristic $0$, the degree equals $\chi_\rho(1)$; in characteristic $p$ the value $\chi_\rho(1)$ is only $\dim V$ modulo $p$.
A character is irreducible if $\rho$ is [[irreducible|FX3NEFYFFN]].

## Remarks

Example: let $S_3$ act on $\C^3$ by permuting coordinates.
The matrix of a permutation $\sigma$ has a $1$ on the diagonal exactly at the points fixed by $\sigma$, so $\chi(\sigma)$ is the number of fixed points of $\sigma$.
This gives $\chi(1) = 3$, $\chi(\text{transposition}) = 1$ and $\chi(\text{3-cycle}) = 0$.
Non-example: let $\Z$ act by shifting on the space of finitely supported sequences.
This space has infinite dimension, so the trace is not defined, and the representation has no character.
Every character satisfies $\chi(hgh^{-1}) = \chi(g)$, so it is constant on each [[conjugacy class|0DEBXAJ5SA]]; such functions are called class functions.
Isomorphic representations have the same character.
For a finite group over $\C$, the converse holds as well.
</document>

<document source="https://tamnd.github.io/math2/node/Z4WVANZPK4.md" tag="Z4WVANZPK4" kind="statement">
# O'Nan–Scott theorem

## Statement

Let $G$ be a finite [[primitive permutation group|Y10S0751KG]] on a set $\Omega$.
Let $N$ be the socle of $G$, the subgroup generated by all minimal [[normal subgroups|0CB0DX5BVV]] of $G$.
Then $N \cong T^k$ for a [[simple group|0YMYMNQ0QZ]] $T$ and an integer $k \ge 1$, and $G$ is of one of the following five types.

1. Affine type: $N$ is an elementary abelian group of order $p^d$ for a prime $p$, and $N$ acts regularly on $\Omega$. Then $\Omega$ can be identified with $\F_p^d$ so that $G = N \rtimes G_0$, where $N$ is the group of translations and $G_0 \le \mathrm{GL}_d(\F_p)$ acts irreducibly on $\F_p^d$.
2. Almost simple type: $N = T$ is nonabelian, and $T \le G \le \Aut(T)$.
3. Diagonal type: $T$ is nonabelian and $k \ge 2$. The [[stabiliser|0AYMP41THN]] in $N$ of a point is a diagonal subgroup $\set{(\alpha_1(t), \dots, \alpha_k(t)) : t \in T}$, where the $\alpha_i$ are automorphisms of $T$. Then $\abs{\Omega} = \abs{T}^{k-1}$.
4. Product type: there are an integer $\ell \ge 2$ and a primitive group $H$ on a set $\Delta$, of almost simple or diagonal type, with these properties. One has $\Omega = \Delta^\ell$ and $G \le H \wr S_\ell$ in its product action on $\Delta^\ell$. The socle is $N = \operatorname{soc}(H)^\ell$.
5. Twisted wreath type: $T$ is nonabelian and $N$ acts regularly on $\Omega$; then $\abs{\Omega} = \abs{T}^k$.

## Remarks

The proof studies the minimal normal subgroups of $G$.
Each one is transitive by primitivity, and it is a direct power $T^m$ of a simple group.
There are at most two minimal normal subgroups, and the type is read off from how a point stabiliser meets $N$ and how $G$ permutes the $k$ factors of $N$.

Examples: the symmetric group $S_n$ on $n \ge 5$ points is of almost simple type with socle $A_n$.
The group of maps $x \mapsto ax + b$ with $a \ne 0$ on $\F_p$ is of affine type with socle the translations.
For $m \ge 5$, the [[wreath product|W1X984PKJ9]] $S_m \wr S_2$ acting on pairs $(i, j)$ with $1 \le i, j \le m$ is of product type.

Books group the types in different ways, and some versions list eight classes instead of five.
The theorem reduces many questions on finite primitive groups to questions on finite simple groups and their automorphism groups.

## Explanations

### L1 human

Primitive permutation groups are the indivisible pieces of all finite permutation groups.
The O'Nan–Scott theorem says that they come in only five shapes.
Each shape is described by a simple group $T$ and the way copies of $T$ sit inside the group.
After this theorem, studying primitive groups mostly means studying simple groups.

### L2 human

The socle of a primitive group is a power $T^k$ of one simple group, and the five types describe how $G$ acts around it.
The affine type is linear algebra over $\F_p$.
The almost simple type is a simple group together with some of its automorphisms.
The other three types are built from several copies of a nonabelian simple group.
Combined with the classification of finite simple groups, the theorem is a main tool in two problems.
One is to find the maximal subgroups of the symmetric and alternating groups, and the other is to bound the orders of primitive groups.
</document>

<document source="https://tamnd.github.io/math2/node/Z4ZFP6FNK2.md" tag="Z4ZFP6FNK2" kind="statement">
# Whittaker–Shannon interpolation formula

## Statement

Let $B > 0$, and let $f \in L^2(\R)$ be a continuous [[band-limited function|P3T104G564]] with bandwidth $B$, that is, its [[Fourier transform|DPVM34ADZV]] vanishes almost everywhere outside $[-B, B]$.
Let $\operatorname{sinc}(t) = \sin(\pi t)/(\pi t)$ for $t \ne 0$ and $\operatorname{sinc}(0) = 1$.
Then
$$f(x) = \sum_{k \in \Z} f\Big( \frac{k}{2B} \Big)\, \operatorname{sinc}(2Bx - k) \quad \text{for every } x \in \R,$$
where the series converges in [[$L^2(\R)$|SASP14MA1Z]] and uniformly on $\R$.

## Remarks

The proof expands $\hat f$, restricted to $[-B, B]$, in a [[Fourier series|G12KXDHPPS]] of period $2B$.
Writing $\hat f(\xi) = \sum_k c_k e^{-2\pi i k \xi/(2B)}$ on $[-B, B]$, Fourier inversion gives $c_k = \frac{1}{2B} f(\frac{k}{2B})$.
Inverting term by term, the inverse transform of $e^{-2\pi i k \xi/(2B)} \mathbf{1}_{[-B, B]}(\xi)$ is $2B \operatorname{sinc}(2Bx - k)$, which gives the formula.
The functions $\sqrt{2B}\, \operatorname{sinc}(2Bx - k)$, $k \in \Z$, form an orthonormal basis of the band-limited functions of bandwidth $B$, so the series converges in $L^2$, with $\sum_k \abs{f(k/(2B))}^2 = 2B \norm{f}_2^2$ by the [[Plancherel theorem|HPZHRZ0502]].
Uniform convergence follows from the bound $\sup \abs{g} \le \sqrt{2B} \norm{g}_2$ for band-limited $g$ of bandwidth $B$.

Example: for $f = \operatorname{sinc}$, $B = \frac12$, the samples are $f(k) = 0$ for $k \ne 0$ and $f(0) = 1$, and the formula reduces to $f(x) = \operatorname{sinc}(x)$.

The sample spacing $1/(2B)$ cannot be enlarged; if the band is wider than $[-B, B]$, different functions can have the same samples, which is aliasing.

## Explanations

### L1 human

A signal with no frequencies above $B$ can be rebuilt exactly from its values at the times $k/(2B)$.
Each sample is replaced by a sinc pulse centred at its time and scaled by its value, and the pulses are added.
Each pulse is $1$ at its own sample time and $0$ at all the others, so the sum has the right values at the sample points.
The band limit forces it to be right everywhere in between.

### L2 human

This is the explicit form of the sampling theorem.
It is also the expansion of $f$ in an orthonormal basis of the Paley–Wiener space, made of translates of one sinc function.
In practice the slow decay of sinc makes truncation errors large, so oversampling and smoother interpolation kernels are used.
</document>

<document source="https://tamnd.github.io/math2/node/Z4ZM26BJHN.md" tag="Z4ZM26BJHN" kind="definition">
# Abstract Wiener space

## Definition

Let $H$ be a real separable [[Hilbert space|HM2HHE3J51]] with inner product $\ip{\cdot}{\cdot}$ and norm $\norm{\cdot}_H$.
Let $E$ be a real [[separable|0Z2P41SVDP]] [[Banach space|06GZ956657]] with dual space $E^*$.
Let $i \colon H \to E$ be an injective [[continuous linear map|03TM135G0N]] with dense image.
For $\ell \in E^*$, let $i^* \ell \in H$ be the vector with $\ip{i^* \ell}{h} = \ell(i h)$ for all $h \in H$.
The triple $(i, H, E)$ is an *abstract Wiener space* if there is a Borel [[probability measure|059TJN0NZ1]] $\mu$ on $E$ with the following property.
For every $\ell \in E^*$, the [[pushforward|3N66TM3TSQ]] $\ell_* \mu$ is the centered [[normal distribution|05AQE0QR0D]] with variance $\norm{i^* \ell}_H^2$.

## Remarks

The measure $\mu$ is unique, and it is called the abstract Wiener measure; it is a centered [[Gaussian measure|P0DV2TCH1N]].
Equivalently, the image under $i$ of the canonical Gaussian [[cylinder set measure|G95YDNFXQN]] of $H$ extends to a countably additive measure on $E$.
Conversely, every centered Gaussian measure on a separable Banach space arises in this way, with $H$ its Cameron–Martin space.

Example: let $E = C_0([0, 1])$ be the [[classical Wiener space|5Z00K7K1HY]] with the sup norm.
Let $H$ be the space of absolutely continuous $h \in E$ with $h' \in L^2$, with $\ip{h}{k} = \int_0^1 h' k' \, dt$, and let $i$ be the inclusion.
With $\mu$ the Wiener measure, take $\ell(\omega) = \omega(t)$.
Then $i^* \ell$ is $s \mapsto \min(s, t)$, whose norm squared is $t$, and indeed $\omega(t)$ has law $N(0, t)$.

Non-example: let $E = H$ be infinite-dimensional and $i$ the identity.
For an orthonormal sequence $(e_k)$, the readings $\ip{x}{e_k}$ would be independent standard normal variables under $\mu$.
Then $\sum_k \ip{x}{e_k}^2 = \infty$ almost surely, which is impossible for $x \in H$, so no such $\mu$ exists.

## Explanations

### L1 human

The Hilbert space fixes the shape of the Gaussian, and the Banach space is large enough to carry the actual measure.
Wiener measure on continuous paths is the model case.
</document>

<document source="https://tamnd.github.io/math2/node/Z50EV9SYNJ.md" tag="Z50EV9SYNJ" kind="definition">
# Primitive ideal

## Definition

Let $R$ be a [[ring|0894J980RK]] with identity.
A [[two-sided ideal|0DDA6CY7GJ]] $P$ of $R$ is a *left primitive ideal* if there is a [[simple|G00C22AEDH]] left $R$-[[module|09VDS2W1E8]] $M$ with
$$P = \operatorname{ann}_R(M) = \set{r \in R : rm = 0 \text{ for all } m \in M}.$$

## Remarks

Equivalently, $P$ is left primitive if and only if the quotient ring $R/P$ is a left [[primitive ring|8F6EHDVJ62]].
Example: in $\Z$, for a prime $p$ the ideal $p\Z$ is the annihilator of the simple module $\Z/p\Z$, so it is primitive.
Non-example: the zero ideal of $\Z$ is prime but not primitive, since $\Z/0 \cong \Z$ is not a field.
In a commutative ring the primitive ideals are exactly the maximal ideals.
The [[Jacobson radical|XA0DCNH7WY]] $J(R)$ is the intersection of all left primitive ideals of $R$.
Right primitive ideals are defined with right modules; the two notions can differ.
</document>

<document source="https://tamnd.github.io/math2/node/Z57CG6W9S6.md" tag="Z57CG6W9S6" kind="example">
# Doob martingale

## Example

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] with a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_n)_{n \ge 0}$, and let $X$ be a real random variable with $E[\abs{X}] < \infty$.
For $n \ge 0$ let
$$M_n = E[X \mid \mathcal{F}_n],$$
the [[conditional expectation|0YAMJRK04E]] of $X$ given $\mathcal{F}_n$.
Then $(M_n)_{n \ge 0}$ is a [[martingale|0CDVK9RQBH]], the Doob martingale of $X$.

Check of the three clauses:

1. $M_n$ is $\mathcal{F}_n$-measurable by the definition of conditional expectation.
2. $E[\abs{M_n}] \le E\bigl[E[\abs{X} \mid \mathcal{F}_n]\bigr] = E[\abs{X}] < \infty$.
3. Since $\mathcal{F}_n \subseteq \mathcal{F}_{n+1}$, the tower property gives $E[M_{n+1} \mid \mathcal{F}_n] = E\bigl[E[X \mid \mathcal{F}_{n+1}] \mid \mathcal{F}_n\bigr] = E[X \mid \mathcal{F}_n] = M_n$.

## Remarks

A Doob martingale is uniformly integrable, and by [[Lévy's upward theorem|W01FD2A8NK]] it converges almost surely and in $L^1$ to $E[X \mid \mathcal{F}_\infty]$, where $\mathcal{F}_\infty$ is the σ-algebra generated by all $\mathcal{F}_n$.
In concentration of measure one takes $X = f(Y_1, \dots, Y_N)$ for independent $Y_i$ and $\mathcal{F}_n = \sigma(Y_1, \dots, Y_n)$.
Then $M_0 = E[X]$ and $M_N = X$, so $X - E[X]$ is a sum of martingale differences, and [[Azuma's inequality|BDA5TTB1F0]] gives [[McDiarmid's inequality|8APFT1RPM4]].

## Explanations

### L1 human

Suppose a random quantity $X$ is revealed bit by bit over time.
At each step, $M_n$ is your best guess of $X$ given what you have seen so far.
Your guess changes as information arrives, but you cannot predict in which direction it will change next.
That is exactly the fair-game property of a martingale.
</document>

<document source="https://tamnd.github.io/math2/node/Z58FRZCF8P.md" tag="Z58FRZCF8P" kind="concept">
# Band matrix

## Summary

A band matrix is a matrix whose nonzero entries all lie in a band around the main diagonal.
The widths of the band below and above the diagonal are the lower and upper bandwidths.
[[Diagonal|044JNC7ZAA]], [[tridiagonal|DARX1KZRHZ]] and [[triangular|GTG8MJGDKC]] matrices are special band matrices.
Algorithms such as Gaussian elimination without pivoting keep the band, so systems with narrow band matrices are cheap to store and to solve.

## Explanations

### L1 human

Many problems link each unknown only to a few nearby unknowns, such as points on a line or a grid.
The matrix of such a problem has zeros everywhere except in a strip along the diagonal.
For example, a $5 \times 5$ matrix with nonzero entries only where the row and column numbers differ by at most $2$ is a band matrix of bandwidth $2$.
One only needs to store the strip, which saves memory and time when the matrix is large.
</document>

<document source="https://tamnd.github.io/math2/node/Z5CCVBME2V.md" tag="Z5CCVBME2V" kind="concept">
# Oracle machine

## Summary

An oracle machine is a [[Turing machine|758EYCN27B]] with an extra device, the oracle, that answers questions of the form "is $n$ in the set $O$?" in a single step.
The set $O$ can be any set of natural numbers, even an uncomputable one.
Oracle machines define relative computability: what can be computed if some set is given for free.
They are used to define Turing reduction, Turing degrees and the Turing jump.

## Explanations

### L1 human

An oracle machine is an ordinary computer program with a phone line to an all-knowing helper.
The helper knows one fixed set $O$ and answers only yes-or-no questions about it.
The program writes a number, calls the helper, and gets the answer immediately.
For example, with the set of halting programs as oracle, a program can decide whether any given program halts, which no ordinary program can do.
</document>

<document source="https://tamnd.github.io/math2/node/Z5DHRKE2Q0.md" tag="Z5DHRKE2Q0" kind="statement">
# Anderson–Kadec theorem

## Statement

Let $E$ be an infinite-dimensional [[separable|0Z2P41SVDP]] real [[Fréchet space|3GAA614673]], for example an infinite-dimensional separable [[Banach space|06GZ956657]].
Then $E$ is [[homeomorphic|0QT00PB5G2]] to the countable product $\R^{\N}$ with the product topology.
In particular, any two such spaces are homeomorphic, and each of them is homeomorphic to the separable [[Hilbert space|HM2HHE3J51]] $\ell^2$.

## Remarks

Kadec proved that all infinite-dimensional separable Banach spaces are homeomorphic, and Anderson proved that $\R^{\N}$ is homeomorphic to $\ell^2$.
The homeomorphisms are far from linear: $\ell^1$ and $\ell^2$ are homeomorphic but not isomorphic as Banach spaces.
Both hypotheses are needed.
In finite dimension, $\R^m$ and $\R^n$ are not homeomorphic for $m \ne n$, by invariance of domain, and neither is homeomorphic to $\R^{\N}$.
Without separability it fails: $\ell^\infty$ is not separable, so it is not homeomorphic to $\ell^2$.

## Explanations

### L1 human

The theorem says that, as far as topology alone is concerned, there is only one separable infinite-dimensional Fréchet space.
Norms, inner products and linear structure distinguish many such spaces, but if one only looks at open sets they all look like the product of countably many lines.
So topology cannot tell $\ell^1$ from $\ell^2$, even though linear algebra can.

### L2 human

The result is the starting point of infinite-dimensional topology.
Since all separable infinite-dimensional Fréchet spaces are homeomorphic, manifolds modelled on any of them can be compared purely topologically.
For Hilbert manifolds the smooth counterpart is the open embedding theorem.
It places every separable Hilbert manifold modelled on an infinite-dimensional separable Hilbert space inside its model space as an open set.
</document>

<document source="https://tamnd.github.io/math2/node/Z5E2GX2B81.md" tag="Z5E2GX2B81" kind="definition">
# Relatively complemented lattice

## Definition

Let $L$ be a [[lattice|WH4SFAACAS]] with [[join and meet|J2T75ATRV5]] $\vee$ and $\wedge$.
For $c \le d$ in $L$, write $[c, d] = \set{x \in L : c \le x \le d}$.
A *relative complement* of $a \in [c, d]$ in $[c, d]$ is an element $b \in [c, d]$ such that
$$a \wedge b = c \quad\text{and}\quad a \vee b = d.$$
The lattice $L$ is *relatively complemented* if for all $c \le d$ in $L$, every element of $[c, d]$ has a relative complement in $[c, d]$.

## Remarks

Equivalently, every interval $[c, d]$ of $L$ is a [[complemented lattice|ZT47BK9Y6K]] with least element $c$ and greatest element $d$.
A relatively complemented lattice with a least and a greatest element is complemented.

Example: the lattice of all subsets of a set $X$.
For $C \subseteq A \subseteq D$, the set $B = C \cup (D \setminus A)$ satisfies $A \cap B = C$ and $A \cup B = D$.

Non-example: the [[pentagon lattice N5|DHC8TZXGYS]] with $0 < a < c < 1$ and $b$ incomparable to $a$ and $c$.
It is complemented, but the interval $[0, c]$ is the chain $0 < a < c$, in which $a$ has no complement.

A complemented [[modular lattice|QX7WZ92QAK]] is relatively complemented: if $x$ is a complement of $a$ and $c \le a \le d$, then $b = c \vee (x \wedge d)$ is a relative complement of $a$ in $[c, d]$.
</document>

<document source="https://tamnd.github.io/math2/node/Z5EETTFWAN.md" tag="Z5EETTFWAN" kind="concept">
# Duality in category theory

## Summary

Duality in category theory is the correspondence between a statement about a [[category|A2DE66937D]] and the statement obtained by reversing all arrows.
A statement holds in the [[opposite category|DAMK8XE1FR]] $\mathcal{C}^{\mathrm{op}}$ exactly when its dual holds in $\mathcal{C}$.
So a theorem proved for all categories gives, for free, its dual theorem.
Pairs of dual notions include [[monomorphism|Y2NP1JFR86]] and [[epimorphism|RVXDSY04KG]], [[initial and terminal objects|EK85VSSRMR]], [[product|S5744QZV4N]] and [[coproduct|4FJ0S71P58]], and [[limit|CHS5AFJGWE]] and [[colimit|BJS1CGTKQ5]].

## Explanations

### L1 human

Many notions in category theory come in pairs that are mirror images of each other.
To pass from one to the other, turn every arrow around and read every composite in the opposite order.
For example, "every arrow out of $X$ is unique" becomes "every arrow into $X$ is unique".
The first describes an initial object, such as the empty set among sets; the second describes a terminal object, such as a one-point set.
Each proof about one notion turns into a proof about its mirror image, so each theorem comes with a second, dual theorem at no extra cost.
</document>

<document source="https://tamnd.github.io/math2/node/Z5G47MQ99B.md" tag="Z5G47MQ99B" kind="concept">
# Projective morphism

## Summary

A projective morphism $X \to Y$ is a morphism that embeds $X$ as a closed subscheme of a projective space over $Y$.
It is the relative form of a projective variety: a family of projective schemes parametrized by $Y$.
Projective morphisms are proper, and most proper morphisms met in practice are projective.

## Explanations

### L1 human

A projective variety is cut out by homogeneous polynomial equations inside projective space.
A projective morphism to $Y$ is the same thing with coefficients that vary over $Y$.
For example, the equations $y^2 z = x^3 + t x z^2 + z^3$ in $\mathbb{P}^2$, with a parameter $t$, describe a family of plane cubic curves over the $t$-line.
The map that sends each point of the family to its value of $t$ is projective.
</document>

<document source="https://tamnd.github.io/math2/node/Z5G5JPT022.md" tag="Z5G5JPT022" kind="definition">
# Chudnovsky algorithm

## Definition

The *Chudnovsky algorithm* computes approximations of [[pi|0C1VYVNV2N]] from the Chudnovsky formula
$$\frac{1}{\pi} = 12\sum_{k=0}^{\infty} \frac{(-1)^k\,(6k)!\,(13591409 + 545140134\,k)}{(3k)!\,(k!)^3\,640320^{3k + 3/2}},$$
where $n!$ is the [[factorial|0XWG0JGDFW]].
To obtain $\pi$ to $D$ decimal digits, it adds the first $N$ terms of this [[series|0AF7X08PCR]] and inverts the sum.
Here $N$ is slightly more than $D/14$.
The partial sum is evaluated by binary splitting.
The rational numbers that form the terms are combined pairwise in a balanced tree, so the work is done with large-integer products.

## Remarks

The ratio of successive terms tends to $-12^3/640320^3 \approx -6.6 \times 10^{-15}$, since $(6k)!/((3k)!(k!)^3)$ grows like a constant times $1728^k k^{-3/2}$ by Stirling's formula; this gives about $14.18$ digits per term.
Example: the $k = 0$ term gives $\pi \approx 640320^{3/2}/(12\cdot 13591409) = 3.14159265358973\ldots$, correct to $13$ decimal places.
Non-example: summing the Leibniz series $\pi/4 = 1 - \frac13 + \frac15 - \cdots$ is not this algorithm and is far slower: its error after $N$ terms is of order $1/N$.
The number $640320^3 = 262537412640768000$ is close to $e^{\pi\sqrt{163}} - 744$, which reflects the [[j-invariant|FEC2XH822R]] value $j\big(\frac{1 + \sqrt{-163}}{2}\big) = -640320^3$.
</document>

<document source="https://tamnd.github.io/math2/node/Z5H8ZZHVMC.md" tag="Z5H8ZZHVMC" kind="problem">
# Change detection

## Problem

Let $X_1, X_2, \dots$ be observed one at a time.
For an unknown change time $\nu \in \set{1, 2, \dots} \cup \set{\infty}$, the variables are independent, $X_n$ has density $f_0$ for $n < \nu$ and density $f_1$ for $n \ge \nu$, where $f_0 \ne f_1$.
Find a [[stopping time|WND87FBAKC]] $T$ of the observations that raises an alarm soon after the change and rarely before it.

Lorden's minimax form measures the delay by
$$\bar D(T) = \sup_{\nu \ge 1} \operatorname*{ess\,sup} E_\nu\left[ (T - \nu + 1)^+ \mid X_1, \dots, X_{\nu - 1} \right],$$
and the task is to minimize $\bar D(T)$ subject to $E_\infty[T] \ge \gamma$ for a given $\gamma > 1$.
Here $E_\nu$ is the [[expectation|0WQNRCNY3M]] when the change occurs at time $\nu$, and $E_\infty$ is the expectation when there is no change.

## Remarks

The problem is a central problem of [[sequential analysis|9CWYNB9003]], with roots in industrial quality control.
E. S. Page proposed the cumulative sum procedure, or CUSUM, for it in 1954.
Gary Lorden introduced the criterion above in 1971 and showed that CUSUM is asymptotically optimal as $\gamma \to \infty$.
George Moustakides proved in 1986 that CUSUM is exactly optimal for this criterion.
In the Bayesian form, studied by Albert Shiryaev, $\nu$ has a geometric prior and the expected delay is minimized subject to a bound on the probability of a false alarm.
There is also an offline form, in which the whole sample is given and the change time is to be estimated.

## Explanations

### L1 human

A machine runs normally, and at some unknown moment it starts to go wrong.
Each new measurement gives a little evidence, and a monitor must decide when to sound the alarm.
Sounding it too early stops a good machine, and sounding it too late wastes good material.
The problem is to find the best trade-off between quick detection and few false alarms.
</document>

<document source="https://tamnd.github.io/math2/node/Z5M5ESDBGK.md" tag="Z5M5ESDBGK" kind="definition">
# 3-manifold

## Definition

A *$3$-manifold* is a topological space $M$ such that:

1. $M$ is [[Hausdorff|0S1FJ8VEX9]];
2. $M$ is second countable;
3. every point of $M$ has an open neighbourhood [[homeomorphic|0QT00PB5G2]] to an open subset of $\R^3$.

Equivalently, it is a [[topological manifold|QQQ9B1DK4P]] of dimension $3$.
A *$3$-manifold with boundary* allows open subsets of the closed half-space $\set{x \in \R^3 : x_3 \ge 0}$ in clause 3, and a *closed* $3$-manifold is a compact one without boundary.

## Remarks

Example: the $3$-torus $T^3 = \R^3 / \Z^3$ is a closed $3$-manifold.
The quotient map $\R^3 \to T^3$ restricted to any open cube of side $1/2$ is a homeomorphism onto an open subset, and $T^3$ is Hausdorff, second countable and compact.

Non-example: the cone $C = (T^2 \times [0, 1]) / (T^2 \times \set{0})$ on the torus fails clause 3 at the cone point $v$.
The local homology group $H_3(C, C \setminus \set{v})$ is $\tilde H_2(T^2) \cong \Z$, which matches $\R^3$, but $H_2(C, C \setminus \set{v}) \cong \tilde H_1(T^2) \cong \Z^2$, while at a point of $\R^3$ this group is $0$.

Other standard examples are the $3$-sphere $S^3$, the [[lens spaces|AZBTMF196E]] and the [[Poincaré homology sphere|NHERCK7C43]].

## Explanations

### L1 human

The definition asks that every point have a small region around it shaped like a piece of ordinary space.
The first two clauses exclude pathological spaces.
Gluing opposite faces of a cube gives an example with no edge or corner anywhere.
A cone on a torus is not an example, because the tip of the cone does not look like a point of space.
</document>

<document source="https://tamnd.github.io/math2/node/Z5SA6A67KF.md" tag="Z5SA6A67KF" kind="concept">
# Zonotope

## Summary

A zonotope is a [[convex polytope|300K6CB1KH]] that is a [[Minkowski sum|7GK5417BJF]] of finitely many line segments.
Equivalently, it is the image of a cube under an affine map.
Zonotopes are centrally symmetric, and all of their faces are again zonotopes.
They are closely tied to [[hyperplane arrangements|N602W0WAVT]].
The face structure of a zonotope is dual to that of a central arrangement with one hyperplane for each generating segment.
Three-dimensional zonotopes are called zonohedra, and limits of zonotopes are [[zonoids|ZMZZ2QRFTZ]].

## Explanations

### L1 human

Take a few sticks pointing in different directions.
Sweep a point along the first stick, then sweep the resulting segment along the second stick, and so on.
The final shape is a zonotope.
Two sticks give a parallelogram; three sticks in the plane give a hexagon with opposite sides parallel; three sticks in space give a box-like parallelepiped.
</document>

<document source="https://tamnd.github.io/math2/node/Z5T0SEF3XA.md" tag="Z5T0SEF3XA" kind="concept">
# Proper cone

## Summary

A proper cone in $\R^n$ is a closed [[convex cone|6VE4KA12VD]] with nonempty interior that contains no line.
Proper cones define the generalized inequalities $x \preceq_K y \iff y - x \in K$ used in conic and semidefinite optimisation.
The nonnegative orthant, the second-order cone and the cone of positive semidefinite matrices are the standard examples.
The [[dual cone|QFY1JR1TK4]] of a proper cone is again proper.

## Explanations

### L1 human

A proper cone is a solid, closed wedge with a sharp tip at the origin.
Solid means it has volume, closed means it contains its boundary, and sharp means it does not contain any full line.
The closed first quadrant of the plane is a proper cone.
A closed half-plane is not, because it contains a full line through the origin.
</document>

<document source="https://tamnd.github.io/math2/node/Z5Y0VH10YR.md" tag="Z5Y0VH10YR" kind="concept">
# Bernstein–Bézier form

## Summary

The Bernstein–Bézier form writes a polynomial in the [[Bernstein basis|0JT92VK307]] adapted to an interval, or to a simplex in several variables.
Its coefficients, the Bézier coefficients, behave like control points.
The polynomial lies between their minimum and maximum, and it agrees with the end coefficients at the endpoints.
This form is numerically stable and is the standard way to store and manipulate polynomial pieces of splines and Bézier curves.

## Explanations

### L1 human

The usual way to write a polynomial uses powers $1, x, x^2, \dots$.
The Bernstein–Bézier form instead uses the blending weights $\binom{n}{k}(1-u)^{n-k}u^k$, where $u$ measures the position in the interval.
Then each coefficient has a geometric meaning, a height that the graph is pulled toward.
For example, on $[0, 1]$ the polynomial $x$ of degree $2$ has Bézier coefficients $0, 1/2, 1$, equally spaced heights along the line.
</document>

<document source="https://tamnd.github.io/math2/node/Z61W394YFA.md" tag="Z61W394YFA" kind="definition">
# Asymptotic expansion from poles of a Mellin transform

## Definition

Let $f \colon (0, \infty) \to \C$ be a function whose [[Mellin transform|N5PVGRJE1T]] $f^*(s) = \int_0^\infty f(x)\, x^{s-1}\, dx$ converges absolutely in a [[fundamental strip|1JAA21FHHQ]] $\alpha < \operatorname{Re} s < \beta$.
Let $\gamma < \alpha$, and assume that $f^*$ extends to a [[meromorphic function|0ZZ7ZYDR8W]] on the strip $\gamma < \operatorname{Re} s < \beta$ with finitely many [[poles|066E7PJCYE]] $\rho_1, \dots, \rho_m$ there.
For each pole $\rho$, write the principal part of $f^*$ at $\rho$ as
$$\sum_{k=0}^{K} \frac{c_{\rho, k}}{(s - \rho)^{k+1}}.$$
The *pole expansion of $f$ at $0$* is
$$\sum_{j=1}^{m} \sum_{k} c_{\rho_j, k}\, \frac{(-1)^k}{k!}\, x^{-\rho_j} (\log x)^k,$$
where the inner sum runs over $0 \le k \le K$ for each pole.
It is the sum of the [[residues|0PHN07KVQT]] of $f^*(s)\, x^{-s}$ at $\rho_1, \dots, \rho_m$.
The expansion at $\infty$ is defined in the same way from the poles to the right of the strip, with the opposite sign.

## Remarks

The justification is the converse mapping theorem.
Assume moreover that $f^*(s) = O(\abs{s}^{-r})$ for some $r > 1$ as $\abs{\operatorname{Im} s} \to \infty$, uniformly in the closed strip $\gamma \le \operatorname{Re} s \le c$ for some $c$ with $\alpha < c < \beta$.
Then, as $x \to 0^+$, $f(x)$ equals the sum above plus $O(x^{-\gamma})$, so this sum is an [[asymptotic expansion|BJQTEPEZ9G]] of $f$ to that order.
The proof writes $f$ by the [[Mellin inversion theorem|XJWC13NY9E]] as an integral over the line $\operatorname{Re} s = c$, moves the line to $\operatorname{Re} s = \gamma$, and collects residues.
The residue of $x^{-s} (s - \rho)^{-k-1}$ at $\rho$ is $x^{-\rho} (-\log x)^k / k!$, which gives the coefficients above.

Example: $f(x) = e^{-x}$ has $f^*(s) = \Gamma(s)$ on $\operatorname{Re} s > 0$.
The poles at $s = -n$ are simple with residue $(-1)^n / n!$, and $\Gamma$ decays exponentially on vertical lines.
So for every $N$ one gets $e^{-x} = \sum_{n=0}^{N} (-1)^n x^n / n! + O(x^{N+1/2})$ as $x \to 0^+$, as Taylor's formula confirms.

Non-example: $f(x) = 1$ for $0 < x \le 1$ and $f(x) = 0$ for $x > 1$ has $f^*(s) = 1/s$ on $\operatorname{Re} s > 0$.
The pole at $0$ gives the term $1$, which is correct for $x < 1$, but $1/s$ decays only like $\abs{s}^{-1}$, so the decay hypothesis fails.
The jump of $f$ at $x = 1$ is what spoils the decay.
</document>

<document source="https://tamnd.github.io/math2/node/Z6F9V4AHHH.md" tag="Z6F9V4AHHH" kind="definition">
# Complete elliptic integral of the first kind

## Definition

Let $k$ be a real number with $0 \le k < 1$.
The *complete elliptic integral of the first kind* is
$$K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1 - k^2 \sin^2\theta}} = \int_0^1 \frac{dt}{\sqrt{(1 - t^2)(1 - k^2 t^2)}}.$$
It is the value at $\varphi = \pi/2$ of the incomplete [[elliptic integral|NXMCYGCB59]] $F(\varphi, k)$ of the first kind.
The *complementary modulus* is $k' = \sqrt{1 - k^2}$, and one writes $K'(k) = K(k')$.

## Remarks

The second integral comes from the first by $t = \sin\theta$; it is an [[improper integral|05KQ3F99BQ]] at $t = 1$ that converges.
Example: $K(0) = \int_0^{\pi/2} d\theta = \pi/2$.
Expanding $(1 - k^2\sin^2\theta)^{-1/2}$ by the binomial series and integrating term by term gives
$$K(k) = \frac{\pi}{2} \sum_{n=0}^{\infty} \left( \frac{(2n)!}{2^{2n} (n!)^2} \right)^2 k^{2n}.$$
Non-example: the formula does not define a finite number at $k = 1$, since $\int_0^{\pi/2} d\theta / \cos\theta$ diverges; indeed $K(k) \to \infty$ as $k \to 1^-$.
Some authors use the parameter $m = k^2$ and write $K(m)$; the value is the same function written in another variable.
The [[arithmetic–geometric mean|E0V3N93DX8]] gives $K(k) = \pi / (2 M(1, k'))$.
</document>

<document source="https://tamnd.github.io/math2/node/Z6JR1GN2RE.md" tag="Z6JR1GN2RE" kind="statement">
# No-retraction theorem

## Statement

Let $n \ge 1$, let $D^n = \set{x \in \R^n : \norm{x} \le 1}$ be the [[closed unit ball|0Q0H13G8ZB]], and let $S^{n-1} = \set{x \in \R^n : \norm{x} = 1}$ be its boundary [[sphere|T62369RFH6]].
There is no [[continuous map|0VGFHTM75Y]] $r \colon D^n \to S^{n-1}$ with $r(x) = x$ for all $x \in S^{n-1}$.

## Remarks

Suppose such a retraction $r$ exists, and let $i \colon S^{n-1} \to D^n$ be the inclusion, so that $r \circ i = \id_{S^{n-1}}$.
Apply [[reduced homology|NJQMWVK3QK]] in degree $n - 1$: the identity of $\tilde{H}_{n-1}(S^{n-1}; \Z) \cong \Z$ factors through $\tilde{H}_{n-1}(D^n; \Z) = 0$, which is impossible.
For $n = 1$ the argument is simply that $D^1 = [-1, 1]$ is connected and $S^0 = \set{-1, 1}$ is not.
For smooth retractions there is also a proof by Sard's theorem: the preimage of a regular value would be a compact $1$-manifold with only one boundary point.

Example of use: the theorem gives the [[Brouwer fixed-point theorem|N7AAJ2MW5T]].
If $f \colon D^n \to D^n$ had no fixed point, the ray from $f(x)$ through $x$ would meet $S^{n-1}$ in a point $r(x)$, and $r$ would be a retraction of $D^n$ onto $S^{n-1}$.

The ball cannot be replaced by any space: the cylinder $S^{n-1} \times [0, 1]$ retracts onto its end $S^{n-1} \times \set{0}$ by $(x, t) \mapsto (x, 0)$.

## Explanations

### L1 human

You cannot push a drum skin onto its rim continuously while holding the rim fixed.
Somewhere the skin would have to tear.

### L2 human

The no-retraction theorem is equivalent to the Brouwer fixed-point theorem, and each implies the other with a short argument.
In homotopy terms, it says that the identity map of $S^{n-1}$ is not null-homotopic, and so it is a statement about the degree of the identity.
Similar retraction arguments, with homology or cohomology, show that the boundary of a compact manifold is never a retract of the manifold.
</document>

<document source="https://tamnd.github.io/math2/node/Z6MF9ZGCMW.md" tag="Z6MF9ZGCMW" kind="statement">
# Fisher–Neyman factorization theorem

## Statement

Let $\set{P_\theta : \theta \in \Theta}$ be a [[statistical model|JZ6R7J36K5]] on a measurable space $(S, \mathcal{S})$ that is dominated by a $\sigma$-finite measure $\mu$: each $P_\theta$ has a [[density|ECDJ8DD9RV]] $f_\theta = dP_\theta/d\mu$.
Let $T \colon S \to \mathcal{T}$ be a measurable map into a measurable space $\mathcal{T}$.
Then $T$ is a [[sufficient statistic|QNMR5F525C]] for $\theta$ if and only if the following holds.
There are measurable functions $g_\theta \colon \mathcal{T} \to [0, \infty)$, for $\theta \in \Theta$, and $h \colon S \to [0, \infty)$ such that, for every $\theta \in \Theta$,
$$f_\theta(x) = g_\theta\left(T(x)\right) h(x) \quad \text{for } \mu\text{-almost every } x \in S.$$

## Remarks

In the discrete case the proof is a direct computation: if the factorization holds, then $P_\theta(X = x \mid T = t) = h(x)/\sum_{y : T(y) = t} h(y)$, which does not depend on $\theta$; conversely one takes $g_\theta(t) = P_\theta(T = t)$ and $h(x) = P(X = x \mid T = T(x))$.
The general case is due to Halmos and Savage.
It first replaces $\mu$ by a probability measure $\lambda = \sum_j c_j P_{\theta_j}$, a countable convex combination equivalent to the whole family.
It then uses Radon–Nikodym derivatives with respect to $\lambda$.
Example: for i.i.d. $\mathcal{N}(\mu, \sigma^2)$ data,
$$f_{(\mu, \sigma^2)}(x) = (2\pi\sigma^2)^{-n/2} \exp\left(-\frac{\sum_i x_i^2 - 2\mu \sum_i x_i + n\mu^2}{2\sigma^2}\right),$$
so $T(x) = \left(\sum_i x_i, \sum_i x_i^2\right)$ is sufficient for $(\mu, \sigma^2)$, with $h = 1$.
Example: for i.i.d. uniform data on $[0, \theta]$, $f_\theta(x) = \theta^{-n}\mathbf{1}\set{\max_i x_i \le \theta}\mathbf{1}\set{\min_i x_i \ge 0}$, so $\max_i x_i$ is sufficient.
The domination hypothesis is needed for the densities $f_\theta$ to exist, so without it the criterion cannot even be stated.

## Explanations

### L1 human

To check whether a summary of the data keeps all the information about a parameter, write down the likelihood.
If the parameter enters the likelihood only through the summary, the summary is sufficient, and conversely.
This turns a question about conditional distributions into simple algebra.

### L2 human

The factorization criterion is the standard way to find sufficient statistics in practice.
For an [[exponential family|EP9VPZ0S6G]] with density $h(x)\exp(\eta(\theta) \cdot T(x) - A(\theta))$, it shows at once that $T$ is sufficient.
It also shows that the likelihood function, up to a factor free of $\theta$, depends on the data only through any sufficient statistic, which links sufficiency with the [[likelihood principle|YR5RAT9HWZ]].
It is the first step towards the [[Rao–Blackwell theorem|Y3TX4J41TP]] and the [[Lehmann–Scheffé theorem|QQZPWDWS29]].
</document>

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# Current

## Definition

Let $U \subseteq \R^n$ be an [[open set|02BH3MCBQB]] and let $0 \le m \le n$.
Let $\mathcal{D}^m(U)$ be the real vector space of smooth [[differential $m$-forms|0E3G0GRPZ6]] on $U$ with compact support in $U$.
A sequence $(\omega_k)$ in $\mathcal{D}^m(U)$ *converges to $0$* if two conditions hold.

1. There is a compact set $K \subseteq U$ that contains the support of every $\omega_k$.
2. Every partial derivative of every coefficient of $\omega_k$ converges to $0$ uniformly on $K$.

An *$m$-dimensional current* on $U$ is a linear map $T \colon \mathcal{D}^m(U) \to \R$ such that $T(\omega_k) \to 0$ whenever $\omega_k \to 0$ in this sense.
The *boundary* of an $m$-current $T$ with $m \ge 1$ is the $(m-1)$-current $\partial T$ given by $\partial T(\omega) = T(d\omega)$.

## Remarks

Example: let $S \subseteq U$ be a compact oriented $m$-dimensional $C^1$ submanifold, possibly with boundary.
Then $[\![S]\!](\omega) = \int_S \omega$ is linear in $\omega$.
It is continuous, since $\abs{\int_S \omega} \le \operatorname{area}(S) \cdot \sup_S \abs{\omega}$.
So $[\![S]\!]$ is an $m$-current, and Stokes' theorem gives $\partial [\![S]\!] = [\![\partial S]\!]$.
Non-example: the map $\omega \mapsto \abs{\int_S \omega}$ is continuous but not linear, so it is not a current.
The $0$-currents on $U$ are the distributions on $U$.
The *mass* of $T$ is the supremum of $T(\omega)$ over the forms $\omega$ with $\abs{\omega(x)} \le 1$ for all $x$, for a suitable norm on $m$-covectors.
</document>

<document source="https://tamnd.github.io/math2/node/Z6QYDSJVFX.md" tag="Z6QYDSJVFX" kind="statement">
# Four-vertex theorem

## Statement

Let $L > 0$ and let $\gamma \colon \R \to \R^2$ be an $L$-periodic curve of class $C^3$, parametrised by [[arc length|00R1JDDV5X]].
Assume that $\gamma$ is injective on $[0, L)$, so that it traces a simple [[closed curve|0D032NSZKY]].
Let $k$ be the signed [[curvature|7JQS6VKFEZ]] of $\gamma$.
A *vertex* of $\gamma$ is a point $s$ with $k'(s) = 0$.
Then $\gamma$ has at least four vertices in $[0, L)$.

## Remarks

The idea is clearest for a convex curve with nonconstant $k$.
Since $k$ is periodic, it has a maximum and a minimum, which are two vertices.
Assume there were no others.
Then the line through the two corresponding points of the curve meets it only there, and $k'$ has a fixed sign on each of the two arcs.
Choose a linear function $f(x) = \ip{x}{v} + c$ that vanishes on that line; then $k'(s) f(\gamma(s))$ has constant sign.
But the plane Frenet formulas give $\int_0^L k'(s) \, \gamma(s) \, ds = 0$ and $\int_0^L k'(s) \, ds = 0$, so the integral of $k' \cdot (f \circ \gamma)$ is $0$, a contradiction.
The general case of simple closed curves needs more work.
Example: the ellipse $x^2/a^2 + y^2/b^2 = 1$ with $a > b > 0$ has exactly four vertices, the ends of its two axes.
The hypothesis that the curve is simple is needed: a limaçon with an inner loop is a closed regular curve with only two vertices.
The theorem was proved for convex curves by Mukhopadhyaya in 1909 and for all simple closed curves by Kneser in 1912.

## Explanations

### L1 human

Take any smooth closed loop in the plane that does not cross itself.
Its curvature cannot just rise once and fall once around the loop.
There are always at least four places where the curvature stops increasing or stops decreasing.
An ellipse shows that four is the best possible: it is most curved at the ends of the long axis and flattest at the ends of the short axis.

### L2 human

When $k$ never vanishes, the vertices of $\gamma$ are exactly the points where its [[evolute|Y3ZNV24V2C]] has zero velocity.
Typically these are cusps of the evolute, as for the four cusps of the evolute of an ellipse.
The proof above uses only the plane Frenet equations $T' = kN$ and $N' = -kT$, integrated over a period.
The theorem is an early example of a global result in differential geometry, in which a local quantity is constrained by the shape of the whole curve.
</document>

<document source="https://tamnd.github.io/math2/node/Z6T00E3JQH.md" tag="Z6T00E3JQH" kind="problem">
# Normal Moore space problem

## Problem

Is every [[normal|0NSR9V8TED]] [[Moore space|68MA0RM5P6]] [[metrizable|091GJ3FYD7]]?

## Remarks

Every metrizable space is a normal Moore space, so the question asks whether the converse holds.
Normality is the key condition: the [[Moore plane|SAR7PBYF07]] is a Moore space that is not normal and not metrizable.
Under Martin's axiom together with the negation of the continuum hypothesis there is a separable normal Moore space that is not metrizable.
So a positive answer cannot be proved in ZFC, the usual axioms of set theory.
Peter Nyikos showed that the product measure extension axiom implies that every normal Moore space is metrizable.
That axiom is consistent if a strongly compact cardinal is consistent.
William Fleissner showed that the positive answer implies the consistency of a measurable cardinal.
So a large cardinal is needed for the positive answer, and the problem is settled only relative to such axioms.

## Explanations

### L1 human

Moore spaces look very much like metric spaces, and normality is one of the main properties of metric spaces.
The question asks whether these two together are enough to make the space metric.
Work on it showed that the answer depends on extra axioms of set theory, which made it a model case for the use of set theory in topology.
</document>

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# Locally compact abelian group

## Definition

A *locally compact abelian group* is a [[topological group|92700QAB5Y]] $G$ such that:

1. the group $G$ is [[abelian|082GHNPJ53]];
2. the topology of $G$ is Hausdorff;
3. the topology of $G$ is [[locally compact|0FE2SB8WVJ]], that is, each point of $G$ has a compact neighbourhood.

## Remarks

Example: $\R^n$ with addition and the usual topology is an LCA group.
It is abelian, Hausdorff, and each point $x$ has the compact neighbourhood $\set{y : \norm{y - x} \le 1}$.
Non-example: the additive group $\Q$ of rational numbers with the topology from $\R$ is abelian and Hausdorff, but it is not locally compact.
Each neighbourhood of $0$ contains a set $(-\varepsilon, \varepsilon) \cap \Q$, hence a sequence of rationals that converges in $\R$ to an irrational number.
This sequence has no subsequence that converges in $\Q$, so no neighbourhood of $0$ is contained in a compact subset of $\Q$.
The usual abbreviation is "LCA group".
Many authors include the Hausdorff condition in the definition of a topological group.
</document>

<document source="https://tamnd.github.io/math2/node/Z6TKYRZ9WX.md" tag="Z6TKYRZ9WX" kind="concept">
# Inherently nonfinitely based algebra

## Summary

A finite algebra is inherently nonfinitely based when no locally finite variety that contains it is [[finitely based|YGYZAQ12MZ]].
This is a strong and robust way of failing to be finitely based: it passes to every finite algebra whose variety contains the given one.
It gives a convenient tool to show that many finite algebras, such as some finite semigroups, are not finitely based.

## Explanations

### L1 human

Some finite algebras cannot be described by finitely many laws.
An inherently nonfinitely based algebra has a stronger defect.
Put it inside any larger, still well-behaved family of algebras, and that family cannot be described by finitely many laws either.
So the defect cannot be repaired by enlarging the setting.
A finite group is never like this, because the variety it generates is locally finite and finitely based.
</document>

<document source="https://tamnd.github.io/math2/node/Z72JWTXM3W.md" tag="Z72JWTXM3W" kind="definition">
# Triebel–Lizorkin space

## Definition

Let $n \ge 1$.
Let $\psi_0 \in \mathcal{S}(\R^n)$, the [[Schwartz space|CG9JEATW3Q]], satisfy $\psi_0(\xi) = 1$ for $\abs{\xi} \le 1$ and $\psi_0(\xi) = 0$ for $\abs{\xi} \ge 2$, and for $j \ge 1$ let $\psi_j(\xi) = \psi_0(2^{-j} \xi) - \psi_0(2^{-j+1} \xi)$.
For a [[tempered distribution|AES2T5JPFP]] $f$, let $\Delta_j f = \mathcal{F}^{-1}(\psi_j \, \mathcal{F} f)$, where $\mathcal{F}$ is the [[Fourier transform|DPVM34ADZV]].
Let $s \in \R$, $0 < p < \infty$ and $0 < q \le \infty$.
The *Triebel–Lizorkin space* $F^s_{p,q}(\R^n)$ is the set of tempered distributions $f$ with
$$\norm{f}_{F^s_{p,q}} = \Big\| \Big( \sum_{j=0}^{\infty} 2^{jsq} \abs{\Delta_j f}^q \Big)^{1/q} \Big\|_{L^p(\R^n)} < \infty,$$
with the inner sum replaced by $\sup_{j \ge 0} 2^{js} \abs{\Delta_j f}$ when $q = \infty$.

## Remarks

A different choice of $\psi_0$ gives the same space with an equivalent quasi-norm, and for $p, q \ge 1$ it is a Banach space.
Compare the [[Besov space|95QBBPRJ9E]] $B^s_{p,q}$, where the $L^p$ norm is taken before the sum over $j$.
For $p = q$ the two definitions coincide, so $F^s_{p,p} = B^s_{p,p}$.

Example: for $1 < p < \infty$, $F^0_{p,2}(\R^n) = L^p(\R^n)$ with equivalent norms, where $L^p$ is the [[Lp space|SASP14MA1Z]].
This is the Littlewood–Paley theorem: $\norm{f}_{L^p}$ is comparable with $\norm{(\sum_j \abs{\Delta_j f}^2)^{1/2}}_{L^p}$.
More generally, $F^k_{p,2}(\R^n)$ is the Sobolev space $W^{k,p}(\R^n)$ for integers $k \ge 0$ and $1 < p < \infty$.

Non-example: a nonzero constant function $c$ is not in any $F^s_{p,q}(\R^n)$ with $p < \infty$.
Here $\Delta_0 c = c$, because $\mathcal{F} c$ is a multiple of the Dirac delta and $\psi_0(0) = 1$, and the constant $\abs{c}$ is not in $L^p(\R^n)$.
</document>

<document source="https://tamnd.github.io/math2/node/Z761N3D19J.md" tag="Z761N3D19J" kind="concept">
# Massey product

## Summary

A Massey product is a secondary operation on cohomology.
It takes three classes $u, v, w$ with $uv = 0$ and $vw = 0$ and produces a set of classes of degree one less than that of $u v w$.
It is defined from the [[cup product|JX47MS1R1H]] on cochains, not just on cohomology.
Massey products detect structure that the [[cohomology ring|RDT4785M4D]] alone does not see, such as the linking of the Borromean rings.

## Explanations

### L1 human

If $uv = 0$ in cohomology, then on cochains the product $a b$ is the coboundary of some cochain $s$.
The same holds for $vw$ with a cochain $t$.
The two ways of filling in, $s$ against $c$ and $a$ against $t$, combine into a new cocycle.
Its class is the Massey product $\langle u, v, w \rangle$, defined up to some choices.
In the Borromean rings, any two of the three rings are unlinked, so in the complement all cup products of the degree-one classes vanish.
But a triple Massey product is nonzero, and it shows that the three rings are linked together.
</document>

<document source="https://tamnd.github.io/math2/node/Z76HD1BZ29.md" tag="Z76HD1BZ29" kind="statement">
# Girsanov theorem

## Statement

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t), P)$ be a filtered probability space, let $T > 0$, and let $B$ be a [[Brownian motion|22735FGJV6]] for $(\mathcal{F}_t)$ under $P$.
Let $\theta$ be a progressively measurable process with $\int_0^T \theta_s^2 \, ds < \infty$ almost surely.
Let $Z = \mathcal{E}\left(\int_0^{\cdot} \theta_s \, dB_s\right)$ be the [[Doléans-Dade exponential|3D88622PBQ]]:
$$Z_t = \exp\left(\int_0^t \theta_s \, dB_s - \frac12 \int_0^t \theta_s^2 \, ds\right), \quad 0 \le t \le T.$$
Assume $E_P[Z_T] = 1$, and let $Q$ be the [[probability measure|059TJN0NZ1]] on $\mathcal{F}_T$ with $dQ = Z_T \, dP$.
Then the process
$$\tilde{B}_t = B_t - \int_0^t \theta_s \, ds, \quad 0 \le t \le T,$$
is a Brownian motion for $(\mathcal{F}_t)_{t \le T}$ under $Q$.

## Remarks

The [[Itô integral|JA95NH206E]] in the exponent makes $Z$ a positive [[local martingale|XC2YPC2BSW]] under $P$, and the hypothesis $E_P[Z_T] = 1$ makes it a true martingale on $[0, T]$.
By Itô's product rule, $Z \tilde{B}$ is a local martingale under $P$, so $\tilde{B}$ is a continuous local martingale under $Q$.
Its quadratic variation is still $t$, so [[Lévy's characterization|8CKDVESXT6]] shows that it is a Brownian motion under $Q$.
Novikov's condition $E_P\left[\exp\left(\tfrac12 \int_0^T \theta_s^2 \, ds\right)\right] < \infty$ implies $E_P[Z_T] = 1$.
Example: for a constant $\theta = \mu$, the density is $Z_T = \exp(\mu B_T - \tfrac12 \mu^2 T)$, and under $Q$ the process $B$ is a Brownian motion with drift $\mu$.
Without $E_P[Z_T] = 1$, the measure $Q$ has total mass less than $1$ and is not a probability measure.

## Explanations

### L1 human

Changing the probabilities of the paths in a suitable way can turn a Brownian motion with drift into a Brownian motion without drift.
The theorem gives the exact weight for each path.
The new and old probability measures agree on which events are impossible.

### L2 human

Girsanov's theorem is the standard way to remove or add a drift in a stochastic differential equation.
In mathematical finance, it gives the risk-neutral measure under which discounted prices are martingales.
It also yields weak solutions of equations $dX = b(X) \, dt + dB$ with a bounded measurable drift $b$, and likelihood ratios for statistics of diffusions.
The theorem extends to general continuous local martingales in place of $B$.
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# Zolotarev polynomials

## Definition

Let $n \ge 2$ be an integer and let $\sigma$ be a real number.
Let $\mathcal{Z}_{n,\sigma}$ be the set of real polynomials of the form
$$p(x) = x^n + \sigma x^{n-1} + q(x),$$
where $q$ is any real polynomial of [[degree|0DPF6WRZW6]] at most $n - 2$.
The *Zolotarev polynomial* $Z_{n,\sigma}$ is the element of $\mathcal{Z}_{n,\sigma}$ with the least value of $\max_{-1 \le x \le 1} \abs{p(x)}$.

## Remarks

The minimiser exists and is unique.
Indeed $Z_{n,\sigma} = g - q^*$, where $g(x) = x^n + \sigma x^{n-1}$ and $q^*$ is the [[best approximation|7N4988HN3P]] of $g$ in the uniform norm on $[-1, 1]$ by polynomials of degree at most $n - 2$.

Example: for $\sigma = 0$ the Zolotarev polynomial is $2^{1-n} T_n$, where $T_n$ is the [[Chebyshev polynomial|M6EAR1JHPP]].
Check: $2^{1-n} T_n$ is monic, its coefficient of $x^{n-1}$ is $0$ because $T_n$ is even or odd, and no monic polynomial of degree $n$ has a smaller maximum on $[-1, 1]$.

Non-example: for $n = 2$ and $\sigma = 0$, the polynomial $x^2$ lies in $\mathcal{Z}_{2,0}$ but is not $Z_{2,0}$.
Its maximum on $[-1, 1]$ is $1$, while $x^2 - 1/2$ has maximum $1/2$.

Some authors normalise the second coefficient as $-n\sigma$ instead of $\sigma$; the family of polynomials is the same.
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# Spectral theory of dynamical systems

## Summary

The spectral theory of dynamical systems studies a [[measure-preserving transformation|556WBDAHNP]] through the spectrum of its [[Koopman operator|37TRG5DBQC]] on $L^2$.
Two systems with unitarily equivalent Koopman operators are called spectrally isomorphic, and properties preserved by this relation are spectral invariants.
Ergodicity, weak mixing, mixing and discrete spectrum are spectral invariants.
The theory is complete for ergodic systems with discrete spectrum, by the [[Halmos–von Neumann theorem|AS18AWFXSB]].
In general, spectral data do not determine a system up to isomorphism.

## Explanations

### L1 human

The idea goes back to Koopman and von Neumann, who described classical mechanics by a linear operator on a Hilbert space, as in quantum mechanics.
One then asks which frequencies the system has: an eigenfunction with eigenvalue $e^{2\pi i \theta}$ is a measurement that turns like a clock at frequency $\theta$.
A rotation of the circle has many such clocks; a strongly chaotic system has none besides the constants.
Spectral data are easy to compute but coarse: the [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] separates systems that the spectrum cannot.
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# Real analytic manifold

## Summary

A real analytic manifold is a [[topological manifold|QQQ9B1DK4P]] with an [[atlas|J7P32JQTVE]] whose transition maps are real [[analytic functions|081WKEDK6C]].
Every real analytic manifold is a [[smooth manifold|F45VQ47GPV]], but the analytic structure is more rigid.
An analytic function on a connected manifold is determined by its values on any nonempty open set.
Spheres, projective spaces, Lie groups and real algebraic manifolds are real analytic.
Every second countable real analytic manifold has a complexification and can be embedded analytically in some $\R^N$.

## Explanations

### L1 human

A manifold is a space that looks locally like $\R^n$, and its coordinate changes say how to pass between local pictures.
In a real analytic manifold these coordinate changes are given locally by convergent power series.
For example, on the circle the angle coordinate on two overlapping arcs differs by a constant on each piece of the overlap, which is analytic.
Functions that are analytic on such a manifold behave like polynomials: if one vanishes on an open set of a connected manifold, it vanishes everywhere.
</document>

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# Monomial ideal

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $S = k[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]].
A *monomial* of $S$ is a product $x^a = x_1^{a_1} \cdots x_n^{a_n}$ with $a = (a_1, \ldots, a_n) \in \N^n$.
An [[ideal|096Z9MP2ME]] $I$ of $S$ is a *monomial ideal* if it is generated by monomials.

## Remarks

An equivalent form: $I$ is a monomial ideal if and only if, for every $f \in I$, each monomial that occurs in $f$ with a nonzero coefficient lies in $I$.
A monomial $x^b$ lies in the monomial ideal generated by monomials $x^{a}$, $a \in A$, if and only if $b - a \in \N^n$ for some $a \in A$.
By [[Dickson's lemma|NQD0Q9S9M7]], every monomial ideal is generated by finitely many monomials, and it has a unique minimal set of monomial generators.

Example: $I = (x^2, xy)$ in $k[x, y]$ is a monomial ideal by definition.
The polynomial $x^3 + x y^5$ lies in $I$, and indeed both of its monomials are multiples of $x^2$ or of $xy$.

Non-example: $I = (x + y)$ in $k[x, y]$ is not a monomial ideal.
It contains $x + y$, but it does not contain the monomial $x$, since every nonzero element of $I$ is a multiple of $x + y$.
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# Interior operator

## Summary

An interior operator on a [[partially ordered set|011C29Q1S1]] is a map that is deflationary, monotone and idempotent.
It is the dual of a [[closure operator|17FVAN6HXX]]: it sends each element to the greatest "open" element below it.
The [[interior|0C38S0KMJK]] of a subset of a topological space is the main example.
Interior operators are also called kernel operators.

## Explanations

### L1 human

An interior operator shrinks an element to the largest element of a special kind inside it.
In the plane, the interior of a closed disc is the open disc: remove the boundary circle.
Taking the interior again changes nothing, and a smaller set has a smaller interior.
These three features define an interior operator.
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<document source="https://tamnd.github.io/math2/node/Z7R10WW158.md" tag="Z7R10WW158" kind="definition">
# Solvable by radicals

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be a nonconstant polynomial.
The polynomial $f$ is *solvable by radicals* over $K$ if there is a [[radical extension|V74WMGQQ9D]] $R/K$ that contains a [[splitting field|02NZZ1J0FH]] of $f$ over $K$.
The equation $f(x) = 0$ is then also called solvable by radicals.

## Remarks

Equivalently, every root of $f$ lies in some radical extension of $K$, since the compositum of finitely many radical extensions inside a common field is radical.
Example: $f = x^3 - 2$ over $\Q$ splits in $R = \Q(\omega, \sqrt[3]{2})$, with $\omega = e^{2\pi i/3}$.
This $R$ is radical, since $\omega^3 = 1 \in \Q$ and $(\sqrt[3]{2})^3 = 2 \in \Q(\omega)$, so $f$ is solvable by radicals.
Non-example: $f = x^5 - 4x + 2$ over $\Q$ is not solvable by radicals, because its Galois group is $S_5$, which is not [[solvable|0P3KZK0FTV]]; see the [[quintic equation|S1WS18QEGJ]].
In characteristic $0$, [[Galois' theorem on solvability|VHCWCFA3MX]] says that $f$ is solvable by radicals if and only if its Galois group is solvable.
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# Golden–Thompson inequality

## Statement

Let $n \ge 1$ and let $A$ and $B$ be $n \times n$ [[Hermitian matrices|0P17NGQQMQ]].
Then
$$\tr e^{A + B} \le \tr\left( e^A e^B \right),$$
where $e^X$ is the [[matrix exponential|020M04FN1H]] and $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

The right side is a positive real number, because $\tr(e^A e^B) = \tr(e^{A/2} e^B e^{A/2})$ and $e^{A/2} e^B e^{A/2}$ is positive definite.
One proof uses the inequality $\tr\big((XY)^{2^k}\big) \le \tr\big(X^{2^k} Y^{2^k}\big)$ for positive definite $X$ and $Y$, which follows from the Cauchy–Schwarz inequality for the trace by induction on $k$.
Put $X = e^{A/2^k}$ and $Y = e^{B/2^k}$ and let $k \to \infty$; the [[Lie product formula|HK3BRMK912]] turns the left side into $\tr e^{A + B}$.
If $AB = BA$, then $e^{A + B} = e^A e^B$ and equality holds.
The naive extension to three Hermitian matrices, with $\abs{\tr(e^A e^B e^C)}$ on the right, is false in general.

## Explanations

### L1 human

For numbers, $e^{a + b} = e^a e^b$.
For Hermitian matrices that do not commute, the two sides differ, but their traces compare in one direction: the trace of $e^{A + B}$ is at most the trace of $e^A e^B$.
This one-sided rule is often enough to bound quantities in physics and probability.

### L2 human

In statistical mechanics, $\tr e^{-\beta H}$ is the partition function of a quantum system with Hamiltonian $H$, so the inequality bounds a partition function by a simpler expression.
In probability, it is used in the approach of Ahlswede and Winter to matrix concentration inequalities.
There one must control the trace of the exponential of a sum of random Hermitian matrices.
Lieb found a version for three matrices, with an integral in place of a simple product.
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# Sard's theorem

## Statement

Let $n, m \ge 1$, let $U \subseteq \R^n$ be open, and let $f \colon U \to \R^m$ be of class $C^k$, where $k \ge \max(n - m + 1, 1)$.
A point $x \in U$ is a [[critical point|0YTCMFB4HC]] of $f$ if the [[Jacobian matrix|047NDTEPCV]] $Df(x)$ has rank less than $m$.
A critical value of $f$ is the image $f(x)$ of a critical point $x$.
Then the set of critical values of $f$ has [[Lebesgue measure|2Z2QVWZ262]] zero in $\R^m$.

## Remarks

When $n < m$, every point is critical, and the theorem says that the image $f(U)$ of a $C^1$ map is a [[null set|VQQM4Y550T]].
In the case $n = m$, the idea is this.
Near a critical point, $f$ squeezes a small cube of side $r$ into a set of volume $o(r^n)$, because $Df(x)$ is singular.
Covering the critical set by small cubes then gives total volume as small as one wants.
The general case uses induction on $n$ and needs the higher smoothness to control the Taylor expansion along the critical set.
Example: $f(x) = x^2$ on $\R$ has the single critical point $0$ and the single critical value $0$.
The smoothness hypothesis cannot be dropped.
Hassler Whitney constructed a $C^1$ function on $\R^2$ that is not constant on a connected arc of critical points, so its critical values fill an interval.

## Explanations

### L1 human

A critical value is a value that a map takes at a point where it flattens out.
Sard's theorem says that a smooth enough map has very few critical values: they form a set of measure zero.
So a value picked at random is almost surely a regular value.

### L2 human

The complement of a null set is dense, so regular values are dense in $\R^m$.
If $y$ is a regular value and $f^{-1}(y)$ is not empty, the implicit function theorem shows that $f^{-1}(y)$ is a submanifold of dimension $n - m$.
This makes Sard's theorem a basic tool of differential topology, for example in transversality arguments and in the definition of the degree of a map.
The same statement holds for smooth maps between manifolds, by working in charts.
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# Successor order

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]], and write $a < b$ when $a \le b$ and $a \ne b$.
A *successor order* on $P$ is a map $\operatorname{succ} \colon P \to P$ such that:

1. $a \le \operatorname{succ}(a)$ for all $a \in P$.
2. If $\operatorname{succ}(a) \le a$, then $a$ is a maximal element of $P$.
3. If $a < b$, then $\operatorname{succ}(a) \le b$.

## Remarks

When $a$ is not maximal, the clauses say that $\operatorname{succ}(a)$ is the least element of $\set{b \in P : a < b}$; when $a$ is maximal, $\operatorname{succ}(a) = a$.
So the map, if it exists, is unique.

Example: $\Z$ with $\operatorname{succ}(n) = n + 1$.
Check: $n \le n + 1$; the inequality $n + 1 \le n$ never holds, so clause 2 is empty; and $n < m$ for integers means $n + 1 \le m$.
On $\N$ this is the [[successor function|0AK9TGSNXY]].

Non-example: $\Q$ with the usual order has no successor order.
No rational is maximal, so clauses 1 and 2 force $a < \operatorname{succ}(a)$; then $b = (a + \operatorname{succ}(a))/2$ satisfies $a < b < \operatorname{succ}(a)$, which breaks clause 3.

The dual notion is a predecessor order.
In Mathlib this is the class `SuccOrder`.
</document>

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# Strongly pseudoconvex domain

## Summary

A strongly pseudoconvex domain is a domain in $\C^n$ with $C^2$ boundary whose [[Levi form|80MTNXWW80]] is positive definite at every boundary point.
It is also called a strictly pseudoconvex domain.
It is the complex analogue of a strictly convex domain.
In this class the analysis of [[several complex variables|BH57D2FBNR]] works best.
The $\bar\partial$-equation has good estimates, and every boundary point is locally convex after a holomorphic change of coordinates.

## Explanations

### L1 human

A strongly pseudoconvex domain bends outwards strictly in every complex direction along its boundary.
The model is the unit ball in $\C^n$.
Small smooth perturbations of the ball stay strongly pseudoconvex, so there are very many such domains.
The bidisc is not one, because its boundary contains flat complex discs, such as $\set{(1, z_2) : \abs{z_2} < 1}$.
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# Polar space

## Summary

A polar space is a geometry of points and lines with one rule.
Each point is collinear with exactly one point or with all points of any line not through it.
The main examples are the points and lines lying on a non-degenerate [[quadric|Q3Q4FF23VG]].
Others come from the [[polarities|727TZQJ4ZA]] of symplectic and Hermitian forms, using the absolute points and totally isotropic lines.
Polar spaces are the geometries of the classical groups other than the linear groups, and the non-degenerate ones of rank $2$ are the [[generalized quadrangles|GQJCPFSYVK]].

## Explanations

### L1 human

Take the quadric surface $x_0 x_1 + x_2 x_3 = 0$ in projective space, which is covered by two families of straight lines.
Its points and the lines lying on it form a polar space.
Pick a line on the surface and a point of the surface off that line.
Then the point is joined by a line of the surface to exactly one point of the line.
In other polar spaces a point can also be joined to all points of a line, and this one-or-all rule turns out to characterize such geometries.
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# Hunt process

## Summary

A Hunt process is a strong Markov process with right-continuous paths that cannot jump at times that can be foreseen.
This class of processes is the standard setting of probabilistic potential theory.
Brownian motion, Lévy processes and Feller processes are Hunt processes.
For a Hunt process, the excessive functions behave like the superharmonic functions of classical potential theory.

## Explanations

### L1 human

A Hunt process is a random motion with three good features.
It forgets its past, even at random times chosen by watching the motion.
Its path has no gaps from the right and has limits from the left.
It never jumps at a moment that one could predict by watching it approach, a property called quasi-left-continuity.
Brownian motion is the basic example.
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# Undecidability of the halting problem

## Statement

Fix a [[Gödel numbering|MFJXPAF0R0]] $e \mapsto M_e$ of all [[Turing machines|758EYCN27B]] with input and output in the natural numbers.
Let $H = \set{(e, x) \in \N^2 : M_e \text{ halts on input } x}$.
Then $H$ is not a [[computable set|PPGD5KW8JC]]: no Turing machine halts on every input $(e, x)$ and outputs $1$ if $(e, x) \in H$ and $0$ otherwise.

## Remarks

Suppose a machine $D$ decides $H$.
Build a machine $N$ that, on input $e$, runs $D$ on $(e, e)$, loops forever if the answer is $1$, and halts if the answer is $0$.
Let $n$ be a number of $N$.
Then $N$ halts on $n$ if and only if $(n, n) \notin H$, that is, if and only if $N$ does not halt on $n$, a contradiction.

The set $H$ is computably enumerable, since a universal machine halts on $(e, x)$ exactly when $M_e$ halts on $x$.
So $H$ is an example of a computably enumerable set that is not computable.
By the same argument, the diagonal set $K = \set{e : M_e \text{ halts on input } e}$ is not computable.

## Explanations

### L1 human

There is no general method to tell, for an arbitrary program and input, whether the program will eventually stop.
One can run the program and wait, but if it has not stopped yet one never knows whether it will.
The proof imagines a perfect halting tester and builds a program that does the opposite of what the tester predicts about itself.

### L2 human

This result goes back to Turing's paper of 1936, and it is the standard first example of an undecidable problem.
Most other undecidability results are proved by reducing the halting problem to them.
Examples are [[Rice's theorem|AKDR5YBBE2]], the undecidability of first-order validity, and the negative solution of [[Hilbert's tenth problem|X8TVJSVTD4]].
In the [[arithmetical hierarchy|7CVJCQKY41]], $H$ is $\Sigma^0_1$-complete.
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# Symmetric and self-reciprocal kernels

## Definition

Let $K \colon (0, \infty) \times (0, \infty) \to \C$ be the [[kernel|9M0NNNFJGM]] of an [[integral transform|YCK709QB06]]
$$(Tf)(x) = \int_0^\infty K(x, y)\, f(y)\, dy.$$
Let $A$ be a class of functions on which $T$ is defined, such as the [[Lebesgue space|7KGF78FGEF]] $L^2(0, \infty)$, with $T(A) \subseteq A$.

1. $K$ is *symmetric* if $K(x, y) = K(y, x)$ for all $x, y > 0$.
2. $K$ is *self-reciprocal* on $A$ (a *Fourier kernel*) if $T(Tf) = f$ for every $f \in A$; that is, $g = Tf$ implies $f = Tg$.

## Remarks

Example: $K(x, y) = \sqrt{2/\pi}\, \cos(xy)$ is symmetric, since $\cos(xy) = \cos(yx)$.
It is self-reciprocal on $L^2(0, \infty)$: this is the inversion theorem for the Fourier cosine transform, which follows from the Fourier inversion theorem applied to the even extension of $f$.
The sine kernel $\sqrt{2/\pi}\, \sin(xy)$ and the Hankel kernel $\sqrt{xy}\, J_\nu(xy)$ for $\nu \ge -1/2$, with $J_\nu$ the Bessel function of the first kind, are also symmetric and self-reciprocal.
Non-example: the Laplace kernel $e^{-xy}$ is symmetric but not self-reciprocal.
For $f(y) = e^{-y}$, $(Tf)(x) = 1/(1 + x)$, and $T(Tf)(x) = \int_0^\infty e^{-xy}/(1 + y)\, dy$ tends to $+\infty$ as $x \to 0^+$, while $f(x) \to 1$.
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<document source="https://tamnd.github.io/math2/node/Z8192111H3.md" tag="Z8192111H3" kind="definition">
# Curve-shortening flow

## Definition

Let $S^1 = \R / \Z$ and let $T > 0$.
Let $\gamma \colon S^1 \times [0, T) \to \R^2$ be continuous and smooth on $S^1 \times (0, T)$.
Assume that for each $t \in (0, T)$ the curve $\gamma_t = \gamma(\cdot, t)$ is an immersed [[plane curve|EWS4WFTA7W]].
Let $\kappa(u, t)$ be the signed [[curvature|7JQS6VKFEZ]] of $\gamma_t$ at $u$.
Let $N(u, t)$ be the unit normal obtained by turning the unit tangent by $+90^\circ$.
The family $\gamma$ is a solution of the *curve-shortening flow* with initial curve $\gamma_0$ if
$$\frac{\partial \gamma}{\partial t}(u, t) = \kappa(u, t)\, N(u, t) \qquad \text{for all } u \in S^1, \ t \in (0, T).$$

## Remarks

Equivalently, $\partial_t \gamma = \partial_s^2 \gamma$, where $s$ is the arc length parameter of $\gamma_t$; this is a nonlinear [[partial differential equation|GC5BDZ6QBC]] of heat type.
The normal velocity is what matters: adding a tangential velocity only changes the parametrization of the curves.
The length $L(t)$ of $\gamma_t$ satisfies $L'(t) = -\int_{\gamma_t} \kappa^2 \, ds$, and the enclosed area $A(t)$ of an embedded counterclockwise curve satisfies $A'(t) = -2\pi$.

Example: the circles $\gamma(u, t) = \sqrt{r_0^2 - 2t}\,(\cos 2\pi u, \sin 2\pi u)$ for $0 \le t < r_0^2 / 2$ form a solution.
For a counterclockwise circle of radius $r$, $\kappa = 1/r$ and $N$ points to the center, and indeed $\partial_t r = -1/r$.

Non-example: the shrinking circles $\gamma(u, t) = (r_0 - t)(\cos 2\pi u, \sin 2\pi u)$ move with normal speed $1$ instead of $1/r$, so they do not solve the equation.
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# Almost periodic sequence

## Summary

An almost periodic sequence is a bounded sequence that comes back arbitrarily close to itself after shifts that occur regularly.
It need not repeat exactly.
It is the discrete version, in the sense of Harald Bohr, of an [[almost periodic function|GZBSV8DYGS]].
Every [[periodic sequence|Q3B6K9HYRG]] is almost periodic, and so is every sum of finitely many sequences $e^{i \lambda n}$.
Almost periodic sequences arise as solutions of difference equations with almost periodic coefficients or forcing.

## Explanations

### L1 human

A periodic sequence returns exactly to itself after a fixed shift.
An almost periodic sequence returns to within any small error you choose, and the good shifts appear at fairly regular gaps.
The sequence $x_n = \cos n$ never repeats exactly, because $2\pi$ is not a rational number.
Still, $n \mapsto \cos(n + p)$ is uniformly close to $\cos n$ whenever the integer $p$ is close to a multiple of $2\pi$, and such integers come again and again.
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# Tensor decomposition

## Definition

Let $F$ be a field, let $d \ge 2$, and let $T \in F^{n_1} \otimes \cdots \otimes F^{n_d}$ be a tensor, that is, an array of size $n_1 \times \cdots \times n_d$ (see [[tensor product|0BF2DE2SYK]]).
A *tensor decomposition* of $T$ is an expression of $T$ in one of the following standard forms.

1. A *canonical polyadic (CP) decomposition* of length $r$ is an identity
$$T = \sum_{i=1}^{r} a_i^{(1)} \otimes a_i^{(2)} \otimes \cdots \otimes a_i^{(d)}, \qquad a_i^{(k)} \in F^{n_k}.$$
2. A *Tucker decomposition* with multilinear size $(r_1, \ldots, r_d)$ is an identity
$$T = \sum_{j_1 = 1}^{r_1} \cdots \sum_{j_d = 1}^{r_d} g_{j_1 \ldots j_d} \, u_{j_1}^{(1)} \otimes \cdots \otimes u_{j_d}^{(d)},$$
where $G = (g_{j_1 \ldots j_d})$ is a core tensor of size $r_1 \times \cdots \times r_d$ and $u_j^{(k)} \in F^{n_k}$.

## Remarks

The smallest length of a CP decomposition of $T$ is the [[tensor rank|W84NTF77M0]] of $T$.
A CP decomposition is a Tucker decomposition whose core is diagonal, with $r_1 = \cdots = r_d = r$.
Example: the $2 \times 2 \times 2$ array with $t_{111} = t_{222} = 1$ and all other entries $0$ has the CP decomposition $e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_2 \otimes e_2$ of length $2$.
Check: the first term has only the entry $(1,1,1)$ equal to $1$, and the second only the entry $(2,2,2)$.
Non-example: $e_1 \otimes e_1 \otimes e_1$ alone is not a CP decomposition of this array, since it misses the entry $t_{222} = 1$.
For $d = 2$, a CP decomposition is a factorization $T = A B^T$ with $r$ columns in $A$ and $B$, and the [[singular value decomposition|000NPTMX6J]] gives one of minimal length.
For $d = 3$, Kruskal's theorem gives conditions under which a CP decomposition of minimal length is essentially unique.
Here essentially unique means unique up to the order of the terms and the scaling of the factors.
This has no analogue for matrices: a minimal factorization $T = A B^T$ of a matrix of rank at least $2$ is never unique in this sense.
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<document source="https://tamnd.github.io/math2/node/Z86VGP5RMG.md" tag="Z86VGP5RMG" kind="statement">
# Word problem for free lattices

## Statement

Let $X$ be a set and let $\mathrm{FL}(X)$ be the [[free lattice|WP20S9NZ15]] on $X$.
For lattice [[terms|6D3WA7N2W4]] $s$ and $t$ over $X$, write $s \le t$ when the value of $s$ is below the value of $t$ in $\mathrm{FL}(X)$.
Then $s \le t$ holds if and only if one of the following holds:

1. $s$ and $t$ are the same letter $x \in X$;
2. $s = s_1 \vee s_2$, and $s_1 \le t$ and $s_2 \le t$;
3. $t = t_1 \wedge t_2$, and $s \le t_1$ and $s \le t_2$;
4. $s$ is a letter $x \in X$, $t = t_1 \vee t_2$, and $x \le t_1$ or $x \le t_2$;
5. $s = s_1 \wedge s_2$, $t$ is a letter $x \in X$, and $s_1 \le x$ or $s_2 \le x$;
6. $s = s_1 \wedge s_2$, $t = t_1 \vee t_2$, and $s_1 \le t$ or $s_2 \le t$ or $s \le t_1$ or $s \le t_2$.

Consequently, equality of two lattice terms in $\mathrm{FL}(X)$ is [[decidable|PPGD5KW8JC]]: $s = t$ in $\mathrm{FL}(X)$ exactly when $s \le t$ and $t \le s$.

## Remarks

In each clause the comparisons on the right involve terms of smaller total length, so the clauses give a recursive algorithm to decide $s \le t$.
The "if" direction holds in every lattice.
The "only if" direction is the content of the theorem; clause 6 is [[Whitman's condition|5TNA2HBG7Y]].
Example: $x \wedge (y \vee z) \le (x \wedge y) \vee (x \wedge z)$ fails in $\mathrm{FL}(\set{x, y, z})$.
By clause 6 it would need one of $x \le (x \wedge y) \vee (x \wedge z)$, $y \vee z \le (x \wedge y) \vee (x \wedge z)$, $x \wedge (y \vee z) \le x \wedge y$ or $x \wedge (y \vee z) \le x \wedge z$, and the clauses refute each of them in turn.
So the distributive law is not an identity of all lattices.

## Explanations

### L1 human

Two lattice formulas, such as $x \wedge (y \vee x)$ and $x$, may look different but always give the same value.
This theorem gives a mechanical test for when this happens.
To compare two formulas, look at their outermost operations and reduce the question to smaller comparisons, until only letters are left.

### L2 human

The word problem asks whether two terms define the same element of a free algebra.
For finitely presented groups it can be undecidable, but for free lattices it has this explicit solution, due to Whitman.
The procedure also decides whether a lattice identity holds in all lattices.
Indeed, $s = t$ holds in every lattice exactly when $s = t$ in the free lattice on the letters of $s$ and $t$.
Whitman also used these rules to show that each element of $\mathrm{FL}(X)$ has a canonical form: a shortest term, unique up to commutativity and associativity.
</document>

<document source="https://tamnd.github.io/math2/node/Z877XDBMQ3.md" tag="Z877XDBMQ3" kind="statement">
# Stone–von Neumann theorem

## Statement

Let $n \ge 1$ and let $H$ be a [[Hilbert space|HM2HHE3J51]].
Let $U$ and $V$ be strongly continuous [[unitary representations|74BPYJNZTM]] of $\R^n$ on $H$ that satisfy the Weyl relations
$$U(s) V(t) = e^{i s \cdot t} \, V(t) U(s) \qquad (s, t \in \R^n).$$
Then $H$ is an orthogonal direct sum of closed subspaces, each invariant under all $U(s)$ and $V(t)$.
On each of them, the pair $(U, V)$ is unitarily equivalent to the Schrödinger pair on $L^2(\R^n)$:
$$(U_0(s) \psi)(x) = \psi(x + s), \qquad (V_0(t) \psi)(x) = e^{i t \cdot x} \psi(x).$$
In particular, every irreducible pair satisfying the Weyl relations is unitarily equivalent to $(U_0, V_0)$.

## Remarks

The pair $(U_0, V_0)$ satisfies the relations: $(U_0(s) V_0(t) \psi)(x) = e^{i t \cdot (x + s)} \psi(x + s) = e^{i s \cdot t} (V_0(t) U_0(s) \psi)(x)$.
The Weyl relations say that $(s, t, z) \mapsto z \, U(s) V(t)$ defines a unitary representation of the [[Heisenberg group|ZZYCV5EX9J]] in which the center acts by scalars.
So the theorem says: for each nontrivial character of the center, the Heisenberg group has exactly one irreducible unitary representation with that central character.
A standard proof considers the operators $\int\int f(s, t) U(s) V(t) \, ds \, dt$, finds a rank-one projection among them, and uses it to build an isometry onto $L^2(\R^n)$.
The exponentiated Weyl form is needed: there are self-adjoint operators $Q$ and $P$ that satisfy $[Q, P] = i$ on a common dense invariant domain but are not equivalent to the Schrödinger pair.
The finite number $n$ of degrees of freedom is also needed.
For infinitely many degrees of freedom there are many inequivalent irreducible representations, a basic phenomenon of quantum field theory.

## Explanations

### L1 human

In quantum mechanics, position and momentum do not commute.
The theorem says that once their commutation law is fixed, there is essentially only one way to realise them by operators.
So the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg are two pictures of the same theory.

### L2 human

In representation theory the theorem classifies the infinite-dimensional irreducible unitary representations of the Heisenberg group.
It is the first case of the Kirillov orbit method, where these representations correspond to the planes in the dual of the Lie algebra.
Its uniqueness statement is what makes the oscillator representation of the metaplectic group possible: each symplectic map gives a second Schrödinger pair, hence an intertwining operator.
</document>

<document source="https://tamnd.github.io/math2/node/Z8CW6Z2K7N.md" tag="Z8CW6Z2K7N" kind="definition">
# Essentially self-adjoint operator

## Definition

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]], and let $A$ be a [[symmetric operator|H971WGMV6C]] on $H$ with dense domain $D(A)$.
Then $A$ is [[closable|3TWMBFQJJ8]], with [[closure|TRC892PJNN]] $\overline{A}$.
The operator $A$ is *essentially self-adjoint* if $\overline{A}$ is self-adjoint, that is, $\overline{A} = (\overline{A})^*$, where ${}^*$ denotes the [[adjoint|86B7HF79PQ]].

## Remarks

The following conditions are equivalent for a symmetric operator $A$:

1. $A$ is essentially self-adjoint;
2. $A$ has exactly one self-adjoint extension;
3. $A^*$ is self-adjoint, and then $A^* = \overline{A}$;
4. the [[kernels|0S23E6YE83]] $\ker(A^* - iI)$ and $\ker(A^* + iI)$ are both $\set{0}$.

Example: on $L^2(\R)$, let $A = -i \, d/dx$ on the smooth functions with compact support.
Then $A^*$ is $-i \, d/dx$ on the Sobolev space $H^1(\R)$.
The equation $A^* u = \pm i u$ means $u' = \mp u$, with solutions $c e^{\mp x}$, which are not in $L^2(\R)$ unless $c = 0$; so condition 4 holds.
Non-example: on $L^2([0,1])$, let $B = -i \, d/dx$ on the smooth functions with compact support in $(0,1)$.
Now $e^{-x}$ and $e^{x}$ lie in $L^2([0,1])$ and solve $B^* u = i u$ and $B^* u = -i u$, so condition 4 fails, and $B$ has many self-adjoint extensions, one for each boundary condition $u(1) = e^{i \theta} u(0)$.
A self-adjoint operator is essentially self-adjoint; a core of a self-adjoint operator $T$ is a domain on which the restriction of $T$ is essentially self-adjoint.
</document>

<document source="https://tamnd.github.io/math2/node/Z8D1CZFTWP.md" tag="Z8D1CZFTWP" kind="statement">
# Euler integral for the gamma function

## Statement

Let $H = \set{s \in \C : \operatorname{Re} s > 0}$, and for $t > 0$ let $t^{s-1} = e^{(s-1)\ln t}$.

1. For every $s \in H$ the integral $\int_0^\infty t^{s-1} e^{-t} \, dt$ converges [[absolutely|0F9VKJPWCA]].
2. The function $G(s) = \int_0^\infty t^{s-1} e^{-t} \, dt$ is [[holomorphic|0NAHXR1FRH]] on $H$.
3. For real $s > 0$, $G(s) = \Gamma(s)$, where $\Gamma$ is the [[gamma function|0V3FZMVB4K]].

So $G$ is the unique holomorphic extension of $\Gamma$ from $(0, \infty)$ to $H$, and $\Gamma(s) = G(s)$ for all $s \in H$.

## Remarks

Absolute convergence follows from $\abs{t^{s-1} e^{-t}} = t^{\sigma - 1} e^{-t}$ with $\sigma = \operatorname{Re} s > 0$, which is the real integrand.
For holomorphy, the truncated integrals $\int_{1/n}^{n}$ are entire in $s$ and converge to $G$ uniformly on each strip $a \le \operatorname{Re} s \le b$ with $0 < a < b$, so the limit is holomorphic.
Uniqueness comes from the [[identity theorem|0YBVKMYX0W]], since $(0, \infty)$ has accumulation points in the connected set $H$.
The integral diverges at $t = 0$ when $\operatorname{Re} s \le 0$; the extension of $\Gamma$ beyond $H$ uses the functional equation $\Gamma(s + 1) = s\,\Gamma(s)$ instead.
In Mathlib the statement is `Complex.Gamma_eq_integral`.

## Explanations

### L1 human

Euler's integral is the most common formula for the gamma function.
It works not only for positive real numbers but for every complex number with positive real part.
For example, at $s = 1$ it gives $\int_0^\infty e^{-t}\,dt = 1 = \Gamma(1)$.

### L2 human

The integral is the gamma function's Mellin transform description: $\Gamma(s)$ is the Mellin transform of $e^{-t}$.
Many identities, such as the formula linking the [[beta function|QHNWVSC6AQ]] with $\Gamma$, start from this integral and a change of variables.
The integral alone only defines $\Gamma$ on the right half-plane; the [[meromorphic|0NSD1TPHR0]] extension to $\C$ comes from the functional equation.
</document>

<document source="https://tamnd.github.io/math2/node/Z8FABSYQTS.md" tag="Z8FABSYQTS" kind="concept">
# Finite-rank operator

## Summary

A finite-rank operator is a bounded linear map between normed spaces whose image is finite-dimensional.
Such operators behave like matrices, even when the spaces are infinite-dimensional.
They are the simplest operators of functional analysis, and many other classes, such as compact and trace class operators, are built as limits of them.

## Explanations

### L1 human

A finite-rank operator takes a possibly huge space and squeezes it into a finite-dimensional piece.
Picture a projector that casts every object of a large room onto one flat wall: the shadows live in a small space.
For a concrete example, on the space of square-summable sequences, the map $(x_1, x_2, x_3, \ldots) \mapsto (x_1 + x_2, 0, 0, \ldots)$ has image the multiples of the first unit vector.
Its image has dimension $1$, so it is an operator of rank $1$.
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<document source="https://tamnd.github.io/math2/node/Z8JJ04X1JX.md" tag="Z8JJ04X1JX" kind="statement">
# Synge's theorem

## Statement

Let $(M, g)$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with positive [[sectional curvature|1X42JAA349]].
Then:

1. If $n$ is even and $M$ is [[orientable|913WEQERBE]], then $M$ is simply connected.
2. If $n$ is odd, then $M$ is orientable.

## Remarks

The idea of the proof: if the conclusion fails, there is a nontrivial free homotopy class of loops along which parallel transport has the required orientation behaviour.
Compactness gives a closed [[geodesic|NSYMQMHPDN]] $\gamma$ of minimal length in this class.
A linear-algebra argument shows that [[parallel transport|TVAGN3D6W5]] once around $\gamma$ fixes a unit vector normal to $\gamma$.
Pushing $\gamma$ in the direction of this parallel field decreases length to second order, by the [[second variation formula|ZB91TH6FKV]] and positive curvature, which contradicts minimality.

Example: the real projective plane $\R P^2$ with the round metric has positive curvature, even dimension and fundamental group of order $2$.
It is not orientable, so this does not contradict part 1.
Its orientability hypothesis is therefore needed.
In odd dimensions, $\R P^3$ is orientable, as part 2 says, but it is not simply connected, so part 1 has no odd-dimensional analogue.

## Explanations

### L1 human

Positive curvature makes loops want to shrink.
Synge's theorem says that for compact spaces of even dimension, every loop can in fact be shrunk to a point, once one excludes the twisted, non-orientable cases.
In odd dimensions, it says the space cannot be twisted like a Möbius band.

### L2 human

Synge's theorem is one of the few known topological obstructions to positive sectional curvature.
For instance, it shows that $\R P^2 \times \R P^2$, whose orientable double cover is not simply connected, has no metric of positive curvature.
The proof is the model for later arguments that combine second variation with the holonomy of a closed geodesic, such as Weinstein's fixed point theorem.
</document>

<document source="https://tamnd.github.io/math2/node/Z8NJ4KHGAT.md" tag="Z8NJ4KHGAT" kind="definition">
# Monogenic function

## Definition

Let $n \ge 2$ and let $\mathbb{R}_{0,n}$ be the real [[Clifford algebra|A38B1QFK20]] generated by $e_1, \ldots, e_n$ with $e_j^2 = -1$ and $e_j e_k = -e_k e_j$ for $j \ne k$.
Let $D = \sum_{j=1}^{n} e_j \dfrac{\partial}{\partial x_j}$ be the [[Dirac operator|3DXGJTEQZH]] on $\R^n$.
Let $U \subset \R^n$ be open and let $f \colon U \to \mathbb{R}_{0,n}$ have continuous first [[partial derivatives|09EGC6F8PT]].
The function $f$ is *left monogenic* on $U$ if
$$D f = \sum_{j=1}^{n} e_j \frac{\partial f}{\partial x_j} = 0 \quad \text{on } U.$$
It is *right monogenic* if $\sum_{j} \dfrac{\partial f}{\partial x_j} e_j = 0$ on $U$.

## Remarks

Example: identify $x \in \R^n$ with $\sum_j x_j e_j$, so that $x^2 = -\abs{x}^2$.
The Cauchy kernel $E(x) = \dfrac{\bar x}{\abs{x}^n}$ with $\bar x = -x$ is left and right monogenic on $\R^n \setminus \set{0}$.
A simpler example is $f(x) = x_1 e_1 - x_2 e_2$, since $Df = e_1 e_1 - e_2 e_2 = -1 + 1 = 0$.

Non-example: $f(x) = x = \sum_j x_j e_j$ is not monogenic, since $Df = \sum_j e_j^2 = -n \ne 0$.

Since $D^2 = -\Delta$, with $\Delta$ the [[Laplacian|0E0B6RA8K0]], each real component of a left monogenic function of class $C^2$ is harmonic.
Many authors use instead the generalized Cauchy–Riemann operator $\partial_{x_0} + \sum_{j=1}^{n} e_j \partial_{x_j}$ on $\R^{n+1}$; for $n = 1$ it gives back holomorphic functions.
</document>

<document source="https://tamnd.github.io/math2/node/Z8R2D33WSM.md" tag="Z8R2D33WSM" kind="example">
# Hopf manifold

## Example

Let $n \ge 2$ and let $\lambda \in \C$ with $0 < \abs{\lambda} < 1$.
Let $W = \C^n \setminus \set{0}$ and let $\Gamma$ be the group of maps $g_k(z) = \lambda^k z$, $k \in \Z$.
The quotient $X = W / \Gamma$ is a (classical) *Hopf manifold*, a compact [[complex manifold|JSGWAN2RJV]] of dimension $n$.

Check that $X$ is a complex manifold.
The action is free: if $\lambda^k z = z$ with $z \ne 0$, then $\lambda^k = 1$, so $k = 0$.
It is [[properly discontinuous|J5X5ED0B2Y]]: a compact $K \subset W$ lies in a shell $\set{r \le \abs{z} \le R}$ with $r > 0$.
The set $g_k(K)$ lies in $\set{\abs{\lambda}^k r \le \abs{z} \le \abs{\lambda}^k R}$, which meets $K$ for only finitely many $k$.
The maps $g_k$ are holomorphic, so the [[quotient|0BR3FY5Q8M]] inherits a complex structure.
It is compact: it is the image of the compact shell $\set{\abs{\lambda} \le \abs{z} \le 1}$.

For $\lambda$ real, the map $z \mapsto (\abs{z}, z/\abs{z})$ identifies $W$ with $(0, \infty) \times S^{2n-1}$, and $\Gamma$ acts only on the first factor.
So $X$ is diffeomorphic to $S^1 \times S^{2n-1}$; the same holds for all $\lambda$.

## Remarks

$X$ is not [[Kähler|X60JG9AEQJ]].
On a compact Kähler manifold of dimension $n$ the class of the Kähler form $\omega$ in $H^2(X; \R)$ is nonzero, since $\int_X \omega^n > 0$.
But by the Künneth formula $H^2(S^1 \times S^{2n-1}; \R) = 0$ for $n \ge 2$, because $2n - 1 \ge 3$.
Also the first [[Betti number|ST71ZV3FES]] of $X$ is $1$, an odd number, which is impossible for a compact Kähler manifold.

For $n = 1$ the same construction gives $\C^* / \lambda^\Z$, an elliptic curve, which is Kähler.
For $n = 2$ one gets the primary [[Hopf surfaces|PNFNGBXRM3]] of diagonal type.

## Explanations

### L1 human

Remove the origin from $\C^n$, and declare two points the same when one is $\lambda$ times the other.
What is left is a thick spherical shell whose inner and outer spheres are glued, which is the product of a circle and a sphere.
This simple space is a compact complex manifold that admits no Kähler metric.
So compact complex geometry is much wider than Kähler geometry, and hence than projective algebraic geometry.
</document>

<document source="https://tamnd.github.io/math2/node/Z8T8YYF5ZZ.md" tag="Z8T8YYF5ZZ" kind="example">
# Fourier series of Bernoulli polynomials

## Example

Let $B_k$ be the $k$-th [[Bernoulli polynomial|BXQKBJJGFG]], and let $k \ge 2$ be an integer.
Then for every $x \in [0, 1]$,
$$B_k(x) = -\frac{k!}{(2\pi i)^k} \sum_{n \in \Z, \, n \ne 0} \frac{e^{2\pi i n x}}{n^k},$$
and the series converges absolutely and uniformly on $[0, 1]$.
So the $1$-periodic function equal to $B_k$ on $[0, 1)$ has the [[Fourier coefficients|5NN2YZZR3V]] $0$ at $n = 0$ and $-\frac{k!}{(2\pi i n)^k}$ at $n \ne 0$, for period $1$.

Check of the coefficients: $\int_0^1 B_k(x) \, dx = 0$ for $k \ge 1$, which gives the coefficient at $n = 0$.
For $n \ne 0$, integration by parts with $B_k' = k B_{k-1}$ and $B_k(1) = B_k(0)$ for $k \ge 2$ gives $\int_0^1 B_k(x) e^{-2\pi i n x} \, dx = \frac{k}{2\pi i n} \int_0^1 B_{k-1}(x) e^{-2\pi i n x} \, dx$.
For $k = 1$, $\int_0^1 (x - \frac{1}{2}) e^{-2\pi i n x} \, dx = -\frac{1}{2\pi i n}$, and induction gives $-\frac{k!}{(2\pi i n)^k}$.
For $k \ge 2$ the coefficients are summable and the periodic extension is continuous, so the series converges uniformly to it.

## Remarks

Grouping $n$ and $-n$ gives real forms; for $k = 2m$ with $m \ge 1$,
$$B_{2m}(x) = (-1)^{m+1} \frac{2 \, (2m)!}{(2\pi)^{2m}} \sum_{n=1}^{\infty} \frac{\cos 2\pi n x}{n^{2m}}.$$
At $x = 0$, where $B_{2m}(0)$ is the Bernoulli number $B_{2m}$, this is Euler's formula $\zeta(2m) = (-1)^{m+1} \frac{(2\pi)^{2m} B_{2m}}{2 \, (2m)!}$ for the [[Riemann zeta function|1XWBY4N0VT]].
For $m = 1$ it gives $\zeta(2) = \frac{\pi^2}{6}$, and for $m = 2$ it gives $\zeta(4) = \frac{\pi^4}{90}$.
For $k = 1$ the same series, $-\sum_{n \ge 1} \frac{\sin 2\pi n x}{\pi n}$, converges to $x - \frac{1}{2}$ only for $0 < x < 1$; at $x = 0$ it gives $0$, the midpoint of the jump.

## Explanations

### L1 human

The Bernoulli polynomials, repeated periodically, have very simple Fourier coefficients: a constant times $1/n^k$.
Reading the formula at a single point turns it into an exact value of a series such as $\sum 1/n^2$ or $\sum 1/n^4$.
This is the cleanest route to the values of the zeta function at even integers.
</document>

<document source="https://tamnd.github.io/math2/node/Z8XMCCZFA4.md" tag="Z8XMCCZFA4" kind="definition">
# Martin's axiom

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]], and read $q \le p$ as "$q$ extends $p$".
Two elements $p, q$ are compatible if some $r \in P$ has $r \le p$ and $r \le q$.

1. An antichain of $P$ is a subset whose elements are pairwise incompatible, and $P$ has the countable chain condition, ccc, if each antichain is countable.
2. A set $D \subseteq P$ is dense if for each $p \in P$ there is $q \in D$ with $q \le p$.
3. A filter on $P$ is a nonempty $G \subseteq P$ that is upward closed and in which any two elements have a common extension in $G$.

Let $\kappa$ be a [[cardinal|0QKPNZKNEA]].
The statement $\mathrm{MA}_\kappa$ concerns each ccc partial order $P$ and each family $\mathcal{D}$ of at most $\kappa$ dense subsets of $P$.
It says that some filter on $P$ meets every member of $\mathcal{D}$.
Martin's axiom, MA, is the statement that $\mathrm{MA}_\kappa$ holds for every $\kappa < 2^{\aleph_0}$, the [[cardinality of the continuum|0WC28W9DNM]].

## Remarks

Example: $\mathrm{MA}_{\aleph_0}$ is provable in ZFC.
Check: list the dense sets as $D_0, D_1, \ldots$, choose $p_0 \ge p_1 \ge \cdots$ with $p_n \in D_n$, and take the elements above some $p_n$.
So the continuum hypothesis implies MA.
Non-example: $\mathrm{MA}_{2^{\aleph_0}}$ is false.
Take the ccc order of finite partial functions from $\omega$ to $2$, ordered by reverse inclusion.
Consider the dense sets "$n$ is in the domain" and "the function differs from $x$", for all $n$ and all $x \in 2^\omega$.
A filter that meets them all would give a total function different from every $x$.
Solovay and Tennenbaum showed that MA together with $2^{\aleph_0} > \aleph_1$ is consistent, if ZFC is consistent.
MA with $2^{\aleph_0} > \aleph_1$ implies Suslin's hypothesis, and MA implies $2^\kappa = 2^{\aleph_0}$ for each infinite $\kappa < 2^{\aleph_0}$.
</document>

<document source="https://tamnd.github.io/math2/node/Z8XNDJSCJS.md" tag="Z8XNDJSCJS" kind="definition">
# Christoffel symbols

## Definition

Let $M$ be a smooth manifold of dimension $n$ with an [[affine connection|JX65KSVHSJ]] $\nabla$.
Let $(U, x^1, \dots, x^n)$ be a chart, and let $\partial_i = \partial / \partial x^i$ be the coordinate vector fields on $U$.
The *Christoffel symbols* of $\nabla$ in this chart are the $n^3$ smooth functions $\Gamma^k_{ij}$ on $U$ determined by
$$\nabla_{\partial_i} \partial_j = \sum_{k=1}^n \Gamma^k_{ij} \, \partial_k \qquad (1 \le i, j \le n).$$

## Remarks

They determine $\nabla$ on $U$: $\nabla_X Y = \sum_k \big( X Y^k + \sum_{i,j} \Gamma^k_{ij} X^i Y^j \big) \partial_k$ for $X = \sum X^i \partial_i$ and $Y = \sum Y^j \partial_j$.
For the Levi-Civita connection of a [[Riemannian metric|Q61VFHCH2J]] with components $g_{ij}$ and inverse matrix $(g^{kl})$,
$$\Gamma^k_{ij} = \tfrac12 \sum_l g^{kl} \big( \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij} \big).$$
Example: for the flat connection of $\R^2$ in polar coordinates $(r, \theta)$, where $g = dr^2 + r^2 d\theta^2$, this formula gives $\Gamma^r_{\theta\theta} = -r$, $\Gamma^\theta_{r\theta} = \Gamma^\theta_{\theta r} = 1/r$, and all other symbols vanish.
Non-example: the $\Gamma^k_{ij}$ are not the components of a tensor field; under a change of chart they transform with an extra term containing second derivatives of the coordinate change.
So they can vanish at a point in one chart and not in another; in Cartesian coordinates the symbols above are all $0$.
The connection is torsion-free exactly when $\Gamma^k_{ij} = \Gamma^k_{ji}$ in every chart.
</document>

<document source="https://tamnd.github.io/math2/node/Z937W8RW4N.md" tag="Z937W8RW4N" kind="definition">
# Congruence lattice

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.
Let $\operatorname{Con} \mathbf{A}$ be the set of all [[congruences|DCSXXT4SER]] on $\mathbf{A}$.
The *congruence lattice* of $\mathbf{A}$ is the partially ordered set $(\operatorname{Con} \mathbf{A}, \subseteq)$.
Its meet and join are
$$\theta \wedge \psi = \theta \cap \psi, \qquad \theta \vee \psi = \operatorname{Cg}(\theta \cup \psi),$$
where $\operatorname{Cg}$ is the [[congruence generated by a relation|QP06QH2Q7C]].

## Remarks

The join $\theta \vee \psi$ is also the [[transitive closure|05J9NWX07X]] of $\theta \cup \psi$, that is, the union of the relations $\theta \circ \psi$, $\theta \circ \psi \circ \theta$, and so on.
$\operatorname{Con} \mathbf{A}$ is a [[complete lattice|D7A0ZGZT5S]] with least element $\Delta_A$ and greatest element $\nabla_A = A \times A$.
It is an [[algebraic lattice|3J0BTXSCVN]], and its [[compact elements|9F7TYES4KE]] are the finitely generated congruences $\operatorname{Cg}(R)$ with $R$ finite.

Example: for the group $\Z/6\Z$, the congruences correspond to the subgroups $\set{0}$, $\set{0, 3}$, $\set{0, 2, 4}$ and $\Z/6\Z$.
So $\operatorname{Con}(\Z/6\Z)$ has four elements, with two incomparable elements in the middle.

Non-example: the union of two congruences need not be their join.
On the group $\Z$, the union of congruence modulo $2$ and modulo $3$ contains $(0, 2)$ and $(2, 5)$ but not $(0, 5)$, so it is not transitive.
</document>

<document source="https://tamnd.github.io/math2/node/Z93KMZ9R37.md" tag="Z93KMZ9R37" kind="statement">
# Reeb stability theorem

## Statement

Let $M$ be a closed connected manifold and let $\mathcal{F}$ be a transversely orientable [[foliation|98HTB7MSPX]] of $M$ of class $C^1$ and codimension $1$.
Assume that $\mathcal{F}$ has a [[compact|0TVJZYB6SH]] leaf $L$ whose [[fundamental group|S1XK3HXHFX]] is finite.
Then every leaf of $\mathcal{F}$ is compact and diffeomorphic to $L$, and there is a fibre bundle $M \to S^1$ whose fibres are the leaves of $\mathcal{F}$.

## Remarks

The theorem is a global consequence of the local Reeb stability theorem.
The local theorem says that a compact leaf with finite holonomy group has a neighbourhood made of leaves that are finite covers of it.
Idea of the proof: since $\pi_1(L)$ is finite and its holonomy acts on a transverse interval preserving orientation, the holonomy is trivial, so nearby leaves are copies of $L$.
The set of points on compact leaves diffeomorphic to $L$ is then open, and a limit argument shows it is closed, so it is all of $M$.

Example: if a closed $3$-manifold $M$ has a transversely orientable codimension one foliation with a leaf homeomorphic to $S^2$, the theorem shows that $M$ fibres over $S^1$ with fibre $S^2$.
The hypothesis on the fundamental group is needed: the [[Reeb foliation|W58NYR1DGR]] of $S^3$ has a compact torus leaf, with infinite fundamental group, while all its other leaves are planes.

## Explanations

### L1 human

A foliation slices a manifold into sheets of one lower dimension.
The theorem says that one closed sheet with almost no loops forces all sheets to be copies of it.
Then the manifold is a stack of these copies arranged around a circle.

### L2 human

The theorem shows that spheres and other leaves with finite fundamental group are rigid inside codimension one foliations.
It is a first step in the study of compact leaves, which leads to Novikov's compact leaf theorem.
The key notion is holonomy, the return map of a transverse interval along loops in a leaf, see the [[holonomy groupoid of a foliation|HMRP135M6K]].
A corollary is that $S^3$ carries no codimension one foliation with a leaf diffeomorphic to $S^2$, since $S^3$ does not fibre over the circle.
</document>

<document source="https://tamnd.github.io/math2/node/Z95Q7B022R.md" tag="Z95Q7B022R" kind="problem">
# Kervaire invariant problem

## Problem

For which integers $n = 4k + 2$ is there a closed framed smooth manifold of dimension $n$ whose [[Kervaire invariant|P4YXE08ZXF]] equals $1$?
Equivalently, in which degrees $n$ do the [[stable homotopy groups of spheres|HGZAYPQBXW]] $\pi^s_n$ contain an element of Kervaire invariant $1$?

## Remarks

Kervaire showed in 1960 that the invariant vanishes in dimension $10$.
Browder showed in 1969 that the invariant can be nonzero only in dimensions $n = 2^j - 2$.
Framed manifolds of Kervaire invariant $1$ exist in dimensions $2$, $6$, $14$, $30$ and $62$.
Hill, Hopkins and Ravenel showed that none exist in dimensions $2^j - 2$ with $j \ge 8$, that is, in dimension $254$ and above.
This left dimension $126$ as the only open case; in 2024 Weinan Lin, Guozhen Wang and Zhouli Xu announced that a manifold of Kervaire invariant $1$ exists in dimension $126$.

## Explanations

### L1 human

The Kervaire invariant is a yes-or-no number attached to certain framed manifolds of dimension $4k + 2$.
The problem asks in which dimensions it can be "yes".
The answer turned out to be a short and strange list: $2$, $6$, $14$, $30$, $62$, and, by the 2024 announcement, $126$.
The question was a guide for decades of work on the homotopy groups of spheres.
</document>

<document source="https://tamnd.github.io/math2/node/Z97HVFD6JE.md" tag="Z97HVFD6JE" kind="concept">
# Parametrix

## Summary

A parametrix of a differential operator is an approximate inverse: an operator whose composites with the given operator differ from the identity by smoothing operators.
For operators on $\R^n$ it is the same as an approximate [[fundamental solution|K2NK9A8M5F]], one whose error is a smooth function instead of zero.
Every [[elliptic operator|MT9P2X85V8]] has a parametrix, which can be built as a [[pseudo-differential operator|FF27N0BWTP]] from the inverse of the principal symbol.
Parametrices are the main tool for proving elliptic regularity and the Fredholm property of elliptic operators on compact manifolds.

## Explanations

### L1 human

Inverting a differential operator exactly is usually hard, and often impossible because the operator has a kernel.
A parametrix gives up exactness but keeps what matters for smoothness: it inverts the operator up to an error that turns every input, however rough, into a smooth output.
Such errors are harmless for questions of regularity, and on a compact manifold they behave like finite-rank errors.
For the operator $-d^2/dx^2$ on the circle, dividing each Fourier coefficient $c_n$ with $n \ne 0$ by $n^2$ is a parametrix, and its error only sees the average of the function.
</document>

<document source="https://tamnd.github.io/math2/node/Z9DN23DAE2.md" tag="Z9DN23DAE2" kind="concept">
# Weil divisor

## Summary

A Weil divisor on a suitable scheme $X$ is a formal sum with integer coefficients of finitely many closed subvarieties of codimension one.
The zeros and poles of a rational function, counted with multiplicity, form a principal Weil divisor.
The quotient of all Weil divisors by the principal ones is the divisor class group, which equals the Picard group on smooth varieties.

## Explanations

### L1 human

On a curve, a divisor is a finite list of points with integer weights, such as $2P - 3Q$.
On a surface, the points are replaced by curves, and in general by subvarieties of codimension one.
A rational function gives a divisor: its zeros with positive weight and its poles with negative weight.
For example, on the projective line, the function $x^2/(x - 1)$ has divisor $2[0] - [1] - [\infty]$.
</document>

<document source="https://tamnd.github.io/math2/node/Z9DYVG5RA3.md" tag="Z9DYVG5RA3" kind="definition">
# Topological manifold

## Definition

Let $n \ge 0$ be an integer and let $M$ be a [[topological space|0CQHFPGJ45]].
$M$ is a *topological manifold of dimension $n$*, or a topological $n$-manifold, if:

1. $M$ is [[Hausdorff|0S1FJ8VEX9]].
2. $M$ is [[second-countable|03PPNCM9GG]].
3. $M$ is locally Euclidean of dimension $n$: each point $x \in M$ has an [[open|02BH3MCBQB]] neighbourhood $U$ [[homeomorphic|0QT00PB5G2]] to an open subset of $\R^n$, the [[Euclidean space|0XPR1E85W0]].

## Remarks

The unit circle $S^1 = \set{(x, y) \in \R^2 : x^2 + y^2 = 1}$ is a $1$-manifold.
It is Hausdorff and second-countable as a subspace of $\R^2$.
The map $(x, y) \mapsto x$ sends the open upper half $\set{y > 0}$ of $S^1$ homeomorphically onto the interval $(-1, 1)$.
The three other open half circles are treated in the same way, and together the four cover $S^1$.

The line with two origins is the quotient of $\R \times \set{0, 1}$ obtained by identifying $(t, 0)$ with $(t, 1)$ for all $t \ne 0$.
It satisfies clauses 2 and 3, but it fails clause 1: the two origins have no disjoint neighbourhoods.

A nonempty manifold has only one dimension, by [[invariance of domain|6X2D0JDFQW]]: a nonempty open subset of $\R^m$ is not homeomorphic to an open subset of $\R^n$ when $m \ne n$.
A pair $(U, \varphi)$ with $\varphi$ a homeomorphism from $U$ onto an open subset of $\R^n$ is called a chart.
A compact manifold without boundary is called a closed manifold.
</document>

<document source="https://tamnd.github.io/math2/node/Z9HNJ529DF.md" tag="Z9HNJ529DF" kind="definition">
# Flag variety

## Definition

Let $k$ be a field, let $V$ be a vector space of dimension $n$ over $k$, and let $0 < d_1 < \cdots < d_m < n$ be integers.
A *flag* of type $(d_1, \ldots, d_m)$ in $V$ is a chain of subspaces $V_1 \subset \cdots \subset V_m$ with $\dim V_i = d_i$.
The *flag variety* $\mathrm{Fl}(d_1, \ldots, d_m; V)$ is the set of these flags.
It is a closed subvariety of the product of [[Grassmannians|CYPA882RB9]] $\prod_i \mathrm{Gr}(d_i, V)$, so it is a [[projective variety|A4H4CZ2V35]].
The *full flag variety* is the case $(d_1, \ldots, d_m) = (1, 2, \ldots, n - 1)$.

## Remarks

The group $\mathrm{GL}(V)$ acts transitively on the flags of a given type, so the flag variety is a [[homogeneous space|85TWR6E75P]].
The stabilizer of a full flag is a [[Borel subgroup|5JSYA1JNCC]], conjugate to the upper triangular matrices, so the full flag variety is $\mathrm{GL}_n / B$.

General form: let $G$ be a connected [[reductive group|2XS84PWE3S]] over an algebraically closed field.
A closed subgroup $P$ of $G$ is *parabolic* if it contains a Borel subgroup.
The *generalized flag variety* is $G / P$, a smooth projective homogeneous variety.

Example: for $m = 1$, the flag variety $\mathrm{Fl}(d_1; V)$ is the Grassmannian $\mathrm{Gr}(d_1, V)$, and for $d_1 = 1$ it is the projective space $\mathbb{P}(V)$.

Non-example: let $T$ be the diagonal torus of $\mathrm{GL}_2$, the stabilizer of the pair of coordinate lines.
The quotient $\mathrm{GL}_2 / T$ is the space of ordered pairs of distinct lines in $k^2$, which is $\mathbb{P}^1 \times \mathbb{P}^1$ minus its diagonal.
It is not projective, since $T$ contains no Borel subgroup, and it is not a flag variety.
</document>

<document source="https://tamnd.github.io/math2/node/Z9TJSZZYG1.md" tag="Z9TJSZZYG1" kind="concept">
# Prime ring

## Summary

A prime ring is a nonzero ring in which $aRb = 0$ forces $a = 0$ or $b = 0$.
Equivalently, the product of two nonzero two-sided ideals is nonzero, that is, the zero ideal is a [[prime ideal|E8K2MN5V1E]].
It is the noncommutative replacement for an integral domain: a commutative ring is prime exactly when it is an integral domain.
Matrix rings over prime rings are prime, so prime rings may have many zero divisors.

## Explanations

### L1 human

In an integral domain, $ab = 0$ forces $a = 0$ or $b = 0$.
Matrix rings fail this, since two nonzero matrices can multiply to zero.
A prime ring asks for something weaker: $a$ times anything times $b$ must not always be zero.
The ring $M_2(\Q)$ is prime even though $e_{11} e_{22} = 0$, because $e_{11} e_{12} e_{22} = e_{12} \ne 0$.
</document>

<document source="https://tamnd.github.io/math2/node/Z9WE435T9G.md" tag="Z9WE435T9G" kind="definition">
# Symbol class

## Definition

Let $n \ge 1$, let $m \in \R$, and let $0 \le \delta \le \rho \le 1$ with $\delta < 1$.
The *symbol class* $S^m_{\rho, \delta}(\R^n \times \R^n)$ is the set of [[smooth functions|0V33BNKZ9S]] $a \colon \R^n \times \R^n \to \C$, $(x, \xi) \mapsto a(x, \xi)$, such that for all multi-indices $\alpha, \beta \in \N^n$ there is a constant $C_{\alpha\beta}$ with
$$\abs{\partial_\xi^\alpha \partial_x^\beta a(x, \xi)} \le C_{\alpha\beta} \, (1 + \abs{\xi})^{m - \rho \abs{\alpha} + \delta \abs{\beta}} \quad \text{for all } x, \xi \in \R^n.$$
Here $\partial_\xi^\alpha$ and $\partial_x^\beta$ are iterated [[partial derivatives|09EGC6F8PT]] and $\abs{\alpha} = \alpha_1 + \cdots + \alpha_n$.
The elements of $S^m_{\rho, \delta}$ are *symbols of order $m$* and *type $(\rho, \delta)$*.
One writes $S^m = S^m_{1, 0}$ and $S^{-\infty} = \bigcap_{m \in \R} S^m$.

## Remarks

Example: let $a(x, \xi) = \sum_{\abs{\gamma} \le k} a_\gamma(x) \xi^\gamma$, where each $a_\gamma$ is smooth and bounded with all its derivatives.
Then $a \in S^k$.
Indeed $\partial_\xi^\alpha \xi^\gamma$ is zero or a monomial of degree $\abs{\gamma} - \abs{\alpha}$, so it is bounded by a constant times $(1 + \abs{\xi})^{k - \abs{\alpha}}$.
Derivatives in $x$ only change the bounded coefficients.
Also $(1 + \abs{\xi}^2)^{s/2} \in S^s$ for every $s \in \R$.

Non-example: $a(x, \xi) = \abs{\xi}$ is not in any symbol class, because it is not smooth at $\xi = 0$.
Similarly $a(\xi) = e^{i \abs{\xi}^2}$ is smooth but lies in no class $S^m_{1,0}$, because its derivatives of order $k$ in $\xi$ grow like $\abs{\xi}^k$.

The classes are nested: $S^m_{\rho, \delta} \subseteq S^{m'}_{\rho, \delta}$ for $m \le m'$.
Products satisfy $S^m_{\rho, \delta} \cdot S^{m'}_{\rho, \delta} \subseteq S^{m + m'}_{\rho, \delta}$, and $\partial_\xi^\alpha$ maps $S^m_{\rho, \delta}$ into $S^{m - \rho\abs{\alpha}}_{\rho, \delta}$.
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<document source="https://tamnd.github.io/math2/node/ZA11N5XPZM.md" tag="ZA11N5XPZM" kind="statement">
# Hurewicz theorem

## Statement

Let $X$ be a [[path-connected|0WB6DH499T]] space with base point $x_0$, and let $n \ge 1$.
Fix a generator $u_n$ of $H_n(S^n; \Z) \cong \Z$.
The *Hurewicz map* $h \colon \pi_n(X, x_0) \to H_n(X; \Z)$ sends the class of $f \colon S^n \to X$ to $f_*(u_n)$.
Here $\pi_n$ is the [[homotopy group|P976WYEFMB]] and $H_n$ is [[singular homology|M0Q5CFJ7ZH]].

1. For $n = 1$, $h$ is surjective and its kernel is the commutator subgroup of the [[fundamental group|S1XK3HXHFX]]. Hence $H_1(X; \Z)$ is the [[abelianisation|0AQZV0905N]] of $\pi_1(X, x_0)$.
2. For $n \ge 2$, assume that $X$ is [[$(n-1)$-connected|1A943BWBQT]]. Then the [[reduced homology|NJQMWVK3QK]] $\tilde H_k(X; \Z)$ vanishes for $k < n$, and $h \colon \pi_n(X, x_0) \to H_n(X; \Z)$ is an isomorphism.

## Remarks

The map $h$ is a homomorphism, because a sum of maps $S^n \to X$ factors through the pinch map $S^n \to S^n \vee S^n$.
For $n \ge 2$, one replaces $X$ by a weakly equivalent CW complex with a single $0$-cell and no cells of dimensions $1$ to $n - 1$.
Then both groups are computed from the $n$-cells and the attaching maps of the $(n+1)$-cells.
Example: $S^n$ is $(n-1)$-connected and $H_n(S^n) \cong \Z$, so $\pi_n(S^n) \cong \Z$, generated by the identity.
The isomorphism stops at the first nonzero degree: $S^2$ is $1$-connected, yet $\pi_3(S^2) \cong \Z$ while $H_3(S^2) = 0$.

## Explanations

### L1 human

Homotopy groups are hard to compute, and homology groups are much easier.
The theorem says that the two agree in the first dimension where something happens.
In dimension one, homology is the fundamental group made commutative.

### L2 human

The theorem gives the first nonzero homotopy group of a simply connected space from its homology.
With the [[Whitehead theorem|6P1E2FJPSD]], it shows that a homology isomorphism between simply connected CW complexes is a homotopy equivalence.
A relative version for pairs of spaces extends it further.
It is also the starting point for Serre's mod $\mathcal{C}$ theory, which compares homotopy and homology modulo classes of abelian groups.
</document>

<document source="https://tamnd.github.io/math2/node/ZA1HC10WM6.md" tag="ZA1HC10WM6" kind="concept">
# Transform pair

## Summary

A transform pair is a function $f$ together with its image $F = Tf$ under an [[integral transform|YCK709QB06]] $T$.
It is often written $f \leftrightarrow F$ or $f(t) \leftrightarrow F(s)$.
Tables of transform pairs are the main working tool of the [[Laplace transform|0V49JEZW06]] and the [[Fourier transform|DPVM34ADZV]].
They let one go back from $F$ to $f$ without computing an [[inverse transform|GC96MQC0PP]].

## Explanations

### L1 human

A transform pair is one line of a dictionary between two languages.
On the left is a function of time, on the right is its transform.
For the Laplace transform, $e^{at} \leftrightarrow 1/(s - a)$ is such a line.
When a calculation ends with $1/(s - a)$, one looks up the line and reads off $e^{at}$.
This only works because the transform is one-to-one on the functions in the table.
</document>

<document source="https://tamnd.github.io/math2/node/ZA405B0Z1S.md" tag="ZA405B0Z1S" kind="concept">
# Dehn surgery

## Summary

Dehn surgery builds a new [[3-manifold|R20TB2KEQQ]] from a [[knot|QYEAZTYGPT]] or [[link|1A5WRCNTM3]] in a given one.
One removes a solid torus around the knot.
Then one glues it back so that its meridian disc caps off a chosen curve on the boundary torus.
For knots in $S^3$ the choices are labelled by a number $p/q \in \Q \cup \set{\infty}$.
By the [[Lickorish–Wallace theorem|1YP4RF1QXV]], every closed orientable $3$-manifold arises by Dehn surgery on a link in $S^3$.

## Explanations

### L1 human

Think of a knotted rope inside space, wrapped in a thin solid tube.
Remove the tube and put a solid tube back in, but twisted.
Now the disc that used to go around the rope closes off a different loop on the tube's surface.
The space you get may be completely new.
For the simplest knot, the round circle, surgery with label $0$ gives the product of a sphere and a circle.
</document>

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# CSP dichotomy theorem

## Statement

Let $\mathbb{B}$ be a finite [[relational structure|MJF9M19E3E]] with finitely many relations.
The problem $\mathrm{CSP}(\mathbb{B})$ takes as input a finite structure $\mathbb{A}$ of the same signature and asks whether there is a [[homomorphism|KBXSZC6BV1]] from $\mathbb{A}$ to $\mathbb{B}$.
A finite structure is a *core* if each of its endomorphisms is an automorphism.
Let $\mathbb{C}$ be the core of $\mathbb{B}$, that is, a core that admits homomorphisms $\mathbb{B} \to \mathbb{C}$ and $\mathbb{C} \to \mathbb{B}$; it is unique up to isomorphism.
A *Taylor polymorphism* of $\mathbb{C}$ is a [[polymorphism|A0K9EZRM7E]] $t$ of $\mathbb{C}$ of some arity $n \ge 2$ that is a [[Taylor operation|QEAMRNRJJD]].
That is, $t(x, \ldots, x) = x$ for all $x$.
Also, for each position $i \le n$, $t$ satisfies an identity $t(u_1, \ldots, u_n) \approx t(v_1, \ldots, v_n)$ with each $u_j, v_j \in \set{x, y}$, $u_i = x$ and $v_i = y$.
Then exactly one of the following holds.

1. $\mathbb{C}$ has a Taylor polymorphism, and $\mathrm{CSP}(\mathbb{B})$ is solvable in polynomial time.
2. $\mathbb{C}$ has no Taylor polymorphism, and $\mathrm{CSP}(\mathbb{B})$ is NP-complete.

In particular, each such constraint satisfaction problem is either in P or NP-complete.

## Remarks

The two algebraic cases exclude each other.
The two complexity outcomes are different only if $\mathrm{P} \ne \mathrm{NP}$.
The hardness half was proved earlier in the [[algebraic approach to constraint satisfaction|3EH4F5H7FT]].
Without a Taylor polymorphism, the polymorphism clone of the core allows a reduction from an NP-complete problem such as 3-colouring.
The tractability half was proved independently by Andrei Bulatov and by Dmitriy Zhuk, with two different polynomial-time algorithms.
This settled the [[Feder–Vardi dichotomy conjecture|WNHZC07XVP]].
By a theorem of Siggers, a finite core has a Taylor polymorphism exactly when it has an idempotent 6-ary polymorphism $s$ of a special form.
This $s$ satisfies $s(x, y, x, z, y, z) = s(y, x, z, x, z, y)$ for all $x, y, z$; see [[Siggers term|JYW1H3J8SX]].

Example: let $\mathbb{B} = K_n$ be the complete graph on $n$ vertices.
Then $\mathrm{CSP}(K_n)$ is the $n$-colouring problem.
For $n = 2$, the majority operation on the two vertices is a polymorphism and a Taylor operation.
So $2$-colouring is in P.
For $n \ge 3$ the graph $K_n$ is a core with no Taylor polymorphism, and $n$-colouring is NP-complete.
More generally, let $H$ be a finite undirected graph.
Then $\mathrm{CSP}(H)$ is in P if $H$ is bipartite or has a loop, and NP-complete otherwise (Hell and Nešetřil).

Passing to the core matters, because a Taylor polymorphism must be idempotent.
For example, the disjoint union of $K_3$ and a single vertex with a loop has no Taylor polymorphism.
Its core is the looped vertex, which has one, and indeed its CSP is trivial: every input maps to the loop.

## Explanations

### L1 human

Many puzzles ask to label the points of a finite input so that given local rules hold, as in map colouring or Sudoku-like constraints.
When the allowed labels and rules are fixed, each such puzzle type is a constraint satisfaction problem.
The theorem says that every such problem type is either solvable by a fast algorithm or as hard as any problem in NP.
There is nothing in between.
Which side a problem falls on is decided by an algebraic test: whether the rules are preserved by a certain kind of "averaging" operation.

### L2 human

The polymorphisms of $\mathbb{B}$ form a clone, and the complexity of $\mathrm{CSP}(\mathbb{B})$ depends only on the identities that this clone satisfies.
The Taylor condition is the weakest nontrivial such identity, and by the cyclic term and Siggers term theorems it can be tested by a single identity.
Earlier special cases include Schaefer's theorem for Boolean structures and the Hell–Nešetřil theorem for undirected graphs.
For infinite templates, and for counting or optimisation versions, separate classification programs exist.
</document>

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# Singular perturbation

## Definition

Let $D$ be a set and let $\varepsilon_0 > 0$.
For each $\varepsilon \in (0, \varepsilon_0)$, let $P_\varepsilon$ be a problem on $D$, such as a [[boundary value problem|TM8VQJWGHZ]] for an [[ordinary differential equation|0NT64JV96A]].
Assume that the data of $P_\varepsilon$ have a limit $P_0$ as $\varepsilon \to 0$.
The problem $P_0$ is the *reduced problem*.
Let $u_\varepsilon \colon D \to \R^n$ be a solution of $P_\varepsilon$.
The family is a *regular perturbation* of $P_0$ if $P_0$ has a solution $u_0$ and $u_\varepsilon \to u_0$ uniformly on $D$ as $\varepsilon \to 0$.
The family is a *singular perturbation* if it is not a regular perturbation.
This means that $P_0$ has no solution, or that $u_\varepsilon$ does not converge uniformly on $D$ to a solution of $P_0$.

## Remarks

Example: $P_\varepsilon$ is $\varepsilon y'' + y' = 0$ on $D = [0, 1]$ with $y(0) = 0$ and $y(1) = 1$.
The solution is $y_\varepsilon(x) = (1 - e^{-x/\varepsilon}) / (1 - e^{-1/\varepsilon})$.
The reduced problem $y' = 0$, $y(0) = 0$, $y(1) = 1$ has no solution.
Indeed $y_\varepsilon(x) \to 1$ for each $x \in (0, 1]$ while $y_\varepsilon(0) = 0$, so the convergence is not uniform.

Non-example: $y' = -y + \varepsilon$ on $[0, 1]$ with $y(0) = 1$.
The solution $y_\varepsilon(x) = \varepsilon + (1 - \varepsilon) e^{-x}$ converges uniformly to $e^{-x}$, the solution of the reduced problem, so the perturbation is regular.

A small parameter multiplying the highest derivative usually makes the problem singular.
Singular problems are treated by the method of matched asymptotic expansions and the method of multiple scales.

## Explanations

### L1 human

A perturbation is regular when the solution at $\varepsilon = 0$ is a good approximation everywhere.
It is singular when this fails somewhere, often near the boundary.
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# Boutet de Monvel's embedding theorem

## Statement

Let $M$ be a compact smooth [[CR manifold|ZC78JKZX9S]] of hypersurface type, of real dimension $2n + 1$ with $n \ge 2$.
Assume that $M$ is strongly pseudoconvex, that is, its [[Levi form|80MTNXWW80]] is definite at every point.
Then there is a smooth CR embedding of $M$ into $\C^N$ for some $N$.

## Remarks

The theorem is due to L. Boutet de Monvel.
The proof uses Kohn's analysis of the tangential Cauchy–Riemann operator $\bar\partial_b$ and of the [[Kohn Laplacian|ZBQ6TVNZZ8]].
On a compact strongly pseudoconvex manifold of real dimension at least $5$, this analysis gives good solvability of $\bar\partial_b$.
This produces many global [[CR functions|JFKYYK2CEB]], and finitely many of them give an embedding.
The boundary of a bounded strongly pseudoconvex domain in $\C^{n+1}$ with smooth boundary is an example where the embedding is given from the start.
The bound on the dimension is necessary: in real dimension $3$ there are compact strongly pseudoconvex CR manifolds with no CR embedding into any $\C^N$, such as [[Rossi's example|1TJXANASFE]].

## Explanations

### L1 human

A CR manifold carries part of the structure of a hypersurface in complex space, but it is defined abstractly, without a surrounding space.
Assume the manifold is compact, curves in the strongly convex way, and has dimension at least five.
Then the theorem says that it really sits inside some complex space.
So in these dimensions the abstract notion gives nothing new.

### L2 human

The result is a global embedding theorem, in contrast with local realisation questions, where Kuranishi and others worked on embedding small pieces.
Combined with results on filling, it shows that such $M$ bound complex spaces with isolated singularities.
The three-dimensional case behaves very differently and is an active subject: there embeddability is a delicate property that is not preserved under small deformations.
</document>

<document source="https://tamnd.github.io/math2/node/ZAJFW1S078.md" tag="ZAJFW1S078" kind="concept">
# Curvature form

## Summary

The curvature form of a [[connection on a principal bundle|NJ21PN5ZGJ]] is a $2$-form on the total space.
Its values lie in the Lie algebra of the structure group.
It measures the failure of the horizontal distribution of the connection to be integrable, and it vanishes exactly when the connection is flat.
For connections on vector bundles it becomes a $2$-form with values in endomorphisms, and for the Levi-Civita connection it recovers the Riemann curvature tensor.
Its invariant polynomials give the characteristic forms of Chern–Weil theory.

## Explanations

### L1 human

A connection tells how to move horizontally from fibre to fibre.
If you move around a tiny loop in the base and lift the loop horizontally, you may come back to a different point of the same fibre.
The curvature form records this failure, loop direction by loop direction.
In electromagnetism, where the group is the circle, the curvature form is the electromagnetic field.
</document>

<document source="https://tamnd.github.io/math2/node/ZANKHGE1YC.md" tag="ZANKHGE1YC" kind="definition">
# Well-founded order

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]], and write $x < y$ when $x \le y$ and $x \ne y$.
The order is *well-founded* when every nonempty subset $A \subseteq P$ has a minimal element, that is, an element $m \in A$ such that no $a \in A$ satisfies $a < m$.

## Remarks

Example: the set $\N^2$ with the product order, $(a, b) \le (c, d)$ when $a \le c$ and $b \le d$, is well-founded.
Check: in a nonempty $A \subseteq \N^2$, take the least first coordinate $a_0$ occurring in $A$, and then the least $b_0$ with $(a_0, b_0) \in A$; no element of $A$ is strictly below $(a_0, b_0)$.

Non-example: the set $\set{1/n : n \ge 1}$ of rational numbers with the usual order is not well-founded, since the whole set has no minimal element.

The order is well-founded exactly when the strict order $<$ is a [[well-founded relation|0DQPNJNPYK]].
With the axiom of dependent choice, the condition is equivalent to the absence of an infinite sequence $x_0 > x_1 > x_2 > \cdots$.
The same definition applies to a preorder, with $x < y$ meaning $x \le y$ and not $y \le x$.
</document>

<document source="https://tamnd.github.io/math2/node/ZARKMD0C0H.md" tag="ZARKMD0C0H" kind="definition">
# Allen–Cahn equation

## Definition

Let $\Omega \subseteq \R^n$ be open and let $\varepsilon > 0$.
Let $W(u) = \frac{1}{4}(1 - u^2)^2$, so that $W'(u) = u^3 - u$.
The *Allen–Cahn equation* is the [[reaction–diffusion equation|TAAHWE917M]] for $u \colon \Omega \times (0, T) \to \R$:
$$u_t = \Delta u - \frac{1}{\varepsilon^2} W'(u) = \Delta u - \frac{1}{\varepsilon^2}(u^3 - u),$$
where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] in $x$.

## Remarks

Example: with $n = 1$, the function $u(x) = \tanh\!\big(x/(\sqrt{2}\,\varepsilon)\big)$ is a stationary solution.
Indeed, with $s = x/(\sqrt{2}\,\varepsilon)$ one finds $u'' = -\varepsilon^{-2}\operatorname{sech}^2(s)\tanh(s) = \varepsilon^{-2}(u^3 - u)$.
Non-example: the Cahn–Hilliard equation $u_t = \Delta(u^3 - u - \varepsilon^2 \Delta u)$ is of fourth order, so it is not of this form.
The equation is the $L^2$ gradient flow of the energy
$$E(u) = \int_\Omega \Big( \frac{1}{2}\abs{\nabla u}^2 + \frac{1}{\varepsilon^2} W(u) \Big)\, dx,$$
so $E$ does not increase along smooth solutions with Neumann boundary conditions.
Some authors scale the equation as $u_t = \varepsilon^2 \Delta u - W'(u)$ instead.
</document>

<document source="https://tamnd.github.io/math2/node/ZARM34FZ6Z.md" tag="ZARM34FZ6Z" kind="concept">
# Moment map

## Summary

A moment map, also called a momentum map, packages the conserved quantities of a symmetry of a [[symplectic manifold|CAN490EPRT]] into one map.
When a [[Lie group|7JBRGCVJRB]] $G$ acts on $(M, \omega)$, a moment map is a map $\mu \colon M \to \mathfrak{g}^*$ whose components generate the action as [[Hamiltonian vector fields|C5EH3B4BEP]].
It is the main tool for symplectic reduction, for convexity theorems and for toric geometry.

## Explanations

### L1 human

In mechanics, a symmetry gives a conserved quantity: translations give linear momentum and rotations give angular momentum.
The moment map collects these quantities into a single map from phase space to the dual of the Lie algebra of the symmetry group.
For the circle rotating the plane about the origin, the moment map is, up to sign and an added constant, half the squared distance to the origin.
Its level sets are the circles that the rotation preserves.
</document>

<document source="https://tamnd.github.io/math2/node/ZAT4TV4030.md" tag="ZAT4TV4030" kind="concept">
# Heat potential

## Summary

A heat potential is the integral of the fundamental solution of the [[heat equation|VP2HN86VAG]] against a measure or a density in space-time.
It plays for the heat equation the role that the Newtonian potential plays for the Laplace equation.
With a density as source, it solves the inhomogeneous heat equation with zero initial data, which is Duhamel's principle.
Heat potentials are the basic objects of the potential theory of the heat equation, with its own notions of capacity and of regular boundary points.

## Explanations

### L1 human

Drop a unit of heat at one point at one moment: the temperature afterwards is the [[heat kernel|KD404MF042]], a Gaussian bump that spreads out as time passes.
If heat is added at many points and many moments, the temperatures from all the drops add up.
The heat potential is this sum, written as an integral.
Unlike the Newtonian potential, it only looks into the past: heat added later cannot affect the temperature now.
</document>

<document source="https://tamnd.github.io/math2/node/ZATS7NV9WD.md" tag="ZATS7NV9WD" kind="concept">
# Linear discriminant analysis

## Summary

Linear discriminant analysis assigns an observation to one of several classes with a rule whose decision boundaries are hyperplanes.
It is the optimal rule when each class has a [[multivariate normal distribution|0XN3K7RSZY]] and all classes share one [[covariance matrix|0V7G6WMFK0]].
In practice the means and the common covariance are estimated from labelled data.
It is closely tied to Fisher's linear discriminant, which also gives a dimension reduction.

## Explanations

### L1 human

Two kinds of flowers differ on average in petal length and width, but the measurements overlap.
Linear discriminant analysis draws a straight line through the plane of measurements and calls every flower on one side the first kind.
The line is placed using the class averages and the shared spread, measured by the [[Mahalanobis distance|P546TSFBK2]].
For one measurement with class means $-1$ and $1$, equal variances and equal sizes, the rule simply looks at the sign of the measurement.
</document>

<document source="https://tamnd.github.io/math2/node/ZAVMXRK4J9.md" tag="ZAVMXRK4J9" kind="definition">
# Subadditive function

## Definition

Let $(A, +)$ be a [[semigroup|0XMEM406JM]], for example $[0, \infty)$, $\R$ or the positive integers under addition.
A [[function|0EPAT51DXQ]] $f \colon A \to \R$ is *subadditive* if
$$f(x + y) \le f(x) + f(y) \quad \text{for all } x, y \in A.$$

## Remarks

Example: $f(x) = \sqrt{x}$ on $[0, \infty)$ is subadditive.
Check: both sides of $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$ are nonnegative, and squaring gives $x + y \le x + y + 2\sqrt{xy}$, which is true.

Non-example: $f(x) = x^2$ on $[0, \infty)$ is not subadditive, since $f(2) = 4 > 2 = f(1) + f(1)$.

The defining condition is a [[functional inequality|YQB2B0EXY8]].
A subadditive sequence is a subadditive function on the positive integers, $a_{m+n} \le a_m + a_n$.
Every concave function $f \colon [0, \infty) \to \R$ with $f(0) \ge 0$ is subadditive.
</document>

<document source="https://tamnd.github.io/math2/node/ZAZFMK7C67.md" tag="ZAZFMK7C67" kind="definition">
# Steiner conic

## Definition

Let $\Pi$ be a [[projective plane|DANFNN8C4Y]].
For a point $U$, let $B(U)$ be the pencil of all lines through $U$.
For a line $g$, let $R(g)$ be the range of all points on $g$.
For a point $U$ and a line $g$ not through $U$, the maps $B(U) \to R(g)$, $m \mapsto m \cap g$, and its inverse $R(g) \to B(U)$, $P \mapsto PU$, are [[perspectivities|QTZWZR1KWQ]].
A *projectivity* $B(U) \to B(V)$ is a composition of finitely many such perspectivities.
Let $U$ and $V$ be distinct points and let $\pi \colon B(U) \to B(V)$ be a projectivity with $\pi(UV) \ne UV$.
The *Steiner conic* of $\pi$ is the set of points
$$C = \set{m \cap \pi(m) : m \in B(U)}.$$

## Remarks

The condition $\pi(UV) \ne UV$ makes $m$ and $\pi(m)$ distinct lines for every $m$, so each intersection is a single point.
The points $U$ and $V$ lie on $C$: $UV \cap \pi(UV) = V$, and the line $m$ with $\pi(m) = UV$ gives $m \cap UV = U$.

Example: in the [[projective plane|ZT9A6JK8QF]] over a field $F$, let $U = [1 : 0 : 0]$ and $V = [0 : 0 : 1]$.
The lines through $U$ are $by + cz = 0$ and the lines through $V$ are $ax + by = 0$, with $(b, c)$ and $(a, b)$ up to scaling.
Let $\pi$ send $by + cz = 0$ to $bx + cy = 0$; this map is linear in the parameters, and over a field such maps are projectivities.
The line $UV$ is $y = 0$, and $\pi$ sends it to $x = 0$, so $\pi(UV) \ne UV$.
Solving $by + cz = 0$ and $bx + cy = 0$ with $t = -c/b$ gives the points $[t^2 : t : 1]$, and $b = 0$ gives $U$.
So $C$ is the conic $xz = y^2$.

Non-example: let $a$ be a line through neither $U$ nor $V$, and let $\pi(m)$ be the line joining $V$ to $m \cap a$.
Then $\pi(UV) = UV$, so the condition fails.
For $m \ne UV$ the point $m \cap \pi(m)$ is $m \cap a$, so these points are the points of the line $a$ other than $a \cap UV$, not a conic.

In a plane over a field, Steiner's theorem says that the Steiner conics are exactly the non-degenerate [[projective conics|R3TMVHTP3A]].
</document>

<document source="https://tamnd.github.io/math2/node/ZB91TH6FKV.md" tag="ZB91TH6FKV" kind="statement">
# First and second variation of arc length

## Statement

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] with Levi-Civita connection $\nabla$ and curvature $R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$ (see [[Riemann curvature tensor|JA29ETCM02]]), and write $\ip{\cdot}{\cdot}$ for $g$.
Let $\gamma \colon [a, b] \to M$ be a smooth curve with $\abs{\gamma'(t)} = 1$ for all $t$.
Let $\Gamma \colon (-\varepsilon, \varepsilon) \times [a, b] \to M$ be a smooth map with $\Gamma(0, t) = \gamma(t)$, let $V(t) = \partial_s \Gamma(0, t)$ be its variation field, and let $L(s)$ be the [[length|0TH78S7GBZ]] of the curve $t \mapsto \Gamma(s, t)$.
Let $D_t$ and $D_s$ be the [[covariant derivatives|5FZDQQHQKM]] along the curves $t \mapsto \Gamma(s, t)$ and $s \mapsto \Gamma(s, t)$.

1. (First variation.) $$L'(0) = \Bigl[ \ip{V}{\gamma'} \Bigr]_{t=a}^{t=b} - \int_a^b \ip{V}{D_t \gamma'} \, dt.$$
2. (Second variation.) If $\gamma$ is a [[geodesic|NSYMQMHPDN]] and $V^\perp = V - \ip{V}{\gamma'}\gamma'$ is the normal part of $V$, then $$L''(0) = \int_a^b \Bigl( \abs{D_t V^\perp}^2 - \ip{R(V^\perp, \gamma')\gamma'}{V^\perp} \Bigr) dt + \Bigl[ \ip{D_s \partial_s \Gamma(0, t)}{\gamma'(t)} \Bigr]_{t=a}^{t=b}.$$

## Remarks

Both formulas come from differentiating $L(s) = \int_a^b \abs{\partial_t \Gamma(s, t)} \, dt$ under the integral sign, using the symmetry $D_s \partial_t \Gamma = D_t \partial_s \Gamma$ and metric compatibility, and integrating by parts.
The second formula also uses $D_s D_t W - D_t D_s W = R(\partial_s \Gamma, \partial_t \Gamma) W$ for vector fields $W$ along $\Gamma$.
For a proper variation, one with $\Gamma(s, a) = \gamma(a)$ and $\Gamma(s, b) = \gamma(b)$ for all $s$, the boundary terms vanish.
So a unit-speed curve is a critical point of length for all proper variations if and only if $D_t \gamma' = 0$, that is, it is a geodesic.
Example: assume all sectional curvatures are at least $k > 0$ and the geodesic $\gamma$ has length greater than $\pi / \sqrt{k}$.
Take $V = \sin\bigl(\pi (t - a)/(b - a)\bigr) E$, with $E$ a parallel unit normal field along $\gamma$.
Then $L''(0) < 0$, so $\gamma$ is not minimizing; this is the core of [[Myers's theorem|FPMJEX3DSJ]].
The second formula needs $\gamma$ to be a geodesic; otherwise extra terms involving $D_t \gamma'$ appear.

## Explanations

### L1 human

Wiggle a curve a little and watch how its length changes.
The first variation formula gives the rate of change: it vanishes for every wiggle that fixes the endpoints exactly when the curve is a geodesic.
The second variation formula gives the second derivative, which tells whether a geodesic is really a shortest path among nearby curves or only a saddle.
Positive curvature makes the second derivative smaller, so it pushes long geodesics to stop being shortest.

### L2 human

The integrand of the second variation, restricted to proper normal variations, is the index form of $\gamma$.
Its null space consists of the [[Jacobi fields|9AHMM41PJ3]] vanishing at the endpoints, so it is nondegenerate exactly when $\gamma(a)$ and $\gamma(b)$ are not [[conjugate|ZPXE5VSZ35]] along $\gamma$.
This is the bridge from curvature to global results such as the theorems of Myers, Synge and Cartan–Hadamard and the Morse index theorem.
Many texts state the same formulas for the energy $E = \tfrac{1}{2}\int \abs{\gamma'}^2 dt$ instead of the length; for unit-speed geodesics and proper normal variations the second variations agree.
</document>

<document source="https://tamnd.github.io/math2/node/ZBFZGA1J5G.md" tag="ZBFZGA1J5G" kind="concept">
# Gowers norm

## Summary

The Gowers norms $\norm{f}_{U^k}$ are norms on complex-valued functions on a finite abelian group, one for each $k \ge 2$.
They average products of $f$ over $k$-dimensional parallelepipeds $x + \omega_1 h_1 + \dots + \omega_k h_k$.
A function with small $U^k$ norm behaves randomly for counting arithmetic progressions of length $k + 1$.
Gowers introduced them for his proof of [[Szemerédi's theorem|B23B0QEMN5]], and they are the basic objects of higher-order Fourier analysis.

## Explanations

### L1 human

The Gowers norm tests whether a function, such as the indicator of a set, looks random from the point of view of additive patterns.
The $U^2$ norm detects correlation with ordinary waves $e^{2\pi i a x / N}$, as in usual Fourier analysis.
The $U^3$ norm also detects quadratic waves $e^{2\pi i a x^2 / N}$, which the $U^2$ norm misses.
If a set has small $U^{k-1}$ norm after subtracting its density, it contains about as many progressions of length $k$ as a random set of the same density.
</document>

<document source="https://tamnd.github.io/math2/node/ZBGT50BW0Z.md" tag="ZBGT50BW0Z" kind="concept">
# Theta correspondence

## Summary

The theta correspondence, or Howe correspondence, matches irreducible representations of the two members of a reductive dual pair inside a symplectic group.
The match is read off from the restriction of the [[oscillator representation|CCR7BAQ5N5]] of the metaplectic group to the product of the two groups.
Roger Howe introduced reductive dual pairs and conjectured that this restriction gives a bijection, now called Howe duality.
Globally, the correspondence is realised by theta lifts of automorphic forms.
These lifts are a classical way to build automorphic forms on one group from those on another.

## Explanations

### L1 human

Take two groups that sit inside a large symplectic group and commute with each other, each being the full commutant of the other.
The oscillator representation, restricted to the pair, breaks into pieces of the form "a representation of the first group times a representation of the second".
The theorem behind the correspondence says that in each piece the first representation determines the second, and conversely.
So the oscillator representation acts as a dictionary between the representations of the two groups.
</document>

<document source="https://tamnd.github.io/math2/node/ZBHGXJF24E.md" tag="ZBHGXJF24E" kind="concept">
# Fundamental groupoid

## Summary

The fundamental [[groupoid|FH3C0T643F]] $\Pi_1(X)$ of a topological space $X$ is a groupoid.
Its objects are the points of $X$, and its morphisms are paths up to [[path homotopy|CCNKA8C711]].
It records the [[fundamental group|S1XK3HXHFX]] at every base point at once, together with the paths that compare them.
It needs no choice of base point, which makes some results, such as versions of the Seifert–van Kampen theorem, easier to state and prove.

## Explanations

### L1 human

The fundamental group looks only at loops from one chosen point.
The fundamental groupoid looks at all paths between all pairs of points, up to deformation with fixed ends.
Paths can be composed when the end of one is the start of the next, and every path can be run backwards.
For a space made of two separate pieces, the groupoid sees both pieces, while a fundamental group sees only the piece of its base point.
</document>

<document source="https://tamnd.github.io/math2/node/ZBKVRMWHMW.md" tag="ZBKVRMWHMW" kind="concept">
# Category of posets

## Summary

The category of posets, written $\mathbf{Pos}$, has the [[partially ordered sets|011C29Q1S1]] as objects and the [[monotone functions|0WFF6AZPBP]] as morphisms.
Its isomorphisms are the [[order isomorphisms|0Y74KN99A5]].
It sits inside the larger category of preordered sets, and it lets order theory use the language of [[category theory|Q1RZT7DV0F]].

## Explanations

### L1 human

The category of posets collects all ordered sets and all maps that respect order.
Two such maps compose to another map that respects order, and the identity map respects order.
For example, $n \mapsto 2n$ from $\N$ to $\N$ and the inclusion of $\N$ into $\Z$ compose to a monotone map from $\N$ to $\Z$.
Treating posets this way makes products, sums and other constructions on orders into special cases of general constructions.
</document>

<document source="https://tamnd.github.io/math2/node/ZBMEVWN100.md" tag="ZBMEVWN100" kind="problem">
# Cauchy's functional equation

## Problem

Find all [[functions|0EPAT51DXQ]] $f \colon \R \to \R$ such that
$$f(x + y) = f(x) + f(y) \quad \text{for all } x, y \in \R.$$

## Remarks

The problem is solved.
Every solution satisfies $f(0) = 0$, $f(-x) = -f(x)$ and $f(qx) = q f(x)$ for every rational $q$, by induction on sums and then division.
So $f(q) = cq$ for all rational $q$, where $c = f(1)$.
If $f$ is also [[continuous|0X3WGX95FE]], then by density of $\Q$ in $\R$ the solution is $f(x) = cx$ for all real $x$.
The same conclusion holds under each of these weaker conditions on a solution $f$:

1. $f$ is continuous at one point;
2. $f$ is [[monotone|0A7KV2WB7A]] on some interval;
3. $f$ is bounded above on some interval;
4. $f$ is measurable for [[Lebesgue measure|2Z2QVWZ262]].

Without such a condition there are other solutions.
Take a [[basis|0KCQ9Y3QKA]] of $\R$ as a vector space over $\Q$, which exists by the [[axiom of choice|0HTGB13FE2]].
Every $\Q$-linear map $\R \to \R$ solves the equation, and choosing its values on the basis freely gives solutions that are not of the form $cx$.
The graph of such a non-linear solution is dense in the plane $\R^2$.

## Explanations

### L1 human

The equation asks for functions that turn sums into sums.
The lines through the origin, $f(x) = cx$, are the obvious answers, and they are the only reasonable ones.
Strange solutions exist only when one allows wild functions, which need the axiom of choice to construct and cannot be drawn.
</document>

<document source="https://tamnd.github.io/math2/node/ZBNQEJ09QE.md" tag="ZBNQEJ09QE" kind="definition">
# Power series in several variables

## Definition

Let $n \ge 1$ and let $\N^n$ be the set of [[multi-indices|0PJXBR6QA3]] $\alpha = (\alpha_1, \dots, \alpha_n)$ with $\alpha_j \ge 0$ integers.
For $w \in \C^n$, write $w^\alpha = w_1^{\alpha_1} \cdots w_n^{\alpha_n}$, with $w_j^0 = 1$.
Let $a \in \C^n$ and let $(c_\alpha)_{\alpha \in \N^n}$ be complex numbers.
The *power series* with centre $a$ and coefficients $c_\alpha$ is the formal sum
$$\sum_{\alpha \in \N^n} c_\alpha (z - a)^\alpha.$$
It *converges* at $z \in \C^n$ if $\sum_{\alpha \in \N^n} \abs{c_\alpha} \abs{(z - a)^\alpha} < \infty$.
Its *sum* at such a point is the sum of the family $(c_\alpha (z - a)^\alpha)_\alpha$, which does not depend on the order of the terms.

## Remarks

Convergence here is [[absolute convergence|0YBDSJARWN]], since the index set $\N^n$ has no preferred order when $n \ge 2$.
Example: for $n = 2$, $a = 0$ and $c_\alpha = 1$ for all $\alpha$, at $z = (1/2, 1/3)$ the sum of absolute values is $\sum_j 2^{-j} \sum_k 3^{-k} = 2 \cdot \tfrac32 = 3 < \infty$, so the series converges, with sum $3$.
Non-example: the same series at $z = (1/2, 1)$ has the terms $2^{-j}$ with $k$ arbitrary, so the sum of absolute values is infinite and the series does not converge there.
Grouping by total degree $\abs{\alpha} = \alpha_1 + \cdots + \alpha_n$ writes the series as $\sum_{m \ge 0} P_m(z - a)$, with $P_m$ a homogeneous polynomial of degree $m$.
For $n = 1$ this is the ordinary [[power series|0MZB06ST2B]], and in that case absolute convergence holds at every point inside the disc of convergence.
</document>

<document source="https://tamnd.github.io/math2/node/ZBQ4KRC98K.md" tag="ZBQ4KRC98K" kind="concept">
# Cartan matrix of an algebra

## Summary

The Cartan matrix of a finite-dimensional algebra records how the simple modules occur as composition factors of the indecomposable [[projective modules|BFFA03GE3V]].
Its rows are indexed by the indecomposable projectives and its columns by the [[simple modules|G00C22AEDH]].
In the modular representation theory of finite groups it is a basic invariant of the group algebra.
It is a different notion from the Cartan matrix of a semisimple Lie algebra, which shares the name.

## Explanations

### L1 human

Over a finite-dimensional algebra, every module is built in layers from simple modules, like a number is built from primes.
The projective modules are the most important building blocks, and each one has a top simple module.
The Cartan matrix is the table that counts how often each simple module appears in the layers of each indecomposable projective module.
For the algebra $K[x]/(x^2)$ there is one simple module and the only indecomposable projective has two layers, so the Cartan matrix is $(2)$.
For a semisimple algebra the projectives are simple and the Cartan matrix is the identity matrix.
</document>

<document source="https://tamnd.github.io/math2/node/ZBQ6TVNZZ8.md" tag="ZBQ6TVNZZ8" kind="concept">
# Kohn Laplacian

## Summary

The Kohn Laplacian is the second order operator $\square_b = \bar\partial_b \bar\partial_b^* + \bar\partial_b^* \bar\partial_b$ on a [[CR manifold|ZC78JKZX9S]], built from the [[tangential Cauchy–Riemann operator|X7GP9BECD8]] $\bar\partial_b$ and its adjoint.
It plays on the boundary of a domain in $\C^n$ the role that the complex Laplacian $\bar\partial \bar\partial^* + \bar\partial^* \bar\partial$ of the [[dbar-Neumann problem|Q4P4R2GDNG]] plays inside.
It is not elliptic, but on strongly pseudoconvex CR manifolds it is hypoelliptic in suitable degrees.
Its study gives the solution of the tangential Cauchy–Riemann equations and the [[Kohn–Rossi cohomology|VC48WDWPV1]].

## Explanations

### L1 human

On a real hypersurface in $\C^n$, such as a sphere, only some of the complex directions are tangent to the surface.
The tangential Cauchy–Riemann operator differentiates only in these directions.
The Kohn Laplacian combines this operator with its adjoint, in the same way as the usual Laplacian combines the gradient with the divergence.
On the Heisenberg group $\C \times \R$, which is a model of the boundary of a ball, it acts on functions as $-Z \bar Z$ for the vector fields $\bar Z = \partial/\partial \bar z - i z\, \partial/\partial t$ and $Z = \partial/\partial z + i \bar z\, \partial/\partial t$.
It differentiates twice in the $z$ directions but only indirectly in the $t$ direction, which is why it is not elliptic.
</document>

<document source="https://tamnd.github.io/math2/node/ZBS1JA62QP.md" tag="ZBS1JA62QP" kind="statement">
# Bernstein's theorem on monotone functions

## Statement

Let $f \colon (0, \infty) \to \R$.
Call $f$ [[completely monotone|VJ1NCNTEHD]] if $f$ is infinitely differentiable and $(-1)^k f^{(k)}(x) \ge 0$ for all integers $k \ge 0$ and all $x > 0$.
Then $f$ is completely monotone if and only if there is a positive Borel measure $\mu$ on $[0, \infty)$ with the following property.
For every $x > 0$ the integral below converges and
$$f(x) = \int_{[0, \infty)} e^{-tx} \, d\mu(t).$$

## Remarks

So the completely monotone functions are exactly the [[Laplace transforms|0V49JEZW06]] of positive measures.
The "if" direction is direct: differentiating under the integral sign gives $(-1)^k f^{(k)}(x) = \int t^k e^{-tx} \, d\mu(t) \ge 0$.
The "only if" direction is the hard part; one proof approximates $f$ by sums built from its derivatives and uses a compactness argument for measures.
Example: $f(x) = 1/x$ is completely monotone on $(0, \infty)$, and $1/x = \int_0^\infty e^{-tx} \, dt$, so $\mu$ is Lebesgue measure on $[0, \infty)$.
By $x \mapsto -x$, the theorem also describes [[absolutely monotonic functions|QQSPWH4NG9]] on $(-\infty, 0)$ as integrals $\int e^{tx} \, d\mu(t)$.

## Explanations

### L1 human

A function whose derivatives alternate in sign forever looks very special.
Bernstein's theorem says that these functions are exactly the mixtures, with nonnegative weights, of decaying [[exponential functions|0J0FCWV32D]] $e^{-tx}$.

### L2 human

The theorem links a condition on derivatives with a positivity condition on a measure.
It is used in probability, where completely monotone functions are Laplace transforms of distributions on $[0, \infty)$, and in the theory of positive definite functions.
If $f$ is also continuous on $[0, \infty)$, then $\mu$ is finite and $\mu([0, \infty)) = f(0)$.
</document>

<document source="https://tamnd.github.io/math2/node/ZBSXSTQM2J.md" tag="ZBSXSTQM2J" kind="concept">
# Radial basis function

## Summary

A radial basis function is a function on $\R^d$ whose value depends only on the distance from a centre point.
Standard examples are the Gaussian $e^{-(\varepsilon r)^2}$, the multiquadric $\sqrt{1 + (\varepsilon r)^2}$ and the [[thin plate spline|P9ZPH7EC8K]] $r^2 \log r$, where $r$ is the distance.
Shifted copies centred at data points are used as building blocks for [[radial basis function interpolation|XZ33P3HBB9]] of scattered data in any dimension.
They also appear as kernels in machine learning and in meshfree methods for differential equations.

## Explanations

### L1 human

Drop a stone in a still pond: the height of the water depends only on the distance from where the stone fell.
A radial basis function has the same symmetry: it looks the same in every direction from its centre.
The Gaussian bump $e^{-\norm{x}^2}$ is the typical example; it equals $1$ at the centre and decays as one moves away.
Placing such bumps at many points and adding them with suitable weights gives flexible surfaces.
The [[definition|3SS5K8MHPK]] makes this precise.
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<document source="https://tamnd.github.io/math2/node/ZBVH2YE50B.md" tag="ZBVH2YE50B" kind="concept">
# Fuchsian equation

## Summary

A Fuchsian equation is a linear differential equation with rational coefficients whose singular points are all [[regular singular points|H96HXG6CA0]].
This includes the point $\infty$ of the Riemann sphere.
Its solutions are multivalued functions of controlled growth, described near each singular point by [[Fuchs' theorem|6EZ400WM94]].
The equation is largely determined by its singular points, its exponents and its [[monodromy|C8HD62AB6Q]].
The hypergeometric equation, the Legendre equation and the Riemann equation are Fuchsian.
See the [[definition|YT2B9KQ0BG]].

## Explanations

### L1 human

A Fuchsian equation has finitely many bad points, and at each of them the solutions only behave like powers and logarithms.
Nothing wild happens anywhere, not even at infinity.
The Gauss hypergeometric equation is the main example, with bad points $0$, $1$ and $\infty$.
The Bessel equation is not Fuchsian, because its solutions oscillate like $\sin z$ near $\infty$.
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<document source="https://tamnd.github.io/math2/node/ZBVJ4B41AX.md" tag="ZBVJ4B41AX" kind="counterexample">
# Hirsch conjecture

## Counterexample

For a [[convex polytope|300K6CB1KH]] $P$, the graph of $P$ has the vertices of $P$ as vertices and the edges of $P$ as edges.
The diameter of $P$ is the largest [[distance|0QV9VNNB4Z]] in this graph between two vertices.
The Hirsch conjecture asserted: if $d \ge 2$ and $P$ is a $d$-dimensional polytope with $n$ [[facets|DPDXN2FJC4]], then the diameter of $P$ is at most $n - d$.
The conjecture is false: in 2010 Francisco Santos constructed a $43$-dimensional polytope with $86$ facets whose diameter is at least $44$, while $n - d = 43$.

## Remarks

Santos's construction first builds a spindle: a polytope with two special vertices whose distance exceeds the dimension.
A known reduction, which adds facets and dimensions in step, then turns it into a counterexample.
For unbounded polyhedra the analogous statement had already been disproved by Victor Klee and David Walkup in 1967.
The bound $n - d$ is attained by the $d$-cubes, which have $n = 2d$ facets and diameter $d$.

The question matters for [[linear programming|VNMXB0J2KA]], since the simplex method walks along edges of the feasible polytope.
So the diameter bounds from below the worst number of steps of any edge-following pivot rule.
Whether the diameter is bounded by a polynomial in $n$ and $d$ remains open.
The best general bound known is quasi-polynomial, of the form $n^{\log_2 d + 2}$, due to Gil Kalai and Daniel Kleitman.

## Explanations

### L1 human

Walk from corner to corner along the edges of a polytope.
The Hirsch conjecture claimed that any two corners are joined by a short walk: at most the number of walls minus the dimension.
For half a century nobody could prove or refute it.
Santos found a polytope in $43$ dimensions where some walk must be longer, so the conjecture is false, though only by a little.
</document>

<document source="https://tamnd.github.io/math2/node/ZBXBQYKP5W.md" tag="ZBXBQYKP5W" kind="definition">
# Littlewood–Paley square function

## Definition

Let $n \ge 1$, and let $\chi$, $\psi$ and $\Delta_j$, $j \in \Z$, be as in the [[Littlewood–Paley decomposition|D1PAT0K446]] on $\R^n$.
Let $f$ be in the [[Lp space|SASP14MA1Z]] $L^p(\R^n)$ for some $1 \le p \le \infty$.
The *Littlewood–Paley square function* of $f$ is the function $Sf \colon \R^n \to [0, \infty]$ given by
$$Sf(x) = \left( \sum_{j \in \Z} \abs{\Delta_j f(x)}^2 \right)^{1/2}.$$

## Remarks

The operator $f \mapsto Sf$ is not linear, but it is sublinear: $S(f + g) \le Sf + Sg$, by the triangle inequality in $\ell^2$.
It depends on the choice of $\chi$, but the $L^p$ norms of square functions built from two choices are comparable for $1 < p < \infty$.

Example: assume moreover that $\chi$ is radial and nonincreasing in $\abs{\xi}$, so that $\psi \ge 0$.
For each $\xi \ne 0$ at most two of the numbers $\psi(2^{-j}\xi)$ are nonzero, and they sum to $1$.
So $\frac{1}{2} \le \sum_j \psi(2^{-j}\xi)^2 \le 1$.
By Plancherel's theorem for the [[Fourier transform|DPVM34ADZV]], $\norm{Sf}_2^2 = \int \sum_j \psi(2^{-j}\xi)^2 \abs{\hat f(\xi)}^2 \, d\xi$, hence $\frac{1}{2}\norm{f}_2^2 \le \norm{Sf}_2^2 \le \norm{f}_2^2$ for $f \in L^2(\R^n)$.

Non-example: the constant function $1$ is not in any $L^p(\R^n)$ with $p < \infty$.
As a tempered distribution its Fourier transform is the point mass at $0$, and $\psi(0) = 0$, so every $\Delta_j 1$ vanishes and the formula gives $S1 = 0$ although $1 \ne 0$.
This is why the comparison of $\norm{Sf}_p$ and $\norm{f}_p$ needs $p < \infty$.
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<document source="https://tamnd.github.io/math2/node/ZBYC3MFW89.md" tag="ZBYC3MFW89" kind="definition">
# Quadratic variation

## Definition

Let $(X_t)_{t \ge 0}$ be a real [[stochastic process|0QH5ENKZM1]].
For a partition $\pi = \{0 = t_0 < t_1 < \dots < t_m = t\}$ of $[0, t]$, let $\norm{\pi} = \max_k (t_k - t_{k-1})$ be its mesh.
The process $X$ has *quadratic variation* $([X]_t)_{t \ge 0}$ if for every $t \ge 0$
$$[X]_t = \lim_{\norm{\pi} \to 0} \sum_{k=1}^{m} \bigl(X_{t_k} - X_{t_{k-1}}\bigr)^2,$$
where the limit is taken in [[probability|0QA1HBD9Z4]] over partitions $\pi$ of $[0, t]$.

## Remarks

A Wiener process $W$ has $[W]_t = t$.
Check: the sum $Q_\pi$ has mean $\sum_k (t_k - t_{k-1}) = t$ and variance $2 \sum_k (t_k - t_{k-1})^2 \le 2t \norm{\pi}$, so $Q_\pi \to t$ in $L^2$ and hence in probability.
A continuously differentiable function $f$ has $[f]_t = 0$, because $\sum_k (f(t_k) - f(t_{k-1}))^2 \le \max\abs{f'}^2 \, t \norm{\pi}$.
For [[fractional Brownian motion|VC6R7XSKRQ]] with Hurst index $H < 1/2$, the sums over uniform partitions tend to infinity, so the limit does not exist.
Every [[semimartingale|MZNFPJJMY9]] has a quadratic variation, given by $[X]_t = X_t^2 - X_0^2 - 2 \int_0^t X_{s-} \, dX_s$.
For a continuous [[local martingale|ZWJ9G89W3T]] $M$, $[M]$ is the unique continuous non-decreasing adapted process with $[M]_0 = 0$ such that $M^2 - [M]$ is a local martingale.
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<document source="https://tamnd.github.io/math2/node/ZBZBKWWEBC.md" tag="ZBZBKWWEBC" kind="definition">
# Normal form

## Definition

Let $A$ be a real $n \times n$ matrix and consider the [[autonomous system|07BNRFQ0PJ]]
$$x' = A x + f_2(x) + f_3(x) + \cdots + f_r(x) + O(\abs{x}^{r+1})$$
near the [[equilibrium point|0G4X36AE0X]] $0$ of $\R^n$.
Here each $f_m$ is a vector field whose components are [[homogeneous polynomials|2C3PAQ7H50]] of degree $m$.
Let $H_m$ be the vector space of such vector fields of degree $m$.
The homological operator $L_A \colon H_m \to H_m$ is $L_A h(x) = Dh(x) \, A x - A h(x)$.
For each $m \ge 2$ choose a complement $G_m$ of the image $L_A(H_m)$ in $H_m$.
The system is in normal form up to order $r$ with respect to these complements if $f_m \in G_m$ for $m = 2, \ldots, r$.

## Remarks

The normal form theorem states that a near-identity polynomial change of coordinates $x = y + h_2(y) + \cdots + h_r(y)$, with $h_m \in H_m$, brings any such system into normal form up to order $r$.
At degree $m$ the change $x = y + h_m(y)$ replaces $f_m$ by $f_m - L_A h_m$ and changes only terms of degree at least $m$, so $f_m$ can be reduced to its component in $G_m$.
If $A = \mathrm{diag}(\lambda_1, \ldots, \lambda_n)$, then $L_A$ maps the monomial vector field $x_1^{k_1} \cdots x_n^{k_n} e_j$ to $(k_1 \lambda_1 + \cdots + k_n \lambda_n - \lambda_j)$ times itself.
So one can take $G_m$ spanned by the resonant monomials, those with $k_1 \lambda_1 + \cdots + k_n \lambda_n = \lambda_j$, and the normal form keeps only resonant terms.
Example: for $\lambda = (1, 2)$, the only resonant monomial of degree $2$ is $x_1^2 e_2$, since $2 \cdot 1 = 2$; the normal form up to order $2$ is $y_1' = y_1$, $y_2' = 2 y_2 + c\, y_1^2$ for a constant $c$.
Non-example: the system $x_1' = x_1 + x_1 x_2$, $x_2' = 2 x_2$ is not in normal form for $\lambda = (1, 2)$, since $x_1 x_2 e_1$ has $1 + 2 \ne 1$ and is not resonant.

## Explanations

### L1 human

A normal form is the result of simplifying the nonlinear terms of an equation by a change of variables, one degree at a time.
The terms that survive are the resonant ones, which reflect relations between the [[eigenvalues|0BQAA2BK3Q]] of the linear part.
</document>

<document source="https://tamnd.github.io/math2/node/ZC179DD2P4.md" tag="ZC179DD2P4" kind="definition">
# Universal bundle

## Definition

Let $G$ be a [[topological group|92700QAB5Y]] and let $p \colon E \to B$ be a [[principal $G$-bundle|8E37E74XVP]].
For a space $X$, let $\operatorname{Prin}_G(X)$ be the set of isomorphism classes of principal $G$-bundles over $X$.
Let $[X, B]$ be the set of [[homotopy|9YA6S6HD9V]] classes of continuous maps $X \to B$.
The bundle $p$ is *universal* if, for every [[CW complex|5W95QFHXPR]] $X$, the map
$$[X, B] \to \operatorname{Prin}_G(X), \quad [f] \mapsto [f^* E],$$
is a bijection, where $f^* E$ is the pullback bundle.

## Remarks

The map is well defined because pullbacks along homotopic maps are isomorphic over CW complexes.
A universal bundle is written $EG \to BG$, and $BG$ is a [[classifying space|WSZ40H5E4T]] of $G$; it is unique up to homotopy equivalence.
A principal $G$-bundle over a CW complex whose total space is [[contractible|R5WP8Z69Q8]] is universal.
Such bundles exist for every topological group $G$, for instance by Milnor's join construction.

Example: $\Z/2\Z$ acts freely on the infinite sphere $S^\infty$ by $x \mapsto -x$, and $S^\infty \to \R P^\infty$ is a principal $\Z/2\Z$-bundle.
Since $S^\infty$ is contractible, this bundle is universal.

Non-example: the bundle $S^1 \to \R P^1$, $x \mapsto \set{x, -x}$, is a principal $\Z/2\Z$-bundle but not universal.
Take $X = \R P^2$: there are two principal $\Z/2\Z$-bundles over $\R P^2$, the trivial one and the double cover $S^2 \to \R P^2$.
But every map $\R P^2 \to \R P^1 \cong S^1$ is null-homotopic, because $H^1(\R P^2; \Z) = 0$, so $[\R P^2, \R P^1]$ has only one element and the map is not a bijection.
</document>

<document source="https://tamnd.github.io/math2/node/ZC4BHVHWE9.md" tag="ZC4BHVHWE9" kind="definition">
# Beth number

## Definition

Assume ZFC, so that each set has a [[cardinal number|0QKPNZKNEA]] and $2^\kappa$ is the cardinality of the [[power set|0BC7KX51Z8]] of a set of cardinality $\kappa$.
The beth numbers $\beth_\alpha$, for [[ordinals|0RDJK7EQYY]] $\alpha$, are defined by [[transfinite recursion|0ZHQMQ2C1N]]:

1. $\beth_0 = \aleph_0$, the cardinality of $\N$, see [[aleph-null|0V2H5PGJ9B]].
2. $\beth_{\alpha + 1} = 2^{\beth_\alpha}$.
3. $\beth_\lambda = \sup_{\beta < \lambda} \beth_\beta$ for each nonzero limit ordinal $\lambda$, the [[supremum|0BTMQ04556]] of the earlier beths.

## Remarks

Example: $\beth_1 = 2^{\aleph_0}$ is the cardinality of $\R$, since $\R$ is in bijection with the power set of $\N$.
$\beth_2 = 2^{2^{\aleph_0}}$ is the cardinality of the power set of $\R$, and also of the set of all functions $\R \to \R$.
Non-example: $\aleph_1$ is a beth number only if $\aleph_1 = \beth_1$, which is the continuum hypothesis and is not provable in ZFC.
By Cantor's theorem $\beth_{\alpha+1} > \beth_\alpha$, and by induction $\beth_\alpha \ge \aleph_\alpha$ for every $\alpha$.
Another notation is $\beth_\alpha = \beth_\alpha(\aleph_0)$; more generally $\beth_0(\kappa) = \kappa$ and $\beth_{\alpha+1}(\kappa) = 2^{\beth_\alpha(\kappa)}$.
</document>

<document source="https://tamnd.github.io/math2/node/ZC78JKZX9S.md" tag="ZC78JKZX9S" kind="concept">
# CR manifold

## Summary

A CR manifold is a smooth real manifold with a distinguished involutive subbundle of its complexified tangent bundle, see the [[definition|A2VXXBYX6A]].
This subbundle plays the role of the holomorphic tangent directions.
The model is a real hypersurface in $\C^n$, such as the sphere $S^{2n-1}$, with the complex tangent directions that lie inside its tangent spaces.
The letters CR stand for Cauchy–Riemann.
A CR structure is what is needed to say which functions satisfy the [[tangential Cauchy–Riemann equations|X7GP9BECD8]].
CR geometry studies the boundaries of domains in [[several complex variables|BH57D2FBNR]], and its main invariant is the [[Levi form|80MTNXWW80]].

## Explanations

### L1 human

The boundary of a domain in $\C^n$ is a real manifold of odd dimension, so it cannot be a complex manifold.
Still, at each of its points some tangent directions are complex lines: those that are tangent to the boundary together with their rotation by $i$.
A CR manifold is a manifold that remembers these complex directions.
For the unit sphere in $\C^2$, the complex tangent directions at $(z_1, z_2)$ form the complex line spanned by $(-\bar z_2, \bar z_1)$.
</document>

<document source="https://tamnd.github.io/math2/node/ZC8DNB75N4.md" tag="ZC8DNB75N4" kind="concept">
# Lawson topology

## Summary

The Lawson topology on a partially ordered set is the coarsest topology that contains both the [[Scott topology|ZR8A03FV67]] and the lower topology.
The Scott topology sees only the "upward" behaviour of the order, so it is rarely Hausdorff.
Adding the lower topology repairs this: on a [[continuous lattice|RXBD58SGXZ]] the Lawson topology is compact and Hausdorff.
It is the main tool for treating domains and continuous lattices as genuine compact spaces.

## Explanations

### L1 human

The Scott topology has only open sets that are closed upward, so a point cannot be separated from the points above it.
The lower topology has the opposite bias: its basic open sets are the points that are not above some fixed elements.
The Lawson topology lets both kinds of sets be open, so it can separate points in both directions.
On the unit interval $[0, 1]$ with its usual order, the proper nonempty Scott open sets are the intervals $(a, 1]$.
The lower topology is generated by the intervals $[0, b)$.
Together they give the usual topology of $[0, 1]$.
</document>

<document source="https://tamnd.github.io/math2/node/ZCB806EV8A.md" tag="ZCB806EV8A" kind="statement">
# Functional equation of the gamma function

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\C$ that is holomorphic on $\C \setminus \set{0, -1, -2, \ldots}$.
For every $z \in \C \setminus \set{0, -1, -2, \ldots}$,
$$\Gamma(z + 1) = z \, \Gamma(z).$$

## Remarks

For $\operatorname{Re} z > 0$, integration by parts in $\Gamma(z + 1) = \int_0^\infty t^{z} e^{-t} \, dt$ gives $z \int_0^\infty t^{z-1} e^{-t} \, dt$; the boundary terms vanish.
Both sides are holomorphic on the connected set $\C \setminus \set{0, -1, -2, \ldots}$, so the identity theorem extends the equation from the half-plane to this set.
Conversely the equation is the usual way to build the extension: for $\operatorname{Re} z > -n$ one sets $\Gamma(z) = \Gamma(z + n)/\big(z (z + 1) \cdots (z + n - 1)\big)$.
With $\Gamma(1) = 1$ the equation gives $\Gamma(n + 1) = n!$, see [[the gamma function extends the factorial|0FDCW77H44]].
For example $\Gamma(\tfrac32) = \tfrac12 \Gamma(\tfrac12) = \tfrac{\sqrt{\pi}}{2}$.
In Mathlib the statement is `Complex.Gamma_add_one`.

## Explanations

### L1 human

The gamma function obeys the same rule as the factorial: moving the argument up by one multiplies the value by the old argument.
For whole numbers this is the rule $(n+1)! = (n+1) \cdot n!$.
The rule holds at every complex number where $\Gamma$ is defined.

### L2 human

The functional equation reduces the values of $\Gamma$ anywhere to its values on a strip of width one, such as $1 \le \operatorname{Re} z < 2$.
It also shows where the poles are: dividing by $z$ at $z = 0$ creates a pole, and each further step to the left creates another.
Together with log-convexity it characterizes $\Gamma$ on $(0, \infty)$, by the [[Bohr–Mollerup theorem|QFF87AXWN2]], and together with boundedness on a strip it characterizes $\Gamma$ on the right half-plane, by [[Wielandt's theorem|WCM87QEEZW]].
</document>

<document source="https://tamnd.github.io/math2/node/ZCBFF6DNA9.md" tag="ZCBFF6DNA9" kind="definition">
# Moishezon manifold

## Definition

Let $X$ be a compact, connected [[complex manifold|JSGWAN2RJV]] of dimension $n$, and let $a(X)$ be its [[algebraic dimension|F0GF0JEZTG]].
Then $X$ is a *Moishezon manifold* if $a(X) = n$, that is, if $X$ has $n$ algebraically independent meromorphic functions.

## Remarks

Example: $\mathbb{CP}^n$ is Moishezon, since $z_1/z_0, \ldots, z_n/z_0$ are $n$ algebraically independent meromorphic functions.
More generally, every smooth projective variety of dimension $n$ is Moishezon, since its field of rational functions has transcendence degree $n$.

Non-example: the diagonal [[Hopf surface|PNFNGBXRM3]] has dimension $2$ but algebraic dimension $1$, so it is not Moishezon.

Since $a(X) \le \dim X$ by [[Siegel's theorem|ENXCWPTG92]], Moishezon manifolds are those with the largest possible algebraic dimension.
Equivalently, $X$ is Moishezon if and only if it is bimeromorphic to a projective manifold.
A Moishezon manifold is projective if and only if it admits a Kähler metric, so non-projective Moishezon manifolds are never [[Kähler|X60JG9AEQJ]].
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<document source="https://tamnd.github.io/math2/node/ZCDCJTTG8V.md" tag="ZCDCJTTG8V" kind="example">
# Hermite equation

## Example

Let $\lambda \in \C$. The *Hermite equation* is
$$y'' - 2 z\, y' + 2 \lambda\, y = 0.$$
Its coefficients $-2z$ and $2 \lambda$ are polynomials, so every point of $\C$ is an [[ordinary point|QK2J038VB4]].
A [[power series solution|08RT9GDDPQ]] $y = \sum_{k \ge 0} a_k z^k$ gives
$$(k + 2)(k + 1)\, a_{k+2} - 2 k\, a_k + 2 \lambda\, a_k = 0, \quad \text{so} \quad a_{k+2} = \frac{2 (k - \lambda)}{(k + 2)(k + 1)}\, a_k.$$
The values $a_0$ and $a_1$ are free, and every solution is an entire function.
If $\lambda = n$ is a nonnegative integer, the recurrence stops at $k = n$ for the series of the same parity as $n$.
This gives a polynomial solution of degree $n$, a multiple of the [[Hermite polynomial|MGACY98E9H]] $H_n$.

## Remarks

For $n = 2$, $H_2(z) = 4 z^2 - 2$; then $y'' = 8$ and $y' = 8 z$, and $8 - 2 z \cdot 8 z + 4 (4 z^2 - 2) = 0$.
The substitution $y = e^{z^2 / 2} u$ links the equation to the quantum harmonic oscillator $u'' + (2 \lambda + 1 - z^2) u = 0$.
The point $\infty$ is an irregular singular point.

## Explanations

### L1 human

The Hermite equation has no bad points in the finite plane, so its solutions are power series that converge everywhere.
For whole numbers $\lambda = n$ one of the series stops and becomes a polynomial.
These Hermite polynomials describe the energy states of a quantum harmonic oscillator.
</document>

<document source="https://tamnd.github.io/math2/node/ZCDWXVMDFN.md" tag="ZCDWXVMDFN" kind="statement">
# Alexandrov's theorem on polyhedra

## Statement

Let $M$ be a [[metric space|0VKD1TP7F3]] that is homeomorphic to the sphere $S^2$.
Assume that the distance between two points of $M$ is the infimum of the lengths of paths joining them.
Suppose that there is a finite set $S \subseteq M$ such that:

1. each point of $M \setminus S$ has a neighbourhood isometric to an open subset of the Euclidean plane;
2. each point $s \in S$ has a neighbourhood isometric to a neighbourhood of the apex of a Euclidean cone with total angle $\theta_s < 2\pi$.

Then there is a three-dimensional [[convex polytope|300K6CB1KH]] $P \subseteq \R^3$, or a doubly covered convex polygon $P$ in a plane, with the following property.
The surface of $P$, with its intrinsic path metric, is isometric to $M$.
This $P$ is unique up to an isometry of $\R^3$, and its vertices are exactly the points of $S$.

## Remarks

Uniqueness follows from a variant of [[Cauchy's rigidity theorem|CJ7KCDDF31]], applied to the two polytopes cut into matching triangles.
Alexandrov proved existence by a continuity argument.
Polytopes with $k$ vertices and metrics with $k$ cone points form spaces of the same dimension.
The map that sends a polytope to its surface metric is injective by the uniqueness part.
An invariance of domain argument then shows that it reaches every metric.
Bobenko and Izmestiev later gave a proof that is closer to a construction, but finding the edges of $P$ from $M$ is not easy in general.

Example: glue two copies of a unit square along their boundaries.
The result is a sphere with four cone points of angle $\pi$, and the theorem returns the doubly covered square itself.

The total angle condition is needed.
By [[Descartes's theorem on total angular defect|PT67C5ANSD]], the defects $2\pi - \theta_s$ of a convex polytope are positive and add up to $4\pi$.
So a metric with a cone point of angle larger than $2\pi$ cannot come from a convex polytope.

## Explanations

### L1 human

Cut out flat polygons and tape them together along their edges into a closed bag.
At each corner the paper angles must add up to less than a full turn.
Alexandrov's theorem says that the bag can be blown up into exactly one convex solid.
The creases of that solid need not be along the tape lines, which is the surprising part.

### L2 human

The theorem characterises the intrinsic metrics of convex polyhedral surfaces: they are exactly the polyhedral metrics on the sphere with positive curvature at every cone point.
It complements [[Cauchy's rigidity theorem|CJ7KCDDF31]], which gives uniqueness when the faces are known.
It has a smooth analogue, the Weyl problem, and it is used in computational geometry to fold a given net into a convex polyhedron.
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<document source="https://tamnd.github.io/math2/node/ZCE333FAWC.md" tag="ZCE333FAWC" kind="definition">
# Regular cardinal

## Definition

Let $\kappa$ be an infinite [[cardinal|0QKPNZKNEA]], identified with the least ordinal of its cardinality.
The cardinal $\kappa$ is regular if its [[cofinality|NTE5C260S1]] is $\kappa$, that is, $\operatorname{cf}(\kappa) = \kappa$.
It is singular if $\operatorname{cf}(\kappa) < \kappa$.

## Remarks

Example: in ZFC, every infinite successor cardinal $\kappa^+$ is regular.
Check: if $\kappa^+$ were the union of fewer than $\kappa^+$ sets each of size less than $\kappa^+$, it would have size at most $\kappa \cdot \kappa = \kappa$, using the axiom of choice.
Non-example: $\aleph_\omega$ is singular, since $\set{\aleph_n : n < \omega}$ is a cofinal subset of order type $\omega$.
Equivalently, in ZFC, $\kappa$ is regular if and only if it is not the union of fewer than $\kappa$ sets each of cardinality less than $\kappa$.
$\aleph_0$ is regular, and a limit cardinal $\aleph_\lambda$ with $\lambda$ a limit ordinal has cofinality $\operatorname{cf}(\lambda)$.
A regular uncountable limit cardinal is called weakly inaccessible; its existence cannot be proved in ZFC, if ZFC is consistent.
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# Mathieu function

## Definition

Let $a$ and $q$ be real numbers.
*Mathieu's equation* is the [[linear differential equation|0QQRP17Z7S]]
$$\frac{d^2 y}{dx^2} + (a - 2q \cos 2x)\, y = 0.$$
Fix $q$.
The values of $a$ for which the equation has a nonzero even solution of [[period|0ATF6Q0633]] $\pi$ or $2\pi$ form an increasing sequence $a_0(q) < a_1(q) < \cdots$.
For each $a_r(q)$, such a solution is unique up to a constant factor.
The *even Mathieu function* $\operatorname{ce}_r(x, q)$ is that solution for $a = a_r(q)$, normalised by $\int_0^{2\pi} \operatorname{ce}_r(x, q)^2\, dx = \pi$.
Its sign is fixed by continuity in $q$ from $\operatorname{ce}_0(x, 0) = \frac{1}{\sqrt 2}$ and $\operatorname{ce}_r(x, 0) = \cos rx$ for $r \ge 1$.
In the same way, odd solutions of period $\pi$ or $2\pi$ exist for an increasing sequence $b_1(q) < b_2(q) < \cdots$ of values of $a$.
The *odd Mathieu function* $\operatorname{se}_r(x, q)$ is the odd solution for $a = b_r(q)$, with the same normalisation and with $\operatorname{se}_r(x, 0) = \sin rx$.

## Remarks

The numbers $a_r(q)$ and $b_r(q)$ are the *characteristic values*; they are eigenvalues of [[Sturm–Liouville|7WV6B2502K]] problems on $[0, \pi]$ or $[0, \frac{\pi}{2}]$ with suitable boundary conditions.
The function $\operatorname{ce}_r$ or $\operatorname{se}_r$ has period $\pi$ when $r$ is even and period $2\pi$ when $r$ is odd.
Example: for $q = 0$ one has $a_r(0) = r^2$ and $\operatorname{ce}_r(x, 0) = \cos rx$ for $r \ge 1$; indeed $y = \cos rx$ satisfies $y'' + r^2 y = 0$ and $\int_0^{2\pi} \cos^2 rx\, dx = \pi$.
For $r = 0$ the constant must be $\frac{1}{\sqrt 2}$, since $\int_0^{2\pi} \frac12\, dx = \pi$.
Non-example: for $q = 0$ and $a = \frac14$, the solution $\cos \frac{x}{2}$ has period $4\pi$, not $\pi$ or $2\pi$, so it is not a Mathieu function in this sense.
For most other values of $a$, [[Floquet theory|BJVZXPB297]] gives a basis of solutions of the form $e^{i\mu x} p(x)$ and $e^{-i\mu x} p(-x)$, with $p$ of period $\pi$ and $\mu$ the characteristic exponent.
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# Amenable group action

## Summary

An action of a group on a set is amenable when the bounded functions on the set carry a [[mean|ZYNY2VFECQ]] that the group leaves invariant.
This is one common meaning; in ergodic theory the phrase also often means simply an action of an amenable group.
This extends amenability from [[groups|CEAT9BESKK]] to group actions: a group is amenable exactly when its action on itself by left translation is amenable.
Every action of an amenable group is amenable, but a non-amenable group can also act amenably, for example with a finite orbit.
Means and Følner sets are the tools that let ergodic theorems pass from $\Z$ to more general groups.

## Explanations

### L1 human

A mean is a way to take an average of every bounded function on a set, even an infinite one.
It must be linear, give $1$ to the constant function $1$, and give a nonnegative answer for nonnegative functions.
The action is amenable if some mean does not change when the group moves the points of the set.
If the group has a finite orbit, averaging over that orbit gives such a mean.
The free group on two generators acting on itself has no such mean, which is the root of paradoxical decompositions.
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# Varifold

## Definition

Let $U \subseteq \R^n$ be an [[open set|02BH3MCBQB]] and let $0 \le m \le n$.
Let $G(n, m)$ be the Grassmannian of $m$-dimensional linear subspaces of $\R^n$, with its usual compact metric topology.
An *$m$-dimensional varifold* in $U$ is a [[Radon measure|HZNEQQBCW4]] $V$ on the locally compact space $U \times G(n, m)$.
The *weight* of $V$ is the Radon measure $\norm{V}$ on $U$ given by $\norm{V}(A) = V(A \times G(n, m))$.

## Remarks

Example: let $M \subseteq U$ be an $m$-dimensional $C^1$ submanifold that is a closed subset of $U$, with tangent plane $T_x M$ at $x \in M$.
Let $\mathcal{H}^m$ be the $m$-dimensional [[Hausdorff measure|Q1TGZK8RN6]], which is surface area on $M$.
Set $V(B) = \mathcal{H}^m(\set{x \in M : (x, T_x M) \in B})$ for Borel sets $B \subseteq U \times G(n, m)$.
This is a Borel measure that is finite on compact sets, since $M$ has locally finite area, so $V$ is a varifold; its weight is $\mathcal{H}^m$ restricted to $M$.
More generally, a [[rectifiable set|7N0GFMW05R]] with a positive density function gives a *rectifiable varifold*.
Non-example: counting measure on a countable dense subset of $U \times G(n, m)$ is not finite on compact sets with interior, so it is not a Radon measure and not a varifold.
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# Minimum-variance unbiased estimator

## Summary

A minimum-variance unbiased estimator is an unbiased estimator with the smallest variance among all unbiased estimators.
It must be smallest at every value of the parameter.
It is also called a uniformly minimum-variance unbiased estimator (UMVUE).
It need not exist, but when it does it is unique with probability one.
The Rao–Blackwell and Lehmann–Scheffé theorems construct it from a complete sufficient statistic.

## Explanations

### L1 human

Among all rules that are right on average, pick the one that scatters least around the truth, whatever the truth is.
That rule is the minimum-variance unbiased estimator.
For coin flips, the proportion of heads is such a rule for the probability of heads.
The requirement "whatever the truth is" is strong: a rule that is best for one value of the parameter but worse for another does not qualify.
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# Lagrange basis polynomials

## Definition

Let $K$ be a [[field|04C5ZBG1WT]], let $n \ge 0$, and let $x_0, \dots, x_n \in K$ be pairwise distinct.
For $j = 0, \dots, n$, the *Lagrange basis polynomial* $\ell_j$ is the polynomial in the [[polynomial ring|0SDC934R52]] $K[x]$ given by
$$\ell_j(x) = \prod_{\substack{0 \le k \le n \\ k \ne j}} \frac{x - x_k}{x_j - x_k}.$$

## Remarks

Each $\ell_j$ has [[degree|0P4JRMSWNH]] $n$, and $\ell_j(x_i) = 1$ if $i = j$ and $\ell_j(x_i) = 0$ if $i \ne j$.
The $\ell_j$ form a [[basis|0KCQ9Y3QKA]] of the space of polynomials of degree at most $n$, and every such $p$ satisfies $p = \sum_{j} p(x_j) \, \ell_j$.
They also satisfy $\sum_j \ell_j = 1$, the case $p = 1$.
With $\ell(x) = \prod_{k=0}^{n} (x - x_k)$, one has $\ell_j(x) = \dfrac{\ell(x)}{(x - x_j) \, \ell'(x_j)}$.

Example: for the nodes $0, 1, 2$ in $\R$, $\ell_1(x) = \frac{(x - 0)(x - 2)}{(1 - 0)(1 - 2)} = -x(x - 2)$.
Check: $\ell_1(0) = 0$, $\ell_1(1) = 1$ and $\ell_1(2) = 0$.

Non-example: for the nodes $0, 0, 1$, the nodes are not distinct, and the factor $\frac{x - x_1}{x_0 - x_1}$ in $\ell_0$ has denominator $0$, so $\ell_0$ is not defined.
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# Immersion

## Definition

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], and let $f \colon M \to N$ be a [[smooth|0V33BNKZ9S]] map.
The map $f$ is an *immersion* if for every point $p \in M$ the [[differential|6P9PXQDSEN]] $df_p \colon T_pM \to T_{f(p)}N$ is injective.

## Remarks

Equivalently, the rank of $df_p$ equals $\dim M$ at every point $p$; in particular $\dim M \le \dim N$.
An immersion that is a homeomorphism onto its image, with the subspace topology, is a *smooth embedding*.
Example: $f \colon \R \to \R^2$, $f(t) = (\cos t, \sin t)$, is an immersion.
Its derivative $f'(t) = (-\sin t, \cos t)$ never vanishes, so $df_t$ is injective for every $t$.
It is not an embedding, because it is not injective.
Non-example: $g(t) = (t^2, t^3)$ is smooth and injective but is not an immersion.
Its derivative $g'(t) = (2t, 3t^2)$ vanishes at $t = 0$, where the image has a cusp.
By the constant rank theorem, near each point an immersion has the local form $(x_1, \dots, x_m) \mapsto (x_1, \dots, x_m, 0, \dots, 0)$ in suitable charts.
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# Concrete category of algebras

## Summary

A concrete category of algebras is a category whose objects are algebras of a fixed signature and whose morphisms are homomorphisms.
It comes with the [[forgetful functor|WV3BPYCHVG]] to sets.
It is the main example of a [[concrete category|FYBRP16KAN]]: objects are sets with structure, and morphisms are functions that respect the structure.
Varieties, quasivarieties and other classes of algebras are studied in this setting.
There, notions such as monomorphism, epimorphism, free object and reflection can be compared with their set-based meanings.

## Explanations

### L1 human

In a concrete category every object has elements and every arrow is an honest function.
Groups with group homomorphisms, rings with ring homomorphisms and lattices with lattice homomorphisms are all of this kind.
Being concrete makes some categorical notions easy to read: an injective homomorphism is a monomorphism, for instance.
Other notions still surprise, since an epimorphism of rings need not be surjective.
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# Primary decomposition

## Summary

A primary decomposition of an ideal writes it as a finite intersection of [[primary ideals|TMTPWVETQK]].
It generalises the factorisation of an integer into prime powers, since $(n) = (p_1^{e_1}) \cap \cdots \cap (p_k^{e_k})$ in $\Z$ when $n = p_1^{e_1} \cdots p_k^{e_k}$.
By the [[Lasker–Noether theorem|5H8Z56KXMS]], every ideal of a Noetherian ring has one, and its [[associated primes|ERFZ6S8X0X]] are unique.
Geometrically, it splits an algebraic set into irreducible components, with extra embedded components that record finer structure.

## Explanations

### L1 human

In rings without unique factorisation of elements, one can still split ideals into simple pieces.
The pieces are primary ideals, and the ideal is their intersection.
For example, in $\Z$ the ideal $(12)$ is $(4) \cap (3)$.
In the polynomial ring $k[x, y]$ the ideal $(x^2, xy)$ is $(x) \cap (x^2, y)$: the line $x = 0$ together with extra structure at the origin.
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# Birational map

## Definition

Let $X$ and $Y$ be [[algebraic varieties|HV6YBS0YDH]] over an algebraically closed field $k$.
A *birational map* $\varphi \colon X \dashrightarrow Y$ is a [[rational map|4D073CCYGV]] that has an inverse: a rational map $\psi \colon Y \dashrightarrow X$ such that $\psi \circ \varphi = \id_X$ and $\varphi \circ \psi = \id_Y$ as rational maps.
Here $\varphi$ and $\psi$ are required to be dominant, so that the compositions are defined.
$X$ and $Y$ are *birational*, or birationally equivalent, if there is a birational map $X \dashrightarrow Y$.

## Remarks

Equivalently, there are nonempty [[open subsets|02BH3MCBQB]] $U \subseteq X$ and $V \subseteq Y$ and an isomorphism of varieties $U \to V$ (a [[morphism|CDS1VWB7A3]] with inverse morphism).
Also equivalently, there is an isomorphism of $k$-algebras between the [[function fields|2D5SS6WM14]] $k(Y)$ and $k(X)$.

Example: $\mathbb{A}^2$ and $\mathbb{P}^2$ are birational, since $\mathbb{A}^2$ is isomorphic to the open subset $\set{z \ne 0}$ of $\mathbb{P}^2$.
They are not isomorphic, since every regular function on $\mathbb{P}^2$ is constant and $\mathbb{A}^2$ has the nonconstant regular function $x$.

Example: the conic $C = V(x^2 + y^2 - 1) \subseteq k^2$, with $\operatorname{char} k \ne 2$, is birational to $\mathbb{A}^1$ through $t \mapsto \left(\frac{1 - t^2}{1 + t^2}, \frac{2t}{1 + t^2}\right)$, whose inverse is $(x, y) \mapsto y/(1 + x)$.

Non-example: the rational map $\mathbb{P}^2 \dashrightarrow \mathbb{P}^1$, $[x : y : z] \mapsto [x : y]$, is dominant but not birational, since $\dim \mathbb{P}^2 = 2 \ne 1 = \dim \mathbb{P}^1$ and birational varieties have the same dimension.
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# Barratt–Priddy–Quillen theorem

## Statement

For $n \ge 1$, let $\Sigma_n$ be the [[symmetric group|0RJMNEN3BJ]] on $n$ letters, and let $\Sigma_\infty = \bigcup_n \Sigma_n$ be the infinite symmetric group, with $B\Sigma_\infty$ its [[classifying space|7XRBR7AV3C]].
Let $QS^0 = \operatorname{colim}_n \Omega^n S^n$, an [[infinite loop space|A7VB2NSGJ2]], and let $Q_0 S^0$ be its path component of the basepoint.
Then there is a natural map $B\Sigma_\infty \to Q_0 S^0$ that induces an isomorphism
$$H_*(B\Sigma_\infty; \Z) \cong H_*(Q_0 S^0; \Z)$$
in [[singular homology|M0Q5CFJ7ZH]].
Equivalently, $QS^0$ is the group completion of the topological monoid $\coprod_{n \ge 0} B\Sigma_n$, and $\Z \times B\Sigma_\infty^+ \simeq QS^0$, where $B\Sigma_\infty^+$ is Quillen's plus construction.

## Remarks

Barratt and Priddy, and independently Quillen, proved the theorem in the early 1970s.
One proof uses the group completion theorem: the homology of the group completion of $M = \coprod_n B\Sigma_n$ is $H_*(M)[\pi_0(M)^{-1}] = \Z[\Z] \otimes H_*(B\Sigma_\infty)$.
The other input is that $\coprod_n B\Sigma_n$ is the free $E_\infty$ space on a point, and that the group completion of the free $E_\infty$ space on $S^0$ is $QS^0$.

At work: taking homotopy groups gives $\pi_k(B\Sigma_\infty^+) \cong \pi_k^s$ for $k \ge 1$, the [[stable homotopy groups of spheres|HGZAYPQBXW]].
For $k = 1$ this says $\pi_1^s \cong \Z/2$, which matches the abelianization $\Sigma_\infty / A_\infty \cong \Z/2$, where $A_\infty$ is the infinite alternating group and the quotient map is the sign of a permutation.

## Explanations

### L1 human

The theorem links two very different objects: the symmetric groups, which permute finite sets, and the stable homotopy groups of spheres.
It says that, after the plus construction, the classifying space of the infinite symmetric group has exactly the stable homotopy groups of spheres as its homotopy groups.
For this reason the stable homotopy of spheres is often described as the algebraic K-theory of the "field with one element", modelled by finite sets.

### L2 human

Since $\pi_1(B\Sigma_\infty^+)$ is the abelianization of $\Sigma_\infty$, the theorem recovers $\pi_1^s \cong \Z/2$, and the higher groups give a description of all stable stems in terms of permutations.
Together with Nakaoka's homological stability for symmetric groups, it computes $H_*(Q_0 S^0)$ from the homology of the groups $\Sigma_n$.
Quillen's analogue for general linear groups over finite fields led to his computation of the algebraic K-theory of finite fields.
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# Sklar's theorem

## Statement

Let $d \ge 2$ and let $H$ be the joint [[cumulative distribution function|0H8H7ZTHW2]] of a random vector $(X_1, \ldots, X_d)$ with [[marginal|0MXT6PRZZ6]] distribution functions $F_1, \ldots, F_d$.

1. There is a [[copula|CAMJRBZANK]] $C$ such that for all $x_1, \ldots, x_d \in \R$,
$$H(x_1, \ldots, x_d) = C(F_1(x_1), \ldots, F_d(x_d)).$$
2. The copula $C$ is uniquely determined on $\operatorname{Ran} F_1 \times \cdots \times \operatorname{Ran} F_d$, where $\operatorname{Ran} F_i$ is the range of $F_i$.
   In particular $C$ is unique when $F_1, \ldots, F_d$ are continuous.
3. Conversely, if $C$ is a $d$-dimensional copula and $F_1, \ldots, F_d$ are distribution functions on $\R$, then $(x_1, \ldots, x_d) \mapsto C(F_1(x_1), \ldots, F_d(x_d))$ is a joint distribution function with marginals $F_1, \ldots, F_d$.

## Remarks

When the $F_i$ are continuous, the proof is short.
Each $U_i = F_i(X_i)$ is uniform on $[0, 1]$, and $X_i \le x_i$ holds exactly when $U_i \le F_i(x_i)$, up to an event of probability $0$.
So the joint distribution function $C$ of $(U_1, \ldots, U_d)$ satisfies part 1.
For part 3, take $U$ with distribution function $C$ and set $X_i = F_i^{-1}(U_i)$ with the generalised inverse $F_i^{-1}(u) = \inf \set{x : F_i(x) \ge u}$.
For discontinuous marginals, part 1 needs an extra randomisation or an extension argument.
Uniqueness then fails off the ranges: for two Bernoulli variables, many copulas give the same $H$.

Example: if $(X_1, X_2)$ is bivariate normal with correlation $\rho$, its copula is the Gaussian copula with parameter $\rho$.
Combining this copula with other marginals produces non-normal distributions with the same dependence.

## Explanations

### L1 human

Sklar's theorem says that a joint distribution is fully described by two separate pieces.
One piece is the list of the individual distributions of the coordinates.
The other piece, the copula, records only how the coordinates depend on each other.
The pieces can be chosen freely and glued back together.

### L2 human

The theorem justifies the two-step modelling used in practice: fit each marginal, then fit a copula to the ranks or to $F_i(X_i)$.
Many measures of dependence, such as Spearman's rho and Kendall's tau, depend only on the copula when the marginals are continuous.
The theorem also shows that the Fréchet–Hoeffding bounds for copulas give bounds for any joint distribution with given marginals.
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# Blocking set

## Summary

A blocking set in a [[projective plane|DANFNN8C4Y]] is a set of points that meets every line.
Any line is a blocking set, so the interesting ones are those that contain no whole line.
The study of their smallest sizes is a main theme of finite geometry, with links to coding theory and to polynomials over finite fields.

## Explanations

### L1 human

Imagine guards placed on points of a finite plane, with the rule that every line must contain a guard.
Putting all guards on one line works, but it is boring.
A blocking set that contains no complete line must be spread out, and it needs more points than a line.
In a plane of order $n$ it needs at least $n + \sqrt{n} + 1$ points.
When $n$ is a square, a smaller copy of a plane of order $\sqrt{n}$ inside the plane reaches this bound.
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# Ehrhart–Macdonald reciprocity

## Statement

Let $P \subseteq \R^n$ be a [[lattice polytope|F39N9NBFPZ]] of dimension $d$, and let $E_P$ be its [[Ehrhart polynomial|9DEG0WRDHR]].
Let $P^\circ$ be the relative interior of $P$, that is, its interior inside its affine hull.
Then for every integer $t \ge 1$,
$$E_P(-t) = (-1)^d \, \#\left(t P^\circ \cap \Z^n\right).$$

## Remarks

One proof triangulates $P$ into lattice simplices and writes $P$ and $P^\circ$ as disjoint unions of half-open simplices.
For a single simplex, the integer points of all dilates are counted with a rational generating function over the cone above the simplex.
The generating functions of a cone and of its interior are related by $z \mapsto 1/z$, and this symmetry gives the sign $(-1)^d$ and the change from $P$ to $P^\circ$.

Example: for $P = [0,1]^2$, $E_P(t) = (t+1)^2$, so $E_P(-t) = (t-1)^2$.
The open square $(0,t)^2$ contains exactly the integer points $(a,b)$ with $1 \le a, b \le t-1$, and there are $(t-1)^2$ of them.
For the triangle with vertices $(0,0)$, $(1,0)$, $(0,1)$, $E_P(-t) = (t-1)(t-2)/2$, which counts the integer points with $a, b \ge 1$ and $a + b \le t - 1$.

Taking $t = 1$ gives the number of interior lattice points of $P$ as $(-1)^d E_P(-1)$.
The statement extends to rational polytopes, where $E_P$ is replaced by the Ehrhart quasi-polynomial.

## Explanations

### L1 human

The Ehrhart polynomial counts grid points in a blown-up polytope, boundary included.
Plugging negative numbers into the same polynomial counts the grid points strictly inside, up to a sign.
So one formula knows both the closed count and the open count.

### L2 human

Reciprocity says that the Ehrhart polynomials of $P$ and of $P^\circ$ are tied by $t \mapsto -t$.
It is one instance of a family of combinatorial reciprocity laws, which also includes reciprocity for order polynomials and for chromatic polynomials.
Together with $E_P(0) = 1$ and the leading coefficient being the volume, it constrains the coefficients of $E_P$.
For example, a lattice polygon $P$ with area $A$ and $b$ lattice points on its boundary has $E_P(t) = A t^2 + \frac{b}{2} t + 1$.
Reciprocity at $t = 1$ then gives $A - \frac{b}{2} + 1$ interior lattice points, which is Pick's theorem.
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# Bareiss algorithm

## Statement

Let $R$ be an [[integral domain|0QVD7WBFGN]] and let $A = (a_{ij})$ be an $n \times n$ matrix over $R$, with $n \ge 2$.
Put $a^{(0)}_{ij} = a_{ij}$.
For $k = 1, \dots, n - 1$ and $k < i, j \le n$, the *Bareiss algorithm* sets
$$a^{(k)}_{ij} = \frac{a^{(k-1)}_{kk} \, a^{(k-1)}_{ij} - a^{(k-1)}_{ik} \, a^{(k-1)}_{kj}}{a^{(k-2)}_{k-1, k-1}},$$
where the denominator is $1$ for $k = 1$.
Assume that the leading principal [[minors|0BT9BYWVRY]] of $A$ of orders $1, \dots, n - 1$ are nonzero.
Then:

1. for each $k$ and each $i, j > k$, the entry $a^{(k)}_{ij}$ equals the determinant of the $(k+1) \times (k+1)$ submatrix of $A$ with rows $1, \dots, k, i$ and columns $1, \dots, k, j$;
2. in particular, every division in the algorithm is exact in $R$, every pivot $a^{(k-1)}_{kk}$ is the leading principal minor of order $k$, and $a^{(n-1)}_{nn} = \det A$.

## Remarks

Clause 1 is proved by induction on $k$ with Sylvester's determinant identity, which expresses a bordered minor of the reduced matrix as a ratio of minors of $A$.
For $k = 1$ it is the direct check $a^{(1)}_{ij} = a_{11} a_{ij} - a_{i1} a_{1j} = \det \begin{pmatrix} a_{11} & a_{1j} \\ a_{i1} & a_{ij} \end{pmatrix}$.
Example: for $A = \begin{pmatrix} 2 & 3 & 1 \\ 4 & 1 & 5 \\ 6 & 2 & 3 \end{pmatrix}$ over $\Z$, step $1$ gives $a^{(1)}_{22} = -10$, $a^{(1)}_{23} = 6$, $a^{(1)}_{32} = -14$, $a^{(1)}_{33} = 0$, and step $2$ gives $a^{(2)}_{33} = \big((-10) \cdot 0 - 6 \cdot (-14)\big)/2 = 42 = \det A$.
Over $\Z$, clause 1 and [[Hadamard's inequality|CGDN1X32NQ]] bound the size of every intermediate entry.
So the bit length of the numbers grows only polynomially, unlike naive fraction-free elimination, where it can double at each step.
If a pivot is zero, rows can be exchanged as in [[Gaussian elimination|0JVJ2WCXZH]], which changes the sign of the [[determinant|00KDZRW4C0]].
The algorithm uses $O(n^3)$ operations in $R$.

## Explanations

### L1 human

To compute a determinant of an integer matrix exactly, one wants to avoid fractions.
Cross-multiplying rows avoids fractions but makes the numbers huge.
The Bareiss algorithm cross-multiplies and then divides by the previous pivot, and a theorem guarantees that this division always leaves no remainder.

### L2 human

It is fraction-free Gaussian elimination: the work is the same order as ordinary elimination, but all intermediate values are minors of the input matrix.
It is a standard method in computer algebra for determinants, ranks and exact solutions of linear systems over $\Z$ or over polynomial rings.
The final pivot is the determinant, and the earlier pivots are the leading principal minors.
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<document source="https://tamnd.github.io/math2/node/ZDCDS6EWGX.md" tag="ZDCDS6EWGX" kind="concept">
# Semilocal ring

## Summary

A semilocal ring is a ring $R$ whose quotient $R/J(R)$ by the [[Jacobson radical|XA0DCNH7WY]] is a [[semisimple ring|NY25HXA478]].
For a [[commutative ring|0310REJJY8]] this is the same as having finitely many maximal ideals, the [[commutative notion|6DYAGW68AV]].
Left Artinian rings and [[local rings|MZ02BKPHJA]] are semilocal.
Semilocal rings are the starting point for [[semiperfect|ZRC2S3ECFK]], [[perfect|49Y9Q949PK]] and [[semiprimary|XVP517QRH5]] rings, which add conditions on $J(R)$.

## Explanations

### L1 human

Dividing a ring by its Jacobson radical removes the part that simple modules cannot see.
For a semilocal ring, what remains is as simple as possible: a finite product of matrix rings over division rings.
For example, the ring of upper triangular $2 \times 2$ matrices over a field is semilocal, since modulo its radical it is a product of two copies of the field.
The integers are not semilocal, since they have infinitely many maximal ideals.
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<document source="https://tamnd.github.io/math2/node/ZDHAMB0H8J.md" tag="ZDHAMB0H8J" kind="definition">
# Manifold of smooth mappings

## Definition

Let $M$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary and $N$ a smooth manifold, and let $C^\infty(M, N)$ be the set of smooth maps $M \to N$.
Fix a [[Riemannian metric|Q61VFHCH2J]] on $N$, and let $\pi \colon TN \to N$ be the [[tangent bundle|DQB1BVQJWD]].
Fix an open neighbourhood $V$ of the zero section in $TN$ on which the [[exponential map|Y7Z0AGY1A7]] is defined.
Choose $V$ so that $(\pi, \exp) \colon V \to N \times N$ is a diffeomorphism onto an open neighbourhood $W$ of the diagonal.
For $f \in C^\infty(M, N)$, let $\Gamma(f^* TN)$ be the space of smooth [[sections|WWX7H5W1RP]] of the pullback bundle.
These are the smooth maps $s \colon M \to TN$ with $s(x) \in T_{f(x)} N$ for all $x$.
With the topology of uniform convergence of all derivatives, $\Gamma(f^* TN)$ is a Fréchet space.
Let $U_f = \set{g \in C^\infty(M, N) : (f(x), g(x)) \in W \text{ for all } x \in M}$ and define $\varphi_f \colon U_f \to \Gamma(f^* TN)$ by
$$\varphi_f(g)(x) = (\pi, \exp)^{-1}(f(x), g(x)).$$
The *manifold of smooth mappings* is $C^\infty(M, N)$ with the smooth [[Fréchet manifold|EBMCFQT0JN]] structure given by the atlas $(U_f, \varphi_f)_{f \in C^\infty(M, N)}$.

## Remarks

Each $\varphi_f$ is a bijection onto an open subset of $\Gamma(f^* TN)$, with inverse $s \mapsto \exp \circ s$, and the chart changes are smooth.
The resulting manifold structure does not depend on the choice of the metric, its topology is the $C^\infty$ topology, and the tangent space at $f$ is $\Gamma(f^* TN)$.

Example: for $M = N = S^1 = \R / 2\pi\Z$ with the flat metric, take $V$ to be the vectors of length less than $\pi$.
For the identity map $f$, the chart sends $g$ near $f$ to the function $s$ with $g(x) = x + s(x)$ and $\abs{s(x)} < \pi$.
So near the identity, the diffeomorphisms of the circle correspond to small smooth functions on the circle.

Non-example: in the same setting, let $f$ be the constant map at a point $y$, and $g$ the constant map at the antipodal point $y + \pi$.
Then $(f(x), g(x)) = (y, y + \pi)$ is a pair at distance $\pi$, so it lies outside $W$, so $g$ is not in the chart domain $U_f$; it lies in a different chart.


When $\dim M \ge 1$ and $\dim N \ge 1$, the model spaces are Fréchet spaces that are not normable, so this atlas is not an atlas of a Banach manifold.
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<document source="https://tamnd.github.io/math2/node/ZDHPQWWCQW.md" tag="ZDHPQWWCQW" kind="concept">
# Harmonic majorant

## Summary

A harmonic majorant of a function is a [[harmonic function|0DEB0JMN0W]] that lies above it everywhere on the domain.
The question whether a [[subharmonic function|Z1MHMGFNFW]] has a harmonic majorant measures how large it is near the boundary of its domain.
For example, a holomorphic function $f$ on the unit disc lies in the [[Hardy space|PNNNFJVF6W]] $H^p$ exactly when $\abs{f}^p$ has a harmonic majorant.
When a harmonic majorant exists, there is a smallest one, the [[least harmonic majorant|G0H97JZRS9]].

## Explanations

### L1 human

Picture the graph of a subharmonic function as a sagging sheet.
A harmonic majorant is a tight, perfectly balanced sheet stretched above it that never dips below it.
On the unit disc, the constant $1$ is a harmonic majorant of $\abs{z}^2$.
On the whole plane, $\abs{z}^2$ has no harmonic majorant at all, because it grows too fast.
</document>

<document source="https://tamnd.github.io/math2/node/ZDJQRKPWGA.md" tag="ZDJQRKPWGA" kind="concept">
# Permutation group

## Summary

A permutation group is a [[subgroup|0X1JXT47GA]] of the [[symmetric group|0515A6D2Y1]] of a set, that is, a set of [[permutations|05RWFKVTJ7]] closed under composition and inverses.
Groups first arose as permutation groups, in the study of the roots of polynomial equations.
By [[Cayley's theorem|0151VYVX27]], every group is isomorphic to a permutation group.
Every [[group action|0WCY42EV9V]] gives a permutation group, the image of the group in the symmetric group of the set.

## Explanations

### L1 human

A permutation group is a collection of ways to shuffle a set of objects that is closed under doing one shuffle after another and under undoing a shuffle.
The symmetries of a square, viewed as shuffles of its four corners, are a good picture.
If the corners are labelled $1, 2, 3, 4$ in order around the square, the quarter turn is the shuffle $1 \to 2 \to 3 \to 4 \to 1$.
The eight symmetries of the square give eight of the $24$ possible shuffles of the corners, and they form a permutation group.
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<document source="https://tamnd.github.io/math2/node/ZDNBCXDGTA.md" tag="ZDNBCXDGTA" kind="definition">
# Continuous lattice

## Definition

Let $L$ be a [[complete lattice|D7A0ZGZT5S]], and let $\ll$ be the [[way-below relation|RDM7J5J9HQ]] on $L$.
The lattice $L$ is *continuous* if every $y \in L$ satisfies
$$y = \bigvee \set{x \in L : x \ll y},$$
where $\bigvee$ denotes the [[supremum|0R3H5PANRQ]].

## Remarks

Since $x \ll y$ implies $x \le y$, the condition says that $y$ is not larger than the supremum of the elements way below it.
Example: $L = [0, 1]$ with the usual order, where $x \ll y$ if and only if $x = 0$ or $x < y$.
For $y > 0$ the supremum of $[0, y)$ is $y$, and for $y = 0$ the set $\set{0}$ has supremum $0$.
Example: every [[algebraic lattice|3J0BTXSCVN]] is continuous, because each compact $c \le y$ satisfies $c \ll y$.
Non-example: the lattice of closed subsets of $[0, 1]$ under inclusion is complete.
But $\varnothing$ is the only element way below $[0, 1]$, so $[0, 1]$ is not the supremum of the elements way below it.
A complete lattice is continuous if and only if $\bigwedge_{j \in J} \bigvee D_j = \bigvee_{f} \bigwedge_{j \in J} f(j)$ for every family $(D_j)_{j \in J}$ of directed subsets, where $f$ runs over the choice functions with $f(j) \in D_j$.
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<document source="https://tamnd.github.io/math2/node/ZDS77Q0F06.md" tag="ZDS77Q0F06" kind="concept">
# Integral equation of the first kind

## Summary

An integral equation of the first kind is a [[linear integral equation|AYZCKG34QK]] in which the unknown function appears only under the integral: $\int K(x, s) \varphi(s) \, ds = f(x)$.
Solving it means inverting an integral operator.
Such equations are typically ill-posed: a solution may fail to exist, and small changes of $f$ can cause large changes of $\varphi$.
Inverse problems, such as recovering a signal from blurred data, lead to equations of the first kind.

## Explanations

### L1 human

One only sees a smoothed version $f$ of the unknown $\varphi$, and one wants $\varphi$ back.
Smoothing destroys fine detail, so going back amplifies noise.
For example, $\int_0^x \varphi(s) \, ds = f(x)$ asks for a function whose integral is $f$.
Its continuous solution is $\varphi = f'$, which exists only when $f$ is continuously differentiable with $f(0) = 0$.
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<document source="https://tamnd.github.io/math2/node/ZDXQRARWT5.md" tag="ZDXQRARWT5" kind="definition">
# Colombeau algebra

## Definition

Let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]], and let $C^\infty(\Omega)$ be the algebra of [[smooth functions|0V33BNKZ9S]] on $\Omega$.
Let $\mathcal{E}(\Omega)$ be the algebra of families $(u_\varepsilon)_{\varepsilon \in (0,1]}$ in $C^\infty(\Omega)$, with pointwise operations.
For a [[compact set|0R6EWFKRCF]] $K \subseteq \Omega$ and a multi-index $\alpha$, let $p_{K,\alpha}(u) = \sup_{x \in K} \abs{\partial^\alpha u(x)}$.

1. A family $(u_\varepsilon)$ is *moderate* if for each $K$ and $\alpha$ there is $N \ge 0$ with $p_{K,\alpha}(u_\varepsilon) = O(\varepsilon^{-N})$ as $\varepsilon \to 0$.
2. A family $(u_\varepsilon)$ is *negligible* if for each $K$, $\alpha$ and $m \ge 0$, $p_{K,\alpha}(u_\varepsilon) = O(\varepsilon^{m})$ as $\varepsilon \to 0$.

The moderate families form a subalgebra $\mathcal{E}_M(\Omega)$, and the negligible families form an [[ideal|096Z9MP2ME]] $\mathcal{N}(\Omega)$ of it.
The *special Colombeau algebra* on $\Omega$ is the [[quotient algebra|0076HWHGHQ]] $\mathcal{G}(\Omega) = \mathcal{E}_M(\Omega) / \mathcal{N}(\Omega)$.

## Remarks

$\mathcal{G}(\Omega)$ is a commutative associative algebra, and the partial derivatives $\partial_j$, applied to each $u_\varepsilon$, pass to the quotient and satisfy the Leibniz rule.
A smooth function $f$ embeds as the class of the constant family $u_\varepsilon = f$, and this embedding keeps products.
On $\R^n$, distributions embed by convolution with a rescaled mollifier $\rho_\varepsilon(x) = \varepsilon^{-n} \rho(x / \varepsilon)$, where $\rho$ is a Schwartz function with $\int \rho = 1$ and all higher moments $0$.
This embedding depends on the choice of $\rho$, and the moment condition makes it agree with the embedding of smooth functions.
The [[Schwartz impossibility theorem|Q0C1H111EN]] shows that the product of continuous functions cannot also be kept; in $\mathcal{G}$ it is kept only up to association.

Example: on $\R$, let $\rho$ be smooth with compact support and $\int \rho = 1$, and let $u_\varepsilon(x) = \varepsilon^{-1} \rho(x / \varepsilon)$.
Then $\sup \abs{\partial^\alpha u_\varepsilon} = \varepsilon^{-1-\alpha} \sup \abs{\rho^{(\alpha)}}$, so the family is moderate.
Its class is associated with the delta: $\int u_\varepsilon \varphi \to \varphi(0)$ for every test function $\varphi$.
Its square is the class of $(u_\varepsilon^2)$.

Non-example: the constant family $u_\varepsilon = e^{1/\varepsilon}$ is not moderate, since $e^{1/\varepsilon}$ grows faster than every power $\varepsilon^{-N}$.
So it defines no element of $\mathcal{G}(\R)$.
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<document source="https://tamnd.github.io/math2/node/ZE166728M3.md" tag="ZE166728M3" kind="definition">
# Potential of a measure

## Definition

Let $n \ge 1$ and let $k \colon \R^n \times \R^n \to (-\infty, +\infty]$ be a lower semicontinuous function, called a *kernel*.
Let $\mu$ be a positive [[Borel measure|DYR1QFNGNJ]] on $\R^n$ with finite total mass and compact [[support|FQTXV6BQZG]].
The *potential of $\mu$ with respect to $k$* is the function $U_k^\mu \colon \R^n \to (-\infty, +\infty]$,
$$U_k^\mu(x) = \int k(x, y) \, d\mu(y).$$

## Remarks

The integral is well defined: a lower semicontinuous function is bounded below on the compact set $\set{x} \times \operatorname{supp} \mu$, and $\mu$ is finite.
Example: for the Dirac measure $\mu = \delta_a$ one gets $U_k^\mu(x) = k(x, a)$.
Example: with the [[Newtonian kernel|6TJTWQ5HCR]] on $\R^3$ and $\mu$ the uniform probability measure on the unit sphere, $U_k^\mu(x) = 1$ for $\abs{x} \le 1$ and $1/\abs{x}$ for $\abs{x} \ge 1$.
Non-example: $k(x, y) = -1/\abs{x - y}$ on $\R^3$ off the diagonal is not a kernel in this sense.
It tends to $-\infty$ at the diagonal, so no value in $(-\infty, +\infty]$ on the diagonal makes it lower semicontinuous.
The potential $U_k^\mu$ is itself lower semicontinuous, by Fatou's lemma.
Standard kernels are the Newtonian kernel, the [[logarithmic kernel|VDZ5FKYC2T]] and the Riesz kernels $\abs{x - y}^{\alpha - n}$ with $0 < \alpha < n$.
Signed measures are handled by writing $\mu = \mu^+ - \mu^-$ where both potentials are finite.
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<document source="https://tamnd.github.io/math2/node/ZE1FSZ6GJF.md" tag="ZE1FSZ6GJF" kind="definition">
# Tikhonov regularization

## Definition

Let $X$ and $Y$ be real or complex [[Hilbert spaces|HM2HHE3J51]], let $K \colon X \to Y$ be a [[bounded linear operator|03TM135G0N]] with [[adjoint|6HSJBJ8TTQ]] $K^*$, let $y \in Y$ and let $\alpha > 0$.
The *Tikhonov regularized solution* of $K x = y$ with parameter $\alpha$ is the unique minimizer $x_\alpha \in X$ of the Tikhonov functional
$$J_\alpha(x) = \norm{K x - y}^2 + \alpha \norm{x}^2.$$
It is given by
$$x_\alpha = (K^* K + \alpha I)^{-1} K^* y.$$
*Tikhonov regularization* is the method that approximates solutions of $K x = y$ by $x_\alpha$, with $\alpha$ chosen according to the noise level of the data.

## Remarks

The operator $K^* K + \alpha I$ satisfies $\ip{(K^* K + \alpha I) x}{x} \ge \alpha \norm{x}^2$, so it is invertible with inverse of norm at most $1/\alpha$; the minimizer is characterized by the normal equation $(K^* K + \alpha I) x = K^* y$.
So $x_\alpha$ exists, is unique, and depends continuously on $y$.

If $y$ lies in the range of $K$, then $x_\alpha \to K^\dagger y$ as $\alpha \to 0$, where $K^\dagger y$ is the solution of smallest norm, given by the [[Moore–Penrose inverse|0YY8VX28JR]].
For noisy data $y^\delta$ with $\norm{y^\delta - y} \le \delta$, the choice $\alpha(\delta) \to 0$ with $\delta^2 / \alpha(\delta) \to 0$ gives $x_{\alpha(\delta)}^\delta \to K^\dagger y$ as $\delta \to 0$.
For compact $K$ with singular system $(\sigma_n, v_n, u_n)$, $x_\alpha = \sum_n \frac{\sigma_n}{\sigma_n^2 + \alpha} \ip{y}{u_n} v_n$, so the unbounded factors $1/\sigma_n$ of [[Picard's criterion|MPB3QF09E5]] are replaced by factors bounded by $1/(2\sqrt{\alpha})$.

Example: $X = Y = \R$ and $K x = k x$ with $k \ne 0$.
Then $J_\alpha(x) = (k x - y)^2 + \alpha x^2$ has derivative $2k(k x - y) + 2\alpha x$, which vanishes at $x_\alpha = k y / (k^2 + \alpha)$, in agreement with the formula.

Non-example: the case $\alpha = 0$ is excluded.
For $K = 0$ on $X = Y = \R$, every $x$ minimizes $\norm{K x - y}^2$, so there is no unique minimizer.

In finite dimensions, with $K$ a data matrix, $x_\alpha$ is the [[ridge regression|3KZW2930V2]] estimator.

## Explanations

### L1 human

The method adds a penalty $\alpha \norm{x}^2$ on the size of the solution to the misfit $\norm{K x - y}^2$, and takes the best compromise.
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<document source="https://tamnd.github.io/math2/node/ZE4QPBC1AF.md" tag="ZE4QPBC1AF" kind="definition">
# Hartley transform

## Definition

Let $f \colon \R \to \C$ be Lebesgue [[integrable|X7GTF3D98F]].
Let $\operatorname{cas}(x) = \cos x + \sin x$.
The *Hartley transform* of $f$ is the function $\mathcal{H} f \colon \R \to \C$ given by
$$(\mathcal{H} f)(\omega) = \frac{1}{\sqrt{2\pi}} \int_\R f(t)\, \operatorname{cas}(\omega t)\, dt.$$

## Remarks

Let $F(\omega) = \frac{1}{\sqrt{2\pi}} \int_\R f(t) e^{-i\omega t}\, dt$ be the unitary [[Fourier transform|DPVM34ADZV]] with angular frequency.
For real $f$, $\operatorname{Re} F(\omega) = \frac{1}{\sqrt{2\pi}} \int f(t) \cos(\omega t)\, dt$ and $\operatorname{Im} F(\omega) = -\frac{1}{\sqrt{2\pi}} \int f(t) \sin(\omega t)\, dt$, so $\mathcal{H} f = \operatorname{Re} F - \operatorname{Im} F$.
In general $\mathcal{H} f(\omega) = \frac{1 + i}{2} F(\omega) + \frac{1 - i}{2} F(-\omega)$.
From this formula and the Fourier inversion theorem, $\mathcal{H}(\mathcal{H} f) = f$ when $f$ and $\mathcal{H} f$ are integrable and $f$ is continuous.
Also $\mathcal{H}$ extends to a unitary operator on $L^2(\R)$ with $\mathcal{H}^2 = \id$.

Example: $f(t) = e^{-t^2/2}$.
Since $f$ is even, $\int f(t) \sin(\omega t)\, dt = 0$, and $\frac{1}{\sqrt{2\pi}} \int e^{-t^2/2} \cos(\omega t)\, dt = e^{-\omega^2/2}$.
So $\mathcal{H} f(\omega) = e^{-\omega^2/2}$.

Non-example: $f(t) = 1$ is not integrable on $\R$, and $\int_\R \operatorname{cas}(\omega t)\, dt$ does not converge for any $\omega$.
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<document source="https://tamnd.github.io/math2/node/ZE6DXEJB8H.md" tag="ZE6DXEJB8H" kind="concept">
# Barrelled space

## Summary

A barrel is a closed, convex, balanced and absorbing set.
A barrelled space is a [[locally convex space|B6GY2KJMWW]] in which every barrel is a neighbourhood of $0$.
Barrelled spaces are the spaces where the uniform boundedness principle (Banach–Steinhaus theorem) holds in its usual form.
Banach spaces, [[Fréchet spaces|3GAA614673]] and [[LF-spaces|7THJSH9SFP]] are barrelled.

## Explanations

### L1 human

A barrel is a set that looks like a closed unit ball: it is closed, convex, symmetric, and big enough in every direction.
In a barrelled space every such set is automatically a genuine neighbourhood of $0$.
This rules out sets that are "thin" in infinitely many directions at once.
The reason one cares is that pointwise bounded families of continuous linear maps are then equicontinuous.
In a Banach space this is the classical uniform boundedness principle, proved with the Baire category theorem.
</document>

<document source="https://tamnd.github.io/math2/node/ZE780BQS42.md" tag="ZE780BQS42" kind="statement">
# Anti-equivalence of affine schemes and rings

## Statement

Let $A$ and $B$ be commutative rings with identity.

1. The natural map $A \to \mathcal{O}_{\operatorname{Spec} A}(\operatorname{Spec} A)$ is a ring isomorphism.
2. The map that sends a [[ring homomorphism|0VTH8KPCG2]] $\varphi \colon A \to B$ to the induced [[morphism of schemes|JXRKH6P1Z4]] $\operatorname{Spec} B \to \operatorname{Spec} A$ is a bijection
$$\Hom_{\mathrm{Ring}}(A, B) \to \Hom_{\mathrm{Sch}}(\operatorname{Spec} B, \operatorname{Spec} A).$$
Its inverse sends a morphism $(f, f^\#)$ to its map on global sections $A \to B$.

Consequently, the [[functor|28NQ0KRHD3]] $\operatorname{Spec}$ is an [[equivalence|K7A04ZBD92]] from the [[opposite category|DAMK8XE1FR]] of commutative rings to the category of [[affine schemes|B3V3XJ2W8G]].
A quasi-inverse is $X \mapsto \mathcal{O}_X(X)$.

## Remarks

Part 1 is a gluing argument: a global section is given locally by fractions $a_i / f_i^{n_i}$ on basic open sets $D(f_i)$ that cover $\operatorname{Spec} A$.
Since the $f_i$ then generate the unit ideal, these fractions glue to a single element of $A$.
For part 2, the stalk maps of a morphism must be local.
This forces $f(\mathfrak{q})$ to be the preimage of $\mathfrak{q}$ under the map on global sections.
Then the sheaf map is determined on each $D(g)$ by localizing.
Example: for a field $k$, the morphisms $\operatorname{Spec} k \to \operatorname{Spec} k[x]$ over $k$ correspond to $k$-algebra maps $k[x] \to k$, that is, to the elements $a \in k$, the $k$-points of the line.
The locality condition is needed.
Let $R$ be a discrete valuation ring with fraction field $K$.
Send the point of $\operatorname{Spec} K$ to the closed point of $\operatorname{Spec} R$, with the inclusion $R \to K$ on sections.
This is a morphism of ringed spaces, but it is not the morphism induced by $R \to K$, which hits the generic point.

## Explanations

### L1 human

Affine schemes and commutative rings carry the same information, with arrows turned around.
A map of spaces $\operatorname{Spec} B \to \operatorname{Spec} A$ is the same thing as a map of rings $A \to B$, which pulls functions back.
So every question about affine schemes can be translated into algebra, and every ring can be thought of as a space.

### L2 human

This is the reason the [[spectrum of a ring|FB45BHHA2J]] is the right local model for algebraic geometry.
It extends to all schemes: for any scheme $X$, morphisms $X \to \operatorname{Spec} A$ correspond to ring maps $A \to \mathcal{O}_X(X)$.
It is used constantly.
For example, it shows that the fiber product of affine schemes is the spectrum of a tensor product.
It also lets one describe a scheme through its functor of points on rings.
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<document source="https://tamnd.github.io/math2/node/ZEA0PEM3B2.md" tag="ZEA0PEM3B2" kind="concept">
# Fundamental class

## Summary

The fundamental class $[M]$ of a closed oriented $n$-manifold $M$ is a class in $H_n(M; \Z)$ that restricts to the chosen local [[orientation|913WEQERBE]] at every point.
It represents the manifold itself as a single homology cycle.
Capping with it gives Poincaré duality, and pairing top-degree cohomology classes with it gives numbers such as degrees of maps and characteristic numbers.

## Explanations

### L1 human

Cut a closed oriented surface into triangles and orient them all the same way.
The sum of all these triangles is a cycle, because each edge appears twice with opposite signs and cancels.
This cycle is the fundamental class: the whole surface counted once.
For a connected closed oriented manifold it generates the top homology group, which is $\Z$.
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<document source="https://tamnd.github.io/math2/node/ZEETZHV69Y.md" tag="ZEETZHV69Y" kind="concept">
# Join

## Summary

The join $X * Y$ of two topological spaces is the space of all segments that run from a point of $X$ to a point of $Y$.
Formally it is $X \times Y \times [0, 1]$ with the end $t = 0$ collapsed onto $X$ and the end $t = 1$ collapsed onto $Y$.
Joins generalise cones and suspensions, and the join of two spheres is again a sphere: $S^m * S^n \cong S^{m+n+1}$.

## Explanations

### L1 human

Place $X$ and $Y$ in space far from each other and draw a straight segment from every point of $X$ to every point of $Y$.
The union of all these segments is the join.
For example, the join of two points is a segment, and the join of a point with a circle is a cone.
The join of two disjoint segments in general position in $\R^3$ is a solid tetrahedron.
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<document source="https://tamnd.github.io/math2/node/ZEH0N3N467.md" tag="ZEH0N3N467" kind="definition">
# Support of a measure

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.
The *support* of $\mu$ is the set
$$\operatorname{supp} \mu = \set{x \in X : \mu(N) > 0 \text{ for every open neighbourhood } N \text{ of } x}.$$

## Remarks

Equivalently, the complement of $\operatorname{supp} \mu$ is the union of all [[open sets|02BH3MCBQB]] $U$ with $\mu(U) = 0$.
So the support is always a closed set.

Example: let $\delta_0$ be the [[Dirac measure|0ACRGYNHSR]] at $0$ on $\R$.
Every [[neighbourhood|0ZCY0Z80B6]] of $0$ contains $0$, so it has measure $1$, and $0 \in \operatorname{supp} \delta_0$.

Non-example: the point $x = 1$ is not in $\operatorname{supp} \delta_0$.
The open neighbourhood $(1/2, 3/2)$ of $1$ does not contain $0$, so its measure is $0$.
Thus $\operatorname{supp} \delta_0 = \set{0}$.

For Lebesgue measure on $\R^n$, the support is all of $\R^n$.
If $X$ is second countable, or if $\mu$ is a Radon measure, then $\mu(X \setminus \operatorname{supp} \mu) = 0$.

## Explanations

### L1 human

The support collects the points where the measure has mass in every small region around them.
Removing the open regions of zero mass leaves exactly the support.
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<document source="https://tamnd.github.io/math2/node/ZEJ9VKKKRG.md" tag="ZEJ9VKKKRG" kind="definition">
# Method of images

## Definition

Let $n \ge 3$ and let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]] with [[boundary|05D5EGMFYV]] $\partial\Omega$.
Let $\sigma_{n-1}$ be the area of the unit sphere, and let $\Phi(z) = \frac{1}{(n-2)\sigma_{n-1}} \abs{z}^{2-n}$ be the [[fundamental solution|BSJ8X2XRN9]] of $-\Delta$.
Fix $x \in \Omega$.
An *image system* for $\Omega$ and $x$ is a finite or countable family of points $x_j \in \R^n \setminus \overline{\Omega}$ and real numbers $c_j$ such that:

1. the series $h(y) = \sum_j c_j \Phi(y - x_j)$ converges to a harmonic function of $y$ on $\Omega$, continuous on $\overline{\Omega}$;
2. $\Phi(y - x) = h(y)$ for every $y \in \partial\Omega$.

The *method of images* is the construction of the [[Green's function|M36SD56MWQ]] of $\Omega$ as $G(x, y) = \Phi(y - x) - h(y)$ from an image system for each $x \in \Omega$.

## Remarks

Example: for the half-space $\Omega = \set{y : y_n > 0}$ and $x = (x', x_n)$, the single point $x_1 = (x', -x_n)$ with $c_1 = 1$ is an image system, since $\abs{y - x} = \abs{y - x_1}$ when $y_n = 0$.
This gives $G(x, y) = \Phi(y - x) - \Phi(y - x_1)$.
For the unit ball and $x \ne 0$, the inverse point $x^* = x/\abs{x}^2$ with $c = \abs{x}^{2-n}$ is an image system.
Non-example: for the half-space, the point $(x', -x_n)$ with $c = -1$ fails clause 2, since then $h = -\Phi(y - x) \ne \Phi(y - x)$ on the boundary.
In the plane the same idea works with $\Phi(z) = \frac{1}{2\pi} \log \frac{1}{\abs{z}}$, where $h$ may also contain an additive constant; for the unit disc this gives the Green's function $\frac{1}{2\pi} \log \abs{\frac{1 - \bar x y}{y - x}}$ in complex notation.
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# Hartogs lemma for subharmonic functions

## Statement

Let $n \ge 2$, let $\Omega \subseteq \R^n$ be an [[open set|02BH3MCBQB]], and let $(u_k)_{k \ge 1}$ be a sequence of [[subharmonic functions|Z1MHMGFNFW]] on $\Omega$.
Assume that the sequence is locally uniformly bounded above: each [[compact set|0R6EWFKRCF]] $L \subseteq \Omega$ has a constant $C_L$ with $u_k \le C_L$ on $L$ for all $k$.
Let $f$ be a continuous real function on $\Omega$ with $\limsup_{k \to \infty} u_k(x) \le f(x)$ for every $x \in \Omega$.
Then for every compact set $K \subseteq \Omega$ and every $\varepsilon > 0$ there is $k_0$ such that
$$u_k(x) \le f(x) + \varepsilon \qquad \text{for all } x \in K \text{ and all } k \ge k_0.$$

## Remarks

Proof idea: fix $x_0 \in K$ and a small radius $r$ such that $f < f(x_0) + \varepsilon/3$ on a closed ball about $x_0$ inside $\Omega$.
By the [[sub-mean value property|BXYQE0Q1ST]], $u_k(y)$ is at most the mean of $u_k$ over the ball $B(y, r)$, for $y$ near $x_0$.
The uniform upper bound allows the reverse form of [[Fatou's lemma|S95TR54TCG]], which bounds the [[limit superior|0T3RZQXDYD]] of these ball means by the mean of $\limsup_k u_k \le f$.
This gives $u_k \le f(x_0) + \varepsilon$ on a neighbourhood of $x_0$ for large $k$, and compactness of $K$ finishes the proof.

The lemma upgrades a pointwise upper bound to a uniform one on compact sets, in the upward direction only.
A typical use: if $u_k \le 0$ in the limit at every point, then $\max_K u_k$ is eventually at most $\varepsilon$ on each compact $K$.
The lower bound is not upgraded in the same way, because subharmonic functions can take very negative values on small sets.

## Explanations

### L1 human

Suppose a sequence of subharmonic functions eventually stays below some continuous ceiling at every single point.
Then, on any compact piece of the domain, the whole sequence eventually stays below the ceiling plus a little, all at once.
Subharmonic functions cannot have thin upward spikes, which is why pointwise control becomes uniform control.

### L2 human

The lemma is named after Hartogs, who used an argument of this kind in his theorem on separately holomorphic functions in several complex variables.
It is a basic compactness tool for subharmonic and plurisubharmonic functions.
Typical uses are bounds on $\frac{1}{k}\log\abs{p_k}$ for sequences of polynomials $p_k$, and the study of upper envelopes of families of subharmonic functions.
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# Many-sorted algebra

## Definition

Let $S$ be a [[set|0SJ3RX9TXA]], whose elements are called *sorts*.
An *$S$-sorted signature* $\Sigma$ is a set of operation symbols, each with a type $s_1 \cdots s_n \to s$, where $n \ge 0$ and $s_1, \dots, s_n, s \in S$.
A *many-sorted $\Sigma$-algebra* $A$ consists of:

1. a set $A_s$ for each sort $s \in S$, the carrier of sort $s$;
2. for each symbol $f$ of type $s_1 \cdots s_n \to s$, a [[function|0EPAT51DXQ]] $f^A \colon A_{s_1} \times \cdots \times A_{s_n} \to A_s$ on the [[Cartesian product|09CEDJYBKP]].

A *homomorphism* $h \colon A \to B$ of $\Sigma$-algebras is a family of maps $h_s \colon A_s \to B_s$ with $h_s(f^A(a_1, \dots, a_n)) = f^B(h_{s_1}(a_1), \dots, h_{s_n}(a_n))$ for every symbol $f$ of type $s_1 \cdots s_n \to s$ and all $a_i \in A_{s_i}$.

## Remarks

Example: let $S = \set{\mathrm{scalar}, \mathrm{vector}}$, with symbols $+ \colon \mathrm{vector}\,\mathrm{vector} \to \mathrm{vector}$, $\cdot \colon \mathrm{scalar}\,\mathrm{vector} \to \mathrm{vector}$, and the field operations on the sort $\mathrm{scalar}$.
A vector space $V$ over a field $K$ gives an algebra with $A_{\mathrm{scalar}} = K$ and $A_{\mathrm{vector}} = V$; each operation sends arguments of the right sorts to the right sort.

Non-example: the same data with a map $K \times V \to K$ in place of scalar multiplication fails clause 2, since the value of $\cdot$ must lie in the carrier of sort $\mathrm{vector}$.

With one sort, a many-sorted algebra is an ordinary algebra.
Birkhoff and Lipson called these structures *heterogeneous algebras*.
Some authors require each carrier $A_s$ to be nonempty.
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# Irreducible matrix

## Definition

Let $n \ge 2$, let $\mathbb{K}$ be a field, and let $A$ be an $n \times n$ [[matrix|0JXZKX526R]] over $\mathbb{K}$.
The matrix $A$ is *reducible* if some $n \times n$ [[permutation matrix|08SP3DF8NB]] $P$ gives
$$P^T A P = \begin{pmatrix} B & C \\ 0 & D \end{pmatrix},$$
where $B$ and $D$ are square matrices of size at least $1$.
The matrix $A$ is *irreducible* if it is not reducible.

## Remarks

Let $G(A)$ be the [[directed graph|0KFYGSHHF5]] with vertices $1, \ldots, n$ and an edge from $i$ to $j$ whenever $a_{ij} \ne 0$.
Then $A$ is irreducible if and only if $G(A)$ is strongly connected.
For a real nonnegative matrix $A$, this is also equivalent to $(I + A)^{n-1}$ having all entries positive.
Example: $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is irreducible, because its graph has the edges $1 \to 2$ and $2 \to 1$.
Non-example: $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is reducible with $P = I$, since it is already block upper triangular with $1 \times 1$ blocks.
Conventions for $1 \times 1$ matrices differ between authors.
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# Variogram

## Summary

The variogram of a spatial random field measures how different the values at two locations are expected to be.
It is a function of the displacement between the two locations.
It is defined as the variance of the increment $Z(s + h) - Z(s)$, and half of it is called the semivariogram.
It exists under weaker assumptions than a covariance function, so it also describes fields without a constant variance, such as the [[Wiener process|22735FGJV6]].
The variogram is the basic input of [[kriging|VP9S3PNJQ0]] and of most of geostatistics.

## Explanations

### L1 human

Measure soil moisture at many points of a field.
Two points a metre apart usually have similar values, while two points a kilometre apart may be very different.
The variogram records, for each distance, the typical squared difference between the values.
It is small near distance zero and grows with the distance, often levelling off at some range beyond which points are unrelated.
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# Dirac operator

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $E$ be a complex [[vector bundle|P4DZM975BV]] over $M$.
Let $D$ be a first-order linear differential operator acting on smooth sections of $E$.
In a local chart $x = (x_1, \ldots, x_n)$, let $(g^{jk}(x))$ be the inverse of the matrix of the metric $g$.
The operator $D$ is a *Dirac operator* if, in every local chart and every local trivialisation of $E$,
$$D^2 = -\sum_{j,k} g^{jk}(x)\frac{\partial^2}{\partial x_j \partial x_k} + (\text{terms of order at most } 1).$$

## Remarks

Equivalently, the principal symbol of $D^2$ is $\abs{\xi}^2_g$ times the identity, so $D^2$ is a generalised [[Laplacian|0E0B6RA8K0]].
It follows that $D$ is an [[elliptic operator|XY8GEQSWPR]].

Example: on $\R^n$ with the flat metric, let $\gamma_1, \ldots, \gamma_n$ be complex $N \times N$ matrices with $\gamma_j\gamma_k + \gamma_k\gamma_j = -2\delta_{jk} I$, and let $D = \sum_j \gamma_j \partial_j$ act on $\C^N$-valued functions.
Then $D^2 = \sum_{j,k}\gamma_j\gamma_k\partial_j\partial_k = \frac{1}{2}\sum_{j,k}(\gamma_j\gamma_k + \gamma_k\gamma_j)\partial_j\partial_k = -\sum_j \partial_j^2$, so $D$ is a Dirac operator.
For $n = 1$ and $\gamma_1 = -i$, this is $D = -i\, d/dx$.

Non-example: on $\R$, the operator $D = d/dx$ has $D^2 = +d^2/dx^2$, whose second-order term has the wrong sign.

On a spin manifold, the Atiyah–Singer Dirac operator acts on the [[spinor bundle|PYYQQKAZ0C]] through Clifford multiplication and the Levi-Civita connection.
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# Δ-complex

## Summary

A $\Delta$-complex is a space built by gluing simplices along their faces.
Each face of a simplex is identified with another simplex by the order-preserving linear map.
It is more flexible than a [[simplicial complex|V8GZ0W1ZZ5]].
A simplex may have several of its faces glued together, and two simplices may share all their vertices.
$\Delta$-complexes are the geometric realizations of semi-simplicial sets, also called $\Delta$-sets, and they are the setting of simplicial homology in Hatcher's textbook.

## Explanations

### L1 human

Take some triangles, some edges and some points, and label the corners of every triangle and edge by $0, 1, 2, \dots$ in order.
Glue edges of triangles to edges, and ends of edges to points, always matching the labels in increasing order.
The torus is obtained from two triangles in this way, with only three edges and one vertex in total.
A simplicial complex would need many more triangles for the torus, since in a simplicial complex distinct triangles cannot share all three corners.
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# Fundamental groupoid

## Definition

Let $X$ be a topological space.
The *fundamental [[groupoid|FH3C0T643F]]* $\Pi_1(X)$ is the [[category|A2DE66937D]] in which:

1. the objects are the points of $X$;
2. a morphism from $x$ to $y$ is a [[path homotopy|CCNKA8C711]] class $[\gamma]$ of paths $\gamma \colon [0, 1] \to X$ with $\gamma(0) = x$ and $\gamma(1) = y$;
3. the composite of $[\alpha] \colon x \to y$ and $[\beta] \colon y \to z$ is $[\alpha \cdot \beta] \colon x \to z$, where $(\alpha \cdot \beta)(s)$ is $\alpha(2s)$ for $s \le 1/2$ and $\beta(2s - 1)$ for $s \ge 1/2$;
4. the identity of $x$ is the class of the constant path at $x$.

## Remarks

Every morphism $[\gamma]$ has the inverse $[\bar\gamma]$, where $\bar\gamma(s) = \gamma(1 - s)$, so $\Pi_1(X)$ is a groupoid.
The group of morphisms from $x$ to itself is the fundamental group $\pi_1(X, x)$.
Example: for a convex set $C \subseteq \R^n$, there is exactly one morphism between any two points, since any two paths with the same ends are path homotopic by the straight-line homotopy.
Non-example: if paths are taken without passing to path homotopy classes, composition is not associative, so clause 3 does not define a category.
Two points lie in the same connected component of $\Pi_1(X)$ exactly when they lie in the same path component of $X$.
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# Kirby calculus

## Definition

A *framed link* in $S^3$ is a [[link|1A5WRCNTM3]] $L = K_1 \cup \cdots \cup K_r$ in $S^3$ with an integer $n_i$, its framing, attached to each component $K_i$.
It determines a closed oriented $3$-manifold $M_L$ by integral [[Dehn surgery|ZA405B0Z1S]] on $L$.
It also determines a compact $4$-manifold $W_L$, obtained from $D^4$ by attaching one $2$-handle along each $K_i$ with framing $n_i$.
This is a [[handle decomposition|9V11N8VVTP]] of $W_L$, and $\partial W_L = M_L$.
*Kirby calculus* is the system of the following moves on framed links, together with isotopy of links:

1. Blow-up and blow-down: add to $L$, or remove from $L$, an unknot with framing $\pm 1$ inside a ball disjoint from the rest of $L$.
2. Handle slide: replace a component $K_i$ by a band sum of $K_i$ with a parallel copy of another component $K_j$.
   The parallel copy is taken with respect to the framing $n_j$, and the framing of the new component changes accordingly.

## Remarks

By the [[Lickorish–Wallace theorem|1YP4RF1QXV]], every closed connected oriented $3$-manifold is $M_L$ for some framed link $L$.
Kirby's theorem states that $M_L$ and $M_{L'}$ are orientation-preserving diffeomorphic if and only if $L$ and $L'$ are related by a finite sequence of the moves above.
A handle slide does not change $W_L$ up to diffeomorphism, while a blow-up replaces $W_L$ by its connected sum with $\pm \C P^2$; both leave $M_L$ unchanged.
Example: the unknot with framing $+1$ describes $S^3$ itself, since a blow-down turns it into the empty link, which describes $S^3$.
Non-example: the unknot with framing $0$ cannot be blown down, because move 1 needs framing $\pm 1$; indeed it describes $S^1 \times S^2$, which is not $S^3$.
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# Lyapunov equation

## Definition

Let $n \ge 1$, let $A$ be an $n \times n$ complex [[matrix|0JXZKX526R]], and let $Q$ be an $n \times n$ [[Hermitian matrix|0P17NGQQMQ]].
Write $A^*$ for the [[conjugate transpose|0DXKE10PHF]] of $A$.

1. The *continuous Lyapunov equation* is $AX + XA^* + Q = 0$ for an unknown $n \times n$ matrix $X$.
2. The *discrete Lyapunov equation* is $AXA^* - X + Q = 0$ for an unknown $n \times n$ matrix $X$.

## Remarks

The continuous equation is the [[Sylvester equation|M9SW4NVP2H]] $AX + XA^* = -Q$.
Many texts in control write it as $A^* P + PA + Q = 0$ instead, which is the same equation with $A^*$ in place of $A$; for real $A$ this reads $A^T P + PA + Q = 0$.
If every eigenvalue of $A$ has negative real part, the continuous equation has the unique solution $X = \int_0^\infty e^{At} Q e^{A^* t} \, dt$, and $X$ is positive definite when $Q$ is.
If every eigenvalue of $A$ has absolute value less than $1$, the discrete equation has the unique solution $X = \sum_{k=0}^{\infty} A^k Q (A^*)^k$.
Example: for $n = 1$, $A = -1$ and $Q = 2$, the continuous equation $-2X + 2 = 0$ gives $X = 1$.
Without the eigenvalue condition a solution may fail to exist: for $n = 1$, $A = 0$ and $Q = 1$, the continuous equation reads $1 = 0$.
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# Functor category

## Definition

Let $\mathcal{C}$ be a small [[category|A2DE66937D]] and let $\mathcal{D}$ be a category.
The *functor category* $[\mathcal{C}, \mathcal{D}]$ is defined as follows:

1. its objects are the [[functors|28NQ0KRHD3]] $\mathcal{C} \to \mathcal{D}$;
2. its morphisms $F \to G$ are the [[natural transformations|KYCPCC57FT]] $\alpha \colon F \Rightarrow G$;
3. the composite of $\alpha \colon F \Rightarrow G$ and $\beta \colon G \Rightarrow H$ is $\beta \circ \alpha$ with components $(\beta \circ \alpha)_A = \beta_A \circ \alpha_A$, and the identity of $F$ has components $1_{F(A)}$.

## Remarks

The composite $\beta \circ \alpha$ is natural, because both of its naturality squares can be pasted together.
It is also written $\mathcal{D}^{\mathcal{C}}$ or $\mathrm{Fun}(\mathcal{C}, \mathcal{D})$.
Smallness of $\mathcal{C}$ ensures that, when $\mathcal{D}$ is locally small, the natural transformations between two functors form a set.
Example: let $G$ be a [[group|G5S1M9]], seen as a category with one object $\ast$ and with the elements of $G$ as morphisms.
A functor $G \to \mathbf{Set}$ is a set $X = F(\ast)$ with a [[group action|0WCY42EV9V]] $g \cdot x = F(g)(x)$, and a natural transformation is a map $h$ with $h(g \cdot x) = g \cdot h(x)$.
So $[G, \mathbf{Set}]$ is the category of $G$-sets.
Non-example: a family of morphisms $\alpha_A \colon F(A) \to G(A)$ that fails one naturality square is not a morphism of $[\mathcal{C}, \mathcal{D}]$.
A morphism of $[\mathcal{C}, \mathcal{D}]$ is an isomorphism exactly when all its components are isomorphisms.
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# Black–Scholes equation

## Summary

The Black–Scholes equation is the linear parabolic equation $V_t + \frac{1}{2}\sigma^2 S^2 V_{SS} + r S V_S - r V = 0$ for the price $V(S, t)$ of a financial option.
It assumes that the price $S$ of the underlying asset follows [[geometric Brownian motion|D48DR46B97]] with volatility $\sigma$.
It also assumes that money earns the constant interest rate $r$.
It is solved backward in time from the payoff at expiry.
A change of variables, starting with $x = \log S$ and reversing time, turns it into the [[heat equation|VP2HN86VAG]].

## Explanations

### L1 human

An option is a contract whose value at a future date $T$ depends on the price of a stock then.
The equation says how much the option is worth today.
The key idea is that a portfolio of the option and the stock can be kept free of risk, so it must grow at the rate $r$.
For a call option with strike $K$, the final value is $\max(S - K, 0)$, and the equation gives the Black–Scholes formula for earlier times.
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# Normed field

## Summary

A normed field is a [[field|04C5ZBG1WT]] with an absolute value, see the [[definition|65HSHAH2Y4]].
An absolute value is a size function that is zero only at $0$, is multiplicative and satisfies the triangle inequality.
The size gives a distance, so a normed field is a [[metric space|0VKD1TP7F3]] and a [[topological field|D7YQW9RG24]].
The real and complex numbers with the usual [[absolute value|0FFXKTKKQC]] and the rationals with a [[p-adic absolute value|WGN82TCEF2]] are the basic examples.
Normed fields are the scalars of [[normed vector spaces|03J5RAKB30]] in general functional analysis.

## Explanations

### L1 human

The absolute value of real numbers tells how big a number is, and $\abs{x - y}$ tells how far apart two numbers are.
A normed field is any field with such a size function that behaves well with products and sums.
The size of a product is the product of the sizes, and the size of a sum is at most the sum of the sizes.
For example, on $\Q$ one may call a number small when it has many factors of $3$: $\abs{9}_3 = 1/9$ and $\abs{1/3}_3 = 3$.
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# Quasi-isometry

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|0VKD1TP7F3]], and let $f \colon X \to Y$ be a map, not necessarily continuous.
The map $f$ is a *quasi-isometric embedding* if there are constants $A \ge 1$ and $B \ge 0$ such that
$$\frac{1}{A} d_X(x, x') - B \le d_Y(f(x), f(x')) \le A \, d_X(x, x') + B \quad \text{for all } x, x' \in X.$$
The map $f$ is a *quasi-isometry* if it is a quasi-isometric embedding and its image is $C$-dense for some $C \ge 0$.
This means that every point of $Y$ lies within distance $C$ of $f(X)$.
The spaces $X$ and $Y$ are *quasi-isometric* if there is a quasi-isometry $X \to Y$.

## Remarks

Example: the inclusion $f \colon \Z \to \R$, with the usual distances.
It satisfies the two inequalities with $A = 1$ and $B = 0$, and each real number lies within $C = 1/2$ of an integer.
So $f$ is a quasi-isometry.

Non-example: the map $f \colon \R \to \R$, $f(x) = x^3$.
It fails the upper inequality: $d(f(n), f(0)) = n^3$ is not bounded by $A n + B$ for large $n$.

A map $f \colon X \to Y$ is a quasi-isometry if and only if it is a quasi-isometric embedding and has a quasi-inverse.
This is a quasi-isometric embedding $g \colon Y \to X$ such that $g \circ f$ and $f \circ g$ are at bounded distance from the identity maps.
Being quasi-isometric is an equivalence relation on metric spaces.
Every bounded metric space is quasi-isometric to a point.
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# Priestley space

## Summary

A Priestley space is a compact topological space with a partial order in which any two points $x \not\le y$ are separated by a clopen upper set.
Priestley spaces are the duals of bounded [[distributive lattices|0HM0HQCK6N]] in [[Priestley duality|SA22SEC75K]].
Forgetting the order, every Priestley space is a [[Stone space|ZV38HA5PAQ]], and a Stone space with the equality order is a Priestley space.
So Priestley spaces extend Stone duality from Boolean algebras to distributive lattices.

## Explanations

### L1 human

A Priestley space carries two structures at once: a compact, very disconnected topology and an order.
One rule ties them together: whenever $x$ is not below $y$, some set that is open, closed and closed upward contains $x$ but not $y$.
A finite poset with the discrete topology is the simplest example.
The clopen upper sets of a Priestley space form a distributive lattice, and every bounded distributive lattice arises in this way.
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# Flexible polyhedron

## Example

A *flexible polyhedron* is a closed polyhedral surface in $\R^3$ that admits a continuous non-rigid motion in which every face stays congruent to itself.
Its edges act as hinges, and the shape changes while each face moves rigidly.
In the language of [[rigidity of frameworks|V5YS9BZX0N]], its framework of vertices and edges, with the faces triangulated, is flexible.

Bricard found flexible octahedra in 1897.
Their surfaces pass through themselves, so they are not boundaries of solids.
Connelly constructed in 1977 the first flexible polyhedron without self-intersections.
A simpler example is Steffen's polyhedron, with $9$ vertices, $21$ edges and $14$ triangular faces.
It satisfies Euler's formula $9 - 21 + 14 = 2$.

These examples are not convex, as they must be.
[[Cauchy's rigidity theorem|CJ7KCDDF31]] shows that no convex polyhedron is flexible.

## Explanations

### L1 human

Imagine a closed surface made of rigid triangles joined by hinges along their edges.
For a convex shape such as a cube cut into triangles, nothing can move.
Some cleverly built nonconvex shapes do wobble: they bend at the hinges while every triangle stays the same.
Such a shape is a flexible polyhedron.
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<document source="https://tamnd.github.io/math2/node/ZGH48BG36Z.md" tag="ZGH48BG36Z" kind="definition">
# Faber polynomials

## Definition

Let $K \subseteq \C$ be a [[compact set|00R0X1WGPX]] with more than one point such that $\widehat{\C} \setminus K$ is simply connected, where $\widehat{\C} = \C \cup \set{\infty}$ is the [[Riemann sphere|0CDMXGPSES]].
Let $\Phi$ be the [[conformal map|0VHNKP1N3K]] of $\widehat{\C} \setminus K$ onto $\set{w : \abs{w} > 1} \cup \set{\infty}$ with $\Phi(\infty) = \infty$ and
$$\Phi(z) = \frac{z}{c} + d_0 + \frac{d_1}{z} + \frac{d_2}{z^2} + \cdots \quad \text{near } \infty,$$
with $c > 0$.
For $n \ge 0$, the *$n$-th Faber polynomial* $F_n$ of $K$ is the polynomial part of the [[Laurent series|0X3XD2ATB0]] of $\Phi(z)^n$ at $\infty$.
That is, $F_n$ is the unique polynomial such that $\Phi(z)^n - F_n(z) \to 0$ as $z \to \infty$.

## Remarks

The map $\Phi$ exists and is unique by the [[Riemann mapping theorem|02WHMEG5ZR]], since $\widehat{\C} \setminus K$ is simply connected and has more than one boundary point.
The constant $c$ is the [[logarithmic capacity|DBQXRTC0VN]] of $K$.
The polynomial $F_n$ has degree $n$ and leading coefficient $c^{-n}$.
Example: for the disc $\abs{z - a} \le r$, $\Phi(z) = (z - a)/r$ and $F_n(z) = ((z - a)/r)^n$.
Example: for $K = [-1, 1]$, $\Phi(z) = z + \sqrt{z^2 - 1}$, and $\Phi^n + \Phi^{-n} = 2 T_n$ with $T_n$ the Chebyshev polynomial; since $\Phi^{-n} \to 0$ at $\infty$, $F_n = 2 T_n$ for $n \ge 1$.
Non-example: the union of two disjoint closed discs does not qualify, because its complement in $\widehat{\C}$ is not simply connected, so there is no map $\Phi$.
With $\Psi = \Phi^{-1}$, the Faber polynomials have the generating function $\Psi'(w)/(\Psi(w) - z) = \sum_{n \ge 0} F_n(z) w^{-n-1}$ for $\abs{w}$ large.
</document>

<document source="https://tamnd.github.io/math2/node/ZGH5RYJTQC.md" tag="ZGH5RYJTQC" kind="definition">
# Orlicz space

## Definition

A *Young function* is a [[convex function|07Q7CNM5WF]] $\Phi \colon [0, \infty) \to [0, \infty)$ with $\Phi(0) = 0$ and $\Phi(t) \to \infty$ as $t \to \infty$.
Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $\Phi$ be a Young function.
For a real or complex [[measurable function|YYNCJSR0PJ]] $f$ on $X$, the *Luxemburg norm* is
$$\norm{f}_\Phi = \inf \set{\lambda > 0 : \int_X \Phi\!\left( \frac{\abs{f}}{\lambda} \right) d\mu \le 1},$$
with $\inf \emptyset = \infty$.
The *Orlicz space* $L^\Phi(\mu)$ is the set of measurable $f$ with $\norm{f}_\Phi < \infty$, modulo equality [[almost everywhere|RARTMA0ZEC]].

## Remarks

Example: for $1 \le p < \infty$, $\Phi(t) = t^p$ is a Young function.
The condition $\int (\abs{f}/\lambda)^p \, d\mu \le 1$ means $\lambda \ge \norm{f}_p$.
So $L^\Phi = L^p$ with $\norm{f}_\Phi = \norm{f}_p$.

Non-example: $\Phi(t) = \sqrt{t}$ is not a Young function, because it is not convex.
On $[0, 1]$ with Lebesgue measure the Luxemburg functional is then $\left( \int \abs{f}^{1/2} \, d\mu \right)^2$.
It fails the triangle inequality: the indicator functions of $[0, \tfrac12]$ and $(\tfrac12, 1]$ each get the value $\tfrac14$, but their sum gets the value $1$.

A function $f$ lies in $L^\Phi$ exactly when $\int_X \Phi(\abs{f}/\lambda) \, d\mu < \infty$ for some $\lambda > 0$.
With the Luxemburg norm, $L^\Phi(\mu)$ is a Banach space.
Some authors allow Young functions with values in $[0, \infty]$; this includes $L^\infty$ as an Orlicz space.
</document>

<document source="https://tamnd.github.io/math2/node/ZGHMHJZQJA.md" tag="ZGHMHJZQJA" kind="definition">
# Parallelizable manifold

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.
A *global frame* on $M$ is a list $V_1, \dots, V_n$ of smooth [[vector fields|0299VCW1TH]] on $M$ with the following property.
For every $p \in M$, the vectors $V_1(p), \dots, V_n(p)$ form a basis of the tangent space $T_pM$.
The manifold $M$ is *parallelizable* if it has a global frame.

## Remarks

Equivalently, the [[tangent bundle|WY6K3P4YCG]] $TM$ is isomorphic, as a smooth vector bundle, to the trivial bundle $M \times \R^n$; a global frame gives the isomorphism $(p, a) \mapsto \sum_i a_i V_i(p)$.
Example: the circle $S^1 \subseteq \R^2$ is parallelizable.
The vector field $V(x, y) = (-y, x)$ is tangent to $S^1$ and nonzero at every point, so it is a global frame for the $1$-dimensional manifold $S^1$.
Every Lie group is parallelizable, by translating a basis of the tangent space at the identity.
Non-example: $S^2$ is not parallelizable.
A global frame would contain a nowhere-vanishing vector field, and by the hairy ball theorem no such field exists on $S^2$.
A parallelizable manifold is orientable, so the Möbius band is not parallelizable either.
</document>

<document source="https://tamnd.github.io/math2/node/ZGNBR2SPZ0.md" tag="ZGNBR2SPZ0" kind="concept">
# Markov chain Monte Carlo

## Summary

Markov chain Monte Carlo (MCMC) is a family of methods to sample from a probability distribution $\pi$.
They run a [[Markov chain|0G48PYM96Q]] whose [[stationary distribution|0KRH5157WP]] is $\pi$.
Averages of a function along the path of the chain then estimate the expectation of that function under $\pi$.
MCMC works even when $\pi$ is known only up to a normalising constant, which makes it a basic tool of Bayesian statistics and statistical physics.
The [[Metropolis–Hastings algorithm|MKHF449E22]] and the Gibbs sampler are the main examples.

## Explanations

### L1 human

Some distributions are too complicated to sample from directly, for example the law of all configurations of a large magnet.
MCMC instead designs a random walk on the configurations that, in the long run, visits each one with the right frequency.
One lets the walk run, records what it sees, and averages.
The price is that successive samples are dependent, so more of them are needed than with independent draws.
</document>

<document source="https://tamnd.github.io/math2/node/ZGYA090JP2.md" tag="ZGYA090JP2" kind="definition">
# Complementary error function

## Definition

Let $\operatorname{erf}$ be the [[error function|7Y56QCZSZJ]].
The *complementary error function* is the function $\operatorname{erfc} \colon \R \to \R$ given by
$$\operatorname{erfc} x = 1 - \operatorname{erf} x.$$

## Remarks

Since $\frac{2}{\sqrt{\pi}} \int_0^\infty e^{-t^2}\, dt = 1$, the definition is equivalent to
$$\operatorname{erfc} x = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2}\, dt.$$

Example: $\operatorname{erfc} 0 = 1$, and $\operatorname{erfc}(-x) = 2 - \operatorname{erfc} x$ because $\operatorname{erf}$ is odd.

Non-example: $x \mapsto 1 + \operatorname{erf} x$ is not $\operatorname{erfc}$; it tends to $2$ rather than $0$ as $x \to \infty$.

For large $x > 0$, $\operatorname{erfc} x \sim \frac{e^{-x^2}}{x\sqrt{\pi}}$.
The definition extends to complex $z$ by $\operatorname{erfc} z = 1 - \operatorname{erf} z$, which is an entire function.
The scaled function $e^{x^2} \operatorname{erfc} x$ is often used in numerical work.
</document>

<document source="https://tamnd.github.io/math2/node/ZH21D5B62C.md" tag="ZH21D5B62C" kind="concept">
# Deligne–Lusztig theory

## Summary

Deligne–Lusztig theory constructs representations of finite [[groups of Lie type|NPEBABQ49M]], such as $GL_n(\F_q)$, from the $\ell$-adic cohomology of certain algebraic varieties.
For each Frobenius-stable [[maximal torus|3JVGZVVTB0]] $T$ and each character $\theta$ of its finite group of points it gives a virtual character $R_T^\theta$.
Every irreducible character of the finite group occurs in some $R_T^\theta$.
The theory, introduced by Deligne and Lusztig in 1976, is the basis of the classification of the irreducible characters of finite reductive groups by Lusztig.

## Explanations

### L1 human

Finite groups like the invertible matrices over a finite field have many representations, and most of them cannot be built by elementary methods such as induction from subgroups.
Deligne and Lusztig found them inside the geometry of special varieties on which the group acts.
The cohomology of these varieties, a tool from algebraic topology adapted to algebraic geometry, carries representations of the finite group.
Taking alternating sums of these gives virtual characters that together contain all irreducible characters.
</document>

<document source="https://tamnd.github.io/math2/node/ZH29VRRRJ6.md" tag="ZH29VRRRJ6" kind="definition">
# Paracompact space

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
An open cover $\mathcal{V}$ of $X$ is a *refinement* of an open cover $\mathcal{U}$ if every $V \in \mathcal{V}$ is contained in some $U \in \mathcal{U}$.
The space $X$ is *paracompact* if every [[open cover|049PXYHW1T]] of $X$ has an open refinement that is a [[locally finite family|DZ297MMGV8]].

## Remarks

Example: $\R$ is paracompact.
Given an open cover $\mathcal{U}$, each compact interval $[n, n + 1]$ is covered by finitely many members of $\mathcal{U}$.
Intersect these members with $(n - 1, n + 2)$; the resulting sets, over all $n \in \Z$, form a locally finite open refinement.

Non-example: the [[long line|1DKCNS63DH]] is not paracompact, although it is Hausdorff and locally homeomorphic to $\R$.

Every [[compact space|0TVJZYB6SH]] is paracompact, since a finite subcover is a locally finite refinement.
Some authors include the Hausdorff condition in the definition of a paracompact space.
</document>

<document source="https://tamnd.github.io/math2/node/ZH2S50NSZ6.md" tag="ZH2S50NSZ6" kind="concept">
# Oscillation theory

## Summary

Oscillation theory studies the zeros of solutions of differential equations, above all of second-order linear equations $(p y')' + q y = 0$.
Its main questions are whether solutions have infinitely many zeros, how the zeros are spaced, and how the zeros of different solutions are related.
The theory started with the comparison and separation theorems of Charles Sturm.
It gives criteria in terms of the coefficients, and it is linked to [[Sturm–Liouville theory|7WV6B2502K]], where the $n$-th eigenfunction has $n - 1$ zeros.

## Explanations

### L1 human

A solution oscillates if it crosses the axis again and again, without end, like a sine wave.
Oscillation theory asks which equations force their solutions to behave like this.
The equation $y'' + y = 0$ is oscillatory, since its solutions $A \sin(x + c)$ have infinitely many zeros.
The equation $y'' - y = 0$ is not: its solutions $A e^x + B e^{-x}$ have at most one zero.
For $y'' + q(x) y = 0$, a large positive $q$ means many oscillations, a phenomenon the Sturm comparison theorem makes precise.
</document>

<document source="https://tamnd.github.io/math2/node/ZH3K0BH68K.md" tag="ZH3K0BH68K" kind="conjecture">
# Graceful tree conjecture

## Statement

Let $G$ be a finite simple [[graph|00C0BR7D7H]] with $m$ edges.
Let $V$ be the vertex set of $G$.
A *graceful labeling* of $G$ is an injective map $f \colon V \to \set{0, 1, \ldots, m}$ with the following property.
The edge labels $\abs{f(u) - f(v)}$, for the edges $uv$ of $G$, are pairwise distinct.
Then the edge labels are exactly $1, 2, \ldots, m$.
The conjecture: every finite [[tree|062D9KXJB1]] has a graceful labeling.

## Remarks

The conjecture is open.
It is also called the Ringel–Kotzig conjecture.
It is known for many families of trees, for example paths, stars and caterpillars, and computer searches have checked it for all trees with few vertices.
A path $v_0 v_1 \cdots v_m$ has the graceful labeling $0, m, 1, m - 1, 2, \ldots$, with edge labels $m, m - 1, \ldots, 1$.
Rosa showed that a graceful labeling of a tree $T$ with $m$ edges gives a decomposition of the complete graph $K_{2m+1}$ into $2m + 1$ copies of $T$.
So the conjecture would imply Ringel's conjecture on such decompositions.
Ringel's conjecture was proved for all large $m$ by Montgomery, Pokrovskiy and Sudakov, by other methods.
Not every graph is graceful: for example the cycle $C_n$ has a graceful labeling only when $n \equiv 0$ or $3 \pmod 4$.

## Explanations

### L1 human

Write distinct numbers from $0$ to $m$ on the vertices of a tree with $m$ edges.
Label each edge by the difference of the numbers at its ends.
The labeling is graceful when the edges get the labels $1, 2, \ldots, m$, each exactly once.
The conjecture says that every tree can be labeled in this way.
It is easy to check for small trees, but no general proof is known.
</document>

<document source="https://tamnd.github.io/math2/node/ZH4V0WHSXT.md" tag="ZH4V0WHSXT" kind="definition">
# Green's function

## Definition

Let $a < b$ and let $p_0, \ldots, p_{n-1}$ be [[continuous functions|0X3WGX95FE]] on $[a, b]$, with $n \ge 1$.
Let $L y = y^{(n)} + p_{n-1} y^{(n-1)} + \cdots + p_0 y$ be the [[linear differential operator|0QQRP17Z7S]] on $[a, b]$.
Let $B_1, \ldots, B_n$ be linear forms of the shape $B_i y = \sum_{j=0}^{n-1} \left( \alpha_{ij}\, y^{(j)}(a) + \beta_{ij}\, y^{(j)}(b) \right)$ with real constants $\alpha_{ij}, \beta_{ij}$.
Assume that the [[boundary value problem|TM8VQJWGHZ]] $L y = 0$, $B_1 y = \cdots = B_n y = 0$ has only the solution $y = 0$.
A *Green's function* of this problem is a function $G \colon [a, b] \times [a, b] \to \R$ with the following property.
For every continuous $f$ on $[a, b]$, the function
$$y(x) = \int_a^b G(x, s)\, f(s)\, ds$$
is a solution of $L y = f$, $B_1 y = \cdots = B_n y = 0$.

## Remarks

Under the assumption, the Green's function exists, and the solution above is the only one.
For fixed $s$, the function $x \mapsto G(x, s)$ solves $L y = 0$ on $[a, s)$ and on $(s, b]$ and satisfies the boundary conditions.
Its derivatives of order up to $n - 2$ are continuous at $x = s$, and its derivative of order $n - 1$ jumps by $1$ there.
Formally, $L_x G(x, s) = \delta(x - s)$.

Example: for $L y = y''$ on $[0, 1]$ with $y(0) = y(1) = 0$, take $G(x, s) = x (s - 1)$ for $x \le s$ and $G(x, s) = s (x - 1)$ for $x \ge s$.
It vanishes at $x = 0$ and $x = 1$, it is linear in $x$ on each side of $s$, and $\partial_x G$ jumps from $s - 1$ to $s$, a jump of $1$.
For $f = 1$ it gives $y(x) = \frac{x (x - 1)}{2}$, and indeed $y'' = 1$ and $y(0) = y(1) = 0$.

Non-example: for $L y = y''$ on $[0, 1]$ with $y'(0) = y'(1) = 0$, the constants solve the homogeneous problem, so the assumption fails.
There is no Green's function: $y'' = 1$ has no solution with these conditions, since $y'(1) - y'(0) = \int_0^1 y'' = 1$.
</document>

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# Smooth morphism of rings

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $A$ be a commutative [[$R$-algebra|DPBJ04JMH8]].
Then $A$ is *smooth* over $R$, and the structure map $R \to A$ is a *smooth morphism*, if both conditions hold:

1. $A$ is a [[finitely presented|764GZHTGV8]] $R$-algebra.
2. $A$ is [[formally smooth|8KPDNVKX7B]] over $R$.

## Remarks

If $A$ is smooth over $R$, then $A$ is [[flat|73RYYEVDKA]] over $R$, and the module of [[Kähler differentials|7HCFARHG7D]] $\Omega_{A/R}$ is a finitely generated projective $A$-module.

Example: the [[polynomial ring|0SDC934R52]] $A = R[x_1, \ldots, x_n]$ is smooth over $R$.
It is finitely presented, with no relations, and it is formally smooth, because a lift is obtained by lifting the images of the variables.

Non-example: let $k$ be a field and $A = k[x_1, x_2, x_3, \ldots]$ in infinitely many variables.
It is formally smooth over $k$ by the same argument, but it is not finitely generated, so clause 1 fails.

Another non-example: $A = k[x, y]/(y^2 - x^3)$ is finitely presented over $k$ but not smooth.
Here $\Omega_{A/k}$ is generated by $dx, dy$ with the one relation $2y \, dy - 3x^2 \, dx = 0$.
At the maximal ideal $\mathfrak m = (x, y)$, this relation vanishes modulo $\mathfrak m$, so $\Omega_{A/k} \otimes_A A/\mathfrak m$ has dimension $2$.
A smooth algebra over a field has differentials that are locally free of rank equal to the local [[Krull dimension|MVGBS3J1P7]], which is $1$ here.
</document>

<document source="https://tamnd.github.io/math2/node/ZHHMDZWA7F.md" tag="ZHHMDZWA7F" kind="concept">
# Relative homotopy group

## Summary

The relative homotopy group $\pi_n(X, A, x_0)$ of a pair $A \subseteq X$ consists of homotopy classes of maps of an $n$-disk into $X$ with boundary in $A$.
It is the homotopy analogue of relative homology.
The relative groups fit into a long exact sequence with the [[homotopy groups|P976WYEFMB]] of $A$ and of $X$.
This sequence is the main tool for computing homotopy groups of fibrations.

## Explanations

### L1 human

A relative homotopy group counts disks in a big space whose rim lies in a smaller subspace.
Two such disks count as the same if one can be deformed into the other while the rim stays in the subspace.
Take the disk $D^2$ with its boundary circle as subspace.
The identity disk cannot be pushed off, and the group is $\Z$, counting how many times the rim winds around the circle.
</document>

<document source="https://tamnd.github.io/math2/node/ZHKS81A7QN.md" tag="ZHKS81A7QN" kind="concept">
# Pesin theory

## Summary

Pesin theory is the study of smooth dynamical systems that are hyperbolic only almost everywhere with respect to an invariant measure.
Its starting point is the [[Oseledets theorem|NJ1PDJ3S4R]], which gives [[Lyapunov exponents|NJ1FJSNK6A]] at almost every point.
When no exponent is zero, the theory builds local stable and unstable manifolds at almost every point.
Their size and the angles between them may vary from point to point.
It links the exponents with entropy, periodic orbits and the existence of physical measures.

## Explanations

### L1 human

In a uniformly hyperbolic system, such as [[Arnold's cat map|BSCZN8Y1AN]], every point sees the same rate of stretching and squeezing.
Most smooth systems are not that regular: the stretching is there on average, but it can be weak and slow to show up near some points.
Pesin theory says that average stretching is enough to recover much of the uniform picture, point by point, for almost every point.
For example, take a smooth map of a surface that preserves area and has one positive and one negative exponent at almost every point.
Then almost every point has a short curve through it that the map squeezes and another one that the map stretches.
The price is that the length of these curves and the angle between them may get small near bad points.
</document>

<document source="https://tamnd.github.io/math2/node/ZHRQ0AT8PM.md" tag="ZHRQ0AT8PM" kind="concept">
# Join-irreducible element

## Summary

An element of a [[lattice|WH4SFAACAS]] is join-irreducible if it is not the bottom element and not the [[join|J2T75ATRV5]] of two smaller elements.
In a finite lattice, every element is a join of join-irreducible elements, so these elements generate the lattice by joins.
For finite [[distributive lattices|0HM0HQCK6N]], the poset of join-irreducible elements determines the whole lattice, which is the content of Birkhoff's representation theorem.
Meet-irreducible elements are defined dually.

## Explanations

### L1 human

A join-irreducible element is one that cannot be split into two smaller pieces by a join.
In the subsets of a set, the join-irreducible elements are the one-element sets: a set with two or more points is the union of two smaller sets.
In the positive integers ordered by divisibility, where join is the least common multiple, they are the prime powers $p^k$ with $k \ge 1$.
For instance $12 = \lcm(4, 3)$ is not join-irreducible, but $8$ is, since every proper divisor of $8$ divides $4$.
</document>

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# Scalar curvature

## Summary

The scalar curvature of a [[Riemannian manifold|7CBT4SM83Y]] is the function obtained by taking the trace of the [[Ricci curvature|4GTNM0WHDH]] with respect to the metric.
It is the simplest curvature invariant: a single number at each point.
It measures how the volume of a small geodesic ball differs from the volume of a Euclidean ball of the same radius.
Questions such as which manifolds admit metrics of positive scalar curvature are a major theme of geometry and topology.

## Explanations

### L1 human

Scalar curvature is the total amount of curving at a point, averaged over all directions.
If it is positive, small balls around the point have slightly less volume than flat balls of the same radius; if negative, slightly more.
For a surface it is twice the Gaussian curvature.
The unit sphere of dimension $n$ has scalar curvature $n(n - 1)$, so the ordinary unit sphere in $\R^3$ has scalar curvature $2$.
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<document source="https://tamnd.github.io/math2/node/ZJ0GBPREMB.md" tag="ZJ0GBPREMB" kind="concept">
# Domain of finite type

## Summary

A domain of finite type is a domain in $\C^n$ with smooth boundary whose boundary has only finite order of contact with complex curves.
The type, introduced by D'Angelo, measures how flat the boundary can be in complex directions.
[[Strongly pseudoconvex domains|Z7RVXBA2KB]] have type $2$, while a boundary that contains a complex disc has infinite type.
For pseudoconvex domains, finite type is the condition under which the $\bar\partial$-Neumann problem gains regularity, by work of Kohn and Catlin.

## Explanations

### L1 human

Take a small holomorphic disc that touches the boundary of the domain at a point.
Measure how closely the disc hugs the boundary there, as an order of tangency.
The type of the boundary point is the largest such order, over all discs.
On the unit sphere a complex line touches only to order $2$, as a tangent line touches a circle.
On the boundary of $\set{\abs{z}^2 + \abs{w}^4 < 1}$ the disc $t \mapsto (1, t)$ touches the boundary point $(1, 0)$ to order $4$.
</document>

<document source="https://tamnd.github.io/math2/node/ZJD91FNTCY.md" tag="ZJD91FNTCY" kind="definition">
# Weighted polynomial approximation

## Definition

Let $w \colon \R \to [0, \infty)$ be a continuous function, called the *weight*, such that $x^k w(x) \to 0$ as $\abs{x} \to \infty$ for every integer $k \ge 0$.
Let $C_w$ be the space of [[continuous functions|0X3WGX95FE]] $f \colon \R \to \R$ with $f(x) w(x) \to 0$ as $\abs{x} \to \infty$, with the seminorm $\norm{f}_w = \sup_{x \in \R} \abs{f(x)} w(x)$.
Let $\mathcal{P}_n$ be the set of real [[polynomial functions|1G0B73WPX8]] of degree at most $n$.
*Weighted polynomial approximation* is the approximation of $f \in C_w$ by elements of $\mathcal{P}_n$, measured by the weighted error
$$E_n(f)_w = \inf\set{\norm{f - p}_w : p \in \mathcal{P}_n} .$$

## Remarks

The decay condition on $w$ ensures that every polynomial lies in $C_w$, so $E_n(f)_w$ is finite.
$E_n(f)_w$ is the [[error of best approximation|YDN0T4MEX0]] of $f$ from $\mathcal{P}_n$ in the seminorm $\norm{\cdot}_w$.
Polynomials are dense in $C_w$ exactly when $E_n(f)_w \to 0$ for every $f \in C_w$; Bernstein's approximation problem asks for which weights this holds.
The same name is used for the weighted $L^p$ error $\inf_p \left( \int_\R \abs{f - p}^p w \, dx \right)^{1/p}$, and for weights on an interval.
Example: $w(x) = e^{-x^2}$ satisfies the decay condition, since $x^k e^{-x^2} \to 0$ for each $k$.
The function $f(x) = \abs{x}$ is continuous with $\abs{x} e^{-x^2} \to 0$, so $f \in C_w$, and $E_n(f)_w \le \norm{f}_w = \sup_x \abs{x} e^{-x^2} = e^{-1/2}/\sqrt{2}$ for every $n$, using $p = 0$.
In the weighted $L^2$ version with the same weight, the best constant approximation of $x^2$ is $c = \int x^2 e^{-x^2} dx \big/ \int e^{-x^2} dx = 1/2$.
Non-example: $w \equiv 1$ fails the decay condition, and then $\norm{p}_w = \infty$ for every nonconstant polynomial $p$.
</document>

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# C0-semigroup

## Definition

Let $X$ be a [[Banach space|06GZ956657]] and let $\mathcal{L}(X)$ be the space of bounded linear operators on $X$.
A *$C_0$-semigroup* (strongly continuous semigroup) on $X$ is a map $T \colon [0, \infty) \to \mathcal{L}(X)$ such that:

1. $T(0) = I$, the identity operator;
2. $T(t + s) = T(t) T(s)$ for all $t, s \ge 0$;
3. $\lim_{t \to 0^+} \norm{T(t) x - x} = 0$ for every $x \in X$.

Its *generator* is the operator $A$ given by $A x = \lim_{t \to 0^+} \frac{T(t) x - x}{t}$, with domain $D(A)$ the set of $x \in X$ for which this limit exists.

## Remarks

Conditions 1 and 2 say that $T$ is a monoid homomorphism from $([0, \infty), +)$ into the multiplicative [[semigroup|0XMEM406JM]] $\mathcal{L}(X)$.

Example: for a bounded operator $A$, $T(t) = e^{tA}$, the [[exponential|020M04FN1H]] series, is a $C_0$-semigroup with generator $A$; here even $\norm{T(t) - I} \to 0$ in [[operator norm|06EKW1AKXF]].

Example: on $X = L^p(\R)$ with $1 \le p < \infty$, the left shifts $(T(t) f)(s) = f(s + t)$ form a $C_0$-semigroup.
Condition 3 holds because translation is continuous in $L^p$ norm for $p < \infty$, which follows by approximation with continuous functions of compact support.
The generator is $A f = f'$ on the Sobolev space $W^{1,p}(\R)$.

Non-example: the same shifts on $L^\infty(\R)$ fail condition 3.
For $f$ the indicator function of $[0, 1]$ and $0 < t < 1$, $T(t) f - f$ equals $1$ in absolute value on $[-t, 0)$, so $\norm{T(t) f - f}_\infty = 1$.

Every $C_0$-semigroup satisfies $\norm{T(t)} \le M e^{\omega t}$ for some constants $M \ge 1$ and $\omega \in \R$, and its generator is closed and densely defined.

## Explanations

### L1 human

A $C_0$-semigroup is a time-evolution made of linear maps that compose correctly and start continuously from the identity.
Its generator is the derivative at time $0$, which plays the role of $A$ in $e^{tA}$.
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# Banach–Mazur distance

## Summary

The Banach–Mazur distance measures how far two [[normed vector spaces|03J5RAKB30]] of the same finite dimension are from being isometric.
It is the smallest possible distortion $\norm{T} \norm{T^{-1}}$ of a [[linear isomorphism|0B8T10CE7N]] $T$ between them.
In geometric terms, it compares the unit balls up to linear maps.
With it, the isometry classes of $n$-dimensional normed spaces form the [[Banach–Mazur compactum|RH1WTN38RQ]].
Many questions of local Banach space theory and asymptotic convex geometry are estimates of this distance.

## Explanations

### L1 human

A norm on $\R^n$ is described by its unit ball, a symmetric convex body.
Two norms are close in the Banach–Mazur sense if a linear change of coordinates makes the unit balls nearly equal.
Then one unit ball fits inside the other, and the other fits inside a slightly enlarged copy of the first.
The distance is $1$ exactly when the two spaces are isometric.
For instance, the square ball of the maximum norm on $\R^2$ and the round disc of the Euclidean norm are at distance $\sqrt{2}$.
The disc fits in the square, and the square fits in the disc enlarged by $\sqrt{2}$.
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<document source="https://tamnd.github.io/math2/node/ZJSW0HP75H.md" tag="ZJSW0HP75H" kind="statement">
# Cone theorem

## Statement

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $X$ be a smooth [[projective variety|A4H4CZ2V35]] over $k$ of dimension $n$.
Let $K_X$ be its canonical divisor (see [[canonical bundle|W05VYY4HBR]]).
Let $N_1(X)$ be the real vector space of $1$-cycles on $X$ modulo numerical equivalence.
Let $\overline{NE}(X) \subseteq N_1(X)$ be the closure of the cone spanned by the classes of curves.
Write $\overline{NE}(X)_{K_X \ge 0} = \set{z \in \overline{NE}(X) : K_X \cdot z \ge 0}$, using the [[intersection pairing|4HV4WTAX9P]].
Then there are countably many rational curves $C_i \subset X$ with
$$0 < -K_X \cdot C_i \le n + 1$$
such that
$$\overline{NE}(X) = \overline{NE}(X)_{K_X \ge 0} + \sum_i \R_{\ge 0} [C_i].$$
Moreover, for every [[ample|RRK8P6TXG2]] divisor $H$ and every $\varepsilon > 0$, only finitely many of the $C_i$ are needed to write
$$\overline{NE}(X) = \overline{NE}(X)_{K_X + \varepsilon H \ge 0} + \sum_{i \text{ finite}} \R_{\ge 0} [C_i].$$

## Remarks

The theorem is due to Mori.
The rational curves come from his bend-and-break method: a curve on which $K_X$ is negative deforms a lot and breaks off rational curves.
The method works first in characteristic $p$ and then passes to characteristic $0$.
Example: on $\mathbb{P}^n$ the space $N_1$ is spanned by the class of a line $\ell$, and $-K \cdot \ell = n + 1$.
So the bound $n + 1$ is sharp, and $\overline{NE}(\mathbb{P}^n) = \R_{\ge 0} [\ell]$.
In characteristic $0$, the theorem extends to log terminal pairs, together with the contraction theorem which contracts each $K_X$-negative extremal ray.

## Explanations

### L1 human

The classes of curves on a variety span a cone in a finite-dimensional real vector space.
The theorem describes the part of this cone where the canonical class is negative.
That part is spanned by the classes of rational curves of bounded degree, one ray at a time, like the edges of a polyhedral cone.

### L2 human

The cone theorem is the first step of the minimal model program: it provides the extremal rays to contract.
With the contraction theorem, each $K_X$-negative extremal ray gives a morphism that contracts exactly the curves in that ray.
For surfaces, these rays are spanned by $(-1)$-curves, by fibres of a ruling, or by a line in $\mathbb{P}^2$.
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# Adams' theorem on Hopf invariant one

## Statement

Let $n \ge 1$.
If there is a continuous map $f \colon S^{4n-1} \to S^{2n}$ whose [[Hopf invariant|TEGTSY7XMX]] $H(f)$ equals $1$, then $2n \in \set{2, 4, 8}$.

## Remarks

Conversely, maps of Hopf invariant one exist for $2n = 2, 4, 8$: they are the [[Hopf fibrations|M5AKNYC4QQ]] $S^3 \to S^2$, $S^7 \to S^4$ and $S^{15} \to S^8$, built from the complex numbers, the quaternions and the octonions.

Adams proved the theorem in 1960 with secondary cohomology operations.
A map of Hopf invariant one gives a two-cell complex $X = S^{2n} \cup_f D^{4n}$ with $Sq^{2n}$ nonzero from $H^{2n}(X; \F_2)$ to $H^{4n}(X; \F_2)$.
For $2n$ not a power of $2$, the Adem relations write $Sq^{2n}$ through lower squares, which vanish on $H^{2n}(X; \F_2)$ because they land in zero groups, so this is impossible.
For $2n = 2^k$ with $k \ge 4$, Adams decomposed $Sq^{2^k}$ through secondary operations.
A shorter proof by Adams and Atiyah uses the Adams operations in complex K-theory.

The theorem has several consequences.
For $m \ge 1$, the sphere $S^{m}$ admits the structure of an H-space only for $m \in \set{1, 3, 7}$.
The space $\R^m$ has the structure of a real division algebra, with bilinear multiplication and no zero divisors, only for $m \in \set{1, 2, 4, 8}$.
For $m \ge 1$, the sphere $S^{m}$ is parallelizable only for $m \in \set{1, 3, 7}$.

## Explanations

### L1 human

The Hopf invariant is an integer attached to a map from a sphere of dimension $4n - 1$ to a sphere of dimension $2n$.
Adams showed that it can equal one only in the three dimensions where the complex numbers, the quaternions and the octonions give an example.
This explains why there are no other number systems like these in which one can divide.

### L2 human

The theorem settles the Hopf invariant one problem.
In the Adams spectral sequence for the sphere at the prime $2$, it says that the classes $h_j$ in $\operatorname{Ext}^{1, 2^j}$ survive only for $j \le 3$.
For $j \ge 4$ there is a nonzero differential $d_2(h_j) = h_0 h_{j-1}^2$.
Through the stable homotopy groups of spheres, it is linked to the image of the J-homomorphism and to the vector fields on spheres problem.
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<document source="https://tamnd.github.io/math2/node/ZK1MHTQGS9.md" tag="ZK1MHTQGS9" kind="concept">
# Generalized Boolean algebra

## Summary

A generalized Boolean algebra is a [[distributive lattice|0HM0HQCK6N]] with a least element $0$ in which any two elements $a$ and $b$ have a difference $a \setminus b$.
This difference is the complement of $a \wedge b$ in the interval $[0, a]$.
It is a [[Boolean algebra|0XADJKM699]] that may lack a greatest element.
Each interval $[0, a]$ in it is a Boolean algebra.
Generalized Boolean algebras correspond to Boolean rings that need not have a unit, in the same way as Boolean algebras correspond to [[Boolean rings|0RCV1ZFP8G]].

## Explanations

### L1 human

Think of the finite subsets of the natural numbers.
You can take unions, intersections and differences of two finite sets and stay finite.
You cannot take the complement of a finite set, because it is infinite.
So the finite sets have set difference but no global complement.
A generalized Boolean algebra is exactly this kind of structure: Boolean algebra with differences, but with no top element required.
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<document source="https://tamnd.github.io/math2/node/ZK55WHXEQX.md" tag="ZK55WHXEQX" kind="example">
# Laplace transform of a periodic function

## Example

Let $P > 0$ and let $f \colon [0, \infty) \to \C$ be [[periodic|089KWVYD69]] with period $P$, that is $f(t + P) = f(t)$ for $t \ge 0$, and integrable on $[0, P]$.
Its [[Laplace transform|05XT5AVJBK]] is
$$F(s) = \frac{1}{1 - e^{-Ps}} \int_0^P f(t) e^{-st}\, dt \quad (\operatorname{Re} s > 0).$$
Check: for $\operatorname{Re} s > 0$ the integral converges absolutely, since $\int_0^\infty \abs{f} e^{-\sigma t}\, dt = \sum_{k \ge 0} e^{-k P \sigma} \int_0^P \abs{f(t)} e^{-\sigma t}\, dt < \infty$ for $\sigma = \operatorname{Re} s$.
Split $[0, \infty)$ into the intervals $[kP, (k + 1)P]$ and substitute $t = u + kP$:
$$F(s) = \sum_{k \ge 0} e^{-kPs} \int_0^P f(u) e^{-su}\, du,$$
a geometric series with ratio $e^{-Ps}$, of modulus less than $1$.
Instance: $f(t) = \abs{\sin t}$ has period $\pi$, and $\int_0^\pi \sin t\, e^{-st}\, dt = \frac{1 + e^{-\pi s}}{s^2 + 1}$.
So
$$\mathcal{L}\{\abs{\sin t}\}(s) = \frac{1 + e^{-\pi s}}{(s^2 + 1)(1 - e^{-\pi s})} = \frac{\coth(\pi s / 2)}{s^2 + 1}.$$

## Explanations

### L1 human

A periodic function repeats the same piece forever.
Each copy of the piece is the first copy delayed by a multiple of the period, and a delay by $kP$ multiplies the transform by $e^{-kPs}$.
Adding up all copies gives a geometric series, so only the transform of one period is needed.
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<document source="https://tamnd.github.io/math2/node/ZKG8D1S566.md" tag="ZKG8D1S566" kind="definition">
# Levi pseudoconvex domain

## Definition

Let $\Omega \subseteq \C^n$ be a domain with $C^2$ boundary, and let $\rho$ be a defining function of $\Omega$.
For $p \in \partial\Omega$, let $H_p$ be the complex tangent space of $\partial\Omega$ at $p$.
The domain $\Omega$ is *Levi pseudoconvex* if the [[Levi form|80MTNXWW80]] of $\partial\Omega$ is positive semidefinite at every boundary point:
$$\sum_{j,k=1}^{n} \frac{\partial^2 \rho}{\partial z_j \partial \bar z_k}(p)\, w_j \bar w_k \ge 0 \quad \text{for all } p \in \partial\Omega \text{ and all } w \in H_p.$$

## Remarks

The condition does not depend on the choice of $\rho$, because another defining function multiplies the Levi form by a positive number.
Example: every convex domain with $C^2$ boundary is Levi pseudoconvex.
Indeed, for a convex domain the real Hessian of $\rho$ at $p$ is positive semidefinite on the real tangent space of $\partial\Omega$ at $p$.
For $w \in H_p$, both $w$ and $i w$ lie in this tangent space, and $L_\rho(p; w)$ is one quarter of the sum of the real Hessian at $w$ and at $i w$.
Non-example: the domain $\set{z \in \C^2 : 1 < \abs{z_1}^2 + \abs{z_2}^2 < 4}$ fails at the inner sphere.
There $\rho(z) = 1 - \abs{z}^2$ is a defining function, and $L_\rho(p; w) = -\abs{w}^2 < 0$ for $w \in H_p$, $w \ne 0$.
When $n = 1$ the space $H_p$ is $\set{0}$, so every planar domain with $C^2$ boundary is Levi pseudoconvex.
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<document source="https://tamnd.github.io/math2/node/ZKJF80DTNB.md" tag="ZKJF80DTNB" kind="statement">
# Holomorphy of the gamma function

## Statement

Let $U = \C \setminus \set{0, -1, -2, \ldots}$.
There is exactly one [[holomorphic function|0NAHXR1FRH]] $\Gamma \colon U \to \C$ that agrees with the [[gamma function|0V3FZMVB4K]] on $(0, \infty)$.

## Remarks

On the half-plane $\operatorname{Re} z > 0$, Euler's integral $\int_0^\infty t^{z-1} e^{-t}\,dt$ is holomorphic, see [[Euler integral for the gamma function|Z8D1CZFTWP]].
On $\operatorname{Re} z > -n$, with $z \notin \set{0, \ldots, -n+1}$, the formula $\Gamma(z + n)/\big(z (z + 1) \cdots (z + n - 1)\big)$ is holomorphic and agrees with the previous one on the overlap; these pieces glue to a holomorphic function on $U$.
Uniqueness follows from the identity theorem, because $U$ is open and connected and $(0, \infty) \subset U$ has accumulation points.
The function cannot be extended holomorphically to any point of $\set{0, -1, -2, \ldots}$, since it has [[poles|QFPFF15PXC]] there.
In Mathlib the statement is `Complex.differentiableAt_Gamma`.

## Explanations

### L1 human

The gamma function, first defined for positive real numbers, extends in a smooth complex way to all complex numbers except $0, -1, -2, \ldots$.
This [[analytic continuation|0M3YQJ81NB]] is unique: there is only one way to do it.

### L2 human

Holomorphy is what lets one prove identities for $\Gamma$ on the real line and then transfer them to complex arguments, by the identity theorem.
The reflection and duplication formulas are usually proved in this way.
Together with the description of the poles, it says that $\Gamma$ is a meromorphic function on $\C$.
</document>

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# Enflo's counterexample to the approximation problem

## Counterexample

There is a separable [[Banach space|06GZ956657]] $X$ that does not have the [[approximation property|8CKE0V6ANK]].
That is, there are a compact set $K \subseteq X$ and $\varepsilon > 0$ with the following property.
Every continuous linear map $T \colon X \to X$ of finite rank satisfies $\norm{Tx - x} \ge \varepsilon$ for some $x \in K$.
Per Enflo constructed such a space; his paper appeared in Acta Mathematica in 1973.

## Remarks

The construction is an explicit but intricate combinatorial one, built from finite-dimensional blocks; it is not short enough to reproduce here.
It has two famous consequences.
First, $X$ has no Schauder basis, because a space with a Schauder basis has the approximation property.
So the basis problem, whether every separable Banach space has a Schauder basis, has a negative answer.
Second, by Grothendieck's work on the approximation property, some compact operator into $X$ is not a norm limit of finite-rank operators.
Later work showed that the spaces $\ell^p$ with $p \ne 2$ contain closed subspaces without the approximation property.

## Explanations

### L1 human

For a long time it was hoped that every reasonable Banach space could be described by coordinates, like the sequence spaces.
It was also hoped that every compact operator was a limit of matrices.
Enflo's space shows that both hopes are false.
It is a separable Banach space where the identity cannot be imitated on compact sets by finite-rank maps.
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<document source="https://tamnd.github.io/math2/node/ZKSPYN2CRJ.md" tag="ZKSPYN2CRJ" kind="definition">
# Sheaf of differentials

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].
Let $\Delta \colon X \to X \times_Y X$ be the diagonal morphism into the [[fiber product|1DZMT4S9FB]].
The morphism $\Delta$ is an immersion, so there is an open subscheme $W \subseteq X \times_Y X$ such that $\Delta$ is a closed immersion of $X$ into $W$.
Let $\mathcal{I} \subseteq \mathcal{O}_W$ be the ideal sheaf of $\Delta(X)$ in $W$.
The *sheaf of relative differentials* of $X$ over $Y$ is
$$\Omega_{X/Y} = \Delta^*(\mathcal{I}/\mathcal{I}^2),$$
a [[quasi-coherent|KPKND0K7W6]] $\mathcal{O}_X$-module.

## Remarks

If $U = \operatorname{Spec} B \subseteq X$ maps into $V = \operatorname{Spec} A \subseteq Y$, then $\Omega_{X/Y}|_U$ is the quasi-coherent sheaf associated to the module of [[Kähler differentials|7HCFARHG7D]] $\Omega_{B/A}$.
There is a universal derivation $d \colon \mathcal{O}_X \to \Omega_{X/Y}$, and $\Omega_{X/Y}$ is generated by the sections $dg$.
If $X$ is a variety over a field $k$, one writes $\Omega_X = \Omega_{X/k}$, the *cotangent sheaf*.

Example: for $X = \mathbb{A}^n_k = \operatorname{Spec} k[x_1, \dots, x_n]$, $\Omega_{X/k}$ is free of rank $n$ with basis $dx_1, \dots, dx_n$.
This matches $\Omega_{B/k}$ for the polynomial ring $B$.

Non-example of local freeness: for the cuspidal curve $X = \operatorname{Spec} k[x, y]/(y^2 - x^3)$ with $\operatorname{char} k \ne 2, 3$, $\Omega_{X/k}$ is generated by $dx, dy$ with the relation $2y \, dy = 3x^2 \, dx$.
Its fiber at the origin has dimension $2$, while the generic rank is $1$, so $\Omega_{X/k}$ is not locally free there.
An irreducible variety $X$ over a perfect field $k$ is smooth over $k$ if and only if $\Omega_{X/k}$ is locally free of rank $\dim X$.
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# Cascade algorithm

## Definition

Let $(h_n)_{n \in \Z}$ be a finite sequence of real numbers with $\sum_n h_n = \sqrt{2}$.
Let $\varphi_0 \in L^2(\R)$ be a starting function, usually the indicator function of $[0, 1)$.
The *cascade algorithm* with filter $(h_n)$ is the iteration
$$\varphi_{i+1}(x) = \sqrt{2} \sum_{n} h_n \, \varphi_i(2x - n), \qquad i = 0, 1, 2, \dots$$
Suppose the sequence $(\varphi_i)$ converges in $L^2(\R)$ to a limit $\varphi$.
Then $\varphi$ satisfies the refinement equation $\varphi(x) = \sqrt{2}\sum_n h_n \varphi(2x - n)$.
So $\varphi$ is the [[scaling function|YH4BERB952]] of the filter.

## Remarks

The condition $\sum_n h_n = \sqrt{2}$ keeps the integral fixed: $\int \varphi_{i+1} = \sqrt{2}\sum_n h_n \cdot \frac{1}{2}\int\varphi_i = \int \varphi_i$.
The values of $\varphi_i$ at the points $k 2^{-i}$ can be computed by repeated upsampling by $2$ and convolution with $\sqrt{2}\,h$.
This is the inverse fast wavelet transform applied to a single coefficient.

Example: for the Haar filter $h_0 = h_1 = 1/\sqrt{2}$ and $\varphi_0 = \mathbf{1}_{[0,1)}$, one step gives $\varphi_1(x) = \mathbf{1}_{[0,1)}(2x) + \mathbf{1}_{[0,1)}(2x - 1) = \mathbf{1}_{[0,1)}(x)$.
So the iteration is constant and converges at once to the Haar scaling function.

Non-example: for the filter $h_0 = h_1 = 1$, the sum is $2 \ne \sqrt{2}$.
From $\varphi_0 = \mathbf{1}_{[0,1)}$ one gets $\varphi_i = 2^{i/2}\mathbf{1}_{[0,1)}$, which does not converge.

For the filters of the Daubechies wavelets, the iterates converge to the scaling function in $L^2(\R)$.
For some filters that satisfy the sum condition, the iteration still fails to converge, so convergence depends on further properties of the filter.

## Explanations

### L1 human

The cascade algorithm turns the self-similarity rule of a scaling function into a repeated computation.
Each round replaces the current shape by a weighted sum of half-size, shifted copies of it.
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# Continuous kernel

## Summary

A continuous kernel is a [[kernel|GSMBXT7FPK]] $K(x, s)$ that is a continuous function of both variables on a closed bounded square $[a, b] \times [a, b]$.
It is the setting of the classical theory of Fredholm and Volterra.
The integral operator with a continuous kernel maps continuous functions to continuous functions and is a [[compact operator|M2NEP1K5AE]] on the space of continuous functions.
Iteration methods, the Liouville–Neumann series and the Fredholm determinant are first developed for such kernels.

## Explanations

### L1 human

A continuous kernel has no jumps and no infinite values.
Since the square is closed and bounded, the kernel is bounded by some number $M$, which makes many estimates easy.
For example, $K(x, s) = e^{x s}$ and $K(x, s) = \min(x, s)$ are continuous kernels on $[0, 1] \times [0, 1]$.
The kernel $1/\abs{x - s}$ is not, because it is infinite on the diagonal.
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<document source="https://tamnd.github.io/math2/node/ZM5VYK2BFB.md" tag="ZM5VYK2BFB" kind="statement">
# Mellin transform under inversion of the variable

## Statement

Let $f \colon (0, \infty) \to \C$ be measurable and let $g(x) = f(1/x)$ for $x > 0$.
Let $s \in \C$.
Then $g$ is [[Mellin convergent|SGZZ4Y0D36]] at $s$ if and only if $f$ is Mellin convergent at $-s$.
In that case
$$\mathcal{M}g(s) = \mathcal{M}f(-s),$$
where $\mathcal{M}$ is the [[Mellin transform|N5PVGRJE1T]].

## Remarks

The proof is the substitution $u = 1/x$, with $dx = -u^{-2} \, du$:
$$\int_0^\infty x^{s-1} f(1/x) \, dx = \int_0^\infty u^{1-s} f(u) \, u^{-2} \, du = \int_0^\infty u^{-s-1} f(u) \, du.$$
The same computation with $\abs{f}$ gives the equivalence of convergence.
So the strip of convergence is reflected in the imaginary axis: if $f$ converges for $\alpha < \operatorname{Re} s < \beta$, then $g$ converges for $-\beta < \operatorname{Re} s < -\alpha$.
Example: $f(x) = e^{-x}$ gives $\int_0^\infty x^{s-1} e^{-1/x} \, dx = \Gamma(-s)$ for $\operatorname{Re} s < 0$.

## Explanations

### L1 human

Replacing $x$ by $1/x$ swaps the roles of $0$ and $\infty$.
On the Mellin side, this simply replaces $s$ by $-s$.

### L2 human

This rule is used with functional equations of the form $\theta(1/x) = \sqrt{x}\, \theta(x)$, where the behaviour at $0$ is read from the behaviour at $\infty$.
It also explains why the Mellin transform of a function and of its reflection have mirror strips of convergence.
Together with the rules for dilations and powers, it shows how the Mellin transform reacts to the maps $x \mapsto a x^{b}$ with $a > 0$ and $b \ne 0$.
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<document source="https://tamnd.github.io/math2/node/ZM6W89YNS7.md" tag="ZM6W89YNS7" kind="statement">
# Frenet–Serret formulas

## Statement

Let $I \subseteq \R$ be an interval and let $\gamma \colon I \to \R^3$ be a curve of class $C^3$ parametrised by [[arc length|00R1JDDV5X]] $s$.
Assume that its [[curvature|7JQS6VKFEZ]] $\kappa(s)$ is positive for all $s \in I$.
Let $T = \gamma'$, $N = T' / \kappa$ and $B = T \times N$, and let $\tau$ be the [[torsion|S4BZ697DSH]] of $\gamma$.
Then $(T(s), N(s), B(s))$ is a positively oriented [[orthonormal basis|0FWJ0VW561]] of $\R^3$ for each $s$, and
$$T' = \kappa N, \qquad N' = -\kappa T + \tau B, \qquad B' = -\tau N.$$

## Remarks

The first formula is the definition of $N$, and the third is the definition of $\tau$.
For the second, differentiate the identities $\ip{N}{N} = 1$, $\ip{N}{T} = 0$ and $\ip{N}{B} = 0$.
They give $\ip{N'}{N} = 0$, $\ip{N'}{T} = -\ip{N}{T'} = -\kappa$ and $\ip{N'}{B} = -\ip{N}{B'} = \tau$.
In matrix form the formulas say that $(T, N, B)' = (T, N, B) A$ with a skew-symmetric matrix $A$ built from $\kappa$ and $\tau$.
For the helix $(a \cos t, a \sin t, bt)$ with $a > 0$, both $\kappa = a / (a^2 + b^2)$ and $\tau = b / (a^2 + b^2)$ are constant.
The hypothesis $\kappa > 0$ is needed, since the normal $N$ is not defined where $\kappa = 0$.
For example, two plane arcs in different planes, smoothly joined through a straight segment, form a curve with $\tau = 0$ wherever $\kappa > 0$ that lies in no plane.

## Explanations

### L1 human

At each point of a space curve there are three natural directions: forward, the way the curve bends, and the direction perpendicular to both.
The Frenet–Serret formulas tell how these three arrows turn as one moves along the curve.
Only two numbers control this turning: the curvature and the torsion.

### L2 human

The formulas form a linear system of ordinary differential equations for the frame $(T, N, B)$ with coefficients $\kappa$ and $\tau$.
This is why curvature and torsion determine the curve up to a rigid motion, by the [[fundamental theorem of space curves|TNYQ5R0NVM]].
They give quick proofs of basic facts.
A curve with $\kappa > 0$ and $\tau = 0$ lies in a plane.
A curve with constant $\kappa > 0$ and $\tau = 0$ is an arc of a circle.
In $\R^n$ there is an analogous frame with $n - 1$ generalised curvatures, under a nondegeneracy hypothesis on the derivatives.
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# Hilbert–Schmidt integral operator

## Definition

Let $(X, \mu)$ be a $\sigma$-finite [[measure space|P15VQS5DJ9]], and let $L^2(X)$ be the [[$L^2$ space|SASP14MA1Z]] of $\mu$.
Let $K \colon X \times X \to \C$ be a measurable function with
$$\int_X \int_X \abs{K(x, y)}^2 \, d\mu(x) \, d\mu(y) < \infty.$$
The *Hilbert–Schmidt integral operator* with kernel $K$ is the operator $T_K \colon L^2(X) \to L^2(X)$ given by
$$(T_K f)(x) = \int_X K(x, y) \, f(y) \, d\mu(y).$$

## Remarks

The operator is well defined: for almost every $x$, $K(x, \cdot) \in L^2(X)$, and the Cauchy–Schwarz inequality gives $\norm{T_K f} \le \norm{K}_{L^2(X \times X)} \norm{f}$.
Its Hilbert–Schmidt norm $\left(\sum_n \norm{T_K e_n}^2\right)^{1/2}$, for any orthonormal basis $(e_n)$, equals $\norm{K}_{L^2(X \times X)}$.

Example: on $X = [0, 1]$ with Lebesgue measure, $K(x, y) = \min(x, y)$ satisfies $\int_0^1 \int_0^1 \min(x, y)^2 \, dx \, dy = 1/6 < \infty$, so $T_K$ is a Hilbert–Schmidt integral operator.

Non-example: on $[0, 1]$, $K(x, y) = \abs{x - y}^{-1/2}$ fails the condition, since $\int_0^1 \int_0^1 \abs{x - y}^{-1} \, dx \, dy = \infty$.
Its operator is still bounded and compact on $L^2[0, 1]$, but it is not a Hilbert–Schmidt operator.

Kernels with the condition above are the [[square-integrable kernels|WWMWX7AD4V]].
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<document source="https://tamnd.github.io/math2/node/ZMCFEEPEKT.md" tag="ZMCFEEPEKT" kind="concept">
# Functional differential equation

## Summary

A functional differential equation relates the derivative of an unknown function at time $t$ to its values at other times.
Often these times fill a whole past interval.
It includes [[ordinary differential equations|0NT64JV96A]], delay equations and equations with distributed memory.
The state at time $t$ is the piece of the solution on an interval $[t - r, t]$, so the phase space is a space of functions.
The theory, developed by Krasovskii, Hale and others, extends existence, uniqueness and stability theory to this setting.

## Explanations

### L1 human

In an ordinary differential equation, the future depends only on the present state.
In a functional differential equation, it may depend on the recent past as well.
For instance, $x'(t) = -\int_{t-1}^{t} x(s) \, ds$ lets the rate of change depend on the average of the last time unit.
To start such an equation, one needs the whole history over that time unit.
</document>

<document source="https://tamnd.github.io/math2/node/ZMDAEW6DHN.md" tag="ZMDAEW6DHN" kind="definition">
# Chevalley group

## Definition

Let $\mathfrak{g}$ be a simple Lie algebra over $\C$ with Cartan subalgebra $\mathfrak{h}$ and root system $\Phi$.
Let $\set{h_1, \dots, h_\ell} \cup \set{e_\alpha : \alpha \in \Phi}$ be a Chevalley basis of $\mathfrak{g}$, and let $\mathfrak{g}_\Z$ be its $\Z$-span.
Let $K$ be a field and let $\mathfrak{g}_K = \mathfrak{g}_\Z \otimes_\Z K$.
For $\alpha \in \Phi$ and $t \in K$, let $x_\alpha(t) = \sum_{k \ge 0} t^k (\operatorname{ad} e_\alpha)^k / k!$, an automorphism of $\mathfrak{g}_K$.
Here $(\operatorname{ad} e_\alpha)^k / k!$ is the reduction of an integer matrix on $\mathfrak{g}_\Z$ and is $0$ for $k \ge 4$.
The (adjoint) *Chevalley group* of type $\Phi$ over $K$ is the subgroup of the [[automorphism group|0K1SCPBNKJ]] of $\mathfrak{g}_K$ generated by all these $x_\alpha(t)$.

## Remarks

It is written $\Phi(K)$, for example $A_n(K)$, $E_8(K)$ or $G_2(q)$ when $K = \F_q$.
The sum defining $x_\alpha(t)$ is finite because $\operatorname{ad} e_\alpha$ is nilpotent, and the Chevalley basis makes its divided powers integral, so the formula makes sense in every characteristic.
Example: for $\Phi$ of type $A_1$, the group $A_1(K)$ is isomorphic to $\mathrm{PSL}_2(K)$.
Non-example: let $K$ be a field of characteristic not $2$.
Then $\mathrm{SL}_2(K)$ is not the adjoint Chevalley group of type $A_1$.
Its centre $\set{\pm I}$ has order $2$, while $A_1(K) \cong \mathrm{PSL}_2(K)$ has trivial centre.
Chevalley proved that $\Phi(K)$ is a [[simple group|0YMYMNQ0QZ]] except for $A_1(\F_2)$, $A_1(\F_3)$, $B_2(\F_2)$ and $G_2(\F_2)$.

## Explanations

### L1 human

A Chevalley group is the group generated by exponentials of root vectors of a simple Lie algebra, rebuilt over an arbitrary field.
</document>

<document source="https://tamnd.github.io/math2/node/ZMEYSAHNX3.md" tag="ZMEYSAHNX3" kind="statement">
# Banach–Zarecki theorem

## Statement

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function.
Then $f$ is [[absolutely continuous|ZT6F4C2YHK]] on $[a, b]$ if and only if the following three conditions hold:

1. $f$ is [[continuous|0P27W9RW0G]] on $[a, b]$;
2. $f$ has [[bounded variation|A8251BHCMP]] on $[a, b]$;
3. $f$ has the [[Luzin N property|WMHMGJD261]].

## Remarks

An absolutely continuous function is continuous and of bounded variation, and covering a null set by small open intervals shows that it has the N property.
For the converse, one shows that if $f$ is differentiable at every point of a measurable set $E$, then the outer measure of $f(E)$ is at most $\int_E \abs{f'}$.
The N property handles the null set where $f'$ fails to exist, and this controls the changes of $f$ over families of short intervals.
No one condition can be dropped.
The [[Cantor function|NPBYYK63FE]] is continuous and monotone but fails the N property.
The function $x \sin(1/x)$ on $[0, 1]$ is continuous with the N property, being smooth away from $0$, but it does not have bounded variation.
A step function is monotone and maps every set to a finite set, but it is not continuous.

## Explanations

### L1 human

The theorem splits absolute continuity into three simpler conditions.
The function must not jump, must not wiggle infinitely much, and must not spread a set of length zero over a set of positive length.

### L2 human

The theorem explains exactly what goes wrong with the [[Cantor function|NPBYYK63FE]] and other [[singular functions|P5AWRD2793]]: they are continuous and monotone, so only the N property can fail.
It is often used to prove absolute continuity of compositions and inverses.
For example, a continuous strictly increasing function is absolutely continuous exactly when it maps null sets to null sets.
</document>

<document source="https://tamnd.github.io/math2/node/ZMPYZ8KD9N.md" tag="ZMPYZ8KD9N" kind="object">
# Griess algebra

## Description

The *Griess algebra* is a real [[vector space|000AYPA9AJ]] $B$ of [[dimension|03GSZ8RTZJ]] $196884$ with a bilinear product $B \times B \to B$.
The product is commutative but not associative.
The algebra also carries a positive definite symmetric bilinear form $(\cdot, \cdot)$ that is associative: $(ab, c) = (a, bc)$ for all $a, b, c \in B$.
Its automorphism group is the [[Monster group|C9VY20TAPT]], the largest [[sporadic group|PZ49Z4CBFZ]].

## Remarks

Robert Griess built this algebra to construct the Monster group, which acts on it by algebra automorphisms.
As a representation of the Monster, $B$ splits as a trivial summand of dimension $1$ and an irreducible summand of dimension $196883$.
The number $196884$ is also the coefficient of $q$ in the expansion of the $j$-function, a coincidence that is part of [[monstrous moonshine|E0Y7Y75T6C]].
In the moonshine module of Frenkel, Lepowsky and Meurman, a vertex operator algebra, the Griess algebra appears as the weight $2$ subspace.

## Explanations

### L1 human

The Griess algebra is a huge space of dimension $196884$ in which two vectors can be multiplied.
It was built to give the Monster group a concrete home: the Monster is exactly the set of symmetries of this algebra.
The multiplication commutes, $ab = ba$, but the order of brackets matters, $(ab)c \ne a(bc)$ in general.
</document>

<document source="https://tamnd.github.io/math2/node/ZMRFSYVWQT.md" tag="ZMRFSYVWQT" kind="statement">
# Bieberbach conjecture

## Statement

Let $f$ be a [[univalent function|R91E83SDBD]] on the unit disc $\mathbb{D}$ with $f(0) = 0$ and $f'(0) = 1$, and write
$$f(z) = z + \sum_{n=2}^{\infty} a_n z^n, \qquad \abs{z} < 1.$$
Then $\abs{a_n} \le n$ for every $n \ge 2$.
If $\abs{a_n} = n$ for some $n \ge 2$, then $f(z) = e^{-i\theta} k(e^{i\theta} z)$ for some $\theta \in \R$, where $k(z) = z/(1-z)^2$ is the [[Koebe function|EZQFWKF03Q]].

## Remarks

Bieberbach proved the case $n = 2$ and stated the general bound as a conjecture.
Loewner proved the case $n = 3$ with his [[differential equation|51G4QJZ345]].
Louis de Branges proved the full statement, which is therefore also called de Branges's theorem.
The conjecture was known to follow from the Robertson conjecture on odd univalent functions, which in turn follows from the Milin conjecture on the logarithmic coefficients of $f$.
De Branges proved the Milin conjecture, using the Loewner equation and an inequality on sums of Jacobi polynomials that had been proved by Askey and Gasper.
The bound is sharp for every $n$ at once, because the Koebe function has $a_n = n$.

## Explanations

### L1 human

Normalise a univalent function on the unit disc so that it starts like $z$.
The theorem says that its $n$-th Taylor coefficient has absolute value at most $n$.
The Koebe function, which maps the disc onto the plane minus a slit, is the extreme case with coefficients exactly $n$.

### L2 human

The case $n = 2$, Bieberbach's inequality $\abs{a_2} \le 2$, is a short consequence of the [[Grönwall area theorem|BYZDAZNHP4]] and implies the [[Koebe quarter theorem|WX0HMW4GT7]] and the distortion theorems.
For larger $n$ the problem resisted for a long time, and partial results such as $\abs{a_n} < e\,n$ and the proofs for small $n$ produced many methods of univalent function theory.
The final proof goes through the stronger Milin inequality and the [[Loewner equation|51G4QJZ345]].
</document>

<document source="https://tamnd.github.io/math2/node/ZMT8JF21R6.md" tag="ZMT8JF21R6" kind="example">
# Runge's phenomenon

## Example

Let $f(x) = \dfrac{1}{1 + 25 x^2}$ on $[-1, 1]$.
For $n \ge 1$ let $p_n$ be the [[interpolation polynomial|SS1YMS2NPC]] of degree at most $n$ of $f$ at the equally spaced nodes $x_i = -1 + 2i/n$, $i = 0, \dots, n$.
Then
$$\max_{x \in [-1, 1]} \abs{f(x) - p_n(x)} \to \infty \quad \text{as } n \to \infty.$$
The error stays small in the middle of the interval but grows without bound near the end points, where $p_n$ oscillates more and more.

The function $f$ is smooth, in fact analytic on all of $\R$, so the failure is not due to a lack of differentiability.
The [[interpolation error formula|HASA2XSG66]] explains it.
The derivatives $f^{(n+1)}$ grow roughly like $(n + 1)! \, 5^{n+1}$ on $[-1, 1]$, because $f$ has complex poles at $\pm i/5$, close to the interval.
This growth outweighs the factor $\frac{1}{(n+1)!} \max \abs{\prod_i (x - x_i)}$ for equally spaced nodes, which is largest near the end points.

With [[Chebyshev nodes|HYK8J56PSH]] instead of equally spaced nodes, the interpolation polynomials of the same $f$ converge [[uniformly|08VZ57PT26]] to $f$ on $[-1, 1]$.

## Explanations

### L1 human

One might think that fitting a polynomial through more and more points of a nice curve always gives a better and better fit.
Runge's example shows that this is false.
For a simple bell-shaped curve and equally spaced points, the fitted polynomial develops wild wiggles near the ends of the interval.
The wiggles get bigger as more points are used.
The cure is to put more points near the ends, as Chebyshev nodes do, or to use piecewise polynomials such as [[splines|84TJ8KQ926]].
</document>

<document source="https://tamnd.github.io/math2/node/ZMY9H2XENE.md" tag="ZMY9H2XENE" kind="concept">
# Nest algebra

## Summary

A nest is a totally ordered and complete family of closed subspaces of a Hilbert space.
The nest algebra of a nest is the algebra of all bounded operators that leave every subspace of the nest invariant.
Nest algebras are the infinite-dimensional analogues of the algebras of upper triangular matrices.
They are the main examples of non-self-adjoint operator algebras, and every nest algebra is a [[reflexive operator algebra|672HNBV803]].

## Explanations

### L1 human

An upper triangular $n \times n$ matrix sends the span of the first $k$ basis vectors into itself, for each $k$.
These spans form a chain of subspaces, each inside the next.
A nest algebra keeps this picture but allows any complete chain of closed subspaces of a Hilbert space, possibly infinite and with no basis behind it.
For example, on $L^2[0, 1]$ the subspaces of functions vanishing on $[0, t]$, for $0 \le t \le 1$, form a nest, and the Volterra integration operator leaves each of them invariant.
</document>

<document source="https://tamnd.github.io/math2/node/ZMZZ2QRFTZ.md" tag="ZMZZ2QRFTZ" kind="concept">
# Zonoid

## Summary

A zonoid is a [[convex body|437FSG3419]] that is a limit, in the [[Hausdorff distance|F79WYYC07C]], of [[zonotopes|Z5SA6A67KF]], that is, of finite Minkowski sums of line segments.
Zonoids are exactly the convex bodies whose support functions are integrals of the functions $x \mapsto \abs{x \cdot v}$, up to a translation.
They appear in the geometry of finite-dimensional normed spaces and in the study of projections of convex bodies.
By Lyapunov's convexity theorem, they also arise as the ranges of nonatomic vector-valued measures.

## Explanations

### L1 human

Adding line segments in the sense of Minkowski gives polytopes with many parallel edges, such as parallelograms, hexagons with parallel opposite sides, or cubes.
A zonoid is what one gets by adding more and more, ever shorter segments, and passing to the limit.
The round disc is a zonoid: it is the limit of regular polygons with $2m$ sides, each of which is a sum of $m$ segments.
Every zonoid is centrally symmetric, so a triangle is never a zonoid.
</document>

<document source="https://tamnd.github.io/math2/node/ZN30Y1ZDB6.md" tag="ZN30Y1ZDB6" kind="concept">
# Discrete Laplace operator

## Summary

The discrete Laplace operator is the analogue of the [[Laplacian|0E0B6RA8K0]] for functions on a grid or on the vertices of a graph.
At each point it adds up the differences between the values at the neighbours and the value at the point.
On the grid $h\Z^d$ it gives the standard finite difference approximation of $\sum_i \partial^2 u / \partial x_i^2$.
On a finite graph it is, up to sign, the [[Laplacian matrix|TN6M6DVZRP]], and it links random walks, electrical networks and discrete harmonic functions.

## Explanations

### L1 human

Ask at each point how much its value differs from the values around it.
If the point sits in a dip, its neighbours are higher, and the discrete Laplacian there is positive.
If it sits on a bump, the discrete Laplacian is negative.
On the integers, $u(m) = m^2$ gives $(m+1)^2 - 2m^2 + (m-1)^2 = 2$ at every point, just as the second derivative of $x^2$ is $2$.
</document>

<document source="https://tamnd.github.io/math2/node/ZN4NMX1DB4.md" tag="ZN4NMX1DB4" kind="statement">
# Ramanujan's master theorem

## Statement

Let $0 < \delta < 1$, and let $C > 0$, $P \in \R$ and $0 \le A < \pi$.
Let $\varphi$ be a [[holomorphic|0NAHXR1FRH]] function on an open set containing the closed half-plane $H = \set{v + iw : v \ge -\delta}$, such that
$$\abs{\varphi(v + iw)} \le C e^{P v + A \abs{w}} \qquad \text{for all } v + iw \in H.$$
Then the series $f(x) = \sum_{k=0}^{\infty} \varphi(k) (-x)^k$ converges for $0 < x < e^{-P}$, and $f$ has an [[analytic continuation|0M3YQJ81NB]] along $(0, \infty)$.
For every $s \in \C$ with $0 < \operatorname{Re} s < \delta$, the [[Mellin transform|N5PVGRJE1T]] of this continuation converges and
$$\int_0^\infty x^{s-1} f(x) \, dx = \frac{\pi}{\sin(\pi s)} \, \varphi(-s).$$

## Remarks

This is the form of the theorem proved by G. H. Hardy.
The idea is to write $f$ as a Mellin–Barnes integral: by the residue theorem, $f(x) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \frac{\pi}{\sin(\pi s)} \varphi(-s) x^{-s} \, ds$ for $0 < c < \delta$, since $\pi / \sin(\pi s)$ has residue $(-1)^k$ at $s = -k$.
The growth bound with $A < \pi$ beats the decay $e^{-\pi \abs{w}}$ of $1/\sin(\pi s)$ and makes the integral converge; Mellin inversion then gives the formula.
Writing $\varphi(k) = \lambda(k) / \Gamma(k + 1)$ and using $\Gamma(s)\Gamma(1 - s) = \pi / \sin(\pi s)$ gives the classical form: if $f(x) = \sum_{k \ge 0} \lambda(k) (-x)^k / k!$, then $\int_0^\infty x^{s-1} f(x) \, dx = \Gamma(s) \lambda(-s)$, under the corresponding conditions on $\lambda$.
Example: $\varphi = 1$ gives $f(x) = 1/(1 + x)$ and $\int_0^\infty x^{s-1}/(1 + x) \, dx = \pi / \sin(\pi s)$ for $0 < \operatorname{Re} s < 1$.
Example: $\lambda = 1$ gives $f(x) = e^{-x}$ and the formula $\int_0^\infty x^{s-1} e^{-x} \, dx = \Gamma(s)$.
The growth condition is needed: $\varphi(z) = \sin(\pi z)$ vanishes at all integers, so $f = 0$, but $\varphi(-s) \ne 0$ for non-integer $s$; here $A = \pi$ and the bound with $A < \pi$ fails.

## Explanations

### L1 human

The theorem computes the Mellin transform of a function from the coefficients of its power series.
If the $k$-th coefficient is $\varphi(k)$ times $(-1)^k$, then the transform is essentially $\varphi$ evaluated at $-s$.
So a formula valid for the whole numbers $k = 0, 1, 2, \dots$ is pushed to negative and complex values.
Ramanujan used it, without a full proof, to evaluate a great many definite integrals.

### L2 human

The growth conditions make the passage from integers to all complex numbers legitimate.
By Carlson's theorem, a function of this type is determined by its values at the non-negative integers.
The proof is an instance of the general Mellin method.
A Mellin–Barnes integral, shifted to the left, gives the power series at $0$, and its inverse transform is the right side.
The theorem yields at once the Mellin transforms of the binomial series $(1 + x)^{-a}$, of Bessel functions and of many hypergeometric functions.
</document>

<document source="https://tamnd.github.io/math2/node/ZN7HJNR2AC.md" tag="ZN7HJNR2AC" kind="concept">
# Doob's h-transform

## Summary

Doob's h-transform changes the transition probabilities of a Markov process by weighting them with a positive excessive function $h$.
The new process, the h-process, behaves like the old one conditioned on an event described by $h$.
Typical events are leaving a domain at a given boundary point or never being killed.
It links positive harmonic functions, the Martin boundary, and conditioned processes.

## Explanations

### L1 human

Take a random motion and a positive function $h$ that never increases on average along the motion.
Tilt the motion so that moves towards places where $h$ is large become more likely, in proportion to $h$.
The tilted motion is the h-transform.
For example, Brownian motion on the half-line $x > 0$, tilted by $h(x) = x$, becomes Brownian motion conditioned never to hit $0$.
</document>

<document source="https://tamnd.github.io/math2/node/ZNC09JXRC5.md" tag="ZNC09JXRC5" kind="definition">
# Kontorovich–Lebedev transform

## Definition

For $y \in \R$ and $x > 0$ let $K_{iy}(x) = \int_0^\infty e^{-x\cosh t} \cos(yt)\, dt$ be the [[modified Bessel function of the second kind|2ET068BZZH]] of order $iy$.
Let $f \colon (0, \infty) \to \C$ be measurable with $\int_0^\infty \abs{f(x)}\, K_0(x)\, dx < \infty$.
The *Kontorovich–Lebedev transform* of $f$ is the function $g \colon \R \to \C$ given by the [[Lebesgue integral|DWJF7ABPR1]]
$$g(y) = \int_0^\infty f(x)\, K_{iy}(x)\, dx.$$

## Remarks

The integral converges absolutely because $\abs{K_{iy}(x)} \le \int_0^\infty e^{-x\cosh t}\, dt = K_0(x)$.
The function $K_{iy}(x)$ is real and even in $y$, so $g$ is even and is usually considered on $(0, \infty)$.
Under suitable conditions on $f$, the inversion formula is
$$f(x) = \frac{2}{\pi^2 x} \int_0^\infty g(y)\, K_{iy}(x)\, \sinh(\pi y)\, y\, dy.$$

Example: $f(x) = 1$.
Since $K_0(x)$ behaves like $-\log x$ near $0$ and like $\sqrt{\pi/(2x)}\, e^{-x}$ at infinity, $K_0$ is integrable and the condition holds.
The formula $\int_0^\infty K_\mu(x)\, dx = \frac{\pi}{2\cos(\pi\mu/2)}$ for $\abs{\operatorname{Re} \mu} < 1$, with $\mu = iy$, gives $g(y) = \frac{\pi}{2\cosh(\pi y / 2)}$.

Non-example: $f(x) = e^x$.
For fixed $y$, $K_{iy}(x) \sim \sqrt{\pi/(2x)}\, e^{-x}$ as $x \to \infty$, so $f(x) K_{iy}(x) \sim \sqrt{\pi/(2x)}$ and the integral diverges at infinity.
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<document source="https://tamnd.github.io/math2/node/ZNCF16AENG.md" tag="ZNCF16AENG" kind="statement">
# Grauert's direct image theorem

## Statement

Let $X$ and $Y$ be [[complex analytic spaces|HXSZA44J5J]], let $f \colon X \to Y$ be a [[proper|15XEBGNGHC]] holomorphic map and let $\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.
For an integer $q \ge 0$, let $R^q f_* \mathcal{F}$ be the sheaf on $Y$ associated with the presheaf $V \mapsto H^q(f^{-1}(V), \mathcal{F})$ of [[sheaf cohomology|PQ6Q143ARJ]] groups.
Then $R^q f_* \mathcal{F}$ is a coherent analytic sheaf on $Y$ for every $q \ge 0$.

## Remarks

When $Y$ is a single point, $R^q f_* \mathcal{F}$ is the vector space $H^q(X, \mathcal{F})$ and $X$ is compact, so the theorem contains the [[Cartan–Serre finiteness theorem|MPNGD4V271]].
The case where $f$ is finite and $q = 0$ is easier and is proved with the [[Weierstrass division theorem|FE83QQS17S]].
The general case is much harder.
Grauert's original proof was long; a later proof by Kiehl and Verdier uses Čech cochains of Stein coverings and compact operators between Fréchet modules.

Example: if $A \subseteq X$ is an analytic subset with its reduced structure, then $f_* \mathcal{O}_A$ is coherent on $Y$, and its support is $f(A)$.
Since supports of coherent sheaves are analytic, this gives [[Remmert's proper mapping theorem|VA2JEWWB61]].

Properness is necessary.
Consider the inclusion $f \colon \C \setminus \set{0} \to \C$.
The stalk of $f_* \mathcal{O}$ at $0$ is the ring of holomorphic functions on punctured discs around $0$, modulo agreement near $0$.
It is not finitely generated over $\mathcal{O}_{\C,0}$.
Indeed it is nonzero and multiplication by $z$ is bijective on it, so Nakayama's lemma rules out finite generation.

## Explanations

### L1 human

Given a proper holomorphic map, one can push a coherent sheaf forward along it and take the cohomology along the fibres.
The theorem says that the result is again a coherent sheaf on the target.
So finiteness survives when one looks at a whole family of compact spaces at once, not only at one compact space.

### L2 human

The theorem is the basic finiteness result of relative complex analytic geometry.
It is used to study families of compact complex spaces, for example to prove semicontinuity of the dimensions of cohomology groups of fibres.
It is the analytic analogue of the coherence of higher direct images of coherent sheaves under proper morphisms of schemes.
</document>

<document source="https://tamnd.github.io/math2/node/ZNDEPVNBBG.md" tag="ZNDEPVNBBG" kind="example">
# Hilbert transform of sinusoids

## Example

Let $\omega > 0$, and define the [[Hilbert transform|NSCEBXM7ZM]] of a bounded continuous function $f$ by the [[principal value|0SYR10HD4R]] at $0$ and at $\infty$
$$Hf(x) = \frac{1}{\pi} \lim_{\varepsilon \to 0^+,\, R \to \infty} \int_{\varepsilon < \abs{u} < R} \frac{f(x - u)}{u}\, du.$$
Then
$$H[\cos(\omega \cdot)](x) = \sin(\omega x), \qquad H[\sin(\omega \cdot)](x) = -\cos(\omega x).$$
Check for the cosine: $\cos(\omega(x - u)) = \cos(\omega x) \cos(\omega u) + \sin(\omega x) \sin(\omega u)$.
On the symmetric set $\varepsilon < \abs{u} < R$, the odd function $\cos(\omega u)/u$ integrates to $0$.
The even function $\sin(\omega u)/u$ integrates to $2 \int_{\varepsilon}^{R} \frac{\sin(\omega u)}{u}\, du$, which tends to $2 \cdot \frac{\pi}{2} = \pi$.
So $H[\cos(\omega \cdot)](x) = \frac{1}{\pi} \cdot \pi \sin(\omega x) = \sin(\omega x)$.
The same computation with $\sin(\omega(x - u)) = \sin(\omega x)\cos(\omega u) - \cos(\omega x)\sin(\omega u)$ gives $-\cos(\omega x)$.
For $\omega < 0$ the signs change, since $\int_0^\infty \frac{\sin(\omega u)}{u}\, du = -\frac{\pi}{2}$.

## Remarks

The result agrees with the Fourier multiplier $-i \sgn(\omega)$ of the Hilbert transform: $H[e^{i\omega x}] = -i \sgn(\omega)\, e^{i\omega x}$.
Applying $H$ twice gives $H^2 \cos = -\cos$, an instance of $H^2 = -\id$.
The function $\cos(\omega x) + i \sin(\omega x) = e^{i\omega x}$ is the analytic signal of $\cos(\omega x)$.

## Explanations

### L1 human

The Hilbert transform shifts every wave by a quarter of a period.
A cosine becomes a sine, and a sine becomes minus a cosine.
This is the property that makes it useful in signal processing, where it turns a real oscillation into its complex partner.
</document>

<document source="https://tamnd.github.io/math2/node/ZNEHMR2CCP.md" tag="ZNEHMR2CCP" kind="statement">
# Fourier transform of the Gaussian

## Statement

Let $n \ge 1$ and $a > 0$, and let $f(x) = e^{-\pi a \abs{x}^2}$ for $x \in \R^n$.
Let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\, dx$ be its [[Fourier transform|DPVM34ADZV]].
Then
$$\hat f(\xi) = a^{-n/2}\, e^{-\pi \abs{\xi}^2 / a} \quad \text{for every } \xi \in \R^n.$$
In particular, for $a = 1$ the function $e^{-\pi \abs{x}^2}$ is its own Fourier transform.

## Remarks

Since $e^{-\pi a \abs{x}^2} = \prod_j e^{-\pi a x_j^2}$, the integral splits into $n$ one-dimensional integrals, so it suffices to take $n = 1$.
For $n = 1$ and $a = 1$, put $g = \hat f$.
By the [[rules for differentiation and multiplication|HHNR31W3WD]], the identity $f'(x) = -2\pi x f(x)$ becomes $g'(\xi) = -2\pi \xi g(\xi)$.
So $g(\xi) = g(0) e^{-\pi \xi^2}$, and $g(0) = \int_\R e^{-\pi x^2}\, dx = 1$ is the Gaussian integral.
The case of general $a$ follows by the change of variables $x \mapsto x / \sqrt{a}$.

The statement shows a trade-off: a narrow Gaussian (large $a$) has a wide transform (variance proportional to $a$).
For every $a > 0$ the product of the two widths is as small as possible, so every Gaussian is an equality case of the Heisenberg inequality.

## Explanations

### L1 human

The bell curve keeps its shape under the Fourier transform.
Squeezing the bell in space spreads it out in frequency by the same factor, and the reverse.
The standard bell $e^{-\pi x^2}$ is exactly fixed.

### L2 human

This computation is the base case for many proofs.
The Fourier inversion theorem, the Plancherel theorem and the solution of the heat equation all use the Gaussian as an approximate identity whose transform is known.
It also gives the characteristic function of the normal distribution after the change of variables $\xi = -t / (2\pi)$.
Gaussians are the extremal functions in the Babenko–Beckner and the Heisenberg inequalities.
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<document source="https://tamnd.github.io/math2/node/ZNMN1YA83C.md" tag="ZNMN1YA83C" kind="statement">
# Finite-dimensional Hausdorff vector topologies are Euclidean

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and give $\mathbb{K}^n$ its Euclidean topology.
Let $E$ be a [[Hausdorff|0S1FJ8VEX9]] [[topological vector space|T32W553W95]] over $\mathbb{K}$ of [[finite dimension|0ZHWKQMVX5]] $n$.
Then every [[linear isomorphism|0B8T10CE7N]] $T \colon \mathbb{K}^n \to E$ is a [[homeomorphism|0QT00PB5G2]].
In particular, a finite-dimensional vector space over $\mathbb{K}$ carries exactly one Hausdorff topology making it a topological vector space.

## Remarks

The map $T(c_1, \ldots, c_n) = c_1 e_1 + \cdots + c_n e_n$ is continuous, because addition and scalar multiplication are continuous in $E$.
For the continuity of $T^{-1}$, let $S$ be the unit sphere of $\mathbb{K}^n$; it is compact, so $T(S)$ is compact and does not contain $0$.
Since $E$ is Hausdorff, there is a balanced neighbourhood $V$ of $0$ disjoint from $T(S)$.
Then $T^{-1}(V)$ is balanced and misses $S$, so it lies in the open unit ball, which gives continuity of $T^{-1}$ at $0$.
Example: all norms on $\R^n$ define the same topology, since each makes $\R^n$ a Hausdorff topological vector space.
The Hausdorff hypothesis is needed: $\R$ with the indiscrete topology is a topological vector space.
The identity map from $\R$ with the Euclidean topology to it is a continuous linear bijection, but not a homeomorphism.
Consequences: every finite-dimensional subspace of a Hausdorff topological vector space is closed, and every linear map from a finite-dimensional Hausdorff topological vector space to any topological vector space is continuous.

## Explanations

### L1 human

On a finite-dimensional space there is only one reasonable way to say that vectors are close.
Whatever Hausdorff topology you choose, as long as adding and scaling are continuous, you get back the usual topology of coordinates.
So in finite dimension, questions of topology reduce to questions about coordinates.

### L2 human

This theorem explains why topology plays no role in finite-dimensional linear algebra and becomes important only in infinite dimension.
It contains the familiar fact that all norms on $\R^n$ are equivalent, and it implies that [[finite-dimensional normed spaces are complete|0ABP2F2ZP7]] and that the [[Heine–Borel theorem holds in them|0PHT6NBYPB]].
A companion result says that a Hausdorff topological vector space with a compact neighbourhood of $0$ must be finite-dimensional.
Together they show that local compactness singles out the finite-dimensional spaces.
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# Hofstadter sequences

## Example

Hofstadter sequences are integer sequences defined by nested [[nonlinear recurrences|7C7FBZGERM]], in which earlier values decide which earlier terms are used.

1. The *Hofstadter $Q$-sequence* is given by $Q(1) = Q(2) = 1$ and $Q(n) = Q(n - Q(n-1)) + Q(n - Q(n-2))$ for $n > 2$. It begins $1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, \dots$.
2. The *Hofstadter $G$-sequence* is given by $G(0) = 0$ and $G(n) = n - G(G(n-1))$ for $n \ge 1$. It begins $0, 1, 1, 2, 3, 3, 4, 4, 5, \dots$.

Check for $Q$: $Q(3) = Q(2) + Q(2) = 2$, $Q(4) = Q(2) + Q(3) = 3$, $Q(5) = Q(2) + Q(3) = 3$, $Q(6) = Q(3) + Q(3) = 4$.
Check for $G$: $G(1) = 1 - G(0) = 1$, $G(2) = 2 - G(1) = 1$, $G(3) = 3 - G(1) = 2$, $G(4) = 4 - G(2) = 3$.
The $G$-sequence has the closed form $G(n) = \lfloor (n+1)/\varphi \rfloor$, where $\varphi$ is the [[golden ratio|0SC8T7548D]].
The $Q$-sequence behaves erratically, and it is not known whether it is defined for all $n$, that is, whether $n - Q(n-1) \ge 1$ and $n - Q(n-2) \ge 1$ always hold.

## Explanations

### L1 human

These sequences come from Douglas Hofstadter's book *Gödel, Escher, Bach*.
In an ordinary recurrence, you always look back a fixed number of steps.
Here, the values of the sequence tell you how far back to look, which makes the rule self-referential.
Some of these sequences turn out to be tame, like $G$, while others, like $Q$, look chaotic and are still not understood.
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# Radical of an ideal

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $I$ be an [[ideal|0MYSA6MYNA]] of $R$.
The *radical* of $I$ is the set
$$\sqrt{I} = \set{r \in R : r^n \in I \text{ for some integer } n \ge 1}.$$

## Remarks

The set $\sqrt{I}$ is an ideal of $R$ that contains $I$: if $a^m \in I$ and $b^n \in I$, then every term of the binomial expansion of $(a + b)^{m + n - 1}$ lies in $I$.
Example: in $\Z$, $\sqrt{12\Z} = 6\Z$, since $6^2 = 36 \in 12\Z$, and if $r^n \in 12\Z$ then the primes $2$ and $3$ divide $r$.
Non-example: $4 \notin \sqrt{12\Z}$, because no power of $4$ is divisible by $3$.
The radical of the zero ideal is the [[nilradical|QQDN6EYQNJ]] of $R$.
An ideal equal to its radical is a [[radical ideal|S4GQN5V54X]].
Other notations are $\operatorname{rad}(I)$ and $r(I)$.
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# Sample mean and sample covariance

## Summary

The sample mean and the sample covariance matrix summarise a sample of vectors by its centre and its spread.
They are the multivariate versions of the [[sample mean|03Y6AQ2S0J]] and the [[sample variance|0NHXGY2C5J]].
They are unbiased estimators of the mean vector and the [[covariance matrix|0V7G6WMFK0]], and they are the starting point of multivariate analysis.

## Explanations

### L1 human

Measure the height and the weight of each person in a group.
The sample mean is the average point: average height and average weight.
The sample covariance matrix records how much each quantity varies, and whether tall people tend to be heavy.
For the two points $(0, 0)$ and $(2, 2)$, the mean is $(1, 1)$, and both coordinates vary together, so the off-diagonal entry is positive.
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# Volume conjecture

## Statement

Let $K$ be a [[hyperbolic knot|K47G2Q24DT]] in $S^3$, so that its complement $S^3 \setminus K$ is a [[hyperbolic 3-manifold|GHVW1Z35BS]] of finite volume $\mathrm{Vol}(S^3 \setminus K)$.
For $N \ge 2$, let $J_N(K; q)$ be the $N$-th colored [[Jones polynomial|4SPC17FPXE]] of $K$, normalised so that its value on the unknot is $1$.
The conjecture states that
$$\lim_{N \to \infty} \frac{2\pi}{N} \log \abs{J_N\left(K; e^{2\pi i / N}\right)} = \mathrm{Vol}(S^3 \setminus K).$$

## Remarks

Kashaev formulated the conjecture in 1997 for invariants he had built from the quantum dilogarithm.
H. Murakami and J. Murakami showed that Kashaev's invariant equals $J_N(K; e^{2\pi i / N})$.
They extended the conjecture to all knots, with the simplicial volume of the complement in place of the hyperbolic volume.
The conjecture is open in general.
It has been verified for the figure-eight knot and for a few other hyperbolic knots.
It would imply that a knot whose colored Jones polynomials all agree with those of the unknot has a complement of simplicial volume zero.

## Explanations

### L1 human

The colored Jones polynomials of a knot come from algebra and quantum physics.
The volume of the knot complement comes from hyperbolic geometry.
The conjecture says that the growth rate of the first, evaluated at special roots of unity, is exactly the second.
It would be a deep bridge between quantum invariants and geometry.
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# Uniform algebra

## Summary

A uniform algebra is a closed subalgebra of the continuous complex functions on a compact Hausdorff space that contains the constants and separates points.
It is a commutative [[Banach algebra|9KYWN94QDN]] with the sup norm.
The main examples come from complex analysis, such as the disc algebra of functions holomorphic inside the disc and continuous up to the boundary.
The [[Stone–Weierstrass theorem|1P8D3MK79H]] says the only uniform algebra on $X$ closed under complex conjugation is $C(X)$ itself.
So the theory of uniform algebras studies the closed subalgebras that are not closed under conjugation.

## Explanations

### L1 human

Start with all continuous functions on a compact space, measured by their largest absolute value.
A uniform algebra is a smaller family that is still closed under sums, products and uniform limits, contains the constants, and can tell any two points apart.
The polynomials in $z$ on the closed unit disc, together with all their uniform limits, form such a family.
It is smaller than all continuous functions: $\overline{z}$ is not a uniform limit of polynomials in $z$.
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# Carathéodory's royal road

## Definition

Let $a < b$, let $G \subseteq \R \times \R^n$ be open, and let $L \colon G \times \R^n \to \R$ be continuous, written $L(t, x, v)$.
For a $C^1$ curve $y \colon [a, b] \to \R^n$ with graph in $G$, let $J(y) = \int_a^b L(t, y(t), y'(t)) \, dt$, a functional of the [[calculus of variations|PQ25NHV0KS]].
Let $\bar x$ be a $C^1$ curve with graph in $G$.
*Carathéodory's royal road* is the following method to show that $\bar x$ minimises $J$ among such curves with the same endpoints.
One looks for a $C^1$ function $S \colon G \to \R$ and a continuous map $\psi \colon G \to \R^n$ such that:

1. $L(t, x, v) - S_t(t, x) - \nabla_x S(t, x) \cdot v \ge 0$ for all $(t, x) \in G$ and $v \in \R^n$;
2. $L(t, x, \psi(t, x)) - S_t(t, x) - \nabla_x S(t, x) \cdot \psi(t, x) = 0$ for all $(t, x) \in G$;
3. $\bar x'(t) = \psi(t, \bar x(t))$ for all $t \in [a, b]$.

The function $L^*(t, x, v) = L(t, x, v) - S_t(t, x) - \nabla_x S(t, x) \cdot v$ is the *equivalent Lagrangian* of $L$ with respect to $S$.

## Remarks

The method works because of this identity, valid for every admissible curve $y$:
$$J(y) = \int_a^b L^*(t, y(t), y'(t)) \, dt + S(b, y(b)) - S(a, y(a)).$$
The boundary terms depend only on the endpoints.
By clauses 1 to 3, the integral of $L^*$ is $\ge 0$ for every $y$ and $= 0$ for $\bar x$, so $J(y) \ge J(\bar x)$.

Clauses 1 and 2 say that $\min_v L^*(t, x, v) = 0$, with the minimum at $v = \psi(t, x)$.
With $H(t, x, p) = \sup_v (p \cdot v - L(t, x, v))$, this means $S_t + H(t, x, \nabla_x S) = 0$, the [[Hamilton–Jacobi equation|5TRXQKP7DY]].

Example: $n = 1$, $G = \R^2$, $L = \frac{1}{2} v^2$, and $\bar x(t) = t$ on $[0, 1]$.
Take $S(t, x) = x - t/2$ and $\psi = 1$.
Then $L^* = \frac{1}{2} v^2 + \frac{1}{2} - v = \frac{1}{2} (v - 1)^2$, which is $\ge 0$ and vanishes at $v = 1$, and $\bar x' = 1$.
So $\bar x(t) = t$ minimises $\int_0^1 \frac{1}{2} y'^2 \, dt$ among $C^1$ curves from $(0, 0)$ to $(1, 1)$.

Non-example: in the same problem, $S = 0$ with $\psi = 1$ fails clause 2, since $L(t, x, 1) = \frac{1}{2} \ne 0$.

## Explanations

### L1 human

The method changes the integrand by a total derivative, which changes the integral only by endpoint terms.
If the new integrand is never negative and is zero along the candidate curve, the candidate wins.
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# Stochastic approximation

## Summary

Stochastic approximation is a family of iterative methods for finding a root or an extremum of a function observed only with random noise.
Each step moves the current guess by a decreasing step size times a noisy measurement.
The first methods were the Robbins–Monro algorithm of 1951, for roots, and the Kiefer–Wolfowitz algorithm of 1952, for maxima.
Stochastic gradient descent, the workhorse of machine learning, is a stochastic approximation method.

## Explanations

### L1 human

Suppose a machine has a dial, and its average output depends on the dial setting, but every single reading is noisy.
To find the setting where the average output equals a target, turn the dial a little against each error you read.
The turns get smaller over time, so the noise averages out while the dial still travels far enough to reach the right place.
For example, to estimate a mean $m$ from samples $Z_1, Z_2, \dots$, the update $\theta_{n+1} = \theta_n + \frac{1}{n} (Z_n - \theta_n)$ with $\theta_1 = 0$ gives exactly the running average of the samples.
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# Control system

## Summary

A control system is a dynamical system whose evolution depends on an input that can be chosen, the control.
For systems governed by [[ordinary differential equations|0NT64JV96A]] it has the form $x' = f(x, u)$, where $x$ is the state and $u$ is the control.
It is the basic object of control theory.
That theory asks which states can be reached, how to steer the system, and how to do so in the best way.

## Explanations

### L1 human

A car is a control system: its position and speed are the state, and the driver chooses the pedals and the steering wheel.
The laws of motion say how the state changes for each choice of the inputs.
The driver wants to reach a goal, such as parking, by choosing the inputs well over time.
A simple mathematical example is $x_1' = x_2$, $x_2' = u$: a point on a line whose acceleration $u$ can be chosen.
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# Observed information

## Definition

Let $\set{f(\cdot\,; \theta) : \theta \in \Theta}$ be a family of densities with $\Theta \subseteq \R^d$ open.
Let $x_1, \dots, x_n$ be observed data, and let
$$\ell(\theta) = \sum_{i=1}^{n} \log f(x_i; \theta)$$
be the log-[[likelihood|0VW9E4QXR6]], assumed twice differentiable in $\theta$.
The *observed information* at $\theta$ is the $d \times d$ matrix
$$J(\theta) = -\nabla^2 \ell(\theta), \qquad J_{jk}(\theta) = -\frac{\partial^2 \ell(\theta)}{\partial \theta_j \, \partial \theta_k},$$
that is, minus the [[Hessian matrix|0693Y2METT]] of the log-likelihood.
For $d = 1$ it is the number $J(\theta) = -\ell''(\theta)$.

## Remarks

It is usually evaluated at the [[maximum likelihood estimate|0FREPFKNS7]] $\hat\theta$, and $J(\hat\theta)$ is then positive semidefinite when $\hat\theta$ is an interior maximum.
If the data are independent with density $f(\cdot\,; \theta)$, then under regularity conditions $E_\theta[J(\theta)] = n I(\theta)$, where $I$ is the [[Fisher information|G1Z3WSTE8Q]].
The inverse $J(\hat\theta)^{-1}$ is a standard estimate of the covariance matrix of $\hat\theta$.

Example: [[Poisson|0XB9GQTN9G]] data with mean $\lambda > 0$ and $t = x_1 + \cdots + x_n > 0$.
Then $\ell(\lambda) = t \log \lambda - n\lambda - \sum_i \log(x_i!)$, so $J(\lambda) = t/\lambda^2$.
At $\hat\lambda = t/n$ this gives $J(\hat\lambda) = n^2/t$.

Non-example: minus the second derivative of the likelihood $L = e^{\ell}$ itself is not the observed information.
In the Poisson example with $n = 1$ and $t = 1$, $-L''(\lambda)$ at $\hat\lambda = 1$ equals $e^{-1}$, while $J(1) = 1$.
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# Generalized method of moments

## Summary

The generalized method of moments (GMM) estimates parameters from moment conditions $E[g(X, \theta_0)] = 0$, possibly more conditions than parameters.
It chooses $\theta$ to make the sample averages of the moment functions as close to zero as possible in a weighted quadratic norm.
It contains the method of moments, least squares and instrumental variables estimation as special cases.
It was introduced in econometrics by Lars Peter Hansen.
It is a standard tool there because it does not need a full model of the data.

## Explanations

### L1 human

Suppose the theory gives you more facts about averages than you have unknown numbers.
With noisy data you usually cannot make all the sample averages match exactly.
GMM picks the parameter that makes the mismatches as small as possible, giving more weight to the averages that are measured more precisely.
For Poisson counts, both the mean and the variance equal the parameter $\lambda$, and GMM combines the two facts into one estimate.
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# Irreducible component

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A subset $Y \subseteq X$, with the subspace topology, is *irreducible* if $Y$ is nonempty and has the following property.
$Y$ is not the union of two [[closed subsets|0ZCK8DQYN1]] of $Y$ that are both different from $Y$.
An *irreducible component* of $X$ is an irreducible subset $Y \subseteq X$ that is maximal: no irreducible subset of $X$ contains $Y$ strictly.

## Remarks

Equivalently, a nonempty space is irreducible when any two nonempty open subsets meet, that is, when every nonempty open subset is dense.
The closure of an irreducible subset is irreducible, so irreducible components are closed.
By Zorn's lemma every irreducible subset lies in an irreducible component, so $X$ is the union of its irreducible components.

Example: the [[algebraic set|VHFMQQYSTN]] $X = V(xy) \subseteq \C^2$ is the union of the axes $V(x)$ and $V(y)$.
Each axis is irreducible, since it is homeomorphic to $\C$ with the topology whose proper closed sets are the finite sets, and two finite sets cannot cover an infinite set.
Neither axis contains the other, and $X$ itself is not irreducible, so the two axes are the irreducible components of $X$.

Non-example: the set $\set{0, 1} \subseteq \C$ is not irreducible, because it is the union of the closed sets $\set{0}$ and $\set{1}$.

In a [[Noetherian topological space|2VQX2E8EAH]], there are only finitely many irreducible components, and none of them lies in the union of the others.
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# Conjugate points

## Summary

Two points on a [[geodesic|NSYMQMHPDN]] are conjugate if a nonzero [[Jacobi field|9AHMM41PJ3]] along the geodesic vanishes at both of them.
Roughly, a family of geodesics leaving the first point refocuses at the second.
They are exactly the critical values of the [[exponential map|Y7Z0AGY1A7]] along the geodesic.
A geodesic stops being length-minimizing after it passes its first conjugate point.

## Explanations

### L1 human

Send out geodesics from a point in nearly the same direction.
If, further on, they come back together to first order at some point, that point is conjugate to the starting point.
On the unit sphere, all geodesics from the north pole meet at the south pole, so the poles are conjugate along every meridian.
In the flat plane, straight lines from a point never meet again, so there are no conjugate points.
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# Enumerative geometry

## Summary

Enumerative geometry is the branch of algebraic geometry that counts geometric objects satisfying given conditions.
Typical questions ask how many lines meet four given lines in space, or how many conics are tangent to five given conics.
Its tools include [[intersection theory|4HV4WTAX9P]], [[Schubert calculus|FWGHMADJR4]] and [[Gromov–Witten invariants|XVJA0JD79P]].
It is a meeting point of algebraic geometry, combinatorics and physics.

## Explanations

### L1 human

How many circles touch three given circles?
The answer, found by Apollonius, is $8$ in general.
Enumerative geometry asks questions of this kind: count the curves or surfaces with prescribed properties.
The answers are expected to be finite numbers that do not change when the given data move a little, as long as they stay general.
Getting them right is subtle, and some classical answers were wrong for a long time.
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# Jacobian variety

## Summary

The Jacobian variety of a compact [[Riemann surface|5K0N8QMV6Q]] of genus $g$ is a [[complex torus|Y7MA6KBE9M]] of dimension $g$.
It is built from the periods of the holomorphic differentials of the surface.
It is a principally polarized [[abelian variety|R7W03TXRS9]].
Its points correspond to divisor classes of degree $0$ on the curve, so it turns questions about divisors into questions about a group.
The [[Abel–Jacobi map|HE8F7KBF42]] maps the curve into its Jacobian.

## Explanations

### L1 human

Integrating the holomorphic differentials of a surface along loops gives a lattice of periods in $\C^g$.
Dividing $\C^g$ by this lattice gives a doughnut-shaped space of dimension $g$, which is the Jacobian.
It is a group, so points of the curve, and formal sums of them, can be added inside it.
For a torus $\C / \Lambda$ of genus $1$, the Jacobian is the torus itself.
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# Progenerator

## Summary

A progenerator over a [[ring|0B1261B5ED]] $R$ is a [[finitely generated|05QY1VEBYD]] [[projective module|BFFA03GE3V]] that is also a generator of the module category.
The free modules $R^n$ with $n \ge 1$ are progenerators.
Progenerators are the modules that control [[Morita equivalence|RD5WKA1VK7]].
A ring $S$ is Morita equivalent to $R$ exactly when it is the endomorphism ring of a progenerator over $R$.
This holds up to the choice of side.

## Explanations

### L1 human

The ring $R$ itself, seen as a module, has three key features.
It is generated by one element, it is projective, and every module is a quotient of a sum of copies of it.
A progenerator is any module that shares these features, with finitely many generators allowed.
For example, $R^3$ is a progenerator, since $R$ is a direct summand of it.
The ring of endomorphisms of a progenerator has the same module theory as $R$.
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# Fourier multiplier

## Summary

A Fourier multiplier is an operator that acts on a function by multiplying its [[Fourier transform|DPVM34ADZV]] by a fixed function $m$, called the symbol.
Differential operators with constant coefficients, the [[Hilbert transform|NSCEBXM7ZM]], the heat semigroup and frequency filters are all of this form.
Every symbol in $L^\infty$ gives a bounded operator on $L^2$, and the bounded operators on $L^2(\R^n)$ that commute with translations are exactly these.
Which symbols give bounded operators on $L^p$ for $p \ne 2$ is a central and hard question of harmonic analysis, studied through [[$L^p$ multipliers|RAGR38Q40B]].
See the [[definition|SD89TZ1PHR]].

## Explanations

### L1 human

An equalizer on a music player raises or lowers each frequency by its own factor.
A Fourier multiplier does exactly that to a function.
It splits the function into frequencies, multiplies the amount of frequency $\xi$ by a number $m(\xi)$, and puts the pieces back together.
For example, $m(\xi) = 1$ for $\abs{\xi} \le 1$ and $0$ otherwise is an ideal low-pass filter that keeps only the low frequencies.
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# Hardy–Littlewood circle method

## Summary

The circle method is an analytic technique for counting the solutions of additive equations.
Examples are writing $N$ as a sum of $s$ $k$-th powers, or as a sum of three primes.
It writes the number of solutions as an integral over the unit circle of a power of an [[exponential sum|97PBMQCPPM]].
The integral is split into major arcs near rationals with small denominators, which give the main term.
The rest, the minor arcs, are shown to contribute less.
It goes back to Hardy and Ramanujan's work on the partition function and was developed by Hardy and Littlewood for [[Waring's problem|7VWGA444K3]].

## Explanations

### L1 human

Say you want to count the ways of writing $N$ as a sum of three cubes.
One builds a sum of waves, one wave for each cube, and raises it to the third power.
Integrating against a wave of frequency $N$ picks out exactly the combinations that add up to $N$.
Near simple fractions such as $0$, $1/2$ or $1/3$ the waves line up and the integrand is large; this gives the main term.
Everywhere else the waves mostly cancel, and the hard work is to prove that this cancellation is strong enough.
For example, this is how Vinogradov proved that every large odd number is a sum of three primes.
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# Skolem–Mahler–Lech theorem

## Statement

Let $K$ be a field of characteristic $0$.
Let $(u_n)_{n \ge 0}$ be a [[solution|0ZV2YKKZ58]] of a [[linear recurrence with constant coefficients|081THG13JZ]] over $K$.
Then the zero set $Z = \set{n \in \N : u_n = 0}$ is the union of a finite set and finitely many [[arithmetic progressions|S70JBRSQZX]] $\set{a + b m : m \in \N}$ with $a \in \N$ and $b \ge 1$.

## Remarks

The standard proof is $p$-adic.
One reduces to a finitely generated subfield, which embeds into the $p$-adic numbers $\Q_p$ for a suitable prime $p$ with all relevant quantities $p$-adic units.
Then for a suitable period $b$ and each residue $a$, the map $m \mapsto u_{a + b m}$ extends to a $p$-adic analytic function on $\Z_p$.
Such a function either vanishes identically or has only finitely many zeros, which gives the progressions and the finite set.
Example: $u_n = 1 + (-1)^n$ solves $u_{n+2} = u_n$, and its zero set is the progression of odd numbers.
The hypothesis of characteristic $0$ is necessary.
Over the field $\F_p(t)$, the sequence $u_n = (1 + t)^n - t^n - 1$ solves a linear recurrence of order $3$.
Its terms vanish exactly when $n$ is a power of $p$, and this infinite set contains no infinite arithmetic progression.
The proof is not effective: it gives no algorithm to find the finite set.
Whether one can decide if a given integer recurrence has a zero at all is the open Skolem problem.

## Explanations

### L1 human

Take a sequence defined by a linear recurrence with fixed coefficients, and ask where it is zero.
The theorem says the answer is never wild: apart from finitely many exceptions, the zeros repeat with a fixed period.
For instance, it can be zero at every third index, but it cannot be zero exactly at the perfect squares.

### L2 human

The theorem is named after Thoralf Skolem, Kurt Mahler and Christer Lech, who proved it in more and more general fields.
The periodic part can be computed, but the finite exceptional set is the hard part.
The proof is a classic use of $p$-adic analysis in number theory, by the same principle as Strassmann's theorem on the zeros of $p$-adic power series.
A recurrence is non-degenerate if no ratio of two distinct roots of its characteristic polynomial is a root of unity.
A common corollary: a non-degenerate solution that is not identically zero has only finitely many zeros.
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<document source="https://tamnd.github.io/math2/node/ZQKA90X325.md" tag="ZQKA90X325" kind="definition">
# Propagation of uncertainty

## Definition

Let $X = (X_1, \ldots, X_n)$ be a random vector with [[mean|0WQNRCNY3M]] $\mu \in \R^n$ and [[covariance matrix|0V7G6WMFK0]] $\Sigma_X$.
Let $f \colon \R^n \to \R^m$ be differentiable at $\mu$, with [[Jacobian matrix|047NDTEPCV]] $J = Df(\mu)$.
The *linear propagation of uncertainty* assigns to $Y = f(X)$ the approximate mean $f(\mu)$ and the approximate covariance matrix
$$\Sigma_Y = J \, \Sigma_X \, J^{T}.$$
For $m = 1$ and uncorrelated $X_i$ with [[variances|0477R9PYAE]] $\sigma_i^2$, this reads
$$\sigma_Y^2 = \sum_{i=1}^{n} \left( \frac{\partial f}{\partial x_i}(\mu) \right)^2 \sigma_i^2.$$
It is the covariance of the first-order Taylor approximation $f(\mu) + J(X - \mu)$.

## Remarks

Example: for $f(x_1, x_2) = x_1 x_2$ and uncorrelated inputs, $\partial f/\partial x_1 = \mu_2$ and $\partial f/\partial x_2 = \mu_1$, so $\sigma_Y^2 = \mu_2^2 \sigma_1^2 + \mu_1^2 \sigma_2^2$, that is $(\sigma_Y/\mu_1\mu_2)^2 = (\sigma_1/\mu_1)^2 + (\sigma_2/\mu_2)^2$ when $\mu_1 \mu_2 \ne 0$.
If $f(x) = Ax + b$ is affine, the formula is exact: $\mathrm{Cov}(AX + b) = A \Sigma_X A^T$.
Non-example: for $f(x) = x^2$ and $X$ normal with mean $0$ and variance $\sigma^2$, the formula gives $\sigma_Y^2 = 0$, while the true variance of $X^2$ is $2\sigma^4$; the linearization fails because $f'(\mu) = 0$.
</document>

<document source="https://tamnd.github.io/math2/node/ZQR56X168K.md" tag="ZQR56X168K" kind="definition">
# Voronoi cell

## Definition

Let $S \subset \R^n$ be a nonempty set with no accumulation point, and let $\abs{x - y}$ denote the [[Euclidean distance|0NPPGHQNXQ]].
The *Voronoi cell* of a point $p \in S$ is
$$V(p) = \set{x \in \R^n : \abs{x - p} \le \abs{x - q} \text{ for all } q \in S}.$$
If $\Lambda$ is a [[lattice|WN9V7WSJKF]] in $\R^n$, the *Voronoi cell of* $\Lambda$ is the cell $V(0)$ of the origin in $S = \Lambda$.

## Remarks

Each condition $\abs{x - p} \le \abs{x - q}$ defines a closed half-space, so $V(p)$ is a closed convex set.
Example: for $\Lambda = \Z^2$, the cell is $V(0) = [-\tfrac12, \tfrac12]^2$.
Indeed, $\abs{x} \le \abs{x - q}$ for $q = (\pm 1, 0)$ and $q = (0, \pm 1)$ gives $\abs{x_1} \le \tfrac12$ and $\abs{x_2} \le \tfrac12$, and these four conditions imply the others.
Non-example: the closed square $[0, 1]^2$ is not the Voronoi cell of $0$ in $\Z^2$, because its point $(1, 1)$ is at distance $0$ from $q = (1, 1)$ and at distance $\sqrt 2$ from $0$.
For a lattice $\Lambda$, the cell $V(0)$ is a convex polytope, symmetric about $0$, its translates $V(0) + v$ with $v \in \Lambda$ cover $\R^n$ with disjoint interiors, and its volume equals $\det \Lambda$.
The cells of all points of $S$ form the Voronoi diagram of $S$.
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# Donaldson theory

## Summary

Donaldson theory studies smooth [[4-manifolds|8H58Q6YJCH]] through moduli spaces of anti-self-dual connections, also called [[instantons|YWBTGC7MFA]], on principal $SU(2)$-bundles over them.
It was started by Simon Donaldson in the early 1980s.
Its first result, [[Donaldson's theorem|34B7K77P31]], restricts the intersection forms of smooth $4$-manifolds.
Its invariants, the Donaldson polynomial invariants, distinguish smooth structures on manifolds that are homeomorphic.

## Explanations

### L1 human

A connection is a rule for comparing directions at nearby points, and its curvature measures how much the rule twists.
On a four-dimensional space, the connections whose curvature has a special symmetry, the instantons, form a new space called a moduli space.
The shape of this moduli space reflects the smooth structure of the original space, not only its topology.
For example, it shows that the topological $4$-manifold with the $E_8$ intersection form cannot be smooth.
The [[Seiberg–Witten invariants|PRDDMSTR4X]] later gave a simpler route to many of the same results.
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<document source="https://tamnd.github.io/math2/node/ZQYMWVT1TT.md" tag="ZQYMWVT1TT" kind="concept">
# Legendre transformation

## Summary

The Legendre transformation turns a strictly [[convex function|07Q7CNM5WF]] $f$ into a new function $f^*$ of the slope $p = f'(x)$.
It records the same information as $f$, but with slopes as the variable instead of points.
Applied twice it gives back $f$, and the derivatives of $f$ and $f^*$ are inverse functions of each other.
It is the classical case of the convex conjugate and is used in mechanics, thermodynamics and optimization; the [[definition|HKPHRSN3ER]] gives the precise formula.

## Explanations

### L1 human

A convex curve can be described by its points, or by its tangent lines.
Each tangent line has a slope $p$ and meets the vertical axis at some height; the Legendre transform $f^*(p)$ is minus that height.
So $f^*$ lists the tangent lines of the graph, one for each slope.
For $f(x) = x^2 / 2$ the tangent at $x$ has slope $p = x$, and one finds $f^*(p) = p^2 / 2$, the same function.
</document>

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# Locally compact abelian group

## Summary

A locally compact abelian group, or LCA group, is a commutative topological group whose topology is Hausdorff and locally compact.
These groups are the natural setting of Fourier analysis.
Each LCA group has a Haar measure and a dual group of characters, and Pontryagin duality says that the group is recovered from its dual.

## Explanations

### L1 human

Think of a group where you can add elements, the order does not matter, and there is a notion of nearness with small compact neighbourhoods.
The real line $\R$ with addition is the basic picture: every point has a closed bounded interval around it.
The integers $\Z$ with the discrete topology, the circle group and every finite abelian group are also examples.
On all of these, one can integrate and expand functions into waves, just as Fourier series expand functions on the circle.
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<document source="https://tamnd.github.io/math2/node/ZR1DPX6VAN.md" tag="ZR1DPX6VAN" kind="definition">
# Total variation diminishing scheme

## Definition

Consider a scalar [[conservation law|8RA7AMWSAF]] $u_t + f(u)_x = 0$ in one space dimension, and a numerical scheme that maps a grid function $u^n = (u_j^n)_{j \in \Z}$ to $u^{n+1} = (u_j^{n+1})_{j \in \Z}$.
The discrete [[total variation|8QA5DEWNYD]] of a grid function $v$ is
$$TV(v) = \sum_{j \in \Z} \abs{v_{j+1} - v_j}.$$
The scheme is *total variation diminishing* (TVD) if $TV(u^{n+1}) \le TV(u^n)$ for every grid function $u^n$ with $TV(u^n) < \infty$.

## Remarks

Example: for $u_t + a u_x = 0$ with $a > 0$, the [[upwind scheme|MTCRZC5QKT]] with $0 \le \nu \le 1$ gives $u_{j+1}^{n+1} - u_j^{n+1} = (1 - \nu)(u_{j+1}^n - u_j^n) + \nu (u_j^n - u_{j-1}^n)$.
By the triangle inequality and a shift of the index, $TV(u^{n+1}) \le (1 - \nu) TV(u^n) + \nu\, TV(u^n) = TV(u^n)$.
Non-example: the [[Lax–Wendroff method|YHZ9BCNDRZ]] with $\nu = 1/2$, applied to the step $u_j^n = 1$ for $j \le 0$ and $u_j^n = 0$ for $j \ge 1$, gives $u_0^{n+1} = 1.125$, $u_1^{n+1} = 0.375$ and $u_j^{n+1} = u_j^n$ for the other $j$.
So $TV(u^{n+1}) = 0.125 + 0.75 + 0.375 = 1.25 > 1 = TV(u^n)$.
Harten's criterion: a scheme that can be written as $u_j^{n+1} = u_j^n + C_{j+1/2} (u_{j+1}^n - u_j^n) - D_{j-1/2} (u_j^n - u_{j-1}^n)$ with $C_{j+1/2} \ge 0$, $D_{j+1/2} \ge 0$ and $C_{j+1/2} + D_{j+1/2} \le 1$ for all $j$ is TVD.
Every monotone conservative scheme is TVD, and every TVD scheme is monotonicity preserving.

## Explanations

### L1 human

A scheme is TVD when the total amount of up-and-down in the solution can only shrink.
This forbids new wiggles near jumps.
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<document source="https://tamnd.github.io/math2/node/ZR6DEE6RBC.md" tag="ZR6DEE6RBC" kind="statement">
# Whitney–Graustein theorem

## Statement

Let a *regular closed curve* be a $1$-periodic map $\gamma \colon \R \to \R^2$ of class $C^1$ with $\gamma'(t) \ne 0$ for all $t$.
Its *turning number* is the degree of the map $t \mapsto \gamma'(t) / \norm{\gamma'(t)}$ from $\R / \Z$ to the unit circle.
A *regular homotopy* between $\gamma_0$ and $\gamma_1$ is a continuous family $(\gamma_u)_{u \in [0, 1]}$ of regular closed curves such that $(u, t) \mapsto \gamma_u'(t)$ is continuous.
Two regular closed curves in the plane are regularly homotopic if and only if they have the same turning number.

## Remarks

The turning number is the [[winding number|0X7SBS8E0F]] of the derivative $\gamma'$ around $0$.
It is invariant under regular homotopy, because it is an integer that depends continuously on $u$.
For the converse, one reparametrises both curves proportionally to arc length.
Their unit tangent maps have the same degree, so they are joined by a [[homotopy|9YA6S6HD9V]] of maps to the circle.
Integrating these tangent fields and correcting the closing condition gives the required family.
Example: the circle $t \mapsto (\cos 2\pi t, \sin 2\pi t)$ has turning number $1$, and the figure eight $t \mapsto (\sin 4\pi t, \sin 2\pi t)$ has turning number $0$.
So these two curves are homotopic as maps but not regularly homotopic.
The theorem was proved by Hassler Whitney in 1937, who credited the result to William Graustein.

## Explanations

### L1 human

Draw a closed loop on paper, and imagine moving it around smoothly without ever making a sharp corner.
The loop may cross itself during the motion, but its velocity must never vanish.
The theorem says that one number decides which loops can be turned into each other this way.
It is the number of full turns that the direction of the pencil makes in one trip around.

### L2 human

The theorem classifies immersions of the circle into the plane up to regular homotopy by the integer turning number.
Smale later extended this classification to immersions of spheres, which led to the eversion of the sphere.
For a regular closed curve of class $C^2$, the turning number equals the total signed curvature divided by $2\pi$.
</document>

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# Drazin inverse

## Definition

Let $F$ be a [[field|04C5ZBG1WT]] and let $A$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $F$.
The *index* of $A$ is the least integer $k \ge 0$ such that $\rank(A^{k+1}) = \rank(A^k)$, where $\rank$ is the [[rank|0ZC9YTXZRY]] and $A^0 = I$.
The *Drazin inverse* of $A$ is the $n \times n$ matrix $A^D$ over $F$ such that, with $k$ the index of $A$:

1. $A^{k+1} A^D = A^k$.
2. $A^D A A^D = A^D$.
3. $A A^D = A^D A$.

There is exactly one matrix with these three properties.

## Remarks

Example: if $A$ is invertible, its index is $0$, and $A^D = A^{-1}$; indeed $A \cdot A^{-1} = I$, $A^{-1} A A^{-1} = A^{-1}$, and $A^{-1}$ commutes with $A$.
Example: if $A$ is [[nilpotent|07BSPZPSMF]] with $A^k = 0$ for its index $k$, then $A^D = 0$; each side of each condition is then $0$, because $A^k = 0$.
Non-example: for $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, of index $2$, the matrix $X = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ is not $A^D$, because $AX = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $XA = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ differ, so clause 3 fails.
If $A = P \begin{pmatrix} C & 0 \\ 0 & N \end{pmatrix} P^{-1}$ with $C$ invertible and $N$ nilpotent, then $A^D = P \begin{pmatrix} C^{-1} & 0 \\ 0 & 0 \end{pmatrix} P^{-1}$.
When the index is at most $1$, the Drazin inverse is called the group inverse and written $A^{\#}$.
</document>

<document source="https://tamnd.github.io/math2/node/ZR7Y5C235P.md" tag="ZR7Y5C235P" kind="statement">
# Łoś–Vaught test

## Statement

Let $T$ be a satisfiable theory in a first-order signature $L$, and let $\kappa$ be an infinite [[cardinal|0EBACQFAZW]] with $\kappa \ge \abs{L}$.
Suppose that $T$ has no finite models and that $T$ is [[$\kappa$-categorical|1JJCGNX240]].
Then $T$ is [[complete|655K8Y7VS8]].

## Remarks

Suppose $T$ is not complete, and pick a sentence $\varphi$ such that neither $\varphi$ nor $\neg \varphi$ follows from $T$.
Then $T \cup \set{\varphi}$ and $T \cup \set{\neg \varphi}$ both have models, and these are infinite.
By the [[Löwenheim–Skolem theorem|484M0TZ1S7]], both have models of size $\kappa$.
These two models are not isomorphic, since one satisfies $\varphi$ and the other does not, which contradicts $\kappa$-categoricity.

Example: the theory of dense linear orders without endpoints has no finite models and is $\aleph_0$-categorical, so it is complete.
Similarly, the theory of algebraically closed fields of a fixed characteristic is complete, since it is categorical in every uncountable cardinal.

The hypothesis on finite models is necessary.
Take the empty theory $T = \emptyset$ in the empty signature, whose models are the nonempty sets.
For each infinite $\kappa$, any two sets of size $\kappa$ are in bijection, so $T$ is $\kappa$-categorical.
But $T$ is not complete: the sentence $\forall x \, \forall y \, (x = y)$ holds in a model of size $1$ and fails in a model of size $2$.

## Explanations

### L1 human

A theory is complete when it decides every statement in its language.
The test says that if all infinite models of one size look the same and there are no finite models, the theory decides everything.
It is a quick way to prove completeness for natural theories such as dense orders or algebraically closed fields.

### L2 human

The test turns a structural fact, uniqueness of models of one cardinality, into a logical fact, completeness.
Completeness of a recursively axiomatized theory implies that it is decidable, so the test also gives decision procedures.
It is often combined with a back-and-forth argument for $\aleph_0$-categoricity, or with a dimension argument, such as transcendence degree, for uncountable categoricity.
Its converse fails: the theory of $(\Z, <)$ is complete but categorical in no infinite cardinal.
</document>

<document source="https://tamnd.github.io/math2/node/ZR8A03FV67.md" tag="ZR8A03FV67" kind="concept">
# Scott topology

## Summary

The Scott topology on a partially ordered set has as open sets the up-closed sets that cannot be reached by a directed supremum from outside.
It turns a [[directed-complete partial order|7TSXP95QN6]] into a topological space whose continuous maps are the Scott-continuous maps.
Its [[specialization preorder|CMYVK9CEKZ]] is the original order, so the order can be recovered from the topology.
It is the topology used in [[domain theory|Q2P5D74Q1V]].

## Explanations

### L1 human

An open set in the Scott topology is an "observable property" of partial information.
If an element has the property, then every better-informed element has it too, so open sets are closed upward.
And if a limit of approximations has the property, then some approximation already has it, since one can only observe finitely much.
For example, in $\N \cup \set{\infty}$, the set $\set{\infty}$ is not Scott-open, because $\infty$ is the supremum of $0, 1, 2, \ldots$, none of which is in the set; but every set $\set{n, n + 1, \ldots, \infty}$ is Scott-open.
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# Feller process

## Definition

Let $E$ be a locally compact Hausdorff space with a countable base.
Let $C_0(E)$ be the [[Banach space|06GZ956657]] of continuous functions $f \colon E \to \R$ that vanish at infinity, with the sup norm.
A *Feller semigroup* on $C_0(E)$ is a family $(T_t)_{t \ge 0}$ of linear maps $T_t \colon C_0(E) \to C_0(E)$ such that:

1. if $f \in C_0(E)$ and $0 \le f \le 1$, then $0 \le T_t f \le 1$;
2. $T_0 = \id$ and $T_{s+t} = T_s T_t$ for all $s, t \ge 0$;
3. $\norm{T_t f - f} \to 0$ as $t \downarrow 0$, for every $f \in C_0(E)$.

Let $(X_t)_{t \ge 0}$ be a time-homogeneous [[Markov process|0B77KZCQ51]] in $E$, and let $\mathbb{E}_x$ be the expectation for the process started at $x$.
The process is a *Feller process* if the formula
$$T_t f(x) = \mathbb{E}_x[f(X_t)]$$
defines a Feller semigroup.

## Remarks

Example: Brownian motion on $\R$ is a Feller process.
Here $T_t f(x) = \int f(y) \frac{1}{\sqrt{2\pi t}} e^{-(y - x)^2/(2t)}\, dy$; this is in $C_0(\R)$ when $f$ is, it preserves $0 \le f \le 1$, and it tends to $f$ uniformly because $f$ is uniformly continuous.
Non-example: on $E = \R$, let the process started at $x$ jump at once to $0$ if $x > 0$ and to $1$ if $x \le 0$, then stay there.
Then $T_t f = f(0)$ on $(0, \infty)$ and $T_t f = f(1)$ on $(-\infty, 0]$, which is not continuous in $x$, so $T_t$ does not map $C_0(E)$ into itself.
Clause 1 allows $T_t 1 < 1$, so the process may be killed; one then adds a cemetery point to $E$.
Every Feller process has a modification with right-continuous paths with left limits, and that modification is strong Markov.
</document>

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# Semiperfect ring

## Summary

A semiperfect ring is a [[semilocal ring|ZDCDS6EWGX]] in which idempotents [[lift|2SRDN8EJG3]] modulo the Jacobson radical.
Equivalently, every finitely generated left module has a [[projective cover|WGYXR97NPK]].
Equivalently again, $1$ is a sum of orthogonal idempotents $e_1, \ldots, e_n$ such that each corner ring $e_i R e_i$ is local.
Left Artinian rings, [[local rings|MZ02BKPHJA]] and [[perfect rings|49Y9Q949PK]] are semiperfect.

## Explanations

### L1 human

A semiperfect ring splits, as a module over itself, into finitely many indecomposable pieces, each of which behaves like a local ring.
This makes its module theory close to that of finite-dimensional algebras.
The ring of $p$-adic integers is semiperfect, being a complete local ring.
The ring of fractions $a/b$ with $b$ prime to $6$ is semilocal but not semiperfect.
Modulo its radical it is $\mathbb{F}_2 \times \mathbb{F}_3$, which has idempotents that do not come from the ring.
</document>

<document source="https://tamnd.github.io/math2/node/ZREF8MTYNW.md" tag="ZREF8MTYNW" kind="definition">
# Dialectica interpretation

## Definition

Let $\mathrm{T}$ be Gödel's system of [[primitive recursive|5NVA2JRHPV]] functionals of finite type.
To each formula $A$ of [[Heyting arithmetic|RXHJPVMPJH]], the *Dialectica interpretation* assigns a formula $A^D = \exists x\, \forall y\, A_D(x, y)$.
Here $x$ and $y$ are finite tuples of variables of finite type, and $A_D$ is quantifier-free.
Write $A^D = \exists x\, \forall y\, A_D(x, y)$ and $B^D = \exists u\, \forall v\, B_D(u, v)$.
The clauses are these.

1. If $A$ is atomic, then $A^D = A_D = A$, with empty tuples $x$ and $y$.
2. $(A \wedge B)^D = \exists x, u\, \forall y, v\, \bigl( A_D(x, y) \wedge B_D(u, v) \bigr)$.
3. $(A \vee B)^D = \exists z, x, u\, \forall y, v\, \bigl( (z = 0 \to A_D(x, y)) \wedge (z \ne 0 \to B_D(u, v)) \bigr)$, where $z$ is a number variable.
4. $(A \to B)^D = \exists f, g\, \forall x, v\, \bigl( A_D(x, f x v) \to B_D(g x, v) \bigr)$.
5. $(\exists z\, A(z))^D = \exists z, x\, \forall y\, A_D(x, y, z)$.
6. $(\forall z\, A(z))^D = \exists f\, \forall z, y\, A_D(f z, y, z)$.

Negation $\neg A$ is read as $A \to (0 = 1)$.

## Remarks

Soundness theorem (Gödel): if Heyting arithmetic proves a closed formula $A$, then there is a tuple $t$ of closed terms of $\mathrm{T}$ such that $\mathrm{T}$ proves $A_D(t, y)$.
Example: for $A = \forall n\, \exists m\, (n < m)$, clauses 5 and 6 give $A^D = \exists f\, \forall n\, (n < f n)$, and the term $f = \lambda n.\, n + 1$ is a witness.
Non-example: let $P(n, k)$ be the primitive recursive relation, given by [[Kleene's T predicate|S9DK13K5PS]], which says that the program $n$ on input $n$ halts within $k$ steps.
The formula $\forall n\, (\exists k\, P(n, k) \vee \neg \exists k\, P(n, k))$ has no witness in $\mathrm{T}$.
A witness would give a computable function $n \mapsto z$ that decides the halting problem.
Combined with a [[double-negation translation|HGJVM49PA4]], it interprets Peano arithmetic in system $\mathrm{T}$.
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# Holomorphically convex domain

## Definition

Let $D \subseteq \C^n$ be an [[open set|02BH3MCBQB]].
Then $D$ is *holomorphically convex* if for every [[compact set|0R6EWFKRCF]] $K \subseteq D$ the [[holomorphically convex hull|CD6SMMZSW8]] $\hat K_D$ is compact.

## Remarks

Since $\hat K_D$ is closed in $D$, the condition says that $\hat K_D$ has no limit points on the boundary of $D$.

Example: a convex open set $D$.
For compact $K \subseteq D$, the hull $\hat K_D$ lies in the convex hull $\operatorname{ch}(K)$, which is compact and contained in $D$.
A subset of $D$ that is closed in $D$ and contained in the compact set $\operatorname{ch}(K) \subseteq D$ is compact.

Non-example: $D = \set{z \in \C^2 : 0 < \abs{z} < 1}$ and $K = \set{\abs{z} = 1/2}$.
By [[Hartogs's extension theorem|FSGES670AZ]] every holomorphic $f$ on $D$ extends to the ball, and by the maximum principle $\abs{f(z)} \le \max_K \abs{f}$ for $0 < \abs{z} \le 1/2$.
So $\hat K_D$ contains $\set{0 < \abs{z} \le 1/2}$, which is not compact.

By the Cartan–Thullen theorem, a domain in $\C^n$ is holomorphically convex if and only if it is a domain of holomorphy.
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# Bochner's theorem

## Statement

Let $G$ be a [[locally compact abelian group|ZR09S9R3AW]] and let $\widehat{G}$ be its [[Pontryagin dual|FMW57WNPDB]].
Let $\varphi \colon G \to \C$ be a continuous function.
Call $\varphi$ [[positive-definite|3PRZ02APFA]] if $\sum_{j,k=1}^{n} c_j \overline{c_k}\, \varphi(x_j - x_k) \ge 0$ for all $n \ge 1$, $x_1, \dots, x_n \in G$ and $c_1, \dots, c_n \in \C$.
Then $\varphi$ is positive-definite if and only if there is a finite positive [[Radon measure|HZNEQQBCW4]] $\mu$ on $\widehat{G}$ with
$$\varphi(x) = \int_{\widehat{G}} \chi(x) \, d\mu(\chi) \quad \text{for all } x \in G.$$
The measure $\mu$ is then unique, and $\mu(\widehat{G}) = \varphi(0)$.

## Remarks

The easy direction: if $\varphi$ has this form, then $\sum_{j,k} c_j \overline{c_k} \varphi(x_j - x_k) = \int_{\widehat{G}} \abs{\sum_j c_j \chi(x_j)}^2 \, d\mu(\chi) \ge 0$.
For the converse, $\varphi$ defines a positive linear functional on $L^1(G)$, hence on the commutative group C*-algebra of $G$.
The spectrum of this algebra is $\widehat{G}$, and the Riesz representation theorem then gives $\mu$.
Example: for $G = \R$, the function $\varphi(x) = e^{-x^2/2}$ is positive-definite, and $\mu$ is the standard normal distribution, since $\int_\R e^{i\xi x} \frac{1}{\sqrt{2\pi}} e^{-\xi^2/2} \, d\xi = e^{-x^2/2}$.
For $G = \R$, the theorem says that the continuous positive-definite functions with $\varphi(0) = 1$ are exactly the [[characteristic functions|0V5B6D6BQH]] of probability distributions.
Continuity is needed: on $\R$ the indicator function of $\set{0}$ is positive-definite but is not the transform of any measure, since such transforms are continuous.

## Explanations

### L1 human

A positive-definite function is one that behaves like a correlation: certain sums built from it are never negative.
Bochner's theorem says that such functions are exactly the averages of waves with nonnegative weights.
In probability, it says which functions are characteristic functions of random variables.

### L2 human

The theorem is a key tool in the proof of the [[Plancherel theorem for abelian groups|V9K9TG0K3J]] and of [[Pontryagin duality|QTVC6F2P9D]].
For $G = \Z$ it is Herglotz's theorem on positive-definite sequences, with measures on the circle.
In the theory of stationary processes it gives the spectral measure of a stationary covariance function.
</document>

<document source="https://tamnd.github.io/math2/node/ZRMWNF10FA.md" tag="ZRMWNF10FA" kind="definition">
# Point estimation

## Definition

Let $(\mathcal{X}, \mathcal{A})$ be a sample space and let $\set{P_\theta : \theta \in \Theta}$ be a [[statistical model|K3AMMSY8DD]] on it.
Let $g \colon \Theta \to D$ be a function, the *estimand*, with values in a measurable space $D$.
A *point estimator* of $g(\theta)$ is a [[statistic|036756NFD6]] $T \colon \mathcal{X} \to D$, that is, a measurable map that does not depend on $\theta$.
For observed data $x \in \mathcal{X}$, the value $T(x)$ is the *point estimate* of $g(\theta)$.
*Point estimation* is the problem of choosing such a $T$ so that $T(X)$ is close to $g(\theta)$ when $X \sim P_\theta$, for every $\theta \in \Theta$.

## Remarks

The general notion of [[estimator|0TTNRP5PFQ]] covers the same object; the word "point" stresses that the output is a single element of $D$, not a set.

Example: in the model of $n$ independent Bernoulli($p$) trials with $\Theta = [0, 1]$ and $g(p) = p$, the map $T(x) = \frac{1}{n}\sum_i x_i$ is a point estimator; it is the [[maximum likelihood estimator|0FREPFKNS7]].

Non-example: the map $T(x) = \frac{1}{n}\sum_i x_i - p$ is not a point estimator of $p$, because it depends on the unknown parameter $p$.
Also, a confidence interval $[T_1(x), T_2(x)]$ is an interval estimator, not a point estimator, since its value is a set, not an element of $D$.

Closeness is measured by a loss function, most often the squared error $(T - g(\theta))^2$, whose expectation is the mean squared error.
</document>

<document source="https://tamnd.github.io/math2/node/ZRV9T8DD05.md" tag="ZRV9T8DD05" kind="statement">
# Jacobi condition

## Statement

Let $a < b$ and let $L(x, y, p)$ be a $C^3$ function on $[a, b] \times \R \times \R$.
Let $J(y) = \int_a^b L(x, y, y') \, dx$ on $C^1([a, b])$, with fixed values $y(a)$ and $y(b)$.
Let $y_0 \in C^2([a, b])$ solve the [[Euler–Lagrange equation|PS0VV149ET]] of $J$.
Assume the strengthened Legendre condition $L_{pp}(x, y_0(x), y_0'(x)) > 0$ for all $x \in [a, b]$.
A point $c \in (a, b]$ is *conjugate* to $a$ if a solution $h \ne 0$ of the [[Jacobi equation|FSD6S7DR8D]] along $y_0$ has $h(a) = h(c) = 0$.

1. (Necessary condition.) If $y_0$ is a [[weak local minimiser|WMPCR723MJ]] of $J$, then no point of the open interval $(a, b)$ is conjugate to $a$.
2. (Sufficient condition.) If no point of $(a, b]$ is conjugate to $a$, then $y_0$ is a strict weak local minimiser of $J$.

## Remarks

Write the [[second variation|4C3BANJ8AH]] as $\int_a^b (P h'^2 + Q h^2) \, dx$, as in the Jacobi equation.
Without conjugate points in $(a, b]$, there is a Jacobi solution $u > 0$ on all of $[a, b]$.
Then $w = -P u' / u$ solves $w' = w^2 / P - Q$.
Adding the zero term $\int_a^b (w h^2)' \, dx$ turns the second variation into $\int_a^b P (h' + w h / P)^2 \, dx$, which is positive for $h \ne 0$.
For the necessary part, a Jacobi solution on $[a, c]$ extended by $0$ gives zero second variation, and smoothing its corner makes the second variation negative.

Example: for $J(y) = \int_0^b (y'^2 - y^2) \, dx$ and $y_0 = 0$, the Jacobi equation is $h'' + h = 0$.
The solution with $h(0) = 0$ is $h = \sin x$, so the first conjugate point is $\pi$.
For $b < \pi$, the extremal $y_0 = 0$ is a strict weak local minimiser.
For $b > \pi$, it is not: with $\varphi(x) = \sin(\pi x / b)$ one gets $J(t\varphi) = \frac{t^2 b}{2}(\pi^2 / b^2 - 1) < 0$ for $t \ne 0$.

The strengthened Legendre condition is needed in both parts.
Conjugate points here generalise [[conjugate points|ZPXE5VSZ35]] along geodesics in Riemannian geometry.

## Explanations

### L1 human

An extremal is a curve that passes the first-derivative test for an integral functional.
Nearby extremals that start at the same point may come back and cross the extremal again; such a crossing point is a conjugate point.
The Jacobi condition says that an extremal can only be a minimiser if this does not happen before its end.
A great circle on a sphere is a good picture: it is the shortest path only until it reaches the antipodal point.

### L2 human

The Legendre condition controls the second variation for test functions that oscillate fast.
The Jacobi condition controls it for all test functions, and it reduces this question to the zeros of one solution of a linear ODE.
Together with the Euler–Lagrange equation and the strengthened Legendre condition, it gives the classical sufficient condition for a weak minimum.
For a strong minimum one adds the Weierstrass condition on the excess function, and the proof embeds the extremal in a field of extremals.
</document>

<document source="https://tamnd.github.io/math2/node/ZS29A7QQC2.md" tag="ZS29A7QQC2" kind="statement">
# Support of a convolution

## Statement

Let $n \ge 1$, and let $f, g \colon \R^n \to \C$ be continuous functions with compact supports $\operatorname{supp} f$ and $\operatorname{supp} g$.
Here the support of a function is the closure of the set where it is not $0$.
Then the [[convolution|2Z29JN3VES]] $f * g$ is continuous, and
$$\operatorname{supp}(f * g) \subseteq \operatorname{supp} f + \operatorname{supp} g = \set{a + b : a \in \operatorname{supp} f, \ b \in \operatorname{supp} g}.$$
In particular $f * g$ has compact support.

## Remarks

If $(f * g)(x) = \int f(y) g(x - y) \, dy \ne 0$, then $f(y) g(x - y) \ne 0$ for some $y$.
So $y \in \operatorname{supp} f$ and $x - y \in \operatorname{supp} g$, and $x = y + (x - y)$ lies in the [[sumset|XNBN31HQWB]] $\operatorname{supp} f + \operatorname{supp} g$.
This sumset is the image of the [[compact set|0R6EWFKRCF]] $\operatorname{supp} f \times \operatorname{supp} g$ under the continuous map $(a, b) \mapsto a + b$, so it is compact and hence closed, and it contains the closure of $\set{f * g \ne 0}$.

Example: if $f$ and $g$ are continuous, positive on $(0, 1)$ and zero elsewhere, then $f * g$ is positive exactly on $(0, 2)$, and $\operatorname{supp}(f * g) = [0, 2] = [0, 1] + [0, 1]$.

The inclusion can be strict, because of cancellation in the integral.
Let $\varphi$ be continuous, positive on $(0, 1)$ and zero elsewhere, and put $\psi = \varphi * \varphi$, supported on $[0, 2]$.
For $f(x) = \varphi(x) - \varphi(x - 2)$ and $g(x) = \varphi(x) + \varphi(x - 2)$ the cross terms cancel and $f * g(x) = \psi(x) - \psi(x - 4)$.
So $\operatorname{supp}(f * g) = [0, 2] \cup [4, 6]$, while $\operatorname{supp} f + \operatorname{supp} g = [0, 6]$.
For integrable functions the same statement holds with essential supports, in the form $\operatorname{supp}(f * g) \subseteq \overline{\operatorname{supp} f + \operatorname{supp} g}$.

## Explanations

### L1 human

If $f$ lives on a set $A$ and $g$ lives on a set $B$, then $f * g$ lives on the set of all sums $a + b$.
For instance, two bumps on $[0, 1]$ give a bump on $[0, 2]$.
So convolution can spread a function out, but only by a controlled amount.

### L2 human

This is why convolution with a [[mollifier|CQMK4YPE1J]] supported in a ball of radius $\varepsilon$ moves the support of a function by at most $\varepsilon$.
Together with [[smoothing by convolution|4P7KPA3Y7R]] it shows that smooth functions with compact support are dense in $L^p$ for $1 \le p < \infty$.
On the line, the [[Titchmarsh convolution theorem|WGY75H4VWW]] gives an equality for the convex hulls of the supports.
</document>

<document source="https://tamnd.github.io/math2/node/ZS2WYH25R0.md" tag="ZS2WYH25R0" kind="definition">
# Ricci flow

## Definition

Let $M$ be a smooth manifold and let $I \subseteq \R$ be an interval.
A *Ricci flow* on $M$ is a family $(g(t))_{t \in I}$ of [[Riemannian metrics|Q61VFHCH2J]] on $M$ with these properties.
The metric $g(t)$ depends smoothly on $t$, and
$$\frac{\partial g}{\partial t}(t) = -2 \operatorname{Ric}\big(g(t)\big) \quad \text{for all } t \in I,$$
where $\operatorname{Ric}(g)$ is the [[Ricci curvature|4GTNM0WHDH]] tensor of $g$.

## Remarks

Example: let $g_0$ be the round metric of the unit sphere $S^n$ with $n \ge 2$, so $\operatorname{Ric}(g_0) = (n - 1) g_0$.
Let $g(t) = (1 - 2(n - 1)t)\, g_0$ for $t < \frac{1}{2(n-1)}$.
The Ricci tensor does not change when the metric is multiplied by a positive constant, so $\operatorname{Ric}(g(t)) = (n - 1) g_0$.
Thus $\partial_t g = -2(n-1) g_0 = -2\operatorname{Ric}(g(t))$, and $g$ is a Ricci flow that shrinks the sphere to a point at $t = \frac{1}{2(n-1)}$.
Non-example: $g(t) = (1 + 2(n - 1)t)\, g_0$ has $\partial_t g = +2\operatorname{Ric}(g(t))$, which has the wrong sign.
A flat metric, such as that of a flat torus, gives a constant Ricci flow $g(t) = g_0$.
In local coordinates the equation reads $\partial_t g_{ij} = -2 R_{ij}$.
</document>

<document source="https://tamnd.github.io/math2/node/ZS5TMX4G8P.md" tag="ZS5TMX4G8P" kind="definition">
# Horn clause

## Definition

A *literal* is an atomic formula or the negation of an atomic formula; it is *positive* in the first case and *negative* in the second.
A *clause* is a finite disjunction $L_1 \vee \cdots \vee L_k$ of literals, as in a [[conjunctive normal form|0B3D4C8YF2]].
A *Horn clause* is a clause in which at most one literal is positive.

## Remarks

A Horn clause with exactly one positive literal is a *definite clause*; one with no positive literal is a *goal clause*.
A definite clause $\neg p_1 \vee \cdots \vee \neg p_n \vee q$ is equivalent to the implication $p_1 \wedge \cdots \wedge p_n \rightarrow q$; for $n = 0$ it is the fact $q$.

Example: $\neg p \vee \neg q \vee r$ is a Horn clause, since only $r$ is positive.

Non-example: $p \vee q$ is not a Horn clause, since it has two positive literals.
No conjunction of Horn clauses is equivalent to it.
The satisfying assignments of a conjunction of Horn clauses are closed under the coordinatewise "and".
But $(1, 0)$ and $(0, 1)$ satisfy $p \vee q$, while $(0, 0)$ does not.

In first-order logic, the atoms may be any atomic formulas, and the clause is read as universally quantified over its variables.
</document>

<document source="https://tamnd.github.io/math2/node/ZS7HP7SGEP.md" tag="ZS7HP7SGEP" kind="definition">
# Inverse Fourier transform

## Definition

Let $n \ge 1$ and let $g \colon \R^n \to \C$ be a Lebesgue [[integrable function|X7GTF3D98F]].
The *inverse Fourier transform* of $g$ is the function $\check g \colon \R^n \to \C$ given by the [[Lebesgue integral|DWJF7ABPR1]]
$$\check g(x) = \int_{\R^n} g(\xi) \, e^{2\pi i \, x \cdot \xi} \, d\xi, \qquad x \in \R^n,$$
where $x \cdot \xi = x_1 \xi_1 + \cdots + x_n \xi_n$.
It is also written $\mathcal{F}^{-1} g$.

## Remarks

This matches the convention $\hat f(\xi) = \int f(x) e^{-2\pi i x \cdot \xi} \, dx$ for the [[Fourier transform|DPVM34ADZV]].
So $\check g(x) = \hat g(-x)$, and $\check g$ is continuous, bounded by $\int \abs{g}$, and tends to $0$ at infinity.
With the convention $\hat f(\xi) = \int f(x) e^{-i x \cdot \xi} \, dx$ the inverse transform carries the factor $(2\pi)^{-n}$.

Example: for $n = 1$ and $g = \mathbf{1}_{[-1/2, 1/2]}$,
$$\check g(x) = \int_{-1/2}^{1/2} e^{2\pi i x \xi} \, d\xi = \frac{\sin(\pi x)}{\pi x} \quad (x \ne 0), \qquad \check g(0) = 1.$$

Non-example: the constant function $g = 1$ on $\R$ is not integrable, and the integral $\int_\R e^{2\pi i x \xi} \, d\xi$ diverges for every $x$.
Its inverse Fourier transform exists only in the sense of tempered distributions, where it is the Dirac delta at $0$.

The name is justified by the [[Fourier inversion theorem|T4Z8YQZKZV]]: if $f$ and $\hat f$ are integrable, then $(\hat f)^\vee = f$ almost everywhere.
</document>

<document source="https://tamnd.github.io/math2/node/ZSA992C21V.md" tag="ZSA992C21V" kind="object">
# Johnson solid

## Description

A *Johnson solid* is a strictly convex polyhedron whose faces are all [[regular polygons|01KJJ9A400]].
It must not be a [[Platonic solid|017E5PRS1Q]], an Archimedean solid, a prism or an antiprism.
Strictly convex means that no two adjacent faces lie in a common plane.
There are exactly $92$ Johnson solids.

Example: the square pyramid with a unit square base and four equilateral triangles is a Johnson solid.
The apex lies above the centre of the base at height $h$ with $h^2 + (\sqrt2/2)^2 = 1$, so $h = \sqrt2/2$, and all eight edges have length $1$.
Its faces are regular, but its apex lies on $4$ edges while each base vertex lies on $3$.
So it is not vertex-transitive, hence not Platonic or Archimedean, and it is not a prism or an antiprism either.

## Remarks

The excluded families are the convex [[uniform polyhedra|BZXHGNGQAX]], whose symmetries act transitively on vertices.
So the Johnson solids are exactly the strictly convex polyhedra with regular faces that are not vertex-transitive.
Norman Johnson published the list of $92$ in 1966, and Victor Zalgaller proved that the list is complete.
Other examples are the pentagonal pyramid, the triangular cupola and the elongated square bipyramid.

## Explanations

### L1 human

Take regular polygons, such as equilateral triangles, squares and regular pentagons, and glue them into a convex solid.
If every corner looks the same, the result is a Platonic solid, an Archimedean solid, a prism or an antiprism.
The Johnson solids are all the other ones, where some corners differ, and there are exactly $92$ of them.
A pyramid with a square base and four equilateral sides is the simplest example.
</document>

<document source="https://tamnd.github.io/math2/node/ZSAMVFPWN6.md" tag="ZSAMVFPWN6" kind="example">
# Conway knot

## Example

The *Conway knot* is the prime [[knot|QYEAZTYGPT]] with $11$ crossings listed as $11n34$ in the knot tables, named after John Horton Conway.
It is obtained from the Kinoshita–Terasaka knot, another $11$-crossing knot, by a mutation.
A mutation cuts out a ball that meets the knot in two arcs, rotates the ball by a half turn, and glues it back.
The result is again a knot, but in general not an equivalent one.

## Remarks

Mutation does not change many [[knot invariants|GZQ5ZE3X6G]].
The Conway knot and the Kinoshita–Terasaka knot have the same [[Jones polynomial|4SPC17FPXE]], the same HOMFLY polynomial, and both have [[Alexander polynomial|JKA25KDPQP]] equal to $1$, like the [[unknot|1TQ06R4Q66]].
They are still different knots: the Conway knot has genus $3$, while the Kinoshita–Terasaka knot has genus $2$, as shown by David Gabai.
Since its Alexander polynomial is $1$, the Conway knot is topologically [[slice|P68G28E4AD]], by a theorem of Michael Freedman.
The Kinoshita–Terasaka knot is smoothly slice.
Lisa Piccirillo proved in 2020 that the Conway knot is not smoothly slice, which answered a question that had been open for about fifty years.

## Explanations

### L1 human

The Conway knot is a knot with eleven crossings that fools many of the standard tests.
Its polynomial invariants agree with those of a close relative, and its Alexander polynomial even agrees with that of the unknot.
It became famous when a graduate student, Lisa Piccirillo, settled whether it bounds a smooth disk in four-dimensional space.
</document>

<document source="https://tamnd.github.io/math2/node/ZSM083M97C.md" tag="ZSM083M97C" kind="concept">
# Hodge structure

## Summary

A Hodge structure of weight $n$ is a lattice, or a rational vector space, with extra data on its complexification.
The complexification splits into pieces $H^{p,q}$ with $p + q = n$, and complex conjugation exchanges $H^{p,q}$ and $H^{q,p}$.
It abstracts the structure that the [[Hodge decomposition|HABCJ4Y0SR]] puts on the cohomology of a compact Kähler manifold.
Hodge structures form an abelian category with tensor products.
They are the linear algebra data behind period maps, variations of Hodge structure and the [[Hodge conjecture|XAEEDWV3PM]].

## Explanations

### L1 human

A Hodge structure is a lattice $H_\Z$ together with a way to split the complex vector space $H_\Z \otimes \C$ into pieces, labelled by pairs of integers.
The pieces must be arranged symmetrically under complex conjugation.
The first example comes from an elliptic curve $E = \C / (\Z + \tau \Z)$, with $\Im \tau > 0$.
Its first cohomology $H^1(E, \Z)$ is a rank $2$ lattice, and over $\C$ it splits into the line of $dz$ and the line of $d\bar z$, a Hodge structure of weight $1$.
The position of the line of $dz$ relative to the lattice recovers $\tau$, and so the curve.
</document>

<document source="https://tamnd.github.io/math2/node/ZSQSPGBN88.md" tag="ZSQSPGBN88" kind="concept">
# Method of images

## Summary

The method of images builds the [[Green's function|M36SD56MWQ]] of a domain from the potential of a point charge by adding fictitious charges outside the domain.
The image charges are chosen so that the total potential is zero on the boundary.
It gives explicit Green's functions for half-spaces, balls, wedges with special angles and strips.
In physics it solves electrostatic problems with grounded conductors.

## Explanations

### L1 human

A point charge near a grounded flat metal plate behaves as if the plate were replaced by an opposite charge at the mirror point behind it.
On the plate the two potentials cancel, so the plate condition holds automatically.
For the half-space $x_3 > 0$ with a unit charge at $x$, the image is a charge $-1$ at $(x_1, x_2, -x_3)$.
For a sphere, the image sits at the inverse point and has a rescaled strength.
</document>

<document source="https://tamnd.github.io/math2/node/ZSTEZ3PRT9.md" tag="ZSTEZ3PRT9" kind="concept">
# Universal class

## Summary

A universal class is a class of structures axiomatized by universal sentences, those of the form "for all $x_1, \ldots, x_n$, a quantifier-free condition holds".
Universal sentences are preserved by [[substructures|676H7EGTKJ]], so every universal class is closed under substructures.
By the [[Łoś–Tarski preservation theorem|9RN0KYXNT1]], an [[elementary class|MEC3ZMC4F1]] is universal exactly when it has this closure property.
Varieties, quasivarieties and universal Horn classes are universal classes.

## Explanations

### L1 human

A universal axiom only checks finite patterns of elements, one tuple at a time.
If a structure passes every such check, then so does every part of it, since a part has fewer tuples to check.
Simple graphs form a universal class: "no vertex is adjacent to itself" and "adjacency is symmetric" are universal.
Fields do not: the axiom "every nonzero element has an inverse" needs an "exists", and indeed the subring $\Z$ of the field $\Q$ is not a field.
</document>

<document source="https://tamnd.github.io/math2/node/ZSVF54TCYP.md" tag="ZSVF54TCYP" kind="concept">
# Maximal subfield

## Summary

Let $D$ be a [[division ring|0EA5BTA0CX]] that is finite-dimensional over its center, a field $K$.
A maximal subfield of $D$ is a subfield that contains $K$ and is not contained in a larger subfield.
Its degree over $K$ always equals the [[degree|NNXSP5KKAC]] of $D$, and it [[splits|NVD7R0T835]] $D$.
So the study of a division algebra reduces in part to the study of the fields it contains.

## Explanations

### L1 human

A division algebra contains many commutative pieces, and some of them are fields.
A maximal subfield is one of these fields that cannot be made larger inside the algebra.
In Hamilton's quaternions over $\R$, the elements $x + yi$ form a copy of $\C$, which is a maximal subfield.
There are infinitely many such copies, one for each line through $0$ in the space spanned by $i, j, k$.
</document>

<document source="https://tamnd.github.io/math2/node/ZSW87QS3TQ.md" tag="ZSW87QS3TQ" kind="definition">
# Lie derivative

## Definition

Let $M$ be a smooth manifold, let $X$ be a smooth vector field on $M$, and let $\varphi_t$ be the local [[flow|06738GGWWK]] of $X$.
Let $T$ be a smooth [[tensor field|WX302T93TW]] on $M$, and let $\varphi_t^* T$ be its pullback by the local diffeomorphism $\varphi_t$.
The *Lie derivative* of $T$ along $X$ is the tensor field $\mathcal{L}_X T$ given at each $p \in M$ by
$$(\mathcal{L}_X T)_p = \frac{d}{dt}\Big|_{t = 0} (\varphi_t^* T)_p.$$

## Remarks

For a smooth function $f$, $\mathcal{L}_X f = X f$, the derivative of $f$ in the direction of $X$.
For a vector field $Y$, $\mathcal{L}_X Y = [X, Y]$, the [[Lie bracket|68FGQEBD6J]].
For a differential form $\omega$, Cartan's formula gives $\mathcal{L}_X \omega = d(\iota_X \omega) + \iota_X d\omega$.
Example: on $\R^2$ let $X = -y \, \partial_x + x \, \partial_y$ and $f = x^2 + y^2$; then $\mathcal{L}_X f = -y \cdot 2x + x \cdot 2y = 0$, as $f$ is invariant under the rotations $\varphi_t$.
Non-example of tensoriality: $\mathcal{L}_X Y$ is not $C^\infty(M)$-linear in $X$, since $\mathcal{L}_{fX} Y = f [X, Y] - (Y f) X$.
So $(\mathcal{L}_X T)_p$ depends on $X$ near $p$, not only on $X_p$; this is the difference with a covariant derivative.
The Lie derivative is a derivation: $\mathcal{L}_X (S \otimes T) = \mathcal{L}_X S \otimes T + S \otimes \mathcal{L}_X T$, and it commutes with contractions.
</document>

<document source="https://tamnd.github.io/math2/node/ZSYCA4EGEZ.md" tag="ZSYCA4EGEZ" kind="concept">
# Lefschetz number

## Summary

Let $f$ be a self-map of a finite CW complex or compact polyhedron.
Its Lefschetz number is the alternating sum of the traces of the maps that $f$ induces on rational homology.
It depends only on the homotopy class of $f$, and for the identity map it is the Euler characteristic.
The [[Lefschetz fixed-point theorem|G44NQXCWT9]] says that if it is not zero, then $f$ has a fixed point.

## Explanations

### L1 human

A map from a space to itself moves the holes of the space around.
For each dimension, the trace measures how much of the homology the map sends back onto itself.
The Lefschetz number adds these traces with alternating signs.
For a constant map on a connected space, only dimension $0$ counts and the Lefschetz number is $1$, which matches the single fixed point.
</document>

<document source="https://tamnd.github.io/math2/node/ZT01G7BTSV.md" tag="ZT01G7BTSV" kind="concept">
# L-infinity space

## Summary

The space $L^\infty(\mu)$ consists of the essentially bounded measurable functions on a measure space, up to equality almost everywhere.
Its norm is the [[essential supremum|WPC2XRMB9Y]] of the absolute value.
It is the case $p = \infty$ of the [[Lp spaces|SASP14MA1Z]] and is a Banach space.
For a $\sigma$-finite measure it is the dual of $L^1(\mu)$.

## Explanations

### L1 human

A function lies in $L^\infty$ when it is bounded once one ignores a set of measure zero.
The norm is the smallest bound that holds outside such a set.
For example, the function on $\R$ equal to $\sin x$ at irrational $x$ and to $x$ at rational $x$ is unbounded, but its $L^\infty$ norm is $1$.
With the counting measure on $\N$ one gets the space $\ell^\infty$ of bounded sequences.
</document>

<document source="https://tamnd.github.io/math2/node/ZT47BK9Y6K.md" tag="ZT47BK9Y6K" kind="concept">
# Complemented lattice

## Summary

A complemented lattice is a [[bounded lattice|72V1Z82M5F]] in which every element $a$ has a complement: an element $b$ with $a \wedge b = 0$ and $a \vee b = 1$.
The lattice of subspaces of a vector space and every [[Boolean algebra|0XADJKM699]] are complemented.
Complements need not be unique; in a distributive lattice they are, and a complemented distributive lattice is the same as a Boolean algebra.
Complemented modular lattices are the setting of projective geometry and of von Neumann's [[continuous geometry|5CET1KWWWH]].

## Explanations

### L1 human

A complement of $a$ is something that has nothing in common with $a$ but fills up everything together with $a$.
For a line through the origin in the plane, any other line through the origin is a complement.
So a complement exists, but there are many choices.
In the lattice of subsets of a set, the only complement of a subset is the set of points outside it.
</document>

<document source="https://tamnd.github.io/math2/node/ZT6F4C2YHK.md" tag="ZT6F4C2YHK" kind="concept">
# Absolute continuity

## Summary

A real function on a closed interval is absolutely continuous when small total length of input intervals forces small total change of the output.
It is stronger than [[uniform continuity|0Q4DR37DVE]] and stronger than [[bounded variation|A8251BHCMP]].
The absolutely continuous functions are exactly the indefinite [[Lebesgue integrals|DWJF7ABPR1]].
So they are the functions for which the [[fundamental theorem of calculus|RX7JSHMP8Z]] holds in Lebesgue's form.
The analogous notion for measures is the [[absolutely continuous measure|P0P21KDV6E]].

## Explanations

### L1 human

Uniform continuity says that one short input interval gives a small change of the output.
Absolute continuity asks the same for many short intervals at once, as long as their total length is small.
Every [[Lipschitz function|0QNVQ4VD9T]], such as $\sin x$, is absolutely continuous.
The [[Cantor function|NPBYYK63FE]] is continuous and increasing but not absolutely continuous: it rises from $0$ to $1$ on a set of total length $0$.
</document>

<document source="https://tamnd.github.io/math2/node/ZT97FC11VA.md" tag="ZT97FC11VA" kind="statement">
# Morley's trisector theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane.
Trisect each of its interior [[angles|0GJAJC6TNZ]], that is, divide it into three equal angles by two rays from the vertex.
For each side, the two trisectors nearest to that side, one from each endpoint of the side, meet at a point.
Then the three points so obtained are the vertices of an equilateral triangle.

## Remarks

Let $R$ be the circumradius and write $A$, $B$, $C$ also for the angles.
The side of the Morley triangle is $8R \sin \frac{A}{3} \sin \frac{B}{3} \sin \frac{C}{3}$, which is symmetric in the angles; this length is one route to a trigonometric proof.
A well-known proof runs backwards: start from an equilateral triangle, attach suitable triangles with angles built from $\frac{A}{3}$, $\frac{B}{3}$, $\frac{C}{3}$, and check that the result is similar to $ABC$.

Example: in an equilateral triangle all angles are $\pi/3$, and the figure is symmetric under rotation by $2\pi/3$ about the centre, so the three points form an equilateral triangle.
The theorem says that this equilateral pattern survives for every triangle, with no symmetry.

## Explanations

### L1 human

Split each corner of any triangle into three equal parts.
Near each side, two of the dividing lines meet.
The three meeting points always form a perfectly equilateral triangle, whatever the shape of the triangle you started with.
The result was found by Frank Morley in 1899.

### L2 human

The theorem is surprising because angle trisection is not a ruler-and-compass operation, see [[trisecting the angle|00J3DCCEK5]], so classical synthetic geometry did not study trisectors.
Proofs use trigonometry or a reverse construction, and Alain Connes gave an algebraic proof using the affine group of a field.
It is a standard example of a deep-looking fact of triangle geometry with an elementary proof.
</document>

<document source="https://tamnd.github.io/math2/node/ZT9A6JK8QF.md" tag="ZT9A6JK8QF" kind="concept">
# Projective space

## Summary

The projective space $\mathbb{P}^n$ over a field $k$ is the set of lines through the origin in $k^{n+1}$.
A point is written in homogeneous coordinates $[x_0 : \dots : x_n]$, defined up to a common nonzero factor.
Projective space adds points at infinity to affine space.
Then, for example, any two distinct lines in the projective plane meet in exactly one point.
It is the ambient space of projective varieties, the compact objects of algebraic geometry.

## Explanations

### L1 human

Stand at the origin of $k^{n+1}$ and look out: each direction you can look in is one point of $\mathbb{P}^n$.
The points $(1, 2, 3)$ and $(2, 4, 6)$ lie on the same line through the origin, so they give the same point $[1 : 2 : 3]$ of the projective plane.
The points with $x_0 \ne 0$ can be scaled to $[1 : a_1 : \dots : a_n]$ and form a copy of $k^n$.
The remaining points, with $x_0 = 0$, are the points at infinity, where parallel lines of $k^n$ meet.
</document>

<document source="https://tamnd.github.io/math2/node/ZT9BSVAFEX.md" tag="ZT9BSVAFEX" kind="definition">
# Pushforward measure

## Definition

Let $(X, \Sigma)$ and $(Y, \mathcal{T})$ be [[measurable spaces|N4KDT83V3S]], let $\mu$ be a [[measure|PR11BZQQPG]] on $(X, \Sigma)$, and let $f \colon X \to Y$ be a [[measurable function|YYNCJSR0PJ]].
The *pushforward* of $\mu$ along $f$ is the map $f_* \mu \colon \mathcal{T} \to [0, \infty]$ given by
$$(f_* \mu)(B) = \mu\big(f^{-1}(B)\big) \quad \text{for } B \in \mathcal{T}.$$

## Remarks

The set $f^{-1}(B)$ lies in $\Sigma$ because $f$ is measurable, so the formula makes sense.
The map $f_* \mu$ is a measure: $f^{-1}(\varnothing) = \varnothing$, and the preimage of a disjoint union of sets $B_n$ is the disjoint union of the sets $f^{-1}(B_n)$.
Other notations are $\mu \circ f^{-1}$ and $f_\# \mu$.
Example: let $\mu$ be the measure on $\R$ with $\mu(A) = $ number of points of $A \cap \set{-1, 1}$, and $f(x) = x^2$.
Then $f_* \mu$ gives mass $2$ to the point $1$, since $f^{-1}(\set{1}) = \set{-1, 1}$.
Non-example: if $f$ is not measurable, some $f^{-1}(B)$ is outside $\Sigma$, and the formula does not define $f_* \mu$ on all of $\mathcal{T}$.
For a measurable $g \colon Y \to [0, \infty]$ the change of variables formula holds:
$$\int_Y g \, d(f_* \mu) = \int_X (g \circ f) \, d\mu.$$
</document>

<document source="https://tamnd.github.io/math2/node/ZTBC6G7DGD.md" tag="ZTBC6G7DGD" kind="statement">
# Faber's theorem on interpolation

## Statement

Let $a < b$.
For each $n \ge 0$, let $x_{n,0} < x_{n,1} < \cdots < x_{n,n}$ be $n + 1$ points of $[a, b]$.
For a [[continuous function|0X3WGX95FE]] $f$ on $[a, b]$, let $p_n(f)$ be the [[Lagrange interpolation|237KR6FRCK]] polynomial of degree at most $n$ that agrees with $f$ at $x_{n,0}, \dots, x_{n,n}$.
Then there is a continuous function $f$ on $[a, b]$ such that $p_n(f)$ does not [[converge uniformly|08VZ57PT26]] to $f$ on $[a, b]$.

## Remarks

Proof idea: the norm of the interpolation map $f \mapsto p_n(f)$ on $C[a, b]$ is the [[Lebesgue constant|7V7KWNK9HN]] $\Lambda_n$ of the $n$-th row of nodes.
For every choice of nodes, $\Lambda_n \ge c \log n$ for some constant $c > 0$, so $\Lambda_n \to \infty$.
By the uniform boundedness principle, $\sup_n \norm{p_n(f)}_\infty = \infty$ for some continuous $f$, and for this $f$ the sequence $p_n(f)$ cannot converge uniformly.

Example: for Chebyshev nodes $\Lambda_n$ grows only like $\frac{2}{\pi} \log n$, which is slow but still unbounded.
So even interpolation at Chebyshev nodes fails for some continuous function, although it converges for every Lipschitz function.

The theorem concerns one fixed array of nodes for all functions.
For each single continuous $f$ there is an array of nodes for which $p_n(f) \to f$ uniformly, for example nodes where $f$ meets its best approximations.

## Explanations

### L1 human

No fixed recipe for choosing interpolation points works for every continuous function.
Whatever points are chosen, some continuous function makes the interpolating polynomials fail to converge to it.
This contrasts with the Weierstrass approximation theorem, which says that good polynomial approximations always exist.

### L2 human

Faber's theorem separates existence of good approximations from their construction by interpolation.
It is a typical application of the uniform boundedness principle with norms that grow like a logarithm, as for divergent Fourier series of continuous functions.
In practice it is mild: for Chebyshev nodes, Lebesgue's lemma gives convergence as soon as $\omega(f, 1/n) \log n \to 0$.
</document>

<document source="https://tamnd.github.io/math2/node/ZTMK6QNKK3.md" tag="ZTMK6QNKK3" kind="definition">
# Input-to-state stability

## Definition

Let $f \colon \R^n \times \R^m \to \R^n$ be locally [[Lipschitz|0AW1E6YTJY]] with $f(0, 0) = 0$.
Consider the control system $\dot x(t) = f(x(t), u(t))$, where the inputs $u \colon [0, \infty) \to \R^m$ are measurable and essentially bounded, with norm $\norm{u}_\infty = \operatorname{ess\,sup}_{t \ge 0} \abs{u(t)}$.
A function $\gamma \colon [0, \infty) \to [0, \infty)$ is of *class $\mathcal{K}$* if it is continuous, strictly increasing and $\gamma(0) = 0$.
A function $\beta \colon [0, \infty) \times [0, \infty) \to [0, \infty)$ is of *class $\mathcal{KL}$* if two conditions hold.
For each $t$, $\beta(\cdot, t)$ is of class $\mathcal{K}$; for each $r$, $\beta(r, \cdot)$ is nonincreasing with limit $0$ at infinity.
The system is *input-to-state stable* (ISS) if there are $\beta$ of class $\mathcal{KL}$ and $\gamma$ of class $\mathcal{K}$ with this property.
For every $x_0 \in \R^n$ and every input $u$, the solution with $x(0) = x_0$ exists for all $t \ge 0$ and satisfies
$$\abs{x(t)} \le \beta(\abs{x_0}, t) + \gamma(\norm{u}_\infty) \quad \text{for all } t \ge 0.$$

## Remarks

Example: $\dot x = -x + u$ on $\R$.
By variation of constants, $x(t) = e^{-t} x_0 + \int_0^t e^{-(t - s)} u(s) \, ds$.
Hence $\abs{x(t)} \le e^{-t} \abs{x_0} + \norm{u}_\infty$, so the system is ISS with $\beta(r, t) = r e^{-t}$ and $\gamma(r) = r$.

Non-example: $\dot x = -x + x u$ on $\R$.
For $u = 0$ the origin is globally asymptotically stable.
But for the constant input $u = 2$ the equation is $\dot x = x$, so $x(t) = e^t x_0$ is unbounded, and no estimate of the ISS form can hold.

Taking $u = 0$ shows that an ISS system has a globally asymptotically stable equilibrium at $0$.
An equivalent characterisation uses an ISS Lyapunov function $V$: a smooth, positive definite, proper function with $\nabla V(x) \cdot f(x, u) \le -\alpha(\abs{x}) + \sigma(\abs{u})$ for some class $\mathcal{K}$ functions $\alpha$ (unbounded) and $\sigma$.

## Explanations

### L1 human

A system is ISS when bounded inputs give bounded states, and when the effect of the initial state fades away with time.
The function $\beta$ measures how fast the start is forgotten, and $\gamma$ measures how strongly the input pushes the state.
</document>

<document source="https://tamnd.github.io/math2/node/ZTTPPXGB30.md" tag="ZTTPPXGB30" kind="definition">
# Definable set

## Definition

Let $L$ be a signature, let $\mathcal{M}$ be an $L$-[[structure|983GZBQYT4]] with domain $M$, and let $A \subseteq M$.
A set $X \subseteq M^n$ is definable with parameters from $A$ when there are an $L$-[[formula|HYJ10AF6RT]] $\varphi(x_1, \dots, x_n, y_1, \dots, y_m)$ and $b_1, \dots, b_m \in A$ such that
$$X = \set{(a_1, \dots, a_n) \in M^n : \mathcal{M} \models \varphi(a_1, \dots, a_n, b_1, \dots, b_m)}.$$
The set $X$ is definable when it is definable with parameters from $M$.
It is definable without parameters, or $\emptyset$-definable, when it is definable with parameters from $\emptyset$.

## Remarks

Example: in the field $(\R, +, \cdot, 0, 1)$, the set $[0, \infty)$ is definable without parameters by $\varphi(x) = \exists y \, (y \cdot y = x)$.
Indeed a real number is a square of a real number exactly when it is nonnegative.

Non-example: in $(\R, <)$, the set $\set{0}$ is not definable without parameters.
The map $\sigma(x) = x + 1$ is an automorphism of $(\R, <)$, and an automorphism maps each $\emptyset$-definable set onto itself; but $\sigma(\set{0}) = \set{1}$.
With the parameter $0$, the formula $x = y_1$ defines $\set{0}$.

The definable subsets of $M^n$ are closed under finite unions, finite intersections and complements, and projections of definable sets are definable.

## Explanations

### L1 human

The definable sets are the sets that the language of the structure can describe.
Allowing parameters means that the description may name finitely many fixed elements.
</document>

<document source="https://tamnd.github.io/math2/node/ZTTZVZWMK9.md" tag="ZTTZVZWMK9" kind="conjecture">
# Zariski's multiplicity conjecture

## Statement

Let $f, g \in \C\{z_0, \dots, z_n\}$ be [[convergent power series|EVPZ7ZG1HB]] without repeated factors, with $f(0) = g(0) = 0$.
The *multiplicity* $\operatorname{mult}_0 f$ is the least degree of a nonzero homogeneous term in the power series expansion of $f$.
Suppose that there is a [[homeomorphism|0QT00PB5G2]] $\phi$ between neighbourhoods of $0$ in $\C^{n+1}$ with $\phi(0) = 0$.
Suppose that $\phi$ maps the [[germ|VZXF7RNFKS]] of $f^{-1}(0)$ at $0$ onto the germ of $g^{-1}(0)$ at $0$.
Then $\operatorname{mult}_0 f = \operatorname{mult}_0 g$.

## Remarks

Zariski asked this question in his 1971 paper "Some open questions in the theory of singularities".
It is open in general.
It holds for plane curves, $n = 1$, where the embedded topological type determines the Puiseux characteristics of the branches and hence the multiplicity.
The multiplicity of $f$ equals the [[multiplicity of the local ring|JX20MNDDXT]] $\C\{z_0, \dots, z_n\} / (f)$, so the conjecture asks whether this algebraic invariant is determined by the embedded topology.
The Milnor number, by contrast, is known to be a topological invariant in this sense.

## Explanations

### L1 human

The multiplicity of a hypersurface at a point counts how many sheets meet there: a smooth point has multiplicity $1$, two crossing lines have multiplicity $2$.
Zariski asked whether this number can be read off from the shape of the hypersurface near the point, as seen by topology, without using its algebra.
For curves in the plane the answer is yes, but in higher dimensions the question is still unanswered.
</document>

<document source="https://tamnd.github.io/math2/node/ZTWTZ8CSGV.md" tag="ZTWTZ8CSGV" kind="statement">
# Hindman's theorem

## Statement

For a set $S$ of positive integers, let $\mathrm{FS}(S)$ be the set of all sums $\sum_{s \in F} s$ where $F$ is a nonempty finite subset of $S$.
Let the positive integers be coloured with finitely many colours.
Then there is an infinite set $S$ of positive integers such that all elements of $\mathrm{FS}(S)$ have the same colour.

## Remarks

A short proof uses the Stone–Čech compactification $\beta\N$, which is a semigroup under an extension of addition.
It finds an idempotent ultrafilter $p$, that is, $p + p = p$.
The colour class that belongs to $p$ then contains $\mathrm{FS}(S)$, where the elements of $S$ are chosen one by one inside sets that belong to $p$.
A set that contains $\mathrm{FS}(S)$ for some infinite $S$ is called an IP set; the theorem says that in every finite colouring some colour class is an IP set.
Taking two elements $x < y$ of $S$, the numbers $x$, $y$ and $x + y$ have the same colour, which gives [[Schur's theorem|V6XJQXZ7E8]].
The sums must use distinct elements.
With repeats allowed, the sums would include both $x$ and $2x$.
Colour $n$ by the parity of the exponent of $2$ in $n$; then $x$ and $2x$ always get different colours.

## Explanations

### L1 human

Colour the positive integers with a few colours.
Hindman's theorem says you can find infinitely many numbers such that every sum of finitely many different ones among them has the same colour.
So one colour contains a whole infinite family closed under such sums.

### L2 human

Hindman's theorem is an infinite, additive analogue of [[Ramsey's theorem|00TQV3F5S4]] and strengthens Schur's theorem.
Its ultrafilter proof was a starting point of the use of the algebra of $\beta\N$ in [[Ramsey theory|G5TWXV565S]].
IP sets also play a role in topological dynamics and ergodic theory, where recurrence along IP sets is studied.
</document>

<document source="https://tamnd.github.io/math2/node/ZTZ1FRRFM8.md" tag="ZTZ1FRRFM8" kind="statement">
# Mellin transform as a Fourier transform in logarithmic variables

## Statement

Let $f \colon (0, \infty) \to \C$ be measurable and let $s = \sigma + i\tau$ with $\sigma, \tau \in \R$.
Define $h_\sigma \colon \R \to \C$ by $h_\sigma(u) = e^{-\sigma u} f(e^{-u})$.
Then $f$ is [[Mellin convergent|SGZZ4Y0D36]] at $s$ if and only if $h_\sigma$ is [[integrable|X7GTF3D98F]] on $\R$.
In that case
$$\mathcal{M}f(\sigma + i\tau) = \widehat{h_\sigma}\left(\frac{\tau}{2\pi}\right),$$
where $\mathcal{M}f$ is the [[Mellin transform|N5PVGRJE1T]] and $\widehat{h}(\xi) = \int_{\R} h(u) e^{-2\pi i u \xi} \, du$ is the [[Fourier transform|DPVM34ADZV]].

## Remarks

The proof is the substitution $x = e^{-u}$, with $dx = -e^{-u} \, du$, which maps $(0, \infty)$ onto $\R$ with reversed orientation.
It gives $\int_0^\infty x^{\sigma - 1} \abs{f(x)} \, dx = \int_{\R} \abs{h_\sigma(u)} \, du$, and
$$\int_0^\infty x^{s-1} f(x) \, dx = \int_{\R} e^{-su} f(e^{-u}) \, du = \int_{\R} h_\sigma(u) e^{-i\tau u} \, du.$$
Example: $f(x) = e^{-x}$ and $\sigma = 1$ give $h_1(u) = e^{-u} e^{-e^{-u}}$, and the statement says $\widehat{h_1}(\tau/2\pi) = \Gamma(1 + i\tau)$.
Other sign conventions use $x = e^{u}$, which replaces $\tau / 2\pi$ by $-\tau / 2\pi$.
For fixed $\sigma$, the map $f \mapsto h_\sigma$ is a bijection, so every theorem on the Fourier transform on $\R$ has a Mellin version on each vertical line.

## Explanations

### L1 human

On a logarithmic scale, multiplication becomes addition, and the powers $x^{s}$ become exponentials $e^{su}$.
So the Mellin transform, which integrates against powers, becomes the Fourier transform, which integrates against exponentials.
The real part of $s$ only adds a fixed weight $e^{-\sigma u}$ to the function.

### L2 human

This dictionary transports results from Fourier analysis.
The Riemann–Lebesgue lemma gives the decay of $\mathcal{M}f(\sigma + i\tau)$ as $\abs{\tau} \to \infty$.
The Fourier inversion theorem and the Plancherel theorem give the Mellin inversion theorem and the Parseval formula for Mellin transforms.
Varying $\sigma$ as well, the same substitution identifies the Mellin transform with a two-sided Laplace transform of $u \mapsto f(e^{-u})$.
It also identifies the strip of Mellin convergence with the strip of convergence of that Laplace transform.
</document>

<document source="https://tamnd.github.io/math2/node/ZV04WWYE2Y.md" tag="ZV04WWYE2Y" kind="definition">
# Artinian ring

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
The ring $R$ is *Artinian* if the set of its [[ideals|0MYSA6MYNA]], ordered by inclusion, satisfies the descending chain condition.
That is, for every chain $I_1 \supseteq I_2 \supseteq \cdots$ of ideals of $R$, there is an index $k$ with $I_n = I_k$ for all $n \ge k$.

## Remarks

Example: $\Z/n\Z$ for $n \ge 1$ is Artinian, because it has only finitely many ideals, so a decreasing chain takes only finitely many values.
Example: $k[x]/(x^2)$ over a field $k$ is Artinian, because every ideal is a $k$-subspace of a $2$-dimensional space.
Non-example: $\Z$ is not Artinian, because $(2) \supsetneq (4) \supsetneq (8) \supsetneq \cdots$ never stops.
Equivalently, $R$ is Artinian when it is an [[Artinian module|YQNZTQGP1X]] over itself, or when every nonempty set of ideals has a minimal element.
For a noncommutative ring one defines left Artinian and right Artinian rings with left and right ideals.

## Explanations

### L1 human

An Artinian ring is a ring with no endless decreasing chain of ideals.
Fields and finite rings are the simplest examples.
</document>

<document source="https://tamnd.github.io/math2/node/ZV239N9H5B.md" tag="ZV239N9H5B" kind="definition">
# Partially well-ordered set

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
The set $P$ is *partially well-ordered* when for every infinite [[sequence|0G9EKM0E15]] $x_1, x_2, x_3, \ldots$ in $P$ there are indices $i < j$ with $x_i \le x_j$.
More generally, let $S$ be a subset of a preordered set $(X, \le)$.
The subset $S$ is partially well-ordered when every infinite sequence in $S$ has indices $i < j$ with $x_i \le x_j$.

## Remarks

So a partially well-ordered set is a partial order that is a [[well-quasi-ordering|NH25G6GC07]].

Example: $\N^2$ with the product order.
Check: from any sequence $(a_n, b_n)$ extract a subsequence on which $a_n$ is non-decreasing, then a further subsequence on which $b_n$ is non-decreasing; two terms of it give $i < j$ with $x_i \le x_j$.

Non-example: $\N$ with the trivial order, where $x \le y$ only when $x = y$, is a partial order but not partially well-ordered: the sequence $0, 1, 2, \ldots$ has no $i < j$ with $x_i \le x_j$.

Every well-ordered set is partially well-ordered.
In Mathlib the subset form is `Set.IsPWO`.
</document>

<document source="https://tamnd.github.io/math2/node/ZV38HA5PAQ.md" tag="ZV38HA5PAQ" kind="concept">
# Stone space

## Summary

A Stone space is a [[compact|0TVJZYB6SH]] [[Hausdorff|0S1FJ8VEX9]] [[totally disconnected|0RJSBR9GW1]] topological space.
Stone spaces are also called Boolean spaces or profinite spaces.
They are exactly the spaces of ultrafilters of Boolean algebras, by [[Stone's representation theorem|S4QJ395PA5]], and [[Stone duality|B9T33D5MJK]] makes the two theories equivalent.
The Cantor set and the profinite groups are standard examples.

## Explanations

### L1 human

A Stone space is compact, so it is "small" in the topological sense, but it is also completely broken into dust: no two points are connected.
Its open sets are generated by sets that are open and closed at the same time.
The Cantor set is the typical picture: it is closed and bounded, yet any two of its points can be split by a gap.
Every finite set with the discrete topology is a Stone space too.
</document>

<document source="https://tamnd.github.io/math2/node/ZV65XMJ9A4.md" tag="ZV65XMJ9A4" kind="concept">
# Kernel density estimation

## Summary

Kernel density estimation estimates an unknown [[probability density|0ZY9DX8FBF]] from a sample by placing a small bump, the kernel, at each observation and averaging the bumps.
The width of the bumps, the bandwidth, controls the trade-off between bias and variance.
It is a basic method of [[nonparametric statistics|VFAJ1EG3X5]] and a smooth alternative to the histogram.

## Explanations

### L1 human

Put a small hill of total area $1/n$ on top of each of the $n$ data points and add the hills.
The result is a smooth curve of total area $1$ that is high where the data are dense.
Narrow hills give a wiggly curve that follows every point; wide hills give a smooth curve that may hide real features.
With the Gaussian bump, the estimate is an average of $n$ normal densities centred at the data points.
</document>

<document source="https://tamnd.github.io/math2/node/ZV7VGE2SQ2.md" tag="ZV7VGE2SQ2" kind="statement">
# Grothendieck spectral sequence

## Statement

Let $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ be [[abelian categories|Y9CZ3MTNF1]], and assume that $\mathcal{A}$ and $\mathcal{B}$ have enough injectives.
Let $G \colon \mathcal{A} \to \mathcal{B}$ and $F \colon \mathcal{B} \to \mathcal{C}$ be left [[exact|S1G61TDW0K]] [[additive functors|KAN07YC9T2]].
Assume that $G$ sends every [[injective object|5QRFTBF4PG]] of $\mathcal{A}$ to an [[$F$-acyclic object|F2CMPBHQ02]] of $\mathcal{B}$.
Then for every object $X$ of $\mathcal{A}$ there is a first quadrant [[spectral sequence|GG8AYV847V]]
$$E_2^{p,q} = (R^p F)(R^q G)(X) \Longrightarrow R^{p+q}(F \circ G)(X),$$
natural in $X$, where $R^n$ denotes the right [[derived functors|NPMC3K6J2F]].

## Remarks

To prove it, take an injective resolution $X \to I^\bullet$ and a Cartan–Eilenberg resolution of the complex $G(I^\bullet)$.
Apply $F$, and compare the two spectral sequences of the resulting double complex.
Because $G(I^n)$ is $F$-acyclic, one of them collapses and computes $R^n(F \circ G)(X)$; the other has the stated $E_2$ page.
Without the acyclicity hypothesis the derived functor of $F \circ G$ is not computed by this double complex, and the statement can fail.
One example is the [[Leray spectral sequence|8BMQ90EJPR]], where $G = f_*$ for a continuous map $f$ and $F$ is the global sections functor.
Another is the [[Lyndon–Hochschild–Serre spectral sequence|VC3QNG6TKR]], where $G$ and $F$ take invariants under a normal subgroup and under the quotient group.
The low degree terms of the spectral sequence give the exact sequence $0 \to R^1 F(G X) \to R^1(F G)(X) \to F(R^1 G(X)) \to R^2 F(G X) \to R^2(F G)(X)$.

## Explanations

### L1 human

If a functor is done in two steps, its derived functors are related to the derived functors of the two steps.
The relation is not a simple formula but a spectral sequence: a table of groups that, after several corrections, gives the answer.
The one condition is that the first step sends the simple building blocks, the injective objects, to objects that the second step treats as simple.

### L2 human

In practice the theorem is used in low degrees, through the five-term exact sequence.
It is also used when the $E_2$ page vanishes outside one row or column, so that $R^n(F \circ G)$ can be read off directly.
In the language of [[derived categories|5ST7NVJFSM]] it says that $R(F \circ G) \cong RF \circ RG$.
</document>

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# Open immersion

## Definition

Let $U$ and $X$ be [[schemes|5A35TYZD02]].
A [[morphism of schemes|JXRKH6P1Z4]] $(f, f^\#) \colon U \to X$ is an *open immersion* if:

1. $f$ is a [[homeomorphism|0QT00PB5G2]] of $U$ onto an [[open subset|02BH3MCBQB]] $V$ of $X$;
2. the induced map $\mathcal{O}_X|_V \to f_* \mathcal{O}_U$ is an isomorphism of sheaves of rings on $V$.

Equivalently, $f$ is an isomorphism of $U$ onto the open subscheme $(V, \mathcal{O}_X|_V)$.

## Remarks

Example: for a ring $A$ and $g \in A$, the morphism $\operatorname{Spec} A_g \to \operatorname{Spec} A$ induced by $A \to A_g$ is an open immersion onto $D(g)$.
On each basic open $D(gh) \subseteq D(g)$ the sheaf map is the isomorphism $A_{gh} \cong (A_g)_h$.
Non-example: for a field $k$, let $f \colon \operatorname{Spec} k \to \operatorname{Spec} k[x]/(x^2)$ come from $k[x]/(x^2) \to k$, $x \mapsto 0$.
Both spaces are one point, so $f$ is a homeomorphism onto an open set and clause 1 holds.
Clause 2 fails: the ring map $k[x]/(x^2) \to k$ is not injective, so it is not an isomorphism.
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# Bilinear interpolation

## Definition

Let $x_1 < x_2$ and $y_1 < y_2$ be real numbers, let $R = [x_1, x_2] \times [y_1, y_2]$, and let $f_{ij} \in \R$ be given values at the corners $(x_i, y_j)$ for $i, j \in \set{1, 2}$.
The *bilinear interpolant* is the unique function $p \colon R \to \R$ of the form $p(x, y) = a + b x + c y + d x y$ with $p(x_i, y_j) = f_{ij}$ for all $i, j$.
In terms of $s = (x - x_1)/(x_2 - x_1)$ and $t = (y - y_1)/(y_2 - y_1)$, it is
$$p(x, y) = (1 - s)(1 - t) f_{11} + s (1 - t) f_{21} + (1 - s) t f_{12} + s t f_{22}.$$
For data on a rectangular grid, *bilinear interpolation* uses this formula on each grid cell.

## Remarks

The formula has the required form and takes the value $f_{ij}$ at each corner, which shows existence.
Uniqueness holds because a function $a + bx + cy + dxy$ vanishing at the four corners has $a = b = c = d = 0$, as one checks on the unit square.
Example: for $f(x, y) = 2 + 3x - y + xy$, the bilinear interpolant on any rectangle is $f$ itself, since $f$ already has the required form.
Non-example: $p(x, y) = x^2$ on the unit square has corner values $0, 1, 0, 1$, but it is not of the form $a + bx + cy + dxy$, so it is not the bilinear interpolant; that interpolant is $x$.
The interpolant is linear along every horizontal and vertical line, but not linear in general: along the diagonal $x = y$ of the unit square, $xy$ restricts to $x^2$.
On a grid, the cellwise interpolants agree on shared edges, so the result is continuous, but its gradient jumps across cell edges in general.

## Explanations

### L1 human

Bilinear interpolation fills in a rectangle from its four corner values by mixing them with weights given by the areas of the opposite sub-rectangles.
It is linear in each variable separately, which is where the name comes from.
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# Montel's theorem

## Statement

Let $U \subseteq \C$ be an open set and let $\mathcal{F}$ be a family of [[holomorphic|0DG0Z72CQD]] functions $U \to \C$.

1. Assume that $\mathcal{F}$ is locally uniformly bounded: for each compact $K \subset U$ there is $M_K$ with $\abs{f(z)} \le M_K$ for all $f \in \mathcal{F}$ and $z \in K$. Then $\mathcal{F}$ is a [[normal family|QWSD2JDRP0]].
2. Assume that $U$ is a [[domain|097KNJFJ7V]] and that there are two distinct complex numbers $a$ and $b$ that no $f \in \mathcal{F}$ takes as a value. Then each sequence in $\mathcal{F}$ has a subsequence that converges uniformly on compact subsets of $U$, either to a holomorphic function or to $\infty$.

## Remarks

Proof idea of part 1: [[Cauchy's estimates|0JJRENWMP6]] bound the derivatives uniformly on each compact set, so the family is equicontinuous there.
The [[Arzelà–Ascoli theorem|0381S8Z5P0]] on an exhausting sequence of compact sets, with a diagonal argument, gives a convergent subsequence.
Part 2, the fundamental normality test, is deeper; one proof uses the [[modular lambda function|4Z1YXT957Z]], as in the proof of [[Picard's little theorem|R2PHM118NG]].
Example: the holomorphic maps $\mathbb{D} \to \mathbb{D}$ of the unit disc form a normal family, since they are bounded by $1$.
In part 2 the limit $\infty$ cannot be excluded: the constant functions $f_n = n$ omit $0$ and $1$ for $n \ge 2$ and tend to $\infty$.
One omitted value is not enough: the functions $e^{nz}$ on $\mathbb{D}$ omit $0$, yet no subsequence converges near $0$.

## Explanations

### L1 human

Montel's theorem says that a family of holomorphic functions cannot be wild if it is bounded.
From any sequence you can always extract a subsequence that converges nicely.
For real functions this is false: $\sin(nx)$ is bounded, but no subsequence converges uniformly.
The second part says that even missing two values is enough to tame the family.

### L2 human

Part 1 is the compactness result behind existence proofs in complex analysis.
In the proof of the [[Riemann mapping theorem|02WHMEG5ZR]], one maximizes $\abs{f'(z_0)}$ over a normal family of injective maps and uses Montel to get a maximizer.
Part 2 is the basis of complex dynamics: the Fatou set of a rational map is the set where its iterates form a normal family.
It also gives a short proof of [[Picard's great theorem|08QA8QTM71]].
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# Topology generated by a family of seminorms

## Summary

A family of [[seminorms|FP5GVCN8GV]] on a vector space defines a topology.
In it, a net converges exactly when it converges for every seminorm of the family.
With this topology the vector space is a [[topological vector space|T32W553W95]], and in fact a [[locally convex space|B6GY2KJMWW]].
This is the standard way to put a topology on function spaces such as spaces of smooth functions, where no single norm is suitable.

## Explanations

### L1 human

Sometimes one way of measuring size is not enough, and one uses many measuring sticks at once.
Two vectors are declared close when they are close for finitely many of the sticks at a time.
For example, on the space of continuous functions on $\R$, the seminorms $p_n(f) = \sup_{\abs{t} \le n} \abs{f(t)}$ give the topology of uniform convergence on bounded intervals.
A sequence $f_k$ converges to $f$ in this topology when $p_n(f_k - f) \to 0$ for each $n$ separately.
The functions $f_k(t) = t / k$ converge to $0$ in this sense, although they do not converge uniformly on all of $\R$.
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<document source="https://tamnd.github.io/math2/node/ZVGM4STCJA.md" tag="ZVGM4STCJA" kind="definition">
# Numerical linear algebra

## Definition

*Numerical linear algebra* is the part of numerical analysis that studies algorithms in finite-precision arithmetic for these problems.
In each, $A$ is a real or complex [[matrix|0VD9ZSWWJG]].

1. solve a linear system $Ax = b$;
2. find $x$ that minimizes $\norm{Ax - b}_2$ (least squares);
3. find the [[eigenvalues|0BQAA2BK3Q]] and eigenvectors of a square matrix $A$;
4. find the [[singular values|09VMDSRXWJ]] and singular vectors of $A$.

An algorithm is judged by its cost, counted in arithmetic operations and memory, and by its accuracy under rounding errors.

## Remarks

An algorithm is *backward stable* when its computed answer is the exact answer for slightly perturbed data.
For a linear system this means $(A + \delta A)\hat x = b$ with $\norm{\delta A}$ of the order of the unit roundoff times $\norm{A}$.
For a backward stable method, the relative error of $\hat x$ is roughly bounded by the [[condition number|AB3S6MHJ8X]] of $A$ times the unit roundoff.
Example: Householder [[QR decomposition|0ERWG4NXQ9]] solves least squares problems in a backward stable way with about $2mn^2 - \tfrac{2}{3}n^3$ operations for an $m \times n$ matrix.
Non-example: computing the exact determinant of an integer matrix with rational arithmetic is a task of computer algebra.
It is not numerical linear algebra, because it does not work in finite precision and has no rounding error to analyse.
Direct methods, such as [[LU decomposition|0RKG5YW66Y]], finish in a fixed number of steps; iterative methods, such as [[power iteration|H4MQD841BR]], improve an approximation step by step.
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# Quiver

## Definition

A *quiver* is a quadruple $Q = (Q_0, Q_1, s, t)$ where:

1. $Q_0$ is a set, whose elements are the *vertices*;
2. $Q_1$ is a set, whose elements are the *arrows*;
3. $s, t \colon Q_1 \to Q_0$ are maps, sending an arrow $a$ to its *source* $s(a)$ and its *target* $t(a)$.

One writes $a \colon i \to j$ when $s(a) = i$ and $t(a) = j$.
The quiver is *finite* if $Q_0$ and $Q_1$ are finite.

## Remarks

Example: the Kronecker quiver has $Q_0 = \set{1, 2}$, $Q_1 = \set{a, b}$ and $s(a) = s(b) = 1$, $t(a) = t(b) = 2$.
It satisfies the definition, though it has two arrows with the same source and target; a simple [[directed graph|0KFYGSHHF5]] would not allow this.
A loop is an arrow $a$ with $s(a) = t(a)$, and it is also allowed.
Non-example: an undirected graph is not a quiver as it stands, since its edges have no source and target, so clause 3 fails.
Choosing an orientation of each edge turns it into a quiver.
A path of length $n \ge 1$ is a sequence of arrows $a_1, \dots, a_n$ with $t(a_r) = s(a_{r+1})$; there is also a trivial path $e_i$ at each vertex $i$.
In Mathlib, `Quiver` is a class on a type of vertices, with a type of arrows for each pair of vertices.
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# Essential supremum

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]] and let $f \colon X \to [-\infty, \infty]$ be a [[measurable function|YYNCJSR0PJ]].
A number $c \in [-\infty, \infty]$ is an *essential upper bound* of $f$ if the set $\set{x \in X : f(x) > c}$ is a [[null set|VQQM4Y550T]].
The *essential supremum* of $f$ is the [[infimum|04KFTA34C5]] of its essential upper bounds in the [[extended real line|0GSZ7B0CGR]]:
$$\operatorname{ess\,sup} f = \inf \set{c \in [-\infty, \infty] : \mu(\set{x \in X : f(x) > c}) = 0}.$$

## Remarks

Example: on $\R$ with Lebesgue measure, let $f = \mathbf{1}_{\Q}$, the indicator function of the rationals.
The set $\set{f > 0} = \Q$ is countable, so it has measure $0$, and $\operatorname{ess\,sup} f \le 0$.
For $c < 0$ the set $\set{f > c}$ is all of $\R$, so $\operatorname{ess\,sup} f = 0$, while $\sup f = 1$.

Non-example: the number $1/2$ is not an essential upper bound of $g(x) = x$ on $[0, 1]$ with Lebesgue measure.
The set $\set{g > 1/2} = (1/2, 1]$ has measure $1/2 \ne 0$.
In fact $\operatorname{ess\,sup} g = 1$.

The essential supremum is itself an essential upper bound, since $\set{f > c}$ is a countable union of the sets $\set{f > c + 1/n}$.
The *essential infimum* is $\operatorname{ess\,inf} f = -\operatorname{ess\,sup}(-f)$.
The quantity $\operatorname{ess\,sup} \abs{f}$ is the norm $\norm{f}_\infty$ of the space [[L-infinity|ZT01G7BTSV]].
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# Arnold diffusion

## Summary

Arnold diffusion is a slow drift of the action variables in a nearly [[integrable system|N0N7CMRCMH]] with three or more degrees of freedom.
Along such an orbit the actions change by an amount of order one, although the perturbation is arbitrarily small.
It shows that the stability given by [[KAM theory|3ZZSXKM16Z]] for most initial conditions does not hold for all of them.
Arnold gave the first example in 1964.

## Explanations

### L1 human

In a nearly integrable system most motions stay on invariant tori, which act like walls.
With two degrees of freedom these walls cut each energy level into separate rooms, so nothing can wander far.
With three or more degrees of freedom the walls are too thin to separate anything, like circles in three-dimensional space.
Then an orbit can slip through the gaps between the tori and, very slowly, travel a long way.
This slow wandering is Arnold diffusion.
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# Partial difference equation

## Definition

Let $d \ge 2$, let $\Omega \subseteq \Z^d$, and let $a_1, \dots, a_r \in \Z^d$ be fixed shift vectors.
Let $F \colon \Omega \times \R^{r} \to \R$ be a given [[function|0EPAT51DXQ]].
A *partial difference equation* for an unknown function $u \colon \Z^d \to \R$ is a condition of the form
$$F\big(m, u(m + a_1), u(m + a_2), \dots, u(m + a_r)\big) = 0 \quad \text{for all } m \in \Omega.$$
A *solution* is a function $u$ that satisfies it for every $m \in \Omega$.

## Remarks

Example: the explicit scheme for the heat equation,
$$u(j, n+1) = u(j, n) + \lambda \big( u(j+1, n) - 2u(j, n) + u(j-1, n) \big), \qquad (j, n) \in \Z \times \N,$$
with a constant $\lambda > 0$, is a partial difference equation with $d = 2$ and shifts $(0, 1), (0, 0), (1, 0), (-1, 0)$.

Non-example: $u(n+2) = u(n+1) + u(n)$ for $u \colon \Z \to \R$ involves only one integer variable, so $d = 1$ and it is an ordinary difference equation.

One may also take $u$ defined only on a subset of $\Z^d$ that contains all the points $m + a_i$, as in the heat scheme on $\Z \times \N$.
For example, the binomial coefficients $c(n, k) = \binom{n}{k}$ on $\N^2$ satisfy $c(n+1, k+1) = c(n, k) + c(n, k+1)$.
The values may also lie in another field or ring instead of $\R$.
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# Kodaira dimension

## Summary

The Kodaira dimension of a smooth projective variety measures how fast the spaces of sections of the powers of its [[canonical bundle|W05VYY4HBR]] grow.
It takes one of the values $-\infty, 0, 1, \dots, n$, where $n$ is the dimension of the variety.
It is a birational invariant, and it is the first coarse invariant in the classification of varieties, as in the [[Enriques–Kodaira classification|1GVF19KFAK]] of surfaces.
Varieties of maximal Kodaira dimension are said to be of general type.

## Explanations

### L1 human

The canonical bundle of a variety is built from its top-degree differential forms.
Count how many independent sections the $m$-th power of this bundle has, and watch how this number grows with $m$.
If it is always zero, the Kodaira dimension is $-\infty$; if it grows like $m^\kappa$, the Kodaira dimension is $\kappa$.
For curves, this gives three classes: the projective line has $-\infty$, elliptic curves have $0$, and curves of genus at least $2$ have $1$.
These three classes match the three geometries: spherical, flat and hyperbolic.
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# Langevin equation

## Summary

The Langevin equation is a stochastic differential equation for the motion of a particle in a fluid.
It combines a deterministic force, a friction force proportional to the velocity, and a random force from collisions with the molecules of the fluid.
The friction and the strength of the noise are tied by the temperature, so that the particle reaches thermal equilibrium.
Without an external force, the velocity is an [[Ornstein–Uhlenbeck process|HZM2ASCDY4]].

## Explanations

### L1 human

A pollen grain in water is pushed by countless water molecules.
On average these pushes slow the grain down, like friction, but they also shake it at random.
The Langevin equation is Newton's law with these two extra forces added.
The random kicks keep the grain moving, while the friction keeps its speed from growing without bound.
In the long run, the velocity of the grain has a normal law whose variance is the temperature divided by the mass, in suitable units.
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# Injective resolution

## Summary

An injective resolution of an object $X$ is an exact sequence $0 \to X \to I^0 \to I^1 \to \cdots$ in which every $I^n$ is injective.
It is the dual of a [[projective resolution|HK2AA1YSQ2]].
Right [[derived functors|NPMC3K6J2F]] of a left exact functor are computed by applying the functor to an injective resolution.
Examples are Ext in the second variable and sheaf cohomology.

## Explanations

### L1 human

An injective object is a target into which maps always extend.
To resolve $X$, embed it into an injective object, then embed the cokernel into another injective object, and so on.
The chain of injective objects remembers $X$ while being easy to map into.
For the integers, $\Z$ sits inside the rationals $\Q$, and the quotient $\Q/\Z$ is again injective.
So $0 \to \Z \to \Q \to \Q/\Z \to 0$ is an injective resolution of $\Z$.
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# Sigma-compact space

## Summary

A σ-compact space is a topological space that is a countable union of compact subspaces.
Euclidean space $\R^n$ is the main example, as the union of the closed balls of radius $n$.
Every σ-compact space is [[Lindelöf|SW0SN1NEJE]], and every [[locally compact|0FE2SB8WVJ]] second-countable space is σ-compact.
The notion lets compactness arguments run one piece at a time.

## Explanations

### L1 human

A compact space is small in a topological sense.
A σ-compact space may be big, but it can be built from countably many compact pieces.
The real line is not compact, but it is the union of the intervals $[-n, n]$ for $n = 1, 2, 3, \ldots$.
So many facts proved on compact intervals can be extended to all of $\R$.
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# Regular and singular points of an analytic set

## Summary

A point of a [[complex analytic variety|GB7M582M5Y]] is regular when the set is a complex submanifold near that point, and singular otherwise.
The regular points form an open dense subset of the variety, and the singular points form a nowhere dense analytic subset.
Singular points are where the local geometry is interesting, and their study is the subject of singularity theory.

## Explanations

### L1 human

Near a regular point, an analytic set looks like a flat piece of $\C^d$, bent smoothly.
At a singular point something goes wrong: several pieces cross, or the set has a sharp tip.
The curve $\set{z_1 z_2 = 0}$ is singular at the origin, where two lines cross.
The cusp $\set{z_1^2 = z_2^3}$ is a single piece, but it is singular at the origin, where it has a sharp point.
All other points of these two curves are regular.
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# Statistical convergence

## Summary

A sequence converges statistically to $L$ when, for every $\varepsilon > 0$, the indices $n$ with $\abs{x_n - L} \ge \varepsilon$ form a set of [[natural density|RVG9Z8NYA6]] zero.
So the sequence may stray from $L$ infinitely often, but only on a sparse set of indices.
Every convergent sequence converges statistically to its limit, but not conversely.
For bounded sequences, statistical convergence to $L$ is equivalent to [[strong Cesàro summability|750MCA763X]] to $L$ with exponent $1$.

## Explanations

### L1 human

Ordinary convergence allows only finitely many terms to be far from the limit.
Statistical convergence allows infinitely many, as long as they become rarer and rarer, so that their proportion among the first $n$ terms tends to $0$.
For example, the sequence that is $1$ at perfect squares and $0$ elsewhere converges statistically to $0$, since there are only about $\sqrt{n}$ squares up to $n$.
It does not converge in the usual sense, because the ones never stop.
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# Birch and Swinnerton-Dyer conjecture

## Statement

Let $E$ be an [[elliptic curve|427N6S6QQW]] over $\Q$ with conductor $N$.
For a prime $p \nmid N$, let $a_p = p + 1 - \#\tilde{E}(\F_p)$, where $\tilde{E}$ is the reduction of $E$ modulo $p$.
For $p \mid N$, let $a_p$ be $1$, $-1$ or $0$ when the reduction of $E$ at $p$ is split multiplicative, non-split multiplicative or additive.
Let
$$L(E, s) = \prod_{p \nmid N} \left(1 - a_p p^{-s} + p^{1 - 2s}\right)^{-1} \prod_{p \mid N} \left(1 - a_p p^{-s}\right)^{-1},$$
which converges for $\mathrm{Re}(s) > 3/2$ and extends to an entire function of $s$ by the [[modularity theorem|CS1NVJBZMH]].
Let $r$ be the rank of the finitely generated abelian group $E(\Q)$ of [[rational points|P7V1BQJPSS]].
Then $L(E, s)$ has a zero of order exactly $r$ at $s = 1$.

## Remarks

The refined form of the conjecture predicts the leading coefficient.
$$\lim_{s \to 1} \frac{L(E, s)}{(s - 1)^r} = \frac{\Omega_E \cdot \mathrm{Reg}_E \cdot \#\text{Ш}(E/\Q) \cdot \prod_{p \mid N} c_p}{(\# E(\Q)_{\mathrm{tors}})^2},$$
where $\Omega_E$ is the real period, $\mathrm{Reg}_E$ the regulator of the canonical height pairing, $\text{Ш}(E/\Q)$ the [[Tate–Shafarevich group|BK9WBMF56G]] (conjectured to be finite), and $c_p$ the Tamagawa numbers.
Status: the conjecture is open; it is one of the Clay Millennium Prize Problems.
Known results: if the order of vanishing of $L(E, s)$ at $s = 1$ is $0$ or $1$, then the rank equals it and $\text{Ш}(E/\Q)$ is finite.
This follows from the [[Gross–Zagier theorem|BX558DTV2D]] together with the work of Kolyvagin.
Coates and Wiles had earlier proved that if $E$ has [[complex multiplication|D3HEJTVY7G]] and $L(E, 1) \ne 0$, then $E(\Q)$ is finite.
Birch and Swinnerton-Dyer formulated the conjecture in the 1960s after computer experiments on the products $\prod_{p \le x} \#\tilde{E}(\F_p)/p$.

## Explanations

### L1 human

The rank measures how many independent rational points an elliptic curve has, but it is hard to compute.
The $L$-function packs together the numbers of points of the curve modulo every prime.
The conjecture says that the rank can be read off from the $L$-function: it is the order of its zero at $s = 1$.
Heuristically, a curve with many rational points tends to have many points modulo $p$, which pushes the Euler product toward zero at $s = 1$.
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# Local martingale

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_t)_{t \ge 0}$.
Let $(X_t)_{t \ge 0}$ be a real [[adapted|YRNFE016FF]] process with càdlàg paths.
$X$ is a *local martingale* if there is a sequence $(\tau_n)_{n \ge 1}$ of [[stopping times|WND87FBAKC]] such that:

1. $\tau_1 \le \tau_2 \le \dots$ almost surely.
2. $\tau_n \to \infty$ almost surely as $n \to \infty$.
3. For each $n$, the [[stopped process|NG35MGYDZR]] $X^{\tau_n} = (X_{t \wedge \tau_n})_{t \ge 0}$ is a [[martingale|0CDVK9RQBH]].

Such a sequence $(\tau_n)$ is called a localising sequence.

## Remarks

Every càdlàg martingale $X$ is a local martingale: take $\tau_n = n$, since a stopped martingale is a martingale by optional stopping.
The deterministic process $X_t = t$ is adapted and continuous but is not a local martingale.
Indeed, clause 3 would give $E[t \wedge \tau_n] = 0$ for all $t$, so $\tau_n = 0$ almost surely, contradicting clause 2.
A local martingale need not be a martingale: if $B$ is a three-dimensional Brownian motion started at $x \ne 0$, then $1/\abs{B_t}$ is a local martingale but not a martingale.
A non-negative local martingale is a [[supermartingale|KYW1PV0SX5]], by Fatou's lemma.
Some authors require clause 3 only for $X^{\tau_n} - X_0$, or for $X^{\tau_n} \mathbf{1}\{\tau_n > 0\}$, so that $X_0$ need not be integrable.
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# Mason–Stothers theorem

## Statement

Let $K$ be a [[field|04C5ZBG1WT]] and let $a, b, c \in K[t]$ be polynomials in the [[polynomial ring|0SDC934R52]] with $a + b = c$.
Assume that $a$, $b$, $c$ are pairwise coprime and that not all of the [[formal derivatives|0EGN1E1JPB]] $a'$, $b'$, $c'$ are zero.
For a nonzero polynomial $f$, let $\operatorname{rad}(f)$ be the product of the distinct monic [[irreducible|03FKX7Z5DC]] factors of $f$.
Then
$$\max\set{\deg a, \deg b, \deg c} \le \deg \operatorname{rad}(abc) - 1.$$

## Remarks

When $K$ is algebraically closed, $\deg \operatorname{rad}(abc)$ is the number of distinct roots of $abc$.
If $K$ has characteristic $0$, the hypothesis on derivatives just says that $a$, $b$, $c$ are not all constant.
A short proof, due to Noah Snyder, studies the Wronskian $ab' - a'b$: it is nonzero, it is divisible by $abc/\operatorname{rad}(abc)$, and its degree is less than $\deg a + \deg b$.
Example: over $\Q$, take $a = t^2$, $b = 2t + 1$ and $c = (t + 1)^2$; then $\operatorname{rad}(abc) = t(2t + 1)(t + 1)$ has degree $3$, and the bound $2 \le 3 - 1$ is sharp.
The derivative hypothesis is necessary: over a field of characteristic $p > 0$, take $a = t^p$, $b = 1$, $c = (t + 1)^p$; then $a' = b' = c' = 0$, the radical has degree $2$, and $\deg c = p > 1$.
A consequence: for $n \ge 3$ there are no pairwise coprime polynomials $x, y, z$ over $\C$, not all constant, with $x^n + y^n = z^n$.

## Explanations

### L1 human

If two polynomials add up to a third, and they share no common factor, then the three together must have many different roots.
More precisely, the number of distinct roots is larger than every one of the degrees.
So you cannot make all three polynomials into high powers of a few simple factors.

### L2 human

The theorem is the polynomial form of the [[abc conjecture|G7JH2NXPC0]], with the degree playing the role of the logarithm of the size and the radical counting distinct prime factors.
In contrast with the integer case, it has a short and complete proof by differentiation.
It gives quick proofs of Fermat's Last Theorem for polynomials and of related results on equations such as $x^2 - y^3 = $ a polynomial.
</document>

<document source="https://tamnd.github.io/math2/node/ZWQ5JBR6XF.md" tag="ZWQ5JBR6XF" kind="problem">
# Brachistochrone problem

## Problem

Let $A$ and $B$ be two points in a vertical plane, with $B$ lower than $A$ and not on the vertical line through $A$.
A particle starts at rest at $A$ and slides without friction along a curve from $A$ to $B$ under constant gravity $g$.
Which curve makes the time of travel from $A$ to $B$ as small as possible?

In coordinates, put $A = (0, 0)$, measure $y$ downwards, and let $B = (x_1, y_1)$ with $x_1, y_1 > 0$.
Energy conservation gives the speed $\sqrt{2 g y}$.
So for a graph $y(x)$ the time is the [[integral functional|JRQZ803Q5Y]]
$$T(y) = \int_0^{x_1} \sqrt{\frac{1 + y'(x)^2}{2 g \, y(x)}} \, dx,$$
to be minimised among curves with $y(0) = 0$ and $y(x_1) = y_1$.

## Remarks

The answer is an arc of a cycloid, the curve traced by a point on a circle that rolls along a horizontal line.
Its parametric form is $x = r(\theta - \sin \theta)$, $y = r(1 - \cos \theta)$, with the radius $r$ fixed by the condition that the curve passes through $B$.
The integrand does not depend on $x$, so the [[Beltrami identity|F7EKTEESQ5]] applies and gives $y (1 + y'^2) = 2r$ for a constant $r$; the cycloid solves this equation.
The fastest curve is not the straight segment; when $x_1 / y_1 > \pi / 2$, the cycloid even dips below the height of $B$ before it rises to $B$.
Johann Bernoulli posed the problem as a public challenge in Acta Eruditorum; solutions by Johann Bernoulli, Jacob Bernoulli, Leibniz and Newton appeared in 1697.
The problem is often taken as the start of the calculus of variations.

## Explanations

### L1 human

A ball rolls from a high point to a lower point off to the side, and one wants the track that gets it there fastest.
The straight track is the shortest but not the fastest.
A track that first drops steeply lets the ball pick up speed early, and the best balance between length and speed is a cycloid.
</document>

<document source="https://tamnd.github.io/math2/node/ZWS0YWGF17.md" tag="ZWS0YWGF17" kind="statement">
# Weak Nullstellensatz

## Statement

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $n \ge 1$.

1. The [[maximal ideals|0PV74TR5FP]] of the [[polynomial ring|0SDC934R52]] $k[x_1, \ldots, x_n]$ are exactly the ideals $\mathfrak{m}_a = (x_1 - a_1, \ldots, x_n - a_n)$ for $a = (a_1, \ldots, a_n) \in k^n$.
2. If $I$ is an [[ideal|096Z9MP2ME]] of $k[x_1, \ldots, x_n]$ with $I \ne k[x_1, \ldots, x_n]$, then there is $a \in k^n$ with $f(a) = 0$ for all $f \in I$.

## Remarks

Each $\mathfrak{m}_a$ is maximal, since it is the kernel of the surjective evaluation map $f \mapsto f(a)$ onto the field $k$.
Conversely, let $\mathfrak{m}$ be maximal; then $k[x_1, \ldots, x_n]/\mathfrak{m}$ is a field finitely generated as a $k$-algebra.
So it is finite over $k$ by [[Zariski's lemma|YBJ2VQHFDQ]], hence equal to $k$ as $k$ is algebraically closed.
If $a_i \in k$ is the image of $x_i$, then $\mathfrak{m} \supseteq \mathfrak{m}_a$, so $\mathfrak{m} = \mathfrak{m}_a$.
Part 2 follows from part 1: a proper ideal lies in a maximal ideal $\mathfrak{m}_a$, and then every $f \in I$ vanishes at $a$.

Example: over $\C$, the ideal $(x^2 + y^2 - 1, x - 1)$ of $\C[x, y]$ is proper and the point $(1, 0)$ is a common zero.

The field must be algebraically closed: over $\R$, the ideal $(x^2 + 1)$ of $\R[x]$ is maximal, but it has no zero in $\R$ and is not of the form $(x - a)$.

## Explanations

### L1 human

Over the complex numbers, a system of polynomial equations has no solution only when the equations imply $1 = 0$ by algebra.
In other words, the points of $\C^n$ are the same thing as the maximal ideals of the polynomial ring.

### L2 human

The weak form is the step from which the full [[Nullstellensatz|D85VZWS9QK]] follows by the Rabinowitsch trick.
It identifies the points of affine space with the maximal ideals of the coordinate ring.
This is the starting point for the maximal spectrum and, later, for schemes.
</document>

<document source="https://tamnd.github.io/math2/node/ZWT5HGRFXX.md" tag="ZWT5HGRFXX" kind="statement">
# Newton–Puiseux theorem

## Statement

Let $k$ be an [[algebraically closed field|02SJG1HC32]] of [[characteristic|1ZB13N78XW]] $0$.
Let $P$ be the field of [[Puiseux series|SENTYNSKSF]] in $X$ over $k$.
Then $P$ is algebraically closed, and it is an [[algebraic closure|0MCXX29HWY]] of the [[field of formal Laurent series|YZWYF7DXTW]] $k((X))$.

In particular, every polynomial in $Y$ of degree $n \ge 1$ with coefficients in $k((X))$ has $n$ roots in $P$, counted with multiplicity.

## Remarks

Proof idea (Newton's method): let $F(Y)$ be a polynomial with coefficients in $P$.
A slope of the [[Newton polygon|H7QNCE2QZJ]] of $F$ for the $X$-adic valuation gives a possible leading exponent $\gamma$ of a root.
A root $c \ne 0$ in $k$ of an associated polynomial gives the leading term $c X^\gamma$.
One then substitutes $Y = X^\gamma(c + Y_1)$ and repeats.
In characteristic $0$ one shows that the denominators of the exponents stop growing, so the process produces a Puiseux series.
A second proof shows that every finite extension of $k((X))$ of degree $n$ is $k((X^{1/n}))$.
Example: the roots of $Y^2 - X^3$ are $\pm X^{3/2}$, and the roots of $Y^2 - X^2(1 + X)$ are $\pm X(1 + X)^{1/2}$, with $(1 + X)^{1/2}$ given by the binomial series.
The characteristic $0$ hypothesis is necessary: over an algebraic closure of $\F_p$, the polynomial $Y^p - Y - X^{-1}$ has no root in the field of Puiseux series.

## Explanations

### L1 human

Solving a polynomial equation in $Y$ whose coefficients are power series in $X$ usually needs fractional powers of $X$.
The theorem says that fractional powers with a common denominator are always enough, when the coefficients come from an algebraically closed field such as $\C$.
For example $Y^2 = X^3$ is solved by $Y = X^{3/2}$.

### L2 human

Geometrically, the theorem parametrizes each branch of a plane algebraic curve $F(X, Y) = 0$ through a point by $X = t^m$, $Y = $ a power series in $t$.
The exponents of the Puiseux expansion are the start of the classification of plane curve singularities.
Over $\C$ the Puiseux series that arise from polynomials converge near $0$, and they describe the branches analytically.
The Newton polygon is the computational tool, and [[Henselian fields|3ACEE4N6NE]] give the general frame for lifting approximate roots.
</document>

<document source="https://tamnd.github.io/math2/node/ZWXW0T3EAC.md" tag="ZWXW0T3EAC" kind="concept">
# Quaternion algebra

## Summary

A quaternion algebra over a field $F$ is a [[central simple algebra|5HRKJBAY2V]] over $F$ of dimension $4$.
When the characteristic of $F$ is not $2$, it has a basis $1, i, j, k$ with $i^2 = a$, $j^2 = b$ and $ij = -ji = k$ for some nonzero $a, b \in F$.
Each one is either a division algebra or isomorphic to the algebra of $2 \times 2$ matrices over $F$.
The class of a quaternion algebra in the Brauer group has order at most $2$.
Quaternion algebras are tied to quadratic forms and the Hilbert symbol.

## Explanations

### L1 human

Hamilton's [[quaternions|0GE4DVQKR8]] are numbers $x + yi + zj + wk$ with $i^2 = j^2 = -1$ and $ij = -ji = k$.
A quaternion algebra keeps this pattern but allows other values for $i^2$ and $j^2$ and other fields of coefficients.
With $i^2 = j^2 = -1$ over $\R$ one gets Hamilton's quaternions, a division algebra.
With $i^2 = j^2 = 1$ over $\R$ one gets an algebra isomorphic to the $2 \times 2$ real matrices, which has zero divisors such as $1 + i$.
</document>

<document source="https://tamnd.github.io/math2/node/ZX8972MACG.md" tag="ZX8972MACG" kind="definition">
# A priori estimate

## Definition

Let $X$ and $Y$ be normed spaces of functions, for example [[Banach spaces|06GZ956657]], and let $\mathcal{F}$ be a class of data $f \in Y$.
Consider a [[partial differential equation|GC5BDZ6QBC]] or boundary value problem $A(u) = f$.
Let $C \colon [0, \infty) \to [0, \infty)$ be a function, often linear, that does not depend on $u$.
An *a priori estimate* for this problem is an inequality $\norm{u}_X \le C(\norm{f}_Y)$.
It must hold for every solution $u \in X$ of $A(u) = f$ with $f \in \mathcal{F}$.
The estimate assumes that $u$ is a solution in $X$; it asserts nothing about the existence of $u$.

## Remarks

Example: let $\Omega \subseteq \R^n$ be bounded and open, $f \in L^2(\Omega)$, and let $u \in H^1_0(\Omega)$ be a weak solution of $-\Delta u = f$.
Testing with $u$ gives $\int_\Omega \abs{\nabla u}^2 = \int_\Omega f u \le \norm{f}_{L^2} \norm{u}_{L^2}$.
By the [[Poincaré inequality|JS2CCR3RNS]], $\norm{u}_{L^2} \le C_P \norm{\nabla u}_{L^2}$, hence $\norm{\nabla u}_{L^2} \le C_P \norm{f}_{L^2}$.
The constant depends only on $\Omega$.

Non-example: a bound $\norm{u}_X \le C(u)$ with a constant that depends on the particular solution is not an a priori estimate.
It gives no uniform control over a family of solutions.

The [[Schauder estimates|HSEKZMAH1V]] and [[Calderón–Zygmund estimates|RYQJH4RYB4]] are a priori estimates in Hölder and $L^p$ norms.
Combined with the Leray–Schauder fixed point method or the method of continuity, a priori estimates yield existence.
</document>

<document source="https://tamnd.github.io/math2/node/ZXES6QGX62.md" tag="ZXES6QGX62" kind="proof">
# Proof of the cancellation laws

## Proof

1. Let $a$, $b$ and $c$ be elements of a [[group|G5S1M9]] $G$ with identity $e$, and suppose that $ab = ac$.
2. Multiply both sides on the left by the [[inverse|0BRHKKP333]] $a^{-1}$. This gives $a^{-1}(ab) = a^{-1}(ac)$.
3. By associativity, $(a^{-1}a)b = (a^{-1}a)c$, so $eb = ec$ and $b = c$.
4. Now suppose that $ba = ca$. Multiply both sides on the right by $a^{-1}$.
5. By associativity, $b(aa^{-1}) = c(aa^{-1})$, so $be = ce$ and $b = c$.

## Idea

Multiply both sides of the equation by the inverse of the common factor, on the same side as that factor.
</document>

<document source="https://tamnd.github.io/math2/node/ZXF5XXT5XM.md" tag="ZXF5XXT5XM" kind="definition">
# Completion of a measure

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
Let $\Sigma_\mu$ be the family of sets $A \cup S$, where $A \in \Sigma$ and $S$ is a subset of some [[null set|VQQM4Y550T]] $N \in \Sigma$.
Define $\bar{\mu} \colon \Sigma_\mu \to [0, \infty]$ by $\bar{\mu}(A \cup S) = \mu(A)$.
The measure space $(X, \Sigma_\mu, \bar{\mu})$ is the *completion* of $(X, \Sigma, \mu)$.

## Remarks

The family $\Sigma_\mu$ is a [[σ-algebra|04WSW8T3SY]] that contains $\Sigma$.
The value $\bar{\mu}(A \cup S)$ does not depend on the choice of $A$ and $S$.
Indeed, if $A \cup S = A' \cup S'$ with $S \subseteq N$ and $S' \subseteq N'$, then $A$ and $A'$ differ only inside the null set $N \cup N'$, so $\mu(A) = \mu(A')$.
Then $\bar{\mu}$ is a [[complete measure|22K8CSQTYM]] that extends $\mu$, and it is the smallest complete extension.
For example, let $X = \set{1, 2, 3}$, $\Sigma = \set{\varnothing, \set{1, 2}, \set{3}, X}$, and $\mu(A) = 1$ if $3 \in A$ and $0$ otherwise.
The null set $\set{1, 2}$ gives $\Sigma_\mu$ equal to the power set of $X$, with $\bar{\mu}(A) = 1$ if $3 \in A$ and $0$ otherwise.
With the same $\Sigma$ and $\nu(A) = 1$ if $1 \in A$ and $0$ otherwise, the only null sets are $\varnothing$ and $\set{3}$.
So $\Sigma_\nu = \Sigma$, and the power set is not the σ-algebra of the completion of $\nu$.
The completion of Lebesgue measure on the Borel sets of $\R$ is Lebesgue measure on the Lebesgue measurable sets.
</document>

<document source="https://tamnd.github.io/math2/node/ZXJ7FJ3WCE.md" tag="ZXJ7FJ3WCE" kind="statement">
# Hellinger–Toeplitz theorem

## Statement

Let $H$ be a [[Hilbert space|HM2HHE3J51]] over $\R$ or $\C$ with inner product $\ip{\cdot}{\cdot}$.
Let $A \colon H \to H$ be a [[linear map|0NRYHJPQG2]] defined on all of $H$.
Assume that
$$\ip{Ax}{y} = \ip{x}{Ay} \qquad \text{for all } x, y \in H.$$
Then $A$ is bounded, that is, $A$ is a [[continuous linear map|03TM135G0N]].

## Remarks

Proof idea: by the [[closed graph theorem|GXM8BX8YM7]], it suffices to show that $A$ is [[closed|VM7B38PA87]].
If $x_n \to x$ and $A x_n \to z$, then for every $y \in H$,
$$\ip{z}{y} = \lim \ip{A x_n}{y} = \lim \ip{x_n}{Ay} = \ip{x}{Ay} = \ip{Ax}{y},$$
so $z = Ax$.

Example: if $A$ is given on $\ell^2$ by a real symmetric infinite matrix and is defined on all of $\ell^2$, the theorem says $A$ is automatically bounded.

Completeness of $H$ is needed.
On the space of finitely supported sequences with the $\ell^2$ inner product, the diagonal map $(x_n) \mapsto (n x_n)$ is defined everywhere and symmetric.
It is unbounded, since it sends the $n$-th unit vector to $n$ times itself.

## Explanations

### L1 human

In quantum mechanics, observables are represented by symmetric operators, and many of them, such as position and momentum, are unbounded.
The theorem says that such an unbounded symmetric operator can never be defined on the whole Hilbert space.
So domains are unavoidable in the theory of unbounded operators.

### L2 human

The theorem is a short application of the closed graph theorem, and it is one reason why the theory of unbounded self-adjoint operators works with dense, proper domains.
It also shows that a bounded operator is self-adjoint exactly when it is symmetric on all of $H$; no extra boundedness hypothesis is needed.
A related form: a linear map $A \colon H \to H$ that has a formal adjoint $B \colon H \to H$, with $\ip{Ax}{y} = \ip{x}{By}$ for all $x, y$, is bounded, by the same proof.
</document>

<document source="https://tamnd.github.io/math2/node/ZXJKSB30WF.md" tag="ZXJKSB30WF" kind="concept">
# Disjoint union topology

## Summary

The disjoint union topology is the natural topology on a disjoint union of topological spaces.
A set is open in it exactly when its part in each piece is open in that piece.
It makes the disjoint union the coproduct in the category of topological spaces.
It is the [[final topology|6ERJRRZK32]] for the inclusions of the pieces.

## Explanations

### L1 human

Put several spaces side by side, with no overlap and no glue between them.
Each piece keeps its own topology, and the pieces do not interact.
A set is open when you can see it as an open set in each piece separately.
For example, the disjoint union of two copies of the interval $(0, 1)$ is a space with two separate parts.
It looks like the subspace $(0, 1) \cup (2, 3)$ of the real line.
</document>

<document source="https://tamnd.github.io/math2/node/ZXK6XM44HS.md" tag="ZXK6XM44HS" kind="statement">
# Absolute continuity of the Lebesgue integral

## Statement

Let $(X, \Sigma, \mu)$ be a measure space and let $f$ be a real or complex [[integrable function|X7GTF3D98F]] on $X$.
Then for every $\varepsilon > 0$ there is $\delta > 0$ with the following property.
For every $A \in \Sigma$ with $\mu(A) < \delta$,
$$\int_A \abs{f} \, d\mu < \varepsilon,$$
where the integral is an [[integral of a non-negative function|DG76G5Z6XW]].

## Remarks

Proof idea: let $f_N = \min(\abs{f}, N)$.
By the [[monotone convergence theorem|G2S6H789RG]], $\int (\abs{f} - f_N) \, d\mu \to 0$, so one can fix $N$ with $\int (\abs{f} - f_N) < \varepsilon / 2$.
Then $\int_A \abs{f} \le \varepsilon / 2 + N \mu(A)$, and $\delta = \varepsilon / (2N)$ works.

Example: on $[a, b]$ with Lebesgue measure, the function $F(x) = \int_a^x f(t) \, dt$ is [[absolutely continuous|5NB63N9W3P]].
Indeed, a finite union of disjoint intervals of total length less than $\delta$ has measure less than $\delta$.

Integrability is needed.
For $f(x) = 1/x$ on $(0, 1]$ with Lebesgue measure, the set $(0, \delta / 2)$ has measure less than $\delta$ but $\int_0^{\delta/2} dx / x = \infty$.

## Explanations

### L1 human

An integrable function cannot hide a lot of area on a tiny set.
If the set is small enough, the area of the function over it is as small as one wants.

### L2 human

In the language of measures, $\nu(A) = \int_A \abs{f} \, d\mu$ defines a finite measure that is absolutely continuous with respect to $\mu$, in the $\varepsilon$-$\delta$ sense.
The Radon–Nikodym theorem gives a converse: every finite measure absolutely continuous with respect to a σ-finite measure $\mu$ has this form.
The uniform version of this property, for a whole family of functions, is part of uniform integrability.
</document>

<document source="https://tamnd.github.io/math2/node/ZXSN8D4MCB.md" tag="ZXSN8D4MCB" kind="concept">
# Reduced scheme

## Summary

A reduced scheme is a [[scheme|5A35TYZD02]] whose rings of functions have no nonzero [[nilpotent elements|0ZB2JG1M9C]].
On a reduced scheme, a function is determined by its values at all points.
Classical varieties are reduced, while non-reduced schemes record extra infinitesimal structure, such as a double point.

## Explanations

### L1 human

On a scheme a function can be nonzero and still have value zero at every point.
This happens when the function is nilpotent, like $x$ in the ring $k[x]/(x^2)$, where $x^2 = 0$.
That scheme is a single point "thickened" in one direction, which remembers a tangent direction.
A reduced scheme is one with no such thickening, like the line $\operatorname{Spec} k[x]$ itself.
</document>

<document source="https://tamnd.github.io/math2/node/ZXVBEPMTCN.md" tag="ZXVBEPMTCN" kind="concept">
# Divisible group

## Summary

A divisible group is an [[abelian group|082GHNPJ53]] in which every element can be divided by every positive integer.
The rational numbers, the real numbers and $\Q/\Z$ are divisible; the integers are not.
Divisible groups are exactly the [[injective|85H7PRHDPZ]] $\Z$-modules, and every divisible group is a direct sum of copies of $\Q$ and of [[Prüfer groups|F0XAMM2PPS]].
Every abelian group embeds in a divisible group, and a divisible subgroup is always a direct summand.

## Explanations

### L1 human

In $\Q$, every number can be halved, divided by three, and so on, and the answer stays in $\Q$.
A divisible group is an abelian group with this property for every element and every positive integer.
In $\Z$ this fails, since $1$ has no half.
On the circle of angles, every angle can be divided by $n$, so the circle group is divisible too.
</document>

<document source="https://tamnd.github.io/math2/node/ZXY0AYKMXZ.md" tag="ZXY0AYKMXZ" kind="statement">
# Maupertuis's principle

## Statement

Let $M$ be a smooth manifold, the configuration space of a mechanical system.
Let $g$ be a Riemannian metric on $M$, so that the kinetic energy at the velocity $v \in T_qM$ is $T = \tfrac12 g_q(v, v)$.
Let $V \colon M \to \R$ be a smooth potential, so that the motions $q(t)$ solve the Euler–Lagrange equations of $L = T - V$.
Fix an energy value $E$ and let $M_E = \set{q \in M : V(q) < E}$.
For a curve $\gamma \colon [a, b] \to M_E$ define the *abbreviated action*
$$S_0[\gamma] = \int_a^b \sqrt{2\,(E - V(\gamma(s)))}\; \norm{\gamma'(s)}_g \, ds.$$
Let $\gamma$ be a regular curve in $M_E$.
Then, up to reparametrisation, $\gamma$ is the path of a motion with total energy $T + V = E$ if and only if the following holds.
The curve $\gamma$ is a critical point of $S_0$ among curves in $M_E$ with the same endpoints.

## Remarks

The functional $S_0$ is the length functional of the *Jacobi metric* $2(E - V)\,g$ on $M_E$.
So the paths of motion of energy $E$ are the [[geodesics|NSYMQMHPDN]] of the Jacobi metric.
This metric is a conformal change of the kinetic metric $g$ of the [[Riemannian manifold|7CBT4SM83Y]] $(M, g)$.
Along a true motion $\norm{\dot q}_g = \sqrt{2(E - V)}$, so $S_0$ equals $\int 2T\,dt = \int p \cdot dq$, the classical form of the abbreviated action.
The proof compares the Euler–Lagrange equations of the length of the Jacobi metric with those of $L$, after the change of time $dt = \norm{\gamma'}_g / \sqrt{2(E - V)}\, ds$.
Unlike [[Hamilton's principle|FAQ9CH2Z76]], the variation fixes the energy and leaves the travel time free.
For a free particle ($V = 0$), the paths are the geodesics of $g$ itself.
The hypothesis $V < E$ is needed: where $V = E$ the Jacobi metric degenerates, and these are the turning points of the motion.

## Explanations

### L1 human

A moving body with fixed total energy picks a path between two points that makes a certain quantity stationary.
This quantity adds up, along the path, the speed the body would have at each point times the length travelled there, and it ignores how long the trip takes.
So the shape of the path can be found without solving for the timing first.

### L2 human

Maupertuis's principle turns a dynamics problem into a geometry problem.
Orbits of energy $E$ become geodesics of the Jacobi metric $2(E - V)g$, and tools of Riemannian geometry, such as curvature and comparison theorems, then give information on the stability of orbits.
It is derived from the [[stationary-action principle|Q104C67P26]] by fixing the energy instead of the time.
</document>

<document source="https://tamnd.github.io/math2/node/ZXZMQM818D.md" tag="ZXZMQM818D" kind="statement">
# Transitivity of integrality

## Statement

Let $A \subseteq B \subseteq C$ be commutative rings.
If $B$ is [[integral|B7X0KFY1S1]] over $A$ and $C$ is integral over $B$, then $C$ is integral over $A$.

## Remarks

Let $c \in C$ and take a monic relation $c^n + b_{n-1} c^{n-1} + \cdots + b_0 = 0$ with $b_i \in B$.
Each $b_i$ is [[integral|1EHN4KPSFS]] over $A$, so $A' = A[b_0, \ldots, b_{n-1}]$ is a [[finitely generated|05QY1VEBYD]] $A$-module.
The relation shows that $A'[c]$ is a finitely generated $A'$-module, so $A'[c]$ is a finitely generated $A$-module.
Hence $c$ is integral over $A$ by the module criterion for integrality.

Example: $\sqrt[4]{2}$ is integral over $\Z[\sqrt{2}]$, as a root of $x^2 - \sqrt{2}$, and $\Z[\sqrt{2}]$ is integral over $\Z$.
So $\sqrt[4]{2}$ is integral over $\Z$; indeed it is a root of $x^4 - 2$.

## Explanations

### L1 human

If each step of a tower of rings only adds roots of monic equations, then the whole tower does too.
This mirrors the fact that an algebraic extension of an algebraic extension of fields is algebraic.

### L2 human

The proof does not build the monic polynomial over $A$ directly; it passes through finite generation of modules.
The same argument shows that the elements of $B$ integral over $A$ form a subring, the [[integral closure|G0V86SJBQW]] of $A$ in $B$.
Transitivity then shows that the integral closure is integrally closed in $B$.
</document>

<document source="https://tamnd.github.io/math2/node/ZY0GDAS488.md" tag="ZY0GDAS488" kind="definition">
# Bäcklund transform

## Definition

Let $E_1$ and $E_2$ be two [[partial differential equations|GC5BDZ6QBC]] for real functions of two variables $(x, y)$.
A *Bäcklund transform* between $E_1$ and $E_2$ is a system of first-order equations
$$v_x = F(x, y, u, v, u_x, u_y), \qquad v_y = G(x, y, u, v, u_x, u_y)$$
with these two properties.

1. The compatibility condition $(v_x)_y = (v_y)_x$ of the system, for an unknown $v$, holds exactly when $u$ satisfies $E_1$.
2. Each pair $(u, v)$ of solutions of the system has $v$ solving $E_2$.

When $E_1 = E_2$, it is called an *auto-Bäcklund transform*.

## Remarks

Example: the Cauchy–Riemann system $v_x = -u_y$, $v_y = u_x$ is an auto-Bäcklund transform of Laplace's equation $u_{xx} + u_{yy} = 0$.
Indeed $(v_x)_y = -u_{yy}$ and $(v_y)_x = u_{xx}$, so the system is compatible exactly when $u_{xx} + u_{yy} = 0$.
Then $v_{xx} + v_{yy} = -u_{yx} + u_{xy} = 0$ for $C^2$ functions, so $v$ is harmonic too.

Example: for a constant $a \ne 0$, the system
$$\Big(\frac{v - u}{2}\Big)_x = a \sin\frac{u + v}{2}, \qquad \Big(\frac{u + v}{2}\Big)_y = \frac{1}{a} \sin\frac{v - u}{2}$$
is an auto-Bäcklund transform of the [[sine-Gordon equation|802QSF9NM3]] $\varphi_{xy} = \sin \varphi$.
With $s = (u + v)/2$ and $d = (v - u)/2$, one gets $d_{xy} = \cos s \sin d$ and $s_{xy} = \cos d \sin s$, so $u_{xy} = \sin(s - d) = \sin u$ and $v_{xy} = \sin(s + d) = \sin v$.

Non-example: the system $v_x = u_x$, $v_y = u_y$ is compatible for every $C^2$ function $u$.
So it imposes no equation on $u$, and it is not a Bäcklund transform between two equations.

## Explanations

### L1 human

A Bäcklund transform is a pair of first-order equations that ties a new function $v$ to a known function $u$.
It can only be solved when $u$ solves a given equation, and then $v$ solves a given equation as well.
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# Riemann solver

## Summary

A Riemann solver is a procedure that computes, exactly or approximately, the solution of a [[Riemann problem|GJWKX22VVE]] for a [[hyperbolic conservation law|XN081NCPF5]].
In numerical schemes it computes the flux between two neighbouring cells from the two cell values.
It is the main ingredient of [[Godunov's scheme|43HZJ7DRT0]] and of the higher-order schemes built on it.
Exact solvers are expensive for systems such as the Euler equations, so approximate solvers such as the [[Roe solver|TNM5CFSZM6]] and the HLL solver are common.
See the [[definition|9PTJWSCFS5]].

## Explanations

### L1 human

At each wall between two cells of a grid, the state on the left differs from the state on the right.
A Riemann solver answers one question: if these two constant states meet at time zero, what flows through the wall just after?
Doing this at every wall tells the scheme how much of each conserved quantity moves from cell to cell.
For the transport equation $u_t + a u_x = 0$ with $a > 0$, the answer is simple: the left state moves through the wall, and the flux is $a u_l$.
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<document source="https://tamnd.github.io/math2/node/ZY5E6ZQZVV.md" tag="ZY5E6ZQZVV" kind="definition">
# Euler–Lagrange equation

## Definition

Let $\Omega \subseteq \R^n$ be an open set and let $L \colon \Omega \times \R \times \R^n \to \R$, written $L(x, z, p)$, be a function of class $C^2$.
Write $L_z = \partial L / \partial z$ and $L_{p_i} = \partial L / \partial p_i$ for its [[partial derivatives|05AXG3QRKR]].
The *Euler–Lagrange equation* of the functional $J(u) = \int_\Omega L(x, u, \nabla u) \, dx$ of the [[calculus of variations|PQ25NHV0KS]] is the second-order equation
$$-\sum_{i=1}^n \frac{\partial}{\partial x_i} \Big( L_{p_i}\big(x, u(x), \nabla u(x)\big) \Big) + L_z\big(x, u(x), \nabla u(x)\big) = 0 \quad \text{in } \Omega$$
for an unknown $u \in C^2(\Omega)$.
In short form it reads $-\operatorname{div} L_p(x, u, \nabla u) + L_z(x, u, \nabla u) = 0$, with the [[divergence|02NSMQVQWK]] taken in $x$.

## Remarks

If $\Omega$ is bounded and $u \in C^2(\overline{\Omega})$ minimises $J$ among $C^2(\overline{\Omega})$ functions with the same boundary values, then $u$ solves the Euler–Lagrange equation.
Indeed $\frac{d}{dt} J(u + t\varphi)\big|_{t=0} = 0$ for every $\varphi \in C^\infty_c(\Omega)$; integrating by parts gives $\int_\Omega (-\operatorname{div} L_p + L_z) \varphi \, dx = 0$ for all such $\varphi$, so the integrand vanishes.
The *weak* form $\int_\Omega \big( L_p(x, u, \nabla u) \cdot \nabla \varphi + L_z(x, u, \nabla u) \varphi \big) \, dx = 0$ for all $\varphi \in C^\infty_c(\Omega)$ makes sense for less regular $u$.

Example: for $L(x, z, p) = \frac{1}{2}\abs{p}^2 - f(x) z$ one has $L_p = p$ and $L_z = -f$, so the Euler–Lagrange equation is Poisson's equation $-\Delta u = f$.

Example: for $n = 1$ and $L(x, z, p) = \sqrt{1 + p^2}$, one gets $\frac{d}{dx}\big(u' / \sqrt{1 + u'^2}\big) = 0$, so $u'$ is constant and the graph is a straight line.

Non-example: a solution of the Euler–Lagrange equation need not be a minimiser.
For $n = 1$, $\Omega = (0, 2\pi)$ and $L = \frac{1}{2}(p^2 - z^2)$, the equation is $u'' + u = 0$, solved by $u = 0$; but $J(t \sin(x/2)) = \frac{t^2}{2} \int_0^{2\pi} \big(\frac{1}{4}\cos^2(x/2) - \sin^2(x/2)\big) \, dx = -\frac{3\pi t^2}{8} < 0 = J(0)$ for $t \ne 0$, and $t\sin(x/2)$ has the same boundary values $0$.

## Explanations

### L1 human

The Euler–Lagrange equation is what one gets by setting the first variation of an integral functional to zero.
Minimisers that are smooth enough must solve it, but solutions need not be minimisers.
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# Category of small categories

## Example

A [[category|A2DE66937D]] is *small* when its objects and its morphisms form sets.
The *category of small categories* $\mathbf{Cat}$ has the small categories as objects and the [[functors|28NQ0KRHD3]] between them as morphisms.
The composite of $F \colon \mathcal{A} \to \mathcal{B}$ and $G \colon \mathcal{B} \to \mathcal{C}$ is the functor $G \circ F$, given on objects and on morphisms by composing the maps.
The identity functor of $\mathcal{A}$ is the identity morphism.
Composition of functors is associative because it is composition of maps on objects and on morphisms, so $\mathbf{Cat}$ is a category.

## Remarks

The empty category is an [[initial object|EK85VSSRMR]], and the category with one object and one morphism is a terminal object.
The [[product|S5744QZV4N]] of $\mathcal{A}$ and $\mathcal{B}$ is the [[product category|AFNE9C4NH6]] $\mathcal{A} \times \mathcal{B}$.
$\mathbf{Cat}$ is [[Cartesian closed|BFBNQ36QS8]]: the exponential object of $\mathcal{A}$ and $\mathcal{B}$ is the [[functor category|1X0WKFQX8K]] $\mathcal{B}^{\mathcal{A}}$.
With [[natural transformations|KYCPCC57FT]] as morphisms between functors, $\mathbf{Cat}$ becomes a [[2-category|CQACXG10JZ]].
The smallness condition avoids a paradox: $\mathbf{Cat}$ itself is not small, so it is not one of its own objects.

## Explanations

### L1 human

Categories can themselves be the objects of a category, with functors as the arrows between them.
To avoid "the set of all sets" style problems, one only takes categories whose objects and arrows form sets.
For example, a [[monoid|0N6R902FG1]] is a category with one object, and a monoid homomorphism is then exactly a functor, so monoids sit inside $\mathbf{Cat}$.
</document>

<document source="https://tamnd.github.io/math2/node/ZY724M2NKQ.md" tag="ZY724M2NKQ" kind="definition">
# Fundamental solution

## Definition

Let $P = \sum_{\abs{\alpha} \le m} a_\alpha(x) \partial^\alpha$ be a [[linear differential operator|6DZETZ5T9G]] on $\R^n$ with smooth coefficients.
A distribution is a continuous linear functional on the space of [[test functions|HS65DW0HXH]] on $\R^n$.
Let $\delta$ be the [[Dirac delta|0KPHENZ5S7]] at the origin, the distribution $\delta(\varphi) = \varphi(0)$.
A *fundamental solution* of $P$ is a distribution $E$ on $\R^n$ with $P E = \delta$ in the sense of distributions.
Explicitly, this means $E\left(P^{t} \varphi\right) = \varphi(0)$ for every test function $\varphi$, where $P^{t} \varphi = \sum_{\abs{\alpha} \le m} (-1)^{\abs{\alpha}} \partial^\alpha (a_\alpha \varphi)$ is the formal transpose of $P$.

## Remarks

Example: on $\R$, the Heaviside function $H$, with $H(x) = 1$ for $x > 0$ and $H(x) = 0$ for $x < 0$, is a fundamental solution of $P = d/dx$.
Indeed, for a test function $\varphi$ one has $H(-\varphi') = -\int_0^\infty \varphi'(x) \, dx = \varphi(0)$.
Example: on $\R^3$, $E(x) = -1/(4 \pi \abs{x})$ is a fundamental solution of the [[Laplacian|0E0B6RA8K0]], and on $\R^2$, $E(x) = \frac{1}{2\pi} \log \abs{x}$ is one.
Non-example: on $\R$, the constant function $1$ is not a fundamental solution of $d/dx$, since its derivative is $0$ and not $\delta$.
A fundamental solution is not unique: adding any distribution $F$ with $P F = 0$ gives another one.
If $P$ has constant coefficients and $f$ is a test function, then $u = E * f$ is smooth and solves $P u = f$.
Every nonzero operator with constant coefficients has a fundamental solution, by the [[Malgrange–Ehrenpreis theorem|3173Y813V7]].

## Explanations

### L1 human

A fundamental solution is a solution of the equation with a unit point source at the origin as right-hand side.
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# Whitehead tower

## Definition

Let $X$ be a [[path-connected space|0WB6DH499T]] with base point, and let $\pi_k$ denote the $k$-th [[homotopy group|P976WYEFMB]].
A *Whitehead tower* of $X$ is a sequence of pointed spaces $X\langle n \rangle$, $n \ge 0$, with $X\langle 0 \rangle = X$.
It comes with pointed maps $q_n \colon X\langle n \rangle \to X\langle n - 1 \rangle$ for $n \ge 1$.
For each $n \ge 0$, the composite $r_n = q_1 \circ \cdots \circ q_n \colon X\langle n \rangle \to X$ must satisfy:

1. $X\langle n \rangle$ is [[$n$-connected|1A943BWBQT]], that is, $\pi_k(X\langle n \rangle) = 0$ for all $k \le n$.
2. $r_n$ induces an isomorphism $\pi_k(X\langle n \rangle) \to \pi_k(X)$ for all $k > n$.

The space $X\langle n \rangle$ is the *$n$-connected cover* of $X$.

## Remarks

For a CW complex $X$ a Whitehead tower exists and each $X\langle n \rangle$ is unique up to weak homotopy equivalence over $X$.
Up to homotopy, $q_n$ can be taken to be a fibration, and then its fibre is an [[Eilenberg–MacLane space|AJHEYJE5XY]] $K(\pi_n(X), n - 1)$.
For $X$ a CW complex, $X\langle 1 \rangle \to X$ is the [[universal cover|Y1YPE6PM52]].
The tower is dual to the [[Postnikov tower|EKGEZGHYBW]]: $X\langle n \rangle$ is the homotopy fibre of $X \to X_n$.

Example: for $X = S^3$, $X\langle 1 \rangle = X\langle 2 \rangle = S^3$, since $S^3$ is $2$-connected.
The cover $S^3\langle 3 \rangle \to S^3$ is a fibration with fibre $K(\Z, 2) = \C P^\infty$.

Non-example: for $X = S^1$, the identity $S^1 \to S^1$ as $X\langle 1 \rangle$ fails clause 1, since $\pi_1(S^1) \cong \Z \ne 0$.
The universal cover $\R \to S^1$ satisfies both clauses.
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# Boundary integral equation for the Dirichlet problem

## Statement

Let $m \in \set{2, 3}$, and let $D \subset \R^m$ be a bounded domain whose boundary $\partial D$ is a connected $C^2$ curve or surface with outward unit normal $\nu$.
Let $\Phi$ be the [[fundamental solution of the Laplacian|BSJ8X2XRN9]] with $-\Delta \Phi = \delta_0$, so $\Phi(x) = \frac{1}{2\pi} \ln \frac{1}{\abs{x}}$ for $m = 2$ and $\Phi(x) = \frac{1}{4\pi \abs{x}}$ for $m = 3$.
For a continuous density $\varphi$ on $\partial D$ put
$$(K\varphi)(x) = \int_{\partial D} \frac{\partial \Phi(x - y)}{\partial \nu(y)} \, \varphi(y) \, ds(y), \quad x \in \partial D.$$
Let $f \colon \partial D \to \R$ be continuous.
Then:

1. The [[integral equation of the second kind|8KJCN49V64]] $\varphi - 2K\varphi = -2f$ has exactly one continuous solution $\varphi$ on $\partial D$.
2. For this $\varphi$, the [[double layer potential|EG8Z3FWV9A]] $u(x) = \int_{\partial D} \frac{\partial \Phi(x - y)}{\partial \nu(y)} \varphi(y) \, ds(y)$, $x \in D$, extends continuously to $\overline{D}$, is [[harmonic|0DEB0JMN0W]] in $D$, and equals $f$ on $\partial D$.

So $u$ is the unique solution of the interior [[Dirichlet problem|8D56GQ4AG8]] with boundary values $f$.

## Remarks

Part 2 follows from the [[jump relations for layer potentials|J58YA8QNNX]]: the limit of the double layer potential from inside is $K\varphi - \frac12 \varphi$, and this equals $f$ exactly when $\varphi - 2K\varphi = -2f$.
For part 1, $K$ is a [[compact operator|M2NEP1K5AE]] on $C(\partial D)$ because its kernel is continuous for $m = 2$ and weakly singular for $m = 3$.
By the [[Fredholm alternative|4HH8C9V8JQ]] it suffices to show that $\varphi - 2K\varphi = 0$ forces $\varphi = 0$.
For such $\varphi$ the double layer potential vanishes in $D$ by the maximum principle.
Its normal derivative is continuous across $\partial D$, so it solves an exterior Neumann problem with zero data and vanishes outside as well.
The jump of the potential across $\partial D$ is $\varphi$, so $\varphi = 0$.

Example: on the unit disc in $\R^2$ one has $\frac{\partial \Phi(x - y)}{\partial \nu(y)} = -\frac{1}{4\pi}$ for $x \ne y$ on the unit circle, so $K\varphi = -\frac{1}{4\pi} \int_{\partial D} \varphi \, ds$.
For $f = 1$ the equation becomes $\varphi + \frac{1}{2\pi} \int \varphi \, ds = -2$, whose solution is the constant $\varphi = -1$, and then $u = 1$ in $D$.

The method turns a problem on the domain $D$ into an equation on its boundary, one dimension lower, which is the basis of boundary element methods.

## Explanations

### L1 human

To find a harmonic function inside a region with given boundary values, one can place a layer of dipoles on the boundary and look for its strength.
The strength must satisfy an integral equation on the boundary alone.
This equation of the second kind always has exactly one solution, which proves that the Dirichlet problem is solvable and gives a way to compute the solution.

### L2 human

The theorem is the classical route, going back to the work of Neumann and Fredholm, from potential theory to [[Fredholm theory|242TFD41KZ]].
Its proof uses the jump relations, the compactness of $K$ and the Fredholm alternative, and uniqueness for the exterior Neumann problem.
For the interior Neumann problem one uses a single layer potential instead, see the [[boundary integral equation for the Neumann problem|CN7NP6R1AQ]].
For the exterior Dirichlet problem the corresponding equation $\varphi + 2K\varphi = 2f$ is not always uniquely solvable, and the ansatz has to be modified.
</document>

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# Riesz basis

## Summary

A Riesz basis of a [[Hilbert space|HM2HHE3J51]] is the image of an [[orthonormal basis|0S9GGTR0EX]] under a bounded invertible operator.
It keeps the main features of an orthonormal basis.
Every vector has a unique expansion, and its length is comparable to the length of its coefficient sequence.
Riesz bases are exactly the bases that are also frames, and they are unconditional [[Schauder bases|528G11SDWR]].
They are the natural notion of a stable basis in sampling theory, wavelets and nonharmonic Fourier analysis.

## Explanations

### L1 human

An orthonormal basis is like a set of perpendicular rulers of length one.
A Riesz basis is what you get after you stretch and tilt the space by a linear map that can be undone.
The rulers are no longer perpendicular or of equal length, but they are not too close to each other either.
So coordinates in a Riesz basis are still unique and still control the length of the vector, up to fixed constants.
</document>

<document source="https://tamnd.github.io/math2/node/ZYG1MEBF4X.md" tag="ZYG1MEBF4X" kind="definition">
# Levitzki radical

## Definition

Let $R$ be a [[ring|0894J980RK]].
A subset $S$ of $R$ is *locally nilpotent* if the following holds for every finite subset $F \subseteq S$.
There is an integer $n \ge 1$ such that every product $f_1 f_2 \cdots f_n$ of $n$ elements of $F$ is $0$.
The *Levitzki radical* $\operatorname{L}(R)$ of $R$ is the sum of all locally nilpotent [[two-sided ideals|0DDA6CY7GJ]] of $R$.

## Remarks

The sum of all locally nilpotent ideals is again locally nilpotent, so $\operatorname{L}(R)$ is the largest locally nilpotent ideal of $R$.
Example: let $R$ be the ring of upper triangular $2 \times 2$ matrices over a field $k$ and $N$ its strictly upper triangular matrices.
$N$ is locally nilpotent since $N^2 = 0$, and $R/N \cong k \times k$ has no nonzero nilpotent elements, so $\operatorname{L}(R) = N$.
Non-example: in $M_2(\Q)$ the set $\set{e_{12}, e_{21}}$ consists of nilpotent elements but is not locally nilpotent, since $(e_{12} e_{21})^m = e_{11} \ne 0$ for all $m$.
Every nilpotent ideal is locally nilpotent and every locally nilpotent ideal is nil, which gives $\operatorname{Nil}_*(R) \subseteq \operatorname{L}(R) \subseteq \operatorname{Nil}^*(R)$ with the [[prime radical|WF117EJ2HF]] and the [[upper nilradical|3G2JQEYBEK]].
</document>

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# Discrete valuation

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
A *discrete valuation* on $K$ is a [[valuation|B85T5VKR3M]] $v \colon K \to \Z \cup \set{\infty}$ such that $v(K^\times) = \Z$.
An element $\pi \in K$ with $v(\pi) = 1$ is a *uniformizer* of $v$.

## Remarks

Example: for a prime $p$, the [[p-adic valuation|04VCPS30ZE]] $v_p$ on $\Q$ is a discrete valuation.
It is a valuation, and $v_p(p^n) = n$ for every $n \in \Z$, so it is onto $\Z$; the prime $p$ is a uniformizer.
Example: for a field $k$, the order of vanishing at $0$ on $k(t)$, given by $v(t^n f/g) = n$ for polynomials $f, g$ with $f(0) g(0) \ne 0$, is a discrete valuation with uniformizer $t$.
Non-example: the trivial valuation on a field, with $v(x) = 0$ for all $x \ne 0$, takes values in $\Z$ but has $v(K^\times) = \set{0} \ne \Z$.
Some texts call a valuation discrete when its value group is isomorphic to $\Z$; dividing by a positive generator then gives a valuation onto $\Z$, called normalized.
Every $x \in K^\times$ is $x = u \pi^{v(x)}$ with $v(u) = 0$.
The set $\set{x : v(x) \ge 0}$ is a [[discrete valuation ring|MGM3CNJW9V]] with maximal ideal $\pi \cdot \set{x : v(x) \ge 0}$.
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<document source="https://tamnd.github.io/math2/node/ZYNY2VFECQ.md" tag="ZYNY2VFECQ" kind="concept">
# Invariant mean

## Summary

An invariant mean on a [[group|G5S1M9]] is a way to average all bounded functions on the group.
The average does not change when the function is translated.
A discrete group has a left-invariant mean exactly when it is [[amenable|CEAT9BESKK]].
Invariant means are the functional form of finitely additive invariant probability measures.

## Explanations

### L1 human

On a finite group one averages a function by adding its values and dividing by the number of elements.
This average does not change when the function is shifted by a group element.
For an infinite group such as the integers, one would like an average of every bounded sequence that ignores shifts.
Such averages exist for the integers, but they cannot be written down explicitly.
For the free group on two generators, no such average exists at all.
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<document source="https://tamnd.github.io/math2/node/ZYS4TTQG2T.md" tag="ZYS4TTQG2T" kind="definition">
# Height of a poset

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
The *height* of $P$ is the supremum of the cardinalities $\abs{C}$ of the [[chains|0S8FE7ZMFM]] $C \subseteq P$.

## Remarks

Example: the subsets of $\set{1, 2, 3}$ ordered by inclusion have height $4$, as shown by the chain $\varnothing \subset \set{1} \subset \set{1, 2} \subset \set{1, 2, 3}$.
A chain of subsets has at most one subset of each size, so no chain is longer.
Non-example: the set $\set{\varnothing, \set{1}, \set{2}, \set{1, 2}}$ has $4$ elements but is not a chain, since $\set{1}$ and $\set{2}$ are incomparable.
Conventions differ: some authors call the height the largest *length* of a chain $x_0 < x_1 < \cdots < x_n$, which is $n$, one less than the number of elements.
In that second convention the height is the Krull dimension of the poset.
The height of an element $x$ is often defined as the height of the down-set $\set{y \in P : y \le x}$, with the same choice of convention.
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# Degree of exactness of a quadrature rule

## Summary

The degree of exactness of a [[quadrature rule|ADSVMCY77A]] is the largest $d$ such that the rule is exact for all polynomials of degree $\le d$.
It is the standard first measure of the quality of a rule: for smooth integrands, a high degree of exactness usually means a small error.
A rule with $n$ nodes has degree of exactness at most $2n - 1$, and Gaussian quadrature reaches this bound.

## Explanations

### L1 human

Polynomials are the simplest functions, so a good integration recipe should at least get them right.
The degree of exactness counts how far up the ladder $1, x, x^2, x^3, \dots$ the recipe is exact.
The midpoint rule is exact for straight lines but not for $x^2$, so its degree is $1$.
Simpson's rule, with three nodes, is exact even for cubics, so its degree is $3$.
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# Thom–Boardman singularities

## Definition

Let $f \colon M \to N$ be a smooth map between [[smooth manifolds|F45VQ47GPV]].
For an integer $i \ge 0$ let
$$\Sigma^{i}(f) = \set{p \in M : \dim \ker df_p = i},$$
where $df_p$ is the [[differential|6P9PXQDSEN]] of $f$ at $p$.
Inductively, let $I = (i_1, \ldots, i_k)$ and suppose that $\Sigma^{I}(f)$ is a submanifold of $M$.
For $j \ge 0$ let $\Sigma^{i_1, \ldots, i_k, j}(f)$ be the set of points of $\Sigma^{I}(f)$ where the restriction $f|_{\Sigma^I(f)}$ has kernel of dimension $j$, that is, $\Sigma^{j}(f|_{\Sigma^I(f)})$.
A point $p$ has *Thom–Boardman symbol* $I$ when $p \in \Sigma^I(f)$, whenever these sets are defined.

## Remarks

Since $\Sigma^{i_1, \ldots, i_k}(f) \subseteq \Sigma^{i_1, \ldots, i_{k-1}}(f)$, the kernel of $f$ restricted to the smaller set lies in the kernel of $f$ restricted to the larger one.
So the symbols are non-increasing: $i_1 \ge i_2 \ge \cdots \ge i_k$.
Boardman constructed submanifolds $\Sigma^I$ of the [[jet space|1QE0Z0756D]] $J^k(M, N)$, depending only on $I$.
Suppose the jet extension $j^k f$ is [[transverse|A2PDZ8GA9C]] to all of them.
Then each $\Sigma^I(f) = (j^k f)^{-1}(\Sigma^I)$ is a submanifold and agrees with the inductive definition.
By Thom's transversality theorem, this holds for a dense set of maps.
If $\dim M = m$ and $\dim N = n$, the subset $\Sigma^i$ of $J^1(M, N)$ has codimension $i(n - m + i)$.
Example: for $f(x, y) = (x, y^3 + xy)$ on $\R^2$, $\Sigma^1(f)$ is the curve $x = -3y^2$.
Parametrising it by $y \mapsto (-3y^2, y)$, the restriction of $f$ is $y \mapsto (-3y^2, -2y^3)$, whose derivative vanishes only at $y = 0$.
So the origin lies in $\Sigma^{1,1}(f)$, and the other points of the curve lie in $\Sigma^{1,0}(f)$.
Non-example: for $f(x, y) = (x, y^3 - x^2 y)$, $\Sigma^1(f)$ is the set $3y^2 = x^2$, a pair of crossing lines.
It is not a submanifold at the origin, so the inductive definition breaks down there; the jet of $f$ is not transverse to $\Sigma^1$.

## Explanations

### L1 human

The symbol of a singular point records how badly a map degenerates, one layer at a time.
The first entry counts the directions crushed by the map.
The next entry counts the directions crushed by the map along the set of points that share the first entry.
The construction is clean only for generic maps, where all these sets are smooth.
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# Pariah groups

## Summary

The pariah groups are the six sporadic simple groups that are not subquotients of the Monster group.
They are the Janko groups $J_1$, $J_3$ and $J_4$, the O'Nan group, the Rudvalis group and the Lyons group.
The other twenty sporadic groups, which are subquotients of the Monster, form the happy family; the name pariah is due to Robert Griess.

## Explanations

### L1 human

The Monster group contains, as pieces of its subgroups, most of the exceptional finite simple groups.
Six exceptional groups do not fit inside it in this way.
They stand apart from the rest, so they are called the pariahs.
</document>

<document source="https://tamnd.github.io/math2/node/ZYZM357DQV.md" tag="ZYZM357DQV" kind="concept">
# hp-FEM

## Summary

The hp-FEM is a version of the [[finite element method|98GZ9DVHTA]] that improves the approximation in two ways at once.
It refines the mesh, which makes the element size $h$ smaller, and it raises the polynomial degree $p$ on the elements.
Small elements are used near singularities of the solution, and high degrees where the solution is smooth.
Take an elliptic problem with piecewise analytic data on a polygon.
Babuška and Guo showed that a well chosen hp-sequence then converges exponentially in the number of unknowns.
See the [[definition|1F7NV6AT2X]].

## Explanations

### L1 human

A standard finite element method uses many small pieces, each with a simple polynomial such as a straight line.
The hp-FEM also lets each piece use a curved polynomial of high degree.
Where the solution is smooth, high degree pays off quickly; near a corner where the solution has a kink in its derivatives, small pieces work better.
Mixing the two well gives errors that drop much faster than with either one alone.
</document>

<document source="https://tamnd.github.io/math2/node/ZZ82BW6CBG.md" tag="ZZ82BW6CBG" kind="definition">
# Elliptic gamma function

## Definition

Let $p, q$ be complex numbers with $\abs{p} < 1$ and $\abs{q} < 1$.
For $z \in \C \setminus \set{0}$ with $p^j q^k z \ne 1$ for all integers $j, k \ge 0$, the *elliptic gamma function* is
$$\Gamma(z; p, q) = \prod_{j, k \ge 0} \frac{1 - p^{j+1} q^{k+1} / z}{1 - p^j q^k z}.$$

## Remarks

With the theta function $\theta(z; p) = (z; p)_\infty\, (p/z; p)_\infty$, built from the [[q-Pochhammer symbol|Y1X8ZZQ7VM]] $(a; p)_\infty = \prod_{j \ge 0}(1 - a p^j)$, the function satisfies
$$\Gamma(qz; p, q) = \theta(z; p)\, \Gamma(z; p, q).$$
Check: in the quotient $\Gamma(qz)/\Gamma(z)$ all factors cancel except the factors with $k = 0$, which give $\prod_{j \ge 0}(1 - p^{j+1}/z)(1 - p^j z) = \theta(z; p)$.

Example: replacing $z$ by $pq/z$ exchanges the numerator and the denominator of the product, so $\Gamma(z; p, q)\, \Gamma(pq/z; p, q) = 1$.

Non-example of a valid argument: $z = 1$ makes the factor $1 - p^0 q^0 z$ vanish, so $\Gamma(z; p, q)$ has a pole at $z = 1$.

The function is symmetric in $p$ and $q$.
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<document source="https://tamnd.github.io/math2/node/ZZD7X71HVN.md" tag="ZZD7X71HVN" kind="definition">
# Lane–Emden equation

## Definition

Let $n \ge 0$ be a real number, called the polytropic index.
The *Lane–Emden equation* of index $n$ is the [[ordinary differential equation|0NT64JV96A]]
$$\frac{1}{\xi^2} \frac{d}{d\xi} \left( \xi^2 \frac{d\theta}{d\xi} \right) + \theta^n = 0$$
for a function $\theta$ of $\xi > 0$, with the initial conditions $\theta(0) = 1$ and $\theta'(0) = 0$.
When $n$ is not an integer, the equation is considered on the interval where $\theta \ge 0$.

## Remarks

For a radial function $u(x) = \theta(\abs{x})$ on $\R^3$, the first term is the [[Laplacian|0E0B6RA8K0]] $\Delta u$, so the equation is $\Delta u + u^n = 0$.
Example: for $n = 0$, $\theta(\xi) = 1 - \xi^2/6$ is the solution.
Indeed, $\xi^2 \theta' = -\xi^3/3$, whose derivative divided by $\xi^2$ is $-1$, and $\theta^0 = 1$.
Non-example: for $n = 1$, the constant $\theta = 1$ is not a solution, since the left side equals $1$.
Closed-form solutions are known for $n = 0$, $n = 1$, where $\theta = \sin \xi / \xi$, and $n = 5$, where $\theta = (1 + \xi^2/3)^{-1/2}$.
In PDE the name Lane–Emden equation, or Lane–Emden–Fowler equation, is also used for $-\Delta u = \abs{u}^{p-1} u$ on a domain of $\R^N$ with $p > 1$.
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<document source="https://tamnd.github.io/math2/node/ZZDMHH8V1K.md" tag="ZZDMHH8V1K" kind="definition">
# Residue calculus for Laplace inversion

## Definition

Let $F$ be a [[meromorphic function|0NSD1TPHR0]] on $\C$ with finitely many poles $s_1, \ldots, s_n$.
Let $\gamma$ be a real number with $\gamma > \operatorname{Re} s_k$ for every $k$.
Assume that there are constants $C > 0$, $p > 0$ and $R_0 > 0$ with $\abs{F(s)} \le C \abs{s}^{-p}$ for all $s$ with $\abs{s} \ge R_0$ and $\operatorname{Re} s \le \gamma$.
*Residue calculus for Laplace inversion* is the evaluation, for $t > 0$, of the [[Bromwich integral|9WRE8KEJJX]] of $F$ along $\operatorname{Re} s = \gamma$ as
$$\frac{1}{2 \pi i} \int_{\gamma - i \infty}^{\gamma + i \infty} e^{s t} F(s) \, ds = \sum_{k=1}^{n} \operatorname{Res}_{s = s_k} \left( e^{s t} F(s) \right),$$
where $\operatorname{Res}$ is the [[residue|0JFGKKWWFK]].
Under these assumptions the identity holds for every $t > 0$.
Let $F$ also be the [[Laplace transform|05XT5AVJBK]] of a function $f$ as in Mellin's inverse formula.
Then the right side is the [[inverse Laplace transform|0JRW4X1MDC]] $f(t)$ at each $t > 0$ where $f$ is continuous.

## Remarks

The identity follows from the [[residue theorem|032HKYS329]] on the region bounded by the segment from $\gamma - i R$ to $\gamma + i R$ and a left arc of radius about $R$.
A Jordan-type estimate shows that the arc contributes $0$ in the limit when $t > 0$.
The hypotheses hold for every rational $F = P/Q$ with $\deg P < \deg Q$.
Example: $F(s) = 1/(s + 1)^2$ has one double pole, and $\operatorname{Res}_{s=-1} e^{s t}/(s+1)^2 = \frac{d}{ds} e^{s t} \big|_{s = -1} = t e^{-t}$.
Non-example: $F(s) = e^{-s}/s$ is not small on left arcs, because $\abs{e^{-s}} = e^{-\operatorname{Re} s}$ grows there.
The residue sum would give $1$ for all $t > 0$, while the true inverse is the delayed step $u(t - 1)$, which is $0$ for $0 < t < 1$.
Transforms that come from partial differential equations often have infinitely many poles.
For them one uses a sequence of contours that avoid the poles, and one gets a series over all poles.
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<document source="https://tamnd.github.io/math2/node/ZZF8XPBG3Z.md" tag="ZZF8XPBG3Z" kind="example">
# Free magma

## Example

Let $X$ be a set.
The *free magma* $M(X)$ on $X$ is defined inductively.
Every $x \in X$ is an element of $M(X)$, and if $u, v \in M(X)$, then the formal product $(u v)$ is an element of $M(X)$.
Two elements are equal only when they are built in the same way.
The [[binary operation|0M418FE7BB]] is $u \cdot v = (u v)$.
So $M(X)$ is a [[magma|9689SFPVE9]].
Its elements are the fully parenthesised nonempty words in $X$, or equivalently the binary trees with leaves labelled by $X$.

It is an [[absolutely free algebra|TTTEV3NWWM]] of the signature with one binary operation.
Check: $M(X)$ is generated by $X$ by construction.
Given a magma $N$ and a map $f \colon X \to N$, define $h(x) = f(x)$ and $h((u v)) = h(u) \cdot h(v)$ by recursion on the construction.
This is well defined because each element has exactly one construction, and it is the only homomorphism that extends $f$.
So $M(X)$ is the [[free algebra|GTS4DRRMZR]] on $X$ of the variety of all magmas.

For $X = \set{x}$, the elements with three leaves are $((x x) x)$ and $(x (x x))$, and they are different.
The number of elements with $n$ leaves is the Catalan number $C_{n-1}$, the number of ways to bracket a product of $n$ factors.
In Mathlib the free magma is `FreeMagma`.

## Explanations

### L1 human

The free magma is what one gets by multiplying letters without any rule at all, not even associativity.
Every way of putting brackets around a product gives a different element.
For one letter $x$ and three factors there are two elements, $(xx)x$ and $x(xx)$; for four factors there are five.
Imposing associativity on it gives the free semigroup of nonempty words.
</document>

<document source="https://tamnd.github.io/math2/node/ZZFFBXDTWN.md" tag="ZZFFBXDTWN" kind="definition">
# Orderable group

## Definition

Let $G$ be a [[group|G5S1M9]].
The group $G$ is *orderable*, or *bi-orderable*, if there is a [[total order|0316AYH1SC]] $\le$ on $G$ such that for all $a, b, c \in G$,
$$a \le b \implies ca \le cb \text{ and } ac \le bc.$$
That is, $(G, \le)$ is a [[linearly ordered group|1P8087YGYQ]] for some order $\le$.

## Remarks

Example: $\Z^2$ under addition is orderable, since the lexicographic order is total and translation invariant.
Non-example: the cyclic group $\Z/2\Z$ is not orderable.
If $g$ is its nonzero element and $0 < g$, then adding $g$ gives $g < g + g = 0$, a contradiction; the case $g < 0$ is the same.
More generally, an orderable group is a [[torsion-free group|VQ4SSKS53D]].
Every torsion-free [[abelian group|071DDFVG4J]] is orderable.
Every [[free group|020Z5SEN27]] is orderable.
A group is orderable if and only if it has a subset $P$ with $PP \subseteq P$, $gPg^{-1} = P$ for each $g \in G$, and $G = P \sqcup \set{e} \sqcup P^{-1}$.
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<document source="https://tamnd.github.io/math2/node/ZZFTPP8JTG.md" tag="ZZFTPP8JTG" kind="definition">
# Inverse mean curvature flow

## Definition

Let $(N, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] of dimension $n + 1$, let $M$ be a closed manifold of dimension $n$, and let $T > 0$.
A family of immersions $F \colon M \times [0, T) \to N$ is a solution of *inverse mean curvature flow* if for every $t \in [0, T)$:

1. the hypersurface $M_t = F(M, t)$ has an outward unit normal $\nu(\cdot, t)$;
2. the [[mean curvature|PA78NFEPCF]] $H(\cdot, t)$ of $M_t$ with respect to $\nu$, the sum of the principal curvatures, is positive everywhere;
3. for all $x \in M$,
$$\frac{\partial F}{\partial t}(x, t) = \frac{1}{H(x, t)}\, \nu(x, t).$$

## Remarks

Example: in $N = \R^{n+1}$ take $M_t$ the sphere of radius $r(t) = r_0 e^{t/n}$ about the origin.
Its mean curvature is $H = n/r$, so the speed required by clause 3 is $r/n$, and indeed $r'(t) = r(t)/n$.
Its area is $e^t$ times the area of $M_0$.
Non-example: a hypersurface with a point where $H \le 0$ fails clause 2, and the flow cannot start from it.
A torus of revolution in $\R^3$ with tube radius $r$ and center circle of radius $R < 2r$ has such points on its inner circle.
Under the flow the area element is multiplied by $e^t$ in general, since its time derivative is $H \cdot (1/H) = 1$ times itself.
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# Teichmüller space

## Summary

The Teichmüller space $T_g$ parametrises the complex structures on a fixed closed surface of genus $g$.
Each point also carries a marking, which remembers the topology up to isotopy.
For $g \ge 2$ it is a [[complex manifold|JSGWAN2RJV]] of dimension $3g - 3$ and is homeomorphic to $\R^{6g - 6}$.
The [[mapping class group|JV5NYG913V]] acts on it properly discontinuously, and the quotient is the [[moduli space of curves|NK9ATH4WB6]] of genus $g$.

## Explanations

### L1 human

A closed surface of genus $g$, like a doughnut with $g$ holes, can be given many different shapes as a [[Riemann surface|5K0N8QMV6Q]].
The moduli space lists these shapes up to isomorphism, but surfaces with extra symmetries make it an orbifold rather than a manifold.
Teichmüller space adds a marking, a way of naming the loops on the surface, which removes this problem.
For genus $1$ a marked torus is $\C / (\Z + \tau \Z)$ with $\tau$ in the [[upper half-plane|X6XGGH5FPW]], and $T_1$ is the upper half-plane itself.
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<document source="https://tamnd.github.io/math2/node/ZZK2GM1V15.md" tag="ZZK2GM1V15" kind="concept">
# Continuous multiplication

## Summary

A set with a binary operation and a topology has continuous multiplication when the operation, seen as a map of two variables, is continuous.
Here the domain carries the [[product topology|0GBG8B49AA]].
This single condition is the common base of [[topological semigroups|QXB6JDG68F]], [[topological monoids|RA56GHB9JZ]] and [[topological groups|92700QAB5Y]].
It asks for joint continuity, which is stronger than continuity in each variable separately.

## Explanations

### L1 human

Think of multiplying two numbers $x$ and $y$.
If you move $x$ and $y$ a little, the product $xy$ also moves only a little.
That is what continuous multiplication means: small changes in both inputs give a small change in the output.
For example, ordinary multiplication on the real line has this property, so $(\R, \cdot)$ has continuous multiplication.
</document>

<document source="https://tamnd.github.io/math2/node/ZZP31F3VPK.md" tag="ZZP31F3VPK" kind="definition">
# Étale cohomology

## Definition

Let $X$ be a [[scheme|5A35TYZD02]].
The *small étale site* $X_{\text{ét}}$ is the [[site|NYE0Z94MME]] defined as follows.

1. Its objects are the [[étale morphisms|H84VHC8PX2]] $U \to X$.
2. Its morphisms are the morphisms $U \to V$ over $X$.
3. A family $\set{U_i \to U}$ is a covering when the union of the images of the $U_i$ is $U$.

An *étale sheaf* on $X$ is a [[sheaf on the site|RT8A8FJ566]] $X_{\text{ét}}$.
The category of étale sheaves of abelian groups on $X$ has enough injectives.
For such a sheaf $F$, the *étale cohomology groups* $H^i_{\text{ét}}(X, F)$, for $i \ge 0$, are the right [[derived functors|NPMC3K6J2F]] of the global section functor $F \mapsto F(X)$.

## Remarks

Example: let $X = \operatorname{Spec} k$ for a field $k$ with separable closure $k^{s}$ and Galois group $G = \Gal(k^{s}/k)$.
Étale $X$-schemes are disjoint unions of spectra of finite separable extensions of $k$.
An étale sheaf $F$ is then the same as the discrete $G$-module $M = \varinjlim F(\operatorname{Spec} k')$ over finite subextensions $k'$.
The global sections are $M^G$, so $H^i_{\text{ét}}(X, F) = H^i(G, M)$ is [[Galois cohomology|EESNTXFWHZ]].
Non-example: if the étale morphisms in clause 1 are replaced by open immersions, one gets the Zariski site.
For an irreducible $X$ and a constant sheaf $A$, every constant sheaf is flasque in the Zariski topology, so $H^i(X, A) = 0$ for all $i > 0$.
Étale cohomology does not have this defect: for a smooth projective curve of genus $g$ over an algebraically closed field and $n$ prime to the characteristic, $H^1_{\text{ét}}(X, \Z/n\Z) \cong (\Z/n\Z)^{2g}$.
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# Stein–Tomas restriction theorem

## Statement

Let $n \ge 2$ and let $S \subseteq \R^n$ be a compact smooth hypersurface, possibly with boundary, whose [[Gaussian curvature|SQVQ5DQT6W]] is nonzero at every point.
Let $\sigma$ be the surface measure on $S$, and let $\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$.
Let
$$1 \le p \le \frac{2(n + 1)}{n + 3}.$$
Then there is a constant $C$ such that
$$\Big( \int_S \abs{\hat f(\xi)}^2 \, d\sigma(\xi) \Big)^{1/2} \le C \norm{f}_{L^p(\R^n)}$$
for every [[Schwartz function|CG9JEATW3Q]] $f$ on $\R^n$.
So the restriction map $f \mapsto \hat f|_S$ extends to a bounded operator from $L^p(\R^n)$ to $L^2(S, \sigma)$.

## Remarks

By duality, the estimate is equivalent to $\norm{\widehat{g \, d\sigma}}_{L^{p'}(\R^n)} \le C \norm{g}_{L^2(S)}$, with $1/p + 1/p' = 1$.
A $TT^*$ argument reduces this to the bound $\norm{f * \hat\sigma}_{L^{p'}} \le C^2 \norm{f}_{L^p}$.
One splits $\hat\sigma$ into dyadic pieces in $\abs{x}$.
Each piece is bounded from $L^1$ to $L^\infty$ by the [[decay of the Fourier transform of surface measure|SYP1ATSTTH]], and on $L^2$ by the size of its Fourier transform.
Interpolating these two bounds, the pieces sum when $p < 2(n+1)/(n+3)$.
Peter Tomas proved the range $p < 2(n + 1)/(n + 3)$, and Elias Stein obtained the endpoint.

Example: for the circle in $\R^2$ the range is $1 \le p \le 6/5$, and for the sphere in $\R^3$ it is $1 \le p \le 4/3$.

The exponent $2(n + 1)/(n + 3)$ is sharp for the $L^2(S)$ norm on the left.
The Knapp example takes $\hat f$ to be a smooth bump adapted to a cap of $S$ of radius $\delta$.
Such a cap lies in a slab of thickness about $\delta^2$.
Comparing both sides as $\delta \to 0$ forces $p \le 2(n+1)/(n+3)$.
Without curvature, for a flat piece of a hyperplane, no estimate with $p > 1$ holds.

## Explanations

### L1 human

For a general function in $L^p$, its Fourier transform is only defined almost everywhere, so its values on a surface of measure zero seem meaningless.
The theorem says that the Fourier transform does have a meaningful restriction to a curved surface, such as a sphere.
This holds with a norm bound when $p$ is close enough to $1$.
Curvature is what makes this work: on a flat plane, it fails.

### L2 human

This is the main known case of the [[Fourier restriction problem|FFCH15ZNZG]] with $L^2$ norm on the surface, and it is sharp in that setting.
It is used to prove Strichartz estimates for the Schrödinger and wave equations, which are restriction estimates for paraboloids and cones.
The [[restriction conjecture|9CE11G48JY]] asks for more general $L^q(S)$ norms and a larger range of $p$.
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# Obstruction theory

## Summary

Obstruction theory studies when a map defined on part of a [[CW complex|5W95QFHXPR]] extends to the whole complex, or when a fibration has a section.
It works by extending one skeleton at a time.
At each step the problem of going one cell dimension higher is measured by a cohomology class, the obstruction.
The method turns geometric extension problems into computations in [[singular cohomology|6N7BC70CX0]] with coefficients in [[homotopy groups|P976WYEFMB]].
Characteristic classes such as the Euler class and the Stiefel–Whitney classes arise as obstructions of this kind.

## Explanations

### L1 human

To extend a map over a CW complex, one builds it cell by cell, in order of dimension.
A map on the boundary sphere of a new cell extends over the cell exactly when it is null-homotopic.
Recording, for each cell, the homotopy class of the map on its boundary gives a cochain, the obstruction cocycle.
If its cohomology class vanishes, one can change the map on lower cells so that every cell of the next dimension can be filled.
For example, the identity of $S^n$ does not extend over the disc $D^{n+1}$, and the obstruction is the generator of $H^{n+1}(D^{n+1}, S^n; \Z) \cong \Z$.
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<document source="https://tamnd.github.io/math2/node/ZZYCV5EX9J.md" tag="ZZYCV5EX9J" kind="example">
# Heisenberg group

## Example

The (real, three-dimensional) *Heisenberg group* $H$ is the group of real upper unitriangular $3 \times 3$ matrices
$$[a, b, c] = \begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}, \qquad a, b, c \in \R,$$
with matrix multiplication.
Multiplying out gives
$$[a, b, c]\,[a', b', c'] = [a + a', \ b + b', \ c + c' + a b'],$$
and $[a, b, c]^{-1} = [-a, -b, -c + ab]$.
So $H$ is $\R^3$ with a polynomial group law, a [[Lie group|7JBRGCVJRB]] of dimension $3$.

Check that $H$ is a [[nilpotent Lie group|DK87YN458D]].
The commutator of $g = [a, b, 0]$ and $g' = [a', b', 0]$ is $g g' g^{-1} g'^{-1} = [0, 0, ab' - a'b]$.
The [[centre|0DZ93493VN]] of $H$ is $Z = \set{[0, 0, c] : c \in \R}$, and all commutators lie in $Z$.
So the lower central series is $H \supset Z \supset \set{1}$, and $H$ is a [[nilpotent group|0Q7MMQHC1P]] of class $2$; it is not abelian.

## Remarks

$H$ is connected and simply connected, being homeomorphic to $\R^3$.
Its Lie algebra has a basis $X, Y, Z$ with $[X, Y] = Z$ and $[X, Z] = [Y, Z] = 0$.
This is the bracket relation of the position and momentum operators of quantum mechanics, up to a constant, which is the origin of the name.
More generally, for $n \ge 1$ the Heisenberg group $H_{2n+1}$ is $\R^n \times \R^n \times \R$ with the law $(x, y, c)(x', y', c') = (x + x', y + y', c + c' + x \cdot y')$, of dimension $2n + 1$.
The matrices $[a, b, c]$ with integer entries form a discrete subgroup $H(\Z)$, and the quotient $H / H(\Z)$ is compact.

## Explanations

### L1 human

The Heisenberg group is the simplest group that is almost, but not quite, commutative.
Two of its elements commute up to an extra shift in the third coordinate, and that shift itself commutes with everything.
In the matrix picture, multiplying two matrices adds the entries above the diagonal, except that the corner entry picks up an extra product $ab'$.
This small failure of commutativity is the algebraic shadow of the uncertainty principle in quantum mechanics.
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